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[ [ "Magic mirrors, dense diameters, Baire category" ], [ "Abstract An old result of Zamfirescu says that for most convex curves $C$ in the plane most points in $R^2$ lie on infinitely many normals to $C$, where most is meant in Baire category sense.", "We strengthen this result by showing that `infinitely many' can be replaced by `contiunuum many' in the statement.", "We present further theorems in the same spirit." ], [ "Introduction", "In a 1982 paper [5] Tudor Zamfirescu proved a remarkable result saying that `most mirrors are magic'.", "For the mathematical formulation let $\\mathcal {C}$ be the set of all closed convex curves in the plane ${\\mathbb {R}^2}$ .", "Fix some $C \\in \\mathcal {C}$ and $z \\in C$ so that the tangent line, $T(z)$ , to $C$ at $z$ is unique, then so is the normal line $N(z)$ to $C$ at $z$ .", "A point $u \\in {\\mathbb {R}^2}$ sees an image of another point $v \\in {\\mathbb {R}^2}$ via $z$ if $u$ and $v$ and $C$ lie on the same side of $T(z)$ and the line $N(z)$ halves the angle $\\angle uzv$ .", "In particular, $u$ sees an image of itself via $z$ if $u \\in N(z)$ and $u$ and $C$ are on the same side of $T(z)$ .", "With the Haussdorf metric $\\mathcal {C}$ becomes a complete metric space.", "It is well-known that the normal $N(z)$ is unique at every point $z \\in C$ for most convex curves $C \\in \\mathcal {C}$ in the Baire category sense, that is, for the elements of a comeagre set of curves in $\\mathcal {C}$ .", "Now the `most mirrors are magic' statement is, precisely, that for most convex curves, most points in ${\\mathbb {R}^2}$ (again in Baire category sense) see infinitely many images of themselves.", "Another theorem from [5] says that for most convex curves, most points in ${\\mathbb {R}^2}$ see infinitely many images of any given point $v \\in {\\mathbb {R}^2}$ .", "Zamfirescu actually proves the existence of countably many images and self-images.", "The purpose of this paper is to show that most mirrors are even more magic.", "Theorem 1.1 For most convex curves, most points in ${\\mathbb {R}^2}$ see continuum many images of themselves.", "Theorem 1.2 For most convex curves $C$ and for every point $v \\in {\\mathbb {R}^2}\\setminus C$ , most points in ${\\mathbb {R}^2}$ see continuum many images of $v$ .", "The condition $v \\notin C$ in the last theorem is used to avoid some trivial complications in the proof.", "The statement holds even for $v \\in C$ .", "Remark.", "Let $C^o$ denote the closed convex set whose boundary is $C$ .", "The above definition of `$u$ sees an image of $v$ via $z\\in C$ ' means that the mirror side of $C$ is the interior one, that is, the segment $uz$ intersects the interior of $C^o$ .", "Theorem REF does not hold when the mirror is on the other side of $C$ because every point in ${\\mathbb {R}^2}\\setminus C^o$ lies on exactly one outer normal halfline to $C$ .", "A statement, analogous to Theorem REF about affine diameters was proved in [2] in 1990, for typical $d$ -dimensional convex bodies for every $d\\ge 2$ .", "The segment $[a,b]$ is an affine diameter of $C \\in \\mathcal {C}$ if there are distinct and parallel tangent lines to $a,b\\in C$ .", "The result in [2] says that for most convex curves $C \\in \\mathcal {C}$ , most points on a fixed affine diameter of $C$ are contained in infinitely many affine diameters of $C$ .", "In this case again we show the existence of continuum many diameters passing through most points in $C^o$ .", "Theorem 1.3 For most convex curves $C \\in \\mathcal {C}$ , most points in $C^o$ lie in continuum many diameters of $C$ .", "Note that every point outside $C^o$ lies on the line of at most one affine diameter as any two affine diameters have a point in common.", "It is not hard to see, actually, that every point outside $C$ lies on a unique affine diameter." ], [ "Plan of proof", "For $C \\in \\mathcal {C}$ let $\\rho (z)$ denote the radius of curvature of $C$ at $z\\in C$ .", "Let $\\mathcal {D}$ denote the family of all convex curves $C \\in \\mathcal {C}$ such that there is a unique tangent line to $C$ at every $z \\in C$ , $\\lbrace z\\in C: \\rho (z)=0\\rbrace $ is dense in $C$ , $\\lbrace z\\in C: \\rho (z)=\\infty \\rbrace $ is dense in $C$ .", "It is well-known, see for instance [6], that $\\mathcal {D}$ is comeagre in $\\mathcal {C}$ .", "We are going to show that every $C \\in \\mathcal {D}$ has the property required in Theorem REF .", "We will need slightly different conditions for Theorems REF and REF .", "But the basic steps of the proofs are the same.", "We explain them in this section in the case of Theorem REF .", "Let $C \\in \\mathcal {D}$ and define, for $z \\in C$ , the halfline $N^+(z)\\subset N(z)$ that starts at $z$ and intersect the interior of $C^o$ .", "Note that every $u \\in {\\mathbb {R}^2}$ lies on some $N^+(z)$ : namely when the farthest point from $u$ on $C$ is $z$ .", "Set $L(u)=\\lbrace z \\in C: u \\in N^+(z)\\rbrace $ and define $H=\\lbrace u \\in {\\mathbb {R}^2}: L(u) \\mbox{ is not perfect}\\rbrace .$ Lemma 2.1 $H$ is a Borel set.", "Write now $u=(u_1,u_2)\\in {\\mathbb {R}^2}$ and define $H^{u_2}=\\lbrace u_1\\in \\mathbb {R}: (u_1,u_2)\\in H\\rbrace $ .", "This is just the section of $H$ on the horizontal line $\\ell (u_2)=\\lbrace (x,y)\\in {\\mathbb {R}^2}: y=u_2\\rbrace $ .", "There are two points $z \\in C$ with $N(z)$ horizontal, so there are at most two exceptional values for $u_2$ where $\\ell (u_2)$ coincides with some $N(z)$ .", "Lemma 2.2 Apart from those exceptional values, $H^{u_2}$ is meagre.", "These two lemmas imply Theorem REF .", "Indeed, deleting the (one or two) exceptional lines from $H$ gives a Borel set $H^{\\prime }$ .", "According to a theorem of Kuratowski (see [3] page 53), if all horizontal sections of the Borel set $H^{\\prime }$ are meagre, then so is $H^{\\prime }$ , and then $H$ itself is meagre.", "So its complement is comeagre, so $L(u)$ is perfect and non-empty for most $u \\in {\\mathbb {R}^2}$ .", "The theorem follows now from the fact that a non-empty and perfect set has continuum many points.", "The proofs of Theorems REF and REF will use the same argument.", "For the proof of Lemma REF we need another lemma that appeared first as Lemma 2 in [4].", "A function $g:[0,1]\\rightarrow {\\mathbb {R}^2}$ is increasing on an interval $I\\subset [0,1]$ (resp.", "decreasing on $I$ ) if every $x,y \\in I$ with $x \\le y$ satisfy $g(x) \\le g(y)$ (resp.", "$g(x)\\ge g(y))$ , and $g$ is monotone in $I$ if it is either increasing or decreasing there.", "For the sake of completeness we present the short proof.", "Lemma 2.3 Assume $g:[0,1]\\rightarrow {\\mathbb {R}^2}$ is continuous and is not monotone in any subinterval of $[0,1]$ .", "Then the set $B=\\lbrace b \\in \\mathbb {R}: \\lbrace x:g(x)=b\\rbrace \\mbox{ is not perfect}\\rbrace $ is meagre.", "Proof of Lemma REF .", "For each $b \\in B$ the level set $\\lbrace x:g(x)=b\\rbrace $ has an isolated point, and so there is an open interval $I_b \\subset [0,1]$ with rational endpoints in which $g(x)=b$ has a unique solution.", "For a given rational interval $(p,q)$ define $B(p,q)=\\lbrace b\\in B: I_b=(p,q)\\rbrace .$ If every $B(p,q)$ is nowhere dense, then we are done since $B$ , as a countable union of nowhere dense sets, is meagre.", "If some $B(p,q)$ is not nowhere dense, then there is a non-empty open interval $I$ in which $B(p,q)$ is dense.", "The line $y=b$ , for a dense subset of $I$ , intersects the graph of $g$ restricted to $(p,q)$ in a single point.", "This implies easily that $g$ is strictly monotone in a subinterval $(p,q)$ , contrary to our assumption.$\\Box $" ], [ "Proof of the lemmas", "Fix $C \\in \\mathcal {D}$ and let $z(\\alpha )$ denote the point $z \\in C$ where the halfline $N^+(z)$ spans angle $\\alpha \\in [0,2\\pi )$ with a fixed unit vector in ${\\mathbb {R}^2}$ .", "This is a parametrization of $C$ with $\\alpha \\in [0,2\\pi ]$ and $z(0)=z(2\\pi )$ .", "We write $C_{\\alpha ,\\beta }$ for the arc $\\lbrace z(\\gamma ): \\alpha < \\gamma < \\beta \\rbrace $ when $0\\le \\alpha < \\beta \\le 2\\pi $ , and the definition is extended, naturally, to the case when $\\alpha < 2\\pi < \\beta $ .", "We always assume that $\\alpha , \\beta $ are rational and $\\beta -\\alpha $ is small, smaller than $0.1$ , say.", "Proof of Lemma REF .", "Note first that the set $K=\\lbrace (u,z)\\in {\\mathbb {R}^2}\\times C: u \\in N^+(z)\\rbrace $ is closed.", "Further, $L(u)$ is not perfect if and only if there is a short arc $C_{\\alpha ,\\beta }$ such that $u \\in N^+(z)$ for a unique $z \\in C_{\\alpha ,\\beta }$ .", "Thus $H = \\bigcup _{\\mbox{ all }C_{\\alpha ,\\beta }} \\lbrace u \\in {\\mathbb {R}^2}: u \\in N^+(z) \\mbox{ for a unique } z \\in C_{\\alpha ,\\beta }\\rbrace .$ Let $p: K \\rightarrow {\\mathbb {R}^2}$ be the projection $p(u,z)=u$ .", "Let $P_{\\alpha ,\\beta }$ be the set of points $u \\in {\\mathbb {R}^2}$ such that there are more than one $z \\in C_{\\alpha ,\\beta }$ with $u\\in N^+(z)$ .", "Then $P_{\\alpha ,\\beta }=\\bigcup _{\\gamma } p(K\\cap ({\\mathbb {R}^2}\\times C_{\\alpha ,\\gamma }))\\cap p(K\\cap ({\\mathbb {R}^2}\\times C_{\\gamma ,\\beta }))$ where the union is taken over all rational $\\gamma $ with $\\alpha <\\gamma < \\beta $ .", "Consequently $H=\\bigcup _{\\mbox{ all }C_{\\alpha ,\\beta }} p(K\\cap ({\\mathbb {R}^2}\\times C_{\\alpha ,\\beta })) \\setminus P_{\\alpha ,\\beta }.$ Since $p(K\\cap ({\\mathbb {R}^2}\\times C_{\\alpha ,\\beta }))$ is $F_{\\sigma }$ for every $\\alpha < \\beta $ , it follows that $H$ is indeed Borel.", "$\\Box $ Proof of Lemma REF .", "The set $z\\in C$ where $N^+(z)$ intersects $\\ell (u_2)$ in a single point consists of one or two open subarcs of $C$ , as one can check easily.", "Let $C_1$ be such an arc.", "It suffices to see that $E=H^{u_2}\\cap \\lbrace u_1\\in \\mathbb {R}: (u_1,u_2)=\\ell (u_2)\\cap N(z) \\mbox{ for some }z \\in C_1\\rbrace $ is meagre, as $H^{u_2}$ either coincides with this set, or is the union of two such sets.", "We may assume that $C_1$ is the graph of a convex function $F: J \\rightarrow \\mathbb {R}$ and $u_2>F(x)$ on $J$ where $J$ is an open interval.", "(This position can be reached after a suitable reflection about a horizontal line.)", "With this notation, $E$ is the set of real numbers $u_1 \\in \\mathbb {R}$ such that the set of points $x\\in J$ for which $(u_1,u_2)\\in N^+(x,F(x))$ is not perfect.", "Then $F^{\\prime }(x)=f(x)$ is continuous and increasing on $J$ .", "Each $z \\in C_1$ is a point $(x,F(x))$ on the graph of $F$ .", "As $\\rho (z) =(1+f(x))^{3/2}/f^{\\prime }(x)$ , $f^{\\prime }$ equals zero resp.", "infinity on a dense set in $J$ .", "The normal $N(z)$ to $z=(x,F(x))$ has equation $(u_2-F(x))f(x)=x-u_1$ , as one checks readily.", "With $g(x)=(u_2-F(x))f(x)-x$ , $g^{\\prime }(x)=-f(x)^2+(u_2-F(x))f^{\\prime }(x)-1$ and so on a dense set in $J$ the value of $g^{\\prime }(x)$ is positive, and on another dense set in $J$ it is negative.", "So $g$ is not monotone in any subinterval of $J$ .", "Lemma REF implies now that $E$ is meagre.", "$\\Box $" ], [ "Proof of Theorem ", "It is known [6] that for most $C \\in \\mathcal {D}$ there is a dense set $E\\subset C$ such that at each point $z \\in E$ the lower curvatures of radii in both directions $\\rho _i^+(z),\\rho _i^-(z)$ vanish and the upper curvatures of radii in both directions $\\rho _s^+(z),\\rho _s^-(z)$ are infinite.", "We let $\\mathcal {D}_1$ denote the set of all $C \\in \\mathcal {D}$ possessing such a dense set $E$ .", "We are going to show that for each $C \\in \\mathcal {D}_1$ , most points see continuum many images of any given point $v \\in {\\mathbb {R}^2}$ , $v\\notin C$ .", "For $z \\in C$ we define the line $R(z)$ as the reflected copy (with respect to $N(z)$ ) of the line through $v$ and $z$ .", "Note that $R(z)$ depends continuously from $z$ .", "Here we need $v \\notin C$ .", "If $u$ sees an image of $v$ via $z$ , then $u \\in R(z)$ .", "More precisely, $u$ sees an image of $v$ via $z$ iff $u,v$ and $C$ are on the same side of $T(z)$ and $u \\in R(z)$ .", "Let $R^+(z)\\subset R(z)$ be the halfline that starts at $z$ and is on the same side of $T(z)$ as $C$ .", "Also, $R^+(z)$ is well defined for all $z \\in C$ .", "As before, $\\ell (u_2)$ is the horizontal line in ${\\mathbb {R}^2}$ whose points have second coordinate equal to $u_2$ .", "Define, for fixed $u_2 \\in \\mathbb {R}$ , $H^{u_2}=\\lbrace u_1 \\in \\mathbb {R}: (u_1,u_2)\\in H\\rbrace $ .", "This is the same as the set of first coordinates of all $u \\in H\\cap \\ell (u_2)$ .", "In the generic case $R(z)$ is not horizontal and so $R(z)\\cap \\ell (u_2)$ is a single point.", "But we have to deal with non-generic situations, that is, when $R(z)$ is horizontal and so coincides with $\\ell (u_2)$ for some $u_2\\in \\mathbb {R}$ .", "Define $Z=\\lbrace z\\in C: R(z) \\mbox{ is horizontal}\\rbrace $ and $U_2=\\lbrace u_2 \\in \\mathbb {R}: \\ell (u_2)=R(z) \\mbox{ for some } z \\in Z\\rbrace $ .", "Both $Z$ and $U_2$ are closed sets and there is a one-to-one correspondence between them given by $z \\leftrightarrow u_2$ iff $R(z)=\\ell (u_2)$ .", "From now on we assume that $Z$ is nowhere dense.", "We will justify this assumption at the end of the proof.", "Then $U_2$ is also nowhere dense.", "$C\\setminus Z$ is open in $C$ and so its connected components $C_1,C_2,\\ldots $ are open arcs in $C$ , and there are at most countably many of them.", "This time we define $L(u,C_i)$ as the set of $z \\in C_i$ via which $u$ sees an image of $v$ .", "Formally, $L(u,C_i)=\\lbrace z \\in C_i: u \\in R^+(z)\\rbrace $ , and define again, for fixed $u_2 \\in \\mathbb {R}$ , $H_i^{u_2}=\\lbrace u_1\\in \\mathbb {R}: L((u_1,u_2),C_i) \\mbox{ is not perfect}\\rbrace .$ A very similar proof shows that $H_i^{u_2}$ is Borel.", "We omit the details, but mention that the condition $v \\notin C$ is needed to show that the corresponding $K=\\lbrace (u,z):\\dots \\rbrace $ is closed.", "Lemma 4.1 For $u_2 \\notin U_2$ the set $H_i^{u_2}$ is meagre.", "Proof.", "With every $u_1 \\in H_i^{u_2}$ we associate a (rational) open arc $C_{\\alpha ,\\beta }$ of $C_i$ such that $u=(u_1,u_2) \\in R(z)$ for a unique $z \\in C_{\\alpha ,\\beta }$ , namely for $z_u$ .", "If the set of $u\\in H_i^{u_2}$ that are associated with $C_{\\alpha ,\\beta }$ is nowhere dense for every rational arc $C_{\\alpha ,\\beta }$ , then we are done as $H_i^{u_2}$ is the countable union of nowhere dense sets.", "So suppose that it is not nowhere dense for some $C_{\\alpha ,\\beta }$ .", "Then there is an open interval $I \\in \\mathbb {R}$ such that $H_i^{u_2}$ is dense in $I$ .", "Choose two distinct points $w^-,w^+$ from $I \\cap H_i^{u_2}$ .", "Then $z_{(w^-,u_2)}$ and $z_{(w^+,u_2)}$ are distinct points and so they are the endpoints of an open subarc $C_{\\gamma ,\\delta }$ of $C_{\\alpha ,\\beta }$ .", "Define the map $h:C_{\\gamma ,\\delta } \\rightarrow I$ by $h(z)=u_1$ when $(u_1,u_2)=\\ell (u_2)\\cap R(z)$ ; $h$ is clearly continuous.", "It is also monotone because its inverse is well-defined on a dense subset $I$ .", "We show next that this is impossible.", "Choose $z_0 \\in C_{\\gamma ,\\delta } \\cap E$ (recall that $E$ is dense in $C$ ).", "Figure: Theorem We fix a new coordinate system in ${\\mathbb {R}^2}$ : the origin coincides with $z_0$ , the $x$ axis with $T(z_0)$ , the tangent line to $C$ at $z_0$ , and the $y$ axis is $N(z_0)$ ; see the figure.", "We assume w.l.o.g.", "that $v_1<0$ and $v_2>0$ where $v=(v_1,v_2)$ .", "A subarc of $C_{\\gamma ,\\delta }$ is the graph of a non-negative convex function $F:[0,\\Delta )\\rightarrow \\mathbb {R}$ such that $F(0)=0$ and $z=z(x)=(x,F(x))$ and $f(x)=F^{\\prime }(x)$ is an increasing function with $f(0)=0$ .", "If the lines $R(z(x))$ and $R(z(0))$ intersect, then they intersect in a single point whose $y$ component is denoted by $y(x)$ .", "Claim 4.2 For every $\\varepsilon >0$ there are $x_1,x_2 \\in (0,\\varepsilon )$ so that $y(x_1)<0$ and $0<y(x_2)<\\varepsilon $ .", "Proof.", "We use the notation of the figure.", "The sine theorem for the triangle with vertices $v,0,z(x)$ implies that $\\phi (x)=x \\sin \\lambda /\|v\|(1+o(1))$ where $o(1)$ is understood when $x \\rightarrow 0$ .", "The slope of the line $R(z(x))$ is $\\tan (\\lambda -\\phi +2\\psi )$ , and $\\tan \\psi (x)=f(x)=x\\cdot \\frac{f(x)-0}{x-0}.$ The liminf and limsup of the last fraction (when $x\\rightarrow 0$ ) is the curvature $\\rho ^+_i(z_0)=0$ and $\\rho ^+_s(z_0)=\\infty $ of $C$ at $z_0$ as $z_0 \\in E$ .", "Consequently for every integer $n>1$ there is $x\\in (0,1/n)$ with $\\tan \\psi (x)< x/n$ and also with $\\tan \\psi (x)>nx$ .", "Then there is $x_1<1/n$ such that $\\lambda /2 < \\lambda -\\phi (x_1) +2\\psi (x_1)< \\lambda $ which implies, after a simple checking, that $y(x_1)<0$ .", "Also, there is $x_2<1/n$ such that $\\lambda -\\phi (x_2) +2\\psi (x_2)> \\lambda +nx_2/2$ .", "A direct computation shows then that $0<y(x_2)<\\varepsilon $ if $n$ is chosen large enough.$\\Box $ We return to the proof of Lemma REF .", "The claim shows that there are $x_1,x_2,x_3 \\in (0,\\Delta )$ with $x_1<x_2<x_3$ such that the line $R(z(x_1))$ and $R(z(x_3))$ strictly separate the origin and the point $R(z_0)\\cap \\ell (u_2)$ while $R(z(x_2))$ does not.", "Writing $z_i=z(x_i), i=1,2,3$ this implies that $z_2$ is between $z_1$ and $z_3$ while $h(z_2)$ is not on the segment $(h(z_1),h(z_3))$ .", "So $h$ is not monotone.$\\Box $ It is evident that $U_2$ , and consequently $U$ , is closed and nowhere dense, so $U$ is meagre.", "The lemma implies, by Kuratowski's theorem, that $H_i\\setminus U$ is meagre.", "It follows that $H_i$ is meagre and then so is $H=\\bigcup _i H_i$ .", "Thus every point in the complement of $H$ sees an image of $v$ via a perfect set in $C$ , except possibly for the points of the meagre set $U$ .", "This perfect set is nonempty, because every point sees an image of $v$ via some $z \\in C$ (for instance by Zamfirescu's result [5]).", "So most points see continuum many images of $v$ .", "Finally we justify the assumption that $Z$ is nowhere dense.", "This is done by choosing the horizontal direction (which is at our liberty) suitably.", "So for a given direction $(\\cos \\theta ,\\sin \\theta )$ write $Z(\\theta )$ for the set of $z\\in C$ such that $R(z)$ is parallel with this direction.", "Every $Z(\\theta )$ is closed and so there is one (actually, many) among them that contains no non-empty open arc of $C$ .", "Choose the corresponding $\\theta $ for the horizontal direction, then $Z=Z(\\theta )$ is nowhere dense.", "$\\Box $" ], [ "Proof of Theorem ", "Write $\\mathcal {C}_1$ for the set of all convex curves $C$ that have a unique tangent at every $z \\in C$ .", "Assume $C \\in \\mathcal {C}_1$ and use the parametrization $z:[0,2\\pi )\\rightarrow C$ as before.", "For $z\\in C$ with $z=z(\\alpha )$ let $z^\\in C$ be the opposite point, that is $z^=z(\\alpha +\\pi )$ .", "It is evident that $z^{*}=z$ .", "Further, $[z,z^]$ is always an affine diameter of $C$ and all affine diameters of $C$ are of this form.", "We need a geometric lemma.", "Lemma 5.1 Most convex curves $C \\in \\mathcal {C}_1$ have the following property: for every $\\varepsilon > 0$ every subarc $C_0$ of $C$ contains points $x,y$ such that $\\frac{\|x-y\|}{\|x^-y^\|} < \\varepsilon .$ The lemma follows from a result in [1], we give a separate proof in the next section.", "From now on we assume that $C \\in \\mathcal {C}_1$ has the property in the lemma.", "We use again the same proof scheme: for $u \\in C^o$ define $L(u)=\\lbrace z\\in C: u\\in [z,z^]\\rbrace $ ; this set is nonempty as one can check easily that every point $u \\in C^o$ lies on at least one affine diameter.", "(This holds for every convex curve, not only for the ones in $\\mathcal {C}_1$ .)", "We set next $H=\\lbrace u \\in C^o: L(u) \\mbox{ is not perfect}\\rbrace $ , and, for fixed $u_2 \\in {\\mathbb {R}^2}$ , $H^{u_2}=H \\cap \\ell (u_2)$ .", "The same proof as in Section 3 shows that $H$ is Borel.", "We claim that $H$ is meagre which implies Theorem REF .", "$C$ has a horizontal affine diameter and we assume w.l.o.g.", "that it lies on the line $\\ell (0)$ .", "To see that $H$ is meagre it suffices to show (by Kuratowski's theorem) that $H^{u_2}$ is meagre as a subset of $\\ell (u_2)$ for $u_2\\ne 0$ .", "We only consider $u_2 \\in \\mathbb {R}$ , $u_2\\ne 0$ with $\\ell (u_2) \\cap C \\ne \\emptyset $ .", "With each $u \\in H^{u_2}$ we associate an isolated point $z_u \\in C$ and a short rational arc $C_{\\alpha ,\\beta }$ such that $z_u$ is the unique $z \\in C_{\\alpha ,\\beta }$ with $u \\in [z,z^]$ .", "We are done if, for each short rational arc $C_{\\alpha ,\\beta }$ , the set of $u \\in H^{u_2}$ that are associated with $C_{\\alpha ,\\beta }$ is nowhere dense.", "So suppose that this fails for some $C_{\\alpha ,\\beta }$ .", "Then there is an open interval $I \\subset \\ell (u_2)$ on which $H^{u_2}$ is dense.", "Choose distinct points $u^-$ and $u^+$ from $I \\cap H^{u_2}$ and let $z^-,z^+$ be the corresponding isolated points on $C_{\\alpha ,\\beta }$ .", "We suppose (by symmetry) that $C_{\\alpha ,\\beta }$ is below the line $\\ell (u_2)$ .", "From now on we consider the subarc $C_0 \\subset C_{\\alpha ,\\beta }$ whose endpoints are $z^-$ and $z^+$ and its opposite arc $C_0^$ .", "We note here that the map $z\\rightarrow z^$ is order preserving on $C_0$ , meaning that if $v\\in C_0$ is between $v_1,v_2 \\in C_0$ , then $v^$ lies between $v_1^$ and $v_2^$ on $C_0^$ .", "Define a map $m:C_0 \\rightarrow \\ell (u_2)$ via $m(z)=\\ell (u_2)\\cap [z,z^]$ ; $m$ is continuous.", "It is one-to-one on a dense subset of $C_0$ which implies that $m$ is order-preserving in the sense that if $v\\in C_0$ is between $v_1,v_2 \\in C_0$ , then $m(v)$ lies between $m(v_1)$ and $m(v_2)$ on $\\ell (u_2)$ .", "We show that this is impossible.", "Using Lemma REF choose two points $v_1,v_2$ on $C_0$ very close to each other so that $\|v_1-v_2\|$ is much shorter than $\|v_1^-v_2^\|$ .", "Then the segment $[v_1,v_2]$ is almost parallel with $[v_1^,v_2^]$ , and the diameters $[v_1,v_1^]$ and $[v_2,v_2^]$ intersect in a point very close to $[v_1,v_2]$ , so this point is below $\\ell (u_2)$ .", "Now apply Lemma REF on the arc between $v_1^$ and $v_2^$ .", "We get points $w_1$ and $w_2$ very close to each other on this arc so that $\|w_1-w_2\|$ is much shorter than $\|w_1^-w_2^\|$ .", "This time the diameters $[w_1,w_1^]$ and $[w_2,w_2^]$ intersect above $\\ell (u_2)$ .", "We assume (by choosing the names $w_1,w_2$ properly) that $v_1^,w_1,w_2,v_2^$ come in this order on $C_0^$ and so $v_1,w_1^,w_2^,v_2$ come in this order on $C_0$ .", "The order of their $m$ -images on $\\ell (u_2)$ is $m(v_1),m(w_2^),m(w_1^),m(v_2)$ .", "Thus indeed, $m$ is not order preserving.$\\Box $" ], [ "Proof of Lemma ", "Given $C \\in \\mathcal {C}_1$ define $A_{k,n}$ as the short arc between $z_k=z(2\\pi k/2n)$ and $z_{k+1}=z(2\\pi (k+1)/2n)$ where $k=0,1,\\ldots ,2n-1$ .", "For positive integers $n,m$ let $\\mathcal {F}_{n,m}$ be the set of all $C \\in \\mathcal {C}_1$ for which there is $A_{k,n}$ such that for all $x,y \\in A_{k,n}$ ($x\\ne y$ ) $\\frac{\|x-y\|}{\|x^-y^\|}\\ge \\frac{1}{m}.$ It is easy to see that $\\mathcal {F}_{n,m}$ is closed in $\\mathcal {C}_1$ , we omit the details.", "We show next that it is nowhere dense.", "Fix a $C \\in \\mathcal {C}_1$ and $\\varepsilon >0$ and let $U(C)$ denote the $\\varepsilon $ -neighbourhood of $C$ .", "We construct another convex curve $\\Gamma \\in \\mathcal {C}_1$ that is contained in $U(C)$ but is not an element of $\\mathcal {F}_{n,m}$ .", "Fix $k\\in \\lbrace 0,1,\\ldots ,n-1\\rbrace $ and consider a fixed arc $A_{k,n}$ and its opposite arc $A^_{k,n}=A_{k+n,n}$ .", "Let $T_k$ be the tangent line to $C$ at $z((k+\\frac{1}{2})\\pi /n)$ and $T_k^$ be the parallel tangent line at $z((k+n+\\frac{1}{2})\\pi /n)$ .", "Translate $T_k^$ a little so that the translated copy intersects $C$ in two points $x_1,y_1$ and the segment $[x_1,y_1]$ lies in $U(C)$ and is much shorter than $[z_{k+n},z_{k+1+n}]$ .", "Similarly translate $T_k$ a little so that the translated copy intersects $C$ in $x_2,y_2$ and $[x_2,y_2]$ lies in $U(C)$ , and is much shorter than $[z_k,z_{k+1}]$ and, most importantly, it is much shorter than $[x_1,y_1]$ , namely, $m\|x_2-y_2\| <\|x_1-y_1\|$ .", "This is clearly possible.", "Now we choose points $w_1$ resp.", "$w_2$ from the caps cut off from $C^o$ by the segment $[x_1,y_1]$ and $[x_2,y_2]$ so that, for $i=1,2$ , the triangles $\\triangle _i=\\textrm {conv}\\lbrace x_i,y_i,w_i\\rbrace $ are homothetic.", "This is possible again.", "Note that $[x_1,w_1]$ and $[x_2,w_2]$ are parallel, and so are $[y_1,w_1]$ and $[y_2,w_2]$ .", "The next target is construct a convex curve $\\Gamma _k$ from $z_k$ to $z_{k+1}$ going through $x_2$ and $y_2$ that lies in $U(C)$ , has a unique tangent at every point, and this tangent coincides with the line through $x_2,w_2$ at $x_2$ and with the line through $y_2,w_2$ at $y_2$ .", "Also, an analogous curve $\\Gamma _{k+n}$ is needed from $z_{k+n}$ to $z_{k+1+n}$ .", "This is quite easy.", "The unique parabola arc connecting $x_2$ to $y_2$ within $\\triangle _2$ that touches the sides $[x_2,w_2]$ at $x_2$ and $[w_2,y_2]$ at $y_2$ is the middle piece of $\\Gamma _k$ .", "To connect this arc by a convex curve to $z_k$ (say) within $U(C)$ choose a point $w\\in C$ on the arc between $z_k$ and $y_2$ so close to $y_2$ that the triangle $\\triangle $ delimited by $T(z)$ , the line through $y_2,w_2$ , and the segment $[y_2,z]$ lies in $U(C)$ .", "The analogous parabola arc in $\\triangle $ gives the next piece of $\\Gamma _k$ , and then add to this piece the subarc of $C$ between $w$ and $z_k$ .", "The middle piece of $\\Gamma _k$ is continued to $z_{k+1}$ the same way.", "The convex curve $\\Gamma _{k+n}$ connecting $z_{k+n}$ to $z_{k+1+n}$ is constructed the same way.", "Note that the tangents to $\\Gamma _k$ at $x_2$ (resp.", "$y_2$ ) are parallel with the tangents to $\\Gamma _{k+n}$ at $x_1$ (and $y_1$ ).", "The curves $\\Gamma _k$ for $k=0,\\ldots ,2n-1$ together form a convex curve $\\Gamma \\in C_1$ .", "It has parallel tangents at $x_1\\in \\Gamma _{k+n}$ and $x_2\\in \\Gamma _k$ , and also at $y_1$ and $y_2$ .", "Thus $[x_1,x_2]$ and $[y_1,y_2]$ are affine diameters of $\\Gamma $ and $m\|x_1-y_1\|<\|x_2-y_2\|$ .", "As this holds for every $k$ , $\\Gamma \\notin \\mathcal {F}_{n,m}$ .", "Thus $\\mathcal {F}_{n,m}$ is indeed nowhere dense.", "It follows that $\\mathcal {C}_2=\\mathcal {C}_1 \\setminus \\bigcup _{n,m}\\mathcal {F}_{n,m}$ is comeagre in $\\mathcal {C}_1$ .", "We show next that every $C \\in \\mathcal {C}_2$ satisfies the requirement of the lemma.", "So we are given $\\varepsilon >0$ and a short subarc $C_0$ of $C$ .", "Take a positive integer $m$ with $1/m<\\varepsilon $ .", "For a suitably large $n$ , $C_0$ contains an arc of the form $A_{k,n}$ .", "As $C \\notin \\mathcal {F}_{n,m}$ , there are distinct points $x,y \\in A_{k,n}$ with $\\frac{\|x-y\|}{\|x^-y^*\|}\\le \\frac{1}{m}< \\varepsilon .$ This finishes the proof.", "$\\Box $ Acknowledgements.", "Research of the first author was partially supported by ERC Advanced Research Grant 267165 (DISCONV), and by Hungarian National Research Grant K 83767.", "The second author was partially supported by the Hungarian National Foundation for Scientific Research Grant K 104178." ] ]	1403.0246
[ [ "Deriving Fa\\`a di Bruno's formula for the derivative of a composite\n function via compositions of integers" ], [ "Abstract We give yet another proof for Fa\\`{a} di Bruno's formula for higher derivatives of composite functions.", "Our proof technique relies on reinterpreting the composition of two power series as the generating function for weighted integer compositions, for which a Fa\\`{a} di Bruno-like formula is quite naturally established." ], [ "Introduction", "According to Faà di Bruno's formula, the $n$ th derivative of a composite function $G\\circ F$ is given by $\\frac{d^n}{dx^n}G(F(x))=\\sum \\frac{n!", "}{b_1!\\cdots b_n!", "}G^{(r)}(F(x))\\prod _{i=1}^n\\Bigl (\\frac{F^{(i)}(x)}{i!", "}\\Bigr )^{b_i},$ where the sum ranges over all different solutions in nonnegative integers $b_1,\\ldots ,b_n$ of $b_1+2b_2+\\cdots +nb_n=n$ and where $r$ is defined as $r=b_1+\\cdots +b_n$ .", "For example, for $n=3$ , the three solutions for $(b_1,b_2,b_3)$ are $(0,0,1)$ , $(1,1,0)$ and $(3,0,0)$ , which correctly yields $G^{\\prime }(F(x))\\cdot F^{\\prime \\prime \\prime }(x) + 3G^{\\prime \\prime }(F(x))\\cdot F^{\\prime }(x)F^{\\prime \\prime }(x)+G^{\\prime \\prime \\prime }(F(x))\\cdot (F^{\\prime }(x))^3$ as third derivative of $G\\circ F$ .", "Many proofs of formula (REF ) have been given, both based on combinatorial arguments — such as via Bell polynomials [3] or set partitions — as well as on analytical; the latter, for example, using Taylor's theorem [7].", "Roman [8] gives a proof based on the umbral calculus.", "Johnson [7] summarizes the historical discoveries and re-discoveries of the formula as well as a variety of different proof techniques.", "Herein, we give (yet) another proof of the formula, one that is based on the combinatorics of integer compositions and a particular interpretation of the composition of power series.", "The essence of our derivation is as follows: First, we consider the number $C_{f,g}(n)$ of (doubly weighted) integer compositions of the positive integer $n$ , for which we derive a closed-form formula; this requires some notation and introduction of terminology, but the derivation and combinatorial interpretation of the formula is quite intuitive.", "Then, for two arbitrary power series $G(x)=\\sum _{n\\ge 0} g_nx^n$ and $F(x)=\\sum _{n\\ge 0}f_nx^n$ , we argue that $G\\circ F$ has a natural interpretation of denoting the generating function $C(x)=\\sum _{n\\ge 0} C_{f,g}(n)x^n$ for $C_{f,g}(n)$ .", "Hence, $\\frac{1}{n!", "}\\frac{d^n}{dx^n}C(0)=C_{f,g}(n)$ .", "This yields formula (REF ) for $x=0$ , but we argue that it is clear that the formula must indeed hold for any $x$ .", "Two remarks are in order: first, as indicated, our derivation does not apply to arbitrary functions $F$ and $G$ , but only to power series.", "While this may be considered a restriction, many interesting functions can indeed be represented as power series (those functions even have a name, real analytical functions).", "We also remark that, throughout, we ignore matters of convergence and treat all series as formal and assume that functions have sufficiently many derivatives.", "Finally, while we think that many derivations of Faà di Bruno's formula given in the literature are similar to the one we outline, we believe the particular approach that we suggest, based on integer compositions and a reinterpretation of the composition of power series, to be novel.Technically, the approach most similar to our own appears to be the one due to Flanders [5], which is, however, conceptually substantially different from our own." ], [ "Integer compositions and partitions", "An integer composition of a positive integer $n$ is a tuple of positive integers $(\\pi _1,\\ldots ,\\pi _k)$ , typically called parts, whose sum is $n$ .", "For example, the eight integer compositions of $n=4$ are $(4),(1,3),(3,1),(2,2),(1,1,2),(1,2,1),(2,1,1),(1,1,1,1).$ An integer partition of $n$ is a tuple of positive integers $(\\pi _1,\\ldots ,\\pi _k)$ whose sum is $n$ and such that $\\pi _1\\ge \\pi _2\\ge \\cdots \\ge \\pi _k$ .", "For instance, there are (only) five integer partitions of $n=4$ , namely $(4),(3,1),(2,2),(2,1,1),(1,1,1,1).$ Both integer compositions and partitions are well-studied objects in combinatorics [2], [6].", "Instead of considering ordinary partitions and compositions as defined, we may consider weighted compositions [1], [4] and partitions, where each part value $\\pi _i\\in \\mathbb {N}=\\lbrace 1,2,3,\\ldots \\rbrace $ may have attributed with it a weight $f(\\pi _i)\\in \\mathbb {R}$ , where $\\mathbb {R}$ denotes the set of real numbers.", "If weights are nonnegative integers, they may be interpreted as colors.", "For instance, when $f(3)=2$ and $f(1)=f(2)=f(4)=f(5)=\\cdots =1$ , then there are ten $f$ -weighted compositions and six $f$ -weighted partitions of $n=4$ .", "These are $(4),(1,3),(1,3^),(3,1),(3^,1),(2,2),(1,1,2),(1,2,1),(2,1,1),(1,1,1,1)$ and $(4),(3,1),(3^,1),(2,2),(2,1,1),(1,1,1,1),$ respectively, where we use a star ($$ ) to differentiate between the two colors of part value 3.", "When weights are nonintegral real numbers, they may simply be interpreted as ordinary `weights' — possibly as probabilities if the range of $f$ is the unit interval $[0,1]$ .", "Let us note that integer partitions of an integer $n$ admit an alternative, equivalent representation.", "Instead of writing a partition of $n$ as a tuple $(\\pi _1,\\ldots ,\\pi _k)$ with $\\pi _1\\ge \\cdots \\ge \\pi _k$ , we may represent it as a tuple $(k_1,\\ldots ,k_n)$ , with $0\\le k_i\\le n$ , for all $i=1,\\ldots ,n$ , whereby $k_i$ denotes the multiplicity of (type) $i\\in \\lbrace 1,2,\\ldots ,n\\rbrace $ in the composition of $n$ .", "For instance, the above five integer partitions of $n=4$ may be represented as $(0,0,0,1),(1,0,1,0),(0,2,0,0),(2,1,0,0),(4,0,0,0).$ Obviously, each such tuple $(k_1,\\ldots ,k_n)$ must satisfy $1\\cdot k_1+2\\cdot k_2+\\cdots +n\\cdot k_n=n$ .", "Assuming that the weighting function $f$ takes on only integral values, for the moment, how many $f$ -weighted integer partitions of $n$ are there?", "Apparently, this number is given by $\\sum _{k_1+2k_2+\\cdots +nk_n=n}f(1)^{k_1}\\dots f(n)^{k_n},$ since the solutions, in positive numbers, of $k_1+2k_2+\\cdots +nk_n=n$ are precisely the integer partitions of $n$ and the product $f(1)^{k_1}\\cdots f(n)^{k_n}$ assigns the different colors to a given partition $(k_1,\\ldots ,k_n)$ .", "How many $f$ -weighted integer compositions of $n$ are there?", "Note that, in the representation $(k_1,\\ldots ,k_n)$ of a partition, $k_1$ denotes the number of `type' 1, $k_2$ denotes the number of `type' 2, ..., and $k_n$ denotes the number of `type' $n$ used in the partition of $n$ .", "Since compositions are ordered partitions, for compositions, we need to distribute the $k_1$ types 1, ..., $k_n$ types $n$ in a sequence of length $(k_1+\\cdots +k_n)$ .", "Therefore, the number of $f$ -weighted integer compositions is simply: $\\sum _{k_1+2k_2+\\cdots +nk_n=n}\\binom{k_1+\\cdots +k_n}{k_1,\\ldots ,k_n}f(1)^{k_1}\\dots f(n)^{k_n},$ where $\\binom{r}{k_1,\\ldots ,k_n}=\\frac{r!", "}{k_1!\\cdots k_n!", "}$ (for $r=k_1+\\cdots +k_n$ ) denote the multinomial coefficients.", "Finally, let us assume that integer partitions/compositions with a given, fixed number $k$ of parts are (additionally) weighted (e.g., colored) by $g(k)$ , for a weighting function $g:\\mathbb {N}\\rightarrow \\mathbb {R}$ .", "For instance, we might double count the $f$ -weighted partitions/compositions with exactly $k_1+\\ldots +k_n=4$ parts (or assign them higher/lower probability).", "Then, the number of $f$ -weighted integer compositions of $n$ where parts are $g$ -weighted is simply given by $C_{f,g}(n)=\\sum _{k_1+2k_2+\\cdots +nk_n=n}\\binom{k_1+\\cdots +k_n}{k_1,\\ldots ,k_n}g(k_1+\\cdots + k_n)\\prod _{i=1}^n f(i)^{k_i}.$ If $f$ and/or $g$ take on nonintegral values, (REF ) and (REF ) denote the total weight of all $f$ -weighted integer partitions/compositions, and (REF ) denotes the total weight of all $f$ -weighted integer compositions of $n$ where parts are $g$ -weighted.", "Henceforth, for brevity, we also call such compositions simply $(f,g)$ -weighted." ], [ "Derivation of Faà di Bruno's formula", "We assume that $F(x)$ and $G(x)$ are the power series $F(x) &= f_0+f_1x^1+f_2x^2+\\ldots = \\sum _{n\\ge 0} f_nx^n,\\\\G(x) &= g_0+g_1x^1+g_2x^2+\\ldots = \\sum _{n\\ge 0}g_nx^n,$ for some real coefficients $f_0,f_1,f_2,\\ldots $ and $g_0,g_1,g_2,\\ldots $ .", "In the remainder, for ease of interpretation, we speak of the $f_n$ and $g_n$ values as if they were nonnegative and integral, but keep in mind that they may be arbitrary real numbers.", "We interpret $F$ and $G$ as follows.", "The function $F$ is the generating function for the number of $f$ -weighted integer compositions of $n$ with exactly one part, whereby $f(n)=f_n$ .", "In fact, the coefficient $f_n$ of $x^n$ of $F(x)$ gives the number of $f$ -weighted integer compositions of $n$ with exactly one part.", "We assume that $f_0=0$ (that is, integer compositions admit only positive integers).", "In the context $G\\circ F$ , we interpret the function $G$ as follows: $G\\circ F$ represents, for $G(x)=x^k$ , the generating function for the number of $f$ -weighted integer compositions with exactly $k$ parts; for $G(x)=a_kx^k$ , it represents the generating function for the number of $f$ -weighted integer compositions with exactly $k$ parts, where $f$ -weighted compositions with $k$ parts are weighted by the factor $a_k$ ; and, finally, for $G(x)=x^j+x^k$ , it represents the generating function for the number of $f$ -weighted integer compositions with either $j$ or $k$ parts (union).", "This interpretation of $G$ , in the context $G\\circ F$ , is a natural interpretation, since, for example, the coefficients of $x^n$ of $(F(x))^2$ have the form $\\sum _{i=0}^nf_{n-i}f_i$ , and all combinations of the number of $f$ -weighted compositions of $n-i$ with one part and the number of $f$ -weighted compositions of $i$ with one part yield the number of $f$ -weighted compositions of $(n-i)+i=n$ with two parts.This is in fact a critical point of our proof; if we interpreted $F(x)$ as the generating function for other combinatorial objects, such as integer partitions, then $G(x)=x^k$ , in the context $G\\circ F$ , could not have the same interpretation as the one we have outlined.", "Then, the interpretation of $(F(x))^k$ follows inductively.", "Similarly, if $(F(x))^k$ denotes the generating function for the number of $f$ -weighted compositions of $n$ with exactly $k$ parts and $(F(x))^j$ denotes the analogous generating function for $j$ parts, then their sum obviously denotes the generating function for $k$ or $j$ parts.", "Hence, to summarize, we interpret $G\\circ F$ as the generating function for the number of $f$ -weighted integer compositions with arbitrary number of parts (recall that the `$+$ ' mean union over number of parts) and where compositions with $k$ parts are weighted by $g(k)=g_k$ .", "Then, by virtue of the definition of generating functions, we know that $\\frac{1}{n!", "}\\frac{d^n}{dx^n}(G\\circ F)(0)$ gives the number of $(f,g)$ -weighted integer compositions of $n$ .", "This is the number $C_{f,g}(n)$ , whence by formula (REF ), we know that $\\frac{1}{n!", "}\\frac{d^n}{dx^n}(G\\circ F)(0) = C_{f,g}(n)= \\sum _{k_1+2k_2+\\cdots +nk_n=n}\\binom{k_1+\\cdots +k_n}{k_1,\\ldots ,k_n}g(k_1+\\cdots + k_n)\\prod _{i=1}^n f(i)^{k_i},$ or, equivalently, $\\frac{d^n}{dx^n}(G\\circ F)(0) = \\sum _{k_1+2k_2+\\cdots +nk_n=n}\\frac{n!", "}{k_1!\\cdots k_n!", "}r!g(r)\\prod _{i=1}^n f(i)^{k_i},$ where we write $r=k_1+\\cdots +k_n$ .", "We note that $f(i) &= \\frac{1}{i!", "}F^{(i)}(0), \\quad \\forall \\:i=1,\\ldots ,n,\\\\r!g(r) &= G^{(r)}(0) = G^{(r)}(F(0)),$ whence we can rewrite (REF ) as $\\frac{d^n}{dx^n}(G\\circ F)(0) = \\sum _{k_1+2k_2+\\cdots +nk_n=n}\\frac{n!", "}{k_1!\\cdots k_n!", "}G^{(r)}(F(0))\\prod _{i=1}^n \\bigl (\\frac{F^{(i)}(0)}{i!", "}\\bigr )^{k_i},$ which is Faà di Bruno's formula (REF ) evaluated at $x=0$ .", "Now, from $(G\\circ F)^{\\prime }(x)=G^{\\prime }(F(x))\\cdot F^{\\prime }(x)$ , and then $(G\\circ F)^{\\prime \\prime }(x)=G^{\\prime \\prime }(F(x))F^{\\prime }(x)+G^{\\prime }(F(x))F^{\\prime \\prime }(x)$ , etc., it is clear that $\\frac{d^n}{dx^n}(G\\circ F)(x)$ is a sum of products of factors $G^{(j)}(F(x))$ and $F^{(m)}(x)$ .", "It is also clear that, whatever the precise form of $\\frac{d^n}{dx^n}(G\\circ F)(x)$ , evaluating it at $x=0$ will simply yield the same sum of products of factors $G^{(j)}(F(x))$ and $F^{(m)}(x)$ , evaluated at $x=0$ .", "Hence, (REF ) must in fact hold for all $x$ , not only for $x=0$ ." ], [ "Discussion", "We argued that $G\\circ F$ has, for arbitrary power series $G$ and $F$ with coefficients $g_n$ and $f_n$ , respectively, a natural interpretation as denoting the generating function for $(f,g)$ -weighted integer compositions, whereby $f(n)=f_n$ and $g(n)=g_n$ , for whose coefficients Faà di Bruno-like formulas quite effortlessly arise." ] ]	1403.0519
[ [ "Explicit metrics for a class of two-dimensional superintegrable systems" ], [ "Abstract We obtain, in local coordinates, the explicit form of the two-dimensional, super-integrable systems of Matveev and Shevchishin involving cubic integrals.", "This enables us to determine for which values of the parameters these systems are indeed globally defined on the two-sphere." ], [ "Introduction", "The study of superintegrable dynamical systems has received many important developments reviewed recently in [6].", "While integrable systems on the cotangent bundle $T^M$ of a $n$ -dimensional manifold, $M$ , require a set of functionally independent observables $(H,Q_1,\\ldots ,Q_{n-1})$ which are all in involution for the Poisson bracket $\\lbrace \\,\\cdot \\,,\\,\\cdot \\,\\rbrace $ , a superintegrable system is made out of $\\nu \\ge n$ functionally independent observables $H,\\quad \\quad Q_1,\\quad \\quad Q_2, \\quad \\cdots \\quad Q_{\\nu -1},$ with the constraints $\\lbrace H,Q_i\\rbrace =0,\\qquad \\hbox{for all\\ }i=1,2,\\ldots ,\\nu -1.$ The maximal value of $\\nu $ is $2n-1$ since the system (REF ) reads $dH(X_{Q_i})=0$ , implying that the span of the Hamiltonian vector fields, $X_{Q_i}$ , is, at each point of $T^M$ , a subspace of the annihilator of the 1-form $dH$ , the latter being of dimension $2n-1$ .", "Let us observe that for two-dimensional manifolds, a superintegrable system is necessarily maximal since $\\nu =3$ .", "As is apparent from [6], the large amount of results for superintegrable models is restricted to quadratically superintegrable ones, which means that the integrals $Q_i$ are either linear or quadratic in the momenta, and the metrics on which these systems are defined are either flat or of constant curvature.", "For manifolds of non constant curvature, Koenigs [3] gave examples of quadratically superintegrable models.", "For some special values of the parameters the metrics happen to be defined on a manifold, $M$ , which is never closed (compact without boundary).", "In their quest for superintegrable systems defined on closed manifolds, Matveev and Shevchishin [4] have given a complete classification of all (local) Riemannian metrics on surfaces of revolution, namely $G=\\frac{dx^2+dy^2}{h_x^2},\\quad \\quad \\quad \\quad h=h(x),\\quad \\quad h_x=\\frac{dh}{dx},$ which have a superintegrable geodesic flow (whose Hamiltonian will henceforth be denoted by $H$ ), with integrals $L=P_y$ and $S$ respectively linear and cubic in momenta, opening the way to the new field of cubically superintegrable models.", "Let us first recall their main results.", "They proved that if the metric $G$ is not of constant curvature, then ${\\cal I}^3(G)$ , the linear span of the cubic integrals, has dimension 4 with a natural basis $L^3,LH,S_1,S_2$ , and with the following structure.", "The map ${\\cal L}:S\\rightarrow \\lbrace L,S\\rbrace $ defines a linear endomorphism of ${\\cal I}^3(g)$ and one of the following possibilities hold: (i) ${\\cal L}$ has purely real eigenvalues $\\pm \\mu $ for some real $\\mu >0$ , then $S_1,\\,S_2$ are the corresponding eigenvectors.", "(ii) ${\\cal L}$ has purely imaginary eigenvalues $\\pm i\\mu $ for some real $\\mu >0$ , then $S_1 \\pm iS_2$ are the corresponding eigenvectors.", "(iii) ${\\cal L}$ has the eigenvalue $\\mu =0$ with one Jordan block of size 3, in this case $\\lbrace L,S_1\\rbrace =\\frac{A_3}{2}\\,L^3+A_1\\,LH,\\quad \\quad \\quad \\quad \\lbrace L,S_2\\rbrace =S_1,$ for some real constants $A_1$ and $A_3$ .", "Superintegrability is then achieved provided the function $h$ be a solution of following non-linear first-order differential equations, namely $\\begin{array}{crcl}(i)& \\quad \\quad h_x(A_0\\,h_x^2+\\mu ^2\\,A_0\\,h^2-A_1\\,h+A_2) &=& \\displaystyle A_3\\,\\frac{\\sin (\\mu \\,x)}{\\mu }+A_4\\,\\cos (\\mu \\,x)\\\\[4mm] (ii)&\\quad \\quad h_x(A_0\\,h_x^2-\\mu ^2\\,A_0\\,h^2-A_1\\,h+A_2) &=& \\displaystyle A_3\\,\\frac{\\sinh (\\mu \\,x)}{\\mu }+A_4\\,\\cosh (\\mu \\,x)\\\\[4mm](iii)& \\quad \\quad h_x(A_0\\,h_x^2-A_1\\,h+A_2) &=& A_3\\,x+A_4\\end{array}$ and the explicit form of the cubic integrals was given in all three cases.", "For instance, when $\\mu =1$ or $\\mu =i$ , their structure is $S_{1,2}=e^{\\pm \\mu y}\\left(a_0(x)\\,P_x^3+a_1(x)\\,P_x^2\\,P_y+a_2(x)\\,P_x\\,P_y^2+a_3(x)\\,P_y^3\\right),$ where the $a_i(x)$ are explicitly expressed in terms of $h$ and its derivatives; see [4].", "For $A_0=0$ these equations are easily integrated and one obtains the Koenigs metrics [3], while the cubic integrals have the reducible structure $S_{1,2}=P_y\\,Q_{1,2}$ where the quadratic integrals $Q_{1,2}$ are precisely those obtained by Koenigs.", "Furthermore it was proved that in the case $(ii)$ , under the conditions $\\mu >0,\\quad \\quad \\quad \\quad A_0>0,\\quad \\quad \\quad \\quad \\mu \\,A_4>\|A_3\|,$ the metric and the cubic integrals are real-analytic and globally defined on ${\\mathbb {S}}^2$ .", "The aim of this article is on the one hand to integrate explicitly the three differential equations in (REF ) and, on the other hand, to determine, by a systematic case study, all special cases which lead to superintegrable models globally defined on simply-connected, closed, Riemann surfaces.", "In Section we analyze the trigonometric case (real eigenvalues), integrating explicitly the differential equation (REF ,$i$ ) to get an explicit local form for the metric and the cubic integrals.", "The global questions are then discussed, and we show that there is no closed manifold, $M$ , on which the superintegrable model under consideration can be defined.", "In Section we investigate the hyperbolic case (purely imaginary eigenvalues).", "Here too, the integration of the differential equation (REF ,$ii$ ) provides an explicit form for both the metric and the cubic integrals.", "The previous results allows the determination of all superintegrable systems globally defined on ${\\mathbb {S}}^2$ , and these are proved in Theorem REF and Theorem REF , namely Theorem 1 The metric $G=\\rho ^2\\,\\frac{dv^2}{D}+\\frac{4D}{P}\\,d\\phi ^2,\\quad \\quad \\quad \\quad v\\in (a,1),\\quad \\quad \\quad \\quad \\phi \\in {\\mathbb {S}}^1,$ with $D=(v-a)(1-v^2),\\quad \\quad P=(v^2-2av+1)^2,\\quad \\quad -\\rho =1+4\\frac{(v-a)D}{P},$ is globally defined on ${\\mathbb {S}}^2$ , as well as the Hamiltonian $H=\\frac{1}{2}\\,G^{ij}P_iP_j=\\frac{1}{2}\\left(\\Pi ^2+\\frac{P}{4D}P_{\\phi }^2\\right),\\quad \\quad \\Pi =\\frac{\\sqrt{D}}{\\rho }\\,P_v,$ iff $a\\in (-1,+1)$ .", "The two cubic integrals $S_1$ and $S_2$ , also globally defined on ${\\mathbb {S}}^2$ , read $S_1=\\cos \\phi \\,{\\cal A}+\\sin \\phi \\,{\\cal B},\\quad \\quad S_2=-\\sin \\phi \\,{\\cal A}+\\cos \\phi \\,{\\cal B},$ where ${\\cal A}=\\Pi ^3-f\\,f^{\\prime \\prime }\\,\\Pi \\,P_{\\phi }^2,\\quad \\quad {\\cal B}=f^{\\prime }\\,\\Pi ^2\\,P_{\\phi }-f\\,(1+f^{\\prime }\\,f^{\\prime \\prime })\\,P_{\\phi }^3,\\quad \\quad \\quad \\quad f=\\sqrt{D}.$ Theorem 2 The metric $G=\\rho ^2\\,\\frac{dx^2}{D}+\\frac{4D}{P}\\,d\\phi ^2,\\quad \\quad \\rho =\\frac{Q}{P},\\quad \\quad x\\in (-1,+1),\\quad \\quad \\phi \\in {\\mathbb {S}}^1,$ with $\\left\\lbrace \\begin{array}{ll}D=(x+m)(1-x^2),& \\\\[4mm]P=\\Big (L_+\\,(1-x^2)+2(m+x)\\Big )\\Big (L_-\\,(1-x^2)+2(m+x)\\Big ),& \\quad \\quad L_{\\pm }=l\\pm \\sqrt{l^2-1},\\\\[4mm]Q=3x^4+4mx^3-6x^2-12mx-4m^2-1,&\\end{array}\\right.$ is globally defined on ${\\mathbb {S}}^2$ , as well as the Hamiltonian $H=\\frac{1}{2}\\,G^{ij}P_iP_j=\\frac{1}{2}\\left(\\Pi ^2+\\frac{P}{4D}P_{\\phi }^2\\right),\\quad \\quad \\Pi =\\frac{\\sqrt{D}}{\\rho }\\,P_x,$ iff $m>1$ , and $l>-1$ .", "The two cubic integrals $S_1$ and $S_2$ , still given by the formulas (REF ) and (REF ), are also globally defined on ${\\mathbb {S}}^2$ .", "In Subsection REF we compare of our results with those of Matveev and Shevchishin [4].", "In particular, for a convenience of the reader, we provide the transition formulas between the coordinates and functions used in [4] and the coordinates and function used in the present paper.", "In Section we analyze the affine case (zero eigenvalue).", "As in the trigonometric case, the system is never defined on closed manifolds but we determine in which cases it is globally defined either on ${\\mathbb {R}}^2$ or on ${\\mathbb {H}}^2$ .", "In Section we draw some conclusions and present some possibly interesting strategy for future developments.", "Acknowledgements: We wish to warmly thank V. Matveev, and J.-P. Michel for their interest in this work, and for enlightening discussions." ], [ "The explicit form of the metric", "The ode (REF ,$i$ ) obtained in [4] is: $h_x\\Big (A_0\\,h_x^2+\\mu ^2\\,A_0\\,h^2-A_1\\,h+A_2\\Big )=A_3\\frac{\\sin (\\mu \\,x)}{\\mu }+A_4\\,\\cos (\\mu \\,x).$ For the Koenigs metrics $A_0=0$ ; we thus must consider here a non-vanishing $A_0$ which can be scaled to 1.", "By a scaling of $x$ we can also set $\\mu =1$ .", "By a translation of $x$ and a scaling of $h$ the right-hand side becomes $\\lambda \\,\\sin x$ , with $\\lambda $ a free real parameter.", "By a translation of $h$ , we can set $A_1=0$ and $A_2=a$ .", "We hence have to solve $h_x(h_x^2+h^2+a)=\\lambda \\,\\sin x,\\quad \\quad \\quad \\quad a\\in {\\mathbb {R}},\\quad \\quad \\lambda \\in {\\mathbb {R}}\\backslash \\lbrace 0\\rbrace .$ Let us regard now $u=h_x$ as a function of the variable $h$ and define $U=u(u^2+h^2+a)\\quad \\quad \\mbox{with}\\quad \\quad \\frac{d^2U}{dx^2}+U=0.$ This last relation, when expressed in terms of the variable $h$ becomes then $\\frac{d}{dh}\\left(u\\,\\frac{dU}{dh}\\right)+u^2+h^2+a=0,\\quad \\quad \\quad \\quad a\\in {\\mathbb {R}},$ and can be integrated, yielding $4hu\\,\\frac{dU}{dh}=c+(u^2+h^2+a)(3u^2-h^2-a).$ Since $U=\\lambda \\,\\sin x$ we have also a first order equation $U^{\\prime 2}=\\lambda ^2-U^2\\quad \\quad \\Rightarrow \\quad \\quad \\left(4hu\\,\\frac{dU}{dh}\\right)^2=16h^2\\,(\\lambda ^2-U^2),$ and upon using (REF ) we obtain a quartic equation for $u$ : $\\Big [c+(u^2+h^2+a)(3\\,u^2-h^2-a)\\Big ]^2=16\\,h^2\\Big [\\lambda ^2-u^2(u^2+h^2+a)^2\\Big ].$ If we define $v=u^2+h^2$ , this equation remains a quartic in $v$ but happens to be linear in $h^2$ .", "Solving for $h^2$ in terms of the variable $v$ , we find $v=u^2+h^2,\\quad \\quad h^2=\\frac{D^{\\prime 2}}{8D},\\quad \\quad D(v)=(v+a)(v^2-a^2+c)+2\\lambda ^2.$ At this stage, it turns out to be convenient to define $f=\\sqrt{D}=\\sqrt{(v+a)(v^2-a^2+c)+2\\lambda ^2}\\quad \\quad \\hbox{and}\\quad \\quad g=2v-f^{\\prime 2}$ where $f^{\\prime }=df/dv$ .", "This allows, once the old coordinates $(x,y)$ have been expressed in terms of the new ones, $(v,y)$ , to get eventually the explicit form of the metric $\\frac{1}{2}G=\\frac{1}{2h_x^2}(dx^2+dy^2)=\\left(\\frac{f^{\\prime \\prime }}{g}\\right)^2dv^2+\\frac{dy^2}{g}$ which gives the Hamiltonian $H\\equiv G^{ij}P_iP_j=\\frac{1}{2}\\left(\\Pi ^2+g\\,P_y^2\\right),\\quad \\quad \\quad \\quad \\Pi =\\frac{g}{f^{\\prime \\prime }}\\,P_v.$" ], [ "The cubic integrals", "They were given in (REF ), as borrowed from [4], and become in our new coordinates with a slight change of notation $S_{\\pm }=e^{\\pm y}\\Big (\\Pi ^3\\mp f^{\\prime }\\,\\Pi ^2\\,P_y+f\\,f^{\\prime \\prime }\\,\\Pi \\,P_y^2\\pm f(1-f^{\\prime }f^{\\prime \\prime })P_y^3\\Big ).$ However due to the relation $\\,dH\\wedge dP_y\\wedge dS_+\\wedge dS_-=0$ , the four observables involved are not functionally independent.", "Indeed, we have $S_+\\,S_-= 8H^3+8a\\,H^2\\,P_y^2+2c\\,H\\,P_y^4-2\\lambda ^2\\,P_y^6,$ so that we may consider two different superintegrable systems ${\\cal I}_+=\\,(H,\\,P_y,\\,S_+)\\quad \\quad \\hbox{and}\\quad \\quad {\\cal I}_-=\\,(H,\\,P_y,\\,S_-).$ Proposition 1 The observables $\\,S_+$ and $S_-$ are integrals and the set $(H,\\,P_y,\\,S_+,\\,S_-)$ generates a Poisson algebra.", "Proof: The Poisson brackets are given by $\\lbrace H,S_{\\pm }\\rbrace =e^{\\pm y}\\,\\frac{g}{f^{\\prime \\prime }}\\,\\Pi \\,P_y^2(\\Pi \\mp f^{\\prime }\\,P_y)\\,\\Big (f\\,f^{\\prime \\prime \\prime }-3(1-f^{\\prime }\\,f^{\\prime \\prime })\\Big ).$ Quite remarkably, the ode $f\\,f^{\\prime \\prime \\prime }-3(1-f^{\\prime }\\,f^{\\prime \\prime })=0$ does linearize upon the substitution $f=\\sqrt{D}$ since we have $2\\Big (f\\,f^{\\prime \\prime \\prime }-3(1-f^{\\prime }\\,f^{\\prime \\prime })\\Big )=D^{\\prime \\prime \\prime }-6=0,$ which gives for $D$ the most general monic polynomial of third degree $D(v)=v^3-s_1\\,v^2+s_2\\,v-s_3,$ whose coefficients are expressed in terms of the symmetric functions of the roots.", "As a matter of fact, the function $D$ already obtained in (REF ) displays exactly 3 parameters $a,c,\\lambda $ .", "Equations (REF ) and (REF ) insure then conservation of both cubic integrals $S_+$ and $S_-$ .", "The Poisson algebra structure follows from the following relations, viz., $\\begin{array}{rcl}\\lbrace S_+,S_-\\rbrace & = & \\displaystyle -16a\\,H^2\\,P_y-8c\\,H\\,P_y^3+12\\lambda ^2\\,P_y^5,\\\\[4mm]S_+\\,S_- & = & \\displaystyle 8H^3+8a\\,H^2\\,P_y^2+2c\\,H\\,P_y^4-2\\lambda ^2\\,P_y^6;\\end{array}$ it is generated by 4 observables in this case.", "$\\quad \\Box $" ], [ "Transformation of the metric and its curvature", "Taking for $D$ the expression (REF ), let us define the following quartic polynomials $P$ and $Q$ , namely $P=8v\\,D-D^{\\prime 2},\\quad \\quad \\quad \\quad Q=2\\,D\\,D^{\\prime \\prime }-D^{\\prime 2}=P+4(v-s_1)D,\\quad \\quad Q^{\\prime }=12\\,D,$ enabling us to write the metric (REF ) in the form $\\frac{1}{2}G=\\rho ^2\\,\\frac{dv^2}{D}+\\frac{4D}{P}\\,dy^2,\\quad \\quad \\quad \\quad \\rho \\equiv \\frac{Q}{P}=1+(v-s_1)\\frac{4D}{P},$ the scalar curvature being given by $R_G=\\frac{1}{4Q^3}\\Big (2PQ\\,W^{\\prime }-(QP^{\\prime }+2PQ^{\\prime })\\,W\\Big ),\\quad \\quad W\\equiv DP^{\\prime }-PD^{\\prime }=8D^2-QD^{\\prime }.$ One should bear in mind the following restrictions: 1.", "The relation $v=u^2+h^2$ requires $v>0$ .", "2.", "For $h$ to be real we must have $D>0.$ 3.", "For the metric $G$ to be Riemannian we need $P>0$ ." ], [ "Global properties", "To study the global geometry of these superintegrable models, we will be using techniques which have proved quite successful in [8] and [9] for integrable models with either a cubic or a quartic integral.", "As emphasized in the Introduction, we will from now on confine considerations to the case of simply connected Riemann surfaces, which, by the Riemann uniformization theorem [5], are conformally related to spaces of constant curvature ${\\mathbb {S}}^2,{\\mathbb {R}}^2,{\\mathbb {H}}^2$ .", "One has first to determine, from the above positivity conditions, the open interval $I\\subset {\\mathbb {R}}$ admissible for the variable $v$ .", "The end-points are singular points for the metric and the possibility of a manifold structure is related to the behavior of the metric at these end-points.", "Either they are true singularities (for instance if the scalar curvature is divergent at these points) or they are apparent singularities (also called coordinate singularities) due to a bad choice of the coordinates as, for instance, $G=dr^2+r^2\\,d\\phi ^2,\\quad \\quad \\quad \\quad r\\in (0,+\\infty ),\\quad \\quad \\phi \\in {\\mathbb {S}}^1,$ for which $r=0$ is an apparent singularity which can be wiped out, using back Cartesian coordinates.", "We will detect true singularities from the scalar curvature: Lemma 1 Let us consider the interval $I=(a,b)$ , allowed for $v$ , i.e., such that $D(v)>0$ and $P(v)>0$ for all $v\\in \\,I$ .", "Suppose that $Q$ has a simple real zero $v_\\,\\in \\,I$ ; then $v=v_$ is a curvature singularity precluding any manifold structure associated with the metric.", "Proof: The relation (REF ) entails that $\\lim _{v\\rightarrow v_}\\,Q^3(v)\\,R_G(v)=-4\\,P(v_)\\,D^2(v_)\\,Q^{\\prime }(v_)$ and the right-hand side of this equation does not vanish.", "The existence of such a curvature singularity for $v_\\in I$ rules out the possibility of a manifold structure.", "$\\quad \\Box $ We will detect non-closedness by Lemma 2 If the variable $v$ takes its values in some interval $I=(a,b)$ and if one of the end-points is a zero of $P$ (and not of $Q$ ), then the manifold having infinite measure, it cannot be closed.", "Proof: Let the allowed interval for $v$ be $I=(a,b)$ .", "The measure of the manifold is $\\mu _G=4\\int _a^b{\\!\\frac{Q(v)}{P^{3/2}(v)}\\,dv}\\int {\\!dy}.$ Now, if $P$ has a zero at one end-point where $Q$ does not vanish, the this integral will be divergent.", "$\\ \\Box $ Let us turn ourselves to the analysis of this first case ($i$ ).", "Given any polynomial $P$ we will use the notation $\\Delta (P)$ for its discriminant.", "The discussion will be organized according to the sign of $\\Delta (D)$ .", "Let us begin with: Proposition 2 If $\\Delta (D)=0$ the superintegrable systems ${\\cal I}_+$ and ${\\cal I}_-$ given by (REF ) are either trivial or are not defined on a closed manifold.", "Proof: If $\\Delta (D)=0$ , we may have first $D=(v-v_0)^3$ .", "The scalar curvature, easily computed using (REF ), is a constant.", "The following theorem, due to Thompson [7], states that for Riemannian spaces of constant curvature, namely $\\ {\\mathbb {S}}^n,\\ {\\mathbb {R}}^n,\\ {\\mathbb {H}}^n\\ $ with $n\\ge 2$ , every (symmetric) Killing-Stäckel tensor of any degree is fully reducible to symmetrized tensor products of the Killing vectors.", "This implies that the cubic integrals are reducible, leaving us with the trivial integrable system $(H,\\,P_y)$ .", "For $\\Delta (D)=0$ we may also have $D=(v-v_0)(v-v_1)^2$ with $v_0\\ne v_1$ , which yields $\\left\\lbrace \\begin{array}{l}P(v)=-(v-v_1)^2\\,p(v),\\quad \\quad p(v)=v^2-2(2v_0+3v_1)v+(2v_0+v_1)^2,\\\\[4mm]\\displaystyle Q=3(v-v_1)^3(v-v_),\\quad \\quad v_=v_0+\\frac{v_0-v_1}{3}.\\end{array}\\right.$ Let us first observe that for the metric $\\frac{1}{2}G=\\frac{9(v-v_)^2}{p(v)^2}\\,\\frac{dv^2}{(v-v_0)}+\\frac{4(v-v_0)}{(-p(v))}\\,dy^2$ to be Riemannian we must have $v>v_0$ and $p(v)<0$ .", "If the roots $w_{\\pm }$ of $p$ are ordered as $w_-<w_+$ , positivity of the metric is achieved iff $v\\in \\, I=(v_0,+\\infty )\\cap (w_-, w_+)$ , the upper bound of $I$ being $w_+$ .", "Since $P(w_+)=0$ and $Q(w_+)\\ne 0$ , the expected manifold cannot be closed by Lemma REF .", "$\\quad \\Box $ Proposition 3 If $\\Delta (D)<0\\,$ the superintegrable systems $\\,{\\cal I}_+$ and $\\,{\\cal I}_-$ given by (REF ) are never globally defined on a closed manifold.", "Proof: If $\\Delta (D)<0\\,$ the polynomial $D$ has only a simple real zero.", "Using new parameters $(a,\\,b)$ we can write $D=(v-v_0)\\Big ((v-a)^2+b^2\\Big ),\\quad \\quad v\\in \\,(v_0,+\\infty ),\\quad \\quad a\\in {\\mathbb {R}},\\quad b\\in {\\mathbb {R}}\\backslash \\lbrace 0\\rbrace ,$ with $\\Delta (D)=-4b^2\\Big ((v_0-a)^2+b^2\\Big )^2$ and, for $P$ and $Q$ , $\\Delta (P)=16384\\,a^2\\Big ((v_0+a)^2+b^2\\Big )^2\\,\\Delta (D),\\quad \\quad \\Delta (Q)=27648\\,b^2\\Big ((v_0-a)^2+b^2\\Big )^2\\,\\Delta (D).$ We must exclude $a=0$ since $P(v)=-(v^2-2v_0v-b^2)^2$ is negative.", "Hence, the previous discriminants are strictly negative, implying that both polynomials $P$ and $Q$ have two simple real zeroes.", "The relation $Q^{\\prime }=12\\,D$ shows that $Q$ is strictly increasing from $Q(v_0)=-[(v_0-a)^2+b^2]$ to $Q(+\\infty )=+\\infty $ , hence there exists a simple zero $v_$ of $Q$ such that $v_>v_0$ while the other one lies to the left of $v_0$ because $Q(-\\infty )=+\\infty $ .", "The polynomial $P$ retains the form $P(v)=-\\Big (v^2-2(v_0+2a)v-a^2-b^2-2av_0\\Big )^2+16a\\Big ((v_0+a)^2+b^2\\Big )v$ showing that for $a<0$ it is never positive as it should; so, we are left with the case $a>0$ .", "From the relations $P(v)=Q(v)+4(v_0+2a-v)\\,D(v),\\quad \\quad \\quad \\quad P^{\\prime }(v)=8\\,D(v)+4(2a+v_0-v)\\,D^{\\prime }(v),$ we see that $P(v_0)$ is strictly negative and that $P^{\\prime }(v)$ is positive from $v=v_0$ to $v=v_0+2a$ .", "Thus $P$ increases to its first zero $v=w_-<v_$ (since $P(v_)=4(2a+v_0-v_)\\,D(v_)>0$ ), is equal to $Q$ for $v=v_0+2a>v_$ , then vanishes at its second zero $w_+$ such that $w_+>v_0+2a$ and, at last, decreases to $-\\infty $ .", "Therefore, we end up with the ordering $v_0<w_-<v_<v_0+2a<w_+.$ So, $D>0$ and $P>0$ iff $v\\in (w_-,w_+)$ , and within this interval $Q$ has a simple zero for $v=v_$ ; hence, by Lemma REF , there is no underlying manifold structure.", "$\\quad \\Box $ Let us conclude this section with Proposition 4 If $\\Delta (D)>0\\,$ the superintegrable systems $\\,{\\cal I}_+$ and $\\,{\\cal I}_-$ given by (REF ) are never globally defined on a closed manifold.", "Proof: Let us order the roots of $D$ according to $0\\le v_0<v_1<v_2$ , so that $D(v)=(v-v_0)(v-v_1)(v-v_2)=v^3-s_1\\,v^2+s_2\\,v-s_3,$ and $D>0$ for $v\\in (v_0,v_1)\\cup (v_2,+\\infty )$ .", "We need to determine now the positivity interval for $P$ .", "Since $\\Delta (P)=4096\\,\\sigma ^2\\Delta (D)>0,\\quad \\quad \\quad \\quad \\sigma =(v_0+v_1)(v_1+v_2)(v_2+v_0)>0,$ there will be either four real simple roots or no real root for $P$ .", "The latter is excluded since $P=8vD-(D^{\\prime })^2$ is negative at the zeroes of $D$ , and positive at those of $D^{\\prime }$ .", "Also, notice that $\\Delta (Q)=-6912\\,\\Delta ^2(D)<0$ implies that $Q$ has two simple real roots and one of them is $v_>v_2$ .", "This is so because $Q(v)=P(v)+4(v-s_1)D(v)$ , which shows that $Q(v_2)=P(v_2)=-(v_0-v_2)^2(v_1-v_2)^2<0$ ; but $Q^{\\prime }=12D$ entails that, for positive $D$ , the function $Q$ is increasing with $Q(+\\infty )=+\\infty $ .", "Hence $v=v_$ is a simple zero of $Q$ , forbidding any manifold structure by Lemma REF .", "The zeroes of $P$ may appear only when $D>0$ .", "Let us consider $v\\in (v_0,v_1)$ .", "Observing that $P(v_0)=-(v_0-v_1)^2(v_0-v_2)^2$ and $P(v_1)=-(v_1-v_0)^2(v_1-v_2)^2$ are negative and that there does exist $v=v_-\\in (v_0,v_1)$ for which $D^{\\prime }(v_-)=0$ , we get $P(v_-)>0$ which implies $v_0<w_0<v_-<w_1<v_1$ , where $(w_0,w_1)$ is the first pair of simple zeroes of $P$ .", "Positivity of both $D$ and $P$ is therefore obtained for $v\\in (w_0,w_1)$ .", "The function $Q$ remains strictly negative for $v\\in \\,[v_0,v_1]$ , and Lemma REF help us conclude that the supposed manifold cannot be closed.", "The remaining two zeroes of $P$ denoted by $w_2<w_3$ must lie in $(v_2,+\\infty )$ .", "Since $Q(v_2)=-(v_2-v_0)^2(v_2-v_1)^2<0$ and then it increases to $Q(+\\infty )=+\\infty $ it will have a simple zero $v=v_>v_2$ , and at this point $P(v_)=4(s_1-v_)D(v_)$ .", "Let us discuss: 1.", "If $v_<s_1$ , we have $P(v_)>0$ , and since $P(+\\infty )=-\\infty $ we get $v_2<w_2<v_<w_3$ .", "The positivity of $D$ and $P$ requires $v\\in (w_2,w_3)$ , and there is no manifold structure since the curvature $R_G$ is singular at $v=v_$ .", "2.", "If $v_\\ge s_1$ , we have $P(v_)<0$ hence $v_2<v_<w_2<w_3$ , and the positivity of $D$ and $P$ requires $v\\in (w_2,w_3)$ .", "Since $Q(w_3)>0$ the supposed manifold cannot be closed by Lemma REF .", "$\\quad \\Box $ We conclude this section by observing that the trigonometric case never leads to superintegrable systems defined on a closed manifold." ], [ "The explicit form of the metric", "The ode (REF ,$ii$ ) obtained in [4] is $h_x(A_0\\,h_x^2-\\mu ^2\\,A_0\\,h^2-A_1\\,h+A_2)=A_3\\frac{\\sinh (\\mu \\,x)}{\\mu }+A_4\\,\\cosh (\\mu \\,x).$ Again, we may put $A_0=1,\\mu =1,A_1=0,A_2=-a$ , but, this time, the right-hand side of the previous equation leads to three different cases we will describe according to $h_x(h_x^2-h^2-a)=\\frac{\\lambda }{2}\\,(e^{x}+\\epsilon \\,e^{-x}),\\quad \\quad \\quad \\quad \\epsilon =0,\\pm 1$ where $\\lambda $ is a free parameter.", "Let us point out that for $\\epsilon =0$ the changes $x \\rightarrow -x$ and $\\lambda \\rightarrow -\\lambda $ show that there is no need to consider $e^{-x}$ in the right-hand side of (REF ).", "With the definitions $u=h_x,\\quad \\quad \\quad \\quad U=u(u^2-h^2-a),\\quad \\quad \\quad \\quad a\\in {\\mathbb {R}},$ we get similarly $U^{\\prime \\prime }-U=0\\quad \\quad \\Rightarrow \\quad \\quad \\frac{d}{dh}\\left(u\\,\\frac{dU}{dh}\\right)-(u^2-h^2-a)=0,$ which can be integrated to yield $4hu\\,\\frac{dU}{dh}=c+(u^2-h^2-a)(3u^2+h^2+a),\\quad \\quad \\quad \\quad c\\in {\\mathbb {R}}.$ Since $\\displaystyle U=\\frac{\\lambda }{2}\\,(e^{x}+\\epsilon \\,e^{-x})$ we also have the first order ode: $U^{\\prime 2}=U^2-\\epsilon \\,\\lambda ^2\\quad \\quad \\Rightarrow \\quad \\quad \\left(4hu\\,\\frac{dU}{dh}\\right)^2=16\\,h^2\\,(U^2-\\epsilon \\lambda ^2),$ which, upon use of (REF ), leaves us with a quartic equation in the variable $u$ .", "Positing $v=h^2-u^2$ , we still have a quartic in $v$ but the $h^2$ dependence is merely linear and we can solve for $h^2$ in terms of the variable $v$ , namely $v=u^2-h^2,\\quad \\quad \\quad \\quad h^2=\\frac{D^{\\prime 2}}{8D},\\quad \\quad \\quad \\quad D(v)=(a-v)(v^2-a^2+c)-2\\epsilon \\lambda ^2,$ giving a result surprisingly similar to the case ($i$ ), except that $v$ needs not be positive.", "Upon defining $f=\\sqrt{D}=\\sqrt{(a-v)(v^2-a^2+c)-2\\epsilon \\lambda ^2}\\quad \\quad \\hbox{and}\\quad \\quad g=f^{\\prime 2}+2v,$ we obtain the metric in the new coordinates $(v,y)$ in the form $\\frac{1}{2}G=\\frac{1}{2h_x^2}(dx^2+dy^2)=\\left(\\frac{f^{\\prime \\prime }}{g}\\right)^2\\,dv^2+\\frac{dy^2}{g},$ together with the Hamiltonian $H\\equiv G^{ij}\\,P_i\\,P_j=\\frac{1}{2}\\left(\\Pi ^2+g\\,P_y^2\\right),\\quad \\quad \\quad \\quad \\Pi =\\frac{g}{f^{\\prime \\prime }}\\,P_v.$" ], [ "The cubic integrals", "They were given in (REF ) and read in our coordinates $S_1=\\cos y\\,{\\cal A}+\\sin y\\,{\\cal B},\\quad \\quad \\quad \\quad S_2=-\\sin y\\,{\\cal A}+\\cos y\\,{\\cal B},$ where ${\\cal A}=\\Pi ^3-f\\,f^{\\prime \\prime }\\,\\Pi \\,P_y^2,\\quad \\quad {\\cal B}=f^{\\prime }\\,\\Pi ^2\\,P_y-f\\,(1+f^{\\prime }\\,f^{\\prime \\prime })\\,P_y^3.$ Proposition 5 The observables $S_1$ and $S_2$ are integrals of the geodesic flow.", "Proof: Let us define the complex object ${\\cal S}=S_1+iS_2=e^{-iy}({\\cal A}+i{\\cal B}).$ The Poisson bracket with the Hamiltonian reads $\\lbrace H,{\\cal S}\\rbrace =-e^{-iy}\\frac{g}{f^{\\prime \\prime }}\\Pi \\,P_y^2(\\Pi +if^{\\prime }\\,P_y)\\Big (f\\,f^{\\prime \\prime \\prime }+3(1+f^{\\prime }\\,f^{\\prime \\prime })\\Big ).$ Again, the transformation $f=\\sqrt{D}$ leads to the following linearization: $2\\Big (f\\,f^{\\prime \\prime \\prime }+3(1+f^{\\prime }\\,f^{\\prime \\prime })\\Big )=D^{\\prime \\prime \\prime }+6=0\\quad \\quad \\Longrightarrow \\quad \\quad D=-(v^3-s_1\\,v^2+s_2\\,v-s_3).$ We conclude via (REF ) and (REF ) that ${\\cal S}$ is an integral.$\\quad \\Box $ As in case ($i$ ) we have $dH\\wedge dP_y\\wedge dS_1\\wedge dS_2=0$ , which shows that these four observables are not functionally independent.", "Indeed, we readily find $S_1^2+S_2^2={\\cal A}^2+{\\cal B}^2=8\\,H^3+8a\\,H^2\\,P_y^2+2c\\,H\\,P_y^4-2\\epsilon \\lambda ^2\\,P_y^6,$ leading us to consider two different superintegrable systems, namely ${\\cal I}_1=(H,\\,P_y,\\,S_1),\\quad \\quad \\quad \\quad {\\cal I}_2=(H,\\,P_y,\\,S_2).$ The Poisson bracket of the two cubic integrals still reduces to a polynomial in the observables $H$ and $P_y$ , viz., $\\lbrace S_1,S_2\\rbrace =-8a\\,H^2\\,P_y-4c\\,H\\,P_y^3+6\\epsilon \\,\\lambda ^2\\,P_y^5,$ as in (REF ) for $S_1^2+S_2^2$ , but this is no longer true for the product $S_1\\,S_2=\\cos (2y)\\,{\\cal A}\\,{\\cal B}+\\sin (2y)\\,\\frac{{\\cal B}^2-{\\cal A}^2}{2}$ which is a new, independent, observable.", "This time, the set $(H,P_y,S_1,S_2)$ of first integrals of the geodesic flow does not generate a Poisson algebra." ], [ "Transformation of the metric and curvature", "Returning to the expression (REF ) of $D$ , let us define the polynomials $P=8v D+D^{\\prime 2},\\quad \\quad \\quad \\quad Q=2DD^{\\prime \\prime }-D^{\\prime 2}=-P-4(v-s_1)D,\\quad \\quad Q^{\\prime }=-12\\,D,$ which readily yield the metric $\\frac{1}{2}G=\\rho ^2\\,\\frac{dv^2}{D}+\\frac{4D}{P}\\,dy^2,\\quad \\quad \\quad \\quad -\\rho \\equiv -\\frac{Q}{P}=1+(v-s_1)\\,\\frac{4D}{P},$ with the restrictions $D>0$ and $P>0$ that ensure its Riemannian signature.", "We notice that the scalar curvature is still given by $R_G=\\frac{1}{4Q^3}\\Big (2PQ\\,W^{\\prime }-(QP^{\\prime }+2PQ^{\\prime })\\,W\\Big ),\\quad \\quad W\\equiv DP^{\\prime }-PD^{\\prime }=8D^2+QD^{\\prime },$ showing that Lemma REF remains valid.", "Lemma 3 Let $I=(-\\infty ,v_0)$ be the allowed interval for $v$ where $v_0$ is a simple zero of $D$ .", "If for all $v\\in \\,I$ one has $P(v)>0$ and $Q(v)>0$ , then the metric exhibits a conical singularity which precludes any manifold structure.", "Proof: Using the relations given in (REF ), when $v\\rightarrow \\,v_0+$ the metric approximates as $\\frac{1}{2}G\\approx \\frac{4}{D^{\\prime }(v_0)}(dr^2+r^2\\,dy^2),\\quad \\quad \\quad \\quad r=\\sqrt{v-v_0} \\rightarrow 0+$ and hence, for this singularity to be apparent, we need to assume $y=\\phi \\in {\\mathbb {S}}^1$ .", "For $v\\rightarrow -\\infty $ we get $\\frac{1}{2}G\\approx dr^2+r^2\\,\\left(\\frac{d\\phi }{3}\\right)^2,\\quad \\quad \\quad \\quad r=\\frac{1}{\\sqrt{-v}}\\rightarrow 0+$ and we cannot have $\\phi /3\\in \\,{\\mathbb {S}}^1$ as well.", "This kind of singularity, called conical, rules out a manifold structure.", "$\\quad \\Box $ For further use we will also prove the general result: Lemma 4 Assume that the metric $G=A(v)\\,dv^2+B(v)\\,d\\phi ^2,\\quad \\quad \\quad \\quad v\\in I=[a,b],\\quad \\quad \\phi \\in {\\mathbb {S}}^1,$ be globally defined on a closed manifold $M$ .", "Then its Euler characteristic is given by $\\chi (M)=\\gamma (b)-\\gamma (a),\\quad \\quad \\quad \\quad \\gamma =-\\frac{B^{\\prime }}{2\\sqrt{A\\,B}}.$ Proof: Using the orthonormal frame $e_1=\\sqrt{A}\\,dv,\\quad \\quad \\quad \\quad e_2=\\sqrt{B}\\,d\\phi ,$ we find that the connection 1-form reads $\\displaystyle \\ \\omega _{12}=\\frac{\\gamma }{\\sqrt{B}}\\ e_2$ , where $\\gamma $ is as in (REF ).", "The curvature 2-form is then given by $R_{12}=d\\omega _{12}=\\frac{\\gamma ^{\\prime }}{\\sqrt{AB}}\\,e_1\\wedge e_2,$ from which we get $\\chi (M)=\\frac{1}{2\\pi }\\int _M{}R_{12}=\\int _I\\,\\gamma ^{\\prime }(v)dv=\\gamma (b)-\\gamma (a),$ which was to be proved.", "$\\quad \\Box $ Let us consider now the global properties of these metrics." ], [ "The global structure for $\\epsilon =0$", "In this section we will keep the notation $D(v)=(a-v)(v^2-a^2+c),\\quad \\quad \\quad \\quad \\Delta (D)=4c^2(a^2-c),$ and organize the discussion according to the values of the discriminant $\\Delta (D)$ of $D$ .", "We will exclude the single case $a=c=0$ since then the scalar curvature vanishes, implying that we loose superintegrability as explained in the proof of Proposition REF ." ], [ "First case: $\\Delta (D)=0$", "We will begin with Proposition 6 There exists no closed manifold for $c=0$ and $a\\ne 0$ .", "Proof: We have, in this case, $D(v)=(a-v)(v^2-a^2),\\quad P(v)=(v-a)^4,\\quad Q(v)=3(v-a)^3(v-v_),\\quad v_=-\\frac{5}{3}\\,a,$ and the metric writes $\\frac{1}{2}G=9\\frac{(v-v_)^2}{(a-v)^4}\\frac{dv^2}{-a-v}+\\frac{4}{3}\\frac{a-v}{(v-v_)}\\,dy^2.$ For $a>0$ we have $D>0$ and $P>0$ iff $v\\in I=(-\\infty ,-a)$ ; but since $v_\\in I$ we get no manifold structure by Lemma REF .", "For $a<0$ the positivity of $G$ is satisfied for $v\\in (-\\infty ,a)\\cap (a,-a)$ .", "In both cases, $a$ is a zero of $P$ but we cannot use Lemma REF because $Q(a)=0$ .", "In fact, the measure of the sought manifold $\\mu _G=12\\int \\frac{(v-v_)}{(v-a)^3}\\,dv\\int \\!dy$ is divergent (since the integrand blows up at $v=a$ ), prohibiting a closed manifold.", "$\\quad \\Box $ Proposition 7 There exists no closed manifold for $c=a^2>0$ .", "Proof: We have now $D(v)=v^2(a-v),\\quad \\quad P(v)=v^2(v-2a)^2,\\quad \\quad Q(v)=3v^3(v-v_),\\quad v_=\\frac{4}{3}\\,a.$ For $a<0$ we have $D>0$ and $P>0$ iff either $v\\in \\,I_1=(2a,a)$ or $v\\in I_2=(-\\infty ,2a)$ .", "In the first interval $Q$ has a simple zero $v=v_$ , and $P(v_)$ and $D(v_)$ do not vanish; in view of Lemma REF we get a curvature singularity.", "As for the second interval, the end-point $v=2a$ is a zero of $P$ where $Q(2a)\\ne 0$ ; hence by Lemma REF , the sought manifold is not closed.", "For $a>0$ we have $I=(-\\infty ,a)$ .", "There will be no curvature singularity since $Q$ never vanishes for $v\\in \\,I$ .", "Since $v=a$ is a simple zero of $D$ such that $P(a)$ and $Q(a)$ are non-zero; we conclude by Lemma REF .", "$\\quad \\Box $" ], [ "Second case: $\\Delta (D)<0$", "Here, we have $D=(a-v)\\Big (v^2+c-a^2\\Big ),\\quad \\quad c>a^2,\\quad \\quad P=(v-w_-)^2(v-w_+)^2,\\quad \\quad w_{\\pm }=a\\pm \\sqrt{c},$ and $Q=-P+4(a-v)D,\\quad \\quad \\quad \\quad Q^{\\prime }=-12\\,D.$ Proposition 8 There exists no closed manifold for $\\Delta (D)<0$ .", "Proof: The positivity of $D$ and $P$ holds for any $\\,v\\in (-\\infty ,w_-)\\cup \\,(w_-,a)$ .", "The second interval is excluded since $Q$ is strictly decreasing and the relations $Q(w_-)=8\\,c^{3/2}\\,(\\sqrt{c}-a)>0,\\quad \\quad \\quad \\quad Q(a)=-c^2<0,$ imply that $Q$ has a simple zero inside the interval $\\,(w_-,a)$ , inducing a curvature singularity as already explained.", "This never happens for $\\,v\\in \\,(-\\infty ,w_-)$ since then $Q(v)>0$ .", "But $w_-$ is a zero of $P$ and $Q(w_-)>0$ ; we conclude by Lemma REF .", "$\\quad \\Box $" ], [ "The case $\\Delta (D)>0$", "This time, $c<a^2$ and we find $\\begin{array}{l}D(v)=(a-v)(v^2-v_0^2),\\quad \\quad \\quad \\quad v_0=\\sqrt{a^2-c},\\\\[4mm]P(v)=\\Big ((v-a)^2-c\\Big )^2,\\quad \\quad \\quad \\quad Q(v)=-P(v)+4(a-v)D(v).\\end{array}$ The parameter $c$ can take its values in the set $(-\\infty ,0)\\cup \\lbrace 0\\rbrace \\cup (0,a^2).$ Let us consider first negative values of $c$ .", "Theorem 1 If $c\\in \\,(-\\infty ,0)$ the superintegrable systems ${\\cal I}_1$ and ${\\cal I}_2$ given in (REF ) are globally defined on ${\\mathbb {S}}^2$ .", "Proof: First of all, we have $P>0$ .", "The ordering of the zeroes of $D$ is $\\,-v_0<a<v_0$ .", "This implies two possible intervals ensuring its positivity: either $v\\in \\,(-\\infty ,-v_0)$ or $v\\in \\,(a,v_0)$ .", "The first case is easily ruled out since $Q$ decreases from $Q(-\\infty )=+\\infty $ to $Q(-v_0)=-P(-v_0)<0$ ; it thus vanishes in the interval and leads to a curvature singularity.", "So let us consider $v\\in \\,(a,v_0)$ .", "Then $Q(a)=-P(a)=-c^2$ is negative, and since $Q$ is decreasing it will remain strictly negative everywhere on the interval.", "Putting $v_0=1$ and performing the transformation $G\\rightarrow 2\\,G$ for convenience, we end up with the explicit form of the metric, namely $G=\\rho ^2\\,\\frac{dv^2}{(v-a)(1-v^2)}+4\\frac{(v-a)(1-v^2)}{(v^2-2av+1)^2}d\\phi ^2,\\quad \\quad v\\in (a,1),\\quad \\quad \\phi \\in \\,{\\mathbb {S}}^1,$ where $a\\in \\,(-1,1),\\quad \\quad \\quad \\quad -\\rho =1+4\\frac{(v-a)^2(1-v^2)}{(v^2-2av+1)^2}.$ Both end-points are apparent singularities because $G(v\\rightarrow 1-)\\sim \\frac{2}{1-a}(dr^2+r^2\\,d\\phi ^2),\\quad \\quad \\quad \\quad r=\\sqrt{1-v},$ and $G(v\\rightarrow a+)\\sim \\frac{4}{1-a^2}(dr^2+r^2\\,d\\phi ^2),\\quad \\quad \\quad \\quad r=\\sqrt{v-a}.$ Let us compute the Euler characteristic.", "Resorting to Lemma REF , we find $\\gamma (v)=\\frac{(1-v^2)^2-4(v-a)^2}{Q(v)}\\quad \\quad \\Longrightarrow \\quad \\quad \\chi (M)=\\gamma (1)-\\gamma (a)=2,$ which proves that the manifold is diffeomorphic to ${\\mathbb {S}}^2$ .", "The measure of this surface is $\\mu _G({\\mathbb {S}}^2)=\\frac{4\\pi }{1+a}.$ Let us investigate now the global status of the integrals $H,P_y,S_1,S_2$ .", "Using (REF ), and referring to the Riemann uniformization theorem, we can write $H=\\frac{1}{2}\\left(\\Pi ^2+P\\,\\frac{P_{\\phi }^2}{4D}\\right)=\\frac{1}{2\\Omega ^2}\\left(P_{\\theta }^2+\\frac{P_{\\phi }^2}{\\sin ^2\\theta }\\right)$ with $t\\equiv \\tan \\frac{\\theta }{2}=\\sqrt{\\frac{(v-a)P}{(1-v^2)}},\\quad \\quad \\quad \\quad \\Omega =\\frac{1-v^2}{P}+v-a,$ and the conformal factor is indeed $C^{\\infty }$ for all $v\\in \\,[a,1]$ .", "To ascertain that the previous integrals are globally defined, we will express them in terms of globally defined quantities, e.g., the $\\mathrm {SO}(3)$ generators on $T^{\\mathbb {S}}^2$ , namely $L_1=-\\sin \\phi \\,P_{\\theta }-\\frac{\\cos \\phi }{\\tan \\theta }\\,P_{\\phi },\\quad \\quad L_2=\\cos \\phi \\,P_{\\theta }-\\frac{\\sin \\phi }{\\tan \\theta }\\,P_{\\phi },\\quad \\quad L_3=P_{\\phi },$ and the constrained coordinates $x^1=\\sin \\theta \\,\\cos \\phi ,\\quad \\quad \\quad \\quad x^2=\\sin \\theta \\sin \\phi ,\\quad \\quad \\quad \\quad x^3=\\cos \\theta .$ The relation $\\Pi =-P_{\\theta }/\\Omega $ and formulas (REF ) and (REF ) yield $S_1=-\\frac{L_2}{\\Omega }\\,\\left(\\Pi ^2-Q\\,\\frac{P_{\\phi }^2}{4D}\\right)+x^2\\,L_3\\,\\left(A\\,\\Pi ^2-B\\,\\frac{P_{\\phi }^2}{4D}\\right),$ where the functions $A,B$ of $\\theta $ retain the form $A=\\frac{D^{\\prime }-\\sqrt{P}\\,\\cos \\theta }{2\\sin \\theta \\,\\sqrt{D}},\\quad \\quad \\quad \\quad B=\\frac{W-Q\\sqrt{P}\\,\\cos \\theta }{2\\sin \\theta \\,\\sqrt{D}}.$ The polynomials $P,\\,Q$ and $W$ are clearly globally defined, as well as the quantities $\\Pi ^2$ and $P_{\\phi }^2/(4D)$ in the Hamiltonian.", "So, it is sufficient to check that the functions $A$ and $B$ are well-behaved near the poles.", "Let us begin with the north-pole ($v\\rightarrow a+$ or $\\theta \\rightarrow 0+$ ) for which we get $\\left\\lbrace \\begin{array}{l}\\displaystyle A=\\frac{\\phi (a)}{2(1-a^2)}-\\frac{\\sin ^2\\theta }{4(1-a^2)^2}+O(\\sin ^4\\theta ),\\\\[4mm]\\displaystyle B=-\\frac{(1-a^2)}{2}\\,\\phi (a)+\\frac{3}{4}\\sin ^2\\theta +O(\\sin ^4\\theta ),\\end{array}\\right.\\quad \\quad \\phi (a)=a^4-2a^2-2a+1,$ while for the south pole ($v\\rightarrow 1-$ or $\\theta \\rightarrow \\pi -$ ) we obtain $\\left\\lbrace \\begin{array}{l}\\displaystyle A=\\frac{\\psi (a)}{2(1-a)}-\\frac{(1-a)^4}{2}\\,\\sin ^2\\theta +O(\\sin ^4\\theta ),\\\\[4mm]\\displaystyle B=-2(1-a)\\,\\psi (a)+6(1-a)^6\\,\\sin ^2\\theta +O(\\sin ^4\\theta ),\\end{array}\\right.\\quad \\quad \\psi (a)=2a^2-4a+1.$ We observe that either $\\phi (a)$ or $\\psi (a)$ may vanish for some $a\\in \\,(0,1)$ , but this does not jeopardize the conclusion.", "For the other integral, due to the relation $S_2=\\lbrace P_{\\phi },S_1\\rbrace =\\frac{L_1}{\\Omega }\\,\\left(\\Pi ^2-Q\\,\\frac{P_{\\phi }^2}{4D}\\right)+x^1\\,L_3\\,\\left(A\\,\\Pi ^2-B\\,\\frac{P_{\\phi }^2}{4D}\\right),$ there is nothing more to check.", "$\\quad \\Box $ Let us consider now the second case where $c$ vanishes.", "Proposition 9 For $c=0$ there exists no closed manifold.", "Proof: The above functions simplify and read $D=-(v+a)(v-a)^2,\\quad \\quad P=(v-a)^4,\\quad \\quad Q=(3v+5a)(v-a)^3,\\quad \\quad a\\ne 0.$ For $a>0$ the positivity of $D$ requires $v\\in \\,I=(-\\infty ,-a)$ , but since $Q$ has a simple zero $\\displaystyle v=-\\frac{5}{3}\\, a\\in \\,I$ , in view of Lemma REF there is no manifold structure.", "For $a<0$ either $v\\in \\,(-\\infty ,a)$ or $v\\in \\,(a,-a)$ ensure the positivity of $D$ .", "But in both cases $P$ vanishes for $v=a$ , and the measure of the would-be manifold $\\mu _G=12\\int \\frac{(v-v_)}{(v-a)^3}\\,dv\\int \\!dy$ is divergent, excluding a closed manifold.", "$\\quad \\Box $ The remaining case is $c\\in (0,a^2)$ .", "The discussion depends strongly on the sign of $a$ .", "Beginning with $a>0$ we have: Proposition 10 For $c\\in (0,\\,a^2)$ and $a<0$ there exists no closed manifold.", "Proof: The two functions $(D,\\,P)$ are now $D(v)=(a-v)(v^2-v_0^2),\\quad v_0=\\sqrt{a^2-c},\\quad P=(v-w_-)^2(v-w_+)^2,\\quad w_{\\pm }=a\\pm \\sqrt{c},$ with the ordering $\\ w_-<a<w_+<-v_0\\ $ .", "The positivity requirements give three possible intervals: $I_1=(-\\infty ,w_-),\\quad \\quad \\quad \\quad I_2=(w_-,a),\\quad \\quad \\quad \\quad I_3=(-v_0,v_0).$ For $v\\in I_1$ we notice that $w_-$ is a zero of $P$ for which $Q(w_-)=4(a-w_-)D(w_-)>0$ , and we conclude by Lemma REF .", "For $v\\in I_2$ since $Q(w_-)>0$ and $Q(a)=-P(a)<0$ , there is a simple zero $v_$ of $Q$ inside $I_2$ ; hence, by Lemma REF , there is no manifold structure.", "For $v\\in I_3$ we have $Q(-v_0)=-P(-v_0)<0$ and then $Q$ decreases to $Q(v_0)$ ; it thus never vanishes and $P>0$ in $I_3$ , opening the possibility of a manifold structure.", "Putting $v_0=1$ and computing the metric brings us back to (REF ).", "$\\quad \\Box $ For $a>0$ we have: Proposition 11 For $c\\in (0,\\,a^2)$ and $a>0$ there exists no closed manifold.", "Proof: The zeros of $D$ and $P$ interlace as follows $\\ w_-<-\|a\|<-v_0<w_+<0<v_0\\ $ giving four possible intervals $I_1=(-\\infty ,w_-),\\quad \\quad I_2=(w_-,-\|a\|),\\quad \\quad I_3=(-v_0,w_+),\\quad \\quad I_4=(w_+,v_0).$ For $v\\in \\,I_1=(-\\infty ,w_-)$ , and since $w_-$ is a zero of $P$ , we use Lemma REF .", "If $v\\in \\,I_2=(w_-,-\|a\|)$ , then $Q$ is strictly decreasing with $Q(w_-)=4(-w_-+a)\\,D(w_-)>0\\quad \\quad \\mbox{and}\\quad \\quad Q(-\|a\|)=-P(-\|a\|)<0,$ so that $Q$ has a simple zero in $I_2$ ; thanks to Lemma REF , there is no manifold structure.", "For $v\\in \\,I_3=(-v_0,w_+)$ or $v\\in \\,I_4=(w_+,v_0)$ , since $w_+$ is a zero of $P$ we invoke again Lemma REF .", "$\\quad \\Box $" ], [ "The global structure for $\\epsilon \\ne 0$", "Let us begin with Proposition 12 If $\\Delta (D)=0$ the superintegrable system is never globally defined on a closed manifold.", "Proof: We may have either $D(v)=(v_0-v)^3$ or $D(v)=(v_0-v)(v-v_1)^2$ with $v_0\\ne v_1$ .", "The first case is ruled out as in Proposition REF since the metric is of constant curvature.", "In the second case we have $\\left\\lbrace \\begin{array}{l}P(v)=(v-v_1)^2\\,p(v),\\quad \\quad \\quad \\quad p(v)=v^2-2(2v_0+3v_1)v+(2v_0+v_1)^2,\\\\[4mm]\\displaystyle Q(v)=3(v-v_1)^3(v-v_),\\quad \\quad v_=v_0+\\frac{v_0-v_1}{3}\\end{array}\\right.$ Let us first consider the case $v_0<v_1$ .", "Then $D$ is positive iff $v\\in I=(-\\infty ,v_0)$ .", "If $\\Delta (p)<0$ then $P>0$ for all $v\\in I$ .", "But, since $v_<v_0$ , there will be a curvature singularity inside $I$ .", "If $\\Delta (p)$ vanishes, we get $p(v)=(v-w_0)^2$ and either $w_0=v_0<0$ or $w_0=2v_0<0$ .", "In the first case there will be a curvature singularity at $\\displaystyle v_=\\frac{4}{3}\\,v_0\\in I$ while, in the second case, the positivity interval becomes $(-\\infty ,w_0)$ ; since $v=w_0$ is a zero of $P$ we use Lemma REF .", "If $\\Delta (p)<0$ we have two real zeroes and $p(v)=(v-w_-)(v-w_+)$ .", "The interval of positivity becomes $I=(-\\infty ,v_0)\\cap (w_-,w_+)$ and since at least one of its end-points will correspond to a zero of $P$ we conclude by Lemma 2.", "Let us then consider the other case $v_0>v_1$ .", "Then $D$ is positive iff $v\\in I=(-\\infty ,v_0)$ .", "If $\\Delta (p)<0$ then $P>0$ for all $v\\in I$ , and we conclude by Lemma REF .", "If $\\Delta (p)=0$ we get $p(v)=(v-w_0)^2$ , and either $w_0=v_0>0$ or $w_0=2v_0>0$ .", "In the first case we remain with $v\\in (-\\infty ,v_0)$ and end up with a conical singularity for $v\\rightarrow -\\infty $ ; in the second case $v\\in (w_0,v_0)$ where $w_0$ is a zero of $P$ , which excludes closedness by Lemma REF .", "If $\\Delta (p)>0$ we have two real zeroes and $p(v)=(v-w_-)(v-w_+)$ .", "The interval of positivity becomes $I=(-\\infty ,v_0)\\cap (w_-,w_+)$ and at least one of its end-points will correspond to a zero of $P$ ; we conclude by Lemma REF .", "$\\quad \\Box $ Let us proceed to Proposition 13 If $\\Delta (D)<0$ the superintegrable systems are never globally defined on a closed manifold.", "Proof: In this case, we can write $D(v)=(v_0-v)[(v-a)^2+b^2],\\quad b\\ne 0,\\quad \\quad Q(v)=-P(v)+4(v_0+2a-v)D(v),$ and $P(v)=p(v)^2-16a[(v_0+a)^2+b^2]v,\\quad \\quad p(v)=v^2-2(v_0+2a)v-a^2-b^2-2av_0.$ We have $D>0$ iff $v\\in I=(-\\infty ,v_0)$ .", "Let us also notice that $\\Delta (P)$ and $\\Delta (Q)$ being negative, $P$ and $Q$ will have two simple real zeroes.", "Since $Q(v_0)<0$ , then $Q$ will have a simple zero $v_<v_0$ .", "If $a=0$ we have $p(v)=(v-w_-)(v-w_+)$ , with the ordering $w_-<w_+$ ; hence $P$ is always positive, but its zeroes may change the interval for $v$ : if $w_-<v_$ the interval for $v$ becomes $(w_-,\\,v_0)$ and then $v_$ is a curvature singularity inside this interval; if $w_->v_$ the interval for $v$ becomes $(w_-,\\,v_0)$ for which Lemma REF applies.", "If $a>0$ , the relation (REF ) tells us that both roots of $P$ must be positive and, since $P(v_0)=\\big ((v_0-a)^2+b^2\\big )^2>0$ , they must lie to the right of $v_0$ .", "The interval for $v$ remains $(-\\infty ,v_0)$ and we conclude by Lemma REF .", "If $a<0$ both roots of $P$ ordered as $w_-<w_+$ must be negative and to the left of $v_0$ .", "The positivity of $P$ will reduce the interval of $v$ either to $(-\\infty ,w_-)$ or to $(w_+,v_0)$ and in both cases Lemma REF allows us to conclude.", "$\\quad \\Box $ Let us end up this section with: Theorem 2 If $\\Delta (D)>0$ one can put $D(v)=-(v-v_0)(v-v_1)(v-v_2)$ with $v_0 < v_1 < v_2$ ; the superintegrable systems $\\ {\\cal I}_1$ and $\\ {\\cal I}_2$ given by (REF ) are indeed globally defined on ${\\mathbb {S}}^2$ iff $v_0+v_2>0$ .", "Proof: Let us define the symmetric polynomials of the roots $s_1,s_2,s_3$ by $D(v)=-(v-v_0)(v-v_1)(v-v_2)=-v^3+s_1\\,v^2-s_2\\,v+s_3.$ The function $D$ is positive iff either $v\\in (-\\infty ,v_0)$ or $v\\in (v_1,v_2)$ .", "Let us first study the polynomial $Q=3v^4-4s_1\\,v^3+\\cdots $ .", "Since $\\Delta (Q)=-6912(v_1-v_0)^4(v_2-v_0)^4(v_2-v_1)^4<0$ we conclude that $Q$ has two simple real zeroes.", "For $v\\in \\,(v_0,v_1)$ the relation $Q^{\\prime }=-12\\,D$ shows that $Q$ increases from $Q(v_0)=-(v_0-v_1)^2(v_0-v_2)^2$ to $Q(v_1)=-(v_1-v_0)^2(v_1-v_2)^2$ ; it then decreases to $Q(v_2)=-(v_2-v_0)^2(v_2-v_1)^2$ so that $Q$ is strictly negative for all $v\\in \\,(v_0,v_2)$ and, since $Q(\\pm \\infty )=+\\infty $ , it will have a simple zero at $v=v_<v_0$ and at $v=\\widetilde{v}_>v_2$ , with the relation $\\displaystyle v_+\\widetilde{v}_=\\frac{4}{3}\\,s_1.$ Let us come back to the first positivity interval for $D$ which is $I=(-\\infty ,v_0)$ .", "As we have already seen, $Q$ has a simple zero $v_\\in I$ .", "Let us prove that $P(v_)>0$ which will be sufficient to ascertain, thanks to Lemma REF , that $v=v_$ is a curvature singularity.", "To this end we use the relation $P(v)=-Q(v) +4(s_1-v)D(v)\\quad \\quad \\Longrightarrow \\quad \\quad P(v_)=4(s_1-v_)\\,D(v_).$ Since $v_<v_0$ we have $D(v_)>0$ and $s_1-v_=\\widetilde{v}_-\\frac{s_1}{3}>v_2-\\frac{s_1}{3}=\\frac{2v_2-v_0-v_1}{3}>0.$ Let us now consider the second positivity interval for $D$ which is $I=(v_1,v_2)$ .", "We find it convenient to define new parameters by $d=\\frac{v_2-v_1}{2}>0,\\quad \\quad \\quad \\quad l=\\frac{v_1+v_2+2v_0}{v_2-v_1}\\in {\\mathbb {R}},\\quad \\quad \\quad \\quad m=\\frac{v_1+v_2-2v_0}{v_2-v_1}>1,$ and a new coordinate, $x$ , by $v=d\\Big (x+\\frac{l+m}{2}\\Big ),\\quad \\quad \\quad \\quad x\\in \\ I=\\,[-1,+1].$ Since $d>0$ we will set $d=1$ .", "It follows that $\\left\\lbrace \\begin{array}{ll}D=(x+m)(1-x^2),& \\\\[4mm]P=\\Big (L_+\\,(1-x^2)+2(m+x)\\Big )\\Big (L_-\\,(1-x^2)+2(m+x)\\Big ),&\\quad \\quad L_{\\pm }=l\\pm \\sqrt{l^2-1},\\\\[4mm]Q=3x^4+4mx^3-6x^2-12mx-4m^2-1,&\\quad \\quad Q^{\\prime }=-12\\,D,\\end{array}\\right.$ and the metric (again up to the change $G\\rightarrow 2\\,G$ ) reads now $G=\\rho ^2\\,\\frac{dx^2}{D}+\\frac{4D}{P}\\,d\\phi ^2,\\quad \\quad \\quad \\quad \\rho =\\frac{Q}{P}.$ For $x\\in \\,I$ the polynomial $Q$ decreases from $Q(-1)=-4(m-1)^2$ to $Q(1)=-4(m+1)^2$ forbidding any curvature singularity.", "It remains to check the positivity of $P$ .", "Its factorized expression shows that for $l\\in \\,[-1,1)$ it has no real root.", "For $l\\ge 1$ it has four simple real roots which lie outside $I$ , and for $l<-1$ two of its real roots are still outside $I$ , the remaining two $x_-<x_+$ being contained in $I$ .", "It follows that $I$ may be reduced to any of the intervals $I_1=(-1,x_-)\\quad \\quad \\mbox{or}\\quad \\quad I_2=(x_-,\\,x_+)\\quad \\quad \\mbox{or}\\quad \\quad I_3=(x_+,1).$ Now, at least one end-point is a zero of $P$ , and by Lemma REF , the expected manifold is not closed.", "So far, we have proved that a manifold can exists iff $\\ l\\in \\,(-1,+\\infty )$ , which translates as $v_0+v_2>0$ .", "Let us study the the behavior of the metric at the end-points of $I$ by setting $x=\\cos \\vartheta $ with $\\vartheta \\in (0,\\pi )$ .", "We find that $G(\\vartheta \\rightarrow 0+)\\approx \\frac{1}{m+1}(d\\vartheta ^2+\\sin ^2\\vartheta \\,d\\phi ^2),\\quad \\quad G(\\vartheta \\rightarrow \\pi -)\\approx \\frac{1}{m-1}(d\\vartheta ^2+\\sin ^2\\vartheta \\,d\\phi ^2),$ and $\\vartheta =0,\\pi $ are indeed apparent singularities.", "From Lemma REF we get $\\gamma =-\\frac{W}{Q\\,\\sqrt{P}}\\quad \\quad W=-(x^2+2x-1+2m)(x^2-2x-1-2m)(x^2+2mx+1)$ which gives $\\chi (M)=\\gamma (1)-\\gamma (-1)=2,$ so that the manifold is actually $M\\cong {\\mathbb {S}}^2$ .", "Returning to the integrals, we will define once more $H=\\frac{1}{2}\\left(\\Pi ^2+P\\,\\frac{P_{\\phi }^2}{4D}\\right)=\\frac{1}{2\\Omega ^2}\\left(P_{\\theta }^2+\\frac{P_{\\phi }^2}{\\sin ^2\\theta }\\right),$ which leads to the relations $\\Omega ^2\\,\\sin ^2\\theta =\\frac{4D}{P},\\quad \\quad \\frac{d\\theta }{\\sin \\theta }=\\frac{F(x)}{(1-x^2)}\\,dx,\\quad \\quad F(x)=\\frac{Q(x)}{2(m+x)\\sqrt{P(x)}},$ from which we deduce $t\\equiv \\tan \\frac{\\theta }{2}=\\exp {\\left(\\int _0^x\\,\\frac{F(u)}{(1-u^2)}\\,du\\right)}.$ We need first to check the behavior of the conformal $\\Omega $ factor at the north pole for $x\\rightarrow 1-$ .", "We have $t=\\sqrt{1-x}\\ T_N(x),\\quad T_N(x)=\\exp {(U(x))},\\quad U(x)=\\int _0^x\\left(\\frac{F(u)}{1+u}-\\frac{F(1)}{2}\\right)\\frac{du}{1-u},$ so that $T_N$ is $C^{\\infty }$ in a neighborhood of $x=+1$ .", "This implies that $\\Omega ^2=\\frac{(1+t^2)^2(m+x)(1+x)}{P(x)\\,T_N^2(x)}$ is also $C^{\\infty }$ in a neighborhood of $x=+1$ .", "At the south pole, i.e., for $x\\rightarrow \\,-1+$ a similar argument works.", "The expression of $S_1$ , in view of $\\Pi =P_{\\theta }/\\Omega $ , is now the following: $S_1=\\frac{L_2}{\\Omega }\\,\\left(-\\Pi ^2+Q\\,\\frac{P_{\\phi }^2}{4D}\\right)+x^2\\,L_3\\,\\left(A\\,\\Pi ^2-B\\,\\frac{P_{\\phi }^2}{4D}\\right)$ with $A=\\frac{D^{\\prime }+\\sqrt{P}\\,\\cos \\theta }{2\\sin \\theta \\,\\sqrt{D}},\\quad \\quad \\quad \\quad B=\\frac{W+Q\\sqrt{P}\\,\\cos \\theta }{2\\sin \\theta \\,\\sqrt{D}},$ giving at the north pole: $\\left\\lbrace \\begin{array}{l}\\displaystyle A=\\frac{1}{\\sqrt{2(m+1)}}\\frac{(l+m+2)}{2\\,T_N(1)}+O(\\sin ^2\\theta ),\\\\[6mm]\\displaystyle B=-\\Big (2(m+1)\\Big )^{3/2}\\frac{(l+m+2)}{2\\,T_N(1)}+O(\\sin ^2\\theta ),\\end{array}\\right.$ where the leading coefficients never vanish since $l+m>0$ .", "To analyze the behavior of $S_1$ at south pole let us define $t=\\frac{1}{\\sqrt{1+x}\\, T_S(x)},\\;T_S(x)=\\exp {(-V(x))},\\;V(x)=\\int _0^x\\left(\\frac{F(u)}{1-u}-\\frac{F(-1)}{2}\\right)\\frac{du}{1+u},$ from which we deduce $\\left\\lbrace \\begin{array}{l}\\displaystyle A=-\\frac{1}{\\sqrt{2(m-1)}}\\frac{(l+m-2)}{2\\,T_S(-1)}+O(\\sin ^2\\theta ),\\\\[6mm]\\displaystyle B=(2(m-1))^{3/2}\\frac{(l+m-2)}{2\\,T_S(-1)}+O(\\sin ^2\\theta ),\\end{array}\\right.$ which are well-behaved.", "For $l+m=2$ the power series expansions begin with $\\sin ^2\\theta $ , a possibility already observed in the proof of Theorem REF .", "As to the integral $S_2$ , the argument given in the proof of Theorem REF works here just as well.", "$\\quad \\Box $" ], [ "Comparison with the results of Matveev and Shevchishin", "In [4] it was stated in Theorem 6.1 that the metric $g=\\frac{dx^2+dy^2}{h_x^2},\\quad \\quad \\quad \\quad h_x=\\frac{dh}{dx},$ where $h$ is a solution of the differential equation (3,ii) with $\\mu =1,\\quad \\quad A_0=1,\\quad \\quad A_1=0,\\quad \\quad A_3=A_4=A_e>0\\quad \\quad \\mbox{and}\\quad \\quad A_2\\in {\\mathbb {R}},$ is globally defined on $S^2$ .", "As we will show in what follows, our results are partly in agreement with this Theorem 6.1.", "Let us first write again the metric in our $(v,\\phi )$ coordinates: $g=\\frac{Q^2}{P^2}\\,\\frac{dv^2}{D}+\\frac{4D}{P}\\,d\\phi ^2,\\quad \\quad \\quad \\phi \\in {\\mathbb {S}}^1,$ where $Q$ and $P$ are deduced from the knowledge of $D$ by the relations given in (REF ).", "So to be able to compare these metrics we have first to notice that $h=\\frac{D^{\\prime }}{2\\sqrt{2D}}\\quad \\quad \\quad D^{\\prime }=\\frac{dD}{dv}, \\quad \\quad \\quad \\quad h_x=u=\\sqrt{h^2+v}=\\sqrt{\\frac{P}{8D}}>0,$ and that $y=\\phi ,\\quad \\quad \\quad \\quad \\frac{dx}{dv}=\\frac{dx}{dh}\\frac{dh}{dv}=\\frac{1}{h_x}\\frac{dh}{dv}=\\frac{Q}{2D\\sqrt{P}}.$ Let us notice that to be riemannian the metric (REF ) requires $D>0$ and $P>0$ and for the transformation $v \\rightarrow x$ to be locally bijective we need $Q$ to have a fixed sign.", "Under the hypotheses of Theorem 6.1 we have, in our notation, $\\epsilon =1,\\quad \\quad \\quad \\quad a=-A_2,\\quad \\quad \\quad \\quad \\lambda =2A_e,$ which gives $D=-(v+A_2)(v^2-A_2^2+c)-8A_e^2,$ where $c$ is a constant of integration which does not appear in the proof of Theorem 6.1 and which can be freely chosen.", "The discriminant of $D$ is $\\Delta (D)=-27\\,\\xi ^2+4A_2(8A_2^2-9c)\\xi +4c^2(A_2^2-c),\\quad \\quad \\quad \\quad \\xi =8A_e^2,$ and the crucial point is that the sign of this discriminant is undefined.", "If $\\Delta (D)>0$ , then Theorem 6.1 of Matveev and Shevchishin agrees with our Theorem 2, and the metric is indeed globally defined on ${\\mathbb {S}}^2$ .", "Nevertheless, if $\\Delta (D)\\le 0$ our Propositions REF and REF show that either curvature singularities or conical singularities rule out any closed manifold.", "Let us mention that we have also found a metric globally defined on ${\\mathbb {S}}^2$ for $\\epsilon =0$ (see Theorem REF ), a case which has not been studied in [4]." ], [ "The affine case", "In this last case, we will prove that there is no closed manifold for the metric.", "However, since the analysis is much simpler we will determine the metrics globally defined either on ${\\mathbb {R}}^2$ or on ${\\mathbb {H}}^2$ ." ], [ "The metric", "The differential equation and the metric are $h_x\\Big (h_x^2+A_1\\,h+A_2\\Big )=A_3\\,x+A_4,\\quad \\quad \\quad \\quad G=\\frac{dx^2+dy^2}{h_x^2},$ see (REF , $iii$ ) and (REF ).", "Differentiating the equation for $h$ gives $\\Big (3\\,h_x^2+A_1 h+A_2\\Big )h_{xx}+A_1\\,h_x^2=A_3,$ and regarding again $u=h_x$ as a function of the new variable $h$ , we rewrite the previous equations as $u(3u^2+A_1 h+A_2)\\frac{du}{dh}=A_3-A_1 u^2.$ Considering the inverse function $h(u)$ we end up with a linear ode, namely $(A_3-A_1 u^2)\\frac{dh}{du}-A_1u\\,h=u(3u^2+A_2).$ Two cases have to be considered: 1.", "If $A_1=0$ then $A_3$ cannot vanish; positing $\\displaystyle \\mu =\\frac{3u^2+A_2}{A_3}$ , the original variable, $x$ , and the metric, $G$ , are now given by $dx=\\mu \\,du,\\quad \\quad \\mu =\\frac{1}{u}\\,\\frac{dh}{du}\\quad \\quad \\Longrightarrow \\quad \\quad G=\\frac{1}{u^2}\\left(\\mu ^2\\,du^2+dy^2\\right).$ Interestingly, the relations $h=h_0+\\frac{A_2}{2A_3}u^2+\\frac{3}{4}\\frac{u^4}{A_3},\\quad \\quad \\quad \\quad A_3\\,x+A_4=A_2\\,u+u^3$ show that we have integrated the ode (REF ) by expressing the function $h$ and the variable $x$ parametrically in terms of $u$ .", "2.", "If $A_1\\ne 0$ we can set $A_1=1$ and, by a shift of $h$ , we may put $A_2=0$ .", "To simplify matters, we will perform the following rescalings: $y\\rightarrow 2y$ , and $G\\rightarrow \\frac{1}{4}G$ .", "This time, we will define $-2\\mu =\\frac{1}{u}\\,\\frac{dh}{du}\\quad \\quad \\Longrightarrow \\quad \\quad G=\\frac{1}{u^2}\\left(\\mu ^2\\,du^2+dy^2\\right),$ and we get two possible solutions for $\\mu $ : $\\mu =1+\\frac{C}{(u^2-A_3)^{3/2}}\\quad \\quad \\mbox{or}\\quad \\quad \\mu =1+\\frac{C}{(A_3-u^2)^{3/2}},$ where $C$ is a real constant of integration." ], [ "Global structure for vanishing $\\mathbf {A_1}$", "We have just seen that $\\displaystyle \\mu =\\frac{3u^2+A_2}{A_3}$ , and must thus discuss two cases separately: 1.", "First case: $A_2=0$ , then we can pose $\\mu =2u^2$ .", "2.", "Second case: $A_2\\ne 0$ , then we can pose $\\mu =1+au^2$ ." ], [ "The case $\\mathbf {A_2=0}$", "The relation (REF ) and the change $u\\rightarrow v=u^2$ yield the metric and Hamiltonian, viz., $G=dv^2+\\frac{dy^2}{v}\\quad \\quad \\Longrightarrow \\quad \\quad H=\\frac{1}{2}(P_v^2+v\\,P_y^2),$ while the cubic integrals read now $S_1=\\frac{2}{3}\\,P_v^3+P_y^2(v\\,P_v+\\frac{y}{2}\\,P_y)$ and $S_2=y\\,S_1-\\left(\\frac{y^2}{4}+\\frac{v^3}{9}\\right)P_y^3-\\frac{2}{3}\\,v^2\\,H\\,P_y.$ This last relation shows that $S_2$ is not algebraically independent, and that the superintegrable system we are considering is just generated by $(H,\\,P_y,\\,S_1)$ .", "Let us mention, for completeness, the following Poisson brackets, namely $\\lbrace P_y,S_1\\rbrace =\\frac{1}{2}\\,P_y^3,\\quad \\quad \\lbrace P_y,S_2\\rbrace =S_1,\\quad \\quad \\lbrace S_1,S_2\\rbrace =\\frac{3}{2}\\,S_2\\,P_y^2.$ Proposition 14 For $A_2=0$ the superintegrable system $(H,P_y,S_1)$ is not globally defined.", "Proof: The Riemannian character of the metric requires $v>0$ and $y\\in {\\mathbb {R}}$ .", "If this metric were defined on a manifold, the scalar curvature would be everywhere defined.", "An easy computation gives for result $\\displaystyle R_G=-\\frac{3}{2v^2}$ which is singular for $v\\rightarrow 0+$ .$\\quad \\Box $" ], [ "The case $\\mathbf {A_2\\ne 0}$", "We have now the Hamiltonian $2H=u^2\\left(\\frac{P_u^2}{\\mu ^2}+P_y^2\\right),\\quad \\quad u>0,\\quad y\\in {\\mathbb {R}},\\quad \\quad \\mu =1+au^2,\\quad a\\in {\\mathbb {R}},$ and the cubic integrals are respectively $S_1=\\frac{2a}{3}\\left(\\frac{u}{\\mu }\\,P_u\\right)^3+P_y\\Big (u\\,P_u\\,P_y+y\\,P_y^2\\Big )$ and $S_2=y\\,S_1-\\frac{1}{2}\\Big (y^2+u^2(1+au^2/3)^2\\Big )P_y^3-\\frac{a}{3}u^2(2+au^2)HP_y.$ The non-trivial Poisson brackets of the observables are then given by $\\lbrace P_y,S_1\\rbrace =P_y^3,\\quad \\quad \\lbrace P_y,S_2\\rbrace =S_1,\\quad \\quad \\lbrace S_1,S_2\\rbrace =3\\,S_2\\,P_y^2\\,+4\\,P_y^3\\,H+\\frac{16}{3}\\,a\\,P_y\\,H^2.$ Proposition 15 For $A_2\\ne 0$ the superintegrable system $(H,P_y,S_1)$ 1. is not globally defined for $a<0$ , 2. is trivial for $a=0$ , 3. is globally defined on $M\\cong {\\mathbb {H}}^2$ for $a>0$ .", "Proof: The scalar curvature reads now $R_G=-\\frac{2}{\\mu ^3}\\,(1+3au^2),\\quad \\quad \\quad \\quad u>0,\\quad y\\in {\\mathbb {R}}.$ If $a<0$ it is singular for $\\displaystyle u_0=\|a\|^{-1/2}$ , and the system cannot be defined on a manifold.", "For $a=0$ the metric reduces to the canonical metric $G({\\mathbb {H}}^2,{\\rm can})=\\frac{du^2+dy^2}{u^2}.$ of the hyperbolic plane ${\\mathbb {H}}^2$ .", "As a consequence of Thompson's theorem, which has been recalled above, $S_1$ and $S_2$ are reducible.", "Of course the set $(H,\\,P_y)$ still remains an integrable system but it is trivial in the sense that it is no longer superintegrable.", "Let us examine the last case for which $a>0$ .", "The change of coordinates $t=u\\left(1+\\frac{a}{3}\\,u^2\\right)\\;\\longmapsto \\;u=\\frac{\\xi ^{1/3}}{a}-\\xi ^{-1/3},\\quad \\quad \\xi (t)=\\frac{3}{2}\\,a^2\\,t+\\sqrt{a^3+\\frac{9}{4}\\,a^4\\,t^2},$ implies that $u(t)$ is $C^{\\infty }$ for all $t\\ge 0$ .", "In these new coordinates the metric becomes $G=\\Omega ^2\\,\\frac{dt^2+dy^2}{t^2}=\\Omega ^2\\,G({\\mathbb {H}}^2,{\\rm can}),\\quad \\quad \\Omega (t)=1+\\frac{a}{3}\\,u^2(t),\\quad \\quad t>0,\\quad y\\in {\\mathbb {R}},$ and, since $\\Omega $ never vanishes, it is globally conformally related to the canonical metric of the hyperbolic plane, $M\\cong {\\mathbb {H}}^2$ .", "Using the generators of $\\mathrm {sl}(2,{\\mathbb {R}})$ on $T^{\\mathbb {H}}^2$ (given in the Appendix) allows us to write the Hamiltonian in the new guise $H=\\frac{t^2}{2\\,\\Omega ^2}\\Big (P_t^2+P_y^2\\Big )=\\frac{1}{2\\,\\Omega ^2}\\Big (M_1^2+M_2^2-M_3^2\\Big ).$ The relations $P_y=M_2+M_3\\quad \\quad \\mbox{and}\\quad \\quad t\\,P_t=\\frac{M_1-x^1\\,P_1}{1+(x^1)^2}$ show that $S_1=\\frac{2a}{3}\\,\\left(\\frac{t\\,P_t}{\\Omega }\\right)^3+P_y^2\\left(\\mu \\,\\frac{t\\,P_t}{\\Omega }+y\\,P_y\\right),\\quad \\quad \\mu (t)=1+a\\,u^2(t),\\quad \\quad a>0,$ is globally defined on $M$ .", "The same is true for $S_2$ (see the relation (REF )).", "$\\quad \\Box $" ], [ "Global structure for non-vanishing $\\mathbf {A_1}$", "In the formula (REF ) let us change $A_3\\rightarrow a$ .", "We have, again, two cases to consider according to $\\epsilon ={\\rm sign}(u^2- a)$ ." ], [ "First case: $\\mathbf {\\epsilon =+1}$", "The metric and the Hamiltonian are given by $G=\\frac{1}{u^2}\\Big (\\mu ^2\\,du^2+dy^2\\Big ),\\quad \\quad H=\\frac{u^2}{2}\\left(\\frac{P_u^2}{\\mu ^2}+P_y^2\\right),\\quad \\quad u^2-a>0,\\quad y\\in \\mathbb {R},$ where $\\mu =1+\\frac{C}{(u^2-a)^{3/2}}.$ The cubic integrals are then $S_1=\\left(\\frac{u}{\\mu }\\,P_u\\right)^3+u(u^2-a)\\,P_u\\,P_y^2-ay\\,P_y^3+2y\\,H\\,P_y$ and $S_2=y\\,S_1+\\frac{1}{2}\\left(a(u^2+y^2)-\\frac{2Cu^2}{\\sqrt{u^2-a}}+\\frac{C^2}{u^2-a}\\right)P_y^3-\\left(u^2+y^2-\\frac{2C}{\\sqrt{u^2-a}}\\right)H\\,P_y.$ The case $C=0$ corresponds to the canonical metric on ${\\mathbb {H}}^2$ , and, as already explained in Proposition REF , the system becomes trivial.", "In the following developments, we will discuss the global properties of our superintegrable system according to the sign of $C \\ne 0$ , rescaling it to $\\pm 1$ .", "Proposition 16 For $C=-1$ the superintegrable system $(H,P_y,S_1)$ is globally defined iff $a<0$ and $\|a\|>1$ , in which case the manifold is $M\\cong {\\mathbb {H}}^2$ .", "Proof: The scalar curvature is $R_G=-\\frac{2}{\\mu ^3}\\left(1+\\frac{(2u^2+a)}{(u^2-a)^{5/2}}\\right).$ For $a\\ge 0$ we must have $u>\\sqrt{a}$ and $R_G$ will be singular for $u_0=\\sqrt{a+1}$ .", "For $a<0$ we must have $u>0$ .", "Then the curvature is singular for $u_0=\\sqrt{1-\\rho }$ if $\\,\\rho =\|a\|\\le 1$ .", "However for $\\rho >1$ the function $\\mu $ no longer vanishes and the curvature remains continuous for all $u\\ge 0$ .", "The metric then reads $G=\\frac{1}{u^2}\\Big (\\mu ^2\\,du^2+dy^2\\Big ),\\quad \\quad \\quad \\quad \\mu =1-\\frac{1}{(\\rho +u^2)^{3/2}},\\quad \\quad u>0,\\quad y\\in {\\mathbb {R}}.$ Let us define the new variable $t=u\\left(1-\\frac{1}{\\rho \\sqrt{\\rho +u^2}}\\right),\\quad \\quad \\quad \\quad u \\in [0,+\\infty )\\longmapsto t \\in [0,+\\infty ).$ Since $\\displaystyle \\mu =\\frac{dt}{du}$ never vanishes, the inverse function $u(t)$ is $C^{\\infty }([0,+\\infty ))$ and the metric can be written as $G=\\Omega ^2\\,G(\\mathbb {H}^2,{\\rm can}),\\quad \\quad \\quad \\quad \\Omega (t)=1-\\frac{1}{\\rho \\sqrt{\\rho +u^2(t)}},\\quad \\quad \\rho >1,$ where the conformal factor $\\Omega (t)$ is $C^{\\infty }$ and never vanishes: the manifold is again $M\\cong {\\mathbb {H}}^2$ .", "The first cubic integral $S_1=\\left(\\frac{t\\,P_t}{\\Omega }\\right)^3+\\mu (t)(\\rho +u^2(t))\\left(\\frac{t\\,P_t}{\\Omega }\\right)P_y^2+\\rho \\,y\\,P_y^3+2y\\,H\\,P_y$ is therefore globally defined (with same argument as in the proof of Proposition (REF )), and (REF ) gives $S_2=y\\,S_1+\\frac{1}{2}\\left(-\\rho (u^2+y^2)+\\frac{2u^2}{\\sqrt{\\rho +u^2}}+\\frac{1}{\\rho +u^2}\\right)P_y^3-\\left(u^2+y^2+\\frac{2}{\\sqrt{\\rho +u^2}}\\right)H\\,P_y,$ showing that this is also true for $S_2$ .", "$\\quad \\Box $ Proposition 17 For $C=+1$ the superintegrable system $(H,P_y,S_1)$ is globally defined either if $a>0$ and the manifold is $M\\cong {\\mathbb {R}}^2$ , or if $a<0$ and $M\\cong {\\mathbb {H}}^2$ .", "Proof: The metric reads now $G=\\frac{1}{u^2}\\Big (\\mu ^2\\,du^2+dy^2\\Big ),\\quad \\quad \\quad \\quad \\mu =1+\\frac{1}{(u^2-a)^{3/2}}.$ Consider first the case $a>0$ for which $u>\\sqrt{a}$ .", "Let us define the new coordinate $t=u\\left(1-\\frac{1}{a\\sqrt{u^2-a}}\\right),\\quad \\quad \\quad \\quad u \\in (\\sqrt{a},+\\infty )\\longmapsto t \\in {\\mathbb {R}}.$ Since, again, $\\displaystyle \\mu =\\frac{dt}{du}$ does not vanish $u(t)$ is $C^{\\infty }({\\mathbb {R}})$ , and the metric $G=\\frac{dt^2+dy^2}{u^2(t)},\\quad \\quad \\quad \\quad t\\in {\\mathbb {R}},\\quad y\\in {\\mathbb {R}},$ turns out to be globally conformally related to the flat metric; the manifold is therefore $M\\cong {\\mathbb {R}}^2$ .", "The cubic integral $S_1=(u(t)\\,P_t)^3+\\mu (t)(u^2(t)-a)(u(t)\\,P_t)\\,P_y^2-ay\\,P_y^3+2y\\,H\\,P_y$ remains hence globally defined, and the same holds true for $S_2$ .", "- For $a=0$ the function $\\displaystyle \\mu =1+\\frac{1}{u^3}$ is no longer even, so we must consider that $u\\in {\\mathbb {R}}$ and the scalar curvature $R_G=2u^6\\,\\frac{(2-u^3)}{(1+u^3)^3}$ is not defined for $u=-1\\ $ ; there is thus no obtainable manifold structure.", "- For $a<0$ we set $\\rho =\|a\|$ and we must take $u>0$ ; we then define the new coordinate $t=u\\left(1+\\frac{1}{\\rho \\sqrt{\\rho +u^2}}\\right),\\quad \\quad \\quad \\quad u \\in (0,+\\infty )\\longmapsto t \\in (0,+\\infty ).$ Since $\\displaystyle \\mu =\\frac{dt}{du}$ never vanishes, the inverse function $u(t)$ is $C^{\\infty }([0,+\\infty ))$ .", "The metric $G=\\Omega ^2\\,\\frac{dt^2+dy^2}{t^2},\\quad \\quad \\Omega (t)=1+\\frac{1}{\\rho \\sqrt{\\rho +u^2}},\\quad \\quad \\rho >0,\\quad t>0,\\quad y\\in {\\mathbb {R}},$ is again globally conformally related to the canonical metric on the manifold $M\\cong {\\mathbb {H}}^2$ .", "The proof that the cubic integrals are also globally defined is the same as in Proposition REF .", "$\\quad \\Box $" ], [ "Second case: $\\mathbf {\\epsilon =-1}$", "The metric and the Hamiltonian are now given by $G=\\frac{1}{u^2}\\Big (\\mu ^2\\,du^2+dy^2\\Big ),\\quad \\quad H=\\frac{u^2}{2}\\left(\\frac{P_u^2}{\\mu ^2}+P_y^2\\right),\\quad \\quad a-u^2>0,\\quad y\\in \\mathbb {R},$ where $\\mu =1+\\frac{C}{(a-u^2)^{3/2}}.$ The scalar curvature reads thus $R_G=-\\frac{2}{\\mu ^3}\\left(1+C\\,\\frac{(2u^2+a)}{(a-u^2)^{5/2}}\\right).$ The cubic integral $S_1$ is the same as in (REF ) while $S_2=y\\,S_1+\\frac{1}{2}\\left(a(u^2+y^2)+\\frac{2Cu^2}{\\sqrt{a-u^2}}+\\frac{C^2}{a-u^2}\\right)P_y^3-\\left(u^2+y^2+\\frac{2C}{\\sqrt{a-u^2}}\\right)H\\,P_y$ is merely obtained by the substitution $C \\rightarrow -C$ .", "Proposition 18 Either for $C=-1$ and $\\ 0<a<1$ or for $C=+1$ the superintegrable system $(H,P_y,S_1)$ is globally defined on the manifold $M\\cong {\\mathbb {H}}^2$ .", "Proof: We must have $a>0$ to ensure $u\\in (0,\\sqrt{a})$ .", "- For $C=-1$ the scalar curvature is singular when $\\mu $ vanishes.", "This happens for $u_0=\\sqrt{a-1}$ and $a\\ge 1$ ; in this case there exists no manifold structure.", "However for $0<a<1$ the function $\\mu $ never vanishes, so we can define $t=-u\\left(1-\\frac{1}{a\\sqrt{a-u^2}}\\right),\\quad \\quad \\quad \\quad u \\in (0,\\sqrt{a}) \\longmapsto t \\in (0,+\\infty ),$ and the inverse function $u(t)$ is in $C^{\\infty }([0,+\\infty ))$ ; this leads to the metric $G=\\Omega ^2\\,G(\\mathbb {H}^2,{\\rm can}),\\quad \\quad \\quad \\quad \\Omega (t)=-1+\\frac{1}{a\\sqrt{a-u^2(t)}},\\quad \\quad 0<a<1,$ where the conformal factor never vanishes; hence, the manifold is again $M\\cong {\\mathbb {H}}^2$ .", "The proof that the cubic integrals are also globally defined is the same as in Proposition REF .", "- For $C=+1$ the function $\\mu =1+\\frac{1}{(a-u^2)^{3/2}}$ never vanishes, implying that the curvature is defined everywhere for $u\\in (0,\\sqrt{a})$ .", "If we define $t=u\\left(1+\\frac{1}{a\\sqrt{a-u^2}}\\right),\\quad \\quad \\quad \\quad u \\in (0,\\sqrt{a}) \\longmapsto t \\in (0,+\\infty ),$ the metric retains the form $G=\\Omega ^2\\,G(\\mathbb {H}^2,{\\rm can}),\\quad \\quad \\quad \\quad \\Omega =1+\\frac{1}{a\\sqrt{a-u^2(t)}},\\quad \\quad a>0,$ where the conformal factor, $\\Omega $ , never vanishes; hence, the manifold is again ${M\\cong \\mathbb {H}}^2$ .", "At last, the proof that the cubic integrals $S_1$ and $S_2$ are also globally defined is the same as in Proposition REF .", "$\\quad \\Box $" ], [ "Conclusion", "We have completed the work initiated by Matveev and Shevchishin in [4] by providing the explicit form of their metrics in local coordinates.", "This allowed us to determine systematically all the cases in which their superintegrable systems can be hosted by a simply-connected, two-dimensional smooth manifold $M$ .", "Let us emphasize that we have achieved, via Theorem REF and Theorem REF , the classification of all these metrics on closed, simply-connected, surfaces, namely on $M\\cong {\\mathbb {S}}^2$ .", "As pointed out in [4] superintegrable systems on a closed manifold should lead to Zoll metrics [1], i.e., to metrics whose geodesics are all closed and of the same length.", "Using the explicit formulas obtained here for the metrics it has been proved by a direct analysis in [10] that all the metrics defined on $S^2$ that we have obtained here are indeed Zoll metrics.", "Generalizing this analysis to closed orbifolds gives either Tannery or Zoll metrics.", "Another obvious line of research would be the generalization of these results to the case of observables of fourth or even higher degree, as well as the challenging problem of their quantization.", "An interesting approach could be to use a well and uniquely defined quantization procedure, in our case the conformally-equivariant quantization [2].", "The latter, from its very definition and construction, could be perfectly fitted to deal with integrable systems on Riemann surfaces." ], [ "Appendix: the hyperbolic plane", "Let us recall that the hyperbolic plane ${\\mathbb {H}}^2=\\lbrace (x^1,\\,x^2,\\,x^3)\\in {\\mathbb {R}}^3\\ \|\\ (x^1)^2+(x^2)^2-(x^3)^2=-1,\\ x^3>0\\rbrace $ may be embedded in $\\mathbb {R}^{2,1}$ as follows $x^1=\\frac{y}{t},\\quad \\quad x^2=\\frac{1}{2t}(t^2+y^2-1),\\quad \\quad x^3=\\frac{1}{2t}(t^2+y^2+1).$ This choice of coordinates leads to the induced metric $G(\\mathbb {H}^2,{\\rm can})=\\frac{dt^2+dy^2}{t^2},\\quad \\quad \\quad \\quad \\quad \\quad t>0,\\quad y\\in {\\mathbb {R}}.$ The generators on $T^({\\mathbb {H}}^2)$ of the group of isometries of ${\\mathbb {H}}^2$ given by $\\begin{array}{l}\\displaystyle M_1=x^2\\,P_3+x^3\\,P_2=t\\,P_t+y\\,P_y, \\\\[3mm]\\displaystyle M_2=x^3\\,P_1+x^1\\,P_3=-ty\\,P_t+\\frac{(1+t^2-y^2)}{2}\\,P_y, \\\\[3mm]\\displaystyle M_3=x^1\\,P_2-x^2\\,P_1=+ty\\,P_t+\\frac{(1-t^2+y^2)}{2}\\,P_y, \\end{array}$ are globally defined and generate, with respect to the Poisson bracket, the Lie algebra $\\mathrm {sl}(2,{\\mathbb {R}})$ , namely $\\lbrace M_1,M_2\\rbrace =-M_3,\\quad \\quad \\quad \\lbrace M_2,M_3\\rbrace =M_1,\\quad \\quad \\quad \\lbrace M_3,M_1\\rbrace =M_2.$" ] ]	1403.0422
[ [ "Automatic classification of time-variable X-ray sources" ], [ "Abstract To maximize the discovery potential of future synoptic surveys, especially in the field of transient science, it will be necessary to use automatic classification to identify some of the astronomical sources.", "The data mining technique of supervised classification is suitable for this problem.", "Here, we present a supervised learning method to automatically classify variable X-ray sources in the second \\textit{XMM-Newton} serendipitous source catalog (2XMMi-DR2).", "Random Forest is our classifier of choice since it is one of the most accurate learning algorithms available.", "Our training set consists of 873 variable sources and their features are derived from time series, spectra, and other multi-wavelength contextual information.", "The 10-fold cross validation accuracy of the training data is ${\\sim}$97% on a seven-class data set.", "We applied the trained classification model to 411 unknown variable 2XMM sources to produce a probabilistically classified catalog.", "Using the classification margin and the Random Forest derived outlier measure, we identified 12 anomalous sources, of which, 2XMM J180658.7$-$500250 appears to be the most unusual source in the sample.", "Its X-ray spectra is suggestive of a ULX but its variability makes it highly unusual.", "Machine-learned classification and anomaly detection will facilitate scientific discoveries in the era of all-sky surveys." ], [ "Introduction", "The identification of variable and transient astrophysical sources will be a major challenge in the near future across all wavelengths.", "The advent of facilities such as the Large Synoptic Survey Telescope (LSST) in optical , the Square Kilometre Array (SKA) in radio and the extended ROentgen Survey with an Imaging Telescope Array (e-ROSITA) in X-rays , will enable the next generation of all-sky time-domain surveys.", "Many types of transients and variable sources are currently known, such as supernovae, cataclysmic variables (CVs), X-ray binaries (XRBs), flare stars, gamma-ray bursts (GRB), tidal disruption flares, and future time-domain surveys will likely uncover novel source types.", "The large number of sources to be surveyed makes identifying interesting transients a challenging task, especially since timely multi-wavelength follow-ups will be critical for fulfilling the transient science goals.", "To this end, we envision that automatic classification will be a crucial part of the processing pipeline .", "Here, we demonstrate the feasibility of using time series and contextual information to automatically classify variable and transient sources.", "We used data from the X-ray Multi-Mirror Mission - Newton (XMM-Newton) because there has not been previous studies on this data set using automatic classification algorithms and because the time series for many of the sources are readily available, thereby enabling us to investigate the efficacy of a classifier built using solely time-domain information.", "Automatic classification is a similar problem across all wavelengths and we expect that the techniques used in this paper can be readily adapted for data sets in other wave-bands.", "The Second XMM-Newton Serendipitous Source Catalog Data Release 2 (2XMMi-DR2) was the largest catalog of X-ray sources at the time it was released, but has since been surpassed by 2XMMi-DR3 and 3XMM.", "In this study, we used 2XMMi-DR2 and kept DR3 as a verification sample.", "There have been previous attempts to classify the unidentified sources in 2XMMi , .", "The traditional method is to cross-match the unknown sources with catalogs in other wavelengths (e.g.", "SDSS, 2MASS) and then use expert knowledge to draw up classification rules.", "For example, one powerful discriminant is the ratio of the optical to X-ray flux for separating active galactic nuclei (AGN) and stars.", "In the scheme used by , sources whose positions coincide with the centres of galaxies are deemed to be AGN.", "Manually selected classification rules often have their basis in science and are usually comprehensible to other experts.", "This method works well when there are only a few pieces of information to be processed (e.g.", "optical to X-ray flux), but becomes intractable when there are many disparate sets of information.", "In machine learning, each piece of information is translated into either a real number or a categorical label known as a feature.", "Machine learned classification excels at finding subtle patterns in data sets with a large number of features.", "Machine learned classification has been used extensively in astronomy.", "In X-ray astronomy, used oblique decision trees to produce a catalog of probabilistically classified X-ray sources from ROSAT.", "Since that study, there have been many advances in automatic classification techniques.", "Ensemble algorithms such as Random Forest (RF) have replaced single decision trees as the state-of-the-art.", "RF has been successfully used in astronomy for the automatic classification of variable stars , and the photometric classification of supernovae .", "In optical astronomy, there are efforts to incorporate automatic classification in the processing pipelines of current and planned surveys , , .", "Feature representation is an important issue in light curve classification.", "Since light curves are rarely observed with exactly the same cadences, they need to be transformed into structured feature sets before different sources can be compared.", "Various light curve feature representations have been used in astronomy.", "For example, transformed the light curves of each Kepler eclipsing binary into a set of 1000 observations by fitting and then interpolating the observations.", "However, this method only works for a very homogenous set of light curves.", "Other studies use a restrictive set of variability measures.", "In , X-ray sources in M31 are placed into two light curve classes - highly variable or outbursts.", "This method has limited descriptive power for the variety of time-variability behaviours.", "In contrast, extracted a large number of features from each light curve in the Hipparcos catalog and used RF and Bayesian networks to automatically classify ${\\sim }$ 6000 unsolved optical variables.", "They achieved a misclassification rate of less than 12% and this is the methodology for feature representation that we have used.", "In this paper, we present the results of using the RF algorithm to classify variable sources in 2XMMi-DR2.", "In Section , we describe the 2XMMi-DR2 data set and the data processing we performed.", "In Section , we describe the RF algorithm.", "In Section we present the classification results using only time-series features and in Section , we show how the classification accuracy increases with the inclusion of contextual features.", "Our main result, a table of probabilistically classified 2XMMi variable sources, is presented in Section .", "In Section we present a method for selecting anomalous sources and briefly describe one of the interesting anomalous source.", "Finally, in Section we discuss the limitations and future prospects of machine learned classification." ], [ "The 2XMM Variable sources", "The 2XMMi-DR2 catalog consists of observations made with the XMM-Newton satellite between 2000 and 2008 and covers a sky area of about 420 deg$^2$ .", "The observations were made using the European Photon Imaging Camera (EPIC) that consists of three CCD cameras – pn, MOS1 and MOS2 – and covers the energy range from 0.2 keV to 12 keV.", "There are $221\\,012$ unique sources in 2XMM-DR2, of which $2\\,267$ were flagged as variable by the XMM processing pipeline .", "The variability test used by the pipeline is a $\\chi ^2$ test against the null hypothesis that the source flux is constant, with the probability threshold set at $10^{-5}$ ." ], [ "Data processing", "In this paper, a detection refers to a light curve in an epoch made by one camera.", "Each detection in our sample has an associated light curve which consists of background subtracted count rates, count rate errors, background count rates, background errors, and time stamps.", "A source can be detected in multiple epochs, and in each epoch there are typically three detections, one by each of the pn, MOS1 and MOS2 cameras.", "The exposure time per detection ranges from a few ks to over 100 ks (Figure REF ).", "The bin widths are in multiples of 10 s and are large enough such that there are a minimum of 18 counts/bin and 5 counts/bin for the pn and MOS detectors respectively.", "To ensure that all the variability in the light curve comes from the source and is not due to background flares or instrumental errors, we filtered out points likely to contain errors.", "First, we removed all points that lie outside the good time intervals (GTIs).", "GTIs are time periods where monitored parameters, such as spacecraft attitude stability and background particle levels, are within acceptable levels.", "Second, we removed all points where the fraction of time exposed, $F_{exp}$ , was $<0.9$ .", "Count rates determined during a low $F_{exp}$ measurement are not reliable.", "Third, we removed points with zero error rates.", "Since an error of zero is not realistic, it indicates some error in the data processing or the observation.", "After the filtering step, we removed sources from the sample with less than 15 data points in the light curve.", "Table REF is a breakdown of the sources in our sample.", "In total, we excluded 983 sources from further considerations.", "Table: The variable 2XMMi sample" ], [ "Classified sample", "For our training set we used the classifications for each discrete variable 2XMMi-DR2 source as determined by Farrell et al.", "(in prep).", "While the classification methodology will be discussed in detail in Farrell et.", "al.", "(in prep), we summarise the process as follows.", "First, the pipeline produced images, spectra, and light curves were manually inspected using the products available on the Leicester Database and Archive Service (LEDAS) webpageshttp://www.ledas.ac.uk/.", "Spurious detections were identified, primarily through examination of the images, and summarily discarded.", "Detections of extended sources were also discarded (e.g.", "supernova remnants, galaxy clusters etc.)", "as any variability detected from these sources within a single XMM-Newton observation would have to be spurious.", "In this manner we discarded 924 out of the original 2,267 variable sources as spurious.", "The nature of the remaining 1,343 real variable sources was determined by searching for matches around the source positions in the SIMBAD astronomical databasehttp://simbad.u-strasbg.fr/simbad/ and the NASA/IPAC Extragalactic Databasehttp://nedwww.ipac.caltech.edu/ (NED), and through a shallow review of the literature.", "The bulk of these sources (44%) were associated with stars, with the rest associated with the centres of galaxies (i.e.", "AGN; 7%), XRBs (6%), CVs (6%), ultraluminous X-ray sources (ULXs; 1%), GRBs (1%), and super soft sources (SSSs; 1%).", "A very small number (representing $\\sim $ 1%) were associated with planets (Jupiter and Saturn), extragalactic globular clusters, and magnetars.", "The remaining sources, comprising 33% of the real variable source sample, did not have a match in either SIMBAD or NED and are thus unidentified.", "The training set thus contains 873 sources in seven classes: AGN, CVs, GRBs, XRBs, SSSs, stars, ULXs, and XRBs, with the unidentified sources not included.", "Table 2 shows a breakdown of the number of sources and detections we have in the classified training set and Figure 2 shows examples of light curves from each class.", "AGN are the central regions of galaxies believed to contain supermassive black holes.", "X-ray emission from AGN is mainly due to inverse Compton scattering and typically follows a power-law spectrum .", "We included different types of stars under the “star” category, including flare stars, binaries, pre-main sequence stars and young stellar objects.", "Late-type flare stars produce X-ray emission from magnetic reconnection in their coronae .", "A CV is a binary system in which a white dwarf accretes from a companion star.", "The typical orbital periods of CVs are between 75 min and 8 hrs.", "CVs can be magnetic (mCV) or non-magnetic; the former are also known as polars or intermediate polars.", "X-ray emission from non-magnetic CVs is mainly due to low temperature thermal plasma emission from shocks formed when material accretes onto the white dwarf.", "In mCVs, the accretion disk is suppressed by the magnetic field and the X-ray emission arises from the boundary of the shock of the collimated accretion flow.", "XRBs are binary systems where the accreting compact object is a black hole or neutron star.", "The donor star in a high-mass XRB is usually a massive O or B-type star, or a Be star while the donor star in a low-mass XRB can be a main-sequence star, a white dwarf or a red giant.", "Both subtypes of XRBs are included in this category.", "ULXs are objects with X-ray luminosities exceeding that of a stellar mass black hole accreting at the Eddington limit.", "They are located within galaxies but not in the nucleus regions.", "SSSs, as their name suggests, are characterised by their extremely soft (peaking at $<0.5$ keV) spectra.", "The accepted paradigm for their nature is that of a white dwarf binary with steady nuclear burning .", "Lastly, the GRBs we are referring to here are afterglow emission from long GRBs, which are believed to be the core collapses of massive stars.", "Figure: Example lightcurves for the seven types of X-ray sources in our training set.Table: Classified sources in the training set" ], [ "Introduction", "The machine learning technique we use here is known as supervised classification .", "Supervised classification uses a set of labelled training examples to construct a prediction model.", "In Section , we described the method used to construct the training set.", "In this section we will explain the classification algorithm in detail." ], [ "Random Forest", "Random Forest (RF) is an ensemble supervised classification algorithm developed by .", "In the training phase, the algorithm builds an ensemble of decision trees.", "Each tree is built using a bootstrap sample from the training set, i.e.", "for $S$ samples in the training set, the algorithm randomly picks $S$ samples with replacement to create the training set for each tree.", "To construct a decision tree, training samples are split at each node (a node is where classification decisions are made) and this process iterates until all the training samples at the node belong to the same class.", "The feature used at each node is the one that produces the highest decrease in Gini impurity, as calculated using the equation: $G = \\displaystyle \\sum _{i=1}^{m}f_k (1-f_k) ,$ where $m$ is the number of classes and $f_k$ is the fraction of sources which belongs to class $k$ .", "The Gini impurity becomes zero when all the sources in a node are of the same type.", "In RF, a small subset of features (typically only a small fraction of the total number of features) are randomly chosen to be considered at each node.", "To predict the class of a new sample, each decision tree in the ensemble votes for a class and the output class is the one with the most votes.", "RF is one of the most accurate classification algorithms available .", "It can handle large datasets with large number of features.", "RF can generalize without overfitting; an overfitted model is one that describes noise rather than the true underlying relationship between features.", "It is also simple to optimize, since there are only two parameters to adjust – the number of trees and the number of variables to use at each node.", "We used the R package randomForest for the experiments performed in this paper.", "Using tuning function tuneRF in the randomForest package, we found that the optimal number of variables to use at each node is 9.", "To find the optimal number of trees, we repeated the experiment with different number of trees, and found 500 trees was optimal." ], [ "Unbalanced training set", "Our training set, as summarized in Table REF , is heavily unbalanced.", "Stars, the most abundant class, outnumber GRB, the rarest class by around 200 to 1.", "Heavily unbalanced training sets can degrade the performance of a machine learned classification algorithm.", "To ameliorate this issue, we oversampled the two most under-represented classes - GRB and SSS, using the SMOTE algorithm .", "SMOTE creates synthetic minority class samples by using the k-nearest neighbours and has been shown to be more robust than simply oversampling the minority class with replacement.", "We used the SMOTE implementation in the DMwR package in R to oversample the GRB class by ten fold and the SSS class by four fold." ], [ "Class membership probabilities", "Class membership probabilities can be more informative then discrete class labels.", "The former provides information on the degree of confidence of the classification, and allows the user to set cutoffs for selecting their class of interest based on the desired level of reliability and completeness.", "RF can provide class membership probabilities in the form of the fraction of votes in the ensemble given for the class.", "In this paper, we report all results as class membership probabilities.", "The variable X-ray source sample allows us to investigate the usefulness of variability information in classification.", "In this section, we describe the time series features we extracted from the X-ray light curves and report on the accuracy of the RF classifier trained using only time series features.", "Table REF is a summary of the general light curve characteristics of each source type.", "Although we cannot arrive at a definitive classification solely using the light curves, variability information can narrow down the potential classes.", "For example, a source with periodic variability is highly unlikely to be an AGN, but it could be a CV or an XRB.", "Based on the expected variability characteristics, we extracted four types of light curve features – periodic features, likelihood of power law decay, flares and statistical features such as fractional variability.", "Table REF is a summary of the time series features.", "We discuss each feature in detail in the following sections.", "Table: Light curve characteristics of each source typeTable: List of time series features used for classification" ], [ "Periodic features", "Some CVs and XRBs display periodicities on timescales of minutes to hours, less than the typical length of our observations , .", "This suggests the frequency domain can inform our classification.", "We used the generalised Lomb-Scargle periodogram from to represent the frequency domain information.", "The advantage of this technique over a conventional Fourier transform is that it can handle unevenly sampled data.", "For evenly sampled light curves, this would be unnecessary.", "However, due to the filtering process, our light curves may be missing data points.", "The generalised periodogram is equivalent to fitting functions of the form $y = a\\cos {\\omega t} + b\\sin {\\omega t} + c$ .", "The inclusion of the offset $c$ makes it more general than the original Lomb-Scargle periodogram .", "Finding the best fit translates to minimizing the squared difference between the data at time $i$ , $y_i$ , and the model $y(t_i)$ represented by the $\\chi ^2$ function: $\\chi ^2 = \\sum \\frac{\\left[ y_i - y(t_i) \\right]^2}{\\sigma _i^2},$ where $\\sigma _i^2$ is the estimated variance at time $i$ .", "The periodogram can be written as: $P(\\omega ) = \\frac{N-1}{2}\\frac{\\chi _0^2 - \\chi ^2(\\omega )}{\\chi _0^2},$ where $\\chi _0$ is the squared deviation of $y_i$ from the mean.", "Equation (REF ) has been normalized by the factor $\\left(N-1\\right)/2$ ($N$ is the number of measurements in the time series) so that if the data are pure noise, then the expected periodogram value is 1.", "This equation has an analytical solution that we used to calculate the periodogram value.", "The false alarm probability (FAP) is also included in our feature set as a way to capture the significance of the periodogram value.", "FAP is calculated using: $\\textrm {FAP} = 1 - \\left[ 1 - \\left(1-\\frac{2P}{N-1}\\right)^{\\frac{N-3}{2}}\\right]^M ,$ where $M$ is the number of peaks in the periodogram.", "This relies on an implicit assumption that the noise in the flux is Gaussian.", "Figure: Amplitude of the best fitting sine function from the Lomb-Scargle periodogram for the training set plotted against the period in seconds.For our classification experiments, we only used the two highest peaks in the periodogram.", "For each peak, we extracted the amplitude of the best fitting sine function ($\\sqrt{a^2+b^2}$ ), the period in seconds and the FAP.", "Figure REF shows a plot of the first two of these three values.", "To ensure the second peak is truly distinct from the first, we eliminated values immediately adjacent to the highest peak and found the second highest peak from the remaining frequency bins." ], [ "Fit to a power-law model", "The identifying feature of a GRB afterglow light curve is a power-law function.", "We fitted a power-law function of the form $y(t)=F_0 \\left(t-t_0\\right)^{-C}$ to the light curves and used the parameters $C$ of the best fitting model as classification features.", "The fitting procedure also determined $F_0$ and $t_0$ but these were not used as classification features.", "We used the curvefit function from the Python package scipy to perform the least squares non-linear fit.", "This process assumes the input errors are Gaussian, which is not always satisfied due to low count rates.", "To circumvent this issue, we binned the data to coarser time bins such that the average number of counts per bin was at least 20.", "To estimate the goodness of fit, we calculated the $\\chi ^2$ statistic using $\\left(y_i-\\hat{y}\\right)^2 / \\sigma _i^2$ , where $\\hat{y}$ is the model estimate of $y_i$ and $\\sigma _i$ is the error after binning.", "The reduced $\\chi ^2$ is another feature for our classifier (Figure REF ).", "Figure: Plot of the reduced χ 2 \\chi ^2 from an power law fit to the light curve, and the decay constant of the fit." ], [ "Flare finding", "X-ray flares are common features in active stars.", "To test for the existence of flares, we decomposed each light curve into a piecewise constant representation and then looked for segments with elevated count rates compared to adjacent segments.", "We used the Bayesian blocks technique to construct the piecewise constant segments .", "This technique is designed for astronomical count data with Poissonian noise and is based on the Bayesian formalism.", "It relies on comparing two hypotheses – the unsegmented hypothesis where the light curve can be described with one rate, and the segmented hypothesis where the light curve is described with two rates.", "The likelihood that the count rate is constant is given by: $L\\left(H_{unseg} \| Data\\right) = \\frac{\\Gamma \\left(A+1\\right)}{\\left(B+1\\right)^{A+1}},$ from Equation (29) in , where $A$ is the number of photons and $B$ is the number of bins.", "On the other hand, the likelihood of the segmented model is: $L\\left(H_{seg} \| Data\\right) = \\frac{\\Gamma \\left(A_1+1\\right)}{\\left(B_1+1\\right)^{A_1+1}} \\times \\frac{\\Gamma \\left(A_2+1\\right)}{\\left(B_2+1\\right)^{A_2+1}},$ where $A_1$ , $B_1$ and $A_2$ , $B_2$ are the number of photons and number of bins in segment one and segment two respectively.", "To compare the two hypotheses, we calculated the odds ratio: $O_{12} = \\frac{L\\left(H_{unseg} \| Data\\right)}{L\\left(H_{seg} \| Data\\right)}.$ If $O_{12}$ is less than one, then a segmented model is favored and the change point that yields the highest likelihood is used to segment the light curve.", "This process is performed recursively and terminates when further segmentation no longer improves the likelihood.", "We then count the number of segments where the count rate is at least three times higher than the expected error compared to the preceding and succeeding segments.", "This is our count of the number of flares in the light curve as shown in Figure REF .", "Figure: Fraction of sources of each type in the training set according to the number of flares found." ], [ "Statistical features", "In addition to the light curve features described in the previous sections, we extracted 16 statistical features that were used by in the classification of variable stars.", "These are general statistical measures that do not depend on the time ordering of the measurements, e.g.", "fractional variability, mean, and standard deviation.", "Detailed descriptions of these features can be found in Table REF ." ], [ "Accuracy on training set", "To evaluate the accuracy of our classifier, we used the method of cross-fold validation.", "We divided the training sample into ten sets, trained with nine sets, used the model created to classify the remaining sample set and then repeated for ten different combinations.", "The overall accuracy is the total number of correctly classified samples divided by the total number of samples in the training set.", "Using only time-series features, the overall accuracy is $\\sim 77\\%$ .", "Figure REF shows the confusion matrix, where the number in each square represents how the detections are classified.", "The sum of each row of the confusion matrix is the total number of detections in that class.", "The numbers in the diagonals are detections that have been correctly classified.", "GRBs, SSSs, and ULXs are the three worst performing classes.", "This is not unexpected since SSSs and ULXs have no distinguishing time series features.", "In contrast, stars, XRBs, and CVs performed relatively well and are usually only confused with each other.", "From Figure REF , it can be seen that XRBs and CVs share semi-periodic temporal behaviour while stars have distinguishing flares.", "It is also worth noting that sources of all types are most likely to be mis-classified as stars.", "Since a significant proportion of our training set are stars, the classifier optimises for accuracy by labelling sources for which it does not have sufficient information as the majority class.", "Figure REF shows a plot of the missed detection rate vs. the false positive rate, known as the Receiver Operating Characteristic (ROC) plot.", "Missed detection is 1 - true positive rate, which is the proportion of samples classified as the actual class; false positive rate is the proportion of samples not of the class but classified as such.", "We created the ROC plot by transforming the results of the multi-class classification into a binary classification, i.e.", "each sample either belongs to the actual class (true positive) or it does not (false positive).", "A well performing classifier should be able to provide a low missed detection rate with a low false positive rate, i.e.", "be on the bottom-left part of the ROC plot.", "Figure REF shows CVs are the best performing source type, even though they only achieved an accuracy of 52%.", "This accuracy is lower than what may be expected since the missed detection rate is only ${\\sim }10\\%$ when the false positive rate is ${\\sim }20\\%$ .", "This is because the test set is unbalanced, a small number of stars mis-classified as CVs would not significantly decrease the accuracy for stars, and therefore would not lead to a high false positive rate.", "The R package randomForest also has the ability to calculate relative feature importance.", "The importance of each feature is estimated by calculating the total decrease in Gini impurity (Equation REF ) from using that feature, averaged over all the trees in the forest.", "Figure REF shows the mean decrease in Gini impurity for the time series features.", "The five most relevant features are, in order of importance: max_slope, powerlaw_goodness_of_fits, median_abs_dev, powerlaw_C, and LombScargle_period1 (all as defined in Table REF ).", "Figure: Confusion matrix from performing 10-fold cross-validation on the training set using the RF classifier with only time-series features.", "The color bar represents the true positive rate.Figure: ROC plot from performing 10-fold cross-validation on the training set using the RF classifier with only time-series features.Figure: Relative importance of the time series features.", "The features are described in detail in Tables .In Section , we showed that time-series features have some discriminative power, but that the classification accuracy is insufficient for practical use.", "In this section, we expand our feature set to include hardness ratios, optical/near infra-red (NIR)/radio cross-matches, proximity to galaxies, and Galactic positions to improve the classification accuracy.", "We begin by describing each of these features and Table REF is a summary of the contextual features used in this paper.", "Table: List of contextual features used for classification" ], [ "Hardness ratios", "Hardness ratio is a crude proxy for the shape of the X-ray spectrum and it has been used with moderate success to classify X-ray sources .", "The XMM-Newton EPIC cameras cover the energy band from 0.2 keV to 12.0 keV.", "The photons gathered are separated into five bands by the 2XMM pipeline, from which four hardness ratios are calculated as follows: $HR_n = (R_{n+1}-R_n)/(R_{n+1}+R_n)$ where $R_n$ is the count rate in the $n$ th energy band (see Table REF for the energy range covered by each band).", "If both bands have count rates within $3\\sigma $ of zero, the resulting hardness ratio can be unpredictable as one is essentially dividing one very small number by another very small number.", "For these cases, we set the hardness ratio to -10.0 as a flag." ], [ "Optical/NIR cross-matches", "For optical and NIR cross-matching, we used the Naval Observatory Merged Astrometric Dataset .", "NOMAD is a conglomeration of various optical photometry and astrometry catalogs and the near-infrared 2MASS catalog.", "To estimate the probability of a chance cross-match, we used the Bayesian method from where we compared the hypothesis that the cross match is genuine to the alternate hypothesis that the source and the optical counterpart are two unrelated sources.", "The ratio of the likelihood of these two hypotheses is known as the Bayes factor, ${B}$ , given by the formula: ${B} = \\frac{2}{\\psi _1^2+\\psi _2^2} exp\\left[-\\frac{\\phi ^2}{2\\left(\\psi _1^2+\\psi _2^2\\right)}\\right],$ where $\\psi _1$ and $\\psi _2$ are the resolution of the two catalogs in arcsec and $\\phi $ is the angular separation between the two sources.", "A high Bayes factor favors the hypothesis that the cross-match is genuine.", "This calculation does not take into account the sky density of the optical sources.", "For our feature set, we included the B, V, J, H, K band magnitudes if a cross match was found, and the corresponding Bayes factor.", "If no cross-match was found, the magnitude was set to 100 as a null flag." ], [ "Radio cross-matches", "We cross-matched the 2XMMi sources with three radio catalogs — the NRAO VLA Sky Survey , the Sydney University Molonglo Sky Survey , and the Second Epoch Molonglo Galactic Plane Survey .", "Together, these catalogs provide all-sky coverage of the radio sky.", "NVSS was a 1.4 GHz radio survey with the Very Large Array covering the entire sky north of declination -40 degrees.", "SUMSS was the counterpart survey with the Molonglo telescope of the southern sky (south of declination -30 degrees) at 843 MHz; MGPS-2 was the Galactic plane radio survey at the same frequency.", "The positional accuracy of NVSS is $<1$ for sources stronger than 15 mJy, and 7 in the survey limit.", "For SUMSS and MGPS-2, the position accuracy is poorer but typically better than 5.", "Since the angular resolution of XMM-Newton EPIC is better than those of NVSS or SUMSS, for our cross-matching we used a $3\\sigma $ search radius based on the radio catalogs.", "We also included the Bayes factor (Equation REF ) to estimate the likelihood of a cross-match.", "The relatively low sky density of radio sources means that a spurious match is unlikely." ], [ "Associations with galaxies", "X-ray sources that correspond to the nuclei of galaxies are likely to be AGN, whilst non-nuclear extragalactic X-ray sources with luminosities of more than $10^{39}$ ergs$^{-1}$ are potential ULX candidates, but can also be foreground stars, XRBs, CVs, or background AGN.", "We cross-matched the 2XMMi sources with the Third Reference Catalog (RC3) of galaxies to find possible galaxy associations.", "RC3 contains more than 23,000 galaxies, including almost all galaxies with apparent diameters greater than 1.", "RC3 contains information on the galaxy center position, the major and minor diameters of the $D_{25}$ isophote (roughly the domain of the galaxy) as well as the position angle.", "We determined $\\alpha $ , the ratio between the angular separation between the source and the galaxy center and the elliptical radius $R_{25}$ .", "If $\\alpha < 1.5$ , then we considered the source to be associated with the galaxy.", "For sources associated with a galaxy, we included $\\alpha $ and the angular separation in the feature set." ], [ "Galactic coordinates", "The last set of features we included is the Galactic position of each source.", "From Figure REF , it can be seen that XRBs are more likely to cluster along the Galactic plane, whilst all other source types are distributed isotropically in Galactic coordinates.", "This motivates the inclusion of Galactic coordinates in the feature set as a way to identify XRBs.", "Figure: Distribution of sources in our training set in Galactic coordinates." ], [ "Accuracy of training set", "As in Section REF , we used 10-fold cross-validation to evaluate the performance of this feature set.", "Using both the time-series and contextual feature sets, the overall accuracy improved significantly from 77% to 97% with the additional features.", "Figures REF and REF show the confusion matrix and the ROC plot, respectively.", "Performance improved across all classes relative to Figures REF and REF .", "However, we misclassified most GRBs as stars (Figure REF ).", "From the ROC plot, it can be seen that we can achieve 100% accuracy for GRBs if we are willing to accept a 5% false positive rate.", "Even though this does not seem high, it would mean an extra ${\\sim }140$ sources misclassified as GRBs in exchange for accurately classifying 9 GRBs.", "We will discuss this issue with minority classes in more detail in Section REF .", "Figure REF shows the relative feature importance of the top 30 features, determined using the mean decrease in Gini impurity.", "X-ray flux and X-ray luminosity (for sources with a galaxy association) are the most informative features, followed by HR3.", "Overall, hardness ratios appear to be highly informative, with all four hardness ratios placed in the top 10 of most informative features.", "On the other hand, time-series features do not rank highly on the list.", "Figure: Confusion matrix from performing 10-fold cross-validation on the training set using the RF classifier with time-series and contextual features.", "The color bar represents the true positive rate.", "The overall accuracy is 97%.Figure: ROC plot from performing 10-fold cross-validation on the training set using the RF classifier with time-series and contextual features.", "Performance across all classes show marked improvement from classification using only time-series features.Figure: Relative importance of the time series and contextual features.", "The features are described in detail in Tables and ." ], [ "Results", "Using the entire training set, we constructed a RF classification model using the method described in Section .", "Then we applied this classification model to the set of unknown 2XMMi variable sources.", "For sources where there are more than one detection, we classified each detection separately and combined the results by averaging the output class membership probabilities.", "Table REF shows the number of unknown sources classified as one of seven classes.", "The majority of the unknown sources are classified as stars.", "We also compiled a downloadable table of the class membership probabilities.", "Table REF shows a portion of that table.", "Table: Unknown variable source classificationTable: 2XMM variable sources classificationFollowing the initial source classification (Farrell et.", "al., in prep), a number of sources in the unknown sample have since been classified in the literature.", "We assessed the accuracy of the classifier by comparing the literature classification to the output of our RF classifier for ${\\sim }12\\%$ of the the unknown sources.", "Confirming the classification for 411 X-ray sources is beyond the scope of this paper.", "We found recently confirmed or tentative classifications for 19 sources and they are listed in Table REF .", "The classifications from our RF classifier agree with the literature classifications in 13 out of 19 cases if we include the two sources that have multiple possible classifications.", "The misclassifications are due to the source belonging to a novel source type, insufficient information, poor data quality, or problems with the classification in the literature.", "Of the six misclassifications, three sources have been classified as ULXs by our RF classifier whilst regarded them as candidate AGN with immediate-mass black holes based on the presence of X-ray variability.", "The criteria used by do not preclude ULXs since they only filtered out sources in known star forming regions and included sources with object type Galaxy shown in the NED databases.", "All three of the sources classified as ULXs are close to a galaxy in RC3 and have X-ray luminosity of between $10^{39}$ and $10^{40}$ ergs/s.", "Here we briefly discuss three of the other misclassifications.", "2XMM J034645.4$+$ 680947: This source is classified as an XRB by our classifier but classified it as a SSS.", "However, there are a few problems with the literature classification.", "only used two hardness ratios in the classification.", "This is coarser than what we have used, which would have resulted in the loss of information.", "There are four observations of this source and the hardness ratio only satisfied the criteria for SSS in the two fainter observations.", "The lack of X-ray flux in the $2-7$ keV band could be a selection effect since the hard emission tends to be undetectable in fainter sources.", "Furthermore, the hardness ratios derived from the 2004 August and 2004 February observations do not classify this source as a SSS.", "We fitted the 2004 August EPIC spectra that were automatically extracted by the XMM pipeline with a Raymond-Smith model , typical for a SSS.", "The best fit parameters are: $N_H = (0.03 \\pm 0.03) \\times 10^{22}$ cm$^{-2}$ , kT = (0.79 $\\pm 0.05) $ keV and $\\chi ^2$ / dof = 175.03/183.", "This is a satisfactory fit, however the temperature is an order of magnitude higher than typical for a SSS .", "From the above arguments, we are skeptical that 2XMM J034645.4$+$ 680947 is a SSS.", "Our RF model classified this source as an XRB, SSS, star and ULX with probabilities $0.349$ , $0.227$ , $0.222$ and $0.16$ respectively.", "This suggests that either we lack sufficient information to classify this source and/or that this source is highly unusual.", "2XMM J060636.4$-$ 694937: This source is classified as an AGN by our classifier but has been confirmed as a classical nova in the Large Magellanic Cloud .", "The observation in our sample occurred during the nova outburst phase.", "Although novae are a subset of CVs we do not have many examples of novae in outburst in the training set, hence to our classifier this is a novel source type.", "This highlights one of the limitations of supervised classification in that the classifier is incapable of recognizing novel classes.", "2XMM J174016.0$-$ 290337: This source is classified as an XRB by our classifier.", "Using only X-ray timing and X-ray spectral data, identified this source as likely to be a symbiotic XRB, a new and rare sub-class of XRBs composed of a late-type giant accreting matter onto a compact object such as a neutron star.", "However, with more optical spectral data, later identified it as an mCV.", "There is an optical counterpart in the XMM-Newton error circle with a spectrum that contains strong Balmer, He I, He II, and Bowen blend emissions, typical of magnetic CVs.", "Similar to the conclusion made by , our classifier favors the interpretation of this source as a XRB, giving it a probability of 0.46, but also gives the probability of this being a CV as 0.29.", "This demonstrates that our classifier is capable of making a conclusion along the same line as an expert in the field using the same information.", "It is worth noting that this is an unusual source and its X-ray properties do not fit with the interpretation of it being an mCV.", "Table: Acknowledgements" ] ]	1403.0188
[ [ "Logarithmic stabilization of the Euler-Bernoulli transmission plate\n equation with locally distributed Kelvin-Voigt damping" ], [ "Abstract In this paper we will study the asymptotic behaviour of the energy decay of a transmission plate equation with locally distributed Kelvin-Voigt feedback.", "Precisly, we shall prove that the energy decay at least logarithmically over the time.", "The originality of this method comes from the fact that using a Carleman estimate for a transmission second order system which will be derived from the plate equation to establish a resolvent estimate which provide, by the famous Burq's result [Bur98], the kind of decay mentionned above." ], [ "Introduction and statement of results", "In recent years, there has been much interest in the stability problems for elastic systems with locally distributed damping.", "Most of the works were devoted to the viscous damping, i.e., the damping is proportional to the velocity (see for instance [7] and [20]).", "Structures with local viscoelasticity arise from use of smart material or passive stabilization of structures.", "However, very little is known about exponential stability for elastic systems with local viscoelastic damping, although there is a fairly deep understanding when the damping is distributed over the entire domain but only for 1-dimension (see [13]).", "To our knowledge, the first paper in this direction was published in 1998 by Liu and Liu [12] where they obtained exponential stability for the Euler-Bernoulli beam equation with local Kelvin-Voigt damping.", "Noting that in our knowledge there are zero results at least for the multi-dimension Euler-Bernoulli plate equation case.", "Consider a clamped elastic domain in $\\mathbb {R}^{n}$ , ($n\\ge 2$ ) which is made of a viscoelastic material with Kelvin-Voigt constitutive relation in which a transmisson effect has been established such a way that the damping is locally effective in only one side the transmission boundary.", "By the Kirchhoff hypothesis, neglecting the rotatory inertia, the transversal vibration (see [8] for the modeling problem) can be described as follows: Let $\\Omega $ and $\\Omega _{1}$ be two open, bounded and connected domains with smooth boundary respectively $\\Gamma $ and $S$ such that $\\Omega _{1}\\subset \\Omega $ and $\\overline{S}\\cap \\overline{\\Gamma }=\\emptyset $ .", "We set also $\\Omega _{2}=\\Omega \\backslash \\overline{\\Omega }_{1}$ which is an open connected domain with boundary $\\partial \\Omega _{2}=\\Gamma \\cup S$ .", "We are going to study the following transmission and boundary value problem $\\left\\lbrace \\begin{array}{lll}\\partial _{t}^{2}u_{1}+\\Delta (c_{1}^{2}\\Delta u_{1}+a.\\Delta \\partial _{t}u_{1})=0&\\textrm {in}&\\Omega _{1}\\times ]0,+\\infty [,\\\\\\partial _{t}^{2}u_{2}+c_{2}^{2}\\Delta ^{2}u_{2}=0&\\textrm {in}&\\Omega _{2}\\times ]0,+\\infty [,\\\\u_{1}=u_{2}&\\textrm {on}&S\\times ]0,+\\infty [,\\\\\\partial _{\\nu }u_{1}=\\partial _{\\nu }u_{2}&\\textrm {on}&S\\times ]0,+\\infty [,\\\\c_{1}\\Delta u_{1}=c_{2}\\Delta u_{2}&\\textrm {on}&S\\times ]0,+\\infty [,\\\\c_{1}\\partial _{\\nu }\\Delta u_{1}=c_{2}\\partial _{\\nu }\\Delta u_{2}&\\textrm {on}&S\\times ]0,+\\infty [,\\\\u_{1}=0&\\textrm {on}&\\Gamma \\times ]0,+\\infty [,\\\\\\Delta u_{1}=0&\\textrm {on}&\\Gamma \\times ]0,+\\infty [,\\\\u_{1}(x,0)=u_{1}^{0}(x),\\;\\partial _{t}u_{1}(x,0)=u_{1}^{1}(x)&\\textrm {in}&\\Omega _{1},\\\\u_{2}(x,0)=u_{2}^{0}(x),\\;\\partial _{t}u_{2}(x,0)=u_{2}^{1}(x)&\\textrm {in}&\\Omega _{2}.\\end{array}\\right.$ Where $\\partial _{\\nu }$ denotes the unit outward normal vector of $\\Omega _{1}$ and $\\Omega $ respectively in $S$ and $\\Gamma $ , $c_{1}$ , $c_{2}$ are strictly positives constants and $a$ is a non negative bounded functions in $\\Omega _{1}$ and we suppose that $a$ vanishing near the boundary $S$ such that there exist a non empty open domain $\\omega \\subset \\Omega _{1}$ such that $a$ is strictly positives in $\\overline{\\omega }$ .", "The energy of a solution of (REF ) at time $t\\ge 0$ is defined by $E(t)=\\frac{1}{2}\\int _{\\Omega _{1}}\\Big (\|\\partial _{t}u_{1}(x,t)\|^{2}+c_{1}^{2}\|\\Delta u_{1}(x,t)\|^{2}\\Big )c_{1}^{-1}\\,\\mathrm {d}x+\\frac{1}{2}\\int _{\\Omega _{2}}\\Big (\|\\partial _{t}u_{2}(x,t)\|^{2}+c_{2}^{2}\|\\Delta u_{2}(x,t)\|^{2}\\Big )c_{2}^{-1}\\,\\mathrm {d}x.$ By Green's formula we can prove that for all $\\;t_{1},\\,t_{2}>0$ we have $E(t_{2})-E(t_{1})=-c_{1}^{-1}\\int _{t_{1}}^{t_{2}}\\!\\!\\!\\int _{\\Omega _{1}}a\|\\Delta \\partial _{t}u_{1}(x,t)\|^{2}\\,\\mathrm {d}x\\,\\mathrm {d}t,$ and this mean that the energy is decreasing over the time.", "We define the operator $\\mathcal {A}$ by $\\mathcal {A}\\left(\\begin{array}{c}u_{1}\\\\u_{2}\\\\v_{1}\\\\v_{2}\\end{array}\\right)=(v_{1},v_{2},-\\Delta (c_{1}^{2}\\Delta u_{1}+a\\Delta v_{1}),-c_{2}^{2}\\Delta ^{2} u_{2})$ in the Hilbert space $\\mathcal {H}=X\\times H$ where $H=H_{1}\\times H_{2}=L^{2}(\\Omega _{1},c_{1}^{-1}\\,\\mathrm {d}x)\\times L^{2}(\\Omega _{2},c_{1}^{-1}\\,\\mathrm {d}x)$ and $\\begin{split}X=\\big \\lbrace (u_{1},u_{2})\\in H\\,:\\,u_{1}\\in H^{2}(\\Omega _{1}),\\,u_{2}\\in H^{2}(\\Omega _{2}),\\,u_{2\|\\Gamma }=0,\\,u_{1\\,\|S}=u_{2\\,\|S},\\partial _{\\nu }u_{1\\,\|S}=\\partial _{\\nu }u_{2\\,\|S}\\big \\rbrace ,\\end{split}$ with domain $\\mathcal {D}(\\mathcal {A})=\\big \\lbrace (u_{1},u_{2},v_{1},v_{2})\\in \\mathcal {H}\\,:\\,(v_{1},v_{2},\\Delta (c_{1}^{2}\\Delta u_{1}+a\\Delta v_{1}),c_{2}^{2}\\Delta ^{2}u_{2})\\in \\mathcal {H},\\,\\Delta u_{2\\,\|\\Gamma }=0,\\\\\\,c_{1}\\Delta u_{1\\,\|S}=c_{2}\\Delta u_{2\\,\|S},\\,c_{1}\\partial _{\\nu }\\Delta u_{1\\,\|S}=c_{2}\\partial _{\\nu }\\Delta u_{2\\,\|S}\\big \\rbrace .$ Now we are able to state our main results Theorem 1.1 There exists $C>0$ such that for every $\\mu \\in \\mathbb {R}$ with $\|\\mu \|$ large, we have $\\Vert (\\mathcal {A}-i\\mu \\,\\mathrm {Id})^{-1}\\Vert _{\\mathcal {L}(\\mathcal {H})}\\le C\\mathrm {e}^{C\|\\mu \|}.$ As an immediate consequence of the previous theorem (see [6] and more recently [4]), we get the following rate of decrease of energy Theorem 1.2 For any $k\\in \\mathbb {N}$ , there exists a constant $C>0$ such that for any initial data $(u_{1}^{0},u_{2}^{0},u_{1}^{1},u_{2}^{1})\\in \\mathcal {D}(\\mathcal {A}^{k})$ , the energy $E(t)$ of the system (REF ) whose solution $u(x,t)$ is starting from $(u_{1}^{0},u_{2}^{0},u_{1}^{1},u_{2}^{1})$ satisfy $E(t)\\le \\frac{C}{(\\ln (2+t))^{2k}}\\Vert (u_{1}^{0},u_{2}^{0},u_{1}^{1},u_{2}^{1})\\Vert _{\\mathcal {D}(\\mathcal {A}^{k})}^{2},\\quad \\forall \\; t>0.$ Remarks 1.1 Under one assumption to the coefficients $c_{1}$ and $c_{2}$ , Ammari and Vodev [3] have proved an exponential stabilization result for the Euler-Bernoulli transmission plate equation with boundary dissipation.", "Again for a transmission model, Ammari and Nicaise [2] have proved, under some geometric condition, an exponential stabilization for a coupled damped wave equation with a damped Kirchhoff plate equation.", "To prove Theorem REF and Theorem REF , we make use the Carleman estimates to obtain information about the resolvent in a boundary domain, the cost is to use phases functions satisfying Hörmander's assumption.", "Albano [1] proved a Carleman estimate for the plate operator, by decomposing the operator as the product of two Schrödinger ones and gives for eatch of them the corresponding Carleman estimate then by making together these two estimates we obtain the result.", "But here we will not need to have a Carleman estimate for the plate equation, namely inspiring from the Albano's decomposition we will derive a second ordre transmission system to which we are going to apply an appropriate Carleman estimates (see section ) for a suitable phases functions, thus we will obtain the resolvent estimate of Theorem REF .", "Theorem REF and Theorem REF are analogous to those of Fathallah [10], in the case of hyperbolic-parabolic coupled system, and Lebeau and Robbiano [15] resuts, in the case of scalar wave equation without transmission, but our method is different from their because it consist to use the Carleman estimates directly for the stationary operator without going through the interpolation inequality.", "For various purposes, several authors have focused to the transmission problems where they needed to find a Carleman estimates near the interface, such as the works of Bellassoued [5] and Fathallah [10] for the stabilization problems and that also of Le Rousseau and Robbiano [17] for a control problem.", "Note that in the case where it has no transmission of the problem (REF ), Theorem REF and Theorem REF remain valid and in this case we need only the classical Carleman estimates (see [15] and [14]).", "In this paper $C$ will always be a generic positive constant whose value may be different from one line to another.", "The outline of this paper is as follow.", "In section we prove the well-Posedness of the problem (REF ), in section we give a global Carleman estimate and we will constract a suitable phases functions and in section we prove the resolvent estimate gived by Theorem REF ." ], [ "Well-Posedness of the problem", "To prove the Well-Posedness of the problem (REF ) we are going to use the semigroups theory.", "Our strategy consiste to write the equations as a Cauchy problem with an operator which generates a semigroup of contractions.", "Throughout this paper, we denote the inner product in the space $H=H_{1}\\times H_{2}$ by $\\left\\langle \\left(\\begin{array}{c}u_{1}\\\\u_{2}\\end{array}\\right),\\left(\\begin{array}{c}v_{1}\\\\v_{2}\\end{array}\\right)\\right\\rangle _{H}=\\int _{\\Omega _{1}}u_{1}(x)\\overline{v_{1}(x)}c_{1}^{-1}\\,\\mathrm {d}x+\\int _{\\Omega _{2}}u_{2}(x)\\overline{v_{2}(x)}c_{2}^{-1}\\,\\mathrm {d}x,$ The Cauchy problem is written in the following form $\\left\\lbrace \\begin{array}{ll}\\partial _{t}\\left(\\begin{array}{c}u_{1}\\\\u_{2}\\\\v_{1}\\\\v_{2}\\end{array}\\right)(t)=\\mathcal {A}\\left(\\begin{array}{c}u_{1}\\\\u_{2}\\\\v_{1}\\\\v_{2}\\end{array}\\right)(t)&t\\in ]0,+\\infty [,\\\\\\left(\\begin{array}{l}u_{1}\\\\u_{2}\\\\v_{1}\\\\v_{2}\\end{array}\\right)(0)=\\left(\\begin{array}{l}u_{1}^{0}\\\\u_{2}^{0}\\\\u_{1}^{1}\\\\u_{2}^{1}\\end{array}\\right).&\\end{array}\\right.$ Now we have to specify the functional space and the domain of the operator $\\mathcal {A}$ .", "In the space $H$ we define the operator $G$ by $G\\left(\\begin{array}{l}u_{1}\\\\u_{2}\\end{array}\\right)=(-c_{1}\\Delta u_{1},-c_{2}\\Delta u_{2})\\qquad \\forall \\;(u_{1},u_{2})\\in \\mathcal {D}(G)$ with domain $\\mathcal {D}(G)=X$ defined in (REF ).", "The space $X$ is equipped with the norm $\\Vert (u_{1},u_{2})\\Vert _{X}=\\Vert G(u_{1},u_{2})\\Vert _{H}$ and we defined the graph norm of $G$ by $\\Vert (u_{1},u_{2})\\Vert _{gr(G)}^{2}=\\Vert (u_{1},u_{2})\\Vert _{H}^{2}+\\Vert G(u_{1},u_{2})\\Vert _{H}^{2}$ then we have the following Proposition 2.1 $(X,\\Vert \\,.\\,\\Vert _{X})$ is a Hilbert space with a norm equivalent to the graph norm of $G$ .", "Proof : It is well known that if $G$ is a colsed operator then $(X,\\Vert \\,.\\,\\Vert _{gr(G)})$ is a Hilbert space.", "Thus to prove the proposition it suffices to show that $G$ is closed and both norms are equivalent.", "By Green's formula and Poincaré inequality it is easy to show that there exists $C>0$ such that $\\left\\langle G\\left(\\begin{array}{l}u_{1}\\\\u_{2}\\end{array}\\right),\\left(\\begin{array}{l}u_{1}\\\\u_{2}\\end{array}\\right)\\right\\rangle _{H}=\\Vert \\nabla u_{1}\\Vert _{L^{2}(\\Omega _{1})}^{2}+\\Vert \\nabla u_{2}\\Vert _{L^{2}(\\Omega _{2})}^{2}\\ge C\\Vert (u_{1},u_{2})\\Vert _{H}^{2}\\quad \\forall \\,(u_{1},u_{2})\\in X.$ Then $G$ is a strictly positive operator and we have $\\Vert G(u_{1},u_{2})\\Vert _{H}.\\Vert (u_{1},u_{2})\\Vert _{H}\\ge \\left\\langle G\\left(\\begin{array}{l}u_{1}\\\\u_{2}\\end{array}\\right),\\left(\\begin{array}{l}u_{1}\\\\u_{2}\\end{array}\\right)\\right\\rangle _{H}\\ge C\\Vert (u_{1},u_{2})\\Vert _{H}^{2}\\quad \\forall \\,(u_{1},u_{2})\\in X$ which prove the equivalence between the two norms.", "Now since $G$ is positive then by in [18], $-G$ is m-dissipative and thus $G$ is a closed operator.", "This completes the proof.", "This last result allows us to properly define the functional space of the operator $\\mathcal {A}$ .", "Proposition 2.2 The two spaces $(X,\\Vert \\,.\\,\\Vert _{2})$ and $(X,\\Vert \\,.\\,\\Vert _{X})$ are algebraically and topologically the same.", "Where we have defined $\\Vert \\,.\\,\\Vert _{2}$ by $\\Vert (u_{1},u_{2})\\Vert _{2}^{2}=\\Vert u_{1}\\Vert _{H^2(\\Omega _{1})}^{2}+\\Vert u_{2}\\Vert _{^{H^2(\\Omega _{2})}}^{2},\\qquad \\forall \\,(u_{1},u_{2})\\in X.$ Proof : We have only to prove that the two norms are equivalent.", "First, we note that $(X,\\Vert \\,.\\,\\Vert _{2})$ is a Hilbert space because $X$ is a closed subspace of $H^{2}(\\Omega _{1}\\cup \\Omega _{2})$ , in addition we have $\\Vert (u_{1},u_{2})\\Vert _{X}^{2}=\\Vert \\Delta u_{1}\\Vert _{L^{2}(\\Omega _{1})}^{2}+\\Vert \\Delta u_{2})\\Vert _{L^{2}(\\Omega _{2})}^{2}\\le C\\Vert (u_{1},u_{2})\\Vert _{2}^{2}\\quad \\forall \\,u\\in X,$ and while $(X,\\Vert \\,.\\,\\Vert _{X})$ is also a Hilbert space, then according to the Banach theorem (see [9]) the two norms are equivalent.", "We set $\\mathcal {H}=X\\times H$ the Hilbert space with the norm $\\Vert (u_{1},u_{2},v_{1},v_{2})\\Vert ^{2}=\\Vert (u_{1},u_{2})\\Vert _{X}^{2}+\\Vert (v_{1},v_{2})\\Vert _{H}^{2}\\qquad \\forall \\, (u_{1},u_{2},v_{1},v_{2})\\in \\mathcal {H},$ and we recall that the domain of the operator $\\mathcal {A}$ is defined by $\\mathcal {D}(\\mathcal {A})=\\big \\lbrace (u_{1},u_{2},v_{1},v_{2})\\in \\mathcal {H}\\,:\\,(v_{1},v_{2},\\Delta (c_{1}^{2}\\Delta u_{1}+a\\Delta v_{1}),c_{2}^{2}\\Delta ^{2}u_{2})\\in \\mathcal {H},\\,\\Delta u_{2\\,\|\\Gamma }=0,\\\\\\,c_{1}\\Delta u_{1\\,\|S}=c_{2}\\Delta u_{2\\,\|S},\\,c_{1}\\partial _{\\nu }\\Delta u_{1\\,\|S}=c_{2}\\partial _{\\nu }\\Delta u_{2\\,\|S}\\big \\rbrace .$ Theorem 2.1 Under the above assumptions, the operator $\\mathcal {A}$ is m-dissipative and especially it generates a strongly semigroup of contractions in $\\mathcal {H}$ .", "Proof : According to Lumer-Phillips theorem (see for exemple [18]) we have only to prove that $\\mathcal {A}$ is m-dissipative.", "Let $(u_{1},u_{2},v_{1},v_{2})\\in \\mathcal {D}(\\mathcal {A})$ then by Green's formula we have $\\mathrm {Re}\\left\\langle \\mathcal {A}\\left(\\begin{array}{l}u_{1}\\\\u_{2}\\\\v_{1}\\\\v_{2}\\end{array}\\right),\\left(\\begin{array}{l}u_{1}\\\\u_{2}\\\\v_{1}\\\\v_{2}\\end{array}\\right)\\right\\rangle _{\\mathcal {H}}&=&\\mathrm {Re}\\left\\langle \\left(\\begin{array}{c}v_{1}\\\\v_{2}\\\\-\\Delta (c_{1}^{2}\\Delta u_{1}+a\\Delta v_{1})\\\\-c_{2}^{2}\\Delta ^{2}u_{2}\\end{array}\\right),\\left(\\begin{array}{l}u_{1}\\\\u_{2}\\\\v_{1}\\\\v_{2}\\end{array}\\right)\\right\\rangle _{\\mathcal {H}}\\\\&=&-c_{1}\\Vert a^{\\frac{1}{2}}\\Delta v_{1}\\Vert _{L^{2}(\\Omega _{1})}^{2}\\le 0.$ This shows that $\\mathcal {A}$ is dissipative.", "Let now $(f_{1},f_{2},g_{1},g_{2})\\in \\mathcal {H}$ and our purpose is to find a couple $(u_{1},u_{2},v_{1},v_{2})\\in \\mathcal {D}(\\mathcal {A})$ such that $\\left(\\mathrm {Id}-\\mathcal {A}\\right)\\left(\\begin{array}{l}u_{1}\\\\u_{2}\\\\v_{1}\\\\v_{2}\\end{array}\\right)=\\left(\\begin{array}{c}u_{1}-v_{1}\\\\u_{2}-v_{2}\\\\v_{1}+\\Delta (c_{1}^{2}\\Delta u_{1}+a\\Delta v_{1})\\\\v_{2}+c_{2}^{2}\\Delta ^{2}u_{2}\\end{array}\\right)=\\left(\\begin{array}{l}f_{1}\\\\f_{2}\\\\g_{1}\\\\g_{2}\\end{array}\\right)$ more explicitly we have to find $(u_{1},u_{2},v_{1},v_{2})\\in \\mathcal {D}(\\mathcal {A})$ such that $\\left\\lbrace \\begin{array}{l}v_{1}=u_{1}-f_{1}\\\\v_{2}=u_{2}-f_{2}\\\\u_{1}+\\Delta ((c_{1}^{2}+a)\\Delta u_{1}-a\\Delta f_{1})=f_{1}+g_{1}\\\\u_{2}+c_{2}^{2}\\Delta ^{2} u_{2}=f_{2}+g_{2}.\\end{array}\\right.$ First note that, by Riesz representation theorem, there exists a unique $(u_{1},u_{2})\\in X=\\mathcal {D}(G)$ such that for all $(\\varphi _{1},\\varphi _{2})\\in X$ we have $\\begin{split}\\langle f_{1}+g_{1},\\varphi _{1}\\rangle _{L^{2}(\\Omega _{1})}+\\langle f_{2}+g_{2},\\varphi _{2}\\rangle _{L^{2}(\\Omega _{2})}+\\langle a\\Delta f_{1},\\Delta \\varphi _{1}\\rangle _{L^{2}(\\Omega _{1})}=\\langle u_{1},\\varphi _{1}\\rangle _{L^{2}(\\Omega _{1})}\\\\+\\langle u_{2},\\varphi _{2}\\rangle _{L^{2}(\\Omega _{2})}+\\langle (c_{1}^{2}+a)\\Delta u_{1},\\Delta \\varphi _{1}\\rangle _{L^{2}(\\Omega _{1})}+c_{2}^{2}\\langle \\Delta u_{2},\\Delta \\varphi _{2}\\rangle _{L^{2}(\\Omega _{2})}.\\end{split}$ In particular for all $(\\varphi _{1},\\varphi _{2})\\in {C}_{c}^{\\infty }(\\Omega _{1})\\times {C}_{c}^{\\infty }(\\Omega _{2})$ the expression (REF ) yields $\\langle \\Delta ((c_{1}^{2}+a)\\Delta u_{1}-a\\Delta f_{1})+(u_{1}-f_{1}-g_{1}),\\varphi _{1}\\rangle _{D^{\\prime }(\\Omega _{1})}=0,\\\\\\langle c_{2}^{2}\\Delta ^{2}u_{2}+(u_{2}-f_{2}-g_{2}),\\varphi _{2}\\rangle _{D^{\\prime }(\\Omega _{2})}=0.$ then we obtain $\\begin{split}u_{1}+\\Delta ((c_{1}^{2}+a)\\Delta u_{1}-a\\Delta f_{1})=f_{1}+g_{1}\\quad \\text{in}\\;L^{2}(\\Omega _{1}),\\\\u_{2}+c_{2}^{2}\\Delta ^{2} u_{2}=f_{2}+g_{2}\\quad \\text{in}\\;L^{2}(\\Omega _{2}).\\end{split}$ Now if we return again to the expression (REF ) then by Green's formula we can write it as follows $\\langle \\Delta ((c_{1}^{2}+a)\\Delta u_{1}-a\\Delta f_{1})+(u_{1}-f_{1}-g_{1}),\\varphi _{1}\\rangle _{L^{2}(\\Omega _{1})}+\\langle c_{2}^{2}\\Delta ^{2}u_{2}+(u_{2}-f_{2}-g_{2}),\\varphi _{2}\\rangle _{L^{2}(\\Omega _{2})}\\\\=-\\langle c_{2}\\Delta u_{2},\\partial _{\\nu }\\varphi _{2}\\rangle _{L^{2}(\\Gamma )}-\\langle c_{1}\\Delta u_{1}\\partial _{\\nu }\\varphi _{1}\\rangle _{L^{2}(S)}+\\langle c_{2}\\Delta u_{2},\\partial _{\\nu }\\varphi _{1}\\rangle _{L^{2}(S)}\\\\+\\langle c_{1}\\partial _{\\nu }\\Delta u_{1},\\varphi _{2}\\rangle _{L^{2}(S)}-\\langle c_{2}\\partial _{\\nu }\\Delta u_{2},\\varphi _{2}\\rangle _{L^{2}(S)}.$ then by (REF ) we get for all $(\\varphi _{1},\\varphi _{2})\\in X$ that $\\langle c_{1}\\partial _{\\nu }\\Delta u_{1}-c_{2}\\partial _{\\nu }\\Delta u_{2},\\varphi _{1}\\rangle _{L^{2}(S)}-\\langle c_{1}\\Delta u_{1}-c_{2}\\Delta u_{2},\\partial _{\\nu }\\varphi _{1}\\rangle _{L^{2}(S)}-\\langle c_{2}\\Delta u_{2},\\partial _{\\nu }\\varphi _{2}\\rangle _{L^{2}(\\Gamma )}=0,$ which yields the following equalities $c_{1}\\Delta u_{1\\,\|S}=c_{2}\\Delta u_{2\\,\|S},\\; c_{1}\\partial _{\\nu }\\Delta u_{1\\,\|S}=c_{2}\\partial _{\\nu }\\Delta u_{2\\,\|S},\\; \\Delta u_{2\\,\|\\Gamma }=0.$ And this concludes the proof.", "One consequence of this last result is that if we assume that $(u_{1}^{0},u_{2}^{0},u_{1}^{1},u_{2}^{1})\\in \\mathcal {D}(\\mathcal {A})$ , there exists a unique solution of (REF ) which can be expressed by means of a semigroup on $\\mathcal {H}$ as follows $\\left(\\begin{array}{c}u_{1}\\\\u_{2}\\\\\\partial _{t}u_{1}\\\\\\partial _{t}u_{2}\\end{array}\\right)=e^{t\\mathcal {A}}\\left(\\begin{array}{c}u_{1}^{0}\\\\u_{2}^{0}\\\\u_{1}^{1}\\\\u_{2}^{1}\\end{array}\\right)$ where $e^{t\\mathcal {A}}$ is the $C_{0}$ -semigroup of contractions generates by the operator $\\mathcal {A}$ .", "And we have the following regularity of the solution $\\left(\\begin{array}{c}u_{1}\\\\u_{2}\\\\\\partial _{t}u_{1}\\\\\\partial _{t}u_{2}\\end{array}\\right)\\in C([0,+\\infty [,\\mathcal {D}(\\mathcal {A}))\\cap C^{1}([0,+\\infty [,\\mathcal {H}).$ And if $(u_{1}^{0},u_{2}^{0},u_{1}^{1},u_{2}^{1})\\in \\mathcal {H}$ , the function $(u_{1}(t),u_{2}(t))$ given by (REF ) is the mild solution of (REF ) and it lives in $C([0,+\\infty [,\\mathcal {H})$ ." ], [ "Carleman estimate", "We consider tow open and disjoint domains $\\mathcal {O}_{1}$ and $\\mathcal {O}_{2}$ in which we define respectively the second order elliptic semi-classical operators $P_{1}=-h^{2}\\Delta -\\alpha _{1}h$ and $P_{2}=-h^{2}\\Delta -\\alpha _{2}h$ with principal symbol $p(x,\\xi )=\|\\xi \|^{2}$ where $h$ is a very small semi-classical parmeter and $\\alpha _{1},\\,\\alpha _{2}\\in \\mathbb {R}$ , and we suppose that $\\partial \\mathcal {O}_{1}=\\gamma \\cup \\gamma _{1}$ , $\\partial \\mathcal {O}_{2}=\\gamma \\cup \\gamma _{2}$ and $\\overline{\\gamma }_{1}\\cap \\overline{\\gamma }_{0}=\\overline{\\gamma }_{2}\\cap \\overline{\\gamma }_{0}=\\emptyset $ .", "Let $\\varphi _{1}\\in {C}^{\\infty }(\\overline{\\mathcal {O}}_{1})$ and $\\varphi _{2}\\in {C}^{\\infty }(\\overline{\\mathcal {O}}_{2})$ tow real value functions.", "We define the two adjoint operators $P_{\\varphi _{1}}=\\mathrm {e}^{\\varphi _{1}/h}P_{1}\\mathrm {e}^{\\varphi _{2}/h}$ and $P_{\\varphi _{1}}=\\mathrm {e}^{\\varphi _{1}/h}P_{1}\\mathrm {e}^{\\varphi _{2}/h}$ of principal symbol respectively $p_{1}(x,\\xi )=p(x,\\xi +i\\nabla \\varphi _{1})$ and $p_{2}(x,\\xi )=p(x,\\xi +i\\nabla \\varphi _{2})$ .", "By denoting $\\partial _{\\nu }$ the unit outward normal vector of $\\mathcal {O}_{1}$ and $\\mathcal {O}_{2}$ respectively in $\\gamma \\cup \\gamma _{1}$ and $\\gamma _{2}$ we assume that the weight function $\\varphi _{1}$ and $\\varphi _{2}$ satisfies $\|\\nabla \\varphi _{1}\|(x)>0,\\;\\forall \\,x\\in \\overline{\\mathcal {O}}_{1}$ and $\|\\nabla \\varphi _{2}\|(x)>0,\\;\\forall \\,x\\in \\overline{\\mathcal {O}}_{2}$ , $\\partial _{\\nu }\\varphi _{1\\,\|\\gamma _{1}}\\ne 0$ and $\\partial _{\\nu }\\varphi _{2\\,\|\\gamma _{2}}<0$ , $\\varphi _{1\\,\|\\gamma }=\\varphi _{2\\,\|\\gamma }$ , $(\\partial _{\\nu }\\varphi _{1})_{\|\\gamma }< 0$ , $(\\partial _{\\nu }\\varphi _{2})_{\|\\gamma }<0$ and $(\\partial _{\\nu }\\varphi _{1})_{\|\\gamma }^{2}-(\\partial _{\\nu }\\varphi _{2})_{\|\\gamma }^{2}>0$ , The sub-ellipticity condition respectively in $\\overline{\\mathcal {O}}_{1}$ and $\\overline{\\mathcal {O}}_{2}$ $\\begin{array}{l}\\forall \\,(x,\\xi )\\in \\overline{\\mathcal {O}}_{1}\\times \\mathbb {R}^{n};\\; p_{\\varphi _{1}}(x,\\xi )=0\\Longrightarrow \\lbrace \\mathrm {Re}(p_{\\varphi _{1}}),\\mathrm {Im}(p_{\\varphi _{1}})\\rbrace (x,\\xi )>0,\\\\\\forall \\,(x,\\xi )\\in \\overline{\\mathcal {O}}_{2}\\times \\mathbb {R}^{n};\\; p_{\\varphi _{2}}(x,\\xi )=0\\Longrightarrow \\lbrace \\mathrm {Re}(p_{\\varphi _{2}}),\\mathrm {Im}(p_{\\varphi _{2}})\\rbrace (x,\\xi )>0.\\end{array}$ The Carleman estimate corresponding to the following transmission boundary value problem $\\left\\lbrace \\begin{array}{ll}\\displaystyle -\\Delta w_{1}-\\frac{\\alpha _{1}}{h}w_{1}=f_{1}&\\text{in }\\mathcal {O}_{1}\\\\\\\\\\displaystyle -\\Delta w_{2}-\\frac{\\alpha _{2}}{h}w_{2}=f_{2}&\\text{in }\\mathcal {O}_{2}\\\\w_{1}=w_{2}+e_{1}&\\text{on }\\gamma \\\\\\partial _{\\nu }w_{1}=\\partial _{\\nu }w_{2}+e_{2}&\\text{on }\\gamma \\\\w_{2}=0&\\text{on }\\gamma _{2}\\end{array}\\right.$ is gived in the following Theorem 3.1 [17] Under the above assumptions on the weight functions $\\varphi _{1}$ and $\\varphi _{2}$ , there exists $h_{0}>0$ and $C>0$ such that $\\begin{split}&h\\Vert \\mathrm {e}^{\\varphi _{1}/h}w_{1}\\Vert _{L^{2}(\\mathcal {O}_{1})}^{2}+h^{3}\\Vert \\mathrm {e}^{\\varphi _{1}/h}\\nabla w_{1}\\Vert _{L^{2}(\\mathcal {O}_{1})}^{2}+h\|\\mathrm {e}^{\\varphi _{1}/h}w_{1}\|_{L^{2}(\\gamma )}^{2}+h^{3}\|\\mathrm {e}^{\\varphi _{1}/h}\\nabla w_{1}\|_{L^{2}(\\gamma )}^{2}+\\\\&h^{3}\|\\mathrm {e}^{\\varphi _{1}/h}\\partial _{\\nu }w_{1}\|_{L^{2}(\\gamma )}^{2}+h\\Vert \\mathrm {e}^{\\varphi _{2}/h}w_{2}\\Vert _{L^{2}(\\mathcal {O}_{2})}^{2}+h^{3}\\Vert \\mathrm {e}^{\\varphi _{2}/h}\\nabla w_{2}\\Vert _{L^{2}(\\mathcal {O}_{2})}^{2}+h\|\\mathrm {e}^{\\varphi _{2}/h}w_{2}\|_{L^{2}(\\gamma )}^{2}+\\\\&h^{3}\|\\mathrm {e}^{\\varphi _{2}/h}\\nabla w_{2}\|_{L^{2}(\\gamma )}^{2}+h^{3}\|\\mathrm {e}^{\\varphi _{2}/h}\\partial _{\\nu }w_{2}\|_{L^{2}(\\gamma )}^{2}\\le C(h^{4}\\Vert \\mathrm {e}^{\\varphi _{1}/h}f_{1}\\Vert _{L^{2}(\\mathcal {O}_{1})}^{2}+h^{4}\\Vert \\mathrm {e}^{\\varphi _{2}/h}f_{2}\\Vert _{L^{2}(\\mathcal {O}_{2})}^{2}+\\\\&h\|\\mathrm {e}^{\\varphi _{1}/h}w_{1}\|_{L^{2}(\\gamma _{1})}^{2}+h^{3}\|\\mathrm {e}^{\\varphi _{2}/h}\\partial _{\\nu }w_{1}\|_{L^{2}(\\gamma _{1})}^{2}+h\|\\mathrm {e}^{\\varphi _{1}/h} e_{1}\|_{L^{2}(\\gamma )}^{2}+h^{3}\|\\mathrm {e}^{\\varphi _{1}/h}\\nabla e_{1}\|_{L^{2}(\\gamma )}^{2}+h^{3}\|\\mathrm {e}^{\\varphi _{1}/h}e_{2}\|_{L^{2}(\\gamma )}^{2})\\end{split}$ for all $w_{1}\\in {C}^{\\infty }(\\overline{O}_{1})$ and $w_{2}\\in {C}^{\\infty }(\\overline{O}_{2})$ satisfing the system (REF ) and $h\\in ]0,h_{0}]$ .", "Remarks 3.1 If the function $w_{1}$ is supported away from $\\gamma _{1}$ the estimate (REF ) is allows true even if we don't assume that $(\\partial _{\\nu }\\varphi _{1})_{\|\\gamma _{1}}\\ne 0$ , while the proof of Theorem REF is local.", "We can not assume that $(\\partial _{\\nu }\\varphi _{1})_{\|\\gamma _{1}}<0$ (it means $\\partial _{\\nu }\\varphi _{1}<0$ in whole $\\partial \\mathcal {O}_{1}$ ), otherwise the weight function attain his global maximum in $\\mathcal {O}_{1}$ and thus our srtategy of the construction of the phases is fails (see below)." ], [ "Weight function's construction", "In this section we will try to find two phases that satisfies the Hörmander's condition except in a finite number of ball where one of them do not satisfies this condition the second does and is strictly greater.", "Note that this result is similar to the Burq's one [6], but here we give a new proof due to F. Laudenbach.", "Then we will adapte this result to our case to constructe a suitable weight functions that will be needed in the following section.", "The main ingredient of this section is the following one.", "Proposition 3.1 Let $\\mathcal {O}$ be a bounded open subset with boundary $\\gamma =\\gamma _{1}\\cup \\gamma _{2}$ where $\\overline{\\gamma }_{1}\\cap \\overline{\\gamma }_{2}=\\emptyset $ , then there exists two real functions $\\psi _{1},\\,\\psi _{2}\\in {C}^{\\infty }(\\mathcal {O})$ and continous on $\\overline{\\mathcal {O}}$ satisfying for $k=1,2$ that $(\\partial _{\\nu }\\psi _{k})_{\|\\gamma _{1}}<0$ and $(\\partial _{\\nu }\\psi _{k})_{\|\\gamma _{2}}>0$ having only degenerate critical points (of finite number) such that when $\\nabla \\psi _{k}=0$ then $\\nabla \\psi _{\\sigma (k)}\\ne 0$ and $\\psi _{\\sigma (k)}>\\psi _{k}$ .", "Where $\\sigma $ is the permutation of the set $\\lbrace 1,2\\rbrace $ different from the identity.", "Remarks 3.2 One consequence of Proposition REF is that for $k=1,2$ we can find a finite number of points $x_{kj_{k}}$ and $j_{k}=1,\\ldots ,N_{k}$ and $\\epsilon >0$ such that $B(x_{kj_{k}},2\\epsilon )\\subset \\overline{\\mathcal {O}}$ and $B(x_{1j_{1}},2\\epsilon )\\cap B(x_{2j_{2}},2\\epsilon )=\\emptyset $ , for all $k=1,2$ and $j_{k}=1,\\ldots ,N_{k}$ and in $B(x_{kj_{k}},2\\epsilon )$ we have $\\psi _{\\sigma (k)}>\\psi _{k}$ (See Figure REF ).", "For $\\lambda >0$ large enough the weight functions $\\varphi _{k}=\\mathrm {e}^{\\lambda \\psi _{k}}$ satisfy the Hörmander's condition in $\\displaystyle U_{k}=\\mathcal {O}\\bigcap \\left(\\bigcup _{j_{k}=1}^{N_{k}}B(x_{kj_{k}},\\epsilon )\\right)^{c}$ .", "Indeed, we have only to prove that for an open bounded subset $U\\in \\mathbb {R}^{n}$ and if $\\psi \\in {C}^{\\infty }(\\overline{U})$ satisfying $\|\\nabla \\psi \|\\ge C$ in $\\overline{U}$ and $\\varphi =\\mathrm {e}^{\\lambda \\psi }$ we have $\\lbrace \\mathrm {Re}(p_{\\varphi }),\\mathrm {Im}(p_{\\varphi })\\rbrace (x,\\xi )\\ge C^{\\prime }$ in $\\overline{U}\\times \\mathbb {R}^{n}$ for $\\lambda >0$ large enough.", "We have $\\left\\lbrace \\begin{array}{c}\\nabla \\varphi =\\lambda \\mathrm {e}^{\\lambda \\psi }\\nabla \\psi \\;\\text{ and }\\;\\varphi ^{\\prime \\prime }=\\mathrm {e}^{\\lambda \\psi }(\\lambda \\nabla \\psi .", "{}^{t}\\nabla \\psi +\\lambda \\psi ^{\\prime \\prime })\\\\p_{\\varphi }(x,\\xi )=0\\Longrightarrow \\langle \\xi ,\\nabla \\varphi \\rangle =0\\text{ and }\|\\xi \|^{2}=\|\\nabla \\varphi \|^{2}\\end{array}\\right.$ then we obtain $\\lbrace \\mathrm {Re}(p_{\\varphi }),\\mathrm {Im}(p_{\\varphi })\\rbrace (x,\\xi )&=&4\\lambda \\mathrm {e}^{\\lambda \\psi }\\,{}^{t}\\xi .\\psi ^{\\prime \\prime }.\\xi +4\\mathrm {e}^{3\\lambda \\psi }(\\lambda ^{4}\|\\nabla \\psi \|^{2}+\\lambda ^{3}\\,{}^{t}\\nabla \\psi .\\psi ^{\\prime \\prime }.\\nabla \\psi )\\\\&=&4\\mathrm {e}^{3\\lambda \\psi }(\\lambda ^{4}\|\\nabla \\psi \|^{2}+O(\\lambda ^{3})).$ Which conclude the result.", "In general, Proposition REF is also true for any smooth manifold with boundary which the latter is the disjoint union of two open and closed submanifolds.", "Figure: The domains of the weight functions ϕ 1 \\varphi _{1} and ψ 1 \\psi _{1} (in yellow and orange), ϕ 2 \\varphi _{2} and ψ 2 \\psi _{2} (in red and orange) where they have not critical points.Proof : While the Morse functions are dense (for the ${C}^{\\infty }$ topology) in the set of ${C}^{\\infty }$ functions then we can find $\\psi _{1}$ a Morse function such that $(\\partial _{\\nu }\\psi _{1})_{\|\\gamma _{1}}<0$ and $(\\partial _{\\nu }\\psi _{1})_{\|\\gamma _{2}}>0$ .", "We can suppose that $\\psi _{1}$ have no local maximum in $\\mathcal {O}$ (The proceeding of the elimination of the maximum is described by Burq [6], we can see also [16] and [11]).", "Let $c$ be a critical point of $\\psi _{1}$ while its index is different from $n$ then we can find a ${C}^{\\infty }$ arc $\\gamma _{c}:[-1,1]\\rightarrow \\Omega $ such that $\\gamma _{c}(0)=c$ and $\\psi _{1}(\\gamma _{c}(1))=\\psi _{1}(\\gamma _{c}(-1))>\\psi _{1}(c)$ .", "We do this construction for all the critical points of $\\psi _{1}$ so that all the arcs are mutually disjoint.", "Hence, this allows us to find a vector field $X$ in $\\mathcal {O}$ , vanishing near the boundary of $\\mathcal {O}$ such that for all critical points $c$ of $\\psi _{1}$ we have $X(\\gamma _{c}(t))=\\stackrel{.", "}{\\gamma }_{c}(t),$ where $\\stackrel{.", "}{\\gamma }$ stand for the time derivative.", "We denote $\\phi _{t}$ its flow: $\\stackrel{.", "}{\\phi _{t}}(x)=X(\\phi _{t}(x)),$ and we set $\\psi _{2}=\\psi _{1}\\circ \\phi _{1}$ , thus $\\psi _{1}$ and $\\psi _{2}$ satisfy the required properties.", "Indeed, since $X\\equiv 0$ near the boundary $\\gamma _{1}$ and $\\gamma _{2}$ which mean that $\\phi _{t}(x)=x$ near $\\gamma _{1}$ and $\\gamma _{2}$ then $\\partial _{\\nu }\\psi _{1\\,\|\\gamma _{1}}=\\partial _{\\nu }\\psi _{2\\,\|\\gamma _{1}}$ and $\\partial _{\\nu }\\psi _{1\\,\|\\gamma _{2}}=\\partial _{\\nu }\\psi _{2\\,\|\\gamma _{2}}$ .", "If $c$ is a critical point of $\\psi _{1}$ then we have $\\psi _{2}(c)=\\psi _{1}(\\gamma _{c}(1))>\\psi _{1}(c)$ , and if $c^{\\prime }$ is a critical point of $\\psi _{2}$ then $c^{\\prime }=\\phi _{-1}(c)$ where $c$ is a critical point of $\\psi _{1}$ and we have $\\psi _{2}(c^{\\prime })=\\psi _{1}(\\phi _{1}\\circ \\phi _{-1}(c))=\\psi _{1}(c)<\\psi _{1}(\\phi _{-1}(c))=\\psi _{1}(c^{\\prime })$ by the construction of $\\gamma _{c}$ .", "Now if we return to our geometric baseline as described in the introduction of this paper then according to Proposition REF and Remark REF and by noting $\\widetilde{\\Omega }_{1}=\\Omega _{1}\\backslash \\overline{B}_{r}$ where $B_{r}$ is an open ball of $\\Omega _{1}$ with radius $r>0$ such that $\\overline{B}_{r}\\subset \\Omega _{1}$ we can find four phases $\\varphi _{1,1}$ , $\\varphi _{1,2}$ , $\\varphi _{2,1}$ and $\\varphi _{2,2}$ verifying the Hörmander's condition respectively in $\\displaystyle U_{1,1}=\\widetilde{\\Omega }_{1}\\bigcap \\left(\\bigcup _{j=1}^{N_{11}}B(x_{11}^{j},\\epsilon )\\right)^{c}$ , $\\displaystyle U_{1,2}=\\widetilde{\\Omega }_{1}\\bigcap \\left(\\bigcup _{j_{2}=1}^{N_{12}}B(x_{12}^{j},\\epsilon )\\right)^{c}$ , $\\displaystyle U_{2,1}=\\Omega _{2}\\bigcap \\left(\\bigcup _{j_{1}=1}^{N_{21}}B(x_{21}^{j},\\epsilon )\\right)^{c}$ and $\\displaystyle U_{2,2}=\\Omega _{2}\\bigcap \\left(\\bigcup _{j_{2}=1}^{N_{22}}B(x_{22}^{j},\\epsilon )\\right)^{c}$ such that $\|\\nabla \\varphi _{1,1}\|>0$ in $U_{1,1}$ , $\|\\nabla \\varphi _{1,2}\|>0$ in $U_{1,2}$ , $\|\\nabla \\varphi _{2,1}\|>0$ in $U_{2,1}$ and $\|\\nabla \\varphi _{2,2}\|>0$ in $U_{2,2}$ , moreover $\\varphi _{1,k}<\\varphi _{1,\\sigma (k)}$ in $B(x_{1k}^{j},2\\epsilon )$ for all $j=1,\\ldots ,N_{1,k}$ and $\\varphi _{2,k}<\\varphi _{2,\\sigma (k)}$ in $B(x_{2k}^{j},2\\epsilon )$ for all $j=1,\\ldots ,N_{2,k}$ .", "Furthermore we have also for all $k=1,2$ $(\\partial _{\\nu }\\varphi _{1,k})_{\|S}<0,\\quad (\\partial _{\\nu }\\varphi _{2,k})_{\|S}<0 \\textrm { and }(\\partial _{\\nu }\\varphi _{2,k})_{\|\\Gamma }<0.$ We can suppose also that $\\varphi _{1,k\\,\|S}=\\varphi _{2,k\\,\|S}$ , and by argument of density we can suppose also that $(\\partial _{\\nu }\\varphi _{1,k})_{\|S}^{2}-(\\partial _{\\nu }\\varphi _{2,k})_{\\,\|S}^{2}>0.$ And this concludes the construction of weight functions that will be used in next section." ], [ "Resolvent estimate", "The purpose of this section is to find an estimate of the resolvent $(\\mathcal {A}-i\\mu \\mathrm {Id})^{-1}$ where $\\mu $ is a real number such that $\|\\mu \|$ is large enough.", "More precisely we prove that $\\Vert (\\mathcal {A}-i\\mu \\mathrm {Id})^{-1}\\Vert _{{L}(\\mathcal {H})}\\le C\\mathrm {e}^{C\|\\mu \|}$ which imply the weak energy decay of the solution of the equation (REF ).", "The main idea consiste to applying the Carleman estimates for a second order elliptic transmission system which is derived from the plate equation and this is what comes from the originality of our work, it means we prove the stability result for a system of fourth order by using an estimate of Carleman of second order only.", "Let $(f_{1},f_{2},g_{1},g_{2})\\in \\mathcal {H}$ and $(u_{1},u_{2},v_{1},v_{2})\\in \\mathcal {D}(\\mathcal {A})$ such that $(\\mathcal {A}-i\\mu \\mathrm {Id})\\left(\\begin{array}{c}u_{1}\\\\u_{2}\\\\v_{1}\\\\v_{2}\\end{array}\\right)=\\left(\\begin{array}{c}f_{1}\\\\f_{2}\\\\g_{1}\\\\g_{2}\\end{array}\\right),$ then we get the following boundary value problem $\\left\\lbrace \\begin{array}{ll}v_{1}-i\\mu u_{1}=f_{1}&\\text{in }\\Omega _{1}\\\\v_{2}-i\\mu u_{2}=f_{2}&\\text{in }\\Omega _{2}\\\\-\\Delta (c_{1}^{2}\\Delta u_{1}+a\\Delta v_{1})-i\\mu v_{1}=g_{1}&\\text{in }\\Omega _{1}\\\\-c_{2}^{2}\\Delta ^{2}u_{2}-i\\mu v_{2}=g_{2}&\\text{in }\\Omega _{2}\\\\u_{1}=u_{2},\\quad \\partial _{\\nu }u_{1}=\\partial _{\\nu }u_{2}&\\text{on }S\\\\c_{1}\\Delta u_{1}=c_{2}\\Delta u_{2},\\quad c_{1}\\partial _{\\nu }\\Delta u_{1}=c_{2}\\partial _{\\nu }\\Delta u_{2}&\\text{on }S\\\\u_{2}=0,\\quad \\Delta u_{2}=0&\\text{on }\\Gamma .\\end{array}\\right.$ Then the solution $(u_{1},u_{2},v_{1},v_{2})$ of (REF ) satisfies $\\left\\lbrace \\begin{array}{ll}v_{1}=i\\mu u_{1}+f_{1}&\\text{in }\\Omega _{1}\\\\v_{2}=i\\mu u_{2}+f_{2}&\\text{in }\\Omega _{2}\\\\\\mu ^{2} u_{1}-\\Delta (c_{1}^{2}\\Delta u_{1}+a\\Delta v_{1})=g_{1}+i\\mu f_{1}&\\text{in }\\Omega _{1}\\\\\\mu ^{2}u_{2}-c_{2}^{2}\\Delta ^{2}u_{2}=g_{2}+i\\mu f_{2}&\\text{in }\\Omega _{2}\\\\u_{1}=u_{2},\\quad \\partial _{\\nu }u_{1}=\\partial _{\\nu }u_{2}&\\text{on }S\\\\c_{1}\\Delta u_{1}=c_{2}\\Delta u_{2},\\quad c_{1}\\partial _{\\nu }\\Delta u_{1}=c_{2}\\partial _{\\nu }\\Delta u_{2}&\\text{on }S\\\\u_{2}=0,\\quad \\Delta u_{2}=0&\\text{on }\\Gamma .\\end{array}\\right.$ This can be rewriten as follows $\\left\\lbrace \\begin{array}{ll}v_{1}=i\\mu u_{1}+f_{1}&\\text{in }\\Omega _{1}\\\\v_{2}=i\\mu u_{2}+f_{2}&\\text{in }\\Omega _{2}\\\\\\displaystyle (-\\Delta -\\frac{\|\\mu \|}{c_{1}})(c_{1}\\Delta u_{1}+\\frac{a}{c_{1}}\\Delta v_{1}-\|\\mu \| u_{1})=\\Phi _{1}=\\frac{1}{c_{1}}g_{1}+i\\frac{\\mu }{c_{1}}f_{1}-a\\frac{\|\\mu \|}{c_{1}^{2}}\\Delta v_{1}&\\text{in }\\Omega _{1}\\\\\\\\\\displaystyle (-\\Delta -\\frac{\|\\mu \|}{c_{2}})(c_{2}\\Delta u_{2}-\|\\mu \|u_{2})=\\Phi _{2}=\\frac{1}{c_{2}}g_{2}+i\\frac{\\mu }{c_{2}}f_{2}&\\text{in }\\Omega _{2}\\\\u_{1}=u_{2},\\quad \\partial _{\\nu }u_{1}=\\partial _{\\nu }u_{2}&\\text{on }S\\\\c_{1}\\Delta u_{1}=c_{2}\\Delta u_{2},\\quad c_{1}\\partial _{\\nu }\\Delta u_{1}=c_{2}\\partial _{\\nu }\\Delta u_{2}&\\text{on }S\\\\u_{2}=0,\\quad \\Delta u_{2}=0&\\text{on }\\Gamma .\\end{array}\\right.$ We set now $w_{1}=c_{1}\\Delta u_{1}-\|\\mu \|u_{1}+\\frac{a}{c_{1}}\\Delta v_{1}\\quad \\text{ and }\\quad w_{2}=c_{2}\\Delta u_{2}-\|\\mu \|u_{2},$ then it easy to show that $w_{1}$ and $w_{2}$ satisfy the following simple transmission problem $\\left\\lbrace \\begin{array}{ll}\\displaystyle -\\Delta w_{1}-\\frac{\|\\mu \|}{c_{1}}w_{1}=\\Phi _{1}&\\text{in }\\Omega _{1}\\\\\\\\\\displaystyle -\\Delta w_{2}-\\frac{\|\\mu \|}{c_{2}}w_{2}=\\Phi _{2}&\\text{in }\\Omega _{2}\\\\w_{1}=w_{2},\\quad \\partial _{\\nu }w_{1}=\\partial _{\\nu }w_{2}&\\text{on }S\\\\w_{2}=0&\\text{on }\\Gamma .\\end{array}\\right.$ We set also $B_{4r}$ a ball of raduis $4r>0$ , such that $a(x)>0$ in $B_{4r}\\subset \\omega $ and we recall the notation gived in the end of the previous section $\\widetilde{\\Omega }_{1}=\\Omega _{1}\\backslash \\overline{B}_{r}$ .", "The most important ingredient of the proof of the resolvent estimate is the following lemma which is essentially a consequence of the Carleman estimate.", "Lemma 4.1 There exist a constant $C>0$ such that for any $(u_{1},u_{2},v_{1},v_{2})\\in \\mathcal {D}(\\mathcal {A})$ solution of (REF ) the following result holds $\\begin{split}\\Vert \\Delta u_{1}\\Vert _{L^{2}(\\Omega _{1})}^{2}+\\Vert \\Delta u_{2}\\Vert _{L^{2}(\\Omega _{2})}^{2}+\\Vert v_{1}\\Vert _{L^{2}(\\Omega _{1})}^{2}+\\Vert v_{2}\\Vert _{L^{2}(\\Omega _{2})}^{2}\\le C\\mathrm {e}^{C\|\\mu \|}\\bigg (\\Vert \\Delta f_{1}\\Vert _{L^{2}(\\Omega _{1})}^{2}\\\\+\\Vert \\Delta f_{2}\\Vert _{L^{2}(\\Omega _{2})}^{2}+\\Vert g_{1}\\Vert _{L^{2}(\\Omega _{1})}^{2}+\\Vert g_{2}\\Vert _{L^{2}(\\Omega _{2})}^{2}+\\int _{\\Omega _{1}}a\|\\Delta v_{1}\|^{2}\\,\\mathrm {d}x+\\int _{B_{4r}}\|u_{1}\|^{2}\\,\\mathrm {d}x\\bigg ),\\end{split}$ for all $\\mu \\in \\mathbb {R}$ large enough.", "Proof : We introduce the cutt-off function $\\chi \\in {C}^{\\infty }(\\Omega _{1})$ by setting $\\chi (x)=\\left\\lbrace \\begin{array}{ll}1&\\text{in }B_{3r}^{c}\\\\0&\\text{in }B_{2r}\\end{array}\\right.$ Next, denote $\\tilde{w}_{1}=\\chi w_{1}$ .", "And by (REF ), one sees that $-\\Delta \\tilde{w}_{1}-\\frac{\|\\mu \|}{c_{1}}\\tilde{w}_{1}=\\widetilde{\\Phi }_{1}=\\chi \\Phi _{1}-[\\Delta ,\\chi ]w_{1}.$ Now keeping the same notations as the previous section and let $\\varphi _{1,1}$ , $\\varphi _{1,2}$ , $\\varphi _{2,1}$ and $\\varphi _{2,2}$ four weight functions that satisfies the conclusion of the section .", "Let $\\chi _{1,1}$ , $\\chi _{1,2}$ , $\\chi _{2,1}$ and $\\chi _{2,2}$ four cut-off functions equal to one respectively in $\\displaystyle \\left(\\bigcup _{j=1}^{N_{11}}B(x_{1j}^{1},2\\epsilon )\\right)^{c}$ , $\\displaystyle \\left(\\bigcup _{j=1}^{N_{12}}B(x_{1j}^{2},2\\epsilon )\\right)^{c}$ , $\\displaystyle \\left(\\bigcup _{j=1}^{N_{21}}B(x_{2j}^{1},2\\epsilon )\\right)^{c}$ and $\\displaystyle \\left(\\bigcup _{j=1}^{N_{22}}B(x_{2j}^{2},2\\epsilon )\\right)^{c}$ and supported respectively in $\\displaystyle \\left(\\bigcup _{j=1}^{N_{11}}B(x_{1j}^{1},\\epsilon )\\right)^{c}$ , $\\displaystyle \\left(\\bigcup _{j=1}^{N_{12}}B(x_{1j}^{2},\\epsilon )\\right)^{c}$ , $\\displaystyle \\left(\\bigcup _{j=1}^{N_{21}}B(x_{2j}^{1},\\epsilon )\\right)^{c}$ and $\\displaystyle \\left(\\bigcup _{j=1}^{N_{22}}B(x_{2j}^{2},\\epsilon )\\right)^{c}$ (in order to eliminate the critical points of the phases functions $\\varphi _{1,1}$ , $\\varphi _{1,2}$ , $\\varphi _{2,1}$ and $\\varphi _{2,2}$ (See Figure REF )).", "We set now $w_{1,1}=\\chi _{1,1}\\tilde{w}_{1}$ , $w_{1,2}=\\chi _{1,2}\\tilde{w}_{1}$ , $w_{2,1}=\\chi _{2,1}w_{2}$ and $w_{2,2}=\\chi _{2,2}w_{2}$ .", "Then from the system (REF ) for $k=1,2$ we obtain $\\left\\lbrace \\begin{array}{ll}\\displaystyle -\\Delta w_{1,k}-\\frac{\|\\mu \|}{c_{1}}w_{1,k}=\\Psi _{1,k}&\\text{in }\\Omega _{1}\\\\\\\\\\displaystyle -\\Delta w_{2,k}-\\frac{\|\\mu \|}{c_{2}}w_{2,k}=\\Psi _{2,k}&\\text{in }\\Omega _{2}\\\\w_{1,k}=w_{2,k},\\quad \\partial _{\\nu }w_{1,k}=\\partial _{\\nu }w_{2,k}&\\text{on }S\\\\w_{2,k}=0&\\text{on }\\Gamma ,\\end{array}\\right.$ where $\\left\\lbrace \\begin{array}{l}\\Psi _{1,k}=\\chi _{1,k}\\widetilde{\\Phi }_{1}-[\\Delta ,\\chi _{1,k}]\\tilde{w}_{1}\\\\\\Psi _{2,k}=\\chi _{2,k}\\Phi _{2}-[\\Delta ,\\chi _{2,k}]w_{2}.\\end{array}\\right.$ Applying now the Carleman estimate gived in the previous section (Theorem REF ) to the system (REF ) for $\\displaystyle h=\\frac{1}{\|\\mu \|}$ then for $k=1,2$ we obtain $\\begin{split}h\\Vert \\mathrm {e}^{\\varphi _{1,k}/h}w_{1,k}\\Vert _{L^{2}(U_{1,k})}^{2}+h^{3}\\Vert \\mathrm {e}^{\\varphi _{1,k}/h}\\nabla w_{1,k}\\Vert _{L^{2}(U_{1,k})}^{2}+h\\Vert \\mathrm {e}^{\\varphi _{2,k}/h}w_{2,k}\\Vert _{L^{2}(U_{2,k})}^{2}+\\\\h^{3}\\Vert \\mathrm {e}^{\\varphi _{2,k}/h}\\nabla w_{2,k}\\Vert _{L^{2}(U_{2,k})}^{2}\\le Ch^{4}(\\Vert \\mathrm {e}^{\\varphi _{1,k}/h}\\Psi _{1,k}\\Vert _{L^{2}(U_{1,k})}^{2}+\\Vert \\mathrm {e}^{\\varphi _{2,k}/h}\\Psi _{2,k}\\Vert _{L^{2}(U_{2,k})}^{2}).\\end{split}$ Relations (REF ) and (REF ) yields $\\begin{split}h\\Vert \\mathrm {e}^{\\varphi _{1,k}/h}w_{1,k}\\Vert _{L^{2}(U_{1,k})}^{2}+h^{3}\\Vert \\mathrm {e}^{\\varphi _{1,k}/h}\\nabla w_{1,k}\\Vert _{L^{2}(U_{1,k})}^{2}+h\\Vert \\mathrm {e}^{\\varphi _{2,k}/h}w_{2,k}\\Vert _{L^{2}(U_{2,k})}^{2}+\\\\h^{3}\\Vert \\mathrm {e}^{\\varphi _{2,k}/h}\\nabla w_{2,k}\\Vert _{L^{2}(U_{2,k})}^{2}\\le Ch^{4}(\\Vert \\mathrm {e}^{\\varphi _{1,k}/h}\\Phi _{1}\\Vert _{L^{2}(U_{1,k})}^{2}+\\Vert \\mathrm {e}^{\\varphi _{2,k}/h}\\Phi _{2}\\Vert _{L^{2}(U_{2,k})}^{2}+\\\\\\Vert \\mathrm {e}^{\\varphi _{1,k}/h}[\\Delta ,\\chi ]w_{1}\\Vert _{L^{2}(U_{1,k})}^{2}+\\Vert \\mathrm {e}^{\\varphi _{1,k}/h}[\\Delta ,\\chi _{1,k}]\\tilde{w}_{1}\\Vert _{L^{2}(U_{1,k})}^{2}+\\Vert \\mathrm {e}^{\\varphi _{2,k}/h}[\\Delta ,\\chi _{2,k}]w_{2}\\Vert _{L^{2}(U_{2,k})}^{2}).\\end{split}$ We addition the two last estimates for $k=1,2$ and using the properties of phases $\\varphi _{1,k}<\\varphi _{1,\\sigma (k)}$ in $\\displaystyle \\left(\\bigcup _{j=1}^{N_{1k}}B(x_{1k}^{j},2\\epsilon )\\right)$ and $\\varphi _{2,k}<\\varphi _{2,\\sigma (k)}$ in $\\displaystyle \\left(\\bigcup _{j=1}^{N_{2k}}B(x_{2k}^{j},2\\epsilon )\\right)$ then we can absorb the terms $[\\Delta ,\\chi _{1,k}]\\tilde{w}_{1}$ and $[\\Delta ,\\chi _{2,k}]w_{2}$ at the right hand side of (REF ) into the left hand side for $h>0$ small.", "More precisly we obtain $\\begin{split}h\\int _{\\widetilde{\\Omega }_{1}}(\\mathrm {e}^{2\\varphi _{1,1}/h}+\\mathrm {e}^{2\\varphi _{1,2}/h})\|\\tilde{w}_{1}\|^{2}\\,\\mathrm {d}x+h\\int _{\\Omega _{2}}(\\mathrm {e}^{2\\varphi _{2,1}/h}+\\mathrm {e}^{2\\varphi _{2,2}/h})\|w_{2}\|^{2}\\,\\mathrm {d}x\\le \\\\Ch^{4}\\bigg (\\int _{\\Omega _{1}}(\\mathrm {e}^{2\\varphi _{1,1}/h}+\\mathrm {e}^{2\\varphi _{1,2}/h})\|\\Phi _{1}\|^{2}\\,\\mathrm {d}x+\\int _{\\Omega _{2}}(\\mathrm {e}^{2\\varphi _{2,1}/h}+\\mathrm {e}^{2\\varphi _{2,2}/h})\|\\Phi _{2}\|^{2}\\,\\mathrm {d}x\\\\+\\int _{\\widetilde{\\Omega }_{1}}(\\mathrm {e}^{2\\varphi _{1,1}/h}+\\mathrm {e}^{2\\varphi _{1,2}/h})\|[\\Delta ,\\chi ]w_{1}\|^{2}\\,\\mathrm {d}x\\bigg ).\\end{split}$ Consequently, by using that $\\Omega _{1}=\\widetilde{\\Omega }_{1}\\cup B_{2r}$ and the expressions of $\\Phi _{1}$ and $\\Phi _{2}$ in (REF ) we see that $\\begin{split}\\int _{\\Omega _{1}}\|w_{1}\|^{2}\\,\\mathrm {d}x+\\int _{\\Omega _{2}}\|w_{2}\|^{2}\\,\\mathrm {d}x\\le C\\mathrm {e}^{C/h}\\bigg (\\int _{\\Omega _{1}}\|f_{1}\|^{2}\\,\\mathrm {d}x+\\int _{\\Omega _{1}}\|g_{1}\|^{2}\\,\\mathrm {d}x+\\int _{\\Omega _{2}}\|f_{2}\|^{2}\\,\\mathrm {d}x\\\\+\\int _{\\Omega _{2}}\|g_{2}\|^{2}\\,\\mathrm {d}x+\\int _{\\Omega _{1}}a\|\\Delta v_{1}\|^{2}\\,\\mathrm {d}x+\\int _{B_{2r}}\|w_{1}\|^{2}\\,\\mathrm {d}x+\\int _{\\widetilde{\\Omega }_{1}}\|[\\Delta ,\\chi ]w_{1}\|^{2}\\,\\mathrm {d}x\\bigg ).\\end{split}$ To accomplish the proof of the lemma we estimate the two last terms in the right hand side of (REF ).", "We set $\\widetilde{\\chi }$ a cutt-off function equal to 1 in a neighborhood of $B_{3r}$ and supported in $B_{4r}$ then we have $(-1+\\Delta )(\\widetilde{\\chi }w_{1})=[\\Delta ,\\widetilde{\\chi }]w_{1}-\\widetilde{\\chi }w_{1}-\\frac{\|\\mu \|}{c_{1}}\\widetilde{\\chi }w_{1}-\\widetilde{\\chi }\\Phi _{1},$ and hence by elliptic estimates (see [19]) we get $\\Vert w_{1}\\Vert _{H^{1}(B_{3r})}^{2}\\!\\!\\!\\!&\\le &\\!\\!\\!C(\\Vert (-1+\\Delta )(\\widetilde{\\chi }w_{1})\\Vert _{H^{-1}(B_{4r})}^{2}+\\Vert w_{1}\\Vert _{L^{2}(B_{4r})}^{2})\\nonumber \\\\&\\le &\\!\\!\\!C(\\Vert \\Phi _{1}\\Vert _{L^{2}(\\Omega _{1})}^{2}+(1+\|\\mu \|^{2})\\Vert w_{1}\\Vert _{L^{2}(B_{4r})}^{2})\\nonumber \\\\&\\le &\\!\\!\\!C\\left(\|\\mu \|^{2}\\Vert f_{1}\\Vert _{L^{2}(\\Omega _{1})}^{2}+\\Vert g_{1}\\Vert _{L^{2}(\\Omega _{1})}^{2}+(1+\|\\mu \|^{2})\\Vert w_{1}\\Vert _{L^{2}(B_{4r})}^{2}+\|\\mu \|^{2}\\int _{\\Omega _{1}}a\|\\Delta v_{1}\|^{2}\\,\\mathrm {d}x\\right).$ Since $\\mathrm {supp}([\\Delta ,\\chi ])\\subset B_{3r}$ we deduce from (REF ) and (REF ) that $\\begin{split}\\int _{B_{2r}}\|w_{1}\|^{2}\\,\\mathrm {d}x+\\int _{\\widetilde{\\Omega }_{1}}\|[\\Delta ,\\chi ]w_{1}\|^{2}\\,\\mathrm {d}x\\le C\\Vert w_{1}\\Vert _{H^{1}(B_{3r})}^{2}\\\\\\le C\\left(\|\\mu \|^{2}\\Vert f_{1}\\Vert _{L^{2}(\\Omega _{1})}^{2}+\\Vert g_{1}\\Vert _{L^{2}(\\Omega _{1})}^{2}+(1+\|\\mu \|^{2})^{2}\\Vert u_{1}\\Vert _{L^{2}(B_{4r})}^{2}+\|\\mu \|^{2}\\int _{\\Omega _{1}}a\|\\Delta v_{1}\|^{2}\\,\\mathrm {d}x\\right).\\end{split}$ On other hand from (REF ) and the transmission conditions we see that $\\begin{split}\\Vert w_{1}\\Vert _{L^{2}(\\Omega _{1})}^{2}+\\Vert w_{2}\\Vert _{L^{2}(\\Omega _{2})}^{2}\\ge \\Vert c_{1}\\Delta u_{1}-\|\\mu \|u_{1}\\Vert _{L^{2}(\\Omega _{1})}^{2}+\\Vert c_{2}\\Delta u_{2}-\|\\mu \|u_{2}\\Vert _{L^{2}(\\Omega _{2})}^{2}-C\\int _{\\Omega _{1}}\\!\\!\\!a\|\\Delta v_{1}\|^{2}\\,\\mathrm {d}x\\\\\\ge -C\\int _{\\Omega _{1}}\\!\\!\\!a\|\\Delta v_{1}\|^{2}\\,\\mathrm {d}x+c_{1}^{2}\\Vert \\Delta u_{1}\\Vert _{L^{2}(\\Omega _{1})}^{2}+c_{2}^{2}\\Vert \\Delta u_{2}\\Vert _{L^{2}(\\Omega _{2})}^{2}+\|\\mu \|^{2}(\\Vert u_{1}\\Vert _{L^{2}(\\Omega _{1})}^{2}+\\Vert u_{2}\\Vert _{L^{2}(\\Omega _{2})}^{2})\\\\+\|\\mu \|(\\Vert \\nabla u_{1}\\Vert _{L^{2}(\\Omega _{1})}^{2}+\\Vert \\nabla u_{2}\\Vert _{L^{2}(\\Omega _{2})}^{2})\\ge \\Vert \\Delta u_{1}\\Vert _{L^{2}(\\Omega _{1})}^{2}+\\Vert \\Delta u_{2}\\Vert _{L^{2}(\\Omega _{2})}^{2}-C\\int _{\\Omega _{1}}\\!\\!\\!a\|\\Delta v_{1}\|^{2}\\,\\mathrm {d}x,\\end{split}$ and by the expression of $v_{1}$ and $v_{2}$ in (REF ) we obtain $\\begin{split}&\\Vert v_{1}\\Vert _{L^{2}(\\Omega _{1})}^{2}\\le \\Vert f_{1}\\Vert _{L^{2}(\\Omega _{1})}^{2}+\|\\mu \|^{2}\\Vert u_{1}\\Vert _{L^{2}(\\Omega _{1})}^{2}\\\\&\\Vert v_{2}\\Vert _{L^{2}(\\Omega _{2})}^{2}\\le \\Vert f_{2}\\Vert _{L^{2}(\\Omega _{2})}^{2}+\|\\mu \|^{2}\\Vert u_{2}\\Vert _{L^{2}(\\Omega _{2})}^{2}.\\end{split}$ Then by combining Proposition REF , and estimates (REF ), (REF ), (REF ) and (REF ) we obtain the results.", "At this step we suppose now that the resolvent estimate (REF ) is not true.", "Then there exist $K_{m}>0$ , $\\mu _{m}\\in \\mathbb {R}$ and a two families $(u_{1,m},u_{2,m},v_{1,m},v_{2,m})\\in \\mathcal {D}(\\mathcal {A})$ and $(f_{1,m},f_{2,m},g_{1,m},g_{2,m})\\in \\mathcal {H}$ , $m=1,2,\\ldots $ such that $\|\\mu _{m}\|\\,\\longrightarrow \\,+\\infty ,\\qquad K_{m}\\,\\longrightarrow \\,+\\infty ,\\qquad \\Vert (u_{1,m},u_{2,m},v_{1,m},v_{2,m})\\Vert _{\\mathcal {H}}=1,$ and $\\mathrm {e}^{K_{m}\|\\mu _{m}\|}(\\mathcal {A}-i\\mu _{m})\\left(\\begin{array}{c}u_{1,m}\\\\u_{2,m}\\\\v_{1,m}\\\\v_{2,m}\\end{array}\\right)=\\left(\\begin{array}{c}f_{1,m}\\\\f_{2,m}\\\\g_{1,m}\\\\g_{2,m}\\end{array}\\right)\\,\\longrightarrow \\,0\\text{ in }\\mathcal {H}.$ This imply that $\\mathrm {e}^{K_{m}\|\\mu _{m}\|}(v_{1,m}-i\\mu _{m}u_{1,m})=f_{1,m}\\,\\longrightarrow \\,0\\text{ in } H^{2}(\\Omega _{1}),\\\\\\mathrm {e}^{K_{m}\|\\mu _{m}\|}(v_{2,m}-i\\mu _{m}u_{2,m})=f_{2,m}\\,\\longrightarrow \\,0\\text{ in } H^{2}(\\Omega _{2}),\\\\\\mathrm {e}^{K_{m}\|\\mu _{m}\|}(-\\Delta (c_{1}^{2}\\Delta u_{1,m}+a\\Delta v_{1,m})-i\\mu _{m}v_{1,m})=g_{1,m}\\,\\longrightarrow \\,0\\text{ in } L^{2}(\\Omega _{1}),\\\\\\mathrm {e}^{K_{m}\|\\mu _{m}\|}(-c_{2}^{2}\\Delta ^{2}u_{2,m}-i\\mu _{m}v_{2,m})=g_{2,m}\\,\\longrightarrow \\,0\\text{ in } L^{2}(\\Omega _{2}).$ From (REF ) and (REF ), we get $\\mathrm {Re}\\left\\langle \\left(\\begin{array}{c}f_{1,m}\\\\f_{2,m}\\\\g_{1,m}\\\\g_{2,m}\\end{array}\\right),\\left(\\begin{array}{c}u_{1,m}\\\\u_{2,m}\\\\v_{1,m}\\\\v_{2,m}\\end{array}\\right)\\right\\rangle _{\\mathcal {H}}=-\\mathrm {e}^{K_{m}\|\\mu _{m}\|}\\int _{\\Omega _{1}}a\|\\Delta v_{1,m}\|^{2}\\,\\mathrm {d}x\\,\\longrightarrow \\,0.$ Then by (REF ) and (REF ), we obtain $\|\\mu _{m}\|^{2}\\mathrm {e}^{\\frac{K_{m}}{2}\|\\mu _{m}\|}\\int _{\\omega }\|\\Delta u_{1,m}\|^{2}\\,\\mathrm {d}x\\,\\longrightarrow \\,0.$ Hence from (REF ) and (REF ) we obtain $\\mathrm {e}^{\\frac{K_{m}}{2}\|\\mu _{m}\|}\\left(\\int _{\\omega }\|\\Delta u_{1,m}\|^{2}\\,\\mathrm {d}x+\\int _{\\omega }\|\\Delta v_{1,m}\|^{2}\\,\\mathrm {d}x\\right)\\,\\longrightarrow \\,0.$ And by (REF ) we have $\\frac{1}{\|\\mu _{m}\|^{2}}\\Vert \\Delta (\\psi .v_{1,m})\\Vert _{L^{2}(\\Omega _{1})}^{2}=O(1),\\quad \\forall \\,\\psi \\in {C}^{\\infty }(\\Omega _{1}).$ Then by multiplying () by $\\mu _{m}^{-1}\\psi .\\overline{v}_{1,m}$ where $\\psi \\in {C}^{\\infty }(\\Omega _{1})$ and $\\mathrm {supp}(\\psi )\\subset \\omega $ we obtain by (REF ) and (REF ) that $\\mathrm {e}^{\\frac{K_{m}}{4}\|\\mu _{m}\|}\\int _{\\omega }\|v_{1,m}\|^{2}\\psi \\,\\mathrm {d}x\\,\\longrightarrow \\,0.$ In particular we obtain that $\\mathrm {e}^{\\frac{K_{m}}{4}\|\\mu _{m}\|}\\int _{B_{4r}}\|v_{1,m}\|^{2}\\,\\mathrm {d}x\\,\\longrightarrow \\,0.$ Then also we get by (REF ) that $\\mathrm {e}^{\\frac{K_{m}}{4}\|\\mu _{m}\|}\\int _{B_{4r}}\|u_{1,m}\|^{2}\\,\\mathrm {d}x\\,\\longrightarrow \\,0.$ Now by applying inequality (REF ) to the system (REF )-() it follows that $\\begin{split}\\Vert \\Delta u_{1,m}\\Vert _{L^{2}(\\Omega _{1})}^{2}+\\Vert \\Delta u_{2,m}\\Vert _{L^{2}(\\Omega _{2})}^{2}+\\Vert v_{1,m}\\Vert _{L^{2}(\\Omega _{1})}^{2}+\\Vert v_{2,m}\\Vert _{L^{2}(\\Omega _{2})}^{2}\\le \\\\C\\mathrm {e}^{C\|\\mu _{m}\|}\\bigg (\\mathrm {e}^{-2K_{m}\|\\mu _{m}\|}\\Big (\\Vert \\Delta f_{1,m}\\Vert _{L^{2}(\\Omega _{1})}^{2}+\\Vert \\Delta f_{2,m}\\Vert _{L^{2}(\\Omega _{2})}^{2}+\\Vert g_{1,m}\\Vert _{L^{2}(\\Omega _{1})}^{2}+\\Vert g_{2,m}\\Vert _{L^{2}(\\Omega _{2})}^{2}\\Big )\\\\+\\mathrm {e}^{-\\frac{K_{m}}{4}\|\\mu _{m}\|}\\left(\\int _{\\Omega _{1}}a\|\\Delta v_{1,m}\|^{2}\\,\\mathrm {d}x+\\int _{B_{4r}}\|u_{1,m}\|^{2}\\,\\mathrm {d}x\\right)\\mathrm {e}^{\\frac{K_{m}}{4}\|\\mu _{m}\|}\\bigg ).\\end{split}$ While the right hand side of (REF ) go to zero as $m\\,\\longrightarrow \\,+\\infty $ by (REF )-(REF ) and estimates (REF ) and (REF ), then we obtain a contradiction with (REF ).", "And this conclude the proof of the resolvent estimate.", "tocsectionReferences" ] ]	1403.0356
[ [ "Unbiased spin-dependent Parton Distribution Functions" ], [ "Abstract We present the first unbiased determination of spin-dependent, or polarized, Parton Distribution Functions (PDFs) of the proton.", "A statistically sound representation of the corresponding uncertainties is achieved by means of the NNPDF methodology: this was formerly developed for unpolarized distributions and is now generalized to the polarized here for the first time.", "The features of the procedure, based on robust statistical tools (Monte Carlo sampling for error propagation, neural networks for PDF parametrization, genetic algorithm for their minimization, and possibly reweighting for including new data samples without refitting), are illustrated in detail.", "Different sets of polarized PDFs are obtained at next-to-leading order accuracy in perturbative quantum chromodynamics, based on both fixed-target inclusive deeply-inelastic scattering data and the most recent polarized collider data.", "A quantitative appraisal on the potential role of future measurements at an Electron-Ion Collider is also presented.", "We study the stability of our results upon the variation of several theoretical and methodological assumptions and we present a detailed investigation of the first moments of our polarized PDFs, compared to other recent analyses.", "We find that the uncertainty on the gluon distribution from available data was substantially underestimated in previous determinations; in particular, we emphasize that a large contribution to the gluon may arise from the unmeasured small-x region, against the common belief that this is actually rather small.", "We demonstrate that an Electron-Ion Collider would provide evidence for a possible large gluon contribution to the nucleon spin, though with a sizable residual uncertainty." ], [ "Abstract", "We present the first unbiased determination of spin-dependent, or polarized, Parton Distribution Functions (PDFs) of the proton.", "A statistically sound representation of the corresponding uncertainties is achieved by means of the NNPDF methodology: this was formerly developed for unpolarized distributions and is now generalized to the polarized here for the first time.", "The features of the procedure, based on robust statistical tools (Monte Carlo sampling for error propagation, neural networks for PDF parameterization, genetic algorithm for their minimization, and possibly reweighting for including new data samples without refitting), are illustrated in detail.", "Different sets of polarized PDFs are obtained at next-to-leading order accuracy in perturbative quantum chromodynamics, based on both fixed-target inclusive deeply-inelastic scattering data and the most recent polarized collider data.", "A quantitative appraisal on the potential role of future measurements at an Electron-Ion Collider is also presented.", "We study the stability of our results upon the variation of several theoretical and methodological assumptions and we present a detailed investigation of the first moments of our polarized PDFs, compared to other recent analyses.", "We find that the uncertainty on the gluon distribution from available data was substantially underestimated in previous determinations; in particular, we emphasize that a large contribution to the gluon may arise from the unmeasured small-x region, against the common belief that this is actually rather small.", "We demonstrate that an Electron-Ion Collider would provide evidence for a possible large gluon contribution to the nucleon spin, though with a sizable residual uncertainty.", "Refereed publications R. D. Ball et al., Polarized Parton Distributions at an Electron-Ion Collider, Phys. Lett.", "B728 (2014) 524 [arXiv:1310.0461] DOI: 10.1016/j.physletb.2013.12.023 R. D. Ball et al., Unbiased determination of polarized parton distributions and their uncertainites, Nucl. Phys.", "B874 (2013) 36 [arXiv:1303.7236] DOI: 10.1016/j.nuclphysb.2013.05.007 M. Anselmino, M. Boglione, U.", "D'Alesio, S. Melis, F. Murgia, E. R. Nocera and A. Prokudin, General Helicity Formalism for Polarized Semi-Inclusive Deep Inelastic Scattering, Phys. Rev.", "D83 (2011) 114019 [arXiv:1101.1011] DOI: 10.1103/PhysRevD.83.114019 Publications in preparation R. D. Ball et al., A first unbiased global extraction of polarized parton distributions Publications in conference proceedings E. R. Nocera, Constraints on polarized parton distributions from open charm and W production data, PoS DIS2013 (2013) 211 [arXiv:1307.0146] E. R. Nocera, Inclusion of $W^\\pm $ single-spin asymmetry data in a polarized PDF determination via Bayesian reweighting, Nuovo Cim.", "C36 (2013) 143-147 [arXiv:1302.6409] DOI: 10.1393/ncc/i2013-11592-4 N. P. Hartland and E. R. Nocera, A Mathematica interface to NNPDFs, Nucl.", "Phys.", "Proc.", "Suppl.", "224 (2013) 54-57 [arXiv:1209.2585] DOI: 10.1016/j.nuclphysbps.2012.11.013 E. R. Nocera, S. Forte, G. Ridolfi and J. Rojo, Unbiased Polarized Parton Distributions and their Uncertainties, Proceedings of 20th International Workshop on Deep-Inelastic Scattering and Related Subjects (DIS 2012), p.937-942 [arXiv:1206.0201] DOI:10.3204/DESY-PROC-2012-02/273 I would like to thank all those people who have supported me in the course of my work as a graduate student towards my Ph.D., particularly during the completion of this Thesis.", "I apologize in advance for missing some of them.", "First of all, I am grateful to my supervisor, prof. Stefano Forte, who set an example to me of a scientist and a teacher.", "In particular, I acknowledge his wide expertise in physics, his patient willingness, his inexhaustible enthusiasm and his illuminating ideas, from which I had the opportunity to benefit (and learn) almost every day in the last three years.", "Second, I am in debt to dr. Juan Rojo, my co-supervisor, for having introduced me to the NNPDF methodology, for continuous assistance with issues about code writing and for encouragement in pursuing my research with dedication.", "I owe my gratitude to him also for the opportunity he will give me to spend a couple of months in Oxford soon.", "To both Stefano and Juan I give my thanks for their training not only in undertaking scientific research, but also in presenting results: in particular, I very much appreciated (and benefitted from) their advices on talks and proceedings prepared for conferences in which I took part as a speaker.", "Besides, I would like to thank prof. Richard D. Ball, who accepted to be the referee of this Thesis, and had carefully reviewed the manuscript: it has much improved thanks to his corrections and suggestions.", "I thank as well the other two external members of the final examination committee, prof. Mauro Anselmino and prof. Giovanni Ridolfi.", "Additional thanks are due to: the Ph.D. school board, for extra financial support provided for housing costs during my stay in Milan; the Laboratorio di Calcolo & Multimedia (LCM) staff, for the computational resources provided to undertake the numerical analyses presented in this Thesis; dr. Francesco Caravaglios, who has allowed for sharing with me not only his office, but also his personal ideas on physics, mathematics, metaphysics, biology, chemistry: though looking weird at first, they finally reveal to me his genius.", "I thank all those people I met during the last three years who also taught me a lot with their experience and knowledge.", "I would like to mention the members of the Department of Physics at the University of Milan, particularly those of the Theoretical Division who participated in the weekly journal club: Daniele Bettinelli, Giuseppe Bozzi, Giancarlo Ferrera, Alessandro Vicini.", "I also thank Mario Raciti for giving me the opportunity to work as a teaching assistant to his course in General Physics for bachelor students in Comunicazione Digitale.", "I aknowledge several physicists I met at conferences and workshops, who raised me in the spin physics community: in particular Alessandro Bacchetta, Elena Boglione, Aurore Courtoy, Isabella Garzia, Francesca Giordano, Delia Hasch, Stefano Melis, Barbara Pasquini, Alexei Prokudin, Marco Radici, Ignazio Scimemi.", "I very much appreciated the friendly and cheerful environment in the Milan Ph.D. school, thanks to all my colleagues fellow students.", "I just mention Alberto, Alice, Elena, Elisa, Rosa, Sofia.", "Special thanks are deserved by Stefano Carrazza, who has rapidly become for me an example of efficiency and hard work and a friend.", "I really enjoyed our discussions not only about physics and NNPDF code, but also about motorbikes and trekking.", "I will never forget our journey from Milan to Marseille to attend DIS2013 nor our excursion to Montenvers, Grand Balcon Nord and Plan de l'Aiguille.", "During my stay in Milan I was housed in Centro Giovanile Pavoniano: I would like to thank the directorate, in particular Fr.", "Giorgio, for his kind hospitality, as well as all other guests who contributed to a friendly and enjoyable atmosphere.", "Finally, I thank my family, in particular my parents, for their support: even though they stopped understanding what I am doing long ago, they have given me the chance to look at the world with curiosity and love.", "Emanuele R. Nocera" ], [ "Introduction", "The investigation of the internal structure of nucleons is an old and intriguing problem which dates back to almost fifty years ago.", "For the past few decades, physicists have been able to describe with increasing details the fundamental particles that constitute protons and neutrons, which actually make up all nuclei and hence most of the visible matter in the Universe.", "This understanding is encapsulated in the Standard Model, supplemented with pertubartive Quantum Chromodynamics (QCD), the field theory which currently describes the strong interaction between the nucleon's fundamental constituents, quarks and gluons.", "It is a remarkable property of QCD, known as confinement, that these are not seen in isolation, but only bound to singlet states of the their respective strong color charge.", "Protons and neutrons are spin one-half bound states.", "Spin is one of the most fundamental concepts in physics, deeply rooted in Poincaré invariance and hence in the structure of space-time itself.", "The elementary constituents of the nucleon carry spin, quarks are spin one-half particles and gluons are spin-one particles.", "It is worth recalling that the discovery of the fact that the proton has structure - and hence really the birth of strong interaction physics - was due to spin, through the measurement of a very unexpected anomalous magnetic moment of the proton by O. Stern and collaborators in 1933 [2].", "After decades of ever more detailed studies of nucleon structure, the understanding of the observed spin of the nucleon in terms of their constituents is a major challenge, far from being succesfully achieved.", "The current picture of the nucleon structure is the result of more than half a century of theoretical and experimental efforts from physicists around the world.", "Even though a detailed historical overview is beyond the scope of this introduction, we find it useful to summarize the main steps in the building of our knowledge on the proton structure, with some emphasis on its spin.", "Quarks were originally introduced in 1963 by Gell-Mann, Ne'eman and Zweig, simply based on symmetry considerations [3], [4], [5], [6], in an attempt to bring order into the large array of strongly-interacting particles observed in experiment.", "In a few words, they recognised that the known hadrons could be associated to some representations of the special unitary $SU(3)$ group.", "This led to the concept of quarks as the building blocks of hadrons.", "Mesons were expected to be quark-antiquark bound states, while baryons were interpreted as bound states of three quarks.", "In Nature there are no indications of the existence of other multiquark states: in order to explain this fundamental evidence and to satisfy the Pauli exclusion principle for baryons, such as the $\\Delta ^{++}$ or the $\\Omega ^-$ which are made up of three quarks of the same flavor, the spin-one-half quarks had to carry a new quantum number [7], later termed colour.", "The modern version of this constituent quark model still successfully describes most of the qualitative features of the baryon spectroscopy.", "A modern realization of Rutherford's experiment has shown us that quarks are real.", "This experiment is the deeply-inelastic scattering (DIS) of electrons (and, later, other leptons, including positrons, muons and neutrinos) off the nucleon, a program that was started in the late 1960's at SLAC [8] (for a review see also Ref. [9]).", "A high-energy lepton interacts with the nucleon, via exchange of a highly virtual gauge boson.", "For a virtuality of $Q^2 > 1$ GeV$^2$ , distances shorter than $0.2$ fm are probed in the proton.", "The early DIS results led to an interpretation as elastic scattering of the lepton off pointlike, spin-one-half, constituents of the nucleon [10], [11], [12], [13], called partons.", "At first, this was understood in the so-called parton model: in this model, the nucleon is observed in the so-called infinite momentum frame, a Lorentz frame in which it is moving with large four-momentum: partons are assumed to move collinearly to the parent hadron, hence their transverse momenta and masses can be neglected.", "Lepton-nucleon scattering is then described in the impulse approximation, i.e.", "partons are treated as free particles and all partons' self-interactions are neglected.", "In the impulse approximation, lepton-nucleon scattering is simply the incoherent sum of lepton interactions with the individual partons in the nucleon, which are carrying a fraction $x$ of its four-momentum.", "These interactions can be computed in perturbation theory, and have to be weighted with the probability that the nucleon contains a parton with the proper value of $x$ .", "This probability, denoted as $f_{q/p}(x)$ , encodes the momentum density of any parton species $q$ , with longitudinal fraction $x$ , in a nucleon $p$ , and is called Parton Distribution Function (PDF).", "This cannot be computed using perturbative theory, since it depends on the non-perturbative process that determines the structure of the nucleon; hence, it has to be determined from the experiment.", "Partons carrying fractional electric charge were subsequently identified with the quarks.", "The existence of gluons was proved indirectly from a missing ($\\sim 50\\%$ ) contribution [14], [15] to the proton momentum not accounted for by the quarks.", "Later on, direct evidence for gluons was found in three-jet production in electron-positron annihilation [16], [17], [18].", "From the observed angular distributions of the jets it became clear that gluons have spin one [19], [20].", "The successful parton interpretation of DIS assumed that partons are almost free (i.e., non-interacting) on the short time scales set by the high virtuality of the exchanged photon.", "This implied that the underlying theory of the strong interactions must actually be relatively weak on short time or, equivalently, distance scales [21].", "In a groundbreaking development, Gross, Wilczek and Politzer showed in 1973 that the non-abelian theory of quarks and gluons, QCD, which had just been developed a few months earlier [22], [23], [24], possessed this remarkable feature of asymptotic freedom [25], [26], a discovery for which they were awarded the 2004 Nobel Prize for Physics.", "The interactions of partons at short distances, while weak in QCD, were then predicted to lead to visible effects in the experimentally measured DIS structure functions known as scaling violations [27], [28].", "These essentially describe the response of the partonic structure of the proton to the resolving power of the virtual photon, set by its virtuality $Q^2$ .", "The greatest triumph of QCD is arguably the prediction of scaling violations, which have been observed experimentally and verified with great precision.", "Deeply-inelastic scattering thus paved the way for QCD.", "Over the following two decades or so, studies of nucleon structure became ever more detailed and precise.", "This was partly due to increased luminosities and energies of lepton machines, eventually culminating in the HERA electron-proton collider [29].", "Also, hadron colliders entered the scene.", "It was realized, again thanks to asymptotic freedom and factorization, which follows from it, that the partonic structure of the nucleon seen in DIS is universal, in the sense that a variety of sufficiently inclusive hadron collider processes, characterized by a large scale, admit a factorized description [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41].", "This offered the possibility of learning about other aspects of nucleon structure (and hence, QCD), for instance about its gluon content which is not primarily accessed in DIS.", "Being known with more precision, nucleon structure also allowed for new physics studies at hadron colliders, the outstanding example perhaps being the discovery of the $W^\\pm $ and $Z^0$ bosons at CERN's $Sp\\bar{p}S$ collider [42], [43], [44].", "The Tevatron and the Large Hadron Collider (LHC) are the most recent continuations of this line of research, which has culminated in the discovery of the Higgs boson [45], [46], announced by ATLAS and CMS collaborations on the $4^{\\mathrm {th}}$ July 2012.", "Concerning spin physics, a milestone in the study of the nucleon was the advent of polarized electron beams in the early seventies [47].", "This later on allowed for DIS measurements with polarized lepton beams and nucleon targets [48], and offered the possibility of studying whether quarks and antiquarks show on average preferred spin directions inside a polarized nucleon.", "The program of polarized DIS has been continuing ever since and it is now a successful branch of particle physics.", "Its most important result is the finding that quark and antiquark spins provide an anomalously small - only about $20\\%-30\\%$ - amount of the proton spin [49], [50], firstly observed by the EMC experiment in the late 1980's.", "This finding, which opened a spin crisis in the understanding of the nucleon structure [51], has raised the interest of physicists in clarifying the potential role played by new candidates to the nucleon's spin, like gluons' polarizations and partons' orbital angular momenta.", "In parallel, there also was a very important line of research on polarization phenomena in hadron-hadron reactions in fixed-target kinematics.", "In particular, unexpectedly large single-transverse spin asymmetries were seen [52], [53], [54], [55], [56].", "In the last decade, the advent of the Relativistic Heavy Ion Collider (RHIC), the first machine to collide polarized proton beams, started to probe the proton spin in new profound ways [57], complementary, but independent, to polarized DIS.", "In particular, more knowledge on the polarization of gluons in the proton and details of the flavor structure of the polarized quarks and antiquarks has been recently achieved, as we will discuss in detail in this Thesis.", "However, despite a flurry of experimental and theoretical activity, a complete and satisfactory understanding of the so-called spin puzzle is still lacking." ], [ "Compelling questions in spin physics", "The information on the proton spin structure is encoded in spin-dependent, or polarized, Parton Distribution Functions (PDFs) of quarks, antiquarks and gluons $\\Delta f_{q/p}(x,Q^2)=f_{q/p}^\\uparrow (x,Q^2) - f_{q/p}^\\downarrow (x,Q^2)\\,\\mbox{,}$ which are the momentum densities of partons $q$ with helicity along ($\\uparrow $ ) or opposite ($\\downarrow $ ) the polarization direction of the parent nucleon $p$ .", "The $Q^2$ dependence of the parton distributions, known as $Q^2$ evolution [58], is quantitatively predictable in perturbative QCD, thanks to asymptotic freedom.", "Physically, it may be thought of the consequence of the fact that partons are observed with higher resolution when they are probed at higher scales; hence it is more likely that a struck quark has radiated one or more gluons so that it is effectively resolved into several partons, each with lower momentum fraction.", "Similarly, a struck quark may have originated from a gluon splitting into a quark-antiquark pair.", "Polarized inclusive, neutral-current, DIS allows one to only access the flavor combinations $\\Delta q^+\\equiv \\Delta q + \\Delta \\bar{q}$ , and the gluon polarization, though the latter is mostly determined indirectly by scaling violations.", "Of particular interest is the singlet quark antiquark combination $\\Delta \\Sigma =\\sum _{q=u,d,s}\\Delta q^+$ , since its integral, known as the singlet axial charge, yields the average of all quark and antiquark contributions to the proton spin: $\\langle S_q\\rangle \\sim \\frac{1}{2}\\int _0^1dx\\Delta \\Sigma (x,Q^2)\\,\\mbox{.", "}$ The anomalously small value observed experimentally for this quantity, from almost three decades of DIS measurement after the EMC result, strenghten the common belief that only about a quarter of the proton spin is carried by quarks and antiquarks.", "The EMC result was followed by an intense scrutiny of the basis of the corresponding theoretical framework, which led to the realization [59], [60] that the perturbative behavior of polarized PDFs deviates from parton model expectations, according to which gluons decouple at large energy scale.", "The almost vanishing value measured by EMC for the singlet axial charge can be explained as a cancellation between a reasonably large quark spin contribution, e.g.", "$\\Delta \\Sigma \\simeq 0.6 - 0.7$ , as expected intuitively, and an anomalous gluon contribution, altering Eq.", "(REF ).", "A large value of the gluon contribution to the proton spin is required to achieve such a cancellation, and QCD predicts that this contribution grows with the energy scale.", "Despite some experimental evidence has suggested that the gluon polarization in the nucleon may be rather small, we emphasize that it is instead still largely uncertain, as we will carefully demonstrate in this Thesis.", "Other candidates for carrying the nucleon spin can come from quark and gluon orbital angular momenta [61], [62], [63] (for a recent discussion on the spin decomposition see also Ref. [64]).", "In any case, the results from polarized inclusive DIS clearly call for further investigations: we summarize in the following some of the outstanding questions to be aswered in spin physics.", "With which accuracy do we know spin-dependent parton distributions?", "The assessment of the singlet and gluon contributions to the proton spin requires in turn a determination of polarized parton distributions from available experimental data.", "In the last decade, several such determinations have been performed at next-to-leading order (NLO) in QCD [65], [66], [67], [68], [69], [70], [71], [72], [73], [74], [75], [76], [77], [78], [79], [80], which is the current state-of-the-art accuracy for polarized fits, mostly based on DIS.", "Some of them also include a significant amount of data other than DIS, namely from semi-inclusive DIS (SIDIS) with identified hadrons in final states [77], [75], [80] or from polarized proton-proton collisions [75].", "However, we notice that they are all based on the standard Hessian methodology for PDF fitting and uncertainty estimation.", "This approach is known [81] to potentially lead to an underestimation of PDF uncertainties, due to the limitations in the linear propagation of errors and, more importantly, to PDF parametrization in terms of fixed functional forms.", "These issues are especially delicate when the experimental information is scarce, like in the case of polarized data.", "In particular, in this Thesis we will clearly demonstrate that a more flexible PDF parametrization is better suited to analyse polarized experimental data without prejudice.", "This will lead to larger, but more faithful, estimates of PDF uncertainties than those obtained in the other available analyses.", "At least, one should conclude that our knowledge of parton's contribution to the nucleon spin is much more uncertain than commonly believed, unless one is willing to make some a priori assumptions on their behavior in the unmeasured kinematic regions.", "The two following questions address in much detail some issues related to quarks and gluons separately.", "How do gluons contribute to the proton spin?", "The interest in an accurate determination of the gluon polarization $\\Delta g(x, Q^2)$ is of particular interest for both phenomenological and theoretical reasons.", "On the phenomenological side, inclusive DIS allows for an indirect determination of the gluon distribution, through scaling violations.", "Since experimental data have a rather limited $Q^2$ lever arm, it follows that $\\Delta g(x, Q^2)$ is only weakly constrained.", "Processes other than inclusive DIS, which receive leading contributions from gluon initiated subprocesses, are better suited to provide direct information on the gluon distribution.", "In particular, these include open-charm production in fixed-target experiments and jet or semi-inclusive production in proton-proton collisions.", "However, the kinematic coverage of these data is limited: hence the integral of the gluon distribution can receive large contributions from the unmeasured region, in particular from the small-$x$ region.", "On the theoretical side, it is a remarkable feature of QCD that the gluon contribution to the nucleon spin may well be significant even at large momentum scales.", "The reason is that the integral of $\\Delta g(x, Q^2)$ evolves as $1/\\alpha _s(Q^2)$ [59], that is, it rises logarithmically with $Q$ .", "This peculiar evolution pattern is a very deep prediction of QCD, related to the so-called axial anomaly.", "It has inspired ideas that a reason for the smallness of the quark spin contribution should be sought in a shielding of the quark spins due to a particular perturbative part of the DIS process $\\gamma ^* g \\rightarrow q\\bar{q}$ [59].", "The associated contributions arise only at order $\\alpha _s(Q^2)$ ; however, the peculiar evolution of the first moment of the polarized gluon distribution would compensate this suppression.", "To be of any practical relevance, a large positive gluon spin contribution, $\\langle \\Delta g\\rangle > 1$ , would be required even at low hadronic scales of a GeV or so.", "A very large polarization of the confining fields inside a nucleon, even though suggested by some nucleon models [82], [83], [84], [85], would be a very puzzling phenomenon and would once again challenge our picture of the nucleon.", "What are the patterns of up, down, and strange quark and antiquark polarizations?", "Inclusive DIS provides information only on the total flavor combinations $\\Delta q^+\\equiv \\Delta q + \\Delta \\bar{q}$ , $q=u,d,s$ .", "Nevertheless, in order to understand the proton helicity structure in detail, one needs to learn about the various quark and antiquark densities, $\\Delta u$ , $\\Delta \\bar{u}$ , $\\Delta d$ , $\\Delta \\bar{d}$ and $\\Delta s$ , $\\Delta \\bar{s}$ separately.", "This also provides an important additional test of the smallness of the quark spin contribution, and could reveal genuine flavor asymmetry $\\Delta \\bar{u}-\\Delta \\bar{d}$ in the proton sea, claimed by some models of nucleon structure [86], [87].", "These predictions are often related to fundamental concepts such as the Pauli principle: since the proton has two valence-$u$ quarks which primarily spin along with the proton spin direction, $u\\bar{u}$ pairs in the sea will tend to have the $u$ quark polarized opposite to the proton.", "Hence, if such pairs are in a spin singlet, one expects $\\Delta \\bar{u}> 0$ and, by the same reasoning, $\\Delta \\bar{d}< 0$ .", "Such questions become all the more exciting due to the fact that rather large unpolarized asymmetries $\\bar{u}-\\bar{d}\\ne 0$ have been observed in DIS and Drell-Yan measurements [88], [89], [90].", "Further fundamental questions concern the strange quark polarization.", "The polarized DIS measurements point to a sizable negative polarization of strange quarks, in line with other observations of significant strange quark effects in nucleon structure.", "What orbital angular momenta do partons carry?", "Quark and gluon orbital angular momenta are the other candidates for the carriers of the proton spin.", "Consequently, theoretical work focused also on these in the years after the spin crisis was announced.", "A conceptual breakthrough was made in the mid 1990s when it was realized [62] that a particular class of off-forward nucleon matrix elements, in which the nucleon has different momentum in the initial and final states, measure total parton angular momentum.", "Simply stated, orbital angular momentum is $\\overrightarrow{r}\\times \\overrightarrow{p}$ , where the operator $r$ can be viewed in Quantum Mechanics as a derivative with respect to momentum transfer.", "Thus, in analogy with the measurement of the Pauli form factor, it takes a finite momentum transfer on the nucleon to access matrix elements with operators containing a factor $r$ .", "It was also shown how these off-forward distributions, referred to as generalized parton distribution functions (GPDs), may be experimentally determined from certain exclusive processes in lepton-nucleon scattering, the prime example being Deeply-Virtual Compton Scattering (DVCS) $\\gamma ^* p\\rightarrow \\gamma p$ [62].", "A major emphasis in current and future experimental activities in lepton scattering is on the DVCS and related reactions.", "What is the role of transverse spin in QCD?", "So far, we have only considered the helicity structure of the nucleon, that is, the partonic structure we find when we probe the nucleon when its spin is aligned with its momentum.", "Experimental probes with transversely polarized nucleons could also be studied, both at fixed-target and collider facilities, and it has been known for a long time now that very interesting spin effects are associated with this in QCD.", "Partly, this is known from theoretical studies which revealed that besides the helicity distributions discussed above, for transverse polarization there is a new set of parton densities, called transversity [91], [92].", "They are defined analogously to Eq.", "(REF ), but now for transversely polarized partons polarized along or opposite to the transversely polarized proton.", "Furthermore, if we allow quarks to have an intrinsic Fermi motion in the nucleon, they can be interpreted in light of more fundamental objects, the so-called Transverse Momentum Dependent parton distribution functions (TMDs) [93], in which the dependence on the intrinsic transverse momentum $k_\\perp $ is made explicit.", "We refer to [94] for a comprehensive review on TMDs and the transverse spin structure of the proton.", "Here, we only mention that the present knowledge of TMDs is comparable to that of PDFs in the early 1970's and very little is known about transversity.", "An intensive experimental campaign is ongoing to take data in polarized SIDIS and to provide a better determination of these distributions [95], [96], [97], [98]." ], [ "Outline of the thesis", "This Thesis addresses the three first questions in the above list, presenting a determination of spin-dependent parton distributions for the proton.", "In particular, two sets are obtained, the first based on inclusive DIS data only, the second also including the most recent data from polarized proton-proton collisions.", "In comparison to other recent analyses, our study is performed within the NNPDF methodology, which makes use of robust statistical tools, including Monte Carlo sampling for error propagation and parametrization of PDFs in terms of neural networks.", "The methodology has been succesfully applied to the unpolarized case [99], [100], [101], [102], [103], [104], [105], [106], [107], [108], [109], [110], [111] with the goal of providing a faithful representation of the PDF underlying probability distribution.", "This is particularly relevant with polarized data, which are rather scarce and, in general, affected by larger uncertainties than those of their unpolarized counterparts.", "Unlike all other standard fits, our parton sets do not suffer from the theoretical bias introduced either by a fixed functional form for PDF parametrization or by quadratic approximation in the Hessian propagation of errors.", "For this reason, we consider our unbiased determination to be crucial for investigating to which extent the common belief that about a quarter of the nucleon spin is carried by quarks and antiquarks, while the gluon contribution is even much smaller, actually holds.", "Our parton determinations are publicly released together with computational tools to use them, including FORTRAN, C++ and Mathematica interfaces.", "Hence they could be used for any phenomenological study of hard scattering processes involving polarized hadrons in initial states.", "We should notice that, in addition to the investigation of the nucleon spin structure, such studies have recently included probes of different beyond-standard-model (BSM) scenarios [112] and possibly the determination of the Higgs boson spin in the diphoton decay channel, by means of the linear polarization of gluons in an unpolarized proton [113].", "In this Thesis, we will not address the study of either TMDs or GPDs, but we notice that the methods ilustrated here apply to the determination of any non-perturbative object from experimental data: hence they could be used to determine such new distributions in the future, in so far as experimental data will reach more and more abundance and accuracy.", "Also, we do not describe either the apparatus which had to be developed to carry out spin physics experiments or the related technical challenges which had to be faced.", "A complete survey on these aspects can be found in Refs.", "[94], [114], [115].", "The outline of this Thesis is as follows.", "Chapter .", "Polarized Deeply-Inelastic Scattering.", "We review the theoretical formalism for the description of DIS with both polarized lepton beams and nuclear targets.", "In particular we derive the expressions for the differential cross-section of the process in terms of polarized structure functions.", "We will restrict our discussion to the contribution arising from the exchange of a virtual photon between the lepton and the nucleon.", "This is indeed sufficient to describe currently available esperimental data, whose energy does not exceed a few hundreds of GeV and none of which come from neutrino beams: hence we do not include either the (suppressed) contribution to neutral-current DIS mediated by a $Z^0$ boson or charged-current DIS mediated by a $W^\\pm $ boson.", "Then, we present the parton model expectations for DIS spin asymmetries introducing helicity-dependent, or polarized, PDFs and we discuss how they are modified in the framework of perturbative QCD.", "We complete our theoretical overview on polarized DIS with a sketch of spin sum rules, a summary of the relevant phenomenological relations between structure functions and measured observables, and the formalism adopted to take into account kinematic target mass corrections.", "Chapter .", "Phenomenology of polarized Parton Distributions.", "We review how a set of PDFs is usually determined from a global fit to experimental data.", "First, we sketch the general strategy for PDF determination and its main theoretical and methodological issues, focusing on those which are peculiar to the polarized case.", "Second, we summarize how some of these problems are addressed within the NNPDF methodology, with the goal of providing a statistically sound determination of PDFs and their uncertainties.", "Finally, we provide an overview on available polarized PDF sets.", "Chapter .", "Unbiased polarized PDFs from inclusive DIS.", "We present the first determination of polarized PDFs based on the NNPDF methodology, NNPDFpol1.0.", "This analysis includes all available data from inclusive, neutral-current, polarized DIS and aims at an unbiased extraction of total quark-antiquark and gluon distributions at NLO accuracy.", "We discuss how the statistical distribution of experimental data is sampled with Monte Carlo generation of pseudodata.", "We provide the details of the QCD analysis and discuss the PDF parametrization in terms of neural networks; we describe the minimization strategy and the peculiarities in the polarized case.", "We present the NNPDFpol1.0 parton set, illustrating its statistical features and its stability upon the variation of several theoretical and methodological assumptions.", "We also compare our results to other recent polarized PDF sets.", "Finally, we discuss phenomenological implications for the spin content of the proton and the test of the Bjorken sum rule.", "The analysis presented in this Chapter has been published by the NNPDF collaboration as a refereed paper [116].", "Chapter .", "Polarized PDFs at an Electron-Ion Collider.", "We investigate the potential impact of inclusive DIS data from a future Electron-Ion Collider (EIC) on the determination of polarized PDFs.", "After briefly motivating our study, we illustrate which EIC pseudodata sets we use in our analysis and how the fitting procedure needs to be optimized.", "Resulting PDFs are compared to NNPDFpol1.0 throughout.", "Finally, we reassess their first moments and we give an estimate of the charm contribution to the $g_1$ structure function of the proton at an EIC.", "The analysis presented in this Chapter has been published by the NNPDF Collaboration as a refereed paper [117].", "Chapter .", "Global determination of unbiased polarized PDFs.", "We extend the analysis presented in Chap.", "in order to include in our parton set, on top of inclusive DIS data, also recent measurements of open-charm production in fixed-target DIS, and of jet and $W$ production in polarized proton-proton collisions.", "Hence, we present the first global determination of polarized PDFs based on the NNPDF methodology: NNPDFpol1.1.", "After motivating our analysis, we review the theoretical description of the new processes and present the features of the relative experimental data we include in our study.", "We then turn to a detailed discussion of the way the NNPDFpol1.1 parton set is obtained via Bayesian reweighting of prior PDF Monte Carlo ensembles, followed by unweighting.", "We also present its main features in comparison to NNPDFpol1.0.", "Finally, we discuss some phenomenological implications for the spin content of the proton, based on our new polarized parton set.", "The analysis discussed in this Chapter has been presented in preliminary form in Refs.", "[118], [119].", "Chapter .", "Conclusions and outlook.", "We will draw our conclusions, highlighting the main results presented in this Thesis.", "We also provide an outlook on future possible developments in the determination of polarized parton distributions within the NNPDF methodology.", "Appendix .", "Statistical estimators.", "We collect the definitions of the statistical estimators used in the NNPDF analyses presented in Chaps. --.", "Despite they were already described in Refs.", "[120], [99], [104], we find it useful to give them for completeness and ease of reference here.", "Appendix .", "A Mathematica interface to NNPDF parton sets.", "We present a package for handling both unpolarized and polarized NNPDF parton sets within a Mathematica notebook file.", "This allows for performig PDF manipulations easily and quickly, thanks to the powerful features of the Mathematica software.", "The package was tailored to the users who are not familiar with FORTRAN or C++ programming codes, on which the standard available PDF interface, LHAPDF [121], [122], is based.", "However, since our Mathematica package includes all the features available in the LHAPDF interface, any user can benefit from the interactive usage of PDFs within Mathematica.", "The Mathematica interface to NNPDF parton sets appeared as a contribution to conference proceedings in Ref. [123].", "Appendix .", "The FONLL scheme for $g_1(x,Q^2)$ up to $\\mathcal {O}(\\alpha _s)$ .", "We collect the relevant explicit formulae for the practical computation of the polarized DIS structure function of the proton, $g_1^p(x,Q^2)$ , within the FONLL approach [124] up to $\\mathcal {O}(\\alpha _s)$ .", "In particular, we will restrict to the heavy charm quark contribution $g_{1}^{p,c}$ to the polarized proton structure function $g_1$ , which might be of interest for studies at an Electron-Ion Collider in the future, as mentioned in Chap.", "." ], [ "Polarized Deeply-Inelastic Scattering", "This Chapter is devoted to a detailed discussion of Deeply-Inelastic Scattering (DIS) with both polarized lepton beams and nuclear targets.", "In particular, we will focus on neutral-current DIS, limited to the kinematic regime in which the exchange of a virtual photon between the lepton and the nucleon provides the leading contribution to the process.", "In Sec.", ", we rederive the expression for the differential cross-section of polarized DIS in terms of polarized structure functions.", "We present the naive parton model expectations for spin asymmetries in Sec.", "REF and we discuss how they should be modified in the framework of QCD in Sec.", "REF .", "We complete our theoretical overview on polarized DIS with a sketch of spin sum rules in Sec. .", "Finally, we summarize the relevant phenomenological relations between structure functions and measured observables in Sec.", ", and the formalism adopted to take into account kinematic target mass corrections in Sec.", "." ], [ "General formalism", "Let us consider the inclusive, neutral-current, inelastic scattering of a polarized lepton (electron or muon) beam off a polarized nucleon target, $l(\\ell ) + N(P) \\rightarrow l^\\prime (\\ell ^\\prime ) + X(P_X)\\,\\mbox{,}$ where the four-momenta of the incoming (outgoing) lepton $l$ ($l^\\prime $ ), the nucleon target $N$ and the undetected final hadronic system $X$ are labelled as $\\ell $ ($\\ell ^\\prime $ ), $P$ , and $P_X$ respectively.", "If the momentum transfer involved in the reaction is much smaller than the $Z^0$ boson mass, as it is customary at polarized DIS facilities, the only sizable contribution to the process is given by the exchange of a virtual photon, see Fig.", "REF .", "Figure: Virtual-photon-exchange contribution to neutral-current DIS.In order to work out the kinematics, we denote the nucleon mass, $M$ , the lepton mass, $m_{\\ell }$ , the covariant spin four-vector of the incoming (outgoing) lepton $s_\\ell $ ($s_{\\ell ^\\prime }$ ) and the spin four-vector of the nucleon, $S$ .", "In the target rest frame, we define the four-momenta to be $\\begin{array}{rcll}\\ell & = & (E, \\mathbf {\\ell }) & \\mbox{incoming lepton,} \\\\\\ell ^{\\prime } & = & (E^{\\prime }, \\mathbf {\\ell ^{\\prime }}) & \\mbox{outgoing lepton,} \\\\P & = & (M, \\mathbf {0}) & \\mbox{proton.", "}\\end{array}$ The deeply inelastic regime is identified by the invariant mass $W$ of the final hadronic system to be much larger than the nucleon mass, namely $W^2=M^2+Q^2\\frac{1-x}{x}\\gg M^2\\,\\mbox{.", "}$ This allows us to neglect all masses and to use the approximation $\\ell ^2 = {\\ell ^{\\prime }}^2 \\approx 0 \\mbox{, } \\ \\ \\ E \\approx \|\\mathbf {\\ell }\| \\mbox{, } \\ \\ \\ E^{\\prime } \\approx \|\\mathbf {\\ell ^{\\prime }}\| \\mbox{. }", "\\ \\ \\ $ Based on these assumptions, only two kinematical variables (besides the centre of mass energy $s=(\\ell + P)^2$ or, alternatively, the lepton beam energy $E$ ) are needed to describe the process in Eq.", "(REF ).", "They can be chosen among the following invariants: $Q^2 & = & -q^2=(\\ell - \\ell ^{\\prime })^2 = 2EE^{\\prime }(1-\\cos \\theta )=4EE^{\\prime } \\sin ^2 \\left(\\frac{\\theta }{2}\\right)\\\\& & \\mbox{the laboratory-frame photon square momentum,}\\nonumber \\\\\\nu & = & E-E^{\\prime } = \\frac{P \\cdot q}{M}\\\\& & \\mbox{the laboratory-frame photon energy,}\\nonumber \\\\x & = & \\frac{Q^2}{2P \\cdot q} = \\frac{Q^2}{2M\\nu }\\\\& & \\mbox{the Bjorken scaling variable,}\\nonumber \\\\y & = & \\frac{P \\cdot q}{P \\cdot \\ell } = \\frac{\\nu }{E}\\\\& & \\mbox{the energy fraction lost by the incoming lepton $\\ell $,}\\nonumber $ where $\\theta $ is the scattering angle between the incoming and the outgoing lepton beams.", "The differential cross-section for lepton-nucleon scattering then reads $d^3\\sigma (\\ell N \\rightarrow \\ell ^{\\prime } X)=\\frac{1}{2s} \\frac{d^3 \\ell ^{\\prime }}{(2\\pi )^3 2E^{\\prime }}\\sum _{s_{\\ell ^{\\prime }}}\\sum _{X} \\int d\\Pi _X\|\\mathcal {M}(\\ell N \\rightarrow \\ell ^{\\prime } X)\|^2\\,\\mbox{,}$ where $\\int d \\Pi _X= \\int \\frac{d^3 \\mathbf {P}_X}{(2\\pi )^3 2E_X } (2\\pi )^4\\delta ^4(P+q-P_X)$ is the phase-space factor for the unmeasured hadronic system and $\|\\mathcal {M}(\\ell N \\rightarrow \\ell ^{\\prime } X)\|^2& = &\\frac{e^4}{q^4}\\left[\\bar{u}(\\ell ,s_{\\ell }) \\gamma ^{\\nu } u (\\ell ^{\\prime }, s_{\\ell ^{\\prime }})\\bar{u}(\\ell ^{\\prime }, s_{\\ell ^{\\prime }})\\gamma ^{\\mu } u (\\ell , s_\\ell )\\right]\\nonumber \\\\& \\times &\\left[\\langle P,S \| J_{\\nu }^{\\dag }(q) \| P_X \\rangle \\langle P_X \| J_{\\mu }(q) \| P,S \\rangle \\right]$ is the squared amplitude including the Fourier transform of the quark electromagnetic current $J^\\mu (q)$ flowing through the hadronic vertex.", "Since we are describing the scattering of polarized leptons on a polarized target, with no measurement of the outgoing lepton polarization nor of the final hadronic system, in Eq.", "(REF ) we must sum over the final lepton spin $s_{\\ell ^{\\prime }}$ and over all final hadrons $X$ , but must not average over the initial lepton spin, nor sum over the nucleon spin.", "It is customary to define the leptonic tensor $L^{\\mu \\nu }=\\sum _{s_{\\ell ^{\\prime }}}\\left[\\bar{u}(\\ell ,s_{\\ell }) \\gamma ^{\\nu } u (\\ell ^{\\prime }, s_{\\ell ^{\\prime }})\\bar{u}(\\ell ^{\\prime }, s_{\\ell ^{\\prime }})\\gamma ^{\\mu } u (\\ell , s_\\ell )\\right]$ and the hadronic tensor $W_{\\mu \\nu }=\\frac{1}{2\\pi }\\sum _X\\int d\\Pi _X\\left[\\langle P,S \| J_{\\nu }^{\\dag }(q) \| P_X \\rangle \\langle P_X \| J_{\\mu }(q) \| P,S \\rangle \\right]$ in order to rewrite Eq.", "(REF ) as $d^3\\sigma =\\frac{1}{2s}\\frac{e^4}{Q^4}2\\pi L^{\\mu \\nu }W_{\\mu \\nu }\\frac{d^3 \\mathbf {\\ell ^{\\prime }}}{(2\\pi )^3 2E^{\\prime }}$ or, in the target rest frame, where $s=2ME$ , and considering $d^3\\ell ^{\\prime }=E^{\\prime 2} dE^{\\prime } d\\Omega $ , $d\\Omega =d\\cos \\theta d\\varphi $ , $\\frac{d^3\\sigma }{d\\Omega dE^{\\prime }}=\\frac{\\alpha _{em}^2}{2MQ^4} \\frac{E^{\\prime }}{E} L^{\\mu \\nu }W_{\\mu \\nu }\\,\\mbox{.", "}$ This is the differential cross-section for finding the scattered lepton in solid angle $d\\Omega $ with energy $(E^\\prime ,E^\\prime +dE^\\prime )$ usually quoted in the literature (see for example Ref. [125]).", "In Eq.", "(REF ), $\\alpha _{em}$ is the fine-structure electromagnetic constant, while $\\varphi $ is the azimuthal angle of the outgoing lepton.", "The variables $E^{\\prime }$ and $\\theta $ are natural ones, in that they are measured in the laboratory frame, by detecting the scattered lepton.", "However, it is more convenient to perform the variable transformation $(E^{\\prime },\\theta ) \\rightarrow (x,y)$ and to express the differential cross-section in terms of the latter quantities as $\\frac{d^3\\sigma }{dx dy d\\varphi }=\\frac{\\alpha ^2_{em} y}{2 Q^4} L^{\\mu \\nu }W_{\\mu \\nu }\\,\\mbox{,}$ since these are gauge-invariant and dimensionless.", "In a completely general way, the leptonic tensor $L^{\\mu \\nu }$ can be decomposed into a symmetric and an antisymmetric part under $\\mu \\leftrightarrow \\nu $ interchange $L_{\\mu \\nu }=L_{\\mu \\nu }^{(S)}(\\ell , \\ell ^{\\prime }) + iL_{\\mu \\nu }^{(A)}(\\ell ,s_{\\ell }, \\ell ^{\\prime })\\,\\mbox{.", "}$ Recalling the identity satisfied by the spinor $u(p,s)$ , for a fermion with polarization vector $s^{\\mu }$ , $u(p,s)\\bar{u}(p,s)=({p} + m) \\frac{1}{2}(1+\\gamma _5 {s})\\,\\mbox{,}$ and summing only on $s_{\\ell ^{\\prime }}$ , the leptonic tensor reads $L_{\\mu \\nu }=\\mbox{Tr}\\left\\lbrace \\gamma _{\\mu } ({\\ell ^{\\prime }}+m_{\\ell })\\gamma _{\\nu } ({\\ell }+m_{\\ell })\\frac{1}{2}(1+\\gamma _5 {s}_{\\ell })\\right\\rbrace \\,\\mbox{.", "}$ Trace computation via Dirac algebra finally leads to (retaining lepton masses) $L_{\\mu \\nu }^{(S)}=2\\left[\\ell _{\\mu } \\ell ^{\\prime }_{\\nu } + \\ell _{\\nu } \\ell ^{\\prime }_{\\mu } - g_{\\mu \\nu } (\\ell \\cdot \\ell ^{\\prime } - m_{\\ell }^2)\\right]\\,\\mbox{,}$ $L_{\\mu \\nu }^{(A)}=2 m_{\\ell } \\epsilon _{\\mu \\nu \\rho \\sigma } s_{\\ell }^{\\rho }(\\ell - \\ell ^{\\prime })^{\\sigma }\\,\\mbox{.", "}$ If the incoming lepton is longitudinally polarized, its spin vector can be expressed as $s_{\\ell }^{\\mu }=\\frac{\\lambda _{\\ell }}{m_{\\ell }} (\|\\mathbf {\\ell }\|, \\hat{\\mathbf {\\ell }} E), \\ \\ \\hat{\\mathbf {\\ell }}=\\frac{\\mathbf {\\ell }}{\|\\mathbf {\\ell }\|}\\,\\mbox{,}$ i.e.", "it is parallel ($\\lambda _\\ell =+1$ ) or antiparallel ($\\lambda _\\ell =-1$ ) to the direction of motion ($\\lambda _\\ell =\\pm 1$ is twice the lepton helicity).", "Then, Eq.", "(REF ) reads $L_{\\mu \\nu }^{(A)}=2 \\lambda _{\\ell } \\epsilon _{\\mu \\nu \\rho \\sigma } \\ell ^{\\rho }(\\ell - \\ell ^{\\prime })^{\\sigma }=2\\lambda _{\\ell } \\epsilon _{\\mu \\nu \\rho \\sigma } \\ell ^{\\rho } q^{\\sigma }\\,\\mbox{.", "}$ Notice that the lepton mass $m_{\\ell }$ appearing in Eq.", "(REF ) has been cancelled by the denominator in Eq.", "(REF ), which refers to a longitudinally polarized lepton.", "In contrast, if it is transversely polarized, that is, $s_{\\ell }^{\\mu }=s_{\\ell \\perp }^{\\mu }$ , no such cancellation occurs and the corresponding contribution is suppressed by a factor $m_{\\ell }/E$ ." ], [ "Hadronic tensor", "The hadronic tensor $W_{\\mu \\nu }$ allows for a decomposition analogous to Eq.", "(REF ), that is $W_{\\mu \\nu }=W_{\\mu \\nu }^{(S)} (q,P)+iW_{\\mu \\nu }^{(A)} (q;P,S)\\,\\mbox{,}$ where the symmetric and antisymmetric parts can be expressed in terms of two pairs of structure functions, $W_1$ , $W_2$ and $G_1$ , $G_2$ , as $\\frac{1}{2M} W_{\\mu \\nu }^{(S)}& = &\\left(-g_{\\mu \\nu } + \\frac{q_{\\mu } q_{\\nu }}{q^2}\\right)W_1(P \\cdot q, q^2)\\nonumber \\\\& + &\\frac{1}{M^2}\\left(P_{\\mu } - \\frac{P \\cdot q}{q^2} q_{\\mu }\\right)\\left(P_{\\nu } - \\frac{P \\cdot q}{q^2} q_{\\nu }\\right)W_2(P \\cdot q, q^2)\\,\\mbox{,}$ $\\frac{1}{2M} W_{\\mu \\nu }^{(A)}& = &\\epsilon _{\\mu \\nu \\rho \\sigma } q^{\\rho }\\Big {\\lbrace }MS^{\\sigma }G_1(P \\cdot q, q^2)\\nonumber \\\\& + &\\frac{1}{M}\\left[P \\cdot q S^{\\sigma } - S \\cdot q P^{\\sigma }\\right]G_2(P \\cdot q, q^2)\\Big {\\rbrace }\\,\\mbox{.", "}$ It is customary to introduce the dimensionless structure functions $F_1(x,Q^2) \\equiv MW_1(\\nu , Q^2)\\,\\mbox{,}\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ F_2(x,Q^2) \\equiv \\nu W_2 (\\nu , Q^2)\\,\\mbox{,}$ $g_1(x,Q^2) \\equiv M^2 \\nu G_1(\\nu , Q^2)\\,\\mbox{,}\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ g_2(x,Q^2) \\equiv M \\nu ^2 G_2 (\\nu , Q^2)\\,\\mbox{,}$ and to rewrite the symmetric and antisymmetric parts of the hadronic tensor as $W_{\\mu \\nu }^{(S)}& = &2\\left(-g_{\\mu \\nu } + \\frac{q_{\\mu } q_{\\nu }}{q^2}\\right)F_1(x, Q^2)\\nonumber \\\\& + &\\frac{2}{P \\cdot q}\\left(P_{\\mu } - \\frac{P \\cdot q}{q^2} q_{\\mu }\\right)\\left(P_{\\nu } - \\frac{P \\cdot q}{q^2} q_{\\nu }\\right)F_2(x, Q^2)\\,\\mbox{,}$ $W_{\\mu \\nu }^{(A)}=\\frac{2M\\epsilon _{\\mu \\nu \\rho \\sigma } q^{\\rho }}{P \\cdot q}\\left\\lbrace S^{\\sigma } g_1 (x, Q^2)+\\left[S^{\\sigma } - \\frac{S \\cdot q}{P \\cdot q} P^{\\sigma }\\right]g_2 (x,Q^2)\\right\\rbrace \\,\\mbox{.", "}$ These expressions give the most general gauge-invariant decompositions of the hadronic tensor for pure electromagnetic, parity conserving, interaction, see e.g [126] and references therein.", "A theoretical description of both neutral- and charged-current DIS at energies of the order of the weak boson masses (or higher) must include parity violating terms in the decomposition of the hadronic tensor.", "Because of them, one no longer has the correspondence that its symmetric part, Eq.", "(REF ), is spin independent and its antisymmetric part, Eq.", "(REF ), is spin dependent.", "Actually, the spin-dependent part of the hadronic tensor becomes a superposition of symmetric and antisymmetric pieces.", "Four more independent structure functions appear in this case, usually called $F_3$ , $g_3$ , $g_4$ and $g_5$ : the first multiplies a term independent from the lepton or nucleon spin four-vector, while the three latter do not.", "Such a general decomposition of the hadronic tensor in DIS, with particular emphasis on the polarized case, can be found in Ref. [127].", "Since experimental data on polarized DIS are taken with electron or muon beams at momentum transfer values not exceeding $Q^2\\sim 100$ GeV$^2$ , we can safely neglect the contribution from a weak boson exchange to describe them properly.", "However, the most general decomposition will be needed in the future to handle neutral-current DIS at high energies, as it may be performed at an Electron-Ion Collider [128], [129], or charged-current DIS with neutrino beams, as it might be available at a neutrino factory [130]." ], [ "Polarized cross-sections differences", "Insertion of Eqs.", "(REF ) and (REF ) into Eq.", "(REF ) yields the expression $\\frac{d^3\\sigma ^{s_{\\ell },S}}{dx dy d \\varphi }=\\frac{\\alpha _{em}^2 y}{2Q^4}\\left[L_{\\mu \\nu }^{(S)} W^{\\mu \\nu (S)} - L_{\\mu \\nu }^{(A)} W^{\\mu \\nu (A)}\\right]\\,\\mbox{,}$ and differences of cross-sections with opposite target helicity states single out the tensor antisymmetric parts $\\frac{d^3\\sigma ^{\\lambda _{\\ell },+S}}{dx dy d\\varphi }-\\frac{d^3\\sigma ^{\\lambda _{\\ell },-S}}{dx dy d\\varphi }=- \\frac{\\alpha _{em}^2 y}{Q^4} L_{\\mu \\nu }^{(A)} W^{\\mu \\nu (A)}\\mbox{ .", "}$ In the target rest frame, using the notation of Fig.", "REF , we parametrize the nucleon spin four-vector as $S^{\\mu }=(0,\\hat{\\mathbf {S}})=(0,\\sin \\alpha \\cos \\beta , \\sin \\alpha \\sin \\beta , \\cos \\alpha )\\,\\mbox{,}$ where we have assumed $\|\\mathbf {S}\|=1$ .", "Taking the direction of the incoming lepton to be along the $z$ -axis, we also have $\\ell ^{\\mu }& =&E(1,0,0,1),\\\\\\ell ^{\\prime \\mu }& = &E^{\\prime }(1, \\sin \\theta \\cos \\varphi , \\sin \\theta \\sin \\varphi , \\cos \\varphi )\\,\\mbox{.", "}$ Figure: Azimuthal and polar angles of thefinal lepton momentum, ℓ ' \\mathbf {\\ell ^{\\prime }}, and the nucleon polarization vector,𝐒\\mathbf {S}.", "The initial lepton moves along the positive zz-axis.Often one defines the (ℓ ^,ℓ ' ^)(\\hat{\\mathbf {\\ell }},\\hat{\\mathbf {\\ell ^{\\prime }}}) leptonplane as the ϕ=0\\varphi =0 plane.Supposing now that the incoming lepton is polarized collinearly to its direction of motion, i.e.", "$\\lambda _{\\ell }=+1$ and $s_{\\ell }^{\\mu }=\\frac{\\ell ^{\\mu }}{m_{\\ell }}$ , we have $L_{\\mu \\nu }^{(A)} W^{\\mu \\nu (A)}& = &-\\frac{8}{\\nu }\\Big {\\lbrace }\\left[(\\ell \\cdot q) (S \\cdot q) - q^2 (S \\cdot q)\\right]g_1 (x, Q^2)\\nonumber \\\\& - &q^2\\left[S \\cdot \\ell - \\frac{(P \\cdot \\ell ) (S \\cdot q)}{P \\cdot q}\\right]g_2 (x, Q^2)\\Big {\\rbrace }\\,\\mbox{.", "}$ Note that, owing to currrent conservation, we have $L_{\\mu \\nu } q^{\\mu } = L_{\\mu \\nu } q^{\\nu } = 0$ and the terms proportional to $q^{\\mu }$ and $q^{\\nu }$ in Eq.", "(REF ) do not contribute when contracted with the leptonic tensor.", "Explicit computation of four-momentum products in this equation yields to a new expression for the differential asymmetry (REF ) $& &\\frac{d^3\\sigma ^{+;+S}}{dx dy d\\phi }-\\frac{d^3\\sigma ^{+;-S}}{dx dy d\\phi }\\nonumber \\\\& = &-\\frac{4 \\alpha _{em}^2}{Q^2} y\\left\\lbrace \\cos \\alpha \\left[\\left(\\frac{E}{\\nu } + \\frac{E^{\\prime }}{\\nu } \\cos \\theta \\right)g_1 (x, Q^2)+\\frac{2 E E^{\\prime }}{\\nu ^2}(\\cos \\theta -1)g_2 (x, Q^2)\\right]\\right.\\nonumber \\\\& + &\\left.\\sin \\alpha \\cos \\phi \\left[\\frac{E^{\\prime }}{\\nu } \\sin \\theta g_1 (x, Q^2)+\\frac{2 E E^{\\prime }}{\\nu ^2} \\sin \\theta g_2 (x, Q^2)\\right]\\right\\rbrace \\,\\mbox{,}$ where $\\phi =\\beta - \\varphi $ is the azimuthal angle between the lepton plane and the $(\\hat{\\mathbf {\\ell }} ,\\hat{\\mathbf {S}})$ plane.", "Notice that the r.h.s.", "of this equation is not expressed in terms of the usual invariants $x$ and $y$ ; to this purpose, let us define the ${\\mathcal {O}}(1/Q)$ quantity $\\gamma \\equiv \\frac{M x}{Q}$ and work out a little algebra to obtain the final expression for the differential polarized cross-section difference Eq.", "(REF ) $& &\\frac{d^3\\sigma ^{+;+S}}{dx dy d\\phi }-\\frac{d^3\\sigma ^{+;-S}}{dx dy d\\phi }\\nonumber \\\\& = &-\\frac{4 \\alpha _{em}^2}{Q^2}\\left\\lbrace \\left[\\left(2-y-\\frac{\\gamma ^2 y^2}{4}\\right)g_1 (x, Q^2)-\\gamma ^2 y g_2 (x, Q^2)\\right]\\cos \\alpha \\right.\\nonumber \\\\& + &\\left.\\sqrt{1-y-\\frac{\\gamma ^2 y^2}{4}}\\left[y g_1 (x, Q^2) + 2 g_2 (x, Q^2)\\right]\\sin \\alpha \\cos \\phi \\right\\rbrace \\,\\mbox{.", "}$ Results obtained so far need a few comments.", "The terms longitudinal and transverse, when speaking about the nucleon polarization, are somewhat ambiguous, insofar as a reference axis is not specified.", "From an experimental point of view, the longitudinal or transverse nucleon polarizations are defined with respect to the lepton beam axis, thus longitudinal (transverse) indicates the direction parallel (orthogonal) to this axis.", "We will use the large arrows $\\Rightarrow $ ($\\Uparrow $ ) to denote these two cases respectively.", "Eq.", "(REF ) refers to the scattering of longitudinally polarized (positive helicity) leptons off a nucleon with positive or negative polarization along an arbitrary direction $\\hat{\\mathbf {S}}$ .", "According to Eqs.", "(REF )-(REF ), the cross-section difference is proportional to $L_{\\mu \\nu }^{(A)}$ , which contains a small factor $m_{\\ell }$ ; as already noticed, this small factor is cancelled by the $1/m_{\\ell }$ factor appearing in the lepton-helicity four-vector, Eq.", "(REF ).", "This would not be the case with transversely polarized leptons, for which one would have $s^{\\mu }=(0,\\hat{\\mathbf {s}})$ , with $\\hat{\\mathbf {s}} \\cdot \\mathbf {\\ell }=0$ .", "Then, transversely polarized leptons lead to tiny cross-section asymmetries of order of $m_\\ell /E$ .", "Eq.", "(REF ) can be specialized to particular cases of the nucleon polarization.", "For longitudinally polarized nucleons, that is $\\hat{\\mathbf {S}} \\parallel \\mathbf {\\ell }$ , one has $\\alpha =0$ and the differential cross-section reads $\\frac{d^3\\sigma ^{+;\\Rightarrow }}{dx dy d\\phi }-\\frac{d^3\\sigma ^{+;\\Leftarrow }}{dx dy d\\phi }=-\\frac{4\\alpha _{em}^2}{Q^2}\\left[\\left(2-y-\\frac{\\gamma ^2 y^2}{2}\\right)g_1 (x,Q^2)-\\gamma ^2 y g_2(x,Q^2)\\right]\\,\\mbox{;}$ for nucleons polarized transversely to the lepton direction, one has $\\alpha =\\pi /2$ and the differential cross-section is $\\frac{d^3\\sigma ^{+;\\Uparrow }}{dx dy d\\phi }-\\frac{d^3\\sigma ^{+;\\Downarrow }}{dx dy d\\phi }=-\\frac{4\\alpha _{em}^2}{Q^2}\\gamma \\sqrt{1-y-\\frac{\\gamma ^2 y^2}{4}}\\left[y g_1(x,Q^2) + 2 g_2(x,Q^2)\\right]\\cos \\varphi \\,\\mbox{.", "}$ In general, the term proportional to $g_2$ is suppressed by a factor $\\gamma $ , Eq.", "(REF ), with respect to the one proportional to $g_1$ : in the Bjorken limit, Eqs.", "(REF )-(REF ) decouple and only $g_1$ is asymptotically relevant.", "We emphasize that, in the case of transverse polarizations, both the $g_1$ and $g_2$ structure functions equally contribute, but the whole cross-section difference is suppressed by the overall factor, Eq.", "(REF ), of order $1/Q$ .", "In the following, we will mostly concentrate on the longitudinally polarized cross-section difference, Eq.", "(REF ).", "From Eq.", "(REF ), it is straightforward to obtain the unpolarized cross-section for inclusive DIS by averaging over spins of the incoming lepton ($s_{\\ell }$ ) and of the nucleon ($S$ ) and by integrating over the azimuthal angle $\\varphi $ .", "It reads $\\frac{d^2 \\sigma ^{unp}}{dx dy}=2\\pi \\frac{1}{2} \\sum _{s_{\\ell }}\\frac{1}{2} \\sum _{S}\\frac{d^2\\sigma ^{s_{\\ell },S}}{dx dy}=\\frac{\\alpha _{em}^2 y}{2Q^4}L_{\\mu \\nu }^{(S)} W^{\\mu \\nu (S)}\\,\\mbox{.", "}$ Finally, the unpolarized cross-section, expressed in terms of the usual unpolarized structure functions $F_1$ and $F_2$ , when neglecting contributions of order $M^2/Q^2$ , is $\\frac{d^2 \\sigma ^{unp}}{dx dy}=\\frac{4 \\pi \\alpha _{em}^2 s}{Q^4}\\left[x y^2 F_1(x, Q^2) + (1-y) F_2 (x, Q^2)\\right]\\,\\mbox{.", "}$" ], [ "Factorization of structure functions", "In the previous Section, we have parametrized the hadronic tensor, which describes the coupling of the virtual photon to the composite nucleon, in terms of four structure functions, namely $F_1$ , $F_2$ and $g_1$ , $g_2$ , see Eqs.", "(REF )-(REF ).", "We have then derived the expression for the differential cross-section asymmetries of longitudinally and transversely polarized nucleons in terms of $g_1$ and $g_2$ , Eqs.", "(REF )-(REF ).", "In principle, by performing DIS experiments with nucleons polarized both longitudinally and transversely, one should learn about the structure functions $g_1$ and $g_2$ , as we will discuss in detail in Sec.", "below.", "In this Section, we would like to provide a description of DIS in the framework of QCD, and in particular give a factorized expression for the structure function $g_1$ .", "Actually, even though QCD is asymptotically free, the computation of any cross-section does involve non-perturbative contributions, since the initial and final states are not the fundamental degrees of freedom of the theory, but compound states of quarks and gluons.", "As we shall see, the factorization theorem allows for the separation of a hard, perturbative and process-dependent part from a low energy, process-independent contribution.", "The latter is given by the Parton Distribution Functions (PDFs), which parametrize our ignorance on the inner structure of the proton.", "In order to deal with the factorized expression for the structure function $g_1$ , we will first provide the leading-order (LO) QCD description of polarized DIS, starting from the naive parton model; we will then give a heuristic development of the next-to-leading order (NLO) perturbative QCD corrections to polarized DIS, focusing on their effects on the $g_1$ structure function." ], [ "Naive parton-model expectations", "The information on the a priori unknown structure of a polarized nucleon is carried by the structure functions $g_1$ and $g_2$ .", "As discussed in Sec.", "REF , they can only be functions of $x$ and $Q^2$ .", "In the naive parton model [10], [11], [12], [13], they allow for simple expressions, since the cross-section for lepton-nucleon scattering is regarded as the incoherent sum of point-like interactions between the lepton and a free, massless parton $\\frac{d^2\\sigma }{dx dy}=\\sum _q e_q^2 f_{q/p}(x) \\frac{d\\hat{\\sigma }}{dy}\\,\\mbox{.", "}$ In this expression, $e_q$ is the fractional charge carried by a parton $q$ , $\\frac{d\\hat{\\sigma }}{dy}$ is the cross-section for the elementary QED process $\\ell q \\rightarrow \\ell ^\\prime q$ , and $f_{q/p}$ is the PDF, the probability density distribution for the momentum fraction $x$ of any parton $q$ in a nucleon $p$ .", "In the simple picture provided by the parton model, PDFs do not depend on the scale $Q^2$ and the structure functions are observed to obey the scaling law $g_{1,2}(x,Q^2)\\rightarrow g_{1,2}(x)$ [10].", "This property is related to the assumption that the transverse momentum of the partons is small.", "In the framework of QCD, however, the radiation of hard gluon from the quarks violates this assumption beyond leading order in pertubation theory, as we will discuss below.", "Of course, the naive parton model predates QCD, but we find it of great value for its intuitive nature.", "If we specialize Eq.", "(REF ) to polarized cross-section asymmetries, we should write $\\frac{d^2\\sigma ^{+;\\Rightarrow }}{dx dy}-\\frac{d^2\\sigma ^{+; \\Leftarrow }}{dx dy}=\\sum _q e_q^2 \\Delta f_{q/p} (x)\\left[\\frac{d\\hat{\\sigma }^{+;+}}{dy}-\\frac{d\\hat{\\sigma }^{+;-}}{dy}\\right]\\,\\mbox{,}$ where $\\frac{d\\hat{\\sigma }^{\\lambda _\\ell ,\\lambda _q}}{dy}$ denotes the elementary cross-section retaining helicity states of both the lepton ($\\lambda _\\ell $ ) and the struck parton ($\\lambda _q$ ).", "We also introduced helicity-dependent, or polarized PDFs, $\\Delta f_{q/p}$ , defined as the momentum densities of partons with spin aligned parallel or antiparallel to the longitudinally polarized parent nucleon: $\\Delta f_{q/p}(x)\\equiv f_{q/p}^{\\uparrow }(x) - f_{q/p}^{\\downarrow }(x)\\,\\mbox{.", "}$ Explicit expressions for the elementary cross-sections appearing in the r.h.s of Eq.", "(REF ) are easily computed at the lowest order in Quantum Electrodynamics (QED): $\\frac{d\\hat{\\sigma }^{+,+}}{dy}=\\frac{4 \\pi \\alpha _{em}^2}{Q^2}\\frac{1}{y}\\,\\mbox{,}\\ \\ \\ \\ \\ \\frac{d\\hat{\\sigma }^{+,-}}{dy}=\\frac{4 \\pi \\alpha _{em}^2}{Q^2}\\frac{(1 - y)^2}{y}\\,\\mbox{.", "}$ Replacing these elementary cross-sections in Eq.", "(REF ) leads to the expression $\\frac{d^2\\sigma ^{+;\\Rightarrow }}{dx dy}-\\frac{d^2\\sigma ^{+; \\Leftarrow }}{dx dy}=\\frac{4 \\pi \\alpha _{em}^2}{Q^2}\\left[\\sum _q e_q^2 \\Delta q (x)(2-y)\\right]\\,\\mbox{,}$ which can be directly compared to Eq.", "(REF ), provided the latter is integrated over the azimuthal angle $\\phi $ .", "Neglecting terms of order $\\mathcal {O}(\\gamma ^2)$ , we finally obtain the naive parton model relations between structure functions $g_1(x)$ , $g_2(x)$ and the polarized distributions $\\Delta f_{q/p}(x)$ : $g_1 (x)& = &\\frac{1}{2} \\sum _q e_q^2 \\Delta f_{q/p}(x)\\,\\mbox{,}\\\\g_2 (x)& = &0\\,\\mbox{.", "}$ These results require a few comments.", "The structure function $g_1$ , Eq.", "(REF ), can be unambigously expressed in terms of quark and antiquark polarized parton distributions.", "Assuming the number of flavors to be $n_f=3$ , we can define the singlet, $\\Delta \\Sigma $ , and nonsinglet triplet, $\\Delta T_3$ , and octet, $\\Delta T_8$ , combinations of polarized quark densities $\\Delta \\Sigma & = &\\Delta u^++\\Delta d^++\\Delta s^+\\,\\mbox{,}\\\\\\Delta T_3& = &\\Delta u^+ -\\Delta d^+\\,\\mbox{,}\\\\\\Delta T_8& = &\\Delta u^+ + \\Delta d^+ -2\\Delta s^+\\,\\mbox{,}$ where $\\Delta q^+=\\Delta q + \\Delta \\bar{q}$ , $q=u,d,s$ are the total parton densities.", "Then, the structure function $g_1$ , Eq (REF ), can be cast into the form $g_1(x)=\\frac{1}{9}\\Delta \\Sigma (x) + \\frac{1}{12}\\Delta T_3(x)+\\frac{1}{36} \\Delta T_8(x)\\,\\mbox{.", "}$ The structure function $g_1$ does not receive any contribution from gluons, yet we shall see in Sec.", "REF that it is not true in the framework of QCD beyond Born approximation.", "The structure function $g_2$ is zero, Eq. ().", "However, non-zero values of $g_2$ can be obtained by allowing the quarks to have an intrinsic Fermi motion inside the nucleon.", "In this case, there is no unambiguous way to calculate $g_2$ in the naive parton model.", "We will not further investigate this issue in this Thesis; a detailed discussion of the problem can be found in Ref. [126].", "The Wilson Operator Product Expansion (OPE) can be applied to the expression of the hadronic tensor $W_{\\mu \\nu }$ in terms of the Fourier transform of the nucleon matrix elements of the elctromagnetic current $J_\\mu (x)$ , Eq.", "(REF ).", "this way, one can give the moments of the structure functions $g_1$ and $g_2$ in terms of hadronic matrix elements of certain operators multiplied by perturbatively calculable Wilson coefficient functions.", "In particular, it can be shown (see e.g.", "[126]) that the first moment of the singlet quark density $a_0=\\int _0^1 dx \\Delta \\Sigma (x)$ is related to the matrix element of the flavor singlet axial current.", "Hence, $a_0$ can be interpreted as the contribution of quarks and antiquarks to the proton's spin, intuitively twice the expectation value of the sum of the $z$ -components of quark and antiquark spins $a_0=2\\langle S_z^{\\mathrm {quarks+antiquarks}}\\rangle \\,\\mbox{.", "}$ Uncritically, one should expect $a_0\\approx 1$ , while in a more realistic relativistic model one finds $a_0\\approx 0.6$ [61].", "In the late 80s, this expectation was found to be in contrast with the anomalously small value measured by the European Muon Collaboration at CERN [49], [50].", "This result could be argued to imply that the sum of the spins carried by the quarks in a proton, $\\langle S_z^{\\mathrm {quarks+antiquarks}}\\rangle $ , was consistent with zero rather than $1/2$ , suggesting a spin crisis in the parton model [51].", "This led to an intense scrutiny of the basis of the theoretical calculation of the structure function $g_1$ and the spin crisis was immediately recognized not to be a fundamental problem, but rather an interesting property of spin structure functions to be understood in terms of QCD.", "We will give a summary of such a description in the following Section." ], [ "QCD corrections and evolution", "The parton model predates the formulation of QCD.", "As soon as QCD is accepted as the theory of strong interactions, with quark and gluon fields as the fundamental fields, one should describe the lepton scattering off partons in the nucleon perturbatively.", "At Born level, the interaction is described by the Feynman diagram in Fig.", "REF -$(a)$ as the tree-level scattering of a quark (or antiquark) off the virtual photon $\\gamma ^$ .", "In this case, quarks are free partons and one recovers the parton model expressions for the structure functions, Eqs.", "(REF )-().", "At $\\mathcal {O}(\\alpha _s)$ , several new contributions appear: the emission of a gluon, Fig.", "REF -$(b)$ , the one-loop correction, Fig.", "REF -$(c)$ , and the process initiated by a gluon which then splits into a quark-antiquark pair, the so-called photon-gluon fusion (PGF) process, Fig.", "REF -$(d)$ .", "The main impact of the QCD interactions is twofold: first, they introduce a mild, calculable, logarithmic $Q^2$ dependence in the parton distributions; second, the correction in Fig.", "REF -$(d)$ generate a contribution to the structure function $g_1$ arising from the polarization of the gluons in the nucleon.", "We shall describe both these effects in the following.", "Figure: Leading contribution (a)(a)and next-to-leading order corrections (b),(c),(d)(b), (c), (d) to DIS." ], [ "Scale dependence of parton distributions", "When including NLO corrections, Fig.", "REF -$(b)$ -$(c)$ , problems arise from the so-called collinear singularities linked to the effective masslessness of quarks.", "The factorization theorem is probed [131] to allow for a separation of the process into a hard and a soft part and for the absorption of the infinity into the soft part (the PDF), which in any case cannot be calculated and has to be determined from experimental data.", "The scale at which the separation is made is called factorization scale $\\mu ^2$ .", "Schematically, one finds terms of the form $\\alpha _s\\ln (Q^2/M)$ , which one splits as follows $\\alpha _s\\ln \\frac{Q^2}{M}=\\alpha _s\\ln \\frac{Q^2}{\\mu ^2}+\\alpha _s\\ln \\frac{\\mu ^2}{M^2}\\,\\mbox{;}$ one then absorbs the first term on the r.h.s.", "of Eq.", "(REF ) into the hard part of the process, and the second term into the soft part.", "The factorization scale $\\mu ^2$ can be chosen arbitrarily and, in exact calculations, physical results must not depend on it.", "In practice, since we never calculate to all orders in perturbation theory, it can make a difference what value we choose, but it turns out that an optimal choice is $\\mu ^2=Q^2$ .", "Consequently, parton distributions no longer obey exact Bjorken scaling, but develop a slow logarithmic dependence on $Q^2$ .", "Actually, if one keeps only the leading-log terms (proportional to $\\alpha _s\\ln (Q^2/\\mu ^2)$ ), one finds that the parton model expressions, Eqs.", "(REF )-(), still hold, provided the replacement $\\Delta f_{q/p}(x) \\longrightarrow \\Delta f_{q/p}(x,Q^2)$ to the $Q^2$ -dependent PDF is made.", "We can think of the scale dependence of PDFs within the following picture.", "As the scale increases, the photon starts to see evidence for the point-like valence quarks within the proton.", "If the quarks were non-interacting, no further structure would be resolved increasing the resolving scale: the Bjorken scaling would set in, and the naive parton model would be satisfactory.", "For this reason, we can consider the naive parton model as the approximation of QCD to Born level.", "However, QCD predicts that on increasing the resolution, one should see that each quark is itself surrounded by a cloud of partons.", "The number of resolved partons which share the proton's momentum increases with the scale.", "The perturbative dependence of the polarized PDFs on the scale $Q^2$ is given by the Altarelli-Parisi evolution equations [58], a set of $(2n_f+1)$ coupled integro-differential equations.", "It is customary to write them in the evolution basis, i.e.", "in terms of linear combinations of the individual parton distributions such that the $(2n_f+1)$ equations maximally decouple from each other.", "To this purpose, we define the polarized gluon distribution $\\Delta g(x,Q^2)$ as in Eq.", "(REF ) and the singlet and nonsinglet quark PDF combinations as $\\Delta q_{\\mathrm {NS}}(x,Q^2)& \\equiv &\\sum _{i=1}^{n_f}\\left(\\frac{e^2_i}{\\langle e^2 \\rangle } - 1\\right)\\left[\\Delta q_i(x,Q^2)+\\Delta \\bar{q}_i(x,Q^2)\\right]\\,\\mbox{,}\\\\\\Delta \\Sigma (x,Q^2)& \\equiv &\\sum _{i=1}^{n_f}\\left[\\Delta q_i(x,Q^2)+\\Delta \\bar{q}_i(x,Q^2)\\right]\\,\\mbox{,}$ where $\\Delta q_i(x,Q^2)$ and $\\Delta \\bar{q}_i(x,Q^2)$ are the scale-dependent quark and antiquark polarized densities of flavor $i$ , also defined according to Eq.", "(REF ).", "The evolution equations are coupled for the singlet quark-antiquark combination and the gluon distribution $\\mu ^2\\frac{\\partial }{\\partial \\mu ^2}\\left(\\begin{array}{c}\\Delta \\Sigma (x,\\mu ^2) \\\\ \\Delta g(x,\\mu ^2)\\end{array}\\right)=\\frac{\\alpha _s(\\mu ^2)}{2\\pi }\\left(\\begin{array}{cc}\\Delta P_{qq}^{\\mathrm {S}} & 2n_f\\Delta P_{qg}^{\\mathrm {S}}\\\\\\Delta P_{qg}^{\\mathrm {S}} & \\Delta P_{gg}^{\\mathrm {S}}\\end{array}\\right)\\otimes \\left(\\begin{array}{c}\\Delta \\Sigma (x,\\mu ^2) \\\\ \\Delta g(x,\\mu ^2)\\end{array}\\right)\\,\\mbox{,}$ while the nonsinglet quark-antiquark combination evolves independently as $\\mu ^2\\frac{\\partial }{\\partial \\ln \\mu ^2}\\Delta q_{\\mathrm {NS}}(x,\\mu ^2)=\\frac{\\alpha _s(\\mu ^2)}{2\\pi }\\Delta P_{qq}^{\\mathrm {NS}}\\otimes \\Delta q_{\\mathrm {NS}}(x,\\mu ^2)\\,\\mbox{.", "}$ In Eqs.", "(REF )-(REF ), $\\Delta P_{ij}^{\\mathrm {S/NS}}$ , $i,j=q,g$ , denotes the singlet/nonsinglet spin-dependent splitting functions for quarks and gluons and $\\otimes $ is the shorthand notation for the convolution product with respect to $x$ $f\\otimes g = \\int _x^1\\frac{dy}{y}f\\left(\\frac{x}{y} \\right) g(y)\\,\\mbox{.", "}$ We notice that Eqs.", "(REF )-(REF ) hold to all orders in perturbative theory, hence splitting functions may be expanded in powers of the strong coupling $\\alpha _s$ : $\\Delta P_{ij}^{p}=\\Delta P_{ij}^{p(0)}(x)+\\frac{\\alpha _s(\\mu ^2)}{2\\pi }\\Delta P_{ij}^{p(1)}(x)+\\mathcal {O}(\\alpha _s^2)\\,\\mbox{,}$ where $p=$ S, NS, and $i,j=q,g$ .", "The splitting functions for polarized PDFs were computed at LO in Ref.", "[58] for the first time, the computation was then extended to NLO in Refs.", "[132], [133], while only partial results are available at NNLO so far [134].", "The solution of the Altareli-Parisi equations may be written as $\\Delta f_i(x,Q^2)=\\sum _j\\Gamma _{ij}(x,\\alpha _s,\\alpha _s^0)\\otimes \\Delta f_j(x,Q_0^2)\\,\\mbox{,}$ where: $f_i=\\Sigma , \\, q_{\\mathrm {NS}}, \\, g$ ; $\\Delta f_j(x,Q_0^2)$ are the corresponding input PDFs, parametrized at an initial scale $Q_0^2$ , to be determined from experimental data; $\\Gamma _{ij}(x,\\alpha _s,\\alpha _s^0)$ are the evolution factors; and we have used the shorthand notation $\\alpha _s\\equiv \\alpha _s(Q^2)\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\alpha _s^0\\equiv \\alpha _s(Q_0^2)\\,\\mbox{.", "}$ The evolution factors also satisfy evolution equations $\\mu ^2\\frac{\\partial }{\\partial \\mu ^2}\\Gamma _{ij}(x,\\alpha _s,\\alpha _s^0)=\\sum _k P_{ik}(x,\\alpha _s)\\otimes \\Gamma _{kj}(x,\\alpha _s,\\alpha _s^0)\\,\\mbox{,}$ with boundary conditions $\\Gamma _{ij}(x,\\alpha _s^0,\\alpha _s^0)=\\delta _{ij}\\delta (1-x)$ .", "The QCD evolution equations are most easily solved using Mellin moments since then all convolutions become simple products, and the equations can be solved in a closed form.", "We refer to [135], [136], [137] for a comprehensive discussion about details concerning such a technique." ], [ "The gluon contribution to the $g_1$ structure function", "Another important consequence of QCD corrections is the rise of a contribution to the $g_1$ structure function from the polarization of gluons in the nucleon, see for instance Fig.", "REF -$(d)$ .", "For this reason, the factorized leading-twist expression for the structure function $g_1$ reads, instead of Eq.", "(REF ), $g_1(x,Q^2)=\\frac{\\langle e^2\\rangle }{2}\\left[C_{\\mathrm {NS}}\\otimes \\Delta q_{\\mathrm {NS}}+C_{\\mathrm {S}}\\otimes \\Delta \\Sigma +2n_f C_g\\otimes \\Delta g\\right]\\,\\mbox{,}$ where $\\langle e^2\\rangle =n_f^{-1}\\sum _{i=1}^{n_f} e_i^2$ is the average charge, with $n_f$ the number of active flavors and $e_i$ their electric charge, and $\\otimes $ denotes the convolution product with respect to $x$ , Eq.", "(REF ).", "Besides, $\\Delta q_{\\mathrm {NS}}$ and $\\Delta \\Sigma $ are the scale-dependent nonsinglet and singlet quark PDF combinations, Eqs.", "(REF )-() and $\\Delta g$ is the gluon PDF.", "Finally, $C_{\\mathrm {NS}}$ , $C_{\\mathrm {S}}$ and $C_g$ are the corresponding coefficient functions related to calculable short-distance cross-sections, for hard photon-quark and photon-gluon cross-sections respectively.", "Coefficient functions are perturbative objects and may be expanded in powers of the strong coupling $\\alpha _s$ $C_p(x,\\alpha _s)=C_p^{(0)}(x)+\\frac{\\alpha _s(\\mu ^2)}{2\\pi }C_p^{(1)}(x)+\\mathcal {O}(\\alpha _s)\\,\\mbox{,}$ with $p=$ S, NS, $g$ and $i,j=q,g$ .", "At the lowest order in $\\alpha _s$ , $C_{\\mathrm {NS}}^{(0)}=C_{\\mathrm {S}}^{(0)}=\\delta (1-x)$ and $C_g^{(0)}=0$ [58], hence the structure function $g_1$ decouples from the gluon contribution, see Eq.", "(REF ), and the parton model prediction, Eq.", "(REF ) is recovered.", "Polarized coefficient functions have been computed up to $\\mathcal {O}(\\alpha _s^2)$ so far [138].", "Notice incidentally that the substitution of Eq.", "(REF ) into Eq.", "(REF ) leads to the relation $g_1(x,Q^2)=\\frac{\\langle e^2\\rangle }{2}\\sum _{j=\\mathrm {NS}, \\Sigma , g} K_j(x,\\alpha _s,\\alpha _s^0)\\otimes \\Delta f_j(x,Q_0^2)\\,\\mbox{,}$ where the hard kernel defined as $K_j(x,\\alpha _s,\\alpha _s^0)=\\sum _{k=\\mathrm {NS},\\mathrm {S},g} C_k(x,\\alpha _s)\\otimes \\Gamma _{ij}(x,\\alpha _s,\\alpha _s^0)$ is completely computable in perturbation theory.", "Hence, in Eq.", "(REF ) we have fully separated the perturbative and the non-perturbative parts entering the structure function $g_1$ .", "Besides, the hard kernels in Eq.", "(REF ) are independent of the particular set of input PDFs, and may thus be computed separately once and for all, suitably interpolated and stored.", "This is of uttermost importance while performing a fit of PDFs to experimental data, as we will further delineate in Sec.", ", since this involves the evaluation of only the one set of convolutions Eq.", "(REF ), which is amenable to computational optimization.", "The gluonic term in the expression of the $g_1$ structure function, see for instance Eq.", "(REF ), can be shown [59], [139], [140] to entail an additional contribution to the singlet axial charge of the form $a_0^{\\mathrm {gluons}}=-n_f\\frac{\\alpha _s(Q^2)}{2\\pi }\\int _0^1dx\\Delta g(x,Q^2)\\,\\mbox{.", "}$ We remind that $n_f$ is the number of light flavors, $u$ , $d$ , $s$ , and heavy flavors are assumed not to contribute.", "Hence, the naive parton model expectation for the axial current $a_0$ , Eq.", "(REF ), should be replaced by $a_0=\\int _0^1dx \\Delta \\Sigma (x,Q^2) + a_0^{\\mathrm {gluons}}\\,\\mbox{.", "}$ At first sight, one should expect that the gluonic term in Eq.", "(REF ) would not survive at large $Q^2$ , since it looks like an $\\alpha _s$ correction which would disappear as the running coupling $\\alpha _s$ vanishes.", "However, the first moment of the gluon contribution $\\int _0^1dx\\Delta g(x,Q^2)$ grows as $[\\alpha _s(Q^2)]^{-1}$ for large values of $Q^2$ , as dictated by Altarelli-Parisi evolution equations, see Eq.", "(REF ).Higher moments are instead decreasing functions of $\\log Q^2$ , falling at a faster rate than for the unpolarized gluon density.", "Hence, the gluon does not decouple from $g_1$ asymptotically and the parton model expression for the structure function $g_1$ , Eq.", "(REF ), is not recovered in perturbative QCD even in the limit $\\alpha _s\\rightarrow 0$ .", "The $Q^2$ behavior of the first moment of the polarized gluon density was originally derived when the QCD evolution equations were first written down in $x$ space [58].", "In fact, the first moment of the polarized gluon splitting function is finite and proportional to the first coefficient of the QCD beta function, which establishes the quoted relation with the running coupling $\\alpha _s(Q^2)$ .", "This relation between the $Q^2$ evolution of the first moment of the polarized gluon density and the running coupling is induced by the axial anomaly [139] corresponding to the QCD version of the anomalous triangle diagram [141], [142].", "As a consequence, the definition of the singlet quark first moment becomes totally ambiguous, because two generic definitions differ by terms of order $\\alpha _s(Q^2)\\int _0^1 dx\\Delta g(x,Q^2)$ .", "For the first moment, what is formally a NLO correction is potentially of the same size.", "Owing to Eq.", "(REF ), the singlet quark first moment, Eq.", "(REF ), defined directly from the structure function $g_1$ and used by the EMC experiment when the spin crisis [51] was announced, does not have to coincide with the constituent quark value, i.e.", "the total fraction of the spin carried by quarks.", "Only for exactly conserved quantities do the corresponding values for constituent and parton quarks have to coincide.", "The first moments of the quark densities are in general only conserved at LO by the QCD evolution, but, due to the axial anomaly, the singlet quark first moment defined from $g_1$ is not conserved in higher orders.", "We conclude that a definition of the singlet quark density $\\Delta \\Sigma (x,Q^2)$ must be carefully specified, as further discussed in Sec.", "REF below.", "The result quoted in Eqs.", "(REF )-(REF ) was advocated to reconcile the EMC result with the theoretical expectation for the proton spin content.", "As explained above, what the EMC experiment actually observed was the singlet axial charge $a_0$ , Eq.", "(REF ).", "The almost vanishing value measured for this quantity can be explained as a cancellation between a reasonably large quark spin contribution, e.g.", "$\\Delta \\Sigma \\simeq 0.6 - 0.7$ , as expected intuitively, and the anomalous gluon contribution.", "However, in order to accomplish this cancellation, one should require a large gluon spin contribution, e.g.", "$\\int _0^1dx\\Delta g(x,Q^2)\\simeq 4$ at $\\langle Q^2\\rangle \\simeq 10$ GeV$^2$ .", "As we have explained, the latter momentum grows indefinitely as $Q^2$ increases, so that in principle such a large value cannot be ruled out.", "Hence, we must carefully investigate with which accuracy we are able to determine each term in Eq.", "(REF ), particularly the gluon contribution, by scrutinizing both the available experimental data and the methodology we use to determine parton distributions from them.", "This is exactly the goal of this Thesis: we will find that the gluon is still largely uncertain, in contrast to somewhat common belief, and that its determination is still a challenge in spin physics." ], [ "Scheme dependence of parton distribution moments", "Beyond leading order, coefficient and splitting functions are no longer universal, hence even though the scale dependence of the structure function $g_1$ is determined uniquely, at least up to higher order corrections, its separation into contributions due to quarks and gluons is scheme dependent (and thus essentially arbitrary).", "The NLO coefficient functions may be modified by a change of the factorization scheme which is partially compensated by a corresponding change in the NLO splitting functions, hence both are required for a consistent NLO computation.", "A comprehensive discussion of scheme dependence can be found in Ref. [114].", "Here, we summarize the main features of the schemes which are commonly used in the analysis of polarized DIS.", "The most popular renormalization scheme is the so-called $\\overline{\\mathrm {MS}}$ scheme [132], [133], in which the first moment of the gluon coefficient function vanishes.", "In this scheme, the gluon density does not contribute to the first moment of the structure function $g_1$ and the scale-dependent singlet axial charge is equal to the singlet quark first moment: $a_0(Q^2)=\\left.\\int _0^1 dx \\Delta \\Sigma (x,Q^2)\\right\|_{\\overline{\\mathrm {MS}}}\\,\\mbox{.", "}$ Also, the first moments of the nonsinglet triplet and octet PDF combinations $a_3=\\int _0^1dx\\Delta T_3(x,Q^2)\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ a_8=\\int _0^1dx\\Delta T_8(x,Q^2)$ are independent of $Q^2$ .", "An alternatively scheme is the so-called Adler-Bardeen scheme [143] defined such that the first moment of the singlet PDF combination is independent of $Q^2$ , thus it can be identified with the total quark helicity.", "The polarized gluon density is defined as in the $\\overline{\\mathrm {MS}}$ scheme, but directly contributes to the singlet axial charge, and consequently to the first moment of $g_1$ , which now reads $a_0(Q^2)=\\left.\\int _0^1\\Delta \\Sigma (x,Q^2)\\right\|_{\\mathrm {AB}}-n_f\\frac{\\alpha _s(Q^2)}{2\\pi }\\int _0^1\\Delta g (x,Q^2)\\,\\mbox{.", "}$ The relation between the first moments of the singlet quark combination in AB and $\\overline{\\mathrm {MS}}$ renormalization schemes is then simply obtained by comparing Eqs.", "(REF )-(REF ).", "As discussed above, the difference is proportional to $\\alpha _s(Q^2)\\int _0^1 dx \\Delta g(x,Q^2)$ and is due to the anomalous nonconservation of the singlet axial current.", "At LO, it is scale-invariant: this implies that the first moment of the polarized gluon distribution increases as $1/\\alpha _s(Q^2)$ with $Q^2$ , hence the gluon contribution in Eq.", "(REF ) is not asymptotically suppressed by powers of $\\alpha _s$ .", "As a consequence, this scheme dependence does not vanish at large $Q^2$ , and the definition of the singlet quark first moment is therefore maximally ambiguous.", "Moments of structure functions are a powerful tool to study some fundamental properties of the nucleon structure, like the total momentum fraction carried by quarks or the total contribution of quark spin to the spin of the nucleon.", "While a complete description of structure functions based on fundamental QCD principles may be unattainable for now, moments of structure functions can be directly compared to rigorous theoretical results, like sum rules, lattice QCD calculations and chiral perturbation theory.", "The light-cone expansion of the current product in Eq.", "(REF ) implies that the $n$ -th moments of the structure functions $g_1$ and $g_2$ , at leading twist are given by [114] $\\int _0^1dx x^{n-1} g_1(x,Q^2)=\\frac{1}{2}\\sum _i\\delta _ia_n^iC_{1,i}^{n}(Q^2,\\alpha _s)\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ n=1,3,5,\\dots $ $\\int _0^1dx x^{n-1}g_2(x,Q^2)=\\frac{1-n}{2n}\\sum _i\\delta _i\\left[a_n^iC_{1,i}^{n}(Q^2,\\alpha _s)-d_n^i C_{2,i}^n(Q^2,\\alpha _s)\\right] \\ \\ \\ \\ \\ \\ n=3,5,7,\\dots $ where the $\\delta _i$ are numerical coefficients, the $C_i^n(Q^2,\\alpha _s)$ are the coefficient functions and the $a_n^i$ and $d_n^i$ are related to the hadronic matrix elements of the local operator.", "The label $i$ indicates what kind of operator is contributing: for flavor-nonsinglet operators, only quark fields an their covariant derivatives occur.", "In the case of the first moment of the $g_1$ structure function, Eq.", "(REF ) can be recast as $\\Gamma _1^{p,n}\\equiv \\int _0^1dx g_1(x,Q^2)=\\frac{1}{12}\\left[C_{\\mathrm {NS}}(Q^2)\\left(\\pm a_3+\\frac{1}{3}a_8\\right)+\\frac{4}{3}C_{\\mathrm {S}}(Q^2)a_0\\right]\\,\\mbox{,}$ where the plus (minus) sign refers to a proton (neutron) target.", "In the above, $a_3$ and $a_8$ are measures of the proton matrix elements of an $SU(3)$ flavor octet of quark axial-vector currents.", "The octet of axial-vector currents is precisely the set of currents that controls the weak $\\beta $ -decays of the neutron and of the spin-$1/2$ hyperons.", "Consequently, $a_3$ and $a_8$ can be expressed in terms of two parameters $F$ and $D$ measured in hyperon $\\beta $ decays [144] $a_3=F+D=1.2701\\pm 0.0025\\,\\mbox{,}\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ a_8=\\frac{1}{\\sqrt{3}}(3F-D)=0.585\\pm 0.025\\,\\mbox{.", "}$ It follows that a measurement of $\\Gamma _1^p(Q^2)$ in polarzed DIS can be interpreted as a measurement of $a_0(Q^2)$ .", "Indeed, Eq.", "(REF ) can be rewritten as $C_{\\mathrm {S}}(Q^2)a_0(Q^2)=9\\Gamma _1^p(Q^2)-\\frac{1}{2}C_{\\mathrm {NS}}(Q^2)(3F+D)\\,\\mbox{.", "}$ Since the two terms on the r.h.s.", "are roughly of the same order, the value of $a_0$ arises from a large cancellation between them (see e.g.", "Ref. [145]).", "The present measured value for $a_0$ is still disturbingly small, as briefly noted at the end of previous Sec.", "REF .", "Finally, in going from the case of a proton to a neutron, $a_0$ and $a_8$ in Eq.", "(REF ) remain unchanged, whereas $a_3$ reverses its sign.", "One thus finds the Bjorken sum rule [146], [147] $\\Gamma _1^p(Q^2)-\\Gamma _1^n(Q^2)=\\frac{1}{6}C_{\\mathrm {NS}}(Q^2)a_3\\,\\mbox{,}$ which was originally derived from current algebra and isospin asymmetry.", "A comparison with experimental data, thus allows for a direct test of isospin, as well as of the predicted scale dependence.", "Furthermore, since the nonsiglet coefficient function $C_{\\mathrm {NS}}$ is known up to three loops [148], Eq.", "(REF ) potentially provides a theoretically very accurate handle on the strong coupling $\\alpha _s$ [67].", "This feature will be discussed in Sec.", "REF in the framework of a determination of an unbiased parton set from inclusive polarized DIS data.", "Finally, we notice that relations like Eq.", "(REF ) can be obtained also for the $g_2$ structure function.", "These include the Burkhardt-Cottingham sum rule [149] and the Efremov-Leader-Teryaev sum rule [150], for a discussion of which we refer to [126], [151].", "Here we only notice that data on $g_2$ are not yet accurate enough for a significant test of them." ], [ "Phenomenology of polarized structure functions", "Experimental information on the structure functions $g_1(x,Q^2)$ and $g_2(x,Q^2)$ is extracted from measured cross-section asymmetries, both longitudinal, $A_\\parallel $ , and transeverse, $A_\\perp $ .", "These are defined by considering longitudinally polarized leptons scattering off a hadronic target, polarized either longitudinally or transversely with respect to the collision axis, and read $A_{\\parallel }=\\frac{d\\sigma ^{\\rightarrow \\Rightarrow }-d\\sigma ^{\\rightarrow \\Leftarrow }}{d\\sigma ^{\\rightarrow \\Rightarrow }+d\\sigma ^{\\rightarrow \\Leftarrow }}\\,\\mbox{;}\\quad A_{\\perp }=\\frac{d\\sigma ^{\\rightarrow \\Uparrow }-d\\sigma ^{\\rightarrow \\Downarrow }}{d\\sigma ^{\\rightarrow \\Uparrow }+d\\sigma ^{\\rightarrow \\Downarrow }}\\,\\mbox{.", "}$ The numerator of these expressions is given by Eqs.", "(REF )-(REF ), while the denominator is twice the unpolarized cross-section, Eq.", "(REF ).", "Inversion of Eqs.", "(REF )-(REF ) gives the explicit relation between the polarized structure functions and the measurable asymmetries Eq.", "(REF ) $g_1(x,Q^2)&=\\frac{F_1(x,Q^2)}{(1+\\gamma ^2)(1+\\eta \\zeta )}\\left[(1+\\gamma \\zeta )\\frac{A_{\\parallel }}{D}-(\\eta -\\gamma )\\frac{A_{\\perp }}{d}\\right]\\,\\mbox{,}\\\\g_2(x,Q^2)&=\\frac{F_1(x,Q^2)}{(1+\\gamma ^2)(1+\\eta \\zeta )}\\left[\\left(\\frac{\\zeta }{\\gamma }-1\\right)\\frac{A_{\\parallel }}{D}+\\left(\\eta +\\frac{1}{\\gamma }\\right)\\frac{A_{\\perp }}{d}\\right]\\,\\mbox{,}$ where we have defined the kinematic factors $d&=\\frac{D\\sqrt{1-y-\\gamma ^2 y^2/4}}{1-y/2}\\,\\mbox{,}\\\\D&=\\frac{1-(1-y)\\epsilon }{1+\\epsilon R(x,Q^2)}\\,\\mbox{,}\\\\\\eta &=\\frac{\\epsilon \\gamma y}{1-\\epsilon (1-y)}\\,\\mbox{,}\\\\\\zeta &=\\frac{\\gamma (1-y/2)}{1+\\gamma ^2 y/2}\\,\\mbox{,}\\\\\\epsilon &= \\frac{4(1-y) - \\gamma ^2 y^2}{2 y^2 + 4 (1-y) + \\gamma ^2 y^2}\\,\\mbox{.", "}$ The unpolarized structure function $F_1$ and unpolarized structure function ratio $R$ which enter the definition of the asymmetries, Eqs.", "(REF )-(), may be expressed in terms of $F_2$ and $F_L$ by $F_1(x,Q^2)&\\equiv &\\frac{F_2(x,Q^2)}{2x\\left[1+R(x,Q^2)\\right]}\\left(1+\\gamma ^2\\right)\\,\\mbox{,}\\\\R(x,Q^2)&\\equiv &\\frac{F_L(x,Q^2)}{F_2(x,Q^2)-F_L(x,Q^2)}\\,\\mbox{.", "}$ The longitudinal and transverse asymmetries are sometimes expressed in terms of the virtual photo-absorption asymmetries $A_1$ and $A_2$ according to $A_\\parallel =D(A_1+\\eta A_2)\\mbox{ ,}\\qquad \\qquad A_\\perp =d(A_2-\\zeta A_1),$ where $A_1(x,Q^2)\\equiv \\frac{\\sigma ^T_{1/2}-\\sigma ^T_{3/2}}{\\sigma ^T_{1/2}+\\sigma ^T_{3/2}}\\mbox{ ,}\\qquad \\qquad A_2(x,Q^2)\\equiv \\frac{2\\sigma ^{TL}}{\\sigma ^T_{1/2}+\\sigma ^T_{3/2}}.$ Recall that $\\sigma ^T_{1/2}$ and $\\sigma ^T_{3/2}$ are cross-sections for the scattering of virtual transversely polarized photons (corresponding to longitudinal lepton polarization) with helicity of the photon-nucleon system equal to 1/2 and 3/2 respectively, and $\\sigma ^{TL}$ denotes the interference term between the transverse and longitudinal photon-nucleon amplitudes.", "In the limit $M^2\\ll Q^2$ Eqs.", "(REF ) reduce to $D=A_\\parallel /A_1$ , $d=A_\\perp /A_2$ , thereby providing a physical interpretation of $d$ and $D$ as depolarization factors.", "Using Eqs.", "(REF ) in Eqs.", "(REF )-() we may express the structure functions in terms of $A_1$ and $A_2$ instead: $g_1(x,Q^2) &= \\frac{F_1(x,Q^2)}{1+\\gamma ^2} \\left[ A_1(x,Q^2)+ \\gamma A_2 (x,Q^2) \\right]\\,\\mbox{,}\\\\g_2(x,Q^2)&=\\frac{F_1(x,Q^2)}{1+\\gamma ^2}\\left[\\frac{A_2}{\\gamma }- A_1\\right]\\,\\mbox{.", "}$ We are interested in the structure function $g_1(x,Q^2)$ , whose moments are proportional to nucleon matrix elements of twist-two longitudinally polarized quark and gluon operators, and therefore can be expressed in terms of longitudinally polarized quark and gluon distributions.", "Using Eqs.", "(REF )-(), we may obtain an expression of it in terms of the two asymmetries $A_{\\parallel }$ , $A_{\\perp }$ , or, using Eqs.", "(REF )-(), in terms of the two asymmetries $A_1$ , $A_2$ .", "Clearly, up to corrections of ${\\mathcal {O}}\\left(M/Q\\right)$ , $g_1$ is fully determined by $A_{\\parallel }$ , which coincides with $A_1$ up to ${\\mathcal {O}}\\left(M/Q\\right)$ terms, while $g_2$ is determined by $A_{\\perp }$ or $A_2$ .", "It follows that, even though in principle a measurement of both asymmetries is necessary for the determination of $g_1$ , in practice most of the information comes from $A_{\\parallel }$ or $A_1$ , with the other asymmetry only providing a relatively small correction unless $Q^2$ is very small.", "It may thus be convenient to express $g_1$ in terms of $A_{\\parallel }$ and $g_2$ $g_1(x,Q^2)=\\frac{F_1(x,Q^2)}{1+\\gamma \\eta }\\frac{A_{\\parallel }}{D}+\\frac{\\gamma (\\gamma -\\eta )}{\\gamma \\eta +1}g_2(x,Q^2)\\,\\mbox{,}$ or, equivalently, in terms of $A_1$ and $g_2$ $g_1(x,Q^2) = A_1(x,Q^2) F_1(x,Q^2) + \\gamma ^2 g_2(x,Q^2)\\,\\mbox{.", "}$ It is then possible to use Eq.", "(REF ) or Eq.", "(REF ) to determine $g_1(x,Q^2)$ from a dedicated measurement of the longitudinal asymmetry, and an independent determination of $g_2(x,Q^2)$ .", "In practice, experimental information on the transverse asymmetry and structure function $g_2$ is scarce [152], [153], [154].", "However, the Wilson expansion for polarized DIS implies that the structure function $g_2$ can be written as the sum of a twist-two and a twist-three contribution [155]: $g_2(x,Q^2)=g_2^{\\mathrm {t2}}(x,Q^2)+g_2^{\\mathrm {t3}}(x,Q^2).$ The twist-two contribution to $g_2$ is simply related to $g_1$ .", "One finds $g_2^{\\mathrm {t2}}(x,Q^2)= -g_1(x,Q^2)+\\int _x^1\\frac{dy}{y} g_1(y,Q^2)$ which in Mellin space becomes $g_2^{\\mathrm {t2}}(N,Q^2)= -\\frac{N-1}{N}g_1(N,Q^2).$ It is important to note that $g_2^{\\mathrm {t3}}$ is not suppressed by a power of $M/Q$ in comparison to $g_2^{\\mathrm {t2}}$ , because in the polarized case the availability of the spin vector allows the construction of an extra scalar invariant.", "Nevertheless, experimental evidence suggests that $g_2^{\\mathrm {t3}}$ is compatible with zero at low scale $Q^2\\sim M^2$ .", "Fits to $g_2^{\\mathrm {t3}}$ [156], [157], as well as theoretical estimates of it [156], [158] support the conclusion that $g_2(x,Q^2)\\approx g_2^{\\mathrm {t2}}(x,Q^2)\\equiv g_2^{\\mathrm {WW}}(x,Q^2)\\,\\mbox{,}$ which is known as the Wandzura-Wilczek [155] relation.", "The effect of such an assumption on a determination of parton distributions from experimental data could be tested to compare to results obtained with the opposite assumption, i.e.", "$g_2(x,Q^2)=0$ .", "We will follow this strategy in our extraction of unbiased polarized PDFs presented in Chap.", "." ], [ "Target mass corrections", "A large part of experimental data in polarized DIS are taken at relatively low values of $Q^2$ , typically a few GeV$^2$ , and at medium-to-large $x$ values.", "In this kinematical region, the target-mass factor $\\gamma $ , Eq.", "(REF ), is of order unity with the finite value of the nucleon mass $M$ , hence contributions which are elsewhere suppressed may play a relevant role.", "These corrections are usually referred to as kinematic higher-twist terms.", "Another source of terms suppressed by inverse powers of $Q^2$ arises from the Wilson expansion of the hadronic tensor, Eqs.", "(REF )-(REF ), i.e.", "from matrix elements of operators of non-leading twist.", "These corrections are referred to as dynamical higher-twist terms.", "A detailed analysis of the effect of twist-3 and twist-4 corrections to $g_1$ , as well as a twist-3 correction to the $g_2$ structure function on the determination of a set of polarized PDFs from inclusive DIS data has been recently presented [78].", "They find that higher-twist terms could have a sizable effect, particularly in the high-$x$ and low-$Q^2$ kinematic region.", "In order to include exactly the effect of kinematic target mass corrections, TMCs henceforth, one has to deal with Eqs.", "(REF )-(REF ), supplemented by some model assumption on $g_2$ , as discussed in Sec. .", "This is required since experimental data on $g_2$ are restricted to a limited range in $(x,Q^2)$ and are affected by large uncertainties.", "Target mass corrections assume simple expressions in Mellin space, where they read [159] $\\tilde{g}_1(N,Q^2)&=&g_1(N,Q^2)+\\frac{M^{2}}{Q^2}\\frac{N(N+1)}{(N+2)^2}\\nonumber \\\\&\\times &\\left[(N+4)g_1(N+2,Q^2)+4\\frac{N+2}{N+1}g_2(N+2,Q^2)\\right]+{\\mathcal {O}}\\left(\\frac{M^2}{Q^2}\\right)^2\\\\\\tilde{g}_2(N,Q^2)&=&g_2(N,Q^2)+\\frac{M^2}{Q^2}\\frac{N(N-1)}{(N+2)^2}\\nonumber \\\\&\\times &\\left[N\\frac{N+2}{N+1}g_2(N+2,Q^2)-g_1(N+2,Q^2)\\right]+{\\mathcal {O}}\\left(\\frac{M^2}{Q^2}\\right)^2\\,\\mbox{.", "}$ Here, we have denoted by $\\tilde{g}_{1,2}(N,Q^2)$ the Mellin space structure functions with TMCs included, while $g_{1,2}(N,Q^2)$ are the structure functions determined in the $M=0$ limit.", "These expressions can be specialized under assumptions for $g_2$ .", "In particular, in the Wandzura-Wilczek case, substituting Eq.", "(REF ) in Eq.", "(REF ) and taking the inverse Mellin transform, we get $\\tilde{g}_1(x,Q^2)=\\frac{1}{2\\pi i}\\int dN\\,x^{-N}\\left[1+\\frac{M^2x^2}{Q^2}\\frac{(N-2)^2(N-1)}{N^2}\\right]g_1(N,Q^2)\\,\\mbox{,}$ where we have shifted $N\\rightarrow N-2$ in the term proportional to $M^2$ .", "Inverting the Mellin transform we then obtain $\\tilde{g}_1(x,Q^2)&=&g_1(x,Q^2)+\\frac{M^2x^2}{Q^2}\\left[-5g_1(x,Q^2)-x\\frac{\\partial g_1(x,Q^2)}{\\partial x}\\right.\\nonumber \\\\& & \\left.+\\int _x^1\\frac{dy}{y}\\left(8g_1(y,Q^2)+4g_1(y,Q^2)\\log \\frac{x}{y}\\right)\\right]\\,\\mbox{.", "}$ Conversely, if we simply set $g_2=0$ , we have $\\tilde{g}_1(x,Q^2)=\\frac{1}{2\\pi i}\\int dN\\,x^{-N}\\left[1+\\frac{M^2x^2}{Q^2}\\frac{(N^2-4)(N-1)}{N^2}\\right]g_1(N,Q^2)\\,\\mbox{,}$ whence $\\tilde{g}_1(x,Q^2)&=&g_1(x,Q^2)+\\frac{m^2x^2}{Q^2}\\left[-g_1(x,Q^2)-x\\frac{\\partial g_1(x,Q^2)}{\\partial x}\\right.\\nonumber \\\\& &\\left.-\\int _x^1\\frac{dy}{y}\\left(4g_1(y,Q^2)+4g_1(y,Q^2)\\log \\frac{x}{y}\\right)\\right]\\,\\mbox{.", "}$ The usefulness of these realtions will be apparent in Chap.", ", where we will use them in an unbiased determination of a polarized parton set based on inclusive polarized DIS data." ], [ "Phenomenology of polarized Parton Distributions", "In this Chapter, we review how a set of parton distribution functions is usually determined from a global fit to experimental data.", "In Sec.", ", we delineate the general strategy for PDF determination and its main theoretical and methodological issues, focusing on those which are peculiar to the polarized case.", "In Sec.", ", we summarize how some of these problems are addressed within the NNPDF methodology, which was developed in recent years to provide a statistically sound determination of parton distributions and their uncertainties.", "In Sec.", ", we finally conclude with an overview on available polarized PDF sets." ], [ "General strategy for ", "In principle, the general strategy for parton fitting can be simply stated.", "Thanks to the factorization theorem, theoretical predictions for the various measured observables are expressed as the convolution between coefficient functions and parton distributions.", "The former are perturbative quantities, computed in field theory at desired accuracy, but different for each partonic subrocess contributing to the observable.", "The latter are non-perturbative objects, but universal, i.e.", "they do not depend on the observable under investigation.", "In order to determine parton distributions from experimental data, they have to be parametrized, usually at an initial energy scale $Q_0^2$ , and then randomly initialized.", "They then need to be evolved up to the energy scale $Q^2$ , relevant for the measurement under investigation, by solving Altarelli-Parisi equations.", "The agreement between measured observables $\\left\\lbrace O_i^{\\mathrm {(exp)}}\\right\\rbrace $ and corresponding theoretical predictions $\\left\\lbrace O_i^{\\mathrm {(th)}}\\right\\rbrace $ is quantified by a figure of merit, usually chosen as the $\\chi ^2$ function, $\\chi ^2=\\sum _{i,j}^{N_{\\mathrm {dat}}}(O_i^{\\mathrm {(exp)}}-O_i^{\\mathrm {(th)}})[\\mathrm {cov}_{ij}](O_j^{\\mathrm {(exp)}}-O_j^{\\mathrm {(th)}})\\,\\mbox{,}$ where $\\mathrm {cov}_{ij}$ is the experimental covariance matrix.", "If data are provided with no correlated systematics, as the case in most polarized measurements, this is the diagonal matrix of the experimental uncertainties.", "The best fit is obtained by minimizing the figure of merit, Eq.", "(REF ), and the corresponding best-fit parameters will finally fix the PDF shape.", "Despite the apparent simplicity of this strategy, the determination of parton distributions from a global set of experimental data, possibly coming from different processes, is a challenging exercise.", "This requires to face several issues, some of which are peculiar to the polarized case.", "We summarize them as follows.", "Lack of experimental data.", "Each observable has its own definition in terms of parton distributions and for this reason a specific observable can constrain or disentangle some distributions and not all of them.", "In principle, several different observables are needed to determine all the $(2n_f+1)$ independent parton components (quark, antiquark and gluon, with $n_f$ the number of active flavors) inside the nucleon.", "If such an information is lacking, either a subset of parton distributions is determined, or general assumptions on the unconstrained PDFs have to be made.", "For instance, the bulk of experimental data to constrain polarized PDFs consists of polarized inclusive DIS data.", "This process only allows for the determination of the total quark distributions $\\Delta u^+=\\Delta u +\\Delta \\bar{u}$ , $\\Delta d^+=\\Delta d +\\Delta \\bar{d}$ , $\\Delta s^+=\\Delta s +\\Delta \\bar{s}$ , and of the gluon $\\Delta g$ (for details, see Chap. ).", "Information on light sea antiquarks is provided either by semi-inclusive DIS with identified hadrons in the final state (mostly pions or kaons) or $W$ production in proton-proton collisions.", "These reactions actually receive leading contributions from partonic subrocesses initiated by $\\bar{u}$ and $\\bar{d}$ antiquarks, hence they will be able to provide information on the corresponding spin-dependent distributions $\\Delta \\bar{u}$ and $\\Delta \\bar{d}$ .", "Besides, it is worth noticing that inclusive DIS indirectly constrains the polarized gluon, $\\Delta g$ , through scaling violations.", "Unfortunately, these turn out to have a mild effect for its determination, due to the small $Q^2$ lever arm of experimental data.", "Observables receiving leading contribution from gluon-initiated partonic subprocesses will be more suited to probe $\\Delta g$ directly.", "They include asymmetries for jet and pion production in proton-proton collisions and for one- or two-hadron and open-charm production in fixed-target lepton-nucleon scattering.", "The theory and phenomenology of these processes will be discussed in more detail in Chap. .", "Finally, we notice that polarized data are less abundant and less accurate than their unpolarized counterparts and have a rather limited kinematic coverage in the $(x,Q^2)$ plane.", "The various experimental processes that provide information on polarized PDFs, together with the corresponding leading partonic subprocesses, PDFs that are being probed and covered kinematic ranges are summarized in Tab.", "REF .", "Table: Summary of polarized processes to determine polarized PDFs.For each of them, we show the leading partonic subprocesses, probed polarizedPDFs, and the ranges of xx and Q 2 Q^2 that become accessible.Processes are separated according to the need of using fragmentation functionsfor their analysis: processes in the upper part of the Table do notinvolve the fragmentation of the struck quark into an observed hadron,while those in the lower part do.", "Theoretical issues.", "Many theoretical subtleties may affect the determination of parton distributions.", "In particular, we distinguish between issues related to the QCD analysis and issues related to the methodology adopted for the fitting of parton distributions.", "The latter especially include the choice of the functional form to parametrize PDFs and of the formalism to propagate uncertainties: they will be addressed separately in the next paragraphs.", "Theoretical details in the QCD analysis concern for instance the treatment of heavy quark mass effects.", "So far, they have been almost completely unaddressed in the analysis of polarized PDFs.", "This is because such effects have been shown to be relatively small on the scale of present-day unpolarized PDF uncertainties [106], which are rather smaller than those of their polarized counterparts.", "Hence, the effects of heavy quark masses hardly emerge in polarized experimental data.", "Another QCD theoretical issue is related to the treatment of higher-twist and nuclear corrections.", "As mentioned in Sec.", ", a large part of experimental data in polarized DIS are taken at relatively low values of $Q^2$ , typically a few GeV$^2$ , and at medium-to-large $x$ values.", "In this kinematic region, such corrections could play a relevant role, as demonstrated in Ref. [78].", "Furthermore, assuming exact $SU(3)$ symmetry, the first moments of the nonsinglet quark combinations can be related to hyperon octet decay constants, see Eqs.", "(REF ).", "However, the violation of $SU(3)$ flavor symmetry is debated in the literature [160], even though a detailed phenomenological analysis seems to support it [161].", "It should be clear that, due to the lack of experimental information, theoretical constraints, such these sum rules, the positivity of measured cross-sections and the integrability of parton distributions, provide a significant input for determining the shape of PDFs in some kinematic regions, as we will discuss in Sec. .", "Finally, we notice that the inclusion of processes involving identified hadrons in final states requires the usage of poorly known fragmentation functions.", "Recent work has emphasized the troubles for all available fragmentation function sets to describe the most updated inclusive charged-particle spectra data at the LHC [162].", "Hence, the inclusion of semi-inclusive DIS and collider pion production data in a global determination of polarized PDFs is likely to introduce, via fragmentation functions, an uncertainty which is diffucult to estimate (though it is usually neglected).", "Functional parametrization.", "The choice of the PDF parametrization is a crucial point.", "In principle, since PDFs represent our ignorance of the non-perturbative nucleon structure, there should be complete freedom in choosing their parametric form.", "However, in order to carry out a parton fit, one needs to choose a particular functional form for the PDFs at the initial scale, usually $x\\Delta f_i(x,Q_0^2)=\\eta _iA_ix^{a_i}(1-x)^{b_i}\\left(1+\\rho _ix^{\\frac{1}{2}}+\\gamma _ix\\right)\\mbox{.", "}$ Some of the parameters in Eq.", "(REF ) can be constrained by barion octet decay constants, by Regge interpretation at small-$x$ , and by constraining to zero parton distributions at $x=1$ .", "Arbitrary assumptions are often made on the parameters to make the fit minimization to converge.", "Of course, as the number of free parameters decreases, the PDF parametrization turns out to be more rigid, thus introducing a bias in the PDF determination.", "Error estimates.", "The Hessian formalism [163], [164] is the most commonly used method for PDF error determination.", "The $\\chi ^2$ function, Eq.", "(REF ), is quadratically expanded about its global minimum $\\Delta \\chi ^2=\\chi ^2-\\chi ^2_0=\\sum _{i=1}^{N_{\\mathrm {par}}}\\sum _{j=1}^{N_{\\mathrm {par}}} H_{ij}(a_i-a_i^0)(a_j-a_j^0)\\,\\mbox{,}$ with $\\chi _0^2=\\chi ^2(S_0)$ and $\\lbrace \\mathbf {a}^0\\rbrace $ the $\\chi ^2$ and the set of $N_{\\mathrm {par}}$ parameters corresponding to the best estimate $S_0$ for the PDF set $\\lbrace \\Delta f\\rbrace $ respectively; $H_{ij}$ is the Hessian matrix element defined as $H_{ij}=\\frac{\\partial ^2\\chi ^2(\\mathbf {\\lbrace a\\rbrace })}{\\partial a_i\\partial a_j}\\,\\mbox{.", "}$ The Hessian matrix, Eq.", "(REF ), has a complete set of $N_{\\mathrm {par}}$ orthonormal eigenvectors $v_{ik}$ with eigenvalues $\\epsilon _k$ defined by $\\sum _{j=1}^{N_{\\mathrm {par}}}H_{ij}\\lbrace \\mathbf {a}^0\\rbrace v_{jk}=\\epsilon _k v_{ik}\\,\\mbox{,}\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\sum _{i=1}^{N_{\\mathrm {par}}} v_{il}v_{jk}=\\delta _{lk}\\,\\mbox{.", "}$ Moving the parameters around their best value, a shift is observed in the $\\chi ^2$ function, $\\Delta \\chi ^2$ .", "Each eigenvector determines a direction in the parameter space along which the $\\chi ^2$ variation about the minimum can be expressed in a natural way.", "However, the fit quality typically deteriorates far more quickly upon variations in some directions than others, hence the eigenvalues $\\epsilon _k$ are distributed over a wide range that covers many orders of magnitude.", "In terms of the diagonalized set of parameters defined with respect to the eigenvectors $v_{ij}$ $z_i=\\sqrt{\\frac{\\epsilon _i}{2}}\\sum _{j=1}^{N_{\\mathrm {par}}}(a_j-a_j^0)v_{ij}$ one obtains $\\Delta \\chi ^2=\\sum _{i=1}^{N_{\\mathrm {par}}}z_i^2$ , i.e.", "the region of acceptable fits around the global minimum is contained inside a hypersphere of radius $\\sqrt{\\Delta \\chi ^2}$ .", "The scheme for the diagonalization procedure is shown in Fig.", "REF .", "Figure: A schematic representation of the transformation from thePDF parameter basis to the orthonormal eigenvector basis as defined byEq. ().", "The figure is taken from Ref.", ".An observable $O$ , depending on the set of parameters $\\lbrace \\mathbf {a}\\rbrace $ via the PDF parametrization, is assumed to be rather well approximated in the neighbourhood of the global minimum by the first term of its Taylor-series expansion.", "The deviation of $O$ from its best estimate $O_0$ is then given by $\\Delta O = O - O_0\\approx \\sum _{i=1}^{N_{\\mathrm {par}}}O_i z_i$ , with $O_i\\equiv \\partial O/\\partial z_i \|_{\\mathbf {z}=\\mathbf {0}}$ .", "For a given variation $\\Delta \\chi ^2$ of the $\\chi ^2$ , the error estimate on the observable $O$ is then evaluated as $\\Delta O = \\left\\lbrace \\sum _{i=1}^{N_{\\mathrm {par}}}\\left[O(S_i^+)-O(S_i^-) \\right]^2\\right\\rbrace ^{\\frac{1}{2}}\\,\\mbox{,}$ where $S_i^\\pm $ are $2\\,N_\\mathrm {par}$ sets of PDFs computed at the two points defined by $z_i^\\pm =\\pm \\frac{\\sqrt{\\Delta \\chi ^2}}{2}$ on the edge of the $N_{\\mathrm {par}}$ -dimensional hypersphere in the $z$ parameter space.", "Together with $S_0$ they form $2N_{\\mathrm {par}}+1$ sets of PDF, that are the ones needed to compute PDFs errors on $S_0$ , and errors on the observable $O$ from Eq.", "(REF ).", "It is worth noticing that the propagation of PDF uncertainties in the Hessian method has been derived under the assumption that a first order, linear approximation is adequate.", "Of course, due to the complicate nature of a global fit, deviations, also from the simple quadratic behavior, Eq.", "(REF ), are inevitable.", "Textbook statistics implies that, if the measurements belong to experimental data sets which are compatible with each others, and linear error propagation holds, one should have $\\Delta \\chi ^2=1$ .", "However, it has been argued that somewhat larger tolerance value $T=\\Delta \\chi ^2$ [165] should be adopted in order for the distribution of $\\chi ^2$ values between different experiments in a global fit to be reasonable.", "The reasonable value of the tolerance $T$ can be determined, for instance, by estimating the range of overall $\\chi ^2$ along each of the eigenvector directions within which a fit to all data sets can be obtained and then averaging the ranges over the $N_{\\mathrm {par}}$ eigenvector directions.", "This method has been investigated in detail in Ref.", "[166] by demanding that indeed $90\\%$ of experiments approximately fall within the $90\\%$ confidence level.", "More refined methods involve the determination of a different tolerance [167] along each Hessian eigenvector (the so-called dynamical tolerance).", "Nevertheless, we should notice that the use of $T>1$ is somewhat controversial, given that there is no rigorous statistical proof for the criterions adopted to estimate it.", "For all these reasons, error estimates based on the Hessian method are not necessarily always accurate.", "A way to estimate PDF uncertainties avoiding quadratic approximation of $\\chi ^2$ is provided by the Lagrange multiplier method [168], [169].", "This is implemented by minimizing a function $\\Psi (\\lbrace a_i\\rbrace ,\\lbrace \\lambda _i\\rbrace )=\\chi ^2(\\lbrace a_i\\rbrace )+\\sum _j\\lambda _jO_j(\\lbrace a_i\\rbrace )$ with respect to the set of PDF parameters $\\lbrace a_i\\rbrace $ for fixed values of the Lagrange multipliers $\\lbrace \\lambda _i\\rbrace $ .", "Each multiplier is related to one specific observable $O_j$ , and the choice $\\lambda _j=0$ corresponds to the best fit $S_0$ .", "By repeating this minimization procedure for many values of $\\lambda _j$ , one can map out precisely how the fit to data deteriorates as the expectation for the observable $O_j$ is forced to change.", "Unlike the Hessian method, the Lagrange multiplier technique does not rely on any assumption regarding the dependence of the $\\chi ^2$ on the parameters of the fit $\\lbrace a_i\\rbrace $ ." ], [ "The NNPDF approach to parton fitting", "In recent years, several polarized PDF sets with uncertainties have been released.", "They slightly differ in the choice of data sets, the form of PDF parametrization, and in several details of the QCD analysis, like the treatment of higher-twist corrections, as will be reviewed with some more detail in Sec.", "below.", "However, they are all based on the standard methodology for PDF fitting, based on a fixed functional parametrization of PDFs and Hessian error estimate.", "This methodology is known [81] to run into difficulties especially when information is scarce, because of the intrinsic bias of fixed parton parametrization.", "This is likely to be particularly the case for polarized PDFs, which rely on data both less abundant and less accurate than their unpolarized counterparts.", "In order to overcome these difficulties, the NNPDF collaboration has proposed and developed a new methodology for PDF determination [99], [100], [101], [102], [103], [104], [105], [106], [107], [108], [109], [110], [111].", "So far, the NNPDF methodology has been successfully adopted to determine unpolarized PDF sets with increasing accuracy, which are now routinely used by the LHC collaborations in their data analysis and for data-theory comparisons.", "The method is based on a Monte Carlo approach, with neural networks used as unbiased interpolants.", "Monte Carlo sampling allows one to evaluate all quantities, such as the uncertainty or the correlation of PDFs, in a statistically sound way, while the use of neural networks provides a robust and flexible parametrization of the parton distributions at the initial scale.", "In the following, we will describe in detail the main features of the NNPDF methodology." ], [ "Monte Carlo sampling of the probability density distribution", "Given a PDF or an observable depending on (polarized) PDFs, $O(\\lbrace \\Delta f_i\\rbrace )$ , its average is given by the integration - in the functional space $V(\\lbrace \\Delta f_i\\rbrace )$ spanned by the parton distributions and weighted by a suitably defined probability measure - of all possible functions describing PDFs at a reference scale: $\\langle O[\\Delta f] \\rangle =\\int _V \\mathcal {D}\\Delta f\\mathcal {P}[\\Delta f]O[\\Delta f]\\,\\mbox{.", "}$ In the NNPDF approach, the probability measure is represented by a Monte Carlo sample in the space of PDFs.", "An ensemble of replicas of the original data set is generated, such that it reproduces the statistical distribution of the experimental data, followed by its projection into the space of PDFs through the fitting procedure.", "Notice that all theoretical assumptions represent a prior for the determination of such probability measure.", "The ensemble in the space of data has to contain all the available experimental information.", "In practice, most data are given with multi-Gaussian probability distributions of statistical and systematic errors, described by a covariance matrix.", "In such cases this is the distribution that will be used to generate the pseudodata.", "However, any other probability distribution can be used if and when required by the experimental data.", "Each replica of the experimental data is a member of the Monte Carlo ensemble and contains as many data points as are originally available.", "Whether the given ensemble has the desired statistical features can be verified by means of statistical standard tests by comparing quantities calculated from it with the original properties of the data.", "Such tests, together with pseudodata generation, will be explicitly discussed in Sec.", "for the case of polarized DIS.", "Here we should notice that the sampling of the underlying probability density distribution of data allows one to circumvent the non-trivial issues related to Hessian propagation of errors.", "Indeed, an ensemble of parton distributions is fitted to pseudodata: this means that, at the end of the fitting procedure, one obtains as many PDFs as the number of replicas $N_{\\mathrm {rep}}$ of the data that were generated.", "The experimental values in each replica will fluctuate according to their distribution in the Monte Carlo ensemble and the best fit PDFs will fluctuate accordingly for each replica.", "Even though individual PDF replicas might fluctuate significantly, averaged quantities like central value and one-sigma error bands are smooth inasmuch as the size of the ensemble increases.", "The advantages of the Monte Carlo methodology are then apparent.", "First, the expectation value for any observable depending on the PDFs or the PDFs themselves can be easily computed as the Monte Carlo average over the PDF ensemble: Eq.", "(REF ) is then replaced by $\\langle O[\\Delta f]\\rangle =\\frac{1}{N_{\\mathrm {rep}}}\\sum _{k=1}^{N_{\\mathrm {rep}}}O[\\Delta f_k]\\,\\mbox{,}$ and similarly uncertainties can be obtained as standard deviations, and so forth.", "Second, non-Gaussian behavior of uncertainties can be tested either at the level of experimental data, by sampling them according to an arbitrary distribution, or at the level of the results in the best fit ensemble of PDFs, by defining proper confidence levels.", "In any case, we do not have to rely on the quadratic assumption, Eq.", "(REF ), made in the Hessian approach.", "Finally, the stability of results upon a change of parametrization can be verified by standard statistical tools, for instance by computing the distance between results in units of their standard deviation.", "Likewise, it is possible to verify that fits performed by removing data from the set have wider error bands but remain compatible within these enlarged uncertainties, or to address how results change within different theoretical assumptions.", "The reliability of the results can thus be assessed directly." ], [ "Neural network parametrization", "The Monte Carlo technique adopted to propagate the experimental error into the space of PDFs is completely independent of the method used to parametrize parton distributions; it might well be used along with standard parametrization, Eq.", "(REF ) [170].", "On the other hand, in order to get a faithful determination of parton distributions, one ought to make sure that the chosen functional form is redundant enough not to introduce a theoretical bias which would artificially reduce parton uncertainty in regions where data do not constrain enough PDFs.", "There are several ways for obtaining such a redundant parametrization.", "One may use some clever polynomial basis, or more refined tools such as self-organising maps [171].", "Within the NNPDF methodology, each parton distribution is parametrized by a neural network, which provide a redundant and minimally biased parametrization.", "The only theoretical assumption is smoothness, a presumed feature of PDFs, which is ensured by the flexibility and adaptability of neural networks.", "In particular, we use feed-forward neural networks [120].", "They are made of a set of interconnected units, called neurons, eventually organized in groups, called layers.", "The state or activation of a given neuron $i$ in a given layer $l$ , $\\xi _i^{(l)}$ , is a real number, determined as a function of the activation of the neurons connected to it, namely those in the previous, $l-1$ , layer.", "Each pair of neurons $(i,j)$ is then connected by a synapsis, characterized by a real number $\\omega _{ij}^{(l-1)}$ , called weight.", "The activation of each neuron $i$ in a given layer $l$ is a function $g$ of the difference between a weighted average of input from neurons in the preceding layer and a threshold $\\theta _i^{(l)}$ : $\\xi _i^{(l)}=g\\left(\\sum _{j=1}^{N_{l-1}}\\omega _{ij}^{(l-1)}\\xi _j^{(l-1)}-\\theta _i^{(l)}\\right)\\,\\mbox{,}$ where $N_{l-1}$ is the number of neurons in the $(l-1)^{\\mathrm {th}}$ layer.", "The input and output vectors are labeled as $\\mathbf {\\xi }^{(1)}$ and $\\mathbf {\\xi }^{(L)}$ respectively, with $L$ the number of layers in the network.", "The activation function $g$ is in general non-linear.", "The simplest example of activation function $g(x)$ is the step function $g(x) = \\Theta (x)$ , which produces binary activation only.", "However, it turns out to be advantageous to use an activation function with two distinct regimes, linear and non-linear, such as the sigmoid $g(x)=\\frac{1}{1-e^{-\\beta x}}\\,\\mbox{.", "}$ This function approaches the step function at large $\\beta $ ; without loss of generality, in the NNPDF methodology we usually take $\\beta =1$ .", "The sigmoid activation function has a linear response when $x\\approx 0$ , and it saturates for large positive or negative arguments.", "If weights and thresholds are such that the sigmoids work on the crossover between linear and saturation regimes, the neural network behaves in a non-linear way.", "Thanks to this non-linear behavior, the neural network is able to reproduce nontrivial functions.", "Basically, multilayer feed-forward neural networks provide a non-linear map between some input $\\mathbf {\\xi }^{(1)}_i$ and output $\\mathbf {\\xi }^{(L)}_j$ variables, parametrized by weights, thresholds and activation function, $\\mathbf {\\xi }^{(L)}=F\\left[\\mathbf {\\xi }^{(1)};\\lbrace \\omega _{ij}^{(l)}\\rbrace ,\\lbrace \\theta _i^{(l)}\\rbrace ;g\\right]\\,\\mbox{.", "}$ For given activation function, the parameters can be tuned in such a way that the neural network reproduces any continuous function.", "The behavior of a neural network is determined by the joint behavior of all its connections and thresholds, and it can thus be built to be redundant, in the sense that modifying, adding or removing a neuron has little impact on the final output.", "Because of these reasons, neural networks can be considered to be robust, unbiased universal approximants.", "In Sec.", ", we will explicitly discuss how polarized PDFs can be parametrized in terms of neural networks.", "In particular, we will discuss their architecture and preprocessing of data to enforce the asymptotic small- and large-$x$ behavior of PDFs." ], [ "Minimization and stopping", "Once each independent PDF is parametrized in terms of neural networks at an inital reference scale $Q_0^2$ , physical observables are computed by convolving hard kernels with PDFs evolved to the scale of the experimental measurements by Altarelli-Parisi evolution.", "The best-fit set of parton distributions is determined by comparing the theoretical computation of the observable with its experimental value, for each Monte Carlo replica.", "This is performed by evaluating a suitable figure of merit, e.g.", "Eq.", "(REF ).", "Both the minimization and the determination of the best-fit in the wide, non-local, space of parameters spanned by the neural network parameters are delicate issues.", "To minimize the error function, a genetic algorithm is used.", "The main advantage of such an algorithm is that it works on a population of solutions, rather than tracing the progress of one point through parameter space.", "Thus, many regions of parameter space are explored simultaneously, thereby lowering the possibility of getting trapped in local minima.", "The basic idea underlying the genetic algorithm is the following.", "Starting from a randomly chosen set of parameters, a pool of possible new sets is generated by mutation of one or more parameters at a time.", "Each new set that has undergone mutation is a mutant.", "A value of the figure of merit that is being minimized corresponds to each set of parameters, so those configurations that fall far away from the minimum can be discarded.", "This procedure is iterated over a sufficiently large number of generations.", "Of course, one has to be sure that the final set of parameters corresponds to acceptable PDFs, i.e.", "they must satisfy theoretical constraints like positivity of cross-sections and sum rules.", "Usually, this requirement is fulfilled by penalizing unacceptable replicas during the minimization by arbitrarily increasing their figure of merit.", "In Sec.", ", we will discuss technical details related to how this issue is faced in the determination of polarized distributions.", "The redundancy of the parametrization also implies another subtle problem, known as overlearning, which happens when neural networks start to fit statistical fluctuations of data, rather than their underlying physical law.", "The solution to this problem is achieved using a cross-validation method to determine a criterion for the fit to stop before entering the overlearning regime.", "Technical details about the specific implemetation of stopping in the polarized case will be extensively discussed in Sec. .", "The main ingredients of the NNPDF methodology described above, namely Monte Carlo sampling of data distribution, neural network parametrization of PDFs, minimization and stopping, are finally sketched in Fig.", "REF .", "Figure: Scheme of the NNPDF methodology for parton fitting." ], [ "Bayesian reweighting of Monte Carlo PDF ensembles", "Monte Carlo sampling of the underlying probability density distribution of data used in the NNPDF methodology allows for applying standard statistical tools to the resulting PDF ensembles.", "Most importantly, Bayesian inference can be exploited to determine the impact on PDFs of new data sets that were not included in a fit.", "The methodology, referred to as reweighting, was presented in detail in Refs.", "[105], [109].", "In short, the main idea underlying reweighting is to assign to each replica in the PDF ensemble a weight which assesses the probability that this replica agrees with new data.", "The expectation value for an observable $O$ , taking into account the new data, is then given by the weighted average $\\langle O[\\Delta f]\\rangle _{\\mathrm {new}} =\\int _V \\mathcal {D}\\Delta f\\mathcal {P}_{\\mathrm {new}}[\\Delta f]O[\\Delta f]=\\frac{1}{N_{\\mathrm {rep}}}\\sum _{k=1}^{N_{\\mathrm {rep}}}w_k O[\\Delta f_k]\\,\\mbox{.", "}$ The weights $w_k$ are computed from the $\\chi ^2$ of the new data to the prediction obtained using a given replica, according to the formula $w_k=\\frac{(\\chi _k^2)^{\\frac{1}{2}(n-1)}e^{-\\frac{1}{2}\\chi _k^2}}{\\frac{1}{N_{\\mathrm {rep}}}\\sum _{k=1}^{N_{\\mathrm {rep}}}(\\chi _k^2)^{\\frac{1}{2}(n-1)}e^{-\\frac{1}{2}\\chi _k^2}}\\,\\mbox{,}$ where $n$ is the number of new data.", "The formula, Eq.", "(REF ), is derived under the assumption that new data have Gaussian errors and that they are statistically independent of the old data.", "By the law of multiplication of probabilities it then follows that $\\mathcal {P}_{\\mathrm {new}}=\\mathcal {N}_\\chi \\mathcal {P}(\\chi \|\\Delta f)\\mathcal {P}_{\\mathrm {old}}(\\Delta f)\\,\\mbox{,}$ where $\\mathcal {N}_\\chi $ is a normalization factor chosen such that $\\sum _{k=1}^{N_{\\mathrm {rep}}}w_k=N_{\\mathrm {rep}}$ and $w_k=\\mathcal {N}_\\chi \\mathcal {P}(\\chi \|\\Delta f_k)\\propto (\\chi _k^2)^{\\frac{1}{2}(n-1)}e^{-\\frac{1}{2}\\chi _k^2}\\,\\mbox{.", "}$ Notice that, after reweighting a given PDF ensemble of $N_{\\mathrm {rep}}$ replicas, the efficiency in describing the distribution of PDFs is no longer the same.", "In fact, the weights give the relative importance of the different replicas, and the replicas with very small weights will become almost irrelevant in ensemble averages.", "The reweighted replicas will thus no longer be as efficient as the old: for a given $N_{\\mathrm {rep}}$ , the accuracy of the representation of the underlying distribution $\\mathcal {P}_{\\mathrm {new}}(\\Delta f)$ will be less than it would be in a new fit.", "This loss of efficiency can be quantified using the Shannon entropy to determine the effective number of replicas left after reweighting: $N_{\\mathrm {eff}}=exp\\left\\lbrace \\frac{1}{N_{\\mathrm {rep}}}\\sum _{k=1}^{N_{\\mathrm {rep}}}w_k\\ln \\left(\\frac{N_{\\mathrm {rep}}}{w_k}\\right)\\right\\rbrace \\,\\mbox{.", "}$ Clearly $0<N_{\\mathrm {eff}}<N_{\\mathrm {rep}}$ : the reweighted fit has the same accuracy as a refit with $N_{\\mathrm {eff}}$ replicas.", "Hence, if $N_{\\mathrm {eff}}$ becomes too low, the reweighting procedure will no longer be reliable, either because the new data contain a lot of information on the PDFs, neccessitating a full refitting, or because the new data are inconsistent with the old.", "Bayesian reweighting allows for the inclusion of new pieces of experimental information in an ensemble of PDFs without performing a new fit.", "This is desirable, particularly when dealing with observables for the computation of which no fast code is available.", "Also notice that, from a conceptual point of view, a determination of a PDF set might be performed by including all data through reweighting of a first reasonable guess, called prior, as suggested in Refs.", "[172], [173].", "Typically, this is the result of a previous PDF fit to data (other than those with which the parton set will be reweighted) or a model based on some theoretical assumptions.", "The reweighting method works fine provided the prior is unbiased, so that one can check that the final results do not depend on the choice of the initial guess.", "We will devote Chap.", "to the discussion of this aspect in a particular case of physical interest: we will reweight a DIS-based parton set with jet and $W$ production data from polarized proton-proton collisions.", "Results obtained via reweighting of existing Monte Carlo PDF sets were shown to be statistically equivalent to those obtained via global refitting: in particular, the method was validated in the unpolarized case by considering inclusive jet and LHC $W$ lepton asymmetry data [105], [109].", "As a further illustration of possible applications of the method, reweighting was used to quantify the impact on unpolarized PDFs of direct photon data [174] and of top quark production data [175], and to study the constraints on nuclear PDFs from LHC pPb data [176].", "Also, notice that the same technique can be straightforwardly applied in the case of Hessian PDF sets [177].", "Once a reweighted PDF set has been determined, it would be interesting to be able to produce a new PDF ensemble with the same probability distribution as a reweighted set, but without the need to include the weight information.", "A method of unweighting has therefore been developed, whereby the new set is constructed by deterministically sampling with replacement the weighted probability distribution [109].", "This means that replicas with a very small weight will no longer appear in the final unweighted set while replicas with large weight will occur repeatedly.", "If the probability for each replica and the probability cumulants are defined as $p_k=\\frac{w_k}{N_{\\mathrm {rep}}}\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ p_k\\equiv p_{k-1}+p_k=\\sum _{j=0}^k p_j\\,\\mbox{,}$ it is possible to quantitatively describe the unweighting procedure.", "Starting with $N_{\\mathrm {rep}}$ replicas with weights $w_k$ , $N_{\\mathrm {rep}}$ new weights $w_k^\\prime $ are determined: $w_k^\\prime =\\sum _{j=1}^{N_{\\mathrm {rep}}^\\prime }\\theta \\left(\\frac{j}{N_{\\mathrm {rep}}^\\prime }-p_{k-1}\\right)\\theta \\left(p_k-\\frac{j}{N_{\\mathrm {rep}}^\\prime }\\right)\\,\\mbox{.", "}$ These weights are therefore either zero or a positive integer.", "By construction they satisfy $N_{\\mathrm {rep}}^\\prime \\equiv \\sum _{k=1}^{N_{\\mathrm {rep}}}w_k^\\prime \\,\\mbox{,}$ i.e.", "the new unweighted set consists of $N_{\\mathrm {rep}}$ replicas, simply constructed by taking $w_k$ copies of the $k$ -th replica, for all $k=1, \\dots , N_{\\mathrm {rep}}$ .", "In Chap.", ", Bayesian reweighting of polarized parton distributions, determined within the NNPDF methodology from inclusive DIS data, will be used to assess the impact of open-charm production data in fixed-target DIS as well as $W$ and jet production data in proton-proton collisions.", "A new polarized parton set will be then determined via unweighting." ], [ "Overview of available polarized parton sets", "First studies of the polarized structure of the nucleon were aimed at an accurate determination of polarized first moments, including detailed uncertainty estimates [143], [66], [67], but did not attempt a determination of a full PDF set.", "This was first proposed in Ref.", "[65], but without uncertainty estimation.", "Several polarized parton sets have been delivered in recent years, all at NLO accuracy, usually updated with the inclusion of new data, theoretical or statistical features.", "In this Section, we review the main features of the presently available polarized PDF sets; the discussion about recent developments in the determination of a polarized parton set based on the NNPDF methodology will be addressed in Chaps.", "--." ], [ "DIS-based fits", "The bulk of experimental information on longitudinally polarized nucleon structure consists of inclusive, photon induced, deep-inelastic scattering with both polarized charged lepton beams and nucleon targets.", "Actually, only data sets coming from this process are included in most of the polarized PDF determinations, as follows.", "BB10.", "The BB10 parton set [76] includes world-available data on the asymmetry $A_1$ or the structure function $g_1$ from polarized inclusive DIS.", "Four independent polarized PDFs are determined there, namely the valence combinations $\\Delta u^-=\\Delta u-\\Delta \\bar{u}$ and $\\Delta d^-=\\Delta d-\\Delta \\bar{d}$ , the gluon, $\\Delta g$ and the quark sea, $\\Delta \\bar{q}$ , assuming symmetric sea, $\\Delta \\bar{u}=\\Delta \\bar{d}=\\Delta \\bar{s}=\\Delta s$ .", "Each PDF is parametrized in terms of a fixed functional form like that of Eq.", "(REF ), but only a subset of them are actually taken to be free, depending on the PDF.", "Also the QCD scale $\\Lambda _{\\mathrm {QCD}}$ is a parameter to be determined in the fit.", "Errors are determined within the standard Hessian approach without any tolerance criterion.", "The analysis is supplemented by the inclusion of experimental systematic uncertainties from different sources, namely data, NMC parametrization of $F_2$ [178] and $R$ parametrization [179] both entering Eq.", "(REF ).", "Also, theoretical systematic uncertainties were estimated from a variation of the factorization and renormalization scale and input parametrization scale.", "Finally, higher twist contributions from heavy flavor Wilson coefficients in fixed-flavor number (FFN) scheme were included in the theoretical QCD analysis.", "The BB10 parton set is publicly available together with a FORTRAN program which allows for computing PDF central values and errors.", "JAM13.", "The JAM13 parton set [78], instead of the asymmetry $A_1$ or the structure function $g_1$ , directly fits the measured longitudinal and transverse asymmetries $A_\\parallel $ and $A_\\perp $ , Eqs.", "(REF ).", "This makes a differencee with most of other analyses, which usually include the information from the $g_1$ structure function extracted from observed asymmetries by each experimental collaboration within different assumptions.", "Besides, a large number of preliminary inclusive DIS data from JLAB are included in the JAM13 study.", "Since these data lie in the large-$x$ and small-$Q^2$ kinematic region, they are expected to be particularly sensitive to target mass and higher-twist effects.", "For this reason, both twist-3 and twist-4 corrections to $g_1$ , as well as a twist-3 correction to the $g_2$ structure function are taken into account.", "Moreover, they consistently apply the nuclear smearing corrections to both the $g_1$ and $g_2$ structure functions for both deuterium and $^3\\mathrm {He}$ , within the framework of the weak binding approximation [180], [181], [182].", "Six independent polarized PDFs are determined, namely the total quark combinations, $\\Delta u^+=\\Delta u+\\Delta \\bar{u}$ and $\\Delta d^+=\\Delta d+\\Delta \\bar{d}$ , the gluon, $\\Delta g$ , and the quark sea, $\\Delta \\bar{q}$ ($q=u, d, s$ ) and assuming a symmetric strangeness $\\Delta s =\\Delta \\bar{s}$ .", "These are parametrized with the functional form Eq.", "(REF ); however, since the experimental piece of information does not allow for a complete determination of all six PDFs above, several additional contraints are adopted.", "In particular, the $\\Delta \\bar{u}$ and $\\Delta \\bar{d}$ parton distributions, which do not contribute directly to the description of inclusive DIS data included in the analysis, are fixed by requiring $\\lim _{x\\rightarrow 0}\\Delta \\bar{q}(x,Q_0^2)=\\frac{1}{2}\\lim _{x\\rightarrow 0}\\Delta q^+(x,Q_0^2)\\,\\mbox{,}$ with $q=u,d$ .", "In addition, in order to avoid unphysical results and provide reasonable values for all distributions, the following constraint is imposed $\\frac{1}{2}\\left(\\left\|\\frac{\\Delta \\bar{q}^{(2)}}{\\Delta \\bar{s}^{(2)}} \\right\|+\\left\|\\frac{\\Delta \\bar{s}^{(2)}}{\\Delta \\bar{q}^{(2)}} \\right\|\\right)=1\\pm 0.25\\,\\mbox{.", "}$ Both Eq.", "(REF ) and Eq.", "(REF ) entail the decrease in the number of free parameters whose values are actually fitted to data.", "The choice of limiting the flexibility of the parametrization given by Eq.", "(REF ) is also made for the gluon, for which the values of the exponents $a_g$ and $b_g$ are somewhat arbitrarily fixed.", "The propagation of uncertainties is performed within the Hessian approach without tolerance criterion.", "ABFR98.", "The ABFR98 parton set [67] is based on a less updated set of inclusive DIS experimental data and provides four polarized PDFs, both the total combinations $\\Delta q^+$ and the gluon $\\Delta g$ , within fixed functional parametrization and Hessian error estimate.", "Despite the ABFR98 is less recent than other analyses discussed here, it should be worth mentioning it at least for two reasons.", "First, this provides polarized PDFs that are fitted in the AB renormalization scheme (see Sec.", "REF ) instead of $\\overline{\\mathrm {MS}}$ used in all other PDF determinations.", "Second, it includes a detailed discussion of theoretical uncertainties originated by neglected higher orders, higher twists, position of heavy quark thresholds, value of the strong coupling, violation of SU(3) flavor symetry and finally uncertainties related to the choice of the functional form." ], [ "Global PDF fits", "In recent years, the knowledge on longitudinally polarized nucleon structure has been supplemented by data coming from processes different from polarized inclusive DIS.", "As outlined in Sec.", ", these include fixed-target SIDIS and hadron or jet production in polarized proton-proton collisions.", "Including these pieces of experimental information in a global NLO analysis is a challenging task, both theoretical and computational, because of the need to deal with hadronic observables.", "They may also depend on the fragmentation of quarks in the final measured hadrons and hence their analysis require the usage of poorly known fragmentation functions.", "Several such global parton sets have been determined in very recent years, as summarized as follows.", "AAC08.", "Besides polarized inclusive DIS data, the AAC08 analysis [73] also includes $\\pi ^0$ production data at RHIC, via a $K$ -factor approximation for the NLO corrections.", "Because these data only provide constraints on the gluon polarization, only four idependent PDFs are determined: they are the same as in the BB10 analysis (symmetric sea is also assumed), but the functional form they use reads $\\Delta f(x,Q_0^2) = \\left[ \\delta x^\\nu - \\kappa (x^\\nu - x^\\mu )\\right]f(x,Q_0^2)\\,\\mbox{,}$ where $\\delta $ , $\\mu $ , $\\nu $ , $\\kappa $ are free parameters to be determined in the fit and $f(x,Q_0^2)$ is the corresponding unpolarized PDF which was taken from Ref. [183].", "For the description of the fragmentation into a pion, the HKNS07 set [184] is used.", "Error estimates are handled via standard Hessian approach, but a tolerance $\\Delta \\chi ^2=12.95$ is assumed in order for the distribution of $\\chi ^2$ values between different experiments in the global fit to be reasonable.", "The AAC08 parton set is publicly available together with a FORTRAN program which allows for computing PDF central values and uncertainties.", "LSS10.", "The LSS10 parton set [77] is based on world-available data from incluisve and semi-inclusive DIS.", "These allow for a determination of light antiquarks.", "Six polarized PDFs are parametrized according to Eq.", "(REF ), namely the total PDF combinations $\\Delta u^+$ and $\\Delta d^+$ , the antiquarks $\\Delta \\bar{u}$ , $\\Delta \\bar{d}$ and $\\Delta \\bar{s}$ and the gluon $\\Delta g$ (for the four latter $\\rho _i=0$ in Eq.", "(REF )).", "The assumption $\\Delta s=\\Delta \\bar{s}$ is also made.", "Hessian error propagation is performed assuming $\\Delta \\chi ^2=1$ .", "Fragmentations functions are taken from the DSS07 analysis [185], [186].", "Similarly to the JAM13 fit, the theoretical QCD analysis takes into account the $1/Q^2$ terms, arising from kinematic target mass corrections and dynamic higher twist corrections, in the expression of the nucleon spin structure function $g_1$ .", "The LSS10 parton set is publicly available together with a FORTRAN program which allows for computing PDF central values.", "DSSV.", "Different PDF sets belong to the DSSV family, due to the remarkable effort put by this collaboration in updating their polarized parton sets with new available data.", "The first global analysis to include polarized collider measurements at RHIC, besides inclusive and semi-inclusive polarized DIS data, was DSSV08 [74], [75].", "Semi-inclusive pion and single-inclusive jet production data were considered in their study.", "This was recently updated by the DSSV+ and DSSV++ fits [187], [188]; in particular, the latter includes the most recent jet production data at RHIC, which were found to constrain the gluon shape in the mid-to-large $x$ region with unprecedented accuracy.", "Six independent polarized PDFs are determined from data, as in the LSS10 analysis, and the DSS07 fragmentation functions are used.", "The uncertainty estimates are provided through the Lagrange multiplier method described in Sec.", "with the conservative assumption $\\Delta \\chi ^2/\\chi ^2=2\\%$ , even though the standard Hessian approach, with $\\Delta \\chi ^2=1$ , is also used for comparison.", "The DSSV08 set is publicly available as 38 hessian eigenvector sets (a set for each minimized parameter for each direction of variation) plus a central value set.", "Table: Main features of the available polarized PDF fits.In Tab.", "REF , we summarize the features of the available polarized parton sets; a detailed comparison of polarized PDF between them and the NNPDF determination will be discussed in Chap.", "." ], [ "Unbiased polarized PDFs from inclusive DIS", "In this Chapter, we present the first determination of polarized parton distributions based on the NNPDF methodology, NNPDFpol1.0.", "This analysis includes all available data from inclusive, neutral-current, polarized DIS and aims at an unbiased extraction of total quark-antiquark and gluon distributions at NLO accuracy.", "In Sec.", "we present the data sets used in the present analysis, and we discuss how their statistical distribution is sampled with Monte Carlo generation of pseudodata.", "We provide details of the QCD analysis in Sec.", ", then we discuss the PDF parametrization in terms of neural networks in Sec.", "; we give particular emphasis on the minimization strategy and its peculiarities in the polarized case.", "The NNPDFpol1.0 parton set is presented in Sec.", ", where we illustrate its statistical features, and its stability upon the variation of several theoretical and methodological assumptions.", "We also compare our results to other recent polarized PDF sets reviewed in Sec. .", "Finally, we discuss phenomenological implications for the spin content of the proton and the test of the Bjorken sum rule in Sec. .", "The analysis presented in this Chapter mostly reproduces Ref.", "[116]." ], [ "Experimental input", "We present the features of the experimental data sets included in the NNPDFpol1.0 analysis and we discuss in detail which piece of information they provide on the polarized structure functions.", "Then, we summarize the construction and the validation of the Monte Carlo pseudodata sample from the input experimental data.", "We consider inclusive, neutral current, lepton-nucleon DIS data coming from-all-over-the-world experiments performed at CERN [50], [189], [190], [191], [192], SLAC [152], [193], [194] and DESY [195], [196].", "These experiments use either electron or muon beams, and either proton or neutron (deuteron or $^3$ He) targets.", "The main features of the data sets included in our analysis are summarized in Tab.", "REF , where we show, for each of them, the number of available data points, the covered kinematic range, and the published observable we use to reconstruct the $g_1$ structure function.", "Their kinematic coverage in the $(x,Q^2)$ plane is also shown in Fig.", "REF .", "[p] Table: NO_CAPTION Experimental data sets included in the NNPDFpol1.0 analysis.", "For each experiment we show the number of data points before and after (in parenthesis) applying kinematic cuts, the covered kinematic range and the measured observable.Experimental data sets included in the NNPDFpol1.0 analysis.", "For each experiment we show the number of data points before and after (in parenthesis) applying kinematic cuts, the covered kinematic range and the measured observable.", "Figure: Experimental data in the (x,Q 2 )(x,Q^2) plane(after kinematic cuts): black points are from CERN experiments , , , , , bluefrom SLAC , , and red fromDESY , .This quantity differs experiment by experiment, since the primary observable can be one of the asymmetries or structure functions discussed in Sec. .", "In the following, we summarize how we reconstruct the $g_1$ structure function from the published experimental observables for individual experiments (labelled as in Tab.", "REF ).", "EMC, SMC, SMClowx, COMPASS, HERMES97.", "All these experiments have performed a measurement of $A_\\parallel $ , then they determined $A_1$ using Eq.", "(REF ), under the assumption $\\eta \\approx 0$ .", "Therefore, the observable published by these experiments actually corresponds to a measurement of $\\frac{A_\\parallel }{D}$ .", "We determine $g_1$ from $\\frac{A_\\parallel }{D}$ using Eq.", "(REF ).", "This is possible because $D$ is completely fixed by Eq.", "() in terms of the unpolarized structure function ratio Eq.", "() and of the kinematics.", "We determine the unpolarized structure function ratio using as primary inputs $F_2$ , for which we use the parametrization of Ref.", "[120], [99], and $F_L$ , which we determine from its expression in terms of parton distributions, using the NLO NNPDF2.1 parton set [107].", "HERMES.", "This experiment has performed a measurement of $A_\\parallel $ , and it published both $A_\\parallel $ and $A_1$ (which is determined using Eq.", "(REF ) and a parametrization of $A_2$ ).", "We use the published values of $A_\\parallel $ , which are closer to the experimentally measured quantity, to determine $g_1$ through Eq.", "(REF ).", "E143.", "This experiment has taken data with three different beam energies, $E_1=29.1$ GeV, $E_2=16.2$ GeV, $E_3=9.7$ GeV.", "For the highest energy both $A_\\parallel $ and $A_\\perp $ are independently measured and $A_1$ is extracted from them using Eq.", "(REF ); for the two lowest energies only $A_\\parallel $ is measured and $A_1$ is extracted from it using Eqs.", "(REF -), while assuming the form Eq.", "(REF ) for $g_2$ .", "The values of $A_1$ obtained with the three beam energies are combined into a single determination of $A_1$ ; radiative corrections are applied at this combination stage.", "Because of this, we should use this combined value of $A_1$ , from which we then determine $g_1$ using Eq.", "(REF ).", "In order to determine $y$ , Eq.", "(REF ), which depends on the beam energy, we use the mean of the three energies.", "E154.", "This experiment measures $A_\\parallel $ and $A_\\perp $ independently, and then extracts a determination of $A_1$ .", "We use these values of $A_1$ to determine $g_1$ by means of Eq.", "(REF ).", "E155.", "This experiment only measures the longitudinal asymmetry $A_\\parallel $ , from which the ratio $g_1/F_1$ is extracted using Eq.", "(REF ) with the Wandzura-Wilczek form of $g_2$ , Eq.", "(REF ).", "In this case, we use these values of $g_1/F_1$ , and we extract $g_1$ using Eq.", "(REF ) for $F_1$ , together with the parametrization of Ref.", "[120], [99] for $F_2$ and the expression in terms of parton distributions and the NLO NNPDF2.1 parton set [107] for $F_L$ .", "All these experiments also provide an extraction of the $g_1$ structure function in their specific framework, based on different assumptions on $g_2$ .", "For this reason, we preferred to use the experimental asymmetries, instead of the corresponding structure functions, which instead we reconstructed on our own.", "We checked that they are consistent with those provided by the experimental collaborations themselves within their uncertainties.", "We have excluded from our analysis all data points with $Q^2 \\le Q^2_{\\mathrm {cut}}=1$ GeV$^2$ , since below such energy scale perturbative QCD cannot be considered reliable.", "A similar choice of cut is usually made in all polarized analyses, specifically in Refs.", "[143], [66], [67], [75], [76], [73].", "Notice that this value of $Q^2_{\\mathrm {cut}}$ is somewhat lower than that adopted in unpolarized fits, where $Q^2_{\\mathrm {cut}}=2$ GeV$^2$ .", "This difference arises not to exclude a large piece of experimental information which, in the polarized case, lies in the small-$Q^2$ region (see Fig.", "REF ).", "We further impose a cut on the squared invariant mass of the hadronic final state $W^2=Q^2(1-x)/x$ in order to remove points which may be affected by sizable dynamical higher-twist corrections.", "The cut is chosen based on a study presented in Ref.", "[197], where higher twist terms were added to the observables, with a coefficient fitted to the data: it was shown there that the higher twist contributions become compatible with zero if one imposes the cut $W^2 \\ge W^2_{\\mathrm {cut}}=6.25$ GeV$^2$ .", "We will follow this choice, which excludes data points with large Bjorken-$x$ at moderate values of the squared momentum transfer $Q^2$ , roughly corresponding to the bottom-right corner of the $(x,Q^2)$ -plane, see Fig.", "REF : in particular, it excludes all available JLAB data [198], [199], [200].", "The number of data points surviving the kinematic cuts for each data set is given in parenthesis in Tab.", "REF .", "As can be seen from Fig.", "REF , the region of the $(x,Q^2)$ -plane where data are available after kinematic cuts is roughly restricted to $4\\cdot 10^{-3}\\lesssim x\\lesssim 0.6$ and 1 GeV$^2\\le Q^2\\lesssim 60$ GeV$^2$ .", "In recent years, the coverage of the low-$x$ region has been improved by a complementary set of SMC data [190] and by the more recent COMPASS data [191], [192], both included in the present analysis.", "In the large-$x$ region, information is provided at rather high $Q^2$ by the same COMPASS data and at lower energy by the latest HERMES measurements [196].", "In the near future, additional polarized inclusive DIS measurements are expected from an update of COMPASS data [201] and from JLAB spin program [78].", "However, the latter will cover the large-$x$ and small $Q^2$ corner, hence their inclusion in a global fit will require a careful treatment of higher-twist corrections, as performed in Ref. [78].", "The data set used in this paper is the same as that of Ref.", "[76], and also the same as the DIS data of the fit of Ref.", "[75], which however has a wider data set beyond inclusive DIS.", "Each experimental collaboration provides uncertainties on the measured quantities listed in the next-to-last column of Tab.", "REF .", "Correlated systematics are only provided by EMC and E143, which give the values of the systematics due to the uncertainty in the beam and target polarizations, while all other experiments do not provide any piece of information on the covariance matrix.", "For each experiment, we determine the uncorrelated uncertainty on $g_1$ by combining the uncertainty on the experimental observable with that of the unpolarized structure function using standard error propagation.", "For EMC and E143 experiments, we also include all available correlated systematics.", "These are provided by both experimental collaborations as a percentage correction to $g_1$ (or, alternatively, to the asymmetry $A_1$ ): we apply the percentage uncertainty on $g_1$ to the structure function determined by us as discussed in Sec. .", "We then construct the covariance matrix $\\mathrm {cov}_{pq}=\\left(\\sum _i\\sigma ^{(c)}_{i,p}\\sigma ^{(c)}_{i,q} + \\delta _{pq} \\sigma ^{(u)}_{p}\\sigma ^{(u)}_{q}\\right)g_{1,p}g_{1,q}\\,\\mbox{,}$ where $p$ and $q$ run over the experimental data points, $g_{1,p}\\equiv g_1(x_p,Q_p^2)$ ($g_{1,q}\\equiv g_1(x_q,Q_q^2)$ ), $\\sigma ^{(c)}_{i,p}$ are the various sources of correlated uncertainties, and $\\sigma ^{(u)}_{p}$ are the uncorrelated uncertainties, which are in turn found as a sum in quadrature of all uncorrelated sources of statistical $\\sigma ^{\\mathrm {(stat)}}_{i,p}$ and systematic $\\sigma ^{\\mathrm {(syst)}}_{i,p}$ uncertainties on each point: $\\left(\\sigma ^{(u)}_{p}\\right)^2=\\sum _i \\left(\\sigma ^{\\mathrm {(stat)}}_{i,p}\\right)^2 +\\sum _j\\left(\\sigma ^{\\mathrm {(syst)}}_{j,p}\\right)^2\\,\\mbox{.", "}$ The correlation matrix is defined as $\\rho _{pq}=\\frac{{\\mathrm {cov}}_{pq}}{\\sigma ^{\\mathrm {(tot)}}_{p}\\sigma ^{\\mathrm {(tot)}}_{q}g_{1,p}g_{1,q}}\\,\\mbox{,}$ where the total uncertainty $\\sigma ^{\\mathrm {(tot)}}_{p}$ on the $p$ -th data point is $\\left(\\sigma ^{\\mathrm {(tot)}}_{p}\\right)^2 =(\\sigma ^{(u)}_{p})^2+\\sum _i\\left(\\sigma ^{(c)}_{i,p}\\right)^2\\,\\mbox{.", "}$ In Tab.", "REF , we show the average experimental uncertainties for each data set, with uncertainties separated into statistical and correlated systematics.", "All values are given as absolute uncertainties and refer to the structure function $g_1$ , which has been reconstructed for each experiment as discussed above.", "As in the case of Tab.", "REF , we provide the values before and after kinematic cuts (if different).", "Table: Averaged statistical, correlated systematic and totaluncertainties before and after (in parenthesis) kinematic cuts for each ofthe experimental sets included in the present analysis.", "Uncorrelated systematicuncertainties are considered as part of the statistical uncertaintyand they are added in quadrature.All values are absolute uncertainties and refer to the structure function g 1 g_1,which has been reconstructed for each experiment as discussed in the text.Details on the number of points andthe kinematics of each data set are provided in Tab.", ".Finally, notice that in both Tabs.", "REF -REF we distinguish between experiments, defined as groups of data which cannot be correlated to each other, and data sets within a given experiment, which could in principle be correlated with each other, as they correspond to measurements of different observables in the same experiment, or measurements of the same observable in different years.", "Even though, in practice, only two experiments provide such correlated systematics (see Tab.", "REF ), this distinction will be useful in the minimization strategy, see Sec.", "below." ], [ "Monte-Carlo generation of the pseudo-data sample", "Error propagation from experimental data to the fit is handled by a Monte Carlo sampling of the probability distribution defined by data, as discussed in Sec. .", "The statistical sample is obtained by generating $N_{\\mathrm {rep}}$ pseudodata replicas, according to a multi-Gaussian distribution centered at the data points and with a covariance equal to that of the original data.", "Explicitly, given an experimental data point $g_{1,p}^{(\\mathrm {exp})}\\equiv g_1(x_p,Q_p^2)$ , we generate $k=1,\\dots ,N_\\mathrm {rep}$ artificial points $g_{1,p}^{(\\mathrm {art}),(k)}$ according to $g_{1,p}^{(\\mathrm {art}),(k)} (x,Q^2)=\\left[1+\\sum _i r_{(c),p}^{(k)}\\sigma _{i,p}^{(c)} + r_{(u),p}^{(k)}\\sigma _p^{(u)}\\right]g_{1,p}^{(\\mathrm {exp})} (x,Q^2)\\,\\mbox{,}$ where $r_{(c),p}^{(k)}$ , $r_{(u),p}^{(k)}$ are univariate Gaussianly distributed random numbers, and $\\sigma _{i,p}^{(c)}$ and $\\sigma _{p}^{(u)}$ are respectively the relative correlated systematic and statistical uncertainty.", "Unlike in the unpolarized case, Eq.", "(REF ) receives no contribution from normalization uncertainties, given that all polarized observables are obtained as cross-section asymmetries.", "The number of Monte Carlo replicas of the data is determined by requiring that the central values, uncertainties and correlations of the original experimental data can be reproduced to a given accuracy by taking averages, variances and covariances over the replica sample.", "A comparison between expectation values and variances of the Monte Carlo set and the corresponding input experimental values as a function of the number of replicas is shown in Fig.", "REF , where we display scatter plots of the central values and errors for samples of $N_{\\mbox{\\scriptsize {rep}}}=10,100$ and 1000 replicas.", "A more quantitative comparison can be performed by defining suitable statistical estimators (see, for example, Appendix B of Ref. [99]).", "In Tabs.", "REF -REF we show the percentage error and the scatter correlation $r$ for central values and errors respectively, whose definition is recalled in appendix .", "The scatter correlation $r$ is, crudely speaking, the correlation between the input value and the value computed from the replica sample.", "We do not compute values for correlations, as these are available for a small number of data points from only two experiments, see Tab.", "REF .", "Some large values of the percentage uncertainty are due to the fact that, for some experiments, $g_1$ can take values which are very close to zero.", "It is clear from both the tables and the plots that a Monte Carlo sample of pseudodata with $N_\\mathrm {rep}=100$ is sufficient to reproduce the mean values and the errors of experimental data to an accuracy which is better than $5\\%$ , while the improvement in going up to $N_\\mathrm {rep}=1000$ is moderate.", "Therefore, we will henceforth use a $N_\\mathrm {rep}=100$ replica sample as a default for our reference fit.", "Figure: Scatter plot of experimental versus artificial Monte Carlo mean central values and absolute uncertainties of polarizedstructure functions computed fromensembles made of N rep =10,100,1000N_{\\mbox{\\scriptsize {rep}}}=10,100,1000 replicas.Table: Table of statistical estimators for the mean value computed fromthe Monte Carlo sample with N rep =10,100,1000N_\\mathrm {rep}=10,100,1000 replicas.Estimators refer to individual experiments and are defined in Appendix B of Ref.", ".Table: Same as Tab.", ", but for errors.We will now briefly outline some details of the QCD analysis of polarized structure functions.", "The observable with which we fit experimental data is the $g_1$ structure function, Eq.", "(REF ), expressed in terms of the PDF combinations in Eqs.", "(REF )-().", "From these relations, supplemented with Eqs.", "(REF )-(REF ), it is clear that neutral-current $g_1$ data only allow for a direct determination of the four polarized PDF combinations $\\Delta \\Sigma $ , $\\Delta T_3$ , $\\Delta T_8$ and $\\Delta g$ : these will form the basis of polarized PDFs to be determined in our analysis.", "In principle, an intrinsic polarized component could also be present for each heavy flavor, as observed in Sec. .", "However, we will neglect it here and assume that heavy quark PDFs are dynamically generated above threshold by (massless) Altarelli-Parisi evolution, in a zero-mass variable-flavor number (ZM-VFN) scheme.", "In such a scheme all heavy quark mass effects are neglected.", "While they can be introduced for instance through the FONLL method [124], these effects have been shown to be relatively small already on the scale of present-day unpolarized PDF uncertainties, and thus are most likely negligible in the polarized case where uncertainties are rather larger.", "We will further comment on this issue in Sec.", "REF and Appendix , where we will sketch how to handle intrinsic charm contribution via the FONLL scheme.", "The proton and neutron PDFs are related to each other by isospin, which we will assume to be exact, thus yielding $\\Delta u^p=\\Delta d^n,\\quad \\Delta d^p=\\Delta u^n, \\quad \\Delta s^p=\\Delta s^n\\,\\mbox{,}$ and likewise for the polarized anti-quarks.", "In the following we will always assume that PDFs refer to the proton.", "As discussed at length in Sec.", "REF , beyond leading order in QCD the first moment of all non-singlet combinations of quark and antiquark distributions are scale independent due to axial current conservation.", "Besides, we enforce $SU(2)$ and $SU(3)$ flavor asymmetry by requiring the first moments of the non-singlet, $C$ -even, combinations, Eqs.", "(REF ), to be fixed by the experimental values of baryon octet decay constants, Eqs.", "(REF ).", "Actually, a much larger uncertainty on the octet axial charge, up to about 30%, is found if SU(3) symmetry is violated [160] with respect to that quoted in Eqs.", "(REF ).", "Even though a detailed phenomenological analysis does not seem to support this conclusion [161], we will take as default this more conservative uncertainty estimation $a_8 = 0.585 \\pm 0.176\\,\\mbox{.", "}$ The impact of replacing this with the more aggressive determination given in Eq.", "(REF ) will be studied in Sec.", "REF .", "Structure functions will be computed in terms of polarized parton distributions using the so-called NNPDF FastKernel method, introduced in Ref. [104].", "In short, in this method the PDFs at scale $Q^2$ are obtained by convoluting the parton distributions at the parametrization scale $Q_0^2$ with a set of Green's functions, which are in turn obtained by solving the QCD evolution equations in Mellin space (below also denoted as $N$ -space).", "These Green's functions are then convoluted with coefficient functions, so that the structure function can be directly expressed in terms of the PDFs at the parametrization scale through suitable kernels $K$ .", "In terms of the polarized PDFs at the input scale (labelled with the subscript 0) we have $g_1^p=\\left\\lbrace K_{g_1, \\Delta \\Sigma }\\otimes \\Delta \\Sigma _0+K_{g_1, \\Delta g} \\otimes \\Delta g_0+K_{g_1, +} \\otimes \\left(\\Delta T_{3,0}+ \\frac{1}{3}\\Delta T_{8,0} \\right)\\right\\rbrace \\,\\mbox{,}$ where the kernels $K_{g_1, \\Delta \\Sigma }$ , $K_{g_1, \\Delta g}$ , $K_{g_1, +}$ take into account both the coefficient functions and $Q^2$ evolution.", "This way of expressing structure functions is amenable to numerical optimization, because all kernels can then be precomputed and stored, and convolutions may be reduced to matrix multiplications by projecting onto a set of suitable basis functions.", "The neutron polarized structure function $g_1^n$ is given in terms of the proton and deuteron ones as $g_1^n = 2\\frac{g_1^d}{1-1.5\\omega _D}-g_1^p\\,\\mbox{,}$ with $\\omega _D=0.05$ the probability that the deuteron is found in a D state.", "Under the assumption of exact isospin symmetry, the expression of $g_1^n$ in terms of parton densities is obtained from Eq.", "(REF ) by interchanging the up and down quark PDFs, which amounts to changing the sign of $\\Delta T_3$ .", "We will assume the values $\\alpha _s(M_Z^2)=0.119$ for the strong coupling constant and $m_c=1.4$ GeV and $m_b=4.75$ GeV for the charm and bottom quark masses respectively.", "We have benchmarked our implementation of the evolution of polarized parton densities up to NLO by cross-checking against the Les Houches polarized PDF evolution benchmark tables [202].$\\endcsname $Note that in Ref.", "[202] the polarized sea PDFs are given incorrectly, and should be $x\\Delta \\bar{u}=-0.045 x^{0.3} (1-x)^7$ and $x\\Delta \\bar{d}=-0.055 x^{0.3} (1-x)^7$ .", "These tables were obtained from a comparison of the HOPPET [203] and PEGASUS [134] evolution codes, which are $x-$ space and $N-$ space codes respectively.", "In order to perform a meaningful comparison, we use the so-called iterated solution of the $N-$ space evolution equations and use the same initial PDFs and running coupling as in [202].", "The relative difference $\\epsilon _{\\mathrm {rel}}$ between our PDF evolution and the benchmark tables of Refs.", "[202] at NLO in the ZM-VFNS are tabulated in Tab.", "REF for various combinations of polarized PDFs: the accuracy of our code is $\\mathcal {O}(10^{-5})$ for all relevant values of $x$ , which is the nominal accuracy of the agreement between HOPPET and PEGASUS.", "Table: Percentage difference between FastKernel perturbativeevolution of polarized PDFs and the Les Houches benchmarktables for differentpolarized PDF combinations at NLO in the ZM-VFNS.Therefore, we can conclude that the accuracy of the polarized PDF evolution in the FastKernel framework is satisfactory for precision phenomenology.", "Finally, we include exactly all kinematic target mass corrections within the formalism presented in Sec. .", "However, we notice that the numerical implementation of Eqs.", "(REF ) or Eq.", "(REF ) is difficult, because of the presence of the first derivative of $g_1$ in the correction term.", "For this reason, we do not factorize TMCs into the hard kernels, as it was done in the unpolarized case, where the first derivative of $F_1$ does not appear [101].", "Rather, we will include target mass effects in an iterative way: we start by performing a fit in which we set $M=0$ and at each iteration of the minimization procedure the target mass corrected $g_1$ structure function is computed by means of Eqs.", "(REF -REF ) using the $g_1$ obtained in the previous minimization step.", "We found that this strategy allows for convergence in a few minimization steps, hence TMCs are properly included when the fit is stopped." ], [ "Fitting strategy", "We will now discuss some details of the fitting strategy adopted in the NNPDFpol1.0 analysis.", "In particular, we describe how PDFs are parametrized in terms of neural networks and how they are trained to experimental data to obtain the optimal fit.", "The main steps of the minimization strategy have been summarized in Sec.", ", and can be found in Ref. [104].", "Here, we emphasize some specific features which were introduced to deal with issues peculiar to the polarized case, in particular with respect to the implementation of theoretical constraints." ], [ "Neural network parametrization", "The four independent polarized PDF flavor combinations in the evolution basis, $\\Delta \\Sigma $ , $\\Delta T_3$ and $\\Delta T_8$ , and the gluon $\\Delta g$ are separately parametrized using a multi-layer feed-forward neural network [108].", "All neural networks have the same architecture, namely 2-5-3-1, which corresponds to 37 free parameters for each PDF, and thus a total of 148 free parameters.", "This is to be compared to about 10-15 free parameters for all other available determinations of polarized PDFs within the standard methodology, see Sec. .", "This parametrization has been explicitly shown to be redundant in the unpolarized case, in that results are unchanged when a smaller neural network architecture is adopted: this ensures that results do not depend on the architecture [108].", "Given that polarized data are much less abundant and affected by much larger uncertainties than their unpolarized counterparts, this architecture is surely adequate in the polarized case too.", "The neural network parametrization is supplemented with a preprocessing function.", "In principle, large enough neural networks can reproduce any functional form given sufficient training time.", "However, the training can be made more efficient by adding a preprocessing step, i.e.", "by multiplying the output of the neural networks by a fixed function.", "The neural network then only fits the deviation from this function, which improves the speed of the minimization procedure if the preprocessing function is suitably chosen.", "We thus write the input PDF basis in terms of preprocessing functions and neural networks ${\\mathrm {NN}}_{\\mathrm {\\Delta pdf}}$ as follows $\\Delta \\Sigma (x,Q_0^2)&=&{(1-x)^{m_1}}{x^{-n_1}}{\\mathrm {NN}}_{\\Delta \\Sigma }(x)\\,\\mbox{,}\\nonumber \\\\\\Delta T_3(x,Q_0^2)&=&A_3{(1-x)^{m_3}}{x^{-n_3}}{\\mathrm {NN}}_{ \\Delta T_3}(x)\\,\\mbox{,}\\nonumber \\\\\\Delta T_8(x,Q_0^2)&=&A_8{(1-x)^{m_8}}{x^{-n_{ \\Delta T_8}}}{\\mathrm {NN}}_{ \\Delta T_3}(x)\\,\\mbox{,}\\\\\\Delta g(x,Q_0^2)&=&{(1-x)^{m_g}}{x^{-n_g}}{\\mathrm {NN}}_{\\Delta g}(x)\\,\\mbox{.", "}\\nonumber $ Of course, one should check that no bias is introduced in the choice of preprocessing functions.", "To this purpose, we first select a reasonable range of values for the large and small-$x$ preprocessing exponents $m$ and $n$ , and produce a PDF determination by choosing for each replica a value of the exponents at random with uniform distribution within this range.", "We then determine effective exponents for each replica, defined as $m_{\\mathrm {eff}}(Q^2)\\equiv \\lim _{x\\rightarrow 1}\\frac{\\ln \\Delta f(x,Q^2) }{\\ln (1-x)}\\,\\mbox{,}$ $n_{\\mathrm {eff}}(Q^2)\\equiv \\lim _{x\\rightarrow 0} \\frac{\\ln \\Delta f(x,Q^2)}{\\ln \\frac{1}{x}}\\,\\mbox{,}$ where $\\Delta f = \\Delta \\Sigma \\mbox{, }\\Delta T_3\\mbox{, }\\Delta T_8\\mbox{, }\\Delta g$ .", "Finally, we check that the range of variation of the preprocessing exponents is wider than the range of effective exponents for each PDF.", "If it is not, we enlarge the range of variation of preprocessing, then repeat the PDF determination, and iterate until the condition is satisfied.", "This ensures that the range of effective large- and small-$x$ exponents found in the fit is not biased, and in particular not restricted, by the range of preprocessing exponents.", "Our final values for the preprocessing exponents are summarized in Tab.", "REF , while the effective exponents obtained in our fit will be discussed in Sec.", "REF .", "Table: Ranges for the small and large xxpreprocessing exponents Eq.", "().It is apparent from Tab.", "REF that the allowed range of preprocessing exponents is rather wider than in the unpolarized case, as a consequence of the limited amount of experimental information.", "The nonsinglet triplet and octet PDF combinations in the parametrization basis, Eq.", "(REF ), $\\Delta T_3$ and $\\Delta T_8$ , are supplemented by a prefactor.", "This is because these PDFs must satisfy the sum rules Eqs.", "(REF ), which are enforced by letting $A_3&=&\\frac{a_3}{\\int _0^1 dx\\,(1-x)^{m_3}x^{-n_3} {\\mathrm {NN}}_{\\Delta T_3}(x)}\\,\\mbox{,}\\nonumber \\\\A_8&=&\\frac{a_8}{\\int _0^1 dx\\, (1-x)^{m_8}x^{-n_8} {\\mathrm {NN}}_{\\Delta T_8}(x) }\\,\\mbox{.", "}$ The integrals are computed numerically each time the parameters of the PDF set are modified.", "The values of $a_3$ and $a_8$ are chosen for each replica as Gaussianly distributed numbers, with central value and width given by the corresponding experimental values, Eqs.", "(REF )-(REF )." ], [ "Genetic algorithm minimization", "As discussed at length in Ref.", "[101] and summarized in Sec.", ", minimization with a neural network parametrization of PDFs must be performed through an algorithm which explores the very wide functional space efficiently.", "This is done by means of a genetic algorithm, which is used to minimize a suitably defined figure of merit, namely the error function [101], $E^{(k)}=\\frac{1}{N_{\\mathrm {dat}}}\\sum _{I,J=1}^{N_{\\mathrm {dat}}}\\left(g_I^{(\\mathrm {art})(k)}-g_I^{(\\mathrm {net})(k)}\\right)\\left(\\left({\\mathrm {cov}}\\right)^{-1}\\right)_{IJ}\\left(g_J^{(\\mathrm {art})(k)}-g_J^{(\\mathrm {net})(k)}\\right)\\,\\mbox{.", "}$ Here $g_I^{\\mathrm {(art)}(k)}$ is the value of the observable $g_I$ at the kinematical point $I$ corresponding to the Monte Carlo replica $k$ , and $g_I^{\\mathrm {(net)}(k)}$ is the same observable computed from the neural network PDFs; the covariance matrix $\\left({\\mathrm {cov}}\\right)_{IJ}$ is defined in Eq.", "(REF ).", "The minimization procedure we adopt follows closely that of Ref.", "[100], to which we refer for a more general discussion.", "Minimization is perfomed by means of a genetic algorithm, which minimizes the figure of merit, Eq.", "(REF ) by generating, at each minimization step, a pool of new neural nets, obtained by randomly mutating the parameters of the starting set, and retaining the configuration which corresponds to the lowest value of the figure of merit.", "The parameters which characterize the behavior of the genetic algorithm are tuned in order to optimize the efficiency of the minimization procedure.", "We essentially rely on previous experience of the development of unpolarized NNPDF sets: in particular, the algorithm is characterized by a mutation rate, which decreases as a function of the number of the algorithm iterations $N_{\\mathrm {ite}}$ according to the law [101] $\\eta _{i,j}=\\eta _{i,j}^{(0)}/N_{\\mathrm {ite}}^{r_\\eta }\\,\\mbox{.", "}$ This way, in the early stages of the training large mutations are allowed, while they become less likely as one approaches the minimum.", "The starting mutation rates are chosen to be larger for PDFs which contain more information.", "We perform two mutations per PDF at each step, with the starting rates given in Tab.", "REF .", "The exponent $r_\\eta $ has been introduced in order to optimally span the whole range of possible beneficial mutations and it is randomized between 0 and 1 at each iteration of the genetic algorithm, as in Ref. [104].", "Table: The initial values of the mutation rates for the twomutations of each PDF.Furthermore, following Ref.", "[104], we let the number of new candidate solutions depend on the stage of the minimization.", "At earlier stages of the minimization, when the number of generations is smaller than $N^{\\mathrm {mut}}$ , we use a large population of mutants, $N_{\\mathrm {mut}}^{a}\\gg 1$ , so a larger space of mutations is being explored.", "At later stages of the minimization, as the minimum is approached, a smaller number of mutations $N_{\\mathrm {mut}}^{b}\\ll N_{\\mathrm {mut}}^{a}$ is used.", "The values of the parameters $N_{\\mathrm {gen}}^{\\mathrm {mut}}$ , $N_{\\mathrm {mut}}^{a}$ and $N_{\\mathrm {mut}}^{b}$ are collected in Tab.", "REF .", "Table: Values of the parameters of the genetic algorithm.Because the minimization procedure stops the fit to all experiments at once, we must make sure that the quality of the fit to different experiments is approximately the same.", "This is nontrivial, because of the variety of experiments and data sets included in the fit.", "Therefore, the figure of merit per data point for a given set is not necessarily a reliable indicator of the quality of the fit to that set, because some experiments may have systematically underestimated or overestimated uncertainties.", "Furthermore, unlike for unpolarized PDF fits, information on the experimental covariance matrix is only available for a small subset of experiments, so for most experiments statistical and systematic errors must be added in quadrature, thereby leading to an overestimate of uncertainties: this leads to a wide spread of values of the figure of merit, whose value depends on the size of the correlated uncertainties which are being treated as uncorrelated.", "A methodology to deal with this situation was developed in Ref. [104].", "The idea is to first determine the optimal value of the figure of merit for each experiment, i.e.", "a set of target values $E_{i}^{\\mathrm {targ}}$ for each of the $i$ experiments, then during the fit give more weight to experiments for which the figure of merit is further away from its target value, and stop to train experiments which have already reached the target value.", "This is done by minimizing, instead of the figure of merit Eq.", "(REF ), the weighted figure of merit $E_{\\mathrm {wt}}^{(k)}=\\frac{1}{N_{\\mathrm {dat}}}\\sum _{j=1}^{N_{\\mathrm {sets}}}p_j^{(k)} N_{\\mathrm {dat},j}E_j^{(k)}\\,\\mbox{,}$ where $E_j^{(k)}$ is the error function for the $j$ -th data set with $N_{{\\mathrm {dat}},j}$ points, and the weights $p_j^{(k)}$ are given by If $E_{i}^{(k)} \\ge E_{i}^{\\mathrm {targ}}$ , then $p_i^{(k)}=\\left( E_{i}^{(k)}/E_{i}^{\\mathrm {targ}}\\right)^n$ , If $E_{i}^{(k)} < E_{i}^{\\mathrm {targ}}$ , then $p_i^{(k)}=0$ , with $n$ a free parameter which essentially determines the amount of weighting.", "In the unpolarized fits of Refs.", "[104], [106], [107], [110] the value $n=2$ was used.", "Here instead we will choose $n=3$ .", "This larger value, determined by trial and error, is justified by the wider spread of figures of merit in the polarized case, which in turn is related to the absence of correlated systematics for most experiments.", "The target values $E_{i}^{\\mathrm {targ}}$ are determined through an iterative procedure: they are set to one at first, then a very long fixed-length fit is run, and the values of $E_{i}$ are taken as targets for a new fit, which is performed until stopping (according to the criterion to be discussed in the following Section).", "The values of $E_{i}$ at the end of this fit are then taken as new targets until convergence is reached, usually after a couple iterations.", "Weighted training stops after the first $N_{\\mathrm {gen}}^{\\mathrm {wt}}$ generations, unless the total error function Eq.", "(REF ) is above some threshold $E^{(k)}\\ge E^{\\mathrm {sw}}$ .", "If it is, weighted training continues until $E^{(k)}$ falls below the threshold value.", "Afterwards, the error function is just the unweighted error function Eq.", "(REF ) computed on experiments.", "This ensures that the figure of merit behaves smoothly in the last stages of training.", "The values for the parameters $N_{\\mathrm {gen}}^{\\mathrm {wt}}$ and $E^{\\mathrm {sw}}$ are also given in Tab.", "REF ." ], [ "Determination of the optimal fit", "Because the neural network parametrization is extremely redundant, it may be able to fit not only the underlying behavior of the PDFs, but also the statistical noise in the data.", "Therefore, the best fit does not necessarily coincide with the absolute minimum of the figure of merit Eq.", "(REF ).", "We thus determine the best fit, as in Refs.", "[100], [101], using a cross-validation method [204]: for each replica, the data are randomly divided in two sets, training and validation, which include a fraction $f_{\\mathrm {tr}}^{(j)}$ and $f_{\\mathrm {val}}^{(j)}=1-f_{\\mathrm {tr}}^{(j)}$ of the data points respectively.", "The figure of merit Eq.", "(REF ) is then computed for both sets.", "The training figure of merit function is minimized through the genetic algorithm, while the validation figure of merit is monitored: when the latter starts increasing while the former still decreases, the fit is stopped.", "This means that the fit is stopped as soon as the neural network is starting to learn the statistical fluctuations of the points, which are different in the training and validation sets, rather than the underlying law which they share.", "In the unpolarized fits of Refs.", "[100], [101], [104], [106], [107], [110] equal training and validation fractions were uniformly chosen, $f_{\\mathrm {tr}}^{(j)}=f_{\\mathrm {val}}^{(j)}=1/2$ .", "However, in this case we have to face the problem that the number of data points is quite small: most experiments include about ten data points (see Tab.", "REF ).", "Hence, it is difficult to achieve a stable minimization if only half of them are actually used for minimization, as we have explicitly verified.", "Therefore, we have chosen to include 80% of the data in the training set, i.e.", "$f_{\\mathrm {tr}}^{(j)}=0.8$ and $f_{\\mathrm {val}}^{(j)}=0.2$ .", "We have explicitly verified that the fit quality which is obtained in this case is comparable to the one achieved when including all data in the training set (i.e.", "with $f_{\\mathrm {tr}}^{(j)}=1.0$ and $f_{\\mathrm {val}}^{(j)}=0.0$ ), but the presence of a nonzero validation set allows for a satisfactory stopping, as we have checked by explicit inspection of the profiles of the figure of merit as a function of training time.", "In practice, in order to implement cross-validation we must determine a stopping criterion, namely, give conditions which must be satisfied in order for the minimization to stop.", "First, we require that the weighted training stage has been completed, i.e., that the genetic algorithm has been run for at least $N_{\\mathrm {gen}}^{\\mathrm {wt}}$ minimization steps.", "Furthermore, we check that all experiments have reached a value of the figure of merit below a minimal threshold $E_{\\mathrm {thr}}$ .", "Note that because stopping can occur only after weighted training has been switched off, and this in turn only happens when the figure of merit falls below the value $E^{\\mathrm {sw}}$ , the total figure of merit must be below this value in order for stopping to be possible.", "We then compute moving averages $\\langle E_{\\mathrm {tr,val}}(i)\\rangle \\equiv \\frac{1}{N_{\\mathrm {smear}}}\\sum _{l=i-N_{\\mathrm {smear}}+1}^iE_{\\mathrm {wt;\\,tr,val}}(l)\\,\\mbox{,}$ of the figure of merit Eq.", "(REF ) for either the training or the validation set at the $l$ -th genetic minimzation step.", "The fit is then stopped if $r_{\\mathrm {tr}} < 1-\\delta _{\\mathrm {tr}}\\quad {\\mathrm {and}}\\quad r_{\\mathrm {val}} > 1+\\delta _{\\mathrm {val}}\\,\\mbox{,}$ where $r_{\\mathrm {tr}}\\equiv \\frac{\\langle E_{\\mathrm {tr}}(i)\\rangle }{\\langle E_{\\mathrm {tr}}(i-\\Delta _{\\mathrm {smear}})\\rangle }\\,\\mbox{,}$ $r_{\\mathrm {val}}\\equiv \\frac{\\langle E_{\\mathrm {val}}(i)\\rangle }{\\langle E_{\\mathrm {val}}(i-\\Delta _{\\mathrm {smear}})\\rangle }\\,\\mbox{.", "}$ The parameter $N_{\\mathrm {smear}}$ determines the width of the moving average; the parameter $\\Delta _{\\mathrm {smear}}$ determines the distance between the two points along the minimization path which are compared in order to determine whether the figure of merit is increasing or decreasing; and the parameters $\\delta _{\\mathrm {tr}}$ , $\\delta _{\\mathrm {val}}$ are the threshold values for the decrease of the training and increase of the validation figure of merit to be deemed significant.", "The optimal value of these parameters should be chosen in such a way that the fit does not stop on a statistical fluctuation, yet it does stop before the fit starts overlearning (i.e.", "learning statistical fluctuation).", "As explained in Ref.", "[104], this is done studying the profiles of the error functions for individual data set and for individual replicas.", "In order to avoid unacceptably long fits, training is stopped anyway when a maximum number of iterations $N_{\\mathrm {gen}}^{\\mathrm {max}}$ is reached, even though the stopping conditions Eqs.", "(REF ) are not satisfied.", "This leads to a small loss of accuracy of the corresponding fits: this is acceptable provided it only happens for a small enough fraction of replicas.", "If a fit stops at $N_{\\mathrm {gen}}^{\\mathrm {max}}$ without the stopping criterion having been satisfied, we also check that the total figure of merit is below the value $E^{\\mathrm {sw}}$ at which weighted training is switched off.", "If it hasn't, we conclude that the specific fit has not converged, and we retrain the same replica, i.e., we perform a new fit to the same data starting with a different random seed.", "This only occurs in about one or two percent of cases.", "The full set of parameters which determine the stopping criterion is given in Tab.", "REF .", "Table: Parameters for the stopping criterion.An example of how the stopping criterion works in practice is shown in Fig.", "REF .", "We display the moving averages Eq.", "(REF ) of the training and validation error functions $\\langle E_{\\mathrm {tr,val}}^{(k)}\\rangle $ , computed with the parameter settings of Tab.", "REF , and plotted as a function of the number of iterations of the genetic algorithm, for a particular replica and for two of the experiments included in the fit.", "The wide fluctuations which are observed in the first part of training, up to the $N_{\\mathrm {gen}}^{\\mathrm {wt}}$ -th generation, are due to the fact that the weights which enter the definition of the figure of merit Eq.", "(REF ) are frequently adjusted.", "Nevertheless, the downwards trend of the figure of merit is clearly visible.", "Once the weighted training is switched off, minimization proceeds smoothly.", "The vertical line denotes the point at which the stopping criterion is satisfied.", "Here, we have let the minimization go on beyond this point, and we clearly see that the minimization has entered an overlearning regime, in which the validation error function $E_{\\mathrm {val}}^{(k)}$ is rising while the training $E_{\\mathrm {tr}}^{(k)}$ is still decreasing.", "Note that the stopping point, which in this particular case occurs at $N_{\\mathrm {gen}}^{\\mathrm {stop}}=5794$ , is determined by verifying that the stopping criteria are satisfied by the total figure of merit, not that of individual experiments shown here.", "The fact that the two different experiments considered here both start overlearning at the same point shows that the weighted training has been effective in synchronizing the fit quality for different experiments.", "Figure: Behaviour of the moving average Eq.", "()of the training and validation figure of merit for two differentdata sets included in a global fit (COMPASS-P andHERMES) as a function of training length.", "The straight vertical lineindicates the point at which the fit stops with the stopping parametersof Tab. .", "The weightedtraining is switched off at N gen wt =5000N_{\\mathrm {gen}}^{\\mathrm {wt}}=5000." ], [ "Theoretical constraints", "Polarized PDFs are only loosely constrained by data, which are scarce and not very accurate.", "Theoretical constraints are thus especially important in reducing the uncertainty on the PDFs.", "We consider in particular positivity and integrability.", "Positivity of the individual cross-sections which enter the polarized asymmetries Eq.", "(REF ) implies that, up to power-suppressed corrections, longitudinal polarized structure functions are bounded by their unpolarized counterparts, i.e.", "$\|g_1(x,Q^2)\| \\le F_1(x,Q^2)\\,\\mbox{.", "}$ At leading order, structure functions are proportional to parton distributions, so imposing Eq.", "(REF ) for any process (and a similar condition on an asymmetry which is sensitive to polarized gluons [205]), would imply $\|\\Delta f_i(x,Q^2)\|\\le f_i(x,Q^2)$ for any pair of unpolarized and polarized PDFs $f$ and $\\Delta f$ , for all quark flavors and gluon $i$ , for all $x$ , and for all $Q^2$ .", "Beyond leading order, the condition Eq.", "(REF ) must still hold, but it does not necessarily imply Eq.", "(REF ).", "Rather, one should then impose at least a number of conditions of the form of Eq.", "(REF ) on physically measurable cross-sections which is equal to the number of independent polarized PDFs.", "For example, in principle one may require that the condition Eq.", "(REF ) is separately satisfied for each flavor, i.e.", "when only contributions from the $i$ -th flavor are included in the polarized and unpolarized structure function: this corresponds to requiring positivity of semi-inclusive structure functions which could in principle be measured (and that fragmentation effects cancel in the ratio).", "A condition on the gluon can be obtained by imposing positivity of the polarized and unpolarized cross-sections for inclusive Higgs production in gluon-proton scattering [205], again measurable in principle if not in practice.", "Because $g_1/F_1\\sim x$ as $x\\rightarrow 0$ [206], the positivity bound Eq.", "(REF ) is only significant at large enough $x\\gtrsim 10^{-2}$ .", "On the other hand, at very large $x$ the NLO corrections to the LO positivity bound become negligible [205], [207].", "Therefore, the NLO positivity bound in practice only differs from its LO counterpart Eq.", "(REF ) in a small region $10^{-2}\\lesssim x\\lesssim 0.3$ , and even there by an amount of rather less that 10% [205], which is negligible in comparison to the size of PDF uncertainties, as we shall see explicitly in Sec. .", "Therefore, we will impose the leading-order positivity bound Eq.", "(REF ) on each flavor combination $\\Delta q_i+\\Delta \\bar{q}_i$ and on the gluon $\\Delta g$ (denoted as $\\Delta f_i$ below).", "We do this by requiring $\|\\Delta f_i(x,Q^2)\| \\le f_i(x,Q^2) + \\sigma _i(x,Q^2)\\,\\mbox{,}$ where $\\sigma _i(x,Q^2)$ is the uncertainty on the corresponding unpolarized PDF combination $f_i(x,Q^2)$ at the kinematic point $(x,Q^2)$ .", "This choice is motivated by two considerations.", "First, it is clearly meaningless to impose positivity of the polarized PDF to an accuracy which is greater than that with which the unpolarized PDF has been determined.", "Second, because the unpolarized PDFs satisfy NLO positivity, they can become negative and thus they may have nodes.", "As a consequence, the LO bound Eq.", "(REF ) would imply that the polarized PDF must vanish at the same point, which would be clearly meaningless.", "As in Ref.", "[104] positivity is imposed during the minimization procedure, thereby guaranteeing that the genetic algorithm only explores the subspace of acceptable physical solutions.", "This is done through a Lagrange multiplier $\\lambda _{\\mathrm {pos}}$ , i.e.", "by computing the polarized PDF at $N_{\\mathrm {dat,pos}}$ fixed kinematic points $(x_p,Q_0^2)$ and then adding to the error function Eq.", "(REF ) a contribution $&&E_{\\mathrm {pos}}^{(k)}={\\lambda _{\\mathrm {pos}}}\\sum _{p=1}^{N_{\\mathrm {dat,pos}}}\\Bigg \\lbrace \\sum _{j=u+\\bar{u},d+\\bar{d},s+\\bar{s},g} \\Theta \\left[\\left\|\\Delta f_j^{\\mathrm {(net)}(k)}(x_p,Q_0^2)\\right\| -\\left(f_j + \\sigma _j\\right)(x_p,Q_0^2) \\right] \\nonumber \\\\&&\\qquad \\qquad \\times \\left[\\left\|\\Delta f_j^{\\mathrm {(net)}(k)}(x_p,Q_0^2)\\right\| -\\left(f_j + \\sigma _j\\right)(x_p,Q_0^2) \\right]\\Bigg \\rbrace \\,\\mbox{.", "}$ This provides a penalty, proportional to the violation of positivity, which enforces Eq.", "(REF ) separately for all the non-zero quark-antiquark combinations.", "The values of the unpolarized PDF combination $f_j(x,Q^2)$ and its uncertainty $\\sigma _j(x,Q^2)$ are computed using the NNPDF2.1 PDF set at NLO [106], while $\\Delta f_j^{\\mathrm {(net)}(k)}$ is the corresponding polarized PDF computed from the neural network parametrization for the $k$ -th replica.", "The polarized and unpolarized PDFs are evaluated at $N_{\\mathrm {dat,pos}}=20$ points with $x$ equally spaced in the interval $x \\in [10^{-2},0.9]\\,\\mbox{.", "}$ Positivity is imposed at the initial scale $Q_0^2=1$ GeV$^2$ since once positivity is enforced at low scales, it is automatically satisfied at larger scales [205], [207].", "After stopping, we finally test the positivity condition Eq.", "(REF ) is satisfied on a grid of $N_{\\mathrm {dat,pos}}=40$ points in the same intervals.", "Replicas for which positivity is violated in one or more points are discarded and retrained.", "In the unpolarized case, in which positivity only played a minor role in constraining PDFs, a fixed value of the Lagrange multiplier $\\lambda _{\\mathrm {pos}}$ was chosen.", "In the polarized case it turns out to be necessary to vary the Lagrange multiplier along the minimization.", "Specifically, we let $\\left\\lbrace \\begin{array}{rcll}\\lambda _{\\mathrm {pos}}& = &\\lambda _{\\mathrm {max}}^{(N_{\\mathrm {gen}}-1)/(N_{\\lambda _{\\mathrm {max}}}-1)} &N_{\\mathrm {gen}} < N_{\\lambda _{\\mathrm {max}}}\\\\\\lambda _{\\mathrm {pos}}& = &\\lambda _{\\mathrm {max}} & N_{\\mathrm {gen}} \\ge N_{\\lambda _{\\mathrm {max}}}.\\end{array}\\right.$ This means that the Lagrange multiplier increases as the minimization proceeds, starting from $\\lambda _{\\mathrm {pos}}=1$ , at the first minimization step, $N_{\\mathrm {gen}}=1$ , up to $\\lambda _{\\mathrm {pos}} = \\lambda _{\\mathrm {max}}\\gg 1$ when $N_{\\mathrm {gen}} = N_{\\lambda _{\\mathrm {max}}}$ .", "After $N_{\\lambda _{\\mathrm {max}}}$ generations $\\lambda _{\\mathrm {pos}}$ is then kept constant to $\\lambda _{\\mathrm {max}}$ .", "The rationale behind this choice is that the genetic algorithm can thus learn experimental data and positivity at different stages of minimization.", "During the early stages, the contribution coming from the modified error function Eq.", "(REF ) is negligible, due to the moderate value of the Lagrange multiplier; hence, the genetic algorithm will mostly learn the basic shape of the PDF driven by experimental data.", "As soon as the minimization proceeds, the contribution coming from the Lagrange multiplier increases, thus ensuring the proper learning of positivity: at this stage, most of the replicas which will not fulfill the positivity bound will be discarded.", "The final values of $N_{\\lambda _{\\mathrm {max}}}=2000$ and $\\lambda _{\\mathrm {max}}=10$ have been determined as follows.", "First of all, we have performed a fit without any positivity constraint and we have observed that data were mostly learnt in about 2000 generations: hence we have taken this value for $N_{\\lambda _{\\mathrm {max}}}$ .", "Then we have tried different values for $\\lambda _{\\mathrm {max}}$ until we managed to reproduce the same $\\chi ^2$ obtained in the previous, positivity unconstrained, fit.", "This ensures that positivity is not learnt to the detriment of the global fit quality.", "Notice that the value of $\\lambda _{\\mathrm {max}}$ is rather small if compared to the analogous Lagrange multiplier used in the unpolarized case [106].", "This depends on the fact that, in this latter case, positivity is learnt at the early stages of minimization, when the error function can be much larger than its asymptotic value: a large Lagrange multiplier is then needed to select the best replicas.", "Also, unpolarized PDFs are quite well constrained by data and positivity is almost automatically fulfilled, except in some restricted kinematic regions; only a few replicas violate positivity and need to be penalized.", "This means that the behavior of the error function Eq.", "(REF ), which governs the fitting procedure, is essentially dominated by data instead of positivity.", "In the polarized case, instead, positivity starts to be effectively implemented only after some minimizaton steps, when the error function has already decreased to a value of a few units.", "Furthermore, we have checked that, at this stage, most of replicas slightly violate the positivity condition Eq.", "(REF ): thus, a too large value of the Lagrange multiplier on the one hand would penalize replicas which are good in reproducing experimental data and only slightly worse in reproducing positivity; on the other, it would promote replicas which fulfill positivity but whose fit to data is quite bad.", "As a consequence of this behavior, the convergence of the minimization algorithm would be harder to reach.", "We also verified that, using a value for the Lagrange multiplier up to $\\lambda _{\\mathrm {pos}}=100$ leads to no significant improvement neither in the fulfillment of positivity requirement nor in the fit quality.", "We will show in detail the effects of the positivity bound Eq.", "(REF ) on the fitted replicas and on polarized PDFs in Sec. .", "Finally, we impose that PDFs are integrable, i.e.", "that they have finite first moments.", "This corresponds to the assumption that the nucleon matrix element of the axial current for the $i$ -th flavor is finite.", "The integrability condition is imposed by computing at each minimization step the integral of each of the polarized PDFs in a given interval, $I(x_1,x_2)=\\int _{x_1}^{x_2} dx~\\Delta q_i(x,Q_0^2) \\,\\qquad \\Delta q_i=\\Delta \\Sigma , \\Delta g, \\Delta T_3, \\Delta T_8$ with $x_1$ and $x_2$ chosen in the small $x$ region, well below the data points, and verifying that in this region the growth of the integral as $x_1$ decreases for fixed $x_2$ is less than logarithmic.", "In practice, we test for the condition $\\frac{I(x_1,x_2)}{I(x_1^\\prime ,x_2)} <\\frac{\\ln \\frac{x_2}{x_1}}{\\ln \\frac{x_2}{x_1\\prime }},$ with $x_1<x_1^\\prime $ .", "Mutations which do not satisfy the condition are rejected during the minimization procedure.", "In our default fit, we chose $x_1=10^{-5}$ , $x_1^\\prime =2\\cdot 10^{-5}$ and $x_2=10^{-4}$ .", "In this Section, we present the first determination of a polarized PDF set based on the NNPDF methodology, NNPDFpol1.0.", "We will first illustrate the statistical features of our PDF fit, then compare the PDFs in our set to those from other recent determinations introduced in Sec. .", "We will also discuss the stability of our results upon the variation of several theoretical and methodological assumptions, namely the treatment of target mass corrections, the use of sum rules to fix the nonsinglet axial charges, the effect of positivity constraints on polarized PDFs, and impact of preprocessing of neural networks on small- and large-$x$ PDF behavior." ], [ "Statistical features", "The statistical features of the NNPDFpol1.0 analysis are summarized in Tabs.", "REF -REF , for the full data set and for individual experiments and sets respectively.", "Table: Statistical estimators for NNPDFpol1.0 withN rep =100N_\\mathrm {rep}=100 replicas.Table: Same as Tab.", ", but for individual experiments.The mean value of the error function, Eq.", "(REF ), $\\langle E\\rangle $ , shown in the tables both for the total, training and validation data sets is the figure of merit for the quality of the fit of each PDF replica to the corresponding data replica.", "The quantity which is actually minimized during the neural network training is this figure of merit for the training set, supplemented by weighting in the early stages of training according to Eq.", "(REF ) and by a Lagrange multiplier to enforce positivity according to Eq.", "(REF ).", "In the table we also show the average over all replicas, $\\langle \\chi _{\\mathrm {tot}}^{2(k)}\\rangle $ , of $\\chi _{\\mathrm {tot}}^{2(k)}$ computed for the $k$ -th replica, which coincides with the figure of merit Eq.", "(REF ), but with the data replica $g_I^{\\mathrm {(art)}(k)}$ replaced by the experimental data $g_I^{\\mathrm {(dat)}}$ .", "We finally show $\\chi ^2_{\\mathrm {tot}}$ , which coincides with the figure of merit Eq.", "(REF ), but again with $g_I^{\\mathrm {(art)}(k)}$ replaced by $g_I^{\\mathrm {(dat)}}$ , and also with $g_I^{(\\mathrm {net})(k)}$ replaced by $\\langle g_I^{(\\mathrm {net})(k)}\\rangle $ , i.e.", "the average of the observable over replicas, which provides our best prediction.", "The average number of iterations of the genetic algorithm at stopping, $\\langle \\mathrm {TL}\\rangle $ , is also given in this table.", "The distribution of $\\chi ^{2(k)}$ , $E_{\\mathrm {tr}}^{(k)}$ , and training lengths among the $N_{\\mathrm {rep}}=100$ replicas are shown in Fig.", "REF and Fig.", "REF respectively.", "Note that the latter has a long tail which causes an accumulation of points at the maximum training length, $N_{\\mathrm {gen}}^{\\mathrm {max}}$ .", "This means that there is a fraction of replicas that do not fulfill the stopping criterion.", "This may cause a loss in accuracy in outlier fits, which however make up fewer than $10\\%$ of the total sample.", "Figure: Distribution of χ 2(k) \\chi ^{2(k)} and E tr (k) E_{\\mathrm {tr}}^{(k)} overthe sample of N rep =100N_{\\mathrm {rep}}=100 replicas.Figure: Distribution of training lengths over the sample ofN rep =100N_{\\mathrm {rep}}=100 replicas.The features of the fit can be summarized as follows: The quality of the central fit, as measured by its $\\chi _{\\mathrm {tot}}^{2}=0.77$ , is good.", "However, this value should be taken with care in view of the fact that uncertainties for all experiments but two are overestimated because the covariance matrix is not available and thus correlations between systematics cannot be properly accounted for.", "This explains the value lower than one for this quantity, which would be very unlikely if it had included correlations.", "The values of $\\chi _{\\mathrm {tot}}^{2}$ and $\\langle E \\rangle $ differ by approximately one unit.", "This is due to the fact that replicas fluctuate within their uncertainty about the experimental data, which in turn are Gaussianly distributed about a true value [120]: it shows that the neural network is correctly reproducing the underlying law thus being closer to the true value.", "This is confirmed by the fact that $\\langle \\chi ^{2(k)}\\rangle $ is of order one.", "The distribution of $\\chi ^2$ for different experiments (also shown as a histogram in Fig.", "REF ) shows sizable differences, and indeed the standard deviation (shown as a dashed line in the plot) about the mean (shown as a solid line) is very large.", "This can be understood as a consequence of the lack of information on the covariance matrix: experiments where large correlated uncertainties are treated as uncorrelated will necessarily have a smaller value of the $\\chi ^2$ .", "Figure: Value of the χ 2 \\chi ^2 per datapoint for the data setsincluded in the NNPDFpol1.0 reference fit,listed in Tab.", ".The horizontal line isthe unweighted average of these χ 2 \\chi ^2 over the data sets andthe black dashed lines give the one-sigma interval about it.The NNPDFpol1.0 parton distributions, computed from a set of $N_{\\mbox{\\scriptsize {rep}}}=100$ replicas, are displayed in Fig.", "REF at the input scale $Q_0^2=1$ GeV$^2$ , in the PDF parametrization basis as a function of $x$ both on a logarithmic and linear scale.", "In Figs.", "REF -REF the same PDFs are plotted in the flavor basis, and compared to other available NLO PDF sets: BB10 [76] and AAC08 [73] in Fig.", "REF , and DSSV08 [75] in Fig.", "REF .", "We do not show a direct comparison to the LSS10 [77] nor JAM13 [78] PDF sets because they are not publicly available.", "We remind from Sec.", "that all these parton determinations are based on somewhat different data sets.", "For instance, BB10 contains purely DIS data and AAC08 contains DIS data supplemented by a few high-$p_T$ -$\\pi ^0$ production data from RHIC: hence they are closely comparable to our PDF determination.", "Instead, the DSSV08 determination includes, on top of DIS data, polarized jet production data, and, more importantly, a large amount of semi-inclusive DIS data which in particular allow for quark-antiquark separation and a more direct handle on strangeness.", "In these plots, NNPDF uncertainties correspond to the nominal one-sigma error bands, while for other PDF sets they are Hessian uncertainties corresponding to the default value assumed in each analysis.", "We remind from Sec.", "that it is assumed to be $T=12.95$ for AAC08, while $T=1$ for all other PDF sets.", "Figure: The NNPDFpol1.0 parton distributions atQ 0 2 =1Q_0^2=1 GeV 2 ^2 in the parametrization basis plotted as a function of xx,on a logarithmic (left) and linear (right) scale.Figure: The NNPDFpol1.0 parton distributions atQ 0 2 =1Q_0^2=1 GeV 2 ^2 in the flavor basis plotted as a function of xx,on a logarithmic (left) and linear (right) scale and compared toBB10 and AAC08 parton sets.Figure: Same as Fig.", ", but compared to DSSV08 parton set.The main conclusions of this comparison can be summarized as follows.", "The central values of the $\\Delta u + \\Delta \\bar{u}$ and the $\\Delta d + \\Delta \\bar{d}$ PDF combinations are in reasonable agreement with those of other parton sets.", "The NNPDFpol1.0 results are in best agreement with DSSV08, in slightly worse agreement with AAC08, and in worst agreement with BB10.", "Uncertainties on these PDFs are generally slightly larger for NNPDFpol1.0 than for other sets, especially DSSV08, which however is based on a much wider data set.", "The NNPDFpol1.0 determination of $\\Delta s +\\Delta \\bar{s}$ is affected by a much larger uncertainty than BB10 and AAC08, for almost all values of $x$ .", "Overall, the AAC08 and BB10 total strange distributions fall well within the NNPDFpol1.0 uncertainty band.", "The NNPDFpol1.0 determination of total strangeness, $\\Delta s +\\Delta \\bar{s}$ is inconsistent at the two sigma level in the medium-to-small $x\\sim 0.1$ region with DSSV08, which is also rather more accurate.", "However, we notice that total strangeness is constrained in the two analyses by rather different experimental information.", "In NNPDFpol1.0, it is determined through its $Q^2$ evolution at different scales together with fixing the first moments of the nonsinglet PDF combinations to the baryonic octet decay constants.", "Conversely, in DSSV08 the total strangeness is mostly determined from semi-inclusive data with strange hadrons in the final states.", "Hence, the flavor combination $\\Delta s+\\Delta \\bar{s}$ is also sensitive to the corresponding fragmentations functions.", "Since these are poorly known, especially for strange hadrons (namely kaons), the result obtained in the DSSV08 analysis is likely to be biased by the form assumed for the fragmentation funcions.", "The gluon PDF is affected by a large uncertainty, rather larger than any other set, especially at small $x$ .", "In particular, the NNPDFpol1.0 polarized gluon distribution is compatible with zero for all values of $x$ .", "At $0.04\\lesssim x\\lesssim 0.2$ , the gluon determination in the DSSV08 parton set benefits from sensitivity to pion and jet production data, which are not included in the other determinations.", "Uncertainties on the PDFs in the regions where no data are available tend to be larger than those of other sets.", "At very large values of $x$ the PDF uncertainty band is largely determined by the positivity constraint, while at small values of $x$ it is prevented to blow up arbitrarily by demanding the integrability of its first moment.", "In Fig.", "REF we compare the structure function $g_1(x,Q^2)$ for proton and neutron, computed using NNPDFpol1.0 (with its one-sigma uncertainty band) to the experimental data included in the fit.", "Experimental data are grouped in bins of $x$ with a logarithmic spacing, while the theoretical prediction and its uncertainty are computed at the central value of each bin.", "The uncertainty band in the NNPDFpol1.0 result is typically smaller than the experimental errors, except at small-$x$ where a much more restricted data set is available; in that region, the uncertainties are comparable.", "Scaling violations of the polarized structure functions are clearly visible, especially for $g_1^p$ , despite the limited range in $Q^2$ .", "Figure: The proton and neutronstructure function g 1 (x,Q 2 )g_1(x,Q^2) displayed as a function of Q 2 Q^2 indifferent bins of xx compared to experimental data.Experimental data are grouped in bins of xx, whileNNPDFpol1.0 results are given at the center of each bin,whose value is given next to each curve.", "In order to improvelegibility,the values of g 1 (x,Q 2 )g_1(x,Q^2)have been shifted by theamount given next to each curve." ], [ "Stability of the results", "Our results have been obtained with a number of theoretical and methodological assumptions, discussed in Secs. -.", "We will now test their stability upon variation of these assumptions." ], [ "Target-mass corrections and $g_2$ .", "We have consistently included in our determination of $g_1$ corrections suppressed by powers of the nucleon mass which are of kinematic origin.", "Thus in particular, we have included target-mass corrections (TMCs) up to first order in ${M^2}/{Q^2}$ .", "Furthermore, both TMCs and the relation between the measured asymmetries and the structure function $g_1$ involve contributions to the structure function $g_2$ proportional to powers of ${M^2}/{Q^2}$ which we include according to Eq.", "(REF ) or Eq.", "(REF ) (see the discussion in Sec. ).", "Our default PDF set is obtained assuming that $g_2$ is given by the Wandzura-Wilczek relation, Eq.", "(REF ).", "In order to assess the impact of these assumptions on our results, we have performed two more PDF determinations.", "In the first, we set $M=0$ consistently everywhere, both in the extraction of the structure functions from the asymmetry data and in our computation of structure functions.", "This thus removes TMCs, and also contributions proportional to $g_2$ .", "In the second, we retain mass effects, but we assume $g_2=0$ .", "The statistical estimators for each of these three fits over the full data set are shown in Tab.", "REF .", "Clearly, all fits are of comparable quality.", "Table: The statistical estimators of Tab.", "(obtained assuming g 2 =g 2 WW g_2=g_2^{\\mbox{\\tiny WW}}) compared to a fit with M=0M=0 orwith g 2 =0g_2=0.Furthermore, in Fig.", "REF we compare the PDFs at the initial scale $Q_0^2$ determined in these fits to our default set: differences are hardly visible.", "This comparison can be made more quantitative by using the distance $d(x,Q^2)$ between different fits, as defined in Appendix (see also Appendix A of Ref. [104]).", "The distance is defined in such a way that if we compare two different samples of $N_{\\mathrm {rep}}$ replicas each extracted from the same distribution, then on average $d=1$ , while if the two samples are extracted from two distributions which differ by one standard deviation, then on average $d=\\sqrt{N_{\\mathrm {rep}}}$ (the difference being due to the fact that the standard deviation of the mean scales as $1/\\sqrt{N_{\\mathrm {rep}}}$ ).", "The distances $d(x,Q^2)$ between central values and uncertainties of the three fits of Tab.", "REF are shown in Fig.", "REF .", "They never exceed $d=4$ , which means less than half a standard deviation for $N_{\\mathrm {rep}}=100$ .", "It is interesting to observe that distances tend to be larger in the large-$x$ region, where the expansion in powers of $M^2/Q^2$ is less accurate, and the effects of dynamical higher twists can become relevant.", "It is reassuring that even in this region the distances are reasonably small.", "We conclude that inclusive DIS data, with our kinematic cuts, do not show sensitivity to finite nucleon mass effects, neither in terms of fit quality, nor in terms of the effect on PDFs.", "Figure: Comparison between the default NNPDFpol1.0PDFs (labeled as g 2 =g 2 WW g_2=g_2^{\\mbox{\\tiny WW}} in the plot),PDFs with M=0M=0 (labeled as noTMCs in the plot) and PDFs with g 2 =0g_2=0;each corresponds to the statistical estimators ofTab.", ".Figure: Distances between each pair of the three sets of PDFsshown in Fig.", "." ], [ "Sum rules", "Our default PDF fit is obtained by assuming that the triplet axial charge $a_3$ is fixed to its value extracted from $\\beta $ decay, Eq.", "(REF ), and that the octet axial charge $a_8$ is fixed to the value of $a_8$ determined from baryon octet decays, but with an inflated uncertainty in order to allow for $SU(3)$ violation, Eq.", "(REF ).", "As discussed after Eq.", "(REF ) uncertainties on them are included by randomizing their values among replicas.", "In order to test the impact of these assumptions, we have produced two more PDF determinations.", "In the first, we have not imposed the triplet sum rule, so in particular $a_3$ is free and determined by the data, instead of being fixed to the value Eq.", "(REF ).", "In the second, we have assumed that the uncertainty on $a_8$ is given by the much smaller value of Eq.", "(REF ).", "Table: The statistical estimators of Tab.", ", but forfits in which the triplet sum rule is not imposed (free a 3 a_3) orin which the octet sum rule is imposed with the smaller uncertaintyEq.", "().The statistical estimators for the total data set for each of these fits are shown in Tab.", "REF .", "Here too, there is no significant difference in fit quality between these fits and the default.", "The distances between PDFs in the default and the free $a_3$ fits are displayed in Fig.", "REF .", "As one may expect, only the triplet is affected significantly: the central value is shifted by about $d \\sim 5$ , i.e.", "about half-$\\sigma $ , in the region $x\\sim 0.3$ , where $x\\Delta T_3$ has a maximum, and also around $x\\sim 0.01$ .", "The uncertainties on the PDFs are very similar in both cases for all PDFs, except $\\Delta T_3$ at small-$x$ : in this case, removing the $a_3$ sum rule results in a moderate increase of the uncertainties; the effect of removing $a_3$ is otherwise negligible.", "The singlet and triplet PDFs for these two fits are compared in Fig.", "REF .", "Figure: Distances between PDFs (central values anduncertainties) for thedefault fit, with a 3 a_3 fixed, and the fit with free a 3 a_3,computed using N rep =100N_{\\mbox{\\tiny rep}}=100replicas from each set.Figure: Comparison of the singlet and triplet PDFs for thedefault fit, with a 3 a_3 fixed, and the fit with free a 3 a_3.The distances between the default and the fit with the smaller uncertainty on $a_8$ are shown in Fig.", "REF .", "In this case, again as expected, the only effect is on the $\\Delta T_8$ uncertainty, which changes in the region $10^{-2}\\lesssim x \\lesssim 10^{-1}$ by up to $d\\sim 6$ (about half a standard deviation): if a more accurate value of $a_8$ is assumed, the determined $\\Delta T_8$ is correspondingly more accurate.", "Central values are unaffected.", "The singlet and octet PDFs for this fit are compared to the default in Fig.", "REF .", "We conclude that the size of the uncertainty on $\\Delta T_8$ has a moderate effect on our fit; on the other hand it is clear that if the octet sum rule were not imposed at all, the uncertainty on the octet and thus on strangeness would increase very significantly, as we have checked explicitly.", "Figure: Distances between PDFs (central values anduncertainties) for the default fit, with a 8 a_8 Eq.", "(),and the fit with the value of a 8 a_8 with smaller uncertainty,Eq.", "().Figure: Comparison of the singlet and octet PDFs for thedefault fit, with a 8 a_8 Eq.", "(), and the fit with thevalue of a 8 a_8 with smaller uncertainty, Eq.", "().We conclude that our fit results are quite stable upon variations of our treatment of both the triplet and the octet sum rules." ], [ "Positivity", "As discussed in Sec.", ", positivity of the individual cross-sections entering the polarized asymmetries Eq.", "(REF ) has been imposed at leading order according to Eq.", "(REF ), using the NLO NNPDF2.1 PDF set [106], separately for the lightest polarized quark PDF combinations $\\Delta u + \\Delta \\bar{u}$ , $\\Delta d +\\Delta \\bar{d}$ , $\\Delta s + \\Delta \\bar{s}$ and for the polarized gluon PDF, by means of a Lagrange multiplier Eq.", "(REF ).", "After stopping, positivity is checked a posteriori and replicas which do not satisfy it are discarded and retrained.", "In Fig.", "REF we compare to the positivity bound for the up, down, strange PDF combinations and gluon PDF a set of $N_{\\mbox{\\tiny rep}}=100$ replicas obtained by enforcing positivity through a Lagrange multiplier, but before the final, a posteriori check.", "Almost all replicas satisfy the constraint, but at least one replica which clearly violates it for the total strangeness combination (and thus will be discarded) is seen.", "Figure: The positivity bound Eq.", "(), comparedto a set of N rep =100N_{\\mbox{\\tiny rep}}=100 replicas (dashedlines).In order to assess the effect of the positivity constraints, we have performed a fit without imposing positivity.", "Because positivity significantly affects PDFs in the region where no data are available, and thus in particular their large-$x$ behavior, preprocessing exponents for this PDF determination had to be determined again using the procedure described in Sec. .", "The values of the large $x$ preprocessing exponents used in the fit without positivity are shown in Tab.", "REF .", "The small $x$ exponents are the same as in the baseline fit, Tab.", "REF .", "Table: Ranges for the large-xxpreprocessing exponents Eq.", "()for the fit in which no positivityis imposed.", "The small-xx exponents are the same as in thebaseline fit Tab.", ".The corresponding estimators are shown in Tab.", "REF .", "Also in this case, we see no significant change in fit quality, with only a slight improvement in $\\chi ^2_{\\mathrm {tot}}$ when the constraint is removed.", "This shows that our PDF parametrization is flexible enough to easily accommodate positivity.", "On the other hand, clearly the positivity bound has a significant impact on PDFs, especially in the large-$x$ region, as shown in Fig.", "REF , where PDFs obtained from this fit are compared to the baseline.", "At small $x$ , instead, the impact of positivity is moderate, because $g_1/F_1\\sim x$ as $x\\rightarrow 0$ [206] so there is no constraint in the limit.", "This in particular implies that there is no significant loss of accuracy in imposing the LO positivity bound, because in the small $x\\lesssim 10^{-2}$ region, where the LO and NLO positivity bounds differ significantly [207] the bound is not significant.", "Table: The statistical estimators of Tab.", "for a fit without positivity constraints.Figure: The NNPDFpol1.0 PDFs with and withoutpositivity constraints compared at the initial parametrization scaleQ 0 2 =1Q_0^2=1 GeV 2 ^2 in the flavor basis." ], [ "Small- and large-$x$ behavior and preprocessing", "The asymptotic behavior of both polarized and unpolarized PDFs for $x$ close to 0 or 1 is not controlled by perturbation theory, because powers of $\\ln \\frac{1}{x}$ and $\\ln (1-x)$ respectively appear in the perturbative coefficients, thereby spoiling the reliability of the perturbative expansion close to the endpoints.", "Non-perturbative effects are also expected to set in eventually (see e.g. [206]).", "For this reason, our fitting procedure makes no assumptions on the large- and small-$x$ behaviors of PDFs, apart from the positivity and integrability constraints discussed in the previous Section.", "It is however necessary to check that no bias is introduced by the preprocessing.", "We do this following the iterative method described in Sec. .", "The outcome of the procedure is the set of exponents Eq.", "(REF ), listed in Tab.", "REF .", "The lack of bias with these choices is explicitly demonstrated in Fig.", "REF , where we plot the 68% confidence level of the distribution of $&\\alpha [\\Delta q(x,Q^2)]=\\frac{\\ln \\Delta q(x,Q^2)}{\\ln \\frac{1}{x}}\\mbox{ ,}\\\\&\\beta [\\Delta q(x,Q^2)]=\\frac{ \\ln \\Delta q(x,Q^2) }{\\ln (1-x)}\\mbox{ ,}$ $\\Delta q=\\Delta \\Sigma \\mbox{, }\\Delta g\\mbox{, }\\Delta T_3\\mbox{, }\\Delta T_8$ , for the default NNPDFpol1.0 $N_{\\mathrm {rep}}=100$ replica set, at $Q^2=Q_0^2=1$ GeV$^2$ , and compare them to the ranges of Tab.", "REF .", "It is apparent that as the endpoints $x=0$ and $x=1$ are approached, the uncertainties on both the small-$x$ and the large-$x$ exponents lie well within the range of the preprocessing exponents for all PDFs, thus confirming that the latter do not introduce any bias.", "Figure: The 68% confidence level of the distribution ofeffective small- and large-xx exponentsEqs.", "()-() for the default N rep =100N_{\\mathrm {rep}}=100replica NNPDFpol1.0 set at Q 0 2 =1Q_0^2=1 GeV 2 ^2, plotted asa functions of xx.", "The range of variation of the preprocessingexponents of Tab.", "is also shown in each case(solid lines)." ], [ "Polarized nucleon structure", "We use the NNPDFpol1.0 parton set to compute the the first moments of the polarized PDFs.", "These are the quantities of greatest physical interest, in that they are directly related to the spin structure of the nucleon, as discussed in Chap. .", "We also assess whether the isotriplet first moment determined within our parton set could provide an unbiased handle on the strong coupling $\\alpha _s$ , via the Bjorken sum rule." ], [ "First moments", "We have computed the first moments $\\langle \\Delta f(Q^2) \\rangle \\equiv \\int _0^1 dx \\, \\Delta f(x,Q^2)$ of each light polarized quark-antiquark, $\\Delta u +\\Delta \\bar{u}$ , $\\Delta d +\\Delta \\bar{d}$ , $\\Delta s +\\Delta \\bar{s}$ , and gluon, $\\Delta g$ , distribution using a sample of $N_\\mathrm {rep}=100$ NNPDFpol1.0 PDF replicas.", "The histogram of the distribution of first moments over the replica sample at $Q_0^2=1$ GeV$^2$ are displayed in Fig.", "REF : they appear to be reasonably approximated by a Gaussian.", "Figure: Distribution of the first moments ofΔu+Δu ¯\\Delta u + \\Delta \\bar{u} (top left), Δd+Δd ¯\\Delta d + \\Delta \\bar{d} (top right),Δs+Δs ¯\\Delta s + \\Delta \\bar{s} (bottom left) and Δg\\Delta g (bottom right)over a set of N rep =100N_\\mathrm {rep}=100 NNPDFpol1.0 PDF replicas.The central value and one-sigma uncertainties of the quark-antiquark combination first moments are listed in Tab.", "REF , while those of the singlet quark combination and the gluon are given in Tab.", "REF .", "Results are compared to those from other parton sets, namely ABFR98 [67], DSSV08 [75], AAC08 [73], BB10 [76] and LSS10 [77].", "Results from other PDF sets are not available for all combinations and scales, because public codes only allow for the computation of first moments in a limited $x$ range, in particular down to a minimum value of $x$ : hence we must rely on published values for the first moments.", "In particular, the DSSV08 and AAC08 results are shown at $Q_0^2=1$ GeV$^2$ , while the BB10 and LSS10 results are shown at $Q^2=4$ GeV$^2$ .", "For ease of reference, in Tab.", "REF the NNPDFpol1.0 values for both scales are shown.", "Table: First moments of the polarized quark distributions at Q 0 2 =1Q_0^2=1GeV 2 ^2; cv denotes the central value, while exp andth denote uncertainties (see text) whose sum in quadratureis given by tot.Table: Same as Tab.", ", but for the totalsinglet quark distribution and the gluon distribution.", "The NNPDFpol1.0results are shown both at Q 0 2 =1Q_0^2=1 GeV 2 ^2 and Q 2 =4Q^2=4 GeV 2 ^2,the ABFR98, DSSV08 and AAC08 results areshown at Q 0 2 =1Q_0^2=1 GeV 2 ^2, and the BB10 and LSS10are shown at Q 2 =4Q^2=4 GeV 2 ^2.In order to compare the results for first moments shown in Tabs.", "REF -REF , it should be understood that the uncertainties shown, and sometimes also the central values, have somewhat different meanings.", "NNPDFpol1.0.", "The exp uncertainty, determined as the standard deviation of the replica sample, is a pure PDF uncertainty: it includes the propagation of the experimental data uncertainties and the uncertainty due to the interpolation and extrapolation.", "ABFR98.", "The central values were obtained in the AB factorization scheme discussed in Sec.", "REF .", "In this scheme, the first moment of the gluon coincides with that in the $\\overline{\\mathrm {MS}}$ scheme used in all other PDF fits presented here, and thus the corresponding value from Ref.", "[67] is shown in Tab.", "REF .", "Conversely, the singlet first moments in the two schemes are different, but are related by the simple relation Eq.", "(REF ).", "In Ref.", "[67] a value of the singlet axial charge $a_0$ in the limit of infinite $Q^2$ was also given.", "In the $\\overline{\\mathrm {MS}}$ , the singlet axial charge and the first moment of $\\Delta \\Sigma $ coincide (see Sec.", "REF ), hence we have determined $\\langle \\Delta \\Sigma \\rangle $ for ABFR98 by evolving down to $Q^2=1$ GeV$^2$ the value of $a_0(\\infty )$ given in Ref.", "[67], at NLO and with $\\alpha _s(M_z)=0.118$ [208] (the impact of the $\\alpha _s$ uncertainty is negligible).", "We have checked that the same result is obtained if $a_0$ is computed as the appropriate linear combination of $\\langle \\Delta \\Sigma \\rangle $ in the AB scheme and the first moment of $\\Delta g$ , Eq.", "(REF ).", "In the ABFR98 study, the exp uncertainty is the Hessian uncertainty on the best fit, and it thus includes the propagated data uncertainty.", "The th uncertainty includes the uncertainty originated by neglected higher orders (estimated by renormalization and factorization scale variations), higher twists, position of heavy quark thresholds, value of the strong coupling, violation of SU(3), and finally uncertainties related to the choice of functional form, estimated by varying the functional form.", "This latter source of theoretical uncertainty corresponds to interpolation and extrapolation uncertainties which are included in the exp for NNPDFpol1.0.", "DSSV08, BB10.", "The central value is obtained by computing the first moment integral of the best-fit with a fixed functional form restricted to the data region, and then supplementing it with a contribution due to the extrapolation in the unmeasured, small-$x$ , region.", "The exp uncertainty in the table is the Hessian uncertainty given by DSSV08 or BB10 on the moment in the measured region, and it thus includes the propagated data uncertainty.", "In both cases, we have determined the th uncertainty shown in the table as the difference between the full first moment quoted by DSSV08 or BB10, and the first moment in the measured region.", "It is thus the contribution from the extrapolation region, which we assume to be $100\\%$ uncertain.", "In both cases, we have computed the truncated first moment in the measured region using publicly available codes, and checked that it coincides with the values quoted by DSSV08 and BB10.", "AAC08.", "The central value is obtained by computing the first moment integral of the best-fit with a fixed functional form, and the exp uncertainty is the Hessian uncertainty on it.", "However, AAC08 uses the tolerance [165] criterion for the determination of Hessian uncertainties, which rescales the $\\Delta \\chi ^2=1$ region by a suitable factor, in order to effectively keep into account also interpolation errors.", "Hence, the exp uncertainties include propagated data uncertainties, as well as uncertainties on the PDF shape.", "LSS10.", "The central value is obtained by computing the first moment of the best fit with a fixed functional form, and the exp uncertainty is the Hessian uncertainty on it.", "Hence it includes the propagated data uncertainty.", "In all cases, the total uncertainty is computed as the sum in quadrature of the exp and th uncertainties.", "Roughly speaking, for LSS10 this includes only the data uncertainties; for DSSV08, and BB10 it also includes extrapolation uncertainties; for AAC08 interpolation uncertainties; for NNPDFpol1.0 both extrapolation and interpolation uncertainties; and for ABFR98 all of the above, but also theoretical (QCD) uncertainties.", "For LSS10 and AAC08, we quote the results obtained from two different fits, both assuming positive- or node-gluon PDF: their spread gives a feeling for the missing uncertainty due to the choice of functional form.", "Remind that the AAC08 results correspond to their Set B which includes, besides DIS data, also RHIC $\\pi ^0$ production data; the DSSV08 fit also includes, on top of these, RHIC jet data and semi-inclusive DIS data; LSS10 includes, beside DIS, also semi-inclusive DIS data.", "All other sets are based on DIS data only.", "Coming now to a comparison of results, we see that for the singlet first moment $\\langle \\Delta \\Sigma \\rangle $ the NNPDFpol1.0 result is consistent within uncertainties with that of other groups.", "The uncertainty on the NNPDFpol1.0 result is comparable, if somewhat larger, to that found whenever the extrapolation uncertainty has been included.", "For individual quark flavors we find excellent agreement in the central values obtained between NNPDFpol1.0 and DSSV08, see Tab.", "REF ; the NNPDFpol1.0 uncertainties are rather larger, but this could also be due to the fact that the data set included in DSSV08 is sensitive to quark-antiquark separation.", "For the gluon first moment $\\langle \\Delta g\\rangle $ , the NNPDFpol1.0 result is characterized by an uncertainty which is much larger than that of any other determination: a factor of three or four larger than ABFR98 and AAC08, ten times larger than BB10, and twenty times larger than DSSV08 and LSS10.", "It is compatible with zero within this large uncertainty.", "We have seen that for the quark singlet, the NNPDFpol1.0 uncertainty is similar to that of groups which include an estimate of extrapolation uncertainties.", "In order to assess the impact of the extrapolation uncertainty for the gluon, we have computed the gluon first moment truncated in the region $x\\in [10^{-3},1]$ : $\\int _{10^{-3}}^1dx\\, \\Delta g(x, Q^2=1 \\mathrm {GeV}^2) = -0.26 \\pm 1.19\\,\\mbox{,}$ to be compared with the result of Tab.", "REF , which is larger by almost a factor four.", "We must conclude that the experimental status of the gluon first moment is still completely uncertain, unless one is willing to make strong theoretical assumptions on the behavior of the polarized gluon at small $x$ , and that previous different conclusions were affected by a significant underestimate of the impact of the bias in the choice of functional form, in the data and especially in the extrapolation region.", "Because of the large uncertainty related to the extrapolation region, only low-$x$ data can improve this situation, such as those which could be collected at a high energy Electron-Ion Collider [209], [128], as we will show in Chap.", "." ], [ "The Bjorken sum rule", "The Bjorken sum rule presented in Sec.", ", Eq.", "(REF ) $\\Gamma _1^{\\mathrm {NS}}\\equiv \\Gamma _1^p\\left(Q^2\\right) - \\Gamma _1^n\\left(Q^2\\right)=\\frac{1}{6}\\Delta C_{\\mathrm {NS}} (\\alpha _s(Q^2)) a_3$ potentially provides a theoretically very accurate handle on the strong coupling $\\alpha _s$ .", "We recall that we have defined the first moment of the proton (neutron) structure function $\\Gamma _1^{p,n}(Q^2)$ in Eq.", "(REF ), the first moment of the nonsinglet triplet PDF combination in the first relation of Eqs.", "(REF ) and $\\Delta C_{\\mathrm {NS}}(\\alpha _s(Q^2))$ is the first moment of the corresponding coefficient function, which is known up to three loops.", "In principle, the truncated isotriplet first moment $\\Gamma _1^{\\mathrm {NS}}\\left(Q^2,x_{\\mathrm {min}}\\right)\\equiv \\int _{x_{\\mathrm {min}}}^1 dx\\left[g_1^p\\left( x,Q^2 \\right) - g_1^n\\left( x,Q^2 \\right) \\right]$ can be extracted from data without any theoretical assumption.", "Given a measurement of $\\Gamma _1^{\\mathrm {NS}}\\left(Q^2,0\\right)$ at a certain scale, the strong coupling can then be extracted from Eq.", "(REF ) using the value of $a_3$ from $\\beta $ decays, while given a measurement of $\\Gamma _1^{\\mathrm {NS}}\\left(Q^2,0\\right)$ at two scales, both $a_3$ and the value of $\\alpha _s$ can be extracted simultaneously.", "In Ref.", "[210], $a_3$ and $\\alpha _s$ where simultaneously determined from a set of nonsinglet truncated moments, both the first and higher moments, by exploiting the scale dependence of the latter [211], with the result $a_3=1.04\\pm 0.13$ and $\\alpha _s(M_z)=0.126^{+0.006}_{-0.014}$ , where the uncertainty is dominated by the data, interpolation and extrapolation, but also includes theoretical QCD uncertainties.", "In this reference, truncated moments were determined from a neural network interpolation of existing data, sufficient for a computation of moments at any scale.", "However, because the small-$x$ behavior of the structure function is only weakly constrained by data, the $x\\rightarrow 0$ extrapolation was done by assuming a powerlike Regge behavior [212].", "The situation within NNPDFpol1.0 can be understood by exploiting the PDF determination in which $a_3$ is not fixed by the triplet sum rule, discussed in Sec.", "REF .", "Using the results of this determination, we find $a_3=\\int _0^1dx\\,\\Delta T_3 (x, Q^2) = 1.19 \\pm 0.22\\,\\mbox{.", "}$ The uncertainty is about twice that of the determination of Ref. [210].", "As mentioned, the latter was obtained from a neural network parametrization of the data with no theoretical assumptions, and based on a methodology which is quite close to that of the NNPDFpol1.0 PDF determination discussed here, the only difference being the assumption of Regge behavior in order to perform the small-$x$ extrapolation.", "This strongly suggests that, as in the case of the gluon distribution discussed above, the uncertainty on the value Eq.", "(REF ) is dominated by the small-$x$ extrapolation.", "To study this effect, in Fig.", "REF we plot the value of the truncated Bjorken sum rule $\\Gamma _1^{\\mathrm {NS}}\\left(Q^2,x_{\\mathrm {min}}\\right)$ Eq.", "(REF ) as a function of the lower limit of integration $x_{\\mathrm {min}}$ at $Q_0^2=1$ GeV$^2$ , along with the asymptotic value $\\Gamma _1^{\\mathrm {NS}}\\left(1 \\mathrm {GeV}^2,0\\right)= 0.16 \\pm 0.03$ which at NLO corresponds to the value of $a_3$ given by Eq.", "(REF ).", "As a consistency check, we also show the same plot for our baseline fit, in which $a_3$ is fixed by the sum rule to the value Eq.", "(REF ).", "It is clear that indeed the uncertainty is completely dominated by the small $x$ extrapolation.", "Figure: The truncatedBjorken sum rule Γ 1 NS Q 2 ,x\\Gamma _1^{\\mathrm {NS}}\\left(Q^2,x\\right)Eq.", "() plotted as a function of xx for Q 2 =1Q^2=1 GeV 2 ^2,for the fit with free a 3 a_3 (left) and for the reference fit with a 3 a_3 fixedto the value Eq.", "() (right).", "In the left plot,the shaded band corresponds to the asymptotic value of the truncated sum rule,Eq.", "(), while in the right plot it corresponds to theexperimental value Eq.", "().We conclude that a determination of $\\alpha _s$ from the Bjorken sum rule is not competitive unless one is willing to make assumptions on the small $x$ behavior of the nonsinglet structure function in the unmeasured region.", "Indeed, it is clear that a determination based on NNPDFpol1.0 would be affected by an uncertainty which is necessarily larger than that found in Ref.", "[210], which is already not competitive.", "The fact that a determination of $\\alpha _s$ from the Bjorken sum rule is not competitive due to small $x$ extrapolation ambiguities was already pointed out in Ref.", "[67], where values of $a_3$ and $\\alpha _s$ similar to those of Ref.", "[210] were obtained." ], [ "Polarized PDFs at an Electron-Ion Collider", "In this Chapter, we investigate the potential impact of inclusive DIS data from a future Electron-Ion Collider (EIC) on the determination of polarized parton distributions.", "After briefly motivating our study in Sec.", ", we illustrate in Sec.", "which EIC pseudodata sets we use in our analysis and in Sec.", "how the fitting procedure described in Sec.", "needs to be optimized.", "Resulting PDFs are presented in Sec.", ", and they are compared to NNPDFpol1.0 throughout.", "Finally, in Sec.", "we reassess the computation of their first moments and we give an estimate of the charm contribution to the $g_1$ structure function.", "The analysis presented in this Chapter is mostly based on Ref.", "[117]." ], [ "Motivation", "As already noticed several times in this Thesis, the bulk of experimental information on longitudinally polarized proton structure comes from inclusive, neutral-current DIS, which allows one to obtain information on the light quark-antiquark combinations $\\Delta q^+\\equiv \\Delta q +\\Delta \\bar{q}$ , $q=u,d,s$ and on the gluon distribution $\\Delta g$ .", "However, presently available DIS data cover only a small kinematic region of momentum fractions and energies $(x,Q^2)$ , as shown in fig.", "REF .", "On the one hand, the lack of experimental information for $x\\lesssim 10^{-3}$ prevents a reliable determination of polarized PDFs at small-$x$ .", "Hence, their first moments will strongly depend on the functional form one assumes for PDF extrapolation to the unmeasured $x$ region.", "On the other hand, the gluon PDF, which is determined by scaling violations, is only weakly constrained, due to the small lever-arm in $Q^2$ of the experimental data.", "Both these limitations were emphasized in Chap.", ", when we have presented the first unbiased set of polarized PDFs, NNPDFpol1.0.", "For these reasons, despite many efforts, both experimental and theoretical, the size of the polarized gluon contribution to the nucleon spin is still largely uncertain, as demonstrated in Sec.", "and in Ref. [187].", "In Sec.", ", we mentioned that other processes, receiving leading partonic contributions from gluon-initiated suprocesses, may provide direct information on the polarized gluon PDF.", "They include open-charm photoproduction data from COMPASS [213] and polarized hadron collider measurements from RHIC [214], [215], [216], [217], [218], specifically semi-inclusive particle and jet production data.", "We explicitly assess their impact on the determination of $\\Delta g$ in Chap.", ", but we note here that all these data are restricted to the medium- and large-$x$ region.", "An EIC [209], [128], [129], with polarized lepton and hadron beams, would allow for a widening of the kinematic region comparable to the one achieved in the unpolarized case with the DESY-HERA experiments H1 and ZEUS [219].", "Note that a Large Hadron-electron Collider (LHeC) [220] would not have the option of polarizing the hadron beam.", "The potential impact of the EIC on the knowledge of the nucleon longitudinal spin structure has been quantitatively assessed in a recent study [221], in which projected neutral-current inclusive DIS and semi-inclusive DIS (SIDIS) artificial data were added to the DSSV+ polarized PDF determination [187]; this study was then extended by also providing an estimate of the impact of charged-current inclusive DIS pseudo-data on the polarized quark-antiquark separation in Ref. [222].", "In view of the fact that a substantially larger gluon uncertainty is found in NNPDFpol1.0 in comparison to previous PDF determinations [75], [77], [73], [76], it is worth repeating the study of the impact of EIC data, but now using NNPDF methodology.", "This is the goal of the study in the present Chapter." ], [ "Inclusive DIS pseudodata from an Electron-Ion Collider", "The realization of an EIC has been proposed for two independent designs so far: the electron Relativistic Heavy Ion Collider (eRHIC) at Brookhaven National Laboratory (BNL) [223] and the Electron Light Ion Collider (ELIC) at Jefferson Laboratory (JLab) [224].", "In both cases, a staged upgrade of the existing facilities has been planned [209], [128], [129], so that an increased center-of-mass energy would be available at each stage.", "Concerning the eRHIC option of an EIC [223], first measurements would be taken by colliding the present RHIC proton beam of energy $E_p=100-250$ GeV with an electron beam of energy $E_e=5$ GeV, while a later stage envisages electron beams with energy up to $E_e=20$ GeV.", "In order to quantitatively assess the impact of future EIC measurements on the determination of polarized PDFs, we have supplemented our QCD analysis presented in Chap.", "and Ref.", "[116] with DIS pseudodata from Ref. [221].", "They consist of three sets of data points at different possible eRHIC electron and proton beam energies, as discussed above.", "These pseudodata were produced by running the PEPSI Monte Carlo (MC) generator [225], assuming momentum transfer $Q^2>1$ GeV$^2$ , squared invariant mass of the virtual photon-proton system $W^2>10$ GeV$^2$ and fractional energy of the virtual photon $0.01\\le y \\le 0.95$ ; they are provided in five (four) bins per logarithmic decade in $x$ ($Q^2$ ).", "For each data set, the $Q^2$ range spans the values from $Q^2_{\\mathrm {min}}=1.39$ GeV$^2$ to $Q^2_{\\mathrm {max}}=781.2$ GeV$^2$ , while the accessible values of momentum fraction $x=Q^2/(sy)$ depend on the available center-of-mass energy, $\\sqrt{s}$ .", "In Tab.", "REF , we summarize, for each data set, the number of pseudodata $N_{\\mathrm {dat}}$ , the electron and proton beam energies $E_e$ , $E_p$ , the corresponding center-of-mass energies $\\sqrt{s}$ , and the smallest and largest accessible values in the momentum fraction range, $x_{\\mathrm {min}}$ and $x_{\\mathrm {max}}$ respectively.", "Table: The three EIC pseudodata sets .For each set we show the number of pointsN dat N_{\\mathrm {dat}}, the electron and proton beam energies E e E_e and E p E_p,the center-of-mass energy s\\sqrt{s}, the kinematic coveragein the momentum fraction xx, and the average absolute statisticaluncertainty 〈δg 1 〉\\langle \\delta g_1 \\rangle .The kinematic coverage of the EIC pseudodata is displayed in Fig.", "REF together with the fixed-target DIS data points dscussed in Chap. .", "The dashed regions show the overall kinematic reach of the EIC data with the two electron beam energies $E_e=5$ GeV or $E_e=20$ GeV, corresponding to each of the two stages at eRHIC.", "It is apparent from Fig.", "REF that EIC data will extend the kinematic coverage significantly, even for the lowest center-of-mass energy.", "In particular, hitherto unreachable small-$x$ values, down to $10^{-4}$ , will be attained, thereby leading to a significant reduction of the uncertainty in the low-$x$ extrapolation region.", "Furthermore, the increased lever-arm in $Q^2$ , for almost all values of $x$ should allow for much more stringent constraints on $\\Delta g(x,Q^2)$ from scaling violations.", "Figure: Kinematic coverage in the (x,Q 2 )(x,Q^2) planefor the fixed-target experimental data included in the NNPDFpol1.0polarized parton fitand the EIC pseudodata from .The shaded bands show the expected kinematic reach ofeach of the two EIC scenarios discussed in the text.The ratio $g_1(x,Q^2)/F_1(x,Q^2)$ is provided in Ref.", "[221] as the inclusive DIS observable, whose relation with the experimentally measured asymmetries was discussed in Sec. .", "The generation of pseudodata assumes a true underlying set of parton distributions: in Ref.", "[221] these are taken to be DSSV+ [187] and MRST [163] polarized and unpolarized PDFs respectively.", "Uncertainties are then determined assuming an integrated luminosity of 10 fb$^{-1}$ , which corresponds to a few months operations for the anticipated luminosities for eRHIC [223], and a $70\\%$ beam polarization.", "Because the DSSV+ polarized gluon has rather more structure than that of NNPDFpol1.0, which is largely compatible with zero, assuming this input shape will allow us to test whether the EIC data are sufficiently accurate to determine the shape of the gluon distribution.", "We reconstruct the $g_1$ polarized structure function from the pseudodata following the same procedure used in Sec.", "for the E155 experiment.", "We provide its average statistical uncertainty in the last column of Tab.", "REF .", "A comparison of these values with the analogous quantities for fixed-target experiments (see Tab.", "REF in Sec. )", "clearly shows that EIC data are expected to be far more precise, with uncertainties reduced up to one order of magnitude.", "No information on the expected systematic uncertainties is available, hence we will ignore them in our present analysis.", "However, we notice that the projected statistical uncertainties set the scale at which one needs to control systematics, which arise from luminosity and polarization measurements, detector acceptance and resolution, and QED radiative corrections.", "We will perform two different fits, corresponding to the two stages envisaged for the eRHIC option of an EIC [223] discussed above, which will be referred to as NNPDFpolEIC-A and NNPDFpolEIC-B.", "The former includes the first two sets of pseudodata listed in Tab.", "REF , while the latter also includes the third set." ], [ "Fit optimization", "The methodology for the determination of polarized PDFs, including their parametrization in terms of neural networks and their minimization through a genetic algorithm, follows the one discussed in detail in Sec. .", "However, due to the accuracy and the kinematic coverage of EIC pseudodata, which are respectively higher and wider in comparison to their fixed-target counterparts, the parameters entering the genetic algorithm and determining its stopping had to be re-tuned.", "In particular, in order to allow the genetic algorithm to explore the space of parameters more efficiently, we have used a large population of mutants and increased the number of weighted training generations to ensure that all data sets are learnt with comparable accuracy.", "The target values of the figure of merit used in the weighted training formula, Eq.", "(REF ), were consistently determined for each of the three EIC pseudodata sets, following the iterative procedure discussed in Sec. .", "As for the stopping criterion, we have modified the values of the width of the moving average $N_{\\mathrm {smear}}$ and the smearing parameter $\\Delta _{\\mathrm {smear}}$ ; for the NNPDFpolEIC-B fit, we have also increased the maximum number of genetic algorithm generations at which the minimization stops if the stopping criterion is not fulfilled.", "Also, equal training and validation fractions were chosen for pseudodata sets, unlike their fixed-target DIS counterparts.", "Indeed, we checked that the EIC pseudodata set size (about fifty points per set) is large enough to ensure fit stability.", "The values of the minimization and stopping parameters used in the NNPDFpolEIC-A and NNPDFpolEIC-B determinations are collected in Tab.", "REF : they can be straightforwardly compared to those used in the NNPDFpol1.0 analysis, see Tabs.", "REF -REF .", "Table: Values of the minimization and stopping parameters entering thefitting algorithm.", "The corresponding values used in the NNPDFpol1.0analysis are quoted in Tabs.", "-.Furthermore, we have redetermined the range in which preprocessing exponents are randomized, since the new information from EIC pseudodata may modify the large- and small-$x$ PDF behavior.", "In Tab.", "REF , we show the values we use for the present fit, which can be compared to NNPDFpol1.0 from Tab.", "REF .", "We have checked that our choice of preprocessing exponents does not bias our fit, according to the procedure discussed in Sec. .", "Table: Ranges for the small- and large-xx preprocessing exponents." ], [ "Results", "We now present our polarized parton sets based on inclusive DIS pseudodata at an EIC discussed in Sec.", ", NNPDFpolEIC-A and NNPDFpolEIC-B.", "First, we discuss their statistical features, then we show the corresponding parton distributions, compared to NNPDFpol1.0.", "All results presented in this section are obtained out of PDF ensembles of $N_{\\mathrm {rep}}=100$ replicas." ], [ "Statistical features", "Various general features of the NNPDFpolEIC-A and NNPDFpolEIC-B PDF determinations are summarized in Tab.", "REF , and can be straightforwardly compared to NNPDFpol1.0, see Tab.", "REF .", "These include: the $\\chi ^2$ per data point of the final best-fit PDF set compared to data (denoted as $\\chi ^2_{\\mathrm {tot}}$ ); the average and standard deviation over the replica sample of the same figure of merit for each replica when compared to the corresponding data replica (denoted as $\\langle E\\rangle \\pm \\sigma _E$ ) computed for the total, training and validation sets; the average and standard deviation of the $\\chi ^2$ of each replica when compared to data (denoted as $\\langle \\chi ^{2(k)}\\rangle $ ); and the average number of iterations of the genetic algorithm at stopping $\\langle \\mathrm {TL}\\rangle $ and its standard deviation over the replica sample.", "All these estimators were introduced in Sec.", "REF and are discussed in detail in Refs.", "[101], [104].", "The distributions of $\\chi ^{2(k)}$ , $E_{\\mathrm {tr}}^{(k)}$ and training lenghts among the $N_{\\mathrm {rep}}=100$ replicas are shown in Fig.", "REF and Fig.", "REF .", "respectively.", "As for the training lenghts, notice the different scale on the horizontal axis for the NNPDFpolEIC-A and NNPDFpolEIC-B fits, consistent with the different allowed maximum number of training generations $N_{\\mathrm {gen}}^{\\mathrm {max}}$ (see Tab.", "REF ).", "Table: Statistical estimators and average training lengths for thetwo fits to EIC pseudodata described in the text,NNPDFpolEIC-A andNNPDFpolEIC-B, withN rep =100N_{\\mathrm {rep}}=100 replicas.", "The corresponding estimators forNNPDFpol1.0 are quoted in Tab.", ".Figure: Distribution of χ 2(k) \\chi ^{2(k)} (upper plots) and E tr (k) E_{\\mathrm {tr}}^{(k)}(lower plots) from a sample of N rep =100N_{\\mathrm {rep}}=100 replicas, for theNNPDFpolEIC-A (left plots) and NNPDFpolEIC-B (rightplots) parton determinations.Figure: Distribution of training lenghts from a sampleof N rep =100N_{\\mathrm {rep}}=100 replicas, for theNNPDFpolEIC-A (left plot) and NNPDFpolEIC-B (rightplot) parton determinations.The fit quality, as measured by $\\chi ^2_{\\mathrm {tot}}$ , is comparable to that of NNPDFpol1.0 ($\\chi ^2_{\\mathrm {tot}}=0.77$ ) for both the NNPDFpolEIC-A ($\\chi ^2_{\\mathrm {tot}}=0.79$ ) and the NNPDFpolEIC-B ($\\chi ^2_{\\mathrm {tot}}=0.86$ ) fits.", "This shows that our fitting procedure can easily accommodate EIC pseudodata.", "The histogram of $\\chi ^2$ values for each data set included in our fits is shown in Fig.", "REF , together with the NNPDFpol1.0 result; the unweighted average $\\langle \\chi ^2\\rangle _{\\mathrm {set}}\\equiv \\frac{1}{N_{\\mathrm {set}}}\\sum _{j=1}^{N_{\\mathrm {set}}}\\chi ^2_{\\mathrm {set,j}}$ and standard deviation over data sets are also shown.", "As already noticed in Sec.", ", $\\chi ^2$ values significantly below one are found as a consequence of the fact that information on correlated systematics is not available for most experiments, and thus statistical and systematic errors are added in quadrature.", "Note that this is not the case for the EIC pseudodata, for which, as mentioned, no systematic uncertainty was included; this may explain the somewhat larger (closer to one) value of the $\\chi ^2$ per data point which is found when the pseudodata are included.", "Figure: Value of the χ 2 \\chi ^2 per data point for the data setsincluded in the NNPDFpolEIC-A (left)and in the NNPDFpolEIC-B (right) fits,compared to NNPDFpol1.0 .The horizontal lines correspond to the unweighted average of theχ 2 \\chi ^2 values shown, and the one-sigma interval about it.The dashed lines refer to NNPDFpolEIC-A(left plot) or NNPDFpolEIC-B (right plot) fits,while the dot-dashed lines refer to NNPDFpol1.0.We notice that EIC pseudodata, which are expected to be rather more precise than fixed-target DIS experimental data, require more training to be properly learned by the neural network.", "This is apparent in the increase in $\\langle TL\\rangle $ in Tab.", "REF when going from NNPDFpol1.0 to NNPDFpolEIC-A and then NNPDFpolEIC-B.", "We checked that the statistical features discussed above do not improve if we run very long fits, up to $N_{\\mathrm {gen}}^{\\mathrm {max}}=50000$ generations, without dynamical stopping.", "In particular, we do not observe a decrease of the $\\chi ^2$ for those experiments whose value exceeds the average by more than one sigma.", "This ensures that these deviations are not due to underlearning, i.e.", "insufficiently long minimization." ], [ "Parton Distributions", "Parton distributions from the NNPDFpolEIC-A and NNPDFpolEIC-B fits are compared to NNPDFpol1.0 in Figs.", "REF -REF respectively.", "In these plots, PDFs are displayed at $Q_0^2=1$ GeV$^2$ as a function of $x$ on a logarithmic scale; all uncertainties shown here are one-sigma bands.", "The positivity bound, obtained from the NNPDF2.3 NLO unpolarized set [110] as discussed in Sec.", ", is also drawn.", "Figure: The NNPDFpolEIC-Aparton distributions at Q 0 2 =1Q_0^2=1 GeV 2 ^2 plotted as a function of xxon a logarithmic scale, compared to NNPDFpol1.0.Figure: Same as Fig.", ", but forNNPDFpolEIC-B, compared to NNPDFpol1.0.The most visible impact of inclusive EIC pseudodata in both our fits is the reduction of PDF uncertainties in the low-$x$ region ($x\\lesssim 10^{-3}$ ) for light flavors and the gluon.", "The size of the effects is different for different PDFs.", "As expected, the most dramatic improvement is seen for the gluon, while uncertainties on light quarks are only reduced by a significant factor in the small-$x$ region.", "The uncertainty on the strange distribution is essentially unaffected: unlike in Ref.", "[221], we find no improvement on strangeness, due to the fact that we do not include semi-inclusive kaon production data, contrary to what was done there.", "When moving from NNPDFpolEIC-A to NNPDFpolEIC-B the gluon uncertainty decreases further, while other PDF uncertainties are basically unchanged.", "In Fig.", "REF , we compare the polarized gluon PDF in our EIC fits to the DSSV08 [75] and NNPDFpol1.0 parton determinations, both at $Q_0^2=1$ GeV$^2$ and $Q^2=10$ GeV$^2$ .", "The DSSV08 uncertainty is the Hessian uncertainty computed assuming $\\Delta \\chi ^2=1$ , which corresponds to the default uncertainty estimate in Ref. [75].", "This choice may lead to somewhat underestimated uncertainties, as discussed at length in Sec. .", "Figure: The polarized gluon PDF Δg(x,Q 0 2 )\\Delta g(x,Q_0^2), at Q 0 2 =1Q_0^2=1GeV 2 ^2 (upper panels) and at Q 2 =10Q^2=10 GeV 2 ^2 (lower panels),in the NNPDFpolEIC PDF sets,compared to DSSV and toNNPDFpol1.0.It is clear that the gluon PDF from our fits including EIC pseudodata is approaching the DSSV08 PDF shape, especially at a lower scale where the corresponding gluon does have some structure, despite the fact that at higher scales, where much of the data is located, perturbative evolution tends to wash out this shape.", "Also, this is more pronounced as more EIC pseudodata are included in our fit, i.e.", "moving from NNPDFpolEIC-A to NNPDFpolEIC-B.", "This means that EIC data would be sufficiently accurate to reveal the polarized gluon structure, if any." ], [ "Phenomenological implications of EIC pseudodata", "In this Section, we use our NNPDFpolEIC-A and NNPDFpolEIC-B parton determinations to reassess the spin content of the proton in the light of future EIC data.", "We also determine the expected contribution of the charm quark to the polarized structure function $g_1$ , focusing on its potential to further pin down the uncertainty of the gluon distribution." ], [ "The spin content of the proton", "It is particularly interesting to examine how the EIC data affect the determination of the first moments of the polarized PDFs $\\Delta f(x,Q^2)$ , Eq.", "(REF ), as they are directly related to the nucleon spin structure.", "We have computed the first moments, Eq.", "(REF ), of the singlet, lightest quark-antiquark combinations and gluon for the NNPDFpolEIC-A and NNPDFpolEIC-B PDF sets.", "The corresponding central values and one-sigma uncertainties at $Q_0^2=1$ GeV$^2$ are shown in Tab.", "REF , compared to NNPDFpol1.0.", "Table: First moments of the polarized quark distributions atQ 0 2 =1Q_0^2=1 GeV 2 ^2 for the fits in the present analysis.The corresponding values for NNPDFpol1.0 are quotedin Tabs.", "-.It is clear that EIC pseudodata reduce all uncertainties significantly.", "Note that moving from NNPDFpolEIC-A to NNPDFpolEIC-B does not improve significantly the uncertainty on quark-antiquark first moments, but it reduces the uncertainty on the gluon first moment by a factor two.", "However, it is worth noticing that, despite a reduction of the uncertainty on the gluon first moment, even for the most accurate NNPDFpolEIC-B fit, the value remains compatible with zero even though the central value is sizable (and negative).", "In order to assess the residual extrapolation uncertainty on the singlet and gluon first moments, we determine the contribution to them from the data range $x\\in [10^{-3},1]$ , i.e.", "$\\langle \\Delta \\Sigma (Q^2)\\rangle _{\\mathrm {TR}}\\equiv \\int _{10^{-3}}^{1}dx\\,\\Delta \\Sigma (x,Q^2)\\mbox{ ,}\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\langle \\Delta g(Q^2)\\rangle _{\\mathrm {TR}}\\equiv \\int _{10^{-3}}^{1}dx\\,\\Delta g(x,Q^2)\\mbox{ .", "}$ The first moments, Eq.", "(REF ), are given in Tab.", "REF at $Q_0^2=1$ GeV$^2$ and $Q^2=10$ GeV$^2$ , where results for central values, uncertainties, and correlation coefficients between the gluon and quark are collected.", "Table: The singlet and gluon truncated first moments andtheir one-sigma uncertainties at Q 2 =1Q^2=1 GeV 2 ^2 and Q 2 =10Q^2=10 GeV 2 ^2for the NNPDFpolEIC-A (left)and NNPDFpolEIC-B (right) PDF sets,compared to NNPDFpol1.0.The correlation coefficient ρ\\rho at Q 2 =10Q^2=10 GeV 2 ^2is also provided.Comparing the results at $Q^2=1$ GeV$^2$ of Tab.", "REF and Tab.", "REF with those in Tabs.", "REF -REF , we see that in the NNPDFpol1.0 PDF determination for the quark singlet combination the uncertainty on the full first moment is about twice as large as that from the measured region, and for the gluon it is about four times as large.", "The difference is due to the extra uncertainty coming from the extrapolation.", "In NNPDFpolEIC-B the corresponding increases are by 20% for the quark and 30% for the gluon, which shows that thanks to EIC data the extrapolation uncertainties would be largely under control.", "The correlation coefficient $\\rho $ significantly decreases upon inclusion of the EIC data: this means that the extra information contained in these data allows for an independent determination of the quark and gluon first moments.", "In Fig.", "REF , we plot the one-sigma confidence region in the $(\\langle \\Delta \\Sigma (Q^2)\\rangle _{\\mathrm {TR}},\\langle \\Delta g(Q^2)\\rangle _{\\mathrm {TR}})$ plane at $Q^2=10$ GeV$^2$ , for NNPDFpolEIC-A, NNPDFpolEIC-B and NNPDFpol1.0.", "Confidence regions are elliptical, since we have assumed that the truncated moments are Gaussianly distributed among the $N_{\\mathrm {rep}}=100$ replicas in the PDF ensemble.", "This is a reasonable assumption for all the three parton sets we are considering here, as shown in Fig.", "REF .", "The main result of our analysis, Fig.", "REF , can be directly compared to Fig.", "8 of Ref.", "[221], which was based on the DSSV framework and is comparable to our NNPDFpolEIC-B results.", "In both analyses EIC pseudodata determine the singlet first moment in the measured region with an uncertainty of about $\\pm 0.05$ .", "Figure: One-sigma confidence region for the quark singlet andgluon first moments in the measured region,Eq. ().", "The values for individual replicas arealso shown.Figure: Distributions of the singlet (left) and gluon (right) truncatedfirst moments at Q 2 =10Q^2=10 GeV 2 ^2 from a set of N rep =100N_{\\mathrm {rep}}=100replicas in the NNPDFpol1.0, NNPDFpolEIC-A andNNPDFpolEIC-B parton ensembles.On the other hand, in Ref.", "[221] the uncertainty on the gluon was found to be about $\\pm 0.02$ , while we get a much larger result of $\\pm 0.30$ .", "One may wonder whether this difference may be due at least in part to the fact that the DSSV fit on which the result of Ref.", "[221] is based also includes jet production and pion production data from RHIC, which may reduce the gluon uncertainty.", "To answer this, we have computed the contribution to the gluon first moment (again at $Q^2=10$ GeV$^2$ ) from the reduced region $0.05\\le x\\le 0.2$ , where the RHIC data are located.", "We find that the uncertainty on the contribution to the gluon first moment in this restricted range is $\\pm 0.083$ using NNPDFpolEIC-B, while it is $\\pm 0.147$ with NNPDFpol1.0 and ${}^{+0.129}_{-0.164}$ with DSSV+ [188].", "We conclude that before the EIC data are added, the uncertainties in NNPDFpol1.0 and DSSV+ are quite similar despite the fact that DSSV+ also includes RHIC data.", "Hence, the larger gluon uncertainty we find for the NNPDFpolEIC-B fit in comparison to Ref.", "[221] is likely to be due to our more flexible PDF parametrization, though some difference might also come from the fact that the SIDIS pseudodata included in Ref.", "[221] provide additional information on the gluon through scaling violations of the fragmentation structure function $g_1^h$ .", "Of course this also introduces an uncertainty related to the fragmentation functions which is difficult to quantify." ], [ "Charm contribution to the $g_1$ structure function", "In the QCD analysis of presently available DIS data, the contribution of heavy quarks to the polarized structure function $g_1$ is usually neglected.", "In both NNPDFpol1.0 and NNPDFpolEIC polarized PDF determinations, heavy quarks are dynamically generated above threshold by (massless) Altarelli-Parisi evolution in the ZM-VFN scheme (see Sec. ).", "An exception to this treatment of heavy quark masses is provided in Ref.", "[76], where charm quark production in the photon-gluon fusion process is treated at LO [226] in the fixed-flavor number (FFN) scheme.", "Intrinsic heavy quark effects are neglected in most analyses of polarized PDFs since they have been shown to be relatively small already on the scale of present-day unpolarized PDF uncertainties [106], which are rather smaller than their polarized counterparts.", "However, as EIC data are expected to be far more accurate than those available so far, effects of finite heavy quark masses could be at least non-negligible.", "The treatment of these effects requires a proper quark mass scheme, for instance the FONLL scheme, firstly introduced in Ref.", "[227] and explicitly extended to DIS in Ref. [124].", "The method is based upon the idea of looking at both the massless and massive scheme calculations as power expansions in the strong coupling constant, and replacing the coefficient of the expansion in the former with their exact massive counterpart in the latter, when available.", "In order to suppress higher order contributions arising in the subtraction term near the threshold region, two prescriptions are proposed in Ref. [124].", "One consists in damping the subtraction term by a threshold factor which differs from unity by power-suppressed terms; the other consists in using a rescaling variable.", "Both prescriptions introduce terms which are formally subleading with respect to the order of the calculation, therefore they do not change its nominal accuracy, but they may in practice improve the perturbative stability and smoothness of the results.", "Figure: The charm contribution g 1 p,c (x,Q 2 )g_1^{p,c}(x,Q^2) to the DIS protonpolarized structure function g 1 p g_1^p as a function of xx atthree different energy scales Q 2 Q^2.Results are shown for both the NNPDFpolEIC-A (left)and NNPDFpolEIC-B (right) parton determinations.Figure: The contour plots for the ratio g 1 p,c (x,Q 2 )/g 1 p (x,Q 2 )g_1^{p,c}(x,Q^2)/g_1^p(x,Q^2)from the fits to EIC pseudodata, NNPDFpolEIC-A (left)and NNPDFpolEIC-B (right).Thanks to its simplicity, the FONLL scheme has been implemented in the FastKernel framework for the determination of unpolarized PDFs based on the NNPDF methodology [106].", "The generalization of the FONLL scheme to the polarized structure function $g_1$ is quite simple: details are given in Appendix .", "For spin-dependent DIS the charm contribution to the structure function $g_1$ , generated through the photon-gluon fusion process, $\\gamma g\\rightarrow c\\bar{c}$ , will very much depend on the currently unknown size of $\\Delta g(x,Q^2)$ at small $x$ .", "For instance, we plot in Fig.", "REF the expectations for the charm contribution $g_1^{p,c}$ to the proton structure function $g_1^p$ , computed from the NNPDFpolEIC-A and NNPDFpolEIC-B parton determinations.", "Results are displayed at three different energy scales, namely $Q^2=2,10,100$ GeV$^2$ .", "In Fig.", "REF , we also show the contour plot for the ratio $g_1^{p,c}(x,Q^2)/g_1^p(x,Q^2)$ for both these fits.", "We conclude that the charm contribution to the proton structure function $g_1^{p,c}$ , though being small, could be as much larger as 10-$20\\%$ of the total $g_1^p$ in the kinematic region probed by an EIC.", "Hence, in order to further pin down the gluon uncertainty from intrinsic charm effects, one should be able to measure its corresponding contribution to the $g_1$ structure function within this accuracy.", "Our result is consistent with that obtained in the DSSV framework presented in Refs. [128].", "In summary, the EIC data would entail a considerable reduction in the uncertainty on the polarized gluon PDF, they would provide first evidence for its possible nontrivial $x$ shape and for its possible large contribution to the nucleon spin.", "However, this goal would be reached with sizable residual uncertainty: hence, the measurement of the charm contribution to the proton structure function $g_1^{p,c}$ , which is directly sensitive to the gluon, might provide more information on the $\\Delta g$ distribution." ], [ "Global determination of unbiased polarized PDFs", "In this Chapter, we present a first global determination of polarized parton distributions based on the NNPDF methodology: NNPDFpol1.1.", "Compared to NNPDFpol1.0, the parton set determined in Chap.", ", NNPDFpol1.1 is obtained using, on top of inclusive DIS data, also data from recent measurements of open-charm production in fixed-target DIS, and of jet and $W$ production in proton-proton collisions.", "After motivating our analysis in Sec.", ", we will review the theoretical description of these processes in Sec. .", "The features of the experimental data included in our analysis are then presented in Sec. .", "In Sec.", ", we discuss how the NNPDFpol1.1 parton set is obtained via Bayesian reweighting of prior PDF Monte Carlo ensembles, followed by unweighting, as outlined in Sec.", "REF .", "We also present its main features in comparison to NNPDFpol1.0 and DSSV08.", "Finally, in Sec.", ", we discuss some phenomenological implications of our new polarized parton set with respect to the spin content of the proton.", "Some of the results presented in this Chapter have appeared in preliminary form in Refs.", "[118], [119]." ], [ "Motivation", "The NNPDFpol1.0 parton set presented in Chap.", "is the first determination of polarized parton distributions based on the NNPDF methodology.", "However, this is based on inclusive, neutral-current, DIS data only, which have two major drawbacks, as pointed out several times in this Thesis.", "First, they do not allow for quark-antiquark separation; second, their kinematic coverage is rather limited, both at small-$x$ and high-$Q^2$ values.", "For this reason, all PDFs are affected by large uncertainties where experimental data are not available.", "Besides, the polarized gluon PDF, determined through scaling violations in DIS, is almost unconstrained because of the rather short $Q^2$ lever arm provided by data.", "As discussed in Sec.", ", one has to resort to processes other than inclusive DIS to obtain further knowledge of polarized parton distributions.", "An impressive set of experimental data have become available in the last years: these include semi-inclusive DIS (SIDIS) data in fixed-target experiments [228], [229], [95], [96], [230], one- or two-hadron and open-charm production data in lepton-nucleon scattering [231], [232], [233], [234], [213], and semi-inclusive particle production [214], [217], [216], high-$p_T$ jet production [218], [215] and parity-violating $W^\\pm $ boson production [235], [236] data in polarized proton-proton collisions at RHIC.", "As already summarized in Tab.", "REF , all these data are expected to probe different aspects of polarized PDFs: semi-inclusive DIS and $W^\\pm $ production data allow one to determine the light quark-antiquark separation, while jet and pion production data in polarized proton-proton collisions, as well as hadron or open-charm electroproduction data in fixed-target experiments, give a handle on the size and the shape of the polarized gluon distribution.", "A theoretical description of the processes corresponding to these data will be given in Sec.", "below.", "Nevertheless, all these measurements fall in the $x$ region already covered by DIS data.", "Given that the uncertainties on the first moments of polarized PDFs, which eventually determine the contribution of each parton to the total proton spin, are already limited by the extrapolation into the unconstrained small-$x$ region (see the discussion in Sec.", "REF ), it is clear that only moderate improvements in these are expected from the addition of new data other than DIS data.", "Only a future high-energy polarized Electron-Ion Collider (EIC) [128], [129], [209] would be likely to probe the small-$x$ regime of PDFs, and thus improve our knowledge of the polarized PDF first moments, as we demonstrated in Chap. .", "Since no further progress is expected by the time an EIC will start to operate, some effort has been devoted to perform global determinations of polarized parton sets, including all available experimental data.", "Presently, only two such sets are available: LSS10 [77], which also includes SIDIS beside inclusive DIS data, and those from the DSSV family [75], [187], [188], which include, on top of these, also inclusive jet and hadron production measurements from polarized proton-proton collisions at RHIC.", "The goal of the analysis of this Chapter is to incorporate in the NNPDF determination of polarized parton distributions the new experimental information provided by some of the processes mentioned above, and thus release the first global polarized PDF set based on the NNPDF methodology: NNPDFpol1.1." ], [ "Theoretical overview of polarized processes other than DIS", "Before addressing our global determination of a set of polarized parton distributions, we provide a theoretical overview of polarized processes other than DIS, available from both fixed-target and collider experiments." ], [ "Semi-inclusive lepton-nucleon scattering\nin fixed-target experiments", "The role of fixed-target lepton-nucleon scattering in determining the spin structure of the nucleon is not restricted to inclusive reactions, as those we have considered so far.", "Actually, more exclusive processes, in which one measures one or more outgoing final state particles, can be used to constrain polarized sea quark distributions and to gain some more knowledge on the polarized gluon distribution.", "In the following, we discuss in turn the potential of each of these processes, namely semi-inclusive DIS and heavy flavor hadron production.", "Semi-inclusive DIS.", "Semi-inclusive DIS (SIDIS) is a DIS process in which a hadron $h$ , originated by the fragmentation of the struck quark, is detected in the final state: $l(\\ell )+N(P)\\rightarrow l^\\prime (\\ell ^\\prime )+h(P_h)+X(P_X)\\,\\mbox{,}$ where we use the same notation adopted in Eq.", "(REF ), supplemented with the four-momentum of the final hadron, $P_h$ .", "Due to the statistical correlation between the flavor of the struck quark and the type of the hadron formed in the fragmentation process, semi-inclusive DIS with identified pions or kaons may provide a handle on the $\\Delta \\bar{u}$ , $\\Delta \\bar{d}$ and $\\Delta \\bar{s}$ parton distributions, respectively [95].", "For example, roughly speaking the presence of a $\\pi ^+$ in the final state indicates that it is likely that a $u$ -quark or a $\\bar{d}$ -antiquark was struck in the scattering, because the $\\pi ^+$ is a $(u\\bar{d})$ bound state.", "The theoretical description of SIDIS, with longitudinally polarized lepton beams, closely follows that of inclusive DIS given in Chap. .", "In analogy to Eq.", "(REF ), the SIDIS differential cross-section can be written as $\\frac{d^6\\sigma }{dxdyd\\phi dzdp_T^2d\\Phi }\\propto L_{\\mu \\nu }W_h^{\\mu \\nu }\\,\\mbox{;}$ here the leptonic tensor, $L_{\\mu \\nu }$ , has exactly the form given in Eq.", "(REF ) with the symmetric and antisymmetric parts of Eqs.", "(REF )-(REF ), while the hadronic tensor, $W_h^{\\mu \\nu }$ , now contains additional degrees of freedom corresponding to the fractional energy $z$ of the final-state hadron, the $p_T$ component of the final hadron three momentum, transverse to that of the virtual photon, and the azimuthal angle $\\Phi $ of the hadron production plane relative to the lepton scattering plane.", "Integration over $\\Phi $ and $p_T^2$ produces the cross-section relevant for the experimental observables.", "These are the longitudinal and transverse asymmetries $A_\\parallel ^h$ and $A_\\perp ^h$ , defined in analogy to Eq.", "(REF ).", "Thanks to factorization, hadronic and leptonic degrees of freedom are separated, hence kinematic factors depending only on $x$ and $y$ (or $Q^2$ ) are carried over directly from inclusive scattering in relating the measured asymmetries to their virtual photo-absorption counterparts, Eq.", "(REF ).", "In particular, $A_1$ and $A_2$ , Eqs.", "(REF ), should now read $A_1^h$ and $A_2^h$ in terms of the SIDIS cross-sections $\\sigma _{1/2,3/2}^h$ of produced hadrons of type $h$ .", "In particular, we obtain $A_1^h(x,Q^2,z)=\\frac{\\sum _q e^2_q\\Delta q(x,Q^2) \\otimes D_q^h(z,Q^2)}{\\sum _q e^2_{q^\\prime }q^\\prime (x,Q^2) \\otimes D_{q^\\prime }^h(z,Q^2)}\\,\\mbox{,}$ where $e_q$ are the quark electric charges, $\\Delta q$ ($q$ ) are the helicity-dependent (-averaged) PDFs, $D_q^h$ is the fragmentation function for the quark $q$ to fragment into a hadron $h$ , and $\\otimes $ denotes the convolution product, Eq.", "(REF ).", "Measurements of spin-dependent asymmetries in SIDIS have been performed by several experimental collaborations, namely SMC [229], HERMES [95] and COMPASS [96], [97], [98], and have been included in some global determinations of polarized parton distributions [77], [75].", "We notice that the analysis of these data requires the usage of fragmentation functions, which have to be determined in turn from experimental data.", "Usually, they are extracted, independently of PDFs, from electron-positron annihilation, proton-proton collisions and possibly SIDIS data (see Ref.", "[237] for a review).", "Despite the considerable experimental and theoretical effort which has gone into the determination of sets of fragmentation functions [238], [239], [240], [184], [241], [185], [186], [242], their knowledge is still rather poor.", "In particular, recent work [162] has emphasized the failure of all available sets of fragmentation functions in describing the most updated inclusive charged-particle spectra data at the LHC.", "For these reasons, when used in a global determination of polarized PDFs including SIDIS data, they are likely to introduce an uncertainty which is difficult to quantify.", "Heavy flavor hadron production.", "Heavy flavor hadron production is clearly sensitive to the shape and the size of the spin-dependent gluon distribution [243], [244].", "In this case, the gluon polarization can be accessed via the photon-gluon fusion (PGF) mechanism, which results in the production of a quark-antiquark pair, see Fig.", "REF -$(d)$ .", "Experimental signatures to tag this partonic subprocess are the production of one or two hadrons with high $p_T$ in the final state, and open-charm events: in this case, the $q\\bar{q}$ pair is required to be a $c\\bar{c}$ pair and an outgoing charmed meson is reconstructed.", "At LO, the virtual photon-nucleon asymmetry for open-charm production, $A_{LL}^{\\gamma N\\rightarrow D^0X}$ , is expressed as $A_{LL}^{\\gamma N \\rightarrow D^0 X}\\equiv \\frac{\\Delta \\sigma _{\\gamma N}}{\\sigma _{\\gamma N}}=\\frac{\\Delta \\hat{\\sigma }_{\\gamma g}\\otimes \\Delta g\\otimes D_c^{D^0}}{\\hat{\\sigma }_{\\gamma g}\\otimes g\\otimes D_c^{D^0}}\\,\\mbox{,}$ where $\\Delta \\hat{\\sigma }_{\\gamma g}$ ($\\hat{\\sigma }_{\\gamma g}$ ) is the spin-dependent (-averaged) partonic cross-section for PGF, $\\gamma ^ g \\rightarrow c\\bar{c}$ , $\\Delta g$ ($g$ ) is the polarized (unpolarized) gluon PDF, and $D_{c}^{D^0}$ is the non-perturbative fragmentation function of a produced charm quark into the observed $D^0$ meson, which is assumed to be spin independent.", "In principle, a measurement of the asymmetry in Eq.", "(REF ) can then provide a direct handle on $\\Delta g$ .", "Measurements of longitudinal spin asymmetries in high-$p_T$ hadron production were performed by HERMES [231], [232] at DESY and by SMC [233] and COMPASS [234] at CERN.", "However, in such a reaction, the measured asymmetries receive contributions not only from pure PGF events, but also from a significant fraction of background events, mainly due to the two competing processes of gluon radiation by QCD Compton scattering ($\\gamma ^q\\rightarrow qg$ ) and photon absorption at the lowest order of DIS ($\\gamma ^q\\rightarrow q$ ).", "At variance with hadron-pair production, open-charm production is free of background, since the PGF subprocess is the main mechanism for producing charm quarks in polarized DIS.", "The proper theoretical description of these processes depends on the virtuality of the probing photon.", "In case of photoproduction, where a quasi-real photon is exchanged, on top of direct contributions [245], [246], [247], [248], one has to include also resolved contributions [249], [250], where the photon fluctuates into a vector meson of the same quantum numbers before the hard scattering with partons in the proton takes place.", "If the virtuality $Q$ of the photon is of $\\mathcal {O}$ (1 GeV) or higher, resolved processes are sufficiently suppressed but the additional momentum scale $Q$ greatly complicates the calculations of phase-space and loop integrals.", "Only few calculations are available for heavy flavor hadron production in case of polarized beams and targets at NLO accuracy [245], [246], [243], [244].", "A complete phenomenological study of charm quark photoproduction in longitudinally polarized lepton-hadron collisions at NLO accuracy has been presented only very recently [251].", "For the first time, both direct and resolved photon contributions have been included there to compute the relevant cross-sections for spin-dependent heavy flavor hadroproduction.", "For this reason, the available data on hadron and open-charm production [231], [232], [233], [234], [213] have not been included in global QCD analyses of polarized parton distributions so far.", "Experimental collaborations have analyzed their data only in terms of the gluon polarization, $\\Delta g(x, Q^2)/g(x, Q^2)$ , under certain simplifying assumptions and based on leading order matrix elements.", "Nonetheless, the results of these exercises, illustrated in Fig.", "REF , are in fairly good agreement with the NLO prediction obtained from our NNPDFpol1.0 analysis.", "We will explicitly show to which extent COMPASS open-charm data can further pin down the polarized gluon uncertainty in Sec.", "REF .", "Figure: The theoretical prediction for the ratio Δg(x,Q 2 )/g(x,Q 2 )\\Delta g(x,Q^2)/g(x,Q^2)computed from the polarized (unpolarized) NNPDFpol1.0(NNPDF2.3) parton sets at NLO, compared to LO determinations fromone- or two-hadron and open-charm production data in fixed-target DISexperiments , , , , ." ], [ "Spin asymmetries in proton-proton collisions", "High-energy collisions from longitudinally polarized proton beams, as available at the Relativistic Heavy-Ion Collider (RHIC), provide a unique way to probe proton spin structure and dynamics [252], [57].", "Typically, the quantities measured at RHIC are spin asymmetries.", "As an example, for collisions of longitudinally polarized proton beams, one defines the double-spin asymmetry for a given process as $A_{LL}=\\frac{\\sigma ^{++}-\\sigma ^{+-}}{\\sigma ^{++}+\\sigma ^{+-}}\\equiv \\frac{\\Delta \\sigma }{\\sigma }\\,\\mbox{,}$ where $\\sigma ^{++}$ ($\\sigma ^{+-}$ ) is the cross-section for the process with equal (opposite) proton beam polarizations.", "As for collisions of unpolarized proton beams, spin-dependent inelastic cross-sections factorize into convolutions of polarized parton distribution functions of the proton and hard-scattering cross-sections describing the spin-dependent interactions of partons.", "Similarly to inclusive DIS, one can write schematically $\\sigma &=&\\sum _{a,b,(c)=q,\\bar{q},g} f_a \\otimes f_b(\\otimes D_c^H) \\otimes \\hat{\\sigma }^{(c)}_{ab}\\,\\mbox{,}\\\\\\Delta \\sigma &=&\\sum _{a,b,(c)=q,\\bar{q},g}\\Delta f_a \\otimes \\Delta f_b(\\otimes D_c^H) \\otimes \\Delta \\hat{\\sigma }^{(c)}_{ab}\\,\\mbox{,}$ for the denominator and the numerator in Eq.", "(REF ) respectively.", "Here, the sum is over all contributing partonic channels $a + b \\rightarrow c (\\rightarrow H) + X$ producing the desired high-$p_T$ , large-invariant mass final state (or detected hadron $H$ ).", "As usual, $\\otimes $ denotes the convolution, Eq.", "(REF ), between unpolarized (polarized) parton distributions, $f_{a,b}$ ($\\Delta f_{a,b}$ ), parton-to-hadron fragmentation function, $D_c^H$ (only for those processes with identified hadrons in final states), and elementary spin-averaged (-dependent) hard partonic cross-section $\\hat{\\sigma }^{(c)}_{ab}$ ($\\Delta \\hat{\\sigma }^{(c)}_{ab}$ ).", "In particular, the spin-dependent partonic cross-section is defined as $\\Delta \\hat{\\sigma }^{(c)}_{ab}\\equiv \\frac{1}{2}\\left[\\hat{\\sigma }_{ab}^{++}-\\hat{\\sigma }_{ab}^{+-} \\right]\\,\\mbox{,}$ the signs denoting the helicity states of the initial partons $a$ , $b$ .", "At RHIC, there are a number of processes which allow for the measurement of spin asymmetries like that in Eq.", "(REF ).", "Depending on the dominant partonic subrocess, they can probe different aspects of the nucleon spin structure.", "For instance, some of them allow for a clean determination of gluon polarizations, while others are more sensitive to quark and antiquark helicity states.", "We will discuss below the main processes for which measurements of spin asymmetries at RHIC are presently available.", "We do not give here details about other measurements possible at RHIC, but not yet performed, such as the observation of high-$p_T$ or prompt photon production, heavy-flavor production and Drell-Yan production of lepton pairs.", "We refer to [57] for a theoretical review on them.", "High-$p_T$ inclusive jet production double-spin asymmetry.", "A clean and theoretically robust process to probe the polarized gluon PDF, $\\Delta g$ , is inclusive jet production, thanks to the dominance of the $gg$ and $qg$ initiated subprocesses in the accessible kinematic range [253], [254] (see also Tab.", "REF ).", "The situation is analogous to the unpolarized case, where inclusive jet production data from the Tevatron and the LHC [255], [256], [257], [258] are instrumental in pinning down the medium- and large-$x$ gluon behavior.", "In polarized collisions, the relevant experimental observable in jet production is the longitudinal double-spin asymmetry defined along Eq.", "(REF ) $A_{LL}^{1jet}=\\frac{\\sigma ^{++}-\\sigma ^{+-}}{\\sigma ^{++}+\\sigma ^{+-}}\\,\\mbox{.", "}$ For dijet production, at LO, the parton kinematics is given by $x_{1}=\\frac{p_T}{2\\sqrt{s}}\\left( e^{\\eta _3} + e^{\\eta _4} \\right)\\,\\mbox{,}\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ x_{2}=\\frac{p_T}{2\\sqrt{s}}\\left( e^{-\\eta _3} + e^{-\\eta _4} \\right)\\mbox{ ,}$ where $p_T$ is the transverse jet momentum, $\\eta _{3,4}$ are the rapidities of the two jets and $\\sqrt{s}$ is the center-of-mass energy.", "In single-inclusive jet production, the underlying Born kinematics is not uniquely determined because the second jet is being integrated out.", "For the illustrative purposes of Fig.", "REF , we will use the following expression to characterize the Born kinematics $x_{1,2}=\\frac{p_T}{\\sqrt{s}} e^{\\pm \\eta } \\, ,$ with $\\eta $ the rapidity of the leading jet, which corresponds to Eq.", "(REF ), provided the incoming partons carry an equal amount of longitudinal momentum and thus $\\eta _3=-\\eta _4$ .", "This is a good approximation at RHIC, due to the limited coverage in rapidity as compared to unpolarized hadron colliders.", "The calculation of both the numerator and the denominator in Eq.", "(REF ) requires the definition of a suitable jet algorithm.", "Also, notice that corrections up to NLO accuracy should be included in the computation of jet cross-sections, through the algorithm for jet reconstruction, since it is only at NLO that the QCD structure of the jet starts to play a role in the theoretical description.", "This would also provide for the first time the possibility of realistically matching the procedures used in experiment to group final-state particles into jets.", "At NLO, a large number of infrared divergencies are found in the computation of virtual and real diagram contributions to the jet cross-section, due to the large number of color-interacting, massless partons involved in the hard-scattering processes.", "It is then necessary to devise a procedure to perform the calculation of the divergent parts and to show their cancellation in the sum which defines any infrared-safe physical observable.", "Several independent methods to calculate any infrared-safe quantity in any kind of hard unpolarized collision are available in the literature [259], [260], [261].", "In particular, the subtraction method of Ref.", "[260] allows for organizing the computation is such a way that the singularities are extracted and canceled by hand, while the remainder may be integrated numerically over phase space.", "This approach has the advantage of being very flexible; it may be used for any infrared-safe observable, with any experimental cut.", "On the other hand, the numerical integration involved turns out to be rather delicate and time-consuming.", "The subtraction method has been used in Ref.", "[262] to develop a computer code that generates partonic events and outputs the momenta of the final-state partons which can be eventually used to define the physical observables for one ore more jet production in proton-proton collisions.", "Such a parton generator is not equivalent to the usual Monte Carlo parton shower programs, since it is the result of a fixed-order QCD calculation.", "The subtraction method and the computer code of Refs.", "[260], [262], supplemented with the proper matrix elements [263], [264], was then extended to the case of polarized proton-proton collisions in Ref. [265].", "The spin-dependent (and spin-averaged) cross-section for single-inclusive high-$p_T$ jet production is also available from Ref. [266].", "In comparison to Ref.", "[265], the approach of Ref.", "[266] uses a largely analytic technique for deriving the relevant partonic cross-sections, which becomes possible if one assumes the jet to be a rather narrow object.", "This assumption is equivalent to the approximation that the cone opening $R$ of the jet is not too large, and hence was termed small-cone approximation (SCA) in [266].", "In the SCA, one systematically expands the partonic cross-sections around $R=0$ .", "The dependence on $R$ is of the form $\\mathcal {A}\\log R+\\mathcal {B}+\\mathcal {O}(R^2)$ : the coefficients $\\mathcal {A}$ and $\\mathcal {B}$ are retained and calculated analytically, whereas the remaining terms $\\mathcal {O}(R^2)$ and beyond are neglected.", "The advantage of this procedure is that it leads to much faster and more efficient computer code, since all singularities arising in intermediate steps have explicitly canceled and are not subject to delicate numerical treatments.", "The SCA was shown [266] to produce results comparable to those of Ref.", "[265] (whose formalism applies to arbitrary cone openings), in the kinematic range where RHIC data are available.", "For these reasons, the code provided in Ref.", "[266] is better suited than that of Ref.", "[265] for the intensive computations required to include jet data in a global QCD fit.", "Semi-inclusive $\\pi ^0$ production double-spin asymmetry.", "In order to constrain the polarized gluon distribution, one can look for high-$p_T$ leading hadrons such as $\\pi ^-$ , $\\pi ^0$ , $\\pi ^+$ , whose production proceeds through the same partonic subprocesses involved in jet production, in particular $qg\\rightarrow qg$ and $gg\\rightarrow qg$ (see Tab.", "REF ).", "However, the hadronization of the struck parton into the final, measured, pion is described by a non-perturbative fragmentation function, $D_c^\\pi $ : this enters the theoretical description of the corresponding double-spin asymmetry, as encompassed in Eqs.", "(REF )-(REF )-().", "Next-to-leading order QCD corrections to the spin-dependent cross-section for single-inclusive hadron production in proton-proton collisions have been computed for the first time in Refs.", "[267], [268].", "As dictated by Eqs.", "(REF )-(), we need to sum over all possible final states in each channel $ab\\rightarrow cX$ , in compliance with the requirement of single-inclusiveness of the cross-section.", "For instance, in case of $qg\\rightarrow qX$ one needs, besides the virtual corrections to $qg\\rightarrow qg$ , three different $2\\rightarrow 3$ reactions: $qg\\rightarrow q(gg)$ , $qg\\rightarrow q(q\\bar{q})$ , $qg\\rightarrow q(q^\\prime \\bar{q}^\\prime )$ (where brackets indicate the unobserved parton pair).", "The combination of all three processes together will allow for obtaining a finite result.", "The two computations presented in Refs.", "[267], [268] differ from each other in the way the integration over the entire phase space of the two unobserved partons, in the $2\\rightarrow 3$ contributions, is carried out.", "In Ref.", "[267], this calculation is performed numerically, by extending the already mentioned subtraction method of Refs.", "[260], [262] to the case of single-hadron production observables.", "A computer code customized to compute any infrared-safe quantity corresponding to one-hadron production at NLO accuracy is then presented as a result.", "Conversely, in Ref.", "[268], the phase-space integration of the $2\\rightarrow 3$ contributions is performed analitically.", "As already noticed in the case of jet production, this has the main advantage to obtain much faster and more efficient computer code, which is better suited for the intensive computations required by a global QCD fit of polarized parton distributions.", "Small-$p_T$ single-spin $W^\\pm $ production asymmetry.", "Production of $W^\\pm $ bosons in high energy collisions from longitudinally polarized proton beams provides an ideal tool for the study of individual helicity states of quarks and antiquarks inside the proton, complementary to, but independent of, SIDIS [57].", "Within the stadard model, the process $\\overrightarrow{p}p\\rightarrow W^\\pm X$ (the arrow denotes the polarized proton beam) is driven by a purely weak interaction which couples left-handed quarks with right-handed antiquarks only ($u_L\\bar{d}_R \\rightarrow W^+$ and $d_L\\bar{u}_R \\rightarrow W^-$ , with some contamination from $s$ , $c$ , $\\bar{s}$ and $\\bar{c}$ , mostly through quark mixing), thus giving rise to a $W$ parity-violating longitudinal single-spin asymmetry, sensitive to $\\Delta q$ and $\\Delta \\bar{q}$ flavor dependence.", "This asymmetry is defined as $A_L\\equiv \\frac{\\sigma ^+ - \\sigma ^-}{\\sigma ^+ + \\sigma ^-}=\\frac{\\Delta \\sigma }{\\sigma }\\,\\mbox{, }$ where $\\sigma ^{+(-)}$ denotes the cross-section for colliding of positive (negative) longitudinally polarized protons off unpolarized protons.", "Notice that this definition differs from that provided in Eq.", "(REF ), since only one of the two proton beams is polarized.", "If we consider the simplest parton-level process $u\\bar{d}\\rightarrow W^+$ at LO, Eq.", "(REF ) will read [57] $A_L^{W^+}\\approx \\frac{\\Delta u(x_1)\\bar{d}(x_2) - \\Delta \\bar{d}(x_1) u(x_2)}{u(x_1)\\bar{d}(x_2) + \\bar{d}(x_1)u(x_2)}\\mbox{, }$ where $x_1$ and $x_2$ are the momentum fractions, carried by quarks and antiquarks, related to $y_W$ , the $W$ boson rapidity relative to the polarized proton, and to $\\sqrt{s}$ , the hadronic center-of-mass energy, by the relation $x_{1,2}=\\frac{M_W}{\\sqrt{s}}e^{\\pm y_W}\\mbox{.", "}$ The measurement of the rapidity distribution of the $W$ bosons thus provide a direct handle on the flavor-separated polarized quark and antiquark distributions.", "Indeed, at large rapidities, $y_W\\gg 0$ , where sea distributions are suppressed because $x_1$ is in the valence region, the asymmetry $A_L^{W^+}$ is given by $\\Delta u/u$ , whereas it approaches $\\Delta \\bar{d}/\\bar{d}$ in the opposite limit, $y_W\\ll 0$ .", "The situation is similar for $W^-$ production, now interchanging the roles of $u$ and $d$ flavors.", "The naive Born-level picture given above needs to be modified to account for additional aspects, both theoretical and experimental [57].", "The former include higher-order perturbative corrections and other allowed initial states, including Cabibbo-suppressed channels.", "Concerning the latter, $W$ bosons are reconstructed through their leptonic decays $W^\\pm \\rightarrow e^\\pm \\nu $ at RHIC, therefore the observed process is actually $pp\\rightarrow \\ell ^\\pm X$ , with the neutrino escaping undetected.", "One will then measure the rapidity distributions of the charged leptons rather than of the $W$ bosons themselves.", "All these issues have been taken into account in the NLO calculation of the cross-section and longitudinal single-spin asymmetry, Eq.", "(REF ), presented in Ref.", "[269] as a computer program: it may be readily used to include experimental spin asymmetry data in a global analysis of polarized parton densities." ], [ "Experimental input", "Among the processes described in Sec.", ", in the present analysis we only consider open-charm production from COMPASS and single-inclusive high-$p_T$ jet and $W$ production from RHIC.", "Actually, the precise knowledge of fragmentation functions plays a minor role in the theoretical description of these processes.", "On the one hand, in the kinematic regime accessed by COMPASS, open-charm production shows only a slight dependence on the fragmentation of the charm quark into a $D$ meson; on the other hand, the theoretical description of jet and $W$ production asymmetries in proton-proton collisions does not involve fragmentation into identified hadrons in the final state.", "We prefer not to include data whose analysis requires the usage of fragmentation functions since these are poorly known objects: hence, they are likely to affect our unbiased PDFs by an uncertainty difficult to quantify, as discussed in Sec.", "REF .", "In Fig.", "REF , we plot the new data points considered in the present analysis, together with inclusive DIS data already included in the fit presented in Chap. .", "More details about leading partonic subprocesses, probed polarized PDFs, and the ranges of $x$ and $Q^2$ that become accessible were already summarized in Tab.", "REF .", "We will discuss below the main features of the new data sets, separately for each process.", "Figure: Kinematic coverage in the (x,Q 2 )(x,Q^2) plane of experimental dataincluded in the NNPDFpol1.1 parton set.New data are listed in the second column.", "Open-charm production at COMPASS.", "The COMPASS collaboration has recently presented experimental results for the photon-nucleon asymmetry $A_{LL}^{\\gamma N \\rightarrow D^0 X}$ , Eq.", "(REF ), obtained by scattering polarized muons of energy $E_{\\mu }=160$ GeV$^2$ (center-of-mass energy roughly $\\sqrt{s}\\sim 18$ GeV$^2$ ) off longitudinally polarized protons or deuterons from a $^6$ LiD or NH$_3$ targets, in the photoproduction regime (photon virtuality roughly $Q^2\\sim 0$ GeV$^2$ ) [213].", "A detailed description of the experimental setup can be found in Ref. [270].", "Three different data sets, each one of $N_{\\mathrm {dat}}=15$ data points, were presented there, depending on which $D^0$ decay mode was assumed to reconstruct the charmed hadron from the observed final states: $D^0\\rightarrow K^-\\pi ^+$ , $D^0\\rightarrow K^-\\pi ^+\\pi ^0$ or $D^0\\rightarrow K^-\\pi ^+\\pi ^+\\pi ^-$ .", "In the following, they will be referred to as COMPASS $K1\\pi $ , COMPASS $K2\\pi $ and COMPASS $K3\\pi $ respectively.", "Experimental correlations between systematic uncertainties are not provided.", "Assuming LO kinematics, the experiment may probe the polarized gluon distribution at medium momentum fraction values, $0.06 \\lesssim x \\lesssim 0.22$ , and at energy scale $Q^2 = 4(m_c^2+p_T^2) \\sim 13$ GeV$^2$ , where $m_c$ is the charm quark mass and $p_{T}$ is the transverse momentum of the produced charmed hadron, see Fig.", "REF .", "High-$p_T$ jet production at STAR and PHENIX.", "Both the STAR and PHENIX experiments at RHIC provided their measurements of the longitudinal double-spin asymmetry for inclusive jet production, Eq.", "(REF ).", "This is obtained by colliding two polarized proton beams at center-of-mass energy $\\sqrt{s}=200$ GeV.", "We refer to [271] and references therein for a detailed description of the RHIC experimental setup.", "Data from STAR are available for the 2005 and 2006 runs and, in a preliminary form, also for the most recent 2009 run; only one set of data is available for PHENIX, corresponding to data taken in 2005.", "In the following, they will be referred to as STAR 1j-05, STAR 1j-06, STAR 1j-09 and PHENIX 1j respectively.", "The features of these data sets, including the number of data points $N_{\\mathrm {dat}}$ , jet-finding algorithm and corresponding cone radius $R$ used for data reconstruction, covered range in integrated rapidity $\\eta $ and intergrated luminosity $\\mathcal {L}$ , are summarized in Tab.", "REF .", "Experimental correlations between systematic uncertainties are not provided.", "Within LO kinematics, jet data roughly cover the range $0.05\\lesssim x \\lesssim 0.2$ and $30\\lesssim p_T^2\\lesssim 800$ GeV$^2$ , see Fig.", "REF .", "Table: Some features of the jet data included in the present analysis:the number of available data points, N dat N_{\\mathrm {dat}}, the algortihm usedfor jet reconstruction, the range over which therapidity η\\eta is integrated and the integrated luminosity, ℒ\\mathcal {L}.", "$W$ boson production at STAR.", "Both the STAR and PHENIX collaborations at RHIC presented first measurements of the parity-violating spin asymmetry $A_L^{W^\\pm }$ , Eq.", "(REF ), based on the 2009 run at $\\sqrt{s}=500$ GeV.", "[235], [236].", "Unfortunately, due to the low integrated luminosity ($\\mathcal {L}=12$ pb$^{-1}$ and $\\mathcal {L}=8.6$ pb$^{-1}$ for STAR and PHENIX respectively), these data will have little impact in determining polarized antiquark flavors, once included in a polarized parton set.", "More interestingly, the STAR collaboration has recently presented preliminary results for the asymmetry $A_L^{W^\\pm }$ , based on data collected in 2012 at $\\sqrt{s}=510$ GeV and with an integrated luminosity $\\mathcal {L}=72$ pb$^{-1}$ [273].", "Two data sets are provided in six bins of the lepton rapidity $\\eta $ and at an integrated lepton transverse momentum $25<p_T<50$ GeV, separately for $W^+$ and $W^-$ : they will be referred to as STAR-$W^+$ and STAR-$W^-$ henceforth.", "Given STAR kinematics, these data sets are likely to constrain light antiquark PDFs roughly in the momentum fraction interval $0.05\\lesssim x \\lesssim 0.4$ and at the energy scale of the $W$ mass (see Fig.", "REF ).", "Besides uncorrelated statistical uncertainties, the measured asymmetries are provided with a correlated systematic uncertainty related to the uncertainty in the beam polarization, given as the $3.4\\%$ of the measured asymmetry.", "Other uncorrelated systematics, due to background and relative luminosity, are estimated to be less than $10\\%$ of statistical errors in the preliminar STAR analysis [273].", "Even though these uncertainties are expected to become smaller in the final data released by STAR, we conservately assume them to be as large as $10\\%$ ." ], [ "Determination of parton distributions", "In this Section, we illustrate how the NNPDFpol1.0 parton set determined in Chap.", "is supplemented with the piece of experimental information discussed above.", "Instead of a global refit of parton distributions including the new data, we will use PDF reweighting, followed by unweighting: this methodology was presented in Refs.", "[105], [109] and its main features were summarized in Sec.", "REF .", "We recall that parton sets determined trough the NNPDF methodology are provided as Monte Carlo ensembles made of equally probable PDF replicas, each fitted to a data replica generated according to the uncertainties and the corresponding correlations measured in the experiments.", "The number of replicas in a given ensemble, $N_{\\mathrm {rep}}$ , is determined by requiring that the central values, uncertainties and correlations of the original experimental data can be reproduced to a given accuracy by taking averages, variances and covariances over the replica sample.", "For the case of polarized PDFs, in Sec.", "REF we determined that a Monte Carlo sample of pseudodata with $N_{\\mathrm {rep}}=100$ replicas is sufficient to reproduce the mean values and the errors of experimental data to an accuracy which is better than $5\\%$ .", "Only a moderate improvement was observed in going up to $N_{\\mathrm {rep}}=1000$ , hence our default choice was $N_{\\mathrm {rep}}=100$ .", "The PDF ensemble forms an accurate representation of the underlying probability distribution of PDFs, conditional on the input data and the particular assumptions (such as the details of the QCD analysis) used in the fit.", "Based on statistical inference and Bayes theorem, PDF reweighting cosists in assigning to each replica in a Monte Carlo ensemble of PDFs, which is referred to as the prior ensemble, a weight which assesses the likelihood that this particular replica agrees with the new data.", "We refer to Sec.", "REF for details about the way the weights are determined from the $\\chi ^2$ of the new data to the prediction obtained using a given replica in the prior.", "The theoretical bases of the reweighting methodology were carefully checked in Refs.", "[105], [109] and it was also shown that results obtained via global refitting or reweighting with new data are statistically equivalent between each other.", "The main limitation of the reweighting method is that the information brought in by new data should be only a moderate correction as compared to the information already included in the prior PDF ensemble.", "This is precisely our present case, since we will add to our fit a few dozens of new hadronic data points, as we discussed in Sec. .", "On the other hand, the reweighting methodology has the main advantage to avoid global refitting: in particular, for each PDF replica in the prior ensemble, the lenghty computation of observables has to be performed only once, instead of at each minimization step required in the case of refitting.", "This is of particular relevance for the polarized case, since the FastKernel method [104], used in Sec.", "to perform fast evolution and fast computation of inclusive DIS observables, has not yet been implemented for the polarized hadronic processes considered here.", "Alternatively, exact NLO calculations could be directly used in a global PDF fit by extending to polarized observables the FastNLO framework [274] and the general-purpose interface APPLgrid [275], but this too is not yet available.", "The main steps of the procedure we follow to determine the NNPDFpol1.1 parton set are described below." ], [ "Construction of the prior PDF ensemble", "Our goal is to include the piece of new exerimental information discussed in Sec.", "into the determination of polarized parton distributions presented in Chap.", "via Bayesian reweighting.", "To this purpose, we have to compute the theoretical predictions for the measured observables, i.e.", "the longitudinal spin asymmetries Eqs.", "(REF )-(REF )-(REF ), based on PDF replicas in NNPDFpol1.0.", "The $\\chi ^2$ of the new data to the prediction obtained using a given replica in NNPDFpol1.0 is then used to compute the weight corresponding to this replica according to Eq.", "(REF ).", "Unfortunately, the NNPDFpol1.0 parton set determined in Chap.", "cannot be used to compute the theoretical predictions for the asymmetries straightforwardly: at NLO accuracy, this computation requires the knowledge of quark and antiquark distributions separately, which are not provided by the NNPDFpol1.0 parton set, because it was determined from a fit to inclusive DIS data only.", "A separation of the polarized quark and antiquark distributions can be achieved in a global fit including SIDIS data, as done in the DSSV08 [75] and LSS10 [77] analyses.", "In principle, we could perform a new, global fit to inclusive and semi-inclusive DIS data and then use the corresponding Monte Carlo ensemble of parton distributions as the prior for the reweighting with collider data presented in Sec. .", "However, as already noticed many times in this thesis, the analysis of SIDIS data requires the usage of poorly known fragmentation functions, which may significantly affect the accuracy of our results.", "For consistency, we should determine a set of fragmentation functions based on the NNPDF methodology and then use this set to perform a fit including SIDIS data, but this is beyond the scope of the present analysis.", "Be that as it may, we circumvent the issue related to the lack of quark-antiquark separation in NNPDFpol1.0 using a different approach, which will be described here for the first time.", "The idea is to supplement the information available in the NNPDFpol1.0 parton set for the $\\Delta u^+$ , $\\Delta d^+$ , $\\Delta s^+$ and $\\Delta g$ distributions with some assumptions on $\\Delta \\bar{u}$ and $\\Delta \\bar{d}$ in order to construct a suitable prior ensemble.", "This is a sensible approach, provided that different ansatz for the $\\Delta \\bar{u}$ and $\\Delta \\bar{d}$ prior distributions lead to the same final result after including the new data by reweighting.", "Indeed, if new data brings in a sufficient amount of information, the final reweighted PDFs will be independent of the original choice of prior [172], [173].", "Of course, it will be essential to show explicitly that this is what happens in our particular situation.", "In principle, we could think of making the quark-antiquark separation for $u$ and $d$ flavors in the prior arbitrarily: for example, we could assign to each replica random values for its $\\Delta u$ , $\\Delta d$ , $\\Delta \\bar{u}$ and $\\Delta \\bar{d}$ distributions, provided their sum reproduces the corresponding total distributions determined in NNPDFpol1.0 and they separately satisfy theoretical constraints (see Sec.", "REF ).", "However, such a prior will lead to highly unefficient reweighting, in the sense that the new underlying PDF probability distribution after reweighting will be sampled by a too small number of effective replicas (for details, see Sec.", "REF ).", "In order to avoid this loss of efficiency, prior ensembles with a huge number of replicas should be produced, but this would be extremely demanding in terms of computational resources.", "Alternatively, the additional information on quark-antiquark separation needed to construct suitable prior ensembles can be obtained from one of the aformentioned fits to SIDIS data.", "we can allow for deviations from the corresponding best-fit $\\Delta \\bar{u}$ and $\\Delta \\bar{d}$ determinations, by supplementing them with additional statistical noise and uncertainties, until the independence of the reweighted results from the prior is achieved.", "The loss of efficiency of these priors will be under control, thus not requiring to be huge, as we will demonstrate below.", "We now discuss in practice how we construct suitable prior ensembles to be afterwards reweighted with the new data discussed in Sec. .", "We supplement the NNPDFpol1.0 parton set with the information on $\\Delta \\bar{u}$ and $\\Delta \\bar{d}$ distributions from the DSSV08 parton fit [75], which includes all available SIDIS data, and we construct a collection of independent prior PDF ensembles.", "First of all, we sample the DSSV08 $\\Delta \\bar{u}$ and $\\Delta \\bar{d}$ parton distributions at a fixed reference scale $Q_0^2=1$ GeV$^2$ .", "We select ten points, half logaritmically and half linearly spaced in the interval of momentum fraction $10^{-3}\\lesssim x \\lesssim 0.4$ , which roughly corresponds to the range covered by SIDIS experimental data relevant for separating quark-antiquark contributions.", "We sample four independent sets of data points assuming the DSSV08 best fit plus one, two, three or four times its nominal $\\Delta \\chi ^2=1$ Hessian uncertainty.", "Separate prior PDF ensembles, labelled as $1\\sigma $ , $2\\sigma $ , $3\\sigma $ and $4\\sigma $ henceforth, will then be constructed for each one of these data sets.", "Of course, a different ansatz could be made on data: the rationale we followed was to increase their uncertainty until independence of the reweighted results from the prior was reached.", "We will explicitly show that this requirement is fulfilled at least by the $3\\sigma $ and $4\\sigma $ prior ensembles at the end of this Section.", "Data points sampled from the DSSV08 fit are then treated, separately for $\\Delta \\bar{u}$ and for $\\Delta \\bar{d}$ , as sets of experimental pseudo-observables.", "Henceforth, they will be labelled as DSSV08$_{U}$ and DSSV08$_{D}$ respectively.", "More precisely, we generate $N_{\\mathrm {rep}}=1000$ replicas of the original DSSV08 pseudodata, following the procedure described in Sec.", ", and then for each individual replica we perform a neural network fit to them: the result gives the $\\Delta \\bar{u}$ and $\\Delta \\bar{d}$ distributions with wich we supplement the NNPDFpol1.0 parton set.", "In order to meaningfully fit the $\\Delta \\bar{u}$ and $\\Delta \\bar{d}$ pseudodata, we need to supplement the input PDF basis given in Sec.", ", namely $\\Delta \\Sigma $ , $\\Delta T_3$ , $\\Delta T_8$ and $\\Delta g$ , with two new linearly independent light quark combinations; we choose them to be the total valence, $\\Delta V$ , and the valence isotriplet, $\\Delta V_3$ , $\\Delta V(x,Q_0^2) &=& \\Delta u^-(x,Q_0^2)+\\Delta d^-(x,Q_0^2)\\,\\mbox{,}\\\\\\Delta V_3(x,Q_0^2) &=& \\Delta u^-(x,Q_0^2)-\\Delta d^-(x,Q_0^2)\\,\\mbox{,}$ where $\\Delta q^-=\\Delta q-\\Delta \\bar{q}\\mbox{, } q=u,d$ .", "In addition, we have assumed that $\\Delta s = \\Delta \\bar{s}$ .", "Even though data are presently insufficient to discriminate between any guess on strage-antistrange distributions and there are actually no theoretical motivations to support a symmetric polarized strangeness, we adopt the choice $\\Delta s =\\Delta \\bar{s}$ as it is usual in all polarized PDF analyses.", "We emphasize that the distribution which is physically meaningful is instead the total strange combination $\\Delta s^+$ which was already determined from inclusive DIS data in NNPDFpol1.0, see Chap .", "Each of the new PDF combinations in Eqs.", "(REF )-() is parametrized as usual by means of a neural network supplemented with a preprocessing polynomial, $\\Delta V(x,Q_0^2)&=&(1-x)^{m_{\\Delta V}} x^{n_{\\Delta V}} \\textrm {NN}_{\\Delta V}(x)\\,\\mbox{,}\\\\\\Delta V_3(x,Q_0^2)&=&(1-x)^{m_{\\Delta V_3}} x^{n_{\\Delta V_3}} \\textrm {NN}_{\\Delta V_3}(x)\\,\\mbox{,}$ where $\\textrm {NN}_{\\Delta \\textrm {pdf}}$ , $\\textrm {pdf}=V, V_3$ , is the output of the neural network and the preprocessing exponents $m,n$ are linearly randomized for each Monte Carlo replica within the ranges given in Tab.", "REF .", "We have checked that our choice of preprocessing exponents does not bias the fit, according to the procedure discussed in Sec. .", "The neural network architecture is the same as in NNPDFpol1.0, namely 2-5-3-1, see Sec.", "REF .", "Note that in terms of the quark PDF input basis, the $\\Delta \\bar{u}$ and $\\Delta \\bar{d}$ distributions which here play the role of pseudo-data are given by the following combinations: $\\Delta \\bar{u}(x,Q_0^2)&=& \\frac{1}{12}\\left( 2\\Delta \\Sigma + 3\\Delta T_3+\\Delta T_8 -3\\Delta V -3\\Delta V_3 \\right)(x,Q_0^2)\\,\\mbox{,}\\\\\\Delta \\bar{d}(x,Q_0^2)&=& \\frac{1}{12}\\left( 2\\Delta \\Sigma - 3\\Delta T_3+\\Delta T_8 -3\\Delta V +3\\Delta V_3 \\right)(x,Q_0^2)\\,\\mbox{.", "}$ For each pseudodata replica $\\Delta \\bar{u}^{(k)}$ , $\\Delta \\bar{d}^{(k)}$ in Eqs.", "(REF )-(), $k=1,\\ldots ,N_{\\mathrm {rep}}$ , we supplement the neural networks for $\\Delta V$ and $\\Delta V_3$ that are being fitted with random replicas from NNPDFpol1.0.", "All the prior PDF ensembles are composed of $N_{\\mathrm {rep}}=1000$ replicas; this larger number of PDF members, in comparison to that used in the analysis presented in Chap.", ", where $N_{\\mathrm {rep}}=100$ , is required to ensure that replicas left after reweighting still describe the underlying PDF probability distribution with sufficient accuracy; $N_{\\mathrm {rep}}=1000$ replicas of NNPDFpol1.0 were generated with this purpose.", "Table: Ranges for the small- and large-xx preprocessing exponentsin Eqs.", "()-().In these fits to pseudodata, the minimization is performed by means of a genetic algorithm, as discussed in Secs.", "REF -REF .", "The implementation of theoretical constraints, both positivity and integrability, also follows consistently the procedure from Ref.", "[116] in order to take care of flavor and antiflavor separation.", "In particular, Eqs.", "(REF )-(REF ) have been enforced by letting $f=u, \\bar{u}, d, \\bar{d}$ separately.", "Note that no additional sum rules affect $\\Delta V$ and $\\Delta V_3$ .", "Following this procedure, we end up with four separate prior PDF ensembles, labeled as $1\\sigma $ , $2\\sigma $ , $3\\sigma $ and $4\\sigma $ , corresponding to the different factors by which the DSSV08 nominal PDF uncertainty has been enlarged.", "The goodness of the pseudodata fits is quantitatively assessed by the $\\chi ^2$ values per data point quoted in Tab.", "REF , which are close to one for both separate and combined DSSV08$_{U}$ and DSSV08$_{D}$ data sets.", "In Fig.", "REF , we show the $x\\Delta \\bar{u}(x,Q_0^2)$ and $x\\Delta \\bar{d}(x,Q_0^2)$ PDFs at the initial energy scale $Q_0^2=1$ GeV$^2$ from the $1\\sigma $ and $4\\sigma $ prior ensembles we have contructed.", "We have checked that the other priors, $2\\sigma $ and $3\\sigma $ , consistently reproduce intermediate results.", "In these plots the positivity bound discussed in Sec.", "and data points sampled from the DSSV08 parton set [75] are also shown.", "Table: The value of the χ tot 2 \\chi ^2_{\\mathrm {tot}} per data point for bothseparate and combined Δu ¯\\Delta \\bar{u} and Δd ¯\\Delta \\bar{d} data setsafter the neural network fit to pseudodata sampled from the DSSV08parton set.Figure: The polarized sea quark distributionsxΔu ¯(x,Q 0 2 )x\\Delta \\bar{u}(x,Q_0^2) (upper plots)and xΔd ¯(x,Q 0 2 )x\\Delta \\bar{d}(x,Q_0^2) (lower plots) at the initial energy scaleQ 0 2 =1Q_0^2=1 GeV 2 ^2 from the neural network fit (full band)to pseudodata sampled from DSSV08 parton set(points with uncertainties).Results are shown for the 1σ1\\sigma (left plots) and4σ4\\sigma (right plots) prior ensembles.The positivity bound from the corresponding unpolarized NNPDF2.3parton set is also shown.Monte Carlo ensembles of polarized parton distributions obtained in this way are equivalent to NNPDFpol1.0 in the $\\Delta q^+=\\Delta q +\\Delta \\bar{q}$ ($q=u, d, s$ ) and $\\Delta g$ sectors, but they are supplemented with quark-antiquark separation, suitable as a starting point for the reweighting procedure.", "Notice that they have exactly the same gluon distribution, since this is not affected by construction.", "In the next sections we quantify the impact on these priors of the new data, and show that results are independent of the specific choice of prior starting from the $3\\sigma $ case." ], [ "Reweighting with new data sets", "We would like to reweight the prior PDF ensembles determined in Sec.", "REF with the data described in Sec. .", "To this purpose, we should compute theoretical predictions for the observables measured in each process under investigation and then compare them to experimental data.", "As explained in Sec.", ", we will then assign to each replica in the PDF ensembles a weight proportional to the $\\chi ^2$ of the new data to the corresponding prediction, given by Eq.", "(REF ).", "Before discussing the impact of each new data set in turn, we summarize a few methodological aspects which apply to all experiments.", "For each of the new processess considered in our analysis, the experimental observable is a longitudinal spin asymmetry, i.e.", "the ratio between cross-sections depending on polarized PDFs (in the numerator) and on unpolarized PDFs (in the denominator), see Eqs.", "(REF )-(REF )-(REF ).", "In these expressions, the numerator will be computed for each polarized replica from the ensembles determined in Sec.", "REF , while the denominator will be evaluated only once using the mean value from the unpolarized NNPDF2.3 parton set [110] at NLO.", "This strategy accounts for the fact that the uncertainty affecting the observed asymmetries is mostly driven by the uncertainty of the polarized parton distributions rather than by that of their unpolarized counterparts, which is actually negligible.", "For each data set we will have to check the effectiveness of the reweighting procedure.", "To this purpose, we will look at the distribution of $\\chi ^2$ per data point among replicas and at its mean value, which is expected to decrease after reweighting.", "Also, we will have to keep under control the loss of accuracy in the description of the underlying PDF probability distribution, by ensuring that the number of replicas left after reweighting, $N_{\\mathrm {eff}}$ , does not become too low.", "In particular, we require that $N_{\\mathrm {eff}}$ should be comparable with the number of replicas, $N_{\\mathrm {rep}}$ , in NNPDFpol1.0, i.e.", "$N_{\\mathrm {eff}}\\sim N_{\\mathrm {rep}}=100$ .", "Indeed, we determined in Sec.", "REF that a Monte Carlo sample of $N_{\\mathrm {rep}}=100$ replicas is sufficient to reproduce the mean values and the errors of experimental data within percent accuracy.", "We may be interested in determining whether the new data are consistent with the old inclusive DIS data.", "To this purpose, for each data set we will evaluate the $\\mathcal {P}(\\alpha )$ distribution, defined by Eq.", "(12) of Ref. [105].", "The parameter $\\alpha $ measures the consistency of the data which are used for reweighting with those included in the prior PDF sets, by providing the factor by which the uncertainty on the new data must be rescaled in order the two sets to be consistent.", "Hence, if the probability density for the parameter $\\alpha $ , $\\mathcal {P}(\\alpha )$ , peaks close to one, one can conclude that new and old data are consistent with each other." ], [ "Open-charm production at COMPASS", "Predictions for the photon-nucleon asymmetry $A_{LL}^{\\gamma N\\rightarrow D^0 X}$ are computed at LO accuracy, using the expressions in Ref.", "[197], based in turn on Ref.", "[243], for both the numerator and the denominator in Eq.", "(REF ).", "Results are compared in Fig.", "REF to COMPASS experimental data, separated into individual decay channels and into three bins of the charmed hadron energy $E_{D^0}$ .", "The curves labelled as DSSV08, AAC08 and BB10 are obtained using the corresponding polarized parton sets [75], [73], [76] and either the CTEQ6 [165] (for DSSV08) or the MRST2004 [276] (for AAC08 and BB10) unpolarized PDF sets.", "The curve labelled as NNPDF is instead computed using the NNPDFpol1.0 parton set.", "Notice that we do not need the prior Monte Carlo ensembles constructed in Sec.", "REF , since we compute the observable at LO and only the polarized gluon appears in Eq.", "(REF ).", "By construction, this distribution is exactly the same in all the parton sets discussed in Sec.", "REF and in NNPDFpol1.0, hence they all give the same prediction for the photon-nucleon asymmetry $A_{LL}^{\\gamma N\\rightarrow D^0 X}$ .", "For all the curves shown in Fig.", "REF , we have used the Peterson parametrization of the fragmentation function $D_c^{D^0}$ [277]; we checked that results are unaffected by the choice of other, slightly different, available parametrizations [278], as noticed in Ref. [251].", "Figure: Experimental double-spin asymmetry for D 0 D^0 mesonphotoproduction A LL γN→D 0 X A_{LL}^{\\gamma N \\rightarrow D^0 X}measured by COMPASS from three decay channelscompared to its LO prediction, Eq.", "(),computed for different PDF sets in three bins of the charmed hadron energyE D 0 E_{D^0} and in five bins of its transverse momentum p T D 0 p_{T}^{D_0}.The agreement between predictions from different PDF sets and experimental data can be quantified by the value of the $\\chi ^2$ per data point, which we compute for both separate and combined COMPASS data sets, see Tab.", "REF .", "The corresponding distribution among NNPDF replicas is shown in the first panel of Fig.", "REF only for the combined data set.", "Notice that, lacking the experimental covariance matrix, the uncertainties which enter the $\\chi ^2$ definition are taken as the sum in quadrature of statistical and systematic uncertainties.", "From Tab.", "REF , it is clear that, even before reweighting, predictions are already in good agreement with experimental data, which however are affected by large errors in comparison to the uncertainty estimate of the asymmetry itself.", "Also, all polarized PDF sets provide a similar description of the data.", "Table: Values of χ 2 /N dat \\chi ^2/N_{\\mathrm {dat}} before reweighting for differentpolarized parton sets.Next, we have quantified the impact of COMPASS open-charm data into NNPDFpol1.0 using Bayesian reweighting.", "In Tab.", "REF , we quote the $\\chi ^2$ per data point after reweighting, $\\chi ^2_{\\mathrm {rw}}/N_{\\mathrm {dat}}$ , while the number of effective replicas left after reweighting, $N_{\\mathrm {eff}}$ , and the modal value of the $\\mathcal {P}(\\alpha )$ distribution, $\\langle \\alpha \\rangle $ , for both separate and combined data sets are collected in Tab.", "REF .", "In Fig.", "REF , we also plot the distribution of $\\chi ^2$ per data point before and after reweighting, and of $\\mathcal {P}(\\alpha )$ for all data sets combined together.", "Figure: Distribution of χ 2 /N dat \\chi ^2/N_{\\mathrm {d}at} for individual replicasbefore (first panel) and after (second panel)reweighting with COMPASS open-charm production data .The shaded region in the first panel corresponds to the central68%68\\% of the distribution.", "The 𝒫(α)\\mathcal {P}(\\alpha ) distribution (third panel)is also shown.All plots refer to the three COMPASS data sets combined together.From Tabs.", "REF -REF and Fig.REF , it is evident that reweighting with COMPASS open-charm data leaves the prior parton set almost unaffected: the $\\chi ^2$ value per data point and its distribution are essentially unchanged after reweighting.", "Also, almost all replicas in the prior ensemble are preserved: this further demonstrates the mild constraining power of COMPASS data sets.", "Furthermore, the observable $A_{LL}^{\\gamma N \\rightarrow D^0 X}$ , Eq.", "(REF ), is compared before and after reweighting in Fig.", "REF , showing unnoticeable differences.", "Finally, the reweighted polarized gluon PDF is drawn in Fig.", "REF together with its one-sigma absolute error and compared to NNPDFpol1.0 [116] at $Q_0^2=1$ GeV$^2$ .", "Again, we notice that the differences are mild.", "Figure: Comparison between the double-spin asymmetry A LL γN→D 0 X A_{LL}^{\\gamma N \\rightarrow D^0 X},Eq.", "(), before and after reweighting with COMPASS open-charmdata .", "Experimental points are also shown.Figure: Comparison between the unweighted and the reweighted polarizedgluon distribution at Q 0 2 =1Q_0^2=1 GeV 2 ^2 (left panel) and theimprovement in its absolute error (right panel).From these results, we can conclude that COMPASS open-charm data lead to a moderate improvement in our knowledge of the polarized gluon PDF, due to their large experimental uncertainties.", "One may wonder whether this conclusion still holds once NLO corrections are taken into account in the computation of the photon-nucleon asymmetry, $A_{LL}^{\\gamma N \\rightarrow D^0 X}$ .", "Such a computation has been recently completed [251] and it was shown that contributions beyond Born-level may significantly affect this asymmetry.", "However, COMPASS experimental data are not only affected by large uncertainties with respect to the corresponding theoretical predictions (both at LO and NLO), but also do not show a clear and stable trend over the covered range of $p_T^{D^0}$ , see Figs.", "REF -REF .", "For this reason, the NLO computation of the photon-nucleon spin asymmetry will not allow for determining the polarized gluon more precisely than its LO counterpart, though in principle the former contains more QCD structure than the latter.", "Hence, we expect that the impact of COMPASS open-charm data on the determination of the polarized gluon PDF will be comparable to that found in our LO analysis, once they will be included in a global NLO QCD fit of parton distributions, as also anticipated in Ref. [251].", "$\\endcsname $We were not able to compute the photon-nucleon asymmetry, $A_{LL}^{\\gamma N \\rightarrow D^0 X}$ , at NLO accuracy because the code developed to this purpose in Ref.", "[251] is not publicly available.", "Qualitatively, the impact of the NLO corrections on this asymmetry can be inferred by comparing our results in Fig.", "REF (at LO) with those in Fig.", "8 of Ref.", "[251] (at NLO): differences are actually hardly noticeable." ], [ "High-$p_T$ jet production at STAR and PHENIX", "Predictions for the longitudinal double-spin asymmetry from single-inclusive jet production, $A_{LL}^{1jet}$ , are shown in Fig.", "REF .", "They are plotted as a function of the transverse jet momentum, $p_T$ , and compared to each experimental data set in Tab.", "REF .", "Predictions are computed at NLO using the code of Ref.", "[266], which was modified to handle NNPDF parton sets.", "We use the same jet algorithm, cone radius and kinematic cuts adopted in the experiment (see Tab.", "REF ).", "In Fig.", "REF , results are shown only for the $1\\sigma $ prior PDF ensemble.", "We have explicitly checked their stability upon the choice of any prior PDF ensemble discussed in Sec.", "REF : the asymmetry shows hardly noticeable differences among different priors, thus proving it is not sensitive to quark-antiquark separation, as expected.", "Figure: Predictions for the double longitudinal spin asymmetry forsingle-inclusive jet production, A LL 1jet A_{LL}^{1jet}, before and after reweightingwith RHIC data.", "Results are obtained from the 1σ1\\sigma prior PDF ensemblediscussed in Sec. .", "Experimental data points are also displayed.Notice the different scale of vertical axis for PHENIX.The $\\chi ^2$ per data point before reweighting for separate and combined data sets is quoted in Tab.", "REF .", "Neither STAR nor PHENIX jet data are provided with experimental covariance matrix, hence we will assume the systematics to be uncorrelated and then sum in quadrature with statistical errors.", "Also, we have to account for the fact that data are taken in bins of $p_T$ , whereas the corresponding theoretical predictions are computed for the center of each bin.", "We estimate the corresponding uncertainty as the maximal variation of the observable within each bin and take that value as a further uncorrelated systematic uncertainty.", "Since STAR data are provided with asymmetric systematic uncertainties, we must take care of symmetrizing them, according to Eqs.", "(7)-(8) in Ref. [99].", "We observe that our predictions are in good agreement with experimental data, as $\\chi ^2/N_{\\mathrm {dat}} \\sim 1$ for all data sets.", "Nevertheless, notice that, except for STAR 1j-09 data set, experimental data points are affected by rather large errors in comparison to the uncertainty on the observable due to the polarized PDFs: hence, this will limit their potential in constraining the polarized gluon distribution.", "Furthermore, similar results are obtained with different choices of the prior PDF ensemble: this strengthens the conclusion that we are looking at a process which is almost insensitive to the quark content of the proton.", "Next, we reweight the various prior ensembles with both separate and combined RHIC inclusive jet production data.", "In Tab.", "REF , we quote the values for the $\\chi ^2$ per data point after reweighting, $\\chi ^2_{\\mathrm {rw}}/N_{\\mathrm {dat}}$ , while the number of effective replicas, $N_{\\mathrm {eff}}$ , and the modal value of the $\\mathcal {P}(\\alpha )$ distribution, $\\langle \\alpha \\rangle $ , are collected in Tab.", "REF .", "In Fig.", "REF , we display, for combined jet data sets, the unweighted distribution of $\\chi ^2/N_{\\mathrm {dat}}$ , the corresponding weighted distribution of $\\chi ^2_{\\mathrm {rw}}/N_{\\mathrm {dat}}$ , and the $\\mathcal {P}(\\alpha )$ distribution.", "In Fig.", "REF we finally compare the asymmetry before and after reweighting.", "Figure: Same as Fig.", ", but for combined RHIC jet dataand for the 1σ1\\sigma prior discussed in Sec.", ".It is clear from Tab.", "REF and Figs.", "REF -REF that jet data from RHIC carry an important piece of experimental information.", "In particular, we observe a substantial improvement in the description of the high precision STAR 1j-09 data set, for which the value of the $\\chi ^2/N_{\\mathrm {dat}}$ decreases from 1.69 to 1.02.", "As we will show below, this improvement translates into significant constraints on the polarized gluon PDF.", "Moreover, the comparison between the $\\chi ^2$ distributions before and after reweighting shows that their peak moves close to one, with a slight narrowing due to the increase in the total number of data points.", "The reweighted observable nicely agrees with experimental data and its uncertainty is reduced with respect to the prior.", "Also, notice that the PDF set loss of accuracy in describing the underlying probability distribution is well under control: the number of replicas left after reweighting, $N_{\\mathrm {eff}}$ , is always larger than $N_{\\mathrm {rep}}=100$ , the value roughly required for NNPDFpol1.0 to reproduce data and errors with percent accuracy.", "Finally, new data sets are consistent with the experimental information already included in the prior, as shown by the $\\mathcal {P}(\\alpha )$ distribution, which is clearly peaked at one (see Fig.", "REF ).", "Nevertheless, we notice that different data sets have different power in improving the fit: the more accurate they are, the more effective they are in constraining the theoretical prediction for the asymmetry $A_{LL}^{1jet}$ and, in turn, the polarized gluon distribution.", "It is clear from Tabs.", "REF -REF that a substantial amount of experimental information comes from the STAR 1j-09 data set.", "We also observe that PHENIX data have a fair impact in improving the fit, due to their large errors: it is likely that they were overestimated, since the modal value of the $\\alpha $ parameter is far below one (see Tab.", "REF ).", "Finally, in Fig.", "REF we compare the polarized gluon PDF from the NNPDFpol1.0 parton set [116] with its counterpart from the $1\\sigma $ prior reweighted with RHIC jet data.", "We also show the corresponding one-sigma absolute error and a comparison between the reweighted results obtained with extremum prior PDF sets, namely $1\\sigma $ and $4\\sigma $ .", "We observe that, in the kinematic range probed at RHIC, the polarized gluon PDF becomes positive and its uncertainty is reduced.", "This effect is consistent with what was found in Ref.", "[188], where these data were included in the DSSV++ parton set.", "Again, we stress that, in the case of jet data, results are independent of the choice of any prior PDF ensemble discussed in Sec.", "REF : the stability of our results is clearly visible from both Tabs.", "REF -REF and from the third panel of Fig.", "REF .", "Hence, results obtained from the reweighting of any prior are equivalent, and can be used indifferently to describe the polarized gluon PDF.", "We have explicitly checked that other PDFs, not shown in Fig.", "REF are unaffected by jet data, as expected.", "Figure: (First panel) Comparison between the polarized gluon PDF fromNNPDFpol1.0 parton set and the result obtainedby reweighting with RHIC jet data.", "(Second panel) the same comparison,but for absolute PDF uncertainty.", "(Third panel) Comparison between thereweighted gluon PDF obtained from the 1σ1\\sigma and the 4σ4\\sigma priorPDF ensembles." ], [ "$W$ boson production at STAR", "Predictions for the longitudinal single-spin asymmetry, Eq.", "(REF ), are computed at NLO using the program of Ref.", "[269], which was consistently modified to handle NNPDF parton sets.", "In Fig.", "REF , we show our predictions for the longitudinal positron (electron) single spin asymmetry $A_L^{e^+}$ ($A_L^{e^-}$ ) from production and decay of $W^{+(-)}$ bosons in bins of the rapidity $\\eta $ , compared to STAR data [273].", "Results are diplayed for the $1\\sigma $ and $4\\sigma $ prior PDF ensembles, and we have checked that intermediate results are obtained from the other priors constructed in Sec.", "REF .", "Unlike open-charm and jet production observables discussed above, the longitudinal single spin asymmetry, Eq.", "(REF ), is sensitive to separate quark and antiquark PDFs.", "For this reason, the corresponding theoretical predictions made from the $4\\sigma $ prior are affected by larger uncertainties than those obtained from the $1\\sigma $ prior.", "Figure: Predictions for the longitudinal positron (upper plots) and electron(lower plots) single spin asymmetry A L e + A_L^{e^+} and A L e - A_L^{e^-}before and after reweighting with STAR data .Results from 1σ1\\sigma (left) and 4σ4\\sigma (right) prior PDF ensemblesare shown.Curves are obtained at NLO with the CHE code .Experimental points are also shown, uncertainties are statistical only.In Tab.", "REF , we summarize the agreement between experimental data and parton sets before reweighting, as quantified by the value of $\\chi ^2/N_{\\mathrm {dat}}$ : as we can see, it is not very good, and all prior PDF sets fail to describe these data sets with sufficient accuracy.", "The discrepancy between data and corresponding theoretical predictions is particularly noticeable for the $W^-$ set.", "This may reflect some tension between semi-inclusive data, included in the DSSV08 global fit, and $W$ production data.", "Indeed, the experimental information included in DSSV08 was inherited by NNPDF prior ensembles, due to the way they were constructed.", "After reweighting with STAR data, we should demonstrate that this dependence has been removed and check that reweighted PDFs properly describe $W^\\pm $ data sets.", "As for the other observables, we proceed to reweight the different prior ensembles with STAR $W$ production data.", "Results are collected in Tabs.", "REF -REF for both separate and combined STAR data sets, and compared to the analogous quantity before reweighting.", "In Fig.", "REF we display, for each prior PDF ensemble, and for combined $W^+$ and $W^-$ STAR data sets, the unweighted distribution of $\\chi ^2/N_{\\mathrm {dat}}$ , the corresponding weighted distribution of $\\chi ^2_{\\mathrm {rw}}/N_{\\mathrm {dat}}$ , and the $\\mathcal {P}(\\alpha )$ distribution.", "Finally, in Fig.", "REF we compare the asymmetry before and after reweighting.", "Figure: Same as Fig.", ", but for combinedW + W^+ and W - W^- STAR data sets and for each prior.We see that, after reweighting, our predictions agree with STAR data, for both the $W^+$ and the $W^-$ final states, but the goodness of this agreement actually depends on the prior.", "In general, we observe that the $\\chi ^2/N_{\\mathrm {dat}}$ value decreases after reweighting (see Tab.", "REF ), and that the $\\chi ^2$ distribution tends to properly peak at one, being more narrow than before reweighting (compare, for each prior PDF ensemble, plots in the first and second column of Fig.", "REF ).", "The value of the effective number of replicas after reweighting, $N_{\\mathrm {eff}}$ , is always larger than $N_{\\mathrm {rep}}=100$ , thus the size of the initial prior sample was large enough.", "The modal value of the $\\mathcal {P}(\\alpha )$ distribution, $\\langle \\alpha \\rangle $ is close to one for the $W^+$ data but almost 1.4 for the $W^{-}$ data, suggesting that in the latter case some experimental uncertainties might be somewhat underestimated.", "This might also explain why the STAR-$W^-$ data set has a much smaller value of $N_{\\mathrm {eff}}$ than STAR-$W^+$ , while we expected the two sets to have a similar impact.", "Finally, the reweighted observable nicely agrees with experimental data and its uncertainty is reduced with respect to the prior, as clearly shown in Fig.", "REF .", "As emphasized in Sec.", "REF , in order to get reliable results we must require the reweighted PDFs to be independent of the prior PDF ensemble.", "In other words, we should discard results which are not stable upon the choice of different prior PDF ensembles and, if needed, we should construct new priors, assuming a different ansatz on the quark-antiquark PDF separation, until this independence is effectively achieved.", "In this case, we can verify that both the $3\\sigma $ and $4\\sigma $ prior PDF ensembles lead to fully equivalent results, as can be seen at the level of the $\\chi ^2$ value per data point after reweighting (see Tab.", "REF ), single-spin asymmetries and PDFs: the explicit comparison of these two distributions from the $3\\sigma $ and $4\\sigma $ samples is shown in Fig.", "REF at $Q_0^2=1$ GeV$^2$ .", "Then, we can conclude that results from both $3\\sigma $ and $4\\sigma $ prior PDF ensembles are statistically indistinguishable and that they can both be used to provide a robust determination of the light antiquark PDFs, $\\Delta \\bar{u}$ and $\\Delta \\bar{d}$ .", "Results displayed in Fig.", "REF refer to the simultaneous reweighting with both $W^+$ and $W^-$ data sets; we have explicitly checked that reweighting with $W^+$ ($W^-$ ) data set separately probe $\\Delta \\bar{d}$ ($\\Delta \\bar{u}$ ) PDF.", "Parton distributions not shown in Fig.", "REF , including strangeness, are not affected by reweighting with $W$ data, as we have explicitly checked.", "The situation is rather different from the unpolarized case, where, instead, $W$ production data also provide some information on strangeness.", "However, we notice that in the unpolarized case Drell-Yan data from both fixed-target experiments and colliders are available: in particular, the former are provided by E605 and E688 experiments at Fermilab [279], [280], [281], [282], while the latter by CDF and D0 at the Tevatron [283], [284], [285] and by ATLAS, CMS and LHCb at the LHC [286], [287], [288].", "These data sets span about three orders of magnitude in the energy scale $Q^2$ , from $Q^2\\sim 20 - 250$ GeV$^2$ for fixed-target experiments, up to the $W$ and $Z$ masses for collider experiments.", "Hence, the effects of the evolution, enhanced by the rather wide $Q^2$ lever-arm of experimental data, combine with the contributions from the Cabibbo-favored partonic subprocesses, initiated by a $s\\bar{c}$ or $\\bar{s}c$ pair: this way, they provide some constraints on the strangeness.", "Of course, since in the polarized case only collider data are available, we cannot observe this effect.", "Figure: Comparison between the Δu ¯\\Delta \\bar{u} (Δd ¯\\Delta \\bar{d})PDF obtained from the reweighting of the 3σ3\\sigma and 4σ4\\sigma priorensembles with STAR WW data at Q 2 =10Q^2=10 GeV 2 ^2." ], [ "Combining COMPASS, STAR and PHENIX data", "The goal of the present analysis is to deliver a polarized parton set including the experimental information coming from the complete piece of information provided by data discussed in Sec. .", "To this purpose, we perform a global reweighting of our prior polarized PDF ensembles generated in Sec.", "REF with all the relevant data from the COMPASS, STAR and PHENIX experiments simultaneously.", "In Tab.", "REF , we show the values of the $\\chi ^2$ per data point before ($\\chi ^2/N_{\\mathrm {dat}}$ ) and after ($\\chi _{\\mathrm {rw}}^2/N_{\\mathrm {dat}}$ ) reweighting for each prior PDF ensemble.", "In Tab.", "REF , we also quote the number of effective replicas left after reweighting, $N_{\\mathrm {eff}}$ , and the modal value of the $\\mathcal {P}(\\alpha )$ distribution.", "In both Tabs.", "REF -REF the corresponding values for separate experiments and data sets are also provided.", "Table: The value of the χ 2 \\chi ^2 per data pointχ 2 /N dat \\chi ^2/N_{\\mathrm {dat}} (χ rw 2 /N dat \\chi ^2_{\\mathrm {rw}}/N_{\\mathrm {dat}})before (after) global reweightingwith all data sets and for each prior PDF ensemble discussed in the text.Table: The number of replicas left after reweighting,N eff N_{\\mathrm {eff}}, and the modal valueof the 𝒫(α)\\mathcal {P}(\\alpha ) distribution, 〈α〉\\langle \\alpha \\rangle .Results refer to global reweightingwith all data sets and for each prior PDF ensemble discussed in the text.As discussed in Sec.", "REF , we must retain only the results which are stable upon the choice of different prior PDF ensembles.", "Looking at the values of the $\\chi ^2$ per data point after reweighting, we argue that this stability is achieved for results obtained starting from the $3\\sigma $ and $4\\sigma $ priors, see Tab.", "REF .", "Conversely, the results obtained from the reweighting of the $1\\sigma $ and $2\\sigma $ prior ensembles must be discarded, since they actually depend on the assumptions on quark-antiquark separation we made for constructing the corresponding priors.", "In order to quantitatively check that the results of the global reweighting of the $3\\sigma $ and $4\\sigma $ samples are indeed statistically equivalent, we compute the distances $d(x,Q^2)$ between the respective PDFs.", "We recall that the distance is a statistical estimator which has the value $d\\sim 1$ when the two samples of $N_{\\mathrm {rep}}$ replicas are extracted from the same underlying distribution, while it is $d=\\sqrt{N_{\\mathrm {rep}}}$ when the two samples are extracted from two distributions which differ on average by one standard deviation (see Sec.", "REF and Appendix for further details).", "The distances are plotted at $Q^2=10$ GeV$^2$ in Fig.", "REF .", "As $d\\sim 2$ , we conclude that the two ensembles obtained from the reweighting of the $3\\sigma $ or $4\\sigma $ priors describe the same underlying PDF probability distribution.", "We choose the reweighted set obtained from the $3\\sigma $ prior for reference.", "Figure: Distances between parton sets obtained via global reweighting of3σ3\\sigma and 4σ4\\sigma prior PDF ensembles at Q 2 =10Q^2=10 GeV 2 ^2.The overall agreement between new data and the corresponding theoretical predictions obtained with this reweighted PDF set is very good, as quantified by the value of the $\\chi ^2$ per data point, $\\chi ^2_{\\mathrm {rw}}/N_{\\mathrm {dat}}=1.02$ .", "The effective number of replicas is $N_{\\mathrm {eff}} \\sim 180$ , so the size of the initial prior ($N_{\\mathrm {rep}}=1000$ ) was large enough even when the information from all data sets is simultaneously combined.", "The modal value of the $\\mathcal {P}(\\alpha )$ distribution is $\\langle \\alpha \\rangle \\sim 1.2 $ , thus quantifying the good agreement between inclusive DIS data in NNPDFpol1.0 and the new data included in NNPDFpol1.1." ], [ "Unweighting: the ", "After global reweighting of the $3\\sigma $ prior PDF ensemble, the unweighting procedure described in Sec.", "is used to produce a polarized PDF set made of $N_{\\mathrm {rep}}=100$ replicas, NNPDFpol1.1, statistically equivalent to the reweighted PDF ensemble, but in which all PDF replicas are equally probable and hence do not need to be used with corresponding weights.", "In comparison to NNPDFpol1.0, the new polarized parton set provides a meaningful determination of sea flavor PDFs $\\Delta \\bar{u}$ and $\\Delta \\bar{d}$ , though based on a small set of $W$ boson production data (but with the advantage to be free of any bias, including poorly known fragmentation functions) and a determination of the gluon PDF $\\Delta g$ which is improved by open-charm and, particularly, jet data.", "In order to study the compatibility of the new data with the DIS sets, included in NNPDFpol1.0, in Tab.", "REF we show the $\\chi ^2$ of each of the experiments included in the NNPDFpol1.0 analysis evaluated with both the old NNPDFpol1.0 and the new NNPDFpol1.1 parton sets.", "We observe that DIS data are described by the two parton sets with comparable accuracy, as we already noticed from the modal value of the $\\mathcal {P}(\\alpha )$ distribution, $\\langle \\alpha \\rangle \\sim 1.2$ (see Tab.", "REF ).", "Table: The χ 2 \\chi ^2 per data point of all the experiments included in theNNPDFpol1.0 analysis evaluated with NNPDFpol1.0 andNNPDFpol1.1 parton setsIn Fig.", "REF , we compare the total PDF combinations $\\Delta q^+\\equiv \\Delta q+\\Delta \\bar{q}$ , $q=u,d,s$ , and the gluon PDF $\\Delta g$ from NNPDFpol1.1 and NNPDFpol1.0 at $Q^2=10$ GeV$^2$ .", "Since the latter is a fit to inclusive DIS data, only these PDFs can be compared meaningfully between the two parton sets.", "In order to quantitatively asses the difference between them, we plot the corresponding distances $d(x,Q)$ at $Q^2=10$ GeV$^2$ in Fig.", "REF .", "We observe that the total quark PDF combinations from the two determinations are statistically equivalent, since the distance for both their central value and uncertainty is not larger than two (Fig.", "REF ) and differences between them are hardly noticeable (Fig.", "REF ).", "On the other hand, the gluon PDF shows significant differences between the two NNPDF parton determinations, in particular in the $x$ region probed by STAR jet data, $0.05\\lesssim x\\lesssim 0.2$ .", "In this region, the gluon from the NNPDFpol1.1 parton set is definitely positive and has a much reduced error band with respect to its NNPDFpol1.0 counterpart.", "The polarized gluon PDF in the two determinations actually samples different underlying probability distributions, which may differ up to one sigma, as the distance grows up to $d\\sim 10$ .", "At lower values of $x$ , where no new data are included, NNPDFpol1.0 and NNPDFpol1.1 are again statistically equivalent, as expected.", "Figure: Comparison between NNPDFpol1.0 and NNPDFpol1.1parton sets at Q 2 =10Q^2=10 GeV 2 ^2.Figure: Distances between NNPDFpol1.0 and NNPDFpol1.1 partondeterminations at Q 2 =10Q^2=10 GeV 2 ^2.In Fig.", "REF , we compare PDFs from NNPDFpol1.1 with those from the global DSSV08 fit [75]: we display $x\\Delta u$ , $x\\Delta d$ , $x\\Delta \\bar{u}$ , $x\\Delta \\bar{d}$ , $x\\Delta s$ and $x\\Delta g$ at $Q^2=10$ GeV$^2$ .", "Uncertainties are nominal one-sigma error bands for NNPDF sets, while they are Hessian uncertainties ($\\Delta \\chi ^2=1$ ) for DSSV08.", "This choice was made in Ref.", "[75] with some caution, since it may lead to underestimated PDF ucertainties in some $(x,Q^2)$ regions, particularly where constraints from experimental data are weak, see also Sec. .", "Figure: The NNPDFpol1.1 parton setcompared to DSSV08 at Q 2 =10Q^2=10 GeV 2 ^2.The main conclusions of the comparison in Fig.", "REF are the following.", "Consistent results are found in the two parton determinations for $\\Delta u$ and $\\Delta d$ PDFs, though the NNPDF uncertainties are slightly larger, especially at small-$x$ values, where experimental data are lacking.", "The NNPDF polarized gluon PDF is in perfect agreement with its DSSV counterpart in the large-$x$ region, $x\\gtrsim 0.2$ , where they show similar uncertainties.", "However, for $x<0.2$ , $\\Delta g$ has a node in the DSSV08 determination, while it is clearly positive from NNPDFpol1.1.", "This result is driven in particular by the most recent and precise jet production data from STAR (labelled as STAR 1j-09 above), which were not available at the time of the original DSSV08 analysis [75] shown in Fig.", "REF .", "Actually, only the data sets labelled as STAR 1j-05 and STAR 1j-06 were included there.", "An update of the DSSV08 fit including also preliminar STAR 1j-09, called DSSV++ [188], pointed to a positive $\\Delta g$ consistent with the result of our analysis.", "Related to the polarized sea quarks, a slight discrepancy is clearly noticeable for the $\\Delta \\bar{u}$ distribution above $x\\sim 3\\cdot 10^{-2}$ between the two parton sets.", "We recall that $W^\\pm $ data were not included in the DSSV08 global fit [75] shown in Fig.", "REF , hence the differences in the $\\Delta \\bar{u}$ distribution may suggest some tension between $W^\\pm $ and SIDIS data.", "This discrepancy may be explained by our poor knowledge of fragmentation functions.", "A shift away the $\\Delta \\bar{u}$ central curve towards positive values, as observed in our analysis, was also found in a preliminary global fit including STAR data in the DSSV framework [289].", "Since $W$ boson production data in the kinematic regime probed by STAR are not sensitive to strangeness, the discrepancy between the NNPDF and DSSV determinations of $\\Delta s$ , already found in NNPDFpol1.0, is still present.", "As discussed in Sec.", ", in the NNPDF analysis the polarized strange PDF is obtained from inclusive DIS data through its $Q^2$ evolution and assumptions about flavor symmetry of the proton sea, enforced by experimentally measured baryon octet decay constants, see Sec. .", "On the other hand, the DSSV08 determination of polarized PDFs also includes semi-inclusive data with identified kaons in final states, which are directly sensitive to strangeness, but are likely to introduce an uncertainty difficult to quantify due to the poor knowledge of the kaon fragmentation function.", "Finally, we made the choice $\\Delta s=\\Delta \\bar{s}$ , like all other polarized analyses [75], [77], [73], [76], [78]: actually, this is not justified by any physical reason, but experimental data do not allow for a determination of $\\Delta s$ and $\\Delta \\bar{s}$ separately." ], [ "Phenomenology of the nucleon spin structure", "In this Section, we use the NNPDFpol1.1 parton set to reevalute the first moments of the polarized PDFs, separately for each quark flavor-antiflavor and for the gluon, in light of the new data sets included in this analysis.", "Then, we produce some predictions for single-hadron production spin asymmetries at RHIC; we compare them with available measured results in order to qualitatively gauge their potential in pinning down the gluon uncertainty." ], [ "The spin content of the proton revisited", "The first moments of the polarized PDFs can be directly related to the fraction of the proton spin carried by partons, as explained in Sec. .", "This reason has especially motivated our efforts in an accurate and unbiased determination of polarized PDF uncertainties.", "In Sec.", "REF , we presented a detailed analysis of the first moments of the total quark PDF combinations, $\\Delta u^+$ , $\\Delta d^+$ , $\\Delta s^+$ and of the gluon PDF, $\\Delta g$ , from a fit to polarized inclusive DIS data only.", "Their potential improvement at a future Electron-Ion Collider was then studied in Sec.", "REF .", "Now, we use the NNPDFpol1.1 parton set to reassess the determination of the first moments of the polarized parton distributions in order to quantify the impact of new data on the proton's spin content.", "We recall thet the (truncated) first moments of the polarized PDFs $\\Delta f(x,Q^2)$ in the region $[x_{\\mathrm {min}},x_{\\mathrm {max}}]$ are defined as $\\langle \\Delta f(Q^2)\\rangle ^{[x_{\\mathrm {min}},x_{\\mathrm {max}}]}=\\int _{x_{\\mathrm {min}}}^{x_{\\mathrm {max}}}dx \\Delta f(x,Q^2)\\,\\mbox{.", "}$ We consider both full moments, i.e.", "$\\langle \\Delta f(Q^2)\\rangle ^{[0,1]}$ , and truncated moments in the $x$ region covered by experimental data, roughly $\\langle \\Delta f(Q^2)\\rangle ^{[10^{-3},1]}$ .", "Let us begin by looking at polarized quarks and antiquarks.", "We compute Eq.", "(REF ) for the total quark-antiquark combinations, i.e.", "$\\Delta f=\\Delta u^+,\\,\\Delta d^+$ , for sea quarks, i.e.", "$\\Delta f=\\Delta \\bar{u},\\,\\Delta \\bar{d}$ , for the polarized strangeness, i.e.", "$\\Delta f=\\Delta s$ , and for the singlet PDF combination, i.e.", "$\\Delta f=\\Delta \\Sigma =\\sum _{q=u,d,s}\\Delta q^+$ .", "The corresponding central values and one-sigma PDF uncertainties obtained from the $N_{\\mathrm {rep}}=100$ replicas of the NNPDFpol1.1 parton set at $Q^2=10$ GeV$^2$ are collected in Tab.", "REF .", "We compare our results to both NNPDFpol1.0 and DSSV08.", "In the latter case, we quote the conservative uncertainty estimate obtained in Ref.", "[75] using the Lagrange multiplier method with $\\Delta \\chi ^2/\\chi ^2=2\\%$ .", "In parenthesis, we also show the uncertainty due to the extrapolation outside the region covered by experimental data, estimated as the difference between the full first moment and its truncated counterpart in the region $[10^{-3},1]$ , quoted in Ref. [75].", "Table: Full and truncated first moments of the polarized quarkdistributions at Q 2 =10Q^2=10 GeV 2 ^2 for the NNPDFpol1.1 setcompared to NNPDFpol1.0and DSSV08 .", "The uncertaintyquoted in parenthesis for DSSV08 is due to the extrapolation inthe unintegrated region as discussed in the text.Table: Full and truncated first moments of the polarized gluondistributions at Q 2 =10Q^2=10 GeV 2 ^2 for the NNPDFpol1.1 setcompared to NNPDFpol1.0and various fits of the DSSV family.", "The uncertaintyquoted in parenthesis for DSSV08 is due to the extrapolation inthe unintegrated region as discussed in the text.Results from Tab.", "REF clearly show that first moments obtained with NNPDFpol1.1 and NNPDFpol1.0 are perfectly consistent with each other, as we already knew from the corresponding agreement at the level of polarized PDFs, see Fig.", "REF .", "Besides, the sensitivity to quark-antiquark separation introduced by $W$ data allows for a reduction of the uncertainty up to $50\\%$ in the NNPDFpol1.1 determination with respect to NNPDFpol1.0.", "The comparison between NNPDF full and truncated moments shows that the relative contribution to the total PDF uncertainty from the small-$x$ extrapolation region is roughly the same in both NNPDFpol1.0 and NNPDFpol1.1, and it is about two times larger than the uncertainty in the measured $x$ region.", "Therefore, we can conclude that the contribution to the total uncertainty from the extrapolation region has reduced by almost a half in NNPDFpol1.1 with respect to NNPDFpol1.0.", "This is likely because the new data, supplemented by the smoothness provided by the neural network parametrization, decrease the number of acceptable small-$x$ behaviors of the polarized quark PDFs.", "Nevertheless, the uncertainty from the small-$x$ extrapolation region is still dominant and it could be finally pinned down only by accurate measurements in this region.", "These may be performed at a future Electron-Ion Collider and they were demonstrated to largely keep under control the extrapolation uncertainties in Sec.", "REF .", "Coming now to the comparison between NNPDFpol1.1 and DSSV08 [75], we notice that truncated first moments are in perfect agreement, both central values and uncertainties.", "Slight differences are found for $\\Delta \\bar{u}$ and $\\Delta \\bar{s}$ , due to the different shape of the corresponding PDFs (see Fig.", "REF ).", "On the other hand, when considering full first moments the NNPDFpol1.1 uncertainties are somewhat larger than those found in the DSSV08 analysis whenever the extrapolation uncertainty is included.", "This is the major effect of our more flexible PDF parametrization, as already discussed in Sec.", "REF .", "Let us now move on to discuss the first moment of the polarized gluon $\\Delta g$ .", "Results for full and truncated moments at $Q^2=10$ GeV$^2$ are presented in Tab.", "REF .", "There, we compare the predictions from NNPDFpol1.1 with those from NNPDFpol1.0, and two fits from the DSSV family: we consider both the original DSSV08 parton set [75] and its update, DSSV++ [188], which includes the same jet production data in NNPDFpol1.1.", "As for quarks, we compute both the full and the truncated moments in the measured region $[10^{-3},1]$ .", "In order to quantify the impact of the RHIC inclusive jet data, we also provide results for the truncated first moment restricted to the region $x \\in [0.05,0.2]$ , which corresponds to the range covered by these data, see Fig.", "REF .", "The results quoted in Tab.", "REF show the substantial improvement in the PDF uncertainties of the gluon first moment in NNPDFpol1.1 as compared to NNPDFpol1.0, due to the constraints on $\\Delta g$ provided by RHIC jet data, see Fig.", "REF .", "This is further illustrated by the truncated first moment in the region covered by these data, $\\langle \\Delta g(Q^2)\\rangle ^{[0.05,0.2]}$ , where the PDF uncertainty is reduced by a factor close to three, and where its central value is clearly positive, almost three sigma away from zero.", "It is clear that the RHIC jet data strongly suggest a positive polarized gluon first moment in the region $x\\in \\left[ 0.05,0.2\\right]$ , unfortunately the absence of other direct constraints outside this region still lead to a quite large value of the gluon full first moment.", "In addition, our results for $\\langle \\Delta g(Q^2)\\rangle ^{[0.05,0.2]}$ , in terms of both central value and uncertainty, turn out to be very close to those obtained in the DSSV++ analysis, which is based on the same set of inclusive jet data [188].", "As in the previous NNPDFpol1.0 analysis, the uncertainty due to the extrapolation outside the region covered by experimental data is substantial and dominates the total uncertainty of the full first moment $\\langle \\Delta g(Q^2)\\rangle ^{[0,1]}$ .", "The only way to further reduce this uncertainty is to provide measurements which probe $\\Delta g$ at smaller values of $x$ than those that are available now.", "In this respect, additional jet data from RHIC taken at higher center-of-mass energy, up to $\\sqrt{s}=500$ GeV$^2$ , may be helpful.", "However, to really pin down the small-$x$ behavior of the polarized PDFs and thus be able to finally reach an accurate determination of $\\langle \\Delta g(Q^2)\\rangle ^{[0,1]}$ , one will have to resort to the Electron-Ion Collider, as quantified in detail in Chap. .", "Our main conclusion on the partons' contribution to the proton spin is then twofold.", "On the one hand, we have found that their first moments are rather well determined in the kinematic region covered by experimental data and in good agreement with the values obtained in DSSV++ [188], the only analysis in which collider data are included.", "In particular, the singlet full first moment is less than a half of the proton spin within its uncertainty (see tab.", "REF ); the gluon first moment is definitely positive, though rather small, in the region constrained by recent STAR jet data, roughly $0.05\\lesssim x \\lesssim 0.2$ .", "On the other hand, we emphasize that the uncertainty on both the singlet and the gluon full first moments coming from the extrapolation to the unmeasured, small-$x$ , region dominates their total uncertainty.", "For this reason, large values of the gluon first moment are not completely ruled out: within our accurate determination of uncertainties, the almost vanishing value for the singlet axial charge observed in the experiment may still be explained as a cancellation between a rather large quark contribution and the anomalous gluon contribution, as discussed in Sec.", "REF .", "Of course, more experimental data, such as those available at a future Electron-Ion Collider, are needed to discriminate which is the behavior of the polarized gluon in the unmeasured small-$x$ region, in particular whether it is actually small, as it is commonly believed." ], [ "Predictions for single-hadron production asymmetries at RHIC", "The parton set presented in Sec.", "REF does not include information from semi-inclusive hadron production spin asymmetries at colliders.", "As discussed in Sec.", "REF , the analysis of these data requires the usage of fragmentation functions, whose poor knowledge entails an additional source of theoretical uncertainty on the extracted PDFs, which is difficult to quantify, (see Sec.", "REF ).", "Nevertheless, in view of the experimental program which is ongoing at RHIC, it is interesting to compare some predictions for these spin asymmetries with the experimental data, which are now available with significant statistics.", "The size of the uncertainties of theoretical predictions will then fix, at least qualitatively, the experimental precision required for data to further pin down the uncertainties on PDFs, once included in a global fit.", "In Fig.", "REF , we show the double-spin asymmetry for single-hadron production in polarized proton-proton collisions, Eq.", "(REF ), compared to experimental measurements from PHENIX.", "In particular, we provide predictions for neutral-pion production at center-of-mass energy $\\sqrt{s}=200$ GeV [217] and $\\sqrt{s}=62.4$ GeV [214], and mid-rapidity ($\|\\eta \|<0.35$ ) charged hadron production at $\\sqrt{s}=62.4$ GeV [290].", "Earlier measurments with neutral-pions [291], [292], [293], with significantly larger uncertainties, are not considered here.", "In Fig.", "REF , we compare the double-spin asymmetry for neutral-pion production at forward rapidity ($0.8<\\eta <2.0$ ) with recent STAR data at $\\sqrt{s}=200$ GeV [216].", "In Fig.", "REF , we also show the predictions for neutral- and charged-pion production spin asymmetries for which data from the PHENIX experiment will be soon available.", "Figure: Predictions for the neutral-pion (upper plots)and charged hadron (lower plots)spin asymmetries computed at NLO accuracy with the NNPDFpol1.1and NNPDF2.3 parton sets, compared to measured data fromPHENIX , , .Figure: (Left panel) Prediction for the neutral-pion spin asymmetrycompared to data measured by STAR .", "(Right panel) Prediction for the neutral- and charged-pion spinasymmetries in the kinematic range accessed by the upcomingPHENIX measurements.", "All theoretical predictions are obtained fromthe NNPDFpol1.1 and NNPDF2.3parton sets at NLO accuracy.The asymmetries are computed as illustrated in Sec.", "REF : the (polarized) numerator in Eq.", "(REF ) is computed for each replica in the NNPDFpol1.1 parton set ($N_{\\mathrm {rep}}=100$ ), while the (unpolarized) denominator is computed only once taking the average PDFs from the NNPDF2.3 [110] parton set at NLO.", "In both the numerator and the denominator, we use the best fit fragmentation functions from the DSS07 set [185].", "The central value and the uncertainty of the prediction are then obtained as the mean and the standard deviations computed from the $N_{\\mathrm {rep}}=100$ results for each replica in the polarized PDF set.", "Hence, the estimated uncertainty on our prediction only takes into account the uncertainty of the polarized PDFs.", "While we expect the uncertainty on unpolarized PDFs has a negligible impact, conversely the additional theoretical uncertainty due to the choice of a set of fragmentation functions may have a rather large impact.", "We finally note that our predictions are made using the code presented in Ref.", "[268], which we have modified to handle NNPDF parton sets.", "Results in Figs.", "REF -REF show that the asymmetry remains very small in the measured $p_T$ range.", "Experimental data are in good agreement with predictions and seems to reinforce the conclusion that the polarized gluon is small in the measured kinematic range.", "However, we notice that, except for PHENIX neutral-pion production at small $p_T$ , experimental uncertainties are rather large in comparison to those of the corresponding theoretical predictions: this is particularly evident for STAR data in Fig.", "REF , which, however, cover a rapidity range larger than that measured by PHENIX.", "The mutual size of experimental and theoretical uncertainties is similar to that observed with COMPASS open-charm data in Sec.", "REF .", "However, in the present case, data show a more definite trend towards a growing asymmetry as $p_T$ increases, partly reproduced by the behavior of the corresponding theoretical prediction.", "For this reason, we expect that data on semi-inclusive particle production presented here will have a moderate impact on pinning down the size of the polarized gluon uncertainty, once included in a global PDF determination." ], [ "Conclusions and outlook", "In this Thesis, we have presented the first unbiased determination of spin-dependent, or polarized, Parton Distribution Functions (PDFs) of the proton.", "These distributions are defined as the momentum densities of partons polarized along or opposite the direction of the parent nucleon and are usually denoted as $\\Delta f(x,Q^2)$ , where $f$ may refer either to individual quark or antiquark flavors, or to a combination of them, or to the gluon.", "Parton distributions depend on both the Bjorken scaling variable $x$ , the fraction of the proton momentum carried by the parton, and on the energy scale $Q^2$ with which the proton is probed.", "While the first is a non-perturbative dependence to be determined from experimental data, the latter is fully predictable in perturbative Quantum Chromodynamics (QCD), the theory of strong interaction.", "In the framework of perturbative QCD, polarized parton distributions are essential ingredients for any phenomenological study of hard scattering processes involving polarized hadrons in initial states.", "The description of these processes in terms of expressions in which the perturbative and the non-perturbative parts are factorized is a powerful success of QCD itself.", "In such a picture, parton distributions are the fundamental objects encoding the information on the inner structure of the nucleon; in particular, polarized parton distributions are related to its spin structure, since their integrals over the Bjorken scaling variable are interpreted as fractions of the proton spin.", "The interest in the determination of polarized PDFs of the nucleon is largely related to the experimental discovery, in the late 80s, that the singlet axial charge of the proton is anomalously small [49], [50], soon followed by the theoretical realization [59], [60] that the perturbative behavior of polarized PDFs deviates from parton model expectations, according to which gluons decouple in the asymptotic limit.", "An accurate determination of polarized PDFs is then needed to precisely assess which fraction of the nucleon spin is carried by quark and gluon spins.", "The residual part of the nucleon spin which possibly would not be accounted for by quarks and gluons may be explained by resorting to their intrinsic Fermi motion and orbital angular momenta [61], [62], [63] (for a recent discussion on the spin decomposition see also Ref. [64]).", "In addition to the investigation of the nucleon's spin structure, polarized PDFs have been recently shown to be useful in the probe of different beyond-standard-model scenarios [112] and in the determination of the Higgs boson spin in the diphoton decay channel, by means of the linear polarization of gluons in an unpolarized proton [113].", "Polarized parton distributions are presently known with much less accuracy than their unpolarized counterparts.", "As pointed out several times in this Thesis, this is mostly due to the experimental data they rely on, which are both less abundant and less accurate than those available in the unpolarized case.", "Several polarized PDF sets have been determined in the last few years [73], [74], [75], [76], [77], [116], [78], [79], [80], but they are all based on the standard Hessian methodology for PDF fitting and uncertainty estimation.", "This approach is known [81] to potentially lead to an underestimation of PDF uncertainties, due to the limitations in the linear propagation of errors and to PDF parametrization in terms of fixed functional forms, both assumed in the standard methodology.", "These issues are especially delicate when the experimental information is scarce, like in the case of polarized data.", "In light of these considerations, an unbiased determination of polarized PDFs is crucial in order to provide an adequate estimate of the uncertainty with which quarks and gluons can actually contribute to the nucleon spin.", "In particular, such a determination allows for scrutinizing the common belief that the anomalous gluon contribution is too small to compensate a reasonably large singlet spin contribution into the almost vanishing axial charge observed in experiments.", "Providing the first unbiased determination of polarized parton distributions has precisely been the goal of the present Thesis." ], [ "Summary of the main results", "In this Thesis, the determination of polarized parton sets has been carried out within the NNPDF methodology.", "This uses a robust set of statistical tools, devised for a statistically sound determination of PDFs and their uncertainties, which include Monte Carlo methods for error propagation, neural networks for PDF parametrization and genetic algorithms for their minimization.", "This methodology has already been successfully applied in the unpolarized case [99], [100], [101], [102], [103], [104], [105], [106], [107], [108], [109], [110], [111], where the NNPDF sets are routinely used by the LHC collaborations in their data analysis and data-theory comparison.", "It has been extended here to polarized PDFs for the first time.", "In more detail, the main achievements presented in this Thesis are summarized below.", "Based on world-available data from polarized inclusive Deep-Inelastic Scattering, we determined a first polarized parton set at next-to-leading order accuracy, NNPDFpol1.0.", "We reviewed in detail the theory and phenomenology of polarized DIS, in particular focusing on the features of the data included in our analysis.", "We discussed how the NNPDF methodology has been adapted to the polarized case and which strategies have been devised to face some issues, like the proper implementation of target mass corrections and positivity constraints in the fitting algorithm.", "Our analysis showed that some PDF uncertainties are likely to be underestimated in other existing determinations, based on the standard methodology, due to their less flexible parametrization.", "This is particularly the case of the gluon, which is left almost unconstrained by inclusive DIS data: hence, its contribution to the nucleon spin is still largely uncertain, unless one makes strong assumptions on the PDF functional form in the small-$x$ ($x\\lesssim 10^{-3}$ ) extrapolation region, where experimental data are presently lacking.", "For the same reason, we also showed that a determination of the strong coupling $\\alpha _s$ from the Bjorken sum rule is not competitive, again because the nonsinglet structure function in the unmeasured small-$x$ region is largely uncertain.", "These conclusions were supported by a careful analysis of the stability of our results upon the variation of a number of theoretical and methodological assumptions, in particular related to the effects of target mass corrections, sum rules, and positivity constraints.", "First, we found that inclusive DIS data, with our kinematic cuts, do not show sensitivity to finite nucleon mass effects, neither in terms of fit quality, nor in terms of the effect on PDFs.", "Second, we concluded that our fit results are quite stable upon variations of the treatment of sum rules dictated by hyperon decays.", "Finally, we emphasized that positivity significantly affects PDFs in the region where no data are available, in particular their large-$x$ behavior.", "The proposed Electron-Ion Collider (EIC) [209], [128], [129] is expected to enlarge the kinematic coverage of data, which is presently rather limited, by at least two orders of magnitude in both $x$ and $Q^2$ .", "This will reduce the uncertainty due to PDF extrapolation to small-$x$ values and will allow for a better determination of the polarized gluon PDF through scaling violations, thanks to a larger $Q^2$ lever arm.", "Using simulated pseudodata for two realistic scenarios at an EIC, with increasing energy of both the lepton and hadron beams, we have studied its potential impact on the determination of polarized PDFs.", "We found that inclusive DIS data at an EIC would entail a considerable reduction in the gluon PDF uncertainty and also provide evidence of a possible large gluon contribution to the nucleon spin, though the latter goal would still be reached with a sizable residual uncertainty.", "The measurement of the charm contribution to the proton structure function, $g_1^{p,c}$ , which is directly sensitive to the gluon, might provide more information on the corresponding distribution.", "We showed that $g_1^{p,c}$ , though being small, could be as much larger as $10-20\\%$ of the total structure function $g_1^p$ in the kinematic region probed by an EIC: hence, in order to further pin down the gluon uncertainty from intrinsic charm effects, one should be able to measure the corresponding contribution to the $g_1$ structure function within this accuracy.", "The Relativistic Heavy Ion Collider (RHIC) is the first facility in the world to collide polarized proton beams.", "Measurements on inclusive jet and $W$ boson production asymmetries have been recently presented: we studied their potential in constraining the polarized gluon and in separating light quark and antiquark PDFs, respectively.", "This new piece of experimental information was included in our polarized parton set by means of Bayesian reweighting of suitable Monte Carlo PDF ensembles [105], [109].", "This method, which consist of updating the underlying PDF probability distribution of a prior ensemble according to the conditional probability for the old PDFs with respect to new data, allows for the inclusion of new data in a PDF set without the need of a global refitting; hence, it could be used to quickly update a PDF set with any new piece of experimental information.", "This way, we were able to provide the first global polarized PDF set obtained within the NNPDF framework, NNPDFpol1.1.", "In comparison to NNPDFpol1.0, the new polarized parton set provides a meaningful determination of sea flavor PDFs $\\Delta \\bar{u}$ and $\\Delta \\bar{d}$ , based on $W$ boson production data (otherwise not determined by inclusive DIS data or determined in SIDIS, but with the bias introduced by poorly known fragmentation functions), and a determination of the gluon PDF $\\Delta g$ which is improved by open-charm and, particularly, jet data.", "We should also notice that, from a conceptual point of view, the methodology we followed to determine this parton set is in itself particularly valuable.", "Indeed, we have explicitly shown how a PDF set can be succesfully obtained by including all data through reweighting of a first unbiased guess, as originally proposed in Refs.", "[172], [173].", "The main conclusion on the partons' contributions to the nucleon spin based on the NNPDFpol1.1 parton set is twofold.", "On the one hand, we have found that PDF first moments are rather well determined in the kinematic region covered by experimental data and are in good agreement with the values obtained in the only available analysis including the same collider data [188].", "In particular, in the region constrained by data, the singlet full first moment is less than a half of the proton spin within its uncertainty, while the gluon first moment is definitely positive, though rather small.", "The determination of the gluon is more accurate in NNPDFpol1.1 than in NNPDFpol1.0, mostly thanks to jet data, located in the region $0.05\\lesssim x \\lesssim 0.2$ .", "On the other hand, we emphasize that the uncertainty on both the singlet and the gluon full first moments coming from the extrapolation to the unmeasured, small-$x$ , region dominates their total uncertainty.", "For this reason, large values of the gluon first moment are not completely ruled out: within our accurate determination of uncertainties, the almost vanishing value for the singlet axial charge observed in the experiment may still be completely explained as a cancellation between a rather large quark and the anomalous gluon contributions.", "We developed a Mathematica package which allows for fast and interactive usage of any available NNPDF parton set, both unpolarized and polarized, see Appendix .", "This interface includes all the features already available through LHAPDF [121], [122], but they can be profitably combined together with those provided by Mathematica.", "The software we developed was tailored to the users who are not familiar with Fortran or C++ languages used by the LHAPDF interface and who can benefit from the more direct usage of PDFs within a Mathematica notebook.", "The NNPDFpol1.0 and NNPDFpol1.1 polarized PDF sets, with $N_{\\mathrm {rep}}=100$ replicas, are publicly available from the NNPDF website http://nnpdf.hepforge.org/ .", "The Mathematica interface, as well as FORTRAN and C++ stand-alone codes for handling these parton distributions, are also available from the same source." ], [ "Future directions", "The NNPDFpol1.1 parton set is based on all the relevant and up-to-date experimental information from deep-inelastic scattering and proton-proton collisions which do not depend on the fragmentation of the struck quark into final observed hadrons.", "Further data are expected from PHENIX and STAR in the upcoming years, which will further improve the accuracy of polarized PDF determinations.", "As further refinements of polarized PDFs will be achieved, they will become more and more appealing for the experimental collaborations to be used in their analysis and for data-theory comparison.", "In this sense, efforts will be devoted to make the NNPDF polarized parton sets the gold standard, as their unpolarized counterparts are quickly becoming.", "In order to obtain additional information on the spin structure of the proton, it will be certainly beneficial to include a wide range of semi-inclusive measurements, namely semi-inclusive DIS in fixed-target experiments [229], [95], [96], [97], [98], and semi-inclusive particle production in polarized collisions at RHIC [217], [214], [290], [216].", "However, a consistent inclusion of these data in a global fit requires first of all the corresponding determination of fragmentation functions using the NNPDF methodology.", "Indeed, available fragmentation function sets [238], [239], [240], [184], [241], [185], [186], [242] suffer from several limitations due to their too rigid parametrization.", "It was recently shown that none of these sets can describe the most updated inclusive charged-particle spectra data at the LHC satisfactorily [162].", "Therefore, a determination of fragmentation functions using the NNPDF methodology is highly desirable by itself, and may be important in various areas of phenomenology [237]; in particular, it will pave the way to use a large data set of semi-inclusive polarized data in future NNPDF analyses.", "Finally, we notice that the methods illustrated here apply to the determination of any non-perturbative object from experimental data.", "Hence, even though a phenomenological study of either TMDs or GPDs was beyond the scope of this Thesis, the NNPDF methodology may be used as well to provide the determination of such distributions in the future, when relevant experimental data will reach more and more abundance and accuracy.", "In summary, in this Thesis not only we have extended the NNPDF framework to the determination of spin-dependent parton sets, but we have also reached the state-of-the-art in our unbiased understanding of the proton's spin content, as allowed by available experimental data.", "Further constraints will be provided by a variety of semi-inclusive measurements, which in turn will require the development of a set of parton fragmentation functions using the NNPDF methodology.", "In the long term, the final word on the spin content of the proton will require brand new facilities such as an Electron-Ion Collider, as we have also extensively discussed in this Thesis.", "Indeed, it could finally bring polarized PDF determinations to a similar level of accuracy as the one reached for their unpolarized counterparts.", "We hope that the NNPDF collaboration will play a leading role in this exciting game." ], [ "Statistical estimators", "In this Appendix, we collect the definitions of the statistical estimators used in the NNPDF analyses presented in Chaps. --.", "Despite they were already described in Refs.", "[120], [99], [104], we find it useful to give them for completeness and ease of reference here.", "In the following, we denote with $\\mathcal {O}$ a generic quantity depending on replicas in a Monte Carlo ensemble of PDFs; it may be a PDF, a linear combination of PDFs, or a physical observable.", "We also denote as $\\langle \\mathcal {O}\\rangle _{\\mathrm {rep}}$ the mean computed over the $N_{\\mathrm {rep}}$ replicas in the ensemble, and as $\\langle \\mathcal {O}\\rangle _{\\mathrm {dat}}$ the mean computed over the $N_{\\mathrm {dat}}$ experimental data for a fixed replica in the ensemble.", "[leftmargin=] Central value $\\left\\langle \\mathcal {O}\\right\\rangle _{\\mathrm {rep}}=\\frac{1}{N_{\\mathrm {rep}}}\\sum _{k=1}^{N_{\\mathrm {rep}}}\\mathcal {O}^{(k)}\\,\\mbox{.", "}$ Variance $\\sigma =\\sqrt{\\left\\langle \\mathcal {O}^2\\right\\rangle _{\\mathrm {rep}}-\\left\\langle \\mathcal {O} \\right\\rangle ^2_{\\mathrm {rep}}}\\,\\mbox{.", "}$ Elements of the correlaton matrix $\\rho _{ij}=\\frac{\\left\\langle \\mathcal {O}_i\\mathcal {O}_j\\right\\rangle _{\\mathrm {rep}}-\\left\\langle \\mathcal {O}_i\\right\\rangle _{\\mathrm {rep}}\\left\\langle \\mathcal {O}_j\\right\\rangle _{\\mathrm {rep}}}{\\sigma _i\\sigma _j}\\,\\mbox{.", "}$ Elements of the covariance matrix $\\mathrm {cov}_{ij}=\\rho _{ij}\\sigma _i\\sigma _j\\,\\mbox{.", "}$ Percentage error over the $N_{\\mathrm {dat}}$ data points $\\left\\langle \\mathrm {PE}\\left[\\langle \\mathcal {O}\\rangle _{\\mathrm {rep}}\\right]\\right\\rangle =\\frac{1}{N_{\\mathrm {dat}}}\\sum _{i=1}^{N_{\\mathrm {dat}}}\\left[\\frac{\\langle \\mathcal {O}_i\\rangle _{\\mathrm {rep}}-\\mathcal {O}_i}{\\mathcal {O}_i}\\right]\\,\\mbox{.", "}$ Scatter correlation between two quantities $r(\\mathcal {O}_1,\\mathcal {O}_2)=\\frac{\\left\\langle \\mathcal {O}_1\\mathcal {O}_2\\right\\rangle _{\\mathrm {dat}}-\\left\\langle \\mathcal {O}_1\\right\\rangle _{\\mathrm {dat}}\\left\\langle \\mathcal {O}_2\\right\\rangle _{\\mathrm {dat}}}{\\sigma _1\\sigma _2}\\,\\mbox{,}$ where $\\mathcal {O}_{1,2}$ may be obtained as averages over Monte Carlo replicas.", "Square distance between central value estimates from two PDF ensembles $d^2\\left(\\langle \\mathcal {O}^{(1)}\\rangle , \\langle \\mathcal {O}^{(2)}\\rangle \\right)=\\frac{[\\langle \\mathcal {O}^{(1)}\\rangle - \\langle \\mathcal {O}^{(2)}\\rangle ]^2}{\\sigma ^2\\left[\\langle \\mathcal {O}^{(1)}\\rangle \\right]+\\sigma ^2\\left[\\langle \\mathcal {O}^{(2)}\\rangle \\right]}\\,\\mbox{,}$ where the variance of the mean is given by $\\sigma ^2[\\langle \\mathcal {O}^{(i)}\\rangle ]=\\frac{1}{N_{\\mathrm {rep}}^{(i)}}\\sigma ^2[\\mathcal {O}^{(i)}]$ in terms of the variance $\\sigma [\\mathcal {O}^{(i)}]$ of the quantities $\\mathcal {O}^{(i)}$ , estimated as the variance of the replica sample, Eq.", "(REF ).", "In our notation $i=1,2$ .", "Square distance between square uncertainty estimates from two PDF ensembles $d^2\\left(\\sigma ^2[\\mathcal {O}^{(1)}], \\sigma ^2[\\mathcal {O}^{(2)}]\\right)=\\frac{\\left(\\bar{\\sigma }_{(1)}^2-\\bar{\\sigma }_{(2)}^2\\right)^2}{\\sigma ^2[\\bar{\\sigma }_{(1)}^2]+\\sigma ^2[\\bar{\\sigma }_{(2)}^2]}\\,\\mbox{,}$ where we have defined $\\bar{\\sigma }_{(i)}^2\\equiv \\sigma ^2[\\mathcal {O}^{(i)}]$ , $i=1,2$ .", "In practice, for small-size replica samples the distances defined in Eqs.", "(REF )-(REF ) display sizable statistical fluctuations.", "In order to stabilize the result, all distances computed in this Thesis are determined as follows: we randomly pick $N^{(i)}_{\\mathrm {rep}}/2$ out of the $N^{(i)}_{\\mathrm {rep}}$ replicas for each of the two subsets.", "The computation of the square distance Eq.", "(REF ) or Eq.", "(REF ) is then repeated for $N_\\mathrm {part}=100$ (randomly generated) choices of $N^{(i)}_{\\mathrm {rep}}/2$ replicas, and the result is averaged: this is sufficient to bring the statistical fluctuations of the distance at the level of a few percent." ], [ "A ", "In this Appendix, we present a package for handling both unpolarized and polarized NNPDF parton sets within a Mathematica notebook file [123].", "This allows for performig PDF manipulations easily and quickly, thanks to the powerful features of the Mathematica software.", "The package was tailored to the users who are not familiar with FORTRAN or C++ programming codes, on which the standard available PDF interface, LHAPDF [121], [122], is based.", "However, since our Mathematica package includes all the features available in the LHAPDF interface, any user can benefit from the interactive usage of PDFs within Mathematica.", "The NNPDF Mathematica package can be downloaded from the NNPDF web page http://nnpdf.hepforge.org/ together with sample notebooks containing a step by step explanation of the NNPDF usage within Mathematica, as well as a variety of examples.", "The procedure to download and run our Mathematica package is rather simple: Download wget http://nnpdf.hepforge.org/math$\\_$ package.tgz Unpack tar -xvzf math$\\_$ package.tgz Run the tutorial mathematica Demo-unpol.nb mathematica Demo-pol.nb The input for our Mathematica package is any .LHgrid file delivered by the NNPDF Collaboration as the final result of a fit.", "These files are publicly available from the NNPDF hepforge website or from the LHAPDF library and should be downloaded separately from the Mathematica package.", "The functions implemented in the package are summarized in Tab. .", "In the following, we briefly demonstrate the NNDPF Mathematica package by examining the NNPDF2.3 parton determination at NLO [110].", "Compute PDF central value and variance.", "We have defined proper functions to keep the computation of PDF central value and variance very easy.", "These built-in functions only need $x$ , $Q^2$ and PDF flavour as input.", "The user can also specify the confidence level to which central value and variance should be computed.", "Make PDF plots.", "Mathematica enables a wide range of plotting options.", "As a few examples, we show the 3D plot and the contour plot of the singlet PDF combination from the NNPDF2.3 parton set at NLO (see Fig.", "REF ).", "Figure: (Left) Simultaneous (x,Q 2 )(x,Q^2) dependence of thesinglet PDF and its one-sigma error band from the NLONNPDF2.3 parton set.", "(Right) Contour plot of the of the singlet PDF from the NNPDF2.3parton set at NLO in the (x,Q 2 )(x,Q^2) plane.", "Perform computations involving PDFs.", "PDF manipulation can be carried out straighforwardly since we have defined functions which handle either single replicas or the whole Monte Carlo ensemble.", "The user can then easily perform any computation which involves PDFs.", "For example, we show in Fig.", "a snapshot of a typical Mathematica notebook in which we use our interface to check the momentum and valence sum rules from the NNPDF2.3 parton set at NLO.", "[p] lX Function Description InitializePDFGrid[path, namegrid] It reads the .LHgrid file into memory specified by namegrid (either unpolarized or polarized) from the location specified by path.", "It also performs the PDF interpolation in the ($x$ ,$Q^2$ ) space by means of a built-in Mathematica interpolation algorithm.", "xPDFcv[x,Q2,f] It returns x times the central value of the PDF with flavor f at a given momentum fraction x and scale Q2 (in GeV$^2$ ).", "Note that f must be an integer, and x and Q2 must be numeric quantities.", "For the unpolarized case, and polarized NNPDFpol1.1 onwards, the LHAPDF convention is used for the flavor f, that is, f=-6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6 corresponds to $\\bar{t}$ , $\\bar{b}$ , $\\bar{c}$ , $\\bar{s}$ , $\\bar{u}$ , $\\bar{d}$ , $g$ , $d$ , $u$ , $s$ , $c$ , $b$ , $t$ .", "For the polarized case, NNPDFpol1.0, the following convention is used for the flavor f: f=0, 1, 2, 3, 4 corresponds to $\\Delta g$ , $\\Delta u +\\Delta \\bar{u}$ , $\\Delta d +\\Delta \\bar{d}$ , $\\Delta s +\\Delta \\bar{s}$ .", "xPDFEnsemble[x,Q2,f] It returns x times the vector of PDF replicas of flavor f at a given momentum fraction x and scale Q2 (in GeV$^2$ ).", "xPDFRep[x,Q2,f,irep] It returns x times the irep PDF replica of flavor f at a given momentum fraction x and scale Q2 (in GeV$^2$ ).", "For irep=0, the mean value over the PDF ensemble is returned.", "xPDF[x,Q2,f] It returns x times the value of the PDF of flavor f at a given momentum fraction x and scale Q2 (in GeV$^2$ ) with its standard deviation.", "xPDFCL[ensemble,x,Q2,f,CL] It returns x times the value of the PDF of flavor f at a given momentum fraction x and scale Q2 (in GeV$^2$ ) with its standard deviation.", "These values are computed over the PDF ensemble specified by ensemble at a confidence level specified by CL.", "The variable ensemble must be a function of x, Q2 and f: it can be either the function xPDFEnsemble or any PDF ensemble defined by the user.", "The variable CL must be a real number between 0 and 1.", "NumberPDF[] It returns the number of PDF members in the set.", "Infoalphas[] It returns information on $\\alpha _s$ evolution used in the QCD analysis.", "alphas[Q2,ipo,imodev] It returns the QCD strong coupling constant $\\alpha _s$ .", "The inputs are: [leftmargin=] Q2: the energy scale $Q^2$ , in GeV$^2$ , at which $\\alpha _s$ is computed; ipo: the perturbative order at which $\\alpha _s$ is computed; 0: LO; 1: NLO; 2: NNLO; imodev: the evolution mode with which $\\alpha _s$ is computed; 0: $\\alpha _s$ is computed as a function of $\\alpha _s$ at the $Z$ mass as given in the .LHgrid file; 1: exact solution of the QCD $\\beta $ -function equation using Runge-Kutta algorithm.", "alphasMZ[] It returns the list of $\\alpha _s$ values at the $Z$ mass used in the QCD analysis for each replica.", "mCharm[], mBottom[], mTop[], mZ[] They return the charm, bottom, top quark mass (in GeV) and the $Z$ boson mass (in GeV) used in the QCD analysis Lam4[], Lam5[] They return the $\\Lambda _{\\mathrm {QCD},4}$ , $\\Lambda _{\\mathrm {QCD},5}$ used in the QCD analysis xMin[], xMax[], Q2Min[], Q2Max[] They return the minimum or maximum value for $x$ or $Q^2$ (in GeV$^2$ ) used in the input .LHgrid grid.", "Description of the functions available in the Mathematica interface to NNPDF partons sets.Description of the functions available in the Mathematica interface to NNPDF partons sets.", "[p] Figure: NO_CAPTION A snapshot from a Mathematica notebook in which the NNPDF interface is used: here momentum and valence sum rules are checked from the NNPDF2.3 parton set at NLO.A snapshot from a Mathematica notebook in which the NNPDF interface is used: here momentum and valence sum rules are checked from the NNPDF2.3 parton set at NLO." ], [ "The FONLL scheme for $g_1(x,Q^2)$ \nup to {{formula:4ba095f9-9c20-4157-b453-eb0c7dd0c6ec}}", "In this Appendix, we collect the relevant explicit formulae for the practical computation of the polarized DIS structure function $g_1(x,Q^2)$ within the FONLL approach [124] up to $\\mathcal {O}(\\alpha _s)$ .", "In particular, we restrict to the heavy charm quark contribution, $g_{1_c}$ , to the polarized proton structure function $g_1$ , which may be of interest for studies at an Electron-Ion Collider, see Sec.", "REF .", "The formulae below extend to the polarized case those collected in Appendix A of Ref. [124].", "For $g_{1_c}$ , up to $\\mathcal {O}(\\alpha _s)$ , the relevant equations, to be compared to Eqs.", "(88)-(92) in Ref.", "[124], are expressed in terms of $n_l=3$ light flavours: $g_{1_c}^{\\mathrm {FONLL}}(x,Q^2)& = &g_{1_c}^{(d)}(x,Q^2) + g_{1_{c}}^{n_l} (x,Q^2)\\,\\mbox{,}\\\\g_{1_c}^{(d)}(x,Q^2)& = &g_{1_c}^{(n_l+1)}(x,Q^2) - g_{1_{c}}^{n_l,0}(x,Q^2)\\,\\mbox{,}\\\\g_{1_c}^{(n_l+1)}(x,Q^2)& = &e_c^2\\int _x^1\\frac{dz}{z}\\left\\lbrace C_q(z,\\alpha _s(Q^2))\\Delta c^+\\left(\\frac{x}{z},Q^2\\right)\\right.\\\\& &\\left.+ \\frac{\\alpha _s(Q^2)}{4\\pi } C_g^{(1)}(z)\\Delta g \\left(\\frac{x}{z},Q^2\\right)\\right\\rbrace \\,\\mbox{,}\\\\g_{1_c}^{(n_l)}(x,Q^2)& = &e_c^2\\frac{\\alpha _s(Q^2)}{4\\pi }\\int _{ax}^1\\frac{dz}{z} H_g^{(1)}(Q^2,m_c^2,z)\\Delta g\\left(\\frac{x}{z}, Q^2 \\right)\\,\\mbox{,}\\\\g_{1_c}^{(n_l,0)}(x,Q^2)& = &e_c^2\\frac{\\alpha _s(Q^2)}{4\\pi }\\int _x^1\\frac{dz}{z} H_g^{(1),0}\\left(\\ln \\left(\\frac{Q^2}{m_c^2} \\right),z \\right)\\Delta g\\left(\\frac{x}{z}, Q^2 \\right)\\,\\mbox{,}$ where the strong coupling $\\alpha _s(Q^2)$ , the polarized gluon distribution $\\Delta g(x,Q^2)$ and the charm combination $\\Delta c^+(x,Q^2)= \\Delta c(x,Q^2) + \\Delta \\bar{c}(x, Q^2)$ are expressed in the same decoupling $n_f=n_l=3$ scheme.", "The charm fractional charge squared is $e_c^2=4/9$ and $a=1+4m_c^2/Q^2$ .", "The massive coefficient functions and their massless limits (labelled with the superscript 0) in Eqs.", "()-() are taken from Ref.", "[294], [138], [295]We notice two misprintings in Ref.", "[294]: Eq.", "(8) should read $sq=\\sqrt{1-4\\frac{z}{1-z}\\frac{m_c^2}{Q^2}}$ and the first term in the second line of Eq.", "(12) should read $-2\\frac{1+z^2}{1-z}\\ln z$ .", "and read: [leftmargin=*] for the gluon $C_{g}^{(1)}(z)& = &T_f\\left[4(2z-1)\\ln \\left(\\frac{1-z}{z} \\right) + 4(3-4z)\\right]\\\\H_g^{(1)}(Q^2,m_c^2,z)& = &T_f\\left[4(2z-1)\\ln \\left(\\frac{1+\\beta }{1-\\beta } \\right) + 4(3-4z)\\beta \\right]\\\\H_g^{(1),0}\\left(\\ln \\left(\\frac{Q^2}{m_c^2} \\right),z \\right)& = &T_f\\left[4(2z-1)\\ln \\left(\\frac{1-z}{z}\\frac{Q^2}{m_c^2} \\right) + 4(3-4z)\\right]$ with $T_f=1/2$ and $\\beta = \\sqrt{1-4\\frac{z}{1-z}\\frac{m_c^2}{Q^2}}$ for the quark $C_q(z,\\alpha _s(Q^2)) = \\delta (1-z) + \\frac{\\alpha _s(Q^2)}{4\\pi } C_q^{(1)}(z)$ with $C_q^{(1)}(z)& = &C_F\\left\\lbrace 4\\left[\\frac{\\ln (1-z)}{1-z}\\right]_+-3\\left[\\frac{1}{1-z}\\right]_+-2(1+z)\\ln (1-z)\\right.\\nonumber \\\\& & \\left.-2\\frac{1+z^2}{1-z}\\ln z +4 +2z +\\delta (1-z)[-4\\zeta (2)-9]\\right\\rbrace $ and $C_F=4/3$ .", "The formulae listed in this Appendix, should be implemented in the FastKernel framework [104], as already succesfully performed in the unpolarized case [106], in order to gauge the impact of heavy charm flavor on polarized PDFs.", "As discussed in Sec.", "REF this may be interesting in future studies at an Electron-Ion Collider." ] ]	1403.0440
[ [ "Stable chaos in fluctuation driven neural circuits" ], [ "Abstract We study the dynamical stability of pulse coupled networks of leaky integrate-and-fire neurons against infinitesimal and finite perturbations.", "In particular, we compare current versus fluctuations driven networks, the former (latter) is realized by considering purely excitatory (inhibitory) sparse neural circuits.", "In the excitatory case the instabilities of the system can be completely captured by an usual linear stability (Lyapunov) analysis, on the other hand the inhibitory networks can display the coexistence of linear and nonlinear instabilities.", "The nonlinear effects are associated to finite amplitude instabilities, which have been characterized in terms of suitable indicators.", "For inhibitory coupling one observes a transition from chaotic to non chaotic dynamics by decreasing the pulse width.", "For sufficiently fast synapses the system, despite showing an erratic evolution, is linearly stable, thus representing a prototypical example of Stable Chaos." ], [ "Introduction", "It is known that cortical neurons in vivo present a high discharge variability, even if stimulated by current injection, in comparison with neurons in vitro [1], [2].", "In particular, these differences are peculiar of pyramidal neurons, while inter-neurons reveal a high neuronal firing variability in both settings [3].", "This variability is usually measured in terms of the coefficient of variation $CV$ of the single neuron inter-spike interval (ISI), defined as the normalized standard deviation of the ISI, i.e, $CV = STD(ISI)/\\langle ISI\\rangle $ [4].", "For cortical pyramidal neurons $CV \\simeq 1.0$ in vivo [1] and $CV < 0.3$ in vitro [2], while for cortical inter-neurons $CV \\simeq 1.0 - 1.2$ [3] in both settings.", "The variability of the spike emissions in vivo resembles a stochastic (Poissonian) process (where $CV=1$ ), however the neural dynamics features cannot be accounted by simple stochastic models [1].", "These phenomena can be instead modelized by considering a deterministically balanced network, where inhibitory and excitatory activity on average compensate one another [5], [6], [7], [8].", "Despite the many papers devoted in the last two decades to this subject, is still unclear which is the dynamical phenomenon responsible for the observed irregular dynamics [9], [10], [11], [12].", "A few authors pointed out the possibility that Stable Chaos [13] could be intimately related to the dynamical behavior of balanced states [14], [15], [16], [17], [18], [19].", "Stable Chaos is a dynamical regime characterized by linear stability (i.e.", "the maximal Lyapunov exponent is negative), yet displaying an erratic behavior over time scales diverging exponentially with the system size.", "Stable Chaos has been discovered in arrays of diffusively coupled discontinuous maps [20] and later observed also in inhibitory neural networks [14].", "This phenomenon is due to the prevalence of nonlinear instabilities over the linear (stable) evolution of the system.", "This leads in diffusively coupled systems to propagation of information (driven by nonlinear effects) and in diluted inhibitory networks to abrupt changes in the firing order of the neurons [13].", "Clear evidences of Stable Chaos have been reported in inhibitory $\\delta $ –coupled networks by considering conductance based models [14] as well as current based models with time delay [15], [16], [17], [18].", "In particular, these analysis focused on the characterization of the time needed for the transient irregular dynamics to relax to the final stable state, the authors convincingly show that these transients diverge exponentially with the system size, a key feature of Stable Chaos.", "Furthermore, in [16], [17] it has been shown that, by considering time extended post-synaptic pulses leads to a transition from stable to regular chaos, where fluctuation driven dynamics is apparently maintained [17].", "In this paper, we would like to compare the dynamics of a balanced network, whose dynamics is driven by fluctuations in the synaptic inputs, with neural networks composed of tonically firing neurons.", "Similar comparisons have been performed in several previous studies [21], [22], however here we would like to focus on the role of nonlinear instabilities and in particular on indicators capable to measure finite amplitude instabilities in such networks.", "The effect of finite perturbations is relevant from the point of view of neuroscience, where the analysis is usually performed at the level of spike trains, and a minimal perturbation corresponds to the removal or addition of a spike.", "This kind of perturbations can produce a detectable modification of the firing rate in vivo in the rat barrel cortex [23].", "This has been reported as the first experimental demonstration of the sensitivity of an intact network to perturbations in vivo, or equivalently of an erratic behavior in neural circuits.", "It is however unclear if this sensitivity should be associated to linear or nonlinear effects.", "In particular the authors in [23] considered a network composed of excitatory and inhibitory neurons, where an extra spike in the excitatory network is soon compensated by an extra spike in the inhibitory network, indicating a sort of balance in the activity of the studied neural circuit.", "The ability of a perturbed balanced network to restore rapidly the steady firing rate has been discussed also in [19] for a minimal model.", "Furthermore, Zillmer et al.", "[16] have shown that a finite perturbation in a stable regime can cause a divergence of the trajectories.", "These further studies, together with the fact that the addition of an extra spike is clearly a finite perturbation from the point of view of dynamical systems, suggest that the results reported in [23] can represent an experimental verification of Stable Chaos.", "Even though all these findings are congruent with the nature of Stable Chaos [13], a careful characterization of this regime in neural networks in terms of finite amplitude indicators is still lacking.", "The only previous study examining this aspect in some details concerns a purely inhibitory recurrent Leaky Integrate-and-Fire (LIF) neural network with an external excitatory drive, which can sustain balanced activity [19].", "Starting from this analysis, which was limited to $\\delta $ -pulses, we have considered an extension the model to finite width pulses.", "Furthermore, we have characterized the linearized evolution via usual Lyapunov exponents and the nonlinear effects in terms of the response of the system to finite perturbations.", "This analysis has been performed by employing previously introduced indicators, like Finite Size Lyapunov Exponents (FSLEs) [24] or the probability that a finite perturbation can be (exponentially) expanded [19], and new indicators capable to capture nonlinear instabilities.", "The paper is organized as follows: Section is devoted to the introduction of the neural network model used through this paper, together with the indicators able to characterize linear and nonlinear instabilities.", "Section presents a comparative study of the linear and nonlinear stability analysis with emphasis on the influence of the pulse-width and of the size of the network on the dynamical behavior.", "Finally, in Sect.", "we discuss our results with respect to the existing literature and we report possible future developments of our research." ], [ "Model and methods", "We will consider a network of $N$ Leaky Integrate-and-Fire (LIF) neurons, where the membrane potential $v_i$ of the $i$ -th neuron evolves as $\\dot{v}_{i}(t)= a-v_{i}(t)+I_i(t)\\, \\quad \\quad i=1,\\cdots , N \\quad ,$ where $a > 1$ is the supra-threshold neuronal excitability, and $I_i$ represents the synaptic current due to the pre-synaptic neurons projecting on the neuron $i$ .", "Whenever a cell reaches the threshold value $v_{th} =1$ a pulse is emitted instantaneously towards all the post-synaptic neurons, and its potential is reset to $v_r = 0$ .", "The synaptic current $I_i(t)= g E_i$ is the superposition of the pre-synaptic pulses $s(t)$ received by the neuron $i$ with synaptic strength $g$ , therefore the expression of the field $E_i$ reads as $E_i(t) = \\frac{1}{K^{\\gamma }} \\sum _{j \\ne i} \\sum _{n\|t_n < t} C_{ij}\\Theta (t-t_n) s(t-t_n) \\quad .$ Where the sum extends to all the spikes emitted in the past in the network, $\\Theta (t-t_n)$ is the Heaviside function and the parameter $\\gamma $ controls the scaling of the normalization factor with the number $K$ of pre-synaptic neurons.", "Proper normalization ensures homeostatic synaptic inputs [25], [26].", "The elements of the $N \\times N$ connectivity matrix $C_{ij}$ are one (zero) in presence (absence) of a connection from the pre-synaptic $j$ -th neuron to the post-synaptic $i$ -th one.", "In this paper we limit our analysis to random sparse networks, where each neuron receives exactly $K$ pre-synaptic connections and this number remains fixed for any system size $N$ .", "The model appearing in Eqs.", "(REF ) and (REF ) is adimensional, the transformation to physical units is discussed in Appendix I.", "By following [5], we assume that the pulses are $\\alpha $ -functions, $s(t)=\\alpha ^2 t \\exp (-\\alpha t)$ , in this case the dynamical evolution of the fields $E_i(t)$ is ruled by the following second order differential equation (ODE): $\\ddot{E}_i(t) +2\\alpha \\dot{E}_i(t)+\\alpha ^2 E_i(t)=\\frac{\\alpha ^2}{K^\\gamma } \\sum _{j \\ne i} \\sum _{n\|t_n < t} C_{ij} \\delta (t-t_n) \\qquad ,$ which can be conveniently rewritten as two first ODEs, as $\\dot{E_i} = P_i - \\alpha E_i, \\qquad \\dot{P_i}=-\\alpha P_i +\\frac{\\alpha ^2}{K^\\gamma } \\sum _{j \\ne i} \\sum _{n\|t_n < t} C_{ij} \\delta (t-t_n) \\ ;$ by introducing the auxiliary field $P_i = \\dot{E}_i -\\alpha E$ .", "The equations (REF ) and (REF ) can be exactly integrated from the time $t=t_n$ , just after the deliver of the $n$ -th pulse, to time $t=t_{n+1}$ corresponding to the emission of the $(n+1)$ -th spike, thus obtaining an event driven map [27], [28] which reads as $E_i(n+1)&=E_i(n) {\\rm e}^{-\\alpha \\tau (n)}+P_i(n)\\tau (n){\\rm e}^{-\\alpha \\tau (n)} \\\\P_i(n+1)&=P_i(n)e^{-\\alpha \\tau (n)}+C_{im} \\frac{\\alpha ^2}{K^\\gamma }\\\\v_{i}(n+1)&=v_i(n)e^{-\\tau (n)}+a(1-e^{-\\tau (n)})+g H_i(n) \\, ,$ where $\\tau (n)= t_{n+1}-t_n$ is the inter-spike interval associated to two successive neuronal firing in the network, which can be determined by solving the transcendental equation $\\tau (n)=\\ln \\left[\\frac{a-v_m(n)}{a+g H_m(n)-1}\\right] \\ ,$ here $m$ identifies the neuron which will fire at time $t_{n+1}$ by reaching the threshold value $v_m(n+1) = 1$ .", "The explicit expression for $H_i(n)$ appearing in equations () and (REF ) is $H_i(n) &=& \\frac{{\\rm e}^{-\\tau (n)} - e^{-\\alpha \\tau (n)}}{\\alpha -1}\\left(E_i(n)+\\frac{P_i(n)}{\\alpha -1} \\right)-\\frac{\\tau (n) e^{-\\alpha \\tau (n)}}{\\alpha -1} P_i(n) \\, .$ The model is now rewritten as a discrete-time map with $3 N -1$ degrees of freedom, since one degree of freedom, $v_m(n+1) =1$ , is lost due to the event driven procedure, which corresponds to perform a Poincaré section any time a neuron fires.", "Our analysis will be devoted to the study of sparse networks, by considering a constant number $K$ of afferent synapses for each neuron, namely $K=20$ .", "Therefore, the normalization factor $K^\\gamma $ appearing in the definition of the pulse amplitude is somehow irrelevant, since here we limit to study a specific value of the in-degree connectivity, without varying $K$ .", "However, to compare with previous studies, we set $\\gamma =1$ for purely excitatory neurons, where $g>0$ , similarly to what done in [29], [30], and $\\gamma =1/2$ for purely inhibitory networks, where $g < 0$ , following the normalization employed in [15], [17], [31], [19].", "The reasons for these different scalings rely on the fact that in the excitatory case, the dynamics of the system are current driven (i.e.", "all neurons are tonically firing even in absence of coupling, being supra-threshold), therefore the synaptic input should be normalized with the number of afferent neurons to maintain an average homeostatic synaptic input [25], [26].", "The situation is different in presence of inhibitory coupling, here the supra-threshold excitability of the single neuron can be balanced by the inhibitory synaptic currents, thus maintaining the neurons in proximity of the firing threshold.", "In this case, the network dynamics is fluctuation driven, because the fluctuations in the synaptic inputs are responsible of the neuronal firing.", "In order to keep the amplitude of the fluctuations of the synaptic current constant, the normalization is now assumed proportional to the square root of the number of the synaptic inputs [11].", "In the present analysis we have tuned the model parameters in order to be in a fluctuation driven regime whenever the inhibitory coupling is considered.", "In particular, we will study, not only the dependence of the dynamics on the pulse shape, but also on the system size, by maintaining a constant number of incoming connections $K$ .", "However, we will not assume that the excitatory external drive (in our case represented by the neuronal excitability $a$ ) will diverge proportionally to $\\sqrt{K}$ , as done in [32], [19], since we are not interested in the emergence of a self-tuned balanced state in the limit $ K \\rightarrow \\infty $ , for $1 << K << N$ [32], [19]." ], [ "Linear Stability Analysis", "To perform the linear stability analysis of the system, we follow the evolution of an infinitesimal perturbation in the tangent space, through the following set of equations obtained from the linearization of the event driven map (REF ,,) $\\delta E_i(n+1) &= e^{-\\alpha \\tau (n)} \\left[ \\delta E_i(n) +\\tau (n) \\delta P_i(n) \\right] \\nonumber \\\\& - e^{-\\alpha \\tau (n)} \\left[\\alpha E_i(n) + (\\alpha \\tau (n)-1) P_i(n) \\right] \\delta \\tau (n)\\,,\\\\\\delta P_i(n+1) &= e^{-\\alpha \\tau (n)} \\left[ \\delta P_i(n)-\\alpha P_i(n) \\delta \\tau (n) \\right]\\, ,\\\\\\delta v_{i}(n+1) &= e^{-\\tau (n)} \\left[ \\delta v_i (n) + (a-v_i(n)) \\delta \\tau (n) \\right] + g \\delta H_i (n)\\nonumber \\\\i & =1,\\dots ,N \\quad ; \\quad \\delta v_m(n+1) \\equiv 0 \\, .$ The boundary condition $\\delta v_m(n+1) \\equiv 0$ is a consequence of the event driven evolution.", "The expression of $\\delta \\tau (n)$ can be computed by differentiating (REF ) and (REF ) $\\delta \\tau (n) =\\tau _v \\delta v_m(n) +\\tau _E\\delta E_m(n)+\\tau _P\\delta P_m(n)\\ ,$ where $\\tau _v:= \\frac{\\partial \\tau }{\\partial v_m} \\quad , \\quad \\tau _E:= \\frac{\\partial \\tau }{\\partial E_m} \\quad , \\quad \\tau _P:= \\frac{\\partial \\tau }{\\partial P_m} \\quad .$ In this paper, we will limit to measure the maximal Lyapunov exponent $\\lambda $ to characterize the linear stability of the studied models.", "This is defined as the the average growth rate of the infinitesimal perturbation $ \\mathbf {\\delta } = (\\delta v_1 \\dots \\delta v_N,\\, \\delta E_1 \\dots \\delta E_N, \\, \\delta P_1 \\dots \\delta P_N),$ through the equation $\\lambda =\\lim _{t\\rightarrow \\infty }\\frac{1}{t}\\log \\frac{\\mid \\mathbf {\\delta } (t)\\mid }{\\mid \\mathbf {\\delta }_0\\mid } \\quad ,$ where $\\mathbf {\\delta }_0$ is the initial perturbation at time zero.", "The evolution of the perturbation $\\mathbf {\\delta }(t)$ has been followed by performing at regular time intervals the rescaling of its amplitude to avoid numerical artifacts, as detailed in [33].", "Furthermore, since our system is time continuous one would expect to have always a zero Lyapunov exponent, which in fact is the maximal Lyapunov if the system is not chaotic.", "However, this does not apply to the event driven map because the evolution is based on a discrete time dynamics, where the motion along the orbit between two successive spikes is no more present due to the performed Poincaré section." ], [ "Finite Size Stability Analysis", "Besides the characterization of the stability of infinitesimal perturbations, we are also interested in analyzing how a perturbation grows according to its amplitude.", "To perform this task several indicators have been introduced in the last years, ranging from Finite Size Lyapunov Exponents (FSLE) [24], [34], [35], [36] to the propagation velocity of finite perturbations [37].", "FSLEs have been mainly employed to charaterize Stable Chaos in spatially extended systems [13] and Collective Chaos in globally coupled systems [38], [39], [40].", "We have performed several tests by employing the usual FSLE definition [36].", "In particular FSLE can be defined by considering an unperturbed trajectory $\\mathbf {x} = (v_1 \\dots v_N,\\, E_1 \\dots E_N, \\, P_1 \\dots P_N)$ and a perturbed trajectory $\\mathbf {x^\\prime } = (v_1^\\prime \\dots v_N^\\prime ,\\, E_1^\\prime \\dots E_N^\\prime , \\, P_1^\\prime \\dots P_N^\\prime )$ , obtained by randomly perturbing all the coordinates (both the fields $E$ and $P$ as well as the membrane potentials) of the generic configuration $\\mathbf {x}$ on the attractor.", "Then we follow the two trajectories in time by measuring their distance $\\Delta (t)= \\parallel \\mathbf {x}(t) - \\mathbf {x^{\\prime }}(t) \\parallel $ , by employing the absolute value norm.", "Whenever $\\Delta (t_k)$ crosses (for the first time) a series of exponentially spaced thresholds $\\theta _k$ , where $\\theta _k = r \\theta _{k-1}$ , the crossing times $t_k$ are registered.", "By averaging the time separation between consecutive crossings over different pairs of trajectories, one obtains the FSLE [36], [24] $\\lambda _F (\\Delta (t_k)) = \\frac{r}{\\langle t_k - t_{k-1} \\rangle }\\qquad ; {\\rm where} \\quad \\Delta (t_k)=\\theta _k$ For small enough thresholds, one recovers the usual maximal Lyapunov exponent, while for large amplitudes, FSLE saturates to zero, since a perturbation cannot be larger than the size of the accessible phase-space.", "In the intermediate range, $\\lambda _F$ tells us how the growth of a perturbation is affected by nonlinearities.", "However, as a general remark, we have noticed that it is extremely difficult to get reliable results from the FSLE analysis, probably because the estimation of $\\lambda _F$ relies on measurements based on single trajectory realizations, which presents huge fluctuations.", "In order to overcome this problem, the single trajectory should be smoothed before estimating the passage times from one threshold to the next one and we observed that the results strongly depend on the adopted smoothing procedure, in particular for the fluctuation driven case.", "Therefore, in order to investigate the growth rate of finite amplitude perturbations we have decided to adopt different indicators rather than the FSLE.", "In particular, an estimation of finite size stability can be obtained by defining the following indicator $D(\\Delta (t)) = \\frac{ d \\left\\langle \\log \\Delta (t) \\right\\rangle }{dt};$ where the average $\\left\\langle \\cdot \\right\\rangle $ is performed over many different initial conditions.", "In the limit $\\Delta (t) \\rightarrow 0$ we expect to recover the maximal Lyapunov exponent $\\lambda $ .", "In order to ensure that the dynamics of the perturbed trajectory will also occur on the attractor associated to the studied dynamics, we have considered extremely small initial perturbations $\\Delta _0 = \\Delta (0) \\simeq 10^{-8}-10^{-10}$ .", "As we will show, after a transient needed for the perturbed trajectory $\\mathbf {x^\\prime }$ to relax to the attractor, $D(\\Delta )$ measures effectively the maximal Lyapunov exponent.", "However, if nonlinear mechanisms are present $D(\\Delta )$ can become larger than $\\lambda $ for finite amplitude perturbations.", "Anyway, analogously to the FSLE, for perturbations of the size of the attractor the indicator $D(\\Delta )$ decays towards zero due to the trajectory folding.", "The studied models present discontinuities of ${\\cal O}(1)$ in the membrane potentials $v_i$ , due to the reset mechanisms, and of ${\\cal O}(\\alpha ^2/K^\\gamma )$ in the fields $P_i$ , due to the pulse arrival.", "In order to reveal, without any ambiguity, the presence of nonlinear instabilities at finite amplitudes, for the estimation of the FSLE and of the indicator $D$ we mainly limit our analysis to the continuous fields $\\lbrace E_i \\rbrace $ .", "In particular, to characterize the finite amplitude instabilities, we consider the following distance between the perturbed and unperturbed orbits $\\Delta ^{(E)}(t) = \\frac{1}{N} \\sum _{i=1}^N \| E_i(t) - E_i^\\prime (t)\|.$ In some cases we have also analyzed the distance $\\Delta ^{(v,E,P)}$ between all the variables associated to the unperturbed and perturbed state with a clear meaning of the adopted symbol.", "Unfortunately, the indicator $D(\\Delta )$ as well as the FSLE cannot be employed in the case of stable chaos, when $\\lambda < 0$ , because in this case small perturbations are quickly damped and one cannot explore the effect of perturbation of growing amplitude by following the dynamics on the attractor.", "In this situation, one should employ different indicators, as done in [34], [41] for coupled map lattices.", "In particular, we proceed as follow, we consider two orbits at an initial distance $\\Delta _0$ and we follow them for a time interval $T$ , then we measure the amplitude of the perturbation at the final time, namely $\\Delta (T)$ .", "We rescale one of the two orbits to a distance $\\Delta _0$ from the other one, keeping the direction of the perturbation unchanged, and we repeat the procedure several times and for several values of $\\Delta _0$ .", "Then, we estimate the finite amplitude growth rate, as $R_T (\\Delta _0) = \\frac{1}{T}\\left\\langle \\log \\frac{ \| \\Delta (T) \| }{\|\\Delta _0\|} \\right\\rangle ,$ where the angular brackets denote the average over a sufficiently large number of repetitions.", "To allow the perturbed orbit to relax on the attractor, we initially perform $\\simeq 10^3$ rescalings, which are not included in the final average.", "However, also this procedure does not guarantee that the attractor is always reached, in particular for very large perturbations.", "Furthermore, the perturbed dynamics is no more constrained to evolve along the tangent space associated to the event driven map.", "As a matter of fact, whenever $\\lambda < 0$ the indicator $R_T(\\Delta _0)$ converges to zero and not to the Lyapunov exponent associated to the discrete time map evolution.", "Finally, following the analysis reported in [19], we consider the probability $P_S(\\Delta _0)$ that a perturbation of amplitude $\\Delta _0$ induces an exponential separation between the reference and perturbed trajectory.", "In particular, we perturb the reference orbit with an initial perturbation $\\Delta _0$ and we follow the evolution of the trajectories for a time span $T$ .", "Whenever $\\Delta (T)$ is larger than a certain threshold $\\theta _L$ this trial contributes to the number of expanding initial perturbations $N_S(\\Delta _0)$ , otherwise is not counted.", "We repeat this procedure $N_T$ times for each perturbation of amplitude $\\Delta _0$ , then $P_S(\\Delta _0)= N_S(\\Delta _0) /N_T$ .", "For the two latter indicators, namely $R_T$ and $P_S$ , we have always employed the total distance $\\Delta ^{(v,E,P)}$ , to confront our findings with the results reported in [19]." ], [ "Results", "As already mentioned, we will compare a current driven excitatory network and a fluctuation driven inhibitory network.", "In particular, the excitatory network is studied in a regime where it presents a collective non trivial partial synchronization [42], [30].", "This state is characterized by quasi-synchronous firing events, as revealed by the raster plot reported in the upper panel of Fig.", "REF , and almost periodic oscillations of the effective current $I_i^{eff}(t) \\equiv a + g E_i(t)$ (see Fig.", "REF , upper panel).", "In this particular case $I_i^{eff} > 1$ therefore the neurons are always supra-threshold.", "In this situation the measure of the $CV$ gives quite low values, namely for the studied case (with $a=1.3$ , $g=0.2$ and $\\alpha =9$ ) $CV \\simeq 0.17$ , similar to pyramidal neurons in vitro.", "Despite this low level of variability in the neuronal dynamics, the sparseness in the matrix connectivity induces chaotic dynamics in the network, which persists even in the thermodynamic limit [30].", "At variance with diluted networks, where the average connectivity scales proportionally to the system size ($K \\propto N^z$ , with $1 \\ge z > 0$ ).", "In this latter case, in the limit $N \\rightarrow \\infty $ the system will recover a regular evolution, similarly to fully coupled networks [28], [29].", "For the inhibitory network, we observe radically different dynamics, this because now $I^{eff}(t)$ oscillates around one, therefore the neurons fire in a quite irregular manner, driven by the fluctuations of the fields $E_i(t)$ , as shown in the lower panels of Figs.", "REF (a) and (b).", "In this case we have examined the dynamics of the model for $a=1.3$ , $g=-0.8$ and different pulse-widths $1/\\alpha $ .", "For $\\alpha \\in [1:5]$ the neuronal dynamics are always quite erratic, being characterized by $CV \\simeq 0.7-1$ (see Fig.", "REF ).", "Narrower pulses (larger $\\alpha $ values) are associated to somehow more regular dynamics and smaller ISI, however we have verified that the ISI and CV saturates to some finite value in the thermodynamic limit (as shown in Fig.", "REF (a) and (b)).", "This suggests that fluctuations will not vanish for $N \\rightarrow \\infty $ and that the system will remain fluctuation driven even in such a limit.", "Furthermore, the two $\\alpha $ -values examined in Figs.", "REF and REF correspond to two different dynamical regimes, further discussed in Sect.", "REF , namely, a chaotic ($\\alpha =3$ ) and a non-chaotic ($\\alpha =5$ ) state.", "Figure: Dependence of the coefficient of variation CVCV (a) and of the inter-spike time intervalISI (b) on the pulse width for the fluctuation driven case.", "The data refer toN=400N = 400 (black circles) and N=1600N=1600 (red squares).The data have been averaged over 10 8 10^8 spikes, once a transient of 10 7 10^7 spikes has been discarder.The other parameters are as in the caption of Fig.", "Figure: Dependence of the coefficient of variation CVCV (a) and of the inter-spike time intervalISI (b) on the size of the network for fluctuation driven networks in tworepresentative situations corresponding to the chaotic (α=3\\alpha =3, black circles) andthe stable chaos (α=5\\alpha = 5, red squares) regimes.", "The reported data have been averagedover 10 8 10^8 spikes, once a transient of 10 7 10^7 spikes has been discarded.The other parameters are as in the caption of Fig.", "." ], [ "Lyapunov analysis", "As previously shown, the fluctuation driven regime is observable for the inhibitory network for all the considered pulse widths.", "In this Subsection we would like to investigate whether such variability is related to a linear instability of infinitesimal perturbations (measured by the maximal Lyapunov exponent $\\lambda $ ) or to other (nonlinear) instabilities present in the system.", "Let us start examining the Lyapunov exponent for such systems, as a first result we observe a strong dependence of $\\lambda $ on the pulse-width (see Fig.", "REF ): the system is chaotic for wide pulses and becomes stable for sufficiently narrow ones.", "These results are in agreement with previously reported results in [16], [17] for an inhibitory network of LIF neurons with delayed synapses.", "In these papers the authors show that chaos can arise only for sufficiently broad pulses, conversely for $\\delta $ -pulses the system is always stable.", "It is worth to notice that the critical $\\alpha $ -value at which occurs the transition to chaos becomes larger as the system size increases, pointing to the question whether the stable regime still exists for finite pulses in the thermodynamic limit or if it is merely a finite size property [17].", "Extensive simulations for sizes of the network up to $N=10,000$ have shown that the stable regime is present even for such a large size (see Fig.REF ).", "Furthermore, we have found an empirical scaling law describing the increase of $\\lambda $ with $N$ , i.e.", "$\\lambda = \\lambda _{\\infty } - c N^{-\\eta }$ where $\\lambda _{\\infty }$ denotes the asymptotic value in the thermodynamic limit and $\\eta $ is the scaling exponent.", "For the two representative cases here studied, the exponent was quite similar, namely $\\eta \\simeq 0.24$ ($\\eta \\simeq 0.22$ ) for $\\alpha =3$ ($\\alpha =5$ ), thus suggesting an universal scaling law for this model when fluctuation driven, with an exponent $\\eta =1/4$ .", "This exponent is different from the one measured for the current driven case, in such situation for sparse connectivity $\\lambda $ converged to its asymptotic value as $1/N$ [30].", "An exponent $\\eta =1$ has been previously measured for coupled map lattices exhibiting spatio-temporal chaos and theoretically justified in the framework of the Kardar-Parisi-Zhang equation [43].", "The scalings we are reporting in this paper are associated to random networks, therefore they demand for a new theoretical analysis.", "Furthermore, the asymptotic values $\\lambda _{\\infty }=0.335(1)$ ($\\lambda _{\\infty }=-0.034(1)$ ) indicate that a critical threshold separating stable from chaotic dynamics persists in the thermodynamic limit.", "Figure: Linear stability analysis of the fluctuation driven state.", "(a) Maximal Lyapunov exponent λ\\lambda as a function of pulse-width α\\alpha ,for two representative system sizes: N=400N = 400 (black circles) andN=1600N = 1600 (red squares); thin dashed lines are drawn for eye guide only.", "(b) Lyapunov exponent as a function of the system sizeNN, for two representative pulse widths: α=3\\alpha = 3 (black circles) and α=5\\alpha = 5 (red squares).Continuous lines correspond to the nonlinear fitting (),which predicts the asymptotic values λ ∞ \\lambda _{\\infty } (thick dashed lines).The fitting parameters entering in Eq.", "() arec=1.08c = 1.08 (c=0.78c = 0.78) and η≃0.24\\eta \\simeq 0.24 (η≃0.22\\eta \\simeq 0.22)for α=3\\alpha = 3 (α=5\\alpha = 5).", "In both figures, λ\\lambda is calculated byintegrating the evolution in the tangent space together with the unperturbed orbit dynamicsduring a time interval equivalent to 10 8 10^8 spikes, after discarding atransient of 10 7 10^7 spikes.", "Remaining parameters as in Fig.", "." ], [ "Finite size perturbation analysis", "Stable chaos in spatially extended systems is due to the propagation of finite amplitude perturbations, while infinitesimal ones are damped.", "In inhibitory neural networks, the origin of Stable Chaos has been ascribed to abrupt changes in the firing order of neurons induced by a discontinuity in the dynamical law, while infinitesimal perturbations leave the order unchanged [14], [13], [17].", "In particular, by examining a conductance based model, in [13] it has been shown that a spike was able to induce a finite perturbation in the evolution of two (not-symmetrically) connected neurons, given that the inhibitory effect of a spike was related to the actual value of the membrane potential of the receiving neuron.", "Therefore two ingredients are needed to observe Stable Chaos in neural models, a non symmetric coupling among neurons, together with the fact that the amplitude of transmitted pulses should depend on the neuron state.", "These requirements are fulfilled also in the present model, despite being current based, since any current based model can be easily transformed in a conductance based one via a nonlinear transformation [5], [44].", "However, the problem is to quantify this effect in terms of some indicator, similarly to what done in spatially extended systems, where Stable Chaos has been characterized in terms of the FSLE and of the velocity of propagation of information [37], [41].", "Figure: FSLE indicator λ F \\lambda _F for the fluctuation driven (black circles) and current driven (red squares) chaotic set-ups.", "An initial perturbation of 10 -9 10^{-9} (10 -7 10^{-7}) is applied to the excitatory (inhibitory) network.", "The distance between the perturbed and unperturbed trajectory Δ (E) \\Delta ^{(E)} is sampled during 300 time units, at fixed time intervals dt=0.2dt = 0.2.", "The sampled curve is smoothed over a sliding window of 20 time units and the resulting curveis used to obtain the times t k t_k at which the system crosses the corresponding thresholds θ k \\theta _k, with r=1r=1(see the definition ().", "This procedure is averaged in the current (fluctuation) drivencase over 5000 (15000) realizations.Thick dashed lines indicate the value of λ\\lambda for each one of the two cases.The current and fluctuation driven cases have been examinedfor the same parameter values reported in Fig.", ", apart thatfor the inhibitory case the inverse of the pulse width isset to α=3\\alpha =3.As a first indicator we consider the FSLE, associated to the norm $\\Delta ^{(E)}$ , the corresponding results are reported in Fig.", "REF for the current and fluctuation driven cases.", "In the former case the FSLE is never larger than the usual Lyapunov exponent $\\lambda $ , with which it coincides over a wide range of perturbation amplitudes.", "In particular, $\\lambda _F (\\Delta ^{(E)}) < \\lambda $ for small amplitudes, due to the fact that initially the perturbation needs a finite time to align along the maximal expanding direction.", "Furthermore, due to the folding mechanism, the perturbation is contracted also for large perturbations of the order of the attractor system size.", "In summary, for current driven dynamics only the instability associated to infinitesimal perturbation is present, as reported also in [40].", "In the fluctuation driven case the situation is quite different as shown in Fig.", "REF , the FSLE essentially coincides with $\\lambda $ for small $\\Delta ^{(E)}$ , but it becomes definitely larger than $\\lambda $ for finite perturbations, revealing a peak around $\\Delta ^{(E)} \\simeq {\\cal O}(1/N)$ .", "These are clear indications that finite amplitude instabilities coexist with infinitesimal ones and they could be in principle even more relevant.", "Figure: Lower panel: Evolution of the average distance <logΔ (E) ><\\log \\Delta ^{(E)}> as a function of time, for thecurrent (red square) and the fluctuation (black circle) driven cases.", "The curvesare obtained by averaging the distances between the perturbed and unperturbed trajectories over 5000(15000) realizations, after applying an initial perturbation of 𝒪(10 -8 ){\\cal O}(10^{-8}).Upper panel: Indicator D (E) D^{(E)} as a function of time for the same cases, calculated as the time derivativeof <logΔ (E) ><\\log \\Delta ^{(E)}>.", "For small perturbations, D (E) D^{(E)} is close to λ\\lambda (thick dashed lines), whileobserving a finite size effect is observable in the fluctuation driven case.The current and fluctuation driven cases have been examinedfor the same parameter values reported in Fig.", ".The estimation of the FSLE, as already mentioned, suffers of several numerical problems in these systems.", "Therefore we decided to consider the indicator $D(\\Delta ^{(E)}(t))$ , for simplicity denoted as $D^{(E)}$ , which is less affected by the single orbit fluctuations, since its estimation is based on the time derivative of the averaged distance $\\left\\langle \\log \\Delta (t) \\right\\rangle $ .", "In Fig.", "REF we report $\\left\\langle \\log \\Delta ^{(E)} (t) \\right\\rangle $ and $D^{(E)}$ as a function of time for a current driven and a fluctuation driven case, in both situations after an initial transient, the indicator $D^{(E)}$ coincides with $\\lambda $ .", "However, in the current driven case it coincides with $\\lambda $ for a very long time before decreasing due to the folding of the trajectories, while in the fluctuation driven situation it becomes soon larger than the maximal Lyapunov exponent and it shows a clear peak at finite amplitudes, before the folding effect sets in.", "The same results are reported in the upper panel of Fig.", "REF as a function of $\\left\\langle \\log \\Delta ^{(E)} (t) \\right\\rangle $ , the peak in the fluctuations driven case is located around $4 \\times 10^{-4}$ thus at a smaller amplitude with respect to the FSLE, despite the system size and parameters are the same in both cases.", "Furthermore, in the lower panel in Fig.", "REF we report the indicator $D(\\Delta ^{v,E,P}(t))$ ($D^{(v,E,P)}$ from now on) estimated for the total distance among the perturbed and unperturbed orbit.", "As expected, the discontinuities present in the evolution of the membrane potentials and of the auxiliary field $P$ due to pulse emission and pulse arrival, induce a small increase on $D^{(v,E,P)}$ with respect to the infinitesimal value $\\lambda $ at finite amplitudes even in the current driven case.", "However, in this case the peak of $D^{(v,E,P)}$ is definitely smaller with respect to the one observed in the fluctuation driven case and it is located at larger perturbations ${\\cal O}(1)$ .", "Similar effects are observable also by considering the FSLE associated to $\\Delta ^{(v,E,P)}$ , data not shown.", "Nevertheless, in order to keep ourselves in a consistent framework, in what follows we will consider the distance between the perturbed and unperturbed continuous fields $\\Delta ^{(E)}$ .", "By choosing this norm, we will avoid the presence of (trivial) peaks due to discontinuities as in the current driven system, but instead, the presence of these peaks will be a genuine indication of nonlinear instabilities, as those present in a fluctuation driven regime.", "Figure: Indicator D(Δ)D(\\Delta ) versus the complete distance Δ (v,E,P) \\Delta ^{(v,E,P)} (lower panel)and versus the distance Δ (E) \\Delta ^{(E)} (upper panel)for the current (red squares) and fluctuation (black circles) driven cases.The curves are obtained with the same procedure described in the caption of Fig.", ".", "In bothpanels, thick dashed lines illustrate the corresponding value of λ\\lambda .The current and fluctuation driven cases have been examinedfor the same parameter values reported in Fig.", ".The indicator $D^{(E)}$ is reported in Fig.", "REF for various system sizes, ranging from $N=400$ to $N=1600$ for the current and fluctuation driven cases.", "We observe that in the current driven case $D^{(E)}$ always gives a value around the corresponding $\\lambda $ at all scales, apart the final saturation effect (see Fig.", "REF ).", "Notice that $\\lambda $ , for these system sizes, strongly depends on $N$ (as shown in [30]), the saturation at the asymptotic value is expected to occur for $N > 5000$ .", "For the fluctuation driven set-up, a peak (larger than $\\lambda $ ) is always present in $D^{(E)}$ at finite amplitudes (see Fig.", "REF ).", "The peak broadens for increasing $N$ extending to larger amplitudes and also its height increases.", "The presence of more neurons in the network renders stronger the finite amplitude effects, while nonlinear instabilities are present at larger and larger perturbation amplitudes.", "Figure: Finite amplitude perturbation analysis for several sizes of the network by using the procedure described in Fig.", "for the distance Δ E \\Delta ^{E} for the current (a) and fluctuation (b) drivensetups.", "In both cases the studied sizes correspond to N=400N=400 (black circles), N=600N=600 (green up-triangles),N=800N=800 (red squares) and N=1600N = 1600 (blue down-triangles), averaged through 7500 realizations.Remaining parameters as in Fig.", ".So far we have considered only chaotic regimes, both in the fluctuation driven and in the current driven case.", "However, even in linearly stable cases the dynamics can be erratic, as shown in Fig.", "REF for the fluctuation driven case corresponding to $\\alpha =5$ for which the maximal Lyapunov is negative at any system size (see Fig.", "REF (b)).", "This kind of erratic behavior, known as Stable Chaos [13], is one the most striking examples of dynamics driven by nonlinear effects, since the linear instabilities are asymptotically damped.", "In this situation neither the FSLE nor the indicator $D(\\Delta )$ can be measured.", "The reason is that, in order to ensure that the dynamics will take place on the associated attractor, finite amplitude perturbations are reached only by starting from very small initial perturbations, which in this case are damped.", "Therefore, we should employ different indicators, namely the finite amplitude growth rate $R_T (\\Delta _0)$ and the probability $P_S(\\Delta _0)$ .", "As shown in Fig.", "REF , for the linearly stable fluctuation driven case corresponding to $\\alpha =5$ , $R_T (\\Delta _0) \\rightarrow 0 $ for sufficiently small perturbations, as expected.", "However $R_T (\\Delta _0)$ becomes soon positive for finite amplitude perturbation and it reveals a large peak $R_T^M$ located at an amplitude $\\Delta _0^M$ .", "For increasing system size $N$ , as shown in Fig.", "REF a linear decrease of $\\Delta _0^M$ with $N$ is clearly observable, while $R_T^M$ reveals a logarithmic increase with $N$ .", "Thus suggesting that this indicator will diverge to infinity in the thermodynamic limit, similarly to the results previously reported in [32], [19].", "However, at variance with these latter studies, in the present context the connectivity remains finite even in the limit $N \\rightarrow \\infty $ .", "The analysis of $P_S(\\Delta _0)$ , reported in Fig.", "REF , reveals that the curve can be well fitted as $P_S(\\Delta _0) = 1 - \\exp (-\\Delta _0/\\beta )^\\mu \\qquad ;$ analogously to what done in [19].", "The parameter $\\beta $ can be considered as a critical amplitude, setting the scale over which nonlinear instabilities take place.", "At variance with the results reported by Monteforte & Wolf in [19], we observe a linear decrease with $N$ of the critical amplitude $\\beta $ (see Fig.", "REF ) and an exponent $\\mu \\simeq 2.3- 2.5$ , depending on the employed system size.", "Instead, Monteforte & Wolf reported a scaling $\\beta \\propto 1/\\sqrt{N}$ and an exponent $\\mu = 1$ .", "Furthermore, we have verified for various continuous $\\alpha $ pulses, with $\\alpha \\in [4;7]$ , that the measured exponent $\\mu $ does not particularly depend on $\\alpha $ .", "The model here studied differs for the shape of the post-synaptic currents from the one examined in [19], where $\\delta $ -pulses have been considered.", "Figure: Characterization of the Stable Chaos regime: finite amplitude instabilities for different network sizes.", "(a) R T R_T indicator as a function of the initial perturbationΔ 0 \\Delta _0.", "(b) Probability P S P_S to observe an exponential increase of the distancebetween a perturbed and an unperturbed orbit versus the initial perturbationΔ 0 \\Delta _0.", "Thick dashed lines refer to the fit to the data with the expressionP S =1-e -(Δ 0 /β) μ P_S = 1 - e^{-(\\Delta _0/\\beta )^{\\mu }}.The studied sizes are N=100N=100 (bluedown-triangle), N=200N=200 (green up-triangles), N=400N=400 (black circles), N=800N = 800(red squares), N=1600N=1600 (magenta diamonds) and N=3200N=3200 (orange right-triangles).", "For eachperturbation Δ 0 \\Delta _0, R T R_T and P S P_S arecalculated after T=5T = 5 time units, threshold defining expanding trajectories θ L =-2\\theta _L = -2 and averaging over N T =5000N_T = 5000 realizations.", "Remaining parameters as reported in Fig.", ".In our opinion, these two latter indicators, $R_T$ and $P_S$ bear essentially the same information: they measure the propensity of a perturbation $\\Delta _0$ to be amplified on a short time scale $T$ .", "This is confirmed by the fact that (as shown in Fig.", "REF ) the values of $\\Delta _0^M$ and $\\beta $ , which set the relevant amplitude scales for the two indicators, both decrease with the same scaling law (namely, $1/N$ ) with the system size.", "A possible explanation for this scaling could be found by assuming that the main source of nonlinear amplification is associated to a spike removal (addition) in the perturbed orbit.", "A missing (extra) spike will perturb, to the leading order, the distance $\\Delta ^{(v,EP)}$ by an amount $\\propto \\alpha ^2\\sqrt{K}/N$ , since the lost (added) post-synaptic pulses are $K$ each of amplitude $\\alpha ^2/\\sqrt{K}$ .", "This argument explains as well the logarithmic increase of $R_T^M$ with the system size.", "Furthermore, the decrease of $\\Delta _0^M$ and $\\beta $ with $N$ seems to indicate that in the thermodynamic limit any perturbation, even infinitesimal, will be amplified.", "This is clearly in contradiction with the fact that the system is linearly stable and it appears to remain stable by increasing $N$ (as shown in Fig.", "REF ).", "In systems exhibiting Stable Chaos, it has been reported many times the fact that the thermodynamic limit and the infinite time limit do not commute [20].", "For finite system size, at sufficiently large times (diverging exponentially with $N$ ) a stable state is always recovered, while if the thermodynamic limit is taken before the infinite time one, the system will remain erratic at any time [13].", "In the present case, it seems that a different non commutativity between the thermodynamic limit and the limit of vanishingly small perturbations is present, similar conclusions have been inferred also in [19].", "Therefore, we can apparently conclude that a fluctuation driven system, which is linearly stable, but presents nonlinear instabilities, will become unstable at any amplitude and time scales in the thermodynamic limit.", "However, one should be extremely careful in deriving any conclusion from these indicators, since they are not dynamical invariant and their values depend not only on the considered variables but also on the employed norm.", "Furthermore, in the present context there is an additional problem related to the meaningful definition of the norm in an infinite space, as that achieved in the thermodynamic limit.", "To understand the limit of applicability of $R_T$ , we have examined this indicator also in the chaotic fluctuation driven case, namely for $\\alpha =3$ .", "Also in this case we observe that $\\Delta _0^M$ will vanish for diverging system size, but with a different scaling law, namely $\\Delta _0^M \\simeq N^{-0.6}$ .", "Furthermore, $R_T^M$ increases with $N$ , but this time it appears to saturate in the thermodynamic limit with a scaling law similar to the one reported in (REF ) for the maximal Lyapunov exponent, more details are reported in the caption of Fig.", "REF .", "Unlike the stable regime, in the chaotic one we cannot justify with the simple spike addition (removal) argument the scaling with $N$ neither for $\\Delta _0^M$ nor for $R_T^M$ .", "It is high probable that in this regime the interactions of the linear and nonlinear instabilities leads to more complicated mechanisms.", "The evolution of the indicator $R_T$ suggests that for increasing $N$ its peak will move down to smaller and smaller amplitude scale.", "However, this result is in contradiction with the behavior of $D^{(E)}$ reported in Fig.", "REF , for this latter indicator the position of the peak is not particularly affected by $N$ .", "In particular, finite amplitude instabilities affect larger and larger scales, contrary to what seen for $R_T$ (see Fig.", "REF ).", "The same behavior is observable for $D^{(v,E,P)}$ , data not shown.", "These contradictory results point out the limits of indicators like $R_T$ and $P_S$ relying on dynamical evolutions not taking place on the attractor of the system.", "Figure: a) Peak position Δ 0 M \\Delta _0^M as a function of NN in a log-logscale for α=3\\alpha = 3 (black circles) and α=5\\alpha = 5 (red squares).", "The continuousline are power law fitting Δ 0 M ∝N -Φ \\Delta _0^M \\propto N^{-\\Phi } , with exponentsΦ=-0.59\\Phi = -0.59 (Φ=-1.05\\Phi = -1.05) for α=3\\alpha = 3 (α=5\\alpha = 5).Inset, maximum value of R T R_T as a function of the number of neurons in the network NNin a log-lin scale.", "The solid lines refer to fittings to the data, namelyR T M =3.09-5.60N -0.27 R^M_T = 3.09 - 5.60 N^{-0.27} for α=3\\alpha = 3;R T M =0.23+0.28log(N)R_T^M = 0.23 + 0.28 \\log (N) for α=5\\alpha = 5.", "R T R_T calculated after a time span T=1T = 1 (T=5)(T = 5)for α=3\\alpha = 3 (α=5)(\\alpha = 5).b) Amplitude scale β\\beta associated to the indicator P S P_S as a function of 1/N1/N.In the inset, β\\beta is reported as a function of α\\alpha for parametervalues associated to non chaotic dynamics.", "In the same rangethe exponent μ∼2.32\\mu \\sim 2.32 (not shown).", "The model parameters refer to thefluctuation driven case studied in Fig. .", "Inset is obtained with N=100N = 100Finally, in order to study the effect of the pulse shape on the finite amplitude behavior as measured by $R_T$ , we proceeded to calculate this indicator for various $\\alpha $ -values.", "As shown in Fig.", "REF , for increasing $\\alpha $ (corresponding to narrower peaks) the position of the maximum $\\Delta _0^M$ moves towards larger amplitudes.", "This effect can be explained by the fact that the maximal Lyapunov exponent decreases with $\\alpha $ (as shown in Fig.", "REF ) and therefore perturbations of bigger and bigger amplitudes are required to destabilize the system for vanishingly pulse width.", "Consistently also the parameter $\\beta $ associated to $P_S$ increases for increasing $\\alpha $ -values, as shown in the inset of Fig.", "REF (b).", "Figure: Finite size instabilities for fluctuation driven dynamics,for different pulse widths.", "(a) R T R_T as a function of the initial perturbation Δ 0 \\Delta _0, for α=2\\alpha = 2 (black circle),α=3\\alpha = 3 (red square), α=4\\alpha = 4 (blue down-triangle), α=5\\alpha = 5 (green up-triangle).The system size is fixed to N=400N=400.", "(b) Peak location in logarithmic scale logΔ 0 M \\log \\Delta _0^M as a function ofthe inverse pulse-width α\\alpha , for two sizes of the network: N=100N=100 (black circles) and N=400N=400 (red squares).The peak positions were found by fitting a quadratic function around the maximum of thefunction R T R_T in (a).", "R T R_T is calculated as described in the caption of Fig.", ".Remaining parameters as in Fig.", "." ], [ "Discussion", "We have investigated the dynamics and stability of current and fluctuation driven neural networks, the former (latter) have been realized as a purely excitatory (inhibitory) pulse coupled network of leaky integrate-and-fire (LIF) neurons with a sparse architecture.", "In particular, we considered random networks with a constant in-degree $K=20$ for any examined size.", "The excitatory network, despite being chaotic, reveals a low spiking variability.", "On the other hand, in the fluctuation driven case the variability is high for any considered pulse width and system size (CV $\\simeq 0.7 -1.0$ ).", "However, a different picture arises when studying the stability of infinitesimal perturbations: the system is chaotic for slow synapses and it becomes stable for sufficiently fast synaptic times ($\\le 4$ ms).", "Furthermore, a chaotic state for the inhibitory network is observable already at small connectivity $K \\sim O(10^1)$ contradicting what reported in [16], where the authors affirmed that a large connectivity is a prerequisite to observe chaotic motion in these models.", "The maximal Lyapunov exponent $\\lambda $ tends towards an asymptotic value for increasing system sizes with a power-law scaling.", "The exponent $\\eta $ associated to this scaling is different in the current (fluctuation) driven case, in particular $\\eta \\simeq 1$ ($\\eta \\simeq 1/4$ ) [30].", "In the fluctuation driven situation the exponent is the same in the chaotic and stable phases.", "The origin of the observed scaling demands for new theoretical analysis, similar to the one developed for spatio-temporal chaotic systems [43].", "Quite astonishingly even in the linearly stable regime an erratic evolution of the network is observable.", "A similar phenomenon has been already observed in several systems ranging from diffusively coupled chaotic maps to neural networks, and it has been identified as stable chaos [13].", "In this context, finite amplitude perturbations are responsible for the erratic behavior observed in the system.", "In diffusively coupled systems this nonlinear instabilities has been characterized in terms of the propagation velocity of the information and of suitable Finite Size Lyapunov Exponents (FSLEs) [37], [41].", "FSLEs have been previously employed in the context of fully coupled neural networks, where they revealed that the origin of the chaotic motion observed in two symmetrical coupled neural populations was due to collective chaos in the mean-field variables driving the single LIF neurons [28].", "In the context of randomly coupled systems the concept of propagation velocity on a lattice looses his sense, while FSLEs reveal serious problems in their numerical implementation.", "However, FSLEs clearly show also in our case that in the current driven case the observed instabilities have a purely linear origin, while in the fluctuation driven situation nonlinear mechanisms are present even when the system is chaotic.", "This analysis is confirmed by a novel indicator we have introduced, namely the local derivative $D(\\Delta )$ of the averaged logarithmic distance $< \\log \\Delta >$ between the reference and the perturbed trajectory.", "This quantity suffers less than the FSLE the trial to trial fluctuations, since it is based on an averaged profile.", "For the fluctuation driven case this indicator is larger than the maximal Lyapunov exponent at finite amplitudes and this effect is present for all the examined system sizes.", "The position of the peak in $D(\\Delta )$ seems not to be particularly influenced by the system size, while the peak itself broadens towards larger amplitudes for increasing $N$ .", "Unfortunately, all these indications cannot tell us if the nonlinear mechanisms are prevailing on the linear ones, but just that the nonlinear effects are present.", "To measure the influence of linear versus nonlinear effects on the system dynamics, novel indicators are required, similar to linear and nonlinear information velocities for diffusively coupled systems [13].", "As a final point we have studied nonlinear instabilities in linearly stable systems emerging in fluctuation driven inhibitory networks for sufficiently narrow postsynaptic currents.", "For the characterization of these instabilities we have employed the average finite amplitude growth rate $R_T(\\Delta _0)$ , measured after a finite time interval $T$ , analogously to what what done in [34], [41], and the probability $P_S(\\Delta _0)$ that an initial perturbation induces an exponential separation between the perturbed and the reference orbits, previoulsy introduced in [19].", "Both these indicators reveal the existence of instabilities associated to finite perturbations, in particular the characteristic amplitude scales associated to these indicators vanish in the thermodynamic limit as $1/N$ .", "Thus suggesting that instabilities in these systems can occur even for infinitesimal perturbations in clear contradiction with the fact that these systems are linearly stable at any system size, as revealed by the Lyapunov analysis.", "This contradiction has lead Monteforte & Wolf to conjecture in [19] that the thermodynamic limit and the limit of of vanishingly small perturbations do not commute in these models.", "Furtermore, we measure a logarithmic divergence with the system size of the peak height of $R_T(\\Delta _0)$ , suggesting that in the thermodynamic limit the value of these indicator will become infinite, similarly to what found in the high connectivity limit for a binary neuronal model in the balanced state [32] and for LIF with $\\delta $ -pulses in [19].", "However, in our study the connectivity remains finite and small in the limit $N \\rightarrow \\infty $ .", "Our opinion, based on the comparison of the indicators $D(\\Delta )$ and $R_T(\\Delta _0)$ performed in a fluctuation driven chaotic situation, is that the above results can be due to the fact that the dynamics considered for the estimation of $R_T(\\Delta _0)$ and $P_S(\\Delta _0)$ do not take place on the attractor of the system.", "This because the indicators are estimated at short times, without allowing the perturbed dynamics to relax onto the attractor.", "The development of new indicators is required to analyze more in depth the phenomenon of Stable Chaos in randomly connected networks." ], [ "Acknowledgments", "We thank A. Politi, S. Luccioli, and J. Berke for useful discussions.", "This work has been supported by the European Commission under the program “Marie Curie Network for Initial Training\", through the project N. 289146, “Neural Engineering Transformative Technologies (NETT)\".", "D. A.-G. would like also to acknowledge the partial support provided by “Departamento Adminsitrativo de Ciencia Tecnologia e Innovacion - Colciencias\" through the program “Doctorados en el exterior - 2013\".", "The LIF model is usually expressed in physical units as follows [45] $\\tau _m \\frac{dV}{d\\hat{t}} = -(V(\\hat{t}) - V_0) + R_m I_{ext} + \\tau _m\\hat{g} \\hat{E} (\\hat{t}) \\qquad ,$ where $R_m$ is the specific mebrane resistance, $\\tau _m$ the membrane time constant, and $V_0$ the resting potential.", "The transformation of the adimensional model (REF ) to (REF ) can be obtained by performing the following set of transformations $V_i=&v_i (V_{th}-V_0)+V_0 \\qquad R_m I_{ext}=&a (V_{th}-V_0)+V_0 \\\\\\hat{g}=&g(V_{th}-V_0)+V_0 \\qquad \\hat{t} =& t\\tau _m \\qquad ,$ where $V_{th}$ is the firing threshold value.", "Notice that $\\hat{\\alpha } = \\alpha /\\tau _m$ and $\\hat{E} = E /\\tau _m$ have the dimensionality of a frequency and $\\hat{g}$ of a potential.", "Realistic values for the introduced parameters, are $\\tau _m = 20$ ms, $V_0 = -60$ mV, $V_{th} = -50$ mV [46].", "The postsynaptic current rise times $1/\\hat{\\alpha }$ employed in this article range from 4 to 20 ms for inhibitory cells, while it is fixed to 2.22 ms for excitatory ones.", "Furthermore, the average neuronal firing rates are of order $\\simeq 50$ Hz ($\\simeq 6$ Hz) for excitatory (inhibitory) networks, which are quite reasonable values for pyramidal neurons (inter-neurons) of the cortex [47], [1], [48], [3]" ] ]	1403.0464
[ [ "Atmospheres and radiating surfaces of neutron stars" ], [ "Abstract The early 21st century witnesses a dramatic rise in the study of thermal radiation of neutron stars.", "Modern space telescopes have provided a wealth of valuable information which, when properly interpreted, can elucidate the physics of superdense matter in the interior of these stars.", "This interpretation is necessarily based on the theory of formation of neutron star thermal spectra, which, in turn, is based on plasma physics and on the understanding of radiative processes in stellar photospheres.", "In this paper, the current status of the theory is reviewed with particular emphasis on neutron stars with strong magnetic fields.", "In addition to the conventional deep (semi-infinite) atmospheres, radiative condensed surfaces of neutron stars and \"thin\" (finite) atmospheres are considered." ], [ "Introduction", "Neutron stars are the most compact of all stars ever observed: with a typical mass $M\\sim (1$ – $2)\\,M_\\odot $ , where $M_\\odot =2\\times 10^{33}$ g is the solar mass, their radius is $R\\approx 10$ – 13 km.", "The mean density of such star is $\\sim 10^{15}$ g cm$^{-3}$, i.e., a few times the typical density of a heavy atomic nucleus $\\rho _0=2.8\\times 10^{14}$ g cm$^{-3}$.", "The density at the neutron-star center can exceed $\\rho _0$ by an order of magnitude.", "Such matter cannot be obtained in a laboratory, and its properties still remain to be clarified.", "Even its composition is not completely known, because neutron stars, despite their name, consist not only of neutrons.", "There are a variety of theoretical models to describe neutron-star matter (see [1] and references therein), and a choice in favor of one of them requires an analysis and interpretation of relevant observational data.", "Therefore, observational manifestations of the neutron stars can be used for verification of theoretical models of matter in extreme conditions [2].", "Conversely, the progress in studying the extreme conditions of matter provides prerequisites for construction of neutron-star models and adequate interpretation of their observations.", "A more general review of these problems is given in [3].", "In this paper, I will consider more closely one of them, namely the formation of thermal electromagnetic radiation of neutron stars.", "Neutron stars are divided into accreting and isolated ones.", "The former ones accrete matter from outside, while an accretion onto the latter ones is negligible.", "There are also transiently accreting neutron stars (X-ray transients), whose active periods (with accretion) alternate with quiescent periods, during which the accretion almost stops.", "The bulk of radiation from the accreting neutron stars is due to the matter being accreted, which forms a circumstellar disk, accretion flows, and a hot boundary layer at the surface.", "At contrast, a significant part of radiation from isolated neutron stars, as well as from the transients in quiescence, appear to originate at the surface or in the atmosphere.", "To interpret this radiation, it is important to know the properties of the envelopes that contribute to the spectrum formation.", "On the other hand, comparison of theoretical predictions with observations may be used to deduce these properties and to verify theoretical models of the dense magnetized plasmas that constitute the envelopes.", "We will consider the outermost envelopes of the neutron stars – their atmospheres.", "A stellar atmosphere is the plasma layer in which the electromagnetic spectrum is formed and from which the radiation escapes into space without significant losses.", "The spectrum contains a valuable information on the chemical composition and temperature of the surface, intensity and geometry of the magnetic field, as well as on the stellar mass and radius.", "In most cases, the density in the atmosphere grows with increasing depth gradually, without a jump, but stars with a very low temperature or a superstrong magnetic field can have a solid or liquid surface.", "Formation of the spectrum with presence of such a surface will also be considered in this paper." ], [ "Masses and radii", "The relation between mass $M$ and radius $R$ of a star is given by a solution of the hydrostatic equilibrium equation for a given equation of state (EOS), that is the dependence of pressure $P$ on density $\\rho $ and temperature $T$ , along with the thermal balance equation.", "The pressure in neutron star interiors is mainly produced by highly degenerate fermions with Fermi energy $\\epsilon _\\mathrm {F}\\gg k_\\mathrm {B}T$ ($k_\\mathrm {B}$ is the Boltzmann constant), therefore one can neglect the $T$ -dependence in calculations of $R(M)$ .", "For the central regions of typical neutron stars, where $\\rho \\gtrsim \\rho _0$ , the EOS and even composition of matter is not well known because of the lack of the precise relativistic many-body theory of strongly interacting particles.", "Instead of the exact theory, there are many approximate models, which give a range of theoretical EOSs and, accordingly, $R(M)$ relations (see, e.g., Chapt.", "6 of [1]).", "For a star to be hydrostatically stable, the density at the stellar center has to increase with increasing mass.", "This condition is satisfied in a certain interval $M_\\mathrm {min}<M<M_\\mathrm {max}$ .", "The minimum neutron-star mass is rather well established, $M_\\mathrm {min}\\approx 0.1\\,M_\\odot $ [4].", "The maximum mass until recently was allowed to lie in a wide range $M_\\mathrm {max}\\sim (1.5$ – $2.5)\\,M_\\odot $ by competing theories (see, e.g., Table 6.1 in Ref.", "[1]), but the discoveries of neutron stars with masses $M=1.97\\pm 0.04\\,M_\\odot $ [5] and $2.01\\pm 0.04\\,M_\\odot $ [6] showed that $M_\\mathrm {max}>2\\,M_\\odot $ .", "Simulations of formation of neutron stars [7], [8] show that $M$ , as a rule, exceeds $M_\\odot $ , the most typical values being in the range (1.2 – $1.6)\\,M_\\odot $ .", "Observations generally agree with these conclusions.", "Masses of several pulsars in double compact-star systems are known with a high accuracy ($\\lesssim 1$ %) due to the measurements of the General Relativity (GR) effects on their orbital parameters.", "All of them lie in the interval from $1.3\\,M_\\odot $ to $2.0\\,M_\\odot $ [9], [5], [6].", "Masses of other neutron stars that have been measured with an accuracy better than 10% cover the range $M_\\odot \\lesssim M \\lesssim 2\\,M_\\odot $ [1], [10].", "Were radius $R$ and mass $M$ known precisely for at least a single neutron star, it would probably ensure selecting one of the nuclear-matter EOSs as the most realistic one.", "However, the current accuracy of measurements of neutron star radii leaves much to be desired." ], [ "Magnetic fields", "Most of the known neutron stars possess strong magnetic fields, unattainable in the terrestrial laboratories.", "Gnedin and Sunyaev [11] pointed out that spectra of such stars can contain the resonant electron-cyclotron line.", "Its detection allows one to obtain magnetic field $B$ by measurement of the cyclotron frequency $\\omega _\\mathrm {c}=eB/(m_\\mathrm {e}c)$ , where $m_\\mathrm {e}$ and $(-e)$ are the electron mass and charge, and $c$ is the speed of light in vacuum (here and hereafter we use the Gaussian system of units).", "The discovery of the cyclotron line in the spectrum of the X-ray pulsar in the binary system Hercules X-1 [12] gave a striking confirmation of this idea.", "About 20 accreting X-ray pulsars are currently known to reveal the electron cyclotron line and sometimes several its harmonics at energies of tens keV, corresponding to $B\\approx (1$ – $4)\\times 10^{12}$ G (e.g., [13], [14], [15], [16]).", "An alternative interpretation of the observed lines was suggested in [17].", "It assumes an anisotropic distribution of electron velocities in a collisionless shock wave with large Lorentz factors (the ratios of the total electron energy to $m_\\mathrm {e}c^2=511$ keV), $\\gamma _\\mathrm {r}\\sim 40$ .", "The radiation frequency of such electrons strongly increases because of the relativistic Doppler effect, which enables an explanation of the observed position of the line by a much weaker field than in the conventional interpretation.", "It was noted in Ref.", "[18] that the small width of the lines (from one to several keV [13]) is difficult to accommodate in this model.", "It also leaves unexplained, why the position of the line is usually almost constant.", "For example, the measured cyclotron energy of the accreting X-ray pulsar A 0535+26 remains virtually constant while its luminosity changes by two orders of magnitude [19].", "On the other hand, most X-ray pulsars do exhibit a dependence, albeit weak, of the observed line frequency on luminosity [20].", "In order to explain this dependence, a model was suggested in [21], assuming that the cyclotron lines are formed by reflection from the stellar surface, irradiated by the accretion column.", "When luminosity increases, the bulk of reflection occurs at lower magnetic latitudes, where the field is weaker than at the pole, therefore the cyclotron frequency becomes smaller.", "This model, however, does not explain the cases where the observed frequency increases with luminosity and, as noted in [22], it does not reproduce X-ray pulses at large luminosities.", "A quantitative description of all observed dependences of the cyclotron frequency on luminosity is developed in Ref.", "[20], based on a physical model of cyclotron-line formation in the accretion column.", "The height of the region above the surface, where the lines are formed, $h\\sim (10^{-3}$ – $10^{-1})R$ , correlates with luminosity, the correlation being positive or negative depending on the luminosity value.", "Then the line is centered at the frequency $\\omega _\\mathrm {c}/(1+h/R)^3$ , where $\\omega _\\mathrm {c}$ is the cyclotron frequency at the base of the accretion column.", "In [22], variations of a polar cap diameter and a beam pattern were additionally taken into account, which has allowed the author to explain variations in the width and depth of the observed lines in addition to their frequencies.", "When cyclotron features are not identified in the spectrum, one has to resort to indirect estimates of the magnetic field.", "For the isolated pulsars, the most widely used estimate is based on the expression $B \\approx 3.2\\times 10^{19} \\,C\\,\\sqrt{\\mathcal {P}\\dot{\\mathcal {P}}}\\textrm {~~G},$ where $\\mathcal {P}$ is the period in seconds, $\\dot{\\mathcal {P}}$ is the period time derivative, and $C$ is a coefficient, which depends on stellar parameters.", "For the rotating magnetic dipole in vacuo [23] $C=R_6^{-3}\\,(\\sin \\alpha )^{-1}\\,\\sqrt{I_{45}}$ , where $R_6\\equiv R/(10^6\\mbox{~cm})$ , $I_{45}$ is the moment of inertia in units of $10^{45}$ g cm$^2$ , and $\\alpha $ is the angle between the magnetic and rotational axes.", "In this case Eq.", "(REF ) gives the magnetic field strength at the pole.", "If $M\\approx (1$ – 2) $M_\\odot $ , then $R_6\\approx 1.0$ – 1.3 and $I_{45}\\approx 1$ – 3 (see [1]).", "For estimates, one usually sets $C=1$ in Eq.", "(REF ) (e.g., [24]).", "A real pulsar strongly differs from a rotating magnetic dipole, because its magnetosphere is filled with plasma, carrying electric charges and currents (see reviews [25], [26], [27], [28] and recent papers [29], [30], [31]).", "According to the model by Beskin et al.", "[32], [33], the magnetodipole radiation is absent beyond the magnetosphere, while the slowdown of rotation is provided by the current energy losses.", "However, the the relation between $B$ and $\\mathcal {P}\\dot{\\mathcal {P}}$ remains similar.", "Results of numerical simulations of plasma behavior in the pulsar magnetosphere can be approximately described by Eq.", "(REF ) with $C\\approx 0.8R_6^{-3}\\,(1+\\sin ^2\\alpha )^{-1/2}\\,\\sqrt{I_{45}}$ [34].", "As shown in [28], this result does not contradict to the model [32], [33].", "Magnetic fields of the ordinary radio pulsars are distributed near $B\\sim 10^{12}$ G [35], the “recycled” millisecond pulsars have $B\\sim (10^{8}$ – $10^{10})$ G [35], [36], [37], and the fields of magnetars much exceed $10^{13}$ G [38], [39].", "According to the most popular point of view, anomalous X-ray pulsars (AXPs) and soft gamma repeaters (SGRs) [38], [39], [40], [41], [42] are magnetars.", "For these objects, the estimate (REF ) most often (although not always) gives $B\\sim 10^{14}$ G, but in order to explain their energy balance, magnetic fields reaching up to $B\\sim 10^{16}$ – $10^{17}$ G in the core at the birth of the star are considered (see [43] and references therein).", "Numerical calculations [44] show that magnetorotational instability in the envelope of a supernova, that is a progenitor of a neutron star, can give rise to nonstationary magnetic fields over $10^{15}$ G. It is assumed that in addition to the poloidal magnetic field at the surface, the magnetars may have much stronger toroidal magnetic field embedded in deeper layers [45], [46].", "Indeed, for a characteristic poloidal component $B_\\mathrm {pol}$ of a neutron-star magnetic field to be stable, a toroidal component $B_\\mathrm {tor}$ must be present, such that, by order of magnitude, $B_\\mathrm {pol}\\lesssim B_\\mathrm {tor} \\lesssim 10^{16}\\mbox{~G}\\sqrt{B_\\mathrm {pol}/(10^{13}\\mbox{~G})}$ [47].", "Meanwhile, there is increasing evidence for the absence of a clear distinction between AXPs and SGRs [48], as well as between these objects and other neutron stars [49], [41], [42].", "There has even appeared the paradoxical name “a low-field magnetar,” applied to those AXPs and SGRs that have $B\\ll 10^{14}$ G (e.g., [50], [51], and references therein).", "For the majority of isolated neutron stars, the magnetic-field estimate (REF ) agrees with other data (e.g., with observed properties of the bow shock nebula in the vicinity of the star [52]).", "For AXPs and SGRs, however, one cannot exclude alternative models, which do not involve superstrong fields but assume weak accretion on a young neutron star with $B\\sim 10^{12}$ G from a circumstellar disk, which could remain after the supernova burst [53], [54], [56], [55].", "There is also a “drift model”, which suggests that the observed AXP and SGR periods equal not to rotation periods but to periods of drift waves, which affect the magnetic-lines curvature and the direction of radiation in the outer parts of magnetospheres of neutron stars with $B\\sim 10^{12}$ G [57], [58].", "Another model suggests that the AXPs and SGRs are not neutron stars at all, but rather massive ($M>M_\\odot $ ) rapidly rotating white dwarfs with $B\\sim 10^8$ – $10^9$ G ([59] and references therein).", "The measured neutron-star magnetic fields are enormous by terrestrial scales, but still far below the theoretical upper limit.", "An order-of-magnitude estimate of this limit can be obtained by equating the gravitational energy of the star to its electromagnetic energy [60].", "For neutron stars, such estimate gives the limiting field $B_\\mathrm {max}\\sim 10^{18}$ – $10^{19}$ G [61].", "Numerical simulations of hydrostatic equilibrium of magnetized neutron stars show that $B_\\mathrm {max}\\lesssim 10^{18}$ G [62], [63], [64], [65].", "Still stronger magnetic fields imply so intense electric currents that their interaction would disrupt the star.", "Note in passing that the highest magnetic field that can be accommodated in quantum electrodynamics (QED) is, by order of magnitude, $[m_\\mathrm {e}^2c^3/(e\\hbar )]\\exp (\\pi ^{3/2}/\\sqrt{\\alpha _\\mathrm {f}})\\approx 10^{42}$ G [66], where $\\alpha _\\mathrm {f}= e^2/(\\hbar c)\\approx 1/137$ is the fine structure constant, and $\\hbar $ is the Planck constant divided by $2\\pi $ .", "We will see below that magnetic fields $B\\gtrsim 10^{11}$ G strongly affect the most important characteristics of neutron-star envelopes.", "These effects are particularly pronounced at radiating surfaces and in atmospheres, which are the main subject of the present review." ], [ "General Relativity effects", "The significance of the GR effects for a star is quantified by the compactness parameter $x_\\mathrm {g}=r_g/R,$ where $r_g=2GM/c^2\\approx 2.95\\,M/M_\\odot \\textrm { km}$ is the Schwarzschild radius, and $G$ is the gravitational constant.", "The compactness parameter of a typical neutron star lies between 1/5 and 1/2, that is not small (for comparison, the Sun has $x_\\mathrm {g}=4.24\\times 10^{-6}$ ).", "Hence, the GR effects are not negligible.", "Two important consequences follow: first, the quantitative theory of neutron stars must be wholly relativistic; second, observations of neutron stars open up a unique opportunity for measuring the GR effects and verification of the GR predictions.", "Figure: Left panel: an illustration of thegravitational light-bending near a neutron star;𝐧\\mathbf {n} is the normal to the surface at a radiatingpoint, 𝐤\\mathbf {k} is the wave vector of an emitted ray in thelocal reference frame, 𝐤 ' \\mathbf {k^{\\prime }} is the wave vector in theobserver's reference frame.", "In addition, the stellarrotation vector Ω\\mathbf {\\Omega } and magnetic moment𝐦\\mathbf {m} are shown.", "The angles formed by the rotation axiswith the magnetic moment (α\\alpha ) and with the line ofsight (ζ\\zeta ) are indicated.Middle panel: wave vectors 𝐤\\mathbf {k}, 𝐤 ' \\mathbf {k^{\\prime }}, andthe magnetic field vector 𝐁\\mathbf {B} in the local referenceframe (x n y n z n )(x_\\mathrm {n} y_\\mathrm {n} z_\\mathrm {n}) with thezz-axis along 𝐧\\mathbf {n} and the xx-axis along theprojection of 𝐁\\mathbf {B} on the surface; θ n \\theta _\\mathrm {n}is the angle between 𝐁\\mathbf {B} and 𝐧\\mathbf {n}, θ k \\theta _kand θ\\theta are the angles between the wave vectors and thenormal, θ B \\theta _B is the angle between the ray andthe magnetic field, and ϕ k \\varphi _k is the azimuth.Right panel: vectors 𝐧\\mathbf {n}, 𝐤\\mathbf {k},𝐤 ' \\mathbf {k^{\\prime }}, and 𝐦 ^≡𝐦/\|𝐦\|\\hat{\\mathbf {m}}\\equiv \\mathbf {m}/\|\\mathbf {m}\| inthe coordinate system (xyz)(xyz) with the zz-axis along theline of sight and the xx-axis along the projection of𝐦\\mathbf {m} on the picture plane; θ m \\theta _\\mathrm {m} is theangle between 𝐦\\mathbf {m} and the line of sight, γ\\gamma isthe angle between 𝐧\\mathbf {n} and 𝐦\\mathbf {m}, and ϕ\\varphi is the azimuth.In GR, gravity at the stellar surface is determined by the equation $g=\\frac{GM}{R^{2}\\,\\sqrt{1-x_\\mathrm {g}}} \\approx \\frac{1.328\\times 10^{14}}{\\sqrt{1-x_\\mathrm {g}}}\\,\\frac{M/M_\\odot }{\\,R_6^{2}}\\textrm { cm s}^{-2}.$ Stellar hydrostatic equilibrium is governed by the Tolman-Oppenheimer-Volkoff equation (corrections due to the rotation and magnetic fields are negligible for the majority of neutron stars): $\\frac{\\mathrm {d}P}{\\mathrm {d}r} =- \\left( 1 + \\frac{P}{\\rho c^2} \\right)\\,\\left( 1 + \\frac{4\\pi r^3 P}{M_r c^2} \\right)\\,\\left( 1 - \\frac{2GM_r}{rc^2} \\right)^{-1/2},$ where $r$ is the radial coordinate measured from the stellar center, and $M_r$ is the mass inside a sphere of radius $r$ .", "The photon frequency, which equals $\\omega $ in the local inertial reference frame, undergoes a redshift to a smaller frequency $\\omega _\\infty $ in the remote observer's reference frame.", "Therefore a thermal spectrum with effective temperature $T_\\mathrm {eff}$ , measured by the remote observer, corresponds to a lower effective temperature $T_\\mathrm {eff}^\\infty = T_\\mathrm {eff}/ (1+z_g),$ where $z_g \\equiv \\omega /\\omega _\\infty -1 = (1-x_\\mathrm {g})^{-1/2} -1$ is the redshift parameter.", "Here and hereafter the symbol $\\infty $ indicates that the given quantity is measured at a large distance from the star and can differ from its value near the surface.", "Along with the radius $R$ that is determined by the equatorial length $2\\pi R$ in the local reference frame, one often considers an apparent radius for a remote observer, $R_\\infty = R \\,(1+z_g).$ With decreasing $R$ , $z_g$ increases so that the apparent radius has a minimum, $\\min R_\\infty \\approx 12$ – 14 km ([1], Chapt. 6).", "The apparent photon luminosity $L_\\mathrm {ph}^\\infty $ and the luminosity in the stellar reference frame $L_\\mathrm {ph}$ are determined by the Stefan-Boltzmann law $L_\\mathrm {ph}^\\infty =4\\pi \\sigma _\\mathrm {SB}\\,R_\\infty ^2\\,(T_\\mathrm {eff}^\\infty )^4,\\quad L_\\mathrm {ph} = 4\\pi \\sigma _\\mathrm {SB}\\,R^2 T_\\mathrm {eff}^4$ with $\\sigma _\\mathrm {SB}= \\pi ^2k_\\mathrm {B}^4/(60\\hbar ^3 c^2)$ .", "According to (REF ) – (REF ), they are interrelated as $L_\\mathrm {ph}^\\infty = (1-x_\\mathrm {g})\\, L_\\mathrm {ph} =L_\\mathrm {ph}/(1+z_g)^2.$ In the absence of the perfect spherical symmetry, it is convenient to define a local effective surface temperature $T_\\mathrm {s}$ by the relation $F_\\mathrm {ph}(\\theta ,\\varphi ) = \\sigma _\\mathrm {SB}T_\\mathrm {s}^4,$ where $F_\\mathrm {ph}$ is the local radial flux density at the surface point, determined by the polar angle ($\\theta $ ) and azimuth ($\\varphi $ ) in the spherical coordinate system.", "Then $L_\\mathrm {ph} =\\int _0^{\\pi }\\sin \\theta \\,\\mathrm {d}\\theta \\int _0^{2\\pi }\\mathrm {d}\\varphi \\,R^2 F_\\mathrm {ph}(\\theta ,\\varphi )\\,.$ The same relation connects the apparent luminosity $L_\\mathrm {ph}^\\infty $ (REF ) with the apparent flux $F_\\mathrm {ph}^\\infty =\\sigma _\\mathrm {SB}\\,(T_\\mathrm {s}^\\infty )^4$ in the remote system, in accord with the relation $T_\\mathrm {s}^\\infty = T_\\mathrm {s}/(1+z_\\mathrm {g})$ analogous to (REF ).", "The expressions (REF ), (REF ) and (REF ) agree with the concepts of the light ray bending and time dilation near a massive body.", "If the angle between the wave vector $\\mathbf {k}$ and the normal to the surface $\\mathbf {n}$ at the emission point is $\\theta _k$ , then the observer receives a photon whose wave vector $\\mathbf {k^{\\prime }}$ makes an angle $\\theta > \\theta _k$ with $\\mathbf {n}$ (Fig.", "REF ).", "The rigorous theory of the influence of the light bending near a star on its observed spectrum has been developed in [67] and cast in a convenient form in [68], [69].", "The simple approximation [70] $\\cos \\theta _k = x_\\mathrm {g}+ (1-x_\\mathrm {g})\\cos \\theta $ is applicable at $x_\\mathrm {g}<0.5$ with an error within a few percent.", "At $\\cos \\theta _k < x_\\mathrm {g}$ , Eq.", "(REF ) gives $\\theta > \\pi /2$ , as if the observer looked behind the neutron-star horizon.", "In particular, for a star with a dipole magnetic field and a sufficiently large inclination angle $\\theta _\\mathrm {m}$ of the dipole moment vector $\\mathbf {m}$ to the line of site, the observer can see the two opposite magnetic poles at once.", "Clearly, such effects should be taken into account while comparing theoretical neutron-star radiation models with observations.", "Let $I_\\omega $ be the specific intensity per unit circular frequency (if $I_\\nu $ is the specific intensity per unit frequency, then $I_\\omega =I_\\nu /(2\\pi )$ ; see [71]).", "A contribution to the observed radiation flux density from a small piece of the surface $\\mathrm {d}\\mathcal {A}$ in the circular frequency interval $[\\omega ,\\omega +\\mathrm {d}\\omega ]$ equals [72], [73] $\\mathrm {d}F_{\\omega _\\infty }^\\infty =I_\\omega (\\mathbf {k})\\,\\cos \\theta _k\\left\| \\frac{\\mathrm {d}\\cos \\theta _k}{\\mathrm {d}\\cos \\theta }\\right\|\\,\\frac{\\mathrm {d}\\mathcal {A}}{D^2}\\,(1-x_\\mathrm {g})\\,\\mathrm {d}\\omega ,$ where $\\mathrm {d}\\omega =(1+z_g)\\,\\mathrm {d}\\omega _\\infty $ .", "Here and hereafter we assume that the rotational velocity of the patch $\\mathrm {d}\\mathcal {A}$ is much smaller than the speed of light.", "If this condition is not satisfied, then the right-hand side of Eq.", "(REF ) should be multiplied by $(\\cos \\tilde{\\theta }_k/\\cos \\theta _k)^4$ , where $\\tilde{\\theta }_k$ is the angle between the surface normal and the wave vector in the reference frame, comoving with the patch $\\mathrm {d}\\mathcal {A}$ at the moment of radiation [72], [73].", "For a spherical star, Eqs.", "(REF ), (REF ) give $F_{\\omega _\\infty }^\\infty =(1-x_\\mathrm {g})^{3/2}\\frac{R^2}{D^2}\\int I_\\omega (\\mathbf {k};\\theta ,\\varphi ) \\,\\cos \\theta _k \\sin \\theta \\,\\mathrm {d}\\theta \\,\\mathrm {d}\\varphi ,$ where the integration is restricted by the condition $\\cos \\theta _k>0$ .", "The magnetic field is also distorted by the space curvature in the GR.", "For the uniform and dipole fields, this distortion is described by Ginzburg & Ozernoi [74].", "In the dipole field case, the magnetic vector is $\\mathbf {B} = B_\\textrm {p}\\,(\\mathbf {n}\\cdot \\hat{\\mathbf {m}})\\,\\mathbf {n}+ B_\\textrm {eq}\\,\\big [(\\mathbf {n}\\cdot \\hat{\\mathbf {m}})\\,\\mathbf {n}- \\hat{\\mathbf {m}}\\big ],$ where $\\hat{\\mathbf {m}}=\\mathbf {m}/\|\\mathbf {m}\|$ is the magnetic axis direction, $B_\\textrm {eq}$ and $B_\\textrm {p}$ are the equatorial and polar field strengths, respectively, and their ratio equals $\\frac{B_\\textrm {eq}}{B_\\textrm {p}}= \\frac{x_\\mathrm {g}^2/2 - (1-x_\\mathrm {g}) \\ln (1-x_\\mathrm {g}) - x_\\mathrm {g}}{[\\ln (1-x_\\mathrm {g})+x_\\mathrm {g}+ x_\\mathrm {g}^2/2] \\sqrt{1-x_\\mathrm {g}}}.$ In the limit of flat geometry ($x_\\mathrm {g}\\rightarrow 0$ ) $B_\\textrm {eq} \\rightarrow B_\\textrm {p}/2$ , but in general $B_\\textrm {eq} / B_\\textrm {p} > 1/2 + x_\\mathrm {g}/8$ .", "Muslimov & Tsygan [75] obtained expansions of the components of a poloidal magnetic field vector $\\mathbf {B}$ over the scalar spherical harmonics near a static neutron star beyond the dipole approximation.", "Equations (REF ) and (REF ) are a particular case of this expansion.", "Petri [76] developed a technique of expansion of electromagnetic fields around a rotating magnetized star over vector spherical harmonics, which allows one to find a solution of the Maxwell equations in the GR for an arbitrary multipole component of the magnetic field.", "In this case, the solutions for a nonrotating star in the GR [75] and for a rotating dipole in the flat geometry [23] are reproduced as particular cases." ], [ "Measuring masses and radii by thermal\nspectrum", "Information on the mass and radius of a neutron star can be obtained from its thermal spectrum.", "To begin with, let us consider the perfect blackbody radiation whose spectrum is described by the Planck function$\\mathcal {B}_{\\omega ,T}$ is the specific intensity of nonpolarized blackbody radiation related to the circular frequency (see [71]).", "$\\mathcal {B}_{\\omega ,T} =\\frac{\\hbar \\omega ^3}{4\\pi ^3c^2}\\,\\frac{1}{\\exp [\\hbar \\omega /k_\\mathrm {B}T]-1},$ and neglect interstellar absorption and nonuniformity of the surface temperature distribution.", "The position of the spectral maximum $\\hbar \\omega _\\mathrm {max}=2.8k_\\mathrm {B}T$ gives us the effective temperature $T_\\mathrm {eff}^\\infty $ , and the measured intensity gives the total flux density $F_\\mathrm {bol}$ that reaches the observer.", "If the star is located at distance $D$ , then its apparent photon luminosity is $L_\\mathrm {ph}^\\infty =4\\pi D^2 F_\\mathrm {bol}$ , and Eq.", "(REF ) yields $R_\\infty $ .", "In reality, comparison of theoretical and measured spectra depends on a larger number of parameters.", "First, the spectrum is modified by absorption in the interstellar matter.", "The effect of the interstellar gas on the X-ray part of the spectrum is approximately described by factor $\\exp [-(N_\\mathrm {H}/10^{21}\\mbox{~cm}^{-2})\\,(\\hbar \\omega /0.16\\mbox{~keV})^{-8/3}]$ , where $N_\\mathrm {H}$ is the hydrogen column density on the line of sight [77].", "Thus one can evaluate $N_\\mathrm {H}$ from an analysis of the spectrum.", "If $D$ is unknown, one can try to evaluate it assuming a typical interstellar gas density for the given Galaxy region and using $D$ as a fitting parameter.", "Second, the temperature distribution can be nonuniform over the stellar surface.", "For example, at contrast to the cold poles of the Earth, the pulsars have heated regions near their magnetic poles, “hot polar caps.” The polar caps of accreting neutron stars with strong magnetic fields are heated by matter flow from a companion star through an accretion disk and accretion column (see [78], [79] and references therein).", "The polar caps of isolated pulsars and magnetars are heated by the current of charged particles, created in the magnetosphere and accelerated by the electric field along the magnetic field lines (see the reviews [25], [80], [28], papers [81], [82], and references therein).", "The thermal spectrum of such neutron stars is sometimes represented as consisting of two components, one of them being related to the heated region and the other to the rest of the surface, each with its own value of the effective temperature and effective apparent radius of the emitting area (e.g., [83]).", "Besides, variable strength and direction of the magnetic field over the surface affect the thermal conductivity of the envelope.", "Hence, the temperature $T_\\mathrm {s}$ of a cooling neutron star outside the polar regions is also nonuniform (see, e.g., [84], [85]).", "Finally, a star is not a perfect blackbody, therefore its radiation spectrum differs from the Planck function.", "Spectral modeling is a complex task, which includes solving equations of hydrostatic equilibrium, energy balance, and radiative transfer (below we will consider it in more detail).", "Coefficients of these equations depend on chemical composition of the atmosphere, effective temperature, gravity, and magnetic field.", "Making different assumptions about the chemical composition, $M$ , $R$ , $T_\\mathrm {eff}$ , and $B$ values, and about distributions of $T_\\mathrm {s}$ and $\\mathbf {B}$ over the surface, one obtains different model spectra.", "Comparison of these spectra with the observed spectrum yields an evaluation of acceptable values of the parameters.", "With the known shape of the spectrum, one can calculate $F_\\mathrm {bol}$ and evaluate $R_\\infty $ using Eq.", "(REF ).", "Identification of spectral features may provide $z_g$ .", "A simultaneous evaluation of $z_g$ and $R_\\infty $ allows one to calculate $M$ from Eqs.", "(REF ), (REF ), (REF ), and (REF ).", "This method of mass and radius evaluation requires a reliable theoretical description of the envelopes that affect the surface temperature and radiation spectrum." ], [ "Neutron-star envelopes", "Not only the superdense core of a neutron star, but also the envelopes are mostly under conditions unavailable in the laboratory.", "By the terrestrial standards, they are characterized by superhigh pressures, densities, temperatures, and magnetic fields.", "The envelopes differ by their composition, phase state, and their role in the evolution and properties of the star.", "In the deepest envelopes, just above the core of a neutron star, matter forms a neutron liquid with immersed atomic nuclei and electrons.", "In these layers, the neutrons and electrons are strongly degenerate, and the nuclei are neutron-rich, that is, their neutron number can be several times larger than the proton number, so that only the huge pressure keeps such nuclei together.", "Electrostatic interaction of the nuclei is so strong that they are arranged in a crystalline lattice, which forms the solid stellar crust.", "There can be a mantle between the crust and the core (though not all of the modern models of the dense nuclear matter predict its existence).", "Atomic nuclei in the mantle take exotic shapes of extended cylinders or planes [86].", "Such matter behaves like liquid crystals [87].", "The neutron-star crust is divided into the inner and outer parts.", "The outer crust is characterized by the absence of free neutrons.", "The boundary lies at the critical neutron-drip density $\\rho _\\mathrm {nd}$ .", "According to current estimates [88], $\\rho _\\mathrm {nd}=4.3\\times 10^{11}$ g cm$^{-3}$.", "With decreasing ion density $n_\\mathrm {i}$ , their electrostatic interaction weakens, and finally a Coulomb liquid becomes thermodynamically stable instead of the crystal.", "The position of the melting boundary, which can be called the bottom of the neutron-star ocean, depends on temperature and chemical composition of the envelope.", "If all the ions in the Coulomb liquid have the same charge $Ze$ and mass $m_\\mathrm {i}= Am_\\mathrm {u}$ , where $m_\\mathrm {u}=1.66\\times 10^{-24}$ g is the atomic mass unit, and if the magnetic field is not too strong, then ion dynamics is determined only by the Coulomb coupling constant $\\Gamma _{\\mathrm {Coul}}$ , that is the typical electrostatic to thermal energy ratio for the ions: $\\Gamma _{\\mathrm {Coul}}= \\frac{(Ze)^2}{a_\\mathrm {i}k_\\mathrm {B}T}= \\frac{22.75\\,Z^2}{T_6}\\,\\left(\\frac{\\rho _6}{A}\\right)^{1/3},$ where $a_\\mathrm {i}=(4\\pi n_\\mathrm {i}/3)^{-1/3}$ , $T_6\\equiv T/(10^6$ K) and $\\rho _6\\equiv \\rho /(10^6$ g cm$^{-3}$).", "Given the strong degeneracy, the electrons are often considered as a uniform negatively charged background.", "In this model, the melting occurs at $\\Gamma _{\\mathrm {Coul}}=175$ [89].", "However, the ion-electron interaction and quantizing magnetic field can shift the melting point by tens percent [89], [90].", "The strong gravity drives rapid separation of chemical elements [91], [92], [93], [94], [95].", "Results of Refs.", "[93], [94], [95] can be combined to find that the characteristic sedimentation time for the impurity ions with mass and charge numbers $A^{\\prime }$ and $Z^{\\prime }$ (that is the time at which the ions pass the pressure scale height $P/\\rho g$ ) in the neutron-star ocean is $t_\\mathrm {sed}\\approx \\frac{46\\,Z^{2.9}{(Z^{\\prime })}^{0.3}A^{-1.8}}{A^{\\prime }-AZ^{\\prime }/Z+\\Delta _T+\\Delta _\\mathrm {C}}\\,\\frac{\\rho _6^{1.3}}{g_{14}^2 T_6^{0.3}} \\mbox{~~days},$ where $g_{14}\\equiv g/(10^{14}$ cm s$^{-2})\\sim $ 1 – 3, $\\Delta _T$ is a thermal correction to the ideal degenerate plasma model [92], [94], and $\\Delta _\\mathrm {C}$ is an electrostatic (Coulomb) correction [94], [95].", "The Coulomb correction $\\Delta _\\mathrm {C}\\sim 10^{-3}-10^{-2}$ dominates in strongly degenerate neutron-star envelopes (at $\\rho \\gtrsim 10^3$ g cm$^{-3}$), and at smaller densities $\\Delta _T\\gtrsim \\Delta _\\mathrm {C}$ .", "Ions with larger $A/Z$ ratios settle faster, while among ions with equal $A/Z$ the heavier ones settle down faster [92], [94], [95].", "It follows from (REF ) that $t_\\mathrm {sed}$ is small compared with the known neutron-star ages, therefore neutron-star envelopes consist of chemically pure layers separated by transition bands of diffusive mixing.", "Especially important is the thermal blanketing envelope that governs the flux density $F_\\mathrm {ph}$ radiated by a cooling star with a given internal temperature $T_\\mathrm {int}$ .", "$F_\\mathrm {ph}$ is mainly regulated by the thermal conductivity in the “sensitivity strip” [96], [97], which plays the role of a “bottleneck” for the heat leakage.", "Position of this strip depends on the stellar parameters $M$ , $R$ , $T_\\mathrm {int}$ , magnetic field, and chemical composition of the envelope.", "Since the heat transport across the magnetic field is hampered, the depth of the sensitivity strip can be different at different places of a star with a strong magnetic field: it lies deeper at the places where the magnetic field is more inclined to the surface [98].", "As a rule, the sensitivity strip embraces the deepest layer of the ocean and the upper part of the crust and lies in the interval $\\rho \\sim 10^5$ – $10^9$ g cm$^{-3}$." ], [ "Atmosphere", "With decreasing density, the ion electrostatic energy and electron Fermi energy eventually become smaller than the kinetic ion energy.", "Then the degenerate Coulomb liquid gives way to a nondegenerate gas.", "The outer gaseous envelope of a star constitutes the atmosphere.", "In this paper, we will consider models of quasistationary atmospheres.", "They describe stellar radiation only in the absence of intense accretion, since otherwise it is formed mainly by an accretion disk or by flows of infalling matter.", "It is important that the sensitivity strip, mentioned in § REF , always lies at large optical depths.", "Therefore radiative transfer in the atmosphere almost does not affect the full thermal flux, so that one can model a spectrum while keeping $F_\\mathrm {ph}$ determined and $T_\\mathrm {s}$ from a simplified model of heat transport in the atmosphere.", "Usually such model is based on the Eddington approximation (e.g., [99]).", "Shibanov et al.", "[100] verified the high accuracy of this approximation for determination of the full thermal flux from neutron stars with strong magnetic fields.", "Atmospheres of ordinary stars are divided into the lower part called photosphere, where radiative transfer dominates, and the the upper atmosphere, whose temperature is determined by processes other than the radiative transfer.", "Usually the upper atmosphere of neutron stars is thought to be absent or negligible.", "Therefore one does not discriminate between the notions of atmosphere and photosphere for the neutron stars.", "In this respect let us note that vacuum polarization in superstrong magnetic fields (see § REF ) makes magnetosphere birefringent, so that the magnetosphere, being thermally decoupled from radiation propagating from the star to the observer, can still affect this radiation.", "Thus the magnetosphere can play the role of an upper atmosphere of a magnetar.", "Geometric depth of an atmosphere is several millimeters in relatively cold neutron stars and centimeters in relatively hot ones.", "These scales can be easily obtained from a simple estimate: as well as for the ordinary stars, a typical depth of a neutron-star photosphere is by order of magnitude slightly larger than the barometric height scale, the latter being equal to $k_\\mathrm {B}T/(m_\\mathrm {i}g)\\approx (0.83/A)\\,(T_6/g_{14})$ cm.", "The photosphere depth to the neutron-star radius ratio is only $\\sim 10^{-6}$ (for comparison, for ordinary stars this ratio is $\\sim 10^{-3}$ ), which allows one to calculate local spectra neglecting the surface curvature.", "The presence of atoms, molecules, and ions with bound states significantly changes the electromagnetic absorption coefficients in the atmosphere, thereby affecting the observed spectra.", "A question arises, whether the processes of particle creation and acceleration near the surface of the pulsars let them to have a partially ionized atmosphere.", "According to canonical pulsar models [24], [25], [26], the magnetosphere is divided in the regions of open and closed field lines, the closed-lines region being filled up by charged particles so that the electric field of the magnetosphere charge in the comoving (rotating) reference frame cancels the electric field arising from the rotation of the magnetized star.", "The photosphere that lies below this part of the magnetosphere is stationary and electroneutral.", "At contrast, there is a strong electric field near the surface in the open-line region.", "This field accelerates the charged particles almost to the speed of light.", "It is not obvious that these processes do not affect the photosphere, therefore quantitative estimates are needed.", "Let us define the column density $y_\\mathrm {col}= \\int _r^\\infty (1+z_g)\\,\\rho (r)\\mathrm {d}r,$ where the factor $(1+z_g)$ takes account of the relativistic scale change in the gravitational field.", "According to [101], in the absence of a strong magnetic field, ultrarelativistic electrons lose their energy mostly to bremsstrahlung at the depth where $y_\\mathrm {col}\\sim 60\\mbox{~g~cm}^{-2}$ .", "As noted by Bogdanov et al.", "[102], such column density is orders of magnitude larger than the typical density of a nonmagnetic neutron-star photosphere.", "Therefore, the effect of the accelerated particles reduces to an additional deep heating.", "The situation changes in a strong magnetic field.", "Electron oscillations driven by the electromagnetic wave are hindered in the directions perpendicular to the magnetic field, which thus decreases the coefficients of electromagnetic wave absorption and scattering by the electrons and atoms (§ REF ).", "Therefore the strong magnetic field “clarifies” the plasma, that is, the same mean (Rosseland [103], [104]) optical depth $\\tau _\\mathrm {R}$ is reached at a larger density.", "For a typical neutron star with $B\\gtrsim 10^{11}$ G, the condition $\\tau _\\mathrm {R}=3/2$ that is required to have $T(r)=T_\\mathrm {eff}$ in the Eddington approximation, is fulfilled at the density [105] $\\rho \\sim B_{12} ~~\\mbox{g~cm$^{-3}$},$ where $B_{12}\\equiv B/(10^{12}$ G).", "Thus the density of the layer where the spectrum is formed increases with growing $B$ .", "At the same time the main mechanism of electron and positron deceleration changes, which is related to Landau quantization (§ REF ).", "In the strong magnetic field, the most effective deceleration mechanism is the magneto-Coulomb interaction, which makes the charged particles colliding with plasma ions to jump to excited Landau levels with subsequent de-excitation through synchrotron radiation [106].", "The magneto-Coulomb deceleration length is inversely proportional to $B$ .", "An estimate [106] of the characteristic depth of the magneto-Coulomb deceleration of ultrarelativistic electrons in the neutron-star atmosphere can be written as $y_\\mathrm {col}\\approx \\big [(\\gamma _\\mathrm {r}/700)\\,Z^2\\,A^{-3}\\,B_{12}^{-2}\\big ]^{0.43}\\,T_6\\mbox{~g~cm}^{-2},$ $\\gamma _\\mathrm {r}\\sim 10^3$ – $10^8$ being the Lorentz factor.", "One can easily see from (REF ) and (REF ) that at $B\\gtrsim 3\\times 10^{12}$ G the electrons are decelerated by emitting high-energy photons in an optically thin layer.", "In this case, the magneto-Coulomb radiation constitutes a nonthermal supplement to the thermal photospheric spectrum of the polar cap.", "At the intermediate magnetic fields $10^{11}$ G $\\lesssim B\\lesssim 3\\times 10^{12}$ G, the braking of the accelerated particles occurs in the photosphere.", "Such polar caps require special photosphere models, where the equations of ionization, energy, and radiative balance would take the braking of charged particles into account.", "The photospheres can have different chemical compositions.", "Before the early 1990s, it was commonly believed that the outer layers of a neutron star consist of iron, as it is the most stable chemical element remaining after the supernova burst that gives birth to a neutron star [107].", "Nevertheless, the outer envelopes of an isolated neutron star may contain hydrogen and helium because of accretion of interstellar matter [108], [109].", "Even if the star is in the ejector regime [110], that is, its rotating magnetosphere throws away the infalling plasma, a small fraction of the plasma still leaks to the surface (see [78] and references therein).", "Because of the rapid separation of ions in the strong gravitational field (§ REF ), an accreted atmosphere can consist entirely of hydrogen.", "In the absence of magnetic field, hydrogen completely fills the photosphere if its column density exceeds $y_\\mathrm {col}\\gtrsim 0.1\\mbox{~g~cm}^{-2}$ .", "In the field $B\\sim 10^{14}$ G this happens at $y_\\mathrm {col}\\gtrsim 10^3\\mbox{~g~cm}^{-2}$ .", "Even in the latter case an accreting mass of $\\sim 10^{-17}M_\\odot $ would suffice.", "But if the accretion occurred at the early stage of the stellar life, when its surface temperature was higher than a few MK, then hydrogen could diffuse into deeper and hotter regions where it would be burnt in thermonuclear reactions [111], leaving helium on the surface [112].", "The same might happen to helium [111], and then the surface would be left with carbon [113], [94].", "Besides, a mechanism of spallation of heavy chemical elements into lighter ones operates in pulsars due to the collisions of the accelerated particles in the open field line regions, which produces lithium, beryllium, and boron isotopes [114].", "Therefore, only an analysis of observations can elucidate chemical composition of a neutron star atmosphere.", "The Coulomb liquid may turn into the gaseous phase abruptly.", "This possibility arises in the situation of a first-order phase transition between the condensed matter and the nondegenerate plasma (see § REF ).", "Then the gaseous layer may be optically thin.", "In the latter case, a neutron star is called naked [115], because its spectrum is formed at a solid or liquid surface uncovered by an atmosphere.", "Although many researchers studied neutron-star atmospheres for tens of years, many unsolved problems still persist, especially when strong magnetic fields and incomplete ionization are present.", "The state of the art of these studies will be considered below." ], [ "Neutron stars with thermal spectra", "In general, a neutron-star spectrum includes contributions caused by different processes beside the thermal emission: for example, processes in pulsar magnetospheres, pulsar nebulae, accretion disk, etc.", "A small part of such spectra allow one to separate the thermal component from the other contributions (see [116], for review).", "Fortunately, their number constantly increases.", "Let us list their main classes." ], [ "X-ray transients", "The X-ray binary systems where a neutron star accretes matter from a less massive star (a Main Sequence star or a white dwarf) are called low-mass X-ray binaries (LMXBs).", "In some of the LMXBs, periods of intense accretion alternate with longer (usually of months, and sometimes years) “periods of quiescence,” when accretion stops and the remaining X-ray radiation comes from the heated surface of the neutron star.", "During the last decade, such soft X-ray transients (SXTs) in quiescence (qLMXBs) yield ever increasing amount of valuable information on the neutron stars.", "Compression of the crust under the weight of newly accreted matter results in deep crustal heating, driven by exothermic nuclear transformations [117], [118].", "These transformations occur in a nonequilibrium layer, whose formation was first studied by Bisnovatyi-Kogan and Chechetkin [119].", "In the review of the same authors [120], this problem is exposed in more detail with applications to different real objects.", "For a given theoretical model of a neutron star, one can calculate the heating curve [121], that is the dependence of the equilibrium accretion-free effective temperature $T_0$ on the accretion rate averaged over a large preceding period of time.", "Comparing the heating curves with a measured $T_0$ value, one can draw conclusions on parameters of a given neutron star and properties of its matter.", "Such analysis has provided restrictions on the mass and composition of the core of the neutron star in SXT SAX J1808.4–3658 [121].", "In [122], [123], a possibility to constrain critical temperatures of proton and neutron superfluidities in the stellar core was demonstrated.", "Prospects of application of such analysis to various classes of X-ray transients are discussed in [124].", "The SXTs that have recently turned into quiescence allow one to probe the state of the neutron-star crust by the decline of $T_\\mathrm {eff}$ .", "Brown et al.", "[125] suggested that during this decline the radiation is fed by the heat that was deposited in the crust in the preceding active period.", "In 2001, SXT KS 1731–260, which was discovered in 1989 by Sunyaev's group [126], turned from the active state into quiescence [127].", "Subsequent observations have provided the cooling rate of the surface of the neutron star in this SXT.", "In 2007, Shternin et al.", "[128] analyzed the 5-year cooling of KS 1731–260 and obtained constraints to the heat conductivity in the neutron-star crust.", "In particular, they showed that the hypothesis on an amorphous state of the crust [129] is incompatible with the observed cooling rate, which means that the crust has a regular crystalline structure.", "Figure REF shows theoretical cooling curves compared to observations of KS 1731–260.", "The theoretical models differs in assumptions on the neutron-star mass, composition of its heat-blanketing envelope, neutron superfluidity in the crust, heat $E_\\mathrm {tot}$ deposited in the crust in the preceding accretion period ($E_{44}\\equiv E_\\mathrm {tot}/10^{44}$ erg), and the equilibrium effective temperature $T_0$ .", "The models 1a, 1c, and 2c were among others described and discussed in [128].", "At that time when only the first 7 observations had been available, it was believed that the thermal relaxation of the crust was over, and $T_0=0.8$ MK [130], which corresponds to the curve 1a in Fig.", "REF .", "Shternin et al.", "[128] were the first to call this paradigm in question.", "They demonstrated that the available observations could be described by the curves 1c ($T_0=6.7\\times 10^5$ K) and 2c ($T_0=6.3\\times 10^5$ K) as well.", "In 2009, new observations of KS 1731–260 were performed, which confirmed that the cooling continues [131].", "The whole set of observations is best described by the model $\\mathrm {1c}^{\\prime }$ (the dot-dashed line in Fig.", "REF ), which only slightly differs from the model 1c and assumes $T_0=0.7$ MK.", "Figure: Theoretical cooling curves for different neutron-star modelscompared with observations of KS 1731–260.", "Theobservational data are from Table 1 ofRef. .", "The blue dots correspond to theobservations used in , and the reddiamond is the new observation.", "The 1σ1\\sigma -errorbars areplotted.", "For the cooling curves, we use the numerical dataand notations from Ref.", ": 1a –M=1.6M ⊙ M=1.6\\,M_\\odot , T 0 =0.8T_0=0.8 MK, E 44 =2.6E_{44}=2.6; 1c –M=1.6M ⊙ M=1.6\\,M_\\odot , T 0 =0.67T_0=0.67 MK, E 44 =2.4E_{44}=2.4; 2c –M=1.4M ⊙ M=1.4\\,M_\\odot , T 0 =0.63T_0=0.63 MK, E 44 =2.4E_{44}=2.4.", "The model 1a,unlike the other three models, assumes an accreted envelopeand a moderate (in terms of ) neutronsuperfluidity in the crust.", "The curve marked 1c ' \\mathrm {1c}^{\\prime }was not shown in .", "It corresponds toM=1.65M ⊙ M=1.65\\,M_\\odot , T 0 =0.7T_0=0.7 MK, E 44 =2E_{44}=2.In 2008, a cooling curve of SXT MXB 1659–29 was constructed for crustal thermal-relaxation stage, which had been observed during 6 years [132].", "This curve generally agreed with the theory.", "In 2012, however, the spectrum suddenly changed, as if the temperature abruptly dropped [133].", "However, the spectral evolution driven by the cooling has already had to reach an equilibrium.", "The observed change of the spectrum can be explained by a change of the line-of-sight hydrogen column density.", "The cause of this change remains unclear.", "Indications to variations of $N_\\mathrm {H}$ were also found in the cooling qLMXB EXO 0748–676 [134].", "Several other qLMXBs have recently turned into quiescence and show signs of thermal relaxation of the neutron-star crust.", "A luminosity decline was even seen during a single 8-hour observation of SXT XTE J1709–267 after the end of the active phase of accretion [135].", "Analyses of observations of some qLMXBs (XTE J1701–462 [136], [137], EXO 0748–676 [138]) confirm the conclusions of Shternin et al.", "[128] on the crystalline structure of the crust and give additional information on the heating and composition of the crust of accreting neutron stars [135], [139], [138].", "In § REF we will discuss the interpretation of the observed qLMXB spectra that underlies such analysis.", "Transiently accreting X-ray pulsars Aql X-1, SAX J1808.4–3658, and IGR J00291+5734 reveal similar properties, but an analysis of their spectral evolution is strongly impeded by the possible presence of a nonthermal component and hot polar caps (see [140], [141], and references therein).", "Their X-ray luminosities in quiescence vary nonmonotonically, as well as those of qLMXBs Cen X-4 [142] and EXO 1745–248 [143].", "The variations of thermal flux that do not conform to the thermal-relaxation scenario may be caused by an accretion on the neutron star, which slows down but does not stop in quiescence [144], [138], [141]." ], [ "Radio pulsars", "There are several normal pulsars whose spectra clearly reveal a thermal component: these are relatively young (of the age $t_\\lesssim 10^5$ years) pulsars J1119–6127, B1706–44, and Vela, and middle-aged ($t_\\sim 10^6$ years) pulsars B0656+14, B1055$-$ 52, and Geminga.", "The spectra of the latter three objects, dubbed “Three Musketeers” [145], are described by a three-component model, which includes a power-law spectrum of magnetospheric origin, a thermal spectrum of hot polar caps, and a thermal spectrum of the rest of the surface [116].", "In most works the thermal components of pulsar spectra is interpreted with the blackbody model, and less often a model of the fully ionized H atmosphere with a predefined surface gravity.", "We will see that both are physically ungrounded.", "Only recently, in Ref.", "[146], the X-ray radiation of PSR J1119–6127 was interpreted using a H atmosphere model with allowance for the incomplete ionization.", "This result will be described in § REF .", "A convenient characteristic of the slowdown of pulsar rotation is the loss rate of the rotational kinetic energy $\\dot{E}_\\mathrm {rot}=-I\\Omega \\dot{\\Omega }$ of a standard rotator with the moment of inertia $I=10^{45}\\mbox{~g~cm}^2$ , typical of neutron stars, where $\\Omega =2\\pi /\\mathcal {P}$ is the angular frequency of the rotation, and $\\dot{\\Omega }$ is its time derivative (see [147]).", "As follows from observations, spectra of millisecond pulsars with $\\dot{E}_\\mathrm {rot}>10^{35}\\mbox{~erg~s}^{-1}$ are mainly nonthermal.", "However, millisecond pulsars PSR J0030+0451, J0437 – 4715, J1024–0719, and J2124–3358, with $\\dot{E}_\\mathrm {rot}\\lesssim 10^{34}\\mbox{~erg~s}^{-1}$ show a thermal spectral component on the nonthermal background.", "In § REF we will consider interpretation of this thermal component based on photosphere models." ], [ "Bursters", "Accreting neutron stars in close binary systems, which produce X-ray bursts with intervals from hours to days, are called bursters.", "The theory of the bursters were formulated in [148] (see also review [149]).", "During intervals between the bursts, a burster's atmosphere does not essentially differ from an atmosphere of a cooling neutron star.", "In such periods, the bulk of the observed X-ray radiation arises from transformation of gravitational energy of the accreting matter into thermal energy.", "The matter, mostly consisting of hydrogen and helium, piles up on the surface and sooner or later (usually during several hours or days) reaches such densities and temperatures that a thermonuclear burst is triggered, which is observed from the Earth as a Type I X-ray burst.Some binaries show Type II X-ray bursts, which recur more frequently than the Type I bursts, typically every several minutes or seconds.", "They may be caused by gravitational instabilities of accreting matter, rather than by thermonuclear reactions [150].", "Some of such bursts last over a minute and are called long X-ray bursts.", "They arise in the periods when the accretion rate is not high, so that the luminosity $L_\\mathrm {ph}$ before the burst does not exceed several percent of the Eddington limit $L_\\mathrm {Edd}$ (§ REF ).", "In this case, the inner part of the accretion disk is a hot ($k_\\mathrm {B}T\\sim 20-30$ keV) flow of matter with an optical thickness about unity.", "It almost does not affect the burst, nor screen it [151].", "As we will see in § REF , the observed spectrum of a burster, its evolution during a long burst, and subsequent relaxation are successfully interpreted with nonmagnetic atmosphere models.", "But if the accretion rate is higher, so that $L_\\mathrm {ph}\\gtrsim 0.1L_\\mathrm {Edd}$ , then the accretion disk is relatively cool and optically thick down to the neutron-star surface.", "In this case, the disk can strongly shield the burst and reprocess its radiation [152], [153], while at the surface a boundary spreading layer is formed.", "The theory of such layer is developed in [154], [155].", "The spreading layer spoils the spectrum so that its usual decomposition becomes ambiguous and needs to be modified, as described in [156], [157]." ], [ "Radio quiet neutron stars", "The discovery of radio quietThis term is rather relative, because some of such objects have revealed radio emission [158], [159].", "neutron stars, whose X-ray spectra are apparently purely thermal, has become an important milestone in astrophysics.", "The radio quiet neutron stars include central compact objects in supernova remnants (CCOs) [160], [161] and X-ray dim isolated neutron stars (XDINSs) [162], [39], [40], [163], [161].", "Exactly seven XDINSs are known since 2001, and they are dubbed “Magnificent Seven” [39].", "Observations have provided stringent upper limits ($\\lesssim 0.1$ mJy) to their radio emission [165].", "XDINSs have longer periods ($>3$ s) than the majority of pulsars, and their magnetic field estimations by Eq.", "(REF ) give, as a rule, rather high values $B \\sim (10^{13}$ – $10^{14})$ G [40], [164].", "It is possible that XDINSs are descendant of magnetars [40], [41], [163].", "About ten CCOs are known to date [166], [161].", "Pulsations have been found in radiation of three of them.", "The periods of these pulsations are rather small (0.1 s to 0.42 s) and very stable.", "This indicates that CCOs have relatively weak magnetic field $B \\sim 10^{11}$ G, at contrast to XDINSs.", "For this reason they are sometimes called “antimagnetars” [166], [161], [167].", "Large amplitudes of the pulsations of some CCOs indicate strongly nonuniform surface temperature distribution.", "To explain it, some authors hypothesized that a superstrong magnetic field might be hidden in the neutron-star crust [168].", "The X-ray source 1RXS J141256.0+792204, which was discovered in 2008 and dubbed Calvera, initially was considered as a possible eighth object with the properties of the “Magnificent Seven” [169].", "However, subsequent observations suggest that its properties are closer to the CCOs.", "In 2013, observations of Calvera at the orbital observatory Chandra provided the period derivative $\\dot{\\mathcal {P}}$ [170].", "According to Eq.", "(REF ), its value corresponds to $B\\approx 4.4\\times 10^{11}$ G. The authors [170] characterize Calvera as an “orphaned CCO,” whose magnetic field is emerging through supernova debris.", "Calvera is also unique in that it is the only energetic pulsar that emits virtually no radio nor gamma radiation, which places constraints on models for particle acceleration in magnetospheres [170]." ], [ "Neutron stars with absorption lines in\ntheir thermal spectra", "CCO 1E 1207.4–5209 has been the first neutron star whose thermal spectrum was found to possess features resembling two broad absorption lines [171].", "The third and fourth spectral lines were reported [172], but their statistical significance was called in question [173].", "It is possible that the complex shape of CCO PSR J0821–4300 may also be due to an absorption line [167].", "Features, which are possibly related to resonant absorption, are also found in spectra of four XDINSs: RX J0720.4–3125 [174], [175], RX J1308.6+2127 (RBS1223) [176], 1RXS J$214303.7+065419$ (RBS1774) [177], [178], [179] and RX J1605.3+3249 [180].", "Possible absorption features were also reported in spectra of two more XDINSs, RX J0806.4–4123 and RX J0420.0–5022 [181], but a confident identification is hampered by uncertainties related to ambiguous spectral background subtraction [164].", "Only the “Walter star” RX J1856.5 – 3754 that was discovered the first of the “Magnificent Seven” [182] has a smooth spectrum without any features in the X-ray range [183].", "An absorption line has been recently found in the spectrum of SGR 0418+5729 [184].", "Its energy varies from $<1$ keV to $\\sim 4$ keV with the rotational phase.", "The authors interpret it as a proton cyclotron line associated with a highly nonuniform magnetic-field distribution between $\\sim 2\\times 10^{14}$ G and $\\sim 10^{15}$ G. The discrepancy with the estimate $B\\sim 6\\times 10^{12}$ G according to Eq.", "(REF ) [51] the authors [184] explain by an absence of a large-scale dipolar component of the superstrong magnetic field (which can be, e.g., contained in spots).", "They reject the electron-cyclotron interpretation on the grounds that it would imply $B\\sim (1-5)\\times 10^{11}$ G, again at odds with the estimate [51].", "Note that the latter contradiction can be resolved in the models [53], [54], [56], [55] that involve a residual accretion torque (§ ).", "There is also no discrepancy if the line has a magnetospheric rather than photospheric origin.", "Similar puzzling lines had been previously observed in gamma-ray bursts of magnetars [185], [186], [48].", "Unlike the radio quiet neutron stars, spectra of the ordinary pulsars were until recently successfully described by a sum of smooth thermal and nonthermal spectral models.", "The first exception is the radio pulsar PSR J1740+1000, in whose X-ray spectrum is found to possess absorption features [187].", "This discovery fills the gap between the spectra of pulsars and radio quiet neutron stars and shows that similar spectral features can be pertinent to different neutron-star classes.", "Currently there is no unambiguous and incontestable theoretical interpretation of the features in neutron-star spectra.", "There were more or less successful attempts to interpret spectra of some of them.", "In § we will consider the interpretations that are based on magnetic neutron-star atmosphere models." ], [ "Which atmosphere can be treated as nonmagnetic?", "The main results of atmosphere modeling are the outgoing radiation spectra.", "Zavlin et al.", "[188] formulated the conditions that allow calculation of a neutron-star spectrum without account of the magnetic field.", "In the theory of stellar atmospheres, interaction of electromagnetic radiation with matter is conventionally described with the use of opacities $\\varkappa $ , that is absorption and scattering cross sections counted per unit mass of the medium.", "Opacities of fully ionized atmospheres do not depend on magnetic field at the frequencies $\\omega $ that are much larger than the electron cyclotron frequency $\\omega _\\mathrm {c}$ , which corresponds to the energy $\\hbar \\omega _\\mathrm {c}\\approx 11.577\\,B_{12}\\mbox{~keV}.$ On this ground, Zavlin et al.", "[188] concluded that for the energies $\\hbar \\omega \\sim (1$ – 10) $k_\\mathrm {B}T$ that correspond to the maximum of a thermal spectrum one can neglect the magnetic-field effects on opacities, if $B\\ll (m_\\mathrm {e}c / \\hbar e)\\,k_\\mathrm {B}T \\sim 10^{10}\\,T_6\\mbox{~G}.$ Strictly speaking, the estimate (REF ) is very relative.", "If the atmosphere contains an appreciable fraction of atoms or ions in bound states, then even a weak magnetic field changes the opacities by spectral line splitting (the Zeeman and Paschen-Back effects).", "Besides, magnetic field polarizes radiation in plasmas [189].", "The Faraday and Hanle effects that are related to the polarization serve as useful tools for studies of the stellar atmospheres and magnetic fields, especially the Sun (see [190], for a review).", "But the bulk of neutron-star thermal radiation is emitted in X-rays, whose polarimetry only begins to develop, therefore one usually neglects such fine effects for the neutron stars.", "Magnetic field drastically affects opacities of partially ionized photospheres, if the electron cyclotron frequency $\\hbar \\omega _\\mathrm {c}$ is comparable to or larger than the electron binding energies $E_\\mathrm {b}$ .", "Because of the high density of neutron-star photospheres, highly excited states do not survive as they have relatively large sizes and low binding energies (the disappearance of bound states with increasing density is called pressure ionization).", "For low-lying electron levels of atoms and positive atomic ions in the absence of a strong magnetic field, the binding energy can be estimated as $E_\\mathrm {b}\\sim (Z+1)^2$ Ry, where $Z$ is the charge of the ion, and $\\mbox{Ry}=m_\\mathrm {e}e^4/(2\\hbar ^2)=13.6057$ eV is the Rydberg constant in energy units.", "Consequently the condition $\\hbar \\omega _\\mathrm {c}\\ll E_\\mathrm {b}$ is fulfilled at $B\\ll B_0\\,(Z+1)^2/2,$ where $B_0 = \\frac{m_\\mathrm {e}^2\\,c\\,e^3}{\\hbar ^3} =2.3505\\times 10^9\\mbox{~G}$ is the atomic unit of magnetic field.", "The conditions (REF ) and (REF ) are fulfilled for most millisecond pulsars and accreting neutron stars." ], [ "Radiative transfer", "A nonmagnetic photosphere of a neutron star does not essentially differ from photospheres of the ordinary stars.", "However, quantitative differences can give rise to specific problems: for instance, the strong gravity results in high density, therefore the plasma nonideality that is usually neglected in stellar atmospheres can become significant.", "Nevertheless, the spectrum that is formed in a nonmagnetic neutron-star photosphere can be calculated using the conventional methods that are described in the classical monograph by Mihalas [104].", "For stationary neutron-star atmospheres, thanks to their small thickness, the approximation of plane-parallel locally uniform layer is quite accurate.", "The local uniformity means that the specific intensity at a given point of the surface can be calculated neglecting the nonuniformity of the flux distribution over the surface, that is, the nonuniformity of $T_\\mathrm {s}$ .", "Almost all models of neutron-star photospheres assume the radiative and local thermodynamic equilibrium (LTE; see [191] for a discussion of this and alternative approximations).", "Under these conditions, it is sufficient to solve a system of three basic equations: equations of radiative transfer, hydrostatic equilibrium, and energy balance.", "The first equation can be written in a plane-parallel layer as (see, e.g., [192]) $\\cos \\theta _k \\frac{\\mathrm {d}I_\\omega (\\hat{\\mathbf {k}})}{\\mathrm {d}y_\\mathrm {col}} =\\varkappa _\\omega I_\\omega -\\int _{(4\\pi )}\\!\\!\\varkappa _\\omega ^\\mathrm {s}(\\hat{\\mathbf {k}}^{\\prime },\\hat{\\mathbf {k}})I_\\omega (\\hat{\\mathbf {k}}^{\\prime }) \\mathrm {d}\\hat{\\mathbf {k}}^{\\prime }- \\varkappa _\\omega ^\\mathrm {a}\\mathcal {B}_{\\omega ,T},$ where $\\hat{\\mathbf {k}}$ is the unit vector along $\\mathbf {k}$ , $\\varkappa _\\omega =\\varkappa _\\omega ^\\mathrm {a}+\\int _{(4\\pi )}\\varkappa _\\omega ^\\mathrm {s}(\\hat{\\mathbf {k}}^{\\prime },\\hat{\\mathbf {k}})\\,\\mathrm {d}\\hat{\\mathbf {k}}^{\\prime }/(4\\pi )$ is the total opacity, $\\varkappa _\\omega ^\\mathrm {a}$ and $\\varkappa _\\omega ^\\mathrm {s}(\\hat{\\mathbf {k}}^{\\prime },\\hat{\\mathbf {k}})$ are its components due to, respectively, the true absorption and the scattering that changes the ray direction from $\\hat{\\mathbf {k}}^{\\prime }$ to $\\hat{\\mathbf {k}}$ , and $\\mathrm {d}\\hat{\\mathbf {k}}^{\\prime } = \\sin \\theta _{k^{\\prime }}\\mathrm {d}\\theta _{k^{\\prime }}\\mathrm {d}\\varphi _{k^{\\prime }}$ is a solid angle element.", "Most studies of the neutron-star photospheres neglect the dependence of $\\varkappa _\\omega ^\\mathrm {s}$ on $\\hat{\\mathbf {k}}^{\\prime }$ and $\\hat{\\mathbf {k}}$ .", "As shown in [193], the inaccuracy that is introduced by this simplification does not exceed 0.3% for the thermal spectral flux of a neutron star at $\\hbar \\omega <1$ keV and reaches a few percent at higher energies.", "For simplicity, in Eq.", "(REF ) we have neglected polarization of radiation and a change of frequency at the scattering.", "In general, the radiative transfer equation includes an integral of $I_\\omega $ not only over angles, but also over frequencies, and contains, with account of polarization, a vector of Stokes parameters instead of $I_\\omega $ , while the scattering cross section is replaced by a matrix.", "A detailed derivation of the transfer equations for polarized radiation is given, e.g., in [192], and solutions of the radiative transfer equation with frequency redistribution are studied in [191].", "The condition of hydrostatic equilibrium follows from Eq.", "(REF ).", "Given that $\|R-r\|\\ll R$ , $\|M-M_r\|\\ll M$ , and $P\\ll \\rho c^2$ in the photosphere, we have $\\frac{\\mathrm {d}P}{\\mathrm {d}y_\\mathrm {col}} = g - g_\\mathrm {rad},$ where (see, e.g., [194]) $g_\\mathrm {rad} &=&\\frac{1}{c}\\,\\frac{\\mathrm {d}}{\\mathrm {d}y_\\mathrm {col}}\\int _0^\\infty \\mathrm {d}\\omega \\int _{(4\\pi )} \\mathrm {d}\\hat{\\mathbf {k}}\\,\\cos ^2\\theta _k\\,I_\\omega (\\hat{\\mathbf {k}})\\nonumber \\\\&\\approx &\\frac{2\\pi }{c}\\int _0^\\infty \\mathrm {d}\\omega \\,\\varkappa _\\omega \\int _0^\\pi \\cos \\theta _k\\, I_\\omega (\\hat{\\mathbf {k}})\\sin \\theta _k\\,\\mathrm {d}\\theta _k.\\hspace{20.0pt}$ The last approximate equality becomes exact for the isotropic scattering.", "The quantity $g_\\mathrm {rad}$ takes account of the radiation pressure that counteracts gravity.", "It becomes appreciable at $T_\\mathrm {eff}\\gtrsim 10^7$ K. Therefore, $g_\\mathrm {rad}$ is usually dropped in calculations of the spectra of the cooler isolated neutron stars, but included in the models of relatively hot bursters.", "Radiative flux of the bursters amply increases during the bursts, thus increasing $g_\\mathrm {rad}$ .", "The critical value of $g_\\mathrm {rad}$ corresponds to the limit of stability, beyond which matter inevitably flows away under the pressure of light.", "In a hot nonmagnetic atmosphere, where the Thomson scattering dominate, the instability appears when the luminosity $L_\\mathrm {ph}$ exceeds the Eddington limit $\\hspace{-20.0pt}L_\\mathrm {Edd} &=& 4\\pi c\\,(1+z_g)\\, {G M m_\\mathrm {p}}/{\\sigma _\\mathrm {T}}\\nonumber \\\\\\hspace{-20.0pt}& \\approx &1.26\\times 10^{38}\\,(1+z_g)\\,({M}/{M_\\odot }) ~~ \\textrm {erg~s}^{-1},$ where $m_\\mathrm {p}$ is the proton mass, and $\\sigma _\\mathrm {T}=\\frac{8\\pi }{3}\\left(\\frac{e^2}{m_\\mathrm {e}c^2}\\right)^2$ is the Thomson cross section, A temperature-dependent relativistic correction to $\\sigma _\\mathrm {T}$ [195] increases $L_\\mathrm {Edd}$ approximately by 7% at typical temperatures $\\sim 3\\times 10^7$ K at the bursters luminosity maximum [151], [194].", "Finally, the energy balance equation in the stationary state expresses the fact that the energy acquired by an elementary volume equals the lost energy.", "The radiative equilibrium assumes that the energy transport through the photosphere is purely radiative, that is, one neglects electron heat conduction and convection, as well as other sources and leaks of heat.", "Under these conditions, the energy balance equation reduces to $\\int _0^\\infty \\mathrm {d}\\omega \\int _{(4\\pi )} I_\\omega (\\hat{\\mathbf {k}})\\,\\cos \\theta _k\\,\\mathrm {d}\\hat{\\mathbf {k}}= F_\\mathrm {ph},$ where $F_\\mathrm {ph}$ is the local flux at the surface that is related to $T_\\mathrm {s}$ according to Eq.", "(REF ).", "Radiation is almost isotropic at large optical depth $\\tau _\\omega =\\int _r^\\infty \\varkappa _\\omega (r^{\\prime }) \\, \\mathrm {d}y_\\mathrm {col}(r^{\\prime }),$ therefore one may restrict to the first two terms of the intensity expansion in spherical functions: $I_\\omega (\\hat{\\mathbf {k}}) = J_\\omega +\\frac{3}{4\\pi }\\mathbf {F}_\\omega \\cdot \\hat{\\mathbf {k}}.$ Here, $J_\\omega = \\frac{1}{4\\pi }\\int _{(4\\pi )}I_\\omega (\\hat{\\mathbf {k}})\\,\\mathrm {d}\\hat{\\mathbf {k}}$ is the mean intensity, averaged over all directions, and $\\mathbf {F}_\\omega = \\int _{(4\\pi )}I_\\omega (\\hat{\\mathbf {k}})\\hat{\\mathbf {k}}\\,\\mathrm {d}\\hat{\\mathbf {k}}$ is the diffusive flux vector.", "Then integro-differential equation (REF ) reduces to a diffusion-type equation for $J_\\omega $ .", "If scattering is isotropic, then in the plane-parallel locally-uniform approximation the stationary diffusion equation has the form $\\frac{\\mathrm {d}^2}{\\mathrm {d}\\tau _\\omega ^2}\\,\\frac{J_\\omega }{3}=\\frac{\\varkappa _\\omega ^\\mathrm {s}}{\\varkappa _\\omega }\\,(J_\\omega - \\mathcal {B}_{\\omega ,T})$ (see [196] for derivation of the diffusion equation from the radiative transfer equation in a more general case).", "Sometimes the diffusion approximation is applied to the entire atmosphere, rather than only to its deep layers.", "In this case, one has to replace $J_\\omega /3$ on the left-hand side of Eq.", "(REF ) by $f_\\omega J_\\omega $ , where $f_\\omega (\\tau _\\omega )$ is the so called Eddington factor [104], which is determined by iterations of the radiative-transfer and energy-balance equations with account of the boundary conditions (see [188] for details).", "In modeling bursters atmospheres, one usually employs Eq.", "(REF ) with the Eddington factor on the left-hand side and an additional term on the right-hand side, a differential Kompaneets operator [197] acting on $J_\\omega $ (see, e.g., [198], [199], [200], [201]).", "The Kompaneets operator describes, in the diffusion approximation, the photon frequency redistribution due to the Compton effect, which cannot be neglected at the high temperatures typical of the bursters.", "In order to close the system of equations of radiative transfer and hydrostatic balance, one needs the EOS and opacities $\\varkappa _\\omega ^\\mathrm {s,a}$ for all densities and temperatures encountered in the photosphere.", "In turn, in order to determine the EOS and opacities, it is necessary to find ionization distribution for the chemical elements that compose the photosphere.", "The basis for solution of these problems is provided by quantum mechanics of all particle types that give a significant contribution to the EOS or opacities.", "In the nonmagnetic neutron-star photospheres, these particles are only the electrons and atomic ions, because molecules do not survive the typical temperatures $T\\gtrsim 3\\times 10^5$ K. We will not consider in detail the calculations of the EOS and opacities in the absence of a strong magnetic field, because they do not basically differ from the ones for the ordinary stellar atmospheres, which have been thoroughly considered, e.g., in the review [202].", "Detailed databases have been developed for them (see [203], for review), the most suitable of which for the neutron-star photospheres are OPAL [204] and OP [205].The OPAL opacities are included in the MESA project [206], and the database OP is available at http://cdsweb.u-strasbg.fr/topbase/TheOP.html In the particular cases where the neutron-star atmosphere consists of hydrogen or helium, all binding energies are smaller than $k_\\mathrm {B}T$ , therefore the approximation of an ideal gas of electrons and atomic nuclei is applicable.", "Systematic studies of neutron-star photospheres of different chemical compositions, from hydrogen to iron, started from the work by Romani [207].", "In the subsequent quarter of century, the nonmagnetic neutron-star photospheres have been studied in many works (see [116] for a review).", "Databases of neutron-star hydrogen photosphere model spectra have been published [188], [208], [209],Models NSA, NSAGRAV, and NSATMOS in the database XSPEC [210].", "and a numerical code for their calculation has been released [193].https://github.com/McPHAC/ A publicly available database of model spectra for the carbon photospheres has been recently published [211].Model CARBATM in the database XSPEC [210].", "In addition, model spectra were calculated for neutron-star photospheres composed of helium, nitrogen, oxygen, iron (e.g., [212], [213], [209], [214]), and mixtures of different elements [208], [213]." ], [ "Atmospheres of bursters", "Burster spectra were calculated by many authors (see, e.g., [151], for references), starting from the pioneering works [215], [216], [153] (see, e.g., [151], for references).", "These calculations as well as observations show that the X-ray spectra of bursters at high luminosities are close to so called diluted blackbody spectrum $F_\\omega \\approx w \\mathcal {B}_{\\omega ,T_\\mathrm {bb}},$ where $\\mathcal {B}_{\\omega ,T}$ is the Planck function (REF ), the parameter $T_\\mathrm {bb}$ is called color temperature, normalization $w$ is a dilution factor, and the ratio $f_\\mathrm {c}=T_\\mathrm {bb}/T_\\mathrm {eff}$ (typically $\\sim 3/2$ ) is called color correction [217], [216], [151].", "The apparent color temperature $T_\\mathrm {bb}^\\infty $ is related to $T_\\mathrm {bb}$ by the relation analogous to (REF ).", "If the luminosity reaches the Eddington limit during a thermonuclear burst, then the photosphere radius $R_\\mathrm {ph}$ first increases, and goes back to the initial value $R$ at the relaxation stage [195].", "Based on this model, Kaminker et al.", "[218] suggested a method of analysis of the Eddington bursts of the bursters and for the first time applied it to obtaining constraints of the parameters of the burster MXB 1728–34.", "Subsequently this method was amended and modernized by other authors (see [151], for references).", "According to Eq.", "(REF ), the bolometric flux equals $F_\\mathrm {bol}=L_\\mathrm {ph}^\\infty /(4\\pi D^2)=\\sigma _\\mathrm {SB}(T_\\mathrm {eff}^\\infty )^4 (R_\\mathrm {ph}^\\infty /D)^2$ .", "But the approximation (REF ) implies $F_\\mathrm {bol}= w\\sigma _\\mathrm {SB}(T_\\mathrm {bb}^\\infty )^4(R_\\mathrm {ph}^\\infty /D)^2$ .", "Therefore, at the late stage of a long burst, when $R_\\mathrm {ph}=R={}$ constant, $w\\propto f_\\mathrm {c}^{-4}$ .", "On the other hand, the dependence of $f_\\mathrm {c}$ on $L_\\mathrm {ph}$ can be obtained from numerical calculations.", "This possibility lies in the basis of the method of studying bursters that was implemented in the series of papers by Suleimanov et al.", "[200], [151].", "The calculations show that $f_\\mathrm {c}$ mainly depends on the ratio $l_\\mathrm {ph}=L_\\mathrm {ph}/L_\\mathrm {Edd}$ , and also on gravity $g$ and chemical composition of the photosphere (mostly on the helium-to-hydrogen fractional abundance, and to a less extent on the content of heavier elements).", "Having approximated the observed spectral normalizations $f_\\mathrm {c}^{-4}(l_\\mathrm {ph})$ by the results of theoretical calculations, one finds the chemical composition that provides an agreement of the theory with observations.", "For this selected composition, one finds the color correction that corresponds to the observed one at different values of $g$ , and thus obtains a curve of allowable values in the $(M,R)$ -plane.", "The point at this curve that satisfies the condition $F_\\mathrm {bol}=l_\\mathrm {ph}F_\\mathrm {Edd}$ , $F_\\mathrm {Edd}=L_\\mathrm {Edd}/[4\\pi D^2(1+z_g)^2]$ being the bolometric flux that corresponds to the Eddington luminosity Eq.", "(REF ), gives an estimate of the mass and radius of the neutron star, if the distance $D$ is known.", "If $D$ is unknown, then this analysis allows one to obtain restrictions on joint values of $M$ , $R$ , and $D$ .", "This method was successfully applied to analyzing the long bursts of bursters 4U 1724–307 [151] and GS 1826–24 [219].", "In both cases, there was a marked agreement of the observed and calculated dependences $f_\\mathrm {c}(l_\\mathrm {ph})$ .", "In [219], the authors have also simulated the light curves, that is, the time dependences of $F_\\mathrm {bol}$ .", "As well as in an earlier work [220], they managed to find the chemical composition of the atmosphere and the accretion rate that give an agreement of the theoretical light curve of each burst and of the intervals between the bursts with observations.", "Thus they obtained an absolute calibration of the luminosity.", "A comparison of the theoretical and observed dependences gives an estimate of the ratio $f_\\mathrm {c}/(1+z_g)$ , which does not depend on the distance $D$ , thus providing additional constraints to the neutron-star mass and radius [151], [219].", "A possible anisotropy of the emission, which modifies the total flux (e.g., because of screening and reflection of a part of radiation by an accretion disk) is equivalent to a multiplication of $D$ by a constant factor, therefore it does not affect the $D$ -independent estimates [219].", "In [221], [222], [223] the authors used a simplified analysis of spectra of bursters, ignoring the dependence $f_\\mathrm {c}(l_\\mathrm {ph})$ , but only assuming that the Eddington luminosity is reached at the “touchdown point,” determined by the maximum of the color temperature.", "This assumption is inaccurate, therefore such simplified analysis fails: it gives considerably lower $R$ values, than the method described above.", "In addition, the authors of [221], [222], [223] analyzed the “short” bursts, for which the theory fails to describe the dependence $f_\\mathrm {c}(l_\\mathrm {ph})$ , and the usual separation of spectral components becomes ambiguous (see § REF ).", "Therefore, the simplified estimates of neutron-star parameters [221], [222], [223] are unreliable (see the discussion in [151]).", "We must note that the current results for the bursters still do leave some open questions.", "First, the estimates for two different sources in [151] and [219] are hard to conciliate: in the case of the H atmosphere model, the former estimate indicates a relatively large neutron-star radius, thus a stiff EOS, whereas the latter gives a constraint, which implies a soft EOS.", "Second, a good agreement between the theory and observations has been achieved only for a restricted decaying part of the lightcurves.", "Third, there is a lack of explanation to different normalizations of spectra for the bursts that have different recurrence times.", "In [219], the authors discuss these uncertainties and possible prospects of their resolution with the aid of future observations." ], [ "Photospheres of isolated neutron stars", "Nonmagnetic atmospheres of isolated neutron stars differ from accreting neutron stars atmospheres, first of all, by a lower effective temperature $T_\\mathrm {s}\\sim 3\\times (10^5$ – $10^6$ ) K, and may be also by chemical composition.", "Examples of spectra of such atmospheres are given in Fig.", "REF .", "If there was absolutely no accretion on a neutron star, then the atmosphere should consist of iron.", "A spectrum of such atmosphere has the maximum in the same wavelength range as the blackbody spectrum, but contains many features caused by bound-bound transitions and photoionization [207], [212], [224], [208].", "Absorption lines and photoionization edges are smeared with increasing $g$ , because the photosphere becomes denser, thus increasing the effects leading to line broadening [225] (for example, fluctuating microfields in the plasma [226]).", "If the atmosphere consists of hydrogen and helium, the spectrum is smooth, but shifted to higher energies compared to the blackbody spectrum at the same effective temperature [207], [188].", "As shown by Zavlin et al.", "[188], this shift is caused by the decrease of light-element opacities according to the law $\\varkappa _\\omega \\propto \\omega ^{-3}$ at $\\hbar \\omega >k_\\mathrm {B}T\\sim 0.1$ keV, which makes photons with larger energies to come from deeper and hotter photosphere layers.", "Zavlin et al.", "[188] payed attention also to the polar diagrams of radiation coming from the atmosphere.", "Unlike the blackbody radiation, it is strongly anisotropic ($I_\\omega (\\hat{\\mathbf {k}})$ quickly decreases at large angles $\\theta _k$ ), and the shape of the polar diagram depends on the frequency $\\omega $ and on the chemical composition of the atmosphere.", "Figure: Radiation energy flux densities as functions of photonenergy E=ℏωE=\\hbar \\omega for a photosphere composed of iron(solid lines), helium (dashed lines), and hydrogen(dot-dashed lines) as compared to the blackbody spectrum(dotted curves) at g 14 =2.43g_{14}=2.43 for different values ofeffective temperature (numbers at the curves correspond tologT eff \\log \\,T_\\mathrm {eff} [K]).", "(Fig.", "3 from , courtesyof J. Pons and ©AAS.", ")Suleimanov & Werner [227] have taken account of the Compton effect on the spectra of isolated neutron stars, using the same technique as for the bursters.", "They have shown that this effect results in a decrease of the high-energy flux at $\\hbar \\omega \\gg 1$ keV for the hydrogen and helium atmospheres.", "It becomes considerable at high effective temperatures $T_\\mathrm {s}> 10^6$ K, where the spectral maximum shifts to the energies $E \\gtrsim 1$ keV.", "This effect makes the spectra of hot hydrogen and helium atmospheres closer to the blackbody spectrum with color correction $f_\\mathrm {c}\\approx 1.6$ – 1.9.", "Papers [224], [228] stand apart, being the only ones where non-LTE calculations were done for a spectrum of an iron neutron-star atmosphere.", "At $T=2\\times 10^5$ K, the difference from the LTE model is about 10% for the flux in the lines and much less in the continuum [224], [208].", "As noted in [208], the difference may be larger at higher temperatures, which turned out to be the case indeed in [228].", "Pons et al.", "[213] performed a thorough study in attempt to describe the observed spectrum of the Walter star RX J1856.5 – 3754 by the nonmagnetic atmosphere models with various chemical compositions.", "It turned out that the hydrogen atmosphere model that reproduces the X-ray part of the spectrum predicts approximately 30 times larger optical luminosity than observed, whereas an iron-atmosphere model corresponds to a too small radius.", "This demonstrates once again that a neutron-star radius estimate strongly depends on the assumptions on its atmosphere.", "Satisfactory results have been obtained for a chemical composition corresponding to the ashes of thermonuclear burning of matter that was accreted on the star at the early stage of its life.", "This model, as well as other models of atmospheres composed of elements heavier than helium, predicted absorption lines in the X-ray spectrum.", "However, subsequent deep X-ray observations with space observatories Chandra [229] and XMM-Newton [83] have not found such lines.", "The failure of the interpretation of the Walter star spectrum with nonmagnetic atmosphere models can be explained by the presence of a strong magnetic field.", "The field is indicated by a nearby nebula glowing in the H$\\alpha $ line [230].", "Such nebulae are found near pulsars, which ionize interstellar hydrogen by shock waves arising from hypersonic pulsar magnetosphere interaction with interstellar medium [52], [231].", "Doubts had initially been cast on the pulsar analogy by the absence of observed pulsations of radiation of this star, but soon such pulsations were discovered [232].", "Interpretation of the Walter star spectrum with magnetic atmosphere models will be considered in § REF .", "The first successful interpretation of an isolated neutron star spectrum based on a nonmagnetic atmosphere model was done in [214].", "The authors showed that the observed X-ray spectrum of the CCO in Cassiopeia A supernova remnant, which appeared around 1680, is well described by a carbon atmosphere model with the effective temperature $T_\\mathrm {eff}\\sim 2\\times 10^6$ K. Subsequent observations revealed that $T_\\mathrm {eff}$ appreciably decreases with time [233], which was explained by the heat-carrying neutrino emission outburst caused by the superfluid transition of neutrons [234], [235].", "At $t_\\approx 330$ yrs this agrees with the cooling theory [97].", "An independent analysis [236] confirmed the decrease of the registered flux, but the authors stressed that the statistical significance of this result is not high and that the same observational data allow other interpretations.", "Recently, a spectrum of one more CCO, residing in supernova remnant HESS J1731–347, was also satisfactorily described by a nonmagnetic carbon atmosphere model [237]." ], [ "Atmospheres of neutron stars in qLMXBs", "Many SXTs reside in globular clusters, whose distances are known with accuracies of 5 – 10%.", "This reduces a major uncertainty that hampers the spectral analysis.", "As we noted in § , spectra of SXTs in quiescence, called qLMXBs, are probably determined by neutron-star thermal emission.", "In early works, these spectra were interpreted with the Planck function, which overestimated the effective temperature and underestimated the effective radius of emitting area.", "However, Rutledge et al.", "[238], [144], [239] found that the nonmagnetic hydrogen atmosphere model provides an explanation to the SXT spectra as caused by radiation from the entire neutron-star surface with acceptable values of the temperature and radius.", "Currently tens qLMXBs in globular clusters are known (they are listed in [240], [241]), and the use of hydrogen atmosphere models for their spectral analysis has become customary.", "For instance, the analysis of the cooling of KS 1731–260 and the other similar objects that was discussed in § REF was based on the measurements of the effective temperature $T_\\mathrm {eff}$ with the use of the models [188] and NSATMOS [209].", "In many works (including [130], [132], [131]), the neutron-star mass and radius were a priori fixed to $M=1.4\\,M_\\odot $ and $R=10$ km, which entrain $g_{14}=2.43$ .", "It was shown in [209], that such fixing of $g$ may strongly bias estimates of the neutron-star parameters (which means, in particular, that the estimates of $T_\\mathrm {eff}$ for KS 1731–260 and MXB 1659–29, quoted in § REF , are unreliable).", "An analysis of thermal spectrum of qLMXB X7 in the globular cluster 47 Tuc, free of such fixing, gave a 90%-confidence area of $M$ and $R$ estimates, which agrees with relatively stiff EOSs of supranuclear matter [209].", "However, the estimates that were obtained in [242] by an analogous analysis for five qLMXBs in globular clusters, although widely scattered, generally better agree with soft EOSs.", "In [243], [244], thermal spectra of two qLMXBs were analyzed using hydrogen and helium atmosphere models.", "It turned out that the former model leads to low estimates of $M$ and $R$ , compatible with the soft EOSs, while the latter yields high values, which require a stiff EOS of superdense matter.", "Thus, despite the progress achieved in recent years, the estimates of neutron-star masses and radii based on the qLMXBs spectral analysis are not yet definitive." ], [ "Photospheres of millisecond pulsars", "Magnetic fields of most millisecond pulsars satisfy the weak-field criteria formulated in § REF .", "Nevertheless, magnetic field does play certain role, because the open field line areas (“polar caps”) may be heated by deceleration of fast particles (see § REF ).", "Therefore, one should take nonuniform temperature distribution into account, while calculating the integral spectrum.", "Models of rotating neutron stars with hot spots were presented in many publications (e.g., [72], [245], [246], and references therein), however most of them used the blackbody radiation model.", "This model is acceptable for a preliminary qualitative description of the spectra and light curves of the millisecond pulsars, but a detailed quantitative analysis must take the photosphere into account.", "Let us consider results of such analyses.", "The nearest and the brightest of the four millisecond pulsars with observed thermal radiation is PSR J0437 – 4715.", "It belongs to a binary system with a 6-billion-year-old white dwarf.", "The low effective temperature of the white dwarf ($\\sim 4000$ K), as well as the brightness of the pulsar and a relatively low intensity of its nonthermal emission favor the analysis of the thermal spectrum.", "Recently, the pulsar's thermal radiation has been extracted from the white-dwarf radiation even in the ultraviolet range [247], although the maximum of the pulsar thermal radiation lies at X-rays.", "Zavlin & Pavlov [248] showed that the thermal X-ray spectrum of PSR J0437 – 4715 can be explained by emission of two hot polar caps with hydrogen photospheres and a nonuniform temperature distribution, which was presented by the authors as a steplike function with a higher value $T\\approx (1$ – $2)\\times 10^6$ K in the central circle of radius $0.2$ – $0.4$ km and a lower value $T\\approx (3$ – $5)\\times 10^5$ K in the surrounding broad ring of radius about several kilometers.", "Subsequent observations of the binary system J0437 – 4715 in spectral ranges from infrared to hard X-rays and their analysis in [249], [102], [250] have generally confirmed the qualitative conclusions of [248].", "In particular, Bogdanov et al.", "[102], [250] reproduced not only the spectrum, but also the light curve of this pulsar at X-rays, using the model of a hydrogen atmosphere with a steplike temperature distribution, supplemented with a power-law component.", "These authors have also explained [251] the power-law spectral component by the Compton scattering of thermal polar-cap photons on energetic electrons in the magnetosphere or in the pulsar wind.", "Thus all the spectral components may have thermal origin.", "Finally, Bogdanov [252] reanalyzed the phase-resolved X-ray spectrum of PSR J0437 – 4715 using the value $M=(1.76\\pm 0.20)\\,M_\\odot $ obtained from radio observations [253], the distance of 156.3 pc measured by radio parallax [254], a nonmagnetic hydrogen atmosphere model NSATMOS [209], and a three-level distribution of $T_\\mathrm {eff}$ around the polar caps.", "As a result, he came to the conclusion that the radius of a neutron star of such mass cannot be smaller than 11 km, which favors the stiff equations of state of supranuclear matter.", "The presence of a hydrogen atmosphere helps one to explain not only the spectrum but also the relatively large pulsed fraction (30 – 50%) in thermal radiation of this and the three other millisecond pulsars with observed thermal components of radiation (PSR J0030+0451, J2124–3358, and J1024–0719).", "According to [250], [116], such strong pulsations may indicate that all similar pulsars have hydrogen atmospheres.", "The measured spectra and light curves of all the four pulsars agree with this assumption [250]." ], [ "Matter in strong magnetic fields", "The conditions of § REF are not satisfied for most of the known isolated neutron stars, therefore magnetic fields drastically affect radiative transfer in their atmospheres.", "Before going on to magnetized atmosphere models, it is useful to consider the magnetic-field effects on their constituent matter." ], [ "Landau quantization", "Motion of charged particles in a magnetic field is quantized in Landau levels [255].", "It means that only longitudinal (parallel to $\\mathbf {B}$ ) momentum of the particle can change continuously.", "Motion of a classical charged particle across magnetic field is restricted to circular orbits, corresponding to a set of discrete quantum states, analogous to the states of a two-dimensional oscillator.", "The complete theoretical description of the quantum mechanics of free electrons in a magnetic field is given in monograph [256].", "It is convenient to characterize magnetic field by its strength in relativistic units, $b$ , and in atomic units, $\\gamma $ : $b&=&{\\hbar \\omega _\\mathrm {c}}/({m_\\mathrm {e}c^2}) = B/B_\\mathrm {QED} ={B_{12}}/{44.14} \\,,\\\\\\gamma &=& B/B_0 = 425.44\\,B_{12}.$ We have already dealt with the atomic unit $B_0$ in § REF .", "The relativistic unit $B_\\mathrm {QED}= m_\\mathrm {e}^2 c^3 / (e\\hbar ) = B_0/\\alpha _\\mathrm {f}^2$ is the critical (Schwinger) field, above which specific QED effects become pronounced.", "In astrophysics, the magnetic field is called strong, if $\\gamma \\gg 1$ , and superstrong, if $b\\gtrsim 1$ .", "In the nonrelativistic theory, the distance between Landau levels equals the cyclotron energy $\\hbar \\omega _\\mathrm {c}$ .", "In the relativistic theory, Landau level energies equal $E_N=m_\\mathrm {e}c^2 \\,(\\sqrt{1+2bN}-1)$ ($N=0,1,2,\\ldots $ ).", "The wave functions that describe an electron in a magnetic field have a characteristic transverse scale $\\sim a_\\mathrm {m}=(\\hbar c/eB)^{1/2}=a_\\mathrm {B}/\\sqrt{\\gamma }$ , where $a_\\mathrm {B}$ is the Bohr radius.", "The momentum projection on the magnetic field remains a good quantum number, therefore we have the Maxwell distribution for longitudinal momenta at thermodynamic equilibrium.", "For transverse motion, however, we have the discrete Boltzmann distribution over $N$ .", "In practice, the Landau quantization becomes important when the electron cyclotron energy $\\hbar \\omega _\\mathrm {c}$ is at least comparable to both the electron Fermi energy $\\epsilon _\\mathrm {F}$ and the characteristic thermal energy $k_\\mathrm {B}T$ .", "If $\\hbar \\omega _\\mathrm {c}$ is appreciably larger than both these energies, then most electrons reside on the ground Landau level in thermodynamic equilibrium, and the field is called strongly quantizing.", "For it to be the case, simultaneous conditions $\\rho <\\rho _B$ and $\\zeta _\\mathrm {e}\\gg 1$ must be fulfilled, where $\\rho _B &=&\\frac{m_\\mathrm {i}}{\\pi ^2\\sqrt{2}\\,a_\\mathrm {m}^3\\,Z}= 7045 \\,\\frac{A}{Z}\\,B_{12}^{3/2}\\text{ \\mbox{g~cm$^{-3}$}},\\\\\\zeta _\\mathrm {e}&=& \\frac{\\hbar \\omega _\\mathrm {c}}{k_\\mathrm {B}T} = 134.34\\,\\frac{B_{12}}{T_6} .$ In the neutron-star atmospheres, these conditions are satisfied, as a rule, at $B\\gtrsim 10^{11}$ G. In the opposite limit $\\zeta _\\mathrm {e}\\ll 1$ , the Landau quantization can be neglected.", "Note that in the magnetospheres, which have lower densities, electrons can condensate on the lowest Landau level even at $B\\sim 10^8$ G because of the violation of the LTE conditions (§ REF ).", "Ions in a neutron-star atmosphere can be treated as nondegenerate and nonrelativistic particles.", "The parameter $\\zeta _\\mathrm {e}$ is replaced for them by $\\zeta _\\mathrm {i}= \\hbar \\omega _\\mathrm {ci}/k_\\mathrm {B}T = 0.0737\\,(Z/A)B_{12}/T_6.$ Here, $\\omega _\\mathrm {ci}=ZeB/(m_\\mathrm {i}c)$ is the ion cyclotron frequency, and $\\hbar \\omega _\\mathrm {ci}=6.35(Z/A)B_{12}$ eV is the ion cyclotron energy.", "In magnetar atmospheres, where $B_{12}\\gtrsim 100$ and $T_6\\lesssim 10$ , the parameter $\\zeta _\\mathrm {i}$ is not small, therefore the Landau quantization of ion motion should be taken into account." ], [ "Interaction with radiation", "The general expression for a differential cross section of absorption of a plane electromagnetic wave by a quantum-mechanical system can be written as (e.g., [257]) $\\mathrm {d}\\sigma =\\frac{4\\pi ^2}{\\omega c}\\left\|\\mathbf {e}\\cdot \\langle f\| \\mathbf {j}_\\mathrm {eff} \| i \\rangle \\right\|^2\\,\\delta (E_f-E_i-\\hbar \\omega )\\,\\mathrm {d}\\nu _f,$ where$\|i\\rangle $ and $\|f\\rangle $ are, respectively, the initial and final states of the system, $\\mathrm {d}\\nu _f$ is the number of final states in the considered energy interval $\\mathrm {d}E_f$ , $\\mathbf {e}$ is the electromagnetic polarization vector, $\\mathbf {j}_\\mathrm {eff}=\\sum _i q_i\\mathrm {e}^{\\mathrm {i}\\mathbf {k}_i}\\dot{\\mathbf {r}}_i$ is the effective electric-current operator, and $\\dot{\\mathbf {r}}_i$ is the velocity operator acting on a particle with charge $q_i$ .", "While calculating the matrix elements $\\langle f\| \\mathbf {j}_\\mathrm {eff} \| i \\rangle $ , it is important to remember that $\\dot{\\mathbf {r}}_i$ is not proportional to the canonical momentum $\\mathbf {p}$ in a magnetic field.", "For the system “electron+proton” interacting with radiation in a constant magnetic field, these matrix elements are derived analytically in [258].", "In the dipole approximation, the cross section of photon interaction with a plasma particle can be expanded in three components corresponding to the longitudinal, right, and left polarizations with respect to the magnetic field (e.g., [189], [259]): $\\sigma (\\omega ,\\theta _B) =\\sum _{\\alpha =-1}^1 \\sigma _\\alpha ^{\\phantom{}}(\\omega )\\,\\,\|e_\\alpha (\\omega ,\\theta _B)\|^2.$ Here, $\\omega $ is the photon frequency, $\\theta _B$ is the angle between $\\mathbf {k}$ and $\\mathbf {B}$ (Fig.", "REF ), and $e_0\\equiv e_z$ and $e_{\\pm 1}\\equiv {(e_x\\pm \\mathrm {i}e_y)}/{\\sqrt{2}}$ are the components of the expansion of the electromagnetic polarization vector $\\mathbf {e}$ in a cyclic basis in the coordinate system with the $z$ -axis along $\\mathbf {B}$ .", "Representation (REF ) is convenient because $\\sigma _\\alpha $ do not depend on $\\theta _B$ .", "Scattering cross-sections in neutron-star photospheres are well known [260], [261], [262].", "For $\\alpha =-1$ , the photon-electron scattering has a resonance at the cyclotron frequency $\\omega _\\mathrm {c}$ .", "Outside a narrow (about the Doppler width) frequency interval around $\\omega _\\mathrm {c}$ , the cross sections for the basic polarizations $\\alpha =0,\\pm 1$ are written as $\\sigma _\\alpha ^\\mathrm {s,e} =\\frac{\\omega ^2}{(\\omega +\\alpha \\omega _\\mathrm {c})^2+\\nu _{\\mathrm {e},\\alpha }^2}\\, \\sigma _\\mathrm {T},$ where $\\sigma _\\mathrm {T}$ is the nonmagnetic Thomson cross section, Eq.", "(REF ), and the effective damping factors $\\nu _{\\mathrm {e},\\alpha }$ are equal to the half of the total rate of spontaneous and collisional decay of the electron state with energy $\\hbar \\omega $ (see [268]).", "The ion cross section looks analogously, $\\sigma _\\alpha ^\\mathrm {s,i} =\\left( \\frac{m_\\mathrm {e}}{m_\\mathrm {i}}\\right)^2\\frac{\\omega ^2\\,Z^4}{(\\omega -\\alpha \\omega _\\mathrm {ci})^2+\\nu _{\\mathrm {i},\\alpha }^2}\\, \\sigma _\\mathrm {T}.$ Unlike the nonmagnetic case, in superstrong fields one cannot neglect the scattering on ions, since $\\sigma _{+1}^\\mathrm {s,i}$ has a resonance at frequency $\\omega _\\mathrm {ci}$ .", "In the absence of magnetic field, absorption of a photon by a free electron is possible only at interaction with a third particle, which takes the difference of the total electron-photon momentum before and after the absorption.", "In a quantizing magnetic field, in addition, also electron transitions between the Landau levels are possible.", "In the nonrelativistic theory, such transitions occur between the equidistant neighboring levels at the frequency $\\omega _\\mathrm {c}$ , which corresponds to the dipole approximation.", "In the relativistic theory, the multipole expansion leads to an appearance of cyclotron harmonics [71].", "Absorption cross-sections at these harmonics were derived in [263] in the Born approximation without allowance for the magnetic quantization of electron motion, and represented in a compact form in [264].", "Allowance for the quantization of electron motion leads to the appearance of cyclotron harmonics in the nonrelativistic theory as well.", "In [265], also in the Born approximation, photon-electron absorption cross-sections were derived for an electron, which moves in a magnetic field and interacts with a nonmoving point charge.", "This model is applicable at $\\omega \\gg \\omega _\\mathrm {ci}$ .", "In the superstrong field of magnetars, the latter condition is unacceptable, therefore one should consider absorption of a photon by the system of finite-mass charged particles, which yields [266], [267] $\\sigma _\\alpha ^\\mathrm {ff}(\\omega )=\\frac{4\\pi e^2}{m_\\mathrm {e}c} \\,\\frac{\\omega ^2\\,\\nu _{\\alpha }^{\\mathrm {ff}}(\\omega )}{(\\omega +\\alpha \\omega _\\mathrm {c})^2 (\\omega -\\alpha \\omega _\\mathrm {ci})^2+\\omega ^2 \\tilde{\\nu }_\\alpha ^2(\\omega )},$ where $\\nu _{\\alpha }^{\\mathrm {ff}}$ is an effective photoabsorption collision frequency, and $\\tilde{\\nu }_\\alpha $ is an effective frequency including also other collisions.", "We see from (REF ) that $\\sigma _{-1}^\\mathrm {ff}$ and $\\sigma _{+1}^\\mathrm {ff}$ have a resonance at the frequencies $\\omega _\\mathrm {c}$ and $\\omega _\\mathrm {ci}$ , respectively.", "Expressions of the effective collision frequencies $\\nu _{\\alpha }^{\\mathrm {ff}}$ and $\\tilde{\\nu }_\\alpha $ in the electron-proton plasma are given in [266].", "One can write $\\nu _{\\alpha }^{\\mathrm {ff}}(\\omega ) =\\frac{4}{3}\\,\\sqrt{\\frac{2\\pi }{m_\\mathrm {e}T}}\\,\\frac{n_\\mathrm {e}\\, e^4}{\\hbar \\omega }\\Lambda _{\\alpha }^{\\mathrm {ff}},$ where $\\Lambda _{\\alpha }^{\\mathrm {ff}}=(\\pi /\\sqrt{3})g_{\\alpha }^{\\mathrm {ff}}$ is a Coulomb logarithm and $g_{\\alpha }^{\\mathrm {ff}}$ is a Gaunt factor, and $g_{-1}^{\\mathrm {ff}}=g_{+1}^{\\mathrm {ff}}$ .", "Without the magnetic field, the Gaunt factor is a smooth function of $\\omega $ .", "A calculation with allowance for the Landau quantization shows, however, that $\\nu _{\\alpha }^{\\mathrm {ff}}(\\omega )$ has peaks at the multiples of the electron and ion cyclotron frequencies for all polarizations $\\alpha $ .", "Free-free absorption in a hydrogen plasma with account of both (electron and ion) types of the cyclotron harmonics has been first calculated in [266], a detailed consideration is given in [267], and a generalization to the case of arbitrary hydrogenlike ions and a discussion of non-Born corrections are presented in [268].", "If $\\omega _\\mathrm {ci}/\\omega \\rightarrow 0$ , then the results of Ref.", "[265] for the electron photoabsorption are reproduced, but one should keep in mind that the ion cyclotron harmonics cannot be obtained by a simple scaling of the electron ones.", "Such scaling was used in neutron-star atmosphere models starting from the work [269] until the publication [266], where it was shown to be qualitatively wrong.", "One can see it in Fig.", "REF , where the electron and ion cyclotron harmonics are shown at equal scales.", "In spite of the choice of the same cyclotron frequency to temperature ratio, the cyclotron peaks in the upper panel are much weaker than in the lower panel.", "Physical reasons and consequences of this fact are discussed in detail in [267].", "It has been also demonstrated [267] that the ion cyclotron harmonics are so weak that they can be neglected in the neutron-star atmospheres.", "Figure: Electron (lower panel) and proton (upper panel) cyclotronharmonics of the Coulomb logarithm for free-free absorptionat ℏω c =5k B T\\hbar \\omega _\\mathrm {c}=5k_\\mathrm {B}T andℏω ci =5k B T\\hbar \\omega _\\mathrm {ci}=5k_\\mathrm {B}T, respectively, for the photonpolarization across the magnetic field.", "Solid lines show theresult of the accurate calculation ofΛ 1 (ω)=Λ -1 (ω)\\Lambda _1(\\omega )=\\Lambda _{-1}(\\omega ) in the Bornapproximation, and dot-dashed line in the upper panel shows theinfinite-proton-mass approximation (in the lower panel iteffectively coincides with the accurate result).", "Forcomparison, the dashed line in the lower panel shows thenonmagnetic Coulomb logarithm." ], [ "Atoms", "As first noticed in [270], atoms with bound states should be much more abundant at $\\gamma \\gg 1$ than at $\\gamma \\lesssim 1$ in a neutron-star atmosphere at the same temperature.", "This difference is caused by the magnetically-induced increase of binding energies and decrease of sizes of atoms in so-called tightly-bound states, which are characterized by electron-charge concentration at short distances to the nucleus.", "Therefore it is important to take account of the bound states and bound-bound transitions in a strong magnetic field even for light-element atmospheres, which would be almost fully ionized in the nonmagnetic case.", "Pioneering works by Loudon, Hasegawa and Howard [271], [272]The papers [271], [272] and some of the works cited below were devoted to the Mott exciton in a magnetized solid, which is equivalent to the problem of a hydrogen atom in a strong magnetic field.", "were at the origin of numerous studies of atoms in strong magnetic fields.", "In most of these studies the authors used the model of an atom with an infinitely heavy (fixed in space) nucleus.", "Their results are summarized in a number of reviews (e.g., [273], [274]).", "The model of an infinitely massive nucleus is too crude to describe the atoms in the strongly magnetized neutron-star atmospheres, but it is a convenient first approximation.", "Therefore, in this section we keep to this model, and postpone going beyond its frames to § REF .", "According to the Thomas-Fermi model, a typical size of an atom with a large nuclear charge $Z_\\mathrm {n}\\gg 1$ is proportional to $\\gamma ^{-2/5}$ in the interval $Z_\\mathrm {n}^{4/3} \\ll \\gamma \\ll Z_\\mathrm {n}^3$ [275].", "At $\\gamma \\gtrsim Z_\\mathrm {n}^3$ , the usual Thomas-Fermi model becomes inapplicable for an atom [276].", "In particular, it cannot describe the difference of the transverse and longitudinal atomic sizes, which becomes huge in such strong fields.", "In this field range, however, a good starting approximation is provided by so called adiabatic approximation, which presents each electron orbital as a product of a Landau function [256], describing free electron motion in the plane transverse to the field, and a function describing a one-dimensional motion of the electron along magnetic field lines in the field of an effective potential, similar to the Coulomb potential truncated at zero [277].", "At $\\gamma \\gg Z_\\mathrm {n}^3$ , all electron shells of the atom are strongly compressed in the directions transverse to the field.", "In the ground state, atomic sizes along and transverse to $\\mathbf {B}$ , respectively, can be estimated as [278] $l_\\perp \\approx \\sqrt{2Z_\\mathrm {n}-1}\\,a_\\mathrm {m},\\quad l_\\Vert \\approx \\frac{Z_\\mathrm {n}^{-1}a_\\mathrm {B}}{\\ln [\\sqrt{\\gamma }/(Z_\\mathrm {n}\\sqrt{2Z_\\mathrm {n}-1})]}.$ In this case, the binding energy $E^{(0)}$ of the ground state increases with increasing $\\mathbf {B}$ approximately as $(\\ln \\gamma )^2$ .", "Here and hereafter, the superscript (0) indicates the approximation of a nonmoving nucleus.", "At $Z_\\mathrm {n}\\gg 1$ and $\\gamma /Z_\\mathrm {n}^3\\rightarrow \\infty $ , the asymptotic estimate reads $E^{(0)}\\sim -Z_\\mathrm {n}\\hbar ^2/(m_\\mathrm {e}l_\\Vert ^2)$ [278].", "However, this asymptote is never reached in practice (see § REF ).", "Particularly many works were devoted to the simplest atom in magnetic field, the H atom.", "Since the electron resides on the ground Landau level $N=0$ in the hydrogen atom at $B>10^9$ G, its spin being directed opposite to the field, a bound state is determined by quantum numbers $s$ and $\\nu $ , where $s = 0,1,2,\\ldots $ corresponds to the electron orbital-momentum projection on the magnetic-field direction, $-\\hbar s$ , and $\\nu =0,1,2,\\ldots $ in the adiabatic approximation is equal to the number of wave-function nodes along this direction.", "The tightly-bound atomic states are characterized by the value $\\nu =0$ , while all non-zero values of $\\nu $ correspond to loosely-bound states.", "Calculations of the hydrogen-atom properties beyond the adiabatic approximation were performed by various methods (variational, discrete-mesh, etc.).", "At $\\gamma \\gg 1$ , the most natural method of calculations is the expansion of the wave function over the Landau orbitals, which constitute a complete orthogonal functional basis in the plane perpendicular to the magnetic field [279].", "Such calculations were done in [279], [280], [281] for the bound states and in [282] also for the continuum states, which allowed one to obtain the oscillator strengths as well as photoionization cross-sections.", "Examples of such cross-sections are presented in Fig.", "REF for the hydrogen atom at rest in strong magnetic fields with account of the finite proton mass.", "The broad peaks correspond to transitions to excited Landau levels $N>0$ , while the narrow peaks and dips near corresponding partial thresholds with $\\hbar \\omega \\approx N\\hbar \\omega _\\mathrm {c}$ are due to resonances related to autoionization of metastable states [282].", "Figure: Logarithm of photoionization cross-section, normalized to theThomson cross section (),log(σ/σ T )\\log (\\sigma /\\sigma _\\mathrm {T}), as function of photon energyℏω\\hbar \\omega for the ground state of the hydrogen atom atrest in magnetic field B=10 11 B=10^{11} G. The curves labelledby “+”, “--”, and “∥\\Vert ” display the cross sectionsfor circular and longitudinal polarizations α=+1\\alpha =+1,-1-1, and 0, respectively, and the curve labelled “⊥\\perp ”is for radiation polarized perpendicular to 𝐁\\mathbf {B}.", "Thewave vector 𝐤\\mathbf {k} is directed along 𝐁\\mathbf {B} forα±1\\alpha \\pm 1 and perpendicular to 𝐁\\mathbf {B} for the othertwo cases.", "(Fig.", "4 from , reproduced with thepermission of ©ESO.", ")Analytical expressions for atomic characteristics are best suited for astrophysical modeling.", "However, the asymptotic estimates at $\\gamma \\gg 1$ do not provide the desirable accuracy.", "For example, the binding energy of the ground-state hydrogen atom at rest, $E^{(0)}_{s\\nu }$ at $s=\\nu =0$ , when calculated in frames of the nonrelativistic quantum mechanics, goes to $(\\ln \\gamma )^2$ Ry in the limit $\\gamma \\rightarrow \\infty $ [271], [277], but this estimate is in error by a factor over 2 at any $B$ values that are encountered in the neutron stars.", "With account of two further terms of the asymptotic expansion [272] $E^{(0)}_{00}\\sim \\ln ^2(\\tilde{\\gamma }/\\ln ^2\\tilde{\\gamma })$ Ry, where $\\tilde{\\gamma }\\approx 0.28\\gamma $ .", "But even this estimate differs from accurate results by 40 – 80 % at $B\\sim 10^{12}$ – $10^{14}$ G. A possible way of solution to this problem consists in constructing analytical approximations to the results of numerical calculations.", "In [283] we gave accurate fitting formulae for many bound states of the hydrogen atom at $B\\lesssim 10^{14}$ G. The energy levels in the infinite-mass approximation have been recently revisited by Popov and Karnakov [284], who obtained analytical expressions, applicable at $B\\gtrsim 10^{11}$ G. Here we will give another approximation for the tightly-bound levels, valid at any $B$ .", "Temporarily ignoring corrections for vacuum polarization (§ REF ) and finite nuclear mass (§ REF ), we present the binding energy as $\\frac{E_{s,0}^{(0)}}{\\mbox{Ry}} = \\frac{(1+s)^{-2}+(1+s)\\, x/a_1 + a_3 x^3 + a_4 x^4 + a_6 x^6}{1 + a_2 x^2 + a_5 x^3 + a_6 x^4},$ where $x=\\ln (1+a_1\\gamma )$ .", "Here, $a_i$ are numerical parameters, which we approximate as functions of $s$ : $a_1 &=& {(0.862+2.5\\,s^2)}/{(1+0.018\\,s^3)} \\,,\\nonumber \\\\a_2 &=& 0.275 +0.1763\\,\\delta _{s,0} + s^{2.5}/6 \\,,\\nonumber \\\\a_3 &=& 0.2775+0.0202\\,s^{2.5},\\qquad \\nonumber \\\\a_4 &=& {0.3157}/{(1+2s)^2}- 0.26\\,\\delta _{s,0},\\nonumber \\\\a_5 &=& 0.0431,\\nonumber \\\\a_6 &=& {2.075\\times 10^{-3}}/{(1+7s^2)^{0.1}}+ 1.062\\times 10^{-4}\\,s^{2.5}\\,.\\nonumber $ Approximation (REF ) accurately reproduces the Zeeman shift of the lowest sublevel of each multiplet in the weak-field limit and the correct asymptote in the strong-field limit.", "Its inexactness is confined within 3% for $s<30$ at $\\gamma >1$ and for $s<5$ at any $\\gamma $ , and within 0.3% for $s=0$ at any $\\gamma $ .", "Binding energies of the loosely-bound states ($\\nu \\geqslant 1$ ) can be evaluated at $\\gamma \\gtrsim 1$ as $E_{s,\\nu }^{(0)} = \\frac{\\mbox{1~Ry}}{(n+\\delta )^2},$ where $&&\\hspace{-20.0pt}n=\\frac{\\nu +1}{2},\\quad \\delta \\approx \\frac{1+s/2}{1+2\\sqrt{\\gamma }+0.077\\gamma }\\quad \\mbox{for odd $\\nu $;}\\nonumber \\\\&&\\hspace{-20.0pt}n=\\frac{\\nu }{2},\\quad \\delta \\approx \\frac{1+s/8}{0.6+1.28\\ln (1+0.7\\gamma ^{1/3})}\\quad \\mbox{for even $\\nu $.", "}\\nonumber $ At $\\gamma \\rightarrow \\infty $ , energies (REF ) tend to those of a field-free H atom ($n^{-2}$ Ry), therefore the loosely-bound states are often called “hydrogenlike” (this picture is broken by vacuum polarization, § REF ).", "In the approximation of an infinite nuclear mass, the energy of any one-electron ion is related to the hydrogen atom energy as $E(Z_\\mathrm {n},B)=Z_\\mathrm {n}^2\\,E(1,B/Z_\\mathrm {n}^2)$ [285].", "Thus one sees, in particular, that the adiabatic approximation for the single-electron ions is applicable at $\\gamma \\gg Z_\\mathrm {n}^2$ , which is a weaker condition than for many-electron atoms.", "Analogous similarity relations exist also for the cross sections of radiative transitions [286].", "However, they are violated if one takes motion across the magnetic field into account.", "Even for an atom at rest, the account of the finite nuclear mass can be important at $s\\ne 0$ .", "These effects will be considered in § REF .", "Binding energies and oscillator strengths of many-electron atoms were successfully calculated with the use of different methods: variational (e.g., [287] and references therein), density-functional [288], [289], [290], Monte Carlo [291], [292], and the Hartree-Fock method [293], [294].", "In the simplest version of the Hartree-Fock method [274], [295], [296], the wave-function basis is constructed from the one-electron wave functions in the adiabatic approximation.", "This method is reliable for calculations of the energies, oscillator strengths, and photoionization cross sections of the helium atom [297].", "But for many-electron atoms the condition of applicability of the adiabatic approximation $\\gamma \\gg Z_\\mathrm {n}^3$ is too restrictive.", "It is overcome in the mesh Hartree-Fock method, where each one-electron orbital is numerically determined as a function of the longitudinal ($z$ ) and radial coordinates on a two-dimensional mesh [298] (see also [299], and references therein), and in the “twice self-consistent” method [300], where a transverse part of each orbital is presented as a linear superposition of the Landau functions with numerically optimized coefficients.", "These works gave a number of important results but were not realized in astrophysical applications.", "In practice, the optimal method for modeling neutron-star atmospheres containing atoms and ions of elements with $2<Z_\\mathrm {n}\\lesssim 10$ proves to be the method by Mori and Hailey [301], where corrections to the adiabatic Hartree approximation are treated by perturbation.", "The latter method can provide an acceptable accuracy at moderate computational expenses." ], [ "Molecules and molecular ions", "Molecular properties in strong magnetic fields have been studied during almost 40 years, but remain insufficiently known.", "Known the best are the properties of diatomic molecules oriented along the field, especially the H$_2$ molecule (see [302], and references therein).", "Lai [303] obtained approximate expressions for its binding energy at $\\gamma \\gtrsim 10^3$ , which grows approximately at the same rate $\\propto (\\ln \\gamma )^2$ as the atomic binding energy.", "In such strong fields, the ground state of this molecule is the state where the spins of both electrons are opposite to the magnetic field and the molecular axis is parallel to it, unlike the weak fields where the ground state is $^1\\Sigma _g$ .", "In moderate fields, the behavior of the molecular terms is quite nontrivial.", "If the molecular axis is parallel to $\\mathbf {B}$ , then the states $^1\\Sigma _g$ and $^3\\Pi _u$ are metastable at $0.18<\\gamma <12.3$ , and decay into the channel $^3\\Sigma _u$ [304].", "It turns out, however, that the molecular orientation along $\\mathbf {B}$ is not optimal in such fields: for example, at $\\gamma =1$ the triplet state of the molecule oriented perpendicular to the field has the lowest energy, and at $\\gamma =10$ the ground state is inclined at 37$^\\circ $ to $\\mathbf {B}$ [305].", "The ion H$_2$$^+$ is well studied, including its arbitrary orientations in a magnetic field (e.g., [306], and references therein).", "An analysis by Khersonskii [307] shows that the abundance of H$_2$$^+$ is very small in neutron-star atmospheres, therefore the these ions are unlikely to affect the observed spectra.", "Strong magnetic fields stabilize the molecule He$_2$ and its ions He$_2$$^{+}$ , He$_2$$^{2+}$ , and He$_2$$^{3+}$ , which do not exist in the absence of the field.", "Mori and Heyl [308] have performed the most complete study of their binding energies in neutron-star atmospheres.", "The ions HeH$^{++}$ , H$_3$$^{++}$ , and other exotic molecular ions, which become stable in the strong magnetic fields, were also considered (see [309], [310], and references therein).", "Having evaluated the ionization equilibrium by the Khersonskii's method [307], one can easily see that the abundance of such ions is extremely small at the densities, temperatures, and magnetic fields characteristic of the neutron stars.", "Therefore, such ions do not affect the thermal spectrum.", "There are rather few results on molecules composed of atoms heavier than He.", "Let us note the paper [311], where the authors applied the density-functional method to calculations of binding energies of various molecules from H$_n$ to Fe$_n$ with $n$ from 1 through 8 at $B$ from $10^{12}$ G to $2\\times 10^{15}$ G. The earlier studies of heavy molecules in strong magnetic fields are discussed in the review by Lai [303].", "All these studies assumed the model of infinitely massive atomic nuclei." ], [ "Relativistic effects", "One can encounter the statement that the use of the nonrelativistic quantum mechanics for calculation of atomic and molecular structure is justified only at $B <B_\\mathrm {QED}$ .", "However, a treatment of the hydrogen atom in strong magnetic fields based on the Dirac equation [312], [313], [314] has not revealed any significant differences from the solution to the same problem based on the Schrödinger equation.", "The reasons for that are clear.", "One can always expand a wave function over a complete basis of two-dimensional functions, such as the set of the Landau functions for all electrons.", "The Landau functions have the same form in the relativistic and nonrelativistic theories [256].", "Coefficients of such expansion are functions of $z$ corresponding to the electron motion along $\\mathbf {B}$ .", "This motion is nonrelativistic for the bound electrons, because the maximal binding energy is much smaller than the electron rest energy $m_\\mathrm {e}c^2=511$ keV.", "Therefore, a system of equations for the functions of $z$ in question can be solved in the nonrelativistic approximation, which thus provides the accurate wave function.", "Nevertheless, there is a specific relativistic effect, which is non-negligible in superstrong fields.", "As noted by Heisenberg and Euler [315], the virtual electron-positron pairs that appear in an electromagnetic field according to the Dirac theory, modify the Maxwell equations.", "This effect is called vacuum polarization.", "To date it has not been observed, but it was studied in many theoretical works, reviewed in detail by Schubert [316].", "A strong electromagnetic field creates a nonzero space charge by acting on the virtual pairs.", "Such charge, in particular, screens the Coulomb interaction between an electron and an atomic nucleus at distances comparable to the Compton wavelength $\\lambda _\\mathrm {C}=2\\pi \\hbar /(m_\\mathrm {e}c)=2\\pi \\alpha _\\mathrm {f}a_\\mathrm {B}$ .", "Shabad and Usov [317], [318] noted that this screening affects the even atomic levels in superstrong magnetic fields, which squeeze the atom so that its size becomes comparable to $\\lambda _\\mathrm {C}$ .", "As a result, instead of the unlimited growth of the binding energies of the tightly-bound states that is predicted by the nonrelativistic theory for unlimited increase of $B$ , these energies ultimately level off.", "For the same reason, the double degeneracy of the loosely-bound states that follows from Eq.", "(REF ) at $\\gamma \\rightarrow \\infty $ , does not realize.", "Machet and Vysotsky [319] have thoroughly studied this effect, confirmed the qualitative conclusions of Shabad and Usov, and obtained more accurate quantitative estimates.", "In particular, according to their results (see also [284]), the effect of the vacuum polarization on the electron binding energies in a nonmoving Coulomb potential can be simulated by replacing the parameter $\\gamma $ to $\\gamma ^\\ast = \\gamma /[1+\\alpha _\\mathrm {f}^3\\gamma /(3\\pi )]$ .", "As a result, the binding energy of the hydrogen atom cannot exceed 1.71 keV at any $B$ ." ], [ "The effects of finite nuclear mass", "An overwhelming majority of studies of atoms in strong magnetic fields assumed the nuclei to be infinitely massive (fixed in space).", "For magnetic neutron-star atmospheres, this approximation is very serious and often an undesirable simplification.", "Let us start with an atom with a nonmoving center of mass.", "The nucleus of a finite mass, as any charged particle, undergoes circular oscillations in the plane perpendicular to $\\mathbf {B}$ .", "In the atom, these oscillations cannot be separated from the electron oscillations, therefore the longitudinal projections of the orbital moments of the electrons and the nucleus are not conserved separately.", "Only their difference is conserved.", "Different atomic quantum numbers correspond to different oscillation energies of the atomic nucleus, multiple of its cyclotron energy.", "As a result, the energy of every level gets an addition, which is non-negligible if the parameter $\\gamma $ is not small compared to the nucleus-to-electron mass ratio.", "For the hydrogen atom and hydrogenlike ions, $\\hbar s$ in Eq.", "(REF ) now corresponds to the difference of longitudinal projections of orbital moments of the atomic nucleus and the electron, and the sum $N+s$ plays role of a nuclear Landau number, $N$ being the electron Landau number.", "For the bound states in strong magnetic fields, $N=0$ , therefore the nuclear oscillatory addition to the energy equals $s\\hbar \\omega _\\mathrm {ci}$ .", "Thus the binding energy of a hydrogen atom at rest is $E_{s\\nu }= E_{s\\nu }^{(0)}(\\gamma ^\\ast ) - \\hbar \\omega _\\mathrm {ci}s,$ where $\\gamma ^\\ast = \\gamma /(1+4.123\\times 10^{-8}\\,\\gamma )$ according to § REF .", "It follows that the number of $s$ values is limited for the bound states.", "In particular, one can easily check using Eqs.", "(REF ) and (REF ) that all bound states have zero moment-to-field projection ($s=0$ ) at $B>6\\times 10^{13}$ G. The account of the finite nuclear mass is more complicated for multielectron atoms.", "Al-Hujaj and Schmelcher [320] have shown that the contribution of the nuclear motion to the binding energy of a non-moving atom equals $\\hbar \\omega _\\mathrm {ci}S(1+\\delta (\\gamma ))$ , where ($-S$ ) is the total magnetic quantum number and $\|\\delta (\\gamma )\|\\ll 1$ .", "Figure: (a) energies, (b) oscillator strengths, and (c)photoionization cross-sections for a hydrogen atom moving inmagnetic field B=2.35×10 12 B=2.35\\times 10^{12} G. Energies of states\|s,0〉\|s,0\\rangle (solid curves) and \|0,ν〉\|0,\\nu \\rangle (dot-dashedcurves) are shown as functions of the transversepseudomomentum K ⊥ K_\\perp (in atomic units).", "The heavy dotson the solid curves are the inflection points atK ⊥ =K c K_\\perp =K_\\mathrm {c}.", "The K ⊥ K_\\perp -dependence ofoscillator strengths (b) is shown for transitions from theground state to the states \|s,0〉\|s,0\\rangle under influence ofradiation with polarization α=+1\\alpha =+1 (solid curves) andα=-1\\alpha =-1 (dashed curves), and also for transitions intostates \|0,ν〉\|0,\\nu \\rangle for α=0\\alpha =0 (dot-dashed curves).Cross sections of photoionization (c) under the influence ofradiation with α=+1\\alpha =+1 (solid curves), α=-1\\alpha =-1(dashed curves), and α=0\\alpha =0 (dot-dashed curves) areshown for the ground state as functions of the photon energyin Ry (the upper x-scale) and keV (the lower x-scale) atK ⊥ =20K_\\perp =20 a.u.", "(the right curve), K ⊥ =200K_\\perp =200 a.u.", "(the middle curve), and K ⊥ =1000K_\\perp =1000 a.u.", "(the left curveof every type).The astrophysical simulations require an account of finite temperatures, hence thermal motion of particles.", "The theory of motion of a system of point charges in a constant magnetic field is reviewed in [321], [322].", "The canonical momentum $\\mathbf {P}$ is not conserved in this motion, but a pseudomomentum $\\mathbf {K}=\\mathbf {P}+(1/2c)\\,\\mathbf {B}\\times \\sum _i q_i \\mathbf {r_i}$ is conserved.", "The pseudomomentum of a single charged particle has a one-to-one correspondence to the position of the guiding center in the $(xy)$ plane, perpendicular to the magnetic field, while a pseudomomentum of an atom or ion equals the sum of pseudomomenta of its constituent particles.", "If the system is electrically neutral as a whole, then all the components of $\\mathbf {K}$ are good quantum numbers.", "For a charged system (an ion), $K^2$ is a good quantum number, while $K_x$ and $K_y$ do not commute.", "The specific effects related to collective motion of a system of charged particles are especially important in a neutron-star atmosphere at $\\gamma \\gg 1$ .", "In particular, so called decentered states may become populated, where an electron is localized mostly in a “magnetic well” aside from the Coulomb center.", "For a hydrogen atom, $\\mathbf {K} = \\mathbf {P} + ({e}/{2c})\\,\\mathbf {B}\\times \\mathbf {R},$ where the vector $\\mathbf {R}$ connects the electron to the proton.", "The studies of this particular case were initiated in the pioneering works [323], [324], [325].", "Numerical calculations of the energy spectrum of the hydrogen atom with account of the effects of motion across a strong magnetic field were performed in [326], [327].", "Probabilities of various radiative transitions were studied in a series of papers ended with [258].", "Figure REF shows the energies, oscillator strengths, and photoionization cross-sections of a hydrogen atom moving in a magnetic field with $\\gamma =1000$ .", "The negative energies in Fig.", "REF a correspond to bound states.", "The reference point is taken to be the sum of the zero-point oscillation energies of free electron and proton, $(\\hbar \\omega _\\mathrm {c}+\\hbar \\omega _\\mathrm {ci})/2$ .", "At small transverse pseudomomenta $K_\\perp $ , the energies of low levels in Fig.", "REF a exceed the binding energy of the field-free hydrogen atom (1 Ry) by an order of magnitude.", "However, the total energy increases with increasing $K_\\perp $ , and it can become positive for the states with $s\\ne 0$ due to the term $\\hbar \\omega _\\mathrm {ci}s$ in Eq.", "(REF ).", "Such states are metastable.", "In essence, they are continuum resonances.", "Note that the transverse atomic velocity equals $\\partial E/\\partial \\mathbf {K}$ , therefore it is maximal at the inflection points ($K_\\perp =K_\\mathrm {c}$ ) on the curves in Fig.", "REF a and decreases with further increase of $K_\\perp $ [327], while the average electron-proton distance continues to increase.", "The atom goes into the decentered state, where the electron and proton are localized near their guiding centers, separated by distance $r_* = (a_\\mathrm {B}^2/\\hbar )K_\\perp /\\gamma $ .", "The dependences of the binding energies on $K_\\perp $ are approximately described at $K_\\perp \\ll K_\\mathrm {c}$ and $K_\\perp \\gg K_\\mathrm {c}$ , respectively, by expressions $\\hspace{-20.0pt}E_{s\\nu }^{(<)} &\\!\\!=&\\!\\!", "E_{s\\nu }^{(0)}- K_\\perp ^2/(2m_\\mathrm {eff}) - \\hbar \\omega _\\mathrm {ci}s,\\\\\\hspace{-20.0pt}E_{s\\nu }^{(>)} &\\!\\!\\!\\!\\!=&\\!\\!", "\\!\\!\\!\\frac{\\mbox{2\\,Ry}}{\\sqrt{\\hat{r}_^{2\\phantom{/}}+ (2\\nu +1)\\, \\hat{r}_^{3/2}+ \\ldots }}- \\hbar \\omega _\\mathrm {ci}s\\,,$ where $\\hat{r}_* \\equiv r_/a_\\mathrm {B}$ and $m_\\mathrm {eff}$ is an effective “transverse mass.” The latter is expressed through the values of $E_{s\\nu }^{(0)}$ for the given and neighboring levels by the perturbation theory [328], [329].", "However, for excited states even a small inaccuracy in $E_{s\\nu }^{(0)}$ may lead to a fatal error in $m_\\mathrm {eff}$ .", "Therefore in practice it is more convenient to use the approximation $m_\\mathrm {eff}\\approx m_\\mathrm {a}\\,[1+(\\gamma /\\gamma _{s\\nu })^{p_{s\\nu }}]$ , where $\\gamma _{s\\nu }$ and $p_{s\\nu }$ are dimensionless parameters, and $m_\\mathrm {a}$ is the true mass of the atom.", "For the tightly bound levels, $\\gamma _{s0}\\approx 6\\times 10^3/(1+2s)^2$ and $p_{s\\nu }\\approx 0.9$ .", "At $B\\lesssim 10^{13}$ G we can approximately describe the energies of the states with $\\nu =0$ at arbitrary $K_\\perp $ , if we replace the ellipsis under the square root in Eq.", "() by the expression $\\hat{r}_/(5+3s)+\\big (\\mbox{2~Ry}/E_{s\\nu }^{(0)}\\big )^2$ , and replace the inflection point $K_\\mathrm {c}$ by intersection of $E_{s\\nu }^{(<)}(K_\\perp )$ with $E_{s\\nu }^{(>)}(K_\\perp )$ .", "At stronger fields or for $\\nu \\ne 0$ , the transition between the centered and decentered states smears, and one has to resort to more complex fitting formulae [283].", "Figure REF b shows oscillator strengths for the main dipole-allowed transitions from the ground state to excited discrete levels as functions of $K_\\perp $ .", "Since the atomic wave-functions are symmetric with respect to the $z$ -inversion for the states with even $\\nu $ , and antisymmetric for odd $\\nu $ , only the transitions that change the parity of $\\nu $ are allowed for the polarization along the field ($\\alpha =0$ ), and only those preserving the parity for the orthogonal polarizations ($\\alpha =\\pm 1$ ).", "For the atom at rest, in the dipole approximation, due to the conservation of the $z$ -projection of the total angular momentum of the system, absorption of a photon with polarization $\\alpha =0,\\pm 1$ results in the change of $s$ by $\\alpha $ .", "This selection rule for a non-moving atom manifests itself in vanishing oscillator strengths at $K_\\perp \\rightarrow 0$ for $s\\ne \\alpha $ .", "In an appropriate coordinate system [324], [327], the symmetry is restored at $K_\\perp \\rightarrow \\infty $ , therefore the transition with $s=\\alpha $ is the only one that survives also in the limit of large pseudomomenta.", "But in the intermediate region of $K_\\perp $ , where the transverse atomic velocity is not small, the cylindrical symmetry is broken, so that transitions to other levels are allowed.", "Thus the corresponding oscillator strengths in Fig.", "REF b have maxima at $K_\\perp \\approx K_\\mathrm {c}$ .", "Analytical approximations for these oscillator strengths are given in [283].", "Figure REF c shows photoionization cross-sections for hydrogen in the ground state as functions of photon energy at three values of $K_\\perp $ .", "The leftward shift of the ionization threshold with increasing $K_\\perp $ corresponds to the decrease of the binding energy that is shown in Fig.", "REF a, while the peaks and dips on the curves are caused by resonances at transitions to metastable states $\|s,\\nu ;K\\rangle $ with positive energies (see [258], for a detailed discussion).", "Quantum-mechanical calculations of the characteristics of the He$^+$ ion that moves in a strong magnetic field are performed in [330], [331].", "The basic difference from the case of a neutral atom is that the the ion motion is restricted by the field in the transverse plane, therefore the values of $K^2$ are quantized [321], [322].", "Clearly, the similarity relations for the ions with nonmoving nuclei (§ REF ) do not hold anymore.", "Currently there is no detailed calculation of binding energies, oscillator strengths, and photoionization cross-sections for atoms and ions other than H and He$^+$ , arbitrarily moving in a strong magnetic field.", "For such species one usually neglects the decentered states and uses a perturbation theory with respect to $K_\\perp $ [328], [329].", "Such approach was realized, e.g., in [301], [297].", "It can be sufficient for simulations of relatively cool atmospheres of moderately magnetized neutron stars.", "Detailed conditions of applicability of the perturbation theory [328], [329] require calculations, but a rough order-of-magnitude estimate can be obtained by requiring that the mean Lorentz force acting on a bound electron because of the atomic thermal motion should be small compared to the Coulomb forces.", "As a result, for an atom with mass $m_\\mathrm {a}=Am_\\mathrm {u}$ we get the condition $k_\\mathrm {B}T/E_\\mathrm {b}\\ll m_\\mathrm {a}/(\\gamma m_\\mathrm {e})\\approx 4A/B_{12}$ , where $E_\\mathrm {b}$ is the atomic ionization energy.", "If $B\\lesssim 10^{13}$ G and $T\\lesssim 10^6$ K, it is well satisfied for low-lying levels of carbon and heavier atoms." ], [ "Equation of state", "Theoretical description of thermodynamics of partially ionized plasmas can be based on either “physical” of “chemical” models (see, e.g., a discussion and references in [332], [333]).", "In the chemical model of plasmas, bound states (atoms, molecules, ions) are treated as separate members of the thermodynamic ensemble, while in the physical model the only members of the ensemble are atomic nuclei and electrons.", "Each of the models can be thermodynamically self-consistent, but the physical model is more relevant from the microscopic point of view, because it does not require a distinction of electrons bound to a given nucleus.", "Such a distinction becomes very ambiguous at high densities, where several nuclei can attract the same electron with comparable forces.", "On the other hand, calculations in frames of the physical model are technically more complicated.", "As a rule, they are based on a diagram expansion, which requires an increase of the number of terms with the density increase.", "For this reason, even the most advanced equation of state for nonmagnetic photospheres that is based on the physical model [334] still restricts to the domain $\\rho \\lesssim 10\\, T_6^3$ g cm$^{-3}$.", "Studies of magnetic neutron-star photospheres, as a rule, are based on the chemical plasma model.", "In this case, the ionization equilibrium is evaluated by minimizing the Helmholtz free energy $F$ given by $F= F_\\mathrm {id}^{(e)} + F_\\mathrm {id}^\\mathrm {(i)} +F_\\mathrm {int} +F_\\mathrm {ex},$ where $F_\\mathrm {id}^{(e)}$ and $F_\\mathrm {id}^{(i)}$ describe the ideal electron and ion gases, $F_\\mathrm {int}$ includes internal degrees of freedom for bound states, and $F_\\mathrm {ex}$ is a nonideal component.", "All thermodynamic functions that are required for modeling a photosphere with a given chemical composition are expressed through derivatives of $F$ over $\\rho $ and $T$ [335].", "According to the Bohr-van Leeuwen theorem,This theorem was proved by different methods in PhD theses by Niels Bohr in 1911 and H.-J.", "van Leeuwen in 1919, and published by the latter in 1921 [336].", "magnetic field does not affect thermodynamics of classical charged particles.", "The situation differs in the quantum mechanics.", "The importance of the quantum effects depends on the parameters $\\zeta _\\mathrm {e}$ () and $\\zeta _\\mathrm {i}$ (REF ).", "We use the equality [335] $F_\\mathrm {id}^{(e)}/V =\\mu _\\mathrm {e} n_\\mathrm {e} - P_\\mathrm {id}^{(e)}$ where $V$ is the volume of the system, and $\\mu _\\mathrm {e}$ , $n_\\mathrm {e}$ , and $P_\\mathrm {id}^\\mathrm {(e)}$ are, respectively, the chemical potential, number density, and pressure in the ideal electron gas model.", "The equation of state is determined by a relation between these quantities, which can be found from relations (e.g., [1], [90]) $\\Bigg \\lbrace \\begin{array}{l}n_\\mathrm {e} \\\\ P_\\mathrm {id}^{(e)}\\end{array}\\Bigg \\rbrace =\\sum _{N,\\sigma }\\frac{(1+2bN)^{1/4}}{\\pi ^{3/2}a_\\mathrm {m}^2\\lambda _\\mathrm {e}}\\Bigg \\lbrace \\begin{array}{l}{\\partial I_{1/2}(\\chi _N,\\tau _N)}/{\\partial \\chi _N}\\\\k_\\mathrm {B}T\\,I_{1/2}^{\\phantom{I}}(\\chi _N,\\tau _N)\\end{array}\\Bigg \\rbrace ,$ where $\\lambda _\\mathrm {e} = [{2\\pi \\hbar ^2}/({ m_\\mathrm {e}k_\\mathrm {B}T})]^{1/2}$ is the thermal de Broglie wavelength, $\\tau _N = k_\\mathrm {B}T/(m_\\mathrm {e}c^2 \\sqrt{1+2bN})$ , $\\chi _N = \\mu _\\mathrm {e} / (k_\\mathrm {B}T) + \\tau _0^{-1} - \\tau _N^{-1}$ , $I_{1/2}(\\chi _N,\\tau _N) \\equiv \\int _0^\\infty \\frac{ \\sqrt{x \\,(1+\\tau _N x/2)}}{ \\exp (x-\\chi _N)+1 }\\,{\\mathrm {d}}x$ is the Fermi-Dirac integral, and the summation is done over all $N$ and all values of spin projections on the magnetic field, $\\hbar \\sigma /2$ , so that $\\sigma =\\pm 1$ for positive $N$ and $\\sigma =-1$ at $N=0$ .", "In a strongly quantizing magnetic field, it is sufficient to retain only the term with $N=0$ in the sums (REF ).", "In this case, the electron Fermi momentum equals $p_\\mathrm {F}=2\\pi ^2a_\\mathrm {m}^2\\hbar n_\\mathrm {e}$ .", "Therefore, with increasing $n_\\mathrm {e}$ at a fixed $B$ , the degenerate electrons begin to fill the first Landau level when $n_\\mathrm {e}$ reaches $n_B=(\\pi ^2\\sqrt{2}\\,a_\\mathrm {m}^3)^{-1}$ .", "This value just corresponds to the density $\\rho _B$ in Eq.", "(REF ).", "The ratio of the Fermi momentum $p_\\mathrm {F}$ in the strongly quantizing field to its nonmagnetic value $\\hbar (3\\pi ^2n_\\mathrm {e})^{1/3}$ equals $[{4\\rho ^2}/{(3\\rho _B^2)}]^{1/3}$ .", "Therefore, the Fermi energy at a given density $\\rho <\\sqrt{3/4}\\,\\rho _B$ becomes smaller with increasing $B$ , that is, a strongly quantizing magnetic field relieves the electron-gas degeneracy.", "For this reason, strongly magnetized neutron-star photospheres remain mostly nondegenerate, as it were in the absence of the field, despite their densities are orders of magnitude higher than the nonmagnetic photosphere densities.", "The free energy of nondegenerate nonrelativistic ions is given by $\\frac{F_\\mathrm {id}^\\mathrm {(i)}}{N_\\mathrm {i}k_\\mathrm {B}T}&=&\\ln \\left(2\\pi \\frac{n_\\mathrm {i}\\lambda _\\mathrm {i}a_\\mathrm {m}^2}{Z}\\right)+ \\ln \\left( 1- \\mathrm {e}^{- \\zeta _\\mathrm {i}}\\right) -1\\nonumber \\\\&& +\\frac{\\zeta _\\mathrm {i}}{2} + \\ln \\Bigg (\\frac{\\sinh [g_\\mathrm {i}\\,\\zeta _\\mathrm {i}(2s_\\mathrm {i}+1)/4] }{ \\sinh (g_\\mathrm {i}\\,\\zeta _\\mathrm {i}/ 4)}\\Bigg ),$ where $\\lambda _\\mathrm {i}= [2\\pi \\hbar ^2 / (m_\\mathrm {i}k_\\mathrm {B}T)]^{1/2}$ is the thermal de Broglie wavelength for the ions, $s_\\mathrm {i}$ is the spin number, and $g_\\mathrm {i}$ is the spin-related g-factor (for instance, $s_\\mathrm {i}=1/2$ and $g_\\mathrm {i}=5.5857$ for the proton).", "All the terms in (REF ) have clear physical meanings.", "At $\\zeta _\\mathrm {i}\\rightarrow 0$ , the first and second terms give together $\\ln (n_\\mathrm {i}\\lambda _\\mathrm {i}^3)$ , which corresponds to the three-dimensional Boltzmann gas.", "The first term corresponds to the one-dimensional Boltzmann gas model at $\\zeta _\\mathrm {i}\\gg 1$ .", "The second-last term in (REF ) gives the total energy $N_\\mathrm {i}\\hbar \\omega _\\mathrm {ci}/2$ of zero-point oscillations transverse to the magnetic field.", "Finally, the last term represents the energy of magnetic moments in a magnetic field.", "The nonideal free-energy part $F_\\mathrm {ex}$ contains the Coulomb and exchange contributions of the electrons and the ions, and the electron-ion polarization energy.", "In the case of incomplete ionization $F_\\mathrm {ex}$ includes also interactions of ions and electrons with atoms and molecules.", "In turn, the interaction between the ions is described differently depending on the phase state of matter.", "The terms that constitute $F_\\mathrm {ex}$ depend on magnetic field only if it quantizes the motion of these interacting particles.", "Here we will not discuss these terms but address an interested reader to the paper [90] and references therein.", "This nonideality is negligible in the neutron-star atmospheres, but it determines the formation of a condensed surface, which will be considered in § REF ." ], [ "Ionization equilibrium", "For photosphere simulations, it is necessary to determine the fractions of different bound states, because they affect the spectral features that are caused by bound-bound and bound-free transitions.", "Solution to this problem is laborious and ambiguous.", "The principal difficulty in the chemical plasma model, namely the necessity to distinguish the bound and free electrons and “attribute” the bound electrons to certain nuclei, becomes especially acute at high densities, where the atomic sizes cannot be anymore neglected with respect to their distances.", "Current approaches to the solution of this problem are based, as a rule, on the concept of so called occupation probabilities of quantum states.", "For example, consider electrons in thermodynamic equilibrium with ions of the $Z$ th chemical element, and let $j$ be the ionization degree of every ion (i.e., the number of lacking electrons), $\\kappa $ is its quantum state, and $E_{j,\\kappa }$ and $g_\\kappa ^{(j)}$ are, respectively, its binding energy and statistical weight.", "An occupation probability $w_{j,\\kappa }$ is an additional statistical weight of the given state under the condition of plasma nonideality, that is under interaction of the ion $(Z,j,\\kappa )$ with surrounding particles, with respect to its weight without such interactions.This ratio is not necessarily less than unity, thus the term “probability” is not quite correct, but we adhere to the traditional terminology.", "As first noted by Fermi [337], occupation probabilities $w_{j,\\kappa }$ cannot be arbitrary but should be consistent with $F_\\mathrm {ex}$ .", "Minimizing $F$ with account of the Landau quantization leads to a system of ionization-equilibrium equations for $n_j\\equiv \\sum _\\kappa n_{j,\\kappa }$ [338], [339] $\\hspace{-20.0pt}\\frac{n_j}{n_{j+1}}&=& n_\\mathrm {e} \\lambda _\\mathrm {e}^3\\,\\,\\frac{\\sinh (\\zeta _j/2)}{\\zeta _j}\\,\\frac{\\zeta _{j+1}}{\\sinh (\\zeta _{j+1}/2)}\\nonumber \\\\&&\\times \\,\\frac{\\tanh (\\zeta _\\mathrm {e}/2)}{\\zeta _\\mathrm {e}} \\,\\frac{\\mathcal {Z}_{\\mathrm {int},j}}{\\mathcal {Z}_{\\mathrm {int},j+1}}\\,\\exp \\left(\\frac{E_{j,\\mathrm {ion}}}{k_\\mathrm {B}T}\\right),$ where $\\mathcal {Z}_{\\mathrm {int},j}=\\sum _\\kappa g_\\kappa ^{(j)}\\,w_{j,\\kappa }\\,\\exp \\left[({E_{j,\\kappa }-E_{j,\\mathrm {gr.st}} })/({ {k_\\mathrm {B}}T})\\right]$ is internal partition function for the $j$ th ion type, $E_{j,\\mathrm {gr.st}}$ is its ground-state binding energy, $E_{j,\\mathrm {ion}}=E_{j,\\mathrm {gr.st}}-E_{j+1,\\mathrm {gr.st}}$ is its ionization energy, and $\\zeta _j$ is the magnetic quantization parameter (REF ).", "Equation (REF ) differs from the usual Saha equation, first, by the terms with $\\zeta _\\mathrm {e}$ and $\\zeta _j$ , representing partition functions for distributions of free electrons and ions over the Landau levels, and second, by the occupation probabilities $w_{j,\\kappa }$ in the expressions for the partition functions $\\mathcal {Z}_{\\mathrm {int},j}$ .", "There were many attempts to find such approximation for the occupation probabilities that best reproduced the real plasma EOS.", "They were discussed, for example, by Hummer and Mihalas [340], who proposed an approximation based on the Inglis-Teller criterion [341] for dissolution of spectral lines because of their smearing due to the Stark shifts in plasma microfields.", "However, the translation of the spectroscopic criterion to thermodynamics is not well grounded.", "It is necessary to clearly distinguish between the disappearance of spectral lines of an atom and the complete destruction of this atom with increasing pressure, as was stressed, e.g., in [342], [343], [344].", "In order to take this difference into account, in [345] we introduced a concept of optical occupation probabilities $\\tilde{w}_{j,\\kappa }$ , which resemble the Hummer-Mihalas occupation probabilities and should be used for calculation of spectral opacities, but differ from the thermodynamic occupation probabilities $w_{j,\\kappa }$ that are used in the EOS calculations.", "Equation (REF ) was applied to modeling partially ionized atmospheres of neutron stars, composed of iron, oxygen, and neon [346], [339], [347], [348].", "The effects related to the finite nuclear masses (§ REF ) were either ignored or treated in the first order of the perturbation theory.", "Since quantum-mechanical characteristics of an atom in a strong magnetic field depend on the transverse pseudomomentum $K_\\perp $ , the atomic distribution over $K_\\perp $ cannot be written in a closed form, and only the distribution over longitudinal momenta $K_z$ remains Maxwellian.", "The first complete account of these effects has been taken in [349] for hydrogen photospheres.", "Let $p_{s\\nu }(K_\\perp )\\,\\mathrm {d}^2K_\\perp $ be the probability of finding a hydrogen atom in the state $\|s,\\nu \\rangle $ in the element $\\mathrm {d}^2K_\\perp $ near $\\mathbf {K}_\\perp $ in the plane of transverse pseudomomenta.", "Then the number of atoms in the element $\\mathrm {d}^3K$ of the pseudomomentum space equals $\\mathrm {d}N(\\mathbf {K}) = N_{s\\nu }\\,\\frac{\\lambda _\\mathrm {a} }{2\\pi \\hbar }\\,\\exp \\left(-\\frac{K_z^2}{2m_\\mathrm {a}k_\\mathrm {B}T}\\right)\\, p_{s\\nu }(K_\\perp ) \\mathrm {d}^3K,$ where $m_\\mathrm {a}$ is the mass of the atom, $\\lambda _\\mathrm {a} = [{2\\pi \\hbar ^2}/({ m_\\mathrm {a} k_\\mathrm {B}T})]^{1/2}$ is its thermal wavelength, and $N_{s\\nu }=\\int \\mathrm {d}N_{s\\nu }(\\mathbf {K})$ is the total number of atoms with given discrete quantum numbers.", "The distribution $N_{s\\nu } p_{s\\nu }(K_\\perp )$ is not known in advance, but should be calculated in a self-consistent way by minimization of the free energy including the nonideal terms.", "It is convenient to define deviations from the Maxwell distribution with the use of generalized occupation probabilities $w_{s\\nu }(K_\\perp )$ .", "Then the atomic contribution $(F_\\mathrm {id}+F_\\mathrm {int})$ to the free energy equals [349] $k_\\mathrm {B}T \\sum _{s\\nu } N_{s\\nu }\\int \\ln \\left[n_{s\\nu } \\lambda _\\mathrm {a}^3\\frac{w_{s\\nu }(K_\\perp ) }{ \\exp (1) \\mathcal {Z}_{s\\nu }}\\right]p_{s\\nu }(K_\\perp )\\,\\mathrm {d}^2K_\\perp ,$ where $\\mathcal {Z}_{s\\nu } =\\frac{ \\lambda _\\mathrm {a}^2 }{ (2\\pi \\hbar ^2) }\\int _0^\\infty w_{s\\nu }(K_\\perp )\\mathrm {e}^{E_{s\\nu }(K_\\perp )/k_\\mathrm {B}T}K_\\perp \\mathrm {d}K_\\perp .$ The nonideal part of the free energy that describes atom-atom and atom-ion interactions and is responsible for the pressure ionization has been calculated in [349] with the use of the hard-sphere model.", "The plasma model included also hydrogen molecules H$_2$ and chains H$_n$ , which become stable in the strong magnetic fields.", "For this purpose, approximate formulae of Lai [303] have been used, which do not take full account of the motion effects, therefore the results of [349] are reliable only when the molecular fraction is small.", "This hydrogen-plasma model underlies thermodynamic calculations of hydrogen photospheres of neutron stars with strong [266] and superstrong [350] magnetic fields.Some results of these calculations are available at http://www.ioffe.ru/astro/NSG/Hmagnet/ Mori and Heyl [308] applied the same approach with slight modifications to strongly magnetized helium plasmas.", "One of the modifications was the use of the plasma microfield distribution from [226] for calculation of $w(K_\\perp )$ .", "Mori and Heyl considered atomic and molecular helium states of different ionization degrees.", "Their treated rotovibrational molecular levels by perturbation theory and considered the dependence of binding energies on orientation of the molecular axis relative to $\\mathbf {B}$ .", "The $K_\\perp $ -dependence of the energy, $E(K_\\perp )$ , was described by an analytical fit, based on an extrapolation of adiabatic calculations at small $K_\\perp $ .", "The motion effects of atomic and molecular ions were not considered." ], [ "Applicability of the LTE approximation", "The models of EOS and ionization balance usually assume that the LTE conditions are satisfied for the atoms and ions.", "In particular, the Boltzmann distribution over the Landau levels is assumed.", "This assumption does not apply for free electrons in the neutron-star atmospheres, if the spontaneous radiative decay rate of excited Landau levels, $\\Gamma _{\\mathrm {r}}= \\frac{4}{3}\\,\\frac{e^2\\omega _\\mathrm {c}^2}{m_\\mathrm {e}c^3}=3.877\\times 10^{15}B_{12}^2\\mbox{ s}^{-1},$ exceeds the rate of their collisional de-excitation.", "In a nonquantizing field, the characteristic frequency of electron-ion Coulomb collisions equals (see, e.g., [351]) $\\Gamma _{\\mathrm {c}}= \\frac{4\\sqrt{2\\pi }n_\\mathrm {i}Z^2 e^4\\Lambda _\\mathrm {c}}{3\\sqrt{m_\\mathrm {e}} (k_\\mathrm {B}T)^{3/2}} =2.2\\times 10^{15}\\,\\frac{Z^2}{A}\\,\\frac{\\rho ^{\\prime }\\Lambda _\\mathrm {c}}{T_6^{3/2}}\\mbox{ s}^{-1},$ where $\\rho ^{\\prime }\\equiv \\rho /$g cm$^{-3}$, and $\\Lambda _\\mathrm {c}$ is a Coulomb logarithm, which weakly depends on $T$ and $\\rho $ and usually has an order of magnitude of 1 – 10.", "In a quantizing field, the electrons are de-excited from the first Landau level by electron-ion Coulomb collisions at the rate $\\hspace{-4.25pt}\\Gamma _{10}&\\!\\!\\!\\!=&\\!\\!\\!\\!\\frac{4\\sqrt{2\\pi }\\,n_\\mathrm {i}Z^2 e^4\\tilde{\\Lambda }_{10}}{\\sqrt{m_\\mathrm {e}} \\,\\,(\\hbar \\omega _\\mathrm {c})^{3/2}} =4.2\\times 10^{12}\\,\\frac{Z^2}{A}\\,\\frac{\\rho ^{\\prime }\\tilde{\\Lambda }_{10}}{B_{12}^{3/2}}\\mbox{~s}^{-1}\\nonumber \\\\&&=4.9\\times 10^{13}\\,\\frac{Z^2}{A}\\frac{\\rho ^{\\prime }\\Lambda _{10}}{B_{12}\\sqrt{T_6}}\\mbox{~s}^{-1},$ where $\\tilde{\\Lambda }_{10}=\\sqrt{\\zeta _\\mathrm {e}}\\Lambda _{10}$ is a new Coulomb logarithm, which has an order of unity at $\\zeta _\\mathrm {e}\\gg 1$ , whereas $\\Lambda _{10}$ has that order at $\\zeta _\\mathrm {e}\\ll 1$ [268].", "Note that the rate of the inverse process of collisional excitation equals $\\Gamma _{01}=\\Gamma _{10}\\,\\mathrm {e}^{-\\zeta _\\mathrm {e}}$ .", "Comparing (REF ) and (REF ), we see that in the weak-field ($B\\lesssim 10^{10}$ G) photospheres of isolated neutron stars, at typical $\\rho \\gtrsim 10^{-3}$ g cm$^{-3}$ and $T_6\\sim 1$ , the LTE conditions are fulfilled (it may not be the case in the magnetosphere due to the lower densities).", "In a strong field ($B\\gtrsim 10^{11}$ G), the LTE is violated, and the fraction of electrons on the excited Landau levels is lower than the Boltzmann value $\\mathrm {e}^{-\\zeta _\\mathrm {e}}$ .", "However, this does not entail any consequence for the atmosphere models, because in the latter case $\\mathrm {e}^{-\\zeta _\\mathrm {e}}$ is vanishingly small.", "For the ions, the spontaneous decay rate of the excited Landau levels $\\Gamma _{\\mathrm {ri}}$ differs from $\\Gamma _{\\mathrm {r}}$ by a factor of $Z\\,(Z m_\\mathrm {e}/m_\\mathrm {i})^3 \\sim 10^{-10}$ .", "The statistical distribution of ions over the Landau levels has been studied in [268].", "The authors showed that the fraction of the ions on the first excited Landau level is accurately given by $\\frac{n_1}{n_0} = \\mathrm {e}^{-\\zeta _\\mathrm {i}} \\,\\frac{ 1 +\\epsilon \\,(\\Gamma _{\\mathrm {ri}}/\\Gamma _\\mathrm {10,i})/(1-\\mathrm {e}^{-\\zeta _\\mathrm {i}}) }{1+ \\Gamma _{\\mathrm {ri}}/\\Gamma _\\mathrm {10,i} +\\epsilon \\,(\\Gamma _{\\mathrm {ri}}/\\Gamma _\\mathrm {10,i})/(\\mathrm {e}^{\\zeta _\\mathrm {i}}-1) } ,$ where $\\epsilon =J_{\\omega }/\\mathcal {B}_{\\omega ,T}$ at $\\omega =\\omega _\\mathrm {ci}$ , and $\\Gamma _\\mathrm {10,i}$ is the collisional frequency of the first level, which differs from Eq.", "(REF ) by the factor $\\sqrt{m_\\mathrm {i}/m_\\mathrm {e}}$ and the value of the Coulomb logarithm.", "The parameter $\\epsilon $ is small in the outer layers of the photospheres, therefore the distribution over the levels is determined by the ratio $\\Gamma _{\\mathrm {ri}}/\\Gamma _\\mathrm {10,i}$ .", "If $\\Gamma _{\\mathrm {ri}}/\\Gamma _\\mathrm {10,i}\\ll 1$ , then the Boltzmann distribution is recovered, that is, the LTE approximation holds; otherwise the excited levels are underpopulated.", "According to [268], $\\frac{\\Gamma _\\mathrm {10,i}}{\\Gamma _{\\mathrm {ri}}}\\sim \\frac{ \\rho ^{\\prime }}{(B_{12}/300)^{7/2}}\\,.$ In the atmospheres and at the radiating surfaces of the ordinary neutron stars this ratio is large, because the denominator is small, and for magnetars with $B\\lesssim 10^{15}$ G the ratio is large because $\\rho ^{\\prime }$ is large (see (REF )).", "Moreover, as shown in [268], even in the outer atmospheres of magnetars, where $\\Gamma _{\\mathrm {ri}}/\\Gamma _\\mathrm {10,i}\\ll 1$ , deviations from the LTE should not affect the spectral modeling.", "The reason is that absorption coefficients are mainly contributed from the second-order quantum transitions that do not change the Landau number $N$ .", "Therefore the depletion of the upper states is unimportant, so that the Kirchhoff law, which holds at the LTE, remains approximately valid also in this case." ], [ "Condensed surface", "Ruderman [352] suggested that a strong magnetic field can stabilize polymer chains directed along the field lines, and that the dipole-dipole attraction of these chains may result in a condensed phase.", "Later works have shown that such chains indeed appear in the fields $B\\sim 10^{12}$ – $10^{13}$ G, but only for the chemical elements lighter than oxygen, and they polymerize into a condensed phase either in superstrong fields, or at relatively low temperatures, the sublimation energy being much smaller than Ruderman assumed (see [353], and references therein).", "From the thermodynamics point of view, the magnetic condensation is nothing but the plasma phase transition caused by the strong electrostatic attraction between the ionized plasma particles.", "This attraction gives a negative contribution to pressure $P_\\mathrm {ex}$ , which is not counterbalanced at low temperatures (at $\\Gamma _{\\mathrm {Coul}}\\gtrsim 1$ ) until the electrons become degenerate with increasing density.", "In the absence of a magnetic field, such phase transitions were studied theoretically since 1930s (see [354], for a review).", "In this case, the temperature of the outer layers of a neutron star $T\\gtrsim (10^5-10^6)$ K exceeds the critical temperature $T_\\mathrm {crit}$ for the plasma phase transition.", "However, we have seen in § REF that a quantizing magnetic field lifts electron degeneracy.", "As a result, $T_\\mathrm {crit}$ increases with increasing $B$ , which may enable such phase transition.", "Lai [303] estimated the condensed-surface density as $\\rho _\\mathrm {s}\\approx 561\\,\\eta \\,A Z^{-3/5}B_{12}^{6/5}\\mbox{~\\mbox{g~cm$^{-3}$}},$ where $\\eta $ is an unknown factor of the order of unity.", "In the ion-sphere model [355], the electrons are replaced by a uniform negative background, and the potential energy per ion is estimated as the electrostatic energy of the ionic interaction with the negative background contained in the sphere of radius $a_\\mathrm {i}=(4\\pi n_\\mathrm {i}/3)^{-1/3}$ .", "By equating $\|P_\\mathrm {ex}\|$ to the pressure of degenerate electrons $P_\\mathrm {e}$ , one obtains Eq.", "(REF ) with $\\eta =1$ .", "This estimate disregards the ion correlation effects, the electron-gas polarizability, and bound state formation.", "Taking account of the electron polarization by different versions of the Thomas-Fermi method, one gets quite different results: for example, the zero-temperature Thomas-Fermi data for a magnetized iron at $10^{10}\\mbox{~G}\\leqslant B\\leqslant 10^{13}$ G [356] can be described by Eq.", "(REF ) with $\\eta \\approx 0.2 +0.01/B_{12}^{0.56}$ , and in a finite-temperature Thomas-Fermi model [357] there is no phase transition at all.", "At $1\\lesssim B_{12}\\lesssim 10^3$ , the EOS of partially ionized, strongly magnetized hydrogen [349] that was described in § REF predicts a phase transition with the critical temperature $T_\\mathrm {crit}\\approx 3\\times 10^5\\,B_{12}^{0.39}$ K and critical density $\\rho _\\mathrm {crit} \\approx 143\\,B_{12}^{1.18}\\mbox{~\\mbox{g~cm$^{-3}$}}$ , which corresponds to $\\eta \\approx 1/4$ .", "With decreasing temperature below $T_\\mathrm {crit}$ , the condensed-phase density increases and tends asymptotically to Eq.", "(REF ) with $\\eta \\approx 1/2$ , while the density of the gaseous phase quickly decreases, and the atmosphere becomes optically thin.", "Lai and Salpeter [358] obtained qualitatively similar results from calculations of density of saturated vapor above the condensed surface, but with 3–4 times lower $T_\\mathrm {crit}$ .", "The quantitative differences may be caused by the less accurate approximate treatment of the molecular contribution in [349], on one hand, and by the less accurate account of the effects of atomic motion across the magnetic field in [358], on the other hand.", "Medin and Lai [353] treated the condensation energy by the density functional method.", "In [359] they calculated the equilibrium density of a saturated vapor of the atoms and polymer chains of helium, carbon, and iron above the respective condensed surfaces at $1\\lesssim B_{12} \\leqslant 10^3$ .", "By equating this density to $\\rho _\\mathrm {s}$ , they found $T_\\mathrm {crit}$ at several $B$ values.", "Unlike previous authors, Medin and Lai [353], [359] have taken a self-consistent account of the electron band structure in the condensed phase.", "Meanwhile, they did not take account of the effects of atomic and molecular motion across the magnetic field in the gaseous phase and treated the excited-states contribution rather roughly.", "They calculated the condensed-surface density assuming that the linear atomic chains, being unchanged as such, form a rectangular lattice in the plane, perpendicular to ${\\mathbf {B}}$ .", "As shown in [90], such evaluated values of $\\rho _\\mathrm {s}$ can be described by Eq.", "(REF ) with $\\eta =0.517+0.24/B_{12}^{1/5}\\pm 0.011$ for carbon and $\\eta =0.55\\pm 0.11$ for iron, and the critical temperature can be evaluated as $T_\\mathrm {crit}\\sim 5\\times 10^4\\,Z^{1/4}\\,B_{12}^{3/4}$ K. For comparison, in the fully-ionized plasma model $T_\\mathrm {crit}\\approx 2.5\\times 10^5\\,Z^{0.85}\\,B_{12}^{0.4}$ K and $\\eta = [1+1.1\\,(T/T_\\mathrm {crit})^5]^{-1}$ .", "Hopefully, the present uncertainty in $\\rho _\\mathrm {s}$ and $T_\\mathrm {crit}$ estimates may be diminished with an analysis of future neutron-star observations.", "When magnetic field increases from $10^{12}$ G to $10^{15}$ G, the cohesive energy, calculated in [359] for the condensed surface, varies monotonically from 0.07 keV to 5 keV for helium, from 0.05 keV to 20 keV for carbon, and from 0.6 keV to 70 keV for iron.", "The power-law interpolation gives order-of-magnitude estimates between these limits.", "The electron work function changes in the same $B$ range from 100 eV to $(600\\pm 50)$ eV.", "With the calculated energy values, the authors [359] determined the conditions of electron and ion emission in the vacuum gap above the polar cap of a pulsar and the conditions of gap formation, and calculated the pulsar death lines on the $\\mathcal {P}$ – $\\dot{\\mathcal {P}}$ plane." ], [ "Radiative transfer in normal modes", "Propagation of electromagnetic waves in magnetized plasmas was studied in many works, the book by Ginzburg [189] being the most complete of them.", "At radiation frequency $\\omega $ much larger than the electron plasma frequency $\\omega _{\\mathrm {pe}}=\\left({4\\pi e^2 n_\\mathrm {e} / m_\\mathrm {e}^\\ast } \\right)^{1/2}$ , where $m_\\mathrm {e}^\\ast \\equiv m_\\mathrm {e}\\sqrt{1+p_\\mathrm {F}^2/(m_\\mathrm {e}c)^2}$ is the effective dynamic mass of an electron at the Fermi surface, the waves propagate in the form of two polarization modes, extraordinary (hereafter denoted by subscript or superscript $j=1$ or X) and ordinary ($j=2$ or O).", "They have different polarization vectors $\\mathbf {e}_j$ and different absorption and scattering coefficients, which depend on the angle $\\theta _B$ (Fig.", "REF ).", "The modes interact with one another through scattering.", "Ventura [260] performed an analysis of the polarization modes in application to the neutron stars from the physics point of view.", "Gnedin and Pavlov [360] formulated the radiative transfer problem in terms of these modes.", "They showed that in the strongly magnetized neutron-star atmospheres, as a rule, except narrow frequency ranges near resonances, a strong Faraday depolarization occurs.", "In this case, it is sufficient to consider specific intensities of the two normal modes instead of the four components of the Stokes vector.", "The radiative transfer equation for these specific intensities is a direct generalization of Eq.", "(REF ) [261]: $&&\\hspace{-20.0pt}\\cos \\theta _k \\frac{\\mathrm {d}I_{\\omega ,j}(\\hat{\\mathbf {k}})}{\\mathrm {d}y_\\mathrm {col}} =\\varkappa _{\\omega ,j}(\\hat{\\mathbf {k}}) I_{\\omega ,j}(\\hat{\\mathbf {k}})-\\frac{1}{2}\\,\\varkappa _{\\omega ,j}^\\mathrm {a}(\\hat{\\mathbf {k}})\\mathcal {B}_{\\omega ,T}\\nonumber \\\\&&-\\sum _{j^{\\prime }=1}^2 \\int _{(4\\pi )}\\varkappa _{\\omega ,j^{\\prime }j}^\\mathrm {s}(\\hat{\\mathbf {k}}^{\\prime },\\hat{\\mathbf {k}})I_{\\omega ,j^{\\prime }}(\\hat{\\mathbf {k}}^{\\prime }) \\,\\mathrm {d}\\hat{\\mathbf {k}}^{\\prime },$ where $\\varkappa _{\\omega ,j}(\\hat{\\mathbf {k}}) \\equiv \\varkappa _{\\omega ,j}^\\mathrm {a}(\\hat{\\mathbf {k}}) + \\sum _{j^{\\prime }=1}^2\\int _{(4\\pi )}\\varkappa _{\\omega ,j^{\\prime }j}^\\mathrm {s}(\\hat{\\mathbf {k}}^{\\prime },\\hat{\\mathbf {k}})\\,\\mathrm {d}\\hat{\\mathbf {k}}^{\\prime }$ .", "The dependence of the opacities $\\varkappa $ on ray directions $(\\hat{\\mathbf {k}},\\hat{\\mathbf {k}}^{\\prime })$ is affected by the magnetic-field direction.", "Therefore, the emission of a magnetized atmosphere, unlike the nonmagnetic one, depends not only on the angle $\\theta _k$ that determines the ray inclination to the stellar surface, but also on the angles $\\theta _\\mathrm {n}$ and $\\varphi _k$ in Fig.", "REF .", "For hydrostatic and energy balance, we can keep Eqs.", "(REF ), (REF ), and (REF ), if we put $I_\\omega =\\sum _{j=1}^2 I_{\\omega ,j}$ by definition.", "The diffusion equation for the normal modes in these approximations was derived in [361], [261].", "For the plane-parallel photosphere it reads [116] $&&\\hspace{-20.0pt}\\frac{\\mathrm {d}}{\\mathrm {d}y_\\mathrm {col}}D_{\\omega ,j}\\frac{\\mathrm {d}}{\\mathrm {d}y_\\mathrm {col}} J_{\\omega ,j} =\\bar{\\varkappa }_{\\omega ,j}^\\mathrm {a}\\,\\left[ J_{\\omega ,j} -\\frac{\\mathcal {B}_{\\omega ,T}}{2} \\right]\\nonumber \\\\&&+\\bar{\\varkappa }_{\\omega ,12}^\\mathrm {s}\\left[J_{\\omega ,j} - J_{\\omega ,3-j} \\right].\\hspace{20.0pt}$ Here, $J_{\\omega ,j} &=& \\frac{1}{4\\pi }\\int _{(4\\pi )}I_{\\omega ,j}(\\hat{\\mathbf {k}})\\,\\mathrm {d}\\hat{\\mathbf {k}},\\nonumber \\\\\\bar{\\varkappa }_{\\omega ,j}^\\mathrm {a} &=& \\frac{1}{4\\pi }\\int _{(4\\pi )}\\varkappa _{\\omega ,12}^\\mathrm {a}\\,\\mathrm {d}\\hat{\\mathbf {k}},\\nonumber \\\\\\bar{\\varkappa }_{\\omega ,j}^\\mathrm {s} &=& \\frac{1}{4\\pi }\\int _{(4\\pi )}\\mathrm {d}\\hat{\\mathbf {k}}^{\\prime }\\int _{(4\\pi )} \\mathrm {d}\\hat{\\mathbf {k}}\\,\\,\\varkappa _{\\omega ,12}^\\mathrm {s}(\\hat{\\mathbf {k}}^{\\prime },\\hat{\\mathbf {k}}) \\, ,\\nonumber $ and the effective diffusion coefficient equals $D_{\\omega ,j} = \\frac{1}{3\\varkappa _{\\omega ,j}^{\\mathrm {eff}}}=\\frac{\\cos ^2\\theta _\\mathrm {n}}{3\\varkappa _{\\omega ,j}^\\Vert } +\\frac{\\sin ^2\\theta _\\mathrm {n}}{3\\varkappa _{\\omega ,j}^\\perp },$ where $\\theta _\\mathrm {n}$ is the angle between $\\mathbf {B}$ and intensity gradient, $\\left\\lbrace \\begin{array}{c}(\\varkappa _j^\\Vert )^{-1}\\\\(\\varkappa _j^{\\perp })^{-1\\rule {0pt}{2ex}}\\end{array}\\right\\rbrace = \\frac{3}{4} \\int _0^\\pi \\left\\lbrace \\begin{array}{c}2\\cos ^2\\theta _B \\\\\\sin ^2\\theta _B\\end{array}\\right\\rbrace \\frac{\\sin \\theta _B\\,\\mathrm {d}\\theta _B}{\\varkappa _j(\\theta _B)}\\,.$ The effective opacity for nonpolarized radiation is $\\varkappa ^{\\mathrm {eff}}={2}/(3D_{\\omega ,1}+3D_{\\omega ,2}).$ The diffusion approximation (REF ) serves as a starting point in an iterative method [362], which allows one to solve the system (REF ) more accurately." ], [ "Plasma polarizability", "In the Cartesian coordinate system with the $z$ -axis along $\\mathbf {B}$ , the plasma dielectric tensor is [189] $\\mathbf {\\varepsilon } = \\mathbf {I} + 4\\pi \\chi =\\left( \\begin{array}{ccc}\\varepsilon _\\perp & i \\varepsilon _\\wedge & 0 \\\\-i\\varepsilon _\\wedge & \\varepsilon _\\perp & 0 \\\\0 & 0 & \\varepsilon _\\Vert \\end{array} \\right),$ where $\\mathbf {I}$ is the unit tensor, $\\chi =\\chi ^\\mathrm {H}+\\mathrm {i}\\chi ^\\mathrm {A}$ is the complex polarizability tensor of plasma, $\\chi ^\\mathrm {H}$ and $\\chi ^\\mathrm {A}$ are its Hermitian and anti-Hermitian parts, respectively.", "Under the assumption that the electrons and ions lose their regular velocity, acquired in an electromagnetic wave, by collisions with an effective frequency $\\nu _\\mathrm {eff}$ independent of the velocities, then the cyclic components of the polarizability tensor are ([189], § 10) $\\chi _\\alpha = -\\frac{1}{4\\pi }\\,\\frac{\\omega _{\\mathrm {pe}}^2}{(\\omega + \\alpha \\omega _\\mathrm {c})\\,(\\omega - \\alpha \\omega _\\mathrm {ci})+\\mathrm {i}\\omega \\nu _\\mathrm {eff}}\\,$ $(\\alpha =0,\\pm 1)$ .", "A more rigorous kinetic theory leads to results which cannot be described by Eq.", "(REF ) with the same frequency $\\nu _\\mathrm {eff}$ for the Hermitian and anti-Hermitian components $\\chi ^\\mathrm {H}_\\alpha $ and $\\chi ^\\mathrm {A}_\\alpha $ ([189], § 6).", "Figure: Absorption coefficients (top panel) andpolarizability coefficientsχ 0 H \\chi ^\\mathrm {H}_{0} (middle panel) and χ ±1 H \\chi ^\\mathrm {H}_{\\pm 1} (bottom panel)in the partially ionized (solid curves) and fully ionized(dot-dashed curves) plasma models atB=3×10 13 B=3\\times 10^{13} G, ρ=1\\rho =1 g cm -3 ^{-3} and T=3.16×10 5 T=3.16\\times 10^5 K.The anti-Hermitian part of the polarizability tensor determines the opacities: $\\varkappa _\\alpha (\\omega )= 4\\pi \\omega \\chi ^\\mathrm {A}_\\alpha (\\omega )/(\\rho c)$ .", "Then the Kramers-Kronig relation gives [363], [364] $\\chi ^\\mathrm {H}_\\alpha (\\omega ) &=&\\frac{c\\rho }{4\\pi ^2\\omega }\\, \\bigg \\lbrace \\!\\int _0^\\omega \\!\\big [\\,\\varkappa _\\alpha (\\omega +\\omega ^{\\prime })- \\varkappa _\\alpha (\\omega -\\omega ^{\\prime })\\,\\big ]\\frac{\\mathrm {d}\\omega ^{\\prime }}{\\omega ^{\\prime }}\\nonumber \\\\&+ & \\int _{2\\omega }^\\infty \\frac{\\varkappa _\\alpha (\\omega ^{\\prime })}{\\omega ^{\\prime }-\\omega }\\,\\mathrm {d}\\omega ^{\\prime }- \\int _0^\\infty \\frac{\\varkappa _{-\\alpha }(\\omega ^{\\prime })}{\\omega ^{\\prime }+\\omega }\\,\\mathrm {d}\\omega ^{\\prime } \\bigg \\rbrace .$ Thus we can calculate the polarizability tensor $\\mathbf {\\chi }$ from the opacities $\\varkappa _\\alpha (\\omega )$ .", "It has been done in [363] for a gas of neutral hydrogen atoms and in [364] for partially ionized hydrogen plasmas.", "Figure REF shows the cyclic components of absorption coefficients, $\\mu _\\alpha =\\rho \\varkappa _\\alpha $ in the top panel, and corresponding polarizability components $\\chi ^\\mathrm {H}_\\alpha $ in the middle and bottom panels, for a partially ionized hydrogen plasma at $B=3\\times 10^{13}$ G, $\\rho =1$ g cm$^{-3}$, and $T=3.16\\times 10^5$ K. In this case, the neutral fraction is 89%.", "For comparison we show the results of an analogous calculation for the fully-ionized plasma model.", "In addition to the proton cyclotron resonance at $\\hbar \\omega =0.19$ keV that is present in both models, the absorption coefficients show rather pronounced features due to atomic transitions in the partially ionized plasma model.", "Most remarkable are the absorption features due to bound-bound transitions at $\\hbar \\omega \\approx 0.2$ –0.3 keV for $\\mu _{+1}$ and the photoionization jump (partly smeared by the magnetic broadening) at $\\hbar \\omega =0.4$ keV for $\\mu _0$ .", "These features have clear imprints on the behavior of $\\chi ^\\mathrm {H}_{+1}$ and $\\chi ^\\mathrm {H}_0$ ." ], [ "Vacuum polarization", "In certain ranges of density $\\rho $ and frequency $\\omega $ , normal-mode properties are dramatically affected by a specific QED effect called vacuum polarization (its other manifestation has already been considered in § REF ).", "The influence of the vacuum polarization on the neutron-star emission has been first evaluated in [365], [366] and studied in detail in the review [367].", "If the vacuum polarization is weak, then it can be linearly added to the plasma polarization.", "Then the complex dielectric tensor can be written as $\\mathbf {\\varepsilon }^{\\prime } = \\mathbf {I} + 4\\pi \\chi + 4\\pi \\chi ^\\mathrm {vac},$ where $\\mathbf {\\chi }^\\mathrm {vac} = (4\\pi )^{-1}\\,\\mathrm {diag}(\\bar{a}, \\bar{a}, \\bar{a}+\\bar{q} )$ is the vacuum polarizability tensor, and diag(...) denotes the diagonal matrix.", "Magnetic susceptibility of vacuum is determined by expression $\\mathbf {\\mu }^{-1} = \\mathbf {I} + \\mathrm {diag}(\\bar{a}, \\bar{a},\\bar{a}+\\bar{m}).$ Adler [368] obtained the vacuum polarizability coefficients $\\bar{a}$ , $\\bar{q}$ , and $\\bar{m}$ that enter Eqs.", "(REF ) and (REF ) in an explicit form at $b\\ll 1$ , Heyl and Hernquist [369] expressed them in terms of special functions in the limits of $b\\ll 1$ and $b\\gg 1$ .", "Kohri and Yamada [370] presented their numerical calculations.", "Finally, in [364] we found simple but accurate expressions $&&\\bar{a}= - \\frac{2\\alpha _\\mathrm {f}}{9\\pi } \\ln \\bigg (1 + \\frac{b^2}{5}\\,\\frac{1+0.25487\\,b^{3/4}}{1+0.75\\,b^{5/4}}\\bigg ),\\\\&&\\bar{q} = \\frac{7\\alpha _\\mathrm {f}}{45\\pi }\\,b^2\\,\\frac{1 + 1.2\\,b}{1 + 1.33\\,b + 0.56\\,b^2},\\\\&&\\bar{m} = - \\frac{\\alpha _\\mathrm {f}}{3\\pi } \\, \\frac{ b^2 }{3.75 + 2.7\\,b^{5/4} + b^2 }.$ The coefficients (REF ) – () are not small at $B\\gtrsim 10^{16}$ G, therefore the vacuum refraction coefficients substantially differ from unity.", "In such strong fields, the vacuum that surrounds a neutron star acts as a lens, distorting its radiation [371], [372], [373].", "At smaller $B$ , the vacuum polarization results in a resonance, which manifests in the coincidence of the normal-mode polarization vectors at a certain frequency, depending on plasma density.", "In the photospheres with $B\\gtrsim 10^{13}$ G, this resonance falls in the range $\\sim 0.1$ – 1 keV and affects the thermal spectrum." ], [ "Polarization vectors of the normal modes", "Shafranov [374] obtained the polarization vectors $\\mathbf {e}_j$ for fully ionized plasmas.", "Ho and Lai [375] presented their convenient expressions in terms of the coefficients $\\varepsilon _\\perp $ , $\\varepsilon _\\Vert $ , $\\varepsilon _\\wedge $ , $\\bar{a}$ , $\\bar{q}$ , and $\\bar{m}$ , including the contributions of electrons, ions, and vacuum polarization.", "In the Cartesian coordinate system ($xyz$ ) with the $z$ -axis along the wave vector $\\mathbf {k}$ and with $\\mathbf {B}$ in the plane $x$ –$z$ , one has $\\mathbf {e}_j=\\left(\\begin{array}{c}e^j_x \\\\ e^j_y \\\\ e^j_z\\end{array}\\right)=\\frac{1}{\\sqrt{1+K_j^2+K_{z,j}^2}} \\,\\left(\\begin{array}{c}\\mathrm {i} K_j \\\\ 1 \\\\ \\mathrm {i} K_{z,j}\\end{array}\\right) ,$ where $&&\\hspace{-20.0pt}K_j = \\beta \\left\\lbrace 1 + (-1)^j \\left[ 1 + \\frac{1}{\\beta ^2}+ \\frac{\\bar{m}}{1+\\bar{a}} \\frac{\\sin ^2\\theta _B}{\\beta ^2}\\right]^{1/2}\\right\\rbrace ,\\hspace{20.0pt}\\\\&&K_{z,j} = - \\frac{(\\varepsilon _\\perp ^{\\prime } - \\varepsilon _\\Vert ^{\\prime }) K_j \\cos \\theta _B + \\varepsilon _\\wedge }{\\varepsilon _\\perp ^{\\prime } \\sin ^2\\theta _B + \\varepsilon _\\Vert ^{\\prime } \\cos ^2\\theta _B }\\, \\sin \\theta _B,\\hspace{20.0pt}\\\\&&\\hspace{-20.0pt}\\beta = \\frac{\\varepsilon _\\Vert ^{\\prime } - \\varepsilon _\\perp ^{\\prime } + \\varepsilon _\\wedge ^2/\\varepsilon _\\perp ^{\\prime } + \\varepsilon _\\Vert ^{\\prime }\\,\\bar{m}/(1+\\bar{a})}{2 \\, \\varepsilon _\\wedge }\\,\\, \\frac{ \\varepsilon _\\perp ^{\\prime }}{\\varepsilon _\\Vert ^{\\prime }}\\,\\,\\frac{\\sin ^2\\theta _B}{\\cos \\theta _B},\\hspace{20.0pt}$ $\\varepsilon _\\perp ^{\\prime } = \\varepsilon _\\perp + \\bar{a}$ , and $\\varepsilon _\\Vert ^{\\prime } = \\varepsilon _\\Vert + \\bar{a} +\\bar{q}$ .", "If the plasma and vacuum polarizabilities are small ($\|\\chi ^\\mathrm {H}_\\alpha \| \\ll (4\\pi )^{-1}$ and $\|\\bar{a}\|,\\bar{q},\|\\bar{m}\| \\ll 1$ ), as usual, $\\beta \\approx \\frac{2\\chi ^\\mathrm {H}_0 - \\chi ^\\mathrm {H}_{+1} - \\chi ^\\mathrm {H}_{-1} +(\\bar{q}+\\bar{m})/(2\\pi )}{2\\,(\\chi ^\\mathrm {H}_{+1} - \\chi ^\\mathrm {H}_{-1}) }\\,\\frac{\\sin ^2\\theta _B}{\\cos \\theta _B} .$" ], [ "Opacities", "In the approximation of isotropic scattering, at a given frequency $\\omega $ , the opacities can be presented in the form $&&\\hspace{-6.99997pt}\\varkappa _j^\\mathrm {a} = \\sum _{\\alpha =-1}^1\|e_{j,\\alpha }(\\theta _B)\|^2 \\,\\frac{\\sigma _\\alpha ^\\mathrm {a}}{m_\\mathrm {i}},\\\\&&\\hspace{-40.0pt}\\varkappa _{jj^{\\prime }}^\\mathrm {s} \\!=\\!\\!", "{\\frac{3}{4}}\\!\\!\\sum _{\\alpha =-1}^1 \\!\\!\|e_{j,\\alpha }(\\theta _B)\|^2 \\,\\frac{\\sigma _\\alpha ^\\mathrm {s}}{m_\\mathrm {i}}\\int _0^\\pi \\!\\!\\!\|e_{j^{\\prime },\\alpha }(\\theta _B^{\\prime })\|^2\\sin \\theta _B^{\\prime }\\,\\mathrm {d}\\theta _B^{\\prime },$ where $\\sigma _\\alpha $ are the cross sections for the three basic polarizations according to Eq.", "(REF ).", "The partial cross sections $\\sigma _\\alpha ^\\mathrm {a,s}$ include contributions of photon interaction with free electrons or ions (free-free transitions) as well as with bound states of atoms and ions (bound-bound and bound-free transitions).", "The latter implies, in particular, averaging of the cross sections of photon and atom absorption over all values of $K_\\perp $ .", "Since the distribution over $K_\\perp $ is continuous for the atoms and discrete for the ions, such averaging for atoms reduces to an integration over $K_\\perp $ , analogous to Eq.", "(REF ), whereas for ions it implies summation with an appropriate statistical weight.", "To date, such calculation has been realized for atoms of hydrogen [266], [350] and helium [308].", "Figure REF presents opacities for the two normal modes propagating at the angle $\\theta _B=10^\\circ $ to the magnetic field under the same physical conditions as in Fig.", "REF .", "One can clearly distinguish the features reflecting the peaks at the ion cyclotron frequency and the resonant atomic frequencies, and the line crossings related to the behavior of the plasma polarizability as function of frequency.", "For comparison, we show also opacities for the fully ionized plasma model under the same conditions.", "They miss the features related to the atomic resonances, and their values is underestimated by orders of magnitude in a wide frequency range.", "Figure: Logarithm of spectral opacities (logϰ j \\log \\varkappa _j) for two normal modes,propagating at the angleθ B =10 ∘ \\theta _B=10^\\circ to the magnetic field lines in ahydrogen plasma atB=3×10 13 B=3\\times 10^{13} G, T=3.16×10 5 T=3.16\\times 10^5 K, ρ=1\\rho =1g cm -3 ^{-3}.", "Solid curves: partially ionized plasma model;dot-dashed curves: fully-ionized plasma model.", "The lower(upper)curve of each type corresponds to the extraordinary(ordinary) wave.", "The arrows indicate the features atresonant frequencies: 1– the ion cyclotron resonanceω=ω ci \\omega =\\omega _\\mathrm {ci}; 2 – energy threshold for a transitionbetween the lowest two levelsℏω=\|E 0,0 (0) -E 1,0 (0) \|\\hbar \\omega =\|E_{0,0}^{(0)}-E_{1,0}^{(0)}\|; 3 – theground-state binding energy ℏω=\|E 0,0 (0) \|\\hbar \\omega =\|E_{0,0}^{(0)}\|; 4– the vacuum resonance." ], [ "Spectra of magnetic photospheres", "Shibanov and coworkers [376] were the first to perform detailed calculations of the spectra of radiation formed in the strongly magnetized neutron-star photospheres, using the fully ionized plasma model, and created a database of magnetic hydrogen spectra [269].Model NSA in the XSPEC database [210].", "They have shown that the spectra of magnetic hydrogen and helium atmospheres are softer than the respective nonmagnetic spectra, but harder than the blackbody spectrum with the same temperature.", "In addition to the spectral energy distribution, these authors have also studied the polar diagram and polarization of the outgoing emission, which proved to be quite nontrivial because of redistribution of energy between the normal modes.", "The thermal radiation of a magnetized photosphere is strongly polarized, and the polarization sharply changes at the cyclotron resonance with increasing frequency.", "At contrast to the isotropic blackbody radiation, radiation of a magnetic photosphere consists of a narrow ($<5^\\circ $ ) pencil beam along the magnetic field and a broad fan beam with typical angles $\\sim 20^\\circ -60^\\circ $ [377] (see also [378]).", "These calculations have thus fully confirmed the early analysis by Gnedin and Sunyaev [11].", "Later, analogous calculations were performed by other research groups [379], [375], [378].", "They paid special attention to manifestations of the ion cyclotron resonance in observed spectra in the presence of superstrong magnetic fields, which was prompted by tentative magnetar discoveries.", "It was shown in [380] that the vacuum polarization leads in the superstrong fields to a conversion of the normal modes, when a photon related to one mode transforms, with certain probability, into a photon of the other mode while crossing a surface with a certain critical density.", "The latter density is related to the photon energy as $\\rho = 0.00964\\,(A/Z)\\,(\\hbar \\omega /\\mbox{keV})^2\\, B_{12}^2/f_B^2 ~\\mbox{g~cm$^{-3}$},$ where $f_B^2= \\alpha _\\mathrm {f}b^2 / [15\\pi (\\bar{q}+\\bar{m})]$ , while $\\bar{q}$ and $\\bar{m}$ are given by Eqs.", "(), (); $f_B$ weakly depends on $B$ , and $f_B\\approx 1$ at $B\\lesssim 10^{14}$ G. The energy $\\hbar \\omega $ in Eq.", "(REF ) corresponds to the line crossing in Fig.", "REF , indicated by arrow 4.", "It follows from Eq.", "(REF ) that in the field of $B\\sim 10^{14}$ G this energy coincides with the ion cyclotron energy at the density where the atmosphere is optically thin for the extraordinary mode, but optically thick for the ordinary mode.", "Under such conditions, the mode conversion strongly suppresses the ion cyclotron feature in the emission spectrum.", "In the first computations of partially ionized photospheres of neutron stars with magnetic fields $B\\sim 10^{12}$ – $10^{13}$ G that were presented in [346] and [339], the properties of the atoms in magnetic fields were calculated by the adiabatic Hartree-Fock method (§ REF ).", "The atomic motion was either ignored [346], or treated approximately by the perturbation theory [339].", "In [364], a hydrogen photosphere model has been constructed beyond the framework of the adiabatic approximation, taking the full account of the partial ionization as well as the atomic motion effects in the strong magnetic fields.", "Figure REF gives an example of radiation spectrum going out of such photosphere with $B=10^{13}$ G. We see a narrow absorption line at the proton cyclotron energy $E=0.063$ keV and the features at higher energies, related to atomic transitions.", "For comparison, a spectrum calculated in the fully-ionized plasma model and the Planck spectrum are shown.", "The comparison shows that the two photospheric models have similar spectral shapes, but the model that allows for the partial ionization has additional features.", "The spectral maximum of both models is shifted to higher energies relative to the Planck maximum.", "This demonstrates that an attempt of interpretation of the hydrogen spectra with the blackbody model would strongly overestimate the effective temperature, while the fully-ionized photosphere model yields a more realistic temperature, but does not reproduce the spectral features caused by atomic transitions.", "Figure: Integral spectra of a hydrogen atmosphere of a neutron starwith M=1.4M ⊙ M=1.4\\,M_\\odot , R=12R=12 km, andwith different effective temperatures T eff T_\\mathrm {eff} (logT eff \\log T_\\mathrm {eff} (K)from 5.5to 6.8 with step 0.1).", "The dashed and solid lines representthe model with a dipole field of strengthB p =10 13 B_\\mathrm {p}=10^{13} G at the pole and oriented along andacross the line of sight, respectively.", "For comparison, thedotted curve shows the model with a constant fieldB=10 13 B=10^{13} G, normal to the surface.Magnetic fields and temperatures of neutron stars vary from one surface point to another.", "In order to reproduce the radiation spectrum that comes to an observer, one can use Eq.", "(REF ).", "The problem is complicated, because the surface distributions of the magnetic field and the temperature are not known in advance.", "As a fiducial model one conventionally employs the relativistic dipole model (REF ), (REF ), while the temperature distribution, consistent with the magnetic-field distribution, is found from calculations of heat transport in neutron-star envelopes (e.g., [84]).", "Results of such calculations, performed in [381], are shown in Fig.", "REF .", "We see that the spectral features are strongly smeared by the averaging over the surface, and the spectrum depends on the magnetic axis orientation $\\theta _\\mathrm {m}$ .", "When the star rotates, the latter dependence leads to pulsations of the measured spectrum.", "Figure: Local spectra at the magnetic pole (solid curve) andequator (dashed curve) for a neutron star with carbonatmosphere, the dipole field with polar strength ofB p =2×10 12 B_\\mathrm {p}=2\\times 10^{12} G (neglecting the relativisticcorrections) and uniform effective temperature3×10 6 3\\times 10^6 K. (Fig.", "20 from ,reproduced with permission of the authors and © OxfordUniversity Press.", ")Mori et al.", "[347], [348] calculated model spectra of neutron-star photospheres composed of the atoms and ions of elements with $Z_\\mathrm {n}\\lesssim 10$ .", "They calculated the quantum-mechanical properties of the atoms and ions by the method of Mori and Hailey [301] and treated the atomic motion effects by the perturbation theory (§ REF ).", "The equation of state and ionization equilibrium were determined by the methods described in § REF , the plasma polarizability was calculated by Eq.", "(REF ), and the opacities were treated according to § REF .", "As an example, Fig.", "REF demonstrates local spectra of the carbon photosphere with magnetic field $B=2\\times 10^{12}$ G, normal to the surface, and the field $B=10^{12}$ G parallel to the surface, which approximately (with account of neither relativistic corrections nor temperature nonuniformity) corresponds to the local spectra at the magnetic pole and equator of a star with a dipole magnetic field.", "By analogy to the case of hydrogen photosphere, the integration over the surface between the pole and equator should smear the spectral features between the two limiting curves shown in the figure.", "The results described in this section have been used to produce databases of spectra of partially ionized, strongly magnetized neutron-star photospheres composed of hydrogen [381] and heavier elements up to neon [348].Models NSMAX and NSMAXG [382] in the database XSPEC [210]." ], [ "Radiation of a naked neutron star", "As we have seen in § REF , the stars with a very low effective temperature and a superstrong magnetic field can have a liquid or solid condensed surface.", "In this case, thermal emission can escape directly from the metallic surface without transformation in a gaseous atmosphere, and then the spectrum is determined by the emission properties of this surface.", "Formation of thermal spectra at a condensed surface of a strongly magnetized neutron star depends on its reflection properties, which were considered in [383], [384], [385], [115], [386], [387], [388].", "The first works [383], [384] gave order-of-magnitude estimates.", "A method of detailed calculation of the reflectivity was proposed in [385] and then was used with some modifications in [385], [115], [386], [387], [388].", "It is as follows.", "First, the normal-mode polarization vectors $\\mathbf {e}_{1,2}^\\mathrm {(t)}$ in the medium under the surface, Eqs.", "(REF ) – (), and the complex refraction coefficients are expressed as functions of the angles $\\theta _k$ and $\\varphi _k$ that determine the direction of a reflected ray (Fig.", "REF ), using the standard dispersion equation for the transmitted wave and the Snell's law.", "Second, the complex electric amplitudes of the incident, reflected, and transmitted waves are expanded over the respective basic polarization vectors $\\mathbf {e}_{1,2}^\\mathrm {(i,r,t)}$ .", "Then the Maxwell boundary conditions yield a system of equations, which determine the coefficients of these expansions.", "These reflected-wave expansion coefficients form the reflection matrix $\\lbrace r_{jj^{\\prime }}\\rbrace $ and determine the surface reflectivity for each incident-wave polarization, $r_{\\omega ,j}=\\sum _{j^{\\prime }}r_{\\omega ,jj^{\\prime }}$ .", "Then the total emissivity $\\varepsilon _\\omega =1-\\frac{1}{2}(r_{\\omega ,1}+r_{\\omega ,2})$ .", "The early works assumed that the ions are firmly fixed at the crystalline lattice sites in the metal.", "In [386], [387], [388] the authors have considered not only this model, but also the opposite limit of free ions.", "It is assumed [386] that the real reflectivity of the surface lies between the limits given by these two models, although this problem has not yet been definitely solved.", "Figure: Emissivity of a condensed iron surface at B=10 13 B=10^{13} G andT=10 6 T=10^6 K, averaged over polarizations, is shown as afunction of energy of a photon emitted at the angleθ k =45 ∘ \\theta _k=45^\\circ , for different magnetic-fieldinclination angles θ n \\theta _\\mathrm {n} and azimuthal anglesϕ k \\varphi _k.", "The thick and thin curves are obtained,respectively, in the models of free and fixed ions.", "Verticaldotted lines mark positions of the characteristic energies:the ion cyclotron energy E ci =ℏω ci E_\\mathrm {ci}=\\hbar \\omega _\\mathrm {ci}, theelectron plasma energy E pe =ℏω pe E_\\mathrm {pe}=\\hbar \\omega _{\\mathrm {pe}}, and thehybrid energy E C E_\\mathrm {C}.Figure REF shows examples of the emissivity $\\varepsilon _\\omega $ , normalized to the blackbody emissivity, as a function of photon energy $E=\\hbar \\omega $ , according to the free- and fixed-ions models, for different values of the angles $\\theta _\\mathrm {n}$ , $\\theta _k$ , and $\\varphi _k$ that are defined in Fig.", "REF .", "The characteristic energies $E_\\mathrm {ci}=\\hbar \\omega _\\mathrm {ci}$ , $E_\\mathrm {pe}=\\hbar \\omega _{\\mathrm {pe}}$ , and $E_\\mathrm {C}=E_\\mathrm {ci}+E_\\mathrm {pe}^2/\\hbar \\omega _\\mathrm {c}$ are marked.", "The spectral features near these energies are explained in [386].", "For instance, the emissivity suppression at $E_\\mathrm {ci}\\lesssim E\\lesssim E_\\mathrm {C}$ is due to the strong damping of one of the two normal modes in the plasma in this energy range.", "In the fixed-ions mode, $\\omega _\\mathrm {ci}\\rightarrow 0$ , therefore there is no kink of the spectrum at $E\\approx E_\\mathrm {ci}$ in this model.", "The results almost coincide in the two alternative models at $E\\gg E_\\mathrm {ci}$ , but strongly differ at $E\\lesssim E_\\mathrm {ci}$ , which may be important for magnetar spectra.", "Near the electron plasma energy $E_\\mathrm {pe}=\\hbar \\omega _{\\mathrm {pe}}$ , there is a resonant absorption, depending on the directions of the incident wave and the magnetic field.", "The local flux density of radiation from a condensed surface is equal to the Planck function $\\mathcal {B}_{\\omega ,T}$ (REF ), multiplied by the normalized emissivity $\\varepsilon _\\omega $ .", "Since $\\varepsilon _\\omega $ depends on the frequency $\\omega $ and on the angles $\\theta _\\mathrm {n}$ , $\\theta _k$ , and $\\varphi _k$ (Fig.", "REF ), thermal radiation depends on the frequency and angles in a nontrivial way.", "In Fig.", "REF , the emissivity is averaged over polarizations.", "But $r_{\\omega ,1}\\ne r_{\\omega ,2}$ , hence the thermal emission of a condensed surface is polarized, the polarization depending in an equally nontrivial way on the frequency and angles.", "For example, the degree of linear polarization can reach tens percent near the frequencies $\\omega _\\mathrm {ci}$ and $\\omega _{\\mathrm {pe}}$ , which makes promising the polarization diagnostics of neutron stars with condensed surfaces.", "Both the intensity and the polarization degree can be evaluated using analytical expressions, which have been constructed in [388] for the reflectivity matrix of a condensed iron surface for $B=10^{12}$ – $10^{14}$ G." ], [ "Thin and layered atmospheres", "Motch, Zavlin, and Haberl [389] suggested that some neutron stars can possess a hydrogen atmosphere of a finite thickness above the solid iron surface.", "If the optical depth of such atmosphere is small for some wavelengths and large for other ones, this should lead to a peculiar spectrum, different from the spectra of thick atmospheres.", "Such spectra were calculated in [390], [391], [392] using simplified boundary conditions for the radiative transfer equation at the inner boundary of the atmosphere.", "More accurate boundary conditions have been suggested in [388], where the authors have taken into account that an extraordinary or ordinary wave, falling from outside on the interface, gives rise to reflected waves of both polarizations, whose intensities add to the respective intensities of the waves emitted by the condensed surface: $I_{\\omega ,j}(\\theta _k,\\varphi ) &=&\\sum _{j^{\\prime }=1,2} r_{\\omega ,jj^{\\prime }}(\\theta _k,\\varphi )\\,I_{\\omega ,j^{\\prime }}(\\pi -\\theta _k,\\varphi )\\nonumber \\\\&&+ \\textstyle \\frac{1}{2}\\big [1-r_{\\omega ,j}(\\theta _k,\\varphi )\\big ]\\,\\mathcal {B}_{\\omega ,T}.$ In Ref.", "[388], the reflectivity matrix was calculated and fitted for linear polarizations, and then converted into the reflectivity matrix $\\lbrace r_{\\omega ,jj^{\\prime }}\\rbrace $ for normal modes pertinent to Eq.", "(REF ), using an approximate relation valid for a sufficiently rarefied photosphere.", "Figure: Comparison of the radiation spectrum of a neutron star witha partially ionized thick hydrogen photosphere (dashed line)with the spectra that are formed at hydrogen columndensities of 1 g cm -2 ^{-2} (dots) and 10 g cm -2 ^{-2} (solid line) overthe iron surface of the star (Fig.", "12 from ,provided by V. F. Suleimanov, reproduced with permission ofthe author and©ESO.", ")In Fig.", "REF we show local spectra of radiation emitted by hydrogen atmospheres of different thicknesses over the iron neutron-star surface with the magnetic field $B=4\\times 10^{13}$ G, normal to the surface with effective temperature $T_\\mathrm {s}=1.2\\times 10^6$ K. The narrow absorption line corresponds to the proton cyclotron resonance in the atmosphere.", "The feature to the right of it is related to atomic transitions (H$_\\mathrm {b-b}$ ).", "It has a large width because of the motion effects (§ REF ).", "This feature is formed mainly at depths $\\sim 2$ g cm$^{-2}$ , that is why it is almost invisible in the spectrum of the thinnest atmosphere that has the column density of 1 g cm$^{-2}$ .", "The kink at $E_\\mathrm {ci}=0.12$ keV corresponds to the ion cyclotron energy of iron, therefore it is absent for the pure hydrogen atmosphere.", "The spectrum of the moderately deep atmosphere (10 g cm$^{-2}$ ) reveals all the three features.", "At high energies ($E\\gtrsim 1$ keV), the spectrum is determined by the condensed-surface emission, because both finite atmospheres are almost transparent at such energies.", "The spectrum of the pure hydrogen atmosphere is harder in this spectral range (cf.", "§ REF ).", "The origin of the thin atmospheres remains hazy.", "Ho et al.", "[390] discussed three possible scenarios.", "First, it is the accretion from the interstellar medium.", "But its rate should be very low, in order to accumulate the hydrogen mass $4\\pi R^2y_\\mathrm {col}\\sim 10^{-20} M_\\odot $ in $\\sim 10^6$ years.", "Another scenario assumes diffusive nuclear burning of a hydrogen layer, fell back soon after the formation of the neutron star [112].", "But this process is too fast at the early cooling epoch, when the star is relatively hot, and would have rapidly consumed all the hydrogen on the surface [393].", "The third possibility is a self-regulating mechanism that is driven by nuclear spallation in collisions with ultrarelativistic particles at the regions of open field lines, which leads to creation of protons and alpha-particles.", "The estimate (REF ) for the penetration depth of the magnetospheric accelerated particles indicates that this process could create a hydrogen layer of the necessary thickness $y_\\mathrm {col}\\sim 1$ g cm$^{-2}$ .", "It is natural to consider also an atmosphere having a helium layer beneath the hydrogen layer.", "Indeed, all three scenarios assume that a hydrogen-helium mixture appears originally at the surface, and the strong gravity quickly separates these two elements.", "Such “sandwich atmosphere” was considered in [391], where the authors showed that its spectrum can have two or three absorption lines in the range $E\\sim (0.2$ – 1) keV at $B\\sim 10^{14}$ G." ], [ "Theoretical interpretation of observed spectra", "As we have seen in § , theoretical models of nonmagnetic atmospheres are successfully applied to analyses of spectra of many neutron stars with relatively weak magnetic fields $B\\lesssim 10^9$ G. There are only a few such examples for the stars with strong magnetic fields.", "They will be discussed in this section.", "At the end of the section we will give a general compilation of modern estimates of masses and radii of neutron stars with weak and strong magnetic fields, based on the photosphere models." ], [ "RX J1856.5–3754", "As we discussed in § REF , there is no satisfactory description of the spectrum of the “Walter star” RX J1856.5–3754 based on nonmagnetic atmosphere models.", "Simple models of magnetic atmospheres also failed to solve this problem.", "It was necessary to explain simultaneously the form of the spectrum in the X-ray and optical ranges that reveal substantially different color temperatures $T_\\mathrm {bb}^\\infty $ , along with the complete absence of absorption lines or other spectral features that was confirmed at a high significance level.", "To solve this problem, Ho [390], [394] involved the model of a partially ionized hydrogen atmosphere of finite thickness above a condensed iron surface with a strong magnetic field.", "He managed to reproduce the measured spectrum of RX J1856.5–3754 in the entire range from X-rays to optical within observational errorbars.", "The best agreement between the theoretical and observed spectra has been achieved at the atmosphere column density $y_\\mathrm {col}=1.2$ g cm$^{-2}$ , $B\\sim (3$ – $4)\\times 10^{12}$ G, $T_\\mathrm {eff}^\\infty =(4.34\\pm 0.03)\\times 10^5$ K, $z_g=0.25\\pm 0.05$ , and $R_\\infty =17.2^{+0.5}_{-0.1}\\,D_{140}$ km.", "Here, the errors are given at the $1\\sigma $ significance level, and $D_{140}\\equiv D/(140$ pc).", "Note that a fit of the observed X-ray spectrum with the Planck function yields a 70% higher temperature and a 3.5 times smaller radius of the emitting surface.", "Such huge difference exposes the importance of a correct physical interpretation of an observed spectrum for evaluation of neutron-star parameters.", "With the aid of expressions (REF ) – (REF ) and Eq.", "(REF ), we obtain from these estimates $T_\\mathrm {eff}=(5.4\\pm 1.1)\\times 10^5$ K, $R=13.8^{+0.9}_{-0.6}\\,D_{140}$ km, and $M=1.68^{+0.22}_{-0.15}\\,D_{140}\\,M_\\odot $ .", "Forgetting for a moment the factor $D_{140}$ , one might conclude that this radius is too large for such mass.", "However, the distance to the star is not very accurately known.", "The value $D=140$ pc was adopted in [390] from [395] and lies between alternative estimates $D\\approx 117$ pc [396] and $D\\approx (160$ – 170) pc [397], [398].", "More recently, a more accurate estimate of the distance was obtained, $D=123^{+11}_{-15}$ pc [399].", "With the latter estimate, we obtain $R=12.1^{+1.3}_{-1.6}$ km and $M=1.48^{+0.16}_{-0.19}\\,M_\\odot $ , which removes all the contradictions.", "Nevertheless, the given interpretation of the spectrum is not indisputable, since it does not agree with the magnetic-field estimate $B\\approx 1.5\\times 10^{13}$ G that has been obtained for this star from Eq.", "(REF ) in [400].", "Using the same thin-atmosphere model, Ho [394] analyzed the light curve of RX J1856.5–3754 and obtained constraints on the angles $\\alpha $ and $\\zeta $ (Fig.", "REF ).", "It turned out that the light curve can be explained if one of these angles is small ($<6^\\circ $ ), while the other angle lies between $20^\\circ $ and $45^\\circ $ .", "In this case, the radio emission around the magnetic poles does not cross the line of sight.", "As noted in [394], this may explain the non-detection of this star as a radio pulsar [165]." ], [ "RBS 1223", "Hambaryan et al.", "[401] analyzed the spectrum of the X-ray source RBS 1223, by a method analogous to the case of RX J1856.5–3754 described in § REF .", "RBS 1223 reveals a complex structure of the X-ray spectrum, which can be described by a wide absorption line centered around $\\hbar \\omega =0.3$ keV, superposed on the Planck spectrum, with the line parameters depending on the stellar rotation phase.", "Using all 2003 – 2007 XMM-Newton observations of this star, the authors [401] obtained a set of X-ray spectra for different rotation phases.", "They tried to interpret these spectra with different models, assuming magnetic fields $B\\sim 10^{13}$ – $10^{14}$ G, different atmosphere compositions, possible presence of a condensed surface and a finite atmosphere.", "Different surface temperature distributions were described by a self-consistent parametric model of Ref. [46].", "As a result, the authors [401] managed to describe the observed spectrum and its rotational phase dependence with the use of the model of the iron surface covered by partially ionized hydrogen atmosphere with $y_\\mathrm {col}\\sim 1$ – 10 g cm$^{-2}$ , with mutually consistent asymmetric bipolar distributions of the magnetic field and the temperature, with the polar values $B_\\textrm {p1}=B_\\textrm {p2}=(0.86\\pm 0.02)\\times 10^{14}$ G, $T_\\mathrm {p1}=1.22^{+0.02}_{-0.05}$ MK, and $T_\\mathrm {p2}=1.15\\pm 0.04$ MK.", "The magnetic field and temperature proved to be rather smoothly distributed over the surface.", "When compared to the theoretical model [46], it implies the absence of a superstrong toroidal component of the crustal magnetic field.", "The integral effective temperature is $T_\\mathrm {eff}\\approx 0.7$ MK.", "The gravitational redshift is estimated to be $z_g=0.16^{+0.03}_{-0.01}$ , which converts into $(M/M_\\odot )/R_6=0.87^{+0.13}_{0.05}$ and suggests a stiff EOS of the neutron-star matter.", "We must note that the paper [401] preceded the work [388], which was discussed in § .", "For this reason, the authors of [401] used rough approximations for the iron-surface emissivity, published before, and simplified boundary conditions for the radiative transfer equations.", "An analysis of the same spectra with the use of the improved results for the emissivity and more accurate boundary conditions, described in § , remains to be done in the future.", "Table: Estimates of neutron-star masses and radii based onatmosphere models." ], [ "1E 1207.4–5209", "The discovery of absorption lines in the spectrum of CCO 1E 1207.4–5209 at energies $E\\sim 0.7\\,N$ keV ($N=1,2,\\ldots $ ) immediately entrained the natural assumption that they are caused by cyclotron harmonics [172].", "As we have seen in § REF , such harmonics can be only electronic, as the ion harmonics are unobservable.", "Therefore, this interpretation implies $B\\approx 7\\times 10^{10}$ G. Mori et al.", "[173] showed that only the first and second lines in the spectrum of 1E 1207.4–5209 are statistically significant, but some authors take also the third and fourth lines into account.", "This hypothesis was developed in [264], where the authors include in the treatment both types of the electron cyclotron harmonics that were discussed in § REF : the quantum oscillations of the Gaunt factor and the relativistic thermal harmonics.", "It is possible that the analogous explanation of the shape of the spectrum may be applied also to CCO PSR J0821–4300 [167].", "Mori et al.", "[347], [348] have critically analyzed the earlier hypotheses about the origin of the absorption lines in the spectrum of 1E 1207.4–5209 and suggested their own explanation.", "They analyzed and rejected such interpretations as the lines of molecular hydrogen ions, helium ions, and also as the cyclotron lines and their harmonics.", "One of the arguments against the latter interpretation is that the fundamental cyclotron line should have much larger depth in the atmosphere spectrum than actually observed.", "Another argument is that the cyclotron lines and harmonics have small widths at a fixed $B$ , therefore their observed width in the integral spectrum is determined by the $B$ distribution.", "Thus their width should be the same, in contradiction to observations [347].", "These arguments were neglected in [264].", "It has to be noted that in [347], as well as in [264], the authors studied the cyclotron harmonics in spectra of fully ionized plasmas.", "The effect of the partial ionization on the model spectrum remains unexplored.", "As an alternative, Mori et al.", "[347], [348] suggested models of atmospheres composed of mid-$Z$ elements.", "An example of such spectrum is shown in Fig.", "REF .", "Its convolution with the telescope point-spread function smears the line groups, producing wide and shallow suppressions of the spectral flux, similar to the observed ones.", "Integration of the local spectrum over the stellar surface, whose necessity we mentioned in § REF , should lead to an additional smearing of the spectral features.", "The authors [348] found that an oxygen atmosphere with magnetic field $B=10^{12}$ G provides a spectrum similar to the observed one.", "However, the constraint $B<3.3\\times 10^{11}$ G that was obtained in [166] disagrees with this model, but rather favors the cyclotron interpretation of the lines.", "Unlike the cases of RX J1856.5–3754 and RBS 1223 that were considered above, there is no published results of a detailed fitting of the observed spectrum of 1E 1207.4–5209 with a theoretical model.", "Thus the applicability of any of them remains hypothetical." ], [ "PSR J1119–6127", "Recently, the partially ionized, strongly magnetized hydrogen photosphere model [381] has been successfully applied to interpret the observations of pulsar J1119–6127 [146], for which the estimate (REF ) gives an atypically high field $B=4\\times 10^{13}$ G. In the X-ray range, it emits pulsed radiation, which has apparently mostly thermal nature.", "At fixed $D=8.4$ kpc and $R=13$ , the bolometric flux gives an estimate of the mean effective temperature $T_\\mathrm {eff}\\approx 1.1$ MK.", "It was difficult to explain, however, the large pulsed fraction ($48\\pm 12$ %) by the thermal emission.", "The authors [146] managed to reproduce the X-ray light curve of this pulsar assuming that one of its magnetic poles is surrounded by a heated area, which occupies 1/3 of the surface, is covered by hydrogen and heated to $1.5$ MK, while the temperature of the opposite polar cap is below $0.9$ MK." ], [ "Masses and radii: the results", "Table REF presents modern estimates of neutron-star masses and radii, obtained from analyses of their thermal spectra with the atmosphere models.", "The estimates that fixed surface gravity in advance are not listed here, because they are strongly biased, as shown, e.g., in [209].", "In most cases determination of the neutron-star radii remains unreliable.", "The estimates are done, as a rule, at a fixed distance $D$ .", "An evaluation of $R$ is also often performed for a fixed mass $M$ .", "In most cases it is stipulated by the fact that a joint evaluation of $R$ and $M$ (let alone $R$ , $M$ , and $D$ ) from the currently available thermal spectra leaves too large uncertainties and almost does not constrain $M$ and $D$ .", "A comparison of the results obtained for the same objects with different assumptions on $D$ values readily shows that a choice of $D$ can drastically affect the $R$ estimate.", "In addition, the estimate of $R$ is strongly affected by assumptions on the photosphere composition, as one can see, for example, from a comparison of the results obtained by assuming hydrogen and helium photospheres for the qLMXBs in globular clusters M28 [243] and M13 [244]." ], [ "Conclusions", "We have considered the main features of neutron-star atmospheres and radiating surfaces and outlined the current state of the theory of the formation of their spectra.", "The observations of bursters and neutron stars in low-mass X-ray binaries are well described by the nonmagnetic atmosphere models and yield ever improving information on the key parameters such as the neutron-star masses, radii, and temperatures.", "The interpretation of observations enters a qualitatively new phase, unbound from the blackbody spectrum or the “canonical model” of neutron stars.", "Absorption lines have been discovered in thermal spectra of strongly magnetized neutron stars.", "On the agenda is their detailed theoretical description, which provides information on the surface composition, temperature and magnetic field distributions.", "Indirectly it yields information on heat transport and electrical conductivity in the crust, neutrino emission, nucleon superfluidity, and proton superconductivity in the core.", "In order to clear up this information, it still remains to solve a number of problems related to the theory of the magnetic atmospheres and radiating surfaces.", "Let us mention just a few of them.", "First, the calculations of the quantum-mechanical properties of atoms and molecules in strong magnetic fields beyond the adiabatic approximation have been so far performed only for atoms with $Z_\\mathrm {n}\\lesssim 10$ and for one- and two-electron molecules and molecular ions.", "The thermal motion effect on these properties has been rigorously treated only for the hydrogen atom and helium ion, and approximately for the heavier atoms.", "It is urgent to treat the finite nuclear mass effects for heavier atoms, molecules, and their ions, including not only binding energies and characteristic sizes, but also cross sections of interaction with radiation.", "This should underlie computations of photospheric ionization equilibrium and opacities, following the technique that is already established for the hydrogen photospheres.", "In the magnetar photospheres, one can anticipate the presence of a substantial fraction of exotic molecules, including polymer chains.", "The properties of such molecules and their ions are poorly known.", "In particular, nearly unknown are their radiative cross sections that are needed for the photosphere modeling.", "Second, the emissivities of condensed magnetized surfaces have been calculated in frames of the two extreme models of free and fixed ions.", "It will be useful to do similar calculations using a more realistic description of ionic bonding in a magnetized condensed matter.", "This should be particularly important in the frequency range $\\omega \\lesssim \\omega _\\mathrm {ci}$ , which is observable for the thermal spectrum in the superstrong magnetic fields.", "Third, the radiative transfer theory, currently used for neutron-star photospheres, implies the electron plasma frequency to be much smaller than photon frequencies.", "In superstrong magnetic fields, this condition is violated in a substantial frequency range.", "Thus the theory of magnetar spectra requires a more general treatment of radiative transfer in a magnetic field.", "In conclusion, I would like to thank my colleagues, with whom I had a pleasure to work on some of the problems described in this review: V.G.", "Bezchastnov, G. Chabrier, W.C.G.", "Ho, D. Lai, Z. Medin, G.G.", "Pavlov, Yu.A.", "Shibanov, V.F.", "Suleimanov, M. van Adelsberg, J. Ventura, K. Werner.", "My special thanks are to Vasily Beskin, Wynn Ho, Alexander Kaminker, Igor Malov, Dmitry Nagirner, Yuri Shibanov, and Valery Suleimanov for useful remarks on preliminary versions of this article.", "This work is partially supported by the Russian Ministry of Education and Science (Agreement 8409, 2012), Russian Foundation for Basic Research (Grant 11-02-00253), Programme for Support of the Leading Scientific Schools of the Russian Federation (Grant NSh–294.2014.2), and PNPS (CNRS/INSU, France).", "tocsectionReferences Haensel P, Potekhin A Y, Yakovlev D G Neutron Stars 1: Equation of State and Structure (New York: Springer, 2007) Fortov V E Phys.", "Usp.", "52 615 (2009) Potekhin A Y Phys.", "Usp.", "53 1235 (2010) Haensel P, Zdunik J L, Douchin F Astron.", "Astrophys.", "385 301 (2002) Demorest P et al.", "Nature 467 1081 (2010) Antoniadis J et al.", "Science 340 448 (2013) Zhang W, Woosley S L, Heger A Astrophys.", "J.", "679 639 (2008) Pejcha O, Thompson T A, Kochanek C S Mon.", "Not.", "R. astron.", "Soc.", "424 1570 (2012) Kramer M, Stairs I H Annu.", "Rev.", "Astron.", "Astrophys.", "46 541 (2008) Lattimer J M Annu.", "Rev.", "Nucl.", "Particle Sci.", "62 485 (2012) Gnedin Yu N, Sunyaev R A Astron.", "Astrophys.", "36 379 (1974) Trümper J et al.", "Astrophys.", "J.", "219 L105 (1978) Coburn W et al.", "Astrophys.", "J.", "580 394 (2002) Rodes-Roca J J et al.", "Astron.", "Astrophys.", "508 395 (2009) Pottschmidt K et al.", "AIP Conf.", "Proc.", "1427 60 (2012) Boldin P A, Tsygankov S S, Lutovinov A A Astron.", "Lett.", "39 375 (2013) Baushev A N, Bisnovatyi-Kogan G S Astron. 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[ [ "Symmetric $q$-deformed KP hierarch" ], [ "Abstract Based on the analytic property of the symmetric $q$-exponent $e_q(x)$, a new symmetric $q$-deformed Kadomtsev-Petviashvili ($q$-KP) hierarchy associated with the symmetric $q$-derivative operator $\\partial_q$ is constructed.", "Furthermore, the symmetric $q$-CKP hierarchy and symmetric $q$-BKP hierarchy are defined.", "Here we also investigate the additional symmetries of the symmetric $q$-KP hierarchy." ], [ "Introduction", "The origin of $q$ -calculus (quantum calculus) [1], [2] traces back to the early 20th century.", "Many mathematicians have important works in the area of $q$ -calculus, $q$ -hypergeometric series and quantum group.", "There are two different forms of $q$ -derivative operators, which are defined respectively by $D_q(f(x))=\\frac{f(qx)-f(x)}{(q-1)x}, \\qquad q\\ne 1 $ and $\\partial _q(f(x))=\\frac{f(qx)-f(q^{-1}x)}{(q-q^{-1})x}, \\qquad q\\ne 1.", "$ The so-called $q$ -deformation of the integrable system (or $q$ -deformed integrable system) started in 1990's by means of the first $q$ -derivative $D_q$ in eq.", "(REF ) instead of usual derivative $\\partial $ with respect to $x$ in the classical system.", "As we know, the $q$ -deformed integrable system reduces to a classical integrable system as $q$ goes to 1.", "Several $q$ -deformed integrable systems have been presented, for example, $q$ -deformation of the KdV hierarchy [4], [5], [3], [6], $q$ -Toda equation [7], $q$ -Calogero-Moser equation [8] and so on.", "The $q$ -deformed Kadomtsev-Petviashvili ($q$ -KP) hierarchy is also a subject of intensive study in the literature from [9] to [17].", "Indeed, it is worth to point out that there exist two variants of the $q$ -deformed integrable system, one belonging to E.Frenkel [3] and another to D.H.Zhang et al.", "[4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17].", "It has been known for some time that different sub-hierarchies of the KP hierarchy can be obtained by adding different reduction conditions on Lax operator $L$ .", "Two important sub-hierarchies of the KP hierarchy are CKP hierarchy [18] through a restriction $L^=-L$ and BKP hierarchy[19] through a restriction $L^=-\\partial L\\partial ^{-1}$ .", "However, to the best of our knowledge, there is no any results on the $q$ -deformed CKP hierarchy and $q$ -deformed BKP hierarchy so far.", "The difficulty to define them is the conjugate operation “$$ ” of $q$ -derivative $D_q$ in eq.", "(REF ).", "In fact, $D_q^\\ne -D_q$ but $D_q^=-D_q\\theta ^{-1}=-\\frac{1}{q}D_{\\frac{1}{q}}$ .", "This paper shows a quite interesting fact as $\\partial _q^=-\\partial _q$ , where the symmetric $q$ -derivative operator $\\partial _q$ is defined by eq.", "(REF ).", "In what follows, we shall fill the gap by constructing the new symmetric $q$ -deformed KP hierarchy based on the symmetric $q$ -derivative operator $\\partial _q$ .", "The paper is organized as follows.", "Some basic results of symmetric $q$ -derivative operator $\\partial _q$ are given in Section 2, and one formula for the symmetric $q$ -exponent $e_q(x)$ is established.", "Then a new symmetric $q$ -KP hierarchy are stated in Sections 3 similarly to the classical KP hierarchy [20], and also symmetric $q$ -CKP hierarchy and symmetric $q$ -BKP hierarchy are given in this section.", "We further study the additional symmetries for the symmetric $q$ -KP hierarchy in Section 4.", "Section 5 is devoted to conclusions and discussions." ], [ "Symmetric quantum calculus", "We give some useful facts about the symmetric $q$ -derivative operator $\\partial _q$ in the form of eq.", "(REF ) based on the literature [2].", "We work in an associative ring of functions which includes a $q$ -variable $x$ and infinite time variables $t_i\\in \\mathbb {R}$ $F={f=f(x; t_1, t_2, t_3,\\cdots ,)}.$ The $q$ -shift operator is defined by $\\theta (f(x))=f(qx).$ Note that $\\theta $ does not commute with $\\partial _q$ .", "Indeed, the relation $(\\partial _q \\theta ^k(f))=q^k\\theta ^k(\\partial _q f), \\qquad k\\in \\mathbb {Z}$ holds.", "The limit of $\\partial _q(f(x))$ as $q$ approaches to 1 is the ordinary differentiation $\\partial _x(f(x)) $ .", "We denote the formal inverse of $\\partial _q$ as $\\partial _q^{-1}$ .", "Proposition 1.", "The conjugate of $\\partial _q$ can be defined as $\\partial _q^=-\\partial _q.$ Proof.", "First step is to prove $\\theta ^=q^{-1}\\theta ^{-1}$ .", "According to the definition, we have $\\partial _q(fg)&=(\\theta f)(\\partial _qg)+(\\partial _qf)(\\theta ^{-1}g) \\\\&=(\\theta g)(\\partial _qf)+(\\partial _qg)(\\theta ^{-1}f).$ Calculating the quantum integration $\\int \\cdot d_qx$ for the above two formulas separately, it follows that $\\int (\\theta f)(\\partial _qg) d_qx&=-\\int (\\partial _qf)(\\theta ^{-1}g) d_qx, \\\\ \\int (\\theta g)(\\partial _qf) d_qx&=-\\int (\\partial _qg)(\\theta ^{-1}f) d_qx.$ Let $g\\rightarrow \\theta ^{-2}g$ in eq.", "(), it now yields $\\int (\\theta ^{-1} g)(\\partial _qf) d_qx=-\\int (\\partial _q\\theta ^{-2}g)(\\theta ^{-1}f) d_qx.$ Comparing it with the eq.", "(REF ), the above equation becomes $\\int (\\theta f)(\\partial _qg) d_qx=\\int (\\partial _q\\theta ^{-2}g)(\\theta ^{-1}f) d_qx.$ It can now be written in the form $<\\theta f, \\partial _qg> = <\\theta ^{-1}f, q^{-2}\\theta ^{-2}\\partial _qg>.$ By letting $g\\rightarrow \\theta ^{-2}g$ and $f\\rightarrow \\theta f$ in the above equation, we find that $<\\theta ^2 f, g> = <f, q^{-2}\\theta ^{-2}g>,$ so one can choose $\\theta ^=q^{-1}\\theta ^{-1}$ .", "We will now proceed to prove $\\partial _q^=-\\partial _q$ .", "Let $f\\rightarrow \\theta ^{-1} f$ and $g\\rightarrow \\theta g$ in the eq.", "(REF ), it now reads $<\\partial _q\\theta ^{-1}f, g> = -<f, \\partial _q\\theta g>.$ This implies $(\\partial _q\\theta )^=-\\partial _q\\theta ^{-1}.$ According to the equation $\\theta ^=q^{-1}\\theta ^{-1}$ , we get $\\partial _q^=-q\\theta \\partial _q\\theta ^{-1}=-\\partial _q.$ $\\square $ The following $q$ -deformed Leibnitz rule holds $\\partial _q^n \\circ f=\\sum _{k\\ge 0}\\binom{n}{k}_q\\theta ^{n-k}(\\partial _q^kf)\\theta ^{-k}\\partial _q^{n-k},\\qquad n\\in {\\mathbb {Z}}$ where the $q$ -number $(n)_q=\\frac{q^n-q^{-n}}{q-q^{-1}}$ and the $q$ -binomial is introduced as $\\binom{n}{0}_q&=1,\\\\\\binom{n}{k}_q&=\\frac{(n)_q(n-1)_q\\cdots (n-k+1)_q}{(1)_q(2)_q\\cdots (k)_q}, \\qquad n\\in \\mathbb {Z}, k\\in \\mathbb {Z}_+.$ To illustrate the $q$ -deformed Leibnitz rule, the following examples are given.", "$\\partial _q \\circ f&=\\theta (f)\\partial _q+(\\partial _q f)\\theta ^{-1},\\\\\\partial _q^2 \\circ f&= (q+q^{-1})\\theta (\\partial _qf)\\theta ^{-1}\\partial _q +\\theta ^2(f)\\partial _q^2+(\\partial _q^2 f)\\theta ^{-2}, \\\\\\partial _q^3\\circ f&=(q^2 +q^{-2}+1)\\theta (\\partial _q^2f)\\theta ^{-2}\\partial _q + (q^2 +q^{-2} +1)\\theta ^2(\\partial _qf)\\theta ^{-1}\\partial ^2_q + (\\partial _q^3 f)\\theta ^{-3}+\\theta ^3(f)\\partial _q^3,\\\\\\partial _q^{-1}\\circ f&=\\theta ^{-1}(f)\\partial ^{-1}_q- \\theta ^{-2}(\\partial _qf)\\theta ^{-1}\\partial ^{-2}_q + \\cdots + (-1)^k \\theta ^{-k-1}(\\partial _q^k f)\\theta ^{-k}\\partial _q^{-k-1}+\\cdots .$ Using the Taylor's formula we can get the following proposition for the symmetric $q$ -exponent $e_q(x)$ , which is crucial to develop the tau function of the symmetric $q$ -KP hierarchy and to research the interaction of $q$ -solitons in the future.", "Proposition 2.", "The $q$ -exponent $e_q(x)$ is defined as $ e_q(x)=\\sum _{n=0}^{\\infty }\\dfrac{x^n}{(n)_q!", "},$ where $(n)_q!=(n)_q (n-1)_q (n-2)_q\\cdots (1)_q,$ then the formula $ e_q(x)=\\exp (\\sum _{k=1}^{\\infty }c_kx^k)$ holds, where $ c_k=\\sum _{i=1}^{k} (-1)^{i-1}\\frac{1}{i} \\sum \\limits _{v_1+v_2+\\cdots +v_i=k \\atop v_1, v_2, \\cdots , v_i \\in \\mathbb {Z}_+}\\frac{1}{(v_1)_q!", "(v_2)_q!", "\\cdots (v_i)_q!", "}.$ Proof.", "From the definition of $e_q(x)$ and Taylor's formula, it follows that $e_q(x)&=1+\\sum _{n=1}^{\\infty }\\dfrac{x^n}{(n)_q!}", "\\\\&=\\exp (\\ln (1+\\sum _{n=1}^{\\infty }\\dfrac{x^n}{(n)_q!}))", "\\\\&=\\exp (\\sum _{i=1}^{\\infty }(-1)^{i-1}\\frac{1}{i}(\\sum _{n=1}^{\\infty }\\dfrac{x^n}{(n)_q!", "})^i)\\\\&=\\exp (\\sum _{k=1}^{\\infty } \\sum _{i=1}^{k} (-1)^{i-1}\\frac{1}{i} \\sum \\limits _{v_1+v_2+\\cdots +v_i=k \\atop v_1, v_2, \\cdots , v_i \\in \\mathbb {Z}_+}\\frac{ x^k}{(v_1)_q!", "(v_2)_q!", "\\cdots (v_i)_q!}", ") \\\\&=\\exp (\\sum _{k=1}^{\\infty }c_kx^k),$ where $c_{k}$ is given by eq.", "(REF ).", "$\\square $ Several explicit forms of $q$ -exponent $e_q(x)$ can be written out as follows.", "$c_1=&1,\\\\c_2=&-\\frac{(q-1)^2}{2(q^2+1)},\\\\c_3=&\\frac{(q-1)^2(q^4-q^3-q^2-q+1)}{3(q^2+1)(q^4+q^2+1)},\\\\c_4=&-\\frac{(q-1)^4(q^4-q^3-2q^2-q+1)}{4(q^2-q+1)(q^6+q^4+q^2+1)},\\\\c_5=&\\frac{(q-1)^4(q^{14}-2q^{13}-2q^{11}+q^{10}-2q^{9}+5q^{8}+q^{7}+5q^{6}-2q^{5}+q^{4}-2q^{3}-2q-1)}{5(q^2+1)(q^2-q+1)(q^6+q^4+q^2+1)(q^8+q^6+q^4+q^2+1)},\\\\c_6=&-\\frac{(q-1)^6((q^{12}+1)(q^{2}-3q+1)+q^{2}(q^{8}+1)(q+1)-4q^{5}(q^{3}-1)(q-1)+2q^7)}{6(q^2-q+1)(q^6+q^4+q^2+1)(q^4-q^3+q^2-q+1)(q^8-q^7+q^6+q^2-q+1)}.$ For the case $ D_q(f(x))=\\frac{f(qx)-f(x)}{(q-1)x}$ and $\\tilde{(n)}_q=\\frac{q^n-1}{q-1}$ , $q$ -exponent function $\\tilde{e}_q(x)$ is defined as $\\tilde{e}_q(x)=\\sum _{n=0}^{\\infty }\\dfrac{x^n}{\\tilde{(n)}_q!", "},$ then $\\tilde{e}_q(x)=\\exp (\\sum _{k=1}^{\\infty }\\tilde{c}_kx^k),$ where $\\tilde{c}_k=\\frac{(1-q)^k}{k(1-q^k)}.$ Recall that the $q$ -exponent function $e_q(x)$ is the eigenfunction of operator $\\partial _q$ , i.e.", "$\\partial _q e_q(x)=e_q(x).$ Furthermore, from $e_q(xz)=\\sum _{n=0}^{\\infty }\\dfrac{(xz)^n}{(n)_q!", "}$ one obtains immediately that the formula $\\partial _q^m e_q(xz)=z^me_q(xz), m=1,2,3,\\cdots ,$ which is useful to define the $q$ -wave function of the symmetric $q$ -KP hierarchy in the following section." ], [ "Symmetric $q$ -deformed KP hierarchy", "Similar to the classical KP hierarchy [19], [20], we will define a new symmetric $q$ -deformed KP hierarchy.", "The Lax operator $L$ of the symmetric $q$ -KP hierarchy is given by $L=\\partial _q+ u_1 +u_2\\partial _q^{-1}+u_3\\partial _q^{-2}+\\cdots .$ where $u_i=u_i(x; t_1, t_2, t_3,\\cdots ,),i=1, 2, 3, \\cdots $ .", "The corresponding Lax equation of the symmetric $q$ -KP hierarchy is defined by $\\dfrac{\\partial L}{\\partial t_n}=[B_n, L], \\ \\ n=1, 2, 3, \\cdots ,$ where the differential part $B_n=(L^n)_+=\\sum \\limits _{i=0}^nb_i\\partial _q^i$ and the integral part $(L^n)_-=L^n-(L^n)_+$ .", "The first few $B_n$ and flow equations in eq.", "(REF ) for dynamical variables $\\lbrace u_1,u_{2},u_{3},\\cdots \\rbrace $ can be written out as follows.", "$B_1 &=\\partial _q+ u_1,\\\\B_2 &=\\partial _q^2+v_1\\partial _q+ v_0,\\\\B_3 &=\\partial _q^3+w_2\\partial _q^2+w_1\\partial _q+ w_0,$ where $L^2=B_2+v_{-1}\\partial _q^{-1}+\\cdots $ and $v_1 &= \\theta (u_1)+ u_1,\\\\v_0 &=(\\partial _qu_1)\\theta ^{-1}+\\theta (u_2)+u_1^2+u_2,\\\\v_{-1} &=(\\partial _qu_2)\\theta ^{-1}+\\theta (u_3)+u_1u_2+u_2\\theta ^{-1}(u_1)+u_3, \\\\w_2&=\\theta (v_1)+ u_1,\\\\w_1&=(\\partial _qv_1)\\theta ^{-1}+\\theta (v_0)+u_1v_1+u_2,\\\\w_0&=(\\partial _qv_0)\\theta ^{-1}+\\theta (v_{-1})+u_1v_0+u_2\\theta ^{-1}(v_1)+u_3.$ The first flow equations are $\\frac{\\partial u_1}{\\partial t_1}= &\\theta (u_2)- u_2,\\\\\\frac{\\partial u_2}{\\partial t_1}= &(\\partial _qu_2)\\theta ^{-1}+\\theta (u_3)+u_1u_2-u_2\\theta ^{-1}(u_1)- u_3,\\\\\\frac{\\partial u_3}{\\partial t_1}= &(\\partial _qu_3)\\theta ^{-1}+\\theta (u_4)+u_1u_3+u_2(\\theta ^{-2}(\\partial _qu_1))\\theta ^{-1}-u_3\\theta ^{-2}(u_1)- u_4,\\\\\\frac{\\partial u_4}{\\partial t_1}= &(\\partial _qu_4)\\theta ^{-1}+\\theta (u_5)+u_1u_4-u_2(\\theta ^{-3}(\\partial _q^2u_1))\\theta ^{-2}-u_4\\theta ^{-3}(u_1)- u_5 \\\\&+(2)_qu_3(\\theta ^{-3}(\\partial _qu_1))\\theta ^{-1}.$ The Lax operator $L$ in eq.", "(REF ) can be generated by a pseudo-difference operator $S=1+ \\sum _{k=1}^{\\infty }s_k\\partial _q^{-k}$ in the following way $L=S \\partial _q S^{-1}.$ Here $S$ is called dressing operator or wave operator of the symmetric $q$ -KP hierarchy.", "Proposition 3.", "Dressing operator $S$ of the symmetric $q$ -KP hierarchy satisfies the Sato equation $\\dfrac{\\partial S}{\\partial t_j}=-(L^j)_-S, \\quad j=1,2, 3, \\cdots .$ Proof.", "From the Lax equation $\\dfrac{\\partial L}{\\partial t_n}=[B_n, L]$ , which is followed by $\\dfrac{\\partial L}{\\partial t_j}&=[B_j, L]=(L^j)_+L-L(L^j)_+\\\\&=(L^j-(L^j)_-)L-L(L^j-(L^j)_-) \\\\&=-(L^j)_-L+L(L^j)_-.$ On the other hand, $\\dfrac{\\partial L}{\\partial t_j}&=\\dfrac{\\partial }{\\partial t_j}(S \\partial _q S^{-1})\\\\&=\\dfrac{\\partial S}{\\partial t_j} \\partial _q S^{-1}+S \\partial _q \\dfrac{\\partial S^{-1}}{\\partial t_j} \\\\&=\\dfrac{\\partial S}{\\partial t_j} S^{-1}S \\partial _q S^{-1}+S \\partial _q (-S^{-1} \\dfrac{\\partial S}{\\partial t_j} S^{-1}) \\\\&=\\dfrac{\\partial S}{\\partial t_j} S^{-1} L- L \\dfrac{\\partial S}{\\partial t_j} S^{-1},$ then $\\dfrac{\\partial L}{\\partial t_j}=-(L^j)_-L+L(L^j)_-=\\dfrac{\\partial S}{\\partial t_j} S^{-1} L- L \\dfrac{\\partial S}{\\partial t_j} S^{-1}.$ The above equation implies that $\\dfrac{\\partial S}{\\partial t_j}S^{-1}=-(L^j)_-, \\quad j=1,2, 3, \\cdots ,$ which ends the proof.", "$\\square $ Definition 1.", "The $q$ -wave function $w_q(x,t;z)$ for the symmetric $q$ -KP hierarchy eq.", "(REF ) with the wave operator $S$ in eq.", "(REF ) is given by $w_q(x,t;z)=S e_q(xz) \\exp ({\\sum _{i=1}^{\\infty }t_iz^i}),$ where $t=(t_1,t_2,t_3,\\cdots )$ .", "Proposition 4.", "The $q$ -wave function $w_q(x,t;z)$ of the symmetric $q$ -KP hierarchy satisfies the following linear $q$ -differential equations $Lw_q &=zw_q, \\partial _{m}w_q=(L^m)_+w_q,$ where $\\partial _{m}=\\frac{\\partial }{\\partial {t_m}}$ .", "Proof.", "Using the equation $\\partial _q e_q(xz)=ze_q(xz)$ , then $Lw_q &=S\\partial _q S^{-1} S e_q(xz) \\exp ({\\sum _{i=1}^{\\infty }t_iz^i})\\\\&=S\\partial _q e_q(xz) \\exp ({\\sum _{i=1}^{\\infty }t_iz^i})\\\\&=zw_q.$ From the Sato equation $\\partial _{m}S=-(L^m)_-S$ , it follows that $\\partial _{m}w_q&=\\partial _{m}(S e_q(xz) \\exp ({\\sum _{i=1}^{\\infty }t_iz^i})) \\\\&=(\\partial _{m}S) e_q(xz) \\exp ({\\sum _{i=1}^{\\infty }t_iz^i})+S e_q(xz) \\exp ({\\sum _{i=1}^{\\infty }t_iz^i})z^m \\\\&=-(L^m)_-Se_q(xz) \\exp ({\\sum _{i=1}^{\\infty }t_iz^i})+S\\partial _q^m e_q(xz) \\exp ({\\sum _{i=1}^{\\infty }t_iz^i}) \\\\&=-(L^m)_-w_q+(L^m)_+w_q \\\\&=(L^m)_+w_q.$ $\\square $ Furthermore, we would like to give the definitions of the symmetric $q$ -CKP hierarchy and the symmetric $q$ -BKP hierarchy respectively to answer the previous question mentioned in the introduction.", "Definition 2.", "Let the operator $L$ in eq.", "(REF ) is the Lax operator for the symmetric $q$ -KP hierarchy associated with eq.", "(REF ), if $L$ satisfies the reduction condition $L^=-L$ , then we call it the symmetric $q$ -CKP hierarchy.", "Definition 3.", "Let the operator $L$ in eq.", "(REF ) is the Lax operator for the symmetric $q$ -KP hierarchy associated with eq.", "(REF ), if $L$ satisfies the reduction condition $L^=-\\theta ^{-\\frac{1}{2}}\\partial _q L\\partial _q^{-1}\\theta ^{\\frac{1}{2}}$ , then it is the symmetric $q$ -BKP hierarchy." ], [ "Additional symmetry of the symmetric $q$ -KP hierarchy", "The another main goal of this note is to consider the additional symmetries of the symmetric $q$ -KP hierarchy.", "First, let us define $\\Gamma _q$ and Orlov-Shulman's $M$ operator as $\\Gamma _q &=\\sum _{i=1}^{\\infty }\\Big (it_i+ic_ix^i\\Big )\\partial _q^{i-1}, \\\\M &= S \\Gamma _q S^{-1},$ where $c_i$ is given by eq.", "(REF ).", "Then the additional flows of the symmetric $q$ -KP hierarchy for each pair {$m,n$ } are defined by $\\dfrac{\\partial S}{\\partial t_{m,n}^}=-(M^mL^n)_-S.$ Proposition 5.", "The additional flows act on $L$ and $M$ of the symmetric $q$ -KP hierarchy as $\\dfrac{\\partial L}{\\partial t_{m,n}^}&=-[(M^mL^n)_-,L], \\\\\\dfrac{\\partial M}{\\partial t_{m,n}^}&=-[(M^mL^n)_-,M].$ Proof.", "By performing the derivative $\\dfrac{\\partial }{\\partial t_{m,n}^}$ on $L=S \\partial _q S^{-1}$ and using the eq.", "(REF ), we observe that $\\dfrac{\\partial L}{\\partial t_{m,n}^}&= \\dfrac{\\partial S}{\\partial t_{m,n}^}\\partial _q S^{-1} + S \\partial _q \\dfrac{\\partial S^{-1} }{\\partial t_{m,n}^} \\\\&=-(M^mL^n)_-S\\partial _q S^{-1}+S \\partial _q (-S^{-1} \\dfrac{\\partial S}{\\partial t_{m,n}^} S^{-1}) \\\\&=-(M^mL^n)_-L+S \\partial _qS^{-1}(M^mL^n)_-\\\\&=-[(M^mL^n)_-,L].$ For the action on $M=S \\Gamma _q S^{-1}$ , there exists similar derivation as $\\dfrac{\\partial L}{\\partial t_{m,n}^}$ , and then $\\dfrac{\\partial M}{\\partial t_{m,n}^}&= \\dfrac{\\partial S}{\\partial t_{m,n}^}\\Gamma _q S^{-1} + S \\Gamma _q \\dfrac{\\partial S^{-1} }{\\partial t_{m,n}^} \\\\&=-(M^mL^n)_-S\\Gamma _q S^{-1}+S \\Gamma _q (-S^{-1} \\dfrac{\\partial S}{\\partial t_{m,n}^} S^{-1}) \\\\&=-(M^mL^n)_-M+S \\Gamma _qS^{-1}(M^mL^n)_-\\\\&=-[(M^mL^n)_-,M].$ In the above calculation, the fact that $\\Gamma _q$ does not depend on the additional flows variables $t_{m,n}^$ has been used.", "$\\square $ Corollary 1.", "$\\dfrac{\\partial L^k}{\\partial t_{m,n}^}&=-[(M^mL^n)_-,L^k], \\\\\\dfrac{\\partial M^k}{\\partial t_{m,n}^}&=-[(M^mL^n)_-,M^k],\\\\\\dfrac{\\partial M^kL^l}{\\partial t_{m,n}^}&=-[(M^mL^n)_-,M^kL^l],\\\\\\dfrac{\\partial M^kL^l}{\\partial t_{n}}&=[B_n,M^kL^l].$ Proof.", "We present only the proof of the first equation here.", "The others can be proved in a similar way.", "$\\dfrac{\\partial L^k}{\\partial t_{m,n}^}&= \\dfrac{\\partial L}{\\partial t_{m,n}^}L^{k-1}+L\\dfrac{\\partial L}{\\partial t_{m,n}^}L^{k-2}+\\cdots +L^{k-2}\\dfrac{\\partial L}{\\partial t_{m,n}^}L+L^{k-1}\\dfrac{\\partial L}{\\partial t_{m,n}^} \\\\&=\\sum _{l=1}^k L^{l-1}\\dfrac{\\partial L}{\\partial t_{m,n}^}L^{k-l} \\\\&=\\sum _{l=1}^k L^{l-1} (- [(M^mL^n)_-,L]) L^{k-l} \\\\&=-[(M^mL^n)_-,L^k],$ where we have used the formula $\\dfrac{\\partial L}{\\partial t_{m,n}^}=-[(M^mL^n)_-,L]$ in the Proposition 5.", "$\\square $ Proposition 6.", "The additional flows ${\\partial _{mn}^}= \\dfrac{\\partial }{\\partial t_{m,n}^}$ commute with the hierarchy $\\partial _k=\\dfrac{\\partial }{\\partial t_k}$ , i.e.", "$[\\partial _{mn}^,\\partial _k]=0,$ thus we call them additional symmetries of the symmetric $q$ -KP hierarchy.", "Proof.", "According to the definition and the Corollary 1, it equals to $[\\partial _{mn}^,\\partial _k]S &=\\partial _{mn}^(\\partial _kS)-\\partial _k(\\partial _{mn}^S) \\\\&=\\partial _{mn}^( -(L^k)_-S)-\\partial _k(-(M^mL^n)_-S) \\\\&=-(\\partial _{mn}^L^k)_-S-(L^k)_-(\\partial _{mn}^ S)+(\\partial _kM^mL^n)_-S+(M^mL^n)_-(\\partial _kS)\\\\&=[(M^mL^n)_-,L^k]_-S+(L^k)_- (M^mL^n)_-S +[(L^k)_+,M^mL^n]_-S-(M^mL^n)_-(L^k)_-S\\\\&=[(M^mL^n)_-,L^k]_-S-[(M^mL^n)_-, (L^k)_+]S +[(L^k)_-,(M^mL^n)_-]S\\\\&=[(M^mL^n)_-,(L^k)_-]_-S+[(L^k)_-,(M^mL^n)_-]S\\\\& =0.$ $[(L^k)_+,(M^mL^n)]_-=[(L^k)_+,(M^mL^n)_-]_-$ and $[(M^mL^n)_-,(L^k)_-]_-=[(M^mL^n)_-,(L^k)_-]$ have been used in the above derivation.", "$\\square $" ], [ "Conclusions and discussions", "To summarize, we have derived the antisymmetric property of $\\partial _q$ in Proposition 1 and a crucial expression of $e_q(x)$ by usual exponential in Proposition 2.", "The analytic property of symmetric $e_q(x)$ in Proposition 2 is used to define the wave function of the symmetric $q$ -KP hierarchy.", "After introducing the dressing operator and the $q$ -wave function of the symmetric $q$ -KP hierarchy in Section 3, we also give the definitions of symmetric $q$ -CKP hierarchy and symmetric $q$ -BKP hierarchy.", "The additional symmetries of the symmetric $q$ -KP hierarchy are obtained in Section 4.", "The above results of this paper show obviously that the symmetric $q$ -KP hierarchy is different with the $q$ -KP hierarchy[8], [9], [10], [11], [12], [13], [14], [15], [16], [17] based on the $D_q(f(x))$ .", "In comparison with the known interesting results of the KP hierarchy [19], [18], [20] and the $q$ -KP hierarchy based on the $D_q(f(x))$ [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], the symmetric $q$ -KP hierarchy defined in this paper deserves further study from several aspects including the tau function and its Hirota bilinear identity, the Hamiltonian structure, the gauge transformation, the symmetry analysis and the interaction of $q$ -solitons.", "Furthermore, it is highly nontrivial to consider above topics of the symmetric $q$ -CKP(or $q$ -BKP) hierarchy because of the reduction condition $L^=-L$ (or $L^=-\\partial _q L \\partial _q^{-1}$ ) and the complexity of the $\\partial _q$ .", "Acknowledgments This work was supported by Erasmus Mundus Action 2 EXPERTS, SMSTC grant no.", "12XD1405000, Fundamental Research Funds for the Central Universities, and NSF grant no.", "11271210, 11201451, 10825101 of China." ] ]	1403.0169
[ [ "Delocalization of electrons by cavity photons in transport through a\n quantum dot molecule" ], [ "Abstract We present new results on cavity-photon-assisted electron transport through two lateral quantum dots embedded in a finite quantum wire.", "The double quantum dot system is weakly connected to two leads and strongly coupled to a single quantized photon cavity mode with initially two linearly polarized photons in the cavity.", "Including the full electron-photon interaction, the transient current controlled by a plunger-gate in the central system is studied by using quantum master equation.", "Without a photon cavity, two resonant current peaks are observed in the range selected for the plunger gate voltage: The ground state peak, and the peak corresponding to the first-excited state.", "The current in the ground state is higher than in the first-excited state due to their different symmetry.", "In a photon cavity with the photon field polarized along or perpendicular to the transport direction, two extra side peaks are found, namely, photon-replica of the ground state and photon-replica of the first-excited state.", "The side-peaks are caused by photon-assisted electron transport, with multiphoton absorption processes for up to three photons during an electron tunneling process.", "The inter-dot tunneling in the ground state can be controlled by the photon cavity in the case of the photon field polarized along the transport direction.", "The electron charge is delocalized from the dots by the photon cavity.", "Furthermore, the current in the photon-induced side-peaks can be strongly enhanced by increasing the electron-photon coupling strength for the case of photons polarized along the transport direction." ], [ "Introduction", "An opto-electronic device provides a different platform of electron transport, namely photon-assisted transport (PAT)[1].", "In the PAT, the energy levels of an electronic system have to match to photon frequency of a radiation source to control the electron motion.", "Therefore, the photon emission and the photon absorption processes play an essential role to enhance electron transport.", "[2] For that purpose, an electrostatic potential produced by a plunger-gate is applied to the electronic system to shift it's energy levels in and out of resonance.", "The plunger-gate is widely used to control charge current [3], thermal current [4], photo-current [5] and spin-dependent current [6] for various quantized systems coupled to photon radiation.", "The PAT controlled by plunger-gate has been investigated to study electrical [7] and optical [8], [9] properties of a double-quantum dot (DQD) system , in which the PAT can be used as a spectroscopic tool in two different regimes defined by a zero [10], and non-zero [11] bias voltage.", "At zero-bias voltage, the DQD works as a proper electron pumping device in which the photon absorption process leads to electron tunneling producing a dc current.", "In the non-zero bias voltage, both the photon absorption and the photon emission processes generate a dc current.", "Recently, both regimes have been realized experimentally in a DQD system at low temperature.", "[12], [13] The most important application of a DQD system in the quantum regime is intended for information storage in a quantum state,[14] quantum-bits for quantum computing, [15], [16] and quantum information processing in two-state system.", "[17] Recent experimental work has focused on using the two lowest energy states contributing to tunneling processes in a DQD working as a two state system: The ground state resonance, and a photon-induced excited state resonance.", "They observed multiphoton absorption processes up to the four-order contributing to the electron transport.", "[13] Based on the above-mentioned considerations, we analyze PAT in serial double quantum dots embedded in a quantum wire.", "The DQD system is connected to two leads and coupled to a photon cavity with linearly polarized photons in the $x$ - and $y$ -directions, where the transport along the quantum wire is in the $x$ -direction.", "A quantum master equation (QME) formalism is utilized to investigate transient transport of electrons controlled by the plunger-gate in the system without and with a single-photon mode.", "[3] Generally, there are two types of QME when characterized according to memory effects, energy-dependent coupling, and the system-leads coupling strength: The Markovian and the non-Markovian QME.", "In the case of the Markovian approximation, the system-leads coupling is assumed weak and independent of energy, memory effect are ignored and most commonly a steady state is sought.", "[18], [19], [20], [21] In the non-Markovian approach, the system is energetically coupled to the leads including memory effect in the system.", "[22], [23], [24] Since we are interested in studying transient transport of electrons in a regime with possible resonances, the non-Markovian model is used in our system.", "[25] In addition, we assume the DQD system to be connected to the leads through a non-zero or small bias window, where the two lowest energy states of the QDQ system can be isolated in the bias window: The ground state and the first-excited state.", "Our model of the DQD system can be seen as a qubit.", "In which the states $\\vert 0\\rangle $ and $\\vert 1\\rangle $ can be represented in terms of the ground state and the first-excited state.", "We will show how the single-photon mode affects the electron transport through both states when located in the bias window and demonstrate the role of photon activated states in the transient current.", "The double serial quantum dot is essential here: The two lowest single-electron states of the dot molecule have very different symmetry.", "The ground state has a symmetric wavefunction, but the excited state has an antisymmetric one.", "The conduction through the ground state is thus higher than through the excited one.", "The “inter-dot tunneling” can be influenced by a photon mode polarized in the transport direction, thus strongly modifying the conduction through the photon replicas of the states in a photon cavity.", "The nontrivial details of this picture will be analyzed in this paper reminding us that the effects rely on the geometry of the system and states beyond the ground state and the first excited one.", "The rest of the paper is organized as follows.", "In Sec.", "we introduce the model to describe the electron transport through a DQD embedded in a quantum wire connected to two leads and a photon cavity.", "Section.", "contains two subsections, the system without and with the photon cavity.", "In the absence of the photon cavity, the transient current through the system controlled by the plunger-gate is demonstrated in the presence of the electron-electron interactions in the DQD system.", "In the photon cavity, the photon-assisted electron transport in the system is presented for a system initially with no electron, but with two linearly polarized photons in the single-photon mode.", "Finally, conclusions are provided in Sec.", "." ], [ "Model and Computational methods", "The aim of this study is to model a photon-assisted electron transport in a DQD system connected to two identical electron reservoirs (lead) and coupled to a single photon mode in a cavity.", "Our first step is to look at the central system, in which electrons are confined in two dimensions.", "We assume a finite quantum wire with hard-wall ends at $x$ = $\\pm L_{x}/2$ with length $L_x=165$ nm.", "It is parabolically confined in the $y$ -direction (perpendicular to the transport direction) with transverse confinement energy $\\hbar \\Omega _0 = 2.0$ meV.", "The embedded quantum dots are modeled by two identical Gaussian potentials in the quantum wire defined as $V_{\\rm DQD}(x,y) = \\sum _{i = 1}^2 V_i\\; \\exp {\\left[ -\\beta _{i}^2\\left( (x-x_i)^2 + y^2\\right) \\right]},$ with quantum-dot strength $V_{1,2} = -2.8$ meV, $x_{1} = 35$ nm, $x_{2} = -35$ nm, and $\\beta _{1,2}$ = $5.0\\times 10^{-2}~{\\rm nm^{-1}}$ such that the radius of each quantum-dot is $R_{\\rm QD} \\approx 20$ nm.", "A sketch of the DQD system under investigation is shown in Fig.", "REF .", "We should mention that the distance between the dots is $L_{\\rm DQD} = 35~{\\rm nm} \\simeq 1.47a_w$ , and each dot is $25~{\\rm nm} = 1.05a_w$ away from the nearest lead, where $a_w$ is the effective magnetic length.", "Figure: (Color online)Schematic diagram depicts the potential representing the DQDembedded in a quantum wire with parameters B=0.1TB = 0.1~{\\rm T},a w =23.8 nm a_{w} = 23.8~{\\rm nm}, and ℏΩ 0 =2.0 meV \\hbar \\Omega _0 = 2.0~{\\rm meV}.The DQD system is in a rectangular photon cavity with a single photon mode.", "The photons in the single photon mode are linearly polarized in the $x$ - or $y$ -directions, meaning that the photon polarization in the cavity is assumed to be parallel or perpendicular to the transport direction with respect to the electric field $\\mathbf {A_{\\rm ph}} = A_{\\rm ph}\\left( a+a^{\\dagger } \\right)\\mathbf {\\hat{e}},$ where $A_{\\rm ph}$ is the amplitude of the photon vector potential, $a^{\\dagger }(a)$ are the creation (annihilation) operators for a photon, respectively, and $\\mathbf {\\hat{e}}$ determines the polarization with $ \\mathbf {\\hat{e}} = \\left\\lbrace \\begin{array}{l l}(e_x,0), & \\quad \\text{ TE$_{011}$}\\\\(0,e_y), & \\quad \\text{ TE$_{101}$},\\end{array} \\right.$ where TE$_{011}$ (TE$_{101}$ ) indicates the parallel (perpendicular) polarized photon in the transport direction, respectively.", "In the following sections, we shall couple the DQD system to both the photon cavity and the leads." ], [ "DQD system coupled to Cavity", "We consider the closed DQD system to be strongly coupled to a photon cavity.", "The many-body (MB) Hamiltonian $H_\\mathrm {S} = H_\\mathrm {DQD} + H_\\mathrm {Cavity} + H_\\mathrm {Int}$ consists of the Hamiltonian for the closed DQD system with the electron-electron interaction $H_\\mathrm {DQD}$ , the free photon cavity Hamiltonian $H_\\mathrm {Cavity}$ , and the Hamiltonian for the electron-photon interaction $H_\\mathrm {Int}$ .", "The DQD system (and the external leads) is placed in an external uniform perpendicular magnetic field $B\\hat{z}$ in the $z$ -direction defining an effective lateral confinement length $a_w = (\\hbar /m^* \\sqrt{(\\omega ^{2}_c + \\Omega ^2_0)})^{1/2}$ , where the effective electron mass is $m^=0.067m_e$ for GaAs material and $\\omega _c = e B/m^c$ is the cyclotron frequency.", "The Hamiltonian for the DQD system in a magnetic field including the electron-electron interaction can be written as $H_\\textrm {DQD} &=& \\sum _{i,j} \\langle \\psi _i \\vert \\left\\lbrace \\frac{\\mathbf {\\pi }_e^2}{2m^}+ V_\\mathrm {DQD} + eV_\\mathrm {pg} \\right\\rbrace \\vert \\psi _j\\rangle \\delta _{i,j} d_i^{\\dagger } d_j \\nonumber \\\\&& + H_\\mathrm {Coul} + H_\\mathrm {Z},$ where $\|\\psi _i\\rangle $ stands for a single-electron states, $V_{\\mathrm {pg}}$ is the the plunger gate potential that shifts the energy levels of the DQD system with respect to the chemical potentials of the leads, $d^\\dagger _{i}$ ($d_{j}$ ) is an operator that creates (annihilates) an electron in the DQD system, respectively.", "Moreover, the canonical momentum is $\\mathbf {\\pi }_e= \\mathbf {p}+\\frac{e}{c}\\mathbf {A}_{\\mathrm {ext}}$ with the kinetic momentum operator $\\mathbf {p}$ , and the vector potential in the Landau gauge $\\mathbf {A}_{\\mathrm {ext}}$ = ($0,-By,0$ ).", "The electron-electron interaction in the central system is given by $H_\\mathrm {Coul} = \\frac{1}{2}\\sum _{ijrs} V_{\\textrm {ijrs}}d_i^{\\dagger }d_j^{\\dagger }d_sd_r,$ with the Coulomb matrix elements $V_{\\textrm {ijrs}}$ .", "[26] The characteristic Coulomb energy is $E_{\\rm C} = e^2/(2\\varepsilon _r a_w) \\approx 2.44~{\\rm meV}$ at $B=0.1$ T with $a_{w} = 23.8~{\\rm nm}$ and $\\varepsilon _r=12.4$ , the dielectric constant of GaAs.", "The characteristic Coulomb energy is greater than the thermal energy of the leads.", "An exact numerical diagonalization method is used here for solving the Coulomb interacting many-electron (ME) Hamiltonian in a truncated Fock space to obtain the ME energy spectrum of the DQD system.", "[27] The Zeeman Hamiltonian shown in the third part of Eq.", "(REF ) describes the interaction between the external magnetic field and the magnetic moment of an electron $H_{\\rm Z}= \\pm \\frac{g^\\ast \\mu _B}{2} B,$ where $\\pm $ stands for $z$ -spin components, $\\mu _B = e\\hbar /2m_{\\rm e}c$ is the Bohr magneton, and the effective Lande $g$ -factor is $g^\\ast = -0.44$ for GaAs.", "In order to investigate photon-assisted electron transport in the DQD system, the electronic system is coupled to a photon cavity.", "The Hamiltonian of the free photon cavity is given by $H_\\textrm {Cavity} = \\hbar \\omega _{\\textrm {ph}} \\hat{N}_{\\rm ph},$ where $\\hbar \\omega _{\\textrm {ph}}$ is the energy of the single mode in the cavity, and $\\hat{N}_{\\rm ph} =a^{\\dagger } a$ is the photon number operator.", "The interaction of the single quantized electromagnetic mode with the electronic system is described by the Hamiltonian including both the diamagnetic and the paramagnetic interactions of photons and electrons $H_\\textrm {Int} &=& g_{\\rm ph}\\sum _{ij}d_i^{\\dagger }d_j\\; g_{ij}\\left\\lbrace a + a^\\dagger \\right\\rbrace \\nonumber \\\\&&+\\frac{g_{\\rm ph}^2}{\\hbar \\Omega _w} \\sum _{i}d_i^{\\dagger }d_i\\left[ \\hat{N}_{\\rm ph} + \\frac{1}{2}\\left( a^\\dagger a^\\dagger + aa + 1 \\right)\\right]$ herein, $g_{\\rm ph} = e A_{\\rm ph} \\Omega _wa_w/c$ is the electron-photon coupling strength, and $g_{ij}$ are dimensionless electron-photon coupling matrix elements.", "[28] Finally, the MB system Hamiltonian $H_{\\rm S}$ is diagonalized in a MB Fock-space $\\lbrace \|\\breve{\\alpha }\\rangle \\rbrace $ to obtain the MB energy spectrum of the DQD system coupled to the photon cavity.", "[29] The diagonalization builds a new interacting MB state basis $\\lbrace \|\\breve{\\nu })\\rbrace $ , in which $\|\\breve{\\nu }) = \\sum _{\\alpha }{\\cal W}_{\\mu \\alpha }\|\\breve{\\alpha }\\rangle $ with ${\\cal W}_{\\mu \\alpha }$ being a unitary transformation matrix.", "The unitary transformation is used to convert the QME and the physical observables from non-interacting MB basis to the interacting MB basis." ], [ "DQD system connected to leads", "The DQD system is connected to two semi-infinite leads with the same width.", "The chemical potential of the lead $l$ is $\\mu _l$ , with $l\\in \\lbrace L,R\\rbrace $ being the left $L$ and the right $R$ lead.", "The the Fermi function in the isolated lead $l$ before coupling to the central system is $f_l\\left( \\epsilon _l(\\mbox{$q$}) \\right) = \\lbrace \\exp [\\epsilon _l(\\mbox{$q$})-\\mu _l]+1\\rbrace ^{-1}$ , where $\\epsilon _l$ is the SE subband energy of the lead $l$ ($\\mbox{$q$}$ is the momentum dummy index.", "[3]) found from the non-interacting ME Hamiltonian of lead $l$ $H_l = \\int d{\\mbox{$q$}}\\, \\epsilon _l(\\mbox{$q$}) {c^\\dagger _{{\\mbox{$q$}}l}}c_{{\\mbox{$q$}}l},$ with ${c^\\dagger _{{\\mbox{$q$}}l}}$ ($c_{{\\mbox{$q$}}l}$ ) the electron creation(annihilation) operator in lead $l$ , respectively.", "[30] In order to instigate electron transport between the subsystems, the DQD system is coupled to the leads with energy dependent coupling coefficients reflecting the geometry of the system $T_{{\\mbox{$q$}i}l} =\\int d\\mathbf {r} d\\mathbf {r^{\\prime }} \\psi _{\\mbox{$q$}l}(\\mathbf {r}^{\\prime })^g_{\\mbox{$q$}il} (\\mathbf {r},{\\bf r^{\\prime }}) \\psi ^\\mathrm {S}_i({\\bf r}).$ An electron may be transferred from a state $\|\\mbox{$q$}\\rangle $ with the wavefunction $\\psi _{\\mbox{$q$}l}(\\mathbf {r}^{\\prime })$ in the leads to a SE state $\|i\\rangle $ with the SE wavefunction $\\psi ^\\mathrm {S}_i({\\bf r})$ in the DQD system and vice versa, where the coupling function is $g_{\\mbox{$q$}il} ({\\bf r},{\\bf r^{\\prime }})$ .", "[25] The coupling coefficients are utilized to construct a time-dependent coupling Hamiltonian in the second quantization language $H_{{\\rm T}l}(t)= \\chi _l(t) \\sum _{i}\\int d{\\mbox{$q$}}\\, \\left[ {c^\\dagger _{{\\mbox{$q$}}l}} T_{{\\mbox{$q$}i}l} d_i+ d^\\dagger _i (T_{{i\\mbox{$q$}l}})^* c_{{\\mbox{$q$}}l}\\right],$ with a time-dependent switching function $ \\chi _l(t) = 1 - 2\\lbrace \\exp [\\alpha _l (t-t_0)] + 1\\rbrace ^{-1}$ with $\\alpha _l = 0.3$ ${\\rm ps}^{-1}$ being a switching parameter.", "After the DQD system is coupled to the leads at $t= 0$ , we calculate the time evolution of the electrons and photons using the density operator and its equation of motion, the Liouville-von Neumann (Lv-N) equation $i\\hbar \\dot{W}(t)= \\left[H(t),W(t)\\right]$ for the whole system.", "As this can not be accomplished we resort to using a projection formalism taking a trace over the Hilbert space of the leads introducing the reduced density operator $\\rho (t)={\\rm Tr}_\\mathrm {L}{\\rm Tr}_\\mathrm {R} W(t)$ with $\\rho (t_0) = \\rho _\\mathrm {S}$[31] and the condition that $W(t<t_0) = \\rho _\\mathrm {L}\\rho _\\mathrm {R}\\rho _\\mathrm {S}$ is the density operator of the total system before the coupling with $\\rho _\\mathrm {S}$ being the density operator of the isolated DQD system.", "[32] The density operator of the leads before the coupling is $\\rho _l= \\exp [{-\\beta (H_l-\\mu _l N_l)}]/{\\rm Tr}_l \\lbrace \\exp [{-\\beta (H_l-\\mu _l N_l)}]\\rbrace $ , where $\\beta = 1/k_{B}T_l$ is the inverse thermal energy, and $N_l$ is the number operator for electrons in the lead $l$ ,[30] The time-dependent mean charge in the central system, the current in the leads are calculated from the reduced density operator as has been detailed in earlier publications.", "[3], [29]" ], [ "Numerical Results", "In this section, we discuss the transport properties through the DQD system controlled by plunger-gate voltage in both cases without a photon cavity and with $x$ - or $y$ -polarized photons in a cavity.", "In order to obtain the PAT, the system has to satisfy the following conditions: the MB energy level spacing has to be greater than the thermal energy $\\Delta E_{\\rm MB} > k_B T$ , and the MB energy level spacing has to be smaller or equal to the photon energy $\\Delta E_{\\rm MB} \\le \\hbar \\omega _{\\rm ph}$ .", "[33] Initially the temperature of the central system is assumed to be $T=0$ K, and the leads are at $T=0.01$ K initially.", "Other physical parameters of the system are presented in table REF .", "Table: Characteristic energy scales of the systemIn addition, we assume the external magnetic field to be $B = 0.1$ T with the effective lateral confinement length $a_w = 23.8$ nm and initially no electron is in the DQD system." ], [ "The DQD system without the photon cavity", "In this section, the properties of the electron transport through the DQD system are presented in the absence of the photon cavity in order to establish a comparison for later results for transport through the system inside a cavity.", "Figure REF (a) shows the energy spectrum of the leads versus wave number $qa_w$ .", "The horizontal black lines are chemical potentials of the left lead $\\mu _L$ and the right lead $\\mu _R$ .", "The chemical potentials are considered to be $\\mu _L = 1.4\\ {\\rm meV}$ and $\\mu _R = 1.3\\ {\\rm meV}$ , implying a small bias voltage $\\Delta \\mu = 0.1$ meV.", "Therefore, the first subband in the parabolic energy spectrum becomes the most active subband contributing to the electron transport process in the energy range [$1.3,1.4$ ] meV.", "In Fig.", "REF (b), the ME energy spectrum of the DQD-system as a function of applied plunger-gate voltage $V_{\\rm pg}$ is shown.", "The energies of two-electron states $N_e = 2$ (2ES, blue dots) are higher than the SE states $N_e = 1$ (1ES, red dots) due to the electron-electron interaction.", "In the absence of the photon cavity, two resonant SE states are situated in the bias window for the range of plunger gate voltage selected here, namely, the ground state resonance and first-excited state resonance (blue squared dots).", "Figure: (Color online)Energy spectra in the case of no photon cavity with magnetic field B=0.1B=0.1 T. (a) SEenergy spectrum in the leads (red) is plotted as a function of scaled wave number qa w q a_w, wherethe chemical potentials are μ L =1.4 meV \\mu _L = 1.4\\ {\\rm meV} and μ R =1.3 meV \\mu _R = 1.3\\ {\\rm meV}(black).", "(b) ME energy spectrum in the central system as a function of plunger gatevoltage V pg V_{\\rm pg} including SE states (1ES, red dots) and two electron states (2ES,blue dots).", "The SE state in the bias window is almost doubly degenerate due to thesmall Zeeman energy.The almost degenerate two spin states of the single-electron ground state are $\|2)$ and $\|3)$ with energies $E_{\\rm 2} = 1.343$ meV and $E_{\\rm 3} = 1.346$ meV.", "These two states get into resonance with the first-subband of the leads at $V_{\\rm pg}^{\\rm G} = 1.2$ mV, where the superscript $\\rm G$ refers to the ground state.", "By tuning the plunger-gate voltage, the two spin states of the first-excited state $\|4)$ and $\|5)$ with energies $E_{\\rm 4} = 1.358$ meV and $E_{\\rm 5} = 1.361$ meV contribute to the electron transport at $V_{\\rm pg}^{\\rm FE} = 0.8$ mV, where the superscript $\\rm FE$ stands for the first-excited state.", "Figure REF displays the left current $I_L$ (red solid) and the right current $I_R$ (dashed blue) through the DQD system.", "Figure: (Color online) The left current I L I_L (red solid) and right current I R I_R (blue dashed) are plotted as a function ofplunger gate voltage V pg V_{\\rm pg} at time t=220t = 220 ps in the case of no photon cavity.Other parameters are B=0.1TB = 0.1~{\\rm T} and Δμ=0.1 meV \\Delta \\mu =0.1~{\\rm meV}.We notice two resonance peaks in the currents: The ground state peak at $V_{\\rm pg}^{\\rm G} = 1.2$ mV and the first-excited state peak at $V_{\\rm pg}^{\\rm FE} = 0.8$ mV.", "The reason for the two current peaks is resonance of the SE states in the DQD system with the first subband energy of the leads.", "An electron in the first-subband of the left lead may tunnel to the state $\|2)$ or $\|3)$ of the DQD system and subsequently tunnel out to the right lead.", "Consequently the ground state peak is observed at $V_{\\rm pg}^{\\rm G}=1.2$ mV.", "In addition, the first-excited state peak reflects a resonance with the states $\|4)$ and $\|5)$ at plunger-gate potential $V_{\\rm pg}^{\\rm FE}=0.8$ mV.", "Figure REF shows the charge density distribution in the DQD system at time $t = 220$ ps (after the initial transient, close to a steady state) in the ground state peak (a), and the first-excited state peak (b).", "In the case of the ground state peak at $V_{\\rm pg}^{\\rm G} = 1.2$ mV, the electron state accumulates in the dots with a strong inter-dot tunneling.", "Therefore, the left and right currents increase in the system.", "But in the case of first-excited state peak at $V_{\\rm pg}^{\\rm FE} = 0.8$ mV, the electron state is strongly localized in the dots without much tunneling between the dots.", "Thus the tunneling between the dots is sufficiently suppressed and the current drops as shown in Fig.", "REF .", "Figure: (Color online) The charge density distribution att=220t = 220 ps in the ground state peak (a) and first-excited state peak (b) shownin Fig.", "in the case with no photon cavity.", "Other parameters areB=0.1B=0.1 T, a w a_{w} = 23.8 nm 23.8~{\\rm nm}, L x L_x = 165 nm = 6.93a w 6.93 a_w, and Δμ=0.1 meV \\Delta \\mu =0.1~{\\rm meV}." ], [ "$x$ -photon polarization (TE{{formula:98f47384-0004-46bf-aaa6-69f821775938}} mode)", "In this section we analyze the electron transport through the DQD system in the presence of an $x$ -polarized single-photon mode with initially two photons in the cavity.", "The photons in the cavity can excite electrons in the DQD system and enhance the electric current, similar to the “classical” PAT case.", "[11] The condition for PAT involving $N_{\\rm ph}$ photon(s) is $\|E_i - E_f\| = N_{\\rm ph}\\hbar \\omega _{\\rm ph}$ ,[34] where $E_i$ ($E_f$ ) is the highest possible initial (lowest possible final) MB energy level of the DQD system, respectively.", "[33] We vary the applied plunger-gate to match $\|E_i - E_f\|$ to the photon energy, thus the PAT is an active process in the system.", "Figure REF shows the MB energy spectrum of the DQD system including the photons with zero-electron states $N_e = 0$ (0ES, green dots) and SE states $N_e = 1$ (1ES, red dots).", "Figure: (Color online) MB Energy spectrum versus the plunger gate voltage V pg V_{\\rm pg}in the case of xx-polarized photon field, where 0ES indicates zero electron states (Ne = 0, green dots),and 1ES stands for single electron states (Ne = 1, red dots).Other parameters are B=0.1B=0.1 T, Δμ=0.1\\Delta \\mu = 0.1 meV, and ℏω ph =0.25 meV \\hbar \\omega _{\\rm ph} = 0.25~{\\rm meV}.In addition to the former states at $V_{\\rm pg}^{\\rm G} = 1.2$ mV and $V_{\\rm pg}^{\\rm FE} = 0.8$ mV, two extra active MB-states are observed in the presence of the photon cavity at $eV_{\\rm pg}^{\\rm G_{\\gamma };FE_{\\gamma }} = eV_{\\rm pg}^{\\rm G;FE} - \\hbar \\omega _{\\rm ph}$ in the bias window (pink squared dots), where the photon energy is $\\hbar \\omega _{\\rm ph} = 0.25$ meV, and $\\rm G_{\\gamma }(\\rm FE_{\\gamma })$ stands for the photon-replica of the ground state(first-excited state), respectively.", "We notice that all states in the bias window are SE states containing only one-electron $N_{\\rm e} = 1$ .", "Figure REF displays the left current $I_L$ (a) and the right current $I_R$ (b) as a function of the plunger-gate voltage $V_{\\rm pg}$ in the presence of the $x$ -polarized photon field at time $t = 220$ ps for different electron-photon coupling strength $g_{\\rm ph} = 0.1~{\\rm meV}$ (blue solid), $0.2~{\\rm meV}$ (green dashed), and $0.3~{\\rm meV}$ (red dotted).", "The positive value of the left current indicates electrons tunneling from the left lead to the DQD system, while the negative value of the right current denotes electrons tunneling from the right lead to the DQD system and vise versa.", "Figure: (Color online) The left current I L I_L (a), and the right current I R I_R (b)versus the plunger gate voltage V pg V_{\\rm pg} in the case of xx-polarized photon field at time t=220 ps t = 220~{\\rm ps}with different electron-photon coupling strength:g ph =0.1g_{\\rm ph} = 0.1 meV (blue solid), 0.20.2 meV (green dashed),and 0.30.3 meV (red dotted).Other parameters are ℏω ph =0.25 meV \\hbar \\omega _{ph} = 0.25~{\\rm meV}, Δμ=0.1 meV \\Delta \\mu =0.1~{\\rm meV}, and B=0.1TB = 0.1~{\\rm T}.In the absence of the photon cavity, two main-peaks are found at $V_{\\rm pg}^{\\rm FEM} = 0.8$ mV and $V_{\\rm pg}^{\\rm GM} = 1.2$ mV as shown in Fig.", "REF .", "In the presence of the photon cavity, two extra side-peaks at $eV_{\\rm pg}^{\\rm GS;FES} = eV_{\\rm pg}^{\\rm GM;FEM} -\\hbar \\omega _{\\rm ph}$ are observed in addition to the original main-peaks at $V_{\\rm pg}^{\\rm GM;FEM}$ .", "The superscripts $\\rm GM$ ($\\rm FEM$ ) refers to the ground states(first-excited state) main-peak, respectively, and $\\rm GS$ ($\\rm FES$ ) stands for photon-induced ground state(first-excited state) side-peak, respectively.", "The side-peaks indicate the PAT, where the system satisfies $e\|V_{\\rm pg}^{\\rm GM;FEM}-V_{\\rm pg}^{\\rm GS;FES}\|\\cong \\hbar \\omega _{\\rm ph} $ .", "[33] The two new side-peaks at $V_{\\rm pg}^{\\rm GS} = 0.95$ mV and $V_{\\rm pg}^{\\rm FES} = 0.55$ mV shown in Fig.", "REF are caused by photon-replica of the ground state and photon-replica of the first-excited state, respectively.", "We find that the separation of the photon replica side-peaks from the original main-peaks corresponds to the photon energy.", "It should be noted that the current in the photon-induced side-peaks is strongly enhanced by increasing the electron-photon coupling strength.", "Thus the photon-induced side-peaks exhibits a PAT process with different photon absorption mechanism from the main-peaks.", "In order to show the dynamics of the PAT process involved in the formation of the photon-induced side-peaks in the left and right current shown in figure REF , we schematically present the photon absorption process in Fig.", "REF .", "Figure: (Color online) Schematic representation of photon-activated resonance energylevels and electron transition by changing the plunger gate voltage V pg V_{\\rm pg} in thephoton-induced first-excited state side-peak at V pg FES =0.55V_{\\rm pg}^{\\rm FES} = 0.55 mV (a),and the photon-induced ground state side-peak at V pg GS =0.95V_{\\rm pg}^{\\rm GS} = 0.95 mV(b) of the Fig.", ".The DQD-system is embedded in a photo cavity with the photon energy ℏω ph \\hbar \\omega _{\\rm ph} and photon contentN ph N_{\\rm ph} in each many-body state.", "The chemical potential difference is eV bias =Δμ=μ L -μ R e V_{\\rm bias} = \\Delta \\mu = \\mu _L - \\mu _R.Figure REF (a) demonstrates the tunneling processes forming the FES at $V_{\\rm pg}^{\\rm FES} = 0.55$ mV.", "The electron from the left or the right lead absorbs two photons and is transferred to the MB states containing two photons $N_{\\rm ph} = 2$ (red solid arrows) situated above the bias window with photon energy $\\hbar \\omega _{\\rm ph}$ .", "The electron tunneling process in the states containing one photon $N_{\\rm ph} = 1$ (blue dashed arrows) and three photons $N_{\\rm ph} = 3$ (green dashed arrows) are very weak.", "Figure REF (b) shows the dynamical mechanism that makes the GS at $V_{\\rm pg}^{\\rm GS} = 0.95$ mV.", "In addition to the electron tunneling in the state containing two photons $N_{\\rm ph} = 2$ (red solid arrows), the tunneling process in one photon state $N_{\\rm ph} = 1$ (blue solid arrows) and three photons $N_{\\rm ph} = 3$ states (green solid arrows) become active.", "The tunneling mechanism here is a multiphoton absorption process with up to three photons with a strong inter-dot tunneling.", "In which the electron is scattered between one, two, and three photon(s) states by absorbing and emitting photon energy $N_{\\rm ph}\\times \\hbar \\omega _{\\rm ph}$ .", "The two photons state here has a shorter lifetime than the two photons states in the FES, because whenever an electron from the left lead tunnels into the two photon states in the DQD system it subsequently directly tunnels out to the right lead.", "To further illustrate the characteristics of the most active MB states in the tunneling process forming the two main current peaks and the two side current peaks in the Fig.", "REF , we present Fig.", "REF which shows the characteristics of the MB states at plunger-gate voltage $V_{\\rm pg}^{\\rm FES} = 0.55$ mV (a), $V_{\\rm pg}^{\\rm FEM} = 0.8$ mV (b), $V_{\\rm pg}^{\\rm GS} = 0.95$ mV (c), and $V_{\\rm pg}^{\\rm GM} = 1.2$ mV (d) in the case of $g_{\\rm ph} = 0.1$ meV.", "Figure REF (a) indicates how the FES is contributed to by the MB states at $V_{\\rm pg}^{\\rm FES} = 0.55$ mV.", "Since there are two photons initially in the cavity, we shall seek the MB-states that contain one, two, and three photon(s), to observe multiple inelastic electron scattering in the states of DQD system at the side-peaks.", "[2] Here, we focus on six MB states, two inactive MB states $\|\\breve{15})$ and $\|\\breve{16})$ in the bias window (blue squared dot) with $N_{\\rm ph} = 1.052$ in each state and the energies $E_{15} = 1.364$ meV and $E_{16} = 1.366$ meV.", "There are four MB state above the bias window: Two photon-activated states $\|\\breve{20})$ and $\|\\breve{21})$ (red squared dot) with $N_{\\rm ph} = 2.073$ in each state and energies $E_{20} = 1.616$ meV and $E_{21} = 1.618$ meV, and two more MB states $\|\\breve{25})$ and $\|\\breve{26})$ (green squared dot) with $N_{\\rm ph} = 3.094$ in each state and energies $E_{25} = 1.867$ meV and $E_{26} = 1.870$ meV.", "We clearly see that the energy difference between the inactive states and the photon-activated states is appropriately equal to the $(N_{\\rm ph,ac}-N_{\\rm ph,in})\\times \\hbar \\omega _{\\rm ph}$ , where $N_{\\rm ph,ac}$ and $N_{\\rm ph,in}$ are the photon number in the photon-activated states and the inactive states, respectively.", "Figure: (Color online) The MB energy spectrum E μ E_{\\mu } (dotted black), the meanelectron number in the MB state \|μ ˘)\|\\breve{\\mu }) (blue dashed line),the mean photon number N ph N_{\\rm ph} (red line) in the case ofxx-polarized field with plunger-gate voltageat (a) photon-induced first-excited side-peak V pg FES =0.55V_{\\rm pg}^{\\rm FES} = 0.55 mV,(b) first-excited main-peak V pg FEM =0.8V_{\\rm pg}^{\\rm FEM} = 0.8 mV,(c) photon-induced ground-state side-peak V pg GS =0.95V_{\\rm pg}^{\\rm GS} = 0.95 mV, and(d) ground-state main-peak V pg GM =1.2V_{\\rm pg}^{\\rm GM} = 1.2 mV of Fig.", "for the case of g ph =0.1g_{\\rm ph} = 0.1 meV (blue solid line).The chemical potentials are μ L =1.4\\mu _{\\rm L} = 1.4 meV and μ L =1.3\\mu _{\\rm L} = 1.3 meV (black line),thus Δμ=0.1 meV \\Delta \\mu =0.1~{\\rm meV}.Other parameters are B=0.1TB = 0.1~{\\rm T}, and ℏω ph =0.25\\hbar \\omega _{\\rm ph} = 0.25 meV.The color of the square is referred to in the text.We observe that the electrons can undergo the following possible tunneling process: An electron from either lead may absorb two photons from the cavity being transferred to two photons states $\|\\breve{20})$ and $\|\\breve{21})$ with absorption energy $E_{20}- E_{15} = (N_{\\rm ph,20} - N_{\\rm ph,15})\\times \\hbar \\omega _{\\rm ph} \\simeq 0.252$ meV or $E_{21}- E_{16} = (N_{\\rm ph,21} - N_{\\rm ph,16})\\times \\hbar \\omega _{\\rm ph} \\simeq 0.252$ meV which is approximately equal to the energy required to transfer an electron from the leads to two photons states as schematically shown in Fig.", "REF (a).", "Therefore, the two photon absorption mechanism dominates here without making electron inelastic scattering to the one and three photon states.", "The electron tunneling process from the leads to the DQD system suggest that the electrons are collected in either individual dot.", "Figure REF (b) shows the MB states of the first-excited state main-peak at $V_{\\rm pg}^{\\rm FEM} = 0.8$ mV.", "There are two inactive states $\|\\breve{11})$ and $\|\\breve{12})$ (blue squared dot) with energies $E_{11} =1.362$ meV and $E_{12} = 1.364$ meV in the bias window ($N_e = 1$ , $N_{\\rm ph} = 0.029$ ) and two photon-activated states $\|\\breve{21})$ , and $\|\\breve{22})$ (red squared dot) with energies $E_{21} = 1.866$ meV and $E_{22} =1.868$ meV above the bias window ($N_e = 1$ , $N_{\\rm ph} = 2.073$ ).", "The photon-activated states that contain two photons are responsible for the electron transport with energy values $E_{21} - E_{11} \\cong (N_{\\rm ph,21}-N_{\\rm ph,11})\\times \\hbar \\omega _{\\rm ph} \\cong 0.504$ meV or $E_{22} - E_{12} \\cong (N_{\\rm ph,22}-N_{\\rm ph,12})\\times \\hbar \\omega _{\\rm ph} \\cong 0.504$ meV.", "Figure REF (c) demonstrates the MB states that participate to the electron transport in the GS at $V_{\\rm pg}^{\\rm GS} = 0.95$ mV.", "The electron transport mechanism here is different from the one for the FES.", "The contributions to the GS is by the following significant MB states: Two active MB states $\|\\breve{9})$ and $\|\\breve{10})$ (blue squared dot) containing $N_{\\rm ph} = 0.976$ with energies $E_{9} = 1.342$ meV and $E_{10} = 1.344$ meV are located in the bias window, two photon-activated states $\|\\breve{14})$ and $\|\\breve{15})$ (red squared dot) have $N_{\\rm ph} = 1.95$ above the bias window with energies $E_{14} = 1.5901$ meV and $E_{15} = 1.5902$ meV, and two more photon-activated states $\|\\breve{19})$ and $\|\\breve{20})$ (green squared dot) contain $N_{\\rm ph} = 2.93$ with energies $E_{19} = 1.838$ meV and $E_{20} = 1.840$ meV.", "Significantly, these six MB states participate in the electron transport with the following important photon absorption processes with inter-dot tunneling as schematically shown previously in Fig.", "REF (b): (1) Electron from either lead absorbs one photon tunneling to the one photon states $\|\\breve{9})$ and $\|\\breve{10})$ , (2) An electron from the left lead absorbs two photons and is transferred to two photons states $\|\\breve{14})$ and $\|\\breve{15})$ with absorption energy $E_{14}- E_{9} = (N_{\\rm ph,14} - N_{\\rm ph,9})\\times \\hbar \\omega _{\\rm ph} \\simeq 0.248$ meV or $E_{15}- E_{10} = (N_{\\rm ph,15} - N_{\\rm ph,10})\\times \\hbar \\omega _{\\rm ph} \\simeq 0.247$ meV which is approximately equal to the energy required to transfer electron from one photon state to two photons states, then the electron tunnels to the right lead emitting photons, (3) Absorbing three photons, an electrons from either the left lead or the right lead transfers to three photons states $\|\\breve{19})$ and $\|\\breve{20})$ with energy $E_{19}- E_{9} = (N_{\\rm ph,19} - N_{\\rm ph,9})\\times \\hbar \\omega _{\\rm ph} \\simeq 0.496$ meV or $E_{20}- E_{10} = (N_{\\rm ph,20} - N_{\\rm ph,10})\\times \\hbar \\omega _{\\rm ph} \\simeq 0.496$ meV that is approximately the energy amount needed to transfer an electron to a three photons state.", "The tunneling processes from the leads to the DQD system and all activated six-MB states suggest that the electrons perform multiple scattering absorption and emission processes between the states in each individual dot with inter-dot tunneling.", "These possible tunneling processes indicate to us that the existence of the FES is caused by multiphoton absorption processes with up to three photons.", "In addition, we should mention that the tunneling process from the left lead to two photon states in the DQD system and the tunnel out to the right lead decreases the dwell time of electron in the central system, while the dwell time of electron in the FES was longer due to charge accumulation in the DQD system.", "Figure REF (d) demonstrates the MB states of the ground state main-peak at $V_{\\rm pg}^{\\rm GM} = 1.2$ mV.", "Four MB states are also important here, two inactive states $\|\\breve{7})$ and $\|\\breve{8})$ (blue squared dot) with energies $E_{7} =1.344$ meV and $E_{8} = 1.346$ meV in the bias window ($N_e = 1$ , $N_{\\rm ph} = 0.029$ ) and two photon-activated states $\|\\breve{15})$ and $\|\\breve{16})$ (red squared dot) with energies $E_{15} = 1.840$ meV and $E_{16} =1.842$ meV above the bias window ($N_e = 1$ , $N_{\\rm ph} = 1.951$ ).", "The energy difference between photon-activated states above the bias window and the inactive states in the bias window satisfies the same rule of the FEM, such that $E_{15} - E_{7} \\cong (N_{\\rm ph,15}-N_{\\rm ph,7})\\times \\hbar \\omega _{\\rm ph} \\cong 0.496$ meV and $E_{16} - E_{8} \\cong (N_{\\rm ph,16}-N_{\\rm ph,8})\\times \\hbar \\omega _{\\rm ph} \\cong 0.496$ meV.", "These results suggest that each photon-activated state above the bias window has two more photons than the inactive states in the bias window at both the main peaks.", "When an electron from the left lead tunnels to the DQD system it absorbs (or forms a quasi-particle with) two photons from the cavity and is transferred to the photon-activated states above the bias window, then it tunnels to the right lead.", "Therefore, both main-peaks are caused by two photons absorption processes in the transport.", "Observing all these photon activated processes it is important to have in mind that the fact that we retained the dia- and the paramagnetic parts of the electron-photon interaction (Eq.", "(REF )) thus allowing for a broader range of transitions possibilities than only the paramagnetic term describes.", "Figure REF indicates the charge density distribution in the DQD system with $x$ -polarized photon field at plunger-gate $V_{\\rm pg}^{\\rm FES} = 0.55$ mV (a), $V_{\\rm pg}^{\\rm FEM} = 0.8$ mV (b), $V_{\\rm pg}^{\\rm GS} = 0.95$ mV (c), and $V_{\\rm pg}^{\\rm GM} = 1.2$ mV (d) of the Fig.", "REF at time $t = 220$ ps and $g_{\\rm ph} = 0.1$ meV.", "Figure: (Color online) The charge density distribution ofthe DQD system with xx-polarized photon field at time 220 ps corresponding to(a) photon-induced first-excited side-peak V pg FES =0.55V_{\\rm pg}^{\\rm FES} = 0.55 mV,(b) first-excited main-peak V pg FEM =0.8V_{\\rm pg}^{\\rm FEM} = 0.8 mV,(c) photon-induced ground-state side-peak V pg GS =0.95V_{\\rm pg}^{\\rm GS} = 0.95 mV, and(d) ground-state main-peak V pg GM =1.2V_{\\rm pg}^{\\rm GM} = 1.2 mV of Fig.", ".for the case of g ph =0.1g_{\\rm ph} = 0.1 meV (blue solid line).", "Other parameters are ℏω ph =0.25 meV \\hbar \\omega _{\\rm ph} =0.25~{\\rm meV}, B=0.1B=0.1 T, a w =23.8 nm a_{w} = 23.8~{\\rm nm}, L x =300 nm L_x = 300~{\\rm nm}, and ℏΩ 0 =2.0 meV \\hbar \\Omega _0 = 2.0~{\\rm meV}.In the FES at $V_{\\rm pg}^{\\rm FES} = 0.55$ mV, the electron charge density forms two peaks which are strongly localized in the dots without inter-dot tunneling shown in Fig.", "REF (a), thus the electron dwell time is increased and the electrons stay longer time in the DQD system.", "In the case of $V_{\\rm pg}^{\\rm FEM} = 0.8$ mV, the electrons are accumulated in the dots with a weak inter-dot tunneling.", "Comparing to the case with no photon cavity Fig.", "REF (a), a slight inter-dot tunneling is observed indicating charge polarization between the dots.", "As a result, the electron charge density is slightly enhanced in the $x$ -polarized photon field, but more importantly it is also slightly delocalized resulting in a higher conductance through the serial dot molecule.", "In the GS at $V_{\\rm pg}^{\\rm GS} = 0.95$ mV, the electron charge density is enhanced and exhibits charge accumulation in the dots with a very strong inter-dot tunneling shown in Fig.", "REF (c) which decreases the electron dwell time in the DQD-system.", "This is the reason why the current in the GS is relatively higher than the current is FES.", "In the case of $V_{\\rm pg}^{\\rm GM} = 1.2$ mV, the electron-photon interactions does not have a big effect on the charge density distribution, the charge distribution of the dots is already overlapping." ], [ "$y$ -photon polarization (TE{{formula:3e15be96-e66f-4492-ac3b-2a93fcf35e0a}} mode)", "In this section, we assume the photon-cavity is linearly polarized in the $y$ -direction with photon energy $\\hbar \\omega _{\\rm ph} = 0.25$ meV and initially two photons in the single photon mode.", "The MB energy spectrum of the DQD-system in the $y$ -polarization is very similar to that in the $x$ -polarization photon mode as shown in Fig.", "REF .", "Two extra MB states are observed with the spin states of the ground state and first-excited state in the bias window, the extra MB states indicate the photon-replica states in the presence of the photon cavity.", "Figure REF shows the left current (a) and the right current (b) at time $t = 220$ ps for different electron-photon coupling strength $g_{\\rm ph} = 0.1~{\\rm meV}$ , (blue solid), $0.2~{\\rm meV}$ (green dashed), and $0.3~{\\rm meV}$ (red dotted).", "Similar to the $x$ -polarized photon field, two extra photon-induced side-peaks at $V_{\\rm pg}^{\\rm FES} = 0.55$ mV and $V_{\\rm pg}^{\\rm GS} = 0.95$ mV are observed with the main-peaks at $V_{\\rm pg}^{\\rm FEM} = 0.8$ mV $V_{\\rm pg}^{\\rm GM} = 1.2$ mV.", "A very weak current enhancement in the photon-induced side-peaks is predicted by increasing the electron-photon coupling strength, while the current enhancement in the photon-induced side-peaks is very strong in the $x$ -polarized photon field shown in Fig.", "REF .", "The weaker effects for the $y$ -polarization are expected since the photon energy is farther from resonance for states describing motion in that direction, i.e.", "the confinement energy in the $y$ -direction is much higher.", "Figure: (Color online) The left current (a) and the right current (b)versus the plunger gate voltage V pg V_{\\rm pg} at time (t=220 ps t = 220~{\\rm ps})in the case of yy-polarized photon field.", "The electron-photon coupling is changed tobe g ph =0.1g_{\\rm ph} = 0.1 meV (blue solid), 0.20.2 meV (green dashed),and 0.30.3 meV (red dotted).Other parameters are ℏω ph =0.25 meV \\hbar \\omega _{\\rm ph} = 0.25~{\\rm meV}, Δμ=0.1 meV \\Delta \\mu =0.1~{\\rm meV},and B=0.1TB = 0.1~{\\rm T}.The characteristics of the MB states in the bias window and above the bias window in the $y$ -polarized photon field are very similar to that in the $x$ -direction shown in Fig.", "REF .", "The main-peaks and the photon-induced side-peaks in the $y$ -polarized photon are contributed to by almost the same absorption processes of the $x$ -polarized photon.", "Figure REF demonstrates the charge density distribution in the DQD system in the case of $y$ -polarized photon field at plunger-gate voltage $V_{\\rm pg}^{\\rm FES} = 0.55$ mV FES (a), $V_{\\rm pg}^{\\rm FEM} = 0.8$ mV FEM (b), $V_{\\rm pg}^{\\rm GS} = 0.95$ mV GS (c), and $V_{\\rm pg}^{\\rm GM} = 1.2$ mV GM (d) shown in the Fig.", "REF , at time $t = 220$ ps and $g_{\\rm ph} = 0.1$ meV.", "In the case of $V_{\\rm pg}^{\\rm FES} = 0.55$ mV (FES), the electrons are strongly localized in the dots with no electron tunneling from the left-dot to the right-dot.", "In FEM at $V_{\\rm pg}^{\\rm FEM} = 0.8$ mV, the electron makes a resonance state localized in each dot without inter-dot tunneling, while a weak inter-dot tunneling was observed in the $x$ -polarized photon field at FEM shown in Fig.", "REF (b).", "Therefore, the electron dwell time in the DQD system in $y$ -polarized photon is longer than that in the $x$ -polarized photon at FEM.", "But at $V_{\\rm pg}^{\\rm GS} = 0.95$ mV (GS), the inter-dot tunneling is very strong and electrons prefer to make inelastic multiple scattering in each dot with inter-dot tunneling.", "In GM at $V_{\\rm pg}^{\\rm GM} = 1.2$ mV, the electrons form a state accumulated in the dots with a strong electron tunneling between the dots similar to the charge density distribution in the $x$ -polarization shown in Fig.", "REF (d).", "Thus, the current in the GM is higher than that in the FES.", "Figure: (Color online) The charge density distribution ofthe DQD system with yy-polarized photon field at time 220 ps corresponding to(a) photon-induced first-excited side-peak V pg FES =0.55V_{\\rm pg}^{\\rm FES} = 0.55 mV,(b) first-excited main-peak V pg FEM =0.8V_{\\rm pg}^{\\rm FEM} = 0.8 mV,(c) photon-induced ground-state side-peak V pg GS =0.95V_{\\rm pg}^{\\rm GS} = 0.95 mV, and(d) ground-state main-peak V pg GM =1.2V_{\\rm pg}^{\\rm GM} = 1.2 mV of Fig.", "for the case of g ph =0.1g_{\\rm ph} = 0.1 meV (blue solid line).", "Other parameters are ℏω ph =0.25 meV \\hbar \\omega _{\\rm ph} =0.25~{\\rm meV}, B=0.1B=0.1 T, a w =23.8 nm a_{w} = 23.8~{\\rm nm}, L x =300 nm L_x = 300~{\\rm nm}, and ℏΩ 0 =2.0 meV \\hbar \\Omega _0 = 2.0~{\\rm meV}." ], [ "Conclusions", "Photon-assisted transient electron transport through a DQD system placed in a photon cavity with initially two linearly polarized photons can be controlled by a plunger-gate voltage.", "The serial double quantum dot molecule is important since the two lowest states of it have very different properties reflected in the fact that the wavefunction of one is symmetric, but antisymmetric of the other.", "We analyzed the electron transport through the system without and with a photon cavity by using a non-Markovian QME formalism.", "In the absence of a photon cavity, two current peaks were found: Ground state peak and a peak due to the first-excited state, originating from resonance energy levels of the DQD system with the first-subband energy of the leads.", "These two states could be used for a qubit in a quantum computer, in which the ground state resonance exhibits a strong inter-dot electron tunneling while the electrons in the first-excited state resonance form a state localized in each dot.", "In the presence of either longitudinally or transversely polarized cavity photon field, two extra side-peaks are found: A peak due to the photon-replica of the ground state, and a photon replica of the first-excited state.", "The appearance of side-peaks is due to PAT of electrons in the DQD system.", "The characteristics of the photon activated MB states have been used to analyze the nature of the PAT.", "The peak due to the photon-replica of ground state is caused by multiphoton absorption processes with up to three photons.", "In the peak caused by the photon replica of the first-excited state, the electrons in the leads are transferred to two-photons states in the DQD system accumulation charge in the individual dots without inter-dot tunneling.", "Furthermore, the current in the photon-induced side-peaks is strongly enhanced by increasing electron-photon coupling strength in the $x$ -polarized of the photon field, while a very slightly enhancement in the photon-induced side-peak current was observed in the $y$ -polarization photon.", "This discrepancy between the polarizations is explained by the anisotropy of the confinement of the central system.", "Change in the photon-electron coupling strength alters the inter-dot tunneling sensitively, altering the conduction through the system.", "To describe this effect properly it is important to include many higher energy states in the system.", "Along the similar line of thought our calculations show that in the present system photon processes of more than one photon are important.", "The fact that we include both the para- and diamagnetic terms in the electron-photon interaction leads to complex photon-electron processes that all contribute to the PAT resonance peaks observed.", "This has to be viewed in light of the common practice to use only the paramagnetic part in two-level systems in order to calculate PAT phenomena.", "In many systems the geometry matters, and the both parts of the interaction can be important for strong enough coupling.", "This work was financially supported by the Icelandic Research and Instruments Funds, the Research Fund of the University of Iceland, the Nordic High Performance Computing facility in Iceland, and the National Science Council in Taiwan through Contract No.", "NSC100-2112-M-239-001-MY3." ] ]	1403.0382
[ [ "Overlap coincidence to strong coincidence in substitution tiling\n dynamics" ], [ "Abstract Overlap coincidence is an equivalent criterion to pure discrete spectrum of the dynamics of self affine tilings.", "In the case of one dimension, strong coincidence on m letter irreducible substitution has been introduced in Dekking (1978) and Arnoux and Ito (2001) which implies that the system is metrically conjugate to a domain exchange.", "However being a domain exchange does not imply the property of pure discrete spectrum of the tiling dynamics.", "The relation between two coincidences has not been established completely.", "In this paper we generalize strong coincidence to higher dimensions and show the implication from overlap coincidence to the new strong coincidence when the associated height group is trivial.", "Furthermore we introduce a new criterion simultaneous coincidence and show the implication from overlap coincidence to the simultaneous coincidence.", "The triviality of the height group is shown in Barge and Kwapisz (2006) and Sing (2006) for 1 dimension irreducible Pisot substitutions." ], [ "Introduction", "The principal aim of this paper is to give a better understanding of pure discrete spectrum of self-affine tiling dynamical systems which have zero-entropy and whose spectral type varies from weakly-mixing to pure discrete.", "The study is strongly motivated by atomic configurations of quasicrystals, which show pure point diffraction.", "Indeed, equivalence of pure point diffraction of quasicrystal structure and pure discrete spectrum of its associated dynamical system is known in quite a general setting.", "If we restrict ourselves to 1-dimension substitutive systems, the problem of pure discrete spectrum can be reformulated using notions of `coincidence' on word combinatorics.", "Many notions of coincidences have been introduced in the study of the dynamical spectrum of self-affine tilings.", "A lot of coincidences among these are proved to be equivalent.", "However relation between strong coincidence of 1-dimension Pisot substitution and other coincidences is not completely understood.", "Overlap coincidence is defined in self-affine tiling and characterizes pure discrete spectrum of the tiling dynamics [19], [14], [13].", "Basically what it means is that every two tiles in the tiling, which overlap after translating by a return vector, have at least one tile in common after some iterations (see subsection REF for the detail).", "This coincidence has been proved to be equivalent with algebraic coincidence [13] and super coincidence [17].", "For an irreducible Pisot substitution in 1-dimension, all of these coincidences are equivalent to the fact that balanced pair algorithm terminates and each balanced pair leads to a coincidence [11], [20].", "This is a combinatorial condition that we can quickly check for a given substitution.", "Strong coincidence has been introduced by [10] in constant length substitution sequences, generalized in unimodular Pisot substitutions by [8], and extended in the case of non-unimodular Pisot substitutions by [17].", "This combinatorial condition guarantees that there is a geometric realization of substitutions which is metrically conjugate to a domain exchange.", "In view of the balanced pair algorithm, strong coincidence only implies that every balanced pair leads to a coincidence, i.e., termination of the balanced pair algorithm seems to be necessary to assume.", "In [11] Hollander and Solomyak proved the termination of the balanced pair algorithm in two letter case, establishing the equivalence between strong coincidence and pure discrete spectrum.", "From this equivalence with the result of Barge and Diamond [5] which guarantees the strong coincidence in two-letter case, we obtained that two-letter irreducible Pisot substitution dynamical systems have pure discrete spectrum.", "However apart from two-letter irreducible Pisot substitution sequences, the relation between the strong coincidence and pure discrete spectrum is not clearly understood.", "This relation is important for an approach towards `Pisot substitution conjecture' [2].", "Recently Nakaishi [16] claimed that the dynamics of irreducible unimodular Pisot substitution sequence satisfying strong coincidence has pure discrete spectrum through domain exchange flow.", "Though we do not yet have a full account of this claim, it indicates that strong coincidence in irreducible unimodular Pisot substitution sequences implies overlap coincidence in irreducible unimodular Pisot substitution tilings in ${\\mathbb {R}}$ .", "This gives us a motivation to look at the other direction.", "As mentioned above, strong coincidence and overlap coincidence are both combinatorial objects which are defined in 1-dimension substitution sequence.", "However, it seems that there is no reason to restrict it to 1-dimension substitutions to see the relationship between these coincidences and seems to be better to transfer the problem into geometric setting.", "So we first generalize strong coincidence in substitution sequences into substitution tiling in ${\\mathbb {R}}^d$ , and give a stronger version `simultaneous coincidence' in §2 and show that overlap coincidence implies simultaneous coincidence in Pisot family substitution tiling in ${\\mathbb {R}}^d$ provided that the associated substitution Delone multi-color set is `admissible' (Def.", "REF ) and the corresponding `height group' (Def.", "REF ) is trivial.", "Since it is shown in [6], [18] that this group is trivial for irreducible Pisot substitutions, we know that every irreducible Pisot substitution having pure discrete spectrum must admit the simultaneous coincidence.", "Here we point out that our higher dimensional generalization of strong coincidence is somewhat too weak condition by itself.", "It must be fulfilled in conjunction with some constraints to make a reasonable connection to other coincidences.", "Indeed, without the constraints of admissibility and trivial height group, for a given substitution tiling, it is always possible to build an associated substitution Delone multi-color set satisfying our extended notion of strong coincidence.", "The first author [1] obtained a converse statement from strong coincidence to overlap coincidence using the extended notion of strong coincidence defined in this paper.", "But in [1], strong coincidence is required for many choices of admissible control points having trivial height group to deduce overlap coincidence.", "It is of interest to minimize the constraints which are sufficient to derive converse direction from strong coincidence to overlap coincidence.", "A tile in ${\\mathbb {R}}^d$ is defined as a pair $T=(A,i)$ where $A=\\mbox{\\rm supp}(T)$ (the support of $T$ ) is a compact set in ${\\mathbb {R}}^d$ which is the closure of its interior, and $i=l(T)\\in \\lbrace 1,\\ldots ,m\\rbrace $ is the color of $T$ .", "We let $g+T = (g+A,i)$ for $g \\in {\\mathbb {R}}^d$ .", "We say that a set $P$ of tiles is a patch if the number of tiles in $P$ is finite and the tiles of $P$ have mutually disjoint interiors.", "A tiling of ${\\mathbb {R}}^d$ is a set ${\\mathcal {T}}$ of tiles such that ${\\mathbb {R}}^d = \\bigcup \\lbrace \\mbox{\\rm supp}(T) : T \\in {\\mathcal {T}}\\rbrace $ and distinct tiles have disjoint interiors.", "Definition 2.1 Let ${\\mathcal {A}}= \\lbrace T_1,\\ldots ,T_m\\rbrace $ be a finite set of tiles in ${\\mathbb {R}}^d$ such that $T_i=(A_i,i)$ ; we will call them prototiles.", "Denote by ${\\mathcal {P}}_{{\\mathcal {A}}}$ the set of non-empty patches.", "We say that $\\Omega : {\\mathcal {A}}\\rightarrow {\\mathcal {P}}_{{\\mathcal {A}}}$ is a tile-substitution (or simply substitution) with a $d\\times d$ expansive matrix $Q$ if there exist finite sets ${\\mathcal {D}}_{ij}\\subset {\\mathbb {R}}^d$ for $i,j \\le m$ such that $\\Omega (T_j)=\\lbrace u+T_i:\\ u\\in {\\mathcal {D}}_{ij},\\ i=1,\\ldots ,m\\rbrace $ with $ Q A_j = \\bigcup _{i=1}^m ({\\mathcal {D}}_{ij}+A_i) \\ \\ \\ \\mbox{for} \\ j\\le m.$ Here all sets in the right-hand side must have disjoint interiors; it is possible for some of the ${\\mathcal {D}}_{ij}$ to be empty.", "The substitution (REF ) is extended to all translates of prototiles by $\\Omega (x+T_j)= Q x + \\Omega (T_j)$ and to patches and tilings by $\\Omega (P)=\\bigcup \\lbrace \\Omega (T):\\ T\\in P\\rbrace $ .", "The substitution $\\Omega $ can be iterated, producing larger and larger patches $\\Omega ^k(P)$ .", "We say that ${\\mathcal {T}}$ is a substitution tiling if ${\\mathcal {T}}$ is a tiling and $\\Omega ({\\mathcal {T}}) = {\\mathcal {T}}$ with some substitution $\\Omega $ .", "In this case, we also say that ${\\mathcal {T}}$ is a fixed point of $\\Omega $ .", "We say that a substitution tiling is primitive if the corresponding substitution matrix $S$ , with $S_{ij}= \\sharp ({\\mathcal {D}}_{ij})$ , is primitive, and irreducible if the characteristic polynomial of $S$ is irreducible.", "We say that ${\\mathcal {T}}$ has finite local complexity (FLC) if $\\forall \\ R > 0$ , $\\exists $ finitely many translational classes of patches whose support lies in some ball of radius $R$ .", "A tiling ${\\mathcal {T}}$ is repetitive if for any compact set $K \\subset {\\mathbb {R}}^d$ , $\\lbrace t \\in {\\mathbb {R}}^d : {\\mathcal {T}}\\cap K = (t + {\\mathcal {T}}) \\cap K\\rbrace $ is relatively dense.", "A repetitive fixed point of a primitive tile-substitution with FLC is called a self-affine tiling.", "Let $\\lambda >1$ be the Perron-Frobenius eigenvalue of the substitution matrix $S$ .", "Let $ D =\\lbrace \\lambda _1,\\ldots ,\\lambda _{d}\\rbrace $ be the set of (real and complex) eigenvalues of $Q$ .", "We say that $Q$ (or the substitution $\\Omega $ ) fulfills a Pisot family if for every $\\lambda \\in D$ and every Galois conjugate $\\lambda ^{\\prime }$ of $\\lambda $ , if $\\lambda ^{\\prime }\\notin D$ , then $\|\\lambda ^{\\prime }\| < 1$ .", "Given a tiling ${\\mathcal {T}}$ in ${\\mathbb {R}}^d$ , we define the tiling space as the orbit closure of ${\\mathcal {T}}$ under the translation action: $X_{{\\mathcal {T}}} =\\overline{\\lbrace -g + {\\mathcal {T}}:\\,g \\in {\\mathbb {R}}^d \\rbrace }$ , in the well-known “local topology”: for a small ${\\mbox{$\\epsilon $}}>0$ two tilings ${\\mathcal {S}}_1,{\\mathcal {S}}_2$ are ${\\mbox{$\\epsilon $}}$ -close if ${\\mathcal {S}}_1$ and ${\\mathcal {S}}_2$ agree on the ball of radius ${\\mbox{$\\epsilon $}}^{-1}$ around the origin, after a translation of size less than ${\\mbox{$\\epsilon $}}$ .", "Then $X_{{\\mathcal {T}}}$ is compact and we get a topological dynamical system $(X_{{\\mathcal {T}}},{\\mathbb {R}}^d)$ where ${\\mathbb {R}}^d$ acts by translations.", "This system is minimal (i.e.", "every orbit is dense) whenever ${\\mathcal {T}}$ is repetitive." ], [ "Point sets", "Recall that a Delone set is a relatively dense and uniformly discrete subset of ${\\mathbb {R}}^d$ .", "We say that $\\mbox{${\\Lambda }$}=(\\Lambda _i)_{i\\le m}$ is a Delone multi-color set in ${\\mathbb {R}}^d$ if each $\\Lambda _i$ is Delone and $\\mbox{\\rm supp}(\\mbox{${\\Lambda }$}):=\\cup _{i=1}^m \\Lambda _i \\subset {\\mathbb {R}}^d$ is Delone.", "A cluster of $\\mbox{${\\Lambda }$}$ is a family of points $\\mbox{\\bf P}= (P_i)_{i \\le m}$ where $P_i \\subset {\\Lambda }_i$ is finite for all $i \\le m$ .", "We say that ${\\Lambda }\\subset {\\mathbb {R}}^d$ is a Meyer set if it is a Delone set and ${\\Lambda }- {\\Lambda }$ is uniformly discrete in ${\\mathbb {R}}^d$ [12].", "The colors of points in the Delone multi-color set have the same meaning as the colors of tiles on tilings.", "Various notions such as primitivity, FLC and repetitivity in point sets are defined in similar way as in tilings.", "Definition 2.2 $\\mbox{${\\Lambda }$}= ({\\Lambda }_i)_{i\\le m}$ is called a substitution Delone multi-color set if $\\mbox{${\\Lambda }$}$ is a Delone multi-color set and there exist an expansive matrix $Q$ and finite sets ${\\mathcal {D}}_{ij}$ for $i,j\\le m$ such that $\\Lambda _i = \\bigcup _{j=1}^m (Q \\Lambda _j + {\\mathcal {D}}_{ij}),\\ \\ \\ i \\le m,$ where the unions on the right-hand side are disjoint.", "For any given substitution Delone multi-colour set $\\mbox{${\\Lambda }$}= (\\Lambda _i)_{i \\le m}$ , we define $\\Phi _{ij} = \\lbrace f : x \\mapsto Qx + a \\, : \\,a \\in {\\mathcal {D}}_{ij}\\rbrace $ .", "Then $\\Phi _{ij}({\\Lambda }_j) = Q {\\Lambda }_j + {\\mathcal {D}}_{ij}$ , where $i \\le m$ .", "We define $\\Phi $ an $m \\times m$ array for which each entry is $\\Phi _{ij}$ .", "For any $k \\in {\\mathbb {Z}}_+$ and $x \\in {\\Lambda }_j$ with $j \\le m$ , we let $\\Phi ^k (x) = \\Phi ^{k-1}((\\Phi _{ij}(x))_{i \\le m})$ .", "For any $k \\in {\\mathbb {Z}}_+$ , $\\Phi ^k (\\mbox{${\\Lambda }$}) = \\mbox{${\\Lambda }$}$ and $\\Phi ^k ({\\Lambda }_j) = \\bigcup _{i \\le m}(Q^k {\\Lambda }_j + ({\\mathcal {D}}^k)_{ij})$ where $({\\mathcal {D}}^k)_{ij} = \\bigcup _{n_1,n_2,\\dots ,n_{(k-1)} \\le m}({\\mathcal {D}}_{in_1} + Q {\\mathcal {D}}_{n_1 n_2} + \\cdots + Q^{k-1} {\\mathcal {D}}_{n_{(k-1)} j}).$ We say that a cluster $\\mbox{\\bf P}$ is legal if it is a translate of a subcluster of a cluster generated from one point of $\\mbox{${\\Lambda }$}$ , that is to say, $a + \\mbox{\\bf P}\\subset \\Phi ^k (x)$ (i.e.", "$a + \\mbox{\\bf P}\\subset \\bigcup _{i=1}^m (Q^k x + ({\\mathcal {D}}^k)_{ij})$ ) for some $k \\in {\\mathbb {Z}}_+$ , $a \\in {\\mathbb {R}}^d$ , $x \\in {\\Lambda }_j$ , and $j \\le m$ .", "Definition 2.3 Let $\\mbox{${\\Lambda }$}$ be a primitive substitution Delone multi-color set in ${\\mathbb {R}}^d$ with an expansive matrix $Q$ .", "We say that $\\mbox{${\\Lambda }$}$ admits an algebraic coincidence if there exist $M \\in {\\mathbb {Z}}_+$ and $\\xi \\in {\\Lambda }_i$ for some $i \\le m$ such that $Q^M \\Xi (\\mbox{${\\Lambda }$}) \\subset {\\Lambda }_i - \\xi $ , where $\\Xi (\\mbox{${\\Lambda }$}) = \\bigcup _{i \\le m} ({\\Lambda }_i - {\\Lambda }_i)$.", "Since $\\mbox{${\\Lambda }$}$ is primitive, there exists $N \\in \\mathbb {N}$ such that $Q^N \\Xi (\\mbox{${\\Lambda }$}) \\subset {\\Lambda }_i - {\\Lambda }_i$ for any $i \\le m$ .", "So $\\mbox{${\\Lambda }$}$ admits an algebraic coincidence if and only if there exists $M \\in \\mathbb {N}$ and $\\xi \\in {\\Lambda }_i$ such that $Q^M ({\\Lambda }_i - {\\Lambda }_i) \\subset {\\Lambda }_i - \\xi $ .", "We say that $\\mbox{${\\Lambda }$}$ is a Pisot family substitution Delone multi-color set if $Q$ fulfills the condition of Pisot family.", "If a self-affine tiling ${\\mathcal {T}}$ is given, we can get an associated substitution Delone multi-color set $\\mbox{${\\Lambda }$}_{{\\mathcal {T}}} = ({\\Lambda }_i)_{i \\le m}$ of ${\\mathcal {T}}$ taking representative points of tiles in the relatively same positions for the same color tiles in the tiling (see [13]).", "On the other hand, if $\\mbox{${\\Lambda }$}$ is a primitive substitution Delone multi-color set for which every $\\mbox{${\\Lambda }$}$ -cluster is legal, then $\\mbox{${\\Lambda }$}+ \\mathcal {A}:= \\lbrace x + T_i : x \\in {\\Lambda }_i, i \\le m \\rbrace $ is a tiling of ${\\mathbb {R}}^d$ , where $\\mathcal {A}$ is the set of prototiles from the associated tile equations (see [14]).", "This bijection establishes a topological conjugacy of $(X_{\\mbox{\\scriptsize ${\\Lambda }$}},{\\mathbb {R}}^d)$ and $(X_{{\\mathcal {T}}},{\\mathbb {R}}^d)$ .", "Note that if we translate representative points of tiles of ${\\mathcal {T}}$ by ${\\Lambda }^{\\prime }_i={\\Lambda }_i-g_i$ , then the set equation will be ${\\Lambda }^{\\prime }_i= \\bigcup _{j=1}^m Q {\\Lambda }^{\\prime }_j + \\mathcal {D}^{\\prime }_{ij}$ with $\\mathcal {D}^{\\prime }_{ij}=\\left\\lbrace d_{ij}-Q g_j+g_i\\ :\\ d_{ij} \\in \\mathcal {D}_{ij}, 1 \\le i,j \\le m \\right\\rbrace $ , or in short, $\\mathcal {D}^{\\prime }_{ij}=\\mathcal {D}_{ij}-Q g_j+g_i$ .", "The corresponding tile equation becomes $Q A^{\\prime }_j= \\bigcup A^{\\prime }_i + \\mathcal {D}^{\\prime }_{ij}$ which is satisfied by $A^{\\prime }_j=A_j+g_j$ .", "So we set $\\mbox{\\rm supp}(T^{\\prime }_j)=A^{\\prime }_j$ and the color of $T^{\\prime }_j$ to be the one of $T_j$ .", "Such modification only translates supports of tiles and corresponding point sets, which causes non essential changes of the description of ${\\mathcal {T}}$ .", "Hereafter, we do not distinguish such changes of reference points and use the same symbols ${\\Lambda }_i$ and $T_i$ .", "Since the representative points of tiles in a tiling ${\\mathcal {T}}$ are taken in the relatively same position for the same type of tiles, for each $i \\le m$ we define a reference point $c_i \\in {\\mathbb {R}}^d$ for the prototile $T_i$ and can take an associated substitution Delone multi-color set $\\mbox{${\\Lambda }$}= \\mbox{${\\Lambda }$}_{{\\mathcal {T}}} = ({\\Lambda }_i)_{i \\le m}$ satisfying ${\\mathcal {T}}= \\lbrace T_i - c_i + u \\ \| \\ u \\in {\\Lambda }_i, i \\le m \\rbrace $ .", "The topological dynamical system $(X_{{\\mathcal {T}}},{\\mathbb {R}}^d)$ has a unique invariant Borel probability measure $\\mu $ when the tiling ${\\mathcal {T}}$ is a self-affine tiling (see [19], [14]).", "The dynamical spectrum of the system $(X_{{\\mathcal {T}}}, {\\mathbb {R}}^d, \\mu )$ refers to the spectrum of the unitary operator $U_x$ arising from the translational action on the space of $L^2$ -functions on $X_{{\\mathcal {T}}}$ .", "We say that the tiling ${\\mathcal {T}}$ has pure discrete dynamical spectrum if the eigenfunctions for the ${\\mathbb {R}}^d$ -action span a dense subspace of $L^2(X_{{\\mathcal {T}}},\\mu )$ [19]." ], [ "Overlap coincidence", "Let $\\Xi ({\\mathcal {T}}) = \\lbrace y \\in {\\mathbb {R}}^d \\ : \\ T = y + S, \\ \\mbox{where $T, S \\in {\\mathcal {T}}$}\\rbrace $ .", "A triple $(u, y, v)$ , with $u + T_i, v + T_j \\in {\\mathcal {T}}$ and $y \\in \\Xi ({\\mathcal {T}})$ , is called an overlap if $ (u+A_i-y)^{\\circ } \\cap (v+A_j)^{\\circ } \\ne \\emptyset , $ where $A_i = \\mbox{\\rm supp}(T_i)$ and $A_j = \\mbox{\\rm supp}(T_j)$ .", "An overlap $(u, y, v)$ is a coincidence if $ \\mbox{$u-y = v$ and $u + T_i, v + T_i \\in {\\mathcal {T}}$ for some $i \\le m$}.$ Let $\\mathcal {O} = (u, y, v)$ be an overlap in ${\\mathcal {T}}$ , we define $\\ell $ -th inflated overlap ${\\Omega }^{\\ell } \\mathcal {O} = \\lbrace (u^{\\prime }, Q^{\\ell } y, v^{\\prime }) \\, :u^{\\prime }+T_k \\in \\Omega ^{\\ell }(u + T_i), v^{\\prime }+T_r \\in \\Omega ^{\\ell }(v+T_j), \\ \\mbox{and $(u^{\\prime },Q^{\\ell }y,v^{\\prime })$is an overlap} \\rbrace .$ Definition 2.4 We say that a self-affine tiling ${\\mathcal {T}}$ admits overlap coincidence if there exists $\\ell \\in {\\mathbb {Z}}_+$ such that for each overlap $\\mathcal {O}$ in ${\\mathcal {T}}$ , ${\\Omega }^{\\ell } \\mathcal {O}$ contains a coincidence.", "Theorem 2.5 [14], [13] Let ${\\mathcal {T}}$ be a self-affine tiling in ${\\mathbb {R}}^d$ for which $\\Xi ({\\mathcal {T}})$ is a Meyer set.", "Then $(X_{{\\mathcal {T}}}, {\\mathbb {R}}^d, \\mu )$ has pure discrete dynamical spectrum if and only if ${\\mathcal {T}}$ admits overlap coincidence.", "Theorem 2.6 [13] Let ${\\mathcal {T}}$ be a self-affine tiling in ${\\mathbb {R}}^d$ for which $\\Xi ({\\mathcal {T}})$ is a Meyer set and $\\mbox{${\\Lambda }$}= \\mbox{${\\Lambda }$}_{{\\mathcal {T}}} = ({\\Lambda }_i)_{i \\le m}$ be an associated substitution Delone multi-color set.", "Then ${\\mathcal {T}}$ admits overlap coincidence if and only if $\\mbox{${\\Lambda }$}$ admits algebraic coincidence.", "A substitution $\\sigma $ over $m$ letters $\\lbrace 1,2,\\dots ,m\\rbrace $ is called primitive, when its substitution matrix $M_{\\sigma }$ is primitive.", "We say that $\\sigma $ is irreducible if the characteristic polynomial of $M_{\\sigma }$ is irreducible, and it is Pisot, if the Perron Frobenius root $\\beta $ of $M_{\\sigma }$ is a Pisot number.", "Then we can construct a bi-infinite sequence which is generated by $\\sigma $ from some fixed letters around the origin and define the natural suspension tiling ${\\mathcal {T}}$ in ${\\mathbb {R}}$ with an expansion factor $\\beta $ arose from $\\sigma $ by associating to the letters of the bi-infinite sequence the intervals whose lengths are given by a left eigenvector of $M_{\\sigma }$ corresponding to $\\beta $ .", "In this case, $\\beta $ is identified with the $1\\times 1$ expansive matrix $Q=(\\beta )$ .", "Let $\\Omega $ be the corresponding tile-substitution and let $\\mathcal {A} = \\lbrace T_1, \\dots , T_m\\rbrace $ be the corresponding prototiles of the intervals whose left end points are all at the origin.", "Taking the left end points of the intervals of tiles in the tiling, we can get an associated substitution Delone multi-color set.", "We can interpret “strong coincidence” of $\\sigma $ into the tiling setting in the following way: Definition 3.1 Let ${\\mathcal {T}}$ be the suspension tiling in ${\\mathbb {R}}$ of a substitution $\\sigma $ .", "We say that ${\\mathcal {T}}$ admits prefix strong coincidence if for any pair of prototiles $\\lbrace T_i, T_j\\rbrace \\subset \\mathcal {A}$ , there exists $L \\in \\mathbb {N}$ such that two supertiles $\\Omega ^L T_i$ and $\\Omega ^L T_j$ have at least one common tile where $\\Omega $ is the tile-substitution.", "We generalize this definition of strong coincidence to ${\\mathbb {R}}^d$ by using the reference points.", "Definition 3.2 Let ${\\mathcal {T}}$ be a self-affine tiling in ${\\mathbb {R}}^d$ .", "Let $c_i$ be the reference point of $T_i$ for $i \\le m$ .", "We say that the set of the reference points is admissible if $\\cap _{i \\le m} (\\mbox{\\rm supp}(T_i)-c_i)$ has non-empty interior.", "Clearly, the left end points of the 1-dimensional suspension tiling form a set of admissible reference points.", "Definition 3.3 Let ${\\mathcal {T}}$ be a self-affine tiling in ${\\mathbb {R}}^d$ with an expansion $Q$ .", "Let $\\mathcal {A} = \\lbrace T_1, \\cdots , T_m\\rbrace $ be the prototile set of ${\\mathcal {T}}$ .", "Let $c_i$ be the reference point of $T_i$ for $ i \\le m$ whose set is admissible.", "Let $\\mbox{${\\Lambda }$}$ be an associated substitution Delone multi-color set for which ${\\mathcal {T}}= \\lbrace u_i + (T_i -c_i) \\ \| \\ u_i \\in {\\Lambda }_i, i \\le m \\rbrace $ .", "If for any $1 \\le i, j \\le m$ , there is a positive integer $L$ that $ \\Omega ^L(T_i - c_i) \\cap \\Omega ^L(T_j - c_j) \\ne \\emptyset ,$ i.e.", "the left hand side contains at least one tile, then we say that $\\mbox{${\\Lambda }$}$ admits strong coincidence.", "Set $\\mathcal {G} :=\\bigcup _{k=0}^{\\infty } Q^{-k} ({\\Lambda }_i-{\\Lambda }_i), \\ \\ \\ \\mbox{for some $i \\le m$}.$ This set is independent of the choice of $i$ .", "Indeed by the primitivity of $\\mbox{${\\Lambda }$}$ , for any $i,j \\le m$ , there exists $n \\in \\mathbb {N}$ that $Q^n({\\Lambda }_i-{\\Lambda }_i) \\subset {\\Lambda }_j-{\\Lambda }_j \\,.$ In plain words, $\\mathcal {G}$ is the set of eventual return vectors, i.e., vectors $v\\in {\\mathbb {R}}^d$ such that there is an $n \\in \\mathbb {N}$ that $Q^n v$ is a return vector of ${\\mathcal {T}}$ .", "Remark 3.4 If $\\mbox{${\\Lambda }$}$ admits strong coincidence, then notice that for any $ i,j \\le m$ , there is a common tile $T_k - c_k + \\eta \\in \\Omega ^L(T_i - c_i) \\cap \\Omega ^L(T_j - c_j)$ for some $\\eta \\in {\\mathbb {R}}^d$ , $L \\in \\mathbb {N}$ and $ k \\le m$ .", "Thus $Q^L {\\Lambda }_i + \\eta \\subset {\\Lambda }_k$ and $Q^L {\\Lambda }_j + \\eta \\subset {\\Lambda }_k$ .", "So $Q^L ({\\Lambda }_i \\cup {\\Lambda }_j)\\subset {\\Lambda }_k -\\eta \\,.$ Thus for any $i,j\\le m$ , $Q^L ({\\Lambda }_i-{\\Lambda }_j) \\subset {\\Lambda }_k-{\\Lambda }_k \\ $ and we obtain ${\\Lambda }-{\\Lambda }\\subset \\mathcal {G}, \\ \\ \\ \\mbox{where} \\ {\\Lambda }= \\cup _{i \\le m} {\\Lambda }_i \\,.$ Remark 3.5 Assume that $\\mbox{${\\Lambda }$}= \\mbox{${\\Lambda }$}_{{\\mathcal {T}}} = ({\\Lambda }_i)_{i \\le m}$ admits algebraic coincidence.", "There is $L\\in \\mathbb {N}$ and $\\eta \\in {\\mathbb {R}}^d$ that $Q^L({\\Lambda }_i-{\\Lambda }_i) \\subset {\\Lambda }_i-\\eta $ for some $i$ .", "This implies $Q^L({\\Lambda }_i-{\\Lambda }_i) -Q^L({\\Lambda }_i-{\\Lambda }_i) \\subset {\\Lambda }_i-{\\Lambda }_i.$ This is equivalent to the fact that the set of eventual return vectors $\\mathcal {G}$ forms an additive group, i.e., $\\mathcal {G}=\\langle \\mathcal {G}\\rangle _{{\\mathbb {Z}}}$ , where $\\langle \\mathcal {G} \\rangle _{{\\mathbb {Z}}}$ is the additive group generated by the elements of $\\mathcal {G}$ .", "Among self-affine tilings, it is interesting to characterize when $\\mathcal {G}=\\langle \\mathcal {G}\\rangle _{{\\mathbb {Z}}}$ holds.", "For e.g., consider the suspension tiling of Thue-Morse substitution $0\\rightarrow 01,\\ 1\\rightarrow 10$ which does not admit overlap coincidence.", "Taking the left end points of the tiles in the tiling, we have ${\\Lambda }_k \\cap [0,\\infty ) = \\left\\lbrace \\sum _{i=0}^{\\ell } b_i 2^i\\ :\\ b_i\\in \\lbrace 0,1\\rbrace , \\ \\sum _{i=0}^{\\ell } b_i\\equiv k \\pmod {2} \\text{ and } \\ell \\in {\\mathbb {Z}}_{\\ge 0}\\right\\rbrace $ with $k \\in \\lbrace 0,1\\rbrace $ .", "One can easily express each element of ${\\Lambda }_1\\cap [0,\\infty )$ as a difference of ${\\Lambda }_0\\cap [0,\\infty )$ .", "So we can show that $\\mathcal {G}={\\mathbb {Z}}[1/2]$ , which forms a group.", "This example also shows that the converse of Remark REF does not hold: ${\\Lambda }-{\\Lambda }\\subset \\mathcal {G}$ does not imply strong coincidence.", "A similar idea using beta-integer with golden mean base works in Example REF , and we can prove that $\\mathcal {G}={\\mathbb {Z}}[(1+\\sqrt{5})/2]$ .", "We do not know yet any example of a self-affine tiling for which $\\mathcal {G}$ is not a group.", "Given a self-affine tiling ${\\mathcal {T}}$ , there are many ways to associate a substitution Delone multi-color set.", "Indeed, Definition REF used the left end points but one may choose other reference points.", "Arnoux-Ito [8] gave the strong coincidence with respect to right end points as well, that is, suffix strong coincidenceIn [8], prefix (resp.", "suffix) strong coincidence is called positive (resp.", "negative) strong coincidence..", "There is a standard way to associate Delone multi-color set to a given self-affine tiling.", "A tile map $\\gamma :{\\mathcal {T}}\\rightarrow {\\mathcal {T}}$ sends a tile $T$ to the one in $\\Omega (T)$ such that $\\gamma (T_1)$ and $\\gamma (T_2)$ are located in the same relative position in $\\Omega (T_1)$ and $\\Omega (T_2)$ whenever $T_1$ and $T_2$ have the same color.", "A control point $c(T)$ of $T\\in {\\mathcal {T}}$ is defined by $ c(T)=\\bigcap _{n=1}^{\\infty } Q^{-n} (\\gamma ^n T).$ The color of the control point $c(T)$ is given by the color of the tile $(T)$ .", "By definition control points of the same color tiles are located in the same relative position and the set of control points $\\mathcal {C}$ is invariant under the expansion by $Q$ , that is, $Q \\mathcal {C} \\subset \\mathcal {C}$ .", "We can choose control points as representative points.", "Let ${\\Lambda }_i$ be chosen to be the control points of tiles in ${\\mathcal {T}}$ of color $i$ .", "Then $\\mathcal {C} = ({\\Lambda }_i)_{i \\le m}$ .", "The set of control points $\\mathcal {C}$ forms a substitution Delone multi-color set.", "Example 3.6 The substitution $\\sigma $ defined by $a\\rightarrow aba, b\\rightarrow bab$ satisfies neither prefix nor suffix strong coincidence.", "However define the tile map $\\gamma $ by choosing $T_b$ in $\\sigma (T_a)$ and the left most $T_b$ in $\\sigma (T_b)$ , then $\\mbox{${\\Lambda }$}$ admits strong coincidence by the choice of the control points.", "Now we introduce a stronger notion than Definition REF : Definition 3.7 Let ${\\mathcal {T}}$ be a self-affine tiling in ${\\mathbb {R}}^d$ with expansion $Q$ .", "Let $\\mathcal {A} = \\lbrace T_1, \\cdots , T_m\\rbrace $ be the prototile set of ${\\mathcal {T}}$ .", "Let $c_i$ be the reference point of $T_i$ for $i \\le m$ whose set is admissible.", "Let $\\mbox{${\\Lambda }$}$ be an associated substitution Delone multi-color set for which ${\\mathcal {T}}= \\lbrace u_i + (T_i -c_i) \\ \| \\ u_i \\in {\\Lambda }_i, i \\le m \\rbrace $ .", "If there is a positive integer $L$ that $ \\bigcap _{i=1}^m \\Omega ^L(T_i - c_i) \\ne \\emptyset ,$ i.e.", "the left hand side contains at least one tile, then we say that $\\mbox{${\\Lambda }$}$ admits simultaneous coincidence.", "Remark 3.8 By the similar argument as Remark REF , if $\\mbox{${\\Lambda }$}$ admits simultaneous coincidence then there exists $L\\in \\mathbb {N}$ , $\\eta \\in {\\mathbb {R}}^d$ and $k \\le m$ such that $ Q^L (\\bigcup _{i =1}^m {\\Lambda }_i)\\subset {\\Lambda }_k -\\eta \\,.$ Remark 3.9 In $d=1$ in Arnoux-Ito's framework of [8], they have chosen the left (or right) end points as control points.", "It is plausible that their definition of geometric substitution acting on broken segments would work by other choices of control points by shifting the origin to the control points located on the 1-dimensional suspension tiling in the expanding line of $M_{\\sigma }$ .", "In this case, a unit segment will grow in prefix and suffix directions at a time, which fits better with the prefix-suffix construction by Canterini and Siegel [9].", "The definition of strong coincidence would naturally be extended in the form to assure that some iterates of every pair of two unit segments share a common segment.", "For a Pisot substitution tiling in ${\\mathbb {R}}$ , one can consider a substitution Delone multi-color set $\\mbox{${\\Lambda }$}$ taking reference points from the left end points of tile intervals.", "The height group of $\\mbox{${\\Lambda }$}$ is defined by Sing [18] which generalizes an idea of Dekking [10] for constant length substitution.", "We extend this definition to general substitution Delone multi-color sets in ${\\mathbb {R}}^d$ .", "Definition 3.10 The height group of $\\mbox{${\\Lambda }$}= ({\\Lambda }_i)_{i \\le m}$ in ${\\mathbb {R}}^d$ is the quotient group $ \\langle {\\Lambda }-{\\Lambda }\\rangle _{{\\mathbb {Z}}} \\ / \\ \\langle {\\Lambda }_i-{\\Lambda }_i \\ \|\\ i\\le m \\rangle _{{\\mathbb {Z}}}, \\ \\ \\ \\mbox{where ${\\Lambda }= \\cup _{i \\le m}{\\Lambda }_i $}\\,.$ Given a tiling ${\\mathcal {T}}$ in ${\\mathbb {R}}^d$ , the height group also depends on the choice of $\\mbox{${\\Lambda }$}$ .", "Example 3.11 Consider again the substitution in Example REF .", "Taking the left end points of $T_a$ and $T_b$ , we have $\\Lambda ={\\Lambda }_a \\cup {\\Lambda }_b = {\\mathbb {Z}}$ , $\\langle {\\Lambda }-{\\Lambda }\\rangle _{{\\mathbb {Z}}}={\\mathbb {Z}}$ , and $\\langle {\\Lambda }_i-{\\Lambda }_i \\ \|\\ i \\in \\lbrace a, b\\rbrace \\rangle _{{\\mathbb {Z}}}=2{\\mathbb {Z}}$ .", "Thus the height group is ${\\mathbb {Z}}/2{\\mathbb {Z}}$ .", "The same is true for the right end points.", "However the third choice in Example REF gives the height group $(2/3){\\mathbb {Z}}/2{\\mathbb {Z}}\\simeq {\\mathbb {Z}}/3{\\mathbb {Z}}$ .", "We note that the height group is not trivial by any choice of admissible reference points.", "In fact, let $c_a$ and $c_b$ be reference points of the prototiles $T_a = ([0,1], a)$ and $T_b= ([0, 1], b)$ in ${\\mathbb {R}}$ .", "To be admissible, $\|c_b - c_a\| < 1$ .", "But if the height group were trivial, then we have $c_b - c_a \\in 2 {\\mathbb {Z}}$ , which shows $c_a=c_b$ and $\\langle {\\Lambda }-{\\Lambda }\\rangle _{{\\mathbb {Z}}}={\\mathbb {Z}}$ ." ], [ "From overlap coincidence to strong coincidence", "A van Hove sequence for ${\\mathbb {R}}^d$ is a sequence $\\mathcal {F}=\\lbrace F_n\\rbrace _{n \\ge 1}$ of bounded measurable subsets of ${\\mathbb {R}}^d$ satisfying $\\lim _{n\\rightarrow \\infty } \\mbox{\\rm Vol}((\\partial F_n)^{+r})/\\mbox{\\rm Vol}(F_n) = 0,~\\mbox{for all}~ r>0, \\ \\ \\ \\\\\\mbox{where} \\ (\\partial F_n)^{+r} = \\lbrace x \\in {\\mathbb {R}}^d : \\mbox{dist} (x, F_n) \\le r \\rbrace \\nonumber .$ For any $\\mathcal {S} \\subset {\\mathcal {T}}$ and van Hove sequence $\\lbrace F_n\\rbrace _{n \\ge 1}$ , we define $ \\mbox{\\rm dens}(\\mathcal {S}) := \\lim _{n \\rightarrow \\infty } \\frac{\\mbox{vol}(\\mathcal {S} \\cap F_n)}{\\mbox{vol}(F_n)}$ if the limit exists.", "Here the density limit is dependent on the van Hove sequence.", "In the case of a self-affine tiling ${\\mathcal {T}}$ , if $\\mathcal {S}$ is a set of all the translates of a patch in ${\\mathcal {T}}$ , the density exists uniformly [14].", "The following theorem shows that overlap coincidence of ${\\mathcal {T}}$ implies strong coincidence of $\\mbox{${\\Lambda }$}_{{\\mathcal {T}}}$ as long as the reference points for the tiling can be chosen to satisfy (REF ).", "This theorem can be proved using various results for pure discrete spectrum such as [19] and [7].", "Theorem 4.1 Let ${\\mathcal {T}}$ be a self-affine tiling in ${\\mathbb {R}}^d$ with expansion $Q$ for which $\\Xi ({\\mathcal {T}})$ is a Meyer set.", "Let $\\mbox{${\\Lambda }$}= \\mbox{${\\Lambda }$}_{{\\mathcal {T}}} = ({\\Lambda }_i)_{i \\le m}$ be an associated admissible substitution Delone multi-color set.", "Assume that ${\\Lambda }-{\\Lambda }\\subset \\langle \\mathcal {G} \\rangle _{{\\mathbb {Z}}},$ where ${\\Lambda }= \\cup _{i \\le m} {\\Lambda }_i$ and $\\mathcal {G} = =\\bigcup _{k=0}^{\\infty } Q^{-k} ({\\Lambda }_i-{\\Lambda }_i)$ for some $i \\le m$ .", "Then overlap coincidence of ${\\mathcal {T}}$ implies strong coincidence of $\\mbox{${\\Lambda }$}$ .", "Furthermore overlap coincidence of ${\\mathcal {T}}$ implies simultaneous coincidence of $\\mbox{${\\Lambda }$}$ .", "Proof.", "For any $ i, j \\le m$ , take $v_i \\in {\\Lambda }_i$ and $v_j \\in {\\Lambda }_j$ .", "Let $\\alpha = v_j - v_i$ .", "By the assumption, $\\alpha \\in {\\Lambda }_j-{\\Lambda }_i \\subset \\langle \\mathcal {G} \\rangle _{{\\mathbb {Z}}}\\,.", "$ From Theorem REF and Remark REF , $\\mathcal {G} = \\langle \\mathcal {G} \\rangle _{{\\mathbb {Z}}}$ .", "Therefore there is $M\\in \\mathbb {N}$ such that $Q^M \\alpha \\in {\\Lambda }_1-{\\Lambda }_1$ .", "Overlap coincidence implies that ${\\rm dens} \\ {\\mathcal {T}}\\cap ({\\mathcal {T}}-Q^n \\alpha ) \\longrightarrow 1$ as $n\\rightarrow \\infty $ [19].", "Since ${\\mathcal {T}}$ has uniform patch frequencies [14], the density is not dependent on the choice of the van Hove sequence.", "So we can take a van Hove sequence $\\lbrace \\mbox{\\rm supp}(\\Omega ^n (T_i -c_i + v_i)) \\rbrace _{n \\ge 1}$ .", "Then we observe that ${\\mathcal {T}}$ admits strong coincidence, otherwise the density is bounded away from one.", "Furthermore, we claim that $\\mbox{${\\Lambda }$}$ admits simultaneous coincidence.", "Fix $k \\le m$ and $v_k \\in {\\Lambda }_k$ .", "For any $i \\le m$ , choose $v_i \\in {\\Lambda }_i$ .", "Let $\\alpha _i = v_k - v_i$ .", "By the admissibility of $\\mbox{${\\Lambda }$}$ , $\\cap _{i \\le m} \\mbox{\\rm supp}(T_i-c_i +v_k)$ has non-empty interior.", "Let $P_n = \\bigcap _{i \\le m} \\mbox{\\rm supp}( \\Omega ^n(T_i - c_i + v_i + \\alpha _i)) \\ \\ \\ \\mbox{for $n \\in \\mathbb {N}$}.$ Then $\\lbrace P_n\\rbrace _{n \\ge 1}$ is a van Hove sequence and $\\Omega ^n (T_k-c_k +v_k)$ is a patch in ${\\mathcal {T}}$ .", "From the overlap coincidence of ${\\mathcal {T}}$ , for each $i \\le m$ $\\frac{{\\rm vol} ( \\Omega ^n (T_k - c_k + v_k) \\cap \\Omega ^n ( T_i - c_i + v_i + \\alpha _i) \\cap P_n) }{{\\rm vol} (P_n) } \\ \\stackrel{n \\rightarrow \\infty }{\\longrightarrow } \\ 1 \\,.$ Thus ${1-\\frac{{\\rm vol} \\left( \\left( \\bigcap _{i \\le m, i \\ne k} \\Omega ^n (T_k - c_k + v_k) \\cap \\Omega ^n ( T_i - c_i + v_i + \\alpha _i ) \\right) \\cap P_n \\right)}{{\\rm vol} (P_n)} } \\\\& \\le &\\sum _{i \\le m, i \\ne k} \\left(1-\\frac{{\\rm vol} \\left( \\Omega ^n (T_k - c_k + v_k) \\cap \\Omega ^n ( T_i -c_i + v_i + \\alpha _i ) \\cap P_n \\right)}{{\\rm vol} (P_n)} \\right) \\ \\longrightarrow \\ 0 \\ \\,.$ It implies that there exists $n \\in \\mathbb {N}$ such that $\\bigcap _{i \\le m, i \\ne k} \\Omega ^n (T_k - c_k + v_k) \\cap \\Omega ^n ( T_i - c_i + v_i + \\alpha _i) \\cap P_n) \\ \\ne \\ \\emptyset \\,.$ Thus $\\bigcap _{i \\le m, i \\ne k} \\Omega ^n (T_k - c_k) \\cap \\Omega ^n ( T_i - c_i) \\ \\ne \\ \\emptyset $ and the claim follows.", "$\\Box $ Corollary 4.2 Let ${\\mathcal {T}}$ be a self-affine tiling in ${\\mathbb {R}}^d$ for which $\\Xi ({\\mathcal {T}})$ is a Meyer set.", "Let $\\mbox{${\\Lambda }$}= \\mbox{${\\Lambda }$}_{{\\mathcal {T}}} = ({\\Lambda }_i)_{i \\le m}$ be an associated admissible substitution Delone multi-color set with a trivial height group.", "Then overlap coincidence of ${\\mathcal {T}}$ implies simultaneous coincidence as well as strong coincidence of $\\mbox{${\\Lambda }$}$ .", "Proof.", "The trivial height group assumption implies ${\\Lambda }-{\\Lambda }\\subset \\langle \\mathcal {G}\\rangle _{{\\mathbb {Z}}}$ .", "$\\Box $ We provide a lemma.", "Lemma 4.3 Let $A$ be a $m\\times m$ integer matrix whose characteristic polynomial is irreducible.", "Then the entries of an eigenvector of $A$ are linearly independent over ${\\mathbb {Q}}$ .", "Proof.", "Let $\\alpha $ be an eigenvalue of $A$ and $(v_1,v_2, \\dots , v_m)\\in {\\mathbb {Q}}(\\alpha )^m$ be a corresponding eigenvector.", "Denote by $\\alpha _i\\ (i=1,\\dots , m)$ the conjugates of $\\alpha $ .", "Applying Galois conjugate map $\\tau _i$ which sends $\\alpha $ to $\\alpha _i$ , we obtain $d$ different eigenvectors $(\\tau _i(v_1),\\dots ,\\tau _i(v_m))$ .", "Since $\\alpha _i\\ (i=1,\\dots , m)$ are all distinct, the corresponding eigenvectors are linearly independent over $.", "So $ B: =(i(vj))$ is invertible.However if $ v1,v2, ..., vm$ are linearlydependent over $ Q$, then the column vectors of $ B$ are linearly dependent over $ which is a contradiction.", "$\\Box $ Now we consider an 1-dim substitution tiling and recall the following lemma from [18] and [6]).", "Lemma 4.4 [18], [6] Let $\\sigma $ be a primitive irreducible substitution and consider its suspension tiling of ${\\mathbb {R}}$ with an expansion factor $\\beta $ , the Perron Frobenius root of the substitution matrix $M_{\\sigma }$ .", "Let $\\mbox{${\\Lambda }$}$ be an associated substitution Delone multi-color set whose points are taken from the left end of tiles in the suspension tiling.", "Then we have $\\langle {\\Lambda }_i - {\\Lambda }_i \\ \| \\ i \\le m \\rangle _{{\\mathbb {Z}}} = \\langle (\\cup _{i \\le m} {\\Lambda }_i) - (\\cup _{i \\le m} {\\Lambda }_i) \\rangle _{{\\mathbb {Z}}}.$ Corollary REF and Lemma REF give a combinatorial result on Pisot substitution sequences: the overlap coincidence implies strong and simultaneous coincidence for 1-dim irreducible Pisot substitutions.", "This result agrees with the earlier results of [6], [4].", "Corollary 4.5 Let $\\sigma $ be the irreducible Pisot substitution over letters $\\lbrace 1,\\dots ,m\\rbrace $ whose natural suspension tiling satisfies overlap coincidence.", "Then there are $L\\in \\mathbb {N}$ and $M\\in \\mathbb {N}$ such that prefixes of length $M$ of $\\sigma ^L(1), \\sigma ^L(2), \\dots , \\sigma ^L(m)$ has the same number of each letter $j$ for $1 \\le j \\le m$ , and all ends with an identical letter.", "Proof.", "As in Lemma REF , we choose the left end points as the set of control points.", "Then it fulfills the conditions of the admissibility and the trivial height group of Theorem REF .", "Theorem REF shows that the suspension tiling satisfies simultaneous coincidence by taking $n$ -th iterates of all prototiles.", "Since the suspension lengths of prototiles form a left eigenvector of $M_{\\sigma }$ , they are linearly independent over ${\\mathbb {Q}}$ by Lemma REF .", "Counting the number of tile types in $n$ -th iterate of each tile up to the tile where simultaneous coincidence occurs, we can claim the result.", "$\\Box $ Example 4.6 We give an example of a substitution tiling for which an associated admissible substitution Delone multi-color set has a trivial height group, but it admits neither overlap coincidence nor strong coincidence (simultaneous coincidence).", "Consider the substitution $\\sigma $ over four letters $\\lbrace a,b,A,B\\rbrace $ defined by $a\\rightarrow aB, b\\rightarrow a,A\\rightarrow Ab, B\\rightarrow A.$ This is ${\\mathbb {Z}}/2{\\mathbb {Z}}$ -extension of Fibonacci substitution.", "The suspension tiling dynamics does not satisfy overlap coincidence as it is computed in [3].", "The suspension lengths of capital letter and non capital letter are the same.", "Introduce a letter to the letter involution $\\tau $ which interchanges capital letter to non capital letter and vice versa.", "Then $\\tau $ and $\\sigma $ commutes from $\\tau (\\sigma ^n(a))=\\sigma ^n(A)$ for all $n \\in \\mathbb {N}$ , we see that $\\sigma $ does not satisfy prefix (nor suffix) strong coincidence.", "However taking the left end points as reference points, the height group is trivial which is confirmed by collecting all return vectors.", "Example 4.7 Consider the substitution $\\sigma $ over six letters $\\lbrace a,b,c,A,B,C\\rbrace $ defined by $a\\rightarrow aB, b\\rightarrow aC, c\\rightarrow a,A\\rightarrow Ab, B\\rightarrow Ac, C\\rightarrow A.$ This is ${\\mathbb {Z}}/2{\\mathbb {Z}}$ -extension of the Rauzy substitution.", "The suspension tiling dynamics of $\\sigma $ satisfies overlap coincidence which is checked by the algorithm in [3].", "By the same reasoning as Example REF , this substitution does not satisfy prefix nor suffix strong coincidence.", "Taking the left end points and collecting all return vectors, we can compute that the height group is equal to ${\\mathbb {Z}}/2{\\mathbb {Z}}$ .", "On the other hand, taking the tile map $\\gamma $ which sends $a\\rightarrow B, b \\rightarrow C, c\\rightarrow a,A\\rightarrow A, B \\rightarrow A, c\\rightarrow A,$ the set of control points is admissible and satisfies simultaneous coincidence.", "One can also check that the height group associate to this control points is trivial.", "So this gives an example of Corollary REF for a reducible substitution.", "Remark 4.8 Nakaishi [16] claimed that unimodular Pisot substitutions with prefix strong coincidence generate pure discrete spectrum of the dynamical system.", "It would mean that strong coincidence implies overlap coincidence in the associated substitution tilings in ${\\mathbb {R}}$ .", "So together with Corollary REF the equivalence between overlap coincidence and strong coincidence could be established.", "We hope directly to observe the result of [16], the implication from strong coincidence to overlap coincidence in the framework of this paper." ], [ "Acknowledgment", "The authors would like to thank the referees for the valuable comments and references.", "This research was supported by Basic Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education, Science and Technology(2010-0011150) and the Japanese Society for the Promotion of Science (JSPS), Grant in aid 21540010.", "The second author is grateful for the support of KIAS for this research." ] ]	1403.0377
[ [ "Milagro Observations of Potential TeV Emitters" ], [ "Abstract This paper reports the results from three targeted searches of Milagro TeV sky maps: two extragalactic point source lists and one pulsar source list.", "The first extragalactic candidate list consists of 709 candidates selected from the Fermi-LAT 2FGL catalog.", "The second extragalactic candidate list contains 31 candidates selected from the TeVCat source catalog that have been detected by imaging atmospheric Cherenkov telescopes (IACTs).", "In both extragalactic candidate lists Mkn 421 was the only source detected by Milagro.", "This paper presents the Milagro TeV flux for Mkn 421 and flux limits for the brighter Fermi-LAT extragalactic sources and for all TeVCat candidates.", "The pulsar list extends a previously published Milagro targeted search for Galactic sources.", "With the 32 new gamma-ray pulsars identified in 2FGL, the number of pulsars that are studied by both Fermi-LAT and Milagro is increased to 52.", "In this sample, we find that the probability of Milagro detecting a TeV emission coincident with a pulsar increases with the GeV flux observed by the Fermi-LAT in the energy range from 0.1 GeV to 100 GeV." ], [ "Introduction", "The Milagro gamma-ray observatory was a water Cherenkov detector located near Los Alamos, New Mexico, USA at latitude $35.9^{\\circ }$ north, longitude $106.7^{\\circ }$ west and altitude 2630 m [4].", "Milagro recorded data from 2001-2008 and was sensitive to extensive air showers initiated by gamma-rays with energies from a few hundred GeV to $\\sim $ 100 TeV.", "Unlike atmospheric Cherenkov telescopes, Milagro had a wide field of view and it was able to monitor the sky with a high duty cycle($>90\\%$ ).", "The Milagro collaboration has performed blind source searches and found a number of TeV sources ([13] and [1] We refer to this as Milagro Galactic Plane Surveys).", "Blind searches for excess events over the full sky have a high probability of picking up random fluctuations.", "Therefore, after trials correction, a full sky blind search is less sensitive than searches using a smaller predefined list of candidates.", "The Fermi Large Area Telescope (Fermi-LAT) collaboration published such a list known as the Bright Source List or 0FGL list [2].", "In a previous publication, Milagro reported a search using 0FGL sources identified as Galactic sources [3], which we will refer to as the Milagro 0FGL paper.", "We report here two Milagro targeted searches for extragalactic sources.", "The first extragalactic candidate list is compiled from the extragalactic sources in the 2FGL catalog[23].", "The analysis presented in this paper looks for the TeV counterparts of these sources.", "The second extragalactic candidate list is made from the TeVCat catalog [25] of extragalactic sources.", "While TeVCat detections may include transient states of variable extragalactic sources, this search looks for long-term time averages by integrating over the full Milagro data set.", "However, it is not appropriate to use the second extragalactic candidate list to perform a population study as it has candidates detected from several instruments with different sensitivities.", "Our previous Milagro 0FGL publication found that the Fermi-LAT bright sources that were measured at or above 3 standard deviations in significance (3$\\sigma $ ) by Milagro were dominated by pulsars and/or their associated pulsar wind nebulae (PWN).", "Therefore, in this paper we extend the previous Galactic search by making a candidate list from the pulsars in the 2FGL source list, and search for TeV emission from the sky locations of gamma-ray pulsars detected by the Fermi-LAT.", "The angular resolution of Milagro $\\left( 0.35^\\circ < \\delta \\theta <1.2^\\circ \\right)$ is not sufficient to distinguish the PWN from the pulsar.", "The first candidate list, which will be referred to as the 2FGL Extragalactic List, is derived from the 2FGL catalog by looking for sources off the Galactic plane ($\|b\| > 10^\\circ $ ) that have no association with pulsars.", "There are 709 Fermi-LAT sources within Milagro's sky coverage ($-7^\\circ <$ DEC$<80^\\circ $ ), of which 72% are associated with blazars.", "Among these blazars 4 are firmly identified as BL LacBL Lac is a type of active galaxy of known to be strongly $\\gamma $ -ray emitting objects [15].", "blazars and 12 are firmly identified as FSRQFlat Spectrum Radio Quasar type of blazars.", "The second extragalactic candidate list, which we will call the TeVCat Extragalactic List, is taken from TeVCat, an online gamma-ray source catalog (http://tevcat.uchicago.edu).", "As of February 8th, 2012 it contained 135 sources, of which 31 were located off the Galactic plane and within Milagro's sky coverage.", "These 31 sources were all detected with Cherenkov telescopes and 23 are identified as BL Lac objects.", "There are 52 sources in the 2FGL catalog associated with pulsars which are in the Milagro's sky coverage.", "Twenty of these pulsars were already considered as candidates in the Milagro 0FGL publication.", "So the third candidate list, which will be called the Pulsar List, consists of only the 32 new pulsars.", "Of these, 17 were identified as pulsars by pulsations seen in Fermi-LAT data and the remaining 15 sources were labeled as pulsars in 2FGL because of their spatial association with known pulsars." ], [ "Spectral Optimizations", "In order to optimize the sensitivity to photon sources, Milagro sky maps are constructed by plotting the location for each event with a weight based on the relative probability of it being due to a primary photon or hadron [4].", "The weight calculation depends on the assumed photon spectrum and can be suboptimal (but not incorrect) if the weight optimization hypothesis is considerably different from the actual source spectrum.", "The weights are therefore optimized separately for two hypotheses.", "For the extragalactic candidate lists, a power law with spectral index $\\alpha $ = -2.0 with a 5 TeV exponential cut-off ($E^{-2.0} e^{- \\frac{E}{5 \\rm {TeV}}}$ ) was assumed.", "This choice reflects the fact that when TeV gamma-rays travel cosmological distances they are attenuated due to interactions with photons from the extragalactic background light [16] with the result that the energy spectrum of extragalactic sources cut off at high energies.", "This spectral assumption is also similar to the power law spectral index and the cut-off energy measured for Mkn 421 and Mkn 501 by [20].", "However, the choice of 5 TeV cut off might reduce the sensitivity of Milagro to the AGNs with lower cut off energies.", "For the Pulsar List, a power law with spectral index $\\alpha = -2.6$ with no TeV cut-off is used, as was done for the previous Milagro 0FGL and Galactic Plane Survey papers." ], [ "Source Detection Technique", "The expected significance at a sky location with no true emission is a Gaussian random variable with mean 0 and unit standard deviation [13].", "A common treatment of N candidate searches is to use a trials correction technique.", "Here one choose a significance threshold, calculate the tail probability (p-value) $\\lambda $ , and adjusts the p-value threshold to $\\frac{\\lambda }{N}$ .", "The purpose of the trials correction is to maintain, at the value $\\lambda $ , the probability of a background fluctuation producing one or more false discoveries among the N searches.", "The False Discovery Rate (FDR) technique discussed in [22] offers some advantages over the trials correction technique.", "Instead of controlling the expected probability of having even one false detection, FDR controls the expected fraction of false discoveries among a set of detections; that is, it controls the contamination fraction of the lists of associations, rather than the probability of a random individual association being acceptedThe required calculations are quite simple and can be implemented in a spreadsheet after the significances of the searches on a list are calculated..", "The key input parameter is again a probability $\\lambda $ , but now $\\lambda $ represents the expected fractional contamination of any announced set of detections.", "Based on this input parameter, the method dynamically adjusts the detection threshold but in a way that depends on the properties of the entire list of search significances (converted into p-values).", "This dynamic adjustment is sensitive to whether the distribution of p-values is flat (as would be expected if there were no detectable sources) or skewed to small p-values (i.e.", "large significances).", "This adjustment lowers the significance threshold for detection if a list is a “target-rich environment” in such a way that the expected fraction of false discoveries among the announced detections remains at the fraction $\\lambda $ .", "In particular, the most significant candidate is required to have a p-value of $\\lambda /N$ just as in the trials-correction method, but the $n$ -th most significant candidate need only have a p-value less than $\\lambda \\times n/N$ .", "As a result, this technique has a higher efficiency for finding real detections, while producing the same results as a trials-correction method in target-poor environments where the only decision is whether to report zero or one detections.", "The method adjusts for both the length of the search list and the distribution of the significances found within the search lists.", "However, we note that as a result, a given candidate location might pass the FDR criteria on one search list, but fail in another.", "We also emphasize that $\\lambda $ controls the expected contamination, i.e.", "averaged over potential lists of associations, not the contamination fraction on a specific listFor example, in an environment with no real sources, one expects to report an empty list $(1-\\lambda \\ )\\times 100 \\%$ of the time, and about $\\lambda \\times 100 \\%$ of the time one would report a list having at least a single (false) candidate.", "The reader is referred to [22] for further details of the methodWe assume that the search points are uncorrelated, as the angular separations between target locations are normally much more widely seperated than the Milagro point spread function..", "In the [3] paper for the Galactic-oriented search with N = 35, a criterion of $3 \\sigma $ was used but it was also found that an FDR criterion of $\\lambda =0.01$ produced the same list of associations.", "Specifying $\\lambda $ rather than a $\\sigma $ threshold also allows using a single criterion for treating each of the search lists.", "The analyses presented in this paper uses $\\lambda =0.01$ for defining a TeV association for all our search lists, but significance thresholds are also tabulated in Table REF for $\\lambda = 0.1,0.05,$ and $0.001$ so that readers can choose the potential contamination level of candidate lists.", "Specific candidates passing looser cuts are also denoted as footnotes to the search list tables." ], [ "Stacking Methodology", "The FDR technique can be used to search for individual candidates with a TeV association.", "A stacking analysis can be used to search for evidence of collective TeV emission among the undetected candidates by studying their mean flux.", "This paper uses the stacking methodology of Section 3 in [21].", "The significance of the stacked flux is given by Equation REF below.", "$\\rm {Significance} = \\frac{\\left< I \\right>}{\\sqrt{\\vee {\\left( \\left< I \\right> \\right)}}},$ where $\\left< I \\right>$ is the weighted average flux as defined in Equation REF below and $ \\vee { \\left< I \\right>}$ is its variance, defined in Equation REF .", "$\\left< I \\right> = \\frac{\\sum {\\frac{I_i}{\\sigma _i^2}}}{\\sum {\\frac{1}{\\sigma _i^2}}}$ $\\vee { \\left< I \\right>} = \\frac{\\sum {\\frac{1}{\\sigma _i^2}}}{\\sum { \\left( \\frac{1}{\\sigma _i^2} \\right)^2}}$ Here $I_i$ is the flux of each candidate and $\\sigma _i$ is the standard deviation of flux of each candidate." ], [ "Flux Calculation", "The flux calculation involves a convolution of the Milagro effective area as a function of energy using an assumed energy spectrum, so the flux has some dependence on the assumed energy spectrum.", "This dependence is greatly reduced when the flux is calculated at the median energy of the detected gamma-ray events at the declination of a source [3].", "Therefore, we report the flux at approximately the median energy.", "Using a similar argument to that in the Milagro 0FGL paper, the flux is derived at 35 TeV for the Pulsar List.", "For the extragalactic spectral assumption the median energy varies between 6 and 11 TeV, and we choose 7 TeV to report the flux for extragalactic source candidates.", "In this paper, we report the flux for the candidates with TeV associations that are identified by the FDR procedure.", "For the remaining candidates we report flux upper limit.", "In all cases, the flux and significance calculations are performed assuming that the target is a point-like source.", "The fluxes are calculated from the excess number of photons above background integrating over a Gaussian point spread functionThe width of the Gaussian point spread function is a function of the estimated energy of each event and varies between $0.3^\\circ $ and $0.7^\\circ $ .", "for a point source at the sky position given by the catalog used to compile the list.", "This approach is similar to that described in the Milagro Galactic Plane Survey papers.", "The upper limits on the flux are determined using the method described in [19] and are based on an upper limit on the number of excess photons with a 95% confidence limit.", "The flux upper limit corresponding to a zero excess is called the expected flux limit.", "The declination dependence of the expected flux limits shown in Figures REF and REF are based on Milagro maps made with the spectral optimizations $dN/dE \\propto E^{-2.6}$ and $dN/dE \\propto E^{-2.0}e^{-\\frac{E}{5 TeV}}$ , respectively.", "The searches presented in this paper did not examine the whole sky.", "Another publication is in progress to produce all-sky flux limits from Milagro.", "Figure: The expected 95% confidence level flux upper limit for Galactic sources corresponding to zero excess derived at 35 TeV for each declination band of the Milagro sky maps made with spectral assumption dN/dE∝E -2.6 dN/dE \\propto E^{-2.6}.Figure: The expected 95% confidence level upper limit on the flux for extragalactic sources corresponding to zero excess derived at 7 TeV for each declination band of the Milagro sky maps made with spectral assumption dN/dE∝E -2.0 e -E 5TeV dN/dE \\propto E^{-2.0}e^{-\\frac{E}{5 TeV}}." ], [ "Results", "In the 2FGL Extragalactic List only 2FGL J1104.4+3812 (also known as Mkn 421) is classified as a source by the FDR procedure with our standard $\\lambda =0.01$ cut.", "Figure REF shows the region in the Milagro sky map around Mkn 421.", "From the 2FGL Extragalactic candidate list the fluxes or the 95% confidence level flux upper limits for the brightest 20% of the 2FGL candidates in the 3 GeV to 10 GeV energy band is given in Table REF .", "From the TeVCat Extragalactic List only Mkn 421 is classified as a source by the FDR procedure with our standard $\\lambda =0.01$ .", "The results with our standard FDR cut of $\\lambda =0.01$ are summarized in Table REF .", "Results from the source search in the Pulsar List are summarized in Table REF .", "In this list, the FDR procedure with $\\lambda =0.01$ classified 3 GeV pulsars (2FGL J2238.4+5902, 2FGL J2030.0+3640 and 2FGL J1928.8+1740c) as having coincident TeV emission.", "Figure REF shows the regions of the Milagro sky maps around those candidates.", "The brighter area near 2FGL J2238.4+5902 (Figure REF ) corresponds to 0FGL J2229.0+6114, and it is also associated with the bright TeV source MGRO J2228+61 [18].", "The Milagro flux at the location of 0FGL J2229.0+6114 was published in the Milagro 0FGL paper.", "Similarly 2FGL J2030.0+3640 (Figure REF ) is located near a brighter area which belongs to 0FGL J2020.8+3649.", "The Milagro flux at this 0FGL source location was also published in the Milagro 0FGL paper.", "Follow-up observations by TeV instruments with better angular resolution could clarify the TeV emission structure in both of these regions; some initial studies of this region have been already done [[5], [14]].", "2FGL J2030.0+3640 also has a spatial association with the Milagro candidate named as C3 in [1].", "Milagro candidate C3 is measured at RA = $307.75^0$ and DEC = $36.52^0$ with extent diameter of $2.8^0$ [1].", "To assess how likely it would be to observe TeV emission coincident with 2FGL sources associated with pulsars if they arose from statistical fluctuations in the Milagro data, we calculate the probability that a set of 32 random points in the Milagro Galactic plane ($\|l\|<10^0$ ) would have 3 or more FDR associations.", "For a simulated background-only sky (consisting of a standard normal significance distribution) the probability of finding 3 or more associations is $1\\times 10^{-6}$ .", "Thus finding 3 associations would be a 4.3 $\\sigma $ fluctuation for random points on a background-only sky.", "As expected, the $\\lambda =0.01$ FDR cut yields no associations 99% of the time with random locations on a random sky (with no real sources).", "However, the probability of finding 3 or more associations from random lists of 32 locations within the actual Milagro Galactic plane $\|l\|<10^0$ (which contains TeV sources) is 0.01.", "This is much higher than for a background-only sky, so that reporting 3 associations in the Milagro Galactic plane data would be a 2.3 $\\sigma $ fluctuation if we were starting from a randomly located candidate list (rather than seeking associations with the 2FGL pulsar list).", "By varying $\\lambda $ the reader can construct alternative target lists with different contamination fractions, to assess how clearly candidates have passed a given association criterion.", "Table REF summarizes the FDR significance thresholds for each of the lists we have searched using $\\lambda =0.1,0.05,0.01$ and $0.001$ .", "The table also gives the significance thresholds which would have resulted from the trials correction technique.", "The comparison between the FDR and the trials corrections thresholds allows assessment of how much the FDR procedure has lowered the significance threshold in response to evidence of associations.", "So far our results have focused on individual candidates with a TeV association.", "We also searched for evidence of collective TeV emission on the candidates that fail the $\\lambda =0.01$ FDR cut by using the stacking method described in section REF .", "We stacked 2FGL Extragalactic candidates in two different ways: first all FDR False 2FGL Extragalactic sources and then the FDR False sources among the brightest 20% in the Fermi-LAT energy band 3-10 GeV.", "These two lists had $0.7 \\sigma $ and $0.6 \\sigma $ significance collectively.", "Stacking of TeV Cat candidates other than Mkn 421 has only a slightly more positive upward fluctuation of $0.9 \\sigma $ .", "The rejected pulsar candidates have a $-0.5 \\sigma $ fluctuation from the background.", "None of these stacking results indicate significant collective gamma-ray emission from the rejected candidates." ], [ "Discussion", "Mkn 421 is the only source that is classified as a TeV source in both extragalactic lists.", "Milagro also observed a signal excess at the sky locations of Mkn 501, TXS 1720+102 and 1ES 0502+675 .", "Their significances are 2.93, 2.84 and 2.53 respectively, which is insufficient to pass our standard FDR cut of $\\lambda =0.01$ .", "Among these three candidates Mkn 501 and 1ES 0502+675 have been already reported as TeV sources in TevCat.", "However TXS 1720+102 has not yet been identified as a source with TeV emission.", "This is a radio quasar type blazar identified at a redshift of 0.732 [10].", "The lowest FDR cut that TXS 1720+102 passes is $\\lambda =0.32$ .", "With this loose FDR cut, three candidates become TeV associations: Mkn 421, Mkn 501 and TXS 1720+102.", "However, the expected contamination of the resulting candidates list is 32% so it is likely that TXS 1720+102 is a background fluctuation.", "While it is hard to advocate a dedicated IACT observation of TXS 1720+102, better observations will be performed by the High Altitude Water Cherenkov (HAWC) survey instrument [6], which is already started to operate at a sensitivity better than Milagro.", "In this paper we presented the TeV flux/flux limit measurements at 32 sky locations of 2FGL sources marked as pulsars.", "At the time we wrote this paper, none of these 2FGL pulsars were reported as detections in the TeVCat or in the H.E.S.S.", "source catalog.", "However, TeV flux upper limits of some these sources have been measured by IACTs.", "For an example, the flux upper limit of PSR J1928+1746 was measured by the VERITAS observatory [7], which is associated with the 2FGL J1928.8+1740c.", "VERITAS observed a $+1.2\\sigma $ significance at this pulsar position and 99% confidence flux upper limit of $2.6\\times 10^{-13}$ cm$^{-2}$ s$^{-1}$ above 1 TeV was measured assuming a power-law spectrum with power law index -2.5.", "Contrasted with this measurement, Milagro measured a $46.41\\pm 11.5\\times 10^{-17}$ photons TeV$^{-1}$ s$^{-1}$ cm$^{-2}$ of flux at 35 TeV from this pulsar position, assuming a power-law spectrum with power law index -2.6.", "The Milagro flux measurement is order of magnitude larger than the VERITAS upper limit.", "This difference may be caused by the wider point spread function of Milagro compared with that of VERITAS ($\\sim 0.11^\\circ $[7]).", "Therefore, the Milagro flux may include some additional diffuse emission or emission from unresolved point sources that is not contained within the VERITAS point spread function.", "We also compared our flux/flux limit measurements with the H.E.S.S.", "Galactic Plane Survey [17], and found that our measurements are consistent with the H.E.S.S.", "measurements.", "The Milagro 0FGL paper reported the Milagro flux/flux limit at the locations of 16 bright Fermi-LAT sources from the 0FGL catalog that were associated with pulsars.", "Among these 16 pulsars, 9 passed the standard FDR cut of $\\lambda =0.01$ .", "0FGL J2055.5+2540, 0FGL J2214.8+3002 and 0FGL J2302.9+443 were categorized as sources with unknown source type and 0FGL J1954.4+2838 was identified as a source with a spatial association with a known supernova remnant.", "In the 2FGL catalog these four sources have been identified as pulsars and only 0FGL J1954.4+2838 passed our standard FDR cut.", "Therefore all together 52 pulsars detected by Fermi-LAT have been observed by Milagro and 13 pulsars were identified with TeV associations.", "We use this sample to study the correlation between GeV and TeV flux.", "Figure REF shows the TeV flux measured by Milagro vs the GeV flux measured by Fermi-LAT for these 52 pulsars.", "Data points marked with red triangles are Milagro upper limits measured at the sky locations of the candidates that failed the $\\lambda =0.01$ FDR cut.", "Blue data points represents the Milagro flux at the sky locations of the candidates that passed the $\\lambda =0.01$ FDR cut.", "The Milagro flux/flux limits used in this plot were derived assuming the targets are point sources.", "However, some of these objects are extended sources, for which the point source flux would underestimate the total flux.", "Geminga is a specific example, as seen in the Milagro 0FGL paper.", "In Figure REF Geminga is circled in red.", "We can also study how the fraction of pulsars with a TeV counterpart changes as a function of the GeV flux.", "We define $F_T$ as the fraction of pulsars that passed our standard FDR cut in a given bin of GeV flux.", "$F_T = \\frac{ \\textnormal {Number of FDR true candidates in a given flux bin} }{ \\textnormal { Total number of candidates in a given flux bin} }$ As shown in Figure REF $F_T$ clearly increases with the Fermi-LAT flux.", "Both the $F_T$ plot and the flux correlation plot strongly prefer a dependence on the GeV flux.", "Therefore we have evidence that pulsars brighter in the GeV energy range are more likely to have a detectable TeV counterpart than pulsars fainter in the GeV energy range.", "Further analysis of the GeV-TeV correlation is in progress and will be published in a follow-up paper.", "Figure: The horizontal axis is the Fermi-LAT flux (photons s -1 ^{-1}cm -2 ^{-2}), integrated over the energy range from 100 MeV to 100 GeV.", "The vertical axis is the Milagro flux derived at 35 TeV ( photons TeV -1 ^{-1}s -1 ^{-1}cm -2 ^{-2}), assuming all targets are point sources.", "Red data points are Milagro upper limits of candidates that failed the λ=0.01\\lambda =0.01 FDR cut.", "Blue data points are the Milagro flux derived for the candidates that passed the λ=0.01\\lambda =0.01 FDR cut.The Milagro flux/flux limits used in this plot were derived assuming the targets are point sources.However, some of these objects are extended sources, for which the point source flux would underestimate the total flux.Geminga is a specific example and it is circled in red.Figure: The fraction F T F_T (see text) of Fermi-LAT pulsars seen by Milagro as a function of half-decade bins of the integrated Fermi-LAT flux ( photons cm -2 s -1 cm^{-2}s^{-1}) in the energy range from 100 MeV to 100 GeV." ], [ "Conclusions", "We present a targeted search for extragalactic sources in the Milagro data using a list of bright 2FGL extragalactic sources and TeV sources from the TeVCat catalog as targets.", "Using the FDR procedure with $\\lambda = 0.01$ , we find that Mkn 421 is the only extragalactic TeV source detected by Milagro.", "There is no evidence of collective TeV emission seen from the remaining extragalactic candidates.", "The analysis performed in the Milagro 0FGL paper has been extended by searching for TeV emission at the locations of 32 additional Fermi-LAT detected pulsars.", "TeV emission has been found associated with three of them: 2FGL J2238.4+5902, 2FGL J2030.0+3640 and 2FGL J1928.8+1740c.", "The first two of these are near bright VHE sources previously reported as being associated with energetic pulsars in the 0FGL catalog.", "They might benefit from a higher spatial resolution TeV follow-up to study the emission structure from the two nearby source regions.", "The pulsar candidates that failed the $\\lambda = 0.01$ FDR cuts were studied in a stacking analysis but did not show any collective TeV emission.", "Finally, we presented evidence that pulsars brighter in the GeV energy range are more likely to have a detectable TeV counterpart." ] ]	1403.0161
[ [ "Combinatorial model for cluster categories of type E" ], [ "Abstract In this paper we give a geometric-combinatorial description of the cluster categories of type E. In particular, we give an explicit geometric description of all cluster tilting objects in the cluster category of type E_6.", "The model we propose here arises from combining two polygons, and it generalises the description of the cluster category of type A and D." ], [ "Introduction", "Caldero-Chapoton-Schiffler defined in [5] categories arising from homotopy classes of paths between two vertices of a regular $(n+3)$ -sided polygon.", "Independently, Buan-Marsh-Reiten-Reineke-Todorov defined cluster categories as certain orbit categories of the bounded derived category of hereditary algebras, see [3].", "The latter are algebras arising from oriented graphs $Q$ with no oriented cycles.", "When $Q$ is an orientation of a Dynkin graph of type $A_n$ , the category constructed in [5] coincides with the one of [3].", "In this paper we model a number of different orbit categories arising from orientations of tree graphs $T_{r,s,t}$ in geometric terms.", "The description we propose here is based on the idea of doubling the set of oriented diagonals in a given regular polygon and combine the dynamics of these two sets in an appropriate way.", "Among the additive categories we can model in this way we find cluster categories of type $T_{r,s,t}$ , as well as other triangulated categories of various Calabi-Yau dimension.", "All these categories arise as mesh categories of a translation quiver whose vertices are single coloured oriented and paired diagonals in a polygon $\\Pi $ .", "We will see that polygons of different sizes yield different categories and these are Calabi-Yau of dimension two only in few exceptional cases.", "The advantage of our approach is that we can model the combinatorics of these various categories in terms of configurations of diagonals in $\\Pi $ .", "For a tree diagram $T_{r,s,t}$ let $\\mathcal {C}_{T_{r,s,t}}$ be cluster category of type $T_{r,s,t}$ defined as the orbit category $\\mathcal {D}^b(\\mathrm { mod } k T_{r,s,t})/\\tau ^{-1}\\Sigma $ .", "Then for $n\\ge \\max \\lbrace r+t+1,r+s+1\\rbrace $ and a regular $(n+3)$ -gon $\\Pi $ , we construct an additive category $\\mathcal {C}^{n+3}_{r,s,t}$ of coloured oriented single and paired diagonals of $\\Pi $ .", "With this notation our first main result can be stated as follows.", "Theorem (REF ) We have the following equivalences of additive categories: $ &\\mathcal {C}^7_{1,2,2}\\rightarrow \\mathcal {C}_{E_6} \\\\&\\mathcal {C}^{10}_{1,2,3}\\rightarrow \\mathcal {C}_{E_7}\\\\&\\mathcal {C}^{16}_{1,2,4}\\rightarrow \\mathcal {C}_{E_8}.$ When $T_{r,s,t}$ is not of Dynkin type we obtain an equivalence from the category $\\mathcal {C}^\\infty _{r,s,t}$ associated to the infinite sided-polygon $\\Pi ^\\infty $ to the full subcategory of the cluster category $\\mathcal {C}_{T_{r,s,t}}$ with indecomposable objects in the transjective component of the AR-quiver of $\\mathcal {C}_{T_{r,s,t}}$ .", "These equivalences enable us to investigate the combinatorics of the cluster category in geometric terms.", "More precisely, we are able to describe all 833 cluster tilting sets of the cluster category of type $E_6,$ $\\mathcal {C}_{E_6}$ , as cluster configuration of six single coloured oriented and paired diagonals in a heptagon.", "The strategy will be to first determine two fundamental families of cluster configurations $\\mathcal {F}_1$ and $\\mathcal {F}_2$ and deduce the remaining cluster tilting sets using the rotation inside $\\Pi $ , induced from the Auslander-Reiten translation $\\tau $ in $\\mathcal {C}_{E_6}$ , as well as a symmetry $\\sigma $ of our model.", "Theorem (REF ) In a heptagon 350 different cluster configurations have one long paired diagonal, and they arise from $\\mathcal {F}_1$ through $\\tau $ .", "483 other cluster configurations arise from $\\mathcal {F}_2$ through $\\sigma $ and $\\tau $ .", "This classification allows us to deduce the following result.", "Theorem (REF ) All 833 cluster tilting sets of $\\mathcal {C}_{E_6}$ can be expressed as configurations of six non-crossing coloured oriented single and paired diagonals inside two heptagons.", "The previous results enable us to deduce geometrical moves describing the mutation process between cluster tilting objects in $\\mathcal {C}_{E_6}$ .", "The geometrical moves we find extend the mutation process of cluster categories of type $A_n$ , as described by Caldero-Chapoton-Schiffler in [5], to the setting of coloured oriented diagonals.", "We will see that in many cases mutations inside $\\mathcal {C}_{E_6}$ correspond to flips of coloured oriented diagonals in $\\Pi $ .", "In the final part of the paper we use the previous results to categorify geometrically cluster algebras of type $F_4$ .", "We also provide a geometric description of the mutation rule.", "Relation to previous work: Geometrical models for cluster categories of other types have been investigated also in [19], [2], [21].", "Moreover, with an appropriate paring of coloured oriented single and paired diagonals in an even sided polygon we recover the categorified geometric realisation of cluster algebras of type $D$ given by Fomin-Zelevinsky in [9].", "The number of clusters in a cluster algebra of finite type was first computed in [9].", "Under the bijection of [3] one deduces the number of cluster tilting sets in the corresponding cluster categories.", "An explicit complete description of the cluster-tilting objects in the cluster category of type $E_6$ and $F_4$ however is new.", "In addition, Fomin-Pylyavskyy used in [8] polygons to describe the cluster algebra structure in certain rings of $\\mathrm {SL}(V)$ -invariants.", "More precisely, they construct invariants determined by tensor graphs associated to diagonals of polygons.", "Using a heptagon, Fomin-Pylyavskyy also model the cluster algebra structure of the homogeneous coordinate ring $\\mathbb {C}[Gr_{3,7}]$ of the affine cone over the Grassmannian $Gr_{3,7}$ of three dimensional subspaces in a seven dimensional complex vector space.", "By a result of Scott, [20], it is known that the ring $\\mathbb {C}[Gr_{3,7}]$ is a cluster algebra type of $E_6$ .", "The approach of [8] however is different then the one we propose here, as it relies on relations satisfied by tensor graphs, called skein relations of tensor graphs.", "Organisation of the article: In Section 2, we state some preliminary results and definitions.", "In particular, we state the fundamental properties of orbit categories and we remind the action of the shift functor on the Auslander-Reiten quiver of $\\mathcal {D}^b(\\mathrm {mod}k Q)$ , for $Q$ an orientation of a simply laced Dynkin diagram.", "In Section 3, we construct the additive category $\\mathcal {C}^{n+3}_{r,s,t}$ associated to a regular (n+3)-gon $\\Pi $ , where $n=\\max \\lbrace r+t+1,r+s+1\\rbrace $ .", "The objects of $\\mathcal {C}^{n+3}_{r,s,t}$ will be single coloured oriented and paired diagonals of $\\Pi $ .", "The morphism spaces are generated by minimal rotations in $\\Pi $ , modulo certain equivalence relations.", "In Section 4 we prove the equivalences of additive categories stated in Theorem REF .", "Further equivalence of additive categories will also be discussed.", "In Proposition REF we show that there is an equivalence between the category $\\mathcal {C}^\\infty _{r,s,t}$ associated to an infinite sided polygon and the full subcategory of the cluster category $\\mathcal {C}_{T_{r,s,t}}$ , whose indecomposable objects belong to the transjective component of the AR-quiver of $\\mathcal {C}_{T_{r,s,t}}$ .", "In Section 5, we describe the combinatorics of $\\mathcal {C}_{E_6}$ geometrically inside a heptagon $\\Pi $ .", "In Theorem REF and Theorem REF we describe all cluster tilting sets of $\\mathcal {C}_{E_6}$ in terms of cluster configurations of coloured oriented diagonals of $\\Pi $ .", "In Proposition REF we also describe all $\\mathrm {Ext}$ -spaces of $\\mathcal {C}_{E_6}$ using curves of coloured oriented diagonals of $\\Pi $ .", "Finally, results concerning the mutation process of cluster configurations will be stated, see Proposition REF .", "In Section 6 we point out further applications of our work.", "In particular we deduce a geometric additive categorification of cluster algebras of type $F_4$ .", "Moreover, we describe how our construction can be used to understand cluster tilting sets inside the cluster categories of type $E_7$ and $E_8$ , as well as cluster tilting sets in the transjective component of the AR-quiver of cluster categories associated to more general tree diagrams $T_{r,s,t}$ ." ], [ "Preliminaries", "Let $k$ be an algebraically closed field and let $Q$ be an acyclic quiver.", "Let $\\mathrm {mod}kQ$ be the abelian category of $k$ -finite dimensional right-modules over the path algebra $kQ$ .", "Let $\\mathcal {D}:=\\mathcal {D}_Q:=\\mathcal {D}^b(\\mathrm {mod} k Q)$ be the bounded derived category of $\\mathrm {mod}kQ$ endowed with the shift functor $\\Sigma :\\mathcal {D}\\rightarrow \\mathcal {D}$ and the Auslander-Reiten translation $\\tau :\\mathcal {D}\\rightarrow \\mathcal {D}$ characterised by $\\mathrm {Hom}_{\\mathcal {D}}(X,-)^\\cong \\mathrm {Hom}_{\\mathcal {D}}(-,\\Sigma \\tau X),$ for all $X\\in \\mathcal {D}$ ." ], [ "Orbit categories of $\\mathcal {D}$", "We are interested in the orbit categories $\\mathcal {C}_Q^p$ of $\\mathcal {D}$ , $p\\in \\mathbb {N}$ , generated by the action of cyclic group generated by the auto-equivalences $F^p:=(\\tau ^{-1}\\Sigma )^p=\\tau ^{-p}\\Sigma ^p$ .", "The objects of $\\mathcal {C}_Q^p$ are the same as the objects of $\\mathcal {D}$ and $\\mathrm {Hom}_{\\mathcal {C}_Q^p}(X,Y):=\\bigoplus _{t\\in \\mathbb {Z}}\\mathrm {Hom}_{\\mathcal {D}}(X,(F^p)^tY).$ Morphisms are composed in a natural way.", "When $p=1$ , $\\mathcal {C}_Q:=\\mathcal {C}^1_Q$ is the cluster category of type $Q$ defined in [3], and independently in [5] in geometric terms for $Q$ of type $A_n$ .", "In all other cases $\\mathcal {C}^p_Q$ is the $p$ -repetitive cluster category studied by the author in [15] for $Q$ of type $A_n$ , and introduced by Zhu in [22] for $Q$ an acyclic quiver." ], [ "Fundamental properties of orbit categories", "Like $\\mathcal {D}$ , the categories $\\mathcal {C}^p_Q$ are Krull-Schmidt and have finite dimensional $\\mathrm {Hom}$ -spaces.", "The categories $\\mathcal {C}^p_Q$ are triangulated categories, and the projection functor $\\pi _i:\\mathcal {D}\\rightarrow \\mathcal {C}^p_Q$ , $i\\in \\mathbb {N}$ is a triangle functor, see [13].", "The induced shift functor is again denoted by $\\Sigma $ .", "Moreover, the categories $\\mathcal {C}^p_Q$ have AR-triangles and the AR-translation $\\tau $ is induced from $\\mathcal {D}$ .", "The categories $\\mathcal {C}^p_Q$ also have the Calabi-Yau property, i.e.", "$(\\tau \\Sigma )^m\\stackrel{\\sim }{\\longrightarrow }\\Sigma ^n $ as triangle functors, here we identify $\\tau \\Sigma $ with the Serre functor of $\\mathcal {C}^p_Q$ .", "In particular, in $\\mathcal {C}_Q$ we have that $n=2$ and $m=1$ , hence $\\mathcal {C}_Q$ is Calabi-Yau of dimension 2.", "In $\\mathcal {C}^p_Q$ we have that $m=2$ and $n=p$ in the above isomorphism of triangle functors, thus $\\mathcal {C}^p_Q$ is said to be a Calabi-Yau category of fractional dimension $\\frac{p}{2}$ .", "Notice that a Calabi-Yau category of fractional dimension $\\frac{p}{2}$ is in general not a Calabi-Yau category of dimension 2.", "In this paper $p\\in \\lbrace 1,2\\rbrace $ , moreover we adopt the convention $\\mathrm {Ext}^i_{{\\mathcal {C}^p}_Q}(X,Y):=\\mathrm {Hom}_{{\\mathcal {C}^p}_Q}(X,\\Sigma ^i Y)$ ." ], [ "Auslander-Reiten quiver of a Krull-Schmidt category", "A stable translation quiver $(\\Gamma ,\\tau )$ in the sense of Riedtmann, [18], is a quiver $\\Gamma $ without loops nor multiple edges, together with a bijective map $\\tau :\\Gamma \\rightarrow \\Gamma $ called translation such that for all vertices $x$ in $\\Gamma $ the set of starting points of arrows which end in $x$ is equal to the set of end points of arrows which start at $\\tau (x)$ .", "For $(\\Gamma ,\\tau )$ one defines the mesh category as the quotient category of the additive path category of $\\Gamma $ by the mesh ideal, see for example [14].", "In particular, the mesh category of $(\\Gamma ,\\tau )$ is an additive category.", "In the next result, let $\\mathbb {Z}Q$ be the repetitive quiver of $Q$ , see [11] for a reminder on this construction.", "Let $\\tau : \\mathbb {Z} Q\\rightarrow \\mathbb {Z} Q$ be the automorphism defined on the vertices $(n,i)$ of $\\mathbb {Z} Q$ by $\\tau (n,i)=(n-1,i)$ , for $n\\in \\mathbb {Z}$ , $i$ a vertex of $Q$ .", "Theorem 2.1 [11] Let $kQ$ be a finite dimensional hereditary $k$ -algebra.", "If $Q$ is an orientation of a simply laced Dynkin graph, $AR(\\mathcal {D}^b(\\mathrm {mod}\\,kQ))$ is isomorphic (as stable translation quiver) to $\\mathbb {Z} Q$ .", "an affine graph, $AR(\\mathcal {D}^b(\\mathrm {mod}\\,kQ))$ splits into components of the form $ \\mathbb {Z} Q$ and $\\mathbb {Z} A_\\infty /r$ , for some $r\\in \\mathbb {N}$ .", "a wild graph, the components of $AR(\\mathcal {D}^b(\\mathrm {mod}\\,kQ))$ are of the form $ \\mathbb {Z} Q$ and $\\mathbb {Z} A_\\infty $ .", "Let $\\mathrm {ind}\\, \\mathcal {D}$ be the full subcategory of $\\mathcal {D}$ of indecomposable objects.", "Theorem 2.2 [11] Let $Q$ be an orientation of a simply laced Dynkin graph.", "The mesh category of $(\\mathbb {Z} Q,\\tau )$ is equivalent to $\\mathrm {ind }\\,\\mathcal {D}$ .", "A first important consequence of this result is that the AR-quiver of $\\mathcal {D}$ is independent of the orientation of $Q$ ." ], [ "Induced action of $\\Sigma $ on {{formula:91dc222f-a8cc-41dd-af1c-185e26f3d061}} ", "In this section let $Q$ be an orientation of a simply-laced Dynkin graph.", "Below we point out some known facts about the induced action of $\\Sigma $ and $\\tau $ on $\\mathbb {Z}Q$ taken from [17], see also [12].", "These considerations, together with Theorem REF , will enable us to determine the precise shape of the AR-quiver of various orbit categories investigated in the sequel.", "The induced action of $\\tau $ on $\\mathbb {Z} Q$ is always an horizontal shift to the left.", "The induced action of $\\Sigma $ on $\\mathbb {Z} A_n$ coincides with a shift of $\\frac{n+1}{2}$ units to the right, composed with a reflection along the horizontal central line of $\\mathbb {Z} A_n$ .", "On $\\mathbb {Z} D_n$ the action of $\\Sigma $ agrees with $\\tau ^{-(n-1)}$ composed with an order two automorphism $\\rho $ defined on $\\mathbb {Z} D_n$ when $n$ is odd.", "While on $\\mathbb {Z} E_6$ the action of $\\Sigma $ coincides with the action of $\\rho \\tau ^{-6}$ where $\\rho $ is an automorphism of order two defined on $\\mathbb {Z} E_6$ .", "Moreover, $\\Sigma $ acts as $\\tau ^{-9}$ on $\\mathbb {Z} E_7$ , and as $\\tau ^{-15}$ on $\\mathbb {Z} E_8$ ." ], [ "The repetitive cluster category $\\mathcal {C}^p_{A_{n}}$", "The geometrical model for the cluster categories of type $E$ we propose in this paper is motivated from the following idea: glue together two copies of AR-quiver of $\\mathcal {C}^2_{A_{n}}$ .", "Let us remind the reader some facts about this category.", "The previous discussion implies that $(AR(\\mathcal {C}^p_{A_{n}}),\\tau )\\cong (\\mathbb {Z} A_{n}/(\\tau ^{-1}\\Sigma )^p,\\tau )$ , for $p\\in \\mathbb {N}$ .", "Moreover, $(AR(\\mathcal {C}^p_{A_{n}}),\\tau )$ can be modelled using diagonals in polygons as done in [15].", "When $p=2$ , the quiver $(AR(\\mathcal {C}^2_{A_{n}}),\\tau )$ can entirely be modelled using oriented diagonals in a regular $(n+3)$ -gon.", "In Figure REF an illustration of this construction is provided for $p=2,n=4$ .", "Figure: AR-quiver of 𝒞 A 4 2 \\mathcal {C}^2_{A_4}." ], [ "Single coloured oriented and paired diagonals in polygons", "Throughout the rest of the paper let $T_{r,s,t}$ be an orientation of a finite connected graph with $r+s+t+1$ vertices and three legs.", "We assume the legs of $T_{r,s,t}$ to have $r$ , resp.", "$s$ , resp.", "$t$ vertices and that one vertex of $T_{r,s,t}$ has three neighbours.", "We say that a tree $T_{r,s,t}$ is symmetric if $s=t$ .", "Unless specified otherwise, for $n\\ge \\mathrm {max}\\lbrace r+s+1;r+t+1\\rbrace $ let $\\Pi $ be a regular $(n+3)$ -gon with vertices numbered in the clockwise order by the group $\\mathbb {Z}/(n+3)\\mathbb {Z}$ .", "For vertices $i,j,k$ of $\\Pi $ we write $i\\le j\\le k$ if $j$ is between $i$ and $k$ in the clockwise order.", "Moreover, we denote by $(i,j)$ the unoriented diagonal of $\\Pi $ joining the vertices $i$ and $j$ and by $[i,j]$ the oriented diagonal of $\\Pi $ starting at $i$ and ending in $j$ .", "We do not consider boundary segments as oriented diagonals." ], [ "Single coloured oriented and paired diagonals of $\\Pi $", "We start describing the geometric construction leading to the modelling of a number of orbit categories of $\\mathcal {D}^b(\\mathrm {mod}\\, k T_{r,s,t})$ arising from orientations of symmetric trees $T_{r,s,t}$ .", "To begin the construction we double the set of oriented diagonals of $\\Pi $ , and distinguish each set with colours using subscripts $R$ , $B$ e.g.", "$[1,3]_R$ is the red diagonal linking the vertex 1 to 3 of $\\Pi $ .", "For every vertex $i$ of $\\Pi $ we form the following $(r+1)$ pairs of coloured oriented diagonals: $[i,i+2]_P=&[i+2,i]_P:=\\lbrace [i,i+2]_R,[i+2,i]_B\\rbrace \\\\[i,i+3]_P=&[i+3,i]_P:=\\lbrace [i,i+3]_R,[i+3,i]_B\\rbrace \\\\&\\dots \\\\[i,i+r+2]_P=&[i+r+2,i]_P:=\\lbrace [i,i+r+2]_R,[i+r+2,i]_B\\rbrace .$ When $r=0$ , we assume that there are no paired diagonals.", "Moreover, let us point out that $[i,j]_P\\ne [j,i]_P$ .", "Next we define a subset of $(r+1)(n+3)$ paired, $s(n+3)$ single red and $t(n+3)$ single blue oriented diagonals of $\\Pi $ as follows: $\\Pi _{r,s,t}:=\\bigg \\lbrace &[i,i+2]_P, \\dots ,[i,i+r+2]_P, \\\\&[i,i+r+3]_R,\\dots ,[i,i+r+s+2]_R, \\\\&[i+r+3,i]_B, \\dots ,[i+r+t+2,i]_B,\\hspace{8.5359pt} \\textrm { i vertex of }\\Pi \\bigg \\rbrace .$ Once coloured oriented diagonals are paired, they stop existing as single coloured oriented diagonals in $\\Pi _{r,s,t}$ .", "Consider the subset of elements of $\\Pi _{r,s,t}$ given by $\\Pi _{r,s,t}\\vert _1:=\\lbrace [1,3]_P, \\dots ,[1,r+3]_P,[1,r+4]_R,\\dots ,[1,r+s+3]_R,[r+4,1]_B, \\dots ,[r+t+3,1]_B\\rbrace $ .", "Then on the one side elements of $\\Pi _{r,s,t}\\vert _1$ are in bijection with the vertices of $T_{r,s,t}$ .", "On the other side, elements of $\\Pi _{r,s,t}\\vert _1$ give rise to a triangulation of a region inside $\\Pi $ homotopic to a regular $(r+s+4)$ -gon, resp.", "to a $(r+t+4)$ -gon.", "In Figure REF an illustration of this situation is provided.", "Coloured oriented diagonals with the same label are identified.", "The black vertices in the figure represent the vertices of $T_{r,s,t}$ .", "The dotted lines are the edges of $T_{r,s,t}$ .", "Vertices on, and edges between, identified paired diagonals give rise to one vertex, and one edge, in $T_{r,s,t}$ .", "Figure: The coloured oriented diagonals in bijectionwith the vertices of T 2,4,1 T_{2,4,1}." ], [ "The automorphisms $\\rho $ and {{formula:9b79691d-9892-4307-9f6c-1a5faefdd79c}}", "Our next aim is to define two automorphisms: $\\rho $ and $\\tau $ , acting on the set of coloured oriented single an paired diagonals of $\\Pi $ associated to a tree $T_{r,s,t}$ .", "The first automorphism is induced from the graph automorphism of a symmetric tree, hence only defined on $\\Pi _{r,t,t}.$ The definition of the second automorphism depends on the parity of the number of sides of $\\Pi $ .", "Let $c\\in \\lbrace R,B,P\\rbrace $ .", "Then we define $\\rho :\\Pi _{r,t,t}\\rightarrow \\Pi _{r,t,t}$ as the automorphism of order two given by $\\rho \\big ([i,j]_c\\big ):={\\left\\lbrace \\begin{array}{ll}[j,i]_B & \\textrm {if } c=R, \\\\[j,i]_R &\\textrm {if } c=B,\\\\[i,j]_P &\\textrm {otherwise.}\\end{array}\\right.", "}$ Figure: The symmetry ρ\\rho .Moreover, we define the automorphism $\\tau :\\Pi _{r,s,t}\\rightarrow \\Pi _{r,s,t}$ as follows: if $s\\ne t$ , then $\\tau ([i,j]_c):=[i-1,j-1]_c.$ If $s=t$ , then $\\tau ([i,j]_c):={\\left\\lbrace \\begin{array}{ll}\\rho ^{(n+3)}\\big ([i-1,j-1]_c\\big ) &\\textrm { if }[i,j]_c\\in \\Pi _{r,t,t}\\vert _1 \\\\[i-1,j-1]_c & \\textrm { otherwise.}\\end{array}\\right.", "}$ Geometrically, the action of $\\tau $ is given by the anticlockwise rotation through $\\frac{2\\pi }{n+3}$ around the centre of $\\Pi $ , on all elements different then the diagonals in $\\Pi _{r,t,t}\\vert _1$ associated to a symmetric tree in a polygon with an odd number of sides.", "On the latter the rotation is followed by the simultaneous change of colour and orientation." ], [ "Minimal clockwise rotations", "Minimal clockwise rotations for unoriented diagonals have been introduced in [5] with the aim of modelling irreducible morphisms in the cluster category $\\mathcal {C}_{A}$ .", "Following the spirit of [5] we now define minimal rotations between diagonals of $\\Pi _{r,s,t}$ .", "Let $k,l$ be non-neighbouring vertices of $\\Pi $ and let $c\\in \\lbrace R, B, P\\rbrace $ .", "Let $\\Pi _{r,s,t}$ be the set of coloured oriented single and paired diagonals in $\\Pi $ associated to an asymmetric tree $T_{r,s,t}$ .", "Then the following three operations between diagonals in $\\Pi _{r,s,t}$ are called minimal clockwise rotation: $[k,l]_c\\rightarrow [k,l+1]_c$ and $[k,l]_c\\rightarrow [k+1,l]_c$ and $@R=0.5pc @C=0.5pc{&&[k,k+r+3]_R[rd]&&\\\\&[k,k+r+2]_P [ru][rd]& &[k+1,k+r+3]_P.", "\\\\&&[k+r+3,k]_B[ur]&&\\\\}$ Next, let $\\Pi _{r,t,t}$ be the set of coloured oriented single and paired diagonals in $\\Pi $ associated to a symmetric tree $T_{r,t,t}$ .", "Then minimal clockwise rotations are defined as before with the following adjustment: when $[k,l]_c $ , $[k,k+r+2]_P$ are in $\\tau (\\Pi _{r,t,t}\\vert _1)$ and $\\Pi $ is odd sided, then we also change simultaneously the colour an the orientation.", "In Figure REF and Figure REF illustrations of the three minimal rotations described above can be found.", "The next remark can be used to model orbit categories arising from orientations of tree graphs $T_{r_1,r_2,\\dots ,r_m}$ where one vertex has $m$ neighbours.", "Remark 3.1 There are three minimal rotations linking the three types of coloured oriented single and paired diagonals of $\\Pi _{r,r,t}$ , namely the red, the blue and the paired ones.", "Starting with oriented diagonals coloured in $m-1$ ways, one can describe the $m$ minimal rotations linking the $m$ types of coloured oriented diagonals with a similar diagram as above." ], [ "Quivers of single coloured oriented\nand paired diagonals of $\\Pi _{r,s,t}$", "Let $\\Gamma ^{n+3}_{r,s,t}$ be the quiver whose vertices are the elements of $\\Pi _{r,s,t}$ .", "An arrow between two vertices of $\\Gamma ^{n+3}_{r,s,t}$ is drawn whenever there is a minimal clockwise rotation linking them.", "No arrow is drawn otherwise.", "Concerning the shape of $\\Gamma ^{n+3}_{r,s,t}$ we remark that $\\Gamma ^{n+3}_{r,s,t}$ always lies on a cylinder, except when $\\Pi $ is odd-sided, and $s=t$ .", "Then we say that $\\Gamma ^{n+3}_{r,t,t}$ lies on a Möbius strip, since the $\\tau $ -orbits of single coloured oriented diagonals are twice as long as the $\\tau $ -orbits of paired diagonals.", "In Figure REF the quiver $(\\Gamma ^{6}_{1,1,1},\\tau )$ of coloured oriented diagonals in a hexagon, as well as the quiver $(\\Gamma ^7_{1,2,2},\\tau )$ associated to a heptagon are illustrated.", "In each quiver we indicate in the last slice the identifications occurring.", "On both quivers the action of $\\tau $ is always given by a shift to the left.", "Figure: The quiver (Γ 1,1,1 6 ,τ)(\\Gamma ^6_{1,1,1},\\tau ) corresponds to the AR-quiver ofthe orbit category 𝒟 b ( mod kD 4 )/Σ 2 \\mathcal {D}^b(\\mathrm {mod}k D_4)/\\Sigma ^2.Figure: The quiver (Γ 1,2,2 7 ,τ)(\\Gamma ^7_{1,2,2},\\tau )corresponds to the AR-quiver ofthe orbit category 𝒟 b ( mod kE 6 )/τ -1 Σ\\mathcal {D}^b(\\mathrm {mod}k E_6)/\\tau ^{-1}\\Sigma ." ], [ "Equivalences of categories", "In this section we show that the construction of $\\Gamma ^{n+3}_{r,t,t}$ and $\\Gamma ^{n+3}_{r,s,t}$ allow us to model geometrically properties of a number of additive categories.", "Let $\\rho $ be the automorphism of $T_{r,t,t}$ and let $n\\ge \\max \\lbrace r+t+1,r+s+1\\rbrace $ .", "Then we can show the following opening result.", "Lemma 4.1 The quiver $(\\Gamma ^{n+3}_{r,s,t},\\tau )$ is a stable translation quiver.", "We have to show three things.", "First, that $\\Gamma ^{n+3}_{r,s,t}$ is connected, has no loops, and is locally finite.", "Second, that for every vertex $v$ of $\\Gamma ^{n+3}_{r,s,t}$ the number of arrows going to $v$ equals the number of arrows leaving $v$ .", "Third, that the map $\\tau $ is bijective.", "All the above properties follow from the construction.", "Let us first consider the symmetric case.", "Then it is not hard to see that when $\\Pi $ is even sided, $\\Gamma ^{n+3}_{r,t,t}\|_R\\cong \\mathbb {Z} A_{r+t+1}/\\tau ^{-(n+3)}$ and $\\Gamma ^{n+3}_{r,t,t}\|_B\\cong \\mathbb {Z} A_{r+t+1}/\\tau ^{-(n+3)}$ since we consider oriented arcs.", "Thus, we deduce that $(\\Gamma ^{n+3}_{r,t,t}\|_R,\\tau \\vert _R)$ and $(\\Gamma ^{n+3}_{r,t,t}\|_B,\\tau \\vert _B)$ are stable translation quivers.", "Forming pairs of coloured oriented diagonals results in gluing these two quivers along $r$ disjoint $\\tau $ -orbits, and the above properties are preserved.", "When $\\Pi $ is odd sided one can check that the modifications in the definition of minimal clockwise rotations and in the definition of $\\tau $ are such that the resulting quiver has the claimed properties.", "Finally, it is not hard to see that also $\\Gamma ^{n+3}_{r,s,t}$ associated to an asymmetric tree is a stable translation quiver.", "Theorem 4.2 Let $\\mathcal {T}$ be an additively finite Krull-Schmidt category.", "Let $\\Gamma $ be a connected component of the AR-quiver of $\\mathcal {T}$ .", "Assume that $\\Gamma \\cong \\mathbb {Z} T_{r,s,t}/\\tau ^{-(n+3)}$ , resp.", "$\\Gamma \\cong \\mathbb {Z} T_{r,t,t}/\\tau ^{-(n+3)}\\rho $ .", "Then there is an isomorphism of stable translation quivers $\\Gamma ^{n+3}_{r,s,t}\\rightarrow \\Gamma ,$ for $\\Gamma ^{n+3}_{r,s,t}$ associated to a regular $(n+3)$ -gon.", "The claim follows from the proof of Lemma REF , since we saw that the following are isomorphisms of stable translation quivers: $\\Gamma ^{n+3}_{r,s,t}\\stackrel{\\simeq }{\\rightarrow } \\mathbb {Z} T_{r,s,t}/\\tau ^{-(n+3)}$ , $\\Gamma ^{n+3}_{r,t,t}\\stackrel{\\simeq }{\\rightarrow } \\mathbb {Z} T_{r,t,t}/\\tau ^{-(n+3)}\\rho ^{(n+3)}$ for all integers $r,s,t$ and $n$ as above." ], [ "Projections", "Our next goal is to define a translation quiver $\\Gamma ^{n+3}_{r,t}$ obtained from ${\\Gamma }^{n+3}_{r,t,t}$ associated to a symmetric tree $T_{r,t,t}$ after folding ${\\Gamma }^{n+3}_{r,t,t}$ along its central line.", "For this consider again the graph automorphism $\\rho $ induced by the map simultaneously changing colour and orientation of the diagonals in $\\Pi _{r,t,t}$ .", "Then the vertices of $\\Gamma ^{n+3}_{r,t}$ are the $\\rho $ -orbits of vertices of $\\Gamma ^{n+3}_{r,t,t}$ , i.e.", "the pairs $\\lbrace [i,i+j]_R, [i+j,i]_B \\rbrace $ , for $i,j$ in vertices of $\\Pi $ .", "The arrows in $\\Gamma ^{n+3}_{r,t}$ are always single and coincide with minimal clockwise rotation around a common vertex of $\\Pi $ linking pairs of coloured oriented diagonals.", "The translation on $\\Gamma ^{n+3}_{r,t}$ is induced from the translation in $\\Gamma ^{n+3}_{r,t,t}$ and given by the anti clockwise rotation through $\\frac{2\\pi }{n+3}$ around the center of $\\Pi $ .", "Clearly, $\\Gamma ^{n+3}_{r,t,t}\\rightarrow \\Gamma ^{n+3}_{r,t}$ is a surjective map of stable translation quivers." ], [ "Cluster categories of type $E_6$ , {{formula:6522517e-d763-4023-afe1-3901bfe56988}} and {{formula:c4defaa7-d656-479c-863f-e6443d6d5af3}}", "As a corollary of Theorem REF we obtain the geometrical modelling of cluster categories $\\mathcal {C}_{T_{r,s,t}}$ where $T_{r,s,t}$ is an arbitrary tree.", "Since the number of connected components of the AR-quiver of $\\mathcal {C}_{T_{r,t,t}}$ varies with the shape of $T_{r,t,t}$ , we proceed considering two cases.", "In this section we focus on tree graphs of Dynkin type, in Section REF the general case will be treated.", "In view of Theorem REF below let $\\mathcal {C}^{n+3}_{r,s,t}$ be the additive category generated by the mesh category of $\\Gamma ^{n+3}_{r,s,t}$ , for $r,s,t,n\\in \\mathbb {N}$ .", "Then we can show the main result of this section.", "Theorem 4.3 We have the following equivalences of additive categories $&\\mathcal {C}^7_{1,2,2}\\rightarrow \\mathcal {C}_{E_6} \\\\&\\mathcal {C}^{10}_{1,2,3}\\rightarrow \\mathcal {C}_{E_7}\\\\&\\mathcal {C}^{16}_{1,2,4}\\rightarrow \\mathcal {C}_{E_8}.$ Since the full subcategories of indecomposable objects of the orbit categories we consider are equivalent to the mesh category of their AR-quiver, we only have to check that there is an isomorphism of stable translation quivers between the AR-quiver of the various orbit categories and the quivers of coloured oriented single and paired diagonals associated to $\\Pi $ .", "This isomorphism then induces the claimed equivalences.", "To do so, the strategy will be to compare the action of $\\tau $ on $\\Gamma ^{n+3}_{r,s,t}$ with the actions of $\\tau $ and of $\\Sigma $ on $\\mathbb {Z} Q$ where $Q$ is an orientation of a simply laced Dynkin diagram, as described in Section REF .", "Below we treat the case $\\mathcal {C}_{E_7}$ , the remaining two claims can be deduced with a similar reasoning.", "From the discussion of Section REF it follows that the AR-quiver of $\\mathcal {C}_{E_7}$ is isomorphic to the quotient graph $\\mathbb {Z}E_7/ \\tau ^{-10}$ .", "On the other side, by definition $\\mathcal {C}^{10}_{1,5,5}$ is the mesh category of $\\Gamma ^{10}_{1,5,5}$ associated to a 10-gon and $\\Gamma ^{10}_{1,5,5}\\cong \\mathbb {Z}E_7/ (\\tau ^{-10})\\cong \\tau ^{-1}\\Sigma $ .", "Corollary 4.4 We also deduce the following equivalences of additive categories $&\\mathcal {C}^{r+4}_{r,0,0}\\rightarrow \\mathcal {C}^{2}_{A_{r+1}} \\\\&\\mathcal {C}^{r+s+4}_{r,s,0}\\rightarrow \\mathcal {C}^{2}_{A_{r+s+1}} \\\\&\\mathcal {C}^{r+t+4}_{r,0,t}\\rightarrow \\mathcal {C}^{2}_{A_{r+t+1}} \\\\&\\mathcal {C}^{r+5}_{r,1,1}\\rightarrow \\mathcal {D}^{b}(\\mathrm {mod}D_{r+3})/\\tau ^{-3}\\Sigma .", "\\\\$ We follow the proof of Theorem REF and observe that for the first claim we consider only paired oriented diagonals of $\\Pi $ .", "Since $[i,j]_P\\ne [j,i]_P$ and the map $\\rho $ is the identity on paired oriented diagonals, we deduce that $\\Gamma ^{r+4}_{r,0,0}$ always lies on a cylinder.", "From Lemma REF we deduce that $\\Gamma ^{r+4}_{r,0,0}\\cong \\mathbb {Z} A_{r+1}/\\tau ^{-(r+3)}$ .", "From the discussion of Section REF we deduce that $\\mathbb {Z} A_{r+1}/\\tau ^{-(r+3)}\\cong \\mathbb {Z} A_{r+1}/\\tau ^{-2}\\Sigma $ which we recognise as the AR-quiver of $\\mathcal {D}^b(\\mathrm { mod } k A_{r+1})/\\tau ^{-2}\\Sigma ^2$ .", "The second and third claim follow in a similar fashion.", "For the last claim we observe that for each vertex of $\\Pi $ the quiver $\\Gamma ^{r+5}_{r,1,1}$ has one red and one blue single coloured oriented diagonal.", "These correspond to the exceptional vertices of a Dynkin diagram of type $D_{r+3}$ .", "Remark 4.5 The equivalences of Theorem REF and Corollary REF allow us to define a shift functor $\\Sigma $ on the categories associated to $\\Pi $ induced by the shift functor $\\Sigma $ defined on the various orbit categories considered above, see also [16].", "In the following however, we will not use the triangulated structure of these categories.", "Let $[i,j]_c$ , $c\\in \\lbrace R,B,P\\rbrace $ , be a coloured oriented single or paired diagonal of a heptagon $\\Pi $ and let $(i,j)$ be the underlying unoriented diagonal.", "Then we deduce the following useful results.", "Corollary 4.6 There is a dense and full functor $\\mathcal {C}_{E_6}&\\rightarrow \\mathcal {C}_{A_{4}}\\\\[i,j]_c&\\mapsto (i,j)$ from the cluster category $\\mathcal {C}_{E_6}$ of type $E_6$ to the cluster category $\\mathcal {C}_{A_4}$ of type $A_4$ .", "Consider the projection $\\pi _1:\\Gamma ^7_{1,2,2}\\rightarrow \\Gamma ^7_{1,2}$ defined in Subsection REF .", "Let $\\Gamma $ be the stable translation quiver of unoriented diagonals of $\\Pi $ , as defined in Caldero-Chapoton-Schiffler's paper [5].", "Then there is a projection $\\pi _2:\\Gamma ^7_{1,2}\\rightarrow \\Gamma $ , which maps $\\lbrace [i,i+j]_R, [i+j,i]_B) \\rbrace $ to the unoriented diagonal $(i,i+j)$ of $\\Pi $ and pairs of arrows in $\\Gamma ^7_{1,2}$ to the one corresponding arrow in $\\Gamma $ .", "We get a surjective map of translation quivers $\\pi _2\\circ \\pi _1:\\Gamma ^7_{1,2,2}\\rightarrow \\Gamma .$ This map then induces a dense and full functor $\\mathcal {C}_{E_6}\\rightarrow \\mathcal {C}_{A_4},$ after identifying $\\mathcal {C}_{E_6}$ , resp.", "$ \\mathcal {C}_{A_4}$ , with the additive category generated by the mesh category of $\\Gamma ^7_{1,2,2}$ , resp.", "$\\Gamma $ .", "In Section we will use the functor $\\mathcal {C}_{E_6}\\rightarrow \\mathcal {C}_{A_{4}}$ to describe all cluster tilting sets of $\\mathcal {C}_{E_6}$ as configurations of coloured oriented diagonals in $\\Pi $ .", "Corollary 4.7 There is a dense and full functor $\\mathcal {C}^{r+4}_{r,0,0}&\\rightarrow \\mathcal {C}_{A_{r+1}}\\\\[i,j]_P&\\mapsto (i,j)$ from the category $\\mathcal {C}^{r+4}_{r,0,0}$ to the cluster category of type $A_{r+1}$ .", "Follows from the proof of Corollary REF ." ], [ "Coloured oriented single and paired diagonals\nand the cluster category of type $D_n$", "The aim of this section is to link our category of coloured oriented single and paired diagonals to the cluster category of type $D_n$ , denoted by $\\mathcal {C}_{D_n}$ .", "We recall that $\\mathcal {C}_{D_n}$ can be modelled geometrically in two equivalent ways.", "One approach arises categorifying the model given by Fomin-Zelevinsky in [9] defined in terms of unoriented diameters and pairs of diagonals in a regular $2n$ -gon.", "The second approach uses tagged arcs in a once punctured disc, as described by Schiffler in [19].", "We can recover both descriptions from our model.", "For the first one, we proceed as follows.", "Let $(\\Gamma _{n-3,n-1,n-1}^{2n}, \\tau )$ be as before and define the quiver $\\Gamma _{D_n}$ having as vertices the (unordered) triples of centrally symmetric diagonals of $\\Pi $ : $&\\lbrace [i,i+2]_P, [i+2+n,i+n]_R, [i+n,i+2+n]_B\\rbrace ,\\\\&\\lbrace [i,i+3]_P, [i+3+n,i+n]_R, [i+n,i+3+n]_B\\rbrace ,\\\\&\\dots \\\\&\\lbrace [i,i+n-1]_P, [i+2n-1,i+n]_R, [i+n,i+2n-1]_B\\rbrace $ together with the oriented single diagonals $[i,n+i]_R, [n+1,i]_B$ of $\\Pi $ , for $1\\le i\\le 2n.$ Notice that $[i,n+i]_R $ and $[n+1,i]_B$ are the central oriented diagonals of $\\Pi .$ The arrows of $\\Gamma _{D_n}$ are induced by the minimal clockwise rotations between diagonals of $\\Pi _{n-3,n-1,n-1}$ , similarly for the translation map.", "This allows us to obtain a surjective map of stable translation quivers $\\Gamma _{n-3,n-1,n-1}^{2n}\\rightarrow \\Gamma _{D_n}$ .", "Dropping the orientations of all coloured diagonals we also obtain the surjective map $\\Gamma _{D_n}\\rightarrow AR(\\mathcal {C}_{D_{n}})$ .", "It follows that there is a dense and full functor $\\mathcal {C}_{n-3,n-1,n-1}^{2n}\\rightarrow \\mathcal {C}_{D_{n}}$ from the mesh category of $\\Gamma _{n-3,n-1,n-1}^{2n}$ , $\\mathcal {C}_{n-3,n-1,n-1}^{2n}$ , to the cluster category $\\mathcal {C}_{D_{n}}.$ In addition, we observe that $\\Gamma _{D_n}\\cong \\mathbb {Z}D_n/(\\tau ^{-1}\\Sigma )^2\\cong AR(\\mathcal {C}^2_{D_{n}})$ , where $\\mathcal {C}^2_{D_{n}}$ is the 2-repetitive cluster category of type $D_n$ defined in Section REF .", "To recover Schiffler's model we need to use a smaller punctured polygon and allow non-contractible loops.", "Defining $\\Gamma ^{n}_{n-3,1,1}$ and $\\mathcal {C}^{n}_{n-3,1,1}$ as before we obtain an isomorphism of stable translation quivers $\\Gamma _{n-3,1,1}^{n}\\stackrel{\\cong }{\\rightarrow }AR(\\mathcal {C}_{D_{n}})$ and the desired equivalence of categories: $\\mathcal {C}^{n}_{n-3,1,1}\\stackrel{\\simeq }{\\rightarrow }\\mathcal {C}_{D_{n}}.$ In Figure REF we illustrate the quiver $\\Gamma ^4_{1,1,1}$ in a regular punctured square.", "Figure: The quiver Γ 1,1,1 4 ≅AR(𝒞 D 4 )\\Gamma ^4_{1,1,1}\\cong AR(\\mathcal {C}_{D_{4}}) in a square." ], [ "Cluster categories associated to trees not of Dynkin type", "When $T_{r,s,t}$ is not an orientation of a simply laced Dynkin graph the AR-quiver of the cluster category $\\mathcal {C}_{T_{r,s,t}}$ splits into various connected components.", "Our next goal is to obtain an isomorphism between the irregular (transjecctive) component of the AR-quiver of $\\mathcal {C}_{T_{r,s,t}}$ and the quiver of coloured oriented single and paired diagonals.", "For this we extend the construction of Section REF and consider and infinite-sided polygon $\\Pi ^\\infty $ .", "Next, let $\\Pi ^\\infty _{r,s,t}$ be the subset of all coloured oriented single and paired diagonals of $\\Pi ^\\infty $ consisting of $r+1$ paired, $s$ red and $t$ blue single coloured oriented diagonals for every vertex of $\\Pi ^\\infty $ : $\\Pi _{r,s,t}:=\\bigg \\lbrace &[i,i+2]_P, \\dots ,[i,i+r+2]_P, \\\\&[i,i+r+3]_R,\\dots ,[i,i+r+s+2]_R, \\\\&[i+r+3,i]_B, \\dots ,[i+r+t+2,i]_B,\\hspace{8.5359pt} \\textrm { i vertex of }\\Pi ^\\infty \\bigg \\rbrace .$ Then we extend the definition of minimal clockwise rotations of Section REF to this setting.", "Thus we obtain a quiver $\\Gamma ^\\infty _{r,s,t}$ , whose vertices are the elements of $\\Pi ^\\infty _{r,s,t}$ and where we link two vertices with an arrow when there is a minimal clockwise rotation between them.", "Moreover, we define $\\tau $ on the elements of $\\Pi ^\\infty _{r,s,t}$ as the anticlockwise rotation around the center of $\\Pi ^\\infty $ induced by the rotation though $\\frac{2\\pi }{n+3}$ in an $(n+3)$ -gon $\\Pi $ letting $n\\rightarrow \\infty $ .", "In this way we turn $(\\Gamma ^\\infty _{r,s,t},\\tau )$ into a stable translation quiver.", "To state the next result, let $\\mathcal {P}$ be the full subcategory of $\\mathcal {C}_{T_{r,s,t}}$ , consisting of $\\tau $ -shifts of indecomposable projective objects in $\\mathcal {C}_{T_{r,s,t}}$ .", "Moreover, denote the additive category generated by the mesh category of $(\\Gamma ^\\infty _{r,s,t},\\tau )$ by $\\mathcal {C}^\\infty _{r,s,t}$ .", "Then we can show the following result.", "Proposition 4.8 The functor $\\varphi :\\mathcal {C}^\\infty _{r,s,t}\\rightarrow \\mathcal {P}$ is an equivalence of additive categories.", "First we observe that $\\Gamma ^\\infty _{r,s,t}\\cong \\mathbb {Z}T_{r,s,t}$ .", "Next, let $\\Pi ^\\infty _{r,s,t}\\vert _1$ be the subset of $\\Pi ^\\infty _{r,s,t}$ consisting of the $r+s+t+1$ single and paired diagonals: $\\lbrace [1,3]_P, \\dots ,[1,r+3]_P,[1,r+4]_R,\\dots ,[1,r+s+3]_R,[r+4,1]_B, \\dots ,[r+t+3,1]_B \\rbrace $ .", "Then, we observe that since $\\Gamma ^\\infty _{r,s,t}$ is a connected stable translation quiver $\\varphi $ induces an injective map between $\\mathrm {Hom}_{C^\\infty _{r,s,t}}(\\tau (D_j),D_i)$ , for $D_i,D_j\\in \\Pi ^\\infty _{r,s,t}$ and $\\mathrm {Hom}_{C_{T_{r,s,t}}}(I_j,\\Sigma (P_i))$ between the indecomposable injective objects and $\\Sigma $ of the projective objects in $\\mathcal {C}_{T_{r,s,t}}$ .", "Thus $\\varphi $ is full.", "Moreover, it is not hard to see that $\\varphi $ is dense and faithful, thus $\\varphi $ is indeed an equivalence of additive categories." ], [ "Combinatorics of the cluster category $\\mathcal {C}_{E_6}$", "Our next aim is to describe the combinatorics of the cluster category $\\mathcal {C}_{E_6}\\cong \\mathcal {C}_{1,2,2}$ inside a heptagon.", "Throughout the chapter let $\\Pi $ be a regular heptagon and let $(\\Gamma ,\\tau ):=(\\Gamma ^7_{1,2,2},\\tau )$ .", "Moreover, let $\\Gamma _{\\Pi }$ be the stable translation quiver having as vertices the unoriented diagonals of $\\Pi $ and arrows given by minimal clockwise rotations, see [5] for details." ], [ "Extension spaces in $\\mathcal {C}_{E_6}$", "The support of $\\mathrm {Hom}(\\tau ^{-1}X,-)$ in $\\mathcal {C}_Q$ is called the front $\\mathrm {Ext}$ -hammock of $X$.", "The support of $\\mathrm {Hom}(-,\\tau X)$ in $\\mathcal {C}_Q$ is called the back $\\mathrm {Ext}$ -hammock of $X$.", "These hammocks can be deduced from the AR-quiver using mesh relations, or using starting and ending functions, see [3].", "For AR-quivers isomorphic to $\\mathbb {Z} Q$ with $Q$ an orientation of a Dynkin graph, the support of $\\mathrm {Hom}(-,-)$ has been described in detail in [1].", "Identify again $\\mathcal {C}_{E_6}$ , resp.", "$ \\mathcal {C}_{A_4}$ with the additive categories generated by the mesh categories of $\\Gamma $ , resp.", "$\\Gamma _{\\Pi }$ .", "In the sequel we view the hammocks inside $\\Gamma $ or $\\Gamma _{\\Pi }$ .", "Moreover, let $\\mathrm {Ext}_{\\Pi }^1(D_X,D_Y):=\\mathrm {Ext}_{\\mathcal {C}_{E_6}}^1(X,Y)$ , for coloured oriented diagonals $D_X,$ $D_Y$ in $\\Pi $ and indecomposable objects $X$ and $Y$ in $\\mathcal {C}_{E_6}$ corresponding to $D_X$ and $D_Y$ by the equivalence of Theorem REF .", "Remark 5.1 As $\\mathcal {C}_{E_6}$ is 2 Calabi-Yau: $D\\mathrm {Ext}_{\\mathcal {C}_{E_6}}^1(X,Y)\\cong \\mathrm {Ext}^1_{\\mathcal {C}_{E_6}}(X,Y), $ for all objects $X,Y$ in $\\mathcal {C}_{E_6}$ .", "Therefore, the back and front $\\mathrm {Ext}$ -hammocks in $\\mathcal {C}_{E_6}$ coincide for all objects.", "On the other hand, in $\\mathcal {C}^2_{A_4}$ the back and front hammocks are disjoint, as the category is not 2-Calabi-Yau." ], [ "Lift of hammocks", "Consider again the projection $\\widetilde{\\pi }:=\\pi _2\\circ \\pi _1:\\Gamma \\rightarrow \\Gamma _{\\Pi }$ defined by $D_{(i,j)}:=\\widetilde{\\pi }(D_X)$ as in Corollary REF .", "For each $D_X$ we define two connected sub-quivers of $\\Gamma $ , $I_1(D_X)$ and $I_2(D_X)$ , as follows.", "The vertices of both $I_1(D_X)$ and $I_2(D_X)$ , are the vertices of $\\Gamma $ in the $\\mathrm {Ext}$ -hammock of $D_X$ in $\\Gamma $ and in the preimage under $\\widetilde{\\pi }$ of the $\\mathrm {Ext}$ -hammock of $D_{(i,j)}$ in $\\Gamma _{\\Pi }$ .", "The arrows of $I_1(D_X)$ and $I_2(D_X)$ coincide with the arrows of $\\Gamma $ .", "Then $I_1(D_X)$ contains the vertex $\\tau ^{-1}(D_X)$ and will be called the front crossing of $D_X$ , $I_2(D_X)$ contains $\\tau (D_X)$ and will be called the back crossing of $D_X$ .", "Note that for all $D_X$ , the sub-quivers $I_1(D_X)$ and $I_2(D_X)$ are disjoint.", "In addition, all coloured oriented diagonals in $\\Pi $ crossing $D_X$ in an interior point of $D_X$ are vertices of $I_1(D_X)\\cup \\rho (I_1(D_X))$ and $I_2(D_X)\\cup \\rho (I_2(D_X))$ .", "See Figure REF , were the vertices of inside the front and back crossings of $D_X$ are in heptagons with bold boundary, for $D_X$ a coloured oriented diagonal in the first slice of $\\Gamma $ .", "Figure: The Ext \\mathrm {Ext}-hammocks of the diagonals withvertices of I 1 I_1 and I 2 I_2 represented inheptagons with bold boundary.In the next result, we assume that $D_X$ is a coloured oriented diagonal of $\\Pi $ in the first slice of $\\Gamma $ .", "This assumption can be dropped using $\\tau $ -shifts, or renumbering the vertices of $\\Pi $ .", "Moreover, we write $\\partial \\Pi $ to indicate the boundary of $\\Pi $ , and for two coloured oriented diagonals $D_X$ and $D_Y$ we say that $D_Y$ enters the smaller region bounded by $D_{X}$ and $\\partial \\Pi $ if the arrow head of $D_Y$ goes to a vertex of $\\partial \\Pi $ inside the region and different from the vertices joined by $D_{X}$ .", "Proposition 5.2 Let $D_{X}, D_{Y}$ be coloured oriented diagonals of $\\Pi .$ Assume $D_X$ is in the first slice of $\\Gamma $ , and that $D_X$ crosses $D_{Y}$ .", "If $D_{X}$ is a paired diagonal, then $\\mathrm {dim}_k(\\mathrm {Ext}^1_{\\Pi }(D_X,D_Y))=1$ .", "If $D_{X}$ is a single diagonal, and $D_{Y}$ enters the smaller region bounded by $D_{X}$ and $\\partial \\Pi $ , then $\\mathrm {dim}_k(\\mathrm {Ext}^1_{\\Pi }(D_X,D_Y))=1$ .", "If $D_{X}$ is paired, $I_1(D_X)$ coincides with $I_1(D_X)\\cup \\rho (I_1(D_X))$ and $I_2(D_X)$ coincides with $I_2(D_X)\\cup \\rho (I_2(D_X))$ , thus the vertices of $I_1(D_X)$ and $I_2(D_X)$ are all the oriented coloured diagonals of $\\Pi $ crossing $D_{X}$ .", "If $D_{X}$ is a single coloured diagonal of $\\Pi $ , we need to distinguish between the diagonals inside $I_i(D_X)$ and $\\rho (I_i(D_X))$ , $i=1,2$ .", "Then we observe that the coloured oriented diagonals in $I_1(D_X)$ and $I_2(D_X)$ are precisely the ones satisfying the assumptions of the proposition." ], [ "Curves of oriented coloured diagonals", "The aim of this section is to divide the $\\mathrm {Ext}$ -hammocks in $\\mathcal {C}_{E_6}$ into curves.", "The reason why we do this is because for each coloured oriented diagonal $D_X$ we want to find a uniform geometric description of the elements inside the $\\mathrm {Ext}$ -hammock of $D_X$ .", "Since the hammocks in $\\mathcal {C}_{E_6}$ are very big, this goal seems hopeless.", "However, dividing the $\\mathrm {Ext}$ -hammock of $D_X$ into smaller sets, allows us to describe the elements of each such set in geometric terms.", "We will call these sets curves.", "Let $X$ be an indecomposable object of $\\mathcal {C}_{E_6}$ and let $D_X$ be the corresponding coloured oriented diagonal viewed as a vertex of $\\Gamma $ .", "For $r\\in \\lbrace 2,4\\rbrace $ , the curves $C_1(D_X),\\ldots ,C_r(D_X)$ of $D_X$ in $\\Gamma $ are $r$ collections of oriented coloured diagonals having non-vanishing extensions with $D_X$ .", "Each collection $C_i(D_X)$ has the shape of a curve in $\\Gamma $ .", "We begin defining the curves of $D_X$ , for $D_X$ in the first slice of $\\Gamma $ .", "For all other vertices $D_X$ of $\\Gamma $ , curves can be defined from the previous ones by $\\tau $ -shifts.", "The first curve of $[1,6]_R$ , denoted by $C_1([1,6]_R)$ , is defined as follows: $C_1([1,6]_R):=&\\lbrace [7,2+i] _c,\\, 0\\le i\\le 3,\\, c\\in \\lbrace R,P\\rbrace \\rbrace \\\\&\\cup \\lbrace [5,7+i] _c,\\, 0\\le i\\le 3,\\,c\\in \\lbrace R,P\\rbrace \\rbrace \\\\&\\cup \\lbrace [6,3]_R\\rbrace .$ The second curve of $[1,6]_R$ , denoted by $C_2([1,6]_R)$ , is defined as follows: $C_2([1,6]_R):=&\\lbrace [5-i,7] _c,\\, 0\\le i\\le 3,\\, c\\in \\lbrace B,P\\rbrace \\rbrace \\\\&\\cup \\lbrace [2,7+i] _c,\\, 0\\le i\\le 3,\\,c\\in \\lbrace B,P\\rbrace \\rbrace \\\\&\\cup \\lbrace [1,4]_B\\rbrace .$ By definition $C_1([1,6]_R)$ is obtained by the sequence of minimal clockwise rotations around the vertices 7, 5, 3 of $\\Pi $ starting in $\\tau ^{-1}([1,6]_R)=[2,7]_R$ and ending with $[6,3]_R$ .", "Dually, $C_2([1,6]_R)$ is obtained by a sequence of minimal anticlockwise rotations around the vertices 7, 2, 4 starting in $\\tau ([1,6]_R)=[5,7]_B$ and ending in $[1,4]_B$ .", "By construction the $\\mathrm {Ext}$ -hammock of $[1,6]_R$ is $C_1([1,6]_R)\\cup C_2([1,6]_R)$ .", "In Figure REF (a) the elements of $C_1([1,6]_R)$ are drawn in the upper half of $\\Gamma $ , while the elements of $C_2([1,6]_R)$ are in the lower half.", "Next, we are going to associate four curves to $[1,5]_R$ .", "The first curve of $[1,5]_R$ , $C_1([1,5]_R)$ , is the set containing coloured oriented diagonals of $\\Pi $ obtained by a sequence of minimal clockwise rotations around the vertices 6, 4, 2 starting in $\\tau ^{-1}([1,5]_R)$ and ending in $[5,2]_R$ .", "The third curve $C_3([1,5]_R)$ is obtained by a sequence of minimal anticlockwise rotations around the vertices 7, 2, 4 starting with $\\tau ([1,5]_R)$ and ending in $[1,4]_B$ .", "Moreover, $C_2([1,5]_R)$ coincides with $C_1([1,6]_R)$ , and $C_4([1,5]_R)$ coincides with $C_2([1,6]_R)$ .", "For $D_X\\in \\lbrace [1,6]_R,[,1,5]_R\\rbrace $ , the curves of $C_1(\\rho (D_X)),\\ldots ,C_r(\\rho (D_X))$ of $\\rho (D_X)$ are defined by $\\rho (C_1(D_X)),\\ldots ,\\rho (C_R(D_X))$ , $r\\in \\lbrace 2,4\\rbrace $ .", "We are left with defining the curves of the paired diagonals $[1,3]_P$ and $[1,4]_P$ .", "For $[1,3]_P$ we have $C_1([1,3]_P)$ given by the set containing both single and paired coloured oriented diagonals obtained by a sequence of minimal rotations in the clockwise order around the vertices 2, 7, 5 starting in $\\tau ^{-1}([1,3]_P)$ and ending in $[5,1]_P$ .", "Similarly $C_2([1,3]_P)$ is obtained by a sequence of minimal anticlockwise rotations around the vertices 2, 4, 5 starting in $\\tau ([1,3]_P)$ and ending in $[6,2]_P$ .", "Next, $C_1([1,4]_P)$ is obtained by a sequence of of minimal clockwise rotations starting in $\\tau ^{-1}([1,4]_P) $ and ending in $[5,1]_P$ .", "$C_3([1,4]_P)$ is obtained rotating in the anticlockwise order $\\tau ([1,4]_P)$ to $[4,7]_P$ .", "Finally, $C_2([1,4]_P)=\\tau ^{-1}(C_1([1,3]_P))$ and $C_4([1,4]_P)=C_2([1,3]_P)$ .", "In Figure REF (a)-(f) we represent the various curves of $D_X$ , for$D_X$ be in the first (and last) slice of $\\Gamma $ .", "The numbers 1, 2, 3, 4 indicate the starting term of the curves $C_1(D_X),\\ldots ,C_4(D_X)$ and the colours of the heptagons indicate the curves they intersect.", "Figure: Decomposition into curves of the Ext-hammocks of the vertices in thefirst slice of Γ\\Gamma ." ], [ "Intersections of curves", "Proposition 5.3 Let $D_X,D_Y$ be vertices of $\\Gamma $ .", "Let $C_1(D_X),\\ldots ,C_r(D_X)$ be the curves of $D_X$ , $r \\in \\lbrace 2,4\\rbrace $ .", "Then $\\mathrm {dim}_k(\\mathrm {Ext}^1_{\\Pi }(D_X,D_Y))$ is equal to the number of curves of $D_X$ intersecting with $D_Y$ in $\\Gamma $ (0 up to 3).", "First, the $\\mathrm {Ext}$ -hammocks in the AR-quiver of $\\mathcal {C}_{E_6}$ are invariant under $\\tau $ -shifts.", "After changing the image of the projective objects of $\\mathrm {mod} k E_6$ in the equivalence of Theorem REF we can assume that $X$ or $Y$ corresponds to a diagonal in the first slice of $\\Gamma $ .", "By remark REF we can treat the cases where $D_X$ , or $D_Y$ belongs to the first slice of $\\Gamma $ in the same way.", "Second, the curves of $D_X$ are by construction such that their intersection points coincide with the vertices $D_Y$ in $\\Gamma $ for which $\\mathrm {dim}_k(\\mathrm {Ext}^1_{\\Pi }(D_X,D_Y))\\ge 1$ .", "From the definition of morphisms in the mesh category of a stable translation quiver, it follows that the number of curves intersecting in $D_Y$ is the dimension of $\\mathrm {Ext}^1_{\\mathcal {C}_{E_6}}(D_X,D_Y).$ Let again $D_X$ be in the first slice of $\\Gamma $ .", "In Figure REF (a)-(f) the dimension of the space $\\mathrm {dim}_k(\\mathrm {Ext}^1_{\\Pi }(D_X,D_Y))$ is expressed by the numbers of colours filling the heptagon containing $D_Y$ .", "If $D_Y$ is in a white heptagon of $\\Gamma $ then $\\mathrm {dim}_k(\\mathrm {Ext}^1_{\\Pi }(D_X,D_Y))=0$ .", "More precisely, the two curves represented in Figure REF (a) and (f) never intersect, and $\\mathrm {dim}_k(\\mathrm {Ext}^1_{\\Pi }(D_X,D_Y))=1$ for $D_Y$ in a coloured heptagon, and $D_X$ in the first slice of $\\Gamma $ .", "In Figure REF (b) and (e) the curve $C_1(D_X)$ intersects $C_2(D_X)$ in two vertices.", "Similarly, for $C_3(D_X)$ and $C_4(D_X)$ .", "The curve $C_2(D_X)$ intersects $C_4(D_X)$ only once.", "The five heptagons where two curves meet have two colours (boundary and interior of the heptagon).", "Then $\\mathrm {dim}_k(\\mathrm {Ext}_{\\Pi }(D_X,D_Y))=2$ for $D_Y$ corresponding to one of these heptagons.", "In Figure REF (c) there are two heptagons where three curves meet.", "They are drawn with three colours, and hence $\\mathrm {dim}_k(\\mathrm {Ext}_{\\Pi }(D_X,D_Y))=3$ for $D_Y$ corresponding to one of these two.", "Moreover, in nine heptagons two curves meet, and they are drawn in two colours.", "Finally, in Figure REF (d) two curves are drawn, and they intersect only in one vertex of $\\Gamma $ ." ], [ "Cluster tilting objects", "Let $Q$ be an orientation of a simply-laced Dynkin graph with $n$ vertices.", "Let $\\mathcal {T}=\\lbrace T_1,T_2,\\dots ,T_n\\rbrace $ be a set of pairwise non isomorphic indecomposable objects of $\\mathcal {C}_Q$ .", "If $\\mathrm {Ext}_{\\mathcal {C}_{Q}}^1(T_i, T_j)=0$ for all $T_i,T_j\\in \\mathcal {T}$ , then one says that $\\mathcal {T}$ is a cluster tilting set of $\\mathcal {C}_{Q}$ .", "A cluster tilting object in $\\mathcal {C}_{Q}$ is the direct sum of all objects of a cluster tilting set in $\\mathcal {C}_Q$ .", "Observe that knowing a cluster tilting objects allows to determines a cluster tilting set and viceversa.", "Moreover, given a cluster tilting set $\\mathcal {T}$ , one says that $\\overline{T}=\\oplus _{j\\ne i}T_j$ , $T_j\\in \\mathcal {T}$ is an almost complete cluster tilting object if there is an indecomposable object $T_i^$ in $\\mathcal {C}_Q$ such that $\\overline{T}\\oplus T_i^$ is a cluster tilting object of $\\mathcal {C}_Q$ .", "The object $T_i^$ is called the complement of $T_i$ .", "The mutation at $i$ of a cluster tilting object $\\mathcal {T}$ in $\\mathcal {C}_Q$ , for $1\\le i\\le n$ , is the operation which replaces the indecomposable summand $T_i$ in $\\oplus _{j=1}^nT_j$ with the complement $T_i^$ of $T_i$ in $\\overline{T}=\\oplus _{j\\ne i}T_j.$ The statements in the next Theorem are shown in [3].", "Theorem 5.4 Let $T$ be a cluster tilting object in $\\mathcal {C}_{Q}$ .", "each almost complete cluster tilting object $\\overline{T}$ in $\\mathcal {C}_{Q}$ has exactly two complements, $T$ and $T^$ .", "If $T$ and $T^$ are complements of $\\overline{T}$ then $\\mathrm {dim}_k(\\mathrm {Ext}_{\\mathcal {C}_{Q}}^1(T,T^))=1.$ On the other side, if $\\mathrm {dim}_k(\\mathrm {Ext}_{\\mathcal {C}_{Q}}^1(T,T^))=1$ , then there is an almost complete cluster tilting object $\\overline{T}$ such that $T$ and $T^$ are complements of $\\overline{T}$ .", "After Proposition 3.8 in [9] and [3] we know that there are 833 cluster tilting sets, hence cluster tilting objects, in $\\mathcal {C}_{E_6}$ .", "From the work of Caldero-Chapoton-Schiffler, see [5], we know that cluster tilting objects of $\\mathcal {C}_{A_n}$ are in bijection with the formal direct sums of diagonals belonging to a maximal collection of non-crossing diagonals in a regular $(n+3)$ -gon.", "In this context mutations corresponds to flips of diagonals.", "More precisely, a flip replaces a diagonal $D_i$ in a given triangulation $\\Delta $ with the unique other diagonal $D_i^$ crossing $D_i$ and completing $\\Delta \\backslash D_i$ to a new triangulation of the regular $(n+3)$ -gon." ], [ "First fundamental family of\ncluster configurations of $\\Pi $", "Our next aim is to describe cluster tilting sets of $\\mathcal {C}_{E_6}$ as configurations of single and paired coloured oriented diagonals in $\\Pi .$ Definition 5.5 A cluster configuration is a family of pairwise different coloured oriented diagonals of $\\Pi $ , $\\mathcal {T}=\\lbrace D_1,D_2,\\ldots ,D_6\\rbrace $ , with the property that $\\mathrm {Ext}_{\\Pi }^1(D_i, D_j)=0$ for all $D_i,D_j\\in \\mathcal {T}$ .", "A coloured oriented diagonal $D_i^$ is called complement of $D_i$ in $\\mathcal {T} $ if $D_i^\\ne D_i$ and $\\mathcal {T}^{\\prime }$ obtained from $\\mathcal {T} $ after replacing ${D_i}$ by ${D_i^}$ is a cluster configuration of $\\Pi $ .", "Consider two heptagons, and a long paired diagonal $L_P:=[i,i+3]_P$ , $i\\in \\mathbb {Z}/7\\mathbb {Z}$ of $\\Pi $ .", "Our next goal is to complete $L_P$ to a set of coloured oriented diagonals inside the two heptagons, giving rise to a cluster configuration of $\\Pi $ .", "For this we remark that $L_P$ divides each $\\Pi $ into the quadrilateral $\\Pi _4$ with boundary vertices $\\lbrace i, i+1,i+2,i+3\\rbrace $ , and the pentagon $\\Pi _5$ with boundary vertices $\\lbrace i, i+3,i+4,i+5,i+6\\rbrace $ , $i\\in \\mathbb {Z}/7\\mathbb {Z}$ .", "In Lemma REF below we will see that triangulating $\\Pi _4$ with a short paired diagonal, and each $\\Pi _5$ with single diagonals of the appropriate colour gives rise to cluster configurations.", "Lemma 5.6 Let $L_P:=[i,i+3]_P$ , $i\\in \\mathbb {Z}/7\\mathbb {Z}$ .", "For $i\\ne 1$ , triangulating each $\\Pi _5$ with single diagonals of the same colour, and $\\Pi _4$ with a short paired diagonal gives a cluster configuration $\\mathcal {T}_{L_P}$ of $\\Pi $ .", "All cluster configurations of $\\Pi $ containing $[j,j+3]_P$ arise as $\\tau ^k(\\mathcal {T}_{L_P})$ , $1\\le j,k\\le 7$ .", "Since $i\\ne 1$ we can assume that the region in $\\Gamma $ outside the $\\mathrm {Ext}$ -hammock of $L_P$ has only blue diagonals below $L_P$ and only red diagonals above $L_P$ .", "Observes that the diagonals outside the $\\mathrm {Ext}$ -hammock are precisely the diagonals involved in triangulations of the two copies of $\\Pi _5$ and $\\Pi _4$ .", "Then chose a short paired diagonal $S_P$ triangulating $\\Pi _4$ with a short paired diagonal.", "Then triangulating a copy of $\\Pi _5$ with only single red diagonals, and triangulating the second copy of $\\Pi _5$ with only single blue diagonals yields a cluster configuration.", "With Proposition REF we deduce that the arcs obtained in this way have no extension in each region above and below $L$ in $\\Gamma $ .", "One can then check that the $\\mathrm {Ext}$ -hammocks in one region do not pass through the other region, nor though $S_P$ .", "Thus, to each red triangulation one can choose a blue triangulations of $\\Pi _5$ , and all choices are possible.", "Similarly, one can complete $\\lbrace L_P, S_P^\\rbrace $ to a cluster configuration, where $S_P^$ is the other short paired diagonal triangulating $\\Pi _4$ .", "Notice that there are no other possibilities to complete $L_P$ to a cluster configuration of $\\Pi $ .", "Next, there are 7 choices for $L_P$ in $\\Gamma $ .", "For each choice of $L_p$ the associated cluster configurations are obtained from the previous by rotation though $\\tau $ .", "Adjustment of the colours-orientations of the single diagonals triangulating $\\Pi _5$ are needed if $L_P$ is the first slice of $\\Gamma $ .", "Notice that the two triangulations of $\\Pi _5$ can be different, and the color is uniquely determined by the position of $L_P$ in $\\Pi $ , resp.", "in $\\Gamma $ .", "In the following we call the cluster configurations given by a long paired diagonal and coloured oriented single and paired diagonals triangulating two copies of $\\Pi _5$ and $\\Pi _4$ , as describe in the first part of Lemma REF , the first fundamental family of cluster configurations.", "We denote this family by $\\mathcal {F}_1$ ." ], [ "Second fundamental family of cluster configurations of $\\Pi $", "We saw in Lemma REF that many cluster configurations correspond to two triangulations of $\\Pi $ .", "Our next goal is to define a second family of cluster configurations describing the remaining cluster tilting set of $\\mathcal {C}_{E_6}$ .", "For this the following general observation is needed.", "For $i\\in \\mathbb {Z}/7\\mathbb {Z}$ , consider the long single red diagonal $L=[i,i+4]_R$ of $\\Pi $ .", "Then $L$ divides $\\Pi $ into the quadrilateral $\\Pi _4:=\\lbrace i+4,i+5,i+6,i\\rbrace $ , and the pentagon $\\Pi _5:=\\lbrace i, i+1,i+2,i+3,i+4\\rbrace $ .", "Let $\\mathcal {T}_L$ be a cluster configuration of $\\Pi $ containing $L$ .", "Then $\\mathcal {T}_L$ necessarily also contains one of the two short single diagonals triangulating $\\Pi _4$ , neighbouring $L$ in $\\Gamma .$ Similarly for $\\rho (L)=[i+4,i]_B$ .", "More precisely, Lemma 5.7 Let $i\\in \\mathbb {Z}/7\\mathbb {Z}$ , $L=[i,i+4]_R$ in $\\Pi $ , and $\\mathcal {T}_L$ be a cluster configuration containing $L$ .", "If $i\\ne 1$ , exactly one of $\\lbrace [i,i+5]_R,[i+6,i+4]_R\\rbrace $ is in $\\mathcal {T}_L$ .", "If $i=1$ , exactly one of $\\lbrace [i,i+5]_R,\\rho ([i+6,i+4]_R)\\rbrace $ is in $\\mathcal {T}_L$ .", "Similarly for $\\rho (L)$ .", "Moreover, in each case the two diagonals are complements to each other.", "Let $i\\ne 1$ and consider the $\\mathrm {Ext}$ -hammock of $L$ in $\\Gamma $ .", "Since $L$ is not in the first slice of $\\Gamma $ the diagonals triangulating $\\Pi _4$ have the same color as $L$ .", "Then one can check that all $\\mathrm {Ext}$ -hammocks of objects outside the $\\mathrm {Ext}$ -hammock of $L$ , which are different from $[i,i-2]_R$ and $[i-1,i-3]_R$ , never contain single diagonals inside the quadrilateral $\\Pi _4$ in $\\Pi .$ Thus, by maximality we deduce that all cluster tilting sets containing $L$ necessarily also contain one of the diagonals inside $\\Pi _4$ .", "Taking one diagonal triangulating $\\Pi _4$ rules out the other, thus the two single diagonals triangulating $\\Pi _4$ are complements to each other.", "For $i=1$ , $L$ is in the first slice of $\\Gamma $ .", "Then one can proceed as before adjusting the colour of the diagonal triangulating $\\Pi _4$ .", "In view of the next result, we point out that the short single diagonals of Lemma REF , triangulating $\\Pi _4$ and neighbouring $L$ in $\\Gamma $ , are displayed in filled light grey heptagons in Figure REF (a)-(n).", "Lemma 5.8 Every six-tuple of diagonals of Figure REF (a)-(n) determines a cluster configuration of $\\Pi $ .", "For each choice of a short single diagonal of Lemma REF triangulating $\\Pi _4$ and neighbouring $L$ in $\\Gamma $ , the claim can be verified by checking that the diagonals in the highlighted heptagons have no extension among each other.", "In the following, we refer to the collection of cluster configurations of Figure REF (a)-(n) as the second fundamental family of cluster configurations of $\\Pi $ , and we denote this family by $\\mathcal {F}_2$ .", "Figure: The second fundamental family of cluster configurations of Π\\Pi ." ], [ "Symmetries in $\\Pi $ leading to cluster\nconfigurations", "We determine two symmetries in $\\Pi $ leading to cluster configurations.", "One symmetry simply switches colours and orientations of the coloured oriented diagonals of a given cluster configuration.", "The second one arises from a left-right symmetry of $\\Gamma $ , and corresponds to a reflection in $\\Pi $ .", "There are two reasons why these symmetries are important.", "First, using these symmetries we can deduce all cluster configurations starting from the sets in $\\mathcal {F}_2$ .", "Second, knowing how a cluster configuration behaves under mutation, allows to understand how the symmetric ones behave.", "Let $c\\in \\lbrace R,B\\rbrace $ .", "For $i\\in \\mathbb {Z}/7\\mathbb {Z}$ let $h_i$ be the line in $\\Pi $ passing through $i$ and the middle point of $i+3,i+4$ .", "On all coloured oriented diagonals different then $[i\\pm 1,i\\mp 1]_c$ , let $\\sigma _i:\\Pi \\rightarrow \\Pi $ be the reflection in $\\Pi $ along $h_i$ followed by a switch of orientation.", "Otherwise, $\\sigma _i([i\\pm 1,i\\mp 1]_c):=\\rho ([i\\pm 1,i\\mp 1]_c).$ In Figure REF we illustrate on the left the cluster configuration (g) and on the right we show the cluster configuration obtained after applying $\\sigma _6$ to it.", "Figure: σ\\sigma -symmetric cluster configurations of Π\\Pi .Lemma 5.9 Let $\\mathcal {T}$ be a cluster configuration belonging to $\\mathcal {F}_2$ .", "Then $\\rho (\\mathcal {T})$ , and $\\sigma _6(\\mathcal {T})$ are also cluster configurations.", "Apply the map $\\rho $ , resp.", "$\\sigma _6$ to the cluster configuration of Lemma REF .", "Because of the shape of the $\\mathrm {Ext}$ -hammocks one indeed produces cluster configurations.", "Notice that the set $\\rho (\\mathcal {T})$ of a cluster configuration $\\mathcal {T}$ is an elements of the $\\tau $ -orbit of $\\mathcal {T}$ , while $\\sigma _i(\\mathcal {T})$ does not belong to any $\\tau $ -orbit of a cluster configuration of $\\mathcal {F}_2$ (nor of $\\mathcal {F}_1$ )." ], [ "Classification of cluster tilting sets of $\\mathcal {C}_{E_6}$ ", "From [9] and [3] we know that there are 833 cluster tilting sets in $\\mathcal {C}_{E_6}$ .", "In the next result we give a complete geometric classification of all cluster tilting sets in $\\mathcal {C}_{E_6}$ in terms of cluster configurations of $\\Pi $ .", "Theorem 5.10 In $\\Pi $ 350 different cluster configurations have one long paired diagonal, and they arise from $\\mathcal {F}_1$ through $\\tau $ .", "483 other cluster configurations arise from $\\mathcal {F}_2$ through $\\sigma $ and $\\tau $ .", "Corollary 5.11 In $\\Pi $ : 224 cluster configurations have precisely one short paired diagonal, 175 cluster configurations have precisely two short paired diagonals, 84 cluster configurations have no paired diagonals of $\\Pi $ .", "All these cluster configurations are different.", "The first part of the claim follows from Lemma REF .", "In fact, we saw that for each long paired diagonal $L_p$ there are 25 ways to triangulate one of the two pentagons $\\Pi _5$ with single coloured diagonals.", "Moreover, there are two ways to triangulate $\\Pi _4$ with short paired diagonals.", "Thus, each $L_p$ gives rise to 50 different cluster configurations.", "Since there are 7 choices for $L_p$ in $\\Pi $ , the first claim follows.", "For the second part of the claim the idea is to consider different cases, depending on the number of short paired diagonals leading to cluster configurations.", "First case: the only paired diagonal of $\\mathcal {T}$ is a short one.", "Then $\\mathcal {T}$ arises from the collection highlighted in (a),(b) or (c) in Figure REF , up to $\\tau $ -shifts and the $\\sigma $ -symmetry of Lemma REF .", "Moreover, after Lemma REF for each coloured oriented diagonal $[i,i-3]_c$ , $c\\in \\lbrace R,B\\rbrace $ in $\\mathcal {T}$ there two possible choices of neighbouring short single diagonals in $\\mathcal {T}$ .", "Consequently, up to $\\tau $ -shifts, the there are 4 different cluster configurations arising from a collection of diagonals as in (a).", "Similarly for (b).", "The collection in (c) gives rise to 8 different cluster configurations up to $\\tau $ -shifts, as there are 4 choices for short single diagonals, and further 4 arise by taking the $\\sigma $ -symmetric case.", "Summing up, the cluster configurations in (a), (b), (c) give rise to 224 different cluster configurations.", "Second case: $\\mathcal {T}$ has exactly two short paired diagonals.", "Then one distinguishes further into (d),(e) and (f) which have single coloured oriented diagonals of the form $[i,i+3]_c$ , $c\\in \\lbrace R,B\\rbrace $ .", "While the cluster configurations in (g),(i),(h) and (l) only contain short single diagonals of the form $[i,i+2]_c$ , $c\\in \\lbrace R,B\\rbrace $ .", "Proceeding as before, taking into account the symmetry $\\sigma $ of Lemma REF one obtains the claimed number.", "In the third case we count the cluster configurations arising from $\\tau $ -shifts of the cluster configurations in (m) and (n) having no paired diagonals.", "As before we deduce that there are 28 the cluster configurations arising from $\\tau $ -shifts of (m) and 56 arising from $\\tau $ -shifts of the 4 cluster configurations in (n).", "Together this gives 84 cluster configurations without paired diagonals." ], [ "The geometry of cluster configurations", "In Section REF we saw that cluster configurations in $\\mathcal {F}_1$ correspond to triangulations with coloured oriented diagonals of two copies of $\\Pi $ .", "Cluster configurations of $\\mathcal {F}_2$ are not as simple to describe.", "In Theorem REF below we can give a general statement concerning the geometry of cluster configurations in $\\Pi $ .", "Theorem 5.12 All 833 cluster tilting sets of $\\mathcal {C}_{E_6}$ can be expressed as configurations of six non-crossing coloured oriented single and paired diagonals inside two heptagons.", "First, given a cluster configurations in $\\mathcal {F}_1\\cup \\mathcal {F}_2$ we divide the coloured oriented diagonals inside two heptagons by colour.", "Paired diagonals appear in both heptagons.", "Then we observe that all cluster configurations in $\\mathcal {F}_1$ and $\\mathcal {F}_2$ are crossing free.", "Moreover, the symmetry $\\sigma $ produces new configurations of diagonals which are again crossing free.", "Taking $\\tau $ -shifts only rotates the entire configurations inside the two heptagons, occasionally switching colours and orientation according to the action of $\\tau $ inside $\\Pi $ .", "Hence the crossing free property is preserved under $\\tau $ -shifts and the claim follows.", "The cluster configuration $\\mathcal {T}=\\lbrace [5,3]_R, [5,2]_R ,[5,1]_P,[5,6]_P, [3,5]_B,[2,5]_B\\rbrace $ is expressed in two heptagons in the center of Figure REF , the numbering of the vertices of one heptagon is highlighted in the figure.", "Paired diagonals appear in both heptagons and have labels.", "The converse statement of Theorem REF is not true, as configurations of non-crossing coloured diagonals different then cluster configurations of $\\Pi $ are not cluster tilting sets of $\\mathcal {C}_{E_6}$ ." ], [ "Mutations of cluster tilting objects in $\\mathcal {C}_{E_6}$", "In this section we will see that in many cases it is possible to deduce the mutation process in $\\mathcal {C}_{E_6}$ from the mutations process inside $\\mathcal {C}_{A_4}$ .", "Moreover, we can deduce the mutation process for the families in $\\mathcal {F}_1$ and $\\mathcal {F}_2$ and taking $\\tau $ -shifts extend it for the remaining cluster configurations.", "In the next definitions we indicate by $\\overline{D}$ the unoriented single diagonal corresponding to a coloured oriented diagonal $D$ of $\\Pi $ .", "Definition 5.13 Let $D_P$ be a paired oriented diagonal and let $\\mathcal {T}$ be a cluster configuration containing $D_P$ .", "The paired diagonal $D_P^$ is the flip-complement of $D_P$ , if $\\overline{D}_P$ and $\\overline{D^}_P$ are related by a flip.", "Definition 5.14 Let $D_S$ be a single coloured oriented diagonal and let $\\mathcal {T}$ be a cluster configuration containing $D_S$ .", "The single coloured oriented diagonal $D_S^$ is the flip-complement of $D_S$ in $\\mathcal {T}$ , if $\\overline{D}_S$ and $\\overline{D^}_S$ are related by a flip and $\\mathcal {T}\\backslash D_S \\cup {D_S^}$ is a cluster configuration.", "Notice that if a flip-complement exists then the colour and orientation is uniquely determined by Lemma REF and Lemma REF .", "Proposition 5.15 Let $D_L$ be a long coloured oriented diagonal of $\\Pi $ giving rise to a cluster configuration.", "If $D_L$ is paired: single coloured diagonals triangulating $\\Pi _5$ , and paired diagonals triangulating $\\Pi _4$ have a flip-complement.", "If $D_L$ is single: single diagonals triangulating $\\Pi _4$ have a flip-complement.", "Let $D_L$ be a long coloured oriented diagonal dividing $\\Pi $ into the quadrilateral $\\Pi _4$ and the pentagon $\\Pi _5$ .", "In Lemma REF we saw that if $D_L$ is paired, triangulating $\\Pi _4$ with a paired diagonal, and two copies of $\\Pi _5$ with single oriented diagonals, always gives a cluster configuration.", "Hence removing a single diagonal of a copy of $\\Pi _5$ or a paired diagonal triangulating $\\Pi _4$ can only be completed to a cluster configuration in two ways, namely with diagonals being flip-complement of each other.", "If $D_L$ is single, the claim follows from Lemma REF .", "Further instances of the mutation process in $\\mathcal {C}_{E_6}$ can be described by flips of coloured oriented diagonals in $\\Pi $ , but not all mutations allow a description of this type.", "This is unsurprising, as for example not all mutations in the cluster algebra $\\mathbb {C}[Gr_{3,7}]$ can be described through Plücker relations, see [20].", "With Figure REF it is not hard to deduce all the remaining mutations occurring.", "For example, the cluster configuration in (a) can be mutated to (b),(c),(e),(l),(m) and to the flip of a diagonal inside one light grey coloured heptagon.", "Similarly, we have a list for the other families.", "Some instances of the more complicated geometric exchanges can be found on the upper pentagon of Figure REF ." ], [ "Exchange graph", "In Figure REF we display a part of the exchange graph of $\\mathcal {C}_{E_6}$ .", "For each heptagon appearing in the figure the numbering of its vertices is as shown on the central heptagon.", "The vertices of the graph correspond to cluster configurations, hence to cluster tilting sets of $\\mathcal {C}_{E_6}$ , edges are drawn when two cluster configurations are related by a single mutation.", "In the two central heptagons of Figure REF the configuration of $\\mathcal {T}=\\lbrace [5,3]_R, [5,2]_R ,[5,1]_P,[5,6]_P, [3,5]_B,[2,5]_B\\rbrace $ is displayed.", "The 8 neighbouring configurations are placed on the vertices of the two central pentagons sharing the vertex corresponding to $\\mathcal {T}$ .", "These 8 sets are obtained from $\\mathcal {T}$ through repeated flips of single diagonals, as described in Proposition REF .", "The vertices of the left pentagon are obtained after mutating $[6,2]_P$ in $\\mathcal {T}$ ." ], [ "Cluster tilted algebras", "Let $T=T_1\\oplus \\dots \\oplus T_n$ be a cluster tilting object of $\\mathcal {C}_{Q}$ , then $\\mathrm {End}_{\\mathcal {C}_Q}(T)$ is the cluster tilted algebra of type $Q.$ The quiver $Q_T$ of $\\mathrm {End}_{\\mathcal {C}_Q}(T)$ has no loops nor 2-cycles and it encodes precisely the exchange matrix of the cluster associated to $T$ , see [4] and [6].", "In $\\mathcal {C}_{A_n}$ the quiver $Q_T$ can be read off from the triangulation $T$ , see [5].", "The vertices of $Q_T$ are the diagonals of the triangulation and an arrow between $D_i$ and $D_j$ is drawn, whenever $D_i$ and $D_j$ bound a common triangle.", "The orientation of the arrow is $D_i\\rightarrow D_j$ , if $D_j$ is linked to $D_i$ by an anticlockwise rotation around the common vertex.", "Extending the definition of $Q_T$ to the case at hand, the quivers corresponding to the cluster tilting sets of Figure REF can be deduced.", "One could also read off the quivers, and relations, directly from $\\Gamma $ by determining the spaces $\\mathrm {Hom}_{\\mathcal {C}_{E_6}}(T,T)$ .", "Figure: Part of the exchange graph for a cluster category of type E 6 E_6.At each vertex of the graph, the diagonals with the same labels are identified." ], [ "Symmetric cluster configurations and the cluster algebra of type ${F_4}$", "Cluster categories arising from valued quivers have been first categorified algebraically in [7].", "The aim of this section is to use the geometric description of the cluster category $\\mathcal {C}_{E_6}$ to categorify geometrically the cluster algebra of type $F_4$ .", "Consider again the map $\\rho $ given by a simultaneous change of colour and orientation of coloured oriented diagonals of $\\Pi $ .", "Definition 6.1 A cluster configuration $\\mathcal {T}$ in $\\Pi $ is $\\rho $ -symmetric if $\\mathcal {T}=\\rho (\\mathcal {T})$ .", "Since we know all cluster configurations of $\\Pi $ , see Theorem REF , we can deduce that there are only three types of $\\rho $ -symmetric cluster configurations in $\\Pi $ .", "First, cluster configurations of $\\Pi $ projecting to triangulations of $\\Pi $ consisting of unoriented arcs through $\\mathcal {C}_{E_6}\\rightarrow \\mathcal {C}_{A_4}$ .", "These arise from $\\mathcal {F}_1$ and from $\\tau $ -shifts of the configurations in Figure REF (d).", "Second, $\\tau $ -shifts of both, the cluster configuration in Figure REF (g) and the $\\sigma $ -symmetric configuration of Figure REF (g').", "Third, $\\tau $ -shifts of the configuration in Figure REF (h).", "We call the first $\\rho $ -symmetric cluster configurations of type T, the second of type C and the third of type L. Moreover, we refer to the double short diagonals in a configuration of type C as middle diagonals.", "In Figure REF configurations of type L and C are illustrated.", "The blue diagonals have opposite orientation with respect to the ones shown in the figure and are omitted.", "The two $\\rho $ -symmetric cluster configurations of type C on the right side of the figure are related through the action of $\\sigma _6$ , already defined in Section REF .", "Figure: ρ\\rho -symmetric cluster configurations of type L and C.Notice that for $\\rho $ -symmetric cluster configurations the quiver $Q_T$ of $\\mathrm {End}_{\\mathcal {C}_{E_6}}(T)$ is symmetric.", "In the next result we show that mutations of cluster tilting objects in $\\mathcal {C}_{E_6}$ preserves the $\\rho $ -symmetry of the cluster configuration.", "Let $\\mathcal {T}_{\\rho }$ be a $\\rho $ -symmetric cluster configuration of $\\Pi $ and denote by $D^$ the unique complement in $\\mathcal {C}_{E_6}$ of $D$ in $\\mathcal {T}_{\\rho }$ .", "Then we observe that $\\rho $ -symmetric cluster configurations always have two $\\rho $ -orbits consisting of paired diagonals and two consist of single diagonals of opposite colour and orientation, thus we can assume $\\mathcal {T}_{\\rho }:=\\lbrace D_{P_1},D_{P_2},D_{S_1},\\rho (D_{S_1}),D_{S_2},\\rho (D_{S_2})\\rbrace $ .", "Keeping the notation as above, we have the following result.", "Proposition 6.2 Let $1\\le k,l\\le 2$ , then $\\mathcal {T}_{\\rho }\\backslash D_{P_k}\\cup D_{P_k}^* \\textrm { and }\\mathcal {T}_{\\rho }\\backslash \\lbrace D_{S_l},\\rho (D_{S_l})\\rbrace \\cup \\lbrace D_{S_l}^,\\rho (D^_{S_l})\\rbrace $ are $\\rho $ -symmetric cluster configurations of $\\Pi $ .", "The claim follows from the mutation rule of $\\mathcal {C}_{E_6}$ and the symmetry of the configurations.", "Moreover, since there are always two paired diagonals and two $\\rho $ -orbits of single diagonals in $\\mathcal {T}_{\\rho }$ we deduce that paired diagonals are exchanged with paired diagonals, and single diagonals with single diagonals.", "We call the $\\rho $ -symmetric cluster configuration obtained from $\\mathcal {T}_{\\rho }$ of Proposition REF the mutation of $\\mathcal {T}_{\\rho }$ at $D_{P_k}$ , resp.", "at $\\lbrace D_{S_l},\\rho (D_{S_l})\\rbrace $, $1\\le k,l\\le 2$ .", "We refer to it by $\\mathcal {T}_{\\rho }^*$ .", "Let $\\mathcal {A}_{F_4}$ be the cluster algebra associated to a root system of type $F_4$ .", "Then we are able to prove the claimed result.", "Proposition 6.3 There is a bijection $\\lbrace \\rho \\textrm {-symmetric cluster configurations in } \\Pi \\rbrace \\rightarrow \\lbrace \\textrm {clusters in }\\mathcal {A}_{F_4}\\rbrace $ compatible with mutations.", "Clearly there is a bijection between the $\\rho $ -orbits of coloured oriented diagonals in $\\Gamma $ and the 24 cluster variables of $\\mathcal {A}_{F_4}$ .", "Proceeding as in the proof of Theorem REF we deduce that there are 105 $\\rho $ -symmetric cluster configurations in $\\Pi $ : 84 are of type T, 14 of type C and 7 of type L. From [9] we know that this number coincides with the number of clusters in the cluster algebra of type $F_4$ , $\\mathcal {A}_{F_4}$ .", "In addition, the mutation rule in $\\mathcal {C}_{E_6}$ induces a unique mutation rule for $\\rho $ -symmetric cluster configurations of $\\Pi $ .", "This can simply be seen performing the mutation case by case.", "The mutation in $\\mathcal {C}_{E_6}$ on $\\rho $ -symmetric cluster configurations then agrees with the mutation of the cluster algebra $\\mathcal {A}_{F_4}$ , since the exchange graph associated to $\\mathcal {A}_{F_4}$ is regular of degree 4, see [9].", "Thus there is essentially just one possible mutation.", "The 84 $\\rho $ -symmetric cluster configurations of type T in $\\mathcal {C}_{E_6}$ are in 2:1 correspondence with the triangulations of $\\Pi $ , thus with the cluster tilting sets of the cluster category of type $\\mathcal {C}_{A_4}$ .", "Geometrically the mutations of $\\rho $ -symmetric cluster configurations in $\\Pi $ are described by the following three moves.", "Let $c\\in \\lbrace R,B\\rbrace $ and let $i$ a vertex of $\\Pi $ .", "We call a triangle bounded by coloured oriented diagonals in $\\Pi $ internal if all its edges are different then boundary edges.", "L-C In a configuration of type L the diagonals $\\lbrace [i+3,i+1]_c,\\rho ([i+3,i+1]_c)\\rbrace $ exchange with $\\lbrace [i-1,i-3]_c,\\rho ([i-1,i-3]_c)\\rbrace $ in a configuration of type C. Similarly $[i+1,i+3]_P\\leftrightarrow [i-3,i-1]_P$ .", "C-T In a configuration of type C the middle diagonals $\\lbrace [i+2,i]_c,\\rho ([i+2,i]_c)\\rbrace $ exchange with $\\lbrace [i,i-3]_c,\\rho ([i,i-3]_c)\\rbrace $ bounding an internal triangle in a configuration of type T. Similarly $[i-2,i]_P\\leftrightarrow [i,i+3]_P$ .", "T-T All diagonals in a configuration of type T not bounding an internal triangle, are exchanged with the usual flip rule.", "The orientation of the new diagonal is uniquely determined by the type of diagonal one exchanges.", "More precisely, the orientation is such that paired diagonals are exchanges with paired diagonals, and single with single.", "Figure: Mutations of ρ\\rho -symmetric cluster tilting configurations.In Figure REF we illustrate a mutation between a $\\rho $ -symmetric cluster configuration of type L and of type C, as well as a mutation between a configuration of type C and type T. The diagonals in dotted lines are complements of each other and as before, paired diagonals are labelled." ], [ "Cluster tilting sets in $\\mathcal {C}_{T_{r,s,t}}$", "The construction of $\\Gamma _{r,s,t}^+$ and $\\Gamma _{r,s,t}^-$ of Section REF was motivated by the following idea: glue two copies of the AR-quiver of $\\mathcal {C}^2_{A_{r+t+1}}$ along two disjoint $\\tau $ -orbits.", "This resulted in pairing diagonals (as in Section REF ).", "In this context one can ask.", "Problem 6.4 How do categorical properties of the original category behave under this gluing operation?", "How do cluster tilting sets behave under this operation?", "The cluster categories $\\mathcal {C}_{E_7}$ and $\\mathcal {C}_{E_7}$ have finitely many cluster tilting set and they can be described as cluster configurations of coloured oriented single and paired diagonals inside a 10-gon, resp.", "a 16-gon, in a similar way as we did in Section REF .", "From these configurations we can identify again a family (denoted by $\\mathcal {F}_1$ previously) of cluster configurations giving rise to triangulations of regions homotopic to a heptagon and a octagon, resp.", "a heptagon and a nonagon, compare with Section REF and Lemma REF .", "In Figure REF a cluster tilting set of $\\mathcal {C}_{E_7}$ arising from a projection of the projective modules in $\\mathrm {mod}k E_7$ is represented.", "Similarly, in Figure REF a cluster tilting set of $\\mathcal {C}_{E_8}$ is represented.", "Figure: A cluster tilting set of 𝒞 E 8 \\mathcal {C}_{E_8}.When $T_{r,s,t}$ is not of Dynkin type, $\\mathcal {C}_{T_{r,s,t}}$ has infinitely many cluster tilting sets.", "With our geometric approach, a number of cluster tilting sets of $\\mathcal {C}_{T_{r,s,t}}$ can be expressed as configurations of coloured oriented single and paired diagonals inside the $(n+3)$ -gon $\\Pi $ , where $n\\ge \\max \\lbrace r+t+1,r+s+1\\rbrace $ .", "Moreover, if one considers the projections $\\mathbb {Z} T_{r,s,t}\\rightarrow \\mathbb {Z} A_{r+t+1}$ and $\\mathbb {Z} T_{r,s,t}\\rightarrow \\mathbb {Z} A_{s+t+1}$ one can describe the $\\mathrm {Ext}$ -hammocks in $\\mathbb {Z} T_{r,s,t}$ using the $\\mathrm {Ext}$ -hammocks in $\\mathbb {Z} A_{r+t+1}$ , resp.", "$\\mathbb {Z} A_{s+t+1}$ , as we did in Subsection REF .", "This enable us to express a number of cluster configurations of $\\mathcal {C}_{T_{r,s,t}}$ as triangulations of a $(s+t+4)$ - and a $(r+t+4)$ -gon.", "Finally, since in type $A$ all quivers obtained through Fomin-Zelevinsky quiver mutations are known, we expect that our model can be used to understand the quivers in the mutation class of an orientation of $T_{r,s,t}$ , as well as parts of the exchange graph of $\\mathcal {C}_{T_{r,s,t}}$ ." ], [ "Almost positive roots", "Consider the set of almost positive roots associated to a root system $\\Phi $ .", "This set, denoted by $\\Phi _{\\ge -1}$ , consists of all positive roots together with all negative simple roots of $\\Phi $ .", "For an initial choice of a triangulation of a regular $(t+3)$ -gon, an explicit bijection between all (unoriented) diagonals of the polygon and the set $\\Phi _{\\ge -1}$ of type $A_t$ was given in [5].", "From Theorem REF , together with [3] it follows that there is a bijection between the vertices of $\\Gamma _{r,s,t}^\\pm $ and the set of almost positive roots $\\Phi _{\\ge -1}$ of the root system of type $E_6$ , $E_7$ and $E_8$ .", "Problem 6.5 Describe geometrically the bijection between the coloured oriented single and paired diagonals in a 7-,10-, resp.16-gon and the almost positive roots of the root system of type $E_6$ , $E_7$ , resp.", "$E_8$ .", "Acknowledgments: I would like to thank Karin Baur, Giovanni Felder and Robert Marsh for all the inspiring discussions we had and for the many very helpful comments.", "I would also like to thank an anonymous referee for helpful suggestions.", "The author was partially supported by the Swiss National Science Foundation Grant Number PDFMP2127430." ] ]	1403.0549
[ [ "Toy models of Universe with an Effective varying $\\Lambda$-Term in Lyra\n Manifold" ], [ "Abstract We are interested by the study of several toy models of the Universe in presence of interacting quintessence DE models.", "Models are considered in the cosmology with an Effective varying $\\Lambda$-Term in Lyra Manifold.", "The motivation of the phenomenological models discussed in this paper is to obtain corresponding models to describe and understand an accelerated expansion of the Universe for the later stage of evolution.", "Phenomenology of the models describes by the phenomenological forms of $\\Lambda(t)$ ($8 \\pi G =c =1$).", "Concerning to the mathematical hardness we discuss results numerically and graphically.", "Obtained results give us hope that proposed models can work as good models for old Universe and in good agreement with observational data." ], [ "Analysis of the observational data shows that our Universe for later stages of evolution indicates accelerated expansion.", "This conclusion is based on the observations of high redshift type SNIa supernovae [1]-[3].", "The last problem is an interesting and important theoretical problem, and the best solutions of this problem based on an assumptions and carries phenomenological character.", "According to the data analysis we accept that in the Universe one of the main components is a Dark Energy and its negative pressure (positive energy density) has enough power to work against gravity and provide accelerated expansion of the Universe.", "To have a balance in Universe the second component known as Dark Matter is considered, which is responsible for the completely other phenomenon known as structure formation.", "According to different estimations Dark Energy occupies about $73\\%$ of the energy of our universe, while dark matter, about $23\\%$ , and usual baryonic matter occupy about $4\\%$ .", "The surveys of clusters of galaxies show that the density of matter is very much less than critical density [4], observations of Cosmic Microwave Background (CMB) anisotropies indicate that the universe is flat and the total energy density is very close to the critical $\\Omega _{tot} \\simeq 1$ [5].", "The simple model for the DE is the cosmological constant with two problems called fine-tuning and coincidence [6].", "These problems have opened ways for alternative models for the Dark energy including a dynamical form of dark energy, as a variable cosmological constant [7]-[8], k-essence model [9]-[10], Chaplygin gas models [11]-[26] to mention a few.", "In recent times were shown that certain type of interaction between DE and DM also could solve mentioned problems.", "On the other hand one can modify the left hand side of Einstein equation and obtain theories such as $f(R)$ theory of the gravity [27]-[34].", "Modifications of these types provide an origin of a fluid identified with dark energy.", "The origin of an accelerated expansion contributed from geometry were considered even before proposed modifications.", "But such theories with different forms of modifications still should pass experimental tests, because they contain ghosts, finite-time future singularities e.t.c, which is the base of other theoretical problems.", "One of the well studied Dark Energy models is a quintessence model [35]-[43], which is a scalar field model described by a field $\\phi $ and $V(\\phi )$ potential and it is the simplest scalar-field scenario without having theoretical problems such as the appearance of ghosts and Laplacian instabilities.", "Energy density and pressure of quintessence DE given as $\\rho _{Q}=\\frac{1}{2}\\dot{\\phi }^{2}+V(\\phi ),$ and $P_{Q}=\\frac{1}{2}\\dot{\\phi }^{2}-V(\\phi ).$ We consider the models of the Universe where an effective energy density and pressure assumed to be given as $\\rho =\\rho _{Q}+\\rho _{b},$ and $P=\\rho _{Q}+\\rho _{b},$ where $\\rho _{b}$ and $P_{b}$ are energy density and pressure of a barotropic fluid which will model DM in Universe with $P_{b}=\\omega _{b}\\rho _{b}$ EoS equation.", "If we will model the background dynamics of the Universe within many component fluid, then we will have $\\rho =\\sum _{i}{\\rho _{i}},$ and $P=\\sum _{i}{P_{i}},$ where $i$ represents the number of components.", "Last assumption is at the heart of the modern theoretical cosmology, and is a starting assumption for all articles in Cosmology.", "It can work particularly for old large scale Universe, where quantum and nonequilibrium effects are not considered.", "Would the last assumption work in early young Universe is an open question, because for early Universe with high energy/small scales quantum effects can have unexpected effects and how the situation should be modified is not clear yet.", "As well as we have other conceptual problems, for instance, we do not know how correctly we can model content of the early Universe (which is also open problem for old Universe).", "As in this work we will consider interaction between components, we would like to have a short discussion on that topic.", "Apart mathematical speculation concerning to the interaction between DE and DM, there is a question concerning to the physics of the interaction $Q$ .", "Since there is not any reason in nature preventing or suppressing a nonminimal coupling between dark energy and dark matter, there may exist interactions between the two components.", "Moreover from observations, no piece of evidence has been so far presented against such interactions.", "Theoretically, several forms of the interaction discussed in recent years, give us possibility to alleviate the coincidence problem, therefore such approach gives a hope and it is one of the active questions considered in literature.", "Despite to the efforts the microscopic nature of the interaction is not clear (up to our knowledge) and considered models based on a phenomenological assumptions.", "The question how the interaction between two components arose is not answered yet.", "One of the assumptions concerning to the interaction between components, is probably the same origin of DE and DM which were from the very begining.", "However it is not clear why they should continue their connection, when they operate on different scales and responsible for different processes.", "Probably the final theory of Quantum Gravity can answer to this question, As the theory is missing and there is a posibility to make an assumption, we think a quantum effect like modified entanglement exist, informing DE and DM that they were the same for very early stages of evolution.", "Below, we would like to present several forms of interaction $Q$ between DE and DM, which are the results of mathematical speculations.", "One group of $Q$ with a general form described as $Q=3H\\sum _{i}{b_{i} \\rho _{i}}+\\sum _{i}{\\gamma _{i}\\dot{\\rho }_{i}},$ where $b_{i}$ and $\\gamma _{i}$ are positive constants.", "Typical value of them is about $0.01\\div 0.03$ .", "In the other group we can include interactions where on the base of classical forms of interactions some modifications are assumed, like $b_{i}$ or $\\gamma _{i}$ to be a function of time $t$ .", "In this paper the problem solving strategy and structure is the following we will assume that the form of the potential $V(\\phi )$ is given $V(\\phi )=V_{0}e^{ \\left[-\\alpha \\phi \\right]}.$ We already made several attempts to consider Cosmologies where $G(t)$ and $\\Lambda (t)$ are varying functions instead of the constants.", "Consideration of varying $G$ and $\\Lambda $ supposed modification of the field equations.", "In this work we will consider cosmological models defined in Lyra manifold with an effective varying $\\Lambda $ term.", "Three different forms for the interaction between a barotropic fluid and a quintessence DE are considered.", "Each form of the interaction suppose an existance of the certain type of the model.", "In this paper we have 6 different models, because depends on the type of the interaction, we also considered a possibility that $\\Lambda $ is a constant.", "Models ordered as Model 1, Model 2 and Model 3 described by the interaction terms $Q$ $Q=3Hb\\rho _{Q}+\\gamma (\\rho _{b}-\\rho _{Q})\\frac{\\dot{\\phi }}{\\phi },$ $Q=3Hb\\rho +\\gamma \\dot{\\rho },$ and $Q=bH^{1-2\\gamma }\\rho _{b}^{\\gamma }\\dot{\\phi }^{2}.$ The second group of the models (Model 4, Model 5 and Model 5) will be described by the same three forms of $Q$ with $\\Lambda (t)$ given as $\\Lambda (t)=H^{2}\\phi ^{-2}+\\delta V(\\phi ),$ where $\\delta $ is a positive constant, $V(\\phi )$ is the potential of the field.", "The first form of the interaction as well as the form of the varying $\\Lambda (t)$ is considered by us in GR with varying $G(t)$ and $\\Lambda (t)$ in Ref.", "[45].", "The second form of the interaction can be considered as one of the classical form of the interaction considered in literature over the years.", "The last form of the interaction were considered in Ref.", "[46] This paper is organized as the follows.", "In section 2 we review the field equations.", "In section 3 we analyse models corresponding to $\\Lambda =const$ case.", "In section 4 we consider three models with varying $\\Lambda (t)$ .", "Finally, in section 5 we give conclusions." ], [ "Field equations [44] that govern our model of consideration are $R_{\\mu \\nu }-\\frac{1}{2}g_{\\mu \\nu }R-\\Lambda g_{\\mu \\nu }+\\frac{3}{2}\\phi _{\\mu }\\phi _{\\nu }-\\frac{3}{4}g_{\\mu \\nu }\\phi ^{\\alpha }\\phi _{\\alpha }=T_{\\mu \\nu }.$ Considering the content of the Universe to be a perfect fluid, we have $T_{\\mu \\nu }=(\\rho +P)u_{\\mu }u_{\\nu }-Pg_{\\mu \\nu },$ where $u_{\\mu }=(1,0,0,0)$ is a 4-velocity of the co-moving observer, satisfying $u_{\\mu }u^{\\mu }=1$ .", "Let $\\phi _{\\mu }$ be a time-like vector field of displacement, $\\phi _{\\mu }=\\left( \\frac{2}{\\sqrt{3}}\\beta ,0,0,0 \\right),$ where $\\beta =\\beta (t)$ is a function of time alone, and the factor $\\frac{2}{\\sqrt{3}}$ is substituted in order to simplify the writing of all the following equations.", "By using FRW metric for a flat Universe, $ds^2=-dt^2+a(t)^2\\left(dr^{2}+r^{2}d\\Omega ^{2}\\right),$ field equations can be reduced to the following Friedmann equations, $3H^{2}-\\beta ^{2}=\\rho +\\Lambda ,$ and $2\\dot{H}+3H^{2}+\\beta ^{2}=-P+\\Lambda ,$ where $H=\\frac{\\dot{a}}{a}$ is the Hubble parameter, and an overdot stands for differentiation with respect to cosmic time $t$ , $d\\Omega ^{2}=d\\theta ^{2}+\\sin ^{2}\\theta d\\phi ^{2}$ , and $a(t)$ represents the scale factor.", "The $\\theta $ and $\\phi $ parameters are the usual azimuthal and polar angles of spherical coordinates, with $0\\le \\theta \\le \\pi $ and $0\\le \\phi <2\\pi $ .", "The coordinates ($t, r,\\theta , \\phi $ ) are called co-moving coordinates.", "The continuity equation reads as, $\\dot{\\rho }+\\dot{\\Lambda }+2\\beta \\dot{\\beta }+3H(\\rho +P+2\\beta ^{2})=0.$ With an assumption that $\\dot{\\rho }+3H(\\rho +P)=0.$ Eq.", "(REF ) will give a link between $\\Lambda $ and $\\beta $ of the following form $\\dot{\\Lambda }+2\\beta \\dot{\\beta }+6H\\beta ^{2}=0.$ To introduce an interaction between DE and DM Eq.", "(REF ) we should mathematically split it into two following equations $\\dot{\\rho }_{DM}+3H(\\rho _{DM}+P_{DM})=Q,$ and $\\dot{\\rho }_{DE}+3H(\\rho _{DE}+P_{DE})=-Q.$ Cosmological parameters of our interest are EoS parameters of each fluid components $\\omega _{i}=P_{i}/\\rho _{i}$ , EoS parameter of composed fluid $\\omega _{tot}=\\frac{P_{m}+P_{\\Lambda } }{\\rho _{m}+\\rho _{\\Lambda }},$ deceleration parameter $q$ , which can be writen as $q=\\frac{1}{2}(1+3\\frac{P}{\\rho } ).$" ], [ "For a complete and full picture we will start our analyse from the models with constant $G$ and $\\Lambda $ .", "Without loss of generality we would like to describe equations allowing us to find dynamics of the models.", "According to the assumption with constant $\\Lambda $ Eq.", "(REF ) will be modified $\\dot{\\rho }+2\\beta \\dot{\\beta }+3H(\\rho +P+2\\beta ^{2})=0.$ and with $\\dot{\\rho }+3H(\\rho +P)=0$ we will obtain that $\\dot{\\beta }+3H\\beta =0.$ The last equation can be integrated very easily and the result is the following $\\beta =\\beta _{0}a^{-3},$ where $a(t)$ is the scale factor and $\\beta _{0}$ is the integration constant.", "In our future calculations we use $\\beta _{0}=1$ initial condition.", "Concerning to the form of the field equations, we need only to assume the form of $Q$ and we will obtain the cosmological solutions.", "Concerning to the mathematical hardness of the problem we will analyse models numerically and investigate graphical behavior of various important cosmological parameters.", "In the following subsections we consider our models with the particular forms of $Q$ considered in Introduction." ], [ "In this section we will pay our attention to the first toy model.", "Within this and other models of this work we would like to examine the behavior of the Universe, to analyse and see if within our assumptions accelerated expansion of the Universe can be observed as well as analyse behavior of important cosmological parameters.", "In this model an interaction between DE and DM are assumed to be $Q=3Hb\\rho _{Q}+\\gamma (\\rho _{b}-\\rho _{Q})\\frac{\\dot{\\phi }}{\\phi },$ where $b$ is positive constant, $\\phi $ is a field, $H$ is the Hubble parameter and $\\rho _{b}$ and $\\rho _{Q}$ represents energy densities of DM and DE.", "Therefore the dynamics of the energy density of the barotropic fluid can be found from $\\dot{\\rho }_{b}+3H \\left(1+\\omega _{b}-\\frac{\\gamma }{3H}\\frac{\\dot{\\phi }}{\\phi } \\right)\\rho _{b}=3H \\left( b-\\frac{\\gamma }{3H}\\frac{\\dot{\\phi }}{\\phi } \\right)\\rho _{Q}.$ Using the same mathematics we can obtain dynamics of DE $\\dot{\\rho }_{Q}+3H \\left( 1+b +\\omega _{Q} -\\frac{\\gamma }{3H} \\frac{\\dot{\\phi }}{\\phi }\\right) \\rho _{Q}=-\\gamma \\frac{\\dot{\\phi }}{\\phi }\\rho _{b}.$ From the graphical analysis of the Hubble parameter and deceleration parameter $q$ we conclude that in the case of a constant $\\Lambda $ the Hubble parameter is a decreasing function, which for later stages of evolution becomes a constant.", "Also we observe that with increasing numerical value of the $\\Lambda $ we increase the numerical value of the Hubble parameter.", "For the deceleration parameter $q$ we see that the transition from the decelerated phase to the accelerated expansion phase in the history of Universe can be seen.", "Moreover, we see that for later stages of evolution $q$ behaves as a constant and its numerical value is in well agreement with observational facts.", "Therefore we can declare that proposed model is in good agreement with the observations.", "Discussed behaviors for the Hubble parameter and deceleration parameter $q$ can be seen in Fig.", "REF .", "We also analyse behavior of $\\omega _{tot}$ and $\\omega _{Q}$ and results can be found in Fig.", "REF .", "We see that that both parameters are a decreasing functions.", "$\\omega _{tot}$ is a positive for early stages of evolution, then it is a negative and for later stages of evolution it is a constant.", "With an increasing the numerical value of the $\\Lambda $ we can satisfy $\\omega _{tot}=-1$ condition i.e for later stages of evolution in the dynamics of the Universe a cosmological constant has an important place.", "Behavior of $\\omega _{Q}$ shows quintessence behavior of DE.", "The parameters of the Model 1 were fixed to obtain the well know fact that $V \\rightarrow 0$ when $t \\rightarrow \\infty $ .", "The field $\\phi $ appears to be an increasing function of time (Fig.", "REF ).", "Figure: Behavior of Hubble parameter HH and qq against tt for the constant Λ\\Lambda .", "Model 1Figure: Behavior of EoS parameter ω tot \\omega _{tot} and ω Q \\omega _{Q} against tt for the constant Λ\\Lambda .", "Model 1Figure: Behavior of filed φ\\phi and potential VV against tt for the constant Λ\\Lambda .", "Model 1" ], [ "In this section we will analyse the second model (Model 2), with the interaction term $Q$ , which has the following form $Q=3Hb\\rho +\\gamma \\dot{\\rho },$ where $\\rho $ is the total energy density.", "The last form can be considered as a classical form of the general form for the interaction, because such interaction were considered intensively in literature.", "In this case our interaction is a function also from time derivative of energy density.", "The dynamics of DM and DE can be found after some mathematical transformations and we have the following forms $(1-\\gamma )\\dot{\\rho }_{b}+3H(1+\\omega _{b}-b)\\rho _{b}=3Hb\\rho _{Q}+\\gamma \\dot{\\rho }_{Q},$ and $(1+\\gamma )\\dot{\\rho }_{Q}+3H(1+\\omega _{Q}+b)\\rho _{Q}=-3Hb\\rho _{b}-\\gamma \\dot{\\rho }_{b}.$ The graphical behavior of the Hubble parameter shows us that it is a decreasing function within the evolution of the Universe.", "Also we see that it is a constant for later stages of evolution, moreover we see that with an appropriate choose of the numerical values of the model parameters we can find an agreement with observational data.", "We also would like to add, that with increasing numerical value of $\\Lambda $ we increase numerical value of the Hubble parameter.", "An investigation of the deceleration parameter $q$ , we found that this model is also in good agreement with observations.", "Behavior of the $q$ also can explain the well known fact that our Universe has transition from $q>0$ phase to the accelerated expansion phase with $q<0$ .", "The combination of several observational data it is concluded that for the our Universe deceleration parameter should be greater than $-1$ which is illustrated for this model.", "In conclusion we would like to indicate that this model is also a good model.", "This analysis is based on the plots of Fig.", "REF .", "For the behavior of $\\omega _{tot}$ which is the EoS parameter for our interacting two component fluid reveals quintessence-like behavior of the Universe for intermediate phases, while it is a positive for early stages of evolution, and for the old Universe it is a cosmological constant for high values of $\\Lambda $ (Fig.", "REF ).", "The decreasing behavior of $\\beta (t)$ is illustrated in Fig.", "REF Figure: Behavior of Hubble parameter HH and qq against tt for the constant Λ\\Lambda .", "Model 2Figure: Behavior of EoS parameter ω tot \\omega _{tot} and ω Q \\omega _{Q} against tt for the constant Λ\\Lambda .", "Model 2Figure: Behavior of β\\beta against tt for the constant Λ\\Lambda .", "Model 2" ], [ "For the third model we assume that interaction between DI and DM can be modeled within $Q=bH^{1-2\\gamma }\\rho _{b}^{\\gamma }\\dot{\\phi }^{2}.$ This type for the interaction is already considered in literature, and we wonder about it role in Lyra manifold.", "Next, we gave differential equations for the dynamics for energy densities.", "For barotropic DM it can be writen as $\\dot{\\rho }_{b}+3H(1+\\omega _{b}-\\frac{b}{3}H^{2(1-\\gamma )}\\rho _{b}^{\\gamma -1}\\dot{\\phi }^{2}),$ and for DE we will have $\\dot{\\rho }_{Q}+3H(1+\\omega _{Q})\\rho _{Q}+bH^{1-2\\gamma }\\rho _{b}^{\\gamma }\\dot{\\phi }^{2}.$ This form of interaction is a nonlinear function from the Hubble parameter, energy density of the barotropic fluid.", "It is a function also from derivative of the field $\\phi $ .", "For illustration we analyse behavior of the Hubble parameter, deceleration parameter $q$ , $\\omega _{tot}$ and $omega_{Q}$ as a function of $b$ , $\\gamma $ with increasing $\\Lambda $ .", "Our analysis shows that this model is in good agreement with observations, therefore in conclusion of this section we would like to mention that considered three interacting models in Lyra manifold can serve as a good models.", "As the starting models they could be generalized and investigated from different corners in order to understand the viability of them.", "In the next section we will consider the varying $\\Lambda (t)$ case.", "The form which we consider in this work is already constructed by us and considered in usual GR for the general case.", "Figure: Behavior of Hubble parameter HH and qq against tt for the constant Λ\\Lambda .", "Model 3Figure: Behavior of EoS parameter ω tot \\omega _{tot} and ω Q \\omega _{Q} against tt for the constant Λ\\Lambda .", "Model 3" ], [ "In this section we will consider three interacting fluid models and will investigate cosmological parameters like the Hubble parameter $H$ , deceleration parameter $q$ , EoS parameters of the total fluid and DE $\\omega _{Q}$ .", "Based on numerical solutions, we will discuss graphical behaviors of the cosmological parameters.", "For the $\\Lambda (t)$ we take a phenomenological form which was considered by us recently.", "The formula of $\\Lambda (t)$ is $\\Lambda (t) = H^{2} \\phi ^{-2}+\\delta V(\\phi ),$ which is a function of the Hubble parameter, potential of the scalar field, and time derivative of the scalar field.", "For the potential we take a simple form $V(\\phi )=e^{[-\\alpha \\phi ]}$ , therefore the form of $\\Lambda (t)$ can be writen also in the following way as only a function of the filed $\\phi $ $\\Lambda (t)=H^{2}\\phi ^{-2}+\\delta e^{[-\\alpha \\phi ]}.$ Therefore, the dynamics of $\\beta $ can be obtained from the following differential equation $2\\beta \\dot{\\beta }+6H\\beta ^{2}+2H\\dot{H}\\phi ^{-2}-2H^{2}\\phi ^{-3}\\dot{\\phi }-\\delta \\alpha e^{[-\\alpha \\phi ]} \\dot{\\phi }=0.$ In forthcoming subsections within three different forms of $Q$ we will investigate the dynamics of the Universe.", "The question of the dynamics for the energy densities of the DE and DM is already discussed in previous section, therefore we will not consider them here and we will start with the comments on the graphical behaviors of the cosmological parameters of the models.", "We will start with the model where $Q=3Hb\\rho _{Q}+\\gamma (\\rho _{b}-\\rho _{Q})\\frac{\\dot{\\phi }}{\\phi }$ ." ], [ "We start the analysis of the model 5 from discussions about graphical behavior of the Hubble parameter and deceleration parameter presented in Fig.", "REF .", "The Hubble parameter is a decreasing function and gets constant value at relatively far future.", "At the top panel we consider three cases corresponding to the behavior of the Hubble parameter.", "From the first plot we see that when $\\gamma =0.02$ , $b=0.01$ and $\\omega _{b}=0.75$ with increasing $\\delta $ we will increase the value of the Hubble parameter.", "The middle plot represents behavior of Hubble parameter as a function of interaction parameters $b$ and $\\gamma $ .", "We see that with an increasing numerical values of the parameters we increase numerical value of the Hubble parameter, when other parameters $\\delta $ and $\\omega _{b}$ are fixed.", "The third plot presents behavior of the Hubble parameter from the $\\omega _{b}$ .", "We see that for later stages of evolution the Hubble parameter is practically does not depend from the $\\omega _{b}$ .", "At the bottom panel we represent graphical behavior of the deceleration parameter.", "We can declare that at early stages of the evolution transition from $q>0$ to $q<0$ can be realised.", "The first plot indicates the strong dependence of the $q$ from the $\\delta $ for $\\gamma =0.02$ , $\\beta =0.01$ and $\\omega _{b}=0.75$ .", "An increasing the numerical value of the $\\delta $ decreases the numerical value of the $q$ .", "We also see that for the later stages of evolution $q$ increases and becomes a constant.", "From the middle plot we can obtain information about behavior of the $q$ as a function of $\\gamma $ and $b$ .", "For early stages of the evolution we do not observe any dependence, which becomes apparent only for the later stages of evolution.", "With an increasing both $\\gamma $ and $b$ we decrease the value of $q$ .", "Finally, the last plot shows $q$ dependence from $\\omega _{b}$ .", "We see almost independent behavior (Fig.", "REF ) from $\\omega _{b}$ .", "We also investigate the behaviors of $\\omega _{tot}$ and $\\omega _{Q}$ and results are presented in Fig.", "REF .", "The behavior of $\\omega _{tot}$ predicts quintessence-like behavior for the Universe.", "Comparision of the results of this model with the Model 1, where constant $\\Lambda $ were assumed, showed that in this model $\\omega _{tot}$ remains strictly above $-1$ .", "Also the value of the $q$ for the later stages of the evolution is higher then in Model 1.", "Figure: Behavior of Hubble parameter HH and deceleration parameter qq against tt for Model 4.Figure: Behavior of EoS parameter ω tot \\omega _{tot} and ω Q \\omega _{Q} against tt for Model 4." ], [ "Here we will analyse the model where the interaction between DE and DM is the form $Q=3Hb\\rho +\\gamma \\dot{\\rho }$ .", "The results of the Model 5 should be compared with Model 2, where instead of the varying $\\Lambda (t)$ the constant $\\Lambda $ were considered.", "The Hubble parameter is a decreasing function and it becomes a constant for the later stages of the evolution.", "Three plots of the top panel of Fig.", "REF give a general idea about the behavior of the Hubble parameter as a function of $\\delta $ (first plot) for $\\gamma =0.02$ , $\\beta =0.01$ and $\\omega _{b}=0.75$ .", "From the middle plot we have information about the Hubble parameter describing dependence of it from the interaction parameters, when the numerical values of the $\\delta $ and $\\omega _{b}$ are taken in advance.", "The last plot gives time evolution of the Hubble parameter as a function from $\\omega _{b}$ .", "The bottom of the same Figure dedicated to the deceleration parameter $q$ .", "Like to the other models, in this case as well, we have the Universe where transition to $q<0$ is possible and we see that it can realised for early stages of evolution.", "To understand behavior of $q$ from the model parameters we considered 3 cases allowing a variation one of the model parameters.", "We see that for later stages of the evolution $q$ from decreasing function becomes an increasing function and for very late stages becomes a constant.", "Moreover, compared with observational facts known about $q$ , we can conclude that this model is also can be considered as a good model.", "Comparision between Model 2 and Model 5, allows us to see that when $\\Lambda $ is a constant, then $\\omega _{tot}$ for later stages is $-1$ , while for the varying model with the same form of the interaction term $\\omega _{tot}>-1$ indicating quintessence-like Universe.", "For both models $\\omega _{tot}$ and $\\omega _{Q}$ are decreasing functions over the time and a constant for later stages of the evolution.", "For the Model 5 numerical value of the $q$ when it is a constant is higher than for the Model 2.", "In the next section we will examine our last model.", "Comparision between it and Model 3 will take a place to see differences between varying $\\Lambda (t)$ and constant $\\Lambda $ cases.", "For all cases to make a real comparison of the models the parameters describing the models assumed to be the same.", "Figure: Behavior of Hubble parameter HH and deceleration parameter qq against tt for Model 5.Figure: Behavior of EoS parameter ω tot \\omega _{tot} and ω Q \\omega _{Q} against tt for Model 5." ], [ "Analysis of the cosmological parameters for the varying $\\Lambda (t)$ and interaction term $Q=bH^{1-2\\gamma }\\rho _{b}^{\\gamma }\\dot{\\phi }^{2}$ presented in Fig.", "REF and Fig.", "REF .", "We conclude that this model as almost the same characters as all other models, therefore we conclude that this model is also a good model.", "We conclude our work in the next section with some thoughts and finalizing obtained result, some observational data information is also given to prove our conclusions.", "Figure: Behavior of Hubble parameter HH and deceleration parameter qq against tt for Model 6.Figure: Behavior of EoS parameter ω tot \\omega _{tot} and ω Q \\omega _{Q} against tt for Model 6." ], [ "Discussion", "In this work we have considered six diffrent models of interacting quintessence DE models, which is one of the well studied scalar field model of Dark Energy.", "We consider a Cosmology with varying $\\Lambda (t)$ in Lyra manifold.", "We take into account modified field equations when $\\Lambda (t)$ is considered.", "Within this background we started to analyse three forms of the interaction terms between DM and DE.", "We also assume that DM can be modeled as a barotropic fluid with $P_{b}=\\omega _{b}\\rho _{b}$ .", "One of the forms of the interaction within the form of $\\Lambda $ we already considered in GR with varying $G(t)$ and $\\Lambda (t)$ .", "The second form of the interaction can be considered as one of the classical forms of the interaction considered in literature and it is a function from total energy density and its time derivative.", "While the last interaction $Q$ by its form is also a relatively new form.", "The construction of the third interaqtion term unit analysis is taken into account.", "It is a function of $H$ , $\\rho _{b}$ and $\\dot{\\phi }^{2}$ .", "According to the $1\\sigma $ level from $H(z)$ data $q\\approx -0.3$ and $H_{0}=68.43\\pm 2.8 \\frac{Km}{sMpc}$ [47].", "On the other hand from data of $SNe Ia$ we have $q \\approx -0.43$ and $H_{0}=69.18 \\pm 0.55\\frac{Km}{sMpc}$ [47].", "Also joint test using $H(z)$ and $SNe Ia$ give $-0.39 \\le q \\le -0.29$ and $H_{0}=68.93 \\pm 0.53\\frac{Km}{sMpc}$ [47].", "Recent astronomical data based on anew infrared camera on the $HST$ gives $H_{0}=73.8 \\pm 2.4\\frac{Km}{sMpc}$ [48].", "The other prob using galactic clusters data suggest $H_{0} = 67 \\pm 3.2\\frac{Km}{sMpc}$ [49].", "Finally, $\\Lambda CDM$ model suggests $q \\rightarrow -1$ and the best fitted parameters of the Ref.", "[50] say that $q \\approx -0.64$ .", "Conclusion of the presented facts is that that generally $q \\ge -1$ .", "Performing analysis of our models we see clearly, that $q \\ge -1$ condition is can be satisfied, moreover we see that the carefull choose of values of the model parametres observational facts can be recovered." ] ]	1403.0109
[ [ "Calder\\'on problem for the p-Laplacian: First order derivative of\n conductivity on the boundary" ], [ "Abstract We recover the gradient of a scalar conductivity defined on a smooth bounded open set in $\\mathbb{R}^d$ from the Dirichlet to Neumann map arising from the $p$-Laplace equation.", "For any boundary point we recover the gradient using Dirichlet data supported on an arbitrarily small neighbourhood of the boundary point.", "We use a Rellich-type identity in the proof.", "Our results are new when $p \\neq 2$.", "In the $p = 2$ case boundary determination plays a role in several methods for recovering the conductivity in the interior." ], [ "Introduction", "Throughout the article we assume $1 < p < \\infty $ , unless explicitly written otherwise.", "We investigate a generalisation of the Calderón problem to the case of the $p$ -Laplace equation: Given a bounded open set $\\Omega \\subset \\mathbb {R}^d$ and a bounded conductivity $\\gamma > 0$ on $\\overline{\\Omega }$ we have the Dirichlet problem $ {\\left\\lbrace \\begin{array}{ll}\\Delta _p^\\gamma (u) = \\operatorname{div}\\left( \\gamma (x) \|\\nabla u\|^{p-2} \\nabla u \\right) = 0 &\\text{in $\\Omega $,} \\\\u = v &\\text{on $\\partial \\Omega $}.\\end{array}\\right.", "}$ For $v \\in W^{1,p}(\\Omega )$ there exists a unique weak solution that satisfies $\\Vert u \\Vert _{W^{1,p}} \\le C \\Vert v \\Vert _{W^{1,p}}$ , where the constant $C$ does not depend on $v$ .", "This is completely standard, see e.g.", "[45].", "The equation arises from the variational problem of minimising the energy $\\int _\\Omega \\gamma \|\\nabla u\|^p \\, d x$ .", "We next define the Dirichlet to Neumann map (hereafter DN map).", "We use the notation $X^{\\prime }$ for the dual space of continuous linear functionals on a Banach space $X$ .", "In the $p=2$ case the DN map gives the stationary measurements of electrical current for given voltage $v$ .", "Definition 1.1 (Dirichlet to Neumann map) Suppose $\\Omega \\subset \\mathbb {R}^d$ is bounded open $C^1$ -set and $0 < \\gamma _0 < \\gamma \\in L^\\infty (\\overline{\\Omega })$ for some constant $\\gamma _0$ .", "The weak DN map $\\Lambda _\\gamma ^w \\colon W^{1,p}(\\Omega )/W^{1,p}_0(\\Omega ) \\rightarrow \\left(W^{1,p}(\\Omega )/W^{1,p}_0(\\Omega )\\right)^{\\prime }$ is defined by $\\left\\langle \\Lambda _\\gamma ^w(v), g \\right\\rangle = \\int _\\Omega \\gamma \|\\nabla u\|^{p-2} \\nabla u \\cdot \\nabla \\tilde{g} \\, d x,$ where $u$ solves the boundary value problem (REF ) with boundary values $v$ , and $\\tilde{g} \\in W^{1,p}(\\Omega )$ with trace $g$ on the boundary.", "The strong DN map is defined pointwise on $\\partial \\Omega $ by the expression $\\Lambda _\\gamma ^s (v)(x_0) = \\gamma (x_0) \|\\nabla u(x_0)\|^{p-2}\\nabla u(x_0) \\cdot \\nu (x_0)$ when it is well-defined.", "We also write $\\Lambda _\\gamma (v)$ when the function $v$ is continuous and defined on a superset of $\\partial \\Omega $ .", "With sufficient regularity we can recover the values of the strong DN map from the weak DN map (see lemma REF ).", "Given knowledge of the DN map we determine $\\nabla \\gamma \|_{\\partial \\Omega }$ , which in the $p=2$ case stands for the gradient of conductivity on the boundary.", "Theorem 1.2 Suppose $\\Omega \\subset \\mathbb {R}^d$ , $d > 1$ , is a bounded open $C^{2,\\beta }$ set for some $0 < \\beta <1$ and the conductivities $\\gamma _1$ and $\\gamma _2$ are bounded from below by a positive constant and continuously differentiable with Hölder-continuous derivatives in $\\overline{\\Omega }$ .", "If $\\Lambda _{\\gamma _1}^w = \\Lambda _{\\gamma _2}^w$ , then $\\nabla \\gamma _1\|_{\\partial \\Omega } = \\nabla \\gamma _2\|_{\\partial \\Omega }$ .", "We use an explicit sequence of boundary values to reconstruct $\\nabla \\gamma (x_0)$ at a boundary point $x_0$ .", "The sequence is supported in arbitrarily small neighbourhood of $x_0$ .", "The boundary values we use were first introduced by Wolff [51] and used by Salo and Zhong [45] to recover the conductivity on the boundary.", "Salo and Zhong [45] show that for all boundary points $x_0 \\in \\partial \\Omega $ there is a sequence of solutions $(u_M)_M$ such that $M^{n-1}N^{1-p}\\int _\\Omega g(x) \|\\nabla u_M\|^p \\, d x \\rightarrow c_p g(x_0)$ as $M, N(M) \\rightarrow \\infty $ , with explicitly computable constant $c_p$ (see equation (REF )).", "They recover the integrals with $g = \\gamma $ from the DN map.", "We recover the integrals with $g = \\alpha \\cdot \\nabla \\gamma $ from a Rellich-type identity, theorem REF , for arbitrary direction $\\alpha \\in \\mathbb {R}^d$ .", "The Calderón problem was first introduced in [14].", "For a review see [50].", "There have been several boundary determination results in the $p = 2$ case.", "Boundary uniqueness for derivatives of the conductivity was first proven by Kohn and Vogelius [30].", "Sylvester and Uhlmann in [49] recovered, for smooth conductivity in smooth domain, conductivity and all its derivatives on the boundary by considering $\\Lambda _\\gamma $ as a pseudodifferential operator.", "Nachman [38] recovered $\\gamma \|_{\\partial \\Omega }$ in Lipschitz domain when $\\gamma \\in W^{1,q}$ , $q > d$ , and its first derivative when $\\gamma \\in W^{2,q}$ , $q > d/2$ .", "Alessandrini [1] used singular solutions with singularity near boundary to recover all derivatives of $\\gamma \|_{\\partial \\Omega }$ with less regularity assumptions on $\\gamma $ .", "There have been several local boundary determination results, e.g.", "[40], [27].", "Other reasonably recent boundary determination results include [2], [13].", "A Rellich identity was used by Brown, Garcia and Zhang [18] to recover the gradient of conductivity on the boundary in the $p=2$ case.", "The Rellich identity was, to the best of our knowledge, introduced in [42].", "Several results (e.g.", "[1]) rely on investigating the difference $\\Lambda _{\\gamma _1} - \\Lambda _{\\gamma _2}$ of DN maps at different conductivities.", "This is difficult in the present setting due to non-linearity of the $p$ -Laplace equation.", "We use Rellich identity, theorem REF , to avoid this problem.", "Electrical impedance tomography has applications in medical (e.g.", "[12]) and industrial imaging (e.g.", "[28]), and geophysics and environmental sciences; see e.g.", "the review [9] and references therein.", "There are practical numerical algorithms for boundary determination in the $p=2$ case, e.g. [39].", "Boundary determination is used in recovering the conductivity in the interior, e.g. [21].", "For example, the algorithm in [47] uses the values of conductivity and its first derivative on the boundary to extend the conductivity, thence applying to conductivities that are not constant on the boundary.", "For more on the $p$ -Laplace equation see e.g.", "[35], [24], [17].", "The equation has applications in e.g.", "image processing [31], fluid mechanics [6], plastic moulding [4], and modelling of sand–piles [5].", "The Calderón problem for the $p$ -Laplace equation was first introduced in [45].", "The authors consider the real and the complex case separately.", "In the complex case they define $p$ -harmonic versions of complex geometrical optics solutions to create highly oscillating functions focused around a given boundary point and use them as Dirichlet data, thus recovering conductivity at the boundary point in question.", "In the real case they replace the $p$ -harmonic CGO solutions with real-valued functions having similar behaviour.", "The real-valued functions were originally introduced by Wolff [51].", "Further progress on the $p$ -Calderón problem was made by the author, Kar and Salo [11].", "In the article we show that we can detect the convex hulls of inclusions, which are regions of significantly higher or lower conductivity, from the DN map.", "Our main tools are the $p$ -harmonic functions of Wolff and a monotonicity inequality.", "Hauer [23] has investigated the DN map related to the $p$ -Laplace equation.", "One method of investigating the Calderón-type inverse problems for non-linear equations is based on studying the Gâteaux derivatives of the map $\\Lambda _\\gamma $ at constant boundary values $a$ .", "In our case this does not work [45]: $\\Lambda _\\gamma (a+tf) = t^{p-1}\\Lambda _\\gamma (f)$ for positive $t$ .", "For $p < 2$ the Gâteaux derivates do not exist and for $p > 2$ the higher derivates fail to exist, or vanish, though one of them might equal $\\Lambda _\\gamma (f)$ .", "Hence nonlinear methods are necessary.", "Other inverse problems related to nonlinear equations similar to the $p$ -Laplace equation have been investigated before; the 1-Laplace equation is used in current density imaging, e.g.", "[29], [37], [25], [26], [46], and the 0-Laplace equation is related to ultrasound modulated electrical impedance tomography, e.g.", "[7], [3], [8], [19].", "In section we introduce our notation and state lemmata.", "In section we restate boundary determination results we use in this article.", "In the final section we reconstruct $\\nabla \\gamma \|_{\\partial \\Omega }$ from the DN map." ], [ "Preliminaries", "We use the following notation: We write $\\partial _\\alpha = \\alpha \\cdot \\nabla $ for vectors $\\alpha $ .", "On the $C^1$ boundary $\\partial \\Omega $ of an open set $\\Omega $ we decompose the gradient of a function $f$ defined in $\\overline{\\Omega }$ as $\\nabla f(x_0) = \\nabla _\\nu f(x_0) + \\nabla _T f(x_0)$ , where $\\nabla _\\nu $ is the normal component of the derivative, i.e.", "the orthogonal projection of the gradient to the normal space of the boundary, and $\\nabla _T$ is the tangential derivative, i.e.", "the orthogonal projection of the gradient on the tangent space of the boundary.", "We have the following identity: $\|\\nabla _\\nu f\|^2 = (\\partial _\\nu f)^2$ .", "We denote the outer unit normal by $\\nu $ .", "We write the Hölder seminorm of order $k$ as $\|f\|_{k,\\beta }$ and the characteristic function of a set $A$ as $\\chi _A(x) ={\\left\\lbrace \\begin{array}{ll}1 \\text{ when } x \\in A \\\\0 \\text{ when } x \\notin A.\\end{array}\\right.", "}$ We need the following elementary inequality: Lemma 2.1 Suppose $0 \\le A,B \\in \\mathbb {R}$ and $2 \\le q < \\infty $ .", "Then for some $C > 0$ we have $\|A^{q-1} - B^{q-1}\| \\le C \\left( A + B \\right)^{q-2} \| A - B\|$ By fundamental theorem of calculus $\|A^{q-1} - B^{q-1}\| = (q-1)\\left\| A-B \\right\|\\left\| \\int _0^1 \\left(A + t(B-A)\\right)^{q-2} \\, d t \\right\|,$ from which the estimate follows by observing that $ 0 \\le A+t(B-A) \\le A+B$ .", "It is well-known that the solutions of the $p$ -Laplace equation are in $C^{1,\\beta }$ .", "For proof of the following lemma see e.g. [33].", "Lemma 2.2 (Regularity result) Suppose $\\Omega $ is a bounded open $C^{1,\\beta _1}$ set with $0 < \\beta _1 \\le 1$ , and suppose the conductivity $0 < \\gamma \\in C^{0,\\beta _2}(\\overline{\\Omega })$ is bounded from above and away from zero.", "Consider the weighted $p$ -Laplace equation (REF ) with boundary values $v \\in C^{1,\\beta _1}(\\partial \\Omega )$ .", "Then the solution $u$ of the weighted $p$ -Laplace equation (REF ) is in $C^{1,\\beta _3}(\\overline{\\Omega })$ for some $\\beta _3 > 0$ .", "In general the solutions are not twice continuously differentiable [36].", "To establish Rellich identity, theorem REF , we use the following interpolation lemma similar to [34]: Lemma 2.3 (Interpolation lemma) Suppose $\\Omega $ is a bounded open set in $\\mathbb {R}^d$ satisfying the measure density condition; that is, suppose there exist $\\delta _0, C > 0$ such that for all $0 < \\delta < \\delta _0$ and for all $y \\in \\overline{\\Omega }$ we have $\|B(y,\\delta )\| \\le C \|B(y,\\delta ) \\cap \\overline{\\Omega }\|.$ Also suppose $f \\in C^\\beta (\\overline{\\Omega })$ for some $\\beta > 0$ .", "Take $1 \\le p < \\infty $ .", "Let $M > 0$ be a constant such that $\|f\|_{0,\\beta } \\le M$ .", "Then $\\Vert f \\Vert _{L^\\infty (\\overline{\\Omega })} \\le C_{d,p,\\beta ,\\Omega }M^{\\frac{d}{d+\\beta p}} \\Vert f\\Vert _{L^p(\\Omega )}^\\frac{\\beta p}{d+ \\beta p}.$ The measure density condition (REF ) is satisfied in a wide variety of situations [22].", "In particular, domains that admit a Sobolev extension theorem satisfy the condition [22].", "Domains of class $C^{1,\\alpha }$ , and more generally Lipschitz domains, certainly admit a Sobolev extension theorem [16], [48].", "A bounded Lipschitz set has a finite number of connected components, and thus also satisfies the condition.", "The proof of the interpolation lemma is almost the same as in .", "Let $y \\in \\overline{\\Omega }$ and write $B_\\delta = B(y,\\delta )$ .", "For $\\delta > 0$ we have $\\Vert f\\Vert _{L^p(\\overline{\\Omega }\\cap B_\\delta )} &\\ge \\Vert f(y)\\Vert _{L^p(\\overline{\\Omega }\\cap B_\\delta )} - \\Vert f(x) - f(y)\\Vert _{L^p(\\overline{\\Omega }\\cap B_\\delta )} \\\\&= \|f(y)\|\\left\|\\overline{\\Omega }\\cap B_\\delta \\right\|^{1/p} - \\left( \\int _{\\overline{\\Omega }\\cap B_\\delta } \|f(x) - f(y)\|^p \\, d x \\right)^{1/p} \\\\&\\ge c_{d,p,\\Omega } \\delta ^{d/p}\|f(y)\| - \|f\|_{0,\\beta }\\left(\\int _{\\overline{\\Omega }\\cap B_\\delta } \|x-y\|^{\\beta p} \\, d x\\right)^{1/p} \\\\&\\ge c_{d,p,\\Omega } \\delta ^{d/p}\|f(y)\| - C_{d,p,\\beta } \\delta ^{\\beta + d/p}M,$ whereby it follows that $\\Vert f \\Vert _{L^\\infty (\\overline{\\Omega })} \\le C_{d,p,\\Omega }\\delta ^{-d/p} \\Vert f\\Vert _{L^p(\\overline{\\Omega })} + C_{d,p,\\beta }\\delta ^\\beta M.$ Choose $\\delta = \\left( \\frac{\\Vert f\\Vert _{L^p(\\overline{\\Omega })}}{M} \\right)^\\frac{p}{d + \\beta p}.$ In our proofs we use the $\\varepsilon $ -perturbed $p$ -Laplace equation.", "Let $\\varepsilon > 0$ and define $u_\\varepsilon $ as the solution of the boundary value problem $ {\\left\\lbrace \\begin{array}{ll}\\operatorname{div}\\left( \\gamma (x) \\left(\|\\nabla u_\\varepsilon \|^2 + \\varepsilon \\right)^\\frac{p-2}{2} \\nabla u_\\varepsilon \\right) = 0 &\\text{in $\\Omega $,} \\\\u_\\varepsilon = v &\\text{on $\\partial \\Omega $}.\\end{array}\\right.", "}$ By calculus of variations there exists a unique solution to (REF ) when $\\Omega $ is a bounded open set and $v \\in W^{1,p}(\\Omega )$ .", "The energy corresponding to (REF ) is $\\int _\\Omega \\gamma \\left(\|\\nabla u_\\varepsilon \|^2 + \\varepsilon \\right)^{p/2} \\, d x.$ Lemma 2.4 (Regularity result for perturbed equation) Suppose $\\Omega $ is a bounded open set with $C^{1,\\beta _1}$ boundary.", "Suppose that boundary values $v \\in C^{1,\\beta _1}(\\partial \\Omega )$ and conductivity $\\gamma $ is Hölder-continuous.", "Then $u_\\varepsilon \\in C^{1,\\beta _2}(\\overline{\\Omega })$ with some $0 < \\beta _2 < 1$ independent of $\\varepsilon $ and the norm $\|u_\\varepsilon \|_{1,\\beta _2}$ has an upper bound independent of $\\varepsilon $ .", "Furthermore, suppose that $\\Omega $ has $C^{2,\\beta _3}$ boundary, $v \\in C^{2,\\beta _3}(\\partial \\Omega )$ and that $\\nabla \\gamma $ is Hölder-continous.", "Then $ u_\\varepsilon \\in C^{2,\\beta _4}(\\overline{\\Omega })$ for some $0 < \\beta _4 < 1$ , which depends on $\\varepsilon $ .", "That $\\nabla u_\\varepsilon $ are Hölder-continuous with $\\beta _2$ and the corresponding norm independent of $\\varepsilon $ follows from [33].", "The $C^2$ –regularity is also standard; see for example [32] or [20].", "Lemma 2.5 (Convergence of the perturbed equations) Suppose $\\Omega $ is a bounded open set with $C^{1,\\beta }$ boundary, $0 < \\beta < 1$ , and boundary values $v$ are in $C^{1,\\beta }$ .", "Also assume that the conductivity $\\gamma $ is Hölder continuous.", "Then $u_\\varepsilon \\rightarrow u$ in $C^{1}(\\overline{\\Omega })$ .", "We first show that the energy (REF ) of the perturbed equation (REF ) converges to the energy of the non-perturbed equation (REF ): The solution $u$ minimises energy, and so we have $&\\int _\\Omega \\gamma \|\\nabla u\|^{p}\\, d x\\\\\\le &\\int _\\Omega \\gamma \|\\nabla u_\\varepsilon \|^{p}\\, d x\\\\\\le &\\int _\\Omega \\gamma \\left(\|\\nabla u_\\varepsilon \|^2+\\varepsilon \\right)^{p/2}\\, d x \\\\\\le &\\int _\\Omega \\gamma \\left(\|\\nabla u\|^2+\\varepsilon \\right)^{p/2}\\, d x \\\\\\le &\\int _\\Omega \\gamma \|\\nabla u\|^{p}\\, d x + C{\\left\\lbrace \\begin{array}{ll}\\int _\\Omega \\varepsilon \\left( 2\|\\nabla u\|^2 + \\varepsilon \\right)^{-1+p/2}\\, d x &\\text{ when } p > 2 \\\\\\int _\\Omega \\varepsilon ^{p/2} \\, d x &\\text{ when } p \\le 2,\\end{array}\\right.", "}$ where the latter integrals vanish as $\\varepsilon \\rightarrow 0$ since $\|\\nabla u\|$ is bounded (lemma REF ).", "We used the fact that $u_\\varepsilon $ minimises the perturbed energy (REF ), and also lemma REF with $q = 1+p/2 > 2$ when $p>2$ , and the fact that for non-negative numbers $(a+b)^q \\le a^q + b^q$ when $0 \\le q = p/2 \\le 1$ , in the final inequality.", "We established that the energy of $u_\\varepsilon $ converges to the energy of $u$ as $\\varepsilon \\rightarrow 0$ .", "Since $\\left\\Vert u_\\varepsilon \\right\\Vert _{L^\\infty }$ are uniformly bounded, it follows that $u_\\varepsilon $ are bounded in $W^{1,p}$ with weight.", "This implies that the sequence has a weakly converging subsequence with limit that has lesser or equal energy than $u$ .", "But $u$ uniquely minimises energy, so $u_\\varepsilon \\rightarrow u$ weakly in weighted $W^{1,p}$ .", "Since the weighted $L^p$ norms of $\\nabla u_\\varepsilon $ converge to weighted $L^p$ norm of $\\nabla u$ , and the sequence converges weakly, we get that $\\nabla u_\\varepsilon \\rightarrow \\nabla u$ strongly in weighted $L^{p}$ space; this is the Radon-Riesz property proven in [41], [43], [44].", "Since the weighted and standard norms are equivalent, $\\nabla u_\\varepsilon \\rightarrow \\nabla u$ strongly in standard $L^p$ .", "Since $\\nabla u_\\varepsilon $ are uniformly Hölder in $\\overline{\\Omega }$ (lemma REF ) we can use the interpolation lemma REF to deduce convergence of $\\nabla u_\\varepsilon $ to $\\nabla u$ in $C(\\overline{\\Omega })$ .", "The same argument holds for $u_\\varepsilon $ , since by Friedrichs-Poincaré inequality $u_\\varepsilon \\rightarrow u$ strongly in $L^p$ ." ], [ "Previous results on boundary determination for $p$ -Laplacian", "We use several results in the paper [45] and restate them here for convenience.", "Suppose $\\Omega \\subset \\mathbb {R}^d$ is an open bounded $C^1$ set and suppose $\\rho \\in C^1(\\mathbb {R}^d)$ is its boundary defining function; that is, $\\Omega = \\lbrace x \\in \\mathbb {R}^d ; \\rho (x) > 0 \\rbrace ,\\quad \\partial \\Omega = \\lbrace x \\in \\mathbb {R}^d ; \\rho (x) = 0 \\rbrace $ and $\\nabla \\rho \\ne 0$ on $\\partial \\Omega $ .", "By translation we may assume that we are trying to recover $\\nabla \\gamma $ at the origin $0 \\in \\partial \\Omega $ , and by rotation and scaling we may assume that $-\\nabla \\rho (0) = \\nu (0) = -e_d,$ the $d$ th coordinate vector.", "We define the map $f \\colon \\Omega \\rightarrow \\mathbb {R}^d_+$ by $f(x^{\\prime },x_d) = (x^{\\prime },\\rho (x))$ for $x = (x^{\\prime },x_d)$ .", "Since $\\rho $ is a $C^1$ function, the map is close to the identity near the origin, and hence invertible in some neighbourhood of the origin.", "We now introduce the $p$ -harmonic oscillating functions of Wolff [51].", "Lemma 3.1 Define the function $h \\colon \\mathbb {R}^d \\rightarrow \\mathbb {R}$ by $h(x)=e^{-x_d}a(x_1)$ , where $a \\colon \\mathbb {R}\\rightarrow \\mathbb {R}$ is the solution to the differential equation $a^{\\prime \\prime }(x_1) + V(a,a^{\\prime })a = 0$ with $V(a,a^{\\prime }) = \\frac{(2p-3)(a^{\\prime })^2 + (p-1)a^2}{(p-1)(a^{\\prime })^2 + a^2}.$ Then $\\Delta _p^1(h)=0$ and the function $a$ is smooth and periodic with period $\\lambda (p)$ , so that $\\int _0^\\lambda a(x_1)\\, d x_1 = 0$ .", "For proof of the lemma see [45].", "We define a smooth positive cutoff function $\\zeta \\in C^\\infty _0(\\mathbb {R}^d)$ , such that $\\zeta (x) = 1$ when $\|x\| < 1/2$ and $\\operatorname{supp}\\zeta \\subset B(0,1)$ .", "We write $v_M(x) = h(Nf(x))\\zeta (Mx),$ where $M$ and $N = N(M)$ are large positive numbers, $M = o(N)$ .", "We define $u_M$ to be the solutions of the initial value problem ${\\left\\lbrace \\begin{array}{ll}\\Delta _p^\\gamma (u_M) = 0 &\\text{in $\\Omega $,} \\\\u_M = v_M &\\text{on $\\partial \\Omega $}.\\end{array}\\right.", "}$ In [45] Salo and Zhong show that $M^{d-1}N^{1-p}\\int _\\Omega \\gamma \|\\nabla v_M\|^p \\, d x \\rightarrow c_p \\gamma (0)$ as $M \\rightarrow \\infty $ .", "The constant $c_p$ is explicit: $c_p = \\frac{K}{p}\\int _{\\mathbb {R}^{d-1}}(\\eta (x^{\\prime },0))^p\\, d x^{\\prime }; \\text{ with } K = \\lambda ^{-1} \\int _0^\\lambda \\left(a^2(t) + (a^{\\prime }(t))^2\\right)^{p/2}\\, d t.$ The proof also holds when $\\gamma $ is replaced by any other function continuous at 0, so as $M \\rightarrow \\infty $ we have $M^{d-1}N^{1-p}\\int _\\Omega \\partial _\\alpha \\gamma \|\\nabla v_M\|^p \\, d x \\rightarrow c_p \\partial _\\alpha \\gamma (0).$ Using [45] we get the following result: Lemma 3.2 Suppose $\\Omega \\subset \\mathbb {R}^d$ is a bounded open set with $C^1$ boundary, $x_0 \\in \\partial \\Omega $ , $d \\ge 2$ , and $g$ is continuous.", "Then $M^{d-1}N^{1-p}\\int _\\Omega g(x) \|\\nabla u_M\|^p \\, d x \\rightarrow c_p g(x_0)$ as $M \\rightarrow \\infty $ ." ], [ "Proof of the main result", "In this section we establish that the quantity $\\int _\\Omega \\partial _\\alpha \\gamma (x) \|\\nabla u\|^p \\, d x$ can be calculated from the measurements represented by the DN map, with the function $u$ solving the weighted $p$ -Laplace equation (REF ) with sufficiently smooth boundary values.", "Then it follows from lemma REF that we can recover $\\partial _\\alpha \\gamma (x_0)$ at any boundary point $x_0 \\in \\partial \\Omega $ .", "The main result, theorem REF , immediately follows.", "Theorem 4.1 (Rellich identity) Suppose $1<p<\\infty $ and that $\\Omega \\subset \\mathbb {R}^d$ is a bounded open set with $C^{2,\\beta }$ boundary for some $0 < \\beta < 1$ .", "Let $0 < \\gamma \\in C^1(\\overline{\\Omega })$ and let $u$ be a weak solution of the equation $\\Delta _p^\\gamma u = 0$ .", "Let $\\alpha \\in \\mathbb {R}^d$ .", "Then $\\int _\\Omega (\\partial _\\alpha \\gamma ) \|\\nabla u\|^p \\, d x = \\int _{\\partial \\Omega } \\gamma (\\alpha \\cdot \\nu ) \|\\nabla u\|^p \\, d S - p\\int _{\\partial \\Omega } (\\partial _\\alpha u) \\gamma \|\\nabla u\|^{p-2} \\partial _\\nu u \\, d S.$ Suppose we have the following identity for the solution $u_\\varepsilon $ of the perturbed $p$ -Laplace equation (REF ): $&\\int _\\Omega (\\partial _\\alpha \\gamma ) \\left( \|\\nabla u_\\varepsilon \|^2 + \\varepsilon \\right)^{p/2} \\, d x \\\\&= \\int _{\\partial \\Omega } \\gamma (\\alpha \\cdot \\nu ) \\left( \|\\nabla u_\\varepsilon \|^2 + \\varepsilon \\right)^{p/2} \\, d S - p\\int _{\\partial \\Omega } \\partial _\\alpha u_\\varepsilon \\gamma \\left( \|\\nabla u_\\varepsilon \|^2 + \\varepsilon \\right)^\\frac{p-2}{2} \\partial _\\nu u_\\varepsilon \\, d S.$ Then, when $\\varepsilon \\rightarrow 0$ , $\\nabla u_\\varepsilon \\rightarrow \\nabla u$ uniformly (by lemma REF ), so we get the claimed identity.", "We now prove the perturbed identity, integrating by parts twice: $\\int _{\\partial \\Omega } &(\\alpha \\cdot \\nu ) \\gamma \\left( \|\\nabla u_\\varepsilon \|^2 + \\varepsilon \\right)^{p/2} \\, d S - \\int _\\Omega (\\partial _\\alpha \\gamma ) \\left(\|\\nabla u_\\varepsilon \|^2 +\\varepsilon \\right)^{p/2} \\, d x \\\\= & \\int _\\Omega \\gamma \\partial _\\alpha \\left( \\left( \|\\nabla u_\\varepsilon \|^2 + \\varepsilon \\right)^{p/2} \\right) \\, d x \\\\= & p\\int _\\Omega \\gamma \\left( \|\\nabla u_\\varepsilon \|^2 + \\varepsilon \\right)^\\frac{p-2}{2} \\nabla u_\\varepsilon \\cdot \\nabla \\partial _\\alpha u_\\varepsilon \\, d x \\\\= & p \\int _{\\partial \\Omega } \\partial _\\alpha u_\\varepsilon \\gamma \\left( \|\\nabla u_\\varepsilon \|^2 + \\varepsilon \\right)^\\frac{p-2}{2} \\partial _\\nu u_\\varepsilon \\, d S \\\\&- p\\int _\\Omega \\operatorname{div}\\left( \\gamma \\left( \|\\nabla u_\\varepsilon \|^2 + \\varepsilon \\right)^\\frac{p-2}{2} \\nabla u_\\varepsilon \\right) \\partial _\\alpha u_\\varepsilon \\, d x \\\\= &p\\int _{\\partial \\Omega } \\partial _\\alpha u_\\varepsilon \\gamma \\left( \|\\nabla u_\\varepsilon \|^2 + \\varepsilon \\right)^\\frac{p-2}{2} \\partial _\\nu u_\\varepsilon \\, d S.$ We need to recover $\\gamma \|_{\\partial \\Omega }$ and $\\nabla u\|_{\\partial \\Omega }$ to recover $\\int _\\Omega \\partial _\\alpha \\gamma \|\\nabla u\|^p \\, d x$ from the DN map.", "Supposing $\\Omega $ is an open set with $C^1$ boundary and $\\gamma $ is continuous, we can recover $\\gamma (x_0)$ for all boundary points $x_0$ .", "This follows directly from [45].", "To calculate $\\partial _\\nu u$ we use $C^{1,\\beta }$ boundary values $v$ so that the solution $u$ of the boundary value problem (REF ) is continuously differentiable up to the boundary (by lemma REF ), so $\\partial _\\nu u$ is defined pointwise.", "Recovering $\\partial _\\nu u$ from the strong definition of the DN map is straightforward, and we do so in lemma REF .", "However, we do not a priori have access to the strong DN map, and must recover it from the weak definition REF .", "This we do in lemma REF , which requires more smoothness on $\\Omega $ and boundary values.", "We have $\\Lambda _\\gamma ^s (v) = \\gamma \|\\nabla u\|^{p-2}\\partial _\\nu u,$ from which by expanding $\\nabla u$ we get the equation $\\gamma \|\\nabla _T u + \\nabla _\\nu u\|^{p-2}\\partial _\\nu u = \\Lambda _\\gamma ^s (v).$ Taking absolute values and reorganising we have $(\|\\nabla _T u\|^2 + \|\\nabla _\\nu u\|^2)^{\\frac{p-2}{2}}\|\\partial _\\nu u\| = \\frac{1}{\\gamma }\|\\Lambda _\\gamma ^s (v)\|.$ We write the left hand side as a function of $\|\\partial _\\nu u\| = \|\\nabla _\\nu u\|= t$ : $F(t) = (\|\\nabla _T u\|^2 + t^2)^{\\frac{p-2}{2}}t = \\frac{1}{\\gamma }\|\\Lambda _\\gamma ^s (v)\|.$ There is a unique $t_0 \\in [0,\\infty [$ with $F(t_0) = \\frac{1}{\\gamma }\|\\Lambda _\\gamma ^s (v)\|$ .", "To see this, observe that $\\lim _{t \\rightarrow 0} F(t) = 0$ , $\\lim _{t \\rightarrow \\infty } F(t) = \\infty $ , and that the function $F$ is strictly increasing and continuous.", "Returning to the original formulation and expanding $\|\\nabla u\|^2$ we therefore get $\\gamma (\|\\nabla _T u\|^2 + t_0^2)^\\frac{p-2}{2} \\partial _\\nu u = \\Lambda _\\gamma ^s (v),$ from which we can solve $\\partial _\\nu u$ .", "We have proven the following lemma: Lemma 4.2 Suppose $\\Omega $ is a bounded open set with $C^{1,\\beta }$ boundary.", "Let the boundary voltage $v \\in C^{1,\\beta }$ be known.", "Then, for any boundary point $x_0 \\in \\partial \\Omega $ , we can compute $\\partial _\\nu u (x_0)$ from the strong DN map with the following algorithm: Solve $t$ from the equation $ (\|\\nabla _T u (x_0)\|^2 + t^2)^{\\frac{p-2}{2}}t = \\frac{1}{\\gamma (x_0)}\|\\Lambda _\\gamma ^s (v)(x_0)\|.", "$ Set $\\partial _\\nu u (x_0) = t\\operatorname{sign}(\\Lambda _\\gamma ^s (v)(x_0))$ .", "The following remark is not used in this paper, but in general locating the points where $\\nabla u = 0$ is interesting for $p$ -harmonic functions $u$ .", "Remark 4.3 We know the boundary points $x_0$ where $\\nabla u(x_0) = 0$ , since $\\partial _\\nu u(x_0) = 0$ if and only if $\\Lambda _\\gamma ^s (v)(x_0) = 0$ , and the Dirichlet data determines $\\nabla _T u$ at boundary points.", "We still need to recover the strong DN map from the weak one.", "Lemma 4.4 Suppose that $\\Omega $ has $C^{2,\\beta }$ boundary, boundary values $v \\in C^{2,\\beta }(\\partial \\Omega )$ and that $\\nabla \\gamma $ is Hölder-continous.", "Then we can recover the pointwise values of the strong DN map $\\gamma (x_0) \\left\|\\nabla u(x_0)\\right\|^{p-2}\\partial _\\nu u (x_0)$ from the weak DN map (see definition REF ) $\\left\\langle \\Lambda _\\gamma ^w(v), g \\right\\rangle = \\int _\\Omega \\gamma \|\\nabla u\|^{p-2} \\nabla u \\cdot \\nabla \\tilde{g} \\, d x.$ For $x_0 \\in \\partial \\Omega $ define test functions $\\eta _\\delta \\colon \\mathbb {R}^d \\rightarrow \\mathbb {R}$ by $\\eta _\\delta (x) = c_d \\delta ^{-d} (\\delta - \|x-x_0\|)\\chi _{B(x_0,\\delta )}$ where $c_d > 0$ is selected so that $\\int _{L} \\eta _\\delta \\, d x = 1$ , where $L$ is the tangent space of $\\partial \\Omega $ at $x_0$ .", "Then $\\eta _\\delta $ lies in all first order Sobolev spaces with $\|\\nabla \\eta _\\delta (x)\| = c_d \\delta ^{-d}$ in $B(x_0,\\delta )$ , and 0 elsewhere.", "Suppose $\\nabla u(x_0)=0$ .", "By Hölder continuity of $\|\\nabla u\|^{p-1}$ we get $\\left\|\\langle \\Lambda _\\gamma ^w (v),\\eta _\\delta \\rangle \\right\| &\\lesssim \\Vert \|\\nabla u\|^{p-2} \\nabla u \\cdot \\nabla \\eta _\\delta \\Vert _{L^1(\\Omega \\cap B(x_0,\\delta ))} \\\\&\\lesssim \\Vert \|\\nabla u\|^{p-1} \|\\nabla \\eta _\\delta \| \\Vert _{L^1(B(x_0,\\delta ))} \\lesssim \\Vert \\delta ^{\\beta _1} \\delta ^{-d} \\Vert _{L^1(\\Omega \\cap B(x_0,\\delta ))} \\\\&\\lesssim \\delta ^{\\beta _1 - d} \\delta ^{d} = \\delta ^{\\beta _1} \\rightarrow 0 = \\Lambda ^s_\\gamma (x_0)$ as $\\delta \\rightarrow 0$ .", "Suppose, then, that $\\nabla u (x_0) \\ne 0$ .", "Since $\\nabla u_\\varepsilon \\rightarrow \\nabla u$ as $\\varepsilon \\rightarrow 0$ , there is $a > 0$ such that $\|\\nabla u_\\varepsilon (x_0)\| \\ge \|\\nabla u (x_0)\|/2$ holds whenever $a > \\varepsilon \\ge 0$ .", "We next prove that the equality $\\gamma (x_0) \\left(\\left\|\\nabla u_\\varepsilon (x_0)\\right\|^2 + \\varepsilon \\right)^\\frac{p-2}{2}\\partial _\\nu u_\\varepsilon (x_0) = \\lim _{\\delta \\rightarrow 0} \\int _\\Omega \\gamma \\left( \|\\nabla u_\\varepsilon \|^2 + \\varepsilon \\right)^\\frac{p-2}{2} \\nabla u_\\varepsilon \\cdot \\nabla \\eta _\\delta \\, d x$ holds.", "By taking the limit $\\varepsilon \\rightarrow 0$ we get the original claim if the rate of convergence in (REF ) is uniform in $\\varepsilon $ .", "By integrating by parts we get ${\\int _\\Omega } \\gamma &\\left( \|\\nabla u_\\varepsilon \|^2 + \\varepsilon \\right)^\\frac{p-2}{2} \\nabla u_\\varepsilon \\cdot \\nabla \\eta _\\delta \\, d x \\\\=&-{\\int _\\Omega } \\operatorname{div}\\left( \\gamma \\left( \|\\nabla u_\\varepsilon \|^2 + \\varepsilon \\right)^\\frac{p-2}{2}\\nabla u_\\varepsilon \\right) \\eta _\\delta \\, d x \\\\&+ {\\int _{\\partial \\Omega }} \\eta _\\delta \\gamma \\left(\\left\|\\nabla u_\\varepsilon \\right\|^2 + \\varepsilon \\right)^\\frac{p-2}{2}\\partial _\\nu u_\\varepsilon \\, d S \\\\= &{\\int _{\\partial \\Omega \\cap B(x_0,\\delta )}} \\eta _\\delta \\gamma \\left(\\left\|\\nabla u_\\varepsilon \\right\|^2 + \\varepsilon \\right)^\\frac{p-2}{2}\\partial _\\nu u_\\varepsilon \\, d S, $ since $u_\\varepsilon $ solves the perturbed equation, and the integral (REF ) equals ${\\int _{\\partial \\Omega \\cap B(x_0,\\delta )}} &\\eta _\\delta \\gamma (x_0) \\left(\\left\|\\nabla u_\\varepsilon (x_0) \\right\|^2 + \\varepsilon \\right)^\\frac{p-2}{2}\\nu (x_0) \\cdot \\nabla u_\\varepsilon (x_0) \\, d S \\\\+& {\\int _{\\partial \\Omega \\cap B(x_0,\\delta )}} \\eta _\\delta \\bigg ( \\gamma \\left(\\left\|\\nabla u_\\varepsilon \\right\|^2 + \\varepsilon \\right)^\\frac{p-2}{2}\\partial _\\nu u_\\varepsilon \\\\&-\\gamma (x_0) \\left(\\left\|\\nabla u_\\varepsilon (x_0) \\right\|^2 + \\varepsilon \\right)^\\frac{p-2}{2} \\partial _\\nu u_\\varepsilon (x_0) \\bigg )\\, d S\\\\=&\\left(\\gamma (x_0) \\left(\\left\|\\nabla u_\\varepsilon (x_0)\\right\|^2 + \\varepsilon \\right)^\\frac{p-2}{2}\\partial _\\nu u_\\varepsilon (x_0) + o_{\\delta \\rightarrow 0}(1)\\right){\\int _{\\partial \\Omega }} \\eta _\\delta \\, d S,$ since $\\gamma $ and $\\nabla u_\\varepsilon $ are continuous.", "By normalisation of the test function this proves the equality (REF ).", "The rate of convergence is uniform in $\\varepsilon $ when $\\delta $ is so small that $\|\\nabla u_\\varepsilon (x)\|$ is bounded away from zero for $x \\in B(x_0,\\delta )$ , since then the following $\\varepsilon $ -indexed family of functions (REF ) are equicontinuous with parameter $\\varepsilon $ : $x \\mapsto \\gamma (x) \\left(\|\\nabla u_\\varepsilon (x)\|^2 +\\varepsilon \\right)^{\\frac{p-2}{2}}\\partial _\\nu u_\\varepsilon (x).$ The family is Hölder-continuous on $\\partial \\Omega $ when $x - x_0$ is so small that $\|\\nabla u_\\varepsilon (x)\| \\ge m > 0$ for some $m$ and all $\\varepsilon < a$ : By lemma REF the functions $x \\mapsto \\nabla u_\\varepsilon (x)$ are equicontinuous with parameter $\\varepsilon $ , and the functions $w \\mapsto \\gamma \\left( \|w\|^2 + \\varepsilon \\right)^\\frac{p-2}{2} w \\cdot \\nu $ are equicontinuous when $\|w\|$ is bounded from above and away from zero.", "Remark 4.5 We have shown that $ \\int _\\Omega (\\partial _\\alpha \\gamma ) \|\\nabla u\|^p \\, d x = \\int _{\\partial \\Omega } \\gamma (\\alpha \\cdot \\nu ) \|\\nabla u\|^p \\, d S - p\\int _{\\partial \\Omega } (\\partial _\\alpha u) \\Lambda _\\gamma ^s (u) \\, d S. $ Since we know $\\gamma \|_{\\partial \\Omega }$ by [45], $\\nabla u\|_{\\partial \\Omega }$ by lemma REF and the strong DN map by lemma REF , we also know $\\int _\\Omega (\\partial _\\alpha \\gamma ) \|\\nabla u\|^p \\, d x.$ Remark 4.6 In using the Rellich identity we need to know the values of the conductivity $\\gamma $ on the entire boundary $\\partial \\Omega $ .", "The gradient $\\nabla u_M$ should be large near $x_0$ and small elsewhere (we have not made this precise), so our proof is essentially local in nature.", "The main theorem REF now follows, since the value of the integral $\\int _\\Omega \\partial _\\alpha \\gamma \|\\nabla u_M\|^p \\, d x$ is known (see remark REF ) and it converges to $\\partial _\\alpha \\gamma (x_0)$ by lemma REF ." ], [ "Acknowledgements", "The research was partly supported by Academy of Finland.", "Part of the research was done during a visit to the Institut Mittag-Leffler (Djursholm, Sweden).", "The author is grateful to Mikko Salo and Joonas Ilmavirta for several discussions." ] ]	1403.0428
[ [ "The jumping phenomenon of the dimensions of Bott-Chern cohomology groups\n and Aeppli cohomology groups" ], [ "Abstract Let $X$ be a compact complex manifold, and let $\\pi: \\mathcal{X} \\rightarrow B$ be a small deformation of $X$, the dimensions of the Bott-Chern cohomology groups $H_{\\rm BC}^{p,q}(X(t))$ and Aeppli cohomology groups $H_{\\rm A}^{p,q}(X(t))$ may vary under this deformation.", "In this paper, we will study the deformation obstructions of a $(p,q)$ class in the central fiber $X$.", "In particular, we obtain an explicit formula for the obstructions and apply this formula to the study of small deformations of the Iwasawa manifold." ], [ "Introduction", "Let $X$ be a compact complex manifold and $\\pi :\\mathcal {X}\\rightarrow B$ be a family of complex manifolds such that $\\pi ^{-1}(0)=X$ .", "Let $X_{t}=\\pi ^{-1}(t)$ denote the fibre of $\\pi $ over the point $t\\in B$ .", "In [9], the author has studied the jumping phenomenon of hodge numbers $h^{p,q}$ of $X$ by studying the deformation obstructions of a $(p,q)$ class in the central fiber $X$ .", "In particular, the author obtained an explicit formula for the obstructions and apply this formula to the study of small deformations of the Iwasawa manifold.", "Besides the Hodge numbers, the dimensions of Bott-Chern cohomology groups and the dimensions of Aeppli cohomology groups are also important invariant of complex structures.", "In [2], D. Angella has studied the small deformations of the Iwasawa manifold and found that the dimensions of Bott-Chern cohomology groups and the dimensions of Aeppli cohomology groups are not deformation invariants.", "In this paper, we will study the Bott-Chern cohomology and Aeppli cohomology by studying the hypercohomology of a complex ${\\mathcal {B}}_{p,q}^{\\bullet }$ constructed in [7].", "In [7], M. Schweitzer proved that $H^{p,q}_{\\rm BC}(X)\\cong {\\mathbb {H}}^{p+q}(X,{\\mathcal {B}}_{p,q}^{\\bullet }),$ and $H^{p,q}_{\\rm A}(X)\\cong {\\mathbb {H}}^{p+q+1}(X,{\\mathcal {B}}_{p+1,q+1}^{\\bullet }).$ As the author did in [9], we will such study the jumping phenomenons from the viewpoint of obstruction theory.", "More precisely, for a certain small deformation $\\mathcal {X}$ of $X$ parameterized by a basis $B$ and a certain class $[\\theta ]$ of the hypercohomology group ${\\mathbb {H}}^{l}(X,{\\mathcal {B}}_{p,q}^{\\bullet })$ , we will try to find out the obstruction to extend it to an element of the relative hypercohomology group group ${\\mathbb {H}}^{l}({\\mathcal {X}},{\\mathcal {B}}_{p,q;{\\mathcal {X}}/B}^{\\bullet })$ .", "We will call those elements which have non trivial obstruction the obstructed elements.", "And then we will see that these elements will play an important role when we study the jumping phenomenon.", "Because we will see that the existence of the obstructed elements is a sufficient condition for the variation of the dimensions of Bott-Chern cohomology and Aeppli cohomology.", "In $§2$ we will summarize the results of M. Schweitzer about Bott-Chern cohomology and Aeppli cohomology, from which we can define the relative Bott-Chern cohomology and Aeppli cohomology on $X_n$ where $X_n$ is the $n$ th order deformation of $\\pi :\\mathcal {X}\\rightarrow B$ .", "We will also introduce some important maps which will be used in the calculation of the obstructions in $§4$ .", "In $§3$ , we will try to explain why we need to consider the obstructed elements.", "The relation between the jumping phenomenon of Bott-Chern cohomology and Aeppli cohomology and the obstructed elements is the following.", "Theorem 1.1 Let $\\pi : \\mathcal {X}\\rightarrow B$ be a small deformation of the central fibre compact complex manifold $X$ .", "Now we consider $\\dim {\\mathbb {H}}^l(X(t),{\\mathcal {B}}_{p,q;t}^{\\bullet })$ as a function of $t\\in B$ .", "It jumps at $t=0$ if there exists an element $[\\theta ]$ either in ${\\mathbb {H}}^l(X,{\\mathcal {B}}_{p,q}^{\\bullet })$ or in ${\\mathbb {H}}^{l-1}(X,{\\mathcal {B}}_{p,q}^{\\bullet }) $ and a minimal natural number $n\\ge 1$ such that the n-th order obstruction $ o_{n}([\\theta ])\\ne 0.$ In $§4$ we will get a formula for the obstruction to the extension we mentioned above.", "Theorem 1.2 Let $\\pi :\\mathcal {X}\\rightarrow B$ be a deformation of $\\pi ^{-1}(0)=X$ , where $X$ is a compact complex manifold.", "Let $\\pi _{n}:X_{n}\\rightarrow B_{n}$ be the $n$ th order deformation of $X$ .", "For arbitrary $[\\theta ]$ belongs to ${\\mathbb {H}}^l(X,{\\mathcal {B}}_{p,q}^{\\bullet })$ , suppose we can extend $[\\theta ]$ to order $n-1$ in ${\\mathbb {H}}^l(X_{n-1},{\\mathcal {B}}_{p,q;X_{n-1}/B_{n-1}}^{\\bullet })$ .", "Denote such element by $[\\theta _{n-1}]$ .", "The obstruction of the extension of $[\\theta ]$ to $n$ th order is given by: $ o_{n}([\\theta ])=-\\partial ^{\\bar{\\partial },{\\mathcal {B}}}_{X_{n-1}/B_{n-1}} \\circ \\kappa _{n} \\circ \\partial ^{{\\mathcal {B}},\\bar{\\partial }}_{X_{n-1}/B_{n-1}}([\\theta _{n-1}])- \\bar{\\partial }^{{\\partial },{\\mathcal {B}}}_{X_{n-1}/B_{n-1}} \\circ \\bar{\\kappa }_{n} \\circ \\bar{\\partial }^{{\\mathcal {B}},{\\partial }}_{X_{n-1}/B_{n-1}}([\\theta _{n-1}]),$ where $\\kappa _{n}$ is the $n$ th order Kodaira-Spencer class and $\\bar{\\kappa }_{n}$ is the $n$ th order Kodaira-Spencer class of the deformation $\\bar{\\pi }:\\bar{{\\mathcal {X}}} \\rightarrow \\bar{B}$ .", "$\\partial ^{\\bar{\\partial },{\\mathcal {B}}}_{X_{n-1}/B_{n-1}}$ , $\\bar{\\partial }^{{\\partial },{\\mathcal {B}}}_{X_{n-1}/B_{n-1}}$ , $\\partial ^{{\\mathcal {B}},\\bar{\\partial }}_{X_{n-1}/B_{n-1}}$ and $\\bar{\\partial }^{{\\mathcal {B}},{\\partial }}_{X_{n-1}/B_{n-1}}$ are the maps defined in $§2£¤$ .", "In $§5$ we will use this formula to study carefully the example given by Iku Nakamura and D. Angella, i.e.", "the small deformation of the Iwasama manifold and discuss some phenomenons." ], [ "The Bott-Chern(Aeppli) Cohomology and Bott-Chern(Aeppli) Hypercohomology ", "All the details of this subsection can be found in [7].", "Let $X$ be a compact complex manifold.", "The Dolbeault cohomology groups $H^{p,q}_{{\\bar{\\partial }}}(X)$ , and more generally the terms $E_r^{p,q}(X)$ in the Frölicher spectral sequence [5], are well-known finite dimensional invariants of the complex manifold $X$ .", "On the other hand, the Bott-Chern and Aeppli cohomologies define additional complex invariants of $X$ given, respectively, by [1], [3] $H^{p,q}_{\\rm BC}(X)={\\ker \\lbrace d\\colon {\\mathcal {A}}^{p,q}(X)\\longrightarrow {\\mathcal {A}}^{p+q+1}(X) \\rbrace \\over {\\rm im}\\,\\lbrace \\partial {\\bar{\\partial }}\\colon {\\mathcal {A}}^{p-1,q-1}(X)\\longrightarrow {\\mathcal {A}}^{p,q}(X) \\rbrace },$ and $H^{p,q}_{\\rm A}(X)={\\ker \\lbrace \\partial {\\bar{\\partial }}\\colon {\\mathcal {A}}^{p,q}(X)\\longrightarrow {\\mathcal {A}}^{p+1,q+1}(X) \\rbrace \\over {\\rm im}\\,\\lbrace \\partial \\colon {\\mathcal {A}}^{p-1,q}(X)\\longrightarrow {\\mathcal {A}}^{p,q}(X) \\rbrace +{\\rm im}\\,\\lbrace {\\bar{\\partial }}\\colon {\\mathcal {A}}^{p,q-1}(X)\\longrightarrow {\\mathcal {A}}^{p,q}(X) \\rbrace }.$ By the Hodge theory developed in [7], all these complex invariants are also finite dimensional and one has the isomorphisms $H^{p,q}_{\\mathrm {A}}(X)\\cong H^{n-q,n-p}_{\\mathrm {\\rm BC}}(X)$ .", "Notice that $H^{q,p}_{\\mathrm {\\rm BC}}(X) \\cong H^{p,q}_{\\mathrm {\\rm BC}}(X)$ by complex conjugation.", "For any $r\\ge 1$ and for any $p,q$ , there are natural maps $H^{p,q}_{\\mathrm {\\rm BC}}(X) \\longrightarrow E_r^{p,q}(X)\\quad \\quad \\quad {\\mbox{\\rm and }}\\quad \\quad \\quad E_r^{p,q}(X) \\longrightarrow H^{p,q}_{\\mathrm {A}}(X).$ Recall that $E_1^{p,q}(X)\\cong H^{p,q}_{{\\bar{\\partial }}}(X)$ and that the terms for $r=\\infty $ provide a decomposition of the de Rham cohomology of the manifold, i.e.", "$H^k_{\\rm dR}(X,\\mathbb {C})\\cong \\oplus _{p+q=k} E_{\\infty }^{p,q}(X)$ .", "From now on we shall denote by $h^{p,q}_{\\mathrm {\\rm BC}}(X)$ the dimension of the cohomology group $H^{p,q}_{\\mathrm {\\rm BC}}(X)$ .", "The Hodge numbers will be denoted simply by $h^{p,q}(X)$ and the Betti numbers by $b_{k}(X)$ .", "For any given $p\\ge 1,q\\ge 1$ , we define the complex of sheaves ${\\mathcal {L}}^{\\bullet }_{p,q}$ by ${\\mathcal {L}}^k_{p,q}=\\bigoplus _{\\begin{array}{c}r+s=k \\\\ r<p,s<q\\end{array}}{\\mathcal {A}}^{r,s}\\quad \\mbox{if }k\\le p+q-2,$ ${\\mathcal {L}}^{k-1}_{p-1,q-1}=\\bigoplus _{\\begin{array}{c}r+s=k \\\\ r\\ge p,s\\ge q\\end{array}}{\\mathcal {A}}^{r,s}\\quad \\mbox{if }k\\ge p+q,$ and the differential: ${\\mathcal {L}}_{p,q}^0 \\stackrel{pr_{{\\mathcal {L}}^1_{p,q}}\\circ d}{\\rightarrow } {\\mathcal {L}}^1_{p,q}\\stackrel{pr_{{\\mathcal {L}}^2_{p,q}}\\circ d}{\\rightarrow } \\ldots \\rightarrow {\\mathcal {L}}^{k-2}_{p,q}\\stackrel{\\partial \\bar{\\partial }}{\\rightarrow }{\\mathcal {L}}^{k-1}_{p,q}\\stackrel{d}{\\rightarrow }{\\mathcal {L}}^k_{p,q}\\stackrel{d}{\\rightarrow }\\ldots $ Then by the above construction, we have the following isomorphisms $H^{p,q}_{\\rm BC}(X)=H^{p+q-1}({\\mathcal {L}}_{p,q}^{\\bullet }(X))\\cong {\\mathbb {H}}^{p+q-1}(X,{\\mathcal {L}}_{p,q}^{\\bullet }),$ $H^{p,q}_{\\rm A}(X)=H^{p+q}({\\mathcal {L}}_{p+1,q+1}^{\\bullet }(X))\\cong {\\mathbb {H}}^{p+q}(X,{\\mathcal {L}}_{p+1,q+1}^{\\bullet }),$ because ${\\mathcal {L}}^k_{p,q}$ are soft.", "We define a sub complex ${\\mathcal {S}}_{p,q}^{\\bullet }$ of ${\\mathcal {L}}_{p,q}^{\\bullet }$ by : $({{\\mathcal {S}}^{\\prime }}_p^{\\bullet },\\partial ):\\; {\\mathcal {O}}\\rightarrow \\Omega ^1 \\rightarrow \\ldots \\rightarrow \\Omega ^{p-1}\\rightarrow 0,\\qquad ({{\\mathcal {S}}^{\\prime \\prime }}_q^{\\bullet },\\bar{\\partial }):\\; \\bar{{\\mathcal {O}}}\\rightarrow \\bar{\\Omega }^{1}\\rightarrow \\ldots \\rightarrow \\bar{\\Omega }^{q-1}\\rightarrow 0,$ ${\\mathcal {S}}_{p,q}^{\\bullet }={{\\mathcal {S}}^{\\prime }}_p^{\\bullet }+{{\\mathcal {S}}^{\\prime \\prime }}_q^{\\bullet }:\\; {\\mathcal {O}}+\\bar{{\\mathcal {O}}} \\rightarrow \\Omega ^1\\oplus \\bar{\\Omega }^{1}\\rightarrow \\ldots \\Omega ^{p-1}\\oplus \\bar{\\Omega }^{p-1}\\rightarrow \\bar{\\Omega }^{p}\\rightarrow \\ldots \\rightarrow \\bar{\\Omega }^{q-1}\\rightarrow 0.$ Note that the inclusion ${\\mathcal {S}}^{\\bullet }\\subset {\\mathcal {L}}^{\\bullet }$ is an quasi-isomorphism [7].", "There is another complex ${\\mathcal {B}}^{\\bullet }_{p,q}$ used in [7] which is defined by: $ {\\mathcal {B}}_{p,q}^{\\bullet } :\\; {(+,-)}{\\rightarrow }{\\mathcal {O}}\\oplus \\bar{{\\mathcal {O}}} \\rightarrow \\Omega ^1\\oplus \\bar{\\Omega }^{1}\\rightarrow \\ldots \\Omega ^{p-1}\\oplus \\bar{\\Omega }^{p-1}\\rightarrow \\bar{\\Omega }^{p}\\rightarrow \\ldots \\rightarrow \\bar{\\Omega }^{q-1}\\rightarrow 0.$ and the following morphism of from ${\\mathcal {B}}^{\\bullet }_{p,q}$ to ${\\mathcal {S}}^{\\bullet }_{p,q}[1]$ is a quasi-isomorphism [7]: $\\begin{array}{ccccccc}\\stackrel{(+,-)}{\\rightarrow }&{\\mathcal {O}}\\oplus \\bar{{\\mathcal {O}}}&\\rightarrow &\\Omega ^1\\oplus \\bar{\\Omega }^{1}&\\rightarrow &\\ldots \\\\\\downarrow &&\\downarrow +&&\\downarrow &&\\\\0&\\rightarrow &{\\mathcal {O}}+\\bar{{\\mathcal {O}}}&\\rightarrow &\\Omega ^1\\oplus \\bar{\\Omega }^{1}&\\rightarrow &\\ldots .\\end{array}$ Therefore we have: $H^{p,q}_{\\rm BC}(X)\\cong {\\mathbb {H}}^{p+q}(X,{\\mathcal {L}}_{p,q}^{\\bullet }[1])\\cong {\\mathbb {H}}^{p+q}(X,{\\mathcal {S}}_{p,q}^{\\bullet }[1])\\cong {\\mathbb {H}}^{p+q}(X,{\\mathcal {B}}_{p,q}^{\\bullet }),$ and $H^{p,q}_{\\rm A}(X)\\cong {\\mathbb {H}}^{p+q}(X,{\\mathcal {L}}_{p+1,q+1}^{\\bullet })\\cong {\\mathbb {H}}^{p+q}(X,{\\mathcal {S}}_{p+1,q+1}^{\\bullet })\\cong {\\mathbb {H}}^{p+q+1}(X,{\\mathcal {B}}_{p+1,q+1}^{\\bullet }),$" ], [ "The Relative Bott-Chern Cohomology and Aeppli Cohomology of $X_n$ ", "Let $\\pi :\\mathcal {X}\\rightarrow B$ be a deformation of $\\pi ^{-1}(0)=X$ , where $X$ is a compact complex manifold.", "For every integer $n\\ge 0$ , denote by $B_{n}=Spec\\,\\mathcal {O}_{B,0}/m_{0}^{n+1}$ the $n$ th order infinitesimal neighborhood of the closed point $0\\in B$ of the base $B$ .", "Let $X_{n}\\subset \\mathcal {X}$ be the complex space over $B_{n}$ .", "Let $\\pi _{n}:X_{n}\\rightarrow B_{n}$ be the $n$ th order deformation of $X$ .", "Denote $\\pi ^{}(m_{0})$ by $\\mathcal {M}_{0}$ .", "If we take the complex conjugation, we have another complex structure of the differential manifold of $\\mathcal {X}$ , we denote this manifold by $\\bar{\\mathcal {X}}$ and $\\pi $ induce a deformation $\\bar{\\pi }:\\bar{\\mathcal {X}}\\rightarrow \\bar{B}$ of $\\bar{X}$ .", "Then we have $\\bar{X}_{n}$ and $\\bar{\\pi }_{n}:\\bar{X}_{n}\\rightarrow \\bar{B}_{n}.$ Let ${\\mathcal {C}}^{\\omega }_B$ be the sheaf of $-valued real analytic functions on $ B$.", "Denote $ OX= (CB)$, $ OX= (CB).$ Let $ m0$ be the maximal idea of $ CB,0$ and$ M0= (m0) $, $ M0= (m0) $ .", "For any sheaf of $ OX$(resp.", "$ OX$) module $ F$.", "Denote $ F=FOX OX $ (resp.", "$ F=FOX OX).$ Let $ OXn= OX,0/ (M0)n+1$£¬ $ OXn= OX,0/ (M0)n+1$.", "For any sheaf of $ OXn$(resp.", "$ OXn$) module $ F$.", "Denote $ F=FOXn OXn$ (resp.", "$ F=FOXn OXn).$$ For any given $p \\ge 1, q \\ge 1$ , We define the complex ${\\mathcal {S}}^{\\bullet }_{X_n/B_n}={\\mathcal {S}}_{p,q;X_n/B_n}^{\\bullet }$ by: $({{\\mathcal {S}}^{\\prime }}_{p;X_n/B_n}^{\\bullet },\\partial _{X_n/B_n} ):\\; {\\mathcal {O}}^{\\omega }_{X_n} \\rightarrow \\Omega _{X_n/B_n}^{1;\\omega } \\rightarrow \\ldots \\rightarrow \\Omega _{X_n/B_n}^{p-1;\\omega }\\rightarrow 0,$ $ ({{\\mathcal {S}}^{\\prime \\prime }}_{q;X_n/B_n}^{\\bullet },\\bar{\\partial }_{X_n/B_n}):\\; {\\mathcal {O}}^{\\omega }_{\\bar{X}_n} \\rightarrow \\bar{\\Omega }_{X_n/B_n}^{1;\\omega }\\rightarrow \\ldots \\rightarrow \\bar{\\Omega }_{X_n/B_n}^{q-1;\\omega }\\rightarrow 0,$ ${\\mathcal {S}}_{p,q;X_n/B_n}^{\\bullet }={{\\mathcal {S}}^{\\prime }}_{p;X_n/B_n}^{\\bullet }+{{\\mathcal {S}}^{\\prime \\prime }}_{q;X_n/B_n}^{\\bullet }:\\; $ ${\\mathcal {O}}^{\\omega }_{X_n}+{{\\mathcal {O}}}_{\\bar{X}_n}^{\\omega } \\rightarrow \\Omega _{X_n/B_n}^{1;^{\\omega }}\\oplus \\bar{\\Omega }_{X_n/B_n}^{1;\\omega }\\rightarrow \\ldots \\Omega _{X_n/B_n}^{p-1;\\omega }\\oplus \\bar{\\Omega }_{X_n/B_n}^{p-1;\\omega }\\rightarrow \\bar{\\Omega }_{X_n/B_n}^{p;\\omega }\\rightarrow \\ldots \\rightarrow \\bar{\\Omega }_{X_n/B_n}^{q-1;\\omega }\\rightarrow 0.$ We can also define ${\\mathcal {B}}^{\\bullet }_{p,q;X_n/B_n}$ by: $ {\\mathcal {B}}_{p,q;X_n/B_n}^{\\bullet } :\\; {\\omega }_{B_n} \\stackrel{(+,-)}{\\rightarrow } {\\mathcal {O}}^{\\omega }_{X_n} \\oplus {\\mathcal {O}}^{\\omega }_{\\bar{X}_n} \\rightarrow \\Omega _{X_n/B_n}^{1;\\omega }\\oplus \\bar{\\Omega }_{X_n/B_n}^{1;\\omega }\\rightarrow \\ldots \\Omega _{X_n/B_n}^{p-1;\\omega }\\oplus \\bar{\\Omega }_{X_n/B_n}^{p-1;\\omega } $ $\\rightarrow \\bar{\\Omega }_{X_n/B_n}^{p;\\omega }\\rightarrow \\ldots \\rightarrow \\bar{\\Omega }_{X_n/B_n}^{q-1;\\omega }\\rightarrow 0,$ where ${\\omega }_{B_n} = \\pi ^{-1} ({\\mathcal {C}}^{\\omega }_{B,0}/(m^{\\omega }_{0})^{n+1}).$ Then the Relative Bott-Chern cohomology and Aeppli cohomology of $X_n$ is defined by $H^{p,q}_{\\rm BC}(X_n/B_n) \\cong {\\mathbb {H}}^{p+q}(X,{\\mathcal {S}}_{p,q;X_n/B_n}^{\\bullet }[1])\\cong {\\mathbb {H}}^{p+q}(X_n,{\\mathcal {B}}_{p,q;X_n/B_n}^{\\bullet }),$ and $H^{p,q}_{\\rm A}(X_n/B_n) \\cong {\\mathbb {H}}^{p+q}(X,{\\mathcal {S}}_{p+1,q+1;X_n/B_n}^{\\bullet })\\cong {\\mathbb {H}}^{p+q+1}(X_n,{\\mathcal {B}}_{p+1,q+1;X_n/B_n}^{\\bullet }).$" ], [ "The Representation of the Relative Bott-Chern Cohomology and Aeppli Cohomology Classes ", "In this subsection we will follow [7] to construct a hypercocycle in $\\check{Z}^{p+q}(X,{\\mathcal {B}}_{p,q}^{\\bullet })$ to represent the relative Bott-Chern cohomology classes.", "Let $[\\theta ]$ be an element of $H^{p,q}_{\\rm BC}(X)$ , represented by a closed $(p,q)$ -form $\\theta $ .", "It is defined in ${\\mathbb {H}}^{p+q}(X,{\\mathcal {L}}_{p,q}[1]^{\\bullet })$ by a hypercocycle, still denoted by $\\theta $ and defined by $\\theta ^{p,q}=\\theta \|_{U_j}$ and $\\theta ^{r,s}=0$ otherwise.", "For gievn $p\\ge 1$ and $q\\ge 1$ , there exists a hypercocycle $w=(c;u^{r,0};v^{0,s})\\in \\check{Z}^{p+q}(X,{\\mathcal {B}}_{p,q}^{\\bullet })$ and an hypercochain $\\alpha =(\\alpha ^{r,s})\\in \\check{C}^{p+q-1}(X,{\\mathcal {L}}_{p,q}[1]^{\\bullet })$ such that $\\theta =\\delta \\!\\!\\!\\check{\\delta }\\alpha +w$ .", "We represent the data in the following table: $\\theta \\longleftrightarrow \\left[\\begin{array}{c\|ccc}\\theta _v^{0,q-1}&&&\\\\\\vdots &&\\alpha ^{r,s}&\\\\\\theta _v^{0,0}&&&\\\\\\hline \\theta _c&\\theta _u^{0,0}&\\cdots &\\theta _u^{p-1,0}\\end{array}\\right]$ The equality $\\theta =\\check{\\delta }\\alpha +w$ corresponds to the following relations: $(\\bigstar )\\left\\lbrace \\begin{array}{rcll}\\theta ^{p,q}&=&\\bar{\\partial } \\alpha ^{p-1,q-1}&\\\\(-1)^{r+s}\\check{\\delta }\\alpha ^{r,s}&=&\\bar{\\partial }\\alpha ^{r,s-1}+\\partial \\alpha ^{r-1,s}&\\forall \\, 1\\le r\\le p-1,\\, 1\\le s\\le q-1\\\\(-1)^s\\check{\\delta }\\alpha ^{0,s}&=&\\bar{\\partial }\\alpha ^{0,s-1}+\\theta _v^{0,s}&\\forall \\, 1\\le s\\le q-1\\\\(-1)^r\\check{\\delta }\\alpha ^{r,0}&=&\\theta _u^{r,0}+\\partial \\alpha ^{r-1,0}&\\forall \\, 1\\le r\\le p-1\\\\\\check{\\delta }\\alpha ^{0,0}&=&\\theta _u^{0,0}+\\theta _v^{0,0}&\\\\\\check{\\delta } \\theta _u^{0,0}&=&\\theta _c&\\end{array}\\right.$ Note that these relations involve relations of the hypercocycles for $\\theta _u$ and $\\theta _v$ : $(-1)^r\\check{\\delta } \\theta _u^{r,0}=\\partial \\theta _u^{r-1,0}\\;\\forall 1\\le r\\le p-1,\\qquad (-1)^s\\check{\\delta } \\theta _v^{0,s}=\\bar{\\partial } \\theta _v^{0,s-1}\\;\\forall 1\\le s\\le q-1.$ If $q=0$ , we simply have: $\\theta \\longleftrightarrow \\left(\\theta _c,\\theta _u^{0,0},\\ldots ,\\theta _u^{p-1,0}\\right)$ with the relations: $\\theta ^{p,0}=\\partial \\theta _u^{p-1,0},\\quad (-1)^r\\check{\\delta } \\theta _u^{r,0}=\\partial \\theta _u^{r-1,0}\\;\\forall 1\\le r\\le p-1,\\quad \\check{\\delta } \\theta _u^{0,0}=\\theta _c.$ Similarly, if $p=0$ , we have $\\theta \\longleftrightarrow \\left(\\theta _c,\\theta _v^{0,0},\\ldots ,\\theta _v^{0,q-1}\\right)$ with the relations: $\\theta ^{0,q}=-\\bar{\\partial } \\theta _v^{0,q-1},\\quad (-1)^s\\check{\\delta } \\theta _v^{0,s}=\\bar{\\partial } \\theta _v^{0,s-1}\\;\\forall 1\\le r\\le p-1,\\quad -\\check{\\delta } \\theta _v^{0,0}=\\theta _c.$ Similarly, let $[\\theta ]$ be an element of $H^{p,q}_{\\rm BC}(X_n/B_n)$ , then it can be represented by a Čech hypercocycle $\\theta _u$ , $\\theta _v$ and $\\theta _c$ of $\\check{Z}^{p+q}(X,{\\mathcal {B}}_{p,q;X_n/B_n}^{\\bullet })$ with the relations: $(-1)^r\\check{\\delta } \\theta _u^{r,0}=\\partial \\theta _u^{r-1,0}\\;\\forall 1\\le r\\le p-1,\\qquad (-1)^s\\check{\\delta } \\theta _v^{0,s}=\\bar{\\partial } \\theta _v^{0,s-1}\\;\\forall 1\\le s\\le q-1.$ $ \\check{\\delta }\\theta _u^{0,0}=\\theta _c,\\qquad -\\check{\\delta } \\theta _v^{0,0}=\\theta _c; $ while for an element $[\\theta ]$ of $H^{p,q}_{\\rm A}(X_n/B_n)$ , it can be represented by a Čech hypercocycle $\\theta _u$ and $\\theta _v$ of $\\check{Z}^{p+q+1}(X,{\\mathcal {B}}_{p+1,q+1;X_n/B_n}^{\\bullet })$ with the relations: $(-1)^r\\check{\\delta } \\theta _u^{r,0}=\\partial \\theta _u^{r-1,0}\\;\\forall 1\\le r\\le p,\\qquad (-1)^s\\check{\\delta } \\theta _v^{0,s}=\\bar{\\partial } \\theta _v^{0,s-1}\\;\\forall 1\\le s\\le q,$ $ \\check{\\delta }\\theta _u^{0,0}=\\theta _c,\\qquad -\\check{\\delta }\\theta _v^{0,0}=\\theta _c.", "$ Before the end of this subsections, we will introduce some important maps which will be used in the computation in $§4$ .", "Define $\\partial ^{\\bar{\\partial },{\\mathcal {B}}}_{X_n/B_n}: H^{\\bullet }(X_n, \\Omega ^{p-1;\\omega }_{X_{n}/B_{n}}) \\rightarrow {\\mathbb {H}}^{\\bullet +p} (X_n,{\\mathcal {B}}_{p,q;X_n/B_n}^{\\bullet }) $ in the following way: Let $[\\theta ]$ be an element of $H^{\\bullet }(X_n, \\Omega ^{p-1;\\omega }_{X_{n}/B_{n}})$ then $\\theta $ can be represented by a cocycle of $\\check{Z}^{\\bullet }(X,\\Omega ^{p-1;\\omega }_{X_{n}/B_{n}})$ , we define $\\partial ^{\\bar{\\partial },{\\mathcal {B}}}_{X_n/B_n}([\\theta ])$ to be the cohomology class associated to the hypercocyle in $\\check{Z}^{p+\\bullet }(X,{\\mathcal {B}}_{p,q;X_n/B_n}^{\\bullet })$ given by $\\theta _u^{p-1,0}=\\theta $ , $\\theta _u^{r,0}=0\\; \\forall 0\\le r\\le p-2$ and $\\theta _v^{0,r}=0\\; \\forall 0\\le r \\le q-1$ , $ \\theta _c=0.", "$ Lemma 2.1 $\\partial ^{\\bar{\\partial },{\\mathcal {B}}}_{X_n/B_n}$ is well defined.", "It is easy to check that the hypercochian given by $\\theta _u$ , $\\theta _v$ and $\\theta _c$ is a hypercocycle.", "On the other hand if there exists a cochain $\\alpha ^{^{\\prime }}$ in $\\check{C}^{\\bullet -1}(X,\\Omega ^{p-1;\\omega }_{X_{n}/B_{n}})$ such that $\\check{\\delta } \\alpha ^{^{\\prime }}= \\theta $ , then if we take a hypercochian $\\alpha $ in $\\check{C}^{p+\\bullet -1}(X,{\\mathcal {B}}_{p,q;X_n/B_n}^{\\bullet })$ given by $\\alpha _u^{p-1,0}=(-1)^{p-1}\\alpha ^{^{\\prime }}$ , $\\alpha _u^{r,0}=0\\; \\forall 0\\le r\\le p-2$ , $\\alpha _v^{0,r}=0\\; \\forall 0\\le r \\le q-1$ and $\\alpha _c=0$ we have $\\delta \\!\\!\\!\\check{\\delta }\\alpha = \\partial ^{\\bar{\\partial },{\\mathcal {B}}}_{X_n/B_n}([\\theta ])$ .", "Therefore $\\partial ^{\\bar{\\partial },{\\mathcal {B}}}_{X_n/B_n}([\\theta ])=0.$ Similarly, we can define $\\bar{\\partial }^{{\\partial },{\\mathcal {B}}}_{X_n/B_n}: H^{\\bullet }(\\bar{X}_n, \\bar{\\Omega }^{q-1;\\omega }_{X_{n}/B_{n}}) \\rightarrow {\\mathbb {H}}^{\\bullet +q} (X_n,{\\mathcal {B}}_{p,q;X_n/B_n}^{\\bullet }) $ in the following way: Let $[\\theta ]$ be an element of $H^{\\bullet }(\\bar{X}_n, \\bar{\\Omega }^{q-1;\\omega }_{X_{n}/B_{n}})$ then $\\theta $ can be represented by a cocycle of $\\check{Z}^{\\bullet }(\\bar{X},\\bar{\\Omega }^{q-1;\\omega }_{X_{n}/B_{n}})$ , we define $\\bar{\\partial }^{{\\partial },{\\mathcal {B}}}_{X_n/B_n}([\\theta ])$ to be the cohomology class associated to the hypercocyle in $\\check{Z}^{q+\\bullet }(X,{\\mathcal {B}}_{p,q;X_n/B_n}^{\\bullet })$ given by $\\theta _v^{0,q-1}=\\theta $ , $\\theta _v^{0,r}=0\\; \\forall 0\\le r\\le q-2$ , $\\theta _u^{r,0}=0\\; \\forall 0\\le r \\le p-1$ and $\\theta _c=0$ .", "This map is also well defined and the proof is just as lemma REF .", "Define $\\partial ^{{\\mathcal {B}},\\bar{\\partial }}_{X_n/B_n}: {\\mathbb {H}}^{\\bullet +p} (X_n,{\\mathcal {B}}_{p,q;X_n/B_n}^{\\bullet }) \\rightarrow H^{\\bullet }(X_n, \\Omega ^{p;\\omega }_{X_{n}/B_{n}}) $ in the following way: Let $[\\theta ]$ be an element of $ {\\mathbb {H}}^{\\bullet +p} (X_n,{\\mathcal {B}}_{p,q;X_n/B_n}^{\\bullet })$ then $\\theta $ can be represented by a hypercocycle of $\\check{Z}^{p+\\bullet }(X,{\\mathcal {B}}_{p,q;X_n/B_n}^{\\bullet })$ , we define ${\\partial }^{{\\mathcal {B}},\\bar{\\partial }}_{X_n/B_n}([\\theta ])$ to be the cohomology class associated to the cocyle in $\\check{Z}^{\\bullet }(X,\\Omega ^{p;\\omega }_{X_{n}/B_{n}})$ given by $\\partial _{X_n/B_n} \\theta _u^{p-1,0}$ .", "Lemma 2.2 $\\partial ^{{\\mathcal {B}}, \\bar{\\partial }}_{X_n/B_n}$ is well defined.", "At first we need to check the cochian given by $\\partial _{X_n/B_n} \\theta _u^{p-1,0}$ is a cocycle.", "In fact, since $\\theta $ is a hypercocycle in $\\check{Z}^{p+\\bullet }(X,{\\mathcal {B}}_{p,q;X_n/B_n}^{\\bullet })$ , we have $(-1)^{p-1}\\check{\\delta } \\theta _u^{p-1,0}= \\partial _{X_n/B_n} \\theta _u^{p-2,0}$ , therefore, $\\check{\\delta } \\partial _{X_n/B_n} \\theta _u^{p-1,0}=(-1)^p\\partial _{X_n/B_n}\\circ \\partial _{X_n/B_n}\\theta _u^{p-2,0}=0$ .", "On the other hand if there exists a cochain $\\alpha $ in $\\check{C}^{p+\\bullet -1}(X,{\\mathcal {B}}_{p,q;X_n/B_n}^{\\bullet })$ such that $\\delta \\!\\!\\!\\check{\\delta }\\alpha = \\theta $ , then if we take a cochian $\\alpha ^{^{\\prime }}$ in $\\check{C}^{\\bullet -1}(X,\\Omega ^{p;\\omega }_{X_{n}/B_{n}})$ given by $\\alpha ^{^{\\prime }} = (-1)^p \\partial _{X_n/B_n} \\alpha _u^{p-1,0}$ , we have $ \\check{\\delta } \\alpha ^{^{\\prime }} = (-1)^p\\check{\\delta } \\partial _{X_n/B_n} \\alpha _u^{p-1,0} = (-1)^{p+1} \\partial _{X_n/B_n} \\check{\\delta } \\alpha _u^{p-1,0} = (-1)^{p+1+p-1}\\partial _{X_n/B_n} \\theta _u^{p-1,0}=\\partial ^{{\\mathcal {B}},\\bar{\\partial },}_{X_n/B_n}([\\theta ])$ .", "Therefore $\\partial ^{\\bar{\\partial },{\\mathcal {B}}}_{X_n/B_n}([\\theta ])=0.$ Similarly, we can define $\\bar{\\partial }^{{\\mathcal {B}},{\\partial }}_{X_n/B_n}: {\\mathbb {H}}^{\\bullet +q} (X_n,{\\mathcal {B}}_{p,q;X_n/B_n}^{\\bullet }) \\rightarrow H^{\\bullet }(\\bar{X}_n, \\Omega ^{q;\\omega }_{X_{n}/B_{n}}) $ in the following way: Let $[\\theta ]$ be an element of $ {\\mathbb {H}}^{\\bullet +q} (X_n,{\\mathcal {B}}_{p,q;X_n/B_n}^{\\bullet })$ then $\\theta $ can be represented by a hypercocycle of $\\check{Z}^{q+\\bullet }(X,{\\mathcal {B}}_{p,q;X_n/B_n}^{\\bullet })$ , we define $\\bar{\\partial }^{{\\mathcal {B}},{\\partial }}_{X_n/B_n}([\\theta ])$ to be the cohomology class associated to the cocyle in $\\check{Z}^{\\bullet }(X,\\bar{\\Omega }^{q;\\omega }_{X_{n}/B_{n}})$ given by $\\bar{\\partial }_{X_n/B_n} \\theta _u^{0,q-1}$ .", "This map is also well defined and the proof is just as lemma REF .", "Remark 2.3 The natural maps from $H^{p,q}_{\\rm BC}(X_n/B_n)$ to $H^{q}(X_n, \\Omega ^{p;\\omega }_{X_{n}/B_{n}})$ and from $H^{q}(X_n, \\Omega ^{p;\\omega }_{X_{n}/B_{n}})$ to $H^{p,q}_{\\rm A}(X_n/B_n)$ mentioned in $§2.1$ respectively are exactly the map: $\\partial ^{{\\mathcal {B}},\\bar{\\partial }}_{X_n/B_n}: {\\mathbb {H}}^{q+p} (X_n,{\\mathcal {B}}_{p,q;X_n/B_n}^{\\bullet })( \\cong H^{p,q}_{\\rm BC}(X_n/B_n)) \\rightarrow H^{q}(X_n, \\Omega ^{p;\\omega }_{X_{n}/B_{n}}), $ $\\partial ^{\\bar{\\partial },{\\mathcal {B}}}_{X_n/B_n}: H^{q}(X_n, \\Omega ^{p;\\omega }_{X_{n}/B_{n}}) \\rightarrow {\\mathbb {H}}^{q+p+1} (X_n,{\\mathcal {B}}_{p+1,q+1;X_n/B_n}^{\\bullet })( \\cong H^{p,q}_{\\rm A}(X_n/B_n)) .", "$ and we denote these maps by $r_{BC,\\bar{\\partial }} $ and $r_{\\bar{\\partial },A} .$ We also denote the following two maps: $\\partial ^{\\bar{\\partial },{\\mathcal {B}}}_{X_n/B_n}: H^{q}(X_n, \\Omega ^{p-1;\\omega }_{X_{n}/B_{n}}) \\rightarrow {\\mathbb {H}}^{q+p} (X_n,{\\mathcal {B}}_{p,q;X_n/B_n}^{\\bullet })( \\cong H^{p,q}_{\\rm BC}(X_n/B_n)), $ $\\partial ^{{\\mathcal {B}},\\bar{\\partial }}_{X_n/B_n}: {\\mathbb {H}}^{q+p+1} (X_n,{\\mathcal {B}}_{p+1,q+1;X_n/B_n}^{\\bullet })( \\cong H^{p,q}_{\\rm A}(X_n/B_n)) \\rightarrow H^{q}(X_n, \\Omega ^{p+1;\\omega }_{X_{n}/B_{n}}).", "$ by $\\partial ^{\\bar{\\partial },BC}_{X_n/B_n}$ and $\\partial ^{A,\\bar{\\partial }}_{X_n/B_n}.$ The following lemma is an important observation which will be used for the computation in $§4$ .", "Lemma 2.4 Let $[\\theta ]$ be an element of $ {\\mathbb {H}}^{l} (X_n,{\\mathcal {B}}_{p,q;X_n/B_n}^{\\bullet })$ which is represented by an element $\\theta $ in $\\check{Z}^{l}(X,{\\mathcal {B}}_{p,q;X_n/B_n}^{\\bullet })$ given by $\\theta _u$ , $\\theta _v$ and $\\theta _c$ , then $\\partial _{X_n/B_n}(\\theta - \\theta _u^{p-1,0})$ is a hypercoboundary.", "The hypercochian $\\partial _{X_n/B_n}(\\theta - \\theta _u^{p-1,0})$ is given by $(\\partial _{X_n/B_n}(\\theta - \\theta _u^{p-1,0}))_u^{r,0}=\\partial _{X_n/B_n} \\theta _u^{r-1,0}, \\forall 0 < r <p-1$ , $ (\\partial _{X_n/B_n}(\\theta - \\theta _u^{p-1,0}))_u^{0,0}=0$ , $(\\partial _{X_n/B_n}(\\theta - \\theta _u^{p-1,0}))_v^{0,r}=0, \\forall 0 \\le r \\le q-1 $ and $(\\partial _{X_n/B_n}(\\theta - \\theta _u^{p-1,0}))_c=0$ .", "Let $\\alpha $ be the hypercochian in $\\check{C}^{l}(X,{\\mathcal {B}}_{p,q;X_n/B_n}^{\\bullet })$ given by $\\alpha _u^{2r,0}=0, \\forall 0 \\le r < p/2, $ $\\alpha _u^{2r-1,0}=\\theta _u^{2r-1,0}, \\forall 0 < r \\le p/2$ , $\\alpha _v^{0,r}=0, \\forall 0 \\le r \\le q-1 $ and $\\alpha _c=0$ and it is easy to see that $\\delta \\!\\!\\!\\check{\\delta }\\alpha =\\partial _{X_n/B_n}(\\theta - \\theta _u^{p-1,0})$ .", "Therefore $\\partial _{X_n/B_n}(\\theta - \\theta _u^{p-1,0})$ is a hypercoboundary.", "Similarly, we have Lemma 2.5 Let $[\\theta ]$ be an element of $ {\\mathbb {H}}^{l} (X_n,{\\mathcal {B}}_{p,q;X_n/B_n}^{\\bullet })$ which is represented by an element $\\theta $ in $\\check{Z}^{l}(X,{\\mathcal {B}}_{p,q;X_n/B_n}^{\\bullet })$ given by $\\theta _u$ , $\\theta _v$ and $\\theta _c$ , then $\\overline{\\partial }_{X_n/B_n}(\\theta - \\theta _v^{0,q-1})$ is a hypercoboundary." ], [ "The Jumping Phenomenon and Obstructions", "There is a Hodge theory also for Bott-Chern and Aeppli cohomologies, see [7].", "More precisely, fixed a Hermitian metric on $X$ , one has that $ H^{\\bullet ,\\bullet }_{\\rm BC}(X) \\;\\simeq \\; \\ker \\tilde{\\Delta }_{\\rm BC} \\qquad \\text{ and }\\qquad H^{\\bullet ,\\bullet }_{\\rm A}(X) \\;\\simeq \\; \\ker \\tilde{\\Delta }_{\\rm A} \\;,$ where $ \\tilde{\\Delta }_{\\rm BC} \\;:=\\;\\left(\\partial \\overline{\\partial }\\right)\\left(\\partial \\overline{\\partial }\\right)^+\\left(\\partial \\overline{\\partial }\\right)^\\left(\\partial \\overline{\\partial }\\right)+\\left(\\overline{\\partial }^\\partial \\right)\\left(\\overline{\\partial }^\\partial \\right)^+\\left(\\overline{\\partial }^\\partial \\right)^\\left(\\overline{\\partial }^\\partial \\right)+\\overline{\\partial }^\\overline{\\partial }+\\partial ^\\partial $ and $ \\tilde{\\Delta }_{\\rm A} \\;:=\\; \\partial \\partial ^+\\overline{\\partial }\\overline{\\partial }^+\\left(\\partial \\overline{\\partial }\\right)^\\left(\\partial \\overline{\\partial }\\right)+\\left(\\partial \\overline{\\partial }\\right)\\left(\\partial \\overline{\\partial }\\right)^+\\left(\\overline{\\partial }\\partial ^\\right)^\\left(\\overline{\\partial }\\partial ^\\right)+\\left(\\overline{\\partial }\\partial ^\\right)\\left(\\overline{\\partial }\\partial ^\\right)^* $ are 4-th order elliptic self-adjoint differential operators.", "In particular, one gets that $ \\dim _H̏^{\\bullet ,\\bullet }_{\\sharp }(X) \\;<\\;+\\infty \\qquad \\text{ for } \\sharp \\in \\left\\lbrace \\overline{\\partial },\\,\\partial ,\\,BC,\\,A\\right\\rbrace \\;.$ Let $\\pi :\\mathcal {X}\\rightarrow B$ be a deformation of $\\pi ^{-1}(0)=X$ , where $X$ is a compact complex manifold and $B$ is a neighborhood of the origin in $.", "Note that $ hp,qBC(X(t))$ and $ hp,qA(X(t))$ are semi-continuous functions of $ t B$ where $ X(t)= -1(t)$ \\cite {Schw}.", "Denotethe $ BC$ operator and the $ A$ on $ X(t)$ by $ BC,t$ and $ A,t$.", "From the prove of the semi-continuity of $ hp,qBC(X(t))$ ($ hp,qA(X(t))$) on \\cite {Schw}, we can see that $ hp,qBC(X(t))$ ( $ hp,qA(X(t))$ )does not jump at the point $ t=0$ if and only if all the $ BC,0$($ A,0$)-harmonic forms on $ X$ can be extendedto relative $ BC,t$($ A,t$)-harmonic forms on a neighborhood of $ 0 B$ which is real analytic in the direction of $ B$, since the $ BC,t$($ A,t$) varies real analytic on $ B$.The above condition is equivalent to the following: all the cohomology classes $ []$ in $ Hp,qBC(X)$($ Hp,qA(X)$) can be extended to a relative $ dt-closed$( $ tt-closed$) forms $ (t)$ such that $ [(t)]0$ on a neighborhood of $ 0 B$ which is real analytic on the direction of $ B$.Therefore in order to study the jumping phenomenon, we need to study the extension obstructions.", "So we need to study the obstructions of the extension of the cohomology classes in $ H(X,Bp,q)$ to a relative cohomology classes in $ H(Xn,Bp,q;Xn/Bn)$.We denote the following complex$$ \\pi ^{-1}(m_0^{\\omega }/(m_0^{\\omega })^{n+1}) \\stackrel{(+,-)}{\\rightarrow } \\mathcal {M}_{0}^{\\omega } / (\\mathcal {M}^{\\omega } _{0})^{n+1}\\otimes {\\mathcal {O}}^{\\omega }_{X_n} \\oplus \\bar{\\mathcal {M}}_{0}^{\\omega } / (\\bar{\\mathcal {M}}^{\\omega } _{0})^{n+1}\\otimes {\\mathcal {O}}^{\\omega }_{\\bar{X}_n} $ $$ M0 / (M 0)n+1Xn/Bn1;M0 / (M 0)n+1Xn/Bn1; $$ ...M0 / (M 0)n+1Xn/Bnp-1;M0 / (M 0)n+1Xn/Bnp-1; $$ M0 / (M 0)n+1Xn/Bnp;...M0 / (M 0)n+1Xn/Bnq-1;0, $$ by $\\mathcal {M}_{0}^{\\omega } / (\\mathcal {M}^{\\omega } _{0})^{n+1}\\otimes {\\mathcal {B}}_{p,q;X_{n-1}/B_{n-1}}^{\\bullet }$ Now we consider the following exact sequences: $ 0 \\rightarrow \\mathcal {M}_{0} / \\mathcal {M}^{n+1}_{0}\\otimes {\\mathcal {B}}_{p,q;X_{n-1}/B_{n-1}}^{\\bullet }\\rightarrow {\\mathcal {B}}_{p,q;X_{n}/B_{n}}^{\\bullet } \\rightarrow {\\mathcal {B}}_{p,q;X_{0}/B_{0}}^{\\bullet }\\rightarrow 0$ which induces a long exact sequence $ 0 \\rightarrow {\\mathbb {H}}^0(X_{n},\\mathcal {M}_{0} / \\mathcal {M}^{n+1}_{0}\\otimes {\\mathcal {B}}_{p,q;X_{n-1}/B_{n-1}}^{\\bullet })\\rightarrow {\\mathbb {H}}^0(X_{n},{\\mathcal {B}}_{p,q;X_{n}/B_{n}}^{\\bullet })\\rightarrow {\\mathbb {H}}^0(X,{\\mathcal {B}}_{p,q;X_{0}/B_{0}}^{\\bullet })$ $ \\rightarrow {\\mathbb {H}}^1(X_{n},\\mathcal {M}_{0} / \\mathcal {M}^{n+1}_{0}\\otimes {\\mathcal {B}}_{p,q;X_{n-1}/B_{n-1}}^{\\bullet }) \\rightarrow ... .$ Let $[\\theta ]$ be a cohomology class in ${\\mathbb {H}}^l(X,{\\mathcal {B}}_{p,q;X_{0}/B_{0}}^{\\bullet })$ .", "The obstruction for the extension of $[\\theta ]$ to a relative cohomology classes in ${\\mathbb {H}}^l(X_{n},{\\mathcal {B}}_{p,q;X_{n}/B_{n}}^{\\bullet })$ comes from the non trivial image of the connecting homomorphism $\\delta ^{}:{\\mathbb {H}}^l(X,{\\mathcal {B}}_{p,q;X_{0}/B_{0}}^{\\bullet })\\rightarrow {\\mathbb {H}}^{l+1}(X_{n},\\mathcal {M}_{0} / \\mathcal {M}^{n+1}_{0}\\otimes {\\mathcal {B}}_{p,q;X_{n-1}/B_{n-1}}^{\\bullet })$ .", "We denote this obstruction by $o_n([\\theta ])$ .", "On the other hand, for a given real direction $\\frac{\\partial }{\\partial x}$ on $B$ , if there exits $n \\in {\\mathbb {N}}$ , such that $o_i([\\theta ])=0, \\forall i \\le n$ and $o_n([\\theta ]) \\ne 0.", "$ Let $\\theta _{n-1}$ be a $n-1$ th order extension of $\\theta $ to a relative cohomology classes in ${\\mathbb {H}}^l(X_{n-1},{\\mathcal {B}}_{p,q;X_{n-1}/B_{n-1}}^{\\bullet })$ .", "$\\delta \\!\\!\\!\\check{\\delta }\\theta _{n-1}=0$ up to order $n-1$ .", "Now if we consider $\\delta \\!\\!\\!\\check{\\delta }\\theta _{n-1}/x^n$ , it is easy to check that $\\delta \\!\\!\\!\\check{\\delta }\\theta _{n-1}/x^n$ is an extension of a non trivial cohomology classes $[\\delta \\!\\!\\!\\check{\\delta }\\theta _{n-1}/x^n(0)]$ in ${\\mathbb {H}}^l(X,{\\mathcal {B}}_{p,q})$ while $[\\delta \\!\\!\\!\\check{\\delta }\\theta _{n-1}/x^n(x_0)]$ is trivial in $X(x_0)$ as a cohomology classes in ${\\mathbb {H}}^l(X(x_0),{\\mathcal {B}}_{p,q;x_0})$ if $x_0 \\ne 0$ .", "From the above discussion, we have the following theorem: Theorem 3.1 Let $\\pi : \\mathcal {X}\\rightarrow B$ be a small deformation of the central fibre compact complex manifold $X$ .", "Now we consider $\\dim {\\mathbb {H}}^l(X(t),{\\mathcal {B}}_{p,q;t}^{\\bullet })$ as a function of $t\\in B$ .", "It jumps at $t=0$ if there exists an element $[\\theta ]$ either in ${\\mathbb {H}}^l(X,{\\mathcal {B}}_{p,q}^{\\bullet })$ or in ${\\mathbb {H}}^{l-1}(X,{\\mathcal {B}}_{p,q}) $ and a minimal natural number $n\\ge 1$ such that the n-th order obstruction $ o_{n}([\\theta ])\\ne 0.$" ], [ "The Formula for the Obstructions", "Since these obstructions we discussed in the previous section are so important when we consider the problem of jumping phenomenon of Bott-Chern cohomology and Aeppli cohomology, we try to find out an explicit calculation for such obstructions in this section.", "As we had done in [9], we need some preparation.", "Cover $X$ by open sets $U_{i}$ such that, for arbitrary $i$ , $U_{i}$ is small enough.", "More precisely, $U_{i}$ is stein and the following exact sequence splits $ 0 \\rightarrow \\pi _{n}^{}(\\Omega _{B_{n}})^{\\omega }(U_{i})\\rightarrow \\Omega _{X_{n}} ^{\\omega }(U_{i})\\rightarrow \\Omega _{X_{n}/B_{n}}^{\\omega }(U_{i})\\rightarrow 0;$ $ 0 \\rightarrow \\bar{\\pi }_{n}^{}(\\Omega _{\\bar{B}_{n}})^{\\omega }(U_{i})\\rightarrow \\Omega _{\\bar{X}_{n}}^{\\omega } (U_{i})\\rightarrow \\bar{\\Omega }_{X_{n}/B_{n}}^{\\omega }(U_{i})\\rightarrow 0.$ So we have a map $\\varphi _{i}: \\Omega _{X_{n}/B_{n}}^{\\omega } (U_{i}) \\oplus \\bar{\\Omega }_{X_{n}/B_{n}} ^{\\omega }(U_{i})\\rightarrow \\Omega _{X_{n}}^{\\omega } (U_{i})\\oplus \\Omega _{\\bar{X}_{n}}^{\\omega }(U_{i})$ , such that $\\varphi _{i}\|_{\\Omega _{X_{n}/B_{n}} ^{\\omega } (U_{i})}(\\Omega _{X_{n}/B_{n}}^{\\omega } (U_{i}))\\oplus \\pi _{n}^{}(\\Omega _{B_{n}})^{\\omega }(U_{i}) \\cong \\Omega _{X_{n}} ^{\\omega }(U_{i}) $ and $\\varphi _{i}\|_{\\bar{\\Omega }_{X_{n}/B_{n}}^{\\omega } (U_{i})}(\\bar{\\Omega }_{X_{n}/B_{n}} ^{\\omega }(U_{i}))\\oplus \\bar{\\pi }_{n}^{}(\\Omega _{\\bar{B}_{n}})^{\\omega }(U_{i}) \\cong \\Omega _{\\bar{X}_{n}} ^{\\omega }(U_{i}) $ .", "This decomposition determines a local decomposition of the exterior differentiation $\\partial _{X_{n}}$ ($\\overline{\\partial }_{X_{n}}$ ) in $\\Omega ^{\\bullet ;\\omega }_{X_{n}}$ (resp.", "$\\bar{\\Omega }^{\\bullet ;\\omega }_{X_{n}}$ )on each $U_{i}$ $ \\partial _{X_{n}}=\\partial ^{i}_{B_{n}}+\\partial ^{i}_{X_{n}/B_{n}} (resp.", "\\partial _{\\bar{X}_{n}}=\\overline{\\partial }^{i}_{{B}_{n}}+\\overline{\\partial }^{i}_{X_{n}/B_{n}}).$ By definition, $\\partial _{X_{n}/B_{n}}$ and $\\overline{\\partial }_{X_{n}/B_{n}}$ are given by $ \\varphi _{i}^{-1}\\circ \\partial ^{i}_{X_{n}/B_{n}}\\circ \\varphi _{i}$ and $ \\varphi _{i}^{-1}\\circ \\overline{\\partial }^{i}_{X_{n}/B_{n}}\\circ \\varphi _{i}$ .", "Denote the set of alternating $q$ -cochains $\\beta $ with values in $\\mathcal {F}$ by $\\check{C}^{q}(\\mathbf {U},\\mathcal {F})$ , i.e.", "to each $q+1$ -tuple, $i_{0}<i_{1}...< i_{q}$ , $\\beta $ assigns a section $\\beta (i_{0},i_{1},..., i_{q})$ of $\\mathcal {F}$ over $U_{i_{0}}\\cap U_{i_{1}} \\cap ... \\cap U_{i_{q}}$ .", "Let us still using $\\varphi _{i}$ denote the following map, $\\varphi _{i}: \\pi _{n}^{}(\\Omega _{B_{n}})^{\\omega } \\wedge \\Omega ^{p;\\omega }_{X_{n}/B_{n}}(U_{i}) \\oplus \\bar{\\pi }_{n}^{}(\\Omega _{\\bar{B}_{n}})^{\\omega } \\wedge \\bar{\\Omega }^{p;\\omega }_{X_{n}/B_{n}}(U_{i} ) & \\rightarrow &\\Omega ^{p+1;\\omega }_{X_{n}}(U_{i}) \\oplus \\Omega ^{p+1;\\omega }_{\\bar{X}_{n}}(U_{i})\\\\\\varphi _{i}(\\omega _{1}\\wedge \\beta _{i_{1}}\\wedge ...\\wedge \\beta _{i_{p}}+\\omega _{2}\\wedge {\\beta ^{^{\\prime }}}_{j_{1}}\\wedge ...\\wedge {\\beta ^{^{\\prime }}}_{j_{p}})& = &\\omega _{1}\\wedge \\varphi _{i}(\\beta _{i_{1}})\\wedge ...\\wedge \\varphi _{i}(\\beta _{i_{p}}) \\\\&&+\\omega _{2}\\wedge \\varphi _{i}({\\beta ^{^{\\prime }}}_{j_{1}})\\wedge ...\\wedge \\varphi _{i}({\\beta ^{^{\\prime }}}_{j_{p}}).$ Define $\\varphi :\\check{C}^{q}(\\mathbf {U},\\pi _{n}^{}(\\Omega _{B_{n}})^{\\omega } \\wedge \\Omega ^{p;\\omega }_{X_{n}/B_{n}} \\oplus \\bar{\\pi }_{n}^{}(\\Omega _{\\bar{B}_{n}})^{\\omega } \\wedge \\bar{\\Omega }^{p;\\omega }_{X_{n}/B_{n}})\\rightarrow \\check{C}^{q}(\\mathbf {U},\\Omega ^{p+1;\\omega }_{{X}_{n}} \\oplus \\Omega ^{p+1;\\omega }_{\\bar{X}_{n}})$ by $ \\varphi (\\beta )(i_{0},i_{1},..., i_{q})=\\varphi _{i_{0}}(\\beta (i_{0},i_{1},...,i_{q})) \\qquad \\forall \\beta \\in \\check{C}^{q}(\\mathbf {U},\\pi _{n}^{}(\\Omega _{B_{n}})^{\\omega } \\wedge \\Omega ^{p;\\omega }_{X_{n}/B_{n}} \\oplus \\bar{\\pi }_{n}^{}(\\Omega _{\\bar{B}_{n}})^{\\omega } \\wedge \\bar{\\Omega }^{p;\\omega }_{X_{n}/B_{n}}),$ where $i_{0}<i_{1}...< i_{q}$ .", "Define the total Lie derivative with respect to $B_{n}$ $ L_{B_{n}}: \\check{C}^{q}(\\mathbf {U},\\Omega ^{p;\\omega }_{X_{n}}\\oplus {\\Omega }^{p;\\omega }_{\\bar{X}_{n}})\\rightarrow \\check{C}^{q}(\\mathbf {U},\\Omega ^{p+1;\\omega }_{X_{n}}\\oplus {\\Omega }^{p+1;\\omega }_{\\bar{X}_{n}})$ by $ L_{B_{n}}(\\beta )(i_{0},i_{1},...,i_{q})=\\partial _{B_{n}}^{i_{0}}(\\beta (i_{0},i_{1},..., i_{q})) \\qquad \\forall \\beta \\in \\check{C}^{q}(\\mathbf {U},\\Omega ^{p;\\omega }_{X_{n}}),$ where $i_{0}<i_{1}...< i_{q}$ (see [4]).", "Define, for each $U_{i}$ the total interior product with respect to $B_{n}$ , $I^{i}:\\Omega ^{p;\\omega }_{X_{n}}(U_{i}) \\oplus \\Omega ^{p;\\omega }_{\\bar{X}_{n}}(U_{i})\\rightarrow \\Omega ^{p;\\omega }_{X_{n}}(U_{i}) \\oplus \\Omega ^{p;\\omega }_{\\bar{X}_{n}}(U_{i})$ by $ I^{i}(\\mu _1 \\partial _{X_n}g_{1}\\wedge \\partial _{X_n}g_{2}\\wedge ...\\wedge \\partial _{X_n}g_{p}+\\mu _2 \\partial _{\\bar{X}_n}{g^{^{\\prime }}}_{1}\\wedge \\partial _{\\bar{X}_n}g^{^{\\prime }}_{2}\\wedge ...\\wedge \\partial _{\\bar{X}_n}g^{^{\\prime }}_{p})= $ $\\mu _1 \\sum _{j=1}^{p}\\partial _{X_n}g_{1}\\wedge ...\\wedge \\partial _{X_n}g_{j-1}\\wedge \\partial ^{i}_{B_{n}}(g_{j})\\wedge \\partial _{X_n}g_{j+1}\\wedge ...\\wedge \\partial _{X_n}g_{p}+ $ $\\mu _2 \\sum _{j=1}^{p}\\partial _{\\bar{X}_n}g^{^{\\prime }}_{1}\\wedge ...\\wedge \\partial _{\\bar{X}_n}g^{^{\\prime }}_{j-1}\\wedge \\partial ^{i}_{B_{n}}(g^{^{\\prime }}_{j})\\wedge \\partial _{\\bar{X}_n}g^{^{\\prime }}_{j+1}\\wedge ...\\wedge \\partial _{\\bar{X}_n}g^{^{\\prime }}_{p}.$ When $p=0$ , we put $I^{i}=0$ (see [4]).", "Define $\\lambda : \\check{C}^{q}(\\mathbf {U},\\Omega ^{p;\\omega }_{X_{n}} \\oplus \\Omega ^{p;\\omega }_{\\bar{X}_{n}})\\rightarrow \\check{C}^{q+1}(\\mathbf {U},\\Omega ^{p;\\omega }_{X_{n}}\\oplus \\Omega ^{p;\\omega }_{\\bar{X}_{n}}) $ by $ (\\lambda \\beta )(i_{0},...,i_{q+1})=(I^{i_{0}}-I^{i_{1}})\\beta (i_{1},...,i_{q+1}) \\qquad \\forall \\beta \\in \\check{C}^{q}(\\mathbf {U},\\Omega ^{p;\\omega }_{X_{n}}\\oplus \\Omega ^{p;\\omega }_{\\bar{X}_{n}}).$ An we have the following lemma, the prove is completely the same as lemma 3.1 in [9]: Lemma 4.1 $\\lambda \\circ \\varphi \\equiv \\delta \\circ \\varphi - \\varphi \\circ \\delta .$ With the above preparation, we are ready to study the jumping phenomenon of the dimensions of Bott-Chern or Aeppli cohomology groups, for arbitrary $[\\theta ]$ belongs to ${\\mathbb {H}}^l(X,{\\mathcal {B}}_{p,q})$ , suppose we can extend $[\\theta ]$ to order $n-1$ in ${\\mathbb {H}}^l(X_{n-1},{\\mathcal {B}}_{p,q;X_{n-1}/B_{n-1}}^{\\bullet }) $ .", "Denote such element by $[\\theta _{n-1}]$ .", "In the following, we try to find out the obstruction of the extension of $[\\theta _{n-1}]$ to $n$ th order.", "Consider the exact sequence $ 0 \\rightarrow \\mathcal {M}^{n}_{0} / \\mathcal {M}^{n+1}_{0}\\otimes {\\mathcal {B}}_{p,q;X_{0}/B_{0}}^{\\bullet }\\rightarrow {\\mathcal {B}}_{p,q;X_{n}/B_{n}}^{\\bullet } \\rightarrow {\\mathcal {B}}_{p,q;X_{n-1}/B_{n-1}}^{\\bullet }\\rightarrow 0$ which induces a long exact sequence $ 0 \\rightarrow {\\mathbb {H}}^0(X_n,\\mathcal {M}^{n}_{0} / \\mathcal {M}^{n+1}_{0}\\otimes {\\mathcal {B}}_{p,q;X_{0}/B_{0}}^{\\bullet })\\rightarrow {\\mathbb {H}}^0(X_{n},{\\mathcal {B}}_{p,q;X_{n}/B_{n}}^{\\bullet })\\rightarrow {\\mathbb {H}}^0(X_{n-1},{\\mathcal {B}}_{p,q;X_{n-1}/B_{n-1}}^{\\bullet })$ $ \\rightarrow {\\mathbb {H}}^1(X_n,\\mathcal {M}^{n}_{0} / \\mathcal {M}^{n+1}_{0}\\otimes {\\mathcal {B}}_{p,q;X_{0}/B_{0}}^{\\bullet }) \\rightarrow ... .$ Let $[\\theta ]$ be a cohomology class in ${\\mathbb {H}}^l(X,{\\mathcal {B}}_{p,q;X_{0}/B_{0}}^{\\bullet })$ .", "The obstruction for $[\\theta _{n-1}]$ comes from the non trivial image of the connecting homomorphism $\\delta ^{}: {\\mathbb {H}}^l(X_{n-1},{\\mathcal {B}}_{p,q;X_{n-1}/B_{n-1}}^{\\bullet }) \\rightarrow {\\mathbb {H}}^{l+1}(X_n,\\mathcal {M}^{n}_{0} / \\mathcal {M}^{n+1}_{0}\\otimes {\\mathcal {B}}_{p,q;X_{0}/B_{0}}^{\\bullet })$ .", "Now we are ready to calculate the formula for the obstructions.", "Let $\\tilde{\\theta }$ be an element of $\\check{C}^{l}(\\mathbf {U},{\\mathcal {B}}_{p,q;X_{n}/B_{n}}^{\\bullet })$ such that its quotient image in $\\check{C}^{l}(\\mathbf {U},{\\mathcal {B}}_{p,q;X_{n-1}/B_{n-1}}^{\\bullet })$ is $\\theta _{n-1}$ .", "Then $\\delta ^{}([\\theta _{n-1}])$ = $[\\delta \\!\\!\\!\\check{\\delta }(\\tilde{\\theta })]$ which is an element of ${\\mathbb {H}}^{l+1}(X_n,\\mathcal {M}_{0}^{n} / \\mathcal {M}^{n+1}_{0}\\otimes {\\mathcal {B}}_{p,q;X_{0}/B_{0}}^{\\bullet }) \\cong \\mathrm {m}_{0}^{n} /\\mathrm {m}^{n+1}_{0} \\otimes {\\mathbb {H}}^{l+1}(X,{\\mathcal {B}}_{p,q;X_{0}/B_{0}}^{\\bullet }).$ Denote $r_{X_{n}}$ the restriction to the space $X_{n}^{\\omega }$ (topological space $X$ with structure sheaf ${\\mathcal {O}}_{X_{n}}^{\\omega }$ ) and denote the following complex $ \\pi ^{-1}(\\Omega _{B_{n}\|B_{n-1}}^{\\omega }) \\stackrel{(+,-)}{\\rightarrow } \\pi _{n-1}^{}(\\Omega _{B_{n}\|B_{n-1}})^{\\omega } \\otimes {\\mathcal {O}}^{\\omega }_{X_n} \\oplus \\bar{\\pi }_{n-1}^{}(\\bar{\\Omega }_{B_{n}\|B_{n-1}})^{\\omega } \\otimes {\\mathcal {O}}^{\\omega }_{\\bar{X}_n} $ $\\rightarrow \\pi _{n-1}^{}(\\Omega _{B_{n}\|B_{n-1}})^{\\omega } \\wedge \\Omega _{X_n/B_n}^{1;\\omega }\\oplus \\bar{\\pi }_{n-1}^{}(\\bar{\\Omega }_{B_{n}\|B_{n-1}})^{\\omega } \\wedge \\bar{\\Omega }_{X_n/B_n}^{1;\\omega }$ $\\rightarrow \\ldots \\pi _{n-1}^{}(\\Omega _{B_{n}\|B_{n-1}})^{\\omega } \\wedge \\Omega _{X_n/B_n}^{p-1;\\omega }\\oplus \\bar{\\pi }_{n-1}^{}(\\bar{\\Omega }_{B_{n}\|B_{n-1}})^{\\omega } \\wedge \\bar{\\Omega }_{X_n/B_n}^{p-1;\\omega }$ $\\rightarrow \\bar{\\pi }_{n-1}^{}(\\bar{\\Omega }_{B_{n}\|B_{n-1}})^{\\omega } \\wedge \\bar{\\Omega }_{X_n/B_n}^{p;\\omega }\\rightarrow \\ldots \\rightarrow \\bar{\\pi }_{n-1}^{}(\\bar{\\Omega }_{B_{n}\|B_{n-1}})^{\\omega } \\wedge \\bar{\\Omega }_{X_n/B_n}^{q-1;\\omega }\\rightarrow 0,$ by $\\pi _{n-1}^{}(\\Omega _{B_{n}\|B_{n-1}})\\wedge {\\mathcal {B}}_{p,q;X_{n-1}/B_{n-1}}^{\\bullet .", "}$ In order to give the obstructions an explicit calculation, we need to consider the following map $ \\rho :{\\mathbb {H}}^{l}(X_n,\\mathcal {M}^{n}_{0}/\\mathcal {M}^{n+1}_{0}\\otimes {\\mathcal {B}}_{p,q;X_{0}/B_{0}}^{\\bullet }) \\rightarrow {\\mathbb {H}}^{l}(X_{n-1},\\pi _{n-1}^{}(\\Omega _{B_{n}\|B_{n-1}})\\wedge {\\mathcal {B}}_{p,q;X_{n-1}/B_{n-1}}^{\\bullet })$ which is defined by $\\rho [\\sigma ]=[ \\varphi ^{-1} \\circ r_{X_{n-1}}\\circ (L_{B_{n}}+L_{\\bar{B}_{n}})\\circ \\varphi (\\sigma )]$ , where $\\varphi ^{-1}$ is the quotient maps: $\\check{C}^{\\bullet }(\\mathbf {U}, \\pi _{n-1}^{}(\\Omega _{B_{n}\|B_{n-1}})^{\\omega } \\wedge (\\Omega ^{p;\\omega }_{X_{n}\|X_{n-1}} \\oplus {\\Omega }^{p;\\omega }_{\\bar{X}_{n}\|\\bar{X}_{n-1}} ))\\rightarrow \\check{C}^{\\bullet }(\\mathbf {U}, \\pi _{n-1}^{}(\\Omega _{B_{n}\|B_{n-1}}) ^{\\omega }\\wedge (\\Omega ^{p;\\omega }_{X_{n-1}/B_{n-1}} \\oplus \\bar{\\Omega }^{p;\\omega }_{X_{n-1}/B_{n-1}} ) ).$ An we have the following lemmas, the proof is completely the same as lemma 3.2 and lemma 3.3 in [9]: Lemma 4.2 The map $\\rho $ is well defined.", "Lemma 4.3 $\\rho ([\\delta \\!\\!\\!\\check{\\delta }(\\tilde{\\theta })])$ is exactly $o_{n}([\\theta ])$ in the previous section.", "Now consider the following exact sequence.", "The connecting homomorphism of the associated long exact sequence gives the Kodaira-Spencer class of order $n$ [8] [1.3.2], $0\\rightarrow \\pi _{n-1}^{}(\\Omega _{B_{n}\|B_{n-1}})^{\\omega }\\rightarrow \\Omega _{X_{n}\|X_{n-1}}^{\\omega }\\rightarrow \\Omega _{X_{n-1}/B_{n-1}}^{\\omega }\\rightarrow 0.", "$ By wedge the above exact sequence with $\\Omega ^{p-1;\\omega }_{X_{n-1}/B_{n-1}}$ , we get a new exact sequence.", "The connecting homomorphism of such exact sequence gives us a map from $H^{q}(X_{n-1},\\Omega ^{p;\\omega }_{X_{n-1}/B_{n-1}})$ to $H^{q+1}(X_{n-1},\\pi ^{*}(\\Omega _{B_{n}\|B_{n-1}})^{\\omega }\\wedge \\Omega ^{p-1;\\omega }_{X_{n-1}/B_{n-1}})$ .", "Denote such map by $\\kappa _{n}$ , for such map is simply the inner product with the Kodaira-Spencer class of order $n$ .", "With the above preparation, we are ready to proof the main theorem of this section.", "Theorem 4.4 Let $\\pi :\\mathcal {X}\\rightarrow B$ be a deformation of $\\pi ^{-1}(0)=X$ , where $X$ is a compact complex manifold.", "Let $\\pi _{n}:X_{n}\\rightarrow B_{n}$ be the $n$ th order deformation of $X$ .", "For arbitrary $[\\theta ]$ belongs to ${\\mathbb {H}}^l(X,{\\mathcal {B}}_{p,q}^{\\bullet })$ , suppose we can extend $[\\theta ]$ to order $n-1$ in $H^l(X_{n-1},{\\mathcal {B}}_{p,q;X_{n-1}/B_{n-1}}^{\\bullet })$ .", "Denote such element by $[\\theta _{n-1}]$ .", "The obstruction of the extension of $[\\theta ]$ to $n$ th order is given by: $ o_{n}([\\theta ])=-\\partial ^{\\bar{\\partial },{\\mathcal {B}}}_{X_{n-1}/B_{n-1}} \\circ \\kappa _{n} \\circ \\partial ^{{\\mathcal {B}},\\bar{\\partial }}_{X_{n-1}/B_{n-1}}([\\theta _{n-1}])- \\bar{\\partial }^{{\\partial },{\\mathcal {B}}}_{X_{n-1}/B_{n-1}} \\circ \\bar{\\kappa }_{n} \\circ \\bar{\\partial }^{{\\mathcal {B}},{\\partial }}_{X_{n-1}/B_{n-1}}([\\theta _{n-1}]),$ where $\\kappa _{n}$ is the $n$ th order Kodaira-Spencer class and $\\bar{\\kappa }_{n}$ is the $n$ th order Kodaira-Spencer class of the deformation $ \\bar{\\pi }: \\bar{{\\mathcal {X}}} \\rightarrow \\bar{B}$ .", "$\\partial ^{\\bar{\\partial },{\\mathcal {B}}}_{X_{n-1}/B_{n-1}}$ , $\\bar{\\partial }^{{\\partial },{\\mathcal {B}}}_{X_{n-1}/B_{n-1}}$ , $\\partial ^{{\\mathcal {B}},\\bar{\\partial }}_{X_{n-1}/B_{n-1}}$ and $\\bar{\\partial }^{{\\mathcal {B}},{\\partial }}_{X_{n-1}/B_{n-1}}$ are the maps defined in $§2$ .", "Note that $o_{n}([\\theta ]) = \\rho \\circ \\delta (\\tilde{\\theta }) = [\\varphi ^{-1} \\circ r_{X_{n-1}}\\circ (L_{B_{n}}+L_{\\bar{B}_{n}} )\\circ \\varphi \\circ \\delta (\\tilde{\\theta })] .$ Because $ (L_{B_{n}}+L_{\\bar{B}_{n}}+\\partial _{X_{n}/B_{n}}+\\overline{\\partial }_{X_{n}/B_{n}} )\\circ \\delta \\!\\!\\!\\check{\\delta }= - \\delta \\!\\!\\!\\check{\\delta }\\circ (L_{B_{n}}+L_{\\bar{B}_{n}}+\\partial _{X_{n}/B_{n}}+\\overline{\\partial }_{X_{n}/B_{n}} ).$ $r_{X_{n-1}}\\circ (L_{B_{n}}+L_{\\bar{B}_{n}} ) \\circ \\varphi \\circ \\delta \\!\\!\\!\\check{\\delta }(\\tilde{\\theta }) & \\equiv & r_{X_{n-1}} \\circ (L_{B_{n}}+L_{\\bar{B}_{n}} )\\circ (\\delta \\!\\!\\!\\check{\\delta }\\circ \\varphi - \\lambda \\circ \\varphi )(\\tilde{\\theta }) \\\\& \\equiv & r_{X_{n-1}} \\circ (L_{B_{n}}+L_{\\bar{B}_{n}} ) \\circ \\delta \\!\\!\\!\\check{\\delta }\\circ \\varphi (\\tilde{\\theta }) \\\\& \\equiv & -r_{X_{n-1}} \\circ (\\partial ^{\\bullet }_{X_{n}/B_{n}} \\circ \\delta \\!\\!\\!\\check{\\delta }+ \\delta \\!\\!\\!\\check{\\delta }\\circ \\partial ^{\\bullet }_{X_{n}/B_{n}}+ \\\\&& \\overline{\\partial }^{\\bullet }_{X_{n}/B_{n}} \\circ \\delta \\!\\!\\!\\check{\\delta }+ \\delta \\!\\!\\!\\check{\\delta }\\circ \\overline{\\partial }^{\\bullet }_{X_{n}/B_{n}} +\\delta \\!\\!\\!\\check{\\delta }\\circ (L_{B_{n}}+L_{\\bar{B}_{n}} ) ) \\circ \\varphi (\\tilde{\\theta }) \\\\& \\equiv & -r_{X_{n-1}} \\circ (\\partial ^{\\bullet }_{X_{n}/B_{n}} \\circ \\delta \\!\\!\\!\\check{\\delta }+ \\delta \\!\\!\\!\\check{\\delta }\\circ \\partial ^{\\bullet }_{X_{n}/B_{n}} + \\overline{\\partial }^{\\bullet }_{X_{n}/B_{n}} \\\\&&\\circ \\delta \\!\\!\\!\\check{\\delta }+ \\delta \\!\\!\\!\\check{\\delta }\\circ \\overline{\\partial }^{\\bullet }_{X_{n}/B_{n}}) \\circ \\varphi (\\tilde{\\theta })- \\delta \\!\\!\\!\\check{\\delta }\\circ r_{X_{n-1}}\\circ (L_{B_{n}}+L_{\\bar{B}_{n}} )\\circ \\varphi (\\tilde{\\theta }).", "\\\\$ Therefore $[\\varphi ^{-1} \\circ r_{X_{n-1}}\\circ (L_{B_{n}}+L_{\\bar{B}_{n}} )\\circ \\varphi \\circ \\delta (\\tilde{\\theta })] & = & [-\\varphi ^{-1} \\circ r_{X_{n-1}} \\circ (\\partial ^{\\bullet }_{X_{n}/B_{n}} \\circ \\delta \\!\\!\\!\\check{\\delta }+ \\delta \\!\\!\\!\\check{\\delta }\\circ \\partial ^{\\bullet }_{X_{n}/B_{n}}\\\\&& + \\overline{\\partial }^{\\bullet }_{X_{n}/B_{n}} \\circ \\delta \\!\\!\\!\\check{\\delta }+ \\delta \\!\\!\\!\\check{\\delta }\\circ \\overline{\\partial }^{\\bullet }_{X_{n}/B_{n}}) \\circ \\varphi (\\tilde{\\theta }) ]\\\\& = & -[ \\partial _{X_{n-1}/B_{n-1}} \\circ \\varphi ^{-1} \\circ r_{X_{n-1}} \\circ \\delta \\!\\!\\!\\check{\\delta }\\circ \\varphi (\\tilde{\\theta }) +\\\\&& \\overline{\\partial }_{X_{n-1}/B_{n-1}} \\circ \\varphi ^{-1} \\circ r_{X_{n-1}} \\circ \\delta \\!\\!\\!\\check{\\delta }\\circ \\varphi (\\tilde{\\theta }) \\\\&& + \\varphi ^{-1} \\circ r_{X_{n-1}} \\circ \\delta \\!\\!\\!\\check{\\delta }\\circ \\varphi (\\widetilde{\\partial _{X_{n-1}/B_{n-1}}(\\theta _{n-1})}) + \\\\&& \\varphi ^{-1} \\circ r_{X_{n-1}} \\circ \\delta \\!\\!\\!\\check{\\delta }\\circ \\varphi (\\widetilde{\\overline{\\partial }_{X_{n-1}/B_{n-1}}(\\theta _{n-1})})\\\\$ Since $(\\varphi ^{-1} \\circ r_{X_{n-1}} \\circ \\delta \\!\\!\\!\\check{\\delta }\\circ \\varphi (\\tilde{\\theta }))^{p-1,0}_u=0$ and $(\\varphi ^{-1} \\circ r_{X_{n-1}} \\circ \\delta \\!\\!\\!\\check{\\delta }\\circ \\varphi (\\tilde{\\theta }))^{0,q-1}_v=0$ , by lemma REF and lemma REF , we know that $[\\partial _{X_{n-1}/B_{n-1}} \\circ \\varphi ^{-1} \\circ r_{X_{n-1}} \\circ \\delta \\!\\!\\!\\check{\\delta }\\circ \\varphi (\\tilde{\\theta })]=0$ and $[\\overline{\\partial }_{X_{n-1}/B_{n-1}} \\circ \\varphi ^{-1} \\circ r_{X_{n-1}} \\circ \\delta \\!\\!\\!\\check{\\delta }\\circ \\varphi (\\tilde{\\theta })]=0$ .", "And from lemma REF and lemma REF , we also know that $[\\varphi ^{-1} \\circ r_{X_{n-1}} \\circ \\delta \\!\\!\\!\\check{\\delta }\\circ \\varphi (\\widetilde{\\partial _{X_{n-1}/B_{n-1}}(\\theta _{n-1})})]&=& [\\varphi ^{-1} \\circ r_{X_{n-1}} \\circ \\delta \\!\\!\\!\\check{\\delta }\\circ \\varphi (\\widetilde{\\partial _{X_{n-1}/B_{n-1}}(\\theta _{n-1} -\\theta _{n-1;u}^{p-1,0})} \\\\&&+\\partial _{X_{n-1}/B_{n-1}}\\theta _{n-1;u}^{p-1,0})]\\\\&=& [\\varphi ^{-1} \\circ r_{X_{n-1}} \\circ \\delta \\!\\!\\!\\check{\\delta }\\circ \\varphi (\\widetilde{\\partial _{X_{n-1}/B_{n-1}}\\theta _{n-1;u}^{p-1,0}})]$ and $[\\varphi ^{-1} \\circ r_{X_{n-1}} \\circ \\delta \\!\\!\\!\\check{\\delta }\\circ \\varphi (\\widetilde{\\overline{\\partial }_{X_{n-1}/B_{n-1}}(\\theta _{n-1})})]&=& [\\varphi ^{-1} \\circ r_{X_{n-1}} \\circ \\delta \\!\\!\\!\\check{\\delta }\\circ \\varphi (\\widetilde{\\overline{\\partial }_{X_{n-1}/B_{n-1}}(\\theta _{n-1}-\\theta _{n-1;v}^{0,q-1})}\\\\ && + \\overline{\\partial }_{X_{n-1}/B_{n-1}}\\theta _{n-1;v}^{0,q-1})]\\\\&=& [\\varphi ^{-1} \\circ r_{X_{n-1}} \\circ \\delta \\!\\!\\!\\check{\\delta }\\circ \\varphi (\\widetilde{\\overline{\\partial }_{X_{n-1}/B_{n-1}}\\theta _{n-1;v}^{0,q-1}})]$ By the definition of the maps: $\\partial ^{\\bar{\\partial },{\\mathcal {B}}}_{X_{n-1}/B_{n-1}}$ , $\\partial ^{{\\mathcal {B}},\\bar{\\partial }}_{X_{n-1}/B_{n-1}}$ and Lemma 3.4 in [9], we have $[\\varphi ^{-1} \\circ r_{X_{n-1}} \\circ \\delta \\!\\!\\!\\check{\\delta }\\circ \\varphi (\\widetilde{\\partial _{X_{n-1}/B_{n-1}}\\theta _{n-1;u}^{p-1,0}})]={\\partial }^{\\bar{\\partial },{\\mathcal {B}}}_{X_{n-1}/B_{n-1}} \\circ {\\kappa }_{n} \\circ {\\partial }^{{\\mathcal {B}},\\bar{\\partial }}_{X_{n-1}/B_{n-1}}([\\theta _{n-1}])$ and similarly, we have $[\\varphi ^{-1} \\circ r_{X_{n-1}} \\circ \\delta \\!\\!\\!\\check{\\delta }\\circ \\varphi (\\widetilde{\\overline{\\partial }_{X_{n-1}/B_{n-1}}\\theta _{n-1;v}^{0,q-1}})]= \\bar{\\partial }^{{\\partial },{\\mathcal {B}}}_{X_{n-1}/B_{n-1}} \\circ \\bar{\\kappa }_{n} \\circ \\bar{\\partial }^{{\\mathcal {B}},{\\partial }}_{X_{n-1}/B_{n-1}}([\\theta _{n-1}])$ So we have: $[\\varphi ^{-1} \\circ r_{X_{n-1}}\\circ (L_{B_{n}}+L_{\\bar{B}_{n}} ) \\circ \\varphi \\circ \\delta (\\tilde{\\theta })]= -{\\partial }^{\\bar{\\partial },{\\mathcal {B}}}_{X_{n-1}/B_{n-1}} \\circ {\\kappa }_{n}\\circ {\\partial }^{{\\mathcal {B}},\\bar{\\partial }}_{X_{n-1}/B_{n-1}}([\\theta _{n-1}])\\\\-\\bar{\\partial }^{{\\partial },{\\mathcal {B}}}_{X_{n-1}/B_{n-1}} \\circ \\bar{\\kappa }_{n} \\circ \\bar{\\partial }^{{\\mathcal {B}},{\\partial }}_{X_{n-1}/B_{n-1}}([\\theta _{n-1}]).$ Apply the above theorem and theorem REF to study the jumping phenomenon of the dimensions of Bott-Chern(Aeppli) cohomology groups.", "We have the following theorems.", "Theorem 4.5 Let $\\pi :\\mathcal {X}\\rightarrow B$ be a deformation of $\\pi ^{-1}(0)=X$ , where $X$ is a compact complex manifold.", "Let $\\pi _{n}:X_{n}\\rightarrow B_{n}$ be the $n$ th order deformation of $X$ .", "If there exists an elements $[\\theta ^1]$ in $H^{p,q}_{\\rm BC}(X)$ or an elements $[\\theta ^2]$ in $H^{p-1,q-1}_{\\rm A}(X)$ and a minimal natural number $n \\ge 1$ such that the $n$ th order obstruction $o_n([\\theta ^1])\\ne 0$ or $o_n([\\theta ^2])\\ne 0$ , then the $h_{p,q}^{\\mathrm {\\rm BC}}(X(t))$ will jump at the point $t=0$ .", "And the formulas for the obstructions are given by: $ o_n([\\theta ^1]) = -{\\partial }^{\\bar{\\partial },{\\mathcal {B}}}_{X_{n-1}/B_{n-1}} \\circ {\\kappa }_{n}\\circ r_{BC,\\bar{\\partial }}([\\theta _{n-1}])\\\\-\\bar{\\partial }^{{\\partial },{\\mathcal {B}}}_{X_{n-1}/B_{n-1}} \\circ \\bar{\\kappa }_{n} \\circ r_{BC,{\\partial }}([\\theta _{n-1}]);$ $ o_n([\\theta ^2]) = - \\partial ^{\\bar{\\partial },BC}_{X_{n-1}/B_{n-1}} \\circ {\\kappa }_{n}\\circ \\partial ^{A,\\bar{\\partial }}_{X_{n-1}/B_{n-1}}([\\theta _{n-1}])\\\\-\\bar{\\partial }^{{\\partial },BC}_{X_{n-1}/B_{n-1}} \\circ \\bar{\\kappa }_{n} \\circ \\bar{\\partial }^{A,{\\partial }}_{X_{n-1}/B_{n-1}}([\\theta _{n-1}]) .$ Theorem 4.6 Let $\\pi :\\mathcal {X}\\rightarrow B$ be a deformation of $\\pi ^{-1}(0)=X$ , where $X$ is a compact complex manifold.", "Let $\\pi _{n}:X_{n}\\rightarrow B_{n}$ be the $n$ th order deformation of $X$ .", "If there exists an elements $[\\theta ^1]$ in $H^{p,q}_{\\rm A}(X)$ or an elements $[\\theta ^2]$ in ${\\mathbb {H}}^{p+q}(X,{\\mathcal {B}}_{p+1,q+1}^{\\bullet })$ and a minimal natural number $n \\ge 1$ such that the $n$ th order obstruction $o_n([\\theta ^1])\\ne 0$ or $o_n([\\theta ^2])\\ne 0$ , then the $h_{p,q}^{\\mathrm {\\rm BC}}(X(t))$ will jump at the point $t=0$ .", "And the formulas for the obstructions are given by: $ o_n([\\theta ^1]) = - \\partial ^{\\bar{\\partial },BC}_{X_{n-1}/B_{n-1}} \\circ {\\kappa }_{n}\\circ \\partial ^{A,\\bar{\\partial }}_{X_{n-1}/B_{n-1}}([\\theta _{n-1}])\\\\-\\bar{\\partial }^{{\\partial },BC}_{X_{n-1}/B_{n-1}} \\circ \\bar{\\kappa }_{n} \\circ \\bar{\\partial }^{A,{\\partial }}_{X_{n-1}/B_{n-1}}([\\theta _{n-1}]) .$ $ o_n([\\theta ^2]) = - r_{\\bar{\\partial },A} \\circ {\\kappa }_{n} \\circ {\\partial }^{{\\mathcal {B}},\\bar{\\partial }}_{X_{n-1}/B_{n-1}}([\\theta _{n-1}])\\\\-r_{{\\partial },A}\\circ \\bar{\\kappa }_{n}\\circ \\bar{\\partial }^{{\\mathcal {B}},{\\partial }}_{X_{n-1}/B_{n-1}}([\\theta _{n-1}]) .$ By these theorems, we can get the following corollaries immediately.", "Corollary 4.7 Let $\\pi :\\mathcal {X} \\rightarrow B$ be a deformation of $\\pi ^{-1}(0)=X$ , where $X$ is a compact complex manifold.", "Suppose that up to order $n$ , the maps $r_{BC,\\bar{\\partial }}: H^{p,q}_{\\rm BC}(X_n/B_n) \\rightarrow H^q(X_n,\\Omega _{X_n/B_n}^{p;\\omega }) $ and $r_{BC,{\\partial }}: H^{p,q}_{\\rm BC}(X_n/B_n) \\rightarrow H^p(\\bar{X}_n,\\bar{\\Omega }_{X_n/B_n}^{q;\\omega }) $ is 0.", "For arbitrary $[\\theta ]$ that belongs to $ H^{p,q}_{\\rm BC}(X) $ , it can be extended to order $n+1$ in $H^{p,q}_{\\rm BC}(X_{n+1}/B_{n+1}).$ This result can be shown by induction on $k$ .", "Suppose that the corollary is proved for $k-1$ , then we can extend $[\\theta ]$ to and element $[\\theta _{k-1}]$ in $H^{p,q}_{\\rm BC}(X_{k-1}/B_{k-1}).$ By Theorem REF , the obstruction for the extension of $[\\theta ]$ to $k$ th order comes from: $ o_k([\\theta ]) = -{\\partial }^{\\bar{\\partial },{\\mathcal {B}}}_{X_{k-1}/B_{k-1}} \\circ {\\kappa }_{n}\\circ r_{BC,\\bar{\\partial }}([\\theta _{k-1}])\\\\-\\bar{\\partial }^{{\\partial },{\\mathcal {B}}}_{X_{k-1}/B_{k-1}} \\circ \\bar{\\kappa }_{n} \\circ r_{BC,{\\partial }}([\\theta _{k-1}]);$ By the assumption, $r_{BC,\\bar{\\partial }}: H^{p,q}_{\\rm BC}(X_{k-1}/B_{k-1}) \\rightarrow H^q(X_{k-1},\\Omega _{X_{k-1}/B_{k-1}}^{p;\\omega }) $ and $r_{BC,{\\partial }}: H^{p,q}_{\\rm BC}(X_{k-1}/B_{k-1}) \\rightarrow H^p(\\bar{X}_{k-1},\\bar{\\Omega }_{X_{k-1}/B_{k-1}}^{q;\\omega }) $ is 0, where $k\\le n+1$ .", "So we have $r_{BC,{\\partial }}([\\theta _{k-1}])=0$ and $r_{BC,{\\bar{\\partial }}}([\\theta _{k-1}])=0$ .", "So the obstruction $o_{k}([\\theta ])$ is trivial which means $[\\theta ]$ can be extended to $k$ th order.", "Since we have $\\partial ^{A,\\bar{\\partial }}_{X_n/B_n}: H^{p,q}_{\\rm A}(X_n/B_n) \\rightarrow H^q(X_n,\\Omega _{X_n/B_n}^{p+1;\\omega }) $ is the composition of $\\partial ^{A,BC}_{X_n/B_n}: H^{p,q}_{\\rm A}(X_n/B_n)\\rightarrow H^{p+1,q}_{\\rm BC}(X_n/B_n)$ and $r_{BC,\\bar{\\partial }}: H^{p+1,q}_{\\rm BC}(X_n/B_n) \\rightarrow H^q(X_n,\\Omega _{X_n/B_n}^{p+1;\\omega }).", "$ With the same proof of the above corollary, we have the following result and we omit the proof.", "Corollary 4.8 Let $\\pi :\\mathcal {X} \\rightarrow B$ be a deformation of $\\pi ^{-1}(0)=X$ , where $X$ is a compact complex manifold.", "Suppose that up to order $n$ , the maps $r_{BC,\\bar{\\partial }}: H^{p+1,q}_{\\rm BC}(X_n/B_n) \\rightarrow H^q(X_n,\\Omega _{X_n/B_n}^{p+1;\\omega }) $ and $r_{BC,{\\partial }}: H^{p,q+1}_{\\rm BC}(X_n/B_n) \\rightarrow H^p(\\bar{X}_n,\\bar{\\Omega }_{X_n/B_n}^{q+1;\\omega }) $ is 0.", "For arbitrary $[\\theta ]$ that belongs to $ H^{p,q}_{\\rm A}(X) $ , it can be extended to order $n+1$ in $H^{p,q}_{\\rm A}(X_{n+1}/B_{n+1}).$" ], [ "An Example", "In this section, we will use the formula in previous section to study the jumping phenomenon of the dimensions of Bott-Chern cohomology groups $h_{\\rm BC}^{p,q}$ and Aeppli cohomology groups $h_{\\rm A}^{p,q}$ of small deformations of Iwasawa manifold.", "It was Kodaira who first calculated small deformations of Iwasawa manifold [6].", "In the first part of this section, let us recall his result.", "Set $G=\\left\\lbrace \\left( \\begin{array}{ccc} 1 & z_{2} & z_{3}\\\\0 & 1 & z_{1}\\\\0 & 0 & 1\\\\\\end{array} \\right); z_{i}\\in \\mathbb {C}\\right\\rbrace \\cong \\mathbb {C}^{3},$ $\\Gamma =\\left\\lbrace \\left( \\begin{array}{ccc} 1 & \\omega _{2} & \\omega _{3}\\\\0 & 1 & \\omega _{1}\\\\0 & 0 & 1\\\\\\end{array} \\right); \\omega _{i}\\in \\mathbb {Z}+\\mathbb {Z}\\sqrt{-1}\\right\\rbrace \\\\.$ The multiplication is defined by $\\left( \\begin{array}{ccc} 1 & z_{2} & z_{3}\\\\0 & 1 & z_{1}\\\\0 & 0 & 1\\\\\\end{array} \\right)\\left( \\begin{array}{ccc} 1 & \\omega _{2} & \\omega _{3}\\\\0 & 1 & \\omega _{1}\\\\0 & 0 & 1\\\\\\end{array} \\right)=\\left( \\begin{array}{ccc} 1 & z_{2}+\\omega _{2} & z_{3}+\\omega _{1}z_{2}+\\omega _{3}\\\\0 & 1 & z_{1}+\\omega _{1}\\\\0 & 0 & 1\\\\\\end{array} \\right).$ $X=G/\\Gamma $ is called Iwasawa manifold.", "We may consider $X=\\mathbb {C}^{3}/\\Gamma $ .", "$g\\in \\Gamma $ operates on $\\mathbb {C}^3$ as follows: $ z^{\\prime }_{1}=z_{1}+\\omega _{1}, \\qquad z^{\\prime }_{2}=z_{2}+\\omega _{2}, \\qquad z^{\\prime }_{3}=z_{3}+\\omega _{1}z_{2}+\\omega _{3},$ where $g=(\\omega _{1},\\omega _{2},\\omega _{3})$ and $z^{\\prime }=z\\cdot g$ .", "There exist holomorphic 1-froms $\\varphi _{1},\\varphi _{2},\\varphi _{3}$ which are linearly independent at every point on $X$ and are given by $\\varphi _{1}=dz_{1},\\qquad \\varphi _{2}=dz_{2}, \\qquad \\varphi _{3}=dz_{3}-z_{1}dz_{2},$ so that $ d\\varphi _{1}=d\\varphi _{2}=0, \\qquad d\\varphi _{3}=-\\varphi _{1}\\wedge \\varphi _{2}.$ On the other hand we have holomorphic vector fields $\\theta _{1},\\theta _{2},\\theta _{3}$ on $X$ given by $ \\theta _{1}=\\frac{\\partial }{\\partial z_{1}},\\qquad \\theta _{2}=\\frac{\\partial }{\\partial z_{2}}+z_{1}\\frac{\\partial }{\\partial z_{3}},\\qquad \\theta _{3}=\\frac{\\partial }{\\partial z_{3}}.$ It is easily seen that $ [\\theta _{1},\\theta _{2}]=-[\\theta _{2},\\theta _{1}]=\\theta _{3}, \\qquad [\\theta _{1},\\theta _{3}]=[\\theta _{2},\\theta _{3}]=0.$ In view of Theorem 3 in [6], $H^{1}(X,\\mathcal {O}_{X})$ is spanned by $\\overline{\\varphi }_{1},\\overline{\\varphi }_{2}$ .", "Since $\\Theta $ is isomorphic to $\\mathcal {O}^{3}$ , $H^{1}(X,T_{X})$ is spanned by $\\theta _{i}\\overline{\\varphi }_{\\lambda }, i=1,2,3, \\lambda =1,2$ .", "Consider the small deformation of $X$ given by $ \\psi (t)=\\sum ^{3}_{i=1}\\sum _{\\lambda =1}^{2}t_{i\\lambda }\\theta _{i}\\overline{\\varphi }_{\\lambda }t-(t_{11}t_{22}-t_{21}t_{12})\\theta _{3}\\overline{\\varphi }_{3}t^{2}.$ We summarize the numerical characters of deformations.", "The deformations are divided into the following three classes, the classes and subclasses of this classification are characterized by the following values of the parameters(all the details can be found in [2]): class (i) $t_{11}=t_{12}=t_{21}=t_{22}=0$ ; class (ii) $D\\left(\\mathbf {t}\\right)=0$ and $\\left(t_{11},\\,t_{12},\\,t_{21},\\,t_{22}\\right)\\ne \\left(0,\\,0,\\,0,\\,0\\right)$ : subclass (ii.a) $D\\left(\\mathbf {t}\\right)=0$ and rank $S=1$ ; subclass (ii.b) $D\\left(\\mathbf {t}\\right)=0$ and rank $S=2$ ; class (iii) $D\\left(\\mathbf {t}\\right)\\ne 0$ : subclass (iii.a) $D\\left(\\mathbf {t}\\right)\\ne 0$ and rank $S=1$ ; subclass (iii.b) $D\\left(\\mathbf {t}\\right)\\ne 0$ and rank $S=2$ ; the matrix $S$ is defined by $ S \\;:=\\;\\left(\\begin{array}{cccc}\\overline{\\sigma _{1\\bar{1}}} & \\overline{\\sigma _{2\\bar{2}}} & \\overline{\\sigma _{1\\bar{2}}} & \\overline{\\sigma _{2\\bar{1}}} \\\\\\sigma _{1\\bar{1}} & \\sigma _{2\\bar{2}} & \\sigma _{2\\bar{1}} & \\sigma _{1\\bar{2}}\\end{array}\\right)$ where $\\sigma _{1\\bar{1}},\\,\\sigma _{1\\bar{2}},\\,\\sigma _{2\\bar{1}},\\,\\sigma _{2\\bar{2}}\\in and $ 12 are complex numbers depending only on $\\mathbf {t}$ such that $ d\\varphi ^3_\\mathbf {t}\\;=:\\; \\sigma _{12}\\,\\varphi ^1_{\\mathbf {t}}\\wedge \\varphi ^2_{\\mathbf {t}}+\\sigma _{1\\bar{1}}\\,\\varphi ^1_{\\mathbf {t}}\\wedge \\bar{\\varphi }^1_{\\mathbf {t}}+\\sigma _{1\\bar{2}}\\,\\varphi ^1_{\\mathbf {t}}\\wedge \\bar{\\varphi }^2_{\\mathbf {t}}+\\sigma _{2\\bar{1}}\\,\\varphi ^2_{\\mathbf {t}}\\wedge \\bar{\\varphi }^1_{\\mathbf {t}}+\\sigma _{2\\bar{2}}\\,\\varphi ^2_{\\mathbf {t}}\\wedge \\bar{\\varphi }^2_{\\mathbf {t}} .$ The first order asymptotic behaviour of $\\sigma _{12},\\,\\sigma _{1\\bar{1}},\\,\\sigma _{1\\bar{2}},\\,\\sigma _{2\\bar{1}},\\,\\sigma _{2\\bar{2}}$ for $\\mathbf {t}$ near 0 is the following: $\\left\\lbrace \\begin{array}{rcl}\\sigma _{12} &=& -1 +\\mathrm {o}\\left(\\left\|\\mathbf {t}\\right\|\\right) \\\\[5pt]\\sigma _{1\\bar{1}} &=& t_{21} +\\mathrm {o}\\left(\\left\|\\mathbf {t}\\right\|\\right) \\\\[5pt]\\sigma _{1\\bar{2}} &=& t_{22} +\\mathrm {o}\\left(\\left\|\\mathbf {t}\\right\|\\right) \\\\[5pt]\\sigma _{2\\bar{1}} &=& -t_{11} +\\mathrm {o}\\left(\\left\|\\mathbf {t}\\right\|\\right) \\\\[5pt]\\sigma _{2\\bar{2}} &=& -t_{12} +\\mathrm {o}\\left(\\left\|\\mathbf {t}\\right\|\\right) \\\\[5pt]\\end{array}\\right.\\qquad \\text{ for } \\qquad \\mathbf {t}\\in \\text{ classes {\\itshape (i)}, {\\itshape (ii)} and {\\itshape (iii)}}\\;.$ The following tables are given by D.Angella in [2].", "Dimensions of the cohomologies of the Iwasawa manifold and of its small deformations: Table: NO_CAPTION Table: NO_CAPTION Table: NO_CAPTION Table: NO_CAPTIONFrom the tables above, we know that the jumping phenomenon happens in $h^{2,0}_{\\rm BC}$ , $h^{0,2}_{\\rm BC}$ and $h^{2,2}_{\\rm BC}$ of Bott-Chern cohomology and symmetrically happens in $h^{3,1}_{\\rm A}$ , $h^{1,3}_{\\rm A}$ and $h^{1,1}_{\\rm A}$ of Aeppli cohomology.", "Now let us explain the jumping phenomenon of the dimensions of Bott-Chern cohomology and Aeppli cohomology by using the obstruction formula.", "From $§4$ in [2], it follows that the Bott-Chern cohomology groups in bi-degree $(2,0),(0,2),(2,2)$ are: $H^{2,0}_{\\rm BC} (X)& =&Span_{\\mathbb {C}}\\lbrace [\\varphi _{1}\\wedge \\varphi _{2}],[\\varphi _{2}\\wedge \\varphi _{3}],[\\varphi _{3}\\wedge \\varphi _{1}]\\rbrace ,\\\\ H^{0,2}_{\\rm BC} (X)& =&Span_{\\mathbb {C}}\\lbrace [\\overline{\\varphi }_{1}\\wedge \\overline{\\varphi }_{2}],[\\overline{\\varphi }_{2}\\wedge \\overline{\\varphi }_{3}],[\\overline{\\varphi }_{3}\\wedge \\overline{\\varphi }_{1}]\\rbrace , \\\\H^{2,2}_{\\rm BC} (X)& =&Span_{\\mathbb {C}}\\lbrace [{\\varphi }_{2}\\wedge {\\varphi }_{3}\\wedge \\overline{\\varphi }_{1}\\wedge \\overline{\\varphi }_{2}], [{\\varphi }_{3}\\wedge {\\varphi }_{1}\\wedge \\overline{\\varphi }_{1}\\wedge \\overline{\\varphi }_{2}],\\\\&&[\\varphi _{1}\\wedge \\varphi _{2}\\wedge \\overline{\\varphi }_{2}\\wedge \\overline{\\varphi }_{3}],[\\varphi _{2}\\wedge \\varphi _{3}\\wedge \\overline{\\varphi }_{2}\\wedge \\overline{\\varphi }_{3}],[\\varphi _{3}\\wedge \\varphi _{1}\\wedge \\overline{\\varphi }_{2}\\wedge \\overline{\\varphi }_{3}],\\\\&&[\\varphi _{1}\\wedge \\varphi _{2}\\wedge \\overline{\\varphi }_{3}\\wedge \\overline{\\varphi }_{1}],[\\varphi _{2}\\wedge \\varphi _{3}\\wedge \\overline{\\varphi }_{3}\\wedge \\overline{\\varphi }_{1}],[\\varphi _{3}\\wedge \\varphi _{1}\\wedge \\overline{\\varphi }_{3}\\wedge \\overline{\\varphi }_{1}]\\rbrace , \\\\$ and the Aeppli cohomology groups in bi-degree $(3,1),(1,3),(1,1)$ are: $H^{3,1}_{\\rm A} (X)& =&Span_{\\mathbb {C}}\\lbrace [\\varphi _{1}\\wedge \\varphi _{2}\\wedge \\varphi _{3}\\wedge \\overline{\\varphi }_{1}],[\\varphi _{1}\\wedge \\varphi _{2}\\wedge \\varphi _{3}\\wedge \\overline{\\varphi }_{2}],[\\varphi _{1}\\wedge \\varphi _{2}\\wedge \\varphi _{3}\\wedge \\overline{\\varphi }_{3}]\\rbrace ,\\\\H^{1,3}_{\\rm A} (X)& =&Span_{\\mathbb {C}}\\lbrace [\\varphi _{1}\\wedge \\overline{\\varphi }_{1}\\wedge \\overline{\\varphi }_{2}\\wedge \\overline{\\varphi }_{3}],[\\varphi _{2}\\wedge \\overline{\\varphi }_{1}\\wedge \\overline{\\varphi }_{2}\\wedge \\overline{\\varphi }_{3}],[\\varphi _{3}\\wedge \\overline{\\varphi }_{1}\\wedge \\overline{\\varphi }_{2}\\wedge \\overline{\\varphi }_{3}]\\rbrace , \\\\H^{1,1}_{\\rm A} (X)& =&Span_{\\mathbb {C}}\\lbrace [\\varphi _{1}\\wedge \\overline{\\varphi }_{1}],[\\varphi _{1}\\wedge \\overline{\\varphi }_{2}],[\\varphi _{1}\\wedge \\overline{\\varphi }_{3}],[\\varphi _{2}\\wedge \\overline{\\varphi }_{1}],\\\\&&[\\varphi _{2}\\wedge \\overline{\\varphi }_{2}],[\\varphi _{2}\\wedge \\overline{\\varphi }_{3}],[\\varphi _{3}\\wedge \\overline{\\varphi }_{1}],[\\varphi _{3}\\wedge \\overline{\\varphi }_{2}]\\rbrace .$ For example, let us first consider $h^{2,0}_{\\rm BC}$ , in the ii) class of deformation.", "The Kodaira-Spencer class of the this deformation is $\\psi _{1}(t)=\\sum ^{3}_{i=1}\\sum _{\\lambda =1}^{2}t_{i\\lambda }\\theta _{i}\\overline{\\varphi }_{\\lambda }$ , and $\\bar{\\psi }_{1}(t)=\\sum ^{3}_{i=1}\\sum _{\\lambda =1}^{2}\\bar{t}_{i\\lambda }\\bar{\\theta }_{i}{\\varphi }_{\\lambda }$ , with $t_{11}t_{22}-t_{21}t_{12}=0$ .", "It is easy to check that $o_{1}(\\varphi _{1}\\wedge \\varphi _{2})=-\\partial (int(\\psi _{1}(t))(\\varphi _{1}\\wedge \\varphi _{2}))-\\bar{\\partial }(int(\\bar{\\psi }_{1}(t))(\\varphi _{1}\\wedge \\varphi _{2}))=0$ , $o_{1}(t_{11}\\varphi _{2}\\wedge \\varphi _{3}-t_{21}\\varphi _{1}\\wedge \\varphi _{3})=-\\partial ((t_{11}t_{22}-t_{21}t_{12})\\varphi _{3}\\wedge \\overline{\\varphi }_{2})=0$ , and $o_{1}(\\varphi _{2}\\wedge \\varphi _{3})=t_{21}\\varphi _{1}\\wedge \\varphi _{2}\\wedge \\overline{\\varphi }_{1}+t_{22}\\varphi _{1}\\wedge \\varphi _{2}\\wedge \\overline{\\varphi }_{2}$ , $o_{1}(\\varphi _{1}\\wedge \\varphi _{3})=t_{11}\\varphi _{1}\\wedge \\varphi _{2}\\wedge \\overline{\\varphi }_{1}+t_{12}\\varphi _{1}\\wedge \\varphi _{2}\\wedge \\overline{\\varphi }_{2}$ .", "Therefore, we have shown that for an element of the subspace $Span_{\\mathbb {C}}\\lbrace [\\varphi _{1}\\wedge \\varphi _{2}],[t_{11}\\varphi _{2}\\wedge \\varphi _{3}-t_{21}\\varphi _{1}\\wedge \\varphi _{3}]\\rbrace $ , the first order obstruction is trivial, while, since $(t_{11},t_{12},t_{21},t_{22})\\ne (0,0,0,0)$ , at least one of the obstruction $o_{1}(\\varphi _{2}\\wedge \\varphi _{3})$ , $o_{1}(\\varphi _{1}\\wedge \\varphi _{3})$ is non trivial which partly explain why the Hodge number $h^{2,0}_{\\rm BC}$ jumps from 3 to 2.", "For another example, let us consider $h^{1,1}_{\\rm A}$ , in the ii) class of deformation.", "It is easy to check that all the first order obstructions of the cohomology classes are trivial.", "However, if we want to study the jumping phenomenon, we also need to consider the obstructions come from ${\\mathbb {H}}^{2}(X,{\\mathcal {B}}_{2,2}^{\\bullet })$ .", "It is easy to check that: $ {\\mathbb {H}}^{2}(X,{\\mathcal {B}}_{2,2}^{\\bullet })=Span_{\\mathbb {C}}\\lbrace [\\varphi _{3}],[\\bar{\\varphi }_{3}]\\rbrace .$ and $ o_1({\\varphi _{3}})=-t_{11}\\varphi _{2}\\wedge \\bar{\\varphi }_{1}-t_{12}\\varphi _{2}\\wedge \\bar{\\varphi }_{2}+t_{21}\\varphi _{1}\\wedge \\bar{\\varphi }_{1}+t_{22}\\varphi _{1}\\wedge \\bar{\\varphi }_{2};$ $ o_1({\\bar{\\varphi }_{3}})=-\\bar{t}_{11}\\bar{\\varphi }_{2}\\wedge {\\varphi }_{1}-\\bar{t}_{12}\\bar{\\varphi }_{2}\\wedge {\\varphi }_{2}+\\bar{t}_{21}\\bar{\\varphi }_{1}\\wedge {\\varphi }_{1}+\\bar{t}_{22}\\bar{\\varphi }_{1}\\wedge {\\varphi }_{2}.$ Note that the first order of $S$ is $\\left(\\begin{array}{cccc}-\\overline{t}_{21} & -\\overline{t}_{12} & \\overline{t}_{22}& \\overline{t}_{11} \\\\-t_{21} & -t_{21} & t_{11} & t_{22}\\end{array}\\right)$ If rank of the first order of $S=1$ , then there exists $c_1,c_2$ such that $ o_1(c_1{\\varphi }_{3} +c_2 \\bar{\\varphi }_{3}) \\ne 0.$ If rank of the first order of $S=2$ , then for all $c_1,c_2$ $ o_1(c_1{\\varphi }_{3} +c_2 \\bar{\\varphi }_{3}) = 0.$ and exactly these obstructions make $h^{1,1}_{\\rm A}$ jumps from 8 to 7 in $(ii.a)$ and from 8 to 6 in $(ii.b)$ .", "In the end of the section, we want to give the following observation as an application of the formula.", "Proposition 5.1 Let $X$ be an non-K$\\ddot{a}$ hler nilpotent complex parallelisable manifold whose dimension is more than 2, and $\\pi :\\mathcal {X}\\rightarrow B$ be the versal deformation family of $X$ .", "Then the number $h_{\\rm A}^{1,1}$ will jump in any neighborhood of $0\\in B$ .", "From the proof of [9] propsition 4.2, we know there exists an element $[\\theta ]$ in $H^0(X,\\Omega _X)$ whose $o_1([\\theta ]) \\ne 0$ .", "It is easy to check that $\\theta $ also represents an element in ${\\mathbb {H}}^2({X,{\\mathcal {B}}_{2,2}}^{\\bullet })$ , let us denote it by $[\\theta ]_{{\\mathcal {B}}}$ and it is also easy to check that $o_1([\\theta ]) = o_1([\\theta ]_{{\\mathcal {B}}})$ in this case.", "Therefore the number $h_{\\rm A}^{1,1}$ will jump in any neighborhood of $0\\in B$ ." ] ]	1403.0285
[ [ "Persistence and failure of mean-field approximations adapted to a class\n of systems of delay-coupled excitable units" ], [ "Abstract We consider the approximations behind the typical mean-field model derived for a class of systems made up of type II excitable units influenced by noise and coupling delays.", "The formulation of the two approximations, referred to as the Gaussian and the quasi-independence approximation, as well as the fashion in which their validity is verified, are adapted to reflect the essential properties of the underlying system.", "It is demonstrated that the failure of the mean-field model associated with the breakdown of the quasi-independence approximation can be predicted by the noise-induced bistability in the dynamics of the mean-field system.", "As for the Gaussian approximation, its violation is related to the increase of noise intensity, but the actual condition for failure can be cast in qualitative, rather than quantitative terms.", "We also discuss how the fulfilment of the mean-field approximations affects the statistics of the first return times for the local and global variables, further exploring the link between the fulfilment of the quasi-independence approximation and certain forms of synchronization between the individual units." ], [ "Background on the exact system", "Validity of MFAs is analyzed in case of a collection of N Fitzhugh-Nagumo excitable units, whose dynamics is set by: dxi=(xi-xi3/3-yi)dt+cNj=1N (xj(t-)-xi)dt dyi=(xi+b)dt+2DdWi, i=1,...N Each unit interacts with every other via diffusive delayed couplings, whereby the coupling strength $c$ and the time-lag $\\tau $ are taken uniform.", "Parameters $\\epsilon =0.01$ and $b=1.05$ are such that the isolated units display excitable behavior, having stable fixed point (FP) as the only attractor.", "The terms $\\sqrt{2D}dW_i$ represent stochastic increments of the independent Wiener processes, viz.", "$dW_i$ satisfy $E(dW_i)=0$ , $E(dW_idW_j)=\\delta _{i,j}dt$ , where $E()$ denotes the expectation over different realizations of the stochastic process.", "Having proposed that the nontrivial conditions for the fulfilment of the MFAs derive from the qualitative properties of the underlying dynamics, we first summarize the typical regimes exhibited by $(x_i(t),y_i(t))$ , beginning with the deterministic case $D=0$ .", "For small $c$ and $\\tau $ , the only attractor of each unit is FP and the dynamics is excitable.", "For larger $c$ and/or larger $\\tau $ , the FP undergoes a Hopf bifurcation and the asymptotic dynamics resides on a stable limit cycle (LC).", "The LC conforms to relaxation oscillations, with two clearly distinguished slow branches, the refractory and the spiking one, and two fast transients in between, cf.", "Fig.", "REF (b) where small noise perturbations are added.", "Small D induces small fluctuations around the attractor of the deterministic dynamics.", "If the latter motion lies on LC, the impact of $D$ is reflected mostly in the fluctuations of phase of the oscillatory dynamics between the different stochastic realizations.", "Apart from the increase of fluctuation amplitudes, enhancing $D$ may give rise to the transition from the stochastically stable FP to the noise induced spiking.", "The latter can appear as nearly periodic or irregular depending on $c, \\tau $ and $D$ .", "It is known that in systems of excitable units subjected to $D$ and $\\tau $ , the length of inter-spike intervals (ISIs) is influenced by the competition between two characteristic time scales [7], [6].", "One is set by the self-oscillation \"period\" $T_0(D)$ obtained for $\\tau =0$ , whereas the other is adjusted with $\\tau $ .", "Loosely speaking, for $\\tau <T_0(D)$ and intermediate $c$ , the noise-led dynamics characterized by $T_0(D)$ prevails over the delay-driven one unless $\\tau $ is commensurate or comparable to $T_0(D)$ .", "This paradigm may carry over to the collective motion due to synchronization of individual units." ], [ "Formulation of MFAs", "The first MFA derives from the strong law of large numbers, by which the sample average $S_N=N^{-1}\\sum \\limits _{i=1}^{N}s_i$ of $N$ independent and identically distributed random variables $s_i$ converges almost surely to the expectation value $E(s_i)$ for $N\\rightarrow \\infty $ .", "How $S_N$ approaches $E(s_i)$ for large, but finite $N$ and finite variances of $s_i$ distributions $\\sigma ^2$ , is specified by the central limit theorem, which implies that $S_N$ follow the normal distribution $\\mathcal {N}(E(s_i),\\sigma ^2/N)$ .", "In our setup, the subsets $\\lbrace x_i(t)\|i=1,\\dots N\\rbrace $ and $\\lbrace y_i(t)\|i=1,\\dots N\\rbrace $ at any given $t$ are obviously not made up of independent variables, but one may still consider the influence of interaction terms negligible if $N$ is sufficiently large.", "The latter is referred to as the quasi-independent approximation (QIA), whose precise formulation is: Approximation 1.", "(QIA) random variables $\\lbrace x_i(t)\|i=1,\\dots N\\rbrace $ and $\\lbrace y_i(t)\|i=1,\\dots N\\rbrace $ for each $t$ and sufficiently large N satisfy the approximate equalities: X(t)1NiN xi(t)E(xi(t)) Y(t)1NiN yi(t)E(yi(t)) On the left of (REF ) are the spatial averages, used to define the global variables $X(t)$ and $Y(t)$ .", "Note that the method implemented in section to test the validity of QIA will reflect the relaxation character of oscillations typical for class II excitable systems.", "The need for the second approximation becomes apparent after carrying out the spatial average and applying the QIA on (REF ).", "The fashion in which the terms $E(x_i^3(t))$ are to be treated is resolved by the Gaussian approximation (GA), given as: Approximation 2.", "(GA) for most time instances $t_0$ , the small random increments $dx_i(t),dy_i(t)$ for $t\\in (t_0,t_0+\\delta t)$ can be computed with sufficiently good accuracy by assuming that the random variables $x_i(t),y_i(t)$ for each $i=1,\\dots N$ and for $t\\in (t_0,t_0+\\delta t)$ are normally distributed around $(E(x_i(t)),E(y_i(t)))\\approx (X(t),Y(t))$ .", "GA is intentionally stated in a weak sense, containing phrases \"sufficiently good accuracy\" and \"for most time instances\".", "The former implies that the approximate solution should have the same qualitative features as the exact one.", "Nevertheless, the phrase \"for most time instances\" is crucial, because it specifically targets the class II excitable systems, being introduced to account for the relaxation character of oscillations, as explicitly demonstrated in section .", "Further note that the GA does not require $\\lbrace x_i(t)\|i=1,\\dots N\\rbrace $ and $\\lbrace y_i(t)\|i=1,\\dots N\\rbrace $ to be Gaussian processes over asymptotically large time intervals, but rather to be Gaussian over small intervals $(t,t+\\delta t)$ , with the latter supposed to hold for most values of $t$ .", "For such $t$ one can express all the higher order moments that appear in the expressions for $dX(t)$ and $dY(t)$ in terms of only the means, viz.", "$X(t)$ and $Y(t)$ , and the second-order moments, including variances $s_x(t)=E(n^2_{x_i}(t)),s_y(t)=E(n^2_{y_i}(t))$ and the covariance $u(t)=E(n_{x_i}(t)n_{y_i}(t))$ , where $n_{x_i}(t)=X(t)-x_i(t), n_{y_i}(t)=Y(t)-y_i(t)$ .", "Here, the QIA is reflected in the fashion in which the spatial and the stochastic averages are related.", "Use of GA in deriving the MF model rests on the notion that the fraction of time where GA fails will not introduce significant differences between the MF and the exact solutions.", "The MF counterpart of (REF ), incorporating QIA and GA, has been derived in [25].", "It constitutes the following system of five deterministic DDE: mx=mx(t)-mx(t)3/3-sx(t)mx(t)-my(t) +c(mx(t-)-mx(t)), my=mx(t)+b, 2sx=sx(t)(1-mx(t)2-sx(t)-c)-u(t) 12sy=u(t)+D, u=(u(t)/)(1-mx(t)2-sx(t)-c)-sy(t)/+sx(t) assuming that the MF solutions satisfy $m_x(t)\\approx X(t), m_y(t)\\approx Y(t)$ .", "Note that some more sophisticated MF approaches [27], [20] adopt the Gaussian decoupling approximation, yet do not require QIA, as their final form accounts for spatial averages of fluctuations of both local and global variables.", "The goal is to first explain the two generic scenarios where both the MFAs hold, outlining the parameter domains where the pertaining local and global dynamics typically occur.", "We also illustrate the case where both the MFAs fail, independently demonstrating that the GA and the QIA are violated.", "As indicated in the Introduction, the methods applied to verify the validity of the MFAs for the oscillatory state are adapted to the essential properties of the class II excitable systems.", "Note that in this section, which contains the numerical results on the exact system, one is primarily concerned with the fulfilment of GA.", "This is done deliberately, given that the breakdown of QIA can be deduced from the dynamics of the MF model, as demonstrated in section .", "Intuitively, one would expect that both the MFAs are satisfied if $c$ and $D$ are small.", "Though this is indeed so, the general conclusion on simultaneous validity of both the MFAs is less straightforward, and should refer to the qualitative properties of the system's dynamics, rather than alluding to certain parameter domains.", "As a preview of this result, it may be stated that the GA and the QIA hold if the local and the global dynamics are characterized by a single attractor of the same type, either a FP or a LC, provided that $D$ is not too large.", "Conversely, if the local and the collective variables yield qualitatively different dynamics or exhibit multistability, the validity of either or both the MFAs is lost.", "Nevertheless, note that the separate conditions for failure of each of the MFAs can be put in a more succinct form, which will be clarified below.", "Figure: (Color online) Impact of noise on validity of GA. (a) and (b) refer to typical scenarios where the GA holds, while (c) concerns one of the cases where it fails.", "For an arbitrary unit, here denoted with index 5, the main frames of (a), (b) and (c) show orbits corresponding to three different stochastic realizations (x 5,r ,y 5,r )(x_{5,r},y_{5,r}) (black solid, dashed and dotted lines), as well as the respective expectations (E(x 5,r ),E(y 5,r ))(E(x_{5,r}),E(y_{5,r})) for an ensemble of 10 realizations, having indicated the latter by the solid red (gray) lines.", "In the insets of (a), (b) and (c) are displayed the appropriate graphic normality tests, such that the red (gray) lines conform to theoretical data sets obeying the Gaussian distribution, whereas the circles reflect the data for x 5,r x_{5,r} actually collected over different realizations.", "The parameter sets are (c,D,τ)=(0.1,0.0002,0.2)(c,D,\\tau )=(0.1,0.0002,0.2) in (a),(c,D,τ)=(0.1,0.0002,2.7)(c,D,\\tau )=(0.1,0.0002,2.7) in (b) and (c,D,τ)=(0.1,0.003,1.5)(c,D,\\tau )=(0.1,0.003,1.5) in (c).The discussion above implies that there are two paradigmatic scenarios where both the GA and the QIA hold.", "By one, the local and the collective dynamics display stochastically stable FP, whereas in the other, the local and the collective dynamics exhibit the stochastically stable LC.", "These two cases are addressed in Fig.", "REF (a) and Fig.", "REF (b), respectively, whereby the intention is to first verify the validity of GA. Before proceeding in this direction, note that the value $c=0.1$ , fixed in both instances, is chosen from an intermediate range to stress that the MFAs' validity extends into such a domain.", "Nonetheless, the data in Fig.", "REF (a) are obtained for small $D_1=0.0002$ and small $\\tau _1=0.2$ , while the setup in Fig.", "REF (b) involves $D_2=D_1$ but the much larger $\\tau _2=2.7$ .", "As an illustration on the qualitative similarity between the individual realizations and the expectations, the main frames of Fig.", "REF (a) and Fig.", "REF (b) show three different stochastic realizations $(x_{5,r},y_{5,r})$ , encoded in black solid, dashed and dotted lines, as well as the expectation values $(E(x_{5,r}),E(y_{5,r}))$ over an ensemble of 10 realizations, having plotted them by the solid red (gray) lines.", "The data are representative for the dynamics of an arbitrary neuron, whereby the particular example refers to the unit $i=5$ .", "The index $r$ accounts for the realizations.", "In case of Fig.", "REF (a), for any $t$ , the expectation closely matches either of the realizations trivially.", "Nevertheless, for the scenario with the LC attractor, the analogous statement holds if $t$ is such that $(E(x_{i,r}(t)),E(y_{i,r}(t))\\approx (X(t),Y(t))$ lies on the slow branches of the given orbit.", "At variance with this, if $(E(x_{i,r}(t)),E(y_{i,r}(t))$ falls onto one of the transients, the expectation departs substantially from the realizations, cf.", "Fig.", "REF (b).", "The two latter points are consistent with the relaxation character of oscillations, and as such are accounted for by the definition of GA.", "Invoking the definition, it follows that the GA's validity is upheld if the number of instances where the expectations closely match the individual realizations strongly prevails over the number of instances where such a correspondence is lost.", "In other words, the GA holds if the expectations preserve the relaxation character of oscillations exhibited by the realizations.", "Though this requirement is qualitative in nature, one may still attempt to attribute it a certain quantitative measure.", "For instance, for the $(c,D_2,\\tau _2)$ parameter set from Fig.", "REF (b), it may be shown that the ratio of points lying on the transients vs those on the slow branches is small ($n_t/n_s\\approx 0.1$ ) over the sufficiently long time period along the trajectory of $(E(x_{i,r}(t)),E(y_{i,r}(t))$ .", "Figure REF (c) refers to the case where GA no longer holds.", "The illustrated local dynamics is obtained for comparably large $D_3=0.003$ , $c=0.1$ and intermediate $\\tau _3=1.5$ , such that the individual stochastic realizations fluctuate around the single LC.", "However, the noise-induced fluctuations are large enough to throw the different realizations out of step, resulting in the strong misalignment between the pertaining oscillation phases.", "Therefore, at variance with Fig.", "REF (b), the expectation substantially departs from each of the realizations at any $t$ .", "For $(c,D_3,\\tau _3)$ , one can no longer interpret $(E(x_{i,r}(t)),E(y_{i,r}(t))$ in terms of clearly discernible slow and fast motions, so that the ratio $n_t/n_s$ cannot be determined.", "Apart from characterizing it by the $n_t/n_s$ ratio, the GA validity has been tested directly for an arbitrary neuron at the given $(c,D,\\tau )$ .", "Having run many different realizations of the processes $x_i(t),y_i(t)$ for the same initial function, we have examined the properties of the distribution of different realizations $x_{i,r}(t_0+\\delta t),y_{i,r}(t_0+\\delta t)$ for small $\\delta t$ , taken to be of the order, in tens or hundreds of iteration steps.", "For the LC dynamics, $(x_{i,r}(t_0),y_{i,r}(t_0))$ has been set on the refractory branch.", "The insets of (a), (b) and (c) display graphic normality tests, where the red lines indicate the theoretical percent of data points that would lie below the given value if obeying the Gaussian distribution, while the blue circles refer to the cumulative distribution of $(x_{5,r},y_{5,r})$ for an ensemble of over 200 realizations.", "Apparently, the distributions corresponding to $(c,D,\\tau )$ in Fig.", "REF (a) and REF (b) are Gaussian, whereas the one for Fig.", "REF (c) is not.", "Results of the graphic tests are corroborated by the standard numerical Shapiro-Wilk method.", "Figure: (Color online) Estimating rate by which the validity of GA deteriorates with increasing DD.", "The plot shows a fraction of stochastic realizations N out /N r N_{out}/N_r in dependence of DD for (c,τ)=(0.1,2.7)(c,\\tau )=(0.1,2.7).", "N out N_{out} accounts for the instances where the representative point escapes the refractory branch of slow motion within the given number of iteration steps T max T_{max}.", "The results converge to the displayed curve as N r N_r is increased.", "The inset refers to the variation of the slope of the curve from the main frame with noise.Having seen that the criterium for validity of GA is primarily qualitative, it is still of interest to find some indication on how the fulfilment of GA deteriorates with variation of the system parameters.", "Naturally, the most relevant question is to assess the rate at which the validity reduces with increasing $D$ for fixed $c$ and $\\tau $ .", "The quantity appropriate to characterize this is determined as follows.", "For very small $D=0.0002$ , we select an arbitrary neuron and fix a point on the refractory branch of its LC orbit.", "Then, a large number of different stochastic realizations $N_r$ for the given parameter set $(D,c,\\tau )=(0.0002,0.1,2.7)$ is run.", "The goal is to find the maximal number of iteration steps $T_{max}$ , for which the representative point in all the realizations still lies on the refractory branch.", "Enhancing $D$ while $T_{max}$ is kept fixed, one naturally encounters realizations where the latter condition is no longer satisfied.", "In Fig.", "REF we demonstrate how the fraction of realizations $N_{out}/N_r$ in which the representative point has escaped the refractory branch in less than or exactly $T_{max}$ steps increases with $D$ .", "Along with allowing one to quantify the gradual loss of GA's validity, this dependence may also be interpreted as an indication on how $D$ gives rise to the number of moments $t$ where $(E(x_{i,r}(t)),E(y_{i,r}(t))$ belong to fast transients, rather than the two slow branches.", "In this context, it is interesting to explain why the curve's slope shows a significant change in behavior around $D_0\\approx 0.0014$ , cf.", "the inset in Fig.", "REF .", "Below $D_0$ , the fluctuations of phase between the different stochastic realizations systematically grow, but the physical picture by which the LC for the expectations $(E(x_{i,r}(t)),E(y_{i,r}(t))$ is described in terms of two pieces of slow motion connected by the two rapid jumps still applies.", "Nevertheless, about $D\\simeq D_0$ , such a picture has to be abandoned, because the LC generated by the expectations no longer matches the phase portrait of individual realizations.", "In particular, the $(E(x_{i,r}(t)),E(y_{i,r}(t))$ cycle lies inside the one for the single realizations, as it fails to reach the latter's spiking branch.", "Once the framework involving qualitative equivalence between the dynamics of realizations and the expectations has been broken, $N_{out}/N_r$ for $D>D_0$ loses its original meaning, but its steady increase reflects the tendency for growing irregularity in the unit's behavior.", "Figure: (Color online) Examining the validity of QIA.", "Consistent with the theorem of large numbers, confirming the validity of GA for the global variables corroborates that QIA holds for the local variables.", "The main frames (insets) of (a), (b) and (c) refer to the graphic normality tests for the collective variables X(t)X(t) (Y(t)Y(t)), whereby the respective parameter sets correspond to those in Fig.", "(a), (b) and (c).", "In (a) and (b) is demonstratedthat X r (t)X_r(t) and Y r (t)Y_r(t) for different stochastic realizations are Gaussian distributed, whereas (c) indicates a substantial departure from the normal distribution in case of Y r (t)Y_r(t).", "(d) illustrates the loss of qualitative analogy between the oscillations characterizing the individual realizations and the expectation for (c,D,τ)(c,D,\\tau ) from (c).We now turn to the analysis on the fulfilment of QIA.", "The intention here is just to briefly mention the two methods that may be used to verify the validity of QIA by examining the exact system, whereas the main point, lying in the ability to predict the failure of QIA solely by the dynamical features of the MF model is left for the next section.", "Considering the exact system, one may either take (i) an indirect approach, derived from a corollary of the QIA formulation, or (ii) the direct approach, based on the notion that approximate synchronization between the units may render them virtually independent.", "Since the more comprehensive discussion on the relation between different types of synchronization and the QIA is provided in subsection REF , we focus on the indirect approach (i).", "One first invokes the central limit theorem, by which for large, but finite $N$ holds that if the local variables are normally distributed for most $t$ , so too are the collective variables.", "Hence, the validity of GA for $X(t)$ and $Y(t)$ should imply that the local variables are independent.", "The normality tests on $X(t)$ and $Y(t)$ are carried out analogously to those for $x_i(t)$ and $y_i(t)$ .", "The main frames (insets) of Fig.", "REF (a), (b) and (c) refer to graphic normality tests on the variable $X(t)$ ($Y(t)$ ) for the parameter sets exactly matching those in Fig.", "REF (a), (b) and (c).", "Figures REF (a) and (b) indicate the validity of GA for $X(t)$ and $Y(t)$ distributions, thereby suggesting that the QIA also applies.", "The positive result in Fig.", "REF (b), which is associated with the oscillatory state, again draws on the main feature of the class II excitable systems.", "An interesting point regarding Fig.", "REF (c) is that the distribution of $X_r(t_0+\\delta t)$ over stochastic realizations conforms to, and the one for $Y_r(t_0+\\delta t)$ sharply deviates from the Gaussian form.", "Such a violation of QIA is mostly found for intermediate $D$ and $\\tau $ .", "Figure REF (d) further illustrates the loss of qualitative analogy between the oscillations for the individual realizations and the expectation, with the latter failing to preserve the relaxation character." ], [ "Predicting the failure of QIA by the dynamics of the MF model", "The aim in this section is to demonstrate how the failure of the QIA is indicated by the dynamics of the MF model.", "To this end, we first present the results of the bifurcation analysis for the approximate system.", "Note that the analysis has not been carried out on the full system (REF ), but rather on its counterpart obtained by retaining the equations for the first moments under the \"adiabatic\" approximation that the evolution of second moments is slow and can be cast as stationary.", "The main reason for this lies in the fact that the reduced system, unlike the original one, is analytically tractable.", "The approximate model is found to display a sequence of supercritical and subcritical Hopf bifurcations, whereby the former (latter) result in creation of a stable (unstable) limit cycle.", "Recall that both types of Hopf bifurcation can further be cast as direct or inverse [28], which refers to whether the fixed point unfolds on the unstable or the stable side, respectively.", "The final expression for the critical time delay in dependence of $c$ and $D$ reads [25], [26]: j=[(-/c)+2j]/, if -2+1/c/0,or j=[-(-/c)+2(j+1)]/, if -2+1/c/<0, where the $+/-$ sign reflects the direct/inverse character of bifurcation, $j=0,1,2,\\dots $ and $\\omega _{\\pm }=\\omega _{\\pm }(c,D), \\kappa =\\kappa (c,D)$ .", "It can be shown by a rather lengthy calculation that the direct (inverse) bifurcations are always supercritical (subcritical) [25].", "The first few branches $j=0,1,\\dots ,6$ of the Hopf bifurcation curves $\\tau _{\\pm }^j(D)$ for the intermediate coupling strength $c=0.1$ are presented in Fig.", "REF .", "In particular, Fig.", "REF (a) is focused on the Hopf curves alone, whereas Fig.", "REF (b) presents a zoom in of Fig.", "REF (a), but also contains additional information, as explained below.", "Note that the presentation scheme in both figures is such that the curves coinciding with the direct (supercritical) Hopf bifurcations are indicated by the black lines, while those corresponding to inverse (subcritical) Hopf bifurcations are plotted by the gray lines.", "Figure: (Color online) (a) First few branches j=0,1,⋯,6j=0,1,\\dots ,6 of the Hopf bifurcation curves τ ± j (D)\\tau _{\\pm }^j(D) for the MF model.", "(b) A close-up view of (a), but including the additional indication on the parameter values where the global fold-cycle bifurcations occur.", "Stability of equilibrium is influenced by a sequence of direct (supercritical) and inverse (subcritical) Hopf bifurcations, shown by the black and gray lines, respectively.", "In (b), the critical values D fc D_{fc} and τ fc \\tau _{fc} for the DD- and τ\\tau -controlled fold-cycle bifurcations are indicated by the solid and the open circle lying at (D,τ)=(D fc ,0)(D,\\tau )=(D_{fc},0) and (D,τ)=(0,τ fc )(D,\\tau )=(0,\\tau _{fc}).", "The dashed line approximately highlights the parameter values above which the dynamics of the MF model always involves a large cycle born via the global bifurcation.", "The bistable regimes emerging due to global bifurcations involve coexistence of FP and LC (instances indicated by triangles) or two LCs (instances indicated by the squares).", "For D>D H D>D_H, the existence of bistable regimes and their form depend on the complex interplay between the local and the global bifurcations.", "Coupling strength is fixed at c=0.1c=0.1.Apart from the local bifurcations which affect the stability of equilibrium, the MF dynamics are influenced in a highly nontrivial fashion by the two global fold-cycle (tangent) bifurcations, one controlled by $D$ and the other by $\\tau $ .", "Note that the direct fold-cycle bifurcation gives rise to a stable large cycle and a saddle cycle.", "The point $(D,\\tau )=(D_{fc},0)$ where the noise alone is sufficient to induce the global bifurcation is indicated by the solid circle in Fig.", "REF .", "In an analogous fashion, the point $(D,\\tau )=(0,\\tau _{fc})$ where solely the delay gives rise to the global bifurcation is denoted by the open circle.", "The dashed line connecting the open and the solid circle approximately highlights the parameter values above which the dynamics of the MF model always involves a large cycle born via the global bifurcation.", "One should caution that in the parameter domains allowing for the local bifurcations, the existence of a large cycle per se does not warrant multistability in the MF dynamics.", "Later on, it is shown that multistability in such domains depends on a complex interplay between the attractors and saddles resulting from the local and global bifurcations.", "Due to global bifurcations, the MF model exhibits two types of bistable regimes, one involving the coexistence between the FP and the LC, and the other characterized by the coexistence of two LCs.", "In the former case, the LC corresponds to a large cycle born in the fold-cycle bifurcation.", "The latter scenario may be realized either by the coaction of the local (direct supercritical) Hopf bifurcation and the global fold-cycle bifurcation, which mainly occurs for $\\tau <\\tau _{fc}$ , or the two cycles may both derive from the fold-cycle bifurcations ($\\tau >\\tau _{fc}$ ).", "In most cases, bistability emerges due to the action of noise, i.e.", "is facilitated by the $D$ -controlled global bifurcation.", "Such regimes are referred to as the noise-induced bistability to distinguish them from the scenario involving the coexistence between the FP and the large cycle born in the $\\tau $ -controlled global bifurcation, which occurs for $\\tau >\\tau _{fc},D<D_{fc}$ .", "Our main point is that the noise-induced bistability in the dynamics of the MF model provides the necessary condition for the failure of QIA, and therefore the failure of MF approximation as a whole.", "In other words, the qualitative features of the dynamics displayed by the MF model can be used to predict in a self-consistent fashion the $(\\tau ,D)$ parameter domains where the QIA is bound to fail.", "Before explaining this point in more detail, we make a remark on why the noise-induced bistability is distinguished from the one owing solely to the $\\tau $ -controlled global bifurcation.", "Though the MF model makes no qualitative distinction between $D$ and $\\tau $ , which are both considered as equally valid bifurcation parameters, the exact system is naturally sensitive to the deterministic/stochastic character of the effects they generate.", "In this context, for $\\tau >\\tau _{fc}$ and sufficiently small noise, the oscillations displayed by the exact system retain their primarily deterministic character and as such satisfy the MF approximation trivially.", "Nonetheless, using the method described in section , we have verified that the stochastic perturbation becomes large enough to compromise the validity of QIA for $D$ fairly close to $D_{fc}$ , the noise intensity marking the onset of the $D$ -controlled global bifurcation in the MF model.", "Figure: (Color online) Bistability exhibited by the approximate model allows one to gain insight into the parameter domains where QIA breaks down.", "In the example provided, the MF dynamics (m x (t),m y (t))(m_x(t),m_y(t)) shows coexistence of the FP, denoted by the orange (light gray) dot, and the LC (black dashed line) born via the global fold-cycle bifurcation.", "Influenced by noise, the typical orbit (X(t),Y(t))(X(t),Y(t)) of the exact system, displayed by the blue (dark gray) solid line, is found to fluctuate between the two attractors of the MF model.", "Failure of GA for the global variables may be considered an indirect evidence of the failure of QIA on the level of local variables.", "The data are obtained for the parameter set (c,D,τ)=(0.1,0.0029,0.3)(c,D,\\tau )=(0.1,0.0029,0.3).Next we show how the noise-induced bistability of the MF model is reflected in the dynamics of the exact system.", "First note that the illustrative examples of parameter values admitting bistability between the FP and the LC are indicated in Fig.", "REF by the triangles, whereas the squares are reserved for the typical cases facilitating coexistence between the two LCs.", "In particular, we have singled out three instances related to bistability between the FP and the LC.", "The point denoted by the solid triangle ($\\blacktriangle $ ) refers to the case bearing no influence from the local Hopf bifurcations, given that $D<D_H$ , where $D_H\\approx 0.0025$ marks the onset of the Hopf bifurcations.", "Nevertheless, in the two examples indicated by the open triangles $()$ , such form of bistability occurs because the equilibrium is stabilized via the inverse subcritical Hopf bifurcation, whereby the unstable cycle born in the Hopf bifurcation acts like a threshold switching between the two stable solutions.", "Implementing the method introduced in section , it has been verified that the QIA is violated in all the three described instances.", "For the case $(c,D,\\tau )=(0.1,0.0029,0.3)$ , we have illustrated the phase portraits corresponding to the two attractors of the MF model and the appropriate orbits for the collective variables of the exact system, see Fig.", "REF .", "The rationale for the failure of QIA rests on the point that the mixed mode of the exact system may be interpreted as stochastic switching between the two attractors of the deterministic MF model.", "Naturally, the ensuing orbits are not normally distributed around the respective averages.", "The analogous explanation also applies for the scenario where the MF model displays coexistence between the two LCs.", "In Fig.", "REF , we have indicated three parameter domains supporting such form of bistability.", "In the cases denoted by the open squares $(\\square )$ , the large cycle from the global bifurcation coexists with the incipient cycle, emerging from the direct supercritical Hopf bifurcation.", "Nonetheless, the solid square $(\\blacksquare )$ points to an instance where the two large cycles coexist, one of them created in the $D$ -controlled, and the other in the $\\tau $ -controlled global bifurcation.", "It has been verified that the QIA breaks down in all of the stated instances.", "One should note that crossing the Hopf bifurcation curves alone does not immediately imply the failure of the QIA.", "Nevertheless, due to interplay with the $D$ -controlled global bifurcation, crossing the curves may become associated with the violation of QIA in two cases, one where the supercritical regime involves bistability of the FP and the LC (inverse subcritical Hopf bifurcation), and the other, which includes coexistence between two LCs (direct supercritical Hopf bifurcation).", "The occurrence of such cases is mostly confined to $\\tau \\lesssim \\tau _{fc}$ , because above $\\tau _{fc}$ the MF dynamics is primarily influenced by the two global bifurcations." ], [ "Fulfilment of MFAs and the statistics of the first return times", "This subsection provides a discussion on some of the corollaries related to the fulfilment of the MFAs.", "Before elaborating on the relation between the synchronization properties and the fulfilment of QIA, we make two auxiliary notes qualifying more closely the terms \"frequency\" and \"phase\" used later on.", "The immediate aim is to show that the effective frequency and phase description of system dynamics may be appropriate if MFAs hold.", "Regarding frequency, we present the results on the distribution of ISIs for $X(t)$ .", "Note that there are two types of collective modes, one where the ISIs are dominated by $T_0(D)$ , which occurs for small and intermediate $D$ under very small $\\tau $ , and the other corresponding to the delay-led dynamics, which is typically seen for small and intermediate $D$ under large $\\tau $ .", "Either way, we have verified that ISIs are normally distributed for an arbitrary stochastic realization under long simulation times.", "In Fig.", "REF (a), the normality test is provided for the more interesting case, showing persistence of Gaussian distribution for the noise-led dynamics under fairly large $D=0.0015$ at $\\tau =0$ .", "Since the analogous conclusion is readily reached for the delay-driven collective mode, one may state that the description of collective motion in terms of the average period (frequency) appears justified if MFAs apply.", "Figure: (Color online) Characterizing the distribution of the return times and the return points for the macroscopic dynamics of the exact system.", "In (a) and (b) are displayed the graphic normality tests respectively indicating that the ISIs and the return points for X(t)X(t) are Gaussian distributed.", "The data refer to the case of noise-led dynamics at (c,D,τ)=(0.1,0.0015,0)(c,D,\\tau )=(0.1,0.0015,0), but one may arrive at qualitatively similar results for the delay-driven dynamics.A question that naturally arises is whether the fulfilment of MFAs implies that the distributions of the return points $P(X_r)$ and $P(Y_r)$ sampled at intervals equal to the average ISI of the macroscopic dynamics are also Gaussian.", "$P(X_r)$ and $P(Y_r)$ are calculated in two steps: one first lets the simulation run for the sufficiently long time to determine the average ISI for $X(t)$ , and then carries on by collecting data on the return points for another very long time period.", "The first point $(X_0,Y_0)$ is chosen to lie on the refractory branch of the LC.", "In Fig.", "REF (b) is displayed the graphic normality test for $P(X_r)$ along an arbitrary stochastic trajectory under the same parameter set as in Fig.", "REF (a).", "The demonstrated normality of distribution indicates that, in statistical sense, the return points remain fairly close to the \"average\" LC.", "From the broader perspective, one may think of this result in the context of building an effective phase description for the collective motion [29], [30], [31].", "Another point of interest is to verify whether the analogous conclusions hold for the local, rather than the global variables.", "Under the same parameter set as in Fig.", "REF (a) and Fig.", "REF (b), one can demonstrate for an arbitrary unit that the ISIs over a very long time series indeed conform to Gaussian distribution if the data from less than $10\\%$ of realizations are discarded.", "However, the return points $P(x_r)$ , sampled at $t_n=nT_s$ , where $T_s$ denotes the average ISI for the given unit, turn out not to be normally distributed.", "This is so because the Gaussian distribution for the local ISIs is comparably broader than the one for the global variables.", "Still, starting off from any point on the refractory or the spiking branch of slow motion, the successive return points, recorded at $T_s$ long intervals, always fall on the \"right\" branch, determined by the location of the initial point.", "Therefore, the above results suggest an interesting point that if the MFAs are satisfied, the use of terms frequency and phase is more appropriate to describe the dynamics of the global, than the local variables." ], [ "Fulfilment of QIA and synchronization", "Having gained insight into the competition between the noise-led and the delay-driven dynamics, as well as the statistical features providing the context for the effective use of terms frequency and phase, we proceed with the analysis of the relation between the synchronization of the individual units and the fulfillment of QIA.", "To begin with, one notes that for being stochastic and excitable in nature, the units cannot exhibit complete synchronization.", "However, the discussion above suggests that it is reasonable to speak of approximate frequency (FS) and phase synchronization (PS) in conditional terms, viz.", "if MFAs are satisfied.", "Presence or absence of these forms of synchronization may give rise to three types of collective states: (i) coherent states where single units display both the approximate FS and PS, (ii) states that exhibit FS, but lack PS and (iii) collective states where approximate FS is not established.", "One may infer the relation between synchronization and QIA by examining the linear interaction terms of the form $c(x_i(t-\\tau )-x_j(t))$ .", "If there is approximate lag-synchronization, the latter become very small, which leaves the neurons virtually independent.", "Therefore, by identifying conditions under which the approximate lag-synchronization is achieved, one effectively looks for the parameter domains where QIA applies.", "We have established that there exist only two scenarios for the approximate lag-synchronization, both of which amount to cases of approximate FS and PS.", "The interaction terms may substantially reduce either (i) for noise-led dynamics at $\\tau \\simeq 0$ , or (ii) for delay-driven dynamics at very large $\\tau \\sim T_0(D)$ .", "A way to characterize the approximate FS for the given parameter set is to calculate the ratio $r=\\Delta T/\\langle \\overline{T_i}\\rangle $ , where $\\Delta T=max\|\\overline{T_i}-\\overline{T_j}\|$ is the maximal difference between the time-averaged ISIs $\\overline{T_i}$ of individual units, whereas $\\langle \\overline{T_i}\\rangle $ denotes the population average $\\langle \\overline{T_i}\\rangle =N^{-1}\\sum \\limits _{i=1}^{N}\\overline{T_i}$ .", "The smaller $r$ becomes, the better FS between the units is achieved.", "The results for $r(c,D)$ plotted in Fig.", "REF (a) refer to the (ii) case at $\\tau =2.7$ .", "We have verified that setting $\\tau =0$ , which corresponds to case (i), yields qualitatively similar results.", "As the main point, note a very large domain where $r$ is small, which indicates the approximate FS.", "Expectedly, for small $c$ and large $D$ , $r$ is seen to rise sharply, implying that FS is lost.", "Figure: (Color online) Focus on identifying the parameter domains that admit frequency and phase synchronization between the single units, which effectively provides an indication of where the QIA holds.", "(a) shows r(c,D)r(c,D) for the delay-driven dynamics at τ=2.7\\tau =2.7.", "In (b) and (c) are illustrated the corresponding I 3 ¯(c,D)\\overline{I_3}(c,D) and I 4 ¯(c,D)\\overline{I_4}(c,D) distributions for τ=2.7\\tau =2.7, respectively.", "Comparing (a), (b) and (c), note the overlap between the regions displaying near-zero values for the rr ratio, I 3 ¯\\overline{I_3} and I 4 ¯\\overline{I_4}.", "The analogous result can be obtained for the noise-driven dynamics at τ=0\\tau =0.The drawback of the method above is that one cannot distinguish whether approximate FS is or is not accompanied by PS.", "To do so, we consider the time-averaged third and fourth order moments of the local potentials $P(x_i(t))$ for the given parameter set, taking the average over a very long stochastic realization.", "Note that if the ergodic hypothesis applied, such an average would equal the one over an ensemble of realizations.", "Nevertheless, whether this holds or not is of marginal significance because the results below are not intended to be rigorous, but should rather provide an illustration on the link between PS and the fulfilment of QIA.", "Therefore, the discussion on the asymptotic distributions here is independent and should by no means be confound with the results from section , which only concern averaging over an ensemble of stochastic realizations.", "As for the moments, the third-order average moment is defined by $\\overline{I_3}=(1/T)\\sum \\limits _{t=1}^{T}I_3(t)$ , where $I_3(t)=\\sum \\limits _{x_i}(x_i-X(t))^3P(x_i(t))$ .", "The analogous relation holds for $\\overline{I_4}$ .", "$P(x_i(t))$ is obtained by dividing the range of possible $x_i$ values into 110 bins $[x,x+\\delta x]$ , whereby one records the fraction of units whose potential falls within the given bin.", "If there is an approximate FS and PS, one expects $x_i$ for most $t$ to be Gaussian distributed around the mean $X(t)$ .", "Then, both $\\overline{I_3}$ and $\\overline{I_4}$ should lie close to zero.", "If there is approximate FS, but PS is lacking, $\\overline{I_3}\\approx 0$ should hold, whereas $\\overline{I_4}$ should substantially depart from zero.", "Finally, if there is no approximate FS, both $\\overline{I_3}$ and $\\overline{I_4}$ are supposed to lie away from zero.", "Results on $\\overline{I_3}$ and $\\overline{I_4}$ at $\\tau =2.7$ for a wide range of $(c,D)$ values, cf.", "Fig.", "REF (b) and REF (c), suggest that domains with approximate FS closely match those with PS.", "Note the overlap between the areas with the smallest $r$ , $\\overline{I_3}\\approx 0$ and $\\overline{I_4}\\approx 0$ in Fig.", "REF (a), REF (b) and REF (c), where QIA should hold." ], [ "Conclusion", "The reduction of computational demand and the possibility of describing the stochastic stability and the stochastic bifurcations can be cast as general reasons for introducing the MF approximate model for an arbitrary set of SDDEs.", "Given the apparent relevance of the MF method, an issue of considerable importance is to be able to determine the domains where such an approach may provide accurate qualitative predictions.", "The approximations behind the MF model are often considered in a simplistic fashion, as if they were completely independent on the class of systems which the model under study belongs to.", "Such a view results by invoking the (stereo)typical requirements for small noise intensity and weak couplings as the main conditions for the validity of the MF model.", "In the present paper, the issue of the MF approximations and their validity is highlighted by taking the example of a system of delay-coupled noisy type II excitable units, represented by the generic Fitzhugh-Nagumo model.", "What we actually show is that, though they contain certain commonly stated elements, the MFAs relevant for the given system also include ingredients that should be precisely adapted to its essential dynamical properties.", "In particular, the inherent features of class II excitable systems, such as relaxation character of oscillations, have been explicitly incorporated into the definitions of the two MFAs we introduced.", "This point is particularly apparent in the definition of the GA, and is further reflected in the fashion in which the validity of both the GA and the QIA has been verified.", "It is found that the requirements for the joint validity of GA and QIA may be expressed in terms of a single qualitative statement, by which the two apply if the local and global dynamics exhibit a unique attractor of the same type, either a FP or a LC, provided that $D$ is not overly large.", "Of the two generic scenarios, the one involving the stochastically stable FP is fairly trivial, whereas the one associated with the LC is more intricate and makes apparent the need for introducing the refined MFAs considered in the paper.", "Focusing on each of the approximations independently, it is shown that validity of GA cannot be explicitly tied to certain parameter domains, but rather comes down to a qualitative requirement for not too large a noise intensity.", "This is the main corollary of the actual statement on the validity of GA, by which GA is satisfied if the qualitative similarity between the individual realizations and the appropriate expectations is maintained for the given parameter set.", "For the oscillatory state, the notion of qualitative similarity effectively refers to the point that the expectations preserve the relaxation character of oscillations.", "In this context, we have attempted to provide some quantitative measure on validity of GA by determining the variation of the $N_{out}/N_r$ ratio with $D$ , see section .", "Nonetheless, our main conclusion regarding the validity of MF approximation is associated with the fulfillment of QIA.", "What we have demonstrated is that the failure of QIA can explicitly be related to the noise-induced bistability of the MF model.", "Such bistable regimes, involving either the coexistence between the FP and the LC or the two LCs, are influenced by the global fold-cycle (tangent) bifurcation controlled via the noise intensity parameter.", "In this fashion, the $(\\tau , D)$ parameter domains where the MF approximation is bound to fail are identified with the domains admitting noise-induced bistability for the MF model's dynamics.", "In other words, the noise-induced bistability of the MF model provides the necessary condition for the failure of the QIA, and thus the MF approximation.", "Note that such parameter domains do not exhaust all the cases where the MF approximation fails, because the breakdown may also be caused by the violation of GA. As for the relationship between the Hopf bifurcation curves determined for the MF model and the failure of MF approximation, we stress that crossing the curves itself does not imply the failure.", "It has already been pointed out that the latter would mean that the MF model could never account for the collective oscillatory states, which is not true.", "Though the asymptotic distribution for the collective variables in the exact system indeed loses the Gaussian property if the curve corresponds to the stochastic Hopf bifurcation, this fact alone has no bearing on the MF approximations we introduced.", "However, in the interplay with the $D$ -controlled global bifurcation, crossing the Hopf bifurcation curves may involve the onset of two different bistable regimes in the MF model, and as such, may contribute to the violation of the QIA, and thereby the MF approximation as a whole.", "It is reasonable to expect that the scope of the conclusion on the relationship between the noise-induced bistability of the MF model and the failure of MF approximation may likely be extended to a broader range of systems, since it draws only on the qualitative properties of the system dynamics.", "For the future research, it would be interesting to examine the refined MFAs and their validity by carrying out the analysis similar to ours in case of the MF models derived for systems exhibiting complex multi-scale oscillations, such as bursting, when subjected to noise and coupling delays.", "This work was supported in part by the Ministry of Education and Science of the Republic of Serbia, under project No.", "171017 and No.", "171015." ] ]	1403.0458
[ [ "An Observed Correlation Between Thermal and Non-Thermal Emission in\n Gamma-Ray Bursts" ], [ "Abstract Recent observations by the $Fermi$ Gamma-ray Space Telescope have confirmed the existence of thermal and non-thermal components in the prompt photon spectra of some Gamma-ray bursts (GRBs).", "Through an analysis of six bright Fermi GRBs, we have discovered a correlation between the observed photospheric and non-thermal $\\gamma$-ray emission components of several GRBs using a physical model that has previously been shown to be a good fit to the Fermi data.", "From the spectral parameters of these fits we find that the characteristic energies, $E_{\\rm p}$ and $kT$, of these two components are correlated via the relation $E_{\\rm p} \\propto T^{\\alpha}$ which varies from GRB to GRB.", "We present an interpretation in which the value of index $\\alpha$ indicates whether the jet is dominated by kinetic or magnetic energy.", "To date, this jet composition parameter has been assumed in the modeling of GRB outflows rather than derived from the data." ], [ "Introduction", "Gamma-ray Bursts (GRBs) are believed to arise from the deaths of massive stars or the coalescence of two compact stellar objects such as neutron stars or black holes.", "The resulting explosion gives rise to an expanding fireball with a jet pointed at the observer but hidden from the observer until the density of radiation and particles in this highly relativistic outflow is low enough for radiation to escape, a region called the photosphere [28].", "While the emission from this fireball is expected to be thermal [15], [31], observations over the past three decades suggest the prompt emission to be highly non-thermal [25], [11], [24], [19], [14], with only a few exceptions [36], [12].", "The conversion of the fireball energy into non-thermal $\\gamma $ -ray radiation involves the acceleration of electrons in the outflow and their subsequent cooling via an emission process such as synchrotron radiation [37], [39].", "Insight into these energy radiation emission processes in GRBs is obtained by comparing the observed $\\gamma $ -ray photon spectra directly to different radiation models.", "The ${\\it Fermi}$ Gamma-ray Space Telescope offers a broad energy range for these comparisons.", "Recent observations [16], [48], [3], [17], [18], [33], [6] show that at least two mechanisms can be present: a non-thermal component that is consistent with synchrotron emission from accelerated electrons in the jet and a typically smaller blackbody contribution from the photosphere.", "This photospheric emission is released when the fireball becomes optically thin so that an observer may see a mixture of thermal and non-thermal emission with different temporal characteristics that, when viewed together, can probe the development and structure of the fireball jet.", "This simple photospheric model has been used to quantitatively interpret several observed correlations such as the Amati correlation [41], [21], [10] We are thus motivated to investigate correlations among spectral parameters derived by fitting the non-thermal component with a synchrotron photon model and the thermal component with a blackbody, an approach developed in previous investigations [5], [6].", "The synchrotron model consists of an accelerated electron distribution, containing a relativistic Maxwellian and a high-energy power law tail that is convolved with the standard synchrotron kernel [6], [5], [34].", "We find that the characteristic energies ($E_{\\rm p}$ for synchrotron and $kT$ for the blackbody) of the synchrotron and blackbody components are highly correlated across all the GRBs in our sample.", "We show that this correlation can be used to address the key question of how the energy of the outflow is distributed, i.e., whether the energy is in a magnetic field or is imparted as kinetic energy to baryons in the jet, and how this energy distribution evolves with time." ], [ "Observations", "The ${\\it Fermi}$ Gamma-ray Burst Monitor (GBM) [27] has detected more than 1200 GRBs since the start of operations on 2008, July 14.", "A smaller number have been seen by the ${\\it Fermi}$ Large Area Telescope (LAT) [2] at energies greater than 100 MeV, but these are particularly interesting because they are among the brightest GRBs and offer the greatest opportunity for spectral analysis across a broad energy range.", "GRBs can last from a few milliseconds to hundreds of seconds or longer and have a variety of temporal profiles, from single spikes to multi-episodic overlapping pulses.", "Single-pulse GRBs exhibit the simplest spectral evolution, providing the “cleanest” signal for fitting physical models to the data [6], [5], [35].", "In this work, we analyze six bright, single-pulse GRBs detected by ${\\it Fermi}$ (see Table REF and Figure REF ) and find correlations between the $E_{\\rm p}$ and $kT$ values within each of these GRBs.", "The GRBs in our sample are GRB 081224A [45], GRB 090719A [42], GRB 100707A [46], GRB 110721A [40], GRB 110920A, GRB 130427A [44].", "The time histories of these GRBs are shown in Figure REF , with vertical dotted lines indicating the time binning used for the analysis of the spectral evolution of each spectral component.", "In a previous analysis [6], the viability of fitting physical models to the ${\\it Fermi}$ GRB data was demonstrated for several GRBs and the spectral evolution of these models over the burst durations was investigated.", "The synchrotron model of [6], was constructed by convolving a shock-accelerated electron distribution of the form $n_{\\rm e}(\\gamma )\\; =\\; n_{0} \\biggl [\\;\\Bigl ( \\hbox{${{\\displaystyle \\gamma \\vphantom{(} }\\over {\\displaystyle \\gamma _{\\hbox{th}} \\vphantom{(} }}$} \\Bigr )^2\\,e^{-\\gamma /\\gamma _{\\hbox{th}}} + \\epsilon \\,\\Bigl ( \\hbox{${{\\displaystyle \\gamma \\vphantom{(} }\\over {\\displaystyle \\gamma _{\\hbox{th}} \\vphantom{(} }}$} \\Bigr )^{-\\delta }\\,\\Theta \\Bigl ( \\hbox{${{\\displaystyle \\gamma \\vphantom{(} }\\over {\\displaystyle \\gamma _{\\hbox{min}} \\vphantom{(} }}$} \\Bigr )\\, \\biggr ]\\,$ with the standard synchrotron kernel [34].", "Here, $n_0$ normalizes the distribution to total number or energy, $\\gamma $ is the electron Lorentz factor in the fluid frame, $\\gamma _{\\hbox{th}}$ is the thermal electron Lorentz factor, $\\gamma _{\\hbox{min}}$ is the minimum electron Lorentz factor of the power-law tail, $\\epsilon $ is the normalization of the power-law, and $\\delta $ is the electron spectral index.", "The function $\\Theta (x)$ is a step function where $\\Theta (x)=0$ for $x<1$ and $\\Theta (x)=1$ for $x>1$ .", "After convolution with the synchrotron kernel, the final fit parameters are the overall normalization of the spectrum, the $\\nu F_{\\nu }$ peak of the spectrum ($E_{\\rm p}$ ), and the electron spectral index, $\\delta $ .", "These fits were found to be as good as those made with the empirical Band function [4] that is the common choice for GRB spectroscopy.", "However, the Band function, being empirical, makes it difficult to deduce a more physical understanding.", "The fits with synchrotron model provide a direct association of the observed spectrum with a physical emission mechanism and therefore the fit parameters can be used to study properties of the GRB jet without ambiguity.", "All of these GRBs were shown to be consistent with a physical model containing both a synchrotron and a blackbody component.", "For five of those GRBs we investigate herein correlations between the previously derived $E_{\\rm p}$ and $kT$ values, and we add to our sample the first pulse of the ultra-bright burst, GRB 130427A, for which a similar analysis has been performed [33].", "GRB 130427A is the brightest GRB detected by ${\\it Fermi}$ to date.", "Although its temporal structure is complex [1], it begins with a bright single pulse that is ideal for our physical modeling, which was used to show that internal shocks cannot explain the observed emission [33].", "GRB 081224A, GRB 110721A, and GRB 130427A were analyzed with GBM and LAT data; the rest of the sample were analyzed with GBM data alone.", "While this sample is limited by the number of bright, single-pulsed GRBs in the ${\\it Fermi}$ data set, this requirement allows reliable interpretation of the fits without confusion from overlapping pulses with different underlying spectra, which is essential to measuring the evolution of the thermal and non-thermal components throughout the duration of the GRB." ], [ "A Correlation Between Spectral Components", "Figure REF shows an example of the spectral evolution of the two separate components.", "A strong correlation is found between $E_{\\rm p}$ and $kT$ , as illustrated in Figures REF & REF .", "A power law of the form $E_{\\rm p}\\propto T^{\\alpha }$ was fit to the $E_{\\rm p}$ , $kT$ pairs of the individual GRBs yielding values of $\\alpha $ ranging from $\\sim $ 1 to 2 (see Table REF ).", "The general temporal trend of both $E_{\\rm p}$ and $kT$ is an evolution from higher to lower energies.", "As can be seen from Figure REF , the evolution of the flux of each component is not necessarily tied to the change in the characteristic energies.", "This is very evident during the rise phase of a pulse during which the flux rises while $E_{\\rm p}$ and $kT$ fall with time.", "However, during the decay phase of the pulse, the flux decreases along with the characteristic energies.", "Table REF lists the ratio of the blackbody flux to the total flux for each burst." ], [ "Interpretation", "To interpret these observations, we assume an emission process in which the thermal and non-thermal emission occur in close proximity to one another with the non-thermal synchrotron emission arising in an optically thin region above the photosphere of the jet.", "The range of the indices observed in the correlation suggests that the relation between the thermal and non-thermal emission varies from burst to burst.", "One way to achieve this is to assume that the composition of GRB outflows vary in their ratio of magnetic content from being magnetically to baryonically dominated.", "In this scenario, the jet dynamics are parameterized by the dependence of the bulk Lorentz factor on the radius as $\\Gamma \\propto R^\\mu $ , from its initial launching radius of $r_0$ until the jet reaches its coasting Lorentz factor $\\eta =L/\\dot{M}c^2$ at the so-called saturation radius $r_{\\rm s}$ , where L is the luminosity and $\\dot{M}$ is the mass outflow rate.", "This will be approximately the jet's Lorentz factor until it is decelerated upon collision with the surrounding medium.", "For magnetically-dominated jets $\\mu \\approx 1/3$ [8], [9], [20], and in the baryonic case $\\mu \\approx 1$ [29].", "Intermediate values correspond to a mix of these components [43], and can be further modified by factors such as the topology of the magnetic field.", "Under these assumptions, there are two regions of interest for which we can define the radial evolution of the bulk Lorentz factor: ${\\Gamma (r)}= \\left\\lbrace \\begin{array}{lll}(r/r_0)^{\\mu } & {\\rm if} & r<r_{\\rm sat}\\\\{\\eta } & {\\rm if} & r_{\\rm sat}<r\\end{array}\\right.$ Here, $r_{\\rm sat}= r_0 \\eta ^{\\frac{1}{\\mu }}$ and is clearly larger when the jet is magnetically dominated.", "The emission of the blackbody is assumed to originate at the photospheric radius ($r_{\\rm ph}$ ), where the optical depth of the jet drops to unity.", "Following [29], the photospheric radius is $\\frac{r_{\\rm ph}}{r_0}=\\left(\\frac{ L\\sigma _{\\rm T}}{8\\pi m_{\\rm p}c^3r_0}\\right)\\frac{1}{\\eta \\Gamma _{\\rm ph}^2}$ where $\\Gamma _{\\rm ph}$ is the Lorentz factor of the outflow at $r_{\\rm ph}$ .", "The value of $\\Gamma _{\\rm ph}$ depends on the magnetic content of the outflow; therefore, $r_{\\rm ph}$ can take on two values, $\\frac{r_{\\rm ph}}{r_0}=\\eta _{\\rm T}^{1/\\mu }\\left\\lbrace \\begin{array}{ll}(\\eta _{\\rm T}/\\eta )^{1/(1+2\\mu )} & {\\rm if~ } \\eta >\\eta _{\\rm T} \\\\(\\eta _{\\rm T}/\\eta )^3 & {\\rm if~ } \\eta <\\eta _{\\rm T}\\end{array}\\right.", ".$ The introduction of the critical Lorentz factor, $\\eta _{\\rm T}=\\left(\\frac{ L\\sigma _{\\rm T}}{8\\pi m_{\\rm p} c^3r_0}\\right)^{{\\mu }/{(1+3\\mu )}}$ provides an important discriminator for the location of the $r_{\\rm ph}$ relative to $r_{\\rm s}$ .", "Outflows with $\\eta =\\eta _{\\rm T}$ have their photospheres at the saturation radius.", "Typical observed Lorentz factors of GRBs derived via different methods indicate values of a few hundred [22], [32].", "In a magnetically dominated ($\\mu =1/3$ ) case, we have $\\eta _{\\rm T}\\simeq 150~L_{53}^{1/6}r_{0,7}^{-1/6}$ .", "For physically relevant values of $L=10^{53} L_{53}$ erg s$^{-1}$ and $r_0=10^7r_{0,7}$ cm, $\\eta _{\\rm T}$ is low compared to observed values for $\\eta $ .", "Therefore, the photosphere is in the acceleration phase for a large segment of the parameter space.", "On the other hand, in baryonic cases ($\\mu =1$ ), $\\eta _{\\rm T}\\simeq 1900~L_{53}^{1/4} r_{0,7}^{-1/4}$ , which is several orders of magnitude higher than observed Lorentz factors.", "Therefore, we assume that magnetically dominated jets have their photospheres in the acceleration phase and baryonically dominated jets have their photospheres in the coasting phase.", "With this critical assumption, we derive two cases for the behaviors of both $E_{\\rm p}$ and $kT$ .", "Close above the photosphere, instabilities in the flow or magnetic field line reconnection can lead to mildly relativistic shocks and accelerate leptons, which in turn emit synchrotron radiation [30], [26].", "The synchrotron peak energy is dependent on the baryon number density $n^{\\prime }_b(r)=L/(4\\pi r^2 m_p c^3 \\Gamma (r) \\eta )$ and the magnetic field $B^{\\prime }\\propto {n^{\\prime }}_b^{1/2}$ .", "The peak synchrotron energy is: $E_{\\rm p}=({3q_e B^{\\prime }_{\\rm ph}}/ {4 \\pi m_e c}) \\gamma _{\\rm e,ph}^2{\\Gamma _{\\rm ph}}$ .", "From this expression we derive the following dependence on the input parameters: $E_{\\rm p} \\propto \\left\\lbrace \\begin{array}{ll}L^{\\frac{3\\mu -1}{4\\mu +2}} \\eta ^{-\\frac{3\\mu -1}{4\\mu +2}}r_0^{\\frac{-5\\mu }{4\\mu +2}} & {\\rm if~ } \\eta >\\eta _{\\rm T}\\\\L^{-1/2} \\eta ^{3} & {\\rm if~ } \\eta <\\eta _{\\rm T}.\\end{array}\\right.$ The acceleration of the jet is assumed to be adiabatic for $r<r_{\\rm sat}$ , leading to a relation between the comoving temperature ($T^{\\prime }$ ) and the comoving volume ($V^{\\prime }$ ) of $T^{\\prime }\\propto V^{\\prime -1/3}$ .", "Since the expansion is along the radial direction of the jet, we have $V^{\\prime }=d^{3}x^{\\prime }=\\Gamma d^{3}x \\equiv 4\\pi \\Gamma r^2dr$ .", "Using Equation REF for $r<r_{\\rm sat}$ , we can write $\\Gamma \\propto r^{\\mu }$ .", "Therefore, the comoving temperature of the sub-dominant thermal component depends on radius as ${T^{\\prime }}\\propto r^{-\\left(\\frac{\\mu +2}{3}\\right)} $ .", "Above the saturation radius, the standard evolution of the temperature is $T^{\\prime }\\propto r^{-2/3}$ .", "At the launching radius ($r_0$ ) the temperature is $T_0=(L/4\\pi r_0^2 a c )^{1/4}$ .", "Therefore, the observed temperature for the two scenarios is: $kT_{\\rm obs}(r_{\\rm ph}) \\propto \\left\\lbrace \\begin{array}{ll}L^{\\frac{14\\mu -5}{12(2\\mu +1)}} \\eta ^{\\frac{2-2\\mu }{6\\mu +3}}r_0^{-\\frac{10\\mu -1}{6(2\\mu +1)}} & {\\rm if~ } \\eta >\\eta _{\\rm T} \\\\L^{-5/12} \\eta ^{8/3} r_0^{1/6} & {\\rm if~ } \\eta <\\eta _{\\rm T} .\\end{array}\\right.$ It is unclear whether the evolution of the photosphere's luminosity or Lorentz factor, or some combination of both, drives the evolution of $E_{\\rm p}$ and $kT$ .", "One natural assumption is that the evolution of the photosphere's luminosity results in the observed variations in $E_{\\rm p}$ and $T$ as a burst proceeds.", "Therefore, considering the two types of jets; magnetically dominated ($r_{\\rm ph}<r_{\\rm s}$ ) and kinetic dominated ($r_{\\rm ph}>r_{\\rm s}$ ) we have two possibilities for the values of $\\alpha $ : in the magnetic case, considering the appropriate powers of $L$ , we have $E_{\\rm p} \\propto T^{\\frac{6(3\\mu -1)}{14\\mu -5}}$ .", "The exponent is singular at $\\mu \\approx 0.36$ , but for values up to $\\mu < 0.6$ (these are the values of $\\mu $ for which the photosphere will occur in the acceleration phase) we are able to explain values of $\\alpha $ from 2 down to 1.4 in the kinetic (baryonic) case we have $E_{\\rm p}\\propto T^{1.2}$ .", "This is observed in some GRBs." ], [ "Discussion", "The analysis of GRBs in the framework of this model can indicate whether the photosphere is in the acceleration or coasting phase, which in turn can be translated to the composition of the jet.", "We find that for exponents close to 2 the jet dynamics are dominated by the magnetic field while exponents close to 1 indicate baryonic jets.", "In our sample of six GRBs observed with ${\\it Fermi}$ , the exponents $\\alpha $ of the relation between the characteristic energies of non-thermal and thermal components (Table REF ) span the range of possible values, showing that energy content of GRB jets ranges from being dominated by the magnetic field to being contained mostly in the kinetic energy of baryons in the jet.", "A possible validation of this interpretation would be the future measurement of polarization in GRBs which will allow for the direct determination of the magnetization of GRB jets [23].", "We note that the lack of a correlation between the ratio of the thermal flux to the total flux with the inferred magnetic content of the jet is puzzling (see Table REF ).", "Naively, it is expected that a photosphere occurring deep in the acceleration phase of the outflow will have its thermal emission be much brighter than the non-thermal emission.", "A possible explanation for the weakness of the observed thermal component has been addressed by several authors [47], [7].", "These works consider the effect of the magnetization parameter ($\\sigma =\\frac{B^2}{4\\pi \\Gamma \\rho c^2}$ ) on the intensity of the thermal component where $\\rho $ is the matter density of the outflow.", "For $\\sigma \\gg 1$ , most of the jet internal energy remains in the advected magnetic field, reducing the intensity of the observed thermal component from the photosphere.", "Another possibility for explaining the lack of correlation of the thermal flux ratios to the different jet modes is to consider that if the non-thermal flux is due to synchrotron following reconnection events above the photosphere, the amount of reconnection may not be simply given by the amount of magnetic energy and by the radius, but may depend also on the degree of tangledness of the field at that radius.", "For reconnection one needs field lines of opposite polarity near each other, and if the degree of randomness is stochastic (as it probably is), this could introduce a randomness in the amount of non-thermal electrons accelerated as well as the synchrotron flux produced.", "However, time-dependent simulations of magnetically dominated outflows in GRBs are not advanced enough to accurately test these assumptions and therefore the reduced intensity of the thermal component is still open to interpretation.", "The ${\\it Fermi}$ GBM collaboration acknowledges support for GBM development, operations and data analysis from NASA in the US and BMWi/DLR in Germany.", "We also thank the anonymous referee for very useful comments that aided in refining this work." ] ]	1403.0374
[ [ "Capillary instability in nanowire geometries" ], [ "Abstract The vapor-liquid-solid (VLS) mechanism has been applied extensively as a framework for growing single-crystal semiconductor nanowires for applications spanning optoelectronic, sensor and energy-related technologies.", "Recent experiments have demonstrated that subtle changes in VLS growth conditions produce a diversity of nanowire morphologies, and result in intricate kinked structures that may yield novel properties.", "These observations have motivated modeling studies that have linked kinking phenomena to processes at the triple line between vapor, liquid and solid phases that cause spontaneous \"tilting\" of the growth direction.", "Here we present atomistic simulations and theoretical analyses that reveal a tilting instability that is intrinsic to nanowire geometries, even in the absence of pronounced anisotropies in solid-liquid interface properties.", "The analysis produces a very simple conclusion: the transition between axisymmetric and tilted triple lines is shown to occur when the triple line geometry satisfies Young's force-balance condition.", "The intrinsic nature of the instability may have broad implications for the design of experimental strategies for controlled growth of crystalline nanowires with complex geometries." ], [ "Acknowledgments ", "This research was supported in part by the US National Science Foundation under Grant No.", "DMR-1105409.", "Use was made of computational resources provided under the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by the National Science Foundation under grant number OCI-1053575.", "T.F.", "was supported for part of this work by a post-doctoral fellowship from the Miller Institute for Basic Research in Science at University of California, Berkeley.", "W.C.C.", "'s effort was supported by a visiting Miller Faculty Fellowship, also from the Miller Institute for Basic Research in Science, during a sabbatical at the University of California, Berkeley.", "Molecular dynamics simulations were performed using the LAMMPS software package.", "[18] To model nanowires wetted by liquid we used the angularly dependent Stillinger-Weber potential for Si.", "[19] First we prepared a simulation block with liquid and diamond-cubic solid phases separated by (111) solid-liquid interface.", "The interface plane was normal to the $z$ direction of the simulation block, while the $x$ and $y$ directions were parallel to crystallographic directions $[1\\overline{1}0]$ and $[11\\overline{2}]$ of the solid phase, respectively.", "The dimensions of the simulation block were $30\\times 50\\times 30$ nm$^{3}$ .", "The phases were equilibrated in a microcanonical (NVE) ensemble for 5 ns leading to a coexistence temperature of around 1682 K. This simulation block was then used to produce input configurations for nanowire-liquid simulations.", "Specifically, we carved out columns with hexagonal crossection and $\\lbrace 11\\overline{2}\\rbrace $ side facets containing solid and liquid phases.", "The sizes of the solid surface facets ranged from 5 to 12 nm.", "To investigate the effects of the droplet size on equilibrium configuration, we varied the amount of liquid phase.", "The columns were then equilibrated during a 5 ns long simulation in a microcanonical .", "We performed up to 30 ns long simulations of equilibrated NW configurations.", "In equilibrium the amounts of solid and liquid phases do not change and the temperature fluctuates around equilibrium value.", "Equilibrium simulations for an isolated droplet on a {111} solid surface, shown in Figure REF b were performed in a microcanonical ensemble for 10 ns.", "[20] The solid slab used in these simulations had dimensions $2.3\\times 6.7\\times 32.9$ nm$^{3}$ parallel to $[1\\overline{1}0]$ , $[111]$ and $[11\\overline{2}]$ crystallographic directions, respectively.", "Figure: Equilibrium wetting configurations of simulated nanowires.a, Nanowire with solid (colored red) and liquid (blue) phasesas identified according to a local crystalline order parameter.", "Thesolid-liquid-vapor contact line has a variable elevation as it traversesthe perimeter of the nanowire: this triple line dips below the facetswith [1 ¯11][\\overline{1}11] and [111 ¯][11\\overline{1}] normals, and abovethe facet with [11 ¯1][1\\overline{1}1] normal.", "Slice b showsthe structure of the solid-liquid interface.", "c, A view ofthe solid liquid interface with semitransparent liquid phase illustratingits faceted nature.Figure: Nanowires with increasing volume of the liquid droplet.", "a,Equilibrium angles depend on the droplet volume, with φ vs \\phi _{vs}increasing with V l V_{l}.", "b, Equilibrium wetting angles ofa liquid droplet on [111][111] surface.", "Spontaneous tilting occurs whennanowire contact angles approach angles indicated in b.Figure: Transition from symmetric to tilted configurations.", "Timesequence of snapshots demonstrating the evolution of the droplet tiltingduring a 24 ns MD simulation.", "Top and middle panels show side andbottom views of the nanowire.", "The middle panel illustrates that thedroplet tilting direction changes with time.", "The bottom panel showsslices going through the nanowire along planes marked in the middlepanel by yellow dashed lines.", "Initially the tilting of the dropletleads to the formation of a large [1 ¯11][\\overline{1}11] solid-liquidinterface facet.", "At later times the droplet wets the side wall ofthe nanowire, with the solid-liquid interface becoming more rounded.Figure: Two Dimensional Model of Nanowire Capillary Instability.a, The angles, φ\\phi , which vary with liquid volume.", "b,The model parameters that represent the degrees of freedom for capillaryshapes: the asymmetric tilt angle α\\alpha and the apparent wettingangle θ s \\theta _{s} appear as coordinates in the free energy surfacesfor different liquid volumes in the middle and bottom figure.", "Fora symmetrical equilibrium configuration (α=0\\alpha =0) φ vs =θ L \\phi _{vs}=\\theta _{L}and φ lv =θ s +π\\phi _{lv}=\\theta _{s}+\\pi /2.", "The free-energy surfaces are plottedfor interface energy ratios γ lv /γ vs =9/10\\gamma _{lv}/\\gamma _{vs}=9/10 and γ sl /γ vs =3/10\\gamma _{sl}/\\gamma _{vs}=3/10.c, Free-energy surface plotted for a volume which is halfof the Young's volume (i.e., the volume at which the φ ij \\phi _{ij}satisfy Young's equation).", "When V l <V l Young V_{l}<V_{l}^{Young} , there isa metastable minimum at a finite θ s \\theta _{s} and zero tilt angle(α=0\\alpha =0), and symmetric saddle points for α≠0\\alpha \\ne 0 .", "Theblue and green lines were computed by steepest descent and representthe most probable path for the system to transform from a symmetricnanowire wetting geometry to a tilted configuration.", "d, Asthe volume increases, the saddle and metastable points approach eachother and converge when V l =V l Young V_{l}=V_{l}^{Young}.", "For liquid volumeslarger than V l Young V_{l}^{Young} the metastable minimum correspondingto a symmetric nanowire geometry disappears, and symmetric configurationsare unstable with respect to tilting.Figure: Capillary forces on triple junctions.", "The net capillary forceson triple junctions for different volumes of the droplet are shownby black arrows.", "These are result of the capillary forces from eachof the three interfaces.", "a, Capillary force is directed normalto the nanowire axis and points inward towards the nanowire for V l <V l Young V_{l}<V_{l}^{Young}.b, Zero force at Young's conditions.", "c, Capillaryforce is directed outward for V l >V l Young V_{l}>V_{l}^{Young}.", "In case acapillary forces push triple junctions towards each other.", "Since tiltingwould increase the separation, it is not favorable.", "On the other hand,in case c a small perturbation from a symmetrical configurationwill result in increasing tilting, since capillary forces try to pullthe two triple junctions apart." ] ]	1403.0152
[ [ "Testing for change-points in long-range dependent time series by means\n of a self-normalized Wilcoxon test" ], [ "Abstract We propose a testing procedure based on the Wilcoxon two-sample test statistic in order to test for change-points in the mean of long-range dependent data.", "We show that the corresponding self-normalized test statistic converges in distribution to a non-degenerate limit under the hypothesis that no change occurred and that it diverges to infinity under the alternative of a change-point with constant height.", "Furthermore, we derive the asymptotic distribution of the self-normalized Wilcoxon test statistic under local alternatives, that is under the assumption that the height of the level shift decreases as the sample size increases.", "Regarding the finite sample performance, simulation results confirm that the self-normalized Wilcoxon test yields a consistent discrimination between hypothesis and alternative and that its empirical size is already close to the significance level for moderate sample sizes." ], [ "Introduction", "We consider a data set generated by a stochastic process $(X_i)_{i\\ge 1}$ , $X_i=\\mu _i+\\varepsilon _i,$ where $(\\mu _i)_{i\\ge 1}$ are unknown constants and where $(\\varepsilon _i)_{i\\ge 1}$ is a stationary, long-range dependent (LRD, in short) process with mean zero and finite variance.", "In particular, we assume that $\\varepsilon _i=G(\\xi _i), \\ i\\ge 1,$ where $(\\xi _i)_{i\\ge 1}$ is a stationary Gaussian process with mean 0, variance 1 and long-range dependence, that is with autocovariance function $\\rho $ satisfying $\\rho (k)\\sim k^{-D}L(k), \\ k\\ge 1,$ where $0<D< 1$ (referred to as long-range dependence (LRD) parameter) and where $L$ is a slowly varying function.", "Furthermore, we suppose that $G:\\mathbb {R}\\longrightarrow \\mathbb {R}$ is a measurable function with $\\operatorname{E}\\left(G(\\xi _i)\\right)=0$ .", "Provided that the previous assumptions hold for the observations $X_1, \\ldots , X_n$ , we wish to test the hypothesis $H: \\mu _1=\\ldots =\\mu _n$ against the alternative $A: \\mu _1=\\ldots =\\mu _k\\ne \\mu _{k+1}=\\ldots =\\mu _n$ for some $k\\in \\left\\lbrace 1, \\ldots , n-1\\right\\rbrace $ .", "Within this setting the location of the change-point is unknown under the alternative.", "In order to motivate our choice of a change-point test, we temporarily assume that the change-point location is known, i.e.", "for a given $k\\in \\left\\lbrace 1, \\ldots , n-1\\right\\rbrace $ we consider the alternative $A_k: \\mu _1=\\ldots =\\mu _k\\ne \\mu _{k+1}=\\ldots =\\mu _n.$ For the test problem $(H, A_k)$ , the Wilcoxon two-sample rank test rejects the hypothesis of no change in the mean for large absolute values of the test statistic $W_{k, n}=\\sum \\limits _{i=1}^k\\sum \\limits _{j=k+1}^n\\left(1_{\\left\\lbrace X_i\\le X_j\\right\\rbrace }-\\frac{1}{2}\\right).$ The Wilcoxon change-point test for the test problem $(H, A)$ is defined by reference to the test statistic $W_{k, n}$ ; see [2].", "It rejects the hypothesis for large values of $\\max \\limits _{1\\le k\\le n-1}\\left\|W_{k, n}\\right\|=\\max \\limits _{1\\le k\\le n-1}\\left\|\\sum \\limits _{i=1}^k\\sum \\limits _{j=k+1}^n\\left(1_{\\left\\lbrace X_i\\le X_j\\right\\rbrace }-\\frac{1}{2}\\right)\\right\|.$ With the objective of calculating the asymptotic distribution of the Wilcoxon test statistic under the null hypothesis, [2] consider the stochastic process $W_n(\\lambda )=\\frac{1}{n d_n}\\sum \\limits _{i=1}^{\\lfloor nr\\rfloor }\\sum \\limits _{j=\\lfloor nr\\rfloor +1}^n\\left(1_{\\left\\lbrace X_i\\le X_j\\right\\rbrace }-\\int _{\\mathbb {R}}F(x)dF(x)\\right), \\ 0\\le \\lambda \\le 1,$ where $d_n$ denotes an appropriate normalization.", "Assuming that $(X_i)_{i\\ge 1}$ has a continuous marginal distribution function $F$ , the asymptotic distribution of $W_n$ can be derived from the empirical process invariance principle of [4] as shown in [2].", "It turns out that both, the limit of $W_n$ and the normalization $d_n$ , depend on the Hermite expansion $1_{\\left\\lbrace G(\\xi _i)\\le x\\right\\rbrace }-F(x)=\\sum \\limits _{q=1}^{\\infty }\\frac{J_q(x)}{q !", "}H_q(\\xi _i),$ where $H_q$ denotes the $q$ -th order Hermite polynomial and where $J_q(x)=\\operatorname{E}\\left(H_q(\\xi _i)1_{\\left\\lbrace G(\\xi _i)\\le x\\right\\rbrace }\\right).$ The scaling factor $d_n$ is defined by $d_n^2=\\operatorname{Var}\\left(\\sum \\limits _{j=1}^nH_m(\\xi _j)\\right),$ where $m$ designates the Hermite rank of the class of functions $\\left\\lbrace 1_{\\left\\lbrace G(\\xi _i)\\le x\\right\\rbrace }-F(x), \\ x\\in \\mathbb {R}\\right\\rbrace $ defined by $m:=\\min \\left\\lbrace q\\ge 1: J_q(x)\\ne 0 \\ \\text{for some} \\ x\\in \\mathbb {R}\\right\\rbrace .$ Presuming the previous conditions hold and the long-range dependence parameter $D$ meets the condition $0<D<\\frac{1}{m}$ , the process $W_n(\\lambda )=\\frac{1}{n d_n}\\sum \\limits _{i=1}^{\\lfloor n\\lambda \\rfloor }\\sum \\limits _{j=\\lfloor n\\lambda \\rfloor +1}^n\\left(1_{\\left\\lbrace X_i\\le X_j\\right\\rbrace }-\\int _{\\mathbb {R}}F(x)dF(x)\\right), \\ 0\\le \\lambda \\le 1,$ converges in distribution to $\\frac{1}{m!", "}(Z_m(\\lambda )-\\lambda Z_m(1))\\int _{\\mathbb {R}}J_m(x)dF(x), \\ 0\\le \\lambda \\le 1,$ where $\\left(Z_m(\\lambda )\\right)_{\\lambda \\in \\left[0, 1\\right]}$ is an $m$ -th order Hermite process, which is self-similar with parameter $H=1-\\frac{mD}{2}\\in \\left(\\frac{1}{2}, 1\\right)$ .", "If $m=1$ , the Hermite process $Z_m$ equals a standard fractional Brownian motion process with Hurst parameter $H=1-\\frac{D}{2}$ .", "We refer to [12] for a general definition of the Hermite process $Z_m$ .", "An application of the continuous mapping theorem to the process $W_n$ yields the asymptotic distribution of the Wilcoxon change-point test.", "More precisely, it has been proved by [2] that under the hypothesis of no change in the mean, the Wilcoxon test statistic $\\frac{1}{n d_n}\\max \\limits _{1\\le k\\le n-1}\\left\|\\sum \\limits _{i=1}^k\\sum \\limits _{j=k+1}^n\\left(1_{\\left\\lbrace X_i\\le X_j\\right\\rbrace }-\\frac{1}{2}\\right)\\right\|$ converges in distribution to $\\sup \\limits _{0\\le \\lambda \\le 1}\\left\|\\frac{1}{m!", "}(Z_m(\\lambda )-\\lambda Z_m(1))\\right\|\\left\|\\int _{\\mathbb {R}}J_m(x)dF(x)\\right\|.$ Furthermore, [3] investigate the asymptotic behaviour of the Wilcoxon change-point test under the alternative with the objective of determining the height of the level shift in such a way that the power of the self-normalized Wilcoxon test is non-trivial.", "For this purpose, they consider local alternatives defined by $A_{\\tau , h_n}:\\mu _i={\\left\\lbrace \\begin{array}{ll}\\mu \\ &\\text{for} \\ i=1, \\ldots , \\lfloor n\\tau \\rfloor \\\\\\mu +h_n \\ &\\text{for} \\ i=\\lfloor n\\tau \\rfloor +1, \\ldots , n,\\end{array}\\right.", "}$ where $0< \\tau < 1$ and where $h_n\\sim c\\frac{d_n}{n}$ , so that under the sequence of local alternatives $A_{\\tau , h_n}$ the height of the level shift decreases if the sample size increases.", "Under the additional assumption that $G(\\xi _i)$ has a continuous distribution function $F$ with bounded density $f$ , this guarantees that under the sequence of alternatives $A_{\\tau , h_n}$ , the process $\\frac{1}{nd_n}\\sum \\limits _{i=1}^{\\lfloor n\\lambda \\rfloor }\\sum \\limits _{j=\\lfloor n\\lambda \\rfloor +1}^n\\left(1_{\\lbrace X_i\\le X_j\\rbrace }-\\frac{1}{2}\\right), \\ 0\\le \\lambda \\le 1,$ converges in distribution to the limit process $\\frac{1}{m!", "}(Z_m(\\lambda )-\\lambda Z_m(1))\\int _{\\mathbb {R}}J_m(x)dF(x)+c\\delta _{\\tau }(\\lambda )\\int _{\\mathbb {R}}f^2(x)dx, \\ 0\\le \\lambda \\le 1,$ where $\\delta _{\\tau }:[0, 1]\\longrightarrow \\mathbb {R}$ is defined by $\\delta _{\\tau }(\\lambda )={\\left\\lbrace \\begin{array}{ll}\\lambda (1-\\tau ) \\ &\\text{for} \\ \\lambda \\le \\tau \\\\(1-\\lambda )\\tau \\ &\\text{for} \\ \\lambda \\ge \\tau \\end{array}\\right.", "}.$ By another application of the continuous mapping theorem it then follows that the Wilcoxon change-point test converges in distribution to a non-degenerate limit process under the sequence of local alternatives $A_{\\tau , h_n}$ ; see [3]." ], [ "Main Results", "An application of the Wilcoxon change-point test to a given data set presupposes that the scaling factor $d_n$ is known.", "Usually this is not the case in statistical practice so that in general the Wilcoxon change-point test as proposed in [2] depends on an unknown normalization.", "As an alternative we propose a normalization that only depends on the given realizations and therefore is referred to as self-normalization.", "The self-normalization approach we consider has originally been established in another context; see [6].", "It has been extended to the change-point testing problem by [10] in order to test for change-points in the mean of short-range dependent time series.", "These authors used the self-normalization method on the Kolmogorov-Smirnov test statistic, in doing so also taking the change-point alternative into account.", "Lobato as well as Shao and Zhang considered weak dependent processes only.", "Following the approach in Shao and Zhang an application to possibly long-range dependent processes was introduced by Shao, who established a self-normalized version of the CUSUM change-point test; see [9].", "As the CUSUM test has the disadvantage of not being robust against possible outliers in the data, an extension of the self-normalization idea to the Wilcoxon test statistic leads to a change-point test that not only has the advantage of avoiding the choice of unknown parameters but also yields a robust alternative to the CUSUM test.", "Given observations $X_1, \\ldots , X_n$ , we consider the rank statistics defined by $R_i=\\operatorname{rank}(X_i)=\\sum \\limits _{j=1}^n1_{\\lbrace X_j\\le X_i\\rbrace }$ for $i=1, \\ldots , n$ .", "An extension of the self-normalization approach to the Wilcoxon change-point test is based on an application of the CUSUM change-point test in terms of the rank statistics $R_i$ .", "Note that due to the identity $\\max \\limits _{k}\\left\|\\sum \\limits _{i=1}^kR_i-\\frac{k}{n}\\sum \\limits _{i=1}^nR_i\\right\|=\\max \\limits _{k}\\left\|\\sum \\limits _{i=1}^k\\sum \\limits _{j=k+1}^n\\left(1_{\\left\\lbrace X_i\\le X_j\\right\\rbrace }-\\frac{1}{2}\\right)\\right\|,$ the CUSUM test statistic of the ranks equals the Wilcoxon change-point test statistic.", "Instead of dividing the test statistic (which is the maximum taken among every possible outcome of the Wilcoxon two-sample rank test) by the unknown quantity $nd_n$ we consider a normalization factor that depends on the location of a potential change-point and which therefore is different for every possible outcome of the Wilcoxon two-sample rank test.", "We define $G_n(k)=\\frac{\\left\|\\sum \\limits _{i=1}^kR_i-\\frac{k}{n}\\sum \\limits _{i=1}^nR_i\\right\|}{\\left\\lbrace \\frac{1}{n}\\sum \\limits _{t=1}^k S_t^2(1,k)+\\frac{1}{n}\\sum \\limits _{t=k+1}^n S_t^2(k+1,n)\\right\\rbrace ^{\\frac{1}{2}}},$ where $&S_{t}(j, k)=\\sum \\limits _{h=j}^t\\left(R_h-\\bar{R}_{j, k}\\right),\\\\&\\bar{R}_{j, k}=\\frac{1}{k-j+1}\\sum \\limits _{t=j}^kR_t.$ The self-normalized Wilcoxon test rejects the hypothesis $H:\\mu _1=\\ldots =\\mu _n$ for large values of the test statistic $T_n(\\tau _1, \\tau _2)=\\sup _{k\\in \\left[\\lfloor n\\tau _1\\rfloor , \\lfloor n\\tau _2\\rfloor \\right]}G_n(k),$ where $0< \\tau _1 <\\tau _2 <1$ .", "Note that the proportion of the data that is included in the calculation of the supremum is restricted by the choice of $\\tau _1$ and $\\tau _2$ .", "This is important as the choice of $\\tau _1$ and $\\tau _2$ influences the properties of the test.", "Structural breaks at the beginning or the end of a sample are hard to detect since there is a lack of information concerning the behaviour of the time series before or after a potential break point.", "Hence, the interval $\\left[\\tau _1, \\tau _2\\right]$ must be small enough for the critical values not to get too large on the one hand, yet large enough to include potential break points on the other hand.", "A common choice is $\\tau _1= 1-\\tau _2=0.15$ ; see [1].", "The following theorem states the asymptotic distribution of the test statistic $T_n(\\tau _1, \\tau _2)$ under the hypothesis of no change in the mean.", "Theorem 1 Suppose that $\\left(X_i\\right)_{i\\ge 1}$ is a stationary process with continuous distribution function $F$ defined by $X_i=\\mu _i+G(\\xi _i)$ for unknown constants $\\left(\\mu _i\\right)_{i\\ge 1}$ and a stationary, long-range dependent Gaussian process $\\left(\\xi _i\\right)_{i\\ge 1}$ with mean 0, variance 1 and LRD parameter $0<D <\\frac{1}{m}$ , where $m$ denotes the Hermite rank of the class of functions $1_{\\left\\lbrace G(\\xi _i)\\le x\\right\\rbrace }-F(x)$ , $x \\in \\mathbb {R}$ .", "Moreover, assume that $\\int _{\\mathbb {R}}J_m(x)dF(x)\\ne 0$ and that $G:\\mathbb {R}\\longrightarrow \\mathbb {R}$ is a measurable function.", "Then, under the hypothesis of no change in the mean, it follows that $T_n(\\tau _1, \\tau _2)~\\overset{\\mathcal {D}}{\\longrightarrow }~T(m,\\tau _1, \\tau _2)$ , where $&T(m, \\tau _1, \\tau _2)=\\sup \\limits _{\\lambda \\in \\left[\\tau _1, \\tau _2\\right]}\\frac{ \\left\|Z_m(\\lambda )-\\lambda Z_m(1)\\right\|}{\\Bigl \\lbrace \\int _0^{\\lambda }\\left(V_m(r; 0, \\lambda )\\right)^2dr+\\int _{\\lambda }^1 \\left(V_m(r; \\lambda , 1)\\right)^2dr\\Bigr \\rbrace ^{\\frac{1}{2}}}\\multicolumn{2}{l}{\\text{with}}\\\\&V_m(r; r_1, r_2)=Z_m(r)-Z_m(r_1)-\\frac{r-r_1}{r_2-r_1}\\left\\lbrace Z_m(r_2)-Z_m(r_1)\\right\\rbrace $ for $r\\in \\left[r_1, r_2\\right]$ , $0< r_1< r_2< 1$ .", "As consistency under fixed alternatives is considered as a fundamental characteristic of appropriate hypothesis testing, we aim at proving Theorem REF , which implies that if there is a change-point in the mean of constant height, the empirical power of the self-normalized Wilcoxon test tends to 1.", "For this purpose, we suppose that under the alternative $X_i={\\left\\lbrace \\begin{array}{ll}\\mu +G(\\xi _i), \\ i=1, \\ldots , k^, \\\\\\mu +\\Delta +G(\\xi _i), \\ i=k^+1, \\ldots , n,\\end{array}\\right.", "}$ where $k^=\\lfloor n\\tau \\rfloor $ and $\\Delta \\ne 0$ is fixed.", "Theorem 2 Suppose that $(\\xi _i)_{i\\ge 1}$ is a stationary, long-range dependent Gaussian process with mean 0, variance 1 and LRD parameter $D$ .", "Moreover, let $G:\\mathbb {R}\\longrightarrow \\mathbb {R}$ be a measurable function and assume that $G(\\xi _i)$ has a continuous distribution function $F$ .", "Given that the parameter $D$ satisfies $0< D<\\frac{1}{m}$ , where $m$ denotes the Hermite rank of the class of functions $1_{\\lbrace G(\\xi _i)\\le x\\rbrace }-F(x)$ , $x\\in \\mathbb {R}$ , $T_n(\\tau _1, \\tau _2)$ diverges in probability to $\\infty $ under fixed alternatives, i.e.", "if $\\left(X_i\\right)_{i\\ge 1}$ satisfies (REF ).", "Furthermore, we wish to study the asymptotic behaviour of the self-normalized Wilcoxon change-point test under local alternatives defined by $A_{\\tau , h_n}(n):\\mu _i={\\left\\lbrace \\begin{array}{ll}\\mu \\ &\\text{for} \\ i=1, \\ldots , \\lfloor n\\tau \\rfloor ,\\\\\\mu +h_n \\ &\\text{for} \\ i=\\lfloor n\\tau \\rfloor +1, \\ldots , n,\\end{array}\\right.", "}$ where $0< \\tau < 1$ and $h_n\\longrightarrow 0$ .", "The following theorem confirms that the self-normalized Wilcoxon test statistic converges to a non-degenerate limit under the sequence of local alternatives $A_{\\tau , h_n}$ .", "Theorem 3 Suppose that $(\\xi _i)_{i\\ge 1}$ is a stationary Gaussian process with mean 0, variance 1 and autocovariance function $\\rho (k)\\sim k^{-D}L(k),$ where $L$ is a slowly varying function and where $0< D<\\frac{1}{m}$ .", "Moreover, let $G:\\mathbb {R}\\longrightarrow \\mathbb {R}$ be a measurable function.", "We assume that $G(\\xi _i)$ has a continuous distribution function $F$ with bounded density $f$ .", "Let $m$ denote the Hermite rank of the class of functions $1_{\\lbrace G(\\xi _i)\\le x\\rbrace }-F(x)$ , $x\\in \\mathbb {R}$ , and suppose that $\\int _{\\mathbb {R}}J_m(x)dF(x)\\ne 0$ .", "Then, under the sequence of alternatives $A_{\\tau , h_n}$ with $h_n\\sim c\\frac{d_n}{n}$ , it follows that $T_n(\\tau _1, \\tau _2)$ converges in distribution to $T(m, \\tau _1, \\tau _2)=\\sup \\limits _{\\lambda \\in \\left[\\tau _1, \\tau _2\\right]}\\frac{\\left\|\\frac{1}{m!", "}\\int _{\\mathbb {R}}J_m(x)dF(x)(Z_m(\\lambda )-\\lambda Z_m(1))+c\\delta _{\\tau }(\\lambda )\\int _{\\mathbb {R}}f^2(x)dx\\right\|}{\\left\\lbrace \\int _0^\\lambda \\left(V_{m, \\tau }(r; 0, \\lambda )\\right)^2dr+\\int _{\\lambda }^1\\left(V_{m, \\tau }(r; \\lambda , 1)\\right)^2dr\\right\\rbrace ^{\\frac{1}{2}}},$ where $V_{m, \\tau }(r; 0, \\lambda )&=\\frac{1}{m!", "}\\int _{\\mathbb {R}}J_m(x)dF(x)\\left(Z_m(r)-\\frac{r}{\\lambda }Z_m(\\lambda )\\right)+c\\int _{\\mathbb {R}}f^2(x)dx\\left(\\delta _{\\tau }(r)-\\frac{r}{\\lambda }\\delta _{\\tau }(\\lambda )\\right),\\\\V_{m, \\tau }(r; \\lambda , 1)&=\\frac{1}{m!", "}\\int _{\\mathbb {R}}J_m(x)dF(x)\\left\\lbrace Z_m(r)-Z_m(\\lambda )-\\frac{r-\\lambda }{1-\\lambda }\\left(Z_m(1)-Z_m(\\lambda )\\right)\\right\\rbrace \\\\&\\quad \\ +c\\int _{\\mathbb {R}}f^2(x)dx\\left(\\delta _{\\tau }(r)-\\frac{1-r}{1-\\lambda }\\delta _{\\tau }(\\lambda )\\right).$" ], [ "Simulation studies", "We will now investigate the finite sample performance of the self-normalized Wilcoxon test statistic.", "For this purpose, we take $G(t) = t$ so that $\\left(X_i\\right)_{i\\ge 1}$ is a Gaussian process.", "Since $G$ is strictly increasing, the Hermite coefficient $J_1(x)$ is not equal to 0 for all $x\\in \\mathbb {R}$ ; see [2].", "Therefore, it holds that $m = 1$ , where $m$ denotes the Hermite rank of $1_{\\left\\lbrace G(\\xi _i)\\le x\\right\\rbrace }-F(x), x\\in \\mathbb {R}$ .", "As a result, $T_n(\\tau _1, \\tau _2)$ has approximately the same distribution as $&\\sup \\limits _{\\lambda \\in \\left[\\tau _1, \\tau _2\\right]}\\frac{ \\left\|B_H(\\lambda )-\\lambda B_H(1)\\right\|}{\\Bigl \\lbrace \\int _0^{\\lambda }\\left(V_H(r; 0, \\lambda )\\right)^2dr+\\int _{\\lambda }^1 \\left(V_H(r; \\lambda , 1)\\right)^2dr\\Bigr \\rbrace ^{\\frac{1}{2}}}\\\\&V_H(r; r_1, r_2)=B_H(r)-B_H(r_1)-\\frac{r-r_1}{r_2-r_1}\\left\\lbrace B_H(r_2)-B_H(r_1)\\right\\rbrace ,$ where $B_H$ is a fractional Brownian motion process with Hurst parameter $H=1-\\frac{D}{2}$ .", "We set critical values on the basis of $10,000$ simulations of fractional Brownian motion time series for different Hurst parameters $H$ and different levels of significance; see Table REF .", "Table: Simulated critical values for the distribution of T(1,τ 1 ,τ 2 )T(1, \\tau _1, \\tau _2) when τ 1 ,τ 2 =0.15,0.85\\left[\\tau _1, \\tau _2\\right]=\\left[0.15, 0.85\\right].", "The sample size is 1000, the number ofreplications is 10,00010, 000.The calculation of the relative frequency of false rejections under the hypothesis is based on $10,000$ realizations of fractional Gaussian noise time series with varying length; see Table REF .", "Table: Level of the self-normalized Wilcoxon change-point test for fractional Gaussian noise time series of length nn with Hurst parameter HH.The level of significance is 5%5\\%.", "The calculations are based on 10,00010,000 simulationruns.The simulation results suggest that the self-normalized Wilcoxon test performs well under the hypothesis since empirical size and asymptotic significance level are already close for moderate sample sizes.", "In particular, it is notable that the size of the self-normalized Wilcoxon change-point test differs considerably from the size of the original Wilcoxon change-point test when $H=0.9$ , that means when we have very strong dependence.", "In that case, the convergence of the Wilcoxon change-point test statistic appears to be rather slow under the hypothesis (see [2], Table 2), whereas the size of the self-normalized Wilcoxon change-point test is still close to the corresponding level of significance.", "We consider fractional Gaussian noise time series with a level shift of height $\\Delta $ after a proportion $\\tau $ of the data in order to analyse the behaviour of the test statistic under the alternative.", "We have done so for several choices of $\\Delta $ and $\\tau $ and for sample sizes $n=100$ and $n = 500$ .", "Table: Empirical power of the self-normalized Wilcoxon change-point testfor fractional Gaussian noise of length n=100n=100 and n=500n=500 with Hurst parameter HH and a level shift in the mean of height Δ\\Delta after a proportion τ=0.5\\tau =0.5.", "The calculations are based on 5,0005,000 simulation runs.Table: Empirical power of the self-normalized Wilcoxon change-point testfor fractional Gaussian noise of length n=100n=100 and n=500n=500 with Hurst parameter HH and a level shift in the mean of height Δ\\Delta after a proportion τ=0.25\\tau =0.25.", "The calculations are based on 5,0005,000 simulation runs.The simulations of the empirical power confirm that the rejection rate becomes higher when $\\Delta $ increases.", "Comparing the empirical power for different Hurst parameters $H$ , we note that the test tends to have less power as $H$ becomes large.", "This seems natural since when there is very strong dependence, i.e.", "$H$ is large, the variance of the series increases, so that it becomes harder to detect a level shift of a fixed height.", "In addition, change-points that are located in the middle of the sample are detected more often than change-points that are located close to the boundary of the testing region determined by $\\left[\\tau _1, \\tau _2\\right]$ .", "Furthermore, Table REF and Table REF show that an increasing sample size goes along with an increase of the empirical power.", "This result confirms that the self-normalized Wilcoxon change-point test yields a consistent discrimination between hypothesis and alternative." ], [ "Proofs", "In order to simplify notation, we write $&J(x)=\\frac{1}{m!", "}J_m(x),\\\\&Z(\\lambda )=Z_m(\\lambda ).$ Proof of Theorem REF .", "The essential step in the proof of Theorem REF is to find a representation for the test statistic $T_n(\\tau _1, \\tau _2)$ as a functional of the Wilcoxon process $W_n(\\lambda )=\\frac{1}{nd_n}\\sum \\limits _{i=1}^{\\lfloor n\\lambda \\rfloor }\\sum \\limits _{j=\\lfloor n\\lambda \\rfloor +1}^n\\left(1_{\\left\\lbrace X_i\\le X_j\\right\\rbrace }-\\frac{1}{2}\\right), \\ 0\\le \\lambda \\le 1.$ For this purpose, rewrite $G_n(k)&=\\frac{\\left\|\\sum \\limits _{i=1}^{k}\\sum \\limits _ {j=k+1}^{n}\\left(1_{\\lbrace X_i\\le X_j\\rbrace }-\\frac{1}{2}\\right)\\right\|}{\\left\\lbrace \\frac{1}{n}\\sum \\limits _{t=1}^k S_t^2(1,k)+\\frac{1}{n}\\sum \\limits _{t=k+1}^n S_t^2(k+1,n)\\right\\rbrace ^{\\frac{1}{2}}}\\\\&=\\frac{\\frac{1}{nd_n}\\left\|\\sum \\limits _{i=1}^{k}\\sum \\limits _ {j=k+1}^{n}\\left(1_{\\lbrace X_i\\le X_j\\rbrace }-\\frac{1}{2}\\right)\\right\|}{\\frac{1}{nd_n}\\left\\lbrace \\frac{1}{n}\\sum \\limits _{t=1}^k S_t^2(1,k)+\\frac{1}{n}\\sum \\limits _{t=k+1}^n S_t^2(k+1,n)\\right\\rbrace ^{\\frac{1}{2}}}.$ As we have $\\frac{1}{nd_n}\\left\|\\sum \\limits _{i=1}^{k}\\sum \\limits _ {j=k+1}^{n}\\left(1_{\\lbrace X_i\\le X_j\\rbrace }-\\frac{1}{2}\\right)\\right\|=\\left\|W_n(\\lambda )\\right\|$ for the numerator of $G_n(k)$ if $k=\\lfloor n\\lambda \\rfloor $ , it remains to show that the denominator of $G_n(k)$ can be represented as a functional of $W_n$ .", "Since $R_i=n+1-\\sum \\limits _{j=1}^n1_{\\left\\lbrace X_i\\le X_j\\right\\rbrace }$ almost surely, it follows that $S_t(1, k)=&-\\sum \\limits _{h=1}^t\\left(\\sum \\limits _{j=1}^n1_{\\lbrace X_h\\le X_j\\rbrace }-\\frac{1}{k}\\sum \\limits _{i=1}^k\\sum \\limits _{j=1}^n1_{\\lbrace X_i\\le X_j\\rbrace }\\right)\\\\=&-\\Biggl \\lbrace \\sum \\limits _{i=1}^t\\sum \\limits _{j=t+1}^n\\left(1_{\\lbrace X_i\\le X_j\\rbrace }-\\frac{1}{2}\\right)+\\sum \\limits _{i=1}^t\\sum \\limits _{j=1}^t\\left(1_{\\lbrace X_i\\le X_j\\rbrace }-\\frac{1}{2}\\right)\\\\&-\\frac{t}{k}\\sum \\limits _{i=1}^k\\sum \\limits _{j=k+1}^n\\left(1_{\\lbrace X_i\\le X_j\\rbrace }-\\frac{1}{2}\\right)-\\frac{t}{k}\\sum \\limits _{i=1}^k\\sum \\limits _{j=1}^k\\left(1_{\\lbrace X_i\\le X_j\\rbrace }-\\frac{1}{2}\\right)\\Biggr \\rbrace $ almost surely.", "Moreover, it is well known that $\\sum \\limits _{i=1}^l\\sum \\limits _{j=1}^l1_{\\left\\lbrace X_i\\le X_j\\right\\rbrace }=\\frac{l(l+1)}{2}.$ Hence, $\\sum \\limits _{i=1}^l\\sum \\limits _{j=1}^l\\left(1_{\\left\\lbrace X_i\\le X_j\\right\\rbrace }-\\frac{1}{2}\\right)=\\frac{l(l+1)}{2}-\\frac{l^2}{2}=\\frac{l}{2},$ so that $S_t(1, k)&=-\\Biggl \\lbrace \\sum \\limits _{i=1}^t\\sum \\limits _{j=t+1}^n\\left(1_{\\lbrace X_i\\le X_j\\rbrace }-\\frac{1}{2}\\right)+\\frac{t}{2}-\\frac{t}{k}\\sum \\limits _{i=1}^k\\sum \\limits _{j=k+1}^n\\left(1_{\\lbrace X_i\\le X_j\\rbrace }-\\frac{1}{2}\\right)-\\frac{t}{k}\\frac{k}{2}\\Biggr \\rbrace \\\\&=-\\Biggl \\lbrace \\sum \\limits _{i=1}^t\\sum \\limits _{j=t+1}^n\\left(1_{\\lbrace X_i\\le X_j\\rbrace }-\\frac{1}{2}\\right)-\\frac{t}{k}\\sum \\limits _{i=1}^k\\sum \\limits _{j=k+1}^n\\left(1_{\\lbrace X_i\\le X_j\\rbrace }-\\frac{1}{2}\\right)\\Biggr \\rbrace $ almost surely.", "Thus, if $\\lambda \\in \\left[\\tau _1, \\tau _2\\right]$ , $&\\int _0^{\\lambda } \\left(\\sum \\limits _{i=1}^{\\lfloor nr\\rfloor }\\sum \\limits _{j=\\lfloor nr\\rfloor +1}^n\\left(1_{\\lbrace X_i\\le X_j\\rbrace }-\\frac{1}{2}\\right)-\\frac{\\lfloor nr\\rfloor }{\\lfloor n\\lambda \\rfloor }\\sum \\limits _{i=1}^{\\lfloor n\\lambda \\rfloor }\\sum \\limits _{j=\\lfloor n\\lambda \\rfloor +1}^n\\left(1_{\\lbrace X_i\\le X_j\\rbrace }-\\frac{1}{2}\\right)\\right)^2dr\\\\&=\\sum \\limits _{t=0}^{\\lfloor n\\lambda \\rfloor }\\int _{\\frac{t}{n}}^{\\frac{t+1}{n}}\\left(\\sum \\limits _{i=1}^{\\lfloor nr\\rfloor }\\sum \\limits _{j=\\lfloor nr\\rfloor +1}^n\\left(1_{\\lbrace X_i\\le X_j\\rbrace }-\\frac{1}{2}\\right)-\\frac{\\lfloor nr\\rfloor }{\\lfloor n\\lambda \\rfloor }\\sum \\limits _{i=1}^{\\lfloor n\\lambda \\rfloor }\\sum \\limits _{j=\\lfloor n\\lambda \\rfloor +1}^n\\left(1_{\\lbrace X_i\\le X_j\\rbrace }-\\frac{1}{2}\\right)\\right)^2dr\\\\&\\quad \\ -\\int _{\\lambda }^{\\frac{\\lfloor n\\lambda \\rfloor +1}{n}}\\left(\\sum \\limits _{i=1}^{\\lfloor nr\\rfloor }\\sum \\limits _{j=\\lfloor nr\\rfloor +1}^n\\left(1_{\\lbrace X_i\\le X_j\\rbrace }-\\frac{1}{2}\\right)-\\frac{\\lfloor nr\\rfloor }{\\lfloor n\\lambda \\rfloor }\\sum \\limits _{i=1}^{\\lfloor n\\lambda \\rfloor }\\sum \\limits _{j=\\lfloor n\\lambda \\rfloor +1}^n\\left(1_{\\lbrace X_i\\le X_j\\rbrace }-\\frac{1}{2}\\right)\\right)^2dr,$ where $\\sum \\limits _{i=1}^{\\lfloor nr\\rfloor }\\sum \\limits _{j=\\lfloor nr\\rfloor +1}^n\\left(1_{\\lbrace X_i\\le X_j\\rbrace }-\\frac{1}{2}\\right)-\\frac{\\lfloor nr\\rfloor }{\\lfloor n\\lambda \\rfloor }\\sum \\limits _{i=1}^{\\lfloor n\\lambda \\rfloor }\\sum \\limits _{j=\\lfloor n\\lambda \\rfloor +1}^n\\left(1_{\\lbrace X_i\\le X_j\\rbrace }-\\frac{1}{2}\\right)=0$ for $r\\in \\left[\\lambda , \\frac{\\lfloor n\\lambda \\rfloor +1}{n}\\right)$ .", "Therefore, the integral over that interval equals 0.", "Consequently, $&\\int _0^{\\lambda } \\left(\\sum \\limits _{i=1}^{\\lfloor nr\\rfloor }\\sum \\limits _{j=\\lfloor nr\\rfloor +1}^n\\left(1_{\\lbrace X_i\\le X_j\\rbrace }-\\frac{1}{2}\\right)-\\frac{\\lfloor nr\\rfloor }{\\lfloor n\\lambda \\rfloor }\\sum \\limits _{i=1}^{\\lfloor n\\lambda \\rfloor }\\sum \\limits _{j=\\lfloor n\\lambda \\rfloor +1}^n\\left(1_{\\lbrace X_i\\le X_j\\rbrace }-\\frac{1}{2}\\right)\\right)^2dr\\\\&=\\sum \\limits _{t=0}^{\\lfloor n\\lambda \\rfloor }\\int _{\\frac{t}{n}}^{\\frac{t+1}{n}}\\left(\\sum \\limits _{i=1}^{\\lfloor nr\\rfloor }\\sum \\limits _{j=\\lfloor nr\\rfloor +1}^n\\left(1_{\\lbrace X_i\\le X_j\\rbrace }-\\frac{1}{2}\\right)-\\frac{\\lfloor nr\\rfloor }{\\lfloor n\\lambda \\rfloor }\\sum \\limits _{i=1}^{\\lfloor n\\lambda \\rfloor }\\sum \\limits _{j=\\lfloor n\\lambda \\rfloor +1}^n\\left(1_{\\lbrace X_i\\le X_j\\rbrace }-\\frac{1}{2}\\right)\\right)^2dr\\\\&=\\frac{1}{n}\\sum \\limits _{t=0}^{k}\\left(\\sum \\limits _{i=1}^{t}\\sum \\limits _{j= t+1}^n\\left(1_{\\lbrace X_i\\le X_j\\rbrace }-\\frac{1}{2}\\right)-\\frac{t}{k}\\sum \\limits _{i=1}^{k}\\sum \\limits _{j=k+1}^n\\left(1_{\\lbrace X_i\\le X_j\\rbrace }-\\frac{1}{2}\\right)\\right)^2\\\\&=\\frac{1}{n}\\sum \\limits _{t=1}^{k} S_t^2(1,k)$ almost surely in case $k=\\lfloor n\\lambda \\rfloor $ .", "For the second term in the denominator of $G_n(k)$ the following equations hold almost surely $S_t(k+1, n)=&-\\Biggr \\lbrace \\sum \\limits _{h=k+1}^t\\left(\\sum \\limits _{j=1}^n1_{\\lbrace X_h\\le X_j\\rbrace }-\\frac{1}{n-k}\\sum \\limits _{i=k+1}^n\\sum \\limits _{j=1}^n1_{\\lbrace X_i\\le X_j\\rbrace }\\right)\\Biggl \\rbrace \\\\=&-\\Biggr \\lbrace \\sum \\limits _{i=1}^t\\sum \\limits _{j=t+1}^n\\left(1_{\\lbrace X_i\\le X_j\\rbrace }-\\frac{1}{2}\\right)+\\sum \\limits _{i=1}^t\\sum \\limits _{j=1}^t\\left(1_{\\lbrace X_i\\le X_j\\rbrace }-\\frac{1}{2}\\right)\\\\&-\\sum \\limits _{i=1}^k\\sum \\limits _{j=k+1}^n\\left(1_{\\lbrace X_i\\le X_j\\rbrace }-\\frac{1}{2}\\right)-\\sum \\limits _{i=1}^k\\sum \\limits _{j=1}^k\\left(1_{\\lbrace X_i\\le X_j\\rbrace }-\\frac{1}{2}\\right)\\\\&-\\frac{t-k}{n-k}\\sum \\limits _{i=k+1}^n\\sum \\limits _{j=1}^k\\left(1_{\\lbrace X_i\\le X_j\\rbrace }-\\frac{1}{2}\\right)-\\frac{t-k}{n-k}\\sum \\limits _{i=k+1}^n\\sum \\limits _{j=k+1}^n\\left(1_{\\lbrace X_i\\le X_j\\rbrace }-\\frac{1}{2}\\right)\\Biggl \\rbrace .$ By (REF ) we get $\\sum \\limits _{i=k+1}^n\\sum \\limits _{j=k+1}^n\\left(1_{\\lbrace X_i\\le X_j\\rbrace }-\\frac{1}{2}\\right)=\\frac{(n-k)(n-k+1)}{2}-\\frac{(n-k)^2}{2}=\\frac{n-k}{2}.$ Furthermore, $1_{\\lbrace X_i\\le X_j\\rbrace }-\\frac{1}{2}=1-1_{\\lbrace X_j< X_i\\rbrace }-\\frac{1}{2}=-\\left(1_{\\lbrace X_j\\le X_i\\rbrace }-\\frac{1}{2}\\right)$ almost surely if $i\\ne j$ .", "This yields $S_t(k+1, n)=&-\\Biggr \\lbrace \\sum \\limits _{i=1}^t\\sum \\limits _{j=t+1}^n\\left(1_{\\lbrace X_i\\le X_j\\rbrace }-\\frac{1}{2}\\right)+\\frac{t}{2}-\\sum \\limits _{i=1}^k\\sum \\limits _{j=k+1}^n\\left(1_{\\lbrace X_i\\le X_j\\rbrace }-\\frac{1}{2}\\right)-\\frac{k}{2}\\\\&-\\frac{t-k}{n-k}\\sum \\limits _{i=k+1}^n\\sum \\limits _{j=1}^k\\left(1_{\\lbrace X_i\\le X_j\\rbrace }-\\frac{1}{2}\\right)-\\frac{t-k}{n-k}\\frac{n-k}{2}\\Biggl \\rbrace \\\\=&-\\Biggr \\lbrace \\sum \\limits _{i=1}^t\\sum \\limits _{j=t+1}^n\\left(1_{\\lbrace X_i\\le X_j\\rbrace }-\\frac{1}{2}\\right)-\\frac{n-t}{n-k}\\sum \\limits _{i=1}^k\\sum \\limits _{j=k+1}^n\\left(1_{\\lbrace X_i\\le X_j\\rbrace }-\\frac{1}{2}\\right)\\Biggl \\rbrace .$ We obtain for $\\lambda \\in \\left[\\tau _1, \\tau _2\\right]$ $&\\int _{\\lambda }^1 \\left(\\sum \\limits _{i=1}^{\\lfloor nr\\rfloor }\\sum \\limits _{j=\\lfloor nr\\rfloor +1}^n\\left(1_{\\lbrace X_i\\le X_j\\rbrace }-\\frac{1}{2}\\right)-\\frac{n-\\lfloor nr\\rfloor }{n-\\lfloor n\\lambda \\rfloor }\\sum \\limits _{i=1}^{\\lfloor n\\lambda \\rfloor }\\sum \\limits _{j=\\lfloor n\\lambda \\rfloor +1}^n\\left(1_{\\lbrace X_i\\le X_j\\rbrace }-\\frac{1}{2}\\right)\\right)^2dr\\\\&=\\sum \\limits _{t=\\lfloor n\\lambda \\rfloor +1}^{n-1}\\int _{\\frac{t}{n}}^{\\frac{t+1}{n}}\\left(\\sum \\limits _{i=1}^{\\lfloor nr\\rfloor }\\sum \\limits _{j=\\lfloor nr\\rfloor +1}^n\\left(1_{\\lbrace X_i\\le X_j\\rbrace }-\\frac{1}{2}\\right)-\\frac{n-\\lfloor nr\\rfloor }{n-\\lfloor n\\lambda \\rfloor }\\sum \\limits _{i=1}^{\\lfloor n\\lambda \\rfloor }\\sum \\limits _{j=\\lfloor n\\lambda \\rfloor +1}^n\\left(1_{\\lbrace X_i\\le X_j\\rbrace }-\\frac{1}{2}\\right)\\right)^2dr\\\\&\\quad \\ +\\int _{\\lambda }^{\\frac{\\lfloor n\\lambda \\rfloor +1}{n}}\\left(\\sum \\limits _{i=1}^{\\lfloor nr\\rfloor }\\sum \\limits _{j=\\lfloor nr\\rfloor +1}^n\\left(1_{\\lbrace X_i\\le X_j\\rbrace }-\\frac{1}{2}\\right)-\\frac{n-\\lfloor nr\\rfloor }{n-\\lfloor n\\lambda \\rfloor }\\sum \\limits _{i=1}^{\\lfloor n\\lambda \\rfloor }\\sum \\limits _{j=\\lfloor n\\lambda \\rfloor +1}^n\\left(1_{\\lbrace X_i\\le X_j\\rbrace }-\\frac{1}{2}\\right)\\right)^2dr$ almost surely, where $\\sum \\limits _{i=1}^{\\lfloor nr\\rfloor }\\sum \\limits _{j=\\lfloor nr\\rfloor +1}^n\\left(1_{\\lbrace X_i\\le X_j\\rbrace }-\\frac{1}{2}\\right)-\\frac{n-\\lfloor nr\\rfloor }{n-\\lfloor n\\lambda \\rfloor }\\sum \\limits _{i=1}^{\\lfloor n\\lambda \\rfloor }\\sum \\limits _{j=\\lfloor n\\lambda \\rfloor +1}^n\\left(1_{\\lbrace X_i\\le X_j\\rbrace }-\\frac{1}{2}\\right)=0$ if $r\\in \\left[\\lambda , \\frac{\\lfloor n\\lambda \\rfloor +1}{n}\\right)$ .", "Therefore, the integral over that interval equals 0.", "For $k=\\lfloor n\\lambda \\rfloor $ this implies $&\\int _{\\lambda }^1 \\left(\\sum \\limits _{i=1}^{\\lfloor nr\\rfloor }\\sum \\limits _{j=\\lfloor nr\\rfloor +1}^n\\left(1_{\\lbrace X_i\\le X_j\\rbrace }-\\frac{1}{2}\\right)-\\frac{n-\\lfloor nr\\rfloor }{n-\\lfloor n\\lambda \\rfloor }\\sum \\limits _{i=1}^{\\lfloor n\\lambda \\rfloor }\\sum \\limits _{j=\\lfloor n\\lambda \\rfloor +1}^n\\left(1_{\\lbrace X_i\\le X_j\\rbrace }-\\frac{1}{2}\\right)\\right)^2dr\\\\&=\\frac{1}{n}\\sum \\limits _{t=k +1}^{n-1}\\left(\\sum \\limits _{i=1}^{t}\\sum \\limits _{j=t+1}^n\\left(1_{\\lbrace X_i\\le X_j\\rbrace }-\\frac{1}{2}\\right)-\\frac{n-t}{n-k}\\sum \\limits _{i=1}^{k}\\sum \\limits _{j=k+1}^n\\left(1_{\\lbrace X_i\\le X_j\\rbrace }-\\frac{1}{2}\\right)\\right)^2\\\\&=\\frac{1}{n}\\sum \\limits _{t=k+1}^{n-1}S_{t}^2(k+1,n)\\\\&=\\frac{1}{n}\\sum \\limits _{t=k+1}^{n}S_{t}^2(k+1,n).$ Due to the previous considerations, the properly normalized denominator of $G_n(k)$ can (almost surely) be represented as follows $&\\frac{1}{nd_n}\\left\\lbrace \\frac{1}{n}\\sum \\limits _{t=1}^{k}S_t^2(1, k)+\\frac{1}{n}\\sum \\limits _{t=k+1}^n S_t^2(k+1,n)\\right\\rbrace ^\\frac{1}{2}\\\\&=\\left\\lbrace \\int _0^{\\lambda }\\left(W_n(r)-\\frac{c_n(r)}{c_n(\\lambda )}W_n(\\lambda )\\right)^2dr+\\int _{\\lambda }^1\\left(W_n(r)-\\frac{1-c_n(r)}{1-c_n(\\lambda )} W_n(\\lambda )\\right)^2dr\\right\\rbrace ^{\\frac{1}{2}},$ where $c_n(\\lambda )=\\frac{\\lfloor n\\lambda \\rfloor }{n}$ for $\\lambda \\in \\left[0, 1\\right]$ .", "All in all, this yields $T_n(\\tau _1, \\tau _2)=\\sup \\limits _{\\lambda \\in \\left[\\tau _1, \\tau _2\\right]}\\frac{\\left\|W_n(\\lambda )\\right\|}{\\left\\lbrace \\int _0^{\\lambda }\\left(W_n(r)-\\frac{c_n(r)}{c_n(\\lambda )}W_n(\\lambda )\\right)^2dr+\\int _{\\lambda }^1\\left(W_n(r)-\\frac{1-c_n(r)}{1-c_n(\\lambda )} W_n(\\lambda )\\right)^2dr\\right\\rbrace ^{\\frac{1}{2}}}.$ The foregoing characterization of the self-normalized Wilcoxon test statistic points out that a representation of $T_n(\\tau _1, \\tau _2)$ as a functional of the process $W_n(\\lambda )=\\frac{1}{n d_n}\\sum \\limits _{i=1}^{\\lfloor n\\lambda \\rfloor }\\sum \\limits _{j=\\lfloor n\\lambda \\rfloor +1}^n\\left(1_{\\left\\lbrace X_i\\le X_j\\right\\rbrace }-\\frac{1}{2}\\right), \\ 0\\le \\lambda \\le 1,$ also depends on the function series $\\left(c_n\\right)_{n\\in \\mathbb {N}}$ in $D\\left[0, 1\\right]$ defined by $c_n(\\lambda )=\\frac{\\lfloor n\\lambda \\rfloor }{n}, \\ 0\\le \\lambda \\le 1$ .", "Since $\\sup \\limits _{\\lambda \\in \\left[0, 1\\right]}\\left\|\\frac{\\lfloor n\\lambda \\rfloor }{n}-\\lambda \\right\|=\\sup \\limits _{\\lambda \\in \\left[0, 1\\right]}\\left(\\lambda -\\frac{\\lfloor n\\lambda \\rfloor }{n}\\right)\\le \\sup \\limits _{\\lambda \\in \\left[0, 1\\right]}\\left(\\lambda -\\frac{ n\\lambda -1}{n}\\right)=\\frac{1}{n}\\longrightarrow 0,$ the sequence $c_n$ , $n\\in \\mathbb {N}$ , converges with respect to the supremum norm to $c\\in D\\left[0, 1\\right]$ defined by $c(\\lambda )=\\lambda $ for $\\lambda \\in \\left[0, 1\\right]$ .", "To simplify subsequent calculations, we treat $c_n$ and $c$ as random variables with values in the closure of $M=\\left\\lbrace f\\in D\\left[0, 1\\right]\\left\|\\right.", "f(\\lambda )=\\frac{\\lfloor n\\lambda \\rfloor }{n} \\ \\text{for some } n\\in \\mathbb {N}, \\ n\\ge \\frac{1}{\\tau _1}\\right\\rbrace .$ Note that $h_n=\\left(\\begin{array}{c}c_n\\\\W_n\\end{array}\\right)\\overset{\\mathcal {D}}{\\longrightarrow }\\left(\\begin{array}{c}c\\\\W_m^\\end{array}\\right),$ where $W_m^(\\lambda )=(Z(\\lambda )-\\lambda Z(1))\\int _{\\mathbb {R}}J(x)dF(x), \\ 0\\le \\lambda \\le 1.$ Obviously, the self-normalized Wilcoxon test statistic can be represented as a functional of the random vector $h_n$ .", "Hence, an application of the continuous mapping theorem just requires the definition of an appropriate function $G:\\overline{M}\\times D\\left[0, 1\\right]\\longrightarrow \\mathbb {R}$ that maps $h_n$ on $T_n(\\tau _1, \\tau _2)=G(h_n)$ .", "For $\\lambda \\in \\left[\\tau _1, \\tau _2\\right]$ consider the function $G_{\\lambda }:\\overline{M}\\times D\\left[0, 1\\right]\\longrightarrow \\mathbb {R}$ that maps an element $h=\\left(h_1, h_2\\right)$ on $\\frac{\\left\|h_2(\\lambda )\\right\|}{\\left\\lbrace \\int _0^{\\lambda }\\left(h_2(r)-\\frac{h_1(r)}{h_1(\\lambda )}h_2(\\lambda )\\right)^2dr+\\int _{\\lambda }^1\\left(h_2(r)-\\frac{1 -h_1(r)}{1-h_1(\\lambda )}h_2(\\lambda )\\right)^2dr\\right\\rbrace ^{\\frac{1}{2}}},$ provided that the function $F:\\overline{M}\\times D\\left[0, 1\\right]\\longrightarrow \\mathbb {R}$ defined by $F(h)=\\inf \\limits _{\\lambda \\in \\left[\\tau _1, \\tau _2\\right]}\\left\\lbrace \\int _0^{\\lambda }\\left(h_2(r)-\\frac{h_1(r)}{h_1(\\lambda )}h_2(\\lambda )\\right)^2dr+\\int _{\\lambda }^1\\left(h_2(r)-\\frac{1 -h_1(r)}{1-h_1(\\lambda )}h_2(\\lambda )\\right)^2dr\\right\\rbrace ^{\\frac{1}{2}}$ does not equal 0 in $h$ .", "Given that $h\\in F^{-1}\\left(\\left\\lbrace 0\\right\\rbrace \\right)$ , we set $G_{\\lambda }(h)=-1$ .", "Since $T_n(\\tau _1, \\tau _2)=\\sup _{\\lambda \\in \\left[\\tau _1, \\tau _2\\right]}G_{\\lambda }(h_n)$ , we intend to apply the continuous mapping theorem to the function $G: \\overline{M}\\times D\\left[0, 1\\right]\\longrightarrow \\mathbb {R}$ , where $G(h)=\\sup _{\\lambda \\in \\left[\\tau _1, \\tau _2\\right]}G_{\\lambda }(h)$ .", "Thus, we have to verify that the function $G$ complies with the requirements of the continuous mapping theorem, i.e.", "we have to prove the following assertions: The function $G$ is measurable with respect to the uniform product metric on $\\overline{M}\\times D\\left[0, 1\\right]$ .", "We have $P(h \\in D_G)=0$ , where $D_G$ denotes the set of discontinuities of $G$ .", "In order to show that $G$ is measurable, we consider the restrictions of $G$ to $\\left(\\overline{M}\\times D\\left[0, 1\\right]\\right)\\setminus F^{-1}(\\lbrace 0\\rbrace )$ and $F^{-1}(\\lbrace 0\\rbrace )$ , respectively.", "Both restrictions are continuous with respect to the uniform metric.", "In particular, both restrictions are Borel measurable.", "Since the restricted domains are Borel measurable subsets of $\\overline{M}\\times D\\left[0, 1\\right]$ , the measurability of the restrictions implies the measurability of $G$ .", "It remains to show that $P(h \\in D_G)=0$ .", "Again, consider the restriction of $G$ to $\\left(\\overline{M}\\times D\\left[0, 1\\right]\\right)\\setminus F^{-1}(\\lbrace 0\\rbrace )$ .", "Because of the continuity of the restriction, $G$ is continuous at every $h\\in \\left(\\overline{M}\\times D\\left[0, 1\\right]\\right)\\setminus F^{-1}(\\lbrace 0\\rbrace )$ as $F^{-1}(\\lbrace 0\\rbrace )$ is a closed subset of $\\overline{M}\\times D\\left[0, 1\\right]$ .", "Therefore, $D_G$ is a subset of $F^{-1}(\\lbrace 0\\rbrace )$ .", "Consequently, it suffices to show that $P(h\\in F^{-1}(\\lbrace 0\\rbrace ))=0$ in order to prove that $P(h\\in D_G)=0$ .", "The random vector $h=(c, W_m^)$ is an element of $F^{-1}(\\lbrace 0\\rbrace )$ if and only if the expression $\\inf \\limits _{\\lambda \\in \\left[\\tau _1, \\tau _2\\right]}\\left\\lbrace \\int _0^{\\lambda }\\left(f(r)-\\frac{r}{\\lambda }f(\\lambda )\\right)^2dr+\\int _{\\lambda }^1\\left(f(r)-\\frac{1-r}{1-\\lambda }f(\\lambda )\\right)^2dr\\right\\rbrace ^{\\frac{1}{2}}$ vanishes when $f=W_m^$ .", "Note that $W_m^(r)-\\frac{r}{\\lambda }W_m^(\\lambda )&=\\int J(x)dF(x)\\left\\lbrace \\left(Z(r)-rZ(1)\\right)-\\frac{r}{\\lambda }\\left(Z(\\lambda )-\\lambda Z(1)\\right)\\right\\rbrace \\\\&=\\int J(x)dF(x)\\left\\lbrace Z(r)-\\frac{r}{\\lambda }Z(\\lambda )\\right\\rbrace $ and $W_m^(r)-\\frac{1-r}{1-\\lambda }W_m^(\\lambda )&=\\int J(x)dF(x)\\left\\lbrace \\left(Z(r)-rZ(1)\\right)-\\frac{1-r}{1-\\lambda }\\left(Z(\\lambda )-\\lambda Z(1)\\right)\\right\\rbrace \\\\&=\\int J(x)dF(x)\\left\\lbrace Z(r)-Z(\\lambda )-\\frac{r-\\lambda }{1-\\lambda }\\left(Z(1)-Z(\\lambda )\\right)\\right\\rbrace .$ Therefore, and as $Z \\in C\\left[0, 1\\right]$ almost surely (see [7]), the term in formula (REF ) vanishes if for some $\\lambda \\in \\left[\\tau _1, \\tau _2\\right]$ $\\left\\lbrace \\int _0^{\\lambda }\\left(V_m(r; 0, \\lambda )\\right)^2dr+\\int _{\\lambda }^1\\left(V_m(r; \\lambda , 1)\\right)^2dr\\right\\rbrace ^{\\frac{1}{2}}=0,$ where $V_m(r; r_1, r_2)=Z(r)-Z(r_1)-\\frac{r-r_1}{r_2-r_1}\\left(Z(r_2)-Z(r_1)\\right).$ It suffices to show that the sample paths of $W_m^$ do not belong to the set of continuous functions $f$ that satisfy $\\left\\lbrace \\int _0^{\\lambda }\\left(f(r)-\\frac{r}{\\lambda }f(\\lambda )\\right)^2dr+\\int _{\\lambda }^1\\left(f(r)-f(\\lambda )-\\frac{r-\\lambda }{1-\\lambda }\\left(f(1)-f(\\lambda )\\right)\\right)^2dr\\right\\rbrace ^{\\frac{1}{2}}=0$ for some $\\lambda \\in \\left[\\tau _1, \\tau _2\\right]$ .", "The above equation only holds if the integrands vanish almost surely on the corresponding intervals.", "In particular, a continuous function $f \\in D\\left[0, 1\\right]$ that meets formula (REF ) satisfies $f(r)=\\frac{1}{\\lambda }f(\\lambda )r$ if $r\\in \\left[0, \\lambda \\right]$ and $f(r)&=f(\\lambda )+\\frac{r-\\lambda }{1-\\lambda }\\left\\lbrace f(1)-f(\\lambda )\\right\\rbrace \\\\&=f(\\lambda )-\\frac{\\lambda }{1-\\lambda }\\left\\lbrace f(1)-f(\\lambda )\\right\\rbrace +\\frac{1}{1-\\lambda }\\left\\lbrace f(1)-f(\\lambda )\\right\\rbrace r$ if $r\\in \\left[\\lambda , 1\\right]$ .", "Consequently, the set of continuous functions which lie in $F^{-1}(\\lbrace 0\\rbrace )$ corresponds to the class of functions $A = & \\Bigl \\lbrace f\\in D\\left[0, 1\\right]\\left\|\\right.", "\\ \\text{for some $\\lambda \\in [\\tau _1, \\tau _2]$ and $a, b \\in \\mathbb {R}$}\\\\&f(r)=\\frac{1}{\\lambda }a r \\ \\text{on $[0, \\lambda ]$} \\ \\text{ and} \\\\&f(r)=a-\\frac{\\lambda }{1-\\lambda }\\left\\lbrace b-a\\right\\rbrace +\\frac{1}{1-\\lambda }\\left\\lbrace b-a\\right\\rbrace r \\ \\text{on $[\\lambda , 1]$}\\Bigr \\rbrace .$ It follows that $P(Z\\in A)=0$ because the sample paths of the Hermite process $Z$ are nowhere differentiable with probability 1 (see [8]), whereas an element in $A$ is differentiable almost everywhere.", "This implies $P(h\\in D_G)=0$ .", "Having verified the preconditions of the continuous mapping theorem we are now able to conclude that the test statistic $T_n(\\tau _1, \\tau _2)$ converges in distribution to $T(m,\\tau _1, \\tau _2)=\\sup \\limits _{\\lambda \\in \\left[\\tau _1, \\tau _2\\right]}\\frac{\\left\|W_m^(\\lambda )\\right\|}{\\left\\lbrace \\int _0^{\\lambda }\\left(W_m^(r)-\\frac{r}{\\lambda }W_m^(\\lambda )\\right)^2dr+\\int _{\\lambda }^1\\left(W_m^(r)-\\frac{1-r}{1-\\lambda } W_m^(\\lambda )\\right)^2dr\\right\\rbrace ^{\\frac{1}{2}}}.$ Due to (REF ) and (REF ), the limit process $T(m, \\tau _1, \\tau _2)$ equals $&\\sup \\limits _{\\lambda \\in \\left[\\tau _1, \\tau _2\\right]}\\frac{ \\left\|Z(\\lambda )-\\lambda Z(1)\\right\|}{\\Bigl \\lbrace \\int _0^{\\lambda }\\left(V_m(r; 0, \\lambda )\\right)^2dr+\\int _{\\lambda }^1 \\left(V_m(r; \\lambda , 1)\\right)^2dr\\Bigr \\rbrace ^{\\frac{1}{2}}}.$ Thus, we have established Theorem REF .", "$ \\Box $ In the proof of Theorem REF we make use of preliminary results stated in Lemma REF , Lemma REF and Corollary REF .", "The line of argument that verifies Lemma REF and Lemma REF is a modification of the proof that establishes Theorem 3.1 in [3].", "Lemma 1 Suppose that $\\left(\\xi _i\\right)_{i\\ge 1}$ is a stationary, long-range dependent Gaussian process with mean 0, variance 1 and LRD parameter $0<D <\\frac{1}{m}$ , where $m$ denotes the Hermite rank of the class of functions $1_{\\left\\lbrace G(\\xi _i)\\le x\\right\\rbrace }-F(x)$ , $x \\in \\mathbb {R}$ .", "Moreover, assume that $\\left(G(\\xi _i)\\right)_{i\\ge 1}$ has a continuous distribution function $F$ and that $G:\\mathbb {R}\\longrightarrow \\mathbb {R}$ is a measurable function.", "Then, if $\\Delta \\in \\mathbb {R}$ , $&\\frac{1}{n^2}\\sum \\limits _{i=1}^{\\lfloor n\\lambda \\rfloor }\\sum \\limits _{j=\\lfloor n\\tau \\rfloor +1}^n1_{\\left\\lbrace G(\\xi _i)\\le G(\\xi _j)+\\Delta \\right\\rbrace }\\overset{P}{\\longrightarrow }\\lambda (1-\\tau )\\int _{\\mathbb {R}}F(x+\\Delta )dF(x),\\\\&\\frac{1}{n^2}\\sum \\limits _{i=1}^{\\lfloor n\\tau \\rfloor }\\sum \\limits _{j=\\lfloor n\\lambda \\rfloor +1}^n1_{\\left\\lbrace G(\\xi _i)\\le G(\\xi _j)+\\Delta \\right\\rbrace }\\overset{P}{\\longrightarrow }\\tau (1-\\lambda )\\int _{\\mathbb {R}}F(x+\\Delta )dF(x)$ for fixed $\\tau $ , uniformly in $\\lambda \\le \\tau $ and $\\lambda \\ge \\tau $ , respectively.", "Proof of Lemma REF .", "We give a proof for the first assertion only as the convergence of the second term follows by an analogous argumentation.", "Let $F_k$ and $F_{k+1, n}$ denote the empirical distribution functions of the first $k$ and last $n-k$ realizations of $G(\\xi _1), \\ldots , G(\\xi _n)$ , i.e.", "$&F_k(x)=\\frac{1}{k}\\sum \\limits _{i=1}^k1_{\\left\\lbrace G(\\xi _i)\\le x\\right\\rbrace },\\\\&F_{k+1, n}(x)=\\frac{1}{n-k}\\sum \\limits _{i=k+1}^n1_{\\left\\lbrace G(\\xi _i)\\le x\\right\\rbrace }.$ For $\\lambda \\le \\tau $ this yields the following representation: $\\sum \\limits _{i=1}^{\\lfloor n\\lambda \\rfloor }\\sum \\limits _{j=\\lfloor n\\tau \\rfloor +1}^n1_{\\left\\lbrace G(\\xi _i)\\le G(\\xi _j)+\\Delta \\right\\rbrace }&=\\left(n-\\lfloor n\\tau \\rfloor \\right)\\lfloor n\\lambda \\rfloor \\frac{1}{n-\\lfloor n\\tau \\rfloor }\\sum \\limits _{j=\\lfloor n\\tau \\rfloor +1}^nF_{\\lfloor n\\lambda \\rfloor }( G(\\xi _j)+\\Delta )\\\\&=\\left(n-\\lfloor n\\tau \\rfloor \\right)\\lfloor n\\lambda \\rfloor \\int _{\\mathbb {R}}F_{\\lfloor n\\lambda \\rfloor }( x+\\Delta )dF_{\\lfloor n \\tau \\rfloor +1, n}(x)$ Since $\\frac{n-\\lfloor n\\tau \\rfloor }{n}\\longrightarrow 1-\\tau $ , it suffices to show that $\\lfloor n\\lambda \\rfloor \\int _{\\mathbb {R}}F_{\\lfloor n\\lambda \\rfloor }( x+\\Delta )dF_{\\lfloor n \\tau \\rfloor +1, n}(x)$ converges to $\\lambda \\int _{\\mathbb {R}}F(x+\\Delta )dF(x)$ .", "For this purpose, we consider the inequality $&\\sup \\limits _{0\\le \\lambda \\le \\tau }\\left\|\\frac{1}{n}\\lfloor n\\lambda \\rfloor \\int _{\\mathbb {R}}F_{\\lfloor n\\lambda \\rfloor }(x+\\Delta )dF_{\\lfloor n\\tau \\rfloor +1, n}(x)-\\lambda \\int _{\\mathbb {R}}F(x+\\Delta )dF(x)\\right\|\\\\&\\le \\sup \\limits _{0\\le \\lambda \\le \\tau }\\left\|\\frac{1}{n}\\int _{\\mathbb {R}}\\lfloor n\\lambda \\rfloor F_{\\lfloor n\\lambda \\rfloor }(x+\\Delta )dF_{\\lfloor n\\tau \\rfloor +1, n}(x)-\\frac{\\lfloor n\\lambda \\rfloor }{n}\\int _{\\mathbb {R}}F(x+\\Delta )dF_{\\lfloor n\\tau \\rfloor +1, n}(x)\\right\|\\\\&\\quad \\ +\\sup \\limits _{0\\le \\lambda \\le \\tau }\\left\|\\frac{\\lfloor n\\lambda \\rfloor }{n}\\int _{\\mathbb {R}}F(x+\\Delta )dF_{\\lfloor n\\tau \\rfloor +1, n}(x)-\\lambda \\int _{\\mathbb {R}}F(x+\\Delta )dF_{\\lfloor n\\tau \\rfloor +1, n}(x)\\right\|\\\\&\\quad \\ +\\sup \\limits _{0\\le \\lambda \\le \\tau }\\left\|\\lambda \\int _{\\mathbb {R}}F(x+\\Delta )dF_{\\lfloor n\\tau \\rfloor +1, n}(x)-\\lambda \\int _{\\mathbb {R}}F(x+\\Delta )dF(x)\\right\| $ and we will show that each of the three terms on its right-hand side converges to 0.", "For the third summand we get $&\\sup \\limits _{0\\le \\lambda \\le \\tau }\\left\|\\lambda \\int _{\\mathbb {R}}F(x+\\Delta )dF_{\\lfloor n\\tau \\rfloor +1, n}(x)-\\lambda \\int _{\\mathbb {R}}F(x+\\Delta )dF(x)\\right\|\\\\&=\\sup \\limits _{0\\le \\lambda \\le \\tau }\\left\|\\lambda \\left(1- \\int _{\\mathbb {R}}F_{\\lfloor n\\tau \\rfloor +1, n}(x)dF(x+\\Delta )- \\left(1-\\int _{\\mathbb {R}}F(x)dF(x+\\Delta )\\right)\\right)\\right\|\\\\&=\\tau \\left\| \\int _{\\mathbb {R}}\\left(F_{\\lfloor n\\tau \\rfloor +1, n}(x)- F(x)\\right)dF(x+\\Delta )\\right\|\\\\&\\le \\tau \\sup \\limits _{x\\in \\mathbb {R}}\\left\| F_{\\lfloor n\\tau \\rfloor +1, n}(x)- F(x)\\right\|$ as a consequence of integration by parts.", "Furthermore, we have $\\sup \\limits _{x\\in \\mathbb {R}}\\left\|F_n(x)-F(x)\\right\|\\longrightarrow 0 \\ a.s.$ by an application of the Glivenko-Cantelli theorem (see [5]) to the stationary and ergodic process $\\left(G(\\xi _i)\\right)_{i\\ge 1}$ .", "So as to deduce an analogous result for $F_{\\lfloor n\\tau \\rfloor +1, n}$ we rewrite $F_{\\lfloor n\\tau \\rfloor +1, n}(x)=\\frac{n}{n-\\lfloor n\\tau \\rfloor }F_n(x)- \\frac{\\lfloor n\\tau \\rfloor }{n-\\lfloor n\\tau \\rfloor }F_{\\lfloor n\\tau \\rfloor }(x)$ and we may therefore conclude $\\left\|F_{\\lfloor n\\tau \\rfloor +1, n}(x)-F(x)\\right\|&\\le \\left\|\\frac{n}{n-\\lfloor n\\tau \\rfloor }\\right\|\\left\|F_{n}(x)-F(x)\\right\|+\\left\|\\frac{\\lfloor n\\tau \\rfloor }{n-\\lfloor n\\tau \\rfloor }\\right\|\\left\|F_{\\lfloor n\\tau \\rfloor }(x)-F(x)\\right\|.$ Thus, $\\sup \\limits _{x\\in \\mathbb {R}}\\left\|F_{\\lfloor n\\tau \\rfloor +1, n}(x)-F(x)\\right\|\\longrightarrow 0 \\ a.s.$ which implies that the third term on the right-hand side of (REF ) converges to 0 almost surely.", "Regarding the second term on the right-hand side of (REF ), we obtain $&\\sup \\limits _{0\\le \\lambda \\le \\tau }\\left\|\\frac{\\lfloor n\\lambda \\rfloor }{n}\\int _{\\mathbb {R}}F(x+\\Delta )dF_{\\lfloor n\\tau \\rfloor +1, n}(x)-\\lambda \\int _{\\mathbb {R}}F(x+\\Delta )dF_{\\lfloor n\\tau \\rfloor +1, n}(x)\\right\|\\\\&=\\sup \\limits _{0\\le \\lambda \\le \\tau }\\left\|\\frac{\\lfloor n\\lambda \\rfloor }{n}-\\lambda \\right\|\\left\|\\int _{\\mathbb {R}}F(x+\\Delta )dF_{\\lfloor n\\tau \\rfloor +1, n}(x)\\right\|.$ The right-hand side of this equation converges to 0 since $\\left\|\\int _{\\mathbb {R}}F(x+\\Delta )dF_{\\lfloor n\\tau \\rfloor +1, n}(x)\\right\|$ is bounded by 1, and as $\\sup \\limits _{0\\le \\lambda \\le \\tau }\\left\|\\frac{\\lfloor n\\lambda \\rfloor }{n}-\\lambda \\right\|\\longrightarrow 0.$ In order to show that the first term in (REF ) converges to 0 as well, we consider the following inequality: $&\\sup \\limits _{0\\le \\lambda \\le \\tau }\\left\|\\frac{1}{n}\\int _{\\mathbb {R}}\\lfloor n\\lambda \\rfloor F_{\\lfloor n\\lambda \\rfloor }(x+\\Delta )dF_{\\lfloor n\\tau \\rfloor +1, n}(x)-\\frac{\\lfloor n\\lambda \\rfloor }{n}\\int _{\\mathbb {R}}F(x+\\Delta )dF_{\\lfloor n\\tau \\rfloor +1, n}(x)\\right\|\\\\&=\\sup \\limits _{0\\le \\lambda \\le \\tau }\\left\|\\frac{1}{n}\\int _{\\mathbb {R}}\\lfloor n\\lambda \\rfloor \\left(F_{\\lfloor n\\lambda \\rfloor }(x+\\Delta )-F(x+\\Delta )\\right)dF_{\\lfloor n\\tau \\rfloor +1, n}(x)\\right\|\\\\&\\le \\sup \\limits _{0\\le \\lambda \\le \\tau }\\left\|\\frac{d_n}{n}\\int _{\\mathbb {R}}d_n^{-1}\\lfloor n\\lambda \\rfloor \\left(F_{\\lfloor n\\lambda \\rfloor }(x+\\Delta )-F(x+\\Delta )\\right)-J(x+\\Delta )Z(\\lambda )dF_{\\lfloor n\\tau \\rfloor +1, n}(x)\\right\|\\\\&\\quad \\ +\\sup \\limits _{0\\le \\lambda \\le \\tau }\\left\|\\frac{d_n}{n}Z(\\lambda )\\int _{\\mathbb {R}}J(x+\\Delta )dF_{\\lfloor n\\tau \\rfloor +1, n}(x)\\right\|$ In what follows, we will prove that both terms on the right-hand side of (REF ) converge to 0.", "For this purpose, we make use of the empirical process non-central limit theorem of [4] which states that $\\left(d_n^{-1}\\lfloor n\\lambda \\rfloor (F_{\\lfloor n\\lambda \\rfloor }(x)-F(x))\\right)_{x\\in \\left[-\\infty , \\infty \\right], \\lambda \\in \\left[0, 1\\right]}\\overset{\\mathcal {D}}{\\longrightarrow }J(x)Z(\\lambda ),$ where $\\overset{\\mathcal {D}}{\\longrightarrow }$ denotes convergence in distribution with respect to the $\\sigma $ -field generated by the open balls in $D\\left(\\left[-\\infty , \\infty \\right]\\times \\left[0, 1\\right]\\right)$ , equipped with the supremum norm.", "Due to the Dudley-Wichura version of Skorohod's representation theorem (see [11], Theorem 2.3.4), we may assume without loss of generality that $\\sup \\limits _{\\lambda , x}\\left\|d_n^{-1}\\lfloor n\\lambda \\rfloor \\left(F_{\\lfloor n\\lambda \\rfloor }(x)-F(x)\\right)-J(x)Z(\\lambda )\\right\|\\longrightarrow 0$ almost surely; see [2].", "As a consequence, the first summand in (REF ) converges to 0 since $&\\sup \\limits _{0\\le \\lambda \\le \\tau }\\left\|\\frac{d_n}{n}\\int _{\\mathbb {R}}d_n^{-1}\\lfloor n\\lambda \\rfloor \\left(F_{\\lfloor n\\lambda \\rfloor }(x+\\Delta )-F(x+\\Delta )\\right)-J(x+\\Delta )Z(\\lambda )dF_{\\lfloor n\\tau \\rfloor +1, n}(x)\\right\|\\\\&=\\frac{d_n}{n}\\sup \\limits _{0\\le \\lambda \\le \\tau , x}\\left\|d_n^{-1}\\lfloor n\\lambda \\rfloor \\left(F_{\\lfloor n\\lambda \\rfloor }(x+\\Delta )-F(x+\\Delta )\\right)-J(x+\\Delta )Z(\\lambda )\\right\|$ and as $\\frac{d_n}{n}$ converges to 0 as well.", "For the second summand we get the following inequality: $\\sup \\limits _{0\\le \\lambda \\le \\tau }\\left\|\\frac{d_n}{n}Z(\\lambda )\\int _{\\mathbb {R}}J(x+\\Delta )dF_{\\lfloor n\\tau \\rfloor +1, n}(x)\\right\|\\le \\frac{d_n}{n}\\sup \\limits _{0\\le \\lambda \\le \\tau }\\left\|Z(\\lambda )\\right\|\\left\|\\int _{\\mathbb {R}}J(x+\\Delta )dF_{\\lfloor n\\tau \\rfloor +1, n}(x)\\right\|$ Note that $J(x)&=\\int _{\\mathbb {R}}1_{\\left\\lbrace G(y)\\le x\\right\\rbrace }H_m(y)\\varphi (y)dy\\\\&=\\int _{\\mathbb {R}}H_m(y)\\varphi (y)dy-\\int _{\\mathbb {R}}1_{\\left\\lbrace x\\le G(y)\\right\\rbrace }H_m(y)\\varphi (y)dy\\\\&=-\\int _{\\mathbb {R}}1_{\\left\\lbrace x\\le G(y)\\right\\rbrace }H_m(y)\\varphi (y)dy,$ where $\\varphi $ denotes the standard normal density function, since $\\int _{\\mathbb {R}}H_m(y)\\varphi (y)dy=0.$ For this reason, we have $\\int _{\\mathbb {R}}J(x+\\Delta )dF_{\\lfloor n\\tau \\rfloor +1, n}(x)&=-\\int _{\\mathbb {R}}\\int _{\\mathbb {R}}1_{\\left\\lbrace x+\\Delta \\le G(y)\\right\\rbrace }H_m(y)\\varphi (y)dy dF_{\\lfloor n\\tau \\rfloor +1, n}(x) \\\\&=-\\int _{\\mathbb {R}}\\int _{\\mathbb {R}}1_{\\left\\lbrace x+\\Delta \\le G(y)\\right\\rbrace }dF_{\\lfloor n\\tau \\rfloor +1, n}(x)H_m(y)\\varphi (y)dy\\\\&=-\\int _{\\mathbb {R}}F_{\\lfloor n\\tau \\rfloor +1, n}(G(y)-\\Delta )H_m(y)\\varphi (y)dy$ and $\\int _{\\mathbb {R}}J(x+\\Delta )dF(x)&=-\\int _{\\mathbb {R}}\\int _{\\mathbb {R}}1_{\\left\\lbrace x+\\Delta \\le G(y)\\right\\rbrace }H_m(y)\\varphi (y)dy dF(x) \\\\&=-\\int _{\\mathbb {R}}\\int _{\\mathbb {R}}1_{\\left\\lbrace x+\\Delta \\le G(y)\\right\\rbrace }dF(x)H_m(y)\\varphi (y)dy\\\\&=-\\int _{\\mathbb {R}}F(G(y)-\\Delta )H_m(y)\\varphi (y)dy.$ Regarding the difference of these terms, we obtain $&\\left\|\\int _{\\mathbb {R}}J(x+\\Delta )dF_{\\lfloor n\\tau \\rfloor +1, n}(x)-\\int _{\\mathbb {R}}J(x+\\Delta )dF(x)\\right\|\\\\&=\\left\|\\int _{\\mathbb {R}}\\left(F(G(y)-\\Delta )-F_{\\lfloor n\\tau \\rfloor +1, n}(G(y)-\\Delta )\\right)H_m(y)\\varphi (y)dy\\right\|\\\\&\\le \\int _{\\mathbb {R}}\\left\|F(G(y)-\\Delta )-F_{\\lfloor n\\tau \\rfloor +1, n}(G(y)-\\Delta )\\right\|\\left\|H_m(y)\\right\|\\varphi (y)dy\\\\&\\le \\sup \\limits _{y\\in \\mathbb {R}}\\left\|F(G(y)-\\Delta )-F_{\\lfloor n\\tau \\rfloor +1, n}(G(y)-\\Delta )\\right\|\\int _{\\mathbb {R}}\\left\|H_m(y)\\right\|\\varphi (y)dy,$ where $\\int _{\\mathbb {R}}\\left\|H_m(y)\\right\|\\varphi (y)dy<\\infty $ because of Hölder's inequality and where $\\sup _{y\\in \\mathbb {R}}\\left\|F(G(y)-\\Delta )-F_{\\lfloor n\\tau \\rfloor +1, n}(G(y)-\\Delta )\\right\|\\longrightarrow 0 \\ a.s.$ by (REF ).", "As a result, $\\int _{\\mathbb {R}}J(x+\\Delta )dF_{\\lfloor n\\tau \\rfloor +1, n}(x)\\overset{\\mathcal {D}}{\\longrightarrow }\\int _{\\mathbb {R}}J(x+\\Delta )dF(x)$ , so that in the end the second summand in (REF ) converges to 0 almost surely, too.", "All in all, the third term on the right-hand side of (REF ) converges to 0 almost surely as it is dominated by the sum of two expressions which both converge to 0 with probability 1.", "This completes the proof of the first assertion in Lemma REF .", "$\\Box $ Corollary 1 Suppose that $\\left(\\xi _i\\right)_{i\\ge 1}$ is a stationary, long-range dependent Gaussian process with mean 0, variance 1 and LRD parameter $0<D <\\frac{1}{m}$ , where $m$ denotes the Hermite rank of the class of functions $1_{\\left\\lbrace G(\\xi _i)\\le x\\right\\rbrace }-F(x)$ , $x \\in \\mathbb {R}$ .", "Moreover, assume that $\\left(G(\\xi _i)\\right)_{i\\ge 1}$ has a continuous distribution function $F$ and that $G:\\mathbb {R}\\longrightarrow \\mathbb {R}$ is a measurable function.", "Then $\\frac{1}{n^2}\\sum \\limits _{i=1}^{\\lfloor n\\tau \\rfloor }\\sum \\limits _{j=\\lfloor n\\tau \\rfloor +1}^n1_{\\left\\lbrace G(\\xi _i)\\le G(\\xi _j)+\\Delta \\right\\rbrace }\\overset{P}{\\longrightarrow }\\tau (1-\\tau )\\int _{\\mathbb {R}}F(x+\\Delta )dF(x)$ for fixed $\\tau $ .", "Proof of Corollary REF .", "Consider the function $G:D\\left[0, \\tau \\right]\\longrightarrow \\mathbb {R}$ , $f\\mapsto f(\\tau )$ .", "As $G$ is continuous with respect to the supremum norm on $D\\left[0, \\tau \\right]$ , Corollary REF follows from Lemma REF and the continuous mapping theorem $\\Box $ Lemma 2 Suppose that $\\left(\\xi _i\\right)_{i\\ge 1}$ is a stationary, long-range dependent Gaussian process with mean 0, variance 1 and LRD parameter $0<D <\\frac{1}{m}$ , where $m$ denotes the Hermite rank of the class of functions $1_{\\left\\lbrace G(\\xi _i)\\le x\\right\\rbrace }-F(x)$ , $x \\in \\mathbb {R}$ .", "Moreover, assume that $\\left(G(\\xi _i)\\right)_{i\\ge 1}$ has a continuous distribution function $F$ and that $G:\\mathbb {R}\\longrightarrow \\mathbb {R}$ is a measurable function.", "Then $\\frac{1}{n^2}\\sum \\limits _{i=1}^{\\lfloor n\\tau \\rfloor }\\sum \\limits _{j=\\lfloor n\\tau \\rfloor +1}^{\\lfloor n\\lambda \\rfloor }1_{\\left\\lbrace G(\\xi _i)\\le G(\\xi _j)+\\Delta \\right\\rbrace }\\overset{P}{\\longrightarrow }\\tau (\\lambda -\\tau )\\int _{\\mathbb {R}}F(x+\\Delta )dF(x)$ for fixed $\\tau $ , uniformly in $\\lambda \\ge \\tau $ .", "Proof of Lemma REF .", "Let $F_{k+1, t}$ denote the empirical distribution function of $G(\\xi _{k+1}), \\ldots , G(\\xi _t)$ , i.e.", "$F_{k+1, t}(x)=\\frac{1}{t-k}\\sum \\limits _{i=k+1}^t1_{\\left\\lbrace G(\\xi _i)\\le x\\right\\rbrace }.$ We may therefore rewrite $\\sum \\limits _{i=1}^{\\lfloor n\\tau \\rfloor }\\sum \\limits _{j=\\lfloor n\\tau \\rfloor +1}^{\\lfloor n\\lambda \\rfloor }1_{\\left\\lbrace G(\\xi _i)\\le G(\\xi _j)+\\Delta \\right\\rbrace }&=\\left(\\lfloor n\\lambda \\rfloor -\\lfloor n\\tau \\rfloor \\right)\\lfloor n\\tau \\rfloor \\frac{1}{\\left(\\lfloor n\\lambda \\rfloor -\\lfloor n\\tau \\rfloor \\right)}\\sum \\limits _{j=\\lfloor n\\tau \\rfloor +1}^{\\lfloor n\\lambda \\rfloor }F_{\\lfloor n\\tau \\rfloor }( G(\\xi _j)+\\Delta )\\\\&=\\left(\\lfloor n\\lambda \\rfloor -\\lfloor n\\tau \\rfloor \\right)\\lfloor n\\tau \\rfloor \\int _{\\mathbb {R}}F_{\\lfloor n\\tau \\rfloor }( x+\\Delta )dF_{\\lfloor n \\tau \\rfloor +1, \\lfloor n\\lambda \\rfloor }(x).$ Furthermore, repeated application of the triangle inequality yields $&\\sup \\limits _{\\tau \\le \\lambda \\le 1}\\left\|\\frac{1}{n}\\left(\\lfloor n\\lambda \\rfloor -\\lfloor n\\tau \\rfloor \\right)\\int _{\\mathbb {R}}F_{\\lfloor n\\tau \\rfloor }(x+\\Delta )dF_{\\lfloor n\\tau \\rfloor +1, \\lfloor n\\lambda \\rfloor }(x)-\\left(\\lambda -\\tau \\right)\\int _{\\mathbb {R}}F(x+\\Delta )dF(x)\\right\|\\\\&\\le \\sup \\limits _{\\tau \\le \\lambda \\le 1}\\left\|\\frac{1}{n}\\left(\\lfloor n\\lambda \\rfloor -\\lfloor n\\tau \\rfloor \\right)\\int _{\\mathbb {R}}\\left(F_{\\lfloor n\\tau \\rfloor }(x+\\Delta )-F(x+\\Delta )\\right)dF_{\\lfloor n\\tau \\rfloor +1, \\lfloor n\\lambda \\rfloor }(x)\\right\|\\\\&\\quad \\ + \\sup \\limits _{\\tau \\le \\lambda \\le 1}\\left\|\\frac{1}{n}\\left(\\lfloor n\\lambda \\rfloor -\\lfloor n\\tau \\rfloor \\right)\\int _{\\mathbb {R}}F(x+\\Delta )dF_{\\lfloor n\\tau \\rfloor +1, \\lfloor n\\lambda \\rfloor }(x)-\\frac{1}{n}\\left(\\lfloor n\\lambda \\rfloor -\\lfloor n\\tau \\rfloor \\right)\\int _{\\mathbb {R}}F(x+\\Delta )dF(x)\\right\|\\\\&\\quad \\ + \\sup \\limits _{\\tau \\le \\lambda \\le 1}\\left\|\\frac{1}{n}\\left(\\lfloor n\\lambda \\rfloor -\\lfloor n\\tau \\rfloor \\right)\\int _{\\mathbb {R}}F(x+\\Delta )dF (x)-\\left(\\lambda -\\tau \\right)\\int _{\\mathbb {R}}F(x+\\Delta )dF (x)\\right\|.$ In order to prove that the stochastic process considered in Lemma REF converges to the given limit process, it is sufficient to show that the expressions on the right-hand side of the above inequality converge to 0.", "We consider each of the three summands separately.", "Apparently, the third term converges to 0 since $&\\sup \\limits _{\\tau \\le \\lambda \\le 1}\\left\|\\frac{1}{n}\\left(\\lfloor n\\lambda \\rfloor -\\lfloor n\\tau \\rfloor \\right)\\int _{\\mathbb {R}}F(x+\\Delta )dF (x)-\\left(\\lambda -\\tau \\right)\\int _{\\mathbb {R}}F(x+\\Delta )dF (x)\\right\|\\\\&=\\sup \\limits _{\\tau \\le \\lambda \\le 1}\\left\|\\frac{1}{n}\\left(\\lfloor n\\lambda \\rfloor -\\lfloor n\\tau \\rfloor \\right)-\\left(\\lambda -\\tau \\right)\\right\|\\int _{\\mathbb {R}}F(x+\\Delta )dF (x)$ and as $\\sup _{\\tau \\le \\lambda \\le 1}\\left\|\\frac{1}{n}\\left(\\lfloor n\\lambda \\rfloor -\\lfloor n\\tau \\rfloor \\right)-\\left(\\lambda -\\tau \\right)\\right\|\\longrightarrow 0$ .", "We have $&\\sup \\limits _{\\tau \\le \\lambda \\le 1}\\left\|\\frac{1}{n}\\left(\\lfloor n\\lambda \\rfloor -\\lfloor n\\tau \\rfloor \\right)\\int _{\\mathbb {R}}\\left(F_{\\lfloor n\\tau \\rfloor }(x+\\Delta )-F(x+\\Delta )\\right)dF_{\\lfloor n\\tau \\rfloor +1, \\lfloor n\\lambda \\rfloor }(x)\\right\|\\\\&\\le \\sup \\limits _{x\\in \\mathbb {R}}\\left\|F_{\\lfloor n\\tau \\rfloor }(x+\\Delta )-F(x+\\Delta )\\right\|\\sup \\limits _{\\tau \\le \\lambda \\le 1}\\left\|\\frac{1}{n}\\left(\\lfloor n\\lambda \\rfloor -\\lfloor n\\tau \\rfloor \\right)\\right\|$ for the first summand.", "As $\\sup _{x\\in \\mathbb {R}}\\left\|F_{\\lfloor n\\tau \\rfloor }(x+\\Delta )-F(x+\\Delta )\\right\|$ converges to 0 almost surely by the Glivenko-Cantelli theorem, so does the right-hand side of the above inequality.", "Finally, consider the second term on the right-hand side of (REF ).", "We have $&\\sup \\limits _{\\tau \\le \\lambda \\le 1}\\left\|\\frac{1}{n}\\left(\\lfloor n\\lambda \\rfloor -\\lfloor n\\tau \\rfloor \\right)\\int _{\\mathbb {R}}F(x+\\Delta )dF_{\\lfloor n\\tau \\rfloor +1, \\lfloor n\\lambda \\rfloor }(x)-\\frac{1}{n}\\left(\\lfloor n\\lambda \\rfloor -\\lfloor n\\tau \\rfloor \\right)\\int _{\\mathbb {R}}F(x+\\Delta )dF(x)\\right\|\\\\&=\\sup \\limits _{\\tau \\le \\lambda \\le 1}\\left\|\\frac{1}{n}\\left(\\lfloor n\\lambda \\rfloor -\\lfloor n\\tau \\rfloor \\right)\\int _{\\mathbb {R}}F(x+\\Delta )d\\left(F_{\\lfloor n\\tau \\rfloor +1, \\lfloor n\\lambda \\rfloor }-F\\right)(x)\\right\|\\\\&=\\sup \\limits _{\\tau \\le \\lambda \\le 1}\\left\|\\frac{1}{n}\\left(\\lfloor n\\lambda \\rfloor -\\lfloor n\\tau \\rfloor \\right)\\int _{\\mathbb {R}}\\left(F_{\\lfloor n\\tau \\rfloor +1, \\lfloor n\\lambda \\rfloor }(x)-F(x)\\right)dF(x+\\Delta )\\right\|\\\\&=\\frac{d_n}{n}\\sup \\limits _{\\tau \\le \\lambda \\le 1}\\left\|\\int _{\\mathbb {R}}d_n^{-1}\\left(\\lfloor n\\lambda \\rfloor -\\lfloor n\\tau \\rfloor \\right)\\left(F_{\\lfloor n\\tau \\rfloor +1, \\lfloor n\\lambda \\rfloor }(x)-F(x)\\right)-J(x)\\left(Z(\\lambda )-Z(\\tau )\\right)dF(x+\\Delta )\\right\|\\\\&\\quad \\ +\\sup \\limits _{\\tau \\le \\lambda \\le 1}\\left\|\\frac{d_n}{n}\\left(Z(\\lambda )-Z(\\tau )\\right)\\int _{\\mathbb {R}}J(x)dF(x+\\Delta )\\right\|\\\\&\\le \\frac{d_n}{n}\\sup \\limits _{\\tau \\le \\lambda \\le 1, x\\in \\mathbb {R}}\\left\|d_n^{-1}\\left(\\lfloor n\\lambda \\rfloor -\\lfloor n\\tau \\rfloor \\right)\\left(F_{\\lfloor n\\tau \\rfloor +1, \\lfloor n\\lambda \\rfloor }(x)-F(x)\\right)-J(x)\\left(Z(\\lambda )-Z(\\tau )\\right)\\right\|\\\\&\\quad \\ +\\frac{d_n}{n}\\sup \\limits _{\\tau \\le \\lambda \\le 1}\\left\|Z(\\lambda )-Z(\\tau )\\right\|\\left\|\\int _{\\mathbb {R}}J(x)dF(x+\\Delta )\\right\|.$ It follows from integration by parts that $&\\int _{\\mathbb {R}}F(x+\\Delta )dF_{\\lfloor n\\tau \\rfloor +1, \\lfloor n\\lambda \\rfloor }(x)-\\int _{\\mathbb {R}}F(x+\\Delta )dF(x)\\\\&=\\int _{\\mathbb {R}}F(x)dF(x+\\Delta )-\\int _{\\mathbb {R}}F_{\\lfloor n\\tau \\rfloor +1, \\lfloor n\\lambda \\rfloor }(x)dF(x+\\Delta ).$ Furthermore, $\\left(\\lfloor n\\lambda \\rfloor -\\lfloor n\\tau \\rfloor \\right)\\left(F_{\\lfloor n\\tau \\rfloor +1, \\lfloor n\\lambda \\rfloor }(x)-F(x)\\right)&=\\sum \\limits _{i=\\lfloor n\\tau \\rfloor +1}^{\\lfloor n\\lambda \\rfloor }1_{\\lbrace G(\\xi _i)\\le x\\rbrace }-\\left(\\lfloor n\\lambda \\rfloor -\\lfloor n\\tau \\rfloor \\right)F(x)\\\\&=\\lfloor n\\lambda \\rfloor F_{\\lfloor n\\lambda \\rfloor }(x)-\\lfloor n\\tau \\rfloor F_{\\lfloor n\\tau \\rfloor }(x)-\\lfloor n\\lambda \\rfloor F(x)+\\lfloor n\\tau \\rfloor F(x)\\\\&=\\lfloor n\\lambda \\rfloor \\left(F_{\\lfloor n\\lambda \\rfloor }(x)-F(x)\\right)-\\lfloor n\\tau \\rfloor \\left(F_{\\lfloor n\\tau \\rfloor }(x)- F(x)\\right).$ As a result, $&\\sup \\limits _{\\tau \\le \\lambda \\le 1, x\\in \\mathbb {R}}\\left\|d_n^{-1}\\left(\\lfloor n\\lambda \\rfloor -\\lfloor n\\tau \\rfloor \\right)\\left(F_{\\lfloor n\\tau \\rfloor +1, \\lfloor n\\lambda \\rfloor }(x)-F(x)\\right)-J(x)\\left(Z(\\lambda )-Z(\\tau )\\right)\\right\|\\\\&=\\sup \\limits _{\\tau \\le \\lambda \\le 1, x\\in \\mathbb {R}}\\left\|d_n^{-1}\\left(\\lfloor n\\lambda \\rfloor \\left(F_{\\lfloor n\\lambda \\rfloor }(x)-F(x)\\right)-\\lfloor n\\tau \\rfloor \\left(F_{\\lfloor n\\tau \\rfloor }(x)- F(x)\\right)\\right)-J(x)\\left(Z(\\lambda )-Z(\\tau )\\right)\\right\|\\\\&\\le \\sup \\limits _{\\tau \\le \\lambda \\le 1, x\\in \\mathbb {R}}\\left\|d_n^{-1}\\lfloor n\\lambda \\rfloor \\left(F_{\\lfloor n\\lambda \\rfloor }(x)-F(x)\\right)-J(x)Z(\\lambda )\\right\|\\\\&\\quad \\ +\\sup \\limits _{\\tau \\le \\lambda \\le 1, x\\in \\mathbb {R}}\\left\|d_n^{-1}\\lfloor n\\tau \\rfloor \\left(F_{\\lfloor n\\tau \\rfloor }(x)- F(x)\\right)-J(x)Z(\\tau )\\right\|.$ Again, we may assume without loss of generality that $\\sup \\limits _{\\lambda , x}\\left\|d_n^{-1}\\lfloor n\\lambda \\rfloor \\left(F_{\\lfloor n\\lambda \\rfloor }(x)-F(x)\\right)-J(x)Z(\\lambda )\\right\|\\longrightarrow 0$ almost surely, as pointed out in the proof of Lemma REF .", "Since $\\frac{d_n}{n}\\longrightarrow 0$ by definition of $d_n$ , we may conclude that the third summand on the right hand side of (REF ) converges to 0, too.", "This completes the proof of Lemma REF .", "$\\Box $ Proof of Theorem REF .", "We have $T_n(\\tau _1, \\tau _2)&=\\sup \\limits _{k\\in \\left[\\lfloor n\\tau _1\\rfloor , \\lfloor n\\tau _2\\rfloor \\right]}G_n(k)\\\\&\\ge G_n(k^),$ where $G_n(k^)=\\frac{\\left\|\\sum \\limits _{i=1}^{k^}\\sum \\limits _{j=k^+1}^n\\left(1_{\\left\\lbrace X_i\\le X_j\\right\\rbrace }-\\frac{1}{2}\\right)\\right\|}{\\left\\lbrace \\frac{1}{n}\\sum \\limits _{t=1}^{k^}S_t^2(1, k^)+\\frac{1}{n}\\sum \\limits _{t=k^+1}^nS_t^2(k^+1, n)\\right\\rbrace ^{\\frac{1}{2}}}$ and where $k^=\\lfloor n\\tau \\rfloor $ denotes the location of the change-point.", "Thus, it suffices to show that $G_n(k^)\\overset{P}{\\longrightarrow }\\infty $ .", "For this purpose, we rewrite $G_n(k^)=\\frac{\\frac{1}{n^2}\\left\|\\sum \\limits _{i=1}^{k^}\\sum \\limits _{j=k^+1}^n\\left(1_{\\left\\lbrace X_i\\le X_j\\right\\rbrace }-\\frac{1}{2}\\right)\\right\|}{\\frac{1}{n^2}\\left\\lbrace \\frac{1}{n}\\sum \\limits _{t=1}^{k^}S_t^2(1, k^)+\\frac{1}{n}\\sum \\limits _{t=k^+1}^nS_t^2(k^+1, n)\\right\\rbrace ^{\\frac{1}{2}}}.$ We will prove that the numerator of $G_n(k^)$ converges to a positive constant, whereas the denominator tends to 0 in order to show divergence to $\\infty $ .", "First, we turn to the denominator, which equals $\\frac{1}{n^2}\\left\\lbrace \\int _0^{\\tau }S_{\\lfloor nr\\rfloor }^2(1, k^)dr+\\int _{\\tau }^1S_{\\lfloor nr\\rfloor }^2(k^+1, n)dr\\right\\rbrace ^{\\frac{1}{2}}.$ Note that for $i\\le k^$ $\\sum \\limits _{j=1}^n1_{\\left\\lbrace X_i\\le X_j\\right\\rbrace }&=\\sum \\limits _{j=1}^{k^}1_{\\left\\lbrace \\mu +G(\\xi _i)\\le \\mu +G(\\xi _j)\\right\\rbrace }+\\sum \\limits _{j=k^+1}^n1_{\\left\\lbrace \\mu +G(\\xi _i)\\le \\mu +G(\\xi _j)+\\Delta \\right\\rbrace }\\\\&=\\sum \\limits _{j=1}^n1_{\\left\\lbrace G(\\xi _i)\\le G(\\xi _j)\\right\\rbrace }+\\sum \\limits _{j=k^+1}^n1_{\\left\\lbrace G(\\xi _j)<G(\\xi _i)\\le G(\\xi _j)+\\Delta \\right\\rbrace }.$ Therefore, $S_t(1, k^)&=-\\sum \\limits _{h=1}^t\\left(\\sum \\limits _{j=1}^n1_{\\left\\lbrace X_h\\le X_j\\right\\rbrace }-\\frac{1}{k^}\\sum \\limits _{i=1}^{k^}\\sum \\limits _{j=1}^n1_{\\left\\lbrace X_i\\le X_j\\right\\rbrace }\\right)\\\\&=-\\sum \\limits _{h=1}^t\\left(\\sum \\limits _{j=1}^n1_{\\left\\lbrace G(\\xi _h)\\le G(\\xi _j)\\right\\rbrace }-\\frac{1}{k^}\\sum \\limits _{i=1}^{k^}\\sum \\limits _{j=1}^n1_{\\left\\lbrace G(\\xi _i)\\le G(\\xi _j)\\right\\rbrace }\\right)\\\\&\\quad \\ -\\sum \\limits _{h=1}^t\\left(\\sum \\limits _{j=k^+1}^n1_{\\left\\lbrace G(\\xi _j)<G(\\xi _h)\\le G(\\xi _j)+\\Delta \\right\\rbrace }-\\frac{1}{k^}\\sum \\limits _{i=1}^{k^}\\sum \\limits _{j=k^+1}^n1_{\\left\\lbrace G(\\xi _j)<G(\\xi _i)\\le G(\\xi _j)+\\Delta \\right\\rbrace }\\right).$ We treat the expression $S_t(1, k^)$ as sum of the following terms $&\\hat{S}_t(1, k^)=-\\sum \\limits _{h=1}^t\\left(\\sum \\limits _{j=1}^n1_{\\left\\lbrace G(\\xi _h)\\le G(\\xi _j)\\right\\rbrace }-\\frac{1}{k^}\\sum \\limits _{i=1}^{k^}\\sum \\limits _{j=1}^n1_{\\left\\lbrace G(\\xi _i)\\le G(\\xi _j)\\right\\rbrace }\\right),\\\\&\\tilde{S}_t(1, k^)=-\\sum \\limits _{h=1}^t\\left(\\sum \\limits _{j=k^+1}^n1_{\\left\\lbrace G(\\xi _j)<G(\\xi _h)\\le G(\\xi _j)+\\Delta \\right\\rbrace }-\\frac{1}{k^}\\sum \\limits _{i=1}^{k^}\\sum \\limits _{j=k^+1}^n1_{\\left\\lbrace G(\\xi _j)<G(\\xi _i)\\le G(\\xi _j)+\\Delta \\right\\rbrace }\\right).$ For the first summand we get $\\hat{S}_t(1, k^)&=-\\sum \\limits _{h=1}^t\\left(\\sum \\limits _{j=1}^n1_{\\left\\lbrace G(\\xi _h)\\le G(\\xi _j)\\right\\rbrace }-\\frac{1}{k^}\\sum \\limits _{i=1}^{k^}\\sum \\limits _{j=1}^n1_{\\left\\lbrace G(\\xi _i)\\le G(\\xi _j)\\right\\rbrace }\\right)\\\\&=-\\sum \\limits _{i=1}^t\\sum \\limits _{j=1}^t1_{\\left\\lbrace G(\\xi _i)\\le G(\\xi _j)\\right\\rbrace }-\\sum \\limits _{i=1}^t\\sum \\limits _{j=t+1}^n1_{\\left\\lbrace G(\\xi _i)\\le G(\\xi _j)\\right\\rbrace }\\\\&\\quad \\ +\\frac{t}{k^}\\sum \\limits _{i=1}^{k^}\\sum \\limits _{j=1}^{k^}1_{\\left\\lbrace G(\\xi _i)\\le G(\\xi _j)\\right\\rbrace } +\\frac{t}{k^}\\sum \\limits _{i=1}^{k^}\\sum \\limits _{j=k^+1}^n1_{\\left\\lbrace G(\\xi _i)\\le G(\\xi _j)\\right\\rbrace }\\\\&=-\\frac{t(t+1)}{2}-\\sum \\limits _{i=1}^t\\sum \\limits _{j=t+1}^n1_{\\left\\lbrace G(\\xi _i)\\le G(\\xi _j)\\right\\rbrace }+\\frac{t}{k^}\\frac{k^(k^+1)}{2} +\\frac{t}{k^}\\sum \\limits _{i=1}^{k^}\\sum \\limits _{j=k^+1}^n1_{\\left\\lbrace G(\\xi _i)\\le G(\\xi _j)\\right\\rbrace }\\\\&=-\\frac{t^2}{2}-\\sum \\limits _{i=1}^t\\sum \\limits _{j=t+1}^n1_{\\left\\lbrace G(\\xi _i)\\le G(\\xi _j)\\right\\rbrace }+\\frac{tk^}{2} +\\frac{t}{k^}\\sum \\limits _{i=1}^{k^}\\sum \\limits _{j=k^+1}^n1_{\\left\\lbrace G(\\xi _i)\\le G(\\xi _j)\\right\\rbrace }.$ We have $&\\frac{1}{n^2}\\sum \\limits _{i=1}^{\\lfloor n\\lambda \\rfloor }\\sum \\limits _{j=\\lfloor n\\lambda \\rfloor +1}^n1_{\\left\\lbrace G(\\xi _i)\\le G(\\xi _j)\\right\\rbrace }\\overset{P}{\\longrightarrow }\\frac{\\lambda (1-\\lambda )}{2}$ uniformly in $\\lambda $ because $\\frac{1}{nd_n}\\sum \\limits _{i=1}^{\\lfloor n\\lambda \\rfloor }\\sum \\limits _{j=\\lfloor n\\lambda \\rfloor +1}^n\\left(1_{\\left\\lbrace G(\\xi _i)\\le G(\\xi _j)\\right\\rbrace }-\\frac{1}{2}\\right)\\overset{\\mathcal {D}}{\\longrightarrow }\\left(Z(\\lambda )-\\lambda Z(1)\\right)\\int _{\\mathbb {R}}J(x)dF(x)$ uniformly in $\\lambda $ by Theorem 1.1 in [2] and as $\\frac{d_n}{n}\\longrightarrow 0$ .", "We may conclude from this and Corollary REF that $\\frac{1}{n^2}\\hat{S}_{\\lfloor n\\lambda \\rfloor }(1, \\lfloor k^\\rfloor )\\overset{P}{\\longrightarrow }0$ uniformly in $\\lambda \\le \\tau $ .", "Because of $1_{\\left\\lbrace G(\\xi _j)<G(\\xi _i)\\le G(\\xi _j)+\\Delta \\right\\rbrace }&=1_{\\left\\lbrace G(\\xi _i)\\le G(\\xi _j)+\\Delta \\right\\rbrace }-1_{\\left\\lbrace G(\\xi _i)\\le G(\\xi _j)\\right\\rbrace },$ the second summand can be written as $\\tilde{S}_{t}(1, {k^})&=-\\sum \\limits _{h=1}^{t}\\left(\\sum \\limits _{j={k^}+1}^n1_{\\left\\lbrace G(\\xi _j)<G(\\xi _h)\\le G(\\xi _j)+\\Delta \\right\\rbrace }-\\frac{1}{{k^}}\\sum \\limits _{i=1}^{{k^}}\\sum \\limits _{j={k^}+1}^n1_{\\left\\lbrace G(\\xi _j)<G(\\xi _i)\\le G(\\xi _j)+\\Delta \\right\\rbrace }\\right)\\\\&=-\\sum \\limits _{i=1}^{t}\\sum \\limits _{j={k^}+1}^n1_{\\left\\lbrace G(\\xi _i)\\le G(\\xi _j)+\\Delta \\right\\rbrace }+\\frac{t}{{k^}}\\sum \\limits _{i=1}^{{k^}}\\sum \\limits _{j={k^}+1}^n1_{\\left\\lbrace G(\\xi _i)\\le G(\\xi _j)+\\Delta \\right\\rbrace }\\\\&\\quad \\ +\\sum \\limits _{i=1}^{t}\\sum \\limits _{j={k^}+1}^n1_{\\left\\lbrace G(\\xi _i)\\le G(\\xi _j)\\right\\rbrace }-\\frac{t}{{k^}}\\sum \\limits _{i=1}^{{k^}}\\sum \\limits _{j={k^}+1}^n1_{\\left\\lbrace G(\\xi _i)\\le G(\\xi _j)\\right\\rbrace }.$ Due to Lemma REF and Corollary REF , $\\frac{1}{n^2}\\tilde{S}_{\\lfloor n\\lambda \\rfloor }(1, k^)$ converges in probability to 0, as well.", "All in all, the previous considerations yield $\\int _{0}^{\\tau }\\left\\lbrace \\frac{1}{n^2}S_{\\lfloor nr\\rfloor }(1, k^)\\right\\rbrace ^2dr\\overset{P}{\\longrightarrow } 0$ as $G:D\\left[0, \\tau \\right]\\longrightarrow \\mathbb {R}$ , $f\\mapsto \\int _{0}^{\\tau }\\left(f(s)\\right)^2ds$ , is continuous with respect to the supremum norm on $D\\left[0, \\tau \\right]$ .", "In analogy to the previous argumentation it can be shown that $\\int _{\\tau }^1\\left\\lbrace \\frac{1}{n^2}S_{\\lfloor nr\\rfloor }(k^+1, n)\\right\\rbrace ^2dr \\overset{P}{\\longrightarrow }0.$ For this purpose, note that, if $i> {k^}$ , $\\sum \\limits _{j=1}^n1_{\\left\\lbrace X_i\\le X_j\\right\\rbrace }&=\\sum \\limits _{j=1}^{k^}1_{\\left\\lbrace \\mu +G(\\xi _i)+\\Delta \\le \\mu +G(\\xi _j)\\right\\rbrace }+\\sum \\limits _{j={k^}+1}^n1_{\\left\\lbrace \\mu +G(\\xi _i)+\\Delta \\le \\mu +G(\\xi _j)+\\Delta \\right\\rbrace }\\\\&=\\sum \\limits _{j=1}^n1_{\\left\\lbrace G(\\xi _i)\\le G(\\xi _j)\\right\\rbrace }-\\sum \\limits _{j=1}^{k^}1_{\\lbrace G(\\xi _i)\\le G(\\xi _j)<G(\\xi _i)+\\Delta \\rbrace }.$ Therefore, $S_t({k^}+1, n)&=-\\sum \\limits _{h={k^}+1}^t\\left(\\sum \\limits _{j=1}^n1_{\\left\\lbrace X_h\\le X_j\\right\\rbrace }-\\frac{1}{n-{k^}}\\sum \\limits _{i={k^}+1}^n\\sum \\limits _{j=1}^n1_{\\left\\lbrace X_i\\le X_j\\right\\rbrace }\\right)\\\\&=-\\sum \\limits _{h={k^}+1}^t\\left(\\sum \\limits _{j=1}^n1_{\\left\\lbrace G(\\xi _h)\\le G(\\xi _j)\\right\\rbrace }-\\frac{1}{n-{k^}}\\sum \\limits _{i={k^}+1}^n\\sum \\limits _{j=1}^n1_{\\left\\lbrace G(\\xi _i)\\le G(\\xi _j)\\right\\rbrace }\\right)\\\\&\\quad \\ +\\sum \\limits _{h={k^}+1}^t\\left(\\sum \\limits _{j=1}^{k^}1_{\\lbrace G(\\xi _h)\\le G(\\xi _j)<G(\\xi _h)+\\Delta \\rbrace }-\\frac{1}{n-{k^}}\\sum \\limits _{i={k^}+1}^n\\sum \\limits _{j=1}^{k^}1_{\\lbrace G(\\xi _i)\\le G(\\xi _j)<G(\\xi _i)+\\Delta \\rbrace }\\right).$ Hence, we consider $S_t({k^}+1, n)$ as sum of the expressions below $&\\hat{S}_t({k^}+1, n)=-\\sum \\limits _{h={k^}+1}^t\\left(\\sum \\limits _{j=1}^n1_{\\left\\lbrace G(\\xi _h)\\le G(\\xi _j)\\right\\rbrace }-\\frac{1}{n-{k^}}\\sum \\limits _{i={k^}+1}^n\\sum \\limits _{j=1}^n1_{\\left\\lbrace G(\\xi _i)\\le G(\\xi _j)\\right\\rbrace }\\right),\\\\&\\tilde{S}_t({k^}+1, n)=\\sum \\limits _{h={k^}+1}^t\\left(\\sum \\limits _{j=1}^{k^}1_{\\left\\lbrace G(\\xi _h)\\le G(\\xi _j)< G(\\xi _h)+\\Delta \\right\\rbrace }-\\frac{1}{n-{k^}}\\sum \\limits _{i={k^}+1}^n\\sum \\limits _{j=1}^{k^}1_{\\left\\lbrace G(\\xi _i)\\le G(\\xi _j)< G(\\xi _i)+\\Delta \\right\\rbrace }\\right).$ The following representation arises from rather simple transformations $\\hat{S}_t({k^}+1, n)&=-\\sum \\limits _{h={k^}+1}^t\\left(\\sum \\limits _{j=1}^n1_{\\lbrace G(\\xi _h)\\le G(\\xi _j)\\rbrace }-\\frac{1}{n-{k^}}\\sum \\limits _{i={k^}+1}^n\\sum \\limits _{j=1}^n1_{\\lbrace G(\\xi _i)\\le G(\\xi _j)\\rbrace }\\right)\\\\&=-\\sum \\limits _{i=1}^t\\sum \\limits _{j=t+1}^n1_{\\lbrace G(\\xi _i)\\le G(\\xi _j)\\rbrace }-\\sum \\limits _{i=1}^t\\sum \\limits _{j=1}^t1_{\\lbrace G(\\xi _i)\\le G(\\xi _j)\\rbrace }\\\\&\\quad \\ +\\sum \\limits _{i=1}^{k^}\\sum \\limits _{j={k^}+1}^n1_{\\lbrace G(\\xi _i)\\le G(\\xi _j)\\rbrace }+\\sum \\limits _{i=1}^{k^}\\sum \\limits _{j=1}^{k^}1_{\\lbrace G(\\xi _i)\\le G(\\xi _j)\\rbrace }\\\\&\\quad \\ +\\frac{t-{k^}}{n-{k^}}\\sum \\limits _{i={k^}+1}^n\\sum \\limits _{j=1}^{k^}1_{\\lbrace G(\\xi _i)\\le G(\\xi _j)\\rbrace }+\\frac{t-{k^}}{n-{k^}}\\sum \\limits _{i={k^}+1}^n\\sum \\limits _{j={k^}+1}^n1_{\\lbrace G(\\xi _i)\\le G(\\xi _j)\\rbrace }\\\\&=-\\sum \\limits _{i=1}^t\\sum \\limits _{j=t+1}^n1_{\\lbrace G(\\xi _i)\\le G(\\xi _j)\\rbrace }-\\frac{t(t+1)}{2} +\\sum \\limits _{i=1}^{k^}\\sum \\limits _{j={k^}+1}^n1_{\\lbrace G(\\xi _i)\\le G(\\xi _j)\\rbrace }+\\frac{{k^}({k^}+1)}{2}\\\\&\\quad \\ +\\frac{t-{k^}}{n-{k^}}\\sum \\limits _{i={k^}+1}^n\\sum \\limits _{j=1}^{k^}\\left(1-1_{\\lbrace G(\\xi _j)\\le G(\\xi _i)\\rbrace }\\right)+\\frac{t-{k^}}{n-{k^}}\\frac{(n-{k^})(n-{k^}+1)}{2}\\\\&=-\\sum \\limits _{i=1}^t\\sum \\limits _{j=t+1}^n1_{\\lbrace G(\\xi _i)\\le G(\\xi _j)\\rbrace }-\\frac{t(t+1)}{2} +\\sum \\limits _{i=1}^{k^}\\sum \\limits _{j={k^}+1}^n1_{\\lbrace G(\\xi _i)\\le G(\\xi _j)\\rbrace }+\\frac{{k^}({k^}+1)}{2}\\\\&\\quad \\ +(t-{k^}){k^}-\\frac{t-{k^}}{n-{k^}}\\sum \\limits _{j=1}^{k^}\\sum \\limits _{i={k^}+1}^n1_{\\lbrace G(\\xi _j)\\le G(\\xi _i)\\rbrace }+\\frac{(t-{k^})(n-{k^}+1)}{2}.$ Based on Lemma REF and Corollary REF , the argumentation that also established $\\frac{1}{n^2}\\hat{S}_{\\lfloor n\\lambda \\rfloor }(1, k^)\\overset{P}{\\longrightarrow }0$ yields $\\frac{1}{n^2}\\hat{S}_{\\lfloor n\\lambda \\rfloor }({k^}+1, n)\\overset{P}{\\longrightarrow }0$ uniformly in $\\lambda \\ge \\tau $ .", "Likewise, it can be shown that $\\frac{1}{n^2}\\tilde{S}_{\\lfloor n\\lambda \\rfloor }({k^}+1, n)\\overset{P}{\\longrightarrow }0$ .", "First of all, we note that $1_{\\left\\lbrace G(\\xi _i)\\le G(\\xi _j)< G(\\xi _i)+\\Delta \\right\\rbrace }&=1_{\\left\\lbrace G(\\xi _j)\\le G(\\xi _i)+\\Delta \\right\\rbrace }-1_{\\left\\lbrace G(\\xi _j)\\le G(\\xi _i)\\right\\rbrace }$ almost surely if $i\\ne j$ .", "Thereby, $\\tilde{S}_t({k^}+1, n)&=\\sum \\limits _{h={k^}+1}^t\\left(\\sum \\limits _{j=1}^{k^}1_{\\lbrace G(\\xi _h)\\le G(\\xi _j)< G(\\xi _h)+\\Delta \\rbrace }-\\frac{1}{n-{k^}}\\sum \\limits _{i={k^}+1}^n\\sum \\limits _{j=1}^{k^}1_{\\lbrace G(\\xi _i)\\le G(\\xi _j)< G(\\xi _i)+\\Delta \\rbrace }\\right)\\\\&=\\sum \\limits _{j=1}^{k^} \\sum \\limits _{i={k^}+1}^t1_{\\left\\lbrace G(\\xi _j)\\le G(\\xi _i)+\\Delta \\right\\rbrace }-\\frac{t-{k^}}{n-{k^}}\\sum \\limits _{j=1}^{k^}\\sum \\limits _{i={k^}+1}^n1_{\\left\\lbrace G(\\xi _j)\\le G(\\xi _i)+\\Delta \\right\\rbrace }\\\\&\\quad \\ -\\sum \\limits _{j=1}^{k^}\\sum \\limits _{i={k^}+1}^t1_{\\left\\lbrace G(\\xi _j)\\le G(\\xi _i)\\right\\rbrace }+\\frac{t-{k^}}{n-{k^}}\\sum \\limits _{j=1}^{k^}\\sum \\limits _{i={k^}+1}^n1_{\\left\\lbrace G(\\xi _j)\\le G(\\xi _i)\\right\\rbrace }.$ As a result, we have $\\frac{1}{n^2}\\tilde{S}_{\\lfloor n\\lambda \\rfloor }({k^}+1, n)\\overset{P}{\\longrightarrow }0$ by Lemma REF and Corollary REF .", "As both terms, $\\frac{1}{n^2}\\hat{S}_{\\lfloor n\\lambda \\rfloor }({k^}+1, n)$ as well as $\\frac{1}{n^2}\\tilde{S}_{\\lfloor n\\lambda \\rfloor }({k^}+1, n)$ , converge in probability to 0 uniformly in $\\lambda \\ge \\tau $ , it follows that $\\int _{\\tau }^{1}\\left\\lbrace \\frac{1}{n^2}S_{\\lfloor n r\\rfloor }(k^+1, n)\\right\\rbrace ^2dr \\overset{P}{\\longrightarrow }0.$ On the basis of the previous considerations we may conclude that the denominator of $G_n(k^)$ converges in probability to 0.", "In order to prove the consistency of the self-normalized Wilcoxon change-point test, it therefore remains to show that the numerator of $G_n(k^)$ , given by $\\frac{1}{n^2}\\left\|\\sum \\limits _{i=1}^{k^}\\sum \\limits _{j=k^+1}^n\\left(1_{\\left\\lbrace X_i\\le X_j\\right\\rbrace }-\\frac{1}{2}\\right)\\right\|,$ converges to a non-negative constant.", "We have $\\frac{1}{n^2}\\left\|\\sum \\limits _{i=1}^{k^}\\sum \\limits _{j=k^+1}^n\\left(1_{\\left\\lbrace X_i\\le X_j\\right\\rbrace }-\\frac{1}{2}\\right)\\right\|&=\\left\|\\frac{1}{n^2}\\sum \\limits _{i=1}^{k^}\\sum \\limits _{j=k^+1}^n1_{\\left\\lbrace G(\\xi _i)\\le G(\\xi _j)+\\Delta \\right\\rbrace }-\\frac{1}{n^2}\\frac{k^(n-k^)}{2}\\right\|.$ Therefore, $\\frac{1}{n^2}\\left\|\\sum \\limits _{i=1}^{k^}\\sum \\limits _{j=k^+1}^n\\left(1_{\\left\\lbrace X_i\\le X_j\\right\\rbrace }-\\frac{1}{2}\\right)\\right\|\\overset{P}{\\longrightarrow }\\tau (1-\\tau )\\int _{\\mathbb {R}}\\left(F(x+\\Delta )-F(x)\\right)dF(x)$ by Corollary REF and since $\\frac{1}{n^2}\\frac{k^(n-k^)}{2}\\longrightarrow \\frac{\\tau (1-\\tau )}{2}$ .", "As the limit in (REF ) does not vanish, $G_n(k^)$ diverges to $\\infty $ and we thus have proved Theorem REF .", "$\\Box $ Proof of Theorem REF .", "Note that because of the corresponding sample path properties of the stochastic process $Z$ , the sample paths of $W_{m, \\tau }^{}(\\lambda )=\\left(Z(\\lambda )-\\lambda Z(1)\\right)\\int _{\\mathbb {R}}J(x)dF(x)+c\\delta _{\\tau }(\\lambda )\\int _{\\mathbb {R}}f^2(x)dx, \\ 0\\le \\lambda \\le 1,$ are almost surely continuous and nowhere differentiable.", "The same argument as in the proof of Theorem REF shows that $T_n(\\tau _1, \\tau _2)$ converges in distribution to $\\sup \\limits _{\\lambda \\in \\left[\\tau _1, \\tau _2\\right]}\\frac{\\left\|W_{m, \\tau }^{}(\\lambda )\\right\|}{\\left\\lbrace \\int _0^{\\lambda }\\left(W_{m, \\tau }^{}(r)-\\frac{r}{\\lambda }W_{m, \\tau }^{}(\\lambda )\\right)^2dr+\\int _{\\lambda }^1\\left(W_{m, \\tau }^{}(r)-\\frac{1-r}{1-\\lambda }W_{m, \\tau }^{}(\\lambda )\\right)^2dr\\right\\rbrace ^{\\frac{1}{2}}}.$ The numerator of the limit process equals $\\left\|\\int _{\\mathbb {R}}J(x)dF(x)\\left(Z(\\lambda )-\\lambda Z(1)\\right)+c\\delta _{\\tau }(\\lambda )\\int _{\\mathbb {R}}f^2(x)dx\\right\|.$ Moreover, for the quantities in the denominator it holds that $W_{m, \\tau }^(r)-\\frac{r}{\\lambda }W_{m, \\tau }^(\\lambda )&=\\left(Z(r)-r Z(1)\\right)\\int _{\\mathbb {R}}J(x)dF(x)+c\\delta _{\\tau }(r)\\int _{\\mathbb {R}}f^2(x)dx\\\\&\\quad \\ -\\frac{r}{\\lambda }\\left(\\left(Z(\\lambda )-\\lambda Z(1)\\right)\\int _{\\mathbb {R}}J(x)dF(x)+c\\delta _{\\tau }(\\lambda )\\int _{\\mathbb {R}}f^2(x)dx\\right)\\\\&=\\int _{\\mathbb {R}}J(x)dF(x)\\left(Z(\\lambda )-\\frac{r}{\\lambda }Z(\\lambda )\\right)+c\\int _{\\mathbb {R}}f^2(x)dx\\left(\\delta _{\\tau }(r)-\\frac{r}{\\lambda }\\delta _{\\tau }(\\lambda )\\right)$ and $W_{m, \\tau }^(r)-\\frac{1-r}{1-\\lambda }W_{m, \\tau }^(\\lambda )&=\\left(Z(r)-r Z(1)\\right)\\int _{\\mathbb {R}}J(x)dF(x)+c\\delta _{\\tau }(r)\\int _{\\mathbb {R}}f^2(x)dx\\\\&\\quad \\ -\\frac{1-r}{1-\\lambda }\\left\\lbrace \\left(Z(\\lambda )-\\lambda Z(1)\\right)\\int _{\\mathbb {R}}J(x)dF(x)+c\\delta _{\\tau }(\\lambda )\\int _{\\mathbb {R}}f^2(x)dx\\right\\rbrace \\\\&=\\int _{\\mathbb {R}}J(x)dF(x)\\left\\lbrace Z(r)-rZ(1)-\\frac{1-r}{1-\\lambda }\\left(Z(\\lambda )-\\lambda Z(1)\\right)\\right\\rbrace \\\\&\\quad \\ +c\\int _{\\mathbb {R}}f^2(x)dx\\left(\\delta _{\\tau }(r)-\\frac{1-r}{1-\\lambda }\\delta _{\\tau }(\\lambda )\\right).$ $\\Box $" ] ]	1403.0265
[ [ "Wong's Equations and Charged Relativistic Particles in Non-Commutative\n Space" ], [ "Abstract In analogy to Wong's equations describing the motion of a charged relativistic point particle in the presence of an external Yang-Mills field, we discuss the motion of such a particle in non-commutative space subject to an external $U_\\star(1)$ gauge field.", "We conclude that the latter equations are only consistent in the case of a constant field strength.", "This formulation, which is based on an action written in Moyal space, provides a coarser level of description than full QED on non-commutative space.", "The results are compared with those obtained from the different Hamiltonian approaches.", "Furthermore, a continuum version for Wong's equations and for the motion of a particle in non-commutative space is derived." ], [ "Introduction", "Over the last twenty years, non-commuting spatial coordinates have appeared in various contexts in the framework of quantum gravity and superstring theories.", "This fact contributed to the motivation for studying classical and quantum theories with a finite or infinite number of degrees of freedom on non-commutative spaces.", "Different mathematical approaches have been pursued and various physical applications have been explored, e.g.", "see references [8], [11], [24], [35], [39] for some partial reviews.", "Beyond the applications, classical and quantum mechanics on non-commutative space are of interest as toy models for field theories which are more difficult to handle, in particular in the case of interactions with gauge fields.", "In this respect, we recall a similar situation concerning the coupling of matter to Yang–Mills fields on ordinary space: A coarser level of description for the latter theories has been proposed by Wong [45] who considered the motion of charged point particles in an external gauge field [6], [7], [9], [15], [16], [26], [27], [44].", "The latter equations allow for various physical applications, e.g.", "to the dynamics of quarks and their interaction with gluons [26], [27].", "Somewhat similar equations, known as Mathisson–Papapetrou–Dixon equations [13], [30], [33] appear in general relativity for a spinning particle in curved space.", "In this spirit, we will consider in the present work the dynamics of a relativistic “point” particle in non-commutative space subject to an external $U_\\star (1)$ gauge field (thereby implementing a suggestion made in an earlier work [38], see also [20] for some related studies based on a first order expansion in the non-commutativity parameter).", "More precisely, we are motivated by the motion in Moyal space, the latter space having been widely discussed as the arena for field theory in non-commutative spaces.", "By proceeding along the lines of Wong's equations (which are discussed in Section ), we derive a set of equations for the dynamics of the particle in Moyal space in Section .", "For the description of the coupling of a particle to a gauge field, the relativistic setting is the most natural one, but our discussions (of Wong's equations in commutative space or of a particle in non-commutative space) could equally well be done within the non-relativistic setting.", "The particular case of a constant field strength is the most tractable one (and the only consistent one for Wong's equations in non-commutative space) and will be considered in some detail in Section .", "Subsequently in Section , we briefly recall the different Hamiltonian approaches which have previously been pursued for the formulation of classical mechanics in non-commutative space and we compare the resulting equations governing the dynamics of particles coupled to an electromagnetic field.", "In the appendix we present a continuum formulation of Wong's equations on a generic space-time manifold, a formulation which readily generalizes to Moyal space.", "For the motion of a relativistic point particle in four-dimensional (commutative or non-commutative) space-time, the following notations will be used.", "The metric tensor is given by $(\\eta _{\\mu \\nu })_{\\mu , \\nu \\in \\lbrace 0, 1,2,3 \\rbrace }= \\operatorname{diag}(1, - 1, -1, -1)$ and we choose the natural system of units ($c\\equiv 1 \\equiv \\hbar $ ).", "The proper time $\\tau $ for the particle is defined (up to an additive constant) by $d\\tau ^2= ds^2,\\qquad \\text{with}\\qquad ds^2 \\equiv dx^{\\mu } dx_{\\mu }= (dt)^2 - (d\\vec{x})^2,$ and for the massive particle we have $ds^2 > 0$ .", "From (REF ) it follows that $\\dot{x}^2=1$ where $\\dot{x}^2 \\equiv \\dot{x}^{\\mu } \\dot{x}_{\\mu } $ and $ \\dot{x}^{\\mu }\\equiv {dx^{\\mu }}/{d\\tau } $ ." ], [ "Reminder on particles in commutative space", "Abelian gauge field in flat space.", "We consider the interaction of a charged massive relativistic particle with an external electromagnetic field given by the $U(1)$ gauge potential $(A^\\mu )$ .", "The motion of this particle along its space-time trajectory $\\tau \\mapsto x^\\mu (\\tau )$ is described by the actionTo be more precise, in the integral (REF ) the variable $\\tau $ is viewed as a purely mathematical parameter which is only identified with proper time after deriving the equations of motion from the action.", "Thus, the relation $\\dot{x}^2=1$ is only to be used at the latter stage.", "$S[x]=-m\\int ds-q\\int dx^\\mu A_\\mu (x(\\tau ))= -m \\int d\\tau \\sqrt{\\dot{x}^2} -q\\int d\\tau \\dot{x}^\\mu A_\\mu (x(\\tau )).$ Here, $q$ denotes the conserved electric charge of the particle associated with the conserved current density $(j^\\mu )$ : $j^\\mu (y)=q\\int d\\tau \\dot{x}^\\mu (\\tau )\\delta ^4(y-x(\\tau )),\\qquad \\partial _\\mu j^\\mu =0,\\qquad \\int d^3yj^0(y)=q.$ We note that the interaction term in the functional (REF ) may be rewritten in terms of the above current according to $q\\int d\\tau \\dot{x}^\\mu A_\\mu (x(\\tau ))=\\int d^4yj^\\mu (y)A_\\mu (y).$ This expression is invariant under infinitesimal gauge transformations, i.e.", "$\\delta _\\lambda A_\\mu = \\partial _\\mu \\lambda $ , thanks to the conservation of the current: $\\delta _\\lambda \\int d^4yj^\\mu A_\\mu =\\int d^4yj^\\mu \\partial _\\mu \\lambda =-\\int d^4y(\\partial _\\mu j^\\mu ) \\lambda =0.$ We note that the parameter $q$ which describes the coupling of the particle to the gauge field might in principle depend on the world line parameter $\\tau $ : if this assumption is made, $q(\\tau )$ appears under the $\\tau $ -integral in (REF ), the current $j^\\mu $ is no longer conserved and the coupling (REF ) is no longer gauge invariant.", "Thus, $q$ is necessarily constant along the path.", "Variation of the action (REF ) with respect to $x^\\mu $ and substitution of the relation $\\dot{x}^2=1$ leads to the familiar equation of motion $m\\ddot{x}^\\mu =q F^{\\mu \\nu }\\dot{x}_\\nu ,\\qquad \\text{with}\\qquad F_{\\mu \\nu } \\equiv \\partial _\\mu A_\\nu - \\partial _\\nu A_\\mu .$ As is well known, the free particle Lagrangian $L_{\\rm free}= -m \\sqrt{\\dot{x}^2}$ which represents the first term of the action (REF ) and which is non-linear in $\\dot{x}^2$ can be replaced by $\\tilde{L}_{\\rm free}=\\frac{m}{2}\\dot{x}^2 $ which is linear in $\\dot{x}^2$ since both Lagrangians yield the same equation of motion.", "Indeed, we will consider the latter Lagrangian in equation (REF ) and in Section where we go over to the Hamiltonian formulation.", "The aim of this work is to generalize this setting to a non-commutative space-time.", "Since a gauge field on non-commutative space entails a non-Abelian structure for the field strength tensor $F_{\\mu \\nu }$ due to the star product, it is worthwhile to understand first the coupling of a spinless particle to a non-Abelian gauge field on commutative space-time.", "Non-Abelian gauge field in flat space.", "We consider a compact Lie group $G$ (e.g.", "$G=\\text{SU}(N)$ for concreteness) with generators $T^a$ satisfying $\\big [T^a, T^b\\big ]= {\\rm i} f^{abc} T^c\\qquad \\text{and}\\qquad \\operatorname{Tr}\\big (T^a T^b\\big )=\\delta ^{ab}.$ Just as for the Abelian gauge field, the source $j_\\mu ^a (x)$ of the non-Abelian gauge field $A_{\\mu }(x) \\equiv A_{\\mu }^a (x)T^a$ (e.g.", "a field theoretic expression like $\\bar{\\psi } \\gamma _\\mu \\hat{T}^a \\psi $ involving a multiplet $\\psi $ of spinor fields) is considered to be given by the current density $j_\\mu ^a (x)$ of a relativistic point particle [6], [7], [15], [16], [26], [27], [44], [45].", "Instead of an electric charge, the particle moving in an external Yang–Mills field is thus assumed to carry a color-charge or isotopic spin $\\vec{q} \\equiv (q^a)_{a=1, \\dots , \\text{dim}G}$ which transforms under the adjoint representation of the structure group $G$ .", "Henceforth, one considers the Lie algebra-valued variable $q(\\tau ) \\equiv q^a(\\tau ) T^a$ which is assumed to be $\\tau $ -dependent.", "The particle is then described in terms of its space-time coordinates $x^\\mu (\\tau )$ and its isotopic spin $q(\\tau )$ , i.e.", "it is referred to with respect to geometric space and to internal space.", "For the moment, we assume $q(\\tau )$ to represent a given non-dynamical (auxiliary) variable and we will comment on a different point of view below.", "Its coupling to an external non-Abelian gauge field $(A_{\\mu })$ is now described by the action $S[x]=-m\\int ds-\\int d\\tau \\dot{x}^\\mu \\operatorname{Tr}\\lbrace q(\\tau )A_\\mu (x(\\tau )) \\rbrace \\nonumber \\\\\\phantom{S[x]}= -m \\int d\\tau \\sqrt{\\dot{x}^2} -\\int d\\tau \\dot{x}^\\mu q^a(\\tau )A^a_\\mu (x(\\tau )).$ The current density (REF ) presently generalizes to a Lie algebra-valued expression $j_\\mu \\equiv j_{\\mu }^aT^a$ given by $j^\\mu (y)=\\int d\\tau \\dot{x}^\\mu (\\tau )q(\\tau )\\delta ^4(y-x(\\tau )),$ and thereby the interaction term in the functional (REF ) can be rewritten as $- \\int d^4y\\operatorname{Tr}(j^\\mu A_\\mu ).$ For an infinitesimal gauge transformation with Lie algebra-valued parameter $\\lambda $ , i.e.", "$\\delta _{\\lambda }A_\\mu = D_\\mu \\lambda \\equiv \\partial _\\mu \\lambda -{\\rm i} g[A_\\mu , \\lambda ]$ , we have $\\delta _\\lambda \\int d^4y\\operatorname{Tr}\\lbrace j^\\mu A_\\mu \\rbrace =\\int d^4y\\operatorname{Tr}\\lbrace j^\\mu D_\\mu \\lambda \\rbrace =-\\int d^4y\\operatorname{Tr}\\lbrace (D_\\mu j^\\mu ) \\lambda \\rbrace .$ Thus, gauge invariance of the action (REF ) requires the current to be covariantly conserved, i.e.", "$D_\\mu j^\\mu =0$ .", "From (REF ) we can deduce by a short calculation that $(D_\\mu j^\\mu )^a (y)=\\int d\\tau \\frac{D q^a}{d\\tau }\\delta ^4(y-x(\\tau )),\\qquad \\text{with}\\qquad \\frac{D q^a}{d\\tau } \\equiv \\frac{d q^a}{d\\tau } -{\\rm i} g \\dot{x}^{\\mu } [A_\\mu (x(\\tau )),q]^a,$ hence $j^\\mu $ is covariantly conserved if the charge $q$ is covariantly constant along the world line: $ {D q^a}/{d\\tau }=0 $ (subsidiary condition).", "Variation of the action with respect to $x^\\mu $ (and use of $\\dot{x}^2=1$ ) yields the equations of motion $m\\ddot{x}^\\mu = \\operatorname{Tr}(q F^{\\mu \\nu })\\dot{x}_\\nu ,\\qquad \\text{where}\\qquad \\frac{D q^a}{d\\tau }=0,$ and $F_{\\mu \\nu } \\equiv \\partial _\\mu A_\\nu - \\partial _\\nu A_\\mu - {\\rm i} g[A_\\mu , A_\\nu ] $ .", "By construction, these equations are invariant under infinitesimal gauge transformations parametrized by $\\lambda (x)$ since the charge $q$ is a non-dynamical (auxiliary) variable transforming as $ \\delta _{\\lambda } q (\\tau ) \\equiv -{\\rm i}g[q(\\tau ), \\lambda (x (\\tau ))]=0$ for all $\\tau $ , hence $\\delta _{\\lambda } \\big (\\operatorname{Tr}(q F_{\\mu \\nu })\\dot{x}^{\\nu } \\big )= -{\\rm i}g\\operatorname{Tr}(q[F_{\\mu \\nu },\\lambda ])\\dot{x}^{\\nu }={\\rm i}g\\operatorname{Tr}([q, \\lambda ]F_{\\mu \\nu })\\dot{x}^{\\nu }=0,\\\\\\delta _{\\lambda } \\left(\\frac{D q}{d\\tau } \\right)= -{\\rm i}g\\dot{x}^{\\mu }[\\delta _{\\lambda } A_{\\mu }, q]= -{\\rm i}g\\left[\\frac{D \\lambda }{d\\tau }, q \\right]= -{\\rm i}g\\frac{D}{d\\tau }[\\lambda , q]=0.$ Equations (REF ), which represent the Lorentz–Yang–Mills force equation for a relativistic particle in an external Yang–Mills field, are known as Wong's equations [45].", "More specifically, the relation $Dq^a /d\\tau =0$ may be viewed as charge transport equation and it is the geometrically natural generalization (to the charge vector $(q^a)$ ) of the constancy of the charge $q$ in electrodynamics.", "A few remarks are in order here.", "We should point out that the equation of motion for $x^{\\mu }$ had already been obtained earlier in curved space by Kerner [25], but the charge transport equation was not established in that setting.", "We refer to the works [6], [7], [26], [27] for a treatment of dynamics involving the Lagrangian $-\\frac{1}{4} \\int d^3 x\\operatorname{Tr}(F^{\\mu \\nu } F_{\\mu \\nu })$ of the gauge field: the latter approach yields the covariant conservation law $D_{\\mu } j^{\\mu }=0$ as a consequence of the equation of motion $D_{\\nu } F^{\\nu \\mu }= j^{\\mu }$ and the relation $[D_{\\mu }, D_{\\nu }] F^{\\nu \\mu }=0$ .", "For a general discussion of the issue of gauge invariance for the coupling of gauge fields to non-dynamical external sources, we refer to [34].", "Equations (REF ) and their classical solutions have been investigated in the literature and applied for instance to the study of the quark gluon plasma [26].", "The particular case of a constant field strength $F_{\\mu \\nu }$ (“uniform fields”) exhibits interesting features [9] which will be commented upon in Section .", "If one regards $(q^a)$ as a dynamical variable which satisfies the equation of motion ${D q^a}/{d\\tau }=0$ and which transforms under gauge variations with the adjoint representation, i.e.", "$\\delta _{\\lambda } q(\\tau )= -{\\rm i}g[q(\\tau ), \\lambda (x (\\tau ))],$ then equations (REF ) are obviously gauge invariant since $F_{\\mu \\nu }$ also transforms with the adjoint representationIf $j^\\mu $ is assumed to transform covariantly, then gauge invariance of the action (REF ) implies $\\partial _\\mu j^\\mu =0$ as has already been noticed in reference [36]..", "However, we emphasize that we started from the action (REF ) to obtain the equation of motion of $x^\\mu $ , the one of $q$ following from the requirement of gauge invariance of the initial actionIn fact, if the charge $q$ is treated as a dynamical variable in the action (REF ), then it amounts to a Lagrange multiplier leading to the equation of motion $\\dot{x}^\\mu A^a_\\mu (x(\\tau ))=0$ which is not gauge invariant.", "We thank the anonymous referee for drawing our attention to this point.. An action which yields the equations (REF ) as equations of motion of both $x^\\mu $ and $q$ has been constructed in the non-relativistic setting in reference [28].", "The relativistic generalization of this approach proceeds as follows.", "(For simplicity, we put the coupling constant $g$ equal to one.)", "One introduces a Lie algebra-valued variable $\\Lambda (\\tau )\\equiv \\Lambda ^a (\\tau ) T^a$ where the functions $\\Lambda ^a (\\tau )$ are Grassmann odd, the charge $q$ being defined as an expression which is bilinear in $\\Lambda $ (i.e.", "a description which is familiar for the spin): $q \\equiv -\\frac{1}{2}[\\Lambda , \\Lambda ]_+,\\qquad \\text{i.e.", "}\\qquad q^a= -\\frac{{\\rm i}}{2}f^{abc} \\Lambda ^b \\Lambda ^c.$ The Lagrangian $L(x, \\dot{x}, \\Lambda , \\dot{\\Lambda }) \\equiv \\frac{m}{2}\\dot{x}^2 + \\frac{{\\rm i}}{2}\\operatorname{Tr}\\left(\\Lambda \\frac{D\\Lambda }{d\\tau }\\right)= \\frac{m}{2}\\dot{x}^2 + \\frac{{\\rm i}}{2}\\operatorname{Tr}(\\Lambda \\dot{\\Lambda }) + \\dot{x}^\\mu \\operatorname{Tr}(q A_\\mu ),$ is invariant under gauge transformations for which $\\delta _{\\lambda }A_\\mu = D_\\mu \\lambda $ and $\\delta _{\\lambda } \\Lambda (\\tau )= -{\\rm i}[\\Lambda (\\tau ), \\lambda (x (\\tau ))],$ which implies the transformation law (REF ).", "Moreover, the Lagrangian leads to the following equations of motion for $x^\\mu $ and $\\Lambda ^a$ : $m \\ddot{x}_\\mu = \\dot{x}^\\nu \\operatorname{Tr}\\lbrace q (\\partial _\\mu A_\\nu - \\partial _\\nu A_\\mu ) \\rbrace + \\operatorname{Tr}(\\dot{q} A_\\mu ),\\qquad 0= \\frac{D \\Lambda ^a}{d\\tau }.$ From (REF ) it follows that $\\frac{D q}{d\\tau }= - \\big [\\Lambda , \\frac{D \\Lambda }{d\\tau }\\big ]_+$ , hence (REF ) implies that $\\frac{D q^a}{d\\tau }=0$ , i.e.", "the charge $q$ is covariantly conserved.", "Substitution of the latter result into the first of equations (REF ) yields the equation of motion $m\\ddot{x}^\\mu =\\operatorname{Tr}(q F^{\\mu \\nu })\\dot{x}_\\nu $ .", "We note that the Hamiltonian associated to the Lagrangian (REF ) reads $H= \\frac{1}{2m}[p^\\mu - \\operatorname{Tr}(q A^\\mu )] [p_\\mu - \\operatorname{Tr}(q A_\\mu )],$ or $H= \\frac{m}{2}\\dot{x}^2$ if expressed in terms of the velocity.", "The Poisson brackets $\\lbrace x^\\mu , p_\\nu \\rbrace = \\delta ^\\mu _\\nu ,\\qquad \\lbrace \\Lambda _a, \\Lambda _b \\rbrace = -{\\rm i} \\delta _{ab},$ which imply the non-Abelian algebra of charges $\\lbrace q_a, q_b \\rbrace = f_{abc} q_c,$ again allow us to recover all previous equations of motion from the evolution equation of dynamical variables $F$ , i.e.", "from $\\dot{F}= \\lbrace F, H \\rbrace $ .", "In terms of the kinematical momentum $\\Pi _\\mu \\equiv p_\\mu - \\operatorname{Tr}(q A_\\mu )$ , the Hamiltonian reads $H= \\frac{1}{2m}\\Pi ^2 $ and the Poisson brackets take the form $\\lbrace x^\\mu , \\Pi _\\nu \\rbrace = \\delta ^\\mu _\\nu ,\\qquad \\lbrace \\Pi _\\mu , \\Pi _\\nu \\rbrace = \\operatorname{Tr}(q F_{\\mu \\nu }),\\qquad \\lbrace q_a, q_b \\rbrace = f_{abc} q_c.$ The dynamical variable $\\Lambda $ which allowed for the Lagrangian formulation is well hidden in the latter equations.", "Continuum formulation of the dynamics.", "The field strength $F_{\\mu \\nu }$ manifests itself physically by the force field, i.e.", "by an exchange of energy and momentum between the charge carrier and the field [40].", "In the framework of field theory, the physical entities are described by local fields, i.e.", "one has a continuum formulation.", "In order to obtain such a formulation for the particle's equations of motion (REF ), we have to integrate these relations over the variable $\\tau $ with a delta function concentrated on the particle's trajectory.", "The resulting expressions then involve the current density $j^{a\\mu }(y)$ defined in equation (REF ) as well as the energy-momentum tensor (density) of the point particle which is given by (see Appendix ) $T^{\\mu \\nu }(y)=\\int d\\tau m{\\dot{x}}^\\mu {\\dot{x}}^\\nu \\delta ^4 (y-x(\\tau ) ).$ More explicitly, by using $\\dot{x}^\\nu \\partial ^y_\\nu \\delta ^4 (y-x(\\tau ) )= - \\dot{x}^\\nu \\partial ^x_\\nu \\delta ^4 (y-x(\\tau ) )= -\\frac{d}{d \\tau }\\delta ^4 (y-x(\\tau ) ),$ we have $\\partial ^y_\\nu T^{\\nu \\mu }(y)= \\int d\\tau m\\dot{x}^\\mu \\dot{x}^\\nu \\partial ^y_\\nu \\delta ^4 (y-x(\\tau ) )= - \\int d\\tau m \\dot{x}^\\mu \\frac{d}{d \\tau } \\delta ^4 (y-x(\\tau ) )\\\\\\phantom{\\partial ^y_\\nu T^{\\nu \\mu }(y)}= \\int d\\tau m\\ddot{x}^\\mu \\delta ^4 (y-x(\\tau ) ),$ and substitution of the particle's equation of motion $ m\\ddot{x}^\\mu = \\operatorname{Tr}(qF^{\\mu \\nu }) \\dot{x}_{\\nu }$ then yields $\\partial ^y_\\nu T^{\\nu \\mu }(y)= \\int d\\tau \\dot{x}_{\\nu } q^a F_a^{\\mu \\nu } (x(\\tau ))\\delta ^4 (y-x(\\tau ) )=F_a^{\\mu \\nu } (y)\\int d\\tau \\dot{x}_{\\nu } q^a\\delta ^4 (y-x(\\tau ) )\\\\\\phantom{\\partial ^y_\\nu T^{\\nu \\mu }(y)}= F_a^{\\mu \\nu } (y)j^a_{\\nu } (y).$ The continuum version of Wong's equations (REF ) thus reads $\\partial _\\nu T^{\\nu \\mu }= \\operatorname{Tr}(F^{\\mu \\nu }j_{\\nu }),\\qquad \\text{where}\\qquad D_{\\mu } j^{\\mu }=0.$ These equations describe the exchange of energy and momentum between the field $F^{\\mu \\nu }$ and the current $j^\\mu $ (i.e.", "the matter).", "They admit an obvious generalization to Moyal space, see equation (REF ) below.", "In Appendix , we show that they also admit a natural extension to curved space (endowed with a metric tensor $(g_{\\mu \\nu } (x))$ ).", "Moreover, we will prove there that they have to hold for arbitrary dynamical matter fields $\\phi $ whose dynamics is described by a generic action $S[\\phi ; g_{\\mu \\nu }, A_{\\mu }^a]$ which is invariant under both gauge transformations and general coordinate transformations ($g_{\\mu \\nu }$ and $A_{\\mu }^a$ representing fixed external fields).", "We note that the expectation values of equations (REF ) viewed as operatorial relations in quantum field theory imply Wong's classical equations of motion for sufficiently localized, quantum “wave-packet” states [9].", "Curved space.", "Finally, we also point out that equations which are somewhat similar to Wong's equations appear in general relativity for a spinning particle in curved space, for which case the contraction of the Riemann tensor $R^\\alpha _{\\hphantom{\\alpha }\\beta \\mu \\nu }$ with the spin tensor $S^{\\mu \\nu }$ plays a role which is similar to the field strength $F^{\\mu \\nu }$ in Yang–Mills theories.", "The explicit form of these equations of motion, which are known as the Mathisson–Papapetrou–Dixon equations [13], [30], [33], is given by $\\frac{\\nabla }{d{\\tau }} (mu^{\\alpha } )+\\frac{1}{2}S^{\\mu \\nu }u^{\\sigma } R^{\\alpha }_{\\hphantom{\\alpha }\\sigma \\mu \\nu }=0,\\qquad \\frac{\\nabla S^{\\alpha \\beta }}{d{\\tau }}=0,$ where $\\frac{\\nabla }{d{\\tau }}$ denotes the covariant derivative along the trajectory and $(u^{\\mu })$ is the particle's four-velocity." ], [ "Lagrangian approach to particles in NC space", "Moyal space and distributions.", "We consider four dimensional Moyal space, i.e.", "we assume that the space-time coordinates fulfill a Heisenberg-type algebra (for a review see [8], [35], [39] and references therein).", "Thus, the star product of functions is defined by $(f \\star g ) (x) \\equiv \\big ({\\rm e}^{\\frac{{\\rm i}}{2}\\theta ^{\\mu \\nu } \\partial _\\mu ^x \\partial _\\nu ^y}f(x) g(y) \\big ) \\big \|_{x=y},$ where the parameters $\\theta ^{\\mu \\nu }= - \\theta ^{\\nu \\mu }$ are constant, and their star commutator is defined by $[f\\stackrel{\\star }{,}g]\\equiv f \\star g - g \\star f$ , which implies that $[x^\\mu \\stackrel{\\star }{,}x^\\nu ]= {\\rm i}\\theta ^{\\mu \\nu }$ .", "In the sequel we will repeatedly use the following fundamental properties of the star product: $\\int d^4xf \\star g= \\int d^4xf \\cdot g,\\qquad \\int d^4xf \\star g \\star h= \\int d^4xh \\star f \\star g.$ An important point to note is that in Moyal space, the integral $\\int d^4x$ plays the role of a traceThis is best seen by employing the Weyl map from operators to functions, and is also the reason for the cyclicity property (REF ).", "Furthermore, when considering gauge fields only the action is gauge invariant, not the Lagrangian., and hence equations of motion must always be derived from the action rather than the Lagrangian.", "For a detailed discussion of the algebras of functions and of distributions on Moyal space in the context of non-commutative spaces and of quantum mechanics in phase space, we refer to [43] and [21], [42], respectively.", "Here, we only note that the star product of the delta distribution $\\delta _y$ (with support in $y$ ) with a function $\\psi $ may be defined by application to a test function $\\varphi $ : $\\langle \\delta _y \\star \\psi ,\\varphi \\rangle \\equiv \\int d^4x (\\delta _y \\star \\psi ) (x)\\varphi (x)=\\int d^4x (\\delta _y \\star \\psi \\star \\varphi ) (x)=\\int d^4x\\delta _y (x) (\\psi \\star \\varphi ) (x)\\\\\\phantom{\\langle \\delta _y \\star \\psi ,\\varphi \\rangle }= (\\psi \\star \\varphi ) (y).$ Hence, the action of the distribution $\\delta _y \\star \\psi $ on the test function $\\varphi $ is equal to the action of the distribution $\\delta _y $ on the test function $\\psi \\star \\varphi $ .", "Similarly, we find $\\langle \\psi \\star \\delta _y,\\varphi \\rangle \\equiv \\int d^4x (\\psi \\star \\delta _y ) (x)\\varphi (x)= \\int d^4x (\\psi \\star \\delta _y \\star \\varphi ) (x)= \\int d^4x (\\delta _y \\star \\varphi \\star \\psi ) (x)\\\\\\phantom{\\langle \\psi \\star \\delta _y,\\varphi \\rangle }= (\\varphi \\star \\psi ) (y).$ The following considerations hold for an arbitrary antisymmetric matrix $(\\theta ^{\\mu \\nu })$ , but for the physical applications it is preferable to assume that $\\theta ^{\\mu 0}=0$ , i.e.", "assume the time to be commuting with the spatial coordinates.", "This choice is motivated by the fact that the parameters $\\theta ^{ij}$ have close analogies with a constant magnetic field both from the algebraic and dynamical points of view [11], and by the fact that a non-commuting time leads to problems with time-ordering in quantum field theory [5].", "Charged particle in Moyal space.", "Since a $U_\\star (1)$ gauge field $(A^\\mu )$ on Moyal space entails a non-Abelian structure for the field strength tensor $(F_{\\mu \\nu })$ due to the star product [8], [35], [39], $F_{\\mu \\nu } \\equiv \\partial _\\mu A_\\nu - \\partial _\\nu A_\\mu - {\\rm i} g [A_\\mu \\stackrel{\\star }{,}A_\\nu ],$ one expects that the treatment of a source for this gauge field as given by a “point” particle should allow for a description of such particles on non-commutative space which is quite similar to the one found by Wong in the case of Yang–Mills theory.", "If the matter content in field theory is given by a spinor field $\\psi $ , then the interaction term with the gauge field reads $\\int d^4yJ^\\mu \\star A_\\mu ,\\qquad \\text{with}\\qquad J^\\mu \\equiv g\\bar{\\psi } \\gamma ^\\mu \\star \\psi ,$ i.e.", "it involves two star products.", "By virtue of the properties (REF ) one of these star products can be dropped under the integral, but not both of them.", "If we consider the particle limit (i.e.", "$J^\\mu $ representing the current density of the particle), it is judicious to maintain the star product between $J^\\mu $ and $A_\\mu $ so as not to hide the non-commutative nature of the underlying space (over which we integrate) and to allow for the use of the cyclic invariance property (REF ) later on in our derivation.", "In fact, the pairing $\\langle J, A\\rangle \\equiv \\int d^4yJ^\\mu \\star A_\\mu $ represents the analogue of the pairing $\\langle j, A\\rangle \\equiv \\int d^4y\\operatorname{Tr}(j^\\mu A_\\mu )$ in Yang–Mills theory on commutative space.", "Accordingly, we will require the action for the particle to be invariant under non-commutative gauge transformations defined at the infinitesimal level by $\\delta _{\\lambda }A_\\mu = D_\\mu \\lambda \\equiv \\partial _\\mu \\lambda - {\\rm i}g[A_\\mu \\stackrel{\\star }{,}\\lambda ]$ where the parameter $\\lambda $ is an arbitrary function.", "As was done for Wong's equations, we assume that the charge $q$ of the relativistic particle in non-commutative space depends on the parameter $\\tau $ parametrizing the particle's world line and that it represents a given non-dynamical variable.", "The coupling of this particle to an external $U_\\star (1)$ gauge field $(A^\\mu )$ can be described by the actionWe note that the coupling of $U_{\\star } (1)$ gauge fields to external currents has also been addressed in the recent work [4].", "$S[x] \\equiv S_{\\rm free} [x] - S_{\\rm int} [x] \\equiv -m \\int d\\tau \\sqrt{\\dot{x}^2} - \\int d^4yJ^\\mu A_\\mu ,$ where $J^\\mu (y) \\equiv \\int d\\tau q(\\tau )\\dot{x}^\\mu (\\tau )\\delta ^4 (y-x(\\tau ) ).$ Keeping in mind the formulation of field theory on non-commutative space [8], [35], [39], we will argue directly with the action rather than the Lagrangian function and require this action to be invariant under non-commutative gauge transformations.", "For such an infinitesimal transformation we have $\\delta _\\lambda \\int d^4yJ^\\mu A_\\mu =\\int d^4yJ^\\mu D_\\mu \\lambda =-\\int d^4y(D_\\mu J^\\mu ) \\star \\lambda .$ Hence, invariance of the action (REF ) under non-commutative gauge transformations requires the current to be covariantly conserved, i.e.", "$D_\\mu J^\\mu =0$ .", "By virtue of equation (REF ) we now infer that $0 \\stackrel{!", "}{=}(D_\\mu J^\\mu )(y)=\\int d\\tau q\\dot{x}^\\mu D^y_\\mu \\delta ^4(y-x(\\tau ))\\nonumber \\\\\\phantom{0 \\stackrel{!", "}{=}(D_\\mu J^\\mu )(y)}=\\int d\\tau q\\dot{x}^\\mu \\left\\lbrace \\partial ^y_\\mu \\delta ^4(y-x(\\tau ))-{\\rm i}g[A_\\mu (y)\\stackrel{\\star }{,}\\delta ^4(y-x(\\tau ))]\\right\\rbrace .$ Equation (REF ) entails that the first term in the last line can be rewritten as $\\int d\\tau q\\dot{x}^\\mu \\partial ^y_\\mu \\delta ^4(y-x(\\tau ))= - \\int d\\tau q\\frac{d}{d\\tau } \\delta ^4(y-x(\\tau ))= \\int d\\tau \\frac{dq}{d\\tau }\\delta ^4(y-x(\\tau )).$ From condition (REF ) it thus follows that the charge $q$ has to be covariantly conserved along the world line in the sense that $0= \\int d\\tau \\frac{Dq}{d\\tau }\\delta ^4(y-x(\\tau )) \\equiv \\int d\\tau \\left\\lbrace \\frac{dq}{d\\tau }\\delta ^4(y-x(\\tau ))-{\\rm i}gq\\dot{x}^\\mu [A_\\mu (y)\\stackrel{\\star }{,}\\delta ^4(y-x(\\tau ))] \\right\\rbrace .$ For later reference, we note that this relation yields the following equality after star multiplication with $A_\\nu (y) \\delta y^{\\nu } $ and integration over $y$ : $\\int d^4y \\int d\\tau \\delta x^{\\nu } (\\tau ) \\frac{dq}{d\\tau }\\delta ^4 (y-x(\\tau )) \\star A_\\nu (y)\\nonumber \\\\\\qquad = {\\rm i} g \\int d^4y \\int d\\tau \\delta x^{\\nu } (\\tau ) q\\dot{x}^\\mu [A_\\mu (y)\\stackrel{\\star }{,}\\delta ^4 (y-x(\\tau ))] \\star A_\\nu (y)\\nonumber \\\\\\qquad = - {\\rm i} g \\int d^4y \\int d\\tau \\delta x^{\\nu } (\\tau ) q\\dot{x}^\\mu \\delta ^4 (y-x(\\tau )) \\star [A_\\mu (y)\\stackrel{\\star }{,}A_\\nu (y)].$ In order to derive the equation of motion for the particle determined by the action (REF ), we vary the latter with respect to $x^\\mu $ .", "The variation of $S_{\\rm free}$ being the same as in commutative space, we only work out the variation of the interaction part $S_{\\rm int}$ , all star products being viewed as functions of the variable $y$ : $\\delta S_{\\rm int}=\\delta \\int d^4y \\int d\\tau \\big \\lbrace \\dot{x}^\\mu q\\delta ^4(y-x(\\tau ))\\star A_\\mu (y) \\big \\rbrace \\\\\\phantom{\\delta S_{\\rm int}}=\\int d^4y \\int d\\tau \\big \\lbrace \\delta \\dot{x}^\\mu q\\delta ^4(y-x(\\tau ))\\star A_\\mu (y) +\\dot{x}^\\mu q\\delta \\big [\\delta ^4(y-x(\\tau ))\\big ] \\star A_\\mu (y)\\big \\rbrace \\\\\\phantom{\\delta S_{\\rm int}}=\\int d^4y \\int d\\tau \\left\\lbrace \\frac{d (\\delta x^\\mu )}{d\\tau }q\\delta ^4(y-x(\\tau ))\\star A_\\mu (y)+\\dot{x}^\\mu q(\\delta x^\\nu )\\partial ^x_\\nu \\delta ^4(y-x(\\tau ))\\star A_\\mu (y)\\right\\rbrace \\\\\\phantom{\\delta S_{\\rm int}}=\\int d^4y \\int d\\tau (\\delta x^\\nu ) \\left\\lbrace - \\frac{d}{d\\tau } \\big [q\\delta ^4(y-x (\\tau ))\\big ]\\star A_\\nu (y) +\\dot{x}^\\mu q\\delta ^4(y-x)\\star \\partial ^y_\\nu A_\\mu (y)\\right\\rbrace .$ By virtue of the product rule, the first term in the last line yields two terms, one involving $\\frac{dq}{d\\tau }$ which can be rewritten using relation (REF ), and one involving $\\frac{d}{d\\tau } \\delta ^4(y-x(\\tau ))$ which can be rewritten using (REF ): $- \\int d^4y \\int d\\tau (\\delta x^\\nu )q\\frac{d}{d\\tau } \\delta ^4(y-x (\\tau )) \\star A_\\nu (y)\\\\\\qquad = \\int d^4y \\int d\\tau (\\delta x^\\nu )q\\dot{x}^\\mu \\big [\\partial ^y_\\mu \\delta ^4(y-x (\\tau )) \\big ] \\star A_\\nu (y)\\\\\\qquad = -\\int d^4y \\int d\\tau (\\delta x^\\nu )q\\dot{x}^\\mu \\delta ^4(y-x (\\tau )) \\star \\partial ^y_\\mu A_\\nu (y).$ Hence, we arrive at $\\delta S_{\\rm int}=\\int d^4y \\int d\\tau (\\delta x^\\nu )\\dot{x}^\\mu q\\delta ^4(y-x (\\tau )) \\star \\big \\lbrace \\partial ^y_\\nu A_\\mu (y) - \\partial ^y_\\mu A_\\nu (y) + {\\rm i}g[A_\\mu (y)\\stackrel{\\star }{,}A_\\nu (y)] \\big \\rbrace \\\\\\phantom{\\delta S_{\\rm int}}=\\int d^4y \\int d\\tau (\\delta x^\\nu )\\dot{x}^\\mu q\\delta ^4(y-x (\\tau )) \\star F_{\\nu \\mu }(y)= \\int d\\tau (\\delta x^\\nu )\\dot{x}^\\mu qF_{\\nu \\mu }(x(\\tau )).$ Note that it is the constraint equation (REF ) following from (REF ) that yields the terms which are quadratic in the gauge field.", "This is very much the same mechanism as in the commutative space calculation which leads to Wong's equations, cf.", "(REF )–(REF ).", "In conclusion, we obtain the following equation of motion for the charged relativistic particle in non-commutative space: $m\\ddot{x}^\\mu =q F^{\\mu \\nu }\\dot{x}_\\nu ,\\qquad \\text{with}\\qquad F_{\\mu \\nu } \\equiv \\partial _\\mu A_\\nu - \\partial _\\nu A_\\mu - {\\rm i} g [A_\\mu \\stackrel{\\star }{,}A_\\nu ].$ We note that the antisymmetry of $F_{\\mu \\nu }$ with respect to its indices implies $0= \\ddot{x}^\\mu \\dot{x}_\\mu = \\frac{1}{2}\\frac{d \\dot{x}^2}{d\\tau },$ which is consistent with $\\dot{x}^2=1$ .", "Consistency of the equation of motion (REF ) requires its gauge invariance, i.e.", "the gauge invariance of its right-hand-side.", "Since $\\delta _{\\lambda } (q F^{\\mu \\nu }\\dot{x}_\\nu )= q(\\delta _{\\lambda } F^{\\mu \\nu }) \\dot{x}_\\nu = - {\\rm i} gq [F^{\\mu \\nu }\\stackrel{\\star }{,}\\lambda ]\\dot{x}_\\nu = gq \\theta ^{\\rho \\sigma } (\\partial _{\\rho } F^{\\mu \\nu }) (\\partial _{\\sigma } \\lambda ) \\dot{x}_\\nu + {\\cal O} \\big (\\theta ^2\\big ),$ the gauge invariance only holds for constant field strengths by contrast to the case of Wong's equation for Yang–Mills fields in commutative space.", "This difference can be traced back to the fact that the analogue of the trace in Yang–Mills theory is given in Moyal space by the integral $\\int d^4x$ : since such an integral does not occur in the differential equation (REF ), the gauge invariance is only realized for constant $F_{\\mu \\nu }$ .", "We will further discuss the latter fields and the comparison with Yang–Mills theory in Section where we will also consider the subsidiary condition for the charge $q$ .", "Here we only note the following points in this respect.", "The equation of motion (REF ) taken for itself is consistent for any constant values of $q$ and $F_{\\mu \\nu }$ .", "Furthermore in non-commutative space the auxiliary variable $q$ cannot be rendered dynamical with a non-trivial gauge transformation law (REF ) for $q(\\tau )$ since the star commutator of $q(\\tau )$ with a gauge transformation parameter $\\lambda (x(\\tau ))$ vanishes.", "In summary, the coupling of a relativistic particle to a gauge field $(A^\\mu )$ is described in general by the Lagrangian $L(x, \\dot{x})= -m \\sqrt{\\dot{x}^2} - q A_\\mu \\dot{x}^\\mu ,\\qquad \\text{or}\\qquad L(x, \\dot{x})= \\frac{m}{2}\\dot{x}^2 + q A_\\mu \\dot{x}^\\mu .$ The non-commutativity of space-time can be implemented in the Lagrangian framework by rewriting the interaction term of the action as an integral $\\int d^4y (J^\\mu \\star A_\\mu ) (y)$ where the current $J^\\mu $ is defined by (REF ) and by requiring this action to be invariant under non-commutative gauge transformations.", "The resulting equation of motion (REF ) is only gauge invariant for constant field strengths.", "Continuum formulation of the dynamics.", "By following the same lines of arguments as for the Lorentz–Yang–Mills force equation (see equations (REF )–(REF )), we can obtain a continuum version of the equation of motion (REF ) by multiplying this equation with $\\delta ^4 (y-x(\\tau )) \\varphi (y)$ , where $\\varphi (y)$ is a suitable test function, and integrating over $\\tau $ and over $y$ .", "More explicitly, by starting from the energy-momentum tensor (REF ) for the point particle and using relation (REF ), we get $\\int d^4y(\\partial _\\nu T^{\\nu \\mu })(y) \\varphi (y)= \\int d^4y\\int d\\tau m\\dot{x}^\\mu \\dot{x}^\\nu \\partial ^y_\\nu \\delta ^4(y-x(\\tau )) \\varphi (y)\\\\\\qquad = - \\int d^4y\\int d\\tau m \\dot{x}^\\mu \\frac{d}{d \\tau } \\delta ^4 (y-x(\\tau )) \\varphi (y)= \\int d^4y\\int d\\tau m\\ddot{x}^\\mu \\delta ^4 (y-x(\\tau )) \\varphi (y).$ Substitution of equation (REF ) then yields the expression $\\int d^4y\\int d\\tau q F^{\\mu \\nu } (x(\\tau )) \\dot{x}_\\nu \\delta ^4 (y-x(\\tau )) \\varphi (y)= \\int d^4y\\int d\\tau qF^{\\mu \\nu } (y) \\dot{x}_\\nu \\delta ^4 (y-x(\\tau )) \\varphi (y)\\\\\\qquad =\\int d^4y(F^{\\mu \\nu } J_\\nu )(y)\\varphi (y),$ where we considered the current density (REF ) in the last line.", "Thus, we have the result $\\partial _\\nu T^{\\nu \\mu }= F^{\\mu \\nu } J_\\nu ,\\qquad \\text{where}\\qquad D_{\\mu } J^{\\mu }=0,$ and these relations are completely analogous to the continuum equations (REF ) which correspond to Wong's equations (apart from the fact that the invariance of (REF ) under non-commutative gauge transformations requires the field strength $F^{\\mu \\nu }$ to be constant)." ], [ "Case of a constant field strength", "We will now discuss the dynamics of charged particles coupled to a constant field strength on the basis of the results obtained in Sections and .", "Indeed this case represents a mathematically tractable and physically interesting application of the general formalism.", "We successively discuss the case of non-Abelian Yang–Mills theory on Minkowski space and the case of a $U_{\\star }(1)$ gauge field on Moyal space while emphasizing the differences that exist for constant field strengths." ], [ "Wong's equations in commutative space", "The case of a “uniform field strength” in Yang–Mills theory has been addressed some time ago by the authors of reference [9], see also [19] for related points.", "Since the Yang–Mills field strength $F_{\\mu \\nu } (x) \\equiv F_{\\mu \\nu }^a (x) T^a$ is not gauge invariant, but transforms under finite gauge transformations as $F_{\\mu \\nu }^{\\prime }= U^{-1} F_{\\mu \\nu } U$ (with $U(x) \\in G=$ structure group), one has to specify first what is meant by a constant field.", "The field $F_{\\mu \\nu }$ is said to be uniform if the gauge field $A_{\\mu } (x) \\equiv A_{\\mu }^a (x) T^a$ at a point $x$ can be related by a gauge transformation to the gauge field $A_{\\mu } (y)$ at any other point $y$ .", "More precisely, for a space-time translation parametrized by $a \\in {\\mathbb {R}}^4$ , there exists a gauge transformation $x\\mapsto U(x; a) \\in G$ such that $A_{\\mu } (x +a)= U^{-1} (x; a) A_{\\mu } (x) U(x; a) + {\\rm i}U^{-1} (x; a) \\partial _{\\mu } U (x; a),$ and thereby $F_{\\mu \\nu } (x + a)= U^{-1} (x; a) F_{\\mu \\nu } (x) U(x; a).$ In this case a gauge may be chosen in which all components of $F_{\\mu \\nu }$ are constant because $F_{\\mu \\nu } $ at the point $x$ can be made equal to its value at some arbitrary point $y$ by transforming it by an appropriate gauge group element.", "Since the field strength $F_{\\mu \\nu } \\equiv \\partial _{\\mu } A_{\\nu } - \\partial _{\\nu } A_{\\mu } + {\\rm i} [A_{\\mu }, A_{\\nu }]$ contains two terms, namely $ \\partial _{\\mu } A_{\\nu } - \\partial _{\\nu } A_{\\mu }$ which has the Abelian form, and $ [A_{\\mu },A_{\\nu }]$ which does not involve derivatives, a constant non-zero field strength $F_{\\mu \\nu }$ can be obtained either from a linear gauge potential (i.e.", "an Abelian-like gauge field), $A_{\\mu }= - \\frac{1}{2}F_{\\mu \\nu } x^{\\nu },\\qquad \\partial _{\\mu } A_{\\nu } - \\partial _{\\nu } A_{\\mu }= F_{\\mu \\nu }= \\operatorname{const},\\qquad [A_{\\mu }, A_{\\nu }]=0,$ or from a constant non-Abelian-like gauge potential, $A_{\\mu }= \\operatorname{const},\\qquad \\partial _{\\mu } A_{\\nu }=0,\\qquad {\\rm i}[A_{\\mu }, A_{\\nu }]= F_{\\mu \\nu }= \\operatorname{const}.$ In fact [9], these two types of potentials exhaust all possibilities for a constant field strength.", "It has been shown for the structure group $\\text{SU}(2)$ that the two types of gauge potentials leading to a same constant field strength are gauge inequivalent and result in physically different behavior when matter interacts with them, e.g.", "the solutions of Wong's equations have completely different properties in both cases.", "An explicit example for $G= \\text{SU}(2)$ is given by a constant magnetic field in $z$ -direction [19]: let $\\sigma _k$ (with $k=1,2,3$ ) denote the Pauli matrices and suppose $F_{0i}=0,\\qquad F_{ij}= \\varepsilon _{ijk} B_k,\\qquad \\text{with}\\qquad (B_k)_{k=1,2,3}= (0,0, 2 \\sigma _3).$ This constant field strength derives from the linear Abelian-like potential $\\vec{A}\\equiv (A_k)_{k=1,2,3}= - \\frac{1}{2}\\vec{x} \\wedge \\vec{B}= (-y, x, 0)\\sigma _3,$ or from a constant non-Abelian-like potential $\\vec{A}= (-\\sigma _2, \\sigma _1, 0)$ ." ], [ "Wong's equations in non-commutative space", "The field strength associated to a $U_{\\star }(1)$ gauge field $(A_{\\mu })$ on Moyal space reads $F_{\\mu \\nu } \\equiv \\partial _{\\mu } A_{\\nu } - \\partial _{\\nu } A_{\\mu } - {\\rm i} g [A_{\\mu }\\stackrel{\\star }{,}A_{\\nu }]= \\partial _{\\mu } A_{\\nu }- \\partial _{\\nu } A_{\\mu } + g \\theta ^{\\rho \\sigma } (\\partial _{\\rho } A_{\\mu })(\\partial _{\\sigma } A_{\\nu }) + {\\cal O}\\big (\\theta ^2\\big ).$ Due to the derivatives appearing in the star commutator, the Abelian-like term $\\partial _{\\mu } A_{\\nu } - \\partial _{\\nu } A_{\\mu }$ and the non-Abelian-like term $- {\\rm i} g [A_{\\mu }\\stackrel{\\star }{,}A_{\\nu }]$ cannot vanish independently of each other: the non-commutative field strength $F_{\\mu \\nu }$ can only be constant for a linear Abelian-like potential.", "More precisely, for $A_\\mu = -\\frac{1}{2}\\bar{B}_{\\mu \\nu } x^\\nu ,$ where the coefficients $\\bar{B}_{\\mu \\nu } \\equiv - \\bar{B}_{\\nu \\mu }$ are constant, we obtain $F_{\\mu \\nu }= \\bar{B}_{\\mu \\nu } - \\frac{g}{4}\\bar{B}_{\\mu \\rho } \\theta ^{\\rho \\sigma } \\bar{B}_{\\sigma \\nu }.$ This field strength is constant, but dependent on the non-commutativity parameters $\\theta ^{\\mu \\nu }$ .", "If we interpret $\\bar{B}_{\\mu \\nu }= \\partial _\\mu A_\\nu - \\partial _\\nu A_\\mu $ as the physical field strength, then (REF ) means that the non-commutativity parameters $\\theta ^{\\mu \\nu }$ modify in general the trajectories of the particle as compared to its motion in commutative spaceWe note that the latter dependence on the non-commutativity parameters can be eliminated mathematically if one assumes that $\\bar{B}_{\\mu \\nu }$ depends in a specific way on the parameters $\\theta ^{\\mu \\nu }$ and some $\\theta $ -independent constants ${B}_{\\mu \\nu }$ .", "To illustrate this point [11], we assume that the only non-vanishing components of $ \\bar{B}_{\\mu \\nu }$ and $\\theta ^{\\mu \\nu }$ are as follows: $\\bar{B}_{12}= - \\bar{B}_{21} \\equiv \\bar{B}, \\theta ^{12}= - \\theta ^{21} \\equiv \\theta $ , i.e.", "$F_{12}= \\bar{B} (1 + \\frac{g}{4}\\theta \\bar{B})= - F_{21}$ .", "If $\\bar{B}$ depends on $\\theta $ and on a $\\theta $ -independent constant $B$ according to $\\bar{B}= \\bar{B} (B; \\theta ) \\equiv \\frac{2}{g \\theta }\\big (\\sqrt{1 + g \\theta B}-1\\big )=B\\left(1-\\frac{g}{4}\\theta B\\right)+{\\cal O}\\big (\\theta ^2\\big ),$ then relation (REF ) implies that the non-commutative field strength $ F_{12} $ is a $\\theta $ -independent constant: $F_{12}= B$ .", ".", "Since the field strength transforms under a finite gauge transformation $U(x) \\in U_{\\star }(1)$ as $F_{\\mu \\nu }= U^{-1} \\star F_{\\mu \\nu } \\star U$ , a constant field $F_{\\mu \\nu }$ is gauge invariant.", "Thus, for constant field strengths the situation is quite different for $ U_{\\star }(1)$ gauge fields and for non-Abelian gauge fields on Minkowski space despite the fact that we encounter the same structure for the gauge transformations, the field strength and the action functional in both cases.", "Let us again come back to the expression (REF ) for the gauge field.", "Substitution of this expression into the subsidiary condition (REF ) yields $0= \\int d\\tau \\big \\lbrace \\dot{q}\\delta ^4(y-x(\\tau )) - {\\rm i}gq\\dot{x}^\\mu [A_\\mu (y)\\stackrel{\\star }{,}\\delta ^4(y-x(\\tau ))] \\big \\rbrace \\nonumber \\\\\\phantom{0}= \\int d\\tau \\big \\lbrace \\dot{q}\\delta ^4(y-x(\\tau )) - \\frac{g}{2}q\\dot{x}^\\mu \\bar{B}_{\\mu \\rho } \\theta ^{\\rho \\sigma }\\partial ^y_{\\sigma } \\delta ^4(y-x(\\tau )) \\big \\rbrace .$ If the matrix $(\\bar{B}_{\\mu \\nu })$ is the inverse of the matrix $(\\theta ^{\\mu \\nu })$ , i.e.", "$\\bar{B}_{\\mu \\rho } \\theta ^{\\rho \\sigma }= \\delta ^{\\sigma }_{\\mu }$ , then $F_{\\mu \\nu }= (1 - \\frac{g}{4})\\bar{B}_{\\mu \\nu }$ and, by virtue of (REF ) and an integration by parts, condition (REF ) takes the form $0= \\int d\\tau \\dot{q}\\delta ^4(y-x(\\tau )) \\left\\lbrace 1 - \\frac{g}{2} \\right\\rbrace .$ The latter relation is obviously satisfied for a constant $q$ .", "In this case, the equation of motion (REF ) for the particle in non-commutative space, i.e.", "$m \\ddot{x}^\\mu = q F^{\\mu \\nu } \\dot{x}_\\nu $ , has the same form as the one of an electrically charged particle in ordinary space.", "This result is analogous to the one obtained for a constant magnetic field in $x^3$ -direction within the Hamiltonian approaches, see equations (REF ) and (REF ) below." ], [ "Hamiltonian approaches to particles in NC space", "To start with, we briefly review the Hamiltonian approaches in commutative space before considering the generalization to the non-commutative setting.", "In the latter setting, we will notice that various approaches yield different results since several expressions which coincide in commutative space no longer agree." ], [ "Reminder on the Poisson bracket approach", "The Hamiltonian formulation of relativistic (as well as non-relativistic) mechanics is based on two inputs (e.g.", "see reference [29]): a Hamiltonian function and a Poisson structure (or equivalently a symplectic structure).", "If one starts from the Lagrangian formulation, the Hamiltonian function is obtained from the Lagrange function by a Legendre transformation.", "E.g.", "the Lagrangian $L (x, \\dot{x})= \\frac{m}{2}\\dot{x}^2 + e A_\\mu \\dot{x}^\\mu $ (involving the constant charge $e$ ) yields the Hamiltonian $H (x,p)= \\frac{1}{2m}(p-eA)^2= \\frac{1}{2m}(p^\\mu -eA^\\mu ) (p_\\mu -eA_\\mu ).$ The trajectories in phase space are parametrized by $\\tau \\mapsto (x (\\tau ), p (\\tau ) ) $ where $\\tau $ denotes a real variable to be identified with proper time after the equations of motion have been derived.", "The Poisson brackets $\\lbrace \\cdot , \\cdot \\rbrace $ of the phase space variables $x^\\mu $ , $p^\\mu $ are chosen in such a way that the evolution equation $\\dot{F}= \\lbrace F, H \\rbrace $ (where $\\dot{F} \\equiv d F /d\\tau $ ) yields the Lagrangian equation of motion for $x^\\mu $ , though written as a system of first order differential equations.", "For instance, if we consider the usual form of the Poisson brackets, i.e.", "the canonical Poisson brackets $\\lbrace x^\\mu , x^\\nu \\rbrace = 0,\\qquad \\lbrace p^\\mu , p^\\nu \\rbrace = 0,\\qquad \\lbrace x^\\mu , p^\\nu \\rbrace = \\eta ^{\\mu \\nu },$ then substitution of $F= x^\\mu $ and $F= p^\\mu $ into $\\dot{F}= \\lbrace F, H \\rbrace $ (with $H$ given by (REF )) yields the system of equations $m \\dot{x}^\\mu = p^\\mu -eA^\\mu ,\\qquad m \\dot{p}^\\mu = e (p_\\nu -eA_\\nu ) \\partial ^{\\mu } A^\\nu ,$ from which we conclude that $m \\ddot{x}^\\mu = \\dot{p}^\\mu -e \\dot{A}^\\mu = \\underbrace{\\frac{e}{m}(p_\\nu -eA_\\nu )}_{=e\\dot{x}_\\nu } \\partial ^\\mu A^\\nu - e\\dot{x}^\\nu \\partial _\\nu A^\\mu = e \\dot{x}_\\nu \\big (\\partial ^\\mu A^\\nu - \\partial ^\\nu A^\\mu \\big ) \\equiv ef^{\\mu \\nu } \\dot{x}_\\nu .$ This equation coincides with the Euler–Lagrange equation for $x^\\mu $ following from the Lagrangian $L (x, \\dot{x})= \\frac{m}{2}\\dot{x}^2 + e A_\\mu \\dot{x}^\\mu $ .", "Concerning the gauge invariance, we emphasize a result [40] which does not seem to be very well known.", "The Hamiltonian (REF ) is not invariant under a gauge transformation $A_\\mu \\rightarrow A_\\mu + \\partial _\\mu \\lambda $ which is quite intriguing.", "However, it is invariant if this transformation is combined with the phase space transformation $(x^\\mu , p^\\mu ) \\rightarrow (x^\\mu , p^\\mu + e \\partial ^\\mu \\lambda )$ , the latter being a canonical transformation since it preserves the fundamental Poisson brackets (REF ).", "Indeed, under this combined transformation the kinematical momentum $\\Pi ^\\mu \\equiv p^\\mu -e A^\\mu $ (which coincides with $m \\dot{x}^\\mu $ ) is invariant.", "We note that the Hamiltonian (REF ) can be rewritten in terms of the variable $\\Pi _\\mu \\equiv p_\\mu - eA_\\mu $ as $H\\equiv \\frac{1}{2m}\\Pi ^2 $ .", "Thereby $H$ has the form of a free particle Hamiltonian, but the Poisson brackets are now modified: from $\\Pi _\\mu = p_\\mu -eA_\\mu $ and (REF ) it follows that $\\lbrace x^\\mu , x^\\nu \\rbrace = 0,\\qquad \\lbrace \\Pi ^\\mu , \\Pi ^\\nu \\rbrace = e f^{\\mu \\nu } (x),\\qquad \\lbrace x^\\mu , \\Pi ^\\nu \\rbrace = \\eta ^{\\mu \\nu },$ with $f^{\\mu \\nu } \\equiv \\partial ^\\mu A^\\nu - \\partial ^\\nu A^\\mu $ .", "Since the electromagnetic field strength $f^{\\mu \\nu }$ is gauge invariant, the latter invariance is manifestly realized in this formulation.", "In summary, the coupling of a charged particle to an electromagnetic field can either be described by the canonical Poisson brackets (REF ) and the minimally coupled Hamiltonian (REF ) or by introducing the field strength into the Poisson brackets (as a non-commutativity of the momenta) and considering a Hamiltonian which has the form of a free particle Hamiltonian." ], [ "Reminder on the symplectic form approach", "If we gather all phase space variables into a vector $\\vec{\\xi } \\equiv ({\\xi }^I) \\equiv (x^0, \\dots ,x^3, p^0, \\dots , p^3)$ , the fundamental Poisson brackets (REF ) read $\\big \\lbrace \\xi ^I, \\xi ^J \\big \\rbrace = \\Omega ^{IJ},\\qquad \\text{with}\\qquad \\big (\\Omega ^{IJ}\\big ) \\equiv \\begin{bmatrix}0 & \\eta ^{\\mu \\nu }\\\\- \\eta ^{\\mu \\nu } & 0\\end{bmatrix}.$ The inverse of the Poisson matrix $\\Omega \\equiv (\\Omega ^{IJ})$ is the matrix with entries $\\omega _{IJ} \\equiv (\\Omega ^{-1})_{IJ}$ which defines the symplectic 2-form $\\omega \\equiv \\frac{1}{2}\\sum \\limits _{I,J}\\omega _{IJ}d{\\xi }^I \\wedge d{\\xi }^J= dp^\\mu \\wedge dx_\\mu ,$ e.g.", "see reference [29] for mathematical details.", "The Hamiltonian equations of motion can be written as $\\dot{\\xi }^I= \\big \\lbrace \\xi ^I, H \\big \\rbrace = \\Omega ^{KJ}\\partial _K \\xi ^I \\partial _J H,\\qquad \\text{i.e.", "}\\qquad \\dot{\\xi }^I= \\Omega ^{IJ}\\partial _J H,$ or equivalently as $\\omega _{IJ}\\dot{\\xi }^J= \\partial _I H $ .", "In terms of the phase space variables $(x^\\mu , \\Pi ^\\mu )$ appearing in the non-canonical Poisson brackets (REF ), the symplectic 2-form (REF ) reads $\\omega = d\\Pi ^\\mu \\wedge dx_\\mu + \\frac{1}{2}e f_{\\mu \\nu }dx^\\mu \\wedge dx^\\nu .$ This formulation of the electromagnetic interaction based on the symplectic 2-form and the evolution equation $\\omega _{IJ}\\dot{\\xi }^J= \\partial _I H $ goes back to the seminal work of Souriau [37]." ], [ "Standard (Poisson bracket) approach to NC space-time", "In order to introduce a non-commutativity for the configuration space, one generally starts from a function $H$ on phase space to which one refers as the Hamiltonian without any reference to a Lagrangian, e.g.", "we can consider the function $H$ given in equation (REF ).", "The non-commutativity of space-time is then implemented by virtue of the Poisson brackets $\\lbrace x^\\mu , x^\\nu \\rbrace = \\theta ^{\\mu \\nu },\\qquad \\lbrace p^\\mu , p^\\nu \\rbrace = 0,\\qquad \\lbrace x^\\mu , p^\\nu \\rbrace = \\eta ^{\\mu \\nu },$ where $\\theta ^{\\mu \\nu }= - \\theta ^{\\nu \\mu }$ is again assumed to be constant.", "(For an overview of the description of non-relativistic charged particles in non-commutative space we refer to [3], [11], [24], the pioneering work being [17], [18], see also [1], [2], [31], [32] for some subsequent early work.", "We also mention that dynamical systems in non-commutative space can be constructed by applying Dirac's treatment of constrained Hamiltonian systems to an appropriate action functional, see [12] and references therein.)", "As in the commutative setting, we gather all phase space variables into a vector $\\vec{\\xi } \\equiv ({\\xi }^I) \\equiv (x^0,\\dots ,x^3, p^0, \\dots , p^3)$ , the fundamental Poisson brackets (REF ) being now given by $\\lbrace \\xi ^I, \\xi ^J \\rbrace = \\Omega ^{IJ},\\qquad \\text{with}\\qquad (\\Omega ^{IJ}) \\equiv \\begin{bmatrix}\\theta ^{\\mu \\nu } & \\eta ^{\\mu \\nu }\\\\- \\eta ^{\\mu \\nu } & 0\\end{bmatrix}.$ Quite generally, the Poisson bracket of two arbitrary functions $F$ , $G$ on phase space reads $\\lbrace F, G \\rbrace \\equiv \\sum \\limits _{I,J} \\Omega ^{IJ}\\partial _I F\\partial _J G= \\theta ^{\\mu \\nu }\\frac{\\partial F}{\\partial {x}^\\mu }\\frac{\\partial G}{\\partial {x}^\\nu } + \\frac{\\partial F}{\\partial {x}^\\mu }\\frac{\\partial G}{\\partial {p}_\\mu } -\\frac{\\partial F}{\\partial {p}^\\mu }\\frac{\\partial G}{\\partial {x}_\\mu }.$ Substitution of $F= x^\\mu $ and $F= p^\\mu $ into $\\dot{F}= \\lbrace F, H \\rbrace $ (with $H$ given by (REF )) yields the system of equations $m \\dot{x}^\\mu = (p_\\nu -eA_\\nu ) (\\eta ^{\\mu \\nu } - e \\theta ^{\\mu \\rho } \\partial _{\\rho } A^\\nu ),\\nonumber \\\\m \\dot{p}^\\mu = e (p_\\nu -eA_\\nu ) \\partial ^{\\mu } A^\\nu .$ In the present case, the phase space transformation $(x^\\mu , p^\\mu ) \\rightarrow (x^\\mu , p^\\mu + e \\partial ^\\mu \\lambda )$ does not represent a canonical transformation since it does not preserve the Poisson brackets (REF ) if $\\theta ^{\\mu \\nu } \\ne 0$ .", "Hence, the resulting Hamiltonian equations of motion (REF ) are not gauge invariant, as has already been pointed out in reference [11] by considering different gauges.", "In the next two subsections, we recall how this problem can be overcome for the particular case of a constant field strength as well as more generally, and we compare with the results obtained in Section from the action involving star products.", "Here, we only note that a non-Abelian structure of the field strength is hidden in equation (REF ).", "To illustrate this point, we consider the particular case where the only non-vanishing components of $ \\theta ^{\\mu \\nu }$ are $\\theta ^{ij}= \\varepsilon ^{ij} \\theta $ (with $i,j \\in \\lbrace 1, 2 \\rbrace $ and $\\varepsilon ^{12} \\equiv -\\varepsilon ^{21} \\equiv 1$ ) and where the only non-vanishing components of $A^\\mu $ are $A^i (x^1, x^2)$ (with $i \\in \\lbrace 1, 2 \\rbrace $ ).", "For this situation which describes a time-independent magnetic field perpendicular to the $x^1 x^2$ -plane, the first of equations (REF ) yields $m \\dot{x}_i= (p_k-eA_k) (\\delta _{ik} - e \\theta \\varepsilon _{ij} \\partial _{j} A_k),$ and implies $m\\frac{d}{dt} (x_i + e \\theta \\varepsilon _{ij} A_j )= (1 + e{\\cal F}_{12})(p_i -eA_i)$ with ${\\cal F}_{12} \\equiv \\partial _1 A_2 - \\partial _2 A_1 +e\\lbrace A_1, A_2 \\rbrace = \\partial _1 A_2 - \\partial _2 A_1 +e\\theta ^{\\rho \\sigma } (\\partial _{\\rho } A_1)(\\partial _{\\sigma } A_2).$ Thus, we find a non-Abelian structure for the generalized field strength, but in the present approach the field ${\\cal F}_{\\mu \\nu } $ is only linear in the non-commutativity parameters in contrast to the field $F_{\\mu \\nu } $ in (REF ) which involves the star commutator $- {\\rm i} [A_\\mu \\stackrel{\\star }{,}A_\\nu ]= \\theta ^{\\rho \\sigma } (\\partial _{\\rho } A_\\mu )(\\partial _{\\sigma } A_\\nu ) +{\\cal O} \\big (\\theta ^2\\big ).$ If the gauge potential is linear in $x$ , the field strengths ${\\cal F}_{\\mu \\nu }$ and $F_{\\mu \\nu }$ as defined by equations (REF ) and (REF ), respectively, coincide with each other (if one identifies the coupling constant $g$ with $e$ ).", "To conclude, we note that (REF ) can be solved for $p_i-eA_i$ in terms of $m \\dot{x}_i$ : the system of first order differential equations (REF ) can then be written as a second order equation for $x^\\mu $ , but the resulting equations of motion are not gauge invariant and they cannot be derived from a Lagrangian [3]." ], [ "Standard approach to NC space-time continued", "The reasoning presented concerning the brackets (REF ) suggests to consider a Hamiltonian which has a free form and to introduce a field strength $B^{\\mu \\nu } (x)$ as a non-commutativity of the momenta, i.e.", "consider phase space variables $(x^\\mu , p^\\mu )$ satisfying the non-canonical Poisson algebra $\\lbrace x^\\mu , x^\\nu \\rbrace = \\theta ^{\\mu \\nu },\\qquad \\lbrace p^\\mu , p^\\nu \\rbrace = e B^{\\mu \\nu },\\qquad \\lbrace x^\\mu , p^\\nu \\rbrace = \\eta ^{\\mu \\nu },$ with $\\theta ^{\\mu \\nu }$ constant.", "As pointed out in reference [22], the Jacobi identities for the algebra (REF ) are only satisfied if the field strength is constant: $\\lbrace x^\\mu , \\lbrace p^\\nu , p^\\lambda \\rbrace \\rbrace + \\lbrace \\text{cyclic permutations of $\\mu $, $\\nu $, $\\lambda $}\\rbrace =e \\theta ^{\\mu \\rho }\\partial _{\\rho } B^{\\nu \\lambda }.$ Thus, the dynamics of a charged particle coupled to a constant field $B^{\\mu \\nu }$ on non-commutative space-time can be described in terms of phase space variables $(x^\\mu , p^\\mu )$ satisfying the non-canonical Poisson algebra (REF ), the Hamiltonian being given by $H (p)= \\frac{1}{2m}p^2= \\frac{1}{2m}p^\\mu p_\\mu $ .", "The Hamiltonian equations of motion $m\\dot{x}^\\mu = p^\\mu ,\\qquad m\\dot{p}^\\mu = e B^{\\mu \\nu } p_\\nu ,$ then imply the second order equation $m \\ddot{x}^\\mu = e B^{\\mu \\nu }\\dot{x}_\\nu .$ This equation of motion for $x^{\\mu }$ coincides with the one that one encounters for $\\theta ^{\\mu \\nu }=0$ since the Hamiltonian only depends on $p$ and not on the coordinates $x^\\mu $ whose Poisson brackets do not vanish.", "However the non-commutativity parameters $\\theta ^{\\mu \\nu }$ appear in quantities like the volume form on phase space which is the 4-fold exterior product of the symplectic form with itself, $dV \\equiv \\frac{1}{4!", "}\\omega ^4= \\frac{1}{\\sqrt{\\text{det}\\Omega }} d\\xi ^1 \\cdots d\\xi ^{8},$ where $\\Omega $ denotes the Poisson matrix and where we suppressed the exterior product symbol." ], [ "“Exotic” (symplectic form) approach to NC space-time", "The Hamiltonian approach to mechanics on non-commutative space based on the simple form (REF ) of the Poisson algebra (in which the Poisson bracket $\\lbrace x^\\mu , p^\\nu \\rbrace $ has the canonical form) has been nicknamed the standard approach.", "As we just recalled, it does not allow for the inclusion of a non-constant field strength.", "By contrast, the so-called exotic approach [18], [24] which is based on a simple form of the symplectic 2-form allows us to describe generic field strengths $ B_{\\mu \\nu } (\\vec{x})$ .", "In this setting, the constant non-commutativity parameters $\\theta ^{\\mu \\nu }$ are introduced into the symplectic 2-formOne may as well consider $\\vec{p}$ -dependent parameters $\\theta ^{\\mu \\nu }$ .", "defined on the phase space parametrized by $(x^\\mu , p^\\mu )$ : $\\omega = dp^\\mu \\wedge dx_\\mu + \\frac{1}{2}e B_{\\mu \\nu }dx^\\mu \\wedge dx^\\nu + \\frac{1}{2}\\theta _{\\mu \\nu }dp^\\mu \\wedge dp^\\nu .$ The Poisson matrix is obtained by the inversion of the symplectic matrix (e.g.", "see reference [41] for the case of a space-time of arbitrary dimension), and therefore it has a more complicated form than the one corresponding to (REF ).", "By way of illustration, we recall the result that one obtains for the simplest instance [18] where one has only two spatial coordinates, i.e.", "$\\vec{x} \\equiv (x_1, x_2)$ : $\\lbrace {x}_1, {x}_2 \\rbrace = \\kappa ^{-1}\\theta ,\\qquad \\lbrace {p}_1, {p}_2 \\rbrace = \\kappa ^{-1}e B,\\qquad \\lbrace {x}_i, {p}_j \\rbrace = \\kappa ^{-1}\\delta _{ij},$ where $\\kappa (\\vec{x}) \\equiv 1 - e \\theta B (\\vec{x}) $ with $\\theta _{12} \\equiv \\theta $ and $B_{12} \\equiv B$ .", "None of the brackets now has a canonical form.", "The equations of motion following from the Hamiltonian $H (\\vec{x}, \\vec{p}) \\equiv \\frac{1}{2m}\\vec{p}^{2} + e V (\\vec{x}) $ read $\\dot{p}_i= e E_i + e B \\varepsilon _{ij} \\dot{x}_j,\\qquad \\text{with}\\qquad E_i \\equiv - \\partial _i V,\\qquad i\\in \\lbrace 1, 2 \\rbrace \\nonumber \\\\m^{\\ast } \\dot{x}_i= p_i - e m \\theta \\varepsilon _{ij} E_j,\\qquad \\text{with}\\qquad m^{\\ast }\\equiv \\kappa m,\\qquad \\kappa (\\vec{x})= 1 - e \\theta B (\\vec{x}),$ where $\\varepsilon _{ij}$ denotes the components of the constant antisymmetric tensor normalized by $\\varepsilon _{12}=1$ .", "The parameter $m^{\\ast }(\\vec{x}) \\equiv m\\kappa (\\vec{x})$ may be viewed as an effective mass depending on the position of the particle.", "Various physical applications of this system of evolution equations have been found in recent years, see [24] and references therein.", "For $V \\equiv 0$ , we have $ p_i= m^{\\ast } \\dot{x}_i=m \\kappa \\dot{x}_i$ , hence $\\dot{p}_i= m \\dot{\\kappa }\\dot{x}_i + m \\kappa \\ddot{x}_i,\\qquad \\text{with}\\qquad \\dot{\\kappa }= -e \\theta \\dot{B}= - e \\theta \\dot{x}_j \\partial _j B.$ Substitution of this expression into the first of equations (REF ) yields a second order differential equation for $x_i$ : $m^{\\ast } (\\vec{x})\\ddot{x}_i= e \\varepsilon _{ij} \\dot{x}_j B^{\\ast },\\qquad \\text{with}\\qquad B^{\\ast } \\equiv B + \\frac{1}{2}m \\theta \\varepsilon _{ij} \\dot{x}_i (\\partial _j B).$ This equation, which looks somewhat exotic, includes a $\\theta $ -dependent term depending on the derivative of the field strength and it involves an $\\vec{x}$ -dependent mass, i.e.", "there is a dependence of parameters on the localization of the particle in the space in which it evolves.", "The expressions in (REF ) simplify greatly in the case of a constant magnetic field: equation (REF ) then reduces to $m \\ddot{x}_i= e\\frac{B}{\\kappa }\\varepsilon _{ij} \\dot{x}_j,\\qquad \\text{with}\\qquad \\kappa = 1 - e \\theta B=\\operatorname{const}.$ As was pointed out earlier [23], this equation of motion coincides with the “standard approach” equation (REF ) after a rescaling of time $t \\rightarrow \\kappa t$ .", "We note that the value $\\check{B} \\equiv \\frac{B}{1 - e \\theta B}$ coincides with the one obtained for a constant magnetic field in two dimensions from the Seiberg–Witten map in non-commutative gauge field theory [11], but it differs from the constant non-commutative field strength $F_{12}\\equiv \\partial _1 A_2 - \\partial _2 A_1 - {\\rm i} e [A_1\\stackrel{\\star }{,}A_2]= \\partial _1 A_2 - \\partial _2 A_1 +e\\lbrace A_1, A_2\\rbrace ,$ e.g.", "in the symmetric gauge $(A_1, A_2)= (- \\frac{B}{2}x_2, \\frac{B}{2}x_1)$ , where one finds $F_{12}= B (1 + \\frac{e \\theta B}{4 \\kappa })$ .", "In conclusion, different Hamiltonian formulations for a charged “point” particle in a non-commutative space lead to different results.", "However, for the special case of a constant magnetic field strength we have seen in the previous two subsections that the different Hamiltonian formulations lead to the same results (or to results that are related to each other by a redefinition of the magnetic field).", "So does the Lagrangian formulation of Section as we have shown in Section ." ], [ "Concluding remarks", "Just as there exist different approaches to the formulation of gauge field theories on non-commutative spaces (e.g.", "the star product approach [39], the approach of spectral triples [10], of matrix models [14], ...), there appear to exist different approaches to the dynamics of relativistic or non-relativistic particles in non-commutative space which are subject to a background gauge field.", "It is plausible that these approaches yield essentially the same results in the particular case of a constant magnetic field, i.e.", "a field strength which does not depend on the non-commuting coordinates.", "The “exotic” (symplectic form) approach to non-commutative space-time can be viewed as an extension of all other approaches to the case of a generic field strength." ], [ "Continuum formulation on a generic manifold", "In this appendix, we show that Wong's equations, as formulated on a generic space-time manifold, admit a simple continuum version.", "Moreover, we will prove that the latter formulation has to hold for arbitrary dynamical matter fields $\\phi $ whose dynamics is described by a generic action $S[\\phi ; g_{\\mu \\nu }, A_{\\mu }^a]$ which is invariant under both gauge transformations and general coordinate transformations ($g_{\\mu \\nu }$ and $A_{\\mu }^a$ representing fixed external fields).", "These arguments generalize to Moyal space in the particular case of a constant field strength.", "Let $M$ be a four dimensional space-time manifold endowed with a fixed metric tensor $(g_{\\mu \\nu })$ of signature $(+,-,-,-)$ .", "We denote the covariant derivative of a tensor field with respect to the Levi-Civita-connection by $\\nabla _{\\mu }$ (e.g.", "$\\nabla _{\\mu } V^{\\nu }= \\partial _{\\mu } V^{\\nu } + \\Gamma ^{\\nu }_{\\mu \\rho } V^{\\rho }$ where the coefficients $ \\Gamma ^{\\nu }_{\\mu \\rho }$ are the Christoffel symbols) and the gauge covariant derivative as before by $D_{\\mu }$ (e.g.", "$\\delta _{\\lambda } A_{\\mu }^a= D_{\\mu } \\lambda ^a \\equiv \\partial _{\\mu } \\lambda ^a - {\\rm i} g [A_{\\mu }, \\lambda ]^a$ for the infinitesimal gauge transformation of the Yang–Mills gauge field $(A_{\\mu }^a)$ ).", "Since we used the notation $\\frac{Dq^a}{d\\tau } \\equiv \\dot{x}^{\\mu } D_{\\mu } q^a$ in the main body of the text, we will write $\\frac{\\nabla V^{\\mu }}{d\\tau } \\equiv \\dot{x}^{\\nu } \\nabla _{\\nu } V^{\\mu }$ for the derivative of the vector field $V^{\\mu } (x(\\tau ))$ along the trajectory $\\tau \\mapsto x(\\tau )$ .", "Lorentz-force and its non-Abelian generalization.", "The Lorentz-force equation on the space-time manifold $M$ reads $m\\frac{\\nabla u^{\\mu }}{d\\tau }=e F^{\\mu }_{\\hphantom{\\mu }\\nu } u^{\\nu },$ where $u^{\\mu } \\equiv \\dot{x}^{\\mu }$ denotes the 4-velocity of the particle of constant charge $q\\equiv e$ and where $F_{\\mu \\nu }$ represents a given electromagnetic field strength.", "This equation of motion follows from the point particle action $S[x]=\\frac{m}{2}\\int d\\tau g_{\\mu \\nu } (x(\\tau ))\\dot{x}^{\\mu }\\dot{x}^{\\nu } +e\\int d\\tau A_{\\mu } (x(\\tau ))\\dot{x}^{\\mu }$ upon variation with respect to $x^{\\mu }$ .", "The natural generalization of (REF ) to non-Abelian Yang–Mills theory is given by Wong's equations as written on the space-time manifold $M$ : $m\\frac{\\nabla ^2 x^{\\mu }}{d\\tau ^2}=q^{a} F^{a \\mu }_{\\hphantom{a \\mu }\\nu }\\dot{x}^{\\nu },\\qquad \\text{where}\\qquad \\frac{D q^{a}}{d\\tau }=0.$ Here, the covariant constancy of the charge-vector $(q^{a})$ represents the geometrically natural generalization of the ordinary constancy of the charge $e$ appearing in the Abelian gauge theory.", "The equation of motion of $x^{\\mu }$ follows from the action functional $S_{\\rm W} [x]=\\frac{m}{2}\\int d\\tau g_{\\mu \\nu } (x(\\tau ))\\dot{x}^{\\mu }\\dot{x}^{\\nu } +\\int d\\tau q^a A^a_{\\mu } (x(\\tau ))\\dot{x}^{\\mu }.$ Continuum formulation.", "The components ${T}^{\\mu \\nu }$ of the energy-momentum tensor (density) and the components of the current density may be defined as functional derivatives of the action, ${T}^{\\mu \\nu }(x) \\equiv 2\\frac{\\delta S_{\\rm W}}{\\delta g_{\\mu \\nu }(x)},\\qquad {j}_{a}^{\\mu }(x) \\equiv \\frac{\\delta S_{\\rm W}}{\\delta A_{\\mu }^{a}(x)}$ so that expression (REF ) implies ${T}^{\\mu \\nu }(y)= \\int d\\tau {\\delta }^{4}(y-x(\\tau ))m \\dot{x}^{\\mu }(\\tau )\\dot{x}^{\\nu }(\\tau ),\\\\{j}^{a\\mu }(y)= \\int d\\tau {\\delta }^{4}(y-x(\\tau ))q^{a}(\\tau )\\dot{x}^{\\mu }(\\tau ).$ We note that the energy-momentum 4-vector is then given by $P^{\\mu }= \\int _{{\\mathbb {R}}^3} d^3 xT^{\\mu 0}$ which yields the standard expressions: $P^{0}= \\int _{{\\mathbb {R}}^3} d^3 xT^{0 0}= m\\dot{x}^0= m\\frac{dt}{d\\tau }= \\frac{m}{\\sqrt{1 - \\vec{v}\\,{}^{2}}},\\qquad P^{i}= m\\dot{x}^i= \\frac{mv^i}{\\sqrt{1 - \\vec{v}\\,{}^{2}}}.$ The 4-divergence of the energy-momentum tensor can be evaluated by substituting the equation of motion $m\\frac{\\nabla ^2x^{\\mu }}{d\\tau ^2}=q^{a} F^{a \\mu }_{\\hphantom{a \\mu }\\nu } \\dot{x}^{\\nu }$ : $\\nabla _{\\mu } {T}^{\\mu \\nu }(y)= \\int d\\tau (\\dot{x}^{\\mu }\\nabla ^y_{\\mu }){\\delta }^{4}(y-x(\\tau )) m\\dot{x}^{\\nu }(\\tau )= \\int d\\tau m\\frac{\\nabla ^{2}x^{\\nu }}{d\\tau ^{2}}(\\tau ){\\delta }^{4}(y-x(\\tau ))\\\\\\phantom{\\nabla _{\\mu } {T}^{\\mu \\nu }(y)}= \\int d\\tau q^{a}(\\tau )F^{a\\nu }_{\\hphantom{a\\nu }\\mu } (x(\\tau ))\\dot{x}^{\\mu }(\\tau ){\\delta }^{4}(y-x(\\tau ))= F^{a\\nu }_{\\hphantom{a\\nu }\\mu } (y){j}^{a\\mu }(y).$ Similarly, substitution of the charge transport equation $\\frac{Dq^{a}}{d\\tau }=0$ into the gauge covariant divergence of the current density gives $D_{\\mu } {j}^{a \\mu }(y)= \\int d\\tau (\\dot{x}^{\\mu }D^y_{\\mu }){\\delta }^{4}(y-x(\\tau ))q^{a}(\\tau )=\\int d\\tau {\\delta }^{4}(y-x(\\tau ))\\frac{Dq^{a}(\\tau )}{d\\tau }= 0.$ Therefore the continuum version of equations (REF ) reads $\\nabla _{\\nu } {T}^{\\nu \\mu }= F^{a\\mu }_{\\hphantom{a\\mu }\\nu } {j}^{a\\nu },\\qquad \\text{where}\\qquad D_{\\mu } {j}^{a\\mu }= 0.$ These relations may be called continuum Lorentz–Yang–Mills equations.", "General derivation of the continuum equations.", "Actually equations (REF ) do not only hold for point particles but in a rather general context as will be shown in the sequel.", "To this end let us consider an arbitrary action functional $S=S[\\phi ; g_{\\mu \\nu },A_{\\mu }^{a}],$ where $(g_{\\mu \\nu })$ and $(A_{\\mu }^{a})$ denote a fixed 4-geometry and Yang–Mills potential respectively, whereas $\\phi $ denotes arbitrary dynamical matter fields.", "Taking the action $S$ to be gauge invariant entails the vanishing of its gauge variation: $0= \\delta _{\\lambda }S=\\int \\left(\\frac{\\delta S}{\\delta \\phi }\\delta _{\\lambda }\\phi +\\frac{\\delta S}{\\delta A_{\\mu }^{a}}\\delta _{\\lambda }A_{\\mu }^{a}\\right).$ Together with the matter field equations of motion $\\delta S/\\delta \\phi =0$ and the gauge variation of the Yang–Mills connection, $\\delta _{\\lambda }A_{\\mu }^{a}=D_{\\mu }\\lambda ^{a}$ , this implies $D_{\\mu } {j}_{a}^{\\mu }=0,\\qquad \\text{where}\\qquad {j}_{a}^{\\mu }(x) \\equiv \\frac{\\delta S}{\\delta A_{\\mu }^{a}(x)},$ i.e.", "the second of equations (REF ).", "The fact that $S$ is geometrically well defined is reflected by its invariance under general coordinate transformations (diffeomorphisms).", "The latter are generated by a generic vector field $\\xi \\equiv \\xi ^{\\mu } \\partial _{\\mu }$ .", "Thus, we have $0= \\delta _{\\xi }S=\\int \\left(\\frac{\\delta S}{\\delta \\phi }\\delta _{\\xi }\\phi +\\frac{\\delta S}{\\delta g_{\\mu \\nu }}\\delta _{\\xi }g_{\\mu \\nu }+\\frac{\\delta S}{\\delta A_{\\mu }^{a}}\\delta _{\\xi }A_{\\mu }^{a}\\right),$ where the matter field equations again imply the vanishing of the first term.", "The metric tensor field and the Yang–Mills connection 1-form $A \\equiv A_{\\mu } dx^{\\mu } \\equiv A_{\\mu }^a T^a dx^{\\mu }$ transform [40] with the Lie derivative with respect to the vector field $\\xi $ : $\\delta _{\\xi }g_{\\mu \\nu }= (L_{\\xi }g)_{\\mu \\nu }= \\nabla _{\\mu }\\xi _{\\nu }+\\nabla _{\\nu }\\xi _{\\mu },\\nonumber \\\\\\delta _{\\xi }A_{\\mu }=(L_{\\xi }A)_{\\mu } \\equiv ((i_{\\xi } d+d i_{\\xi }) A)_{\\mu }= (i_{\\xi } (dA- {\\rm i}\\frac{g}{2}[A,A]) - {\\rm i} g [A, i_{\\xi } A] + d i_{\\xi } A)_{\\mu }\\nonumber \\\\\\phantom{\\delta _{\\xi }A_{\\mu }}= \\xi ^{\\nu }F_{\\nu \\mu }+D_{\\mu }(\\xi ^{\\nu }A_{\\nu }).$ Here, $i_{\\xi }$ denotes the inner product of differential forms with the vector field $\\xi $ and $F \\equiv dA- {\\rm i}\\frac{g}{2}[A,A] \\equiv \\frac{1}{2} F_{\\mu \\nu } dx^{\\mu } \\wedge dx^{\\nu } $ the Yang–Mills curvature 2-form.", "Substitution of the variations (REF ) into (REF ) and use of relation (REF ) now yields $\\nabla _{\\mu } {T}^{\\mu \\nu }= F^{a\\nu }_{\\hphantom{a\\nu }\\mu }{j}_{a}^{\\mu },\\qquad \\text{where}\\qquad {T}^{\\mu \\nu }(x) \\equiv 2\\frac{\\delta S}{\\delta g_{\\mu \\nu }(x)},$ i.e.", "the first of equations (REF ), thereby completing the proof of our claim." ], [ "Acknowledgements", "D.B.", "is a recipient of an APART fellowship of the Austrian Academy of Sciences, and is also grateful for the hospitality of the theory division of LANL and its partial financial support.", "F.G. wishes to thank Fabien Vignes-Tourneret for a useful discussion on the Moyal algebra.", "We wish to thank the anonymous referees as well as the editors for their pertinent and constructive comments which contributed to the clarification of several points, as well as for pointing out several relevant references.", "[1]Referencesref" ] ]	1403.0255
[ [ "Walking Technipions in a Holographic Model" ], [ "Abstract We calculate masses of the technipions in the walking technicolor model with the anomalous dimension gamma_m =1, based on a holographic model which has a naturally light technidilaton phi as a composite Higgs with mass m_phi simeq 125 GeV.", "The one-family model (with 4 weak-doublets) is taken as a concrete example in such a framework, with the inputs being F_pi=v/2 simeq 123 GeV and m_phi simeq 125 GeV as well as gamma_m=1.", "It is shown that technipion masses are enhanced by the large anomalous dimension to typically O(1) TeV.", "We find a correlation between the technipion masses and S^{(TC)}, the S parameter arising only from the technicolor sector.", "The current LHC data on the technipion mass limit thus constrains S^{(TC)} to be not as large as O(1), giving a direct constraint on the technicolor model building.", "This is a new constraint on the technicolor sector alone quite independent of other sector connected by the extended-technicolor-type interactions, in sharp contrast to the conventional S parameter constraint from the precision electroweak measurements." ], [ "Introduction", "The mystery of the origin of the masses of the fundamental particles is one of the most important issues to be revealed in elementary particle physics.", "Even though the property of the 125 GeV scalar boson discovered at the LHC seems to be quite consistent with that of the Higgs boson in the Standard Model (SM) so far, there are several reasons for possible existence of the physics beyond the SM, such as the dynamical origin of the mass of the Higgs itself.", "Technicolor (TC) [1], [2] is an attractive candidate for such alternatives.", "Phenomenologically viable TC models, Walking TC (WTC) models [3], [4], based on the approximately scale invariant dynamics (ladder Schwinger-Dyson (SD) equation), having a large mass anomalous dimension, $\\gamma _m \\simeq 1$ , predicted existence of a light composite scalar boson as a pseudo Nambu-Goldstone (NG) boson associated with the spontaneous breaking of the approximate scale invariance.", "That is called the technidilaton, which can be identified with the 125 GeV boson discovered at the LHC [5], [6],[7].", "Recent lattice studies [8] actually indicate existence of a light flavor-singlet scalar bound state in the QCD with large number of massless flavors $N_f=8$ , which is a candidate theory for the walking technicolor with anomalous dimension near unity as was suggested by several lattice results [9].", "Such a light scalar could be a candidate for the technidilaton.", "(Note that a similar light scalar was also found on the lattice $N_f=12$ QCD [10], which may be a generic feature of the conformal dynamics, though not walking.)", "Here we note that as we have repeatedly emphasized [11][5], [6], [7], the technidilaton cannot be exactly massless: Although the scale symmetry is spontaneously broken by the condensate of the technifermion bilinear operator which is non-singlet under the scale transformation, it is at the same time explicitly broken by the same condensate, giving rise to the nonperturbative scale anomaly even if the coupling is non-running in the perturbative sense (The coupling runs by the condensate formation due to nonpertuabative dynamics).", "Thus it is this non-zero technidilaton mass $M_\\phi $ , arising from the noperturbative scale anomaly due to the chiral condensate in WTC, that can dynamically explain the origin of the Higgs mass which is left mysterious in SM.", "This is somewhat analogous to the $\\eta ^\\prime $ meson in the ordinary QCD, where the chiral $U(1)_A$ symmetry is spontaneously broken by the condensate and explicitly broken also by the anomaly (the chiral $U(1)_A$ anomaly) and hence cannot be exactly massless: Nevertheless it is regarded as a pseudo NG boson a la Witten-Veneziano having a parametrically massless limit i.e., the large $N_c$ limit (with fixed ratio $N_f/N_c (\\ll 1)$ , Veneziano limit) in a way that the chiral $U(1)_A$ anomaly tends to zero (only as a limit, not exactly zero).", "The mass of the technidilaton as a pseudo-NG boson comes from the nonperturbative trace anomaly $\\theta _\\mu ^\\mu \\ne 0$ due to the chiral condensate and can be estimated through the partially conserved dilatation current (PCDC) relation [4].", "A precise ladder evaluation of $M_\\phi F_\\phi $ based on this PCDC relation reads [12]: $(M_\\phi F_\\phi )^2 =- 4 \\langle \\theta _\\mu ^\\mu \\rangle \\simeq 0.154 \\cdot N_f N_{\\rm TC}\\cdot m_D^4 \\simeq \\left(2.5 \\cdot v_{\\rm EW}^2\\right)^2 \\cdot \\left[(8/N_f)(4/N_{\\rm TC})\\right]$ , where $v_{\\rm EW}^2=(246\\, {\\rm GeV})^2= N_D F_\\pi ^2 \\simeq 0.028 \\cdot N_f N_{\\rm TC} \\cdot m_D^2$ (Pagels-Stokar formula), with $N_D (=N_f/2)$ being the number of the electroweak doublets for $SU(N_{\\rm TC})$ gauge theory.", "Thus the mass of the LHC Higgs, $M_\\phi \\simeq 125\\, {\\rm GeV} \\simeq v_{\\rm EW}/2$ , can be obtained, when we take $v_{\\rm EW}/F_\\phi =2F_\\pi /F_\\phi \\simeq 1/5=0.2$ ($v_{\\rm EW}= 2 F_\\pi $ for $N_{\\rm TC}=4, N_f=8$ (Farhi-Susskind one-family model [13]).", "Amazingly, this value of $F_\\phi $ turned out to be consistent with the LHC Higgs data [5].", "Note the scaling $M_\\phi /v_{\\rm EW} \\sim 1/\\sqrt{N_f N_{\\rm TC}}$ and $F_\\phi /v_{\\rm EW} \\sim (N_{\\rm TC} N_f)^0$ , which is a generic result independent of the ladder approximation.", "This implies existence of a limit, so-called Veneziano limit, namely the large $N_{\\rm TC}$ limit with $N_f/N_{\\rm TC}=$ fixed ($\\gg 1$ , though) (so as to be close to the conformal window), where the technidilaton has a vanishing mass $M_\\phi /v_{\\rm EW}\\rightarrow 0$ in such a way that the nonperturbative scale anomaly in units of the weak scale vanishes $\\langle \\theta _\\mu ^\\mu \\rangle /v_{\\rm EW}^4 \\sim 1/(N_f N_{\\rm TC}) \\rightarrow 0$ in that limit.", "Thus a light technidilaton $M_\\phi /v_{\\rm EW} \\ll 1$ is naturally realized near such a Veneziano limit as is the case of the walking regime of the large-$N_f$ QCD [14] 4One might think that such a large $N_f$ (and $N_{\\rm TC}$ ) would result in the so-called $S$ parameter problem.", "We shall later discuss that it is not necessarily the case.", "5More specific computation [15] via the ladder Bethe-Salpeter (BS) equation combined with the ladder SD equation in the large $N_f$ QCD implies $M_\\phi \\sim 4 F_\\pi $ in the walking regime.", "Although it is much lighter than the techni-vector/axialvector with mass $\\sim 12 F_\\pi $ , it implies $M_\\phi \\simeq 500\\, {\\rm GeV}$ [11] in the one-family model, which is still somewhat larger than the LHC Higgs.", "Such a ladder BS calculation as it stands corresponds to the flavor non-singlet scalar mass, totally ignoring the full gluodynamics such as the mixing with the glueball and effects of the axial $U(1)_A$ anomaly (instanton effects).", "Inclusion of such full gluodynamics will further lower the flavor-singlet scalar meson mass [14].", ".", "More recently, in a holographic WTC model [16] which we are based on this paper, it was shown [7] that with the holographic parameter $G$ corresponding to the gluon condensate, we have $M_\\phi /(4\\pi F_\\pi ) \\simeq 3/(2\\sqrt{N_c}) (1+G)^{-1} \\rightarrow 0$ with $F_\\phi /F_\\pi \\simeq \\sqrt{2N_f}$ as $G\\rightarrow \\infty $ .", "This implies that the scale symmetry is parametrically realized in a way that the nonperturbative scale anomaly does vanish: $\\langle \\theta _\\mu ^\\mu \\rangle = - (M_\\phi ^2 F_\\phi ^2)/4 \\sim v_{\\rm EW}^4 /(N_f N_c) (1 +G)^{-2} \\rightarrow 0$ , in the strong gluon condensate limit $G \\rightarrow \\infty $ due to an additional factor $1/(1+G)^{-2}$ (besides the factor $1/(N_f N_c)$ ).", "It was also shown [7] that in this $G \\rightarrow \\infty $ limit the technidilaton behaves as a NG boson much lighter than other bound sates such as the techni-$\\rho $ and techni-$a_1$ : $M_\\phi /M_\\rho , M_\\phi /M_{a_1}\\rightarrow 0$ .", "This is indeed analogous to the flavor-singlet $\\eta ^\\prime $ meson which parametrically behaves a NG boson of the axial $U(1)_A$ symmetry a la Witten-Veneziano in the large $N_c$ limit, as we mentioned above.", "In fact it naturally realizes $M_\\phi \\simeq F_\\pi \\simeq 125$ GeV for $G\\simeq 10$ in the one-family model [7].", "Besides the lattice studies mentioned above, similar arguments for realizing such a parametrically light dilaton are given in somewhat different holographic contexts [17].", "Once the technidilaton mass is tuned to be 125 GeV, the holography determines the technidilaton couplings (essentially controlled by the decay constant $F_\\phi $ ) to the SM particles, which nicely reproduce the present LHC data for the 125 GeV Higgs, where the best fit value of the technidilaton decay constant in the case of one-family model is $v_{\\rm EW}/F_\\phi = 0.2-0.4 \\, (N_{\\rm TC} =4,3)$ (depending on the electroweak singlet technifermion numbers) [7].", "In this paper, based on the holographic model of Ref.", "[16], [7], we study another phenomenological issue of the generic WTC, the technipions, which are the left-over (pseudo) NG bosons besides the (fictitious) NG bosons absorbed into SM gauge bosons.", "They exist in a large class of the WTC having large $N_f$ , $N_f >2$ and will be a smoking gun of this class of WTC in the future LHC.", "As a concrete realization of the WTC, we here consider the Farhi-Susskind one-family model [13], inspired by the lattice studies on $N_f=8$ QCD, which as already mentioned suggest existence of a light flavor-singlet scalar [8] and the walking behavior as well [9].", "The model consists of $N_{\\rm TC}$ copies of a whole generation of the SM fermions, in such a way that the TC sector of the model is $SU(N_{\\rm TC})$ gauge theory with four weak-doublets, namely eight fundamental Dirac fermions $N_f=2 N_D=8$ .", "The global chiral symmetry breaking pattern is then $SU(8)_L \\times SU(8)_R/SU(8)_V$ , resulting in the emergence of 63 NG bosons.", "Three of them are eaten by the SM weak gauge bosons, while 60 technipions remain as physical states.", "All the technipions become massive through the explicit breaking of the chiral $SU(8)_L \\times SU(8)_R$ symmetry due to the SM gauge interactions and extended TC (ETC) gauge interactions, and thereby become pseudo NG bosons.", "Estimation of the masses of these technipions are very important for the studies of collider phenomenologies, though it is a challenging task due to the need of non-perturbative calculations.", "This paper is the first attempt to compute the technipion masses using models of holographic dual.", "In Ref.", "[18], the mass of the charged technipions originating from the electromagnetic interaction and also the analogous mass of the colored technipions were studied in the large $N_f$ QCD by using the BS equation with the improved ladder approximation with two-loop running coupling.", "Further in Ref.", "[19], all the masses of technipions in the one-family model were estimated in the ladder analysis.", "However, it should be noted that the results obtained by using the improved ladder analysis, though qualitatively good, have ambiguities in quantitative estimate which come from their systematic uncertainty originating from the approximation itself.", "Also, recent lattice result [9] indicates that $SU(3)$ gauge theory with 8 fundamental fermions (which is exactly the case for the one-family TC with $N_{\\rm TC}=3$ ) possesses walking nature, which is contrasted with the improved ladder analysis showing that it is deep inside the chiral symmetry breaking phase without walking signals.", "We would need other nonpertubative method having a different systematic uncertainty to make the quantitative estimate more reliable to be compared with the experiments.", "Besides from-the-first-principle calculations on the lattice, one such a method would be the holographic approach which is based on the gauge-gravity duality [20] and is more useful for the phenomenological studies in the sense that desirable values of phenomenologically relevant quantities can be easily obtained by tuning input parameters of the effective holographic model.", "In the application to the WTC, based on the popular (bottom-up) holographic QCD [21], [22], with setting the anomalous dimension $\\gamma _m=0$ through the bulk scalar mass parameter, we shall engineer the walking theory by implementing the large anomalous dimension $\\gamma _m=1$ instead of $\\gamma _m=0$ [23], [24], [16], [7].", "Particularly in the model having the technidilaton [16], [7], we can tune the input holographic parameters, besides the mass anomalous dimension, so as to adjust the physical quantities such as the mass of the composite Higgs (technidilaton), the weak scale, as well as the $S$ parameter to the experimental values.", "Once those holographic parameters are fixed, other phenomenological quantities which can be calculated from the holographic model become predictions.", "In Ref.", "[25], by using the same holographic model as Refs.", "[16], [7], the hadronic leading order contributions in QCD and WTC to the anomalous magnetic moment ($g-2$ ) of leptons were calculated.", "It was shown that, in the case of the real-life QCD, the known QCD contributions to $g-2$ of leptons are correctly reproduced, then it was applied to the calculation of the contribution from the WTC dynamics.", "Here we adopt a similar approach for the estimation of the masses of the technipions in the one-family model of WTC, based on the first order perturbation of the explicit chiral symmetry breaking by the “weak gauge couplings” of SM gauge interactions and the ETC gauge interactions (Dashen's formula), while the full nonperturbative contributions of WTC sector are included by the holography (or its effective theory).", "This is the same strategy as the QCD estimate of the $\\pi ^+ - \\pi ^0$ mass difference, where the explicit chiral symmetry breaking is given by the QED lowest order coupling, while the full QCD nonperturbative contributions are estimated through the current correlators by various method like ladder, holography, lattice etc..", "It will be shown that technipion masses in the one-family model are enhanced due to the walking dynamics to the order of typically $O(1)$ TeV, qualitatively the same as the previous estimate [19], with somewhat larger value.", "The large enhancement of the technipion mass has long been noted to be a generic feature of the large anomalous dimension [26], and concretely shown in the explicit walking dynamics with $\\gamma _m=1$ based on the ladder SD equation [3] and in the large $N_f$ QCD with $\\gamma _m \\simeq 1$ [18].", "Striking fact is that although the explicit chiral symmetry breakings are formally very small due to the “weak gauge couplings”, the nonperturbative contributions from the WTC sector lift all the technipions masses to the TeV region so that they all lose the nature of the “pseudo NG bosons”.", "This is actually a universal feature of the dynamics with large anomalous dimension, “amplification of the symmetry violation” [2], as dramatically shown in the top quark condensate model [27], based on the Nambu-Jona-Lasinio model with large anomalous dimension $\\gamma _m =2$ .", "Note that although the left-over light spectrum are just three exact NG bosons absorbed into $W/Z$ bosons, our theory is completely different from the $N_f=2$ model with the symmetry breaking of $SU(2)_L \\times SU(2)_R/SU(2)_V$ .", "In fact the three exact NG bosons as well as the technidilaton are composite of the linear combinations of all the $N_f=8$ technifermions.", "Here it is to be noted that the possible back reaction of the SM sector to the technidilaton mass through these weak gauge couplings (top loop and EW gauge boson loops), which is of potentially large quadratic divergence, actually was computed in the effective theory for WTC (dilaton chiral perturbation theory) coupled weakly to the SM sector, the result being negligibly small due to the largeness of $F_\\phi $ (see Section III B of Ref. [6]).", "We also show that there is a correlation between the technipion masses and $S^{({\\rm TC})}$ which is the magnitude of the contribution to the $S$ parameter only from the TC sector, and that the latter cannot be as large as $O(1)$ due to the constraints from the currently available LHC data on the masses of the technipions.", "This is a new constraint on the TC dynamics alone, quite independently of the conventional $S$ parameter constraint from the precision electroweak measurements which may involve not only the TC sector but the large contributions from other sector through the ETC interactions in such a way that they could largely cancel each other, as suggested in the Higgsless models [28].", "This paper is organized as follows.", "In the next section, after the one-family model is briefly reviewed, the holographic model formulated in Refs.", "[16], [7] is applied for the calculation of the masses of the technipions in the one-family model.", "Constraints from the currently available LHC data, as well as implications for the future collider phenomenology are discussed in Sec. .", "Section is devoted to the summary of the paper.", "In Appendix , as a check of reliability of our calculations, we monitor the same holographic method by applying the estimation of the masses of colored technipions to the $\\pi ^+ - \\pi ^0$ mass difference in the real-life QCD.", "In Appendix , the current correlator obtained from the holographic calculation is compared to that obtained from ladder BS calculation, the result being consistent each other." ], [ "Holographic Estimate of Technipion Masses", "In the Farhi-Susskind one-family model [13], eight flavors of technifermions (techniquarks $Q_c$ and technileptons $L$ ) are introduced: $Q_c\\equiv (U_c, D_c)^T$ (where $c=r,g,b$ is the QCD color charge) and $L \\equiv (N, E)^T$ , all having $SU(N_{\\rm TC}$ ) charge, which are further embedded in a larger extended TC group, say $SU(N_{\\rm TC}+3)$ , by involving three generations of SM fermions.", "The chiral symmetry therefore is $SU(8)_L \\times SU(8)_R$ , which is broken by the technifermion condensation $\\langle \\bar{F}F \\rangle \\ne 0$ ($F=Q, L$ ) down to $SU(8)_V$ , resulting in the emergence of 63 NG bosons.", "Three of them are eaten by $W$ and $Z$ bosons, while other 60 remain as physical states.", "Those are called technipions.", "Technipions obtain their masses through the explicit breaking effects (such as SM gauge interactions and extended TC four-fermion interactions), and become pseudo NG bosons.", "The technipions are classified by the isospin and QCD color charges, which are listed in Table REF together with the currents coupled to them, where the notation follows the original literature [13].", "For construction of the chiral Lagrangian described by those technipions, readers may refer to Ref. [19].", "Table: The technipions and their color and isospin representation, as well as associated currents in the one-family model .Here λ a \\lambda _a (a=1,⋯,8a=1,\\cdots ,8) are the Gell-Mann matrices,τ i \\tau ^i SU(2)SU(2) generators normalized asτ i =σ i /2\\tau ^i=\\sigma ^i/2 (i=1,2,3i=1,2,3) with the Pauli matrices σ i \\sigma ^i, andthe label cc stands for the QCD color index c=r,g,bc=r,g,b.The holographic model proposed in Ref.", "[16] is based on a bottom-up approach for holographic dual of QCD (“hard-wall model”) [21], [22], with the input of the anomalous dimension $\\gamma _m =0$ of QCD case being simply replaced by $\\gamma _m =1$ , the value expected in the walking theory.", "The model incorporates $SU(N_f)_L \\times SU(N_f)_R$ gauge theory defined on the five-dimensional anti-de Sitter (AdS) space-time, which is characterized by the metric $ds^2= g_{MN} dx^M dx^N = \\left(L/z \\right)^2\\big (\\eta _{\\mu \\nu }dx^\\mu dx^\\nu -dz^2\\big )$ with $\\eta _{\\mu \\nu }={\\rm diag}[1, -1, -1,-1]$ .", "Here, $M$ and $N$ ($\\mu $ and $\\nu $ ) represent five-dimensional (four-dimensional) Lorentz indices, and $L$ denotes the curvature radius of the AdS background.", "The fifth direction, denoted as $z$ , is compactified on an interval, $ \\epsilon \\le z \\le z_m $ , where $z=\\epsilon $ (which will be taken to be 0 after all calculations are done) is the location of the ultraviolet (UV) brane while $z=z_m$ is that of the infrared (IR) brane.", "The model introduces a bulk scalar $\\Phi _S$ which transforms as a bifundamental representation field under the $SU(N_{f})_L \\times SU(N_{f})_R$ gauge symmetry; a field which is dual to the (techni-)fermion bilinear operator $\\bar{F} F$ .", "The mass parameter for $\\Phi _S$ , $m_{\\Phi _S}$ , is thus holographically related to $\\gamma _m$ as $m_{\\Phi _S}^2=- (3-\\gamma _m)(1+ \\gamma _m)/L^2$ .", "When we apply the model for the calculations of physical quantities in WTC models, we take $\\gamma _m = 1$ .", "In addition, as in Ref.", "[16], [7] we include another bulk scalar, $\\Phi _G$ , dual to the (techni-)gluon condensation operator $ G_{\\mu \\nu }^2$ , which has vanishing mass parameter since its conformal dimension is taken to be 4 6Note that in our holographic model based on the popular static hard wall model [21], [22], having the IR brane at $z_m$ fixed by hand (which explicitly breaks the conformal invariance in the five-dimension), the dilaton/radion for stabilizing the IR brane at $z_m$ as that discussed in some holographic models [29] is set to have a large mass of order ${\\mathcal {O}}(1/z_m)=$ several TeV's (See Table REF ) and is irrelevant to our discussions of the technidilaton.", "In fact our technidilaton [16], [7] is identified as a bound state of technifermion and anti-technifermion, which holographically corresponds to the ground state in Kaluza-Klein (KK) modes for the flavor-singlet part of the bulk scalar $\\Phi _S$ , in sharp contrast to the radion and dilaton in other holographic WTC models[17], [30].", "Actually, the flavor-singlet part in $\\Phi _S$ mixes with a glueball-like scalar from $\\Phi _G$ .", "However, as shown in Ref.", "[16], the mixing turns out to be negligible when one requires the present holographic model to reproduce the UV asymptotic behaviors of current correlators in OPE.", "Moreover, the lowest glueball as the lowest KK mode of $\\Phi _G$ was explicitly computed to be near 20 TeV, much heavier than the 125 GeV technidilaton as the fermionic bound state from $\\Phi _S$ in the walking case ($\\gamma _m=1,\\, G \\simeq 10$ ) [7], which is in sharp contrast to the QCD case ($\\gamma _m =0, \\,G\\simeq 0.25$ ) where both glueball (from $\\Phi _G$ ) and the flavor-singlet fermionic bound state (from $\\Phi _S$ ) are comparably heavy $\\simeq 1.2-1.3$ GeV (and may be strongly mixed) [25].", ".", "Thanks to this additional explicit bulk scalar field $\\Phi _G$ , this holographic model is the only model which naturally improves the matching with the OPE of the underlying theory (QCD and WTC) for current correlators so as to reproduce gluonic $1/Q^4$ term.", "This term is clearly distinguished from the same $1/Q^4$ term from chiral condensate in the case of WTC with $\\gamma _m=1$ The action of the model is given as [16] $S_5 = S_{\\rm bulk} + S_{\\rm UV} + S_{\\rm IR}\\,, $ where $S_{\\rm bulk}$ denotes the five-dimensional bulk action, $S_{\\rm bulk}&=&\\int d^4 x \\int _\\epsilon ^{z_m} dz\\sqrt{g}\\frac{1}{g_5^2} \\, e^{c_G g_5^2 \\Phi _G}\\Bigg [\\frac{1}{2} \\partial _M \\Phi _G \\partial ^M \\Phi _G\\nonumber \\\\&&+ {\\rm Tr}[D_M \\Phi _S^\\dag D^M \\Phi _S - m_{\\Phi _S}^2 \\Phi _S^\\dag \\Phi _S ]\\nonumber \\\\&&- \\frac{1}{4} {\\rm Tr}[L_{MN}L^{MN} + R_{MN}R^{MN}]\\Bigg ]\\,, $ and $S_{\\rm UV, IR}$ are the boundary actions which are given in Ref. [7].", "The covariant derivative acting on $\\Phi _S$ in Eq.", "(REF ) is defined as $D_M\\Phi _S=\\partial _M \\Phi _S+iL_M\\Phi _S-i\\Phi _S R_M$ , where $L_M(R_M)\\equiv L_M^a(R_M^a) T^a$ with $L_M (R_M)$ being the five-dimensional gauge fields and $T^a$ being the generators of $SU(N_{f})$ which are normalized as ${\\rm Tr}[T^a T^b]=\\delta ^{ab}$ .", "$L(R)_{MN}$ is the five-dimensional field strength which is defined as $L(R)_{MN} = \\partial _M L(R)_N - \\partial _N L(R)_M- i [ L(R)_M, L(R)_N ]$ , and $g$ is defined as $g={\\rm det}[g_{MN}]= (L/z)^{10}$ .", "The five-dimensional vector and axial-vector gauge fields $V_M$ and $A_M$ are defined as $ V_M = (L_M + R_M)/\\sqrt{2}$ and $A_M = (L_M-R_M)/\\sqrt{2}$ .", "It is convenient to work with the gauge-fixing $V_z=A_z\\equiv 0$ and take the boundary conditions $V_\\mu (x,\\epsilon )=v_\\mu (x)$ , $A_\\mu (x,\\epsilon )=a_\\mu (x)$ and $\\partial _z V_\\mu (x,z)\|_{z=z_m}=\\partial _z A_\\mu (x,z)\|_{z=z_m}= 0$ , where $v_\\mu (x)$ and $a_\\mu (x)$ correspond to sources for the vector and axial-vector currents, respectively.", "We solve the equations of motion for (the transversely polarized components of) the Fourier transformed fields, $V_\\mu (q,z)=v_\\mu (q) V(q,z)$ and $A_\\mu (x,z)=a_\\mu (q) A(q,z)$ , where $V(q,z)$ and $A(q,z)$ denotes the profile functions for the bulk vector and axial-vector gauge fields.", "We then substitute the solutions back into the action in Eq.", "(REF ), to obtain the generating functional $W[v_\\mu , a_\\mu ]$ holographically dual to WTC.", "Evaluating the UV asymptotic behaviors of the vector and axial-vector current correlators $\\Pi _V(Q^2)$ and $\\Pi _A(Q^2)$ with $Q^2\\equiv -q^2$ , we can thus fix the gauge coupling $g_5$ and the parameter $c_G$ appearing in the action to match the asymptotic forms with the expressions expected from the operator product expansion (OPE): [16] $\\frac{ L}{g_5^2} = \\frac{N_{\\rm TC}}{12\\pi ^2}\\,, \\qquad c_G = - \\frac{N_{\\rm TC}}{192 \\pi ^3}\\,,$ After the value of $\\gamma _m$ is taken to be 1, and a specific number for $N_{\\rm TC}$ is fixed, remaining parameters in the holographic model are $\\xi $ , $z_m$ , and $G$ , where $\\xi $ and $G$ parametrize the IR values for the vacuum expectation values of the bulk scalars $\\Phi _S$ and $\\Phi _G$ , $v_S(z=z_m)$ and $v_{\\chi _G}(z=z_m)$ [16]: $v_S(z_m) &=& \\frac{\\xi }{L}\\,, \\nonumber \\\\v_{\\chi _G}(z_m) &=& 1 + G\\,,$ with $v_{\\chi _G} = \\langle \\chi _G(x,z) \\rangle \\equiv \\langle e^{c_G g_5^2\\Phi _G(x,z)/2 } \\rangle $ .", "Hence, once three physical quantities are chosen to fix these parameters, we calculate all other quantities related to WTC models.", "We shall choose the technipion decay constant, $F_\\pi $ , the technidilaton mass, $M_\\phi $ , and the $S$ parameter (actually $S^{({\\rm TC})}$ coming from only the TC sector, but denoted as $S$ hereafter) as those three.", "$F_\\pi $ is taken to be $F_\\pi =123$ GeV so that it reproduces the electroweak (EW) scale $v_{\\rm EW}^2 = N_D F_\\pi ^2 = (246\\ {\\rm GeV})^2$ with $N_D=4$ , where $N_D$ is the number of the EW doublets exist in the model.", "The technidilaton mass $M_\\phi $ is taken to be $M_\\phi =125$ GeV to be identified with the LHC Higgs boson [7].", "As for the $S$ parameter, we take several values, namely $S=(0.1,0.3,1.0)$ for our study.", "This is because, although $S=0.1$ is a phenomenologically viable benchmark value, there is a possibility that even if the WTC dynamics itself produces a somewhat large value of $S$ , contributions coming from other part of the model (such as the extended TC interactions) could partially cancel it in a way similar to the concept of fermion-delocalization effect studied in Higgsless models [28].", "The values of parameters ($\\xi $ , $z_m$ , $G$ ) which reproduce the above mentioned three physical quantities ($F_\\pi , M_\\phi , S$ ) for the cases of $N_{\\rm TC}=3,4,$ and 5 are summarized in Table REF .", "Table: Parameter sets which reproduce F π =123F_\\pi =123 GeV, M φ =125M_\\phi =125 GeV, and S=(0.1,0.3,1.0)S=(0.1,\\, 0.3,\\, 1.0)In the following subsections, we estimate the masses of the technipions with the parameter sets listed there." ], [ "Color-singlet technipion masses", "The color-singlet technipions $P^0 \\sim (\\bar{Q} \\gamma _5 Q - 3 \\bar{L} \\gamma _5 L)$ and $P^{i=1,2,3} \\sim ( \\bar{Q} \\gamma _5 \\sigma ^i Q - 3 \\bar{L} \\gamma _5 \\sigma ^i L)$ listed in Table REF obtain their masses through the extended TC-induced four-fermion interaction, ${\\cal L}_{\\rm 4-fermi}^{\\rm ETC} (\\Lambda _{\\rm ETC})= \\frac{1}{\\Lambda _{\\rm ETC}^2}\\left(\\bar{Q} Q \\bar{L}L - \\bar{Q} \\gamma _5 \\sigma ^i Q \\bar{L} \\gamma _5 \\sigma ^i L\\right)\\,.$ The masses can be estimated by using the current algebra as $m_{P^{i,0}}^2= \\frac{1}{F_\\pi ^2}\\langle 0 \|[{\\bf Q}_{P^{i,0}}, [{\\bf Q}_{P^{i,0}}, {\\cal L}_{\\rm 4-fermi}^{\\rm ETC}(\\Lambda _{\\rm ETC})]]\|0 \\rangle \\,,$ where ${\\bf Q}_{P^{i,0}}$ denotes the chiral charges defined as ${\\bf Q}_{P^{i,0}}= \\int d^3 x \\, J^0_{P^{i,0}}(x)$ with the corresponding currents $J^\\mu _{P^{0,i}}(x)$ listed in Table REF .", "The $P^i$ and $P^0$ masses are thus evaluated to be [19] $m_{P^0}^2 &=& \\frac{5}{2} \\frac{\\langle 0\| \\bar{F}F \|0 \\rangle ^2_{\\Lambda _{\\rm ETC}}}{F_\\pi ^2 \\Lambda _{\\rm ETC}^2}\\,, \\nonumber \\\\m_{P^i}^2 &=& 4 \\frac{\\langle 0\| \\bar{F}F \|0 \\rangle ^2_{\\Lambda _{\\rm ETC}}}{F_\\pi ^2 \\Lambda _{\\rm ETC}^2}\\,, $ where we used $\\langle 0\| \\bar{L}L \|0 \\rangle =1/3 \\langle 0\| \\bar{Q}Q \|0 \\rangle \\equiv \\langle 0\| \\bar{F}F \|0 \\rangle $ .", "The ratio $m_{P^0}^2/m_{P^i}^2 =5/8$ is a salient prediction of the one-family model, independently of the walking dynamics.", "Table: The predicted values of the color-singlet technipion masses forN TC =3,4N_{\\rm TC}=3,4 and 5 with F π =123F_\\pi =123 GeV, M φ =125M_\\phi =125 GeV and S=(0.1,0.3,1.0)S=(0.1,0.3,1.0) fixed.The present holographic model gives a formula for the technifermion condensate $\\langle 0\| \\bar{F}F \|0 \\rangle $ renormalized at the extended TC scale $\\Lambda _{\\rm ETC}$ as [16], [7] $\\langle 0\| \\bar{F}F \|0 \\rangle _{\\Lambda _{\\rm ETC}}= - \\frac{\\sqrt{3}N_{\\rm TC}}{12\\pi ^2} \\frac{\\Lambda _{\\rm ETC} \\xi (1+G)}{z_m^2}\\,.$ This allows us to express the $P^{i,0}$ masses in Eq.", "(REF ) as follows: $m_{P^0} &=& \\sqrt{\\frac{5}{2}} \\frac{\\sqrt{3} N_{\\rm TC}}{12 \\pi ^2 F_\\pi } \\frac{\\xi (1+G)}{ z_m^2}\\,, \\nonumber \\\\m_{P^i} &=& 2 \\frac{\\sqrt{3} N_{\\rm TC}}{12 \\pi ^2 F_\\pi } \\frac{\\xi (1+G)}{z_m^2}\\,.", "$ Using the parameter set given in Table REF , we thus calculate the $P^{i,0}$ masses to obtain the numbers listed in Table REF 7These $P^{0,i}$ mass values are somewhat larger than those obtained in Ref.", "[19] based on estimate with help of the (improved) ladder SD analysis.", "This is because of the difference of the size of intrinsic mass scale obtained in both approaches: In the present holographic model, a typical hadron mass scale such as dynamical mass of technifermions $m_F$ is predicted to be $\\simeq 4 \\pi F_\\pi ={\\cal O}({\\rm TeV})$ , while $m_F$ estimated by the ladder approximation tends to get smaller than 1 TeV.", "The larger $m_F$ gives rise to the larger size of hadron spectrum, except the technidilaton protected by the scale symmetry [7].", ".", "With smaller $\\xi $ values as in Table REF , which ensures the presence of the light technidilaton [7], the technipion decay constant $F_\\pi $ can be approximated to be $F_\\pi \\simeq \\sqrt{\\frac{N_{\\rm TC}}{12\\pi ^2}} \\frac{\\xi (1+G)}{z_m}\\,.$ Putting this into Eq.", "(REF ) thus leads to the approximate formula for the $P^{i,0}$ masses: $m_{P^0} &\\simeq & \\sqrt{\\frac{5}{2}} \\frac{\\sqrt{N_{\\rm TC}}}{2\\pi z_m}\\,, \\nonumber \\\\m_{P^i} &\\simeq & 2 \\frac{\\sqrt{N_{\\rm TC}}}{2\\pi z_m}\\,, $ by which one can check that the numbers listed in Table REF are well reproduced." ], [ "Color-triplet and -octet technipion masses", "The color-octet and -triplet technipions $\\theta _a^{i(0)} \\sim \\bar{Q} \\gamma _5 \\lambda _a \\sigma ^{i}({\\bf 1}_{2\\times 2}) Q$ and $T_c^{i(0)} \\sim \\bar{Q}_c \\gamma _5 \\sigma ^i ({\\bf 1}_{2 \\times 2})L$ listed in Table REF acquire the masses by QCD gluon interactions 8This is analogous to the electromagnetic effects on the QCD pion.", "In Appendix , we apply the same method used here for the calculation of $\\pi ^+ - \\pi ^0$ mass difference, and show that holographic calculation reproduce the experimental value to a good accuracy.", ".", "The masses can be estimated by assuming the one-gluon exchange contribution, given as an integration over the momentum carried by the vector and axial-vector correlators $\\Pi _{V,A}$ : $m_{3,8}^2 = \\frac{3 C_{3,8}}{4\\pi F_\\pi ^2} \\int _0^\\infty dQ^2 \\, \\alpha _s(Q^2) \\left[ \\Pi _V(Q^2) - \\Pi _A(Q^2) \\right]\\,,$ with the group factor $C_{3(8)}=4/3(3)$ for the color-triplet (octet) technipion, $Q\\equiv \\sqrt{-q^2}$ being the Euclidean momentum.", "Again the ratio $m_3/m_8=4/9$ is a salient prediction of the one-family model, independently of the walking dynamics.", "In Eq.", "(REF ) we have incorporated the $Q^2$ -dependence of the QCD gauge coupling $\\alpha _s$ .", "Table: The predicted values of the color-triplet (m 3 m_3) and -octet (m 8 m_8) technipion masses forN TC =3,4N_{\\rm TC}=3,4 and 5 with F π =123F_\\pi =123 GeV, M φ =125M_\\phi =125 GeV and S=(0.1,0.3,1.0)S=(0.1,0.3,1.0) fixed.The present holographic model gives the formulas for $\\Pi _{V}(Q^2)$ and $\\Pi _{A}(Q^2)$ as [16], [7] $\\Pi _{V(A)}(Q^2) = \\frac{N_{\\rm TC}}{12\\pi ^2} \\frac{\\partial _z V(A)(Q^2, z)}{z} \\Bigg \|_{z=\\epsilon \\rightarrow 0}\\,.", "$ The vector and axial-vector profile functions $V(Q^2,z)$ and $A(Q^2,z)$ are determined by solving the following equations: $&&\\left[- Q^2 + \\omega ^{-1}(z) \\partial _z \\omega (z) \\partial _z\\right] V(Q^2, z) = 0\\,, \\nonumber \\\\&&\\left[- Q^2 + \\omega ^{-1}(z) \\partial _z \\omega (z) \\partial _z - 2 \\left( \\frac{L}{z} \\right)^2 [v_S(z)]^2\\right]A(Q^2, z) = 0\\,,$ with the boundary conditions $V(Q^2,z)\|_{z=\\epsilon \\rightarrow 0}=A(Q^2,z)\|_{z=\\epsilon \\rightarrow 0}=1$ .", "In Eq.", "(REF ) the vacuum expectation value $v_S(z)$ and a function $\\omega (z)$ are given as [16], [7] $v_S(z) &=& \\frac{\\xi (1+G)}{L} \\frac{(z/z_m)^2}{1+G(z/z_m)^4}\\frac{\\log (z/\\epsilon )}{\\log (z_m/\\epsilon )}\\,, \\nonumber \\\\\\omega (z) &=&\\frac{L}{z} \\left( 1 + G \\left( \\frac{z}{z_m} \\right)^4 \\right)^2\\,.$ Thus $(\\Pi _V -\\Pi _A)$ in Eq.", "(REF ) is evaluated as a function of the holographic parameters, $\\xi , G$ and the IR position $z_m$ .", "(For details, see Refs.", "[16], [7].)", "Using the parameter sets given in Table REF , we thus estimate the colored technipion masses for $S=(0.1, 0.3, 1.0)$ to obtain the values given in Table REF 9In Ref.", "[19] the colored technipion masses were estimated to be somewhat smaller than those listed in Table REF .", "The estimate in Ref.", "[19] was based on an assumption that $(\\Pi _V-\\Pi _A)$ in the integrand in Eq.", "(REF ) is dominated in the UV region and hence can be replaced with the OPE expressions from the UV cutoff down to some IR scale $\\sim 4 \\pi F_\\pi $ .", "As shown in the present study, however, such an assumption results in underestimate of the masses.", "This is the main reason for the difference between the estimated size of colored technipion masses in the present study and Ref. [19].", "Apart from this point, the present holographic estimation of the mass of all the technipions is roughly consistent with the ladder estimate of Ref.", "[19], as long as compared with the parameter choice corresponding to the ladder estimate.", ".", "In evaluating the integral over $Q^2$ in Eq.", "(REF ) we have introduced the UV cutoff $\\Lambda _{\\rm UV}^2 = (4\\times 10^6\\ {\\rm GeV})^2$ , which is consistent with the current bound from flavor changing neutral current [31].", "As for the QCD gauge coupling $\\alpha _s(Q^2)$ , the one-loop running coupling is used with taking 5, 6, and $(6+2N_{\\rm TC})$ flavors as number of active colored fermions in regions $Q^2 < m_t^2$ , $m_t^2 < Q^2 < (4\\pi F_\\pi )^2$ , and $(4\\pi F_\\pi )^2 < Q^2$ , respectively, where $(4\\pi F_\\pi )$ corresponds to the size of the dynamical mass scale of the technifermions estimated from the present holographic model [16], [7].", "The value of the coupling at the $Z$ boson mass scale, $\\alpha _s(m_Z^2)=0.1182$ [32], is used as input, and an infrared regularization is introduced in such away that $\\alpha _s(Q^2)$ takes constant value $\\alpha _s(Q^2=1\\, {\\rm GeV}^2)$ in the region $Q^2 < 1\\, {\\rm GeV}^2$ .", "We have checked that infrared regularization dependence is negligible for the estimation of technipion masses.", "In Fig.", "REF , we show $(\\Pi _V(Q^2)-\\Pi _A(Q^2))$ calculated from Eq.", "(REF ) for the case of $N_{\\rm TC} = 4$ .", "Figure: Π V (Q 2 )-Π A (Q 2 )\\Pi _V(Q^2)-\\Pi _A(Q^2) calculated from the holographic WTC model for the case of N TC =4N_{\\rm TC} = 4.", "Solid, dashed, and dotted curves correspond to the cases of S=0.1S=0.1, 0.30.3, and 1.01.0, respectively.General tendencies of $(\\Pi _V(Q^2)-\\Pi _A(Q^2))$ for the case of $N_{\\rm TC} = 3$ and 5 are similar to the case of $N_{\\rm TC} = 4$ .10In Appendix , the current correlator obtained from the holographic calculation is compared to that obtained from ladder BS calculation.", "In the figure, solid, dashed, and dotted curves correspond to the cases of $S=0.1$ , $0.3$ , and $1.0$ , respectively.", "Since $(\\Pi _V(Q^2)-\\Pi _A(Q^2))$ at $Q^2=0$ is equal to $F_\\pi ^2$ , which is adjusted to be $(123\\ {\\rm GeV})^2$ by tuning parameters so that the model reproduces $v_{\\rm EW} = 2 F_\\pi = 246$ GeV, all the curves take the same value at $Q^2=0$ .", "However, slopes of each curve at $Q^2=0$ are different since it is proportional to the magnitude of the $S$ parameter, $S = - 4 \\pi N_D \\frac{d}{d Q^2} (\\Pi _V(Q^2) - \\Pi _A(Q^2)) \\Bigg \|_{Q^2 =0}\\,.$ From this, we understand the general tendency of the above results, namely the smaller the value of the $S$ parameter, heavier the masses of technipions due to the larger contribution in the integral of Eq.", "(REF ).", "This is telling us the important thing.", "As we wrote earlier in this section, the EW precision measurements do not necessarily require that the contribution to the $S$ parameter from the TC dynamical sector, denoted as $S^{({\\rm TC})}$ before, is small, because the ultimate value of the $S$ parameter depends on how the TC sector is embedded in the whole model together with the SM fields.", "On the other hand, the correlation between the slope of $(\\Pi _V(Q^2)-\\Pi _A(Q^2))$ at $Q^2=0$ and the masss of colored technipions shown here rather directly constrain the contribution to the $S$ parameter from the TC dynamics itself if the experimental lower bound of the mass of the colored technipion become larger.", "In the next section, we will show that the current LHC bounds on the masses of technipions are already placing constraint that the contribution to the $S$ parameter from the TC sector should no be as large as $O(1)$ .", "To summarize, we have calculated the masses of the technipions in the one-family TC model, and shown that those are rather heavy, raging from $\\sim 1$ TeV to $4.6$ TeV depending on the $N_{\\rm TC}$ and the input value of the $S$ parameter.", "One thing which should be noted here is that the existence of heavy technipions does not necessarily mean that the scale of explicit breaking of chiral symmetry is also large.", "Indeed, the reason why we obtained large technipion masses here is that the contribution from the UV energy scale is enhanced due to the large mass anomalous dimension $\\gamma _m = 1$ though the explicit breaking itself is rather small perturbation.", "And accordingly, we expect the walking behavior persists all the way down to the intrinsic dynamical scale of the TC model, without decoupling of any degree of freedom of the technifermion." ], [ "LHC phenomenology", "As we have estimated the masses of technipions in the one-family model, we discuss collider phenomenologies related to the technipions in this section.", "First we discuss the constraints from the currently available LHC data, and then briefly mention implications for the future collider phenomenology." ], [ "Color-octet technipion $\\theta _a^0$", "The relevant cross sections and partial decay widths for two-body decays are read off from Ref. [19].", "For the reference values for the mass of $\\theta _a^0$ given in Table REF , we compute the total width of $\\theta _a^0$ and branching ratios to obtain the numbers shown in Table REF .", "Table REF implies that the digluon and $t\\bar{t}$ events at the LHC may provide good probes for the discovery of $\\theta _a^0$ .", "In Table REF we show the predicted signal strengths of $\\theta _a^0$ at the 8 TeV LHC for $gg$ and $t\\bar{t}$ channels for the reference values of the $\\theta _a^0$ masses listed in Table REF .", "These signals can be constrained by the current LHC limits on searches for new resonances in the dijet and $t\\bar{t}$ channels [33], [34] as shown in Table REF .", "Thus the current LHC data, especially on the $t \\bar{t}$ channel, have already excluded the color-octet technipion $\\theta _a^0$ with the mass $m_{\\theta _a^0}\\simeq 1.5$ for $(N_{\\rm TC}, S)= (3, 1.0), (4, 1.0)$ and with the mass $\\simeq 1.6$ TeV for $(N_{\\rm TC}, S)=(5, 1.0)$ ." ], [ "Color-singlet technipion $P^0$", "The formulas for relevant cross sections and partial decay widths are read off from Ref. [19].", "For the predicted values for the mass of $P^0$ listed in Table REF , we calculate the total width of $P^0$ and branching ratios to obtain the values shown in Table REF .", "From this, one can see that the digluon and $t\\bar{t}$ events at the LHC may be channels for discovery of $P^0$ .", "For each channel the predicted signal strengths of $P^0$ at the 8 TeV LHC for the reference values of the mass listed in Table REF are displayed in Table REF .", "In comparison with the current LHC limits listed in Table REF , we thus see that the color-singlet technipion $P^0$ has already been excluded for the masses 720, 760 and 790 GeV corresponding to the cases of $(N_{\\rm TC}, S)=(3, 1.0), (4, 1.0), (5, 1.0)$ , respectively." ], [ "Color-triplet technipions $T_c^{0,i}$", "The LHC experiments have placed stringent constraints on the scalar leptoquarks $({\\rm LQ}_{1,2,3})$ [35], in which the most stringent bound on the mass has been set on the second generation leptoquark ${\\rm LQ}_2$ as $m_{{\\rm LQ}_2} \\ge 1070 \\, {\\rm GeV}\\,, $ with 100% branching ratio for the decay to $\\mu \\nu _\\mu + 2j$ being assumed.", "Though the coupling property of the color-triplet technipion $T_c^{0,i}$ depends highly on modeling of the extended TC, we may apply the above strongest bound on the $T_c^{0,i}$ masses.", "Comparing the reference values of $T_c^{0,i}$ masses listed in Table REF with the bound in Eq.", "(REF ), we thus see that the current LHC data have excluded the color-triplet technipions $T_c^{0,i}$ with the masses at around 1.0 – 1.1 TeV, corresponding to the cases of $(N_{\\rm TC}, S)=(3,1.0), (4, 1.0), (5, 1.0)$ ." ], [ "Implications for technirho searches", "A typical signature of the dynamical EW symmetry breaking scenario at hadron colliders is the existence of the new vector particle, called the technirho, which is an analogue of the rho meson in QCD.", "In the case of one-family model, there are various kinds of technirho mesons from the viewpoint of the SM gauge representation, in a similar way as technipions have various SM gauge representations.", "In general, technirho mesons are produced through the mixing with the SM gauge boson which is produced by the Drell-Yan (DY) process, though how they decay depend on the mass relation among technirho mesons and technipions.", "The mass of technirho meson can be calculated by the holographic method as $M_{\\rho } \\simeq (3.6,\\, 2.1,\\, 1.1) \\ {\\rm TeV}\\ \\ {\\rm for}\\ \\ S = (0.1,\\, 0.3,\\, 1.0).$ ($N_{\\rm TC}$ dependence is negligible.)", "Due to the large enhancement of the technipion masses (see Tables REF and REF ), decay channels of technirho mesons to a pair of technipions are closed, therefore, they decay to SM particles or a SM particle plus one technipion.", "In the case of the iso-triplet technirho mesons ($\\rho ^{\\pm , 3}$ ), they dominantly decay to a pair of gauge bosons or one gauge boson plus one technipion like $pp & \\stackrel{\\rm DY}{\\rightarrow } & \\rho ^\\pm \\rightarrow W^\\pm + P^3 \\quad {\\rm or} \\quad Z + P^\\pm \\,, \\nonumber \\\\pp & \\stackrel{\\rm DY}{\\rightarrow } & \\rho ^3 \\rightarrow W^\\pm + P^\\mp \\,.$ In the case of the iso-singlet and color-singlet ($\\rho ^0$ )/color-octet technirho mesons ($\\rho _8^0$ ), there is a possibility that those technirhos dominantly decay to the technidilaton $(\\phi )$ and the photon/gluon like $pp & \\stackrel{\\rm DY}{\\rightarrow } & \\rho ^0 \\rightarrow \\phi + \\gamma \\,, \\nonumber \\\\pp & \\stackrel{\\rm DY}{\\rightarrow } & \\rho ^0_8 \\rightarrow \\phi + g\\,.$ The detailed study of collider phenomenologies will be presented in the future publications." ], [ "Summary", "In this paper, we calculated the masses of the technipions in the one-family WTC model based on a holographic approach, which is known to be successful in the case of QCD.", "It was shown that technipion masses are enhanced due to the walking dynamics to as large as ${\\cal O} (1)$ TeV, somewhat larger than the previous estimates [19].", "Constraints from the currently available LHC data, as well as implications for the future collider phenomenology were also discussed.", "In particular, we found a correlation between the technipion masses and the technicolor contribution to the $S$ parameter, $S^{({\\rm TC})}$ , which gives a constraint on the WTC model building solely from the current LHC data on the technipion mass limit: $S^{({\\rm TC})}$ should not be as large as ${\\cal O} (1)$ .", "This is a new constraint on the contribution to the $S$ parameter from the technicolor dynamics alone, in contrast to the full $S$ parameter constraint from the precision electroweak measurements.", "We would like to thank Koji Terashi for useful information and discussions.", "This work was supported by the JSPS Grant-in-Aid for Scientific Research (S) #22224003 and (C) #23540300 (K.Y.", ")." ], [ "$\\pi ^+ - \\pi ^0$ mass difference in the real-life QCD", "Here, we apply the same method used in subsection REF for the calculation of the $\\pi ^+ - \\pi ^0$ mass difference in the real-life QCD.", "The dominant part of the mass difference between $\\pi ^+$ and $\\pi ^0$ comes from the explicit breaking of the chiral symmetry due to the electromagnetic interaction.", "The formula for this mass difference is quite similar to the one in Eq.", "(REF ): we just have to replace $\\alpha _s(Q^2) \\rightarrow \\alpha _{\\rm EM} \\simeq 1/137$ , $C_{3,8} \\rightarrow 1$ , and identify $F_\\pi $ as the pion decay constant $f_\\pi \\simeq 92$ MeV: $\\Delta m_\\pi ^2 & \\equiv &m_{\\pi ^+}^2-m_{\\pi ^0}^2\\nonumber \\\\&=& \\frac{3\\, \\alpha _{\\rm EM}}{4\\pi f_\\pi ^2} \\int _0^\\infty dQ^2 \\left[ \\Pi _V(Q^2) - \\Pi _A(Q^2) \\right]\\,.$ Parameters of the holographic model are chosen so that it reproduces correct asymptotic behavior of the current correlators and experimental values of physical quantities.", "The optimal choice of the parameters is [16]: $\\gamma _m = 0,\\ \\ \\ \\xi =3.1,\\ \\ \\ G=0.25,\\ \\ \\ z_m^{-1}=347\\ {\\rm MeV}.$ $\\Pi _V(Q^2) - \\Pi _A(Q^2)$ obtained from the holographic calculation with the above input parameters is shown in Fig.", "REF .", "Figure: Π V (Q 2 )-Π A (Q 2 )\\Pi _V(Q^2)-\\Pi _A(Q^2) in the real-life QCD obtained from the holographic calculation with the input parameters shown in Eq.", "().By inserting this $\\Pi _V(Q^2) - \\Pi _A(Q^2)$ into Eq.", "(REF ), we obtain $\\Delta m_\\pi ^2 \\simeq (32\\ {\\rm MeV})^2$ , in reasonable agreement with experimental value $\\Delta m_\\pi ^2 \\simeq (35\\ {\\rm MeV})^2$ [32].", "We have also calculated $\\Delta m_\\pi ^2$ with taking $G=0$ so that we could see the effect of gluon condensation, though the result was almost the same as in the case of $G=0.25$ .", "This is reasonable since gluon condensation effect is expected to contribute to $\\Pi _{V}(Q^2)$ and $\\Pi _{A}(Q^2)$ in the same way, so the effect cancels in the integration of Eq.", "(REF ).", "The result can be translated to the form $\\Delta m_\\pi \\equiv m_{\\pi ^+}-m_{\\pi ^0} = \\frac{\\Delta m_\\pi ^2}{m_{\\pi ^+}+m_{\\pi ^0}} \\simeq 3.7\\ {\\rm MeV}$ , where we used experimental values of $m_{\\pi ^+}+m_{\\pi ^0}$ in the denominator.", "This is compared to the result in Ref.", "[21] obtained by using the same holographic model except that, in their calculation, no gluonic-condensation effect is incorporated.", "They obtained $\\Delta m_\\pi \\simeq 3.6 - 4.0\\ {\\rm MeV}$ depending on the choice of input parameter $\\xi $ , which is in reasonable agreement with our result." ], [ "Comparison to the ladder BS calculation", "In this appendix, we compare $\\Pi _V(Q^2)-\\Pi _A(Q^2)$ calculated from the holographic WTC model with that calculated from the inhomogeneous BS equation with the improved ladder approximation in Ref. [18].", "In Fig.", "REF , we show $\\Pi _V(Q^2)-\\Pi _A(Q^2)$ for the case of $N_{\\rm TC} = 3$ (solid, dashed, and dotted (red) curves correspond to the cases of $S=0.1$ , $0.3$ , and $1.0$ ) obtained from the holographic calculation, along with the one calculated from ladder BS equation (dashed-dotted (blue) curve).", "Both the vertical and the horizontal axes are normalized by $F_\\pi ^2$ .", "Figure: Π V (Q 2 )-Π A (Q 2 )\\Pi _V(Q^2)-\\Pi _A(Q^2) calculated from the holographic WTC model for the case of N TC =3N_{\\rm TC} = 3, compared with that calculated from the inhomogeneous BS equation with the improved ladder approximation in Ref. .", "Solid, dashed, and dotted (red) curves correspond to the cases of S=0.1S=0.1, 0.30.3, and 1.01.0 obtained from the holographic calculation, while dashed-dotted (blue) curve is the one calculated from ladder BS equation.", "Both the vertical and the horizontal axes are normalized by F π 2 F_\\pi ^2$\\Pi _V(Q^2)-\\Pi _A(Q^2)$ obtained from ladder BS calculation shown in the figure is same as that shown in Fig.", "6 in Ref. [18].", "See Ref.", "[18] for detailed explanation for the calculation.", "From the figure, we see that the one calculated from the ladder BS equation is similar to the one calculated from holographic method with $S=1.0$ , and each gives the contribution to the mass of the color-triplet technipion 1.2 and 1.1 TeV, respectively.", "This similarity can be understood from the fact that the ladder BS calculation show that the contribution to the $S$ parameter from one EW doublet (denoted as $\\hat{S}$ ) is about $0.3$ [18], while $S=1.0$ in one-family model means $\\hat{S}=0.25$ since one-family model has four EW doublet.", "The slope of $\\Pi _V(Q^2)-\\Pi _A(Q^2)$ is proportional to the magnitude of $\\hat{S}$ , thus the similarity between ladder calculation and the holographic calculation with $S=1.0$ ($\\hat{S}=0.25$ ) is a confirmation of consistency between two calculation methods." ] ]	1403.0467
[ [ "A Primal Dual Active Set with Continuation Algorithm for the\n \\ell^0-Regularized Optimization Problem" ], [ "Abstract We develop a primal dual active set with continuation algorithm for solving the \\ell^0-regularized least-squares problem that frequently arises in compressed sensing.", "The algorithm couples the the primal dual active set method with a continuation strategy on the regularization parameter.", "At each inner iteration, it first identifies the active set from both primal and dual variables, and then updates the primal variable by solving a (typically small) least-squares problem defined on the active set, from which the dual variable can be updated explicitly.", "Under certain conditions on the sensing matrix, i.e., mutual incoherence property or restricted isometry property, and the noise level, the finite step global convergence of the algorithm is established.", "Extensive numerical examples are presented to illustrate the efficiency and accuracy of the algorithm and the convergence analysis." ], [ "Introduction", "Over the last ten years, compressed sensing [7], [12] has received a lot of attention amongst engineers, statisticians and mathematicians due to its broad range of potential applications.", "Mathematically it can be formulated as the following $\\ell ^0$ optimization problem: $\\begin{aligned}& \\min _{x \\in \\mathbb {R}^{p}}\\Vert x\\Vert _{0},\\\\\\textrm {subject to}\\ \\ & \\ \\ \\Vert \\Psi x -y\\Vert _{2}\\le \\epsilon ,\\end{aligned}$ where the sensing matrix $\\Psi \\in \\mathbb {R}^{n\\times p}$ with $p\\gg n$ has normalized column vectors (i.e., $\\Vert \\psi _i\\Vert =1$ , $i=1,\\cdots ,p$ ), $\\epsilon \\ge 0$ is the noise level, and $\\Vert x\\Vert _{0}$ denotes the the number of nonzero components in the vector $x$ .", "Due to the discrete structure of the term $\\Vert x\\Vert _0$ , it is very challenging to develop an efficient algorithm to accurately solve the model (REF ).", "Hence, approximate methods for the model (REF ), especially greedy heuristics and convex relaxation, are very popular in practice.", "In greedy algorithms, including orthogonal matching pursuit [31], stagewise orthogonal matching pursuit [15], regularized orthogonal matching pursuit [26], CoSaMP [25], subspace pursuit [11], and greedy gradient pursuit [5] etc., one first identifies the support of the sought-for signal, i.e., the locations of (one or more) nonzero components, iteratively based on the current dual variable (correlation), and then updates the components on the support by solving a least-squares problem.", "There are also several variants of the greedy heuristics, e.g., (accelerated) iterative hard thresholding [3], [4] and hard thresholding pursuit [16], which are based on the sum of the current primal and dual variable.", "In contrast, basis pursuit finds one minimizer of a convex relaxation problem [9], [34], for which a wide variety of convex optimization algorithms can be conveniently applied; see [2], [10], [30], [35] for a comprehensive overview and the references therein.", "Besides greedy methods and convex relaxation, the “Lagrange” counterpart of (REF ) (or equivalently, the $\\ell ^0$ -regularized minimization problem), which reads $\\min _{x \\in \\mathbb {R}^{p}} J_\\lambda (x) = \\tfrac{1}{2}\\Vert \\Psi x - y\\Vert ^2 + \\lambda \\Vert x\\Vert _0,$ has been very popular in many applications, e.g., model selection, statistical regression, and image restoration.", "In the model (REF ), $\\lambda >0$ is a regularization parameter, controlling the sparsity level of the regularized solution.", "Due to the nonconvexity and discontinuity of the function $\\Vert x\\Vert _0$ , the relation between problems (REF ) and (REF ) is not self evident.", "We shall show that under certain assumptions on the sensing matrix $\\Psi $ and the noise level $\\epsilon $ (and with $\\lambda $ chosen properly), the support of the solution to (REF ) coincides with that of the true signal, cf.", "Theorem REF .", "Very recently, the existence and a characterization of global minimizers to (REF ) were established in [22], [27].", "However, it is still very challenging to develop globally convergent algorithms for efficiently solving problem (REF ) in view of its nonconvexity and nonsmoothness.", "Nonetheless, due to its broad range of applications, several algorithms have been developed to find an approximate solution to problem (REF ), including iterative hard thresholding [4], forward backward splitting [1], penalty decomposition [23] and stochastic continuation [32], [33], to name just a few.", "Theoretically, these algorithms can at best have a local convergence.", "Very recently, in [21], [22], based on a coordinatewise characterization of the global minimizers, a novel primal dual active set (PDAS) algorithm was developed to solve problem (REF ).", "The extensive simulation studies in [22] indicate that when coupled with a continuation technique, the PDAS algorithm merits a global convergence property.", "We note that the PDAS can at best converge to a coordinatewise minimizer.", "However, if the support of the coordinatewise minimizer is small and the sensing matrix $\\Psi $ satisfies certain mild conditions, then its active set is contained in the support of the true signal, cf.", "Lemma REF .", "Hence, the support of the minimizer will coincide with that of the true signal if we choose the parameter $\\lambda $ properly (and thus control the size of the active set) during the iteration.", "This motivates the use of a continuation strategy on the parameter $\\lambda $ .", "The resulting PDAS continuation (PDASC) algorithm extends the PDAS developed in [22].", "In this work, we provide a convergence analysis of the PDASC algorithm under commonly used assumptions on the sensing matrix $\\Psi $ for the analysis of existing algorithms, i.e., mutual incoherence property and restricted isometry property.", "The convergence analysis relies essentially on a novel characterization of the evolution of the active set during the primal-dual active set iterations.", "To the best of our knowledge, this represents the first work on the global convergence of an algorithm for problem (REF ), without using a knowledge of the exact sparsity level.", "The rest of the paper is organized as follows.", "In Section , we describe the problem setting, collect basic estimates, and provide a refined characterization of a coordinatewise minimizer.", "In Section , we give the complete algorithm, discuss the parameter choices, and provide a convergence analysis.", "Finally, in Section , several numerical examples are provided to illustrate the efficiency of the algorithm and the convergence theory." ], [ "Regularized $\\ell ^0$ -minimization", "In this section, we describe the problem setting, and derive basic estimates, which are essential for the convergence analysis.", "Further, we give sufficient conditions for a coordinatewise minimizer to be a global minimizer." ], [ "Problem setting", "Suppose that the true signal $x^$ has $T$ nonzero components with its active set (indices of nonzero components) denoted by $A^$ , i.e., $T=\|A^\|$ and the noisy data $y$ is formed by $y = \\sum _{i\\in {A}^} x_i^\\psi _i + \\eta .$ We assume that the noise vector $\\eta $ satisfies $\\Vert \\eta \\Vert \\le \\epsilon $ , with $\\epsilon \\ge 0$ being the noise level.", "Further, we let $S = \\lbrace 1,2,...,p\\rbrace \\quad \\mbox{and}\\quad I^ = S\\backslash {A}^.$ For any index set ${A}\\subseteq S$ , we denote by $x_{A}\\in \\mathbb {R}^{\|{A}\|}$ (respectively $\\Psi _{{A}} \\in \\mathbb {R}^{n\\times \|{A}\|}$ ) the subvector of $x$ (respectively the submatrix of $\\Psi $ ) whose indices (respectively column indices) appear in ${A}$ .", "Last, we denote by $x^o$ the oracle solution defined by $x^o = \\Psi _{{A}^}^\\dag y,$ where $\\Psi _A^\\dag $ denotes the pseudoinverse of the submatrix $\\Psi _A$ , i.e., $\\Psi _A^\\dag =(\\Psi ^t_{A}\\Psi _{A})^{-1}\\Psi ^t_{{A}}$ if $\\Psi ^t_{A}\\Psi _{A}$ is invertible.", "In compressive sensing, there are two assumptions, i.e., mutual incoherence property (MIP) [13] and restricted isometry property (RIP) [8], on the sensing matrix $\\Psi $ that are frequently used for the convergence analysis of sparse recovery algorithms.", "The MIP relies on the fact that the mutual coherence (MC) $\\nu $ of sensing matrix $\\Psi $ is small, where the mutual coherence (MC) $\\nu $ of $\\Psi $ is defined by $\\nu = \\max \\limits _{1\\le i, j \\le p, i\\ne j} \|\\psi _i^t\\psi _j\|.$ A sensing matrix $\\Psi $ is said to satisfy RIP of level $s$ if there exists a constant $\\delta \\in (0,1)$ such that $(1-\\delta )\\Vert x\\Vert ^2 \\le \\Vert \\Psi x\\Vert ^2 \\le (1+\\delta )\\Vert x\\Vert ^2, \\,\\, \\forall x\\in \\mathbb {R}^p \\mbox{ with } \\Vert x\\Vert _0 \\le s,$ and we denote by $\\delta _s$ the smallest constant with respect to the sparsity level $s$ .", "We note that the mutual coherence $\\nu $ can be easily computed, but the RIP constant $\\delta _s$ is nontrivial to evaluate.", "The next lemma gives basic estimates under the MIP condition.", "Lemma 2.1 Let $A$ and $B$ be disjoint subsets of $S$ .", "Then $\\begin{aligned}\\Vert \\Psi ^t_A y\\Vert _{\\ell ^\\infty }&\\le \\Vert y\\Vert ,\\\\\\Vert \\Psi ^t_B\\Psi _A x_A\\Vert _{\\ell ^\\infty }& \\le \|A\|\\nu \\Vert x_A\\Vert _{\\ell ^\\infty },\\\\\\Vert (\\Psi _A^t\\Psi _A)^{-1} x_A\\Vert _{\\ell ^\\infty } &\\le \\frac{\\Vert x_A\\Vert _{\\ell ^\\infty }}{1 - (\|A\|-1)\\nu } \\quad \\mbox{if} \\quad (\|A\| -1)\\nu <1.\\end{aligned}$ If $A = \\emptyset $ , then the estimates are trivial.", "Hence we will assume $A$ is nonempty.", "For any $i\\in A$ , $\|\\psi _i^t y\| \\le \\Vert \\psi _i\\Vert \\Vert y\\Vert \\le \\Vert y\\Vert .$ This shows the first inequality.", "Next, for any $i\\in B$ , $\|\\psi _i^t\\Psi _A x_A\| = \|\\sum _{j\\in A} \\psi _i^t\\psi _jx_j\|\\le \\sum _{j\\in A}\|\\psi _i^t\\psi _j\|\|x_j\|\\le \|A\|\\nu \\Vert x_A\\Vert _{\\ell ^\\infty }.$ This shows the second assertion.", "To prove the last estimate, we follow the proof strategy of [36], i.e., applying a Neumann series method.", "First we note that $\\Psi _A^t\\Psi _A$ has a unit diagonal because all columns of $\\Psi $ are normalized.", "So the off-diagonal part $\\Phi $ satisfies $\\Psi _A^\\Psi _A = E_{\|A\|} + \\Phi ,$ where $E_{\|A\|}$ is an identity matrix.", "Each column of the matrix $\\Phi $ lists the inner products between one column of $\\Psi _A$ and the remaining $\|A\|-1$ columns.", "By the definition of the mutual coherence $\\nu $ and the operator norm of a matrix $\\Vert \\Phi \\Vert _{\\ell ^\\infty ,\\ell ^\\infty } = \\max _{k\\in A}\\sum _{j\\in A\\setminus \\lbrace k\\rbrace } \|\\psi _j^t\\psi _k\|\\le (\|A\|-1)\\nu .$ Whenever $\\Vert \\Phi \\Vert _{\\ell ^\\infty ,\\ell ^\\infty }<1$ , the Neumann series $\\sum _{k=0}^\\infty (-\\Phi )^k$ converges to the inverse $(E_{\|A\|}+\\Phi )^{-1}$ .", "Hence, we may compute $\\begin{aligned}\\Vert (\\Psi _A^\\Psi _A)^{-1}\\Vert _{\\ell ^\\infty ,\\ell ^\\infty } & = \\Vert (E_{\|A\|}+\\Phi )^{-1}\\Vert _{\\ell ^\\infty ,\\ell ^\\infty }= \\Vert \\sum _{k=0}^\\infty (-\\Phi )^k\\Vert _{\\ell ^\\infty ,\\ell ^\\infty }\\\\&\\le \\sum _{k=0}^\\infty \\Vert \\Phi \\Vert _{\\ell ^\\infty ,\\ell ^\\infty }^k=\\frac{1}{1-\\Vert \\Phi \\Vert _{\\ell ^\\infty ,\\ell ^\\infty }} \\le \\frac{1}{1-(\|A\|-1)\\nu }.\\end{aligned}$ The desired estimate now follows immediately.", "The following lemma collects some estimates on the RIP constant $\\delta _s$ ; see [25] and [11] for the proofs.", "Lemma 2.2 Let $A$ and $B$ be disjoint subsets of $S$ .", "Then $\\begin{aligned}\\Vert \\Psi _A^t\\Psi _A x_A\\Vert &\\gtreqqless (1\\mp \\delta _{\|A\|})\\Vert x_A\\Vert ,\\\\\\Vert (\\Psi _A^t\\Psi _A)^{-1} x_A\\Vert &\\gtreqqless \\frac{1}{1\\pm \\delta _{\|A\|}}\\Vert x_A\\Vert ,\\\\\\Vert \\Psi _A^t\\Psi _B\\Vert &\\le \\delta _{\|A\|+\|B\|},\\\\\\Vert \\Psi _A^\\dag y\\Vert & \\le \\frac{1}{\\sqrt{1-\\delta _{\|A\|}}}\\Vert y\\Vert ,\\\\\\delta _{s} & \\le \\delta _{s^{\\prime }}, \\mbox{ if } s<s^{\\prime }.", "\\end{aligned}$ The next lemma gives some crucial estimates for one-step primal dual active set iteration on the active set $A$ .", "These estimates provide upper bounds on the dual variable $d=\\Psi ^t(y-\\Psi x)$ and the error $\\bar{x}_A=x_A-x^_A$ on the active set $A$ .", "They will play an essential role for subsequent analysis, including the convergence of the algorithm.", "Lemma 2.3 For any set ${A}\\subseteq S$ with $\|{A}\|\\le T$ , let ${B} = {A}^\\setminus {A}$ and $I=S\\setminus {A}$ , and consider the following primal dual iteration on $A$ $x_{{A}} = \\Psi _{{A}}^\\dag y, \\quad x_{{I}} = 0,\\quad d = \\Psi ^t( y -\\Psi x).$ Then the quantities $\\bar{x}_A\\equiv x_{{A}} - x^_{{A}}$ and $d$ satisfy the following estimates.", "(a) If $\\nu < 1/(T-1)$ , then $d_A = 0 $ and $\\begin{aligned}\\Vert \\bar{x}_A\\Vert _{\\ell ^\\infty }& \\le \\frac{1}{1-(\|A\| -1)\\nu }\\left(\|B\|\\nu \\Vert x_B^\\Vert _{\\ell ^\\infty } + \\epsilon \\right),\\\\ \|d_{j}\| & \\ge \|x_j^\| - \\Vert x_B^\\Vert _{\\ell ^\\infty }(\|B\|-1)\\nu - \\epsilon - \|A\|\\nu \\Vert \\bar{x}_A\\Vert _{\\ell ^\\infty }, \\quad \\forall j\\in B,\\\\ \|d_j\| &\\le \|B\|\\nu \\Vert x_B^\\Vert _{\\ell ^\\infty } + \\epsilon + \|A\|\\nu \\Vert \\bar{x}_A\\Vert _{\\ell ^\\infty }, \\quad \\forall j\\in I^\\cap I.", "\\end{aligned}$ (b) If the RIP is satisfied for sparsity level $s:=\\max \\lbrace \|A\| + \|B\|, T+1\\rbrace $ , then $d_A = 0$ and $\\begin{aligned}\\Vert \\bar{x}_{A}\\Vert & \\le \\frac{\\delta _{\|{A}\| + \|{B}\|}}{1-\\delta _{\|{A}\|}}\\Vert x^_{B}\\Vert + \\frac{1}{\\sqrt{1-\\delta _{\|{A}\|}}}\\epsilon ,\\\\ \|d_{j}\| & \\ge \|x^_j\| - \\delta _{\|B\|}\\Vert x_{{B}}^\\Vert - \\epsilon - \\delta _{\|A\|+1}\\Vert \\bar{x}_{{A}}\\Vert , \\quad \\forall j\\in B,\\\\\|d_j\|&\\le \\delta _{\|B\|+1}\\Vert x_{{B}}^\\Vert + \\epsilon + \\delta _{\|A\|+1}\\Vert \\bar{x}_{{A}}\\Vert , \\quad \\forall j\\in {I}^\\cap {I},\\end{aligned}$ We show only the estimates under the RIP condition and using Lemma REF , and that for the MIP condition follows similarly from Lemma REF .", "If $A= \\emptyset $ , then all the estimates clearly hold.", "In the case $A\\ne \\emptyset $ , then by the assumption, $\\Psi _A^t\\Psi _A$ is invertible.", "By the definition of the update $x_A$ and the data $y$ we deduce that $d_A = \\Psi _A^t( y - \\Psi _A x_A) = 0,$ and $\\begin{aligned}\\bar{x}_{{A}} &= (\\Psi ^t_{A}\\Psi _{A})^{-1}\\Psi ^t_{{A}} (\\Psi _{{A}^} x_{{A}^}^ + \\eta - \\Psi _{{A}} x_{{A}}^)\\\\&= (\\Psi ^t_{A}\\Psi _{A})^{-1}\\Psi ^t_A (\\Psi _{B} x^_{B} + \\eta ).\\end{aligned}$ Consequently, by Lemma REF and the triangle inequality, there holds $\\begin{aligned}\\Vert \\bar{x}_A\\Vert & \\le \\frac{1}{1-\\delta _{\|A\|}}\\Vert \\Psi ^t_A\\Psi _B x_B^\\Vert +\\Vert \\Psi _A^\\dag \\eta \\Vert \\\\& \\le \\frac{1}{1-\\delta _{\|A\|}}\\delta _{\|A\|+\|B\|}\\Vert x_B^\\Vert + \\frac{1}{\\sqrt{1-\\delta _{\|A\|}}}\\epsilon .\\end{aligned}$ It follows from the definition of the dual variable $d$ , i.e., $d_j = \\psi ^t_j(y - \\Psi _{A}x_{A}) = \\psi ^t_j(\\Psi _{B} x^_{{B}} + \\eta - \\Psi _{A} \\bar{x}_{A}),$ Lemma REF , and $\\psi _j^t\\psi _j=1$ that for any $j\\in B$ , there holds $\\begin{aligned}\|d_j\| & = \|\\psi ^t_j\\psi _j x_j^ + \\psi _j^t(\\Psi _{B\\setminus \\lbrace j\\rbrace } x_{B\\setminus \\lbrace j\\rbrace }^* + \\eta - \\Psi _A\\bar{x}_A)\|\\\\& \\ge \|x_j^\| - (\|\\psi _j^t\\Psi _{B\\setminus \\lbrace j\\rbrace } x_{B\\setminus \\lbrace j\\rbrace }^\| + \|\\psi _j^t\\eta \| + \|\\psi _j^t\\Psi _A\\bar{x}_A\|)\\\\& \\ge \|x_j^\| - \\delta _{\|B\|}\\Vert x_B^\\Vert - \\epsilon - \\delta _{\|A\|+1}\\Vert \\bar{x}_{A}\\Vert .\\end{aligned}$ Similarly, for any $j\\in I^\\cap I $ , there holds $\|d_j\| \\le \\delta _{\|B\|+1} \\Vert x^_B\\Vert +\\epsilon + \\delta _{\|A\|+1}\\Vert \\bar{x}_A\\Vert .$ This completes the proof of the lemma." ], [ "Coordinatewise minimizer", "Next we characterize minimizers to problem (REF ).", "Due to the nonconvexity and discontinuity of the function $\\Vert x\\Vert _0$ , the classical theory [20] on the existence of a Lagrange multiplier cannot be applied directly to show the equivalence between problem (REF ) and the Lagrange counterpart (REF ).", "Nonetheless, both formulations aim at recovering the true sparse signal $x^$ , and thus we expect that they are closely related to each other.", "We shall establish below that with the parameter $\\lambda $ properly chosen, the oracle solution $x^o$ is the only global minimizer of problem (REF ), and as a consequence, we derive directly the equivalence between problems (REF ) and (REF ).", "To this end, we first characterize the minimizers of problem (REF ).", "Since the cost function $J_\\lambda (x)$ is nonconvex and discontinuous, instead of a global minimizer, we study its coordinatewise minimizers, following [37].", "A vector $x=(x_1,x_2, \\dots ,x_p)^t\\in \\mathbb {R}^p$ is called a coordinatewise minimizer to $J_\\lambda (x)$ if it is the minimum along each coordinate direction, i.e., $x_i \\in \\mathop \\mathrm {arg}\\min \\limits _{t\\in \\mathbb {R}} J_\\lambda (x_1,...,x_{i-1},t,x_{i+1},...,x_p).$ The necessary and sufficient condition for a coordinatewise minimizer $x$ is given by [21], [22]: $x_i \\in S^{\\ell ^0}_\\lambda (x_i + d_i)\\quad \\forall i\\in S,$ where $d = \\Psi ^t(y - \\Psi x)$ denotes the dual variable, and $S^{\\ell ^0}_\\lambda $ is the hard thresholding operator defined by $S^{\\ell ^0}_{\\lambda }(v) \\left\\lbrace \\begin{array}{ll} =0, & \|v\| < \\sqrt{2\\lambda }, \\\\[1.3ex]\\in \\lbrace 0, \\mbox{sgn}(v)\\sqrt{2\\lambda }\\rbrace , & \|v\| = \\sqrt{2\\lambda }, \\\\[1.3ex]= v, & \|v\| > \\sqrt{2\\lambda }.", "\\end{array}\\right.$ The condition (REF ) can be equivalently written as $\\left\\lbrace \\begin{array}{l}\|x_i + d_i\| > \\sqrt{2\\lambda } \\Rightarrow d_i = 0,\\\\[1.3ex]\|x_i + d_i\| < \\sqrt{2\\lambda } \\Rightarrow x_i = 0,\\\\[1.3ex]\|x_i + d_i\| = \\sqrt{2\\lambda } \\Rightarrow x_i = 0 \\mbox{ or } d_i =0.\\end{array}\\right.$ Consequently, with the active set ${A} = \\lbrace i: x_i\\ne 0\\rbrace $ , there holds $\\min \\limits _{i\\in {A}} \|x_i\| \\ge \\sqrt{2\\lambda } \\ge \\Vert d\\Vert _{\\ell ^\\infty }.$ It is known that any coordinatewise minimizer $x$ is a local minimizer [22].", "To further analyze the coordinatewise minimizer, we need the following assumption on the noise level $\\epsilon $ : Assumption 2.1 The noise level $\\epsilon $ is small in the sense $\\epsilon \\le \\beta \\min _{i\\in A^}\|x_i^\|$ , $0\\le \\beta <1/2$ .", "The next lemma gives an interesting characterization of the active set of the coordinatewise minimizer.", "Lemma 2.4 Let Assumption REF hold, and $x$ be a coordinatewise minimizer with support ${A}$ and $\|A\|\\le T$ .", "If either (a) $\\nu < (1-2\\beta )/(3T - 1)$ or (b) $\\delta \\triangleq \\delta _{2T} \\le (1-2\\beta )/(2\\sqrt{T}+1)$ holds, then ${A}\\subseteq {A}^$ .", "Let $I=S\\setminus A$ .", "Since $x$ is a coordinatewise minimizer, it follows from (REF ) that $x_A = \\Psi _A^\\dag y,\\quad x_I=0, \\quad d = \\Psi ^t(y - \\Psi x).$ We prove the assertions by means of contradiction.", "Assume the contrary, i.e., $A\\nsubseteq A^$ .", "We let $B=A^\\setminus A$ , which is nonempty by assumption, and denote by $i_{A} \\in \\lbrace i\\in I: {\|x_i^\| }= \\Vert x^_B\\Vert _{\\ell ^\\infty }\\rbrace $ .", "Then $i_A\\in B$ .", "Further by (REF ), there holds $\|d_{i_A}\| \\le \\Vert d\\Vert _{\\ell ^\\infty } \\le \\min _{i\\in A}\|x_i\| \\le \\min _{i\\in A\\backslash A^} \|x_i\| \\le \\Vert \\bar{x}_A\\Vert _{\\ell ^\\infty } \\le \\Vert \\bar{x}_A\\Vert .$ Now we discuss the two cases separately.", "Case (a).", "By Lemma REF , $\\epsilon \\le \\beta \\min _{i\\in A^}\|x_i^\| \\le \\beta \\Vert x_B^\\Vert _{\\ell ^\\infty }$ from Assumption REF and the choice of the index $i_A$ , we have $\\begin{aligned}\\Vert \\bar{x}_A\\Vert _{\\ell ^\\infty } &\\le \\frac{1}{1-(\|A\| -1)\\nu }\\left(\|B\|\\nu \\Vert x_B^\\Vert _{\\ell ^\\infty } + \\epsilon \\right) \\\\& \\le \\frac{1}{1-(\|A\|-1)\\nu }(\|B\|\\nu +\\beta )\\Vert x_B^\\Vert _{\\ell ^\\infty },\\\\\|d_{i_A}\| & \\ge \|x_{i_A}^\| - \\Vert x_B^\\Vert _{\\ell ^\\infty }(\|B\|-1)\\nu - \\epsilon - \|A\|\\nu \\Vert \\bar{x}_A\\Vert _{\\ell ^\\infty }\\\\& \\ge \\Vert x_B^\\Vert _{\\ell ^\\infty }\\left(1-(\|B\|-1)\\nu - \\beta - \|A\|\\nu \\frac{1}{1-(\|A\|-1)\\nu }(\|B\|\\nu +\\beta )\\right).\\end{aligned}$ Consequently, we deduce $\\begin{aligned}\|d_{i_A}\| - \\Vert \\bar{x}_A\\Vert _{\\ell ^\\infty } &\\ge \\frac{\\Vert x^_B\\Vert _{\\ell ^\\infty }}{1 - (\|A\|-1)\\nu }\\left[ 1- (\|A\| + 2\|B\|)\\nu - (\|A\|+\|B\|)\\nu ^2 + 2\\nu +\\nu ^2 - \\beta (\\nu +2)\\right]\\\\& \\ge \\frac{\\Vert x^_B\\Vert _{\\ell ^\\infty }}{1-(\|A\|-1)\\nu }\\left[1-3T\\nu + \\nu -2\\beta + \\nu (1 - \\beta - 2T\\nu )\\right]\\\\& \\ge \\frac{\\Vert x^_B\\Vert _{\\ell ^\\infty }}{1-(\|A\|-1)\\nu }\\left[1-(3T-1)\\nu -2\\beta \\right]>0,\\end{aligned}$ under assumption (a) $\\nu <(1-2\\beta )/(3T-1)$ .", "This leads to a contradiction to (REF ).", "Case (b).", "By assumption, $\|A\| + \|B\|\\le 2T$ and by Lemma REF , there hold $\\begin{aligned}\\Vert \\bar{x}_A\\Vert &\\le \\frac{\\delta }{1-\\delta }\\Vert x^_B\\Vert + \\frac{1}{\\sqrt{1-\\delta }}\\epsilon \\\\&\\le \\frac{\\delta }{1-\\delta }\\Vert x^_B\\Vert + \\frac{1}{1-\\delta }\\epsilon ,\\\\\|d_{i_A}\|& \\ge \|x^_{i_A}\| - \\delta \\Vert x_{{B}}^\\Vert - \\epsilon - \\delta \\Vert \\bar{x}_{{A}}\\Vert \\\\& \\ge \|x^_{i_A}\| - \\frac{\\delta }{1-\\delta }\\Vert x_B^\\Vert - \\frac{1}{1-\\delta }\\epsilon .\\end{aligned}$ Consequently, with the assumption on $\\epsilon $ and $\\delta <\\frac{1-2\\beta }{2\\sqrt{T}+1}$ , we get $\\begin{aligned}\|d_{i_A}\| - \\Vert \\bar{x}_A\\Vert &\\ge \|x^_{i_A}\| - \\frac{2\\delta }{1-\\delta }\\Vert x_B^\\Vert - \\frac{2}{1-\\delta }\\epsilon \\\\&\\ge \|x^_{i_A}\|\\left(1 - \\frac{2\\sqrt{T}\\delta + 2\\beta }{1-\\delta }\\right) > 0,\\end{aligned}$ which is also a contradiction to (REF ).", "This completes the proof of the lemma.", "From Lemma REF , it follows if the support size of the active set of the coordinatewise minimizer can be controlled, then we may obtain information of the true active set $A^$ .", "However, a local minimizer generally does not yield such information; see following result.", "The proof can be found also in [27], but we include it here for completeness.", "Proposition 2.1 for any given index set $A\\subseteq S$ , the solution $x$ to the least-squares problem $\\mathrm {min}_{\\mathrm {supp}(x)\\subseteq A} \\Vert \\Psi x - y\\Vert $ is a local minimizer.", "Let $\\tau = \\min \\lbrace \|x_i\|: x_i\\ne 0\\rbrace $ .", "Then for any small perturbation $h$ in the sense $\\Vert h\\Vert _{\\ell ^\\infty }<\\tau $ , we have $x_i \\ne 0\\rightarrow x_i + h_i \\ne 0$ .", "Now we show that $x$ is a local minimizer.", "To see this, we consider two cases.", "First consider the case $\\mathrm {supp}(h)\\subseteq A$ .", "By the definition of $x$ , and $\\Vert x\\Vert _0 \\le \\Vert x+h\\Vert _0$ , we deduce $\\begin{aligned}J_\\lambda (x+h) &= \\tfrac{1}{2}\\Vert \\Psi (x+h) - y\\Vert ^2 + \\lambda \\Vert x+h\\Vert _0\\\\& \\ge \\tfrac{1}{2}\\Vert \\Psi x - y\\Vert ^2 + \\lambda \\Vert x\\Vert _0 = J_\\lambda (x).\\end{aligned}$ Alternatively, if $\\mathrm {supp}(h)\\nsubseteq A$ , then $\\Vert x+h\\Vert _0 \\ge \\Vert x\\Vert _0 + 1$ .", "Since $\\lim \\limits _{\\Vert h\\Vert \\rightarrow 0}\\Vert \\Psi (x+h) - y\\Vert = \\Vert \\Psi x - y\\Vert ,$ we again have $J_\\lambda (x+h) > J_\\lambda (x)$ for sufficiently small $h$ .", "This completes the proof of the proposition.", "Now we can study global minimizers to problem (REF ).", "For any $\\lambda > 0$ , there exists a global minimizer $x_\\lambda $ to problem (REF ) [22].", "Further, the following monotonicity relation holds [19][18].", "Lemma 2.5 For $\\lambda _1>\\lambda _2>0$ , there holds $\\Vert x_{\\lambda _1}\\Vert _0 \\le \\Vert x_{\\lambda _2}\\Vert _0.$ If the noise level $\\epsilon $ is sufficiently small, and the parameter $\\lambda $ is properly chosen, the oracle solution $x^o$ is the only global minimizer to $J_\\lambda (x)$ , cf.", "Theorem REF , which in particular implies the equivalence between the two formulations (REF ) and (REF ); see Remark REF below.", "Theorem 2.1 Let Assumption REF hold.", "(a) Suppose $\\nu < (1-2\\beta )/(3T - 1)$ and $\\beta \\le (1- 2(T-1)\\nu )/({T+3})$ , and let $\\xi = \\frac{1 - 2(T-1)\\nu - 2\\beta - \\beta ^2}{2T}\\min _{i\\in A^}\|x_i^\|^2.$ Then for any $\\lambda \\in (\\epsilon ^2/2, \\xi )$ , $x^o$ is the only global minimizer to $J_\\lambda (x)$ .", "(b) Suppose $\\delta \\triangleq \\delta _{2T} \\le (1-2\\beta )/({2\\sqrt{T}+1})$ and $\\beta \\le (1-2\\delta - \\delta ^2)/4$ , and let $\\xi = \\left[\\frac{1}{2}(1-\\delta ) - \\frac{\\delta ^2}{1-\\delta } - \\frac{\\beta }{\\sqrt{1-\\delta }} -\\frac{1}{2}\\beta ^2\\right]\\min _{i\\in A^}\|x_i^\|^2.$ Then for any $\\lambda \\in (\\epsilon ^2/2, \\xi )$ , $x^o$ is the only global minimizer to $J_\\lambda (x)$ .", "Let $x$ be a global minimizer to problem (REF ), and its support be $A$ .", "It suffices to show $A=A^$ .", "If $\|A\| \\ge T+1$ , then by the choice of $\\lambda $ , we deduce $J_\\lambda (x) \\ge \\lambda (T+1) > \\lambda T + \\tfrac{1}{2}\\epsilon ^2 \\ge J_\\lambda (x^o),$ which contradicts the minimizing property of $x$ .", "Hence, $\|A\| \\le T$ .", "Since a global minimizer is always a coordinatewise minimizer, by Lemma REF , we deduce $A\\subseteq A^$ .", "If $A\\ne A^$ , then $B = A^\\backslash A$ is nonempty.", "By the global minimizing property of $x$ , there holds $x = \\Psi _A^\\dag y$ .", "Using the notation $\\bar{x}_A$ from Lemma REF , we have $J_\\lambda (x) = \\tfrac{1}{2}\\Vert \\Psi _B x^_B + \\eta - \\Psi _A \\bar{x}_A\\Vert ^2 + \\lambda \|A\|.$ Now we consider the cases of the MIP and RIP separately.", "Case (a): Let $i_{A} \\in \\lbrace i\\in A^c: {\|x_i^\| }= \\Vert x^_B\\Vert _{\\ell ^\\infty }\\rbrace $ , then $i_A\\in B$ and $\|x_{i_A}^\| = \\Vert x^_B\\Vert _{\\ell ^\\infty }$ .", "Hence, by Lemma REF , there holds $\\begin{aligned}\\tfrac{1}{2}\\Vert \\psi _{i_A}x^_{i_A} &+ \\Psi _{B\\backslash \\lbrace i_A\\rbrace } x^_{B\\backslash \\lbrace i_A\\rbrace } + \\eta - \\Psi _A \\bar{x}_A\\Vert ^2 \\\\\\ge & \\tfrac{1}{2}\|x_{i_A}^\|^2 - \|x_{i_A}^\|\\left(\|\\langle \\psi _{i_A},\\Psi _{B\\backslash \\lbrace i_A\\rbrace } x^_{B\\backslash \\lbrace i_A\\rbrace }\\rangle \| + \|\\langle \\psi _{i_A}, \\eta \\rangle \| + \|\\langle \\psi _{i_A}, \\Psi _A \\bar{x}_A\\rangle \|\\right)\\\\\\ge & \\tfrac{1}{2}\|x_{i_A}^\|^2 - \|x_{i_A}^\|\\left((\|B\|-1)\\nu \|x_{i_A}^\| + \\epsilon + \\frac{\|A\|\\nu }{1 - (\|A\| -1)\\nu }(\|B\|\\nu \|x_{i_A}^\| + \\epsilon ) \\right).\\end{aligned}$ Now with $\\epsilon <\\beta \\min _{i\\in A^} \|x_i^\|\\le \\beta \|x_{i_A}^\|$ from Assumption (REF ), we deduce $\\begin{aligned}J_\\lambda (x) \\ge & \|x_{i_A}^\|^2\\left(\\frac{1}{2} - \\left((\|B\|-1)\\nu + \\beta + \\frac{\|A\|\\nu }{1 - (\|A\| -1)\\nu }(\|B\|\\nu + \\beta ) \\right)\\right) + \\lambda \|A\| \\\\= & \|x_{i_A}^\|^2\\left(\\frac{1}{2} - (T-1)\\nu -\\beta \\right) + \|x_{i_A}^\|^2 \|A\|\\nu \\left(1 - \\frac{\|B\|\\nu + \\beta }{1- (\|A\|-1)\\nu }\\right) +\\lambda \|A\| \\\\\\ge & \|x_{i_A}^\|^2\\left(\\frac{1}{2} - (T-1)\\nu -\\beta \\right),\\end{aligned}$ where the last inequality follows from $(\|A\| + \|B\|-1)\\nu + \\beta < 1 $ .", "By Assumption REF , there holds $\\epsilon ^2/2 \\le \\beta ^2/2\\min _{i\\in A^}\|x_i^\|^2$ .", "Now by the assumption $\\beta \\le \\left(1- 2(T-1)\\nu \\right)/(T+3)$ , we deduce $(T+1)\\beta ^2 + 2\\beta <(T+3)\\beta \\le 1-2(T-1)\\nu $ , and hence $T\\beta ^2 <1-2(T-1)\\nu -2\\beta -\\beta ^2$ .", "Together with the definition of $\\xi $ , this implies $\\xi >\\epsilon ^2/2$ .", "Further, by the choice of the parameter $\\lambda $ , i.e., $\\lambda \\in (\\epsilon ^2/2,\\xi )$ , there holds $J_\\lambda (x) - J_\\lambda (x^o) \\ge \\left[\\frac{1}{2} - (T-1)\\nu - \\beta -\\frac{1}{2}\\beta ^2\\right]\\min _{i\\in A^}\|x_i^\|^2 - \\lambda T >0,$ which contradicts the optimality of $x$ .", "Case (b): It follows from (REF ) that $\\begin{aligned}J_\\lambda (x)& \\ge \\tfrac{1}{2}\\Vert \\Psi _B x_B^\\Vert ^2 - \|\\langle \\eta , \\Psi _B x_B^\\rangle \| - \|\\langle x_B^,\\Psi _B^t\\Psi _A \\bar{x}_A\\rangle \| + \\lambda \|A\|\\\\&\\ge \\Vert \\Psi _B x_B^\\Vert (\\tfrac{1}{2}\\Vert \\Psi _B x_B^\\Vert - \\epsilon ) - \\Vert x_B^\\Vert \\delta \\Vert \\bar{x}_A\\Vert + \\lambda \|A\|.\\end{aligned}$ By Assumption REF and the assumptions on $\\beta $ and $\\delta $ , we deduce $\\sqrt{1-\\delta }\\Vert x_B^\\Vert \\ge \\epsilon $ .", "Now in view of the monotonicity of the function $t(t/2 - \\epsilon )$ for $t\\ge \\epsilon $ , and the inequality $\\Vert \\Psi _B x_B^\\Vert \\ge \\sqrt{1-\\delta }\\Vert x_B^\\Vert $ from the definition of the RIP constant $\\delta $ , we have $\\Vert \\Psi _B x_B^\\Vert (\\tfrac{1}{2}\\Vert \\Psi _B x_B^\\Vert - \\epsilon ) \\ge \\sqrt{1-\\delta }\\Vert x_B^\\Vert (\\tfrac{1}{2}\\sqrt{1-\\delta }\\Vert x_B^\\Vert - \\epsilon ).$ Thus by Lemma REF , we deduce $\\begin{aligned}J_\\lambda (x)& \\ge \\frac{1-\\delta }{2}\\Vert x_B^\\Vert ^2 - \\epsilon \\sqrt{1-\\delta }\\Vert x_B^\\Vert - \\Vert x_B^\\Vert \\left(\\frac{\\delta ^2}{1-\\delta }\\Vert x_B^\\Vert + \\frac{\\delta }{\\sqrt{1-\\delta }}\\epsilon \\right) + \\lambda \|A\|\\\\& = \\frac{1-\\delta }{2}\\Vert x_B^\\Vert ^2 - \\frac{1}{\\sqrt{1-\\delta }}\\epsilon \\Vert x_B^\\Vert - \\Vert x_B^\\Vert ^2\\frac{\\delta ^2}{1-\\delta } + \\lambda \|A\|\\\\&\\ge \\Vert x_B^\\Vert ^2 \\left[\\frac{1-\\delta }{2} - \\frac{\\delta ^2}{1-\\delta } - \\frac{\\beta }{\\sqrt{1-\\delta }}\\right] + \\lambda \|A\|,\\end{aligned}$ where the last line follows from $\\epsilon < \\beta \\Vert x_B^\\Vert $ , in view of Assumption REF .", "Appealing again to Assumption REF , $\\epsilon ^2/2\\le \\beta ^2\\min _{i\\in A^}\|x_i^\|^2/2\\le \\beta ^2\\Vert x^_B\\Vert ^2/2$ .", "Next it follows from the assumption $ \\beta \\le (1-\\delta -\\delta ^2)/4$ that the inequality $\\begin{aligned}\\beta ^2 + \\frac{\\beta }{\\sqrt{1-\\delta }} & \\le \\frac{\\beta ^2 +\\beta }{\\sqrt{1-\\delta }} \\le \\frac{2\\beta }{1-\\delta }\\\\& \\le \\frac{1-2\\delta -\\delta ^2}{2(1-\\delta )} = \\frac{1-\\delta }{2} - \\frac{\\delta ^2}{1-\\delta }\\end{aligned}$ holds.", "This together with the definition of $\\xi $ yields $\\xi >\\epsilon ^2/2$ .", "Further, the choice of $\\lambda \\in (\\epsilon ^2/2,\\xi )$ implies $J_\\lambda (x) - J_\\lambda (x^o) \\ge \\Vert x_B^\\Vert ^2 \\left[\\frac{1-\\delta }{2} - \\frac{\\delta ^2}{1-\\delta } - \\frac{\\beta }{\\sqrt{1-\\delta }} - \\frac{1}{2}\\beta ^2\\right] - \\lambda \|B\| > 0,$ which again leads to a contradiction.", "This completes the proof of the theorem.", "Proposition 2.2 Let the conditions in Theorem REF hold.", "Then the oracle solution $x^o$ is a minimizer of (REF ).", "Moreover, the support to any solution of problem (REF ) is $A^$ .", "First we observe that there exists a solution $\\bar{x}$ to problem (REF ) with $\|\\mbox{supp}(\\bar{x})\|\\le T$ by noticing that the true solution $x^$ satisfies $\\Vert \\Psi x^ - y\\Vert \\le \\epsilon $ and $\\Vert x^\\Vert _0\\le T$ .", "Clearly, for any minimizer $\\bar{x}$ to problem (REF ) with support $\|A\|\\le T$ , then $\\Psi ^\\dag _A y$ is also a minimizer with $\\Vert \\Psi \\Psi _A^\\dag y - y\\Vert \\le \\Vert \\Psi \\bar{x} - y\\Vert $ .", "Now if there is a minimizer $\\bar{x}$ with $A \\ne A^$ , by repeating the arguments in the proof of Theorem REF , we deduce $\\tfrac{1}{2}\\Vert \\Psi \\Psi _A^\\dag y - y \\Vert ^2 + \\lambda \\Vert \\Psi _A^\\dag y\\Vert _0 = J_\\lambda (\\Psi _A^\\dag y) > J_\\lambda (x^o)=\\tfrac{1}{2}\\epsilon ^2 + \\lambda T \\Rightarrow \\Vert \\Psi \\bar{x} - y\\Vert > \\epsilon ,$ which leads a contradiction to the assumption that $\\bar{x}$ is a minimizer to problem (REF ).", "Hence, any minimizer of (REF ) has a support $A^$ , and thus the oracle solution $x^o$ is a minimizer.", "Remark 2.1 Due to the nonconvex structure of problem (REF ), the equivalence between problem (REF ) and its “Lagrange” version (REF ) is generally not clear.", "However under certain assumptions, their equivalence can be obtained, cf.", "Theorem REF and Proposition REF .", "Further, we note that very recently, the equivalence between (REF ) and the following constrained sparsity problem $\\min \\Vert \\Psi x - y\\Vert \\quad \\mbox{subject to }\\quad \\Vert x\\Vert _0 \\le T$ was discussed in [28]." ], [ "Primal-dual active set method with continuation", "In this section, we present the primal-dual active set with continuation (PDASC) algorithm, and establish its finite step convergence property." ], [ "The PDASC algorithm", "The PDASC algorithm combines the strengthes of the PDAS algorithm [22] and the continuation technique.", "The complete procedure is described in Algorithm REF .", "The PDAS algorithm (the inner loop) first determines the active set from the primal and dual variables, then update the primal variable by solving a least-squares problem on the active set, and finally update the dual variable explicitly.", "It is well known that for convex optimization problems the PDAS algorithm can be interpreted as the semismooth Newton method [20].", "Thus the algorithm merits a local superlinear convergence, and it reaches convergence with a good initial guess.", "In contrast, the continuation technique on the regularization parameter $\\lambda $ allows one to control the size of the active set $A$ , and thus the active set of the coordinatewise minimizer lies within the true active set $A^$ .", "For example, for the choice of the parameter $\\lambda _0\\ge \\Vert \\Psi ^t y\\Vert ^2_{\\ell ^\\infty }/2$ , $x(\\lambda _0)=0$ is the unique global minimizer to the function $J_{\\lambda _0}$ , and the active set $A$ is empty.", "[hbt!]", "Primal dual active set with continuation (PDASC) algorithm [1] Set $\\lambda _0 \\ge \\frac{1}{2} \\Vert \\Psi ^t y\\Vert ^2_{\\ell ^\\infty }$ , ${A}(\\lambda _0) = \\emptyset $ , $x(\\lambda _0) =0$ and $d(\\lambda _0)= \\Psi ^t y$ , $\\rho \\in (0,1)$ , $J_{max}\\in \\mathbb {N}$ .", "$k=1,2,...$ Let $\\lambda _k = \\rho \\lambda _{k-1}$ , ${A}_0 = {A}(\\lambda _{k-1})$ , $(x^0,d^0) = (x(\\lambda _{k-1}), d(\\lambda _{k-1}))$ .", "$j=1,2,..., J_{max}$ Compute the active and inactive sets ${A}_j$ and ${I}_j$ : ${A}_j = \\left\\lbrace i: \|x^{j-1}_i+ d^{j-1}_i\| > \\sqrt{2\\lambda _k}\\right\\rbrace \\quad \\mbox{and}\\quad {I}_j = {A}_j^c.$ Check stopping criterion ${A}_j = {A}_{j-1}$ .", "Update the primal and dual variables $x^j$ and $d^j$ respectively by $\\left\\lbrace \\begin{array}{l}x_{{I}_j}^{j} = 0, \\\\[1.2ex]\\Psi _{{{A}}_j}^t \\Psi _{{{A}}_j} x_{{{A}}_j}^{j} = \\Psi _{{{A}}_j}^t y, \\\\[1.2ex]d^{j} = \\Psi ^t(\\Psi x^j - y).\\end{array}\\right.$ blueSet $\\widetilde{j}=\\min (J_{max},j)$ , and ${A}(\\lambda _k) = \\left\\lbrace i: \|x^{\\widetilde{j}}_i+ d^{\\widetilde{j}}_i\| > \\sqrt{2\\lambda _k}\\right\\rbrace $ and $(x(\\lambda _{k}),d(\\lambda _{k})) = (x^{\\widetilde{j}},d^{\\widetilde{j}})$ .", "Check stopping criterion: $\\Vert \\Psi x(\\lambda _k) - y\\Vert \\le \\epsilon $ .", "In the algorithm, there are a number of free parameters: the starting value $\\lambda _0$ for the parameter $\\lambda $ , the decreasing factor $\\rho \\in (0,1)$ (for $\\lambda $ ), and the maximum number $J_{max}$ of iterations for the inner PDAS loop.", "Further, one needs to set the stopping criteria at lines 6 and 10.", "Below we discuss their choices.", "The choice of initial value $\\lambda _0$ is not important.", "For any choice $\\lambda _0\\ge \\Vert \\Psi ^t y\\Vert ^2_{\\ell ^\\infty }/2$ , $x=0$ is the unique global minimizer, and $A=\\emptyset $ .", "Both the decreasing factor $\\rho $ and the iteration number $J_{max}$ affect the accuracy and efficiency of the algorithm: Larger $\\rho $ and $J_{max}$ values make the algorithm have better exact support recovery probability but take more computing time.", "Numerically, $\\rho $ is determined by the number of grid points for the parameter $\\lambda $ .", "Specifically, given an initial value $\\lambda _0 \\ge \\Vert \\Psi ^t y\\Vert ^2_{\\ell ^\\infty }/2$ and a small constant $\\lambda _{min}$ , e.g., $\\mbox{1e-15}\\lambda _0$ , the interval $[\\lambda _{min},\\lambda _0]$ is divided into $N$ equally distributed subintervals in the logarithmic scale.", "A large $N$ implies a large decreasing factor $\\rho $ .", "The choice $J_{max}=1$ generally works well, which is also covered in the convergence theory in Theorems REF and REF below.", "The stopping criterion for each $\\lambda $ -problem in Algorithm REF is either $A_j = A_{j-1}$ or $j = J_{max}$ , instead of the standard criterion $A_j = A_{j-1}$ for active set type algorithms.", "The condition $j = J_{max}$ is very important for nonconvex problems.", "This is motivated by the following empirical observation: When the true signal $x^$ does not have a strong decay property, e.g., 0-1 signal, the inner PDAS loop (for each $\\lambda $ -problem) may never reach the condition ${A}_j = {A}_{j-1}$ within finite steps; see the example below.", "Example 3.1 In this example, we illustrate the convergence of the PDAS algorithm.", "Let $-1<\\mu <0$ , ${A}^ = \\lbrace 1,2\\rbrace $ , and $\\Psi _1 = \\frac{1}{\\sqrt{1+\\mu ^2}}(1,\\mu ,0,...,0)^t,\\ \\ \\Psi _2 = \\frac{1}{\\sqrt{1+\\mu ^2}}(\\mu , 1, 0,...,0)^t, \\ \\ x^_1 = x^_2 = 1.$ In the absence of data noise $\\eta $ , the data $y$ is given by $y = \\frac{1}{\\sqrt{1+\\mu ^2}}(1+\\mu , 1+\\mu ,0,...,0)^t.$ Now we let $\\sqrt{2\\lambda }\\in (\\frac{(1+\\mu )^2}{1+\\mu ^2},\\frac{(1-\\mu ^2)^2}{(1+\\mu ^2)^2})$ , the initial guess ${A}_1 = \\lbrace 1\\rbrace $ .", "Then direct computation yields $\\begin{aligned}x^1 &= \\frac{1}{1+\\mu ^2}((1+\\mu )^2, 0)^t,\\\\y - \\Psi x^1 & = \\frac{1 - \\mu ^2}{(\\sqrt{1+ \\mu ^2})^3}(-\\mu , 1,0,...,0)^t,\\\\d^1 &= \\frac{1}{(1+\\mu ^2)^2}(0, (1-\\mu ^2)^2)^t.\\end{aligned}$ Hence $d^1_2 > \\sqrt{2\\lambda }> x^1_1$ , and ${A}_2 = \\lbrace 2\\rbrace $ .", "Similarly, we have ${A}_3 = \\lbrace 1\\rbrace = {A}_1$ , which implies that the algorithm simply alternates between the two sets $\\lbrace 1\\rbrace $ and $\\lbrace 2\\rbrace $ and will never reach the stopping condition ${A}_k = {A}_{k+1}$ .", "The stopping condition at line 10 of Algorithm REF is a discrete analogue of the discrepancy principle.", "This rule is well established in the inverse problem community for selecting an appropriate regularization parameter [18].", "The rationale behind the rule is that one cannot expect the reconstruction to be more accurate than the data accuracy in terms of the discrepancy.", "In the PDASC algorithm, if the active set is always contained in the true active set $A^$ throughout the iteration, then the discrepancy principle can always be satisfied for some $\\lambda _k$ , and the solution $x(\\lambda _k)$ resembles closely the oracle solution $x^o$ ." ], [ "Convergence analysis", "Now we discuss the convergence of Algorithm REF .", "We shall discuss the cases of the MIP and RIP conditions separately.", "The general proof strategy is as follows.", "It essentially relies on the control of the active set during the iteration and the certain monotonicity relation of the active set $A(\\lambda _k)$ (via the continuation technique).", "In particular, we introduce two auxiliary sets $G_{\\lambda ,s_1}$ and $G_{\\lambda ,s_2}$ , cf.", "(REF ) below, to precisely characterize the evolution of the active set $A$ during the PDASC iteration.", "First we consider the MIP case.", "We begin with an elementary observation: under the assumption $\\nu <(1-2\\beta )/(2T-1)$ of the mutual coherence parameter $\\nu $ , there holds $(2T-1)\\nu + 2\\beta <1$ .", "Lemma 3.1 If $\\nu < (1-2\\beta )/(2T - 1)$ , then for any $\\rho \\in ( ((2T-1)\\nu +2\\beta )^2,1)$ there exist $s_1,s_2\\in (1/(1-T\\nu + \\nu -\\beta ),1/(T\\nu + \\beta ))$ , $s_1>s_2$ , such that $s_2 = 1+ (T\\nu - \\nu +\\beta )s_1$ and $\\rho =s_2^2/s_1^2$ .", "By the assumption $v<(1-2\\beta )/(2T-1)$ , $T\\nu + \\beta < 1 - T\\nu + \\nu -\\beta $ .", "Hence for any $s_1\\in (1/(1-T\\nu + \\nu -\\beta ),1/(T\\nu + \\beta ))$ , there holds $s_1>1+(T\\nu -\\nu +\\beta )s_1 \\quad \\mbox{and}\\quad 1+(T\\nu -\\nu +\\beta )s_1>\\frac{1}{1-T\\nu +\\nu -\\beta },$ i.e., $\\frac{1}{T\\nu + \\beta } >s_1 > 1+ (T\\nu - \\nu +\\beta )s_1 > \\frac{1}{1-T\\nu + \\nu -\\beta }.$ Upon letting $s_2 = 1+ (T\\nu - \\nu +\\beta )s_1$ , we deduce $\\frac{1}{T\\nu + \\beta } >s_1 > s_2 > \\frac{1}{1-T\\nu + \\nu -\\beta }.$ Now the monotonicity of the function $f(s_1)=s_2/s_1$ over the interval $(1/(1-T\\nu +\\nu -\\beta ),1/(T\\nu + \\beta ))$ , and the identities $\\begin{aligned}&\\frac{1 + (T\\nu - \\nu +\\beta )/(T\\nu + \\beta )}{1/(T\\nu + \\beta )} = (2T-1)\\nu + 2\\beta ,\\\\&\\frac{1 + (T\\nu - \\nu +\\beta )/(1-T\\nu + \\nu -\\beta )}{1/(1-T\\nu + \\nu -\\beta )} = 1,\\\\\\end{aligned}$ imply that there exists an $s_1$ in the internal such that $s_2/s_1 = \\sqrt{\\rho }$ for any $\\rho \\in ( ((2T-1)\\nu +2\\beta )^2,1)$ .", "Next for any $\\lambda >0$ and $s>0$ , we denote by $G_{\\lambda ,s} \\triangleq \\left\\lbrace i: \|x_i^\| \\ge \\sqrt{2\\lambda }s\\right\\rbrace .$ The set $G_{\\lambda ,s}$ characterizes the true sparse signal $x^$ (via level sets).", "The lemma below provides an important monotonicity relation on the active set $A_k$ during the iteration, which is essential for showing the finite step convergence of the algorithm in Theorem REF below.", "Lemma 3.2 Let Assumption REF hold, $\\nu < (1-2\\beta )/({2T - 1})$ , $\\rho \\in ( ((2T-1)\\nu +2\\beta )^2,1)$ , and $s_1$ and $s_2$ be defined in Lemma REF .", "If $G_{\\lambda ,s_1}\\subseteq A_k\\subseteq A^$ , then $G_{\\lambda ,s_2}\\subseteq A_{k+1}\\subseteq A^$ .", "Let $A = A_k$ , $B = A^\\backslash A$ .", "By Lemma REF , we have $\\begin{aligned}\|x_i\| &\\ge \|x_i^\| - \\Vert \\bar{x}_A\\Vert _{\\ell ^\\infty } \\ge \|x_i^\| - \\frac{\|B\|\\nu \\Vert x_B^\\Vert _{\\ell ^\\infty } + \\epsilon }{1 - (\|A\|-1)\\nu }, \\quad \\forall i\\in A,\\\\\|d_j\| &\\le \|B\|\\nu \\left(1 + \\frac{\|A\|\\nu }{1- (\|A\| -1)\\nu }\\right) \\Vert x_B^\\Vert _{\\ell ^\\infty } + \\epsilon \\left(1 + \\frac{\|A\|\\nu }{1- (\|A\| -1)\\nu }\\right), \\quad \\forall \\,\\, j\\in I^,\\\\\|d_i\|&\\ge \|x_i^\|+ \\nu \\Vert x_B^\\Vert _{\\ell ^\\infty } - \|B\|\\nu \\left(1 + \\frac{\|A\|\\nu }{1- (\|A\| -1)\\nu }\\right) \\Vert x_B^\\Vert _{\\ell ^\\infty } - \\epsilon \\left(1 + \\frac{\|A\|\\nu }{1- (\|A\| -1)\\nu }\\right), \\quad \\forall \\,\\, i\\in B.\\end{aligned}$ Using the fact $\\epsilon \\le \\beta \\min _{i\\in A^}\|x_i^\|\\le \\beta \\Vert x_B^\\Vert _{\\ell ^\\infty }$ from Assumption REF and the trivial inequality $\\frac{\|B\|\\nu + \\beta }{1- T\\nu +\\nu + \|B\|\\nu } \\le \\frac{T\\nu + \\beta }{1+\\nu }$ , we arrive at $\\begin{aligned}\|B\|\\nu \\left(1 + \\frac{\|A\|\\nu }{1- (\|A\| -1)\\nu }\\right) &\\Vert x_B^\\Vert _{\\ell ^\\infty } + \\epsilon \\left(1 + \\frac{\|A\|\\nu }{1- (\|A\| -1)\\nu }\\right) \\\\\\le & \\frac{\|B\|\\nu + \\beta }{1- T\\nu +\\nu + \|B\|\\nu }(1+\\nu )\\Vert x_B^\\Vert _{\\ell ^\\infty }\\le (T\\nu +\\beta )\\Vert x_B^\\Vert _{\\ell ^\\infty }.\\end{aligned}$ Consequently, $\\begin{aligned}\|d_j\|&\\le (T\\nu + \\beta )\\Vert x_B^\\Vert _{\\ell ^\\infty }, \\quad \\forall j\\in I^, \\\\\|d_i\|&\\ge \|x_i^\| - (T\\nu - \\nu +\\beta )\\Vert x_B^\\Vert _{\\ell ^\\infty }, \\quad \\forall i\\in B.\\end{aligned}$ It follows from the assumption $G_{\\lambda ,s_1}\\subseteq A = A_k$ that $\\Vert x^_B\\Vert _{\\ell ^\\infty } < s_1\\sqrt{2\\lambda }$ .", "Then for all $j\\in I^$ , we have $\|d_j\| < s_1(T\\nu + \\beta )\\sqrt{2\\lambda } < \\sqrt{2\\lambda },$ i.e., $j\\in I_{k+1}$ .", "This shows $A_{k+1}\\subseteq A^$ .", "For any $i\\in I \\cap G_{\\lambda ,s_2}$ , we have $\|d_i\| > s_2\\sqrt{2\\lambda } - (T\\nu - \\nu +\\beta )s_1 \\sqrt{2\\lambda } \\ge \\sqrt{2\\lambda },$ This implies $i\\in A_{k+1}$ by (REF ).", "It remains to show that for any $i\\in A\\cap G_{\\lambda ,s_2}$ , $i\\in A_{k+1}$ .", "Clearly, if $A = \\emptyset $ , the assertion holds.", "Otherwise $\\begin{aligned}\|x_i\| & \\ge \|x_i^\| - \\frac{\|B\|\\nu +\\beta }{1 - (\|A\|-1)\\nu }\\Vert x_B^\\Vert _{\\ell ^\\infty } \\\\& > s_2\\sqrt{2\\lambda } - (T\\nu - \\nu +\\beta )s_1 \\sqrt{2\\lambda } \\ge \\sqrt{2\\lambda },\\end{aligned}$ where the last line follows from the elementary inequality $\\frac{\|B\|\\nu + \\beta }{1 - (\|A\|-1)\\nu } \\le T\\nu - \\nu +\\beta .$ This together with (REF ) also implies $i\\in A_{k+1}$ .", "This concludes the proof of the lemma.", "Now we can state the convergence result.", "Theorem 3.1 Let Assumption REF hold, and $\\nu < (1-2\\beta )/({2T - 1})$ .", "Then for any $\\rho \\in (((2T-1)\\nu + 2\\beta )^2,1)$ , Algorithm REF converges in finite steps.", "For each $\\lambda _k$ -problem, we denote by $A_{k,0}$ and $A_{k,\\diamond }$ the active set for the initial guess and the last inner step (i.e., $A(\\lambda _k)$ in Algorithm REF ), respectively.", "Now with $s_1$ and $s_2$ from Lemma REF , there holds $G_{\\lambda ,s_1}\\subset G_{\\lambda ,s_2}$ , and using Lemma REF , for any index $k$ before the stopping criterion at line 10 of Algorithm REF is reached, there hold $G_{\\lambda _k,s_1} \\subseteq A_{k,0}\\quad \\mbox{and}\\quad G_{\\lambda _k,s_2}\\subseteq A_{k,\\diamond }.$ Note that for $k=0$ , $G_{\\lambda _0,s_2} = \\emptyset $ and thus the assertion holds.", "To see this, it suffices to check $\\Vert x^\\Vert _{\\ell ^\\infty } < s_2 \\Vert \\Psi ^t y\\Vert _{\\ell ^\\infty }$ .", "By Lemma REF and the inequality $s_2 > 1/({1-T\\nu + \\nu -\\beta })$ we obtain that $\\begin{aligned}\\Vert \\Psi ^t y\\Vert _{\\ell ^\\infty } &\\ge \\Vert \\Psi _{A^}^t\\Psi _{A^} x^_{A^}\\Vert _{\\ell ^\\infty } -\\Vert \\Psi ^t \\eta \\Vert _{\\ell ^\\infty } \\\\&\\ge (1 - (T-1)\\nu )\\Vert x^\\Vert _{\\ell ^\\infty } - \\epsilon >\\Vert x^\\Vert _{\\ell ^\\infty }/s_2.\\end{aligned}$ Now for $k>0$ , it follows by mathematical induction and the relation $A_{k,\\diamond } = A_{k+1,0}$ .", "It follows from (REF ) that during the iteration, the active set $A_{k,\\diamond }$ always lies in $A^$ .", "Further, for $k$ sufficiently large, by Lemma REF , the stopping criterion at line 10 must be reached and thus the algorithm terminates; otherwise $A^ \\subseteq G_{\\lambda _k,s_1},$ then the stopping criterion at line 10 is satisfied, which leads to a contradiction.", "Next we turn to the convergence of Algorithm REF under the RIP condition.", "Let $1 - (2\\sqrt{T} + 1)\\delta > 2\\beta $ , an argument analogous to Lemma REF implies that for any $\\sqrt{\\rho }\\in ((2\\delta \\sqrt{T} + 2\\beta )/({1-\\delta }),1)$ there exist $s_1$ and $s_2$ such that $\\frac{1-\\delta }{\\delta \\sqrt{T} + \\beta }> s_1 > s_2 >\\frac{1-\\delta }{1 - \\delta -\\delta \\sqrt{T} - \\beta }, \\quad s_2 = 1+\\frac{\\delta \\sqrt{T} + \\beta }{1-\\delta }s_1 , \\quad \\frac{s_2}{s_1} = \\sqrt{\\rho }.$ The next result is an analogue of Lemma REF .", "Lemma 3.3 Let Assumption REF hold, $\\delta \\triangleq \\delta _{T+1} \\le (1-2\\beta )/(2\\sqrt{T}+1)$ , and $\\sqrt{\\rho }\\in ((2\\delta \\sqrt{T} + 2\\beta )/({1-\\delta }),1)$ .", "Let $s_1$ and $s_2$ are defined by (REF ).", "If $G_{\\lambda ,s_1}\\subseteq A_k\\subseteq A^$ , then $G_{\\lambda ,s_2}\\subseteq A_{k+1}\\subseteq A^$ .", "Let $A = A_k$ , $B = A^\\backslash A$ .", "Using the notation in Lemma REF , we have $\\begin{aligned}\|x_i\| &\\ge \|x_i^\| - \\Vert \\bar{x}_A\\Vert \\ge \|x_i^\| - \\frac{\\delta \\Vert x_B^\\Vert + \\epsilon }{1 -\\delta } , \\quad \\forall i\\in A,\\\\\|d_j\| &\\le \\delta \\Vert x_B^\\Vert + \\epsilon + \\delta \\Vert \\bar{x}_A\\Vert \\le \\frac{\\delta \\Vert x_B^\\Vert + \\epsilon }{1-\\delta }, \\quad \\forall j\\in I^,\\\\\|d_i\|&\\ge \|x_i^\| - \\delta \\Vert x_B^\\Vert - \\epsilon - \\delta \\Vert \\bar{x}_A\\Vert \\ge \|x_i^\|-\\frac{\\delta \\Vert x_B^\\Vert + \\epsilon }{1-\\delta }, \\quad \\forall i\\in B.\\end{aligned}$ By the assumption $G_{\\lambda ,s_1}\\subseteq A_k$ , we have $\\Vert x_B^\\Vert _{\\ell ^\\infty } <s_1\\sqrt{2\\lambda }$ .", "Now using the relation $s_1 < (1-\\delta )/({\\delta \\sqrt{T} + \\beta })$ and Assumption REF , we deduce $\\frac{\\delta \\Vert x_B^\\Vert + \\epsilon }{1-\\delta } \\le \\frac{\\delta \\sqrt{T} + \\beta }{1 - \\delta }\\Vert x_B^\\Vert _{\\ell ^\\infty } < \\sqrt{2\\lambda }.$ Thus for $j\\in I^$ , $\|d_i\|<\\sqrt{2\\lambda }$ , i.e., $A_{k+1}\\subset A^$ .", "Similarly, using the relations $s_2 = 1+s_1(\\delta \\sqrt{T} + \\beta )/({1-\\delta })$ and $s_1>(1-\\delta )/(1 - \\delta -\\delta \\sqrt{T} - \\beta )$ , we arrive at that for any $i\\in G_{\\lambda ,s_2}$ , there holds $\|x_i^\| - \\frac{\\delta \\Vert x_B^\\Vert + \\epsilon }{1-\\delta } > s_2\\sqrt{2\\lambda } - \\frac{\\delta \\sqrt{T}+ \\beta }{1 - \\delta }s_1\\sqrt{2\\lambda } = \\sqrt{2\\lambda }.", "$ This implies that for $i\\in G_{\\lambda ,s_2}\\cap A$ , $\|x_i\|>\\sqrt{2\\lambda }$ , and for $i\\in G_{\\lambda ,s_2}\\cap I$ , $\|d_i\|>\\sqrt{2\\lambda }$ .", "Consequently, (REF ) yields the desired relation $(G_{\\lambda ,s_2}\\cap A)\\subseteq A_{k+1}$ , and this concludes the proof of the lemma.", "Now we can state the convergence of Algorithm REF under the RIP assumption.", "The proof is similar to that for Theorem REF , and hence omitted.", "Theorem 3.2 Let Assumption REF hold, and $\\delta \\triangleq \\delta _{T+1} \\le (1-2\\beta )/({2\\sqrt{T}+1})$ .", "Then for any $\\sqrt{\\rho } \\in \\left((2\\delta \\sqrt{T} + 2\\beta )/({1-\\delta }),1\\right)$ , Algorithm REF converges in finite steps.", "Remark 3.1 Theorems REF and REF indicate that Algorithm REF converges in finite steps, and the active set $A(\\lambda _k)$ remains a subset of the true active set $A^$ .", "Corollary 3.1 Let the assumptions in Theorem REF hold.", "Then Algorithm REF terminates at the oracle solution $x^o$ .", "First, we note the monotonicity relation $A(\\lambda _k)\\subset A^$ before the stopping criterion at line 10 of Algorithm REF is reached.", "For any $A\\subsetneq A^$ , let $x = \\Psi _A^\\dag y$ .", "Then by the argument in the proof of Theorem REF , we have $J_\\lambda (x) = \\tfrac{1}{2}\\Vert \\Psi x - y\\Vert ^2 + \\lambda \|A\| > \\tfrac{1}{2}\\epsilon ^2 + \\lambda T \\Rightarrow \\Vert \\Psi x - y\\Vert > \\epsilon ,$ which implies that the stopping criterion at line 10 in Algorithm REF cannot be satisfied until the oracle solution $x^o$ is reached." ], [ "Connections with other algorithms", "Now we discuss the connections of Algorithm REF with two existing greedy methods, i.e., orthogonal matching pursuit (OMP) and hard thresholding pursuit (HTP)." ], [ "Connection with the OMP.", "To prove the convergence of Algorithm REF , we require either the MIP condition ($\\nu < (1-2\\beta )/({2T - 1})$ ) or the RIP condition ($\\delta _{T+1} \\le (1-2\\beta )/({2\\sqrt{T}+1})$ ) on the sensing matrix $\\Psi $ .", "These assumptions have been used to analyze the OMP before: MIP appeared in [6] and RIP appeared in [17].", "Further, for the OMP, the MIP assumption is fairly sharp, but the RIP assumption can be improved [38], [24].", "Our convergence analysis under these assumptions, unsurprisingly, follows the same line of thought as that for the OMP, in that we require the active set $A(\\lambda _k)$ always lies in the true active set $A^$ during the iteration.", "However, we note that this requirement is unnecessary for the PDASC, since the active set can move inside and outside the true active set $A^$ during the iteration.", "The numerical examples in section below confirm this observation.", "This makes the PDASC much more flexible than the OMP." ], [ "Connection with the HTP.", "Actually, the HTP due to Foucart [16] can be viewed a primal-dual active set method in the $T$ -version, i.e., at each iteration, the active set is chosen by the first $T$ -component for both primal and dual variables.", "This is equivalent to a variable regularization parameter $\\lambda $ , where $\\sqrt{2\\lambda }$ is set to the $T$ -th components of $\|x^k\| + \|d^k\|$ at each iteration.", "Naturally, one can also apply a continuation strategy on the parameter $T$ ." ], [ "Numerical tests", "In this section we present numerical examples to illustrate the efficiency and accuracy of the proposed PDASC algorithm.", "The sensing matrix $\\Psi $ is of size $n\\times p$ , the true solution $x^$ is a $T$ -sparse signal with an active set $A^$ .", "The dynamical range $R$ of the true signal $x^$ is defined by $R = M/m$ , with $M = \\max \\lbrace \|x^{}_{i}:i\\in A^{}\|\\rbrace $ and $m= \\min \\lbrace \|x^{}_{i}\|:i\\in A^{}\\rbrace =1$ .", "The data $y$ is generated by $y = \\Psi x^* + \\eta ,$ where $\\eta $ denotes the measurement noise, with each entry $\\eta _i$ following the Gaussian distribution $N(0,\\sigma ^2)$ with mean zero and standard deviation $\\sigma $ .", "The exact noise level $\\epsilon $ is given by $\\epsilon =\\Vert \\eta \\Vert _2$ .", "In Algorithm REF , we always take $\\lambda _{0} = \\Vert \\Psi ^t y\\Vert _{\\ell ^\\infty }$ , and $\\lambda _{min} = \\mbox{1e-15}\\lambda _{0}$ .", "The choice of the number of grid points $N$ and the maximum number $J_{max}$ of inner iterations will be specified later." ], [ "The behavior of the PDASC algorithm", "First we study the influence of the parameters in the PDASC algorithm on the exact recovery probability.", "To this end, we fix $\\Psi $ to be a $500\\times 1000$ random Gaussian matrix, and $\\sigma = \\mbox{1e-2}$ .", "All the results are computed based on 100 independent realizations of the problem setup.", "To this end, we consider the following three settings: $J_{max} =5$ , and varying $N$ ; see Fig.", "REF (a).", "$N=100$ , and varying $J_{max}$ ; see Fig.", "REF (b).", "$N=100$ , $J_{max}=5$ , and an approximate noise level $\\bar{\\epsilon }$ ; see Fig.", "REF (c).", "We observe that the influence of the parameters $N$ and $J_{max}$ is very mild on the exact support recovery probability.", "In particular, a reasonably small value for these parameters (e.g.", "$N=50$ , $J_{max} =1$ ) is sufficient for accurately recovering the exact active set $A^$ .", "Unsurprisingly, a very small value of $N$ can degrade the accuracy of active set recovery greatly, due to insufficient resolution of the solution path.", "In practice, the exact noise level $\\epsilon $ is not always available, and often only a rough estimate $\\bar{\\epsilon }$ is provided.", "The use of the estimate $\\bar{\\epsilon }$ in place of the exact one $\\epsilon $ in Algorithm REF may sacrifice the recovery probability.", "Hence it is important to study the sensitivity of Algorithm REF with respect to the variation of the parameter $\\epsilon $ .", "We observe from Fig.", "REF (c) that the parameter $\\epsilon $ does not affect the recovery probability much, unless the estimate $\\bar{\\epsilon }$ is grossly erroneous.", "Figure: The influence of the algorithmic parameters (NN, J max J_{max} and ϵ\\epsilon )on the exact recovery probability.To gain further insight into the PDASC algorithm, in Fig.", "REF , we show the evolution of the active set (for simplicity let $A_k = A(\\lambda _k)$ ) .", "It is observed that the active set $A_k$ can generally move both “inside” and “outside” of the true active set $A^$ .", "This is in sharp contrast to the OMP, where the size of the active set is monotone during the iteration.", "The flexible change in the active set might be essential for the efficiency of the algorithm.", "This observation is valid for random Gaussian, random Bernoulli and partial DCT sensing matrices.", "Figure: Numerical results for random Gaussian (top row, R=100R=100, n=500n=500, p=1000p=1000, σ=1e-3\\sigma =\\mbox{1e-3}),random Bernoulli (middle row, R=1000R = 1000, n=2 10 n = 2^{10}, p=2 12 p = 2^{12}, σ=1e-3\\sigma =\\mbox{1e-3})and partial DCT (bottom row, R=1000R = 1000, n=2 11 n = 2^{11}, p=2 13 p = 2^{13}, σ=1e-3\\sigma =\\mbox{1e-3}) sensing matrix.The parameters NN and J max J_{max} are set to N=50N=50 and J max =1J_{max}=1, respectively.For each $\\lambda _k$ , with $x({\\lambda _{k-1}})$ ($x({\\lambda _0})=0$ ) as the initial guess, the PDASC generally reaches convergence within a few iterations, cf.", "Fig.", "REF , which is observed for random Gaussian, random Bernoulli and partial DCT sensing matrices.", "This is attributed to the local superlinear convergence of the PDAS algorithm.", "Hence, when coupled with the continuation strategy, the PDASC procedure is very efficient.", "Figure: Number of iterations of PDASC at each λ k \\lambda _k for random Gaussian (top left with R=1000R=1000, n=500n=500, p=1000p=1000, T=200T= 200, σ=1e-3\\sigma =\\mbox{1e-3}),random Bernoulli (top right with R=1000R = 1000, n=2 10 n = 2^{10}, p=2 12 p = 2^{12}, T=2 8 T = 2^{8}, σ=1e-3\\sigma =\\mbox{1e-3})and partial DCT (bottom with R=1000R = 1000, n=2 11 n = 2^{11}, p=2 13 p = 2^{13}, T=2 8 T = 2^{8}, σ=1e-3\\sigma =\\mbox{1e-3}) sensing matrix.The parameters NN and J max J_{max} are set to N=50N=50 and J max =5J_{max}=5, respectively." ], [ "Comparison with existing algorithms", "In this part, we compare Algorithm REF with six state-of-the-art algorithms in the compressive sensing literature, including orthogonal matching pursuit (OMP) [31], greedy gradient pursuit (GreedyGP) [5], accelerated iterative hard thresholding (AIHT) [3], hard thresholding pursuit (HTP) [16], compressive sampling matching pursuit (CoSaMP) [25] and homotopy algorithm [14], [29].", "First, we consider the exact support recovery probability, i.e., the percentage of the reconstructions whose support agrees with the true active set $A^*$ .", "To this end, we fix the sensing matrix $\\Psi $ as a $500\\times 1000$ random Gaussian matrix, $\\sigma =\\mbox{1e-3}$ , $(N,J_{max})=(100,5) $ or $(50,1)$ , and all results are computed from 100 independent realizations of the problem setup.", "Since the different dynamical range may give different results, we take $R=1$ , 10, 1e3, 1e5 as four exemplary values.", "The numerical results are summarized in Fig.", "REF .", "We observe that when the dynamical range $R$ is not very small, the proposed PDASC algorithm with $(N,J_{max})=(100,5)$ has a better exact support recovery probability, and that with the choice $(N,J_{max})$ is largely comparable with other algorithms.", "Figure: The exact support recovery probability for four different dynamical ranges: R=1,10,10 3 R=1,\\ 10,\\ 10^3, and 10 5 10^5.To further illustrate the accuracy and efficiency of the proposed PDASC algorithm, we compare it with other greedy methods in terms of CPU time and reconstruction error.", "To this end, we fix $\\sigma =\\mbox{1e-2}$ , $(N,J_{max})=(100,5)$ or $(50,1)$ .", "The numerical results for random Gaussian, random Bernoulli and partial DCT sensing matrices with different parameter tuples $(R,n,p,T)$ are shown in Tables REF -REF , respectively.", "The results in the tables are computed from 10 independent realizations of the problem setup.", "It is observed that the PDASC algorithm yields reconstructions that are comparable with that by other methods, but usually with less computing time.", "Further, we observe that it scales better with the problem size than other algorithms.", "Table: Numerical results (CPU time and errors) for medium-scale problems, with random Gaussiansensing matrix Ψ\\Psi , of size p=10000,15000,20000,25000,30000p = 10000,\\ 15000,\\ 20000,\\ 25000,\\ 30000, n=⌊p/4⌋n = \\lfloor p/4\\rfloor ,T=⌊n/3⌋T= \\lfloor n/3\\rfloor .", "The dynamical range RR is R=1000R=1000, and the noise variance σ\\sigma isσ=1e-2\\sigma =\\mbox{1e-2}.Table: Numerical results (CPU time and errors) for medium-scale problems, with random Bernoullisensing matrix Ψ\\Psi , of size p=10000,15000,20000,25000,30000p = 10000,\\ 15000,\\ 20000,\\ 25000,\\ 30000, n=⌊p/4⌋n = \\lfloor p/4\\rfloor ,T=⌊n/4⌋T= \\lfloor n/4\\rfloor .", "The dynamical range RR is R=10R=10, and the noise variance σ\\sigma isσ=1e-2\\sigma =\\mbox{1e-2}.Table: Numerical results (CPU time and errors) for large-scale problems, with partial DCTsensing matrix Ψ\\Psi , of size p=2 13 p = 2^{13}, 2 14 2^{14}, 2 15 2^{15}, 2 16 2^{16}, 2 17 2^{17}, n=⌊p/4⌋n = \\lfloor p/4\\rfloor ,T=⌊n/3⌋T= \\lfloor n/3\\rfloor .", "The dynamical range RR is R=100R=100, and the noise variance σ\\sigma isσ=1e-2\\sigma =\\mbox{1e-2}.Lastly, we consider one-dimensional signals and two-dimensional images.", "In this case the explicit form of the sensing matrix $\\Psi $ may be not available, hence the least-squares step (for updating the primal variable) at line 7 of Algorithm REF can only be solved by an iterative method.", "We employ the conjugate gradient (CG) method to solve the least-squares problem inexactly.", "The initial guess for the CG method for the $\\lambda _k$ -problem is the solution $x(\\lambda _{k-1})$ , and the stopping criterion for the CG method is as follows: either the number of CG iterations is greater than 2 or the residual is below a given tolerance $\\mbox{1e-5}\\epsilon $ .", "For the one-dimensional signal, the sampling matrix $\\Psi $ is of size $665\\times 1024$ , and it consists of applying a partial FFT and an inverse wavelet transform, and the signal under wavelet transformation has 247 nonzero entries and $\\sigma =\\mbox{1e-4}$ , $N=50$ , $J_{\\max }=1$ .", "The results are shown in Fig.", "REF and Table REF .", "The reconstructions by all the methods, except the AIHT and CoSaMP, are visually very appealing and in excellent agreement with the exact solution.", "The reconstructions by the AIHT and CoSaMP suffer from pronounced oscillations.", "This is further confirmed by the PSNR values which is defined as $\\textit {PSNR}=10\\cdot \\log \\frac{V^2}{MSE}$ where $V$ is the maximum absolute value of the reconstruction and the true solution, and $MSE$ is the mean squared error of the reconstruction, cf.", "Table REF .", "For the two-dimensional MRI image, the sampling matrix $\\Psi $ amounts to a partial FFT and an inverse wavelet transform of size $1657\\times 4096$ .", "The image under wavelet transformation has 792 nonzero entries and $\\sigma =\\mbox{1e-4}$ , $N=50$ , and $J_{\\max }=1$ .", "The numerical results are shown in Fig.", "REF and Table REF .", "The observation for the one-dimensional signal remains largely valid: except the CoSaMP, all other methods can yield almost identical reconstructions within similar computational efforts.", "Therefore, the proposed PDASC algorithm is competitive with state-of-the-art algorithms.", "Figure: Reconstruction results of 1 dimension signal.Figure: Reconstruction results of two-dimensional phantom images.Table: One-dimensional signal: n=665n=665, p=1024p=1024, T=247T=247, σ=1e-4\\sigma =\\mbox{1e-4}.Table: Two-dimensional image, n=1657n=1657, p=4096p=4096, T=792T=792, σ=1e-4\\sigma =\\mbox{1e-4}." ], [ "Conclusion", "We have developed an efficient and accurate primal-dual active set with continuation algorithm for the $\\ell ^0$ penalized least-squares problem arising in compressive sensing.", "It combines the fast local convergence of the active set technique and the globalizing property of the continuation technique.", "The global finite step convergence of the algorithm was established under the mutual incoherence property or restricted isometry property on the sensing matrix.", "Our extensive numerical results indicate that the proposed algorithm is competitive in comparison with state-of-the-art algorithms in terms of efficiency, accuracy and exact recovery probability, without a knowledge of the exact sparsity level." ], [ "Acknowledgement", "The research of B. Jin is supported by NSF Grant DMS-1319052, and that of X. Lu is partially supported by National Science Foundation of China No.", "11101316 and No.", "91230108." ] ]	1403.0515
[ [ "A note on bi-linear multipliers" ], [ "Abstract In this paper we prove that if $\\chi_{_E}(\\xi-\\eta)-$ the indicator function of measurable set $E\\subseteq \\mathbb{R}^d,$ is a bi-linear multiplier symbol for exponents $p,q,r$ satisfying the H\\\"{o}lder's condition $\\frac{1}{p}+\\frac{1}{q}=\\frac{1}{r}$ and exactly one of $p,q,$ or $r'=\\frac{r}{r-1}$ is less than $2,$ then $E$ is equivalent to an open subset of $\\mathbb{R}^d.$" ], [ "Introduction and statement of results", "The remarkable work of M. Lacey and C. Thiele , [1] on boundedness of the bi-linear Hilbert transform motivated a lot of research in the area of Euclidean harmonic analysis.", "For $f,g$ in $\\mathcal {S}({\\mathbb {R}})-$ the Schwartz class on ${\\mathbb {R}}$ , the bi-linear Hilbert transform is defined by $H(f, g)(x) = p.v.", "\\int _{{\\mathbb {R}}} f(x-y)g(x+y)\\frac{dy}{y}.$ Or equivalently, $H(f,g)(x)= -i \\int _{{\\mathbb {R}}}\\int _{{\\mathbb {R}}}\\hat{f}(\\xi )\\hat{g}(\\eta ) \\it {sgn}(\\xi -\\eta )e^{2\\pi i x(\\xi +\\eta )}d\\xi d\\eta ,$ where $\\hat{}$ denotes the Fourier transform and $\\it {sgn}(\\xi )=\\left\\lbrace \\begin{array}{ll}1, & \\hbox{$\\xi >0$} \\\\0, & \\hbox{$\\xi =0$}\\\\-1, & \\hbox{$\\xi <0$.", "}\\end{array}\\right.$ We would like to remark that the bi-linear Hilbert transform is invariant under the operations of simultaneous translation, dilation, and modulation.", "The modulation in-variance is a subtle property shared by the bi-linear Hilbert transform and poses additional difficulties while proving suitable $L^p-$ estimates for the operator.", "It is not difficult to convince ourselves that the classical approach of Littlewood-Paley decomposition, which is a useful technique to handle singular integral operators, is not quite helpful to deal with operators having modulation in-variance property.", "In papers and [1] M. Lacey and C. Thiele developed techniques very useful to handle such operators.", "These techniques are commonly referred to as the time-frequency techniques.", "In seminal papers they proved the following $L^p-$ estimates for the bi-linear Hilbert transform.", "Theorem 1.1 , [1] Let $1<p,q\\le \\infty $ and $\\frac{2}{3}<r <\\infty $ be such that $\\frac{1}{p}+\\frac{1}{q}=\\frac{1}{r}$ .", "Then for all functions $f,g \\in \\mathcal {S}({\\mathbb {R}})$ , there exists a constant $C>0$ such that $\\Vert H(f,g)\\Vert _{L^{r}({\\mathbb {R}})}\\le C \\Vert f\\Vert _{L^{p}({\\mathbb {R}})}\\Vert g\\Vert _{L^{q}({\\mathbb {R}})}.$ In this paper we are interested in studying bi-linear multiplier operators having modulation in-variance property.", "The bi-linear multiplier operators in general are defined as follows: Let $m(\\xi -\\eta )$ be a bounded measurable function on ${\\mathbb {R}}^d$ and $(p,q,r),~0<p,q,r\\le \\infty $ be a triplet of exponents.", "Consider the bi-linear operator $T_m$ initially defined for functions $f$ and $g$ in a suitable dense class by $T_{m}(f,g)(x)= \\int _{{\\mathbb {R}}^d}\\int _{{\\mathbb {R}}^d}\\hat{f}(\\xi )\\hat{g}(\\eta ) m(\\xi -\\eta )e^{2\\pi i x\\cdot (\\xi +\\eta )}d\\xi d\\eta .$ We say that $T_m$ is a bi-linear multiplier operator for the triplet $(p,q,r)$ if $T_m$ extends to a bounded operator from $L^p({\\mathbb {R}}^d)\\times L^q({\\mathbb {R}}^d)$ into $L^r({\\mathbb {R}}^d),$ i.e.", "there exists a constant $C>0,$ independent of functions $f$ and $g,$ such that $\\Vert T_m(f,g)\\Vert _{L^r({\\mathbb {R}}^d)}\\le C \\Vert f\\Vert _{L^p({\\mathbb {R}}^d)} \\Vert g\\Vert _{L^q({\\mathbb {R}}^d)}.$ The bounded function $m$ is said to be a bi-linear multiplier symbol for the triplet $(p,q,r)$ if the corresponding operator $T_m$ is a bi-linear multiplier operator for $(p,q,r).$ We denote by $\\mathcal {M}_{p,q}^{r}({\\mathbb {R}}^d)$ the space of all bi-linear multiplier symbols for the triplet $(p,q,r).$ Further, the norm of $m\\in \\mathcal {M}_{p,q}^{r}({\\mathbb {R}}^d)$ is defined to be the norm of the corresponding bi-linear multiplier operator $T_m$ from $L^p({\\mathbb {R}}^d)\\times L^q({\\mathbb {R}}^d)$ into $L^r({\\mathbb {R}}^d),$ i.e.", "$\\Vert m\\Vert _{\\mathcal {M}_{p,q}^{r}({\\mathbb {R}}^d)}=\\Vert T_m\\Vert _{L^p({\\mathbb {R}}^d)\\times L^q({\\mathbb {R}}^d)\\rightarrow L^r({\\mathbb {R}}^d)}.$ The bi-linear multiplier symbols on the Torus group $d$ and discrete group ${\\mathbb {Z}}^d$ are defined similarly.", "The space of bi-linear multiplier symbols on $d$ and ${\\mathbb {Z}}^d$ will be denoted by $\\mathcal {M}_{p,q}^{r}(d)$ and $\\mathcal {M}_{p,q}^{r}({\\mathbb {Z}}^d)$ respectively.", "Remark 1.2 Unless specified otherwise, we shall always assume that exponents $p,q,r$ satisfy $0< p,q,r\\le \\infty $ and the Hölder's condition $\\frac{1}{p}+\\frac{1}{q}=\\frac{1}{r}.$ Now we describe some important properties of bi-linear multipliers.", "Some of these properties will be used later in the paper.", "Proposition 1.3 Let $p,q,r$ be exponents satisfying the Hölder's condition.", "Then bi-linear multiplier symbols satisfy the following properties: If $c\\in and $ m, m1,$ and $ m2$ are in $ Mp,qr(Rd),$ then so are $ c m$ and $ m1+m2.$ Moreover,$ cmMp,qr(Rd)=\|c\| mMp,qr(Rd) $ and $ m1+m2Mp,qr(Rd)C(m1Mp,qr(Rd) +m2Mp,qr(Rd)).$\\item If $ mMp,qr(Rd)$ and $ Rd,$ then $ m(.", ")=m(.-)$ is in $ Mp,qr(Rd)$ with same norm as $ m.$\\item If $ mMp,qr(Rd)$ and $ >0,$ then $ m(.)=m(.", ")$ is in $ Mp,qr(Rd)$ with same norm as $ m.$\\item If $ mMp,qr(Rd)$ and $ A=(ai,j)dd$ is an orthogonal matrix acting on $ Rd,$ then $ m(A.", ")$ is in $ Mp,qr(Rd)$ with same norm as $ m.$\\item If $ mMp,qr(Rd)$ and $ hL1(Rd),$ then the convolution $ mhMp,qr(Rd),$ provided $ r1.$ Moreover, we have $ mhMp,qr(Rd)hL1(Rd) mMp,qr(Rd).$\\item If $ mj$ is a sequence of bi-linear multiplier symbols in $ Mp,qr(Rd)$ such that the norm$ mjMp,qr(Rd)C$ for some fixed constant $ C>0$ uniformly for all $ j=1, 2, ....$ Further assume that $ mj$ are uniformly bounded by a bounded measurable function on $ Rd$, then if $ mjm$ a.e.", "as $ j,$ we havethat $ mMp,qr(Rd)$ and $ mMp,qr(Rd)C.$$ Remark 1.4 We would like to remark that corresponding properties hold true for bi-linear multiplier symbols on $d$ and ${\\mathbb {Z}}^d.$ Statement of the result In this paper we study some particular type of bi-linear multiplier symbols, namely those which are indicator functions of measurable sets.", "In general, there is no effective method to decide that the indicator function of a measurable set is a bi-linear multiplier symbol for some exponents.", "An important example in this direction is given by $\\chi _{_I}(\\xi -\\eta ),$ for an interval $~I\\subset {\\mathbb {R}}.$ This is a consequence of boundedness of the bi-linear Hilbert transform that $\\chi _{_I}(\\xi -\\eta )$ is a bi-linear multiplier symbol for all exponents satisfying the conditions of Theorem REF .", "In the current paper, we study structural properties (in the sense of measure theory) of sets whose indicator functions give rise to bi-linear multiplier symbols.", "The motivation for this paper comes from the beautiful work of V. Lebedev and A. Olevskiĭ [2] on the classical Fouirer multipliers.", "We will prove an analogue of their result in the context of bi-linear multiplier operators.", "In order to describe V. Lebedev and A. Olevskiĭ's result, we need the following definition.", "Definition 1.5 We say that measurable sets $E$ and $E^{\\prime }$ are equivalent if the symmetric difference $E\\Delta E^{\\prime }$ has Lebesgue measure zero.", "Theorem 1.6 [2] Let $E\\subseteq {\\mathbb {R}}^d$ be a measurable set and $p\\ne 2.$ If $\\chi _{_E}-$ the indicator function of $E,$ is an $L^p-$ multiplier, i.e., the linear operator $f\\rightarrow (\\chi _{_E}\\hat{f}\\check{)},~f\\in L^2({\\mathbb {R}}^d)\\cap L^p({\\mathbb {R}}^d)$ extends boundedly from $L^p({\\mathbb {R}}^d)$ into itself, then $E$ is equivalent to an open set in ${\\mathbb {R}}^d.$ This theorem tells us that the structure of set whose indicator function is an $L^p-$ multiplier, $p\\ne 2,$ cannot be very complicated in the sense of measure theory.", "As an immediate consequence of this, we see that the indicator function of a nowhere dense set of positive Lebesgue measure is never an $L^p-$ multiplier for $p\\ne 2.$ As mentioned previously, in this paper our aim is to prove an analogue of Theorem REF in the context of bi-linear multipliers.", "In particular, we prove the following result: Theorem 1.7 Let $E$ be a non-empty measurable subset of ${\\mathbb {R}}^d$ and $p,q,r$ be exponents such that $1\\le p,q,r\\le \\infty ,~\\frac{1}{p}+\\frac{1}{q}=\\frac{1}{r}$ and exactly one of $p,q,$ or $r^{\\prime }=\\frac{r}{r-1}$ is less than $2.$ Suppose that $\\chi _{_E}(\\xi -\\eta )$ is a bi-linear multiplier symbol for the triplet $(p,q,r)$ , then $E$ is equivalent to an open subset of ${\\mathbb {R}}^d.$ Remark 1.8 In the case of classical Fourier multipliers $p=2$ plays a special role by virtue of the Plancherel theorem.", "In a sharp contrast to this, in the theory of bi-linear mutipliers there is absolutely no easy way by which one can test that a given bounded function $m(\\xi -\\eta )$ is a bi-linear multiplier symbol for a given triplet of exponents.", "The range of exponents covered by Theorem REF falls in the complement of what is commonly known as local $L^2-$ range of exponents.", "The local $L^2-$ range consists of exponents satisfying $2\\le p,q,r^{\\prime }\\le 2.$ Generally, it is believed that in this range of exponents most of the bi-linear multiplier operators are well behaved as far as the boundedness is concerned.", "Basic results and proof of Theorem REF In this section first we provide some basic definitions and results which would be required to complete the proof of Theorem REF .", "Definition 2.1 Let $E$ be a measurable set of ${\\mathbb {R}}^d.$ Then we say that a point $x\\in {\\mathbb {R}}^d$ is a density point for $E$ if $\\lim \\limits _{t\\rightarrow 0}\\frac{\|B(x,t)\\cap E\|}{\|B(x,t)\|}=1,$ where $B(x,t)$ denotes the Euclidean ball of radius $t>0$ centered at $x\\in {\\mathbb {R}}^n$ and $\|.\|$ denotes the Lebesgue measure of a set.", "The set of all density points for the set $E$ is denoted by $E^d.$ The set $\\partial _e E=\\overline{E^d}\\cap \\overline{(E^c)^d}$ is referred to as the essential boundary of $E,$ where $E^c$ denotes the complement of $E.$ Lemma 2.2 If $E\\subseteq {\\mathbb {R}}^d$ is a measurable set, then $E$ and $E^c$ are both equivalent to open sets if and only if $\\partial _e E-$ the essential boundary of $E,$ has Lebesgue measure zero.", "This lemma is easy to verify and hence its proof is not included here.", "Next, we describe an important lemma proved by V. Lebedev and A. Olevskiĭ in [2].", "Lemma 2.3 [2] Let $E\\subseteq {\\mathbb {R}}^d$ be a measurable set such that $\|\\partial _e E\|>0.$ Then for every $N\\in {\\mathbb {N}}$ and for every subset $A\\subseteq A_N=\\lbrace 1,2,\\dots ,N\\rbrace ,$ there exist $x_0, h\\in {\\mathbb {R}}^d$ such that the arithmetic progression $x_n=x_0+nh,~n\\in A_N$ satisfies the following conditions $x_n\\in E^d,~~\\text{if}~~n\\in A~~\\text{and}~~x_n\\in (E^c)^d,~\\text{if}~~n\\in A_N\\setminus A.$ This lemma plays a crucial role in the proof of Theorem REF .", "Since, we have exploited the methodology of [2] in order to prove the main result of this paper, Lemma REF is a key tool for the current paper as well.", "Finally, we would require de Leeuw's type transference result for bi-linear multipliers.", "This would allow us to restrict bi-linear multiplier symbols defined on ${\\mathbb {R}}^d$ to the discrete group ${\\mathbb {Z}}^d.$ As a consequence of this, we will have to deal with some bi-linear multiplier operators on $d.$ In the context of this paper, it turns out that working with operators on $d$ is simpler compared to working with the original operator defined on ${\\mathbb {R}}^d.$ This approach helps us in proving some good estimates on bi-linear operators under consideration and eventually helps us completing the proof of our main result.", "The following is a bi-linear analogue of the celebrated transference result proved by de Leeuw for the classical Fourier multipliers.", "Theorem 2.4 Let $m(\\xi -\\eta ) \\in \\mathcal {M}_{p,q}^{r}({\\mathbb {R}}^d)$ be a regulated function.", "Then $m\\upharpoonright _{{\\mathbb {Z}}^d}$ belongs to $\\mathcal {M}_{p,q}^{r}(d)$ with norm bounded by a constant multiple of $\\Vert m \\Vert _{ \\mathcal {M}_{p,q}^{r}({\\mathbb {R}}^d)}.$ We are now in a position to prove the main result of this paper.", "Proof of Theorem REF Notice that the same property holds for the complement $E^c,$ i.e., the indicator function $\\chi _{_{E^c}}$ is a bi-linear multiplier symbol for the triplet $(p,q,r).$ Therefore, in the view of Lemma REF , it is enough to show that the essential boundary $\\partial _e E$ has Lebesgue measure zero.", "The proof is given by contradiction.", "Therefore, suppose on the contrary that $\|\\partial _e E\|>0.$ Let $N\\in {\\mathbb {N}}$ be fixed and $\\lbrace \\epsilon _n\\rbrace _{n=1}^N$ be a random sequence of 0 and $1.$ Then as an application of Lemma REF we know that there exists $x_0, h\\in {\\mathbb {R}}^d$ such that $x_n=x_0+nh \\in \\left\\lbrace \\begin{array}{ll}E^d, & \\hbox{$\\epsilon _n=1$} \\\\(E^c)^d, & \\hbox{$\\epsilon _n=0$.", "}\\end{array}\\right.$ For $t>0,$ we consider the following function $m_t(x)&=&\\frac{1}{\|B(x,t)\|}\\int _{B(x,t)}\\chi _{_E}(y)dy\\\\&=& \\frac{\|B(x,t)\\cap E\|}{\|B(x,t)\|}.$ Observe that $m_t=\\chi _{_E}\\ast A_t,$ where $A_t(.", ")=\\frac{1}{\|B(0,t)\|}\\chi _{_{B(0,t)}}(.", ").$ Clearly, for all $t$ the norm $\\Vert A_t\\Vert _{L^{1}({\\mathbb {R}}^d)}=1.$ Since $r\\ge 1,$ by Proposision REF we obtain that $m_t\\in \\mathcal {M}_{p,q}^r({\\mathbb {R}}^d).$ Moreover $\\Vert m_t\\Vert _{\\mathcal {M}_{p,q}^r({\\mathbb {R}}^d)}\\le \\Vert \\chi _{_E}\\Vert _{\\mathcal {M}_{p,q}^r({\\mathbb {R}}^d)},~\\forall t>0.$ Let $e_1=(1,0,\\dots ,0)$ be the unit vector of the $x_1-$ axes and $\\psi $ be an affine mapping of ${\\mathbb {R}}^d$ which maps the vector $ne_1$ to $x_n.$ At this point we invoke Proposition REF once again and obtain that the composition $m_t\\circ \\psi \\in {\\mathcal {M}_{p,q}^r}({\\mathbb {R}}^d)$ with uniform norm with respect to the parameter $t.$ Recall the definition of density points and notice that $m_t(x_n)\\rightarrow \\epsilon _n$ as $t\\rightarrow 0,$ which is the same as $m_t\\circ \\psi (ne_1)\\rightarrow \\epsilon _n$ as $t\\rightarrow 0.$ Now we identify the set $\\lbrace ne_1:n\\in {\\mathbb {Z}}\\rbrace $ with the discrete group ${\\mathbb {Z}}$ and invoke de Leeuw's type transference theorem for bi-linear multipliers, namely Theorem REF from .", "This yields that $m_t\\circ \\psi \\upharpoonright _{{\\mathbb {Z}}}-$ the restriction of $m_t\\circ \\psi $ to ${\\mathbb {Z}},$ belongs to $\\mathcal {M}_{p,q}^r($ and $\\Vert m_t\\circ \\psi \\upharpoonright _{{\\mathbb {Z}}}\\Vert _{\\mathcal {M}_{p,q}^r(}\\le C\\Vert m_t\\Vert _{\\mathcal {M}_{p,q}^r({\\mathbb {R}}^d)}~\\forall t>0.$ Since the above estimate is uniform with respect to $t>0,$ a standard limiting argument (see Proposition REF ) allows us to conclude that the sequence $\\lbrace \\epsilon _n\\rbrace _{n=1}^N$ is a bi-linear multiplier symbol for the triplet $(p,q,r),$ i.e., $\\lbrace \\epsilon _n\\rbrace \\in {\\mathcal {M}_{p,q}^r}(.$ Moreover the norm is independent of $N\\in {\\mathbb {N}}.$ We shall show that this leads to a contradiction.", "First, we will prove that exponent $p$ or $q$ cannot be smaller than $2.$ Since $\\lbrace \\epsilon _n\\rbrace \\in {\\mathcal {M}_{p,q}^r}($ with multiplier norm uniformly bounded in $N\\in {\\mathbb {N}},$ we use Khintchine's inequality to deduce that the bi-linear Littlewood-Paley operator $S:(P,Q)\\rightarrow \\left(\\sum _n \|S_n(P,Q)\|^2\\right)^{\\frac{1}{2}}$ is bounded from $L^p(\\times L^q($ into $L^r(,$ where $P$ and $Q$ are trigonometric polynomials and $S_n$ is the bi-linear multiplier operator on $ with corresponding bi-linear multiplier symbol $ {n}.$ But, we already know that $ p,q2$ is a necessary condition for the boundedness of the bi-linear Littlewood-Paley operator $ S$ (see the paper by P.~Mohanty and S.~Shrivastava~\\cite {ms2} for a proof of this assertion) and hence we get a contradiction if either of $ p$ and $ q$ is less than $ 2.$$ Next, we assume that $r^{\\prime }<2,$ and show that this assumption also gives a contradiction.", "Let $p,q,r$ be exponents such that $p,q,r>2.$ Since the sequence $\\lbrace \\epsilon _n\\rbrace \\in {\\mathcal {M}_{p,q}^r}($ with multiplier norm uniformly bounded in $N\\in {\\mathbb {N}},$ we have that $\\Vert S_{\\Gamma }(P,Q)\\Vert _{L^{r}(} \\le C \\Vert P\\Vert _{L^{p}(} \\Vert Q\\Vert _{L^{q}(},$ where $S_{\\Gamma }$ is the bi-linear multiplier operator on $ with corresponding bi-linear multiplier symbol $ {n}.$$ Take $Q\\equiv 1$ on $ and consider\\begin{eqnarray}S_\\Gamma (P,Q)(x)&=&\\sum _{m,l\\in {\\mathbb {Z}}} \\hat{P}(m)\\hat{Q}(l) \\epsilon _{l-m} e^{2 \\pi i (m+l)x}\\\\&=&\\sum _{m\\in {\\mathbb {Z}}} \\hat{P}(m)\\epsilon _{-m} e^{2 \\pi i mx}.\\end{eqnarray}Therefore, we obtain that the linear operator $ T:fnZ n P(n) e2 i n$ is bounded from $ Lp($ into $ Lr(,$ i.e., we have\\begin{eqnarray}\\Vert T P\\Vert _{L^r(}\\le C \\Vert P\\Vert _{L^p(}.\\end{eqnarray}This leads to a contradiction as we know that the inequality~(\\ref {lin1}) does not hold true for all choices of $ {n}n=1N, NN$ as $ r>2.$\\\\This completes the proof of Theorem~\\ref {result}.\\begin{thebibliography}{10}\\bibitem {oscar} Blasco, O., {\\it Bi-linear multipliers and transference,} Int.", "J.", "Math.", "Math.", "Sci.", "2005, no.", "4, 545--554.\\end{thebibliography}\\bibitem {de} de Leeuw, K., {\\it On L_p-multipliers,} Ann.", "of Math.", "(2) 81 1965 364--379.\\bibitem {g} Grafakos, L.; Torres, Rodolfo H., {\\it Multilinear Calderón-Zygmund theory,} Adv.", "Math.", "165 (2002), no.", "1, 124--164.\\bibitem {lt1} Lacey, M.; Thiele, C.,{\\it L^p estimates on the bi-linear Hilbert transform for 2<p<\\infty ,} Ann.", "of Math.", "(2) 146 (1997), no.", "3, 693--724.$ Lacey, M.; Thiele, C., On Calderon's conjecture, Ann.", "of Math.", "(2) 149 (1999), no.", "2, 475–496.", "Lebedev, V.; Olevskiĭ, A., Idempotents of Fourier multiplier algebra, Geom.", "Funct.", "Anal.", "4 (1994), no.", "5, 539–544 Mohanty, P.; Shrivastava, S., A note on the bi-linear Littlewood-Paley square function, Proc.", "Amer.", "Math.", "Soc.", "138 (2010), no.", "6, 2095–2098.", "Mohanty, P.; Shrivastava, S., Bi-linear Littlewood-Paley for circle and transference, Publ.", "Mat.", "55 (2011), no.", "2, 501–519." ], [ "Basic results and proof of Theorem ", "In this section first we provide some basic definitions and results which would be required to complete the proof of Theorem REF .", "Definition 2.1 Let $E$ be a measurable set of ${\\mathbb {R}}^d.$ Then we say that a point $x\\in {\\mathbb {R}}^d$ is a density point for $E$ if $\\lim \\limits _{t\\rightarrow 0}\\frac{\|B(x,t)\\cap E\|}{\|B(x,t)\|}=1,$ where $B(x,t)$ denotes the Euclidean ball of radius $t>0$ centered at $x\\in {\\mathbb {R}}^n$ and $\|.\|$ denotes the Lebesgue measure of a set.", "The set of all density points for the set $E$ is denoted by $E^d.$ The set $\\partial _e E=\\overline{E^d}\\cap \\overline{(E^c)^d}$ is referred to as the essential boundary of $E,$ where $E^c$ denotes the complement of $E.$ Lemma 2.2 If $E\\subseteq {\\mathbb {R}}^d$ is a measurable set, then $E$ and $E^c$ are both equivalent to open sets if and only if $\\partial _e E-$ the essential boundary of $E,$ has Lebesgue measure zero.", "This lemma is easy to verify and hence its proof is not included here.", "Next, we describe an important lemma proved by V. Lebedev and A. Olevskiĭ in [2].", "Lemma 2.3 [2] Let $E\\subseteq {\\mathbb {R}}^d$ be a measurable set such that $\|\\partial _e E\|>0.$ Then for every $N\\in {\\mathbb {N}}$ and for every subset $A\\subseteq A_N=\\lbrace 1,2,\\dots ,N\\rbrace ,$ there exist $x_0, h\\in {\\mathbb {R}}^d$ such that the arithmetic progression $x_n=x_0+nh,~n\\in A_N$ satisfies the following conditions $x_n\\in E^d,~~\\text{if}~~n\\in A~~\\text{and}~~x_n\\in (E^c)^d,~\\text{if}~~n\\in A_N\\setminus A.$ This lemma plays a crucial role in the proof of Theorem REF .", "Since, we have exploited the methodology of [2] in order to prove the main result of this paper, Lemma REF is a key tool for the current paper as well.", "Finally, we would require de Leeuw's type transference result for bi-linear multipliers.", "This would allow us to restrict bi-linear multiplier symbols defined on ${\\mathbb {R}}^d$ to the discrete group ${\\mathbb {Z}}^d.$ As a consequence of this, we will have to deal with some bi-linear multiplier operators on $d.$ In the context of this paper, it turns out that working with operators on $d$ is simpler compared to working with the original operator defined on ${\\mathbb {R}}^d.$ This approach helps us in proving some good estimates on bi-linear operators under consideration and eventually helps us completing the proof of our main result.", "The following is a bi-linear analogue of the celebrated transference result proved by de Leeuw for the classical Fourier multipliers.", "Theorem 2.4 Let $m(\\xi -\\eta ) \\in \\mathcal {M}_{p,q}^{r}({\\mathbb {R}}^d)$ be a regulated function.", "Then $m\\upharpoonright _{{\\mathbb {Z}}^d}$ belongs to $\\mathcal {M}_{p,q}^{r}(d)$ with norm bounded by a constant multiple of $\\Vert m \\Vert _{ \\mathcal {M}_{p,q}^{r}({\\mathbb {R}}^d)}.$ We are now in a position to prove the main result of this paper." ], [ "Proof of Theorem ", "Notice that the same property holds for the complement $E^c,$ i.e., the indicator function $\\chi _{_{E^c}}$ is a bi-linear multiplier symbol for the triplet $(p,q,r).$ Therefore, in the view of Lemma REF , it is enough to show that the essential boundary $\\partial _e E$ has Lebesgue measure zero.", "The proof is given by contradiction.", "Therefore, suppose on the contrary that $\|\\partial _e E\|>0.$ Let $N\\in {\\mathbb {N}}$ be fixed and $\\lbrace \\epsilon _n\\rbrace _{n=1}^N$ be a random sequence of 0 and $1.$ Then as an application of Lemma REF we know that there exists $x_0, h\\in {\\mathbb {R}}^d$ such that $x_n=x_0+nh \\in \\left\\lbrace \\begin{array}{ll}E^d, & \\hbox{$\\epsilon _n=1$} \\\\(E^c)^d, & \\hbox{$\\epsilon _n=0$.", "}\\end{array}\\right.$ For $t>0,$ we consider the following function $m_t(x)&=&\\frac{1}{\|B(x,t)\|}\\int _{B(x,t)}\\chi _{_E}(y)dy\\\\&=& \\frac{\|B(x,t)\\cap E\|}{\|B(x,t)\|}.$ Observe that $m_t=\\chi _{_E}\\ast A_t,$ where $A_t(.", ")=\\frac{1}{\|B(0,t)\|}\\chi _{_{B(0,t)}}(.", ").$ Clearly, for all $t$ the norm $\\Vert A_t\\Vert _{L^{1}({\\mathbb {R}}^d)}=1.$ Since $r\\ge 1,$ by Proposision REF we obtain that $m_t\\in \\mathcal {M}_{p,q}^r({\\mathbb {R}}^d).$ Moreover $\\Vert m_t\\Vert _{\\mathcal {M}_{p,q}^r({\\mathbb {R}}^d)}\\le \\Vert \\chi _{_E}\\Vert _{\\mathcal {M}_{p,q}^r({\\mathbb {R}}^d)},~\\forall t>0.$ Let $e_1=(1,0,\\dots ,0)$ be the unit vector of the $x_1-$ axes and $\\psi $ be an affine mapping of ${\\mathbb {R}}^d$ which maps the vector $ne_1$ to $x_n.$ At this point we invoke Proposition REF once again and obtain that the composition $m_t\\circ \\psi \\in {\\mathcal {M}_{p,q}^r}({\\mathbb {R}}^d)$ with uniform norm with respect to the parameter $t.$ Recall the definition of density points and notice that $m_t(x_n)\\rightarrow \\epsilon _n$ as $t\\rightarrow 0,$ which is the same as $m_t\\circ \\psi (ne_1)\\rightarrow \\epsilon _n$ as $t\\rightarrow 0.$ Now we identify the set $\\lbrace ne_1:n\\in {\\mathbb {Z}}\\rbrace $ with the discrete group ${\\mathbb {Z}}$ and invoke de Leeuw's type transference theorem for bi-linear multipliers, namely Theorem REF from .", "This yields that $m_t\\circ \\psi \\upharpoonright _{{\\mathbb {Z}}}-$ the restriction of $m_t\\circ \\psi $ to ${\\mathbb {Z}},$ belongs to $\\mathcal {M}_{p,q}^r($ and $\\Vert m_t\\circ \\psi \\upharpoonright _{{\\mathbb {Z}}}\\Vert _{\\mathcal {M}_{p,q}^r(}\\le C\\Vert m_t\\Vert _{\\mathcal {M}_{p,q}^r({\\mathbb {R}}^d)}~\\forall t>0.$ Since the above estimate is uniform with respect to $t>0,$ a standard limiting argument (see Proposition REF ) allows us to conclude that the sequence $\\lbrace \\epsilon _n\\rbrace _{n=1}^N$ is a bi-linear multiplier symbol for the triplet $(p,q,r),$ i.e., $\\lbrace \\epsilon _n\\rbrace \\in {\\mathcal {M}_{p,q}^r}(.$ Moreover the norm is independent of $N\\in {\\mathbb {N}}.$ We shall show that this leads to a contradiction.", "First, we will prove that exponent $p$ or $q$ cannot be smaller than $2.$ Since $\\lbrace \\epsilon _n\\rbrace \\in {\\mathcal {M}_{p,q}^r}($ with multiplier norm uniformly bounded in $N\\in {\\mathbb {N}},$ we use Khintchine's inequality to deduce that the bi-linear Littlewood-Paley operator $S:(P,Q)\\rightarrow \\left(\\sum _n \|S_n(P,Q)\|^2\\right)^{\\frac{1}{2}}$ is bounded from $L^p(\\times L^q($ into $L^r(,$ where $P$ and $Q$ are trigonometric polynomials and $S_n$ is the bi-linear multiplier operator on $ with corresponding bi-linear multiplier symbol $ {n}.$ But, we already know that $ p,q2$ is a necessary condition for the boundedness of the bi-linear Littlewood-Paley operator $ S$ (see the paper by P.~Mohanty and S.~Shrivastava~\\cite {ms2} for a proof of this assertion) and hence we get a contradiction if either of $ p$ and $ q$ is less than $ 2.$$ Next, we assume that $r^{\\prime }<2,$ and show that this assumption also gives a contradiction.", "Let $p,q,r$ be exponents such that $p,q,r>2.$ Since the sequence $\\lbrace \\epsilon _n\\rbrace \\in {\\mathcal {M}_{p,q}^r}($ with multiplier norm uniformly bounded in $N\\in {\\mathbb {N}},$ we have that $\\Vert S_{\\Gamma }(P,Q)\\Vert _{L^{r}(} \\le C \\Vert P\\Vert _{L^{p}(} \\Vert Q\\Vert _{L^{q}(},$ where $S_{\\Gamma }$ is the bi-linear multiplier operator on $ with corresponding bi-linear multiplier symbol $ {n}.$$ Take $Q\\equiv 1$ on $ and consider\\begin{eqnarray}S_\\Gamma (P,Q)(x)&=&\\sum _{m,l\\in {\\mathbb {Z}}} \\hat{P}(m)\\hat{Q}(l) \\epsilon _{l-m} e^{2 \\pi i (m+l)x}\\\\&=&\\sum _{m\\in {\\mathbb {Z}}} \\hat{P}(m)\\epsilon _{-m} e^{2 \\pi i mx}.\\end{eqnarray}Therefore, we obtain that the linear operator $ T:fnZ n P(n) e2 i n$ is bounded from $ Lp($ into $ Lr(,$ i.e., we have\\begin{eqnarray}\\Vert T P\\Vert _{L^r(}\\le C \\Vert P\\Vert _{L^p(}.\\end{eqnarray}This leads to a contradiction as we know that the inequality~(\\ref {lin1}) does not hold true for all choices of $ {n}n=1N, NN$ as $ r>2.$\\\\This completes the proof of Theorem~\\ref {result}.\\begin{thebibliography}{10}\\bibitem {oscar} Blasco, O., {\\it Bi-linear multipliers and transference,} Int.", "J.", "Math.", "Math.", "Sci.", "2005, no.", "4, 545--554.\\end{thebibliography}\\bibitem {de} de Leeuw, K., {\\it On L_p-multipliers,} Ann.", "of Math.", "(2) 81 1965 364--379.\\bibitem {g} Grafakos, L.; Torres, Rodolfo H., {\\it Multilinear Calderón-Zygmund theory,} Adv.", "Math.", "165 (2002), no.", "1, 124--164.\\bibitem {lt1} Lacey, M.; Thiele, C.,{\\it L^p estimates on the bi-linear Hilbert transform for 2<p<\\infty ,} Ann.", "of Math.", "(2) 146 (1997), no.", "3, 693--724.$ Lacey, M.; Thiele, C., On Calderon's conjecture, Ann.", "of Math.", "(2) 149 (1999), no.", "2, 475–496.", "Lebedev, V.; Olevskiĭ, A., Idempotents of Fourier multiplier algebra, Geom.", "Funct.", "Anal.", "4 (1994), no.", "5, 539–544 Mohanty, P.; Shrivastava, S., A note on the bi-linear Littlewood-Paley square function, Proc.", "Amer.", "Math.", "Soc.", "138 (2010), no.", "6, 2095–2098.", "Mohanty, P.; Shrivastava, S., Bi-linear Littlewood-Paley for circle and transference, Publ.", "Mat.", "55 (2011), no.", "2, 501–519." ] ]	1403.0078
[ [ "Effect of magnetic field on the velocity autocorrelation and the caging\n of particles in two-dimensional Yukawa liquids" ], [ "Abstract We investigate the effect of an external magnetic field on the velocity autocorrelation function and the \"caging\" of the particles in a two-dimensional strongly coupled Yukawa liquid, via numerical simulations.", "The influence of the coupling strength on the position of the dominant peak in the frequency spectrum of the velocity autocorrelation function confirms the onset of a joint effect of the magnetic field and strong correlations at high coupling.", "Our molecular dynamics simulations quantify the decorrelation of the particles' surroundings - the magnetic field is found to increase significantly the caging time, which reaches values well beyond the timescale of plasma oscillations.", "The observation of the increased caging time is in accordance with findings that the magnetic field decreases diffusion in similar systems." ], [ "Introduction", "Strongly coupled plasmas [1] comprise a large class of physical systems, in which the ratio of the inter-particle potential energy to the kinetic energy, expressed by the coupling parameter $\\Gamma $ , (largely) exceeds 1.", "Dusty plasmas [2] are a notable type of strongly coupled many-particle systems that appear both in astrophysical environments and can as well be realized in laboratory.", "In laboratory settings dust particles can grow in a reactive plasma environment, or can be externally introduced into non-reactive (typically noble gas) discharge plasmas.", "In this latter case both three-dimensional and two dimensional particle configurations can be realized.", "Microgravity conditions favor three-dimensional settings, while in the presence of gravity lower-dimensional configurations are routinely formed.", "In typical laboratory setups (a radio-frequency driven plasma source with parallel, horizontal electrodes) two-dimensional layers of particles can be realized [3], the position of the dust particle layer is determined by the balance of the major forces acting on the particles, which are usually the electrostatic force, gravitational force, and ion drag force.", "Additional forces, e.g.", "the thermophoretic force [4] can change the particle configuration drastically, and make it possible to realize three-dimensional structures (Yukawa balls) in the presence of a thermal gradient of the background gas [5].", "A wide variety of physical phenomena taking place in two-dimensional particle layers – e.g.", "crystal formation and melting [6], transport processes [7], as well as the propagation of waves [8] – have been thoroughly investigated in experiments, by theoretical approaches, and via simulation methods.", "Besides the crystallized phase, the strongly coupled liquid phase of dusty plasmas has been receiving a lot of attention.", "It is an important property of this phase that the surroundings of individual particles are “quasi-stable” for a certain time, in contrast to the solid and gaseous limiting cases, where the time of particle localization, respectively, is infinite and extremely short.", "This property of the liquid phase is the basis of several features of strongly coupled plasmas [9]: among other properties, the Coulomb one-component plasma in the $\\Gamma \\gtrsim 50$ domain was found (i) to exhibit a shear viscosity that follows an Arrhenius type behavior, with an activation energy related to the “binding energy” of the particles in the cages, and (ii) to obey the Stokes-Einstein relation, characteristics for dense fluids.", "At strong coupling the particles oscillate in local minima of the potential landscape, which itself, changes on the time scale of particle diffusion.", "This difference of the time scales for the plasma oscillations and diffusion serves as the basis of the Quasi-Localized Charge Approximation [10], that allows calculation of the dispersion relations of collective excitations from static properties of the system (pair correlation).", "Quantitative data for the localization time have been obtained for Coulomb and Yukawa liquids in [11] by Molecular Dynamics simulations, using a technique of [12], that allows tracing of the changes of the neighborhoods of the particles.", "The simulation results have confirmed that at high coupling the particles spend several oscillation cycles in local minima of the potential surface without experiencing substantial changes in their surroundings.", "The effect of magnetic fields on strongly coupled dusty plasmas became an important topic in the last few years [13], [14], [15], [16], [17].", "Theoretical and simulation studies have demonstrated the formation of magnetoplasmons and their higher harmonics in strongly coupled Coulomb and Yukawa systems [13].", "Detailed studies of the impact of the magnetic field on the collective excitations and the self-diffusion have been presented, respectively, in [14] and [17].", "The effect of magnetic field on binary Yukawa systems has been studied in [18].", "Another line of research focuses on systems of superparamagnetic particles [19].", "Experiments, aimed at the realization of magnetized dusty plasmas, have faced, however, serious difficulties, as the external magnetic fields cause a significant perturbation to the plasma itself (like filamentation) before affecting the dynamics of the dust system [20].", "An alternative method to investigate magnetic field effects was suggested in [21], based on the equivalence of the magnetic Lorentz force and the Coriolis inertial force acting on moving objects when they are viewed in a rotating reference frame.", "Experimental realization of a rotating dusty plasma has confirmed the theoretical predictions and has proven the formation of magnetoplasmons [22] in the “magnetized” dusty plasma.", "We note that in magnetized strongly coupled plasmas many of the effects are qualitatively different from those observed and well known for weakly coupled plasmas, due to the interplay of magnetization and strong correlation effects.", "In this paper we investigate the effect of an external, homogeneous magnetic field on the behavior of the velocity autocorrelation function (VACF) in the time and frequency domains, as well as on the caging of the particles, in two-dimensional strongly coupled Yukawa liquids.", "These phenomena are investigated using Molecular Dynamics simulations.", "The model and the simulation techniques are described in Sec.", "II.", "In Sec.", "III we present and analyze the simulation results, while Sec.", "IV gives a short summary of the work." ], [ "Model and simulation method", "We investigate the effect of the magnetic field on many-particle systems, in which particles interact via a screened Coulomb (Debye-Hückel, or Yukawa) potential: $\\phi (r) = \\frac{Q}{4 \\pi \\varepsilon _0} \\frac{\\exp (-r/\\lambda _D)}{r},$ where $Q$ is the charge of the particles and $\\lambda _D$ is the screening (Debye) length.", "The ratio of the inter-particle potential energy to the thermal energy is expressed by the coupling parameter $\\Gamma = \\frac{Q^2}{4 \\pi \\varepsilon _0 a k_B T},$ where $T$ is temperature.", "We introduce the screening parameter $\\kappa = a / \\lambda _D$ , where $a = (1/\\pi n)^{-1/2}$ is the two-dimensional Wigner-Seitz radius and $n$ is the areal number density of the particles.", "In particular, we investigate the effect of the magnetic field on the velocity autocorrelation function (VACF) of the particles and the cage correlation function that quantifies the relation of the localization time of the particles to the timescale of plasma oscillations.", "We apply the Molecular Dynamics (MD) simulation method to describe the motion of the particles governed by the Newtonian equation of motion.", "For the integration of the equation of motion that accounts for the presence of the magnetic field we use the method described in [23].", "The number of particles is fixed at $N$ = 4000 (at $N$ = 1000 in the calculations of cage correlations, see later) and we use a quadratic simulation box.", "The particles move in the $(x,y)$ plane and the magnetic field is assumed to be homogeneous and directed perpendicularly to the two-dimensional layer of the particles, i.e.", "${\\bf B} = (0,0,B)$ .", "The strength of the magnetic field is expressed in terms of $\\beta = \\frac{\\omega _c}{\\omega _p},$ where $\\omega _c = Q B / m$ is the cyclotron frequency and $\\omega _p= \\sqrt{n Q^2 / 2 \\varepsilon m a}$ is the nominal 2D plasma frequency.", "We note that the Larmor radius becomes smaller than the WS radius at magnetic fields $\\beta \\gtrsim 0.1.$ The velocity autocorrelation function is defined as (see e.g.", "[24]): $A_{vv}(t) = \\langle {\\bf v}(t) {\\bf v}(0) \\rangle ,$ while its normalized value (giving $A_{vv}(0) =1$ ) is given by: $\\overline{A}_{vv}(t) = \\frac{\\langle {\\bf v}(t) {\\bf v}(0) \\rangle }{\\langle {\\bf v}(0) {\\bf v}(0) \\rangle }.$ The Fourier transform of the VACF is defined as $A_{vv}(\\omega ) = \\int _0^\\infty A_{vv}(t) {\\rm e}^{i\\omega t} {\\rm d}t,$ and is calculated by replacing the upper limit of the integration with a time $t_{max}$ , for which $A_{vv}(t)\\cong 0$ at $t > t_{max}$ .", "Taking the time integral of the VACF the self-diffusion coefficient of the particles can be calculated: $D = \\frac{1}{2} \\int _0^\\infty A_{vv}(t) {\\rm d}t.$ We note, however, that previous studies have shown that this integral may be divergent for 2D system, at certain range of parameters [25], where calculations, as well as experimental measurements of the self diffusion coefficient, based on the mean square displacement (MSD) of the particles have both shown superdiffusion, MSD$\\propto t^\\alpha $ , with $\\alpha > 1$ [26].", "The Fourier transform of the VACF is known to be connetcted with the longitudinal and transverse fluctuations in the system [24].", "Therefore we also calculate the respective fluctuation spectra $L(k,\\omega )$ and $T(k,\\omega )$ , for a discrete set of wave numbers $k = m (2 \\pi / L) = m k_{min},~m=1,2,...$ , accommodated by the simulation box of edge length $L$ .", "To accomplish this calculation we collect data during each time step of the simulation for the microscopic currents $\\lambda (k,t)= \\sum _j v_{j x}(t) \\exp \\bigl [ i k x_j(t) \\bigr ], \\nonumber \\\\\\tau (k,t)= \\sum _j v_{j y}(t) \\exp \\bigl [ i k x_j(t) \\bigr ],$ where $x_j$ and $v_j$ are the position and velocity of the $j$ -th particle.", "These data sequences are subsequently Fourier analyzed to yield, e.g.", "$L(k,\\omega )$ as: $L(k,\\omega ) = \\frac{1}{2 \\pi N} \\lim _{\\Delta T \\rightarrow \\infty }\\frac{1}{\\Delta T} \| \\lambda (k,\\omega ) \|^2,$ where $\\Delta T$ is the length of data recording period and $\\lambda (k,\\omega ) = {\\cal {F}} \\bigl [ \\lambda (k,t) \\bigr ]$ is the Fourier transform of (REF ).", "Calculation of $T(k,\\omega )$ proceeds in the same way.", "We note that the collective modes show up as peaks in these current fluctuation spectra.", "To quantify the time-dependence of the correlation of the particles' surroundings we adopt the method proposed in [12] and used in [11] for the investigation of strongly coupled Coulomb and Yukawa liquids.", "We use a generalized neighbor list $\\ell _i$ for particle $i$ , $\\ell _i = \\lbrace f_{i,1},f_{i,2},...,f_{i,N} \\rbrace $ .", "Due to the sixfold symmetry we always find the six closest neighbors of particle $i$ and the $f$ -s corresponding to these particles are set to a value 1, while all other $f$ -s are set to 0.", "The similarity between the surroundings of the particles at $t$ =0 and $t>0$ is measured by the list correlation function: $C_{\\ell }(t) = {{\\langle \\ell _i(t) \\ell _i(0) \\rangle } \\over {\\langle \\ell _i(0)^2 \\rangle }},$ where $\\langle \\cdot \\rangle $ denotes averaging over particles and initial times.", "$C_{\\ell }(t=0) = 1$ , and $C_{\\ell }(t)$ is a monotonically decaying function (provided that averaging is sufficient).", "The number of particles that have left the original cage of particle $i$ at time $t$ can be determined as $n_i^{\\rm out} (t) = \|\\ell _i(0)^2\| - \\ell _i(0) \\ell _i(t),$ where the first term gives the number of particles around particle $i$ at $t$ = 0 (that, actually, equals to six, in our case), while the second term gives the number of `original' particles that remained in the surrounding after time $t$ elapsed.", "As the next step an integer value $c$ is defined, which is the number of the 'original' neighbors that have to leave the cage before we say that the cage has undergone a 'substantial change'.", "The cage correlation function $C_{\\rm cage}^{c}(t)$ can be calculated by averaging over particles and initial times, of the function $\\Theta (c - n_i^{\\rm out})$ , i.e.", "$C_{\\rm cage}^{c}(t) =\\langle \\Theta (c - n_i^{\\rm out} (0,t)) \\rangle .$ Here $\\Theta $ is the Heaviside function.", "We calculate the cage correlation functions for $c=3$ , meaning that half of the original neighbors leave the cage.", "We adopt the definition of the caging time introduced in [11], according to which $t_{\\rm cage}$ is defined as the time when $C_{\\rm cage}^{3}$ decays to a value 0.1." ], [ "Velocity autocorrelation", "The general effect of the external magnetic field on the velocity autocorrelation function is illustrated in Fig.", "REF (a), at $\\Gamma =120$ and $\\kappa =2$ .", "At $\\beta =0$ the VACF exhibits a few oscillations, which is a fingerprint of localized oscillations of the particles.", "In the magnetized case the dominant frequency is clearly increased, as well as the values of the extrema of the oscillatory VACF.", "The Fourier transform of the VACF, $A_{vv}(\\omega )$ , shown in Fig.", "REF (b) exhibits characteristic changes when the magnetic field is applied.", "In the $\\beta =0$ case the spectrum exhibits a single peak at about $\\omega / \\omega _p \\approx 0.48$ .", "This peak is known the be related to the longitudinal current fluctuations (see e.g.", "[24]), and this frequency indeed corresponds to the plateau of the dispersion relation of the longitudinal mode, shown in Fig.", "REF (a).", "At $\\beta >0$ the formation of a magnetoplasmon shifts the peak position to a higher value, $\\omega / \\omega _p \\approx 0.74$ , following the change of the character of the mode dispersion curve, plotted in Fig.", "REF (b).", "The low frequency part of $A_{vv}(\\omega )$ , seen in Fig.", "REF (b), is depleted at $\\beta >0$ with respect to the $\\beta =0$ case, and we also observe the formation of a small peak at $\\omega / \\omega _p = 0.5$ , corresponding to the cyclotron frequency of the dust particles.", "We note that, according to (REF ) and (REF ), $D = \\frac{1}{2} A_{vv}(\\omega =0)$ .", "Evaluation of $A_{vv}(\\omega =0)$ from Fig.", "REF (b) is ambigous, in accordance with the possible divergence of the Green-Kubo integral (REF ) already quoted.", "Plotting the normalized VACF for the unmagnetized case (see Fig.", "REF (c)) confirms that $\\overline{A}_{vv}(t)$ decays as $t^{-1}$ at long times, making (REF ) divergent (if we assume that the decay rate is maintained up to infinitely long time).", "It is not the topic of the present paper to investigate this effects further, and, accordingly, we shall not discuss the $\\omega \\rightarrow 0$ behavior of $A_{vv}(\\omega )$ .", "Fig.", "REF (a) presents a series of normalized VACF-s for increasing magnetization, at fixed $\\Gamma =120$ and $\\kappa =1$ .", "The data show that the behavior of $A_{vv}(t)$ is significantly altered with the introduction of the magnetic field.", "The dominant frequency (easily observed by eye) increases with increasing $\\beta $ , and the oscillations of the VACF-s persist for an increasingly longer time when the strength of the magnetic field is increased.", "Fig.", "REF (b) shows the respective $A_{vv}(\\omega )$ functions in the frequency domain, where the dominant frequency shows up as a definite peak.", "In [13] it has been shown that at high coupling the dominant frequency in the longitudinal fluctuation spectrum takes a value $\\omega _1^2 = \\omega _c^2 + 2 \\omega _E^2 = \\beta ^2 \\omega _p^2 + 2 \\omega _E^2,$ where $\\omega _E$ is the Einstein frequency, defined as the oscillation frequency of a test particle in a frozen environment.", "At $\\kappa =1$ we have $\\omega _E \\cong 0.52 \\omega _p$ (see Fig.", "19(b) of Ref.", "[27], note, however, that the numerical data of the same paper, given by eq.", "(54) are false).", "The positions of the peaks observed in Fig.", "REF (b), in comparison with the theoretical prediction given above, are listed in Table 1.", "We find a very good agreement (only a few % deviation) between the two sets of data, confirming the theoretical arguments of [13], according to which the dominant oscillation frequency forms due to a combined effect of magnetic field and strong correlations, lifting the fundamental frequency above the cyclotron frequency $\\omega _c = \\beta \\omega _p$ .", "Next, we investigate the effect of the coupling strength on the normalized VACF-s and their Fourier transform, at a fixed value of normalized magnetic field, $\\beta =0.5$ .", "The data are displayed in Fig.", "REF , for $\\kappa =1$ .", "The time-domain data show only a more persisting correlation at higher coupling, however, the frequency spectrum $A_{vv}(\\omega )$ shows an upwards shift of the dominant peak with lowering $\\Gamma $ .", "The relation (REF ), discussed above, holds only for a high coupling.", "In the limit of vanishing correlations (weakly coupled plasma limit, $\\Gamma \\rightarrow 0$ ) the frequency of the resulting (upper) hybrid mode in a magnetized plasma (where the direction of propagation is perpendicular to the direction of the magnetic field) is known to turn into the Random Phase Approximation (RPA) value (for details see [28]), $\\omega _2^2 = \\omega _c^2 + \\omega _p^2 = \\omega _p^2 (\\beta ^2+1).$ At $\\beta =0.5$ , the frequency defined by the above equation is $\\omega _2 \\cong 1.19 \\omega _p$ .", "This frequency value is not reached by our simulation data at decreasing coupling, due to the significant broadening of the frequency spectrum, as indicated in Fig.", "REF , for the conditions $\\Gamma =10$ , $\\kappa =2$ , and $\\beta =0.5$ .", "Here the magnetoplasmon becomes hardly recognizable at reduced wave numbers $ka \\gtrsim 2.5$ , where the fluctuation spectrum is practically featureless.", "Table: The dependence of the frequency of the dominant peak observed in A vv (ω)A_{vv}(\\omega ), as function of the normalized magnetic field β\\beta , for Γ=120\\Gamma =120 and κ=1\\kappa =1.Figure: (color online) Cage correlation functions: the effect of Γ\\Gamma at (a) β=0\\beta =0 and (b) β=0.5\\beta =0.5.Figure: (color online) The dependence of the caging time (a) on the magnetic field strength at Γ=20\\Gamma =20 and 120, and (b) on the coupling strength Γ\\Gamma at β=0\\beta =0 and 0.5." ], [ "Caging", "The dependence of the cage correlation function $C_{\\rm cage}^3(t)$ , defined by eq.", "(REF ), on the system parameters, is analyzed in Fig.", "REF .", "The cage correlation functions have been calculated for a series of $\\Gamma $ values, for the unmagnetized case ($\\beta =0$ ) and for a moderate value of the magnetization, $\\beta =0.5$ .", "The data are shown for $\\kappa =2$ , the behavior is similar at other values of screening.", "Comparison of the data for these two cases, shown in Fig.", "REF (a) and (b), respectively, reveals the increase of the caging time with increasing magnetic field.", "This behavior can easily be understood by the decreasing Larmor radius of the particles, that becomes a fraction of the interparticle distance at the highest $\\beta $ values considered.", "Finally, we show the dependence of the caging time, defined earlier as $C_{\\rm cage}^3(t_{\\rm cage}) =0.1$ , in Fig.", "REF (a) on the magnetic field (at $\\kappa =2$ and coupling values $\\Gamma =120$ and 20), and in Fig.", "REF (b) on $\\Gamma $ (at fixed $\\kappa =2$ and magnetic field strengths $\\beta =0$ and 0.5).", "Fig.", "REF (a) reveals an approximately three times increase of the caging time when $\\beta $ is increased from 0 to 1, at both values of coupling.", "At the parameter pair $\\Gamma =120$ and $\\kappa =2$ we reach $\\omega _p t_{\\rm cage} \\cong 400$ at $\\beta =1.2$ .", "Now we estimate how many oscillations caged particles execute during this time.", "One oscillation cycle of a caged particle in the strong coupling domain corresponds to $\\omega _1 t \\cong 2 \\pi $ , where $\\omega _1$ is defined by Eq.", "(REF ).", "For a number of oscillation cycles, $N_{osc}$ , within the cage $\\omega _1 t_{\\rm cage} \\cong 2 \\pi N_{osc}$ holds.", "from this, $N_{osc} = \\frac{1}{2\\pi } (\\omega _p t_{\\rm cage}) \\frac{\\omega _1}{\\omega _p} =\\frac{1}{2\\pi } (\\omega _p t_{\\rm cage}) \\sqrt{ \\beta ^2 + 2\\frac{\\omega _E^2}{\\omega _p^2}}.$ For $\\kappa =2$ we have $\\omega _E / \\omega _p \\cong 0.32$ (according to Ref.", "[27]), so at $\\beta =0$ and $\\beta =0.5$ , respectively, $N_{osc} \\cong 0.051 (\\omega _p t_{\\rm cage})$ , and $N_{osc} \\cong 0.107 (\\omega _p t_{\\rm cage})$ .", "Selected values, corresponding to the data shown in Fig.", "REF (b), are given in Table II.", "The data unambiguously confirm that in the $\\Gamma \\gg 1$ domain the particles carry out several oscillation cycles within their cages, before the potential landscape changes due to the diffusion of the particles.", "In magnetized systems $N_{osc}$ increases because of two reasons: (i) due to the reduction of the diffusion with increasing magnetic field, and (ii) due to the increasing oscillation frequency in the strong coupling domain.", "This effect has important consequences in determining the properties of strongly coupled plasmas [9].", "Table: Number of oscillations cycles of caged particles, N osc N_{osc}, at selected Γ\\Gamma , β\\beta parameter pairs, at κ=2\\kappa =2." ], [ "Summary", "We have investigated the effect of a homogeneous magnetic field on the velocity autocorrelation function and the caging phenomenon in two-dimensional Yukawa liquids in the strong coupling domain.", "The velocity autocorrelation functions have been analyzed both in the time and frequency domains.", "The dominant peak in $A_{vv}(\\omega )$ , related to the longitudinal current fluctuations in the liquid, was shown to be formed at high coupling at a frequency defined by a joint effect of cyclotron motion and strong inter-particle correlations [13].", "Towards lower coupling values the position of the peak was found to shift upwards, towards the RPA limit, which, however, was not reached due to the broadening of the peak.", "The caging time of the particles, of which the relation to the plasma oscillation cycles is of paramount importance in determining the liquid state properties of the plasma, was found to increase significantly with the applied magnetic field.", "Experimental verification of our computational results should be possible in future rotating dusty plasma experiments [22].", "This work has been supported by the Grants OTKA K-105476 and MES RK 1137/GF-1.", "We thank Prof. M. Bonitz for useful discussions." ] ]	1403.0232
[ [ "Generalization of the Schwarz-Christoffel mapping to multiply connected\n polygonal domains" ], [ "Abstract A generalization of the Schwarz-Christoffel mapping to multiply connected polygonal domains is obtained by making a combined use of two preimage domains, namely, a rectilinear slit domain and a bounded circular domain.", "The conformal mapping from the circular domain to the polygonal region is written as an indefinite integral whose integrand consists of a product of powers of the Schottky-Klein prime functions, which is the same irrespective of the preimage slit domain, and a prefactor function that depends on the choice of the rectilinear slit domain.", "A detailed derivation of the mapping formula is given for the case where the preimage slit domain is the upper half-plane with radial slits.", "Representation formulae for other canonical slit domains are also obtained but they are more cumbersome in that the prefactor function contains arbitrary parameters in the interior of the circular domain." ], [ "Introduction", "The Schwarz-Christoffel mapping to polygonal domains is an important result in the theory of complex-valued functions and one that finds numerous applications in applied mathematics, physics, and engineering [1].", "The applicability of Schwarz-Christoffel formula is nonetheless limited by the fact that it pertains only to simply connected polygonal domains.", "Therefore, there has long been considerable interest in extending the Schwarz-Christoffel transformation to multiply connected polygonal domains.", "Recently, explicit formulae for generalized Schwarz-Christoffel mappings from a circular domain to both bounded and unbounded multiply connected polygonal regions have been obtained using two equivalent approaches.", "Using reflection arguments, DeLillo and collaborators [2], [3] derived infinite product formulae for the generalized Schwarz-Christoffel mappings, whereas Crowdy [4], [5] used a function-theoretical approach to obtain a mapping formula in terms of the Schottky-Klein prime function associated with the circular domain.", "(The equivalence of the two methods was established in [3].)", "It is important to point out that both these formulations entail an implicit choice of a multiply connected rectilinear slit domain in an auxiliary preimage plane, whereby each rectilinear boundary segment of this preimage domain is mapped to a polygonal boundary (see below).", "The introduction of a preimage domain with rectilinear boundaries makes it possible to write down an explicit expression for the derivative of the desired conformal mapping, which can then be integrated to yield a generalised Schwarz-Christoffel formula.", "It is thus clear that different choices of rectilinear slit domains in the auxiliary plane will result in different formulae for the generalised Schwarz-Christoffel mapping.", "Therefore, a derivation of alternative representation formulae is of some interest.", "In this paper a general formalism to construct conformal mappings of multiply connected polygonal domains is presented.", "The method applies to both bounded and unbounded polygonal domains but emphasis will be given to the bounded case as formulated in §.", "A key ingredient in the approach described herein is the introduction of a multiply connected rectilinear slit domain in an auxiliary $\\lambda $ -plane, such that each rectilinear boundary in the $\\lambda $ -plane is mapped to a polygonal boundary in the $z$ -plane.", "It is advantageous to have rectilinear boundaries in the preimage domain because the derivative of the respective conformal mapping has constant argument on each of domain's boundaries.", "To derive an expression for the mapping derivative, it is expedient to introduce a slit map from a bounded multiply connected circular domain in an auxiliary $\\zeta $ -plane to the rectilinear slit domain in the $\\lambda $ -plane.", "Using the properties of the Schottky-Klein prime function associated with the circular domain, as summarized in § and §, it is then possible to obtain an explicit expression for the derivative of the mapping from the circular domain to the polygonal region.", "The desired conformal mapping is then written in final form as an indefinite integral whose integrand consists of i) a product of powers of the Schottky-Klein prime functions and ii) a prefactor function, $S(\\zeta )$ , that depends on the choice of the rectilinear slit domain in the $\\lambda $ -plane.", "A detailed derivation of the prefactor $S(\\zeta )$ is given in § for the case where the rectilinear slit domain consists of the upper half-plane cut along radial segments.", "This preimage domain turns out to be the most convenient one not only in that it naturally extends to multiply connected domains the usual treatment of simply connected polygonal regions but most importantly because it yields a simpler formula for the generalized Schwarz-Christoffel mapping.", "As further illustration of the method presented here, the corresponding formulae for two other canonical slit domains (including the case considered by Crowdy [4]) are derived in §.", "These alternative representations have, however, the inconvenience of containing certain arbitrary parameters in the interior of the circular domain.", "For completeness, the case of unbounded polygonal domains is briefly discussed in §, after which our main results and conclusions are summarized in §.", "Let us consider a target region $D_z$ in the $z$ -plane that is a bounded $(M+1)$ -connected polygonal domain.", "The outer boundary polygon is denoted by $P_0$ and the $M$ inner polygons by $P_j$ , $j=1,...,M$ ; see figure REF .", "Let the vertices at polygon $P_j$ , $j=0, 1,...,M$ , be denoted by $z_k^{(j)}$ , $k=1,...,n_j$ , and let $\\pi \\alpha _{k}^{(j)}\\in [0,2\\pi ]$ be the interior angles at the respective vertices.", "Following the notation of Driscoll & Trefethen [1], we write $\\pi \\alpha _k^{(j)}=\\pi (\\beta _k^{(j)}+1), \\qquad k=1,...,n_j,$ where $\\pi \\beta _k^{(j)}$ represents the turning angle at vertex $z_k^{(j)}$ , so that the parameters $\\beta _k^{(j)}$ must satisfy the following relations: $\\sum _{k=1}^{n_0}\\beta _k^{(0)}=-2, \\qquad \\sum _{k=1}^{n_j}\\beta _k^{(j)}=2, \\quad j=1,...,M.$ Figure: ζ\\zeta -plane" ], [ "Radial slit domains in the upper half-plane", "Let us now consider a domain $D_\\lambda $ in a subsidiary $\\lambda $ -plane consisting of the upper half-plane with $M$ slits pointing towards the origin.", "Denote by $L_0$ the real axis in the $\\lambda $ -plane and by $L_j$ , $j=1,...,M$ , the $M$ radial slits; see figure REF .", "We seek a conformal mapping $z(\\lambda )$ from the radial slit domain $D_\\lambda $ in the upper half-$\\lambda $ -plane to the polygonal domain $D_z$ in the $z$ -plane, where the real $\\lambda $ -axis is mapped to the outer polygon $P_0$ and each radial slit $L_j$ is mapped to an inner polygon $P_j$ .", "If we denote by $\\lambda _{k}^{(j)}$ the preimages in the $\\lambda $ -plane of the vertices $z_k^{(j)}$ on polygon $P_j$ , $j=0,1,...,M$ , then $z(\\lambda )$ must have a branch point at $\\lambda =\\lambda _{k}^{(j)}$ such that $\\frac{dz}{d\\lambda }\\approx \\textrm {constant}\\cdot (\\lambda -\\lambda _{k}^{(j)})^{\\beta _k^{(j)}} \\quad \\mbox{for}\\quad \\lambda \\rightarrow \\lambda _{k}^{(j)}.$ Furthermore, the derivative $dz/d\\lambda $ must have constant arguments on each boundary segment in the $\\lambda $ -plane, that is, $\\arg \\left[\\frac{dz}{d\\lambda }\\right]= \\textrm {const.}", "\\quad \\mbox{for}\\quad \\lambda \\in L_j, \\ j=0,1,...,M.$ This follows from the fact that both $\\arg [d\\lambda ]$ and $\\arg [dz]$ are constant on the respective boundaries in the $\\lambda $ - and $z$ -planes.", "In the case of simply connected polygonal regions, conditions (REF ) and (REF ) can be easily satisfied by writing $dz/d\\lambda $ as a product of monomials of the type $(\\lambda -\\lambda _{k}^{(j)})^{\\beta _k^{(j)}}$ , leading to the well-known Schwarz-Christoffel formula [6].", "But for multiply connected domains condition (REF ) represents a major obstacle in deriving an explicit formula for $dz/d\\lambda $ , for no longer is it obvious how to construct a function that has constant argument on the multiple rectilinear boundaries of the domain $D_\\lambda $ .", "This obstacle can nonetheless be overcome by introducing a conformal mapping, $\\lambda (\\zeta )$ , from a circular domain $D_\\zeta $ in an auxiliary $\\zeta $ -plane to the slit domain $D_\\lambda $ .", "This allows us to write an explicit expression for the derivative $dz/d\\lambda $ in terms of the $\\zeta $ variable, as will be seen in §." ], [ "Bounded circular domains", "Let $D_\\zeta $ be a circular domain in the $\\zeta $ -plane consisting of the unit disc with $M$ smaller nonoverlapping disks excised from it.", "Label the unit circle by $C_0$ and the $M$ inner circular boundaries by $C_1,...,C_M$ , and denote the centre and radius of the circle $C_j$ by $\\delta _j$ and $q_j$ , respectively.", "For convenience, we introduce the notation $\\delta _0=0$ and $q_0=1$ for the unit circle.", "A schematic of $D_\\zeta $ is shown in figure REF .", "Now let $\\lambda (\\zeta )$ be a conformal mapping from the circular domain $D_\\zeta $ to the domain $D_\\lambda $ defined above, where the unit circle, is mapped to the real axis in the $\\lambda $ -plane and the interior circles $C_j$ map to the slits $L_j$ , $j=1,...,M$ .", "Furthermore, let the point $\\zeta =1$ map to the origin in the $\\lambda $ -plane and the point $\\zeta =-1$ map to infinity; see figures REF and REF .", "In an abuse of notation, we shall write $z(\\zeta )\\equiv z(\\lambda (\\zeta ))$ for the associated conformal mapping from $D_\\zeta $ to the polygonal region $D_z$ , where the unit circle $C_0$ maps to the outer polygonal boundary $P_0$ and the interior circles $C_j$ map to the inner polygons $P_j$ ; see figures REF and REF .", "If we denote by $a_{k}^{(j)}$ the preimages in the $\\zeta $ -plane of the vertices $z_k^{(j)}$ on polygon $P_j$ , then condition (REF ) can be recast as ${z_\\lambda }(\\zeta )\\approx \\textrm {constant}\\cdot (\\zeta -a_{k}^{(j)})^{\\beta _k^{(j)}} \\quad \\mbox{for}\\quad \\zeta \\rightarrow a_{k}^{(j)},$ where ${z_\\lambda }(\\zeta )$ denotes the derivative $dz/d\\lambda $ expressed in terms of the $\\zeta $ -variable.", "Similarly, the requirement (REF ) becomes $\\arg \\left[{z_\\lambda }(\\zeta )\\right]= \\textrm {const.}", "\\quad \\mbox{for}\\quad \\zeta \\in C_j, \\ j=0,1,...,M.$ As first noticed by Crowdy [4], albeit in a somewhat different and less general formulation, it is possible to write an explicit expression for $z_\\lambda (\\zeta )$ that satisfies conditions (REF ) and (REF ) by exploiting the properties of the Schottky-Klein prime function associated with the circular domain $D_\\zeta $ .", "Once an expression for $z_\\lambda (\\zeta )$ is obtained, a corresponding expression for $z_\\zeta (\\zeta )$ follows from the chain rule which can then be integrated yielding the desired conformal mapping $z(\\zeta )$ , as will be shown in §.", "Prior to this, however, a brief overview of the Schottky-Klein prime function is in order.", "Consider the bounded circular domain $D_\\zeta $ defined above; see figure REF .", "Introduce the following Möbius maps: $\\theta _j(\\zeta )=\\delta _j+\\frac{q_j^2 \\zeta }{1-\\bar{\\delta }_j\\zeta }, \\quad j=0,1,...M,$ where a bar denotes complex conjugation.", "For $\\zeta \\in C_j$ , it is easy to establish the following relations: ${\\zeta }={\\theta _j}(1/\\bar{\\zeta }) \\qquad \\Longleftrightarrow \\qquad \\bar{\\zeta }=\\bar{\\theta }_j(1/\\zeta ),$ where in the second identity we introduced the conjugate function $\\bar{\\theta }(\\zeta )=\\overline{\\theta (\\bar{\\zeta })}$ .", "Now let $C^{\\prime }_j$ , $j=1,...,M$ , denote the reflection of the circle $C_j$ in the unit circle $C_0$ .", "One can readily verify that every $\\theta _j(\\zeta )$ maps the interior of the circle $C_j^{\\prime }$ onto the exterior of the circle $C_j$ .", "Thus, the set $\\Theta $ consisting of all compositions of the maps $\\theta _j(\\zeta )$ , $j=1,...,M$ , and their inverses defines a classical Schottky group [7].", "The region in the complex plane exterior to all of the circles $C_j$ and $C_j^{\\prime }$ is called the fundamental region, $F$ , of the Scotkky group $\\Theta $ .", "Given a Scotkky group $\\Theta $ , we can associate a Schottky-Klein prime function, denoted by $\\omega (\\zeta ,\\alpha )$ , for any two distinct points $\\zeta $ and $\\alpha $ in the fundamental region $F$ .", "The Schottky-Klein prime function has the following infinite product representation [7]: $\\omega (\\zeta ,\\alpha )=(\\zeta -\\alpha )\\prod _{\\theta \\in \\Theta ^{\\prime \\prime }} \\frac{(\\zeta -\\theta (\\alpha ))(\\alpha -\\theta (\\zeta ))}{(\\zeta -\\theta (\\zeta ))(\\alpha -\\theta (\\alpha ))},$ where the subset $\\Theta ^{\\prime \\prime } \\subset \\Theta $ consists of all compositions of the maps $\\theta _j(\\zeta )$ and $\\theta ^{-1}_j(\\zeta )$ , excluding the identity and taking only an element or its inverse (but not both).", "For example, if $\\theta _1(\\theta ^{-1}_2(\\zeta ))$ is included in the set $\\Theta ^{\\prime \\prime }$ , then $\\theta _2(\\theta ^{-1}_1(\\zeta ))$ must be excluded.", "Let us now recall some useful functional identities involving the Schottky-Klein prime function.", "Firstly, note that by definition the Schottky-Klein prime function is antisymmetric in its two arguments: $\\omega (\\zeta ,\\gamma )=-\\omega (\\gamma ,\\zeta ).$ Secondly, for the Schottky-Klein prime function associated with the circular domain $D_\\zeta $ the following functional relation $\\bar{\\omega }\\left(1/\\zeta ,1/\\gamma \\right)=-\\frac{1}{\\zeta \\gamma }\\omega (\\zeta ,\\gamma )$ holds [8].", "A third important relation of the Schottky-Klein prime function is as follows.", "Let $\\zeta _1$ , $\\zeta _2$ , $\\gamma _1$ , and $\\gamma _2$ be four points in $F$ , then we have $\\frac{\\omega (\\theta _j(\\zeta _1),\\gamma _1)}{\\omega (\\theta _j(\\zeta _2),\\gamma _2)}=\\frac{\\beta _j(\\gamma _1,\\gamma _2)}{\\beta _j(\\zeta _1,\\zeta _2)}\\left(\\frac{1-\\bar{\\delta }_j\\zeta _2}{1-\\bar{\\delta }_j\\zeta _1}\\right)\\frac{\\omega (\\zeta _1,\\gamma _1)}{\\omega (\\zeta _2,\\gamma _2)},$ where $\\beta _j(\\zeta ,\\gamma )=\\exp \\left[2 \\pi \\mathrm {i} (v_j(\\zeta )- v_j(\\gamma ))\\right], \\quad j=1,...,M.$ Equation (REF ) follows from a related expression given in ch.", "12 of Baker [7] for the ratio ${\\omega (\\theta _j(\\zeta ),\\gamma )}/{\\omega (\\zeta ,\\gamma )}$ ; see also related discussion in the monograph by Hejhal [9].", "Here the functions $\\lbrace v_j(\\zeta )~\|~j=1,...,M\\rbrace $ are the $M$ integrals of the first kind of the Riemann surface associated with the domain $D_\\zeta $ .", "These are analytic but not single-valued functions everywhere in $F$ .", "(They can be made single-valued by inserting cuts connecting each pair of circles $C_j$ and $C_j^{\\prime }$ ; see Baker [7].)", "Each function $v_j(\\zeta )$ has constant imaginary part on the circles $C_1,...,C_M$ , and zero imaginary part on the unit circle $C_0$ , that is, ${\\rm Im}[v_j(\\zeta )]=Q_{jl} \\qquad \\mbox{for}\\qquad \\zeta \\in C_l, \\ l=0,1,...,M,$ where $Q_{jl}$ is a real constant, with $Q_{j0}=0$ [10].", "As particular cases of (REF ) we have $\\frac{\\omega (\\theta _j(\\zeta ),\\gamma _1)}{\\omega (\\theta _j(\\zeta ),\\gamma _2)}=\\beta _j(\\gamma _1,\\gamma _2)\\frac{\\omega (\\zeta ,\\gamma _1)}{\\omega (\\zeta ,\\gamma _2)},$ $\\frac{\\omega (\\theta _j(\\zeta _1),\\gamma )}{\\omega (\\theta _j(\\zeta _2),\\gamma )}=\\frac{1}{\\beta _j(\\zeta _1,\\zeta _2)}\\left(\\frac{1-\\bar{\\delta }_j\\zeta _2}{1-\\bar{\\delta }_j\\zeta _1}\\right)\\frac{\\omega (\\zeta _1,\\gamma )}{\\omega (\\zeta _2,\\gamma )}.$ We also note for later use that for $\\zeta _1, \\zeta _2\\in C_l$ , $l=0,1,...,M$ , the function $\\beta _j(\\zeta ,\\gamma )$ satisfies the following relations $\|\\beta _j(\\zeta _1,\\zeta _2)\|&=1, \\\\\\beta _j(1/\\bar{\\zeta }_1,1/\\bar{\\zeta }_2)&={\\beta _j(\\zeta _1,\\zeta _2)}.$ Identity (REF ) follows immediately from (REF ) and (REF ).", "To derive (), first notice that from (REF ) we have $v_j(\\zeta )=\\overline{ v}_j(1/ \\zeta ),$ for $ \\zeta \\in C_0$ and everywhere else in $F$ by analytic continuation.", "On the other hand, for $\\zeta \\in C_l$ , $l=1,...,M$ , relation (REF ) implies that $v_j(\\zeta )- v_j(1/\\bar{\\zeta })&=2 \\mathrm {i} Q_{jl},$ where we have used (REF ).", "This, together with (REF ), implies ().", "Note furthermore that relations (REF ) and (REF ) trivially hold for $j=0$ , if we define $\\beta _0(\\gamma _1,\\gamma _2)\\equiv 1.$" ], [ "Radial slit maps", "In this section, two general classes of functions are defined as ratios of Schottky-Klein prime functions (or of products thereof) in such a way that they have constant arguments on the circles $C_j$ , $j=0,1,...,M$ .", "Because of this property, which will be extensively used in §, these functions represent radial slit maps defined on the circular domain $D_\\zeta $ .", "Here there are two cases to consider depending on whether the image radial slit domain is bounded or unbounded." ], [ "Bounded radial slit domains", "First define the functions $F_j(\\zeta ;\\zeta _1,\\zeta _2)=\\frac{\\omega (\\zeta ,\\zeta _1)}{\\omega (\\zeta ,\\zeta _2)}, \\qquad \\zeta _1,\\zeta _2\\in C_j, \\quad j=0,1,...,M.$ (These functions were introduced by Crowdy [4] as two separate classes of functions; here we adopt a somewhat different notation and give a unified treatment of them.)", "An important property of the functions above is that they have constant argument on all circles $C_l$ , $l=0,1,...,M$ .", "To see this, note that for $\\zeta \\in C_l$ one has $F_j(\\zeta ;\\zeta _1,\\zeta _2)&=\\frac{{\\omega ({\\theta _l}(1/\\bar{\\zeta }),\\zeta _1)}}{{\\omega ({\\theta _l}(1/\\bar{\\zeta }),\\zeta _2)}}&& \\mbox{[from (\\ref {eq:zbar})]}\\cr &={\\beta _l (\\zeta _1,\\zeta _2)}\\frac{\\omega (1/\\bar{\\zeta }, \\zeta _1)}{\\omega (1/\\bar{\\zeta }, \\zeta _2)}&& \\mbox{[from (\\ref {idrat2})]}\\cr &={\\beta _l (\\zeta _1,\\zeta _2)}\\frac{\\omega ({\\theta _j}(1/\\bar{\\zeta }_1),1/\\bar{\\zeta })}{\\omega ({\\theta _j}(1/\\bar{\\zeta }_2),1/\\bar{\\zeta })}&& \\mbox{[from (\\ref {refid0}) and (\\ref {eq:zbar})]}\\cr &=\\frac{\\beta _l (\\zeta _1,\\zeta _2) }{{\\beta _j(1/\\bar{\\zeta }_1,1/\\bar{\\zeta }_2)}}\\left(\\frac{1-\\bar{\\delta }_j/\\bar{\\zeta }_2}{1-\\bar{\\delta }_j/\\bar{\\zeta }_1}\\right)\\frac{\\omega (1/\\bar{\\zeta },1/\\bar{\\zeta }_1)}{\\omega (1/\\bar{\\zeta },1/\\bar{\\zeta }_2)}&& \\mbox{[from (\\ref {idrat3})]}\\cr &=\\frac{\\beta _l (\\zeta _1,\\zeta _2) }{{\\beta _j(\\zeta _1,\\zeta _2)}}\\left(\\frac{1-\\bar{\\delta }_j/\\bar{\\zeta }_2}{1-\\bar{\\delta }_j/\\bar{\\zeta }_1}\\right)\\frac{\\omega (1/\\bar{\\zeta },1/\\bar{\\zeta }_1)}{\\omega (1/\\bar{\\zeta },1/\\bar{\\zeta }_2)}.&& \\mbox{[from (\\ref {eq:barbeta})]}$ Taking complex conjugate and using (REF ) yields $\\overline{F_j(\\zeta ;\\zeta _1,\\zeta _2)}&=\\frac{\\overline{\\beta _l (\\zeta _1,\\zeta _2)}}{\\overline{\\beta _j(\\zeta _1,\\zeta _2)}}\\left(\\frac{\\zeta _2-\\delta _j}{\\zeta _1-\\delta _j}\\right)F_j(\\zeta ;\\zeta _1,\\zeta _2),$ which implies in view of (REF ) that $\\arg \\left[F_j(\\zeta ;\\zeta _1,\\zeta _2)\\right]= \\mbox{const.}", "\\quad \\mbox{for}\\quad \\zeta \\in C_l, \\ l=0,1,...,M.$ Since $F_j(\\zeta ;\\zeta _1,\\zeta _2)$ has constant argument on $C_l$ , $l=0,1,...,M$ , a simple zero at $\\zeta =\\zeta _1\\in C_j$ , and a simple pole at $\\zeta =\\zeta _2\\in C_j$ , it immediately follows that this function maps the circular domain $D_\\zeta $ onto a half-plane punctured along $M$ radial segments, where the circle $C_j$ is mapped to the axis containing the origin whose direction has argument $\\arg [F_j(\\zeta ;\\zeta _1,\\zeta _2)]$ and the other circles $C_l$ , $l\\ne j$ , are mapped to the slits.", "An alternative formula for this mapping in terms of an infinite product was obtained by DeLillo and Kropf [11].", "From the preceding discussion it is then clear that the conformal mapping defined by $\\lambda (\\zeta )&= -\\mathrm {i}\\, F_0(\\zeta ;1,-1)\\cr &= -\\mathrm {i}\\frac{\\omega (\\zeta ,1)}{\\omega (\\zeta ,-1)}$ maps the circular domain $D_\\zeta $ onto the upper half-$\\lambda $ -plane with $M$ radial slits excised from it, where the points $\\zeta =1$ and $\\zeta =-1$ are respectively mapped to the origin and infinity in the $\\lambda $ -plane, the unit circle maps to the real axis, and the inner circles map to the radial slits; see figures REF and REF .", "For later use, we quote here the derivative of mapping (REF ): $\\frac{d\\lambda }{d\\zeta } =-\\mathrm {i}\\frac{ \\omega _\\zeta (\\zeta ,1)\\omega (\\zeta , -1)-\\omega _\\zeta (\\zeta ,-1)\\omega (\\zeta , 1)}{ \\omega (\\zeta ,-1)^2},$ where $\\omega _\\zeta (\\zeta ,\\alpha )$ denotes the derivative of $\\omega (\\zeta ,\\alpha )$ with respect to its first argument." ], [ "Unbounded radial slit domains", "Now consider a second class of functions defined by the following ratio: $Q(\\zeta ;\\alpha ,\\beta )=\\frac{\\omega (\\zeta ,\\alpha )\\omega (\\zeta ,\\bar{\\alpha }^{-1})}{\\omega (\\zeta ,\\beta )\\omega (\\zeta ,\\bar{\\beta }^{\\,-1})},$ where $\\alpha $ and $\\beta $ are two arbitrary points in $D_\\zeta $ .", "Using arguments analogous to those employed in the §REF , it is easy to verify [4], [10] that $\\arg \\left[Q(\\zeta ;\\alpha ,\\beta )\\right]= \\mbox{const.}", "\\quad \\mbox{for}\\quad \\zeta \\in C_j, \\ j=0,1,...,M.$ It then follows that the mapping $\\lambda =Q(\\zeta ;\\alpha ,\\beta )$ conformally maps $D_\\zeta $ to the entire $\\lambda $ -plane cut along $M+1$ radial slits, where the point $\\zeta =\\alpha $ is mapped to the origin in the $\\lambda $ -plane, the point $\\zeta =\\beta $ is mapped to infinity, and each circle $C_j$ , $j=0,1,...,M$ , is mapped to a radial slit in the $\\lambda $ -plane.", "The functions $F_j(\\zeta ;\\zeta _1,\\zeta _2)$ and $Q(\\zeta ;\\alpha ,\\beta )$ defined above play an important role in constructing conformal mappings to multiply connected polygonal domains, as will become evident in the next section." ], [ "Conformal mappings to bounded polygonal domains", "In this section, we construct an explicit formula for the conformal mappings, $z(\\zeta )$ , from the bounded circular domain $D_\\zeta $ to a bounded multiply connected polygonal domain $D_z$ , using as an auxiliary tool the slit map $\\lambda (\\zeta )$ from $D_\\zeta $ to the upper half-$\\lambda $ -plane with $M$ radial slits.", "As explained in §REF , we first need to obtain an expression for the derivative $z_\\lambda (\\zeta )$ such that it has: i) the appropriate branch point at the prevertices $a_k^{(j)}$ and ii) constant argument on the circles $C_j$ .", "To this end, let $\\lbrace \\gamma _{1}^{(j)},\\gamma _{2}^{(j)} \\in C_j \| j=0,1,...,M\\rbrace $ be a set of arbitrary points on the circles $C_j$ .", "Using (REF ), (REF ) and (REF ), it is not difficult to show that the functions $\\omega (\\zeta ,\\gamma _{1}^{(0)}) \\omega (\\zeta ,\\gamma _{2}^{(0)}) \\prod _{k=1}^{n_0} \\left[\\omega (\\zeta ,a_{k}^{(0)})\\right]^{\\beta _k^{(0)}}$ and $\\frac{\\prod _{k=1}^{n_j} \\left[\\omega (\\zeta ,a_{k}^{(j)})\\right]^{\\beta _k^{(j)}}}{\\omega (\\zeta ,\\gamma _{1}^{(j)}) \\omega (\\zeta ,\\gamma _{2}^{(j)})}, \\qquad j=1,...,M,$ all have constant arguments on the circles $C_l$ , $l=0,1,...,M$ .", "Let us then write $\\frac{d z}{d\\lambda }=\\mathcal {B}R(\\zeta )\\left\\lbrace \\frac{ \\omega (\\zeta ,\\gamma _1^{(0)})\\omega (\\zeta , \\gamma _2^{(0)})}{\\prod _{j=1}^{M} \\omega (\\zeta ,\\gamma _{1}^{(j)}) \\omega (\\zeta ,\\gamma _{2}^{(j)})}\\prod _{j=0}^{M} {\\prod _{k=1}^{n_j} \\left[\\omega (\\zeta ,a_{k}^{(j)})\\right]^{\\beta _k^{(j)}}}\\right\\rbrace .$ where $\\mathcal {B}$ is a complex constant and $R(\\zeta )$ is a correction function to be determined later.", "Because $\\omega (\\zeta ,\\gamma )$ has a simple zero in $\\zeta =\\gamma $ , it is clear that $z_\\lambda (\\zeta )$ has the right branch point at $\\zeta =a_k^{(j)}$ ; see (REF ).", "Furthermore, it follows from the preceding discussion that (REF ) will have constant argument on the circles $C_j$ , so long as the function $R(\\zeta )$ does so.", "This requirement, together with additional constraints on $z_\\lambda (\\zeta )$ concerning the location of its zeros and poles, dictates the specific form of $R(\\zeta )$ , as shown next.", "First note that the points $\\lbrace \\gamma _{1}^{(j)},\\gamma _{2}^{(j)}\\in C_j\|j=1,...,M\\rbrace $ appearing in (REF ) must correspond to the preimages in the $\\zeta $ -plane of the end points of the respective slits in the $\\lambda $ -plane.", "This follows from the fact that $d\\lambda /d\\zeta $ vanishes at the preimages of the slit end points, whereas $dz/d\\lambda $ does not; hence, $z_\\lambda (\\zeta )$ must have simple poles at these points.", "More specifically, the points $\\gamma _{1}^{(j)}$ and $\\gamma _{2}^{(j)}$ , for $j=1,...,M$ , are obtained by computing the roots (on the circle $C_j$ ) of the following equation: $\\omega _\\zeta (\\zeta ,1)\\omega (\\zeta , -1)-\\omega _\\zeta (\\zeta ,-1)\\omega (\\zeta , 1)=0,$ which yields the zeros of $d\\lambda /d\\zeta $ ; see (REF ).", "Note, furthermore, that since $\\gamma _{1}^{(0)}$ and $\\gamma _{2}^{(0)}$ are arbitrary points on the unit circle at which $z_\\lambda (\\zeta )$ is nonzero, the terms containing these points in the numerator of (REF ) must be cancelled out by an appropriate choice of the function $R(\\zeta )$ .", "In addition, $R(\\zeta )$ must also produce a double zero for $z_\\lambda (\\zeta )$ at $\\zeta =-1$ , since $d\\lambda /d\\zeta $ has a double pole at this point; see (REF ).", "These requirements can be satisfied by choosing $R(\\zeta )$ of the form $R(\\zeta ) &= \\frac{\\omega (\\zeta ,-1)^2}{\\omega (\\zeta ,\\gamma _1^{(0)})\\omega (\\zeta ,\\gamma _2^{(0)})},$ which clearly has constant argument on the circles $C_j$ ; see (REF ).", "Inserting (REF ) into (REF ) yields $\\frac{d z}{d\\lambda }=\\mathcal {B}\\frac{ \\omega (\\zeta ,-1)^2}{\\prod _{j=1}^{M} \\omega (\\zeta ,\\gamma _{1}^{(j)}) \\omega (\\zeta ,\\gamma _{2}^{(j)})}\\prod _{j=0}^{M} {\\prod _{k=1}^{n_j} \\left[\\omega (\\zeta ,a_{k}^{(j)})\\right]^{\\beta _k^{(j)}}},$ which combined with (REF ) gives $\\frac{dz}{d \\zeta }&=\\mathcal {B} S(\\zeta )\\prod _{j=0}^{M} \\prod _{k=1}^{n_j} \\left[\\omega (\\zeta ,a_{k}^{(j)})\\right]^{\\beta _k^{(j)}},$ where $S(\\zeta )=\\frac{ \\omega _\\zeta (\\zeta ,1)\\omega (\\zeta , -1)-\\omega _\\zeta (\\zeta ,-1)\\omega (\\zeta ,1)}{\\prod _{j=1}^{M}\\omega (\\zeta ,\\gamma _{1}^{(j)}) \\omega (\\zeta ,\\gamma _{2}^{(j)})}.$ In (REF ) a constant factor ($-\\mathrm {i})$ has been absorbed into $\\cal B$ .", "Upon integrating (REF ), one then finds that the desired mapping $z(\\zeta )$ is given by $z(\\zeta )&=\\mathcal {A} +\\mathcal {B}\\int ^\\zeta S(\\zeta ^{\\prime }) \\prod _{j=0}^{M} \\prod _{k=1}^{n_j} \\left[\\omega (\\zeta ^{\\prime },a_{k}^{(j)})\\right]^{\\beta _k^{(j)}} d\\zeta ^{\\prime }.$ where $\\mathcal {A}$ and $\\mathcal {B}$ are complex constants.", "Recall that the points $\\lbrace \\gamma _{1}^{(j)}, \\gamma _{2}^{(j)}\\in C_j\|j=1,...,M\\rbrace $ appearing in the prefactor function $S(\\zeta )$ in (REF ) are to be determined by solving equation (REF ), which in turn depends on the conformal moduli $\\lbrace \\delta _j, q_j\|j=1,...,M$ } of the domain $D_\\zeta $ .", "For a given polygonal domain $D_z$ , the conformal moduli of the domain $D_\\zeta $ are not known a priori and must be determined simultaneously with the other parameters appearing in (REF ), namely, the prevertices $\\lbrace a_k^{(j)}\\in C_j\|j=0,1,...,M; k=1,...,n_j\\rbrace $ and the constant $\\mathcal {B}$ .", "Solving this parameter problem [1] in general is a very difficult task.", "Fortunately, in many applications, the specific details of the target polygonal domain (e.g., the areas and centroids of the polygons $P_j$ , and the lengths of their respective edges) need not be known a priori.", "In such cases, one can freely specify the domain $D_\\zeta $ , all prevertices on the unit circle, and all but two prevertices on each inner circle $C_j$ , and then solve the reduced parameter problem associated with the orientation of the various polygons and the univalence of the mapping function $z(\\zeta )$ .", "For instance, formulae (REF ) and (REF ) were recently used by Green & Vasconcelos [12] to construct a conformal mapping from the circular domain $D_\\zeta $ to a degenerate polygonal domain consisting of a horizontal strip with $M$ vertical slits in its interior (this conformal mapping corresponds to the complex potential for multiple steady bubbles in a Hele-Shaw channel)." ], [ "Representation formulae using other canonical slit domains", "The procedure described in the previous section can be readily extended to other rectilinear slit domains $D_\\lambda $ in the subsidiary $\\lambda $ -plane, so long as the corresponding slit map $\\lambda (\\zeta )$ is known explicitly.", "For each choice of domain $D_\\lambda $ , a specific formula results for the prefactor $S(\\zeta )$ appearing in conformal mapping (REF ).", "As an illustration of our procedure, we derive below the respective expressions for the function $S(\\zeta )$ associated with two of the five canonical slit domains listed in the book of Nehari [13], namely: i) a circular disk with $M$ concentric circular-arc slits; and ii) an unbounded radial slit domain obtained by excising from the entire plane $M+1$ rectilinear slits pointing toward the origin.", "(Other canonical rectilinear slit domains can be treated in similar manner.)", "Let us first discuss the case of an auxiliary slit domain consisting of a disk with concentric circular-arc slits, as originally considered by Crowdy [4].", "Here the function $\\eta (\\zeta )=\\frac{ \\omega (\\zeta ,\\alpha )}{\|\\alpha \| \\omega (\\zeta ,\\bar{\\alpha }^{-1})},$ for $\\alpha \\in D_\\zeta $ , maps the circular domain $D_\\zeta $ onto the unit disc with $M$ concentric circular slits, where the point $\\zeta =\\alpha $ maps to the origin in the $\\eta $ -plane [4].", "Thus, the logarithmic transformation $\\lambda &=\\log \\eta (\\zeta ),$ with an appropriate choice of branch cut from $\\zeta =\\alpha $ to $\\zeta =1$ , maps $D_\\zeta $ to a domain $D_\\lambda $ in the $\\lambda $ -plane consisting of a semi-infinite strip bounded from the right by the line ${\\rm Re}[\\lambda ]=0$ and containing $M$ vertical slits in its interior, where the unit circle $C_0$ is mapped to the vertical edge of the strip and the circles $C_j$ , $j=1,...,M$ , are mapped to the vertical slits.", "As before, the points $\\lbrace \\gamma _{1}^{(j)},\\gamma _{2}^{(j)}\\in C_j\|j=1,...,M\\rbrace $ correspond to the preimages in the $\\zeta $ -plane of the end points of the slits in the $\\lambda $ -plane, so that $z_\\lambda (\\zeta )$ has simple poles at these points.", "Since the point $\\zeta =\\alpha $ is a logarithmic singularity of the slit map $\\lambda (\\zeta )$ , then $z_\\lambda (\\zeta )$ must have a simple zero at this point.", "Starting from (REF ) and in light of the preceding discussion, one readily concludes that in this case the correction function $R(\\zeta )$ can be chosen as $R(\\zeta )&= \\frac{\\omega (\\zeta ,\\alpha )\\omega (\\zeta ,\\bar{\\alpha }^{-1})}{\\omega (\\zeta ,\\gamma _1^{(0)})^2},$ which has constant argument on the circles $C_j$ , as follows from (REF ) and from the fact that $\\bar{\\gamma }_1^{(0)}=1/\\gamma _1^{(0)}$ .", "Inserting (REF ) into (REF ) and setting $\\gamma _2^{(0)}=\\gamma _1^{(0)}$ (recall that these points are arbitrary), we obtain $\\frac{dz}{d\\lambda }=\\mathcal {B}\\frac{ \\omega (\\zeta ,\\alpha )\\omega (\\zeta ,\\bar{\\alpha }^{-1})}{\\prod _{j=1}^{M} \\omega (\\zeta ,\\gamma _{1}^{(j)}) \\omega (\\zeta ,\\gamma _{2}^{(j)})}\\prod _{j=0}^{M} {\\prod _{k=1}^{n_j} \\left[\\omega (\\zeta ,a_{k}^{(j)})\\right]^{\\beta _k^{(j)}}},$ Using (REF ) and (REF ), one finds that the derivative $z_\\zeta (\\zeta )$ can be rewritten as in (REF ), where the prefactor function $S(\\zeta )$ now reads $S(\\zeta )=\\frac{ \\omega _\\zeta (\\zeta ,\\alpha )\\omega (\\zeta ,\\bar{\\alpha }^{-1})-\\omega _\\zeta (\\zeta ,\\bar{\\alpha }^{-1})\\omega (\\zeta , \\alpha )}{\\prod _{j=1}^{M} \\omega (\\zeta ,\\gamma _{1}^{(j)}) \\omega (\\zeta ,\\gamma _{2}^{(j)})},$ which recovers the result obtained by Crowdy [4].", "As a further illustration of the generality of our approach, consider now the case where the domain $D_\\lambda $ consists of the entire $\\lambda $ -plane with $M+1$ radial slits, denoted by $L_j$ , $j=0,1,...M$ .", "Recall that the corresponding slit map $\\lambda (\\zeta )$ in this case is given by (REF ) which has a simple pole at $\\zeta =\\beta $ ; hence $z_\\lambda (\\zeta )$ must have a double zero at this point.", "Note furthermore that the points $\\lbrace \\gamma _{1}^{(j)},\\gamma _{2}^{(j)}\\in C_j\|j=0,1,...,M\\rbrace $ must correspond to the preimages of the end points of the respective slits $L_j$ .", "This implies, in particular, that the prime functions containing the points $\\gamma _{1}^{(0)}$ and $\\gamma _{2}^{(0)}$ in the numerator of (REF ) must be replaced with identical terms in the denominator, since $z_\\lambda (\\zeta )$ must now have simple poles at these two points, as well as at $\\gamma _{1}^{(j)}$ and $\\gamma _{2}^{(j)}$ , $j=1,...,M$ .", "This can be accomplished with an appropriate choice of the function $R(\\zeta )$ , which must also produce the required double zero at $\\zeta =\\beta $ .", "Indeed, these requirements can be satisfied by choosing $R(\\zeta )&= \\left[\\frac{\\omega (\\zeta ,\\beta )\\omega (\\zeta ,\\bar{\\beta }^{-1})}{\\omega (\\zeta ,\\gamma _0^{(1)})\\omega (\\zeta ,\\gamma _0^{(2)})}\\right]^2.$ After inserting (REF ) into (REF ) and applying the chain rule, one finds that the prefactor $S(\\zeta )$ for this case is given by $S(\\zeta )=\\frac{ T(\\zeta )}{\\prod _{j=0}^{M} \\omega (\\zeta ,\\gamma _{1}^{(j)}) \\omega (\\zeta ,\\gamma _{2}^{(j)})},$ where $T(\\zeta )&= \\omega _\\zeta (\\zeta ,\\alpha )\\left[\\omega (\\zeta , \\beta )\\omega (\\zeta ,\\bar{\\alpha }^{-1})\\omega (\\zeta , \\bar{\\beta }^{-1})\\right]-\\omega _\\zeta (\\zeta ,\\beta )\\left[\\omega (\\zeta , \\alpha )\\omega (\\zeta ,\\bar{\\alpha }^{-1})\\omega (\\zeta , \\bar{\\beta }^{-1})\\right]\\cr &+\\omega _\\zeta (\\zeta ,\\bar{\\alpha }^{-1})\\left[\\omega (\\zeta ,\\alpha )\\omega (\\zeta , \\beta )\\omega (\\zeta , \\bar{\\beta }^{-1})\\right]-\\omega _\\zeta (\\zeta ,\\bar{\\beta }^{-1})\\left[\\omega (\\zeta , \\alpha )\\omega (\\zeta , \\beta )\\omega (\\zeta ,\\bar{\\alpha }^{-1})\\right].$ Similar expressions for the prefactor function $S(\\zeta )$ pertaining to other canonical slit domains can be readily obtained, but further details will not be presented here.", "It is to be emphasized that, in contrast to formula (REF ) for the upper half-plane with radial slits, the prefactor functions $S(\\zeta )$ obtained for other canonical slit domains have arbitrary parameters, e.g., the point $\\alpha $ in (REF ) and the points $\\alpha $ and $\\beta $ in (REF ), in the interior of the domain $D_\\zeta $ .", "To avoid this extra unnecessary complication, the formulation given in § should be preferred in applications; see § for further discussion on this point." ], [ "Conformal mappings to unbounded polygonal domains", "In this section we consider, for completeness, the problem of conformal mappings from the bounded circular domain $D_\\zeta $ to unbounded multiply connected polygonal regions, using the upper half-$\\lambda $ -plane with $M$ radial slits as our auxiliary rectilinear slit domain.", "Let the target domain $D_z$ in the $z$ -plane be the unbounded region exterior to $M+1$ nonoverlapping polygons $P_j$ , $j=0,1,...,M$ .", "We shall adopt the same notation as in Sec.", "to designate the vertices of the polygonal boundaries and the corresponding turning angles.", "Notice, however, that now we have $\\sum _{k=1}^{n_j}\\beta _k^{(j)}=2,\\qquad j=0,1,...,M.$ Here we wish to obtain a conformal mapping, $z(\\zeta )$ , from a bounded circular domain $D_\\zeta $ to the unbounded polygonal region $D_z$ , where each circle $C_j$ , $j=0,1,...,M$ , is mapped to a polygonal boundary $P_j$ and the point $\\zeta =\\zeta _\\infty $ is mapped to infinity.", "Employing a procedure similar to that used in § for bounded polygonal domains, analogous formula for the mapping of unbounded polygonal regions can be readily obtained.", "The starting point for constructing the desired mapping is the equation $\\frac{dz}{d\\lambda }=\\mathcal {B}R(\\zeta )\\frac{\\prod _{j=0}^{M} \\prod _{k=1}^{n_j} \\left[\\omega (\\zeta ,a_{k}^{(j)})\\right]^{\\beta _k^{(j)}}}{\\prod _{j=0}^{M} \\omega (\\zeta ,\\gamma _{1}^{(j)}) \\omega (\\zeta ,\\gamma _{2}^{(j)})},$ which is the counterpart of expression (REF ) used for bounded polygonal domains.", "Notice that in contrast with (REF ), the prime functions containing the points $\\gamma _{1}^{(0)}$ and $\\gamma _{2}^{(0)}$ appear in the denominator of (REF ) because now $\\sum _{k=1}^{n_0}\\beta _k^{(0)}=2$ .", "As before, the points $\\lbrace \\gamma _1^{(j)},\\gamma _2^{(j)}\\in C_j\|j=1,...,M\\rbrace $ are identified with the preimages in the $\\zeta $ -plane of the end points of the $M$ slits in the $\\lambda $ -plane, whereas $\\gamma _1^{(0)}$ and $\\gamma _2^{(0)}$ are arbitrary points on $C_0$ at our disposal.", "It is also clear from previous discussions that $z_\\lambda (\\zeta )$ must have a double pole at $\\zeta =\\zeta _\\infty $ and a simple zero at $\\zeta =-1$ .", "These requirements can be enforced by choosing $R(\\zeta )&= \\left[\\frac{\\omega (\\zeta ,-1)\\omega (\\zeta ,\\gamma _1^{(0)})}{\\omega (\\zeta ,\\zeta _\\infty )\\omega (\\zeta ,1/\\bar{\\zeta }_\\infty )}\\right]^2.$ After inserting this into (REF ) and setting $\\gamma _1^{(0)}=\\gamma _2^{(0)}$ , one finds $\\frac{dz}{d\\lambda }=\\mathcal {B}\\frac{\\omega (\\zeta ,-1)^2}{\\left[\\omega (\\zeta ,\\zeta _\\infty )\\omega (\\zeta ,1/\\bar{\\zeta }_\\infty )\\right]^2\\prod _{j=1}^{M} \\omega (\\zeta ,\\gamma _{1}^{(j)}) \\omega (\\zeta ,\\gamma _{2}^{(j)})}\\prod _{j=0}^{M} \\prod _{k=1}^{n_j} \\left[\\omega (\\zeta ,a_{k}^{(j)})\\right]^{\\beta _k^{(j)}}.$ Upon using (REF ), this becomes $\\frac{dz}{d \\zeta }=\\mathcal {B} S_\\infty (\\zeta )\\prod _{j=0}^{M} \\prod _{k=1}^{n_j} \\left[\\omega (\\zeta ,a_{k}^{(j)})\\right]^{\\beta _k^{(j)}},$ where $S_\\infty (\\zeta ) =\\frac{S(\\zeta )}{\\left[\\omega (\\zeta ,\\zeta _\\infty )\\omega (\\zeta ,1/\\bar{\\zeta }_\\infty )\\right]^2},$ with $S(\\zeta )$ as given in (REF ).", "After integrating (REF ), one finds that the conformal mapping $z(\\zeta )$ from $D_\\zeta $ to an unbounded multiply connected polygonal region is given by the same integral expression (REF ) obtained for the case of bounded polygonal domains, the only difference being that the prefactor is now given by the function $S_\\infty (\\zeta )$ shown in (REF ).", "This property was first noticed by Crowdy [5], who obtained a conformal mapping from $D_\\zeta $ to an unbounded polygonal region by implicitly considering an auxiliary rectilinear slit domain consisting of a semi-infinite strip with vertical slits.", "As shown above, relation (REF ) holds irrespective of the choice of the rectilinear slit domain used to construct the corresponding mapping formulas for bounded and unbounded multiply connected polygonal domains." ], [ "Discussion", "A general framework has been presented for constructing conformal mappings from a bounded circular domain $D_\\zeta $ to a multiply connected polygonal region $D_z$ (either bounded or unbounded).", "A key ingredient in our scheme is the introduction of a conformal mapping from $D_\\zeta $ to a rectilinear slit domain $D_\\lambda $ in a subsidiary $\\lambda $ -plane.", "This allows us to write an explicit formula for the derivative $z_\\lambda (\\zeta )$ , and hence for $z_\\zeta (\\zeta )$ , in terms of the Schottky-Klein prime function associated with the domain $D_\\zeta $ .", "After integration, the desired conformal mapping $z(\\zeta )$ is then obtained as an indefinite integral whose integrand consists of a product of powers of the Schottky-Klein prime functions and a prefactor function $S(\\zeta )$ that depends on the choice of the rectilinear slit domain $D_\\lambda $ .", "An explicit formula for $S(\\zeta )$ was derived by first considering the case where the rectilinear slit domain $D_\\lambda $ consists of the upper half-plane with radial slits.", "The generality of our approach was subsequently demonstrated by obtaining alternative formulae for the prefactor function $S(\\zeta )$ pertaining to other canonical slit domains.", "For a given polygonal domain $D_z$ , these various formulas (once their associated parameters have been determined) provide different representations of the same conformal mapping $z(\\zeta )$ .", "It is to be noted, however, that the formula for $S(\\zeta )$ obtained by considering the upper half-plane with radial slits is arguably the simplest one, in the sense that the only unknown parameters are the zeros of the slit map $\\lambda (\\zeta )$ , which can be numerically computed once the domain $D_\\zeta $ is specified.", "By contrast, the corresponding formulae for $S(\\zeta )$ obtained for other canonical slit domains have, in addition, one or more arbitrary parameters inside the domain $D_\\zeta $ .", "Although the function $S(\\zeta )$ does not ultimately depend on the values of these parameters (except for an overall factor independent of $\\zeta $ ; see Crowdy [14]), the existence of arbitrary parameters inside $D_\\zeta $ may present an additional (and unnecessary) source of complication.", "This is particularly true in the case that a given target polygonal domain $D_z$ is specified, for here the conformal moduli of the domain $D_\\zeta $ are not known a priori and hence the arbitrary parameters cannot be fixed beforehand.", "In light of the foregoing discussion, it can be argued that the mapping formula derived in § using the upper half-plane with radial slits should be viewed as the natural extension to multiply connected polygonal domains of the standard Schwarz-Christoffel mapping from the upper half-plane to a simply connected polygonal region.", "It should also be preferable in applications because of its simplicity.", "In this context, it is worth noting that this mapping formula was recently employed by Green & Vasconcelos [12] to construct exact solutions for multiple bubbles steadily moving in a Hele-Shaw channel.", "It is thus hoped that other problems involving multiply connected polygonal domains may be conveniently tackled with the formalism presented here." ], [ "Acknowledgments", "The author wishes to thank C. C. Green and D. G. Crowdy for helpful discussions.", "He is also appreciative of the hospitality of the Department of Mathematics at Imperial College London (ICL), where this research was carried out.", "Financial support from a scholarship from the Conselho Nacional de Desenvolvimento Cientifico e Tecnologico (Brazil) for a sabbatical stay at ICL is acknowledged." ] ]	1403.0423
[ [ "Quantum delayed-choice experiment with a beam splitter in a quantum\n superposition" ], [ "Abstract A quantum system can behave as a wave or as a particle, depending on the experimental arrangement.", "When for example measuring a photon using a Mach-Zehnder interferometer, the photon acts as a wave if the second beam-splitter is inserted, but as a particle if this beam-splitter is omitted.", "The decision of whether or not to insert this beam-splitter can be made after the photon has entered the interferometer, as in Wheeler's famous delayed-choice thought experiment.", "In recent quantum versions of this experiment, this decision is controlled by a quantum ancilla, while the beam splitter is itself still a classical object.", "Here we propose and realize a variant of the quantum delayed-choice experiment.", "We configure a superconducting quantum circuit as a Ramsey interferometer, where the element that acts as the first beam-splitter can be put in a quantum superposition of its active and inactive states, as verified by the negative values of its Wigner function.", "We show that this enables the wave and particle aspects of the system to be observed with a single setup, without involving an ancilla that is not itself a part of the interferometer.", "We also study the transition of this quantum beam-splitter from a quantum to a classical object due to decoherence, as observed by monitoring the interferometer output." ], [ "Quantum circuit and qubit-resonator interaction control", "The quantum circuit used to implement our quantum delayed-choice experiment consists of two phase qubits coupled to a superconducting coplanar waveguide resonator, as shown in Fig.", "S.1.", "The sample, as described in Ref.", "1, involves four qubits, two of which are not used in our experiment and not shown in the circuit diagram.", "The resonator has a fixed frequency of $6.205$ GHz, while the frequency of each qubit is tunable through a flux bias coil.", "Neglecting the effects of higher levels of the qubits, the qubit-resonator interaction Hamiltonian in the interaction picture is given by $H=\\hbar \\sum _{j=1}^2\\Omega _j\\left( e^{i\\Delta _jt}\\sigma _j^{+}a+e^{-i\\Delta _jt}\\sigma _j^{-}a^{\\dagger }\\right),(S1)$ where $\\sigma _j^{+}$ and $\\sigma _j^{-}$ are the raising and lowering operators for qubit $j$ , $a$ and $a^{\\dagger }$ the annihilation and creation operators of the field in the resonator, and $\\Omega _j$ is the coupling strength between qubit $j$ and the resonator with the detuning $\\Delta _j$ .", "The Hamiltonian describes the coherent energy exchange between the qubits and the resonator.", "When $\\Delta _j\\gg \\Omega _j$ the photon transfer probability between qubit $j$ with the resonator is negligible, implying their interaction is effectively switched off.", "On the other hand, when $\\Delta _j=0$ and $\\Delta _k\\gg \\Omega _k$ ($j,k=1,2$ and $j\\ne k$ ) the interaction between qubit $j$ and the resonator is described by the Jaynes-Cummings Hamiltonian.", "The qubit frequency tunability allows us to freely control the qubit-resonator interaction.", "Figure: (Color online) Circuit schematic.", "Two phase qubits arecoupled to a superconducting coplanar waveguide resonator viacapacitors.", "The detuning between each qubit and the resonator isadjusted through a flux bias coil, enabling the relevantqubit-resonator interaction to be effectively switched on and off.External microwave pulses are coupled to the qubits through the fluxbias coil, and to the resonator through the capacitor on the leftside of the resonator.", "The couplingstrengths of the resonator to the two qubits are 2π×17.52\\pi \\times 17.5 MHz and 2π×17.72\\pi \\times 17.7 MHz, respectively.", "The energy relaxation times forthese two qubits are 520 ns and 560 ns, respectively, and theRamsey dephasing times for both qubits are about 150 ns.", "Theenergy relaxation time of the resonator is 3.0 μ\\mu s withoutmeasurable dephasing." ], [ "The first Ramsey pulse with a coherent field in the\nresonator", "The coherent field state $\\left\| \\alpha \\right\\rangle $ stored in the resonator can be expressed as a superposition of photon number states: $\\left\| \\alpha \\right\\rangle =\\sum _{n=0}^\\infty C_n\\left\|n\\right\\rangle $ , where $C_n=\\exp \\left( -\\left\| \\alpha \\right\|^2/2\\right) \\alpha ^n/\\sqrt{n!}", "$ is the probability amplitude for having $n$ photons.", "The Rabi oscillation frequency associated with the photon-number state $\\left\| n\\right\\rangle $ is $\\sqrt{n}\\Omega $ .", "After an interaction time $t_\\alpha =\\pi /(4\\left\| \\alpha \\right\|\\Omega )$ the state of the qubit-resonator system evolves to $\\left\| \\Psi \\right\\rangle =\\sum _{n=0}^\\infty [ C_n\\cos (\\sqrt{n}\\Omega t_\\alpha )\| g\\rangle -iC_{n+1}\\sin (\\sqrt{n+1}\\Omega t_\\alpha ) \| e\\rangle ] \|n\\rangle .\\\\ (S2) $ The coherent field has a Poissonian photon-number distribution, with the mean photon number $\\stackrel{-}{n}=\\left\| \\alpha \\right\| ^2$ and variance $\\Delta n=\\left\| \\alpha \\right\| $ .", "When the field amplitude is large, $\\Delta n$ is much smaller than $\\stackrel{-}{n}$ .", "In this case, we have $C_{n+1}/C_n=\\alpha /\\sqrt{n+1}\\simeq e^{i\\vartheta }$ and $\\sqrt{n}\\Omega t_\\alpha \\simeq \\left\| \\alpha \\right\| \\Omega t_\\alpha =\\pi /4$ , where $\\vartheta $ is the argument of the complex amplitude $\\alpha $ .", "For $\\vartheta =0$ , the total state $\\left\| \\Psi \\right\\rangle $ is approximately a product of the qubit state $(\\left\| g\\right\\rangle -i\\left\| e\\right\\rangle )/\\sqrt{2}$ with the field state $\\left\|\\alpha \\right\\rangle $ ." ], [ "Experimental sequence for Ramsey interference", "The state $\\left\| \\psi _{b,i}\\right\\rangle $ of Eq.", "(3) of the main text with $\\varphi \\ne 0$ , $\\pi /2$ is generated by coherently pumping photons into the resonator one by one through the ancilla qubit, emploiting an algorithm theoretically proposed in Ref.", "2.", "Experimental implementation of this algorithm in a superconducting resonator involves alternative, well-controlled qubit drive operations and qubit-resonator swap operations, as detailedly described in Refs.", "3 and 4 and illustrated in Fig.", "S2.", "Here the qubit drive operation is achieved by applying a resonant microwave pulse through the flux bias coil.", "To decrease the reasonable cutoff in the Fock-state expansion, we first generate the supersotion state $\\mathcal {N}(\\cos \\varphi \\left\| \\alpha /2\\right\\rangle -\\sin \\varphi \\left\|-\\alpha /2\\right\\rangle )$ with $\\alpha =2$ (the cutoff is $n=4$ ), and then displace it in phase space by an amount $\\alpha /2$ , achieving the state $\\left\| \\psi _{b,i}\\right\\rangle $ .", "The displacement operation is performed using a microwave pulse capacitively coupled to the resonator.", "For $\\varphi =0$ , we directly generate $\\left\| \\psi _{b,i}\\right\\rangle $ from the vacuum state by performing the displacement operation $D(\\alpha )$ .", "We produce the states $\\left\| \\psi _{b,i}\\right\\rangle $ for $\\varphi =0$ , $\\pi /8$ , $\\pi /4$ , $3\\pi /8$ , $\\pi /2$ , with the fidelities being $0.898\\pm 0.023$ , $0.702\\pm 0.021$ , $0.726\\pm 0.028$ , $0.760\\pm 0.017$ , and $0.992\\pm 0.004$ , respectively.", "After preparation of $\\left\| \\psi _{b,i}\\right\\rangle $ , we achieve the first pulse R$_1$ by turning on the interaction between the test qubit and the resonator for a time t$_\\alpha $ .", "For the observation of the Ramsey signal displayed in Fig.", "2a of the main text, R$_1$ is directly followed by a phase-tunable, on-resonance microwave$\\pi /2$ pulse R$_2(\\theta )$ applied to the qubit.", "For different $\\varphi $ , the Ramsey interference pattern is constructed by measuring the resulting excited-state probability of the test qubit as a function of $\\theta $ .", "To obtain the result shown in Fig.", "2b of the main text, after R$_1$ an aditional iSwap gate is applied to the resonator field and the ancilla, as shown in Fig.", "S.2.", "The wave and particle behaviors of the test qubit is post-selected by detecting the state of the ancilla." ], [ "Measurement of Wigner functions", "To reconstruct the WF of the reduced density operator of the resonator field without reading-out the state of the test qubit, we perform the displacement operation $D(-\\chi )$ after the second Ramsey pulse, and then let the ancilla qubit initially in the ground state interact with the resonator field for a variable time $\\tau $ , followed by the measurement of the state of the ancilla.", "The measured probability $P_e(\\tau )$ for the ancilla being in the excited state, as a function of $\\tau $ , is used to infer the diagonal elements of the displaced density matrix of the resonator field and hence the value of the WF at point $\\chi $ in phase space [$3,4$ ].", "To map out the WF of the resonator field associated with the states $\\left\| g\\right\\rangle $ and $\\left\| e\\right\\rangle $ of the test qubit, we perform the joint qubit-resonator tomography, which requires reading-out both the test and ancilla qubits simultaneously, as described in Ref.", "5.", "In Fig.", "S3, we present the measured WFs of the field state in the resonator for $\\varphi =\\pi /8$ , $3\\pi /8$ .", "Panel a (d) shows the WFs when the qubit state is traced out, while b (e) and c (f) exhibit the WFs associated with the outcomes $\\left\| g\\right\\rangle $ and $\\left\| e\\right\\rangle $ for $\\varphi =\\pi /8$ ($3\\pi /8$ ), respectively.", "In each panel, the simulated and measured WFs are shown in the upper and lower rows, respectively.", "Experimental imperfections are not included in numerical simulation.", "As expected, due to the quantum coherence between the present and absent states of the QBS the WF for each case exhibits a nonclassical feature around $\\chi =1$ .", "When the qubit state is measured, the quantum interference and hence the negativity of the WF are enhanced.", "These results, together with those shown in Fig.", "2 of the main text, reveal the quantum nature of the QBS for a wide range of the parameter $\\varphi $ ." ], [ "Observing transition from quantum to classical beam splitter", "Another benefit of this experimental implementation is that it allows the observation of the transition from a quantum to a classical beam splitter.", "To demonstrate this, we now delay the interaction between the test qubit and the resonator for a time $T$ after the QBS has been prepared in the cat state $\\left\| \\psi _{b,i}\\right\\rangle $ .", "Then, just before the test qubit-resonator interaction, the field density operator is given by $[6]$ $\\rho _b &=&\\mathcal {N}^2[\\cos ^2\\varphi \|\\alpha ^{\\prime }\\rangle \\langle \\alpha ^{\\prime }\|+\\sin ^2\\varphi \|0\\rangle \\langle 0\|\\\\&&-\\frac{1}{2}e^{-\|\\alpha \|^2(1-e^{-\\gamma T})/2}\\sin (2\\varphi )(\|\\alpha ^{\\prime }\\rangle \\langle 0\|+\|0\\rangle \\langle \\alpha ^{\\prime }\|)],(S3) $ where $\\alpha ^{\\prime }=\\alpha e^{-\\gamma T/2}$ and $\\gamma $ is the decay rate of a photon in the resonator; we have ignored imperfections in the cat state preparation.", "In our experimental setup the single-photon lifetime is $\\tau =1/\\gamma \\simeq 3.0~\\mu $ s. The qubit-resonator interaction time is set to $t_{\\alpha ^{\\prime }}=\\pi /\\left( 4\\alpha ^{\\prime }\\Omega \\right) $ , so that after R$_2(\\theta )$ the state component $\|\\alpha ^{\\prime }\\rangle \|g\\rangle $ approximately evolves to $\|\\alpha ^{\\prime }\\rangle \|\\psi _w\\rangle $ , while $\|0\\rangle \|g\\rangle $ evolves to $\|0\\rangle \|\\psi _p\\rangle $ .", "Figure: (Color online) Transition from a quantum to a classicalbeam splitter.", "(a) Measured Ramsey signals for different delays TT,defined as the interval between the end of the cat state preparationand the start of the qubit-resonator interaction.", "The parameters arethe same as those in Fig. S3.", "(b) Absolute value of thenegative-valued minimum of the WF of the QBS state after R 2 _2,without reading out the test qubit state, displayed versus TT.", "Thedamping of the quantum coherence of the QBS is characterized by thedecrease in the negativity of the WF.", "Error bars indicate thestatistical variance.", "The line is a master-equation simulationtaking the prepared cat state as the initial state for thedecoherence evolution.", "The resonator single-photon lifetime isaround 3.0 μ\\mu s, with negligible pure dephasing.In Fig.", "S4(a) we display the measured Ramsey interference signals for $\\varphi =\\pi /4$ with different delays $T$ , showing that the fringe contrast is insensitive to the field decay, as expected.", "However, the quantum coherence between the active and inactive states of the QBS degrades much faster due to decoherence.", "Figure S4(b) shows the negative-valued minimum value of the WF as a function of delay $T$ , with the WF measured after R$_2(\\theta )$ but without reading out the test qubit state.", "The blue symbols are the measured results, whereas the red curve represents the simulated decay, taking the prepared cat state as the initial state.", "For $\\gamma T=1/3$ , the amplitude of the coherent field is reduced by only $15\\%$ , but the absolute value of the minimum of the WF decays almost to zero, revealing the quick damping of the quantum coherence, which can be defined as the sum of the off-diagonal elements of the third term of Eq.", "(S3) in the photon-number representation $[7]$ .", "Equation (S3) predicts that the quantum coherence is shrunk by a factor of about 0.51 after this delay.", "With a cat state of larger size and higher fidelity, as has been achieved in $[3,4,8,9]$ , one could observe further decay of the quantum coherence of the QBS." ] ]	1403.0203
[ [ "Equidistribution of saddle periodic points for Henon-type automorphisms\n of C^k" ], [ "Abstract In this paper, we prove the equidistribution of saddle periodic points for Henon-type automorphisms of C^k with respect to it equilibrium measure.", "A general strategy to obtain equidistribution properties in any dimension is presented.", "It is based on our recent theory of densities for positive closed currents.", "Several fine properties of dynamical currents are also proved." ], [ "Introduction", "Let $f$ be a polynomial automorphism of $k$ .", "We extend it to $\\mathbb {P}^k$ as a bi-rational self-map that we still denote by $f$ .", "Let $I_+$ denote the indeterminacy set of $f$ and $I_-$ the one for the inverse $f^{-1}$ of $f$ .", "They are contained in the hyperplane at infinity $H_\\infty :=\\mathbb {P}^k\\setminus k$ .", "We assume that $f$ is not an automorphism of $\\mathbb {P}^k$ because otherwise its dynamics is elementary.", "So the indeterminacy sets $I_+$ and $I_-$ are non-empty.", "The following notion was introduced by the second author in [24].", "Definition 1.1 We say that $f$ is a Hénon map or a regular automorphism if $I_+\\cap I_-=\\varnothing .$ The interesting point here is that the last condition is quite simple to check and Hénon automorphisms form a rich family of non-uniformly hyperbolic dynamical systems for which we can develop a satisfactory theory.", "In dimension 2, all dynamically interesting automorphisms of 2 are conjugated to Hénon maps, see Friedland-Milnor [21] and also [15], [19], [20].", "Assume now that $f$ is a Hénon map on $k$ .", "We first recall some basic properties of $f$ and refer to the papers by de Thélin [6] and the authors [8], [12], [24] for details.", "Let $d_\\pm $ denote the algebraic degrees of $f^{\\pm 1}$ .", "There is an integer $1\\le p\\le k-1$ such that $\\dim I_+=k-p-1$ , $\\dim I_-=p-1$ and $d_+^p=d_-^{k-p}\\ge 2$ .", "We define the main dynamical degree of $f$ as $d:=d_+^p=d_-^{k-p}$ .", "This is also the main dynamical degree of $f^{-1}$ and the topological entropies of $f$ and $f^{-1}$ are both equal to $\\log d$ .", "The restrictions of $f$ and its inverse to the hyperplane at infinity $H_\\infty $ satisfy $f(H_\\infty \\setminus I_+)=f(I_-)=I_- \\quad \\text{and}\\quad f^{-1}(H_\\infty \\setminus I_-)=f^{-1}(I_+)=I_+.$ Let $K_+$ (resp.", "$K_-$ ) be the set of points $z\\in k$ such that the orbits $(f^n(z))_{n\\ge 0}$ (resp.", "$(f^{-n}(z))_{n\\ge 0}$ ) are bounded in $k$ .", "They are closed in $k$ and we have $\\overline{K}_\\pm =K_\\pm \\cup I_\\pm .$ The indeterminacy sets $I_-$ and $I_+$ are attracting respectively for $f$ and $f^{-1}$ .", "Their basins are equal to $\\mathbb {P}^k\\setminus \\overline{K}_+$ and $\\mathbb {P}^k\\setminus \\overline{K}_-$ .", "The intersection $K:=K_+\\cap K_-$ is compact in $k$ .", "It is the set of points $z$ whose entire orbits $(f^n(z))_{n\\in \\mathbb {Z}}$ are bounded in $k$ .", "Let $\\omega _{\\rm FS}$ denote the Fubini-Study $(1,1)$ -form on $\\mathbb {P}^k$ so normalized that the integral of the top power $\\omega _{\\rm FS}^k$ on $\\mathbb {P}^k$ is 1.", "The following weak limits exist $\\tau _\\pm :=\\lim _{n\\rightarrow \\infty } d_\\pm ^{-n} (f^{\\pm n})^(\\omega _{\\rm FS})$ and define two positiveThroughout the paper, the positivity of $(p,p)$ -currents is in the strong sense.", "closed $(1,1)$ -currents of mass 1 on $\\mathbb {P}^k$ .", "We have $f^(\\tau _+)=d_+\\tau _+$ and $f_(\\tau _-)=d_-\\tau _-$ .", "The currents $\\tau _+$ and $\\tau _-$ have Hölder continuous local potentials outside $I_+$ and $I_-$ respectively.", "They are called the Green $(1,1)$ -currents of $f$ and $f^{-1}$ .", "The positive closed currents $T_+:=\\tau _+^p$ and $T_-:=\\tau _-^{k-p}$ are respectively the main Green currents of $f$ and $f^{-1}$ .", "The current $T_+$ is the unique positive closed $(p,p)$ -current of mass 1 in $\\mathbb {P}^k$ with support in $\\overline{K}_+$ and the current $T_-$ is the unique positive closed $(k-p,k-p)$ -current of mass 1 in $\\mathbb {P}^k$ with support in $\\overline{K}_-$ .", "The wedge-product $\\mu :=T_+\\wedge T_-=\\tau _+^p\\wedge \\tau _-^{k-p}$ is a well-defined invariant probability measure with support in $K$ .", "It turns out that $\\mu $ is the unique invariant probability measure of maximal entropy $\\log d$ of $f$ and $f^{-1}$ .", "The measure $\\mu $ is moreover exponentially mixing and hyperbolic.", "It is called the Green measure or equilibrium measure of $f$ and $f^{-1}$ .", "In this paper, we give the proof that saddle periodic points are equidistributed with respect to $\\mu $ .", "Denote by $P_n$ the set of periodic points of period $n$ of $f$ in $k$ and $SP_n$ the set of those which are saddles.", "For any number $0<\\epsilon <1$ , denote by $SP_n^\\epsilon $ the set of saddle periodic points $a$ of period $n$ in $k$ such that the differential $Df^n(a)$ admits exactly $p$ eigenvalues of modulus larger than $(d_+-\\epsilon )^{n/2}$ and $k-p$ eigenvalues of modulus smaller than $(d_--\\epsilon )^{-n/2}$ .", "Here the eigenvalues are counted with multiplicity.", "They do not depend on the choice of coordinate system on $k$ .", "We have the following theorem, see Bedford-Lyubich-Smillie [2] for the case of dimension $k=2$ and [23] for a $p$ -adic version independently obtained by Lee.", "The main result by Lee may offer an arithmetic approach this this problem by taking $p\\rightarrow \\infty $ .", "Theorem 1.2 Let $f,d, \\mu ,P_n,SP_n$ and $SP_n^\\epsilon $ be as above.", "Let $Q_n$ be either $P_n, SP_n$ or $SP_n^\\epsilon $ .", "Then $ d^{-n} \\sum _{a\\in Q_n} \\delta _a\\rightarrow \\mu $ as $n$ goes to infinity, where $\\delta _a$ denotes the Dirac mass at $a$ .", "The proof of this result is developed in the rest of the paper.", "A key point is the use of our theory of densities of positive closed currents developed in [14].", "We refer to that paper for basic notations and results concerning tangent cones, the notion of density and the intersection of currents in a weak sense.", "We will describe below our strategy which, as far as we know, is the first approach to obtain the equidistribution of periodic points for a non-uniformly hyperbolic holomorphic system with arbitrary numbers of stable and unstable directions.", "The main ideas are quite general and can be adapted to other meromorphic dynamical systems and other questions, see also Remark below.", "Let $\\Delta $ denote the diagonal of $\\mathbb {P}^k\\times \\mathbb {P}^k$ and $\\Gamma _n$ denote the compactification of the graph of $f^n$ in $\\mathbb {P}^k\\times \\mathbb {P}^k$ .", "The set $P_n$ can be identified with the intersection of $\\Gamma _n$ and $\\Delta $ in $k\\times k$ .", "The dynamical system associated with the map $F:=(f,f^{-1})$ on $\\mathbb {P}^k\\times \\mathbb {P}^k$ is similar to the one associated with Hénon-type maps on $\\mathbb {P}^k$ .", "It was used by the first author in [8] in order to obtain the exponential mixing of $\\mu $ on $k$ .", "Observe that $\\Gamma _n$ is the pull-back of $\\Delta $ or $\\Gamma _1$ by $F^{n/2}$ or $F^{(n-1)/2}$ .", "So a property similar to the uniqueness of the main Green currents mentioned above allows us to prove that the positive closed $(k,k)$ -current $d^{-n}[\\Gamma _n]$ converges to the main Green current of $F$ which is equal to $T_+\\otimes T_-$ .", "Therefore, since the measure $\\mu =T_+\\wedge T_-$ can be identified with $[\\Delta ]\\wedge (T_+\\otimes T_-)$ , Theorem REF is equivalent to $\\lim _{n\\rightarrow \\infty } [\\Delta ]\\wedge d^{-n}[\\Gamma _n]=[\\Delta ]\\wedge \\lim _{n\\rightarrow \\infty } d^{-n}[\\Gamma _n]$ on $k\\times k$ .", "So our result requires the development of a good intersection theory in any dimension.", "The typical difficulty is illustrated in the following example.", "Consider $\\Delta ^{\\prime }$ the unit disc in $\\lbrace 0\\rbrace \\subset 2$ and $\\Gamma _n^{\\prime }$ the graph of the function $x\\mapsto x^{d^n}$ over $\\Delta ^{\\prime }$ .", "The currents $d^{-n}[\\Gamma _n^{\\prime }]$ converge to a current on the vertical boundary of the unit bidisc in 2 while their intersection with $[\\Delta ^{\\prime }]$ is the Dirac mass at 0.", "So we have $\\lim _{n\\rightarrow \\infty } [\\Delta ^{\\prime }]\\wedge d^{-n}[\\Gamma _n^{\\prime }]\\ne [\\Delta ^{\\prime }]\\wedge \\lim _{n\\rightarrow \\infty } d^{-n}[\\Gamma _n^{\\prime }].$ We see in this example that $\\Gamma _n^{\\prime }$ is tangent to $\\Delta ^{\\prime }$ at 0 with maximal order.", "We can perturb $\\Gamma _n^{\\prime }$ in order to get manifolds which intersect $\\Delta ^{\\prime }$ transversally but the limit of their intersections with $\\Delta ^{\\prime }$ is still equal to the Dirac mass at 0.", "In fact, this phenomenon is due to the property that some tangent lines to $\\Gamma _n^{\\prime }$ are too close to tangent lines to $\\Delta ^{\\prime }$ .", "It is not difficult to construct a map $f$ such that $\\Gamma _n$ is tangent or almost tangent to $\\Delta $ at some points for every $n$ .", "In order to handle the main difficulty in our problem, the strategy is to show that the almost tangencies become negligible when $n$ tends to infinity.", "This property is translated in our study into the fact that a suitable density for positive closed currents vanishes.", "Then, a geometric approach developed in [7] allows us to obtain the result.", "We will give the details in the second part of this article.", "We explain now the notion of density of currents in the dynamical setting and then develop the theory in the general setting of arbitrary positive closed currents.", "Let ${\\rm Gr}(\\mathbb {P}^k\\times \\mathbb {P}^k,k)$ denote the Grassmannian bundle over $\\mathbb {P}^k\\times \\mathbb {P}^k$ where each point corresponds to a pair $(x,[v])$ of a point $x\\in \\mathbb {P}^k\\times \\mathbb {P}^k$ and the direction $[v]$ of a simple tangent $k$ -vector $v$ of $\\mathbb {P}^k\\times \\mathbb {P}^k$ at $x$ .", "Let $\\widetilde{\\Gamma }_n$ denote the set of points $(x,[v])$ in ${\\rm Gr}(\\mathbb {P}^k\\times \\mathbb {P}^k,k)$ with $x\\in \\Gamma _n$ and $v$ a $k$ -vector not transverse to $\\Gamma _n$ at $x$ .", "Let $\\widehat{\\Delta }$ denote the lift of $\\Delta $ to ${\\rm Gr}(\\mathbb {P}^k\\times \\mathbb {P}^k,k)$ , i.e.", "the set of points $(x,[v])$ with $x\\in \\Delta $ and $v$ tangent to $\\Delta $ .", "The intersection $\\widetilde{\\Gamma }_n\\cap \\widehat{\\Delta }$ corresponds to the non-transverse points of intersection between $\\Gamma _n$ and $\\Delta $ .", "Note that $\\dim \\widetilde{\\Gamma }_n+\\dim \\widehat{\\Delta }$ is smaller than the dimension of ${\\rm Gr}(\\mathbb {P}^k\\times \\mathbb {P}^k,k)$ and the intersection of subvarieties of such dimensions are generically empty.", "Analogous construction can be done for the manifolds $\\Gamma _n^{\\prime }$ and $\\Delta ^{\\prime }$ given above.", "We show that the currents $d^{-n}[\\widetilde{\\Gamma }_n]$ cluster on some positive closed current $\\widetilde{_+ on {\\rm Gr}(\\mathbb {P}^k\\times \\mathbb {P}^k,k).", "It can be obtained from the current +:=T_+\\otimes T_- on \\mathbb {P}^k\\times \\mathbb {P}^k in a similar way as in the construction of \\widetilde{\\Gamma }_n.", "One can also construct \\widetilde{_+ by lifting + to a positive closed current \\widehat{_+ of the same dimension in {\\rm Gr}(\\mathbb {P}^k\\times \\mathbb {P}^k) and then transforming it into \\widetilde{_+ via some incidence manifold.Using a theorem due to de Thélin \\cite {deThelin2} on the hyperbolicity of \\mu we show that the density between \\widetilde{_+ and \\widehat{\\Delta } vanishes.", "This property says that almost tangencies are negligible when n goes to infinity.", "The above example with \\Gamma _n^{\\prime } and \\Delta ^{\\prime } is an illustration of the opposite situation.", "}Let us be more precise.", "The vanishing of the density between \\widetilde{_+ and \\widehat{\\Delta } implies that the mass of \\widetilde{\\Gamma }_n in a small enough neighbourhood of \\widehat{\\Delta } is smaller than \\epsilon d^n for any given small constant 0<\\epsilon <1 when n is large enough.", "It follows that for some projection close to the projection (x,y)\\mapsto x-y from k\\times k to k the size of the ramification locus of \\Gamma _n over a small neighbourhood of 0\\in k is smaller than \\epsilon d^n.", "On the other hand, with respect to this projection, \\Gamma _n is a ramified covering over k of degree approximatively d^n.", "An argument à la Hurwitz permits to construct almost d^n graphs \\Gamma _n^{(j)} contained in \\Gamma _n over a small neighbourhood of 0\\in k.}Each graph \\Gamma _n^{(j)} intersects \\Delta at a unique point corresponding to a periodic point of f. If a sequence of such graphs converges in the sense of currents, it converges uniformly.", "Therefore, we control the convergence of a large part of the intersection \\Gamma _n\\cap \\Delta thanks to the convergence of \\Gamma _n.", "This together with some standard arguments imply the identity\\lim _{n\\rightarrow \\infty } [\\Delta ]\\wedge d^{-n}[\\Gamma _n]=[\\Delta ]\\wedge \\lim _{n\\rightarrow \\infty } d^{-n}[\\Gamma _n]which is equivalent to Theorem \\ref {th_main}.", "}The dynamical setting enters into the picture, first because in the above arguments we use that the graphs \\Gamma _n are horizontal with respect to the projection (x,y)\\mapsto x-y.", "Near the diagonal \\Delta , they are contained in a fixed box along \\Delta .", "The other more serious point concerns the delicate computation of the density between \\widetilde{_+ and \\widehat{\\Delta }.Roughly speaking, we want to show that \\widetilde{_+ is not concentrated near \\widehat{\\Delta } as we can observe positive closed currents near a point with positive Lelong number.", "}For \\mu -almost every point z\\in k, denote by E_s(z) and E_u(z) the stable and unstable tangent subspaces for f and \\mu at x.", "Since \\mu is hyperbolic with p positive Lyapounov exponents and k-p negative ones, we have \\dim E_s(z)=k-p, \\dim E_u(z)=p and E_s(z)\\cap E_u(z)=\\varnothing .", "Denote by \\Pi the canonical projection from {\\rm Gr}(\\mathbb {P}^k\\times \\mathbb {P}^k,k) to \\mathbb {P}^k\\times \\mathbb {P}^k.", "Let \\mathbf {G}(z) be the set of points (z,z,[v]) in \\Pi ^{-1}(z,z) such that v is not transverse to the vector spaceE_s(z)\\times E_u(z).", "Since the last vector space is transverse to \\Delta , the varieties \\mathbf {G}(z) and \\widehat{\\Delta } are disjoint.", "}We can show that the intersection of \\widetilde{_+ and \\Pi ^{-1}(\\Delta ) is, in a weak sense of currents defined later, equal to the average, with respect to \\mu , of the currents of integration on \\mathbf {G}(z).", "This delicate property is basically due to the hyperbolicity of \\mu and Oseledec^{\\prime }s theorem.", "We see roughly that \\widetilde{ crosses \\Pi ^{-1}(\\Delta ) through the varieties \\mathbf {G}(z) which are disjoint from \\widehat{\\Delta }.", "This is the key point to get the vanishing of the mentioned density between \\widetilde{_+ and \\widehat{\\Delta }.", "The proof requires also several geometric properties of \\widetilde{_+ and \\widehat{_+.", "They are of independent interest.", "}}}}}}}\\medskip {\\bf Acknowledgements. }", "The paper was partially written during the visit of thefirst author at the Shanghai Center for Mathematical Sciences.", "He would like to thank the institute, Yi-Jun Yao and Weiping Zhang fortheir great hospitality.", "}$" ], [ "Preliminary on currents and cohomology", "In this section, we will give some properties of positive closed currents and some properties of the action of meromorphic maps on currents and on Hodge cohomology.", "They will be used to overcome technical difficulties in the proof of our main result.", "Some of them are of independent interest.", "Though the strategy should work in a quite general setting, it is already involved for Hénon maps on $k$ .", "So we will limit ourself to the simplest situation required for Hénon maps and refer to [9], [10] for the case of general meromorphic maps on compact Kähler manifolds.", "Slicing theory for currents.", "We will discuss slicing theory in the case of positive closed currents.", "It will be applied later to woven positive closed currents.", "Recall that Federer's slicing theory can be applied to positive closed currents and to currents defined by complex varieties which are not necessarily closed, see Federer [18].", "The later case allows to extend the slicing theory to woven currents, not necessarily closed, that we will introduce in the next section.", "Let $\\pi :V\\rightarrow W$ be a proper surjective holomorphic map between Kähler manifolds of dimension $l$ and $m$ respectively with $l\\ge m$ .", "Let $T$ be a positive closed $(p,p)$ -current on $V$ of bi-dimension at least equal to $(m,m)$ , i.e.", "$p\\le l-m$ .", "Then for almost every $y\\in W$ , the slice $\\langle T\|\\pi \|y\\rangle $ exists and is a positive closed current of bi-dimension $(l-p-m,l-p-m)$ and of bi-degree $(p+m,p+m)$ with support in $\\pi ^{-1}(y)$ .", "This current is obtained as the limit of a sequence of currents of the form $T\\wedge \\pi ^(\\phi _n)$ , where $(\\phi _n)$ is a suitable sequence of positive $(m,m)$ -forms on $W$ which converges weakly to the Dirac mass at $y$ .", "Moreover, if $\\phi $ is a continuous $(m,m)$ -form on $W$ and $\\psi $ is a continuous $(l-p-m,l-p-m)$ -form with compact support on $V$ then $\\langle T,\\pi ^(\\phi )\\wedge \\psi \\rangle = \\int _{y\\in W} \\langle T\|\\pi \|y\\rangle (\\psi ) \\phi (y).$ Assume now that $\\pi $ is proper on a closed subset $G$ of $V$ which contains the support of $T$ .", "We developed in [13] a simpler slicing theory with some advantages that we recall below.", "Let $\\psi $ be an $(l-p-m,l-p-m)$ -form of class $\\mathcal {C}^2$ on $V$ .", "Assume it is a real form; otherwise, we can consider separately its real and imaginary parts.", "Then $\\langle T\|\\pi \|y\\rangle (\\psi )$ is equal almost everywhere on $W$ to a d.s.h.", "function, i.e.", "locally a difference of two psh functions, see Remark 2.2.6 in [13].", "If the mass of $T$ is bounded and $\\psi $ is fixed, we can take these local psh functions in a suitable compact family of psh functions.", "Here is a consequence of this property.", "Proposition 2.1 Let $V, W,\\pi $ and $G$ be as above.", "Let $T_n$ be a sequence of positive closed $(p,p)$ -currents on $V$ converging to a positive closed $(p,p)$ -current $T$ with $p\\le l-m$ .", "Assume that all $T_n$ are supported by $G$ .", "Then there is a subsequence $T_{n_i}$ such that for almost every $y\\in W$ we have $\\lim _{i\\rightarrow \\infty } \\langle T_{n_i}\|\\pi \|y\\rangle =\\langle T\|\\pi \|y\\rangle .$ Recall that the map embedding psh functions in $L^1_{loc}$ is compact.", "From the d.s.h.", "property we just mentioned, for every $\\mathcal {C}^2$ test function $\\psi $ , the sequence of d.s.h.", "functions $\\langle T_{n}\|\\pi \|y\\rangle (\\psi )$ on $W$ converges to $\\langle T\|\\pi \|y\\rangle (\\psi )$ in $L^1_{loc}$ .", "Recall also that any convergent sequence in $L^1_{loc}$ admits a subsequence which converges almost everywhere.", "We deduce from the property of slicing mentioned above that for a given $\\psi $ in $\\mathcal {C}^2$ , there is a subsequence $T_{n_i}$ such that we have almost everywhere on $W$ $\\lim _{i\\rightarrow \\infty } \\langle T_{n_i}\|\\pi \|y\\rangle (\\psi ) =\\langle T\|\\pi \|y\\rangle (\\psi ).$ Using the standard diagonal process, we can have such a property of a suitable sequence $(n_i)$ for any fixed countable family of test $\\mathcal {C}^2$ forms $\\psi $ .", "Choose such a family which is dense in the space $\\mathcal {C}^0$ .", "The last identity applied to $\\psi $ in this family implies the identity in the proposition.", "Our slicing theory does not apply to non-closed currents.", "Consider the case where $T$ is the current of integration on a complex variety $\\Lambda $ of dimension $l-p$ , immersed in $V$ , whose $2(l-p)$ -dimensional volume is finite in each compact subset of $V$ .", "Then for almost every $y\\in W$ the intersection $\\Lambda \\cap \\pi ^{-1}(y)$ is either empty or a variety of dimension $l-p-m$ with finite $2(l-p-m)$ -dimensional volume in each compact subset of $V$ and the intersection is transverse at almost every point.", "Moreover, the slice $\\langle T\|\\pi \|y\\rangle $ in the classical sense of Federer and is equal to the current of integration on $\\Lambda \\cap \\pi ^{-1}(y)$ .", "For simplicity, we only consider generic $y$ satisfying these properties.", "Note that in the case where the restriction of $\\pi $ to $V$ has rank strictly smaller than $m$ , then for almost every $y$ we have $\\langle T\|\\pi \|y\\rangle =0$ .", "We describe now a situation which will be used later.", "Consider the case where $V=V^{\\prime }\\times \\mathbb {P}^k$ with $V^{\\prime }$ compact Kähler of dimension $l-k$ .", "If $H$ is a generic projective subspace of dimension $r$ in $\\mathbb {P}^k$ , we will define the intersection of $T$ with $V^{\\prime }\\times H$ , in the slicing theory sense.", "Let $\\mathbb {G}$ denote the Grassmannian which parametrizes the family of such $H$ .", "For $\\xi \\in \\mathbb {G}$ , denote by $H_\\xi $ the corresponding projective subspace in $\\mathbb {P}^k$ .", "Let $\\Sigma $ be the incidence manifold defined by $\\Sigma :=\\lbrace (x,\\xi ) \\text{ with } x\\in H_\\xi \\subset \\mathbb {P}^k \\text{ and } \\xi \\in \\mathbb {G}\\rbrace \\subset \\mathbb {P}^k\\times \\mathbb {G}.$ Let $\\pi _1,\\pi _2$ denote the natural projections from $V^{\\prime }\\times \\Sigma $ onto $V=V^{\\prime }\\times \\mathbb {P}^k$ and $\\mathbb {G}$ .", "They are submersions.", "So the positive closed current $\\pi _1^(T)$ is well-defined.", "Define the intersection $T\\wedge [V^{\\prime }\\times H_\\xi ]$ as the push-forward of the slice $\\langle \\pi _1^(T)\|\\pi _2\| \\xi \\rangle $ by $\\pi _1$ to $V$ .", "This slice exists for generic $\\xi \\in \\mathbb {G}$ .", "We will be concerned with the case where $V^{\\prime }$ is a product of $\\mathbb {P}^k$ and a Grassmannian bundle over $\\mathbb {P}^k$ .", "Some general norm controls on pull-back operators.", "The manifolds involved in our study are bi-rational to products of projective spaces and Grassmannians.", "We will sometimes reduce the problem to the case of such products where the structure of Hodge cohomology groups is simpler.", "The following proposition allows us to do this reduction.", "It is a direct consequence of the main theorem in [10].", "Proposition 2.2 Let $\\pi :X^{\\prime }\\rightarrow X$ be a bi-meromorphic map between two compact Kähler manifolds of dimension $k$ .", "Let $f,g:X\\rightarrow X$ and $f^{\\prime }:X^{\\prime }\\rightarrow X^{\\prime }$ be dominant meromorphic maps.", "Assume that $f$ and $f^{\\prime }$ are conjugated: $f\\circ \\pi =\\pi \\circ f^{\\prime }$ .", "Let $f^$ and $g^$ (resp.", "${f^{\\prime }}^$ ) denote the pull-back operators associated with $f$ and $g$ (resp.", "$f^{\\prime }$ ) acting on the Hodge cohomology group $H^{q,q}(X,$ (resp.", "$H^{q,q}(X^{\\prime },$ ) for an integer $0\\le q\\le k$ .", "Then there is a constant $A\\ge 1$ depending only on $X,X^{\\prime },\\pi $ and the norms on Hodge cohomologies of $X,X^{\\prime }$ such that $A^{-1}\\Vert {f^{\\prime }}^\\Vert \\le \\Vert f^\\Vert \\le A\\Vert {f^{\\prime }}^\\Vert \\quad \\text{and}\\quad \\Vert f^\\circ g^\\Vert \\le A\\Vert f^\\Vert \\Vert g^\\Vert .$ Note that on a compact Kähler manifold with a fixed Kähler metric the mass of a positive closed current depends only on its cohomology class.", "So the last proposition is equivalent to a mass control for these currents under the action of meromorphic maps.", "We can apply it to $f^n, g^n$ and ${f^{\\prime }}^n$ instead of $f,g$ and $f^{\\prime }$ keeping the same constant $A$ .", "We will also need the following proposition in order to control the mass of a current, see [10] for a more general result.", "Proposition 2.3 Let $f:X\\rightarrow X$ be a dominant meromorphic map on a compact Kähler manifold of dimension $k$ .", "Let $X_0$ be a non-empty Zariski open subset of $X$ such that $f$ defines a bi-holomorphic map between $X_0$ and its image.", "Let $T$ be a positive closed $(q,q)$ -current of mass 1 on $X$ with $0\\le q\\le k$ .", "Then the pull-back $(f_{\|X_0})^(T)$ of $T$ to $X_0$ has finite mass.", "Denote by $f^\\bullet (T)$ its extension by 0 to $X$ .", "Then $f^\\bullet (T)$ is a positive closed $(q,q)$ -current.", "Moreover, its mass is bounded by the norm of $f^$ on $H^{q,q}(X,$ times a constant $A$ which only depends on the Kähler metric on $X$ and the norm on $H^{q,q}(X,$ .", "Note that $f^\\bullet (T)$ depends on the choice of $X_0$ but in applications we often have a standard choice.", "We can apply the proposition to $f^n$ instead of $f$ with the same constant $A$ .", "This constant does not depend on the choice of $X_0$ .", "Note also that we have $(f^n)^\\bullet =(f^\\bullet )^n$ when $f$ defines an automorphism of $X_0$ .", "Tensor products of currents.", "We will need criteria to check that a positive closed current is the tensor product of two other ones.", "Let $X$ and $Y$ be compact Kähler manifolds of dimension $k$ and $l$ .", "For simplicity, assume that $H^{r,s}(X,=0$ when $r\\ne s$ .", "Let $\\pi _X,\\pi _Y$ be the canonical projections from $X\\times Y$ onto its factors $X$ and $Y$ respectively.", "Fix Kähler forms $\\omega _X,\\omega _Y$ on $X$ and $Y$ .", "We have the following result.", "Proposition 2.4 Let $T$ be a positive closed current on $X\\times Y$ of bi-dimension $(s,s)$ .", "Assume there is an integer $0\\le r\\le \\min (k,s)$ such that $T\\wedge \\pi _X^(\\omega _X^{r+1})=0$ and $T\\wedge \\pi _Y^(\\omega _Y^{s-r+1})=0$ .", "Assume also that $R:=(\\pi _X)_\\big (T\\wedge \\pi _Y^(\\omega _Y^{s-r})\\big )$ is an extremal positive closed current of bi-dimension $(r,r)$ on $X$ .", "Then there is a positive closed current $S$ of bi-dimension $(s-r,s-r)$ on $Y$ such that $T=R\\otimes S$ .", "Let $x$ and $y$ denote coordinates on $X$ and $Y$ .", "Observe that each smooth $(s,s)$ -form on $X\\times Y$ can be written as a finite linear combination of forms of the following type or their conjugates: $\\Phi :=h(x,y)\\alpha (x)\\wedge \\beta (y)\\wedge \\Omega (x)\\wedge \\Theta (y),$ where $h$ is a smooth positive function on $X\\times Y$ , $\\alpha $ is a $(u,0)$ -form on $X$ , $\\beta $ is a $(0,u)$ -form on $Y$ , $\\Omega $ is a smooth positive $(v,v)$ -form on $X$ and $\\Theta $ is a smooth positive form of bi-degree $(s-u-v,s-u-v)$ on $Y$ .", "We first prove the following claim.", "Claim.", "We have $\\langle T,\\Phi \\rangle =0$ when $u\\ne 0$ or $v\\ne r$ .", "Observe that since $T$ is positive and $\\omega _X$ is strictly positive, the hypotheses on $T$ imply that for any $(r+1,r+1)$ -form $\\phi $ on $X$ , we have $T\\wedge \\pi _X^(\\phi )=0$ .", "We then deduce that the same identity also holds for every $(u,u)$ -form $\\phi $ with $u\\ge r+1$ .", "Similarly, we have $T\\wedge \\pi _Y^(\\psi )=0$ for any $(v,v)$ -form $\\psi $ on $Y$ with $v\\ge s-r+1$ .", "Observe also that for any constant $\\lambda > 0$ , the forms $i^{u^2} \\big [\\lambda \\alpha (x)\\wedge \\overline{\\alpha (x)}+\\lambda ^{-1}\\overline{\\beta (y)}\\wedge \\beta (y)\\pm 2{\\rm Re}\\big (\\alpha (x)\\wedge \\beta (y)\\big )\\big ]$ and $i^{u^2} \\big [\\lambda \\alpha (x)\\wedge \\overline{\\alpha (x)}+\\lambda ^{-1}\\overline{\\beta (y)}\\wedge \\beta (y)\\pm 2{\\rm Im}\\big (\\alpha (x)\\wedge \\beta (y)\\big )\\big ]$ are weakly positive.", "Since $T,\\Omega ,\\Theta $ are positive and $h$ is bounded, we can bound $\|\\langle T,\\Phi \\rangle \|$ by a constant times $\\lambda \\big \\langle T, i^{u^2} \\alpha (x)\\wedge \\overline{\\alpha (x)}\\wedge \\Omega (x)\\wedge \\Theta (y)\\big \\rangle + \\lambda ^{-1} \\big \\langle T, i^{u^2} \\overline{\\beta (y)}\\wedge \\beta (y)\\wedge \\Omega (x)\\wedge \\Theta (y)\\big \\rangle .$ For a good choice of $\\lambda $ , the last expression is equal to $ \\big \\langle T, i^{u^2} \\alpha (x)\\wedge \\overline{\\alpha (x)}\\wedge \\Omega (x)\\wedge \\Theta (y)\\big \\rangle ^{1/2}\\big \\langle T, i^{u^2} \\overline{\\beta (y)}\\wedge \\beta (y)\\wedge \\Omega (x)\\wedge \\Theta (y)\\big \\rangle ^{1/2}.$ It is not difficult to see that the second factor vanishes when $v\\ne r$ and the first factor vanishes when $v=r$ and $u\\ne 0$ .", "This completes the proof of the claim.", "For a bi-degree reason, we easily deduce from the claim that $T$ vanishes on $\\phi (x)\\wedge \\psi (y)$ for all smooth $(u,v)$ -form $\\phi $ on $X$ and $(s-u,s-v)$ -form $\\psi $ on $Y$ provided that $(u,v)\\ne (r,r)$ .", "Consider now a smooth $(r,r)$ -form $\\phi $ on $X$ and a smooth $(s-r,s-r)$ -form $\\psi $ on $Y$ .", "Observe that $d(T\\wedge \\psi (y))=T\\wedge d\\psi (y)=0$ thanks to the last observation.", "So if $\\psi $ is positive, then $T\\wedge \\psi (y)$ is a positive closed current.", "Its horizontal dimension with respect to the projection $\\pi _Y$ is 0 in the sense that $T\\wedge \\psi (y)\\wedge \\omega _Y(y)=0$ .", "Therefore, we can prove as in Lemma 3.3 in [14] that $T\\wedge \\psi (y)$ can be disintegrated into positive closed currents on $X\\times \\lbrace y\\rbrace $ with respect to a positive measure on $Y$ .", "The push-forward of $T\\wedge \\psi (y)$ to $X$ is bounded by a constant times $R$ since $\\psi $ is bounded by a constant times $\\omega _Y^{s-r}$ .", "Since $R$ is extremal, the above currents on $X\\times \\lbrace y\\rbrace $ are proportional to $R$ .", "The property holds without the positivity of $\\psi $ since we can always write $\\psi $ as a linear combination of positive forms.", "We deduce from the last property that $\\langle T,\\phi (x)\\wedge \\psi (y)\\rangle = \\langle R,\\phi \\rangle S(\\psi ),$ where $S$ is some continuous linear form, i.e.", "a current of bi-dimension $(s-r,s-r)$ on $Y$ .", "The last identity also holds for $(u,v)\\ne (r,r)$ since in this case its both sides vanish.", "We deduce that $T=R\\otimes S$ because the vector space generated by the forms $\\phi (x)\\wedge \\psi (y)$ is dense in the space of test $(s,s)$ -forms on $X\\times Y$ .", "Since $T$ is positive closed, it is easy to check that $S$ is also positive and closed.", "This completes the proof of the proposition.", "In the dynamical setting, we will use the proposition below in order to check the hypotheses of the last result.", "Let $f:X\\rightarrow X$ and $g:Y\\rightarrow Y$ be bi-meromorphic maps.", "Assume that $X,Y$ are homogeneous and there are dense Zariski open sets $X_0\\subset X$ and $Y_0\\subset Y$ such that $f,g$ are automorphisms of $X_0$ and $Y_0$ respectively.", "Define the bi-meromorphic map $h:X\\times Y\\rightarrow X\\times Y$ by $h(x,y):=(f(x),g(y))$ .", "Assume there are constants $d>\\delta >1$ and an integer $1\\le p\\le k-1$ such that $\\Vert (f^n)^\\Vert =O(\\delta ^n)$ on $H^{q,q}(X,$ for $q\\ne p$ and $\\Vert (f^n)^\\Vert =O(d^n)$ on $H^{p,p}(X,$ as $n\\rightarrow \\infty $ .", "Fix also a constant $d^{\\prime }>1$ such that $\\Vert (g^n)^\\Vert =O({d^{\\prime }}^n)$ on $H^{q,q}(Y,$ for every $q$ .", "Proposition 2.5 Let $S_n$ be a sequence of positive closed currents of bi-dimension $(s,s)$ on $X\\times Y$ with bounded mass.", "Let $T$ be a limit value of the sequence of currents $d^{-n}{d^{\\prime }}^{-n} (h^n)^\\bullet (S_n)$ .", "Then we have $T\\wedge \\pi _X^(\\omega _X^{r+1})=0$ and $T\\wedge \\pi _Y^(\\omega _Y^{s-r+1})=0$ for $r:=k-p$ .", "A $(q,q)$ -class in the cohomology group of a compact Kähler manifold is said to be pseudo-effective if it contains a positive closed current.", "So we can define a partial order relation between real classes: we have $c\\le c^{\\prime }$ if $c^{\\prime }-c$ is pseudo-effective.", "We also say that a class is strictly positive if it is larger than or equal to the class of a strictly positive closed form.", "Recall also that on homogeneous manifolds, any positive closed currents can be approximated by smooth positive closed forms in the same cohomology class.", "These forms can be obtained using a convolution with holomorphic automorphisms close to the identity, see e.g.", "[22].", "Recall also the Künneth formula in our case where $H^{r,s}(X,=0$ for $r\\ne s$ , see [28].", "We have the following canonical decomposition of the Hodge cohomology on $X\\times Y$ $H^{q,q}(X\\times Y,=\\sum _r H^{r,r}(X,\\times H^{q-r,q-r}(Y,.$ Here, for simplicity, we set $H^{r,r}(X,=0$ if either $r<0$ or $r>\\dim X$ and we apply the same convention to all manifolds.", "Proof of Proposition REF .", "Since $(h^n)^$ preserves the Künneth decomposition for $X\\times Y$ , it is not difficult to see that $\\Vert (h^n)^\\Vert =O(d^n{d^{\\prime }}^n)$ on the Hodge cohomology of $X\\times Y$ .", "Note that $(h^n)^\\bullet $ is not compatible with the action of $h^n$ on cohomology.", "Since $X\\times Y$ is homogeneous, $S_n$ can be approximated by smooth positive forms $S_n^{(j)}$ in the same cohomology class.", "We deduce that $d^{-n}{d^{\\prime }}^{-n} (h^n)^\\bullet (S_n)$ is smaller than or equal to all limit values of $d^{-n}{d^{\\prime }}^{-n} (h^n)^(S_n^{(j)})$ when $j\\rightarrow \\infty $ .", "If $c_n$ denotes the cohomology class of $S_n$ , the class of $d^{-n}{d^{\\prime }}^{-n} (h^n)^\\bullet (S_n)$ is smaller than or equal to $d^{-n}{d^{\\prime }}^{-n} (h^n)^(c_n)$ .", "So we only have to check that all limit values of $d^{-n}{d^{\\prime }}^{-n} (h^n)^(c_n)$ belong to the component $H^{p,p}(X,\\times H^{r+l-s,r+l-s}(Y,$ .", "Since $(h^n)^$ preserve the Künneth decomposition, it is enough to verify that the sequence of operators $d^{-n}{d^{\\prime }}^{-n} (h^n)^$ converges to 0 on $H^{q,q}(X,\\times H^{q^{\\prime },q^{\\prime }}(Y,$ for $q\\ne p$ and for every $q^{\\prime }$ .", "This is clear because $(h^n)^$ is the product of the operator $(f^n)^$ acting on $H^{q,q}(X,$ and the operator $(g^n)^$ acting on $H^{q^{\\prime },q^{\\prime }}(Y,$ .", "The norm of first operator is equal to $O(\\delta ^n)$ and the norm of the second one is $O({d^{\\prime }}^n)$ .", "The result follows.", "$\\square $ Some norm control on pull-back operators.", "Let $f:X\\rightarrow X$ be a bi-meromorphic map on a compact Kähler manifold $X$ of dimension $k$ .", "Let $X_0$ be a non-empty Zariski open subset of $X$ such that $f$ is a bi-holomorphism between $X_0$ and its image.", "Assume that $H^{r,s}(X,=0$ for $r\\ne s$ .", "Let $1\\le p\\le k-1$ , $n_0\\ge 1$ be integers and let $1\\le \\delta <d$ be real numbers such that $\\Vert (f^n)^\\Vert =O(\\delta ^n)$ on $H^{q,q}(X,$ for $q\\ge p+1$ and $(f^{n_0})^(c_0)\\le d^{n_0}c_0$ for some strictly positive class $c_0$ in $H^{p,p}(X,$ .", "Note that the last condition implies that $\\Vert (f^n)^\\Vert =O(d^n)$ on $H^{p,p}(X,$ .", "Let $Z$ be another homogeneous compact Kähler manifold of dimension $m$ .", "Fix a Kähler form $\\omega _Z$ on $Z$ and consider on $X\\times Z$ the Kähler form $\\pi _X^(\\omega _X)+\\pi _Z^(\\omega _Z)$ , where $\\pi _X,\\pi _Z:X\\times Z\\rightarrow X, Z$ are the canonical projections.", "Let $\\widehat{f}:X\\times Z\\rightarrow X\\times Z$ be a bi-meromorphic map which is also a bi-holomorphic map between $X_0\\times Z$ and $f(X_0)\\times Z$ .", "Assume that $f\\circ \\pi _X=\\pi _X\\circ \\widehat{f}$ .", "So $\\widehat{f}$ preserves the vertical fibration associated with $\\pi _X$ .", "Assume finally that for $x\\in X_0$ , the restriction of $\\widehat{f}$ to $\\lbrace x\\rbrace \\times Z$ is a bi-holomorphic map onto $\\lbrace f(x)\\rbrace \\times Z$ whose action on Hodge cohomology is the identity.", "For the last property, we identify both $\\lbrace x\\rbrace \\times Z$ and $\\lbrace f(x)\\rbrace \\times Z$ to $Z$ in the canonical way.", "The property automatically holds when $Z$ is a Grassmannian.", "This is a situation we will consider later.", "Here is an important proposition that we will need.", "Recall that Proposition REF allows to use this result for manifolds which are bi-meromorphic to $X\\times Z$ .", "Proposition 2.6 Let $\\delta ^{\\prime }$ be any constant such that $\\delta <\\delta ^{\\prime }<d$ .", "Then $\\Vert (\\widehat{f}^n)^\\Vert =O(\\delta ^{\\prime n})$ on $H^{q,q}(X\\times Z,$ for $q>p+m$ .", "There is an integer $n_1\\ge 1$ such that $(\\widehat{f}^{n_1})^(\\widehat{c}_1)\\le d^{n_1}\\widehat{c}_1$ for some strictly positive class $\\widehat{c}_1$ in $H^{p+m,p+m}(X\\times Z,$ .", "In particular, we have $\\Vert (\\widehat{f}^n)^\\Vert =O(d^n)$ on $H^{p+m,p+m}(X\\times Z,$ .", "Moreover, if $\\Vert (f^n)^\\Vert =O(\\delta ^n)$ on $H^{q,q}(X,$ for $q<p$ then $\\Vert (\\widehat{f}^n)^\\Vert =O(d^n)$ on $H^{q,q}(X\\times Z,$ for every $q$ .", "Observe that from the last assertion in Proposition REF , we only need to consider integers $n$ which are divisible by $n_0$ .", "Therefore, replacing $f,\\widehat{f}, d,\\delta ,\\delta ^{\\prime }$ with $f^{n_0},\\widehat{f}^{n_0},d^{n_0},\\delta ^{n_0}$ and $\\delta ^{\\prime n_0}$ allows to assume that $n_0=1$ .", "We need the following lemma that can be applied to $\\widehat{f}$ instead of $f$ .", "Lemma 2.7 The operator $f^$ preserves the cone of pseudo-effective classes in $H^{q,q}(X,$ for every $q$ .", "If $c$ and $c^{\\prime }$ are pseudo-effective classes, not necessarily of the same bi-degree, then for every $n\\ge 0$ $(f^n)^(c)\\le (f^)^n(c) \\qquad \\mbox{and} \\qquad (f^n)^(cc^{\\prime })\\le (f^n)^(c)(f^n)^(c^{\\prime }).$ Let $c$ be a pseudo-effective class.", "Since $X$ is homogeneous, it can be represented by a smooth positive closed form $\\alpha $ .", "Observe that $f^(\\alpha )$ is a positive closed $L^1$ -form which represents the class $f^(c)$ .", "In general, $f^(\\alpha )$ may have singularities along the indeterminacy set of $f$ .", "So the first assertion in the lemma is clear.", "Let $f^{\\prime }:X\\rightarrow X$ be another dominant meromorphic map.", "We first prove that $(f\\circ f^{\\prime })^(c)\\le f^{\\prime }(f^(c))$ for any pseudo-effective class $c$ .", "Applying inductively this inequality to $f^{\\prime }=f,f^2,\\ldots ,f^{n-1}$ gives the first inequality in the lemma.", "Let $\\alpha $ be a smooth positive closed form in the class $c$ .", "Then $(f\\circ f^{\\prime })^(c)$ is the class of the positive closed $L^1$ -form $(f\\circ f^{\\prime })^(\\alpha )$ and $f^(c)$ is the class of the positive closed $L^1$ -form $\\beta :=f^(\\alpha )$ .", "The above $L^1$ -forms are smooth on a suitable Zariski open subset of $X$ .", "Let $\\beta _n$ be smooth positive forms in the class $f^(c)$ which converge to $\\beta $ .", "Then $f^{\\prime }(\\beta _n)$ are positive closed $L^1$ -forms in the class $f^{\\prime }(f^(c))$ .", "Their masses depend only on their cohomology classes and hence are independent of $n$ .", "Extracting a subsequence allows to assume that $f^{\\prime }(\\beta _n)$ converge to some positive closed current $\\gamma $ .", "We can obtain $\\beta _n$ from $\\beta $ using a convolution with holomorphic automorphisms close to the identity as mentioned above.", "We get that $\\beta _n$ converge to $\\beta $ locally uniformly outside the singularities of $\\beta $ .", "It follows that $\\gamma $ is equal to $f^{\\prime }(\\beta )$ on a Zariski open subset of $X$ .", "Since $f^{\\prime }(\\beta )$ is equal to the $L^1$ -form $(f\\circ f^{\\prime })^(\\alpha )$ on a Zariski open set, we deduce that $\\gamma \\ge (f\\circ f^{\\prime })^(\\alpha )$ since $\\gamma $ may have a singular part supported by a subvariety of $X$ .", "Thus $(f\\circ f^{\\prime })^(c)\\le f^{\\prime }(f^(c))$ .", "This implies the first inequality in the lemma.", "For the last inequality in the lemma, we can for simplicity assume that $n=1$ .", "Consider a smooth positive closed form $\\theta $ in the class $c^{\\prime }$ .", "So $f^(\\alpha \\wedge \\theta )$ represents the class $f^(cc^{\\prime })$ and $\\beta _n\\wedge f^(\\theta )$ represents $f^(c)f^(c^{\\prime })$ .", "So any limit value of the sequence $\\beta _n\\wedge f^(\\theta )$ represents $f^(c)f^(c^{\\prime })$ .", "Such a limit value is equal to $ f^(\\alpha )\\wedge f^(\\theta )$ on a Zariski open set and hence equal to $f^(\\alpha \\wedge \\theta )$ on a Zariski open set.", "The last current is an $L^1$ -form and has no mass on proper analytic subsets of $X$ .", "We conclude that the considered limit value is at least equal to $f^(\\alpha \\wedge \\theta )$ .", "The last inequality in the lemma follows.", "Recall that we assumed $n_0=1$ .", "So the last lemma implies that $(f^n)^(c_0)\\le d^nc_0$ for every $n\\ge 1$ .", "Denote by $c_X$ and $c_Z$ the classes of $\\omega _X$ and $\\omega _Z$ .", "Their powers are strictly positive classes.", "Fix a constant $\\delta _1$ such that $\\delta <\\delta _1<\\delta ^{\\prime }$ .", "Replacing $f$ with a power of $f$ allows to assume that $f^(c_X^q)\\le \\delta _1c_X^q$ for $q>p$ .", "We also have the same inequality for $q<p$ when we assume that $\\Vert (f^n)^\\Vert =O(\\delta ^n)$ on $H^{q,q}(X,$ for $q<p$ .", "Recall that we assume that $H^{r,s}(X,=0$ for $r\\ne s$ .", "By Künneth formula, we have the following canonical decomposition of the Hodge cohomology on $X\\times Z$ $H^{q,q}(X\\times Z,=\\sum _r H^{r,r}(X,\\times H^{q-r,q-r}(Z,.$ In general, the above decomposition is not invariant under the action of $\\widehat{f}^$ .", "Define $E_{q,s}:=\\sum _{r\\ge s} H^{r,r}(X,\\times H^{q-r,q-r}(Z,.$ So we have a decreasing sequence of vector spaces with $E_{q,0}=H^{q,q}(X\\times Z,$ .", "We can show that $E_{q,s}$ is invariant under $\\widehat{f}^$ .", "Lemma 2.8 There is a constant $A>0$ such that $\\widehat{f}^\\big (c_X^s\\otimes c_Z^{q-s}\\big )\\le \\delta _1 (c_X^s\\otimes c_Z^{q-s}) + A \\sum _{j=1}^{q-s} c_X^{s+j}\\otimes c_Z^{q-s-j}$ for all $p+1\\le s\\le k$ and $0\\le q-s\\le m$ .", "If $f^(c_X^s)\\le \\delta _1 c_X^s$ for $s<p$ , then the above estimate also holds for $s< p$ .", "We prove the first assertion.", "The second one is obtained in the same way.", "We first show that $\\widehat{f}^\\big (c_X^s\\otimes c_Z^{q-s}\\big )$ belongs to $E_{q,s}$ .", "For this purpose, it is enough to check that the cup-product of this class with $\\pi _X^(c)$ vanishes for any class $c$ in $H^{k-s+1,k-s+1}(X,$ .", "Moreover, we only have to check the last property for $c$ pseudo-effective because such classes generate $H^{k-s+1,k-s+1}(X,$ .", "Since the considered class is pseudo-effective, we only need to show that its cup-product with $\\pi _X^(c)$ is negative or zero.", "Define $\\widetilde{c}_X:=\\pi _X^(c_X)$ and $\\widetilde{c}_Z:=\\pi _Z^(c_Z)$ .", "By Lemma REF , we have $\\widehat{f}^\\big (c_X^s\\otimes c_Z^{q-s}\\big ) &\\le & \\widehat{f}^(\\widetilde{c}_X^s)\\widehat{f}^(\\widetilde{c}_Z^{q-s})=\\pi _X^(f^(c_X^s))\\widehat{f}^(\\widetilde{c}_Z^{q-s})\\\\&\\le & \\delta _1 \\pi _X^(c_X^s)\\widehat{f}^(c_Z^{q-s}).$ The cup-product of the factor $\\pi _X^(c_X^s)$ with $\\pi _X^(c)$ vanishes for a bi-degree reason.", "It follows that $\\widehat{f}^\\big (c_X^s\\otimes c_Z^{q-s}\\big )$ belongs to $E_{q,s}$ .", "Since $c_X$ and $c_Z$ are Kähler classes, any real class in $E_{q,s+1}$ can be bounded by the last term of the inequality in the lemma provided that $A$ is large enough.", "Therefore, in order to obtain the result, we only need to check that $\\pi _X^(c_X^s)\\widehat{f}^(c_Z^{q-s})$ is equal to $c_X^s\\otimes c_Z^{q-s}$ plus a class in $E_{q,s+1}$ .", "By Poincaré's duality, if $\\kappa _X$ and $\\kappa _Z$ are classes in $H^{k-s,k-s}(X,$ and $H^{m-q+s,m-q+s}(Z,$ respectively with $c_X^s\\kappa _X=1$ and $c_Z^{q-s}\\kappa _Z=1$ , it suffices to show that $\\pi _X^(c_X^s)\\widehat{f}^(c_Z^{q-s})\\pi _X^(\\kappa _X)\\pi _Z^(\\kappa _Z)=1.$ But this identity is clear because $\\pi _X^(c_X^s)\\pi _X^(\\kappa _X)$ can be represented by a generic fiber of $\\pi _X$ and the restriction of $\\widehat{f}$ to generic fibers of $\\pi _X$ acts trivially on the cohomology of these fibers.", "The lemma follows.", "Using exactly the same arguments gives us the following lemma.", "Lemma 2.9 There is a constant $A>0$ such that $\\widehat{f}^\\big (c_X^p\\otimes c_Z^{q-p}\\big )\\le d (c_X^p\\otimes c_Z^{q-p}) + A \\sum _{j=1}^{q-p} c_X^{p+j}\\otimes c_Z^{q-p-j}$ for $0\\le q-p\\le m$ .", "End of the proof of Proposition REF .", "Recall that we assumed $n_0=0$ and $f^(c_X^s)\\le \\delta _1 c_X^s$ for $s>p$ .", "For the last assertion in the proposition, we can also assume that $f^(c_X^s)\\le \\delta _1 c_X^s$ for $s<p$ .", "Claim.", "We have $\\Vert (\\widehat{f}^n)^(c_X^s\\otimes c_Z^{q-s})\\Vert =O(d^n)$ for $s=p$ and $\\Vert (\\widehat{f}^n)^(c_X^s\\otimes c_Z^{q-s})\\Vert =O(n^M\\delta _1^n)$ for $p+1\\le s\\le k$ and $0\\le q-s\\le m$ with a suitable integer $M$ .", "For the last assertion in the proposition, we have $\\Vert (\\widehat{f}^n)^(c_X^s\\otimes c_Z^{q-s})\\Vert =O(d^n)$ for every $s$ .", "It is not difficult to deduce the first and third assertions in the proposition from the claim and Künneth decomposition of cohomology on $X\\times Z$ .", "For the first assertion, we use that $H^{q,q}(X\\times Z,=E_{q,p+1}$ when $q>p+m$ .", "We prove now the claim.", "By Lemma REF , it is enough to check the same estimates for $(\\widehat{f}^)^n$ instead of $(\\widehat{f}^n)^$ .", "We will obtain these estimates using a decreasing induction on $s$ .", "We can consider that the case $s=k+1$ is trivial because $c_X^{k+1}=0$ for a bi-degree reason.", "So assume that our above claim is true for $s+1,s+2,\\ldots $ instead of $s$ with $0\\le s\\le k$ and we prove it for $s$ .", "We only consider the case $s\\le p-1$ and the other cases can be obtained in the same way.", "By Lemma REF , we have $(\\widehat{f}^)^N(c_X^s\\otimes c_Z^{q-s})\\le \\delta _1 (\\widehat{f}^)^{N-1}\\big (c_X^s\\otimes c_Z^{q-s}\\big ) +A \\sum _{j=1}^{q-s} (\\widehat{f}^)^{N-1}\\big (c_X^{s+j}\\otimes c_Z^{q-s-j}\\big ).$ By induction hypothesis, the norm of the last sum is smaller than a constant times $d^{N-1}$ .", "Since $\\delta _1<d$ , the last inequality applied to $N=n,n-1,\\ldots ,1$ implies that $\\Vert (\\widehat{f}^)^n(c_X^s\\otimes c_Z^{q-s})\\Vert =O(d^n)$ .", "The claim follows.", "It remains to prove the second assertion in the proposition.", "Observe first that $H^{p+m,p+m}(X\\times Z,=E_{p+m,p}$ .", "Therefore, any combination of $c_X^s\\otimes c_Z^{p+m-s}$ with strictly positive coefficients and with $p\\le s\\le \\min (k,p+m)$ is a strictly positive class in $H^{p+m,p+m}(X\\times Z,$ .", "Define $\\widehat{c}_1:=c_X^p\\otimes c_Z^m+\\sum _{j=1}^{\\min (k-p,m)} A_jc_X^{p+j}\\otimes c_Z^{m-j},$ where $A_j$ are constants large enough such that $A_j\\ll A_{j+1}$ .", "It is not difficult to deduce from Lemmas REF and REF that $\\widehat{c}_1$ satisfies the proposition.", "Here, we can take $n_1=1$ but we already replaced twice $f$ with an iterate.", "$\\square $ Main examples.", "We describe now the main examples that will be considered later.", "Let $X$ be a compact Kähler manifold of dimension $k$ which is a projective space or the product of two projective spaces.", "So $X$ is homogeneous and $H^{r,s}(X,=0$ for $r\\ne s$ .", "Let $\\omega _X$ be a Kähler form on $X$ and denote by $c_X$ its cohomology class.", "Let $f:X\\rightarrow X$ be a bi-rational map and let $d,p,\\delta $ be as above such that $f^(c_X^p)\\le dc_X^p$ and $f^(c_X^q)\\le \\delta c_X^q$ for $q\\ne p$ .", "Denote by ${\\rm Gr}(X,k-p)$ the space of points $(x,[v])$ , where $x$ is a point in $X$ and $[v]$ is the direction of a simple complex tangent $(k-p)$ -vector of $X$ at $x$ .", "The natural projection from ${\\rm Gr}(X,k-p)$ to $X$ defines a fibration whose fibers are isomorphic to the Grassmannian $\\mathbb {G}$ of vector subspaces of dimension $k-p$ in $k$ .", "We can lift $f$ to a bi-rational map $\\widehat{f}:{\\rm Gr}(X,k-p)\\rightarrow {\\rm Gr}(X,k-p)$ by $\\widehat{f}(x,[v]):=(f(x), [f_(v)])$ for $x$ in a suitable Zariski open subset of $X$ .", "Let $\\widetilde{f}$ denote the lift of $\\widehat{f}$ to ${\\rm Gr}({\\rm Gr}(X,k-p),k-p)$ which is defined in the same way.", "Proposition 2.10 We have $\\Vert (\\widehat{f}^n)^\\Vert =O(d^n)$ on the Hodge cohomology groups of ${\\rm Gr}(X,k-p)$ .", "If $K$ is the dimension of ${\\rm Gr}({\\rm Gr}(\\mathbb {P}^k,k-p),k-p)$ , we also have $\\Vert (\\widetilde{f}^n)^\\Vert =O(d^n)$ on the Hodge cohomology group of ${\\rm Gr}({\\rm Gr}(\\mathbb {P}^k,k-p),k-p)$ of bi-degree $(q,q)$ for every $q\\ge K-k+p$ .", "Over a chart $k\\subset X$ , the fibration ${\\rm Gr}(X,k-p)$ can be identified in a natural way with $k\\times \\mathbb {G}$ .", "So ${\\rm Gr}(X,k-p)$ is bi-rational to $X\\times \\mathbb {G}$ .", "By Proposition REF , we can consider $\\widehat{f}$ as a bi-rational map of $X\\times \\mathbb {G}$ .", "The first assertion in the proposition is a direct consequence of the last assertion in Proposition REF .", "The second assertion in Proposition REF can be applied to $f^{\\prime }:=\\widehat{f}$ and $X^{\\prime }:=X\\times \\mathbb {G}$ as we will see below.", "The manifold ${\\rm Gr}({\\rm Gr}(\\mathbb {P}^k,k-p),k-p)$ is bi-rational to $X^{\\prime }\\times \\mathbb {G}^{\\prime }$ , where $\\mathbb {G}^{\\prime }$ is a Grassmannian.", "In order to see this point, it is enough to identify some Zariski open subset of $X^{\\prime }$ with a Zariski open subset of a complex Euclidean space.", "So we can consider $\\widetilde{f}$ as a map on $X^{\\prime }\\times \\mathbb {G}^{\\prime }$ which preserves the natural fibration over $X^{\\prime }$ .", "Applying Proposition REF to $f^{\\prime },\\widetilde{f}, X^{\\prime }\\times \\mathbb {G}^{\\prime }$ instead of $f,\\widehat{f}, X\\times Z$ gives the result.", "Examples 2.11 Let $f$ be a Hénon automorphism on $k$ that we extend to a bi-rational map on $\\mathbb {P}^k$ .", "Let $d_\\pm , d, p$ be as in the introduction.", "So the operator $f^$ on $H^{q,q}(\\mathbb {P}^k,$ is just the multiplication by $d_+^q$ for $q\\le p$ and by $d_-^{k-q}$ for $q\\ge p$ .", "Define $\\delta :=\\max (d_+^{p-1},d_-^{k-p-1})$ .", "We have $1\\le \\delta < d$ .", "So the last proposition can be applied to $f$ and its lifts to ${\\rm Gr}(\\mathbb {P}^k,k-p)$ and ${\\rm Gr}({\\rm Gr}(\\mathbb {P}^k,k-p),k-p)$ .", "Consider the bi-rational map $F=(f,f^{-1})$ on $\\mathbb {P}^k\\times \\mathbb {P}^k$ .", "If $\\omega _{\\rm FS}$ denotes the Fubini-Study form on $\\mathbb {P}^k$ and $\\pi _1,\\pi _2$ denote the projections from $\\mathbb {P}^k\\times \\mathbb {P}^k$ onto its factors, we consider on $\\mathbb {P}^k\\times \\mathbb {P}^k$ the Kähler metric $\\pi _1^(\\omega _{\\rm FS})+\\pi _2^(\\omega _{\\rm FS})$ .", "Let $c_{\\rm FS}$ denote the class of $\\omega _{\\rm FS}$ .", "We have $F^(c_{\\rm FS}^p\\otimes c_{\\rm FS}^{k-p})=d^2 (c_{\\rm FS}^p\\otimes c_{\\rm FS}^{k-p})$ and $F^(c_{\\rm FS}^r\\otimes c_{\\rm FS}^s)\\le d\\delta (c_{\\rm FS}^r\\otimes c_{\\rm FS}^s)$ for $(r,s)\\ne (p,k-p)$ .", "So we can apply Proposition REF to $F,d^2,d\\delta ,k$ instead of $f,d,\\delta ,p$" ], [ "Woven currents and tame currents", "This section contains some geometric properties of positive closed currents that we will use in our study of Hénon maps.", "We discuss the notions of woven and tame currents which have an independent interest.", "Laminar currents in dimension 2 were introduced and studied by Bedford-Lyubich-Smillie [1], see also Sullivan [27].", "Woven currents and laminar currents in higher dimension were introduced by the first author of the present paper in [8].", "If two Riemann surfaces in 2 are the limits of two sequences of Riemann surfaces $\\Gamma _n$ and $\\Gamma _n^{\\prime }$ with $\\Gamma _n\\cap \\Gamma _n^{\\prime }=\\varnothing $ , then their intersection is either empty or also a Riemann surface.", "In higher dimension and codimension, this property is no longer true.", "This is one of the main difficulties with woven and laminar currents in higher dimension.", "In order to simplify the exposition, we will work on a projective manifold $V$ of dimension $l$ .", "We can also extend the theory to currents on non-compact manifolds.", "Fix a Kähler form $\\omega $ on $V$ and consider the Kähler metric on $V$ induced by $\\omega $ .", "Measurable webs, woven currents and standard refinement.", "We introduce here some basic notions.", "Let $0\\le r\\le l$ be an integer.", "Denote by ${\\rm Lam}_r(V)$ the set of positive $(l-r,l-r)$ -currents that can be written as a finite or countable sum $S=\\sum [\\Lambda _i]$ , where the $\\Lambda _i$ 's are irreducible analytic sets of dimension $r$ , immersed in $V$ , such that $\\sum {\\rm vol}(\\Lambda _i)<\\infty $ .", "Here $[\\Lambda _i]$ denotes the current of integration on $\\Lambda _i$ and ${\\rm vol}(\\Lambda _i)$ denotes the $2r$ -dimensional volume of $\\Lambda _i$ that can be computed using Wirtinger's theorem by ${\\rm vol}(\\Lambda _i)={1\\over r!}", "\\int _{\\Lambda _i} \\omega ^r.$ We say that $S$ is a lame.", "Note that we don't assume that the $\\Lambda _i$ 's are disjoint.", "Moreover, given a current $S$ as above the decomposition $S=\\sum [\\Lambda _i]$ is not unique.", "The reason to consider here finite or countable sums is to gain flexibility in working with woven and laminar currents.", "We can identify ${\\rm Lam}_r(V)$ with a subset of the family $\\mathcal {P}_r(V)$ of (strongly) positive currents of bi-dimension $(r,r)$ on $V$ .", "The later is a metric space endowed with the following family of distances ${\\rm dist}_\\alpha (T,T^{\\prime }):=\\sup _{\\Vert \\varphi \\Vert _{\\mathcal {C}^\\alpha }\\le 1} \|\\langle T-T^{\\prime },\\varphi \\rangle \|.$ The induced topology on $\\mathcal {P}_r(V)$ is the same for any $\\alpha >0$ and coincides with the weak topology on currents.", "Define ${\\rm Lam}_r^(V):={\\rm Lam}_r(V)\\setminus \\lbrace 0\\rbrace $ and $\\mathcal {P}_r^(V):=\\mathcal {P}_r(V)\\setminus \\lbrace 0\\rbrace $ .", "Using the local description below of woven currents, we will see in Lemma REF that ${\\rm Lam}_r^(V)$ is a universally measurable set, i.e.", "it is measurable with respect to all Borel probability measures on $\\mathcal {P}_r(V)$ .", "Definition 3.1 We call measurable $l$ -web any positive Borel measure $\\nu $ on $\\mathcal {P}_r^(V)$ without mass outside ${\\rm Lam}_r^(V)$ such that $\\int _{{\\rm Lam}_r^(V)} \\Vert S\\Vert d\\nu (S)<\\infty .$ A current $T$ of bi-dimension $(r,r)$ on $V$ is said to be woven if there is a measurable $r$ -web $\\nu $ (which is called a measurable $r$ -web associated with $T$ ) such that $T:= \\int _{{\\rm Lam}_r^(V)} S d\\nu (S)$ or equivalently $\\langle T,\\varphi \\rangle := \\int _{{\\rm Lam}_r^(V)} \\langle S,\\varphi \\rangle d\\nu (S)$ for any test continuous $(r,r)$ -form $\\varphi $ on $V$ .", "Note that a woven current may be associated with different measurable webs, e.g.", "the Fubini-Study form $\\omega _{\\rm FS}$ on $\\mathbb {P}^k$ can be obtained as an average of hypersurfaces of degree $d$ for any positive integer $d$ as shown in the following example.", "Example 3.2 Let $\\mathbb {U}(k+1)$ denote the unitary group which acts naturally on $\\mathbb {P}^k$ .", "If $\\sigma $ is the Haar measure on $\\mathbb {U}(k+1)$ and $H$ is a subvariety of dimension $r$ and of degree $d$ of $\\mathbb {P}^k$ , we have the following identity in the sense of currents $\\omega _{\\rm FS}^{k-r}=d^{-1}\\int _{\\tau \\in \\mathbb {U}(k+1)} \\tau _[H]d\\sigma (\\tau ).$ This identity says that $\\omega _{\\rm FS}^{k-r}$ can be written as an average of currents of integration on subvarieties of degree $d$ and of dimension $r$ .", "So it is a woven current.", "Note that lames can be divided into smaller ones giving different webs associated with the same current.", "If $\\nu $ is a measurable $r$ -web, it has finite mass outside any neighbourhood of 0 in $\\mathcal {P}_r(V)$ .", "Indeed, outside any neighbourhood of 0, currents in $\\mathcal {P}_r(V)$ have mass bounded from below by a strictly positive constant.", "Let $\\Sigma _r$ be the set of $(R,S)$ in ${\\rm Lam}^_r(V)\\times {\\rm Lam}^_r(V)$ such that $R\\le S$ .", "Denote by $\\pi $ and $\\pi ^{\\prime }$ the natural projections $(R,S)\\mapsto R$ and $(R,S)\\mapsto S$ .", "Definition 3.3 Let $\\nu _1$ and $\\nu _2$ be two measurable $r$ -webs on $V$ .", "We say that $\\nu _1$ is a refinement of $\\nu _2$ and write $\\nu _2\\prec \\nu _1$ if there is a positive measure $\\nu _{12}$ on $\\Sigma _r$ such that $\\nu _1=\\pi _(\\nu _{12})$ .", "If $\\nu _{12}^{\\prime }$ is the restriction of $\\nu _{12}$ to the complement of a neighbourhood of 0 in ${\\rm Lam}_r(V)$ , then $\\pi ^{\\prime }_(\\nu _{12}^{\\prime })$ is absolutely continuous with respect to $\\nu _2$ .", "For $\\nu _2$ -almost every $S$ denote by $\\nu _{12}^S:=\\langle \\nu _{12}\|\\pi ^{\\prime }\|S\\rangle $ the conditional measure of $\\nu _{12}$ with respect to the fiber of $\\pi ^{\\prime }$ over the point $S$ .", "We identify it to a measure on ${\\rm Lam}_r^(V)$ .", "Then $\\nu _{12}^S$ is a finite or countable sum of Dirac masses and defines a measurable $l$ -web associated with $S$ .", "Two measurable $r$ -webs are equivalent if they admit a common refinement, see also Lemma REF below.", "Roughly speaking, in order to get a refinement $\\nu _1$ of $\\nu _2$ we decompose the lames $S$ of $\\nu _2$ into a finite or countable number of smaller ones using the conditional measures $\\nu _{12}^S$ .", "For example, when $\\nu _2$ is a Dirac mass at a point $R$ , then $\\nu _{12}^S=0$ for $S\\ne R$ and $\\nu _{12}^R=\\nu _{12}$ .", "In this case, $\\nu _{12}^R$ is identified with $\\nu _1$ .", "We get a decomposition of $R$ into a finite or countable sum of lames.", "The general case can be deduced from this case by taking an average with respect to $\\nu _2$ .", "Lemma 3.4 Let $\\nu _1$ and $\\nu _2$ be measurable $r$ -webs on $V$ .", "Assume that $\\nu _2\\prec \\nu _1$ .", "Then they are associated with the same current.", "If $\\nu _3$ is another measurable $r$ -web such that $\\nu _3\\prec \\nu _2$ , then $\\nu _3\\prec \\nu _1$ .", "The properties in Definition REF imply that $\\int _R R d\\nu _1(R)=\\int _{R,S} R d\\nu _{12}(R,S)=\\int _S \\Big (\\int _R R d\\nu _{12}^S(R)\\Big ) d\\nu _2(S)=\\int _S Sd\\nu _2(S).$ This gives the first assertion in the lemma.", "Denote by $\\nu _{12}$ the measure in Definition REF and $\\nu _{23}$ the similar one associated with $\\nu _2$ and $\\nu _3$ .", "If we identify each fiber $\\lbrace S\\rbrace \\times {\\rm Lam}_r^(V)$ of $\\pi ^{\\prime }$ with ${\\rm Lam}_r^(V)$ , we can define a measure $\\nu _{13}$ on ${\\rm Lam}_r^(V)\\times {\\rm Lam}_r^(V)$ by their conditional measures with respect to $\\nu _3$ $\\langle \\nu _{13}\|\\pi ^{\\prime }\|S\\rangle :=\\int _R \\nu _{12}^R d \\nu _{23}^S(R).$ It is not difficult to check that $\\nu _{13}$ induces the relation $\\nu _3\\prec \\nu _1$ .", "Indeed, one can consider the case where $\\nu _3$ is a Dirac mass and obtain the general case by taking an average.", "The abstract formalism on woven currents introduced above does not require a choice of local coordinates.", "It does not depend on the metric on the manifold.", "So it offers a convenient setting to work with different local coordinate systems and other operations on woven currents, e.g.", "the lifting of currents to Grassmannian bundles.", "However, in order to get a more precise picture on woven currents and to construct measurable webs, we will work in convenient local coordinates.", "Up to a choice of local coordinate systems, we give now a uniform way to decompose a measurable web into an infinite sum of $s$ -elementary webs with $s=1,2,\\ldots $ Approximation and refinement for local analytic sets.", "Let $z=(z_1,\\ldots ,z_l)$ be local holomorphic coordinates with $\|z_i\|<4$ .", "Write $z_i=x_i+\\sqrt{-1} y_i$ with $x_i,y_i\\in \\mathbb {R}$ .", "We consider here measurable webs $\\nu $ with lames inside the cube $\\mathbb {U}:=\\lbrace \|x_i\|,\|y_i\|\\le 1\\rbrace =[-1,1]^{2l}$ .", "Let $\\Lambda $ be an irreducible analytic set of dimension $r$ , not necessarily closed, immersed in the cube and with finite $2r$ -dimensional volume.", "Then there is a projection onto $r$ coordinates among $z_1,\\ldots , z_l$ whose generic fibers intersect $\\Lambda $ in finite or countable sets.", "For simplicity, assume that the projection $z\\mapsto (z_1,\\ldots ,z_r)$ satisfies this property.", "Otherwise, in order to get the same construction for all varieties, we choose the convenient projection with the smallest lexicographical index.", "For each $s\\ge 1$ , consider the real hyperplanes $\\lbrace x_i=j2^{-s}\\rbrace $ and $\\lbrace y_i=j2^{-s}\\rbrace $ with $j\\in \\mathbb {Z}$ and $i\\le r$ that will be called separating hyperplanes (there are also separating hyperplanes associated with the other projections).", "They divide the cube $[-2,2]^{2l}$ into $2^{2r(s+2)}$ (closed) towers.", "Two towers are said to be adjacent if they have common points.", "In particular, a tower is adjacent to itself.", "The union of the towers which are adjacent to a given tower is called a fat tower.", "The union of towers which are adjacent to one of the towers in the last fat tower is called a very fat tower.", "So fat and very fat towers are just extensions in the horizontal directions of ordinary towers.", "Consider complex manifolds which are graphs over the basis of a tower that can be extended to graphs over the basis of the associated very fat tower (we don't ask the later graphs to be defined over the boundary of the basis of the very fat tower).", "We call them $s$ -elementary lames.", "The extension of a graph in a tower to a graph in the corresponding fat or very fat tower is called the fat or very fat extension respectively.", "We will work with the family of $s$ -elementary lames in the box $\\mathbb {U}:=[-1,1]^{2l}$ .", "Observe that this family of is compact with respect to the local uniform convergence topology on holomorphic graphs.", "A finite positive measure on this compact set is called an $s$ -elementary web in $\\mathbb {U}$ .", "Considering the fat and very fat extensions of graphs is just a technical point which allows us to avoid the possible bad behavior of graphs near the vertical boundary of a tower.", "Denote by $\\nu ^{[s]}_\\Lambda $ the maximal $s$ -elementary web whose associated current is smaller than or equal to $[\\Lambda ]$ .", "Denote also by $[\\Lambda ^{[s]}]$ this current.", "The measure $\\nu ^{[s]}_\\Lambda $ is a sum of Dirac masses.", "It gives an approximation of $\\Lambda $ : we have $[\\Lambda ]-[\\Lambda ^{[s]}]\\rightarrow 0$ in the mass norm as $s\\rightarrow \\infty $ .", "Note that $[\\Lambda ]-[\\Lambda ^{[s]}]$ is a lame, i.e.", "a point in ${\\rm Lam}_r(V)$ .", "If it is not zero, the sum of the Dirac mass at this point and $\\nu _\\Lambda ^{[s]}$ is denoted by $\\nu _\\Lambda ^{(s)}$ .", "Otherwise, define $\\nu _\\Lambda ^{(s)}:=\\nu _\\Lambda ^{[s]}$ .", "So $\\nu _\\Lambda ^{(s)}$ is a measure associated with $[\\Lambda ]$ and gives a refinement of $[\\Lambda ]$ .", "Finally, we observe that the construction depends only on the current $[\\Lambda ]$ .", "More precisely, if we remove from $\\Lambda $ a closed subset of zero $2r$ -dimensional measure, we obtain another analytic set associated with the same current.", "Our construction gives the same approximation and refinement.", "Approximation and refinement for local lames.", "The construction extends without difficulty to any lame $S=\\sum [\\Lambda _i]$ in $\\mathbb {U}$ such that the above condition on the projection $z\\mapsto (z_1,\\ldots , z_r)$ is satisfied for each component $\\Lambda _i$ .", "We define $\\nu ^{[s]}_S$ as the maximal $s$ -elementary web whose associated current, denoted by $S^{[s]}$ , is smaller than or equal to $S$ .", "The current $S-S^{[s]}$ is a lame, i.e.", "a point in ${\\rm Lam}_r(V)$ .", "If it is not zero, the sum of the Dirac mass at this point and $\\nu _S^{[s]}$ is denoted by $\\nu _S^{(s)}$ .", "Otherwise, define $\\nu _S^{(s)}:=\\nu _S^{[s]}$ .", "This is a refinement of $S$ .", "We extend the construction to an arbitrary lame $S=\\sum [\\Lambda _i]$ in $\\mathbb {U}$ .", "Denote by $S_1$ the sum of $[\\Lambda _i]$ which satisfy the above condition on the projection $z\\mapsto (z_1,\\ldots , z_r)$ .", "The above construction can be applied to $S_1$ .", "Denote by $S_2$ the sum of the remaining $[\\Lambda _i]$ satisfying the similar condition for the next projection $z\\mapsto (z_{i_1},\\ldots ,z_{i_r})$ with respect to the lexicographical index order.", "We do the similar construction for the new projection and repeat it again for the other projections respecting always the lexicographical order.", "With the notations similar to the ones given above, define $\\nu _S^{[s]}:=\\sum _i \\nu _{S_i}^{[s]} \\quad \\text{and} \\quad \\nu _S^{(s)}:=\\sum _i \\nu _{S_i}^{(s)}.$ They are respectively the $s$ -approximation and $s$ -refinement of $S$ .", "They do not depend on the choice of the decomposition $S=\\sum [\\Lambda _i]$ .", "Approximation and refinement for global lames.", "Fix a covering of $V$ by a finite number of cubes $\\mathbb {U}_1,\\ldots ,\\mathbb {U}_N$ as above.", "Assume that $S$ is a lame in $V$ .", "We can decompose it into local lames $S=S_1+\\cdots +S_N$ , where $S_1$ is the restriction of $S$ to $\\mathbb {U}_1$ and by induction $S_i$ is the restriction to $\\mathbb {U}_i$ of $S-S_1-\\cdots -S_{i-1}$ .", "We then apply the above construction to each $S_i$ in $\\mathbb {U}_i$ and obtain the webs $\\nu ^{(s)}_{S_i}$ and $\\nu ^{[s]}_{S_i}$ .", "Define $\\nu ^{(s)}_S:=\\nu ^{(s)}_{S_1}+\\cdots +\\nu ^{(s)}_{S_N} \\quad \\text{and}\\quad \\nu ^{[s]}_S:=\\nu ^{[s]}_{S_1}+\\cdots +\\nu ^{[s]}_{S_N}.$ The sum of $N$ $s$ -elementary webs on $\\mathbb {U}_1,\\ldots ,\\mathbb {U}_N$ respectively is called an $s$ -elementary web on $V$ .", "If $S^{[s]}$ is the current associated with $\\nu _S^{[s]}$ , then $S-S^{[s]}\\rightarrow 0$ in the mass norm.", "Approximation and refinement for global webs.", "We can apply the same method to refine or approximate all woven currents.", "If $T$ is such a current and $\\nu $ is an associated web, define using the above notations $\\nu ^{(s)}:=\\int _{{\\rm Lam}_r^(V)} \\nu ^{(s)}_S d\\nu (S) \\quad \\text{and}\\quad \\nu ^{[s]}:=\\int _{{\\rm Lam}_r^(V)} \\nu ^{[s]}_S d\\nu (S).$ We say that $\\nu ^{(s)}$ and $\\nu ^{[s]}$ are respectively the standard $s$ -refinement and the standard $s$ -approximation of $\\nu $ .", "If $T^{[s]}$ is the current associated with $\\nu _T^{[s]}$ , then $T-T^{[s]}\\rightarrow 0$ in the mass norm.", "Here are two applications of the construction.", "Lemma 3.5 The set ${\\rm Lam}_r^(V)$ is universally measurable.", "Using a finite number of boxes $\\mathbb {U}_1,\\ldots ,\\mathbb {U}_N$ as above, it is not difficult to reduce the problem to the set of lames $S$ with supports in a box $\\overline{\\mathbb {U}}$ such that the projection $\\pi (z):=(z_1,\\ldots , z_r)$ is of maximal rank on each component of $S$ .", "Denote by $\\mathcal {L}$ the set of such lames $S$ with mass bounded by a fixed constant $M$ .", "It is enough to show that $\\mathcal {L}$ is universally measurable.", "Using the decomposition of lames into elementary ones as above, we see that $\\mathcal {L}$ is also the set of currents of the forms $S=\\sum _{s\\ge 1} S_s$ , where $S_s$ is a finite sum of $s$ -elementary lames and $\\Vert S\\Vert \\le M$ .", "Note that such a current $S$ is associated with infinitely many different decompositions into elementary lames.", "Denote by $\\mathcal {L}_s$ the set of currents which are equal to a sum of at most $2^{2rs}M$ $s$ -elementary lames.", "This is a compact set of currents.", "Consider the infinite product space $\\Pi _{s\\ge 1} \\mathcal {L}_s$ endowed with the natural product topology.", "Let $\\mathcal {L}^{\\prime }$ be the subset of points $(S_1,S_2,\\ldots )$ in this space such that $\\Vert S\\Vert \\le M$ .", "The last condition is equivalent to $\\sum _{s=1}^n\\Vert S_s\\Vert \\le M$ for every $n\\ge 1$ .", "So $\\mathcal {L}^{\\prime }$ is a Borel set and the map $(S_1,S_2,\\ldots )\\mapsto S:=\\sum S_s$ from $\\mathcal {L}^{\\prime }$ to $\\mathcal {L}$ is continuous and surjective.", "Therefore, the image $\\mathcal {L}$ of this map is universally measurable, see [3].", "The lemma follows.", "Lemma 3.6 Let $\\nu _1,\\nu _2$ and $\\nu _3$ be measurable $r$ -webs on $V$ .", "If $\\nu _3\\prec \\nu _1$ and $\\nu _3\\prec \\nu _2$ , then there is a measurable $r$ -web $\\nu _0$ such that $\\nu _1\\prec \\nu _0$ and $\\nu _2\\prec \\nu _0$ .", "In particular, if $\\nu _1$ and $\\nu _2$ are equivalent to $\\nu _3$ in the sense of Definition REF , then $\\nu _1$ and $\\nu _2$ are also equivalent.", "Using a covering of $V$ by cubes $\\mathbb {U}_1,\\ldots , \\mathbb {U}_N$ as above, we can reduce the problem to the case where $T$ is a current on the cube $[-1,1]^{2l}$ of $l$ .", "Using the above construction with the similar notations, we have $\\nu _3^{(s)}\\prec \\nu _1^{(s)}$ , $\\nu _3^{(s)}\\prec \\nu _2^{(s)}$ , $\\nu _1^{[s]}\\le \\nu _3^{[s]}$ and $\\nu _2^{[s]}\\le \\nu _3^{[s]}$ .", "The last two inequalities are not in general equalities because some $s$ -elementary lames in $\\nu _3^{[s]}$ may not be lames of $\\nu _1^{[s]}$ and $\\nu _2^{[s]}$ after the refinement.", "The measurable webs $\\nu _i^{(s)}$ are all associated with $T$ .", "If $T_i^{[s]}$ is the woven current associated with $\\nu _i^{[s]}$ , then $T-T_i^{[s]}$ is a woven current which tends to 0 as $s\\rightarrow \\infty $ .", "Write $ \\nu _i^{[s]}=h_i \\nu _3^{[s]}$ with $0\\le h_i\\le 1$ .", "Define $h:=\\min (h_1,h_2)$ , $\\vartheta ^{[s]}:=h\\nu _3^{[s]}$ and denote by $T^{[s]}$ the woven current associated with $\\vartheta ^{[s]}$ .", "Since $1-h\\le (1-h_1)+(1-h_2)$ , the woven current $T-T^{[s]}$ tends to 0 as $s\\rightarrow \\infty $ .", "For a fixed integer $s_0$ large enough, define $\\vartheta \\langle 1\\rangle :=\\vartheta ^{[s_0]}$ and $\\nu _i\\langle 1\\rangle :=\\nu ^{(s_0)}_i-\\vartheta ^{[s_0]}$ for $i=1,2,3$ .", "We can see $\\nu _i^{(s_0)}$ as refinements of $\\nu _i$ satisfying the same hypotheses on $\\nu _i$ .", "The web $\\vartheta \\langle 1\\rangle $ is approximately a common refinement of $\\nu _i$ .", "With $s_0$ large enough, the mass of the woven current associated with $\\nu _i\\langle 1\\rangle $ , i.e.", "the error of the approximation, which does not depend on $i$ , is smaller than $1/2$ .", "We repeat the above construction in order to refine approximately $\\nu _i\\langle 1\\rangle $ .", "We obtain a web $\\vartheta \\langle 2\\rangle $ such that the woven current associated with $\\nu _i\\langle 1\\rangle $ is approched by the one associated with $\\vartheta \\langle 2\\rangle $ : the difference of these currents is associated with three measurable webs $\\nu _i\\langle 2\\rangle $ and has a mass smaller than $1/4$ .", "By induction, we obtain sequences $\\nu _i\\langle m\\rangle $ and $\\vartheta \\langle m\\rangle $ such that the mass of the woven current associated with $\\nu _i\\langle m\\rangle $ is smaller than $2^{-m}$ .", "By construction, the web $\\nu _0:=\\sum \\vartheta \\langle m\\rangle $ refines all $\\nu _1$ , $\\nu _2$ and $\\nu _3$ .", "This completes the proof of the lemma.", "Weakly laminar, laminar and tame currents.", "We introduce now currents with stronger geometric properties.", "Definition 3.7 A woven current $T$ of bi-dimension $(r,r)$ on $V$ is weakly laminarThe terminology is changed with respect to the one in [8].", "if it admits a measurable $r$ -web $\\nu $ such that for $\\nu \\times \\nu $ -almost every pair of lames $S=\\sum [\\Lambda _i]$ and $S^{\\prime }=\\sum [\\Lambda _j^{\\prime }]$ either $\\Lambda _i\\cap \\Lambda _j^{\\prime }=\\varnothing $ or $\\Lambda _i\\cap \\Lambda _j^{\\prime }$ is open in $\\Lambda _i$ and in $\\Lambda _j^{\\prime }$ for all $i,j$ .", "We say that $T$ is laminar if there is a measurable $r$ -web $\\nu $ associated with $T$ and a measurable subset $A$ of ${\\rm Lam}_r^(V)$ such that $\\nu =0$ outside $A$ and for all $S=\\sum [\\Lambda _i]$ and $S^{\\prime }=\\sum [\\Lambda _j^{\\prime }]$ in $A$ either $\\Lambda _i\\cap \\Lambda _j^{\\prime }=\\varnothing $ or $\\Lambda _i\\cap \\Lambda _j^{\\prime }$ is open in $\\Lambda _i$ and in $\\Lambda _j^{\\prime }$ for all $i,j$ .", "We say that $T$ is completely weakly laminar or completely laminar if the above corresponding property holds for all measurable webs $\\nu $ associated with $T$ .", "A priori, the pairs of lames satisfying the condition for the weak laminarity property form a subset in ${\\rm Lam}_r^(V)\\times {\\rm Lam}_r^(V)$ which is not necessarily of the product form $A\\times A$ .", "So laminar currents are weakly laminar.", "One can show that the converse holds when $r=\\dim V-1$ .", "If $\\Lambda _1$ and $\\Lambda _2$ are two manifolds of dimension $r$ such that $\\Lambda _1\\cap \\Lambda _2$ is non-empty and of dimension $<r$ then $[\\Lambda _1]+[\\Lambda _2]$ is laminar but it is not completely weakly laminar.", "To see this point, we can consider the web which is the sum of the Dirac masses at $[\\Lambda _1]$ and at $[\\Lambda _2\\setminus \\Lambda _1]$ .", "We have the following proposition which was obtained by Dujardin for $(1,1)$ -currents on manifolds of dimension 2 [16].", "Proposition 3.8 Let $T$ be a woven positive closed $(p,p)$ -current on $V$ .", "Let $K$ be a compact subset of $V$ such that $T$ has no mass on $K$ .", "Assume that outside $K$ the current $T$ can be locally written as a wedge-product of positive closed $(1,1)$ -currents with continuous potentials.", "If $2p\\le l$ , we assume moreover that $T\\wedge T=0$ on $X\\setminus K$ .", "Then $T$ is completely weakly laminar.", "Observe that the case $2p>l$ can be reduced to the case $2p\\le l$ by replacing $V$ by $V\\times V$ and $T$ by $T\\otimes [V]$ in $V\\times V$ .", "Assume that $2p\\le l$ .", "Let $\\nu $ be a measurable web associated with $T$ .", "We have to show that $T$ is weakly laminar with respect to $\\nu $ .", "We can refine $\\nu $ in order to assume that $\\nu $ -almost every lame is defined by an irreducible manifold which does not intersect $K$ .", "Assume that $T$ is not completely weakly laminar and set $r:=l-p$ .", "Then for a suitable $\\nu $ , there is a subset ${\\cal W}$ of ${\\rm Lam}_r^(V)\\times {\\rm Lam}_r^(V)$ of positive $\\nu \\times \\nu $ measure such that for every $([Y],[Z])$ in ${\\cal W}$ we have $Y\\cap Z\\ne \\varnothing $ and $\\dim Y\\cap Z=s$ for some integer $0\\le s\\le r-1$ .", "We can refine $\\nu $ and reduce ${\\cal W}$ in order to assume that all these sets $Y$ and $Z$ are closed submanifolds of a fixed open subset $\\mathbb {U}$ of $V\\setminus K$ , as in the above local description of woven currents.", "The tangent cone of $Y\\times Z$ with respect to the diagonal $\\Delta $ of $V\\times V$ is a non-empty variety.", "We deduce from the definition of tangent currents that no tangent current of $T\\otimes T$ along $\\Delta $ vanishes over $\\mathbb {U}$ .", "Here we identify $\\mathbb {U}$ with an open subset of $\\Delta $ .", "Recall that such a tangent current is a positive closed current on the normal bundle to $\\Delta $ .", "It can be obtained locally as a limit value of the images of $T\\otimes T$ by a sequence of dilations in the normal directions to $\\Delta $ , see [14].", "On the other hand, by Theorem 5.10 in [14], over $\\mathbb {U}$ , this tangent cone should be the pull-back of the current $T\\wedge T$ to the normal vector bundle to $\\Delta $ .", "This contradicts the hypothesis that $T\\wedge T=0$ .", "The proposition follows.", "Let ${\\rm Gr}(V,r)$ denote the Grassmannian bundle over $V$ which is the set of points $(x,[v])$ , where $x$ is a point in $V$ and $[v]$ is the direction of a simple complex tangent $r$ -vector of $V$ at $x$ .", "If $S$ is a current in ${\\rm Lam}_r(V)$ , write $S=\\sum [\\Lambda _i]$ .", "We can lift each $\\Lambda _i$ to ${\\rm Gr}(V,r)$ by considering the set $\\widehat{\\Lambda }_i$ of points $(x,[v])$ with $x$ a regular point in $\\Lambda _i$ and $v$ tangent to $\\Lambda _i$ at $x$ .", "If $\\sum \\Vert \\widehat{\\Lambda }_i\\Vert $ is finite, $\\widehat{S}:=\\sum [\\widehat{\\Lambda }_i]$ is a current in ${\\rm Lam}_r({\\rm Gr}(V,r))$ and we say that $\\widehat{S}$ is the lift of $S$ to ${\\rm Gr}(V,r)$ .", "It does not depend on the choice of the decomposition $S=\\sum [\\Lambda _i]$ .", "Let $T$ be a woven positive closed current of bi-dimension $(r,r)$ and let $\\nu $ be a measurable $r$ -web associated with $T$ .", "We have $T=\\int _{{\\rm Lam}_r^(V)} S d\\nu (S).$ Assume that $\\nu $ -almost every $S$ admits a lift to ${\\rm Gr}(V,r)$ .", "We can always have this property by refining $\\nu $ .", "If the integral $\\int _{{\\rm Lam}_r^(V)} \\Vert \\widehat{S}\\Vert d\\nu (S)$ is finite, the current $\\widehat{T}:= \\int _{{\\rm Lam}_r^(V)} \\widehat{S} d\\nu (S)$ is well-defined and is called a lift of $T$ to ${\\rm Gr}(V,r)$ .", "It may depend on the choice of the measurable web $\\nu $ .", "The push-forward of $\\widehat{T}$ to $V$ is always equal to $T$ .", "Definition 3.9 We say that $T$ is almost tame if it admits a measurable web $\\nu $ as above with $\\int _{{\\rm Lam}_r^(V)} \\Vert \\widehat{S}\\Vert d\\nu (S)$ finite and if there is a positive closed current $\\widehat{T}^{\\prime }$ on ${\\rm Gr}(V,r)$ such that $\\widehat{T}^{\\prime }\\ge \\widehat{T}$ and the push-forward of $\\widehat{T}^{\\prime }$ to $V$ is equal to $T$ .", "We say that $T$ is tame if we can choose $\\widehat{T}^{\\prime }$ equal to $\\widehat{T}$ , i.e.", "the last current is closed.", "The above measurable web $\\nu $ is said to be almost tame or tame respectively.", "Note that almost tame currents are necessarily closed.", "Example 3.10 Consider the situation in Example REF .", "Set $\\Omega _{k-r}:=\\omega _{\\rm FS}^{k-r}$ .", "If $\\widehat{H}$ is the lift of $H$ to ${\\rm Gr}(\\mathbb {P}^k,r)$ , since the action of $\\mathbb {U}(k+1)$ extends canonically to ${\\rm Gr}(\\mathbb {P}^k,r)$ , the positive closed current $\\widehat{\\Omega }_{k-r}:=d^{-1}\\int _{\\tau \\in \\mathbb {U}(k+1)} \\tau _[\\widehat{H}]d\\sigma (\\tau )$ is a lift of $\\Omega _{k-r}$ to ${\\rm Gr}(\\mathbb {P}^k,r)$ .", "It is invariant under the action of $\\mathbb {U}(k+1)$ .", "Since ${\\rm Gr}(\\mathbb {P}^k,r)$ is a homogeneous space which is the quotient of $\\mathbb {U}(k+1)$ by a subgroup, $\\widehat{\\Omega }_{k-r}$ is a smooth form.", "For $d=1$ , we call $\\widehat{\\Omega }_{k-r}$ the standard lift of $\\Omega _{k-r}$ to ${\\rm Gr}(\\mathbb {P}^k,r)$ .", "Note that if $H$ is a smooth hypersurface of degree $d$ of $\\mathbb {P}^k$ , then the lift $\\widehat{H}$ of $H$ to ${\\rm Gr}(\\mathbb {P}^k,k-1)$ has volume of order $d^2$ .", "This is the reason why the limit currents of varieties are not woven in general.", "Let $\\pi :V\\rightarrow W$ be a holomorphic submersion onto a compact complex manifold $W$ .", "If $S=\\sum [\\Lambda _i]$ is as above write $S=S_1+S_2$ with $S_1:=\\sum _1 [\\Lambda _i]$ and $S_2:=\\sum _2 [\\Lambda _i]$ , where $\\sum _1$ is taken over the $\\Lambda _i$ 's such that the restriction of $\\pi $ to $\\Lambda _i$ is generically of maximal rank and $\\sum _2$ is taken over the other $\\Lambda _i$ 's.", "Let $T=\\int _{{\\rm Lam}^_r(V)} [S] d\\nu (S)$ be a woven positive closed current of bi-dimension $(r,r)$ associated with a measurable web $\\nu $ .", "Write $T=T_1+T_2$ with $T_i:=\\int _{{\\rm Lam}_r^(V)} [S_i] d\\nu (S)$ .", "We have the following lemma that can be extended to the case where $\\pi $ is a dominant meromorphic map.", "Lemma 3.11 Let $T,\\nu ,T_1,T_2$ be as above.", "If $\\nu $ is almost tame, then $T_1$ and $T_2$ are closed.", "We use the notations introduced above.", "Denote by $Z$ the analytic set of points $(x,[v])$ in ${\\rm Gr}(V,r)$ such that $v$ is not transverse to the fiber of $\\pi $ through $x$ .", "Let $\\widehat{T}_2$ be the restriction of $\\widehat{T}^{\\prime }$ to $Z$ .", "This is a positive closed current.", "Note that for every lame $S=\\sum [\\Lambda _i]$ , we have $\\widehat{\\Lambda }_i \\subset Z$ if and only if the rank of $\\pi $ on $\\Lambda _i$ is not maximal.", "Define $\\widehat{T}_1:=\\widehat{T}^{\\prime }-\\widehat{T}_2$ .", "This current is also positive and closed.", "By Definition REF , the push-forwards of $\\widehat{T}^{\\prime }$ and $\\widehat{T}$ to $V$ are both equal to $T$ .", "Therefore, $T_i$ is the push-forward of $\\widehat{T}_i$ to $V$ .", "The lemma follows.", "Woven currents as limits of analytic sets.", "In dynamics, woven currents are often constructed as limits of currents of integration on analytic sets.", "The following result was obtained in [7].", "Theorem 3.12 Let $\\Gamma _n$ be a sequence of analytic subsets of pure dimension $r$ in a projective manifold $V$ and let $d_n$ be positive numbers such $d_n^{-1}[\\Gamma _n]$ converge to a current $T$ .", "Let $\\widehat{\\Gamma }_n$ be the lift of $\\Gamma _n$ to ${\\rm Gr}(V,r)$ .", "Assume that the $2r$ -dimensional volume of $\\widehat{\\Gamma }_n$ is bounded by $cd_n$ for some constant $c>0$ .", "Then $T$ is woven.", "Note that we can lift the regular part of $\\Gamma _n$ to ${\\rm Gr}(V,r)$ and its compactification is an analytic subset of ${\\rm Gr}(V,r)$ that we still denote by $\\widehat{\\Gamma }_n$ .", "Sketch of the proof.", "Since $V$ is projective, it can be embedded in a projective space.", "For simplicity, we can assume that $V$ is the projective space $\\mathbb {P}^k$ and $d_n$ is the degree of $\\Gamma _n$ .", "Fix a generic central projection $\\pi :\\mathbb {P}^k\\setminus I\\rightarrow \\mathbb {P}^r$ , where $I$ is a projective subspace of dimension $k-r-1$ of $\\mathbb {P}^k$ and $\\mathbb {P}^r$ is identified with a projective subspace in $\\mathbb {P}^k\\setminus I$ .", "If $z$ is a point in $\\mathbb {P}^k\\setminus I$ , then $\\pi (z)$ is the intersection of $\\mathbb {P}^r$ with the projective subspace of dimension $k-r$ containing $I$ and $z$ .", "If $z_0$ is a generic point in $\\mathbb {P}^r$ and $U$ is a small neighbourhood of $z_0$ , we can show that $\\pi ^{-1}(U)\\cap \\Gamma _n$ contains almost $d_n$ graphs over $U$ for $n$ large enough ($d_n$ is the maximal number one can have).", "This is the consequence of the property that the set of ramification of $\\pi $ restricted to $\\Gamma _n$ is small enough over $U$ .", "We will see in Propositions REF and REF below similar situations.", "The control of the ramification is obtained from the hypothesis on the volume $\\widehat{\\Gamma }_n$ using Fubini theorem and a generic choice of $\\pi $ , $z_0$ .", "The limits of the obtained graphs as $n\\rightarrow \\infty $ form a part of $T$ .", "We have to cover $\\mathbb {P}^r$ with such open sets $U$ with different sizes in order to construct a complete measurable web associated with $T$ .", "For the details, see [7].", "$\\square $ The following result can be deduced from the proof of the above theorem.", "Proposition 3.13 There is an increasing sequence of integers $(n_i)$ and measurable $r$ -webs $\\nu _{n_i}$ and $\\nu $ associated with $d_{n_i}^{-1}[\\Gamma _{n_i}]$ and $T$ such that $\\nu _{n_i}\\rightarrow \\nu $ in the weak sense of measures on ${\\rm Lam}_r^(V)$ .", "Moreover, we can write $\\nu _{n_i}=\\sum _{s\\ge 1} \\nu _{n_i}[s]$ and $\\nu =\\sum _{s\\ge 1} \\nu [s]$ such that $\\nu _{n_i}[s]$ and $\\nu [s]$ are $s$ -elementary webs and $\\nu _{n_i}[s]\\rightarrow \\nu [s]$ in the weak sense of measures on ${\\rm Lam}^_r(V)$ .", "Proposition 3.14 Let $\\Gamma _n, \\widehat{\\Gamma }_n$ and $T$ be as in the last theorem.", "Assume that the $2r$ -dimensional volume of the lift of $\\widehat{\\Gamma }_n$ to ${\\rm Gr}({\\rm Gr}(V,r),r)$ is bounded by $cd_n$ for some constant $c>0$ .", "Then $T$ is almost tame.", "Let $S$ be a cluster value of the sequence $d_n^{-1}[\\widehat{\\Gamma }_n]$ .", "Applying Theorem REF to $\\widehat{\\Gamma }_n$ implies that $S$ is woven.", "Here, in order to check the hypotheses of that theorem, we need to lift $\\Gamma _n$ twice.", "With the construction explained above, we see that the elementary lames of $S$ are obtained as limits of open subsets of $\\widehat{\\Gamma }_n$ .", "So the elementary lames whose projections on $V$ are of dimension $r$ are the lifts of some varieties in $V$ to ${\\rm Gr}(V,r)$ .", "For the other elementary lames, their projections on $V$ vanish in the sense of currents.", "It follows that for a suitable measurable web, the lift of $T$ to ${\\rm Gr}(V,r)$ is bounded by $S$ .", "Since $S$ is closed, the current $T$ is almost tame.", "Theorem REF can be extended to some local situation.", "We will need some steps in the proof of such local version that we recall below.", "Let $\\mathbb {B}_r$ denote the unit ball and $\\rho \\mathbb {B}_r$ the ball of center 0 and of radius $\\rho $ in $r$ .", "Consider an analytic subset $\\Gamma $ of pure dimension $r$ of $3\\mathbb {B}_r\\times 3\\mathbb {B}_s$ , not necessarily irreducible, which is contained in $3\\mathbb {B}_r\\times 2\\mathbb {B}_s$ .", "For simplicity assume that $\\Gamma $ is smooth.", "So the natural projection from $\\Gamma $ onto $3\\mathbb {B}_r$ defines a ramified covering and we denote by $d$ its degree.", "The ramified locus is a divisor of $\\Gamma $ with integer coefficients.", "Its push-forward to $3\\mathbb {B}_r$ is a divisor with integer coefficients on $3\\mathbb {B}_r$ .", "The positive closed $(1,1)$ -current associated with this divisor is denoted by $[P]$ and is called the postcritical current.", "Observe that when $[P]=0$ the set $\\Gamma $ is a union of $d$ graphs over $3\\mathbb {B}_r$ .", "The following result gives us a more quantitative property.", "Proposition 3.15 There is a constant $c>0$ independent of $\\Gamma $ and $d$ such that $\\Gamma \\cap (2\\mathbb {B}_r\\times 3\\mathbb {B}_s)$ contains at least $d-c\\Vert P\\Vert $ graphs over $2\\mathbb {B}_r$ .", "The case of dimension $r=1$ is just a consequence of Riemann-Hurwitz's formula.", "The general case is reduced to the dimension 1 case by slicing $3\\mathbb {B}_r$ by lines through the origin.", "We then stick graphs of dimension 1 in order to get graphs of dimension $r$ over $2\\mathbb {B}_r$ using the main result in [25], see [7] for details.", "Note that the proposition still holds for $\\Gamma $ singular but the postcritical current $[P]$ has to be defined differently.", "The current $[P]$ depends strongly on the coordinate system we use.", "Therefore, in the general dynamical setting, we need a more subtle version of the last proposition.", "Define $\\mathbb {U}:=4\\mathbb {B}_r\\times 3\\mathbb {B}_s$ and assume now that $\\Gamma $ is a smooth analytic subset of pure dimension $r$ of $\\mathbb {U}$ which is contained in $4\\mathbb {B}_r\\times \\mathbb {B}_s$ and is a ramified covering of degree $d$ over $4\\mathbb {B}_r$ .", "Let ${\\rm Gr}(\\mathbb {U},s)$ denote the set of point $(z,[v])$ where $z$ is a point in $\\mathbb {U}$ and $[v]$ is the direction of a simple complex tangent $s$ -vector $v$ of $\\mathbb {U}$ at $z$ .", "It can be identified with the product of $\\mathbb {U}$ with the Grassmannian $\\mathbb {G}$ parametrizing the family of complex linear subspaces of dimension $s$ in ${r+s}$ through a fixed point.", "We have $\\dim \\mathbb {G}=rs$ and $\\dim {\\rm Gr}(\\mathbb {U},s)=rs+r+s$ .", "We are interested in linear subspaces close enough to the vertical ones.", "More precisely, we consider linear subspaces parallel to a space of equation $z^{\\prime }=Az^{\\prime \\prime } \\quad \\text{with} \\quad z=(z^{\\prime },z^{\\prime \\prime })\\in r\\times s,$ where $A$ is a complex $r\\times s$ -matrix whose coefficients have modulus smaller than 1.", "The family of those matrices is identified with an open set $\\mathbb {G}^\\star $ in $\\mathbb {G}$ .", "Define ${\\rm Gr}(\\mathbb {U},s)^\\star :=\\mathbb {U}\\times \\mathbb {G}^\\star $ .", "Denote by $\\widetilde{\\Gamma }$ the set of points $(z,[v])\\in {\\rm Gr}(\\mathbb {U},s)$ such that $z\\in \\Gamma $ and $v$ is not transverse to the tangent space of $\\Gamma $ at $z$ .", "So $\\widetilde{\\Gamma }$ is an analytic subset of ${\\rm Gr}(\\mathbb {U},s)$ of pure dimension $rs+r-1$ .", "Indeed, it is not difficult to see that the restriction of $\\widetilde{\\Gamma }$ to a fiber $\\lbrace z\\rbrace \\times \\mathbb {G}$ is a hypersurface of this fiber.", "Define $\\widetilde{\\Gamma }^\\star :=\\widetilde{\\Gamma }\\cap {\\rm Gr}(\\mathbb {U},s)^\\star $ .", "Proposition 3.16 There is a constant $c>0$ independent of $\\Gamma $ and $d$ such that $\\Gamma \\cap (\\mathbb {B}_r\\times 3\\mathbb {B}_s)$ contains at least $d-c\\Vert \\widetilde{\\Gamma }^\\star \\Vert $ graphs over $\\mathbb {B}_r$ .", "Fix a constant $\\delta >0$ small enough depending only on $r$ and $s$ .", "We will consider the projections $\\pi _A:r\\times s\\rightarrow r$ given by $\\pi _A(z):=z^{\\prime }-Az^{\\prime \\prime }$ with $\\Vert A\\Vert \\le \\delta $ .", "We will apply Proposition REF to the coordinate system $z_A=(z^{\\prime }_A,z^{\\prime \\prime }_A):=(z^{\\prime }-Az^{\\prime \\prime },z^{\\prime \\prime })$ instead of $(z^{\\prime },z^{\\prime \\prime })$ .", "We will add the letter $A$ in the above notations when we use these coordinates.", "In these coordinates, $\\pi _A$ is just the natural projection on the first $r$ coordinates.", "Since $\\delta $ is small, the following lemma is clear by continuity.", "Lemma 3.17 The restriction of $\\Gamma $ to $3\\mathbb {B}_r^A\\times 3\\mathbb {B}_s^A$ is a ramified covering of degree $d$ over $3\\mathbb {B}_r^A$ and is contained in $3\\mathbb {B}_r^A\\times 2\\mathbb {B}_s^A$ .", "If $Z$ is a graph over $2\\mathbb {B}_r^A$ contained in $2\\mathbb {B}_r^A\\times 2\\mathbb {B}_s^A$ then its restriction to $\\mathbb {B}_r\\times 3\\mathbb {B}_s$ is also a graph over $\\mathbb {B}_r$ .", "End of the proof of Proposition REF .", "Using the last lemma and Proposition REF , we only need to check that there is a matrix $A$ with $\\Vert A\\Vert \\le \\delta $ such that the mass of the associated poscritical current $[P_A]$ is smaller than a constant times the mass of $\\widetilde{\\Gamma }^\\star $ .", "By Fubini's theorem, there is a matrix $A$ with $\\Vert A\\Vert \\le \\delta $ such that the mass of $\\widetilde{\\Gamma }\\cap (\\mathbb {U}\\times \\lbrace A\\rbrace )$ is smaller than a constant times $\\Vert \\widetilde{\\Gamma }^\\star \\Vert $ , where the points in the last intersection are counted with multiplicity.", "It is enough now to observe that $P_A$ is equal on $3\\mathbb {B}_r^A$ to the image of $\\widetilde{\\Gamma }\\cap (\\mathbb {U}\\times \\lbrace A\\rbrace )$ by $\\pi _A$ .", "This completes the proof of the proposition.", "$\\square $ Slicing theory for woven currents.", "In Section , we already discussed slicing for positive closed currents and for varieties.", "Since woven currents are generated by pieces of varieties, the theory extends without difficulty to them.", "Let $\\pi :V\\rightarrow W$ and $l,m$ be as in the beginning of Section .", "Assume for simplicity that $V$ and $W$ are projective manifolds.", "Let $T$ be a woven current on $V$ of bi-dimension $(r,r)$ associated with a web $\\nu $ .", "It follows from Federer's slicing theory for flat currents that for almost every $y\\in W$ the slice $\\langle T\|\\pi \|y\\rangle $ exists and we have $\\langle T\|\\pi \|y\\rangle =\\int _{{\\rm Lam}_r^(V)} \\langle S\|\\pi \|y\\rangle d\\nu (S).$ Note that the family of $y$ satisfying the above property depends not only on $T$ but also on the choice of $\\nu $ .", "When the above identity holds for $y$ , we say that the web $\\nu $ is compatible with the slice $\\langle T\|\\pi \|y\\rangle $ .", "A necessary condition for $\\nu $ to be compatible with $\\langle T\|\\pi \|y\\rangle $ is that $\\nu $ -almost every lame $S$ either is disjoint from $\\pi ^{-1}(y)$ or intersects $\\pi ^{-1}(y)$ transversally at almost every point of intersection, see also Section .", "We describe now the situation that will be used later.", "Let $\\Gamma _n, T$ and $d_n$ be as in Theorem REF .", "Choose the covering of $V$ by cubes $\\mathbb {U}_1,\\ldots ,\\mathbb {U}_N$ as in Section 2.", "Proposition 3.18 There is an increasing sequence of integers $(n_i)$ such that for almost every $y\\in W$ the slices $\\langle T\|\\pi \|y\\rangle $ and $d_{n_i}^{-1}\\langle [\\Gamma _{n_i}]\|\\pi \|y\\rangle $ are well-defined and $d_{n_i}^{-1}\\langle [\\Gamma _{n_i}]\|\\pi \|y\\rangle \\rightarrow \\langle T\|\\pi \|y\\rangle $ .", "For almost every $y\\in W$ there are measurable webs $\\nu _{n_i}$ , $\\nu _{n_i}[s]$ , $\\nu $ , $\\nu [s]$ , depending on $y$ , which satisfy Proposition REF and such that $\\nu _{n_i}$ and $\\nu $ are compatible with the above slices.", "Moreover, for all $i,s$ and for every graph $\\Lambda $ corresponding to a point in the support of $\\nu _{n_i}[s]$ or $\\nu [s]$ , the fat extension of $\\Lambda $ either is disjoint from $\\pi ^{-1}(y)$ or intersects $\\pi ^{-1}(y)$ transversally.", "The first assertion is given by Proposition REF .", "By replacing $(n_i)$ with a suitable subsequence and using Proposition REF , we obtain $\\nu _{n_i}$ , $\\nu _{n_i}[s]$ , $\\nu $ , $\\nu [s]$ satisfying this proposition.", "Then for almost every $y$ , the webs $\\nu $ and $\\nu _{n_i}$ are compatible with $\\langle T\|\\pi \|y\\rangle $ and $\\langle [\\Gamma _{n_i}]\|\\pi \|y\\rangle $ .", "We explain how to modify the above webs in order to get the last property in the proposition.", "The modification depends on $y$ .", "Recall that almost every lame used here either is disjoint from $\\pi ^{-1}(y)$ or intersects $\\pi ^{-1}(y)$ transversally at almost every point of intersection.", "We want in particular to remove the second word \"almost\" in the last sentence.", "For this purpose, we can simply remove from each lame a suitable proper analytic subset and this does not modify the currents.", "However, since we want the lames to be elementary, the problem is slightly more delicate.", "Observe that any elementary lame of order $s$ can be divided into elementary lames of order $s+1$ and gives us a refinement.", "Fix a constant $\\epsilon _1>0$ small enough and consider the set $K$ of elementary lames $\\Lambda $ of order 1 such that either the distance between the very fat extension of $\\Lambda $ and $\\pi ^{-1}(y)$ is larger than $\\epsilon _1$ or this very fat extension intersects $\\pi ^{-1}(y)$ transversally and the angle between them is at least equal to $\\epsilon _1$ .", "This is a compact subset of ${\\rm Lam}^_r(V)$ .", "We replace $\\nu {[1]}$ with its restriction $\\nu [1]_{\|K}$ to $K$ .", "We also refine $\\nu [1]-\\nu [1]_{\|K}$ into a 2-elementary web that we add to $\\nu [2]$ .", "It is not difficult to find positive measures $\\nu _{n_i}[1]^{\\prime }\\le \\nu _{n_i}[1]$ such that $\\nu _{n_i}[1]^{\\prime }\\rightarrow \\nu [1]_{\|K}$ and for every $\\Lambda $ in the support of $\\nu _{n_i}^{\\prime }$ the fat extension of $\\Lambda $ either is disjoint from $\\pi ^{-1}(y)$ or intersects $\\pi ^{-1}(y)$ transversally.", "We do a similar modification to $\\nu _{n_i}[1]$ and $\\nu _{n_i}[2]$ : we replace $\\nu _{n_i}[1]$ with $\\nu _{n_i}[1]^{\\prime }$ and add to $\\nu _{n_i}[2]$ the refinement of order 2 of $\\nu _{n_i}[1]-\\nu _{n_i}[1]^{\\prime }$ .", "This completes the construction of $\\nu [1]$ and $\\nu _{n_i}[1]$ .", "We repeat now the same modification for $\\nu [2]$ and $\\nu _{n_i}[2]$ using another $\\epsilon _2$ small enough and the refinement of lames of order 2 into lames of order 3.", "We obtain the final webs $\\nu [2]$ and $\\nu _{n_i}[2]$ .", "A simple induction allows to obtain the webs satisfying the proposition.", "If we choose in each step the constant $\\epsilon _s$ small enough, it is not difficult to insure that there is no mass lost after an infinite number of steps, that is, the currents associated with $\\nu $ , $\\nu _{n_i}$ , after modifications, are still equal to $T$ , $d_{n_i}^{-1}[\\Gamma _{n_i}]$ and compatible with the slicing by $\\pi ^{-1}(y)$ .", "This ends the proof of the proposition.", "We come back now to the situation in Section where $V=V^{\\prime }\\times \\mathbb {P}^k$ .", "We use the notations introduced there.", "Let $T$ be a woven current on $V$ of bi-dimension $(r,r)$ associated with a web $\\nu $ .", "Then for almost every $\\xi \\in \\mathbb {G}$ the intersection $T\\wedge [V^{\\prime }\\times H_\\xi ]$ exists and we have $T\\wedge [V^{\\prime }\\times H_\\xi ]=\\int _{{\\rm Lam}_r^(V)} S\\wedge [V^{\\prime }\\times H_\\xi ]d\\nu (S).$ The family of $\\xi $ satisfying the last identity depends on the choice of $\\nu $ .", "When the above identity holds for $\\xi $ , we say that the lamination $\\nu $ is compatible with the intersection $T\\wedge [V^{\\prime }\\times H_\\xi ]$ .", "A necessary condition for $\\nu $ to be compatible with $T\\wedge [V^{\\prime }\\times H_\\xi ]$ is that $\\nu $ -almost every lame $S$ either is disjoint from $V^{\\prime }\\times H_\\xi $ or intersects $V^{\\prime }\\times H_\\xi $ transversally at almost every point of intersection.", "We have the following direct consequence of Proposition REF and Proposition REF .", "Corollary 3.19 Let $T$ and $\\Gamma _n$ be as in Proposition REF .", "Then there is an increasing sequence of integers $(n_i)$ such that for almost every $\\xi \\in \\mathbb {G}$ the intersections $T\\wedge [V^{\\prime }\\times H_\\xi ]$ and $d_{n_i}^{-1}[\\Gamma _{n_i}]\\wedge [V^{\\prime }\\times H_\\xi ]$ are well-defined and we have $d_{n_i}^{-1}[\\Gamma _{n_i}]\\wedge [V^{\\prime }\\times H_\\xi ]\\rightarrow T\\wedge [V^{\\prime }\\times H_\\xi ]$ .", "For almost every $\\xi \\in \\mathbb {G}$ , there are measurable webs $\\nu _{n_i}$ , $\\nu _{n_i}[s]$ , $\\nu $ , $\\nu [s]$ , depending on $\\xi $ , which satisfy Proposition REF and such that $\\nu _{n_i}$ , $\\nu $ are compatible with the above intersections.", "Moreover, for all $i,s$ and for every graph $\\Lambda $ corresponding to a point in the support of $\\nu _{n_i}[s]$ or $\\nu [s]$ , the fat extension of $\\Lambda $ either is disjoint from $V^{\\prime }\\times H_\\xi $ or intersects $V^{\\prime }\\times H_\\xi $ transversally.", "We will also need the following lemma.", "Lemma 3.20 Let $\\nu _n, \\nu $ be $s$ -elementary webs and let $T_n,T$ be the associated woven currents.", "Assume that $\\nu _n\\rightarrow \\nu $ .", "Let $H$ be a projective subspace of $\\mathbb {P}^k$ .", "Assume that for $\\nu $ -almost every graph $\\Lambda $ , the fat extension of $\\Lambda $ either is disjoint from $V^{\\prime }\\times H$ or intersects $V^{\\prime }\\times H$ transversally.", "Assume also that the intersections $T_n\\wedge [V^{\\prime }\\times H]$ and $T\\wedge [V^{\\prime }\\times H]$ exist and are compatible with the above webs.", "Then any limit value of $T_n\\wedge [V^{\\prime }\\times H]$ is larger than or equal to $T\\wedge [V^{\\prime }\\times H]$ .", "We prove the lemma assuming only that $\\lim \\nu _n\\ge \\nu $ .", "This allows to replace $\\nu $ with its restrictions to suitable compact sets in order to get the above condition on $\\Lambda $ for every graph $\\Lambda $ corresponding to a point in the support of $\\nu $ .", "Let $\\nu _n^{\\prime }$ be the restriction of $\\nu _n$ to a small enough neighbourhood of the support of $\\nu $ .", "We have $\\lim \\nu _n^{\\prime }\\ge \\nu $ and $\\nu _n^{\\prime }\\le \\nu _n$ .", "Observe that if a graph $\\Lambda ^{\\prime }$ is close enough to the graph $\\Lambda $ in the lemma then the intersection $\\Lambda ^{\\prime }\\cap (V^{\\prime }\\times H)$ is transverse and close to $\\Lambda \\cap (V^{\\prime }\\times H)$ .", "It follows that for $n$ large enough, the current obtained by intersecting the lames of $\\nu _n^{\\prime }$ with $[V^{\\prime }\\times H]$ is close to $T\\wedge [V^{\\prime }\\times H]$ .", "Since $\\nu _n^{\\prime }\\le \\nu _n$ and $\\lim \\nu _n^{\\prime }\\ge \\nu $ the lemma follows." ], [ "Green currents of Hénon maps", "In this section, we give some crucial properties of Green currents associated with Hénon maps that we need for the proof of the main theorem.", "Let $f$ be a Hénon map on $k\\subset \\mathbb {P}^k$ as in the introduction.", "Define $T_+:=\\tau _+^p$ and $T_-:=\\tau _-^{k-p}$ .", "They are positive closed currents of mass 1 respectively of bi-degree $(p,p)$ and $(k-p,k-p)$ with support in $\\overline{K}_+$ and $\\overline{K}_-$ .", "We have $f^(T_+)=dT_+$ and $f_(T_-)=d T_-$ .", "For dimension reason $T_\\pm $ have no mass on $I_\\pm $ and then they have no mass on the hyperplane at infinity $H_\\infty $ because $\\overline{K}_\\pm =K_\\pm \\cup I_\\pm $ .", "The following result was obtained in [12] using the theory of super-potentials.", "Theorem 4.1 The current $T_+$ is the unique positive closed $(p,p)$ -current of mass 1 supported by $\\overline{K}_+$ .", "In particular, it is an extremal positive closed $(p,p)$ -current on $\\mathbb {P}^k$ .", "Moreover, if $S$ is a positive closed $(p,p)$ -current of mass 1, smooth in a neighbourhood of $I_-$ , then $d^{-n}(f^n)^(S)$ converges to $T_+$ as $n\\rightarrow \\infty $ .", "Note that a similar version of the above theorem holds for $T_-,\\overline{K}_-$ and $f^{-1}$ .", "The following theorem generalizes some results in [1] for dimension $k=2$ and [7] for higher dimension.", "Theorem 4.2 The currents $T_+$ and $T_-$ are completely weakly laminar and tame.", "We will prove the result for $T_+$ .", "The case of $T_-$ can be obtained in the same way.", "Let $L$ be a linear subspace of dimension $k-p$ of $\\mathbb {P}^k$ which does not intersect $I_-$ .", "It follows from Theorem REF that $d^{-n} [f^{-n}(L)]$ converges to $T_+$ .", "Denote by $L_n$ the compactification of $f^{-n}(L)\\cap k$ and let $\\widehat{L},\\widehat{L}_n$ be respectively the lifts of $L,L_n$ to ${\\rm Gr}(\\mathbb {P}^k,k-p)$ , see the notations in Section and the end of Section .", "We have $\\widehat{L}_n=\\widehat{f}^{-n}(\\widehat{L})$ in the Zariski open subset $k\\times \\mathbb {G}$ of ${\\rm Gr}(\\mathbb {P}^k,k-p)$ .", "The varieties $\\widehat{L}$ and $\\widehat{L}_n$ are irreducible and not contained in the complement of $k\\times \\mathbb {G}$ .", "The properties of $f^n$ described in Example REF imply that the volume of $\\widehat{L}_n$ is equal to $O(d^n)$ .", "We used here that the mass of a positive closed current, in particular the volume of an analytic set, only depends on its cohomology class.", "In the same example, we see that the volume of the lift of $\\widehat{L}_n$ to ${\\rm Gr}({\\rm Gr}(\\mathbb {P}^k,k-p),k-p)$ is also equal to $O(d^n)$ .", "Therefore, by Theorem REF and Proposition REF , the Green current $T_+$ and the cluster values of $d^{-n}[\\widehat{L}_n]$ are almost tame woven positive closed currents.", "Let $S$ be such a cluster value on $k\\times \\mathbb {G}$ .", "Since it has finite mass, its extension by 0 is a positive closed current on ${\\rm Gr}(\\mathbb {P}^k,k-p)$ .", "Recall that we can construct a measurable web associated with $S$ such that its lames are elementary and obtained as limits of open subsets of $\\widehat{L}_n$ .", "Write $S=S^{\\prime }+S^{\\prime \\prime }$ , where $S^{\\prime }$ (resp.", "$S^{\\prime \\prime }$ ) is formed by lames whose projections on $k$ are non-degenerate (resp.", "degenerate), i.e.", "of dimension $k-p$ (resp.", "smaller than $k-p$ ).", "So for a suitable measurable web, $S^{\\prime }$ is a lift of $T_+$ to ${\\rm Gr}(\\mathbb {P}^k,k-p)$ .", "By Lemma REF , $S^{\\prime }$ and $S^{\\prime \\prime }$ are closed.", "We want to prove that $T_+$ is tame.", "For this purpose, it is sufficient to check that $S^{\\prime \\prime }=0$ .", "Consider the family $\\mathcal {F}$ of all currents $S^{\\prime \\prime }$ obtained as above for different cluster values $S$ .", "These currents have a bounded mass.", "If $S$ is the limit of a sequence $d^{-n_j} [\\widehat{L}_{n_j}]$ and $S_1$ is a cluster value of $d^{-n_j+1} [\\widehat{L}_{n_j-1}]$ then $d^{-1}\\widehat{f}^\\bullet (S_1)=S$ on $k\\times \\mathbb {G}$ and hence on ${\\rm Gr}(\\mathbb {P}^k,k-p)$ .", "It follows that there is a current $S_1^{\\prime \\prime }$ in $\\mathcal {F}$ such that $S^{\\prime \\prime }=d^{-1}\\widehat{f}^\\bullet (S_1^{\\prime \\prime })$ .", "By induction, there are currents $S_n^{\\prime \\prime }$ in $\\mathcal {F}$ such that $d^{-n}(\\widehat{f}^n)^\\bullet (S_n^{\\prime \\prime })=S^{\\prime \\prime }$ .", "The h-dimension of currents in $\\mathcal {F}$ with respect to the projection on $\\mathbb {P}^k$ is smaller than $k-p$ .", "Therefore, their cohomology classes are in $E_{q,p+1}$ for some $q$ , see Section for the notation.", "The claim in the proof of Proposition REF implies that the norm of $(\\widehat{f}^n)^$ on $E_{q,p+1}$ is equal to $o(d^n)$ .", "The relation between $S^{\\prime \\prime }$ and $S_n^{\\prime \\prime }$ and the fact that $S_n^{\\prime \\prime }$ have a bounded mass imply that $S^{\\prime \\prime }=0$ .", "So $T_+$ is tame.", "It remains to prove that $T_+$ is completely weakly laminar.", "Recall that $T_+=\\tau _+^p$ and $\\tau _+$ has local continuous potentials outside $I_+$ .", "Moreover, $T_+$ has no mass on $I_+$ and $\\tau _+^{p+1}=0$ outside $I_+$ , see [24] for details.", "Proposition REF implies the result.", "Recall that the measure $\\mu $ is mixing and hyperbolic, see [5], [8].", "It admits $p$ positive and $k-p$ negative Lyapounov exponents.", "For $\\mu $ almost every point $z$ denote by $E_u(z)$ the unstable tangent subspace of $\\mathbb {P}^k$ at the point $z$ and $E_s(z)$ the stable one.", "So we have $\\dim E_u(z)=p$ and $\\dim E_s(z)=k-p$ .", "We will denote by $[E_s(z)]$ the direction of the complex tangent $(k-p)$ -vectors at $z$ defining $E_s(z)$ .", "The set of points $(z,[E_s(z)])$ in ${\\rm Gr}(\\mathbb {P}^k,k-p)$ can be seen as a measurable graph over $\\mu $ -almost every point in the support of $\\mu $ .", "So we can lift $\\mu $ to a probability measure $\\mu _+$ on this graph that we call stable Oseledec measure associated with $f$ and $\\mu $ .", "Since the stable bundle is invariant and $\\mu $ is mixing, $\\mu _+$ is also invariant under $\\widehat{f}$ and mixing.", "We can construct in the same way the unstable Oseledec measure $\\mu _-$ associated with $f$ and $\\mu $ .", "It is a probability measure on the set of points $(z,[E_u(z)])$ in ${\\rm Gr}(\\mathbb {P}^k,p)$ .", "The following result characterizes Oseledec measures.", "We will use it for $q=0$ and for the form $\\widehat{\\Omega }_p$ defined in Example REF in order to prove the intersection properties we need.", "Recall that over $k$ we identify ${\\rm Gr}(\\mathbb {P}^k,k-p)$ with $k\\times \\mathbb {G}$ .", "Let $m$ denote the dimension of $\\mathbb {G}$ .", "Let $\\pi :{\\rm Gr}(\\mathbb {P}^k,k-p)\\rightarrow \\mathbb {P}^k$ denote the canonical projection.", "Proposition 4.3 Let $q$ be an integer such that $0\\le q\\le p$ .", "Let $\\alpha $ be a smooth positive closed form of bi-dimension $(k-p+q,k-p+q)$ on ${\\rm Gr}(\\mathbb {P}^k,k-p)$ .", "Let $c$ be the mass of the current $\\pi _(\\alpha )$ .", "Then $d_+^{-(p-q)n}(\\widehat{f}^n)^* (\\alpha )\\wedge \\pi ^(\\tau _+^q\\wedge T_-)$ defines a positive measure of mass $c$ on $k\\times \\mathbb {G}$ which converges to $c\\mu _+$ as $n\\rightarrow \\infty $ .", "Since $\\pi $ is a submersion, $\\pi _(\\alpha )$ is a smooth positive closed $(p-q,p-q)$ -form.", "Since $(\\widehat{f}^n)^* (\\alpha )$ is smooth on $k\\times \\mathbb {G}$ , the wedge-product in the proposition is well-defined and is a positive measure.", "Its mass is equal to the mass of its push-forward to $\\mathbb {P}^k$ .", "This push-forward is equal to $d_+^{-(p-q)n}(f^n)^* \\pi _(\\alpha )\\wedge \\tau _+^q\\wedge T_-$ .", "Since $(f^n)^ \\pi _(\\alpha )$ is smooth on the support of $T_-$ , the last wedge-product on $k$ coincides with the same wedge-product on $\\mathbb {P}^k$ .", "Since $\\tau _+$ and $T$ are of mass 1 and $d_+^{-(p-q)n}(f^n)^$ is equal to the identity on $H^{p-q,p-q}(\\mathbb {P}^k,$ , it is not difficult to see that the mass of the considered measure is equal to the mass $c$ of $\\pi _(\\alpha )$ .", "For the proof of the last proposition, we will use a decreasing induction on $q$ .", "The following lemma proves the case $q=p$ .", "The main ingredient here is Oseledec's theorem.", "Lemma 4.4 Let $\\alpha $ be a smooth closed $(m,m)$ -form on ${\\rm Gr}(\\mathbb {P}^k,k-p)$ .", "Then we have $(\\widehat{f}^n)^ (\\alpha )\\wedge \\pi ^(\\mu )\\rightarrow c\\mu _+$ on $k\\times \\mathbb {G}$ , where $c$ is the constant equal to $\\pi _(\\alpha )$ .", "Since $\\alpha $ can be written as a combination of smooth positive closed forms, we can assume that it is positive.", "Observe that $\\pi _(\\alpha )$ is a closed $(0,0)$ -current.", "So it is defined by the above constant $c$ .", "For simplicity, assume that $c=1$ .", "We deduce that for any probability measure $\\nu $ on $k$ the positive measure $\\alpha \\wedge \\pi ^(\\nu )$ is of mass 1 or equivalently the push-forward of $\\alpha \\wedge \\pi ^(\\nu )$ to $k$ is equal to $\\nu $ .", "In particular, the mass of $\\alpha \\wedge [\\pi ^{-1}(z)]$ is 1 for every $z\\in k$ .", "Let $\\widehat{\\varphi }$ be a smooth test function on ${\\rm Gr}(\\mathbb {P}^k,k-p)$ .", "We have to show that $\\big \\langle (\\widehat{f}^n)^\\bullet (\\alpha )\\wedge \\pi ^(\\mu ),\\widehat{\\varphi }\\big \\rangle \\rightarrow \\langle \\mu _+,\\widehat{\\varphi }\\rangle .$ Define for $\\mu $ -almost every $z$ the function $\\psi (z):=\\widehat{\\varphi }(z,[E_s(z)])$ .", "Define also $\\widehat{\\psi }:=\\psi \\circ \\pi $ .", "For $\\mu $ -almost every point $z$ , denote by $\\Sigma (z)$ the set of points $(z,[v])$ in $\\pi ^{-1}(z)$ such that $v$ is not transverse to $E_u(z)$ .", "It is a hypersurface in $\\pi ^{-1}(z)$ and hence of Lebesgue measure 0 there.", "By Oseledec's theorem [29], if $(z,[v])$ is out of $\\Sigma (z)$ and $z_n:=f^{-n}(z)$ , then the distance between $\\widehat{f}^{-n}(z,[v])$ and $(z_n,[E_s(z_n)])$ tends to 0.", "It follows that $\\widehat{\\varphi }\\circ \\widehat{f}^{-n}-\\widehat{\\psi }\\circ \\widehat{f}^{-n}$ tends to 0 outside the union of $\\Sigma (z)$ .", "Since this function is bounded, using that $\\alpha $ is smooth, we deduce that $\\big \\langle \\alpha \\wedge \\pi ^(\\mu ), \\widehat{\\varphi }\\circ \\widehat{f}^{-n}-\\widehat{\\psi }\\circ \\widehat{f}^{-n}\\big \\rangle \\rightarrow 0.$ This together with the following identities imply the result.", "We have since $\\mu $ is invariant $\\big \\langle \\alpha \\wedge \\pi ^(\\mu ), \\widehat{\\varphi }\\circ \\widehat{f}^{-n}\\big \\rangle = \\big \\langle (\\widehat{f}^n)^\\bullet \\big (\\alpha \\wedge \\pi ^(\\mu )\\big ),\\widehat{\\varphi }\\big \\rangle = \\big \\langle (\\widehat{f}^n)^\\bullet (\\alpha )\\wedge \\pi ^(\\mu ),\\widehat{\\varphi }\\big \\rangle $ and since the push-forward of $\\alpha \\wedge \\pi ^(\\mu )$ to $k$ is equal to $\\mu $ $\\big \\langle \\alpha \\wedge \\pi ^(\\mu ), \\widehat{\\psi }\\circ \\widehat{f}^{-n}\\big \\rangle = \\big \\langle \\alpha \\wedge \\pi ^(\\mu ), \\pi ^(\\psi \\circ f^{-n})\\big \\rangle = \\langle \\mu , \\psi \\circ f^{-n}\\rangle =\\langle \\mu ,\\psi \\rangle .$ The last integral is also equal to $\\langle \\mu _+,\\widehat{\\varphi }\\rangle $ .", "The proposition follows.", "Lemma 4.5 Let $\\widehat{\\mu }$ be a positive measure on $k\\times \\mathbb {G}$ .", "Assume there is a negative current $U$ of bi-dimension $(1,1)$ on $k\\times \\mathbb {G}$ such that $\\widehat{\\mu }-\\mu _+={dd^c}U$ and $\\pi _(U)=0$ .", "Then $\\widehat{\\mu }=\\mu _+$ .", "By hypotheses, we have $\\pi _(\\widehat{\\mu })=\\pi _(\\mu _+)$ .", "Therefore, $\\widehat{\\mu }$ is a probability measure and $\\pi _(\\widehat{\\mu })=\\pi _(\\mu _+)=\\mu $ .", "Thus, we can write $\\widehat{\\mu }=\\int \\widehat{\\mu }_z d\\mu (z)$ and $\\mu _+=\\int \\mu _{+,z} d\\mu (z)$ , where $\\widehat{\\mu }_z$ and $\\mu _{+,z}$ are respectively the conditional measures of $\\widehat{\\mu }$ and $\\mu _+$ with respect to $\\pi $ and $\\mu $ .", "Note that $\\widehat{\\mu }_z$ and $\\mu _{+,z}$ are probability measures on $\\pi ^{-1}(z)$ which are defined for $\\mu $ -almost every $z$ .", "It is enough to prove that $\\widehat{\\mu }_z=\\mu _{+,z}$ .", "Since $\\pi _(U)=0$ , as in Section 3 of [14], we obtain that $U$ is a vertical current with respect to $\\pi $ in the sense that it can be decomposed into currents on fibers of $\\pi $ .", "More precisely, there is a positive measure $\\mu ^{\\prime }$ on $k$ and for $\\mu ^{\\prime }$ -almost every point $z\\in k$ there is a negative current $U_z$ on $\\pi ^{-1}(z)$ such that $U=\\int U_z d\\mu ^{\\prime }(z)$ .", "Write $\\mu ^{\\prime }=\\mu ^{\\prime }_r+\\mu ^{\\prime }_s$ with $\\mu ^{\\prime }_r$ absolutely continuous and $\\mu ^{\\prime }_s$ singular with respect to $\\mu $ .", "Multiplying $U_z$ by a suitable constant depending on $z$ , possibly by 0, allows to assume that $\\mu ^{\\prime }_r=\\mu $ .", "Claim.", "We have ${dd^c}U_z=\\widehat{\\mu }_z-\\mu _{+,z}$ for $\\mu $ -almost every $z$ and ${dd^c}U_z=0$ for $\\mu ^{\\prime }_s$ -almost every $z$ .", "Assuming the claim, we first complete the proof of the lemma.", "By definition, $\\mu _{+,z}$ is a Dirac mass.", "For simplicity, we use a local coordinate system $x$ on $\\pi ^{-1}(z)$ so that $\\mu _{+,z}$ is the Dirac mass $\\delta _0$ at 0.", "We only have to check that $\\widehat{\\mu }_z\\ge \\delta _0$ .", "Choose a negative function $\\varphi $ with support in a small neighbourhood of 0 which is smooth outside 0 and equal to $\\log \\Vert x\\Vert $ near 0.", "Define $\\varphi _n:=\\max (\\varphi ,-n)$ .", "Assume that the inequality $\\widehat{\\mu }_z\\ge \\delta _0$ does not hold.", "Since $\\widehat{\\mu }_z$ and $\\mu _{+,z}$ have the same mass, we easily deduce that $\\langle \\widehat{\\mu }_z-\\mu _{+,z},\\varphi _n\\rangle \\rightarrow \\infty $ as $n\\rightarrow \\infty $ .", "On the other hand, we have $\\langle \\widehat{\\mu }_z-\\mu _{+,z},\\varphi _n\\rangle =\\langle {dd^c}U_z,\\varphi _n\\rangle =\\langle U_z,{dd^c}\\varphi _n\\rangle .$ The last integral is bounded above because $U_z$ is negative and ${dd^c}\\varphi _n$ is positive in a fixed neighbourhood of 0 and smooth outside this neighbourhood.", "This is a contradiction.", "It remains to prove the claim.", "Define $V_z:={dd^c}U_z-\\widehat{\\mu }_z+\\mu _{+,z}$ for $\\mu $ -almost every $z$ and $V_z:={dd^c}U_z$ for $\\mu ^{\\prime }_s$ -almost every $z$ .", "We have to show that $V_z=0$ for $\\mu ^{\\prime }$ -almost every $z$ .", "Consider a dense sequence $\\phi _n$ in the space of smooth test functions with compact supports in $k\\times \\mathbb {G}$ .", "It is enough to check for each $\\phi :=\\phi _n$ that $\\langle V_z,\\phi \\rangle =0$ for $\\mu ^{\\prime }$ -almost every $z$ or equivalently for any smooth function $\\chi (z)$ with compact support in $k$ $\\int \\langle V_z,\\phi \\rangle \\chi (z) d\\mu ^{\\prime }(z)=0.$ By definition of $V_z$ , the last integral is equal to $\\big \\langle {dd^c}U-\\widehat{\\mu }+\\mu _+,(\\chi \\circ \\pi )\\phi \\big \\rangle $ .", "Therefore, it vanishes by hypotheses.", "This completes the proof of the lemma.", "Lemma 4.6 Let $S$ be a positive closed $(q,q)$ -current on ${\\rm Gr}(\\mathbb {P}^k,k-p)$ .", "Let $\\gamma $ be a continuous negative form of bi-dimension $(k-p+q+1,k-p+q+1)$ on a neighbourhood of $\\pi ^{-1}(\\overline{K}_-)$ .", "Then the mass of $d^{-n}(\\widehat{f}^n)^\\bullet (S\\wedge \\gamma )\\wedge \\pi ^(T_-)$ on $k\\times \\mathbb {G}$ is bounded by a constant independent of $n$ .", "Observe that the form $(\\widehat{f}^n)^(\\gamma )$ is defined and continuous on a neighbourhood of $\\pi ^{-1}(K_-)$ .", "Moreover, $T_-$ is a power of a positive closed $(1,1)$ -current with continuous potential on $k$ .", "Therefore, the wedge-product in the lemma is well-defined on $k\\times \\mathbb {G}$ .", "Choose a smooth positive closed form $\\theta $ on ${\\rm Gr}(\\mathbb {P}^k,k-p)$ such that $\\gamma \\ge -\\theta $ on a neighbourhood of $\\pi ^{-1}(\\overline{K}_-)$ which contains the support of $\\pi ^(T_-)$ .", "Replacing $\\gamma $ with $-\\theta $ allows to assume that $\\gamma $ is negative closed and smooth on ${\\rm Gr}(\\mathbb {P}^k,k-p)$ .", "It is now clear that the mass of $d^{-n}(\\widehat{f}^n)^\\bullet (S\\wedge \\gamma )$ is bounded since $d^{-n}(\\widehat{f}^n)^$ is bounded on the Hodge cohomology of ${\\rm Gr}(\\mathbb {P}^k,k-p)$ , see Example REF .", "It follows that the mass of $d^{-n}(\\widehat{f}^n)^\\bullet (S\\wedge \\gamma )\\wedge \\pi ^(T_-)$ is also bounded.", "To see this last point, it is enough to use that $T_-=\\tau _-^{k-p}$ and to approximate $\\tau _-$ by a sequence of smooth positive closed $(1,1)$ -forms on $\\mathbb {P}^k$ with decreasing potentials.", "Recall that the mass of a positive closed current on a compact Kähler manifold depends only on its cohomology class.", "The lemma follows.", "End of the proof of Proposition REF .", "We use a decreasing induction on $q$ .", "The case $q=p$ was considered in Lemma REF .", "Assume the proposition for $q+1$ with some $0\\le q\\le p-1$ .", "We show it for $q$ .", "For simplicity assume that the mass $c$ of $\\pi _(\\alpha )$ is 1.", "Observe that the bi-dimension of $\\alpha $ is at most equal to $(k-1,k-1)$ .", "Therefore, Leray's spectral theory applied to the fibration by $\\pi $ gives a formula similar to the Künneth formula in the product case and implies that the cohomology class $\\lbrace \\alpha \\rbrace $ belongs to the ideal generated by $\\pi ^(\\oplus _{r\\ge 1}H^{r,r}(\\mathbb {P}^k,)$ , see [28].", "Since the class $\\lbrace \\tau _+\\rbrace $ and its powers generate the ideal $\\oplus _{r\\ge 1} H^{r,r}(\\mathbb {P}^k,$ , we deduce that $\\lbrace \\alpha \\rbrace $ can be written as the cup-product of $\\lbrace \\pi ^(\\tau _+)\\rbrace $ with the class of some smooth closed form $\\beta $ of the right bi-degree.", "We can assume that $\\beta $ is positive since we can always reduce the problem to this case by considering linear combinations of positive forms.", "The condition $c=1$ implies that the mass of $\\pi _(\\beta )$ is 1.", "Since $\\tau _+$ is of mass 1, it belongs to the class of the Fubini-Study form $\\omega _{\\rm FS}$ .", "We deduce that $\\alpha $ is cohomologous to $\\pi ^(\\omega _{\\rm FS})\\wedge \\beta $ .", "In particular, there is a smooth form $\\gamma ^{\\prime }$ such that $\\alpha =\\pi ^(\\omega _{\\rm FS})\\wedge \\beta +{dd^c}\\gamma ^{\\prime }$ .", "Adding to $\\gamma ^{\\prime }$ a suitable negative closed form allows to assume that $\\gamma ^{\\prime }$ is negative.", "There is also a quasi-psh function $u$ such that $\\omega _{\\rm FS}=\\tau _+-{dd^c}u$ .", "Since $\\tau _+$ has continuous potential outside $I_+$ , the function $u$ is continuous outside $I_+$ .", "Adding to $u$ a suitable constant allows to assume that it is positive on a neighbourhood of $\\overline{K}_-$ .", "Therefore, we have $\\alpha =\\pi ^(\\tau _+)\\wedge \\beta +{dd^c}\\gamma $ with $\\gamma :=\\gamma ^{\\prime }-(u\\circ \\pi )\\beta $ .", "Using that $d_+^{-1}f^(\\tau _+)=\\tau _+$ and $\\widehat{f}^\\bullet \\circ \\pi ^=\\pi ^\\circ f^\\bullet $ , we obtain $d_+^{-(p-q)n}(\\widehat{f}^n)^\\bullet (\\alpha )\\wedge \\pi ^(\\tau _+^q\\wedge T_-) =d_+^{-(p-q-1)n}(\\widehat{f}^n)^\\bullet (\\beta )\\wedge \\pi ^(\\tau _+^{q+1}\\wedge T_-)+ {dd^c}U_n$ with $U_n:=d^{-n}(\\widehat{f}^n)^\\bullet \\big (\\pi ^(\\tau _+^q)\\wedge \\gamma \\big )\\wedge \\pi ^(T_-)$ .", "The induction hypothesis implies that the first term in the last sum converges to $\\mu _+$ .", "By Lemma REF , the negative currents $U_n$ have bounded masses.", "So we can extract convergent subsequences.", "We claim that the sequence of measures $U_n\\wedge \\pi ^(\\omega _{\\rm FS})$ converges to 0.", "This property implies that all cluster values $U$ of $U_n$ satisfy $\\pi _(U)=0$ and Lemma REF gives the result.", "It remains to prove the claim.", "As in the proof of Lemma REF , we can assume that $\\gamma $ is a negative closed smooth form.", "Since $\\widehat{f}^n$ is an automorphism of $k\\times \\mathbb {G}$ , the mass of the measure $U_n\\wedge \\pi ^(\\omega _{\\rm FS})$ is equal to the mass of its image by $\\widehat{f}^n$ , i.e.", "the mass of $\\pi ^(\\tau _+^q)\\wedge \\gamma \\wedge d^{-n}(\\widehat{f}^n)_\\bullet \\pi ^(T_-\\wedge \\omega _{\\rm FS})=\\pi ^(\\tau _+^q)\\wedge \\gamma \\wedge \\pi ^\\big [d^{-n}(f^n)_\\bullet (T_-\\wedge \\omega _{\\rm FS})\\big ].$ Since $(f^n)_$ acts on $H^{k-p+1,k-p+1}(\\mathbb {P}^k,$ as the multiplication by $d_+^{(p-1)n}$ , the current $d^{-n}(f^n)_\\bullet (T_-\\wedge \\omega _{\\rm FS})$ tends to 0 as $n\\rightarrow \\infty $ .", "The result follows.", "$\\square $ Remark 4.7 We don't know in general if $d^{-n}(\\widehat{f}^n)^\\bullet \\big (\\alpha \\wedge \\pi ^(\\tau _+^q)\\big )$ converges to a constant times a canonical lift of $T_+$ to ${\\rm Gr}(\\mathbb {P}^k,k-p)$ even for $q=0$ or $q=p$ .", "We don't know if $T_+$ admits a unique lift to ${\\rm Gr}(\\mathbb {P}^k,k-p)$ .", "This is true when $p=1$ because in this case we can show that $T_+$ is completely laminar." ], [ "Equidistribution of saddle periodic points", "Let $F:k\\times k\\rightarrow k\\times k$ be the polynomial automorphism defined in Example REF .", "We extend it to a bi-rational map of $\\mathbb {P}^k\\times \\mathbb {P}^k$ .", "Recall that Proposition REF can be applied to $F$ .", "The indeterminacy sets of $F$ and $F^{-1}$ are $(I_+\\times \\mathbb {P}^k)\\cup (\\mathbb {P}^k\\times I_-)$ and $(I_-\\times \\mathbb {P}^k)\\cup (\\mathbb {P}^k\\times I_+)$ respectively.", "Its dynamics is similar to the one of Hénon maps.", "Let $\\pi _1,\\pi _2:\\mathbb {P}^k\\times \\mathbb {P}^k\\rightarrow \\mathbb {P}^k$ denote the canonical projections.", "The following result is proved in the same way as for Theorem REF .", "We can also deduce it from that theorem in $\\mathbb {P}^k$ and Propositions REF and REF .", "Theorem 5.1 Let $S$ be a positive closed $(k,k)$ -current on $\\mathbb {P}^k\\times \\mathbb {P}^k$ .", "Assume that the support of $S$ does not intersect $I_-\\times I_+$ .", "Then $d^{-2n} (F^n)^\\bullet (S)$ converges to $c+$ as $n\\rightarrow \\infty $ , where $+:=T_+\\otimes T_-$ and $c:=\\big \\langle S,\\pi _1^(\\omega _{\\rm FS}^{k-p})\\wedge \\pi _2^(\\omega _{\\rm FS}^p)\\big \\rangle $ .", "A similar result holds for $F^{-1}$ .", "We will use the following corollary.", "Corollary 5.2 Let $\\Gamma _n$ denote the closure of the graph of $f^n$ in $\\mathbb {P}^k\\times \\mathbb {P}^k$ .", "Then the sequence of positive closed $(k,k)$ -currents $d^{-n}[\\Gamma _n]$ converges to $+$ as $n\\rightarrow \\infty $ .", "If $n$ is even, we can write $d^{-n}[\\Gamma _n]=d^{-n}(F^{n/2})^\\bullet [\\Delta ]$ .", "Otherwise, write $d^{-n}[\\Gamma _n]=d^{-n+1}(F^{(n-1)/2})^\\bullet (d^{-1}[\\Gamma _1])$ .", "Since $f$ is a Hénon map, it is not difficult to check that $\\Delta $ and $\\Gamma _n$ do not intersect $I_-\\times I_+$ for $n\\ge 0$ .", "Therefore, we can apply Theorem REF .", "Since $\\Gamma _n$ is the closure of the graph of $f^n$ , the constant $c$ there is equal to $d^{-n}\\int _{\\Gamma _n} \\pi _1^(\\omega _{\\rm FS}^{k-p})\\wedge \\pi _2^(\\omega _{\\rm FS}^p)=d^{-n}\\int _{\\mathbb {P}^k}\\omega _{\\rm FS}^{k-p}\\wedge (f^n)^(\\omega _{\\rm FS}^p).$ The last expression defines the mass of the current $d^{-n}(f^n)^(\\omega _{\\rm FS}^p)$ which is equal to 1 because the action of $f^$ on $H^{p,p}(\\mathbb {P}^k,$ is the multiplication by $d$ .", "The corollary follows.", "Denote by $\\widehat{\\Gamma }_n$ and $\\widehat{\\Delta }$ the lifts of $\\Gamma _n$ and $\\Delta $ to ${\\rm Gr}(\\mathbb {P}^k\\times \\mathbb {P}^k,k)$ .", "Let $\\widehat{F}$ denote the canonical lift of $F$ to ${\\rm Gr}(\\mathbb {P}^k\\times \\mathbb {P}^k,k)$ .", "We have $\\widehat{\\Gamma }_n=\\widehat{F}^{-n/2}(\\widehat{\\Delta })$ if $n$ is even and $\\widehat{\\Gamma }_n=\\widehat{F}^{-(n-1)/2}(\\widehat{\\Gamma }_1)$ if $n$ is odd.", "Let $(n_i)$ be an increasing sequence of integers such that $d^{-n_i}[\\widehat{\\Gamma }_{n_i}]$ converges to a positive closed $(k,k)$ -current $\\widehat{_+.", "As in the proof of Theorem \\ref {th_lam_Green}, we obtain the following result.", "}\\begin{proposition} The current + is completely weakly laminar and tame.The current \\widehat{_+ is a lift of + to {\\rm Gr}(\\mathbb {P}^k\\times \\mathbb {P}^k,k).", "In particular, we have \\Pi _(\\widehat{_+)=+, where\\Pi :{\\rm Gr}(\\mathbb {P}^k\\times \\mathbb {P}^k,k)\\rightarrow \\mathbb {P}^k\\times \\mathbb {P}^k denotes the canonical projection.", "}}Note that over k\\times k, {\\rm Gr}(\\mathbb {P}^k\\times \\mathbb {P}^k,k) can be identified with the product of k\\times k with a Grassmannian \\mathbf {G}.Recall that for \\mu -almost every point z we denoted by E_u(z) the unstable tangent subspace and E_s(z) the stable one.", "Denote by \\widehat{\\mu }^\\Delta the lift of \\mu to the set of points (z,z,[E_s(z)\\times E_u(z)]) in {\\rm Gr}(\\mathbb {P}^k\\times \\mathbb {P}^k,k).", "Here is a key point in the proof of our main result.\\end{proposition}$ Proposition 5.3 We have that $\\widehat{_+\\curlywedge \\Pi ^[\\Delta ] is well-defined and equal to \\widehat{\\mu }^\\Delta .", "}$ The existence of the last wedge-product means that There is a unique tangent current of $\\widehat{_+ along \\Pi ^{-1}(\\Delta ).", "More precisely, when one dilates local coordinates in the normal directions to \\Pi ^{-1}(\\Delta ), the image of \\widehat{_+ converges to a unique positive closed current on the normal vector bundle of \\Pi ^{-1}(\\Delta ) independently of the choice of coordinates.", "The limit is called tangent current.\\item [(b)] The tangential h-dimension of \\widehat{_+ along \\Pi ^{-1}(\\Delta ) is minimal, i.e.", "0 in our case.", "Equivalently, in our case, the tangent current can be decomposed into currents of integration on fibers of the above normal vector bundle.", "}In such a situation, the tangent current is the pull-back to the above normal vector bundle of a positive measure which is by definition the weak intersection \\widehat{_+\\curlywedge \\Pi ^[\\Delta ].We refer to \\cite {DS12} for details.", "}The uniqueness of this tangent current requires the laminar properties of \\widehat{_+ and other dynamical arguments.", "We will analyze \\widehat{_+ using various projections.We first prove the property on the tangential h-dimension of \\widehat{_+.", "See \\cite {DS12} for the density \\kappa _r of positive closed currents.", "It describes the cohomology classes of the tangent currents along a submanifold, according to their behavior along various directions.", "Our study in \\cite {DS12} shows that tangent currents are not unique in general but they are in the same cohomology class of the normal vector bundle to the submanifold.", "We then obtain the cohomology classes \\kappa _r on the submanifold using Leray^{\\prime }s theory.", "}\\begin{lemma} We have \\kappa _r(\\widehat{_+,\\Pi ^{-1}(\\Delta ))=0 if r>0, i.e.", "the tangential h-dimension of \\widehat{_+ along \\Pi ^{-1}(\\Delta ) is 0.", "}Assume there is an integer r\\ge 1 such that \\kappa _r(\\widehat{_+,\\Pi ^{-1}(\\Delta ))\\ne 0.", "Choose r maximal satisfying this property.", "Recall that such a maximal r is called the tangential h-dimension of \\widehat{_+ along \\Pi ^{-1}(\\Delta ).By Lemma 3.8 in \\cite {DS12}, \\kappa _r(\\widehat{_+,\\Pi ^{-1}(\\Delta )) is a pseudo-effective cohomology class of bi-dimension (r,r) on {\\rm Gr}(\\mathbb {P}^k\\times \\mathbb {P}^k,k) that can be represented by a positive closed current in {\\rm supp}(+)\\cap \\Pi ^{-1}(\\Delta ).Observe that the intersection {\\rm supp}(+)\\cap \\Pi ^{-1}(\\Delta ) has a compact projection in k\\times k.So we only need to work in k\\times k\\times \\mathbf {G}.", "}Define \\widehat{_{+m}:=d^{-2m} (F^m)^\\bullet (\\widehat{_+).", "Since \\widehat{F} is an automorphism on k\\times k\\times \\mathbf {G}, we have\\kappa _r(\\widehat{_+,\\Pi ^{-1}[\\Delta ])=\\kappa _r(\\widehat{_{+m},d^{-2m}\\Pi ^[\\Gamma _{-2m}]).", "Theorem \\ref {th_F_equi} applied to F^{-1} implies that d^{-2m}[\\Gamma _{-2m}] converges to -:=T_-\\otimes T_+.", "So d^{-2m}\\Pi ^[\\Gamma _{-2m}] converges to \\Pi ^(-).Let \\widehat{_+^{\\prime } denote a limit value of the sequence\\widehat{_{+m}.", "Since \\Pi ^(-) is the wedge-product of (1,1)-currents with continuous potentials, the intersection \\widehat{_+^{\\prime }\\wedge \\Pi ^(-) is a well-defined positive measure.", "It follows from the theory of densities that the density dimension between \\widehat{_+^{\\prime } and \\Pi ^(-) is zero.", "Corollary 5.8 in \\cite {DS12} on the upper semi-continuity of densities implies that\\kappa _r(\\widehat{_{+m},d^{-m}\\Pi ^[\\Gamma _{-m}]) tends to 0.", "This is a contradiction.", "The lemma follows.", "}If E_z and E_w are two tangent subspaces at z and w in \\mathbb {P}^k, of dimension respectively k-p and p, then E_z\\times E_w is a tangent subspace of \\mathbb {P}^k\\times \\mathbb {P}^k at (z,w), of dimension k. This construction induces a natural embedding of {\\rm Gr}(\\mathbb {P}^k,k-p)\\times {\\rm Gr}(\\mathbb {P}^k,p) into {\\rm Gr}(\\mathbb {P}^k\\times \\mathbb {P}^k,k).", "For simplicity, we identify {\\rm Gr}(\\mathbb {P}^k,k-p)\\times {\\rm Gr}(\\mathbb {P}^k,p) with its image.", "Since F is a product map, {\\rm Gr}(\\mathbb {P}^k,k-p)\\times {\\rm Gr}(\\mathbb {P}^k,p) can be seen as an invariant submanifold of {\\rm Gr}(\\mathbb {P}^k\\times \\mathbb {P}^k,k).", "}\\begin{lemma} The current \\widehat{_+ is supported by {\\rm Gr}(\\mathbb {P}^k,k-p)\\times {\\rm Gr}(\\mathbb {P}^k,p).", "}By Proposition \\ref {prop_F_laminar}, there is a measurable web associated with + which can be lifted to a measurable web of \\widehat{_+ in {\\rm Gr}(\\mathbb {P}^k\\times \\mathbb {P}^k,k).Since +=T_+\\otimes T_-, we have +\\wedge \\pi _1^(\\omega _{\\rm FS}^{k-p+1})=0 and +\\wedge \\pi _2^(\\omega _{\\rm FS}^{p+1})=0.", "Therefore, if L is a generic lame of+, its tangent space at a regular point is the product of a tangent subspace of dimension k-p of the first factor \\mathbb {P}^k and a tangent subspace of dimension p of the second one.", "Sothe regular part of L is locally a product of a manifold of dimension k-p in \\mathbb {P}^k and another of dimension p.It follows that \\widehat{L} is contained in {\\rm Gr}(\\mathbb {P}^k,k-p)\\times {\\rm Gr}(\\mathbb {P}^k,p).The lemma follows.", "}We will see later that the currents \\widehat{T}_+, \\widehat{T}_- in the following lemma are lifts of T_+, T_- to {\\rm Gr}(\\mathbb {P}^k,k-p) and {\\rm Gr}(\\mathbb {P}^k,p) respectively, see Section \\ref {section_woven} for the definition.\\end{lemma}\\begin{lemma} Let \\widehat{_s and \\widehat{_u denote the push-forward of \\widehat{_+ to {\\rm Gr}(\\mathbb {P}^k,k-p)\\times \\mathbb {P}^k and \\mathbb {P}^k\\times {\\rm Gr}(\\mathbb {P}^k,p) respectively.", "Then there are positive closed currents \\widehat{T}_+ on {\\rm Gr}(\\mathbb {P}^k,k-p) and \\widehat{T}_- on {\\rm Gr}(\\mathbb {P}^k,p) such that\\widehat{_s=\\widehat{T}_+\\otimes T_- and \\widehat{_u=T_+\\otimes \\widehat{T}_-.", "}We prove the existence of \\widehat{T}_+.", "The case of \\widehat{T}_- can be obtained in the same way.Let \\widehat{_{+,n} be a limit value of d^{-n_i+2n} [\\widehat{\\Gamma }_{n_i-2n}] when i\\rightarrow \\infty .", "Then \\widehat{_{+,n} is also supported by {\\rm Gr}(\\mathbb {P}^k,k-p)\\times {\\rm Gr}(\\mathbb {P}^k,p) and \\widehat{_+=d^{-2n} (\\widehat{F}^n)^\\bullet (\\widehat{_{+,n}).Let \\widehat{_{s,n} denote the push-forward of \\widehat{_{+,n} to {\\rm Gr}(\\mathbb {P}^k,k-p)\\times \\mathbb {P}^k.Define h:=(\\widehat{f},f^{-1}) the product map on {\\rm Gr}(\\mathbb {P}^k,k-p)\\times \\mathbb {P}^k.", "We have \\widehat{_s=d^{-2n} (h^n)^\\bullet (\\widehat{_{s,n}).", "Recall that {\\rm Gr}(\\mathbb {P}^k,k-p) is bi-rational to \\mathbb {P}^k\\times \\mathbb {G}.", "We still denote by \\widehat{f} the canonical lift of f to\\mathbb {P}^k\\times \\mathbb {G}.", "}Recall that d^{-n}[\\Gamma _n] converges to T_+\\otimes T_- which is supported in \\overline{K}_+\\times \\overline{K}_-.", "Therefore,the support of \\widehat{_+ is contained in the fibers over \\overline{K}_+\\times \\overline{K}_-.We can apply Proposition \\ref {prop_product_map} to \\widehat{_{s,n} and then Proposition\\ref {prop_current_tensor} (we have to permute the factors of X\\times Y in those propositions).The currentR defined as in Proposition \\ref {prop_current_tensor} is supported by \\overline{K}_-.", "Theorem \\ref {th_unique_Green} applied to f^{-1} implies that R is necessarily a multiple of T_-.", "Since T_- is extremal, Proposition \\ref {prop_current_tensor} implies the result.", "}Denote by \\Pi _s, \\Pi _u, \\pi _s and \\pi _u the canonical projections from {\\rm Gr}(\\mathbb {P}^k,k-p)\\times {\\rm Gr}(\\mathbb {P}^k,p) onto {\\rm Gr}(\\mathbb {P}^k,k-p)\\times \\mathbb {P}^k, \\mathbb {P}^k\\times {\\rm Gr}(\\mathbb {P}^k,p), {\\rm Gr}(\\mathbb {P}^k,k-p) and {\\rm Gr}(\\mathbb {P}^k,p) respectively.", "The following lemma shows that\\widehat{T}_+, \\widehat{T}_- are lifts of T_+, T_- to {\\rm Gr}(\\mathbb {P}^k,k-p) and {\\rm Gr}(\\mathbb {P}^k,p) respectively.", "}}}\\begin{lemma} Let \\widehat{\\Omega }_p be the standard lift of \\omega _{\\rm FS}^p to {\\rm Gr}(\\mathbb {P}^k,k-p) defined in Example \\ref {ex_Fubini_bis}.", "Then d^{-n_i} (\\widehat{f}^{n_i})^(\\widehat{\\Omega }_p) converges to\\widehat{T}_+ and a similar result holds for\\widehat{T}_-.\\end{lemma}Let L be a generic projective subspace of dimension k-p in \\mathbb {P}^k.By definition of \\widehat{\\Omega }_p,we only have to prove that d^{-n_i}(\\widehat{f}^{n_i})^\\bullet [\\widehat{L}] converges to \\widehat{T}_+.We will identify {\\rm Gr}(\\mathbb {P}^k\\times \\mathbb {P}^k,k) with \\mathbb {P}^k\\times \\mathbb {P}^k\\times \\mathbf {G} via the natural bi-rational map and we still denote by \\Pi the projection onto \\mathbb {P}^k\\times \\mathbb {P}^k.", "}Fix a generic L as in Corollary \\ref {cor_inter_woven} (we change the notation H_\\xi by L).", "So up to extracting a subsequence of (n_i), we can assume that the intersections d^{-n_i}[\\widehat{\\Gamma }_{n_i}]\\wedge [\\mathbb {P}^k\\times L\\times \\mathbf {G}] and \\widehat{_+\\wedge [\\mathbb {P}^k\\times L\\times \\mathbf {G}] are well-defined and d^{-n_i}[\\widehat{\\Gamma }_{n_i}]\\wedge [\\mathbb {P}^k\\times L\\times \\mathbf {G}] converges to \\widehat{_+\\wedge [\\mathbb {P}^k\\times L\\times \\mathbf {G}].Moreover, we can associate to d^{-n_i}[\\widehat{\\Gamma }_{n_i}] and \\widehat{_+ the measurable webs which are described in that corollary.", "We also denote them by \\nu _{n_i}, \\nu _{n_i}[s], \\nu and \\nu [s].", "}\\medskip {\\bf Claim.}", "Any limit value S of d^{-n_i}(\\widehat{f}^{n_i})^\\bullet [\\widehat{L}] is larger than or equal to \\widehat{T}_+.", "}\\medskip }Assuming the claim, we first complete the proof of the lemma.We have seen in the proof of Theorem \\ref {th_lam_Green} that S is a lift of T_+ to {\\rm Gr}(\\mathbb {P}^k,k-p)and the projection of a generic lame of S to \\mathbb {P}^k is non-degenerate, i.e.", "locally of maximal dimension k-p. On the other hand, \\Pi _(\\widehat{_+) is the limit of d^{-n}[\\Gamma _n] which is equal to +=T_+\\otimes T_-.", "We deduce thatthe push-forward of \\widehat{_s to \\mathbb {P}^k\\times \\mathbb {P}^k is also equal to + and hencethe push-forward of \\widehat{T}_+ to \\mathbb {P}^k is equal to T_+.", "Since the same property holds for S,the current S-\\widehat{T}_+, which is positive according to the claim, has horizontal dimension <k-p with respect to the projection onto \\mathbb {P}^k.", "But since generic lames of S have horizontal dimension k-p, we necessarily have S-\\widehat{T}_+=0.The lemma follows.\\hfill \\square }\\medskip {\\bf Proof of the claim.}", "The idea is to show that we can approach \\widehat{T}_+ by lames in \\widehat{f}^{-n_i}(\\widehat{L}) or more precisely by currents smaller than or equal to d^{-n_i}(\\widehat{f}^{n_i})^\\bullet [\\widehat{L}].Define L_{n_i} as the closure in \\mathbb {P}^k of f^{-n_i}(L)\\cap k and L_{(n_i)} the closure in \\mathbb {P}^k\\times \\mathbb {P}^k of \\pi _2^{-1}(L)\\cap \\Gamma _{n_i}\\cap k\\times k. The last analytic set is the family of points (z,f^{n_i}(z)) with z\\in f^{-n_i}(L)\\cap k. So we have \\pi _1(L_{(n_i)})=L_{n_i}.", "Define also \\widehat{L}_{n_i} as the lift of L_{n_i} to {\\rm Gr}(\\mathbb {P}^k,k-p) and \\mathcal {L}_{(n_i)}:=\\Pi ^{-1}(L_{(n_i)})\\cap \\widehat{\\Gamma }_{n_i}.", "}The discussion before the claim on the intersection of currents with \\mathbb {P}^k\\times L\\times \\mathbf {G} implies that d^{-n_i}[\\mathcal {L}_{(n_i)}]\\rightarrow \\widehat{_+\\wedge [\\mathbb {P}^k\\times L\\times \\mathbf {G}].", "The last intersection is supported by {\\rm Gr}(\\mathbb {P}^k,k-p)\\times {\\rm Gr}(\\mathbb {P}^k,p).By Lemma \\ref {lemma_Ts},its push-forward to {\\rm Gr}(\\mathbb {P}^k,k-p)\\times \\mathbb {P}^k is equal to\\widehat{T}_+\\otimes ([L]\\wedge T_-).", "Therefore, its push-forward to {\\rm Gr}(\\mathbb {P}^k,k-p) is equal to the one of\\widehat{T}_+\\otimes ([L]\\wedge T_-)and hence equal to \\widehat{T}_+ because [L]\\wedge T_- is a probability measure.So in order to obtain the claim, it is enough to approach \\widehat{_+\\wedge [\\mathbb {P}^k\\times L\\times \\mathbf {G}] by woven currents on{\\rm Gr}(\\mathbb {P}^k,k-p)\\times {\\rm Gr}(\\mathbb {P}^k,p) whose push-forwards to {\\rm Gr}(\\mathbb {P}^k,k-p) are bounded by d^{-n_i} [\\widehat{L}_{n_i}].", "}For this purpose, we cannot directly use the currents d^{-n_i}[\\mathcal {L}_{(n_i)}] because they are not supported by {\\rm Gr}(\\mathbb {P}^k,k-p)\\times {\\rm Gr}(\\mathbb {P}^k,p) and they admit no natural push-forward to {\\rm Gr}(\\mathbb {P}^k,k-p).", "Recall that we however have d^{-n_i}[\\mathcal {L}_{(n_i)}]\\rightarrow \\widehat{_+\\wedge [\\mathbb {P}^k\\times L\\times \\mathbf {G}].The idea now is to modify the lames of d^{-n_i}[\\mathcal {L}_{(n_i)}] without changing their limits in order to get convenient lames in {\\rm Gr}(\\mathbb {P}^k,k-p)\\times {\\rm Gr}(\\mathbb {P}^k,p).", "We have to make sure that the push-forwards of those lames to {\\rm Gr}(\\mathbb {P}^k,k-p) satisfy the desired property.", "}Let \\widehat{\\Lambda } be a graph corresponding to a generic point with respect to the measure \\nu [s] for some s.It is contained in {\\rm Gr}(\\mathbb {P}^k,k-p)\\times {\\rm Gr}(\\mathbb {P}^k,p).Since \\widehat{_+ is a lift of +, the projection \\Lambda :=\\Pi (\\widehat{\\Lambda }) of \\widehat{\\Lambda } to \\mathbb {P}^k\\times \\mathbb {P}^k corresponds to a lame of +.We can refine the webs if necessary in order to get \\Lambda smooth and contained in k\\times k.Since +=T_+\\otimes T_-, we have seen that \\Lambda is locally a product \\Lambda _+\\times \\Lambda _- of a manifold \\Lambda _+ of dimension k-p in \\mathbb {P}^k with another \\Lambda _- of dimension p. We can also write locally\\widehat{\\Lambda }=\\widehat{\\Lambda }_+\\times \\widehat{\\Lambda }_-, where \\widehat{\\Lambda }_+,\\widehat{\\Lambda }_- are the lifts of \\Lambda _+,\\Lambda _- to {\\rm Gr}(\\mathbb {P}^k,k-p) and {\\rm Gr}(\\mathbb {P}^k,p) respectively.", "}The projection \\widehat{\\Lambda }_s:=\\Pi _s(\\widehat{\\Lambda }) of \\widehat{\\Lambda } to {\\rm Gr}(\\mathbb {P}^k,k-p)\\times \\mathbb {P}^k is locally \\widehat{\\Lambda }_+\\times \\Lambda _- and corresponds to a lame of \\widehat{_s.", "We see that the projection of \\widehat{\\Lambda }\\cap (\\mathbb {P}^k\\times L\\times \\mathbf {G}) and the projection of \\widehat{\\Lambda }_s\\cap ({\\rm Gr}(\\mathbb {P}^k,k-p)\\times L) to {\\rm Gr}(\\mathbb {P}^k,k-p) are both equal to the lift of \\pi _1(\\Lambda \\cap \\pi _2^{-1}(L)) to {\\rm Gr}(\\mathbb {P}^k,k-p).", "So each connected component \\lambda of \\widehat{\\Lambda }\\cap (\\mathbb {P}^k\\times L\\times \\mathbf {G}) is the product of the lift of \\pi _1(\\Pi (\\lambda )) to {\\rm Gr}(\\mathbb {P}^k,k-p) with a point in {\\rm Gr}(\\mathbb {P}^k,p).", "}Consider now a graph \\widehat{\\Lambda }^{\\prime } corresponding to a generic point with respect to \\nu _{n_i}[s] which is close enough to \\widehat{\\Lambda }.", "It is an open subset of \\widehat{\\Gamma }_{n_i}.", "Recall that the fat extension of \\widehat{\\Lambda }^{\\prime } is also close to the fat extension of \\widehat{\\Lambda }.", "This insures that the approximation below is uniform on the unextended lames.Consider a connected component \\widehat{\\lambda }^{\\prime } of \\widehat{\\Lambda }^{\\prime }\\cap [\\mathbb {P}^k\\times L\\times \\mathbf {G}].", "This is an open subset of \\mathcal {L}_{(n_i)}.We deduce from the above properties of \\widehat{\\Lambda } that \\widehat{\\lambda }^{\\prime } can be approximated by the product of the lift of \\pi _1(\\Pi (\\widehat{\\lambda }^{\\prime })) to {\\rm Gr}(\\mathbb {P}^k,k-p) with a point in {\\rm Gr}(\\mathbb {P}^k,p).We need here the fact that \\mathbb {P}^k\\times L\\times \\mathbf {G} is transverse to the fat extension of \\widehat{\\Lambda } which is guaranteed by Corollary \\ref {cor_inter_woven} used just before the statement of the claim.", "}Denote by \\widehat{\\lambda }^{\\prime \\prime } the above product which is an approximation of \\widehat{\\lambda }^{\\prime }.", "The choice is not unique but the projection of \\widehat{\\lambda }^{\\prime \\prime } in {\\rm Gr}(\\mathbb {P}^k,k-p) is always the lift of \\pi _1(\\Pi (\\widehat{\\lambda }^{\\prime })).Observe that since \\pi _1(\\Pi (\\widehat{\\lambda }^{\\prime })) is an open subset of the variety L_{n_i}, the projection of \\widehat{\\lambda }^{\\prime \\prime } to {\\rm Gr}(\\mathbb {P}^k,k-p) is a lame in \\widehat{L}_{n_i}.The approximation can be controlled uniformly on graphs \\widehat{\\Lambda }^{\\prime } close enough to \\widehat{\\Lambda }.", "Therefore, we can apply Lemma \\ref {lemma_cv_inter} to \\nu _{n_i}[s] and \\nu [s] for each s.By considering the projection onto {\\rm Gr}(\\mathbb {P}^k,k-p), we deduce that every limit value of d^{-n_i}[\\widehat{L}_{n_i}] is larger than or equal to \\widehat{T}_+.", "This is the claim.\\hfill \\square }\\medskip }}}{\\bf End of the proof of Proposition \\ref {prop_a_weak_intersection}.", "}Recall that when \\widehat{_+\\curlywedge \\Pi ^[\\Delta ] is well-defined, \\widehat{_+ admits a unique tangent current along \\Pi ^{-1}(\\Delta ) and the above intersection is the so-called shadow of this tangent current onto \\Pi ^{-1}(\\Delta ), see \\cite {DS12}.", "Let \\widehat{\\nu } denote the shadow of a tangent current of \\widehat{_+ along \\Pi ^{-1}(\\Delta ).By Lemma \\ref {lemma_kappa_T+}, this is a positive measure with support in {\\rm Gr}(\\mathbb {P}^k,k-p)\\times {\\rm Gr}(\\mathbb {P}^k,p).It is enough to prove that \\widehat{\\nu }=\\widehat{\\mu }^\\Delta .", "Indeed, in this case all tangential currents are necessarily vertical and equal to the pull-back of \\widehat{\\mu }^\\Delta .", "}Define \\mu ^\\Delta :=+\\wedge [\\Delta ].Since +=T_+\\otimes T_-, it is not difficult to see that \\mu ^\\Delta =(\\pi _{1\|\\Delta })^(\\mu ), where \\mu :=T_+\\wedge T_- is the Green measure of f.Since \\Pi _(\\widehat{_+)=+, we deduce that \\Pi _(\\widehat{\\nu })=\\mu ^\\Delta .", "In particular, \\widehat{\\nu } is a probability measure.", "By definition of \\widehat{\\mu }^\\Delta , we also have \\Pi _(\\widehat{\\mu }^\\Delta )=\\mu ^\\Delta .", "So we need to prove that for \\mu -almost every z\\in \\mathbb {P}^k the conditional measures\\langle \\widehat{\\nu }\|\\Pi \|(z,z)\\rangle and \\langle \\widehat{\\mu }^\\Delta \|\\Pi \|(z,z)\\rangle of \\widehat{\\nu } and \\widehat{\\mu }^\\Delta with respect to \\Pi are equal.", "Note that the second conditional measure is the Dirac mass at the point (z,z,[E_s(z)\\times E_u(z)]) in the fiber \\lbrace (z,z)\\rbrace \\times \\mathbf {G} of \\Pi .", "}Since \\widehat{_s=(\\Pi _s)_(\\widehat{_+), if \\Pi _+:{\\rm Gr}(\\mathbb {P}^k,k-p)\\times \\mathbb {P}^k\\rightarrow \\mathbb {P}^k\\times \\mathbb {P}^k denotes the canonical projection, then (\\Pi _s)_(\\widehat{\\nu })=\\widehat{\\mu }_s, where \\widehat{\\mu }_s:=\\widehat{_s\\curlywedge \\Pi _+^[\\Delta ]provided that the last intersection exists.Observe that \\Pi _+^{-1}(\\Delta ) is the graph of the canonical projection \\pi :{\\rm Gr}(\\mathbb {P}^k,k-p)\\rightarrow \\mathbb {P}^k.Since \\widehat{_s=\\widehat{T}_s\\otimes T_- and T_- is a power of a positive closed (1,1)-current with continuous potentials on k, it is not difficult to see that the last intersection exists and is equal to\\widehat{T}_s\\wedge \\pi ^(T_-) if we identify \\Pi _+^{-1}(\\Delta ) with {\\rm Gr}(\\mathbb {P}^k,k-p) in the canonical way, see Lemma 5.11 in \\cite {DS12}.By Lemma \\ref {lemma_Ts_lim} and Proposition \\ref {prop_main_inter} with q=0 and \\alpha =\\widehat{\\Omega }_p, the last measure is equal to \\mu _+.The constant c in this proposition is 1 since \\widehat{\\nu } is a probability measure.", "}We deduce that the conditional measure \\langle \\widehat{\\mu }_s \|\\Pi _+\|(z,z)\\rangle is equal to the Dirac mass at the point ((z,[E_s(z)]),z).", "It follows that the push-forward of \\langle \\widehat{\\nu }\|\\Pi \|(z,z)\\rangle to {\\rm Gr}(\\mathbb {P}^k,k-p)\\times \\mathbb {P}^k is equal to the Dirac mass at the point((z,[E_s(z)]),z).", "In the same way, we obtain that the push-forward of \\langle \\widehat{\\nu }\|\\Pi \|(z,z)\\rangle to \\mathbb {P}^k\\times {\\rm Gr}(\\mathbb {P}^k,p) is equal to the Dirac mass at the point(z,(z,[E_u(z)])).", "We conclude that \\langle \\widehat{\\nu }\|\\Pi \|(z,z)\\rangle is equal to the Dirac mass at (z,z,[E_s(z)\\times E_u(z)]).This completes the proof of the proposition.\\hfill \\square }\\medskip }Recall that if \\Gamma is an analytic subset of dimension k in \\mathbb {P}^k\\times \\mathbb {P}^k we define \\widetilde{\\Gamma }as the closure of the set of points (x,[v]) in {\\rm Gr}(\\mathbb {P}^k\\times \\mathbb {P}^k,k) with x in the smooth part of \\Gamma and v a non-zero complex tangent k-vector to \\mathbb {P}^k\\times \\mathbb {P}^k at x which is not transverse to \\Gamma .", "Let \\Sigma denote the set of points (x,[v],[w]), where x\\in \\mathbb {P}^k\\times \\mathbb {P}^k, v and w are non-zero complex tangent k-vectors of \\mathbb {P}^k\\times \\mathbb {P}^k at x.", "This is a smooth complex manifold.", "Over k\\times k, we can identify it with k\\times k\\times \\mathbf {G}\\times \\mathbf {G}.Denote by \\Sigma ^{\\prime } the analytic subset of points (x,[v],[w])\\in \\Sigma such that v is not transverse to w. Over k\\times k, it can be identified with the product of k\\times k with a hypersurface of \\mathbf {G}\\times \\mathbf {G}, possibly with singularities.", "}Let p_1 and p_2 denote the projections from \\Sigma to {\\rm Gr}(\\mathbb {P}^k\\times \\mathbb {P}^k,k) given byp_1(x,[v],[w]):=(x,[v]) and p_2(x,[v],[w]):=(x,[w]).", "So p_1 and p_2 define two fibrations on \\Sigma which are locally trivial with fibers isomorphic to \\mathbf {G}.", "They also define two fibrations on \\Sigma ^{\\prime } which are locally trivial but the fibers may be singular.We have\\widetilde{\\Gamma }=p_1\\big (p_2^{-1}(\\widehat{\\Gamma })\\cap \\Sigma ^{\\prime }\\big ).Define \\widetilde{_+ as the limit of d^{-n_i}[\\widetilde{\\Gamma }_{n_i}].", "The last formula implies that this limit exists and is equal to\\widetilde{_+=(p_1)_\\big (p_2^(\\widehat{_+)\\wedge [\\Sigma ^{\\prime }]\\big ).The last wedge-product exists because the restriction of p_2 to \\Sigma ^{\\prime } is a (possibly singular) fibration, i.e.", "locally a product.", "}We now prove the crucial point in our approach for the main theorem.", "It says that \\Gamma _n is mostly transverse to \\Delta when n\\rightarrow \\infty .", "}\\begin{corollary} The density between \\widetilde{_+ and \\widehat{\\Delta } is zero.", "}By Proposition \\ref {prop_a_weak_intersection}, the current \\widehat{_+ admits a unique tangent current along \\Pi ^{-1}(\\Delta ) whose shadow on \\Pi ^{-1}(\\Delta ) is \\widehat{\\mu }^\\Delta .", "Recall that this tangent current can be defined as the limit of the images of \\widehat{_+ under local holomorphic dilations in the normal directions to \\Pi ^{-1}(\\Delta ).In the present situation, we can work over k\\times k where we can identify \\Sigma and \\Sigma ^{\\prime } with products of k\\times k with some varieties.", "This allows us to use the obvious dilations given in a coordinate system on k\\times k.}So if P:\\Sigma \\rightarrow \\mathbb {P}^k\\times \\mathbb {P}^k is the canonical projection,we can also use the dilations in the normal directions to P^{-1}(\\Delta ) induced by the obvious dilations in the normal directions of \\Delta in k\\times k. Therefore, we deduce fromProposition \\ref {prop_a_weak_intersection} that(p_2^(\\widehat{_+)\\wedge [\\Sigma ^{\\prime }]) \\curlywedge [P^{-1}(\\Delta )]=p_2^(\\widehat{\\mu }^\\Delta )\\wedge [\\Sigma ^{\\prime }].Using the above transform \\widehat{_+\\mapsto \\widetilde{_+, we obtain\\widetilde{_+\\curlywedge [\\Pi ^{-1}(\\Delta )]=(p_1)_\\big (p_2^(\\widehat{_+)\\wedge [\\Sigma ^{\\prime }] \\big ) \\curlywedge [\\Pi ^{-1}(\\Delta )]=(p_1)_\\big (p_2^(\\widehat{\\mu }^\\Delta )\\wedge [\\Sigma ^{\\prime }]\\big )on {\\rm Gr}(\\mathbb {P}^k\\times \\mathbb {P}^k,k).", "}The right hand side of the last identity is a positive closed current with support in \\Pi ^{-1}(\\Delta ).", "We denote it by R.In order to compute R, we first replace \\widehat{\\mu }^\\Delta with a Dirac mass \\delta at a point (x,[v]) in {\\rm Gr}(\\mathbb {P}^k\\times \\mathbb {P}^k,k) with x\\in k\\times k. We identify {\\rm Gr}(\\mathbb {P}^k\\times \\mathbb {P}^k,k) with k\\times k\\times \\mathbf {G} over k\\times k.If \\mathbf {G}(v) denotes the hypersurface of [w]\\in \\mathbf {G} with w not transverse to v, then (p_1)_\\big (p_2^(\\delta )\\wedge [\\Sigma ^{\\prime }]\\big ) is equal to the current of integration on \\lbrace x\\rbrace \\times \\mathbf {G}(v).", "}For \\mu -almost every z, we denote by \\mathbf {G}(z) the hypersurface \\mathbf {G}(v) with a vector v defining E_s(z)\\times E_u(z).", "The last vector space is identified with a subspace of the tangent space of \\mathbb {P}^k\\times \\mathbb {P}^k at (z,z).Using the definition of \\widehat{\\mu }^\\Delta ,we deduce that the above current R is equal toR=\\int [\\lbrace (z,z)\\rbrace \\times \\mathbf {G}(z)] d\\mu (z).The tangent current to \\widetilde{_+ along \\Pi ^{-1}(\\Delta ) is the pull-back of R to the normal vector bundle of \\Pi ^{-1}(\\Delta ) in {\\rm Gr}(\\mathbb {P}^k\\times \\mathbb {P}^k,k).", "We denote it by S.}Observe that since E_s(z)\\cap E_u(z)=\\lbrace 0\\rbrace , the two vector spaces E_s(z)\\times E_u(z) and \\Delta \\cap k\\times k only intersect at the point (z,z).", "SoE_s(z)\\times E_u(z) is transverse to \\Delta andeach manifold \\lbrace (z,z)\\rbrace \\times \\mathbf {G}(z) is disjoint from \\widehat{\\Delta }.", "Hence, the tangent current of S along \\widehat{\\Delta } is zero.", "Therefore, Proposition 4.13in \\cite {DS12} implies that the tangent current of\\widetilde{_+ along \\widehat{\\Delta } is zero.", "This completes the proof of the corollary.", "}In what follows, denote by (x,y) the standard coordinates in k\\times k. Define also z:=x-y and w:=x.", "So the diagonal \\Delta is given by x=y or by z=0.", "}\\begin{lemma} Let R_0>0 be a fixed constant.", "Thenif R>0 is a constant large enough then \\Gamma _n\\cap \\lbrace \\Vert z\\Vert \\le R_0\\rbrace is contained in \\lbrace \\Vert w\\Vert < R\\rbrace for every n\\ge 1.\\end{lemma}Fix a neighbourhood U_+ of I_+ and a neighbourhood U_- of I_- small enough such that \\overline{U}_+\\cap \\overline{U}_-=\\varnothing .", "Since I_+ is attractive for f^{-1}, we can choose U_+ so that f(\\mathbb {P}^k\\setminus U_+)\\subset \\mathbb {P}^k\\setminus U_+.", "We can also assume that f^{-1}(\\mathbb {P}^k\\setminus U_-)\\subset \\mathbb {P}^k\\setminus U_-.Fix a constant R>0 large enough.", "Consider a point (x,y) such that \\Vert z\\Vert \\le R_0 and \\Vert w\\Vert \\ge R. We have \\Vert x\\Vert \\ge R-R_0, \\Vert y\\Vert \\ge R-R_0 and \\Vert x-y\\Vert \\le R_0.", "Therefore, either x\\notin U_+ or y\\notin U_-.", "We have to show that (x,y)\\notin \\Gamma _n.", "Assume without loss of generality that x\\notin U_+.", "It suffices to prove that y\\ne f^n(x).", "}Let f_+ denote the homogeneous part of maximal degree d_+ of f. Then the closure of \\lbrace f_+=0\\rbrace is an analytic subset of \\mathbb {P}^k whose intersection with the hyperplane at infinity is equal to I_+.", "For x out of a neighbourhood of this analytic set, we have that \\Vert f(x)\\Vert \\ge c\\Vert x\\Vert ^{d_+} for some constant c>0.", "In particular, this holds for x as above when the neighbourhood of \\lbrace f_+=0\\rbrace is small enough.", "So we have \\Vert f(x)\\Vert >2\\Vert x\\Vert .", "This together with the inclusion f(\\mathbb {P}^k\\setminus U_+)\\subset \\mathbb {P}^k\\setminus U_+ imply by induction that \\Vert f^n(x)\\Vert \\ge 2^n\\Vert x\\Vert >\\Vert y\\Vert .", "The lemma follows.", "}For simplicity, using a linear change of coordinates, we can assume that R=1 satisfies the above lemma for some constant R_0>0.", "Define z^{\\prime }:=\\lambda z and w^{\\prime }:=w for some large constant \\lambda >4/R_0 that will be made precise later.", "We will apply Proposition \\ref {prop_branch_bis} to the restriction of \\Gamma _n in the domain \\mathbb {U}:=4\\mathbb {B}_k\\times 3\\mathbb {B}_k with respect to the coordinates (z^{\\prime },w^{\\prime }).", "}\\begin{lemma} Let d_n denote the number of periodic points of period n of f in k counted with multiplicity.", "Thenthe restriction of \\Gamma _n to \\mathbb {U} is contained in 4\\mathbb {B}_k\\times \\mathbb {B}_k and is a ramified covering of degree d_nover the factor 4\\mathbb {B}_k.", "Moreover, we have d_n=d^n+o(d^n).\\end{lemma}It is clear that \\Gamma _n\\cap \\mathbb {U}\\subset 4\\mathbb {B}_k\\times \\mathbb {B}_k so \\Gamma _n\\cap \\mathbb {U} is a ramified covering over 4\\mathbb {B}_k.Its degree is equal to the number of points in the intersection \\Gamma _n\\cap \\Delta \\cap (k\\times k) counted with multiplicity.", "So this degree is equal to d_n.\\end{corollary}Let \\pi :k\\times k\\rightarrow k denote the projection (x,y)\\mapsto x-y.", "If \\nu is a smooth probability measure with compact support in a small neighbourhood of 0 in k then d_n is the mass of the measure [\\Gamma _n]\\wedge \\pi ^(\\nu ).Since d^{-n}[\\Gamma _n]\\rightarrow +, the sequence d^{-n}d_n converges to the mass of +\\wedge \\pi ^(\\nu ).In particular, this mass does not depend on the choice of \\nu .", "In order to compute this mass, we take a sequence of measures \\nu _n converging to the Dirac mass at 0.", "Since+=T_+\\otimes T_-and T_\\pm are powers of positive closed (1,1)-currents with continuous potentials, +\\wedge \\pi ^(\\nu _n) convergeto (T_+\\otimes T_-)\\wedge [\\Delta ] which is the probability measure \\mu ^\\Delta .", "We conclude that d^{-n}d_n\\rightarrow 1.", "This completes the proof of the lemma.", "}Recall that we are using the coordinate systems z^{\\prime }:=\\lambda z=\\lambda (x-y) and w^{\\prime }:=w:=x in k\\times k.}\\begin{lemma} Let 0<\\delta <1 be a fixed constant.", "Fix also a constant \\lambda >0 large enough depending on \\delta .", "Then for n large enough, \\Gamma _n\\cap (\\mathbb {B}_k\\times 3\\mathbb {B}_k) admits at least (1-\\delta ^2)d^n connected components which are graphs over \\mathbb {B}_k.\\end{lemma}We want to apply Proposition \\ref {prop_branch_bis}.", "Fix an n large enough and define \\Gamma :=\\Gamma _n\\cap \\mathbb {U}.", "If \\widetilde{\\Gamma }^\\star is defined as in Proposition \\ref {prop_branch_bis}, we have to show that d^{-n}\\Vert \\widetilde{\\Gamma }^\\star \\Vert is as small as we want when \\lambda is large enough.", "}We first consider \\Gamma _n and \\Delta in the coordinate system (z,w) and recall that \\Gamma _n\\cap \\lbrace \\Vert z\\Vert \\le R_0\\rbrace is contained in \\lbrace \\Vert w\\Vert \\le 1\\rbrace .", "Define \\mathbb {U}_0:=\\lbrace \\Vert z\\Vert <4, \\Vert w\\Vert <3\\rbrace and \\Delta _0:=\\Delta \\cap \\mathbb {U}_0.Recall also that \\Pi :{\\rm Gr}(\\mathbb {P}^k\\times \\mathbb {P}^k,k)\\rightarrow \\mathbb {P}^k\\times \\mathbb {P}^k is the canonical projection and \\Pi ^{-1}(\\mathbb {U}_0) is identified with the product \\mathbb {U}_0\\times \\mathbf {G}, where \\mathbf {G} is the Grassmannian of linear subspaces of dimension k in k\\times k. Consider the square matrix A with complex coefficients of size k\\times k. The linear subspace z=Aw corresponds to a point in \\mathbf {G}.", "So we can use A for affine coordinates of a Zariski open subset \\mathbf {G}_0 of \\mathbf {G}.", "In these coordinates, the lift \\widehat{\\Delta }_0 of \\Delta _0 to {\\rm Gr}(\\mathbb {P}^k\\times \\mathbb {P}^k,k) is identified with \\Delta _0\\times \\lbrace 0\\rbrace .", "}The coordinate change (z^{\\prime },w^{\\prime })=(\\lambda z,w) on \\mathbb {U}_0 induces the coordinate change (z^{\\prime },w^{\\prime },A^{\\prime }):=(\\lambda z, w, \\lambda A) on \\mathbb {U}_0\\times \\mathbf {G}_0 and can be seen as the dilation along the normal directions to \\widehat{\\Delta }_0, as we have used when we defined the tangent currents in \\cite {DS12}.", "By Corollary \\ref {cor_density_T}, the density between \\widetilde{_+ and \\widehat{\\Delta } is zero.", "Thus, since \\lambda is large enough, the mass of \\widetilde{_+ in \\mathbb {W}:=\\lbrace \\Vert z^{\\prime }\\Vert < 4, \\Vert w^{\\prime }\\Vert < 3,\\Vert A\\Vert < 1\\rbrace with respect to the standard Euclidean metric associated with these coordinates, is as small as we want.By definition of \\widehat{_+ and \\widetilde{_+, we have d^{-n_i}[\\widetilde{\\Gamma }_{n_i}]\\rightarrow \\widetilde{_+.Therefore, the mass of d^{-n_i}[\\widetilde{\\Gamma }_{n_i}] in \\mathbb {W} is as small as we want when i is large enough.", "The property holds for every choice of \\widehat{_+.", "Since n is large enough, we deduce that for \\Gamma ,\\widetilde{\\Gamma }^\\star defined above, d^{-n}\\Vert \\widetilde{\\Gamma }^\\star \\Vert is as small as we want.", "This completes the proof of the lemma.", "}Recall that \\pi _1,\\pi _2 are the canonical projections from \\mathbb {P}^k\\times \\mathbb {P}^k onto its factors.Let \\Gamma ^{(j)}_n denote one of the graphs obtained in Lemma \\ref {lemma_branch_Gamma}.It is the graph of f^n over the domain \\pi _1(\\Gamma ^{(j)}_n).It intersects \\Delta at a unique point (a^{(j)}_n,a^{(j)}_n), where a^{(j)}_n is a periodic point of period n. The eigenvalues of the differential Df^n(a_n^{(j)}) of f^n at a^{(j)}_n do not depend on the local coordinate system at a_n^{(j)}.", "Fix a constant 0<\\epsilon <1 as in the introduction.", "}\\begin{proposition} Let \\delta and \\lambda be as in Lemma \\ref {lemma_branch_Gamma}.", "Then for n large enough there are at least(1-\\delta )d^n graphs \\Gamma _n^{(j)} such that Df^n(a_n^{(j)}) admits exactly p eigenvalues with modulus \\ge (d_+-\\epsilon )^{n/2} and k-p eigenvalues with modulus \\le (d_--\\epsilon )^{-n/2}, counted with multiplicity.\\end{proposition}Using the action of f^n on the cohomology of \\mathbb {P}^k, we havefor 1\\le q\\le pd^{-n}\\int _{\\Gamma _n} \\pi _1^(\\omega _{\\rm FS}^{k-p+q})\\wedge \\pi _2^(\\omega _{\\rm FS}^{p-q})=d^{-n}\\int _{\\mathbb {P}^k} \\omega _{\\rm FS}^{k-p+q}\\wedge (f^n)^(\\omega _{\\rm FS}^{p-q}) \\le d_+^{-n}and for 1\\le q\\le k-pd^{-n}\\int _{\\Gamma _n} \\pi _1^(\\omega _{\\rm FS}^{k-p-q})\\wedge \\pi _2^(\\omega _{\\rm FS}^{p+q})=d^{-n}\\int _{\\mathbb {P}^k} (f^n)_(\\omega _{\\rm FS}^{k-p-q})\\wedge \\omega _{\\rm FS}^{p+q}\\le d_-^{-n}.", "}Observe that the graphs \\Gamma ^{(j)}_n are contained in a fixed compact subset of k\\times k. Moreover, on any fixed compact subset of k, the standard Kähler form i{\\partial \\overline{\\partial }}\\Vert x\\Vert ^2 is comparable with \\omega _{\\rm FS}.", "Therefore, there are at least (1-\\delta )d^n graphs \\Gamma _n^{(j)} such that for 1\\le q\\le p\\int _{\\Gamma _n^{(j)}} (i{\\partial \\overline{\\partial }}\\Vert x\\Vert ^2)^{k-p+q}\\wedge (i{\\partial \\overline{\\partial }}\\Vert y\\Vert ^2)^{p-q} \\le c d_+^{-n}and for 1\\le q\\le k-p\\int _{\\Gamma _n^{(j)}} (i{\\partial \\overline{\\partial }}\\Vert x\\Vert ^2)^{k-p-q}\\wedge (i{\\partial \\overline{\\partial }}\\Vert y\\Vert ^2)^{p+q} \\le c d_-^{-n},where c>0 is a constant depending on \\delta ,\\lambda but independent of n. In what follows, we only consider the graphs satisfying these estimates.", "}It is convenient now to work with the coordinates (z,w)=(x-y,x).", "Since (z,w)=(\\lambda ^{-1}z^{\\prime },w^{\\prime }), \\Gamma _n^{(j)} is a graph in \\lambda ^{-1}\\mathbb {B}_k\\times \\mathbb {B}_k over \\lambda ^{-1}\\mathbb {B}_k.", "Denote by h:\\lambda ^{-1}\\mathbb {B}_k\\rightarrow \\Gamma _n^{(j)} the canonical map.", "}\\medskip {\\bf Claim.}", "There is a constant c_1>0 depending on \\delta ,\\lambda such that for 1\\le q\\le p\\big \\Vert h^\\big [(i{\\partial \\overline{\\partial }}\\Vert x\\Vert ^2)^{k-p+q}\\big ]\\wedge h^\\big [(i{\\partial \\overline{\\partial }}\\Vert y\\Vert ^2)^{p-q}\\big ]\\big \\Vert _0\\le c_1d_+^{-n}and for 1\\le q\\le k-p\\big \\Vert h^\\big [ (i{\\partial \\overline{\\partial }}\\Vert x\\Vert ^2)^{k-p-q}\\big ]\\wedge h^\\big [(i{\\partial \\overline{\\partial }}\\Vert y\\Vert ^2)^{p+q}\\big ] \\big \\Vert _0\\le c_1d_-^{-n},where \\Vert \\cdot \\Vert _a denotes the norm of a cotangent vector of maximal bi-degree (k,k) at the point a.", "}\\medskip }Assuming the claim, we first complete the proof of the proposition.Define for simplicity D:=Df^n(a^{(j)}_n).Let \\gamma _1,\\ldots ,\\gamma _k denote the eigenvalues of D ordered so that \|\\gamma _1\|\\le \\cdots \\le \|\\gamma _k\|.Let l be an integer such that \|\\gamma _i\|\\le 1 for i\\le k-l and \|\\gamma _i\|\\ge 1 for i>k-l. We have either l\\le p or k-l\\le k-p.", "Replacing f with f^{-1} if necessary, we can assume that l\\le p.}The operator D induces a linear automorphism, denoted by D^, of the cotangent space of k at a_n^{(j)} and its exterior powers.Let v be a unitary positive cotangent vector of bi-degree (p-q,p-q)which is an eigenvector associated with the eigenvalue \|\\gamma _{k-p+q+1}\|^{2}\\ldots \|\\gamma _k\|^{2} for some 1\\le q\\le p. Since i{\\partial \\overline{\\partial }}\\Vert y\\Vert ^2 is strictly positive,v is bounded by a constant times i{\\partial \\overline{\\partial }}\\Vert y\\Vert ^2.We deduce from the claim that\\big \\Vert h^\\big [(i{\\partial \\overline{\\partial }}\\Vert x\\Vert ^2)^{k-p+q}\\big ]\\wedge (\\pi _2\\circ h)^(v)\\big \\Vert _0\\le c_2d_+^{-n}for some constant c_2>0.", "}Consider also the push-forward operator (\\pi _1\\circ h)_* on forms of maximal bi-degree (k,k) at 0.This is the multiplication by the real Jacobian of the differential of (\\pi _1\\circ h)^{-1} at a_n^{(j)}.", "This Jacobian is equal to \|\\gamma _1-1\|^{2}\\ldots \|\\gamma _k-1\|^{2} because (\\pi _1\\circ h)^{-1}(x)=f^n(x)-x.", "Consider the value of this operator acting at the vector inside the sign \\Vert \\ \\Vert _0 of the last inequality.", "We get\\big \\Vert (i{\\partial \\overline{\\partial }}\\Vert x\\Vert ^2)^{k-p+q}\\wedge D^(v)\\big \\Vert _{a_n^{(j)}}\\le c_2d_+^{-n}\|\\gamma _1-1\|^{2}\\ldots \|\\gamma _k-1\|^{2}.Hence\|\\gamma _{k-p+q+1}\|^2\\ldots \|\\gamma _k\|^2 \\big \\Vert (i{\\partial \\overline{\\partial }}\\Vert x\\Vert ^2)^{k-p+q}\\wedge v\\big \\Vert _{a_n^{(j)}}\\le c_2d_+^{-n}\|\\gamma _1-1\|^{2}\\ldots \|\\gamma _k-1\|^{2}.Since v is unitary, we deduce that\|\\gamma _{k-p+q+1}\|^2\\ldots \|\\gamma _k\|^2 \\le c_3d_+^{-n}\|\\gamma _1-1\|^{2}\\ldots \|\\gamma _k-1\|^{2}for some constant c_3>0.\\end{lemma}Consider first the case l<p and take q=p-l.", "The choice of l impliesthat \|\\gamma _i-1\|\\le 2 for i\\le k-l and \|\\gamma _i-1\|\\le 2\|\\gamma _i\| for i\\ge k-l+1.", "This contradicts the inequality obtained above when n is large enough.", "So we have l=p.", "By taking q=1, we deduce from the same inequality that \|\\gamma _{k-p+1}\|\\ge c_4d_+^{n/2} for some constant c_4>0.", "Finally, since l=p, we can apply the same arguments to f^{-1} instead of f and obtain that\|\\gamma _{k-p}^{-1}\|\\ge c_4d_-^{n/2} since the eigenvalues of Df^{-1}(a_n^{(j)}) are \\gamma _1^{-1},\\ldots ,\\gamma _k^{-1}.The proposition follows.", "}{\\bf Proof of the claim.}", "We prove the first estimate.", "The second one is obtained in the same way.", "We haveh^\\big [(i{\\partial \\overline{\\partial }}\\Vert x\\Vert ^2)^{k-p+q}\\big ]\\wedge h^\\big [(i{\\partial \\overline{\\partial }}\\Vert y\\Vert ^2)^{p-q}\\big ]= h^\\big [(i{\\partial \\overline{\\partial }}\\Vert x\\Vert ^2)^{k-p+q}\\wedge (i{\\partial \\overline{\\partial }}\\Vert y\\Vert ^2)^{p-q}\\big ].Denote by \\Theta this form.", "Since it is positive of maximal bi-degree, we can write\\Theta (z)=\\varphi (z) (idz_1\\wedge d\\overline{z}_1)\\wedge \\ldots \\wedge (idz_k\\wedge d\\overline{z}_k),where \\varphi (z) is a positive function.We have to show that \\varphi (0)\\le c_1d_+^{-n} for some constant c_1>0.", "}The estimate given just before the claim implies that the integral of \\Theta on \\lambda ^{-1}\\mathbb {B}_k is smaller than cd_+^{-n}.", "Therefore, it is enough to check that \\varphi is a psh function.Observe that \\Theta is a finite sum of forms of type(i{\\partial \\overline{\\partial }}\|g_1\|^2)\\wedge \\ldots \\wedge (i{\\partial \\overline{\\partial }}\|g_k\|^2),where g_1,\\ldots ,g_k are holomorphic functions.", "The last form is equal to\|{\\rm Jac}(g_1,\\ldots ,g_k)\|^2(idz_1\\wedge d\\overline{z}_1)\\wedge \\ldots \\wedge (idz_k\\wedge d\\overline{z}_k),where {\\rm Jac} denotes the complex Jacobian of a holomorphic map.It is now clear that \\varphi is psh.", "The claim follows.\\hfill \\square }}}\\medskip {\\bf End of the proof of Theorem \\ref {th_main}.", "}Define\\mu _n:=d^{-n}\\sum _{a\\in Q_n} \\delta _a \\quad \\mbox{and} \\quad \\mu _n^\\Delta :=d^{-n}\\sum _{a\\in Q_n} \\delta _{(a,a)},where Q_n is as in the statement of the theorem and \\delta _{(a,a)} denotes the Dirac mass at the point (a,a) in \\Delta .", "By Lemma \\ref {lemma_degree_Gamma}, the mass of \\mu _n is bounded and any limit value of \\mu _n is of mass at most equal to 1.", "So in order to prove the theorem, it is enough to consider the case of smallest sets Q_n, i.e.", "Q_n= SP_n^\\epsilon .", "}If (n_i) is an increasing sequence of integers such that \\mu _{n_i} converges to a measure \\mu ^{\\prime }, we only have to show that \\mu ^{\\prime }\\ge \\mu .It is more convenient to work on \\Delta .", "Denote by \\mu ^{\\prime \\Delta } the limit of \\mu ^\\Delta _{n_i}.", "We have to check that \\mu ^{\\prime \\Delta }\\ge \\mu ^\\Delta .For this purpose, it suffices to construct a positive measure \\mu ^{\\prime \\prime \\Delta } such that \\mu ^{\\prime \\prime \\Delta }\\le \\mu ^\\Delta , \\mu ^{\\prime \\prime \\Delta }\\le \\mu ^{\\prime \\Delta } and \\Vert \\mu ^{\\prime \\prime \\Delta }\\Vert \\ge 1-\\delta for every fixed constant 0<\\delta <1.", "}Let \\lambda >0 be the constant satisfying Proposition \\ref {prop_branch_Gamma}.", "Define\\mathbb {S}_{n_i}:=d^{-n_i}\\sum _j [\\Gamma _{n_i}^{(j)}].By extracting a subsequence, we can assume that \\mathbb {S}_{n_i} converges to a current \\mathbb {S}.Since \\Gamma _{n_i}^{(j)} are graphs of bounded holomorphic maps, we have\\mathbb {S}_{n_i}\\wedge [\\Delta ] and \\mathbb {S}\\wedge [\\Delta ] are well-defined and \\mathbb {S}_{n_i}\\wedge [\\Delta ]\\rightarrow \\mathbb {S}\\wedge [\\Delta ].The last convergence is guaranteed by the fact that if a sequence of bounded graphs converges in the sense of currents then it also converges locally uniformly.", "}Define \\mu ^{\\prime \\prime \\Delta }:=\\mathbb {S}\\wedge [\\Delta ].", "Since d^{-n}[\\Gamma _n]\\rightarrow +, we have \\mathbb {S}\\le +.", "It follows that \\mu ^{\\prime \\prime \\Delta }\\le \\mu ^\\Delta .By Proposition \\ref {prop_branch_Gamma}, \\mathbb {S}_{n_i}\\wedge [\\Delta ] is a positive measure smaller than or equal to \\mu _{n_i}^\\Delta .Thus, \\mu ^{\\prime \\prime \\Delta }\\le \\mu ^{\\prime \\Delta }.", "Proposition \\ref {prop_branch_Gamma} also implies that the mass of \\mu ^{\\prime \\prime \\Delta } is at least equal to 1-\\delta .", "This completes the proof of the theorem.\\hfill \\square }\\begin{remark} \\rm The current \\mathbb {S} is constituted by lames of +=T_+\\otimes T_-.", "So its lames are locally a product of two manifolds.", "This property induces a product structure of \\mu by stable and unstable manifolds.\\end{remark}}\\begin{remark}\\rm Let L_+ and L_- be analytic subsets of \\mathbb {P}^k of pure dimension k-p and p respectively.", "Assume that L_+\\cap I_-=\\varnothing and L_-\\cap I_+=\\varnothing .", "Using the same method we can show that the points in f^{-n}(L_+)\\cap f^n(L_-) are equidistributed with respect to \\mu as n\\rightarrow \\infty , see also \\cite {DS6,Dujardin}.", "For the proof, we replace the graphs \\Gamma _n with f^{-n}(L_+)\\times f^n(L_-)=F^{-n}(L_+\\times L_-).", "The problem is simpler because the last analytic sets are already products of varieties and their lifts to {\\rm Gr}(\\mathbb {P}^k\\times \\mathbb {P}^k,k) are contained in {\\rm Gr}(\\mathbb {P}^k,k-p)\\times {\\rm Gr}(\\mathbb {P}^k,p).\\end{remark}\\end{lemma}}}}}$ T.-C. Dinh, UPMC Univ Paris 06, UMR 7586, Institut de Mathématiques de Jussieu, F-75005 Paris, France.", "DMA, UMR 8553, Ecole Normale Supérieure, 45 rue d'Ulm, 75005 Paris, France.", "[email protected], http://www.math.jussieu.fr/$\\sim $ dinh N. Sibony, Université Paris-Sud, Mathématique - Bâtiment 425, 91405 Orsay, France.", "[email protected]" ] ]	1403.0070
[ [ "Beyond Q-Resolution and Prenex Form: A Proof System for Quantified\n Constraint Satisfaction" ], [ "Abstract We consider the quantified constraint satisfaction problem (QCSP) which is to decide, given a structure and a first-order sentence (not assumed here to be in prenex form) built from conjunction and quantification, whether or not the sentence is true on the structure.", "We present a proof system for certifying the falsity of QCSP instances and develop its basic theory; for instance, we provide an algorithmic interpretation of its behavior.", "Our proof system places the established Q-resolution proof system in a broader context, and also allows us to derive QCSP tractability results." ], [ "The study of propositional proof systems for certifying the unsatisfiability of quantifier-free propositional formulas is supported by multiple motivations [3], [15].", "First, the desire to have an efficiently verifiable certificate of a formula's unsatisfiability is a natural and basic one, and indeed the field of propositional proof complexity studies, for various proof systems, whether and when succinct proofs exist for unsatisfiable formulas.", "Next, theorem provers are typically based on such proof systems, and so insight into the behavior of proof systems can yield insight into the behavior of theorem provers.", "Also, algorithms that perform search to determine the satisfiability of formulas can typically be shown to implicitly generate proofs in a proof system, and thus lower bounds on proof size translate to lower bounds on the running time of such algorithms.", "Finally, algorithms that check for unsatisfiability proofs of various restricted forms have been shown to yield tractable cases of the propositional satisfiability problem and related problems (see for example [1], [2]).", "In recent years, increasing attention has been directed towards the study of quantified proof systems that certify the falsity of quantified propositional formulas, which study is also pursued with the motivations similar to those outlined for the quantifier-free case.", "Indeed, the development of so-called QBF solvers, which determine the truth of quantified propositional formulas, has become an active research theme, and the study of quantified proof systems is pursued as a way to understand their behavior, as well as to explore the space of potential certificate formats for verifying their correctness on particular input instances [14].", "Q-resolution [4] is a quantified proof system that can be viewed as a quantified analog of resolution, one of the best-known and most customarily considered propositional proof systems.", "In the context of quantified propositional logic, Q-resolution is a heavily studied and basic proof system on which others are built and to which others are routinely compared, as well as a point of departure for the discussion of suitable certificate formats (see [11], [16], [12] for examples).", "However, the Q-resolution proof system has intrinsic shortcomings.", "First, it is only applicable to quantified propositional sentences that are in prenex form, that is, where all quantifiers appear in front.", "While it is certainly true that an arbitrary given quantified propositional formula may be efficiently prenexed, the process of prenexing is not canonical: intuitively, it involves choosing a total order of variables consistent with the partial order given by the input formula.", "As argued by Egly, Seidl, and Woltran [10], this may disrupt the original formula structure, “artificially extend” the scopes of quantifiers, and generate dependencies among variables that were not originally present, unnecessarily increasing the expense of solving; we refer the reader to their article for a contemporary discussion of this issue.Let us remark that using so-called dependency schemes is a potential way to cope with such introduced dependencies in a prenex formula [16].", "A second shortcoming of Q-resolution is that it is only defined in the propositional setting, despite that some scenarios may be more naturally and cleanly modelled by allowing variables to be quantified over domains of size greater than two." ], [ "In this article, we introduce a proof system that directly overcomes both of the identified shortcomings of Q-resolution and that, in a sense made precise, generalizes Q-resolution.", "We here define the quantified constraint satisfaction problem (QCSP) to be the problem of deciding, given a relational structure $\\mathbf {B}$ and a first-order sentence $\\phi $ (not necessarily in prenex form) built from the conjunction connective ($\\wedge $ ) and the two quantifiers ($\\forall $ , $\\exists $ ), whether or not the sentence is true on the structure.", "To permit different variables to have different domains, we formalize the QCSP using multi-sorted first-order logic.", "Our proof system (Section ) allows for the certification of falsity of QCSP instances.", "While Q-resolution provides rules for deriving clauses from a given quantified propositional formula, our proof system provides rules for deriving what we call constraints at various formula locations of a given QCSP instance; here, a constraint $(V, F)$ is a set $V$ of variables paired with a set $F$ of assignments, each defined on $V$ .", "A formula location $i$ paired with a constraint is called a judgement; a proof in our system is a sequence of judgements where each is derived from the previous ones via the rules.", "Crucially, we formulate and prove a key lemma (Lemma REF ) that shows (essentially) that if a judgement $(i, V, F)$ is derivable from a QCSP instance $(\\phi , \\mathbf {B})$ , then there exists a formula $\\psi (V)$ that “defines” the constraint $(V, F)$ over $\\mathbf {B}$ , such that $\\psi (V)$ can be conjoined to the input sentence $\\phi $ at location $i$ while preserving logical equivalence.", "This key lemma is then swiftly deployed to establish soundness and completeness of our proof system (Theorem REF ).", "We view the formulation of our proof system and of this key lemma as conceptual contributions.", "They offer a broader, deeper, and more general perspective on Q-resolution and what it means for a clause to be derivable by Q-resolution: we show (in a sense made precise) that each clause derivable by Q-resolution is derivable by our proof system (see Theorem REF ).", "This yields a clear and transparent proof of the soundness of Q-resolution which, interestingly, is carried out in the setting of first-order logic, despite the result concerning propositional logic.", "In order to relate our proof system to Q-resolution, we give a proof system for certain quantified propositional formulas (Section ) and prove that this second proof system is a faithful propositional interpretation of our QCSP proof system (Theorem REF ).", "We also provide an algorithmic interpretation of this second proof system.", "In particular, we give a nondeterministic search algorithm such that traces of execution that result in certifying falsity correspond to refutations in the proof system (Section REF ).", "As a consequence, the proof system yields a basis for establishing running-time lower bounds for any deterministic algorithm which instantiates the non-deterministic choices of our search algorithm.", "In the final section of the article (Section ), we present and study a notion of consistency for the QCSP that is naturally induced by our proof system.", "In the context of constraint satisfaction, a consistency notion is a condition which is necessary for the satisfiability of an instance and which can typically be efficiently checked.", "An example used in practice is arc consistency, and understanding when various forms of consistency provide an exact characterization of satisfiability (that is, when consistency is sufficient for satisfiability in addition to being necessary) has been a central theme in the tractability theory of constraint satisfaction [1], [2], [8].", "Atserias, Kolaitis, and Vardi [1] showed that checking for $k$ -consistency, an oft-considered consistency notion, can be viewed as detecting the absence of a proof of unsatisfiability having width at most $k$ , in a natural proof system (the width of a proof is the maximum number of variables appearing in a line of the proof); Kolaitis and Vardi [13] also characterized $k$ -consistency algebraically as whether or not Duplicator can win a natural Spoiler-Duplicator pebble game in the spirit of Ehrenfeucht-Fraïssé games.", "Inspired by these connections, we directly define a QCSP instance to be $k$ -judge-consistent if it has no unsatisfiability proof (in our proof system) of width at most $k$ ; and, we then present an algebraic, Ehrenfeucht-Fraïssé-style characterization of $k$ -judge-consistency (Theorem REF ).", "As an application of this algebraic characterization, we prove that (in a sense made precise) any case of the QCSP that lies in the tractable regime of a recent dichotomy theorem [7], is tractable via checking for $k$ -judge-consistency.Let us remark that this dichotomy theorem has since been generalized [5].", "That is, within the framework considered by that dichotomy, if a class of QCSP instances is tractable at all, it is tractable via $k$ -judge consistency.", "We remark that earlier work [6] presents algebraically a notion of consistency for the QCSP, but no corresponding proof system was explicitly presented; the notion of $k$ -judgement consistency can be straightforwardly verified to imply the notion of consistency in this earlier article.", "To sum up, this article presents a proof system for non-prenex quantified formulas.", "Our proof system is based on highly natural and simple rules, and its utility is witnessed by its connections to Q-resolution and by our presentation of a consistency notion that it induces, which allows for the establishment of tractability results.", "We hope that this proof system will serve as a point of reference and foundation for the future study of solvers and certificates for non-prenex formulas.", "One particular possibility for future work is to compare this proof system to others that are defined on non-prenex formulas, such as those discussed and studied by Egly [9].", "When $f$ is a mapping, we use $f \\upharpoonright S$ to denote its restriction to a set $S$ ; we use $f[s \\rightarrow b]$ to denote the extension of $f$ that maps $s$ to $b$ .", "We extend this notation to sets of mappings in the natural fashion." ], [ "We assume basic familiarity with the syntax and semantics of first-order logic.", "For the sake of broad applicability, in this article, we work with multi-sorted relational first-order logic, formalized here as follows.", "A signature is a pair $(\\sigma , \\mathcal {S})$ where $\\mathcal {S}$ is a set of sorts and $\\sigma $ is a set of relation symbols; each relation symbol $R \\in \\sigma $ has an associated arity $\\mathsf {ar}(R)$ which is an element of $\\mathcal {S}^$ .", "Each variable $v$ has an associated sort $s(v) \\in \\mathcal {S}$ ; an atom is a formula $R(v_1, \\ldots , v_k)$ where $R \\in \\sigma $ and $s(v_1) \\ldots s(v_k) = \\mathsf {ar}(R)$ .", "A structure $\\mathbf {B}$ on signature $(\\sigma , \\mathcal {S})$ consists of an $\\mathcal {S}$ -indexed family $B = \\lbrace B_s ~\|~ s \\in \\mathcal {S}\\rbrace $ of sets, called the universe of $\\mathbf {B}$ , and, for each symbol $R \\in \\sigma $ , an interpretation $R^{\\mathbf {B}} \\subseteq B_{\\mathsf {ar}(R)}$ .", "Here, for a word $w = w_1 \\ldots w_k \\in \\mathcal {S}^$ , we use $B_w$ to denote the product $B_{w_1} \\times \\cdots \\times B_{w_k}$ .", "Suppose that $V$ is a set of variables where each variable $v \\in V$ has an associated sort $s(v)$ ; by a mapping $f: V \\rightarrow B$ , we mean a mapping that sends each $v \\in V$ to an element $f(v) \\in B_{s(v)}$ .", "When $\\phi $ is a formula, we use $\\mathsf {free}(\\phi )$ to denote the set containing the free variables of $\\phi $ .", "The width of a formula $\\phi $ is the maximum of $\|\\mathsf {free}(\\psi )\|$ over all subformulas $\\psi $ of $\\phi $ .", "A quantified-conjunctive formula (for short, qc-formula) is a formula over a signature built from atoms on the signature, conjunction ($\\wedge $ ), and the two quantifiers ($\\forall $ , $\\exists $ ).", "Note that we permit conjunction of arbitrary arity.", "As expected, a qc-sentence is a qc-formula $\\phi $ such that $\\mathsf {free}(\\phi ) = \\emptyset $ .", "We allow conjunction of arbitrary (finite) arity, in formulas.", "We will use $\\top $ to denote a sentence that is always true; this is considered to be a qc-sentence.", "A relation $P$ is qc-definable over a structure $\\mathbf {B}$ if there exists a qc-formula $\\phi (v_1, \\ldots , v_k)$ such that $P \\subseteq B_{s(v_1) \\ldots s(v_k)}$ and $P$ contains a tuple $(b_1, \\ldots , b_k)$ if and only if $\\mathbf {B}, b_1, \\ldots , b_k \\models \\phi (v_1, \\ldots , v_k)$ .", "Note that by the notation $\\mathbf {B}, b_1, \\ldots , b_k \\models \\phi (v_1, \\ldots , v_k)$ , we mean that the structure $\\mathbf {B}$ and the mapping taking each $v_i$ to $b_i$ satisfy $\\phi $ .", "We sometimes use $\\equiv $ to indicate logical equivalence of two formulas.", "We define the QCSP to be the problem of deciding, given a QCSP instance, which is a pair $(\\phi , \\mathbf {B})$ where $\\phi $ is a qc-sentence and $\\mathbf {B}$ is a structure that both have the same signature, whether or not $\\mathbf {B}\\models \\phi $ .", "In this section, we present our proof system for the QCSP, and establish some basic properties thereof, including soundness and completeness.", "Let $(\\phi , \\mathbf {B})$ be a QCSP instance, and conceive of $\\phi $ as a tree.", "The proof system will allow us to derive what we call constraints at the various nodes of the tree.", "To facilitate the discussion, we will assume that each qc-sentence $\\phi $ has, associated with it, a set $I_{\\phi }$ of indices that contains one index for each subformula occurrence of $\\phi $ , that is, for each node of the tree corresponding for $\\phi $ .", "Let us remark that (in general) the collection of constraints derivable at an occurrence of a subformula does not depend only on the subformula, but also on the subformula's location in the full formula $\\phi $ .", "When $i$ is an index, we use $\\phi (i)$ to denote the actual subformula of the subformula occurrence corresponding to $i$ ; we will also refer to $i$ as a location.", "Figure: Formula discussed in Examples and .Example Consider the qc-sentence $\\phi = \\exists x \\forall y (E(x,y) \\wedge (\\exists x E(x,y)))$ .", "(See Figure REF .)", "When viewed as a tree, this formula has 6 nodes.", "We may index them naturally according to the depth-first search order: we could take the index set $\\lbrace 1, \\ldots , 6 \\rbrace $ where $\\phi (4) = \\phi (6) = E(x,y)$ , $\\phi (5) = \\exists x \\phi (6)$ , $\\phi (3) = \\phi (4) \\wedge \\phi (5)$ , $\\phi (2) = \\forall y \\phi (3)$ , and $\\phi (1) = \\exists x \\phi (2)$ .", "$\\Box $ We say that an index $i$ is a parent of an index $j$ , and also that $j$ is a child of $i$ , if, in viewing the formula $\\phi $ as a tree, the root of the subformula occurrence of $i$ is the parent of the root of the subformula occurrence of $j$ .", "Note that, when this holds, the formula $\\phi (i)$ either is of the form $Q v \\phi (j)$ where $Q$ is a quantifier and $v$ is a variable, or is a conjunction where $\\phi (j)$ appears as a conjunct.", "As examples, with respect to the qc-sentence and indexing in Example REF , index 3 has two children, namely, 4 and 5, and index 3 has one parent, namely, 2.", "Definition Let $(\\phi , \\mathbf {B})$ be a QCSP instance.", "A constraint (on $(\\phi , \\mathbf {B})$ ) is a pair $(V, F)$ where $V$ is a set of variables occurring in $\\phi $ , and $F$ is a set of mappings from $V$ to $B$ .", "A judgement (on $(\\phi , \\mathbf {B})$ ) is a triple $(i, V, F)$ where $i \\in I_{\\phi }$ and $(V, F)$ is a constraint with $V \\subseteq \\mathsf {free}(\\phi (i))$ ; it is empty if $F = \\emptyset $ .", "Here, we use the convention that (relative to a QCSP instance) there is exactly one map $e: \\emptyset \\rightarrow B$ defined on the empty set, so there are two constraints whose variable set is the empty set: the constraint $(\\emptyset , \\emptyset )$ , and the constraint $(\\emptyset , \\lbrace e \\rbrace )$ where $e$ is the aforementioned map.", "When $(U_1, F_1)$ , $(U_2, F_2)$ are two constraints on the same QCSP instance, we define the join of $F_1$ and $F_2$ , denoted by $F_1 \\bowtie F_2$ , to be the set $\\lbrace f: U_1 \\cup U_2 \\rightarrow B ~\|~(f \\upharpoonright U_1) \\in F_1, (f \\upharpoonright U_2) \\in F_2 \\rbrace .$ When $(U, F)$ is a constraint and $y$ is a variable in $U$ , we use $\\epsilon _y F$ to denote the set $\\lbrace f: U \\setminus \\lbrace y \\rbrace \\rightarrow B ~\|~\\textup {for each $ b Bs(y)$, it holds that $ f[y b] F$}\\rbrace .$ The operator $\\epsilon _y$ will be used to eliminate a universally quantified variable $y$ .", "Dually, in the following definition, projection can be used to cope with existential quantification.", "Definition A judgement proof on $(\\phi , \\mathbf {B})$ is a finite sequence of judgements, each of which has one of the following types: Table: NO_CAPTIONWe say that a judgement $(i, V, F)$ is derivable if there exists a judgement proof that contains the judgement.", "The width of a judgement $(i, V, F)$ is $\|V\|$ .", "The width of a judgement proof is the maximum width over all of its judgements, and the length of a judgement proof is the number of judgements that it contains.", "$\\Box $ Let us emphasize that, by definition, a judgement proof is a finite sequence of judgements, and by definition, in order for a triple $(i, V, F)$ to be a judgement, it must hold that all variables in $V$ are free variables of $\\phi (i)$ .", "Consequently, upward flow can only be applied to a judgement $(j, V, F)$ if all variables in $V$ are free variables of $\\phi (i)$ , where $i$ is the parent of $j$ ; an analogous comment holds for downward flow.", "Example Let $\\phi $ be the qc-sentence from Example REF (shown in Figure REF ), considered as a sentence over signature $(\\lbrace E \\rbrace , \\lbrace e, u \\rbrace )$ with $\\mathsf {ar}(E) = eu$ and where $s(x) = e$ and $s(y) = u$ .", "Define $\\mathbf {B}$ to be a structure over this signature having universe $B$ defined by $B_e = \\lbrace a, b, c \\rbrace $ and $B_u = \\lbrace d, e, f \\rbrace $ , and where $E^{\\mathbf {B}} = \\lbrace (a, d), (a, e), (a, f), (b, e), (c, f) \\rbrace $ .", "To offer a feel of the proof system, we give some examples of derivable judgements.", "Let $F_E$ be the set of assignments from $\\lbrace x, y \\rbrace $ to $B$ that satisfy $E(x,y)$ (over $\\mathbf {B}$ ).", "By (atom), we may derive the judgement $(4, \\lbrace x, y \\rbrace , F_E)$ .", "By (upward flow), we may then derive the judgement $(3, \\lbrace x, y \\rbrace , F_E)$ .", "By ($\\forall $ -elimination), we may then derive the judgement $(2, \\lbrace x \\rbrace , G)$ , where $G$ contains the single map that takes $x$ to $a$ .", "By applying (downward flow) twice, we may then derive the judgement $(4, \\lbrace x \\rbrace , G)$ .", "By (atom), we may also derive the judgement $(6, \\lbrace x, y \\rbrace , F_E)$ .", "By (projection), we may then derive the judgement $(6, \\lbrace x \\rbrace , H)$ , where $H$ contains the maps taking $x$ to $a$ , $b$ , and $c$ , respectively.", "Let us remark that, even though $\\phi (4) = \\phi (6)$ and we derived the judgement $(4, \\lbrace x \\rbrace , G)$ , it is not possible to derive the judgement $(6, \\lbrace x \\rbrace , G)$ .", "(This can be verified by Lemma REF , to be presented next, and the observation that $\\mathbf {B}\\models \\phi $ .)", "$\\Box $ We now prove soundness and completeness of our proof system; we first establish a lemma, which indicates what it means for a judgement to be derivable.", "When $\\phi $ is a qc-formula with index set $I$ , and $\\lbrace \\theta _i \\rbrace _{i \\in I}$ is a family of formulas, we use $\\phi ^{+\\theta }$ to denote the formula obtained from $\\phi $ by replacing, at each location $i$ , the subformula $\\phi (i)$ by $\\phi (i) \\wedge \\theta _i$ .", "Formally, we define $\\phi ^{+\\theta }$ by induction.", "When $\\phi (i)$ is an atom, we define $\\phi ^{+\\theta }(i) = \\phi (i) \\wedge \\theta _i$ .", "When $\\phi (i) = \\phi (j) \\wedge \\phi (k)$ , we define $\\phi ^{+\\theta }(i) = \\phi ^{+\\theta }(j) \\wedge \\phi ^{+\\theta }(k) \\wedge \\theta _i$ .", "When $\\phi (i) = Qv \\phi (j)$ , we define $\\phi ^{+\\theta }(i) = (Q v \\phi ^{+\\theta }(j)) \\wedge \\theta _i$ .", "We define $\\phi ^{+\\theta }$ to be $\\phi ^{+\\theta }(r)$ where $r$ is the root index of $\\phi $ (that is, where $r$ is such that $\\phi (r) = \\phi $ ).", "Lemma Let $(\\phi , \\mathbf {B})$ be a QCSP instance.", "For every derivable judgement $(i, V, F)$ , there exists a qc-formula $\\psi $ such that $\\mathsf {free}(\\psi ) = V$ ; for each $f: V \\rightarrow B$ , it holds that $f \\in F$ if and only if $\\mathbf {B}, f \\models \\psi $ ; and for any family $\\lbrace \\theta _i \\rbrace _{i \\in I}$ of formulas, it holds that $\\phi ^{+\\theta }$ entails $\\phi ^{+\\theta }[i \\stackrel{\\wedge }{\\rightarrow }\\psi ]$ (and hence that $\\phi ^{+\\theta } \\equiv \\phi ^{+\\theta }[i \\stackrel{\\wedge }{\\rightarrow }\\psi ]$ ).", "Here, $\\phi ^{+\\theta }[i \\stackrel{\\wedge }{\\rightarrow }\\psi ]$ denotes the formula where, at location $i$ , the subformula $\\phi ^{+\\theta }(i)$ is replaced with $\\phi ^{+\\theta }(i) \\wedge \\psi $ .", "Moreover, if one has a judgement proof of width at most $k$ , then each of the formulas $\\psi $ produced for its judgements has $\\mathsf {width}(\\psi ) \\le k$ .", "We consider the different types of judgements, and use the notation from Definition REF .", "In each case, the claim on the width is straightforwardly verified.", "In the case of (atom), we take $\\psi = \\phi (i)$ ; the formulas $\\phi ^{+\\theta }$ and $\\phi ^{+\\theta }[i \\stackrel{\\wedge }{\\rightarrow }\\psi ]$ are logically equivalent since $\\phi (i) \\equiv \\phi (i) \\wedge \\phi (i)$ .", "In the case of (projection), by induction, we have that $\\phi ^{+\\theta }$ entails $\\phi ^{+\\theta }[i \\stackrel{\\wedge }{\\rightarrow }\\psi ^{\\prime }]$ where $\\psi ^{\\prime }$ is the formula for the judgement $(i, V, F)$ .", "We take $\\psi = \\exists v_1 \\ldots \\exists v_m \\psi ^{\\prime }$ , where $v_1, \\ldots , v_m$ is a listing of the elements in $V \\setminus U$ .", "We have that $\\phi ^{+\\theta }[i \\stackrel{\\wedge }{\\rightarrow }\\psi ^{\\prime }]$ entails $\\phi ^{+\\theta }[i \\stackrel{\\wedge }{\\rightarrow }\\psi ]$ , and hence by transitivity of the entailment relation that $\\phi ^{+\\theta }$ entails $\\phi ^{+\\theta }[i \\stackrel{\\wedge }{\\rightarrow }\\psi ]$ .", "In the case of (join), by induction, we have (for any family $\\lbrace \\theta _j \\rbrace _{j \\in I}$ ) that $\\phi ^{+\\theta }$ entails both $\\phi ^{+\\theta }[i \\stackrel{\\wedge }{\\rightarrow }\\psi _1]$ and $\\phi ^{+\\theta }[i \\stackrel{\\wedge }{\\rightarrow }\\psi _2]$ where $\\psi _1$ and $\\psi _2$ are the formulas corresponding to the judgements $(i, U_1, F_1)$ and $(i, U_2, F_2)$ .", "We take $\\psi = \\psi _1 \\wedge \\psi _2$ .", "Fix a family $\\lbrace \\theta _j \\rbrace _{j \\in I}$ , and define $\\lbrace \\theta ^{\\prime }_j \\rbrace _{j \\in I}$ to be the family that has $\\theta ^{\\prime }_i = \\theta _i \\wedge \\psi _1$ , and is everywhere else equal to $\\lbrace \\theta _j \\rbrace _{j \\in I}$ .", "We have that $\\phi ^{+\\theta }$ entails $\\phi ^{+\\theta }[i \\stackrel{\\wedge }{\\rightarrow }\\psi _1] \\equiv \\phi ^{+\\theta ^{\\prime }}$ , and that $\\phi ^{+\\theta ^{\\prime }}$ entails $\\phi ^{+\\theta ^{\\prime }}[i \\stackrel{\\wedge }{\\rightarrow }\\psi _2]$ .", "It follows that $\\phi ^{+\\theta }$ entails $\\phi ^{+\\theta ^{\\prime }}[i \\stackrel{\\wedge }{\\rightarrow }\\psi _2] \\equiv \\phi ^{+\\theta }[i \\stackrel{\\wedge }{\\rightarrow }(\\psi _1 \\wedge \\psi _2)]$ .", "In the case of ($\\forall $ -elimination), by induction, we have that $\\phi ^{+\\theta }$ entails $\\phi ^{+\\theta }[j \\stackrel{\\wedge }{\\rightarrow }\\psi ^{\\prime }]$ , where $\\psi ^{\\prime }$ is the formula for the judgement $(j, V, F)$ .", "We take $\\psi = \\forall y \\psi ^{\\prime }$ .", "We claim that the formula $\\phi ^{+\\theta }[j \\stackrel{\\wedge }{\\rightarrow }\\psi ^{\\prime }]$ is logically equivalent to $\\phi ^{+\\theta }[i \\stackrel{\\wedge }{\\rightarrow }\\psi ]$ , which suffices to give that $\\phi ^{+\\theta }$ entails $\\phi ^{+\\theta }[i \\stackrel{\\wedge }{\\rightarrow }\\psi ]$ .", "This is because the subformula of $\\phi ^{+\\theta }[j \\stackrel{\\wedge }{\\rightarrow }\\psi ^{\\prime }]$ at location $i$ is logically equivalent to $(\\forall y (\\phi ^{+\\theta }(j) \\wedge \\psi ^{\\prime })) \\wedge \\theta _i$ which is logicically equivalent to $(\\forall y \\phi ^{+\\theta }(j)) \\wedge (\\forall y \\psi ^{\\prime }) \\wedge \\theta _i$ .", "In the cases of (upward flow) and (downward flow), we take $\\psi $ to be equal to the formula that is given to us by the previous judgement.", "It is straightforwardly verified that $\\phi ^{+\\theta }[i \\stackrel{\\wedge }{\\rightarrow }\\psi ]$ and $\\phi ^{+\\theta }[j \\stackrel{\\wedge }{\\rightarrow }\\psi ]$ are logically equivalent.", "Theorem Let $(\\phi , \\mathbf {B})$ be a QCSP instance.", "An empty judgement on $(\\phi , \\mathbf {B})$ is derivable if and only if $\\mathbf {B}\\lnot \\models \\phi $ .", "This theorem is proved in the following way.", "The forward direction follows immediately from the previous lemma.", "For the backward direction, we show by induction that, for each location $i$ , there exists a derivable judgement for which the formula $\\psi $ given by the previous lemma is equal to $\\phi (i)$ !", "Suppose that an empty judgement $(i, V, F)$ is derivable.", "Then by invoking the (projection) rule, the empty judgement $(i, \\emptyset , \\emptyset )$ is derivable.", "Define $\\lbrace \\theta _j \\rbrace _{j \\in I}$ so that $\\theta _j$ is the true formula $\\top $ for each $j \\in I$ ; then, invoking Lemma REF , we have that there exists a qc-formula $\\psi $ that is false on $\\mathbf {B}$ (that is, $\\mathbf {B}\\lnot \\models \\psi $ ) and such that $\\phi $ entails $\\phi [i \\stackrel{\\wedge }{\\rightarrow }\\psi ]$ .", "We have that $\\mathbf {B}\\lnot \\models \\phi [i \\stackrel{\\wedge }{\\rightarrow }\\psi ]$ , and hence (since $\\phi $ entails $\\phi [i \\stackrel{\\wedge }{\\rightarrow }\\psi ]$ ) we have that $\\mathbf {B}\\lnot \\models \\phi $ .", "Suppose that $\\mathbf {B}\\lnot \\models \\phi $ .", "We claim that for each location $i$ , there exists a derivable judgement $(i, V, F)$ where the corresponding formula $\\psi $ , given by the proof of Lemma REF , is equal to $\\phi (i)$ .", "This suffices, as then the root location $r$ has a derivable judgement $(r, V, F)$ such that $F = \\emptyset $ .", "We establish the claim by induction.", "When $\\phi (i)$ is an atom, we use the judgement given by (atom) in the proof system (Definition REF ).", "When $\\phi (i)$ is a conjunction, let $j$ and $k$ be the children of $i$ , so that $\\phi (i) = \\phi (j) \\wedge \\phi (k)$ .", "Let $(j, V_j, F_j)$ and $(k, V_k, F_k)$ be the derivable judgements given by induction.", "By (upward flow) in the proof system, we have that $(i, V_j, F_j)$ and $(i, V_k, F_k)$ are derivable judgements; by invoking (join), we obtain the desired judgement.", "When $\\phi (i)$ begins with existential quantification, let $j$ be the child of $i$ , and denote $\\phi (i) = \\exists x \\phi (j)$ .", "Let $(j, V, F)$ be the derivable judgement given by induction; by applying the rule (projection) to obtain a constraint on $V \\setminus \\lbrace x \\rbrace $ and then the rule (upward flow), we obtain the desired derivation.", "When $\\phi (i)$ begins with universal quantification, let $j$ be the child of $i$ , and denote $\\phi (i) = \\forall y \\phi (j)$ .", "Let $(j, V, F)$ be the derivable judgement given by induction; by the ($\\forall $ -elimination) rule, we obtain the desired derivation." ], [ "Propositional proof system", "In this section, we introduce a different proof system, which is a propositional interpretation of the QCSP proof system.", "For differentiation, we refer to judgements and judgement proofs as defined in the previous section as constraint judgements and constraint judgement proofs.", "A literal is a propositional variable $v$ or the negation $\\overline{v}$ thereof.", "Two literals are complementary if one is a variable $v$ and the other is $\\overline{v}$ ; each is said to be the complement of the other.", "A clause is a disjunction of literals that contains, for each variable, at most one literal on the variable; a clause is sometimes viewed as the set of the literals that it contains.", "A clause is empty if it does not contain any literals.", "The variables of a clause are simply the variables that underlie the clause's literals, and the set of variables of a clause $\\alpha $ is denoted by $\\mathsf {vars}(\\alpha )$ .", "A clause $\\gamma $ is a resolvent of two propositional clauses $\\alpha $ and $\\beta $ if there exists a literal $L \\in \\alpha $ such that its complement $M$ is in $\\beta $ , and $\\gamma = (\\alpha \\setminus \\lbrace L \\rbrace ) \\cup (\\beta \\setminus \\lbrace M \\rbrace )$ .", "A clause $\\gamma $ is falsified by a propositional assignment $a$ if $a$ is defined on $\\mathsf {vars}(\\gamma )$ and each literal in $\\gamma $ evalutes to false under $a$ .", "We define a QCBF instance to be a propositional formula not having free variables that is built from clauses, conjunction, and universal and existential quantification over propositional variables.", "As with QCSP instance, we assume that each QCBF instance $\\psi $ has an associated index set that contains an index for each subformula of $\\psi $ .", "Note that a clause is not considered to have any subformulas, other than itself.", "As an example, consider the QCBF instance $\\exists x \\forall y \\exists z((\\overline{y} \\vee z) \\wedge (y \\vee \\overline{z} \\vee x))$ .", "This formula would have 6 indices: one for each of the two clauses, one for the conjunction of the two clauses, and one for each of the quantifiers.", "Let $\\psi $ be a QCBF instance.", "A clause judgement (on $\\psi $ ) is a pair $(i, \\alpha )$ where $i \\in I_{\\psi }$ and $\\alpha $ is a clause with $\\mathsf {vars}(\\alpha ) \\subseteq \\mathsf {free}(\\psi (i))$ ; a clause judgement $(i, \\alpha )$ is empty if $\\alpha $ is empty.", "Definition A clause judgement proof on a QCBF instance $\\psi $ is a finite sequence of clause judgements, each of which has one of the following types: Table: NO_CAPTIONWe say that a clause judgement $(i, \\alpha )$ is derivable if there exists a clause judgement proof that contains the clause judgement.", "$\\Box $ The width of a clause judgement $(i, \\alpha )$ is $\|\\mathsf {vars}(\\alpha )\|$ .", "The width of a clause judgement proof is the maximum width over all of its clause judgements; the length of a clause judgement proof is the number of judgements that it contains.", "In a clause judgement proof, we refer to judgements that are not derived by the rules (upward flow) and (downward flow) as non-flow judgements.", "Let us emphasize that we allow resolution over both existential and universal variables, and the resolvent must be non-tautological, because it must be a clause (for our definition of clause)." ], [ "Relationship to the QCSP proof system", "We now define the notion of a QCSP translation of a QCBF instance $\\psi $ , which intuitively is a QCSP instance that behaves just like $\\psi $ .", "When discussing QCSP translations, we will be concerned with structures $\\mathbf {B}$ that have just one sort $s$ with $B_s = \\lbrace 0, 1 \\rbrace $ ; we slightly abuse notation and simply write $B = \\lbrace 0, 1 \\rbrace $ .", "Definition When $\\psi $ is a QCBF instance, define a QCSP translation of $\\psi $ to be a QCSP instance $(\\phi , \\mathbf {B})$ where $\\mathbf {B}$ is a one-sorted structure with $B = \\lbrace 0, 1 \\rbrace $ and where $\\phi $ is obtainable from $\\psi $ by replacing each clause $\\gamma $ having variables $v_1, \\ldots , v_k$ with an atom $R(v_1, \\ldots , v_k)$ such that $R^{\\mathbf {B}} = \\lbrace (f(v_1), \\ldots , f(v_k)) ~\|~\\mbox{$f: \\lbrace v_1, \\ldots , v_k \\rbrace \\rightarrow \\lbrace 0, 1 \\rbrace $ satisfies $\\gamma $} \\rbrace ;$ we typically assume that $I_{\\phi } = I_{\\psi }$ and that each subformula of $\\phi $ has the same index as the natural corresponding subformula of $\\psi $ .", "Note that when $\\psi $ is a QCBF instance and $(\\phi , \\mathbf {B})$ is a QCSP translation thereof, it can be immediately verified, by induction, that for each index $i$ , an assignment $g$ to $\\lbrace 0, 1 \\rbrace $ that is defined on $\\mathsf {free}(\\psi (i)) = \\mathsf {free}(\\phi (i))$ satisfies $\\psi (i)$ if and only if it satisfies $\\phi (i)$ .", "In particular, we have that $\\psi $ is true if and only if $\\phi $ is true on $\\mathbf {B}$ .", "We prove that our clause judgement proof system is a faithful interpretation of our QCSP proof system, as made precise by the following theorem.", "Theorem Let $\\psi $ be a QCBF instance and let $(\\phi , \\mathbf {B})$ be a QCSP translation of $\\psi $ .", "For each clause judgement $(i, \\alpha )$ that is derivable from $\\psi $ , there exists a constraint judgement $(i, \\mathsf {vars}(\\alpha ), F)$ derivable from $(\\phi , \\mathbf {B})$ such that the unique $g: \\mathsf {vars}(\\alpha ) \\rightarrow \\lbrace 0, 1 \\rbrace $ that does not satisfy $\\alpha $ is not in $F$ .", "The other way around, for each constraint judgement $(i, V, F)$ that is derivable from $(\\phi , \\mathbf {B})$ , and for each mapping $g: V \\rightarrow \\lbrace 0, 1 \\rbrace $ with $g \\notin F$ , there exists a clause judgement $(i, \\alpha )$ derivable from $\\psi $ where $\\mathsf {vars}(\\alpha ) \\subseteq V$ and $g$ does not satisfy $\\alpha $ .", "Consequently, an empty clause judgement is derivable from $\\psi $ if and only if an empty constraint judgement is derivable from $(\\phi ,\\mathbf {B})$ .", "The proof of this theorem is provided in Section ." ], [ "Simulation of Q-resolution", "We now show that our clause judgement proof system simulates Q-resolution [4], as made precise by the following theorem.", "Theorem Let $\\psi $ be a QCBF instance in prenex form, whose quantifier-free part is a conjunction of clauses with index $c$ .", "If a clause $\\gamma $ is derivable from $\\psi $ by Q-resolution, then the clause judgement $(c, \\gamma )$ is derivable from $\\psi $ by the clause judgement proof system.", "It is straightforwardly verified that each clause derivable by Q-resolution from $\\psi $ is contained in the smallest set $\\mathcal {C}$ of clauses satisfying the following recursive definition: Each clause $\\alpha $ appearing in $\\psi $ is in $\\mathcal {C}$ .", "$\\mathcal {C}$ is closed under taking resolvents.", "If $\\alpha \\in \\mathcal {C}$ and $y \\in \\mathsf {vars}(\\alpha )$ is universally quantified and is the first variable in $\\mathsf {vars}(\\alpha )$ to be quantified on the unique path from $c$ to the root of $\\psi $ , then $\\alpha \\setminus \\lbrace y, \\overline{y} \\rbrace $ is in $\\mathcal {C}$ .", "It suffices to show, then, that for each $\\alpha \\in \\mathcal {C}$ , the clause judgement $(c, \\alpha )$ is derivable.", "We consider the three types of clauses according to the just-given recursive definition.", "For each clause $\\alpha $ appearing in $\\psi $ , the clause judgement $(c, \\alpha )$ is derivable by applying the (clause) rule at the location of $\\alpha $ , followed by one application of the (upward flow) rule.", "For a clause that is a resolvent of two other clauses, one can simply apply the (resolve) rule.", "Finally, suppose that $(c, \\alpha )$ is derivable and that $y \\in \\mathsf {vars}(\\alpha )$ satisfies the described condition.", "We need to show that $\\alpha \\setminus \\lbrace y, \\overline{y} \\rbrace $ is derivable.", "Let $j$ be the first location where $y$ is quantified when walking from $c$ to the root, and let $K$ be the set of nodes appearing on the unique path from $c$ to the child of $j$ (inclusive).", "By the definition of $\\mathcal {C}$ , no variable in $\\mathsf {vars}(\\alpha )$ is quantified at a location in $K$ and $\\mathsf {vars}(\\alpha ) \\subseteq \\mathsf {free}(\\psi (k))$ for each $k \\in K$ ; hence, (upward flow) can be applied repeatedly to derive $(k, \\alpha )$ for each $k \\in K$ .", "By applying ($\\forall $ -removal) at the child of $j$ , we obtain $(j, \\alpha \\setminus \\lbrace y, \\overline{y} \\rbrace $ ; then, (downward flow) can be applied repeatedly to derive $(c, \\alpha \\setminus \\lbrace y, \\overline{y} \\rbrace $ .", "The soundness of Q-resolution (derivability of an empty clause implies falsehood) is thus a consequence of this theorem, Theorem REF , and Theorem REF ." ], [ "Algorithmic interpretation", "We define the following notions relative to a QCBF instance $\\psi $ .", "We view $\\psi $ as a rooted tree.", "When $i, j \\in I_{\\psi }$ , we write $i \\le _{\\psi } j$ if $i$ is an ancestor of $j$ , that is, if $i$ occurs on the unique path from $j$ to the root; we write $i <_{\\psi } j$ if $i \\le _{\\psi } j$ and $i \\ne j$ .", "We define a located variable to be a pair $(i, u)$ where $i \\in I_{\\psi }$ is an index, and $u$ is a variable that is quantified at location $i$ ; this pair is a $\\forall $ -located variable if $u$ is universally quantified at location $i$ .", "We say that index $j \\in I_{\\psi }$ follows a located variable $(i, u)$ if $i <_{\\psi } j$ and for each index $k \\in I_{\\psi }$ with $i <_{\\psi } k \\le _{\\psi } j$ , it holds that $u \\in \\mathsf {free}(\\phi (k))$ .", "We say that a located variable $(j, v)$ follows a located variable $(i, u)$ if $j$ follows $(i, u)$ .", "We say that an index $j$ or a located variable $(j, v)$ follows a set $S$ of located variables if $j$ follows each located variable in $S$ .", "A set $S$ of located variables is coherent if for any two distinct elements $(i, u), (j, v) \\in S$ , one follows the other (that is, either $(i, u)$ follows $(j, v)$ or $(j, v)$ follows $(i, u)$ ).", "When $S$ is a set of located variables, we use $\\mathsf {vars}(S)$ to denote the set of variables occurring in the located variables in $S$ .", "Observe that when a set $S$ of located variables is coherent, no variable occurs in two distinct located variables in $S$ , and so $\|S\| = \|\\mathsf {vars}(S)\|$ .", "We now present a nondeterministic, recursive algorithm that, in a sense to be made precise, corresponds to the proof system.", "At each point in time, the algorithm maintains a set $S$ of coherent variables; actions it may perform include branching on a located variable $(i, u)$ such that adding $(i, u)$ to $S$ is still coherent, and, nondeterministically setting a $\\forall $ -located variable $(i, y)$ that follows $S$ .", "The algorithm returns either the false value $F$ or the indeterminate value $\\bot $ .", "On these two values, we define the operation $\\vee $ by $F \\vee F = F$ and $\\bot \\vee F = F \\vee \\bot = \\bot \\vee \\bot =\\bot $ .", "Intuitively speaking, the algorithm returns the indeterminate value $\\bot $ when a nondeterministically selected action cannot be carried out.", "We assume that when the algorithm is first invoked on a given QCBF instance, the set $S$ is initially assigned to the empty set.", "Algorithm Detect_Falsity( QCBF instance $\\psi $ , coherent set $S$ , assignment $a:\\mathsf {vars}(S)\\rightarrow \\lbrace 0,1\\rbrace $ ) { Select nondeterministically and perform one of the following: (falsify) check if there exists a location $i$ following $S$ such that $\\psi (i)$ is a clause falsified by $a$ with $\\mathsf {vars}(\\psi (i))=\\mathsf {vars}(S)$ ; if so, return F, else return $\\bot $ ; (Q-branch) check if there exists a located variable $(i,u)\\notin \\hspace{0.1pt}S$ such that $S\\cup \\lbrace (i,u)\\rbrace $ is coherent; if not, return $\\bot $ , else: - nondeterministically select such a located variable $(i,u)$ ; - nondeterministically pick subsets $S_{0},S_{1}\\subseteq \\hspace{0.1pt}S$ with $S_{0}\\cup \\hspace{0.1pt}S_{1}=S$ ; - return Detect-Falsity($\\psi ,S_{0}\\cup \\lbrace (i,u)\\rbrace ,(a\\upharpoonright \\mathsf {vars}(S_{0}))[u\\rightarrow 0]$ ) $\\vee $ Detect-Falsity($\\psi ,S_{1}\\cup \\lbrace (i,u)\\rbrace ,(a\\upharpoonright \\mathsf {vars}(S_{1}))[u\\rightarrow 1]$ ) ($\\forall $ -branch) check if there exists a $\\forall $ -located variable $(i,y)$ that follows $S$ ; if not, return $\\bot $ , else: - nondeterministically select such a $\\forall $ -located variable $(i,y)$ ; - nondeterministically pick a value $b\\in \\lbrace 0,1\\rbrace $ ; - return Detect-Falsity($\\psi ,S\\cup \\lbrace (i,y)\\rbrace ,a[y\\rightarrow \\vspace{0.1pt}b]$ ) } Relative to a clause judgement proof, we employ the following terminology.", "A clause judgement $(j, \\beta )$ that is derived using a previous judgement $(i, \\alpha )$ is said to be a successor of $(i, \\alpha )$ ; also, $(i, \\alpha )$ is said to be a predecessor of $(j, \\beta )$ .", "So, a clause judgement derived using the (clause) rule has 0 predecessors, one derived using the (resolve) rule has 2 predecessors, and one derived using one of the other rules has 1 predecessor.", "We say that a clause judgement proof is tree-like if each clause judgement has at most one successor; in this case, each clause judgement $(i, \\alpha )$ naturally induces a tree (defined recursively) where: each node is labelled with a clause judgement; the root is labelled with $(i, \\alpha )$ ; and, for each predecessor of the clause judgement $(i, \\alpha )$ , the root has a child which is the tree of the predecessor.", "We formalize the notion of a trace of the nondeterministic algorithm.", "A trace of a QCBF instance $\\psi $ is a rooted tree where: each node has a label $(S, a)$ where $S$ is a coherent set of located variables and $a: \\mathsf {vars}(S) \\rightarrow \\lbrace 0, 1 \\rbrace $ is an assignment; each node has 0, 1, or 2 children; when a node has 2 children and label $(S, a)$ , the labels of the two children could be generated by the (Q-branch) step from $(S, a)$ ; when a node has 1 child and label $(S, a)$ , the label of the child could be generated by the ($\\forall $ -branch) step from $(S, a)$ ; when a node has 0 children and label $(S, a)$ , the node has an associated index $i$ that follows $S$ and such that $\\psi (i)$ is a clause falsified by $a$ with $\\mathsf {vars}(\\psi (i)) = \\mathsf {vars}(S)$ .", "We take it as evident that this notion of trace properly formalizes the recursion trees that the algorithm generates.", "Let $e$ denote the unique assignment from $\\emptyset $ to $\\lbrace 0, 1 \\rbrace $ .", "We now show that, up to polynomial-time computable translations, tree-like clause judgement proofs of an empty clause correspond precisely to traces having root label $(\\emptyset , e)$ .", "Theorem Let $\\psi $ be a QCBF instance; let $n \\ge 1$ .", "There exists a tree-like clause judgement proof $P$ (viewed as a tree) of an empty clause with $n$ non-flow judgements if and only if there exists a trace $T$ whose root has label $(\\emptyset , e)$ and having $n$ nodes.", "Moreover, both implied translations (from proof to trace, and from trace to proof) can be computed in polynomial time.", "The proof of this theorem is provided in Section ." ], [ "Algebraic characterization of $k$ -judge-consistency", "We will assume that all structures under discussion in this section are finite, in that each structure's universe is finite.", "Definition Let $k \\ge 1$ .", "A QCSP instance $(\\phi , \\mathbf {B})$ is $k$ -judge-consistent if there does not exist a judgement proof of width less than or equal to $k$ that contains an empty judgement.", "Definition Let $(\\phi , \\mathbf {B})$ be a QCSP instance, where $\\phi $ is a qc-formula with index set $I$ , and let $k \\ge 1$ .", "A $k$ -constraint system $P$ provides, for each $i \\in I$ and each $V \\subseteq \\mathsf {free}(\\phi (i))$ with $\|V\| \\le k$ , a non-empty set $P[i, V]$ of maps from $V$ to $B$ satisfying the following four properties: $(\\alpha )$ If $\\phi (i)$ is an atom $R(v_1, \\ldots , v_m)$ with $V = \\lbrace v_1, \\ldots , v_m \\rbrace $ , then $P[i, \\lbrace v_1, \\ldots , v_m \\rbrace ] \\subseteq \\lbrace f: \\lbrace v_1, \\ldots , v_m \\rbrace \\rightarrow B ~\|~ (f(v_1), \\ldots , f(v_m)) \\in R^{\\mathbf {B}} \\rbrace $ $(\\pi )$ If $U \\subseteq V$ , then $P[i, U] = (P[i, V] \\upharpoonright U)$ .", "$(\\lambda )$ If $j$ is a child of $i$ and $V \\subseteq \\mathsf {free}(\\phi (j))$ , then $P[i, V] = P[j, V]$ .", "$(\\epsilon )$ If $j$ is a child of $i$ , $\\phi (i) = \\forall y \\phi (j)$ , $U$ is a subset of $\\mathsf {free}(\\phi (j))$ with $\|U\| \\le k$ and $y \\in U$ , and $V = U \\setminus \\lbrace y \\rbrace $ , then $P[i, V] \\subseteq \\epsilon _y( P[j, U] )$ .", "We show that the existence of a $k$ -constraint system characterizes $k$ -judge consistency.", "Theorem Let $(\\phi , \\mathbf {B})$ be a QCSP instance.", "There exists a $k$ -constraint system $P$ for the instance if and only if the instance is $k$ -judge-consistent.", "A proof of this theorem can be found in Section .", "Theorem For each $k \\ge 1$ , there exists a polynomial-time algorithm that, given a QCSP instance $(\\phi , \\mathbf {B})$ , decides if the instance is $k$ -judge-consistent.", "We begin by describing the algorithm, which decides, given a QCSP instance $(\\phi , \\mathbf {B})$ , whether or not there exists a $k$ -constraint system (this property is equivalent to $k$ -judge-consistency by Theorem REF ).", "Throughout, $i$ and $j$ will always denote indices from $I_{\\phi }$ .", "For each $i \\in I_{\\phi }$ and $V \\subseteq \\mathsf {free}(\\phi (i))$ with $\|V\| \\le k$ , the algorithm initializes $Q[i, V]$ to be $\\lbrace f: \\lbrace v_1, \\ldots , v_m \\rbrace \\rightarrow B ~\|~ (f(v_1), \\ldots , f(v_m)) \\in R^{\\mathbf {B}} \\rbrace $ in the case that $\\phi (i)$ is an atom $R(v_1, \\ldots , v_m)$ and $V = \\lbrace v_1, \\ldots , v_m \\rbrace $ , and otherwise initializes $Q[i, V]$ to be the set of all maps from $V$ to $B$ .", "The algorithm then iteratively performs the following rules (which parallel properties $(\\pi )$ , $(\\lambda )$ , and $(\\epsilon )$ in the definition of $k$ -constraint system) until no changes can be made to $Q$ : When $i$ is an index and $U \\subseteq V \\subseteq \\mathsf {free}(\\phi (i))$ with $\|V\| \\le k$ , assign to $Q[i, U]$ the value $Q[i, U] \\cap (Q[i, V] \\upharpoonright U)$ and then assign to $Q[i, V]$ the value $\\lbrace f \\in Q[i, V] ~\|~ (f \\upharpoonright U) \\in Q[i, U] \\rbrace $ .", "If $j$ is a child of $i$ , $V \\subseteq \\mathsf {free}(\\phi (i)) \\cap \\mathsf {free}(\\phi (j))$ , and $\|V\| \\le k$ , assign to each of $Q[i, V]$ , $Q[j, V]$ the value $Q[i, V] \\cap Q[j, V]$ .", "If $j$ is a child of $i$ , $\\phi (i) = \\forall y \\phi (j)$ , and $U$ is a set of variables with $y \\in U \\subseteq \\mathsf {free}(\\phi (j))$ and $\|U\| \\le k$ , then assign to $Q[i, U \\setminus \\lbrace y \\rbrace ]$ the value $Q[i, U \\setminus \\lbrace y \\rbrace ] \\cap \\epsilon _y(P[j, U])$ .", "This algorithm runs in polynomial time: there are polynomially many pairs $(i, V)$ for which $Q[i, V]$ is initialized and used, and each $Q[i, V]$ contains (at most) polynomially many maps: when $\|V\| \\le k$ , the number of maps from $V$ to $B$ is polynomial.", "(Here, when we say polynomial, we mean as a function of the input length.)", "Applying the three given rules can be done in polynomial time; each time they are applied, the sets $Q[i, V]$ may only decrease in size.", "Hence, the process of repeatedly applying the three rules until no changes are possible terminates in polynomial time.", "We now explain why the instance is $k$ -judge-consistent if and only if no set $Q[i, V]$ is empty, which suffices to give the theorem.", "It is straightforward to verify that, for any $k$ -constraint-system $P$ , the invariant $P[i, V] \\subseteq Q[i, V]$ is maintained by the algorithm.", "Hence, when the algorithm terminates, if any set $Q[i, V]$ is empty, then there does not exist a $k$ -constraint system $P$ .", "It is also straightforward to verify that, when the algorithm terminates, the four properties in the definition of $k$ -constraint system hold on $Q$ .", "(As an example, consider property ($\\lambda $ ).", "Suppose that $j$ is a child of $i$ and that $V \\subseteq \\mathsf {free}(\\phi (i)) \\cap \\mathsf {free}(\\phi (j))$ .", "When the algorithm terminates, since the second rule can no longer be applied it must hold that $Q[i,V] = Q[j,V] = Q[i,V] \\cap Q[j,V]$ .)", "Hence, if the algorithm terminates without any empty set $Q[i, V]$ , it holds that $Q$ is a $k$ -constraint system.", "We can upper bound the number of iterations that the algorithm performs on an instance $(\\phi , \\mathbf {B})$ in the following way.", "Let $n$ be the maximum number of free variables, over all subformulas of $\\phi $ .", "For each index $i$ of $\\phi $ , the algorithm maintains, for each $V \\subseteq \\mathsf {free}(\\phi (i))$ with $\|V\| \\le k$ , a set of mappings from $V$ to $B$ .", "The size of such a set is at most $\|B\|^{\|V\|}$ .", "In each iteration, each such set of mappings can only have mappings deleted from it.", "The number of iterations is thus upper bounded by the number of mappings that can occur in such sets of mappings, which is $\|I_{\\phi }\|( {n \\atopwithdelims ()k} \|B\|^k + {n \\atopwithdelims ()k-1}\|B\|^{k-1} + \\cdots +{n \\atopwithdelims ()0} \|B\|^0 )$ .", "We now show that checking for $k$ -judge-consistency gives a way to decide a set of prenex qc-sentences that is tractable via the dichotomy theorem on so-called prefixed graphs [7].", "In particular, we prove this in the setting where relation symbols have bounded arity.", "Let us refer to the width notion defined in that previous work [7] as elimination width.", "Define the Q-width of a prenex qc-sentence $\\phi $ to be the maximum of its elimination width and $\\max _R \|\\mathsf {ar}(R)\|$ (where this maximum ranges over all relation symbols $R$ appearing in $\\phi $ ).", "Theorem Let $k \\ge 1$ .", "Suppose that $\\phi $ is a prenex qc-sentence with Q-width $k$ (or less).", "For any finite structure $\\mathbf {B}$ , it holds that $(\\phi , \\mathbf {B})$ is $k$ -judge-consistent if and only if $\\mathbf {B}\\models \\phi $ .", "(Intuitively, this says that checking for $k$ -judge consistency is a decision procedure for QCSP instances involving $\\phi $ .)", "This theorem, in conjunction with Theorem REF , immediately implies that for any set $\\Phi $ of qc-sentences having Q-width bounded by a constant $k$ , checking for $k$ -judge-consistency is a uniform polynomial-time procedure that decides any QCSP instance $(\\phi , \\mathbf {B})$ where $\\phi \\in \\Phi $ and $\\mathbf {B}$ is finite.", "Hence, in the setting of bounded arity, checking for $k$ -judge-consistency is a generic reasoning procedure that correctly decides the tractable cases of QCSP identified by the work on elimination width.", "In order to establish this theorem, we first prove a lemma.", "Lemma Suppose that the QCSP instance $(\\theta , \\mathbf {B})$ is $k$ -judge-consistent, that $\\mathbf {B}$ is a finite structure, and that $\\theta ^{\\prime }$ is a qc-sentence obtained from $\\theta $ by applying one of the following three syntactic transformations to a subformula of $\\theta $ : $\\bigwedge _{i \\in I} \\phi _i \\leadsto (\\bigwedge _{j \\in J} \\phi _j) \\wedge (\\bigwedge _{k \\in K} \\phi _k)$ , where $I$ is the disjoint union of $J$ and $K$ $Qv (\\phi \\wedge \\psi ) \\leadsto (Qv \\phi ) \\wedge \\psi $ where $v \\notin \\mathsf {free}(\\psi )$ $\\forall y \\bigwedge _{i \\in I} \\phi _i \\leadsto \\bigwedge _{i \\in I} (\\forall y \\phi _i)$ Then, the QCSP instance $(\\theta ^{\\prime }, \\mathbf {B})$ is $k$ -judge-consistent.", "By Theorem REF , it suffices to show that if $(\\theta , \\mathbf {B})$ has a $k$ -constraint system $P$ , then $(\\theta ^{\\prime }, \\mathbf {B})$ does as well.", "We consider each of the three cases.", "Case (1): We define a $k$ -constraint system $P^{\\prime }$ for $(\\theta ^{\\prime }, \\mathbf {B})$ in the following way.", "Relative to the transformation, let $i$ denote the index of $\\phi _i$ in both $\\theta $ and $\\theta ^{\\prime }$ ; let $c$ denote the index of $\\bigwedge _{i \\in I} \\phi _i$ in $\\theta $ and of $(\\bigwedge _{j \\in J} \\phi _j) \\wedge (\\bigwedge _{k \\in K} \\phi _k)$ in $\\theta ^{\\prime }$ ; let $a$ be the index of $\\bigwedge _{j \\in J} \\phi _j$ in $\\theta ^{\\prime }$ ; and let $b$ be the index of $\\bigwedge _{k \\in K} \\phi _k$ in $\\theta ^{\\prime }$ .", "For each other subformula occurrence in $\\theta ^{\\prime }$ , there is a corresponding subformula occurrence in $\\theta $ ; we will assume that these two corresponding subformula occurrences share the same index.", "We now describe how to define $P^{\\prime }$ .", "Whenever discussing $P^{\\prime }[d, V]$ , it will hold that $d$ is an index of $\\theta ^{\\prime }$ , and we assume that $V \\subseteq \\mathsf {free}(\\theta ^{\\prime }(d))$ and $\|V\| \\le k$ .", "We define $P^{\\prime }[i, V]$ as $P[i, V]$ .", "We define $P^{\\prime }[c, V]$ as $P[c, V]$ .", "We define $P^{\\prime }[a, V]$ as $P[c, V]$ , and similarly we define $P^{\\prime }[b, V]$ as $P[c, V]$ .", "For each other index $\\ell $ of $\\theta ^{\\prime }$ , we define $P^{\\prime }[\\ell , V]$ as $P[\\ell , V]$ .", "It is straightforward to verify that $P^{\\prime }$ is a $k$ -constraint system.", "Case (2): We proceed as in the previous case; we define a $k$ -constraint system $P^{\\prime }$ for $(\\theta ^{\\prime }, \\mathbf {B})$ .", "Relative to the transformation, let $a$ denote the index of $Qv (\\phi \\wedge \\psi )$ in $\\theta $ ; let $b$ denote the index of the subformula $Qv \\phi $ in $\\theta ^{\\prime }$ .", "We define $P^{\\prime }[b, V]$ as $P[a, V]$ .", "For each other subformula occurrence of $\\theta ^{\\prime }$ with index $\\ell $ , there exists a corresponding subformula occurrence of $\\theta $ which we assume to also have index $\\ell $ .", "We define $P^{\\prime }[\\ell , V]$ as $P[\\ell , V]$ .", "It is straightforward to verify that $P^{\\prime }$ is a $k$ -constraint system.", "Case (3): We proceed as in the previous cases; we define a $k$ -constraint system $P^{\\prime }$ for $(\\theta ^{\\prime }, \\mathbf {B})$ .", "Let $d$ denote the index of $\\forall y \\bigwedge _{i \\in I} \\phi _i$ in $\\theta $ , and also the index of $\\bigwedge _{i \\in I} (\\forall y \\phi _i)$ in $\\theta ^{\\prime }$ .", "Let $i$ denote the index of $\\phi _i$ in both $\\theta $ and $\\theta ^{\\prime }$ .", "Let $c$ denote the index of $\\bigwedge _{i \\in I} \\phi _i$ in $\\theta $ , and let $i^{\\prime }$ denote the index of $\\forall y \\phi _i$ in $\\theta ^{\\prime }$ .", "For each $V \\subseteq \\mathsf {free}(\\forall y \\phi _i)$ with $\|V\| \\le k$ , define $P^{\\prime }[i^{\\prime }, V]$ to be $P[d, V]$ .", "Elsewhere, define $P^{\\prime }$ to be equal to $P$ (each other index of $\\theta ^{\\prime }$ corresponds to an index of $\\theta $ ).", "It is straightforward to verify that $P^{\\prime }$ is a $k$ -constraint system.", "In the region of interest, the property $(\\epsilon )$ can be verified as follows.", "Suppose that $U \\subseteq \\mathsf {free}(\\phi (i))$ has $\|U\| \\le k$ and $y \\in U$ , and that $V = U \\setminus \\lbrace y \\rbrace $ .", "Then $P[d, V] \\subseteq \\epsilon _y(P[c, U]) = \\epsilon _y(P[i, U])$ since $P$ is a $k$ -constraint system.", "As $P^{\\prime }[i^{\\prime }, V] = P[d, V]$ by our definition of $P^{\\prime }$ , it follows that $P^{\\prime }[i^{\\prime }, V] \\subseteq \\epsilon _y(P[i, U])$ .", "(Theorem REF ) Suppose that the instance $(\\phi , \\mathbf {B})$ is not $k$ -judge-consistent.", "Then, by definition, there exists a judgement proof for the instance containing an empty judgement, implying that $\\mathbf {B}\\lnot \\models \\phi $ by Theorem REF .", "For the other direction, suppose that $\\mathbf {B}\\lnot \\models \\phi $ .", "From the definition of elimination width (defined as width in [7]), it can straightforwardly be verified by induction on the number of variables in $\\phi $ that $\\phi $ can be transformed to a sentence $\\phi ^{\\prime }$ having width less than or equal to $k$ , via the three syntactic transformations of Lemma REF .", "As these three syntactic transformations preserve logical equivalence, we have $\\mathbf {B}\\lnot \\models \\phi ^{\\prime }$ .", "By Theorem REF , an empty judgement is derivable; by the proof of this theorem, there is a judgement proof with the empty judgement whose width is equal to the width of $\\phi ^{\\prime }$ .", "Since the width of $\\phi ^{\\prime }$ is less than or equal to $k$ , we thus obtain a judgement proof of the empty judgement having width less than or equal to $k$ , so by definition, $(\\phi ^{\\prime }, \\mathbf {B})$ is not $k$ -judge-consistent.", "By appeal to Lemma REF , $(\\phi , \\mathbf {B})$ is not $k$ -judge-consistent." ], [ "Acknowledgements", "The author thanks Moritz Müller and Friedrich Slivovsky for useful comments.", "This work was supported by the Spanish project TIN2013-46181-C2-2-R, by the Basque project GIU12/26, and by the Basque grant UFI11/45." ], [ "Proof of Theorem ", "The theorem follows directly from the following two theorems.", "Theorem Let $\\psi $ be a QCBF instance and let $(\\phi , \\mathbf {B})$ be a QCSP translation of $\\psi $ .", "For each clause judgement proof of $\\psi $ having length $s$ and width $w$ , there exists a constraint judgement proof of $(\\phi , \\mathbf {B})$ having length $\\le 2s$ and width $\\le w+1$ such that: each clause judgement $(i, \\alpha )$ appearing in the first proof has the entailment property that there exists a constraint judgement in the second proof of the form $(i, \\mathsf {vars}(\\alpha ), F)$ such that each $f \\in F$ satisfies $\\alpha $ (equivalently, the unique $g: \\mathsf {vars}(\\alpha ) \\rightarrow \\lbrace 0, 1 \\rbrace $ that does not satisfy $\\alpha $ is not in $F$ ).", "A direct consequence of this theorem is that if the original clause judgement proof contains an empty clause, then the produced constraint judgement proof contains an empty constraint.", "We prove this by induction on $s$ .", "Given a clause judgement proof $P$ of length $s + 1$ we create a constraint judgement proof in the following way.", "Apply induction to the clause judgement proof consisting of the first $s$ judgements in $P$ ; this gives a constraint judgement proof $P^{\\prime }$ .", "We then need to show how to augment $P^{\\prime }$ .", "We consider cases, depending on the rule used to derive the last judgement of $P$ .", "We use the notation of Definition REF .", "In the case of (clause) deriving $(i, \\alpha )$ , apply (atom) at location $i$ .", "In the case of (resolve) deriving $(i, \\gamma )$ from $(i, \\alpha )$ and $(i, \\beta )$ , let $v$ be the variable underlying the complementary literals that are eliminated from $\\alpha $ and $\\beta $ to obtain $\\gamma $ .", "The rule (join) is applied to the constraint judgements corresponding to $(i, \\alpha )$ and $(i, \\beta )$ to obtain a new judgement, and then (projection) is used to remove the variable $v$ from that new judgement.", "In the case of (upward flow) or (downward flow), the same rule is applied to the corresponding constraint judgement.", "In the case of ($\\forall $ -removal), the rule ($\\forall $ -elimination) is applied to the corresponding constraint judgement.", "In the case (resolve), two new constraint judgements are produced, and in all other cases, one new constraint judgement is produced; hence, the claim on the length is correct.", "In the case (resolve), the width of the first constraint judgement produced is one more than the width of the corresponding clause judgement, and the width of the second constraint judgement produced is equal to the width of the corresponding clause judgement; in all other cases, the new constraint judgement produced has width equal to that of the corresponding clause judgement.", "Hence, the claim on the width is correct.", "In each case, it is straightforward to verify the claimed entailment property.", "As an example, we verify the claimed entailment property in the case of (resolve).", "Suppose that (resolve) derives $(i, \\gamma )$ from $(i, \\alpha )$ and $(i, \\beta )$ .", "Let $g_{\\alpha }: \\mathsf {vars}(\\alpha ) \\rightarrow \\lbrace 0, 1 \\rbrace $ , $g_{\\beta }: \\mathsf {vars}(\\beta ) \\rightarrow \\lbrace 0, 1 \\rbrace $ , and $g_{\\gamma }: \\mathsf {vars}(\\gamma ) \\rightarrow \\lbrace 0, 1 \\rbrace $ be assignments not satisfying $\\alpha $ , $\\beta $ , and $\\gamma $ , respectively.", "Let $v$ be the variable such that $\\mathsf {vars}(\\gamma ) = (\\mathsf {vars}(\\alpha ) \\cup \\mathsf {vars}(\\beta )) \\setminus \\lbrace v \\rbrace $ .", "We assume without loss of generality that $g_{\\alpha }(v) = 0$ and that $g_{\\beta }(v) = 1$ .", "Let $(i, \\mathsf {vars}(\\alpha ), F_{\\alpha })$ and $(i, \\mathsf {vars}(\\beta ), F_{\\beta })$ be the constraint judgements for $(i, \\alpha )$ and $(i, \\beta )$ , respectively; we have $g_{\\alpha } \\notin F_{\\alpha }$ and $g_{\\beta } \\notin F_{\\beta }$ .", "Consider the constraint judgement $(i, \\mathsf {vars}(\\alpha ) \\cup \\mathsf {vars}(\\beta ), F_{\\alpha } \\bowtie F_{\\beta })$ obtained by applying (join) to these two constraint judgements.", "By definition of the join $\\bowtie $ , neither $g_{\\alpha }$ not $g_{\\beta }$ has an extension defined on $\\mathsf {vars}(\\alpha ) \\cup \\mathsf {vars}(\\beta )$ that is contained in $F_{\\alpha } \\bowtie F_{\\beta }$ .", "Next, consider the constraint judgement $(i, \\mathsf {vars}(\\gamma ), (F_{\\alpha } \\bowtie F_{\\beta }) \\upharpoonright \\mathsf {vars}(\\gamma ))$ obtained from the previous one by applying (projection).", "We claim that $g_{\\gamma } \\notin (F_{\\alpha } \\bowtie F_{\\beta }) \\upharpoonright \\mathsf {vars}(\\gamma )$ .", "Suppose not, for a contradiction; then there exists an extension $g^{\\prime }_{\\gamma }$ of $g_{\\gamma }$ which is contained in $F_{\\alpha } \\bowtie F_{\\beta }$ .", "If $g^{\\prime }_{\\gamma }(v) = 0$ , then $g^{\\prime }_{\\gamma }$ is an extension of $g_{\\alpha }$ , but $g^{\\prime }_{\\gamma } \\in F_{\\alpha } \\bowtie F_{\\beta }$ contradicts $g_{\\alpha } \\notin F_{\\alpha }$ ; analogously, If $g^{\\prime }_{\\gamma }(v) = 1$ , then $g^{\\prime }_{\\gamma }$ is an extension of $g_{\\beta }$ , but $g^{\\prime }_{\\gamma } \\in F_{\\alpha } \\bowtie F_{\\beta }$ contradicts $g_{\\beta } \\notin F_{\\beta }$ .", "Theorem Let $\\psi $ be a QCBF instance and let $(\\phi , \\mathbf {B})$ be a QCSP translation of $\\psi $ .", "For each constraint judgement proof of $(\\phi , \\mathbf {B})$ having length $s$ and width $w$ , there exists a clause judgement proof of $\\psi $ of length $\\le s \\cdot \\max (w2^{w-1}, 1)$ and width $\\le w$ such that: each constraint judgement $(i, V, F)$ appearing in the first proof has the entailment property that, for each mapping $g: V \\rightarrow \\lbrace 0, 1 \\rbrace $ with $g \\notin F$ , there exists a clause judgement $(i, \\alpha )$ with $\\mathsf {vars}(\\alpha ) \\subseteq V$ in the second proof where $\\alpha $ is not satisfied by $g$ .", "A direct consequence of this theorem is that if a constraint judgement proof of $(\\phi , \\mathbf {B})$ having length $s$ and width $w$ contains an empty constraint, it may be augmented by one constraint judgement to contain an empty constraint of the form $(i, \\emptyset , \\emptyset )$ , and then the theorem yields that there is a clause judgement proof having an empty clause of length $\\le (s+1) \\cdot \\max (w2^{w-1}, 1)$ and width $\\le w$ .", "We proceed as in the proof of the previous theorem.", "We prove this by induction on $s$ .", "Given a constraint judgement proof $P$ of length $s+1$ , we create a clause judgement proof $P^{\\prime }$ by applying induction to $P$ with the last constraint judgement removed; we then explain how to augment the resulting clause judgement proof $P^{\\prime }$ so that the last constraint judgement of $P$ has a corresponding clause judgement with the properties given in the theorem statement.", "We consider cases depending on the rule used to derive the last constraint judgement of $P$ ; we use the notation of Definition REF .", "In the case of (atom) deriving $(i, V, F)$ , apply (clause) at location $i$ .", "In the case of (projection) deriving $(i, U, F \\upharpoonright U)$ from $(i, V, F)$ , we first explain how to obtain the clause judgements in the case that $\|V\| = \|U\| + 1$ .", "Let $v$ be the variable such that $U \\cup \\lbrace v \\rbrace = V$ .", "For each clause judgement $(i, \\alpha )$ with $\\mathsf {vars}(\\alpha ) \\subseteq U$ that can be obtained by resolving two clause judgements in $P^{\\prime }$ on the variable $v$ , include the clause judgement in the proof.", "The maximum number of clause judgements that we can add in this fashion is the number of clauses on $(w-1)$ variables, that is, $2^{w-1}$ .", "In the general case where $U \\subseteq V$ , we may proceed by applying the described procedure $\|V\| - \|U\|$ many times.", "Since $\|V\| - \|U\| \\le w$ , the total number of clause judgements that will be added can be upper bounded by $w 2^{w-1}$ .", "In the case of (join), no clause judgement needs to be added.", "This is because of the following.", "Suppose the constraint judgement $(i, U_1 \\cup U_2, F_1 \\bowtie F_2)$ is obtained by applying (join) to $(i, U_1, F_1)$ and $(i, U_2, F_2)$ .", "For each mapping $g: U_1 \\cup U_2 \\rightarrow \\lbrace 0, 1 \\rbrace $ with $g \\notin F_1 \\bowtie F_2$ , it holds (by definition of $\\bowtie $ ) that either $g \\upharpoonright U_1 \\notin F_1$ or $g \\upharpoonright U_2 \\notin F_2$ .", "In the case of ($\\forall $ -elimination) deriving $(i, V \\setminus \\lbrace y \\rbrace , \\epsilon _y F)$ from $(j, V, F)$ , take all clause judgements $(j, \\alpha )$ where $y \\in \\mathsf {vars}(\\alpha ) \\subseteq V$ , and apply ($\\forall $ -removal) to each of these clause judgements.", "In the case of (upward flow) or (downward flow), the same rule is applied to the corresponding clause judgement.", "In each case, the clause judgements produced have width less than or equal to $w$ .", "We now consider the number of clause judgements produced in each case.", "This number is 1 in the cases (atom), (upward flow), and (downward flow), and is 0 in the case (join).", "In the case of (projection), we argued that this number is less than or equal to $w 2^{w-1}$ .", "In the case of ($\\forall $ -elimination), since this rule can only be applied if $w \\ge 1$ and at most $2^{w-1}$ clauses are generated, we can also bound this number by $w 2^{w-1}$ .", "In each case, it is straightforward to verify the claimed entailment property." ], [ "Proof of Theorem ", "The theorem follows directly from the following two theorems.", "Theorem Let $\\psi $ be a QCBF instance.", "Given a tree-like clause judgement proof $P$ (viewed as a tree) of an empty clause, there exists a trace $E$ whose root has label $(\\emptyset , e)$ and where the number of nodes in $E$ is equal to the number of non-flow judgements in $P$ .", "(Here, we use $e$ to denote the unique assignment from $\\emptyset $ to $\\lbrace 0, 1 \\rbrace $ .)", "Also, the translation from $P$ to $E$ is polynomial-time computable.", "We prove the following result, which yields the theorem.", "Suppose that $P$ is a tree-like clause judgement proof, viewed as a tree; using $(i, \\alpha )$ to denote the clause judgement at the root of $P$ , there exists a trace $E$ whose number of nodes is equal to the number of non-flow judgements in $P$ , and whose root has label $(S, a)$ , such that the following two conditions hold: For each $v \\in \\mathsf {vars}(\\alpha )$ , the set $S$ contains the located variable $(j, v)$ where $j$ is the first location above $i$ where $v$ is quantified.", "The assignment $a$ is the unique assignment on $\\mathsf {vars}(\\alpha )$ that falsifies $\\alpha $ .", "We prove this result by induction on the structure of $P$ , describing directly how to construct $E$ .", "We consider cases depending on how the clause judgement at the root of $P$ was derived; we use the notation of Definition REF .", "In the case of (clause), let $E$ consist of a single node having label $(S, a)$ , where $(S, a)$ is the unique pair satisfying the two conditions.", "In the case of (resolve), suppose that $(i, \\gamma )$ is the clause judgement at the root of $P$ and that $(i, \\gamma )$ is derived as a resolvent of $\\alpha $ and $\\beta $ via clause judgements $(i, \\alpha )$ and $(i, \\beta )$ .", "Suppose that $v \\in \\alpha $ and $\\overline{v} \\in \\beta $ are the complementary literals such that $\\gamma = (\\alpha \\setminus \\lbrace v \\rbrace ) \\cup (\\beta \\setminus \\lbrace \\overline{v} \\rbrace )$ .", "Take the trace whose root has label $(U, c)$ where $U$ is the union of $S$ and $T$ but without the located variable containing $v$ , and where $c$ is the unique assignment on $\\mathsf {vars}(U) = \\mathsf {vars}(\\gamma )$ that falsifies $\\gamma $ .", "Since $i$ follows $S$ and $i$ follows $T$ , we have that $i$ follows $U$ , and we have that $(S, a)$ and $(T, b)$ could be generated from $(U, c)$ via a (Q-branch) step.", "In the case of ($\\forall $ -removal), suppose that the clause judgement at the root of $P$ has the form $(i, \\alpha \\setminus \\lbrace y, \\overline{y} \\rbrace )$ and is derived from $(j, \\alpha )$ where $\\phi (i) = \\forall y \\phi (j)$ .", "If $\\alpha \\cap \\lbrace y, \\overline{y} \\rbrace = \\emptyset $ , then the trace $E$ can be taken to be the trace given by induction.", "Otherwise, take the trace for $(j, \\alpha )$ given by induction, and let $(T, a)$ denote its root node label.", "Set $S$ to be $T$ , but with the located variable for $y$ removed.", "We have that $(T, a)$ could be derived from $(S, a \\upharpoonright \\mathsf {vars}(S))$ by a ($\\forall $ -branch) step; hence, we may take the trace obtained from the trace for $(j, \\alpha )$ by adding on the top a new root node with label $(S, a \\upharpoonright \\mathsf {vars}(S))$ .", "In the case of (upward flow) or (downward flow), we simply take the trace given by induction.", "This preserves condition (1): if $i$ is the parent of $j$ in $\\phi $ and $(i, \\alpha )$ and $(j, \\alpha )$ are clause judgements in $P$ , then $\\mathsf {vars}(\\alpha ) \\subseteq \\mathsf {free}(\\psi (i)) \\cap \\mathsf {free}(\\psi (j))$ and so no variable in $\\mathsf {vars}(\\alpha )$ is quantified at location $i$ (nor $j$ ).", "Theorem Let $\\psi $ be a QCBF instance.", "Given a trace $T$ with root node label $(\\emptyset , e)$ , there exists a tree-like clause judgement proof $P$ of an empty clause where the number of non-flow nodes in $P$ (viewed as a tree) is equal to the number of nodes in $T$ .", "Also, the translation from $T$ to $P$ is polynomial-time computable.", "We prove the following result which implies the theorem: for any trace $T$ with root node label $(S, a)$ , there exists a tree-like clause judgement proof $P$ ending in $(i, \\alpha )$ where the number of nodes in $P$ and $T$ are related as in the theorem statement, and such that the following two conditions hold: $i$ follows $S$ .", "$\\mathsf {vars}(S) = \\mathsf {vars}(\\alpha )$ and $a$ is the unique assignment on $\\mathsf {vars}(\\alpha )$ that falsifies $\\alpha $ .", "We prove the result by induction; we consider cases depending on the type of the root node of $T$ , that is, depending on how many children the root node of $T$ has.", "If the root node of $T$ is a leaf, the result is clear from the definition of trace.", "If the root node of $T$ has one child, let $(S \\cup \\lbrace (j, y) \\rbrace , a[y \\rightarrow b])$ be the label of the child of the root node.", "By induction, there exists a tree-like clause judgement proof ending with $(k, \\beta )$ where $k$ follows $S \\cup \\lbrace (j, y) \\rbrace $ and $a[y \\rightarrow b]$ is the unique assignment on $\\mathsf {vars}(\\beta )$ that falsifies $\\beta $ .", "Since $(j, y)$ follows $S$ and $k$ follows $(j, y)$ , by applying (upward flow), we may obtain a tree-like clause judgement proof ending with $(c, \\beta )$ where $c$ is the child of $j$ .", "Then, as $\\psi (j) = \\forall y \\psi (c)$ , we can apply ($\\forall $ -removal) to $(c, \\beta )$ to obtain the desired clause judgement proof.", "If the root node of $T$ has two children, let $(S_0 \\cup \\lbrace (j,u)\\rbrace , (a \\upharpoonright \\mathsf {vars}(S_0))[u \\rightarrow a_0])$ and $(S_1 \\cup \\lbrace (j,u)\\rbrace , (a \\upharpoonright \\mathsf {vars}(S_1))[u \\rightarrow a_1])$ be the labels of the children; we have $a_0, a_1 \\in \\lbrace 0, 1 \\rbrace $ and $a_0 \\ne a_1$ .", "By induction, we have tree-like clause judgement proofs ending with $(k_0, \\beta _0)$ and $(k_1, \\beta _1)$ where $k_0$ follows $S_0 \\cup \\lbrace (j, u) \\rbrace $ and $(a \\upharpoonright \\mathsf {vars}(S_0))[u \\rightarrow a_0]$ is the unique assignment on $\\mathsf {vars}(\\beta _0)$ that falsifies $\\beta _0$ ; and similarly, $k_1$ follows $S_1 \\cup \\lbrace (j, u) \\rbrace $ and $(a \\upharpoonright \\mathsf {vars}(S_1))[u \\rightarrow a_1]$ is the unique assignment on $\\mathsf {vars}(\\beta _1)$ that falsifies $\\beta _1$ .", "By applying (upward flow), we obtain tree-like clause judgement proofs ending with $(\\ell _0, \\beta _0)$ and $(\\ell _1, \\beta _1)$ where $\\ell _0$ is the child of the lowest location in $S_0 \\cup \\lbrace (j,u) \\rbrace $ , and $\\ell _1$ is the child of the lowest location in $S_1 \\cup \\lbrace (j,u) \\rbrace $ .", "Let $m$ be the child of the lowest location in $S \\cup \\lbrace (j,u) \\rbrace $ .", "At least one of $\\ell _0$ , $\\ell _1$ is equal to $m$ (since $S_0 \\cup S_1 = S$ ).", "If one of $\\ell _0$ , $\\ell _1$ is not equal to $m$ , say $\\ell _b$ , we may apply (downward flow) to obtain a clause judgement proof $(m, \\beta _b)$ ; this is because $\\mathsf {vars}(\\beta _b)$ is free in every location between $m$ and $\\ell _b$ (inclusive), as $S \\cup \\lbrace (j, u) \\rbrace $ is coherent.", "We hence obtain clause judgement proofs for $(m, \\beta _0)$ and for $(m, \\beta _1)$ .", "Apply (resolve) to these to obtain the desired clause judgement proof." ], [ "Proof of Theorem ", "Theorem REF follows immediately from the two lemmas presented in this section.", "Lemma Let $(\\phi , \\mathbf {B})$ be a QCSP instance.", "If there exists a $k$ -constraint system $P$ for the instance, then the instance is $k$ -judge-consistent.", "We show, by induction on the proof structure, that if $(i, V, F)$ with $\|V\| \\le k$ is a derivable judgement, then $P[i, V] \\subseteq F$ .", "We consider cases based on which rule was used to derive $(i, V, F)$ .", "In the case of (atom), we have $P[i, V] \\subseteq F$ by property $(\\alpha )$ .", "In the case of (projection), we suppose that $(i, V, F)$ is a previous judgement with $P[i, V] \\subseteq F$ , and that the judgement of interest has the form $(i, U, F \\upharpoonright U)$ , where $U \\subseteq V$ .", "We have $P[i, U] = (P[i, V] \\upharpoonright U) \\subseteq (F \\upharpoonright U)$ , where the equality holds by property $(\\pi )$ .", "In the case of (join), we suppose that $(i, U_1, F_1)$ and $(i, U_2, F_2)$ are previous judgements with $P[i, U_1] \\subseteq F_1$ and $P[i, U_2] \\subseteq F_2$ , and that the judgement of interest is $(i, U_1 \\cup U_2, F_1 \\bowtie F_2)$ .", "By property $(\\pi )$ , we have that $P[i, U_1 \\cup U_2] \\upharpoonright U_1 = P[i, U_1]$ and $P[i, U_1 \\cup U_2] \\upharpoonright U_2 = P[i, U_2]$ .", "It follows that $P[i, U_1 \\cup U_2] \\subseteq P[i, U_1] \\bowtie P[i, U_2]\\subseteq F_1 \\bowtie F_2$ .", "The cases of (upward flow) and (downward flow) follow immediately from property $(\\lambda )$ .", "In the case of ($\\forall $ -elimination), we suppose that $(j, V, F)$ is a previous judgement with $P[j, V] \\subseteq F$ , and that the judgement of interest is $(i, V \\setminus \\lbrace y \\rbrace , \\epsilon _y F)$ where $i$ is the parent of $j$ , and $\\phi (i) = \\forall y \\phi (j)$ .", "We have $P[i, V \\setminus \\lbrace y \\rbrace ] \\subseteq \\epsilon _y( P[j, V] )\\subseteq \\epsilon _y(F)$ , where the first containment holds by property $(\\epsilon )$ .", "Definition Let $k \\ge 1$ .", "A structure $\\mathbf {B}$ is $k$ -behaved if for each $i$ with $1 \\le i \\le k$ , there are finitely many relations of arity $i$ that are qc-definable over $\\mathbf {B}$ .", "Lemma Let $k \\ge 1$ .", "Let $(\\phi , \\mathbf {B})$ be a QCSP instance where $\\mathbf {B}$ is $k$ -behaved.", "If the instance is $k$ -judge-consistent, then then there exists a $k$ -constraint system $P$ for the instance.", "Relative to a QCSP instance, we say that a judgement is $k$ -derivable if there exists a judgement proof of width less than or equal to $k$ that contains the judgement.", "Let $I$ be an index set for $\\phi $ .", "Let us say that a $k$ -derivable judgement $(i, V, F)$ is minimal if for all sets $G$ such that the judgement $(i, V, G)$ is $k$ -derivable, it holds that $G \\subseteq F$ implies $G = F$ .", "We claim that, when $i \\in I$ and $V \\subseteq \\mathsf {free}(\\phi (i))$ with $\|V\| \\le k$ , there is a unique minimal $k$ -derivable judgement $(i, V, F)$ .", "The existence of a minimal $k$ -derivable judgement follows from Lemma REF and the $k$ -behavedness of $\\mathbf {B}$ .", "To establish uniqueness, suppose for a contradiction that $(i, V, F_1)$ and $(i, V, F_2)$ are both minimal $k$ -derivable judgements and $F_1 \\ne F_2$ .", "By the definition of minimal, we have $F_1 \\lnot \\subseteq F_2$ and $F_2 \\lnot \\subseteq F_1$ , so $F_1 \\cup F_2 \\lnot \\subseteq F_1$ and $F_1 \\cup F_2 \\lnot \\subseteq F_2$ .", "By the (join) rule, the judgement $(i, V, F_1 \\bowtie F_2)$ is $k$ -derivable; since here $F_1 \\bowtie F_2 = F_1 \\cap F_2$ , we obtain a contradiction.", "For all $i \\in I$ and $V \\subseteq \\mathsf {free}(\\phi (i))$ , we define $P[i, V]$ so that $(i, V, P[i, V])$ is the unique minimal $k$ -derivable judgement involving $i$ and $V$ .", "We confirm that $P$ is a $k$ -constraint system by verifying that it satisfies each of the four properties of the definition of $k$ -constraint system.", "In discussing each of the properties, we use the notation of Definition REF .", "Property $(\\alpha )$ follows immediately from the (atom) rule.", "For property $(\\pi )$ , suppose that $U \\subseteq V$ .", "We have that $(i, U, P[i, U])$ and $(i, V, P[i, V])$ are $k$ -derivable.", "It follows that $(i, V, F_V)$ and $(i, U, F_U)$ are $k$ -derivable, where $F_V = P[i, U] \\bowtie P[i, V]$ and $F_U = F_V \\upharpoonright U$ .", "We have $F_V \\subseteq P[i, V]$ and $F_U \\subseteq P[i, U]$ ; it follows, by definition of $P$ , that $F_V = P[i, V]$ and $F_U = P[i, U]$ .", "Since $F_U = F_V \\upharpoonright U$ , we have $P[i, U] = P[i, V] \\upharpoonright U$ .", "For property $(\\lambda )$ , suppose that $j$ is a child of $i$ with $V \\subseteq \\mathsf {free}(\\phi (j))$ .", "We have that $(i, V, P[i, V])$ and $(j, V, P[j, V])$ are $k$ -derivable.", "By the (downward flow) and (upward flow) rules, we obtain that $(i, V, P[j, V])$ and $(j, V, P[i, V])$ are $k$ -derivable.", "By definition of $P$ , we obtain that $P[i, V] \\subseteq P[j, V]$ and $P[j, V] \\subseteq P[i, V]$ and hence $P[i, V] = P[j, V]$ .", "For property $(\\epsilon )$ , suppose that $j$ is a child of $i$ , $\\phi (i) = \\forall y \\phi (j)$ , $U$ is a subset of $\\mathsf {free}(\\phi (j))$ with $\|U\| \\le k$ and $y \\in U$ , and $V = U \\setminus \\lbrace y \\rbrace $ .", "That $P[i, V] \\subseteq \\epsilon _y( P[j, U] )$ follows immediately from applying the ($\\forall $ -elimination) rule to the $k$ -derivable judgement $(j, U, P[j, U])$ ." ] ]	1403.0222
[ [ "B fields in OB stars (BOB): The discovery of a magnetic field in a\n multiple system in the Trifid Nebula, one of the youngest star forming\n regions" ], [ "Abstract Recent magnetic field surveys in O- and B-type stars revealed that about 10% of the core-hydrogen-burning massive stars host large-scale magnetic fields.", "The physical origin of these fields is highly debated.", "To identify and model the physical processes responsible for the generation of magnetic fields in massive stars, it is important to establish whether magnetic massive stars are found in very young star-forming regions or whether they are formed in close interacting binary systems.", "In the framework of our ESO Large Program, we carried out low-resolution spectropolarimetric observations with FORS2 in 2013 April of the three most massive central stars in the Trifid nebula, HD164492A, HD164492C, and HD164492D.", "These observations indicated a strong longitudinal magnetic field of about 500-600G in the poorly studied component HD164492C.", "To confirm this detection, we used HARPS in spectropolarimetric mode on two consecutive nights in 2013 June.", "Our HARPS observations confirmed the longitudinal magnetic field in HD164492C.", "Furthermore, the HARPS observations revealed that HD164492C cannot be considered as a single star as it possesses one or two companions.", "The spectral appearance indicates that the primary is most likely of spectral type B1-B1.5V.", "Since in both observing nights most spectral lines appear blended, it is currently unclear which components are magnetic.", "Long-term monitoring using high-resolution spectropolarimetry is necessary to separate the contribution of each component to the magnetic signal.", "Given the location of the system HD164492C in one of the youngest star formation regions, this system can be considered as a Rosetta Stone for our understanding of the origin of magnetic fields in massive stars." ], [ "Introduction", "Magnetic fields have fundamental effects not only on the evolution of massive stars, on their rotation, and on the structure, dynamics, and heating of their radiatively-driven winds, but also on their final display as supernova or gamma-ray burst.", "About a few dozen massive magnetic stars are currently known, just enough to establish the fraction of magnetic, core-hydrogen burning stars to be of the order of 8% (Grunhut et al.", "[12]), which appears to be similar to that of intermediate-mass stars.", "While it is established that the magnetic fields in massive stars are not dynamo-supported, but stable with decay times exceeding the stellar lifetime, their origin is highly debated.", "The two main competing ideas are that the fields are either “fossil” remnants of the Galactic ISM field that are amplified during the collapse of a magnetised gas cloud (e.g.", "Price & Bate [27]), or that they are formed in a dramatic close-binary interaction, i.e., in a merger of two stars or a dynamical mass transfer event (e.g.", "Ferrario et al.", "[10]).", "The intermediate-mass stars show a magnetic fraction during their pre-main sequence evolution similar to Herbig stars (e.g., Hubrig et al.", "[16]; Hubrig et al.", "[17] and references therein; Alecian et al.", "[1]) as their main-sequence descendants, which may speak for fossil fields.", "On the other hand, there are almost no close binaries amongst the magnetic intermediate-mass main-sequence Ap-type stars (e.g.", "Carrier et al.", "[5]).", "Since this is expected if they were merger products, this argues for the binary hypothesis of the field origin.", "We present our spectropolarimetric observations of three massive stars in the Trifid Nebula in the framework of our “B fields in OB stars” (BOB) collaboration.", "This nebula is a very young ($\\lesssim 10^6$ yrs) and active site of star formation containing a rich population of young stellar objects (YSOs) and protostars (e.g.", "Cernicharo et al.", "[7]).", "The spectacular, well-known optical Hii region provides an ideal place for investigating the onset of star birth and triggered star formation.", "This large nebula is ionised by the O7.5 Vz star HD 164492A (Sota et al.", "[32]), which is the central object of the multiple system ADS 10991, containing at least seven components (A to G; Kohoutek et al.", "[22]).", "Due to the faintness of the components HD 164492B and HD 164492E-G (all have visual magnitudes fainter than 10.6), we only searched for a magnetic field for the three most massive components, HD 164492A, C, and D." ], [ "Magnetic field measurements using FORS 2 spectropolarimetry", "Three components of the system ADS 10991, HD 164492A, C, and D, were observed with the FOcal Reducer low dispersion Spectrograph (FORS 2) mounted on the 8 m Antu telescope of the VLT on 2013 April 9.", "This multi-mode instrument is equipped with polarisation-analysing optics comprising super-achromatic half-wave and quarter-wave phase retarder plates, and a Wollaston prism with a beam divergence of 22$$ in standard-resolution mode.", "We used the GRISM 600B and the narrowest available slit width of 0$$ 4 to obtain a spectral resolving power of $R\\sim 2000$ .", "The observed spectral range from 3250 to 6215 Å includes all Balmer lines apart from H$\\alpha $ , and numerous Hei lines.", "The position angle of the retarder waveplate was changed from $+45^{\\circ }$ to $-45^{\\circ }$ and vice versa every second exposure, i.e., we executed the sequence $-45^{\\circ }$$+45^{\\circ }$ , $+45^{\\circ }$$-45^{\\circ }$ , $-45^{\\circ }$$+45^{\\circ }$ , etc.", "up to 6–8 times.", "For the observations we used a non-standard readout mode with low gain (200kHz,1$\\times $ 1,low), which provides a broader dynamic range, hence allowed us to reach a higher signal-to-noise ratio (S/N) in the individual spectra.", "We achieved a S/N of 1670 per pixel for the component HD 164492A, 1490 for HD 164492C, and 835 for HD 164492D in the final integral spectra.", "The integral spectra of all three components in the spectral region around H$\\gamma $ are presented in the left panel of Fig.", "REF .", "While the spectrum of component HD 164492A corresponds to that of a typical O7-type star, HD 164492C appears to be an early-B type star, and the spectrum of HD 164492D displays numerous emission lines similar to those observed in Herbig Be stars (e.g.", "Herbig [13]).", "The mean longitudinal magnetic field is the average over the stellar hemisphere visible at the time of observation of the component of the magnetic field parallel to the line of sight, weighted by the local emerging spectral line intensity.", "The determination of the mean longitudinal magnetic field using low-resolution FORS 1/2 spectropolarimetry has been described in detail by two different groups: Bagnulo et al.", "([2], [3]) and Hubrig et al.", "([14], [15]).", "To identify systematic differences that might exist when the FORS 2 data is treated by different groups, the mean longitudinal magnetic field, $\\left< B_{\\rm z}\\right>$ , was derived in all three stars by each group separately, using independent reduction packages.", "No magnetic field at a significance level of 3$\\sigma $ was detected in HD 164492A and HD 164492D.", "For HD 164492C, using a set of IRAF and IDL routines based on the recipes described by Bagnulo et al.", "([3]) and Fossati et al.", "(in prep.", "), we determined a mean longitudinal magnetic field of $\\left<B_{\\rm z}\\right>_{\\rm all}=523\\pm 37$ G measured using the whole spectrum and $\\left<B_{\\rm z}\\right>_{\\rm hyd}=600\\pm 54$ G using only the hydrogen lines, i.e.", "the magnetic field is discovered in this component at a significance level higher than 10$\\sigma $ .", "These measurements are consistent within a few tens of Gauss with those obtained using the software package developed by Hubrig et al.", "([14], [15]): $\\left<B_{\\rm z}\\right>_{\\rm all}=472\\pm 44$ G and $\\left<B_{\\rm z}\\right>_{\\rm hyd}=576\\pm 60$ G. To illustrate the strong magnetic field in HD 164492C, we present in the right panel of Fig.", "REF the Stokes $I$ and $V$ spectra in which a distinct Zeeman feature is observed at the position of the H$\\beta $ line." ], [ "Magnetic field measurements using HARPS", "To further investigate the spectral appearance and behaviour of the magnetic field in HD 164492C, we acquired two additional spectropolarimetric observations with the HARPS polarimeter (Snik et al.", "[31]) attached to ESO's 3.6 m telescope (La Silla, Chile) on two consecutive nights at the beginning of 2013 June.", "The polarimetric spectra with a $S/N$ of about 350 per pixel in the Stokes $I$ spectra and a resolving power of $R = 115\\,000$ cover the spectral range 3780–6910 Å, with a small gap between 5259 and 5337 Å.", "Each observation was split into four subexposures, obtained with different orientations of the quarter-wave retarder plate relative to the beam splitter of the circular polarimeter.", "Again, the reduction and magnetic field measurements were carried out using independent software packages developed for the treatment of HARPS data.", "Within the first package, the reduction and calibration was performed using the HARPS data reduction pipeline available at the 3.6 m telescope in Chile.", "The normalisation of the spectra to the continuum level consisted of several steps described in detail by Hubrig et al.", "([17]).", "The Stokes $I$ and $V$ parameters were derived following the ratio method described by Donati et al.", "([8]), and null polarisation spectra were calculated by combining the subexposures in such a way that polarisation cancels out.", "These steps ensure that the data contain no spurious signals (e.g.", "Ilyin [18]).", "Within the second software package, we reduced and calibrated the data with the REDUCE package (Piskunov & Valenti [26]), which performs an optimal extraction of the echelle orders after several standard steps, such as bias subtraction, flat-fielding, and cosmic-ray removal.", "The wavelength calibration and the continuum normalisation were treated using standard techniques.", "Figure: II, VV, and NN SVD profiles obtained for HD 164492C for both nights.The VV and NN profiles were expanded bya factor of 75 and shifted upwards for better visibility.Figure: II, VV, and NN LSD profiles (from bottom to top) obtained for HD 164492C.The VV and NN profiles were expanded bya factor of 75 and shifted upwards for better visibility.One software package used to study the magnetic field in HD 164492C, the so-called multi-line Singular Value Decomposition (SVD) method for Stokes Profile Reconstruction was recently introduced by Carroll et al.", "([6]).", "The results obtained with the SVD method using about 80 lines in the line mask, including He lines and avoiding hydrogen and telluric lines, are presented in Fig.", "REF .", "The line mask was constructed using the VALD database (e.g.", "Kupka et al.", "[23]).", "Observations on both nights show definite detections with a false-alarm probability (FAP) lower than $10^{-10}$ .", "For the second software package we used the LSD technique (Donati et al.", "[8]; Kochukhov et al.", "[21]).", "The details of the analysis procedure can be found in Makaganiuk et al.", "([24]) and Fossati et al.", "([11]).", "About 170 lines were used in the line mask, but a test with the 80 lines used for the SVD program package did not show any significant difference in the results.", "Fig.", "REF shows that, similar to the SVD treatment, observations on both nights show definite detections with FAPs lower than $10^{-10}$ .", "Figure: Average profiles (except for the Mgii 4481 line)calculated using several unblended lines: five Siiii lines, ten Oii lines, and four Nii lines.For comparison, we also present the average of ten Hei lines.The emission feature on the red sideof the Nii profile belongsto the Oiii 5006.8 Å line.For each element, the upper spectrum presents the difference betweenthe average profiles from both nights.", "The apparent features in these difference spectra are mostlydue to the variation of the radial velocity from one night to the next.Figure: The HST WFPC2 image of HD 164492C and its surroundings obtained during a short 30.8 s exposure in the F502N filter.North is up and east to the left.The image size is ∼5 '' ×35\\sim 5^{\\prime \\prime }\\times 35.Equatorial coordinates are shown, where the vertical axis is DEC and the horizontal axis is RA.The contours are on logarithmic scale.The letters C, D, and C2 refer to the notations of X-ray sourcesas detected by Rho et al.", "() using observations with the Chandra X-ray observatory.The changes in the shape of the Stokes $I$ and $V$ profiles during two consecutive nights suggest that HD 164492C is either a spectroscopically variable star because of chemical spots or it is a double-lined spectroscopic binary in which the lines of the primary and the secondary appear strongly blended on both nights.", "The presence of the weak Heii 4686 Å line in the central strongest component indicates a spectral type B1 (for normal He content) or B1.5 (for enhanced He), implying an effective temperature of about 24 000–26 000 K (Nieva [25]).", "Moreover, the He lines appear in normal strength for the $T_{\\rm eff}$ estimated from the metal lines.", "Therefore a He-strong star can be excluded.", "To better understand the origin of the variability of this system we investigated the behaviour of a few individual elements.", "In Fig.", "REF , we display the average profiles of the best clean lines identified in the HARPS spectra.", "Remarkably, both the Siiii and Mgii profiles exhibit several absorption peaks in this plot.", "In the same figure, the upper spectrum presents the difference between the average profiles from both nights for each element.", "The detected features in these difference spectra are mostly caused by the variation of the radial velocity from one night to the next, indicating that HD 164492C cannot be considered as a single star.", "According to the archive HST WFPC2 image of HD 164492C and its surroundings presented in Fig.", "REF , the source C coinciding with HD 164492C has an elongated shape, implying that we should see at least two stars in our HARPS spectra.", "If we assume that we observe a binary system, a reasonable interpretation of the observed line profiles might be that the main absorption peak corresponds to one star with $v\\,\\sin \\,i=55$ km s$^{-1}$ that is superimposed on a broad-lined star with $v\\,\\sin \\,i$ of about 140 km s$^{-1}$ .", "This second fast-rotating star would contribute to the left and right wings of the profiles presented in Fig.", "REF .", "It is possible that this star also has surface chemical patches, since profiles of different elements look somewhat different.", "On the other hand, the relatively rapid radial velocity variation in the spectrum of the primary ($\\sim $ 4 km s$^{-1}$ per day) indicates that this star should have a close companion.", "Thus, we expect that the system HD 164492C is at least a triple system.", "Figure: II, VV, and NN profiles obtained for HD 164492C during the first night.Top panel: LSD profiles using metal lines.Bottom panel: SVD profiles using Siiii lines.The VV and NN profiles were expanded byfactors of 25 and shifted upwards for better visibility.Additional absorption peaks are indicated by arrows.In the top panel of Fig.", "REF , we present $I$ , $V$ , and $N$ LSD profiles obtained with the line mask excluding broad Hei lines.", "In the bottom panel we show the SVD analysis where only 24 Siiii lines are included.", "In the SVD Stokes $I$ Siiii line profile, the blend in the red wing of the primary appears double-peaked, while one more peak in the blue wing of the primary is best visible in the LSD Stokes $I$ profile.", "The measurement of the strength of the mean longitudinal magnetic field using the first-order moment method is described by Rees & Semel ([28]) and requires measuring the equivalent width of Stokes $I$ .", "Because of the strong blending of the components, we are unable to estimate the strength of the detected magnetic field.", "Given the complex configuration and shape of the Stokes profiles, it is currently also impossible to conclude exactly which components possess a magnetic field.", "The detection of a significant Stokes $V$ signature in the SVD and LSD profiles at about $-$ 100 km s$^{-1}$ and $+$ 150 km s$^{-1}$ from the line core of the primary suggests that more than one component might hold a magnetic field.", "The Stokes $V$ profile obtained on June 3 is well centred on the position of the primary.", "Assuming that HD 164492C is a single star, with the SVD and LSD methods we obtain results very similar to those obtained with FORS 2: between 500 and 700 G for the first night, and 400 and 600 G for the second night." ], [ "Discussion", "Using FORS 2 and HARPS in the framework of our ESO Large Program 191.D-0255, we detected a magnetic field in the poorly studied system HD 164492C.", "Although HD 164492C appears to be a multiple system, the multiplicity configuration is currently unclear and cannot be better elucidated without additional observations.", "X-ray emission from HD 164492C is firmly detected using Chandra observations, but is blended with a nearby unidentified X-ray source (component C2; Rho et al.", "[29]).", "The total X-ray luminosity of these two marginally spatially resolved sources is $2\\times 10^{32}$ erg s$^{-1}$ , with both components having similar X-ray brightness.", "The component C2 shows X-ray variability and is harder in X-rays than HD 164492C.", "To identify and model the physical processes that are responsible for the generation of magnetic fields in massive stars, it is important to understand the formation mechanism of magnetic massive stars.", "Although the Trifid Nebula has often been studied, its distance is not accurately known.", "Rho et al.", "([30]) reviewed literature values between 1.68 and 2.84 kpc and adopted a distance of about 1.7 kpc.", "Cambrésy et al.", "([4]) found $2.7\\pm 0.5$ kpc in their analysis of new near- and mid-infrared data.", "Torii et al.", "([33]) used both 1.7 and 2.7 kpc in their discussion.", "Even shorter distances of 816 pc and 1093 pc were estimated by Kharchenko et al.", "([19], [20]) by combining proper-motion data with optical and near-infrared photometry in their cluster analysis, respectively.", "The spatial distribution of the components of the multiple system ADS 10991 and the photometric study by Kohoutek et al.", "([22]), which revealed almost the same E(B-V) values for components A–C, suggest that these components build a physical system in the nucleus of the Trifid nebula.", "The age of the Trifid Nebula is only a few 0.1 Myr according to Cernicharo et al.", "([7]), who considered the spatial extent of the Hii region.", "The age of the cluster M20 and the time interval of the star formation in this cluster, of which the system HD 164492 is a member, can probably be larger, of the order of 1 Myr.", "Torii et al.", "([33]) argued that the formation of first-generation stars in the Trifid nebula, including the main ionising O7.5 star HD 164492A, was triggered by the collision of two molecular clouds on a short time-scale of $\\sim $ 1 Myr.", "New insights into the understanding of massive star formation in the Trifid Nebula can be expected from a recently established large project based on infrared and X-ray observations of 20 massive star-forming regions, among them the Trifid Nebula (Feigelson et al.", "[9]).", "The presented first detection of a magnetic massive multiple system in one of the youngest star-forming regions implies that this system may play a pivotal role in our understanding of the origin of magnetic fields in massive stars.", "Future spectropolarimetric monitoring of this system is urgently needed to better characterise the components, their orbital parameters, and the magnetic field topology.", "We thank J. Maíz Appellániz, A. de Koter, A. Herrero, and F. Schneider for useful comments.", "T.M.", "acknowledges financial support from Belspo for contract PRODEX GAIA-DPAC." ] ]	1403.0491
[ [ "Behavior recognition and analysis in smart environments for\n context-aware applications" ], [ "Abstract Providing accurate/suitable information on behaviors in sma\\-rt environments is a challenging and crucial task in pervasive computing where context-awareness and pro-activity are of fundamental importance.", "Behavioral identifications enable to abstract higher-level concepts that are interesting to applications.", "This work proposes the unified logical-based framework to recognize and analyze behavioral specifications understood as a formal logic language that avoids ambiguity typical for natural languages.", "Automatically discovering behaviors from sensory data streams as formal specifications is of fundamental importance to build seamless human-computer interactions.", "Thus, the knowledge about environment behaviors expressed in terms of temporal logic formulas constitutes a base for the reactive and precise reasoning processes to support trustworthy, unambiguous and pro-active decisions for applications that are smart and context-aware." ], [ "Introduction", "Nowadays smart spaces are filled with different sensors and sensor-like equipments.", "A sensor is a device that detects events or changes from a physical environment, that is a devise which is sensitive to a physical stimulus.", "These sensors might constitute the IoT spaces (Internet of Things) in which objects with unique identifiers create their own scenarios and interactions.", "On the other hand, the decisive feature of smart spaces is context-awareness which stands for the capabilities to examine changes in the environment and to react to these changes adequately.", "Important aspects of context might be: where you are, who you are with, and what resources are nearby.", "In other words, context is “...any information that can be used to characterize the situation of an entity.", "An entity is a person, place, or object that is considered relevant to the interaction between a user and an application, including the user and applications themselves” [7].", "In software engineering context-awareness means sensing and reacting on the environment.", "Sensing and context understanding are necessary and of critical importance to pro-active decisions which should be interpreted into domain-relevant concepts and situations.", "Formal logic allows assertions about actions and behaviors using accurate and precise notations, eliminating ambiguity common to other languages.", "“Logic has simple, unambiguous syntax and semantics.", "It is thus ideally suited to the task of specifying information systems” [5] showing the form of an argument to be valid or invalid.", "Knowledge about arguments enable achieving clear thinking and relevant arguments.", "The contribution of this paper is a novel and unified logical-based framework to deploy automatic methods for the behavior recognition and its reliable knowledge representation through the formalism of temporal logic.", "It allows to support reactive analysis of logical satisfiability, in order to obtain trustworthy decisions for the dynamically changing smart environment.", "Decisions of a system are transparent for users/inhabitants and satisfy the assumption of context-awareness and pro-activity.", "It is demonstrated that this logical framework is expressive enough.", "It is also demonstrated that on-line logical reasoning is suitable for sensor data streams.", "The semantic tableaux method for temporal logic as a reasoning procedure is considered.", "The architecture of a software system (see Figure REF ) is proposed, as well as algorithms (see Algorithm and Algorithm ) to generate and interpret logical specifications.", "The simple yet illustrative examples are provided, see Formulas (REF ) and (REF ) for Algorithm and the discussed example for Algorithm at the end of Section , as well as related motivating examples in Section .", "To the best of our knowledge, this paper presents the first formal study of both the reactive behavior recognition and deductive-oriented analysis for context-aware applications over sensor networks.", "On the other hand, this research opens some new directions, especially related to implementation and experiments.", "There are many works considering behavior analysis in pervasive computing.", "A survey for human activity is provided in the fundamental work [1].", "Features, representations, classification models, and datasets are surveyed.", "Work is comprehensive and discusses many important aspects of the domain.", "This paper refers to single-layered approaches as considered in [1].", "A survey of activity recognition for wearable sensors is provided in work [15].", "A taxonomy according to aspects of response time and learning scheme is introduced.", "A couple of systems are qualitatively compared due the mentioned aspects, as well as some other ones.", "Formal logic approaches, except for the fuzzy logic, are not considered.", "Behavior recognition in smart homes is a topic in work [6], whose approach influenced in some way this paper, however, models base on Hidden Markov Models, which constitutes a different approach in comparison to this one.", "In work [3] a hierarchical framework for human activity recognition is presented, however, the framework focuses on video based activity recognition.", "The method of rather manual transformation into logical rules, is done in an off-line manner, and reasoning based on the resolution is proposed.", "The aim is to discover a semantic gap between the low level (data) and the high level (human understanding).", "In work [4] a similar approach is presented but formalization is based on a adaptation of temporal relations from the Allen's temporal interval logic, and the reasoning process is not considered.", "Apart from the issue of a hierarchical approach, these works influence this paper in such a way that the formalization of the observed (human) activities is made on the basis of formal logic.", "This paper follows work [12] which concerns on-the-fly modeling logical specifications and observing behaviors of users/inhabitants, in other words, logical specifications are understood as knowledge about user preferences.", "Work [8] proposes patterns for a property specification and is considered in a more detailed way in the following Sections of this paper, especially when discussing the so called learning-based approach.", "Work [16] discusses possibilities of using temporal logic and model checking for the recognition of human activities.", "This paper is relatively close to the work, however, a deduction based approach is proposed.", "The novel aspects are unified logical framework, basing on a purely logical approach, and deductive-based reasoning processes to obtain pro-active decisions." ], [ "Preliminaries", "A context model that consists of three layers is shown in Figure REF , c.f.", "also [12].", "Figure: A three-layer context model for a smart environmentIt contains different sensor devices which are distributed in the whole physical area.", "It also refers to the concept of Ambient Intelligence (AmI), i.e.", "electronic devices that are sensitive and responsive to the presence of humans/inhabitants.", "Smart applications must both understand context, that is be context-awareness, and provide pro-activity, that is act in advance to deal with future situations, especially negative or difficult ones.", "A context-aware system is able to adapt its operations to the current context without explicit user intervention.", "Figure: Context-awareness and pro-activity of appsThe dynamic nature of context models' analysis is shown in Figure REF , i.e.", "supplementing Figure REF , where different phases are repeated periodically to sense behaviors and to generate proper system's reactions enabling context-awareness and pro-activity of applications which operate in a smart environment.", "Temporal Logic, and Propositional Linear Time Temporal Logic PLTL considered here, is a branch of formal logic with statements whose valuations depend on time flows [18].", "The reasoning method of semantic tableaux is well known in classical logic but it can be applied in temporal logic [9], [11].", "The method provides truth trees.", "The branch of a tree is a set of nodes/formulas connecting a node with a descendant.", "Semantic tableaux is also a decision procedure providing, through open branches (that is, not containing complementary pair/pairs of atomic formulas, e.g.", "$f$ and $\\lnot f$ ) and closed branches (that is, containing complementary pair/pairs of atomic formulas), the binary answer Yes-No as a result of an inquiry.", "If $F$ is an examined formula and $\\Delta $ is a truth tree build for a formula, then the semantic tableaux method gives answers to the following questions related to the satisfiability problem: formula $F$ is not satisfied iff the finished $\\Delta (F)$ is closed; formula $F$ is satisfiable iff the finished $\\Delta (F)$ is open; formula $F$ is always valid iff finished $\\Delta (\\lnot F)$ is closed.", "The proof follows directly from the semantic tableaux method." ], [ "Motivating examples", "Let us consider some examples to illustrate the approach and provide some motivation.", "A basic distinction two approaches regarding method of building logical specification is introduced: model-based – the case occurs when logical specifications (models) for context-aware systems are prepared in advance; in other words, the initial specification is not empty, but new events may affect a particular specification leading to its modification, it can be used in a decision/reasoning process without any change, but logical specification can also be dynamically expanded/rebuilt when the system operates, see the evacuation example below; learning-based – the case occurs when logical specifications are build on-line, that is in real-time, during normal operation of a a context-aware system; in other words, the initial specification is initially empty, and when new events occur, logical specification is built/rebuilt, and at any time it can be used in decision-making processes see work [12] or the second example below.", "The first example discusses an evacuation situation, i.e.", "people are located inside risk areas (e.g.", "buildings or sport stadiums) and a dangerous situation occurs.", "Context-aware and smart systems should help inhabitants/people by providing trustworthy information about evacuation paths.", "The evacuation plan, expressed as a logical specification $\\Sigma $ , and understood as a set of temporal logic formulas, must be prepared in advance.", "(This is a reverse situation comparing other hypothetical cases where logical specifications might be built on-line i.e.", "when the system operates.)", "Formulas describe possible and recommended actions/transitions during the evacuation process.", "After the evacuation process has been started and is being carried out, dynamically changing situations, e.g.", "fire on a passage, may require extension of $\\Sigma $ introducing new formulas describing new situations.", "It is done by software agents observing changes in particular areas.", "(Graph-based description might contains nodes with different attributes, such as entrances to corridors or staircases and edges that connect different areas.)", "Figure: The product of reasoning – sample truth treesLet $V=\\lbrace \\ldots , v10, v11, \\ldots , p110, p115, p116, \\ldots \\rbrace $ are places and passages of a building for which the evacuation plan is to be prepared.", "$\\Sigma =\\lbrace \\ldots , v10 \\Rightarrow \\Diamond p110, \\ldots \\rbrace $ is a (small) fragment of the evacuation plan expressed in terms of temporal logic formulas.", "When new objects appear in place $v10$ ($v10$ is satisfied), then the reasoning process starts, see Figure REF .a.", "The open branch (blue$\\circ $ ) of the tree provides literals, that is atomic formulas or their negations, $v10$ and $p110$ that satisfy the initial formula that consists of satisfied $v10$ and conjunction of all formulas that belongs to $\\Sigma $ .", "It allows to identify formula $v10 \\Rightarrow \\Diamond p110$ that describes the next supporting people action for a particular place, as a part of an evacuation process.", "Another situation is shown in Figure REF .b.", "Let $\\Sigma =\\lbrace \\ldots , ((v11 \\Rightarrow \\Diamond p115) \\vee (v11 \\Rightarrow \\Diamond p116)), \\ldots \\rbrace $ is another fragment of an evacuation plan showing the choice of escape routes.", "New objects which appear in place $v11$ involve the reasoning process that provides through two open branches, two subsets of literals $v11$ and $p115$ , and also $v11$ and $p116$ .", "It means that two different actions are possible, i.e.", "$v11 \\Rightarrow \\Diamond p115$ or $v11 \\Rightarrow \\Diamond p116$ .", "In the last case, see Figure REF .c, the extension of logical specification $\\Sigma $ is discussed.", "Supposing that the dynamically changing situation, e.g.", "fire, forces the closure of passage $p115$ .", "It leads to the need of extending the logical specification by a new formula $\\Box (\\lnot p115)$ , i.e.", "$\\Sigma := \\Sigma \\cup \\lbrace \\Box (\\lnot p115)\\rbrace $ .", "Thus, every reasoning process for $p115$ leads to the closed branch (blue$\\times $ ), i.e.", "the contradiction.", "It means that the “fired” passage will never be proposed as an action for the evacuation procedure.", "This example is also discussed in a more formal way after Algorithm in Section .", "The above considerations should be supplemented with the following information.", "The accepted decomposition procedure in Figure REF , as well as labeling, refers to the first-order predicate calculus provided in [9].", "In some cases, the outer operator $\\Box $ is omitted to simplify considerations/formulas, in other words, for example, one should write down $\\Box (v11 \\Rightarrow \\Diamond p116)$ , however, the well-known rules of generalization/particularization justify the simplified notation.", "Reasoning engines have become more available in recent years, c.f.", "[17], however, selection of an appropriate existing prover is not in the scope of this paper.", "Another example might refer to the situation when logical specification $\\Sigma $ , interpreted as knowledge about user/inhabitant behaviors, is built on-line, i.e.", "initially $\\Sigma =\\emptyset $ , and then, observing present users' behaviors, new temporal logic formulas for particular objects/users are added to set $\\Sigma $ .", "Work [8] discusses methods of obtaining logical specifications from a natural language.", "The method is based on pattern recognition.", "The consideration in this paper provides a method/idea to obtain logical specifications from a (technical) language of physical sensors/signals which is less complex when comparing it to a natural language.", "Some sample patterns for a “sensor language” are provided in Figure REF .", "Figure: Sample PLTL patterns for events pp, qq, rr, etc.Registered (atomic) events for every user might comprise a label for a physical node (e.g.", "the presence in a node) and time for the event occurrence (i.e.", "the time stamp), e.g.", "$\\langle p210,t2014.08.14.21.56.00 \\rangle $ .", "These elementary events are translated into logical specifications when analyzing time of the events and employing (predefined) patterns.", "If logical specification $\\Sigma $ is built, then the pro-active decision might be taken when new event, say $gt$ , occurs and is considered as a kind of trigger.", "Triggering is an important aspect for this case.", "$\\Sigma $ , in fact (past) behaviors, is now interpreted as user's preferences to support a new action of a user.", "The entire input formula for the reasoning process might comprise conjunction of satisfied $gt$ and conjunctions $C(\\cdot )$ of the $\\Sigma $ formulas, i.e.", "cumulatively $gt \\wedge C(\\Sigma )$ .", "The reasoning process, and its sub-instances, might be performed in a similar way as in the previous case shown in Figure REF ." ], [ "System architecture", "The architecture of a proposed system embodied in its well-identified components is briefly discussed in this Section.", "It allows to understand how the system works, and what are the basic functionalities and services of particular components.", "An overall architecture of systems for both model- and learning-based approaches is shown in Figure REF .", "Figure: An overall architecture of systems(flows: solid lines – both approaches,dashed line – only model-based approach,dotted line – only learning-based approach)Signals are gathered (see tracking/sensing in Figures REF and REF ) from an environment by Signal Manager.", "Then signals are interpreted by Signal Interpreter producing temporal logic formulas generated by an algorithm such as Algorithm , that is translating events to logical formulas.", "If once massive amounts of data are processed (the learning-based approach) then formulas flow to Specification Manger that stores the basic logical specification $\\Sigma $ , that is the current logical model of a system.", "If single data is processed (rather the model-based but also possible in the case of the learning-based approach) then a formula/formulas are provided to the Reasoning Engine.", "The second input for the Reasoning Engine component is logical specification $\\Sigma $ .", "The component performs logical reasoning using the semantic tableaux method, however, the resolution-based reasoning is also possible.", "The output is information, for example, basing on Corollary , which is interpreted by Result Interpreter.", "It provides two outputs.", "The first one allows to update, if necessary, the current logical specification $\\Sigma $ (stored in Specification Manager) by Specification Updater through deleting or adding some new formulas.", "The second one allows Action Provider to supply (see reacting/influencing in Figures REF and REF ) signals to the environment.", "Flows in Figure REF are not labeled (except specification $\\Sigma $ and formulas/formula $f$ ) since they would require precise definitions of the flowing data.", "On the other hand, their meanings seem intuitive.", "Some brief and overall information on methods of Reasoning Engine basing on the semantic tableaux method, see also Section and Corollary , is shown in Table REF .", "Table: Methods of Reasoning Engine$C(\\Sigma )$ means a conjunction of all formulas constituting logical specification $\\Sigma $ , in other words, a set of formulas $\\Sigma $ are interpreted (preprocessed) inside Reasoning Engine as a conjunction of formulas $C(\\Sigma )$ .", "$f$ is a single formula provided by Signal Interpreter.", "The reasoning process may comprise many methods and aspects that follow from the input data/formulas, see formulation in Table REF , as well as the assumed reasoning method (truth trees), for example, examining satisfiability of the possessed specification, which happens if a new formula is added to a specification, whether a property can be inferred from a specification using deductive approach, etc." ], [ "Building and managing specifications", "Discovering formal specifications automatically from sensory data streams is discussed below.", "The process of building logical specifications should be considered from a broader point of view which follows from the taxonomy discussed at the beginning of Section .", "The introduction of a method for building logical specifications, the physical world, or smart environment, is formally described over a graph structure.", "An attributed graph $G$ is a tuple $\\langle V, E, N, \\alpha , S, \\beta \\rangle $ , where $\\langle V,E\\rangle $ is a directed graph with a set of vertices $V$ and a set of edges or lines $E$ , $N$ is a set of labels/names, $\\alpha : V \\rightarrow N$ is a function that labels vertices, $S$ is a set of labels/sensors, and $\\beta : V \\rightarrow 2^{S}$ is a function that labels vertices.", "A smart environment $En$ is an attributed graph as defined above.", "$N$ are commonly used (informal) names for vertices, or nodes (for example: a gate, a crossroad, a staircase, a classroom, etc), if necessary.", "$S$ are sensors located in a node that detects or measures a physical property and records, indicates, or otherwise responds to it (for example: tactile sensors, temperature, humidity and light sensors, chemical sensors, bio-sensors, etc).", "This approach enables the gathering of multiple sensory data in a single node, if necessary.", "For example, on the basis of formal logic, it can be illustrated by a formula $s_1 \\wedge s_2 \\wedge \\lnot s_3 \\wedge s_4$ where $s_1, s_2, s_3, s_4 \\in S$ , and they are responsible for reading sensory data available in a particular node $v_i \\in V$ , say there are the following four data: temperature exceeded, humidity exceeded, high levels of light, and vibration, respectively.", "However, to simplify the consideration in the rest of the paper the existence of a single sensor in every node is assumed, and it is always the object presence sensor/detector that also identifies this object.", "Let us consider a set of users/inhabitants $O=\\lbrace o_{1},o_{2},... \\rbrace $ that operate in a smart environment.", "These users are identified on-line, i.e.", "when the system operates, and have unique identifiers.", "The problem of objects'/users'/inhabitants' unambiguous identification is a well-known question and it may be done in different ways, for example by using RFID, PDA devices, biometric data, image scanning, pattern recognition, and others.", "The issue of users'/inhabitants' identification is not discussed here.", "Events basing on the object presence detection in nodes are registered and the time-stamp for every event is also registered.", "An event $b_i$ is a triple that belongs to $\\langle O,V,T \\rangle $ , where $O$ is a set of identified users/inhabitants, $V$ is a node of a network, and $T$ is a set of time stamps.", "A behavior $B$ of a smart environment is a set of events $\\lbrace b_{1}, b_{2}, \\dots , b_i, \\ldots \\rbrace $ .", "For example, $b_{i}=\\langle idEmily,p0018,t2015.02.11.09.30.15 \\rangle $ means that the presence of the $idEmily$ object is observed at the physical point/area/node $p0018$ of the environment, and the time stamp assigned to this event is $t2015.02.11.09.30.15$ .", "Let us note that all nodes that occur in events, or in a behavior, are equivalent to vertices that occur in an attributed graph, or a smart environment.", "The following notation is introduced.", "Let $b_{i}.o_{j}$ be an object $o_{j}$ that belongs to an event $b_i$ , and $b_{i}.v_{k}$ is a node $v_{k}$ that belongs to an event $b_i$ , etc.", "The algorithm for building logical specifications for every object registered in a smart environment is given as Algorithm .", "[htb] Building logical specifications for objects $O$ [1] Input: Output: (New) behavior $B$ (non-empty) Logical specifications $L_{i=1,\\ldots }$ Divide $B$ into subsets $B_{i=1,\\ldots }$ for every object $o_{i=1,\\ldots }$ every $B_i$ $L_i := \\emptyset $ initiating specification for $o_i$ $\\forall v \\in G$ $v \\notin \\lbrace b_{i}.v_{j} : b_i \\in B_i \\wedge j>0 \\rbrace $ $L_i := L_i \\cup \\lbrace \\Box \\lnot (G.v) \\rbrace $ saf Form list $h=[h_1,\\ldots ,h_n]$ from set $B_{i}$ ; Sort list $h$ ascending by time stamps; $l:=1$ ; $k:=l$ ; $(h_{k}.v = h_{l}.v)\\wedge (l<n)$ $l:= l+1$ ; $h_{k}.v = h_{l}.v$ $L_i := L_i \\cup \\lbrace \\Diamond (h_{k}.v) \\rbrace $ liv1 $h_{k}.v \\ne h_{l}.v$ $L_i := L_i \\cup \\lbrace \\Box (h_{k}.v \\Rightarrow \\Diamond (h_{l}.v)) \\rbrace $ liv2 $l=n$ Logical specification $\\Sigma $ , or $L_{i}$ , is a set of syntactically correct temporal logic formulas.", "The algorithm bases on the analysis of all events that occur in a smart environment.", "The algorithm is explained with the remarks given below.", "Separate specifications for each object are built (line ); Every system should be described using both safety and liveness properties [2]; It is tested which nodes are not involved in registered events (line ); The most general form for safety (informally: nothing bad will ever happen) is $\\Box \\lnot (p)$ , i.e.", "some nodes might be never visited (line , labeled “saf”); one can consider the absence pattern in terms of Figure REF ; Auxiliary lists (lines and ) are created for events that occur for an object; List $h$ consists of at least one element (line ); The repeat loop allows to find all sequences of events following each other (lines from to ); The inner loop allows to skip to a different node/event, if any (lines from to ); The most general form for liveness (informally: something good will happen) is $\\Box (q \\Rightarrow \\Diamond r)$ or $\\Diamond r$ , i.e.", "some nodes are visited (lines or , labeled “liv1” or “liv2”, respectively); one can consider the existence or response patterns in terms of Figure REF , respectively; The existence pattern can occur at most once (line ); Summing up, temporal logic formulas are produced in three places of the algorithm which are labeled by “saf”, “liv1”, and “liv2”.", "Let us consider the illustrative example for Algorithm .", "Nodes for a smart environment are $\\lbrace e2, s03, s07, s08, \\rbrace $ , i.e.", "three labeled nodes/vertices.", "The considered objects/users $O=\\lbrace \\ldots ,o_{5},\\ldots \\rbrace $ .", "A behavior, that is registered events, is $B=\\lbrace \\langle o5,s03,t2015.02.12.09.30.15\\rangle ,\\nonumber \\\\\\langle o5,s08,t2015.02.12.09.32.40\\rangle ,\\nonumber \\\\\\langle o5,s08,t2015.02.12.09.33.30\\rangle ,\\nonumber \\\\\\langle o5,s08,t2015.02.12.09.34.20\\rangle ,\\nonumber \\\\\\langle o5,s07,t2015.02.12.09.35.20\\rangle ,\\nonumber \\\\\\langle o5,s07,t2015.02.12.11.37.15\\rangle \\rbrace $ The algorithm produces the following logical specification $L_i=\\lbrace \\Box \\lnot (e2),\\Box (s03 \\Rightarrow \\Diamond s08),\\Box (s08 \\Rightarrow \\Diamond s07) \\rbrace $ Every logical specification can be used for the reasoning process as shown in Figure REF , or in Figure REF as another example of a truth tree for Formula (REF ), where conjunction of all sub-formulas are analyzed.", "Figure: Another example of a truth treeMany different methods, as well as deductive systems, for truth trees and semantic tableaux are discussed in work [10] that might help to operate and manipulate efficiently and effectively with truth trees.", "The more general remarks for Algorithm are given below.", "The algorithm produces logical specifications $L_{i}$ for every object that operates in a smart environment; It should be stressed again that, to simplify considerations, the one-sensor case (the object detection) is discussed, in other words, Formula (REF ) might be replaced by a single atomic sub-formula $s_1$ as an example, or, in terms of the algorithm, by $h_{k}.v$ as an example; The more general issue is the question when the algorithm should operate, for example, whenever it is required (on demand) or at “the end of a day” (whatever it means), this is an open question for future work; Another open issue is the question of what happens when an “old” specification, i.e.", "specification obtained as a result of the previous execution, is summed, if necessary, with specifications of the current execution, then one should examine the entire specification using decision procedures mentioned at the end of Section , as Corollary , to discover open and closed branches; The sketch for the algorithm that unifies, if necessary, all specifications obtained from Algorithm is given as Algorithm , of course, there is no problem to prepare the reverse algorithm, that separates logical specifications due to each object.", "[htb] Building logical specification for smart env.", "$En$ [1] Input: Output: Logical specifications $L_{i=1,\\ldots ,n}$ (for object $o_{i=1,\\ldots ,n}$ ) Logical specification $\\Sigma $ every $L_i$ $\\forall f \\in L_i$ attribute formula $f$ uniquely due to object $o_i$ $\\Sigma := \\bigcup _{i=1}^{n} L_i$ The following two statements are valid.", "The time complexity for Algorithm is expressed by $\\mathcal {O}(o \\cdot n)$ , where $o$ is the number of objects that operate in a smart environment, and $n$ is a number of events registered for each object.", "If a set of all objects and a set of all events are finite, then Algorithm always terminates.", "The main, outer loop depends on a number of objects $o$ .", "The inner, repeat loop depends on a number of events $n$ .", "Other operations (assignment) and loops (limited number of iterations) give constant costs.", "Thus, the time complexity of Algorithm is linearly dependent on the numbers of objects and events.", "The number of objects is finite (the for loop), the number of vertices is limited (the inner for loop), as well as the number of registered events is limited (the inner repeat loop), thus, the algorithm always terminates.", "Let us supplement this Section with Algorithm that illustrates more formally considerations following Figure REF .", "[htb] Managing and interpreting truth trees (sketch) [1] Input: Output: Logical specification $\\Sigma $ ; new formula $f$ Truth tree $\\Delta $ ; logical specification $\\Sigma $ ; $Open$ ; $L:=$ formulas $\\Sigma $ that refer to the same object as $f$ refers; Build truth tree $\\Delta $ for a combined formula $f \\wedge C(L)$ ; $R:=$ select branches of $\\Delta $ with literals from formula $f$ ; $Open:=$ select open branches from $R$ ; $Closed:=$ select closed branches from $R$ ; .....", "If necessary, remove/modify formulas from specification $\\Sigma $ basing on literals which belong to $f$ and $Closed$ ; $\\Sigma := \\Sigma \\cup \\lbrace f\\rbrace $ the new basic specification; ..... Analyze nodes from $Open$ to provide new actions; It gives an idea how both Reasoning Engine and Result Interpretation, shown in Figure REF , work.", "The $f \\wedge C(\\Sigma )$ case, see Table REF , is taken into account.", "It is assumed that initially $\\Sigma $ contains no contradiction.", "$Closed$ is a set of closed branches of a tree and constitutes a base for further modification of the basic logical specification $\\Sigma $ , if necessary, removing formulas that contradict with a newly introduced formula.", "$Closed^{\\prime }$ is a set of all literals extracted from $Closed$ .", "$Open$ is a set of open branches of a tree and constitutes a base for selecting satisfiable graph nodes.", "$Open^{\\prime }$ is a set of all literals extracted from $Open$ .", "If necessary, specification $\\Sigma $ is modified, see lines –, to remove contradictory formulas from a specification.", "This operation is performed using literals which belong to $Closed$ /$Closed^{\\prime }$ (contradictory literals) and $f$ (new formulas, perhaps influencing the basic specification $\\Sigma $ through introducing contradictions, if any), see the example and the last subcase given below.", "Analyzing open branches $Open$ to provide actions for a system, see line , is a standard procedure, see the example and all subcases given below.", "The illustrative example to supplement both Algorithm and informal considerations succeeding Figure REF is now provided.", "For the (Figure) REF .a subcase, $\\Sigma =\\lbrace \\ldots , v10 \\Rightarrow \\Diamond p110, \\ldots \\rbrace $ and $f=v10$ .", "Then $Open^{\\prime }=\\lbrace v10,p110\\rbrace $ provides literals that allow to find the appropriate formula in $\\Sigma $ , that is formula $v10 \\Rightarrow \\Diamond p110$ .", "For the REF .b subcase, $\\Sigma =\\lbrace \\ldots , ((v11 \\Rightarrow \\Diamond p115) \\vee (v11 \\Rightarrow \\Diamond p116)), \\ldots \\rbrace $ and $f=v11$ .", "Then $Open^{\\prime }=\\lbrace \\lbrace v11,p115\\rbrace ,\\lbrace v11,p116\\rbrace \\rbrace $ provides literals leading to formula $((v11 \\Rightarrow \\Diamond p115) \\vee (v11 \\Rightarrow \\Diamond p116))$ describes formula showing two equivalent movements (passages $p115$ or $p116$ ).", "For the REF .c subcase, $f=\\Box (\\lnot p115)$ .", "Then $Open^{\\prime }=\\lbrace v11,p116\\rbrace $ and $Closed^{\\prime }=\\lbrace \\ldots ,v11,p115,\\dots \\rbrace $ .", "On one hand, $Open^{\\prime }$ allows to point passage $p116$ .", "On the other hand, $Closed^{\\prime }$ , showing literals $v11,p115$ , allows to modify a formula as a result of the passage elimination (fire), that is to replace $((v11 \\Rightarrow \\Diamond p115) \\vee (v11 \\Rightarrow \\Diamond p116))$ by $(v11 \\Rightarrow \\Diamond p116)$ .", "Then the resulting specification is $\\Sigma = \\lbrace \\Box (\\lnot p115), \\ldots , v11, (v11 \\Rightarrow \\Diamond p116)\\rbrace $ .", "Summing up, encoding behaviors to logical specifications is a natural process that can be applied to context-aware systems.", "There are two different approaches mentioned in the beginning of Section .", "Some other studies that refer to the implementation and application aspects are open research questions.", "For example, the form of a formula located in the root of truth trees, that is the disjunction of sub-formulas (the choice between alternatives) or conjunction of sub-formulas (satisfiability, contradiction).", "Another example is a method for storing formulas, as well as an idea to register multiplicity of formulas/events to introduce additional information about the event popularity.", "Logical specifications, encoding registered behaviors, can be interpreted as preferences understood as a priority in selection.", "Thus, gathering knowledge about preferences is also expressed as logical formulas." ], [ "Concusion", "This paper presents a method for behavior discovery as well as the logical satisfiability-oriented reactive analysis for smart and sensor-based environments to support context-aware and pro-active decisions.", "This approach constructs the process for building logical specifications that fulfill the recognition process providing behavioral specification in terms of temporal logic formulas.", "The proposed unified logical framework is focused on sensor based activity recognition.", "Future works should cover more detailed algorithms, architecture of a multi-agent system and detailed use cases.", "Considering graph representations and transformations [14], [13] is encouraging for efficient implementation and deploying with presented here logical-oriented approach.", "More comparison study with other existing methods and more theoretical and experimental evaluations are required for future work." ] ]	1403.0185
[ [ "Satellites of Radio AGN in SDSS: Insights into AGN Triggering and\n Feedback" ], [ "Abstract We study the effects of radio jets on galaxies in their vicinity (satellites) and the role of satellites in triggering radio-loud active galactic nuclei (AGNs).", "The study compares the aggregate properties of satellites of a sample of 7,220 radio AGNs at z < 0.3 (identified by Best & Heckman 2012 from the SDSS and NVSS+FIRST surveys) to the satellites of a control sample of radio-quiet galaxies, which are matched in redshift, color, luminosity, and axis ratio, as well as by environment type: field galaxies, cluster members and brightest cluster galaxies (BCGs).", "Remarkably, we find that radio AGNs exhibit on average a 50% excess (17{\\sigma} significance) in the number of satellites within 100 kpc even though the cluster membership was controlled for (e.g., radio BCGs have more satellites than radio-quiet BCGs, etc.).", "Satellite excess is not confirmed for high-excitation sources, which are only 2% of radio AGN.", "Extra satellites may be responsible for raising the probability for hot gas AGN accretion via tidal effects or may otherwise enhance the intensity or duration of the radio-emitting phase.", "Furthermore, we find that the incidence of radio AGNs among potential hosts (massive ellipticals) is similar for field galaxies and for non-BCG cluster members, suggesting that AGN fueling depends primarily on conditions in the host halo rather than the parent, cluster halo.", "Regarding feedback, we find that radio AGNs, either high or low excitation, have no detectable effect on star formation in their satellites, as neither induced star formation nor star formation quenching is present in more than ~1% of radio AGN." ], [ "Introduction", "Active galactic nuclei (AGN) are believed to be powered by two fundamentally different modes of accretion.", "Optically selected and luminous radio-loud AGN (R-AGN) are powered by the radiatively-efficient accretion of gas onto a central supermassive black hole (SMBH) at a rate of one to ten percent of the Eddington limit.", "In contrast the majority of R-AGN, which have a low radio luminosity, accrete gas in a radiatively inefficient manner at a rate below one percent of the Eddington limit [9].", "The mode of accretion may also be related to galaxy morphology.", "While AGN in general are hosted by galaxies that span a range of morphological types, the hosts of R-AGN are most often massive ellipticals (e.g., Ekers & Ekers 1973, Best at al.", "2005b, Kauffmann et al.", "2008).", "This difference in R-AGN accretion processes could be a result of differences in the fueling mechanism.", "Optical and luminous radio AGN may be triggered by mergers and one-on-one interactions (Heckman et al.", "1986, Barnes & Hernquist 1996) which could supply the large quantities of cold gas required to maintain a high accretion rate.", "Low luminosity R-AGN may be triggered by the accretion of hot gas from the surrounding halo (Fabian 1994, Allen et al.", "2006), although there could be additional triggering mechanisms [42].", "The key to understanding these triggering mechanisms therefore lies in the kpc to Mpc-scale environments of R-AGN, which is populated by satellite galaxies.", "Many studies have examined the relationship between R-AGN incidence and the environment.", "[12] found that R-AGN are more frequently found in central group and cluster galaxies when compared to galaxies of similar stellar mass.", "[8] studied a small sample of 91 R-AGN and found that the fraction of R-AGN shows little dependence on the Mpc-scale local galaxy density, but is dependent on the number of galaxies in the group in which the R-AGN is located, while the slightly larger sample of 212 R-AGN of [41] showed a slight increase in R-AGN incidence with density at the Mpc scale.", "[11] and [42] both found a strong dependence of R-AGN activity on the stellar mass of the host galaxy, but when this is accounted for, R-AGN activity still has a dependence on density at the Mpc scale.", "[36] found that the environments of R-AGN at the $\\sim $ 100 kpc scale are roughly twice as dense as those of radio-quiet AGN or radio-quiet galaxies of similar mass.", "In contrast to these studies, [53] found that at the $\\sim $ 100 kpc scale only powerful R-AGN have higher clustering than radio-quiet galaxies of the same mass, and that for scales greater than about 160 kpc the clustering of R-AGN is similar to that of radio-quiet galaxies.", "The aforementioned studies find an increase in the R-AGN incidence rate in denser, cluster environments.", "This should not be surprising considering that R-AGN likely feed on the hot gas that is plentiful in clusters.", "However, it must be acknowledged that neither all ellipticals or brightest cluster galaxies (BCGs) are radio loud, nor are all massive field ellipticals radio quiet.", "The approach taken in the current study is therefore different as it separates the general environment (BCG, cluster member, or field galaxy) from the small-scale environment ($<100$ kpc).", "We compare the small-scale environments (i.e.", "the satellite populations) of a statistically large sample of R-AGN and a matching control sample of radio-quiet galaxies.", "Control galaxies are chosen not only to match the mass, type, and star formation history of an R-AGN but also the large scale environment it is found in, such that the controls of field R-AGN are also field galaxies, the controls of cluster member R-AGN are also cluster members, and the controls of BCG R-AGN are also BCGs.", "This careful matching allows us to compare the satellite populations of R-AGN and otherwise similar radio-quiet galaxies to determine why only a subset of galaxies with similar environments become R-AGN.", "In our study of satellite populations we will separately focus on radio AGN with high accretion rates.", "The two modes of R-AGN accretion can be distinguished with emission line ratios [38].", "[14] defined an `excitation index' composed of four optical emission line ratios which can be used to separate R-AGN with a low accretion rate, which are known as low excitation radio galaxies (LERGs) from those with a high accretion rate, known as high excitation radio galaxies (HERGs).", "At most radio luminosities HERGs comprise only a few percent of the total R-AGN population, although they become more common at the highest radio luminosities [9].", "Many studies employ the Fanaroff-Riley classification, which classifies R-AGN based on their radio morphology [29].", "Luminous R-AGN, which tend to be edge brightened, are known as Fanaroff-Riley type 2 (FR2) sources, while core-brightened lower-luminosity sources are referred to as Fanaroff-Riley type 1 (FR1) sources.", "Although LERGs and HERGs often exhibit FR1 and FR2 morphology, respectively, [38] demonstrated that a number of FR2 sources have low-excitation spectra (i.e.", "are LERGs).", "Since the LERG & HERG classifications are more closely related to accretion mode and do not require high resolution radio maps, we adopt their use in our study, as determined by [9].", "In addition to exploring the role of satellites in R-AGN triggering, the second major goal of this study is to understand if and to what extent satellites can be affected through feedback from the powerful jets that are the hallmark of R-AGN.", "These highly collimated jets originate in the nucleus of the AGN at relativistic speeds and may extend for kiloparsecs or even megaparsecs beyond the host galaxy (Tremblay et al.", "2010, Schoenmakers et al.", "2000).", "Radio AGN deposit most of their energy into the interstellar or intergalactic medium kinetically via their high-velocity jets (see Fabian 2012 for a recent review).", "Such interactions may be responsible for quenching star formation in the host galaxy, and there is some evidence that R-AGN may lead to quenching of star formation in their satellite galaxies.", "[44] found that satellite galaxies in the projected jet paths of FR2 sources are redder than satellites outside the jet path, but no such trend was found for the satellites of FR1 sources.", "This suggests that FR2 jets tend to quench star formation in satellite galaxies while FR1 jets do not.", "We will revisit these results in our study by using a much larger sample of R-AGN and by contrasting entire satellite populations of R-AGN (regardless of proximity to the jet) to those of the radio-quiet control sample.", "Somewhat paradoxically, radio jets are also considered as the mechanism behind possible positive feedback, both in the host galaxy and in adjacent satellite galaxies.", "The idea is that interactions with jets may induce star formation by driving shocks into dense clouds which then collapse [4].", "Such AGN-driven star formation may have been important in building up the massive spheroid in the early phases of galaxy formation [45].", "The radio jets of both HERGs and LERGs are believed to be capable of producing positive feedback outside of the host as well.", "The powerful radio jets associated with HERGs pierce external clouds in the IGM, but triggered star formation may proceed in their slowly expanding radio lobes (De Young 1981, Bicknell et al.", "2000).", "Since the jets of LERGs are generally less powerful, they may be capable of inducing star formation even in head-on collisions [49].", "A few candidates of such jet-induced star formation in the vicinity of R-AGN have been observed.", "The star forming region `09.6' in the eastern lobe of the nearby FR 2 source 3C 285 was first observed by [48].", "Chandra observations confirm that this region is indeed experiencing a starburst phase [33], while a satellite galaxy of the FR2 source PKS2250–41, which is not currently in the projected jet path, also shows evidence of jet-induced star formation [35].", "Induced star formation as a result of interactions with weak R-AGN has also been observed: one example is the LERG Centaurus A.", "[20] recently showed that the youngest stars in the inner filament of Centaurus A are only a few Myrs old and are probably the result of the shock induced collapse of a molecular cloud.", "Downstream from the inner star-forming filament, the radio jet interacts with an H I cloud and produces another filament of recently-formed stars [31].", "Minkowski's Object (MO) is another example of LERG-induced star formation.", "This star forming region is 15 kpc away from its host R-AGN NGC 541.", "Strong UV and H$\\alpha $ emission are indicative of the starburst nature of this object [49].", "Simulations by [30] have reproduced the observational characteristics of MO, including the star formation rate (SFR) of 0.3 M$_$ yr$^{-1}$ .", "While it is possible that MO is an example of star formation being reignited in an already star-forming galaxy, this seems unlikely.", "Although [21] could not rule out an underlying old stellar population in MO, an HI cloud downstream from MO was observed, which suggests that unlike Centaurus A the neutral hydrogen cooled out of a warm and clumpy IGM and then collapsed.", "While there is observational evidence that in some individual cases intense star formation may have been triggered by R-AGN jets, it is not clear how common this phenomenon is and whether it can be positively stated that the star formation is indeed the result of jet interactions.", "Furthermore, it is not clear if this induced star formation leads to the emergence of pristine, new satellites or whether it is an enhancement in already existing, star-forming satellites.", "As in the case of star formation quenching, we will approach this question by studying the colors of satellites of R-AGN and a matched control sample of radio quiet galaxies.", "In both the study of quenching and the induction of star formation in satellites we will pay special attention to HERGs in which either of these processes may be more pronounced or more common.", "The layout of our paper is as follows.", "In Section 2, samples are presented, and in Section 3 we describe our method for selecting the control sample of radio quiet galaxies as well as generating distributions of satellite properties.", "Our results are presented in Section 4, while in Sections 5 and 6 we discuss the implications of our findings.", "In this work the cosmological parameters adopted are $\\Omega _m\\,=\\,0.27$ , $\\Omega _\\Lambda \\,=\\,0.73$ , and $H_0\\,=\\,71\\, \\textrm {km}\\,\\textrm {s}^{-1}\\,\\textrm {Mpc}^{-1}$ ." ], [ "Construction of Samples", "We have two primary samples: a sample of R-AGN drawn from [9] and a control sample of matching radio quiet galaxies that we have assembled.", "Each galaxy in the control sample was chosen to have the same type of environment as its counterpart in the R-AGN sample.", "Control galaxies were also chosen to approximately match the galaxy mass and star formation histories of the R-AGN by selecting control galaxies that provide the closest match in $r$ magnitude, $u-r$ color, redshift, and axis ratio.", "We then compare the satellites of R-AGN and radio quiet samples.", "We use the term `satellite' to refer to nearby galaxies that are physically associated with the hosts in our samples, while `neighbors' refer to candidate satellites, which includes foreground or background objects not associated with the host.", "This distinction is necessary because candidate satellites are faint and are generally lacking spectroscopic redshifts.", "We have drawn neighbors for each of these two samples, and to correct for background contamination we have also drawn neighbor-like objects from two sets of offset positions.", "Details of these samples are given below." ], [ "R-AGN sample", "[9] have assembled a sample of 18,286 radio galaxies by combining the 7th data release of the SDSS spectroscopic sample (DR7; Abazajian et al.", "2009) with the National Radio Astronomy Observatory (NRAO) Very Large Array (VLA) Sky Survey (NVSS; Condon et al.", "1998) and the Faint Images of the Radio Sky at Twenty centimeters (FIRST; Becker, White & Helfand 1995) survey.", "Radio emission may arise from AGN or star formation activity, so [9] used a combination of three methods to separate R-AGN from star-forming galaxies.", "The first method is a comparison of the 4000 Å break strength to the ratio of radio luminosity per stellar mass.", "Since R-AGN have enhanced radio luminosity relative to star-forming galaxies, they are thus separable in this plane [11].", "The second method is the emission-line diagnostic or `BPT' diagram (Baldwin et al.", "1981, Kauffmann et al.", "2003) which compares the ratio of [O III] 5007 Å and H$\\beta $ line fluxes to the ratio of [NII] 6584 Å and H$\\alpha $ .", "Only 30% of the galaxies in the radio source sample could be classified using the BPT diagram: the remainder lacked detections or limits in at least one line.", "The final method is a comparison of the H$\\alpha $ line luminosity to the radio luminosity.", "This method operates under the assumption that in star-forming galaxies both the H$\\alpha $ line and radio luminosities are a consequence of star formation and thus correlated, while the radio luminosities of R-AGN are offset because their radio luminosities are boosted by AGN activity.", "Differing classifications are sometimes given by these three methods.", "In these instances a final classification was chosen based on the properties of the galaxies in each class: see Table A1 of [9].", "Figure: Histogram showing the distribution of R-AGN redshifts.", "Vertical lines at zz = 0.1, 0.15, and 0.2 indicate divisions used for the redshift ranges.The magnitude range of our sample is 14.5 $<$ r $<$ 17.77, matching the limits of the SDSS main galaxy spectroscopic sample [46].", "Many of the R-AGN in [9] are fainter than this limit and lie at redshifts greater than $z=0.3$ .", "We are not interested in higher redshift R-AGN since we cannot draw control samples for them.", "We imposed a lower redshift limit at $z=0.04$ to avoid excessive background contamination in the 100 kpc radius used to obtain neighbors.", "Thus the final redshift range of our sample is 0.04 $<$ z $<$ 0.3, and Figure REF shows this redshift distribution.", "At the lower limit a radius of 100 kpc corresponds to an angular radius of 21.", "Only 28 R-AGN are removed from the sample as a result of this lower redshift limit.", "We used the photometry of the eighth SDSS data release (DR8) [2] rather than DR7 because photometric redshifts, used to reduce non-satellite contamination in neighbors, tend to be more accurate in DR8 than in DR7 (see Appendix ).", "Also, in previous data releases the outer regions of large galaxies were oversubtracted, which affected the photometry of these galaxies as well as faint nearby objects [2].", "Improved sky subtraction procedures in DR8 have mostly addressed this issue.", "Fifty-six objects listed as R-AGN in [9] were either not present or were misidentified in DR8, so our final R-AGN sample consists of 7,220 galaxies.", "Figure REF is a color-magnitude diagram that shows the relation of the R-AGN sample, shown in black, to SDSS galaxies in the same redshift range.", "Also plotted is the control sample, the selection of which is discussed below.", "Colors and magnitudes in this figure have been $k$ -corrected with the $k$ -correction provided by [18].", "This figure shows that the R-AGN hosts fall along the red sequence, with R-AGN favoring luminous galaxies.", "Figure: Color-magnitude diagram comparing the R-AGN and control samples with SDSS galaxies in the same redshift range.", "In this figure, M r M_r and u-ru-r have been kk-corrected.", "Contours containing 90%, 50%, and 25% of the samples are shown.", "R-AGN in the sample tend to be hosted by luminous galaxies on the red sequence.", "The properties of galaxies in the control sample closely match those of the R-AGN.One of the criteria for producing the control sample is to match the type of environment in which the galaxy is located.", "Three cluster membership categories were used: field galaxy, cluster member, and BCG.", "For each of the R-AGN we performed a nearest-neighbor search to the BCGs reported in the galaxy cluster catalog of [32], which is based on DR7.", "The clusters in this catalog are in the redshift range $0.1<z<0.55$ , and the list of additional clusters at $z<0.1$ , available from the author's web pagehttp://home.fnal.gov/$\\sim $ jghao/gmbcg_sdss_catalog.html, allow us to assign cluster membership for all R-AGN in the sample.", "We classified as BCGs all R-AGN whose DR8 coordinates were within 1.5 kpc of the BCG catalog coordinates, and found that our sample of R-AGN contains 949 BCGs or 13% of the total.", "Those R-AGN that were between 1.5 kpc and 1.5 Mpc away from the nearest BCG and had a difference in spectroscopic redshift less than z $=$ 0.01 were classified as cluster members.", "We have chosen 1.5 Mpc as a typical virial radius ($r_{200}$ ) of clusters.", "Of our sample, 436 R-AGN or 6% of the total were classified as cluster members, and the mean distance from cluster members to the BCG of the host cluster is 700 kpc.", "The remaining 5,835 R-AGN or 81% of the sample were classified as field galaxies.", "While R-AGN have a preference for rich environments as demonstrated in previous studies, the majority of radio AGN are nevertheless not in clusters.", "Most (86%) of the R-AGN in the sample have been classified as either HERGs (209) or LERGs (6,005) by [9].", "The remaining 1,006 were not classified because they lacked emission lines.", "Table REF gives the number of HERGs, LERGs, and unclassified galaxies in each environment type.", "Interestingly, HERGs are most common among the field R-AGN (3.4%) and rarer among cluster members (1.1%).", "They are very rare among BCGs (0.4%).", "This result supports the scenario in which HERGs arise from cold gas accretion, which is more abundant in the field and is in agreement with the findings of [42], who found that LERGs show an increasing rate of incidence with the local density, while HERGs show a decrease, as do optically-selected AGN.", "We will return to the incidence rate of R-AGNs in different environments in Section REF .", "rcccccccc 9 Number of R-AGN per accretion mode and redshift bin for each environment type.", "Environment HERG LERG Unclassified $0.04<z<0.1$ $0.1<z<0.15$ $0.15<z<0.2$ $0.2<z<0.3$ Total Field2004,8457901,1151,4881,7111,5215,835 Member5 368 6339129163105436 BCG 479215353185258453949 Total2096,0051,0061,2071,8022,1322,0797,220 Columns 2 and 3 give the number of R-AGN with the stated accretion mode (from Best & Heckman 2012) in each type of environment, while column 4 lists those whose accretion mode was not identified.", "Columns 5-8 present the number of R-AGN in each redshift bin, while column 9 presents the number of R-AGN per environment type." ], [ "Control sample (non R-AGN)", "For each R-AGN, we have selected a control galaxy which is not an R-AGN (or is below the limits of radio surveys) but has the same cluster membership status and similar overall physical properties as its counterpart R-AGN.", "We considered as control candidates those galaxies in DR8 that are within the FIRST footprint ($4^h<RA<20^h$ ) and therefore subject to classification by [9].", "The area of DR8 within the FIRST footprint is identical to that of DR7, so the control candidates are also subject to cluster member classification with the [32] catalog.", "This provides a pool of 842,296 control candidates.", "Each control galaxy was selected to closely match the $r$ magnitude, $u-r$ color, redshift, and axis ratio of its corresponding R-AGN.", "These properties were used for selection because the redshift and $r$ magnitude give the optical luminosity, which together with $u-r$ are a proxy for the stellar mass of the galaxy while the $u-r$ color is a proxy for the star formation history.", "Radio AGN are predominantly found in spheroidal early-type galaxies (ellipticals) rather than among the flattened early types containing stellar disks (S0s) (e.g., Ekers & Ekers 1973, Véron-Cetty & Véron 2001).", "The two varieties of early type galaxies have similar colors and overlapping absolute magnitude ranges (e.g., Cheng et al.", "2011).", "The inclusion of the axis ratio as a matching criterion helps ensure that the control sample is of the correct morphological type [17].", "The selection was done by minimizing the metric, $\\small R \\,= \\,\\sqrt{\\left(\\frac{\\Delta z}{0.21}\\right)^2+\\left(\\frac{\\Delta r}{2.99}\\right)^2+\\left(\\frac{\\Delta ( u-r)}{1.96}\\right)^2+\\left(\\frac{\\Delta ( b/a)}{0.49}\\right)^2},$ where the quantities $\\Delta z$ , $\\Delta r$ , $\\Delta (u-r)$ , and $\\Delta (b/a)$ are the differences between the R-AGN and control candidate parameters, and the denominators are scaling factors that correspond to the 95 percentile ranges in redshift, $r$ magnitude, $u-r$ color, and axis ratio (b/a).", "The control sample is plotted together with the R-AGN sample in Fig.", "REF , and it can be seen that the control sample closely matches the color-magnitude distribution of the R-AGN sample.", "The means and standard deviations of the differences between the control and R-AGN galaxy properties also demonstrate the success of our control sample selection.", "The mean difference for $r$ is $\\delta r=0.0076$ ($\\pm $ 0.065), for redshift the difference is $\\delta z=-2.97\\times 10^{-4}$ ($\\pm 4.6\\times 10^{-3}$ ), for $u-r$ the difference is $\\delta (u-r)=-0.0026$ ($\\pm $ 0.05), and for axis ratio the difference is $\\delta (b/a)=6.07\\times 10^{-6}$ ($\\pm 1.1\\times 10^{-2}$ ).", "In other words, the standard deviations are roughly 2% of the 95 percentile range used for the scaling factors in the metric.", "Some of the control galaxies are best matches to more than one of the R-AGN in the sample: this occurs for only 11% of the control galaxies." ], [ "Satellites of R-AGN and control samples", "In this study, we have examined all satellites within a circular region centered on the host galaxies in our samples regardless of their position angle with respect to the jet.", "For the study of feedback on the satellite population it may appear better to focus only on satellites lying close to the jet.", "However, jet directions are not available for the majority of the sample.", "Furthermore, the satellite PKS2250-41 [35] shows evidence of jet-induced star formation even though it is not currently in the jet path, justifying our approach.", "The radius of the search circle was determined by examining the projected lengths of the 782 R-AGN jets in the list provided by [39].", "Of the R-AGN in this list, 90% have jets shorter than 100 kpc.", "A search radius greater than 100 kpc would allow us to include the end points of a few more R-AGN at the cost of a large increase in unassociated neighbors.", "We restricted our analysis to satellites that lie within 100 kpc (in projection) of the galaxies in our samples.", "An inner boundary at 5 kpc was also used to avoid spurious sources which may arise as a result of shredding of the host galaxy by the SDSS pipeline.", "Neighbors include objects classified as galaxies by the DR8 pipeline that are brighter than $r\\,=\\,22$ , close to the SDSS magnitude limit of $r\\,=\\,22.2$ .", "We make no requirement that the magnitudes of neighbors be fainter than that of the R-AGN or control galaxy, but we find that only in a few instances are neighbors more luminous than their target galaxies.", "We derive properties of the satellite population by subtracting the distribution of sources selected in the same way as neighbors, but in parts of the sky that are offset (two degrees in RA) from the host (R-AGN or its control).", "To make this procedure more robust, we first remove from candidate neighbors those with significantly discrepant photometric redshifts.", "Considering the large uncertainties of available photometric redshifts (Appendix A) we apply a very conservative cut such that the candidate neighbors are removed if their photometric redshift is more than 0.2 greater than the spectroscopic redshift of the host.", "In this study we examine the co-added distribution of satellite properties for R-AGN, and, separately, for their radio quiet counterparts, which span a redshift range from 0.04 to 0.3.", "As a result, at the low redshift limit we probe galaxies down to $M_r$ = $-14.3$ , which shifts to $M_r$ = $-18.8$ at the upper redshift limit.", "To account for the resulting Malmquist bias we first construct various satellite distribution functions in smaller redshift ranges, in which the bias is greatly reduced.", "After we confirm that different redshift slices have similar distributions of satellite properties, we average them.", "The four redshift ranges are 0.04 $<\\,z\\,<$ 0.1, 0.1 $<\\,z\\,<$ 0.15, 0.15 $<\\,z\\,<$ 0.2, and 0.2 $<\\,z\\,<$ 0.3.", "The distribution of R-AGN redshifts as well as the redshift range divisions are shown in Fig.", "REF while Table REF gives the number of R-AGN per environment type in each redshift range.", "For a given redshift range, the number of neighbors per bin of a distribution are given by, $N_R\\,=\\,\\textrm {Number of R-AGN neighbors},$ $N_c\\,=\\,\\textrm {Number of control neighbors},$ $B_R\\,=\\,\\textrm {Number of R-AGN offset objects},$ $B_c\\,=\\,\\textrm {Number of control offset objects}.$ Table gives the number of neighbors and offset objects per redshift bin in each environment.", "This table shows that roughly half of neighbors are background objects.", "The number of satellites per host around R-AGN is given by subtracting the offsets from the number of R-AGN satellites and dividing by the number of primary galaxies in that redshift range, $N_{sat,i}\\,=\\,\\frac{N_R-B_R}{N_{host}}.$ To get the error in the number of satellites per host, we use, assuming normal approximation of counting errors: $\\sigma _{sat,i}\\,=\\,\\frac{\\sqrt{N_R+B_R}}{N_{host}}.$ The same method is used to calculate the number of satellites per control galaxy.", "However, since in 11% of cases the same control is chosen for more than one R-AGN the errors calculated in the above way, which assumes independent measurements, will be underestimated for control galaxies by 5%.", "This is a small effect and we do not correct for it.", "This procedure gives the nearly volume-complete distributions of R-AGN and control satellites separately for each of the four redshift ranges.", "We then combined the four distributions with weighted averaging, and the resulting distribution for the R-AGN sample is given by $D_{final,R}\\,=\\,\\frac{\\sum _{i=1}^{4}D_{R,i}/\\sigma ^2_{R,i}}{\\sum _{i=1}^{4}1/\\sigma _{R,i}^2},$ and the error is given by $\\sigma _{final,R}\\,=\\,\\frac{1}{\\sum _{i=1}^{4}1/\\sigma _{R,i}^2}.$ The final distribution for the control sample is computed in the same manner.", "rcccccccc[h] 6 Number of neighbors and offsets per redshift bin brighter than $M_r=-17$ .", "Neighbors with discrepant redshifts are not included in these counts.", "Environment Number Bin 1 Bin 2 Bin 3 Bin 4 $N_R$ 10,0237,8997,3375,815 Field $N_C$ 9,2036,0016,7894,825 $B_R$ 5,8203,9353,3862,485 $B_C$ 5,9064,1083,5252,490 $N_R$ 458867915543 Cluster $N_C$ 437784692454 Member $B_R$ 183321308191 $B_C$ 247367298164 $N_R$ 7071,6901,7322,605 BCG $N_C$ 6581,5031,5772,330 $B_R$ 246535528712 $B_C$ 203549487709 These redshift bins are the same as those in Table REF ." ], [ "Results", "To investigate R-AGN triggering and the effects of R-AGN jets on satellite galaxies, we compare the populations of R-AGN and control satellites.", "This is done by calculating and comparing distributions of satellites of R-AGN and control galaxies as functions of projected distance from the host, satellite luminosity, and satellite color.", "We have also examined the ratios of distributions of R-AGN and control satellites." ], [ "Distribution of projected distances of satellites from the host", "Figure REF shows distributions of projected radial distances of satellites from their hosts.", "Field galaxies are in panel a, cluster members in panel b, BCGs in panel c, and HERGs, regardless of environment, in panel d. At all redshifts, our satellite samples are complete for satellites brighter than $M_r \\,=\\, -19$ , so we adopt this absolute magnitude cut for the radial distance distributions.", "It should be noted that 90% of the HERGs are field galaxies.", "As expected, BCGs have more satellites than either field galaxies or non-BCG cluster members, regardless of whether they host an R-AGN or not.", "The satellite populations of field galaxies and non-BCG cluster members are similar for a given type of host.", "However, what is remarkable is that the mean number of satellites differs between R-AGN and radio-quiet galaxies, and it does so in all environments.", "Furthermore, this excess of satellites around R-AGN is present over the entire 100 kpc projected radius that we explore.", "The situation is ambiguous for the case of HERGs, where no obvious excess is visible.", "What we can confidently say is that the excess seen in panels a, b, and c reflects differences in the satellite populations of LERGs and not HERGs.", "Figure: Histograms of the distance distributions of satellites for which M r <-19M_r < -19, our completeness limit in the highest redshift bin.", "The process of filtering by photometric redshift, subtracting the offset, and combining redshift ranges with weighted averaging was used to create these distance distributions.", "Distributions in black are for R-AGN satellites, while those in orange are for the satellites of control galaxies.", "Panels a, b, & c show the results for field galaxies, cluster members, and BCGs, respectively, while panel d shows the results for HERGs.", "These plots show that R-AGN in all environments have more satellites within 100 kpc than radio-quiet galaxies." ], [ "Absolute magnitude distribution of satellites", "We construct satellite luminosity distributions to further characterize the differences in the satellite populations of R-AGN versus the control sample.", "Absolute magnitudes were computed using the spectroscopic redshifts of the host galaxies and have been $k$ corrected using the corrections provided by [18].", "The resulting $M_r$ distributions are shown in Fig.", "REF with the results for the three environments in panels a, b, & c, and HERG satellites in panel d. Figure: Histograms of the absolute magnitude distributions of satellites for which M r ≤-17M_r\\le -17, our completeness limit in the lowest redshift bin.", "Panels a-d are as in figure .", "Distributions of R-AGN satellites are shown in black, with control satellites in orange.", "As with Figure , these plots show that R-AGN in all environments have more nearby satellites than radio-quiet galaxies, and that this excess is not restricted to satellites of some luminosity.Unlike the luminosity function of field galaxies, the $M_r$ distributions of satellites in Fig.", "REF have a peak, which is to be expected for a flux-limited sample [16].", "The distribution of BCG satellites peaks at a brighter magnitude than field galaxies or cluster members.", "Since BCGs are generally massive and luminous, their satellites tend to be more luminous as well.", "The distributions of the satellites of R-AGN and control galaxies are quite similar among themselves, for all environments, with peaks at similar positions.", "In terms of the average number of satellites per host that are brighter than $M_r=-17$ , we find that field R-AGN have 2.14 ($\\pm 0.03$ ) satellites per host, and field control galaxies have 1.46 ($\\pm 0.03$ ).", "For cluster members, R-AGN have 3.47 ($\\pm 0.12$ ) and control galaxies have 2.61 ($\\pm 0.11$ ) satellites per host, while R-AGN that are BCGs have 4.30 ($\\pm 0.09$ ) and control BCGs have 3.72 ($\\pm 0.08$ ) satellites per host.", "We find that HERGs have 1.43 ($\\pm 0.15$ ) satellites per host on average, while the control galaxies of HERGs have 1.48 ($\\pm 0.13$ ).", "This formally confirms that the excess is very significant in all environments.", "However, no excess is formally found for HERGs." ], [ "Satellite colors", "We compared the $g-r$ color distributions of R-AGN and control satellites to provide context for the examination of the feedback effects of R-AGN jets on nearby satellites (Section REF ).", "Although $u$ is more sensitive to recent star formation, we chose to use the $g-r$ colors for satellites since many of them have poor $u$ -band photometry.", "We again used an absolute magnitude cut at $M_r\\,=\\, -19$ to probe the same part of the satellite population at all redshifts.", "The $k$ -correction of [18] was also applied to the $g-r$ colors, and the analysis was restricted to satellites for which $\\sigma (g-r) < 0.1$ mag ($\\sim $ 90% of total).", "Histograms of these distributions are shown in Fig.", "REF .", "Figure: Distributions of kk-corrected g-rg-r colors for R-AGN and control satellites.", "These plots are for satellites for which σ g 2 +σ r 2 <0.1\\sqrt{\\sigma _g^2+\\sigma _r^2}< 0.1 and M r <-19M_r< -19.", "Panels a-d are as in figure .", "The majority of satellites are red.In general the distributions of both the satellites of R-AGN and control galaxies peak near $g-r=0.8$ , indicating that satellites in general are red irrespective of R-AGN.", "In other words, the satellite population around these massive galaxies is mostly quenched.", "Whether R-AGN feedback plays any active role in this quenching cannot be well established from these distributions and is deferred to Section REF .", "The distribution for HERG satellites is noisy but is consistent with no excess satellites around HERGs, as seen in panel d of Fig.", "REF where the same $M_r$ cut was applied." ], [ "Characterization of the excess satellite population", "From the number histograms alone it is difficult to see if the population of excess R-AGN satellites has a preferred separation from the host, a preferred luminosity, or color.", "This is because the absolute excess will appear stronger where the distribution is higher.", "We have therefore examined the ratio of the distributions of R-AGN and control satellites.", "Ratio plots of the $M_r$ , color, and distance distributions for the field sample are shown in Fig.", "REF , with error bars computed via simple error propagation.", "Individual bins with errors greater than 50% are not shown.", "We have only shown plots for the field sample because it contains the most R-AGN: similar results are seen for the member and BCG samples.", "Figure: Ratios of the distributions of satellites for field R-AGN and their control, radio-quiet galaxies.", "Panel a shows the ratio of luminosity distributions, panel b the color distributions, and panel c the distance distributions.", "The ratio is typically ∼\\sim 1.5 with a possible weak dependence on brightness and color.", "See text for more details.All three distributions in Fig.", "REF are roughly flat at 1.5, which indicates that field R-AGN have $\\sim $ 50% more satellites than control galaxies.", "In other words, the satellite populations of R-AGN are scaled up versions of the satellite population around radio-quiet galaxies.", "We may be seeing a slight preference of the excess population for brighter and redder satellites.", "The former is not surprising as any possible influence on AGN activity would presumably be more likely for more massive satellites.", "The latter can be explained if some of the blue satellites are preferentially further from a galaxy in real distance than their projected distance would suggest and consequently do not contribute to excess population." ], [ "R-AGN fraction", "Previous studies (e.g.", "Best et al.", "2005b, Van Velzen et al.", "2012) have shown that the fraction of galaxies that are R-AGN (the R-AGN incidence rate) is an increasing function of the stellar mass of the host.", "We revisit these results by including two new aspects in the analysis.", "First, we obtain incidence rates separately for R-AGN in each type of environment (BCGs, cluster members, and field).", "Second, we consider the incidence rate to be the R-AGN fraction in the eligible galaxy population, not among all galaxies in a given mass (i.e., luminosity) bin.", "The idea is that one is primarily interested in R-AGN incidence among the galaxies that could (and perhaps did or will) host an R-AGN, so we define the eligible population to occupy the same part of the parameter space in redshift, magnitude, color, and axis ratio as galaxies that are current R-AGN hosts.", "Each R-AGN in the sample defines a position in the parameter space around which we determine the R-AGN incidence.", "We take the vicinity to be $R<0.2$ (Eq.", "1) and count the R-AGN used to probe the parameter space as 0.5, which leads to less bias than either not counting it or giving it a full count.", "These fractions are then binned by $M_r$ , and the median of each bin is shown in Fig.", "REF .", "Bins with fewer than five R-AGN were omitted.", "Figure REF shows that while the fraction of eligible galaxies that currently host an R-AGN is generally low, with an overall average of 6%, the incidence increases from close to zero to a high near $\\sim $ 20% for the most luminous galaxies.", "The incidence is generally higher in BCGs, as might be expected, while the incidence in cluster members is only slightly greater than that of field galaxies.", "This suggests that fueling is primarily a locally determined process and that being in a more massive cluster halo confers modest additional benefits, unless a galaxy sits in the center of the cluster halo, as is the case for BCGs.", "Figure: Fraction of galaxies that are R-AGN, for different environments.", "The fraction is calculated as the median of individual incidence estimates.", "These estimates are the ratio of galaxies that are R-AGN among galaxies that have similar properties (redshift, magnitude, color, axis ratio), i.e.", "galaxies eligible to be R-AGN.", "The R-AGN fraction is higher in BCG's, with cluster members having a slightly higher incidence than field galaxies.", "[11] show a similar relation for R-AGN incidence but as a function of stellar mass in panel a of their Fig.", "2.", "The mean NVSS radio luminosity of our sample is $2\\times 10^{24}$ W Hz$^{-1}$ at 1.4 GHz, so our results are best compared to the middle curve of panel a.", "The R-AGN incidence of [11] is expressed as a fraction of all galaxies from SDSS DR2 without regard for the galaxies' likelihood of hosting R-AGN, while we have only considered galaxies from the same type of environment that are likely to host R-AGN.", "Even so, considering that most R-AGN are field galaxies and the majority of massive galaxies are eligible as R-AGN, the results for the field galaxies and cluster members in our sample follow a trend similar to that of [11] for $10^{24}$ W Hz$^{-1}$ at 1.4 GHz.", "[12] found that at a stellar mass of $\\sim 5\\times 10^{11}M_$ the R-AGN incidence in BCGs is only slightly greater than that of other galaxies, although for masses lower than $\\sim \\times 10^{11}M_$ the R-AGN incidence in BCGs is over an order of magnitude greater than that of other galaxies.", "This is in agreement with what we find in Fig REF .", "[12] also found that except for group and cluster galaxies within $0.2r_{200}$ of the center of the system, the R-AGN incidence among non-BCG group and cluster galaxies is similar to that of field galaxies.", "This was based on a comparison of the R-AGN sample defined by [10] to all galaxies in DR4 within the same redshift range as the [10] R-AGN sample, regardless of these galaxies' likelihood of hosting an R-AGN.", "These results are in agreement with our finding that field and cluster member R-AGN have similar incidence rates." ], [ "The role of satellite populations in R-AGN triggering", "We have found that in all environments R-AGN have more nearby satellites than radio-quiet galaxies, but our results are not a simple confirmation of what other studies have found regarding the R-AGN clustering.", "[52], [40], [24], and [36] find that R-AGN are more clustered than optical AGN or radio-quiet galaxies.", "The control samples of [40], [24], and [36] were even chosen to match the stellar masses of the R-AGN.", "However, the results of these studies can be interpreted simply to mean that R-AGN prefer denser, cluster environments or, judging by the results in Section REF that show that R-AGN incidence among eligible cluster members and field galaxies is the same, that the type of galaxies that typically host R-AGN (massive ellipticals) are more clustered, a well known result [25].", "By including cluster membership in the selection criteria of control galaxies we have added crucial new information: the satellite population of R-AGN is richer, whether the R-AGN is a field galaxy, cluster member, or the very BCG.", "It is now widely believed that LERGs are triggered by the cooling of small amounts of gas from the hot halos in which they reside [9].", "Since $\\sim $ 80% of the R-AGN in our sample are LERGs, our finding that R-AGN in all environments have an excess of satellites suggests that the availability of hot halo gas is not the only prerequisite for triggering activity in LERGs.", "For example, the dark matter halo masses of the BCGs in our R-AGN and control samples are similar, which would suggest that they have similar quantities of hot halo gas available.", "The difference is that BCGs that host R-AGN have more nearby satellites, which therefore probably play some role in triggering.", "This argument applies to field galaxies and cluster members as well, since the matched pairs of R-AGN and control galaxies have similar halo masses and therefore presumably similar amounts of hot halo gas.", "Again, the difference between R-AGN and radio-quiet galaxies in these environments is the number of nearby satellites.", "Results in Section suggest that the triggering is facilitated by satellites with a wide range of masses and at various projected distances from the host." ], [ "Dark matter halo bias", "While it is tempting to interpret the excess of satellites as related to the presence of R-AGN, we must explore the possibility that even if the control sample is matched in stellar mass, it may be systematically different in halo mass.", "If the abundance of satellites follows the halo mass rather than the stellar mass, the excess of satellites may result from a mismatch in halo masses.", "We have used the dark matter halo catalog of [54] to investigate this possibility.", "This catalog presents two dark matter halo masses for each galaxy: one calculated based on the ranking of group characteristic luminosity, and the other based on the ranking of group stellar mass: we used those based on the group stellar mass.", "This catalog uses the fourth SDSS data release (DR4) and extends out to $z=0.2$ , so only about 1,400 matched pairs of R-AGN and control galaxies were found.", "The mean of the log of R-AGN masses is 0.1 dex greater than that of control galaxies.", "Is this modest systematic offset in halo masses sufficient to explain a factor of 1.5 difference in satellite population?", "Figure REF shows the number of satellites as a function of halo mass for the field control sample in the redshift range $0.15<z<0.2$ .", "There is a trend that galaxies in more massive halos indeed have more satellites.", "However, this rise is not so steep.", "In order to bring the number of satellites from 1.5 (the number observed on average around control sample galaxies) to 2.1 (the number for R-AGN hosts) the mismatch in halo masses would have to be 0.96 dex (a factor of 9) while in reality it is only 0.1 dex (25%).", "In other words, the excess of R-AGN satellites cannot simply be a result of R-AGN having dark matter halos that are systematically more massive than those of control galaxies.", "Figure: Distribution of satellite counts as a function of halo mass for the field sample of control galaxies in the redshift range 0.15<z<0.20.15<z<0.2.", "The solid line is the least-squares fit to the distribution, and the dashed horizontal lines show the average number of satellites for R-AGN (2.1) and control galaxies (1.5).", "For the excess R-AGN satellites to be a result of mismatched halo masses, a difference of 0.96 dex (dotted vertical lines) between R-AGN and controls is necessary.", "Since the actual difference is only 0.1 dex, R-AGN must genuinely have more satellites within 100 kpc than radio-quiet galaxies." ], [ "The effects of R-AGN feedback on the satellite population", "In addition to the triggering of R-AGN, we also wish to examine the satellite population with the goal of understanding the effects of R-AGN interactions (presumably that of the jet) with satellites.", "In the discussion that follows, we use color distributions of satellites to examine two scenarios.", "In the first, R-AGN jets frequently quench star formation in their satellites, while in the second interactions with R-AGN jets lead to induced star formation.", "We study the effects of R-AGN feedback by comparing the difference between $g-r$ color distributions of R-AGN and control satellites.", "Let us first examine how feedback would modify the difference in color distribution between R-AGN and control galaxies in the absence of an overall excess of satellites around R-AGN.", "If neither quenching nor induced star formation are present, the difference would be zero, as in panel a of Fig.", "REF .", "If interactions commonly lead to quenching, there would be a deficit of blue R-AGN satellites accompanied by an increase in the number of red R-AGN satellites relative to the control distribution, as shown in panel b.", "Figure: Schematic plots illustrating the effects of R-AGN feedback on the difference color distribution between R-AGN and control galaxies, assuming no net difference in the number of satellites.", "Panel a shows the case where neither quenching nor induced star formation as a result of jet interactions are common in satellites.", "Panel b shows the result of star formation being quenched in many R-AGN satellites as a result of jet interactions, leading to a deficit of bluer satellites that is matched by an excess of red ones, while panel c shows the result of radio jets inducing star formation in gas clouds that would not otherwise form stars, producing a net excess of blue satellites.", "Panel d shows the result of radio jets enhancing star formation in satellites that are already actively forming stars.Figure: Schematic plots illustrating the effects of R-AGN feedback on the difference color distribution between R-AGN and control galaxies, for scenario in which R-AGN have more red satellites than control galaxies.", "Panel a shows the case where neither quenching nor induced star formation as a result of jet interactions are common in satellites.", "Panel b shows the result of star formation being quenched in many R-AGN satellites as a result of jet interactions, leading to a deficit of bluer satellites that is matched by an excess of red ones, while panel c shows the result of radio jets inducing star formation in gas clouds that would not otherwise form stars, producing a net excess of blue satellites.", "Panel d shows the result of radio jets enhancing star formation in satellites that are already actively forming stars.Next, we consider two scenarios of induced star formation.", "In the first, radio jets induce star formation in gas clouds that would not otherwise be detected as galaxies, i.e.", "induced star formation forms new galaxies.", "Induced star formation would cause R-AGN to have an excess of blue satellites, as shown in panel c. Faint satellite galaxies that increase in luminosity as a result of induced star formation would also lead to an excess of blue satellites.", "In the other scenario, interactions with R-AGN jets may enhance the SFR in satellites that already form stars, making them bluer (panel d).", "In this scenario there is no net excess, but some moderately blue galaxies become bluer, causing a deficit at moderate value colors and an excess at very blue colors.", "However, from Section we know that there is an intrinsic excess of satellites around R-AGN, so the scenarios described above need to be modified to take that into account, as shown in Figure REF .", "As we have seen, this excess is primarily red simply because most satellites are red, so we obtain modified scenarios by adding a red bump to the scenarios described above.", "Now these modified scenarios can be compared to the observations.", "Figure: Plots of the difference between the color distributions of R-AGN and control satellites: panels a-c are for the three environment types and panel d is for HERGs.", "The dotted vertical line at g-r=0.7g-r=0.7 marks the split between blue and red satellites.", "Panels a-c resemble panel a of Fig , indicating that R-AGN neither quench nor induce star formation in their satellites.", "The point labelled MO in panel b shows what would be 5% occurrence of satellites with the color of Minkowski's Object.", "The actual value is much closer to zero, which suggests that R-AGN rarely induce star formation in external gas clouds.", "Quenching as a result of jet interactions is also uncommon.Figure REF shows the difference plots for the three environment types as well as the HERG sample.", "The vertical dotted line at $g-r=0.7$ indicates the split between blue and red satellites such that these plots can be easily compared to Fig.", "REF .", "Panels a$-$ c of Fig.", "REF resemble panel a of Fig.", "REF , with no blue deficit accompanying the red bump.", "This indicates that in any environment, LERGs play no role in quenching star formation in satellites.", "[44] examined whether R-AGN quench star formation in satellite galaxies that lie in the projected path of the radio jets.", "These authors manually examined the satellites of a small sample of 21 FR1 and 72 FR2 sources and classified them, based on their spatial relation to the radio jets as lying inside or outside the jet path.", "It was found that the $u-r$ color distribution of satellites in the path of FR1 radio jets is similar to that of satellites outside the path while satellites in the jet path of FR2s are redder than satellites outside the path.", "[44] therefore concluded that only the high-power FR2 galaxies quench star formation in satellites.", "If we assume that most of the HERGs in our sample present FR2 morphology, then we may compare our findings for HERG satellites to the results of [44].", "Panel d of Fig.", "REF shows that, like R-AGN in all environments, HERGs have an excess of red satellites.", "This excess is not accompanied by a deficit of blue R-AGN satellites, as the sum of the bins bluer than $g-r=0.7$ is 0.04 ($\\pm 0.07$ ).", "Our results indicate that high accretion R-AGN, like low accretion ones, at best only rarely quench star formation in nearby satellites.", "This non-detection of quenching is in apparent disagreement with the results of [44], but note that we have considered quenching on entire satellite populations, while [44] only investigated satellites in the path of the jet.", "If R-AGN frequently induced star formation in their satellites, we would expect an excess of blue satellites between $0\\lesssim g-r\\lesssim 0.6$ in Fig.", "REF .", "Such an excess is not seen for R-AGN in any environment, nor for HERGs.", "The field sample places a limit on the incidence of star formation induction to no more than 1%.", "This provides evidence that star formation is rarely induced by interactions with R-AGN jets.", "We have included in panel b of Fig.", "REF a point showing what would be 5% occurrence of satellites with the color of Minkowski's Object.", "The actual value is consistent with zero, which indicates that R-AGN rarely induce star formation in external gas clouds, nor do they lead to any significant enhancement in existing gas-rich satellites." ], [ "Summary", "We have studied R-AGN triggering by comparing the satellite populations of R-AGN and a control sample of radio-quiet galaxies, which was matched to the R-AGN in luminosity, color, redshift, axis ratio, and cluster membership.", "We also compared these two satellite populations to search for evidence of quenching in satellites as a result of jet interactions as well as jet-induced star formation in satellites.", "We separately analyzed HERGs to determine whether either the triggering or the feedback behaves differently for powerful radio AGN.", "Our conclusions can be summarized as follows: The incidence rate of R-AGN, which we define as the fraction of R-AGN among the galaxies whose properties are such that they could host an R-AGN and are found in a similar environment, is a strong function of luminosity, as found in previous studies that defined the incidence rate in a less restrictive way (e.g., with respect to all galaxies in a mass bin).", "We find that the incidence rate of field R-AGN and cluster member R-AGN is comparable (4% for field, 6% for members on average), while that of BCGs is significantly higher at the same luminosity (14% on average).", "The relative similarity of the incidence rates of field and non-BCG cluster members suggests that some previous studies that find that R-AGN are more clustered on cluster scales ( 1 Mpc) either derive that signal from BCGs, which are indeed more likely to be R-AGN, or from the fact that the type of galaxies that host R-AGN (massive ellipticals) are more likely to be found in clusters.", "The incidence of R-AGN is consequently primarily dependent on the availability of fuel (hot gas) locally, in the individual halos, rather than in the cluster halo.", "LERG R-AGNs belonging to a given environment type (field, cluster member, BCG) have an excess of satellites out to at least 100 kpc in projected distance compared to similar radio-quiet galaxies in respective environments.", "This excess is on average 50% for field R-AGN.", "For many galaxies this excess results in having a massive satellite vs. not having any.", "This finding is similar to the [36] result of an overdensity of R-AGN with respect to optical, radio-quiet AGN, with the important difference that we control for the general type of environment.", "Thus, we find that even R-AGN BCGs have more galaxies in their vicinity than radio-quiet BCGs.", "The excess is not the result of a mismatch in halo masses, but suggests a genuine relation between satellites and R-AGN triggering.", "Because the excess of red satellites is not accompanied by a deficit of blue satellites, we conclude that neither HERGs nor LERGs are instrumental in quenching star formation in their satellites.", "Similarly, since no excess of blue R-AGN satellites is observed we conclude that LERGs and HERGs rarely ($<1\\%$ of cases) induce star formation in satellites.", "Our results shine new light on the scenario in which HERGs are found in sparse environments and are fueled by the accretion of cold gas, while LERGs prefer dense environments and accrete small quantities of hot halo gas.", "We find that the relevant environment is on small scales ($\\sim 100$ kpc).", "Thus, the fueling of LERGs either becomes more probable, or made more extensive and/or stronger by the excess of nearby satellites, which explains for example why not every BCG hosts an R-AGN, despite the dense cluster environment.", "Alternatively, excess satellites may trace an increased infall rate onto hosts that also feeds the gas to the black hole.", "We will address these possibilities in future work.", "Finally, our results regarding the feedback in satellite population have implications on the role of R-AGN feedback in general." ], [ "Acknowledgements", "We thank the referee for constructive comments and suggestions.", "This study uses data from the SDSS Archive.", "Funding for SDSS-III has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, and the U.S. Department of Energy Office of Science.", "The SDSS-III web site is http://www.sdss3.org/.", "SDSS-III is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS-III Collaboration including the University of Arizona, the Brazilian Participation Group, Brookhaven National Laboratory, University of Cambridge, Carnegie Mellon University, University of Florida, the French Participation Group, the German Participation Group, Harvard University, the Instituto de Astrofisica de Canarias, the Michigan State/Notre Dame/JINA Participation Group, Johns Hopkins University, Lawrence Berkeley National Laboratory, Max Planck Institute for Astrophysics, Max Planck Institute for Extraterrestrial Physics, New Mexico State University, New York University, Ohio State University, Pennsylvania State University, University of Portsmouth, Princeton University, the Spanish Participation Group, University of Tokyo, University of Utah, Vanderbilt University, University of Virginia, University of Washington, and Yale University.", "This research made use of the “K-corrections calculator” service available at http://kcor.sai.msu.ru/" ], [ "Photometric Redshifts", "In our study, photometric redshifts (photo-z's) are used to reduce contamination from faint background galaxies.", "A number of photometric redshifts are available in the SDSS database.", "The primary photometric redshifts from DR7, which are stored as Photoz in the DR7 Catalog Archive Server (CAS), are calculated using a technique that combines template fitting with an empirical fitting procedure to estimate the redshift [1].", "For the eighth data release, this method was also used with the inclusion of the galaxy's inclination angle to reduce systematic bias [55].", "These redshifts are stored in the DR8 CAS as Photoz.", "An alternative approach using the random forest technique described in [15] was used to estimate the photometric redshifts which are stored as PhotozRF.", "In order to determine which photometric redshift is the most consistently accurate, we compared the three photometric redshifts from DR7 and DR8 to the spectroscopic redshifts (spec-z's) of a sample of galaxies from the Deep Extragalactic Evolutionary Probe 2 (DEEP2) survey [22].", "We began with the unique redshift catalog from the DEEP2 Data Release 3 (DR3), of which only galaxies with a reliable redshift and within the Extended Groth Strip (EGS) were used.", "These galaxies were then searched for in both DR7 and DR8, and 3,600 were found to have photometric redshifts in both data releases.", "We used the same magnitude cut of $r$ = 22 as was used for the galaxies from SDSS, which left 1,897 galaxies.", "The spectroscopic redshifts of these galaxies from DEEP2 together with the three photometric redshifts are plotted in Fig.", "REF .", "Panel $a$ shows the photometric redshifts reported in DR7, panel $b$ shows the primary photometric redshifts from DR8, while panel $c$ plots the DR8 photometric redshifts derived using the random forest technique.", "In all panels, the spectroscopic redshift is plotted on the abscissa.", "Figure REF shows that for the sample as a whole, all three methods for determining photometric redshifts tend to underestimate redshifts, and this is more pronounced for DR7.", "The scatter between photometric and spectroscopic redshifts is large at about 0.15 dex and is not symmetric as objects at redshifts greater than $\\sim $ 1 have systematically underestimated redshifts.", "Because of this large asymmetric scatter, we decided to use photometric redshifts to only remove objects with the most discrepant (large) redshifts.", "We used the following process to select which photometric redshift to use.", "To simplify the analysis we treat our target galaxies as if they were at a redshift of $z=0.15$ , the mean redshift of the R-AGN sample.", "We then compute the fraction of galaxies in the background (spec-z $>0.15$ ) of this $z=0.15$ “galaxy\" that are identified as background galaxies by their photometric redshifts.", "The large scatter in photometric redshifts means that many galaxies with a photo-z $>0.15$ are in fact at a spec-z $<0.15$ , so we add a 0.15 dex margin and consider as background galaxies those for which photo-z $>0.3$ .", "Of the 1,793 galaxies with spec-z $>$ 0.15, the DR7 Photoz was greater than 0.3 for 663 galaxies, or 37% of the total.", "For the DR8 Photoz, 1,241 galaxies had an estimated redshift greater than 0.3 or 69% of the total, while the DR8 PhotozRF estimated a redshift greater than 0.3 for 1,106 of the galaxies, or 62% of the total.", "We therefore conclude that Photoz from DR8 is the most useful for identifying background galaxies whose redshift is greater than the $z$ = 0.3 redshift limit of our sample of radio galaxies.", "Figure: Photometric redshifts from SDSS versus the spectroscopically determined redshift from DEEP2 (EGS field), plotted with a one-to-one line.", "(aa) Photoz from DR7, (bb) Photoz from DR8, (cc) PhotozRF from DR8.", "Photometric redshifts from DR7 tend to be more overestimated than those from DR8." ] ]	1403.0003
[ [ "Molecular observations of comets C/2012 S1 (ISON) and C/2013 R1\n (Lovejoy): HNC/HCN ratios and upper limits to PH3" ], [ "Abstract We present molecular observations carried out with the IRAM 30m telescope at wavelengths around 1.15 mm towards the Oort cloud comets C/2012 S1 (ISON) and C/2013 R1 (Lovejoy) when they were at 0.6 and 1 au, respectively, from the Sun.", "We detect HCN, HNC, and CH3OH in both comets, together with the ion HCO+ in comet ISON and a few weak unidentified lines in comet Lovejoy, one of which might be assigned to methylamine (CH3NH2).", "The monitoring of the HCN J = 3-2 line showed a tenfold enhancement in comet ISON on November 14.4 UT due to an outburst of activity whose exact origin is unknown, although it might be related to some break-up of the nucleus.", "The set of CH3OH lines observed was used to derive the kinetic temperature in the coma, 90 K in comet ISON and 60 K in comet Lovejoy.", "The HNC/HCN ratios derived, 0.18 in ISON and 0.05 in Lovejoy, are similar to those found in most previous comets and are consistent with an enhancement of HNC as the comet approaches the Sun.", "Phosphine (PH3) was also searched for unsuccessfully in both comets through its fundamental 1-0 transition, and 3 sigma upper limits corresponding to PH3/H2O ratios 4-10 times above the solar P/O elemental ratio were derived." ], [ "Introduction", "Radio spectroscopic observations of comets during their visit to the inner solar system have allowed to detect a wide variety of molecules in their coma (e.g., 2002).", "These observations have provided significant constraints on the chemical nature of comets coming from the two main solar system reservoirs, the Oort cloud and the Kuiper belt, whose composition is expected to reflect to some extent that of the regions of the protosolar nebula where they were once formed.", "Two bright comets coming from the Oort cloud approached the Sun in late 2013, allowing us to perform sensitive radio spectroscopic observations and to probe their volatile content.", "C/2012 S1 (ISON) –hereafter ISON– was discovered on September 2012 at 6.3 au from the Sun using a 0.4-m telescope of the International Scientific Optical Network ( 2012).", "It is a sungrazing comet, which at perihelion, on 2013 November 28.8 UT, passed at just 0.012 au from the Sun (MPEC 2013-Q27).", "Its orbital elements are consistent with a dynamically new comet, with fresh ices not previously irradiated by sunlight.", "A worldwide observational campaign has extensively followed this comet from heliocentric distances beyond 4 au ( 2013; 2013) to disappearance around perihelion ( 2014).", "C/2013 R1 (Lovejoy) –hereafter Lovejoy– was discovered in September 2013 at $r_h$ = 1.94 au by Terry Lovejoy using a 0.2-m telescope ( 2013).", "This comet reached perihelion on 2013 December 22.7 UT.", "According to its orbital elements (MPEC 2014-D13), this is not its first perihelion passage.", "In this Letter we report IRAM 30m spectroscopic observations of the comets ISON and Lovejoy carried out when they were at heliocentric distances of $\\sim $ 0.6 and $\\sim $ 1 au, respectively." ], [ "Observations", "The observations of comets ISON and Lovejoy were carried out with the IRAM 30m telescope during the period 13-16 November 2013.", "At these dates (before perihelion for both comets) ISON spanned a heliocentric distance of 0.67-0.58 au and a geocentric distance of 0.93-0.89 au while Lovejoy was at 1.09-1.06 au from the Sun and 0.43-0.41 au from the Earth.", "The position of the comets was tracked using the orbital elements from JPL HorizonsSee http://ssd.jpl.nasa.gov/horizons.cgi.", "The EMIR 230 GHz dual polarization receiver ( 2012) and the FTS spectrometer ( 2012) were used to obtain spectra in the frequency ranges 249.0-256.7 GHz and 264.7-272.4 GHz with a spectral resolution of 0.2 MHz ($\\sim $ 0.23 km s$^{-1}$ if expressed as equivalent radial velocity).", "Important molecular lines such as HCN $J$ = 3-2, HNC $J$ = 3-2, HCO$^+$ $J$ = 3-2, PH$_3$ 1$_0$ -0$_0$ , and various CH$_3$ OH rotational transitions fall within the spectral range covered.", "Most of the observations were carried out using the wobbler-switching observing mode, with the secondary mirror nutating by $\\pm $ 90$^{\\prime \\prime }$ at a rate of 0.5 Hz.", "Pointing and focus were regularly checked on Mars and nearby quasars.", "Weather conditions were rather poor during 13 and 14 November, with 6-12 mm of precipitable water vapour (pwv), quite good during 15 November (pwv 1-3 mm), and excellent during 16 November (pwv $<$ 1 mm).", "The half-power beam width (HPBW) of the IRAM 30m telescope at the observed frequencies ranges from 8.9 to 9.8$^{\\prime \\prime }$ , and the pointing error is typically lower than 2$^{\\prime \\prime }$ .", "Line intensities were converted from antenna temperature $T_A^*$ to main beam brightness temperature $T_{\\rm mb}$ by dividing by $B_{\\rm eff}$ /$F_{\\rm eff}$ (e.g., 1997), where $B_{\\rm eff}$ is in the range 0.50-0.54 and $F_{\\rm eff}$ is 0.88 at the observed frequencies.", "The data were reduced using the software GILDASSee http://www.iram.fr/IRAMFR/GILDAS.", "The bright HCN $J$ = 3-2 line was observed to monitor cometary activity and to locate the position of maximum molecular emission.", "In both comets the maximum intensity of the HCN $J$ = 3-2 line was found slightly offset from the presumed position of the comet nucleus, at offsets, in (RA, Dec), of ($-8^{\\prime \\prime }, +4^{\\prime \\prime }$ ) for ISON (i.e., in the direction of the tail) and of ($0^{\\prime \\prime }, +5^{\\prime \\prime }$ ) in the case of Lovejoy.", "Due to the more favourable weather conditions during 15 and 16 November, the highest quality data for both comets were acquired during these dates, on November 15.4 UT for ISON and November 16.4 UT for Lovejoy.", "The $T_{\\rm mb}$ rms noise levels reached, per 0.2 MHz channel, were 0.013-0.017 K for ISON and 0.008-0.010 K for Lovejoy, after averaging the two polarizations." ], [ "HCN monitoring", "A strong variation of the HCN $J$ = 3-2 intensity was observed in comet ISON during November 13.4-16.4 UT, with a tenfold intensity enhancement from Nov. 13.4 to 14.4 UT and a progressive decline afterwards (see Fig.", "REF and Table REF ).", "The outburst of activity of comet ISON on Nov. 14 was reported by various teams observing at different wavelengths.", "The dramatic increase of the production rate of HCN reported by (2013) and presented here was matched by enhancements in the production rates of other molecules such as OH, CN, and C$_2$ ( 2013; 2013a), an increase in the visual brightness, and the appearance of wings in the coma, which may suggest that the outburst was caused by some splitting of the nucleus ( 2013).", "Another outburst of activity was reported on Nov. 19 by (2013b).", "Whether these outbursts were caused by nucleus splitting, delayed sublimation (e.g., 2009), a change in the orientation of the rotation axis, or some other reason is not clear.", "In the case of the comet Lovejoy, the intensity of the HCN $J$ = 3-2 line remained nearly constant (within 10 %) from Nov. 13.4 to 16.4 UT (see Fig.", "REF and Table REF ).", "Table: HCN JJ = 3-2 line parameters and HCN production rates" ] ]	1403.0463
[ [ "Dynamics and control of fast ion crystal splitting in segmented Paul\n traps" ], [ "Abstract We theoretically investigate the process of splitting two-ion crystals in segmented Paul traps, i.e.", "the structural transition from two ions confined in a common well to ions confined in separate wells.", "The precise control of this process by application of suitable voltage ramps to the trap segments is non-trivial, as the harmonic confinement transiently vanishes during the process.", "This makes the ions strongly susceptible to background electric field noise, and to static offset fields in the direction of the trap axis.", "We analyze the reasons why large energy transfers can occur, which are impulsive acceleration, the presence of residual background fields and enhanced anomalous heating.", "For the impulsive acceleration, we identify the diabatic and adiabatic regimes, which are characterized by different scaling behavior of the energy transfer with respect to time.", "We propose a suitable control scheme based on experimentally accessible parameters.", "Simulations are used to verify both the high sensitivity of the splitting result and the performance of our control scheme.", "Finally, we analyze the impact of trap geometry parameters on the crystal splitting process." ], [ "Introduction", "Linear crystals of ions trapped in linear Paul traps have allowed for ground-breaking experiments in the fields of quantum computation, quantum simulation and precision measurements [1].", "Segmented, micro-structured Paul trap arrays have been proposed as a future hardware platform for scalable quantum information experiments [2].", "Small groups of ions are trapped separately from each other, such that precise manipulation of the qubits can be accomplished.", "Experimental protocols then require ion shuttling operations, in addition to laser- or microwave-driven logic gates.", "Essential shuttling operations are splitting and merging of linear ion crystals.", "It is important that they are fast on the typical timescale for quantum gates of 10-100$\\mu $ s, and in order to allow for gate operations or readout after the splitting, a low energy transfer is required.", "Shuttling of trapped ions in segmented traps has been realized within a few oscillation cycles of the harmonic trap by time-dependent control of the trap voltages [3], [4], at energy transfers below one motional quantum.", "Crystal splitting in a segmented trap was first demonstrated in Ref.", "[5], at energy transfers of about 140 phonons within a splitting time of 10 ms. With optimizations, splitting has been included to the set of methods for quantum computing, e.g.", "for quantum teleportation [6] and entanglement purification [7].", "Currently, the best reported result is a gain of about two vibrational quanta per ion at a time duration of 55 $\\mu $ s [4].", "The experimental challenge for the control of this process is given by the fact that the harmonic part of the electrostatic trap potential has to change its sign during this process and therefore has to cross zero.", "This situation of weak confinement reduces the attainable speed and potentially increases the final motional excitation.", "In order to make the process more robust and faster, it is desirable to achieve a large quartic component of the axial trapping potential.", "Trap geometries tailored to improve splitting performance were investigated in [8].", "Optimized geometry parameters for surface electrodes traps were derived in Ref.", "[9].", "In Ref.", "[10], robust splitting operations on slow timescales were carried out by means of real-time observation of the ion positions and feedback on the segment voltages.", "In this work, we analyze the splitting process with the aim of achieving low energy transfers in segmented miniaturized Paul traps.", "We reduce our analysis to the process of splitting ion crystals, as the process of merging ion crystals is merely the time reversed process.", "Furthermore, we restrict ourselves to the case of two ions.", "For splitting and merging processes with several ions, the general procedures and conclusions are still valid.", "The manuscript is organized as follows: In Sec.", ", we introduce the formalism for describing the electrostatic potentials during the splitting operations and the equilibrium positions of the ions, and we analyze the dependence of the equilibrium positions on the control parameters.", "In Sec.", ", we give a detailed explanation of the possible reasons for high energy transfers.", "Based on these considerations, a procedure for the design of suitable voltage ramps is given in Sec.", ".", "In Sec.", ", we analyze the performance of these ramps by numerical simulations.", "Finally, in Sec.", ", we compare typical examples for trap geometries and discuss the implication for ion splitting.", "We desire to split a two-ion crystal residing at center segment $C$ along the trap axis $x$ , to obtain two ions stored in separated potential wells at the position of the splitting segments $S$ neighboring $C$ , see Fig.", "REF .", "Figure: The process of ion crystal splitting.", "It is shown schematically how two ions are moved from the initial center segment CC to different destination segments S R,L S_{R,L} by changing a confining electrostatic potential from a) a strong harmonic confining potential (α>0\\alpha >0) via b) a predominantly quartic potential (α≈0\\alpha \\approx 0) to c) a double-well potential (α<0\\alpha <0).", "The external potential is determined by the voltages applied to the respective electrodes.", "The equilibrium positions are sketched as dashed lines.", "The outer electrodes OO facilitate the splitting process by increasing the transient quartic confinement and offer the possibility to cancel a possible axial background field by application of a differential voltage.", "The color coding of the segments and the corresponding voltages is used throughout the manuscript.Note that we consider only the spatial dimension along the trap axis, as we assume that tight radial confinement persists throughout the process and the ions are always located on the rf node of the trap.", "Typical distances between segments range between 50 and 500 $\\mu $ m, while the initial ion distance is 2-4 $\\mu $ m. The total external electrostatic potential along the trap axis can be written as $\\Phi (x)\\approx \\beta ~x^4+\\alpha ~x^2+\\gamma ~x$ where the coefficients $\\alpha ,\\beta ,\\gamma $ are given by the the trap geometry and the voltages applied to the trap segments.", "This Taylor approximation is valid as long as the the ions are located sufficiently close to $x=0$ , which is the center of the $C$ segment.", "Throughout the splitting process, the external potential is changing from a single well potential $\\alpha _i>0$ to a double well potential $\\alpha _f<0$ , crossing the critical point (CP) at $\\alpha =0$ .", "Note that $\\beta >0$ is required to guarantee confinement at $\\alpha \\le 0$ .", "The approximation of Eq.", "REF holds for $\\alpha \\ge 0$ and for $\\alpha \\lesssim 0$ as long as the separation of the two potential wells is small compared to the width of segment $C$ .", "When the distance of the ions from the center of the $C$ segment becomes comparable to the width of the segment, anharmonic terms of order $>4$ contribute significantly to the total potential.", "These are not taken into account here since the outcome of the splitting process is determined around the CP, as will be pointed out in the following sections.", "Furthermore, beyond the CP, the distance of the separated wells is still increasing monotonically for decreasing $\\alpha $ as long as the variation $\\beta $ is sufficiently small, and the corresponding trap frequencies in these wells are monotonically increasing.", "For studies which require precision beyond the CP, the higher order terms can be taken into account numerically.", "A cubic term does not contribute to the potential if the trap is sufficiently symmetric along the trap axis.", "Including Coulomb repulsion, the total electrostatic potential of a two-ion crystal at a center-of-mass position $x_0$ and distance $d$ is given by $\\Phi _{tot}(x_0,d)=\\Phi (x_0+d/2)+\\Phi (x_0-d/2)+\\frac{\\kappa }{d},$ with $\\kappa =e/4\\pi \\epsilon _0$ .", "At the CP, the harmonic confinement vanishes, and a weak residual confinement is maintained by the interplay between Coulomb repulsion and quartic part of the external potential.", "It is therefore desirable to maximize $\\beta $ at the CP.", "For a given trap geometry, the attainable $\\beta $ is limited by the voltage range which can be applied to the trap electrodes The maximum voltage is ultimately limited by the electric breakdown threshold.", "In practice, as precisely controlled time-dependent voltage waveforms are to be applied to the trap segments, the voltage range will be determined by the electrical design, where one faces a trade-off between voltage range and output bandwidth [11], [12]..", "The coefficients of the potential Eq.", "REF are given by the segment bias voltages and the electrostatic properties of the trap: $\\alpha &=&U_C~\\alpha _C+U_S\\alpha _S+U_O\\alpha _O \\\\\\beta &=&U_C~\\beta _C+U_S\\beta _S+U_O\\beta _O \\\\\\gamma &=&\\Delta U_S\\gamma _S+\\Delta U_O\\gamma _O+\\gamma ^{\\prime }$ An offset parameter $\\gamma ^{\\prime }$ is introduced for taking trap non-idealities – leading to a symmetry breaking force along the trap axis – into account, see Sec.", "REF .", "In contrast to the symmetric quadratic and quartic contributions, the asymmetric tilt potential is controlled by the differential voltages $\\Delta U_{S,O}$ between the corresponding left and right electrodes of the respective pair.", "The segment coefficients are given by Taylor expansions of the standard potentials $\\phi _n(x)$ , which are the dimensionless electrostatic potentials along the trap axis if a +1V bias is applied to segment $n$ and all other segments are grounded [13], [14]: $\\phi _{n,m}(x)=\\phi _n\\vert _{x_0^{(m)}}+\\phi _n^{\\prime }\\vert _{x_0^{(m)}}\\delta x+\\frac{1}{2}\\phi _n^{\\prime \\prime }\\vert _{x_0^{(m)}}\\delta x^2+\\frac{1}{24}\\phi _n^{(4)}\\vert _{x_0^{(m)}}\\delta x^4+\\mathcal {O}\\left(\\delta x^6\\right).$ with $\\delta x=x-x_0^{(m)}$ , i.e.", "the Taylor expansions are carried out at center of segment $m$ , $x_0^{(m)}$ .", "The coefficients for Eqs.", "REF ,, are obtained for $m=C,n=C,S,O$ : $\\alpha _n=\\frac{1}{2}f_n\\phi _{n,C}^{\\prime \\prime }(0), \\qquad \\beta _n=\\frac{1}{24}f_n\\phi _{n,C}^{(4)}(0), \\qquad \\gamma _n=f_n\\phi _{n,C}^{\\prime }(0),$ with $f_{C}=1$ and $f_{S,O}=2$ accounting for two $S,O$ segments acting symmetrically at $x=0$ .", "Note that $\\gamma _C=0$ by definition.", "In the following, for numerical calculations, we use the specific geometry parameters of a three dimensional microstructured segmented ion trap A as detailed in Sec..", "There, other traps and their geometry parameters are listed and analyzed as well." ], [ "Equilibrium positions", "We consider two ions of mass $m$ and charge $e$ , with their equilibrium positions given by the center-of-mass $x_0$ and the equilibrium distance $d$ : $x_{L,R}=x_0\\pm d/2,$ determined by minimizing of the total electrostatic potential Eq.", "REF .", "The confinement is characterized by the local trap frequency, which is given by the curvature of of the external potential at the ion positions: $\\omega =\\sqrt{\\frac{e}{m}\\Phi ^{\\prime \\prime }(x_{L,R})}.$ The extremal points of the external potential Eq.", "REF are given by $x_0^{(0)}&=&\\frac{\\alpha }{3^{1/3}\\zeta }-\\frac{\\zeta }{2\\cdot 3^{2/3}\\beta } \\\\x_0^{(\\pm )}&=&\\frac{(i\\sqrt{3}\\pm 1)\\alpha }{2\\cdot 3^{1/3}\\zeta }+\\frac{(1\\mp i\\sqrt{3})\\zeta }{4\\cdot 3^{2/3}\\beta } \\\\$ where $\\zeta (\\alpha ,\\beta ,\\gamma )=\\left(9\\beta ^2\\gamma +\\sqrt{3}\\sqrt{8\\alpha ^3\\beta ^3+27\\beta ^4\\gamma ^2}\\right)^{1/3}.$ Initially, at $\\alpha =\\alpha _i$ , the confining harmonic part of the external potential and the Coulomb repulsion are dominant, thus we can neglect the quartic potential.", "The trap frequency is then given by $\\omega ^2=2\\alpha e/m$ at an ion distance of $d=\\left(\\kappa /\\alpha \\right)^{1/3}$ .", "At the CP, $\\alpha =0$ , and without tilt, $\\gamma =0$ , the ion distance is determined by quartic confinement and Coulomb repulsion: $d_{CP}=\\left(2\\kappa /\\beta \\right)^{1/5}.$ The Coulomb repulsion pushes the ions away from the trap center (where the curvature of the external potential vanishes), such that a residual harmonic confinement persists because of the quartic term.", "The minimum trap frequency during the splitting process is thus given by [8] $\\omega _{CP}=\\beta ^{3/10}\\left(3e/m\\right)^{1/2}\\left(2\\kappa \\right)^{1/5}.$ Near the CP, the equilibrium distance can be computed from a perturbative expression up to second order: $d(\\alpha )\\approx d_{CP}-\\frac{1}{5}\\left(\\frac{16}{\\beta ^4\\kappa }\\right)^{1/5}\\alpha +\\frac{2}{25}\\left(\\frac{4}{\\beta ^7\\kappa ^3}\\right)^{1/5}\\alpha ^2,$ for $\\vert \\alpha \\vert \\ll \\beta d_{CP}^2$ and $\\vert \\alpha \\vert \\ll \\kappa d_{CP}^{-3}$ .", "The center-of-mass position of the ion crystal near the critical point to first order in the tilt parameter $\\gamma $ is: $x_0(\\alpha ,\\gamma )\\approx \\gamma \\left(-\\frac{1}{3\\cdot 2^{2/5} \\beta ^{3/5} \\kappa ^{2/5}}-\\frac{2^{1/5}}{45\\cdot \\beta ^{6/5} \\kappa ^{4/5}}\\alpha +\\frac{26\\cdot 2^{4/5}}{675\\beta ^{9/5} \\kappa ^{6/5}}\\alpha ^2\\right)$ If the ions are sufficiently separated, $\\alpha \\ll 0$ , the Coulomb repulsion can be neglected and the equilibrium positions approximately coincide with the extrema of the external potential: $d_f=\\sqrt{-2 \\alpha _f/\\beta }$ and the final trap frequency is given by $\\omega _f^2=-4\\alpha _f e/m$ ." ], [ "Critical tilt value", "A static background force along the trap axis can to keep the ions confined in one common potential well throughout the splitting process.", "We make use of the external potential minima Eqs.", "REF to obtain an estimate for the tilt parameter $\\tilde{\\gamma }$ , beyond which the splitting ceases to work.", "In the following, we assume $\\gamma >0$ .", "Figure: Critically tilted potential, see text such that the Coulomb repulsion fails to push the right ion across the saddle point.In the presence of a nonzero potential tilt, an imperfect bifurcation occurs, i.e.", "the second potential well opens up at $\\tilde{\\alpha }<0$ , see Fig.", "REF c).", "We obtain a scaling law for $\\tilde{\\gamma }$ by calculating at which tilt parameter the original potential well is deep enough to keep both mutually repelling ions confined, see Fig.", "REF .", "The saddle point where the second potential well opens can be found by solving $x_{0,c}^{(0)}=x_{0,c}^{(+)}$ for $\\tilde{\\alpha }$ , yielding $\\tilde{\\alpha }=-{\\textstyle \\frac{3}{2}}\\beta ^{1/3}\\vert \\gamma \\vert ^{2/3}$ .", "From this we obtain its position For $\\gamma \\ge 0$ , $x_0^{(0)}$ corresponds to the left potential minimum which always exists, and for $\\alpha <\\tilde{\\alpha }<0$ , $x_0^{(+)}$ corresponds to the right potential minimum and $x_0^{(-)}$ corresponds to the maximum of the separation barrier.", "By contrast, for $\\gamma <0$ , $x_0^{(0)}$ corresponds to the right potential minimum, and for $\\alpha <0<\\tilde{\\alpha }$ , $x_0^{(+)}$ corresponds to the left minimum.", "to be $x_{0,c}^{(+,0)}={\\textstyle \\frac{1}{2}}\\left(\\gamma /\\beta \\right)^{1/3}$ .", "At $\\tilde{\\alpha }$ , the left potential minimum is located at twice the distance from the origin $x_{0,c}^{(-)}=-\\left(\\gamma /\\beta \\right)^{1/3}$ .", "The potential attains the same value as on the saddle point $V(x_{0,c}^{(+,0)})$ at the position $\\tilde{x}_{c}^{(+)}=-{\\textstyle \\frac{3}{2}}\\left(\\gamma /\\beta \\right)^{1/3}$ .", "The depth of the potential well defined by the saddle point when the right well opens is therefore $\\Delta V_{c}=V(x_{0,c}^{(-)})-V(x_{0,c}^{(+,0)})=\\frac{27}{16}\\left(\\frac{\\gamma ^4}{\\beta }\\right)^{1/3}.$ We can now define a criterion which determines whether the ions are actually separated by comparing the Coulomb potential to the depth of the initial well at the CP, Eq.", "REF : If the Coulomb repulsion pushes the right ion beyond the saddle point $x_{0,c}^{(+,0)}$ , it will end up in the right potential well, otherwise the two ions will stay in the left well.", "Thus, the Coulomb energy at an ion distance of $x_{0,c}^{(+,0)}-\\tilde{x}_{c}^{(+)}$ has to be larger than the well depth $\\Delta V_{c}$ .", "These considerations lead to a critical tilt value of $\\tilde{\\gamma } < \\pm ~C_{\\gamma }\\left(\\kappa ^3 {\\beta ^2}\\right)^{1/5}.$ Despite the fact that the situation depicted Fig.", "REF does not actually occur, as the external force at the saddle point vanishes and therefore cannot balance the Coulomb force, the obtained scaling behavior is confirmed by numerical calculations, revealing a prefactor of $C_{\\gamma }=$ 1.06.", "The result Eq.", "REF enables us to determine the required degree of precision by which the background axial field has to be corrected.", "For this calculation, only the geometry parameters $\\beta _{C,S,O}$ are needed.", "Furthermore, the sensitivity decreases as $\\beta ^{2/5}$ , which directly characterizes the gain in robustness when the accessible voltage range is enhanced.", "For trap A (Sec.", "), we derive a value of $\\tilde{\\gamma }\\approx 3V/m$ , corresponding to the requirement to set $\\Delta U_O$ more accurately than about 9 mV." ], [ "Impulsive acceleration at the critical point", "A naïve approach towards crystal splitting is the linear interpolation between two voltage sets pertaining to a single well and a double well, leading to a constant variation rate of the harmonic coefficient $\\alpha $ .", "As this does not involve a dedicated control of the ion distance, it is equivalent to a rapid sweep across a structural transition of the ion crystal.", "This leads to an unfavorable power-law scaling of the energy transfer with respect to the sweep time [15], which prevents attaining adiabaticity.", "In the following, we derive an approximation for the energy transfer, assuming the variation of $\\alpha $ around the CP to be uniform.", "We consider the energy transfer to be caused by impulsive displacement: At the CP, the equilibrium distance changes most rapidly, while the confinement - and therefore the restoring forces - are reduced.", "Fig.", "REF a) shows that the situation corresponds to a harmonic oscillator which is suddenly dragged at uniform speed, causing displacement and therefore a gain in potential energy.", "Within the characteristic timescale set by half a the trap oscillation cycle $\\tau _{CP}=\\pi /\\omega _{CP}$ , this yields the displacement: $\\delta d_{CP} &\\approx & \\dot{d}_{CP} \\;\\xi \\; \\tau _{CP}/2\\\\&\\approx & \\left.\\frac{\\partial d}{\\partial \\alpha }\\right\|_{CP} \\dot{\\alpha }_{CP} \\;\\xi \\; \\tau _{CP}/2\\\\&\\approx & \\left(\\beta _{CP}^4~\\kappa \\right)^{-1/5} \\dot{\\alpha }_{CP} \\;\\xi \\; \\tau _{CP}/2,$ where Eq.", "REF was used in the last line.", "The factor $\\xi $ accounts for the fact that the trap frequency increases beyond the CP, such that the restoring forces set in before $\\tau _{CP}$ and the resulting displacement is reduced.", "This sudden displacement mechanism is sketched in Fig.", "REF a).", "The potential energy of an ion is consequently increased by $\\delta E &= & \\frac{1}{2}m\\omega _{CP}^2 \\left( \\delta d_{CP}/2\\right)^2 \\\\&= & \\frac{\\pi ^2}{8} \\;\\xi ^2 \\;m \\left(\\beta _{CP}^4 ~\\kappa \\right)^{-2/5} \\dot{\\alpha }^2_{CP},$ which serves as an approximation of the final energy transfer.", "For a sufficiently small $\\vert \\dot{\\alpha }\\vert _{CP}$ , adiabaticity sets in and the energy transfer scales exponentially with the splitting time.", "The reason for this is that the Coulomb repulsion serves to push the ions outwards, providing smooth variation of the equilibrium distance as compared to discontinuous behavior of the minima of the external potential, see Fig.", "REF b).", "It therefore leads to rapid, but continuous variation of the equilibrium positions with $\\alpha $ .", "The onset of the adiabatic regime is identified by comparing displacement $\\delta d_{CP}$ to the change of the equilibrium distance within $\\tau _{CP}$ below the CP (see Fig.", "REF a)), which means that the ion acceleration around the CP is sufficiently slow to prevent sudden displacement.", "We therefore compare the acceleration $\\ddot{d}_{CP}$ to the reference acceleration $d_{CP}\\omega _{CP}^2$ , yielding the adiabaticity parameter $\\chi &=&\\frac{\\ddot{d}_{CP}}{d_{CP}\\omega _{CP}^2} \\\\&=&\\frac{4}{25}\\frac{m}{3e}2^{-1/5}\\beta _{CP}^{-9/5}\\kappa ^{-6/5}\\dot{\\alpha }_{CP}^2$ In the adiabatic regime, $\\chi <1$ , the energy transfer is given by: $\\delta E^{\\prime }\\approx \\delta E \\exp \\left[c^2 \\left(1-\\frac{1}{\\chi }\\right)\\right]$ Numerical simulations are carried out for different constant values for $\\beta $ and a linear variation of $\\alpha $ around the CP.", "The results are shown in Fig.", "REF b).", "It can be seen that the approximations Eqs.", "REF ,REF hold over a wide range of splitting times and quartic coefficients, and that large energy transfers in the regime of 10$^4$ -10$^6$ phonons are readily obtained.", "The simulations yield a value of $\\xi ^2\\approx $ 0.1.", "We conclude that in this regime, the energy transfer depends only on the ion mass, the variation rate of $\\alpha $ and the quartic confinement at the CP.", "As can be seen from the simulation results, still large energy transfers are obtained at the onset of adiabaticity, such that splitting at energy transfers on the single phonon level would require splitting times on the order of several hundreds of $\\mu $ s. As we will show in further sections, this problem can be overcome using ramps that ensure a small ion acceleration $\\ddot{d}_{CP}$ at the CP.", "Note that $\\ddot{d}_{CP}=\\left.\\frac{\\partial ^2 d}{\\partial \\alpha ^2}\\right\|_{CP}\\dot{\\alpha }_{CP}^2+\\left.\\frac{\\partial d}{\\partial \\alpha }\\right\|_{CP}\\ddot{\\alpha }_{CP}.$ For sufficiently uniform variation of $\\alpha $ , the second term can generally be neglected, such that by using Eq.", "REF , we obtain $\\ddot{d}_{CP}=\\frac{2}{25}\\left(\\frac{4}{\\beta _{CP}^7\\kappa ^3}\\right)^{1/5}\\dot{\\alpha }_{CP}^2.$ Thus, the energy transfer can be reduced by ensuring a small variation rate of $\\alpha $ at the CP." ], [ "Uncompensated potential tilt", "A residual static force along the trap axis, expressed by the coefficient $\\gamma ^{\\prime }$ in Eq.", ", can originate from stray charges, laser induced charging of the trap [16], trap geometry imperfections or residual ponderomotive forces along the trap axis.", "The behavior of the equilibrium positions in the presence of an imperfectly compensated tilt, shown in Fig.", "REF , reveals a discontinuity for the critical $\\tilde{\\gamma }$ , leading to diverging acceleration.", "The divergence of the acceleration impedes us to perform the splitting process adiabatically for $\\vert \\gamma \\vert \\lesssim \\tilde{\\gamma }$ , i.e.", "the voltages can not be changed sufficiently slow to suppress motional excitation.", "Thus, one might encounter the situation that the tilt is sufficiently well compensated to allow for splitting, but sufficiently low excitations cannot be obtained irrespectively of the splitting time and other control parameters.", "For small tilt parameters, $\\vert \\gamma \\vert \\ll \\tilde{\\gamma }$ , we can employ the perturbative expressions Eqs.", "REF , REF of the equilibrium positions to obtain $\\frac{\\partial ^2 x_{R,L}}{\\partial \\alpha ^2}=\\frac{\\partial ^2 x_0}{\\partial \\alpha ^2}\\pm \\frac{1}{2}\\frac{\\partial ^2 d}{\\partial \\alpha ^2}=\\gamma \\frac{52\\cdot 2^{4/5}}{675\\beta ^{9/5} \\kappa ^{6/5}}\\pm \\frac{2}{25}\\left(\\frac{4}{\\beta ^7\\kappa ^3}\\right)^{1/5}$ We can estimate the tilt parameter at which the acceleration of one of the ions is twofold compared to the tilt-free case determined by Eq.", "REF to be about 67% of the critical tilt $\\tilde{\\gamma }$ .", "Due to the divergence of the acceleration at $\\tilde{\\gamma }$ , we can expect the actual acceleration at this tilt value to be substantially larger, we thus conclude that a residual tilt $\\vert \\gamma \\vert \\ll \\tilde{\\gamma }$ is required to realize crystal splitting at low motional excitation.", "A possible experimental scheme for this has been demonstrated in [10]: The separation process is performed on a slow (second) timescale under continuous Doppler cooling and detection.", "The ion positions are extracted from the camera image, and a deviation of the center-of-mass from the initial value is restored by automatic adjustment of the outer electrode differential voltage $\\Delta U_O$ ." ], [ "Anomalous heating at the critical point", "Microstructured ion traps exhibit anomalous heating, i.e.", "the mean phonon number increases due to thermalization with the electrodes at a timescale much faster than predicted by the assumption that only Johnson-Nyquist noise is present [17].", "This process can be modeled as $\\dot{\\bar{n}}=\\Gamma _h$ , with the heating rate $\\Gamma _h(\\omega )= S_E(\\omega ) e^2 /4m\\hbar \\omega $ where the spectral electric-field-noise density $S_E$ depends on the trap frequency $\\omega $ .", "A polynomial decrease $S_E\\propto \\omega ^{-a}$ is often assumed, where experimentally determined values for the exponent $a$ range from 0.5 to 2.5.", "Additionally, peaked features might arise in the noise spectrum which are caused by technical sources.", "Moreover, the absolute values of the heating rates strongly depend on the properties of the electrode surfaces.", "Typical values at trap frequencies in the 1 MHz regime range from 0.1 to tens of phonons per millisecond.", "As the trap frequency is strongly decreased around the CP, we can expect a significant amount of excess energy after the splitting caused by anomalous heating, increasing for longer splitting durations.", "We model this contribution by integrating over a time dependent heating rate: $\\Delta \\bar{n}_{th}=\\int _0^T \\Gamma _h\\left(\\omega (t)\\right) dt.$ For the simulations that follow we will employ an experimentally determined relation for trap A (Sec. )", "which is $\\Gamma _h(\\omega )\\approx 6.3\\cdot \\left(\\omega /2\\pi \\textrm {MHz}\\right)^{-1.81} ms^{-1}$ .", "This does not depend on the geometry of trap A but on the properties of our trap apparatus.", "In the case of imperfect control of the ion distance around the CP, Sec.", "REF , or in the presence of an uncompensated tilt, Sec.", "REF , one will attempt to reduce the motional excitation by splitting very slowly.", "This might however be unsuccessful as anomalous heating will strongly contribute to the energy gain at large splitting times.", "Experimental procedures for ensuring a sufficient degree of control are therefore ultimately required." ], [ "Voltage ramps", "In this section we explain our scheme for designing voltage ramps for the splitting process.", "Our intention is to provide a scheme which can be applied any given trap geometry.", "We do explicitly not rely on the precise knowledge of the electrostatic trap potentials, but rather on quantities which can be measured with reasonable effort.", "Furthermore, we describe how a single voltage level can be used as a tuning parameter to achieve the optimum result.", "Our scheme assumes that the tilt potential is perfectly compensated, $\\gamma =0$ .", "We proceed as follows: We first describe how the segment voltages are supposed to vary with the harmonicity parameter $\\alpha $ , where we simply fix voltage levels on a small set of mesh points.", "We then show how this is used in conjunction with a chosen distance-versus-time and available distance-versus-$\\alpha $ information to obtain time-domain voltage ramps which can be employed in the experiment." ], [ "Static voltage sets", "The calculation of suitable voltage ramps relies on the signs and on the magnitude ordering of the geometry parameters.", "In Table REF we list values for several different microstructured traps.", "We assume that any reasonable segmented trap geometry will exhibit similar characteristics.", "From the results of Sec.", ", it is clear that we desire a large positive value of $\\beta _{CP}$ .", "We assume that the voltages which can be applied to the segments are limited by hardware constraints to the symmetric maximum/minimum values $\\pm U_{lim}$ .", "To achieve the largest possible $\\beta $ at the CP, we begin the splitting protocol by ramping the $O$ segments to $+U_{lim}$ , keep them at constant bias during around the CP, and ramp them back to zero bias after the splitting.", "The CP is defined by the condition $\\alpha =0$ , which is accomplished by suitable voltages $U_{C,S}$ .", "This leaves one degree of freedom, which can be eliminated by maximizing $\\beta _{CP}$ .", "We solve Eq.", "REF for $U_C$ : $U_C=\\frac{1}{\\alpha _C}\\left(\\alpha -\\alpha _O U_O-\\alpha _S U_S\\right).", "$ The largest possible $\\beta _{CP}$ is then given by inserting this result into Eq.", "and setting $U_O^{(CP)}=+U_{lim},U_S^{(CP)}=-U_{lim}$ : $\\max _{U_C,U_S}\\beta _{CP}=\\left(\\beta _O+\\frac{\\beta _C}{\\alpha _C}\\alpha _S-\\beta _S-\\frac{\\beta _C}{\\alpha _C}\\alpha _O\\right)U_{lim}$ Static splitting voltage sets are obtained by fixing the initial, CP and final voltage configurations and interpolating between these.", "The procedure consists of the following steps: Determine the initial $\\alpha _i>0$ from Eq.", "REF using the initial voltages $U_C^{(i)}<0$ V, $U_S^{(i)}=U_O^{(i)}=0$ V. Choose the voltages at the CP such that the maximum $\\beta _{CP}$ is attained, by setting $U_O^{(CP)}=+U_{lim},U_S^{(CP)}=-U_{lim}$ and $U_C^{(CP)}$ from Eq.", "REF for $\\alpha =0$ .", "If the geometry parameters are such that $U_C^{(CP)}$ exceeds $\\pm U_{lim}$ , set $U_C^{(CP)}=-U_{lim}$ and obtain $U_S^{(CP)}$ solving Eq.", "REF for $U_S$ rather than $U_C$ .", "If the magnitude of $U_S^{(CP)}$ is chosen smaller than $U_{lim}$ , this leads to smaller values of $\\beta _{CP}$ and a larger ion separation at the CP.", "This offers the possibility for well-controlled studies of the dependence of the splitting process on the quartic confinement at the CP..", "Determine the desired final voltages.", "We choose $U_C^{(f)}=0$ V, $U_S^{(f)}=U_S^{(CP)}=-U_{lim}$ and $U_O^{(f)}=0$ V. This choice is convenient when $U_C^{(i)}\\approx -U_{lim}$ and ensures that the ions are finally kept close to the respective centers of the $S$ segments with a trap frequency similar to the initial one.", "Obtain $\\alpha _f$ from Eq.", "REF .", "For approaching the CP, $\\alpha _i\\ge \\alpha >0$ , set $U_S(\\alpha ) = \\left(1-\\frac{\\alpha }{\\alpha _i}\\right) U_S^{(CP)}$ and $U_O(\\alpha ) = \\left\\lbrace \\begin{array}{ll}{2\\left(1-\\frac{\\alpha }{\\alpha _i}\\right) U_{lim}\\qquad \\alpha > \\frac{\\alpha _i}{2} \\\\ U_{lim}\\qquad \\ \\ \\alpha \\le \\frac{\\alpha _i}{2}\\\\}\\end{array}and obtain \\right.U_C(\\alpha ) from Eq.", "\\ref {eq:UCfromUS}.\\item Beyond the CP, 0\\ge \\alpha \\ge \\alpha _f, set\\begin{equation}U_S(\\alpha )=-U_{lim}\\end{equation}and\\begin{equation}U_O(\\alpha ) = \\left\\lbrace \\begin{array}{ll}{U_{lim}\\qquad \\alpha > \\frac{\\alpha _f}{2} \\\\2\\left(1-\\frac{\\alpha }{\\alpha _f}\\right) U_{lim} \\ \\ \\ \\ \\alpha \\le \\frac{\\alpha _f}{2} \\\\}\\end{array}and obtain \\right.U_C(\\alpha ) from Eq.", "\\ref {eq:UCfromUS}.\\end{equation}$ Figure: Voltage ramp transfer to the time domain: A predefined time-to-distance function d(t)d(t) shown in panel a) is used in conjunction with α\\alpha -to-distance information α(d)\\alpha (d) shown in b) to determine the time-dependent electrode voltages U n (t)U_n(t) using the static voltage sets U n (α)U_n(\\alpha ) from panel c).", "The resulting ramps U n (t)U_n(t) are shown in d).", "The dashed curves are corresponding to the case when the voltage ramps are calculated according to the presented method, but realistic trap potentials from simulations are used to determine d f d_f and d(α)d(\\alpha ).", "The dashed arrows exemplify how a specific value U C U_C is obtained.Time domain ramps We now show how to design suitable time-domain voltage ramps $U_n(t)$ that will assure well-controlled splitting.", "It has been shown in Sec.", "REF that a small value of the acceleration at the CP, $\\ddot{d}_{CP}$ , is required for achieving a low energy transfer.", "This in turn is guaranteed by well-controlled variation of of the distance $d(t)$ throughout the splitting process.", "As $d(\\alpha )$ is monotonically decreasing with $\\alpha $ , it can be inverted to obtain $\\alpha (d)$ which is used to compute the final voltage ramp as $U_n(\\alpha (d(t)))$ (see Fig.", "REF .", ").", "Possible choices for $d(t)$ are a sine-squared ramp $d(t)=d_i+\\left(d_f-d_i\\right)\\sin ^2\\left(\\frac{\\pi t}{2T}\\right)$ or a polynomial ramp $d(t)=d_i+\\left(d_f-d_i\\right)\\left(-10\\frac{t^3}{T^3}+15\\frac{t^4}{T^4}-6\\frac{t^5}{T^5}\\right)$ Both ramps fulfill $d(0)=d_i,d(T)=d_f,\\dot{d}(0)=\\dot{d}(T)=0$ .", "The polynomial ramp, used in the following, additionally fulfills $\\ddot{d}(0)=\\ddot{d}(T)=0$ , while the second derivative of the sine-squared ramp displays discontinuities.", "However, these features presumably play no role in experiments, as the voltage ramps are generally subject to discretization and filtering.", "Different methods can be employed for the determination of $d(\\alpha )$ : The equilibrium distance can be computed by employing realistic trap potentials from simulation data, using the voltage configuration pertaining to a given $\\alpha $ as determined by the static voltage sets $U_n(\\alpha )$ .", "This method requires the simulated potentials to match the actual trap potential with great precision.", "The equilibrium distance can be computed using values from calibration measurements for the coefficients $\\alpha _n,\\beta _n$ .", "This circumvents the need for simulations and accounts for parameter drifts.", "It yields only valid values for distances which are small compared to the electrode width, however we will show in Sec.", "that this procedure yields useful voltage ramps.", "Ion distances can be measured by imaging the ion crystal on a camera, while voltages configurations for decreasing $\\alpha $ values are applied.", "This is the most direct method, and it benefits from the availability of a precise gauge of imaging magnification from measurements of the trap frequency.", "Simulation results In order to analyze the sensitivity of the splitting process and the performance of our ramp design protocol, we numerically solve the classical equations of motion.", "For the time- and energy-scales and potential shapes under consideration, we expect quantum effects to play no significant role.", "For the case of single-ion shuttling, the occurrence of quantum effects is thoroughly discussed in Ref.", "[18].", "We perform the simulations using either the Taylor approximation of the potentials or the realistic potentials from electrostatic simulations [14] for trap A, which is similar to that described in Ref.", "[19].", "The voltage ramps $U_i(t)$ are used in conjunction with the potentials to yield the equations of motion for the ion positions $x_1<x_2$ .", "Employing the Taylor approximation potential Eq.", "REF , these read $-m\\ddot{x}_{1,2}=4\\beta (t) x_{1,2}^3+2\\alpha (t) x_{1,2}+\\gamma \\pm \\frac{\\kappa }{(x_2-x_1)^2},$ where the coefficients are given by using the voltage ramps in Eqs.", "REF , ,.", "For realistic trap potentials, we obtain $-m\\ddot{x}_{1,2}=\\sum _{n=C,S,O} U_n(t) \\left.", "\\frac{d\\phi _n}{dx}\\right\|_{x_{1,2}}\\pm \\frac{\\kappa }{(x_2-x_1)^2}\\\\$ The possibility to perform the simulations with approximate and realistic potentials serves the purpose of verifying the performance of the voltage ramps.", "These are determined purely by trap properties around the CP, which are conveniently accessible by measurements.", "More precisely, the time-domain voltage ramps are based on a $d(\\alpha )$ dependency given by the Taylor approximation potential according to Fig.", "REF , while the resulting energy transfer pertaining to these ramps can be obtained from simulations using realistic potentials.", "Note that a nonzero tilt can be present in the simulations based on the realistic potentials by summing separately over electrodes $O_L$ and $O_R$ and adding the differential voltage $\\pm \\Delta U_O$ given by $\\gamma /\\gamma _O$ accordingly.", "The calculations presented here employ the mass of $^{40}$ Ca$^+$ ions which we use in our experiments, and all simulations were performed for a limiting voltage range $U_{lim}=10$ V. Eqs.", "REF or REF are solved numerically using the NDSolve package from Mathematica, with the ions starting at rest.", "The final oscillation of each ion around its equilibrium position is analyzed and yields the energy transfer expressed as the mean phonon number $\\bar{n}=\\Delta E/\\hbar \\omega _f$ .", "We distinguish several regimes of laser-ion interaction: i) If the vibrational excitation becomes so large that the average Doppler shift per oscillation cycle exceeds the natural linewidth of a cycling transition, ion detection by counting resonance fluorescence photons will be impaired.", "ii) Measurement of the energy transfer i.e.", "by probing on a stimulated Raman transition [3] typically requires mean phonon numbers below about 300. iii) The Lamb-Dicke regime of laser-ion interaction, where coherent dynamics on resolved sidebands can be driven [20] is typically attained below about 10 phonons.", "The borders between these regimes depend on the trap frequency, ion mass and the specific atomic transitions to be driven, thus the regimes are indicated as broad gray bands in Fig.", "REF .", "Note that if final excitations in the measurable regime are obtained, an electrical counter kick can be applied for bringing the oscillation to rest [3].", "Dependence on splitting time We first analyze the dependence of the energy transfer on the duration of the splitting process $T$ , the result is shown in Fig.", "REF .", "The calculation is carried out for the ideal case of perfectly compensated potential tilt.", "We see that the final excitation becomes sufficiently low to remain in the Lamb-Dicke regime for typical laser-ion interaction settings at times larger than about 40 $\\mu s$ , which clearly outperforms the naïve approach of voltage interpolation from Sec.", "REF .", "We also take into account increased anomalous heating around the CP by employing the averaged heating rate according to Eq.", "REF .", "We see that for our specific heating rates, the limit of about one phonon per ion can not be overcome, but as the anomalous heating contribution is scaling as $1/T$ , the splitting result becomes rather insensitive with respect to the precise choice of the $T$ beyond $T=$ 50 $\\mu s$ .", "The simulation results verify our approach of calculating the voltage ramps using the Taylor approximated potentials.", "One recognizes that the resulting energy transfer in this case is larger by a factor of about two throughout the entire range of splitting durations.", "As can be seen from Fig.", "REF , this is due to the fact that the Taylor expansion leads to an incorrect voltage set pertaining to the CP, which in turn leads to uncontrolled acceleration as explained in Sec.", "REF .", "The discrepancy becomes irrelevant for splitting times larger than $T=$ 60 $\\mu $ s. At around 60 to $70~\\mu $ s the oscillatory excitation becomes smaller than $\\bar{n}=0.1$ , corresponding to the limit we can currently resolve in our experiment.", "The slight inaccuracy for low phonon numbers is due to numerical artifacts.", "Even lower energy transfers at shorter $T$ could possibly be achieved by ramp engineering, i.e.", "by the application of shortcut-to-adiabaticity approaches [18], [21].", "Figure: Energy transfer versus splitting time: Oscillatory (red) and thermal excitation (blue), and the sum of both (black) versus the splitting duration TT.", "The solid lines correspond to the calculation using the Taylor approximation, the dashed lines correspond to the full potential calculation, see text.", "Grey bands seperate different regimes of laser-ion interaction, see text.", "The thermal excitation was deduced from experimental heating rate data according to Sec.", ".", "The inset shows the trap frequency (black) and the corresponding heating rate (red) as a function of normalized time during the splitting process.", "Sensitivity analysis Figure: Mean coherent excitation as a function of the offset voltage at the center segment at the CP (a) and the tilt force γ\\gamma (b).", "The tilt voltage +ΔU O +\\Delta U_O is applied to the right outer segment and -ΔU O -\\Delta U_O is applied to the left outer segment.", "The mean phonon number for the right ion is depicted by dashed lines and by solid lines for the left ion.", "The curves correspond to different splitting times: T=60μsT = 60\\mu s (green), T=40μsT = 40\\mu s (black), T=20μsT = 20\\mu s (red).", "The critical tilt is at γ ˜=3\\tilde{\\gamma }=3~V/m.Two crucial parameters for the splitting operation are the offset voltage at the CP $\\Delta U_C^{(CP)}$ and the potential tilt $\\gamma $ .", "Small variations of these parameters lead to strong coherent excitations as shown in Fig.", "REF .", "The CP voltage offset $\\Delta U_C^{(CP)}$ serves both for modeling and compensation of inaccuracies of the trap potentials, leading to a wrongly determined CP voltage configuration and therefore to increased acceleration.", "It is implemented into the simulations by just adding it to $U_C^{(CP)}$ as determined by Eq.", "REF in the calculation of the static voltage sets.", "We see that even for sufficiently slow splitting, the Lamb-Dicke regime can only be attained if this voltage offset, and therefore the CP voltages in general, are correct within a window of about 20 mV, on the other hand it becomes clear that this voltage serves as convenient fine tuning parameter.", "The minimum excitation does not occur at $\\Delta U_C^{(CP)}=0$ , but is slightly shifted to positive values.", "This can be understood by considering that $\\vert \\dot{\\alpha }\\vert _{CP}$ is increased for any $\\Delta U_C^{(CP)} \\ne 0$ , but $\\ddot{\\alpha }_{CP}$ is decreased for $\\Delta U_C^{(CP)} >0$ .", "With $\\partial d/\\partial \\alpha $ , the second term in Eq.", "REF leads to a reduced total acceleration for small positive $\\Delta U_C^{(CP)}$ .", "Larger values again lead to increased acceleration because of a smaller $\\beta _{CP}$ value.", "All other calculations in this work are done using $\\Delta U_C^{(CP)}=0$ .", "For the case of an uncompensated tilt $\\gamma ^{\\prime }$ , we observe an even stronger dependence of the energy transfer.", "Fine tuning of the voltage difference on the outer segments $\\Delta U_O$ on the sub-mV level is required to reach the single phonon regime.", "Moreover, we observe that moderate uncompensated potential tilts reduce the energy transfer to one of the ions, as its CP acceleration is reduced by a more smooth $x(\\alpha )$ dependence.", "This might be of interest for specific applications where only the energy transfer to one of the ions is of importance.", "Dependence on the limiting voltage Figure: Dependence on the voltage limit: Oscillatory excitation as a function of the maximum voltage on the outer segments with all other limiting voltages remaining unchanged.", "The curves correspond to different splitting times: T=40μsT = 40\\mu s (green), T=30μsT = 30\\mu s (black), T=20μsT = 20\\mu s (red).Finally we study the dependence of the energy transfer on the limiting voltage $U_{lim}$ .", "We find that by increasing the voltage limit, beyond $U_{lim}=10$ V used so far, we can obtain lower coherent excitations as shown in Fig.", "REF .", "For this simulation, only the maximum voltage on the outer segments (max $U_O$ ) is increased and all other limits remain unchanged.", "We infer that by increasing the voltage limit on these electrodes up to about 50 V, one can reduce the mean phonon number by a factor of $\\approx 8$ for $T=60\\mu $ s. For lower splitting durations the enhancing factor becomes slightly smaller.", "Trap geometry optimization We have been showing in Sec.", "that the outcome of a crystal splitting operation is strongly determined by magnitude of the quartic confinement coefficient at the CP $\\beta _{CP}$ from Eq.", "REF .", "We thus investigate the effect of the trap geometry on the coefficients $\\alpha _n,\\beta _n,\\gamma _n$ from Eqs.", "REF .", "We calculate the realistic potentials from electrostatic simulations [14] to infer the geometry parameters according to Eq.", "REF .", "In particular, six different traps designs were studied, four of which are three-dimensional and two are surface-electrode traps.", "The results are shown in Tab.", "REF .", "The calculations are carried out for a generic simplified geometry shown in Fig.", "REF d), which is essentially determined by the segment width $w$ , the slit height $h$ and the spacer thickness $d$ for the three-dimensional traps.", "Trap A ,B[19] and C[13] are similar segmented micro-structured ion traps .", "Trap B is subdivided into a loading region of larger geometry, B (wide), and a narrow processing region, B (narrow).", "The data for trap C pertains to a wedge segment of $w=100\\mu $ m surrounded by larger segments.", "Trap D is a segmented planar ion trap [22], the calculations are performed at a distance of $100~\\mu $ m between the ion and the surface.", "Trap D2 is a planar ion trap featuring a segmented ground plane, otherwise identical to trap D. Trap A was used for all simulations in section .", "Table: Comparison of trap geometry parameters for different linear segmented Paul traps.", "Letters A to D denote different traps which are operated at various institutes, see text.", "Note that γ C =0\\gamma _C=0 by definition.", "The trap frequency at the critical point is specified for U lim U_{lim}=10V and 40 ^{40}Ca + ^+ ions.For trap A and B (wide) we calculate similar parameters, however the minimum trap frequency during the splitting is larger for trap A.", "Trap B (narrow) exhibits the highest minimum trap frequency of the six geometries as the total dimensions of this section of the trap are rather small.", "The wedge segment in trap C helps to increase the minimum trap frequency but choosing an overall smaller size seems to be a more favorable solution.", "The planar trap D has a similar minimum trap frequency as trap B (wide) and is also suitable for splitting ion crystals.", "The segmentation of the ground plane of this trap (D2) offers an enhanced $\\alpha _C$ , i.e.", "a large trap frequency.", "The calculations show however that for a segmentation of the center electrode, the potentials become more anharmonic and the Taylor approximation Eq.", "REF breaks down.", "Thus, the sign and magnitude ordering of the coefficients might be different from the other geometries, therefore the geometry parameters and the ion height above the surface should be carefully chosen to allow for successful splitting operations.", "Figure: Calculated geometry parameters α n ,β n ,γ n \\alpha _n,\\beta _n,\\gamma _n and the maximum β CP \\beta _{CP} at the critical point for a linear segmented Paul trap with dimensions h=400μh=400~\\mu m, d=250μd=250~\\mu m as a function of the segment width ww.", "The color code is as above: blue - C, red - S, green - O.", "The limiting voltage for the electrodes is U lim =10VU_{lim}=10V.For trap A we calculated the geometry parameters for varying segment width $w$ , the result is shown in Fig.", "REF .", "We analyze the dependance of all potential coefficients on $w$ with parameters $h$ and $d$ held constant.", "For splitting operations the optimum segment width would be at about $w=125\\mu $ m, while the actual segment width of the trap is $w=200\\mu $ m. We could therefore obtain a roughly twofold increase of $\\beta _{CP}$ bought at the expense of a reduced trap frequency for ion storage due to the reduced $\\alpha _C$ coefficient.", "Finally, we investigate the dependence of $\\beta _{CP}$ on the overall trap geometry size.", "We therefore pick trap parameters $h$ and $d$ from the range of typical values and determine the optimum segment width $w$ for these.", "Defining the effective trap size $d_{eff}=\\left(w^2+h^2+d^2\\right)^{1/2}$ , we find a scaling behavior of $\\beta _{CP}\\approx 2.2 \\cdot 10^{24} V \\cdot d_{eff}^{-4}$ , i.e.", "the best attainable value for the quartic confinement coefficient scales as the inverse fourth power with the effective trap size, which is the similar to the presumed distance scaling law for anomalous heating [17].", "We conclude that for a trap architecture aiming at shuttling-based scalable quantum information, the considerations presented here should be incorporated into the design process to facilitate crystal splitting operations.", "Conclusion We have pointed out the pitfalls for ion crystal splitting: Uncontrolled separation and uncompensated background fields lead to enhanced acceleration of the ions when the single well potential is transformed into a double well, which would require splitting times in the millisecond range to keep the motional excitation near the single phonon level.", "This in turn leads to strong anomalous heating due to the reduced confinement during the splitting process.", "We presented a framework to design voltage ramps which allow for coping with these problems.", "The scheme does only rely on measured calibration data which is obtained for the initial situation, where the ions are tightly confined in a single potential well.", "We carried out simulations, which elucidate the energy transfer mechanisms, and verify the performance of our scheme for the voltage ramp calculation.", "We showed that excitations near the single phonon level can be obtained for the specific trap apparatus we use.", "Furthermore, we analyzed the suitability of different trap geometries for ion crystal splitting by means of electrostatic simulations.", "We concluded that crystal splitting becomes easier for smaller trap structures, and that dedicated optimization of the geometry can be helpful.", "In future work, we envisage to analyze how crystal splitting can be performed on faster timescales by using shortcut-to-adiabaticity approaches, with an emphasis on robustness against experimental imperfections.", "Acknowledgments We thank René Gerritsma and Georg Jacob for proofreading the manuscript.", "This research was funded by the Office of the Director of National Intelligence (ODNI), Intelligence Advanced Research Projects Activity (IARPA), through the Army Research Office grant W911NF-10-1-0284.", "All statements of fact, opinion or conclusions contained herein are those of the authors and should not be construed as representing the official views or policies of IARPA, the ODNI, or the US Government.", "CTS acknowledges support from the German Federal Ministry for Education and Research (BMBF) via the Alexander von Humboldt Foundation." ], [ "Simulation results", "In order to analyze the sensitivity of the splitting process and the performance of our ramp design protocol, we numerically solve the classical equations of motion.", "For the time- and energy-scales and potential shapes under consideration, we expect quantum effects to play no significant role.", "For the case of single-ion shuttling, the occurrence of quantum effects is thoroughly discussed in Ref.", "[18].", "We perform the simulations using either the Taylor approximation of the potentials or the realistic potentials from electrostatic simulations [14] for trap A, which is similar to that described in Ref.", "[19].", "The voltage ramps $U_i(t)$ are used in conjunction with the potentials to yield the equations of motion for the ion positions $x_1<x_2$ .", "Employing the Taylor approximation potential Eq.", "REF , these read $-m\\ddot{x}_{1,2}=4\\beta (t) x_{1,2}^3+2\\alpha (t) x_{1,2}+\\gamma \\pm \\frac{\\kappa }{(x_2-x_1)^2},$ where the coefficients are given by using the voltage ramps in Eqs.", "REF , ,.", "For realistic trap potentials, we obtain $-m\\ddot{x}_{1,2}=\\sum _{n=C,S,O} U_n(t) \\left.", "\\frac{d\\phi _n}{dx}\\right\|_{x_{1,2}}\\pm \\frac{\\kappa }{(x_2-x_1)^2}\\\\$ The possibility to perform the simulations with approximate and realistic potentials serves the purpose of verifying the performance of the voltage ramps.", "These are determined purely by trap properties around the CP, which are conveniently accessible by measurements.", "More precisely, the time-domain voltage ramps are based on a $d(\\alpha )$ dependency given by the Taylor approximation potential according to Fig.", "REF , while the resulting energy transfer pertaining to these ramps can be obtained from simulations using realistic potentials.", "Note that a nonzero tilt can be present in the simulations based on the realistic potentials by summing separately over electrodes $O_L$ and $O_R$ and adding the differential voltage $\\pm \\Delta U_O$ given by $\\gamma /\\gamma _O$ accordingly.", "The calculations presented here employ the mass of $^{40}$ Ca$^+$ ions which we use in our experiments, and all simulations were performed for a limiting voltage range $U_{lim}=10$ V. Eqs.", "REF or REF are solved numerically using the NDSolve package from Mathematica, with the ions starting at rest.", "The final oscillation of each ion around its equilibrium position is analyzed and yields the energy transfer expressed as the mean phonon number $\\bar{n}=\\Delta E/\\hbar \\omega _f$ .", "We distinguish several regimes of laser-ion interaction: i) If the vibrational excitation becomes so large that the average Doppler shift per oscillation cycle exceeds the natural linewidth of a cycling transition, ion detection by counting resonance fluorescence photons will be impaired.", "ii) Measurement of the energy transfer i.e.", "by probing on a stimulated Raman transition [3] typically requires mean phonon numbers below about 300. iii) The Lamb-Dicke regime of laser-ion interaction, where coherent dynamics on resolved sidebands can be driven [20] is typically attained below about 10 phonons.", "The borders between these regimes depend on the trap frequency, ion mass and the specific atomic transitions to be driven, thus the regimes are indicated as broad gray bands in Fig.", "REF .", "Note that if final excitations in the measurable regime are obtained, an electrical counter kick can be applied for bringing the oscillation to rest [3]." ], [ "Dependence on splitting time", "We first analyze the dependence of the energy transfer on the duration of the splitting process $T$ , the result is shown in Fig.", "REF .", "The calculation is carried out for the ideal case of perfectly compensated potential tilt.", "We see that the final excitation becomes sufficiently low to remain in the Lamb-Dicke regime for typical laser-ion interaction settings at times larger than about 40 $\\mu s$ , which clearly outperforms the naïve approach of voltage interpolation from Sec.", "REF .", "We also take into account increased anomalous heating around the CP by employing the averaged heating rate according to Eq.", "REF .", "We see that for our specific heating rates, the limit of about one phonon per ion can not be overcome, but as the anomalous heating contribution is scaling as $1/T$ , the splitting result becomes rather insensitive with respect to the precise choice of the $T$ beyond $T=$ 50 $\\mu s$ .", "The simulation results verify our approach of calculating the voltage ramps using the Taylor approximated potentials.", "One recognizes that the resulting energy transfer in this case is larger by a factor of about two throughout the entire range of splitting durations.", "As can be seen from Fig.", "REF , this is due to the fact that the Taylor expansion leads to an incorrect voltage set pertaining to the CP, which in turn leads to uncontrolled acceleration as explained in Sec.", "REF .", "The discrepancy becomes irrelevant for splitting times larger than $T=$ 60 $\\mu $ s. At around 60 to $70~\\mu $ s the oscillatory excitation becomes smaller than $\\bar{n}=0.1$ , corresponding to the limit we can currently resolve in our experiment.", "The slight inaccuracy for low phonon numbers is due to numerical artifacts.", "Even lower energy transfers at shorter $T$ could possibly be achieved by ramp engineering, i.e.", "by the application of shortcut-to-adiabaticity approaches [18], [21].", "Figure: Energy transfer versus splitting time: Oscillatory (red) and thermal excitation (blue), and the sum of both (black) versus the splitting duration TT.", "The solid lines correspond to the calculation using the Taylor approximation, the dashed lines correspond to the full potential calculation, see text.", "Grey bands seperate different regimes of laser-ion interaction, see text.", "The thermal excitation was deduced from experimental heating rate data according to Sec.", ".", "The inset shows the trap frequency (black) and the corresponding heating rate (red) as a function of normalized time during the splitting process." ], [ "Sensitivity analysis", "Two crucial parameters for the splitting operation are the offset voltage at the CP $\\Delta U_C^{(CP)}$ and the potential tilt $\\gamma $ .", "Small variations of these parameters lead to strong coherent excitations as shown in Fig.", "REF .", "The CP voltage offset $\\Delta U_C^{(CP)}$ serves both for modeling and compensation of inaccuracies of the trap potentials, leading to a wrongly determined CP voltage configuration and therefore to increased acceleration.", "It is implemented into the simulations by just adding it to $U_C^{(CP)}$ as determined by Eq.", "REF in the calculation of the static voltage sets.", "We see that even for sufficiently slow splitting, the Lamb-Dicke regime can only be attained if this voltage offset, and therefore the CP voltages in general, are correct within a window of about 20 mV, on the other hand it becomes clear that this voltage serves as convenient fine tuning parameter.", "The minimum excitation does not occur at $\\Delta U_C^{(CP)}=0$ , but is slightly shifted to positive values.", "This can be understood by considering that $\\vert \\dot{\\alpha }\\vert _{CP}$ is increased for any $\\Delta U_C^{(CP)} \\ne 0$ , but $\\ddot{\\alpha }_{CP}$ is decreased for $\\Delta U_C^{(CP)} >0$ .", "With $\\partial d/\\partial \\alpha $ , the second term in Eq.", "REF leads to a reduced total acceleration for small positive $\\Delta U_C^{(CP)}$ .", "Larger values again lead to increased acceleration because of a smaller $\\beta _{CP}$ value.", "All other calculations in this work are done using $\\Delta U_C^{(CP)}=0$ .", "For the case of an uncompensated tilt $\\gamma ^{\\prime }$ , we observe an even stronger dependence of the energy transfer.", "Fine tuning of the voltage difference on the outer segments $\\Delta U_O$ on the sub-mV level is required to reach the single phonon regime.", "Moreover, we observe that moderate uncompensated potential tilts reduce the energy transfer to one of the ions, as its CP acceleration is reduced by a more smooth $x(\\alpha )$ dependence.", "This might be of interest for specific applications where only the energy transfer to one of the ions is of importance." ], [ "Dependence on the limiting voltage", "Finally we study the dependence of the energy transfer on the limiting voltage $U_{lim}$ .", "We find that by increasing the voltage limit, beyond $U_{lim}=10$ V used so far, we can obtain lower coherent excitations as shown in Fig.", "REF .", "For this simulation, only the maximum voltage on the outer segments (max $U_O$ ) is increased and all other limits remain unchanged.", "We infer that by increasing the voltage limit on these electrodes up to about 50 V, one can reduce the mean phonon number by a factor of $\\approx 8$ for $T=60\\mu $ s. For lower splitting durations the enhancing factor becomes slightly smaller." ], [ "Trap geometry optimization", "We have been showing in Sec.", "that the outcome of a crystal splitting operation is strongly determined by magnitude of the quartic confinement coefficient at the CP $\\beta _{CP}$ from Eq.", "REF .", "We thus investigate the effect of the trap geometry on the coefficients $\\alpha _n,\\beta _n,\\gamma _n$ from Eqs.", "REF .", "We calculate the realistic potentials from electrostatic simulations [14] to infer the geometry parameters according to Eq.", "REF .", "In particular, six different traps designs were studied, four of which are three-dimensional and two are surface-electrode traps.", "The results are shown in Tab.", "REF .", "The calculations are carried out for a generic simplified geometry shown in Fig.", "REF d), which is essentially determined by the segment width $w$ , the slit height $h$ and the spacer thickness $d$ for the three-dimensional traps.", "Trap A ,B[19] and C[13] are similar segmented micro-structured ion traps .", "Trap B is subdivided into a loading region of larger geometry, B (wide), and a narrow processing region, B (narrow).", "The data for trap C pertains to a wedge segment of $w=100\\mu $ m surrounded by larger segments.", "Trap D is a segmented planar ion trap [22], the calculations are performed at a distance of $100~\\mu $ m between the ion and the surface.", "Trap D2 is a planar ion trap featuring a segmented ground plane, otherwise identical to trap D. Trap A was used for all simulations in section .", "Table: Comparison of trap geometry parameters for different linear segmented Paul traps.", "Letters A to D denote different traps which are operated at various institutes, see text.", "Note that γ C =0\\gamma _C=0 by definition.", "The trap frequency at the critical point is specified for U lim U_{lim}=10V and 40 ^{40}Ca + ^+ ions.For trap A and B (wide) we calculate similar parameters, however the minimum trap frequency during the splitting is larger for trap A.", "Trap B (narrow) exhibits the highest minimum trap frequency of the six geometries as the total dimensions of this section of the trap are rather small.", "The wedge segment in trap C helps to increase the minimum trap frequency but choosing an overall smaller size seems to be a more favorable solution.", "The planar trap D has a similar minimum trap frequency as trap B (wide) and is also suitable for splitting ion crystals.", "The segmentation of the ground plane of this trap (D2) offers an enhanced $\\alpha _C$ , i.e.", "a large trap frequency.", "The calculations show however that for a segmentation of the center electrode, the potentials become more anharmonic and the Taylor approximation Eq.", "REF breaks down.", "Thus, the sign and magnitude ordering of the coefficients might be different from the other geometries, therefore the geometry parameters and the ion height above the surface should be carefully chosen to allow for successful splitting operations.", "Figure: Calculated geometry parameters α n ,β n ,γ n \\alpha _n,\\beta _n,\\gamma _n and the maximum β CP \\beta _{CP} at the critical point for a linear segmented Paul trap with dimensions h=400μh=400~\\mu m, d=250μd=250~\\mu m as a function of the segment width ww.", "The color code is as above: blue - C, red - S, green - O.", "The limiting voltage for the electrodes is U lim =10VU_{lim}=10V.For trap A we calculated the geometry parameters for varying segment width $w$ , the result is shown in Fig.", "REF .", "We analyze the dependance of all potential coefficients on $w$ with parameters $h$ and $d$ held constant.", "For splitting operations the optimum segment width would be at about $w=125\\mu $ m, while the actual segment width of the trap is $w=200\\mu $ m. We could therefore obtain a roughly twofold increase of $\\beta _{CP}$ bought at the expense of a reduced trap frequency for ion storage due to the reduced $\\alpha _C$ coefficient.", "Finally, we investigate the dependence of $\\beta _{CP}$ on the overall trap geometry size.", "We therefore pick trap parameters $h$ and $d$ from the range of typical values and determine the optimum segment width $w$ for these.", "Defining the effective trap size $d_{eff}=\\left(w^2+h^2+d^2\\right)^{1/2}$ , we find a scaling behavior of $\\beta _{CP}\\approx 2.2 \\cdot 10^{24} V \\cdot d_{eff}^{-4}$ , i.e.", "the best attainable value for the quartic confinement coefficient scales as the inverse fourth power with the effective trap size, which is the similar to the presumed distance scaling law for anomalous heating [17].", "We conclude that for a trap architecture aiming at shuttling-based scalable quantum information, the considerations presented here should be incorporated into the design process to facilitate crystal splitting operations." ], [ "Conclusion", "We have pointed out the pitfalls for ion crystal splitting: Uncontrolled separation and uncompensated background fields lead to enhanced acceleration of the ions when the single well potential is transformed into a double well, which would require splitting times in the millisecond range to keep the motional excitation near the single phonon level.", "This in turn leads to strong anomalous heating due to the reduced confinement during the splitting process.", "We presented a framework to design voltage ramps which allow for coping with these problems.", "The scheme does only rely on measured calibration data which is obtained for the initial situation, where the ions are tightly confined in a single potential well.", "We carried out simulations, which elucidate the energy transfer mechanisms, and verify the performance of our scheme for the voltage ramp calculation.", "We showed that excitations near the single phonon level can be obtained for the specific trap apparatus we use.", "Furthermore, we analyzed the suitability of different trap geometries for ion crystal splitting by means of electrostatic simulations.", "We concluded that crystal splitting becomes easier for smaller trap structures, and that dedicated optimization of the geometry can be helpful.", "In future work, we envisage to analyze how crystal splitting can be performed on faster timescales by using shortcut-to-adiabaticity approaches, with an emphasis on robustness against experimental imperfections." ], [ "Acknowledgments", "We thank René Gerritsma and Georg Jacob for proofreading the manuscript.", "This research was funded by the Office of the Director of National Intelligence (ODNI), Intelligence Advanced Research Projects Activity (IARPA), through the Army Research Office grant W911NF-10-1-0284.", "All statements of fact, opinion or conclusions contained herein are those of the authors and should not be construed as representing the official views or policies of IARPA, the ODNI, or the US Government.", "CTS acknowledges support from the German Federal Ministry for Education and Research (BMBF) via the Alexander von Humboldt Foundation." ] ]	1403.0097
[ [ "Blow up of solutions of semilinear heat equations in non radial domains\n of $\\mathbb R^2$" ], [ "Abstract We consider the semilinear heat equation \\begin{equation}\\label{problemAbstract}\\left\\{\\begin{array}{ll}v_t-\\Delta v= \|v\|^{p-1}v & \\mbox{in}\\Omega\\times (0,T)\\\\ v=0 & \\mbox{on}\\partial \\Omega\\times (0,T)\\\\ v(0)=v_0 & \\mbox{in}\\Omega \\end{array}\\right.\\tag{$\\mathcal P_p$} \\end{equation} where $p>1$, $\\Omega$ is a smooth bounded domain of $\\mathbb R^2$, $T\\in (0,+\\infty]$ and $v_0$ belongs to a suitable space.", "We give general conditions for a family $u_p$ of sign-changing stationary solutions of \\eqref{problemAbstract}, under which the solution of \\eqref{problemAbstract} with initial value $v_0=\\lambda u_p$ blows up in finite time if $\|\\lambda-1\|>0$ is sufficiently small and $p$ is sufficiently large.", "Since for $\\lambda=1$ the solution is global, this shows that, in general, the set of the initial conditions for which the solution is global is not star-shaped with respect to the origin.", "In previous paper by Dickstein, Pacella and Sciunzi this phenomenon has already been observed in the case when the domain is a ball and the sign changing stationary solution is radially symmetric.", "Our conditions are more general and we provide examples of stationary solutions $u_p$ which are not radial and exhibit the same behavior." ], [ "Introduction", "We consider the nonlinear heat equation $\\left\\lbrace \\begin{array}{ll}v_t-\\Delta v= \|v\|^{p-1}v & \\mbox{ in }\\Omega \\times (0,T)\\\\v=0 & \\mbox{ on }\\partial \\Omega \\times (0,T)\\\\v(0)=v_0 & \\mbox{ in }\\Omega \\end{array}\\right.$ where $\\Omega \\subset \\mathbb {R}^N$ , $N\\in \\mathbb {N}$ , $N\\ge 1$ is a smooth bounded domain, $p>1$ , $T\\in (0, +\\infty ]$ and $v_0\\in C_0(\\Omega )$ , where $C_0(\\Omega )=\\lbrace v\\in C(\\bar{\\Omega }),\\ v=0 \\mbox{ on } \\partial \\Omega \\rbrace .$ It is well known that the initial value problem (REF ) is locally well-posed in $C_0(\\Omega )$ and admits both nontrivial global solutions and blow-up solutions.", "Denoting with $T_{v_0}$ the maximal existence time of the solution of (REF ), we define the set of initial conditions for which the corresponding solution is global, i.e.", "$\\mathcal {G}=\\lbrace v_0\\in C_0(\\Omega ),\\ T_{v_0}=+\\infty \\rbrace $ and its complementary set of initial conditions for which the corresponding solution blows-up in finite time: $\\mathcal {B}=\\lbrace v_0\\in C_0(\\Omega ),\\ T_{v_0}<+\\infty \\rbrace .$ For a fixed $w\\in C_0(\\Omega )$ , $w\\ne 0$ and $v_0=\\lambda w$ , $\\lambda \\in \\mathbb {R}$ , it is known that if $\|\\lambda \|$ is small enough then $v_0\\in \\mathcal {G}$ , while if $\|\\lambda \|$ is sufficiently large then $v_0\\in \\mathcal {B}$ .", "Moreover, for any $N\\ge 1$ , considering nonnegative initial data, it can be easily proved that the set $\\mathcal {G}^+=\\lbrace v_0\\in \\mathcal {G},\\ v_0\\ge 0\\rbrace $ is star-shaped with respect to 0 (indeed it is convex, see [8]).", "On the contrary when the initial condition changes sign $\\mathcal {G}$ may not be star-shaped.", "Indeed for $N\\ge 3$ , in [1] and [7] it has been shown that there exists $p^<p_S$ , with $p_S=\\frac{N+2}{N-2}$ , such that $\\forall \\ p \\in \\ (p^,p_S)$ the elliptic problem $\\left\\lbrace \\begin{array}{ll}-\\Delta u_p= \|u_p\|^{p-1}u_p & \\mbox{ in }\\Omega \\\\u_p=0 & \\mbox{ on }\\partial \\Omega ,\\end{array}\\right.$ admits a sign a changing solution $u_p$ for which there exists $\\epsilon >0$ such that if $0<\|1-\\lambda \|<\\epsilon $ then $\\lambda u_p\\in \\mathcal {B}$ .", "More precisely this result has been proved first in [1] when $\\Omega $ is the unit ball and $u_p$ is any sign-changing radial solution of (REF ), and then in [7] for general domains $\\Omega $ and sign-changing solutions $u_p$ of (REF ) (assuming w.l.g.", "that $\\Vert u_p^+\\Vert _{\\infty }=\\Vert u_p\\Vert _{\\infty }$ ), satisfying the following conditions $\\int _{\\Omega }\|\\nabla u_p\|^2 dx\\rightarrow 2S^{\\frac{N}{2}}\\qquad \\mathrm {(a)}$ $\\left\|\\frac{\\min u_p}{\\max u_p}\\right\|\\rightarrow 0\\qquad \\mathrm {(b)}$ as $p\\rightarrow p^$ , where $S$ is the best Sobolev constant for the embedding of $H^1_0(\\Omega )$ into $L^{2^}(\\Omega )$ .", "When $N=1$ such a result does not hold since it is easy to see that for any sign changing solution $u_p$ of (REF ), $\\lambda u_p\\in \\mathcal {G}$ for $\|\\lambda \|<1$ and $\\lambda u_p\\in \\mathcal {B}$ for $\|\\lambda \|>1$ .", "The case $N=2$ was left open in the papers [1] and [7], mainly because there is not a critical Sobolev exponent in this dimension so that the conditions and results of these papers are meaningless.", "Recently inspired by [5], [6], Dickstein, Pacella and Sciunzi in [4] succeeded to prove a blow up theorem similar to the one in [1], considering radial sign changing stationary solutions $u_p$ of (REF ) in the unit ball for large exponents $p$ .", "Indeed, the condition $p\\rightarrow +\\infty $ in dimension $N=2$ can be considered to be the natural extension of the condition $p\\rightarrow p_S$ for $N\\ge 3$ .", "In this paper we consider again the case $N=2$ but the bounded domain $\\Omega $ is not necessarily a ball.", "We deal with sign-changing solutions $u_p$ of (REF ) with the following two properties: $ \\exists C>0, \\mbox{ such that }\\ \\ p\\int _{\\Omega } \|\\nabla u_p\|^{2}dx \\le C,\\qquad \\mathrm {(A)}$ $ \\lim _{R\\rightarrow +\\infty }\\lim _{p\\rightarrow +\\infty } \\mathcal {S}_{p,R}=0,\\qquad \\mathrm {(B)}$ where, for $R>0$ , $\\mathcal {S}_{p,R}:=\\sup \\left\\lbrace \\left\|\\frac{u_p(y)}{u_p(x_p^+)}\\right\|^{p-1}\\ : \\ y\\in \\Omega , \\ \|y-x_p^+\|> R\\mu _p^+\\right\\rbrace ,$ for $x_p^+$ such that $\|u_p(x_p^+)\|=\\Vert u_p\\Vert _{\\infty }$ and $\\mu _p^+:=\\frac{1}{\\sqrt{p\|u_p(x_p^+)\|^{p-1}}}$ .", "Our main result is the following Theorem 1.1 Let $N=2$ and $u_p$ be a family of sign-changing solutions of problem (REF ) satisfying $(A)$ and $(B)$ .", "Then, up to a subsequence, there exists $p^>1$ such that for $p>p^$ there exists $\\epsilon =\\epsilon (p) >0$ such that if $0<\|1-\\lambda \|<\\epsilon $ , then $\\lambda u_p\\in \\mathcal {B}.$ A few comments on conditions $(A)$ and $(B)$ are needed.", "The first one is an estimate of the asymptotic behavior, as $p\\rightarrow +\\infty $ , of the energy of the solutions.", "It is satisfied by different kinds of sign changing solutions (see [2], [3], [5]), in particular by the radial ones in the ball (see [4], [6]).", "The condition $(B)$ is more peculiar and essentially concerns the asymptotic behavior of $\|u_p(y)\|^{p-1}$ for points $y$ which are not too close to $x_p^+$ ; note that $p\|u_p(y)\|^{p-1}$ can also be divergent since $\\liminf _{p\\rightarrow +\\infty }\\Vert u_p\\Vert _{\\infty }\\ge 1$ .", "It is satisfied, in particular, by sign changing radial solutions $u_{p,\\mathcal {K}}$ of (REF ) having any fixed number $\\mathcal {K}$ of nodal regions (see Section for details).", "But it also holds for a class of sign changing solutions in more general domains as we show in the next theorem.", "Theorem 1.2 Let $\\Omega \\subset \\mathbb {R}^2$ be a smooth bounded domain, containing the origin, invariant under the action of a cyclic group $G$ of rotations about the origin with $ \|G\|\\ge me$ ($\|G\|$ is the order of $G$ ), for a certain $m>0$ .", "Let $(u_p)$ be a family of sign changing $G$ -symmetric solutions of (REF ) such that $ p\\int _{\\Omega }\|\\nabla u_p\|^2 dx\\le \\alpha \\,8\\pi e,\\ \\mbox{ for $p$ large}$ and for some $\\alpha <m+1$ .", "Then $u_p$ satisfies $(A)$ and $(B)$ up to a subsequence.", "Remark 1.3 The existence of sign-changing solutions of (REF ) satisfying the assumption (REF ) of Theorem REF has been proved in [2] for $m\\ge 4$ and for large $p$ .", "Putting together Theorem REF and Theorem REF one gets the extension of the blow-up result of [4] to other symmetric domains.", "Note that this in particular includes the case of sign changing radial solutions in a ball, providing so a different proof of the result of [4] which strongly relied on the radial symmetry.", "The proof of Theorem REF follows the same strategy as the analogous results in [1], [7], [4] being a consequence of the following proposition, which is a particular case of [1] Proposition 1.4 Let $u$ be a sign changing solution of (REF ) and let $\\varphi _1$ be a positive eigenvector of the self-adjoint operator $L$ given by $L\\varphi =-\\Delta \\varphi -p\|u\|^{p-1}\\varphi $ , for $\\varphi \\in H^2(\\Omega )\\cap H^1_0(\\Omega )$ .", "Assume that $ \\int _{\\Omega }u\\varphi _1\\ne 0.$ Then there exists $\\epsilon >0$ such that if $0<\|1-\\lambda \|<\\epsilon $ , then $\\lambda u\\in \\mathcal {B}$ .", "More precisely we will show that under the assumptions $(A)$ and $(B)$ condition (REF ) is satisfied for $p$ large (see Theorem REF in the following).", "This proof is based on rescaling arguments about the maximum point of $u_p$ : using the properties of the solution of the limit problem, we analyze the asymptotic behavior as $p\\rightarrow +\\infty $ of the rescaled solutions and of the rescaled first eigenfunction of the linearized operator at $u_p$ .", "In this analysis the assumption $(B)$ is crucial.", "To get Theorem REF we prove a more general result which shows that condition $(B)$ holds for a quite general class of solutions $u_p$ of (REF ) (see Lemma REF and Theorem REF ).", "The paper is organized as follows.", "In Section we collect some properties satisfied by any family of solutions $u_p$ under condition $(A)$ and we give a characterization of condition $(B)$ .", "In Section we carry out an asymptotic spectral analysis under assumption $(A)$ and $(B)$ , studying the asymptotic behavior of the first eigenvalue and of the first eigenfunction of the linearized operator at $u_p$ .", "Section is devoted to the proof of Theorem REF via rescaling arguments.", "In Section we find a sufficient condition which ensures the validity of property $(B)$ (see Lemma REF ) and we select a general class of solutions to (REF ) which satisfies this sufficient condition (see Theorem REF ).", "In Section we prove Theorem REF .", "Finally in Section we show that also the sign-changing radial solutions in the unit ball satisfy the assumptions of Theorem REF ." ], [ "Preliminary results", "We fix some notation.", "For a given family $(u_p)$ of sign-changing stationary solutions of (REF ) we denote by $NL_p$ the nodal line of $u_p$ , $x_p^{\\pm }$ a maximum/minimum point in $\\Omega $ of $u_p$ , i.e.", "$u_p(x_p^{\\pm })=\\pm \\Vert u_p^{\\pm }\\Vert _{\\infty }$ , $\\mathcal {N}_p^{\\pm }:=\\lbrace x\\in \\Omega \\ :\\ u_p^{\\pm }(x)\\ne 0\\rbrace \\subset \\Omega $ denote the positive/negative domain of $u_p$ , $\\mu _p^{\\pm }:=\\frac{1}{\\sqrt{p\|u_p(x_p^{\\pm })\|^{p-1}}}$ , $\\widetilde{\\Omega }_p:=\\widetilde{\\Omega }_p^{+}=\\frac{\\Omega -x_p^{+}}{\\mu _p^{+}}=\\lbrace x\\in \\mathbb {R}^2: x_p^{+}+\\mu _p^{+}x\\in \\Omega \\rbrace $ , $\\widetilde{\\mathcal {N}_p^+}:=\\frac{\\mathcal {N}_p^+-x_p^{+}}{\\mu _p^{+}}=\\lbrace x\\in \\mathbb {R}^2: x_p^{+}+\\mu _p^{+}x\\in \\mathcal {N}_p^+\\rbrace $ , $d(x,A):=dist (x,A)$ , for any $x\\in \\mathbb {R}^2, A\\subset \\mathbb {R}^2$ .", "We assume w.l.o.g.", "that $\\Vert u_p\\Vert _{\\infty }=\\Vert u_p^+\\Vert _{\\infty }$ .", "In the next two lemmas we collect some useful properties holding under condition $(A)$ .", "Lemma 2.1 Let $(u_p)$ be a family of solutions to (REF ) and assume that $(A)$ holds.", "Then $\\Vert u_p\\Vert _{L^{\\infty }(\\Omega )}\\le C$ $\\liminf _{p\\rightarrow +\\infty } u_p(x_p^{\\pm })\\ge 1$ $\\mu _p^{\\pm }\\rightarrow 0 \\ \\mbox{ as }\\ p\\rightarrow +\\infty $ It is well known, see for instance [3] or [5].", "Lemma 2.2 Let $(u_p)$ be a family of solutions to (REF ) and assume that $(A)$ holds.", "Then, up to a subsequence, the rescaled function $v_p^+(x):=\\frac{p}{u_p(x_p^+)}(u_p(x_p^++\\mu _p^+ x)-u_p(x_p^+))$ defined on $\\widetilde{\\Omega }_{p}$ converges to $U$ in $C^1_{loc}(\\mathbb {R}^2)$ , where $U(x)=\\log \\left(\\frac{1}{1+\\frac{1}{8} \|x\|^2}\\right)^2$ is the solution of the Liouville problem $\\left\\lbrace \\begin{array}{lr}-\\Delta U=e^{U}\\ \\mbox{ in }\\ \\mathbb {R}^2\\\\U\\le 0, \\ U(0)=0\\ \\mbox{ and }\\ \\int _{\\mathbb {R}^2}e^{U}=8\\pi .\\end{array}\\right.$ Moreover $\\frac{d(x_p^+, NL_p)}{\\mu _p^{+}}\\rightarrow +\\infty \\ \\mbox{ as }\\ p\\rightarrow +\\infty $ and $\\frac{d(x_p^+, \\partial \\Omega )}{\\mu _p^{+}}\\rightarrow +\\infty \\ \\mbox{ as }\\ p\\rightarrow +\\infty .$ It is well known, see for instance [3] Observe that, under condition $(A)$ , by (REF ), for any $R>0$ there exists $p_R>1$ such that the set $\\lbrace y\\in \\Omega , \\ \|y-x_p^+\|> R\\mu _p^+ \\rbrace \\ne \\emptyset $ for $p\\ge p_R$ .", "As a consequence for any $R>0$ the value $\\mathcal {S}_{p,R}:=\\sup \\left\\lbrace \\left\|\\frac{u_p(y)}{u_p(x_p^+)}\\right\|^{p-1}\\ : \\ y\\in \\Omega , \\ \|y-x_p^+\|> R\\mu _p^+\\right\\rbrace $ in the definition of condition $(B)$ is well-defined for $p\\ge p_R$ .", "Next we give a characterization of condition $(B)$ : Proposition 2.3 Assume that $u_p$ is a family of sign-changing solutions to (REF ) which satisfies condition $(A)$ .", "Then for any $R>0$ there exists $p_R>1$ such that the set $\\lbrace y\\in \\mathcal {N}_p^+, \\ \|y-x_p^+\|> R\\mu _p^+ \\rbrace \\ne \\emptyset $ for $p\\ge p_R$ and so $ \\mathcal {M}_{p,R}:=\\sup \\left\\lbrace \\left\|\\frac{u_p(y)}{u_p(x_p^+)}\\right\|^{p-1}\\ : \\ y\\in \\mathcal {N}_p^+, \\ \|y-x_p^+\|> R\\mu _p^+\\right\\rbrace $ is well defined.", "Moreover condition $(B)$ is equivalent to $\\left\\lbrace \\begin{array}{lr}\\displaystyle { \\lim _{p\\rightarrow +\\infty }}\\frac{\\Vert u_p^-\\Vert _{L^{\\infty }(\\Omega )}^{p-1}}{\\Vert u_p^+\\Vert _{L^{\\infty }(\\Omega )}^{p-1}}= 0 & \\qquad (B1)\\\\\\\\\\displaystyle {\\lim _{R\\rightarrow +\\infty }}\\lim _{p\\rightarrow +\\infty } \\mathcal {M}_{p,R}=0 & \\qquad (B2)\\end{array}\\right.$ For $R>0$ and $p>1$ define the set ${\\Omega }_{p,R}:=\\left\\lbrace y\\in \\Omega , \\ \|y-x_p^+\|> R\\mu _p^+\\right\\rbrace \\ (\\subseteq \\Omega ).$ In order to prove the equivalence it is enough to show that for any $R>0$ there exists $p_R>1$ such that ${\\Omega }_{p,R}= \\left({\\Omega }_{p,R}\\cap \\mathcal {N}_p^+\\right) \\cup \\mathcal {N}_p^-\\cup NL_p$ for $p\\ge p_R$ and the union is disjoint.", "Indeed (REF ) implies that $\\mathcal {S}_{p,R}=\\max \\left\\lbrace \\mathcal {M}_{p,R}, \\ \\frac{\\Vert u_p^-\\Vert _{L^{\\infty }(\\Omega )}^{p-1}}{\\Vert u_p^+\\Vert _{L^{\\infty }(\\Omega )}^{p-1}}\\right\\rbrace $ and so the conclusion.", "By definition $x_p^+\\in \\mathcal {N}_p^+$ , moreover by (REF ) and (REF ) $\\frac{d(x_p^+, NL_p)}{\\mu _p^+}\\rightarrow +\\infty $ and $\\frac{d(x_p^+, \\partial \\Omega )}{\\mu _p^+}\\rightarrow +\\infty $ as $p\\rightarrow +\\infty $ and so for any $R>0$ there exists $p_R>1$ such that $\\frac{d(x_p^+, NL_p)}{\\mu _p^+}\\ge 2R$ and $\\frac{d(x_p^+, \\partial \\Omega )}{\\mu _p^+}\\ge 2R$ for $p\\ge p_R$ , which implies that $B_{R\\mu _p^+}(x_p^+)\\subset \\mathcal {N}_p^+$ for $p\\ge p_R$ .", "As a consequence $\\left({\\Omega }_{p,R}\\cap \\mathcal {N}_p^-\\right)=\\mathcal {N}_p^-, \\ \\left({\\Omega }_{p,R}\\cap NL_p\\right)= NL_p$ and the set $\\left({\\Omega }_{p,R}\\cap \\mathcal {N}_p^+\\right)\\ne \\emptyset $ for $p\\ge p_R$ .", "Hence (REF ) follows from the fact that ${\\Omega }_{p,R}=\\left({\\Omega }_{p,R}\\cap \\mathcal {N}_p^+\\right)\\cup \\left({\\Omega }_{p,R}\\cap \\mathcal {N}_p^-\\right) \\cup \\left({\\Omega }_{p,R}\\cap NL_p\\right)$ and the union is disjoint.", "Remark 2.4 Condition $(B1)$ can be equivalently written as $\\lim _{p\\rightarrow +\\infty }\\frac{\\mu _p^+}{\\mu _p^-}=0.$" ], [ "Linearization of the limit problem", "In [4] the linearization at $U$ of the Liouville problem (REF ) has been studied.", "In the following we recall the main results.", "For $v\\in H^2(\\mathbb {R}^2)$ we define the linearized operator by $L^(v):=-\\Delta v -e^{U}v.$ We consider the Rayleigh functional $\\mathcal {R}^: H^1(\\mathbb {R}^2)\\rightarrow \\mathbb {R}$ $\\mathcal {R}^(w):=\\int _{\\mathbb {R}^2}\\left(\|\\nabla w\|^2-e^{U}w^2 \\right)\\ dx$ and define $\\lambda _1^:=\\inf _{ \\begin{array}{c} w\\in H^1(\\mathbb {R}^2)\\\\ \\Vert w\\Vert _{L^2(\\mathbb {R}^2)}=1\\end{array}}\\mathcal {R}^(w).$ Proposition 3.1 We have the following i) $(-\\infty < )\\ \\lambda _1^<0$ ; ii) every minimizing sequence of (REF ) has a subsequence strongly converging in $L^2(\\mathbb {R}^2)$ to a minimizer; iii) there exists a unique positive minimizer $\\varphi _1^$ to (REF ), which is radial and radially non-increasing; iv) $\\lambda _1^$ is an eigenvalue of $L^$ and $\\varphi _1^$ is an eigenvector associated to $\\lambda _1^$ .", "Moreover $\\varphi _1^\\in L^{\\infty }(\\mathbb {R}^2)$ .", "See [4]." ], [ "Linearization of the Lane-Emden problem", "In this section we consider the linearization of the Lane-Emden problem and study its connections with the linearization $L^$ of the Liouville problem.", "We define the linearized operator at $u_p$ of the Lane-Emden problem in $\\Omega $ $L_p(v):=-\\Delta v-p \|u_p\|^{p-1}v,\\ \\ \\ v\\in H^2(\\Omega )\\cap H^1_0(\\Omega ).$ Let $\\lambda _{1,p}$ and $\\varphi _{1,p}$ be respectively the first eigenvalue and the first eigenfunction (normalized in $L^2$ ) of the operator $L_p$ .", "It is well known that $\\lambda _{1,p}< 0,$ $\\forall \\ p>1$ and that $\\varphi _{1,p}\\ge 0$ , moreover $\\Vert \\varphi _{1,p}\\Vert _{L^2(\\Omega )}=1$ .", "Rescaling $\\widetilde{\\varphi }_{1,p}(x):= \\mu _p^+ \\ \\varphi _{1,p}(x_p^++\\mu _p^+ x), \\ \\ x\\in \\widetilde{\\Omega }_p$ and setting $\\widetilde{\\lambda }_{1,p}:= \\left(\\mu _p^+\\right)^2\\, \\lambda _{1,p}$ then, it is easy to see that $\\widetilde{\\varphi }_{1,p}$ and $\\widetilde{\\lambda }_{1,p}$ are respectively the first eigenfunction and first eigenvalue of the linear operator $\\widetilde{L}_p$ in $\\widetilde{\\Omega }_p$ with homogeneous Dirichlet boundary conditions, defined as $\\widetilde{L}_p v:=-\\Delta v- V_p(x)\\, v,\\ \\ \\ \\ \\ v\\in H^2( \\widetilde{\\Omega }_p )\\cap H^1_0( \\widetilde{\\Omega }_p ),$ where $V_p(x):=\\left\|\\frac{u_p(x_p^++\\mu _p^+x)}{u_p(x_p^+)} \\right\|^{p-1}=\\left\| 1+\\frac{v_p^+(x)}{p} \\right\|^{p-1}\\ \\ \\ \\ \\ \\mbox{($v_p^+$ is the function defined in Lemma \\ref {convergenzaVp})}.$ Observe that $\\Vert \\widetilde{\\varphi }_{1,p}\\Vert _{L^2(\\widetilde{\\Omega }_p)}=\\Vert \\varphi _{1,p}\\Vert _{L^2(\\Omega )}=1$ .", "Lemma 3.2 Up to a subsequence $\\int _{\\widetilde{\\Omega }_p}\\left(e^U- V_p\\right)\\widetilde{\\varphi }_{1,p}^2\\rightarrow 0 \\ \\mbox{ as }\\ p\\rightarrow +\\infty .$ For $R>0$ we divide the integral in the following way $\\int _{\\widetilde{\\Omega }_p}\\left(e^U- V_p\\right)\\widetilde{\\varphi }_{1,p}^2=\\underbrace{\\int _{\\widetilde{\\Omega }_p\\cap \\lbrace \|x\|\\le R\\rbrace }\\left(e^U- V_p\\right)\\widetilde{\\varphi }_{1,p}^2}_{A_{p,R}}+\\underbrace{\\int _{\\widetilde{\\Omega }_p\\cap \\lbrace \|x\|> R\\rbrace }\\left(e^U- V_p\\right)\\widetilde{\\varphi }_{1,p}^2}_{B_{p,R}}.$ Now $A_{p,R}=\\int _{\\widetilde{\\Omega }_p\\cap \\lbrace \|x\|\\le R\\rbrace }\\left(e^U- V_p\\right)\\widetilde{\\varphi }_{1,p}^2\\rightarrow 0\\ \\ \\mbox{ as }\\ p\\rightarrow +\\infty , \\ \\mbox{ for all }\\ R>0,$ since, up to a subsequence, $v_p^+\\rightarrow U$ in $C^1_{loc}(\\mathbb {R}^2)$ as $p\\rightarrow +\\infty $ (see Lemma REF ) and so $V_p\\rightarrow e^U$ uniformly in $B_R(0)$ , up to a subsequence.", "On the other hand $B_{p,R} &=& \\int _{\\widetilde{\\Omega }_p\\cap \\lbrace \|x\|> R\\rbrace }\\left(e^U- V_p\\right)\\widetilde{\\varphi }_{1,p}^2=\\underbrace{\\int _{\\widetilde{\\Omega }_p\\cap \\lbrace \|x\|> R\\rbrace }e^U \\widetilde{\\varphi }_{1,p}^2}_{C_{p,R}}-\\underbrace{\\int _{\\widetilde{\\Omega }_p\\cap \\lbrace \|x\|> R\\rbrace }V_p\\widetilde{\\varphi }_{1,p}^2}_{D_{p,R}}.$ Following [4] we estimate $(0\\le )\\ \\ C_{p,R}= \\int _{\\widetilde{\\Omega }_p\\cap \\lbrace \|x\|> R\\rbrace }e^U \\widetilde{\\varphi }_{1,p}^2\\le \\sup _{\|x\|>R}e^{U(x)} \\int _{\\widetilde{\\Omega }_p}\\widetilde{\\varphi }_{1,p}^2=\\sup _{\|x\|>R}e^{U(x)}=\\sup _{\|x\|>R}\\left( \\frac{1}{1+\\frac{\|x\|^2}{8}}\\right)^2\\le \\frac{64}{R^2}.$ Last we estimate $(0\\le )\\ \\ D_{p,R}= \\int _{\\widetilde{\\Omega }_p\\cap \\lbrace \|x\|> R\\rbrace }V_p\\widetilde{\\varphi }_{1,p}^2&\\le & \\sup _{\\widetilde{\\Omega }_p\\cap \\lbrace \|x\|> R\\rbrace } V_p(x)\\ \\int _{\\widetilde{\\Omega }_p} \\widetilde{\\varphi }_{1,p}^2\\\\&=&\\sup _{\\widetilde{\\Omega }_p\\cap \\lbrace \|x\|> R\\rbrace } V_p(x)\\\\&=& \\sup _{\\widetilde{\\Omega }_p\\cap \\lbrace \|x\|> R\\rbrace }\\left\| \\frac{u_p(x_p^+ +\\mu _p^+ x)}{u_p(x_p^+)}\\right\|^{p-1}\\\\&=& \\sup _{\\Omega \\cap \\lbrace \|y-x_p^+\|> R\\mu _p^+\\rbrace }\\left\| \\frac{u_p(y)}{u_p(x_p^+)}\\right\|^{p-1}\\\\&=& \\mathcal {S}_{p,R},$ so assumption $(B)$ implies that $\\lim _{R\\rightarrow +\\infty }\\lim _{p\\rightarrow +\\infty } D_{p,R}=0,$ and this concludes the proof.", "Theorem 3.3 We have, up to a subsequence, that $\\widetilde{\\lambda }_{1,p}\\rightarrow \\lambda _1^ \\ \\ \\mbox{ as }\\ p\\rightarrow +\\infty .$ We divide the proof into two steps.", "Step 1.", "We show that, up to a subsequence, for $\\epsilon >0$ $\\lambda _1^\\le \\widetilde{\\lambda }_{1,p} + \\epsilon \\ \\ \\mbox{ for $p$ sufficiently large}.$ $\\lambda _1^ &\\le & \\int _{\\mathbb {R}^2}\\left( \|\\nabla \\widetilde{\\varphi }_{1,p}\|^2-e^{U}\\ \\widetilde{\\varphi }_{1,p}^2\\right)\\nonumber \\\\& =& \\int _{\\widetilde{\\Omega }_p} V_p\\widetilde{\\varphi }_{1,p}^2 + \\widetilde{\\lambda }_{1,p}\\int _{\\widetilde{\\Omega }_p}\\widetilde{\\varphi }_{1,p}^2-\\int _{\\widetilde{\\Omega }_p}e^{U}\\ \\widetilde{\\varphi }_{1,p}^2\\nonumber \\\\& =& \\widetilde{\\lambda }_{1,p}- \\int _{\\widetilde{\\Omega }_p}\\left(e^U- V_p\\right)\\widetilde{\\varphi }_{1,p}^2$ and so the conclusion follows by Lemma REF .", "Step 2.", "We show that, up to a subsequence, for $\\epsilon >0$ $\\widetilde{\\lambda }_{1,p}\\le \\lambda _1^* + \\epsilon \\ \\ \\mbox{ for $p$ sufficiently large}.$ The proof is similar to the one in [4], we repeat it for completeness.", "For $R>0$ , let us consider a cut-off regular function $\\psi _R(x)=\\psi _R(r)$ such that $\\left\\lbrace \\begin{array}{lr}0\\le \\psi _R\\le 1\\\\\\psi _R=1 \\ \\mbox{ for }\\ r\\le R\\\\\\psi _R=0\\ \\mbox{ for }\\ r\\ge 2 R\\\\\|\\nabla \\psi _R\|\\le 2/R\\end{array}\\right.$ and let us set $ w_R:=\\frac{\\psi _R\\varphi _1^}{\\Vert \\psi _R\\varphi _1^\\Vert _{L^2(\\mathbb {R}^2)}}.$ Hence, from the variational characterization of $\\widetilde{\\lambda }_{1,p}$ we deduce that $\\widetilde{\\lambda }_{1,p} &\\le & \\int _{\\mathbb {R}^2}\\left(\|\\nabla w_R\|^2- V_p(x)w_R^2\\right)dx\\nonumber \\\\&=& \\int _{\\mathbb {R}^2}\\left(\|\\nabla w_R\|^2- e^{U(x)}w_R^2\\right)dx+ \\int _{\\mathbb {R}^2}\\left(e^{U(x)}- V_p(x)\\right)w_R^2dx.$ Since $w_R\\rightarrow \\varphi _1^$ in $H^1(\\mathbb {R}^2)$ as $R\\rightarrow +\\infty $ , it is easy to see that given $\\epsilon >0$ we can fix $R>0$ such that $\\int _{\\mathbb {R}^2}\\left(\|\\nabla w_R\|^2- e^{U(x)}w_R^2\\right)dx\\le \\lambda _1^+\\frac{\\epsilon }{2}.$ For such a fixed value of $R$ we can argue similarly as in the proof of (REF ) in Lemma REF to obtain that, up to a subsequence in $p$ , $\\int _{\\mathbb {R}^2}\\left(e^{U(x)}- V_p(x)\\right)w_R^2dx\\le \\frac{\\epsilon }{2}$ for $p$ large enough.", "Hence the proof of Step 2 follows from (REF ), (REF ) and (REF ).", "Corollary 3.4 Up to a subsequence $\\widetilde{\\varphi }_{1,p}\\rightarrow \\varphi _1^\\ \\mbox{ in }\\ L^2(\\mathbb {R}^2)\\ \\mbox{ as }\\ p\\rightarrow +\\infty .$ By Lemma REF and Theorem REF we have that, passing to a subsequence $\\lambda _{1}^-\\mathcal {R}^(\\widetilde{\\varphi }_{1,p})= (\\lambda _{1}^-\\widetilde{\\lambda }_{1,p})+\\widetilde{\\lambda }_{1,p}-\\mathcal {R}^(\\widetilde{\\varphi }_{1,p})= (\\lambda _{1}^-\\widetilde{\\lambda }_{1,p})+\\int _{\\mathbb {R}^2}\\left(e^{U(x)}-V_p(x) \\right)\\widetilde{\\varphi }_{1,p}^2\\rightarrow 0$ as $p\\rightarrow +\\infty $ , namely $\\widetilde{\\varphi }_{1,p}$ is a minimizing sequence for (REF ) and so the result follows from points ii) and iii) of Proposition REF ." ], [ "Proof of Theorem ", "The proof of Theorem REF follows the same strategy as in [1], [7], [4] and it is a consequence of Proposition REF , which is a particular case of Theorem 2.3 in [1].", "Hence, to obtain Theorem REF , it is enough to prove the following: Theorem 4.1 Let $u_p$ be a family sign changing solutions to (REF ) which satisfies conditions $(A)$ and $(B)$ .", "Then there exists $p^>1$ such that up to a subsequence, for $p>p^$ $\\int _{\\Omega }u_p\\varphi _{1,p} >0,$ where $\\varphi _{1,p}$ is the first positive eigenfunction of the linearized operator $L_p$ at $u_p$ .", "Since by an easy computation $\\int _{\\Omega }u_p\\varphi _{1,p}=\\frac{p-1}{-\\lambda _{1,p}}\\int _{\\Omega }\|u_p\|^{p-1}u_p\\varphi _{1,p}$ (see [4]), we can study the sign of $\\int _{\\Omega }\|u_p\|^{p-1}u_p\\varphi _{1,p},$ which is the same as the sign of $\\frac{1}{u_p(x_p^+)^p\\mu _p^+} \\int _{\\Omega }\|u_p\|^{p-1}u_p\\varphi _{1,p}.$ We show that, up to a subsequence, $\\frac{1}{u_p(x_p^+)^p\\mu _p^+}\\int _{\\Omega }\|u_p\|^{p-1}u_p\\varphi _{1,p}\\rightarrow \\int _{\\mathbb {R}^2}e^U \\varphi _1^\\ (>0)\\ \\ \\mbox{ as }\\ p\\rightarrow +\\infty $ from which the conclusion follows.", "In order to prove (REF ) we change the variable and, for any $R>0$ , we split the integral in the following way $\\frac{1}{u_p(x_p^+)^p\\mu _p^+}\\int _{\\Omega }\|u_p\|^{p-1}u_p\\varphi _{1,p} &=&\\frac{1}{u_p(x_p^+)^p}\\int _{\\widetilde{\\Omega }_p}\|u_p(x_p^++\\mu _p^+x)\|^{p-1}u_p(x_p^++\\mu _p^+x)\\widetilde{\\varphi }_{1,p}(x)dx\\\\&=& \\underbrace{\\frac{1}{u_p(x_p^+)^p}\\int _{\\widetilde{\\Omega }_p\\cap \\lbrace \|x\|\\le R\\rbrace }\|u_p(x_p^++\\mu _p^+x)\|^{p-1}u_p(x_p^++\\mu _p^+x)\\widetilde{\\varphi }_{1,p}(x)dx}_{E_{p,R}}\\\\&&+ \\underbrace{\\frac{1}{u_p(x_p^+)^p}\\int _{\\widetilde{\\Omega }_p\\cap \\lbrace \|x\|> R\\rbrace }\|u_p(x_p^++\\mu _p^+x)\|^{p-1}u_p(x_p^++\\mu _p^+x)\\widetilde{\\varphi }_{1,p}(x)dx}_{F_{p,R}}$ By Hölder inequality, the convergence of $v_p^+$ to $U$ in $C^1_{loc}(\\mathbb {R}^2)$ up to a subsequence (see Lemma REF ) and Corollary REF we have, for $R>0$ fixed: $\\left\|E_{p,R}-\\int _{\\lbrace \|x\|\\le R\\rbrace }e^U\\varphi _1^\\right\| &\\le &\\int _{\\widetilde{\\Omega }_p\\cap \\lbrace \|x\|\\le R\\rbrace }\\left\| \\frac{u_p(x_p^++\\mu _p^+x)}{u_p(x_p^+)} \\right\|^{p}\|\\widetilde{\\varphi }_{1,p}(x)-\\varphi _1^(x)\|dx\\\\&& + \\int _{\\lbrace \|x\|\\le R\\rbrace }\\varphi _1^(x)\\left\|\\left\|1+\\frac{v_p^+(x)}{p} \\right\|^{p-1}\\left( 1+\\frac{v_p^+(x)}{p} \\right) -e^U(x)\\right\|dx\\\\&\\le & \\Vert \\widetilde{\\varphi }_{1,p}-\\varphi _1^\\Vert _{L^2(\\mathbb {R}^2)} \\left[\\int _{\\widetilde{\\Omega }_p\\cap \\lbrace \|x\|\\le R\\rbrace }\\left\| 1+\\frac{v_p^+(x)}{p} \\right\|^{2p}\\right]^{\\frac{1}{2}}\\\\&& +\\sup _{\\lbrace \|x\|\\le R\\rbrace } \\left\|\\left\|1+\\frac{v_p^+(x)}{p} \\right\|^{p-1}\\left( 1+\\frac{v_p^+(x)}{p} \\right) -e^U(x)\\right\| \\int _{\\lbrace \|x\|\\le R\\rbrace }\\varphi _1^(x) dx\\\\& \\rightarrow & 0,$ as $p\\rightarrow +\\infty $ , up to a subsequence.", "For $R$ sufficiently large the term $\\int _{\\lbrace \|x\|> R\\rbrace }e^U\\varphi _1^dx$ may be made arbitrary small since $e^U\\in L^1(\\mathbb {R}^2)$ and $\\varphi _1^$ is bounded (Proposition REF - iv)).", "Using Hölder inequality, $\\Vert \\widetilde{\\varphi }_{1,p}\\Vert _{L^2(\\widetilde{\\Omega }_p)}=1 $ , assumption $(A)$ and (REF ) we have $\|F_{p,R}\|^2&\\le & \\Vert \\widetilde{\\varphi }_{1,p}\\Vert ^2_{L^2(\\widetilde{\\Omega }_p)}\\int _{\\widetilde{\\Omega }_p\\cap \\lbrace \|x\|> R\\rbrace }\\left\|\\frac{u_p(x_p^++\\mu _p^+x)}{u_p(x_p^+)}\\right\|^{2p}dx\\\\&= & \\int _{\\widetilde{\\Omega }_p\\cap \\lbrace \|x\|> R\\rbrace }\\left\|\\frac{u_p(x_p^++\\mu _p^+x)}{u_p(x_p^+)}\\right\|^{2p}dx\\\\&= &\\frac{1}{u_p(x_p^+)^2} \\int _{\\Omega \\cap \\lbrace \|y-x_p^+\|>R\\mu _p^+\\rbrace }\\frac{p\|u_p(y)\|^{2p}}{u_p(x_p^+)^{p-1}}dy\\\\&\\le & \\frac{p\\int _{\\Omega }\|u_p\|^{p+1}}{u_p(x_p^+)^2}\\left[ \\sup _{\\Omega \\cap \\lbrace \|y-x_p^+\|>R\\mu _p^+\\rbrace }\\left\| \\frac{u_p(y)}{u_p(x_p^+)}\\right\|^{p-1}\\ \\right]\\\\& \\stackrel{(A)+(\\ref {maxBoundedAwayFromZero})}{ \\le } & C \\mathcal {S}_{p,R}.$ And so by assumption $(B)$ we have $\\lim _{R\\rightarrow +\\infty }\\lim _{p\\rightarrow +\\infty }F_{p,R}=0.$" ], [ "A sufficient condition for $(B)$", "Next property is a sufficient condition for $(B)$ : Lemma 5.1 Assume that there exists $C>0$ such that $\|x-x_p^+\|^2 p\|u_p(x)\|^{p-1}\\le C$ for all $p$ sufficiently large and all $x\\in \\Omega $ .", "Then condition $(B)$ holds true up to a subsequence in $p$ .", "Let $R>0$ fixed and let $y\\in \\Omega $ , $\|y-x_p^+\|> R\\mu _p^+$ , then for $p$ large, by (REF ) $\\left\|\\frac{u_p(y)}{u_p(x_p^+)}\\right\|^{p-1}= \\frac{p\|u_p(y)\|^{p-1}}{p\|u_p(x_p^+)\|^{p-1}}\\stackrel{(\\ref {P_3^1})}{\\le }\\frac{C}{\|y-x_p^+\|^2}\\frac{1}{p\|u_p(x_p^+)\|^{p-1}}\\le \\frac{C}{R^2(\\mu _p^+)^2}\\frac{1}{p\|u_p(x_p^+)\|^{p-1}}=\\frac{C}{R^2}$ hence $0\\le \\ \\limsup _{p\\rightarrow +\\infty }\\mathcal {S}_{p,R}\\le \\frac{C}{R^2}$ and $(B)$ follows, up to a subsequence in $p$ , passing to the limit as $R\\rightarrow +\\infty $ .", "Condition (REF ) is a special case of a more general result that has been proved in [3] for any family $(u_p)$ of solutions to (REF ) under condition $(A)$ and which we recall here.", "Given $n\\in \\mathbb {N}\\setminus \\lbrace 0\\rbrace $ families of points $(x_{i,p})$ , $i=1,\\ldots ,n$ in $\\Omega $ such that $p\|u_p(x_{i,p})\|^{p-1}\\rightarrow +\\infty \\ \\mbox{ as }\\ p\\rightarrow +\\infty ,$ which implies in particular $\\liminf _{p\\rightarrow +\\infty } u_p(x_{i,p})\\ge 1,$ we define the parameters $\\mu _{i,p}$ by $\\mu _{i,p}^{-2}=p \|u_p(x_{i,p})\|^{p-1},\\ \\mbox{ for all }\\ i=1,\\ldots ,n,$ and we introduce the following properties: $(\\mathcal {P}_1^n)$ For any $i,j\\in \\lbrace 1,\\ldots ,n\\rbrace $ , $i\\ne j$ , $\\lim _{p\\rightarrow +\\infty }\\frac{\|x_{i,p}-x_{j,p}\|}{\\mu _{i,p}}=+\\infty .$ $(\\mathcal {P}_2^n)$ For any $i=1,\\ldots ,n$ , $v_{i,p}(x):=\\frac{p}{u_p(x_{i,p})}(u_p(x_{i,p}+\\mu _{i,p}x)-u_p(x_{i,p}))\\longrightarrow U(x)$ in $C^1_{loc}(\\mathbb {R}^2)$ as $p\\rightarrow +\\infty $ , where $U(x)=\\log \\left(\\frac{1}{1+\\frac{1}{8} \|x\|^2}\\right)^2$ is the solution of $-\\Delta U=e^{U}$ in $\\mathbb {R}^2$ , $U\\le 0$ , $U(0)=0$ and $\\int _{\\mathbb {R}^2}e^{U}=8\\pi $ .", "Moreover $\\frac{d(x_{i,p},\\partial \\Omega )}{\\mu _{i,p}}\\rightarrow +\\infty \\quad \\mbox{ and } \\quad \\frac{d(x_{i,p},NL_p)}{\\mu _{i,p}}\\rightarrow +\\infty \\ \\mbox{ as }\\ p\\rightarrow +\\infty .$ $(\\mathcal {P}_3^n)$ There exists $C>0$ such that $p \\min _{i=1,\\ldots ,n} \|x-x_{i,p}\|^2 \|u_p(x)\|^{p-1}\\le C$ for all $p$ sufficiently large and all $x\\in \\Omega $ .", "Proposition 5.2 ([3]) Let $(u_p)$ be a family of solutions to (REF ) and assume that $(A)$ holds.", "Then there exist $k\\in \\mathbb {N}\\setminus \\lbrace 0\\rbrace $ and $k$ families of points $(x_{i,p})$ in $\\Omega $ $i=1,\\ldots , k$ such that $x_{1,p}=x_p^+$ and, after passing to a sequence, $(\\mathcal {P}_1^k)$ , $(\\mathcal {P}_2^k)$ , and $(\\mathcal {P}_3^k)$ hold.", "Moreover, given any family of points $x_{k+1,p}$ , it is impossible to extract a new sequence from the previous one such that $(\\mathcal {P}_1^{k+1})$ , $(\\mathcal {P}_2^{k+1})$ , and $(\\mathcal {P}_3^{k+1})$ hold with the sequences $(x_{i,p})$ , $i=1,\\ldots ,k+1$ .", "At last, we have $\\sqrt{p}u_p\\rightarrow 0\\quad \\textrm { in $ 1loc({p xi,p, i=1,..., k}) $ as $ p+$.", "}$$$ Theorem 5.3 Let $(u_p)$ be a family of solutions to (REF ) which satisfies condition $(A)$ .", "Let $k\\in \\mathbb {N}\\setminus \\lbrace 0\\rbrace $ be the maximal number of families of points $(x_{i,p})$ , $i=1,\\ldots , k$ , for which $(P^k_1)$ , $(P^k_2)$ and $(P^k_3)$ hold up to a subsequence (as in Proposition REF ).", "If $k=1$ then condition $(B)$ holds true up to a subsequence in $p$ .", "By $(A)$ Proposition REF applies.", "Just observe that $x_{1,p}=x_p^+$ and so when $k=1$ property $(\\mathcal {P}_3^k)$ reduces to (REF ) and so the conclusion follows by Lemma REF .", "In the following we use Theorem REF to obtain condition $(B)$ for suitable classes of solutions." ], [ "Proof of Theorem ", "Before proving Theorem REF we observe that the existence of sign changing stationary solutions $u_p$ to (REF ) satisfying assumptions (REF ) and (REF ) has been proved for $m\\ge 4$ in [2] for $p$ large.", "The proof uses the fact that the energy is decreasing along non constant solutions, and relies on constructing a suitable initial condition $v_0$ for problem (REF ) such that any stationary solution in the corresponding $\\omega $ -limit set satisfies the energy estimate (REF ).", "This construction can be done for $p$ large even without any symmetry assumption on $\\Omega $ (see [2] for details).", "Anyway when $\\Omega $ is a simply connected $G$ -symmetric smooth bounded domain with $\|G\|\\ge m$ also some qualitative properties of $u_p$ under condition (REF ) may be obtained (for instance the nodal line does not touch $\\partial \\Omega $ , it does not pass through the origin, etc, as shown in [2]).", "Then, in [3] a deeper asymptotic analysis of $u_p$ as $p\\rightarrow +\\infty $ has been done, showing concentration in the origin and a bubble tower behavior, when $\\Omega $ is a simply connected $G$ -symmetric smooth bounded domain with $\|G\|\\ge m e$ .", "Here we do not require $\\Omega $ to be simply connected.", "Clearly assumption (REF ) is a special case of condition $(A)$ , hence in particular Proposition REF holds.", "As before we assume w.l.o.g.", "that $\\Vert u_p\\Vert _{\\infty }=\\Vert u_p^+\\Vert _{\\infty }$ .", "The proof of Theorem REF follows then from the following Proposition 6.1 Let $\\Omega \\subset \\mathbb {R}^2$ be a smooth bounded domain, $O\\in \\Omega $ , invariant under the action of a cyclic group $G$ of rotations about the origin which satisfies (REF ) for a certain $m>0$ .", "Let $(u_p)$ be a family of sign changing $G$ -symmetric stationary solutions of (REF ) which satisfies (REF ).", "Then condition $(B)$ is satisfied up to a subsequence.", "As we will see Proposition REF is a consequence of the general sufficient condition in Theorem REF .", "Hence in order to prove it we only need to show that $k=1$ , where the number $k$ is the maximal number of families of points $(x_{i,p})$ , $i=1,\\ldots , k$ , for which $(P^k_1)$ , $(P^k_2)$ and $(P^k_3)$ hold, up to a subsequence, as in Proposition REF .", "When $m=4$ the result has been already proved in [3].", "Here we show the general case (see also [3]).", "We start with the following: Lemma 6.2 Let $\\Omega \\subset \\mathbb {R}^2$ be a smooth bounded domain, $O\\in \\Omega $ , invariant under the action of a cyclic group $G$ of rotations about the origin which satisfies (REF ) for a certain $m>0$ .", "Let $(u_p)$ be a family of sign changing $G$ -symmetric stationary solutions of (REF ) which satisfies (REF ).", "Let $k, x_{i,p}$ and $\\mu _{i,p}$ for $i=1,\\ldots , k$ be as in Proposition REF .", "Then $\\frac{\|x_{i,p}\|}{\\mu _{i,p}} \\ \\mbox{ is bounded.", "}$ The proof is similar to the one of [3].", "Let us fix $i\\in \\lbrace 1,\\ldots , k\\rbrace $ .", "In order to simplify the notation we drop the dependence on $i$ namely we set $x_{p}:=x_{i,p}$ and $\\mu _{p}:=\\mu _{i,p}$ .", "Without loss of generality we can assume that either $(x_{p})_p\\subset \\mathcal {N}_p^{+}$ or $(x_{p})_p\\subset \\mathcal {N}_p^{-}$ .", "We prove the result in the case $(x_{p})_p\\subset \\mathcal {N}_p^{+}$ , the other case being similar.", "Let $h:=\|G\|$ , ($\\mathbb {N}\\setminus \\lbrace 0\\rbrace \\ni h\\ge me$ ) and let us denote by $g^j$ , $j=0,\\dots , h-1$ , the elements of $G$ .", "We consider the rescaled nodal domains $\\widetilde{\\mathcal {N}_{p}^+}^{j} :=\\lbrace x\\in \\mathbb {R}^2\\ : \\ \\mu _p x +g^jx_p\\in \\mathcal {N}_p^+\\rbrace ,\\ \\ j=0,\\dots , h-1,$ and the rescaled functions $z_{p}^{j,+}(x): \\widetilde{\\mathcal {N}_{p}^+}^{j} \\rightarrow \\mathbb {R}$ defined by $ z_{p}^{j,+}(x):=\\frac{p}{u_{p}^+(x_{p})}\\left( u_{p}^+(\\mu _{p} x+g^jx_{p})-u_{p}^+(x_{p}) \\right), \\ \\ j=0,\\dots , h-1.$ Observe that, since $\\Omega $ is $G$ -invariant, $g^j x_p\\in \\Omega $ for any $j=0,\\dots , h-1$ .", "Moreover $u_{p}$ is $G$ -symmetric and $x_p$ satisfies (REF ), hence it's not difficult to see from $(\\mathcal {P}_2^{k})$ that each function $z_{p}^{j,+}$ converges to $U$ in $C^1_{loc}(\\mathbb {R}^2)$ , as $p\\rightarrow \\infty $ and $8\\pi =\\int _{\\mathbb {R}^2}e^{U}dx$ (see also [3]).", "Assume by contradiction that there exists a sequence $p_n\\rightarrow +\\infty $ such that $\\frac{\|x_{p_n}\|}{\\mu _{p_n}}\\rightarrow + \\infty $ .", "Let $d_n:=\|g^j x_{p_n}-g^{j+1}x_{p_n}\|,\\quad j=0,..,h-1.$ Then, since the $h$ distinct points $g^j x_{p_n}$ , $j=0,\\ldots , h-1$ , are the vertices of a regular polygon centered in $O$ , $d_n=2\\widetilde{d}_n \\sin {\\frac{\\pi }{h}}$ , where $\\widetilde{d}_n:=\|g^jx_{p_n}\|\\equiv \|x_{p_n}\|$ , $j=0,..,h-1$ .", "Hence $\\frac{d_n}{\\mu _{p_n}}\\rightarrow +\\infty .$ Let $R_{n}:=\\min \\left\\lbrace \\frac{d_n}{3},\\frac{d(x_{p_n},\\partial \\Omega )}{2}, \\frac{d(x_{p_n}, NL_{p_n})}{2}\\right\\rbrace ,$ then by (REF ) and (REF ) $\\frac{R_n}{\\mu _{p_n}}\\rightarrow +\\infty ,$ moreover, by construction, $& B_{R_n}(g^j x_{p_n})\\subseteq \\mathcal {N}_{p_n}^+, \\ \\ \\mbox{ for }\\ j=0,\\dots ,h-1 \\\\&B_{R_n}(g^j x_{p_n})\\cap B_{R_n}(g^l x_{p_n}) =\\emptyset ,\\ \\ \\mbox{ for }j\\ne l. $ Using (REF ), the convergence of $z_{p_n}^{j,+}$ to $U$ , (REF ) and Fatou's lemma, we have $8\\pi &=&\\int _{\\mathbb {R}^2}e^{U}dx\\nonumber \\\\&\\stackrel{\\textrm {Fatou + conv.", "of v_{p_n}^j} + (\\ref {invadeR2})}{\\le }& \\lim _n \\int _{ B_{\\frac{R_n}{\\mu _{p_n}}}(0) } e^{z_{p_n}^{j,+} + (p_n+1)\\left(\\log {\\left\|1+\\frac{z_{p_n}^{j,+}}{p_n}\\right\|}-\\frac{z_{p_n}^{j,+}}{(p_n+1)}\\right)}dx\\nonumber \\\\&= &\\lim _n \\int _{B_{\\frac{R_n}{\\mu _{p_n}}}(0)}\\left\|1+\\frac{z_{p_n}^{j,+}(x)}{p_n} \\right\|^{(p_n+1)}dx\\nonumber \\\\&=& \\lim _n \\int _{B_{\\frac{R_n}{\\mu _{p_n}}}(0)}\\left\|\\frac{ u^+_{p_n}(\\mu _{p_n} x+g^jx_{p_n})}{ u^+_{p_n}(x_{p_n})}dx \\right\|^{(p_n+1)}dx\\nonumber \\\\&=& \\lim _n \\int _{B_{R_n}(g^jx_{p_n})}\\frac{\\left\| u^+_{p_n}(x)\\right\|^{(p_n+1)}}{(\\mu _{p_n})^2 \\left\|u^+_{p_n}(x_{p_n})\\right\|^{(p_n+1)}}dx\\nonumber \\\\&=&\\lim _n \\frac{p_n}{\\left\|u^+_{p_n}(x_{p_n})\\right\|^2} \\int _{B_{R_n}(g^jx_{p_n})} \\left\| u^+_{p_n}(x)\\right\|^{(p_n+1)}dx\\nonumber \\\\&\\stackrel{(\\ref {seqmaggioridiuno})}{\\le }&\\lim _n p_n\\int _{B_{R_n}(g^jx_{p_n})} \\left\| u^+_{p_n}(x)\\right\|^{(p_n+1)}dx.$ Summing on $j=0,\\dots , h-1$ , using (), (REF ) and assumption (REF ) we get: $h\\cdot 8\\pi &\\le &\\lim _n\\ p_n \\sum _{j=0}^{h-1} \\int _{B_{R_n}(g^jx_{p_n})} \\left\| u^+_{p_n}(x)\\right\|^{(p_n+1)}dx\\\\&\\stackrel{(\\ref {palleDisgiunte}) + (\\ref {contenutoInOmega}) }{\\le }&\\lim _n\\ p_n\\int _{\\mathcal {N}_{p_n}^+} \\left\| u_{p_n}(x)\\right\|^{(p_n+1)}dx\\\\&=&\\lim _n\\ \\left( p_n\\int _{\\Omega } \\left\| u_{p_n}(x)\\right\|^{(p_n+1)}dx - p_n\\int _{\\mathcal {N}_{p_n}^-} \\left\| u_{p_n}(x)\\right\|^{(p_n+1)}dx \\right)\\\\&\\stackrel{{\\footnotesize \\mbox{\\cite [Lemma 3.1]{DeMarchisIanniPacella2}}}}{\\le } &\\lim _n\\ p_n\\int _{\\Omega } \\left\| u_{p_n}(x)\\right\|^{(p_n+1)}dx -\\ 8\\pi e\\\\&\\stackrel{(\\ref {energiaLimitata})}{\\le }& (\\alpha -1)\\ 8\\pi e\\\\&< & m\\ 8\\pi e$ which gives a contradiction with (REF ).", "Last using Lemma REF we can prove that the number $k$ in Proposition REF is equal to one.", "Lemma 6.3 Let $\\Omega \\subset \\mathbb {R}^2$ be a smooth bounded domain, $O\\in \\Omega $ , invariant under the action of a cyclic group $G$ of rotations about the origin which satisfies (REF ) for a certain $m>0$ .", "Let $(u_p)$ be a family of sign changing $G$ -symmetric stationary solutions of (REF ) which satisfies (REF ).", "Let $k$ be, as in Proposition REF , the maximal number of families of points $(x_{i,p})$ , $i=1,\\ldots , k$ , for which, after passing to a subsequence, $(P^k_1)$ , $(P^k_2)$ and $(P^k_3)$ hold.", "Then $k=1.$ The proof is the same as in [3], we repeat it for completeness.", "Let us assume by contradiction that $k > 1$ and set $x^+_p=x_{1,p}$ .", "For a family $(x_{j,p})$ , $j\\in \\lbrace 2,\\ldots , k\\rbrace $ by Lemma REF , there exists $C>0$ such that $\\frac{\|x_{1,p}\|}{\\mu _{1,p}}\\le C\\quad \\textrm {and}\\quad \\frac{\|x_{j,p}\|}{\\mu _{j,p}}\\le C.$ Thus, since by definition $\\mu ^+_p=\\mu _{1,p}\\le \\mu _{j,p}$ , also $\\frac{\|x_{1,p}\|}{\\mu _{j,p}}\\le C.$ Hence $\\frac{\|x_{1,p}-x_{j,p}\|}{\\mu _{j,p}}\\le \\frac{\|x_{1,p}\|+\|x_{j,p}\|}{\\mu _{j,p}}\\le C\\quad \\textrm {as $ +$},$$which contradicts $ (P1k)$.$ A special case in Theorem REF : the radial solutions In this section we show that, when the domain $\\Omega $ is the unit ball in $\\mathbb {R}^2$ , the unique (up to a sign) radial solution $u_{p,\\mathcal {K}}$ of (REF ) with $\\mathcal {K}\\ge 2$ nodal regions satisfies conditions $(A)$ and $(B)$ .", "Thus Theorem REF applies to $u_{p,\\mathcal {K}}$ , namely we re-obtain the result already known from [4] through a different proof which does not rely on radial arguments.", "Let us fix the number of nodal regions $\\mathcal {K}\\ge 2$ .", "As before we assume w.l.o.g.", "that $\\Vert u_{p,\\mathcal {K}}\\Vert _{\\infty }=\\Vert u_{p,\\mathcal {K}}^+\\Vert _{\\infty }$ .", "The main result is the following: Proposition 7.1 Let $\\Omega $ be the unit ball in $\\mathbb {R}^2$ and for $\\mathcal {K}\\ge 2$ let $u_{p,\\mathcal {K}}$ be the unique radial solution of (REF ) with $\\mathcal {K}$ nodal domains.", "Then there exists $m (=m(\\mathcal {K}))$ $>0$ for which the assumptions of Theorem REF are satisfied.", "In [4] it has been proved that $u_{p,\\mathcal {K}}$ satisfies assumption $(A)$ (by extending the arguments employed in [2] for the case with two nodal regions).", "Hence there exists $C(=C(\\mathcal {K}))$ such that $p\\int _{\\Omega }\|u_{p,\\mathcal {K}}\|^{p+1}dx\\le C.$ Let us define $\\alpha :=\\frac{C}{8\\pi e}$ and let $\\bar{m}>0$ be such that $\\bar{m}>\\frac{C}{8\\pi e} -1.$ Let $G$ be a cyclic group of rotations about the origin such that $\|G\|\\ge \\bar{m}e$ .", "Of course the unit ball is $G$ invariant, moreover, since $u_{p,\\mathcal {K}}$ is radial, it is in particular $G$ -symmetric and so we have proved that (REF ) and (REF ) hold true with $m=\\bar{m}$ .", "We conclude the section with some more consideration on condition $(B)$ in the radial case.", "It is easy to show that (see [4] for the proof) $\\Vert u_{p,\\mathcal {K}}\\Vert _{\\infty }=u_{p,\\mathcal {K}}(0)\\ (>0)$ , namely $x_p^+\\equiv 0$ .", "Hence condition $(B)$ in this radial case reads as follows: $ \\lim _{R\\rightarrow +\\infty }\\lim _{p\\rightarrow +\\infty } S_{p,R}=0,\\qquad \\mathrm {(B)}$ where, for $R>0$ , $S_{p,R}:=\\sup \\left\\lbrace \\left\|\\frac{u_{p,\\mathcal {K}}(r)}{u_{p,\\mathcal {K}}(0)}\\right\|^{p-1}\\ : \\ \\ R\\mu _p^+<r<1 \\right\\rbrace $ and $u_{p,\\mathcal {K}}(r)=u_{p,\\mathcal {K}}(\|x\|)$ , $r=\|x\|$ .", "In addition to the general characterization in Proposition REF , it is easy to prove in the radial case, also the following characterization of condition $(B)$ : Proposition 7.2 Let $\\Omega $ be the unit ball in $\\mathbb {R}^2$ and for $\\mathcal {K}\\ge 2$ let $u_{p,\\mathcal {K}}$ be the unique radial solution of (REF ) with $\\mathcal {K}$ nodal domains.", "Set $0< r_{p,1} <r_{p,2}<\\ldots <r_{p,\\mathcal {K}-1} <1 $ the nodal radii of $u_{p,\\mathcal {K}}(r)$ .", "Then for any $R>0$ there exists $p_R>1$ such that the set $\\lbrace R\\mu _p^+<r<r_{p,1} \\rbrace \\ne \\emptyset $ for $p\\ge p_R$ and so $ \\mathcal {M}^{\\prime }_{p,R}:=\\sup \\left\\lbrace \\left\|\\frac{u_{p,\\mathcal {K}}(r)}{u_{p,\\mathcal {K}}(0)}\\right\|^{p-1}\\ : \\ R\\mu _p^+<r< r_{p,1} \\right\\rbrace $ is well defined.", "Moreover condition $(B)$ is equivalent to $\\left\\lbrace \\begin{array}{lr}\\displaystyle { \\lim _{p\\rightarrow +\\infty }}\\sup _{\\lbrace r_{p,1}<r<1\\rbrace }\\frac{ \|u_{p,\\mathcal {K}}(r)\|^{p-1}}{u_p(0)^{p-1}}= 0 & \\qquad (B1^{\\prime })\\\\\\\\\\displaystyle {\\lim _{R\\rightarrow +\\infty }}\\lim _{p\\rightarrow +\\infty } \\mathcal {M}^{\\prime }_{p,R}=0 & \\qquad (B2^{\\prime })\\end{array}\\right.$ Remark 7.3 Observe that $(B1^{\\prime })$ was already known, indeed in [4] the authors proved that $\\sup _{\\lbrace r_{p,1}<r<1\\rbrace }\\frac{ \|u_{p,\\mathcal {K}}(r)\|}{u_{p,\\mathcal {K}}(0)}\\longrightarrow \\alpha <\\frac{1}{2}\\ \\mbox{ as }\\ p\\rightarrow +\\infty .$ Before proving Theorem REF we observe that the existence of sign changing stationary solutions $u_p$ to (REF ) satisfying assumptions (REF ) and (REF ) has been proved for $m\\ge 4$ in [2] for $p$ large.", "The proof uses the fact that the energy is decreasing along non constant solutions, and relies on constructing a suitable initial condition $v_0$ for problem (REF ) such that any stationary solution in the corresponding $\\omega $ -limit set satisfies the energy estimate (REF ).", "This construction can be done for $p$ large even without any symmetry assumption on $\\Omega $ (see [2] for details).", "Anyway when $\\Omega $ is a simply connected $G$ -symmetric smooth bounded domain with $\|G\|\\ge m$ also some qualitative properties of $u_p$ under condition (REF ) may be obtained (for instance the nodal line does not touch $\\partial \\Omega $ , it does not pass through the origin, etc, as shown in [2]).", "Then, in [3] a deeper asymptotic analysis of $u_p$ as $p\\rightarrow +\\infty $ has been done, showing concentration in the origin and a bubble tower behavior, when $\\Omega $ is a simply connected $G$ -symmetric smooth bounded domain with $\|G\|\\ge m e$ .", "Here we do not require $\\Omega $ to be simply connected.", "Clearly assumption (REF ) is a special case of condition $(A)$ , hence in particular Proposition REF holds.", "As before we assume w.l.o.g.", "that $\\Vert u_p\\Vert _{\\infty }=\\Vert u_p^+\\Vert _{\\infty }$ .", "The proof of Theorem REF follows then from the following Proposition 6.1 Let $\\Omega \\subset \\mathbb {R}^2$ be a smooth bounded domain, $O\\in \\Omega $ , invariant under the action of a cyclic group $G$ of rotations about the origin which satisfies (REF ) for a certain $m>0$ .", "Let $(u_p)$ be a family of sign changing $G$ -symmetric stationary solutions of (REF ) which satisfies (REF ).", "Then condition $(B)$ is satisfied up to a subsequence.", "As we will see Proposition REF is a consequence of the general sufficient condition in Theorem REF .", "Hence in order to prove it we only need to show that $k=1$ , where the number $k$ is the maximal number of families of points $(x_{i,p})$ , $i=1,\\ldots , k$ , for which $(P^k_1)$ , $(P^k_2)$ and $(P^k_3)$ hold, up to a subsequence, as in Proposition REF .", "When $m=4$ the result has been already proved in [3].", "Here we show the general case (see also [3]).", "We start with the following: Lemma 6.2 Let $\\Omega \\subset \\mathbb {R}^2$ be a smooth bounded domain, $O\\in \\Omega $ , invariant under the action of a cyclic group $G$ of rotations about the origin which satisfies (REF ) for a certain $m>0$ .", "Let $(u_p)$ be a family of sign changing $G$ -symmetric stationary solutions of (REF ) which satisfies (REF ).", "Let $k, x_{i,p}$ and $\\mu _{i,p}$ for $i=1,\\ldots , k$ be as in Proposition REF .", "Then $\\frac{\|x_{i,p}\|}{\\mu _{i,p}} \\ \\mbox{ is bounded.", "}$ The proof is similar to the one of [3].", "Let us fix $i\\in \\lbrace 1,\\ldots , k\\rbrace $ .", "In order to simplify the notation we drop the dependence on $i$ namely we set $x_{p}:=x_{i,p}$ and $\\mu _{p}:=\\mu _{i,p}$ .", "Without loss of generality we can assume that either $(x_{p})_p\\subset \\mathcal {N}_p^{+}$ or $(x_{p})_p\\subset \\mathcal {N}_p^{-}$ .", "We prove the result in the case $(x_{p})_p\\subset \\mathcal {N}_p^{+}$ , the other case being similar.", "Let $h:=\|G\|$ , ($\\mathbb {N}\\setminus \\lbrace 0\\rbrace \\ni h\\ge me$ ) and let us denote by $g^j$ , $j=0,\\dots , h-1$ , the elements of $G$ .", "We consider the rescaled nodal domains $\\widetilde{\\mathcal {N}_{p}^+}^{j} :=\\lbrace x\\in \\mathbb {R}^2\\ : \\ \\mu _p x +g^jx_p\\in \\mathcal {N}_p^+\\rbrace ,\\ \\ j=0,\\dots , h-1,$ and the rescaled functions $z_{p}^{j,+}(x): \\widetilde{\\mathcal {N}_{p}^+}^{j} \\rightarrow \\mathbb {R}$ defined by $ z_{p}^{j,+}(x):=\\frac{p}{u_{p}^+(x_{p})}\\left( u_{p}^+(\\mu _{p} x+g^jx_{p})-u_{p}^+(x_{p}) \\right), \\ \\ j=0,\\dots , h-1.$ Observe that, since $\\Omega $ is $G$ -invariant, $g^j x_p\\in \\Omega $ for any $j=0,\\dots , h-1$ .", "Moreover $u_{p}$ is $G$ -symmetric and $x_p$ satisfies (REF ), hence it's not difficult to see from $(\\mathcal {P}_2^{k})$ that each function $z_{p}^{j,+}$ converges to $U$ in $C^1_{loc}(\\mathbb {R}^2)$ , as $p\\rightarrow \\infty $ and $8\\pi =\\int _{\\mathbb {R}^2}e^{U}dx$ (see also [3]).", "Assume by contradiction that there exists a sequence $p_n\\rightarrow +\\infty $ such that $\\frac{\|x_{p_n}\|}{\\mu _{p_n}}\\rightarrow + \\infty $ .", "Let $d_n:=\|g^j x_{p_n}-g^{j+1}x_{p_n}\|,\\quad j=0,..,h-1.$ Then, since the $h$ distinct points $g^j x_{p_n}$ , $j=0,\\ldots , h-1$ , are the vertices of a regular polygon centered in $O$ , $d_n=2\\widetilde{d}_n \\sin {\\frac{\\pi }{h}}$ , where $\\widetilde{d}_n:=\|g^jx_{p_n}\|\\equiv \|x_{p_n}\|$ , $j=0,..,h-1$ .", "Hence $\\frac{d_n}{\\mu _{p_n}}\\rightarrow +\\infty .$ Let $R_{n}:=\\min \\left\\lbrace \\frac{d_n}{3},\\frac{d(x_{p_n},\\partial \\Omega )}{2}, \\frac{d(x_{p_n}, NL_{p_n})}{2}\\right\\rbrace ,$ then by (REF ) and (REF ) $\\frac{R_n}{\\mu _{p_n}}\\rightarrow +\\infty ,$ moreover, by construction, $& B_{R_n}(g^j x_{p_n})\\subseteq \\mathcal {N}_{p_n}^+, \\ \\ \\mbox{ for }\\ j=0,\\dots ,h-1 \\\\&B_{R_n}(g^j x_{p_n})\\cap B_{R_n}(g^l x_{p_n}) =\\emptyset ,\\ \\ \\mbox{ for }j\\ne l. $ Using (REF ), the convergence of $z_{p_n}^{j,+}$ to $U$ , (REF ) and Fatou's lemma, we have $8\\pi &=&\\int _{\\mathbb {R}^2}e^{U}dx\\nonumber \\\\&\\stackrel{\\textrm {Fatou + conv.", "of v_{p_n}^j} + (\\ref {invadeR2})}{\\le }& \\lim _n \\int _{ B_{\\frac{R_n}{\\mu _{p_n}}}(0) } e^{z_{p_n}^{j,+} + (p_n+1)\\left(\\log {\\left\|1+\\frac{z_{p_n}^{j,+}}{p_n}\\right\|}-\\frac{z_{p_n}^{j,+}}{(p_n+1)}\\right)}dx\\nonumber \\\\&= &\\lim _n \\int _{B_{\\frac{R_n}{\\mu _{p_n}}}(0)}\\left\|1+\\frac{z_{p_n}^{j,+}(x)}{p_n} \\right\|^{(p_n+1)}dx\\nonumber \\\\&=& \\lim _n \\int _{B_{\\frac{R_n}{\\mu _{p_n}}}(0)}\\left\|\\frac{ u^+_{p_n}(\\mu _{p_n} x+g^jx_{p_n})}{ u^+_{p_n}(x_{p_n})}dx \\right\|^{(p_n+1)}dx\\nonumber \\\\&=& \\lim _n \\int _{B_{R_n}(g^jx_{p_n})}\\frac{\\left\| u^+_{p_n}(x)\\right\|^{(p_n+1)}}{(\\mu _{p_n})^2 \\left\|u^+_{p_n}(x_{p_n})\\right\|^{(p_n+1)}}dx\\nonumber \\\\&=&\\lim _n \\frac{p_n}{\\left\|u^+_{p_n}(x_{p_n})\\right\|^2} \\int _{B_{R_n}(g^jx_{p_n})} \\left\| u^+_{p_n}(x)\\right\|^{(p_n+1)}dx\\nonumber \\\\&\\stackrel{(\\ref {seqmaggioridiuno})}{\\le }&\\lim _n p_n\\int _{B_{R_n}(g^jx_{p_n})} \\left\| u^+_{p_n}(x)\\right\|^{(p_n+1)}dx.$ Summing on $j=0,\\dots , h-1$ , using (), (REF ) and assumption (REF ) we get: $h\\cdot 8\\pi &\\le &\\lim _n\\ p_n \\sum _{j=0}^{h-1} \\int _{B_{R_n}(g^jx_{p_n})} \\left\| u^+_{p_n}(x)\\right\|^{(p_n+1)}dx\\\\&\\stackrel{(\\ref {palleDisgiunte}) + (\\ref {contenutoInOmega}) }{\\le }&\\lim _n\\ p_n\\int _{\\mathcal {N}_{p_n}^+} \\left\| u_{p_n}(x)\\right\|^{(p_n+1)}dx\\\\&=&\\lim _n\\ \\left( p_n\\int _{\\Omega } \\left\| u_{p_n}(x)\\right\|^{(p_n+1)}dx - p_n\\int _{\\mathcal {N}_{p_n}^-} \\left\| u_{p_n}(x)\\right\|^{(p_n+1)}dx \\right)\\\\&\\stackrel{{\\footnotesize \\mbox{\\cite [Lemma 3.1]{DeMarchisIanniPacella2}}}}{\\le } &\\lim _n\\ p_n\\int _{\\Omega } \\left\| u_{p_n}(x)\\right\|^{(p_n+1)}dx -\\ 8\\pi e\\\\&\\stackrel{(\\ref {energiaLimitata})}{\\le }& (\\alpha -1)\\ 8\\pi e\\\\&< & m\\ 8\\pi e$ which gives a contradiction with (REF ).", "Last using Lemma REF we can prove that the number $k$ in Proposition REF is equal to one.", "Lemma 6.3 Let $\\Omega \\subset \\mathbb {R}^2$ be a smooth bounded domain, $O\\in \\Omega $ , invariant under the action of a cyclic group $G$ of rotations about the origin which satisfies (REF ) for a certain $m>0$ .", "Let $(u_p)$ be a family of sign changing $G$ -symmetric stationary solutions of (REF ) which satisfies (REF ).", "Let $k$ be, as in Proposition REF , the maximal number of families of points $(x_{i,p})$ , $i=1,\\ldots , k$ , for which, after passing to a subsequence, $(P^k_1)$ , $(P^k_2)$ and $(P^k_3)$ hold.", "Then $k=1.$ The proof is the same as in [3], we repeat it for completeness.", "Let us assume by contradiction that $k > 1$ and set $x^+_p=x_{1,p}$ .", "For a family $(x_{j,p})$ , $j\\in \\lbrace 2,\\ldots , k\\rbrace $ by Lemma REF , there exists $C>0$ such that $\\frac{\|x_{1,p}\|}{\\mu _{1,p}}\\le C\\quad \\textrm {and}\\quad \\frac{\|x_{j,p}\|}{\\mu _{j,p}}\\le C.$ Thus, since by definition $\\mu ^+_p=\\mu _{1,p}\\le \\mu _{j,p}$ , also $\\frac{\|x_{1,p}\|}{\\mu _{j,p}}\\le C.$ Hence $\\frac{\|x_{1,p}-x_{j,p}\|}{\\mu _{j,p}}\\le \\frac{\|x_{1,p}\|+\|x_{j,p}\|}{\\mu _{j,p}}\\le C\\quad \\textrm {as $ +$},$$which contradicts $ (P1k)$.$ A special case in Theorem REF : the radial solutions In this section we show that, when the domain $\\Omega $ is the unit ball in $\\mathbb {R}^2$ , the unique (up to a sign) radial solution $u_{p,\\mathcal {K}}$ of (REF ) with $\\mathcal {K}\\ge 2$ nodal regions satisfies conditions $(A)$ and $(B)$ .", "Thus Theorem REF applies to $u_{p,\\mathcal {K}}$ , namely we re-obtain the result already known from [4] through a different proof which does not rely on radial arguments.", "Let us fix the number of nodal regions $\\mathcal {K}\\ge 2$ .", "As before we assume w.l.o.g.", "that $\\Vert u_{p,\\mathcal {K}}\\Vert _{\\infty }=\\Vert u_{p,\\mathcal {K}}^+\\Vert _{\\infty }$ .", "The main result is the following: Proposition 7.1 Let $\\Omega $ be the unit ball in $\\mathbb {R}^2$ and for $\\mathcal {K}\\ge 2$ let $u_{p,\\mathcal {K}}$ be the unique radial solution of (REF ) with $\\mathcal {K}$ nodal domains.", "Then there exists $m (=m(\\mathcal {K}))$ $>0$ for which the assumptions of Theorem REF are satisfied.", "In [4] it has been proved that $u_{p,\\mathcal {K}}$ satisfies assumption $(A)$ (by extending the arguments employed in [2] for the case with two nodal regions).", "Hence there exists $C(=C(\\mathcal {K}))$ such that $p\\int _{\\Omega }\|u_{p,\\mathcal {K}}\|^{p+1}dx\\le C.$ Let us define $\\alpha :=\\frac{C}{8\\pi e}$ and let $\\bar{m}>0$ be such that $\\bar{m}>\\frac{C}{8\\pi e} -1.$ Let $G$ be a cyclic group of rotations about the origin such that $\|G\|\\ge \\bar{m}e$ .", "Of course the unit ball is $G$ invariant, moreover, since $u_{p,\\mathcal {K}}$ is radial, it is in particular $G$ -symmetric and so we have proved that (REF ) and (REF ) hold true with $m=\\bar{m}$ .", "We conclude the section with some more consideration on condition $(B)$ in the radial case.", "It is easy to show that (see [4] for the proof) $\\Vert u_{p,\\mathcal {K}}\\Vert _{\\infty }=u_{p,\\mathcal {K}}(0)\\ (>0)$ , namely $x_p^+\\equiv 0$ .", "Hence condition $(B)$ in this radial case reads as follows: $ \\lim _{R\\rightarrow +\\infty }\\lim _{p\\rightarrow +\\infty } S_{p,R}=0,\\qquad \\mathrm {(B)}$ where, for $R>0$ , $S_{p,R}:=\\sup \\left\\lbrace \\left\|\\frac{u_{p,\\mathcal {K}}(r)}{u_{p,\\mathcal {K}}(0)}\\right\|^{p-1}\\ : \\ \\ R\\mu _p^+<r<1 \\right\\rbrace $ and $u_{p,\\mathcal {K}}(r)=u_{p,\\mathcal {K}}(\|x\|)$ , $r=\|x\|$ .", "In addition to the general characterization in Proposition REF , it is easy to prove in the radial case, also the following characterization of condition $(B)$ : Proposition 7.2 Let $\\Omega $ be the unit ball in $\\mathbb {R}^2$ and for $\\mathcal {K}\\ge 2$ let $u_{p,\\mathcal {K}}$ be the unique radial solution of (REF ) with $\\mathcal {K}$ nodal domains.", "Set $0< r_{p,1} <r_{p,2}<\\ldots <r_{p,\\mathcal {K}-1} <1 $ the nodal radii of $u_{p,\\mathcal {K}}(r)$ .", "Then for any $R>0$ there exists $p_R>1$ such that the set $\\lbrace R\\mu _p^+<r<r_{p,1} \\rbrace \\ne \\emptyset $ for $p\\ge p_R$ and so $ \\mathcal {M}^{\\prime }_{p,R}:=\\sup \\left\\lbrace \\left\|\\frac{u_{p,\\mathcal {K}}(r)}{u_{p,\\mathcal {K}}(0)}\\right\|^{p-1}\\ : \\ R\\mu _p^+<r< r_{p,1} \\right\\rbrace $ is well defined.", "Moreover condition $(B)$ is equivalent to $\\left\\lbrace \\begin{array}{lr}\\displaystyle { \\lim _{p\\rightarrow +\\infty }}\\sup _{\\lbrace r_{p,1}<r<1\\rbrace }\\frac{ \|u_{p,\\mathcal {K}}(r)\|^{p-1}}{u_p(0)^{p-1}}= 0 & \\qquad (B1^{\\prime })\\\\\\\\\\displaystyle {\\lim _{R\\rightarrow +\\infty }}\\lim _{p\\rightarrow +\\infty } \\mathcal {M}^{\\prime }_{p,R}=0 & \\qquad (B2^{\\prime })\\end{array}\\right.$ Remark 7.3 Observe that $(B1^{\\prime })$ was already known, indeed in [4] the authors proved that $\\sup _{\\lbrace r_{p,1}<r<1\\rbrace }\\frac{ \|u_{p,\\mathcal {K}}(r)\|}{u_{p,\\mathcal {K}}(0)}\\longrightarrow \\alpha <\\frac{1}{2}\\ \\mbox{ as }\\ p\\rightarrow +\\infty .$" ], [ "A special case in Theorem ", "In this section we show that, when the domain $\\Omega $ is the unit ball in $\\mathbb {R}^2$ , the unique (up to a sign) radial solution $u_{p,\\mathcal {K}}$ of (REF ) with $\\mathcal {K}\\ge 2$ nodal regions satisfies conditions $(A)$ and $(B)$ .", "Thus Theorem REF applies to $u_{p,\\mathcal {K}}$ , namely we re-obtain the result already known from [4] through a different proof which does not rely on radial arguments.", "Let us fix the number of nodal regions $\\mathcal {K}\\ge 2$ .", "As before we assume w.l.o.g.", "that $\\Vert u_{p,\\mathcal {K}}\\Vert _{\\infty }=\\Vert u_{p,\\mathcal {K}}^+\\Vert _{\\infty }$ .", "The main result is the following: Proposition 7.1 Let $\\Omega $ be the unit ball in $\\mathbb {R}^2$ and for $\\mathcal {K}\\ge 2$ let $u_{p,\\mathcal {K}}$ be the unique radial solution of (REF ) with $\\mathcal {K}$ nodal domains.", "Then there exists $m (=m(\\mathcal {K}))$ $>0$ for which the assumptions of Theorem REF are satisfied.", "In [4] it has been proved that $u_{p,\\mathcal {K}}$ satisfies assumption $(A)$ (by extending the arguments employed in [2] for the case with two nodal regions).", "Hence there exists $C(=C(\\mathcal {K}))$ such that $p\\int _{\\Omega }\|u_{p,\\mathcal {K}}\|^{p+1}dx\\le C.$ Let us define $\\alpha :=\\frac{C}{8\\pi e}$ and let $\\bar{m}>0$ be such that $\\bar{m}>\\frac{C}{8\\pi e} -1.$ Let $G$ be a cyclic group of rotations about the origin such that $\|G\|\\ge \\bar{m}e$ .", "Of course the unit ball is $G$ invariant, moreover, since $u_{p,\\mathcal {K}}$ is radial, it is in particular $G$ -symmetric and so we have proved that (REF ) and (REF ) hold true with $m=\\bar{m}$ .", "We conclude the section with some more consideration on condition $(B)$ in the radial case.", "It is easy to show that (see [4] for the proof) $\\Vert u_{p,\\mathcal {K}}\\Vert _{\\infty }=u_{p,\\mathcal {K}}(0)\\ (>0)$ , namely $x_p^+\\equiv 0$ .", "Hence condition $(B)$ in this radial case reads as follows: $ \\lim _{R\\rightarrow +\\infty }\\lim _{p\\rightarrow +\\infty } S_{p,R}=0,\\qquad \\mathrm {(B)}$ where, for $R>0$ , $S_{p,R}:=\\sup \\left\\lbrace \\left\|\\frac{u_{p,\\mathcal {K}}(r)}{u_{p,\\mathcal {K}}(0)}\\right\|^{p-1}\\ : \\ \\ R\\mu _p^+<r<1 \\right\\rbrace $ and $u_{p,\\mathcal {K}}(r)=u_{p,\\mathcal {K}}(\|x\|)$ , $r=\|x\|$ .", "In addition to the general characterization in Proposition REF , it is easy to prove in the radial case, also the following characterization of condition $(B)$ : Proposition 7.2 Let $\\Omega $ be the unit ball in $\\mathbb {R}^2$ and for $\\mathcal {K}\\ge 2$ let $u_{p,\\mathcal {K}}$ be the unique radial solution of (REF ) with $\\mathcal {K}$ nodal domains.", "Set $0< r_{p,1} <r_{p,2}<\\ldots <r_{p,\\mathcal {K}-1} <1 $ the nodal radii of $u_{p,\\mathcal {K}}(r)$ .", "Then for any $R>0$ there exists $p_R>1$ such that the set $\\lbrace R\\mu _p^+<r<r_{p,1} \\rbrace \\ne \\emptyset $ for $p\\ge p_R$ and so $ \\mathcal {M}^{\\prime }_{p,R}:=\\sup \\left\\lbrace \\left\|\\frac{u_{p,\\mathcal {K}}(r)}{u_{p,\\mathcal {K}}(0)}\\right\|^{p-1}\\ : \\ R\\mu _p^+<r< r_{p,1} \\right\\rbrace $ is well defined.", "Moreover condition $(B)$ is equivalent to $\\left\\lbrace \\begin{array}{lr}\\displaystyle { \\lim _{p\\rightarrow +\\infty }}\\sup _{\\lbrace r_{p,1}<r<1\\rbrace }\\frac{ \|u_{p,\\mathcal {K}}(r)\|^{p-1}}{u_p(0)^{p-1}}= 0 & \\qquad (B1^{\\prime })\\\\\\\\\\displaystyle {\\lim _{R\\rightarrow +\\infty }}\\lim _{p\\rightarrow +\\infty } \\mathcal {M}^{\\prime }_{p,R}=0 & \\qquad (B2^{\\prime })\\end{array}\\right.$ Remark 7.3 Observe that $(B1^{\\prime })$ was already known, indeed in [4] the authors proved that $\\sup _{\\lbrace r_{p,1}<r<1\\rbrace }\\frac{ \|u_{p,\\mathcal {K}}(r)\|}{u_{p,\\mathcal {K}}(0)}\\longrightarrow \\alpha <\\frac{1}{2}\\ \\mbox{ as }\\ p\\rightarrow +\\infty .$" ] ]	1403.0115
[ [ "Entanglement entropy of the $\\nu=1/2$ composite fermion non-Fermi liquid\n state" ], [ "Abstract The so-called ``non-Fermi liquid'' behavior is very common in strongly correlated systems.", "However, its operational definition in terms of ``what it is not'' is a major obstacle against theoretical understanding of this fascinating correlated state.", "Recently there has been much interest in entanglement entropy as a theoretical tool to study non-Fermi liquids.", "So far explicit calculations have been limited to models without direct experimental realizations.", "Here we focus on a two dimensional electron fluid under magnetic field and filling fraction $\\nu=1/2$, which is believed to be a non-Fermi liquid state.", "Using the composite fermion (CF) wave-function which captures the $\\nu=1/2$ state very accurately, we compute the second R\\'enyi entropy using variational Monte-Carlo technique and an efficient parallel algorithm.", "We find the entanglement entropy scales as $L\\log L$ with the length of the boundary $L$ as it does for free fermions, albeit with a pre-factor twice that of the free fermion.", "We contrast the results against theoretical conjectures and discuss the implications of the results." ], [ "Entanglement entropy of the $\\nu =1/2$ composite fermion non-Fermi liquid state.", "Junping Shao Department of Physics, Cornell University, Ithaca, NY 14853, USA Department of Physics, Binghamton University, Binghamton, NY 13902, USA Eun-Ah Kim Department of Physics, Cornell University, Ithaca, NY 14853, USA F.D.M.", "Haldane Department of Physics, Princeton University, Princeton, NJ 08544, USA Edward H. Rezayi Department of Physics, California State University Los Angeles, Los Angeles, CA 90032, USA The so-called “non-Fermi liquid” behavior is very common in strongly correlated systems.", "However, its operational definition in terms of “what it is not” is a major obstacle against theoretical understanding of this fascinating correlated state.", "Recently there has been much interest in entanglement entropy as a theoretical tool to study non-Fermi liquids.", "So far explicit calculations have been limited to models without direct experimental realizations.", "Here we focus on a two dimensional electron fluid under magnetic field and filling fraction $\\nu =1/2$ , which is believed to be a non-Fermi liquid state.", "Using the composite fermion (CF) wave-function which captures the $\\nu =1/2$ state very accurately, we compute the second Rényi entropy using variational Monte-Carlo technique and an efficient parallel algorithm.", "We find the entanglement entropy scales as $L\\log L$ with the length of the boundary $L$ as it does for free fermions, albeit with a pre-factor twice that of the free fermion.", "We contrast the results against theoretical conjectures and discuss the implications of the results.", "Despite its ubiquity in strongly correlated materials, the metallic `non-Fermi liquid' behavior has been challenging to characterize theoretically.", "At the phenomenological level, non-Fermi liquid behavior is defined by a metallic system exhibiting physical properties that are qualitatively inconsistent with Landau's Fermi-liquid theory.", "Examples of non-Fermi liquid metals include the strange-metal phase of the high T$_c$ cuprates[1], systems near a metallic quantum critical point[2], [3], [4] and two-dimensional electron system subject to a magnetic field at filling $\\nu =1/2$ (often referred to as Fermi-liquid-like state) [5], [6], [7].", "However, there are many ways in which a system can deviate from a normal Fermi-liquid, such as diverging effective mass, vanishing quasiparticle weight, and anomalous transport[8], [3], [9], [10], [11], [12] and little is known about how different forms of deviation can be related.", "Hence the theoretical challenge of addressing a problem without a weakly interacting quasiparticle description has been compounded by the lack of a measure that can be used to define and classify non-Fermi liquids.", "Here we turn to a quantum information measure that is sensitive to entangled nature of many-body wave-functions: the bi-partite entanglement entropy.", "For gapped systems, the entanglement entropy of the reduced density matrix $\\rho _{A}\\equiv \\mathrm {Tr}_{B}\|\\Psi \\rangle \\langle \\Psi \|$ of a subsystem $A$ with respect to its complement $B$ for a given ground state wave-function $ \\left\|\\Psi \\right\\rangle $ is widely believed to follow the area law, i.e., asymptotically proportional to the contact area of two subsystems, with rigorous arguments for lattice systems [13], [14].", "On the other hand, an explicit formula for a multiplicative logarithmic correction to the area law was suggested by PhysRevLett.96.100503 based on the Widom conjecture[16] and numerically confirmed in Ref.", "[17] for free fermions at dimensions $d>1$ .", "This dramatic violation of the area law for free fermions is in stark contrast to the area law found for critical bosons[14] up to subleading corrections[18], [19] and shows how Fermionic statistics by itself drives long-range entanglement in the presence of a Fermi surface.", "A key question is whether non-perturbative strong correlation effects would further enhance bi-partite entanglement entropy.", "Since the explicit form of bi-partite entanglement entropy found in Refs.", "[15], [17] follows from exact results on non-interacting one-dimensional Fermion systems associated each points in Fermi surface[20], strong interactions are likely to cause corrections to this explicit form.", "So far the only explicit results available for strongly interacting fermions at $d>1$ is by vishwanath for Gutzwiller projected two-dimensional (2D) fermi-surface which is a candidate ground state wave-function for a critical spin-liquid with spinon Fermi surface.", "Their variational Monte Carlo calculation of second Rényi entropy $S_2$ showed little change in both the functional dependence on $L_A$ the linear dimension of the subsystem $A$ (i.e., $S_2\\propto L_A\\log L_A$ ) and the coefficient upon projection.", "Following this numerical work, swinglesenthil argued that the entanglement entropy of certain non-Fermi liquid states would be given by the free fermion formula of Ref. [15].", "In this letter we focus on the composite fermion wave-function for the half-filled Landau level $\\nu =1/2$ state proposed by PhysRevLett.72.900 as a test case of non-Fermi liquids.", "Though it is a candidate wave-function, it is an electronic wave-function with strong numerical, theoretical and experimental support as a ground state wave-function capturing the observed $\\nu =1/2$ non-Fermi liquid state.", "First, the structure factor calculated with this wave-function shows a good agreement with the structure factor obtained from the exact ground state wave-function [23].", "Further, the wave-function is supported by field theoretical studies of fermions coupled to Chern-Simons gauge theory [8], [24]as it describes a state with diverging effective mass for fermions with flux attachment.", "Finally, the $\\nu =1/2$ state is experimentally established to be a non-Fermi liquid state with a Fermi surface supporting anomalous sound propagation [5], [6], [7], in agreement with expectations of Refs.", "[8], [24].", "We calculate second Rényi entropy $S_2$ using variational Monte Carlo techniques implementing an algorithm improved from those used in Refs.", "[21], [25].", "We will focus on the comparison between subsystem linear dimension $L_A$ dependence of $S_2$ for free fermions and for the CF wave-function.", "Rényi entanglement entropy and Widom formula.– The second Rényi entropy is defined as $S_{2}\\equiv -\\ln \\left[\\mathrm {Tr}_{A}\\left\\lbrace \\rho _{A}^{2}\\right\\rbrace \\right],$ where $\\rho _{A}\\equiv \\mathrm {Tr}_{B}\|\\Psi \\rangle \\langle \\Psi \|$ is the reduced density matrix of region $A$ .", "$S_2$ has become a quantity of growing interest as a measure of bi-partite entanglement since a convenient scheme for calculating $S_2$ using variational Monte Carlo technique was shown in Ref. [26].", "For free fermions the leading $L_A$ dependence of second Rényi entropy is given by[15], [17] $S_2&=\\dfrac{3}{4}c(\\mu )L_A\\log L_A+o(L_A\\log L_A), \\nonumber \\\\&c(\\mu )=\\dfrac{\\log 2}{\\pi ^{2}(2\\pi )^{d-1}}\\int _{\\partial \\Omega }dS_{x}\\int _{\\partial \\Gamma }dS_{k}\|\\mathbf {n}_{k}\\cdot \\mathbf {n}_{x}\|,$ where $\\mu $ is the chemical potential, $\\Omega $ is the real space region $A$ and $\\partial \\Gamma $ is the Fermi surface.", "$\\mathbf {n}_{x}$ and $\\mathbf {n}_{k}$ denote the normal vectors on the spatial boundary $\\partial \\Omega $ and the Fermi surface respectively.", "For Eq.", "(REF ), the linear dimension of the system is scaled to unity.", "In this work, we will consider 37 fermions in 2D occupying momenta shown in Fig.", "REF for both free fermions and for a $\\nu =1/2$ composite fermion non-Fermi liquid.", "A straight forward evaluation of $c(\\mu )$ for the Fermi surface shown in Fig.", "REF and a square-shaped region $A$ results in an asymptotic form for the second Rényi entropy $S_{2, {\\rm Widom}}\\sim (0.134)\\lambda \\log \\lambda $ as a prediction based on “Widom formula” Eq.", "(REF ).", "From here on we use the dimensionless quantity $\\lambda \\equiv k_F L_A$ , where $k_F$ is the radius of the Fermi surface.", "Figure: Fermi surface for N=37N=37 particles in 2D.", "The set of momenta are shown as blue points.", "Red circle denotes the Fermi surface ∂Γ\\partial \\Gamma of radius k F ≈10π/37k_F \\approx \\sqrt{10\\pi /37}.Monte Carlo evaluation of $S_2$ .– In order to calculate the Rényi entropy $S_2$ for the $\\nu =1/2$ CF wave-function, we use the scheme of Ref.", "[26] and consider two copies of the system to evaluate the expectation value of the SWAP operator which is related to $S_2$ as follows: $&e^{-S_{2}}\\nonumber \\\\&=\\sum _{\\beta _{1},\\beta _{2}}\\sum _{\\alpha _{1},\\alpha _{2}}\\langle \\beta _{2}\|\\langle \\alpha _{1}\|\\Psi \\!\\rangle \\!\\langle \\Psi \|\\alpha _{1}\\rangle \|\\beta _{1}\\rangle \\langle \\beta _{1}\|\\langle \\alpha _{2}\|\\Psi \\rangle \\langle \\Psi \|\\alpha _{2}\\rangle \|\\beta _{2}\\rangle \\nonumber \\\\&\\equiv \\left\\langle {\\rm SWAP}_A\\right\\rangle .$ Here $\\alpha _i$ and $\\beta _i$ , with $i=1,2$ for the two copies, are real space coordinates within each copy of subregions, i.e., $\\alpha _1\\in A_1,\\beta _1\\in B_1$ and $\\alpha _2\\in A_2,\\beta _2\\in B_2$ .", "Below we calculate the expectation value for the model wave-function by sampling the wave-function over the two copies, introducing a “particle number trick” which improves the computing time and allows for parallelization compared to the previous calculation of $S_2$ for itinerant fermions[25] .", "Particle number trick.– Compared to the case of positive definite spin wave-functions studied in Ref.", "[26], itinerant fermion systems come with two major challenges against evaluation of $\\langle {\\rm SWAP}_A\\rangle $ : (1) the wave-function is not positive definite, (2) the number of fermions in the region $A$ fluctuates.", "The first issue had been partially mitigated in Ref.", "[21] using the so-called “sign trick” exactly factorizing $\\langle {\\rm SWAP}_A\\rangle $ into a product of two terms each concerning only magnitude or only sign.", "On the other hand, the fermion number fluctuation was not an issue in Ref.", "[21] as the Gutzwiller projector ensured one fermion per site.", "mcminis dealt with the fermion number fluctuation for free fermions and for Slater-Jastrow trial wave-functions by discarding the Monte Carlo moves that result in different particle numbers for the region $A$ in the two copies.", "In this approach, the running time for free fermions scales as $O(N^3)$ as a function of the number of particles $N$ when combined with fast single-rank updates of the Slater determinant.", "However, we found a direct adaptation of the algorithm used in Ref.", "[25] for the CF wave-function to be prohibitively slow.", "This is mainly due to the fact that a change to the position of a single composite fermion changes all the elements of the determinant and hence we cannot benefit from the fast single-rank updates of the Slater determinant.", "This motivated us to develop an alternate way of dealing with the particle number fluctuation issue.", "We show in [27] that $\\langle {\\rm SWAP}_A\\rangle $ can be further exactly factorized into contributions from sectors of fixed particle numbers in each subregion.", "Evaluating the contributions from each sector separately not only effectively reduces the size of the space that needs to be sampled for each contribution but it also allows parallelization.", "Our algorithm takes about $10^4$ CPU hours per data point, compared to $10^5$ CPU hours per data point using the algorithm of Ref. [25].", "In addition the restriction of the sampled space achieved using the particle number trick reduces the error bars in a way similar to the \"ratio trick\" for lattice models proposed by melko2010.", "Implementing the particle trick algorithm on a cluster of 100 processors, we were able to obtain data for 20 different values of $l\\equiv L_A/L$ with 37 composite fermions in days instead of weeks.", "Wave-function.– The wave-function for the spin polarized $\\nu =1/2$ can be written as a determinant of fermions in zero field times a $\\nu =1/2$ bosonic Laughlin state [23], [28], which we write as $\\det _{i,j}t_i({\\bf d}_j)\|\\Psi _L^{1/2}\\rangle ,$ where $t$ is a single electron translation operator, and ${\\bf d}$ is a displacement satisfying $N_\\Phi {\\bf d}\\in m{\\bf L}_1+n{\\bf L}_2$ , $m$ and $n$ are integers, and $N_\\Phi $ is the magnetic flux quanta.", "The unit cell of the torus is specified by ${\\bf L}_1$ and ${\\bf L}_2$ , which spans an area equal to $2\\pi N_\\Phi \\ell ^2_B$ , where $\\ell _B$ is the magnetic length.", "The displacements are given in terms of wave-vectors of composite fermions by $ k_a=\\epsilon _{ab}d^b/\\ell _B^2$ .", "Acting with the determinant, the holomorphic part of the coherent state wave-function can be written as: ${\\big (}\\sum _{P=1}^{N!}", "(-1)^P F_P(z_1,\\ldots ,z_N){\\big )} F_{CM}(\\sum _i(z_i-\\bar{d})) \\nonumber \\\\F_P(\\lbrace z_i\\rbrace )=\\prod _{i<j}\\sigma (z_i-z_j+d_{P(i)}-d_{P(j)})^2 \\prod _ie^{d^_{P(i)}z_i}$ where $F_{CM}(z)=\\sigma (z)^2$ is the center of mass wave-function, $d=(d_x+id_y)/\\sqrt{2}\\ell _B$ , and $z=(x+iy)/\\sqrt{2}\\ell _B$ are complex distances and coordinates.", "Note that in the definition of complex quantities, which are dimensionless, we include a $\\sqrt{2}$ factor.", "The function $\\sigma (z)$ is the Weierstrass $\\sigma $ function which in terms of the Jacobi $\\vartheta $ function is: $\\sigma (z)={\\vartheta _1(\\kappa z;\\tau )\\over \\kappa \\vartheta _1^\\prime (0;\\tau ) }\\exp (i(\\kappa z)^2/\\pi (\\tau -\\tau ^)).$ Here $\\kappa =\\pi /L_1$ , $L=(L_x+L_y)/\\sqrt{2}\\ell _B$ is the linear dimension of the system with $L^_1L_2-L^_2L_1=2\\pi i N_\\Phi $ , and $\\tau =L_2/L_1$ is the modular parameter of the torus.", "The wave-function in Eq.", "(REF ) cannot be cast as a determinant and therefore proves inconvenient for Monte-Carlo calculations.", "Instead, we present a new expression that is inspired by a construction due to Jain and Kamilla[29], [30].", "These authors treat the translation operators as $c$ -numbers and take the Jastrow factors inside the determinant.", "Doing so reduces their action on $\\sigma (z_i-z_k)$ (the relative part) from $2(N-1)$ to $N-1$ .", "To compensate we double each $d_i(t_j)$ .", "$F_{CF}=\\det _{i,j} (e^{d^_j z_i} \\prod _{k(\\ne i)}\\sigma (z_i-z_k+2(d_j-\\bar{d}))\\times \\nonumber \\\\F_{CM}(\\sum _i(z_i-\\bar{d})).$ To obtain the full expression of the wave-functions a non-holomorphic exponential factor $e^{-\\sum _i z_iz^_i/2}$ has to be included.", "For convenience we have made a specific choice of the zeros of the center of mass part of the wave-function[31].", "This resolves the two-fold topological degeneracy of the state.", "For the $\\nu =1/2$ state in a magnetic field it is non-trivial to decide how to fill the “Fermi sea”, as the usual kinetic energy is completely quenched due to the magnetic field.", "One of us has proposed a model Hamilton[32], which can be used to find the composite fermion Fermi sea.", "$H_{\\rm ``kin^{\\prime \\prime }} \\equiv \\frac{\\hbar ^2}{2mN}\\sum _{r<s}\|{\\bf k}_r-{\\bf k}_s\|^2$ where $m$ is the electron mass.", "The above “kinetic energy” is independent of uniform boost (the so called k-invariance) as is the energy of the model wave-function.", "The invariance under uniform boost guarantees $F_1=-1$[33], [34], [35], which is the only nonzero Fermi liquid parameter of the model.", "Hence by choosing the momenta ${\\bf k}_r$ 's that minimize this “kinetic energy”, the wave-function effectively describes a state with diverging effective mass as indicated by the non-perturbative effect of electron coupling to the fluctuating Chern-Simons gauge field in field theoretic studies[8], [24].", "For particular values of $N$ , including $N=37$ , we consider the set of momenta minimizing $H_{\\rm ``kin^{\\prime \\prime }} $ to be same as the set of momenta filling the Fermi sea for free fermions minimizing the usual kinetic energy.", "Hence we can use the same set of momenta shown in Fig.", "REF for both the composite and free fermion wave-functions.", "The pair correlation function calculated using the composite fermion wavefunction Eq.", "(REF ) with this choice of momenta show Fermi-Liquid-like $2k_F$ oscillations [23].", "Figure: Plot of S 2 /λS_2/\\lambda as a function of λ\\lambda .", "Red corresponds to free fermions, while blue corresponds to composite fermions at half-filling in the lowest Landau level.", "Error bars indicate 95% confidence intervals.", "Dashed lines are best linear fits of S 2 /λS_2/\\lambda to the form a+clogλa+c \\log \\lambda , where aa and cc are fit parameters.Results.- - We obtained the second Rényi entropy $S_2$ as a function of $\\lambda $ for both the free fermion gas and the composite fermion trial wave-function for $\\nu =1/2$ state.", "Here $L_A$ is the linear dimension of the system and $k_F=\\sqrt{10\\pi /37}$ is the radius of the Fermi surface.", "The results and the error bar are shown in Fig.", "REF .", "Our results for the free fermion gas shown in red are in good agreement with previous numerical results[17], [25] and hence demonstrate reliability of our algorithm using the particle number trick.", "The ratio $S_2/\\lambda $ is linear in $\\log \\lambda $ and the linear fit results in 95% confidence level shown in the red dashed line in Fig.", "REF follows $S_2[\\Psi _0] =(0.135\\pm 0.01) \\lambda \\log \\lambda $ where $\\Psi _0$ denotes the free Fermi gas wave-function for the set of momenta shown in Fig.", "REF .", "Comparing the results Eq.", "(REF ) to Eq.", "(REF ) we again confirm the validity of the Widom formula Eq.", "(REF ) for free fermions.", "For the $\\nu =1/2$ non-Fermi liquid state, the results shown in blue in Fig.", "REF again exhibit linear dependence of $S_2/{\\lambda }$ on $\\log {\\lambda }$ , i.e.", "multiplicative logarithmic violation of the area law.", "However, the linear fit again at 95% confidence level: $S_2[\\Psi _{\\nu =1/2}]= {(0.27\\pm 0.02) \\lambda \\log \\lambda }$ reveals that the coefficient is no longer given by the “Widom formula”.", "The steeper slope for the non-Fermi liquid state is evident even from the raw data.", "Curiously, the coefficient of the multiplicative logarithmic correction term for $\\nu =1/2$ non-Fermi liquid is not only larger compared to that for free fermions, but it appears to be double the value expected from the “Widom formula” Eq.", "(REF ).", "Discussion.– In this letter we introduced a particle number trick which can speed up variational Monte Carlo calculation of Rényi entropies via parallelization and used the improved algorithm to calculate the second Rényi entropy $S_2$ of the $\\nu =1/2$ composite fermion non-Fermi liquid state captured by the trial wave-function.", "We found the multiplicative logarithmic violation of the area law with the same functional dependence on the linear dimension of the subregion, i.e., $S_2\\propto {\\lambda \\log \\lambda }$ , as is the case for free fermions but with a coefficient that is roughly double what is found for free fermions.", "Our results support the conjecture that $S_2\\propto {\\lambda \\log \\lambda }$ might be the strongest form of area law violation in 2D made in Ref. [22].", "However against the conjecture made in [36] regarding universality of the Widom formula, the strong enhancement of the entanglement entropy captured by coefficient doubling in our results revealed a violation of the Widom formula for the $\\nu =1/2$ non-Fermi liquid state.", "Our explicit calculation of entanglement entropy for an established non-Fermi liquid state raises a plethora of interesting questions.", "One question is how generic are such factors of order one change in the coefficient of $L_A\\log L_A$ term in the entanglement entropy of strongly correlated fermions forming a non-Fermi liquid state.", "There is little literature on this coefficient for interacting fermions.", "While a Gutzwiller projected Fermi surface constrained to maintain one fermion per site showed little difference from free fermions[21], Slater-Jastrow wave-functions showed small changes in the coefficient[25].", "However, while Slater-Jastrow wave-functions are frequently used as a way of building in correlation effects to wave-functions and used to model Fermi liquids, it is not clear if the effect of the Jastrow factor is perturbative as the $S_2$ result of Ref.", "[25] does not extrapolate to free fermion result in the limit of fermion residue $Z\\rightarrow 1$ .", "It will be interesting to use the particle number trick on a d-wave metal wave-function which is proposed to be stabilized by a ring-exchange Hamiltonian[37] in this context.", "Another question is whether it is possible to gain analytic insight into the enhancement of entanglement due to strong correlation we found, building on the field theory literature on the $\\nu =1/2$ state.", "Finally, the investigation of entanglement spectra may reveal more dramatic differences between the free fermions and $\\nu =1/2$ non-Fermi liquid state.", "Acknowledgements We thank Tarun Grover, Roger Melko, Nick Read and Abolhassan Vaezi, for helpful discussions.", "F.D.M.H.", "and E.H.R.", "were supported by DOE Grant DE-SC0002140, E.-A.K.", "was supported by NSF CAREER grant DMR 0955822, J.S.", "was supported by NSF-DMR 0955822 during his internship at Cornell University.", "This work used the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by National Science Foundation grant number OCI-1053575 and Computational Center for Nanotechnology Innovations (CCNI) through NYSTAR.", "F.D.M.H.", "also acknowledges support from the W. M. Keck Foundation.", "This work was also partially supported by a grant from the Simons Foundation (#267510 to F. D. M. Haldane) for Sabbatical Leave support." ] ]	1403.0577
[ [ "Majorana fermion exchange in strictly one dimensional structures" ], [ "Abstract It is generally thought that adiabatic exchange of two identical particles is impossible in one spatial dimension.", "Here we describe a simple protocol that permits adiabatic exchange of two Majorana fermions in a one-dimensional topological superconductor wire.", "The exchange relies on the concept of \"Majorana shuttle\" whereby a $\\pi$ domain wall in the superconducting order parameter which hosts a pair of ancillary Majoranas delivers one zero mode across the wire while the other one tunnels in the opposite direction.", "The method requires some tuning of parameters and does not, therefore, enjoy the full topological protection.", "The resulting exchange statistics, however, remains non-Abelian for a wide range of parameters that characterize the exchange." ], [ "Effect of disorder and Berry matrix evaluation", "In order to address the stability of our proposed exchange protocol we have performed additional numerical simulations of the Kitaev chain with a $\\pi $ domain wall in the presence of disorder in the on-site potential and other parameters.", "To this end we add a disorder term ${\\cal H_{\\rm dis}}=\\sum _j \\delta \\mu _j c^\\dagger _j c_{j}$ to the lattice Hamiltonian ${\\cal H_{\\rm latt}}$ defined in Eq.", "(12) of the manuscript.", "Here $\\delta \\mu _j$ is a random potential uniformly distributed in the interval $(-w,w)$ .", "The characteristic spectra of the Hamiltonian as a function of the domain wall position $M$ are shown in Fig.", "REF for various values of disorder strength $w$ .", "We observe that weak disorder $w=\\Delta /5=0.02t$ has no visible effect on the zero modes.", "Even relatively strong disorder $w=\\Delta =0.1t$ has only modest effect on the zero modes.", "Only in the dirty limit, i.e.", "when $w$ significantly exceeds $\\Delta $ , we find a notable splitting between the zero mode energies.", "We have also investigated the effect of disorder in the hopping amplitudes $t$ and the pairing potential amplitude $\\Delta $ with similar results.", "Of particular interest is our result in Fig.", "REF showing that phase disorder in $\\Delta _{j,j+1}=\\Delta _0e^{i\\delta \\phi _{j,j+1}}$ has no visible effect on the zero modes, despite the fact that complex order parameter breaks the time reversal symmetry of the problem that protects the doublet of the ancillary majoranas.", "From these simulations it appears that the time reversal symmetry must only be preserved on average to protect the Majorana zero modes residing at the domain wall.", "Although we do not fully understand the fundamental basis for this result we have verified it to be true numerically for many different realizations of the phase disorder.", "Results of our simulations with various types of disorder indicate that the exchange protocol proposed in our manuscript is robust.", "Given the system with a specific realization of disorder one can tune a single parameter, such as the global chemical potential $\\mu $ , until the splitting between the zero modes is minimized, as indicated in Fig.", "REF .", "This is possible so long as the disorder strength does not significantly exceed the size of the superconducting gap $\\Delta $ .", "Once again, such a limitation on the disorder strength is common to all 1D realizations of Majorana zero modes: too strong a disorder would ultimately destroy the topological phase.", "In addition to the quasiparticle spectra, which we showed to support our low-energy effective theory, it is possible to obtain a more detailed characterization of the Majorana exchange from the numerical simulation of the Kitaev chain.", "Below, we directly evaluate the non-Abelian Berry matrix introduced by Wilczek and Zee [1] which describes the unitary evolution of the zero modes in the degenerate ground state manifold.", "According to Ref.", "zee1, the solution of the time-dependent Schrödinger equation $i{\\partial \\psi \\over \\partial t}=H(t)\\psi $ for degenerate states $\\psi _a$ in the adiabatic limit is given by $\\psi _a(t)=U_{ab}(t)\\psi _b(0).$ Here $U_{ab}(t)=\\exp {\\left[\\int _0^t\\langle \\eta _a(\\tau )\|\\dot{\\eta }_b(\\tau )\\rangle d\\tau +\\ln {\\langle \\eta _a(t)\|{\\eta }_b(0)\\rangle }\\right]}$ represents the unitary evolution operator (the “Berry matrix”) and $\\eta _b(t)$ are instantaneous degenerate eigenstates of $H(t)$ whose dependence on $t$ is chosen to be smooth and to satisfy the initial condition $\\eta _a(0)=\\psi _a(0)$ .", "Our numerical diagonalization of the Kitaev chain Hamiltonian yields the instantaneous eigenstates $\\eta _{1,2}(t)$ associated with the Majorana zero modes for the domain wall position $M$ at time $t$ .", "We can use these to evaluate $U_{ab}(t)$ from Eq.", "(REF ) and thus ascertain the unitary evolution of the Majorana zero modes described by $\\psi _a(t)$ .", "In practice, we always encounter a small energy splitting between the zero modes due to the finite size of our system and potentially other effects such as the small undesirable coupling $t^{\\prime }_2$ .", "In the presence of a non-zero energy splitting we choose the instantaneous eigenstates $\\eta _{1,2}(t)$ as linear combinations of Figure: Majorana wavefunction amplitudes \|η 1,2 (t)\| 2 \|\\eta _{1,2}(t)\|^2 as chosen for the Berry matrix evaluation in the Kitaev chain with a π\\pi domain wall.", "As the domain wall moves from 0 to 50, the Majorana mode γ 1 \\gamma _1 continuously moves from 0 to 50.", "When the domain wall passes through site 25, the other Majorana mode (γ 2 \\gamma _2) teleports from the left end to the right end.the two low-energy states that (i) obey the desired initial condition (i.e.", "represent two Majorana particles at the ends of the wire), and (ii) evolve smoothly as we move the domain wall while their wavefunctions remain Majorana throughout the process.", "An example of the wavefunctions chosen in this way is given in Fig.", "REF .", "With this choice of basis we typically find that the integral in the exponent of Eq.", "(REF ) vanishes while the second (monodromy) term is nonzero.", "Together this gives $U_{ab}(t)\\simeq \\begin{pmatrix}0 & 1 \\\\-1 & 0\\end{pmatrix},$ to within the numerical accuracy of our calculation.", "Once again this confirms that the two Majorana zero modes exchange in the process and that the exchange satisfies the rules of the Ising braid group given in Eq.", "(1) of the manuscript.", "Importantly, the result in Eq.", "(REF ) remains valid in the presence of moderate amounts of disorder.", "Figure: The effect of phase disorder on the spectrum of the Kitaev chain with a π\\pi domain wall.", "Random phase δφ j,j+1 \\delta \\phi _{j,j+1} is uniformly distributed between (-w φ ,w φ )(-w_\\phi ,w_\\phi )." ], [ "Generalized exchange protocol", "In strictly one dimension the Majorana coupling path indicated in Fig.", "2a in the $\\pi $ domain wall protocol implements an exchange between $\\gamma _1$ and $\\gamma _2$ .", "This path is not the only one that achieves braiding.", "In this section, we discuss other paths in the parameter space of the Hamiltonian (6) that can bring about braiding.", "Before the general discussion, we consider another specific coupling path of the same four Majorana fermions ($\\gamma _1,\\ \\gamma _2,\\ \\Gamma _1,\\Gamma _2$ ) designed to achieve braiding of $\\gamma _1$ and $\\gamma _2$ .", "As we shall argue below this path can be implemented by the setup displayed in Fig.", "3e.", "The path we consider consists of two steps and is depicted in Fig.", "2b.", "We start from $H=i\\epsilon \\Gamma _1\\Gamma _2$ ($\\theta =0,\\ \\varphi =0$ ) and let the coupling vary from $\\theta =0$ to $\\theta =\\pi $ as $\\varphi =0$ remains fixed.", "As shown in Fig.", "2b, the point on the unit sphere moves straight from the north pole to the south pole.", "According to Eq.", "(9) this produces the following evolution of the zero modes $\\gamma _1 \\rightarrow -\\gamma _1,\\quad \\gamma _2 \\rightarrow \\gamma _2.$ Now the coupling is at the south pole and we wish to return back to the north pole along a different trajectory characterized $\\varphi =\\pi /4$ .", "We thus perform a basis rotation from $(-\\gamma _1,\\gamma _2)$ to a new basis $(-\\gamma _1-\\gamma _2,-\\gamma _1+\\gamma _2)/\\sqrt{2}$ which corresponds to the zero modes given in Eq.", "(9) for $\\theta =\\pi $ and $\\varphi =\\pi /4$ .", "Now we can let $\\theta $ vary from $\\pi $ to 0, so the system returns back to the north pole.", "The zero modes are $\\cos \\theta \\frac{\\gamma _1+\\gamma _2}{\\sqrt{2}}-\\sin \\theta \\Gamma _1,\\quad \\frac{-\\gamma _1+\\gamma _2}{\\sqrt{2}}.$ As $\\theta $ varies from $\\pi $ to 0, we thus have $\\gamma _1+\\gamma _2\\rightarrow -\\gamma _1-\\gamma _2, \\quad \\gamma _1-\\gamma _2\\rightarrow \\gamma _1-\\gamma _2$ Performing finally another rotation in $\\varphi $ by $-\\pi /4$ to return to the original basis we obtain for the overall evolution of $\\gamma _1$ and $\\gamma _2$ $\\gamma _1 &\\rightarrow -\\gamma _1 \\rightarrow \\gamma _2, \\\\\\gamma _2 &\\rightarrow \\gamma _2 \\rightarrow -\\gamma _1,$ showing that the two Majoranas have indeed exchanged in this process.", "To understand how this exchange protocol can be implemented using the device depicted in Fig.", "3e it is easiest to consider the process in reverse.", "As we further elaborate in Appendix C below it is easy to see that twisting the superconducting phase $\\phi _1$ in the upper half of the wire by $2\\pi $ corresponds to the trajectory on the unit sphere going from the north to the south pole with $\\varphi =\\pi /4$ .", "The underlying physics is captured by Eq.", "(REF ) below.", "Now the problem is that although the physical Kitaev Hamiltonian is mapped back onto itself under such a $2\\pi $ phase twist, the effective Majorana Hamiltonian $H(s)$ is mapped to $-H(s)$ .", "This is because the definition of the Majorana operators involves $e^{i\\phi /2}$ and is therefore not single valued in $\\phi $ .", "One can deal with this issue by a redefinition of the Majorana operators in the upper segment of the wire, as discussed e.g.", "in Ref.", "halperin1.", "One can, alternately, imagine undoing the phase twist performed on $\\gamma _1$ by twisting the phase of a short segment of the wire very close to its top end by $-2\\pi $ .", "This has no effect on $\\gamma _2$ and can be pictured as going back to the north pole along the $\\varphi =0$ line.", "Since $\\gamma _2$ is not involved in this last (imagined) step it has no effect on braiding and simply implements the transformation back to the original basis.", "This completes the path indicated in Fig.", "2b (taken in reverse).", "Comparing the two braiding processes in Fig.", "2, we find that they have one feature in common.", "When the paths are projected onto the unit sphere, the covered areas are $\\pi /2$ .", "In the following, we will prove that for an arbitrary closed coupling path, $\\gamma _1$ and $\\gamma _2$ exchange if the path begins at one of the poles and if the covered area is $\\pi /2$ .", "Following the method employed in Ref.", "Couplingbraiding, we compute Berry's phase of the ground states after a coupling cycle.", "Accumulation of the Berry phase can be regarded as the result of the braiding operation[4].", "Let us rewrite the coupling Hamiltonian in Eq.", "(6) in another economical way $H=i(X \\gamma _1 + Y \\gamma _2 + Z \\Gamma _1)\\Gamma _2, $ where $X=E\\sin \\theta \\cos \\varphi $ , $Y=E\\sin \\theta \\sin \\varphi $ , and $Z=E\\cos \\theta $ .", "When the coupling is off, the four Majorana fermions possess zero energy.", "The ground state has four-fold degeneracy and can be represented by $\| 0 \\rangle ,\\ c^\\dag \| 0 \\rangle ,\\ d^\\dag \| 0 \\rangle ,\\ d^\\dag c^\\dag \| 0 \\rangle $ , where each fermionic operator is formed by two Majorana operators $c=(\\gamma _1-i\\gamma _2)/2$ and $d=(\\Gamma _1-i\\Gamma _2)/2$ .", "The coupling Hamiltonian can be rewritten in this fermionic basis $H=E\\begin{pmatrix}Z & 0 & 0 & -X-iY \\\\0 & Z & -X+iY & 0 \\\\0 & -X-iY & -Z & 0 \\\\-X+iY & 0 & 0 & -Z\\end{pmatrix}.$ Due to the conservation of fermionic parity, two blocks of the Hamiltonian with different parities can be discussed separately $H_{\\rm {even}}=H_{\\rm {odd}}^*=\\begin{pmatrix}Z & -X-iY \\\\-X+iY & -Z\\end{pmatrix}.$ Turning on the coupling changes the ground state degeneracy from four-fold to two-fold.", "The two ground states with energy $-E$ in the even and odd parity sectors are given by $\| \\rm {e} \\rangle &=\\frac{1}{\\sqrt{2E(E-Z)}}\\begin{pmatrix}-E+Z \\\\-X+Yi\\end{pmatrix},\\quad \\\\\| \\rm {o} \\rangle &=\\frac{1}{\\sqrt{2E(E-Z)}}\\begin{pmatrix}-E+Z \\\\-X-Yi\\end{pmatrix}.$ Now we introduce differential forms to compute the Berry phases.", "The Berry connections ($\\langle \\Psi \|d\| \\Psi \| \\rangle $ ) in even and odd parity sectors are simply written as differential one-forms[5] $A_{\\rm {even}}=-A_{\\rm {odd}}=-\\frac{i(XdY-YdX)}{2E(E-Z)},$ and the Berry curvatures ($d\\langle \\Psi \|d\| \\Psi \| \\rangle $ ), which are differential two-forms, are given by $dA_{\\rm {even}}&=\\frac{i}{2E^3}(ZdX\\wedge dY+XdY\\wedge dZ+ Y dZ\\wedge dX)\\nonumber \\\\&=\\frac{i}{2}\\sin \\theta d\\theta \\wedge d \\varphi .$ We note that $E$ is not constant so $dE^2=2EdE=2XdX+2YdY+2ZdZ$ .", "After performing a closed loop operation, the original ground states gain extra Berry phases $\| e^{\\prime } \\rangle &=\\exp \\big ( \\oint _\\mathcal {C}A_{\\rm {even}} \\big )\| e \\rangle =\\exp \\big ( \\int dA_{\\rm {even}} \\big )\| e \\rangle , \\nonumber \\\\\| o^{\\prime } \\rangle &=\\exp \\big ( \\oint _\\mathcal {C}A_{\\rm {odd}} \\big )\| o \\rangle =\\exp \\big ( \\int dA_{\\rm {odd}} \\big )\| o \\rangle .", "$ On the one hand, the line integrals become surface integrals by Stokes' theorem so $2i\\int dA_{\\rm {even}}=-2i\\int dA_{\\rm {odd}}=\\int \\sin \\theta d\\theta \\wedge d\\varphi $ is the area covered by the coupling path on the unit sphere.", "On the other hand, at the beginning of the process $\\theta =0$ so the initial ground states are given by $\| \\rm {e} \\rangle =\| 0 \\rangle ,\\quad \| \\rm {o} \\rangle =c^\\dagger \| 0 \\rangle $ Ref.", "PhysRevLett.86.268 shows that when $\\gamma _1$ and $\\gamma _2$ braiding occurs, $\| \\rm {e}^{\\prime } \\rangle =e^{i\\pi /4}\| \\rm {e} \\rangle ,\\quad \| \\rm {o}^{\\prime } \\rangle =e^{-i\\pi /4}\| \\rm {o} \\rangle .$ Using the relation between the final ground states $\| o^{\\prime } \\rangle =(\\gamma _1^{\\prime }+i\\gamma _2^{\\prime })\| e^{\\prime } \\rangle $ , we have $\\gamma _1^{\\prime }=\\gamma _2$ and $\\gamma _2^{\\prime }=-\\gamma _1$ .", "Therefore, to achieve braiding between $\\gamma _1$ and $\\gamma _2$ in the coupling process, the area $\\int \\sin \\theta d\\theta \\wedge d\\varphi =\\pi /2$ is required by comparing the Berry phases in Eq.", "(REF )." ], [ "Majorana Josephson junction", "When Majorana modes are present in a Josephson junction, it is known that the current phase relation has an anomalous $4 \\pi $ periodicity.", "[6] We consider a similar junction device as illustrated by Fig.", "3e.", "However, the $4\\pi $ periodicity is absent in this device due to the proximity of other Majorana zero modes at the two ends of the wire.", "In this situation a $2\\pi $ phase twist in the bottom half of the wire is equivalent to a braiding operation of Majorana modes $\\gamma _1$ and $\\gamma _2$ located at the ends.", "As we demonstrate below this occurs when $\\gamma _1$ and $\\gamma _2$ couple to the same linear combination of the Majorana modes located at the junction.", "The interplay of four Majorana fermions achieves the braiding operation.", "In the following we denote the Majorana fermion located on the upper (lower) part of the junction by $\\alpha $ ($\\beta $ ).", "To demonstrate the braiding operation, let $\\phi _1=0$ remain fixed as $\\phi _2=\\phi $ varies from 0 to $2\\pi $ .", "The coupling between the two Majorana modes is given by $iE\\cos (\\phi /2)\\Gamma _1\\Gamma _2$ .", "As $\\phi =0,\\ 2\\pi $ , the two junction Majorana coupling suppresses the coupling effect between the edges ($\\gamma _1,\\gamma _2$ ) and junction ($\\Gamma _2$ ).", "As $\\phi =\\pi $ , the edge and junction coupling dominates in the absence of the junction coupling.", "The low-energy effective Hamiltonian can be written as $H_J^{2\\pi }(\\phi )=i E \\big ( \\epsilon \\sin (\\phi /2) \\frac{\\gamma _1+\\gamma _2}{\\sqrt{2}}+\\cos (\\phi /2)\\Gamma _1)\\Gamma _2,$ where $\\Gamma _1=-\\frac{\\alpha -\\beta }{\\sqrt{2}}$ , $\\Gamma _2=\\frac{\\alpha +\\beta }{\\sqrt{2}}$ .", "$E(\\phi )$ and $\\epsilon $ are positive constants.", "Tuning of a single parameter (such as the wire chemical potential $\\mu $ ) is required to achieve symmetric coupling to a single linear combination of the junction Majoranas.", "When the phase $\\phi $ varies, the wavefunctions of $\\beta $ and $\\gamma _2$ change according to $\\beta (\\phi )=e^{i\\phi /2}c^\\dag _{2j}+e^{-i\\phi /2}c_{2j},\\quad \\gamma _2(\\phi )=e^{i\\phi /2}c^\\dag _{2b}+e^{-i\\phi /2}c_{2b},$ where $c_{2j}$ and $c_{2b}$ are $\\phi $ independent fermionic operators representing $\\beta $ and $\\gamma _2$ respectively.", "After the $2\\pi $ twist of the bottom half of the wire, $\\beta (2\\pi )=-\\beta (0)$ and $\\gamma _2(2\\pi )=-\\gamma _2(0)$ .", "On the other hand, the evolution of the two zero energy modes of $H_J^{2\\pi }(\\phi )$ can be written as a function of $\\phi $ $\\frac{\\gamma _1-\\gamma _2}{\\sqrt{2}},\\quad \\cos (\\phi /2) \\frac{\\gamma _1+\\gamma _2}{\\sqrt{2}} - \\epsilon \\sin (\\phi /2)\\Gamma _1 $ The second Majorana operator is unnormalized for simplicity.", "Following the same line or argument as in Sec.", "above, after $2\\pi $ rotation, we find that $\\gamma _1 &\\rightarrow -\\gamma _2(2\\pi )=\\gamma _2(0)\\\\\\gamma _2 &\\rightarrow -\\gamma _1$ We thus observe that braiding of $\\gamma _1$ and $\\gamma _2$ can be achieved by twisting the phase in one half of the wire by $2\\pi $ .", "In this process the system comes back to the original ground state (up to an important overall phase).", "This is unlike the Josephson effect with $4\\pi $ periodicity.", "In the following, we compare these two different processes.", "Figure: The lowest energy eigenvalues of ℋ latt \\cal H_{\\rm latt} in Eq.", "(12) with Δ=0.1t\\Delta =0.1t and N=50N=50.For the $4\\pi $ Josephson junction, we can imagine that the two ends of the chain are at infinity so $\\gamma _1$ and $\\gamma _2$ are completely decoupled and remain exact zero modes throughout.", "In this regard, the Hamiltonian is given by only the coupling of two Majorana fermions at the junction $H_J^{4\\pi }(\\phi )=iE\\cos (\\phi /2)\\beta \\alpha $ Both of these two processes start from the same ground states ($\\phi =0$ ) obeying $(\\beta +i\\alpha )\| G_{2\\pi ,4\\pi } \\rangle =0$ with the energy $-E$ .", "We follow the evolution of the two gapped Majorana operators.", "In the $4\\pi $ junction regime, the two gapped Majorana modes $(\\alpha ,\\ \\beta )$ stay in the same form.", "In the braiding regime the evolution of the Majorana modes is given by $\\epsilon \\sin (\\phi /2) \\frac{\\gamma _1+\\gamma _2}{\\sqrt{2}}+\\cos (\\phi /2)\\Gamma _1,\\ \\Gamma _2,$ The final modes at $\\phi =2\\pi $ , which are $\\Gamma _1\\rightarrow -\\Gamma _1$ and $\\Gamma _2\\rightarrow \\Gamma _2$ , imply $\\alpha \\rightarrow \\beta $ and $\\beta \\rightarrow \\alpha $ .", "On the other hand, due to the $2\\pi $ phase rotation, $\\beta \\rightarrow -\\beta $ in both cases.", "The ground states in the braiding protocol and the $4\\pi $ Josephson junction satisfy two different equations $-i(\\beta +i\\alpha )\| G^{\\prime }_{2\\pi } \\rangle =0,\\quad -(\\beta -i\\alpha )\| G^{\\prime }_{4\\pi } \\rangle =0$ At the same time, the Hamiltonian evolves to the original Hamiltonian $H_J^{2\\pi ,4\\pi }=-iE\\beta \\alpha (2\\pi )=iE\\beta \\alpha $ The ground state of the braiding Hamiltonian stays the same with the energy $-E$ but the ground state of the $4\\pi $ Josephson junction evolves to the excited state with the energy $E$ .", "Thus, the presence or absence of the coupling to Majorana fermions at the ends of the wire brings two completely different outcomes — braiding and $4\\pi $ periodicity, respectively.", "We now support the above analysis by a detailed calculation using the lattice model $\\cal H_{\\rm latt}$ of the TSC chain defined in Eq.", "(12).", "We show that the Hamiltonian $H^{2\\pi }_J$ in Eq.", "(REF ) indeed describes the low-energy degrees of freedom in the one-dimensional TSC modulo a small correction that we argue is unimportant for the outcome of the braiding operation.", "To this end we numerically solve the Hamiltonian $\\cal H_{\\rm latt}$ with the order parameter distribution given by $\\Delta _{j,j+1}=\\Biggl \\lbrace \\begin{matrix}\\Delta ,\\quad \\quad \\ j<M \\\\0,\\quad \\quad \\ j=M \\\\\\Delta e^{i\\phi },\\ \\ j>M \\\\\\end{matrix},$ where $M=N/2$ (taking $N$ even) and the other parameters are the same as in Fig.", "3.", "As illustrated by Fig.", "REF the energy eigenvalues of $\\cal H_{\\rm latt}$ as a function of $\\phi $ show gapped and gapless properties in two distinct situations.", "In Fig.", "REF a, two of the Majorana modes remain at zero energy and the absence of the energy level crossing indicates the absence of the ground state switching.", "This low-energy spectrum is consistent with the form expected for $H_J^{2\\pi }(\\phi )$ and we thus tentatively conclude that braiding occurs for these parameters.", "The energy levels in Fig.", "REF b show a significant splitting between the zero modes.", "This indicates that additional terms not included in $H_J^{2\\pi }(\\phi )$ are present in the low-energy theory.", "In this situation we do not expect braiding to occur; rather, if the phase twist is implemented relatively fast we expect a transition to the excited state and the resulting $4\\pi $ -periodic behavior.", "To further confirm that the effective low-energy theory of $\\cal H_{\\rm latt}$ is described by $H_J^{2\\pi }$ , we now compute the effective couplings between the four Majorana fermions $(\\gamma _1,\\ \\gamma _2,\\ \\alpha ,\\ \\beta )$ .", "First, imagine a cut in the middle of the chain, that is, we let $t=\\Delta =0$ on the link between sites $M$ and $M+1$ in $\\cal H_{\\rm latt}$ .", "Next, by using the values of the parameters in Fig.", "REF a and then solving the eigenvalue problem, we obtain the wavefunctions of the four zero energy modes at the ends of the two separated chains.", "We note that $\\gamma _2$ and $\\beta $ , which are $\\phi $ -dependent, must be computed each time while $\\phi $ varies.", "Second, restoring non-zero $t$ on the middle link we sandwich $\\cal H_{\\rm latt}$ between these four Majorana mode wavefunctions and obtain the relevant couplings.", "These are shown in Fig.", "REF c. On the basis of these results, the effective low-energy Hamiltonian can be written as a function of $\\phi $ $H_{\\rm eff}(\\phi )=i\\frac{E}{2} & \\big (\\epsilon ^{\\prime } \\gamma _1\\alpha + \\epsilon ^{\\prime }\\gamma _2 \\beta + \\epsilon \\sin (\\phi /2)\\gamma _1\\beta \\nonumber \\\\&+ \\epsilon \\sin (\\phi /2) \\gamma _2 \\alpha + 2 \\cos (\\phi /2)\\beta \\alpha \\big )$ We note that $\\epsilon ^{\\prime } \\sim \\epsilon \\ll 1$ because the coupling between $\\beta $ and $\\alpha $ is usually much larger than couplings between distant Majoranas.", "The effective Hamiltonian can be rewritten in an economical way $H_{\\rm eff}(\\phi )=H^{2\\pi }_J(\\phi )+i\\frac{E}{2}(\\epsilon ^{\\prime }-\\epsilon \\sin (\\phi /2))( \\gamma _1\\alpha +\\gamma _2\\beta ) $ Thus, the low-energy sector of the Kitaev lattice Hamiltonian with a phase twist is described by $H_J^{2\\pi }$ with an extra term.", "When $\\epsilon ^{\\prime }\\ne \\epsilon $ the extra term produces a splitting between the zero modes $\\delta E\\simeq E\|\\epsilon ^{\\prime }-\\epsilon \|$ for $\\phi =\\pi $ .", "We can thus identify the extra terms in $H_{\\rm eff}(\\phi )$ as being responsible for the behavior indicated in Fig.", "REF b.", "As in the case of the Majorana shuttle, $\\epsilon ^{\\prime }$ and $\\epsilon $ depend on the system parameters.", "Fig.", "REF a shows that by tuning a single parameter, e.g.", "the chemical potential $\\mu $ , we can achieve the situation in which $\\epsilon ^{\\prime }=\\epsilon $ and Majorana zero modes remain robust during the process." ] ]	1403.0033
[ [ "Geometry of the Hopf Bundle and spin-weighted Harmonics" ], [ "Abstract We demonstrate that it is conceptually and computationally favorable to regard spin-weighted spherical harmonics as vector valued functions on the total space $SO(3)$ of the Hopf bundle, satisfying a covariance condition with respect to the gauge group $U(1)$ of this bundle.", "A key role is played by the invariant connection form of the principle Hopf bundle, known to physicists from the geometry behind magnetic monopoles." ], [ "Introduction", "Spin-weighted spherical harmonics often play an important role in the mathematical analysis of physical problems, in particular in gravitational physics because of the tensor character of the metric field.", "A prominent example is the study of the polarization anisotropies of the cosmic microwave background.", "(For reviews see [1], [2].)", "In contrast to the standard spherical harmonics, spin-weighted spherical ones, with a non-trivial spin weight, can not be considered globally as functions on the 2-sphere.", "Similar to magnetic monopoles, one has to use at least two patches, together with a $U(1)$ -valued transition function, for an appropriate treatment.", "The Hopf bundle provides again the most suitable geometrical framework for a conceptually satisfactory intrinsic treatment.", "Moreover, we shall see that some of the rather involved calculations in standard treatments, as for instance in [3], can be avoided." ], [ "Some differential geometric tools", "For the benefit of readers not so familiar with the differential geometry of connections in principle fiber bundles, we provide in this section some background material that is used afterwards." ], [ "Invariant connection in a homogeneous principle fiber bundle", "The Hopf bundle $SO(3)(S^2,SO(2))$ is a special case of a homogeneous principal fiber bundle $G(G/H,H)$ over $M=G/H$ for Lie groups $H\\subset G$ .", "We denote their Lie algebras by $\\mathcal {H}$ and $\\mathcal {G}$ , respectively.", "Let $\\theta $ be the Maurer-Cartan form (canonical 1-form) on $G$ .", "We recall that this $\\mathcal {G}$ -valued 1-form associates by definition to every left-invariant vector field $X$ on $G$ its value $X(e)$ for the unit element $e\\in G$ .", "$\\theta $ is left-invariant, $L_g^{\\ast }\\theta =\\theta $ , and behaves under right-translations $R_g$ according to $R_g^{\\ast }\\theta =Ad(g^{-1})\\theta \\quad \\mbox{for all} \\quad g\\in G.$ We assume that the homogeneous space $M=G/H$ is reductive, i.e., that there is a decomposition $\\mathcal {G}=\\mathcal {H}\\oplus \\mathcal {M} \\quad \\mbox{with}\\quad Ad(H) \\mathcal {M}\\subset \\mathcal {M}.$ With respect to this direct decomposition, the $\\mathcal {H}$ -component $A$ of $\\theta $ defines a connection on the principal bundle $G(G/H,H)$ , which is also invariant under left translations by the structure group $H$ .", "This connection of the Hopf bundle will play a key role in our study of spin harmonics.", "We give a simple proof of the well-known fact that $A$ is indeed a left-invariant connection form.", "(This will also fix some of our notation that largely follows that of [4]).)", "a) Let $X\\in \\mathcal {H}$ , and $X^{\\sharp }$ the corresponding fundamental vector field on $G,\\, X^{\\sharp }(g)=dL_g(X)$ .", "Then we have, using the definition of the Maurer-Cartan form $\\theta (X^{\\sharp })=X$ , $A(X^{\\sharp })=\\theta (X^{\\sharp })_{\\mathcal {H}}=X.$ b) From (REF ) we deduce the second characterizing property of a connection form $R_h^{\\ast } A=Ad(h^{-1})\\circ A.$ Indeed, with the decomposition (REF ) for which we denote the projection on $\\mathcal {H}$ by $pr_{\\mathcal {H}}$ , we have for a vector field $Y$ on $G$ according to (REF ) $ (R_h^{\\ast }A)(Y)=A(R_{h\\ast }Y))= pr_{\\mathcal {H}}\\circ \\theta (R_{h\\ast }Y)=pr_{\\mathcal {H}}(Ad(h^{-1})\\,\\theta (Y))=Ad(h^{-1})\\, A(Y),$ where we used in the last step that $Ad(h)$ commutes with the projection $pr_{\\mathcal {H}}$ , because $\\mathcal {H}$ and $\\mathcal {M}$ are invariant under $Ad(h)$ .", "The left-invariance of the connection form $A$ follows from: $ L_h^{\\ast } A=(L_h^{\\ast }\\circ pr_{\\mathcal {H}})\\theta = (pr_{\\mathcal {H}} \\circ L_h^{\\ast })\\theta = pr_{\\mathcal {H}}(\\theta )=A .$" ], [ "Properties of the covariant differential associated to $A$", "We first recall the definition of the (exterior) covariant differential $D$ associated to a connection form $A$ on a principal bundle $P(M,H)$ .", "This acts on vector-valued differential $k$ -forms $\\phi $ on $P$ that are horizontal and satisfy the following covariance condition with respect to the right action $R_h$ (leaving fibers over a point of the base manifold $M$ invariant): $R_h^{\\ast } \\phi =\\rho (h^{-1}) \\phi .$ Here $\\rho $ is a representation of the structure group $H$ in the target space of $\\phi $ .", "By definition, $D\\phi $ is the horizontal projection of the exterior differential $d\\phi $ .", "This geometrical definition can be translated to the well-known formula $D\\phi =d\\phi +\\rho _{\\ast }(A)\\wedge \\phi ,$ where $\\rho _{\\ast }$ denotes the induced representation of $\\mathcal {H}$ .", "A crucial property of $D$ is that it maps horizontal differential $k$ -forms of type $\\rho $ into horizontal $(k+1)$ -forms of the same type $\\rho $ .", "A further property, used later, is that for a left-invariant connection in the homogeneous $H$ - principle bundle $G(G/H,H)$ the differential $D$ commutes with $L_g^{\\ast }, \\,g\\in G$ : $D\\circ L_g^{\\ast }=L_g^{\\ast } \\circ D \\quad \\mbox{for a left-invariant} \\: A.$ Indeed, from (REF ) we obtain $ L_g^{\\ast }(D\\phi )=d(L_g^{\\ast }\\phi )+L_g^{\\ast }(\\rho _{\\ast }(A))\\wedge L_g^{\\ast }\\phi , $ and (REF ) follows from $L_g^{\\ast } A= A$ , since $L_g^{\\ast }(\\rho _{\\ast }(A) )= \\rho _{\\ast }( L_g^{\\ast }A)=\\rho _{\\ast }(A)$ .", "At this point we specialize these considerations to the Hopf bundle.", "A basis of left-invariant 1-forms on $SO(3)$ in terms of the Euler angles $(\\varphi ,\\vartheta ,\\psi )$ is given by $\\theta ^1 &=& -\\sin \\vartheta \\cos \\psi \\, d\\varphi +\\sin \\psi \\,d\\vartheta ,\\nonumber \\\\\\theta ^2 &=& \\sin \\vartheta \\sin \\psi \\,d\\varphi + \\cos \\psi \\,d\\vartheta , \\nonumber \\\\\\theta ^3 &=& \\cos \\vartheta \\, d\\varphi +d\\psi .$ The forms ${\\theta ^i}$ satisfy the Maurer-Cartan equations: $d\\theta ^1 +\\theta ^2\\wedge \\theta ^3=0, \\quad \\mbox{and cyclic permutations}.$ The dual basis of left-invariant vector fields is $e_1 &=& -\\frac{\\cos \\psi }{\\sin \\vartheta }\\,\\partial _\\varphi +\\sin \\psi \\,\\partial _\\vartheta +\\cot \\vartheta \\cos \\psi \\,\\partial _\\psi , \\nonumber \\\\e_2 &=& \\frac{\\sin \\psi }{\\sin \\vartheta }\\,\\partial _\\varphi + \\cos \\psi \\,\\partial _\\vartheta - \\cot \\vartheta \\sin \\psi \\,\\partial _\\psi , \\nonumber \\\\e_3 &=& \\partial _\\psi .$ These are the fundamental vector fields on $SO(3)$ generated by right action on the standard basis ${I_k}$ of the Lie algebra of $SO(3)$ , defined by the 1-parameter subgroups about the three orthogonal axis of $\\mathbb {R}^3$ .", "So $e_k=I_k^{\\sharp }$ , and the Maurer-Cartan form $\\theta $ on $SO(3)$ satisfies $\\theta (e_k)=I_k, \\quad \\mbox{hence} \\quad \\theta =\\sum _{k} \\theta ^k I_k.$ Obviously, the left-invariant connection form of the Hopf bundle is given by $A=\\theta ^3=\\cos \\vartheta \\,d\\varphi + d\\psi .$ The two vector fields $e_1, e_2$ are therefore horizontal.", "Furthermore, the horizontal linear combinations $(e_1 \\mp i e_2)/\\sqrt{2}$ are of type $\\pm 1$ .", "The same is true for the dual linear combinations $(\\theta ^1 \\pm i \\theta ^2)/\\sqrt{2}$ .", "The vector field $e_3$ is vertical, and is a fundamental vector field of the bundle with $A(e_3)=1$ .", "For a horizontal form $\\phi $ of type (spin) $s$ (i.e., the representation (REF ) below) the covariant differential $D$ is given by $D\\phi =d\\phi +isA\\wedge \\phi .$ We know that this preserves the type $s$ , and that $D$ commutes with $L_g^{\\ast },\\,g\\in SO(3)$ , since $A$ is left-invariant.", "In passing we note that the curvature $F$ belonging to $A$ is given by $F=dA=-\\sin \\vartheta \\,d\\vartheta \\wedge d\\varphi $ , thus projects on the negative of the volume form of $S^2$ .", "This is an example of a monopole field with monopole number (first Chern number) -2.", "We now return to the general situation.", "Consider the matrix elements $\\rho _{ik}(g)$ of a representation of $G$ relative to a basis in the representation space $V$ , and let $f_{ik}(g)=\\rho _{ik}(g^{-1}).$ These functions obviously satisfy the covariance condition (REF ) (with the restriction of $\\rho $ to $H$ ), and therefore $D f_{ik}$ is defined.", "Under a left translation $L_a, \\, a\\in G$ , we find $ (L_a^{\\ast } f_{ik})(g)=f_{ik}(ag)=\\rho _{ik}(g^{-1} a^{-1})=\\sum _l \\rho _{il}(g^{-1})\\rho _{lk}(a^{-1})=\\sum _l f_{il}(g) \\rho _{lk}(a^{-1}), $ thus the transformation law $L^{\\ast }_{g^{-1}} (f_{ik})=\\sum _{l} f_{il}\\,\\rho _{lk}(g), \\quad g\\in G.$ In other words, for fixed $i$ the functions $\\lbrace f_{ik}\\rbrace $ transform according to the representation $\\rho $ .", "From (REF ) it follows that the 1-forms $D f_{ik}$ satisfy the same transformation law under the left-action by $G$ .", "Next, we translate the transformation law (REF ) to the pull-backs of $f_{ik}$ by a local section $\\sigma :\\,M\\rightarrow G$ of the (non-trivial) principal bundle $ G(M=G/H,H)$ .", "Note that $M$ is naturally a $G$ -manifold: $g\\in G$ acts on an element $aH\\in G/H$ by left multiplication.", "Consider again a horizontal form $\\phi $ , satisfying the covariance condition (REF ), and let $\\phi _\\sigma (x):=\\phi (\\sigma (x))$ (see the commuting diagram below).", "$\\begin{picture}(8,3)\\put (0,2.5){G} \\put (2.2,0.5){\\vector (-2,3){1.1}}\\put (4,2.5){\\mathcal {H}} \\put (0.8,2.1){\\vector (2,-3){1.1}}\\put (2,0){G/H} \\put (4,1.2){\\phi _\\sigma }\\put (3,0.5){\\vector (2,3){1.1}}\\put (1.5,2.7){\\vector (1,0){2}} \\put (1,1){ \\pi }\\put (1.9,1.5){ \\sigma } \\put (2.5,2.75){\\phi }\\end{picture}$ An element $g\\in G$ transforms $\\phi _\\sigma $ according to $\\phi _\\sigma (g\\cdot x)=\\phi \\circ \\sigma (g\\cdot x)$ , where $g\\cdot x$ denotes the action of $g$ on $x\\in M$ .", "We factorize $\\sigma (g^{-1}\\cdot x)$ as follows $ \\sigma (g^{-1}\\cdot x) =g^{-1} \\sigma (x) \\Bigl (\\underbrace{\\sigma (x)^{-1}g\\sigma (g^{-1}\\cdot x)}_{\\in H} \\Bigr )=:g^{-1}\\sigma (x)\\, h_{(g,x)}.$ Using the covariance property, we obtain $\\phi _{\\sigma }(g^{-1}\\cdot x)=\\rho ( h^{-1}_{(g,x)} )\\phi (g^{-1}\\sigma (x))=\\rho ( h^{-1}_{(g,x)} )(L^{\\ast }_{g^{-1}}\\phi )(\\sigma (x)), $ so $\\phi _{\\sigma }(g^{-1}\\cdot x)=\\rho ( h^{-1}_{(g,x)} )(L^{\\ast }_{g^{-1}}\\phi )_{\\sigma }(x).$ We use this equation in the pull-back of (REF ), where $\\phi $ becomes the column vector $f_{ik}$ for fixed $i$ (the representation $\\rho (g)$ affects only the second index).", "If $f_{ik}$ now denotes for simplicity its pull-back $\\sigma ^{\\ast }f_{ik}=f_{ik}\\circ \\sigma $ , we obtain the same equation, and (REF ) becomes $f_{ik}(g^{-1}\\cdot x)=\\sum _j\\rho ( h^{-1}_{(g,x)} )_{ij}\\sum _l f_{jl}\\,\\rho _{lk}(g).$ (The representation $\\rho $ of $H$ in the first factor on the right affects only the first index of $f_{ik}$ .)", "Application to the Hopf bundle.", "For the Hopf bundle the irreducible representations of $G=SO(3)$ are $(2l+1)$ -dimensional, usually denoted by $D^{(l)}$ .", "Thus $f^{(l)}_{sm}=D^{(l)}_{sm}(g^{-1}), \\quad g\\in SO(3).$ In this case the structure group $H$ is the Abelian group $SO(2)\\cong U(1)$ , with the 1-dimensional irreducible representations $\\rho ^{(s)}(h)=e^{is\\alpha } h, \\quad h=e^{i\\alpha },\\, s\\in \\mathbb {Z}.$ Equation (REF ) becomes with the notation $h_{(g,x)}=:e^{i\\alpha (g,x)}$ $f^{(l)}_{sm}(g^{-1}\\cdot x)=e^{-is\\alpha (g,x)}\\sum _{m^{\\prime }} f^{(l)}_{sm^{\\prime }} (x)\\,D^{(l)}_{m^{\\prime }m}(g).$ Conversely, it is easy to show that this transformation law uniquely determines $f^{(l)}_{sm}$ , up to an $m$ -independent normalization constant.", "This remark will later turn out to be useful." ], [ "Applications to spin-weighted vector-valued differential forms", "In this section we apply the previous results to spin-weighted differential forms.", "Important formulae for spin-harmonics are obtained without much effort." ], [ "Action of $D$ on spin-weighted functions", "The pull-back of (REF ) with the ('standard') local section $\\sigma $ : polar angles $(\\vartheta ,\\varphi )$ of $S^2 \\mapsto (\\varphi ,\\vartheta ,\\psi =0)$ : Euler angles for $SO(3)$ , becomes $\\sigma ^{\\ast }(D\\phi )=d\\phi _\\sigma +isA_\\sigma \\wedge \\phi _\\sigma , $ where $\\phi _\\sigma =\\sigma ^\\ast \\phi ,\\,A_\\sigma =\\sigma ^\\ast A=\\cos \\vartheta \\,d\\varphi $ .", "For $\\sigma ^\\ast (D\\phi )$ we write $D\\phi _\\sigma $ .", "In what follows in this subsection we drop for simplicity the pull-back index $\\sigma $ .", "Then we formally obtain equation (REF ), but with the 1-form $A=\\cos \\vartheta \\,d\\varphi $ on $S^2$ .", "The spin-weighted spherical harmonics $_{s}Y_{lm}$ on $S^2$ (minus the poles) are defined by $\\Bigl (\\frac{4\\pi }{2l+1}\\Bigr )^{1/2}\\, _{-s}Y_{lm} (n):=f^{(l)}_{sm}\\circ \\sigma =D^{(l)}_{sm}(\\sigma ^{-1}(n)).$ We are interested in an explicit form of their covariant differential.", "This is just a special case of the following formula for a vector-valued function $_{s}\\phi $ of type $s$ that follows from (REF ) $D(_{s}\\phi ) = d(_{s}\\phi ) +is\\cos \\vartheta \\,_{s}\\phi \\,d\\varphi = \\partial _\\vartheta \\, (_{s}\\phi )\\,d\\vartheta +(\\partial _\\varphi +is\\cos \\vartheta )\\, _{s}\\phi \\,d\\varphi .$ In terms of the basis $\\theta ^{\\pm }=(d\\vartheta \\pm i \\sin \\vartheta \\,d\\varphi )/\\sqrt{2}$ , we obtain the important formula $-\\sqrt{2} D(_{s}\\phi )= {(\\partial \\!\\!\\!\\!\\!\\!\\;\\diagup }^{\\ast }\\,_{s}\\phi ) \\,\\theta ^{+} +{(\\partial \\!\\!\\!\\!\\!\\!\\;\\diagup }\\,_s\\phi )\\, \\theta ^{-},$ with ${\\partial \\!\\!\\!\\!\\!\\!\\;\\diagup } &=& -\\Bigl (\\partial _\\vartheta +\\frac{i}{\\sin \\vartheta }\\partial _\\varphi \\Bigr ) +s\\cot \\vartheta , \\nonumber \\\\{\\partial \\!\\!\\!\\!\\!\\!\\;\\diagup }^{\\ast } &=& -\\Bigl (\\partial _\\vartheta -\\frac{i}{\\sin \\vartheta }\\partial _\\varphi \\Bigr ) -s\\cot \\vartheta .$ Since $\\theta ^{(\\pm )}$ are of type $\\pm 1$ and $D(_{s}Y_{lm} )$ of type $s$ , we conclude that ${\\partial \\!\\!\\!\\!\\!\\!\\;\\diagup }\\,_{s}Y_{lm}$ is of type $s+1$ and ${\\partial \\!\\!\\!\\!\\!\\!\\;\\diagup }^{\\ast }\\,_{s}Y_{lm} $ of type $s-1$ .", "(More precisely, one should say that these objects are pull-backs of fields on $SO(3)$ of the stated types.)", "Together with the remark after equation (REF ), it follows that ${\\partial \\!\\!\\!\\!\\!\\!\\;\\diagup }\\,_{s}Y_{lm}$ must be proportional to $_{s+1}Y_{lm}$ and ${\\partial \\!\\!\\!\\!\\!\\!\\;\\diagup }^{\\ast }\\,_{s}Y_{lm} $ proportional to $_{s-1}Y_{lm}$ , with proportionality constants that are independent of $m$ .", "These can thus be determined by considering the simple special case $m=0$ .", "Using the following two recursion relations of the associated Legendre functions $P^m_l(x)$ : $ (1-x^2)\\frac{d}{dx} P_l^m(x)=(l+1)xP_l^m(x)-(l-m+1) P_{l+1}^m(x), $ $ \\sqrt{1-x^2} \\frac{dP_l^m}{dx} = P_l^{m+1} - \\frac{mx}{\\sqrt{1-x^2}} P_l^m $ (see, e.g., [5]), one readily obtains the well-known formulae ${\\partial \\!\\!\\!\\!\\!\\!\\;\\diagup }\\,_{s}Y_{lm} &=& \\sqrt{(l-s)(l+s+1)}\\, _{s+1}Y_{lm}, \\nonumber \\\\{\\partial \\!\\!\\!\\!\\!\\!\\;\\diagup }^{\\ast }\\,_{s}Y_{lm} &=& -\\sqrt{(l+s)(l-s+1)}\\, _{s-1}Y_{lm}.$ We emphasize, that our derivation of these equations does not involve any lengthy calculations." ], [ "Splicing of the Hopf bundle with the frame bundle", "We denote with $\\omega $ the Levi-Civita connection form on the frame bundle $F(S^2,SO(2))$ .", "Relative to the dual $(\\pm )$ -bases ${\\theta ^{(\\pm )}}$ and ${e^{(\\pm )}}$ the matrix of $\\omega $ is given by $(\\omega )=\\left(\\begin{array}{ll}iA & \\quad 0 \\\\0 & \\quad -iA\\end{array}\\right).$ This can easily be checked: Since the metric is $g=\\theta ^{(+)}\\theta ^{(-)}$ , i.e., $g_{\\pm \\pm }=0, \\, g_{+-}=g_{-+}=1$ , the symmetry condition $\\omega _{+-}=-\\omega _{-+}$ holds, and the first structure equation is satisfied.", "For a tensor field $T$ on the 2-sphere, with components $T^{\\alpha _1\\alpha _2...\\alpha _p}_{\\beta _1\\beta _2...\\beta _q}=:T^{(\\alpha )}_{(\\beta )}$ , relative to the $(\\pm )$ -basis, the covariant Levi-Civita derivative $D^{LC}$ has the components $(D^{LC}T)^{(\\alpha )}_{(\\beta )} = d\\,T^{(\\alpha )}_{(\\beta )} +s\\, i\\,A \\,T^{(\\alpha )}_{(\\beta )},$ where the integer $s$ is determined as follows: each upper index $\\pm $ adds $\\pm 1$ to $s$ and each such lower index subtracts $\\pm 1$ .", "Comparison of (REF ) with (REF ) and (REF ) shows that the directional derivative $\\nabla _X$ belonging to the Levi-Civita derivative for a tensor field $T$ is related to that of $D$ as follows: $(\\nabla _{e^{(\\pm )}}T)^{(\\alpha )}_{(\\beta )}= (D_{e^{(\\pm )}}T)^{(\\alpha )}_{(\\beta )}= -\\frac{1}{\\sqrt{2}}\\left\\lbrace \\begin{array}{l}{\\partial \\!\\!\\!\\!\\!\\!\\;\\diagup }^{\\ast }T^{(\\alpha )}_{(\\beta )} \\\\\\unknown.", "{}{\\partial \\!\\!\\!\\!\\!\\!\\;\\diagup }T^{(\\alpha )}_{(\\beta )} \\,,\\end{array}\\right.$ with $s$ defined in connection with (REF ).", "Since equation (REF ) shows that $D^{LC}$ is closely related to the differential $D$ discussed previously, although the frame bundle over $S^2$ and the Hopf bundle are quite different objects.", "For this reason it is natural to consider the splicing of the two bundles, with the combined connection $A\\oplus \\omega $ .", "(A precise definition of this construction is given in [6].)", "The corresponding covariant differential will be denoted by $\\mathcal {D}$ .", "For a horizontal vector- valued differential form $\\phi $ of type $\\rho ^s\\times \\rho ^{LC}$ of the structure group $U(1)\\times U(1)$ we have $\\mathcal {D} = d\\phi +(\\rho ^s\\times \\rho ^{LC})_{\\ast } (A\\oplus \\omega )\\wedge \\phi =d\\phi +[\\rho ^s_{\\ast }(A)\\otimes 1 + 1 \\otimes \\rho ^{LC}_{\\ast }(\\omega )]\\wedge \\phi .$ The connection (REF ) shows for instance that the second term on the right may sometimes vanish.", "In some special cases only one summand in the square bracket will not be 0.", "For example, one sees that $\\mathcal {D} \\,e^{(\\pm )}=0,\\quad \\mathcal {D}\\, \\theta ^{(\\pm )}=0.$ Here is another typical application: Consider a function $f_{\\pm s}$ of type $\\pm s$ .", "Then $ \\mathcal {D} (f_{\\pm s} \\underbrace{e^{(\\pm )} \\otimes \\cdot \\cdot \\cdot \\otimes e^{(\\pm )}}_{\\mbox{s factors}} ) =D^{LC} (f_{\\pm s}\\, e^{(\\pm )} \\otimes \\cdot \\cdot \\cdot \\otimes e^{(\\pm )})=\\mathcal {D} (f_{\\pm s})\\, e^{(\\pm )} \\otimes \\cdot \\cdot \\cdot \\otimes e^{(\\pm )} $ or $D^{LC} (f_{\\pm s}\\, e^{(\\pm )} \\otimes \\cdot \\cdot \\cdot \\otimes e^{(\\pm )})=(d f_{\\pm s} \\pm is A f_{\\pm s})\\,( e^{(\\pm )} \\otimes \\cdot \\cdot \\cdot \\otimes e^{(\\pm )}).$ An equivalent relationship was proven in [3].", "This reference is a classic source for spin weighted spherical harmonics.", "These are also well described in Appendix 4 of [1]." ] ]	1403.0480
[ [ "Positronium resonance contribution to the electron g-2" ], [ "Abstract Recently a few authors pointed out that the positroniums give rise to an extra contribution to the electron $g-2$ with the magnitude comparable to the $O(\\alpha^5)$ perturbative effect, which is not taken into account in the perturbative calculation up to $O(\\alpha^5)$.", "Here, we scrutinize how the positronium resonances contribute to the electron $g-2$ through the vacuum polarization function, and conclude that there is no additional sizable $O(\\alpha^5)$ contribution to the electron $g-2$." ], [ "Introduction", "Recently, Ref.", "[1] pointed out that the positroniums in the vector channel give an additional contribution to the electron $g-2$ , $a_e$ , which cannot be captured by the perturbative analysis up to $O(\\alpha ^5)$ [2].", "Subsequently, Ref.", "[3] checked the calculation in Ref.", "[1] and presented an updated value for such a contribution.", "They concluded that the correction is independent of the perturbative contribution and has the same order of magnitude as the $O(\\alpha ^5)$ perturbative contribution and thus affects to the comparison of the experiment and the theory of $a_e$ .", "The assertion in Refs.", "[1], [3] seems to be gradually attaining the consensus in the community of particle phenomenology.", "During the preparation of this article, the two papers [4], [5] presented a negative conclusion on the results in Refs.", "[1], [3].", "In such a circumstance, this article attempts to scrutinize the current issue from the basic of the quantum field theory.", "The consideration in full order QED in Sec.", "shows that Refs.", "[1], [3] do not dealt with the contribution of the positronium resonances.", "The proper identification of such a contribution immediately shows that there is no contribution to $a_e$ from the positronium resonances with the size found in Refs.", "[1], [3].", "In Sec.", ", we start with summarizing the question to be addressed here and present the answer to it.", "Section discusses the connection of this paper with those of the precedence works [1], [3], [4], [5].", "It turns out that the analysis perspective itself, which provides a more convincing approach to the question, is quite different from that in Refs.", "[1], [3], [4], [5]." ], [ "Positronium contribution", "Following Ref.", "[1], we restrict our attention to the type of the QED contribution induced through the vacuum polarization to the electron $g-2$ as shown in Fig.", "REF .", "In order to disentangle the confusion, we first marshal the question itself to be addressed here : How large is the contribution from the positroniums in the vector channel to the electron $g-2$ through Fig.", "REF .", "To this end, we start with reconsidering full-order QED contribution to the two-point function of the electromagnetic current $j_\\mu \\equiv -e\\,\\overline{\\psi } \\gamma _\\mu \\psi $ in the QED with the electron only, which suffices for the succeeding discussion $&\\int d^4 x\\,e^{{\\rm i}q \\cdot x}\\,\\left<0\\right\|T j_\\mu (x) j^\\nu (0)\\left\|0\\right>_{\\rm 1PI}={\\rm i}\\left(\\delta _\\mu ^{\\ \\nu } q^2 - q_\\mu q^\\nu \\right) \\Pi (q^2)\\,.$ The renormalized function will be obtained by $\\Pi _{\\rm R}(q^2) = \\Pi (q^2) - \\Pi (0)$ .", "Since QED does not have any nontrivial classical gauge configurations such as instantons, we can identify which set of Feynman diagrams, e.g.", "an infinite series of ladder diagrams, is associated with the quantum dynamics relevant to the phenomenon of one's interest.", "It is worthwhile to recall some basic features of the state space and $\\Pi (q^2)$ .", "The physical space of QED is spanned by stable one-particle states and multi-particle states composed of them.", "Since confinement does not occur and no stable bound state exists in QED, the only one-particle states are photon, electron and positron.", "Every multi-particle state consists of photon(s) and electron(s).", "The vacuum polarization function $\\Pi (q^2)$ defined in Eq.", "(REF ) is analytic on the surface obtained from two complex planes by braiding on the branch cuts.", "Each of the branch cuts is associated with a multi-particle state $\\left\|\\Psi ;\\,\\left\\lbrace {\\bf q}_j,\\,\\lambda _j\\right\\rbrace _j\\right>$ ($\\lambda _j$ denotes the polarization.)", "that couples non-trivially to the electromagnetic current $j_\\mu $ ; $\\left<0\\right\| j_\\mu (0)\\left\|\\Psi ;\\,\\left\\lbrace {\\bf q}_j,\\,\\lambda _j\\right\\rbrace _j\\right> \\ne 0$ .", "The examples of such multi-particle states are multi-photons, $3 \\gamma $ , $5 \\gamma $ , or $e^- e^+$ , $e^- e^+ \\gamma $ , etc.", "The kinematics involved in the matrix element, say, $\\left<0\\right\| j_\\mu (0)\\left\|3\\gamma ;\\,\\left\\lbrace {\\bf q}_j,\\,\\lambda _j\\right\\rbrace _{j=1,2,3}\\right>$ can be found in Ref. [6].", "With this analytic structure of $\\Pi (q^2)$ in our mind, we derive the dispersion relation for $\\Pi _{\\rm R}(q^2) /q^2$ after introducing the infrared regulator so that the branch cuts of multi-photons start from infinitesimally small constant $s_0 > 0$ $&\\frac{\\Pi _{\\rm R}(q^2 + {\\rm i}\\epsilon )}{q^2 + {\\rm i}\\,\\epsilon }=-\\frac{1}{\\pi }\\int _{0+}^\\infty \\frac{ds}{s}\\,\\frac{{\\rm Im}\\,\\Pi _{\\rm R}(s + {\\rm i}\\,0)}{q^2 - s + {\\rm i}\\epsilon }\\,.", "$ This together with Eq.", "(REF ) immediately yields the expression for the contribution to $a_e$ of the type in FIG.", "REF as a superposition of the contribution $a_e(s)$ from the vector boson with mass squared $s$ weighted by ${\\rm Im}\\,\\Pi _{\\rm R}(s + {\\rm i}\\,0)$ $a_e[{\\rm vp}] =\\int _{0+}^\\infty \\frac{ds}{s}\\,{\\rm Im}\\,\\Pi _{\\rm R}(s + {\\rm i}\\,0)\\,a_e(s)\\,.$ In fact, the branch cuts associated with the multi-photons are overlooked in FIG.", "3 of Ref. [1].", "Instead, $\\Pi (q^2)$ is supposed to have complex poles.", "However, complex poles are just the concepts that are often introduced temporarily in particle phenomenology for the purpose to calculate the total decay width and make comparison with the experiments.", "The imaginary part of a complex pole, the decay width, depends upon one's definition.", "The requirement of the gauge independence, for instance, may motivate to choose a more favorable one [7].", "Theoretically, the unitarity is assured only if $\\Pi (q^2)$ can receive nontrivial contribution from the states $\\left\|\\Psi ;\\,\\left\\lbrace {\\bf q}_j,\\,\\lambda _j\\right\\rbrace _j\\right>$ such that $\\left<0\\right\| j_\\mu (0)\\left\|\\Psi ;\\,\\left\\lbrace {\\bf q}_j,\\,\\lambda _j\\right\\rbrace _j\\right>\\ne 0$ , which results in producing the branch cuts of $\\Pi (q^2)$ .", "Figure: A typical positronium contribution tothe vacuum polarization function.The blob part denotesthe one-particle irreducible correlation functionof four electromagnetic currents.Now, Eq.", "(REF ) together with the above remark on the analytic property of $\\Pi _{\\rm R}(q^2)$ immediately allows to identify which type of diagrams contains the positronium contribution.", "The ortho-positronium contribution is associated with ${\\rm Im}\\,\\Pi _{\\rm R}(s)$ originating from $(2n + 3) \\gamma $ states ($n = 0,\\,1,\\,2,\\cdots $ ).", "This is because the component involved in the state $j_\\mu (0) \\left\|0\\right>$ that can be considered as the positronium ground state, say, should be concentrated in the region centered at $\\sqrt{s} \\simeq 2 m_e - \\alpha ^2 m_e /4$ with the narrow width $\\propto \\alpha ^6 m_e$ so that its overlap with each multi-state containing $e^- e^+$ is vanishing small.", "Therefore, the physical ortho-positronium resonance contribution comes from, say, the diagram in FIG.", "REF , but with the vacuum polarization part replaced by a particular form shown in FIG.", "REF , which is distinctly different than that dealt with in Refs.", "[1], [3] and essentially that in Refs.", "[4], [5].", "Such diagrams at the leading-order will be $O(\\alpha ^7)$ if at least one photon must be exchanged in each of the two light-by-light scattering amplitudes in FIG.", "REF to form a positronium.", "The smallness of such a contribution will be speculated from the result [10] for the $O(\\alpha ^5)$ contribution to $a_e$ caused by the diagrams of the type in FIG.", "REF with the one-loop light-by-light scattering subdiagrams, which belong to Set I(j) according to the classification scheme of $O(\\alpha ^5)$ diagrams [8], [9] $a_e[{\\rm I(j)}, e\\ {\\rm only}]= 0.000\\ 3950\\ (87) \\left(\\frac{\\alpha }{\\pi }\\right)^5\\,.$ This is quite smaller than the dominant tenth-order contribution which is found to have magnitude $O(1) \\times (\\alpha /\\pi )^5$ [2].", "The resonance contribution starting at $O(\\alpha ^7)$ will be further suppressed by the factor $\\left(\\alpha /\\pi \\right)^2$ .", "In this paper, we see how the physical positronium resonance contributes to the electron $g-2$ through the diagram in FIG.", "REF .", "Needless to say, there is no systematic way to single out the resonance contribution by separating it from the continuum contribution.", "Properly speaking, what is done above is to correctly identify a set of the diagrams which contains the contribution of positronium resonances in the vector channel." ], [ "Conclusion and discussion", "The plain consideration in Sec.", "enabled us to correctly identify a set of diagrams that contain the positronium resonance contribution, and leads the conclusion that there is no additional nonperturbative correction to the electron $g-2$ of the same size as the perturbative correction of $O(\\alpha ^5)$ as was first pointed out in Ref. [1].", "Here, we discuss the connection of our founding with the precedence works concerning with the current issue.", "Obviously, the difference of this work from Refs.", "[1], [3], [4], [5] stems from the fact that we never neglect the instability of the positroniums and deal with the proper space of states in QED.", "The question to be addressed in this article is defined at the beginning of Sec.", ", and we obtain a definite answer to it.", "In contrast, the other works seems to cast the following question by neglecting the unstable character of positroniums: Is the QED dynamics which are mainly concerned with the formation of positroniums give rise to an additional nonperturbative contribution to the electron $g-2$ ?", "Instead of chasing the details of the discussions in Refs.", "[1], [3], [4], [5], we discuss the following points in the rest of the paper: On one hand, one focuses on some coulombic dynamics nonperturbatively.", "On the other hand, one wishes to forbid the decay of the bound state, which cannot be realized just from the perturbative order counting.", "We examine theoretically what concrete approximation reconciles these seemingly contradictory situations.", "There is no local field theory that reproduces only the approximation to the two-point function, i.e.", "the connected diagram contribution.", "It is thus inevitable to deal with another types of contribution which unstabilize positroniums, and to examine the current issue by working with the state space as described in Sec. .", "Figure: Connected diagram contribution tothe two point function of the electromagnetic currentsin lattice QED.An arrowed dotted line representsthe inverse of the electron Dirac operator,D[A] -1 (x,y)D[A]^{-1}(x,\\,y).𝒬 QED \\left<\\mathcal {Q}\\right>_{\\rm QED} denotesthe vacuum expectation value of the quantity 𝒬\\mathcal {Q} in QED.The second term contributes only if the positionsof the two current operators coincide with each other.We adopt the lattice regulariation for QED just because the terminology in the framework of the lattice field theory is used in the following discussion.", "Figure REF corresponds to the (gauge-invariant) nonperturbative approximation to the vacuum polarization function (REF ) taken in Refs.", "[1], [3], [4], [5].", "Each arrowed dashed line in that figure is not the fermion propagator in the perturbation theory, but the inverse of the Dirac operator of the electron, $D[A]^{-1}(x,\\,y)$ , under a given gauge potential $A$ .", "The symbol $\\left<\\mathcal {Q}[A]\\right>_{\\rm QED}$ denotes the vacuum expectation value (VEV) of a quantity $\\mathcal {Q}[A]$ that depends only on the gauge potential in QED It is not necessary to work in Euclidean space unless one attempts to simulate the system.", "$\\left<\\mathcal {Q}[A]\\right>_{\\rm QED} =&\\,\\frac{1}{Z_{\\rm QED}}\\int d\\psi \\,d\\overline{\\psi }\\,dA\\,\\exp \\left({\\rm i}S_{\\rm QED}\\left[A,\\,\\psi ,\\,\\overline{\\psi }\\right]\\right)\\mathcal {Q}[A] \\nonumber \\\\=&\\,\\frac{1}{Z_{\\rm QED}}\\int dA\\,\\exp \\left({\\rm i}S_{\\rm G}[A]\\right){\\rm det}\\left(- D[A]\\right)\\,\\mathcal {Q}[A] \\,,\\nonumber \\\\S_{\\rm QED}\\left[A,\\,\\psi ,\\,\\overline{\\psi }\\right] =& S_{\\rm G}[A]+ a^4 \\sum _{x} \\overline{\\psi }(x) ({\\rm i}D[A] \\psi )(x)\\,,\\nonumber \\\\S_{\\rm G}[A]=&\\,a^4 \\sum _{x}\\left\\lbrace - \\frac{1}{4}\\,F_{\\mu \\nu }(x)\\,F^{\\mu \\nu }(x)+ \\frac{1}{2\\lambda } \\left(\\partial \\hspace{-0.5pt}\\smash{\\ast }\\right.\\hspace{-5.0pt}_{\\mu } A^\\mu \\right)^2 \\,,\\nonumber \\\\Z_{\\rm QED} =&\\int d\\psi \\,d\\overline{\\psi }\\,dA\\,\\exp \\left({\\rm i}S_{\\rm QED}\\left[A,\\,\\psi ,\\,\\overline{\\psi }\\right]\\right)\\,,$ where“$a$ ” denotes the lattice spacing, $F_{\\mu \\nu }(x)$ in the noncompact formulation of the lattice QED takes the familiar form $F_{\\mu \\nu }(x) = \\partial _\\mu A_\\nu (x) - \\partial _\\nu A_\\mu (x)$ but with the forward difference $a\\,{\\partial _{\\mu }} f(x) \\equiv f(x + a \\widehat{\\mu }) - f(x)$ , and $\\partial \\hspace{-0.5pt}\\smash{\\ast }$ $ denotes the backward difference operator;$ a $\\ast $ f(x) f(x) - f(x - a )$.The tadpole diagram in FIG.~\\ref {fig:ConnDiagLatticeQED}appears because $ D[A]$ involves theWilson line, $ e-ie a A(x)$, which parallel-transports backthe variable at $ x + a $ to $ x$.$ Figure: Disconnected diagram contribution tothe correlation function of two electromagnetic currents.The diagram in FIG.", "appearsas a part of this type of diagrams.Noting that the VEV of the operator $O\\left[\\psi ,\\,\\overline{\\psi },\\,A\\right]$ depending on the electron fields can be converted to that of $\\widehat{O}[A]$ defined by $\\widehat{O}[A] \\equiv \\left[O\\left[\\frac{1}{a^4} \\frac{\\partial ^L}{{\\rm i}\\partial \\overline{\\eta }},\\,\\frac{1}{a^4} \\frac{\\partial ^R}{{\\rm i}\\partial \\eta },\\,A\\right]\\exp \\left(a^4 \\sum _x a^4 \\sum _y{\\rm i}\\overline{\\eta }(x)\\,D[A]^{-1}(x,\\,y)\\,{\\rm i}\\eta (y)\\right)\\right]_{\\eta ,\\,\\overline{\\eta } \\rightarrow 0}\\,,$ with the left (right) derivative $\\partial ^L /\\partial \\overline{\\eta }(x)$ ($\\partial ^R /\\partial \\eta (y)$ ), the total contribution to the full correlation function of two electromagnetic currents $&\\int d^4 x\\,e^{{\\rm i}q \\cdot x}\\,\\left<0\\right\|T j_\\mu (x) j^\\nu (0)\\left\|0\\right>={\\rm i}\\left(\\delta _\\mu ^{\\ \\nu } q^2 - q_\\mu q^\\nu \\right) J(q^2)\\,,$ is found to be given by the sum of the connected diagram in FIG.", "REF and the disconnected diagram shown in FIG.", "REF , which also contains the contribution of the one-particle reducible diagrams.", "A simple diagrammatic consideration enables to express $J(q^2)$ in term of $\\Pi (q^2)$ $J(q^2) = \\frac{\\Pi (q^2)}{1 + \\Pi (q^2)}\\,.$ We recall that the disconnected diagram contains the diagrams responsible to the decay of the positroniums.", "The simulation of the connected diagram in FIG.", "REF will allow to measure the masses of the pseudo-bound states in the vector channel that are absolutely stable.", "However, the connected diagram contains, say, the ladder-type photon exchange only in the “$t$ -channel”, where the“$s$ -channel” is taken to be in the direction of the injected momentum in FIG.", "REF .", "If we cut the subdiagrams with four external fermion lines out of a perturbative diagram of the type in FIG.", "REF and embed it again into the rest of the original after rotating it by 90 degrees in a clockwise direction, we will obtain a diagram of the type in FIG.", "REF .", "This indicates that there is no local Lagrangian density that reproduces the contribution from the connected diagram and no contribution from disconnected diagram.", "The situation should be contrasted with the case in which the decay process caused by the weak interaction is neglected.", "Then, there exists a local field theory that describes the system with the weak interaction switched off The weak interaction will be switched off by letting the VEV of Higgs doublet $v$ and the electron yukawa coupling $y_e$ going to 0 with the electron mass $m_e = y_e v /\\sqrt{2}$ fixed finite., and we can construct the state space at the zeroth-order of the approximation.", "A more concrete and familiar example is the description of hadron physics where the zeroth-order is approximated by the world with QCD only and the corrections due to QED The electromagnetic correction to the meson masses can also be incorporated perturbatively as in Ref. [11].", "and the dynamics of weak interactions are managed perturbatively.", "If no local field theory describing the zeroth-order approximation exists, we cannot proceed with the calculation of $a_e$ with use of the dispersion relation as Eq.", "(REF ) which relies on the analytic property of $\\Pi (q^2)$ , the existence of the state space and unitarity.", "Hence, we have to tackle with the current issue using the state space described as in Sec.", "and Eq.", "(REF ).", "The connected diagram in FIG.", "REF gives a significant contribution to the electron $g-2$ .", "Eq.", "(REF ) implies that such a contribution comes from the intermediate states of $e^- e^+$ , $e^- e^+ \\gamma $ , etc.", "and can be calculated by means of perturbation.", "The author thanks G. Mishima for letting him know full details of Ref.", "[1] and the basic of Bethe-Salpeter amplitude." ] ]	1403.0416
[ [ "On the probability distribution function of the mass surface density of\n molecular clouds I" ], [ "Abstract The probability distribution function (PDF) of the mass surface density is an essential characteristic of the structure of molecular clouds or the interstellar medium in general.", "Observations of the PDF of molecular clouds indicate a composition of a broad distribution around the maximum and a decreasing tail at high mass surface densities.", "The first component is attributed to the random distribution of gas which is modeled using a log-normal function while the second component is attributed to condensed structures modeled using a simple power-law.", "The aim of this paper is to provide an analytical model of the PDF of condensed structures which can be used by observers to extract information about the condensations.", "The condensed structures are considered to be either spheres or cylinders with a truncated radial density profile at cloud radius r_cl.", "The assumed profile is of the form rho(r)=rho_c/(1+(r/r_0)^2)^{n/2} for arbitrary power n where rho_c and r_0 are the central density and the inner radius, respectively.", "An implicit function is obtained which either truncates (sphere) or has a pole (cylinder) at maximal mass surface density.", "The PDF of spherical condensations and the asymptotic PDF of cylinders in the limit of infinite overdensity rho_c/rho(r_cl) flattens for steeper density profiles and has a power law asymptote at low and high mass surface densities and a well defined maximum.", "The power index of the asymptote Sigma^(-gamma) of the logarithmic PDF (Sigma x P(Sigma)) in the limit of high mass surface densities is given by gamma = (n+1)/(n-1)-1 (spheres) or by gamma=n/(n-1)-1 (cylinders in the limit of infinite overdensity)." ], [ "Introduction", "Observations of the structure of molecular clouds provide insights about the physical processes in the cold dense phase of the interstellar medium and will give us a better understanding how they evolve and eventually form stars.", "They are furthermore essential to test theoretical models of the origin of the stellar mass function [52], [10], [50], [27], [11], [30] and the star formation rate [42], [51], [26], [15] which are both thought to be related to the density structure of a turbulent molecular gas.", "The high resolution and sensitivity of modern telescopes allows a detailed analysis of the 1-point statistic or probability distribution function (PDF) of the mass surface density of molecular cloud gas.", "They are obtained using either the reddening of stars [34], [25], [44], [35], [2] or more recently the infrared emission of dust grains [28], [29], [59], [60], [58], [56].", "Despite the complexity of the molecular clouds the observed PDFs of the mass surface density show very similar properties.", "They all are characterized by a broad peak and a tail at high mass surface densities approximately given by a power law.", "The PDF at low mass surface densities around the broad peak is attributed to randomly moving gas commonly referred to as 'turbulence' while the tail is attributed to self-gravitating cloud structures.", "The relative amount of the two different components seems to be related to the star formation activity in the cloud as discussed by [34].", "While non-star forming clouds as the 'Coalsack' or the 'Lupus V' region show only a very low or no evidence of a tail the PDFs of star forming clouds as 'Taurus' or 'Orion' are characterized by a strong tail with no clear separation between the two components.", "The observations seem to be broadly consistent with current simulations of turbulent molecular clouds.", "Turbulence would naturally create a wide range of densities and simulations of driven isothermal turbulence have shown that the corresponding PDF has a log-normal form [63], [49], [53], a result which has been confirmed analytically [46].", "The projection of the density of those simulations has also been found to be closely log-normal in shape [47], [64], [18], [6].", "Deviations are expected for non isothermal turbulence which produces higher probabilities at low or high densities [57], [53], [43].", "More recent simulations of forced turbulence also show depending on the assumed forcing for the PDF of the volume density a deviation from the log-normal function with enhanced probabilities at low densities [17], [40], [14].", "The functional form is as shown by [14] approximately described by a statistical function proposed by [31].", "Simulations of the time evolution of molecular clouds have shown that at late stage the PDF of the volume density would develop a tail-like structure at high density values [39], [8], [65].", "The same behavior is also seen in the PDF of the mass surface density [4], [41], [16].", "Currently, observed PDFs are analyzed using a log-normal function for the peak and a simple power law for the tail, respectively.", "The log-normal function allows a first estimate of the density contrast of the volume density in a turbulent medium based on the fundamental relation of the statistical properties of the mass surface density and the ones of the volume density as provided by [21] and also by [7], [6].", "The interpretation of the tail is frequently based upon a simple power law density profile $\\rho (r)\\propto r^{-n}$ of spheres where the PDF of the mass surface density is also a power law.", "In case of the logarithmic PDF ($\\Sigma P(\\Sigma )=P(\\ln \\Sigma )$ )The PDF of the logarithmic values of the mass surface density is referred to as logarithmic PDF while the PDF of the absolute values of the mass surface density as linear PDF.", "the corresponding power law would be $\\Sigma ^{-\\gamma }$ with $\\gamma =2/(n-1)$ [41], [16].", "Applying this relation the slope of the tail of the PDF of a number of star forming molecular clouds indicates a radial density profile with a power index $n\\sim 2$ [58] as expected for collapsing clouds.", "A different approach has been chosen by [35] who also applied a log-normal function to the tail.", "However, the analytical functions show partly strong deviations to the observed curves.", "Most of the PDFs published by [34] reveal a tail at low mass surface densities below the peak which cannot be explained in terms of the simple analytical function.", "The tail at high mass surface densities has several features which are not expected using simple power law profiles of the radial density.", "Foremost, the tail is restricted to a certain range of mass surface densities.", "For a number of PDFs published by [34] the tail indicates a strong cutoff or a strong change of the slope around $A_V=6-10~{\\rm mag}$ .", "The interpretation of the tail is further complicated by the observational facts that condensed clouds are generally located on a certain background level and that they are restricted to small regions within the cloud complex as e.g.", "in case of the Rosette molecular cloud [59].", "The tail is therefore not necessarily a simple power law nor directly related to the radial density profile.", "Furthermore, the analytical functions do not provide a physical explanation for the peak position of the PDF which occurs in case of a number of molecular clouds around $A_V\\sim 1~{\\rm mag}$ .", "In this and the following papers an analytical physical model of the global PDF of molecular clouds is developed which resembles the main observed features and is meant to derive basic physical parameters of star forming molecular clouds as the pressure and the density contrast in the turbulent gas.", "This paper focuses on the 1-point statistical properties of individual condensed structures, assumed to be spheres and cylinders.", "In Sect.", "2 an analytical solution of the mass surface density and the corresponding PDFs for the considered shapes is presented which is based on a truncated analytical density profile widely used in astrophysical problems.", "In Sect.", "3 the properties of the PDFs are discussed and asymptotes for low and high mass surface densities are provided.", "Also studied is the location of the maximum position of the logarithmic and linear PDF.", "A summary is given in Sect. 4.", "The technical details can be found in the appendices.", "Let us assume for the condensed structures a simple analytical density profile given by $\\rho (r) = \\frac{\\rho _{\\rm c}}{(1+(r/r_{0})^2)^{n/2}},$ where $\\rho _{\\rm c}$ is the density in the cloud center and $r_{\\rm 0}$ the inner radius.", "The density profile has a flat part in the inner region $(r\\ll r_0)$ which approaches asymptotically a power law $\\rho \\propto r^{-n}$ at large radii $(r\\gg r_0)$ .", "In studies of stellar clusters this inner radius is frequently referred to as `core radius' [36], [37], [38].", "The analytical profile is used in astrophysics for its convenience and as generalization of physical density profiles to model the stellar surface brightness of Globular clusters (e. g. [13], [12]) and more recently the dust emission of dense filaments (e. g. [3], [45], [33]).", "We make another reasonable assumption that the profile is truncated at radius $r_{\\rm cl}$ as expected if the clouds are cold structures embedded in warmer gas.", "In case of pressure equilibrium the gas pressure at the outer boundary of the condensed structure would be equal to the pressure $p_{\\rm ext}$ in the surrounding medium.", "We refer to the inverse of the density ratio $\\rho (r_{\\rm cl})/\\rho _{\\rm c}=q$ of the density at cloud radius and cloud center as `overdensity'.", "In case of isothermal clouds this ratio is identical with the term 'overpressure' used to characterize the gravitational state of self-gravitating structures in previous studies of pressurized clouds [22], [19], [24], [23].", "Specific profiles with certain values of the power $n$ are known solutions for the physical problem of self-gravitating gaseous clouds.", "The profile for $n=5$ is valid for a gaseous sphere where the temperature of the gas is regulated by the adiabatic law with a ratio 1.2 of the two specific heats [61], [32].", "This density profile has been applied to describe the surface brightness profiles of globular clusters and is known as Plummer-model [54], [55].", "The density profiles which correspond to the power indices $n=2$ , 3, and 4 are related to profiles of isothermal self-gravitating clouds.", "The profile with $n=4$ is the exact solution for a self-gravitating isothermal cylinder [62], [48].", "The radial density profile of pressurized isothermal self-gravitating spheres, referred to as Bonnor-Ebert sphere [9], [5], cannot be expressed through a simple analytical formula.", "However, the profile for spherical clouds up to an overdensity $\\sim 100$ is in excellent agreement with the analytical profile with $n=3$ .", "The profile with $n=3$ is also identical with the well known King model without truncation [36], [37].", "The radial density profile of spheres with higher overdensity might be approximated with a profile where $n=2$ .", "The statistical properties of isothermal clouds are analyzed in more detail in a forthcoming paper [20]." ], [ "Mass surface density profiles", "The mass surface density profile of a truncated analytical density profile (Eq.", "REF ) of a sphere or a cylinder seen at inclination angle $i$ where $i=0^\\circ $ refers to a cylinder seen edge-on is given by $\\Sigma _n (r) = \\frac{2}{\\cos ^\\beta i}\\int \\limits _0^{r_{\\rm cl}\\sqrt{1-(r/r_{\\rm cl})^2}} {\\rm d}l\\frac{\\rho _{\\rm c}}{(1+(r^2+l^2)/r_{\\rm 0}^2)^{n/2}}.$ where $\\beta =0$ for spheres and $\\beta =1$ for cylinders.", "In the following it is convenient to define a parameter $y_n = (1-q^{2/n})(1-x^2),$ where $x=0\\le r/r_{\\rm cl}\\le 1$ is the normalized impact parameter where $r$ is the projected radius and $r_{\\rm cl}$ the cloud radius.", "The profile of the mass surface density can then be written as $\\Sigma _{n} (x) = \\frac{2}{\\cos ^\\beta i} r_0\\rho _c q^{\\frac{n-1}{n}}(1-y_n)^{\\frac{1-n}{2}}\\int \\limits _0^{u_{\\rm max}} \\frac{{\\rm d}u}{(1+u^2)^{n/2}},$ where $u_{\\rm max} = \\sqrt{\\frac{y_n}{1-y_n}}.$ In case of isothermal self-gravitating pressurized clouds the product of inner radius and central density is proportional to $\\sqrt{p_{\\rm ext}}$ and is given by $r_0 \\rho _c = \\sqrt{\\frac{\\xi _n p_{\\rm ext}}{4\\pi G}}\\frac{1}{\\sqrt{q}} ,$ where $p_{\\rm ext}$ is the external pressure, $G$ the gravitational constant and where $\\xi _2=2$ , $\\xi _3=8.63$ , and $\\xi _4 = 8$ [20].", "It is convenient to consider the normalized mass surface density $X_n = \\Sigma _n \\cos ^\\beta i \\,(2\\rho _{\\rm c}r_0)^{-1}q^{-\\frac{n-1}{n}},$ which depends only on the parameter $y_n$ .", "The functional dependence is shown in Fig.", "REF .", "Figure: Normalized mass surface density X n X_n as function of 1-y n 1-y_nfor truncated density profiles as given in Eq.", "for variouspower indices nn ranging from 0 to 10.", "The thicker linescorrespond to n=0n=0, 1, and 3.", "For a fixed pressure ratio qq the mass surfacedensities for a given power index nn vary from the central value at 1-y n =q 2/n 1-y_n=q^{2/n} tozero at the edge of the cloud where 1-y n =11-y_n=1.", "The central values of the normalized mass surfacedensity for various pressure ratios qq are shown as thin gray lines.Profiles of the normalized mass surface density for a number of different assumptions for the power index $n$ of a truncated density profile are shown in Fig.", "REF .", "The method used to derive the profiles is described in Sects.", "REF and REF .", "The profiles for $n=1$ , 2, 3, and 4 are simple analytical functions given in App.", "REF .", "At an overdensity of 10 the inner part of the profiles of cylinders and spheres closely resembles a Gaussian approximation with a width as given in Sect.", "REF .", "Figure: Profiles of the normalized mass surface density X n X_nof a truncated density profile as given in Eq.", "for threeassumptions of the pressure ratio q=p ext /p c q=p_{\\rm ext}/p_{\\rm c}.The maxima or the curves are labeled with the corresponding power indexnn.", "The profiles for n=1n=1 and n=3n=3 are enhanced with a thicker line.The dashed lines are Gaussian approximations (Eq. )", "where the varianceis obtained using Eq.", ".The profile for n=0n=0 need to be considered as an asymptote." ], [ "PDF of the mass surface density", "The PDF of the mass surface density can be given as an implicit function of $y_n$ .", "In the limit $y_n \\rightarrow 1$ and $y_n \\rightarrow 0$ also explicit expressions of the asymptotic behavior of the PDF can be given and are discussed.", "The PDF for the mass surface density is given by $P( \\Sigma _{n} (x)) = \\frac{{\\rm d}P}{{\\rm d} \\Sigma _{n} }= - P(r) \\left(\\frac{{\\rm d} \\Sigma _{n} }{{\\rm d}r}\\right)^{-1},$ where $P(r)$ is the probability to measure a mass surface density at impact radius $r$ .", "For a sphere this is given by $P(r) = 2\\pi r / (\\pi r_{\\rm sph}^2)=2x/r_{\\rm sph}$ and for a cylinder $P(r) = 1/r_{\\rm cyl}$ .", "It is convenient to consider the PDF of the normalized mass surface density as defined in Eq.", "REF which is given by $P(X_n) = P(\\Sigma _n)\\frac{1}{\\cos ^\\beta i}( 2 r_0 \\rho _{\\rm c}) q^{\\frac{n-1}{n}}.$ with $\\beta =0$ for spheres and $\\beta =1$ for cylinders.", "By taking the derivative of Eq.", "REF it is straightforward to show that $\\frac{{\\rm d} X_n }{{\\rm d}r} = -(1-q^{2/n})\\frac{x}{r_{\\rm cl}} \\frac{1+ (n-1)\\sqrt{y_n}X_n}{\\sqrt{y_n}(1-y_n)}.$ For spheres we obtain the implicit function $P_{\\rm sph}(X_n) = \\frac{2}{1-q^{2/n}}\\frac{\\sqrt{y_n}(1-y_n)}{1+(n-1) \\sqrt{y_n} X_n(y_n)}.$ As we see the normalized PDF $(1-q^{2/n})P_{\\rm sph}(X_n)$ is an implicit function of the parameter $y_n$ alone.", "The corresponding PDF for cylinders is $P_{\\rm cyl}(X_n) = \\frac{1}{2x}P_{\\rm sph}(X_n),$ where the normalized impact parameter can be expressed through $x = \\sqrt{\\frac{1-y_n-q^{2/n}}{1-q^{2/n}}}.$ The PDF has a pole at the maximum mass surface density ($x=0$ ) or where $y_n = 1-q^{2/n}$ (Fig.", "REF ).", "Because of the pole for cylinders we have therefore not a generalized form of the PDF that depends only on the parameter $y_n$ .", "However, to obtain an expression which allows a similar discussion in the following section for spheres and cylinders we can consider the asymptotic PDF for infinitely high overdensity.", "In the limit $1-y_n\\gg q^{2/n}$ the impact parameter behaves as $x \\approx \\sqrt{1-y_n}.$ We consider therefore the following asymptotic PDF $P^{(a)}_{\\rm cyl}(X_n) = \\frac{1}{1-q^{2/n}}\\frac{\\sqrt{y_n}\\sqrt{1-y_n}}{1+(n-1) \\sqrt{y_n} X_n(y_n)}.$ As shown in App.", "the asymptotic PDF provides for all $q$ the correct asymptote in the limit of small mass surface densities and the correct asymptotic behavior at large mass surface densities for $q\\rightarrow 0$ .", "The PDFs of spheres and the asymptotic PDFs of cylinders with a truncated analytical density profile for various different assumptions for the power $n$ and the pressure ratio $q$ are shown in Fig.", "REF .", "They are truncated at the highest mass surface density.", "The exact asymptotes for high and low mass surface densities also shown in the figure are derived in Sect. .", "The maximum position is discussed in more detail in Sect. .", "The curves have a functional form that depends only on the power $n$ of the radial density profile.", "They are truncated at the central mass surface density." ], [ "Asymptotes at high and low mass surface densities", "At low mass surface densities the PDF approaches asymptotically a power law $P(\\Sigma )\\propto \\Sigma $ .", "The asymptotic behavior at high mass surface densities depends on the steepness of the radial density profile.", "For $n>1$ the PDF approaches asymptotically a power law given by $P_{\\rm sph}(X_n) \\propto X_n^{-\\frac{n+1}{n-1}},\\quad P_{\\rm cyl}^{(a)}(X_n)\\propto X_n^{-\\frac{n}{n-1}}.$ As can be seen in the figure the asymptote is a better representation of the PDF at high mass surface densities for steeper density profiles.", "In the limit of large power $n$ the PDF at high mass surface densities becomes $P(X_n)\\propto X_n^{-1}$ as expected for a source with a Gaussian density profile (Sect.", "REF ).", "For $n=1$ the PDF at high mass surface density is an exponential function $P_{\\rm sph}(X_n)\\propto e^{-2 X_1}, \\quad P_{\\rm cyl}^{(a)}(X_n) \\propto e^{-X_1}.$ For clouds with $n<1$ the PDF is limited to a maximum mass surface density given by $X_n=1/(1-n)$ .", "In the neighborhood of this maximum value the PDF varies strongly with mass surface density.", "In the limit of $n=0$ the PDF becomes identical to the PDF of a homogeneous sphere or cylinder.", "The power law asymptote at high mass surface densities may be used to infer the power $n$ of the radial density profile.", "For spheres a power law $P_{\\rm sph}\\propto \\Sigma ^{-\\alpha }$ would indicate a power index $n=(\\alpha +1)/(\\alpha -1)$ of the radial density profile as can be derived for simple power law density profiles [41], [16].", "As shown in Fig.", "REF the asymptotic behavior at large mass surface densities in case of cylinders is only established for sufficiently high overdensities.", "For example, a profile with $n=4$ which is consistent with the density profile of self-gravitating isothermal cylinders the PDF for overdensities as high as 100 has no apparent power law at high mass surface densities.", "For cylinders with high overpressure the PDF at the pole is approximately given by $P_{\\rm cyl}(X_n)\\propto X_n^{-\\frac{n}{n-1}}/\\sqrt{1-(X_n/X_n(0))^{2/(n-1)}}$ A power law is only established for the range $[X_n]_{\\rm max}\\ll X_n\\ll X_n(0)$ where $[X_n]_{\\rm max}$ is the mass surface density at the PDF maximum (Sect.", "REF ).", "Figure: PDF of the normalized mass surface density X n X_n of cylindrical clouds havinga density distribution with n=1n=1 (gray lines) and n=4n=4 (black lines).", "The clouds have an overdensity (1/q1/q)of either 10 or 100.", "The long dashed curves show the asymptote for infinite high overdensity.", "The dotted lines markthe asymptotic value at the poles.", "The short dashed lines give the asymptotic values in the limit of high and lowvalues of X n X_n.Figure: PDF of the normalized mass surface density X n X_n in case of spheres (top figure)and the asymptotic PDF in case of cylinders (bottom figure) with a truncated radial density profilegiven by Eq.", "for various assumptions of the power nn.The thin lines showat which mass surface density X n X_n at given density ratio q=ρ(r cl )/ρ c q=\\rho (r_{\\rm cl})/\\rho _{\\rm c} the PDF correspondingto a certain power nn truncates (for spheres) or has a pole (for cylinders).The lines are labeled with log 10 q\\log _{10} q.", "The dashed lines showthe asymptotic behavior at large and small X n X_n." ], [ "The maxima of the asymptotic PDF", "As we see in Fig.", "REF the PDF of spheres and the asymptotic PDF of cylinders with the analytical density profile have well defined maxima at $[X_n]_{\\rm max}$ which depend only on the power $n$ .", "This allows a simple interpretation of the observed mass surface density at the peak position in terms of the normalization factor $2r_0\\rho _{\\rm c}q^{(n-1)/n}$ for given $n$ using the definition Eq.", "REF .", "In case of isothermal clouds the maximum position can be used to infer the pressure in the ambient medium.", "Table: Maxima positions of the linear and logarithmic PDFFigure: Maxima position of the PDF of the parameter y n y_n and thenormalized mass surface density X n X_n for spheres (top) and cylinders (bottom) as functionof the power nn (dotted curves).", "Black lines correspond to the linear PDF and the gray linesto logarithmic PDF.", "The solid curves are polynomial fits to the accurate position values.", "Thevalues for isothermal self-gravitating spheres (n=2n=2 and n=3n=3) and cylinders (n=4n=4)are shown as small diamonds.", "The asymptotic behaviorof [X n ] max =1/(1-n)[X_n]_{\\rm max}=1/(1-n) for n→0n\\rightarrow 0 for cylinders is shown as dashed line.Asymptotes described in Sect.", "are shown as dashed lines.Related to the maximum position is a parameter $[y_n]_{\\rm max}$ .", "The functional dependence of the maximum position $[y_n]_{\\rm max}$ and $[X_n]_{\\rm max}$ of the linear and logarithmic asymptotic PDF on the power $n$ is shown in Fig.", "REF .", "The curves are derived by solving the conditional equations given in App. .", "Selected values are given in Tab.", "REF .", "As shown in Sect.", "REF , in the limit $n\\rightarrow 0$ the PDFs become the ones of homogeneous spheres and cylinders where the maximum value is related to the central mass surface density.", "In this limit we have therefore $[y_n]_{\\rm max}\\rightarrow 1$ and $[X_n ]_{\\rm max}=1$ .", "In case of cylinders the normalized mass surface density at PDF maximum in the limit of flat density profiles is approximately given by $[X_n]_{\\rm max}=1/(1-n)$ .", "The parameter $[y_n]_{\\rm max}$ provides for given power $n$ an estimate of the corresponding impact parameter as function of the pressure ratio $q$ based on Eq.", "REF .", "Below a minimum overpressure $(1-[y_n]_{\\rm max})^{-n/2}$ the maximum coincides with the central mass surface density.", "At higher overpressures the impact parameter moves outwards and reaches asymptotically a maximum value $x = \\sqrt{1-[y_n]_{\\rm max}}$ .", "If we consider for example the logarithmic PDF of a sphere with $n=3$ the minimum overdensity with $x=0$ is $q^{-1} = \\left(1-\\left[\\sqrt{2}-1\\right]\\right)^{-3/2} \\approx 2.23,$ where the parameter $[y_n]_{\\rm max}$ is taken from Tab.", "REF .", "In the limit $q\\rightarrow 0$ the impact parameter at the PDF maximum becomes $x = \\sqrt{1-\\left[\\sqrt{2}-1\\right]} \\approx 0.765.$ As the maximum position $[y_n]_{\\rm max}$ decreases with $n$ the impact parameter related to the PDF maximum is larger for steeper density profiles and reaches $x=1$ for $n\\rightarrow \\infty $ .", "In case of the logarithmic PDF the mass surface density related to the PDF maximum is larger in respect to the linear PDF and corresponds to a smaller impact parameter.", "The dependence of the maxima of the linear PDF in the limit $n\\rightarrow \\infty $ can be described by simple power laws as shown in App.", "REF .", "The asymptotes for $[y_n]_{\\rm max}$ and $[X_n]_{\\rm max}$ provide as shown in Fig.", "REF good results for spheres above $n\\approx 3$ and for cylinders above $n\\approx 1$ .", "Table: Coefficients of the polynomial approximation of the maximum position of the asymptotic PDFThe ratio $\\frac{[\\Sigma _n]_{\\rm cut}}{[\\Sigma _n]_{\\rm max}}=\\frac{[X_n]_{\\rm cut}}{[X_n]_{\\rm max}}$ between the central mass surface density $[X_n]_{\\rm cut}$ and the mass surface at the PDF maximum can be used to infer the pressure ratio $q$ for given power $n$ .", "As an example we consider a critical stable sphere which radial density profile is close to the analytical profile with $n=3$ .", "The critical stable sphere has a density ratio of $q\\approx 1/14.04$ Applying Eq.", "REF we find that the corresponding central mass surface density is given by $[X_3]_{\\rm cut} = q_{\\rm crit}^{-\\frac{2}{3}}\\sqrt{1-q_{\\rm crit}^{2/3}} \\approx 5.296.$ For critical stable spheres the ratio between the cutoff and the maximum of the logarithmic PDF is therefore given by $\\frac{[X_3]_{\\rm cut}}{[X_3]_{\\rm max}} = q_{\\rm crit}^{-\\frac{2}{3}}\\sqrt{1-q_{\\rm crit}^{2/3}}\\frac{2\\sqrt{2}-2}{\\sqrt{2\\sqrt{2}-2}}\\approx 4.821,$ where the maximum position is taken from Tab.", "REF .", "The functional dependence of the maximum position on the power $n$ can over a large range be well fitted by a polynomial function $[y_n]_{\\rm max} &=& \\sum _{i=1}^{N_{\\rm max}} a_i (\\ln n)^i, \\\\\\ [X_n]_{\\rm max} &=& \\sum _{i=1}^{N_{\\rm max}} a_i (\\ln n)^i,$ where $N_{\\rm max}=6$ for spheres and $N_{\\rm max}=5$ for cylinders.", "The coefficients are listed in Tab.", "REF ." ], [ "Summary and conclusion", "The study summarizes a series of properties of the PDF of the mass surface density of spherical and cylindrical structures having an analytical radial density profile $\\rho =\\rho _{\\rm c}/(1+(r/r_0)^2)^{n/2}$ where $\\rho _c$ is the central density and $r_0$ the inner radius.", "The profiles are assumed to be truncated at a cloud radius $r_{\\rm cl}$ as expected for cold structures embedded in a considerably warmer medium.", "The results are therefore applicable to individual condensed structures in star forming molecular clouds.", "The PDF for given geometry is determined by the power $n$ , the density ratio $q=\\rho (r_{\\rm cl})/\\rho _{\\rm c}$ , the product $\\rho _{\\rm c}r_{0}$ , and, in case of a cylinder, the inclination angle $i$ .", "It is convenient to describe the properties of the PDF in terms of the unit free mass surface density defined by $X_n = \\Sigma _n \\frac{\\cos ^\\beta i}{2\\rho _{\\rm c} r_0}q^{-\\frac{n-1}{n}},$ where $\\beta =0$ for spheres and $\\beta =1$ for cylinders.", "The properties are: For given geometry and power index $n$ the normalized PDF $(1-q^{2/n})P(X_n)$ is a unique curve expressed through a simple implicit function of the parameter $y_n = (1-q^{2/n})(1-x^2)$ where $x = r/r_{\\rm cl}$ is the normalized impact parameter.", "At the central mass surface density $X_n(0)$ the PDF of spheres has a sharp cut-off while the PDF of cylinders has a pole.", "At high overdensities the PDF has a well defined maximum at fixed $[X_n]_{\\rm max}$ .", "At mass surface densities which are small relative to the maximum position the asymptotic PDF approaches asymptotically a power law $P(X_{\\rm n})\\propto X_n$ .", "In the limit of high overdensities the PDFs approach for $n>1$ at mass surface densities above the peak asymptotically power laws.", "They are given by $P(X_n)\\propto X_n^{-\\frac{n+1}{n-1}}$ in case of spheres and, with the exception of the pole, by $P(X_n)\\propto X^{-\\frac{n}{n-1}}$ in case of cylinders.", "For given overdensity the asymptote is a better approximation for steeper density profiles (larger $n$ ).", "For $n<1$ the PDF has a strong cutoff and is limited to a maximum mass surface density $X_n\\le 1/(1-n)$ .", "The slope of the PDF at high mass surface densities can also be obtained assuming a simple power law profile for spherical clouds (e.g.", "[41], [16]).", "But it should be considered that this profile only is an asymptotic behavior in the limit of high overdensities and seems more appropriate for collapsing clouds while most condensations might not be in such a state.", "As shown in the paper in general the shape of the PDF is not a power law.", "Further, the profile would produce a nonphysical high probability at low mass surface densities.", "The derived properties are related to background subtracted structures within molecular clouds.", "They are therefore not directly applicable to measurements of the global PDF of molecular clouds which is a statistical mean of different properties not only of the condensed structures but the surrounding medium as well.", "For instance the tail at high mass surface densities seen in the PDF of star forming molecular clouds may have different physical explanations.", "It also need to be considered that the functional form of the PDF is affected by an additional background.", "In case of filaments the situation is furthermore complicated through a possible variation of the inclination angle.", "Those problems are addressed in a following paper [20] based on isothermal self-gravitating pressurized spheres and cylinders.", "This work was supported by grants from the Natural Sciences and Engineering Research Council of Canada and the Canadian Space Agency.", "The author likes to thank Prof. P. G. Martin and Dr. Richard Tuffs for his support, Quang Nguyen Luong for reading the manuscript and his helpful comments, and the unknown referee for the suggestions." ], [ "Case $n\\ge 1$", "In case $n>1$ the integral can be expressed through the incomplete and complete beta function $\\int \\limits _0^{u_{\\rm max}} \\frac{{\\rm d}u}{(1+u^2)^{\\frac{n}{2}}} &=& \\frac{1}{2}{\\rm B}\\left(\\frac{n}{2}-\\frac{1}{2},\\frac{1}{2}\\right)\\nonumber \\\\&&\\times \\bigg \\lbrace 1-I_{1-y_n}\\left(\\frac{n}{2}-\\frac{1}{2},\\frac{1}{2}\\right)\\bigg \\rbrace ,$ with the condition $a,b>0$ where the normalized incomplete beta function is given by $I_\\xi (a,b) = \\frac{1}{B(a,b)}\\int _0^\\xi {\\rm d}t\\,t^{a-1}(1-t)^{b-1}.$ The beta function is equal to ${\\rm B} (a,b) = \\frac{\\Gamma (a)\\Gamma (b)}{\\Gamma (a+b)},$ where $\\Gamma (x)$ is the $\\Gamma $ -function given by $\\Gamma (x) = \\int _0^{\\infty }{\\rm d}t\\,t^{x-1} e^{-t}.$ In Sect.", "the asymptotic behavior of $I_\\xi (a,b)$ with $a=(n-1)/2$ and $b=0.5$ is discussed.", "In general the asymptotic behavior is better for higher powers of $a$ .", "The mass surface density for given external pressure and overdensity through the cloud center for $n>1$ is $\\Sigma _n(0) = \\frac{r_0\\rho _c}{\\cos ^\\beta i} \\,{\\rm B}\\left(\\frac{n-1}{2},\\frac{1}{2}\\right)\\left\\lbrace 1-I_{q^{2/n}}\\left(\\frac{n-1}{2},\\frac{1}{2}\\right)\\right\\rbrace .$ In the limit of high overdensity ($q\\rightarrow 0$ ) the central mass surface density becomes the asymptotic value $\\Sigma _n (0) \\approx \\frac{r_0\\rho _c}{\\cos ^\\beta i} \\frac{\\Gamma (\\frac{n-1}{2})\\Gamma (\\frac{1}{2})}{\\Gamma (\\frac{n}{2})}.$" ], [ "Case $n<1$", "To estimate the profile for $n<1$ we can transform the integral to $\\int \\limits _0^{u_{\\rm max}}\\frac{{\\rm d}u}{(1+u^2)^{\\frac{n}{2}}} &=& \\frac{1}{n-1} \\Bigg \\lbrace n \\int \\limits _0^{u_{\\rm max}}\\frac{{\\rm d}u}{(1+u^2)^{\\frac{n}{2}+1}}\\nonumber \\\\&&- \\frac{u_{\\rm max}}{(1+u_{\\rm max}^2)^{\\frac{n}{2}}}\\Bigg \\rbrace .$ The integral can then be calculated using the complete and incomplete beta function as in Eq.", "REF .", "The mass surface density profile becomes $\\Sigma _{n} (x) &=& \\frac{2}{\\cos ^\\beta i} r_0\\rho _c q^{\\frac{n-1}{n}}\\frac{1}{1-n}\\Bigg \\lbrace \\sqrt{y_n}\\nonumber \\\\&&-n (1-y_n)^{\\frac{1-n}{2}} \\int \\limits _0^{u_{\\rm max}} \\frac{{\\rm d}u}{(1+u^2)^{\\frac{n}{2}+1}}\\Bigg \\rbrace .$ In the limit of high overdensity the central mass surface density approaches asymptotically a maximum value given by $\\Sigma _n (0) = \\frac{2}{\\cos ^\\beta i} r_0\\rho _c q^{\\frac{n-1}{n}}\\frac{1}{1-n}.$" ], [ "Analytical profiles of the mass surface densities", "The mass surface density profiles of the truncated analytical density profile for any natural number $n=1,2,...$ can be expressed through simple analytical functions.", "For $n=1$ , 2, 3, and 4 the profiles are for example given by $\\Sigma _1 (x) &=& \\frac{2}{\\cos ^\\beta i}(r_0\\rho _c) \\ln \\left[\\frac{1+\\sqrt{y_1}}{\\sqrt{1-y_1}}\\right],\\\\\\Sigma _{2} (x) &=& \\frac{2}{\\cos ^\\beta i} \\rho _c r_0 q^{\\frac{1}{2}} \\frac{1}{\\sqrt{1-y_{2}}}\\tan ^{-1}\\sqrt{\\frac{y_2}{1-y_2}},\\\\\\Sigma _{3} (x) &=& \\frac{2}{\\cos ^\\beta i} \\rho _c r_0 q^{\\frac{2}{3}}\\frac{1}{1-y_3}\\sqrt{y_3},\\\\\\Sigma _{4} (x) &=& \\frac{2}{\\cos ^\\beta i}\\rho _c r_0 q^{\\frac{3}{4}}\\frac{1}{(1-y_4)^{3/2}}\\frac{1}{2}\\nonumber \\\\&&\\times \\Bigg \\lbrace \\sqrt{y_4}\\sqrt{1-y_4} + \\tan ^{-1}\\sqrt{\\frac{y_4}{1-y_4}}\\Bigg \\rbrace ,$ where $y_n = (1-x^2)(1-q^{2/n})$ .", "The profiles of higher orders can be derived by applying successively the integral transform Eq.", "REF .", "The profile $n=4$ applies for isothermal self-gravitating pressurized cylinders.", "For cylinders exist a maximum mass line density given by $[M/l]_{\\rm max}=2K/G$ where $G$ is the gravitational constant.", "$K$ is a constant given by $K= kT / (\\mu m_{\\rm H})$ where $T$ is the effective temperature, $k$ the Boltzmann constant, $\\mu $ the mean molecular weight and $m_{\\rm H}$ is the atomic mass of hydrogen.", "If we replace the pressure ratio with $q = (1-f)^2$ where $f=(M/l)/[M/l]_{\\rm max}\\le 1$ is the normalized mass line density we obtain the expression given in the work of [23].", "The profile for $n=3$ closely describes the profile of Bonnor-Ebert spheres with overdensities less than $\\sim 100$ [20]." ], [ "Gaussian approximation ($n\\rightarrow \\infty $ )", "Under certain circumstances the inner region of the profile can be approximated by a Gaussian function as will be shown in the following where the width is related to physical parameters as the overdensity $q^{-1}$ and the inner radius $r_0$ .", "The density profile can in general be expressed through $\\rho (x) = \\rho _{\\rm c} e^{-\\frac{n}{2}\\ln \\left[1+(x r_{\\rm cl}/r_0)^2\\right]},$ where $\\frac{r^2_{\\rm cl}}{r^2_0} = q^{-\\frac{2}{n}}\\left(1-q^{\\frac{2}{n}}\\right).$ Where $(x r_{\\rm cl}/r_0)^2\\ll 1$ we can linearize the logarithm using $\\ln [1+(x r_{\\rm cl}/r_0)^2]\\approx (x r_{\\rm cl}/r_0)^2$ and we obtain a Gaussian density profile $\\rho (x) = \\rho _{\\rm c} e^{-\\frac{1}{2}\\left(\\frac{x}{\\sigma _\\rho }\\right)^2},$ where the variance is given by $\\sigma _{\\rho }^2 = \\frac{1}{n}\\frac{r^2_{0}}{r^2_{\\rm cl}} = \\frac{1}{n}\\frac{q^{\\frac{2}{n}}}{1-q^{\\frac{2}{n}}}.$ The approximation improves with power $n$ .", "In case of a pressure ratio $q$ the density profile becomes approximately a Gaussian function for all impact parameters if $n\\gg -2\\ln q / \\ln 2$ or $q^{\\frac{2}{n}}\\gg \\frac{1}{2}$ .", "For large $n$ the variance becomes $\\sigma _\\rho ^2 \\rightarrow -\\frac{1}{2 \\ln q},$ which decreases slowly with overdensity.", "In a similar approach we can derive the asymptotic profile of the mass surface density.", "Considering the same condition for $n$ as above we obtain for example for $(1-y_n)^{\\frac{1-n}{2}} = q^{\\frac{1-n}{n}} e^{-\\frac{1}{2} \\left(\\frac{x}{\\sigma _{\\Sigma }}\\right)^{2}},$ also a Gaussian form where the variance of the mass surface density is given by $\\sigma ^2_{\\Sigma } = \\frac{1}{{n-1}}\\frac{r_{0}^2}{r_{\\rm cl}^2}=\\frac{1}{n-1}\\frac{q^{\\frac{2}{n}}}{1-q^{\\frac{2}{n}}},$ where $\\sigma _{\\Sigma }\\approx \\sigma _\\rho =\\sigma $ for large $n$ .", "The central region ($x\\ll r_0/r_{\\rm cl}$ ) of the profile is therefore approximately given by a Gaussian function $\\Sigma _n \\approx \\rho _{\\rm c}r_0e^{-\\frac{1}{2}\\left(\\frac{x}{\\sigma }\\right)^2} {\\rm B}\\left(\\frac{n-1}{2},\\frac{1}{2}\\right)\\left(1-I_{q^{\\frac{2}{n}}}\\left(\\frac{n-1}{2},\\frac{1}{2}\\right)\\right).$ The approximation provides the exact central mass surface density.", "We want to consider the case of high overdensity and large power $n$ so that the contribution of the incomplete beta function in the central region of the cloud becomes negligible (see Sect. ).", "In the limit of large $n$ the beta function becomes $\\frac{\\Gamma \\left(\\frac{n-1}{2}\\right)\\Gamma \\left(\\frac{1}{2}\\right)}{\\Gamma \\left(\\frac{n}{2}\\right)}\\rightarrow \\sqrt{\\frac{2\\pi }{n}}.$ Replacing $n$ through the variance of the density profile we obtain for the asymptotic profile for given overdensity $\\Sigma _\\infty (x) \\rightarrow \\rho _{\\rm c} \\sqrt{2\\pi }\\sigma r_{\\rm cl}\\,e^{-\\frac{1}{2}\\left(\\frac{x}{\\sigma }\\right)^2} = \\Sigma _\\infty (0) e^{-\\frac{1}{2}\\left(\\frac{x}{\\sigma }\\right)^2}.$" ], [ "Asymptotes in the limit $y_n\\rightarrow 0$ (low {{formula:d51b98b6-58ec-43b1-944c-0ccfc33d9537}} )", "In the limit of large impact parameters ($x\\rightarrow 1$ ) it follows that $y_n\\rightarrow 0$ and consequently $u_{\\rm max} \\approx \\sqrt{y_n}\\ll 1$ .", "The integrand in Eq.", "REF is approximately a constant so that the unit-free mass surface density becomes $X_n \\approx \\sqrt{y_n}.$ From Eq.", "REF we find directly the corresponding asymptotic behavior for spheres which is given by $P_{\\rm sph}(X_n) \\approx \\frac{2}{1-q^{2/n}} X_n.$ For cylinders we find from Eq.", "REF $P_{\\rm cyl}(X_n)\\approx P_{\\rm cyl}^{(a)}(X_n) \\approx \\frac{1}{1-q^{2/n}}X_n.$ For $q^{2/n}\\rightarrow 0$ we obtain the asymptote of the probability function of homogeneous spheres or cylinders.", "In the limit of low impact parameter ($x\\rightarrow 0$ ) and high overdensity ($q\\ll 1$ ) we have $y_n\\rightarrow 1-q^{2/n}\\sim 1$ .", "The normalized mass surface density is then approximately given by $X_n \\sim (1-y_n)^{\\frac{1-n}{2}} \\zeta _n\\gg 1,$ where $\\zeta _n=0.5 \\,{\\rm B}((n-1)/2,1/2)$ .", "We can use this relation to replace $1-y_n$ to obtain an expression of the PDF as function of the mass surface density.", "For the PDF of spheres we find that at high mass surface densities the PDF approaches asymptotically a power law given by $P_{\\rm sph}(X_n) \\sim \\frac{1}{n-1}\\frac{2}{1-q^{2/n}}X_n^{-\\frac{n+1}{n-1}} \\zeta _{n}^{\\frac{2}{n-1}}.$ For the asymptotic PDF of cylinders we find $P^{(a)}_{\\rm cyl}(X_n) \\sim \\frac{1}{n-1}\\frac{1}{1-q^{2/n}}X_n^{-\\frac{n}{n-1}} \\zeta _{n}^{\\frac{1}{n-1}}.$ Replacing $1-y_n$ in the Eq.", "REF with the above expression for the mass surface density provides the asymptotic behavior of the PDF of cylinders at the pole given by $P_{\\rm cyl}(X_n) \\sim \\frac{1}{n-1}\\frac{1}{\\sqrt{1-q^{2/n}}} \\frac{X_n^{-\\frac{n}{n-1}}\\zeta ^{\\frac{1}{n-1}}}{\\sqrt{1-(X_n/X_n(0))^{\\frac{2}{n-1}}}},$ where $X_n(0) = q^{\\frac{1-n}{n}}\\zeta _n$ is the central mass surface density.", "A power law is only established for cylinders with sufficiently high overpressure so that $(X_n/ X_n(0))^{2/(n-1)}\\ll 1$ ." ], [ "Asymptotes for $n=1$", "In the limit of large $y_1$ the mass surface density behaves as $X_1 \\sim \\ln \\frac{2}{\\sqrt{1-y_1}}.$ Replacing $1-y_1 \\approx 4 \\,e^{-2 X_1}$ in Eq.", "REF and in Eq.", "REF provides the asymptotes $P_{\\rm sph}(X_1) \\approx \\frac{2}{1-q^2}(1-y_n) \\sim \\frac{8}{1-q^2} e^{-2 X_1}$ for and $P^{(a)}_{\\rm cyl}(X_1) \\approx \\frac{1}{1-q^2}\\sqrt{1-y_n} \\sim \\frac{2}{1-q^2} e^{-X_1}.$ For the special case $n=1$ the PDF at high mass surface densities is therefore approximately described by a simple exponential function.", "The asymptotic behavior of the PDF for a cylinder including the region at the pole is $P_{\\rm cyl}(X_1) \\approx \\frac{2}{\\sqrt{1-q^2}}\\frac{e^{-X_1}}{\\sqrt{1-e^{-2(X_1(0)-X_1)}}},$ where $X_1(0) = \\ln (2 q^{-1})$ ." ], [ "Asymptotes for $n<1$", "As pointed out in Sect.", "REF for $n<1$ the mass surface density has a maximum possible value.", "In the limit $y_n\\rightarrow 1$ we have $X_n \\approx \\frac{1}{1-n}\\sqrt{y_n}\\le \\frac{1}{1-n}.$ As can be shown we have $P_{\\rm sph}(X_n)\\rightarrow 0$ and $P_{\\rm cyl}^{(a)}(X_n)\\rightarrow 0$ for $X_n\\rightarrow 1/(1-n)$ ." ], [ "Asymptote in the limit $n\\rightarrow \\infty $", "As we have seen in the previous section in the limit of high $n$ the power law slope at high mass surface densities approaches asymptotically $\\alpha =1$ .", "The corresponding PDF can be directly obtained from Eq.", "REF .", "Replacing $n-1$ by the standard deviation of the Gaussian approximation we get for spheres with high overdensity in the limit of $n\\rightarrow \\infty $ $P_{\\rm sph}(X_n) \\sim 2 \\sigma ^2 X_{n}^{-1}.$ The same result is obtained from Eq.", "REF by deriving the corresponding derivative and using Eq.", "REF .", "For the asymptotic PDF of cylinders Eq.", "REF $P^{(a)}_{\\rm cyl}(X_n) \\sim \\sigma ^2 X_{n}^{-1}.$" ], [ "Asymptote in the limit $n\\rightarrow 0$", "In the limit $n\\rightarrow 0$ it follows from Eq.", "REF for the mass surface density $X_n \\rightarrow \\frac{1}{1-n} \\sqrt{y_n}.$ In case of spheres the PDF of the mass surface density becomes $P_{\\rm sph}(X_n) \\approx \\frac{2}{1-q^{2/n}}\\sqrt{y_n}\\rightarrow 2\\sqrt{1-x^2},$ which is the PDF of homogeneous spheres.", "Likewise, we find for the asymptotic PDF of cylindrical clouds in the limit $n\\rightarrow 0$ that $P^{(a)}_{\\rm cyl}(X_n) \\approx P_{\\rm cyl}(X_n)\\rightarrow \\frac{\\sqrt{1-x^2}}{x},$ which is the PDF of a homogeneous cylinder." ], [ "Condition for PDF maxima", "The maxima position were derived for both linear and logarithmic PDFs of spheres and cylinders.", "For cylinders the asymptotic PDF as defined in Eq.", "REF was considered." ], [ "For linear values ($P(\\Sigma _n)$ )", "The condition for maxima of the linear PDF is given by $\\frac{{\\rm d}P}{{\\rm d}y_n}(X_n) = 0.$ This leads to $X_n y_n^{\\frac{3}{2}}(n^2-1)+ y_n (n+2) - 1 = 0$ in case spheres and to $X_n y_n^{\\frac{3}{2}}n (n-1) +y_n (n+1)-1 = 0$ for cylinders.", "For $n=1$ and $n=3$ the maxima are simple analytical expressions listed in Tab.", "REF .", "In the limit $n\\gg 1$ the condition for maxima of the linear PDF is equal for spheres and cylinders and is given by $X_n y_n^{\\frac{3}{2}} n^2 + y_n n -1 = 0.$ As $y_n\\rightarrow 0$ for $n\\rightarrow \\infty $ it follows from Eq.", "REF and Eq.", "REF that the mass surface density behaves approximately as $X_n \\sim e^{\\frac{n}{2}y_n}\\frac{1}{2}\\sqrt{\\frac{2\\pi }{n}} P(\\chi ^2,1),$ where $P(\\chi ^2,1)$ is the PDF of the $\\chi ^2$ -distribution and where $\\chi ^2 \\rightarrow n y_n.$ In the limit of $n\\gg 1$ the condition for maxima becomes a function of $n y_n=C$ where $C$ is a constant.", "Solving $e^{C/2}\\sqrt{\\frac{\\pi }{2}} P(C,1) C^{\\frac{3}{2}}+C-1 = 0$ provides $C\\approx 0.58404$ .", "The maxima position is therefore approximately given by $[y_n]_{\\rm max} &=& C/n, \\\\{[X_{n}]}_{\\rm max} &=& e^{C/2}\\sqrt{\\frac{\\pi }{2n}}P(C,1).$" ], [ "For logarithmic values ($\\Sigma _n P(\\Sigma _n)$ )", "The maxima of the PDF of logarithm values are given by the condition $\\frac{{\\rm d} }{{\\rm d}y_n}[X_n P(X_n)] =0.$ This leads to $2X_n^2 y_n (n-1) - X_n \\left((n-1) \\sqrt{y_n}+\\frac{1-3 y_n}{\\sqrt{y_n}}\\right) = 1$ in case of spheres and to $X_n^2 y_n (n-1) - X_n\\left((n-1)\\sqrt{y_n}+\\frac{1-2 y_n}{\\sqrt{y_n}}\\right) =1$ in case of cylinders.", "For $n=3$ the maxima positions are again simple analytical expressions given in Tab.", "REF ." ], [ "Asymptotic behavior of the incomplete beta function", "To derive the asymptotic behavior of the incomplete beta function in Eq.", "REF for the mass surface density in the limit $n\\gg 1$ we consider the approximation (Eq.", "26.5.20, of [1]) $I_\\xi (a,b) &\\sim & 1 - P(\\chi ^2,\\nu ), \\nonumber \\\\&=& \\left(2^{\\frac{\\nu }{2}}\\Gamma \\left(\\frac{\\nu }{2}\\right)\\right)^{-1}\\int \\limits _{\\chi ^2}^{\\infty }{\\rm d}t\\,t^{\\frac{\\nu }{2}-1}\\,e^{-\\frac{t}{2}},$ where $P(\\chi ^2,\\nu )$ is the $\\chi ^2$ distribution function of $\\nu $ events where $\\chi ^2 & = & (a+b-1)(1-\\xi )(3-\\xi ) - (1-\\xi )(b-1),\\\\\\nu &=&2 b.$ In our case we have $a=(n-1)/2$ and $b=1/2$ .", "The incomplete beta function is then approximately given by $I_{1-y_n}\\left(\\frac{n-1}{2},\\frac{1}{2}\\right) \\sim \\frac{1}{\\sqrt{2\\pi }}\\int \\limits _{\\chi ^2}^{\\infty }{\\rm d}t\\,t^{-\\frac{1}{2}} e^{-\\frac{t}{2}},$ where $\\chi ^2 =\\left(\\frac{n-2}{2}\\right)y_n (2+y_n) + y_n\\frac{1}{2}.$ For given power $n$ the approximation improves with increasing $\\xi $ .", "In the limit $y_n\\rightarrow 0$ we can use the replacement $(1-y_n)^{\\frac{1-n}{2}} \\sim e^{\\frac{n-1}{2}y_n},$ so that the mass surface density becomes $\\Sigma _n \\sim 2 r_0\\rho _{\\rm c} q^{\\frac{n-1}{n}}e^{-\\frac{1-n}{2}y_n}\\frac{1}{2}{\\rm B}\\left(\\frac{n-1}{2},\\frac{1}{2}\\right)P(\\chi ^2,1).$ In the limit $n\\rightarrow \\infty $ we have $\\chi ^2 \\approx n y_n \\rightarrow - 2 \\,(1-x^2)\\,\\ln q,$ so that the beta function becomes independent of $n$ ." ], [ "Incomplete beta function for $b=1/2$", "The power law approximation of the PDF at large mass surface densities for $n>1$ as presented in Sect.", "are valid for negligible contribution of the incomplete beta function to the mass surface density.", "We have seen in Fig.", "REF that the power law is only a good representation for large mass surface densities and that the approximation of the PDF improves for larger $n$ .", "Fig.", "REF shows the value of the incomplete beta function as given in Eq.", "REF for different assumptions for the powers $n$ and the density ratio $q$ .", "As we see the value of the incomplete beta function for given $q$ decreases for larger $n$ .", "In the limit $n\\rightarrow \\infty $ for given $q$ we obtain the asymptotic value of the incomplete beta function given in the previous section.", "In the limit $n\\rightarrow 1$ we have $I_{1-y_n}\\left(\\frac{n-1}{2},\\frac{1}{2}\\right) \\rightarrow 1.$ Figure: Incomplete beta function I ξ (a,b)I_\\xi (a,b) (black contours)for a=(n-1)/2a=(n-1)/2 and b=1/2b=1/2 as function of ξ=1-y n \\xi =1-y_n.The lines are labeled with the corresponding power nn.", "The line for a power n=3n=3is emphasized through a thick line.", "The gray linescorrespond to ξ=q 2/n \\xi =q^{2/n} for fixed density ratio qq and varying power nn.", "The linesare labeled with log 10 q\\log _{10} q.", "The gray dashed lines are obtained using the approximationof the incomplete beta function (Eq. ).", "The asymptotic value of the incompletebeta function for given qq in the limit n→∞n\\rightarrow \\infty is indicated through dotted lines.The filled circle corresponds to a power n=3n=3 and an overdensityq crit -1 =14.04q_{\\rm crit}^{-1} = 14.04 of a critical stable sphere." ] ]	1403.0454
[ [ "On the functional Blaschke-Santalo inequality" ], [ "Abstract In this paper, using functional Steiner symmetrizations, we show that Meyer and Pajor's proof of the Blaschke-Santalo inequality can be extended to the functional setting." ], [ "Introduction", "For a convex body $K\\subset \\mathbb {R}^n$ and a point $z\\in \\mathbb {R}^n$ , the polar body $K^z$ of $K$ with respect to $z$ is the convex set defined by $ K^{z}=\\lbrace y\\in \\mathbb {R}^n: \\langle y-z,x-z\\rangle \\le 1\\;{\\rm for}\\;{\\rm every}\\;x\\in K\\rbrace $ .", "The Santaló point $s(K)$ of $K$ is a point for which $V_n(K^{s(K)})=\\min _{z\\in int(K)}V_n(K^{z})$ , where $V_n(K)$ denotes the volume of set $K$ .", "The Blaschke-Santaló inequality [4], [18], [19] states that $V_n(K)V_n(K^{s(K)})\\le V_n(B_2^n)^2$ , where $B_2^n$ is the Euclidean ball.", "For a log-concave function $f:\\mathbb {R}^n\\rightarrow [0,\\infty )$ and a point $z\\in \\mathbb {R}^n$ , its polar with respect to $z$ is defined by $f^{z}(y)=\\inf _{x\\in \\mathbb {R}^n}\\frac{e^{-\\langle x-z,y-z\\rangle }}{f(x)}$ .", "The Santaló point $s(f)$ of $f$ is the point $z_0$ satisfying $\\int f^{z_0}=\\inf _{z\\in \\mathbb {R}^n}\\int f^{z}$ .", "The functional Blaschke-Santaló inequality of log-concave functions is the analogue of Blaschke-Santaló inequality of convex bodies.", "Theorem 1.1 (Artstein, Klartag, Milman).", "Let $f:\\mathbb {R}^n\\rightarrow [0,+\\infty )$ be a log-concave function such that $0<\\int f<\\infty $ .", "Then, $\\int _{\\mathbb {R}^n} f\\int _{\\mathbb {R}^n} f^{s(f)}\\le (2\\pi )^n$ with equality holds exactly for Gaussians.", "When $f$ is even, the functional Blaschke-Santaló inequality follows from an earlier inequality of Ball [2]; and in [9], Fradelizi and Meyer proved something more general (see also [11]).", "Lutwak and Zhang [13] and Lutwak et al.", "[14] gave other very different forms of the Blaschke-Santaló inequality.", "In this paper, we give a more general result than Theorem REF , which becomes into a special case of $\\lambda =1/2$ in Theorem REF .", "Theorem 1.2 Let $f:\\mathbb {R}^n\\rightarrow [0,+\\infty )$ be a log-concave function such that $0<\\int f<\\infty $ .", "Let $H$ be an affine hyperplane and let $H_+$ and $H_-$ denote two closed half-spaces bounded by $H$ .", "If $\\lambda \\in (0,1)$ satisfies $\\lambda \\int _{\\mathbb {R}^n}f=\\int _{H_+}f$ .", "Then there exists $z\\in H$ such that $\\int _{\\mathbb {R}^n}f\\int _{\\mathbb {R}^n}f^z\\le \\frac{1}{4\\lambda (1-\\lambda )}(2\\pi )^n.$ In [12], Lehec proved a very general functional version for non-negative Borel functions, Theorem REF is a particular case of result of Lehec.", "Lehec's proof is by induction on the dimension, and the proof is by functional Steiner symmetrizations.", "In fact, Mayer and Pajor [15] have proved the Blaschke-Santaló inequality for convex bodies, here we show that Meyer and Pajor's proof of the Blaschke-Santaló inequality can be extended to the functional setting.", "It has recently come to our attention that in a remark of [9], Fradelizi and Meyer expressed the same idea to prove the functional Blaschke-Santaló inequality." ], [ "Notations and background materials", "Let $\|\\cdot \|$ denote the Euclidean norm.", "Let ${\\rm int} A$ denote the interior of $A\\subset \\mathbb {R}^n$ .", "Let ${\\rm cl} A$ denote the closure of $A$ .", "Let ${\\rm dim} A$ denote the dimension of $A$ .", "A set $C\\subset \\mathbb {R}^n$ is called a convex cone if $C$ is convex and nonempty and if $x\\in C$ , $\\lambda \\ge 0$ implies $\\lambda x\\in C$ .", "We define $C^{\\ast }:=\\lbrace x\\in \\mathbb {R}^n:\\langle x,y\\rangle \\le 0\\;\\;{\\rm for}\\;{\\rm all}\\;y\\in C\\rbrace $ and call this the dual cone of $C$ .", "For a non-empty convex set $K\\subset \\mathbb {R}^n$ and an affine hyperplane $H$ with unit normal vector $u$ , the Steiner symmetrization $S_H K$ of $K$ with respect to $H$ is defined as $S_H K:=\\lbrace x^{\\prime }+\\frac{1}{2}(t_1-t_2)u:\\;x^{\\prime }\\in P_H(K),\\;t_i\\in I_K(x^{\\prime })\\;{\\rm for}\\;i=1,2\\rbrace $ , where $P_H(K):=\\lbrace x^{\\prime }\\in H:\\;x^{\\prime }+tu\\in K\\; {\\rm for}\\;{\\rm some}\\;t\\in \\mathbb {R}\\rbrace $ is the projection of $K$ onto $H$ and $I_K(x^{\\prime }):=\\lbrace t\\in \\mathbb {R}:\\;x^{\\prime }+tu\\in K\\rbrace $ .", "Let $\\bar{\\mathbb {R}}=\\mathbb {R}\\cup \\lbrace -\\infty ,\\infty \\rbrace $ .", "For a given function $f:\\mathbb {R}^n\\rightarrow \\bar{\\mathbb {R}}$ and for $\\alpha \\in \\bar{\\mathbb {R}}$ we use the abbreviation $\\lbrace f=\\alpha \\rbrace :=\\lbrace x\\in \\mathbb {R}^n:f(x)=\\alpha \\rbrace $ , and $\\lbrace f\\le \\alpha \\rbrace $ , $\\lbrace f<\\alpha \\rbrace $ etc.", "are defined similarly.", "A function $f:\\mathbb {R}^n\\rightarrow \\bar{\\mathbb {R}}$ is called proper if $\\lbrace f=-\\infty \\rbrace =\\emptyset $ and $\\lbrace f=\\infty \\rbrace \\ne \\mathbb {R}^n$ .", "A function $\\phi $ is called convex if $\\phi $ is proper and $\\phi (\\alpha x+(1-\\alpha )y)\\le \\alpha \\phi (x)+(1-\\alpha )\\phi (y)$ for all $x,y\\in \\mathbb {R}^n$ and for any $0\\le \\lambda \\le 1$ .", "A function $f$ is called log-concave if $f=e^{-\\phi }$ , where $\\phi $ is a convex function.", "A function $f:\\mathbb {R}^n\\rightarrow \\mathbb {R}\\cup \\lbrace +\\infty \\rbrace $ is called coercive if $\\lim _{\|x\|\\rightarrow +\\infty }f(x)=+\\infty $ .", "A function $f$ is called symmetric about $H$ if for any $x^{\\prime }\\in H$ and $t\\in \\mathbb {R}$ , $f(x^{\\prime }+tu)=f(x^{\\prime }-tu)$ .", "A function $f: \\mathbb {R}^n\\rightarrow \\mathbb {R}$ is called unconditional about $z$ if $f(x_1-z_1,\\dots ,x_n-z_n)=f(\|x_1-z_1\|,\\dots ,\|x_n-z_n\|)$ for every $(x_1,\\dots ,x_n)\\in \\mathbb {R}^n$ .", "If $z=0$ , then $f$ is called unconditional.", "The effective domain of convex function $\\phi $ is the nonempty set ${\\rm dom}\\phi :=\\lbrace \\phi <\\infty \\rbrace $ .", "The support of function $f$ is the set ${\\rm supp}f:=\\lbrace f\\ne 0\\rbrace $ .", "For log-concave function $f=e^{-\\phi }$ , it is clear that ${\\rm supp}f={\\rm dom}\\phi $ .", "The nonempty set ${\\rm epi}\\phi :=\\lbrace (x,r)\\in \\mathbb {R}^n\\times \\mathbb {R}:\\;r\\ge \\phi (x)\\rbrace $ denote the epigraph of convex function $\\phi $ .", "For an affine subspace $G$ of $\\mathbb {R}^n$ , let $G^{\\perp }$ denote the orthogonal complement of $G$ , we have $G^{\\bot }=\\lbrace x\\in \\mathbb {R}^n: \\langle x, y-y^{\\prime }\\rangle =0\\; {\\rm for\\;every\\;}y, y^{\\prime }\\in G\\rbrace $ .", "The Santaló point $s_{G}(f)$ of $f$ about $G$ is a point satisfying $\\int f^{s_{G}(f)}=\\inf _{z\\in G}\\int f^z$ .", "Let $f$ be a log-concave function such that $0<\\int f<\\infty $ , and let $H_+$ and $H_-$ be two half-spaces bounded by an affine hyperplane $H$ ; let $0<\\lambda <1$ ; we shall say that $H$ is $\\lambda $ -separating for $f$ if $\\int _{H_+}f\\int _{H_-}f=\\lambda (1-\\lambda )\\left(\\int _{\\mathbb {R}^n}f\\right)^2$ and when $\\lambda =1/2$ , we shall say that $H$ is medial for $f$ .", "For a function $\\phi : \\mathbb {R}^n\\rightarrow \\bar{\\mathbb {R}}$ , its Legendre transform about $z$ is defined by $\\mathcal {L}^{z}\\phi (y)=\\sup _{x\\in \\mathbb {R}^n}[\\langle x-z,y-z\\rangle -\\phi (x)]$ .", "If $f(x)=e^{-\\phi (x)}$ , where $\\phi (x)$ is a convex function, then $f^{z}(y)=e^{-\\mathcal {L}^{z}\\phi (y)}$ .", "Since $\\mathcal {L}^z(\\mathcal {L}^z\\phi )=\\phi $ for a convex function $\\phi $ , $(f^z)^z=f$ .", "If $z=0$ , we shall use the simpler notation $\\mathcal {L}$ for $\\mathcal {L}^{0}$ .", "Given two functions $f,g:\\mathbb {R}^n\\rightarrow [0,\\infty )$ , their Asplund product is defined by $(f\\star g)(x)=\\sup _{x_1+x_2=x}f(x_1)g(x_2)$ .", "The $\\lambda $ -homothety of a function $f$ is defined as $(\\lambda \\cdot f)(x)=f^{\\lambda }(\\frac{x}{\\lambda })$ .", "Then, the classical Prékopa inequality (see Prékopa [16], [17]) can be stated as follows: Given $f,g:\\mathbb {R}^n\\rightarrow [0,+\\infty )$ and $0<\\lambda <1$ , $\\int (\\lambda \\cdot f)\\star ((1-\\lambda )\\cdot g)\\ge \\left(\\int f\\right)^{\\lambda }\\left(\\int g\\right)^{1-\\lambda }$ .", "The following lemma, as a particular case of a result due to Ball [3], was proved by Meyer and Pajor in [15].", "Lemma 2.1 [15] Let $f_0$ , $f_1$ , $f_2: \\mathbb {R}^{+}\\rightarrow \\mathbb {R}^{+}$ be three functions such that $0<\\int ^{+\\infty }_{0}f_i<\\infty ,\\;i=0,1,2$ , they are continuous and suppose that $f_0\\left(\\frac{2xy}{x+y}\\right)\\ge f_1(x)^{\\frac{y}{x+y}}f_2(y)^{\\frac{x}{x+y}}$ for every $x,y>0$ .", "Then one has $\\frac{1}{\\int ^{+\\infty }_{0}f_0(t)dt}\\le \\frac{1}{2}\\left(\\frac{1}{\\int ^{+\\infty }_{0}f_1(t)dt}+\\frac{1}{\\int ^{+\\infty }_{0}f_2(t)dt}\\right).$" ], [ "The functional Steiner symmetrization", "The familiar definition of Steiner symmetrization for a nonnegative measurable function $f$ can be stated as following (see [5], [6], [7], [8]): Definition 1 For a measurable function $f:\\mathbb {R}^n\\rightarrow [0,+\\infty )$ and an affine hyperplane $H\\subset \\mathbb {R}^n$ , let $m$ denote the Lebesgue measure, if $m(\\lbrace f>t\\rbrace )<+\\infty $ for all $t>0$ , then its Steiner symmetrization is defined as $S_H f(x)=\\int _{0}^{\\infty }\\mathcal {X}_{S_H \\lbrace f>t\\rbrace }(x)dt,$ where $\\mathcal {X}_{A}$ denotes the characteristic function of set $A$ .", "Next, we give a approach of defining Steiner symmetrization for coercive convex functions by the Steiner symmetrization of epigraphs.", "A similar functional steiner symmetrization is defined in a remark of AKM's paper [1] and studied in an article by Lehec [10].", "The idea of our definition is same as the given definition in a remark at the end of an article by Fradelizi and Meyer [9].", "Definition 2 For a coercive convex function $\\phi $ and an affine hyperplane $H\\subset \\mathbb {R}^n$ , we define the Steiner symmetrization $S_H \\phi $ of $\\phi $ with respect to $H$ as a function satisfying ${\\rm epi}(S_H \\phi )=S_{\\widetilde{H}}({\\rm cl}\\;{\\rm epi} \\phi ),$ where $\\widetilde{H}=\\lbrace (x^{\\prime },s)\\in \\mathbb {R}^{n+1}:x^{\\prime }\\in H\\rbrace $ is an affine hyperplane in $\\mathbb {R}^{n+1}$ .", "Remark 1 (i) By Definition REF , for an integrable log-concave function $f=e^{-\\phi }$ , the Steiner symmetrization of $f$ can be defined as $S_H f:=e^{-(S_H \\phi )}$ .", "If we define $S_H f$ by Definition REF , then $S_H f$ still satisfies (REF ).", "Thus, for integrable log-concave functions, the two definitions are essentially same.", "(ii) By Definition REF , for a given $x^{\\prime }\\in H$ and any $s\\in \\mathbb {R}$ , we have $V_1\\left(\\lbrace (S_H\\phi )(x^{\\prime }+tu)<s\\rbrace \\right)=V_1\\left(\\lbrace \\phi (x^{\\prime }+tu)<s\\rbrace \\right)$ .", "By the Fubini's theorem, we have $\\int _{\\mathbb {R}}(S_H f)(x^{\\prime }+tu)dt=\\int _{\\mathbb {R}}f(x^{\\prime }+tu)dt.$ Similarly, $\\int _{\\mathbb {R}^n} S_H f=\\int _{\\mathbb {R}^n} f$ is also established.", "Proposition 1 For a coercive convex function $\\phi $ and an affine hyperplane $H\\subset \\mathbb {R}^n$ with outer unit normal vector $u$ , then $S_H \\phi $ has the following properties.", "(i) $S_H \\phi $ is a closed coercive convex function and symmetric about $H$ .", "(ii) Let $H_1$ and $H_2$ be two orthogonal hyperplanes in $\\mathbb {R}^n$ , then $S_{H_2}(S_{H_1} \\phi )$ is symmetric about both $H_1$ and $H_2$ .", "(iii) For any given $x^{\\prime }\\in H$ and $t\\in \\mathbb {R}$ , let $\\phi _1(t):=\\phi (x^{\\prime }+tu)$ and $(S\\phi _1)(t):=(S_H\\phi )(x^{\\prime }+tu)$ , then $(S\\phi _1)(t)$ satisfies one of the following three cases.", "1).", "$(S\\phi _1)(t)=\\phi _1(t_1)=\\phi _1(t_1-2t)$ for some $t_1\\in \\mathbb {R}$ .", "2).", "$(S\\phi _1)(t)=\\phi _1(t_0-2t)\\ge \\lim _{t\\rightarrow t_0,\\;t<t_0}\\phi _1(t)$ for some $t_0\\in \\mathbb {R}$ .", "3).", "$(S\\phi _1)(t)=\\phi _1(t_0+2t)\\ge \\lim _{t\\rightarrow t_0,\\;t>t_0}\\phi _1(t)$ for some $t_0\\in \\mathbb {R}$ .", "(i) By the fact that $\\phi $ is convex if and only if ${\\rm epi}\\phi $ is convex, since $\\phi $ is convex, ${\\rm epi} \\phi $ is a convex subset of $\\mathbb {R}^{n+1}$ .", "Since the closure of a convex set is convex, and the Steiner symmetrization of a convex set is also convex, by (REF ), ${\\rm epi}(S_H \\phi )$ is a convex subset of $\\mathbb {R}^{n+1}$ .", "Therefore, $S_H \\phi $ is a convex function.", "By Definition REF , it is clear that $S_H \\phi $ is closed, coercive and symmetric with respect to $H$ .", "(ii) Since ${\\rm epi}(S_{H_2}(S_{H_1} \\phi ))$ is symmetric about both $\\widetilde{H_1}$ and $\\widetilde{H_2}$ , where $\\widetilde{H_i}=\\lbrace (x^{\\prime },s)\\in \\mathbb {R}^{n+1}:x^{\\prime }\\in H_i\\rbrace $ ($i=1,2$ ), $S_{H_2}(S_{H_1} \\phi )$ is symmetric about both $H_1$ and $H_2$ .", "(iii) If ${\\rm dom}\\phi _1=\\mathbb {R}$ , by (REF ) in Definition REF , we have ${\\rm epi}(S\\phi _1)=S_{\\widetilde{H}}({\\rm cl}\\;{\\rm epi}\\phi _1).$ Thus there exists some $t_1\\in \\mathbb {R}$ satisfying $(S\\phi _1)(t)=\\phi _1(t_1)=\\phi _1(t_1-2t).$ If ${\\rm dom}\\phi _1\\ne \\mathbb {R}$ , then there exist eight cases for ${\\rm dom}\\phi _1$ : 1) $[\\alpha ,\\beta ]$ ; 2) $(\\alpha ,\\beta )$ ; 3) $(\\alpha ,\\beta ]$ ; 4) $[\\alpha ,\\beta )$ ; 5) $(-\\infty ,\\beta ]$ ; 6) $(-\\infty ,\\beta )$ ; 7) $[\\alpha ,+\\infty )$ ; 8) $(\\alpha ,+\\infty )$ .", "Here, we only prove our conclusion for ${\\rm dom}\\phi _1=(\\alpha ,\\beta )$ .", "By the same method we can prove our conclusion for other cases.", "For ${\\rm dom}\\phi _1=(\\alpha ,\\beta )$ , by Definition REF , it is clear that $(S\\phi _1)(t)=+\\infty $ for $\|t\|\\ge \\frac{\\beta -\\alpha }{2}$ .", "If $\|t\|<\\frac{\\beta -\\alpha }{2}$ , let $\\lim _{x\\rightarrow \\alpha ,\\;x>\\alpha }\\phi _1(x)=b_1,\\;\\;\\lim _{x\\rightarrow \\beta ,\\;x<\\beta }\\phi _1(x)=b_2$ , then we consider the following four cases.", "(a) If $b_1=b_2=+\\infty $ , then by (REF ), there exists some $t_1\\in \\mathbb {R}$ satisfying (REF ).", "(b) If $b_1<+\\infty ,\\;\\; b_2=+\\infty $ , then there exists $\\gamma \\in (\\alpha ,\\beta )$ such that $\\phi _1(\\gamma )=b_1$ .", "Then by (REF ), for $\|t\|<\\frac{\\gamma -\\alpha }{2}$ , (REF ) is established, for $\|t\|\\ge \\frac{\\gamma -\\alpha }{2}$ , we have $(S\\phi _1)(t)=\\phi _1(\\alpha +2t)\\ge b_1$ .", "(c) If $b_1=+\\infty ,\\;\\;b_2<+\\infty $ , then there exists $\\gamma \\in (\\alpha ,\\beta )$ such that $\\phi _1(\\gamma )=b_2$ .", "Then by (REF ), for $\|t\|<\\frac{\\beta -\\gamma }{2}$ , (REF ) is established, for $\|t\|\\ge \\frac{\\gamma -\\alpha }{2}$ , we have $(S\\phi _1)(t)=\\phi _1(\\beta -2t)\\ge b_2$ .", "(d) If $b_1<\\infty ,\\;\\;b_2<+\\infty $ , we consider three cases.", "If $b_1=b_2$ , then (REF ) is established.", "If $b_1>b_2$ , the proof is same as in (c).", "If $b_1<b_2$ , the proof is same as in (b).", "This completes the proof." ], [ "The proofs of theorems", "In order to prove theorems stated in the introduction, we have to establish the following six lemmas: Lemma 4.1 If $f$ be a log-concave function such that $0<\\int f<\\infty $ , then the function $F$ defined by $F(z):=\\int _{\\mathbb {R}^n}f^{z}(x)dx$ has the following properties.", "(i) $F(z)$ is a coercive convex function on $\\mathbb {R}^n$ and is strictly convex on ${\\rm int}\\;{\\rm dom}F$ ; (ii) If $f(x)$ is even about $z_0$ , then $F(z)$ is also even about $z_0$ .", "(i) Step 1.", "We shall prove $F$ is coercive.", "Let $f=e^{-\\phi }$ , for any given $z\\in \\mathbb {R}^n$ and $r>0$ , we have $F(z)=\\int _{\\mathbb {R}^n}f^{z}(x+z)dx\\ge \\int _{rB_2^n}f^{z}(x+z)dx=\\int _{rB_2^n}e^{-\\mathcal {L}\\phi (x)+\\langle x,z\\rangle }dx.$ Since $f=e^{-\\phi }$ is integrable, there is $\\gamma >0$ and $h\\in \\mathbb {R}$ such that $\\phi (x)\\ge \\gamma \\sum _{i=1}^{n}\|x_i\|+h\\;\\;{\\rm for}\\;{\\rm any}\\;x\\in \\mathbb {R}^n.$ Thus, for $y\\in \\gamma B_\\infty ^n$ , where $B_\\infty ^n=\\lbrace x\\in \\mathbb {R}^n:\|x_i\|\\le 1,i=1,\\dots ,n\\rbrace $ , $\\mathcal {L}\\phi (y)\\le \\sup _{x\\in \\mathbb {R}^n}[\\langle y,x\\rangle -\\gamma \\sum _{i=1}^{n}\|x_i\|-h]\\le -h$ .", "Let $rB_2^n\\subset \\frac{1}{2}\\gamma B_\\infty ^n$ , we have $rB_2^n\\subset {\\rm int}({\\rm dom}\\mathcal {L}\\phi )$ .", "Since function $g(x):\\;=e^{-\\mathcal {L}\\phi (x)}$ is continuous on $rB_2^n$ .", "Thus, there exists $m>0$ such that $g(x)\\ge m$ for any $x\\in rB_2^n$ .", "Therefore, $\\int _{rB_2^n}e^{-\\mathcal {L}\\phi (x)+\\langle x,z\\rangle }dx\\ge m\\int _{rB_2^n}e^{\\langle x,z\\rangle }dx.$ For any $z\\in \\mathbb {R}^n$ and $\|z\|\\ge 1$ , let $z^{\\prime }=\\frac{r}{2}\\frac{z}{\|z\|}$ , we get a closed half-space $H^{+}=\\lbrace x\\in \\mathbb {R}^n: \\langle x-z^{\\prime },z\\rangle \\ge 0\\rbrace $ .", "For any $x\\in H^{+}$ , we have $\\langle x,z\\rangle \\ge \\langle z^{\\prime }, z\\rangle =\\frac{r}{2}\|z\|$ .", "Therefore, $\\int _{rB_2^n}e^{\\langle x,z\\rangle }dx&\\ge &\\int _{(rB_2^n)\\cap H^{+}}e^{\\frac{r\|z\|}{2}}dx=V_n((rB_2^n)\\cap H^{+})e^{\\frac{r\|z\|}{2}}.$ Since $V_n((rB_2^n)\\cap H^{+})$ is a positive constant independent of $z$ , by (REF ), (REF ) and (REF ), $F(z)$ is coercive.", "Step 2.", "We shall prove that $F$ is convex and is strictly convex on ${\\rm int}\\;{\\rm dom}F$ .", "First, we prove $F(z)$ is proper.", "It is clear that $F(z)>-\\infty $ for any $z\\in \\mathbb {R}^n$ .", "The following claim shows that $\\lbrace F=\\infty \\rbrace \\ne \\mathbb {R}^n$ .", "Claim 1 For any $z\\in {\\rm int}\\;{\\rm supp}f$ , $F(z)<\\infty $ .", "Proof of Claim REF .", "For any $z\\in {\\rm int}\\;{\\rm supp}f$ , there is a closed ball $z+rB_2^n\\subset {\\rm supp}f$ .", "Since ${\\rm supp}f={\\rm dom}\\phi $ , there is $M\\in \\mathbb {R}$ such that $M=\\sup \\lbrace \\phi (y): y\\in z+rB_2^n\\rbrace $ .", "Thus, we have $f^z(x)\\le \\exp \\lbrace -\\sup _{y\\in (z+rB_2^n)}[\\langle x-z,y-z\\rangle -\\phi (y)]\\rbrace \\le e^M \\cdot e^{-r\|x-z\|^2}.$ Therefore, $\\int _{\\mathbb {R}^n} f^z(x)dx\\le e^M\\int _{\\mathbb {R}^n}e^{-r\|x-z\|^2}dx<\\infty .$ $\\Box $ For any $z_1,z_2\\in \\mathbb {R}^n$ and $\\alpha \\in (0,1)$ .", "Let $f=e^{-\\phi }$ , we have $F(z)=\\int _{\\mathbb {R}^n}e^{-\\mathcal {L}\\phi (x)+\\langle x,z\\rangle }dx$ .", "Since $g_x(z):=e^{-\\mathcal {L}\\phi (x)+\\langle x,z\\rangle }$ is a convex function about $z$ , we have $F(\\alpha z_1+(1-\\alpha )z_2)\\le \\alpha F(z_1)+(1-\\alpha )F(z_2).$ If $z_1,z_2\\in {\\rm int}\\;{\\rm dom}F$ and $z_1\\ne z_2$ , then inequality (REF ) is a strict inequality.", "Thus $F(z)$ is strictly convex on ${\\rm int}\\;{\\rm dom}F$ .", "(ii) Since $f(x)$ is even about $z_0$ , $f(z_0+x)=f(z_0-x)$ for any $x\\in \\mathbb {R}^n$ .", "For any $z\\in \\mathbb {R}^n$ , we have $F(z_0+z)=\\int _{\\mathbb {R}^n}f^{z_0+z}(x)dx=\\int _{\\mathbb {R}^n}f^{z_0-z}(-x+2z_0)dx=F(z_0-z).$ This completes the proof.", "Remark 2 By Lemma REF , if $f$ is even about $z_0$ , then $s(f)=z_0$ .", "Lemma 4.2 Let $f$ be a log-concave function such that $0<\\int f<\\infty $ , and let $G\\subset \\mathbb {R}^n$ be an affine subspace satisfying $G\\cap {\\rm int}\\;{\\rm supp}f\\ne \\emptyset $ .", "Then there exists a unique point $z_0\\in G$ satisfying the following two equivalent claims.", "(i) $F(z_0)=\\min \\lbrace F(z); z\\in G\\rbrace $ , where $F(z):=\\int _{\\mathbb {R}^n}f^{z}(x)dx$ .", "(ii) ${\\rm grad}F(z_0)=\\int _{\\mathbb {R}^n} x f^{z_0}(x+z_0)dx\\in G^{\\bot }$ .", "By Lemma REF , $F$ is coercive and strictly convex on ${\\rm int}\\;{\\rm dom}F$ , thus there is a unique minimal point $z_0=s_G(f)$ on $G$ .", "Let $f=e^{-\\phi }$ , then $F(z)=\\int _{\\mathbb {R}^n}e^{-\\mathcal {L}\\phi (x)+\\langle x,z\\rangle }dx$ .", "By the dominated convergence theorem, we have ${\\rm grad} F(z)=\\int _{\\mathbb {R}^n}xe^{-\\mathcal {L}\\phi (x)+\\langle x,z\\rangle }dx=\\int _{\\mathbb {R}^n}xf^z(x+z)dx$ .", "Next, we prove the equivalence of (i) and (ii).", "Let $\\eta _1,\\dots ,\\eta _m\\;(m<n)$ be an orthonormal basis of $G$ and let $\\eta _{m+1},\\dots ,\\eta _n$ be an orthonormal basis of $G^{\\perp }$ .", "Let $z=\\sum _{i=1}^{n}z_i\\eta _i$ , since $z_0=s_G(f)\\in G$ , we have $\\left.\\frac{\\partial F(z)}{\\partial z_i}\\right\|_{z=z_0}=\\lim _{t\\rightarrow 0}\\frac{F(z_0+t\\eta _i)-F(z_0)}{t}=0,\\;\\;i=1,\\dots ,m$ .", "Hence, ${\\rm grad} F(z_0)\\in G^{\\bot }$ .", "On the other hand, if ${\\rm grad}F(z_0)\\in G^{\\bot }$ , then $\\left.\\frac{\\partial F(z)}{\\partial z_i}\\right\|_{z=z_0}=0,\\;i=1,\\dots ,m$ .", "Since $F(z)$ is strictly convex on $G\\cap {\\rm int}\\;{\\rm dom}F$ , $z_0$ is the unique minimal point on $G$ .", "Remark 3 In Lemma REF , if $G=\\mathbb {R}^n$ , then the lemma shows that the Santaló point $s(f)$ of $f$ is the barycenter of the function $f^{s(f)}$ .", "Lemma 4.3 Let $f$ be a log-concave function such $0<\\int f<\\infty $ .", "Let $G\\subset \\mathbb {R}^n$ be an affine subspace satisfying $G\\cap {\\rm int}\\;{\\rm supp}f\\ne \\emptyset $ and $z=s_G(f)$ .", "Let $H$ be an affine hyperplane such that $G\\subset H$ and let $g$ be the function defined by $g^{z}=S_H (f^{z})$ .", "Then we have $s_G(g)=z=s_G(f)$ .", "It may be supposed that $z=s_G(f)=0$ , $H=\\lbrace (x_1,\\cdots ,x_n)\\in \\mathbb {R}^n:x_n=0\\rbrace $ and $G=\\lbrace (x_1,\\cdots ,x_n)\\in \\mathbb {R}^n:x_{m+1}=\\cdots =x_{n}=0\\rbrace $ for some $m$ , $1\\le m\\le n-1$ .", "By Lemma REF , we have $\\int _{\\mathbb {R}^n}xf^{0}(x)dx\\in G^{\\bot }$ .", "Let $f^{0}_{x^{\\prime }}(t):=f^{0}(x^{\\prime }+tu)$ for any $x^{\\prime }\\in H$ , where $u$ is the unit normal vector of $H$ .", "Thus, $\\int _{H}x_i\\left(\\int _{\\mathbb {R}}f^{0}_{x^{\\prime }}(t)dt\\right)dx^{\\prime }=0\\;\\;\\textrm {for} \\;\\;1\\le i\\le m$ .", "By $g^{0}=S_H(f^{0})$ and (REF ), for every $x^{\\prime }\\in H$ , $\\int _{\\mathbb {R}}f^{0}_{x^{\\prime }}(t)=\\int _{\\mathbb {R}}g^0_{x^{\\prime }}(t)$ .", "Thus, $\\int _{H}x_i\\left(\\int _{\\mathbb {R}}g^{0}_{x^{\\prime }}(t)dt\\right)dx^{\\prime }=0\\;\\;\\textrm {for} \\;\\;1\\le i\\le m$ , which conversely gives $\\int _{\\mathbb {R}^n}x g^{0}(x)dx\\in G^{\\bot }$ .", "Thus, by Lemma REF again, we obtain $s_G(g)=0=s_G(f)$ .", "Lemma 4.4 For a log-concave function $f$ such that $0<\\int f<\\infty $ , if $f$ is symmetric about some affine hyperplane $H$ , then, for any $z\\in H$ , $f^{z}$ is also symmetric about $H$ .", "Let $u$ be the unit normal vector of $H$ .", "For any $x^{\\prime },y^{\\prime }\\in H$ and $s,t\\in \\mathbb {R}$ , since $f(x^{\\prime }+su)=f(x^{\\prime }-su)$ , we have $f^{z}(y^{\\prime }+tu)&=& \\inf _{x^{\\prime }+su\\in \\mathbb {R}^n}\\frac{\\exp \\lbrace -\\langle y^{\\prime }+tu-z,x^{\\prime }+su-z\\rangle \\rbrace }{f(x^{\\prime }+su)}\\nonumber \\\\&=&\\inf _{x^{\\prime }+su\\in \\mathbb {R}^n}\\frac{\\exp \\lbrace -\\langle y^{\\prime }-z-tu, x^{\\prime }-z-su\\rangle \\rbrace }{f(x^{\\prime }-su)}=f^{z}(y^{\\prime }-tu).$ This completes the proof.", "Lemma 4.5 Let $f$ be a log-concave function such that $0<\\int f<\\infty $ and let $H$ be an affine hyperplane satisfying $H\\cap {\\rm int}\\;{\\rm supp}f\\ne \\emptyset $ and $z\\in H\\cap {\\rm int}\\;{\\rm supp}f$ ; let $\\lambda $ , $0<\\lambda <1$ such that $H$ is $\\lambda $ -separating for $f^{z}$ .", "Then $\\int _{\\mathbb {R}^n}(S_H f)^{z}\\ge 4\\lambda (1-\\lambda )\\int _{\\mathbb {R}^n}f^{z}.$ It may be supposed that $z=0$ and $H=\\lbrace (x_1,\\dots ,x_n):x_n=0\\rbrace $ .", "For $y^{\\prime }\\in H$ and $s\\in \\mathbb {R}$ , let $(y^{\\prime },s)$ denote $y^{\\prime }+su$ , where $u$ is a unit normal vector of $H$ .", "For $f^0$ and $s\\in \\mathbb {R}$ , we define a new function $f^0_{(s)}(y^{\\prime }):=f^0(y^{\\prime },s),\\;{\\rm for\\; any}\\;y^{\\prime }\\in H.$ Next we shall prove that for any $y^{\\prime }\\in H$ and $s,t>0$ $\\left(\\frac{t}{s+t}\\cdot f_{(s)}^{0}\\right)\\star \\left(\\frac{s}{s+t}\\cdot f_{(-t)}^{0}\\right)(y^{\\prime })\\le (S_H f)^{0}_{(\\frac{2st}{s+t})}(y^{\\prime }).$ Claim 2 For any $x^{\\prime }\\in H$ and $w\\in \\mathbb {R}$ , if $(S_Hf)(x^{\\prime }+wu)>0$ , then there is some $w_1\\in \\mathbb {R}$ such that $(S_H f)(x^{\\prime }+wu)\\le f(x^{\\prime }+w_1u)$ and $(S_Hf)(x^{\\prime }+wu)\\le f(x^{\\prime }+(w_1-2w)u)$ .", "Proof of Claim REF .", "Let $f=e^{-\\phi }$ , since $(S_H f)(x^{\\prime }+wu)>0$ , then $(S_H\\phi )(x^{\\prime }+wu)<+\\infty $ .", "By Proposition REF (iii), there is $w_1\\in \\mathbb {R}$ such that $(S_H\\phi )(x^{\\prime }+wu)\\ge \\phi (x^{\\prime }+w_1u)$ and $(S_H\\phi )(x^{\\prime }+wu)\\ge \\phi (x^{\\prime }+(w_1-2w)u)$ , here we assume $\\phi (x^{\\prime }+w_1u)$ or $\\phi (x^{\\prime }+(w_1-2w)u)$ equals the limit in Proposition REF (iii), which doesn't affect our proof.", "Hence the claim follows.$\\Box $ For any $y_1^{\\prime }$ , $y_2^{\\prime }\\in H$ such that $y^{\\prime }=y_1^{\\prime }+y_2^{\\prime }$ , we have $(S_H f)_{ (\\frac{2st}{s+t})}^{0}(y^{\\prime })&=&\\inf _{(x^{\\prime },w)\\in H\\times \\mathbb {R}}\\frac{\\exp \\lbrace -\\langle (y^{\\prime },\\frac{2st}{s+t}),(x^{\\prime },w)\\rangle \\rbrace }{(S_Hf)(x^{\\prime },w)}\\nonumber \\\\&\\ge &\\inf _{(x^{\\prime },w)\\in H\\times \\mathbb {R}}\\frac{\\exp \\lbrace -\\langle (y^{\\prime },\\frac{2st}{s+t}),(x^{\\prime },w)\\rangle \\rbrace }{f(x^{\\prime },w_1)^{\\frac{t}{s+t}}f(x^{\\prime },w_1-2w)^{\\frac{s}{s+t}}}\\nonumber \\\\&\\ge &\\inf _{(x^{\\prime },w)\\in H\\times \\mathbb {R}}\\frac{\\exp \\lbrace -\\frac{t}{s+t}\\langle (\\frac{s+t}{t}y_1^{\\prime },s),(x^{\\prime },w_1)\\rangle \\rbrace }{f(x^{\\prime },w_1)^{\\frac{t}{s+t}}}\\nonumber \\\\&&\\times \\inf _{(x^{\\prime },w)\\in H\\times \\mathbb {R}}\\frac{\\exp \\lbrace -\\frac{s}{s+t}\\langle (\\frac{s+t}{s}y_2^{\\prime },-t),(x^{\\prime },w_1-2w)\\rangle \\rbrace }{f(x^{\\prime },w_1-2w)^{\\frac{s}{s+t}}}\\nonumber \\\\&\\ge &f^0\\left(\\frac{s+t}{t}y_1^{\\prime },s\\right)^{\\frac{t}{s+t}}f^0\\left(\\frac{s+t}{s}y_2^{\\prime },-t\\right)^{\\frac{s}{s+t}},$ where the first inequality is by Claim REF , and the second inequality is by $\\inf (AB)\\ge (\\inf A)(\\inf B)$ , and last inequality is by the definition of the polar of functions.", "Since $y^{\\prime }_1$ and $y^{\\prime }_2$ are arbitrary, we get (REF ).", "Let $F_0(w)=\\int _{H}(S_H f)^{0}_{(w)}$ , $F_1(s)=\\int _{H}f^{0}_{(s)}$ and $F_2(t)=\\int _{H}f^{0}_{(-t)}$ .", "By the Prékopa inequality and (REF ), we have $F_0(\\frac{2st}{s+t})\\ge F_1(s)^{\\frac{t}{s+t}}F_2(t)^{\\frac{s}{s+t}}\\;{\\rm for}\\;{\\rm every}\\;s,t>0.$ Now, by Proposition REF (i) and Lemma REF , $(S_H f)^{0}$ is symmetric about $H$ , we have $\\int _{0}^{+\\infty }F_0=\\frac{1}{2}\\int _{\\mathbb {R}^n}(S_H f)^0$ and since $H$ is $\\lambda $ -separating for $f^{0}$ , we have $\\left(\\int _{0}^{+\\infty }F_1\\right)\\left(\\int _{0}^{+\\infty }F_2\\right)=\\lambda (1-\\lambda )\\left(\\int _{\\mathbb {R}^n}f^0\\right)^2$ .", "Since $F_0$ , $F_1$ , $F_2:[0,+\\infty )\\rightarrow \\mathbb {R}^{+}$ satisfy the hypothesis of Lemma REF , and by definitions of $F_1$ and $F_2$ , one has $\\int _{0}^{+\\infty }F_1+\\int _{0}^{+\\infty }F_2=\\int _{\\mathbb {R}^n}f^0$ , thus, by Lemma REF $\\frac{2}{\\int _{\\mathbb {R}^n}(S_Hf)^0}\\le \\frac{1}{2}\\left(\\frac{1}{\\int _{0}^{+\\infty }F_1}+\\frac{1}{\\int _{0}^{+\\infty }F_2}\\right)=\\frac{1}{2\\lambda (1-\\lambda )\\int _{\\mathbb {R}^n}f^0}.$ This gives the desired inequality.", "Lemma 4.6 If $f$ is an integrable, unconditional, log-concave function, then $\\int _{\\mathbb {R}^n}f\\int _{\\mathbb {R}^n}f^{0}\\le (2\\pi )^n$ .", "Let $f_1=f$ , $f_2=f^{0}$ and $f_3=e^{-\\frac{\|x\|^2}{2}}$ , then $f_1$ , $f_2$ and $f_3$ are unconditional.", "Thus we have $\\int _{\\mathbb {R}^n}f_j=2^n\\int _{\\mathbb {R}_+^n}f_j,\\;\\;j=1,2,3$ .", "For $(y_1,\\dots ,y_n)\\in \\mathbb {R}^n$ , we define $g_i(y_1,\\dots ,y_n)=f_i(e^{y_1},\\dots ,e^{y_n})e^{\\sum _{i=1}^{n}y_i}$ .", "We get $\\int _{\\mathbb {R}_+^n}f_j=\\int _{\\mathbb {R}^n}g_j$ , and for every $s,t\\in \\mathbb {R}^n$ , $g_1(s)g_2(t)\\le g_3\\left(\\frac{s+t}{2}\\right)^2$ .", "Hence $\\int _{\\mathbb {R}^n}f\\int _{\\mathbb {R}^n}f^{0}\\le (2\\pi )^n$ follows from Prékopa inequality.", "Proof of Theorem REF .", "We proceed by $n$ successive Steiner symmetrizations until we get an unconditional log-concave function.", "Let $u_1\\in S^{n-1}$ , $u_1$ orthogonal to $H=H_1$ and let $(u_i)_{i=2}^{n}\\subset S^{n-1}$ such that $(u_1,\\dots ,u_n)$ form an orthonormal basis for $\\mathbb {R}^n$ .", "Let $z_1=s_{H_1}(f)$ and define a log-concave function $f_1$ by the identity $f_1^{z_1}=S_{H_1}(f^{z_1})$ .", "Then $\\int f_1^{z_1}=\\int f^{z_1}$ .", "By Proposition REF (i) and Lemma REF , $f_1$ is symmetric about $H_1$ and by Lemma REF , applied to $f^{z_1}$ , $z=z_1$ and $H=H_1$ , $\\lambda $ -separating for $f=(f^{z_1})^{z_1}$ , we get $\\int _{\\mathbb {R}^n}f_1\\ge 4\\lambda (1-\\lambda )\\int _{\\mathbb {R}^n}f$ and thus $\\int f_1\\int f_1^{z_1}\\ge 4\\lambda (1-\\lambda )\\int f\\int f^{z_1}$ .", "Choose now the hyperplane $H_2$ , orthogonal to $u_2$ , and medial for $f_1$ and define $z_2=s_{(H_1\\cap H_2)}(f_1)$ .", "By Lemma REF we have $z_1=s_{H_1}(f)=s_{H_1}(f_1)$ , we get $\\int f_1^{z_2}=\\min _{z\\in H_1\\cap H_2}\\int f_1^{z}\\ge \\min _{z\\in H_1}\\int f_1^{z}=\\int f_1^{z_1}$ .", "We define now a new log-concave function $f_2$ by the identity $f_2^{z_2}=S_{H_2}(f_1^{z_2})$ .", "By Proposition REF (ii) and Lemma REF , $f_2$ is symmetric about both $H_1$ and $H_2$ .", "Since $H_2$ is medial for $f_1$ , we get by Lemma REF applied to $f_1^{z_2}$ , $z=z_2$ and $H=H_2$ that $\\int f_2\\ge \\int f_1$ .", "Moreover, we have $\\int f_2^{z_2}=\\int S_{H_2}(f_1^{z_2})=\\int f_1^{z_2}\\ge \\int f_1^{z_1}$ .", "It follows that $\\int f_2\\int f_2^{z_2}\\ge \\int f_1\\int f_1^{z_1}$ .", "We continue this procedure by choosing hyperplanes $H_2,\\dots ,H_n$ , points $z_2,\\dots ,z_n$ , and defining log-concave functions $f_2,\\dots ,f_n$ such that for $2\\le i\\le n$ , we have (i) $H_i$ is medial for $f_{i-1}$ and orthogonal to $u_i$ ; (ii) $z_i=s_{(H_1\\cap H_2\\cap \\dots \\cap H_i)}(f_{i-1})$ ; (iii) $f_i^{z_i}=S_{H_i}(f_{i-1}^{z_i})$ .", "From (ii) (iii) and Lemma REF , we have $z_i=s_{(H_1\\cap \\dots \\cap H_i)}(f_{i-1})=s_{(H_1\\cap \\dots \\cap H_i)}(f_i)$ .", "Choosing $H_{i+1}$ , $z_{i+1}$ , $f_{i+1}$ according to (i) (ii) (iii), we get thus $\\int f_{i+1}^{z_{i+1}}=\\int S_{H_{i+1}}(f_{i}^{z_{i+1}})=\\int f_{i}^{z_{i+1}} \\ge \\int f_i^{s_{(H_1\\cap \\dots \\cap H_i)}(f_i)}=\\int f_i^{z_i}$ .", "Now, Lemma REF applied to $f_i^{z_{i+1}}$ , $z=z_{i+1}$ and $H_{i+1}$ , medial for $f_i=(f_i^{z_{i+1}})^{z_{i+1}}$ , gives $\\int f_{i+1}\\ge \\int f_i$ .", "Thus, $\\int f_i\\int f_i^{z_i}$ is an increasing sequence, for $2\\le i\\le n$ .", "Therefore, we have $4\\lambda (1-\\lambda )\\int f\\int f^{z_1} \\le \\int f_1\\int f_1^{z_1}\\le \\dots \\le \\int f_n\\int f_n^{z_n}$ .", "From Proposition REF (ii), $f_n$ is an unconditional function about $z_n$ and $z_n\\in H_1\\cap H_2\\cap \\dots \\cap H_n$ is a center of symmetry for $f_n$ .", "By Lemma REF , we have $\\int f\\int f^{z_1}\\le \\frac{(2\\pi )^n}{4\\lambda (1-\\lambda )}$ , this concludes the proof.$\\Box $" ] ]	1403.0299
[ [ "Blind and fully constrained unmixing of hyperspectral images" ], [ "Abstract This paper addresses the problem of blind and fully constrained unmixing of hyperspectral images.", "Unmixing is performed without the use of any dictionary, and assumes that the number of constituent materials in the scene and their spectral signatures are unknown.", "The estimated abundances satisfy the desired sum-to-one and nonnegativity constraints.", "Two models with increasing complexity are developed to achieve this challenging task, depending on how noise interacts with hyperspectral data.", "The first one leads to a convex optimization problem, and is solved with the Alternating Direction Method of Multipliers.", "The second one accounts for signal-dependent noise, and is addressed with a Reweighted Least Squares algorithm.", "Experiments on synthetic and real data demonstrate the effectiveness of our approach." ], [ "Introduction", "Hyperspectral imaging is a continuously growing area of remote sensing, which has received considerable attention in the last decade.", "Hyperspectral data provide spectral images over hundreds of narrow and adjacent bands, coupled with a high spectral resolution.", "These characteristics are suitable for detection and classification of surfaces and chemical elements in the observed images.", "Applications include land use analysis, pollution monitoring, wide-area reconnaissance, and field surveillance, to cite a few.", "When unmixing hyperspectral images [1], two types of pixels can be distinguished: the pure pixels and the mixed ones.", "Each pure pixel, also called endmember, contains the spectral signature of a constituent material in the scene, whereas a mixed pixel consists of a mixture of the endmembers.", "The fraction of each endmember in a mixed pixel is called abundance.", "Three consecutive tasks are usually required for unmixing: determining the number of endmembers, extracting the spectral signature of the endmembers, and estimating their abundances for every pixel in the scene.", "Several algorithms have been proposed to perform each stage separately.", "Virtual Dimensionality (VD) [2], followed by N-FINDR [3] and FCLS [4] is among the most widely used processing pipeline.", "Alternative methods jointly performs (part of) these tasks in order to solve the blind source separation problem [5], [6], [7], [8].", "In order to introduce our approach, we shall now describe the noise-free case first.", "Consider the linear mixing model where a mixed pixel is expressed as a linear combination of the endmembers weighted by their fractional abundances.", "In matrix form, we simply have: $\\widetilde{S}= R{A}$ where $\\widetilde{S}= (\\tilde{s}_{1},\\ldots , \\tilde{s}_{N}) $ , $R= (r_{1},\\ldots , r_{M})$ , ${A} = ({a}_{1},\\ldots , {a}_{M})^{\\top }$ , $\\tilde{s}_{j}$ is the $L$ -dimensional spectrum of the $j$ -th pixel, $L$ is the number of frequency bands, $r_{i}$ is ${L}$ -dimensional spectrum of the $i$ -th endmember, $M$ is the number of endmembers, ${a}_i$ is the ${N}$ -dimensional abundance map of the $i$ -th endmember, and $N$ is the number of pixels in the image.", "Model (REF ) means that the $(i,j)$ -th entry ${A}_{ij}$ of matrix $A$ represents the abundance of the endmember $r_{i}$ in pixel $\\tilde{s}_{j}$ .", "The abundances obey the nonnegativity and sum-to-one constraints: $A_{ij} \\ge 0 $ for all $i$ and $j$ , and $ \\sum _{i=1}^M A_{ij} = 1$ for all $j$ .", "Note that the tilde placed over symbols refers to noise-free data and all vectors are column vectors.", "In this study, we shall assume that the endmembers are unknown but present in the scene.", "Let $\\omega $ be a subset of $N^{\\prime }$ indexes in $\\lbrace 1, \\ldots ,N\\rbrace $ that contains at least the column index of each endmember.", "Under these assumptions, and without loss of generality, we observe that the mixing model (REF ) can be reformulated as follows $\\widetilde{S}= \\widetilde{S}_{\\omega }X$ where $\\widetilde{S}_\\omega = (\\tilde{s}_{{\\omega }_1},\\ldots , \\tilde{s}_{{\\omega }_{N^{\\prime }}}) $ denotes the restriction of $\\widetilde{S}$ to its columns indexed by $\\omega $ , and ${X} = ({x}_{1},\\ldots , {x}_{N^{\\prime }})^{\\top }$ is the abundance matrix.", "Similarly as above, $X_{ij}$ is the abundance of $\\tilde{s}_{{\\omega }_i}$ in $\\tilde{s}_j$ .", "On the one hand, if $\\tilde{s}_{\\omega {_i}}$ is an endmember, ${x}_i$ has non-zero entries and represents the corresponding abundance map.", "On the other hand, if $\\tilde{s}_{\\omega {_i}}$ is a mixed pixel, ${x_i}$ has all its elements equal to zero.", "As a consequence, $X$ admits $N^{\\prime }-M$ rows of zeros, the other rows being equal to rows of $A$ .", "This means that $X$ allows to identify the endmembers in $\\widetilde{S}$ through its non-zero rows, which is an interesting property to be exploited in the case where the endmembers are unknown.", "Let us now turn to the more realistic situation where some noise corrupts the observations.", "In this case, model (REF ) becomes $ S= \\widetilde{S}+ E= \\widetilde{S}_{\\omega }X+ E$ where $S$ denotes the available data and $E$ the noise.", "The aim of this paper is to derive two unmixing approaches with increasing complexity, depending on how noise is to be handled.", "These methods are blind in the sense that the endmembers and their cardinality are unknown.", "The first one considers the approximate model $S\\approx S_{\\omega }X+ E$ Compared to (REF ), we thus assume that noise does not dramatically affect factorization of the mixing process, which is valid for very high signal-to-noise ratio (SNR).", "With this approach, we shall look for a few columns of $S_{\\omega }$ that can effectively represent the whole scene.", "This strategy subserves a blind and self-dependent framework.", "It departs from methods based on a preselected dictionary of endmembers estimated from other experimental conditions, and thus do not accurately represent the endmembers in $R$ .", "In order to estimate the abundance matrix $X$ , we use prior information.", "First, we impose that the estimated abundances obey the non-negativity and sum-to-one constraints, namely, $X_{ij} \\ge 0 $ for all $(i,j)$ , and $ \\sum _{i=1}^N X_{ij} = 1$ for all $j$ .", "In addition, as discussed above, the algorithm has to force rows of $X$ to be zero vectors in order to identify the endmembers.", "Because the locations and the cardinality of the endmembers are unknown, the set of candidates has to be sufficiently large, that is, $N^{\\prime } \\gg M$ .", "We thus expect many rows in $X$ to be equal to zero.", "To promote this effect, the so-called Group Lasso $\\ell _{2,1}$ -norm regularization can be employed [9].", "Because model (REF ) becomes a poor approximation of model (REF ) as the noise power increases, we shall also propose an alternative strategy to solve the unmixing problem based on the exact model (REF ).", "The first approach leads to a convex optimization problem that can be solved with the Alternating Direction Method of Multipliers (ADMM) [10].", "The second one takes the noise in $S_{\\omega }$ into account, which results in a non-convex and heteroscedastic optimization problem.", "The latter will be solved with an Iterative Reweighted Least Squares (IRLS) algorithm.", "Few models of the form (REF ) have been studied in the literature [11], [12], [13].", "These last three works assume that $S_{\\omega }$ is noise-free.", "Moreover, in [11], the authors use an $\\ell _\\infty $ -norm rather than the $\\ell _{2,1}$ -norm regularization considered here.", "In [12], the authors derive a Matching Pursuit approach [14] in order to estimate the endmembers.", "A similar technique is considered in [13], but the authors do not assume that the endmembers are present in the scene and use a predefined dictionary.", "Finally, note that the non-negativity and sum-to-one constraints are not considered in [11], [12].", "The rest of this paper is organized as follows.", "Sections and respectively describe the unmixing models (REF ) and (REF ), and the corresponding estimation methods.", "Section provides experimental results on synthetic and real data.", "Finally, Section concludes this paper.", "The aim of this section is to derive the estimation method for model (REF ), and finally define each step of the ADMM run to get the solution.", "In this approximate model, we assume that the noise $E$ is Gaussian independent and identically distributed, with zero mean and possibly unknown variance $\\sigma ^2$ , that is, $E_{k,i}\\sim \\mathcal {N}(0,\\sigma ^2)$ .", "The negative log-likelihood for model (REF ) is given by $\\mathcal {L}(X) =\\frac{NL}{2}\\log (2\\pi ) +\\frac{NL}{2}\\log (\\sigma ^2) + \\frac{1}{2\\sigma ^2} \\Vert S-S_{\\omega }X\\Vert ^2_F $ The Maximum Likelihood (ML) estimate, namely, the minimizer of $\\mathcal {L}(X)$ , is the solution of the usual Least Squares (LS) approximation problem $\\min _{X}\\Vert S-S_{\\omega }X\\Vert ^2_F$ .", "Since model (REF ) follows from an approximation of model (REF ), the relevance of this LS fidelity term is essentially to ensure that $S_{\\omega }X$ matches $S$ .", "The unmixing problem under investigation, however, requires that $X$ only has a few rows different from zero, in addition to the non-negativity and sum-to-one constraints.", "This leads to following convex optimization problem $ \\begin{array}{ll}\\min _{X} & \\frac{1}{2}\\Vert S-S_{\\omega }X\\Vert _\\text{F}^2+ \\mu \\sum _{k=1}^N \\Vert x_{k}\\Vert _2 \\\\\\text{subject to} & X_{ij} \\ge 0 \\quad \\forall \\, i,j \\\\& \\sum _{i=1}^N X_{ij} = 1 \\quad \\forall \\, j\\end{array}$ with $\\mu \\ge 0$ a regularization parameter and $x_k$ the $k$ -th row of $X$ .", "The Group Lasso regularization term induces sparsity in the estimated abundance matrix at the group level [9], by possibly driving all the entries in several rows $x_k$ of $X$ to zero.", "It is worth noting that when $\\mu =0$ and $S_{\\omega }=S$ , the solution of problem (REF ) is the identity matrix $X=I$ .", "It follows that the efficiency of the approach relies on the $\\ell _{2,1}$ -norm regularization function." ], [ "ADMM algorithm", "The solution of problem (REF ) can be obtained in a simple and flexible manner using the ADMM algorithm [10].", "We consider the canonical form $ \\begin{array}{ll}\\min _{X,Z} & \\frac{1}{2}\\Vert S-S_{\\omega }X\\Vert _\\text{F}^2+ \\mu \\sum _{k=1}^N \\Vert z_{k}\\Vert _2 + {\\mathcal {I}}(Z) \\\\\\text{subject to} & AX+ BZ= C\\end{array}$ with $A= \\left(\\begin{array}{c} I\\, \\\\ 1^ \\top \\end{array}\\right),\\;B= \\left(\\begin{array}{c} -I\\\\ \\,0^\\top \\end{array}\\right),\\;C= \\left(\\begin{array}{c} 0\\, \\\\ 1^\\top \\end{array}\\right), \\nonumber $ where ${\\mathcal {I}}$ is the indicator of the positive orthant guarantying the positivity constraint, that is, ${\\mathcal {I}}(Z)= 0$ if $Z\\succeq 0$ and $+\\infty $ otherwise.", "The equality constraint imposes the consensus $X=Z$ and the sum-to-one constraint.", "In matrix form, the augmented Lagrangian for problem (REF ) is given by [15] $\\mathcal {L}_\\rho (X,Z,\\Lambda ) = \\frac{1}{2}\\Vert S-S_{\\omega }X\\Vert _F^2+ \\mu \\sum _{k=1}^N \\Vert z_{k}\\Vert _2 +{\\mathcal {I}}(Z)+\\text{trace}(\\Lambda ^\\top (AX+ BZ- C))+\\frac{\\rho }{2}\\, \\Vert AX+ BZ- C\\Vert _\\text{F}^2$ where $\\Lambda $ is the matrix of Lagrange multipliers, $\\mu $ and $\\rho $ are positive regularization and penalty parameters, respectively.", "The flexibility of the ADMM lies in the fact that it splits the initial variable $X$ into two variables, $X$ and $Z$ , and equivalently the initial problem into two subproblems.", "At iteration $k+1$ , the ADMM algorithm is outlined by three sequential steps:" ], [ "Minimization of $\\mathcal {L}_\\rho (X,Z^k,\\Lambda ^k)$ with respect to {{formula:a907fe8a-61ec-4c70-b272-cb56b9d540c4}}", "This step takes into account the previous estimates of $Z$ and $\\Lambda $ .", "The augmented Lagrangian is quadratic in terms of $X$ .", "As a result, the solution has an analytical expression that is obtained by setting the gradient of $\\mathcal {L}_\\rho (X,Z^k,\\Lambda ^k)$ to zero: $X^{k+1} = ({S_{\\omega }}^\\top S_{\\omega } +\\rho A^\\top A)^{-1}(S_{\\omega }^\\top S- A^\\top [\\Lambda ^k+\\rho \\,(BZ^k-C)])$" ], [ "Minimization of $\\mathcal {L}_\\rho (X^{k+1},Z,\\Lambda ^k)$ with respect to {{formula:fd36b179-c2fe-465a-a5ae-bb0b8e062b49}}", "After removing the terms that are independent of $Z$ , the minimization of $L_\\rho (X^{k+1},Z,\\Lambda ^k)$ with respect to $Z$ reduces to solving the following problem: $\\begin{array}{ll}\\min _{Z} & \\mu \\sum _{k=1}^N \\Vert z_{k}\\Vert _2 +\\text{trace}(\\Lambda ^\\top BZ)+\\frac{\\rho }{2} \\Vert AX+ BZ- C\\Vert _\\text{F}^2\\\\\\text{subject to} & Z\\succeq 0\\end{array}$ This minimization step can be split into $N$ problems given the structure of matrices $A$ and $B$ , one for each row of $Z$ , that is, $\\begin{array}{ll}\\min _{z} & \\frac{1}{2} \\Vert z-v\\Vert _2^2 + \\alpha \\Vert z\\Vert _2 + {\\mathcal {I}}(z)\\end{array}$ where $v= x+\\rho ^{-1} \\lambda $ , $\\alpha = \\rho ^{-1}\\mu $ , $\\lambda $ , $x$ and $z$ correspond to a row in $\\Lambda $ , $X$ and $Z$ respectively.", "The minimization problem (REF ) admits a unique solution, given by the proximity operator [16] of $f(z)=\\alpha \\Vert z\\Vert _2 + {\\mathcal {I}}(z)$ $\\left\\lbrace \\begin{array}{ll}z^\\ast = 0& \\text{if } \\Vert (v)_+\\Vert _2 < \\alpha \\\\z^\\ast = \\left(1 - \\frac{\\alpha }{\\Vert (v)_+\\Vert _2}\\right) (v)_+ & \\text{otherwise}\\end{array}\\right.$ where $(\\cdot )_+ = \\max (0,\\cdot )$ .", "On the one hand, the proximity operator of $f_1(z)=\\alpha \\Vert z\\Vert _2$ is the Multidimensional Shrinkage Thresholding Operator (MiSTO) [17].", "On the other hand, the proximity operator of the indicator function $f_2(z)={\\mathcal {I}}(z)$ is the projection onto the positive orthant.", "The proximity operator of $f(z)$ in (REF ), that we refer to as Positively constrained MiSTO, is an extension of both previous operators.", "The solution is of the form $\\text{prox}_{f}=\\text{prox}_{f_1}\\circ \\text{prox}_{f_2}$ , that is, the thresholding of the projection.", "Operator (REF ) was recently used in [18].", "A proof for this operator can be found in the Appendix." ], [ "Update of the Lagrange multipliers $\\Lambda $", "Update of the Lagrange multipliers is carried out at the end of each iteration.", "$\\Lambda ^{k+1}$ represents the running sum of residuals.", "It gives an insight on the convergence of the algorithm.", "As $k$ tends to infinity, the primal residual tends to zero and $\\Lambda ^{k+1}$ converges to the dual optimal point.", "$\\Lambda ^{k+1} = \\Lambda ^{k} +\\rho (AX^{k+1}+BZ^{k+1}-C).$ As suggested in [10], a reasonable stopping criteria is that the primal and dual residuals must be smaller than some tolerance thresholds, namely, $\\Vert AX^{k+1}+BZ^{k+1}-C\\Vert _2 \\le \\epsilon _{\\text{pri}} \\quad \\text{and} \\quad \\Vert \\rho A^\\top B(Z^{k+1}-Z^k)\\Vert _2 \\le \\epsilon _{\\text{dual}}$ The pseudocode for the so-called GLUP method is provided by Algorithm REF .", "It is worth emphasizing that the main difference between the ADMM steps developed in GLUP and those in [13] arises in the ADMM variable splitting.", "The global problem in [13] is decomposed into three subproblems: the least squares minimization, the Group Lasso regularization, and projection on the positive orthant.", "A consequence is that three ADMM variables are used instead of two, which leads to additional steps.", "In addition, the sum-to-one constraint is not considered in [13].", ": $X= \\text{GLUP}(S,S_{\\omega },\\rho ,\\mu )$ [1] Precompute $A$ , $B$ , and $C$ Initialize $Z= 0$ and $\\Lambda = 0$ $Q= ({S_{\\omega }}^\\top S_{\\omega } +\\rho A^\\top A)^{-1}$ $\\Vert R\\Vert _2 \\ge \\epsilon _{\\text{pri}}$ or $\\Vert P\\Vert _2 \\ge \\epsilon _{\\text{dual}}$ $X= Q(S_{\\omega }^\\top S- A^\\top (\\Lambda +\\rho \\,[BZ-C]))$ $Z^{\\text{old}} = Z$ $i=1\\cdots N^{\\prime }$ $v_i = ((x_i)^{\\top } +\\rho ^{-1} \\lambda _i)_{+}$ $ \\Vert v_i\\Vert _2 < \\rho ^{-1}\\mu $ $ z_i= 0$ $z_i = \\left(1 - \\frac{\\mu }{\\rho \\Vert v_i\\Vert _2}\\right) v_i $ $R= AX+ BZ- C$ $P= \\rho AB(Z-Z_{\\text{old}})$ $ \\Lambda = \\Lambda +\\rho (AX+BZ-C) $" ], [ "Model description", "We now turn to the more realistic model (REF ).", "Let $E_{\\omega }$ and $I_{\\omega }$ be the $L$ -by-$N^{\\prime }$ and $N$ -by-$N^{\\prime }$ restrictions of $E$ and $I$ to the columns indexed by $\\omega $ , respectively.", "The noisy mixing model (REF ) is given by $S= (S_{\\omega }- E_{\\omega })X+ E= S_{\\omega }X+E(I-I_{\\omega }X)$ This model belongs to the family of heteroscedastic regression [19], where the variance of the additive noise depends on $X$ .", "Let us define the matrix $C(X)$ as $C(X)=(I- I_{\\omega }X)^{\\top }(I- I_{\\omega }X)$ It follows that $\\text{vec}(E(I-I_{\\omega }X))\\sim \\mathcal {N}(0,\\sigma ^2C(X)\\otimes I)$ where $\\otimes $ represents the Kronecker product of matrices, and $\\text{vec}(\\cdot )$ is the operator that stacks the columns of a matrix on top of each other.", "The presence of $X$ in the expression of the noise variance has consequences on the negative log-likelihood of model $(\\ref {model2})$ , which no longer leads to the LS approximation problem $\\begin{split}\\mathcal {L}(X,\\sigma ^2) &= \\frac{1}{2}\\log \|\\sigma ^2C(X)\\otimes I\|+\\frac{1}{2} \\text{vec}(S-S_{\\omega }X)^{\\top }(\\sigma ^2C(X)\\otimes I)^{-1} \\text{vec}(S-S_{\\omega }X) \\\\&=\\frac{L}{2}\\log \|\\sigma ^2C(X)\| + \\frac{1}{2} \\text{trace}((S-S_{\\omega }X) (\\sigma ^2C(X))^{-1} (S-S_{\\omega }X)^{\\top }) \\\\&=\\frac{L}{2}\\log \|\\sigma ^2C(X)\| + \\frac{1}{2} \\Vert S-S_{\\omega }X\\Vert ^2_{(\\sigma ^2C(X))^{-1}}\\end{split}$ The ML estimate for problem (REF ) with the Group Lasso regularization, nonnegativity and sum-to-one constraints yields the following constrained optimization problem: $\\begin{array}{ll}\\min _{X, \\sigma ^2} & \\frac{L}{2}\\log \|\\sigma ^2C(X)\| + \\frac{1}{2} \\Vert S-S_{\\omega }X\\Vert ^2_{(\\sigma ^2C(X))^{-1}} + \\mu \\sum _{k=1}^N \\Vert x_{k}\\Vert _2 \\\\\\text{subject to} & X_{ij} \\ge 0 \\quad \\forall \\, i,j\\\\& \\sum _{i=1}^N X_{ij} = 1 \\quad \\forall \\, j\\end{array}$" ], [ "Alternating ADMM algorithm", "Problem (REF ) is not convex and requires the estimation of $\\sigma ^2$ .", "The second term in the objective function is closely related to Iteratively Reweighted Least Squares (IRLS) algorithms used as a solution in heteroscedastic models [20].", "Note that, in IRLS algorithms, $(S-S_{\\omega }X) $ in equation (REF ) is usually substituted by $(S-S_{\\omega }X) ^{\\top }$ .", "This has consequences on the $X$ minimization step.", "In IRLS, the estimation process is carried out in two steps.", "The first step consists of updating weights, which are usually set to be inversely proportional to variances.", "The second step is the calculation of the LS estimator using the updated weights.", "Many strategies can be used to estimate the variances for the weight matrix, see for example [21], [22], [19].", "The resolution of problem (REF ) with respect to $\\sigma ^2$ for fixed $X$ gives $\\sigma ^2(X)= \\frac{1}{NL}\\text{trace}((S-S_{\\omega }X)\\,C(X)^{-1} (S-S_{\\omega }X)^{\\top })$ Let $W(X) = \\sigma ^2(X)C(X)$ denote the weight matrix of the least squares term in (REF ).", "To solve problem (REF ) with respect to $\\sigma ^2$ and $X$ , we propose to proceed iteratively.", "Let $X^{k}$ be the solution of the previous iteration.", "The first step consists of calculating $W(X^k)$ using equations (REF ) and (REF ).", "In the second step, this updated weight matrix is used to estimate $X^{k+1}$ as follows $\\begin{array}{ll}\\min _{X} & \\frac{1}{2} \\Vert S-S_{\\omega }X\\Vert ^2_{(W^k)^{-1}} + \\mu \\sum _{k=1}^N \\Vert x_{k}\\Vert _2 \\\\\\text{subject to} & X_{ij} \\ge 0 \\quad \\forall \\,i,j \\\\& \\sum _{i=1}^N X_{ij} = 1 \\quad \\forall \\, j\\end{array}$ where $W^k = W(X^k)$ .", "Given $W^k$ , problem (REF ) reduces to a weighted version of GLUP (REF ) due to the weighted norm in the first term.", "The ADMM solution developed in section can be adapted to solve the optimization problem (REF ).", "Minimizing the augmented Lagrangian with respect to $Z$ , and updating the Lagrange multipliers, can by carried out exactly as in Section .", "For concision, only the $X$ -minimization step is described hereafter." ], [ "Minimization of $\\mathcal {L}_\\rho (X,Z^k,\\Lambda ^k)$ with respect to {{formula:2f6856c5-f613-4704-8b6f-dd0da8fc78f4}}", "Omitting the terms that do not depend on $X$ , the minimization of the augmented Lagrangian $\\mathcal {L}_\\rho (X,Z^k,\\Lambda ^k)$ with respect to $X$ leads to $\\begin{array}{ll}\\min _{X} & \\frac{1}{2} \\Vert S-S_{\\omega }X\\Vert ^2_{(W^k)^{-1}} +\\text{trace}(\\Lambda ^\\top (AX)) +\\frac{\\rho }{2} \\Vert AX+BZ-C\\Vert ^2_{\\text{F}}\\end{array}$ Problem (REF ) is quadratic in $X$ and admits an analytical solution obtained by setting the gradient to zero.", "This amounts to solving the Sylvester equation, which has an analytic solution [23] $S_{\\omega }^{\\top }S_{\\omega }X(W^k)^{-1} +\\rho A^{\\top }AX= S_{\\omega }^{\\top }S(W^k)^{-1} - \\rho A^{\\top }\\left(BZ^k-C+\\frac{\\Lambda ^k}{\\rho }\\right)$ Problem (REF ) is not convex.", "An alternating optimization algorithm is more likely to converge to local minima with worse accuracy than the convex version.", "For this reason, we suggest, as a warm start, to initialize NGLUP with GLUP estimate.", "Algorithm REF provides the pseudocode for NGLUP.", "The algorithm contains two main loops.", "The inner loop aims at finding the solution of problem (REF ), whereas the outer loop updates the least-square weight matrix.", ": $X= \\text{NGLUP}(S,S_{\\omega },\\rho ^\\circ ,\\mu ^\\circ ,\\rho ,\\mu )$ [1] Precompute $A$ , $B$ , and $C$ Initialize $X=\\text{GLUP}(S,S_{\\omega },\\rho ^\\circ ,\\mu ^\\circ )$ , $Z=X$ , $\\Lambda =0$ $\\Vert X-X_{\\text{old}} \\Vert _2 \\ge \\epsilon _\\text{tol}$ $C(X)=(I- I_{\\omega }X)^{\\top }(I- I_{\\omega }X)$ $\\sigma ^2(X)= \\frac{1}{NL}\\text{trace}((S-S_{\\omega }X)C(X)^{-1} (S-S_{\\omega }X)^{\\top })$ $W(X)=\\sigma ^2(X)C(X)$ $X^{\\text{old}}=X$ , $J = 1$ ($\\Vert R\\Vert _2 \\ge \\epsilon _{\\text{pri}}$ or $\\Vert P\\Vert _2 \\ge \\epsilon _{\\text{dual}}$ ) and ($J \\le J_{\\text{max}}$ ) $X= $ solution of Sylvester equation (REF ) $Z_{\\text{old}}=Z$ $i=1\\cdots N^{\\prime }$ $v_i = ((x_i)^{\\top } +\\rho ^{-1} \\lambda _i)_{+}$ $ \\Vert v_i\\Vert _2 < \\rho ^{-1}\\mu $ $ z_i= 0$ $z_i = \\left(1 - \\frac{\\mu }{\\rho \\Vert v_i\\Vert _2}\\right) v_i $ $R= AX+ BZ- C$ $P= \\rho AB(Z-Z_{\\text{old}})$ $ \\Lambda = \\Lambda +\\rho (AX+BZ-C) $ $J = J + 1$" ], [ "Synthetic Data", "The performance of GLUP and NGLUP were evaluated using synthetic data.", "We used seven endmembers with 420 spectral samples extracted from the USGS library.", "Figure REF shows the reflectance of the endmembers.", "The spectral mutual coherence between two spectra is defined as $\\theta _{ij}=\\frac{\\langle s_i,s_j\\rangle }{\\Vert s_i\\Vert \\Vert s_j\\Vert }$ .", "The average mutual coherence of the eight endmembers was $\\theta _{\\text{avg}}=0.9171$ .", "The abundances were generated based on a Dirichlet distribution with unit parameter, as a consequence of which the resulting abundances obeyed the non-negativity and sum-to-one constraint, and were uniformly distributed over this simplex.", "Figure: Reflectance of selected endmembers from USGS Library.First, we used three endmembers to generate an hyperspectral data set containing 100 pixels with a SNR of 50 dB.", "The pure pixels were indexed by integers 1–3 for simplicity, the mixed pixels being indexed by integers 4–100.", "We run GLUP algorithm using all the observations ($S_{\\omega }=S$ ) with $\\mu =10$ and $\\rho =100$ .", "The primal and dual tolerances were set to $10^{-5}$ .", "Figure REF shows the mean of each row $x_k$ of the estimated abundance matrix $\\hat{X}$ .", "We observe that the first three pixels can be identified as the endmembers since the mean values of the first three rows are clearly different from zero.", "GLUP was able to provide this result in $4.73$ secondsMachine specifications: 2.2 GHz Intel Core i7 processor and 8 GB RAM with a Root Mean Square Error (RMSE), defined as $\\frac{1}{N^2}\\Vert \\hat{X}-X\\Vert _{\\text{F}}^2$ , equal to $0.0049$ .", "We tested NGLUP in less favorable conditions by increasing the number of endmembers and decreasing the SNR.", "To this end, 7 endmembers were used to generate 93 mixed pixels.", "Data were corrupted with an additive Gaussian noise with a SNR of 20 dB.", "We tested the algorithm for a maximum number of inner iterations $J_{\\text{max}}=1$ , 10 and 100.", "We found that NGLUP converged to the same solution even when the number of inner iterations $J$ was equal to 1.", "For this reason, only one inner iteration per outer iteration was used for the rest of the experiments.", "The running time of the algorithm was 45 seconds.", "Figure REF shows the mean value of each row $x_k$ of the abundance matrix $\\hat{X}$ estimated by GLUP and NGLUP algorithms.", "In both cases, the 7 largest mean values correspond to the 7 endmembers.", "As expected, NGLUP converged to a sparser and more accurate solution than GLUP.", "Figure: Mean value of each row of X ^\\hat{X} estimated with GLUP, obtained with 100 pixels and SNR=50\\text{SNR}=50 dB.Figure: Mean value of each row of X ^\\hat{X}, obtained with 100 pixels and SNR=20\\text{SNR}=20 dB: GLUP (left), NGLUP (right).Table: Probability of detecting M ^\\hat{M} endmembers, using synthetic data generated with M=7M=7 endmembers.We repeated this simulation 100 times.", "For each realization, we examined the number of mean values of the rows of $\\hat{X}$ that were larger than a predefined threshold equal to $0.01$ .", "We considered this value $\\hat{M}$ as the estimated number of endmembers in the scene.", "Table REF provides the probability of detecting $\\hat{M}$ endmembers with our two approaches, given synthetic data generated with $M=7$ endmembers.", "The same task was performed using Virtual Dimensionality (VD) [2].", "We compared the results of NGLUP with those of VD, the probability of false alarm of VD being set to $10^{-3}$ .", "Table REF shows that NGLUP was able to identify the presence of 7 endmembers in $98\\%$ (resp., $96\\%$ ) of the cases with an SNR of 30 dB (resp.", "20 dB).", "VD only identified 2 endmembers in most cases.", "Even with higher values of the SNR, VD did not identify the correct number of endmembers.", "This is due to the fact that VD has asymptotic convergence, and thus requires a very large number of observations in order to converge.", "This explains the poor performance of VD compared to NGLUP." ], [ "Real data", "In this section, we shall evaluate the performance of NGLUP using real hyperspectral data.", "The tests were performed on the so-called images of Pavia University,Available at http://www.ehu.es/ccwintco/index.php/Home provided by the ROSIS imaging spectrometer.", "The scene has a spatial dimension of $610\\times 715$ , that is, a total of $207,400$ pixels with a spatial resolution of $3.7$ meters per pixel.", "Each pixel is composed of 102 spectral samples over the range 430-860 nm.", "Given the high dimension of this data set, a subset $S$ of 300 pixels was randomly selected from the available observations.", "NGLUP was run with $S_w=S$ .", "Based on this subset of observations, we selected those few that best described the whole scene.", "The estimated abundance matrix had a few rows different from zero, pointing out the candidate endmembers.", "Nevertheless, due to the extensive presence of redundant spectra, some rows revealed several occurrences of the same endmember.", "An additional step was thus required to remove redundant spectra among the endmembers determined by our algorithm.", "Several procedures have been proposed in the literature to perform this task.", "For example, in [11], the authors suggest to use K-means clustering in order to choose a subset of independent observations.", "In our experiments, we found it sufficient to impose a maximum value of $0.95$ for the mutual coherence among the estimated endmembers.", "Redundant endmembers according to this criterion were discarded.", "With Pavia University data, this rule gave us 5 distinct endmembers.", "We assumed that these endmembers, obtained from a small subset of the observations, were valid for the whole scene.", "This assumption can be justified by the fact that the image has lots of homogeneous surfaces where the spectral variability is negligible.", "We then used these endmembers and applied the Fully Constrained Least Squares (FCLS) algorithm on the whole data set.", "Figure REF shows the abundance map for every endmember determined by NGLUP.", "The maps successfully describe the urban features of the scene and highlight its topography.", "They cast the pixels as combinations of meadow, tree, shadow, roof, painted metal sheet (with asphalt).", "Finally, we compared the performance of NGLUP with N-FINDR.", "Using the same subset of observations as NGLUP, we determined 5 endmembers with N-FINDR.", "Then, we applied FCLS on the whole image with the endmembers determined by N-FINDR.", "Table REF shows the RMSE, the maximum and average spectral angles obtained for both methods.", "This comparison shows that NGLUP outperformed N-FINDR.", "Figure: Painted metalTable: Performance of NGLUP and N-FINDR for Pavia University data set" ], [ "Conclusion and perspectives", "In this work, we presented two approaches for blind and fully constrained unmixing.", "Both methods are based on mixing models with increasing complexity, and allow to simultaneously determine the endmembers and estimate their local abundance in the scene.", "Compared to the first model called GLUP, the second model NGLUP explicitly considers that endmembers present in the scene are corrupted by noise.", "Experiments on synthetic and real data demonstrated the excellent performance of both approaches.", "Future work includes their extension to an online framework, which would allow to reduce their complexity and to make them adaptive to changing environmental conditions.", "Since problem (REF ) is convex, we simply have to check the validity of the solution in the two cases $\\Vert (v)_+\\Vert _2 > \\alpha $ and $\\Vert (v)_+\\Vert _2 < \\alpha $ .", "Let $f_0(z)=\\frac{1}{2} \\Vert z-v\\Vert _2^2 + \\alpha \\Vert z\\Vert _2$ .", "For $\\Vert (v)_+\\Vert _2 > \\alpha $ , the gradient of $f_0$ is given by $\\nabla f_0(z^\\ast ) = \\left( 1 + \\frac{\\alpha }{\\Vert z^\\ast \\Vert _2}\\right)z^\\ast - v$ Replacing by the appropriate expression from (REF ) yields $&\\nabla f_0(z^\\ast ) = (v)_+ - v\\ge 0 \\\\&z^\\ast _i \\cdot \\nabla f_0(z^\\ast )_i \\propto ((v)_+)_i \\cdot ((v)_+ - v)_i = 0$ These two conditions correspond the optimality conditions, which means that $z\\succeq 0$ is a solution for the constrained problem.", "For more details, refer to section 4.2.3 in [24].", "For the second case, note that for every $z\\succeq 0$ , we have $\\sum _i z_iv_i \\le \\sum _i z_i(v_i)_+ \\le \\Vert z\\Vert _2 \\cdot \\Vert (v)_+\\Vert _2$ It follows that $\\begin{split}f_0(z)-f_0(0) & = \\frac{1}{2}\\sum _i z_i^2 - \\sum _i z_iv_i +\\alpha \\Vert z\\Vert _2 \\\\& \\ge \\frac{1}{2}\\Vert z\\Vert _2^2 -\\Vert z\\Vert _2 \\cdot \\Vert (v)_+\\Vert _2 +\\alpha \\Vert z\\Vert _2 \\\\& \\ge \\frac{1}{2}\\Vert z\\Vert _2^2 +\\Vert z\\Vert _2 ( \\alpha - \\Vert (v)_+\\Vert _2 ) \\end{split}$ This proves that for $\\Vert (v)_+\\Vert _2 \\le \\alpha $ , the minimum is reached for $z^\\ast = 0$ ." ] ]	1403.0289
[ [ "Object Tracking via Non-Euclidean Geometry: A Grassmann Approach" ], [ "Abstract A robust visual tracking system requires an object appearance model that is able to handle occlusion, pose, and illumination variations in the video stream.", "This can be difficult to accomplish when the model is trained using only a single image.", "In this paper, we first propose a tracking approach based on affine subspaces (constructed from several images) which are able to accommodate the abovementioned variations.", "We use affine subspaces not only to represent the object, but also the candidate areas that the object may occupy.", "We furthermore propose a novel approach to measure affine subspace-to-subspace distance via the use of non-Euclidean geometry of Grassmann manifolds.", "The tracking problem is then considered as an inference task in a Markov Chain Monte Carlo framework via particle filtering.", "Quantitative evaluation on challenging video sequences indicates that the proposed approach obtains considerably better performance than several recent state-of-the-art methods such as Tracking-Learning-Detection and MILtrack." ], [ "Introduction", "Visual tracking is a fundamental task in many computer vision applications including event analysis, visual surveillance, human behaviour analysis, and video retrieval [18].", "It is a challenging problem, mainly because the appearance of tracked objects changes over time.", "Designing an appearance model that is robust against intrinsic object variations (shape deformation and pose changes) and extrinsic variations (camera motion, occlusion, illumination changes) has attracted a large body of work [4], [24].", "Rather than relying on object models based on a single training image, more robust models can be obtained through the use of several images, as evidenced by the recent surge of interest in object recognition techniques based on image-set matching.", "Among the many approaches to image-set matching, superior discrimination accuracy, as well as increased robustness to practical issues (such as pose and illumination variations), can be achieved by modelling image-sets as linear subspaces [10], [11], [12], [20], [21], [22].", "In spite of the above observations, we believe modelling via linear spaces is not completely adequate for object tracking.", "We note that all linear subspaces of one specific order have a common origin.", "As such, linear subspaces are theoretically robust against translation, meaning a linear subspace extracted from a set of points does not change if the points are shifted equally.", "While the resulting robustness against small shifts is attractive for object recognition purposes, the task of tracking is to generally maintain precise locations of objects.", "To account for the above problem, in this paper we first propose to model objects, as well as candidate areas that the objects may occupy, through the use of generalised linear subspaces, affine subspaces, where the origin of subspaces can be varied.", "As a result, the tracking problem can be seen as finding the most similar affine subspace in a given frame to the object's affine subspace.", "We furthermore propose a novel approach to measure distances between affine subspaces, via the use of non-Euclidean geometry of Grassmann manifolds, in combination with Mahalanobis distance between the origins of the subspaces.", "See Fig.", "REF for a conceptual illustration of our proposed distance measure.", "Figure: Difference between point-to-subspace and subspace-to-subspace distance measurement approaches.", "(a) Three groups of images, with each image represented as a point in space;the first group (top-left) contains three consecutive object images (frames 1, 2 and 3) used for generating the object model;the second group (bottom-left) contains tracked object images from frames t-2t-2 and t-1t-1;the third group (right) contains three candidate object regions from frame tt.", "(b) Subspace generated based on object images from frames 1, 2 and 3, represented as a dashed line;the minimum point-to-subspace distance can result in selecting the wrong candidate region (wrong location).", "(c) Generated subspaces, represented as points on a Grassmann manifold;the top-left subspace represents the object model;each of the remaining subspaces was generated by using tracked object images from frames t-2t-2 and t-1t-1,with the addition of a unique candidate region from frame tt;using subspace-to-subspace distance is more likely to result in selecting the correct candidate region.To the best of our knowledge, this is the first time that appearance is modelled by affine subspaces for object tracking.", "The proposed approach is somewhat related to adaptive subspace tracking [13], [19].", "Ho [13] represent an object as a point in a linear subspace, which is constantly updated.", "As the subspace was computed using only recent tracking results, the tracker may drift if large appearance changes occur.", "In addition, the location of the tracked object is inferred via measuring point-to-subspace distance, which is in contrast to the proposed method, where a more robust subspace-to-subspace distance is used.", "Ross [19] improved tracking robustness against large appearance changes by modelling objects in a low-dimensional subspace, updated incrementally using all preceding frames.", "Their method also involves a point-to-subspace distance measurement to localise the object.", "The proposed method should not be confused with subspace learning on Grassmann manifolds proposed by Wang [25].", "More specifically, in [25] an online subspace learning scheme using Grassmann manifold geometry is devised to learn/update the subspace of object appearances.", "In contrast to the proposed method, they also consider the point-to-subspace distance to localise objects." ], [ "Proposed Affine Subspace Tracker (AST)", "The proposed Affine Subspace Tracker (AST) is comprised of four components, overviewed below.", "A block diagram of the proposed tracker is shown in Fig.", "REF .", "Motion Estimation.", "This component takes into account the history of object motion in previous frames and creates a set of candidates as to where the object might be found in the new frame.", "To this end, it parameterises the motion of the object between consecutive frames as a distribution via particle filter framework [2].", "Particle filters are sequential Monte Carlo methods and use a set of points to represent the distribution.", "As a result, instead of scanning the whole of the new frame to find the object, only highly probable locations will be examined.", "Candidate Subspaces.", "This module encodes the appearance of a candidate (associated to a particle filter) by an affine subspace $A_{i}^{(t)}$ .", "This is achieved by taking into account the history of tracked images and learning the origin $\\mu _{i}^{(t)}$ and basis $U_{i}^{(t)}$ of $A_{i}^{(t)}$ for each particle.", "Decision Making.", "This module measures the likelihood of each candidate subspace $A_{i}^{(t)}$ to the stored object models in the bag $\\mathcal {M}$ .", "Since object models are encoded by affine subspaces as well, this module determines the similarity between affine subspaces.", "The most similar candidate subspace to the bag $\\mathcal {M}$ is selected as the result of tracking.", "Bag of Models.", "This module keeps a history of previously seen objects in a bag.", "This is primarily driven by the fact that a more robust and flexible tracker can be attained if a history of variations in the object appearance is kept [15].", "To understand the benefit of the bag of models, assume a person tracking is desired where the appearance of whole body is encoded as an object model.", "Moreover, assume at some point of time only the upper body of person is visible (due to partial occlusion) and the tracker has successfully learned the new appearance.", "If the tracking system is only aware of the very last seen appearance (upper-body in our example), upon termination of occlusion, the tracker is likely to lose the object.", "Keeping a set of models (in our example both upper-body and whole body) can help the tracking system to cope with drastic changes.", "Each of the components is elucidated in the following subsections.", "Figure: Block diagram for the proposed Affine Subspace Tracker (AST)." ], [ "Motion Estimation", "In the proposed framework, we are aiming to obtain the location $x \\in \\mathcal {X}$ , $y \\in \\mathcal {Y}$ and the scale $s \\in \\mathcal {S}$ of an object in frame $t$ based on prior knowledge about previous frames.", "A blind search in the space of $\\mathcal {X}-\\mathcal {Y}-\\mathcal {S}$ is obviously inefficient, since not all possible combinations of $x$ , $y$ and $s$ are plausible.", "To efficiently search the $\\mathcal {X}-\\mathcal {Y}-\\mathcal {S}$ space, we use a sequential Monte Carlo method known as the Condensation algorithm [14] to determine which combinations in the $\\mathcal {X}-\\mathcal {Y}-\\mathcal {S}$ space are most probable at time $t$ .", "The key idea is to represent the $\\mathcal {X}-\\mathcal {Y}-\\mathcal {S}$ space by a density function and estimate it through a set of random samples (also known as particles).", "As the number of particles becomes large, the condensation method approaches the optimal Bayesian estimate of density function (combinations in the $\\mathcal {X}-\\mathcal {Y}-\\mathcal {S}$ space).", "Below, we briefly describe how the condensation algorithm is used within the proposed tracking approach.", "Let $\\mathcal {Z}^{(t)} = \\left(x^{(t)},y^{(t)},s^{(t)}\\right)$ denote a particle at time $t$ .", "By the virtue of the principle of importance sampling [2], the density of $\\mathcal {X}-\\mathcal {Y}-\\mathcal {S}$ space (or most probable candidates) at time $t$ is estimated as a set of $N$ particles $\\lbrace \\mathcal {Z}_{i}^{(t)}\\rbrace _{i=1}^{N}$ using previous particles $\\lbrace \\mathcal {Z}^{(t-1)}_{i}\\rbrace _{i=1}^{N}$ and their associated weights $\\lbrace w^{(t-1)}_{i}\\rbrace _{i=1}^{N}$ with $\\sum \\nolimits _{i=1}^N w^{(t-1)}_{i} = 1$.", "For now we assume the associated weights of particles are known and later discuss how they can be determined.", "In the condensation algorithm, to generate $\\lbrace \\mathcal {Z}_{i}^{(t)}\\rbrace _{i=1}^{N}$, $\\lbrace \\mathcal {Z}^{(t-1)}_{i}\\rbrace _{i=1}^{N}$ is first sampled (with replacement) $N$ times.", "The probability of choosing a given element $\\mathcal {Z}^{(t-1)}_{i}$ is equal to the associated weight $w^{(t-1)}_{i}$.", "Therefore, the particles with high weights might be selected several times, leading to identical copies of elements in the new set.", "Others with relatively low weights may not be chosen at all.", "Next, each chosen element undergoes an independent Brownian motion step.", "Here, the Brownian motion of a particle is modelled by a Gaussian distribution with a diagonal covariance matrix.", "As a result, for a chosen particle $\\mathcal {Z}^{(t-1)}_{\\ast }$ from the first step of condensation algorithm, a new particle $\\mathcal {Z}^{(t)}_{\\ast }$ is obtained as a random sample of $\\mathcal {N}\\left(\\mathcal {Z}^{(t-1)}_,\\Sigma \\right)$ where $\\mathcal {N}\\left(\\mu ,\\Sigma \\right)$ denotes a Gaussian distribution with mean $\\mu $ and covariance $\\Sigma $.", "The covariance $\\Sigma $ governs the speed of motion, and is a constant parameter over time in our framework." ], [ "Candidate Templates", "To accommodate variations in object appearance, this module models the appearance of particlesWe loosely use “particle appearance” to mean the appearance of a candidate template described by a particle.", "by affine subspaces (see Fig.", "REF for a conceptual example).", "An affine subspace is a subset of Euclidean space [23], formally described by a 2-tuple {$\\mu ,U$ } as: ${A} = \\left\\lbrace {z} \\in \\mathbb {R}^{D}: {z} = \\mu + {U} {y} \\right\\rbrace $ where $\\mu \\in \\mathbb {R}^{D}$ and ${U} \\in \\mathbb {R}^{D \\times n}$ are origin and basis of the subspace, respectively.", "Let ${I}(\\mathcal {Z}^{(t)}_{\\ast },t)$ denote the vector representation of an $N_{1} \\times N_{2}$ patch extracted from frame $t$ by considering the values of particle $\\mathcal {Z}^{(t)}_{\\ast }$.", "That is, frame $t$ is first scaled appropriately based on the value $s^{(t)}_\\ast $ and then a patch of $N_{1} \\times N_{2}$ pixels with the top left corner located at $\\left(x^{(t)}_\\ast ,y^{(t)}_\\ast \\right)$ is extracted.", "Figure: In the proposed approach, object appearance is modelled by an affine subspace.", "An affine subspace is uniquely described by its origin μ\\mu and basis UU.", "Here, μ\\mu and basis UU are obtained by computing mean and eigenbasis of a set of object images.The appearance model for $\\mathcal {Z}^{(t)}_{\\ast }$ is generated from a set of $P+1$ images by considering $P$ previous results of tracking.", "More specifically, let $\\widehat{\\mathcal {Z}}^{(t)}$ denote the result of tracking at time $t$ , $\\widehat{\\mathcal {Z}}^{(t)}$ is the most similar particle to the bag of models at time $t$ .", "Then set $\\mathbb {B}_{\\mathcal {Z}_{\\ast }}^{(t)} \\mbox{~=~}\\left\\lbrace {I}(\\widehat{\\mathcal {Z}}^{(t-P)},t \\mbox{~--~} P),{I}(\\widehat{\\mathcal {Z}}^{(t-P+1)},t \\mbox{~--~} P+1),\\cdots , {I}(\\mathcal {Z}^{(t)}_{\\ast },t) \\right\\rbrace $ is used to obtain the appearance model for particle $\\mathcal {Z}^{(t)}_{\\ast }$.", "More specifically, the origin of affine subspace associated to $\\mathcal {Z}^{(t)}_{\\ast }$ is the mean of $\\mathbb {B}_{\\mathcal {Z}_{\\ast }}^{(t)}$.", "The basis is obtained by computing the Singular Value Decomposition (SVD) of $\\mathbb {B}_{\\mathcal {Z}_{\\ast }}^{(t)}$ and choosing the $n$ dominant left-singular vectors." ], [ "Bag of Models", "Although affine subspaces accommodate object changes along with a set of images, to produce a robust tracker, the object's model should be able to reflect the appearance changes during the tracking process.", "Accordingly, we propose to keep a set of object models $m_{j}=\\lbrace \\mu _{j},U_{j}\\rbrace $ for coping with deformations, pose variations, occlusions, and other variations of the object during tracking.", "Fig.", "REF shows two frames with a tracked object, the bag models used to localise the object, and the recent images of the image set used to generate each bag model.", "Figure: (a) Two examples of a frame with a tracked object.", "(b) The first eigenbasis of ten sample template bags.", "(c) The recent frame in each of the 10 image sets used to generate the templates.A bag $\\mathcal {M} = \\lbrace m_{1},\\cdots ,m_{k}\\rbrace $ is defined as a set of $k$ object models, each $m_j$ is an affine subspace learned during the tracking process.", "The bag is updated every $W$ frames (see Fig.", "REF ) by replacing the oldest model with the latest learned model ( latest result of tracking specified by $\\widehat{\\mathcal {Z}}^{(t)}$).", "The size of bag $k$ determines the memory of the tracking system.", "Thus, a large bag with several models might be required to track an object in a challenging scenario.", "In our experiments, a bag of size 10 with the updating rate $W = 5$ is used in all experiments.", "Figure: The model extraction procedure involves a sliding window update scheme.The template is learned from a set of P consecutive frames.Template update occurs every W frames.Having a set of models at our disposal, we will next address how the similarity between a particle's appearance and the bag can be determined." ], [ "Decision Making", "Given the previously learned affine subspaces as the input to this module, the aim is to find the nearest affine subspace to the bag templates.", "Although the minimal Euclidean distance is the simplest distance measure between two affine subspaces (the minimum distance of any pair of points of the two subspaces), this measure does not form a metric [5] and it does not consider the angular distance between affine subspaces, which can be a useful discriminator [16].", "However, the angular distance ignores the origin of affine subspaces and simplifies the problem to a linear subspace case, which we wish to avoid.", "To address the above limitations, we propose a distance measure with the following form: $\\operatorname{dist}({A}_{i},{A}_{j})\\mbox{~=~}\\operatorname{dist}_G\\left({U}_{i},{U}_{j}\\right)+\\alpha ({\\mu }_{i} - {\\mu }_{j})^T{M}({\\mu }_{i}-{\\mu }_{j})$ where $\\operatorname{dist}_G$ is the Geodesic distance between two points on a Grassmann manifold [7], $({\\mu }_{i}-{\\mu }_{j})^T{M}({\\mu }_{i}-{\\mu }_{j})$ is the Mahalanobis distance between origins of ${A}_{i}$ and ${A}_{j}$ , and $\\alpha $ is a mixing weight.", "The components in the proposed distance are described below.", "A Grassmann manifold (a special type of Riemannian manifold) is defined as the space of all $n$ -dimensional linear subspaces of $\\mathbb {R}^D$ for $0<n<D$.", "A point on Grassmann manifold $\\mathcal {G}_{D,n}$ is represented by an orthonormal basis through a $D \\times n$ matrix.", "The length of the shortest smooth curve connecting two points on a manifold is known as the geodesic distance.", "For Grassmann manifolds, the geodesic distance is given by: $\\operatorname{dist}_G\\left({X},{Y}\\right)=\\Vert \\Theta \\Vert _2$ where $\\Theta =[\\theta _1,\\theta _2,\\cdots ,\\theta _n]$ is the principal angle vector, $\\cos (\\theta _l)=\\max _{{x} \\in {X},{y} \\in {Y}}{x}^T {y} = {x}_l^T{y}_l$ subject to $\\left\\Vert {x} \\right\\Vert \\mbox{~=~} \\left\\Vert {x} \\right\\Vert \\mbox{~=~} 1$, ${x}^T {x}_i \\mbox{~=~} {y}^T {y}_i \\mbox{~=~} 0$, $i \\mbox{~=~} 1, \\ldots , l-1$.", "The principal angles have the property of $\\theta _i \\in [0, \\pi /2]$ and can be computed through the SVD of ${X}^T {Y}$ [7].", "We note that the linear combination of a Grassmann distance (distance between linear subspaces) and Mahalanobis distance (between origins) of two affine subspaces has roots in probabilistic subspace distances [9].", "More specifically, consider two normal distributions $\\mathcal {N}_1\\left({\\mu }_1,{C}_1\\right)$ and $\\mathcal {N}_2\\left({\\mu }_2,{C}_2\\right)$ with ${C}_{i} = \\sigma ^{2}\\mathbb {I} + {U}_{i}{U}_{i}^T$ as the covariance matrix, and ${\\mu }_{i}$ as the mean vector.", "The symmetric Kullback-Leibler (KL) distance between $\\mathcal {N}_1$ and $\\mathcal {N}_2$ under orthonormality condition (${U}_{i}^T {U}_{i}=\\mathbb {I}_{n}$) results in: $J_{KL} & \\mbox{=} &\\frac{1}{2\\sigma ^2}({\\mu }_{1} \\mbox{--} {\\mu }_{2})^T\\left(2\\mathbb {I}_{D} \\mbox{~--~} {U}_{1}{U}_{1}^T \\mbox{~--~} {U}_{2} {U}_{2}^T\\right)({\\mu }_{1} \\mbox{--} {\\mu }_{2}) \\nonumber \\\\& & + ~ \\frac{1}{2\\sigma ^2(\\sigma ^2+1)}\\left(2n - 2\\mbox{tr}({U}_{1}^T {U}_{2}{U}_{2}^T {U}_{1})\\right)$ The term $\\mbox{tr}({U}_{1}^T {U}_{2}{U}_{2}^T {U}_{1})$ in $J_{KL}$ is identified as the projection distance on Grassmann manifold $\\mathcal {G}_{D,n}$ (defined as $\\operatorname{dist}_{Proj}\\left({U}_1,{U}_2\\right)=\\Vert sin(\\Theta )\\Vert _2$) [9], and the term $({\\mu }_{1}-{\\mu }_{2})^T\\left(2\\mathbb {I}_{D} - {U}_{1}{U}_{1}^T - {U}_{2} {U}_{2}^T\\right)({\\mu }_{1}-{\\mu }_{2})$ is the Mahalanobis distance with ${M} = 2\\mathbb {I}_{D} - {U}_{1}{U}_{1}^T - {U}_{2} {U}_{2}^T$.", "Since the geodesic distance is a more natural choice for measuring lengths on Grassmann manifolds (compared to the projection distance), we have elected to combine it with the Mahalanobis distance from (REF ), resulting in the following instantiation of the general form given in Eqn.", "(REF ): $\\operatorname{dist}({A}_{i},{A}_{j}) \\hspace{-6.375pt}& \\mbox{=} &\\hspace{-6.375pt} \\operatorname{dist}_{G}({U}_{i},{U}_{j})\\nonumber \\\\& &\\hspace{-6.375pt} + ~\\alpha ({\\mu }_{i} \\mbox{~--~} {\\mu }_{j})^T\\left(2\\mathbb {I}_{D} \\mbox{~--~} {U}_{i}{U}_{i}^T \\mbox{~--~} {U}_{j} {U}_{j}^T\\right)({\\mu }_{i} \\mbox{~--~} {\\mu }_{j})$ We measure the likelihood of a candidate subspace $A_{i}^{(t)}$ , given template $m_{j}$ , as follows: $p \\left( A_{i}^{(t)}\|m_{j} \\right) = \\exp \\left(\\frac{-\\operatorname{dist}(A_{i}^{(t)},m_{j})}{\\sigma }\\right) $ where $\\sigma $ indicates the standard deviation of the likelihood function and is a parameter in the tracking framework.", "The likelihoods are normalised such that $\\sum _{i=1}^{N}p(A_{i}^{(t)}\|m_{j})=1$.", "To measure the likelihood between a candidate affine subspace $A_{i}^{(t)}$ and bag $\\mathcal {M}$, the individual likelihoods between $A_{i}^{(t)}$ and bag templates $m_{j}$ should be integrated.", "Based on [17], we opt for the sum rule: $p(A_{i}^{(t)}\|\\mathcal {M}) = \\sum \\nolimits _j^k{p(A_{i}^{(t)}\|m_{j})}$ The object state is then estimated as: $\\widehat{\\mathcal {Z}}^{(t)} = \\mathcal {Z}_j^{(t)},~~~~~\\mbox{where}~~~j = \\underset{i}{\\operatorname{argmax}}~~p(A_{i}^{(t)}\|\\mathcal {M})$ [!b] : Affine Subspace Tracking [1] New frame, a set of updated candidate object states from the last frame, and the previous $\\small {P-1}$ estimated object states $\\lbrace \\widehat{\\mathcal {Z}}^{(\\tau )}\\rbrace _{\\tau = t-P+1}^{t-1}$ Initialisation: $t = 1:P$ Set the initial object state $\\widehat{\\mathcal {Z}}^{(t)}$ in the first $\\small P$ frames.", "Use a single state to indicate the location.", "Begin: Select candidate object states according to the dynamic model $\\lbrace \\mathcal {Z}_{i}^{(t)}\\rbrace _{i=1}^{N}$ For each sample, extract the corresponding image patch For each $\\mathcal {Z}^{(t)}_{i}$ do: Generate the affine subspace $A_{i}^{(t)}\\lbrace \\mu _{i}^{(t)},U_{i}^{(t)}\\rbrace $ based on image regions corresponding to $\\mathcal {Z}^{(t)}_{i}$ and $\\lbrace \\widehat{\\mathcal {Z}}^{(\\tau )}\\rbrace _{\\tau = t-P+1}^{t-1}$ Calculate the likelihoods given each template in the bag by Eqn.", "(REF ) Compute the final likelihoods using Eqn.", "(REF ) Determine the object state $\\widehat{\\mathcal {Z}}^{(t)}$ by Maximum Likelihood (ML) estimation Update the existing candidate object states according to their probabilities [14] current object state $\\widehat{\\mathcal {Z}}^{(t)}$" ], [ "Computational Complexity", "The computational complexity of the proposed tracking framework can be associated with generating a new model and comparing a target candidate with a model.", "The model generation step requires $O(D^3+2Dn)$ operations.", "Computing the geodesic distance between two points on $G_{D,n}$ requires $O((D+1)n^2+n^3)$ operations.", "Therefore, comparing an affine subspace candidate against each bag template needs $O((2n+3)D^2+(n^2+1)D+n^3+n^2)$ operations." ], [ "Experiments", "In this section we evaluate and analyse the performance of the proposed AST method using eight publicly available videosThe videos and the corresponding ground truth are available at http://vision.ucsd.edu/bbabenko/project_miltrack.shtml consisting of two main tracking tasks: face and object tracking.", "The sequences are: Occluded Face [1], Occluded Face 2 [4], Girl [6], Tiger 1 [4], Tiger 2 [4], Coke Can [4], Surfer [4], and Coupon Book [4].", "Example frames from several videos are shown in Fig.", "REF .", "Each video is composed of 8-bit grayscale images, resized to $320 \\times 240$ pixels.", "We used the raw pixel values as image features.", "For the sake of computational efficiency in the affine subspace representation, we resized each candidate image region to $32 \\times 32$, and the number of eigenvectors ($n$ ) used in all experiments is set to three.", "Furthermore, we only consider 2D translation and scaling in the motion modelling component.", "The batch size ($W$) for the template update is set to five as a trade-off between computational efficiency and effectiveness of modelling appearance change during fast motion.", "We evaluated the proposed tracker based on (i) average center location error, and (ii) precision [4].", "Precision shows the percentage of frames for which the estimated object location is within a threshold distance of the ground truth.", "Following [4], we use a fixed threshold of 20 pixels.", "To contrast the effect of affine subspace modelling against linear subspaces, we assessed the performance of the AST tracker against a tracker that only exploits linear subspaces, , an AST where $\\mu = 0$ for all models.", "The results, in terms of center location errors, are shown in Table REF.", "The proposed AST method significantly outperforms the linear subspaces approach, thereby confirming our idea of affine subspace modelling.", "Table: Performance comparison between tracking based on affine and linear subspaces,in terms of average center location errors (pixels)." ], [ "Quantitative Comparison", "To assess and contrast the performance of AST tracker against state-of-the-art methods, we consider six methods, here.", "The competitors are: fragment-based tracker (FragTrack) [1], multiple instance boosting-based tracker (MILTrack) [4], [3], online Adaboost (OAB) [8], tracking-learning-detection (TLD) [15], incremental visual tracking (IVT) [19], and Sparsity-based Collaborative Model tracker (SCM) [26].", "We use the publicly available source codes for FragTrackhttp://www.cs.technion.ac.il/amita/fragtrack/fragtrack.htm, MILTrackhttp://vision.ucsd.edu/ bbabenko/project_miltrack.shtml, OAB[2], TLDhttp://info.ee.surrey.ac.uk/Personal/Z.Kalal/, IVThttp://www.cs.toronto.edu/dross/ivt/ and SCMhttp://ice.dlut.edu.cn/lu/Project/cvpr12_scm/cvpr12_scm.htm.", "Tables REF and REF show the performance in terms of precision and location error, respectively, for the proposed AST method as well as the competing trackers.", "Fig.", "REF shows resulting bounding boxes for several frames from the Surfer, Coupon Book, Occluded Face 2 and Girl sequences.", "On average, the proposed AST method obtains notably better performance than the competing trackers, with TLD being the second best tracker.", "Table: Precision at a fixed threshold of 20, as per .Best performance is indicated by \\ast ,while second best is indicated by **\\ast \\ast .The higher the precision, the better." ], [ "Qualitative Comparison", "Heavy occlusions.", "Occlusion is one of the major issues in object tracking.", "Trackers such as SCM, FragTrack and IVT are designed to resolve this problem.", "Other trackers, including TLD, MIL and OAB, are less successful in handling occlusions, especially at frames 271, 529 and 741 of the Occluded Face sequence, and frames 176, 432 and 607 of Occluded Face 2.", "SCM can obtain good performance mainly as it is capable of handling partial occlusions via a patch-based model.", "The proposed AST approach can tolerate occlusions to some extent, thanks to the properties of the appearance model.", "One prime example is Occluded Face 2, where AST accurately localised the severely occluded object at frame 730.", "Pose Variations.", "On the Tiger 2 sequence, most trackers, including SCM, IVT and FragTrack, fail to track the object from the early frames onwards.", "On Tiger 2, the proposed AST approach can accurately follow the object at frames 207 and 271 when all the other trackers have failed.", "In addition, compared to the other trackers, the proposed approach partly handles motion blurring (frame 344), where the blurring is a side-effect of rapid pose variations.", "On Tiger 1, although TLD obtains the best performance, AST can successfully locate (in contrast to the other trackers) the object at frames 204 and 249, which are subject to occlusion and severe illumination changes.", "Rotations.", "The Girl and Surfer sequences include drastic out-of-plane and in-plane rotations.", "On Surfer, FragTrack and SCM fail to track from the start.", "The proposed AST approach consistently tracked the surfer and outperforms the other trackers.", "On Girl, the IVT, OAB, and FragTrack methods fail to track in many frames.", "While IVT is able to track in the beginning, it fails after frame 230.", "The AST approach manages to track the correct person throughout the whole sequence, especially towards the end where the other trackers fail due to heavy occlusion.", "Illumination changes.", "The Coke Can sequence consists of dramatic illumination changes.", "FragTrack fails from frame 20 where the first signs of illumination changes appear.", "IVT and OAB fail from frame 40 where the frames include both severe illumination changes and slight motion blur.", "MILTrack fails after frame 179 where a part of the object is almost faded by the light.", "Since affine subspaces accommodate robustness to the illumination changes, the proposed AST approach can accurately locate the object throughout the whole sequence.", "Imposters/Distractors.", "The Coupon Book sequence contains a severe appearance change, as well as an imposter book to distract the tracker.", "FragTrack and TLD fail mainly where the imposter book appears.", "AST successfully tracks the correct book with notably better accuracy than the other methods." ], [ "Main Findings and Future Directions", "In this paper we investigated the problem of object tracking in a video stream where object appearance can drastically change due to factors such as occlusions and/or variations in illumination and pose.", "The selection of subspaces for target representation purposes, in addition to a regular subspace update, are mainly driven by the need for an adaptive object template reflecting appearance changes.", "We argued that modelling the appearance by affine subspaces and applying this notion on both the object templates and the query data leads to more robustness.", "Furthermore, we maintain a record of $k$ previously observed templates for a more robust tracker.", "We also presented a novel subspace-to-subspace measurement approach by reformulating the problem over Grassmann manifolds, which provides the target representation with more robustness against intrinsic and extrinsic variations.", "Finally, the tracking problem was considered as an inference task in a Markov Chain Monte Carlo framework using particle filters to propagate sample distributions over time.", "Comparative evaluation on challenging video sequences against several state-of-the-art trackers show that the proposed AST approach obtains superior accuracy, effectiveness and consistency, with respect to illumination changes, partial occlusions, and various appearance changes.", "Unlike the other methods, AST involves no training phase.", "There are several challenges, such as drifts and motion blurring, that need to be addressed.", "A solution to drifts could be to formulate the update process in a semi-supervised fashion in addition to including a training stage for the detector.", "Future research directions also include an enhancement to the updating scheme by measuring the effectiveness of a new learned model before adding it to the bag of models.", "To resolve the motion blurring issues, we can enhance the framework by introducing blur-driven models and particle filter distributions.", "Furthermore, an interesting extension would be multi-object tracking and how to join multiple object models." ], [ "Acknowledgements", "NICTA is funded by the Australian Government through the Department of Communications and the Australian Research Council through the ICT Centre of Excellence program." ] ]	1403.0309
[ [ "A hypergeometric basis for the Alpert multiresolution analysis" ], [ "Abstract We construct an explicit orthonormal basis of piecewise ${}_{i+1}F_{i}$ hypergeometric polynomials for the Alpert multiresolution analysis.", "The Fourier transform of each basis function is written in terms of ${}_2F_3$ hypergeometric functions.", "Moreover, the entries in the matrix equation connecting the wavelets with the scaling functions are shown to be balanced ${}_4 F_3$ hypergeometric functions evaluated at $1$, which allows to compute them recursively via three-term recurrence relations.", "The above results lead to a variety of new interesting identities and orthogonality relations reminiscent to classical identities of higher-order hypergeometric functions and orthogonality relations of Wigner $6j$-symbols." ], [ "Introduction", "Wavelet theory has had proved useful in many areas of mathematics and engineering such as functional analysis, Fourier analysis, and signal processing.", "The Alpert multiresolution analysis is associated with the spline spaces of piecewise polynomials of degree at most $n$ , discontinuous at the integers.", "The generators for this multiresolution analysis are the Legendre polynomials, restricted and scaled to $[0,1)$ and set to zero otherwise.", "Using the symmetry inherent in this system, Alpert [1], [2] constructed a set of wavelet functions and then used them to analyze various integral operators (see also [3]).", "This basis has been used in [6] with the moment interpolating technique to construct smooth multiwavelets.", "An interesting problem from both computational and theoretical point of view is to find a wavelet basis which can be written in terms of explicit formulas.", "In [12] an analysis was performed on the coefficients in the refinement equation satisfied by the modified Legendre polynomials and it was shown that the entries in these matrices could be written as multiples of certain generalized Jacobi polynomials evaluated at $1/2$ and that these entries satisfy generalized eigenvalue equations.", "Moreover a new basis of wavelets was implicitly introduced through the matrix equation connecting the wavelets to the scaling functions by considering upper triangular matrices with positive diagonal entries.", "In the present paper we provide explicit formulas for these wavelets, their Fourier transforms, and related matrix coefficients in terms of hypergeometric functions.", "We give a different construction and direct proofs of the characteristic properties of this new basis thus making the paper self-contained.", "Our results imply new identities between higher-order hypergeometric functions and suggest interesting connections to representation theory.", "The paper is organized as follows.", "In Section we set the notation and review the elements of multiresolution analysis needed for the sequel.", "In Section we postulate the orthogonality and symmetry conditions which characterize the wavelet functions.", "We show that these properties are satisfied by a sequence of piecewise polynomials supported on $(-1,1)$ , which on $(-1,0)$ and $(0,1)$ can be written as ${}_{i+1}F_{i}$ hypergeometric functions.", "We also exhibit families of differential equations satisfied by these functions.", "In Section we explain the differences between the wavelets constructed here and the ones introduced by Alpert.", "We also prove that the entries in the matrices relating our wavelets to the scaling functions can be written as balanced ${}_4 F_3$ hypergeometric functions evaluated at 1.", "In particular, these formulas imply that the matrices are upper triangular with positive diagonal entries, thus relating the formulas here to the implicit construction in [12].", "We also indicate how these results are reminiscent to classical identities of higher-order hypergeometric functions and orthogonality relations of Wigner $6j$ -symbols.", "In Section we derive a simple closed formula for the Fourier transform of these wavelets in terms of ${}_2F_3$ hypergeometric functions and we write associated differential equations for them.", "Finally, in Section , we give recurrence formulas for the entries in the matrices of the wavelet equation." ], [ "Preliminaries", "Let $\\phi _0,\\dots ,\\phi _r$ be compactly supported $L^2$ -functions, and suppose that $V_0 = {\\rm cl}_{L^2}\\,{\\rm span}\\lbrace \\phi _i({\\cdot }-j): i = 0,1,\\dots ,r,\\ j\\in {\\mathbb {Z}}\\rbrace $ .", "Then $V_0$ is called a finitely generated shift invariant (FSI) space.", "Let $(V_p)_{p\\in {\\mathbb {Z}}}$ be given by $V_p = \\lbrace \\phi (2^p{\\cdot }): \\phi \\in V_0\\rbrace $ .", "Each space $V_p$ may be thought of as approximating $L^2$ at a different resolution depending on the value of $p$ .", "The sequence $(V_p)$ is called a multiresolution analysis (MRA) [5], [11], [13] generated by $\\phi _0,\\dots ,\\phi _r$ if (a) the spaces are nested, $\\cdots \\subset V_{-1}\\subset V_0\\subset V_1\\subset \\cdots $ , and (b) the generators $\\phi _0,\\dots ,\\phi _r$ and their integer translates form a Riesz basis for $V_0$ .", "Because of (a) and (b) above, we can write $V_{j+1} = V_j \\oplus W_j, \\qquad \\forall j\\in {\\mathbb {Z}}.$ The space $W_0$ is called the wavelet space, and if $\\psi _0,\\dots ,\\psi _r$ generate a shift-invariant basis for $W_0$ , then these functions are called wavelet functions.", "If, in addition, $\\phi _0,\\dots ,\\phi _r$ and their integer translates form an orthogonal basis for $V_0$ , then $(V_p)$ is called an orthogonal MRA.", "Let $S_{-1}^n$ be the space of polynomial splines of degree $n$ continuous except perhaps at the integers, and set $V_0^n = S_{-1}^n \\cap L^2({\\mathbb {R}})$ .", "With $V_p^n$ as above these spaces form a multiresolution analysis.", "If $n=0$ the multiresolution analysis obtained is associated with the Haar wavelet while for $n>0$ they were introduced by Alpert [1], [2].", "Let $\\phi _j(t)= \\hat{p}_j(2t-1)\\chi _{[0,1)}(t)$ where $\\hat{p}_j(t)$ is the Legendre polynomial [18] of degree $j$ orthonormal on $[-1,1]$ with positive leading coefficient i.e.", "$\\hat{p}_j(t)=k_j t^j +\\text{lower degree terms}$ , with $k_j>0$ and $\\int _{-1}^1 \\hat{p}_j(t)\\hat{p}_k(t)dt=\\delta _{k,j}.$ These polynomials have the following representation in terms of a ${}_2 F_1$ hypergeometric function [18], $\\hat{p}_n(t) = \\frac{\\sqrt{2n+1}}{\\sqrt{2}}\\,{}_2 F_{1}\\left({-n,\\ n+1\\atop 1}; \\ \\frac{1-t}{2}\\right),$ where formally, ${}_p F_{q}\\left({a_1,\\ \\dots \\ a_p\\atop b_1,\\ \\dots \\ b_q}; \\ t\\right)= \\sum _{i=0}^{\\infty }\\frac{(a_1)_i\\dots (a_p)_i}{(b_1)_i\\dots (b_q)_i(1)_i}t^i$ with $(a)_0=1$ and $(a)_i = a(a+1)\\ldots (a+i-1)$ for $i>0$ .", "Since one of the numerator parameters in the definition of $\\hat{p}_n$ is a negative integer, the series in equation (REF ) has only finitely many terms.", "With the normalization taken above $\|\|\\phi _j\|\|^2_{{L^2({\\mathbb {R}})}}=\\frac{1}{2}$ , and $\\Phi _n=\\left[\\begin{matrix}\\phi _0&\\cdots &\\phi _n\\end{matrix}\\right]^T$ and its integer translates form an orthogonal basis for $V_0$ .", "For the convenience in later computations we set $P_n(t)=\\left[\\begin{matrix}\\hat{p}_0(t)\\\\\\vdots \\\\\\hat{p}_n(t)\\end{matrix}\\right].$ Equation (REF ) implies the existence of the refinement equation, $\\Phi _n\\left(\\frac{t}{2}\\right)=C^n_{-1}\\Phi _n(t)+C^n_1\\Phi _n(t-1),$ see [5], [12].", "In order to exploit the symmetry of the Legendre polynomials we shift $t\\rightarrow t+1$ so that $\\Phi _n\\left(\\frac{t+1}{2}\\right)&=P_n(t)\\chi _{[-1,1)}(t)=C^n_{-1}\\Phi _n(t+1)+C^n_1\\Phi _n(t)\\nonumber \\\\&=C^n_{-1}P_n(2t+1)\\chi _{[-1,0)}(t)+C^n_1P_n(2t-1)\\chi _{[0,1)}(t).$" ], [ "Construction of the functions", "We want to construct a set $\\lbrace h^n_0,\\ldots ,h^n_n\\rbrace $ of functions such that, On $[-1,0)$ and $[0,1)$ , $h^n_i$ is a polynomial of degree at most $n$ , $\\int _{-1}^1 t^s h^n_i(t)dt=0$ , $0\\le s\\le n,\\ 0\\le i\\le n$ , $\\int _{-1}^1 h^n_i(t) h^n_j(t)dt=2\\delta _{i,j}, \\quad 0\\le i, j\\le n$ , $h^n_i(-t)=(-1)^{i+1} h^n_i(t)$ for $t\\in (0,1)$ , and $\\int _0^1 t^s h^n_i(t) dt =0,\\quad s< i$ .", "We will show below that these functions have a hypergeometric representation and use them in Theorem REF to construct a wavelet basis for ${L^2({\\mathbb {R}})}$ .", "Theorem 1 For $0\\le t<1$ , we have $&h_{n-i}^n(t)=\\sum _{k=0}^n d^i_{n,k} t^k, \\qquad \\text{if $i=0$ or $i$ odd} \\\\&h_{n-i}^n(t)=\\sum _{k=0}^{n-1} d^{i-1}_{n-1,k} t^k, \\qquad \\text{for $i>0$ even,}$ where $d^i_{n,k}&=(-1)^n\\frac{\\sqrt{2n-2i+1}}{(-n)_i}\\frac{(-n)_k(n-i+1)_k}{(1)_k(1)_k}\\nonumber \\\\&\\times \\prod _{m=0}^{\\frac{i-1}{2}}(n+k+1-2m)\\prod _{m=0}^{\\frac{i-3}{2}}(n-k-1-2m),\\ 0\\le k\\le n,\\ i=0,\\ \\text{or}\\ i\\ \\text{odd.", "}$ We extend $h_{n-i}^n(t)$ on $[-1,0)$ using (iv).", "In formula (REF ) and throughout the paper we use the convention that an empty product is equal to 1.", "Remark 2 We note that conditions (i) through (iv) are the same as those imposed by Alpert in his wavelet construction.", "To fix a basis he imposes more vanishing moments on $[-1,1)$ (see [2]).", "We were not able to find a hypergeometric representation for his functions.", "Remark 3 From the above formula it follows that the polynomials $\\frac{(-n)_i h^n_{n-i}(t)}{\\sqrt{2n-2i+1}}$ for $i=0$ or $i$ odd and $\\frac{(-n+1)_{i-1} h^n_{n-i}(t)}{\\sqrt{2n-2i+1}}$ for $i>0$ even have integer coefficients.", "We prove this theorem after developing a few lemmas.", "We begin with, Lemma 4 For $i\\le n$ $\\sum _{k=0}^{n}\\frac{(-n)_k}{k!}\\binom{n-i+k}{n-i}\\frac{1}{x+k}=\\frac{n!", "(-x+1)_{n-i}}{(n-i)!", "(x)_{n+1}}.$ In particular, when $x=n-i+1$ we have $\\sum _{k=0}^{n}\\frac{(-n)_k}{k!}\\binom{n-i+k}{k}\\frac{1}{n-i+k+1}=(-1)^{n+i}\\frac{n!", "(n-i)!}{(2n-i+1)!", "}.$ Consider a polynomial $f(x)$ of degree at most $n$ .", "The Lagrange interpolation formula at the points $x_k=-k$ , $k=0,1,\\dots , n$ gives $(x)_{n+1}\\sum _{k=0}^{n}\\frac{(-1)^k}{k!(n-k)!", "}\\frac{f(-k)}{x+k}=f(x),$ or equivalently $\\sum _{k=0}^{n}\\frac{(-n)_k}{k!", "}\\frac{f(-k)}{x+k}=\\frac{n!f(x)}{(x)_{n+1}}.$ Applying the above formula with $f(x)=\\binom{n-i-x}{n-i}$ we find $\\sum _{k=0}^{n}\\frac{(-n)_k}{k!}\\binom{n-i+k}{n-i}\\frac{1}{x+k}=\\frac{n!", "(-x+1)_{n-i}}{(n-i)!", "(x)_{n+1}},$ which gives the result.", "It is easy to see, Lemma 5 For $k\\ge n-i+2$ and $n-k$ odd $ d^i_{n,k}=0$ .", "Also Lemma 6 If $i+l<n$ and $l\\ge -1$ then for any polynomial $p(k)$ with deg $p\\le i+l$ , $\\sum _{k=0}^n\\frac{(-n)_k(n-i+1)_k}{(1)_k(l+2)_k}p(k)=0.$ By linearity, it is enough to show (REF ) for the polynomials $p(k,r)=k(k-1)\\cdots (k-r+1)$ with $0\\le r\\le i+l$ .", "If the common factor in the numerator and denominator are canceled and the change of variable $k=r+m$ made, the above sum is equal to $\\sum _{k=0}^n\\frac{(-n)_k(n-i+1)_k}{(1)_k(l+2)_k}p(k,r)&=c_{n,r,i,l}\\sum _{m=0}^{n-r}\\frac{(-n+r)_m(n-i+r+1)_m}{(1)_m(r+l+2)_m}\\\\&=c_{n,r,i,l}\\,{}_2 F_{1}\\left({-n+r,\\ n-i+r+1\\atop r+l+2}; \\ 1\\right)\\\\&=c_{n,r,i,l}\\frac{(-n+i+l+1)_{n-r}}{(r+l+2)_{n-r}}=0,$ where $c_{n,r,i,l}=\\frac{(-n)_{r}(n-i+1)_{r}}{(l+2)_{r}}$ .", "The Chu-Vandermonde identity was used to obtain the last line in the above equation.", "Note first that (iv) implies that (iii) will automatically hold if $i$ and $j$ have opposite parity (i.e.", "one of them is odd and the other one is even).", "This shows that the constructions of the two sets of functions $\\lbrace h^{n}_{n-i}\\rbrace _{i\\text{-odd}}$ and $\\lbrace h^{n}_{n-i}\\rbrace _{i\\text{-even}}$ are independent of each other.", "Moreover, we have $\\int _{-1}^1 t^s h^n_{n-i}(t) dt=((-1)^{n-i+s+1}+1)\\int _{0}^1 t^s h^n_{n-i}(t) dt.$ It is easy to see now that if the functions $\\lbrace h^{n}_{n-i}\\rbrace _{i\\text{-odd}}$ defined by (REF ) and (iv) satisfy (i), (ii), (iii) and (v), then the functions $\\lbrace h^{n}_{n-i}\\rbrace _{i>0\\text{ even}}$ defined by () and (iv) also satisfy (i), (ii), (iii) and (v) (because $h^{n}_{n-i}(t)=h^{n-1}_{n-1-(i-1)}(t)$ for $i>0$ even and $t\\in (0,1)$ ).", "If $i=0$ , then (REF ) and (REF ) show that $h^n_n(t)=(-1)^{n}\\sqrt{2}\\hat{p}_n(1-2t)$ for $t\\in [0,1)$ , where $\\hat{p}_n(t)$ is Legendre polynomial (REF ) and the required properties of $h^n_n(t)$ follow easily from the properties of $\\hat{p}_n(t)$ .", "It remains to show that for $i$ odd the functions defined by (REF ) satisfy the required conditions, so we fix $i$ odd below.", "Formula (REF ) yields $\\int _0^1 t^s h^n_{n-i}(t)dt =\\sum _{k=0}^n d_{n,k}^i \\frac{1}{k+s+1}=c_{n,i}\\sum _{k=0}^n\\frac{(-n)_k(n-i+1)_k}{(1)_k(s+2)_k}q(k),$ where $c_{n,i}=(-1)^n\\frac{\\sqrt{2n-2i+1}}{(-n)_i(s+1)!", "}$ and $q(k)=\\prod _{j=1}^s(k+j)\\prod _{m=0}^{\\frac{i-1}{2}}(n+k+1-2m)\\prod _{m=0}^{\\frac{i-3}{2}}(n-k-1-2m).$ For $s<n-i$ the degree of $q(k)$ is $s+i< n$ and we deduce from Lemma REF that the right-hand side of (REF ) is equal to zero which gives (v).", "For (ii), we need to show only that $\\int _{0}^1 t^s h^n_{n-i}(t) dt=0$ when $s>n-i$ and $s$ and $n-i$ have opposite parity (the rest follows from (REF ) and (v)).", "For such $s$ we find $\\int _{0}^1 t^s h^n_{n-i}(t) dt=\\sum _{k=0}^n d^i_{n,k} \\frac{1}{k+s+1}.$ Note that $k+s+1$ cancels one of the factors in $\\prod _{m=0}^{\\frac{i-1}{2}}(n+k+1-2m)$ so the right-hand side of (REF ) is equal to zero from Lemma REF with $l=-1$ .", "This completes the proof of (ii).", "Finally, we want to show that (iii) holds when $i$ and $j$ are odd.", "Suppose first that $i$ and $j$ are odd and $i< j$ .", "Using (v) we see that $\\int _0^1h^n_{n-j}(t)h^n_{n-i}(t)dt&=\\sum _{k=n-i}^nd_{n,k}^j\\int _0^1 t^k h^n_{n-i}(t)dt\\\\&=\\sum _{n-k\\ \\text{even}}d_{n,k}^j\\int _0^1 t^k h^n_{n-i}(t)dt+\\sum _{n-k\\ \\text{odd}}d_{n,k}^j\\int _0^1 t^k h^n_{n-i}(t)dt.$ In the sums on the last line of the above equation $n-i\\le k \\le n$ .", "Since in the second sum $n-k$ is odd, Lemma REF shows that $d_{n,k}^j=0$ while in the first sum $k>n-i$ and $k$ and $n-i$ are of opposite parity so the discussion after (REF ) shows that the integral is equal to zero.", "We now show that $\\int _0^1h^n_{n-i}(t)^2 dt=1.$ With the substitution of (REF ) we find from above that $\\int _0^1h^n_{n-i}(t)^2 dt&=d^i_{n,n-i}\\int _0^1t^{n-i}h^n_{n-i}(t)dt\\\\&=\\frac{(-1)^n\\sqrt{2n-2i+1}d^i_{n,n-i}}{(-n)_i}\\sum _{k=0}^n\\frac{(-n)_k(n-i+1)_k}{(1)_k(1)_k}\\frac{\\pi _i(k)}{n-i+k+1},$ where $\\pi (k)$ is the polynomial in $k$ of degree $i$ given by $\\pi _i(k)=\\prod _{m=0}^{\\frac{i-1}{2}}(n+k+1-2m)\\prod _{m=0}^{\\frac{i-3}{2}}(n-k-1-2m).$ If we add and subtract $\\pi _i(-n+i-1)/(n-i+k+1))$ to the summand then using Lemma REF with $s=-1$ and the fact that $(\\pi _i(k)-\\pi _i(-n+i-1))/(n-i+k+1))$ is a polynomial of degree $i-1$ in $k$ we find $\\int _0^1h^n_{n-i}(t)^2dt=(-1)^{i+n}\\frac{(2n-i+1)!}{n!(n-i)!", "}\\sum _{k=0}^n\\frac{(-n)_k(n-i+1)_k}{(1)_k(1)_k(n-i+k+1)}.$ The proof now follows from equation (REF ) in Lemma REF .", "Remark 7 From the explicit formulas given in Theorem REF it is clear that $h_{i}^n(t)$ are hypergeometric functions although some care is needed in the hypergeometric representation since some of the coefficients vanish (see Lemma REF ).", "If we set $n_{\\epsilon }=n+\\epsilon $ , then for $i>0$ odd we can rewrite formula (REF ) as $\\begin{split}h_{n-i}^n(t)=&\\frac{(-1)^n\\sqrt{2n-2i+1}\\,(n+1)}{(-n)_i}\\prod _{m=0}^{\\frac{i-3}{2}}(n-1-2m)^2\\\\&\\qquad \\times \\lim _{\\epsilon \\rightarrow 0}{}_{i+2}F_{i+1}\\left(\\begin{matrix} -n,\\; \\alpha _1,\\; \\beta _1 \\\\1, \\;\\alpha _0,\\; \\beta _0 \\end{matrix}\\,; t\\right),\\end{split}$ where $\\alpha _1=\\lbrace n_{\\epsilon }-i+1,n_{\\epsilon }-i+3,\\dots ,n_{\\epsilon }+2\\rbrace $ , $\\alpha _0=\\lbrace n_{\\epsilon }-i+2,n_{\\epsilon }-i+4,\\dots ,n_{\\epsilon }+1\\rbrace $ , $\\beta _1=\\lbrace -n_{\\epsilon }+2,-n_{\\epsilon }+4,\\dots ,-n_{\\epsilon }+i-1\\rbrace $ , $\\beta _0=\\lbrace -n_{\\epsilon }+1,-n_{\\epsilon }+3,\\dots ,-n_{\\epsilon }+i-2\\rbrace $ .", "A similar formula can be written for $i$ even using ().", "Remark 8 From the theory of generalized hypergeometric functions we know that the function $F={}_i F_{j}\\left({a_1,\\ \\dots \\ a_i\\atop b_1,\\ \\dots \\ b_j}; \\ t\\right)$ satisfies the differential equation $\\left[D(D+b_1-1)\\cdots (D+b_{j}-1)-t(D+a_1)\\cdots (D+a_{i})\\right]F=0$ where $D=t\\frac{d}{dt}$ , see [17].", "Thus if $i$ is odd, $t\\in (0,1)$ we can use (REF ) to see that $h_{n-i}^n(t)$ satisfies the differential equation $\\big [t(1-t)\\frac{d^2}{dt^2}+(1-(i+2)t)\\frac{d}{dt}+(n-i+1)(n+2)\\big ]\\mathcal {L}_i h_{n-i}^n(t)=0,$ where $\\mathcal {L}_i=\\prod _{m=0}^{\\frac{i-3}{2}}(D-n+2m)\\prod _{m=0}^{\\frac{i-1}{2}}(D+n-i+1+2m).$ The use of equation (REF ) shows that $\\mathcal {L}_i h_{n-i}^n(t)&=(-1)^{n+\\frac{i-1}{2}}(-n-1)_i\\sqrt{2n-2i+1}\\,{}_2 F_{1}\\left({-n+i-1,\\ n+2\\atop 1}; \\ t\\right)\\\\&=c_i \\hat{p}^{0,i}_{n-i+1}(1-2t),$ where $\\hat{p}^{0,i}_{n-i+1}(t)$ is the orthonormal Jacobi polynomial on $[-1,1]$ .", "This implies that for fixed $i$ , the functions $\\mathcal {L}_i h_{n-i}^n(t)$ are orthogonal on $[0,1)$ for $n\\ge i$ .", "Using [15] and the specific form of the hypergeometric representation in (REF ), it is possible to derive even lower-order differential equations for $h_{n-i}^n(t)$ , but they also correspond to generalized eigenvalue equations." ], [ "Wavelets for the Alpert multiresolution", "As mentioned in section 2 the functions $\\Phi _n$ generate a multiresolution analysis with $V_0={\\rm cl}_{{L^2({\\mathbb {R}})}}{\\rm span}\\lbrace \\phi _j(\\cdot -i),\\ i\\in {\\mathbb {Z}},\\ j=0\\ldots ,n\\rbrace $ where the function $\\phi _j(\\frac{t+1}{2})$ when restricted to $[-1,1)$ is the normalized Legendre polynomial $\\hat{p}_j(t)$ .", "In equation (REF ) the wavelet space $W_0$ is the orthogonal complement of $V_0$ in $V_1$ .", "We look for a set $\\Psi _n=[\\psi ^n_0,\\cdots , \\psi ^n_n]^T$ of functions in ${L^2({\\mathbb {R}})}$ , each supported on $[0,1)$ , whose integer translates provide a basis for $W_0$ .", "Using properties (i)-(iv) of the functions $h^{n}_{i}(t)$ and a standard argument (see for instance [2]) we see that if we choose $\\psi ^n_j(t)=h^n_j(2t-1)\\chi _{[0,1)}(t),$ then we obtain a basis for ${L^2({\\mathbb {R}})}$ by dilations and translations.", "Theorem 9 The set of functions $\\lbrace 2^{\\frac{k}{2}} \\psi ^n_j(2^k\\cdot -i),i,k\\in {\\mathbb {Z}}, j=0,\\ldots ,n \\rbrace $ forms an orthonormal, compactly supported, piecewise polynomial basis for ${L^2({\\mathbb {R}})}$ .", "On the interval $[0,1)$ we have Theorem 10 The set of functions $\\lbrace \\sqrt{2}\\phi _j,\\ 2^{\\frac{k}{2}} \\psi ^n_j(2^k\\cdot -i),k\\in {\\mathbb {Z}}^+,\\ i=0,\\ldots , 2^k-1,\\ j=0,\\ldots ,n \\rbrace $ forms an orthonormal, piecewise polynomial basis for $L^2([0,1])$ .", "From the theory of multiresolution analysis, there are $(n+1)\\times (n+1)$ matrices $D^n_{-1}$ and $D^n_1$ such that the following equations $\\Psi _n\\left(\\frac{t+1}{2}\\right)&=D^n_{-1}\\Phi _n(t+1)+D^n_1\\Phi _n(t)\\nonumber \\\\&=D^n_{-1}P_n(2t+1)\\chi _{[-1,0)}(t)+D^n_1P_n(2t-1)\\chi _{[0,1)}(t)$ hold, where $D^n_1=2\\int _0^1\\Psi _n\\left(\\frac{t+1}{2}\\right) P_n(2t-1) dt,$ and $D^n_{-1}=2\\int _{-1}^0\\Psi _n\\left(\\frac{t+1}{2}\\right)P_n(2t+1) dt.$ Thus $(D^n_1)_{i,j}=2\\int _0^1\\psi ^n_i\\left(\\frac{t+1}{2}\\right)\\hat{p}_j(2t-1) dt=2\\int _0^1 h^n_i(t)\\hat{p}_j(2t-1) dt,\\ 0\\le i,j\\le n .$ These formulas and property (v) show that $D^n_1$ is an upper triangular matrix and as we will see below the diagonal entries are positive.", "From the orthogonality properties of the above functions we find $4I_n=D_{-1}^n(D_{-1}^n)^T+D_{1}^n(D_{1}^n)^T.$ We note that the right-hand side of the above equation differs by a factor of 2 from equation (5) in [12].", "This is due to the fact that we have normalized the components of $\\Psi _n$ to be orthonormal.", "The symmetry properties of the wavelet functions give $(D^n_{-1})_{i,j}=(-1)^{i+j+1}(D^n_1)_{i,j},$ which combined with (REF ) leads to the orthogonality relations $0=((-1)^{i+k}+1)\\sum _{j=i}^n(D^n_1)_{i,j}(D^n_1)_{k,j}, \\ k<i,$ and $2=\\sum _{j=i}^n(D^n_1)_{i,j}(D^n_1)_{i,j}.$ Using the representation developed above for $\\psi ^n_j$ we will show, Theorem 11 The nonzero entries in $D^n_1$ are given as follows: $(D^n_1)_{n,n}=\\sqrt{2},$ for $i$ odd and $j\\le i$ , $(D^n_1)_{n-i,n-j}&=c_{n,i,j}(-1)^{n+j}\\sqrt{2(n-i)+1}\\sqrt{2(n-j)+1}\\nonumber \\\\&\\times {}_4 F_{3}\\left({-\\frac{i-j}{2},\\ \\frac{j-i+1}{2},\\ n-\\frac{i+j-1}{2},\\ n-\\frac{i+j}{2}+1\\atop n-\\frac{i}{2}+\\frac{3}{2},\\ -\\frac{i}{2},\\ n-i+\\frac{3}{2}}; \\ 1\\right),$ where $c_{n,i,j}&=2^{i/2}\\,\\frac{i!!(n-i+\\frac{3}{2})_{\\frac{i-1}{2}}(-n+j)_{n-i}(n-j+1)_{n-i}}{(2n-i+1)!", "},$ while for $i$ even and positive and $j\\le i$ , $(D^n_1)_{n-i,n-j}=(D^{n-1}_1)_{n-i,n-j}.$ Remark 12 Note that the ${}_4 F_3$ hypergeometric functions are balanced and satisfy the orthogonality equations (REF )-(REF ).", "From the explicit formulas above we see that the diagonal entries are positive.", "As shown in [12] the orthogonality, upper triangularity, and positivity of the diagonal entries uniquely specify the matrix $D^n_1$ .", "We prove the result for $i=0$ and $i$ odd since for $i>0$ even the formula follows from the properties of the wavelets.", "For $i=0$ or $i$ odd equation (REF ) in Theorem REF and condition (v) give, $(D^n_1)_{n-i,n-j}=(-1)^{n-j}\\sqrt{2}\\sqrt{2(n-j)+1}\\sum _{s=n-i}^{n-j}\\sum _{k=0}^n\\frac{(-n+j)_s(n-j+1)_s}{(1)_s(1)_s}\\frac{d^i_{n,k}}{k+s+1}.$ Lemma REF yields $\\sum _{k=0}^n\\frac{(-n)_k(n-i+1)_k}{(1)_k(1)_k}\\frac{1}{k+s+1}=\\frac{n!(-s)_{n-i}}{(n-i)!", "(s+1)_{n+1}}=(-1)^{i}\\frac{(-n)_{i}(-s)_{n-i}}{(s+1)_{n+1}}$ which coupled with Lemma REF shows that $\\sum _{k=0}^n\\frac{d^i_{n,k}}{k+s+1}=\\frac{(-1)^{n+i}\\sqrt{2n-2i+1}\\,(-s)_{n-i}}{(s+1)_{n+1}}\\prod _{m=0}^{\\frac{i-1}{2}}(n-s-2m)\\prod _{m=0}^{\\frac{i-3}{2}}(n+s-2m).$ Thus $(D^n_1)_{n-i,n-j}&=(-1)^{n+j}\\sqrt{2}\\sqrt{2(n-i)+1}\\sqrt{2(n-j)+1}\\sum _{s=n-i}^{n-j}\\frac{(-n+j)_s(n-j+1)_s}{(s+n+1)!(s-n+i)!", "}\\\\&\\times \\prod _{m=0}^{\\frac{i-1}{2}}(n-s-2m)\\prod _{m=0}^{\\frac{i-3}{2}}(n+s-2m),$ where the fact that $\\frac{(-s)_{n-i}}{(1)_s(1)_s(s+1)_{n+1}}=\\frac{(-1)^{n-i}}{(s+n+1)!(s-n+i)!", "}$ has been used.", "Making the change of variable $s=n-i+l$ yields, $(D^n_1)_{n-i,n-j}&=(-1)^{n+j}\\sqrt{2}\\sqrt{2(n-i)+1}\\sqrt{2(n-j)+1}\\frac{(-n+j)_{n-i}(n-j+1)_{n-i}}{(2n-i+1)!", "}\\\\&\\times \\sum _{l=0}^{i-j}\\frac{(j-i)_l(2n-i-j+1)_l}{(2n-i+2)_l l!", "}\\prod _{m=0}^{\\frac{i-1}{2}}(i-l-2m)\\prod _{m=0}^{\\frac{i-3}{2}}(2n-i+l-2m).$ Note that the terms in the above sum are zero if $l$ is odd (here $i$ is odd) since the first product starts from $i-l$ which is positive and even and ends up with $-l+1$ which is negative or zero.", "Using the identity $(a)_{2l}=2^{2l}\\left(\\frac{a}{2}\\right)_l\\left(\\frac{a+1}{2}\\right)_l$ for $(j-i)_{2l}$ , $(2l)!$ , $(2n-i+2)_{2l}$ and $(2n-i-j+1)_{2l}$ together with the identities, $&\\frac{\\prod _{m=0}^{\\frac{i-1}{2}}(i-2l-2m)}{(\\frac{1}{2})_l}=\\frac{i!!", "}{(\\frac{-i}{2})_l}$ and $&\\frac{\\prod _{m=0}^{\\frac{i-3}{2}}(2n-i+2l-2m)}{(n-\\frac{i}{2}+1)_l}=2^{\\frac{i-1}{2}}\\frac{(n-i+\\frac{3}{2})_{\\frac{i-1}{2}}}{(n-i+\\frac{3}{2})_l}$ give the result.", "Because of the way that the indices enter in the above formulas it is not so simple to obtain recurrence relations.", "For this reason we obtain another representation.", "Lemma 13 Suppose $j<i$ .", "For $i$ odd and $j$ even, $(D^n_1)_{n-i,n-j}&=(-1)^{\\frac{i-j+1}{2}}\\frac{(j+1)!!(i-j-2)!!", "}{2^{i/2}(n-\\frac{i+j-1}{2})_{\\frac{i+1}{2}}}\\sqrt{2(n-i)+1}\\sqrt{2(n-j)+1}\\nonumber \\\\&\\times {}_4 F_{3}\\left({-\\frac{j}{2},\\ \\frac{j-i+1}{2},\\ n+\\frac{3-j}{2},\\ -n+\\frac{i+j}{2}\\atop 1,\\ \\frac{1}{2},\\ \\frac{3}{2}}; \\ 1\\right),$ for $i,\\ j$ odd, $&(D^n_1)_{n-i,n-j}=(-1)^{\\frac{i-j}{2}+1}\\frac{(j)!!(i-j-1)!!", "}{2^{\\frac{i+2}{2}}(n-\\frac{i+j-2}{2})_{\\frac{i+1}{2}}}\\sqrt{2(n-i)+1}\\sqrt{2(n-j)+1}\\nonumber \\\\&\\quad \\times (2n-j+2)(j+1)\\,{}_4 F_{3}\\left({-\\frac{j-1}{2},\\ \\frac{j-i}{2}+1,\\ n-\\frac{j}{2}+2,\\ -n+\\frac{i+j+1}{2}\\atop 2,\\ \\frac{3}{2},\\ \\frac{3}{2}}; \\ 1\\right),$ while for the remaining coefficients use equation (REF ).", "We only consider the case when $i$ is odd and $j$ is even.", "The Whipple transformation of a balanced ${}_4F_3$ hypergeometric function is the following $&{}_4 F_{3}\\left({-n,\\ x,\\ y,\\ z\\atop u,\\ v,\\ w}; \\ 1\\right)\\\\&=\\frac{(1-v+z-n)_n(1-w+z-n)_n}{(v)_n(w)_n}{}_4 F_{3}\\left({-n,\\ u-x,\\ u-y,\\ z\\atop u,\\ 1-v+z-n,\\ 1-w+z-n}; \\ 1\\right).$ Thus with $i$ odd, $j$ even, $n=\\frac{i-j-1}{2}$ , $u=-i/2$ , and $z= n-\\frac{i+j}{2}+1$ , we find $&{}_4 F_{3}\\left({-\\frac{i-j}{2},\\ \\frac{j-i+1}{2},\\ n-\\frac{i+j-1}{2},\\ n-\\frac{i+j}{2}+1\\atop n-\\frac{i}{2}+\\frac{3}{2},\\ -\\frac{i}{2},\\ n-i+\\frac{3}{2}}; \\ 1\\right)\\\\&=\\frac{(1)_{\\frac{i-j-1}{2}}(-\\frac{i}{2}+1)_{\\frac{i-j-1}{2}}}{(n-\\frac{i-3}{2})_{\\frac{i-j-1}{2}}(n-i+\\frac{3}{2})_{\\frac{i-j-1}{2}}}{}_4 F_{3}\\left({-\\frac{j}{2},\\ \\frac{j-i+1}{2},\\ -n+\\frac{j-1}{2},\\ n-\\frac{i+j}{2}+1\\atop 1,\\ -\\frac{i}{2},\\ -\\frac{i}{2}+1}; \\ 1\\right).$ An application of this formula to equation (REF ) and the identity $\\left(n-\\frac{i+j-1}{2}\\right)_{\\frac{j+2}{2}}\\left(n-\\frac{i-3}{2}\\right)_{\\frac{i-j-1}{2}}=\\left(n-\\frac{i+j-1}{2}\\right)_{\\frac{i+1}{2}},$ yields $(D^n_1)_{n-i,n-j}&=(-1)^{j-i}\\sqrt{2}\\frac{2^{\\frac{i-3-2j}{2}}i!!(1)_{\\frac{i-j-1}{2}}(-\\frac{i}{2}+1)_{\\frac{i-j-1}{2}}}{(n-\\frac{i+j-1}{2})_{\\frac{i+1}{2}}(i-j)!", "}\\sqrt{2(n-i)+1}\\sqrt{2(n-j)+1}\\nonumber \\\\&\\times {}_4 F_{3}\\left({-\\frac{j}{2},\\ \\frac{j-i+1}{2},\\ -n+\\frac{j-1}{2},\\ n-\\frac{i+j}{2}+1\\atop 1,\\ -\\frac{i}{2},\\ -\\frac{i}{2}+1}; \\ 1\\right).$ Another use of Whipple's transformation with $n= \\frac{j}{2}$ , $z=\\frac{j-i+1}{2}$ , and $u=1$ gives the result.", "Remark 14 Besides the orthogonality relations (REF )-(REF ), equations (REF ) and (REF ) imply the relations, $2I_{n+1}=C^n_{-1}{C^n_{-1}}^T+C^n_1{C^n_1}^T,$ and $C^n_{-1} {D^n_{-1}}^T + C^n_1 {D^n_{1}}^T=0.$ Using the symmetry property (REF ) and equation (15) of [12] we see that that the matrix $A$ composed of the even rows of $C^n_1$ and the odd rows of $\\frac{D^n_1}{\\sqrt{2}}$ or vice versa is unitary which yields orthogonality relations among the entries of $C^n_1$ , and $\\frac{D^n_1}{\\sqrt{2}}$ .", "It is interesting to compare these orthogonality relations with other known orthogonality relations for ${}_4F_3$ series, and in particular with the orthogonality of the Wigner $6j$ -Symbols and Racah polynomials, see [19] or the book [16].", "However, we could not relate the orthogonality relations above to this theory.", "Providing a Lie-theoretic interpretation of these new orthogonality equations is a very interesting problem.", "Another challenging problem is to connect the orthogonality relations here to an appropriate extension of the Fields and Wimp expansion formula [10]." ], [ "Fourier Transform", "An important tool in wavelet theory is the Fourier transform given by, $\\hat{\\psi }^n_k(\\theta )=\\int _{0}^1\\psi ^n_k(t)e^{-i\\theta t} dt.$ In order to compute this Fourier transform we will focus on $\\hat{h}^n_k(\\theta )=\\int _{-1}^1 h^n_k(t)e^{-i\\theta t} dt,$ since equation (REF ) implies $\\hat{\\psi }^{n}_{k}(\\theta )=\\frac{e^{-i\\theta /2}}{2}\\hat{h}^n_k(\\theta /2)$ .", "Theorem 15 For $j$ odd $\\hat{h}^n_k(\\theta )$ is given by $\\hat{h}^n_{n-j}(\\theta )=&\\frac{(-1)^{\\frac{j+1}{2}}2^{j+1}(\\frac{j+1}{2})!(n+1)!(n+2)!}{(j+2)!(2n+3)!(n+\\frac{3}{2}-\\frac{j}{2})!", "}\\sqrt{2n-2j+1}(-i\\theta )^{n+2}\\nonumber \\\\&\\qquad \\times {}_2 F_{3}\\left({\\frac{n+3}{2},\\ \\frac{n+4}{2}\\atop \\frac{j+4}{2},\\ n+\\frac{5}{2},\\ n+\\frac{5-j}{2}}; \\ -\\frac{\\theta ^2}{4}\\right),$ while for $j$ even, $\\hat{h}^n_{n-j}(\\theta )=\\hat{h}^{n-1}_{n-j}(\\theta )$ .", "Remark 16 It is remarkable that while the wavelet functions are limits of higher-order hypergeometric functions, their Fourier transforms have a simple closed formula in terms of ${}_2F_3$ series.", "Since $h^n_k(t)$ is a polynomial of degree $n$ on $[0,1)$ , Euler's formula and integration by parts show that $\\hat{h}^n_{n-j}(\\theta )$ can be written in terms of $\\sin {\\theta }$ and $\\cos {\\theta }$ multiplied by polynomials in $1/\\theta $ of degree $n+1$ .", "This is also true of the wavelets constructed by Alpert et al.", "We prove the formula for $j=0$ or $j$ odd since for $j$ even and positive the result follows from equation ().", "From the symmetry properties of $h^n_{n-j}$ we find $\\int _{-1}^1 h^n_{n-j}(t)f(t)dt=\\int _{0}^1[(-1)^{n-j+1}f(-t)+f(t)] h^n_{n-j}(t)dt,$ so that $\\hat{h}^n_{n-j}(\\theta )=\\int _{0}^1[(-1)^{n-j+1}e^{i\\theta t}+e^{-i\\theta t}] h^n_{n-j}(t)dt.$ From property (ii), $\\int _{-1}^1 h^n_k(t) t^s dt=0,$ for $0\\le s\\le n$ .", "Thus only the moments for $s>n$ and $s\\equiv n-j+1\\mod {2}$ are nonzero.", "For such $s$ we find $&\\int _0^1 h^n_{n-j}(t) t^s dt=\\sum _{k=0}^n d^j_{n,k}\\frac{1}{k+s+1}\\\\&=(-1)^n\\frac{\\sqrt{2n-2j+1}}{(-n)_j}\\sum _{k=0}^n\\frac{(-n)_k(n-j+1)_k}{(k!", ")^2}\\frac{\\prod _{m=0}^{\\frac{j-1}{2}}(n+k+1-2m)\\prod _{m=0}^{\\frac{i-3}{2}}(n-k-1-2m)}{k+s+1}.$ By Lemma REF , $k$ can be replaced by $-(s+1)$ in the above products which gives $\\int _0^1 h^n_{n-j}(t) t^s dt&=(-1)^n\\frac{\\sqrt{2n-2j+1}}{(-n)_j}\\prod _{m=0}^{\\frac{j-1}{2}}(n-s-2m)\\prod _{m=0}^{\\frac{j-3}{2}}(n+s-2m)\\\\&\\times \\sum _{k=0}^n\\frac{(-n)_k(n-j+1)_k}{(k!", ")^2}\\frac{1}{k+s+1}.$ The use of Lemma REF with $x=s+1$ shows that the above sum is equal to $\\frac{(-n)_j(-1)^j(-s)_{n-j}}{(s+1)_{n+1}}$ so that $\\int _0^1 h^n_{n-j}(t) t^s dt=(-1)^{n+j}\\frac{\\sqrt{2n-2j+1}(-s)_{n-j}}{(s+1)_{n+1}}\\prod _{m=0}^{\\frac{j-1}{2}}(n-s-2m)\\prod _{m=0}^{\\frac{j-3}{2}}(n+s-2m).$ The identity $\\frac{(-s)_{n-j}(-1)^{n+j}}{s!}=\\frac{1}{(s-n+j)!", "}$ allows the above integral to be rewritten as $\\int _0^1 h^n_{n-j}(t)\\frac{t^s}{s!}", "dt=\\frac{\\sqrt{2n-2j+1}}{(s-n+j)!", "(s+1)_{n+1}}\\prod _{m=0}^{\\frac{j-1}{2}}(n-s-2m)\\prod _{m=0}^{\\frac{j-3}{2}}(n+s-2m).$ For $j>0$ the change of variables $s=n+2+2r,\\ r=0,1,\\ldots $ yields $&\\int _{-1}^1 h^n_{n-j}(t)\\frac{t^{n+2+2r}}{(n+2+2r)!}", "dt\\\\&=\\frac{2^{j+1}(-1)^{\\frac{j+1}{2}}\\sqrt{2n-2j+1}}{(2+2r+j)!", "(n+3+2r)_{n+1}}\\prod _{m=0}^{\\frac{j-1}{2}}(m+r+1)\\prod _{m=0}^{\\frac{j-3}{2}}(n+1+r-m).$ This can be simplified using $\\frac{1}{(2+j+2r)!}=\\frac{1}{(2+j)!(3+j)_{2r}}=\\frac{1}{(2+j)!", "(\\frac{3+j}{2})_r(\\frac{4+j}{2})_r 2^{2r}}$ , $\\frac{1}{(n+3+2r)_{n+1}}=\\frac{(n+2)!}{(2n+3)!", "}\\frac{(\\frac{n+3}{2})_r(\\frac{n+4}{2})_{r}}{(n+2)_r(n+\\frac{5}{2})_r},$ $\\prod _{m=0}^{\\frac{j-1}{2}}(m+r+1)=\\frac{(\\frac{j+1}{2}+r)!}{r!}=\\frac{(\\frac{j+1}{2})!(\\frac{j+3}{2})_r}{r!", "},$ and $\\prod _{m=0}^{\\frac{j-3}{2}}(n+r+1-m)=\\frac{(n+1+r)!}{(n+\\frac{3}{2}-\\frac{j}{2}+r)!}=\\frac{(n+1)!(n+2)_r}{(n+\\frac{3}{2}-\\frac{j}{2})!", "(n+\\frac{5}{2}-\\frac{j}{2})_r},$ to obtain $&\\int _{-1}^1 h^n_{n-j}(t)\\frac{t^{n+2+2r}}{(n+2+2r)!}", "dt\\\\\\nonumber \\qquad &=\\frac{(-1)^{\\frac{j+1}{2}}2^{j+1}(\\frac{j+1}{2})!(n+1)!(n+2)!}{(j+2)!(2n+3)!(n+\\frac{3}{2}-\\frac{j}{2})!}\\sqrt{2n-2j+1}\\frac{(\\frac{n+3}{2})_r(\\frac{n+4}{2})_{r}}{r!", "(\\frac{4+j}{2})_r(n+\\frac{5}{2})_r(n+\\frac{5}{2}-\\frac{j}{2})_r}\\frac{1}{4^r}.$ Thus $\\hat{h}^n_{n-j}(\\theta )&=\\sum _{s=0}^{\\infty }\\int _{-1}^1 h^n_{n-j}(t)(-i\\theta )^s \\frac{t^s}{s!}", "dt\\\\&=(-i\\theta )^{n+2}\\sum _{r=0}^{\\infty }(-i\\theta )^{2r}\\int _{-1}^1 h^n_{n-j}(t)\\frac{t^{n+2+2r}}{(n+2+2r)!}", "dt$ and the result for $j$ odd is obtained by the substitution of (REF ) in the above formula.", "For $j=0$ the substitution $s=n+1+2r$ in equation (REF ) shows that $\\hat{h}^n_n=\\hat{h}^{n-1}_n$ which completes the proof.", "Remark 17 From the differential equation given in Remark REF and (REF ) we obtain differential equations for $\\hat{h}^n_{n-j}$ .", "With $D_{\\theta }:=\\frac{\\theta }{2}\\frac{d}{d\\theta }=\\theta ^2\\frac{d}{d\\theta ^2}$ and since $D_{\\theta }\\theta ^n f(\\theta )=\\frac{n}{2} \\theta ^n f(\\theta )+\\theta ^n D_{\\theta }f(\\theta )$ we find for $j$ odd $&\\bigg [\\left(D_{\\theta }-\\frac{n+2}{2}\\right)\\left(D_{\\theta }+\\frac{n+1}{2}\\right)\\left(D_{\\theta }+\\frac{j-n}{2}\\right)\\left(D_{\\theta }+\\frac{n+1-j}{2}\\right)\\\\\\nonumber &+\\frac{\\theta ^2}{4}\\left(D_{\\theta }+\\frac{1}{2}\\right)\\left(D_{\\theta }+1\\right)\\bigg ]\\hat{h}^n_{n-j}(\\theta )=0,$ while for $j$ even replace $n$ by $n-1$ and $j$ by $j-1$ in the above formula.", "Remark 18 The formula for $\\hat{h}^n_{n-j}$ allows the development of asymptotic formulas for large $n$ and $j$ .", "We will not systematically explore this here but be content to give a simple example in the case when $j=tn$ , $0<t\\le 1$ .", "In this case for $j$ odd $\\hat{h}^n_{(1-t)n}(\\theta )=&\\frac{(-1)^{\\frac{tn+1}{2}}2^{tn+1}(\\frac{tn+1}{2})!(n+1)!(n+2)!}{(tn+2)!(2n+3)!(n+\\frac{3}{2}-\\frac{tn}{2})!", "}\\sqrt{2n-2tn+1}(-i\\theta )^{n+2}\\nonumber \\\\&\\qquad \\times {}_2 F_{3}\\left({\\frac{n+3}{2},\\ \\frac{n+4}{2}\\atop \\frac{tn+4}{2},\\ n+\\frac{5}{2},\\ n+\\frac{5-tn}{2}}; \\ -\\frac{\\theta ^2}{4}\\right).$ The use of Stirling's formula (with the help of Mathematica) shows that $&\\frac{(\\frac{tn+1}{2})!(n+1)!(n+2)!}{(tn+2)!(2n+3)!(n+\\frac{3}{2}-\\frac{tn}{2})!", "}\\nonumber \\\\&=e^n(4n)^{-n}\\left(1-\\frac{t}{2}\\right)^{-n(1-\\frac{t}{2})-2}\\frac{(2t)^{-\\frac{nt}{2}}}{16\\sqrt{2}t^{3/2}n^3}\\left(1-\\frac{104-12t+27t^2}{24(2-t)nt}+O\\left(\\frac{1}{n^2}\\right)\\right).$ Since the hypergeometric function can be expanded as ${}_2 F_{3}\\left({\\frac{n+3}{2},\\ \\frac{n+4}{2}\\atop \\frac{tn+4}{2},\\ n+\\frac{5}{2},\\ n+\\frac{5-tn}{2}}; \\ -\\frac{\\theta ^2}{4}\\right)=1-\\frac{\\theta ^2}{4nt(2-t)}+O\\left(\\frac{1}{n^2}\\right),$ we find $\\hat{h}^n_{(1-t)n}(\\theta )=\\frac{(-1)^{\\frac{tn+1}{2}}(-i\\theta )^{n+2}e^n\\sqrt{2n(1-t)+1}}{2^{n+3/2} n^{n+3}t^{\\frac{nt+3}{2}} (2-t)^{n(1-\\frac{t}{2})+2}}\\left(1-\\frac{104-12t+27t^2+6\\theta ^2}{24(2-t)tn}+O\\left(\\frac{1}{n^2}\\right)\\right).$" ], [ "Recurrence Formulas", "The formulas for the entries in $D^n_1$ given by Lemma REF allow simple recurrences in $n$ and $i$ .", "To this end we use formula (3.7.8) in [4] for balanced ${}_4 F_3$ series: $&\\frac{b(e-a)(f-a)(g-a)}{a-b-1}(F(a-,b+)-F)\\\\\\nonumber &-\\frac{a(e-b)(f-b)(g-b)}{b-a-1}(F(a+,b-)-F)+cd(a-b)F=0,$ where $F={}_4 F_{3}\\left({a,\\ b,\\ c,\\ d\\atop e,\\ f,\\ g}; \\ 1\\right)$ , $F(a+,b-)={}_4 F_{3}\\left({a+1,\\ b-1,\\ c,\\ d\\atop e,\\ f,\\ g}; \\ 1\\right)$ and $F(a-,b+)={}_4 F_{3}\\left({a-1,\\ b+1,\\ c,\\ d\\atop e,\\ f,\\ g}; \\ 1\\right)$ .", "From this equation we find: Theorem 19 The entries of the matrix $D^{n}_1$ satisfy the following recurrence relations $(D^n_1)_{n-i+2,n-j}+k^1_{n,i,j}(D^n_1)_{n-i,n-j}+k^2_{n,i,j}(D^n_1)_{n-i-2,n-j}=0,$ where $i$ is odd, $j<i-2$ , $k^1_{n,i,j}&= h\\left(\\left(cd(a-b)-\\frac{b(e - a)(f - a)(g - a)}{(a - b - 1)}\\right)\\frac{(1+a-b)}{a(e-b)(f-b)(g-b)}-1\\right),\\\\k^2_{n,i,j}&= l\\frac{b(e - a)(f - a)(g - a)(1+a-b)}{(a - b - 1)a(e-b)(f-b)(g-b)},$ and $&a=\\frac{j-i+1}{2},\\quad b= -n+\\frac{i+j}{2},\\quad c=-\\frac{j}{2},\\quad d=n+\\frac{3-j}{2}, \\quad e=1,\\quad f=\\frac{1}{2},\\\\&g=\\frac{3}{2},\\quad h=\\frac{\\sqrt{2(n-i)+5}(-2n+i+j-1)}{\\sqrt{2(n-i)+1}(i-j-2)}, \\\\&l=\\frac{\\sqrt{2(n-i)+5}(2n-i-j-1)(2n-i-j+1)}{\\sqrt{2(n-i)-3}(i-j)(i-j-2)}.$ If $j$ is even, the recurrence relation (REF ) follows from equations (REF ) and (REF ).", "If $j$ is odd, we use equations (REF ) and (REF ) to deduce that (REF ) holds with $k^1$ and $k^2$ defined by equations (REF )-() where $a=\\frac{j-i+2}{2}$ , $b= -n+\\frac{i+j+1}{2}$ , $c=-\\frac{j-1}{2}$ , $d=n+\\frac{4-j}{2}$ , $e=2$ , $f=\\frac{3}{2}$ , $g=\\frac{3}{2}$ , $h=\\frac{\\sqrt{2(n-i)+5}(-2n+i+j-2)}{\\sqrt{2(n-i)+1}(i-j-1)}$ , and $l=\\frac{\\sqrt{2(n-i)+5}(2n-i-j)(2n-i-j+2)}{\\sqrt{2(n-i)-3}(i-j+1)(i-j-1)}$ .", "Although $a$ , $b$ , $c$ , $d$ , $e$ , $f$ , $g$ , $h$ and $l$ are different in this case, they lead to the same formulas for $k^1$ and $k^2$ .", "Likewise we can obtain a recurrence relation in $n$ .", "Theorem 20 The entries of the matrix $D^{n}_1$ satisfy the following recurrence relations $(D^{n+1}_1)_{n-i+1,n-j+1}+k^1_{n,i,j}(D^n_1)_{n-i,n-j}+k^2_{n,i,j}(D^{n-1}_1)_{n-i-1,n-j-1}=0,$ where $i$ is odd, $j<i<n$ , $k^1$ and $k^2$ are given in equations (REF )-(), and $&a=n+\\frac{3-j}{2},\\quad b= -n+\\frac{i+j}{2},\\quad c=-\\frac{j}{2},\\quad d=\\frac{j-i+1}{2},\\quad e=1,\\quad f=\\frac{1}{2},\\\\&g=\\frac{3}{2}, \\quad h=\\frac{\\sqrt{2(n-i)+3}\\sqrt{2(n-j)+3}(2n-i-j+1)}{\\sqrt{2(n-i)+1}\\sqrt{2(n-j)+1}(2n-j+2)},\\\\&l=\\frac{\\sqrt{2(n-i)+3}\\sqrt{2(n-j)+3}(2n-i-j-1)(2n-i-j+1)}{\\sqrt{2(n-i)-1}\\sqrt{2(n-j)-1}(2n-j)(2n-j+2)}.$" ], [ "Acknowledgements", "We would like to thank a referee for carefully reading the manuscript and the observation that the recurrence coefficients in equations (REF ) and (REF ) are the same for $j$ even and odd." ] ]	1403.0483
[ [ "The Erd\\H{o}s-Gy\\'arf\\'as problem on generalized Ramsey numbers" ], [ "Abstract Fix positive integers $p$ and $q$ with $2 \\leq q \\leq {p \\choose 2}$.", "An edge-coloring of the complete graph $K_n$ is said to be a $(p, q)$-coloring if every $K_p$ receives at least $q$ different colors.", "The function $f(n, p, q)$ is the minimum number of colors that are needed for $K_n$ to have a $(p,q)$-coloring.", "This function was introduced by Erd\\H{o}s and Shelah about 40 years ago, but Erd\\H{o}s and Gy\\'arf\\'as were the first to study the function in a systematic way.", "They proved that $f(n, p, p)$ is polynomial in $n$ and asked to determine the maximum $q$, depending on $p$, for which $f(n,p,q)$ is subpolynomial in $n$.", "We prove that the answer is $p-1$." ], [ "Introduction", "The Ramsey number $r_k(p)$ is the smallest natural number $n$ such that every $k$ -coloring of the edges of the complete graph $K_n$ contains a monochromatic $K_p$ .", "The existence of $r_k(3)$ was first shown by Schur [13] in 1916 in his work on Fermat's Last Theorem and it is known that $r_k(3)$ is at least exponential in $k$ and at most a multiple of $k!$ .", "It is a central problem in graph Ramsey theory to close the gap between the lower and upper bound, with connections to various problems in combinatorics, geometry, number theory, theoretical computer science and information theory (see, e.g., [9], [10]).", "The following natural generalization of the Ramsey function was first introduced by Erdős and Shelah [3], [4] and studied in depth by Erdős and Gyárfás [5].", "Let $p$ and $q$ be positive integers with $2 \\le q \\le \\binom{p}{2}$ .", "An edge-coloring of the complete graph $K_n$ is said to be a $(p, q)$ -coloring if every $K_p$ receives at least $q$ different colors.", "The function $f(n, p, q)$ is the minimum number of colors that are needed for $K_n$ to have a $(p,q)$ -coloring.", "To see that this is indeed a generalization of the usual Ramsey function, note that $f(n, p, 2)$ is the minimum number of colors needed to guarantee that no $K_p$ is monochromatic.", "That is, $f(n, p, 2)$ is the inverse of the Ramsey function $r_k(p)$ and so we have $c^{\\prime } \\frac{\\log n}{\\log \\log n} \\le f(n, 3, 2) \\le c \\log n.$ Erdős and Gyárfás [5] proved a number of interesting results about the function $f(n,p,q)$ , demonstrating how the function falls off from being equal to $\\binom{n}{2}$ when $q = \\binom{p}{2}$ to being at most logarithmic when $q = 2$ .", "In so doing, they determined ranges of $p$ and $q$ where the function $f(n, p, q)$ is linear in $n$ , where it is quadratic in $n$ and where it is asymptotically equal to $\\binom{n}{2}$ .", "Many of these results were subsequently sharpened by Sárközy and Selkow [11], [12].", "One simple observation made by Erdős and Gyárfás is that $f(n, p, p)$ is always polynomial in $n$ .", "To see this, it is sufficient to note that if a coloring uses fewer than $n^{1/(p-2)} - 1$ colors then it necessarily contains a $K_p$ which uses at most $p - 1$ colors.", "For $p = 3$ , this is easy to see since one only needs that some vertex has at least two neighbors in the same color.", "For $p = 4$ , we have that any vertex will have at least $n^{1/2}$ neighbors in some fixed color.", "But, since there are fewer than $n^{1/2} - 1$ colors on this neighborhood of size at least $n^{1/2}$ , the case $p = 3$ implies that it contains a triangle with at most two colors.", "The general case follows similarly.", "Erdős and Gyárfás [5] asked whether this result is best possible, that is, whether $q = p$ is the smallest value of $q$ for which $f(n, p, q)$ is polynomial in $n$ .", "For $p = 3$ , this is certainly true, since we know that $f(n, 3, 2) \\le c \\log n$ .", "However, for general $p$ , they were only able to show that $f(n, p, \\lceil \\log p \\rceil )$ is subpolynomial, where here and throughout the paper we use $\\log $ to denote the logarithm taken base 2.", "This left the question of determining whether $f(n, p, p - 1)$ is subpolynomial wide open, even for $p = 4$ .", "The first progress on this question was made by Mubayi [8], who found an elegant construction which implies that $f(n, 4, 3) \\le e^{c \\sqrt{\\log n}}$ .", "This construction was also used by Eichhorn and Mubayi [2] to demonstrate that $f(n, 5, 4) \\le e^{c \\sqrt{\\log n}}$ .", "More generally, they used the construction to show that $f(n, p, 2 \\lceil \\log p \\rceil - 2)$ is subpolynomial for all $p \\ge 5$ .", "In this paper, we answer the question of Erdős and Gyárfás in the positive for all $p$ .", "That is, we prove that $f(n, p, p -1)$ is subpolynomial for all $p$ .", "Quantitatively, our main theorem is the following.", "Theorem 1.1 For all natural numbers $p \\ge 4$ and $n \\ge 1$ , $f(n, p, p-1) \\le 2^{16p (\\log n)^{1 - 1/(p-2)} \\log \\log n}.$ In Section , we define our $(p,p-1)$ -coloring by a recursive procedure.", "We begin by reviewing Mubayi's $(4,3)$ -coloring, as it is the base case of our recursion.", "The formal proof of the fact that our coloring is indeed a $(p,p-1)$ -coloring is quite technical and thus we first give an outline of the proof in Section .", "Then, in Section , we establish some properties of the coloring.", "Finally, in Section , we prove that the coloring given in Section is a $(p,p-1)$ -coloring.", "We will conclude with some further remarks.", "Notation.", "For vectors $v\\in X^{t_{1}+t_{2}},v_{1}\\in X^{t_{1}},v_{2}\\in X^{t_{2}}$ , we will often use the notation $v=(v_{1},v_{2}),$ in order to indicate that the $i$ -th coordinate of $v$ is equal to the $i$ -th coordinate of $v_{1}$ for $1\\le i\\le t_{1}$ and the $(t_{1}+j)$ -th coordinate of $v$ is equal to the $j$ -th coordinate of $v_{2}$ for $1\\le j\\le t_{2}$ .", "We will use similar notation for several vectors.", "Throughout the paper, $\\log $ denotes the base 2 logarithm.", "For the sake of clarity of presentation, we systematically omit floor and ceiling signs whenever they are not essential." ], [ "The coloring construction", "The purpose of this section is to define the coloring used to prove Theorem REF .", "The coloring can be considered as a generalization of (a variant of) Mubayi's $(4,3)$ -coloring.", "We therefore first introduce this coloring and then redefine it in a way that can be naturally extended.", "We then present the coloring used to prove Theorem REF .", "As it is a rather involved recursive definition, we give an example to illustrate it.", "We conclude the section by establishing a bound on the number of colors used in this coloring.", "In the following sections, we will show that this coloring is a $(p,p-1)$ -coloring, completing the proof." ], [ "Mubayi's $(4,3)$ -coloring", "Let $N=m^{t}$ for some integers $m$ and $t$ .", "Suppose that we are given two distinct vectors $v,w\\in [m]^{t}$ of the form $v=(v_{1},\\ldots ,v_{t})$ and $w=(w_{1},\\ldots ,w_{t})$ .", "Define $c(v,w)=\\big (\\lbrace v_{i},w_{i}\\rbrace ,a_{1},\\ldots ,a_{t}\\big ),$ where $i$ is the least coordinate in which $v_{i}\\ne w_{i}$ and $a_{j}=0$ if $v_{j}=w_{j}$ and $a_{j}=1$ if $v_{j}\\ne w_{j}$ .", "If $v=w$ , define $c(v,v)=0.$ Note that $c$ is a symmetric function.", "This is a variant of Mubayi's coloring and can be proved to be a $(p,p-1)$ -coloring for small values of $p$ .", "One might suspect that this is a $(p,p-1)$ -coloring for large integers $p$ as well, but, unfortunately, it fails to be a $(26,25)$ -coloring (and a $(p,p-1)$ -coloring for all $p\\ge 26$ ) for the following reason.", "Consider the set $\\lbrace 1,2,3\\rbrace ^{3}$ .", "This set has $3^{3}=27$ elements and at most $3\\cdot 2^{3}=24$ colors are used in coloring this set.", "Therefore, we can find 26 vertices with at most 24 colors within the set.", "Moreover, for every fixed $p$ and large enough $N$ , letting $s=\\sqrt{\\log p}$ , the set $S=\\lbrace 1,2,\\ldots ,2^{s}\\rbrace ^{s}$ has cardinality $2^{s^{2}}=p$ and uses at most ${2^{s} \\atopwithdelims ()2}2^{s}<2^{3s}=2^{3\\sqrt{\\log p}}$ colors and, for large enough $m$ and $t$ , is a subset of $[m]^{t}$ .", "Hence, this edge-coloring of the complete graph on $[N]$ fails to be a $(p,2^{3\\sqrt{\\log p}})$ -coloring." ], [ "Redefining Mubayi's coloring", "Before proceeding further, let us redefine the coloring given above from a slightly different perspective.", "We do this to motivate the $(p,p-1)$ -coloring which we use to establish Theorem REF .", "Let $m=2^{r_{1}}$ and, abusing notation, identify the set $[m]$ with $\\lbrace 0,1\\rbrace ^{r_{1}}$ .", "Let $r_{2}=r_{1}t$ for some positive integer $t$ .", "Suppose that we are given two vectors $v,w\\in [m]^{t}=\\lbrace 0,1\\rbrace ^{r_{1}t}$ .", "We decompose $v$ as $v=(v_{1}^{(1)},\\ldots ,v_{t}^{(1)})$ , where $v_{i}^{(1)}\\in \\lbrace 0,1\\rbrace ^{r_{1}}$ for $i=1,2,\\ldots ,t$ and similarly decompose $w$ .", "The function $c$ was defined as follows: $c(v,w)=\\big (\\lbrace v_{i}^{(1)},w_{i}^{(1)}\\rbrace ,a_{1},\\ldots ,a_{t}\\big ),$ where $i$ is the least coordinate in which $v_{i}^{(1)}\\ne w_{i}^{(1)}$ and, for $j=1,2,\\ldots ,t$ , $a_j$ represents whether $v_{j}^{(1)}=w_{j}^{(1)}$ or not.", "If $v=w$ , then $c(v,v)=0$ .", "Define $h_{1}$ as the first coordinate of $c$ .", "That is, $h_{1}(v,w)=\\lbrace v_{i}^{(1)},w_{i}^{(1)}\\rbrace $ (we let $h_{1}(v,v)=0$ for convenience).", "Note that $h_{1}$ takes a pair of vectors of length $r_{2}=r_{1}t$ as input and outputs a pair of vectors of length $r_{1}$ .", "For two vectors $x,y\\in \\lbrace 0,1\\rbrace ^{r_{1}}$ of the form $x=(x_{1},\\ldots ,x_{r_{1}})$ , $y=(y_{1},\\ldots ,y_{r_{1}})$ , define the function $h_{0}$ as follows.", "We have $h_{0}(x,x)=0$ for each $x$ and, if $x \\ne y$ , then $h_{0}(x,y)=\\lbrace x_{i},y_{i}\\rbrace ,$ where $i$ is the minimum index for which $x_{i}\\ne y_{i}$ .", "Since all $x_{i}$ and $y_{i}$ are either 0 or 1, there are only two possible outcomes for $h_{0}$ , 0 if the two vectors are equal and $\\lbrace 0,1\\rbrace $ if they are not equal.", "Note that $h_{0}$ takes a pair of vectors of length $r_{1}$ as input and outputs a pair of vectors of length $r_{0}=1$ .", "Thus, both $h_{1}$ and $h_{0}$ are functions which record the first `block' that is different.", "The difference between the two functions lies in their interpretation of `block': for $h_{1}$ it is a subvector of length $r_{1}$ and for $h_{0}$ it is a subvector of length $r_{0}$ .", "Summarizing, we see that $c$ is equivalent to the coloring $c^{\\prime }$ given by $c^{\\prime }(v,w)=\\Big (h_{1}(v,w),h_{0}(v_{1}^{(1)},w_{1}^{(1)}),\\ldots ,h_{0}(v_{t}^{(1)},w_{t}^{(1)})\\Big ).$ Informally, we first decompose the given pair of vectors $v$ and $w$ into subvectors of length $r_{2}$ and apply $h_{1}$ (we observe only a single subvector in this case since $v$ and $w$ themselves are vectors of length $r_{2}$ ).", "Then we decompose $v$ and $w$ into subvectors of length $r_{1}$ and apply $h_{0}$ to each corresponding pair of subvectors of $v$ and $w$ ." ], [ "Definition of the coloring", "In this section, we generalize the construction given in the previous section to obtain a $(p,p-1)$ -coloring.", "For a positive integer $\\alpha $ , we will describe the coloring as an edge-coloring of the complete graph over the vertex set $\\lbrace 0,1\\rbrace ^{\\alpha }$ .", "Let $r_{0},r_{1},\\ldots $ be a sequence of positive integers such that $r_{0}=1$ and $r_{d-1}$ divides $r_{d}$ for all $d\\ge 1$ .", "For a set of indices $I$ , let $\\pi _{I}$ be the canonical projection map from $\\lbrace 0,1\\rbrace ^{\\alpha }$ to $\\lbrace 0,1\\rbrace ^{I}$ .", "We will write $\\pi _{i}$ instead of $\\pi _{[i]}$ for convenience.", "Thus $\\pi _{i}$ is the projection map to the first $i$ coordinates.", "The key idea in the construction is to understand vectors at several different resolutions.", "Suppose that we are given two vectors $v,w\\in \\lbrace 0,1\\rbrace ^{\\alpha }$ .", "For $d\\ge 0$ , let $a_d$ and $b_d$ be integers satisfying $a_{d}\\ge 0$ and $1\\le b_{d}\\le r_{d}$ such that $\\alpha =a_{d}r_{d}+b_{d}$ .", "Let $v=\\Big (v_{1}^{(d)},v_{2}^{(d)},\\ldots ,v_{a_{d}+1}^{(d)}\\Big ),$ where $v_{i}^{(d)}\\in \\lbrace 0,1\\rbrace ^{r_{d}}$ for $i=1,2,\\ldots ,a_{d}$ and $v_{a_{d}+1}^{(d)}\\in \\lbrace 0,1\\rbrace ^{b_{d}}$ .", "We refer to the vectors $v_{i}^{(d)}$ as blocks of resolution d. We similarly decompose $w$ as $w=\\big (w_{1}^{(d)},w_{2}^{(d)},\\ldots ,w_{a_{d}}^{(d)},w_{a_{d}+1}^{(d)}\\big )$ for $d\\ge 0$ .", "We first define two auxiliary families of functions $\\eta _{d}$ and $\\xi _{d}$ .", "For $d\\ge 0$ , if $v\\ne w$ , let $\\eta _{d}(v,w)=\\Big (i,\\lbrace v_{i}^{(d)},w_{i}^{(d)}\\rbrace \\Big ),$ where $i$ is the minimum index such that $v_{i}^{(d)}\\ne w_{i}^{(d)}$ .", "If $v=w$ , let $\\eta _{d}(v,v)=0.$ Note that $\\eta _{d}$ is a symmetric function.", "Further note that $\\eta _{d}$ is slightly different from $h_{d}$ defined in the previous subsection since we add an additional coordinate which records the index $i$ as well.", "The main theorem is valid even if we do not add this index, but we choose to add it as it simplifies the proof.", "We refer the reader to Subsection REF for a further discussion of this point.", "For $d\\ge 0$ , let $\\xi _{d}(v,w)=\\Big (\\eta _{d}\\big (v_{1}^{(d+1)},w_{1}^{(d+1)}\\big ),\\ldots ,\\eta _{d}\\big (v_{a_{d+1}+1}^{(d+1)},w_{a_{d+1}+1}^{(d+1)}\\big )\\Big ).$ Note that the function $\\xi _{d}$ decomposes the vectors into blocks of resolution $d+1$ and outputs a vector containing information about blocks of resolution $d$ .", "For $d\\ge 0$ , let $c_{d}=\\xi _{d}\\times \\xi _{d-1}\\times \\ldots \\times \\xi _{0}.$ Note that the coloring $c_{d}$ depends on the choice of the parameters $r_{0},r_{1},\\ldots ,r_{d+1}$ .", "We prove our main theorem in two steps: we first estimate the number of colors and then prove that it is a $(p,p-1)$ -coloring.", "Theorem 2.1 Let $p$ and $\\beta $ be fixed positive integers with $\\beta \\ne 1$ .", "For the choice $r_{i}=\\beta ^{i}$ for $0\\le i\\le p+1$ , the edge-coloring $c_{p}$ of the complete graph on $n=2^{\\beta ^{p+1}}$ vertices uses at most $2^{4(\\log n)^{1-1/(p+1)}\\log \\log n}$ colors.", "Theorem 2.2 Let $p$ and $\\alpha $ be fixed positive integers.", "Then, for every choice of parameters $r_{1},\\ldots ,r_{p+1}$ , the edge-coloring $c_{p}$ is a $(p+3,p+2)$ -coloring of the complete graph on the vertex set $\\lbrace 0,1\\rbrace ^{\\alpha }$ .", "For integers $n$ of the form $n=2^{\\beta ^{p+1}}$ , Theorem REF follows from Theorems REF and REF .", "For general $n \\ge p+3 \\ge 4$ , first notice that if $n^2 < 2^{16p(\\log n)^{1-1/(p+1)}\\log \\log n}$ , then the statement is trivially true, as we may color each edge with different colors.", "Hence, we may assume that the inequality does not hold, from which it follows that $ 2\\log n \\ge 16p(\\log n)^{1-1/(p+1)}\\log \\log n \\ge 16p(\\log n)^{1-1/(p+1)}$ and $n \\ge 2^{(8p)^{p+1}}$ .", "Hence, there exists an integer of the form $2^{\\beta ^{p+1}}$ which is at most $n^{(1+1/8p)^{p+1}} \\le n^2$ .", "Therefore, there exists a $(p+3,p+2)$ -coloring of the complete graph on the vertex set $[n]$ using at most $2^{4(2\\log n)^{1-1/(p+1)}\\log (2\\log n)}\\le 2^{4 \\cdot 2(\\log n)^{1-1/(p+1)} (1+\\log \\log n)}\\le 2^{16(\\log n)^{1-1/(p+1)}\\log \\log n}$ colors (in the second inequality we used the fact that $\\log \\log n\\ge \\log \\log 4 \\ge 1$ ).", "Thus we obtain Theorem REF .", "Theorem REF is proved in Subsection REF , while Theorem REF is proved in Section and builds on the two sections leading up to it." ], [ "Example", "Let us illustrate the coloring by working out a small example.", "Suppose that $r_{1}=2$ and $r_{2}=4$ .", "Let $v=(0,0,1,0,1,1,0)$ and $w=(0,0,1,1,1,0,0)$ be vectors in $\\lbrace 0,1\\rbrace ^{7}$ .", "Then $v & =(0,0,1,0,1,1,0)=\\Big (\\textrm {`}0,0 \\textrm {^{\\prime }}, \\textrm {`}1,0 \\textrm {^{\\prime }}, \\textrm {`}1,1 \\textrm {^{\\prime }}, \\textrm {`}0 \\textrm {^{\\prime }}\\Big )=\\Big (\\textrm {`}0,0,1,0 \\textrm {^{\\prime }},\\textrm {`}1,1,0 \\textrm {^{\\prime }}\\Big ),$ where the quotation marks indicate the blocks of each resolution.", "Similarly, $w & =(0,0,1,1,1,0,0)=\\Big (\\textrm {`}0,0\\textrm {^{\\prime }},\\textrm {`}1,1\\textrm {^{\\prime }},\\textrm {`}1,0\\textrm {^{\\prime }},\\textrm {`}0\\textrm {^{\\prime }}\\Big )=\\Big (\\textrm {`}0,0,1,1\\textrm {^{\\prime }},\\textrm {`}1,0,0\\textrm {^{\\prime }}\\Big ).$ The function $\\eta _{0}$ records the first pair of blocks of resolution 0 which are different.", "So $\\eta _{0}(v,w)=(4,\\lbrace 0,1\\rbrace ),$ where the value of the first coordinate, 4, indicates that $v$ and $w$ first differ in the fourth coordinate.", "Similarly, the function $\\eta _{1}$ will record the first pair of blocks of resolution 1 which are different.", "So $\\eta _{1}(v,w)=\\Big (2,\\lbrace (1,0),(1,1)\\rbrace \\Big ).$ Computing $\\xi _{0}$ and $\\xi _{1}$ involves one more step.", "To compute $\\xi _{0}$ , we apply $\\eta _{0}$ to each pair of blocks of resolution 1.", "Therefore, $\\xi _{0}(v,w) & =\\Big (\\eta _{0}\\big ((0,0),(0,0)\\big ),\\eta _{0}\\big ((1,0),(1,1)\\big ),\\eta _{0}\\big ((1,1),(1,0)\\big ),\\eta _{0}\\big ((0),(0)\\big )\\Big )\\\\& =\\big (0,(2,\\lbrace 0,1\\rbrace ),(2,\\lbrace 1,0\\rbrace ),0\\big ),$ which is a vector of length four.", "Similarly, to compute $\\xi _{1}$ , we apply $\\eta _{1}$ to each pair of blocks of resolution 2.", "Therefore, $\\xi _{1}(v,w) & =\\Big (\\eta _{1}\\big ((0,0,1,0),(0,0,1,1)\\big ),\\eta _{1}\\big ((1,1,0),(1,0,0)\\big )\\Big )\\\\& =\\left(\\Big (2,\\big \\lbrace (1,0),(1,1)\\big \\rbrace \\Big ),\\Big (1,\\big \\lbrace (1,1),(1,0)\\big \\rbrace \\Big )\\right),$ which is a vector of length two." ], [ "Number of colors", "In this subsection, we prove Theorem REF .", "[Proof of Theorem REF ] Recall that $\\beta $ is a positive integer greater than 1 and $r_{d}=\\beta ^{d}$ for $0\\le d\\le p+1$ .", "Let $\\alpha =\\beta ^{p+1}$ .", "The goal here is to give an upper bound on the number of colors in the edge-coloring $c_p$ of the complete graph with vertex set $\\lbrace 0,1\\rbrace ^{\\alpha }=\\lbrace 0,1\\rbrace ^{\\beta ^{p+1}}$ .", "First, for $0\\le d\\le p$ , the function $\\eta _{d}$ outputs either zero or an index and a pair of distinct blocks of resolution $d$ .", "Hence, there are at most $1+\\alpha \\cdot 2^{r_{d}}(2^{r_d} -1) \\le \\alpha 2^{2\\beta ^{d}}$ possible outcomes for the function $\\eta _{d}$ .", "Second, for $0\\le d\\le p$ , the function $\\xi _{d}$ is a product of $\\frac{\\alpha }{r_{d+1}}=\\beta ^{p-d}$ outcomes of $\\eta _{d}$ .", "Hence, there are at most $\\big (\\alpha \\cdot 2^{2\\beta ^{d}}\\big )^{\\beta ^{p-d}}= \\beta ^{(p+1)\\beta ^{p-d}}\\cdot 2^{2\\beta ^{p}}$ possible outcomes for the function $\\xi _{d}$ .", "Since $c_{p}$ is defined as $\\xi _{p}\\times \\xi _{p-1}\\times \\cdots \\times \\xi _{0}$ , the total number of colors used in $c_{p}$ is at most $\\prod _{d=0}^{p} \\left(\\beta ^{(p+1)\\beta ^{p-d}}\\cdot 2^{2\\beta ^{p}}\\right)\\le \\beta ^{2(p+1)\\beta ^{p}} 2^{2(p+1)\\beta ^p} \\le 2^{4(p+1)\\beta ^p\\log \\beta }.$ Let $n=2^{\\alpha }=2^{\\beta ^{p+1}}$ and note that $\\beta ^{p}=(\\log n)^{1-1/(p+1)}$ and $\\log \\beta =\\frac{1}{p+1}\\log \\log n$ .", "Thus, we have colored the edges of the complete graph on $n$ vertices using at most $2^{4(\\log n)^{1-1/(p+1)}\\log \\log n}$ colors, as claimed in Theorem REF .", "As we saw in Subsection REF , for large enough $q$ , Mubayi's coloring (which is similar to $c_{1}$ ) is not a $(q,q-1)$ -coloring or even a $(q,q^{\\varepsilon })$ -coloring for any fixed $\\varepsilon >0$ .", "Similarly, we can see that the same is true for the coloring $c_{p}$ for every fixed $p$ (we will briefly describe the proof of this fact in Subsection REF ).", "This explains why we need to consider $c_{p}$ with an increasing value of $p$ ." ], [ "Outline of proof", "In this section, we outline the proof of Theorem REF .", "Assume that we want to prove that the edge-coloring of the complete graph on the vertex set $\\lbrace 0,1\\rbrace ^{\\alpha }$ given by $c_{p}$ is a $(p+3,p+2)$ -coloring.", "We will use induction on $\\alpha $ to prove the stronger statement that the coloring is a $(q, q-1)$ coloring for all $q \\le p +3$ .", "To illustrate a simple case, assume that we are about to prove it for $\\alpha =r_{p+1}$ and have proved it for all smaller values of $\\alpha $ .", "Let $S\\subset \\lbrace 0,1\\rbrace ^{\\alpha }$ be a given set of size at most $p+3$ .", "We wish to show that the edges of $S$ receive at least $\|S\| - 1$ distinct colors.", "Let $\\alpha ^{\\prime } = r_{p+1} - r_p$ .", "For two vectors $v,w\\in S$ satisfying $v\\ne w$ , let $v=(v^{\\prime },v^{\\prime \\prime })$ and $w=(w^{\\prime },w^{\\prime \\prime })$ where $v^{\\prime },w^{\\prime }\\in \\lbrace 0,1\\rbrace ^{\\alpha ^{\\prime }}$ and $v^{\\prime \\prime },w^{\\prime \\prime }\\in \\lbrace 0,1\\rbrace ^{\\alpha -\\alpha ^{\\prime }} = \\lbrace 0,1\\rbrace ^{r_{p}}$ .", "Note that since $\\alpha ^{\\prime } = r_{p+1}-r_p$ is divisible by $r_{p}$ , the first $\\frac{\\alpha ^{\\prime }}{r_p}$ blocks of resolution $p$ of $v$ are identical to those of $v^{\\prime }$ and a similar fact holds for $w$ and $w^{\\prime }$ .", "If $v^{\\prime }=w^{\\prime }$ then, by the observation above, the first $\\frac{\\alpha ^{\\prime }}{r_p}$ coordinates of $\\xi _{p-1}$ are all zero.", "On the other hand, if $v^{\\prime }\\ne w^{\\prime }$ , then the first block of resolution $p$ on which $v$ and $w$ differ is one of the first $\\frac{\\alpha ^{\\prime }}{r_p}$ blocks.", "Hence, in this case, at least one of the first $\\frac{\\alpha ^{\\prime }}{r_p}$ coordinates of $\\xi _{p-1}$ is non-zero.", "Thus, if we define sets $\\Lambda _{I}$ and $\\Lambda _{E}$ as $\\Lambda _{I}=\\big \\lbrace c_{p}(v,w)\\,:\\, v^{\\prime }\\ne w^{\\prime },\\, v,w\\in S\\big \\rbrace $ and $\\Lambda _{E}=\\big \\lbrace c_{p}(v,w)\\,:\\, v^{\\prime }=w^{\\prime },\\, v\\ne w,\\, v,w\\in S\\big \\rbrace ,$ then we have $\\Lambda _{I}\\cap \\Lambda _{E}=\\emptyset $ .", "Hence, it suffices to prove that $\|\\Lambda _{I}\|+\|\\Lambda _{E}\|\\ge \|S\|-1$ .", "The index `I' stands for inherited colors and `E' stands for emerging colors.", "The coloring $c_{p}$ contains more information than necessary to prove that the number of colors is large.", "Hence, we consider only part of the coloring $c_{p}$ .", "The part of the coloring that we consider for $\\Lambda _{I}$ and $\\Lambda _{E}$ will be different, as we would like to highlight different aspects of our coloring depending on the situation.", "Define the sets $C_{I}$ and $C_{E}$ as $C_{I}=\\Big \\lbrace \\big (c_{p}(v^{\\prime },w^{\\prime }),\\eta _{p-1}(v^{\\prime \\prime },w^{\\prime \\prime })\\big )\\,:\\, v^{\\prime } \\ne w^{\\prime },\\, v,w\\in S\\Big \\rbrace $ and $C_{E}=\\Big \\lbrace \\lbrace v^{\\prime \\prime },w^{\\prime \\prime }\\rbrace \\,:\\, v^{\\prime }=w^{\\prime },\\, v^{\\prime \\prime } \\ne w^{\\prime \\prime },\\, v,w\\in S\\Big \\rbrace .$ We claim here without proof that $\|C_{I}\|\\le \|\\Lambda _{I}\|$ and $\|C_{E}\|\\le \|\\Lambda _{E}\|$ .", "Abusing notation, for two vectors $v,w\\in S$ , we will from now on refer to the color between $v$ and $w$ as the corresponding `color' in $C_{I}$ or $C_{E}$ .", "It now suffices to prove that $\|C_{I}\|+\|C_{E}\|\\ge \|S\|-1$ .", "To analyze the colors in $C_{I}$ and $C_{E}$ , we take a step back and consider the first $\\alpha ^{\\prime }$ coordinates of the vectors in $S$ .", "Let $S^{\\prime }=\\pi _{\\alpha ^{\\prime }}(S)$ .", "Note that $S^{\\prime }$ is the collection of vectors $v^{\\prime }$ in the notation above.", "There is a certain `branching phenomenon' of vectors and colors.", "For a vector $v^{\\prime }\\in S^{\\prime }$ , let $T_{v^{\\prime }}=\\lbrace v\\,:\\,\\pi _{\\alpha ^{\\prime }}(v)=v^{\\prime },\\, v\\in S\\rbrace $ .", "Hence, $T_{v^{\\prime }}$ is the set of vectors in $S$ whose first $\\alpha ^{\\prime }$ coordinates are equal to $v^{\\prime }$ .", "Note that $\\sum _{v^{\\prime }\\in S^{\\prime }}\|T_{v^{\\prime }}\|=\|S\|.$ Consider two vectors $v,w\\in S$ .", "If $v$ and $w$ are both in the same set $T_{v^{\\prime }}$ , then the color between $v$ and $w$ belongs to $C_{E}$ and if they are in different sets, then the color between $v$ and $w$ belongs to $C_{I}$ .", "For a color $c\\in C_{I}$ , note that the first coordinate of $c$ is of the form $c_{p}(v^{\\prime },w^{\\prime })$ for two vectors $v^{\\prime },w^{\\prime }\\in S^{\\prime }$ .", "Further note that $c_{p}(v^{\\prime },w^{\\prime })$ is the color of an edge that lies within $S^{\\prime }$ .", "Hence, $c$ is a `branch' of some color of an edge that lies within $S^{\\prime }$ .", "In particular, by induction on $\\alpha $ , we see that $\|C_{I}\|\\ge \|S^{\\prime }\|-1.$ For a color $c\\in C_{E}$ , let $\\mu _{c}$ be the number of (unordered) pairs of vectors $v,w$ such that $c$ is the color between $v$ and $w$ .", "We have the following equation $\\sum _{c\\in C_{E}}\\mu _{c}=\\sum _{v^{\\prime }\\in S^{\\prime }}{\|T_{v^{\\prime }}\| \\atopwithdelims ()2}\\ge \\sum _{v^{\\prime }\\in S^{\\prime }}(\|T_{v^{\\prime }}\|-1).$ Let us first consider the simple case when $\\mu _{c}=1$ for all $c\\in C_{E}$ (that is, there are no overlaps between the emerging colors).", "In this case, we have $\|C_{E}\|=\\sum _{c\\in C_{E}}\\mu _{c}$ .", "By (REF ), we have $\|C_{I}\|+\|C_{E}\| & \\ge (\|S^{\\prime }\|-1)+\|C_{E}\|=(\|S^{\\prime }\|-1)+\\sum _{c\\in C_{E}}\\mu _{c},$ which by (REF ) and (REF ) is at least $(\|S^{\\prime }\|-1)+\\sum _{v^{\\prime }\\in S^{\\prime }}(\|T_{v^{\\prime }}\|-1)=\\big (\\sum _{v^{\\prime }\\in S^{\\prime }}\|T_{v^{\\prime }}\|\\big )-1=\|S\|-1$ and thus the conclusion follows for the case when $\\mu _c =1$ for all $c \\in C_E$ .", "However, there might be some overlap between the emerging colors.", "Note that there are $\|C_{E}\|$ emerging colors instead of the $\\sum _{c\\in C_{E}}\\mu _{c}$ which we obtain by counting with multiplicity.", "Thus, there are $\\sum _{c\\in C_{E}}(\\mu _{c}-1)$ `lost' emerging colors.", "Our key lemma asserts that every lost emerging color will be accounted for by contributions towards $\|C_{I}\|$ .", "Formally, we will improve (REF ) and obtain the following inequality $\|C_{I}\|\\ge (\|S^{\\prime }\|-1)+\\sum _{c\\in C_{E}}(\\mu _{c}-1).$ Given this inequality, we will have $\|C_{I}\|+\|C_{E}\|\\ge (\|S^{\\prime }\|-1)+\\sum _{c\\in C_{E}}(\\mu _{c}-1)+\|C_{E}\|=(\|S^{\\prime }\|-1)+\\sum _{c\\in C_{E}}\\mu _{c},$ which, as above, implies that $\|C_{I}\|+\|C_{E}\|\\ge \|S\|-1$ .", "We conclude this section with a sketch of the proof of (REF ).", "To see this, we further study the branching of the colors.", "Define $C_{B}$ as the set of colors that appear within the set $S^{\\prime }$ , that is, $C_{B}=\\big \\lbrace c_{p}(v^{\\prime },w^{\\prime })\\,:\\, v^{\\prime },w^{\\prime }\\in S^{\\prime }\\big \\rbrace ,$ where the index `B' stands for base colors.", "Every color $c\\in C_{I}$ is of the form $c=(c^{\\prime },?", ")$ , where $c^{\\prime }\\in C_{B}$ and the question mark `?'", "stands for an unspecified coordinate.", "Thus, we immediately have at least $\|C_{B}\|$ colors in $C_{I}$ (this is the content of Equation (REF )).", "Now take a color $c^{\\prime \\prime }=\\lbrace v^{\\prime \\prime },w^{\\prime \\prime }\\rbrace \\in C_{E}$ and suppose that $c^{\\prime \\prime }$ has multiplicity $\\mu _{c^{\\prime \\prime }}$ .", "Then there exist vectors $x_{i}\\in S^{\\prime }$ for $i=1,2,\\ldots ,\\mu _{c^{\\prime \\prime }}$ such that $c^{\\prime \\prime }$ is the color between $(x_{i},v^{\\prime \\prime })$ and $(x_{i},w^{\\prime \\prime })$ .", "Consider the colors of the two pairs $\\big ((x_{1},v^{\\prime \\prime }),(x_{2},v^{\\prime \\prime })\\big )$ and $\\big ((x_{1},v^{\\prime \\prime }),(x_{2},w^{\\prime \\prime })\\big )$ in $C_I$ .", "These are $\\big (c_{p}(x_{1},x_{2}),\\eta _{p-1}(v^{\\prime \\prime },v^{\\prime \\prime })\\big ) & =(c_{1,2},0)\\in C_{I} \\quad \\textrm {and}\\\\\\big (c_{p}(x_{1},x_{2}),\\eta _{p-1}(v^{\\prime \\prime },w^{\\prime \\prime })\\big ) & =\\big (c_{1,2},\\eta _{p-1}(c^{\\prime \\prime })\\big )\\in C_{I},$ respectively, where $c_{1,2}\\in C_{B}$ (here we abuse notation and define $\\eta _{p-1}(c^{\\prime \\prime })=\\eta _{p-1}(v^{\\prime \\prime },w^{\\prime \\prime })$ , which is allowed since the right-hand-side is symmetric in the two input coordinates).", "Note that by the inductive hypothesis, there are at least $\\mu _{c^{\\prime \\prime }}-1$ distinct colors of the form $c_{i,j}$ for distinct pairs of indices $i$ and $j$ .", "Hence, by considering these colors, we add colors of the types $(c_{i,j},0)$ and $(c_{i,j},\\eta _{p-1}(c^{\\prime \\prime }))$ for at least $\\mu _{c^{\\prime \\prime }}-1$ distinct colors $c_{i,j}\\in C_{B}$ .", "Even if one of these two colors equals the color $(c_{i,j},?", ")$ counted above, we have added at least $\\mu _{c^{\\prime \\prime }}-1$ colors to $C_{I}$ by considering the color $c^{\\prime \\prime }\\in C_{E}$ .", "Now consider another color $c_{1}^{\\prime \\prime }\\in C_{E}$ .", "This color adds a further $\\mu _{c_{1}^{\\prime \\prime }}-1$ colors to $C_{I}$ as long as $\\eta _{p-1}(c^{\\prime \\prime })\\ne \\eta _{p-1}(c_{1}^{\\prime \\prime })$ .", "Therefore, if we can somehow guarantee that $\\eta _{p-1}(c^{\\prime \\prime })$ is distinct for all $c^{\\prime \\prime }$ , then we have $\|C_{I}\|\\ge \|C_{B}\|+\\sum _{c\\in C_{E}}(\\mu _{c}-1),$ which proves (REF ), since $\|C_{B}\|\\ge \|S^{\\prime }\|-1$ by the inductive hypothesis.", "Hence, it would be helpful to have distinct $\\eta _{p-1}(c^{\\prime \\prime })$ for each $c^{\\prime \\prime }\\in C_{E}$ .", "Even though we cannot always guarantee this, we can show that there exists a resolution in which the corresponding fact does hold.", "This will be explained in more detail in Section ." ], [ "Properties of the coloring", "In this section, we collect some useful facts about the coloring functions $c_{d}$ .", "Before listing these properties, we introduce the formal framework that we will use to describe them." ], [ "Refinement of functions", "For a function $f\\,:\\, A\\rightarrow B$ , let $\\Pi _{f}=\\lbrace f^{-1}(b)\\,:\\, b\\in f(A)\\rbrace $ .", "Thus, $\\Pi _{f}$ is a partition of $A$ into sets whose elements map by $f$ to the same element in $B$ .", "For two functions $f$ and $g$ defined over the same domain, we say that $f$ refines $g$ if $\\Pi _{f}$ is a refinement of $\\Pi _{g}$ .", "This definition is equivalent to saying that $f(a)=f(a^{\\prime })$ implies that $g(a)=g(a^{\\prime })$ and is also equivalent to saying that there exists a function $h$ for which $g=h\\circ f$ .", "The term $f$ refines $g$ is also referred to as $g$ factors through $f$ in category theory.", "This formalizes the concept that $f$ contains more information than $g$ .", "For two functions $f$ and $g$ defined over the same domain $A$ , let $f\\times g$ be the function defined over $A$ where $(f\\times g)(a)=(f(a),g(a))$ .", "The following proposition collects several basic properties of refinements of functions which will be useful in the proof of the main theorem.", "Proposition 4.1 Let $f_{1},f_{2},f_{3}$ and $f_{4}$ be functions defined over the domain $A$ .", "(i) (Identity) $f_1$ refines $f_1$ .", "(ii) (Transitivity) If $f_{1}$ refines $f_{2}$ and $f_{2}$ refines $f_{3}$ , then $f_{1}$ refines $f_{3}$ .", "(iii) If $f_1$ refines $f_3$ , then $f_{1}\\times f_{2}$ refines $f_{3}$ .", "(iv) If $f_1$ refines both $f_2$ and $f_3$ , then $f_{1}$ refines $f_{2}\\times f_{3}$ .", "(v) If $f_1$ refines $f_3$ and $f_2$ refines $f_4$ , then $f_{1}\\times f_{2}$ refines $f_{3}\\times f_{4}$ .", "(vi) If $f_{1}$ refines $f_{2}$ , then, for all $A^{\\prime }\\subset A$ , we have $\|f_{1}(A^{\\prime })\|\\ge \|f_{2}(A^{\\prime })\|$ .", "Let $\\Pi _{i}=\\Pi _{f_{i}}$ for $i=1,2,3$ .", "(i) This is trivial since $\\Pi _{1}$ refines $\\Pi _{1}$ .", "(ii) If $f_{1}$ refines $f_{2}$ and $f_{2}$ refines $f_{3}$ , then $\\Pi _{1}$ refines $\\Pi _{2}$ and $\\Pi _{2}$ refines $\\Pi _{3}$ .", "Therefore, $\\Pi _{1}$ refines $\\Pi _{3}$ and $f_{1}$ refines $f_{3}$ .", "(iii) Since $f_1 \\times f_2$ clearly refines $f_1$ , this follows from (ii).", "(iv) If $f_{1}(a) = f_{1}(a^{\\prime })$ , then $f_{2}(a)=f_{2}(a^{\\prime })$ and $f_{3}(a)=f_{3}(a^{\\prime })$ .", "Hence, $(f_2 \\times f_3)(a) = (f_2 \\times f_3)(a^{\\prime })$ and we conclude that $f_1$ refines $f_2 \\times f_3$ .", "(v) By (iii), $f_1 \\times f_2$ refines both $f_3$ and $f_4$ .", "Therefore, by (iv), $f_1 \\times f_2$ refines $f_3 \\times f_4$ .", "(vi) For $i=1,2$ , let $\\Pi _{i}\\vert _{A^{\\prime }}=\\lbrace X\\cap A^{\\prime }\\,:\\, X\\in \\Pi _{i},X\\cap A^{\\prime }\\ne \\emptyset \\rbrace $ and note that $\|f_{i}(A^{\\prime })\|=\\Big \|\\Pi _{i}\\vert _{A^{\\prime }}\\Big \|$ .", "Since $\\Pi _{1}$ is a refinement of $\\Pi _{2}$ , we see that $\\Pi _{1}\\vert _{A^{\\prime }}$ is a refinement of $\\Pi _{2}\\vert _{A^{\\prime }}$ .", "Therefore, it follows that $\|f_{1}(A^{\\prime })\|=\\Big \|\\Pi _{1}\\vert _{A^{\\prime }}\\Big \|\\ge \\Big \|\\Pi _{2}\\vert _{A^{\\prime }}\\Big \|=\|f_{2}(A^{\\prime })\|,$ as required.", "Refinements arise in our proof because we often consider colorings with less information than the full coloring.", "In the outline above, we considered several different sets of colors, namely, $\\Lambda _I$ , $\\Lambda _E$ , $C_I$ and $C_E$ and we claimed without proof that $\|C_I\| \\le \|\\Lambda _I\|$ and $\|C_E\| \\le \|\\Lambda _E\|$ .", "If we can show that $\\Lambda _I$ is a refinement of $C_I$ and $\\Lambda _E$ is a refinement of $C_E$ , then these inequalities follow from Proposition REF (vi) above." ], [ "Properties of the coloring", "We developed our formal framework for a rigorous treatment of the following two lemmas.", "It may be helpful at this stage to recall the definitions of $\\eta _d$ , $\\xi _d$ and $c_{d}$ from Subsection REF .", "Lemma 4.2 Suppose that $\\alpha $ , $\\alpha ^{\\prime }$ and $d$ are integers with $d \\ge 0$ and $1 \\le \\alpha ^{\\prime }\\le \\alpha $ .", "Then the following hold (where all functions are considered as defined over $\\lbrace 0,1\\rbrace ^{\\alpha }\\times \\lbrace 0,1\\rbrace ^{\\alpha }$ ): (i) $\\eta _{d}$ refines $\\eta _{d}\\circ (\\pi _{\\alpha ^{\\prime }}\\times \\pi _{\\alpha ^{\\prime }})$ .", "(ii) $\\xi _{d}$ refines $\\xi _{d}\\circ (\\pi _{\\alpha ^{\\prime }}\\times \\pi _{\\alpha ^{\\prime }})$ .", "(iii) $c_{d}$ refines $c_{d}\\circ (\\pi _{\\alpha ^{\\prime }}\\times \\pi _{\\alpha ^{\\prime }})$ .", "The case $\\alpha ^{\\prime }=\\alpha $ is trivial so we assume that $\\alpha ^{\\prime }<\\alpha $ .", "(i) Let $v$ and $w$ be vectors in $\\lbrace 0,1\\rbrace ^{\\alpha }$ and let $v^{\\prime }=\\pi _{\\alpha ^{\\prime }}(v)$ and $w^{\\prime }=\\pi _{\\alpha ^{\\prime }}(w)$ .", "We will show that one can compute the value of $\\eta _{d}(v^{\\prime },w^{\\prime })$ based only on the value of $\\eta _{d}(v,w)$ .", "This clearly implies the desired conclusion.", "If $\\eta _{d}(v,w)=0$ , then $v=w$ and it follows that $\\eta _{d}(v^{\\prime },w^{\\prime })=0$ .", "Assume then that $\\eta _{d}(v,w)=(i,\\lbrace v_{i}^{(d)},w_{i}^{(d)}\\rbrace )$ for some index $i$ and blocks $v_{i}^{(d)},w_{i}^{(d)}$ of resolution $d$ .", "Let $j$ be the first coordinate in which the two vectors $v_{i}^{(d)}$ and $w_{i}^{(d)}$ differ.", "Then the first coordinate $x$ (note that $1\\le x\\le \\alpha $ ) in which $v$ and $w$ differ is $x=(i-1)\\cdot r_{d}+j$ and satisfies $(i-1)\\cdot r_{d}<x\\le \\min \\lbrace i\\cdot r_{d},\\alpha \\rbrace .$ Note that the values of $i$ and $j$ can be deduced from $\\eta _{d}(v,w)$ and hence $x$ can as well.", "It thus suffices to verify that $\\eta _{d}(v^{\\prime },w^{\\prime })$ can be computed using only $\\alpha , \\alpha ^{\\prime }, r_d$ , $x$ , $i$ , $v_{i}^{(d)}$ and $w_{i}^{(d)}$ .", "If $\\alpha ^{\\prime }>i\\cdot r_{d}$ , then we have $\\eta _{d}(v^{\\prime },w^{\\prime })=\\eta _{d}(v,w)=(i,\\lbrace v_i^{(d)}, w_i^{(d)}\\rbrace )$ and the claim is true.", "On the other hand, if $\\alpha ^{\\prime }\\le i\\cdot r_{d}$ , then there are two cases.", "If $\\alpha ^{\\prime }<x$ , then we have $v^{\\prime }=w^{\\prime }$ .", "Therefore, $\\eta _{d}(v^{\\prime },w^{\\prime })=0$ and the claim holds for this case as well.", "The final case is when $x\\le \\alpha ^{\\prime }\\le i\\cdot r_{d}$ .", "In this case, we see that $\\eta _{d}(v^{\\prime },w^{\\prime })=\\left(i, \\Big \\lbrace \\pi _{[\\alpha ^{\\prime }-(i-1)r_{d}]}(v_{i}^{(d)}),\\pi _{[\\alpha ^{\\prime }-(i-1)r_{d}]}(w_{i}^{(d)}) \\Big \\rbrace \\right)$ and the claim holds.", "(ii) Let $v$ and $w$ be two vectors in $\\lbrace 0,1\\rbrace ^{\\alpha }$ .", "Then $\\xi _{d}(v,w)=\\big (\\eta _{d}(v_{1}^{(d+1)},w_{1}^{(d+1)}),\\eta _{d}(v_{2}^{(d+1)},w_{2}^{(d+1)}),\\ldots ,\\eta _{d}(v_{a+1}^{(d+1)},w_{a+1}^{(d+1)})\\big ),$ for some integer $a\\ge 0$ .", "Let $v^{\\prime }=\\pi _{\\alpha ^{\\prime }}(v)$ and $w^{\\prime }=\\pi _{\\alpha ^{\\prime }}(w)$ .", "Suppose that $(j-1)r_{d+1}<\\alpha ^{\\prime }\\le jr_{d+1}$ .", "Then note that the $j$ -th block of resolution $d+1$ of $v^{\\prime }$ is $\\pi _{[\\alpha ^{\\prime }-(j-1)r_{d+1}]}(v_{j}^{(d+1)})$ and that of $w^{\\prime }$ is $\\pi _{[\\alpha ^{\\prime }-(j-1)r_{d+1}]}(w_{j}^{(d+1)})$ .", "Then $\\xi _{d}(v^{\\prime },w^{\\prime })$ consists of $j$ coordinates, where for $1\\le i<j$ the $i$ -th coordinate is identical to the $i$ -th coordinate of $\\xi _{d}(v,w)$ and, for $i=j$ , the $j$ -th coordinate is $\\eta _{d}\\circ (\\pi _{[\\alpha ^{\\prime }-(j-1)r_{d+1}]}\\times \\pi _{[\\alpha ^{\\prime }-(j-1)r_{d+1}]})(v_{j}^{(d+1)},w_{j}^{(d+1)}).$ Thus the function $\\xi _{d}$ refines $\\xi _{d}\\circ (\\pi _{\\alpha ^{\\prime }}\\times \\pi _{\\alpha ^{\\prime }})$ coordinate by coordinate (by part (i) of this lemma).", "Hence, by Proposition REF (v), we see that $\\xi _{d}$ refines $\\xi _{d}\\circ (\\pi _{\\alpha ^{\\prime }}\\times \\pi _{\\alpha ^{\\prime }})$ .", "(iii) This follows from $c_{d}=\\xi _{d}\\times \\cdots \\times \\xi _{0}$ , part (ii) of this lemma and Proposition REF (v).", "Lemma REF seems intuitively obvious and might even seem trivial at first sight, but a moment's thought reveals the fact that it is nontrivial.", "To see this, consider the function $ h_d(v,w) = \\lbrace v_i^{(d)}, w_i^{(d)}\\rbrace , $ which is the projection to the second coordinate of $\\eta _d(v,w)$ .", "Then the function $h_d$ fails to satisfy Lemma REF (i).", "Moreover, if the functions $\\xi _d$ and $c_d$ were built using $h_d$ instead of $\\eta _d$ , these would also fail to satisfy the claim of Lemma REF .", "The next lemma completes the proof of one of the promised claims, namely, that $\\Lambda _I$ (or, rather, a generalization thereof) refines $C_I$ .", "Lemma 4.3 Suppose that positive integers $d,p,\\alpha $ and $\\alpha ^{\\prime }$ are given such that $1\\le d\\le p+1$ and $\\alpha ^{\\prime }$ is the maximum integer less than $\\alpha $ divisible by $r_{d}$ .", "Let $\\gamma _{d}$ be the function which takes a pair of vectors $v,w\\in \\lbrace 0,1\\rbrace ^{\\alpha }$ as input and outputs $\\gamma _{d}(v,w)=(c_{p}(v^{\\prime },w^{\\prime }),\\eta _{d-1}(v^{\\prime \\prime },w^{\\prime \\prime })),$ where $v=(v^{\\prime },v^{\\prime \\prime })$ and $w=(w^{\\prime },w^{\\prime \\prime })$ for $v^{\\prime },w^{\\prime }\\in \\lbrace 0,1\\rbrace ^{\\alpha ^{\\prime }}$ and $v^{\\prime \\prime },w^{\\prime \\prime }\\in \\lbrace 0,1\\rbrace ^{\\alpha -\\alpha ^{\\prime }}$ .", "Then $c_{p}\\vert _{\\lbrace 0,1\\rbrace ^{\\alpha }\\times \\lbrace 0,1\\rbrace ^{\\alpha }}$ refines $\\gamma _{d}$ .", "For brevity, we restrict the functions to the set $\\lbrace 0,1\\rbrace ^{\\alpha }\\times \\lbrace 0,1\\rbrace ^{\\alpha }$ throughout the proof.", "By Lemma REF (iii), we know that $c_{p}$ refines $c_{p}\\circ (\\pi _{\\alpha ^{\\prime }}\\times \\pi _{\\alpha ^{\\prime }})$ and hence $c_{p}$ refines the first coordinate of $\\gamma _{d}$ .", "On the other hand, since $\\alpha ^{\\prime }$ is the maximum integer less than $\\alpha $ divisible by $r_{d}$ , the term $\\eta _{d-1}(v^{\\prime \\prime },w^{\\prime \\prime })$ forms the last coordinate of the vector $\\xi _{d-1}(v,w)$ .", "Hence, by Proposition REF (iii), $\\xi _{d-1}$ refines $\\eta _{d-1}(v^{\\prime \\prime },w^{\\prime \\prime })$ .", "By the definition of $c_{p}$ and Proposition REF (iii), we know that $c_{p}$ refines $\\xi _{d-1}$ .", "Therefore, by transitivity (Proposition REF (ii)), we see that $c_{p}$ refines $\\eta _{d-1}(v^{\\prime \\prime },w^{\\prime \\prime })$ .", "Thus, $c_{p}$ refines both coordinates of $\\gamma _{d}$ and hence, by Proposition REF (iv), we see that $c_{p}$ refines $\\gamma _{d}$ ." ], [ "Proof of the main theorem", "In this section we prove Theorem REF , which asserts that for all $\\alpha \\ge 1$ and $p\\ge 1$ , the edge-coloring of the complete graph on the vertex set $\\lbrace 0,1\\rbrace ^{\\alpha }$ given by $c_{p}$ is a $(p+3,p+2)$ -coloring.", "We will prove by induction on $\\alpha $ that every set $S$ with $\|S\| \\le p+3$ receives at least $\|S\| - 1$ distinct colors.", "The base case is when $\\alpha \\le r_{p}$ .", "In this case, for two distinct vectors $v,w\\in \\lbrace 0,1\\rbrace ^{\\alpha }$ , we have $\\xi _{p}(v,w)=\\big (\\eta _{p}(v,w)\\big )=\\big ((1,\\lbrace v,w\\rbrace )\\big ).$ Hence, for a given set $S\\subset \\lbrace 0,1\\rbrace ^{\\alpha }$ , the edges within this set are all colored with distinct colors, thereby implying that at least ${\|S\| \\atopwithdelims ()2}\\ge \|S\|-1$ colors are used.", "Now suppose that $\\alpha >r_{p}$ is given and the claim has been proved for all smaller values of $\\alpha $ .", "Let $S\\subset \\lbrace 0,1\\rbrace ^{\\alpha }$ be a given set with $\|S\|\\le p+3$ .", "For each $1\\le d\\le p$ , let $\\alpha _{d}$ be the largest integer less than $\\alpha $ which is divisible by $r_{d}$ .", "Note that since $r_{d-1}$ divides $r_{d}$ for all $1\\le d\\le p$ , we have $\\alpha _{p}\\le \\alpha _{p-1}\\le \\cdots \\le \\alpha _{1}.$ For $1\\le d\\le p$ , define sets $\\Lambda _{I}^{(d)}$ and $\\Lambda _{E}^{(d)}$ as $\\Lambda _{I}^{(d)}=\\big \\lbrace c_{p}(v,w)\\,:\\,\\pi _{\\alpha _{d}}(v)\\ne \\pi _{\\alpha _{d}}(w),\\, v,w\\in S\\big \\rbrace $ and $\\Lambda _{E}^{(d)}=\\big \\lbrace c_{p}(v,w)\\,:\\,\\pi _{\\alpha _{d}}(v)=\\pi _{\\alpha _{d}}(w),\\, v\\ne w,\\, v,w\\in S\\big \\rbrace .$ Since $\\alpha _{d}$ is divisible by $r_{d}$ , if $\\pi _{\\alpha _{d}}(v)=\\pi _{\\alpha _{d}}(w)$ , then the first $\\mbox{$\\frac{\\alpha _{d}}{r_{d}}$}$ coordinates of $\\xi _{d-1}(v,w)$ will all be zero.", "On the other hand, if $\\pi _{\\alpha _{d}}(v)\\ne \\pi _{\\alpha _{d}}(w)$ , then this is not the case.", "Since $\\xi _{d-1}$ is part of $c_{p}$ , this implies that $\\Lambda _{I}^{(d)}\\cap \\Lambda _{E}^{(d)}=\\emptyset $ .", "Hence, for all $d$ , the number of colors within $S$ is exactly $\|\\Lambda _{I}^{(d)}\|+\|\\Lambda _{E}^{(d)}\|$ .", "It therefore suffices to prove that $\|\\Lambda _{I}^{(d)}\|+\|\\Lambda _{E}^{(d)}\|\\ge \|S\|-1$ for some index $d$ .", "We would like to extract only the important information from the colors in $\\Lambda _{I}^{(d)}$ and $\\Lambda _{E}^{(d)}$ .", "For each $1\\le d\\le p$ and a given pair of vectors $v,w\\in S$ , let $v=(v_{d}^{\\prime },v_{d}^{\\prime \\prime })$ and $w=(w_{d}^{\\prime },w_{d}^{\\prime \\prime })$ for $v_{d}^{\\prime },w_{d}^{\\prime }\\in \\lbrace 0,1\\rbrace ^{\\alpha _{d}}$ and $v_{d}^{\\prime \\prime },w_{d}^{\\prime \\prime }\\in \\lbrace 0,1\\rbrace ^{\\alpha -\\alpha _{d}}$ .", "Define the sets $C_{I}^{(d)}$ and $C_{E}^{(d)}$ as $C_{I}^{(d)}=\\Big \\lbrace \\big (c_{p}(v_{d}^{\\prime },w_{d}^{\\prime }),\\eta _{d-1}(v_{d}^{\\prime \\prime },w_{d}^{\\prime \\prime })\\big )\\,:\\, v_{d}^{\\prime } \\ne w_{d}^{\\prime },\\, v,w\\in S\\Big \\rbrace $ and $C_{E}^{(d)}=\\Big \\lbrace \\lbrace v_{d}^{\\prime \\prime },w_{d}^{\\prime \\prime }\\rbrace \\,:\\, v_{d}^{\\prime }=w_{d}^{\\prime },\\, v_{d}^{\\prime \\prime } \\ne w_{d}^{\\prime \\prime },\\, v,w\\in S\\Big \\rbrace .$ By Lemma REF and Proposition REF (vi), we see that $\|C_{I}^{(d)}\|\\le \|\\Lambda _{I}^{(d)}\|$ .", "We also have $\|C_{E}^{(d)}\|\\le \|\\Lambda _{E}^{(d)}\|$ .", "To see this, suppose that a color $\\lbrace v_{d}^{\\prime \\prime },w_{d}^{\\prime \\prime }\\rbrace \\in C_{E}^{(d)}$ comes from a pair of vectors $v=(v_{d}^{\\prime },v_{d}^{\\prime \\prime })$ and $w=(w_{d}^{\\prime },w_{d}^{\\prime \\prime })$ in $S$ .", "Since $v_{d}^{\\prime }=w_{d}^{\\prime }$ and $\\alpha _{d}$ is divisible by $r_{d}$ , the function $\\eta _{d}$ applied to the last pair of blocks of resolution $d+1$ of $v$ and $w$ is equal to $(i,\\lbrace v_{d}^{\\prime \\prime },w_{d}^{\\prime \\prime }\\rbrace )$ for some integer $i$ .", "Therefore, the last coordinate of $\\xi _{d}(v,w)$ has value $(i, \\lbrace v_{d}^{\\prime \\prime },w_{d}^{\\prime \\prime }\\rbrace )$ .", "This implies that $\|C_{E}^{(d)}\|\\le \|\\Lambda _{E}^{(d)}\|$ .", "Hence, it now suffices to prove that $\|C_{I}^{(d)}\|+\|C_{E}^{(d)}\|\\ge \|S\|-1$ for some index $1\\le d\\le p$ .", "Assume for the sake of contradiction that we have $\|C_{I}^{(d)}\|+\|C_{E}^{(d)}\|\\le \|S\|-2$ for all $1\\le d\\le p$ .", "The following is the key ingredient in our proof.", "Claim 5.1 If $\|C_{I}^{(p)}\|+\|C_{E}^{(p)}\|\\le \|S\|-2$ , then there exists an index $d$ such that $\\eta _{d-1}(c)$ is distinct for each $c\\in C_{E}^{(d)}$ .", "The proof of this claim will be given later.", "Let $d$ be the index guaranteed by this claim and let $C_{I}=C_{I}^{(d)},\\, C_{E}=C_{E}^{(d)}$ .", "Abusing notation, for two vectors $v,w\\in S$ , we will from now on refer to the color between $v$ and $w$ as the corresponding `color' in $C_{I}$ or $C_{E}$ .", "Let $S^{\\prime }=\\pi _{\\alpha _{d}}(S)$ and, for a vector $v^{\\prime }\\in S^{\\prime }$ , let $T_{v^{\\prime }}=\\lbrace v\\,:\\,\\pi _{\\alpha _{d}}(v)=v^{\\prime },\\, v\\in S\\rbrace $ .", "Note that the sets $T_{v^{\\prime }}$ form a partition of $S$ .", "Therefore, $\\sum _{v^{\\prime }\\in S^{\\prime }}\|T_{v^{\\prime }}\|=\|S\|.$ Let $C_{B}$ be the set of colors which appear within the set $S^{\\prime }$ under the coloring $c_{p}$ .", "Since $S^{\\prime }\\subset \\lbrace 0,1\\rbrace ^{\\alpha _{d}}$ and $\\alpha _d < \\alpha $ , the inductive hypothesis implies that $\|C_{B}\|\\ge \|S^{\\prime }\|-1.$ For a color $c\\in C_{E}$ , let $\\mu _{c}$ be the number of (unordered) pairs of vectors $v,w$ such that $c$ is the color between $v$ and $w$ .", "Note that $\\sum _{c\\in C_{E}}\\mu _{c}=\\sum _{v^{\\prime }\\in S^{\\prime }}{\|T_{v^{\\prime }}\| \\atopwithdelims ()2}\\ge \\sum _{v^{\\prime }\\in S^{\\prime }}(\|T_{v^{\\prime }}\|-1).$ Together with the three equations above, the following bound on $\|C_I\|$ , whose proof we defer for a moment, yields a contradiction.", "$ \|C_{I}\|\\ge \|C_{B}\|+\\sum _{c\\in C_{E}}(\\mu _{c}-1).$ Indeed, if this inequality holds, then, by (REF ), (REF ) and (REF ), respectively, we have $\|C_{I}\| + \|C_E\| &\\ge \\left((\|S^{\\prime }\| - 1) + \\sum _{c \\in C_E} (\\mu _c - 1) \\right) + \|C_E\|= (\|S^{\\prime }\| - 1) + \\sum _{c \\in C_E} \\mu _c \\\\&\\ge (\|S^{\\prime }\| - 1) + \\sum _{v^{\\prime } \\in S^{\\prime }} (\|T_{v^{\\prime }}\| - 1)= \\left(\\sum _{v^{\\prime } \\in S^{\\prime }} \|T_{v^{\\prime }}\| \\right) - 1.$ By (REF ), we see that the right hand side is equal to $\|S\| - 1$ .", "Therefore, we obtain $\|C_I\| + \|C_E\| \\ge \|S\| -1$ , which contradicts the assumption that $\|C_I\| + \|C_E\| \\le \|S\| - 2$ .", "To prove (REF ), we examine the interaction between the three sets of colors $C_I$ , $C_B$ and $C_E$ .", "Note that each color $c\\in C_{I}$ is of the form $c=(c^{\\prime },?", ")$ for some $c^{\\prime }\\in C_{B}$ , where the question mark `?'", "stands for an unspecified coordinate.", "This fact already gives the trivial bound $\|C_{I}\| \\ge \|C_{B}\|$ .", "To obtain (REF ), we improve this inequality by considering the `?'", "part of the color and its relation to colors in $C_E$ .", "Take a color $c^{\\prime \\prime }=\\lbrace v^{\\prime \\prime },w^{\\prime \\prime }\\rbrace \\in C_{E}$ and suppose that $c^{\\prime \\prime }$ has multiplicity $\\mu _{c^{\\prime \\prime }} \\ge 2$ .", "Then there exist vectors $x, y \\in S^{\\prime }$ such that $(x,v^{\\prime \\prime }), (x, w^{\\prime \\prime }) \\in T_{x}$ and $(y,v^{\\prime \\prime }), (y, w^{\\prime \\prime }) \\in T_y$ .", "Consider the color of the pairs $\\big ((x,v^{\\prime \\prime }),(y,v^{\\prime \\prime })\\big )$ and $\\big ((x,v^{\\prime \\prime }),(y,w^{\\prime \\prime })\\big )$ in $C_{I}$ .", "These colors are of the form $\\big (c_{p}(x,y),\\eta _{d-1}(v^{\\prime \\prime },v^{\\prime \\prime })\\big ) & =(c_p(x,y),0)\\in C_{I} \\quad \\textrm {and}\\\\\\big (c_{p}(x,y),\\eta _{d-1}(v^{\\prime \\prime },w^{\\prime \\prime })\\big ) & =\\big (c_p(x,y),\\eta _{d-1}(c^{\\prime \\prime })\\big )\\in C_{I}.$ Here we abuse notation and define $\\eta _{d-1}(c^{\\prime \\prime })=\\eta _{d-1}(v^{\\prime \\prime },w^{\\prime \\prime })$ , which is allowed since the right-hand-side is symmetric in the two input coordinates.", "Therefore, having a color $c^{\\prime \\prime }$ with $\\mu _{c^{\\prime \\prime }} \\ge 2$ already implies that $\|C_I\| \\ge \|C_B\| + 1$ .", "We carefully analyze the gain coming from these pairs for each color in $C_E$ .", "To this end, for each $x \\in S^{\\prime }$ , we define $ C_{E, x} = \\Big \\lbrace \\lbrace v^{\\prime \\prime }, w^{\\prime \\prime }\\rbrace \\,:\\, (x,v^{\\prime \\prime }), (x,w^{\\prime \\prime }) \\in T_{x}, \\,\\, v^{\\prime \\prime } \\ne w^{\\prime \\prime } \\Big \\rbrace .", "$ For each $c^{\\prime } \\in C_B$ , we will count the number of colors of the form $(c^{\\prime }, ?)", "\\in C_I$ .", "There are two cases.", "Case 1 : For all $x , y \\in S^{\\prime }$ with $c_p(x,y) = c^{\\prime }$ , $C_{E,x} \\cap C_{E, y} = \\emptyset $ .", "Apply the trivial bound asserting that there is at least one color of the form $(c^{\\prime }, ?", ")$ in $C_I$ .", "Case 2 : There exists a pair $x , y \\in S^{\\prime }$ with $c_p(x,y) = c^{\\prime }$ such that $C_{E,x} \\cap C_{E, y} \\ne \\emptyset $ .", "If we have $c^{\\prime \\prime } \\in C_{E,x} \\cap C_{E, y}$ for some $x, y \\in S^{\\prime }$ with $c_p(x,y) = c^{\\prime }$ , then, by the observation above, we have both $(c^{\\prime }, 0)$ and $(c^{\\prime }, \\eta _{d-1}(c^{\\prime \\prime }))$ in $C_I$ .", "This shows that the number of colors in $C_I$ of the form $(c^{\\prime }, ?", ")$ is at least $ \\left\|\\lbrace (c^{\\prime }, 0)\\rbrace \\cup \\left\\lbrace (c^{\\prime }, \\eta _{d-1}(c^{\\prime \\prime })) \\,:\\, \\exists x,y \\in S^{\\prime },\\,c_p(x,y) = c^{\\prime },\\,\\, c^{\\prime \\prime } \\in C_{E,x} \\cap C_{E, y} \\right\\rbrace \\right\|.", "$ By Claim REF , the function $\\eta _{d-1}$ is injective on $C_E$ and thus the above number is equal to $ 1 + \\left\| \\left\\lbrace c^{\\prime \\prime } \\,:\\, \\exists x,y \\in S^{\\prime },\\,c_p(x,y) = c^{\\prime },\\,\\, c^{\\prime \\prime } \\in C_{E,x} \\cap C_{E, y} \\right\\rbrace \\right\|.", "$ By combining cases 1 and 2, we see that the number of colors in $C_I$ satisfies $\|C_I\| \\ge & \\,\\, \|C_B\| + \\sum _{c^{\\prime } \\in C_B} \\left\| \\left\\lbrace c^{\\prime \\prime } \\,:\\, \\exists x,y \\in S^{\\prime },\\,c_p(x,y) = c^{\\prime },\\,\\, c^{\\prime \\prime } \\in C_{E,x} \\cap C_{E, y} \\right\\rbrace \\right\| \\\\=\\,\\,&\\,\\, \|C_B\| + \\sum _{c^{\\prime \\prime } \\in C_E} \\left\| \\left\\lbrace c^{\\prime } \\,:\\, \\exists x,y \\in S^{\\prime },\\,c_p(x,y) = c^{\\prime },\\,\\, c^{\\prime \\prime } \\in C_{E,x} \\cap C_{E, y} \\right\\rbrace \\right\|.$ For a fixed color $c^{\\prime \\prime } \\in C_E$ , there are precisely $\\mu _{c^{\\prime \\prime }}$ vectors $x \\in S^{\\prime }$ for which the color $c^{\\prime \\prime }$ is in $C_{E,x}$ .", "Hence, by the induction hypothesis, for each fixed $c^{\\prime \\prime }$ , we have $ \\left\| \\left\\lbrace c^{\\prime } \\,:\\, \\exists x,y \\in S^{\\prime },\\,c_p(x,y) = c^{\\prime },\\,\\, c^{\\prime \\prime } \\in C_{E,x} \\cap C_{E, y} \\right\\rbrace \\right\|\\ge \\mu _{c^{\\prime \\prime }} - 1.", "$ Thus we obtain $ \|C_I\| \\ge \|C_B\| + \\sum _{c^{\\prime \\prime } \\in C_E} (\\mu _{c^{\\prime \\prime }} - 1), $ which is (REF )." ], [ "Proof of Claim ", "Claim REF asserts that there exists an index $d$ such that $\\eta _{d-1}(c)$ is distinct for each $c\\in C_{E}^{(d)}$ .", "It will be useful to consider the function $h_{d}$ , which is defined as follows: for distinct vectors $v$ and $w$ , define $h_{d}(v,w)=\\lbrace v_{i}^{(d)},w_{i}^{(d)}\\rbrace ,$ where $v_{i}^{(d)},w_{i}^{(d)}$ are the first pair of blocks of resolution $d$ for which $v_{i}^{(d)}\\ne w_{i}^{(d)}$ .", "Also, define $h_{d}(v,v)=0$ for all vectors $v$ .", "Note that we can also define $h_{d}$ over unordered pairs $\\lbrace v,w\\rbrace $ of vectors as $h_{d}(\\lbrace v,w\\rbrace ) = h_{d}(v,w)$ , since $h_d(v,w) = h_d(w,v)$ for all pairs $v$ and $w$ .", "Throughout the subsection, by abusing notation, we will be applying $h_d$ to both ordered and unordered pairs without further explanation.", "Recall that $\\eta _{d}(v,w)=(i,\\lbrace v_{i}^{(d)},w_{i}^{(d)}\\rbrace )$ and $\\eta _{d}(v,v)=0$ and, therefore, $\\eta _{d}$ refines $h_{d}$ (both considered as functions over the domain $C_{E}^{(d)}$ ).", "Hence, to prove the claim, it suffices to prove that $h_{d-1}(c)$ is distinct for each $c\\in C_{E}^{(d)}$ .", "Another important observation is that for all $1\\le d\\le p$ , we can redefine the sets $C_{E}^{(d)}$ as $C_{E}^{(d)}=\\big \\lbrace h_{d}(v,w)\\,:\\,\\pi _{\\alpha _{d}}(v)=\\pi _{\\alpha _{d}}(w),\\, v\\ne w,\\, v,w\\in S\\big \\rbrace .$ We first prove that there is a certain monotonicity between the sets $C_{E}^{(d)}$ for $1\\le d\\le p$ .", "Claim 5.2 For all $d$ satisfying $2\\le d\\le p$ , there exists an injective map $\\jmath _{d}\\,:\\, C_{E}^{(d-1)}\\rightarrow C_{E}^{(d)}$ which maps $\\lbrace x,y\\rbrace \\in C_{E}^{(d-1)}$ to $\\jmath _{d}(x,y)=\\Big \\lbrace (v,x),(v,y)\\Big \\rbrace \\in C_{E}^{(d)},$ for some vector $v\\in \\lbrace 0,1\\rbrace ^{\\alpha _{d-1}-\\alpha _{d}}$ depending on the color $\\lbrace x,y\\rbrace $ .", "Furthermore, $h_{d-1}\\circ \\jmath _{d}$ is the identity map on $C_{E}^{(d-1)}$ .", "Take a color $\\lbrace x,y\\rbrace \\in C_{E}^{(d-1)}$ and assume that $\\lbrace x,y\\rbrace =h_{d-1}(v_{x},v_{y})$ for $v_{x},v_{y}\\in S$ .", "By the definition of $C_{E}^{(d-1)}$ , we may take $v_{x}$ and $v_{y}$ of the form $v_{x}=(v_{0},x)\\quad \\textrm {and}\\quad v_{y}=(v_{0},y),$ for some vector $v_{0}\\in \\lbrace 0,1\\rbrace ^{\\alpha _{d-1}}$ .", "Fix an arbitrary such pair $(v_{x},v_{y})$ for each $\\lbrace x,y\\rbrace \\in C_{E}^{(d-1)}$ .", "Let $v_{0}=(v_{1},v_{2})$ for $v_{1}\\in \\lbrace 0,1\\rbrace ^{\\alpha _{d}}$ and $v_{2}\\in \\lbrace 0,1\\rbrace ^{\\alpha _{d-1}-\\alpha _{d}}$ .", "Then $v_{x}=(v_{1},v_{2},x)$ and $v_{y}=(v_{1},v_{2},y)$ .", "Since $\\pi _{\\alpha _{d}}(v_{x})=v_{1}=\\pi _{\\alpha _{d}}(v_{y}),$ we see that $h_{d}(v_{x},v_{y})=\\Big \\lbrace (v_{2},x),(v_{2},y)\\Big \\rbrace \\in C_{E}^{(d)}.$ Define $\\jmath _{d}(x,y)=h_{d}(v_{x},v_{y})$ and note that the range of $\\jmath _{d}$ is indeed $C_{E}^{(d)}$ .", "Moreover, since $v_{2}$ is a vector of length $\\alpha _{d-1}-\\alpha _{d}$ which is divisible by $r_{d-1}$ , we see that $ h_{d-1}(\\jmath _{d}(x,y)) = h_{d-1}\\Big ( (v_2,x), (v_2,y) \\Big ) = \\lbrace x,y\\rbrace .$ The claim follows.", "In particular, Claim REF implies that $\|C_{E}^{(1)}\|\\le \|C_{E}^{(2)}\|\\le \\cdots \\le \|C_{E}^{(p)}\|.$ If $\|C_{E}^{(1)}\|\\le 1$ , then $d=1$ trivially satisfies the required condition.", "Hence, we may assume that $\|C_{E}^{(1)}\|\\ge 2$ .", "On the other hand, recall that we are assuming that $\|C_{I}^{(p)}\|+\|C_{E}^{(p)}\|\\le \|S\|-2\\le p+1$ .", "If $\|C_{I}^{(p)}\|=0$ , then there exists at most one element $v_p \\in \\pi _{\\alpha _{p}}(S)$ and all elements of $S$ are of the form $(v_p,x)$ for some $x \\in \\lbrace 0,1\\rbrace ^{\\alpha - \\alpha _p}$ .", "But then $ \|C_E^{(p)}\| \\ge {\|S\| \\atopwithdelims ()2} \\ge \|S\| - 1,$ contradicting our assumption.", "Therefore, we may assume that $\|C_{I}^{(p)}\|\\ge 1$ , from which it follows that $\|C_{E}^{(p)}\|\\le p$ .", "Hence, $2\\le \|C_{E}^{(1)}\|\\le \|C_{E}^{(2)}\|\\le \\cdots \\le \|C_{E}^{(p)}\|\\le p.$ If $p=1$ , this is impossible.", "If $p\\ge 2$ , then, by the pigeonhole principle, there exists an index $d$ such that $\|C_{E}^{(d-1)}\|=\|C_{E}^{(d)}\|$ .", "For this index, the map $\\jmath _{d}$ defined in Claim REF becomes a bijection.", "Then, since $h_{d-1}\\circ \\jmath _{d}$ is the identity map on $C_{E}^{(d-1)}$ , we see that $h_{d-1}(c)$ are distinct for all $c\\in C_{E}^{(d)}$ .", "This proves the claim." ], [ "Better than $(p+3,p+2)$ -coloring", "Let $r=\\sqrt{\\frac{p+4}{2}}$ .", "We can in fact prove that $c_{p}$ is a $\\left(p+\\lfloor r\\rfloor +1,p+\\lfloor r\\rfloor \\right)$ -coloring.", "This improvement comes from exploiting the slackness of the inequality (REF ) used in Subsection REF .", "To see this, we replace the bound on $S$ by $\|S\|\\le p+r+1$ in the proof given above.", "Since we have already proved the result for $\|S\| \\le p+3$ , we may assume that $\|S\| \\ge p+4$ .", "If $\|C^{(p)}_{I}\|\\ge r-1$ , then we have $\|C^{(p)}_{E}\|\\le \|S\|-2-\|C^{(p)}_{I}\|\\le p$ and we can proceed as in the proof above.", "We may therefore assume that $\|C^{(p)}_{I}\|<r-1$ .", "Let $S_p =\\pi _{\\alpha _p}(S)$ .", "Then, since $\|S_p\|-1\\le \|C^{(p)}_{I}\|<r-1,$ we know that $\|S_p\|<r$ .", "Since $\\sum _{v\\in S_p}\|\\pi _{\\alpha _{p}}^{-1}(v)\|=\|S\|,$ there exists a $v\\in S_p$ such that $\|\\pi _{\\alpha _{p}}^{-1}(v)\|\\ge \\frac{\|S\|}{\|S_{p}\|}$ .", "Note that every pair of vectors $w_{1},w_{2}\\in \\pi _{\\alpha _{p}}^{-1}(v)$ gives a distinct emerging color.", "Moreover, by the inductive hypothesis, we have at least $\|S_p\|-1$ inherited colors.", "Hence, the total number of colors in the coloring $c_{p}$ within the set $S$ is at least $\|S_p\|-1+{\|\\pi _{\\alpha _{p}}^{-1}(v)\| \\atopwithdelims ()2}\\ge \|S_p\|-1+\\frac{1}{2}\\frac{\|S\|}{\|S_{p}\|}\\left(\\frac{\|S\|}{\|S_{p}\|}-1\\right),$ which, since $ \|S_{p}\|< r=\\sqrt{\\frac{p+4}{2}}\\le \\sqrt{\\frac{\|S\|}{2}},$ is minimized when $\|S_{p}\|$ is maximized.", "Thus the number of colors within the set $S$ is at least $\\sqrt{\\frac{\|S\|}{2}}-1+\|S\|-\\sqrt{\\frac{\|S\|}{2}}=\|S\|-1.$ This concludes the proof." ], [ "Using fewer colors", "Recall that the coloring $c_{p}$ was built from the functions $\\eta _{d}(v,w)=\\big (i,\\lbrace v_{i}^{(d)},w_{i}^{(d)}\\rbrace \\big ),$ where $i$ is the minimum index for which $v_{i}^{(d)}\\ne w_{i}^{(d)}$ .", "The function $\\eta _{d}$ can in fact be replaced by the function $h_{d}(v,w)=\\big \\lbrace v_{i}^{(d)},w_{i}^{(d)}\\big \\rbrace $ (note that this is the function used in Section REF ).", "In other words, even if we replace all occurrences of $\\eta _{d}$ with $h_{d}$ in the definition of $c_{p}$ , we can still show that $c_{p}$ is a $(p+3,p+2)$ -coloring.", "Moreover, there exists a constant $a_p$ such that the coloring of the complete graph on $n$ vertices defined in this way uses only $2^{a_{p} (\\log n)^{1-1/(p+1)}}$ colors.", "That is, we gain a $\\log \\log n$ factor in the exponent compared to Theorem REF .", "The tradeoff is that the proof is now more complicated, the chief difficulty being to find an appropriate analogue of Lemma REF which works when $\\eta _d$ is replaced by $h_d$ ." ], [ "Top-down approach", "There is another way to understand our coloring as a generalization of Mubayi's coloring.", "Recall that Mubayi's coloring is given as follows: for two vectors $v,w\\in [m]^{t}$ satisfying $v=(v_{1},\\ldots ,v_{t})$ and $w=(w_{1},\\ldots ,w_{t})$ , let $c(v,w)=\\big (\\lbrace v_{i},w_{i}\\rbrace ,a_{1},a_{2},\\ldots ,a_{t}\\big ),$ where $i$ is the minimum index for which $v_{i}\\ne w_{i}$ and $a_{j}=0$ if $v_{j}=w_{j}$ and $a_{j}=1$ if $v_{j}\\ne w_{j}$ .", "Suppose that we are given positive integers $t_{1}$ and $t_{2}$ .", "For two vectors $v,w\\in [m]^{t_{1}t_{2}}$ , let $v=(v_{1}^{(1)},\\ldots ,v_{t_{2}}^{(1)})$ and $w=(w_{1}^{(1)},\\ldots ,w_{t_{2}}^{(1)})$ for vectors $v_{i}^{(1)}\\in [m]^{t_{1}}$ and $w_{i}^{(1)}\\in [m]^{t_{1}}$ .", "Define the coloring $c^{(2)}$ as $c^{(2)}(v,w)=\\big (\\lbrace v_{i}^{(1)},w_{i}^{(1)}\\rbrace ,c(v_{1}^{(1)},w_{1}^{(1)}),\\ldots ,c(v_{t_{2}}^{(1)},w_{t_{2}}^{(1)})\\big ),$ where $i$ is the minimum index for which $v_i^{(1)} \\ne w_i^{(1)}$ .", "Note that this can also be understood as a variant of $c$ , where we record more information in the $(a_{1},\\ldots ,a_{t})$ part of the vector (this is a `top-down' approach and the previous definition is a `bottom-up' approach).", "The coloring $c^{(2)}$ is essentially equivalent to $c_2$ defined in Section REF above and can be further generalized to give a coloring corresponding to $c_p$ for $p \\ge 3$ .", "However, the proof again becomes more technical for this choice of definition.", "One advantage of defining the coloring using this top-down approach is that it becomes easier to see why the coloring $c_p$ on $K_{n_2}$ contains the coloring $c_p$ on $K_{n_1}$ , where $n_1 < n_2$ , as an induced coloring.", "To see this in the example above, suppose that $n_1 = m^{t_1 t_2}$ and $n_2 = n^{s_1 s_2}$ for $m \\le n$ , $t_1 \\le s_1$ and $t_2 \\le s_2$ .", "Then the natural injection from $[m]$ to $[n]$ extends to an injection from $[m]^{t_1}$ to $[n]^{s_1}$ and then to an injection from $[m]^{t_1 t_2}$ to $[n]^{s_1 s_2}$ .", "This injection shows that the coloring $c^{(2)}$ on $K_{n_2}$ contains the coloring $c^{(2)}$ on $K_{n_1}$ as an induced coloring.", "As in Section REF , it then follows that $c^{(2)}$ (and thus $c_2$ ) fails to be a $(q,q^{\\varepsilon })$ -coloring for large enough $q$ .", "Similarly, for all fixed $p \\ge 3$ , we can show that $c_p$ fails to be a $(q,q^{\\varepsilon })$ -coloring for large enough $q$ ." ], [ "Stronger properties", "We can show (see [1]) that Mubayi's coloring, discussed in Section REF , actually has the following stronger property: for every pair of colors, the graph whose edge set is the union of these two color classes has chromatic number at most three (previously, we only established the fact that the clique number is at most three).", "We suspect that this property can be generalized.", "Question 6.1 Let $p \\ge 4$ be an integer.", "Does there exist an edge-coloring of the complete graph $K_n$ with $n^{o(1)}$ colors such that the union of every $p-1$ color classes has chromatic number at most $p$ ?", "We do not know whether our coloring has this property or not." ], [ "Lower bound", "Some work has also been done on the lower bound for $f(n,p,p-1)$ .", "As mentioned in the introduction, for $p = 3$ it is known that $c^{\\prime }\\frac{\\log n}{\\log \\log n} \\le f(n,3,2) \\le c \\log n$ .", "For $p = 4$ , the gap between the lower and upper bounds is much wider.", "The well-known bound $r_k(4) \\le k^{ck}$ on the multicolor Ramsey number of $K_4$ translates to $f(n,4,3) \\ge c\\frac{\\log n}{\\log \\log n}$ , while Mubayi's coloring gives an upper bound of $f(n,4,3) \\le e^{c\\sqrt{\\log n}}$ .", "The lower bound has been improved, first by Kostochka and Mubayi [7], to $f(n,4,3) \\ge c\\frac{\\log n}{\\log \\log \\log n}$ and then, by Fox and Sudakov [6], to $f(n,4,3) \\ge c\\log n$ , which is the current best known bound.", "For $p \\ge 5$ , we can obtain a similar lower bound from the following formula, valid for all $p$ and $q$ .", "$ f\\Big (n f(n,p-1,q-1), p, q\\Big ) \\ge f(n,p-1,q-1).$ To prove this formula, put $N= nf(n,p-1,q-1)$ and consider an edge-coloring of $K_N$ with fewer than $f(n,p-1,q-1)$ colors.", "It suffices to show that there exists a set of $p$ vertices which uses at most $q-1$ colors on its edges.", "If $f(n,p-1,q-1)=1$ , then the inequality above is trivially true.", "If not, then for a fixed vertex $v$ , there exists a set $V$ of at least $\\left\\lceil \\frac{N-1}{f(n,p-1,q-1) - 1} \\right\\rceil \\ge n$ vertices adjacent to $v$ by the same color.", "Since the edges within the set $V$ are colored by fewer than $f(n,p-1,q-1)$ colors, the definition of $f(n,p-1,q-1)$ implies that we can find a set $X$ of $p-1$ vertices with at most $q-2$ colors used on its edges.", "It follows that the set $X \\cup \\lbrace v\\rbrace $ is a set of $p$ vertices with at most $q-1$ colors used on its edges.", "The claim follows.", "From (REF ) and the lower bound $f(n,4,3) \\ge c \\log n$ , one can deduce that $f(n,p,p-1) \\ge (1 + o(1)) f(n,4,3) \\ge (c + o(1)) \\log n$ for all $p \\ge 5$ .", "On the other hand, since the best known upper bound on $f(n,p,p-1)$ is $f(n,p,p-1) \\le 2^{16p (\\log n)^{1 - 1/(p-2)} \\log \\log n},$ the gap between the upper and lower bounds gets wider as $p$ gets larger.", "It would be interesting to know whether either bound can be substantially improved.", "In particular, the following question seems important.", "Question 6.2 For $p \\ge 5$ , can we give better lower bounds on $f(n,p,p-1)$ than the one which follows from $f(n,4,3)$ ?" ] ]	1403.0250
[ [ "Characterization and Control of Quantum Spin Chains and Rings" ], [ "Abstract Information flow in quantum spin networks is considered.", "Two types of control -- temporal bang-bang switching control and control by varying spatial degrees of freedom -- are explored and shown to be effective in speeding up information transfer and increasing transfer fidelities.", "The control is model-based and therefore relies on accurate knowledge of the system parameters.", "An efficient protocol for simultaneous identification of the coupling strength and the exact number of spins in a chain is presented." ], [ "Introduction", "Nature, at a fundamental level, is governed by the laws of quantum mechanics.", "Until recently quantum phenomena were mostly studied by physicists but significant advances in theory and technology are increasingly pushing quantum phenomena into the realm of engineering, as building blocks for novel technologies and applications from chemistry to computing.", "Among the interesting applications are spin networks.", "The latter have many potential applications including spintronics and as networks for transmitting quantum information between processing nodes on a chip, for example.", "It is the latter application that is considered in this paper.", "Information in quantum spin networks is encoded in quantum states and its propagation is governed by the Schrodinger equation.", "Recent work has shown that this leads to new phenomena such as the emergence of anti-gravity centers for spin chains [1].", "The dynamic behavior of even simple spin networks is complicated.", "For example, an excitation created at one end of a linear chain of spins, does not propagate in a classical fashion, hopping between neighboring spins from one end to the next.", "Rather the excitation creates a wavepacket, which is a superposition of eigenstates of the Hamiltonian, that evolves, dispersing and refocusing.", "Information transfer from one spin to another in a spin network is therefore not straightforward.", "Neglecting environmental decoherence, quantum transport is determined by the Hamiltonian of a system $H$ via the Schrodinger equation $\\imath \\hbar \\frac{\\partial }{\\partial t}\\Psi ({\\bf x},t) = H \\Psi ({\\bf x},t),$ where $\\imath =\\sqrt{-1}$ and $\\hbar $ is the Planck constant divided by $2\\pi $ .", "We shall choose units such that $\\hbar =1$ .", "Expressing $\\Psi ({\\bf x},0)$ as a linear combination of eigenfunctions $\\phi _n({\\bf x})$ of the Hamiltonian, $\\Psi ({\\bf x},0) = \\sum _n c_n \\phi _n({\\bf x}),$ where $H\\phi _n({\\bf x})=E_n\\phi _n({\\bf x})$ and $E_n$ is a real number corresponding to the energy of $\\phi _n({\\bf x})$ , we see immediately that for a static Hamiltonian $H$ $\\Psi ({\\bf x},t) = e^{-\\imath H t} \\Psi ({\\bf x},0)= \\sum _n c_n e^{-\\imath E_n t} \\phi _n({\\bf x}).$ $\\Psi ({\\bf x},t)$ governs the evolution of quantum states and the propagation of information encoded in it." ], [ "Quantum Spin Networks", "A quantum spin network for our purposes is simply a collection of $N$ spin$-\\tfrac{1}{2}$ particles arranged in space with some of coupling between spins specified by an interaction Hamiltonian $H = \\sum _{m,n=1}^{N} J_{mn}\\left(\\sigma ^x_m \\sigma ^x_{n} + \\sigma ^y_m \\sigma ^y_{n}+ \\epsilon \\sigma ^z_m\\sigma ^z_{n} \\right).$ $\\epsilon $ is a constant that depends on the type of interaction with $\\epsilon =0$ for XX coupling and $\\epsilon =1$ for Heisenberg coupling, being common.", "$J_{mn}$ is the strength of the coupling between spin $m$ and spin $n$ , usually proportional to the cubic power of the physical distance between the two spins.", "The factors $\\sigma ^{x,y,z}_i$ denote the single spin Pauli operators $\\sigma ^x= \\begin{pmatrix} 0 & 1 \\\\ 1 & 0 \\end{pmatrix}, \\quad \\sigma ^y= \\begin{pmatrix} 0 & -\\imath \\\\ \\imath & 0 \\end{pmatrix}, \\quad \\sigma ^z= \\begin{pmatrix} 1 & 0 \\\\ 0 & -1 \\end{pmatrix}.$ $n$ indicates the position of the spin the operator is acting on $\\sigma ^{x,y,z}_n =I_{2\\times 2} \\otimes \\ldots \\otimes I_{2 \\times 2} \\otimes \\sigma ^{x,y,z}\\otimes I_{2\\times 2}\\otimes \\ldots \\otimes I_{2 \\times 2},$ where the factor $\\sigma ^{x,y,z}$ occupies the $n$ th position among the $N$ factors.", "The system Hilbert space on which $H$ acts is conveniently taken as $\\mathcal {H}:=\\mathbb {C}^{2^N}$ .", "We restrict our attention in this paper to the single excitation subspace, i.e., it is assumed that the total number of excitations in the network is one.", "The state space of the network is then spanned by the subset of $N$ single excitation quantum states $\\lbrace \| n \\rangle :n=1,\\dots ,N\\rbrace $ , where $\| n \\rangle =\| \\uparrow \\uparrow \\ldots \\uparrow \\downarrow \\uparrow \\ldots \\uparrow \\uparrow \\rangle $ indicates that the excitation is localized at spin $n$ .", "However, unlike in a classical network the system can be in any superposition of these basis states.", "The natural coupling among the spins allows an excitation at $n$ to drift towards an excitation at $m$ , but the fidelity is limited by $\\begin{split}p_{mn}(t) &= \\left\| \\langle m \| e^{-\\imath H_1 t} \| n \\rangle \\right\|^2 \\\\&= \\left\| \\sum _{k=0}^{\\tilde{N}} \\langle m \|\\Pi _k\| n \\rangle e^{-\\imath \\lambda _k t}\\right\| ^2\\\\&\\le \\left(\\sum _{k=0}^{\\tilde{N}} \\left\| \\langle m \|\\Pi _k\| n \\rangle \\right\|\\right)^2=:p_{mn}^,\\end{split}$ where $\\Pi _k$ is the projector onto the $k$ th eigenspace of the Hamiltonian $H = \\sum _k \\lambda _k \\Pi _k$ and the $\\Pi _k, k=1,\\dots , N,$ correspond to the single excitation subspace $\\mathcal {H}_1$ .", "$p_{mn}^$ , also referred to as Information Transfer Capacity (ITC), is an upper bound on $p_{mn}(t)$ , which is attainable if there exist a time $t\\ge 0$ such that $e^{-\\imath \\lambda _k t} = s_k e^{i\\phi } \\qquad \\forall k \\mbox{ s.t. }", "s_k \\ne 0,$ where $s_k = \\mathop {\\rm Sgn}(\\langle i \| \\Pi _k \| j \\rangle ) \\in \\lbrace 0,\\pm 1\\rbrace $ is a sign factor and $\\phi $ an arbitrary global phase factor.", "Terms with $s_k=0$ correspond to eigenspaces that have no overlap with the initial or final state and can be ignored.", "Restricting ourselves to the set $K^{\\prime }$ of indices for which $s_k=\\pm 1$ $s_k = \\exp \\left[-\\imath \\pi \\left(2n_k + \\tfrac{1}{2}(s_k-1)\\right) \\right]\\qquad \\forall k\\in K^{\\prime }$ where $n_k$ is an integer.", "Inserting this into (REF ), taking the logarithm and dividing by $-\\imath $ $\\lambda _k t = 2\\pi n_k + \\tfrac{\\pi }{2}(s_k-1)-\\phi \\qquad \\forall k \\in K^{\\prime }.$ To obtain meaningful constraints independent of the arbitrary phase $\\phi $ we subtract the equations $(\\lambda _k-\\lambda _\\ell ) t = 2\\pi (n_k-n_\\ell ) + \\tfrac{\\pi }{2}(s_k-s_\\ell )\\qquad \\forall k,\\ell \\in K^{\\prime }.$ These conditions are necessary and sufficient for attainability and physical, involving only differences of the eigenvalues, which are observable and independent of arbitrary phases.", "Broadly speaking the bounds are attainable in principle, i.e., we can get arbitrarily close to achieving the maximum transition probability simply by waiting the right amount of time $t$ , if the transition frequencies $\\omega _{1k}$ are not rationally dependent.", "In the context of spin chains an attainable upper bound of 1 for the transition probability $m\\rightarrow n$ means that the network is capable for transferring an excitation from node $m$ to node $n$ with fidelity arbitrarily close to 1.", "Previous work has shown that the maximum information transfer capacity between the end spins in a chain with uniform coupling between nearest neighbors is typically 1 regardless of the length of the chain.", "However, unlike in the classical case, the maximum transfer capacity from an end spin to a spin in the middle of the chain is far less than 1.", "Thus, it is possible for an excitation to move from one end of the chain to the other without passing through the center." ], [ "Control of Information Transfer", "The information transfer capacities of spin networks are interesting from a theoretical point of view in terms of understanding information flow in quantum networks and the restrictions resulting from quantum mechanical evolution.", "However, even if perfect state transfer between two nodes in a spin network is possible, in practice the time required to achieve a sufficiently high fidelity can be very long.", "For example, for a spin chain with only five nodes and uniform Heisenberg coupling, computation of the maximum information transfer capacity and attainability considerations show that state transfer from 1 to 5 can be achieved with arbitrarily high fidelity.", "However, when the transfer time is limited to say 1000 time steps in units of $J^{-1}$ then the attainable fidelity is only around 90%.", "This is where control becomes relevant.", "Previous work has shown that even bang-bang switching control of a local perturbation can significantly speed up information transfer between two ends of a chain with Heisenberg coupling [2].", "The type of control employed in this work was very simple, requiring no more than bang-bang switching of a fixed local perturbation $H_C$ to the Hamiltonian so that the evolution of the system was governed by the original system Hamiltonian $H_0$ when the control was turned off, and by the perturbed Hamiltonian $H_0+H_C$ when the control was turned on.", "The control perturbations $H_C$ considered in [2] involved local perturbation of the coupling $J_{12}$ between the first pair of spins.", "As manipulating the coupling strength between spins can be difficult to achieve we can alternatively consider a perturbation that involves local detuning.", "Lie algebra considerations show that for a chain with Heisenberg coupling a simple detuning perturbation of the form $H_C=\\sigma _1^z$ is sufficient for full controllability of the system on the single excitation subspace [3].", "Optimizing the switching times as described in [2] shows that we can achieve high-fidelity state transfer.", "Fig.", "REF shows an example of an optimal switching sequence and the corresponding transfer probability $p_{17}(t)$ as a function of time.", "We see that we can indeed achieve constructive interference or refocusing of the excitation at the target time at the desired end node.", "For the purposes of better illustration of the dynamical evolution we have chosen a short chain but the procedure is effective for chains with hundreds of nodes, although the resulting trajectories are extremely complex.", "Figure: Example of switching control and corresponding evolution oftransfer probability p 17 (t)p_{17}(t) for Heisenberg spin chain of length 7.A perhaps even more interesting example for control is information transfer between nodes in a network beyond what is possible without control according to the maximum information transfer capacity.", "A simple example of this type is information transfer between nodes in a ring.", "For this arrangement of spins it can be shown that the transfer fidelity between nodes is limited and usually strictly less than 1.", "In particular this is the case for information transfer from node 1 to 4, for example, in a ring with seven nodes.", "In this case we can derive analytic expressions for the maximum information transfer capacity, which show that $p_{14}(t)$ is strictly less than 1 [4].", "Unfortunately, unlike for chains, permutation symmetry of the nodes in a ring shows that applying a control of type $\\sigma _1^{z}$ to the first spin is not sufficient for controllability.", "For example, the operation of swapping spins $k$ and $N+2-k$ commutes both with $H_0$ and $H_C$ and is therefore a symmetry.", "There are two further symmetries and the Lie algebra dimension is only 17, and numerical test of the optimization show that the attainable fidelity is limited, consistent with the dynamical constraints imposed by the symmetries and dark states [5].", "An alternative to temporal on-off switching control is the application of spatially distributed static perturbations.", "In this case we use the spatial degrees of freedom to control the flow of information.", "We choose the control Hamiltonian to be $H=\\operatorname{diag}(c_1,\\ldots ,c_N)$ and optimize the spatial biases $c_n$ to maximize the transfer fidelity from the initial node to the target node at a certain target time.", "We can choose a fixed target time $T$ and find ${\\bf c}=(c_n)$ to maximize $\|\\langle m \| \\exp [-\\imath T (H_0+H_C({\\bf c}))] \| n \\rangle \|^2,$ or we can let $T$ vary within a certain range.", "An example of the resulting biases and evolution of the transfer probability for transfer from node 1 to node 4 and 5, respectively, is shown in Fig.", "REF .", "The transfer fidelities are on the order of $1-10^{-4}$ .", "The solutions are not unique.", "The top graph shows larger biases with less variation across nodes, which may be preferable from a practical point of view but has the disadvantage of resulting in faster oscillations.", "They are therefore less robust with regard to variations in the readout time.", "The bottom graph shows another solution with a large bias on one node, which suppresses rapid oscillations in the transfer probability $p_{15}(t)$ and suggests greater robustness with regard to readout mistiming.", "Figure: Example of spatial bias control and corresponding evolution oftransfer probability p 14 (t)p_{14}(t) (top) and p 15 (t)p_{15}(t) (bottom)for Heisenberg spin ring of length 7.", "The red lines in the graphshow the corresponding transfer probabilities without control." ], [ "Identification of System Parameters", "The examples show that both types of control can be very effective with suitable optimization.", "However, the control design in both cases is model-based, and in practice the model parameters for spin networks will often at best be known approximately and may vary, in particular for engineered systems subject to fabrication tolerances.", "Previous work on control suggests reasonable robustness of optimal controls with regard to model uncertainties [2], but experimental system identification is expected to be crucial for the success of quantum control of engineered spin networks.", "In practice there are numerous uncertainties ranging from the network topology, to the precise number of spins in the network and the strength of the coupling between connected nodes.", "System identification for quantum systems, including spin networks, has rapidly become a hot topic generating numerous papers [6] — [16].", "However, these papers make numerous assumptions on the available resources, e.g., requiring prior knowledge of the precise topology, including the exact number of spins and high-precision quantum state tomography, resources and knowledge that may not be available in practice.", "In the following we consider a simple protocol to identify both the number of spins in the network and the coupling strength for rings with uniform coupling using a simple measurement protocol that requires only a small number of binary-outcome measurements on fixed spin in the network.", "We assume uniform coupling strength and that we can initialize the system in state $\| 1 \\rangle $ and measure it in the same state $\| 1 \\rangle $ at time $t \\in [0,T]$ , resulting in a measurement outcome of 0 or 1 only.", "$T$ must be sufficiently large to capture the complete dynamics of this measurement trace.", "The Hamiltonian of the system is parametrized as an $N \\times N$ matrix $H_1(N,J) = \\begin{pmatrix}0 & J & \\ldots & 0 & 0 & 0 & \\ldots & 0 & J \\\\J & 0 & \\ldots & 0 & 0 & 0 & \\ldots & 0 & 0 \\\\\\vdots & \\vdots & \\ddots & \\vdots & \\vdots & \\vdots & & \\vdots & \\vdots \\\\0 & 0 & \\ldots & 0 & J & 0 & \\ldots & 0 & 0 \\\\0 & 0 & \\ldots & J & 0 & J & \\ldots & 0 & 0 \\\\0 & 0 & \\ldots & 0 & J & 0 & \\ldots & 0 & 0\\\\\\vdots & \\vdots & & \\vdots & \\vdots & \\vdots & \\ddots & \\vdots & \\vdots \\\\0 & 0 & \\ldots & 0 & 0 & 0 & \\ldots & 0 & J \\\\J & 0 & \\ldots & 0 & 0 & 0 & \\ldots & J & 0\\end{pmatrix}$ The probability of measuring 1 at time $t$ is $\\theta (N,J,t) = \|\\langle 1 \|e^{-iH_1(N,J)t}\| 1 \\rangle \|^2.$ So given $M$ measurement results $E$ for times $t_1,\\dots ,t_m$ , each repeated $R_m$ times, the probability of $H_1(N,J)$ being the correct Hamiltonian is given by the binomial distribution $P(H_1(N,J)\|E) = \\nonumber \\\\\\prod _m {A_m \\atopwithdelims ()R_m} \\theta (N,J,t_m)^{A_m}(1-\\theta (N,J,t_m)(t_m))^{R_m-A_m}$ where $A_m$ is the number of 1 measurements at time $t_m$ .", "For numerical reasons use instead the log likelihood $L(N,J\|E) = -\\log P(H_1(N,J\|E)).$ To estimate the parameters $N$ and $J$ , we iteratively take $M$ measurements at times $t_m$ sampled in the time interval $[0,T]$ using the Hammersley low-discrepancy sequence.", "We then sample $L$ over a given parameter domain $\\lbrace N_{\\min }, N_{\\min }+1, \\dots , N_{\\max }\\rbrace \\times [J_{\\min },J_{\\max }]$ with $K$ sample points per ring-size parameter $N$ .", "Initially $K$ points $p_k$ are sampled uniformly for each size $N$ in the coupling strength parameter interval $[J_{\\min },J_{\\max }]$ .", "$L$ is evaluated at each sample point $p_k$ .", "Then the $K$ samples are resampled according to the sampling density function $D(j) = \\frac{1}{2} (p_{k+1} - p_{k-1}) L(N, j\|E).$ where $k$ is chosen such that $p_k$ is closest to $j$ , $p_{-1} = p_0 =0$ and $p_{K+1} = p_K = T$ at the interval boundaries.", "Iteratively additional measurements are taken by continuing the Hammersley sequence on the measurement trace, updating the $L$ values at the sample points to update the log-likelihood for the new measurements, and then resample the coupling strength intervals.", "This is repeated until the sampled log-likelihood over the parameter domain has a clear peak according to the sample points taken.", "Then the $N$ value with the largest $L$ value for the sampled points is selected and a simple 1D maximization strategy, such as hill-climbing is used to find the coupling strength for which the log-likelihood is largest.", "The samples per coupling strength interval $K$ can typically be small, say about 50 as the resampling process moves them towards the highest peaks in the interval and tracks these peaks.", "It is typically also sufficient to only take a few measurements $M$ , say 10 per iteration, with a low number of repetitions $R$ , say 10.", "After a few iterations (about 10), the log-likelihood has a clear peak and ring size as well as coupling strength can then easily be estimated to high accuracy.", "Fig.", "REF shows an example for a ring with 6 spins and coupling strength $0.666$ .", "After 10 iterations, adding new sample times for the measurements, each measured 10 times, and then the final optimization for the coupling strength resulted in estimating the ring size clearly to be 6 with a coupling strength of $0.666083$ , i.e.", "an error of $8.3 \\times 10^5$ .", "Figure: Network identification example for a ring with 6 nodes andcoupling strength 0.6660.666.", "Top: log-likelihood sample positions in theparameter domain {5,6,⋯,15}×[.5,1.5]\\lbrace 5,6,\\dots ,15\\rbrace \\times [.5,1.5], after 10iterations of taking 10 time samples, repeated 10 times, showing aclear peak close to the exact parameter values.", "Bottom: log-likelihoodfunction over the coupling strength parameter domain [.5,1.5][.5,1.5] forN=6N=6." ], [ "Conclusion", "We have demonstrated a bang-bang and a bias control scheme for optimizing the information transfer between nodes in simple quantum spin networks, either ensuring maximal transfer in limited time or enabling maximal transfer against the natural dynamics of the network.", "We have further presented a scheme to simultaneously estimate the network size and coupling strength for simple networks with uniform coupling using an efficient sampling strategy to take measurements as well as sampling the parameter domain.", "Future work will extend the control and characterization schemes to more general and realistic quantum networks." ] ]	1403.0226
[ [ "Probing reionization using quasar near-zones at redshift z ~ 6" ], [ "Abstract Using hydrodynamical simulations coupled to a radiative transfer code, we study the additional heating effects in the intergalactic medium (IGM) produced by $z\\sim 6$ quasars in their near-zones.", "If helium is predominantly in HeII to begin with, both normalization ($T_0$) and slope ($\\gamma$) of the IGM effective equation-of-state get modified by the excess ionization from the quasars.", "Using the available constraints on $T_0$ at $z\\sim6$, we discuss implications for the nature and epoch of HI and HeII reionization.", "We study the extent of the HeIII region as a function of quasar age and show, for a typical inferred age of $z \\sim 6$ quasars (i.e.", "$\\sim 10^8$ yrs), it extends up to 80% of the HI proximity region.", "For these long lifetimes, the heating effects can be detected even when all the HI lines from the proximity region are used.", "Using the flux and curvature probability distribution functions (PDFs), we study the statistical detectability of heating effects as a function of initial physical conditions in the IGM.", "For the present sample size, cosmic variance dominates the flux PDF.", "The curvature statistics is more suited to capturing the heating effects beyond the cosmic variance, even if the sample size is half of what is presently available." ], [ "Introduction", "Unravelling the process of reionization, which signals the end of the `dark ages' of our universe, is one of the current challenges of observational and theoretical cosmology.", "Two major milestones in the reionization history of the universe are those of hydrogen (H $\\rm \\scriptstyle I$ ) and singly ionized helium (He $\\rm \\scriptstyle II$ ).", "Study of the evolution of hydrogen reionization combines observational evidences from various sources; optical probes include the Gunn-Peterson absorption troughs [25] in the spectra of high-redshift bright sources such as (a) quasars [16], [74], [47], (b) Lyman-$\\alpha $ emitters [34], [65], [51], [48] and (c) $\\gamma $ -ray bursts [70], [37], [31], [59].", "The Thomson scattering optical depth measurements from the Cosmic Microwave Background (CMB) temperature and polarization power spectra are consistent with an instantaneous reionization at redshift $z \\sim 11$ [39], [52], [38], which may be interpreted as an estimate of the mean reionization redshift.", "At radio frequencies, the redshifted 21-cm hyperfine line of neutral hydrogen promises a unique three-dimensional mapping of the epoch of reionization (EoR) of hydrogen [23].", "All the available observations at present are consistent with an extended H $\\rm \\scriptstyle I$ reionization history that probably began at $z \\sim 15$ and ended around $z \\sim 6$ [78], [11], [12], [54], [55], [43], [44].", "The current observational probes of He $\\rm \\scriptstyle II$ reionization include measuring the Gunn-Peterson absorption troughs in the He $\\rm \\scriptstyle II$ Lyman-$\\alpha $ forest [32], [79], [57], [63], [76], [66].", "These observations suggest that the EoR of He $\\rm \\scriptstyle II$ is close to $z \\sim 2.7$ .", "The reionization of He $\\rm \\scriptstyle II$ also leaves a thermal imprint on the hydrogen Lyman-$\\alpha $ forest due to the additional heating effect on the velocity widths of the Lyman-$\\alpha $ lines [28].", "The thermal evolution of the intergalactic medium (IGM) from $2 \\le z \\le 4.8$ has been probed using the observations of the Lyman-$\\alpha $ forest [58], [61], [41].", "The velocity widths of the hydrogen Lyman-$\\alpha $ forest lines seem to exhibit a sudden increase between redshifts $z \\sim 3.5$ and 3, which may represent evidence for the reionization of He $\\rm \\scriptstyle II$ .", "The inferred temperature measurements, taken in conjunction with the adiabatic cooling expected to occur after the reionization of hydrogen, also constrain the EoR of hydrogen to below $z \\sim 9$ [67].", "Recently, [4] reported measurements of the IGM temperature from $2 \\le z \\le 4.8$ using the curvature statistic to quantify the temperature; their observations indicated gradual heating of the IGM from $z \\sim 4.4$ towards lower redshifts, in contrast to the adiabatic cooling expected in single-step models of reionization.", "These measurements are consistent with an extended epoch of He $\\rm \\scriptstyle II$ reionization starting probably at $z \\gtrsim 4.4$ and terminating around $z \\sim 3$ .", "Helium is expected to be singly ionized around the same time as the hydrogen gets ionized, and first-generation galaxies are believed to be the likely sources for completion of hydrogen and He $\\rm \\scriptstyle I$ reionization.", "In the single-step model of reionization, it is believed that massive, metal-free Population III stars [49], [71] may have provided the hard photons required for He $\\rm \\scriptstyle II$ reionization.", "In this model, a population of metal-free (Pop III) stars are required at redshifts $z > 6$ to reionize both H $\\rm \\scriptstyle I$ and He $\\rm \\scriptstyle II$ .", "In the absence of a strong ionizing background for He $\\rm \\scriptstyle II$ , it may recombine and hence to be reionized again at a lower redshift.", "Therefore, probes of intergalactic He $\\rm \\scriptstyle II$ are important for understanding the role of Population III stars in the early reionization of He $\\rm \\scriptstyle II$ and setting up a He $\\rm \\scriptstyle II$ ionizing background prior to the quasar era (i.e.", "$z \\sim 6$ ).", "Recently, there are indications of the presence of Population III stars even as late as $z \\sim 3$ possibly due to inefficient transport of heavy elements and/or poor mixing that leave pockets of pristine gas even in chemically evolved galaxies [33], [69], [30], [10].", "If, on the other hand, reionization took place as a two-step process (hydrogen first and He $\\rm \\scriptstyle II$ later), quasarsStrictly speaking, the term `quasar' is reserved for describing radio-loud quasi-stellar objects.", "However, as frequently done in the literature, we will use the term `quasars' in this paper to indicate quasi-stellar objects, irrespective of their radio properties.", "are believed to be the most likely candidates for reionization of He $\\rm \\scriptstyle II$ since their spectra are sufficiently hard.", "However, the number density of bright quasars peaks at $z \\sim 2 - 3$ and decreases rapidly above $z \\sim 4$ [2], [40].", "Hence, in the two-step model of reionization, the final stages of He $\\rm \\scriptstyle II$ reionization are expected to coincide with the peak of the quasar activity at $z \\sim 2 - 3$ .", "Quasar proximity zonesHere, and in what follows, the term “proximity zone” or “H $\\rm \\scriptstyle I$ proximity zone” describes the region in the vicinity of the quasar where the ionizing flux from the quasar dominates the background flux., where the excess ionization by the quasar allows the measurement of the velocity width of the Lyman-$\\alpha $ line, have been used to probe the thermal state of the IGM at $z \\sim 6$ [6].", "This, in turn, can be used to probe the role of quasars in He $\\rm \\scriptstyle II$ reionization.", "The IGM temperature in the near-zoneHere, and in what follows, the term “near zone” refers to the region in the vicinity of the quasar within the He $\\rm \\scriptstyle III$ ionization front, where the heating effects are significant.", "is influenced by both the existing background radiation as well as the additional radiation from the quasar itself.", "A first measurement of the near-zone temperature around a quasar at redshift 6 has been reported in [6] using Keck/HIRES data in combination with hydrodynamical simulations.", "Recently, an additional source of heating has been observed in the ionized near-zones of high-redshift quasars at $z \\sim 6$ , which is attributed [5] to the initial stages of helium reionization around that redshift, since the excess heating can be easily accounted for if the He $\\rm \\scriptstyle II$ is ionized by the quasar.", "The inferred excess temperature in the quasar near-zone can be used to place constraints on the epoch of H $\\rm \\scriptstyle I$ reionization [14], [56].", "In this paper, we explore several aspects of the additional heating effect in the near-zones of quasars at $z \\sim 6$ using the results of high-resolution hydrodynamical (SPH) simulations with gadget-2 [64], and the ionization correction done using a 1D radiative transfer code which we have developed.", "The gas temperature in the general IGM is given by the assumed equation of state [28] and computed self-consistently for the near-zone of the quasar.", "We first validate our simulations by computing the additional temperature in the near zone for different initial equations of state of the general IGM, and different assumed values of the He $\\rm \\scriptstyle II$ fraction prior to the active quasar phase.", "We obtain the expected relationship between the excess temperature and the initial He $\\rm \\scriptstyle II$ fraction in the quasar near-zone, and also find a connection between the magnitude of the steepening of the equation of state and the initial He $\\rm \\scriptstyle II$ fraction.", "We then use our simulation results to measure the size of the region in the near-zone heated by the quasar in comparison to the H $\\rm \\scriptstyle I$ proximity zone, as a function of the age of the quasar.", "We also validate the usage of the flux and curvature statistics to measure the increased temperature in the near-zone of the quasar, and, in particular, address the effect of cosmic variance.", "For the flux statistics tests, we employ a number of pixels typical of the sample sizes in available observations of quasar near-zones at redshifts $\\sim 6$ .", "Using the Kolmogorov-Smirnov (KS) statistic to quantify the effect of the additional heating, and examining its variation with the parameters of the equation of state, $T_0$ and $\\gamma $ , we establish the connection between the thermal evolution of the IGM following the reionization of hydrogen, and the detectability of the additional heating in the quasar near-zone.", "We also consider the possible dependence of the detectability of the additional heating effect on the assumed values of the background (metagalactic) photoionization rate of He $\\rm \\scriptstyle II$ , which translates into varying the He $\\rm \\scriptstyle II$ fraction in the near-zone of the quasar.", "This allows a connection to the effect of Population III stars on reionizing He $\\rm \\scriptstyle II$ at redshifts $z > 6$ (which constrains the initial He $\\rm \\scriptstyle II$ fraction in the quasar near-zone) in single-step reionization scenarios.", "The paper is organized as follows: In Sec.", ", we describe our hydrodynamical simulations and the numerical formalism for obtaining the simulated spectra in the quasar near-zone.", "In Sec.", ", we provide a validation of our simulations by computing the excess temperature in the quasar near-zone for different values of the equation of state normalization, and the initial He $\\rm \\scriptstyle II$ fraction, with comparison to the measured average temperature [5] in seven quasar near-zones at redshift $\\sim 6$ .", "We also describe the modification to the initial equation of state of the IGM due to the additional heating, and its dependence on the initial He $\\rm \\scriptstyle II$ fraction in the quasar near-zone.", "In Sec.", ", we describe the results obtained from our calculations as regards (a) the extent of the region around the quasar within which the additional heating is expected to contribute significantly, (b) the dependence of the additional heating effect in the near-zone on the initial equation of state of the IGM, quantified by the flux and curvature statistics, and (c) the dependence of the heating effect on the initial He $\\rm \\scriptstyle II$ fraction in the near-zone, which is related to the single-step reionization by Population III stars.", "We then summarize our findings in a brief concluding section.", "Throughout this article, we assume the cosmological parameters $\\Omega _m =0.26$ , $\\Omega _{\\Lambda } = 0.74$ , $\\Omega _b h^2 = 0.024$ , $h = 0.72$ , $\\sigma _8= 0.85$ , and $n_s = 0.95$ , which are consistent with the third-year WMAP and Lyman-$\\alpha $ forest data [62], [72].", "The helium fraction by mass is taken to be 0.24 [50].", "We perform cosmological hydrodynamical simulations using the parallel smoothed-particle hydrodynamics (SPH) code gadget-2 [64].", "We use two sets of simulations in this work: the lower resolution simulation contains $256^3$ each of gas and dark matter particles in a periodic box of size $60 h^{-1}$ comoving Mpc, and the high resolution simulation contains $512^3$ each of gas and dark matter particles in a periodic box of size $10 h^{-1}$ comoving Mpc.", "In both cases, the gravitational softening length is 1/30th of the mean interparticle spacing, and initial conditions are generated following the transfer function of Eisenstein and Hu [15].", "Both sets of simulations are started at $z = 99$ .", "Output baryonic density and velocity fields are generated at redshift $z \\sim 6$ .", "Recently, it has been shown that when AGN feedback effects are taken into account in simulations, one finds that quasar host galaxies at redshifts $\\sim 6$ are not `special' [17].", "It is now recognized that the existence of overdensities in the quasar near-zone can influence the background H $\\rm \\scriptstyle I$ photoionization rate measurements using the proximity effect [60], [24], [19], but the thermal effects of choosing the quasar in a random position as compared to locating them in a high density environment may be minor [56].", "Observationally, [75] find no evidence of an overdensity of i-dropout galaxies around three $z \\sim 6$ quasars, [36] find only two out of five quasar fields showing any evidence of overdensity, and [3], studying the environment of a redshift 5.72 quasar, find no enhancement of Lyman-$\\alpha $ emitters in the surroundings, compared to the blank fields.", "For most part of this work, we make the implicit assumption that quasars are not “special” and hence do not arise preferentially in biased regions.", "However, we come back to this point and provide a qualitative discussion of the effects of locating the quasars in biased regions, in Sec.", "REF .", "Lines of sight are extracted randomly in each simulation box at redshift 6, and the density and velocity fields along each line-of-sight is obtained.", "From the density grid of baryons in the simulation box, we compute the (physical) number densities of hydrogen and helium, $n_{\\rm {H}}$ and $n_{\\rm {He}}$ (assuming the mass fraction $Y = 0.24$ of helium) and then solve the equilibrium photoionization equations for H $\\rm \\scriptstyle I$ , He $\\rm \\scriptstyle I$ and He $\\rm \\scriptstyle II$ .", "Here, we explicitly assume that the universe is already reionized and the IGM, assumed to be optically thin, is in photoionization equilibrium with the background.", "The background ionizing radiation is assumed to follow the optically thin photoionization rates of hydrogen and helium as predicted by the “quasars + galaxies” Haardt-Madau background at redshift $\\sim 6$ , i.e.", "Table 3 of [26].", "The value of the background H $\\rm \\scriptstyle I$ photoionization rate considered here is consistent at the 1$\\sigma $ level with the results of the simulations of [8] and the observations of quasar proximity zone sizes in [77].", "It is slightly higher than the value ($1.57 \\pm 0.62) \\times 10^{-13} {\\rm {s}}^{-1}$ , measured by [9] using quasar proximity effects.", "The background He $\\rm \\scriptstyle II$ photoionization rate, $\\Gamma ^{\\rm bg}_{\\rm HeII}$ , is known to have large fluctuations even at $z \\sim 3$ due to the small number of ionizing sources within the characteristic mean free path of ionizing photons [18], [21], [35].", "At $z \\sim 6$ , this effect is expected to be severe, and the $\\Gamma ^{\\rm bg}_{\\rm HeII}$ we use is very small and should be treated as representative only.", "Later, we study the effect of varying this parameter on the results obtained.", "In the absence of additional radiation from the quasars, we assign the gas temperature to each pixel by using the equation of state of the photoionized IGM [28] with the normalization temperature $T_0 = 10^4$ K, and the slope $\\gamma =1.3$ .", "In principle, $T_0$ and $\\gamma $ at a given epoch can be fixed by comparing model predictions with observations.", "Later, we also explore some models with physically motivated ranges in $T_0$ and $\\gamma $ and draw conclusions regarding the epoch of H $\\rm \\scriptstyle I$ reionization.", "We now evolve of temperatures and ion densities of hydrogen and helium (caused by ionization due to the quasar as well as the metagalactic background) along a line of sight with the quasar placed at the first gridpoint.", "The four parameters, the temperature (obtained by using the equation of state) and the ion densities of H $\\rm \\scriptstyle I$ , He $\\rm \\scriptstyle II$ and He $\\rm \\scriptstyle III$ (obtained under the equilibrium conditions with the photoionization rates from the background, i.e.", "without contribution from the quasar) are incorporated as initial conditions.", "The luminosity of the quasar at the Lyman edge, $L_{\\rm HI}$ , is computed from the magnitude $M_{\\rm AB} = -26.67$ at 1450 Å (a typical magnitude for a luminous quasar at redshift $\\sim 6$ ).", "We assume the broken power law spectral index of $f_{\\nu } \\propto \\nu ^{-0.5}, 1050$ Å $< \\lambda < 1450$ Å, and $f_{\\nu } \\propto \\nu ^{-1.5}$ for $\\lambda < 1050$ Å.", "Hence, for the frequencies of interest, $f_{\\nu } \\propto \\nu ^{-\\alpha _s}$ where $\\alpha _s = 1.5$ ; the assumed spectral index is consistent with the inferred measurements [77] from observations of high-redshift quasar proximity zone sizes.", "These parameters are then used to derive the quasar contribution to the photoionization rates for H $\\rm \\scriptstyle I$ , He $\\rm \\scriptstyle I$ and He $\\rm \\scriptstyle II$ respectively.", "Since hydrogen is assumed to be highly ionized prior to the quasar being `switched on', the H $\\rm \\scriptstyle I$ ionization front from the quasar travels effectively at the speed of light.", "The region in the vicinity of the quasar in which the additional heating effects are expected to be significant may be characterized by the extent of the He $\\rm \\scriptstyle III$ region.", "To calculate the extent of this region, we track the location of the He $\\rm \\scriptstyle II$ ionization front.", "To do this, we use the relativistic equation of propagation of the ionization front modified to include the effects of optical depth: $\\frac{dR}{dt} = c\\left(\\frac{\\dot{N}_{\\rm eff} - 4 \\pi R^3 n_{\\rm He III} n_e \\alpha _{\\rm HeIII}/3}{\\dot{N}_{\\rm eff} + 4 \\pi R^2 f_{\\rm HeII} n_{\\rm He} c - 4 \\pi R^3 n_{\\rm He III} n_e \\alpha _{\\rm HeIII}/3}\\right)$ where $\\dot{N}_{\\rm eff} = \\dot{N} e^{-\\tau _{\\rm HeII}}$ with $\\dot{N}$ being the rate of production of He $\\rm \\scriptstyle II$ -ionizing photons, and $\\tau $ being the optical depth at the He $\\rm \\scriptstyle II$ edge at the distance $R$ .", "The above equation is analogous to that used by [29] for the case of stellar Stromgren spheres, in which the optical depth effects are incorporated.", "Using the above equation, we can compute the time required by the He $\\rm \\scriptstyle II$ front to reach a particular gridpoint under consideration.", "We can also compute the distance $R$ reached by the front after a time $t_Q$ , where $t_Q$ is the lifetime of the quasar.", "This distance $R_{\\rm He } = R(t = t_Q)$ is defined to be the location of the He $\\rm \\scriptstyle II$ ionization front (or radius of the He $\\rm \\scriptstyle III$ ionized sphere) at the end of the quasar lifetime.", "We use this distance $R_{\\rm He}$ to quantify the extent of the region in which additional heating effects are expected to be important, later in Sec.", "REF .", "Our numerical procedure is described in detail in Appendix .", "For the evolution of the species densities and temperatures, we closely follow [7].", "The radiative transfer implementation differs from [7] as regards the tracking of the ionization front.", "We have validated the front locations and speeds with Fig.", "5 of [42], and the effect of the front propagation on the size of the near-zones is described in Sec.", "REF ." ], [ "Profile generation and statistics", "We define the redshift grid along a line-of sight, using: $x(z) = \\int _0^z d_H (z^{\\prime }) dz^{\\prime }$ where $d_H (z) = c(\\dot{a}/a)^{-1}$ is the Hubble distance and $a$ is the scale factor.", "Once we know the ion densities and gas temperatures at each pixel, following [13], the Lyman$-{\\alpha }$ optical depth due to hydrogen at every redshift $z_0$ can be computed as: $\\tau _{\\rm \\alpha }(z_0) &=& \\frac{c I_{\\alpha }}{\\sqrt{\\pi }} \\int dx \\frac{n_{\\rm HI}(x, z(x))}{b[x,z(x)] [1 + z(x)]} \\nonumber \\\\&& \\quad \\times \\ V\\left\\lbrace \\alpha , \\frac{c[z(x) -z_0]}{b[x,z(x)](1+z_0)} + \\frac{v[x,z(x)]}{b[x,z(x)]} \\right\\rbrace $ where $b[x,z(x)] = \\sqrt{2 k_B T[x,z(x)]/m_{\\rm H}}$ is the thermal $b$ -parameter for hydrogen, $V$ is the Voigt profile function, in which the damping coefficient is $6.265 \\times 10^8 {\\rm {s}}^{-1}$ , and $I_{\\alpha } = 4.48 \\times 10^{-18}$ cm$^{2}$ is related to the absorption cross-section $\\sigma _{\\alpha }$ for the Lyman-$\\alpha $ photons: $\\sigma _{\\alpha } (\\nu ) = \\frac{c I_{\\alpha }}{b \\sqrt{\\pi }} V \\left[\\alpha ,\\frac{c(\\nu - \\nu _{\\alpha })}{b \\nu _{\\alpha }} \\right]$ where $\\nu _{\\alpha }$ is the hydrogen Lyman-$\\alpha $ frequency which corresponds to the wavelength 1215.67 Å.", "Using the above expression for the Lyman-$\\alpha $ optical depth, the simulated spectra are generated using $F = {\\rm {exp}}(-{\\tau _{\\alpha }})$ for the flux $F$ at each pixel.Though we do not convolve the spectra with instrumental broadening, this effect is expected to be negligible as compared to the thermal broadening effect which we are interested in.", "We mimic the noise by adding Gaussian distributed noise having a signal-to-noise ratio (SNR) 21, equal to a typical SNR achieved for for $z \\sim 6$ quasars with available instruments.", "We generate spectra for a number of such lines of sight for the statistical analyses.", "We consider two statistical indicators of the effect of the additional heating in this work : (a) the flux PDF statistics and (b) the curvature statistics.", "Note that [5] have used the cumulative distribution of velocity widths of Lyman-$\\alpha $ lines obtained with Voigt profile fitting, to measure the temperature.", "However, unlike in the case of low redshift Lyman-$\\alpha $ forest absorption, one will not be able to use higher Lyman-series lines to constrain the number of Voigt profile components.", "Hence, the derived $b$ -distribution need not be well constrained.", "Therefore, in the present analysis, we explore the possibility of using the curvature statistics, that does not involve Voigt profile decomposition, to quantify the detectability of additional heating.", "Section REF contains detailed descriptions of the flux and curvature statistics used to investigate the heating effect." ], [ "Excess heating in the quasar near-zones", "In this section, we describe a validation of the numerical procedure by computing the additional heating effect and comparing it to the measured value of the average excess temperature in the near-zones of quasars in [5].", "In particular, we investigate the effects of varying the normalization of the initial equation of state, and also the He $\\rm \\scriptstyle II$ fraction in the vicinity of the quasar before the quasar is switched on.", "We explore how the combination of these parameters may be used to place possible constraints on the redshift of H $\\rm \\scriptstyle I$ reionization as well as single-step reionization models where He $\\rm \\scriptstyle II$ is also ionized by massive stars.", "For this purpose, we employ the results of the $512^3$ , $10 h^{-1}$ comoving Mpc box simulation with the quasar having a luminosity correponding to $M_{\\rm AB} = -26.67$ at 1450 Å, and a lifetime of 100 Myr.", "The initial equation of state parameters and the background photoionization rates are varied and the resulting final values of temperature as a function of $(1 + \\delta )$ , where, $\\delta $ is the overdensity, are computed." ], [ "Modifications to equation of state", "In Fig.", "REF , we have plotted the $T-(1+\\delta )$ relation prior to and after additional heating by the quasar.", "We have chosen three different normalizations of the initial equation of state: $T_0 =$ 8000, 10000 and 12000 K, keeping the slope $\\gamma = 1.3$ fixed.", "The range in $\\delta $ plotted is from low to mildly nonlinear overdensities, and is representative of the range that contributes significantly to the intergalactic Lyman-$\\alpha $ absorption seen in quasar spectra.", "For each value of $T_0$ considered, the parameter $\\Gamma ^{\\rm bg}_{\\rm HeII}$ is varied from $10^{4}$ HM12 to HM12, where HM12($\\ =\\ 4.42 \\times 10^{-19}{\\rm {s}^{-1}}$ ) is the value of the background He $\\rm \\scriptstyle II$ photoionization rate computed by [26].", "This is equivalent to varying the initial He $\\rm \\scriptstyle II$ fraction in the vicinity of the quasar from $x_{\\rm HeII} \\sim 0.05$ to $x_{\\rm HeII} \\sim 1$ .", "We first describe the basic trends which are apparent in all the figures: (a) For all values of $\\Gamma ^{\\rm bg}_{\\rm HeII}$ under consideration, there is an increase in the temperature.", "When $\\Gamma ^{\\rm bg}_{\\rm HeII}$ is higher (i.e.", "the initial $x_{\\rm HeII}$ is close to 0.05), the temperature enhancement is less.", "Also, irrespective of $\\Gamma ^{\\rm bg}_{\\rm HeII}$ , the heated `equations of state' approach each other at high densities where the effects of recombination keep the He $\\rm \\scriptstyle II$ fraction high, and hence the gas is heated to a higher temperature.", "Therefore, for higher $\\Gamma ^{\\rm bg}_{\\rm HeII}$ , the measured value of $\\gamma $ also becomes large (the “heated” equation of state acquires a steeper slope).", "(b) When $\\Gamma ^{\\rm bg}_{\\rm HeII}$ is very small (i.e.", "the initial $x_{\\rm HeII}$ is close to 1), there is a uniform rise in temperature over the whole range of $\\delta $ under consideration, i.e.", "we find a $\\delta $ -independent heating.", "This leads to the equation of state being shifted upward (i.e.", "only enhancement in $T_0$ ) with a negligible change in the slope.", "If indeed a major part of He $\\rm \\scriptstyle II$ is ionized at $z \\sim 6$ by the quasars, then our findings suggest that the H $\\rm \\scriptstyle I$ gas will still have some memory of the H $\\rm \\scriptstyle I$ reionization.", "Figure: The initial equation of state and the effect of the additional heating for different values of the background metagalactic photoionization rate, Γ HeII bg \\Gamma ^{\\rm bg}_{\\rm HeII} (in s -1 ^{-1}).", "The normalization of the equation of state, T 0 T_0 is varied from 8000 - 12000 K. For each value of T 0 T_0, Γ HeII bg \\Gamma ^{\\rm bg}_{\\rm HeII} is varied from 10 4 10^{4} HM12 to HM12, where HM12 is the Haardt-Madau background photoionization rate.", "This is equivalent to varying the initial He II \\rm \\scriptstyle II fraction in the quasar vicinity from x HeII ∼0.05x_{\\rm HeII} \\sim 0.05 to x HeII ∼1x_{\\rm HeII} \\sim 1.", "In each figure, the asterisk with the error bar shows the measured average temperature in the near-zones of the seven redshift ∼6\\sim 6 quasars considered in .To summarize, there are two simultaneous trends which occur in the equation of state due to the decrease in $x_{\\rm HeII}$ : (a) a decrease in the normalization shift, and (b) an increase in the slope.", "We now consider these two trends separately, i.e.", "we explore the individual change in the parameters $T_0$ and $\\gamma $ ($\\Delta T_0$ and $\\Delta \\gamma $ ) when the value of $x_{\\rm HeII}$ is changed.", "For each initial value of $T_0$ (8000 K, 10000 K and 12000 K), we plot the change in temperature at the mean density, $\\Delta T_0$ against $x_{\\rm HeII}$ for the five different values of $x_{\\rm HeII}$ under consideration.", "This is shown in Fig.", "REF .", "It can be seen that $\\Delta T_0 \\propto x_{\\rm HeII}$ for all values of the initial $T_0$ .", "This is in line with the analytic formulation provided in [22] where it is argued that $\\Delta T \\propto x_{\\rm HeII}$ (initial), where $\\Delta T$ is the difference between the initial and heated temperatures.", "If we consider a fixed value of $x_{\\rm HeII}$ , for a higher initial $T_0$ , the value of the $\\Delta T_0$ is lower.", "This, again, is consistent with our previous findings that regions which are already `heated' can be additionally heated only to a limited extent.", "Figure: The variation of ΔT 0 \\Delta T_0 with the initial x HeII x_{\\rm HeII}.", "The relationship is linear, with the ΔT\\Delta T at a fixed x HeII x_{\\rm HeII} increasing with decrease in initial temperature.We now investigate the corresponding relationship for the case of the change in $\\gamma $ , i.e the $\\Delta \\gamma -x_{\\rm HeII}$ relation.", "For this, we plot the difference $\\Delta \\gamma $ between the slopes of the `heated' and `initial' equations of state, against $x_{\\rm HeII}$ , for the five different values of $x_{\\rm HeII}$ under consideration.", "This is done for each initial value of $T_0$ (8000 K, 10000 K and 12000 K).", "The results are shown in Fig.", "REF .", "As expected, there is negligible change in $\\gamma $ when the He $\\rm \\scriptstyle II$ fraction is close to 1.", "We also note that for a fixed value of $x_{\\rm HeII}$ , the value of $\\Delta \\gamma $ is higher when the initial $T_0$ is lower.", "However, we see that the value of $\\Delta \\gamma $ reaches a maximum of about 0.1 at the lowest He $\\rm \\scriptstyle II$ fraction and initial $T_0$ that we consider.", "The reason for this flattening is as follows: At high enough densities, all the curves in Fig.", "REF are constrained to follow the top curve due to recombination effects.", "At lower values of density, each curve in Fig.", "REF is shifted upward with respect to the initial equation of state, and the magnitude of this shift increases with increase in the value of $x_{\\rm HeII}$ .", "However, for low enough values of $x_{\\rm HeII}$ , both the `right top point' (which is constrained due to recombination effects) and the `left bottom point' (which is anchored close to the initial equation of state) are asymptotically fixed.", "This brings the slope to a near-saturation, which leads to the flattening out of $\\Delta \\gamma $ .", "The maximum change in slope is greater if the shift in the overall normalization is higher, which happens if the initial $T_0$ is lower.", "Hence, the maximum value of $\\Delta \\gamma $ decreases with increase in the initial $T_0$ , as we see in Fig.", "REF .", "Our $\\Delta \\gamma -x_{\\rm HeII}$ relation above is analogous to the $\\Delta T -x_{\\rm HeII}$ noted in the literature.", "We infer that the value $\\Delta \\gamma \\sim 0.1$ is representative of the maximum increase in the slope of the equation of state that may be achieved in physically feasible reionization scenarios.", "Figure: The variation of Δγ\\Delta \\gamma with the initial x HeII x_{\\rm HeII}.", "The Δγ\\Delta \\gamma at a fixed x HeII x_{\\rm HeII} increases with decrease in initial temperature, but reaches a maximum of about 0.1 at the lowest He II \\rm \\scriptstyle II fractions under consideration.We speculate that the shifting upwards of the equation of state (which arises when the initial $x_{\\rm HeII}$ values are high), may be easier to detect observationally than the (maximum) slope change of $\\lesssim 0.1$ (which occurs when the initial $x_{\\rm HeII}$ values are low).", "This also depends on how sensitive the statistical test used for distinguishability, is to the steepness of the equation of state, as compared to how sensitive it is to an overall increase in normalization.", "We will find, in the subsequent sections, that the curvature statistic is more sensitive to the expected shift $\\Delta T_0 \\sim 1000 - 5000$ K in the normalization of the equation of state, than to the expected change $\\lesssim 0.1$ in its slope." ], [ "Implications of temperature measurements", "We now compare the results of our simulations with the available observations.", "At present, with a limited number of $z \\sim 6$ quasars that are observed at high spectral resolution, constraints on the slope of the equation of state may be difficult.", "However, $T_0$ can be measured [5].", "In what follows, we try to get constraints on the $\\Gamma ^{\\rm bg}_{\\rm HeII}$ using the available $T_0$ measurements.", "The measured average temperature (log $T$ (in K)$ \\ =\\ 4.21^{+0.06}_{-0.07}$ ) in quasar near-zones at redshift $\\sim 6$ [5] is indicated by the asterisk with error bar in each plot of Fig.", "REF .", "We note the following: (a) If the initial equation of state has $T_0 = 8000$ K (a lower initial temperature), then the temperatures are lower than the $1\\sigma $ lower bound on the measurement for all the $\\Gamma ^{\\rm bg}_{\\rm HeII}$ values under consideration.", "Thus it may be possible to rule out the corresponding reionization histories leading to this temperature prior to the switching on of the quasar.", "The temperature $T_0 = 8000$ K arises, for example, if we assume the instantaneous reionization followed by adiabatic cooling and compression, when the redshift of reionization of hydrogen is at $z_{\\rm re} = 11$ with its associated temperature being $T_{\\rm re} \\sim 25000$ K. (b) However, if the initial $T_0 = 10000$ K, then the $\\Gamma ^{\\rm bg}_{\\rm HeII}$ is constrained to $\\lesssim 10^{-18}$ s$^{-1}$ , which corresponds to $x_{\\rm HeII} \\gtrsim 0.96$ , in order to be consistent with the measurements.", "The value of $T_0 = 10000$ K is, in turn consistent, with the reionization of H $\\rm \\scriptstyle I$ at $z_{\\rm re} = 11$ and $T_{\\rm re} \\sim 30000$ K. These are physically acceptable redshifts and temperatures of H $\\rm \\scriptstyle I$ reionization.", "(c) If the initial equation of state, on the other hand, has $T_0 = 12000$ K (a higher initial temperature), then the $\\Gamma ^{\\rm bg}_{\\rm HeII}$ value is constrained to $\\lesssim 10^{-16}$ s$^{-1}$ , which corresponds to $x_{\\rm HeII} \\gtrsim 0.26$ , in order to be consistent with the measured temperature.", "The values of the initial $T_0 = 12000$ K and $\\gamma = 1.3$ are difficult to reproduce with simple reionization models involving only adiabatic cooling and compression, but may arise in more complex models involving external sources of heating etc.", "In this case, the temperature measurement may be consistent with single-step models of reionization.", "It is to be noted that the additional heating effect is smaller for the case of higher initial $T_0$ than for the lower case.", "This leads to the curves being closer to each other in the bottom panel of Figure REF .", "In fact, this effect can be quantified using the curvature statistics by performing a Kolmogorov-Smirnov test between the `initial' and `heated' spectra, which we do and describe further in Section REF .", "In this way, the exercise presented above validates our procedure and also captures the dependence of the heating to (a) $T_0$ , which connects up the heating effect to the epoch of hydrogen reionization in two-step models, and (b) $\\Gamma ^{\\rm bg}_{\\rm HeII}$ , which connects to the possibility of single-step reionization of both H $\\rm \\scriptstyle I$ and He $\\rm \\scriptstyle II$ .", "In any case, the prevalence of sufficiently hard sources at high redshifts substantially increases the $\\Gamma ^{\\rm bg}_{\\rm HeII}$ value and hence affects the temperature in the near-zone.", "In the following sections, we quantify each of these effects, and also relate them to the detectability of the additional heating using statistical analyses." ], [ "Results", "In the previous section, we have described in detail the modifications to the equation of state that occur due to the effect of the additional heating.", "We have also investigated the implications of the measured temperature in the near-zones of the quasars on the values of the various parameters of the IGM at that epoch.", "These point to constraints on both, the epoch of reionization of H $\\rm \\scriptstyle I$ as well as single-step models of reionization.", "In the present section, we shall describe the main results of our simulations with respect to : (a) the relative extent of the He-heated region around quasars, compared to the H $\\rm \\scriptstyle I$ proximity zone, as a function of the age of the quasar, (b) the detectability of the additional heating effects as quantified by the flux and curvature statistics, and (c) implications for the detectability of additional heating in single-step reionization scenarios." ], [ "Extent of additional heating around quasars", "As the Lyman$-\\alpha $ absorption from the general IGM at redshift 6 is optically thick, a profile analysis to estimate the gas temperature can be performed only in the quasar's proximity zone.", "In this zone, the H $\\rm \\scriptstyle I$ gas is highly ionized due to the excess ionization from the quasar.", "However, the fraction of this gas which is influenced by additional heat from the He $\\rm \\scriptstyle II$ ionization by the quasar depends on where the He $\\rm \\scriptstyle II$ front is located.", "This depends both on the quasar lifetime $t_Q$ , as well as the line-of-sight optical depth for the He $\\rm \\scriptstyle II$ ionizing photons.", "If the He $\\rm \\scriptstyle III$ front does not reach the edge of the H $\\rm \\scriptstyle I$ proximity zone for some reason, it would lead to dilution in the statistical tests to measure excess temperature.", "In order to provide estimates on the front location and the H $\\rm \\scriptstyle I$ proximity zone, a larger box-size (which includes these regions which are typically of the order of 8-9 proper Mpc) is required.", "Therefore, in this section, we address this issue using the lower resolution $256^3$ , $60 h^{-1}$ comoving Mpc box simulation with the initial equation of state having $T_0 = 10^4$ K, $\\gamma = 1.3$ , and the quasar luminosity corresponding to $M_{\\rm AB} = -26.67$ at 1450 Å.", "Using Eq.", "(REF ), the equation of propagation of the He $\\rm \\scriptstyle II$ ionization front that takes into account optical depth effects, we calculate the location $R_{\\rm He} =R(t_Q) $ of the front at the end of the quasar lifetime $t_Q$ .", "The He $\\rm \\scriptstyle II$ front location is computed for 50 random lines-of-sight extracted in the simulation box.", "We repeat the computation for two different values of $t_Q$ , 10 Myr and 100 MyrThe assumed lifetimes of the quasar considered are indicative; at redshifts $z \\sim 6$ , measurements have placed the lifetimes of quasars at $\\ge 10^7$ years [27], [73]., and the results are plotted in Fig.", "REF .", "It can be seen that the extent of the He $\\rm \\scriptstyle III$ region (where additional heating of He, etc.", "are expected to be significant) increases as the quasar lifetime is increased, going up to about 8-8.5 proper Mpc from the quasar in a time interval of 100 Myr.", "The blue vertical line shows the maximum extent of the He $\\rm \\scriptstyle III$ region for a given $t_Q$ which occurs in the limit of zero optical depth.", "This is computed by setting $\\tau _{\\rm HeII} = 0$ in Eq.", "(REF ), so that $\\dot{N} _{\\rm eff} = \\dot{N}$ , where $\\dot{N}$ is the rate of production of ionizing photons from the quasar.", "For quasar lifetimes of the order of 10 Myr, the optical depth effects are negligible and the mean location of the front is close to the maximum value that occurs in the limit of zero optical depth.", "For $t_Q \\sim 100$ Myr, the front is able to travel a greater distance, but the optical depth effects begin to be important, and, on an average, the front reaches $\\gtrsim $ 80% of the maximum distance in about 66% cases.", "We now consider the relative extent of the He $\\rm \\scriptstyle III$ region with respect to the H $\\rm \\scriptstyle I$ proximity zone of the quasar.", "Since one looks for the signatures of additional heating in the full H $\\rm \\scriptstyle I$ proximity zone of the quasar, it is important to quantify the extent of the region within this proximity zone in which additional heating effects due to ionization of He $\\rm \\scriptstyle II$ are significant.", "The H $\\rm \\scriptstyle I$ proximity zone, $R_{\\rm H}$ , is defined through the relation $\\Gamma _{\\rm HI}^{QSO} (R_{\\rm H}) = \\Gamma ^{\\rm bg}_{\\rm HI}$ .", "The maximum value of $R_{\\rm H}$ for the quasar luminosity under consideration and the background $\\Gamma ^{\\rm bg}_{\\rm HI}$ , is $\\sim 14$ proper Mpc from the quasar.", "The distance $R_{\\rm He}$ is defined as $R_{\\rm He} = R(t = t_Q)$ using Eq.", "(REF ) with the optical depth effect taken into account.", "The ratio $R_{\\rm He}/R_{\\rm H}$ , representing the relative extent of the He $\\rm \\scriptstyle III$ region within $R_{\\rm H}$ , is plotted as histograms in Fig.", "REF for the 50 lines-of-sight considered.", "It can be seen that this ratio is about $30-35 \\%$ for quasar lifetimes of the order of 10 Myr, but increases to about $80 \\%$ for a quasar lifetime of $\\sim 100$ Myr.", "This illustrates that the He $\\rm \\scriptstyle II$ front covers about $80 \\%$ of the H $\\rm \\scriptstyle I$ proximity zone of the quasar for $t_Q \\sim 100$ Myr.", "Figure: The extent of the He III \\rm \\scriptstyle III zone R He R_{\\rm He} for quasar lifetimes of 10 Myr (top panel), and 100 Myr (bottom panel).", "Each histogram comprises a total of 50 lines-of-sight.", "The blue vertical line shows the location of the He II \\rm \\scriptstyle II front when the effect of optical depth is neglected, which represents the maximum extent of the He III \\rm \\scriptstyle III region for the given time.Figure: The relative extent of the He III \\rm \\scriptstyle III zone with respect to the H I\\rm \\scriptstyle I proximity zone, R He /R H R_{\\rm He}/R_{\\rm H}, for quasar lifetimes of 10 Myr (top panel), and 100 Myr (bottom panel).", "Each histogram comprises 50 lines-of-sight.", "As the quasar lifetime is increased, the relative extent of the He III \\rm \\scriptstyle III region also increases.", "For t Q ∼10 8 t_Q \\sim 10^8 years (the typical inferred lifetime of the z∼6z \\sim 6 quasar), more than 80% of the H I\\rm \\scriptstyle I proximity zone is heated in 78% of the sightlines.The above result is closely connected with a related phenomenon of the “saturation” or equilibrium value of the temperature in the region in which the heating effect is important.", "This saturation effect is seen as an increase in the temperature in a fairly distance-independent manner so that an equilibrium value is reached, after which there is little or no increase in the temperature over the timescales of interest for almost all gridpoints in the He $\\rm \\scriptstyle III$ region under consideration.", "This places a maximum bound on the temperature which the IGM may be heated to with ionization of both H $\\rm \\scriptstyle I$ and He $\\rm \\scriptstyle II$ .", "This effect is reminiscent of the corresponding phenomenon in the interstellar medium where one finds the maximum temperatures to be $T_{\\rm HI} \\sim 20000$ K when H $\\rm \\scriptstyle I$ is ionized and $T_{\\rm HeII} \\sim 40000$ K when both H $\\rm \\scriptstyle I$ and He $\\rm \\scriptstyle II$ are ionized; the exact values vary according to the detailed physics and optically thick/thin cases, but these numbers provide reasonable upper limits.", "In our present case the saturation is found to be achieved when the lifetime of the quasar is sufficiently high, $\\sim 100$ Myr.", "Since the helium front covers about $80 \\%$ of the H $\\rm \\scriptstyle I$ proximity zone within this time, the additional heating effect extends into a larger region and consequently, the rise in temperature is much more apparent, and fairly independent of distance.", "In contrast, for a quasar lifetime of 10 Myr, only about $30-35 \\%$ of the H $\\rm \\scriptstyle I$ proximity zone near the quasar is influenced by the additional heating and it is possible that some of the pixels inside these regions have not yet reached the saturation in temperature.", "This means that for sufficiently long time scales ($\\sim $ 100 Myr), the additional heating depends more on the initial IGM parameters and less on the distance from the quasar and the gas density.", "This turns out be important for the discussion in the following sections." ], [ "Flux statistics and dependence on equation of state", "In this section, we will explore some statistical tests to understand the sensitivity of the additional heating effect to the parameters of the general intergalactic medium at that epoch.", "For this purpose, we use the results of $512^3$ simulation box, which has a resolution of 2.65 km/s per pixel, and consider a quasar having a luminosity corresponding to $M_{\\rm AB} = -26.67$ at 1450 Å, and a lifetime of 100 Myr.", "We consider two statistics which are both based on the observed hydrogen Lyman-$\\alpha $ spectrum in order to quantify the additional heating effect, and the dependence on the equation of state parameters, $T_0$ and $\\gamma $ : (a) the probability distribution function (PDF) of the flux, and (b) the PDF of the flux curvature.", "We also consider the two-dimensional flux-curvature distribution.", "We probe cosmic variance by using the same set of parameters, but different sets of lines-of-sight.", "The fiducial equation of state used for this purpose is $T_0 = 10^4$ K, $\\gamma = 1.3$ .", "The background $\\Gamma ^{\\rm bg}_{\\rm HI}$ , $\\Gamma ^{\\rm bg}_{\\rm HeI}$ and $\\Gamma ^{\\rm bg}_{\\rm HeII}$ values correspond to those given by HM12 [26] at redshift 6.", "The transmitted flux in the Lyman-$\\alpha $ forest is sensitive to both, the temperature as well as the ionization state of hydrogen and therefore, to isolate the effect of additional heating around the quasar, we require the breaking of this degeneracy.", "For our chosen background photoionization rates, the spectrum when the quasar is not present is dark and hence featureless at redshift 6.", "Hence, it is impossible to compare the flux obtained from this spectrum with that when the quasar is present.", "Hence, we instead isolate the heating effect by generating a control sample (with the same initial conditions) of spectra with the temperature given by the initial equation of state and the ionization state being the same as that when the quasar is present.", "In other words, there is no He-related heating in the “control” sample.", "Gaussian distributed noise is added to both the “control\" and the “heated\" spectra with a signal-to-noise ratio 21, mimicking the typical values in the observed HIRES quasar spectra.", "For all the statistical analyses, we replicate the typical sample size (total number of pixels) used in the observational studies of the $z \\sim 6$ quasars till now, since the spectral resolution in the observations is close to the resolution in our simulations.", "To take into account any distance-dependent effects, it may also be desirable to use a longer line-of-sight obtained by splicing together shorter sightlines available in the simulation box.", "However, we have seen in the previous section that for quasar lifetimes of the order of 100 Myr, the temperatures reach equilibrium and the heating effect becomes fairly independent of distance from the quasar.", "To illustrate this statistically, we implemented the numerical routine for the fiducial equation of state parameters, $T_0 = 10^4$ K and $\\gamma = 1.3$ for a line-of-sight having length $40 h^{-1}$ comoving Mpc (constructed by splicing together four lines-of-sight of length $10 h^{-1}$ comoving Mpc each having 512 pixels), with the quasar lifetime of 100 Myr.", "The generated sample spectra, both heated (red) and control (black) are plotted in Fig.", "REF .", "Five such lines-of-sight were considered (so that the total sample size, $(2048 \\times 5) \\ {\\rm {pixels}} \\times 2.65$ km/s per pixel $\\sim 7$ quasars $\\times \\ 3500$ km/s per quasar), and the flux PDF was generated for both the heated and the control spectra.", "The flux PDFs for the control and the heated sample were compared using the Kolmogorov-Smirnov (KS) statistic, and they were found to be distinguishable with $94.5 \\%$ confidence.", "This shows that the distinguishability of the samples is fairly independent of distance from the quasar if the quasar lifetime is of the order of 100 Myr.", "We also noticed that the temperature enhancement is fairly independent of the distance of the pixel from the quasar, for this case.", "On the other hand, if the same exercise is repeated for a quasar lifetime of 10 Myr, it is found that the sample with additional heating resembles the control sample very strongly and the two flux PDF distributions are distinguishable only at the $15 \\%$ level.", "This is to be expected since, as we have seen in Sec.", "REF , the helium front travels to only about $30-35\\%$ of the hydrogen near-zone in this lifetime and hence the additional heating effect is confined to a small part of the line-of-sight under consideration.", "It is assumed that the quasar shines with constant flux during the entire lifetime for the purpose of the simulations.", "For long timescales ($\\gtrsim $ few Myr), the quasar light curves cannot be constrained using direct observations.", "However, these and subsequent results depend upon the integrated thermal effects throughout the active lifetime of the quasar.", "The luminosity of the quasar used in the simulations is to be taken an estimate of the average luminosity of the quasar throughout its active lifetime.", "We thus infer that for sufficiently long timescales of $\\sim $ 100 Myr, the actual location of the pixel with respect to the quasar may not be as relevant as other parameters such as the initial equation of state as far as the heating effect is concerned.", "For this reason, in all the further statistical studies, we will use 20 lines-of-sight of length $10 h^{-1}$ Mpc each comprising 512 pixels, which replicates the sample size in the observations of the quasar spectra We are not concerned with the spatial density correlations in the present study..", "Figure: Sample spectra, both heated (red dashed line) and control (black solid line) for a line-of-sight of having 2048 pixels drawn through the simulation box.", "The quasar lifetime is 100 Myr and the flux PDFs of the two samples are distinguishable with 94.594.5 % confidence." ], [ "Flux PDF statistics", "We compare two samples of 20 lines-of-sight each having 512 pixelsDue to the limited box size of the simulation, about 20 pixels at the extreme of the box have slight errors in the Lyman-$\\alpha $ optical depth due to the incompleteness of the integral in the Voigt profile generation.", "For the statistical tests, therefore, we discard these 20 pixels (equivalent to about 50 km/s) at the extreme of the box.", "for the “control” and “heated” spectra generated, using the Kolmogorov-Smirnov (KS) statistic.", "Note that apart from the additional heating, all other parameters of the heated model are identical to the “control” one.", "The results for the cumulative flux distributions are plotted below in Fig.", "REF and Fig.", "REF , along with the KS statistics `$d$ ' (the maximum separation between the two cumulative probability distributions) and `prob' (the probability that the two samples come from the same parent distribution) in each case.", "In Fig.", "REF , the temperature at mean density is fixed at $T_0 = 10^4$ K and $\\gamma $ is increased from 1.1 to 1.5.", "It can be seen, that the distributions for the samples with and without additional heating may be distinguished with $\\sim 100\\%$ confidence when $\\gamma = 1.1$ , but only with $69.78\\%$ confidence when $\\gamma = 1.5$ .", "Hence, a higher slope of the initial equation of state leads to a greater resemblance to the control sample.", "In Fig.", "REF , the slope is fixed at $\\gamma =1.3$ and $T_0$ is varied from 8000 K to 12000 K. The flux PDFs for the sample with and without additional heating are distinguishable at the $99.79\\%$ level when $T_0 =$ 8000 K, but only at the $87.13 \\%$ level when $T_0 =$ 12000 K. Hence, if the initial $T_0$ is larger, the distinguishability of the two samples becomes poorer.", "Figure: Comparison of the cumulative flux PDF of the spectrum with and without additional heating by the quasar for a sample of 20 lines-of-sight each having 512 pixels, drawn through the simulation box.", "The temperature at mean density is taken as 10 4 ^4 K with the slope being varied from 1.1 (top), 1.3 (middle) and 1.5 (bottom).", "The values of the KS statistics dd and prob are indicated on each panel.", "It can be seen that the distinguishability of the heated and non-heated spectra goes down as the slope of the equation of state is increased.", "With the smallest slope of 1.1, the spectra for the two cases are completely distinguishable even with 20 lines-of-sight.Figure: Same as Figure , with the temperature at mean density being varied from 8000 K (top), 10000 K (middle) and 12000 K (bottom).", "The distinguishability of the heated and non-heated spectra goes down as the value of the temperature at mean density is increased.", "With the smallest temperature of 8000 K, the spectra for the two cases are distinguishable at the 99.79%99.79\\% level even with 20 lines-of-sight.We conclude that we are able to distinguish between the heated and control samples using 20 lines-of-sight and the flux PDF, and the extent of the distinguishability is sensitive to the initial parameters ($T_0$ and $\\gamma $ ) of the equation of state.", "However, among these 20 lines-of-sight, we find that the statistical difference in the inferred flux PDF due to cosmic variance is greater than the difference introduced by additional heating from the quasar.", "This is summarized in Fig.", "REF where we have plotted the cumulative probability distribution for two subsamples each from the control and the heated distributions.", "Each sample comprises 5120 pixels (10 lines-of-sight).", "It can be seen that the effect of additional heating on the flux PDF is within the cosmic variance of the individual samples.", "Hence, we infer that the flux PDF alone is not very sensitive to the additional heating effect, but may be more sensitive to the H $\\rm \\scriptstyle I$ ionizing radiation, which is the purpose for which it is traditionally used.", "Figure: The cumulative probability distribution of the flux, for two samples each of control and heated spectra.", "Each sample comprises 10 lines-of-sight (5120 pixels).", "The blue dashed and green dot-dot-dot-dashed curves represent the heated samples and the black solid and red dot-dashed curves represent the control ones.", "It can be seen that the effect of the additional heating is within the cosmic variance of the individual samples." ], [ "Curvature statistics", "The flux PDF statistic points to a connection between the heating effect and the initial equation of state.", "However, the difference is within the individual cosmic variance of the samples, making it difficult for the technique to be used in practice to identify a given spectrum as being “heated” or not.", "In order to address this effect and also to isolate the effect of the additional heating from the ionization information (both of which are captured in the flux), we consider here a alternative statistic, to characterize the spectra.", "In the literature, this has been done in several ways: (a) by using the $b$ -distribution from Voigt profile fitting to the mock spectra [5], (b) by using wavelets [68] or (c) by using the curvature parameter [4].", "Unlike in the case of low-redshift Lyman-$\\alpha $ forest absorption, the $b$ parameter need not be well constrained as one will not be able to use higher Lyman series lines.", "In this section, we explore the usage of the curvature parameter, to analyse the heating effect statistically.", "Following [4], the curvature parameter can be defined as: $\\kappa = \\left\|\\frac{F^{\\prime \\prime }}{(1 + (F^{\\prime })^2)^{3/2}}\\right\|$ where $F$ is the normalized fluxOur $\\kappa $ corresponds to $\|\\kappa \|$ of [4]..", "The binned average of the curvature at a given flux, together with simulations, are used to measure the IGM temperature without resorting to Voigt profile fitting techniques by [4].", "As pointed out by these authors, the denominator of the above expression is essentially unity and hence only the double derivative of the flux contributes to the curvature.", "We follow [4] where the flux (and all its derivatives) are measured with respect to the velocity grid in km/s.", "We evaluate the curvature parameter for both, the control and the heated spectra.", "In addition to the KS statistic for the flux, described in the previous subsections, we now also use the KS statistic for the $\\kappa $ distribution and use the two dimensional KS statistic to compare the joint flux-$\\kappa $ distributions.", "In this way, the effect of the additional heating may be quantified.", "We begin by calibrating the effect of the curvature statistic.", "To do this, we consider the fiducial equation of state, having parameters $T_0 = 10^4 \\ \\rm {K}, \\gamma = 1.3$ , and a single line-of-sight (512 pixels).", "We first generate noise-free spectra along the line-of-sight for both “control” and “heated” cases, and compute the curvature values for both of these.", "Noise is then added to both the control and heated samples, and the curvature values are again computed.", "Now, the control and the heated samples are statistically compared (using the KS test) with respect to the flux PDF, the curvature, and the joint flux-$\\kappa $ distributions for both the cases, i.e.", "with and without noise added to the spectra.", "We find that when no noise is added to either the “control” or the “heated” spectra, then the three KS probabilities are 0.752 (for flux PDF alone), 0.002 (for $\\kappa $ aloneHere, and in what follows, we disregard the pixels having flux values greater than 0.9 or less than 0.1, for all curvature statistics.", "This is done following [4], to avoid both, saturated pixels at low flux as well as uncertainties in the curvature values at high flux.)", "and 0.021 (for the 2d KS test).", "This confirms that the curvature parameter is far better able to distinguish between the heated and the control samples than the flux PDF.", "This is to be expected since the curvature parameter directly captures the effect of thermal broadening.", "On the other hand, when noise is added to both the “control” and “heated” spectra, then the above three probabilities become 0.316 (for flux PDF alone), 0.768 (for $\\kappa $ alone) and 0.529 (for the 2d KS test).", "These values (also summarized in Table REF ) indicate that the curvature statistic is strongly influenced by the noise in the spectrum, which washes out the distinguishability of the control and the heated spectra.", "This has also been noted previously by [4].", "Table: This table indicates the KS test probabilities for the non-noise added and the noise added spectra.", "The KS test is performed between the control and the heated samples of 512 pixels each.", "The last row indicates the probability values for the two-dimensional KS test of the flux-κ\\kappa joint distribution.", "It can be seen that noise significantly affects the value of prob for the curvature statistic.Since the noise significantly dominates the curvature statistic, in order for the efficient usage of the curvature statistic, it is important to smooth the noisy spectrum before applying this statistic.", "In [4], this is achieved by fitting the raw spectra with a smoothly varying $b$ -spline and the curvature is computed from the smoothed spectra.", "In this work, we convolve the noisy spectra with a Gaussian filter having a specific smoothing velocity width and vary the width until the convolved spectrum best matches the ideal, non-noise added spectrum.", "The results of this exercise are illustrated in Figs.", "REF and REF .", "In Fig.", "REF , the top panel shows the 2D scatter plot of the $\\kappa $ -flux joint distribution for the control sample, with and without noise added to the spectrum.", "The bottom panel shows the noisy 2D distribution convolved with a Gaussian smoothing filter of 10 km/s, compared to the noise-free distribution.", "The figure shows that the noise is efficiently convolved out by smoothing with the Gaussian filter, since the convolved scatter plot closely resembles the original, non-noise added plot.", "We now fine-tune the value of the smoothing velocity until the convolved distribution most closely matches the ideal non-noise added distribution, and plots for different smoothing velocities of 3, 5, 7 and 8 km/s are in Figure REF .", "It is seen that a smoothing velocity of 7 km/s most closely matches the non-noise added distribution and hence we adopt it for the subsequent analysis.", "This is also apparent from the plot in Fig.", "REF which illustrates the pixel dependence of the flux and the curvature parameter for the three cases : no noise, noise added, and noise convolved with the 7 km/s Gaussian filter.", "We also note that the curvature parameter values we obtain are consistent (at the same order-of-magnitude) with those in [4]As an aside, we have found that smoothing with a moving boxcar distribution for different boxcar widths does not produce the systematic effects noted in Fig.", "REF and hence, the convolution with the Gaussian filter is preferred over the moving boxcar to smooth the distribution.", ".", "Figure: The top panel shows the 2D scatter plots of the flux-κ\\kappa distribution in the non-noise added (ideal) case (red plus signs), and the noise added case (green crosses).", "The distributions are significantly different.", "In the bottom panel, the non-noise added (ideal) distribution (red plus signs) is shown along with the noisy spectrum convolved with a 10 km/s filter (green crosses).", "The figure shows that it is indeed possible to approach the ideal 2D distribution when the noise is convolved out with a smoothing velocity.Figure: The average κ\\kappa and the associated error in different flux bins are plotted versus flux (for the control spectrum).", "These plots show the approach of the convolved flux-κ\\kappa distribution (dashed lines) to the non-noise added (ideal) distribution (solid lines) using different smoothing velocities, 3 km/s, 5 km/s, 7 km/s and 8 km/s from top left to bottom right.", "The blue dotted lines indicate the limits of the range in flux used for all the curvature statistics (0.1≤ Flux ≤0.90.1 \\le \\rm Flux \\le 0.9).", "It can be seen that the smoothing velocity of 7 km/s (bottom left) most closely resembles the ideal distribution.Figure: Top panel: The (control) quasar spectrum for the three cases : no noise added, noise added and noise convolved with the Gaussian filter of 7 km/s.", "It may be seen that the convolution with the Gaussian filter closely approximates the ideal distribution.", "Lower three panels: The curvature parameter as a function of pixel for three cases from top to bottom : no noise added, noise added, and noise convolved with the Gaussian filter of 7 km/s, for both the heated and the control samples along a line-of-sight.We now vary the equation of state, and the 2d KS test between flux and $\\kappa $ for 512 pixels (1 line-of-sight) yields the values in the second column of Table REF .", "It can be seen that the trend of greater distinguishability with smaller $T_0$ and $\\gamma $ , which we found for the flux PDF case, is reproduced for the case of the curvature statistic as well.", "The curvature statistic can effectively distinguish between the control and heated spectra for different equations of state even with a sample of 512 pixels (a single line-of-sight).", "The prob values for a sample of five lines-of-sight are also provided in the last column of Table REF .", "This shows that the distinguishability of the samples crosses the 90% level with a sample of 5 sightlines (equivalent to using two quasar spectra) for all equations of state under consideration.", "If we use 20 lines-of-sight, the control and heated spectra are completely distinguishable (to less than about one part in $10^{8}$ ) for all equations of state under consideration.", "Table: This table indicates the two-dimensional KS test probabilities of the flux-κ\\kappa joint distribution for different equations of state with a sample of 512 pixels (1 line-of-sight) and 2560 pixels (5 lines-of-sight).", "The KS test is performed between the control and the heated samples.", "It can be seen that the distinguishability of the samples decreases as T 0 T_0 and/or γ\\gamma are increased, quantifying the dependence of the additional heating effect on the initial equation of state.", "Note that all the background photoionization rates are fixed at the HM12 values.In order to explore the extent of the effect of cosmic variance on our results, we consider now our fiducial equation of state and compare the cumulative probability distributions of the curvature statistic for two control subsamples, each of 10 sightlines, and two “heated” subsamples, each again of 10 sightlines.", "The resulting plot is shown in Fig.", "REF .", "The blue dashed and green dot-dot-dot-dashed curves represent the heated samples and the black solid and red dot-dashed curves represent the control samples.", "It may be clearly seen that the heating effect is well above the “cosmic variances” of the individual samples; this figure may be compared to the previous Fig.", "REF where the opposite effect was noted.", "Hence, we conclude that the curvature statistic will be able to distinguish the “non-heated” and “heated” spectra over and above their internal cosmic variance even when we use a sample size as limited as what is available today.", "Figure: The cumulative probability distribution of the curvature statistic, κ\\kappa , for two samples each of control and heated spectra.", "Each sample comprises 10 lines-of-sight (5120 pixels).", "The blue dashed and green dot-dot-dot-dashed curves represent the heated samples and the black solid and red dot-dashed curves represent the control ones.", "It can be seen that the effect of the additional heating is well above the cosmic variance of the individual samples.", "This figure may be compared with Fig.", "where the opposite effect was noted." ], [ "Dependencies on single-step reionization by Population III stars", "In the preceding sections, we have statistically quantified the dependence of the heating effect on the equation of state parameters ($T_0$ and $\\gamma $ ).", "In standard two-step reionization scenarios, these two parameters may be mapped to the redshift of hydrogen reionization, and the associated IGM temperature at that redshift.", "In this section, we briefly consider the effects of our study on constraining single-step models of reionization.", "In Sec.", ", we illustrated the effects of changing the $\\Gamma ^{\\rm bg}_{\\rm HeII}$ photoionization rate on the temperature-density distribution, for different initial values of the normalization of the equation of state, $T_0$ .", "We also indicated which combinations of these two parameters produced results which were consistent with those measured in the near-zones of the $z \\sim 6$ quasars [5].", "It was found that when the $\\Gamma ^{\\rm bg}_{\\rm HeII}$ was small (or when the initial $x_{\\rm HeII}$ was high), $T_0$ showed the maximum increase with no apparent change in $\\gamma $ .", "However, as the $\\Gamma ^{\\rm bg}_{\\rm HeII}$ became higher, while the increase in temperature was moderate, we found that the equation of state became steeper (i.e.", "$\\gamma $ became higher).", "As the curvature statistics uses the whole spectra, it should be sensitive to changes in both $T_0$ and $\\gamma $ .", "Therefore, we now discuss how the detectability of the heating effect depends on the initial value of $x_{\\rm HeII}$ .", "This, in turn, can be connected to early reionization of both H $\\rm \\scriptstyle I$ and He $\\rm \\scriptstyle II$ by massive stars in single-step models [71], [78], [11], [12].", "In the single-step model of reionization, Population III stars reionize both H $\\rm \\scriptstyle I$ and He $\\rm \\scriptstyle II$ at redshifts $z > 6$ .", "In some single-step models [71], the fraction of helium in He $\\rm \\scriptstyle III$ may hence reach about $60 \\%$ by $z \\sim 5.6$ , which translates into $x_{\\rm HeII}$ being only of the order of $\\sim 0.4$ .", "In order to investigate the effect of a lower initial $x_{\\rm HeII}$ in the quasar near-zone, we consider different values of the metagalactic background $\\Gamma ^{\\rm bg}_{\\rm HeII}$ which translates into varying the initial He $\\rm \\scriptstyle II$ fraction, $x_{\\rm HeII}$ , and investigate the detectability of the additional heating to the variation of $x_{\\rm HeII}$ .", "The fiducial equation of state parameters, $T_0 = 10^4$ K, and $\\gamma = 1.3$ are used in this study.", "For each value of $\\Gamma ^{\\rm bg}_{\\rm HeII}$ which we consider, we generate “control” and “heated” spectra, then these two samples are compared using the 2d Kolmogorov-Smirnov statistic.", "The results are indicated in Table REF .", "The table shows that the effect of the additional heating is more apparent if the initial fraction of $x_{\\rm HeII}$ is greater.", "This is to be expected from the qualitative indications in Fig.", "REF , since a greater $x_{\\rm HeII}$ fraction leads to a higher final (heated) temperature, and hence a greater difference between the control and the heated samples.", "The argument may be reversed to provide constraints on the metagalactic He $\\rm \\scriptstyle II$ background required before the quasar is turned on, in order for the the additional heating effect to be detected at a particular level.", "For example, with all other parameters being equivalent, if the additional heating effect is to be detected with greater than 75 % confidence, then the initial He $\\rm \\scriptstyle II$ fraction in the vicinity of the quasar is constrained to $\\gtrsim 0.74$ , which, in turn, constrains the $\\Gamma ^{\\rm bg}_{\\rm HeII}$ to $\\lesssim 10^{-17}$ .", "Consequently, we infer that in single-step models of reionization where the $x_{\\rm HeII}$ in the quasar vicinity takes very small values, the additional heating effect may be considerably less detectable than in two-step models, which allow for a greater He $\\rm \\scriptstyle II$ fraction in the quasar near-zone.", "Table: This table indicates the two-dimensional KS test probabilities of the flux-κ\\kappa joint distribution for different initial He II \\rm \\scriptstyle II fractions with a sample of 512 pixels (1 line-of-sight).", "The KS test is performed between the control and the heated samples.", "It can be seen that the distinguishability of the samples decreases if the initial He II \\rm \\scriptstyle II fraction is lower (or equivalently, if the He II \\rm \\scriptstyle II metagalactic background is higher), thus quantifying the dependence of the additional heating effect on the initial He II \\rm \\scriptstyle II fraction.", "In the above table, the initial equation of state is fixed at the fiducial value (T 0 =10000T_0 = 10000 K, γ=1.3\\gamma = 1.3.", ")Hence, we have effectively probed the sensitivity of the curvature statistic to the initial He $\\rm \\scriptstyle II$ fraction in the vicinity of the quasar.", "However, as we saw in Sec.", ", the change in the He $\\rm \\scriptstyle II$ fraction leads to both, a moderate increase in temperature as well as a steepening of the slope.", "In the preceding subsections while discussing the curvature statistics, we have kept the value of the $\\Gamma ^{\\rm bg}_{\\rm HeII}$ fixed at the HM12 value.", "This, as we have seen in Sec.", ", leads to a shift in the overall equation of state with no apparent change in slope.", "Hence, by performing the KS test in the previous subsections, we have equivalently captured the sensitivity of the curvature statistics to a change in the overall normalization, and seen that an overall normalization shift may be readily distinguished even with a sample of 512 pixels.", "Here, we briefly indicate the complementary effect, i.e.", "the sensitivity of the curvature statistic to a change in the slope alone.", "Note that this effect would not be captured in the previous tests with the curvature statistics, since the small (HM12) value of $\\Gamma ^{\\rm bg}_{\\rm HeII}$ considered therein, ensured that the slope change between the control and heated samples was negligible.", "We have seen in Fig.", "REF in Sec.", "that the maximum expected change in the initial $\\gamma = 1.3$ , for the lowest initial temperatures and He $\\rm \\scriptstyle II$ fractions under consideration, is of the order of $\\Delta \\gamma = 0.1.$ We find that a sample of 512 pixels can distinguish $\\Delta \\gamma = \\pm 0.1$ with only about 7% - 39 % confidence.", "If one also takes into account the observational and other sources of errors, we speculate that the allowed change ($\\sim 0.1$ ) in $\\gamma $ may be more difficult to detect statistically than the allowed change ($\\sim 1000 - 5000$ K) in $T_0$ .", "This also shows us that the curvature statistic is more sensitive to the detection of the change in the normalization than to the change in the slope of the equation of state." ], [ "Influence of other effects", "Here, we provide discussions of the other factors that may also influence the observed spectra in the quasar near-zones, and an analysis on their significance for this study.", "Figure: Cumulative probability distribution along a sightline, for the curvature for the halo and field locations of the quasars.", "The median values of curvature are lower in the case of quasars residing in biased locations, corresponding to higher temperatures both for the initial and heated spectra.", "Quasars in biased regions: In order to explore the effects of locating quasars in biased regions, we extracted spectra by placing the quasar at the most massive halo (mass $\\sim 1.49 \\times 10^{11} h^{-1} M_{\\odot }$ ) in the simulation box.", "The baryonic overdensities around the quasar are found to be $\\sim 10-50$ , and hence, on applying the equation of state, the initial temperatures are $\\sim 20000 - 35000$ K. The curvature median values for biased locations of the quasar are slightly lower than those for unbiased locations, as plotted in Fig.", "REF .", "From the plots in Fig.", "REF , we speculate that there arises a degeneracy between the heating effects and the overdensities in the vicinity of the quasars when the quasars are modelled in biased locations.", "Temperatures of $T_0 \\sim 20000$ K at redshift 6 arise in models where reionization occurs relatively late, $z_{\\rm re} \\lesssim 8$ .", "Therefore, the density enhancement and consequent temperature enhancement in biased locations may lead to estimates of a later epoch of reionization, than if the quasars are modelled in unbiased regions.", "A caveat to this discussion is the assumption of the initial equation of state being valid even at large overdensities of 10-50 that arise in biased regions.", "A detailed treatment including the effects of shocks, etc.", "may be required to estimate the initial temperatures in these cases.", "However, this effect is expected to be minor, as indicated by the findings of [56].", "Three-dimensional effects and the environment: Our simulations use a 1D treatment of radiative transfer.", "In reality, the quasar radiates in 3D with some finite opening angle.", "However, since the observations of quasar spectra and the Lyman-$\\alpha $ forest are always along a line-of-sight or a set of several sightlines, the line-of-sight treatment of radiative transfer is adequate for producing simulated spectra and for the further statistical analyses.", "Recently, [46] have detected the presence of galaxy overdensities in the environment of four $z \\sim 6$ quasars.", "As stellar spectra are generally soft, there is negligible emission above 4 Ryd from galaxies, which is required for the ionization of He $\\rm \\scriptstyle II$ .", "In other words, while the value of $\\Gamma _{\\rm HI}$ may be changed slightly, there is negligible contribution to $\\Gamma _{\\rm He II}$ from the galaxies.", "Since the dominant contribution to the heating effect comes from the ionization of He $\\rm \\scriptstyle II$ , the heating effects and their detectability are influenced very little by the galaxies in the quasar environment.", "Variations in background HI ionizing flux: We assume a uniform value of $\\Gamma ^{\\rm bg}_{\\rm HI}$ in the general IGM prior to the “switching on” of the quasar.", "Close to reionization, the value of $\\Gamma ^{\\rm bg}_{\\rm HI}$ may show spatial variations.", "However, our results do not change significantly since they are not sensitive to the actual value of $\\Gamma ^{\\rm bg}_{\\rm HI}$ .", "Essentially, in the near-zone, the quasar ionizing flux dominates the background (by about a factor of 100 or more).", "Hence, small fluctuations in the initial $\\Gamma ^{\\rm bg}_{\\rm HI}$ are not expected to have a significant impact on the results and statistical analyses." ], [ "Summary of main results", "In this paper, we used detailed hydrodynamical simulations to provide an analysis of the features associated with the heating due to the ionization of He $\\rm \\scriptstyle II$ in the near-zones of high-redshift quasars, and their implications for constraining the epochs of H $\\rm \\scriptstyle I$ and He $\\rm \\scriptstyle II$ reionization.", "Our main findings may be summarized as follows: We have seen that the measured temperature [5] in the quasar near-zones arises from a combination of two effects : the initial He $\\rm \\scriptstyle II$ fraction in the quasar vicinity, and the normalization of the initial equation of state of the IGM.", "If the initial temperature at mean density is $\\lesssim 8000$ K, the measured temperature in the quasar near-zones is higher than that expected for the allowed range of initial He $\\rm \\scriptstyle II$ fractions ($x_{\\rm He II} = 0.04 - 1$ ) in the quasar vicinity.", "This shows that the temperature measurement can be used to place constraints on (a) the epoch and temperature of hydrogen reionization, and (b) single-step models of reionization that predict the initial He $\\rm \\scriptstyle II$ fraction.", "We recover the expected linear relationship of $\\Delta T_0$ increasing with the initial helium fraction $x_{\\rm HeII}$ .", "Akin to the $\\Delta T- x_{\\rm HeII}$ relation discussed in the literature [22], we also demonstrate a $\\Delta \\gamma - x_{\\rm He II}$ relation, which shows a decrease in $\\Delta \\gamma $ with increasing $x_{\\rm HeII}$ and a flattening out at the lowest $x_{\\rm HeII}$ values, thus illustrating the steepening of the equation of state with decrease in the He $\\rm \\scriptstyle II$ fraction in the quasar vicinity.", "Observationally, this steepening effect, which persists even for high initial temperatures where $\\Delta T_0$ is low, may also be used to constrain the near-zone He $\\rm \\scriptstyle II$ fraction.", "However, the maximum expected increase in the slope may be more difficult to detect observationally than the expected shift in the overall normalization.", "Optical depth effects are coupled to the propagation of the ionization front in the radiative transfer, so that we obtain a handle on the extent of the near-zone of He $\\rm \\scriptstyle III$ , where the additional heating is expected to contribute significantly.", "If the quasar age is $\\sim 100$ Myr, more than 80% of the H $\\rm \\scriptstyle I$ proximity zone is heated in 78% of the sightlines.", "The heated fraction of the H $\\rm \\scriptstyle I$ proximity zone is only about 30% - 35% for quasar lifetimes of $\\sim 10$ Myr.", "This indicates that including the entire extent of the H $\\rm \\scriptstyle I$ proximity zone for the temperature enhancement may result in some dilution of the statistics when the quasar lifetimes are short.", "However, considering the entire proximity zone of H $\\rm \\scriptstyle I$ is a valid approximation if the quasar lifetimes are longer, $\\gtrsim 100$ Myr.", "This is also the timescale for the saturation of the heating effect, making it fairly independent of distance.", "We have quantified the effect of additional heating by using the flux PDF and curvature statistics to compare the real spectra to the simulated spectra without heating.", "We have noted that the sensitivity of the curvature statistic to the noise in the spectra may be effectively removed by smoothing with a Gaussian filter with a velocity width of 7 km/s.", "Both these statistics indicate that a higher value of $T_0$ and/or $\\gamma $ leads to less detectability of the effect of additional heating.", "This connects the additional heating due to He $\\rm \\scriptstyle II$ reionization, to the epoch of hydrogen reionization.", "We find that the curvature statistic provides far more effective distinguishability of the heating effect, which is over and above the cosmic variance of individual samples of 10 lines-of-sight each having 512 pixels (chosen to match the typical sample sizes available in observations of seven quasars at redshift $\\sim 6$ ).", "We also find that the detectability of the heating effect is dependent on the initial He $\\rm \\scriptstyle II$ fraction in the quasar vicinity, with a greater He $\\rm \\scriptstyle II$ fraction leading to greater detectability." ], [ "Acknowledgements", "The research of HP is supported by the Shyama Prasad Mukherjee research grant of the Council of Scientific and Industrial Research (CSIR), India.", "The hydrodynamical simulations were performed using the Cetus and Perseus clusters of the IUCAA High Performance Computing Centre.", "HP thanks Jayanti Prasad and Vikram Khaire for helpful discussions.", "We thank George Becker, James Bolton, Martin Haehnelt, T. Padmanabhan, Patrick Petitjean and David Syphers for useful comments on the manuscript.", "We thank the anonymous referee for helpful suggestions that improved the quality of the presentation." ], [ "Calculational details", "In this appendix, we present the details of the calculations which were described briefly in Section .", "The initial conditions are described by photoionization equilibrium with the background, this system of equations is given by: $n_{\\rm {HI}} \\Gamma ^{\\rm bg}_{\\rm {HI}} &=& n_{\\rm {HII}} n_e \\alpha _{\\rm {HII}} \\nonumber \\\\n_{\\rm {HeI}} \\Gamma ^{\\rm bg}_{\\rm {HeI}} + n_{\\rm {HeIII}} n_e \\alpha _{\\rm {HeIII}} &=&n_{\\rm {HeII}} \\Gamma ^{\\rm bg}_{\\rm {HeII}} + n_{\\rm {HeII}} n_e \\alpha _{\\rm {HeII}}\\nonumber \\\\n_{\\rm {HeII}} \\Gamma ^{\\rm bg}_{\\rm {HeII}} &=& n_{\\rm {HeIII}} n_e \\alpha _{\\rm {HeIII}}$ with the boundary conditions that: $n_{\\rm HI} + n_{\\rm HII} = n_{\\rm H}$ , $n_{\\rm HeI} + n_{\\rm HeII} + n_{\\rm HeIII} = n_{\\rm He}$ , and $n_{\\rm HII} +n_{\\rm HeII} + 2 n_{\\rm HeIII} = n_{e}$ .", "Here, $\\Gamma ^{\\rm bg}_{\\rm x}$ represents the photoionization rate of species `x' from the background ionizing radiation assumed [26], the $\\alpha $ 's are the radiative recombination rate coefficients, and the $n$ 's represent the (proper) number densities.", "The background photoionization rates are given by (in ${\\rm {s}}^{-1}$ ): $\\Gamma ^{\\rm bg}_{\\rm HI} = 2.30 \\times 10^{-13} ; \\ \\Gamma ^{\\rm bg}_{\\rm HeI} = 1.54 \\times 10^{-13}; \\nonumber \\\\\\Gamma ^{\\rm bg}_{\\rm HeII} = 4.42 \\times 10^{-19} .$ The temperatures are assigned to each pixel by the equation of state: $T(x,z) = T_0(z) [1 + \\delta (x)]^{\\gamma - 1}$ where, $T_0$ is the normalization temperature, $\\delta (x)$ is the overdensity at the pixel and $\\gamma $ is the slope of the equation of state.", "Our numerical procedure now involves solving the system of four differential equations for the temperature evolution and hydrogen and helium ion densities evolution: $\\frac{d n_{\\rm {HII}}}{dt} &=& n_{\\rm {HI}} \\Gamma _{\\rm {HI}} - n_{\\rm {HII}} n_e\\alpha _{\\rm {HII}} - 3H(t)n_{\\rm HII}\\nonumber \\\\\\frac{d n_{\\rm {HeII}}}{dt} &=& n_{\\rm {HeI}} \\Gamma _{\\rm {HeI}} + n_{\\rm {HeIII}}n_e \\alpha _{\\rm {HeIII}} \\nonumber \\\\\\qquad &-& n_{\\rm {HeII}} \\Gamma _{\\rm {HeII}} -n_{\\rm {HeII}} n_e\\alpha _{\\rm {HeII}} - 3H(t)n_{\\rm HeII} \\nonumber \\\\\\frac{d n_{\\rm {HeIII}}}{dt} &=& n_{\\rm {HeII}} \\Gamma _{\\rm {HeII}} -n_{\\rm {HeIII}} n_e \\alpha _{\\rm {HeIII}} - 3H(t)n_{\\rm HeIII} \\nonumber \\\\\\frac{dT}{dt} &=& \\frac{2}{3 k_B n_{tot}} [H_{tot}(n_i) -C(n_i,T)] \\nonumber \\\\&&\\qquad - 2 H(t) T - \\frac{T}{n_{tot}}\\frac{d n_{tot}}{dt}$ In the above equations, $\\Gamma _{\\rm x}$ represents the photoionization rates of species `x' (contributed both by the quasar as well as the background in the near-zone, and by the background alone, for the far zone).", "The adiabatic index is $5/3$ , and $H_{tot}(n_i)$ and $C(n_i,T)$ represent the total photoheating rate per unit volume, and radiative cooling function respectively.", "The Hubble parameter is $H(t)$ , and $n_{tot} = n_{\\rm H} + n_{\\rm He} + n_e$ is the total number density of particles of different species.", "The term $- 2 H(t) T$ in the temperature evolution equation represents the contribution of the expansion of the universe to the adiabatic cooling of the gas.", "We ignore the contribution from $- 3H(t)n$ in the evolution of the species densities, since the ionization time scales under consideration are much smaller than $H^{-1}(t)$ .", "The last term $-T (d n_{tot}/dt) /n_{tot}$ represents the correction due to species evolution.", "This correction is only about 1 part in $10^3$ at the highest temperatures, but the effect is expected to be important in the initial stages of evolution.", "The photoionization rates from the quasar at a distance $R$ are given by: $\\Gamma _{\\rm HI}^{QSO} (R) &=& \\int _{\\nu _{\\rm HI}}^{\\infty } \\frac{L_{\\nu }}{4 \\pi R^2 h \\nu } \\sigma _{\\rm HI} (\\nu ) \\exp (-\\tau _{\\rm HI}) \\ d \\nu ; \\nonumber \\\\\\Gamma _{\\rm HeI}^{QSO} (R) &=& \\int _{\\nu _{\\rm HeI}}^{\\infty } \\frac{L_{\\nu }}{4\\pi R^2 h \\nu } \\sigma _{\\rm HeI} (\\nu ) \\exp (-\\tau _{\\rm HeI}) \\ d \\nu \\nonumber \\\\\\Gamma _{\\rm HeII}^{QSO} (R) &=& \\int _{\\nu _{\\rm HeII}}^{\\infty } \\frac{L_{\\nu }}{4\\pi R^2 h \\nu } \\sigma _{\\rm HeII} (\\nu ) \\exp (-\\tau _{\\rm HeII}) \\ d \\nu $ where $L_{\\nu } = L_{\\rm HI} (\\nu /\\nu _{\\rm HI})^{-\\alpha _s}$ .", "The $\\sigma (\\nu )$ 's denote the photoionization cross-sections for H $\\rm \\scriptstyle I$ , He $\\rm \\scriptstyle I$ and He $\\rm \\scriptstyle II$ respectively and the $\\tau $ 's are the corresponding optical depths, calculated as $\\tau _{\\rm x}(R) &=& \\sum _{i = 1}^{n(R)}[n_{\\rm HI} (i) \\sigma _{\\rm HI} (\\nu _{\\rm x}) + n_{\\rm HeI} (i) \\sigma _{\\rm HeI} (\\nu _{\\rm x}) \\nonumber \\\\&+& n_{\\rm HeII} (i) \\sigma _{\\rm HeII} (\\nu _{\\rm x})] l $ where, $l$ is the pixel size and $\\nu _{\\rm x}$ is the ionization edge of species $\\rm {x} =$ H $\\rm \\scriptstyle I$ , He $\\rm \\scriptstyle I$ or He $\\rm \\scriptstyle II$ .", "For simplicity of computation, we only consider the optical depth at the ionization edge of the relevant species in the photoionization rate.", "The sum is over all the pixels up to the $n(R)$ th pixel which is at the distance $R$ from the quasar.", "The total photoionization rate is obtained by adding the contributions from the quasar [Eq.", "(REF )] and the metagalactic background [Eq.", "(REF )].", "Recombination rates are as given in [20], [1] and [45] for H $\\rm \\scriptstyle II$ , He $\\rm \\scriptstyle II$ (including dielectronic recombination) and He $\\rm \\scriptstyle III$ .", "We use case A recombination coefficients here as they have been found to be the appropriate choice for comparison with hydrodynamical simulations of quasar near-zones [7].", "The details are as follows: Case A recombination coefficients (in $\\rm {cm}^3 \\rm {s}^{-1}$ ): (a) $\\alpha _{\\rm HII} = 6.28 \\times 10^{-11} T^{-0.5} (T/1000)^{-0.2} (1 +(10^{-6} T)^{0.7})^{-1}$ (b) $\\alpha _{\\rm HeII} = 1.5 \\times 10^{-10} T^{-0.6353}$ (c) $\\alpha _{\\rm HeIII} = 3.3 \\times 10^{-10} T^{-0.5} (T/1000)^{-0.2} (1 + (2.5\\times 10^{-7} T)^{0.7})^{-1} $ Dielectronic recombination coefficient for helium (in $\\rm {cm}^3 \\rm {s}^{-1}$ ): (a) $\\alpha _{\\rm HeII}^d = 1.93 \\times 10^{-3} T^{-1.5} \\exp (-470000/T) (1 +0.3 \\exp (-94000/T))$ To analyze the photo-heating, we use the background heating rates as given in [26] at redshift $\\sim 6$ (in ergs s$^{-1}$ ): $&& E^{\\rm bg}_{\\rm HI} = 1.5824 \\times 10^{-24} ; \\ E^{\\rm bg}_{\\rm HeI} = 1.792 \\times 10^{-24};\\nonumber \\\\&& E^{\\rm bg}_{\\rm HeII} = 4.304 \\times 10^{-29} \\,.$ We add to the above background heating rates, the additional heating rate due to the quasar with the previously mentioned luminosity and spectral index, given by: $E_{\\rm HI}^{QSO} (R) &=& \\int _{\\nu _{\\rm HI}}^{\\infty } \\frac{L_{\\nu } h (\\nu -\\nu _{\\rm HI})}{4 \\pi R^2 h \\nu } \\sigma _{\\rm HI} (\\nu ) \\exp (-\\tau _{\\rm HI}) \\ d \\nu ;\\nonumber \\\\E_{\\rm HeI}^{QSO} (R) &=& \\int _{\\nu _{\\rm HeI}}^{\\infty } \\frac{L_{\\nu } h (\\nu -\\nu _{\\rm HeI}) }{4 \\pi R^2 h \\nu } \\sigma _{\\rm HeI} (\\nu ) \\exp (-\\tau _{\\rm HeI})\\ d \\nu \\nonumber \\\\E_{\\rm HeII}^{QSO} (R) &=& \\int _{\\nu _{\\rm HeII}}^{\\infty } \\frac{L_{\\nu } h (\\nu - \\nu _{\\rm HeII})}{4 \\pi R^2 h \\nu } \\sigma _{\\rm HeII} (\\nu ) \\exp (-\\tau _{\\rm HeII}) \\ d \\nu \\nonumber \\\\$ At any distance $R$ from the quasar, the total photoheating rate per unit volume, $H_{tot} (R)$ , is given by $H_{tot} (R) = \\sum _{i} n_i [E_i^{\\rm bg} +E_i^{QSO}(R)]$ where the sum is over $i =$ H $\\rm \\scriptstyle I$ , He $\\rm \\scriptstyle I$ and He $\\rm \\scriptstyle II$ .", "The cooling function consists of contributions from (a) bremsstrahlung and (b) recombination.", "We use the corresponding expressions as given by [20], [1] and [45] for H $\\rm \\scriptstyle II$ , He $\\rm \\scriptstyle II$ and He $\\rm \\scriptstyle III$ , including a contribution from the dielectronic recombination of He $\\rm \\scriptstyle II$ .", "Collisional ionization and its associated cooling are ignored since, for the range of temperatures and densities considered here, their magnitudes are negligible as compared to the photoionization and the cooling rates by recombination and bremsstrahlung respectively, which we have considered here.", "The details are: Recombination cooling rates (in erg $\\rm {cm}^{-3} \\rm {s}^{-1}$ ): (a) $\\Lambda _{\\rm HII} = 2.82 \\times 10^{-26} T^{0.3} ( 1 + 3.54 \\times 10^{-6}T)^{-1} n_{\\rm HII} n_e$ (b) $\\Lambda _{\\rm HeII} = 1.55 \\times 10^{-26} T^{0.3647} n_{\\rm HeII} n_e $ (c) $\\Lambda _{\\rm HeIII} = 1.49 \\times 10^{-25} T^{0.3} (1 + 0.885 \\times 10^{-6} T)^{-1} n_{\\rm HeIII} n_e$ Dielectronic recombination cooling rate for helium (in erg $\\rm {cm}^{-3} \\rm {s}^{-1}$ ): (a) $\\Lambda _{\\rm HeII}^d = 1.24 \\times 10^{-13} T^{-1.5} \\exp (-470000/T) (1 +0.3 \\exp (-94000/T)) n_{\\rm HeII} n_e$ Bremsstrahlung (in erg $\\rm {cm}^{-3} \\rm {s}^{-1}$ ): $\\Lambda _b = 1.43 \\times 10^{-27} T^{0.5} \\ g_{\\rm ff} n_e (n_{\\rm HII} + n_{\\rm HeII} + 4 n_{\\rm HeIII})$ where, the Gaunt factor $g_{\\rm ff}$ is given by $g_{\\rm ff} = 1.1 + 0.34\\exp (-(5.5.", "- \\rm {log}_{10} T)^2/3)$ ." ], [ "Description of the code", "The algorithmic procedure is as outlined in Fig.", "REF .", "First, a number $N$ lines of sight are extracted randomly in our simulation box at redshift 6.", "For each line of sight, the density and velocity fields, $\\delta _b$ , and $v_b$ of the baryonic particles are obtained.", "The equilibrium ion number densities and temperature are found under the assumption of photoionization with the metagalactic background and equation of state, by solving Eq.", "(REF ) using the Newton-Raphson technique with the routine NEWT in Numerical Recipes [53].", "The inputs to the code at this stage are $T_0$ , $\\gamma $ and the background photoionization rates.", "Figure: Flowchart describing the numerical scheme.Next, the line-of-sight is gridded into $n$ equispaced intervals with the length of each interval being equal to the average pixel size in the simulation, and the quasar is placed at the first gridpoint.", "The inputs are the luminosity of the quasar at the Lyman-edge, $L_{\\rm HI}$ , the spectral index $\\alpha _s$ and the quasar lifetime $t_Q$ .", "The start time of evolution of the thermal and ionization state of a gridpoint interval is decided by the time at which the He $\\rm \\scriptstyle II$ ionization front reaches that gridpoint, which is calculated from Eq.", "(REF ) using the known distance to the gridpointStrictly speaking, one should evolve the gridpoint even if the He $\\rm \\scriptstyle II$ front has not reached it, to account for the Hubble expansion.", "However, we do not do this since the time scales under present consideration are much shorter than $H^{-1}(t)$ ..", "The initial conditions are the equilibrium species fractions and temperatures found previously.", "The four rate equations in Eq.", "(REF ) are now solved using a FORTRAN90 code based on the ODEINT routine of the Numerical Recipes [53].", "The ion densities and temperatures at each gridpoint interval are evolved with a time-step $\\Delta t$ , which is dynamic in nature, being inversely proportional to the rate of ionizing photons at the distance of the gridpoint; a typical value being $\\Delta t\\sim 10^6$ s. We follow the approach of [7] in that when the relative change in the electron number density falls below $10^{-12}$ , the ion fractions are solved for assuming photoionization equilibrium and a larger time-step is considered.", "In case the He $\\rm \\scriptstyle II$ ionization front has not yet reached a particular gridpoint within the quasar lifetime, the temperatures and ion densities are solved for assuming photoionization equilibrium with no contribution from the quasar to $\\Gamma _{\\rm HeII}$ and $E_{\\rm HeII}$ .", "Hence, those gridpoints located beyond the He $\\rm \\scriptstyle II$ front do not “see” the quasar as far as photoionization of He $\\rm \\scriptstyle II$ and the resulting gas heating are concerned.", "In this way, the location of the He $\\rm \\scriptstyle II$ ionization front at the end of the quasar lifetime is also known.", "The final values of number densities of different species and the temperature, for each gridpoint, are then used to update the optical depth values at the ionization edges, and the location of the He $\\rm \\scriptstyle II$ ionization front.", "Once these values are passed to the next gridpoint, the process is repeated until the end of the line-of-sight is reached.", "The temperature and neutral hydrogen density at each pixel are used to generate the simulated spectrum along that line-of-sight, by defining the redshift grid as described in the previous section.", "Note that in this procedure, the optical depth value contributes to the determination of the location of the ionization front, which determines the start time of the next gridpoint and its consequent evolution, which in turn contributes to the optical depth for the further gridpoints under consideration.", "Hence, if the (integrated) optical depth effect becomes large enough so that the front is “stopped”, the subsequent gridpoints do not “see” the ionization and heating photons from the quasar, and are ionized and heated by the background alone.", "Finally, the combined set of all the gridpoints at each line of sight, and the number of lines of sight extracted in the simulation box are used to obtain the flux statistics.", "In our simulations, we do not use the realistic quasar continuum to generate spectra.", "Hence, all the artificial effects coming from the issues related to continuum fitting will not be present in our analysis." ] ]	1403.0221
[ [ "Prospects for CW and LP operation of the European XFEL in hard X-ray\n regime" ], [ "Abstract The European XFEL will operate nominally at 17.5 GeV in SP (short pulse) mode with 0.65 ms long bunch train and 10 Hz repetition rate.", "A possible upgrade of the linac to CW (continuous wave) or LP (long pulse) modes with a corresponding reduction of electron beam energy is under discussion since many years.", "Recent successes in the dedicated R&D program allow to forecast a technical feasibility of such an upgrade in the foreseeable future.", "One of the challenges is to provide sub-Angstrom FEL operation in CW and LP modes.", "In this paper we perform a preliminary analysis of a possible operation of the European XFEL in the hard X-ray regime in CW and LP modes with the energies of 7 GeV and 10 GeV, respectively.", "We consider lasing in the baseline XFEL undulator as well as in a new undulator with a reduced period.", "We show that, with reasonable requirements on electron beam quality, lasing on the fundamental will be possible in sub-Angstrom regime.", "As an option for generation of brilliant photon beams at short wavelengths we also consider harmonic lasing that has recently attracted a significant attention." ], [ "Introduction", "Successful operation of X-ray free electron lasers (FELs) [1], [2], [3], based on self-amplified spontaneous emission (SASE) principle [4], opens up new horizons for photon science.", "The European XFEL [5] will be the first hard X-ray FEL user facility based on superconducting accelerator technology, and will provide unprecedented average brilliance of photon beams.", "The XFEL linac will operate nominally at 17.5 GeV in a burst mode with up to 2700 bunches within a 0.65 ms long bunch train and 10 Hz repetition rate.", "Even though the RF pulses are much longer than those available at X-ray FEL facilities, based on normal conducting accelerators, in the context of this paper we will call this SP (short pulse) mode of operation.", "In order to cope with high repetition rate within a pulse train, special efforts are being made to develop fast X-ray instrumentation [6].", "Still, many user experiments would strongly profit from increasing distance between X-ray pulses while lengthening pulse trains and keeping total number of X-ray pulses unchanged (or increased).", "Such a regime would require an operation of the accelerator with much longer RF pulses (LP, or long pulse mode), or even in CW (continuous wave) mode as a limit.", "A possible upgrade of the XFEL linac to CW or LP modes with a corresponding reduction of electron beam energy is under discussion since many years [7].", "Recent successes in the dedicated R&D program [8] allow to forecast a technical feasibility of such an upgrade in the foreseeable future.", "One of the main challenges of CW upgrade is to provide sub-Ångstrom FEL operation which is, obviously, more difficult with lower electron energies.", "One can consider improving the electron beam quality as well as reducing the undulator period as possible measures.", "An additional possibility is a harmonic lasing [18], [19], [20], [21], [22], [23] that has recently attracted a significant attention [23], [24].", "Harmonic lasing can extend operating range of an X-ray FEL facility and provide brilliant photon beams of high energies for user experiments.", "In this paper we briefly discuss all these possibilities.", "Note that the goal of this paper is not to define precisely the parameter range for operation of the European XFEL after CW upgrade, but to stimulate further intensive studies aiming at definition of optimal regimes." ], [ "CW upgrade of the linac", "A possible upgrade of the XFEL linac to CW or LP modes holds a great potential for a further improvement of X-ray FEL user operation, including a more comfortable (for experiments) time structure, higher average brilliance, improved stability etc.", "The drawbacks are a somewhat smaller peak brilliance and a reduced photon energy range, both due to a lower electron beam energy.", "Both disadvantages can, however, be minimized by an improvement of the electron beam quality and application of advanced FEL techniques.", "Moreover, one can keep a possibility to relatively quickly switch between SP and CW modes thus greatly improving the flexibility of the user facility.", "For a CW upgrade of the linac, the following main measures will be needed [8], [9]: i) Upgrade of the cryogenic plant with the aim to approximately double its capacity; ii) Installation of new RF power sources: compact Inductive Output Tubes (IOTs); iii) Exchange of the first 17 accelerator modules by the new ones (including a larger diameter 2-phase helium tube, new HOM couplers etc.)", "designed for operation in CW mode with a relatively high gradient (up to 16 MV/m).", "This ensures that the beam formation system (up to the last bunch compressor) operates with a similar energy profile as it does in SP mode.", "Then 12 old accelerating modules are relocated to the end of the linac; iiii) Installation of a new injector generating a high-brightness electron beam in CW mode.", "Table: Main assumptions for CW/LP upgrade of the XFEL facility.The first two items can be realized in a straightforward way; the third one is based on the steady progress of the TESLA technology [11] and is not particularly challenging.", "Until recently the main uncertainty was connected with the absence of CW injectors providing a sufficient quality of electron beams.", "However, last year there was an experimental demonstration of small emittances (for charges below 100 pC) at a CW photoinjector using a DC gun followed immediately by acceleration with superconducting cavities [12].", "The measured parameters are already sufficient for considering this kind of injector as a candidate for CW upgrade of the XFEL linac (although the operation would be limited to low charge scenarios).", "As an alternative one can consider a superconducting RF gun that can potentially produce also larger charge bunches with low emittances (the progress reports can be found in [13], [14]) or even a normal conducting RF gun [15].", "In the latter case a special regime can, in principle, be organized when a continuous sequence of short RF pulses is used [16] instead of powering the gun in true CW mode.", "In this paper we do not present a comprehensive technical description of the CW upgrade, it will be published elswhere [10].", "Here we only summarize some technical details in Table REF .", "To predict a possible electron energy range in CW and LP modes, and to test the XFEL cryomodules in these regimes, a series of measurements is being performed at DESY [8], [9].", "The measurements demonstrate stable behavior of the modules in these regimes, and allow to conservatively predict that the energy can reach 7 GeV in CW mode, and 10 GeV in LP mode with 35% duty factor [8], [9].", "Recent measurements [17] of a cryomodule equipped with large grain Nb cavities and improved HOM couplers demonstrated even better performance, and allow for more optimistic forecasts (as it is reflected in Table 1).", "Moreover, all these measurements have been done with pre-series XFEL cryomodules which have not yet reached an ultimate performance.", "In other words, one can hope for higher electron energies after CW upgrade.", "Nevertheless, in this paper we conservatively consider electron energy range between 7 GeV and 10 GeV." ], [ "Harmonic lasing", "Apart from the standard regime of the FEL operation, namely lasing at the fundamental wavelength to saturation, in this paper we will also consider harmonic lasing as an option for reaching short wavelengths.", "Harmonic lasing in single-pass high-gain FELs [18], [19], [20], [21], [22], [23] is the radiative instability at an odd harmonic of the planar undulator developing independently from lasing at the fundamental.", "Contrary to nonlinear harmonic generation (which is driven by the fundamental in the vicinity of saturation), harmonic lasing can provide much more intense, stable, and narrow-band FEL beam if the fundamental is suppressed.", "The most attractive feature of saturated harmonic lasing is that the brilliance of a harmonic is comparable to that of the fundamental.", "Indeed, a good estimate for the saturation efficiency is $\\lambda _{\\mathrm {w}}/(h L_{\\mathrm {sat}})$ , where $\\lambda _{\\mathrm {w}}$ is the undulator period, $L_{\\mathrm {sat}}$ is the saturation length, and $h$ is harmonic number.", "At the same time, the relative rms bandwidth has the same scaling.", "If we consider lasing at the same wavelength on the fundamental and on a harmonic (with the retuned undulator parameter $K$ ), transverse coherence properties are about the same since they are mainly defined by emittance-to-wavelength ratio.", "Thus, also the brilliance is about the same in both cases.", "In many cases, however, the saturation length for harmonics can be shorter than that of the fundamental at the same wavelength.", "As a consequence, for a given undulator length one can reach saturation on harmonics at a shorter wavelength.", "It was shown in a recent study [23] that the 3rd and even the 5th harmonic lasing in X-ray FELs is much more robust than usually thought, and can be widely used at the present level of accelerator and FEL technology.", "For a successful harmonic lasing the fundamental mode must be suppressed.", "A possible method to disrupt the fundamental without affecting the third harmonic lasing was suggested in [21]: one can use $2\\pi /3$ (or $4\\pi /3$ ) phase shifters between undulator modules.", "It was found out, however, that this method is inefficient in the case of a SASE FEL (see [23], [24] for more details).", "One can consider an alternation of $2\\pi /3$ and $4\\pi /3$ phase shifters [22] but this variation of the method works also not sufficiently well in many practical cases.", "In [23] another modification of the phase shifters method was proposed which suggests piecewise distribution of phase shifters $2\\pi /3$ and $4\\pi /3$ , and works better in practical situations [24].", "One can also consider a random distribution of phase shifters $2\\pi /3$ and $4\\pi /3$ [25] although one needs a very larger number of phase shifters in this case.", "The suppression of the fundamental by using a spectral filter in a chicane installed between two parts of the undulator was also proposed in [23].", "The conclusion in [23] was that the best way of suppression is the combination of phase shifters and a filter.", "In the context of CW operation the use of the filter may have limitations due to power losses in it (even though the filtering is done well below FEL saturation).", "Thus, the bunch repetition rate might be limited depending on charge and wavelength.", "In this case one can use a new method for suppression [26]: switching between the 5th and the 3rd harmonics.", "As we will see in the next Section, their performance is comparable in the considered parameter range.", "Thus, the following trick is possible.", "Imagine, we aim at lasing at 1 Å.", "We tune the first part of the undulator to the resonance with 5 Å, so that we are interested in the 5th harmonic lasing.", "The fundamental and the third harmonic are suppressed by the piecewise combination of (some of the) phase shifters $2\\pi /5$ , $4\\pi /5$ , $6\\pi /5$ , and $8\\pi /5$ such that they stay well below saturation in the first part of the undulator.", "Then, in the second part we reduce parameter $K$ such that the resonance at 3 Å is achieved.", "Now the fifth harmonic from the first part continues to get amplified as the third harmonic (while the first and the third ones are off resonance).", "The fundamental in the second part is suppressed with the help of piecewise distribution of phase shifters $2\\pi /3$ and $4\\pi /3$ .", "If necessary, one can later switch back to a resonance with 5 Å, and so on.", "Also, a use of many pieces with 5th and 3rd harmonic lasing without phase shifters might be possible.", "Moreover, the scheme can be generalized to the case of even higher harmonics.", "More detailed description of the method as well as numerical simulations will be published elsewhere [26].", "Note also that the FEL bandwidth in this scheme is defined by the 5th harmonic lasing, while the saturation power - by the third harmonic lasing, i.e.", "the spectral power and brilliance will be higher than in the case of lasing only on one harmonic or only on the fundamental.", "This is similar to the concepts of the harmonic lasing self-seeded FEL [23], [27] and pSASE [28], [29]." ], [ "Lasing in the baseline undulator", "In the following we will consider the range of beam energies from 7 to 10 GeV, assuming that the former can be achieved in CW mode, and the latter - in LP mode with about 35% duty factor [8].", "The hard X-ray undulators SASE1 and SASE2 of the European XFEL have 4 cm period, and the largest K-value of 3.9 is achieved at the gap of 10 mm.", "The net magnetic length of the undulator is 175 m. Our task is to define the range of achievable photon energies depending on electron beam quality.", "We consider lasing on the fundamental as well as on the 3rd and the 5th harmonics.", "The formulas from [23] are used to calculate the saturation length.", "We assume that the peak current is 5 kA in all considered cases (different compression scenarios we leave for future studies).", "We optimize beta-function in the undulator for the shortest gain length.", "However, when the optimum beta is smaller than 15 m (which we assume as a technical limit), we set it to 15 m. Figure: Energy spread versus normalized emittance for which the saturation in SASE1 undulator is possible at 1 Å(upper graphs), 0.75 Å(middle graphs), and 0.5 Å(lower graphs).", "Thebeam energy is 10 GeV (left column) and 7 GeV (right column), and the peak current is 5 kA.", "Beta-function is optimized for thehighest gain when the optimum is larger than 15 m, otherwise beta-function is 15 m.Solid, dash, and dot curves correspond to the 1st, the 3rd, and the 5th harmonic lasing, correspondingly.In Fig.", "REF we present energy spread versus normalized emittance for which the saturation is possible at 1 Å, 0.75 Å, and 0.5 Å for the electron energies of 7 and 10 GeV.", "The corresponding photon energy range is 12.4–24.8 keV.", "In the case of 10 GeV the lasing at 1 Å is not possible on the fifth harmonic because K is not sufficiently large.", "However, lasing to saturation on the fundamental and on the 3rd harmonic is possible practically for any reasonable beam quality.", "Resonance at 0.5 Å cannot be achieved on the fundamental, but lasing to saturation on the 3rd and on the 5th harmonics is possible for a sufficiently bright electron beam.", "In the case of 7 GeV the resonance at 1 Å and at shorter wavelengths is not possible on the fundamental.", "However, lasing to saturation on the 3rd and the 5th harmonics in sub-Ångstrom regime is possible (although at 0.5 Å it would require extremely bright electron beams).", "Figure: Range of photon energies accessible with the fundamental (between the solid lines), the third harmonic (between the dash lines),and the fifth harmonic (between the dot lines).", "Upper left: ϵ n =0.4μ\\epsilon _{\\mathrm {n}}=0.4 \\ \\mu m, σ ℰ =1\\sigma _{\\cal {E}} = 1 MeV,upper right: ϵ n =0.4μ\\epsilon _{\\mathrm {n}}=0.4 \\ \\mu m, σ ℰ =2\\sigma _{\\cal {E}} = 2 MeV,lower left: ϵ n =0.8μ\\epsilon _{\\mathrm {n}}=0.8 \\ \\mu m, σ ℰ =1\\sigma _{\\cal {E}} = 1 MeV,lower right: ϵ n =0.8μ\\epsilon _{\\mathrm {n}}=0.8 \\ \\mu m, σ ℰ =2\\sigma _{\\cal {E}} = 2 MeV.The undulator period is 4 cm, maximum KK is 3.9, the peak current is 5 kA.", "Beta-function is optimized for thehighest gain when the optimum is larger than 15 m, otherwise beta-function is 15 m.We can also calculate photon energy range that can be achieved in the considered electron energy range depending on electron beam quality.", "In Fig.", "REF we present the results for four different combinations of slice emittance and energy spread ranging from 0.4 $\\mu $ m and 1 MeV (upper left plot) to 0.8 $\\mu $ m and 2 MeV (lower right plot).", "One can see that harmonics have a significant advantage over the fundamental only if the electron beam is bright enough.", "One can notice that, for example, lasing on the 5th harmonic is not possible for the most pessimistic parameter set.", "Finally, let us note that the baseline undulator with a relatively large period and large K value has an advantage of a big tunability range, as one can see from Fig.", "REF .", "Moreover, one can keep the usual operation range after switching back to the SP mode.", "Another advantage is that keeping a relatively large gap (10 mm) is favorable in the context of CW operation with a high average power of the electron beam." ], [ "Undulator with a shorter period", "When electron energy is decreased, a natural step towards reaching short wavelengths is a choice of an undulator with a shorter period.", "An obvious disadvantage of such a solution is that the tunability range is reduced.", "In order to keep it large, one should reduce an undulator gap thus increasing maximum K value.", "There are, however, limitations due to stronger wakefields and a necessity to keep a large enough aperture for transportation of a high average power beam.", "In this Section we consider an undulator with the period of 3 cm and a slightly reduced gap of 8 mm.", "The maximum K value, that is achieved at a closed gap, equals 3.", "In Fig.", "REF we show the photon energy range for different sets of emittance and energy spread, ranging from 0.4 $\\mu $ m and 1 MeV (upper left plot) to 0.8 $\\mu $ m and 2 MeV (lower right plot).", "Comparing Fig.", "REF with Fig.", "REF , we can observe that there is a shift towards higher photon energies and a reduction of the range (in other words, an increase of the highest achievable photon energies is smaller than an increase of the lowest ones).", "One can also notice that the fifth harmonic lasing is only possible for the most optimistic parameter set.", "A possible advantage of using a new undulator is that one can foresee more phase shifters already at the design stage.", "In particular, one can simultaneously introduce phase shifters between undulator modules and integrate them into a module design [24].", "One can consider different scenarios for modification of the undulator system of the European XFEL in the context of the CW upgrade of the linac: keeping old undulators, installing new undulators instead of both SASE1 and SASE2, or only instead of one of them.", "Note that there is also a potential option of the facility extension with new undulator beamlines.", "Figure: Range of photon energies accessible with the fundamental (between the solid lines), the third harmonic (between the dash lines),and the fifth harmonic (between the dot lines).", "Upper left: ϵ n =0.4μ\\epsilon _{\\mathrm {n}}=0.4 \\ \\mu m, σ ℰ =1\\sigma _{\\cal {E}} = 1 MeV,upper right: ϵ n =0.4μ\\epsilon _{\\mathrm {n}}=0.4 \\ \\mu m, σ ℰ =2\\sigma _{\\cal {E}} = 2 MeV,lower left: ϵ n =0.8μ\\epsilon _{\\mathrm {n}}=0.8 \\ \\mu m, σ ℰ =1\\sigma _{\\cal {E}} = 1 MeV,lower right: ϵ n =0.8μ\\epsilon _{\\mathrm {n}}=0.8 \\ \\mu m, σ ℰ =2\\sigma _{\\cal {E}} = 2 MeV.The undulator period is 3 cm, maximum KK is 3, the peak current is 5 kA.", "Beta-function is optimized for thehighest gain when the optimum is larger than 15 m, otherwise beta-function is 15 m." ], [ "Conclusion", "In this paper we have considered scenarios for operation of the European XFEL in hard X-ray regime after a possible CW upgrade.", "We can conclude that operation in sub-Ångstrom regime is possible even at a conservatively assumed beam energy of 7 GeV provided that bright enough electron beams are available.", "Recent achievements in technology of high-brightness CW injectors support this assumption, although further studies on optimization of compression and transport of these beams are required.", "Recent tests of prototypes of XFEL cryomodules indicate a possibility that a higher electron energy (about 9 GeV) can be achieved in CW mode what will make sub-Ångstrom operation of the European XFEL in this mode much easier.", "Correspondingly, operation in LP mode with a high duty factor will bring the energy into a very comfortable range, above 10 GeV.", "We have not discussed the properties of the FEL beam in the case of a reduced energy of the linac.", "As we have already mentioned, such a reduction would result in a smaller peak power (and peak brilliance).", "Energy dependence of output parameters of the optimized SASE FEL can be estimated with the help of the results of Ref. [30].", "After choice of operation points and a detailed optimization of electron beam parameters it will be possible to perform detailed calculations of the main properties of the FEL radiation in the same way as it was done in [31]." ] ]	1403.0465
[ [ "Weak solution of the non-perturbative renormalization group equation to\n describe the dynamical chiral symmetry breaking" ], [ "Abstract We analyze the dynamical chiral symmetry breaking (D$\\chi$SB) in the Nambu-Jona-Lasinio (NJL) model by using the non-perturbative renormalization group (NPRG) equation.", "The equation takes a form of two-dimensional partial differential equation for the multi-fermion effective interactions $V(x,t)$ where $x$ is $\\bar\\psi\\psi$ operator and $t$ is the logarithm of the renormalization scale.", "The D$\\chi$SB occurs due to the quantum corrections, which means it emerges at some finite $t_{\\rm c}$ in the mid of integrating the equation with respect to $t$.", "At $t_{\\rm c}$ some singularities suddenly appear in $V$ which is compulsory in the spontaneous symmetry breakdown.", "Therefore there is no solution of the equation beyond $t_{\\rm c}$.", "We newly introduce the notion of weak solution to get the global solution including the infrared limit $t\\rightarrow \\infty$ and investigate its properties.", "The obtained weak solution is global and unique, and it perfectly describes the physically correct vacuum even in case of the first order phase transition appearing in finite density medium.", "The key logic of deduction is that the weak solution we defined automatically convexifies the effective potential when treating the singularities." ], [ "Introduction", "The dynamical chiral symmetry breaking (D$\\chi $ SB) has been the central issue of the elementary particle physics since it was initially founded by Nambu and Jona-Lasinio [20], [21].", "In short, all the variety of elementary particles comes out of the special pattern of the spontaneous chiral symmetry breaking of some unified gauge theory.", "For the quarks, the strong interactions described by QCD breaks the chiral symmetry at the infrared scale, where the QCD gauge coupling constant becomes enough strong.", "Recently QCD in high temperature and density has drawn strong attention since there are fruitful new phases expected, including the chiral symmetry restoration and the color superconductivity.", "The D$\\chi $ SB is highly non-perturbative phenomena and cannot be treated by the perturbation theory.", "Thus there have been various non-perturbative approaches to analyze it such as the lattice simulation and the Schwinger-Dyson (SD) approaches.", "Here we use the non-perturbative renormalization group (NPRG) method originated from the Kadanoff and Wilson's [26], [17] idea.", "The NPRG approaches share good features; it has no sign problem at finite chemical potential, which has suffered the lattice simulation seriously; it can improve the approximation systematically so as to reduce the gauge dependence of physical quantities [6], [7], which is the essential problem of the SD approaches.", "Now we state the central subject of this paper.", "In the NPRG method, we integrate the path integral of the theory from the micro scale to the macro scale, slice by slice, and obtain the differential equation of the effective action with respect to the logarithmic renormalization scale $t$ , which is defined by $t =-\\log (\\Lambda /\\Lambda _0),$ where $\\Lambda _0$ is the bare cut off scale and $\\Lambda $ is the renormalization scale.", "In the simplest approximation, the effective action is expressed in terms of the field polynomials without derivatives in addition to the normalized kinetic terms.", "Then the coupling constants $C_i$ , the coefficients of these operator polynomials, are functions of $t$ .", "The change of these coupling constants are given by functions of the coupling constants themselves, $\\frac{d}{dt} C_i(t) = \\beta _i\\left(C(t)\\right),$ where the right-hand side is called the $\\beta $ -function.", "This differential equation is the renormalization group equation (RGE).", "The $\\beta $ -function is evaluated by the one-loop diagrams only and the propagators constituting the loop are limited to have the sliced momentum.", "Therefore its evaluation is so simple that only the angular integration should be done to leave the total solid angle factor, which is called the shell mode integration.", "Here we consider the four-fermi coupling constant $G$ .", "The four-fermi interactions play an essential role in the total framework of the D$\\chi $ SB analysis.", "The $\\beta $ -function for the dimensionless four-fermi coupling constant rescaled by the renormalization scale $\\Lambda $ , $\\tilde{G}(\\equiv \\Lambda ^2 G)$ , is given by the following Feynman diagrams, $\\frac{d}{dt}\\tilde{G}&=-2 \\tilde{G} +\\ \\ \\begin{minipage}[c]{0.5}\\includegraphics [scale=1.0]{ladder_beta_4fermi_gauge.pdf}\\end{minipage} \\!\\!,$ where the first term in the right-hand side represents the canonical scaling due to the operator dimension, the solid lines are quarks, the wavy lines are gluons and the above loops represent the shell mode integration.", "It is evaluated as follows [2], [5], [4], $\\frac{d}{dt}\\tilde{G} &= -2\\tilde{G}+\\frac{1}{2\\pi ^2}\\left(\\tilde{G}+(3+\\xi )C_2 \\pi \\alpha _{\\rm s}\\right)^2,$ where $\\alpha _{\\rm s}$ is the gauge coupling constant squared as usual, $C_2$ is the second Casimir invariant of the quark representation and $\\xi $ is the gauge fixing parameter in the standard $R_\\xi $ gauge.", "Due to the chiral symmetry, the RGE for four-fermi operator is closed by itself, no contribution from higher dimensional operators allowed.", "Consider first the Nambu-Jona-Lasinio (NJL) model as an effective infrared model of QCD.", "Then the $\\beta $ -function is simplified to give $\\frac{d}{dt} \\tilde{G} &= -2\\tilde{G}+\\frac{1}{2\\pi ^2}\\tilde{G}^2,$ and it is depicted in Fig.", "REF , where the arrows show the direction of renormalization group flows towards the infrared.", "Note that there are two fixed points, zeros of the $\\beta $ -function, at 0 and $\\tilde{G}_{\\rm c}=4\\pi ^2$ , and they are infrared (stable) fixed point and ultraviolet (unstable) fixed point respectively.", "It is readily seen that there are two phases, the strong phase and the weak phase, and the critical coupling constant dividing the phase is $\\tilde{G}_{\\rm c}$ .", "If the initial coupling constant is larger than $\\tilde{G}_{\\rm c}$ , flows go to the positive infinite, otherwise they approaches to the origin.", "We are sure that this is the quickest argument ever to derive the critical coupling constant in the NJL model and the result is equal to the mean field approximation.", "Figure: The β\\beta function for the four-fermi coupling constant G ˜\\tilde{G} in the NJL model.The weak phase has no problem here.", "The flows approaching to the origin do not mean that the four-fermi interactions vanish at the infrared.", "Since the variable here is the dimensionless coupling constant, the four-fermi interactions look vanishing due to the multiplication factor of $\\exp (-2t)$ .", "As for the dimensionful variable, it just stops running to give some constant size infrared interactions determined by the initial coupling constant.", "On the other hand, the strong phase has a serious problem.", "Equation (REF ) is easily integrated to give the following solution, $\\tilde{G}(t) = \\frac{\\tilde{G}_{\\rm c}\\tilde{G}_0}{\\tilde{G}_0-\\left(\\tilde{G}_0-\\tilde{G}_{\\rm c}\\right){\\rm e}^{2t}} ,$ where $\\tilde{G}_0$ is the initial coupling constant (Fig.", "REF ).", "This solution is a blowup solution and it diverges at a finite $t_{\\rm c}$ defined by, $t_{\\rm c}= \\frac{1}{2}\\log \\frac{\\tilde{G}_0}{\\tilde{G}_0-\\tilde{G}_{\\rm c}}.$ Because of this blowup behavior, we cannot obtain the solution after $t_{\\rm c}$ , that is, there is no global solution in Eq.", "(REF ) for $\\tilde{G}_{\\rm 0}>\\tilde{G}_{\\rm c}$ .", "Figure: Blowup behavior of the running four-fermi coupling constant G ˜\\tilde{G}.This is not a fake nor an insufficientness of our treatment.", "This must happen necessarily [3], [8].", "The reason is the following.", "The NJL model is expected to have the D$\\chi $ SB with strong coupling constant $\\tilde{G}_0 > \\tilde{G}_{\\rm c}$ .", "This is the quantum effects, that is, it is caused by the quantum loop corrections.", "The NPRG method calculates the effective interactions by adding the quantum loop corrections from micro to macro, step by step.", "Then the initially symmetric theory shows up quite a different effective interactions exhibiting the spontaneous mass generation, at some intermediate stage of integration of the NPRG equation.", "Figure: Second order phase transition.The schematic view of the chiral condensates or the dynamical mass as a function of the renormalization scale $t$ is drawn in Fig.", "REF , where at $t_{\\rm c}$ the D$\\chi $ SB occurs.", "This figure resembles the temperature dependence of the spontaneous magnetization if the low and high $t$ are reversed.", "Then consider what happens if $t$ approaches $t_{\\rm c}$ from the origin.", "The correlation length of spins ($\\bar{\\psi }\\psi $ operators) must diverge and also the magnetization susceptibility diverges.", "The susceptibility is proportional to the total spin fluctuation $\\langle (\\bar{\\psi }\\psi )^2\\rangle $ .", "This is nothing but the four-fermi interactions.", "Therefore the diverging behavior of the four-fermi interactions itself is a normal and naive phenomenon that just tells us we are approaching the second order phase transition point.", "Figure: The β\\beta function for the four-fermi coupling constant G ˜\\tilde{G} in gauge theories.The same story holds for QCD case.", "The $\\beta $ -function of the four-fermi coupling constant now depends on the gauge coupling constant as shown in Fig.", "REF ($\\xi =0, C_2=1$ case).", "Switching on the gauge interactions, the gauge coupling constant becomes large towards the infrared, the parabola function moves up, and finally after $\\alpha _{\\rm s}>\\alpha _{\\rm s}^{\\rm c}=\\pi /3$ , it has no zero at all.", "Accordingly, the stable and unstable fixed points approach to each other, and they are merged to pair-annihilate.", "After this annihilation, total space belongs to the strong phase and the four-fermi coupling constant moves to positive infinite.", "This is our NPRG view of how QCD breaks the chiral symmetry in the infrared, and also it explains the peculiar phase diagram of the so-called strong QED (or the gauged NJL model) as is depicted in Fig.REF from the RGE point of view[2], [5], [4].", "Figure: Phase structure of strong QED in α s -G ˜\\alpha _{\\rm s} - \\tilde{G} plane.Now the problem is how to calculate the infrared quantities, such as the chiral condensates and the dynamical mass.", "They reside all beyond the blowup point.", "We stress here that this blowup is irrelevant to the approximation scheme.", "Any theory, approximated or not, as far as it exhibits the D$\\chi $ SB, must encounter the spontaneous breaking in the mid of the renormalization, since it is produced by the sufficient quantum corrections.", "The D$\\chi $ SB must be expressed by the singular effective interactions of fermions.", "Therefore, any model providing the D$\\chi $ SB must meet singularities in the mid of renormalization.", "This is inevitable.", "Many NPRG analyses of the D$\\chi $ SB have been performed by introducing the bosonization of the multi-fermi interactions [1], [2], [5], [14], which is called the auxiliary field method or the Hubbard-Stratonovich transformation.", "This type of approaches may avoid the blowup singularity and has worked well indeed.", "However it must contain additional difficulties or ambiguities in how to treat and approximate the system of bosonic degrees of freedom.", "We take the other way around.", "We do not introduce the bosonic degrees of freedom and instead we extend the notion of the solution of NPRG equation.", "First we go back to the original NPRG equation without expansion in polynomials of field operators.", "Then it is a partial differential equation (PDE) of at least two variable, the operator and renormalization scale $t$ .", "The blowup behavior is reexamined in the next section and it is made clear that even the solution of PDE must encounter the corresponding singularity at $t=t_{\\rm c}$ .", "Thus the origin of singularity is irrelevant to the Taylor expansion itself of PDE.", "Even if the PDE type of RGE is adopted, we cannot have any global solution.", "Then we introduce the notion of weak solution of PDE [13].", "It allows solution with singularities and we may define the global solution including the infrared limit.", "This paper is organized as follows.", "In Sect.", "2, taking the NJL model as an example, we briefly review the Wegner-Houghton equation, an approximation of NPRG equation.", "In Sect.", "3, the notion of weak solution is introduced and a practical method to construct the weak solution is explained in Sect. 4.", "In Sect.", "5, we solve the NJL model to explicitly construct the weak solution.", "In Sect.", "6, we discuss the effect of the bare mass of quarks, which is necessary to define the chiral order parameters, and we construct the Legendre effective potential.", "In Sect.", "7, the method of weak solution is applied to the first order phase transition case in the finite chemical potential NJL model.", "The convexity of the effective potential given by the weak solution is fully discussed.", "Finally we summarize this paper in Sect. 8.", "To conclude, we will prove that the weak solution of NPRG equation is perfectly successful, giving the global solution to calculate the infrared physical quantities.", "Most impressive is the case of the first order phase transition.", "There the weak solution correctly picks up the lowest free energy vacuum among the multi locally stable vacua, through the procedure that the effective potential is automatically convexified by the weak solution." ], [ "Non-perturbative renormalization group", "In this section we introduce the non-perturbative renormalization group by using the Wegner-Houghton (WH) equation.", "For simplicity, we take the Nambu–Jona-Lasinio (NJL) model with a simplified discrete chiral symmetry, which is regarded as a low energy effective model of QCD explaining the D$\\chi $ SB.", "The Lagrangian density is given by $\\mathcal {L}&=\\bar{\\psi }{{\\hfil /\\hfil \\crcr \\partial }} \\psi -\\frac{G_0}{2} (\\bar{\\psi }\\psi )^2,$ where $\\psi $ and $\\bar{\\psi }$ are the quark field and the antiquark field, respectively.", "There is no continuous chiral symmetry.", "However, the Lagrangian is invariant under the following discrete chiral ($\\gamma _5$ ) transformation, $\\psi \\rightarrow \\gamma _5\\psi ,\\ \\bar{\\psi }\\rightarrow -\\bar{\\psi }\\gamma _5.$ This discrete chiral symmetry forbids the mass term $m\\bar{\\psi }\\psi $ and the chiral condensate $\\langle \\bar{\\psi }\\psi \\rangle $ as well as the usual continuous chiral symmetry.", "It is first clarified by Nambu and Jona-Lasinio [20], [21] that for the strong enough four-fermi coupling constant $G_0$ , it exhibits D$\\chi $ SB.", "Using the mean field approximation, the critical coupling constant $G_{\\rm c}$ is found to be $4\\pi ^2/\\Lambda _0^2$ , where $\\Lambda _0$ is the ultraviolet cutoff scale.", "Note that the mean field approximation is equivalent to the self-consistency equation limited to the large-$N$ leading diagrams, where $N$ is the number of quark flavors.", "In the NPRG approach, a central object is the Wilsonian effective action $S_{\\rm eff}[\\phi ;\\Lambda ]$ defined by integrating out the microscopic degrees of freedom $\\phi _{\\rm H}$ with momenta higher than the renormalization scale $\\Lambda $ , $\\int \\!\\mathcal {D}\\phi _{\\rm H} e^{-S_0[\\phi _{\\rm L},\\phi _{\\rm H};\\Lambda _0]}= e^{-S_{\\rm eff} [\\phi _{\\rm L};\\Lambda ]},$ where $S_0[\\phi ;\\Lambda _0]$ is the bare action with the ultraviolet cutoff scale $\\Lambda _0$ and $\\phi $ generically denotes all relevant fields.", "The renormalization scale $\\Lambda $ is parametrized by dimensionless variable $t$ as defined in Eq.", "(REF ), $\\Lambda (t) = \\Lambda _0 e^{-t}.$ The $t$ -dependence of the effective action $S_{\\rm eff}[\\phi ;t]$ is given by the NPRG equation, which is the following functional differential equation, $\\frac{d}{dt} S_{\\rm eff} [\\phi ;t] =\\beta _{\\rm WH}\\left[\\frac{\\delta S_{\\rm eff}}{\\delta \\phi },\\frac{\\delta ^2 S_{\\rm eff}}{\\delta \\phi ^2};t\\right],$ which is also called the Wegner-Houghton (WH) equation [24] (see Ref.", "[10] for the detail form of $\\beta _{\\rm WH}$ ).", "The right-handed side called the $\\beta $ -function is actually evaluated by the one-loop diagram exactly.", "No higher loops do not contribute.", "This functional differential equation should be solved as the initial value problem, where the initial condition refers to the bare action, $S_{\\rm eff} [\\phi ;0] = S_0[\\phi ] .$ Solving the equation towards the infrared ($t \\rightarrow \\infty $ ) and we get the macro effective action from which physical quantities such as the chiral condensate and the dynamical quark mass are obtained.", "The WH equation itself is exact, but it cannot be solved exactly, since it has infinite degrees of freedom.", "Here we apply the WH equation to the NJL model.", "As an approximation, we restrict the full interaction space of the effective action $S_{\\rm eff}[\\psi ,\\bar{\\psi };t]$ into the subspace most relevant to D$\\chi $ SB.", "We adopt the so-called local potential approximation where any quantum corrections to the derivative interactions are ignored.", "Furthermore, we set the local potential as a function only of the scalar fermion-bilinear field, $x=\\bar{\\psi }\\psi $ .", "Then our effective action takes the following form, $S_{\\rm eff}[\\psi ,\\bar{\\psi };t]&=\\int \\!d^4x \\left\\lbrace \\bar{\\psi }{{\\hfil /\\hfil \\crcr \\partial }}\\psi - V(\\bar{\\psi }\\psi ;t)\\right\\rbrace .$ The potential term $V(x;t)$ , where $x$ denotes $\\bar{\\psi }\\psi $ (integrated over space time), is called the fermion potential here, whose initial condition is set to be $V(x;t=0)=\\frac{G_0}{2}x^2$ according to the NJL Lagrangian (REF ).", "When evaluating the $\\beta $ -function we adopt an additional approximation of ignoring the large-$N$ non-leading part in $\\beta _{\\rm WH}$ .", "Then, the NPRG equation for the fermion potential in the large-$N$ leading is given by the following partial differential equation, $\\partial _t V(x;t)=\\frac{\\Lambda ^4}{4\\pi ^2}\\log \\left(1+\\frac{1}{\\Lambda ^2}(\\partial _x V)^2\\right)\\equiv - F(\\partial _x V;t).$ Hereafter we sometimes use a quick notation to save space like $\\partial _t=\\partial /\\partial t$ etc.", "The renormalization scale $\\Lambda $ , the momentum cutoff, is defined by referring to the length of four-dimensional Euclidean momentum $p_\\mu $ , that is, $\\sum _{\\mu =1}^{4} p_\\mu ^2 \\le \\Lambda ^2$ .", "This is called the four-dimensional cutoff.", "The right-hand side of equation is nothing but the simple trace-log formula of the Gaussian integral of the shell modes and it corresponds to the one-loop corrections, though it is exact.", "Note that the approximate NPRG equation above is known to be equivalent to the mean field calculation [2], [5], [4].", "Now we introduce the mass function $M(x;t)$ , the first derivative of the fermion potential, $M(x;t)=\\partial _x V(x;t) .$ The value of the mass function at the origin is the coefficient of mass operator $\\bar{\\psi }\\psi $ in the effective action, and this is why we call it the mass function.", "Note that it is still a function of $x$ , that is, a function of operator, or more rigorously, a function of the bilinear Grassmannian variable $x$ .", "The chiral symmetry transformation defined in Eq.", "(REF ) is represented by the reflection, $x \\longrightarrow -x.$ The NPRG equation (REF ) preserves this reflection symmetry and the initial condition also respects it.", "Accordingly, the fermion potential at any time $t$ satisfies the reflection symmetry, $V(-x;t)=V(x;t) ,$ and therefore the mass function satisfies, $M(-x;t)=-M(x;t) ,$ and it is an odd function of $x$ .", "Then the mass function must vanish at the origin, $M(0;t)=0,$ assuming it is well-defined at least.", "This means there is no spontaneous mass generation and the phase is chiral symmetric.", "Then what actually happens in case when D$\\chi $ SB and the spontaneous mass generation occurs?", "We have to abandon well-definedness of the mass function at the origin, that is, the fermion potential $V$ must lose its differentiability at the origin.", "Then we can have the following situation, $M(0+;t) = - M(0-;t) \\ne 0,$ that is, the mass function has a jump at the origin, still keeping its odd function property.", "This is the spontaneous mass generation.", "To explain clearly that this gives us the dynamically generated mass indeed, we need the notion of the effective potential, which will be introduced in Sec.", "6.", "Now we understand what we should deduce.", "At some finite $t_{\\rm c}$ , the fermion potential becomes non-analytic at the origin, and it generates a finite gap in the mass function.", "The gap continues to increase and finally gives the dynamical quark mass at the infrared.", "Just before $t_{\\rm c}$ , the second derivative of $V$ must become large and increases towards infinite at $t_{\\rm c}$ .", "The second derivative of $V$ at the origin gives the four-fermion interactions at the scale $t$ , and this divergent behavior is nothing but the blowup solution we find in the expanded NPRG equation in Eq.", "(REF ) in case of $G_0>G_{\\rm c}$ [3], [8].", "See the plot (d) in Fig.", "REF , which is exactly what we like to realize as a solution for the mass function.", "Before $t_{\\rm c}$ the total function is a smooth analytic function, whereas after $t_{\\rm c}$ there is a finite jump at the origin which shows the spontaneous mass generation.", "Such a solution, however, is not allowed as a solution of the partial differential equation (PDE) in Eq.", "(REF ), since the solution of the partial differential equation must satisfy the equation at every point.", "Therefore, the differentiability at any point ($x,t$ ) is mandatory, while our expecting solution must break it, though only at the origin.", "To cure this situation, we have to modify the original equation so that differentiability of function $V$ might not be necessary.", "This is the notion of weak solution, which is defined by the solution of the weak equation, and it will be the subject of the next section." ], [ "Weak solution and the Rankine-Hugoniot condition", "In this section, we first derive the partial differential equation for the mass function and define the weak solution of it [23], [19].", "Then we construct the weak solution explicitly using the standard method of the characteristics [13] in the next section.", "We start with the following partial differential equation for $V(x;t)$ , $\\partial _t V(x;t) = -F(\\partial _x V, x;t).$ This is an extended system of the NJL-model NPRG equation (REF ), where the $\\beta $ -function $F$ contains explicit dependence on $x$ .", "The reason why we include this extension is that the gauge theory analysis requires this additional $x$ dependence, and also the total system becomes more symmetric and better interpreted in terms of the classical mechanics.", "Differentiating the PDE (REF ) with respect to $x$ , we obtain the following PDE for the mass function, $\\partial _t M(x;t)& = -\\partial _x F(M(x;t),x;t) \\\\& =-\\frac{\\partial F}{\\partial M} \\cdot \\frac{\\partial M}{\\partial x}- \\frac{\\partial F}{\\partial x} .$ Here it should be noted that we have used somewhat subtle notations that two forms of the partial derivative of $F$ with respect to $x$ have different meanings as for treatment of variable $M$ .", "This equation belongs to a class of PDE called the conservation law type since it has a form of conservation law in two-dimensional space time.", "This class contains the famous Burgers' equation for non-linear wave without viscosity [9].", "We integrate the above equation convoluted with a smooth and bounded test function $\\varphi (x;t)$ , and then by the partial integration we have $\\int _{0}^{\\infty }\\!", "dt \\int _{- \\infty }^{\\infty } \\!", "dx\\left(M \\frac{\\partial \\varphi }{\\partial t}+F\\frac{\\partial \\varphi }{\\partial x}\\right)+ \\int _{- \\infty }^{\\infty }& dx~\\left.", "M \\,\\varphi \\right\|_{t=0} =0.$ Note that the above equation has no direct derivative of function $M(x;t)$ .", "Now we define the weak solution as follows; the weak solution $M(x;t)$ satisfies Eq.", "(REF ) for any smooth and bounded test function $\\varphi $ .", "The weak solution $M(x;t)$ does not have to be differentiated, thus it can have some non-differentiable point.", "Any strong solution satisfying the original PDE (REF ) at every point is in fact the weak solution.", "The inverse does not hold.", "In this sense, the weak solution is a wider notion than the strong solution.", "Then even in case when the strong solution does not have the global solution, the weak solution can be global and then we can calculate infrared physical quantities using the weak solution.", "In general, the weak solution may not be unique.", "It also may depend on the initial condition.", "We will prove the weak solution of our NPRG equation is in fact unique and global, and we can successfully obtain the infrared physics without any ambiguity.", "Here we state what is the weak solution without explicit proof (see e.g.", "Ref.", "[13] for detailed argument).", "We suppose a weak solution has some jump singularities at some finite number of points and otherwise it is smooth, that is, it is piecewisely differentiable.", "Then almost everywhere other than the singularity, the weak solution must satisfy the strong equation (REF ) .", "This is the first property of the weak solution and thus it is not so different from the strong solution.", "Figure: Rankine-Hugoniot conditionSecond, what is required at the singularity?", "As is shown in Fig.", "REF , we look into the neighborhood of a singularity where there is a jump discontinuity on a curve $x=S(t)$ , and values of $M$ at the left-hand side and right-hand side of the singularity is denoted by $M_{\\rm L}$ and $M_{\\rm R}$ respectively.", "Remember that the original equation (REF ) is regarded as the conservation law where the charge density is $M(x,t)$ and the current is $F(M(x;t),t)$ .", "Consider a small boxed region $dS \\times dt$ in Fig.", "REF and evaluate the total charge in the section of length $dS$ .", "The conservation law requires that the change of total charge must be balanced with the difference between total flow-in and flow-out.", "We have, $\\left(M^{\\rm (L)}-M^{\\rm (R)}\\right) dS(t) =\\left[F(M^{\\rm (L)})-F(M^{\\rm (R)}) \\right] dt.$ This is nothing but the integrated form of the conservation law.", "The above condition (REF ) is called the Rankine-Hugoniot condition [22], [15], [16], and all discontinuity must satisfy it.", "This is the second property which the weak solution must satisfy, and no more.", "This condition actually determines the equation of motion of the singularity position by the first order differential equation.", "It needs one initial condition to determine the motion of the discontinuity point, and it is actually given by the spontaneous generation of the singularity as is seen later.", "The Rankine-Hugoniot condition for the mass function $M$ is converted to the condition for the fermion potential $V$ .", "In Fig.", "REF , we denote the fermion potential at each side of the singularity by $V^{\\rm (L)}$ and $V^{\\rm (R)}$ respectively.", "Then we evaluate its difference between bottom left point and top right point of the $dt \\times dS$ box as follows, $dV^{\\rm (L, R)}=\\left.\\frac{\\partial V }{\\partial x}\\right\|^{\\rm (L,R)} dS+ \\left.\\frac{\\partial V }{\\partial t}\\right\|^{\\rm (L,R)} dt=M^{\\rm (L,R)} dS -F(M^{\\rm (L,R)}) dt.$ We have the following relation given by the RH condition (REF ), $dV^{\\rm (L)} - dV^{\\rm (R)} = (M^{\\rm (L)}-M^{\\rm (R)})dS- (F(M^{\\rm (L)}) - F(M^{\\rm (R)}))dt =0.$ Therefore the fermion potential develops equally between left and right side of the singularity.", "Taking account of the fact that there is no singularity in the beginning in case of our physical initial condition, that is, any singularity starts in the mid of $t$ -development, we conclude $V^{\\rm (L)} = V^{\\rm (R)}.$ Thus the fermion potential function $V(x;t)$ is continuous everywhere as far as it is the weak solution of the PDE.", "This form of the RH condition helps us much to obtain the weak solution in the next section." ], [ "Method of characteristics", "In this section, we construct the weak solution using the standard textbook method of characteristics.", "The characteristics is a curve on two-dimensional world ($x,t$ ) and is denoted by $x=\\bar{x}(t)$ .", "Consider the mass function on this curve, $\\bar{M}(t)=M(\\bar{x};t)$ , and we calculate its derivative, $\\dfrac{d\\bar{M}(t)}{dt}&= \\dfrac{\\partial M(\\bar{x}(t);t)}{\\partial \\bar{x}} \\dfrac{d\\bar{x}(t)}{dt}+\\dfrac{\\partial M(\\bar{x}(t);t)}{\\partial t} \\\\&=\\dfrac{\\partial M(\\bar{x}(t);t)}{\\partial \\bar{x}} \\dfrac{d\\bar{x}(t)}{dt}- \\dfrac{\\partial F(\\bar{M}, \\bar{x}(t);t)}{\\partial \\bar{M}} \\dfrac{\\partial M(\\bar{x}(t);t)}{\\partial \\bar{x}}- \\dfrac{\\partial F(\\bar{M}, \\bar{x}(t);t)}{\\partial \\bar{x}}.$ We take the characteristics curve to satisfy the following differential equation, $\\dfrac{d\\bar{x}(t)}{dt} & =\\dfrac{\\partial F(\\bar{M}, \\bar{x};t)}{\\partial \\bar{M}},$ then the mass function on it satisfies, $\\dfrac{d\\bar{M}(t)}{dt}= -\\dfrac{\\partial F(\\bar{M}, \\bar{x};t)}{\\partial \\bar{x}}.$ This coupled pair of ordinary differential equations (REF ) and (REF ) are equivalent to the original two-dimensional partial differential equation (REF ).", "The initial conditions for these equations are set by, $\\bar{x}(t=0)&=x_0, \\\\\\bar{M}(t=0)&=\\left.\\partial _x V(x;t)\\right\|_{x=x_0,t=0} = M_0,$ where $x_0$ is a parameter discriminating characteristics and $M_0$ is the initial value of $M$ at $x=x_0$ .", "When explicitly indicating the initial value of the characteristic, we use $\\bar{x}(t;x_0)$ .", "The characteristics will give us the strong solution of PDE.", "Take a characteristics starting at $x_0$ , we solve the coupled ODE, and then we get the characteristics $\\bar{x}(t; x_0)$ as shown in Fig.", "REF (a), and the mass function on it, $\\bar{M}(\\bar{x}(t);t)$ .", "Figure: Characteristics.Here we change our viewpoint.", "The coupled ODE in equations (REF ) and (REF ) is regarded as the canonical equation of motion of a kinematical system with the coordinate $\\bar{x}$ , the momentum $\\bar{M}$ and the time-dependent Hamiltonian $F(\\bar{M}, \\bar{x}; t)$ .", "We plot two variables $\\bar{x}(t)$ and $\\bar{M}(t)$ in Fig.", "REF (b) and this is nothing but the phase space orbit of the Hamiltonian system.", "The fermion potential $V$ also has its counterpart in mechanics.", "Consider the fermion potential on the characteristics, $\\bar{V}(t) = V(\\bar{x}(t);t)$ , and we calculate its derivative, $\\frac{d \\bar{V}(t)}{dt}=\\dfrac{\\partial V(\\bar{x}(t);t)}{\\partial \\bar{x}} \\dfrac{d\\bar{x}}{dt}+ \\dfrac{\\partial V(\\bar{x}(t);t)}{\\partial t}= \\bar{M} \\frac{d \\bar{x} }{dt} -F(\\bar{M};t).$ The right-hand side is the Lagrangian of this mechanical system.", "Then the fermion potential is obtained by integration, $V(\\bar{x}(t);t) = V(x_0;t=0) + \\int _0^t \\left[\\bar{M} \\frac{d\\bar{x}}{dt} -F(\\bar{M};t)\\right]dt.$ Thus this is the action of the system as a function of the final time $t$ and the coordinate variable at the final time $\\bar{x}(t)$ , plus the initial value.", "Then the original PDE in Eq.", "(REF ) is what is called the Hamilton-Jacobi equation in this kinematical system.", "The starting point of this particle is controlled by the parameter $x_0$ , and its initial momentum is the initial mass function $M(x_0;t=0)$ .", "Then we consider all orbits at once as shown in Fig.", "REF , which is now the motion of string on the two-dimensional world $x \\times M$ .", "The string never crosses to itself, since the equation of particle motion uniquely determines the phase space orbit, thus no crossing of orbits is allowed.", "The shape of the string at $t$ gives the mass function $M(x;t)$ , the strong solution of original PDE.", "Figure: Phase space orbit of particles and motion of stringThere is nothing singular nor explosive blowup so far.", "The string motion gives the strong solution, locally.", "The point here is that it may not give the global solution as it is, when the string is folded and cannot define a function of $x$ .", "The string itself is parametrized by $x_0$ , and $M(\\bar{x}(t;x_0);t)$ is a unique function of $x_0$ .", "However when the characteristics are crossing to each other, we have multi $x_0$ for single value of $\\bar{x}(t)$ .", "This situation is intuitively stated as that we have “multi-valued function” $M(x;t)$ at $t > t_{\\rm c}$ .", "We note that the string motion here with the proper initial condition has no singularity at all up to the infinite time $t=\\infty $ .", "The string neither breaks up nor shrinks.", "The string $M(\\bar{x}(t;x_0);t)$ as a function of $x_0$ is always continuous.", "This fact will assure that the weak solution we define now is unique and satisfies the so-called entropy condition.", "Normally the multi-folded structure is born as three-folded leaves, and in the example in section 7, two three-folded structures meet together to become five-folded.", "Figure: World surface made of string motion.Let us see a typical plot for the NJL model (super critical situation, $G_0=1.005 G_{\\rm c}$ ) in Fig.", "REF , where the string motion and characteristics for $0<t<5$ is shown to make a surface.", "The initial string is a straight line given by the initial mass function, $G_0 x$ , and it develops to generate the self-folding structure.", "After a finite $t_{\\rm c}$ the surface is 3-folded.", "Simultaneously the characteristics pass over the origin.", "All leaves of this surface are the strong solution at least locally.", "Now we construct the weak solution by patch-working leaves to define a single valued function $M(x;t)$ of $x$ at any time $t$ .", "After the patch-working, there appear discontinuity singularities.", "How to cut and patch is determined by the RH condition.", "Figure: Geometrical interpretation of RH condition.We pick up a three folded sector and draw $V(x;t)$ and $M(x;t)$ as multi-valued functions of $x$ in Fig.", "REF (a) and (b) respectively.", "The fermion potential has a characteristic shape of multi-folded structure, which is called swallowtail in the mechanics or spinodal decomposition in the thermodynamics.", "The mass function $M(x;t)$ is a fixed $t$ section of the surface in Fig.", "REF .", "The weak solution must satisfy the continuity of $V(x;t)$ according to the RH condition (REF ), and it is uniquely determined to give the solid line as drawn in Fig.", "REF (a).", "Note that this selection can also be stated as that we take the maximum possible branch among the candidates.", "This weak solution gives a solution for $M(x;t)$ as is drawn in Fig.", "REF (b).", "The points, A, B, L and R, in (a) and (b) are corresponding to each other, where L and R in (a) are the same.", "The vertical line is the cut we take for patch working and the mass function has a discontinuity from point L to R. We integrate this multi-valued function along the string path, and we obtain $\\int _{\\rm L}^{\\rm R} Mdx = \\int _{\\rm L}^{\\rm R}\\frac{\\partial V}{\\partial x} dx=V^{\\rm (R)}-V^{\\rm (A)}+V^{\\rm (A)}-V^{\\rm (B)}+V^{\\rm (B)}-V^{\\rm (L)}= V^{\\rm (R)}-V^{\\rm (L)}=0,$ where we have used the RH condition in the form of Eq.", "(REF ).", "This assures that the total (signed) area surrounded by the string and the cut (vertical line) must vanish.", "In other words the two areas of left and right-hand side of the cut are equal to each other [25].", "These geometrical representation of the RH condition is convenient to determine the weak solution from the string motion diagram $M(x;t)$ .", "When folding is generated, we just cut and patch so that we might keep the equal area rule.", "We should stress here that the equal area rule is not a condition for the weak solution but just a resultant feature referring to the strong solutions in the backyard.", "These multi-leave strong solutions are not seen by the weak solution, and thus it is impossible to refer to the backyard solutions in order to get the next time step.", "The target solution in the renormalization group equation is the effective interactions at a scale $t$ and there is absolutely no information for the backyard strong solutions.", "In fact, the original RH condition (REF ) is equivalent to the conservation of the difference $V^{\\rm (R)}-V^{\\rm (L)}$ in Eq.", "(REF ).", "After adding the fact that any singularity is generated in the mid of $t$ development, we have the continuity $V^{\\rm (R)}=V^{\\rm (L)}$ and consequently the equal area rule.", "Original condition (REF ) or (REF ) is enough to determine the weak solution uniquely.", "Therefore we do not have to do any access to the backyard solutions to solve the renormalization group equation.", "The renormalization group equation determines its weak solution by itself, without recourse to the multi-leave solutions.", "The above equal area rule may remind the reader of the Maxwell's rule in treating the coexisting situation, for example, the liquid and the gas, where the vertical line in Fig.", "REF (b) looks like the coexisting isothermal line in the $p(=x)$ -$V(=M)$ phase diagram.", "However the Maxwell's rule has to refer to the backyard thermodynamic functions which are usually not well established in the equilibrium thermodynamics.", "In this sense the Maxwell's rule is to be classified as a phenomenological rule.", "Nevertheless we may assign such jump line between L and R the coexisting states of L and R states with the bulk domain structure.", "We do not discuss this viewpoint more since it is out of the scope of this paper.", "As is proved in Eq.", "(REF ), the fermion potential is the action up to the initial constant in the mechanics analogy, and in fact the selection of leaves satisfying the RH condition will be done so that the maximum action principle holds, which is directly related to the minimum free energy principle as is seen later." ], [ "Weak solution of the NJL model", "We construct the weak solution for the NJL model explicitly.", "The Hamiltonian $F$ has no explicit dependence on $x$ , that is, the system is translationally invariant, $F(M,x;t) = - \\frac{\\Lambda ^4}{4\\pi ^2}\\log \\left(1+\\frac{M^2}{\\Lambda ^2}\\right).$ Therefore the momentum $M$ is conserved in the time development and the characteristics $\\bar{x}(t)$ are the contour of $M$ .", "Then the string motion is driven by horizontal move of each point as shown in Fig.", "REF .", "The initial position of the string is given by the bare four-fermi interactions, $M(x;0) = \\partial _x V(x;0) = G_0 x,$ which is a straight line.", "Figure: String motion.The velocity of each point of string is given by, $\\dfrac{d\\bar{x}(t)}{dt}= \\dfrac{\\partial F(\\bar{M},x;t) }{\\partial \\bar{M}}= - \\dfrac{\\Lambda ^4}{4\\pi ^2} \\dfrac{2 M }{\\Lambda ^2 + M^2},$ and it has an opposite sign against the momentum $M$ .", "The velocity is suppressed for large $M$ region and it will vanish for large $t$ (small $\\Lambda $ ).", "The schematic motion of string is drawn in Fig.", "REF in case of super critical phase, where the string starts being folded after a finite $t_c$ .", "Now we solve the characteristics by integrating Eq.", "(REF ) with the initial conditions, $\\bar{x}(0;x_0)=x_0\\ , \\ \\ \\bar{M}(x(0);0) = G_0 x_0 =M_0,$ and we get, $\\bar{x}(t;x_0) & =x_0 + \\int ^t_0 \\frac{d\\bar{x}}{dt} dt=x_0 -\\frac{2M_0\\Lambda ^4_0}{4\\pi ^2}\\int ^t_0 \\frac{{\\rm e}^{-4t}}{\\Lambda _0^2 {\\rm e}^{-2t} +M_0^2} dt\\\\&= x_0+\\frac{\\Lambda _0^2 M_0}{4\\pi ^2}\\left[{\\rm e}^{-2t} - 1-\\frac{M_0^2}{\\Lambda _0^2}\\log \\frac{\\Lambda _0^2{\\rm e}^{-2t} +M_0^2}{\\Lambda _0^2+M_0^2}\\right] \\\\&\\equiv C_t(x_0).$ If $t$ is enough small, the function $C_t$ is a monotonic function and then we can express the initial value $x_0$ by a function of the final value $x(t)$ using the inverse of function $C_t$ , $x_0 = C^{-1}_t(x).$ Finally we obtain the solution by using this inverse function as follows, $M(x;t)= M(C^{-1}_t(x);0)= G_0 C_t^{-1}(x).$ The solution in Eq.", "(REF ) is the strong solution as far as the inverse function $C^{-1}_t$ can be defined.", "This situation will be broken at finite $t$ for strong enough coupling constant $G_0$ .", "The monotonicity of function $C_t$ breaks at the origin first.", "We evaluate the derivative at the origin, $\\left.", "\\frac{dC_t (x_0)}{dx_0}\\right\|_{x_0=0}=\\left.", "\\frac{\\partial \\bar{x}(t;x_0)}{\\partial x_0}\\right\|_{x_0=0}=1+\\frac{\\Lambda _0^2G_0}{4\\pi ^2} \\left({\\rm e}^{-2t}-1\\right).$ This becomes negative at a finite $t_{\\rm c}$ , $t_{\\rm c}=\\frac{1}{2} \\log \\frac{G_0\\Lambda _0^2}{G_0\\Lambda _0^2-4\\pi ^2},$ for strong bare coupling constant $G_0 > G_c$ , $G_{\\rm c}\\Lambda _0^2=4\\pi ^2\\ .$ This result is equal to the argument in Sec.", "1 using the blowup solution.", "Finally we calculate the fermion potential exactly by integrating Eq.", "(REF ), $V(x;t)&= V(x_0;0) +\\int _0^t\\frac{dV(\\bar{x}(t);t)}{dt} dt \\\\&= \\dfrac{1}{2}G_0 x_0^2 +\\frac{3}{4} M_0 \\left(x - x_0 \\right)-\\frac{\\Lambda _0^4}{16\\pi ^2}\\left[{\\rm e}^{-4t} \\log \\left(1+\\frac{M_0^2}{\\Lambda _0^2} {\\rm e}^{2t}\\right)-\\log \\left(1+\\frac{M_0^2}{\\Lambda _0^2}\\right)\\right]\\\\&= \\dfrac{1}{2}G_0 x_0^2 +\\frac{\\Lambda _0^2}{16\\pi ^2}\\Bigg \\lbrace \\left[3M_0^2 \\left({\\rm e}^{-2t} -1-\\frac{M_0^2}{\\Lambda _0^2}\\log \\frac{\\Lambda _0^2 {\\rm e}^{-2t}+M_0^2}{\\Lambda _0^2+M_0^2}\\right)\\right] \\\\&\\qquad -\\Lambda _0^2\\left[{\\rm e}^{-4t} \\log \\left(1+\\frac{M_0^2}{\\Lambda _0^2} {\\rm e}^{2t}\\right)-\\log \\left(1+\\frac{M_0^2}{\\Lambda _0^2}\\right)\\right]\\Bigg \\rbrace \\ ,$ where $x_0$ and $M_0=G_0 x_0$ should be understood as a function of $x$ through the inverse function $C_t^{-1}$ in Eq.", "(REF ).", "The fermion potential is also multi-valued where the inverse function $C_t^{-1}$ is multi-valued.", "Figure: (a) Characteristics.", "(b) String motion, i.e., set of local strong solutions of mass function.", "(c) Characteristics selected by the RH condition and discontinuity at the red line.", "(d) Weak solution of mass function.", "Discontinuity at the origin to satisfy RH condition.We show the numerical plots of these results in Fig.", "REF in case of the super critical interaction, $G_0=1.005 G_{\\rm c}$ and $t_{\\rm c}=2.65$ , and explain how to do the patch work to construct the weak solution.", "Hereafter all the dimensional variables are rescaled by $\\Lambda _0$ to be dimensionless.", "The characteristics are plotted in Fig.", "REF (a), and we find they will start crossing to each other after $t>t_{\\rm c}$ .", "The over populated region enlarges with 3-folded structure.", "The string motion is drawn in Fig.", "REF (b), where we see it starts folding after $t_{\\rm c}$ .", "Then we have to cut and patch after $t_{\\rm c}$ .", "First, note that the original chiral symmetry of the system assures the odd function property of the mass function $M(-x;t) = -M(x;t)$ even after folding.", "The RH condition of the form in Eq.", "(REF ) tells us that the cut out area should have the same area always.", "Then the only way to satisfy RH condition is to put the discontinuity at the origin and adopt the up and down leaves, without using the mid leaf.", "Thus, we obtained the weak solution, which are drawn in Fig.", "REF (c)/(d).", "This weak solution is uniquely obtained for any initial four-fermi coupling constant and it is always global up to the infrared limit $t\\rightarrow \\infty $ .", "From the conservation law point of view, the symmetry breaking and the RH solution after that are understood as follows.", "The current $F$ is always negative and it is larger for larger $\|M\|$ .", "Then the charge $M$ accumulates towards the origin with increasing slope $\\partial _x M\|_0$ since the current at the origin is vanishing.", "At $t_{\\rm c}$ , the slope becomes infinite, the singularity appears and we start defining discontiuous solution.", "Now the charge flows in to the backyard and flows out from the backyard at the origin, ignoring the vanishing current condition there.", "This is the RH solution.", "We answer here for the possible question for the necessity of the weak solution.", "If we agree that there is a singularity in $M$ in fact, but only at the origin, then according to the total symmetry $M(-x)=M(x)$ , we may solve the system only for the half space $(0,\\infty )$ .", "This is true.", "We demonstrated that the numerical analysis of the PDE in $(0,\\infty )$ with the free end boundary condition at $0+$ could work actually and $M(0+)$ is obtained to grow up after $t>t_{\\rm c}$ [6].", "However this way of getting the dynamical mass is really tedious and its reliability is not proved.", "Moreover, most important is that such trick does not work for the first order phase transition case in Sec.7, where the singularity appears off the origin, pairwisely, and then they move.", "In such case the weak solution is the only method to reliably calculate the physical quantities." ], [ "Bare mass and the Legendre effective potential", "In the previous section, we have obtained the D$\\chi $ SB weak solution of the mass function $M(x;t)$ which has a finite jump at the origin.", "In this section we argue how to calculate the chiral order parameters, the dynamical mass of quark and the chiral condensate, in the infrared limit.", "The physical mass of quark should be defined by $M(0;\\infty )$ and it is actually not well-defined even by the weak solution itself.", "This is a common issue of the spontaneous symmetry breaking and we take a standard method of adding the bare mass term $m_0\\bar{\\psi }\\psi $ explicitly breaking the chiral symmetry to the bare action and investigate what happens.", "The addition of bare mass term does not modify our system of NPRG at all.", "It just changes the initial condition of the fermion potential as $V(x;0) = \\dfrac{G_0}{2} x^2 + m_0 x,$ and the initial mass function takes the following form, $M(x;0) = G_0 x + m_0.$ As is stressed in Eq.", "(REF ), the NJL system is translationally invariant with respect to $x$ .", "We shift the origin of $x$ coordinate as, $x \\longrightarrow x - m_0/G_0\\ ,$ and completely the same calculation holds for the massless case in the previous section.", "Therefore the weak solution with bare mass is obtained from that of massless case by shifting $x$ argument as follows, $M(x;t,;m_0)= M(x+m_0/G_0;t,;m_0=0).$ As for the fermion potential, we have to take account of the global shift of the bare potential in addition to the shift of $x$ , $V(x;t;m_0) = V(x+m_0/G_0;t;m_0=0) - \\dfrac{m_0^2}{2G_0}$ This looks miraculous considering that the RGE system with bare mass is completely different from massless case, all operators are coupled together.", "Figure: RG evolution of the physical masses in m 0 →0m_0\\rightarrow 0 and m 0 =0m_0=0.The NPRG equation given by Eq.", "() (μ=0\\mu =0, G=1.7G c G=1.7G_{\\rm c})is used.Now the mass function at the origin is well-defined and we can define the physical mass by the following chiral limit, $M_{\\rm phys}(t)= \\lim _{m_0\\rightarrow +0} \\left.M(x;t,;m_0)\\right\|_{x=0}.$ Fig.", "REF shows the RG evolution of the physical mass for $m_0\\ne 0$ and its chiral limit.", "The physical mass in the chiral limit shows the second order phase transition due to the singular behavior of the mass function at the origin, while the physical mass at $m_0\\ne 0$ shows the cross over behavior.", "The reader may consider that the weak-solution method is not necessary if we consider $m_0\\ne 0$ from the very beginning.", "This statement is absolutely wrong.", "First of all, note that the mass function is a function of operator $x$ totally describing the effective interactions.", "Even with the bare mass included, the mass function still has completely the same singularity.", "Singularity does not disappear, and it is just moved away from the origin.", "Actually, Ref.", "[6] shows that the Taylor expansion method to solve the PDE (REF ) does not work with the small bare mass.", "Second, if we are interested only in the origin of the mass function, then there seems to be no singularity as just drawn in Fig.", "REF .", "However this is the case of the second order phase transition.", "Such disappearance of singularity does not hold for the first order phase transition, which will be explicitly argued in the next section.", "The difference here is that the second order phase transition is directly related to the symmetry breakdown and therefore is sensitively affected by the explicit breaking term, while the first order phase transition is not related at all to the symmetry breaking and therefore is not altered much by the explicit breaking term.", "These points will be clear in the next section.", "In order to evaluate the chiral condensate and the Legendre effective potential for it, we first introduce the source term $j\\bar{\\psi }\\psi $ in the bare action.", "Then the initial condition of the fermion potential is $V(x;t=0;j)&= m_0 x + \\frac{G_0}{2} x^2 +jx.$ The free energy as a function of $j$ is given by the value of $V$ at the origin, $W(j;t)=V(x=0;t;j),$ and the chiral condensate is defined by, $\\phi (j;t)\\equiv \\langle \\bar{\\psi }\\psi \\rangle _j=\\frac{\\partial W(j;t)}{\\partial j}.$ Note that these quantities are all defined at renormalization scale $t$ .", "We define the Legendre effective potential of the chiral condensate as follows, $V_{\\rm L} (\\phi ;t)= j\\phi (j;t) -W(j;t),$ where $j$ in the right-hand side is considered as a function of $\\phi $ through the inverse of Eq.", "(REF ).", "We have the standard relation, $\\dfrac{\\partial V_{\\rm L}}{\\partial \\phi }=j.$ As seen previously, because of the translational invariance of the system with respect to $x$ , the $j$ -dependence of $V$ is determined by applying Eq.", "(REF ), $V( x;t;j)=V( x+j/G_0;t;j=0)-\\frac{m_0 j}{G_0}-\\frac{j^2}{2G_0},$ Thus the free energy and the chiral condensate are calculated through the quantities with $j=0$ as follows, $W(j;t) &= V( x=0;t;j)= V(j/G_0; t;j=0)-\\frac{m_0 j}{G_0}-\\frac{j^2}{2G_0}, \\\\\\phi (j;t)& =\\frac{1}{G_0} \\left[M(j/G_0;t;j=0)-m_0-j\\right].$ Therefore, we conclude that the chiral condensate function $\\phi (j;t)$ has the same structure of multi-valuedness as the mass function, which is shown in Fig.", "REF (a).", "Figure: Weak solution of the chiral condensate and the Legendre effective potential V L (φ)V_L(\\phi ).We have obtained the weak solution of mass function $M(x;t)$ , which determines the weak solution of the chiral condensate $\\phi (j;t)$ .", "The weak solution of the mass function is easily obtained by the equal area rule.", "This rule does work as well for the weak solution of the chiral condensate, since the difference between functions of $M$ and $G_0 \\phi $ are well-defined single valued function, $m_0 + j$ , and it does not contribute to the total area.", "We denote the discontinuity of $\\phi $ at the singularity by $\\phi ^{\\rm (L)}$ and $\\phi ^{\\rm (R)}$ , which are values on each side of the singularity, respectively.", "The behavior of the Legendre effective potential in the corresponding multi-valued sector is drawn in Fig.", "REF (b).", "Note that between $\\phi ^{\\rm (L)}$ and $\\phi ^{\\rm (R)}$ , the function $j(\\phi )$ is not monotonic and therefore the Legendre effective potential in that region breaks the convexity (monotonicity of the derivative).", "We evaluate the jump of the Legendre effective potential at the singularity ($j=j_{\\rm s}$ ), $\\Delta V_{\\rm L} &= V_{\\rm L}(\\phi ^{\\rm (L)}) - V_{\\rm L}(\\phi ^{\\rm (R)})= j _{\\rm s}(\\phi ^{\\rm (L)}-\\phi ^{\\rm (R)} ) -(W^{\\rm (L)} - W^{\\rm (R)}) \\\\&= j_{\\rm s} (\\phi ^{\\rm (L)}-\\phi ^{\\rm (R)} ) -(V^{\\rm (L)} - V^{\\rm (R)})= j _{\\rm s}(\\phi ^{\\rm (L)}-\\phi ^{\\rm (R)} ) ,$ where we have used the continuity of the fermion potential $V$ proved in Eq.", "(REF ) which assures the continuity of the free energy $W$ at the singularity.", "Taking account of the fact that $j$ is equal to the derivative of the Legendre effective potential at both of $\\phi ^{\\rm (L)}$ and $\\phi ^{\\rm (R)}$ , the condition in Eq.", "(REF ) means that there is a common tangent line between these two points.", "Therefore the weak solution of the Legendre effective potential is the envelope of it.", "In other words the weak solution condition automatically convexifies the Legendre effective potential.", "Any patch working of function $\\phi (j)$ in Fig.", "REF (a) does assure the monotonic change of j as a function of $\\phi $ .", "However it is not enough to have the convexified effective potential.", "If we do a wrong patch work (wrong placing of $j_{\\rm s}$ ) violating the RH condition, then the fermion potential $V$ has a discontinuity, and accordingly the Legendre effective potential $V_{\\rm L}$ is not even uniquely determined as a function of $\\phi $ .", "In this paper we do not demonstrate to calculate these potentials for the zero density NJL model, instead we make plots for the finite density model in the next section.", "By examining the plots there, readers may easily understand what happens for the second order phase transition case of the zero density NJL, since it is simpler and more straightforward.", "The automatic convexification proves the greatest feature of the weak solution, although it is not so obvious in case of the second order phase transition, where the convexified effective potential just shows up as that with a flat bottom connecting the symmetry breaking pair of vacua.", "In case of the first order phase transition, however, this is crucial, since the convexified effective potential correctly selects the globally lowest free energy vacuum even when there are meta-stable vacua around.", "We will demonstrate it in the next section." ], [ "First order phase transition at finite chemical potential", "In this section we investigate the first order phase transition observed at finite chemical potential ($\\mu \\ne 0$ ).", "The first order phase transition is highly non-trivial compared to the second order phase transition because the RG evolution of the physical mass has a finite jump even with the bare mass $m_0\\ne 0$ (as shown in Fig.", "REF ).", "Therefore the bare mass does not help at all.", "Moreover the effective potential has the multi-local minima and most important is whether the NPRG solution can pick up the physically correct, globally lowest free energy vacuum or not.", "We show that the weak solution at finite chemical potential is uniquely constructed, and the effective potential obtained is automatically “convexified\", which assures the solution is the physically correct lowest free energy vacuum.", "Figure: RG evolutions of the physical mass at finite density.We consider the finite density NJL model.", "The chemical potential $\\mu $ is introduced by adding the term $\\mu \\bar{\\psi }\\gamma _0\\psi $ to the bare Lagrangian (REF ).", "To make the NPRG equation best appropriate for the future finite temperature calculation, we use the spacial three dimensional momentum cutoff: $\\sum _{i=1}^3 p_i^2\\le \\Lambda ^2$ to define the renormalization scale.", "Calculation of $\\beta $ -function for the Wilsonian effective potential proceeds as follows.", "The $\\beta $ -function is given by the shell mode path integration, ${\\partial _t} V(x;t)=\\frac{1}{2 d t} \\int _{-\\infty }^{\\infty } d {p_{\\scalebox {0.6}{0}}} \\int \\limits _{\\Lambda -\\Lambda d t \\le \|{\\mathbf {p}}\| < \\Lambda } \\frac{d^3 p}{(2\\pi )^4} ~{\\rm tr} \\log O \\ ,$ where the inverse propagator matrix $O$ is given by $O= \\left(\\begin{array}{cc}V_{{\\psi }^{\\rm T} {\\psi }} & -i {{\\hfil /\\hfil \\crcr p}}^{\\rm \\scalebox {0.6}{T}}-\\mu \\gamma ^{0}+V_{{\\psi }^{\\rm T} {\\bar{\\psi }^{\\rm T}}} \\\\[2pt]-i {{\\hfil /\\hfil \\crcr p}} -\\mu \\gamma ^{0}+V_{\\bar{\\psi } \\psi } & V_{\\bar{\\psi } {\\bar{\\psi }^{\\rm T}}}\\end{array}\\right)\\ ,\\ \\ V_{{\\bar{\\psi }^{\\rm T}} {\\psi }} = \\frac{\\delta ^2 V}{\\delta {\\bar{\\psi }^{\\rm T}} \\delta {{\\psi }}},\\ etc.$ We omit the large-$N$ non-leading terms (diagonal elements in $O$ ) and we get ${\\partial _t} V(x;t)& = \\frac{\\Lambda ^3}{2 \\pi ^3} \\int _{-\\infty }^{\\infty } d{p_{\\scalebox {0.6}{0}}}~\\log \\bigl \\lbrace { (p_{\\scalebox {0.6}{0}}}-i\\mu )^2+\\Lambda ^2+({\\partial _x} V)^2 \\bigr \\rbrace \\\\&= \\frac{\\Lambda ^3}{\\pi ^2}\\left[\\sqrt{\\Lambda ^2+M^2}+\\left(\|\\mu \|-\\sqrt{\\Lambda ^2+M^2}\\right)\\cdot \\Theta \\left(\|\\mu \|-\\sqrt{\\Lambda ^2+M^2}\\right)+C\\right],$ where $\\Theta (x)$ is the Heaviside step function and $C$ is a divergent constant independent of $\\mu $ , $\\Lambda $ and $M$ , and thus we ignore it here.", "The RGE for the mass function is obtained as $\\begin{split}{\\partial _t} M(x;t) &= \\frac{\\Lambda ^3}{2 \\pi ^3}\\int _{-\\infty }^{\\infty } d {p_{\\scalebox {0.6}{0}}}~ \\frac{2 M \\partial _x M}{ {p_{\\scalebox {0.6}{0}}}^2-2 i \\mu {p_{\\scalebox {0.6}{0}}} -\\mu ^2 + \\Lambda ^2 + M^2 }=(\\partial _x M) \\frac{\\Lambda ^3 M \\Theta (\\sqrt{\\Lambda ^2+M^2}-\\mu )}{ \\pi ^2 \\sqrt{\\Lambda ^2+M^2} } .\\end{split}$ The translational invariance with respect to $x$ still holds.", "Note that due to the change of the cutoff scheme the critical coupling constant at vanishing chemical potential changes, and we now have $G_{\\rm c}=2\\pi ^2/\\Lambda _0^2$ .", "The characteristic curve is given by the following equation of motion, $\\frac{d\\bar{x} (t)}{dt}=-\\frac{\\Lambda ^3 M}{\\pi ^2\\sqrt{\\Lambda ^2+M^2} }\\ \\Theta \\left(\\sqrt{\\Lambda ^2+M^2}-\\mu \\right).$ The particle motion stops suddenly at some renormalization scale due to the $\\Theta (x)$ factor and the smaller the momentum $M$ , the earlier the particle stops.", "Figure: (a) Characteristics.", "(b) Position of discontinuity determined by the RH condition.Now we solve the system using the characteristics and construct the weak solution in order.", "In Fig.", "REF , we show the characteristics for the finite density NJL model with $m_0=0$ , $\\mu =0.7.$ and $G_0=1.7G_{\\rm c}$ .", "The characteristics start folding at $t_{\\rm c}=0.41$ at some finite $x$ place, and it occurs at two places simultaneously.", "This is well seen in Fig.", "REF , where the bare mass is included to be $m_0=0.01\\Lambda $ and we plot the mass function $M$ in the left most column, the fermion potential $V$ in the mid column and the Legendre effective potential $V_{\\rm L}$ in the right most column respectively.", "Note that we set a finite bare mass, and accordingly the mass function and the fermion potential are translated to the negative $x$ side as derived in Eq.", "(REF ).", "If there is no bare mass at all, the chiral symmetry assures these two singularity starts symmetrically keeping the odd function property of the mass function.", "This situation is quite different from the second order phase transition at vanishing density.", "There, the appearance of the singularity means the D$\\chi $ SB and it occurs at the origin.", "Now this first emergence of the folding does not mean the chiral symmetry breaking, and singularities are pairwisely generated.", "It just means the convexity of the Legendre effective potential is broken at some finite $x$ positions.", "It should be noted that this is not the emergence of the meta stable state yet.", "The pair of singularities grows up and we define the weak solution of the mass function $M$ by using the equal area rule just as before which is depicted by the dashed line in the mass function.", "As for the fermion potential $V$ , the weak solution, which must be continuous, just takes the upper envelope.", "It is proved easily that there is no other way of making the continuous function.", "This selection is equivalent to the maximum potential principle, and it is related to the maximal action principle in the viewpoint of the kinematical system.", "This variational property of solution can be directly related to another type of weak solution, the viscosity solution [11], [12], which will be treated elsewhere.", "As time goes by, the singularity points move towards the symmetry center at $x_{\\rm c}=-m_0/G_0$ , which is the origin in case $m_0=0$ .", "In the mid of this movement, between $t=0.4$ and $0.5$ , the three folded branches at the right-hand side crosses the origin $x=0$ .", "This happens exactly when there appears a meta stable state, a local minimum, if we faithfully adopt all the multi-valued branches.", "This is seen in Eq.", "(REF ), where in the above situation, the source $j$ can reach zero on a local (unused) strong solution $M$ , and it means the derivative of the Legendre effective potential calculated from the multi-valued $M$ vanishes there.", "The pair of singularities reaches the symmetry center $x_{\\rm c}$ at $t=0.72$ .", "Then the singularities are merged into one singularity on the symmetry center, and it does stop and will not move forever.", "This is the only way of satisfying the RH condition, which is understandable by the equal area rule.", "Here note that there are four regions contributing the area.", "After this merge of pair of singularities, all the functions $M, V, V_{\\rm L}$ given by the weak solution are very much like those of the second order phase transition after the breakdown.", "There are no effects of meta stable state near the origin.", "In Fig.", "REF , the special time $t=0.5615$ is picked up.", "This is the time of the first order phase transition for these specific parameters of the model.", "It is characterized by the time that the right singularity crosses the origin.", "The vanishing derivative point of the Legendre effective potential jumps from the central well to the right-hand side well as seen in the corresponding figure (4c).", "At this phase transition point the chiral condensate does jump although the bare mass is included to explicitly break the chiral symmetry, which is drawn in Fig.REF .", "Thus the bare mass does not help us to avoid the jump of the physical quantities in $t$ -development in the first order phase transition case, and even in that case the weak solution successfully describes this jump behavior.", "We may in principle regularize this jump by limiting the total degrees of freedom to be finite.", "Then there cannot be any singularity in all functions and the jump becomes just a quick change of the physical value.", "In this case not only the second order phase transition but also the first order phase transition are smoothed out to be the crossover.", "Note that most essential point of these procedures is that the phase transition is correctly described so that the lowest free energy vacuum is automatically selected.", "This selection is done by the weak solution, not by hand.", "The weak solution is unique, that is, starting with the regular and smooth function as the initial condition, developing by time, encountering singularities, then it automatically treat the multi-folding structure to pick up a single valued function, and the resultant solution is physically correct.", "The move of singularities are drawn in Fig.REF , where note that the symmetry center is the origin ($m_0=0)$ .", "Pairwisely generated singularities move towards the origin, merged into one and it stays at the origin.", "The characteristics are all moving in to the discontinuity, thus it satisfies the so-called entropy condition.", "Figure: RG evolution of physical quantities and the weak solution for the finite density NJLwith non-zero bare mass(G 0 =1.7G c G_0=1.7 G_{\\rm c}, m 0 =0.01Λ 0 m_0=0.01\\Lambda _0, μ=0.7\\mu =0.7,t=0.3~t=0.3, 0.40.4, 0.50.5, 0.56150.5615, ∞\\infty ).", "(a) The mass function M(x)M(x).", "(b) The fermion potential V(x)V(x).", "(c) The Legendre effective potential V L (φ)V_{\\rm L}(\\phi ).The solid lines denote the weak solution, and the dashed lines denote the local strong solutionsdiscarded.The straight solid line in (c) corresponds to the jump in MM and is actually the envelope." ], [ "Summary", "In this paper, we have introduced the weak solution to define the singular D$\\chi $ SB solution of NPRG equation that can predict physical quantities such as the physical quark mass and the chiral condensate.", "The weak solution satisfies the integral form of the partial differential equation.", "Specifically we have evaluated the weak solution of the large-$N$ NPRG equation of the NJL model for the mass function which is the first derivative of the fermion potential with respect to the scalar bilinear-fermion field $\\bar{\\psi }\\psi $ .", "Also we applied our method to the finite density NJL model where the first order phase transition appears to give highly non-trivial situation compared to the second order phase transition case of vanishing density.", "There is no global solution for our NPRG equation unless we do not consider the weak solution, as far as it describes the D$\\chi $ SB due to the quantum corrections, that is, the mass function must encounter singularities at a finite scale.", "The weak solution helped us perfectly to define the global solution up to the infrared limit of the infinite time where we can evaluate physical quantities.", "The weak solution is uniquely obtained, and there is no ambiguity or approximation to get it.", "The Rankine-Hugoniot condition assuring the weak solution is modified into the equal area rule, by which we can easily obtain the weak solution from the multi-folded local strong solutions.", "However here must be some questions.", "Since our partial differential equation is the renormalization group equation and must be solved with the initial condition only, that is, the effective action at a time.", "After it has the discontinuity, it does not know the discarded strong solutions behind.", "The effective interactions must determine its development with the information given by the weak solution only.", "Therefore there can be no notion of “area” referring to all multi-valued leaves.", "This is true.", "We should note again that the equal area rule is a result given by the weak solution, not an assumption to determine the weak solution.", "We just use it in order to have the weak solution easily.", "In fact, the Rankine-Hugoniot condition is written down as continuity of $V$ or conservation law for $M$ , and it is the local condition without referring to the strong solutions behind and it can determine the solution without recourse to the discarded strong solutions.", "Thus the weak solution with singularity can develop by itself without any additional information.", "The weak solution defined in this paper has a perfect feature to describe the physically correct vacuum even when there are multi meta stable local minima given by all the local strong solutions in the Legendre effective potential.", "The basic logic to achieve this is that the weak solution successfully convexifies the Legendre effective potential, and it assures that the lowest global minimum determines the vacuum.", "In this paper we have worked our only the NJL type models with large-$N$ leading approximation.", "We can go ahead to define the weak solution of the NPRG equation for gauge theories even with non-ladder (large-$N$ non-leading) effects.", "For example, the large-$N$ leading NPRG partial differential equation is defined by, $\\partial _t V(x;t) &=\\frac{\\Lambda ^4}{4\\pi ^2}\\ln \\left[1+\\Lambda ^{-2}\\left(\\partial _x V+(3+\\xi )\\frac{ C_2\\pi \\alpha _{\\rm s}}{\\Lambda ^2} \\, x\\right)^2\\right]\\ ,$ which has the same type of PDE analyzed in this paper with translationally non-invariant Hamiltonian $F$ .", "Also the extension including large-$N$ non-leading diagrams has been done to give $\\begin{split}\\partial _t V(x;t)&=\\frac{\\Lambda ^4}{4\\pi ^2}\\log \\left[1+\\frac{B^2}{\\Lambda ^2}\\right]+\\frac{\\Lambda ^4}{8\\pi ^2}\\log \\left[\\frac{\\Lambda ^2+B^2}{\\Lambda ^2+{\\color {black} (\\partial _x V)^2}}+\\frac{3\\Lambda ^2G^2}{(\\Lambda ^2+{\\color {black} (\\partial _x V)^2})^2}\\right] \\\\&\\quad +\\frac{\\Lambda ^4}{4\\pi ^2}\\log \\left[1+\\xi \\frac{\\partial _x V\\, G}{\\Lambda ^2+{\\color {black} (\\partial _x V)^2}}\\right],\\end{split}$ where $B=\\partial _x V+2C_2\\pi \\alpha _{\\rm s}x/\\Lambda ^2$ and $G=2 C_2 \\pi \\alpha _{\\rm s}x/\\Lambda ^2$[6], [7].", "This is also manageable by our weak solution method.", "Application to gauge theories are reported in the forthcoming paper.", "Figure: Central engine of Dχ\\chi SB.Finally we mention about the driving engine of the D$\\chi $ SB taking the NJL model as an example.", "Let us see the neighborhood of the origin, $M \\simeq 0$ , in Eq.", "(REF ) and we have the approximated Hamiltonian $F$ as follows, $F(M,x;t) = - \\frac{\\Lambda ^4}{4\\pi ^2}\\log \\left(1+\\frac{M^2}{\\Lambda ^2}\\right)\\simeq - \\frac{\\Lambda ^2}{4\\pi ^2} M^2 = - \\frac{\\Lambda _0^2 e^{-2t}}{4\\pi ^2} M^2\\ .$ Therefore this particle is a usual classical particle but with negative and time varying mass of ${2\\pi ^2 e^{2t}}/{\\Lambda _0^2}$ [6] .", "The particle motion becomes gradually slow due to the exponential increase of mass.", "The string near the origin is approximated by a straight line.", "The velocity is proportional to the momentum, with negative sign though, and the string near the origin just turns counter-clockwisely until it stops (Fig.", "REF ).", "This is the central engine of generating the singularity and of bringing the D$\\chi $ SB.", "Note that the D$\\chi $ SB occurs exactly when the string at the origin becomes vertical.", "Then the derivative of the mass function $M(x;t_{\\rm c})$ with respect to $x$ becomes divergent at the origin.", "We denote this slope by $\\tilde{G}(t)$ and it is evaluated as $\\tilde{G}(t) = \\frac{M(\\bar{x}(t);t)}{\\bar{x}(t)} e^{-2t}\\Lambda _0^2\\ .$ The RG equation for $\\tilde{G}(t)$ reads $\\frac{d}{dt}\\tilde{G(t)}=-\\left(2+\\frac{1}{\\bar{x}}\\frac{d\\bar{x}}{dt}\\right)\\tilde{G}=-\\left(2+\\frac{1}{\\bar{x}}\\frac{dF(M,x;t)}{dM}\\right)\\tilde{G}= -2\\tilde{G}+ \\frac{1}{2\\pi ^2}\\tilde{G}^2.$ This is nothing but the RG equation of the four-fermi interactions obtained in Eq.", "(REF ), whose blow up solution encounters the singularity at $t_{\\rm c}$ .", "However if we work with the motion of string, the verticality of string does not mean anything singular; it is just vertical.", "Some authors define inverse four-fermi coupling constant $g(t)\\equiv 1/\\tilde{G(t)}$ , which satisfies the following RG equation[18], $\\frac{d}{dt}g(t) = + 2g - \\frac{1}{2\\pi ^2}.$ This equation has the global solution $g(t) = \\frac{1}{4\\pi ^2} + \\left(g(0) - \\frac{1}{4\\pi ^2}\\right)e^{2t}\\ .$ In the strong phase ($g < 1/4\\pi ^2$ ), $g(t)$ crosses the origin at $t_{\\rm c}$ and goes to the negative region without any singularity.", "It has a global solution up to $t=\\infty $ .", "This tricky procedure is not authorized by itself.", "However, it is perfectly right if we consider the string motion and just use the inverse slope instead of the slope to describe the string near the origin.", "Sometimes in case of finite density media, the inverse coupling constant $g(t)$ goes to the negative region and returns back to the positive region again.", "Even in such case, it is understood to have given the right result, that is, the chiral symmetry is not broken in the macro effective theory.", "In gauge theories the same engine does exist but there is a little difference.", "The gluons are massless and have no intrinsic scale, and the engine does not stop at the infrared.", "Therefore, the string rotates infinitely around the origin, which corresponds to the infinitely many unphysical solutions encountered in the SD type analyses.", "This singularity generation mechanism quite resembles to the generation of a shock and its expansion procedure described by the non-linear wave equation of the Burgers' [9].", "Also our NJL weak solution satisfies the so-called entropy condition and it is understandable as a physical shock.", "We have done this argument using the NJL ($\\mu =0$ ) model near the origin.", "However it can be applied as well to any neighborhood of the singularity birth place.", "Thus we demonstrated that the D$\\chi $ SB is the spontaneous generation and growth of a shock describable by the weak solution of PDE.", "The generation and growth of a non-moving shock describes the “continuous” behavior in the second order phase transition.", "On the other hand, a moving shock realizes the “jump” behavior of the first order phase transition.", "We stress again that this is the first demonstration and successful application of the weak solution for the non-perturbative renormalization group equation.", "Why does it work so well?", "One intuitive plausible argument follows.", "The definition of the weak solution is given by some integration equality without recourse to the singularities of the target function.", "Our target functions are effective interactions which are to be path integrated with operators.", "Therefore such functions do not have to be regular as they are, and it may be enough that they satisfy necessary equations as a form of integration convoluted with smooth and bounded test functions.", "In the introduction we claimed that encountering the singularity in the mid of renormalization group solution is inevitable in any, approximated or not, model of the D$\\chi $ SB.", "If the PDE system here is the continuum approximation of the originally discrete many body system, then we may add higher derivative terms, for example, the dissipation term, as improvement of approximation, and such dissipation may regulate the singularity.", "This is well known in case of the Burgers' equation.", "In our PDE of the non-perturbative renormalization group equation, though, there may appear higher derivative contribution in improving the approximation, but it cannot help at all, because our singularity is intrinsic and is not a matter of approximation.", "There may be another way out by changing the system drastically.", "If we invent some regularization method to limit the number of degrees of freedom $N$ to be finite, the spontaneous symmetry breaking cannot occur and all effective interactions and thermodynamic functions are singularity free.", "Then, after all calculations, we take the infinite $N$ limit to get the physical results.", "However there has not been known any good method of realizing finite degrees of freedom regularization in formulating the non-perturbative renormalization group equation.", "We have found a way of limiting the depth of quantum corrections, and the D$\\chi $ SB is regularized.", "After obtaining the thermodynamic potentials we take the infinite depth limit, and finally reach the right functions with singularities.", "We will report it in a separate article." ], [ "Acknowledgements", "The authors greatly appreciate stimulating discussion with Yasuhiro Fujii, and also helpful lectures by Prof. Akitaka Matsumura who told us how to define and construct the weak solution, which initiated our work in this paper." ] ]	1403.0174
[ [ "Observations and Implications of Large-Amplitude Longitudinal\n Oscillations in a Solar Filament" ], [ "Abstract On 20 August 2010 an energetic disturbance triggered large-amplitude longitudinal oscillations in a nearby filament.", "The triggering mechanism appears to be episodic jets connecting the energetic event with the filament threads.", "In the present work we analyze this periodic motion in a large fraction of the filament to characterize the underlying physics of the oscillation as well as the filament properties.", "The results support our previous theoretical conclusions that the restoring force of large-amplitude longitudinal oscillations is solar gravity, and the damping mechanism is the ongoing accumulation of mass onto the oscillating threads.", "Based on our previous work, we used the fitted parameters to determine the magnitude and radius of curvature of the dipped magnetic field along the filament, as well as the mass accretion rate onto the filament threads.", "These derived properties are nearly uniform along the filament, indicating a remarkable degree of cohesiveness throughout the filament channel.", "Moreover, the estimated mass accretion rate implies that the footpoint heating responsible for the thread formation, according to the thermal nonequilibrium model, agrees with previous coronal heating estimates.", "We estimate the magnitude of the energy released in the nearby event by studying the dynamic response of the filament threads, and discuss the implications of our study for filament structure and heating." ], [ "Introduction", "High-cadence H$\\alpha $ observations of large-amplitude longitudinal (LAL) oscillations in solar filaments were first reported by [11].", "These oscillations consist of rapid motions of the plasma along the filament, with displacements comparable to the filament length.", "Since then, a few more events have been identified [10], [35], [18], [39].", "In all of the observed oscillations the period ranges from 0.7 to 2.7 hours, with velocity amplitudes from 30 to $100~\\mathrm {km~s^{-1}}$ .", "The accelerations are considerable, in many cases comparable to the solar gravitational acceleration.", "In addition, the oscillations are always triggered by a small energetic event close to the filament.", "Explaining the LAL oscillations is very challenging because the restoring force responsible for the huge acceleration must be very strong.", "The energy of the oscillation is also enormous, because the filament is massive and large velocities are generated.", "However, the motions damp quickly in a few periods, implying a very effective damping mechanism.", "Several models have been proposed to explain the restoring force and damping mechanism of the LAL oscillations [31], but most do not successfully describe the thread motions.", "Recently, we studied the oscillations of threads forming the basic components of a filament [23] in a 3D sheared arcade [8].", "In this model, the threads reside in large-scale dips on highly sheared field lines within the overall magnetic structure.", "We found that the restoring force is mainly gravity, and the pressure forces are small [22].", "This result has been confirmed with numerical simulations by [18] and [40].", "We also studied analytically the normal modes of plasma on a dipped field line, and found that the oscillation generally involves the superposition of two components: gravity-driven modes and slow modes associated with pressure gradients [21].", "However, for typical filament plasma properties the gravity-driven modes dominate.", "This type of oscillation resembles the motion of a gravity-driven pendulum, where the frequency depends only on the solar gravity and the field-line dip curvature.", "We estimated the minimum value of the magnetic field at the dips and found agreement with previous estimates and observed values.", "Additionally, this study revealed a new method for measuring the radius of curvature of the filament dips.", "These studies demonstrated that the LAL oscillations are strongly related to the filament-channel geometry.", "We identified the damping of the LAL oscillations as a natural consequence of the thermal nonequilibrium process most likely responsible for the formation and maintenance of the cool filament threads [22].", "In this process, the localized footpoint heating produces chromospheric evaporation and subsequent collapse of the evaporated mass into cool condensations in the corona [2], [12].", "As long as the heating remains steady, the threads continuously accrete mass and grow in length.", "Non-adiabatic effects (i.e., thermal conduction and optically thin radiation) also contribute weakly to the damping [22], [40].", "The damping times obtained with this model generally agree with the observed values.", "Because LAL oscillations in filaments are observed to occur after small energetic events nearby, we speculated that local heating and/or flows from the energetic event triggers and drives oscillations in filament threads magnetically linked to the event site [22].", "[40] modeled the effects of two types of perturbations on a filament thread – impulsive heating at one leg of the loop and impulsive momentum deposition – and determined that both can cause oscillations.", "In this work we study a LAL filament oscillation, its trigger, and subsequent damping observed on 20 August 2010 by the Atmospheric Imaging Assembly (AIA) instrument on the Solar Dynamics Observatory (SDO) [15].", "We also utilize co-temporal magnetograms from the Helioseismic and Magnetic Imager (HMI) instrument on SDO [27], to provide the overall magnetic context and local connectivity between the filament and the triggering event.", "The oscillation properties yield the geometry of the magnetic field supporting the cool plasma and the mass accretion rate.", "This information leads to fundamental conclusions about the filament structure, as well as the dynamics and energetics of the triggering event.", "Some preliminary results of this work were shown in [13].", "In § we describe the data and sequence of events, while § discusses the event that triggered the oscillations.", "In § our methodology for measuring the filament oscillation and deriving key parameters is described.", "§ presents the results and examines their implications for the oscillation mechanism, mass accretion, and energization.", "Based on these results, we describe the most likely filament structure in §.", "Our findings and conclusions are summarized in §." ], [ "Observations and event overview", "AIA/SDO and HMI/SDO observe the entire solar disk with regular, high temporal cadence.", "AIA obtains images in seven EUV filters at 12-s cadence, and in two UV filters at 24-s cadence.", "HMI takes polarized full-disk images of the Stokes $I$ and $V$ parameters, and the data pipeline yields line-of-sight (LOS) magnetograms, Dopplergrams, and intensity images at a 45-s cadence.", "The events considered in this investigation occurred in a large filament channel centered in the southern-hemisphere active region (AR) NOAA 11100, which was studied in detail for a different time period by [37] and [36].", "For an overview of the AR and the filament, including H$\\alpha $ images and a HMI LOS magnetogram, we refer the reader to Figure 1 of [36].", "On 20 August 2010, the decayed active region was located east of central meridian in the southern hemisphere at about $-27^\\circ $ latitude.", "A long filament cuts diagonally through the AR, and consists of two segments in the southeastern and northwestern regions of the AR.", "Our analysis is focused on the northwestern part of the filament outlined by the $160^{\\prime \\prime } \\times 160^{\\prime \\prime }$ region shown in Figure REF in the AIA 171 Å channel at 4 times between 18:00 and 20:23 UT.", "Therefore, all further references in this paper to “the filament\" apply to the northwest segment only.", "The analyzed temporal sequence started at 15:00 UT August 20 and ended at 05:00 UT August 21.", "SDO takes science data almost continuously, interrupted only by brief eclipses and calibration sequences, which were taken into account in the analysis.", "The AIA and HMI images were transformed to the same spatial scale and co-aligned with the SolarSoft routine “aia_prep.pro” and the mapping routines developed by D. Zarro.", "The filament is visible as a dark band in H$\\alpha $ and in most of the AIA EUV filters (171 Å, 193 Å, 211 Å and 304 Å).", "The dark band corresponds to the region where the cool plasma resides, characterized by increased absorption of background emission and an emission deficit in coronal lines (see Figure REF ).", "In addition, the edges of the dark band clearly exhibit emission in these lines.", "[28], [29] and [26] studied this absorption/emission pattern in filaments, and found that the absorption pattern resembles the H$\\alpha $ morphology while the bright emission comes from the prominence-corona transition region (PCTR).", "In fact our thermal nonequilibrium model predicted this bright emission from the PCTR at both ends of filament threads [23].", "A movie of AIA 171 Å images of the region shown in Figure REF is available in the online journal, showing the sequence of events during the 3 hours when most of the interesting activity occurs.", "At the beginning of the observation interval (15:00 UT) the motions inside the filament channel resemble the well-known counterstreaming [41], [1], with different parts of the structure moving out of phase.", "Figure REF (a) shows a representative example of this initial stage, although the counterstreaming displacements are not evident in this still image.", "Around 18:10 UT a narrow collimated flow of plasma appears near the south end of the filament (see §).", "In Figure REF (b) this jet is clearly visible inside the overplotted dashed box and bright plasma is seen over the entire filament.", "By 19:00 UT the outflow from the jet source has ceased and the filament oscillates with a very coherent, large-scale motion.", "In this first hour after the jet onset, the flow appears to push the existing filament threads toward the northwest, particularly in the southeastern portion of the filament (Figure REF (c)).", "The filament continues oscillating (Figure REF (d)) until the end of the temporal sequence.", "After 19:00 UT the filament oscillations are evident in all AIA filters.", "As is obvious from the movie, however, not all sections of the filament oscillate: the central portion remains at rest throughout the event, while the east and west segments oscillate (see §).", "The oscillation damps quickly at first but continues throughout the 14-hour observation.", "At 21:20 UT an eruption occurs outside our studied region but close to the NW end of the oscillating filament, centered at $\\lbrace x \\sim 60^{\\prime \\prime },y\\sim -420^{\\prime \\prime }\\rbrace $ .", "The erupting filament appears to come from the SW extension of the same large-scale filament channel under study, but only the closest oscillating filament threads (i.e., slits 33-34) are affected significantly.", "For maximum clarity we focus on the oscillations produced by the jet, before the nearby eruption.", "Figure: Time sequence of 171 Å images containing the analyzed filament and jets.", "(a) Just before the impulsive event.", "b) Fully developed jet flows visible in emission.", "The dashed box outlines the slit used to study the jet in §.", "c) The SE part of the filament is clearly excited and displaced toward the north.", "d) The oscillating filament.", "White circles indicate the 36 positions along the filament selected for analysis; for clarity only positions 1 and 36 are numbered.", "White boxes outline three examples of slits at position 1 as described in §.", "A movie of the AIA 171 Å images of the region shown is available in the online journal." ], [ "Trigger Analysis", "Around 18:10 UT very collimated plasma flows start to emanate from $\\lbrace x \\sim -135^{\\prime \\prime }, y \\sim -440^{\\prime \\prime }\\rbrace $ , ejecting plasma toward the SE end of the filament situated $\\sim 70^{\\prime \\prime }$ ($\\sim $ 51 Mm) away from the source (see Figure REF and accompanying movie).", "This jet is clearly visible in emission in several AIA EUV filters (171 Å, 193 Å, 131 Å, 304 Å, 211 Å, and 94 Å), indicating that the multi-thermal flow includes very hot plasma.", "At the same time, a secondary jet appears to come from a position north of the primary jet source ($\\lbrace x \\sim -130^{\\prime \\prime }, y \\sim -420^{\\prime \\prime }\\rbrace $ ).", "The presence of filamentary cool plasma between these source locations makes it difficult to determine whether the secondary jet really is independent, or whether the primary jet is merely obscured by the filament between the two positions.", "In either case, these streams merge rapidly and proceed as one highly structured flow along the filament channel.", "A detailed analysis of the jet is beyond the scope of this work, and would not affect the present analysis and interpretation of the oscillations.", "Therefore, in subsequent discussions we treat the jet as a single entity originating at the southernmost location.", "Initially the outflow is narrowly focused and linear (see movie).", "Therefore, we placed an artificial slit over the image (indicated by the dashed box in Figure REF (b)) and extracted the intensities along this slit as a function of time between 18:00 UT and 18:30 UT.", "The intensity signals were logarithmized and subject to a third-order polynomial detrending, which allowed us to improve the signal-to-noise ratio.", "We avoided temporal smoothing to preserve the original time resolution of the AIA data [33].", "The resulting time-distance diagram is shown in Figure REF , where the bottom panel displays the same image as the top panel with linear best fits to the diagonal intensity structures overplotted as green dashed lines.", "The derived outflow speeds are given for each fitted structure.", "The flow is episodic with a few minutes between successive ejections.", "The flow speed reaches $\\sim 95~ \\mathrm {km~s^{-1}}$ in the first episode, and decreases significantly in subsequent episodes.", "These values are consistent with the speeds of different jets in the same filament channel analyzed by [36], although those jets did not produce filament oscillations.", "Figure: Time-distance diagrams of the episodic initiation jet.", "a) AIA 171 Å intensity extracted from the dashed box indicated in Fig.", "(b) on the y-axis; time is given on the xx-axis (starting at 18:00 UT).", "b) Same as a) with linear fits of several propagating fronts associated with individual ejection episodes overplotted in green.", "The red dotted line gives the approximate y-location of the jet origin, with plasma emission propagating either northward or southward from this position along the slit.", "The numbers give the estimated speed from these fits in km s -1 \\mathrm {km~s^{-1}}Most emission structures in Figure REF are tilted upward, indicating that the flow is moving northward, although at the bottom of the Figure a few flows are moving southward.", "The red dotted line marks the line of demarcation between these cotemporal flows.", "Several of the jets described in [36] are also clearly bi-directional.", "Applying the same analysis method to the AIA 193 Å, 335 Å and 304 Å images gives very similar time-distance diagrams as Figure REF and the same values for the speeds of the fronts.", "Therefore we conclude that the flow speed is independent of temperature, and that we are measuring the bulk speed of the plasma ejected from the source.", "These speeds are projected onto the image plane, however, so the low southward values are most likely from flows that are primarily vertical.", "Furthermore, the apparent acceleration of the northward flows might indicate a smooth transition from vertical to horizontal field-line geometry, as we expect from the magnetic structure of the sheared arcade model.", "In the absence of Doppler information derived from emission line profiles, however, we can only speculate about the true velocities.", "The most compelling interpretation of these observations is that the bi-directional jets are reconnection outflows originating in the low corona.", "If the reconnection occurred in the chromosphere or photosphere, the downward jet would not be seen.", "[36] found evidence for flux cancellation associated with their bi-directional jets observed near the same filament channel, providing support for reconnection as the driver.", "However, we studied the HMI LOS magnetograms at our jet source regions and found no unambiguous evidence for changes in magnetic flux density, neither for mutual flux cancellation nor flux emergence.", "The photospheric flux density values are very close to the observed noise level in the filament channel, and fluctuate constantly on short time scales (from a few minutes down to the 45-s cadence of the magnetograms).", "The initial jet flows are visible in emission in all AIA EUV filters, but after the impulsive phase when the threads are perturbed by the hot flows, there is not enough hot plasma to produce emission in the 94 Å filtergram.", "Later in the initiation event the jet path becomes less collimated and more turbulent, with some of the bright features reversing direction (see Figure REF movie).", "Further analysis and interpretation of the spatial and energetic relationship between the jet and the filament threads is discussed in §REF , §REF , and §." ], [ "Methodology: oscillation analysis", "The LAL oscillation event is clearly visible in H$\\alpha $ and the AIA/SDO filters 171 Å, 193 Å, and 131 Å, but harder to discern in the 304 Å passband.", "The ground-based H$\\alpha $ images are useful for context, but they have lower spatial resolution and cadence than the AIA/SDO data.", "Therefore we identified and parameterized the oscillations from the EUV images, placing slits centered along the filament at different angles with respect to it.", "We have chosen this method because it is not possible to identify and track individual threads or features from the images.", "We determined the boundaries of the filament by eye, following the dark band in the 171 Å images, and placed the center of each slit at the midpoint of the dark band.", "As shown in Figure REF (d), we selected 36 slits whose centers are separated by 6 pixels (3.6 arcsecs) along the total (projected) filament length of 126 arcsecs (92 Mm).", "The slits are numbered from 1 to 36 starting from the southeast end of the filament and ending at the northwest end.", "The filament is composed of two quasi-linear segments: the eastern part is the longest, containing the first 28 slits, while the short western part is tilted $66^\\circ $ with respect to the other and contains the last 8 slits.", "Each slit is 3 pixels (1.8 arcsecs) wide and 300 pixels (180 arcsecs) long.", "The intensity was averaged over the 3 transverse pixels and the resulting intensity distribution along the slit was plotted as a function of time, as the primary tool for analysis of the oscillations (Figures REF -REF ).", "Note that we utilized the full 12-s cadence of AIA data.", "The best EUV passband for this analysis is 171 Å because the dark and bright regions have the highest contrast with very sharp profiles.", "The filament slice seen in each 171 Å slit appears as a dark band surrounded by two bright bands.", "This pattern allows us to clearly identify the motions of the filament threads.", "Figure REF shows 171 Å time-distance diagrams for 3 slits centered at the same labeled position 1, differing only in the angle between the slit and the main direction of the filament.", "Although the oscillation is clear in all three panels, the oscillatory pattern in the time-distance diagrams depends strongly on this angle.", "In Figure REF (a), where the slit angle is $11^\\circ $ , several threads with different alignments contribute to the emission and the oscillation is only clear between $t=$ 20:00 and 22:00 UT.", "Furthermore, this slit is not situated at the equilibrium position of the oscillation and is not aligned with the thread motion.", "Thus the threads enter and leave the slit as they oscillate, so only parts of the oscillation are seen in the panel.", "In Figure REF (b), the slit is tilted $21^\\circ $ relative to the filament axis and oscillations are very clear from $t=$ 18:25 UT onward.", "At this angle the dark band is seen throughout the observation, and the bright emission is symmetric around the cool plasma.", "Moreover, the dark band and adjacent bright emission oscillate together, indicating that the PCTR moves with the filament threads.", "In Figure REF (c) the slit angle is $31^\\circ $ and the oscillations are clearly visible, but their amplitude is smaller than in Figure REF (b).", "Thus, at this angle the slit contains the equilibrium position of the oscillation, but the co-alignment with the actual motion of the threads is poorer than in Figure REF (b).", "Therefore we conclude that, of the three examples at position 1 along the filament, the $21^\\circ $ slit is aligned best with the oscillating threads.", "This figure only compares the results of three different angles between the slit and the filament axis for one particular slit position.", "Overall we analyzed the oscillations by making time-distance diagrams for angles ranging from $0^\\circ $ to $40^\\circ $ at $1^\\circ $ intervals, for the first 28 positions along the filament.", "For the remaining 8 positions it was necessary to increase the angular range to $0^\\circ $ - $90^\\circ $ in order to observe the oscillations.", "From the resulting 1840 time-distance diagrams, we selected the best slit direction for every position along the filament, by choosing the strongest oscillation pattern: a clear dark band sandwiched between 2 distinct bright emission regions, the longest possible time interval over which the oscillations were measurable, and the maximum possible displacement amplitude.", "Figure: Time-distance diagrams from SDO/AIA 171 Å images in slit position 1 on the filament with angles (a) 11 ∘ 11^\\circ , (b) 21 ∘ 21^\\circ , and (c) 31 ∘ 31^\\circ .", "The slits used to generate these three diagrams are shown on Fig.", "(d).", "Data gaps are due to eclipses and calibration sequences.", "The plot begins on 20 August and ends 21 August 2010.After creating time-distance diagrams for all 36 positions along the filament, we found that the filament only oscillated at positions 1-6, 33, and 34, while the remainder of the filament remained stationary.", "Figures REF -REF depict the best time-distance diagrams obtained with our method for the 8 oscillating positions.", "In Figure REF the oscillation is very clear most of the time, but in the cases plotted in Figures REF and REF the oscillation is clear only between $t$ = 18:00 UT and 22:00 UT.", "Plasma from the nearby filament eruption (§) blurs the oscillations in Figure REF , but they reappear after $t$ = 23:00 UT.", "In all cases, a transient intensity increase related to the triggering appears around $t$ = 18:12 UT; then the threads start oscillating, reaching their maximum displacement by t $\\sim $ 19:00 UT.", "This initial oscillation phase is very complex in all cases: several threads can be discerned as dark bands capped at both ends by thinner bright bands, but it is impossible to follow a clear and continuous dark band immediately after the maximum displacement has been reached.", "However, we were able to identify the initial oscillation by the bright emission.", "After the first half period a clear dark band oscillates but damps quickly, so the displacement has been reduced considerably within a few periods.", "It is important to note that, in the 6 cases shown in Figures REF and REF , we can distinguish oscillations nearly to the end of the temporal sequence.", "The movie of Figure REF shows that part of the nearby erupted filament passes over all slit positions from the NW to the SE, taking one hour to travel across the field of view.", "This eruption produced visible brightening in slits 33 and 34 (Fig.", "REF ) at $t \\sim $ 22:00 UT, and excited a new damped oscillation in the NW section of the filament.", "In Figures REF and REF , however, we see only cool (dark) plasma extending below the dark band during $t$ = 22:00 - 24:00 UT, and the oscillations in the SE section of the filament were not disrupted.", "Figure: Time-distance diagrams of the best slits at positions 1 (a), 2 (b), and 3 (c).", "The data of the positions of the center of the dark band are plotted as orange points.", "The fitted functions are also plotted as red and green lines corresponding to the exponentially decaying sinusoid and the Bessel function respectively.Figure: As Fig.", "for the slit positions 4 (a), 5 (b), and 6 (c).Figure: As Fig.", "for the slit positions 33 (a), and 34 (b).We selected the position of the minimum emission intensity in the dark band at every time from the time-distance diagrams.", "For each of the 8 best slits, we selected an initial data point ($t_i$ ,$s_i$ ) along the dark band and found the coordinates ($t_i$ ,$s_{i,min}$ ) at which the intensity is minimized.", "We then moved to the next point in time ($t_{i+1}$ , $s_{i,min}$ ) and scanned through the spatial range ($s_{i,min}-\\epsilon $ /2, $s_{i,min}+\\epsilon $ /2), where $\\epsilon $ is of the order of the width of the dark region, to find the minimum-intensity location ($t_{i+1},s_{i+1,min}$ ).", "This process was repeated for the entire oscillation.", "Hence the coordinates of the filament thread are given by the set of points ($t_j,s_{j,min}$ ) with minimum intensity along the oscillating curve at times when the oscillation is clear.", "In Figures REF - REF the resulting ($t_j,s_{j,min}$ ) data are plotted as orange points.", "To characterize the oscillations quantitatively, we fit these data with two functions applicable to oscillating threads without and with mass accretion.", "For the constant-mass assumption we used an exponentially decreasing sinusoid of the form $s(t)=s_0 + A_\\mathrm {Exp} e^{-(t-t_0)/\\tau } \\cos \\left[ \\omega (t-t_0) +\\phi _0 \\right] +d_0 (t-t_0)~,$ where $s_0, t_0, \\phi _0$ , and $d_0$ are derived from the best fit.", "Here $s$ is a position along the slit, $t_0$ is the time of maximum displacement with respect to the oscillation center, $s_0$ is the oscillation center position at $t_0$ , $\\phi _0$ is the initial phase of the oscillation, and $d_0$ is the drift velocity.", "The central position as a function of time is then $s_0 + d_0 (t-t_0)$ .", "This drift velocity accounts for slow displacements not associated with oscillations.", "However, in all fits $d_0$ is very small and the central position remains at $s_0$ .", "The results of the fits of the exponentially decaying sinusoid are plotted in Figures REF - REF as red curves.", "[22] found that, if a filament thread continuously accretes mass, then its oscillation is described by a Bessel function instead of a sinusoid.", "Therefore we also fit the minimum intensity locations vs. time with $s(t) = s_0 + A_\\mathrm {Bes} J_0\\left[ \\omega (t-t_0) + \\psi _0 \\right] e^{-(t-t_0)/\\tau _w}+d_0 (t-t_0)~,$ where $\\psi _0$ is related to the mass accretion rate (see §§REF and REF ) and $\\tau _w$ is the weak damping time possibly associated with small, non-adiabatic energy losses.", "In Figures REF - REF the results of the Bessel function fits are shown by green lines.", "The functions of Equations (REF ) and (REF ) and the uncertainties in their parameters have been fitted to the data using IDL's CURVEFIT routine, which yields a nonlinear least-squares fit to a function of an arbitrary number of parameters.", "We also propagated errors to obtain the uncertainties of the derived quantities of the directly fitted parameters.", "Visually the two functional fits to the data are very good (see Figures REF - REF ).", "Both have the same number of free parameters, so their relative goodness can be assessed by comparing the standard deviation $\\sigma ^2$ of each function [3].", "Both fits have similar, reduced $\\sigma ^2$ values $\\le 13~\\mathrm {arcsecs}^2$ for all 8 cases.", "Thus, the deviation of the data with respect to the fitted functions is less than 3.6 arcsecs, indicating that the fits are equally good.", "However, as demonstrated below, the modified Bessel function of Equation REF provides better fits to the initial plasma dynamics and damping, and to the longevity of the oscillations.", "As noted earlier, the initial phase is very confusing and the dark band is not clear.", "Therefore we extrapolated the fitted functions from later times into this phase, and we tracked the behavior of the bright plasma to derive the initial thread displacements.", "We found that the modified Bessel function follows the initial motion of the bright plasma much better than the sinusoid-exponential; in particular, the predicted amplitude of the displacement is much closer to the observed values (see §REF ).", "In slit 1, for example, the first maximum displacement is $\\sim $ 50 arcsecs with respect to the central position of the oscillation, whereas the displacement of the next maximum is $\\sim $ 20 arcsecs, a 60 $\\%$ reduction.", "In spite of this strong damping, the oscillations remain visible until the end of the temporal domain.", "Because the sinusoidal-exponential fit imposes the same damping at all times, this approximation cannot match both the initial strong damping and the long-term persistence of the oscillations.", "In contrast, the modified Bessel fit can account for both the strong damping at the onset and the weak damping of the later oscillations.", "This conclusion applies equally to the other 7 slits, except for the interval where a secondary excitation occurs and a new damped oscillation is excited in slits 33 and 34 (see Figure REF )." ], [ "Oscillation direction", "With the method described in § we determined the main direction of oscillation at each slit position along the filament.", "As shown in Figure REF (a), the oscillation angles are in the range $\\theta = 15^\\circ -35^\\circ $ , with a mean value of $25^\\circ $ , in the first six positions corresponding to the SE end, and $\\theta =63^\\circ $ and $81^\\circ $ in the last two positions, corresponding to the NW end.", "The change in angle between adjacent slit orientations was chosen to be $1^\\circ $ for a given slit center position, so we consider the uncertainty of the angle to be $\\pm 1^\\circ $ .", "The filament appears to contain ensembles of threads with similar orientations locally.", "Our method of measuring the orientation selects the thread or set of threads with the clearest oscillatory pattern.", "Therefore we expect that the orientation of the measured thread is applicable to all threads close to that slit.", "The angles estimated at the first six positions agree with published measurements of the magnetic-field orientation in prominences [16], [17], [9], [32], [6], [20].", "The filament structure undergoes an abrupt southward shift around slit 28, such that the NW segment of the filament apparently protrudes laterally from the PIL (see Figure REF ).", "The angles between the plasma motion in slits 33-34 and the main filament axis are $-3^\\circ $ and $14^\\circ $ , indicating that the oscillating threads are almost parallel to the SE segment of the filament and not aligned with the protrusion.", "This suggests that the NW end of the filament might be a barb [25], but it is difficult to determine from the available images and the HMI magnetograms whether this substructure is associated with a parasitic polarity or exhibits other definitive characteristics of a barb.", "To support the threads in their observed orientation, the magnetic field in this segment must be mostly horizontal, as proposed in some models of filament barbs [4], [34], [7]." ], [ "Oscillation initiation", "The energetic event occurs at one edge of the filament channel containing the filament of interest, approximately 51 Mm from the closest visible thread end.", "The relative timing of the oscillation onset at different positions along the filament provides clues as to the mechanism whereby the impulsive energy release triggers and drives the oscillations.", "Two scenarios are possible: serial and parallel activation.", "In the “serial\" picture, a disturbance reaches the closest threads first, then propagates along the filament exciting sequentially the farther threads.", "In the “parallel\" picture, the hot plasma travels simultaneously along many field lines within the filament channel, initiating oscillations when each stream within the jet reaches the thread supported by that field line.", "The SE footpoints of these field lines are connected to the source site, and the jet flows take longer to reach the more distant threads.", "Our analysis provides enough evidence to discriminate between these scenarios, and to select the most appropriate one.", "The onset times for the oscillations can indicate how the trigger reaches the threads.", "A good signature of this time of impact, in the time-distance diagrams, is the first observed brightening of the hot emission immediately south of the dark band (e.g., the cool thread), which occurs immediately before the oscillations start.", "As shown in Figure REF (b), the time of initial brightening increases slightly with slit position, indicating that the triggering first occurs closest to slit 1 and within a few minutes reaches more distant locations along the filament.", "However, a propagating disturbance moving from the SE to the NW part of the filament is not evident, as would be expected in the serial scenario.", "More importantly, the middle part of the filament remains at rest throughout the observation period; this is completely inconsistent with serial activation.", "The fact that we observe brightenings in the time-distance plots exactly where and when the threads begin oscillating strongly suggests that parallel activation is taking place.", "These brightenings occur when the hot jet flows reach the threads, and the oscillation starts exactly at this time.", "Close to the source the jet is highly collimated, but the larger-scale magnetic geometry guides this flow to cover a large fraction of the filament channel.", "Therefore we conclude that the jet excites the threads as predicted by the “parallel” picture.", "The directly fitted parameter $t_0$ indicates the time of maximum thread displacement, and thus should also be related to the oscillation onset time.", "However, $t_0$ depends on the velocity and mass of the threads at the moment when the oscillation was triggered.", "For example, filament plasma moving in the opposite direction to the trigger flow, due to counterstreaming [41], [1], could delay the time of maximum displacement.", "In fact $t_0$ is earlier for slits 2, 3, 33, and 34 than at slit 1, which is closest to the energetic event.", "Thus, $t_0$ alone is not a good indicator of the time when the trigger reaches the threads.", "We estimated a lower limit on the arrival time by computing the straight distance from the jet origin to the slit positions and dividing by the jet-flow speed.", "The arrival times (solid line) plotted in Figure REF (b) are computed with the maximum jet-flow speed of $95~\\mathrm {km~ s^{-1}}$ determined in §.", "The derived and observed arrival times agree well, indicating that the jet is responsible for the excitation of the thread oscillations.", "The last two positions do not fit the $95~\\mathrm {km~ s^{-1}}$ prediction as well.", "One possible explanation is that small uncertainties in the determination of the flow speed and the path length will produce greater differences in the arrival times for the more distant positions.", "As shown in Fig.", "REF , each episode of jetting is slower than its predecessor.", "Our analysis of the relative timings also indicates that the first episode excites the oscillations, while the later episodes do not significantly perturb the already moving threads." ], [ "Periods", "Both functional fits yield similar values for the oscillation period $P$ along each slit, ranging from 0.7 to 0.86 hours with a mean value of 0.82 hours.", "Furthermore the period is nearly uniform along the filament (Figure REF (c)), with the exception of slit 33, and agrees with previous observations of LAL oscillations.", "Although the oscillation period exhibits little variation, the oscillations are not completely in phase (see movie), indicating that the filament threads oscillate quasi-independently.", "Thus, the estimated periods reflect the local characteristics of the filament, which must be relatively uniform." ], [ "Maximum Displacement and Speed", "The range of displacement amplitudes for the exponential-sinusoid fit is 6 - 21 Mm, and for the modified Bessel fit from 17 to 40 Mm (Figure REF (d)).", "The Bessel function fit yields the largest amplitudes, and is more consistent with the observations during the initial phase, as discussed in §.", "The maximum speeds plotted in Figure REF (e) range between 10 and $41~\\mathrm {km~s^{-1}}$ for the exponential-sinusoid fit and from 17 to $47~\\mathrm {km~s^{-1}}$ for the Bessel fit.", "As with the periods, both fits yield similar speeds.", "The error bars of the amplitudes of the Bessel fits for slits 4 and 6 are very large, because the uncertainties are very sensitive to small changes in the data fitted.", "In these two cases the nominal displacement values are similar to the others, however, so the fit appears to be adequate.", "The maximum displacement and speed depend on the energy provided by the triggering event, which is located near the SE end of the filament (see §).", "Because the distance between the trigger site and the slit positions along the filament increases with slit number, one might expect the oscillation amplitude to decrease with slit number.", "However, we have not found any evidence of such a decrease.", "In fact the smallest speed is in position 3, close to the trigger, whereas the values at the far NW positions are similar to those at the SE positions.We speculate that the triggering flows follow the magnetic field lines that are confined in the filament channel.", "These field lines do not expand or diverge, much resulting relatively little attenuation of the jet kinetic energy." ], [ "Damping times", "The first three time-distance diagrams (Figure REF ) exhibit clear oscillations throughout the observing period, which can be fitted by the modified Bessel function alone (weak damping at later times).", "However, in slits 4-34 the quality of the data is reduced and the late oscillations are harder to measure.", "Although Figures REF and REF also show signs of thread motions throughout, there are fewer data points (orange dots) and poorer fits at later times because we could not identify a coherent oscillatory pattern then (see §).", "Therefore, the strong and weak damping components are most easily separable in the first 3 slits, yielding the most reliable estimates of the weak damping time ($\\tau _w \\sim 6-9$ hr).", "Independent of the quality of the Bessel function fit, the first period of the oscillation is very strongly damped.", "We computed the associated damping time of the first stage of the oscillation by adjusting a decaying exponential between the first peak at $t_0$ and the second peak, one period later at $t_0 + P$ .", "In § we noted that the displacement is reduced by 60% in the first period, which yields a damping time comparable to the oscillation period ($e^{-P/\\tau } \\approx e^{-1} \\approx 0.4$ ).", "Figure REF (f) shows the resulting strong damping times at all 8 slit positions.", "Thus, in the initial stage of the oscillation $\\tau _\\mathrm {strong} \\sim P ~.$ Figure REF (f) also shows the damping times of the sinusoid-exponential fit (Eq.", "REF ), $\\tau $ , which range from 1.0 to 2.4 hours; the damping time per period is $\\tau / P=$ 1.3-2.8.", "Because the exponential-sinusoid damping time is larger than the strong damping time of the modified Bessel function fit, suppressing the oscillations far too quickly, we conclude that the sinusoid-exponential fit cannot account for the initial strong damping.", "It is interesting to note that both damping times are almost uniform along the filament.", "The rapid damping implies that the associated damping mechanism must be very efficient for the LAL oscillations.", "We discuss this mechanism in terms of the thermal nonequilibrium model in §REF ." ], [ "Radius of curvature", "Assuming that the restoring force is the projected gravity along the dipped field lines supporting the prominence plasma, the angular frequency of an oscillating thread is $\\omega =\\frac{2 \\pi }{P}=\\sqrt{\\frac{g_0}{R}}~,$ where $g_0$ is the solar gravity and $R$ is the radius of curvature of the dips [22], [21].", "With this expression we obtained the radius of curvature of the dipped field lines of the observed filament using the calculated oscillation periods.", "In Figure REF these radii are plotted for different positions along the filament; the range of values is $R= 43-66~\\mathrm {Mm}$ .", "The absence of any clear dependence of the magnitude on location suggests that the geometry of the field lines supporting the filament plasma was more or less uniform along the channel in the regions where oscillations were observed.", "Figure REF displays a 3D representation of the magnetic field structure inferred from our model at the 8 best slit positions, with a LOS HMI magnetogram as the background.", "The yellow curves representing the dipped parts of the field lines that support the oscillating threads are drawn with the orientations and radii of curvature given in Figures REF (a) and REF .", "However, for clarity we placed the bottom positions of all dips 10 Mm above the photosphere.", "The threads oscillate around $s_0 + d_0~(t - t_0)$ (from Equations REF and REF ), but $d_0$ is very small or negligible according the function fits.", "Thus the threads oscillate around a fixed position, $s_0$ .", "We also assumed that the bottom of the dips are the equilibrium positions of the thread displacements.", "These displacements are not symmetric with respect to the center position, as shown in Figures REF - REF : maximum elongation occurs on the side of the thread farthest from the jet, and only reaches approximately half of this amplitude on the same side as the jet due to the strong damping.", "The approximate position where the energetic event occurs (brightening) is indicated by a red dome, and the direction of the jet as a red cone.", "Figure: Derived radius of curvature, RR, of the dipped field lines supporting the threads (Eq.", ").", "Diamonds represent quantities derived from the exponential-sinusoidal function fits (Eq.", "), asterisks represent quantities derived from Bessel function fits (Eq.", ").Figure: The 3D geometry inferred from our analysis viewed from above (top) and the side (bottom).", "The grayscale plane shows a SDO/HMI magnetogram taken at 18:00 UT.", "The continuous white line over the magnetogram is the magnetic PIL of the region.", "The yellow lines represent the magnetic field lines supporting the filament plasma at the 8 best slit positions.", "Their radii of curvature correspond to the values of Fig.", ".", "The red dome symbolizes the origin of the initiation jets and the red cone indicates the direction of the jet flows.", "All of the dipped field lines are plotted with the bottom of the dips 10 Mm above the photosphere; this height was arbitrarily chosen to yield a clear view of the dips on the bottom panel." ], [ "Minimum magnetic-field strength", "We computed the minimum magnetic-field strength at the field line dips, assuming that the magnetic tension should at least balance the weight of the threads.", "Previously we found the following expression for the minimum value of the magnetic field as function of the electron number density $n_\\mathrm {e}$ [22]: $B[\\mathrm {\\mathrm {G}}] \\ge 26\\, \\left(\\frac{n_\\mathrm {e}}{10^{11}~\\mathrm {cm}^{-3}}\\right)^{1/2}\\,P[\\mathrm {hours}]~.$ The electron density in this filament was not measured, so we assume typical values in the range $10^{10} -10^{11}~\\mathrm {cm^{-3}}$ [14].", "We consider the electron density to be the most important source of uncertainty in the estimated magnetic field, because the uncertainties associated with the fit are smaller than the above density range.", "With this consideration Equation REF yields $B[\\mathrm {\\mathrm {G}}] \\ge \\left( 17 \\pm 9 \\right) \\,P[\\mathrm {hours}]~.$ Because the oscillation period is nearly uniform along the filament, the variation of the magnetic field along the filament is smaller than the uncertainty associated with the density.", "Thus, the minimum magnetic field strength along the filament is $14 \\pm 8~\\mathrm {G}$ , which is consistent with typical directly measured field strengths [24]." ], [ "Mass accretion rate", "In our model [22], the LAL oscillations are damped by continuous mass accretion onto the filament threads at rate $\\alpha $ , and the oscillations are described by the Bessel function of Equation (REF ).", "The phase $\\psi _0$ is related to the mass accretion rate as $\\psi _0=\\omega m_0/\\alpha $ , where $m_0$ is the mass of the thread at $t_0$ and $\\psi _0$ and $\\omega $ are derived from the Bessel function fit.", "To obtain $m_0$ , we assume that the thread is a cylinder of length $l$ , radius $r$ , and uniform electron density $n_\\mathrm {e}$ .", "Thus, $m_0=1.27 m_\\mathrm {p} n_\\mathrm {e} \\pi r^2 l$ , where $1.27 m_\\mathrm {p}$ is the average coronal particle mass and $m_\\mathrm {p}$ is the proton mass.", "Taking a typical thread radius $r=100~\\mathrm {km}$ [19] we obtain $m_0 \\mathrm {(kg)} = 6.67 \\times 10^6~\\left(\\frac{n_\\mathrm {e}}{10^{11}~\\mathrm {cm^{-3}}}\\right) ~l \\mathrm {(Mm)}~.$ The thread length is estimated to be the slit-aligned length of the central dark (absorption) region of Figures REF - REF , measured at the second period after the maximum displacement time in all cases.", "This procedure overestimates the real length of the thread because we are not considering the PCTR at both ends of each thread (which should be thin), and ignoring the fact that the 171 Å dark region probably contains several threads along the LOS.", "As for the minimum magnetic-field strength, the mass $m_0$ depends on an assumed electron density range.", "Thus $m_0=\\left( 4 \\pm 3 \\right) \\times 10^6~l \\mathrm {(Mm)}~.$ The thread length, $l$ , is quite uniform along the filament, so the variation of $m_0$ along the filament is smaller than its uncertainty.", "With this mass estimate and the measurement of $l$ , the mass accretion rate is estimated to be $\\alpha =\\left(36 \\pm 27 \\right) ~\\times 10^6 ~\\mathrm {kg ~hr^{-1}}~,$ which is consistent with the theoretical values predicted by our thermal nonequilibrium model with steady heating [12], [23].", "The theoretical accretion rates are slightly smaller than the observed values, primarily because the present observational study has significant uncertainties in the thread length, radius, and density.", "In addition, these differences could be associated with small temporal and spatial variations in the heating at the footpoints.", "Consequently we consider the factor-of-2 agreement between the observed and predicted mass accretion rates to be quite good." ], [ "Jet energy estimate", "As discussed in §, the filament oscillations are triggered by a small jet close to the southeast end of the filament.", "A highly collimated flow of hot plasma emanates from this small area at a projected speed of $\\sim 95~\\mathrm {km~s^{-1}}$ , yielding oscillating filament plasma that reaches speeds of $\\sim 50$ km s$^{-1}$ .", "We estimate the energy released to the filament by the jet event by computing the kinetic energy of the oscillations: $E=1/2 M_\\mathrm {osc} v^2$ , where $M_\\mathrm {osc}$ is the oscillating thread mass and $v$ is the averaged velocity amplitude of the oscillation ($\\sim 30~\\mathrm {km~s^{-1}}$ ).", "Because only 8 of 36 slit positions exhibit periodic motions, we estimate that only $8/36$ ($\\sim $ 22%) of the total filament mass oscillates.", "With these considerations the energy imparted by the jet to the filament oscillations is $E=10^{26}\\left(\\frac{M}{10^{14} g}\\right)~\\mathrm {erg}~.$ For a typical prominence mass $M=10^{12}-10^{15} ~ g$ [14], the energy of the oscillating filament is predicted to be $E=10^{24}-10^{27}~\\mathrm {erg}$ .", "The energy transferred to the filament by the jet is an unknown fraction of the total energy released by the initiating event, however, so $E$ places a lower limit on the trigger energy.", "[40] carried out numerical experiments generating LAL oscillations by impulsive heating at one footpoint of a loop, and found that the energy of the thread oscillations is only 4% of the impulsive energy release.", "In our case the initiating event manifests several key characteristics of a microflare, most notably the presence of a jet, the duration, and the energy range [5], [30]." ], [ "Filament structure", "Our results provide important clues about the structure of the entire filament channel, which is sketched in Figure REF .", "As shown in § the filament channel is curved, forming a semi-circular structure.", "In order to reproduce the magnetic field direction at the slits and the hemispheric rules, the chirality of the filament should be sinistral, in agreement with [36] who found that the barbs are left bearing.", "The NW substructure traversed by slits 28-36 is most likely a barb pointing to a small, negative parasitic polarity within the mainly positive region.", "The PIL in this part of the active region is highly irregular, so it is difficult to discern the main direction of the PIL there.", "However, a more detailed study of this feature is outside the scope of this paper.", "The cartoon of Figure REF is consistent with the 3D reconstructed field of Figure REF , in terms of the orientation and location of the dipped field lines in the oscillating segments.", "At the western side of the channel, Figure REF depicts a region not analyzed in this paper: a filament segment that erupts around 21:40 UT.", "The oscillations in slits 33-34 are disrupted by the eruption, whereas the oscillations in slits 1-6 are largely unaffected.", "Therefore we expect that the field lines supporting the barb are linked to this erupting region, while the field lines supporting the SE part of the filament (slits 1-6) probably are not connected to the erupting filament segment.", "Figure: Schematic picture of the magnetic connections between the jet (red star) and selected filament threads (striped regions showing the thread directions).", "The PIL is indicated by a dashed line; the broad orange ribbon is the filament channel.", "Field lines (black and blue solid lines) supporting white striped areas carry jet plasma from the energetic event to the oscillating threads, while purple striped areas represent threads along field lines that are not connected to the jet and hence do not oscillate.The position of the jet is crucial because it determines the magnetic field connectivity to the oscillating threads.", "We assume that the field lines supporting these threads are magnetically connected at one footpoint with the jet source.", "The source position appears to be very close to the PIL, but the inherent uncertainty in the magnetograms makes it difficult to determine whether the jet site is north or south of the PIL.", "Given the southern location of the filament, the filament chirality, the predominance of inverse polarity in filaments, and the observed direction of the threads, however, we conclude that the jet source is most likely situated north of the PIL where the background field is predominantly negative." ], [ "Summary and Conclusions", "In this work we have studied the LAL oscillations of a filament and the associated trigger, using observations and theory to determine key properties of the filament and trigger.", "The LAL oscillations are displacements of low $\\beta $ plasma supported by the magnetic field, so the direction of motion reflects the direction of the local magnetic field.", "Our results agree with the few direct measurements of the orientation of the magnetic field in filament threads with respect to the associated PIL.", "Thus, the oscillation analysis described in this work is a novel tool to determine the orientation of the filament magnetic field.", "We determined fundamental characteristics of the LAL oscillations and the triggering flows by fitting curves to the SDO/AIA time-distance diagrams.", "We used a exponentially decaying sinusoid and a modified Bessel function to fit the oscillation data, respectively representing constant-mass and mass-accreting solutions.", "Both fits generally match the data well, but the Bessel function fits the initial stage of the oscillation significantly better.", "Therefore mass accretion is likely to play a major role in damping the oscillations rapidly.", "We conclude that our earlier model for the damped LAL oscillations [22], [21] accurately explains the behavior of this filament.", "Using our earlier analytic approximations, we determined the radius of curvature and minimum strength of the magnetic field lines that support the filament plasma, and inferred the magnetic structure of the oscillating portions of the filament (Figures REF and REF ).", "We found that the geometry varies little along the filament, demonstrating that the different parts of the filament form a quasi-coherent structure whose origin and subsequent evolution remain linked.", "The resulting structure, as well as the presence of LAL oscillations, are compatible with the two leading magnetic-structure models — the flux rope and the sheared arcade — which predict that the bulk of the filament plasma resides in the dips [24].", "Although both models predict dipped field lines that can host oscillating threads, the distribution of cool mass condensed in these dips through thermal nonequilibrium has been studied in depth only for the sheared arcade model.", "The thermal nonequilibrium model for filament mass formation predicts that the existing threads accrete material at the same rate as the chromospheric evaporation rate, as long as the standard coronal heating is localized at the footpoints.", "We established previously that this continuous accretion of mass is responsible for the strong damping of the LAL oscillations [22].", "The mass accretion rate of the filament threads computed in the present study agrees with the predictions of our thermal nonequilibrium model [12], [21], and hence with the well-established quiet-Sun coronal heating rate of [38].", "Based on our results, we suggest that LAL oscillations provide a new opportunity for constraining coronal heating models beyond the usual analyses of AR coronal loops.", "We propose the following general picture of the event and the filament structure.", "A reconnection process takes place close to PIL at the northern side.", "The resulting jet plasma flows along the filament channel field lines at a projected speed of $\\sim 95~\\mathrm {km~s^{-1}}$ .", "These field lines only connect with some parts of the filament, such that the flow reaches the SE part and the NW barb.", "Threads in these regions are pushed by the hot flows, then oscillate in the dips with a motion resembling a pendulum.", "Other complex physical phenomena could take place when the hot flows reach the cool plasma, but a detailed study of this interaction is beyond the scope of this work.", "The restoring force of the oscillations is the projected gravity along the dips, as our model predicts.", "Continuous, localized coronal heating produces evaporation of chromospheric plasma that accretes onto the already formed filament threads.", "This mass accretion is responsible for the initial strong damping of the oscillations.", "More observations of LAL oscillations in filaments and the associated triggering events, together with detailed simulations of the response of filament threads to hot flows, are needed to improve our understanding of this intriguing phenomenon.", "Additional theoretical modeling of LAL oscillations also will advance the use of seismology to probe the ambient physical conditions in filaments, as demonstrated here.", "Further analyses of LAL oscillations would benefit as well from greater understanding of the coronal heating mechanism, as the likely driver of mass accretion onto filament threads.", "We anticipate significant progress on these questions to be made in the near future by the combined capabilities of SDO, IRIS, and the upcoming Solar Orbiter mission.", "ML gratefully acknowledge partial financial support by the Spanish Ministry of Economy through projects AYA2011-24808 and CSD2007-00050.", "This work contributes to the deliverables identified in FP7 European Research Council grant agreement 277829, “Magnetic Connectivity through the Solar Partially Ionized Atmosphere\", whose PI is E. Khomenko.", "KK acknowledges support for this work by a co-op agreement between the Catholic University of America and NASA Goddard Space Flight Center, sponsored by NASA's Heliophysics LWS and SR programs.", "KM gratefully acknowledges funding from the National Science Foundation via grant # 0962619.", "SDO is a mission for NASA's Living With a Star program.", "HG, JK, and TK also thank the LWS TR&T Program for support.", "We also thank S. Antiochos, I. Arregui, A. Asensio-Ramos, J. L. Ballester, C. R. DeVore, A. Díaz, A. Lopez-Ariste, F. Moreno-Insertis, R. Oliver, D. Orozco-Suarez, and J. Terradas for helpful discussions and suggestions." ] ]	1403.0381
[ [ "Flexible Time and Ether in One-dimensional Cellular Automata" ], [ "Abstract Flexible Time is a new formalism for calculations about one-dimensional cellular automata.", "It unifies the states of a finite number of cells into a single object, even if they occur at different times.", "This gives greater flexibility to handle the structures that occur in the development of a cellular automaton.", "An ether is a periodic pattern of cells that arises in some cellular automata from almost all random initial configurations.", "In this thesis, the formalism is developed in detail and then applied to the problem of ether formation.", "For the elementary cellular automaton Rule 54, a partial result is proved: There is a fragment of the ether that arises with probability 1 from every random initial configuration and is propagated with probability 1 to any later time.", "This is a strong argument that the ether under Rule 54 indeed arises from almost all input configurations." ], [ "List of Symbols", "chapterSets and Functions 10ex0ex$\\mathbb {N}$ [pg:integer-sets]Set of positive integers REF 10ex0ex$\\mathbb {N}_0$ [pg:integer-sets]Set of nonnegative integers REF 10ex0ex$B^A$ [pg:func-A-B]Set of functions from $A$ to $B$ REF 10ex0ex$A \\subset B$ [pg:subset-notation]$A$ is proper subset of $B$ REF 10ex0ex$A \\subseteq B$ [pg:subset-notation]$A$ is subset or equal to $B$ REF 10ex0ex$\\mathop {\\mathrm {dom}}f$ [pg:domain-func]Domain of the function $f$ REF chapterSequences 10ex0ex$\\lambda $ [pg:empty-sequence]Empty sequence REF 10ex0ex$A^$ [eq:A-star]Kleene closure of $A$ REF 10ex0ex$\\mathopen \|s\\mathclose \|$ [pg:length]Length of sequence $s$ REF chapterSpace-time 10ex0ex$p_T$ [pg:space-time-point]Time component of point $p$ REF 10ex0ex$p_X$ [pg:space-time-point]Space component of point $p$ REF 10ex0ex$T$ [eq:unit-vectors]Unit vector in temporal direction REF 10ex0ex$X$ [eq:unit-vectors]Unit vector in spatial direction REF 10ex0ex$N(p, r)$ [eq:nb-domain]Neighbourhood domain of point $p$ REF 10ex0ex$I_t(i, j)$ [def:intervals]Interval domain at time $t$ REF chapterProcesses 10ex0ex$\\mathcal {P}$ [def:cellular-process]Set of cellular processes with states in $\\Sigma $ REF 10ex0ex$\\mathop {\\mathrm {dom}}\\pi $ [eq:domain]Domain of process $\\pi $ REF 10ex0ex$\\pi \\mathrel \\mathrm {comp}\\psi $ [def:compatible]$\\pi $ is compatible with $\\psi $ REF 10ex0ex$\\pi \|_S$ [eq:process-restriction]Restriction of process $\\pi $ to set $S$ REF 10ex0ex$\\pi ^{(t)}$ [def:cellular-process]Time slice at time $t$ of process $\\pi $ REF 10ex0ex$\\nu (p, w)$ [eq:neighbourhood]Neighbourhood process for point $p$ REF 10ex0ex$S(p, \\pi )$ [eq:values]Set of possible states for the point $p$ REF 10ex0ex$\\Delta \\pi $ [pg:determined]Events determined by $\\pi $ under rule $\\varphi $ REF 10ex0ex$\\mathop {\\mathrm {cl}}\\nolimits \\pi $ [def:closure]Closure of process $\\pi $ under rule $\\varphi $ REF 10ex0ex$\\mathop {\\mathrm {cl}}\\nolimits ^{(t)} \\pi $ [def:closure]Closure at time $t$ of process $\\pi $ REF 10ex0ex$[p] \\pi $ [eq:shifted]Process $\\pi $ , shifted by $p$ REF 10ex0ex$\\pi \\mathrel {\\prec \\succ }\\psi $ [def:spatial-arrangement]$\\pi $ is left of $\\psi $ REF 10ex0ex$\\psi \\supseteq _L \\pi $ [eq:left-extension]$\\psi $ is left extension of $\\pi $ REF 10ex0ex$\\pi \\subseteq _R \\psi $ [eq:right-extension]$\\psi $ is right extension of $\\pi $ REF chapterSituations 10ex0ex$\\mathcal {S}$ [def:situation]Set of situations with state set $\\Sigma $ REF 10ex0ex$\\delta (s)$ [pg:size-vector]Size vector of situation $s$ REF 10ex0ex$\\mathrm {pr}(a)$ [def:process-of]Process of situation $a$ REF 10ex0ex$\\mathrm {pr}_{a}(b)$ [eq:relative-process]Process of situation $b$ , shifted by $a$ REF 10ex0ex$[p]$ [def:situation-notation]Displacement by distance $p$ REF 10ex0ex$[0]$ [pg:empty-situation]Empty situation 10ex0ex$a \\mathrel {{\\backslash }\\!", "{\\backslash }}x$ [def:end-factors]$a$ is left factor of $x$ REF 10ex0ex$x \\mathrel {{/}\\!", "{/}}a$ [def:end-factors]$a$ is right factor of $x$ REF 10ex0ex$\\mathbin {\\langle b\\rangle }$ [def:overlap]Overlap at situation $b$ REF 10ex0ex$a \\sim b$ [def:equivalent]$a$ is equivalent to $b$ REF chapterReactions 10ex0ex$a \\rightarrow b$ [def:reaction]$a$ reacts to $b$ REF 10ex0ex$a \\rightarrow _R b$ [pg:reaction]The reaction $a \\rightarrow b$ is element of the set $R$ REF 10ex0ex$\\mathop {\\mathrm {dom}}R$ [pg:r-domain]Domain of the reaction set $R$ REF 10ex0ex$\\hat{a}$ [def:determined-situation]Interval situation determined by $a$ REF 10ex0ex$a_L$ [def:boundary-interval]Leftmost minimal separating interval in $a$ REF 10ex0ex$a_R$ [def:boundary-interval]Rightmost minimal separating interval in $a$ REF 10ex0ex$+_a$ , $-_a$ [def:slope-operators]Slope operators for $a$ REF 10ex0ex$\\mathcal {A}_\\varphi $ [def:achronal]Set of achronal situations for $\\varphi $ REF 10ex0ex$\\mathcal {A}_{\\varphi +}$ , $\\mathcal {A}_{\\varphi -}$ [def:achronal]Slopes for $\\varphi $ REF 10ex0ex$R_+$ , $R_-$ [def:slope-subsystem]Slope subsystems of the reaction system $R$ REF" ], [ "Introduction", "Cellular automata came into being as an almost brutally simplified model of information processing in a physical medium.", "John von Neumann invented cellular automata (together with Stanislaw Ulam) as part of his work on self-reproducing systems [31].", "He needed a simplified physical universe in which he could construct a model which captures the essential properties of self-reproduction in a biological organism.", "This model universe had to be simple enough that a single person could reason about it while using only paper and pencil and no mechanical aid.", "It also had to consist of simple components in order to make sure that self-reproduction was a property of the simulated organism and not already built into the physics of its universe.", "These requirements lead to several simplifications.", "The first one is that only discrete parameters could be used, especially no real numbers.", "Time in a cellular automaton therefore runs in discrete steps, like the ticks of a clock.", "Space is reduced to a rectangular grid.", "It consists of the points of the $n$ -dimensional grid $\\mathbb {Z}^n$ : we speak then of an $n$ -dimensional cellular automaton.", "The second simplification concerns the interior of the universe.", "It must be possible to describe the self-reproducing organism completely with a finite number of symbols.", "The world simulated by the automaton consists of objects at the lattice points, which are called cells.", "A cell can be in one of several states, and there could be only finitely many of them if it was possible to write down a configuration of the automaton.", "The cells are thought as small information-processing machines, representing atoms, electrical components or possibly biological cells in a tissue.", "For the purpose of von Neumann, the simulated universe of the cellular automaton would contain only a finite number of cells that simulated the self-reproducing system.", "All other cells were in a special, quiet state, which stood for a kind of vacuum that remained unchanged.", "Activity was always caused by cells in other states.", "The physics of this model universe is one of local interaction.", "The purpose of the model universes of cellular automata is the simulation of an object of moderate size.", "Therefore the world view of cellular automata is based on Newtonian physics.", "It especially ignores General Relativity.", "The physical laws in a cellular automaton are then the same at every point and for every time step.", "In order to make it possible that they can be described completely, they also have to allow a finite description.", "Therefore the state of a cell in the next time step cannot depend on the states of all the cells in the automaton.", "This leads to the idea that the state of a cell in the next time step should only depend on the states of the cells in its direct neighbourhood.", "Such a neighbourhood contains only the cells at an Euclidean distance less than or equal to a given constant $r$ .", "(Note that with this definition a cell is always part of its neighbourhood.)", "The number of cells in a neighbourhood is then always finite.", "The number of combinations of states that these cells can have together is also finite: therefore the behaviour of each cell can be described by a finite table.", "It maps each state of the neighbourhood of a cell to the state of the cell in the next time step.", "The neighbourhoods of all points in $\\mathbb {Z}^n$ , and therefore those of the cells in the cellular automaton, look the same: therefore it is possible to specify the behaviour of the whole system of cells with a single finite rule.", "The number $r$ is called the radius of the cellular automaton.", "A beneficial side effect is that no point of $\\mathbb {Z}^n$ is special, as it is in Newtonian physics.", "Another side effect is that signal transmission in the universe of cellular automata always has a finite speed.", "This is what John von Neumann did, and he started to develop a self-reproducing system in a specific two-dimensional cellular automaton.", "A second important step in the history of cellular automata was the invention of the “Game of Life” by John H. Conway in 1970 [31].", "It is a two-dimensional cellular automaton with two states and an especially simple rule and became soon very popular.", "This was the time when computers with a graphics display started to become accessible to many people.", "A development that is important in the context of the present work is that some of them did run “Life” systematically with initial configurations that were chosen at random.", "The usual method is to chose a probability $p$ and then let the computer initialise independently every cell with probability $p$ in state 1 and with probability $1 - p$ in state 0.", "State 0 is the quiet state in Life.", "Then after several time steps of evolutionThe word “evolution” means different things in different contexts.", "Here I use it in the wider sense of “development over time”, not in the narrow sense of Darwinian evolution.", "There is also research on cellular automata where the subject is the Darwinian evolution of transition rules for a specific purpose.", "(See e. g. Mitchell, Crutchfield and Hraber [46] or the review by Mitchell [45].)", "Nevertheless the use of “evolution” in the wider sense is also established in cellular automata theory.", "The word “evolution” has more specific associations than e. g. “development”, therefore I use it here.", "stable patterns often emerge.", "Some of them stay unchanged, others oscillate with a period of 2, seldom more, time steps, and a few move through the two-dimensional cellular space.", "This set a precedent, and random initial configurations have become a standard tool that is used when one wants to get an overview of the behaviour of an unknown cellular automaton.", "So when in 1983 Stephen Wolfram [59] wanted to survey the possible behaviours of cellular automata, he too tested them on random initial configurations.", "The plan to survey the possible behaviours of cellular automata had a side effect that influences cellular automata theory until now.", "Wolfram worked with one-dimensional cellular automata, and in order to have a subset of manageable size, he chose the set of automata with two states and radius 1, the simplest class at all from which one can expect nontrivial behaviour.", "Wolfram called them “elementary cellular automata”.", "There are 256 of them, but if one views automata as equivalent if they differ only by an interchange of the states 0 and 1 or of left and right, only 88 types of behaviour remain.", "(See Li and Packard [33].)", "There has been earlier research on elementary cellular automata, but Wolfram gave this class a name and introduced a system of code numbers with which one can refer to their rules, and they have stayed in the centre of research since then.", "The choice of one-dimensional automata is also in another sense advantageous, since their behaviour can be easily displayed in a two-dimensional diagram with one space and one time dimension.", "This makes communication about their behaviour much more direct than that about two-dimensional automata.", "In contrast to John von Neumann's rule and “Life”, the set of elementary cellular automata contains rules in which there is no state that can be considered quiet.", "With some of the rules, the cellular automata stay chaotic when started from a random initial configuration.", "But with others, a similar phenomenon as with “Life” occurs.", "After some time, particles appear and move or stay on a simple background, but here the background does not consist of a region of cells all in the same state.", "Instead, the regions between the particles consist of a spatially periodic pattern.", "After several steps of evolution of the cellular automaton, the same pattern occurs again.", "When the evolution of the cellular automaton is displayed as a space-time diagram, the background looks like a wallpaper pattern.", "Nowadays, such a periodic background pattern that arises from almost every random initial configuration is often called an “ether”, in analogy to the ether concept of pre-relativistic physics.", "There, as in cellular automata, the ether is a background in which particles and signals move.", "While this thesis focuses on ether formation under Rule 54, we will now take a larger perspective and look for examples of ether formation among the elementary cellular automata in general.", "Table: A survey of elementary cellular automata (Part1).Table: A survey of elementary cellular automata (Part2).Table: A survey of elementary cellular automata (Part3).For this we need a criterion that tells us whether a cellular automaton has an ether.", "To my knowledge there is however no general definition in the literature under which conditions a cellular automaton has an ether.", "As a working condition for the following small survey, we will use the following definition, based on Rule 54: If the evolution diagram of a typical random initial configuration contains regions that are periodic in space and time, and these regions grow over time, then this cellular automaton has an ether.$\\endcsname $Thus an ether may be an uniform pattern consisting of a single cell, like the quiet state under some rules.", "I do not exclude it because one of the properties of the ether is that it is the background on which particles move.", "In this aspect, an uniform ether is not different from other ethers.", "Not excluding the uniform case also makes the definition simpler.", "With it we can check empirically for all elementary cellular automata whether they have an ether.", "This is done in Table REF –REF .", "The diagrams in this figure show the evolution of a random initial configuration under all equivalence classes of elementary cellular automata.", "All evolution diagrams use the same initial configuration, one in which each cell is with probability $\\frac{1}{2}$ in state 0 and with probability $\\frac{1}{2}$ in state 1.", "For this probability distribution the behaviour of the cellular automaton is unchanged when state 0 is exchanged with 1 in its transition rule, or left with right.", "We will therefore show only the evolution of one of these related rules, namely that with the lowest code number.", "For some of the rules, e. g. Rule 110, a larger evolution diagram than that one shown here is needed to clarify whether they have an ether.", "But if we do this, we see that only the cellular automata with the code numbers 9, 14, 25, 37, 54, 57, 62, 110, 142 and 184 (and those equivalent to them) support an ether.", "Their evolution diagrams are marked in Table REF –REF with an E." ], [ "The Problem of Ether Formation", "The formation of an ether in some cellular automata has been mentioned by some authors as an open question, but to my knowledge no one has published a solution.", "Bruno Martin [34] writes, “By observing space-time diagrams of the rule 54 on a random configuration, we always see a kind of background with space and time period 4 [...].", "The background apparition is usually very fast (less than ten iterations are enough) and still mysterious, we have no explanation of this phenomenon.” As a repeating pattern, the ether is the simplest structure in a cellular automaton that can grow to an unlimited size.", "The problem of ether formation is therefore the simplest question about an important form of self-organisation in cellular automata, namely the autonomous emergence of large structures.", "Self-organisation and the construction of complex patterns and machines in a cellular automaton have in common that they involve the synchronised behaviour of many cells.", "To understand them one first needs a language in which one can express the behaviour of a large number of interacting cells over time.", "In an earlier publication [51] I have called this language “Flexible Time”." ], [ "Motivation", "A basic idea behind the formalism of Flexible Time is that it generalises the way in which a finite part of a configuration of a cellular automaton is written.", "For a one-dimensional cellular automaton, which is an infinite sequence of cells, a configuration is simply an infinite sequence of cell states.", "A single cell state is usually written as a symbol, often a number, and then one naturally writes a finite region of the cellular automaton as a string of symbols.", "Then a sequence like 011101110111 is a possible content of a region in a cellular automaton for which the state set contains the values 0 and 1.", "Mathematics has already developed notations and theories to work with such strings of symbols efficiently.", "One simple notational device is the use of exponents to express repetition, which allows us to express the previous string as $(0111)^3$ .", "There is also a whole theory of formal languages to handle such strings.", "I want to be able to use such methods.", "Another ingredient is selective interest.", "A natural way to understand a complex system is to decompose it into subsystems and first try to understand them first.", "However, a cellular automaton is a model of a physical (or biological or information processing, &ct.) system.", "In such a system there are lots of processes that start and end independently of each other—lots of organisms that are born and die at any moment in time, or lots of tasks that are completed independently.", "There is no global synchronisation.", "In a world in which information travels with finite speed, the starting time of a process is only influenced by the processes in its direct neighbourhood.", "And if it consists itself of subprocesses, which are also loosely coupled, then they may finish at different times, and it makes no sense to speak of the “end time” of the main process of which they are parts.", "A notation for cellular automata that allows one to focus on the behaviour of arbitrary subprocess has to take that into account." ], [ "The Formalism of Flexible Time", "My idea to solve to the questions implied here is influenced by the concepts of Relativity theory.", "We will now give up the thought that there must be a globally determined time.", "Instead, when a complex process consists of subprocesses that end at different times in different places, then the end times of the subprocesses together form the end time of the process.", "They are, all together, viewed as a single moment in time.", "With this concept of time it is no longer necessary to work with configurations of infinite size.", "We will instead specify the content of a finite region of the cellular automaton, namely that where a specific process starts, and are then able to compute the generalised end time of this process and the states of the cells at this time.", "There is a mathematical object that specifies the location of some cells, both in space and in time, together with their states.", "I call it a situation, and it is a generalisation of the finite sequence of cell states that I mentioned before.", "Situations have in common with cell state sequences that they are formally strings of symbols; therefore the familiar concepts for words in a formal language can be applied to them.", "There is another kind of mathematical object, the reaction, which we will use to specify such a process.", "It is a pair of situations, one for the beginning and one for the end of the process.", "The set of all reactions for a cellular automaton provides the same information as the transition rule.", "Reactions and situations are thus an alternative means to reason about the behaviour of a cellular automaton.", "Together they form the formalism of Flexible Time.", "We have thus the following (approximate) equivalences: Table: NO_CAPTION" ], [ "Aims and Methods", "The intent of this thesis is to construct a general framework in which one can solve questions about collective behaviour in one-dimensional cellular automata.", "It should be a language that is adaptable to a wide range of questions.", "This way I want to provide a step forward towards the solution of the last one among Stephen Wolfram's “Twenty Problems in the Theory of Cellular Automata” [61], What higher-level descriptions of information processing in cellular automata can be given?", "My approach to address Wolfram's question is to invent a language in which one can describe more easily the components of a cellular automaton's information processing system.", "For a concrete problem this allows to create a vocabulary of space-time patterns and their interactions.", "With this language, increasingly larger structures could be described and understood, until one would understand the behaviour of a large and complex information-processing system.", "In its current state the language is not so powerful.", "It does however allow to express with situations the patterns that one can see in space-time diagrams more or less directly, and then to express and prove general theorems about them.", "The development of such a tool is easier with a concrete application in mind.", "Therefore, a second aim of this work is to find an explanation why there is an ether in the elementary cellular automaton with rule code 54.", "I have chosen ether formation because it is a case of self-organisation and therefore interesting in its own right.", "Furthermore it is the simplest case of self-organisation in cellular automata that I am aware of.", "Rule 54 has a relatively simple ether that arises early in the evolution of a random initial configuration.", "Nevertheless this cellular automaton is far from trivial: With Rule 54 one can e. g. compute arbitrary Boolean functions [36].", "This makes it more probable that the results about ether formation in Rule 54 and the methods to derive them carry over to other interesting cellular automata.", "The thesis is a kind of sequel to my article [51]: There, a description of Flexible Time in the context of a specific cellular automaton was given, but its general theory was missing.", "Here I provide a theoretical justification for the formalism, together with a study of ether formation for a specific cellular automaton.", "A more general theory of structure formation in one-dimensional cellular automata is something I would like to be able to do at a later time." ], [ "Requirements on the Theory", "At a very basic level it has always been difficult to express the phenomena in a cellular automaton in an understandable way.", "This is true especially if emphasis is placed on concrete interactions, like the collision of two particles.", "All authors use pictures in some way.", "This becomes however difficult once larger structures are involved and the details of such diagrams would become smaller and smaller.", "The aim of this work is therefore to construct a formalism that expresses the behaviour of a large mass of cells in a cellular automaton.", "We need to express the evolution of a cellular automaton in a way that corresponds to the structures that one can see in the two-dimensional space-time diagrams, and which is at the same way able to represent arbitrarily large structures.", "My goal was to find a language in which one can describe the behaviour of cellular automata in an algebraic notation—a kind of cellular automata evolution program that is run by the human brain.", "The formalism is therefore completely algebraic.", "No pictures are necessary to specify details.", "They are however still useful for clarification and to get ideas.", "It is an important requirement that the formalism is not bound to a specific moment in time.", "A pattern that we see in a space-time diagram usually extends over a longer period of time.", "We see it as a two-dimensional form, and pieces that appear to us as connected may belong to different times.", "A formalism that restricts us to snapshots of the cellular automaton at specific moments in time can not display this.", "I have therefore developed a formalism that allows us to jump forward and backward in time." ], [ "Two Kinds of Mathematics", "This work is also an attempt to support a return, after years in which computer experiments dominated, to the idea that the behaviour of cells in a cellular automaton should be something that can be comprehended with the help of pencil and paper alone.", "I will call these methods here “traditional mathematics”.", "The methods of computer experiments and of traditional mathematics have different aims and result in different kinds of understanding.", "The result of a computer experiment is knowledge about a single case.", "A result of traditional mathematics is a theorem about an infinity of cases.", "One may say that, since the majority of the questions asked in scientific research are about “all cases” of a certain kind, traditional mathematics is the only way to answer them.", "However, the requirement of traditional mathematics always to work with an infinity of cases at once is a severe restriction.", "In contrast, computer experiments can be done even in cases where there is not enough understanding of the question to apply the methods of traditional mathematics.", "The restricted nature of traditional mathematics also holds a great promise.", "If it works, traditional mathematics has results that automatically apply to a great range of cases.", "This is because the very restrictedness of its methods makes sure that its results have only a small number of preconditions.", "This automatism suggests a method that combines some of the advantages of both approaches: If one takes a phenomenon that has been found empirically in a small number of cases and finds a proof for it, then it will automatically tell us something about an infinity of other cases.", "This is what I do here with the ether in Rule 54." ], [ "Previous Work", "This work is an extension of the ideas presented in [51], where the formalism of Flexible Time was introduced for the case of Rule 110.", "An application of the formalism to Rule 54 was presented in [52], where it was also shown how the ether and particles were represented with situations and reactions." ], [ "Thanks", "I want to thank Andrew Adamatzky, Genaro J. MartÃnez and Lars Immisch for their help in getting me to Bristol.", "I also want to thank my thesis supervisors Andrew Adamatzky, Rob Laister and Tony Solomonides for general support and for comments on earlier stages of this work.", "The comments on this thesis during the various examinations also proved extremely helpful." ], [ "Notation", "Since the purpose of this thesis is to introduce a new mathematical language, I have to introduce many new words and notations.", "When reading the thesis, it may be difficult to keep an overview of all these concepts.", "All newly introduced words are therefore listed in the index at the end, and on page REF there is a list of all symbolic notations used in the text, together with short explanations.", "The new concepts themselves are introduced and explained step by step in the following chapters.", "But first we have to clarify the notations for some basic and well-known concepts which are written differently by different authors." ], [ "Sets and Functions", "The set of positive integers is $\\mathbb {N}= \\lbrace 1, 2, 3, \\dots \\rbrace $ , and the set of non-negative integers is $\\mathbb {N}_0 = \\mathbb {N}\\cup \\lbrace 0 \\rbrace $ .", "If $A$ and $B$ are two sets, then $B^A$ is the set of functions from $A$ to $B$ .", "Therefore we can say of a function $f\\colon A \\rightarrow B$ that $f \\in B^A$ .", "The set $A$ is then called the domain of $f$ , and we will write it $\\mathop {\\mathrm {dom}}f$ .", "Thus for the specific function $f$ just mentioned we have $\\mathop {\\mathrm {dom}}f = A$ .", "There are two conventions in use for the symbol of set inclusion, $\\subset $ .", "They differ in the case where the two sets to be compared are equal.", "Here we will use the convention that $A \\subset B$ means that $A$ is a proper subset of $B$ .", "If we want to include the case that $A = B$ , we write $A \\subseteq B$ .", "A useful property of functions between sets is monotonity.", "Let $F$ be a function that maps subsets of $A$ to subsets of $B$ .", "Then we will say that $F$ is monotone if $F(a) \\subseteq F(a^{\\prime })$ whenever $a\\subseteq a^{\\prime }$ .", "Similarly, a property $P$ of subsets of a set $A$ is monotone if, when $a \\subseteq A$ has property $P$ and $a^{\\prime }$ is a set with $a \\subseteq a^{\\prime } \\subseteq A$ , then $a^{\\prime }$ has property $P$ .", "Let now $F$ be a monotone function $F$ that maps subsets of $A$ to subsets of $A$ .", "If it also has the property that $a \\subseteq F(a)$ and $F(F(a)) = F(a)$ for all $a\\subseteq A$ , then $F$ is called a closure operator.", "(The last two concepts are taken from order theory [11].)" ], [ "Sequences", "We will work very often with finite sequences of arbitrary objects.", "Let $A$ be a set.", "An $A$ -sequence of length $\\ell $ is then an $\\ell $ -tuple of elements of $A$ .", "As it is usual in formal language theory, we may write a sequence as a formal product of its elements.", "So if $a = (\\alpha _1, \\dots , \\alpha _\\ell )$ is an element of $A^\\ell $ , then it can also be written as $\\alpha _1 \\dots \\alpha _\\ell $ .", "This automatically leads to the notion of a product of $A$ -sequences, defined by concatenation.", "If $b = \\beta _1 \\dots \\beta _m \\in A^m$ is another $A$ -sequence, then their product is $a b = \\alpha _1 \\dots \\alpha _\\ell \\beta _1 \\dots \\beta _m\\in A^{\\ell + m}\\,.$ It is easier to work with sequences if one does not always have to refer to its elements.", "We therefore introduce now a small arithmetic for sequences, beginning with the product just defined.", "It introduces a semigroup structure in the set of all sequences, therefore it is natural to introduce an empty sequence.", "It is written $\\lambda $ and will be used a lot, albeit mostly under another name.", "Then, since it is useful to have a product of an $A$ -sequence and an element of $A$ , we identify $A$ with $A^1$ , the set of 1-tuples.", "There is also $A^0$ , the set that only contains $\\lambda $ .", "With these notations we can introduce a name for the set of all $A$ -sequences.", "It is called the Kleene closure [26] of $A$ and has the algebraic structure of a monoid, $A^ = \\bigcup _{\\ell \\ge 0} A^\\ell \\,.$ We will also use other notations and notions that are related to products, without making much fuss about it.", "One example for this is the use of exponents, another the concept of the decomposition of a sequence: If $a \\in A^$ and there are $b$ , $c \\in A^$ with $a =b c$ , then we will speak of this equation as the decomposition of $a$ into $b$ and $c$ .", "We will use this as a way to introduce the variables $b$ and $c$ without explicitly mentioning that they are elements of $A^$ .", "Note also that if e. g. $a = b c$ and of the two factors of $a$ only $b$ is known, then this already determines $c$ .", "We will use this as a way to introduce $c$ .", "A concept that we have already used implicitly is the length of a sequence.", "We will now introduce a notation for it: if $a \\in A^\\ell $ , then its length is $\\mathopen \|a\\mathclose \| = \\ell $ .", "We will often use the fact that the length of a product $a b$ is $\\mathopen \|a b\\mathclose \| =\\mathopen \|a\\mathclose \| + \\mathopen \|b\\mathclose \|$ ." ], [ "Sequences and Functions", "The identification of $A$ -sequences with tuples makes another simplification possible.", "A function $f \\colon A^n \\rightarrow B$ can be viewed as taking $n$ parameters from the set $A$ and mapping them to an element of the set $B$ .", "In this case we will encounter expressions of the form $f(\\alpha _1,\\dots , \\alpha _n)$ , where $\\alpha _1$ , ..., $\\alpha _n$ are elements of $A$ .", "But with the definitions above, $f$ is also a function that maps $A$ -sequences of length $n$ to $B$ .", "Then we can use expressions of the form $f(a)$ instead, with an $a \\in A^n$ , for example with $a =\\alpha _1 \\dots \\alpha _n$ .", "This simplification becomes especially useful if $f$ is the transition function of a one-dimensional cellular automaton.", "It is especially convenient if $a$ is a product of sequences, say $a = b c$ with $b \\in A^\\ell $ and $c \\in A^{n-\\ell }$ .", "We can then write terms like $f(b c)$ instead of much more voluminous expressions like $f(\\beta _1, \\dots ,\\beta _\\ell , \\gamma _1, \\dots , \\gamma _{n-\\ell })$ ." ], [ "Background", "In this chapter I describe in greater detail how this thesis relates to cellular automata research in general.", "I also describe the relation of this work to other kinds of research that inspired it and how they influenced it." ], [ "Structures in Cellular Automata", "There are several approaches in use by with which researchers try to get an understanding of the space-time structures that occur in the evolution of one-dimensional cellular automata.", "To give an overview I will now describe some of these works.", "Besides the projects that are directly concerned with pattern formation I will also describe research that has the description of patterns as its main theme." ], [ "Turing's Work", "The ancestor of all mathematical research about pattern formation is certainly Alan Turing's paper on the chemical basis of morphogenesis [57].", "From the viewpoint of cellular automata, there are may similarities: Turing worked with a ring of cells that have only knowledge of their direct neighbours, he stressed the necessity of a randomised initial configuration, and he found that his setup created periodic patterns.", "He even did a computational (but not computer) experiment.", "On the other hand, his time parameter was continuous, and the state of his cells was characterised by two or three continuous parameter.", "The view that cellular automata are a good model to study pattern formation still had to wait for some time." ], [ "Triangles", "For any research on pattern formation it is useful to find a kind of structure that occurs in many cellular automata.", "This enhances the probability that the results of the research are applicable to many kinds of automata.", "Different approaches on pattern formation can therefore be classified by the kinds of patterns on which they concentrate.", "Triangular structures appear in many one-dimensional cellular automata when they are run from random initial configurations.", "It is therefore natural to use them as the building block for the description of more complex structures.", "There is one such approach that uses triangles as building blocks for larger structures [37], [43].", "It currently concentrates on Rule 110.", "This is one of the cellular automata that have been studied in great detail.", "It has a very complex behaviour and became even more interesting after Matthew Cook had proved its support for universal computation [6].", "The triangles in Rule 110 are the building blocks of larger structures.", "They are therefore represented by “tiles”, which are subsets of the two-dimensional plane.", "The development of the cellular automaton can therefore be understood by a covering of the two-dimensional plane without a gap.", "From the work with these tiles one can therefore derive the possible periodic patterns in a cellular automaton, especially candidates for the ether and for particles [40]." ], [ "Tilings", "A tiling approach somewhat similar to this is used by Ollinger and Richard [48], [53] to express the interactions of particles under Rule 110.", "It uses this approach to express the behaviour of the cellular automaton in terms of particles and collisions.", "There are “tiles” which represent pieces of the ether, others which represent the movement of a particle over a finite amount of time, and others that represent the collision of two or more particles.", "A tiling of the two-dimensional plane that corresponds to the space-time diagram of a cellular automaton is then represented in an abstract form by “a planar map whose vertices are labeled by collisions and edges by particles” [53].", "These graphs are then used to represent complex interactions between particles, especially by Richard [53] to understand Rule 110.", "The method is however applicable to cellular automata in general." ], [ "Replicating Patterns", "Another specialised approach to express the large-scale structure for a specific class of cellular automata concerns those rules which support replication.", "This class is a subclass of those rules that have a quiet state.", "In them one can look at localised patterns that consist of a finite number of cells in non-quiet states, while all the other cells are quiet.", "Replication then occurs in rules under which a small localised pattern in an initial configuration later reappears as several copies.", "These too then replicate, and the evolution of such a pattern in a one-dimensional cellular automaton generates a fractal-like structure, a generalisation of the Sierpiǹski triangle.", "Gravner and Griffeath [21] give a formal definition for replication in one-dimensional automata and then search among other things for replicating patterns under Rule 22.", "In another paper, by Gravner, Gliner and Pelfrey [20], several transition rules are investigated for their replicating patterns." ], [ "Domains and Defects", "In several articles the configuration of the cellular automaton is decomposed into regions with a regular structure and defects between them [5], [27], [29], [30].", "When the initial configuration is chosen at random, the defects usually take a random walk.", "From time to time two of them collide and annihilate each other, which enlarges the regular regions.", "This way the state of the cellular automaton becomes more ordered over time, a phenomenon that has some similarities with ether formation.", "There is a more theoretical view of these phenomena, in a paper by Eloranta [15], which yields rigorous results in a simpler case.", "In it, the set of states of the cellular automaton is divided into two subsets, $S$ and $T$ , such that the next state of a cell the neighbours of which are all elements of $S$ is another element of $S$ , and the same is true for $T$ .", "The author then investigates the behaviour of the boundary between a region of $S$ cells and a region of $T$ cells in which the cell states were chosen at random.", "He finds that the boundary moves either deterministically with maximal speed or it is a random walk, and that it is possible to give explicit, albeit complicated expressions for the speed of the walk.", "A similar pattern of self-organisation occurs in cellular automata with an ether and particles, as in Rule 54 [4] and Rule 110 [32].", "In these automata an ether forms that is disrupted by particles; the particles move and collide and sometimes destroy each other.", "While the transition rule of these automata is deterministic, the number of particles behaves nevertheless in these automata as if the collisions and decays occurred at random.", "In both papers a power law is found by the computer simulations.", "In the paper by Li and Nordahl [32] it concerns the dependence of the density of particles over time, while Boccara, Nasser and Roger [4] measure the density of a specific particle.", "How does one define particles and background?", "Mostly it was obvious to the researchers, but there are systematic approaches.", "The method of “computational mechanics” by Crutchfield and Hanson [7], [8], [23] is a systematic approach that allows, among other things, to divide the configuration of the cellular automaton into regular domains and the domain walls between them.", "A domain wall may move, therefore particles count as domain walls.", "A domain is in the simplest case a spatially periodic pattern that is preserved by the transition rule, so the ether counts as a domain.", "There also exist more complex domains, and the authors have found a way to identify them mechanically by a program.", "Then it is possible to create another finite automaton that classifies the cells as belonging either to a domain or one of the domain wall.", "This allows to show simplified pictures of the often very complicated evolution diagrams.", "Further research in this direction has been done by Marcus Pivato [49], [50].", "Here, too, the aim is to divide the cellular evolution into different regions with different behaviour, again in the form of patterns and defects, but with finer subdivisions." ], [ "Grouping and Supercells", "Another method to describe large-scale structures simply ignores the structures that arise in the evolution of the cellular automaton.", "It uses “grouping” operations for the classification of cellular automata [13], [14], [34], [42], [47].", "In it the cells of the automaton are arranged in blocks of $n$ cells, and one then considers the cellular automaton that consists of these “supercells”.", "One also considers transition rules that aggregate several's time steps into one.", "This way one can establish equivalences between automata and introduce a partial order between them in terms of the complicatedness of their behaviour.", "Among these works the most elaborate is the work of Delorme, Mazoyer, Ollinger and Theyssier [13], [14].", "In it the authors give a formal definition for the generalised grouping operations and then prove theorems about them in an abstract way.", "They also define three concrete grouping operations, find some equivalence classes of one-dimensional automata under these operations and prove how they are related in the partial order defined by the grouping operation." ], [ "Global Behaviour", "All this work with local structures in cellular evolution has also as its goal the understanding of cellular automata and to classify them by their behaviour.", "The first approach of this kind that found greater resonance was Wolfram's [60] classification.", "It divides the cellular automata into four classes according to the behaviour they show when starting from a random initial configuration—in other words, by their ability for self-organisation.", "However, this classification scheme is not decidable, as Culik and Yu [9] showed.", "Another point is that only four classes provide only a very small amount of information about the cellular automata—especially because the automata with nontrivial behaviour end up in only two of them.", "For this and other reasons, the business of finding classification schemes for cellular automata is still going on actively.", "A recent survey [35] lists 18 different classification schemes, just for the elementary cellular automata." ], [ "Physics as Metaphor and Model", "As we have seen in the introduction, a cellular automaton can be understood as the simulation of a physical system.", "The nature of this system is however the subject of some confusion: Is it Newtonian or is it relativistic—and what role does such an old-fashioned concept as the ether play?" ], [ "Newtonian and Relativistic Physics", "Since a cellular automaton is only a rough approximation to a physical system, we have a certain amount of freedom in our interpretation.", "We can choose what kind of physics our cellular automaton should resemble.", "The formalism of Flexible Time is an attempt to bring a relativistic interpretation into the cellular automata, which have before mostly interpreted in a Newtonian fashion.", "A sign of the Newtonian viewpoint is the existence of an universal clock.", "In the usual formulation, a one-dimensional cellular automaton consists of an infinite line of cells, and they evolve in discrete time steps.", "Time passes therefore at every point in the same way.", "The central point of Relativity, on the other hand, is the finite maximal speed with which signals can propagate.", "In a cellular automaton we also have a finite maximal speed: It is given by the radius of the transition rule.", "If the transition rule has radius $r$ , then the state of a cell can influence in the next time step only the cells at most $r$ places to the left or the right.", "The analogy has been known for a long time: In the context of the Game of Life, this maximal speed has already been called by J. H. Conway the “speed of light” [17].", "We can use the analogy to let the cellular automaton play the role of the universe of Special Relativity.", "We can take this analogy a step further.", "As in Relativity, when there is no global concept of time, causality becomes important.", "For cellular automata, causality can become the question, “If I change the state of one cell in the initial configuration, which cells change their state in later time steps?” This has been asked e. g. by Wolfram [61].", "In this thesis, the dual question becomes important, “If the states of only a finite number of cells are known in a cellular automaton, the states of which other cells can be determined from this knowledge?” This question will lead to the concept of the closure in Definition REF .", "We can maintain the standpoint that the set of all cells at a given time is not such an important concept.", "After all, each cell knows only about a finite number of its direct neighbours.", "The concept of a configuration, consisting of all cells at a time step, is therefore nothing which one is forced to use.", "We have, as in Relativity, a freedom to choose which events we consider as occurring at the same time.", "In Relativity, they form a “space-like” set.", "In Flexible Time, we will speak of achronal situations.", "We only have the requirement that the events that can influence each other causally cannot be part of the same time slice.$\\endcsname $This requirement is broken a little bit in achronal situations, but it is correct in the large scale.", "Then we have the flexibility that allows us to follow more easily the structures that occur in the evolution of a cellular automaton." ], [ "Space-time", "Another important concept that became popular through Relativity is that of space-time.", "We will used heavily the freedom that it provides.", "The space-time viewpoint for cellular automata is actually quite old.", "An early example occurs in Konrad Zuse's article about Calculating Space [62].", "Here, in Figures 9 and 10, the author uses a mode of display in which events from different times are displayed together.", "This way the movement of a particle can be shown, even though it extends over several time steps.", "This is however an informal use of a flexible time; I have not seen diagrams of the same style elsewhere.", "There is however an example where events from different times occur naturally during a computation of a cellular automaton.", "William Gosper [18] uses such a scheme to compute the evolution under the Game of Life (or another two-dimensional cellular automaton) in a faster way.", "One could view his scheme as a form of Flexible Time in two dimensions—albeit one in which all situations are based on squares with an edge length that is a power of 2.", "This work was an important inspiration for me." ], [ "The Ether and Other Muddled Metaphors", "There still remains the question which role the old-fashioned concept of the ether plays in such highly modern physics.", "A part of the answer is that the word is already in use: The name “ether” has apparently been introduced by Matthew Cook [6] for the regular pattern in Rule 110, and it has been used by other authors too.", "We can however take the concept of the ether a bit more seriously, as the physicists of the 19th century did.", "For them, the ether formed a background on which signals travel.", "The ether was however specifically invented to support the transmission of waves, for which there is no analogue in the context of cellular automata.", "We do have particles that move in the ether, but there is no ether at the place where the particle is located.", "To have a true ether, we would need an analogy to ether vibrations as they were thought to occur in the physical ether.", "To my knowledge, nobody has attempted such an analogy.", "I therefore believe that we should take the analogy to the ether—in contrast to that with Relativity—not too seriously.", "If one keeps this in mind, even an inexact metaphor can serve as support for the intuition and help to find names for the phenomena that occur in cellular automata.", "In case of the ether, I have done so in Chapter , where I speak of “pure” and “disturbed” ether (the disturbances being the particles), thus using exactly that analogy I just have rejected as being not exact.", "Another incongruent use of physical metaphors is the use of the word “particle”.", "It is nowadays a common word for a localised structure in a cellular automaton that moves with a constant speed.", "The name is especially used for a localised structure that stays in its place, like the static structures that occur under Rule 54 [4].", "One of the earliest uses must be again Zuse [62], who explicitly set out to simulate physical particles with cellular automata.", "The particle metaphor is nowadays used by many authors (and also in this work), but it is not a faithful image of, say, elementary particles, or Newtonian idealised point particles.", "Among the features that are generally missing are an analogy to mass or impulse, or to any kind of conservation theorems.", "What remains is a kind of “topological” image of physical particles, in which the particles move in straight lines and interact only when they collide, but there are no general laws about that what is the result of the collision.", "Once again this is a metaphor that should not be followed too far." ], [ "Relation to Logic and Language", "As the subject of this thesis is the construction of a language for easier mathematical reasoning, I have to name other projects that are related to mathematical languages and their construction.", "Most of them have provided context or direction for this project." ], [ "Combinatory Logic", "The structure of the resulting reaction system has some similarities with the systems used in Combinatory Logic [10], [24].", "One of the motivation that lead to the introduction of Combinatory Logic was the analysis of the substitution process in formulas [10].", "The formal process with which a term is substituted for a variable in a formula is quite complicated, especially if the formula may contain free and bound variables.", "In Combinatory Logic, the substitution process is decomposed into elementary steps, which consists of purely textual substitutions.", "(See e. g.", "[24].)", "This concentration on elementary, textual substitutions in Combinatory Logic served as a model for the development of Flexible Time.", "Especially the concepts of applying a reaction (Definition REF ) and confluence (Theorem REF ) have their similarities in Combinatory Logic." ], [ "Development of a Language", "There are many predecessors for the idea to develop a language that helps us to think more efficiently about cellular automata.", "Among the first, and certainly the most illustrious, was Gottfried Wilhelm Leibniz with his project of a “universal characteristic”.", "This was to be an ideal language and a general symbolic method or both, because there is a certain ambiguity in Leibniz's writings [55].", "In the first interpretation, the language should consist of “signs which process a determinate content and exactly correspond in their structure to the analysis of thought” [55], in the second it would be a symbolic calculus, an “instrument to reason” [55].", "In this second aspect, Leibniz's work is widely seen as a predecessor to formal logic.", "(A rare concrete example for his thoughts about formal reasoning looks to modern eyes like a formalisation of set inclusion or propositional calculus [12].)", "The first aspect emphasises the idea the signs of the language should correspond to the concepts of thought in a simple way.", "This is an aspect that is mostly ignored in the theory of formal systems, but not by Leibniz: His symbolism for integration was clearly designed with the intention in mind to find symbols that aid thought.", "The resulting mathematical language has always been seen as widely superior to Newton's version [12].However, to my knowledge, Leibniz seems not to have understood the formalism of calculus as a part of his project of finding an universal characteristic.", "So far the current thesis could be seen as a part of Leibniz's project, but there are differences.", "First, Leibniz had viewed the signs of his universal characteristic as the most primitive concepts, and believed that they could be found once and for all.", "Second, he imagined his universal language as something complete, encompassing all human knowledge.", "A growing language, intended for a small subuniverse of mathematics, would not be his intention.", "In order to find a model for this kind of project we need to look into a direction that at first seems to be completely unrelated: the construction of languages for the communication with extraterrestrials.", "There is a program outlined by Lancelot Hogben [25] on how to establish communication with an extraterrestrial civilisation via radio signals.", "To establish a means of communication, and a common vocabulary of concepts, “lessons” are sent out to the extraterrestrials, starting with numbers and arithmetic and then building up on this base increasingly complex concepts.", "Hogben's paper is only a sketch of such a program.", "The most elaborate implementation is certainly Hans Freudenthal's “Lincos” [16], in which he introduces step by step the concepts for mathematics, time, basic human behaviour and elementary physics.", "But this description is misleading in one point: Freudenthal's primary interest was to create a logical language that was actually usable for communication, and in order to do this he used interstellar communication as an example problem.", "This is then the point where a project like Freudenthal's becomes a model for works like this thesis.", "We have here namely an example for a language that grows step by step from examples, which is never complete, and which at every step of its development can only access a limited set of concepts.", "It also sets an example by requiring a concrete example to let the language grow." ], [ "Influence on this Work", "As Freudenthal needed a communication problem to develop a language for communication, we will need a self-organisation problem to develop a language about self-organisation and structures.", "Another lesson from Freudenthal's work is to let the language evolve step by step, from simple to complex concepts.", "For this thesis this means that at the beginning the concepts are quite general and are valid for every one-dimensional cellular automaton.", "Step by step, by the amount that we learn about the theory and its abilities, the range of the definitions and theorems becomes more restricted, but in exchange they become more powerful, until finally they offer insight into ether formation under Rule 54." ], [ "Cellular Evolution", "This chapter starts with the definition of the basic terms that are needed to speak about one-dimensional automata, and then introduces concepts that captures their development over time.", "It finishes with theorems about that what can be said about the development of a cellular automaton when only a part of its cells is known." ], [ "One-dimensional Cellular Automata", "Imagine the cellular automaton as a physical object.", "It consists of an infinite row of cells.", "The positions of the cells are integers.", "The cells are simple machines with a finite amount of memory, and they are all equal.", "Two cells may differ only by the content of their memory.", "The possible states of a cell are elements of the finite state set $\\Sigma $ .", "There is a function $c \\colon \\mathbb {Z}\\rightarrow \\Sigma $ that maps the position of a cell to its state.", "Such a function is called here a configuration of the cellular automaton.", "The set of all configurations of a cellular automaton is therefore the set $\\Sigma ^\\mathbb {Z}$ .", "Cellular automata evolve over time.", "Time for cellular automata is discrete and the time coordinate takes integer values.", "We speak of time steps.", "The behaviour of a cellular automaton is given by a local transition rule $\\varphi $ with radius $r$ , $\\varphi \\colon \\Sigma ^{2r+1} \\rightarrow \\Sigma \\,.$ It maps the neighbourhood of a cell to the state of it one time step later.", "To see in which way, we need a notion for the collection of all configurations of a cellular automaton at all time steps.", "I call such a collection the evolution sequence of the cellular automaton.In the theory of dynamical systems this is often called the orbit.", "But the definition of this term seems to vary between authors.", "For Alligood, Sauer and Yorke [2], the orbit is a set, while for Strogatz [56] it is a sequence.", "However, in the case of cellular automata a sequence is the more natural choice.", "It is an infinite sequence $(c_t)_{t \\ge 0}$ of configurations.", "We will only consider evolution sequences that belong to a specific local transition rule $\\varphi $ .", "In such an evolution the configuration $c_0$ must be specified in advance: it is the initial configuration of the sequence.", "Each later configuration $c_t$ , with $t \\ge 1$ , depends by the global transition rule $c_t(x) = \\varphi (c_{t-1}(x - r), \\dots , c_{t-1}(x + r))\\qquad \\text{for all $x \\in \\mathbb {Z}$}.$ on its predecessor configuration $c_{t-1}$ ." ], [ "Conventions", "We can specify a one-dimensional cellular automaton completely by specifying $\\Sigma $ , $r$ and $\\varphi $ .", "Since we will work almost always with one specific cellular automaton at a time, we will from now on keep $\\Sigma $ , $\\varphi $ and $r$ fixed and not refer to it in most of the notation.", "Furthermore, since the subject of this thesis are one-dimensional automata, we will from now on in most cases omit the adjective “one-dimensional”." ], [ "Radius Invariance", "It is possible that two different local transition rules lead to the same global rule.", "A pair of such equivalent transition rules is easy to construct: For a given local transition rule $\\varphi $ with radius $r$ , let $\\varphi ^{\\prime }\\colon \\Sigma ^{2r^{\\prime } +1} \\rightarrow \\Sigma $ be a rule with radius $r^{\\prime } >r$ such that $\\varphi ^{\\prime }(\\sigma _{-r^{\\prime }}, \\dots , \\sigma _{r^{\\prime }}) =\\varphi (\\sigma _{-r}, \\dots , \\sigma _r)\\qquad \\text{for all $\\sigma _{-r^{\\prime }}, \\dots , \\sigma _{r^{\\prime }} \\in \\Sigma $.", "}$ Then $\\varphi ^{\\prime }$ , which ignores the states of the additional cells, has the same global transition rule as $\\varphi $ .", "We call such a $\\varphi ^{\\prime }$ the extension of $\\varphi $ to the radius $r^{\\prime }$ .", "It is easy to see that if two local transition rules lead to the same global rule, then one must be the extension of the other one.", "The centre of our interest in a cellular automaton is the behaviour of its cells, not its local transition rule.", "Therefore we will view here cellular automata with the same global transition rule as equivalent, since they have the same evolutions.", "Nevertheless the local transition rule provides an easy way to specify the properties of a cellular automaton.", "So we will use it, but we will require that the properties and functions defined for cellular automata are invariant of the radius of its local transition rule, in the following sense: Definition 3.1 (Radius Invariance) A property of a cellular automaton is radius-invariant when it is true for a local transition rule $\\varphi $ if and only if it is true for all its extensions.", "What is then the radius of the cellular automaton itself?", "We will here allow that a cellular automaton has more than one radius: A number $r$ is a radius for a cellular automaton if there exists a local transition rule with radius $r$ that generates its global transition function." ], [ "Cellular Processes", "At this point, our only tool to analyse the concrete behaviour of a cellular automaton—i. e.", "when its initial configuration $c_0$ is given—is its evolution sequence $(c_t)_{t \\ge 0}$ .", "But this is for many applications not enough.", "It requires the knowledge of infinitely many cell states, which is too much when our interest is only on the development of a specific localised pattern.", "A better way of formalisation for cellular evolution is inspired by the way cellular evolution is usually shown in pictures." ], [ "Space-time Diagrams", "The evolution of a cellular automaton is in general shown by a diagram like that in Figure REF .", "Figure: Space-time diagram of an evolution under Rule 54.It is a rectangular array in which the states of the cells are shown by squares of different colours.", "In this picture, which shows the evolution of an elementary cellular automaton, the colours are white for cells in state 0 and black for cells in state 1.", "This colour convention is kept—sometimes in a modified way—in all the other pictures of cellular evolution that occur in this thesis.", "The place of the square in the diagram specifies its place in space and time.", "The $x$ -coordinate of a square determines its location in space, and its horizontal position marks the point in time to which it refers.", "In this thesis I use the physicist's convention for space-time diagrams in which time runs upward.$\\endcsname $There is also a strong tradition to draw the diagrams with time running downward.", "It probably has its origin in the time when one-dimensional cellular were simulated in a computer and then printed with a line printer, with one letter for every cell state and the cells of one time step in a line.", "Then the natural way to display the evolution is to print them in the way they are computed, and in the resulting diagram time runs downward.", "(For an example see Wolfram [59].)", "For some reason, the physicist's convention is also the preferred convention used by French authors on cellular automata.", "Each row in such a diagram is then part of a configuration, and the partial image of configuration $c_{t+1}$ is directly above that of $c_t$ .", "Sometimes we will not need a whole rectangle and draw therefore only a subset of its squares." ], [ "Explanation of Figure ", "The diagram in Figure REF then displays the evolution of a random initial configuration under Rule 54.", "Since it displays a the evolution of an infinite line of cells, there is no wraparound and, in contrast to many other space-time diagrams, the leftmost cell in each time step is not the right neighbour of its rightmost cell." ], [ "Events and Space-time", "We will now introduce a mathematical equivalent to the space-time diagrams, namely the concept of cellular processes.", "It will be able to display any behaviour of the cells and is not restricted to the case that the cells follow a transitions rule.", "The first step in defining a mathematical analog of an space-time diagram is to find a representation for a single square.", "A square has a position and a colour, and the colour represents a state.", "The position has a space and a time component, and both are integers.", "Therefore the following definition is reasonable.", "Definition 3.2 (Cellular Event) A cellular event is a pair $(p, \\sigma ) \\in \\mathbb {Z}^2 \\times \\Sigma $ .", "It consists of a position $p$ and a state $\\sigma $ .", "The first component of $p$ is its time coordinate and the second its space coordinate.", "The pair $(p, \\sigma )$ will be usually written $[p]\\sigma $ .", "With this definition, negative time values are explicitly allowed.", "We will need them later, when situations are introduced in Definition REF , because for them it is natural to refer to events with negative time values.", "Viewed from our current standpoint, this gives us the choice to let the evolution of the cellular automaton begin at an arbitrary time step, not just at time 0.", "The name “event” has been taken from Relativity theory.", "There it stands for a point of four-dimensional space-time (see Wald [58]).", "I have here extended it to mean “space${}+{}$ time${}+{}$ physical conditions at this point”, first because I have not found another word for this idea, and second because it then harmonises with the use of “event” in everyday language.", "Using another word from Relativity [58], we will call the position $p$ of an event $[p] \\sigma $ a space-time point.", "The convention that the first component of a space-time point is the time and the second the space coordinate is also from Relativity.", "It extends in a natural way to all other cases where an element of $\\mathbb {Z}^2$ is used for the same purpose in the context of cellular automata.", "I use here concepts from Relativity because Relativity theory has already well-developed concepts to treat space and time in an unified way.", "With the notations $p_T = t \\qquad \\text{and}\\qquad p_X = x$ we will refer to the components of a space-time point $p \\in \\mathbb {Z}^2$ with $p = (t, x)$ .", "The use of capital letters for this purpose in unusual, but lower case $t$ is already used as time variable and occurs also as index letter.", "As a kind of inverse to the component notation we will need the unit vectors of space-time, $T= (1, 0) \\qquad \\text{and}\\qquad X= (0, 1),$ especially to refer to differences between space-time points in a more abstract way.", "A final set of conventions refer to the “point” component of an event $[p] \\sigma $ .", "If $p = (t,x)$ , we may write $[p]$ as $[t, x]$ .", "If $t = 0$ , we may abbreviate $[t, x]$ to $[x]$ .", "We also write $[p][q]$ for $[p + q]$ .", "I list them here only for reasons of completeness.", "They will become useful later in the context of situations." ], [ "Processes", "Next we need a mathematical object that resembles a whole space-time diagram.", "There are two possibilities.", "We may interpret the diagram as the picture of a function that maps a space-time point to a cell state.", "Then the correct way to represent a space-time diagram would be a map from a subset of $\\mathbb {Z}^2$ to $\\Sigma $ .", "We will however also need to express unions and intersections of cellular processes, something that is easier to express if a cellular process were a set of events.", "Therefore I will introduce now a concept that intends to unite the good properties of functions and sets.There is a viewpoint in mathematics that functions are sets, but it is apparently not shared by everyone.", "Therefore I do here the unification explicitly.", "(I had used the other approach in my previous paper [51].)", "Let $A$ and $B$ be two sets.", "I call a set $F \\subseteq A \\times B$ function-like if there is a set $D \\subseteq A$ and a function $f\\colon D \\rightarrow B$ such that $F = \\lbrace \\, (a, f(a)) \\colon a \\in D \\,\\rbrace \\,.$ In notation we will treat function-like sets like functions.", "The term $F(a)$ stands for the element $b \\in B$ for which $(a, b) \\in F$ .", "There is exactly one such $b$ because $F$ is function-like.", "The domain of $F$ is the set $\\mathop {\\mathrm {dom}}F = \\lbrace \\, a \\colon \\exists b \\in B \\colon (a, b) \\in F \\,\\rbrace \\,.$ For the $F$ of equation (REF ) we have $\\mathop {\\mathrm {dom}}F = D$ .", "The restriction of $F$ to a set $A^{\\prime } \\subseteq A$ is the set $F\|_{A^{\\prime }} = \\lbrace \\, (a, b) \\in F \\colon a \\in A^{\\prime } \\,\\rbrace ,$ which is also function-like.", "Then we can define cellular processes as a special kind of function-like sets.", "Together with the cellular processes we define also a short notation for the subset of all events at a certain time.", "Definition 3.3 (Cellular Process) A cellular process is a function-like set of cellular events.", "The set of all cellular processes is called $\\mathcal {P}$ .", "If $\\pi \\in \\mathcal {P}$ and $t \\in \\mathbb {Z}$ , then its restriction to events at time $t$ , its time slice, is the cellular process $\\pi ^{(t)} = \\lbrace \\, ([t, x] \\sigma \\in \\pi \\colon x \\in \\mathbb {Z} \\,\\rbrace \\,.$" ], [ "Compatibility", "Next we consider the set-theoretic operations for cellular processes.", "Here we must know whether the result of a set-theoretic operation applied to one or more cellular processes is again a cellular process.", "This is no problem with subset formation and intersection: since the subset of a cellular process is again a cellular process, the intersection of two processes is a process too.", "The only exception is the union of cellular processes.", "It is not always a function-like set.", "An exception may occur when two cellular processes $\\pi $ , $\\theta \\in \\mathcal {P}$ have domains that overlap in a point $p$ .", "It is then possible that there are events $[p]\\sigma \\in \\pi $ and $[p]\\tau \\in \\theta $ with $\\sigma \\ne \\tau $ .", "Then the set $\\pi \\cup \\theta $ exists, but it is no longer function-like.", "If it were, there would be a function $f \\colon \\mathop {\\mathrm {dom}}\\pi \\cup \\mathop {\\mathrm {dom}}\\theta \\rightarrow \\Sigma $ with $f(p)= \\sigma $ and $f(p) = \\tau $ at the same time, which is impossible.", "If this does not happen, we say that $\\pi $ and $\\theta $ are compatible: Definition 3.4 (Compatibility) Two cellular processes $\\pi $ , $\\theta \\in \\mathcal {P}$ are compatible if $\\pi (p) = \\theta (p)\\qquad \\text{for all $p \\in \\mathop {\\mathrm {dom}}\\pi \\cap \\mathop {\\mathrm {dom}}\\theta $.", "}$ We write this as $\\pi \\mathrel \\mathrm {comp}\\theta $ .", "The question of compatibility plays an important role in this theory of cellular processes.", "In the rest of this text we must check very often whether a certain construction is possible, and if it is not, the cause is almost always incompatibility." ], [ "Evolution", "We will now define what it means when a cellular process follows a transition rule.", "The construction that will be defined at the end must generalise the way in which an evolution sequence depends on its initial configuration.", "This is because currently the only way the behaviour of a cellular automaton is formally defined is via the evolution sequence.", "As an intermediate step and to verify later the definition, we will translate now evolution sequences and configurations into the language of cellular processes.", "For this, let $(c_t)_{t \\ge 0}$ be an evolution sequence of a cellular automaton.", "Then there exists a cellular process $\\gamma = \\lbrace \\, [t, x]c_t(x) \\colon t, x \\in \\mathbb {Z} \\,\\rbrace $ that contains all the information in $(c_t)_{t \\ge 0}$ .", "The information of every configuration $c_t$ in the sequence is contained in the time slice $\\gamma ^{(t)} = \\lbrace \\, [t, x]c_t(x) \\colon x \\in \\mathbb {Z} \\,\\rbrace $ of $\\gamma $ .", "Our task is then to find a construction that, among other things, extends the initial time slice $\\gamma ^{(0)}$ to the whole process $\\gamma $ , in the same way as the global transition rule (REF ) extends the initial configuration $c_0$ to $(c_t)_{t \\ge 0}$ .", "We will call this construction the closure of a process, because it will turn out to be a closure operator as defined on page REF .", "The closure will be defined in two steps.", "First we consider the case of a single event.", "Given a process $\\pi $ and a point $p$ , what does it mean that we can reconstruct the state of the event at $p$ from $\\pi $ ?", "If this is the case, we say that the event at $p$ is determined by $\\pi $ .", "What this means exactly will be described in Definition REF .", "As a second step we consider the events that are determined by $\\pi $ , together with the events that are determined by $\\pi $ and them, and so on: together they form the closure of $\\pi $ .", "It will turn out that not every process has a closure.", "The result of the second step is Definition REF .$\\endcsname $The construction introduced here has some similarity with the use of “tiling constraints” to specify the space-time pattern of the cell states in a one-dimensional cellular automaton by Ollinger and Richard [48]." ], [ "Determined Events", "Let $p = (t, x)$ be a space-time point.", "Given a cellular process $\\pi \\in \\mathcal {P}$ and a transition rule $\\varphi $ of radius $r$ , what could be the state of the event at $p$ ?", "We will answer this question first for the case of $\\pi = \\gamma $ , with $\\gamma $ as in (REF ).", "In $\\gamma $ , the states of an event at time $t > 0$ depend, by the global transition rule, on the events at time $t - 1$ .", "We will then say that the events of $\\gamma \\setminus \\gamma ^{(0)}$ are determined by $\\gamma $ .", "From this we will now distill concepts that tell us how a transition rule $\\varphi $ acts on a cellular process.", "The first goal is then to express the global transition rule for evolution sequences in a form that is meaningful for processes like $\\gamma $ .", "The point $p = (t, x)$ is the coordinate of the cell at position $x$ and time $t$ .", "The state of a cell at time $t$ depends on the states of the cells in its neighbourhood at time $t - 1$ .", "So we must consider the neighbourhood of the point $p - T= (t - 1, x)$ to compute the state of the event at $p$ ." ], [ "Neighbourhoods", "For easier notation we will now first describe the neighbourhood of the point $p$ instead of that of $p -T$ .", "We begin with the neighbourhood of a cell as a set of space-time points, without reference to a cellular process.", "Since the transition rule has radius $r$ , the cell in the cellular automaton at position $x$ has a neighbourhood that consists of the cells at positions $x -r$ , ..., $x + r$ .", "At a time $t$ , these cells are located at the space-time points $(t, x - r)$ , ..., $(t, x + r)$ .", "The central cell itself is located at $(t, x)$ , or $p$ .", "Therefore we can say that the neighbourhood of the point $p$ is consists of the points $p - r X$ , ..., $p + r X$ .", "To refer to it we introduce the following definition.", "Definition 3.5 (Neighbourhood Domain) Let $p \\in \\mathbb {Z}^2$ and $r \\in \\mathbb {N}_0$ .", "The neighbourhood domain of $p$ with radius $r$ is the set $N(p, r) = \\lbrace p - r X, \\dots , p + r X\\rbrace \\,.$ Next we must find an expression for the states of those events in $\\gamma $ that are located at the points of $N(p, r)$ .", "We need them not just as a set, but also in their natural order.", "Therefore we express them as the cellular process $\\nu (p, w)$ , defined below.", "In the same way that we can write $[p] \\sigma \\in \\gamma $ to express the fact that in the process $\\gamma $ the event at point $p$ has state $\\sigma $ , we will write $\\nu (p, \\omega _{-r} \\dots \\omega _r) \\subseteq \\gamma $ to express the fact that in $\\gamma $ the events at the points $p - rX$ , ..., $p + r X$ have, respectively, the states $\\omega _{-r}$ , ..., $\\omega _r$ .", "Definition 3.6 (Neighbourhood Process) Let $w = \\omega _{-r} \\dots \\omega _r \\in \\Sigma ^{2r+1}$ .", "The neighbourhood process for $w$ at $p$ is the cellular process $\\nu (p, w) =\\lbrace [p - r X]\\omega _{-r}, \\dots , [p + r X]\\omega _r \\rbrace \\,.$" ], [ "The Transition Rule", "We now return to the computation of the state of $\\gamma $ at $p$ .", "With neighbourhood processes we can express the global transition rule (REF ) for evolution sequences in a new way for cellular processes like for $\\gamma $ .", "A direct translation of (REF ) uses the fact that $\\gamma (t, x) = c_t(x)$ for all $t \\in \\mathbb {N}_0$ and $x \\in \\mathbb {Z}$ .", "We now replace all terms like $c_t(x)$ with terms of the form $\\gamma (t,x)$ and get the formula $\\gamma (t, x) = \\varphi (\\gamma (t-1, x-r), \\dots , \\gamma (t-1, x+r)),$ which is valid for all $t > 0$ and $x \\in \\mathbb {Z}$ .", "With neighbourhood processes this becomes the condition, $\\text{if}\\quad \\nu ((t-1, x), \\omega _{-r} \\dots \\omega _r) \\subseteq \\gamma ,\\quad \\text{then}\\quad \\gamma (t, x) = \\varphi (\\omega _{-r}, \\dots , \\omega _r),$ which can be shortened by using $p = (t, x)$ again and by setting $w =\\omega _{-r} \\dots \\omega _r$ .", "Then it becomes the requirement that $\\text{if}\\quad w \\in \\Sigma ^{2r+1}\\quad \\text{and}\\quad \\nu (p - T, w) \\subseteq \\gamma ,\\quad \\text{then}\\quad \\gamma (p) = \\varphi (w)\\,.$ This is is a formulation of the global transition rule for the process $\\gamma $ .", "It is valid for all $p \\in \\mathbb {Z}^2$ with $p_T > 0$ ." ], [ "Arbitrary Processes", "Now we return to an arbitrary process $\\pi $ .", "We ask which state we should expect for the event at $(t, x)$ , given the information in $\\pi $ .", "To do this we will view $\\pi $ as a window into the evolution of a cellular automaton that follows rule $\\varphi $ .", "If $w = \\omega _{-r} \\dots \\omega _r$ and $\\nu ((t - 1, x), w) \\subseteq \\pi $ , then at the time $t - 1$ the states of the cells in the neighbourhood of the cell at $x$ are $\\omega _{-r}$ , ..., $\\omega _r$ .", "At time $t$ , the state of the cell at $x$ must then be $\\varphi (w)$ .", "This is then the expected state for the event at $(t, x)$ .", "When however $N(p, r) \\lnot \\subseteq \\mathop {\\mathrm {dom}}\\pi $ , then we have not enough information about the evolution to find the state for $p$ in this way.", "Instead we can find a set of possible states: If there is a process $\\pi ^{\\prime } \\supseteq \\pi $ such that $\\nu (p - T, w) \\subseteq \\pi ^{\\prime }$ , then $\\nu (p - T, w)$ is a possible neighbourhood for $p - T$ , and $\\varphi (w)$ is a possible state for the event at $p$ .", "The set of possible states for the event at $p$ is therefore $\\lbrace \\, \\varphi (w) \\colon w \\in \\Sigma ^{2r+1},\\exists \\pi ^{\\prime } \\subseteq \\mathcal {P} \\colon \\nu (p - T, w) \\subseteq \\pi ^{\\prime } \\supseteq \\pi \\,\\rbrace \\,.$ However, as stated here this definition involves an infinite number of processes $\\pi ^{\\prime }$ , which is bad for actual computations.", "We avoid this by choosing only those $\\pi ^{\\prime }$ that contain only as many additional points that $\\nu (p - T, w) \\subseteq \\pi $ .", "Then $\\nu (p, - T, w)$ is compatible with $\\pi $ (Figure REF ).", "This then leads to the following definition, in which $\\pi ^{\\prime }$ does no longer occur explicitly.", "Figure: Determining the possible states for the event at pp.Definition 3.7 (Set of Possible States) Let $\\pi \\in \\mathcal {P}$ be a cellular process and $\\varphi $ be a transition rule for $\\Sigma $ of radius $r$ .", "The set of possible states for the event at $p$ is $S(p, \\pi ) =\\lbrace \\, \\varphi (w) \\colon w \\in \\Sigma ^{2r + 1},\\nu (p - T, w) \\mathrel \\mathrm {comp}\\pi \\,\\rbrace \\,.$ If there is only one possible state for the event at $p$ , then it is determined.", "This is expressed in the following definition.", "Definition 3.8 (Determined Events) Let $\\varphi \\colon \\Sigma ^{2r+1} \\rightarrow \\Sigma $ be a transition rule and $\\pi \\in \\mathcal {P}$ a process.", "If $S(p, \\pi ) = \\lbrace \\sigma \\rbrace ,$ then both the event $[p] \\sigma $ and the point $p$ are determined by $\\pi $ .", "The set of all events that are determined by $\\pi $ is $\\Delta \\pi $ ." ], [ "Non-constant Transition Rules", "Note that if $\\varphi $ is a constant function, then every point of $\\mathbb {Z}^2$ is determined.", "We will therefore restrict the following definitions and theorems to non-constant transition functions, in order to avoid this unintuitive property.", "The following lemma summarises useful properties of determinedness that are only true for a transition rule that is non-constant.", "The statements of this lemma are also the reason why the set $N(p, r)$ gets a special name.", "Lemma 3.9 (Events Determined by a Time Slice) Let $\\varphi $ be a non-constant transition rule for $\\Sigma $ and $\\pi \\in \\mathcal {P}$ a process.", "Then: A point $p$ is determined by $\\pi $ if and only if it is determined by $\\pi \|_{N(p - T, r)}$ .", "If $p$ is determined by $\\pi $ , the set $\\pi \|_{N(p - T, r)}$ is nonempty.", "For every time $t$ , the set $\\Delta \\pi ^{(t)}$ consists only of events at time $t + 1$ .", "The first statement of this lemma expresses again the fact that determinedness is a local property and relies only on a finite number of events.", "The second statement is about causality.", "For it we need to have the view of a cellular automaton as a physical system governed by the “physics” $\\varphi $ .", "Then the events of $\\pi \|_{N(p - T, r)}$ can be understood as the “cause” of the event at $p$ .", "The second statement of the theorem then states that an event at time $t$ which is determined by $\\pi $ is always caused by an event at time $t - 1$ .", "It is also a statement about the maximal speed with which information is transmitted: the state of a cell at time $t$ can only be caused by the cells at most $r$ positions to its left or right.", "The third statement is tailored for its use in connection with the closure of a process, which will be defined next.", "[Proof of the lemma] For the proof of the first statement we note that for every $w \\in \\Sigma ^{2r+1}$ the domain of the neighbourhood process $\\nu (p, w)$ is $N(p, r)$ .", "Therefore the set $S(p, \\pi )$ does actually depend only on $\\pi \|_{N(p - T, r)}$ .", "The knowledge of this part of $\\pi $ is therefore also enough to find out whether $p$ is determined.", "To show the second statement we prove its converse.", "Assume that $\\pi \|_{N(p - T, r)}$ is empty.", "Then every neighbourhood process $\\nu (p - T, w)$ in (REF ) is compatible to $\\pi $ .", "Since $\\varphi $ is non-constant, the set $S(p, \\pi )$ has more than one element.", "Therefore the point $p$ is then not determined by $\\pi $ .", "The third statement then follows from the second." ], [ "The Closure", "Now we can extend the global transition rule (REF ) from $\\gamma $ to arbitrary cellular processes.", "Similar to the way an evolution sequence is generated by always computing the configuration for time $t$ from the configuration for time $t - 1$ , the closure of a cellular process is created from time slices, each of them depending on the previous one, that are finally put together.", "There is however no direct analog to the initial configuration.", "To understand what is meant with the closure of a cellular process, imagine that cells are multicoloured lights that can be switched on or off.", "If a light is switched on, it has one of a finite set of colours.", "A cellular automaton is then an infinite line of such lights, and the colours represent the states of its cells.", "A cellular process $\\pi $ is then a certain light pattern, a rule when to switch the lights on and with which colours.", "The closure of $\\pi $ is another light pattern, where the lights are switched on not only when it is required by $\\pi $ but also depending on the lights that were switched on at the previous time step.", "When the lights switched on at the previous time step determine the state of a cell in the current time step, then this cell is also switched on, besides the cells that are required by $\\pi $ to be switched on.", "A conflict between these two rules is possible: It can happen that the light pattern prescribes one colour for a cell at a certain time and $\\pi $ describes another.", "Then we will say that for this $\\pi $ the closure does not exist.", "To be practical this procedure must have a starting time.", "We will require that there was a time when no event of $\\pi $ happened; otherwise $\\pi $ will have no closure.", "In the following definition we therefore call a cellular process $\\pi $ quiet before $t_0$ if $\\pi ^{(t)} = \\emptyset $ for all $t < t_0$ .", "Definition 3.10 (Closure) Let $\\varphi $ be a non-constant transition rule for $\\Sigma $ and $\\pi \\in \\mathcal {P}$ be a cellular process that is quiet before $t_0$ .", "The closure of $\\pi $ at time $t$ under $\\varphi $ is the process $\\mathop {\\mathrm {cl}}\\nolimits ^{(t)} \\pi $ .", "It exists always when $t \\le t_0$ .", "When $t > t_0$ , the process $\\mathop {\\mathrm {cl}}\\nolimits ^{(t)} \\pi $ exists if $\\mathop {\\mathrm {cl}}\\nolimits ^{(t-1)} \\pi $ exists and $\\pi ^{(t)}$ is compatible with $\\Delta \\mathop {\\mathrm {cl}}\\nolimits ^{(t-1)} \\pi $ .", "It is defined by the recursion $\\mathop {\\mathrm {cl}}\\nolimits ^{(t)} \\pi ={\\left\\lbrace \\begin{array}{ll}\\pi ^{(t)} & \\text{if $t \\le t_0$,} \\\\\\pi ^{(t)} \\cup \\Delta \\mathop {\\mathrm {cl}}\\nolimits ^{(t-1)} \\pi & \\text{if $t > t_0$.}\\end{array}\\right.", "}$ The closure of $\\pi $ under $\\varphi $ exists if all time slices $\\mathop {\\mathrm {cl}}\\nolimits ^{(t)} \\pi $ exist.", "It is the process $\\mathop {\\mathrm {cl}}\\nolimits \\pi = \\bigcup _{t \\in \\mathbb {Z}} \\mathop {\\mathrm {cl}}\\nolimits ^{(t)} \\pi \\,.$ An incidental result of equation (REF ) is that always $(\\mathop {\\mathrm {cl}}\\nolimits \\pi )^{(t)} = \\mathop {\\mathrm {cl}}\\nolimits ^{(t)} \\pi $ .", "This is the way the operator $\\mathop {\\mathrm {cl}}\\nolimits ^{(t)}$ fits into the formalism of time slices of Definition REF .", "There is one act of choice in this definition.", "If a process $\\pi $ is quiet before $t_0$ , then it is also quiet before any time $t^{\\prime }_0 <t_0$ .", "Therefore one can also compute the closure of $\\pi $ using $t^{\\prime }_0$ as starting time.", "But this has no influence on $\\mathop {\\mathrm {cl}}\\nolimits \\pi $ .", "To see this, assume that $\\psi $ is the closure of $\\pi $ as computed with $t_0$ and $\\psi ^{\\prime }$ the closure of $\\pi $ as computed with $t^{\\prime }_0$ .", "Then for $t \\le t_0$ , the first case of (REF ) applies to the computation of $\\psi $ , and we have $\\psi ^{(t)} =\\pi ^{(t)} = \\emptyset $ .", "For $t \\le t^{\\prime }_0$ it also applies to the computation of $\\psi ^{\\prime }$ , so we have then $\\psi ^{\\prime (t)} = \\pi ^{(t)} =\\emptyset $ .", "For $t^{\\prime }_0 < t \\le t_0$ , the second case of (REF ) applies and we can see by induction that then $\\psi ^{\\prime (t)} = \\emptyset $ : If $\\psi ^{\\prime (t-1)} = \\emptyset $ , then $\\psi ^{\\prime (t)}= \\pi ^{(t)} \\cup \\Delta \\psi ^{\\prime (t-1)}= \\emptyset \\cup \\Delta \\emptyset = \\emptyset \\,.$ Therefore for $t \\le t_0$ , $\\psi ^{(t)} = \\psi ^{\\prime (t)}$ .", "At later times, the second part of (REF ) comes into play for both $\\psi $ and $\\psi ^{\\prime }$ to construct the next time slice.", "Therefore for $t > t_0$ the time slice $\\psi ^{(t)}$ exists if and only if $\\psi ^{\\prime (t)}$ exists, and when one of them exists, then $\\psi ^{(t)}= \\psi ^{\\prime (t)}$ .", "This shows then that $\\psi $ exists if and only $\\psi ^{\\prime }$ exists, and if they exist, they are equal.", "In other words, the choice of $t_0$ has no influence on the closure.", "Most properties of the closure are proved by inductions in the style of this proof.", "In it one can also see why the definition of the closure was restricted to non-constant transition rules: This restriction ensured in (REF ) that $\\Delta \\emptyset = \\emptyset $ and thus removed unnecessary complexity." ], [ "Properties of the Closure", "We now will prove some properties of the closure that either become useful later or will provide insight about this concept." ], [ "Shift Invariance", "Like the physical laws which they imitate, the laws of a cellular automaton are independent of an absolute location in space and time.", "In the next chapter we will use this property for a simplification of the formalism; at this point we are mainly concerned with a way to express it for cellular automata.", "Figure: A shifted cellular process.We can express it by the concept of a space-time shifted process (Figure REF ).", "The notation $[p] \\pi $ has been chosen in harmony to the other uses of square brackets in this text.", "Definition 3.11 (Space-time Shift) Let $\\pi \\in \\mathcal {P}$ be a process and $p \\in \\mathbb {Z}^2$ be a space-time point.", "We write for the copy of $\\pi $ that is shifted by $p$ , $[p] \\pi = \\lbrace \\, [p + q] \\sigma \\colon [q] \\sigma \\in \\pi \\,\\rbrace \\,.$ A property of cellular processes is shift-invariant if it is true for a process $\\pi $ if and only if it is true for $[p] \\pi $ .", "A function $F \\colon \\mathcal {P} \\rightarrow \\mathcal {P}$ between cellular processes is shift-invariant if $F([p] \\pi ) = [p]F(\\pi )$ for all $p$ and $\\pi $ .", "Then we can say that $\\Delta $ and $\\mathop {\\mathrm {cl}}\\nolimits $ are two shift-invariant functions.", "In a more informal way we will also say that determinedness is a shift-invariant notion, meaning that $\\pi $ determines $[p]\\sigma $ if and only if $[q] \\pi $ determines $[q + p] \\sigma $ .", "I do not give here a formal proof for these facts: they can easily be verified from their definitions, by checking that they use only differences between space-time points and no absolute coordinates." ], [ "Radius Invariance", "As requested at the beginning of this chapter, the concepts introduced here are radius-invariant.", "Neither the closure nor the set of determined events of a process are dependent of the radius of the transition rule.", "To verify this, the following lemma is sufficient, since the closure is defined with help of the set of determined events, and determined events are defined with help of the set of possible states for a point.", "In the proof we have to distinguish between the sets $S(p, \\pi )$ for the transition rules $\\varphi $ and $\\varphi $ ; we will therefore write the two sets of possible states as $S_\\varphi (p, \\pi )$ and $S_{\\varphi ^{\\prime }}(p,\\pi )$ .", "Lemma 3.12 (Possible States are Radius Invariant) Let $\\varphi \\colon \\Sigma ^{2r+1} \\rightarrow \\Sigma $ be a transition rule and $\\varphi ^{\\prime } \\colon \\Sigma ^{2r^{\\prime }+1} \\rightarrow \\Sigma $ an extension of $\\varphi $ with radius $r^{\\prime } > r$ .", "Let $\\pi \\in \\mathcal {P}$ be a cellular process and $p \\in \\mathbb {Z}^2$ .", "Then $S_\\varphi (p, \\pi ) = S_{\\varphi ^{\\prime }}(p, \\pi )$ .", "Assume that $\\sigma \\in S_\\varphi (p, \\pi )$ .", "Then there are states $\\omega _{-r}$ , ..., $\\omega _r \\in \\Sigma $ such that $\\varphi (\\omega _{-r}, \\dots , \\omega _r) = \\sigma $ and $\\nu (p - T,\\omega _{-r} \\dots \\omega _r) \\mathrel \\mathrm {comp}\\pi $ .", "Now choose the cell states $\\omega _{-r^{\\prime }}$ , ..., $\\omega _{-r-1}$ and $\\omega _{r+1}$ , ...$\\omega _{r^{\\prime }} \\in \\Sigma $ in the following way: If $p - T+ i X\\in \\mathop {\\mathrm {dom}}\\pi $ , then $\\omega _i = \\pi (p - T+ i X)$ ; otherwise $\\omega _i$ is arbitrary.", "Then $\\nu (p - T, \\omega _{-r^{\\prime }} \\dots \\omega _{r^{\\prime }})$ is compatible with $\\pi $ .", "Since $\\varphi ^{\\prime }$ is an extension of $\\varphi $ , we have also $\\varphi ^{\\prime }(\\omega _{-r^{\\prime }} \\dots \\omega _{r^{\\prime }}) = \\varphi (\\omega _{-r}, \\dots , \\omega _r) = \\sigma $ .", "Therefore $\\sigma \\in S_{\\varphi ^{\\prime }}(p, \\pi )$ , which in turn proves that $S_\\varphi (p, \\pi ) \\subseteq S_{\\varphi ^{\\prime }}(p, \\pi )$ .", "Assume that $\\sigma \\in S_{\\varphi ^{\\prime }}(p, \\pi )$ .", "This means that there are states $\\omega _{-r^{\\prime }}$ , ..., $\\omega _{r^{\\prime }} \\in \\Sigma $ such that $\\varphi ^{\\prime }(\\omega _{-r^{\\prime }}, \\dots , \\omega _{r^{\\prime }}) = \\sigma $ and $\\nu (p - T,\\omega _{-r^{\\prime }} \\dots \\omega _{r^{\\prime }}) \\mathrel \\mathrm {comp}\\pi $ .", "Then $\\varphi (\\omega _{-r},\\dots , \\omega _r) = \\sigma $ because $\\varphi ^{\\prime }$ is an extension of $\\varphi $ , and $\\nu (p - T, \\omega _{-r} \\dots \\omega _r)$ is compatible with $\\pi $ because $\\nu (p - T, \\omega _{-r} \\dots \\omega _r)$ is a subset of $\\nu (p - T, \\omega _{-r^{\\prime }} \\dots \\omega _{r^{\\prime }})$ .", "This then proves that $\\sigma \\in S_\\varphi (p, \\pi )$ , and therefore that $S_{\\varphi ^{\\prime }}(p, \\pi ) \\subseteq S_\\varphi (p, \\pi )$ ." ], [ "Monotony", "We return for a moment to the view of a cellular process as a partial description for an evolution of a cellular automaton.", "Finding the closure can then be seen as reconstructing an evolution from incomplete information.", "More information should then result in a larger reconstruction.", "So we will expect that the closure of the superset of a process is a superset of its closure, or, in other words, that the closure operator defines a monotone function.", "This property is used very often.", "Its proof begins with the proof of the same property for determinateness.", "Lemma 3.13 (Determinateness is Monotone) Let $\\pi \\subseteq \\psi \\in \\mathcal {P}$ be two processes and $\\varphi $ be a non-constant transition rule.", "Then $\\Delta \\pi \\subseteq \\Delta \\psi $ .", "Let $[p]\\sigma $ be determined by $\\pi $ .", "Then $S(p, \\pi ) = \\lbrace \\sigma \\rbrace $ .", "Since $\\psi \\supseteq \\pi $ , the requirement that $\\nu (p- T, w) \\mathrel \\mathrm {comp}\\psi $ is a stronger restriction on $w$ than the requirement that $\\nu (p - T, w) \\mathrel \\mathrm {comp}\\pi $ .", "So we must have $S(p, \\psi ) \\subseteq S(p, \\pi )$ and therefore $S(p,\\psi ) \\subseteq \\lbrace p\\rbrace $ .", "On the other hand, $S(p, \\psi )$ has at least one element.", "So $[p]\\sigma $ is determined also by $\\psi $ .", "All properties of the closure involve questions of its existence, therefore also this one.", "The theorem below expresses the intuitive notion that more requirements on the behaviour of a cellular automaton make it more likely that they are inconsistent and cannot be satisfied by the evolution of a cellular automaton.", "Theorem 3.14 (Closure is Monotone) Let $\\pi \\subseteq \\psi \\in \\mathcal {P}$ be two processes and $\\varphi $ be a non-constant transition rule.", "If $\\mathop {\\mathrm {cl}}\\nolimits \\psi $ exists, then $\\mathop {\\mathrm {cl}}\\nolimits \\pi $ exists and $\\mathop {\\mathrm {cl}}\\nolimits \\pi \\subseteq \\mathop {\\mathrm {cl}}\\nolimits \\psi $ .", "Assume that $\\mathop {\\mathrm {cl}}\\nolimits \\psi $ exists.", "We will say that the theorem is true for time $t$ if $\\mathop {\\mathrm {cl}}\\nolimits ^{(t)} \\pi $ exists and $\\mathop {\\mathrm {cl}}\\nolimits ^{(t)} \\pi \\subseteq \\mathop {\\mathrm {cl}}\\nolimits ^{(t)} \\psi $ .", "Since $\\mathop {\\mathrm {cl}}\\nolimits \\psi $ exists, there must be a $t_0 \\in \\mathbb {Z}$ such that $\\psi $ is quiet before $t_0$ .", "Then $\\pi $ is quiet before $t_0$ too, because $\\pi \\subseteq \\psi $ .", "Therefore $\\mathop {\\mathrm {cl}}\\nolimits ^{(t)} \\pi =\\pi ^{(t)} \\subseteq \\psi ^{(t)} = \\mathop {\\mathrm {cl}}\\nolimits ^{(t)} \\psi $ for all $t\\le t_0$ .", "So the theorem is true for every $t \\le t_0$ .", "Let now $t > t_0$ and assume that the theorem is true for $t - 1$ .", "Then $\\mathop {\\mathrm {cl}}\\nolimits ^{(t-1)} \\pi $ exists.", "Because $\\mathop {\\mathrm {cl}}\\nolimits ^{(t-1)} \\pi \\subseteq \\mathop {\\mathrm {cl}}\\nolimits ^{(t-1)} \\psi $ , we have $\\Delta \\mathop {\\mathrm {cl}}\\nolimits ^{(t-1)} \\pi \\subseteq \\Delta \\mathop {\\mathrm {cl}}\\nolimits ^{(t-1)} \\psi $ with Lemma REF .", "Because $\\pi ^{(t)} \\subseteq \\psi ^{(t)}$ , we have $\\pi ^{(t)} \\cup \\Delta \\mathop {\\mathrm {cl}}\\nolimits ^{(t-1)} \\pi \\subseteq \\psi ^{(t)} \\cup \\Delta \\mathop {\\mathrm {cl}}\\nolimits ^{(t-1)} \\psi ,$ but only as an inclusion between sets of events.", "We have not yet proved that these sets are cellular processes.", "The right side of (REF ) is however the cellular process $\\mathop {\\mathrm {cl}}\\nolimits ^{(t)} \\psi $ , and therefore the left side, as its subset, must also be a process.", "This then means that $\\pi ^{(t)}$ is compatible with $\\Delta \\mathop {\\mathrm {cl}}\\nolimits ^{(t-1)} \\pi $ and that therefore $\\mathop {\\mathrm {cl}}\\nolimits ^{(t)} \\pi $ exists.", "Since the left side of (REF ) is then $\\mathop {\\mathrm {cl}}\\nolimits ^{(t)} \\pi $ , while its right side is $\\mathop {\\mathrm {cl}}\\nolimits ^{(t)} \\psi $ , we have proved that $\\mathop {\\mathrm {cl}}\\nolimits ^{(t)} \\pi \\subseteq \\mathop {\\mathrm {cl}}\\nolimits ^{(t)} \\psi $ .", "Therefore the theorem is true for time $t$ if it is true for time $t - 1$ when $t> t_0$ .", "So we have shown by induction that the theorem is true for all times $t$ , and therefore true in general." ], [ "Closure and Evolution Sequence", "As the final task of this chapter we now verify what we required of the closure at the beginning, when we motivated its construction: The process $\\gamma $ of (REF ), the translation of the evolution sequence $(c_t)_{t \\ge 0}$ , is the closure of the cellular process for its initial configuration, $\\gamma ^{(0)}$ .", "We will prove a bit more: The following lemma shows that for every time $t \\ge 0$ , the time slice $\\gamma ^{(t)}$ determines the following time slice $\\gamma ^{(t+1)}$ in the same way that the configuration $c_t$ determines the following configuration $c_{t+1}$ .", "The lemma is then the analog of the transition rule (REF ) for cellular processes.", "Lemma 3.15 (Global Transition Rule for Processes) Let $\\varphi $ be a non-constant transition rule and let $\\gamma $ be as in (REF ).", "Then for all $t \\ge 0$ , $\\gamma ^{(t+1)} = \\Delta \\gamma ^{(t)}\\,.$ By Lemma REF , the only events that can possibly be determined by $\\gamma ^{(t)}$ have a time coordinate of $t + 1$ .", "It only remains to prove that $\\gamma ^{(t)}$ determines all events $[t+1, x] c_{t+1}(x)$ with $x \\in \\mathbb {Z}$ .", "Let $p = (t + 1, x)$ .", "To know whether this point is determined we have to find $S(p, \\gamma ^{(t)})$ .", "Because $\\mathop {\\mathrm {dom}}\\nu (p - T, w) \\subset \\mathop {\\mathrm {dom}}\\gamma ^{(t)}$ for all $w \\in \\Sigma ^{2r+1}$ , the process $\\nu (p - T, w)$ is compatible with $\\gamma ^{(t)}$ if and only if it is a subset of $\\gamma ^{(t)}$ .", "So we must find all $w \\in \\Sigma ^{2r+1}$ that satisfy $\\nu (p - T, w) =\\lbrace [t, x - r]c_t(x - r), \\dots , [t, x + r]c_t(x + r) \\rbrace ,$ where $r$ is the radius of $\\varphi $ .", "This equation has one solution, $w = c_t(x - r) \\dots c_t(x + r)$ .", "So the set $S(p, \\gamma ^{(t)})$ has exactly one element, which means that the event at $p$ is determined.", "Its state is $\\varphi (w)$ , which is equal to $c_{t+1}(t + 1, x)$ by the global transition rule (REF ).", "This proves the lemma.", "Using this lemma we can then easily see that $\\mathop {\\mathrm {cl}}\\nolimits ^{(t)}\\gamma ^{(0)} = \\gamma ^{(t)}$ for $t \\ge 0$ and that $\\mathop {\\mathrm {cl}}\\nolimits ^{(t)}\\gamma ^{(0)}$ is empty for $t < 0$ .", "This then shows that $\\gamma $ is indeed generated by its initial time slice $\\gamma ^{(0)}$ ." ], [ "Summary", "In this chapter we have formalised the concept of the evolution of a cellular automaton in a new way, in order to be able to understand the fate of a localised arrangement of cells.", "The goal was to have a notation that treats events at different times on an equal footing.", "The starting point was the description of cellular automata with configurations and evolution sequences.", "This is a natural way to understand the cellular automaton as a machine that evolves over time.", "The transition rule then is a description of the law that governs the behaviour of the cells.", "This method to describe the behaviour of cellular automata was then decomposed into its components.", "The configurations became sets of cellular events.", "The transition rule was expressed in a radius-invariant way as the function that generates the set of determined events for a process.", "The evolution sequence became the closure of a process.", "The concept of closure helps us to express how information propagates in a cellular automaton.", "We have seen that the closure operator is monotone, which will help us to reason in an abstract way about cellular processes.", "We have seen how the evolution of a configuration is expressed with cellular processes.", "We have introduced the concept of radius-invariance.", "The closure operator, as the new form of the transition rule, is radius-invariant even for a cellular process of finite size.", "We have therefore extended the concept of the initial configuration to an arbitrary set of cells at arbitrary times.", "Some cellular processes however have no closure and it is not yet clear how to construct processes that have a closure.", "This question will be answered in Chapter with the concept of achronal situations." ], [ "Reaction Systems", "One of the goals of this thesis is to find a way in which we can express the laws of large-scale behaviour in a cellular automaton.", "A “law of large-scale behaviour” is here any statement that involves an arbitrarily large number of cells.", "The triangles below in Figure REF are an intuitive example: One knows that the exact number of white cells in the triangle's base does not matter.", "It could become arbitrarily large, and the same kind of triangular shape would result.", "We need to express this kind of intuitive law—and much more complex laws—in a formal way.$\\endcsname $The specific law that is expressed in Figure REF is expressed in Table REF .", "Other examples are the laws of ether formation, like Lemma REF .", "Figure: Triangles as computations under Rule 54.In this chapter we will define a formalism with which we can express instances of such large-scale laws in terms of the input and result of a computation.", "The computation is then represented by the ordered pair of input and output; intermediate steps are ignored.The concepts introduced here rely to a great extent on the virtual state machine (VSM) introduced by Christopher G. Langton [28].", "In contrast to the VSM, which is understood by Langton as a long-lived entity, a reaction (defined below) always refers to a finite time span.", "One could then understand a reaction as the description of a single computational step in the existence of a VSM." ], [ "Situations", "We will take here the viewpoint that we build the computer inside the universe represented by the cellular automaton.", "The transition rule is kept fixed; it represents the physics of that world.", "In a computer, we distinguish between the data and the computing machine.", "In the cellular automaton, the data and the machine are both cell states.", "A very simple example for the way data and computation interact are the triangular structures found under Rule 54 and other cellular automata (Figure REF ).", "Here we may view the cell configuration at the initial time as the input; it consists of two cells in state 1 that surround a sequence of zeros.", "The highlighted cells in the figure are then the computation initiated by this input.", "How long the computation lasts depends directly on the number of zeros in the input.", "We can view therefore it as a kind of counter, or a loop that counts downward.", "Another viewpoint, since the triangles have a different shape depending on whether the number of zeros is odd or even, is to understand the triangles as programs that test for parity.", "When we now look at these triangular processes in terms of the cells involved, we see that the computation takes a different amount of time at different locations.", "It lasts longer at the center of the triangle than at its margins.", "Since we have viewed the initial interval of cells as the input of the computation, we will take the other sides of the triangle as its result.", "It then consists roughly of the black cells at the boundaries of the triangles, together with their direct neighbours.", "Later, in Figure REF , the input and result of this kind of computation will be shown explicitly.", "So the formalism of Flexible Time makes no difference between the data and the machine and represents them in a single mathematical object.", "In a similar vein, input and result of a computation are the same kind of mathematical object.", "This is so because it must be possible that the result of one computation is the input of another computation." ], [ "Properties of Situations", "The mathematical objects that represent the input or result of a cellular computation are called here situations; they are defined below.", "But before we can write down the formal definition, we will collect the properties a situation must have.", "(a) A situation specifies a finite cellular process.", "To specify the input or the result of a computation in a cellular automaton means to express requirements on the states of cells: at a specified time a cell must be in a specified state.", "Put together, these requirements are the events of a cellular process.", "It is a finite process because the input and output of a computation always have a finite number of bits.", "(b) A situation may specify events at different times.", "With the finite speed at which signals travel in a cellular automaton, exact synchronisation of the components in the computer is difficult, and different parts of a computation may end at a different time.", "We must therefore allow that the events of a situation belong to different times.", "(c) A situation specifies a sequence of events.", "The cells in a one-dimensional cellular automaton have a natural order from left to right.", "Situations generalise finite sets of adjacent cells together with their states.", "Therefore it is good for our intuition if the events in a situation also form a sequence.", "It especially allows us to use the formalism of finite sequences for situations.", "I will now sketch the way in which we get a meaningful left-to-right arrangement for the events of a situation, starting from the order in a set of adjacent cells.", "Take a situation that specifies the states of such a set of cells and use it as the input of a computation.", "In general, the output situation of this computation specifies events at different times.", "They have no longer a natural left-to-right order, but if the duration of the computation is short enough, there remains an approximate order.", "As information travels with finite speed in a cellular automaton, each event in the result has only a few events in the input that are near enough to have caused it.", "So we can arrange the events in the output approximately from left to right with help of the input events.", "Later we will make this idea precise; we will then have a correspondence between the order of the input of any computation and that of its output.", "(d) A situation has a size vector.", "This vector is an analog to the length $\\mathopen \|u\\mathclose \|$ of a finite sequence $u$ .", "Imagine that the sequence $u$ consists of letters written on grid paper, with the $\\mathopen \|u\\mathclose \|$ letters of $u$ on the squares numbered 0, ..., $\\mathopen \|u\\mathclose \|- 1$ .", "Now consider $u v$ , the product of $u$ with some other sequence $v$ .", "In it, when it is written down the same way, the letters of $v$ will occupy the positions $\\mathopen \|u\\mathclose \|$ , ..., $\\mathopen \|u\\mathclose \| + \\mathopen \|v\\mathclose \| - 1$ , instead of 0, ..., $\\mathopen \|v\\mathclose \| - 1$ , as it would have been if $v$ alone were written.", "So the length of a sequence $u$ marks the point behind $u$ , and this point is the starting point of the second factor in a product involving $u$ .", "The same happens with situations, except that its size vector is an element of $\\mathbb {Z}^2$ and that it can be chosen arbitrarily.", "We then can define a notion of product for situations similar to that for sequences; the arbitrariness of the size vector makes it possible that there are gaps between the event sequences of the factors.", "The following is then a situation: a sequence of cellular events together with a size vector.", "Definition 4.1 (Situations) A situation of length $\\ell $ with states in $\\Sigma $ is a pair $((e_0, \\dots , e_{\\ell -1}), p) \\in (\\mathbb {Z}^2 \\times \\Sigma )^\\ell \\times \\mathbb {Z}^2$ for which the set $\\lbrace \\, e_i \\colon 0 \\le i < \\ell \\,\\rbrace $ is a cellular process.", "The set of all situations of length $\\ell $ is $\\mathcal {S}_\\ell $ .", "A situation in general is an element of the set $\\mathcal {S} = \\bigcup _{\\ell \\ge 0} \\mathcal {S}_\\ell \\,.$ The event sequence of the situation $((e_0, \\dots ,e_{\\ell -1}), p)$ is then the tuple of events, $(e_0, \\dots ,e_{\\ell -1})$ , and its size vector is the point $p$ .", "A third property of the situation that we required before, namely a cellular process associated to it, is introduced in the following definition.", "It also specifies a notation for the size vector that does not require to spell out a situation as a pair.", "Definition 4.2 (Components of a Situation) Let $a = ((e_0, \\dots , e_{\\ell -1}), p)$ be a situation.", "The process of $a$ is the cellular process $\\mathrm {pr}(a) = \\lbrace \\, e_i \\colon 0 \\le i < \\ell \\,\\rbrace \\,.$ For $p$ , the size vector of $a$ , we write $\\delta (a)$ .", "The symbol for the size vector, $\\delta $ , should remind of another way to view a situation.", "We can view a situation $a$ as having a left end at the coordinate origin an a left end at $\\delta (a)$ ; then $\\delta (a)$ is the difference between the two ends of a situation.", "Therefore the symbol.", "With the notations of Definition REF we can already express a convenient shorthand notation.", "Let $a$ and $b$ be two situations.", "The process of $b$ , shifted by $a$ is then $\\mathrm {pr}_{a}(b) = [\\delta (a)]\\mathrm {pr}(b)\\,.$ It will become useful once we have defined the product of situations." ], [ "Path Notation", "Let $a$ be a situation of length $\\ell $ .", "If we want to express it in full detail, we currently have to write it in the form $a = (([p_0]\\alpha _0, \\dots , [p_{\\ell -1}]\\alpha _{\\ell -1}), p_\\ell ),$ with $p_i \\in \\mathbb {Z}^2$ and $\\alpha _i \\in \\Sigma $ for all $i$ .", "The use of expressions of this kind for longer calculations and proofs would however soon become quite cumbersome.", "Therefore we now introduce a shorter form.", "It will fulfil the remaining requirement on situations and provide a way to treat situations in the same manner as finite sequences.", "In the full-developed formalism we will then refer to the properties of a situation $a$ only with help of the new notation, the process $\\mathrm {pr}(a)$ and the size vector $\\delta (a)$ and no longer refer to terms of the form (REF ) directly.", "The new notation uses a relative notation for the locations of the events in a situation.", "Definition 4.3 (Path Notation) Let $a$ be a situation written in the form (REF ).", "Let $\\tilde{p}_0 = p_0$ and $\\tilde{p}_i = p_i - (p_{i-1} + X)$ for $i > 0$ .", "The long path notation for $a$ is then $a = [\\tilde{p}_0]\\alpha _0[\\tilde{p}_1]\\alpha _1 \\dots [\\tilde{p}_{\\ell -1}]\\alpha _{\\ell -1}[\\tilde{p}_\\ell ],$ The terms $[\\tilde{p}_i]$ are the displacements of $a$ .", "The short path notation of $a$ is similar to this, but all terms $[\\tilde{p}_i]$ with $\\tilde{p}_i = (0, 0)$ are removed from it.", "An exception is the case of $\\ell = 0$ : a situation $a = [\\tilde{p}_0]$ cannot be shortened.", "It is by definition already in short path notation.", "One can understand the path notation as the description of a writing process.", "In it, symbols for the cell states are written into a square grid similar to a space-time diagram.", "After writing a symbol into the square at point $p_{i-1}$ the cursor is at $p_{i-1} + X$ .", "The displacement $[\\tilde{p}_i]$ is then the amount of extra movement before the next symbol can be written down.", "This explains the occurrence of the unit vector $X$ in the definition of $\\tilde{p}_i$ above.", "It also explains the formula $p_i = \\sum _{j = 0}^i \\tilde{p}_j + j X\\qquad \\text{for $i = 0$, \\dots , $\\ell $}$ that converts the displacements of the long path notations back into absolute positions.", "(Note that $p_0$ , ..., $p_{\\ell - 1}$ are locations of events, while $p_\\ell $ is the size vector!)", "The abbreviations for $[\\tilde{p}_i]$ that were defined before in the context of cellular events are also allowed for situations.", "So we can write $[(t, x)]$ as $[t, x]$ , and $[0, x]$ as $[x]$ .", "Sometimes we will use a “mixed” path notation, with not all $[0]$ -terms omitted.", "As an example of how this works, let us look at a cellular automaton with state set $\\Sigma = \\lbrace 0, 1 \\rbrace $ .", "We assume that at time $t = 0$ the cells at position $x = 0$ , 1, 2 and 3 are in the states 1, 0, 0 and 1.", "We now want to express this information with a situation.", "For this we start with a cellular process.", "We know the cellular events $[0, 0]1$ , $[0, 1]0$ , $[0, 2] 0$ and $[0, 3] 1$ .", "Using the abbreviation convention for the positions of cellular events, we can write the process that contains them as $\\pi = \\lbrace [0] 1, [1] 0, [2] 0, [3] 1 \\rbrace \\,.$ A natural way to write $\\pi $ as a situation in the form (REF ) is $a = (( [0] 1, [1] 0, [2] 0, [3] 1), (0, 4))\\,.$ In a situation, the events of $\\pi $ must be arranged in a sequence; we have here chosen the most natural one, an arrangement from left to right by their $x$ -positions.", "For the size vector of the situation we have chosen the point $(0, 4)$ , one position to the right of the last event in the event sequence of $a$ .", "If then $a$ is written in the long path notation (REF ), it becomes $a = [0]1 [0]0 [0]0 [0]1 [0]\\,.$ We now see that the choice of $a$ in (REF ) was natural: all displacements in the new notation become $[0]$ .", "We can remove them all, and this leads to the short path notation for $a$ , namely 1001.", "(The similarity between the short path notation and the notation for finite sequences is intended.)" ], [ "Products", "The path notation leads to a natural definition for the product of two situations.", "For the concatenation of situations in the following definition we employ the convention that $[p][q] = [p + q]$ that was already introduced for cells: Here is where it becomes useful.", "Definition 4.4 (Product) Let $a$ , $b \\in \\mathcal {S}$ be two situations.", "We get their product by concatenating the long path notation for $a$ with the long path notation for $b$ .", "The product exists if the resulting expression is a situation.", "The product of $a$ and $b$ is written $a b$ .", "To understand this definition, let $a = [\\tilde{p}_0]\\alpha _0 \\dots \\alpha _{\\ell -1}[\\tilde{p}_\\ell ]\\qquad \\text{and}\\qquad b = [\\tilde{q}_0]\\beta _0 \\dots \\beta _{m-1} [\\tilde{q}_m]$ be two processes in long path notation.", "Then their product, if it exists, has the long path notation $\\begin{aligned}[b]ab&= [\\tilde{p}_0]\\alpha _0 \\dots \\alpha _{\\ell -1}[\\tilde{p}_\\ell ][\\tilde{q}_0]\\beta _0 \\dots \\beta _{m-1} [\\tilde{q}_m]\\\\&= [\\tilde{p}_0]\\alpha _0 \\dots \\alpha _{\\ell -1}[\\tilde{p}_\\ell +\\tilde{q}_0]\\beta _0 \\dots \\beta _{m-1} [\\tilde{q}_m]\\,.\\end{aligned}$ The first line in this equation is that what we get when we simply concatenate the path notations for $a$ and $b$ .", "The second line is that what we get after applying the convention.", "We now return to the pair notation for situations to find the process of $a b$ .", "We assume that $a$ is as in (REF ), and $b$ is similar, so that we have $a &= (([p_0]\\alpha _0, \\dots , [p_{\\ell -1}]\\alpha _{\\ell -1}), p_\\ell ),\\\\b &= (([q_0]\\beta _0, \\dots , [q_{m-1}]\\beta _{m-1}), q_m)\\,.$ In the second equation the $q_i$ are related to the $\\tilde{q}_i$ in the same way as the $p_i$ to the $\\tilde{p}_i$ in Definition REF .", "Then the product of $a$ and $b$ has the form $\\begin{aligned}[b]ab &=(([p_0]\\alpha _0, \\dots , [p_{\\ell -1}]\\alpha _{\\ell -1},\\\\&\\qquad [p_\\ell + q_0]\\beta _0, \\dots , [p_\\ell + q_{m-1}]\\beta _{m-1}),p_\\ell + q_m)\\,.\\end{aligned}$ We can then translate this formula into a lemma that describes the product in a more abstract form.", "Lemma 4.5 (Properties of the Product) Let $a$ , $b \\in \\mathcal {S}$ be two situations.", "If $\\mathrm {pr}(a)$ is compatible to $\\mathrm {pr}_{a}(b)$ , then the product $a b$ exists.", "Its process and size vector are $\\mathrm {pr}(a b) = \\mathrm {pr}(a) \\cup \\mathrm {pr}_{a}(b)\\qquad \\text{and}\\qquad \\delta (a b) = \\delta (a) + \\delta (b)\\,.$ The event sequence in (REF ) contains the events in $\\mathrm {pr}(a)$ together with the events in $\\mathrm {pr}(b)$ , but the latter shifted by the size vector $p_\\ell $ of $a$ .", "The set of these events is therefore $\\mathrm {pr}(a) \\cup [\\delta (a)]\\mathrm {pr}(b)$ , or $\\mathrm {pr}(a) \\cup \\mathrm {pr}_{a}(b)$ in the notation of (REF ).", "It is a cellular process if $\\mathrm {pr}(a)$ is compatible to $\\mathrm {pr}_{a}(b)$ , and if this is true, then $a b$ is a situation.", "The equations in (REF ) can then be read directly from (REF ).", "The left equation in (REF ) is the chain rule for situations.", "It is the reason why the notation $\\mathrm {pr}_{a}(b)$ was introduced in (REF ).", "Together with the product for situations we get the usual notations that are related to it.", "Among them are exponentiation, Kleene closure and other constructions that were already described for sequences.", "In contrast to ordinary sequences we must however be careful whether a product actually exists.", "The set of situations therefore does not form a semigroup.", "Nevertheless it has a neutral element of multiplication.", "As we can infer from (REF ), it is the situation $[0]$ .", "We will speak of it as the empty situation.", "A very important subset of $\\mathcal {P}$ is the set $\\lbrace \\, [0] \\sigma \\colon \\sigma \\in \\Sigma \\,\\rbrace ^$ .", "It contains all those situations that, when written down in short path notation, look like elements of $\\Sigma ^$ .", "Therefore we will introduce no special symbol for them but call this set of situation also $\\Sigma ^$ .", "It will be always clear from the context which set is meant.", "As a means to distinguish the elements of $\\Sigma ^$ from other situations we will use for them, and only for them, the length notation $\\mathopen \|\\,\\cdot \\,\\mathclose \|$ of finite sequences." ], [ "Induction Proofs with Situations", "The product of situations is also important because it allows induction proofs.", "Assume e. g. that a property $P$ of situations is “closed under non-empty multiplication”: This shall mean that if two non-empty situations $a$ , $b \\in \\mathcal {S}$ have the property $P$ and $a b$ exist, then it has property $P$ .", "Then if all situations in $\\Sigma ^$ have property $P$ , all elements of $\\mathcal {S}$ have it.", "This is in fact an induction over the length of the situations.", "This specific form of induction requires however that the factors $a$ and $b$ are nonempty, which is not always easy to check.", "Therefore we will use another, more complex induction principle.", "Before we can express it, we have to handle an ambiguity of the notation for situations.", "We will have to split a situation $s$ into the product of the three terms $a$ , $[p]$ and $b$ .", "But because of the convention that $[p_1][p_2]$ is equal to $[p_1 + p_2]$ , we could also split $s$ into $a[p - q]$ , $[q]$ and $b$ , for any $q \\in \\mathbb {Z}$ .", "So we cannot unambiguously say that the displacement $[p]$ is a factor of the situation $s$ .", "This problem is solved with the help of the long path notation (REF ): in it the terms $[\\tilde{p}_i]$ are unambiguous.", "Definition 4.6 (Honest Decomposition) Let $s \\in \\mathcal {S}$ be a situation with the long path notation $s = [p_0] \\sigma _0 \\dots \\sigma _{\\ell - 1} [p_\\ell ]$ .", "A decomposition $s = a [p] b$ of $s$ is honest if there is an index $i$ such that $a &= [p_0] \\sigma _0 \\dots \\sigma _{i - 1} [0],&p &= p_i,&b &= [0] \\sigma _{i} \\dots \\sigma _{\\ell - 1} [p_\\ell ]\\,.$ The cases of $a = [0]$ or $b = [0]$ are here explicitly allowed.", "They refer to the one-sided decompositions $s = [p] b$ and $s = a [p]$ .", "With this definition we can now describe the new induction principle.", "Theorem 4.7 (Induction over Displacements) Let $P$ be a property of the elements of $\\mathcal {S}$ .", "Assume that for all $s \\in \\mathcal {S}$ , if $s \\in \\Sigma ^$ , then $P$ is true for $s$ , if $s = a [p] b$ is a honest decomposition with $p \\ne (0,0)$ and $P$ is true for $a$ and $b$ , then $P$ is true for $s$ .", "Then $P$ is true for all elements of $\\mathcal {S}$ .", "For this proof we define the number of nontrivial displacements in a situation $s$ as the number of displacements $[p_i]$ with $p_i \\ne (0, 0)$ that occur in its long path notation.", "We write this number as $d(s)$ .", "The proof is then an induction over $d(s)$ .", "If $d(s) = 0$ , then $s \\in \\Sigma ^$ , and $P$ is true for $s$ .", "If $d(s) > 0$ , then a term $[p]$ with $p \\ne (0, 0)$ occurs in the long path notation of $s$ .", "So there is a honest decomposition $s = a[p] b$ .", "We then have $d(s) > d(a) + d(b)$ , so $P$ is true for $a$ and $b$ by induction.", "Therefore $P$ is also true for $s$ ." ], [ "Reactions", "A reaction represents a computation in a cellular automaton.", "It consists of two situations that are related to each other by a cellular process.", "The first situation is the input of the computation.", "Its events start the activity of the cellular automaton; the activity itself is represented by the cellular process that is the closure of the input situation; the result of the computation is represented by the second situation of the reaction.", "Its process must lie completely inside the closure of the input process: This means that all events of the output are determined by the input via the transition rule.", "Its size vector must be the same as that of the input situation: This will allow us to replace the input of a reaction with its output when the input is part of a larger situation.See below at Theorem REF for more details.", "Otherwise the choice of the second situation is arbitrary.", "Reactions were introduced in [51].", "They have their name from the arrow with which reactions are written here, because it reminds of chemical reactions.", "Definition 4.8 (Reactions) Let $\\varphi $ be a transition rule for $\\Sigma $ and $a$ , $b \\in \\mathcal {S}$ be two situations.", "If $\\mathrm {pr}(b) \\subseteq \\mathop {\\mathrm {cl}}\\nolimits \\mathrm {pr}(a)\\qquad \\text{and}\\qquad \\delta (a) = \\delta (b),$ then the pair $(a, b)$ is a reaction for $\\varphi $ .", "For $(a, b)$ we will usually write $a \\rightarrow b$ .", "Figure: The processes involved in the reaction a→ba \\rightarrow b.Sometimes we will use the more general term of an abstract reaction.", "This is a pair $(a, b)$ with $\\delta (a) = \\delta (b)$ in which $a$ is compatible with $b$ .", "We will also use the formula $a \\rightarrow b$ as a proposition.", "Then it expresses the fact that there is a reaction $(a, b)$ .", "It may be a reaction for $\\varphi $ or an abstract reaction, depending on the context.", "The processes that belong to this reaction are shown in Figure REF .", "Often, when the situations that are part of a reaction are complex, they will be drawn separately, as in Figure REF below.", "This diagram is annotated with the names of the situations and not the processes, as in the previous figure: we will choose whichever is appropriate.", "Figure: Another way to display the reaction ofFigure ." ], [ "Sets of Reactions", "In order to be able to calculate with them, we will now consider reactions that belong to a set.", "For this let $S \\subseteq \\mathcal {S}$ be a set of situations and $R \\subseteq S \\times S$ a set of reactions between its elements.", "The set $S$ is then the domain of reaction setdomain of $R$ .", "We use a special notation for reactions that belong to a set.", "If $(a,b) \\in R$ , we write this as $a \\rightarrow _R b$ .", "If $R$ is known from context, we may write it even as $a \\rightarrow b$ .", "As before, an expression $a \\rightarrow _R b$ may be used as the proposition.", "It then means that the pair $(a, b)$ is an element of $R$ .", "As a set of pairs, $R$ is a binary relation on $S$ .", "The reaction sets that we use for the understanding of cellular automata are mainly pre-orders.", "In order theory, a binary relation is called a pre-order if it is transitive and reflexive [11].", "We recapitulate what this means: $R$ is transitive if $a \\rightarrow _R c$ whenever $a \\rightarrow _R b$ and $b \\rightarrow _R c$ .", "$R$ is reflexive if $a \\rightarrow _R a$ for all $a \\in R$ .", "In a set of reaction that is a pre-order, reflexivity allows to reconstruct the domain by $\\mathop {\\mathrm {dom}}R =\\lbrace \\, a \\in \\mathcal {S} \\colon a \\rightarrow _R a \\,\\rbrace \\,.$ Therefore for a set of reactions that is a pre-order it is not necessary to specify the domain separately from the reaction set.", "There is another small fact that is useful in its own right: If a set $R$ of reactions is a pre-order, then every situation in its domain has a closure.", "This is because for every $a \\in \\mathop {\\mathrm {dom}}R$ there is a reaction $a \\rightarrow _R a$ , by the transitivity of $R$ , and the definition of reactions requires that $a$ then has a closure." ], [ "Reaction Systems", "Transitivity of a reaction set allows to form a chain of reactions, each using as input the result of the previous one, and combine them into a single reaction that computes the result of the last reaction in the chain from the input of the first one.", "We now introduce another way to create new reactions from old ones, one that is specific to cellular automata.", "It reflects the local nature of the interactions between the cells.", "Definition 4.9 (Application of a Reaction) We call the reaction $xay \\rightarrow xby$ , where $a$ , $b$ , $x$ and $y$ are situations, the application of $a \\rightarrow b$ on $xay$ .", "Figure: Application of a triangle reaction under Rule 54.One example for the application of the reaction $a \\rightarrow b$ to the situation $x a y$ is shown in Figure REF .", "The events of $\\mathrm {pr}(x a y)$ are displayed as squares with thick frames, like Figure: NO_CAPTION$\\mathrm {pr}(x b y)$ are displayed in darker colours, like Figure: NO_CAPTION$\\mathrm {pr}(x a y)$ .", "In both colour sets, the lighter and the darker, the brighter squares represent cells in state 0 and the darker squares, cells in state 1.", "One can also see from this diagram that the process $\\mathrm {pr}(x b y)$ is a subset of the closure of $\\mathrm {pr}(x a y)$ .", "The reaction $a \\rightarrow b$ , displayed in the centre of the diagram, is a triangle reaction.See also Figure REF .", "This kind of reaction will be formally defined later, in Definition REF .", "The base line of its triangle consists of the squares with frames but in lighter colours at the bottom of the diagram; the events belonging to it are all in state 0.", "The other two sides of the triangle are part of $\\mathrm {pr}_{x}(b)$ ; they are shown as squares with frames, but with lightly coloured interior, and they are partially in the states 0 and 1.", "There is also an operation of applying a reaction to a situation: It generates a reaction $x a y \\rightarrow x b y$ from a reaction $a \\rightarrow b$ and a situation $x a y$ .", "We will show now that under reasonable conditions on $x$ , $y$ , $a$ and $b$ application is always possible.", "This is done in two steps, because the definition of application in the form it was stated above uses too many variables at once.", "Instead of working with a reaction between situations that consist of three factors, we will first work with reactions between products of two factors.", "Lemma 4.10 (Parallel Processing) Let $a \\rightarrow a^{\\prime }$ and $b \\rightarrow b^{\\prime }$ be reactions for $\\varphi $ .", "Assume that $\\mathop {\\mathrm {cl}}\\nolimits \\mathrm {pr}(a b)$ exists.", "Then $a b \\rightarrow a^{\\prime } b^{\\prime }$ is a reaction for $\\varphi $ .", "It is clear that $\\delta (a b) = \\delta (a^{\\prime } b^{\\prime })$ .", "So it remains to prove that $\\mathrm {pr}(a^{\\prime } b^{\\prime }) \\subseteq \\mathop {\\mathrm {cl}}\\nolimits \\mathrm {pr}(a b)$ .", "We have $\\mathrm {pr}(a^{\\prime }b^{\\prime }) = \\mathrm {pr}(a^{\\prime }) \\cup \\mathrm {pr}_{a^{\\prime }}(b^{\\prime })$ by the chain rule (REF ).", "Therefore the proof of the lemma is complete if we show that $\\mathrm {pr}(a^{\\prime }) \\subseteq \\mathop {\\mathrm {cl}}\\nolimits \\mathrm {pr}(a b)$ and $\\mathrm {pr}_{a^{\\prime }}(b^{\\prime }) \\subseteq \\mathop {\\mathrm {cl}}\\nolimits \\mathrm {pr}(a b)$ .", "Because $a \\rightarrow a^{\\prime }$ is a reaction, we have $\\mathrm {pr}(a^{\\prime }) \\subseteq \\mathop {\\mathrm {cl}}\\nolimits \\mathrm {pr}(a)$ .", "Since $\\mathrm {pr}(a) \\subseteq \\mathrm {pr}(a b)$ , we have $\\mathop {\\mathrm {cl}}\\nolimits \\mathrm {pr}(a) \\subseteq \\mathop {\\mathrm {cl}}\\nolimits \\mathrm {pr}(a b)$ by monotony of the closure (Theorem REF ).", "Therefore $\\mathrm {pr}(a^{\\prime })\\subseteq \\mathop {\\mathrm {cl}}\\nolimits \\mathrm {pr}(a b)$ .", "Because $b \\rightarrow b^{\\prime }$ is a reaction, we have $\\mathrm {pr}(b^{\\prime }) \\subseteq \\mathop {\\mathrm {cl}}\\nolimits \\mathrm {pr}(b)$ .", "The closure is shift-invariant, therefore $\\mathrm {pr}_{a^{\\prime }}(b^{\\prime }) \\subseteq \\mathop {\\mathrm {cl}}\\nolimits \\mathrm {pr}_{a^{\\prime }}(b)$ .", "Now $\\delta (a) =\\delta (a^{\\prime })$ because $a \\rightarrow a^{\\prime }$ is a reaction, so we have $\\mathrm {pr}_{a^{\\prime }}(b^{\\prime }) \\subseteq \\mathop {\\mathrm {cl}}\\nolimits \\mathrm {pr}_{a}(b)$ .", "Since $\\mathrm {pr}_{a}(b)\\subseteq \\mathrm {pr}(a b)$ , we have $\\mathop {\\mathrm {cl}}\\nolimits \\mathrm {pr}_{a}(b) \\subseteq \\mathop {\\mathrm {cl}}\\nolimits \\mathrm {pr}(a b)$ , again by monotony of the closure.", "Therefore $\\mathrm {pr}_{a^{\\prime }}(b^{\\prime })\\subseteq \\mathop {\\mathrm {cl}}\\nolimits \\mathrm {pr}(a b)$ .", "In this proof the condition $\\delta (a) = \\delta (a^{\\prime })$ played a crucial role in keeping the processes $\\mathrm {pr}_{a}(b)$ and $\\mathrm {pr}_{a^{\\prime }}(b^{\\prime })$ at the same position.", "This is why it appeared in the definition of reactions.", "Theorem 4.11 (Applying Creates a Reaction) Let $\\varphi $ be a transition rule for $\\Sigma $ .", "If $a \\rightarrow b$ is a reaction for $\\varphi $ and there are $x$ , $y \\in \\mathcal {S}$ for which $\\mathop {\\mathrm {cl}}\\nolimits (x a y)$ exists, then $x a y \\rightarrow x b y$ is a reaction for $\\varphi $ .", "(Figure REF .)", "Figure: Applying a→ba \\rightarrow b to xayx a y.Since $\\mathop {\\mathrm {cl}}\\nolimits \\mathrm {pr}(xay)$ exists, the processes $\\mathop {\\mathrm {cl}}\\nolimits \\mathrm {pr}(x)$ and $\\mathop {\\mathrm {cl}}\\nolimits \\mathrm {pr}_{a x}(y)$ exist by Theorem REF .", "This means that $x \\rightarrow x$ and $y \\rightarrow y$ are reactions for $\\varphi $ .", "By Lemma REF , $x a \\rightarrow x b$ is a reaction for $\\varphi $ because $x \\rightarrow x$ and $a \\rightarrow b$ are, and $x a y \\rightarrow xb y$ is a reaction for $\\varphi $ because $x a \\rightarrow x b$ and $y \\rightarrow y$ are.", "Now we will introduce a name for the property of a set of reactions that the operation of application in it is freely possible.", "Note that in its definition there are no explicit restrictions on the situations $x$ and $y$ at the sides of $a$ ; there is however the implicit restriction that $x a y$ must be an element of $R$ .", "Definition 4.12 (Closed under Application) Let $\\Sigma $ be a set.", "Let $R$ be a set of reactions with $\\mathop {\\mathrm {dom}}R\\subseteq \\mathcal {S}$ .", "If for all reactions $a \\rightarrow _R b$ and for all situations $x$ , $y \\in \\mathcal {S}$ , with $x a y \\in R$ there is a reaction $x a y \\rightarrow _R xb y$ , then $R$ is closed under application.", "Theorem REF also expresses that the reactions in $R$ are local in scope: When $R$ is closed under application, it depends only on the initial situation $a$ and not on the situations $x$ and $y$ around it, whether a reaction $x a y \\rightarrow _R x b y$ is possible.", "Now we can finally introduce the central concept of this thesis.", "Reaction systems will serve as a replacement of the evolution sequence defined in Chapter for the understanding of cellular automata.", "Definition 4.13 (Reaction System) Let $\\varphi $ be a transition rule.", "A reaction system for $\\varphi $ is a set of reactions for $\\varphi $ that is a pre-order and closed under application.", "Similarly, an abstract reaction system is a set of abstract reactions that is a pre-order and closed under application.", "The set of all reactions for a given transition rule is obviously a reaction system, but we will usually need smaller ones.", "The operation of applying a reaction to a situation will allow us to define a large set of reactions with the help of a small set of local reactions.", "Therefore we define now how a small set of reactions and a set of situations together generate a reaction system.", "As it is common with generated sets in mathematics, a large part of the work with generated reaction systems is about deriving properties of the whole system from those of the set of generators.", "Definition 4.14 (Generated Reaction System) Let $S \\subseteq \\mathcal {S}$ be a set of situations and $G$ a set of reactions.", "Let $R$ be the smallest reaction system with $S \\subseteq \\mathop {\\mathrm {dom}}R$ and $G \\subseteq R$ (i. e. no proper subset of $R$ has this property).", "Then $R$ is the reaction system generated by $G$ from $S$ .", "The set $S$ is the set of generating situations for $R$ , and $G$ is the set of generating reactions for $R$ .", "We are mainly interested in non-abstract reaction systems.", "An abstract reaction system is usually created from a reaction system for a rule $\\varphi $ in order to have a system that is easier to handle.", "With Theorem REF we see that a reaction system for $\\varphi $ can be generated from an arbitrary reaction set $G$ and a set $S$ of situations for which the only requirement is that all its elements must have a closure under $\\varphi $ .", "If we know this, we can work with the reaction system in a quasi-algebraic way, without referring to the closure again." ], [ "Summary", "In this chapter we have introduced situations and reactions.", "They are, in a manner of speaking, the substantives and basic propositions of the new language.", "Much effort has been done to establish an intuitive notation for situations.", "We have then seen how to construct larger situations from smaller situations by multiplication, and larger reactions from smaller reactions by the concatenation of applications.", "This made it possible to define a reaction system in terms of a small number of situations and reactions.", "A small set of generating reactions then defines the reactions of a large set of situations, in the same way as the local transition rule defines the behaviour of a cellular automaton." ], [ "Interval-preserving Automata", "In this and the next chapter we will show how to construct a reaction system for a one-dimensional cellular automata from its transition rule.", "But since the behaviour of cellular automata varies greatly, we will consider here a subclass for which it is not too complex in a geometrical sense.", "This subclass of interval-preserving rules contains however the complex elementary cellular automata rule 54 and the computationally universal rule 110 [6]; therefore no restriction on the computational complexity of cellular automata is apparent if one restricts one's view to interval-preserving transition rules." ], [ "Intervals", "Before we can describe what interval preservation shall mean, we must define intervals and develop a notation for them and their arrangement in space-time.", "We will develop it first in the context of cellular processes and then, a bit later, for situations.", "An interval consists of a finite number of cells that are positioned without a gap.", "In the space-time viewpoint of cellular processes it is also bound to a specific moment in time.", "We define intervals together with a notation for their domain.We do not use a square bracket notation analogous to the notation $[i, j]$ for intervals on the real line: This would lead to too much optical confusion with the other uses of square brackets in this text.", "Definition 5.1 (Intervals) An interval domain at time $t \\in \\mathbb {Z}$ is a set of points of the form $I_t(i, j) = \\lbrace \\, (t, x): i \\le x < j \\,\\rbrace $ with $i$ , $j \\in \\mathbb {Z}$ and $i \\le j$ (Figure REF ).", "An interval process at time $t$ is a cellular process whose domain is an interval domain.", "Figure: An interval process at time tt.Thus the set $I_t(i, j)$ stands for the interval domain at time $t$ that reaches from the cell position $i$ to the cell position $j$ (but excludes it).", "We allow that an interval is empty: This happens if an interval has a domain of the form $I_t(i, j)$ with $i = j$ .", "In this case the time $t$ is no longer determined by the set $I_t(i, j)$ .", "We will then use the convention that the empty set is an interval at any time.", "Now consider two compatible intervals that belong to the same time.", "A nice property of intervals is that there is only a limited number of ways in which these intervals can lie with respect to each other.", "Three kinds of spatial arrangement are especially important.", "In the following definition we will introduce notations for them.", "One arrangement occurs if neither of the two intervals is a proper subset of the other one: then one of them must be at the left of the other one, if we allow overlap.", "The other two arrangements occur when one of the intervals is the left or the right end of the other interval.", "While the three notations below are intended especially for the use with intervals, their definitions are meaningful for any cellular process.", "Definition 5.2 (Spatial Arrangement of Processes) Let $\\pi , \\psi \\in \\mathcal {P}$ be two processes.", "$\\pi $ is left of $\\psi $ , written $\\pi \\mathrel {\\prec \\succ }\\psi $ , if for all $p \\in \\mathop {\\mathrm {dom}}\\pi $ there is a $\\xi \\ge 0$ such that $p + \\xi X\\in \\mathop {\\mathrm {dom}}\\psi $ , for all $q \\in \\mathop {\\mathrm {dom}}\\psi $ there is a $\\xi \\ge 0$ such that $q - \\xi X\\in \\mathop {\\mathrm {dom}}\\pi $ , and $\\pi $ is compatible with $\\psi $ .", "$\\psi $ is a left extension of $\\pi $ , written $\\psi \\supseteq _L \\pi $ , if $\\psi \\supseteq \\pi \\qquad \\text{and}\\qquad \\psi \\mathrel {\\prec \\succ }\\pi \\,.$ $\\psi $ is a right extension of $\\pi $ , written $\\pi \\subseteq _R \\psi $ , if $\\pi \\subseteq \\psi \\qquad \\text{and}\\qquad \\pi \\mathrel {\\prec \\succ }\\psi \\,.$ The expression “left of” is here used in an inclusive sense, such that always $\\pi \\mathrel {\\prec \\succ }\\pi $ .", "The relation $\\pi \\mathrel {\\prec \\succ }\\psi $ is always true when $\\pi $ or $\\psi $ are empty, and $\\psi \\supseteq _L \\pi $ and $\\pi \\subseteq _R \\psi $ are always true when $\\pi $ is empty.", "If two nonempty interval processes are related by $\\mathrel {\\prec \\succ }$ , $\\supseteq _L$ or $\\subseteq _R$ , they always occur at the same time.", "Figure: Spatial orientation of intervals.In the case of two intervals the relations of definition REF have an especially simple form (Figure REF ).", "To see this, let $\\pi _1$ and $\\pi _2$ be two intervals at time $t$ with $\\mathop {\\mathrm {dom}}\\pi _1 = I_t(i_1, j_1)$ and $\\mathop {\\mathrm {dom}}\\pi _2 = I_t(i_2, j_2)$ .", "Then, $\\pi _1 \\mathrel {\\prec \\succ }\\pi _2 & \\qquad \\text{iff}\\qquad \\pi _1 \\mathrel \\mathrm {comp}\\pi _2 && \\quad \\text{and}\\quad i_1 \\le i_2,\\ j_1 \\le j_2, \\\\\\pi _1 \\supseteq _L \\pi _2 & \\qquad \\text{iff}\\qquad \\pi _1 \\supseteq \\pi _2 && \\quad \\text{and}\\quad i_1 = i_2,\\ j_1 \\le j_2, \\\\\\pi _1 \\subseteq _R \\pi _2 & \\qquad \\text{iff}\\qquad \\pi _1 \\subseteq \\pi _2 &&\\quad \\text{and}\\quad i_1 \\le i_2,\\ j_1 = j_2\\,.$ Another connection between the relations in Definition REF is the following lemma.", "It describes $\\mathrel {\\prec \\succ }$ in terms of $\\subseteq _R$ and $\\supseteq _L$ .", "Lemma 5.3 Let $\\pi $ , $\\psi \\in \\mathcal {P}$ be two cellular processes.", "Then $\\pi \\mathrel {\\prec \\succ }\\psi $ is equivalent to $\\pi \\subseteq _R \\pi \\cup \\psi \\supseteq _L \\psi $ .", "It can be seen directly from the definition that $\\pi \\subseteq _R\\pi \\cup \\psi $ is equivalent to $\\pi \\mathrel {\\prec \\succ }\\pi \\cup \\psi $ , and that $\\pi \\cup \\psi \\supseteq _L \\psi $ is equivalent to $\\pi \\cup \\psi \\mathrel {\\prec \\succ }\\psi $ .", "Therefore $\\pi \\subseteq _R \\pi \\cup \\psi \\supseteq _L\\psi $ is true if and only if $\\pi \\mathrel {\\prec \\succ }\\pi \\cup \\psi \\mathrel {\\prec \\succ }\\psi $ .", "Assume $\\pi \\mathrel {\\prec \\succ }\\pi \\cup \\psi \\mathrel {\\prec \\succ }\\psi $ : If $p \\in \\mathop {\\mathrm {dom}}\\pi $ , then there is a $\\xi \\ge 0$ with $p + \\xi X\\in \\mathop {\\mathrm {dom}}\\psi $ because $\\pi \\cup \\psi \\mathrel {\\prec \\succ }\\psi $ , and if $p \\in \\mathop {\\mathrm {dom}}\\psi $ , then there is a $\\xi \\ge 0$ with $p - \\xi X\\in \\mathop {\\mathrm {dom}}\\pi $ because $\\pi \\mathrel {\\prec \\succ }\\pi \\cup \\psi $ ; together this shows $\\pi \\mathrel {\\prec \\succ }\\psi $ .", "Assume $\\pi \\mathrel {\\prec \\succ }\\psi $ : Then $\\pi $ and $\\psi $ are compatible, and therefore the process $\\pi \\cup \\psi $ exists.", "To check whether $\\pi \\mathrel {\\prec \\succ }\\pi \\cup \\psi $ is true, we only have to check that for $p \\in \\mathop {\\mathrm {dom}}\\psi $ there is a $\\xi \\ge 0$ such that $p - \\xi X\\in \\mathop {\\mathrm {dom}}\\pi $ , but that is true because $\\pi \\mathrel {\\prec \\succ }\\psi $ .", "The same way we can show that $\\pi \\cup \\psi \\mathrel {\\prec \\succ }\\psi $ .", "Together this proves that $\\pi \\mathrel {\\prec \\succ }\\pi \\cup \\psi \\mathrel {\\prec \\succ }\\psi $ ." ], [ "Interval Situations", "Since a sequence $u \\in \\Sigma ^$ of cell states is interpreted as a situation, its process $\\mathrm {pr}(u)$ is an interval process.", "More general, the process $\\mathrm {pr}([t, x]u)$ has the domain $I_t(x, x + \\mathopen \|u\\mathclose \|)$ .", "Therefore every interval process can be written as $\\mathrm {pr}([p]u)$ with an appropriate $p \\in \\mathbb {Z}^2$ and $u \\in \\Sigma ^$ .", "This leads to the following definition for the set of situations that represent interval processes.", "Definition 5.4 (Interval Situations) An interval situation with states in $\\Sigma $ is a situation $[p]u \\in \\mathcal {S}$ with $p \\in \\mathbb {Z}^2$ and $u \\in \\Sigma ^$ .", "For the following calculation we will need a notation that mirrors the notations for interval processes in the language of situations.", "First we introduce a notation for the left and right ends of a situation, in analogy to $\\subseteq _R$ and $\\supseteq _L$ .", "If $a$ is the left or right end of $x$ , then it is a factor of it; therefore I have chosen symbols for these concepts that remind of division operators.This notation is also influenced by the alternative notation $m \\mathrel \\backslash n$ for “$m$ divides $n$ ” by Knuth, Graham and Patashnik [19].", "Definition 5.5 (Left and Right Factors) Let $a$ and $x \\in \\mathcal {S}$ be situations.", "If there is a situation $x^{\\prime }$ such that $a x^{\\prime } = x$ , then $a$ is a left factor of $x$ .", "We will write this as $a \\mathrel {{\\backslash }\\!", "{\\backslash }}x$ .", "If there is a situation $x^{\\prime }$ such that $x = x^{\\prime } a$ , then $a$ is a right factor of $x$ .", "We will write this as $x \\mathrel {{/}\\!", "{/}}a$ .", "Next we need a notation for overlapping situations.", "Here I have chosen the symbol $\\mathbin {\\langle b\\rangle }$ , in analogy to the symbol $\\diamond $ for the overlapping of two strings that is used by Harold V. McIntosh [44].", "Definition 5.6 (Overlap) Let $b \\in \\mathcal {S}$ be a situation.", "Then the displacement $\\mathbin {\\langle b\\rangle } = [-\\delta (b)]$ is the overlap operator for $b$ .", "This notation is subject to a convention: We will use $\\mathbin {\\langle b\\rangle }$ only for a situation that contains $b$ as a factor.", "This means that: If $x \\mathbin {\\langle b\\rangle } y \\in \\mathcal {S}$ , then $x \\mathrel {{/}\\!", "{/}}b$ and $b\\mathrel {{\\backslash }\\!", "{\\backslash }}y$ , if $x \\mathbin {\\langle b\\rangle } \\in \\mathcal {S}$ , then $x \\mathrel {{/}\\!", "{/}}b$ , and if $\\mathbin {\\langle b\\rangle } y \\in \\mathcal {S}$ , then $b \\mathrel {{\\backslash }\\!", "{\\backslash }}y$ .", "The term $\\mathbin {\\langle b\\rangle }$ will never be used at its own to indicate a situation.", "With the convention for overlap operators we can state a chain rule for situations with overlap, a special case of the chain rule in (REF ).", "In contrast to the general case, this equation has also a kind of converse, with intersection instead of union: $\\mathrm {pr}(x \\mathbin {\\langle b\\rangle } y) &= \\mathrm {pr}(x) \\cup \\mathrm {pr}_{x \\mathbin {\\langle b\\rangle }}(y), \\\\\\mathrm {pr}_{x \\mathbin {\\langle b\\rangle }}(b) &=\\mathrm {pr}(x) \\cap \\mathrm {pr}_{x \\mathbin {\\langle b\\rangle }}(y)\\,.$ In almost all cases where we use $\\mathbin {\\langle b\\rangle }$ , the situation $b$ will be an interval.", "If the situations $x$ and $y$ in the term $x \\mathbin {\\langle b\\rangle } y$ are also intervals, then the relations $\\mathrel {\\prec \\succ }$ , $\\supseteq _L$ and $\\subseteq _R$ between their processes can be expressed by the overlap operator.", "Lemma 5.7 (Overlap and Spatial Arrangement of Intervals) Let $x$ , $y$ and $b \\in \\Sigma ^$ be intervals.", "Then $\\mathrm {pr}(x) &\\mathrel {\\prec \\succ }\\mathrm {pr}_{x \\mathbin {\\langle b\\rangle }}(y) && \\quad \\text{iff}\\quad x \\mathrel {{/}\\!", "{/}}b \\mathrel {{\\backslash }\\!", "{\\backslash }}y, \\\\\\mathrm {pr}(x) &\\supseteq _L \\mathrm {pr}_{x \\mathbin {\\langle b\\rangle }}(b) && \\quad \\text{iff}\\quad x \\mathrel {{/}\\!", "{/}}b, \\\\\\mathrm {pr}(b) &\\subseteq _R \\mathrm {pr}(y) && \\quad \\text{iff}\\quad b \\mathrel {{\\backslash }\\!", "{\\backslash }}y\\,.$ These relations can be derived with the help of the equivalences in ().", "We will prove (REF ) first.", "Assume that $\\mathrm {pr}(x) \\mathrel {\\prec \\succ }\\mathrm {pr}_{x \\mathbin {\\langle b\\rangle }}(y)$ .", "Then $\\mathrm {pr}(x)$ is compatible with $\\mathrm {pr}_{x\\mathbin {\\langle b\\rangle }}(y)$ , and therefore the union of these processes exists.", "This union is according to (REF ) the process $\\mathrm {pr}(x\\mathbin {\\langle b\\rangle } y)$ .", "Therefore the situation $x \\mathbin {\\langle b\\rangle } y$ exists, and this is, according to our convention, equivalent to $x \\mathrel {{/}\\!", "{/}}b\\mathrel {{\\backslash }\\!", "{\\backslash }}y$ .", "For the opposite direction, assume that $x \\mathrel {{/}\\!", "{/}}b \\mathrel {{\\backslash }\\!", "{\\backslash }}y$ is true and therefore $x \\mathbin {\\langle b\\rangle } y$ exists.", "We will use here the equivalence (REF ).", "We have $\\mathop {\\mathrm {dom}}\\mathrm {pr}(x) = I_0(0, \\mathopen \|x\\mathclose \|), \\qquad \\mathop {\\mathrm {dom}}\\mathrm {pr}_{x \\mathbin {\\langle b\\rangle }}(y)= I_0(\\mathopen \|x\\mathclose \| - \\mathopen \|b\\mathclose \|, \\mathopen \|x\\mathclose \| - \\mathopen \|b\\mathclose \| + \\mathopen \|y\\mathclose \|).$ To apply (REF ), we have to show that $\\mathrm {pr}(x)$ is compatible with $\\mathrm {pr}_{x \\mathbin {\\langle b\\rangle }}(y)$ and that $0 \\le \\mathopen \|x\\mathclose \| -\\mathopen \|b\\mathclose \|$ and $\\mathopen \|x\\mathclose \| \\le \\mathopen \|x\\mathclose \| - \\mathopen \|b\\mathclose \| + \\mathopen \|y\\mathclose \|$ .", "The first condition is true because $\\mathrm {pr}(x) \\cup \\mathrm {pr}_{\\mathbin {\\langle b\\rangle }}(y)$ exists.", "The second condition is equivalent to $\\mathopen \|x\\mathclose \| \\ge \\mathopen \|b\\mathclose \| \\le \\mathopen \|y\\mathclose \|$ .", "It is true because $b$ is, according to the convention for $\\mathbin {\\langle b\\rangle }$ , the common part of $x$ and $y$ .", "Therefore $\\mathrm {pr}(x) \\mathrel {\\prec \\succ }\\mathrm {pr}_{x \\mathbin {\\langle b\\rangle }}(y)$ .", "The other two equivalences are special cases of the first one, with $y = b$ or $x = b$ and are proved in a similar way." ], [ "Borrowing an Interval", "We will use the overlap operator for making situations and reactions more readable.", "We will split a situation into an equivalent situation that consists of overlapping parts; then we apply reactions to the parts and put the parts together again.", "For this procedure we need a notion of equivalence under which two equivalent situations initiate the same computation.", "The following definition does this.", "Definition 5.8 (Equivalent Situations) Let $a$ , $b \\in \\mathcal {S}$ be two situations.", "If $\\mathrm {pr}(a) = \\mathrm {pr}(b)\\qquad \\text{and}\\qquad \\delta (a) = \\delta (b),$ we say that $a$ is equivalent to $b$ and write it as $a \\sim b$ .", "With this definition we can express $x \\mathrel {{/}\\!", "{/}}b$ as $x \\sim x\\mathbin {\\langle b\\rangle } b$ and $b \\mathrel {{\\backslash }\\!", "{\\backslash }}y$ as $y \\sim b \\mathbin {\\langle b\\rangle } y$ .", "We will use this equivalence from time to time to split situations into overlapping parts.", "To see that equivalent situations cause the same reactions we note that if $a \\sim b$ and the closure of $\\mathrm {pr}(a)$ exists, then there is a reaction $a \\rightarrow b$ , with no other requirements on the transition rule.", "So if $b \\rightarrow x$ is a reaction for $\\varphi $ , then $a \\rightarrow x$ is also a reaction for $\\varphi $ .", "Equivalence is symmetric, therefore the converse is also true and the set of situations that start from $a$ is the same as the set of reactions that start from $b$ .", "The following derivation then illustrates the work with overlapping situations: Assume that the reaction system $R$ contains a reaction $by \\rightarrow _R b y^{\\prime }$ and that there is a situation $x \\in \\mathop {\\mathrm {dom}}R$ that ends with $b$ .", "If also $x y \\in \\mathop {\\mathrm {dom}}R$ , then there is also a reaction $x y\\rightarrow _R x y^{\\prime }$ .", "We could prove this by introducing a situation $x^{\\prime }$ such that $x = x^{\\prime } b$ and then applying the reaction $b y \\rightarrow _R b y^{\\prime }$ to $x^{\\prime } b y$ , but there is a notationally shorter way: We will then instead say that $x \\mathrel {{/}\\!", "{/}}b$ and write the following chain of reactions, withot the need to introduce $x^{\\prime }$ .", "$x y \\sim x \\mathbin {\\langle b\\rangle } b y\\rightarrow x \\mathbin {\\langle b\\rangle } b y^{\\prime }\\sim x y^{\\prime }\\,.$ With this technique the descriptions of longer chains of reactions become considerably shorter.", "An example for it occurs in (REF ).", "Note that this derivation only shows that $x y \\rightarrow x y^{\\prime }$ is a reaction for $\\varphi $ , not that it belongs to $R$ .", "This must be verified separately.", "Finding a reaction result is however often the more difficult part, especially if the derivation is long." ], [ "Interval Preservation", "Like many concepts we need the concept of interval preservation in two forms.", "One is a global form that applies to a transition rule, the other a localised form that applies to a single interval.", "The localised form is defined for situations and not processes, because that is the form where we need it in Lemma REF .", "Definition 5.9 (Interval Preservation) A transition rule $\\varphi $ for $\\Sigma $ is interval-preserving if for all interval processes $\\pi \\in \\mathcal {P}$ the process $\\Delta \\pi $ is an interval.", "Let $u \\in \\Sigma ^$ be an interval situation.", "If $\\Delta \\mathrm {pr}(u)$ is an interval, then $\\varphi $ is interval-preserving for $u$.", "Interval-preserving rules are never constant functions: If $\\varphi $ is constant, then the domain of $\\Delta \\pi $ is $\\mathbb {Z}^2$ for every process $\\pi $ , and it is therefore never an interval.", "So we do not specify explicitly for interval-preserving rules that they are non-constant.", "The simple behaviour of interval-preserving rules, announced at the beginning of this chapter, becomes visible when we look at the closure of an interval process.", "Lemma 5.10 (Closure of an Interval) If a process is an interval, then its closure exists under an interval-preserving transition rule and all its time slices are then intervals.", "Let $\\pi $ be an interval process at time $t_0$ and $\\varphi $ the transition rule.", "Then $\\pi $ is quiet before $t_0$ , and we have $\\mathop {\\mathrm {cl}}\\nolimits ^{(t)} \\pi = \\emptyset $ for $t < t_0$ and $\\mathop {\\mathrm {cl}}\\nolimits ^{(t_0)}\\pi = \\pi ^{(t_0)}$ : these time slices are intervals.", "If $t > t_0$ , then $\\pi ^{(t)} = \\emptyset $ and equation (REF ) becomes $\\mathop {\\mathrm {cl}}\\nolimits ^{(t)} \\pi =\\Delta \\mathop {\\mathrm {cl}}\\nolimits ^{(t-1)} \\pi $ .", "This means that if $\\mathop {\\mathrm {cl}}\\nolimits ^{(t-1)} \\pi $ is an interval, then $\\mathop {\\mathrm {cl}}\\nolimits ^{(t)} \\pi $ is also an interval because $\\varphi $ is interval-preserving.", "Therefore all $\\mathop {\\mathrm {cl}}\\nolimits ^{(t)} \\pi $ with $t > t_0$ are intervals by induction, and $\\mathop {\\mathrm {cl}}\\nolimits \\pi $ exists.", "We can see an example for such a closure in Figure REF .", "It is part of a larger background process, the same process as in Figure REF .", "The initial interval is shown by the squares Figure: NO_CAPTIONFigure: NO_CAPTIONFigure: NO_CAPTION Figure: The closure of an interval under Rule 54 as part of theevolution of a random initial configuration." ], [ "Testing for Interval Preservation", "The following theorem shows that it is possible to determine in finite time whether a transition rule is interval-preserving.", "Theorem 5.11 (Interval Preservation is Local) A transition rule $\\varphi $ of radius $r$ is interval-preserving if and only if it is interval-preserving for all $u \\in \\Sigma ^$ with $\\mathopen \|u\\mathclose \| \\le 2r + 1$ .", "Table: Tests for interval preservation (Part 1).Table: Tests for interval preservation (Part 2).Table: Tests for interval preservation (Part 3).With this theorem we can easily find out which of the elementary cellular automata are interval-preserving.", "This is done in Table REF –REF .", "Each row in this table describes one transition rule.", "It contains a list of evolution diagrams, one for each interval of maximal length 3.", "If a transition rule is interval-preserving, the top rows of all its diagrams must be intervals.", "As before, only one of the maximally four equivalent transition rules is shown.", "Rule 0 is omitted because it is constant and therefore cannot be interval-preserving.", "We then find that most of the elementary cellular automata are interval-preserving: Therefore, for better visual recognition only the non-preserving rules are marked with a “no” in the last column.", "We can then conclude that all elementary cellular automata except 0, 4–11, 32, 40, 130, 138, 160 and 168 are interval-preserving.", "The tables here were generated by a program; a more detailed description how one can check by hand whether a transition rule is interval-preserving appears in Section .", "[Proof of Theorem REF ] We need only to prove that if $\\varphi $ is interval-preserving for all $u \\in \\Sigma ^$ with $\\mathopen \|u\\mathclose \| \\le 2r + 1$ , then it is also interval-preserving for every $v \\in \\Sigma ^$ with $\\mathopen \|v\\mathclose \| > 2r +1$ .", "Let us therefore write $v = \\nu _0 \\dots \\nu _{\\ell -1} \\in \\Sigma ^\\ell $ with $\\ell \\ge 2r + 1$ and let $\\psi _i = \\mathrm {pr}(v)\|_{I_0(i - r, i + r)}$ be, for $-r \\le i < \\ell + r$ , a sequence of maximally $2r + 1$ events of $\\mathrm {pr}(v)$ , centred at $(0, i)$ .", "Then every point that is determined by $\\mathrm {pr}(v)$ is determined by some of the $\\psi _i$ , since interval-preserving rules are non-constant.", "There are three different shapes of $\\psi _i$ , depending on $i$ : $\\psi _i ={\\left\\lbrace \\begin{array}{ll}\\mathrm {pr}([0]\\nu _0 \\dots \\nu _{i+r})& \\text{if $-r \\le i < r$,} \\\\\\mathrm {pr}([i-r]\\nu _{i-r} \\dots \\nu _{i+r})& \\text{if $r \\le i < \\ell - r$,} \\\\\\mathrm {pr}([i-r]\\nu _{i-r} \\dots \\nu _{\\ell -1})& \\text{if $\\ell - r \\le i < \\ell + r$.}\\end{array}\\right.", "}$ Let us call them left, central and right $\\psi _i$ (Figure REF ).", "If $-r \\le i < r$ , then $\\psi _i\\subseteq \\psi _r$ , so all events that are determined by a left $\\psi _i$ are already determined by $\\psi _r$ .", "If $\\ell - r \\le i <\\ell + r$ , then $\\psi _i \\subseteq \\psi _{\\ell - 1 - r}$ , so all events determined by a right $\\psi _i$ are already determined by $\\psi _{\\ell - 1 - r}$ .", "Therefore all events determined by $\\mathrm {pr}(v)$ are determined by at least one of the central $\\psi _i$ .", "Figure: The different shapes of ψ i \\psi _i.All central $\\psi _i$ are intervals of length $2r + 1$ .", "Under a rule with radius $r$ , a central $\\psi _i$ therefore determines the point $(1, i)$ .", "Since $\\varphi $ is interval-preserving, the interval determined by $\\psi _i$ must therefore contain an event at $(1, i)$ .", "The set of events determined by $\\mathrm {pr}(v)$ is a union of such intervals, one for each $i$ with $r \\le i < \\ell -r$ , and is therefore itself an interval.", "Therefore $\\varphi $ is also interval-preserving for all $u$ with $\\mathopen \|u\\mathclose \| \\ge 2r + 1$ ." ], [ "Separating Intervals", "Let now $\\varphi $ be interval-preserving.", "We will now introduce a method to compute the states of cells at the next time step only for parts of an interval.", "For this we need as a technical tool a special kind of intervals, the “separating intervals”, that serve as a boundary between the different parts of an interval.", "Definition 5.12 (Separating Intervals) An interval process $\\pi \\in \\mathcal {P}$ is separating if for all processes $\\psi $ , $\\psi _1$ , $\\psi _2 \\in \\mathcal {P}$ with $\\psi = \\psi _1 \\cup \\psi _2$ and $\\psi _1 \\supseteq _L \\pi \\subseteq _R \\psi _2$ , $\\Delta \\psi _1 \\cup \\Delta \\psi _2 = \\Delta \\psi \\qquad \\text{and}\\qquad \\Delta \\psi _1 \\cap \\Delta \\psi _2 = \\Delta \\pi \\,.$ An interval situation $a \\in \\mathcal {S}$ is separating if $\\mathrm {pr}(a)$ is separating.", "Note here that if $\\psi _1 \\supseteq _L \\pi $ and $\\pi \\subseteq _R\\psi _2$ , then $\\psi _1 \\cap \\psi _2 = \\pi $ .", "So an equivalent formulation of (REF ) is that under the conditions of Definition REF , $\\Delta \\psi _1 \\cup \\Delta \\psi _2 =\\Delta (\\psi _1 \\cup \\psi _2)\\quad \\text{and}\\quad \\Delta \\psi _1 \\cap \\Delta \\psi _2 =\\Delta (\\psi _1 \\cap \\psi _2)\\,.$ With separating intervals we can split an interval process into parts and evolve them independently.", "They solve the following question: Assume that we have an interval $\\psi $ that consists of two parts, $\\psi _1$ and $\\psi _2$ , such that $\\psi = \\psi _1 \\cup \\psi _2$ .", "(The parts may overlap.)", "Is it then possible to compute $\\Delta \\psi _1$ and $\\Delta \\psi _2$ independently such that their union is $\\Delta \\psi $ , or does something get lost?", "The answer is: Yes, it is possible, if there is a separating interval $\\pi $ with $\\psi _1\\supseteq _L \\pi \\subseteq _R \\psi _2$ .", "By the first part of (REF ), no element of $\\Delta \\psi $ is omitted by the separate computation from $\\psi _1$ and $\\psi _2$ , while the second part requests that the contents of $\\Delta \\psi _1$ and $\\Delta \\psi _2$ are independent of each other.", "It is actually a restricted form of independence because the process $\\pi $ is subset of $\\psi _1$ and of $\\psi _2$ .", "The elements of $\\Delta \\pi $ can therefore be computed from both processes, but no other element of $\\Delta \\psi _1$ can be computed from $\\psi _2$ and no other element of $\\Delta \\psi _2$ can be computed from $\\psi _1$ because of the second part of (REF ).", "In Definition REF , the processes $\\psi _1$ and $\\psi _2$ are arranged from left to right.", "We naturally should then expect that then $\\Delta \\psi _1$ too is at the left of $\\Delta \\psi _2$ .", "The following lemma proves this for $\\psi _1$ and $\\psi _2$ that are intervals.", "It also shows that if $\\Delta \\psi _1$ and $\\Delta \\psi _2$ overlap and $\\psi _1 \\mathrel {\\prec \\succ }\\psi _2$ , then the right boundary of $\\Delta \\psi _1$ and the left boundary of $\\Delta \\psi _2$ depend only on $\\pi $ .", "Lemma 5.13 (Order Preservation for Overlapping Intervals) Let $\\varphi $ be an interval-preserving transition rule for $\\Sigma $ and let $\\pi $ , $\\psi _1$ , $\\psi _2 \\in \\mathcal {P}$ be three intervals with $\\psi _1 \\supseteq _L \\pi \\subseteq _R \\psi _2$ , where $\\pi $ is a separating interval.", "Then $\\Delta \\psi _1 \\mathrel {\\prec \\succ }\\Delta \\psi _2$ .", "Moreover, if $\\Delta \\psi _1$ and $\\Delta \\psi _2$ are nonempty and $\\mathop {\\mathrm {dom}}\\Delta \\psi _1 &= I_t(i^{\\prime }_1, j^{\\prime }_1), &\\mathop {\\mathrm {dom}}\\Delta \\psi _2 &= I_t(i^{\\prime }_2, j^{\\prime }_2),$ then $i^{\\prime }_1 \\le i^{\\prime }_2 \\le j^{\\prime }_1 \\le j^{\\prime }_2$ , and the numbers $i^{\\prime }_2$ and $j^{\\prime }_1$ depend only on $\\pi $ .", "The difficulty in the following proof is that $\\Delta \\pi $ may be empty, so we cannot prove the lemma by proving that $\\Delta \\psi _1 \\supseteq _L \\Delta \\pi \\subseteq _R \\Delta \\psi _2$ .", "Instead we must work with the interval boundaries.", "We will assume that the four processes $\\psi _1$ , $\\psi _2$ , $\\Delta \\psi _1$ and $\\Delta \\psi _2$ are all nonempty, because otherwise the lemma would be trivially true.", "We will now view $\\pi $ as a constant and $\\psi _1$ and $\\psi _2$ as variables.", "For all the other quantities in the proof we will keep track whether they depend on $\\psi _1$ or $\\psi _2$ .", "Let now $\\mathop {\\mathrm {dom}}\\psi _1 &= I_t(i_1, j_1), &\\mathop {\\mathrm {dom}}\\psi _2 &= I_t(i_2, j_2)\\,.$ Then the numbers $i_1$ , $j_1$ , ..., $i^{\\prime }_2$ , $j^{\\prime }_2$ in (REF ) and (REF ) are uniquely determined by $\\psi _1$ respectively $\\psi _2$ .", "Because we have $\\psi _1 \\supseteq _L \\pi \\subseteq _R \\psi _2$ , we must have $i_1 \\le i_2 \\le j_1 \\le j_2$ .", "The intersection of $\\psi _1$ and $\\psi _2$ is $\\pi $ , therefore $\\mathop {\\mathrm {dom}}\\pi = I_0(i_2, j_1)$ , so $i_2$ and $j_1$ are constants.", "This also means that if $\\psi _1$ is a large set, $i_1$ must be a small number, and if $\\psi _2$ is large, then $j_2$ is a large number.", "The set $\\Delta \\pi $ may be empty, so we cannot give an expression for $\\mathop {\\mathrm {dom}}\\Delta \\pi $ similar to those in (REF ).", "But the number of elements in $\\Delta \\pi $ is a well-defined constant, and we will call it $\\ell $ .", "We have always $\\mathop {\\mathrm {dom}}\\Delta \\psi _1 &\\supseteq I_{t+1}(i_1 - r, j_1 + r),&\\mathop {\\mathrm {dom}}\\Delta \\psi _2 &\\subseteq I_{t+1}(i_2 + r, j_2 - r)\\,.$ Assume now that $j_2 > j_1 + 2r$ , where $r$ is the radius of $\\varphi $ .", "Then the rightmost elements of $\\Delta \\psi _1 \\cup \\Delta \\psi _2$ must belong to $\\Delta \\psi _2$ .", "This means that $\\Delta \\psi _1$ is at the left of $\\Delta \\psi _2$ and they overlap at $\\ell $ events.", "So we must have $i^{\\prime }_1 \\le i^{\\prime }_2 \\le j^{\\prime }_1\\le j^{\\prime }_2$ and $i^{\\prime }_2 + \\ell = j^{\\prime }_1$ .", "If we keep $\\psi _2$ fixed but let $\\psi _1$ vary arbitrarily, the condition $i^{\\prime }_2 + \\ell = j^{\\prime }_1$ must always be true.", "This means that $j^{\\prime }_1$ stays the same for all values of $\\psi _1$ , and because it only depends on $\\psi _1$ , it must be a constant.", "The same way, by assuming $i_1 < i_2 - 2r$ , we can see that $i^{\\prime }_2$ is a constant.", "So we have always $i^{\\prime }_2 \\le j^{\\prime }_1$ , and $i^{\\prime }_2$ and $j^{\\prime }_1$ are constants.", "For the validity of $\\Delta \\psi _1 \\cap \\Delta \\psi _2 = \\Delta \\pi $ it is then necessary that $i^{\\prime }_1 \\le i^{\\prime }_2\\le j^{\\prime }_1 \\le j^{\\prime }_2$ is true in general, which means that $\\psi _1\\mathrel {\\prec \\succ }\\psi _2$ .", "A corollary of this lemma shows what happens if we split $\\pi $ into two intervals.", "Corollary 5.14 (Separation by Bounded Intervals) Let $\\varphi $ be an interval-preserving transition rule for $\\Sigma $ .", "Let $\\pi _1$ , $\\pi _2 \\in \\mathcal {P}$ be separating intervals and $\\psi _1$ , $\\psi _2 \\in \\mathcal {P}$ be intervals with $\\psi _1 \\supseteq _L \\pi _1 \\mathrel {\\prec \\succ }\\pi _2 \\subseteq _R \\psi _2\\,.$ Then $\\Delta \\psi _1 \\mathrel {\\prec \\succ }\\Delta \\psi _2$ .", "Let $\\pi $ be an arbitrary interval process that reaches from the left end of $\\pi _1$ to the right end of $\\pi _2$ , such that we have $\\pi _1 \\subseteq _R \\pi \\supseteq _L \\pi _2$ .", "Then $\\pi \\cup \\psi _2$ is an interval, and we have $\\psi _1\\supseteq _L \\pi _1 \\subseteq _R \\pi \\cup \\psi _2$ .", "So we can apply Lemma REF and get $\\Delta \\psi _1 \\mathrel {\\prec \\succ }\\Delta (\\pi \\cup \\psi _2)$ .", "Because $\\pi _2$ is separating and $\\pi \\supseteq _L \\pi _2 \\subseteq _R \\psi _2$ , we have $\\Delta (\\pi \\cup \\psi _2) = \\Delta \\pi \\cup \\Delta \\psi _2$ .", "We derive from the resulting relation $\\Delta \\psi _1 \\mathrel {\\prec \\succ }\\Delta \\pi \\cup \\Delta \\psi _2$ with Lemma REF the relation $\\Delta \\psi _1 \\subseteq _R \\Delta \\psi _1 \\cup \\Delta \\pi \\cup \\Delta \\psi _2$ , and this leads to $\\Delta \\psi _1 \\subseteq _R \\Delta \\psi _1 \\cup \\Delta \\psi _2$ .", "The same way we can also prove $\\Delta \\psi _1 \\cup \\Delta \\psi _2 \\supseteq _L \\Delta \\psi _2$ .", "These two relations together are equivalent to $\\Delta \\psi _1 \\mathrel {\\prec \\succ }\\Delta \\psi _2$ , again by Lemma REF ." ], [ "The Set of Separating Intervals", "All this assumes that separating intervals exist.", "We need to make that certain and would also like to get an overview about which intervals are separating.", "This we will do in two steps.", "First we will prove that being a separating interval is a monotone property: an interval that contains a separating interval as a subset is itself separating.", "Then we will show that under a rule of radius $r$ , every interval of exactly $2r$ cells is separating.", "From this we can then conclude that every interval of at least $2r$ cells is separating.", "Lemma 5.15 (Being Separating is Monotone) Let $\\varphi $ be an interval-preserving transition rule and let $\\pi \\subseteq \\pi ^{\\prime } \\in \\mathcal {P}$ be two intervals.", "If $\\pi $ is separating, then $\\pi ^{\\prime }$ is separating too.", "We will first prove the lemma for the case that $\\pi \\subseteq _R\\pi ^{\\prime }$ .", "For this, let $\\psi ^{\\prime }_1$ , $\\psi ^{\\prime }_2$ , $\\psi ^{\\prime } \\in \\mathcal {P}$ be any processes with $\\psi ^{\\prime }_1 \\supseteq _L \\pi ^{\\prime }\\subseteq _R \\psi ^{\\prime }_2$ and $\\psi ^{\\prime }_1 \\cup \\psi ^{\\prime }_2 = \\psi ^{\\prime }$ .", "Let $\\psi _1= (\\psi ^{\\prime }_1 \\setminus \\pi ^{\\prime }) \\cup \\pi $ ; then $\\psi _1 \\supseteq _L \\pi \\subseteq _R \\psi ^{\\prime }_2$ and $\\psi _1 \\cup \\psi ^{\\prime }_2 = \\psi ^{\\prime }$ .", "We have $\\Delta \\psi ^{\\prime }_1 = \\Delta (\\psi _1 \\cup \\pi ^{\\prime }) =\\Delta \\psi _1 \\cup \\Delta \\pi ^{\\prime }$ because $\\pi $ is separating.", "Therefore, $\\begin{alignedat}[b]{2}\\Delta \\psi ^{\\prime }_1 \\cup \\Delta \\psi ^{\\prime }_2&= \\Delta \\psi _1 \\cup \\Delta \\pi ^{\\prime } \\cup \\Delta \\psi ^{\\prime }_2 \\\\&= \\Delta \\psi _1 \\cup \\Delta \\psi ^{\\prime }_2&&\\qquad \\text{because $\\Delta \\pi ^{\\prime } \\subseteq \\Delta \\psi ^{\\prime }_2$} \\\\&= \\Delta (\\psi _1 \\cup \\psi ^{\\prime })&&\\qquad \\text{because $\\psi _1 \\supseteq _L \\pi \\subseteq _R \\psi ^{\\prime }_2$} \\\\&= \\Delta \\psi ^{\\prime }\\,.&&\\qquad \\text{because $\\psi _1 \\subseteq \\psi $}\\end{alignedat}$ On the other hand, $\\begin{alignedat}[b]{2}\\Delta \\psi ^{\\prime }_1 \\cap \\Delta \\psi ^{\\prime }_2&= (\\Delta \\psi _1 \\cup \\Delta \\pi ^{\\prime })\\cap \\Delta \\psi ^{\\prime }_2 \\\\&= (\\Delta \\psi _1 \\cap \\Delta \\psi ^{\\prime }_2)\\cup \\Delta \\pi ^{\\prime }&&\\qquad \\text{because $\\Delta \\pi ^{\\prime } \\subseteq \\Delta \\psi ^{\\prime }_2$} \\\\&= \\Delta \\pi \\cup \\Delta \\pi ^{\\prime }&&\\qquad \\text{because $\\psi _1 \\supseteq _L \\pi \\subseteq _R \\psi ^{\\prime }_2$} \\\\&= \\Delta \\pi ^{\\prime }\\,.&&\\qquad \\text{because $\\Delta \\pi \\subseteq \\Delta \\pi ^{\\prime }$}\\\\\\end{alignedat}$ This proves the lemma in the case that $\\pi \\subseteq _R \\pi ^{\\prime }$ .", "The same kind of argument works when $\\pi ^{\\prime } \\supseteq _L \\pi $ .", "In the general case we note that if $\\pi \\subseteq \\pi ^{\\prime }$ , then there is always an interval process $\\pi ^{\\prime \\prime }$ such that $\\pi \\subseteq _R \\pi ^{\\prime \\prime }$ and $\\pi ^{\\prime } \\supseteq _L \\pi ^{\\prime \\prime }$ .", "This reduces the general case to the two other cases.", "Lemma 5.16 (Existence of Separating Intervals) Under a transition rule with radius $r$ , every interval that consists of at least $2r$ events is separating.", "The proof makes use of the fact that determinedness is a local property: if $\\theta $ is a process, then a point $p$ is determined by $\\theta $ if and only if it is determined by $\\theta \|_{N(p - T, r)}$ .", "We have seen this in Lemma REF .", "The set $N(p - T, r)$ is the neighbourhood domain of $p$ for the previous time step, defined in (REF ).", "Let $\\pi $ be the separating interval and let $\\psi $ , $\\psi _1$ and $\\psi _2$ be as in Definition REF .", "We know already that $\\Delta \\psi _1 \\cup \\Delta \\psi _2\\subseteq \\Delta \\psi $ and $\\Delta \\pi \\subseteq \\Delta \\psi _1 \\cap \\Delta \\psi _2$ because $\\Delta $ is monotone (Lemma REF ).", "So it remains to prove $\\Delta \\psi \\subseteq \\Delta \\psi _1 \\cup \\Delta \\psi _2\\qquad \\text{and}\\qquad \\Delta \\psi _1 \\cap \\Delta \\psi _2\\subseteq \\Delta \\pi \\,.$ Let now $p\\in \\mathop {\\mathrm {dom}}\\Delta \\psi $ be an arbitrary point and let $N_p$ stand for $N(p - T, r)$ .", "The proof of (REF ) then relies on the fact that $N_p$ is an interval domain of length $2r + 1$ , but $\\mathop {\\mathrm {dom}}\\pi $ has at least $2r$ points.", "The points of ${N_p} \\setminus \\mathop {\\mathrm {dom}}\\pi $ must therefore be either completely at the left or completely at the right of $\\pi $ .", "In the first case we have $(\\psi _2 \\setminus \\pi )\|_{N_p} = \\emptyset $ , in the second, $(\\psi _1 \\setminus \\pi )\|_{N_p} = \\emptyset $ .", "Therefore, $\\psi _2\|_{N_p} = \\pi \|_{N_p}\\qquad \\text{or}\\qquad \\psi _1\|_{N_p} = \\pi \|_{N_p}\\,.$ Assume first that $p \\in \\mathop {\\mathrm {dom}}\\Delta \\psi $ .", "Because $\\psi \|_{N_p}$ is equal to $\\psi _1\|_{N_p} \\cup \\psi _2\|_{N_p}$ , and with equation (REF ), the process $\\psi \|_{N_p}$ must either be equal to $\\pi \|_{N_p} \\cup \\psi _2\|_{N_p} =\\psi _2\|_{N_p}$ or $\\psi _1\|_{N_p} \\cup \\pi \|_{N_p} = \\psi _1\|_{N_p}$ or both.", "In the first case, $p \\in \\mathop {\\mathrm {dom}}\\Delta \\psi _1$ , in the second case, $p \\in \\mathop {\\mathrm {dom}}\\Delta \\psi _2$ .", "This proves $\\Delta \\psi \\subseteq \\Delta \\psi _1 \\cup \\Delta \\psi _2$ .", "Assume now that $p \\in \\mathop {\\mathrm {dom}}(\\Delta \\psi _1 \\cap \\Delta \\psi _2)$ .", "Then $p$ depends on $\\psi _1\|_{N_p}$ and on $\\psi _2\|_{N_p}$ .", "One of these processes is equal to $\\pi \|_{N_p}$ by (REF ), therefore $p \\in \\mathop {\\mathrm {dom}}\\Delta (\\pi \|_{N_p})$ .", "This proves $\\Delta \\psi _1 \\cap \\Delta \\psi _2 \\subseteq \\Delta \\pi $ .", "A transition rule may also have separating intervals of less than $2r$ elements.", "So to get an overview about the separating intervals of a specific transition rule, we should know the set of its minimal separating intervals.", "We will actually need three kinds of minimal intervals, defined below.", "Definition 5.17 (Minimal Separating Interval) A separating interval process is left minimal if no events can be removed from its right side without making it non-separating.", "It is right minimal if no events can be removed from its left side without making it non-separating.", "It is minimal if it is both left minimal and right minimal.", "Left and right minimal intervals occur as the boundaries of a separating interval: If $\\psi $ is a separating interval, then there are intervals $\\pi _1$ and $\\pi _2$ such that $\\pi _1 \\subseteq _R \\psi \\supseteq _L\\pi _2$ .", "The processes $\\pi _1$ and $\\pi _2$ can be viewed as the “left and right end” of $\\psi $ .", "The shortest interval $\\pi _1$ that is still separating is then a left minimal interval, and the shortest separating interval $\\pi _2$ is a right minimal interval." ], [ "Example: The Elementary Cellular Automata", "For the elementary cellular automata there exists a simple test to find the minimal separating intervals.", "The main reason for this is the small radius of their transition rules.", "This means, with Lemma REF , that we only need to check whether there are intervals of length 0 and 1 that are separating.", "The case of a separating interval of length 0 does indeed occur, and it means that the cells never interact.", "There are two rules that have this property: Rule 51, which lets the cells alternate between the states 0 and 1, and the identity function, Rule 204.", "For the other rules we must check whether there are intervals that consist of a single event and are separating.", "We will call such an interval $\\pi $ and place it at the origin; it consists of the single event $[0, 0] \\sigma $ .", "As we have seen in the proof of Lemma REF , we need only to verify that for all cellular processes $\\psi _1$ , $\\psi _2$ with $\\psi _1 \\supseteq _L \\pi \\subseteq _R \\psi _2$ we have $\\Delta (\\psi _1 \\cup \\psi _2) \\subseteq \\Delta \\psi _1 \\cup \\Delta \\psi _2$ and $\\Delta \\psi _1 \\cap \\Delta \\psi _2 \\subseteq \\Delta \\pi $ in order to prove that $\\pi $ is a separating interval.", "We will use the convention that at time 0, the cell at position $i$ is in state $\\xi _i$ ; we have then $\\sigma = \\xi _0$ .", "For an arbitrary event at time 1 we will write $e$ : its state is $\\eta $ and its location, $x$ , such that we have $e = [1, x]\\eta $ .", "Since $\\varphi $ has radius 1, the state of the event $e$ can only depend on $\\xi _{x -1}$ , $\\xi _x$ and $\\xi _{x + 1}$ : This will be important in the following derivation.", "Let now $e$ be element of $\\Delta (\\psi _1 \\cup \\psi _2)$ .", "Then, if $x <0$ , we have $e \\in \\Delta \\psi _1$ and if $x > 1$ , we have $e \\in \\Delta \\psi _2$ .", "The remaining case, $x = 0$ , is the key to finding a necessary condition for $\\pi $ : If the state of $e$ depends both on events in $\\psi _1$ and $\\psi _2$ , then $e$ cannot be an element of $\\Delta \\psi _1 \\cup \\Delta \\psi _2$ , and $\\pi $ cannot be separating.", "This can only occur when $\\varphi (\\xi _{-1}, \\sigma , \\xi _1)$ , the state of $e$ , depends on both $\\xi _{-1}$ and $\\xi _1$ .", "Therefore a necessary condition for $\\pi $ being separating is that such a dependency does not happen.", "We can ensure that by requiring that at least one of the following two equations is true: $\\forall \\xi _{-1} \\in \\Sigma &\\colon &\\varphi (\\xi _{-1}, \\sigma , 0) &= \\varphi (\\xi _{-1}, \\sigma , 1), \\\\\\forall \\xi _1 \\in \\Sigma &\\colon &\\varphi (0, \\sigma , \\xi _1) &= \\varphi (1, \\sigma , \\xi _1)\\,.$ Here the first equation means that the cell to the right has no influence on the next state of a cell in state $\\sigma $ , and the second, that the cell at the left has no influence on the next state.", "The conditions (REF ) are also sufficient.", "To show this, we will first prove that $\\Delta (\\psi _1 \\cup \\psi _2)\\subseteq \\Delta \\psi _1 \\cup \\Delta \\psi _2$ .", "In the proof we will assume that $e \\in \\Delta (\\psi _1 \\cup \\psi _2)$ .", "As we have seen before, if $x < 0$ , then $e \\in \\Delta \\psi _1$ and if $x > 1$ , then $e\\in \\Delta \\psi _2$ .", "In the remaining case of $x = 0$ , at least one of the two conditions in (REF ) must be true.", "If (REF ) is true, then $\\eta $ depends only on $\\xi _{-1}$ and $\\sigma $ , and therefore $e \\in \\Delta \\psi _1$ .", "If () is true, then $\\eta $ depends only on $\\sigma $ and $\\xi _1$ , and $e \\in \\Delta \\psi _2$ .", "This shows that $\\Delta (\\psi _1 \\cup \\psi _2) \\subseteq \\Delta \\psi _1 \\cup \\Delta \\psi _2$ .", "Next we prove that $\\Delta \\psi _1 \\cap \\Delta \\psi _2 \\subseteq \\Delta \\pi $ .", "We will assume that $e \\in \\Delta \\psi _1 \\cap \\Delta \\psi _2$ .", "Since $\\varphi $ has radius 1, the only possible values for $x$ are then $-1$ , 0 and 1.", "In the case of $x = -1$ we use the fact that $e$ is an element of $\\Delta \\psi _2$ .", "This means that we have $\\varphi (\\xi _{-2},\\xi _{-1}, \\sigma ) = \\eta $ for all $\\xi _{-2}$ , $\\xi _{-1} \\in \\Sigma $ and that therefore $e \\in \\Delta \\pi $ .", "In the case of $x = 0$ , the condition $e \\in \\Delta \\psi _1$ requires that (REF ) is true, and $e \\in \\Delta \\psi _2$ requires that () is true.", "Taken together, the two conditions imply that $\\varphi (\\xi _{-1}, \\sigma ,\\xi _1) = \\eta $ for all $\\xi _{-1}$ , $\\xi _1 \\in \\Sigma $ and that we have here again $e \\in \\Delta \\pi $ .", "The case $x = 1$ can be handled in a similar way as $x = -1$ .", "This shows that $\\Delta \\psi _1 \\cap \\Delta \\psi _2 \\subseteq \\Delta \\pi $ .", "We therefore have now proved that (REF ) are necessary and sufficient conditions that the interval $\\pi $ is separating.", "The tests (REF ) have been done by a program for all interval-preserving elementary cellular automata different from 51 and 204.", "The results are shown in Table REF .", "If none of the length 1 intervals are separating, then the intervals of length 2 are the minimal separating intervals.", "If an interval $\\pi $ of length 1 is minimally separating, then only those intervals of length 2 are separating that do not contain $\\pi $ .", "So if, e. g., 0 is a separating interval but 1 is not separating, then the only minimal separating interval is 11.", "This argument explains the second row of Table REF ; the other rows are explained similarly.", "Table: Separating intervals of the interval-preserving elementarycellular automata.In Section and in the context of Rule 54, we will see a related method to find the minimal separating intervals of a transition rule.", "It uses the characteristic reactions, which are defined next." ], [ "Characteristic Reactions", "We will now introduce the concept of characteristic reactions as a way to express the relation between an interval $\\pi $ and the set $\\Delta \\pi $ in the language of situations and reactions.", "The characteristic reaction specifies the location and the dimensions of $\\Delta \\pi $ in relation to $\\pi $ in a kind of shorthand.", "We will introduce it in two steps.", "The first step is for the case that $\\Delta \\pi $ is nonempty.", "Then the notion of the location of $\\Delta \\pi $ has an obvious meaning.", "If $\\pi $ is separating and it is part of a longer interval, then the cells of $\\Delta \\pi $ serve as a separator between the events determined by the cells left of $\\pi $ from those determined by the cells right of $\\pi $ .", "This property is important for proofs about the behaviour of reaction systems, so we would like to have it for all of the separating intervals.", "We will use it in the second step to extend the notion of the characteristic reaction such that for a separating interval $\\pi $ we can speak of the “location” of $\\Delta \\pi $ even then when $\\Delta \\pi $ is empty.", "The characteristic reactions themselves will however not become part of the final reaction system; they are only a tool to define it." ], [ "Construction of the Characteristic Reactions", "We can express the property of a rule $\\varphi $ to be interval-preserving in the following way: If $a \\in \\Sigma ^$ , then there are $i \\in \\mathbb {Z}$ and $\\hat{a} \\in \\Sigma ^$ such that $\\Delta \\mathrm {pr}(a) = \\mathrm {pr}([1, i]\\hat{a})$ .", "The interval situation $\\hat{a}$ is then always uniquely determined by $a$ .", "So we define Definition 5.18 (Determined Interval) Let $\\varphi $ be an interval-preserving transition rule for $\\Sigma $ and let $a \\in \\Sigma ^$ .", "Then the situation $\\hat{a} \\in \\Sigma ^$ for which there is an $i \\in \\mathbb {Z}$ such that $\\Delta \\mathrm {pr}(a) = \\mathrm {pr}([1, i] \\hat{a}),$ is the determined interval of $a$ under $\\varphi $ .", "The “hat” accent of $\\hat{a}$ should remind of the operator $\\Delta $ .", "In contrast to $\\hat{a}$ , the number $i$ is only then uniquely determined when $\\hat{a} \\ne [0]$ , because only then $\\Delta \\mathrm {pr}(a) \\ne \\emptyset $ .", "In this case we can express the relation between $\\mathrm {pr}(a)$ and $\\Delta \\mathrm {pr}(a)$ by a reaction.", "This definition is important because it specifies the numbers $i$ and $j$ , which will be needed later to define the actual reaction system.", "Figure: The two sides of the characteristicreaction (), overlayed.Definition 5.19 (Characteristic Reactions, Preliminary Form) Let $\\varphi $ be an interval-preserving transition rule for $\\Sigma $ .", "Let $a \\in \\Sigma ^$ be an interval with $\\hat{a} \\ne [0]$ and let $\\Delta \\mathrm {pr}(a) = \\mathrm {pr}([1, i] \\hat{a})$ .", "Then the reaction $a \\rightarrow [1, i] \\hat{a} [-1, j]$ with $j = \\mathopen \|a\\mathclose \| - i - \\mathopen \|\\hat{a}\\mathclose \|$ , is the characteristic reaction for $a$ under $\\varphi $ .", "(Figure REF .)", "Now we extend this definition to interval situations $a$ for which $\\hat{a}$ can be empty—but in this case $a$ must be a separating interval.", "We can do this by using other separating intervals as “test functions”.", "The following lemma is there to show that the final definition of separating intervals extends the preliminary definition.", "Theorem 5.20 (Intervals as Reactions) Let $\\varphi $ be an interval-preserving transition rule for $\\Sigma $ and $a \\in \\Sigma ^$ a separating interval for $\\varphi $ .", "Then there exist $i$ , $j \\in \\mathbb {Z}$ such that for all $x \\in \\Sigma ^$ with $\\widehat{x a} \\ne [0]$ there is an $i^{\\prime } \\in \\mathbb {Z}$ such that the characteristic reaction for $x a$ is $xa \\rightarrow [1, i^{\\prime }] \\widehat{x a} [-1, j],$ and for all $y \\in \\Sigma ^$ with $\\widehat{a y} \\ne [0]$ there is a $j^{\\prime } \\in \\mathbb {Z}$ such that the characteristic reaction for $a y$ is $ay \\rightarrow [1, i] \\widehat{a y} [-1, j^{\\prime }].$ If also $\\hat{a} \\ne [0]$ , then $a$ has the characteristic reaction $a \\rightarrow [1, i] \\hat{a} [-1, j]$ , the same as in (REF ).", "We will use in this proof that if $u \\rightarrow [1, i] \\hat{u} [-1, j]$ is a characteristic reaction, then $\\mathop {\\mathrm {dom}}\\Delta \\mathrm {pr}(u) = I_1(i,\\mathopen \|u\\mathclose \| - j)$ : The process $\\Delta \\mathrm {pr}(u)$ reaches from $i$ cells to the left of $\\mathrm {pr}(u)$ to $j$ cells to the right of $\\mathrm {pr}(u)$ , one time step later.", "To apply Lemma REF we first define processes $\\pi $ , $\\psi _1$ and $\\psi _2$ with $\\psi _1 \\supseteq _L \\pi \\subseteq _R \\psi _2$ , namely $\\psi _1 = \\mathrm {pr}_{[-\\mathopen \|x\\mathclose \|]}(x a), \\qquad \\pi = \\mathrm {pr}(a), \\qquad \\psi _2 = \\mathrm {pr}(ay)\\,.$ In order to verify the inclusion $\\psi _1 \\supseteq _L \\pi $ we note that we have $\\mathrm {pr}_{[-\\mathopen \|x\\mathclose \|]}(x a) = \\mathrm {pr}_{[-\\mathopen \|x\\mathclose \|]}(x) \\cup \\mathrm {pr}_{[-\\mathopen \|x\\mathclose \|]x}(a)$ with the chain rule, and $\\mathrm {pr}_{[-\\mathopen \|x\\mathclose \|]x}(a)= \\mathrm {pr}(a)$ because $\\delta ([-\\mathopen \|x\\mathclose \|]x) = (0, 0)$ .", "Therefore $\\mathrm {pr}_{[-\\mathopen \|x\\mathclose \|]}(x a) = \\mathrm {pr}(x) \\cup \\mathrm {pr}(a) \\supseteq _L \\mathrm {pr}(a)$ , which means that $\\psi _1 \\supseteq _L \\pi $ .", "This explains also why the shift of $\\mathrm {pr}(xa)$ to the left with $[-\\mathopen \|x\\mathclose \|]$ is necessary.", "The other inclusion, $\\pi \\subseteq _R \\psi _2$ , can also be verified by an application of the chain rule.", "The domains of the processes $\\psi _1$ and $\\psi _2$ are then $\\mathop {\\mathrm {dom}}\\psi _1 = I_0(-\\mathopen \|x\\mathclose \|, \\mathopen \|a\\mathclose \|), \\qquad \\mathop {\\mathrm {dom}}\\psi _2 = I_0(0, \\mathopen \|ay\\mathclose \|),$ and the events determined by them are located at $\\mathop {\\mathrm {dom}}\\Delta \\psi _1 = I_1(-\\mathopen \|x\\mathclose \| + i^{\\prime }, \\mathopen \|a\\mathclose \| - j), \\qquad \\mathop {\\mathrm {dom}}\\Delta \\psi _2 = I_1(i, \\mathopen \|ay\\mathclose \| - j^{\\prime })\\,.$ as we can see from the characteristic reactions (REF ) and from the remark before the proof.", "For the process $\\psi _1$ we have to keep in mind that it, and therefore also $\\Delta \\psi _1$ , is shifted to the left by $\\mathopen \|x\\mathclose \|$ cells.", "Then we know by Lemma REF that the boundaries of the intervals in (REF ) are arranged in the form $-\\mathopen \|x\\mathclose \| + i^{\\prime } \\le i \\le \\mathopen \|a\\mathclose \| - j \\le \\mathopen \|ay\\mathclose \| - j^{\\prime }$ and that $i$ and $\\mathopen \|a\\mathclose \| - j$ only depend on $a$ .", "Therefore the variables $i$ and $j$ only depend on $a$ .", "Assume now that $\\hat{a} \\ne [0]$ and therefore $\\Delta \\pi \\ne \\emptyset $ .", "Then $\\mathop {\\mathrm {dom}}\\pi = I_0(0, \\mathopen \|a\\mathclose \|)$ and $\\mathop {\\mathrm {dom}}\\Delta \\pi = I_1( i, \\mathopen \|a\\mathclose \| - j)$ , which means that the characteristic reaction for $a$ (in the preliminary form of Definition REF ) must be $a \\rightarrow [1, i] \\hat{a}[-1, j]$ .", "This theorem then justifies the following definition of the characteristic reaction for separating intervals.", "Definition 5.21 (Characteristic Reactions, Final Form) Let $a \\in \\Sigma ^$ be a separating interval situation.", "If there are intervals $x$ , $y \\in \\Sigma ^$ with $\\widehat{x a}\\ne [0] \\ne \\widehat{a y}$ , and they have the characteristic reactions $x a \\rightarrow [1, i^{\\prime }] \\widehat{x a} [-1, j]$ and $a y \\rightarrow [1, i] \\widehat{a y} \\, [-1, j^{\\prime }]$ , then the reaction $a \\rightarrow [1, i] \\hat{a} [-1, j]$ is the characteristic reaction for $a$ under $\\varphi $ .", "The following lemma expresses the monotony of the closure in terms of characteristic reaction.", "It shows that in the previous definition $\\hat{a}$ is always a part of $\\widehat{x a}$ and $\\widehat{a y}$ , and how to recover it.", "Lemma 5.22 (Reactions of Separating Intervals) Let $a \\in \\Sigma ^$ be a separating interval under an interval-preserving transition rule, and let $x$ , $y\\in \\Sigma ^$ .", "Then $\\widehat{x a} \\mathrel {{/}\\!", "{/}}\\hat{a}$ and $\\hat{a} \\mathrel {{\\backslash }\\!", "{\\backslash }}\\widehat{a y}$ .", "Let $\\varphi $ be the transition rule.", "We will prove only the first equivalence, the second one is its mirror image and the proof is similar.", "Assume that the characteristic reaction of $a$ and $x a$ are $a \\rightarrow [1, i] \\hat{a} [-1, j]\\qquad \\text{and}\\qquad x a \\rightarrow [1, i^{\\prime }] \\widehat{x a} [-1, j]\\,.$ We will show first that $\\mathrm {pr}_{x [1, i]}(\\hat{a}) \\subseteq \\mathrm {pr}_{[1,i^{\\prime }]}(\\widehat{x a})$ .", "We see from (REF ) that $\\mathrm {pr}_{[1, i]}(\\hat{a}) = \\Delta \\mathrm {pr}(a)$ and $\\Delta \\mathrm {pr}(x a) =\\mathrm {pr}_{[1, i^{\\prime }]}(\\widehat{x a})$ .", "With the chain rule we get $\\mathrm {pr}_{x}(a) \\subseteq \\mathrm {pr}(xa)$ , by the monotony of the $\\Delta $ operator, $\\Delta \\mathrm {pr}_{x}(a) \\subseteq \\Delta \\mathrm {pr}(xa)$ , and by putting these relations together we get $\\mathrm {pr}_{x [1, i]}(\\hat{a}) = \\Delta \\mathrm {pr}_{x}(a) \\subseteq \\Delta \\mathrm {pr}(xa) = \\mathrm {pr}_{[1, i^{\\prime }]}(\\widehat{x a})\\,.$ Next we show that $\\delta (x [1, i]) = \\delta ([1, i^{\\prime }] \\widehat{x a}\\mathbin {\\langle \\hat{a}\\rangle })$ .", "We see from the characteristic reaction for $a$ that $i = \\mathopen \|a\\mathclose \| - \\mathopen \|\\hat{a}\\mathclose \| - j$ and from the characteristic reaction for $x a$ that $j = \\mathopen \|x a\\mathclose \| - i^{\\prime } - \\mathopen \|\\widehat{x a}\\mathclose \|$ .", "Therefore $i = \\mathopen \|a\\mathclose \| - \\mathopen \|\\hat{a}\\mathclose \| - \\mathopen \|x a\\mathclose \| + i^{\\prime } + \\mathopen \|\\widehat{x a}\\mathclose \| =i^{\\prime } + \\mathopen \|\\widehat{x a}\\mathclose \| - \\mathopen \|x\\mathclose \| - \\mathopen \|\\hat{a}\\mathclose \|$ , and $\\delta (x[1, i]) = (1, \\mathopen \|x\\mathclose \| + i) = (1, i^{\\prime } + \\mathopen \|\\widehat{x a}\\mathclose \| -\\mathopen \|\\hat{a}\\mathclose \|) = \\delta ([1, i^{\\prime }] \\widehat{x a} \\mathbin {\\langle \\hat{a}\\rangle })$ .", "This means that $\\begin{aligned}[b]\\mathrm {pr}_{[1, i^{\\prime }]}(\\widehat{x a})&= \\mathrm {pr}_{[1, i^{\\prime }]}(\\widehat{x a}) \\cup \\mathrm {pr}_{x [1, i]}(\\hat{a}) \\\\&= \\mathrm {pr}_{[1, i^{\\prime }]}(\\widehat{x a}) \\cup \\mathrm {pr}_{[1, i^{\\prime }] \\widehat{x a}}(\\mathbin {\\langle \\hat{a}\\rangle } \\hat{a})= \\mathrm {pr}_{[1, i^{\\prime }]}(\\widehat{x a} \\mathbin {\\langle \\hat{a}\\rangle } \\hat{a}),\\end{aligned}$ which also means that $\\widehat{x a} \\mathrel {{/}\\!", "{/}}\\hat{a}$ ." ], [ "Summary", "In this chapter we were concerned with interval behaviour and the left-to-right arrangement of intervals.", "The goal of this was to find a subset of the transition rules that harmonise with the definition of reaction systems in the previous chapter.", "We have seen that interval-preserving transition rules are such a subset.", "Interval preservation is a useful property because intervals already play an important role in cellular automata: The transition rule is expressed in terms of intervals.", "Intervals are conceptually simple cellular processes: Their domain can be expressed with three numbers, and their content can be expressed in a natural way as a sequence of cell states.", "They are therefore easy to express with situations.", "Interval preservation then also puts a limit on the complicatedness of the closure of an interval.", "Every time slice of it is an interval, so it does not become more complicated over time.", "The complexity of the behaviour of an interval-preserving cellular automaton is therefore confined to the interior of this closure.", "A specific result of this chapter was the usefulness of separating intervals.", "They form the boundaries between different regions in a cellular automaton that do not influence each other in the next time step.", "This makes them useful for the selective evolution of an initial configuration of cells that we define in the next chapter.", "We have learned how to express intervals and their interactions both in terms of processes and of situations.", "We have seen that it is possible to test for interval preservation of a transition rule in a finite number of steps.", "It is also possible to find the minimal separating intervals of a transition rule in a finite number of steps.", "As a technical tool to express the properties of separating intervals we have introduced characteristic reactions.", "They express which events are determined by a separating interval and where the zones of influence are for the cells at the left and the right of the separating interval." ], [ "A Local Reaction System", "Although they do characterise interval-preserving transition rules, characteristic reactions cannot be used as generators of a reaction system without unpleasant side effects.", "I mean the following: Let $a_1 \\rightarrow [1, i_1] \\hat{a}_1 [-1, j_1]$ and $a_2 \\rightarrow [1, i_2] \\hat{a}_2 [-1, j_2]$ be the characteristic reactions of separating intervals.", "If we apply them in sequence to the situation $a_1 a_2$ , then we get $\\begin{aligned}[b]a_1 a_2&\\rightarrow [1, i_1] \\hat{a}_1 [-1, j_1] a_2 \\\\&\\rightarrow [1, i_1] \\hat{a}_1 [-1, j_1][1, i_2] \\hat{a}_2 [-1, j_2] \\\\&= [1, i_1] \\hat{a}_1 [j_1 + i_2] \\hat{a}_2 [-1, j_2]\\,.\\end{aligned}$ So, unless $j_1 + i_2 \\le 0$ , which is usually not the case, the result of this reaction is not an interval but contains a gap.", "If we had used the characteristic reaction of $a_1 a_2$ directly, we would have encountered no gap: $a_1 a_2\\rightarrow [1, i_1] \\widehat{a_1 a_2} [-1, j_2]\\,.$ (In this reaction the coefficients $i_1$ and $j_2$ occur because of Theorem REF .)", "No characteristic reaction could have recovered the missing piece, even when starting from $[1, i_1] \\hat{a}_1 [-1, j_1] a_2$ : if we had applied a characteristic reaction to $\\hat{a}_1$ or parts of it, it would only yield events at time step 2.", "But if we had preserved in the first step of (REF ) some events at the right end of $a_1$ , then the gap in the interval could have been avoided.", "We will now define a new kind of reaction that accomplishes this." ], [ "Reactions for Separating Intervals", "The parts of a separating interval that must be preserved in a reaction are the minimal separating intervals of Definition REF .", "We now introduce a notation for the situations that correspond to them.", "Definition 6.1 (Leftmost and Rightmost Intervals) Let $a \\in \\mathcal {S}$ be a separating interval.", "Let $a_L$ be the shortest separating interval for which $a_L\\mathrel {{\\backslash }\\!", "{\\backslash }}a$ .", "Let $a_R$ be the shortest separating interval for which $a \\mathrel {{/}\\!", "{/}}a_R$ .", "Then $a_L$ and $a_R$ are the leftmost and rightmost minimal separating intervals of $a$ .", "If an interval $a$ is minimally separating, then $a_L = a_R = a$ .", "On the other hand, if $a$ is separating, then $(a_L)_R$ and $(a_R)_L$ are minimal separating intervals, but not necessarily identical.", "Other properties of $a_L$ and $a_R$ are the subject of the following lemma.", "Lemma 6.2 (Extending a Separating Interval) Let $a$ , $x \\in \\Sigma ^$ be interval situations, of which $a$ is separating.", "Then $(a x)_L &= a_L, & (a x)_R &= (a_R x)_R, \\\\(x a)_R &= a_R, & (x a)_L &= (x a_L)_L\\,.$ Only the equations of the first line need to be proved.", "We get $(ax)_L$ by removing events from the right part of $a x$ as long as the result is still separating.", "Since $a$ is separating, we can remove $x$ completely.", "This proves $(a x)_L = a_L$ .", "The rightmost separating interval in $a x$ of which we know must is $a_R x$ because we do not know whether $x$ is separating.", "Therefore the rightmost separating part of $a x$ must be the rightmost separating part of $a_R x$ .", "This proves $(a x)_R = (a_R x)_R$ .", "The second piece of the definition is a notation for a specific kind of displacement terms.", "Definition 6.3 (Slope Operators) Let $a \\in \\Sigma ^$ be a separating interval with characteristic reaction $a \\rightarrow [1, i] \\hat{a} [-1, j]$ .", "Then the left and right slope operators of $a$ are ${+_a} = [1, i -\\mathopen \|a_L\\mathclose \|]\\qquad \\text{and}\\qquad {-_a} = [-1, j -\\mathopen \|a_R\\mathclose \|]\\,.$ The slope operators are defined in this way because then we have $\\delta (a) = \\delta (a_L +_a a -_a a_R)\\qquad \\text{and}\\qquad \\Delta \\mathrm {pr}(a) = \\mathrm {pr}_{a +_a}(\\hat{a})\\,.$ Taken together these equations imply that $a \\rightarrow a_L +_a a -_a a_R$ is a reaction for $\\varphi $ .", "The two conditions can be verified with the help of Definitions REF and REF .", "The slope operators are no addition operators, and $+_a$ is not the inverse of $-_a$ .", "Their symbols have been chosen to stay in harmony with the operators $\\oplus _k$ and $\\ominus _k$ that were already introduced in [51]; here they will reappear in Definition REF .", "Other notations that have no relations to addition, like $\\uparrow _a$ and $\\downarrow _a$ , were considered but rejected.", "In the case of an arrow notation, the main reason was that there is no complete agreement whether the future or the past is “up” (see p. REF ) and a notation that is agnostic in this aspect is therefore preferable.", "Another point is that we will need to distinguish between two kinds of slope operators; the existence of both encircled and not encircled plus and minus symbols in is therefore another reason to use them as the notation for slopes." ], [ "A System of Interval Reactions", "Now we can define the new reaction system.", "It will later become a part of the “local reaction system” of Definition REF , but is conceptually a bit simpler.", "It is however complex enough to show essential features and motivate the extensions.", "For this system let $\\varphi $ be an interval-preserving transition rule for $\\Sigma $ .", "Let $R$ be the reaction system that is generated by $\\Sigma ^$ and the following reactions, where $u \\in \\Sigma ^$ may be any separating interval, $u &\\rightarrow _R u_L +_u \\hat{u} -_u u_R, \\\\\\hat{u} -_u u +_u \\hat{u} &\\rightarrow _R \\hat{u}&&\\quad \\text{if $u$ is minimally separating.", "}$ The diagrams for these reactions can be seen in Figure REF .", "The reactions are shown as parts of a larger situation, which is displayed in grey.", "Figure: A system of interval reactions.In the first diagram, which shows reaction (REF ), the interval $u$ is replaced by the interval $\\hat{u}$ that is determined by it, but the left and right ends of $u$ are kept for the use in later reactions.", "The second diagram shows reaction ().", "Its left side is somewhat difficult to display: the situation $\\hat{u} -_u u +_u \\hat{u}$ begins with $\\hat{u}$ , followed by $u$ , and then, because $\\delta (\\hat{u} -_u u +_u) = (0, 0)$ , the same interval $\\hat{u}$ occurs again.", "The reaction then eliminates the $u$ interval.", "To prove that $\\delta (\\hat{u} -_u u +_u {}) = (0, 0)$ , we only have to notice that $u$ is a minimal separating interval and that therefore $u_L= u_R = u$ .", "Then the first condition in (REF ) becomes $\\delta (u +_u \\hat{u} -_u u) = \\delta (u)$ .", "From this follows $\\delta (u +_u \\hat{u} -_u {}) = (0, 0)$ , which is equivalent to the assertion." ], [ "How it works", "With this reaction system we can avoid the problems we had with characteristic reactions.", "To see how this works, let $b$ be a minimal separating interval.", "Instead of $a_1$ and $a_2$ as before, we now consider the separating intervals $a_1 b$ and $b a_2$ , which overlap in $b$ .", "Because of this we have $(a_1 b)_L = (b a_2)_R =b$ by Lemma REF .", "We can then apply a reaction of the form (REF ) on $a_1 b$ ; the result is $\\begin{aligned}[b]a_1 b&\\rightarrow _R (a_1 b)_L +_{a_1 b} \\widehat{a_1 b} -_{a_1b} (a_1 b)_R \\\\&= (a_1 b)_L +_{a_1} \\widehat{a_1 b} -_b b\\,.\\end{aligned}$ In the same way $b a_2$ reacts to $b +_b \\widehat{b a_2} -_{a_2} (ba_2)_R$ .", "We now evolve the left part of the interval $a_1 b a_2$ first, as we did in (REF ).", "In the following computation the parts of the formulas that change in the next step are underlined, to make it more readable.", "Then we get, $\\begin{aligned}[b]\\underline{a_1 b} a_2& \\rightarrow _R (a_1 b)_L +_{a_1} \\widehat{a_1 b} -_b\\underline{b a_2} \\\\& \\rightarrow _R (a_1 b)_L +_{a_1} \\underline{\\widehat{a_1 b}} -_b b+_b \\underline{\\widehat{b a_2}} -_{a_2} (b a_2)_R \\\\& \\sim (a_1 b)_L +_{a_1} \\widehat{a_1 b} \\mathbin {\\langle \\hat{b}\\rangle }\\underline{\\hat{b} -_b b +_b \\hat{b}} \\mathbin {\\langle \\hat{b}\\rangle }\\widehat{b a_2} -_{a_2} (b a_2)_R \\\\& \\rightarrow _R (a_1 b)_L +_b\\underline{\\widehat{a_1 b} \\mathbin {\\langle \\hat{b}\\rangle } \\hat{b}} \\mathbin {\\langle \\hat{b}\\rangle }\\widehat{b a_2} -_{a_2} (b a_2)_R \\\\& \\sim (a_1 b)_L +_b \\widehat{a_1 b} \\mathbin {\\langle \\hat{b}\\rangle }\\widehat{b a_2} -_{a_2} (b a_2)_R\\,.\\end{aligned}$ We can then see that $\\widehat{a_1 b} \\mathbin {\\langle \\hat{b}\\rangle } \\widehat{b a_2}$ , the part of the reaction result that belongs to time step 1, is now an interval.", "If instead we apply rule (REF ) directly to $a_1 b a_2$ , then we get the reaction $a_1 b a_2 \\rightarrow _R (a_1 b)_L +_b\\widehat{a_1 b a_2} -_{a_2} (b a_2)_R$ .", "By comparing its result with the result of the previous computation we see also that $\\widehat{a_1b} \\mathbin {\\langle \\hat{b}\\rangle } \\widehat{b a_2} \\sim \\widehat{a_1 b a_2}$ ." ], [ "Well-Behaved Transition Rules", "There were two ideas in the previous section that motivated the jump from characteristic reactions to the reaction system (REF ): It should be possible to reach all elements of the closure of a situation with reactions, and one should be able to do it by applying reactions to this situation in any order.", "We will make these vague concepts later precise as covering property and confluence and prove them at the end if this chapter.", "The proofs however are valid only for a subclass of the interval-preserving transition rules.", "This class of well-behaved transition rules, which is defined next, is introduced mainly for convenience.", "It was found by trial and error, trying to exclude special cases that would make proofs and concepts too complex, while keeping the theory applicable for Rules 54 and 110.", "Definition 6.4 (Well-Behaved) A transition rule $\\varphi $ on $\\Sigma $ is well-behaved if $\\varphi $ is interval-preserving, if $\\pi \\in \\mathcal {P}$ is a non-separating interval, then $\\Delta \\pi = \\emptyset $ , if $\\pi \\in \\mathcal {P}$ is a minimal separating interval, then $\\Delta \\pi $ is either a minimal separating interval or non-separating, and the empty interval is not separating.", "Condition 2 in this definition is necessary for the proof of Lemma REF .", "It ensures that all reactions that start from intervals and compute new events, i. e. those of the form (REF ), do indeed start from separating intervals.", "We do not need to consider very short intervals as special cases.", "Condition 2 is a completeness property for reaction system (REF ) and for the systems that will be later derived from it.", "The condition is always true for elementary cellular automata, but it can become false for radii greater than 1.", "We will now construct a counterexample for $r = 2$ and $\\Sigma = \\lbrace 0, 1, 2\\rbrace $ .", "Its transition rule is $\\varphi (\\sigma _{-2}, \\sigma _{-1}, \\sigma _0, \\sigma _1, \\sigma _2) =\\left\\lbrace \\begin{array}{l@{\\quad }l}0 & \\text{if $\\sigma _0 = 0$,} \\\\\\max \\lbrace \\sigma _{-2}, \\dots , \\sigma _2\\rbrace & \\text{otherwise.", "}\\end{array}\\right.$ Then the interval $\\pi = \\mathrm {pr}(0)$ provides a contradiction.", "It is not separating, since the state of the cell at location $-1$ can influence the next state of the cell at location 1 and vice versa, but we also have $\\Delta \\pi = \\mathrm {pr}([1, 0] 0)$ .", "It is this crossover influence between cells that Condition 2 prevents.", "Condition 3 is necessary in the context of achronal situations (Definition REF below).", "It concerns situations of the form $a +_a \\hat{a}$ or $\\hat{b} -_b b$ , with $a$ and $b$ minimal separating intervals.", "These situations arise frequently in reaction system (REF ) and other systems that have a reaction $u \\rightarrow u_L +_u \\hat{u} -_u u_R$ .", "In the result of this reaction, the interval $\\hat{u}$ has $\\widehat{u_L}$ as its left end and $\\widehat{u_R}$ as its right end; so with $a = u_L$ and $b = u_R$ we can say that $u_L +_u \\hat{u} -_u u_R$ begins with $a +_a \\hat{a}$ and ends with $\\hat{b} -_b b$ .", "The condition then ensures that a reaction of the same type as before, when applied to $\\hat{u}$ , does not destroy $\\hat{a}$ and $\\hat{b}$ .", "This is because such a reaction, when applied to an interval, leaves its left and right minimal separating intervals intact: The interval $\\hat{a}$ is by condition 3 not longer than a minimal separating interval and is therefore part of $\\hat{u}_L$ , and $\\hat{b}$ is for the same reason a part of $\\hat{u}_R$ .", "Here a counterexample occurs with an elementary cellular automaton, Rule 1.", "We have found in Table REF that the interval 1 is minimally separating for this rule, but $\\Delta \\mathrm {pr}(1)$ is the interval $\\mathrm {pr}([1, -1] 000)$ , as we can see in Table REF .", "We have thus an interval $\\pi $ for which $\\Delta \\pi $ consists of three events; such intervals are never minimally separating for elementary cellular automata.", "The purpose of Condition 3 is to exclude rules with separating intervals that have such excessive influence.", "Condition 4 is an intuitively obvious requirement on separating intervals, but it is violated by transition rules of radius 0.", "These are rules in which the state of a cell does only depend on the state of a single cell at the previous time step.", "Excluding them from consideration therefore is no loss." ], [ "Achronal Situations", "The reaction products in the system (REF ) have a specific form, a generalisation of intervals, for which we will now give a definition.", "The set is called “achronal” because these situations, like the intervals, consist of events that belong almost to the same time.", "We think of the events in them as arranged from left to right, not in a temporal sequence.", "Achronal situation also have in common with intervals that every achronal situation has a closure and can therefore be the starting point of a reaction.", "This will be proved later, in Theorem REF .", "Definition 6.5 (Achronal Situations) The set of achronal situations for an interval-preserving transition rule $\\varphi $ is the set $\\mathcal {A}_\\varphi \\subset \\mathcal {S}$ .", "It is defined recursively in the following way: A situation $s \\in \\mathcal {S}$ is an element of $\\mathcal {A}_\\varphi $ if and only if $s \\in \\Sigma ^$ , or $s = y b +_b \\hat{b} x$ , with $y b$ , $\\hat{b} x \\in \\mathcal {A}_\\varphi $ and $b \\in \\Sigma ^$ minimally separating, or $s = x \\hat{b} -_b b y$ , with $x \\hat{b}$ , $b y \\in \\mathcal {A}_\\varphi $ and $b \\in \\Sigma ^$ minimally separating.", "We will also use two subsets of $\\mathcal {A}_\\varphi $ .", "The set $\\mathcal {A}_{\\varphi +}$ consists of those elements of $\\mathcal {A}_\\varphi $ that are constructed only with the $+$ operators, and the set $\\mathcal {A}_{\\varphi -}$ consists of those elements of $\\mathcal {A}_\\varphi $ that are constructed only with the $-$ operators.", "These sets are called the positive and negative slopes.", "Similarly, the terms $\\hat{b} -_b b$ and $b +_b \\hat{b}$ in Definition REF are called generating slopes." ], [ "Use of Slopes", "The positive and negative slopes provide a notation with which we can name the different parts of a situation.", "Later, in Lemma REF , we will see that every situation can react into a situation that is the product of a positive and negative slope.", "Figure: A triangle and its slopes under Rule 54.Figure REF is an example.", "Here the left triangle of Figure REF is expressed as a reaction.", "It begins with an interval situation and ends with a situation that consists of a positive and a negative slope.", "(This example will be continued with Figure REF .)", "The generating slopes are important because the reactions that transform positive generating slopes into positive, or negative generating slopes into negative generating slopes, are among the building blocks for the reaction system associated to a transition rule, which is described in Definition REF ." ], [ "Induction", "If we view the recursive construction of the achronal situation as a sequential process, then the intervals are created at its beginning, and every other achronal situation $s$ has either a decomposition $s = y b +_b \\hat{b} x$ with $y b$ and $\\hat{b}x$ constructed earlier, or a decomposition $s = x \\hat{b} -_b b y$ with $x \\hat{b}$ and $b y$ constructed earlier.", "We have therefore an induction principle for achronal situations: Lemma 6.6 (Achronal Induction) Let $\\varphi $ be an interval-preserving transition rule.", "Let $S \\subseteq \\mathcal {A}_\\varphi $ be a set of situations where $\\Sigma ^ \\subseteq S$ , if $y b$ , $\\hat{b} x \\in S$ , where $b$ is a minimal separating interval, then $y b +_b \\hat{b} x \\in S$ , and if $x \\hat{b}$ , $b y \\in S$ , where $b$ is a minimal separating interval, then $x \\hat{b} -_b b y \\in S$ .", "Then $S = \\mathcal {A}_\\varphi $ .", "$\\Box $ This induction principle uses the operators $+_b$ and $-_b$ .", "Situations are however defined in terms of displacements, not in terms of slope operators.", "Therefore it is not yet clear whether we can, when given an achronal situation, reconstruct the slope operators with which it was constructed.", "The following lemma shows that the answer is “yes, but it is not completely trivial”.", "Lemma 6.7 (Slope Operators) Let $\\varphi $ be a well-behaved transition rule and let $s \\in \\mathcal {A}_\\varphi $ be an achronal situation with a honest decomposition $s = x[p]y$ in which $p \\ne (0, 0)$ .", "Then there is either $[p] = +_b$ with $x \\mathrel {{/}\\!", "{/}}b$ , where $b$ is a minimal separating interval, or $[p] = -_b$ with $b \\mathrel {{\\backslash }\\!", "{\\backslash }}y$ , where $b$ is a minimal separating interval, or $[p] = +_{b_1} -_{b_2}$ with $x \\mathrel {{/}\\!", "{/}}b_1$ and $b_2\\mathrel {{\\backslash }\\!", "{\\backslash }}y$ , where $b_1$ and $b_2$ are minimal separating intervals and $\\hat{b}_1 = \\hat{b}_2 = [0]$ .", "We see from the definition of $\\mathcal {A}_\\varphi $ that $s$ can be written as a sequence of cell states and slope operators.", "In this proof we will call this sequence the symbol sequence for $s$ .", "In the symbol sequence for $s$ , every one of the symbols either contributes to $x$ , to $[p]$ or to $y$ .", "The symbols that contribute to $[p]$ can only be slope operators.", "They form a subsequence of maximal length in the symbol sequence; it is maximal because the decomposition is honest.", "If an operator $+_b$ contributes to $[p]$ , then the interval $b$ must appear at the left of it in the symbol sequence.", "Because $\\varphi $ is well-behaved, $b$ is never empty.", "Therefore $+_b$ can only appear at the left end of the sequence of slope operators that contribute to $[p]$ .", "At its right side it must be followed by $\\hat{b}$ , but only if $\\hat{b} \\ne [0]$ ; in that case $+_b$ is the only factor of $[p]$ .", "For the same reason $-_b$ can only appear at the right end of $[p]$ ; and if $\\hat{b} \\ne [0]$ , then $-_b$ is the only factor of $[p]$ .", "Therefore $[p]$ is a product of at most two slope operators in a prescribed order.", "Since $p \\ne (0, 0)$ , at least one of them must appear.", "This leads to the three cases of the lemma.", "It is clear that all of these three cases can occur.", "They can easily be distinguished: we have either $\\delta (p)_T = +1$ , $-1$ or 0.", "Therefore “the number of slope operators”More exactly, this number is the minimal number of slope operators with which a situation can be written.", "Ambiguous cases are possible: if $+_{b_1}= [1, 0]$ and $-_{b_2} = [-1, 0]$ , then $b_1 b_2 = b_1 +_{b_1}-_{b_2} b_2$ , and this is in fact an equality, not just an equivalence.", "in a situation is a well-defined concept, and induction over this number is possible.", "It will be the most common form of induction used in this text." ], [ "Achronal Situation Occur Naturally", "In the proof of Lemma REF we have seen that every achronal situation can be written as a sequence of elements of $\\Sigma $ together with slope operators: Every $-_b$ must be surrounded by $\\hat{b}$ at the left and $b$ at the right and every $+_b$ must be surrounded by $b$ at the left and $\\hat{b}$ at the right.", "Whether a situation is achronal therefore depends only on the terms next to the slope operators.", "This means that if the two situations $s_1 x$ and $x s_2$ are achronal, their “overlapping product” $s_1 x s_2$ is also achronal.", "The converse is not always true, but at least when the common part of the two situations is a separating interval: Lemma 6.8 (Splitting at Separating Intervals) Let $\\varphi $ be a well-behaved transition rule for $\\Sigma $ .", "Let $s_1$ , $b$ , $s_2 \\in \\mathcal {S}$ , where $b$ is a separating interval for $\\varphi $ .", "Then if $s_1 b s_2 \\in \\mathcal {A}_\\varphi $ , then $s_1 b \\in \\mathcal {A}_\\varphi $ and $b s_2 \\in \\mathcal {A}_\\varphi $ .", "Let $s = s_1 b s_2$ .", "We perform an induction over the number of slope operators in $s$ .", "If $s \\in \\Sigma ^$ , the lemma is obviously true.", "Otherwise $s$ has at least one slope operator, either $+_a$ or $-_a$ .", "Assume now that $s = x \\hat{a} -_a a y$ is a decomposition of $s$ with $x \\hat{a}$ , $a y \\in \\mathcal {A}_\\varphi $ , where $a$ is minimally separating.", "Since $b$ is a nonempty interval, it must be part of either $x \\hat{a}$ or of $a y$ .", "If $b$ is part of $ x \\hat{a}$ , then there is a situation $s^{\\prime }_2$ such that $x \\hat{a} = s_1 b s^{\\prime }_2$ .", "Since $s_1 b s^{\\prime }_2$ has fewer slope operators than $s$ , the induction hypothesis can be applied to it, and therefore $s_1 b \\in \\mathcal {A}_\\varphi $ and $b s^{\\prime }_2 \\in \\mathcal {A}_\\varphi $ .", "Because $\\varphi $ is well-behaved, the interval $\\hat{a}$ is not longer than a separating interval; therefore $b s^{\\prime }_2,$ which contains the separating $b$ interval as its factor, cannot be just a part of $\\hat{a}$ .", "So there must be a situation $z \\in \\mathcal {S}$ such that $b s^{\\prime }_2 = z \\hat{a}$ .", "Then $b s^{\\prime }_2 -_aa y = z \\hat{a} -_a a y$ and therefore $b s^{\\prime }_2 -_a a y \\in \\mathcal {A}_\\varphi $ .", "If $b$ is part of $a y$ , then there is a situation $s^{\\prime }_1$ such that $a y = s^{\\prime }_1 b s_2$ .", "Then $s^{\\prime }_1 b \\in \\mathcal {A}_\\varphi $ and $b s_2 \\in \\mathcal {A}_\\varphi $ by induction.", "When dividing up $a y$ , the situation $a$ must become a part of $s^{\\prime }_1 b$ because it is the leftmost minimal separating interval of this situation and $s^{\\prime }_1 b$ contains already the separating interval $b$ : the interval $a$ could not have been cut into pieces.", "Therefore $x \\hat{a} -_a s^{\\prime }_1 b \\in \\mathcal {A}_\\varphi $ .", "So if $s = x \\hat{a} -_a a y$ , then we could divide $s$ either into $s_1 b$ and $b s^{\\prime }_2 -_a a y$ or into $x \\hat{a} -_a s^{\\prime }_1 b$ and $bs_2$ .", "If $s = y a +_a \\hat{a} x$ , then there are similar decompositions for it; they can be found by a mirror image of this argument.", "With the methods developed so far we can now show that achronal situations occur naturally in the reaction system (REF ).", "Lemma 6.9 (Achronal Domain) Let $R$ be the interval reaction system of (REF ).", "Then $\\mathop {\\mathrm {dom}}R \\subseteq \\mathcal {A}_\\varphi $ .", "We will show that the generating reactions (REF ) transform achronal situations into achronal situations.", "As the initial situations of $R$ are intervals and therefore obviously achronal, this will prove that all situations in $\\mathop {\\mathrm {dom}}R$ are achronal.", "Let $xuy \\in \\mathcal {A}_\\varphi $ , where $u$ is a separating interval.", "Then $x u_L$ , $u$ and $u_R y$ are achronal by Lemma REF .", "One can see directly that $u_L +_u \\hat{u} -_u u_R$ is achronal.", "Therefore $x u_L +_u \\hat{u}-_u u_R y \\in \\mathcal {A}_\\varphi $ , again by Lemma REF .", "This proves that the reaction $u \\rightarrow _R u_L +_u \\hat{u} -_u u_R$ preserves achronality.", "Let now $x \\hat{u} -_u u +_u \\hat{u} y \\in \\mathcal {A}_\\varphi $ , where $u$ is a minimal separating interval.", "Then $x \\hat{u}$ and $\\hat{u} y$ are achronal by by Lemma REF .", "Therefore we have $x \\hat{u} y \\in \\mathcal {A}_\\varphi $ .", "This proves that the reaction $\\hat{u} -_u u +_u \\hat{u} \\rightarrow _R \\hat{u}$ preserves achronality." ], [ "Closure", "A preference for symmetry now leads to another question: If all reactions results are achronal situations, can we then also extend the set of input situations of the reaction system in (REF ) from $\\Sigma ^$ to $\\mathcal {A}_\\varphi $ ?", "The following theorem shows that this is possible for a subset of the achronal situations.", "For this we have to introduce a new concept.", "It represents the intuitive notion that the events of a situation are arranged approximately from left to right.", "To express the concept for a situation $s$ we consider the honest decompositions of $s$ of the form $s = s_1 [p_1] u_1 [q_1] s_2 [p_2] u_2 [q_2] s_3,$ in which $u_1$ and $u_2$ are intervals and $p_1$ , $q_1$ , $p_2$ , $q_2\\ne (0, 0)$ .", "We then write $\\pi _1 = \\mathrm {pr}_{s_1[p_1]}(u_1)$ and $\\pi _2 =\\mathrm {pr}_{s_1[p_1]u_1[q_1]s_2[p_2]}(u_2)$ for the processes that belong to $u_1$ and $u_2$ (Figure REF ).", "Figure: The situation ss in ().", "Theprocesses belonging to u 1 u_1 and u 2 u_2 occur at the same time.Now consider the decompositions of the form (REF ) the processes $\\pi _1$ and $\\pi _2$ belong to the same time.", "If for every decomposition of $s$ of this form we have $\\pi _1 \\mathrel {\\prec \\succ }\\pi _2$ , then the situation $s$ is ordered.", "Theorem 6.10 (Closure of Achronal Situations) Let $\\varphi $ be a well-behaved transition rule for $\\Sigma $ and $s \\in \\mathcal {A}_\\varphi $ be ordered.", "Then $\\mathop {\\mathrm {cl}}\\nolimits \\mathrm {pr}(s)$ exists.It is this theorem for which we need the fact that achronal situations are ordered: A counterexample in the reaction system for Rule 54 is the situation $000 \\oplus 01 \\oplus _2 1\\ominus _2 10$ , written in the notation (REF ).", "It is not ordered but achronal, and it has no closure.", "It is enough if we restrict the proof of the theorem to the case where $s$ is a balanced situation.", "This shall mean that $\\delta (s)_T= 0$ , that $\\mathrm {pr}(s)^{(t)} = \\emptyset $ for all $t \\ge 1$ and that $s$ is either an interval or there is a decomposition $s = \\hat{a} -_a x+_b \\hat{b}$ with minimal separating intervals $a$ and $b$ .", "(See Figure REF .", "Note that $\\hat{a}$ or $\\hat{b}$ may be empty.)", "Figure: A balanced situation.Every situation can be extended to a balanced situation; if that situation has a closure, then the original situation has a closure too.", "First we will however prove the existence of the closure for the simplest nontrivial balanced situations.", "This result is then used as a stepping stone for the proof of Theorem REF .", "Lemma 6.11 (Closure of Simple Balanced Situation) Let $\\varphi $ be a well-behaved transition rule for $\\Sigma $ and let $s= \\hat{a} -_a u +_b \\hat{b}$ be an achronal situation for $\\Sigma $ in which $u$ is an interval and $a$ , $b$ are minimal separating intervals.", "Then $s \\rightarrow \\hat{u}$ is a reaction for $\\varphi $ , and $\\mathop {\\mathrm {cl}}\\nolimits ^{(0)}\\mathrm {pr}(s) = \\mathrm {pr}(\\hat{u})$ .", "Figure: The situations in Lemma .A diagram of the processes for $s$ and $\\hat{u}$ can be seen in Figure REF .", "The process belonging to $\\hat{u}$ overlaps with those of $\\hat{a}$ and $\\hat{b}$ .", "To compute the closure of $\\mathrm {pr}(s)$ we must express $s$ in the language of cellular processes.", "Let therefore $\\pi = \\mathrm {pr}(s)$ ; its components are then $\\alpha ^{\\prime } = \\mathrm {pr}(\\hat{a}),\\qquad \\mu = \\mathrm {pr}_{\\hat{a} -_a}(u),\\qquad \\beta ^{\\prime } = \\mathrm {pr}_{\\hat{a} -_a u +_b}(b),$ such that $\\pi = \\alpha ^{\\prime } \\cup \\mu \\cup \\beta ^{\\prime }$ .", "The process $\\pi $ then consists of the time slices $\\pi ^{(-1)} = \\mu $ and $\\pi ^{(0)} =\\alpha ^{\\prime } \\cup \\beta ^{\\prime }$ .", "Since $s$ is an achronal situation, we must have $a \\mathrel {{\\backslash }\\!", "{\\backslash }}u \\mathrel {{/}\\!", "{/}}b$ .", "We will therefore also need names for the end intervals of $\\mu $ .", "They are $\\alpha = \\mathrm {pr}_{\\hat{a} -_a}(a)\\qquad \\text{and}\\qquad \\beta = \\mathrm {pr}_{\\hat{a} -_a u \\mathbin {\\langle b\\rangle }}(b).$ Then we can say that $\\mu $ begins with $\\alpha $ and ends with $\\beta $ , such that we have $\\alpha \\subseteq _R \\mu \\supseteq _L\\beta $ (Figure REF ).", "Figure: The processes related to the situations inFigure .The main part of the proof then consists of a computation of the space-time locations of all these processes.", "For this we let the characteristic reactions of $a$ and $b$ be $a \\rightarrow [1, i] \\hat{a} [-1, j^{\\prime }]\\qquad \\text{and}\\qquad b \\rightarrow [1, i^{\\prime }] \\hat{b} [-1, j]\\,.$ By Theorem REF , we must then have $u\\rightarrow [1, i] \\hat{u} [-1, j]$ as characteristic reaction for $u$ .", "This is because $u$ begins with $a$ , and therefore its characteristic reaction shares its left displacement term $[1, i]$ with that of $a$ , and $u$ ends with $b$ and therefore its characteristic reaction shares its right displacement term $[-1, j]$ with that of $b$ .", "This “bounding” of the location of $\\hat{u}$ by $a$ and $b$ is the core of the proof.", "Because of the left-oriented structure of the formalism it however does not become directly visible in the following calculations.", "Before we start with the calculations proper, we will determine short expressions for the values of $\\delta (\\hat{a} -_a)$ and $\\delta (\\hat{a} -_a u)$ , terms that will occur at several places.", "For the first term we begin with the equation $-_a = [-1, j^{\\prime } -\\mathopen \|a\\mathclose \|]$ , which follows from Definition REF .", "Then we can see that $\\delta (\\hat{a} -_a) = (0, \\mathopen \|\\hat{a}\\mathclose \|) + (-1, j^{\\prime } - \\mathopen \|a\\mathclose \|) = (-1,\\mathopen \|\\hat{a}\\mathclose \| + j^{\\prime } - \\mathopen \|a\\mathclose \|)$ .", "We now use a relation derived from the characteristic reaction for $a$ to simplify that term.", "The left and the right side of a reaction must have the same size vector, and this means for the characteristic reaction for $a$ that $\\mathopen \|a\\mathclose \| = i+ \\mathopen \|\\hat{a}\\mathclose \| + j^{\\prime }$ .", "Using that we see that $\\delta (\\hat{a} -_a) =(-1, -i)$ .", "With this result we get an expression for the second size vector, $\\delta (\\hat{a} -_a u) = (-1, -i) + (0, \\mathopen \|u\\mathclose \|) = (-1,\\mathopen \|u\\mathclose \| - i)$ .", "It too can be brought into a form that is more useful later, this time with the equation $\\mathopen \|u\\mathclose \| = i + \\mathopen \|\\hat{u}\\mathclose \| + j$ that is derived from the characteristic reaction of $u$ .", "The result is $(\\mathopen \|\\hat{u}\\mathclose \| + j)$ , such that we have $\\delta (\\hat{a} -_a) = (-1, -i)\\qquad \\text{and}\\qquad \\delta (\\hat{a} -_a u) = (0, \\mathopen \|\\hat{u}\\mathclose \| + j)\\,.$ Next we will show that $\\alpha ^{\\prime } = \\Delta \\alpha $ and $\\beta ^{\\prime } =\\Delta \\beta $ .", "To find a term that expresses $\\Delta \\alpha $ in terms of situations, we use the characteristic reaction for $a$ .", "We can read it as saying that the set of events determined by the process $\\mathrm {pr}(a)$ is $\\mathrm {pr}([1, i] \\hat{a})$ .", "(See Definition REF .)", "This is also valid for shifted versions of $\\mathrm {pr}(a)$ , so the set of events determined by $\\alpha = \\mathrm {pr}_{\\hat{a} -_a}(a)$ must be $\\Delta \\alpha =\\mathrm {pr}_{\\hat{a} -_a}([1, i] \\hat{a})$ .", "The position of $\\hat{a}$ in this term is the sum of two displacements, $\\delta (\\hat{a} -_a)$ and $(1,i)$ .", "Since $\\delta (\\hat{a} -_a) + (1, i) = (0, 0)$ , we have therefore $\\Delta \\alpha = \\mathrm {pr}(\\hat{a})= \\alpha ^{\\prime }\\,.$ For the same reason, this time with the characteristic reaction of $\\hat{b}$ , the set of events determined by $\\beta = \\mathrm {pr}_{\\hat{a} -_a u\\mathbin {\\langle b\\rangle }}(\\hat{b})$ must be $\\Delta \\beta = \\mathrm {pr}_{\\hat{a} -_a u\\mathbin {\\langle b\\rangle }}([1, i^{\\prime }]\\hat{b})$ .", "So we must compute $\\delta (\\hat{a} -_a u\\mathbin {\\langle b\\rangle }) + (1, i^{\\prime })$ to find the position of $\\hat{b}$ in this term: Then we get $\\delta (\\hat{a} -_a u \\mathbin {\\langle b\\rangle }) + (1, i^{\\prime }) = (-1, \\mathopen \|\\hat{u}\\mathclose \| + j) + (0, -\\mathopen \|b\\mathclose \|) + (1, i^{\\prime }) = (0, \\mathopen \|\\hat{u}\\mathclose \| + j - \\mathopen \|b\\mathclose \|+ i^{\\prime })$ .", "We use the equation $\\mathopen \|b\\mathclose \| = i^{\\prime } + \\mathopen \|\\hat{b}\\mathclose \| + j$ , which is derived from the characteristic reaction for $b$ , to simplify the result of this computation to $(0, \\mathopen \|\\hat{u}\\mathclose \| - \\mathopen \|\\hat{b}\\mathclose \|)$ .", "Therefore we have $\\Delta \\beta = \\mathrm {pr}([0, \\mathopen \|\\hat{u}\\mathclose \| - \\mathopen \|\\hat{b}\\mathclose \|] \\hat{b})\\,.$ To find the location of $\\beta ^{\\prime } = \\mathrm {pr}_{\\hat{a} -_a u +_b}(\\hat{b})$ we use the fact that $+_b = [1, i^{\\prime } - \\mathopen \|b\\mathclose \|]$ .", "(See Definition REF ).", "Then we can calculate $\\delta (\\hat{a} -_a u +_b) = (-1, \\mathopen \|\\hat{u}\\mathclose \| + j) + (1, i^{\\prime } -\\mathopen \|b\\mathclose \|) = (0, \\mathopen \|\\hat{u}\\mathclose \| + j + i^{\\prime } - \\mathopen \|b\\mathclose \|)$ and simplify the result in the same way as before to $(0, \\mathopen \|\\hat{u}\\mathclose \| - \\mathopen \|\\hat{b}\\mathclose \|)$ .", "Therefore we get $\\beta ^{\\prime } = \\mathrm {pr}([0, \\mathopen \|\\hat{u}\\mathclose \| - \\mathopen \|\\hat{b}\\mathclose \|] \\hat{b}),$ which shows that $\\Delta \\beta = \\beta ^{\\prime }$ .", "With this data we can compute the time slices of $\\mathop {\\mathrm {cl}}\\nolimits \\pi $ and therefore show that $\\pi $ actually has a closure.", "We have already seen that $\\pi ^{(-1)} = \\mu $ , $\\pi ^{(0)} = \\alpha ^{\\prime } \\cup \\beta ^{\\prime }$ , and that $\\pi ^{(t)} = \\emptyset $ for all other values of $t$ .", "Therefore, applying the definition (REF ) of the closure we get [3] $\\mathop {\\mathrm {cl}}\\nolimits ^{(t)} \\pi &= \\pi ^{(t)} &&= \\emptyset &\\qquad &\\text{for $t < -1$,}\\\\\\mathop {\\mathrm {cl}}\\nolimits ^{(-1)} \\pi &= \\pi ^{(-1)} &&= \\mu , \\\\\\mathop {\\mathrm {cl}}\\nolimits ^{(0)} \\pi &= \\pi ^{(0)} \\cup \\Delta \\mathop {\\mathrm {cl}}\\nolimits ^{(-1)} \\pi \\\\&= (\\alpha ^{\\prime } \\cup \\beta ^{\\prime }) \\cup \\Delta \\mu &&= \\Delta \\mu , \\\\\\mathop {\\mathrm {cl}}\\nolimits ^{(t)} \\pi &= \\pi ^{(t)} \\cup \\Delta \\mathop {\\mathrm {cl}}\\nolimits ^{(t - 1)} \\pi &&= \\Delta \\mathop {\\mathrm {cl}}\\nolimits ^{(t - 1)} \\pi &&\\text{for $t \\ge 1$.", "}$ Only the third equation must be explained.", "It is true because $\\alpha ^{\\prime } = \\Delta \\alpha $ and $\\beta ^{\\prime } = \\Delta \\beta $ .", "Since $\\alpha $ and $\\beta $ are subsets of $\\mu $ , we must then have $\\alpha ^{\\prime } \\subseteq \\Delta \\mu \\supseteq \\beta ^{\\prime }$ by the monotony of the $\\Delta $ operator.", "This then proves that $\\alpha ^{\\prime } \\cup \\beta ^{\\prime }$ is compatible with $\\Delta \\mu $ and that $\\mathop {\\mathrm {cl}}\\nolimits ^{(0)} \\pi $ actually exists.", "Together these equations show that $\\mathop {\\mathrm {cl}}\\nolimits \\pi $ exists.", "It remains to prove that $s \\rightarrow \\hat{u}$ is a reaction for $\\varphi $ .", "We have to show that $\\delta (s) = \\delta (\\hat{u})$ and that $\\mathrm {pr}(\\hat{u}) \\subseteq \\mathop {\\mathrm {cl}}\\nolimits \\mathrm {pr}(s)$ .", "We know already that $\\delta (\\hat{a} -_a u+_b) = (0, \\mathopen \|\\hat{u}\\mathclose \| - \\mathopen \|\\hat{b}\\mathclose \|)$ ; therefore $\\delta (s) =\\delta (\\hat{a} -_a u +_b \\hat{b}) = (0, \\mathopen \|\\hat{u}\\mathclose \|) = \\delta (\\hat{u})$ .", "To prove that $\\mathrm {pr}(\\hat{u}) \\subseteq \\mathop {\\mathrm {cl}}\\nolimits \\mathrm {pr}(s)$ we will now show that $\\mathrm {pr}(\\hat{u}) = \\Delta \\mu $ .", "For this we use the fact that the set of events determined by $\\mu = \\mathrm {pr}_{\\hat{a} -_a}(u)$ is the process $\\Delta \\mu = \\mathrm {pr}_{\\hat{a} -_a}([-1, -i] u)$ .", "Then, since $\\delta (\\hat{a} -_a) + (-1, -i) = (0, 0)$ , we must have $\\mathrm {pr}(\\hat{u})= \\Delta \\mu $ .", "Now we can apply the result of (REF ) that $\\Delta \\mu = \\mathop {\\mathrm {cl}}\\nolimits ^{(0)} \\pi $ and see that $\\mathrm {pr}(\\hat{u}) \\subseteq \\mathop {\\mathrm {cl}}\\nolimits \\mathrm {pr}(s)$ .", "This then concludes the proof that $s \\rightarrow \\hat{u}$ is a reaction; it also proves that $\\mathop {\\mathrm {cl}}\\nolimits ^{(0)} \\pi = \\mathrm {pr}(\\hat{u})$ .", "[Proof of Theorem REF ] As explained above, we restrict our case to balanced intervals.", "For a situation $a$ , we will call the first time $t$ for which $\\mathrm {pr}(a)^{(t)} \\ne \\emptyset $ the starting time of $a$ .", "A balanced situation has then a starting time $t \\le 0$ , and the balanced situations with starting time 0 are the intervals.", "Since intervals have a closure, it is therefore enough to show that if $t< 0$ and every balanced situation with starting time $t + 1$ has closure, then every situation with starting time $t$ has a closure too.", "We do this in the following way.", "A process $s$ with starting time $t_0$ has by definition the time slices of the closure $\\mathop {\\mathrm {cl}}\\nolimits ^{(t)} \\mathrm {pr}(s)$ for every $t \\le t_0$ , with $\\mathop {\\mathrm {cl}}\\nolimits ^{(t_0)} \\mathrm {pr}(s) = \\mathrm {pr}(s)^{(t_0)}$ .", "We will then show that for every such $s$ exists another situation $s^{\\prime }$ with starting time $t_0 + 1$ such that $\\mathop {\\mathrm {cl}}\\nolimits ^{(t_0 + 1)} \\mathrm {pr}(s) = \\mathrm {pr}(s^{\\prime })^{(t_0 +1)}$ .", "The closure of $s^{\\prime }$ exists by induction, and we have, as before, $\\mathop {\\mathrm {cl}}\\nolimits ^{(t_0 + 1)} \\mathrm {pr}(s^{\\prime }) = \\mathrm {pr}(s^{\\prime })^{(t_0 + 1)}$ .", "Therefore, $\\mathop {\\mathrm {cl}}\\nolimits \\mathrm {pr}(s) = \\bigcup _{t \\le t_0} \\mathrm {pr}(s)^{(t)} \\cup \\bigcup _{t > t_0} \\mathop {\\mathrm {cl}}\\nolimits \\mathrm {pr}(s^{\\prime })^{(t)},$ so the closure of $s$ exists then.", "Now we must isolate in $s$ the factors that contribute to $\\mathrm {pr}(s)^{(t_0)}$ .", "For this we will use the decomposition $s = s_0 -_{a_1} u_1 +_{b_1} s_1 \\dots s_{\\ell -1} -_{a_\\ell } u_\\ell +_{b_\\ell } s_\\ell \\,.$ Figure: The situation ss in ().in which the $u_i$ are the intervals that belong to time $t_0$ .", "More precisely, we write $\\pi _i = \\mathrm {pr}_{s_0 \\dots s_{i_1} +_{a_i}}(u_i)$ for the process that belongs to $u_i$ and require that $\\mathrm {pr}(s)^{(t_0)} = \\bigcup _{i=1}^\\ell \\pi _\\ell $ .", "The situations $s_i$ are arbitrary and need not be intervals.", "Nevertheless, since $s$ is achronal, every situation $s_i$ with $i <\\ell $ ends with $\\hat{a}_{i+1}$ , and every $s_i$ with $i > 0$ begins with $\\hat{b}_i$ .", "Therefore $s \\sim s_0 \\mathbin {\\langle \\hat{a}_1\\rangle }\\underline{\\hat{a}_1 -_{a_1} u_1 +_{b_1} \\hat{b}_1}\\mathbin {\\langle \\hat{b}_1\\rangle } s_1 \\dots s_{\\ell -1} \\mathbin {\\langle \\hat{a}_\\ell \\rangle }\\underline{\\hat{a}_\\ell -_{a_\\ell } u_\\ell +_{b_\\ell } \\hat{b}_\\ell }\\mathbin {\\langle \\hat{b}_\\ell \\rangle } s_\\ell \\,.$ We can now apply Lemma REF to the underlined factors in this equation.", "In the current context it says that $\\Delta \\pi _i = \\mathrm {pr}_{s_0 \\dots s_{i-1} \\mathbin {\\langle \\hat{a}_i\\rangle }}(\\hat{u}_i)$ for every $i$ .", "We therefore get the situation $s^{\\prime }$ by replacing the underlined factors in $s$ with the intervals $\\hat{u}_i$ , $s^{\\prime } \\sim s_0 \\mathbin {\\langle \\hat{a}_1\\rangle }\\hat{u}_1\\mathbin {\\langle \\hat{b}_1\\rangle } s_1 \\dots s_{\\ell -1} \\mathbin {\\langle \\hat{a}_\\ell \\rangle }\\hat{u}_\\ell \\mathbin {\\langle \\hat{b}_\\ell \\rangle } s_\\ell \\,.$ In fact the situation $s^{\\prime }$ is what we get when we resolve the overlaps at the right side of the previous equation.", "This is always possible because every $\\hat{u}_i$ begins with $\\hat{a}_i$ and ends with $\\hat{b}_i$ .", "Figure: The left end of Figure ,with the factors of ss and s ' s^{\\prime } overlayed.We also have to check whether one of the processes $\\Delta \\pi _i$ intersects with other parts of $\\mathrm {pr}(s)^{(t_0+1)}$ .", "But $\\Delta \\pi _i$ may have a non-empty intersection only with the processes belonging to $s_{i-1}$ and $s_i$ .", "This is because $s$ is ordered: The intervals of $\\mathrm {pr}(s)^{(t_0+1)}$ to the left of $\\hat{a}_i$ must all belong to a situation $s_k$ or $u_k$ with $k \\le i - 1$ .", "Therefore such an interval may extend to the right at most as far as the right end of $\\hat{a}_i$ .", "Similarly, the intervals of $\\mathrm {pr}(s)^{(t_0+1)}$ to the right of $\\hat{b}_i$ must all belong to a situation $s_k$ or $u_k$ with $i \\le k$ and therefore extend to the left at most to the left end of $\\hat{b}_i$ .", "The left-to-right arrangement of the intervals of $s$ is therefore preserved in $s^{\\prime }$ , with the intervals $\\hat{u}_i$ inserted in the gaps between the $s_i$ .", "This shows that $s^{\\prime }$ is ordered.", "So $\\Delta \\mathrm {pr}(s)^{(t_0)}$ is compatible with $\\mathrm {pr}(s)^{(t_0 + 1)}$ , and $\\mathop {\\mathrm {cl}}\\nolimits ^{(t_0 +1)} \\mathrm {pr}(s)$ exists and is equal to $\\mathrm {pr}(s^{\\prime })^{(t_0 +1)}$ .", "Therefore we can define now a reaction system that has $\\mathcal {A}_\\varphi $ as its domain." ], [ "The Local Reaction System", "The preliminary reaction system (REF ) has the disadvantage that its set of generating reactions is infinite.", "We cannot specify them in a list in the same way as we can do this with a transition rule.", "We will now make the local nature of the interactions in a cellular automaton more visible by decomposing the generating reactions of (REF ) into a finite number of reactions that involve only a finite number of events.", "The following lemma specifies these reactions and shows how they generate the reactions of (REF ).", "Lemma 6.12 (Local Generators) Let $R$ be a reaction system that contains for all separating intervals $u$ and all $\\sigma \\in \\Sigma $ the reactions $u &\\rightarrow _R u_L +_u \\hat{u} -_u u_R,&&\\text{if $u$ is minimal,}\\\\\\hat{u} -_u u\\sigma &\\rightarrow _R \\widehat{u\\sigma } -_{u\\sigma } (u\\sigma )_R,&&\\text{if $u$ is right minimal,} \\\\\\sigma u +_u \\hat{u}& \\rightarrow _R (\\sigma u)_L +_{\\sigma u} \\widehat{\\sigma u},&&\\text{if $u$ is left minimal.", "}$ Then $R$ contains for all separating intervals $v$ , not just those that are minimally separating, the reaction $v \\rightarrow _R v_L +_v\\hat{v} -_v v_R$ .", "The diagrams for these reactions are shown in Figure REF ; it uses the same conventions as Figure REF .", "Figure: Reactions to generate an interval.The intention behind the definitions is that reaction (REF ) is used to generate a new interval one time step in the future—but this time one of minimal length—and that () and () then are used to expand it to the left and the right.", "We prove the lemma by induction over the length of $v$ : A separating interval $v$ is either minimally separating or there exist a separating interval $w \\in \\Sigma ^$ and a state $\\sigma \\in \\Sigma $ such that $v = w \\sigma $ or $v = \\sigma w$ .", "If $v$ is minimal, then there is by (REF ) a reaction $v \\rightarrow _R v_L +_v \\hat{v} -_v v_R$ .", "If $v = w \\sigma $ , then there is by induction a reaction $w \\rightarrow _Rw_L +_w \\hat{w} -_w w_R$ .", "We apply it to $v$ and get $v\\rightarrow _R w_L +_w \\hat{w} -_w w_R \\sigma $ .", "Now let $x \\in \\Sigma ^$ such that $w = x w_R$ .", "Since $w_R$ is separating, there is by Lemma REF an $x^{\\prime } \\in \\Sigma ^$ such that $\\widehat{x w_R} = x^{\\prime } \\widehat{w_R}$ .", "Therefore $v \\rightarrow _Rw_L +_w x^{\\prime } \\widehat{w_R} -_w w_R \\sigma $ is a reaction in $R$ .", "Then, since $w_R$ is right minimal and ${-_w}= {-_{w_R}}$ , we can apply the reaction () with $u = w_R$ to the result of (REF ) and get $w_L +_w x^{\\prime } \\widehat{w_R} -_w w_R \\sigma \\rightarrow _R w_L +_w x^{\\prime } \\widehat{w_R \\sigma }-_{w_R \\sigma } (w_R \\sigma )_R\\,.$ We must now interpret the result of this reaction in terms of $v$ .", "We have $w_L = v_L$ and $(w_R \\sigma )_R = (w \\sigma )_R = v_R$ by Lemma REF , which also means that ${+_w} = {+_v}$ and ${-_{w_R \\sigma }} ={-_v}$ .", "For the middle term of the reaction result we apply again Lemma REF : Since $w_R \\sigma $ is separating and $x w_R \\sigma = w \\sigma = v$ , we must have $x^{\\prime }\\widehat{w_R \\sigma } = \\hat{v}$ .", "Therefore the result of (REF ) is $v_L +_v \\hat{v} -_vv_R$ .", "Putting everything together we show this way that $v \\rightarrow _Rv_L +_v \\hat{v} -_v v_R$ if $v = w \\sigma $ .", "If $v = w \\sigma $ , a similar argument can be used.", "Definition 6.13 (Local Reaction System) Let $\\varphi $ be a well-behaved transition rule.", "Let $\\Phi $ be the reaction system generated by the ordered situations in $\\mathcal {A}_\\varphi $ and the following reactions, for all separating $u\\in \\Sigma ^$ and $\\sigma \\in \\Sigma $ , $u &\\rightarrow _\\Phi u +_u \\hat{u} -_u u,&&\\text{if $u$ is minimal,} \\\\\\hat{u} -_u u +_u \\hat{u} &\\rightarrow _\\Phi \\hat{u},&&\\text{if $u$ is minimal,} \\\\\\sigma u +_u \\hat{u}& \\rightarrow _\\Phi (\\sigma u)_L +_{\\sigma u} \\widehat{\\sigma u},&&\\text{if $u$ is left minimal,} \\\\\\hat{u} -_u u\\sigma &\\rightarrow _\\Phi \\widehat{u\\sigma } -_{u\\sigma } (u\\sigma )_R,&&\\text{if $u$ is right minimal.", "}$ This reaction system is called the local reaction system for $\\varphi $ .", "For completeness, the diagrams for these reactions are also shown, in Figure REF .", "Figure: Generators of the local reaction system.In the rest of this chapter we will prove the following properties of the local reaction system.", "Theorem 6.14 (Properties of Local Reaction Systems) Let $\\varphi $ be a well-behaved transition rule and $\\Phi $ the local reaction system for $\\varphi $ .", "Then $\\Phi $ has the following properties: (Covering, Figure REF ).", "If $s \\in \\mathop {\\mathrm {dom}}\\Phi $ and $[p] \\sigma \\in \\mathop {\\mathrm {cl}}\\nolimits \\mathrm {pr}(s)$ , then there is a reaction $s\\rightarrow _\\Phi v$ such that $[p] \\sigma \\in \\mathrm {pr}(v)$ .", "(Confluence, Figure REF ).", "If there are reactions $a \\rightarrow _\\Phi b_1$ and $a \\rightarrow _\\Phi b_2$ , then there is a situation $c \\in \\mathop {\\mathrm {dom}}\\Phi $ such that $b_1 \\rightarrow _\\Phi c$ and $b_2\\rightarrow _\\Phi c$ .", "The first property is proved in Theorem REF , the second property in Theorem REF ." ], [ "Covering", "The property that is the subject of this section is a kind of converse to the definition of reactions with help of the closure: Given a reaction system $R$ , do we have $\\mathop {\\mathrm {cl}}\\nolimits \\mathrm {pr}(a) = \\bigcup \\lbrace \\, \\mathrm {pr}(b): a \\rightarrow _R b \\,\\rbrace $ for a situation $a \\in \\mathop {\\mathrm {dom}}R$ ?", "If this is true, then we say that $R$ covers the closure of $a$ .", "If $R$ covers the closure of every $a \\in \\mathop {\\mathrm {dom}}R$ , then no information about the cellular automaton gets lost when switching from the work with closures to the work with reaction systems.", "The most important case is of course the local reaction system $\\Phi $ .", "We will now prove the closure property in a slightly different form, by asking whether a specific event belongs to the closure of $a$ .", "The simplest case occurs when the initial situation $a$ itself an interval.", "We can then express Lemma REF for well-behaved transition rules in terms of reactions.", "Lemma 6.15 (Intervals are Covering) Let $\\varphi $ be a well-behaved transition rule.", "Let $R$ be a reaction system for $\\varphi $ where for every separating interval $a \\in \\Sigma ^$ with characteristic reaction $a \\rightarrow [1, i] \\hat{a} [-1, j]$ there is a reaction $a \\rightarrow _R a_+ \\hat{a} a_-$ with $\\delta (a_+) = (1, i)$ and $\\delta (a_-) = (-1, j)$ .", "Then for every interval $u \\in \\mathop {\\mathrm {dom}}R$ and every event $[p]\\sigma \\in \\mathop {\\mathrm {cl}}\\nolimits \\mathrm {pr}(u)$ there is a reaction $u \\rightarrow _R v$ with $[p]\\sigma \\in \\mathrm {pr}(v)$ .", "We will prove the following assertion for every $t \\ge 0$ : If there is a reaction $u \\rightarrow _R u_+ a u_-$ with $\\mathop {\\mathrm {cl}}\\nolimits ^{(t)} \\mathrm {pr}(u) =\\mathrm {pr}_{u_+}(a)$ , then there is a reaction $u \\rightarrow _R u^{\\prime }_+ a^{\\prime } u^{\\prime }_-$ with $\\mathop {\\mathrm {cl}}\\nolimits ^{(t + 1)} \\mathrm {pr}(u) = \\mathrm {pr}_{u^{\\prime }_+}(a^{\\prime })$ .", "Since $cl_\\varphi ^{(0)} \\mathrm {pr}(u) = \\mathrm {pr}(u)$ , we know then by induction that for all $t \\ge 0$ there is a reaction $u \\rightarrow _R v$ with $\\mathop {\\mathrm {cl}}\\nolimits ^{(t)} \\mathrm {pr}(u) \\subseteq \\mathrm {pr}(v)$ , which proves the lemma.", "Assume now that $u \\rightarrow _R u_+ a u_-$ with $\\mathop {\\mathrm {cl}}\\nolimits ^{(t)} \\mathrm {pr}(u) =\\mathrm {pr}_{u_+}(a)$ .", "If $a$ is separating, then there is a reaction (REF ) for it.", "Then $\\Delta \\mathrm {pr}(a) = \\mathrm {pr}([1, i] \\hat{a})$ , and therefore $\\mathop {\\mathrm {cl}}\\nolimits ^{(t + 1)}\\mathrm {pr}(u) = \\Delta (\\mathop {\\mathrm {cl}}\\nolimits ^{(t)} \\mathrm {pr}(u)) = \\Delta \\mathrm {pr}_{u_+}(a) = \\mathrm {pr}_{u_+}([1, i] \\hat{a})$ .", "So if we set $u^{\\prime }_+ = u_+a_+$ , $a^{\\prime } = \\hat{a}$ and $u_- = a_- u_-$ , the assertion is true for a separating $a$ .", "If $a$ is non-separating, especially empty, then $\\Delta \\mathrm {pr}(a) =\\emptyset $ by assumption.", "This also means that $\\mathop {\\mathrm {cl}}\\nolimits ^{(t + 1)}\\mathrm {pr}(u) = \\Delta (\\mathop {\\mathrm {cl}}\\nolimits ^{(t)} \\mathrm {pr}(u)) = \\Delta \\mathrm {pr}_{u_+}(a) = \\emptyset $ .", "So we may choose $u^{\\prime }_+ = u_+ a$ , $a^{\\prime } =[0]$ , and $u^{\\prime }_- = u_-$ to fulfil the initial assertion of this proof.", "The local reaction system is then a specific case of the reaction system in the previous lemma, so we get: Lemma 6.16 (Covering) Let $\\varphi $ be a well-behaved transition rule.", "Let $R$ be a reaction system for $\\varphi $ which has for every separating interval $a \\in \\Sigma ^$ a reaction $a \\rightarrow _R a_L +_a \\hat{a} -_a a_R\\,.$ Then for every interval $u \\in \\mathop {\\mathrm {dom}}R$ and every event $[p]\\sigma \\in \\mathop {\\mathrm {cl}}\\nolimits \\mathrm {pr}(u)$ there is a reaction $u \\rightarrow _R v$ with $[p]\\sigma \\in \\mathrm {pr}(v)$ .", "Let $a \\rightarrow [1, i] \\hat{a} [-1, j]$ be the characteristic reaction for $a$ in (REF ).", "Then $\\delta (a_L+_a) = (0, \\mathopen \|a_L\\mathclose \|) + (1, i - \\mathopen \|a_L\\mathclose \|) = (1, i)$ and $\\delta (-_a a_R) = (-1, j - \\mathopen \|a_R\\mathclose \|) + (0, \\mathopen \|a_R\\mathclose \|) = (-1,j)$ .", "Therefore we can apply Lemma REF , which finishes the proof.", "With this lemma we can prove covering for the general case (Figure REF ).", "Theorem 6.17 (Covering by Achronal Situations) Let $\\Phi $ be the local reaction system for a well-behaved transition rule $\\varphi $ and let $s \\in \\mathop {\\mathrm {dom}}\\Phi $ .", "Then for all events $[p] \\sigma \\in \\mathop {\\mathrm {cl}}\\nolimits \\mathrm {pr}(s)$ there is a reaction $s\\rightarrow _\\Phi v$ with $[p]\\sigma \\in \\mathrm {pr}(v)$ .", "Figure: A reaction that covers the point pp.Assume that $a = a_1 u a_2$ with $u \\in \\Sigma ^$ separating.", "Then there is a reaction $a \\rightarrow _\\Phi a_1 +_u \\hat{u} -_u a_2$ , and if $[p]\\sigma \\notin \\mathop {\\mathrm {cl}}\\nolimits \\mathrm {pr}(a_1 +_u \\hat{u} -_u a_2)$ , we must have $[p] \\sigma \\in pr_{a_1}(u)$ .", "Assume that $a = a_1 \\hat{u} -_u u +_u \\hat{u} a_2$ with $u \\in \\Sigma ^$ minimally separating.", "Then there is a reaction $a \\rightarrow _\\Phi a_1 \\hat{u} a_2$ , and if $[p]\\sigma \\notin \\mathop {\\mathrm {cl}}\\nolimits \\mathrm {pr}(a_1 \\hat{u}a_2)$ , we must have $[p] \\sigma \\in pr_{a_1 \\hat{u} -_u}(u)$ .", "These two types of reactions generate all reactions in $\\Phi $ , so we see: If $a \\rightarrow _\\Phi b$ and $[p] \\sigma \\notin \\mathop {\\mathrm {cl}}\\nolimits \\mathrm {pr}(b)$ , then there must be a reaction $a \\rightarrow _\\Phi b^{\\prime }$ with $[p] \\sigma \\in \\mathrm {pr}(b^{\\prime })$ .", "Now there is for every $a \\in \\mathop {\\mathrm {dom}}\\Phi $ a reaction $a \\rightarrow _\\Phi b$ with $b =u_1 +_{u_1} \\dots +_{u_{k-1}} u_k +_{u_k}v -_{w_\\ell } w_\\ell -_{w_{\\ell -1}} \\dots -_{w_1} w_1\\,.$ in which $v$ is an interval and the $u_i$ and $w_i$ are minimal separating intervals.", "This can shown in an analogous way to the proof of Lemma REF below.", "With this definition we have $\\mathop {\\mathrm {cl}}\\nolimits \\mathrm {pr}(b) = \\mathrm {pr}(b) \\cup \\mathop {\\mathrm {cl}}\\nolimits \\mathrm {pr}_{u_1 +_{u_1}\\dots +_{u_\\ell }}(v)$ .", "This is so because $\\varphi $ is well-behaved and therefore the $u_i$ and $v_i$ provide no additional events to the closure of $b$ .We have e. g. $\\Delta \\mathrm {pr}(u_1)\\subseteq \\mathrm {pr}_{u_1 +_{u_1}}(u_2)$ by the third property of Definition REF .", "Now $v$ , as an interval, is covering, so there is either $[p] \\sigma \\in \\mathop {\\mathrm {cl}}\\nolimits \\mathrm {pr}(b)$ ; then a reaction $v \\rightarrow _\\Phi c$ can be applied to $b$ in order to cover $[p] \\sigma $ .", "Or, by the argument outlined above, there is directly a reaction $a \\rightarrow _\\varphi b^{\\prime }$ with $[p]\\sigma \\in \\mathrm {pr}(b^{\\prime })$ .", "In either case $[p] \\sigma $ is covered." ], [ "Confluence", "I have borrowed the notion of confluence from the theory of term rewriting systems, especially the lambda calculus [1].", "If a term rewriting system is confluent and a term $a$ can be transformed by one rule of that system to a term $b_1$ and by another rule to a term $b_2$ , then there is a term $c$ in that system that serves as a unifying target for $b_1$ and $b_2$ : there is a rule that transforms $b_1$ to $c$ and another rule that transforms $b_2$ to $c$ .", "Figure: Two confluent reactions.In the language of reaction systems this means: A reaction system $R$ is confluent if for every pair of reactions $a \\rightarrow _R b_1$ and $a\\rightarrow _R b_2$ there is a situation $c \\in \\mathop {\\mathrm {dom}}R$ such that $b_1 \\rightarrow _Rc$ and $b_2\\rightarrow _R c$ (Figure REF ).", "We will now show that this is also true for the local reaction system $\\Phi $ .", "The situation $c$ will then have a specific form, which we will specify with help of a subset of $\\Phi $ .", "Definition 6.18 (Slope Subsystems) Let $R \\subseteq \\mathcal {A}_\\varphi \\times \\mathcal {A}_\\varphi $ be a reaction system for the transition rule $\\varphi $ .", "Then the subsystems $R_+ = R \\cap (\\mathcal {A}_{\\varphi +} \\times \\mathcal {A}_{\\varphi +})\\qquad \\text{and}\\qquad R_- = R \\cap (\\mathcal {A}_{\\varphi -} \\times \\mathcal {A}_{\\varphi -})$ of $R$ are the systems of positive and negative slope reactions.", "The elements of $\\mathop {\\mathrm {dom}}R_-$ and $\\mathop {\\mathrm {dom}}R_+$ are the positive and negative slopes.", "The slope subsystems of $R$ therefore consist only of reactions between slopes of the same kind.", "The system $R_+$ consists of all reactions in $R$ that transform a positive slope into a positive slope, while $R_-$ consists of all the reactions that transform negative slopes into negative slopes.", "(Note that every interval situation is a negative and a positive slope and that therefore the reactions between intervals belong to both subsystems.)", "The following lemma about slope subsystems is also important in its own right.", "Lemma 6.19 (Slope Decomposition) Let $\\Phi $ be a local reaction system and $a \\in \\Phi $ .", "Then there exist situations $a_+ \\in \\mathop {\\mathrm {dom}}\\Phi _+$ and $a_- \\in \\mathop {\\mathrm {dom}}\\Phi _-$ such that $a \\rightarrow _\\Phi a_+ a_-$ .", "If $a$ is not the product of an element of $\\mathop {\\mathrm {dom}}\\Phi _+$ with an element of $\\mathop {\\mathrm {dom}}\\Phi _-$ , then it must contain a $-_b$ operator left of an $+_c$ operator.", "Especially there must be a pair of $-_b$ and $+_c$ operator that are separated only by an interval.", "This means that there must be a decomposition $a = a_1 \\hat{b}_1-_{b_1} u +_{b_2} \\hat{b}_2 a_2$ , where $u$ is an interval, $b_1$ and $b_2$ are minimal separating intervals and $b_1 = u_L$ and $b_2 =u_R$ .", "Then, since $u$ is a separating interval, there is a reaction $u \\rightarrow _\\Phi b_1 +_{b_1} \\hat{u} -_{b_2} b_2$ .", "The application of this reaction to $a$ results in $a \\rightarrow _\\Phi a_1 \\underline{\\hat{b}_1 -_{b_1}b_1 +_{b_1} \\hat{u}} -_{b_2} b_2+_{b_2} \\hat{b}_2 a_2$ We concentrate now on the underlined part of the reaction result.", "Because the result is achronal, there must be an interval $u^{\\prime }$ such that $\\hat{u} = \\hat{b}_1 u^{\\prime }$ .", "This means that we can apply a reaction of the form () to $\\hat{b}_1 -_{b_1} b_1 +_{b_1}\\hat{u}$ : We get then the reaction $\\hat{b}_1 -_{b_1} b_1 +_{b_1} \\hat{u} = \\hat{b}_1 -_{b_1} b_1 +_{b_1} \\hat{b}_1 u^{\\prime } \\rightarrow _\\Phi \\hat{b}_1 u^{\\prime } =\\hat{u}$ .", "Applying this reaction to the result of (REF ) results in $a \\rightarrow _\\Phi a_1 \\underline{\\hat{u} -_{b_2} b_2+_{b_2} \\hat{b}_2} a_2$ With the same kind of argument we can show that there is a reaction that replaces the underlined part of this reaction with $\\hat{u}$ , resulting in $a \\rightarrow _\\Phi a_1 \\hat{u} a_2$ .", "This reaction has removed one $-_{b_1}$ and one $+_{b_2}$ operator from $a$ .", "Repeating this removes all pairs of $-_b$ and $+_{b^{\\prime }}$ from $a$ .", "The result is a reaction $a \\rightarrow _\\Phi a_+ a_-$ to a situation of the required form.", "The situation $c$ that makes the two reactions $a \\rightarrow _\\Phi b_1$ and $a \\rightarrow _\\varphi b_2$ confluent will be constructed step by step in an induction proof.", "The following proposition is a technical lemma that will be used in Lemma REF to transform $b_1$ and $b_2$ into successively better approximations of $c$ .", "Lemma 6.20 (Creation of a Minimal Separating Boundary) Let $a \\in \\mathop {\\mathrm {dom}}\\Phi $ , where $\\Phi $ is a local reaction system.", "Then at least one of the following cases occurs: $\\delta (a)_T \\le 0$ and there is a reaction $a \\rightarrow _\\Phi a^{\\prime }-_u u$ , where $u$ is a right minimal separating interval, $\\delta (a)_T \\ge 0$ and there is a reaction $a \\rightarrow _\\Phi u+_u a^{\\prime }$ , where $u$ is a left minimal separating interval, or $\\delta (a)_T = 0$ , there is a reaction $a \\rightarrow _\\Phi v$ to a non-separating interval $v$ , and $\\mathop {\\mathrm {cl}}\\nolimits ^{(t)} \\mathrm {pr}(a) = \\emptyset $ for all $t \\ge 1$ .", "Let $\\delta (a)_T \\le 0$ .", "Because of Lemma REF there is a reaction $a \\rightarrow _\\Phi a_+ a_-$ with $a_+ \\in \\mathop {\\mathrm {dom}}\\Phi _+$ and $a_- \\in \\mathop {\\mathrm {dom}}\\Phi _-$ .", "If $\\delta (a_-)_T < 0$ , there must be a decomposition $a_- = a^{\\prime }_--_v v x$ , where $v$ is a minimal separating interval and $x$ an interval.", "Since $v x$ is separating, there is by Lemma REF a reaction $v x \\rightarrow _\\Phi (v x)_L+_{v x} \\widehat{v x} -_{v x} (v x)_R$ .", "So we have a reaction $a \\rightarrow _\\Phi a_+ a^{\\prime }_- -_v (v x)_L +_{v x}\\widehat{v x} -_{v x} (v x)_R$ and case 1 occurs with $u = (vx)_R$ .", "If $\\delta (a_-)_T = 0$ , we must also have $\\delta (a_+)_T = 0$ because $\\delta (a_+)_T + \\delta (a_-)_T = \\delta (a)_T \\le 0$ while $\\delta (a_+) \\ge 0$ .", "Then $a_+ a_-$ is an interval.", "If $\\mathop {\\mathrm {cl}}\\nolimits ^{(1)} \\mathrm {pr}(a) \\ne \\emptyset $ , then there must be a reaction $a_+ a_- \\rightarrow _\\Phi (a_+ a_-)_L +_{a_+ a_-}\\widehat{a_+ a_-} -_{a_+ a_-} (a_+ a_-)_R$ ; then case 1 occurs with $u = (a_+ a_-)_R$ .", "If $\\mathop {\\mathrm {cl}}\\nolimits ^{(1)} \\mathrm {pr}(a) = \\emptyset $ , then $a_+ a_-$ must be a non-separating interval because $\\varphi $ is well-behaved; then case 3 occurs with $v = a_+ a_-$ .", "The case of $\\delta (a)_T \\ge 0$ is handled in a mirror-symmetric way.", "We will now show a slightly stronger form of confluence, in order to get a good induction proof.", "In the following lemma we will say that two situations $x_1$ and $x_2$ are equal until time $t$ if for all $\\tau \\le t$ we have $\\mathrm {pr}(x_1)^{(\\tau )} = pr(x_2)^{(\\tau )}$ .", "Lemma 6.21 (Approximated Confluence) Let $\\Phi $ be a local reaction system.", "If there are reactions $a \\rightarrow _\\Phi b_1$ and $a \\rightarrow _\\Phi b_2$ and the situations $b_1$ and $b_2$ are equal before time $t$ , then there are situations $c_1$ , $c_2 \\in \\mathop {\\mathrm {dom}}\\Phi $ that are equal until time $t +1 $ , and reactions $a \\rightarrow _\\Phi c_1$ , $b_1 \\rightarrow _\\Phi c_1$ , $a\\rightarrow _\\Phi c_2$ and $b_2 \\rightarrow _\\Phi c_2$ .", "In the following proof, $\\mathcal {S}_{+-}$ is the set $\\lbrace \\, a_+a_-\\colon a_+ \\in \\Phi _+, a_- \\in \\Phi _- \\,\\rbrace $ .", "Let $b_1$ and $b_2$ be equal before time $t$ .", "If both $\\mathrm {pr}(b_1)^{(t)}$ and $\\mathrm {pr}(b_1)^{(t)}$ are empty, the lemma is trivially true, so we assume from now on that this is not the case.", "We know already that $\\delta (b_1) = \\delta (b_2)$ .", "With Lemma REF we can also assume that $b_1$ and $b_2$ are elements of $\\mathcal {S}_{+-}$ .", "If $b_1$ and $b_2$ are equal until time $t$ , then there are situations $x$ , $y$ , $b^{\\prime }_1$ and $b^{\\prime }_2 \\in \\mathop {\\mathrm {dom}}\\Phi $ such that $b_1= x b^{\\prime }_1 y$ and $b_2 = x b_2 y$ , and $t$ is the minimum of $\\delta (x)$ and $\\delta (x b_1)$ .", "Since $\\delta (b_1)$ and $\\delta (b_2)$ are equal, $\\delta (b^{\\prime }_1)$ and $\\delta (b^{\\prime }_2)$ are equal too and also elements of $\\mathcal {S}_{+-}$ .", "If $\\delta (b^{\\prime }_1)_T = \\delta (b^{\\prime }_2)_T > 0$ , then there are by Lemma REF two reactions $b^{\\prime }_1 \\rightarrow u_1+_{u_1} b^{\\prime \\prime }_1$ and $b^{\\prime }_2 \\rightarrow u_2 +_{u_2} b^{\\prime \\prime }_2$ , with intervals $u_1$ and $u_2$ .", "We may assume without loss of generality that $\\mathopen \|u_1\\mathclose \| \\le \\mathopen \|u_2\\mathclose \|$ .", "Then $\\mathrm {pr}_{x}(u_1) \\subseteq \\mathop {\\mathrm {cl}}\\nolimits \\mathrm {pr}(a)$ and $\\mathrm {pr}_{x}(u_2) \\subseteq \\mathop {\\mathrm {cl}}\\nolimits \\mathrm {pr}(a)$ , therefore $\\mathrm {pr}_{x}(u_2)\|_{\\mathop {\\mathrm {dom}}\\mathrm {pr}_{x}(u_1)} = \\mathrm {pr}_{x}(u_1)$ .", "So $\\mathrm {pr}_{x}(u_1) \\subseteq \\mathrm {pr}_{x}(u_2)$ , and if $u_1 \\ne u_2$ , then $u_1$ is an initial segment of $u_2$ , which is impossible because $u_2$ is already a right minimal separating interval.", "So we must have $u_1 = u_2$ .", "Then we can set $c_1= x u_1 +_{u_1} b^{\\prime \\prime }_1 y$ and $c_2 = x u_1 +_{u_1} b^{\\prime \\prime }_2 y$ ; these situations are equal until time $t + 1$ .", "The same kind of argument works if $\\delta (b^{\\prime }_1) = \\delta (b^{\\prime }_2) < 0$ .", "Now we assume that $\\delta (b^{\\prime }_1)_T = \\delta (b^{\\prime }_2)_T = 0$ .", "If $b^{\\prime }_1$ and $b_2$ are intervals, then they must be equal, by an argument similar to that in the previous paragraph.", "We can then set $c_1 =c_2 = b_2$ .", "Otherwise, if $b_1$ is not an interval, it must still be an element of $\\mathcal {S}_{+-}$ , so there must be a reaction $b^{\\prime }_1 \\rightarrow u_++_{u_+} b^{\\prime \\prime }_1 -_{u_-} u_-$ , with separating intervals $u_+$ and $u_-$ .", "But this means that $b_2$ cannot react to a non-separating interval $v$ : If this were the case, the process $\\mathrm {pr}_{x}(u_+)$ would be a part of $\\mathrm {pr}_{x}(v)$ , but $\\mathrm {pr}_{x}(u_+)$ is a separating interval and therefore cannot be part of a non-separating interval.", "So there must be a reaction $b^{\\prime }_2 \\rightarrow u_+ +_{u_+} b^{\\prime \\prime }_2 -_{u_-} u_-$ , where the “re-use” of $u_+$ and $u_-$ can be justified as in the previous paragraphs.", "In this case we have $c_1 = x u_+ +_{u_+} b^{\\prime \\prime }_1 -_{u_-}u_-$ and $c_2 = x u_+ +_{u_+} b^{\\prime \\prime }_2 -_{u_-} u_-$ .", "A similar argument can be used when $b_2$ is not an interval.", "This concludes the proof.", "Theorem 6.22 (Confluence) If there are reactions $a \\rightarrow _\\Phi b_1$ and $a \\rightarrow _\\Phi b_2$ , then there is a situation $c \\in \\mathop {\\mathrm {dom}}\\Phi $ such that $b_1 \\rightarrow _\\Phi c$ and $b_2 \\rightarrow _\\Phi c$ .", "We apply the induction steps outlined in Lemma REF .", "Since $b_1$ and $b_2$ are finite, there is certainly a time $t_0$ such that $b_1$ and $b_2$ are equal until $t_0$ .", "By repeated application of the lemma we get the four sequences of reactions $a &\\rightarrow _\\Phi c_{1, k},\\qquad &b_1 &\\rightarrow _\\Phi c_{1, k}, \\\\a &\\rightarrow _\\Phi c_{2, k},&b_2 &\\rightarrow _\\Phi c_{2, k},$ with $c_{1, k}$ and $c_{2, k}$ equal until time $k$ .", "The only remaining question is whether this process stops after a finite number of steps.", "To see this this we note that if $\\mathrm {pr}(b_1)^{(t)} = \\mathrm {pr}(b_1)^{(t)} =\\emptyset $ for all time steps $t > t_0$ , then the same is true for $c_{1, k}$ and $c_{2, k}$ , and for arbitrary $k$ .", "This can be verified by following the constructions in Lemma REF and Lemma REF ." ], [ "Summary", "In this chapter we have found a way to construct a reaction system from a transition rule.", "Separating intervals played an important role.", "They allowed us to construct the set of achronal situations; and an easily recognisable subset of them, the ordered achronal sets, were shown to have always a closure.", "We have therefore found a subset of situations that generalises intervals but nevertheless consist of events at different times.", "To prove this we had to restrict the set of transition rules a bit further, from interval-preserving to well-behaved rules.", "I expect that this restriction is only temporary and later may be loosened to allow for an extension of achronal sets to a larger class of transition rules.", "For the moment we have nevertheless the definition of a reaction system that is usable for “naturally occurring” transition rules, like Rule 54 in the next chapter.", "This local reaction system was introduced and shown to have useful properties.", "Since it has the covering property, all information that can be found with help of the closure operator can also be found with reactions.", "We are therefore no longer dependent on processes to derive results on cellular automata." ], [ "Rule 54", "Up to now we have worked with cellular automata only in an abstract way.", "Now we will introduce a concrete cellular automaton which already has a complex behaviour.", "The aim of this chapter is then to demonstrate the concepts of the previous chapters for an elementary cellular automaton, Rule 54.", "It also shows how structures of intermediate complexity manifest in the context of Flexible Time." ], [ "Elementary Cellular Automata", "Rule 54 arises in the context of the elementary cellular automata.", "We have seen in the introduction that they are the one-dimensional cellular automata with radius 1 and $\\Sigma = \\lbrace 0,1\\rbrace $ and that Stephen Wolfram [59] has provided an enumeration scheme for them.", "In Wolfram's enumeration scheme we interpret the state set $\\Sigma $ as a set of integers.", "There is a number $s$ such that $\\Sigma = \\lbrace 0,\\dots , s - 1\\rbrace $ , and $\\Sigma $ can be seen as the set of digits for base $s$ .", "A sequence of such digits is then an integer.", "Then we can view every state of the neighbourhood of a cell as a number with $2r +1$ digits, the code number for the neighbourhood.", "If we then enumerate the results of $\\varphi $ applied to every neighbourhood $w \\in \\Sigma ^{2r+1}$ by the code number of $w$ , the transition rule itself is another number under base $s$ , this time with $s^{2r+1}$ digits.", "This number is the code number for the function $\\varphi $ .", "Definition 7.1 (Code numbers) Let $\\Sigma = \\lbrace 0, \\dots , s-1 \\rbrace \\subseteq \\mathbb {N}_0$ , $r \\in \\mathbb {N}$ , and let $\\varphi \\colon \\Sigma ^{2r+1} \\rightarrow \\Sigma $ be a transition rule.", "For any sequence $w = \\omega _0 \\dots \\omega _{2r} \\in \\Sigma ^{2r+1}$ , let $c(w) = \\Sigma _{i=0}^{2r} \\omega _i s^i$ .", "Then the code number for $\\varphi $ is $\\sum _{u \\in \\Sigma ^d}\\varphi (u)s^{c(u)}\\,.$" ], [ "Cellular Processes as Diagrams", "We need to determine the local reaction system for Rule 54.", "These computations involve some cellular processes, and the easiest way to write them down is as a rectangular diagram—especially since these cellular processes will contain events from at most two different time steps.", "Such a diagram may have the shape ${\\begin{matrix} &\\tau \\\\ \\sigma _0 & \\sigma _1& \\sigma _2 \\end{matrix}}$ .", "This specific diagram describes a cellular process in which the cells in the states $\\sigma _0$ , $\\sigma _1$ and $\\sigma _2$ belong to time step 0 and the cell in state $\\tau $ belongs to time step 1.", "In such diagrams the leftmost event in the bottom line always belongs to the space-time point $(0, 0)$ , therefore the process can be written in the set notation as $\\lbrace [0, 0]\\sigma _0, [0,1]\\sigma _1, [0, 2] \\sigma _2, [1, 1]\\tau \\rbrace $ ." ], [ "Basic Properties of Rule 54", "I have chosen Rule 54 because it has some complex behaviour [4], [36], and it is a relatively simple rule in which an ether appears.", "An example for ether generation is Figure REF .", "Rule 54 has the transition rule $\\varphi _{54}(s) =\\left\\lbrace \\begin{array}{r@{\\quad }l}1 & \\text{for $s \\in \\lbrace (0,0,1), (1,0,0), (0,1,0), (1,0,1)\\rbrace $,}\\\\0 & \\text{otherwise.", "}\\end{array}\\right.$ Note that $\\varphi _{54}$ is symmetric under the interchange of left and right.", "Figure: “Rule icon” for Rule 54.The rule can be described in a diagram in Figure REF .", "The diagram displays each of the eight possible 3-cell neighbourhoods together with the next state of the central cell.", "This diagram has been sometimes called the “Rule Icon”For example by [54] and in the “Wolfram Atlas”—see http://atlas.wolfram.com/01/01/54/ for Rule 54..", "The description in (REF ) is for a human reader (in contrast to a computer) difficult to memorise.", "A simpler description is the following slogan, which can be verified from Figure REF .", "“$\\varphi _{54}(w) = 1$ if $w$ contains at least one 1, except if the cells in state 1 touch.” Here we say that two cells “touch” if they are direct neighbours.", "Thus the two cells in state 1 touch in the neighbourhood $(1, 1,0)$ , but not in the neighbourhood $(1, 0, 1)$ ." ], [ "Interval Preservation", "Next we must check whether Rule 54 is interval-preserving.", "To do this, we must test for all intervals $w\\in \\Sigma ^k$ with $k \\le 3$ whether $\\Delta \\mathrm {pr}(w)$ is an interval under the transition rule $\\varphi _{54}$ .", "If this is true, then $\\varphi _{54}$ is interval-preserving by Theorem REF .", "To do this we need a practical way to compute all the events determined by an interval $w$ .", "Among them are the events that can be found directly by the transition rule, when applied to the intervals of length $2r + 1$ , together with those events that are determined by smaller cellular processes.", "The transition rule implies that an interval $w$ of length $2r + 1$ determines the event $[1, r] \\varphi (w)$ .", "We are now interested in all subsets of $\\mathrm {pr}(w)$ that already determine the event $[1, r] \\varphi (w)$ .", "Since $\\mathrm {pr}(w)$ is an interval, every subset of $\\mathrm {pr}(w)$ can be extended to an interval by adding events.", "Therefore it is enough to search for the interval subsets of $\\mathrm {pr}(w)$ that determine the event $[1, r] \\varphi (w)$ .", "These intervals can be found by an application of Definition REF : If there is a decomposition $w = x w^{\\prime }y$ such that the value of $\\varphi (x w^{\\prime } y)$ is independent of the contents of $x$ and $y$ , then $\\mathrm {pr}_{x}(w^{\\prime })$ already determines $[1, r]\\varphi (w)$ .This means that the application of the transition rule must lead to the same result $\\varphi (x^{\\prime } w^{\\prime } y^{\\prime }) = \\varphi (w)$ for all intervals $x^{\\prime }$ , $y^{\\prime } \\in \\Sigma ^$ with $\\mathopen \|x^{\\prime }\\mathclose \| = \\mathopen \|x\\mathclose \|$ and $\\mathopen \|y^{\\prime }\\mathclose \| = \\mathopen \|y\\mathclose \|$ .", "This also means that the interval $\\mathrm {pr}(w^{\\prime })$ determines the event $[1, r - \\mathopen \|x\\mathclose \|] \\varphi (w)$ .", "With this method we can find all the intervals of length $\\le 2r + 1$ that determine an event under the transition rule $\\varphi $ .", "All this can then be expressed by a rule: If $\\varphi (x w y) =\\sigma $ for all $x \\in \\Sigma ^k$ and $y \\in \\Sigma ^\\ell $ , then $w$ determines the event $[1, r - k] \\sigma $ .", "We will now find these reactions for Rule 54.", "Rule 54 has the following cases where this rule can be applied: There are two cases with $k = 1$ and $\\ell = 0$ , namely $\\varphi (001) = \\varphi (101) = 1$ and $\\varphi (011) = \\varphi (111) = 0$ .", "The first equation shows that the interval 01 determines the event $[1, 0] 1$ and the second that 11 determines $[1, 0] 0$ .", "There are two cases with $k = 0$ and $\\ell = 1$ , namely $\\varphi (100) = \\varphi (101) = 1$ and $\\varphi (110) = \\varphi (111) = 0$ .", "The first equation proves that the interval 10 determines the event $[1, 1] 1$ and the second that 11 determines $[1, 1] 0$ .", "With these arguments we have found four new rules to find a determined event that belongs to an interval.", "With them it is now possible to find the next state of the middle cell for all intervals of length 3 that begin with 10 or 11 and for those that end with 01 or 11.", "There remain the reactions for the neighbourhoods that cannot be shortened in this way, namely 000 and 010.", "These neighbourhoods determine the events $[1, 1] 0$ and $[1, 1] 1$ , respectively.", "To the other intervals we can apply one of the four new reactions to get the state of the middle cell one time step later.", "Therefore we have now the six cases where an interval determines an event, $000 &\\text{ determines } [1, 1] 0, & {\\begin{matrix} &0\\\\ 0&0&0 \\end{matrix}} \\\\01 &\\text{ determines } [1, 0] 1, & {\\begin{matrix} 1\\\\ 0&1 \\end{matrix}} \\\\11 &\\text{ determines } [1, 0] 0, & {\\begin{matrix} 0\\\\ 1&1 \\end{matrix}} \\\\010 &\\text{ determines } [1, 1] 1, & {\\begin{matrix} &1\\\\ 0&1&0 \\end{matrix}} \\\\10 &\\text{ determines } [1, 1] 1, & {\\begin{matrix} &1\\\\ 1&0&\\phantom{1} \\end{matrix}} \\\\11 &\\text{ determines } [1, 1] 0\\,.", "& {\\begin{matrix} &0\\\\ 1&1&\\phantom{1} \\end{matrix}}$ With these rules we can compute the events determined by an interval.", "At the right they are visualised with space-time diagrams.", "In them, the bottom line contains the process of $w$ , and on top of it there is the event that is determined by it.", "Each of these diagrams, when applied to an interval, gives us the identity of one event that is determined by this interval.", "No we can prove that $\\varphi _{54}$ is interval-preserving.", "For this we take all intervals whose length is at most $2r + 1 = 3$ and apply graphically all reactions to them that can be applied.", "We then get the following diagrams; all of them have $\\mathrm {pr}(w)$ as their bottom row and $\\Delta \\mathrm {pr}(w)$ as the top row.", "$\\begin{array}[b]{8c}{\\begin{matrix} \\phantom{1}\\\\ 0 \\end{matrix}} &{\\begin{matrix} \\phantom{1}\\\\ 1 \\end{matrix}} &&&{\\begin{matrix} \\phantom{1}\\\\ 0&0 \\end{matrix}} &{\\begin{matrix} 1\\\\ 0&1 \\end{matrix}} &{\\begin{matrix} &1\\\\ 1&0 \\end{matrix}} &{\\begin{matrix} 0&0\\\\ 1&1 \\end{matrix}} \\\\[3ex]{\\begin{matrix} &0\\\\ 0& 0& 0 \\end{matrix}} &{\\begin{matrix} &1\\\\ 0& 0& 1 \\end{matrix}} &{\\begin{matrix} 1&1&1\\\\ 0& 1& 0 \\end{matrix}} &{\\begin{matrix} &0&0\\\\ 0& 1& 1 \\end{matrix}} &{\\begin{matrix} &1\\\\ 1& 0& 0 \\end{matrix}} &{\\begin{matrix} &1\\\\ 1& 0& 1 \\end{matrix}} &{\\begin{matrix} 0&0&1\\\\ 1& 1& 0 \\end{matrix}} &{\\begin{matrix} 0&0&0\\\\ 1& 1& 1 \\end{matrix}}\\end{array}$ They show that $\\varphi _{54}$ is interval-preserving for the intervals of length $\\le 3$ ; by Theorem REF it must then preserve all intervals." ], [ "Characteristic Reactions", "The diagrams in (REF ) allow us now to write down the characteristic reactions for all those intervals that determine at least one event.", "Their characteristic reactions are $01 & \\rightarrow [1, 0] 1 [-1, 1], \\\\10 & \\rightarrow [1, 1] 1 [-1, 0], \\\\11 & \\rightarrow [1, 0] 00 [-1, 0], \\\\000 & \\rightarrow [1, 1] 0 [-1, 1], \\\\010 & \\rightarrow [1, 0] 111 [-1, 0]\\,.$ Figure: “Invariant Rule Icon” for Rule 54.Together, these reactions allow us to derive all the events that are determined by a given cellular process.", "We can write them as an alternative form of the rule icon of Figure REF , one that in contrast to it does no longer depend explicitly on the radius.", "This diagram is shown in Figure REF .", "As we will see later, the minimal separating intervals of Rule 54 are the intervals of length 2.", "For the first three of them we have just determined the characteristic reactions.", "The characteristic reaction for the last interval, 00, can now be determined according to Definition REF .", "The characteristic reaction for the interval $a = 00$ must have the form $a \\rightarrow [1, i] \\hat{a} [-1, j],$ and we must now determine $i$ , $j$ and $\\hat{a}$ .", "This can be done with help of reaction ().", "If we extend $a$ to the left with 0, we get the reaction $0a \\rightarrow [1, 1] 0 [-1, 1]$ , therefore we must have $j = 1$ .", "And if we extend $a$ to the right with 0, we get $a0 \\rightarrow [1, 1] 0 [-1, 1]$ and therefore $i = 1$ .", "The only value for $\\hat{a}$ for which $\\delta (a) = \\delta ([1, i] \\hat{a} [-1, j])$ is $a = [0]$ , therefore the characteristic reaction for 00 is $00 \\rightarrow [1, 1] [-1, 1]\\,.$ Next we can see that the intervals of length 1 are not separating.", "This is because we cannot construct a characteristic reaction for them.", "For the “interval” consisting of a cell in state 0, we would have $i = 0$ in reaction (REF ) because of the characteristic reaction (REF ) for 01 but $i = 1$ because of reaction (REF ).", "To verify that the cell in state 1 does not form a separating interval, note that (REF ) requires $j = 1$ while () requires $j = 0$ .", "Figure: Separating intervals and their characteristic reactions forRule 54.On the other hand, Lemma REF shows that the intervals of length 2 are separating, so they must be the minimal separating intervals.", "They and their characteristic reactions are shown in Figure REF ." ], [ "The Local Reaction System", "With the characteristic reactions for the minimal separating intervals we can now determine the structure of the achronal situations for Rule 54." ], [ "Notation", "According to Definition REF , the set of achronal situations is known when the generating slopes $\\hat{b}-_b b$ and $b +_b \\hat{b}$ are known for all minimal separating intervals $b$ .", "For concrete calculations, the repetition of $b$ in these terms becomes however easily annoying.", "Therefore we will first introduce another, related, notation.", "It is a variant of the slope operators of Definition REF .", "Definition 7.2 (Concrete Slope Operators) Let $i \\in \\mathbb {Z}$ .", "Then we write $\\ominus _i = [-1, -i] \\quad \\text{and}\\quad \\oplus _i = [1, -i]\\,.$ If $r$ is the radius of the transition rule, we will use the abbreviations $\\ominus $ for $\\ominus _r$ and $\\oplus $ for $\\oplus _r$ .", "This notation had been introduced in [51] and was already used in [52]." ], [ "Achronal Situations", "Now we will derive the generating slopes $\\hat{b} -_b b$ and $b +_b \\hat{b}$ from the characteristic reactions of Figure REF .", "They are listed in Table REF .", "Its first column contains the characteristic reactions, and the other columns contain the generating slopes derived from them.", "Table: Characteristic reactions and generating slopes of Rule 54.First we need expressions for the slope operators $+_b$ and $-_b$ in terms of the new operators of Definition REF .", "We assume here, as before, that the characteristic reaction for every minimal separating interval $b$ is $b \\rightarrow [1, i] \\hat{b} [-1, j]$ .", "When we then use the notation of Definition REF , the two kinds of slope operators are related by the equations $+_b &= [1, i - \\mathopen \|b\\mathclose \|] = \\oplus _{\\mathopen \|b\\mathclose \| - i} \\\\-_b &= [-1, j - \\mathopen \|b\\mathclose \|] = \\ominus _{\\mathopen \|b\\mathclose \| - j}$ Here we have used the fact that $b$ is a minimal separating interval and that therefore $b_L = b_R = b$ .", "With the equations in (REF ) we now can derive the entries is the second and third column of Table REF from the characteristic reactions in the first columns.", "This derivation consists of two steps.", "The first is finding the values of $b$ and $\\hat{b}$ : They can be read of the characteristic reactions in the first column.", "The second step consists of finding the slope operators $+_b$ and $-_b$ .", "I will now show this in detail for the second column.", "We see from the first column of Table REF that there are two kinds of characteristic reactions, namely those where the reaction product starts with $[1, 1]$ and those where it starts with $[1, 0]$ .", "In the first case there is $i = 1$ and therefore $+_b = \\oplus _{2 - 1}= \\oplus $ , and that is why in the second column of Table REF the first and the third entry contains a $\\oplus $ operator.", "In the second case there is $i = 0$ and $+_b =\\oplus _{2 - 0} = \\oplus _2$ , and therefore the last second and fourth entry in the second column in Table REF contain a $\\oplus _2$ operator.", "The same way we can derive the third column of Table REF ." ], [ "Generating Reactions", "Now, to complete the description of the local reaction system for Rule 54, we need to find its generating reactions.", "They are defined in Definition REF and consist of two subsets: those that are associated to the minimal separating intervals, and those that are found by extending a minimal separating interval to the left or to the right.", "(a) The first subset consists of the reactions $b \\rightarrow _\\Phi b+_b \\hat{b} -_b b$ and $\\hat{b} -_b b +_b \\hat{b} \\rightarrow _\\Phi \\hat{b}$ for all minimal separating intervals $b$ .", "We do already know that $+_{00} &= +_{10} = \\oplus , &\\qquad +_{01} &= +_{11} = \\oplus _2, \\\\-_{00} &= -_{01} = \\ominus , &-_{10} &= -_{11} = \\ominus _2$ and that $\\widehat{00} = [0], \\qquad \\widehat{01} = \\widehat{10} = 1, \\qquad \\widehat{11} = 00\\,.$ With this we can calculate the reactions of the form $b \\rightarrow _\\Phi b+_b \\hat{b} -_b b$ in the following way, $00 &\\rightarrow _\\Phi 00 +_{00} \\widehat{00} -_{00} 00&&= 00 \\oplus \\ominus 00, \\\\01 &\\rightarrow _\\Phi 01 +_{01} \\widehat{01} -_{01} 01&&= 01 \\oplus _2 1 \\ominus 01, \\\\10 &\\rightarrow _\\Phi 10 +_{10} \\widehat{10} -_{10} 10&&= 10 \\oplus 1 \\ominus _2 10, \\\\11 &\\rightarrow _\\Phi 11 +_{11} \\widehat{11} -_{11} 11&&= 11 \\oplus _2 00 \\ominus _2 11,$ and the reactions of the form $\\hat{b} -_b b +_b \\hat{b}\\rightarrow _\\Phi \\hat{b}$ in the following way, $\\ominus 00 \\oplus =\\widehat{00} -_{00} 00 +_{00} \\widehat{00}&\\rightarrow _\\Phi \\widehat{00} = [0], \\\\1 \\ominus 01 \\oplus _2 1 =\\widehat{01} -_{01} 01 +_{01} \\widehat{01}&\\rightarrow _\\Phi \\widehat{01} = 1, \\\\1 \\ominus _2 10 \\oplus 1 =\\widehat{10} -_{10} 10 +_{10} \\widehat{10}&\\rightarrow _\\Phi \\widehat{10} = 1, \\\\00 \\ominus _2 11 \\oplus _2 00 =\\widehat{11} -_{11} 11 +_{11} \\widehat{11}&\\rightarrow _\\Phi \\widehat{11} = 11\\,.$ The results of these two sets of calculations are collected in the left and right bottom fields of Table REF , respectively.", "(b) The second subset of the reactions in Definition REF consists of reactions of the form $\\sigma b +_b \\hat{b} \\rightarrow _\\Phi (\\sigma b)_L +_{\\sigma b}\\widehat{\\sigma b}$ and $\\hat{b} -_b b\\sigma \\rightarrow _\\Phi \\widehat{b\\sigma } -_{b\\sigma } (b\\sigma )_R$ with $\\sigma \\in \\Sigma $ , where $b$ is a left or right minimal separating interval, respectively.", "We will now concentrate on the second type of reactions, which is sufficient because Rule 54 is symmetric.", "To compute the reactions $\\hat{b} -_b b\\sigma \\rightarrow _\\Phi \\widehat{b\\sigma } -_{b\\sigma } (b\\sigma )_R$ for all right minimal separating intervals $b$ we need to know $\\widehat{b \\sigma }$ , $(b\\sigma )_R$ and $-_{b \\sigma }$ for every right minimal interval $b$ and every $\\sigma \\in \\Sigma $ .", "And for this we first need to know the set of right minimal intervals for Rule 54.", "The easiest way to do it is to start in greater generality and to determine the values of $b_L$ and $b_R$ for every separating interval $b$ .", "In case of Rule 54 this is simple: Since every interval of length 2 is a minimal separating interval, $b_L$ consists of the two leftmost events in $b$ and $b_R$ if the two rightmost events in $b$ .", "(If $b$ is separating, it must contain a minimal separating interval and is therefore at least two cells long.)", "For this reason the set of left and right minimal separating intervals under Rule 54 is both times $\\Sigma ^2$ .", "Next we need the characteristic reactions for all elements of $\\Sigma ^3$ .", "For them we need, in turn, to know the cells determined by all the intervals in $\\Sigma ^3$ .", "They can be determined from (REF ) and are $\\begin{array}[b]{8c}{\\begin{matrix} &0\\\\ 0&0&0 \\end{matrix}} &{\\begin{matrix} &1\\\\ 0&0&1 \\end{matrix}} &{\\begin{matrix} 1&1&1\\\\ 0&1&0 \\end{matrix}} &{\\begin{matrix} 1&0&0\\\\ 0&1&1 \\end{matrix}} &{\\begin{matrix} &1\\\\ 1&0&0 \\end{matrix}} &{\\begin{matrix} &1\\\\ 1&0&1 \\end{matrix}} &{\\begin{matrix} 0&0&1\\\\ 1&1&0 \\end{matrix}} &{\\begin{matrix} 0&0&0\\\\ 1&1&1 \\end{matrix}}\\,.\\end{array}$ These diagrams are also a short description of the connection between $b$ and $\\hat{b}$ for all $b \\in \\Sigma ^3$ .", "We can derive the values of $\\hat{b}$ for $b \\in \\Sigma ^3$ directly from them, $\\begin{aligned}[b]\\widehat{000} &= 0, & \\widehat{010} &= 111, &\\widehat{100} &= 1, & \\widehat{110} &= 001, \\\\\\widehat{001} &= 1, & \\widehat{011} &= 100, &\\widehat{101} &= 1, & \\widehat{111} &= 000\\,.\\end{aligned}$ The values of $\\hat{b}$ for all $b \\in \\Sigma ^2$ are already listed in (REF ).", "To compute $(b \\sigma )_R$ and $-_{b \\sigma }$ we use the relations $(\\sigma _1 \\sigma _2 \\sigma _3)_R &= \\sigma _2 \\sigma _3, &-_{\\sigma _1 \\sigma _2 \\sigma _3} &= -_{\\sigma _2 \\sigma _3}$ for all $\\sigma _1$ , $\\sigma _2$ , $\\sigma _3 \\in \\Sigma $ ; for the second relation the values of $-_{\\sigma _2 \\sigma _3}$ can be found in ().", "With this data we can now calculate the reactions of the type $\\hat{b} -_b b\\sigma \\rightarrow _\\Phi \\widehat{b\\sigma }-_{b\\sigma } (b\\sigma )_R$ in the following way, $\\ominus 000 =\\widehat{00} -_{000} 000 &\\rightarrow _\\Phi \\widehat{000} -_{000} (000)_R= 0 \\ominus 00, \\\\\\ominus 001 =\\widehat{00} -_{001} 001 &\\rightarrow _\\Phi \\widehat{001} -_{001} (001)_R= 1 \\ominus 01, \\\\1 \\ominus 010 =\\widehat{01} -_{010} 010 &\\rightarrow _\\Phi \\widehat{010} -_{010} (010)_R= 111 \\ominus _2 10, \\\\1 \\ominus 011 =\\widehat{01} -_{011} 011 &\\rightarrow _\\Phi \\widehat{011} -_{011} (011)_R= 100 \\ominus _2 11, \\\\1 \\ominus _2 100 =\\widehat{10} -_{100} 100 &\\rightarrow _\\Phi \\widehat{100} -_{100} (100)_R= 1 \\ominus 00, \\\\1 \\ominus _2 101 =\\widehat{10} -_{101} 101 &\\rightarrow _\\Phi \\widehat{101} -_{101} (101)_R= 1 \\ominus 01, \\\\00 \\ominus _2 110 =\\widehat{11} -_{110} 110 &\\rightarrow _\\Phi \\widehat{110} -_{110} (110)_R= 001 \\ominus _2 10, \\\\00 \\ominus _2 111 =\\widehat{11} -_{111} 111 &\\rightarrow _\\Phi \\widehat{111} -_{111} (111)_R= 000 \\ominus _2 11\\,.$ The reactions that are found in this calculation are shown in Table REF , both as formulas and as diagrams.", "Table: Diagrams of the generator reactions for Rule 54.Together with the reactions that are their left-to right mirror images they form the upper part of the “Reactions” section in Table REF .", "Table: Generator reactions of the local reaction system for Rule 54.This completes the calculation of the local reaction system for Rule 54.", "The result is a new form of the transition rule $\\varphi _{54}$ ." ], [ "Understanding Rule 54 Better", "While $\\varphi _{54}$ describes how a cell's neighbourhood influences its state in the next time step, each reaction in the local reaction system describes the relation between a separating interval $\\pi $ and the interval $\\Delta \\pi $ that is determined by it.", "The generating slopes in Table REF describe the relations between the boundaries of $\\pi $ and $\\Delta \\pi $ : The slope $00 \\oplus $ means that if $\\pi $ begins with 00, then $\\Delta \\pi $ reaches one cell to the right of the left end of $\\pi $ ; the slope $10 \\oplus 1$ tells that if $\\pi $ begins with 10, then the left end of $\\Delta \\pi $ reaches to the same position, but its leftmost event must be in state 1, and so on.", "Even this localised knowledge helps us to understand the behaviour of Rule 54 better.", "For an example we use the task of finding the closure of an interval, something that we had already begun in Figure REF .", "We can now express the closure of an interval with a reaction $u \\rightarrow _\\Phi a_+ a_-$ , where $a_+$ is a positive and $a_-$ a negative slope.", "The reaction system $\\Phi $ has been constructed in such a way that there is a reaction in which the situations $a_+$ and $a_-$ form the boundaries of the triangle, which we will now assume.", "Then the process of $a_+$ consists of the leftmost separating intervals of each time slice of $\\mathop {\\mathrm {cl}}\\nolimits \\mathrm {pr}(u)$ .", "The analysis of the previous paragraph then helps us to understand better the ragged boundaries of the closure in Figure REF .", "Now let us add an event to the left side of the interval at the base of Figure REF .", "Then its closure grows too.", "The kind of growth, and how it depends on the added event, tells us how a change in the initial configuration is propagated to later time steps.", "For Rule 54, this is expressed by the reactions in the top half of the “Reactions” section of Table REF .", "The reactions at the left side of the table show what happens when a cell is added to the right, and those at the right side of the table show what happens when a cell is added to the left.", "The influence of the added event varies greatly depending on the states of the cells at the end of the original interval.", "We see from one pair of reactions, $010 \\oplus 1 \\rightarrow 01 \\oplus _2 111$ and $110 \\oplus 1\\rightarrow 11 \\oplus _2 001$ , that when the left side of the original interval is 00, the added event adds two events in the next time step; on the other hand, another pair, $001 \\oplus _2 1 \\rightarrow 00 \\oplus 1$ and $101 \\oplus _2 1\\rightarrow 10 \\oplus 1$ , proves that the closure may also stay unchanged.", "(It is peculiar to Rule 54 that the state of the added event has no influence on the number of cells that are added in the next time step, only on their states.)", "We have therefore found for each cellular automaton a specific pattern of influence, described by the generators of the local reaction system." ], [ "Summary", "In this chapter we have seen how the local reaction system is computed for a concrete rule.", "In calculations with a concrete system, brevity is an advantage and redundancy is annoying.", "Therefore we used in this chapter the symbols $\\ominus _k$ and $\\oplus _k$ instead of $-_a$ and $+_a$ for the display of the resulting reaction system.", "In spite of this the computation of the reaction system may appear to be quite long and complex, with all the explanations given.", "If one leaves them out, it is however possible to do the whole work described here on a single piece of paper.", "Nevertheless the resulting system in Table REF looks somewhat voluminous when compared with the original description of Rule 54 in (REF ).", "This is caused, among other things, by the requirement that a local reaction system covers the full closure of each of its situations.", "If we drop this requirement, then we can create for special purposes reaction systems that are easier to describe and more powerful.", "One of them will be constructed in the next chapter.", "The advantage of the large size of Table REF is however that it provides additional information about the way in which information travels in the cellular automaton.", "This was not directly visible from the transition rule." ], [ "Appendix: Rule 110", "Here I will give another example and construct the local reaction system for another elementary cellular automaton, Rule 110.", "The derivation will be much more sketchy, but it should also show how the calculation of a concrete local reaction system can be done in a relatively small space." ], [ "Nature of the Rule", "We will use as the initial description of the rule an icon similar to that of Figure REF .", "We see especially that Rule 110 is an asymmetric rule, in contrast to Rule 54.", "Figure: “Rule icon” for Rule 110.To understand Rule 110 better, we will now find a slogan for it, as we had done for Rule 54.", "This time the slogan becomes especially simple if we take the states 0 and 1 as Boolean values.", "Then we can write, “$\\varphi _{110}(\\sigma _{-1}, \\sigma _0, \\sigma _1) = \\sigma _0\\mathbin \\mathbf {xor} \\sigma _1$ , except that $\\varphi _{110}(0, 1, 1) =1$ .”" ], [ "Graphical Evolution", "With Figure REF we will now search for the cases where less than three cells determine the state of the central cell in the next time step.", "From the slogan we know that the value of this cell depends only on the interval consisting of the central cell and its neighbour, except when that interval is 11.", "For the other cases we can write this as the diagrams ${\\begin{matrix} 0\\\\ 0&0 \\end{matrix}}$ , ${\\begin{matrix} 1\\\\ 0&1 \\end{matrix}}$ and ${\\begin{matrix} 1\\\\ 1&0 \\end{matrix}}$ , as we have done in (REF ).", "There is also a fourth case when two adjacent cells determine the next state of a cell: In the interval 01, the next state of the right cell is always 1.", "Thus we get the following diagrams, $\\begin{array}[b]{4c}{\\begin{matrix} 0\\\\ 0&0 \\end{matrix}} &{\\begin{matrix} 1\\\\ 0&1 \\end{matrix}} &{\\begin{matrix} 1\\\\ 1&0 \\end{matrix}} &{\\begin{matrix} &1\\\\ 0&1 \\end{matrix}}\\,.\\end{array}$ We can also see from Figure REF that there are no shorter intervals that determine an event.", "To get all the diagrams that are needed for a graphical evolution we have to add the diagram for the interval 111, because the next state of its central cell cannot be derived otherwise; then we get $\\begin{array}[b]{6c}{\\begin{matrix} 0\\\\ 0&0 \\end{matrix}} &{\\begin{matrix} 1\\\\ 0&1 \\end{matrix}} &{\\begin{matrix} 1\\\\ 1&0 \\end{matrix}} &{\\begin{matrix} &1\\\\ 0&1 \\end{matrix}} &{\\begin{matrix} &0\\\\ 1&1&1 \\end{matrix}}\\,.\\end{array}$ These diagrams are then enough to find for every cellular process the events that are determined by it." ], [ "Minimal Intervals", "We have seen in Table REF that Rule 110 is interval-preserving, and from Table REF that its minimal separating intervals are 0 and 11.", "For the theory we will however need also the intervals that can occur as rightmost or leftmost minimal intervals.", "There is an additional leftmost minimal interval, 10, and another rightmost minimal interval, 01." ], [ "Characteristic Reactions", "For these intervals we now determine the events that are determined by them, using the diagrams of (REF ).", "We get another list of diagrams, $\\begin{array}[b]{4c}{\\begin{matrix} &\\phantom{1}\\\\ 0 \\end{matrix}} &{\\begin{matrix} &\\phantom{1}\\\\ 1&1 \\end{matrix}} &{\\begin{matrix} 1&1\\\\ 0&1 \\end{matrix}} &{\\begin{matrix} 1\\\\ 1&0 \\end{matrix}}\\,.\\end{array}$ The characteristic reactions for the minimally separating intervals are then $0 &\\rightarrow [1, 0] [-1, 1], &11 &\\rightarrow [1, 1] [-1, 1], \\\\01 &\\rightarrow [1, 0] 11 [-1, 0], &10 &\\rightarrow [1, 0] 1 [-1, 1]\\,.$ We find the coefficients in the first reaction once we realise that the next state of a cell in state 0 is always the state of the cell at its right.", "Therefore, the boundary between the cells that are determined by the left side and those determined by the right side is at the left of the cell in state 0, and this is reflected in the reaction.", "For the second reaction we must note that the next state for the left cell in the interval 11 depends on information at the left, and the state of the right cell depends on information from the right.", "The last two reactions in (REF ) can be read directly from the diagrams." ], [ "Generating Slopes", "From the characteristic reactions we derive the generating slopes.", "For the positive slopes we use the left coefficients of the characteristic reactions that belong to the left minimal separating intervals.", "We then get the situations $0\\oplus $ , $11 \\oplus $ and $01 \\oplus _2 1$ .", "For the negative slopes we use the rightmost minimal separating intervals and the coefficients of the right side of the characteristic reactions and get $\\ominus _0 0$ , $\\ominus 11$ and $11\\ominus _2 01$ ." ], [ "Reactions", "The generating reactions can also be derived in a graphical manner, especially the slope reactions.", "We will give here only a few examples.", "One example is the reaction $00 \\oplus \\rightarrow 0 \\oplus 0$ .", "First we write down the diagram for the input situation, ${\\begin{matrix} \\phantom{1}\\\\0&0 \\end{matrix}}$ .", "This we extend with help of the rule for graphical evolution and get ${\\begin{matrix} 0&\\\\ 0&0 \\end{matrix}}$ .", "We now mark on this diagram the rightmost positive slope.", "Then we get ${\\begin{matrix} \\colorbox {gray!45}{\\scriptstyle 0}\\\\ \\colorbox {gray!45}{\\scriptstyle 0}&0 \\end{matrix}}$ , and when we translate the sequence of marked events back into a positive slope, it is $0 \\oplus 0$ .", "In a similar way the reaction $010 \\oplus _2 1 \\rightarrow 0 \\oplus 11$ is derived.", "Here the diagram for the initial situation is ${\\begin{matrix} &1\\\\0&1&0 \\end{matrix}}$ .", "We extend it to ${\\begin{matrix} 1&1\\\\0&1&0 \\end{matrix}}$ and mark in it the rightmost positive slope.", "The result is ${\\begin{matrix} \\colorbox {gray!45}{\\scriptstyle 1}&\\colorbox {gray!45}{\\scriptstyle 1}\\\\\\colorbox {gray!45}{\\scriptstyle 0}&1&0 \\end{matrix}}$ , and the marked events belong to the situation $0 \\oplus 11$ .", "The reactions in the bottom of Table REF are derived more easily algebraically.", "For the bottom left block in that table we need to find reactions of the form $u \\rightarrow u +_u \\hat{u} -_uu$ , as in (REF ), where $u$ is a minimal separating interval.", "In the case of the interval 11, we must therefore find generating slopes $u +_u \\hat{u}$ and $u +_u \\hat{u}$ , with $u = 11$ .", "We have already seen that these generating slopes are $11 \\oplus $ and $\\ominus 11$ .", "This leads to the reaction $11 \\rightarrow 11 \\oplus \\ominus 11$ .", "For the bottom right block we need to find in the same way reactions of the form $\\hat{u} -_u u +_u \\hat{u} \\rightarrow _\\Phi \\hat{u}$ , as in , again for the minimal blocking intervals.", "In the case of the interval 11 we get the reaction $\\ominus 11 \\oplus \\rightarrow [0]$ , since $\\widehat{11} = [0]$ .", "Table: Generator reactions of the local reaction system for Rule 110." ], [ "Results", "All these results are summarised in Table REF .", "It is clearly visible that this local reaction system has a completely different structure than that of Rule 54.", "But it is not yet clear what exactly this difference means." ], [ "The Ether", "Now we will use the reaction system we found in the previous chapter to describe ether formation under Rule 54.", "As a first step towards this goal we must find a description of the ether in terms of situations and reactions.", "We will take a more general point of view and explain how periodic structures in a cellular automaton are described in the formalism of Flexible Time.", "First we introduce some concepts for simple periodic structures that occur in one-dimensional cellular automata.", "We will define them in general; then we introduce terms for the special forms they have under Rule 54.", "At the end of the chapter we will use them to describe why the ether under Rule 54 occurs spontaneously and why it is stable." ], [ "Triangles", "As yest we know only the generating reactions of a local reaction system.", "All of them involve only a small number of cellular events.", "In order to understand the large-scale behaviour of a cellular automaton we need then to find reactions that involve a larger number of events.", "And in order to find general laws for the behaviour of the cellular automaton that can be expressed with a theorem or a formula, we need to find sets of situations that all behave in a similar way." ], [ "Families of Situations", "This section is about the simplest form of such a general law, namely about the evolution of repeated patterns in the initial configuration of a cellular automaton.", "In case of Rule 54 we are interested in sequences of cells in the same state together with a boundary of a single cell in the opposite state, such as 011110 and 1000001.", "We can then decompose the initial configurations into such sequences, which overlap at their boundaries.", "A cell sequence $\\dots 100011110 \\dots $ can then be decomposed into 10001 and 011110, and when it is part of an interval situation it can be written as the overlapping product $10001 \\mathbin {\\langle 01\\rangle } 011110$ .", "The boundary cells in the opposite state are included here because they make the decomposition uniquely determined, and also because the reactions that originate from them lead also to situations that are useful in the model for ether formation at the end of this chapter.", "For a different kind of problem another decomposition might be more useful.", "In the following definition this kind of decomposition is formalised and generalised, as much as we can do without making it difficult to handle.", "We especially drop the requirement that the repeated pattern must always be an interval: this would be unimportant for most calculations with situations.", "Definition 8.1 (Family of Situations) Let $R$ be a reaction system and $a$ , $x$ , $b \\in \\mathop {\\mathrm {dom}}R$ be situations.", "Then the set $\\lbrace \\, a x^k b \\colon k \\ge 0 \\,\\rbrace $ is a family of situations in $R$ .", "The representation (REF ) is not the only one for a family of situations.", "For all constants $n$ , $k_0 \\in \\mathbb {N}_0$ we have the equivalences $\\lbrace \\, a x^k b \\colon k \\ge k_0 \\,\\rbrace &= \\lbrace \\, (a x^{k_0}) x^k b \\colon k \\ge 0 \\,\\rbrace , \\\\\\lbrace \\, a x^{n k + k_0} b \\colon k \\ge 0 \\,\\rbrace &= \\lbrace \\, (a x^{k_0}) (x^n)^k b \\colon k \\ge 0 \\,\\rbrace ,$ therefore the left sides of these equations are also valid representations of situation families.", "Now we make use of the first of these equations and introduce names for the two families of initial intervals described above: $F_0 = \\lbrace \\, 10^k1 \\colon k \\ge 1 \\,\\rbrace \\qquad \\text{and}\\qquad F_1 = \\lbrace \\, 01^k0 \\colon k \\ge 1 \\,\\rbrace \\,.$" ], [ "Layers", "Our goal was to find general laws for the evolution of situation families like $F_0$ and $F_1$ .", "We will now find the reactions that describe the evolution of the elements of such a situation family over a single time step.", "Most of these reactions, namely those that start from an element of a situation family that is larger than a certain minimal size, have a similar form.", "This general form for a set of reactions is given in the next definition.", "The elements of these layer reactions will be used as building blocks for other reactions that extend over several time steps, therefore their name.", "Figure: A layer reaction.Definition 8.2 (Layer Reactions) Let $R$ be a reaction system and let $A_0 = \\lbrace \\, a_0 x_0^k b_0\\colon k \\ge 0 \\,\\rbrace $ and $A_1 = \\lbrace \\, a_1 x_1^k b_1 \\colon k \\ge 0 \\,\\rbrace $ be two families of situations in $\\mathop {\\mathrm {dom}}R$ .", "A set of reactions (Figure REF ) $\\lbrace \\, a_0 x_0^k b_0 \\rightarrow _R y_+ a_1 x_1^k b_1 y_-\\colon k \\ge 0 \\,\\rbrace ,$ with $y_+$ , $y_- \\in \\mathop {\\mathrm {dom}}R$ is then a family of layer reactions from $A_0$ to $A_1$ .", "In order to make the definition not unnecessarily specific, the requirement that the situation families $A_0$ and $A_1$ consist of intervals is not part of it.", "In the case that $A_0$ and $A_1$ are actually families of intervals, we have usually $y_+ \\in \\mathop {\\mathrm {dom}}R_+$ and $y_- \\in \\mathop {\\mathrm {dom}}R_-$ .", "Then $y_+$ represents a step into the future and $y_-$ the corresponding movement back to the past.", "The following lemmas can be used to construct a family of layer reactions from simpler reactions.", "We will use them to find layer reactions for the interval families $F_0$ and $F_1$ .", "Lemma 8.3 (Repeatable Reactions) Let $R$ be a reaction system.", "Then, $\\text{if}\\qquad a x &\\rightarrow _R y a,\\qquad \\text{then}\\qquad a x^k \\rightarrow _R y^k a&&\\text{for all $k \\ge 0$,} \\\\\\text{if}\\qquad x a &\\rightarrow _R a y,\\qquad \\text{then}\\qquad x^k a \\rightarrow _R a y^k&&\\text{for all $k \\ge 0$.", "}$ The proof is by induction.", "The first equation is trivially true for $k = 0$ .", "Now assume that $k \\ge 0$ and $a x^k \\rightarrow _R x^k a$ .", "Then there is a chain of reactions $a x^{k+1} = a x^k x \\rightarrow _R y^k a x\\rightarrow _R y^k y a = y^{k+1} a$ .", "Therefore we have by induction $a x^k\\rightarrow y^k a$ for every $k \\ge 0$ .", "The second equation is proved in the same way.", "Lemma 8.4 (Generators of Layer Reactions) Let $R$ be a reaction system which contains the reactions $a \\rightarrow _R a^{\\prime } c, \\qquad c x \\rightarrow _R y c \\qquad \\text{and}\\qquad c b \\rightarrow _R b^{\\prime }\\,.$ Then $R$ has the family of layer reactions $\\lbrace \\, a x^k b \\rightarrow _R a^{\\prime }y^k b^{\\prime } \\colon k \\ge 0 \\,\\rbrace $ .", "By Lemma REF , applied to $c x \\rightarrow _R y c$ , we have for every $k \\ge 0$ a reaction $c x^k \\rightarrow _R y^k c$ .", "We have then the chain of reactions $a x^k b \\rightarrow _R a^{\\prime } c x^k b \\rightarrow _R a^{\\prime } y^kc b \\rightarrow _R a^{\\prime } y^k b^{\\prime }$ , which proves the lemma.", "The layer reactions for the interval families $F_0$ and $F_1$ are then $L_0 &= \\lbrace \\, 1 0^{k+2}1 &&\\rightarrow _\\Phi 10 \\oplus 1 0^k 1 \\ominus 01&&\\colon k\\ge 0 \\,\\rbrace , \\\\L_1 &= \\lbrace \\, 01^k0 &&\\rightarrow _\\Phi 01 \\oplus _2 1 0^k 1 \\ominus _2 10&&\\colon k\\ge 2 \\,\\rbrace \\,.$ In the terminology of Definition REF , the set $L_0$ is a family of layer reactions from $F_0$ to $F_0 \\cup \\lbrace 11 \\rbrace $ , and $L_1$ is a family of layer reactions from $F_1$ to $F_0$ .", "The set $F_0 \\cup \\lbrace 11 \\rbrace $ is indeed a situation family: It is equal to $\\lbrace \\, 1 0^k 1\\colon k \\ge 0 \\,\\rbrace $ .", "For the proof of the first formula we use the reactions $1 0^2 \\rightarrow _\\Phi 10 \\oplus 1 (\\ominus 00), \\quad (\\ominus 00) 0 \\rightarrow _\\Phi 0 (\\ominus 00), \\quad (\\ominus 00) 1 \\rightarrow _\\Phi 1 \\ominus 01.$ The common part $c = \\ominus 00$ of these reaction is put in parentheses for better legibility.", "Then we can see that $a = 1 0^2$ , $a^{\\prime } = 10 \\oplus 1$ , $x = y = 0$ , $b = 1$ and $b^{\\prime } = 1 \\ominus 01$ in the terminology of Lemma REF .", "Therefore there is a family of layer reactions $\\lbrace \\, 1 0^2 0^k 1 \\rightarrow _\\Phi 10 \\oplus 10^k 1 \\ominus 01 \\colon k \\ge 0 \\,\\rbrace $ under Rule 54, the same as in (REF ).", "The second formula is derived from the reactions $01^2 \\rightarrow _\\Phi 01 \\oplus _2 1 (00 \\ominus _2 11), \\quad (00 \\ominus _2 11) 1 \\rightarrow _\\Phi 0 (00 \\ominus _2 11), \\\\(00 \\ominus _2 11) 0 \\rightarrow _\\Phi 0^2 1 \\ominus _2 10.$ The common part of these reactions is once again put into parentheses.", "We can see from these reactions that $a = 01^2$ , $a^{\\prime } = 01 \\oplus _2 1$ , $x = 1$ , $y = b = 0$ and $b^{\\prime } = 0^2 1 \\ominus _2 10$ in the terminology of Lemma REF .", "Therefore there is a family of reactions $\\lbrace \\, 01^2 1^k 0 \\rightarrow _\\Phi 01 \\oplus _2 1 0^k 0^2 1 \\ominus _210 \\colon k \\ge 0 \\,\\rbrace $ under Rule 54, the same as in ()." ], [ "Triangles", "As a final step in this subproject of finding general laws for the evolution of finitely often repeated patterns in the initial configuration, we now trace their evolution over as many time steps as possible.", "This is done with triangle reactions.", "An example for a triangle reaction under Rule 54 is shown in Figure REF .", "It is a copy of Figure REF , but now it shows the reaction in the new notation, as $1 0^{13} 1\\rightarrow _\\Phi (10\\oplus )^7 1 (\\ominus 01)^7$ .", "The evolution diagram at the left side shows the closure of the process for initial situation $1 0^{13}1$ .", "The two diagrams at the right side show the processes of the input and the result situation of the triangle reaction: they are the edges of the triangle process at the right.", "If we view the triangle as a temporal process, then its input situation shows two “particles”, the interval situations 10 and 01, that are positioned at the boundaries of a sequence of 11 cells in state 0; the result situation shows how the particles move towards each other until they collide.", "This similarity to the collision of macroscopic particles makes triangle reactions a promising tool for understanding the behaviour of cellular automata.In the typical behaviour of Rule 54, the structures 10 and 01 are very short-lived [4].", "For it and other cellular automata, the word “particle” means therefore generally larger, longer-lived structures.", "Nevertheless, the interpretation given here looks like a useful generalisation.", "Figure: Triangle process and triangle reaction under Rule 54.The following definition of triangle reactions harmonises with the definition of layer reactions.", "As before, the definition ignores the temporal aspect of the triangle reactions: In the cases that interest us here most, the situations $a$ , $x$ and $b$ are intervals, while $y_+$ is an element of $\\mathop {\\mathrm {dom}}R_+$ and $y_-$ of $\\mathop {\\mathrm {dom}}R_-$ .", "Definition 8.5 (Triangle Reactions) Let $R$ be a reaction system.", "Let $A = \\lbrace \\, a x^k b \\colon k \\ge 0 \\,\\rbrace \\subseteq \\mathop {\\mathrm {dom}}R$ be a family of situations.", "A family of triangle reactions for $A$ is a set of reactions (Figure REF ), $\\lbrace \\, a x^k b \\rightarrow _R y_+^k c y_-^k \\colon k \\ge 0 \\,\\rbrace \\,.$ Figure: Generic form of a triangle reaction.As before with layers, there are many equivalent forms to represent triangle reactions.", "We are especially interested in the apparently more general form $\\lbrace \\, a x^{kn + i} b \\rightarrow _Ry_+^{k+k_0} c y_-^{k+k_0} \\colon k \\ge 0 \\,\\rbrace $ with $n$ , $i$ and $k_0 \\in \\mathbb {N}_0$ .", "This set of reactions can be brought into the same form as (REF ) by writing it as $\\lbrace \\, (a x^{i}) (x^n)^k b \\rightarrow _R y_+^k \\,(y_+^{k_0} c y_-^{k_0})\\,y_-^k \\colon k \\ge 0 \\,\\rbrace $ and so be shown to be equivalent to it.", "We can then bring the reaction of Figure REF into the form (REF ) by writing it as $1 0^{2 \\times 6+ 1} 1 \\rightarrow _\\Phi (10 \\oplus )^{6+1} 1 (\\ominus 01)^{6+1}$ .", "Triangle reactions can be derived from layer reactions.", "If $A = \\lbrace \\, ax^k b \\colon k \\ge 1 \\,\\rbrace $ is a family of situations and $L$ is a family of layer reactions from $A$ to $A \\cup \\lbrace ab \\rbrace $ , then there is a family of triangle reactions for $A$ .", "The following lemma shows why.", "(The requirement in the lemma that there is a reaction $a b \\rightarrow _Rc$ is no restriction, since one can always set $c = a b$ .)", "Lemma 8.6 (Generators of Triangle Reactions) Let $R$ be a reaction system.", "If there is a reaction $a b \\rightarrow _Rc$ and a family of layer reactions $L = \\lbrace \\, a x^{k+1} b \\rightarrow _R y_+ a x^k b y_- \\colon k \\ge 0 \\,\\rbrace ,$ then there exists in $R$ a family of triangle reactions $\\lbrace \\, a x^k b \\rightarrow _R y_+^k c y_-^k \\colon k \\ge 0 \\,\\rbrace \\,.$ Let a $k \\ge 0$ be given.", "We can then apply for $k$ times one of the reactions of $L$ , first to the situation $a x^k b$ , and then always to the result of the previous reaction.", "Then we have found a reaction $a x^k b \\rightarrow _R y_+^k a b y_-^k$ .", "Next we apply the reaction $a b \\rightarrow _Rc$ to the result of this reaction and get $ax^k b \\rightarrow _R y_+^k c y_-^k$ .", "An interesting phenomenon occurs when the layer reaction consumes more that one copy of the repeated pattern $x$ .", "Then the one family of layer reactions splits into several sets of triangle reactions.", "This occurs especially in the family of layer reactions $L_0$ for Rule 54.", "Lemma 8.7 (Multiple Triangle Reactions) Let $R$ be a reaction system that contains a family of layer reactions, $L = \\lbrace \\, a x^{k+n} b \\rightarrow _R y_+ a x^k b y_- \\colon k \\ge 0 \\,\\rbrace ,$ and the reactions $a x^{i} b \\rightarrow _R c_i$ for $i = 0$ , ..., $n -1$ .", "Then $R$ contains for every value of $i$ a family of triangle reactions, $\\lbrace \\, a x^{kn + i} b \\rightarrow _R y_+^k c_i y_-^k \\colon k \\ge 0 \\,\\rbrace \\,.$ We will write the terms $x^k$ and $x^{k+n}$ in differing ways, depending on the value of $k$ .", "If there is a $j \\in \\mathbb {N}_0$ such that $k = jn + i$ and $i < n$ , then $x^k = (x^n)^j$ and $x^{k+n} = x^i(x^n)^j$ .", "Therefore $L$ is the disjoint union of $n$ families of layer reactions, $L_i = \\lbrace \\, (a x^i) (x^n)^j b \\rightarrow _R y_+ a (x^n)^j b y_-\\colon j \\ge 0 \\,\\rbrace \\qquad \\text{for $i = 0$, \\dots , $n - 1$}.$ Then we can apply the previous lemma to each of the reactions families $L_i$ .", "We get for every value of $i$ a family of triangle reactions, $\\lbrace \\, (a x^i) (x^n)^k b \\rightarrow _R y_+^k c_i y_-^k \\colon j\\ge 0 \\,\\rbrace $ , the same as in (REF ).", "Now we will derive triangles and the triangle reaction for Rule 54.", "Among the two families of layer reactions only $L_0$ fulfils the requirement of Lemma REF .", "So we set $L_0 = \\lbrace \\, 1 0^{k+2}1 \\rightarrow _\\Phi 10 \\oplus 1 0^k 1 \\ominus 01 \\colon k\\ge 0 \\,\\rbrace $ for $L$ and choose as the finishing reactions $a x^{i} b \\rightarrow _Rc_i$ the two reactions $11 \\rightarrow _\\Phi 11$ and $101 \\rightarrow _\\Phi 10 \\oplus 1 \\ominus 01$ .", "Then we can apply the lemma and get the following families of triangle reactions, $T_{00} &= \\lbrace \\, 1 0^{2k} 1 && \\rightarrow _\\Phi (10 \\oplus )^k 11 (\\ominus 01)^k&&\\colon k \\ge 0 \\,\\rbrace , \\\\T_{01} &= \\lbrace \\, 1 0^{2k + 1} 1 && \\rightarrow _\\Phi (10 \\oplus )^k (10 \\oplus 1 \\ominus 01) (\\ominus 01)^k &&\\colon k \\ge 0 \\,\\rbrace \\,.$ We can view these two families of reactions as an improved version of the layer reactions $L_0$ , and as their replacement.", "For $L_1$ no such replacement has been found, therefore the general laws for Rule 54 that we have found with the methods of this section are $T_{00}$ , $T_{01}$ and $L_1$ .", "They are listed in Table REF .", "Table: Large-scale reactions under Rule 54" ], [ "What Is the Ether?", "Now we can try to understand ether formation under Rule 54.", "For this we need to express explicitly what the ether is.", "We will do this first in terms of configurations and then in terms of situations and reactions.", "In Figure REF at the beginning we have seen an example of ether formation.", "In it we can now recognise how, when starting from a random initial configuration, large regions that consist of a regular pattern begin to form.", "We see that this pattern consists at alternating times of a configuration in which blocks of three cells in state 0 are separated by one cell in state 1, and of a configuration in which three cells in state 1 are separated by one cell in state 0.", "After two time steps the same patterns arise again, but but shifted to the side by a distance of two cells, such that a true repetition only occurs after 4 time steps.", "We will speak of the two configurations as the phases of the ether, in analogy to the usage of MartÃnez et al.", "[40] for Rule 110.", "We also see that there are several such ether regions in Figure REF .", "They are separated by larger structures between them, or sometimes just by phase differences.", "This is a typical phenomenon in ether formation: when starting from a random configuration one almost never gets a “pure”, or empty ether.", "Instead one gets large regions with a regular pattern that are separated by disturbances.", "Over time the ether regions coalesce and the distance between the disturbances increases.", "This has already been noted by Boccara, Nasser and Roger [4].", "We will therefore first describe an empty ether and then turn to the more realistic case of an ether with disturbances." ], [ "The Empty Ether", "A configuration of empty ether then consists of an infinite repetition of one of the patterns $0^3 1$ and $1^3 0$ .", "In contrast to configurations, situations are always finite.", "We could now represent a finite part of such an ether configuration by a situation of the form $(0^3 1)^k$ or $(1^3 0)^k$ , depending on the phase.", "We choose however the family of situations $\\lbrace \\, 1 (0^3 1)^k\\colon k \\ge 0 \\,\\rbrace $ as our standard representative of an ether configuration.", "The reason for the choice of this phase and for adding a 1 at the left is that then the situation $1 0^3 1$ is an element of the family and that this situation is the input of one of the triangle reactions of the family $T_{01}$ , namely $1 0^3 1 \\rightarrow _\\Phi (10 \\oplus )^2 1 (\\ominus 01)^2$ .", "In order to get a notation in which no overlapping situations occur we will write this reaction in a slightly different form,The right site of the reaction will be later decomposed into the situations 1, $(0 \\oplus 1)^2$ and $(\\ominus 01)^2$ .", "These situations do not overlap, even if two of their processes, namely $\\mathrm {pr}_{1}((0 \\oplus 1)^2)$ and $\\mathrm {pr}_{1 (0 \\oplus 1)^2}((\\ominus 01)^2)$ , do overlap.", "$1 0^3 1 \\rightarrow _\\Phi 1 (0 \\oplus 1)^2 (\\ominus 01)^2\\,.$ The next step to construct a description of the empty ether is then to find reactions similar to (REF ) for all intervals of the form $1 (0^3 1)^k$ .", "To do this we need the following lemma, which provides a kind of converse to the reaction (REF ).", "Lemma 8.8 (Converse Triangle Reaction) $1 (\\ominus 01)^2 (0 \\oplus 1)^2 \\rightarrow _\\Phi 1 0^3 1\\,.$ First we show that $1 \\ominus 010 \\oplus 1 \\rightarrow _\\Phi 1^3$ .", "This is true because $\\underline{1 \\ominus 010} \\oplus 1\\rightarrow _\\Phi 11 \\underline{1 \\ominus _2 10 \\oplus 1}\\rightarrow _\\Phi 111;$ the terms that change in the next reaction step are underlined.", "We have then also shown that $1 (\\ominus 01)^2 (0 \\oplus 1)^2 \\rightarrow _\\Phi 1 \\ominus 0 1^3 1 \\oplus 1$ and only must verify that the reaction product reacts to $1 0^3 1$ : $\\underline{1 \\ominus 01} 1 \\underline{10 \\oplus 1}&\\rightarrow _\\Phi 1 \\underline{00 \\ominus _2 111} \\oplus _2 001 \\\\&\\rightarrow _\\Phi 10 \\underline{00 \\ominus _2 11 \\oplus _2 00} 1\\rightarrow _\\Phi 10001\\,.$ This proves the lemma.", "We now combine (REF ) and (REF ) to a reaction in which the interval $10^3 1$ does no longer occur, $1 (\\ominus 01)^2 (0 \\oplus 1)^2\\rightarrow _\\Phi 1 (0 \\oplus 1)^2 (\\ominus 01)^2\\,.$ The reaction (REF ) and this reaction are the building blocks for the larger reactions that originate from the $1(0^3 1)^k$ terms.", "In order to express them better we introduce the abbreviations $\\varepsilon _- = \\ominus 01\\qquad \\text{and}\\qquad \\varepsilon _+ = 0 \\oplus 1\\,.$ Note that these terms are not achronal.", "In the formalism they can therefore appear only as factors of situations in $\\mathop {\\mathrm {dom}}\\Phi $ , not as terms in their own right.", "With the abbreviations of (REF ) we will now summarise the reactions that characterise the empty ether.", "Definition 8.9 (Basic Ether Reactions) The basic ether reactions for Rule 54 are $1 0^3 1 \\rightarrow _\\Phi 1 \\varepsilon _+^2 \\varepsilon _-^2\\qquad \\text{and}\\qquad 1 \\varepsilon _+^2 \\varepsilon _-^2 \\rightarrow _\\Phi 1\\varepsilon _+^2 \\varepsilon _-^2\\,.$ The basic ether reactions are shown in Figure REF .", "The cells of the ether situations are shown in black and white, and the background in grey tones is a finite part of the empty ether.", "Figure: Basic ether reactions on a background of empty ether.From these two reactions we will now derive two sets of reactions that involve larger parts of the empty ether.", "The set one describes the evolution of larger segments of the initial configuration, and the second one is a description of the evolution of a part of the empty ether.", "Theorem 8.10 (Ether Reactions) $1 (0^3 1)^k & \\rightarrow _\\Phi 1 \\varepsilon _+^{2k} \\varepsilon _-^{2k}&&\\text{for $k \\ge 0$,}\\\\1 \\varepsilon _-^{2k} \\varepsilon _+^{2\\ell }& \\rightarrow _\\Phi 1 \\varepsilon _+^{2\\ell } \\varepsilon _-^{2k}&&\\text{for $k$, $\\ell \\ge 0$.", "}$ We need for this proof a reaction that is derived from the reaction at the right side of (REF ) with the help of Lemma REF .", "In order to apply this lemma to the reaction $1 \\varepsilon _-^2\\varepsilon _+^2 \\rightarrow _\\Phi 1 \\varepsilon _+^2 \\varepsilon _-^2$ we need to write it in the form $x a \\rightarrow _\\Phi a y$ .", "We do this by resolving the $\\varepsilon _-$ terms at the left side of the reaction.", "Then we get $1\\ominus 01 \\ominus 0 (1 \\varepsilon _+^2) \\rightarrow _\\Phi (1 \\varepsilon _+^2)\\varepsilon _-^2$ , with the repeated factor $a$ put in parentheses to emphasise it.", "Now we can apply () and get a reaction $(1 \\ominus 01 \\ominus 0)^k 1 \\varepsilon _+^2 \\rightarrow _\\Phi 1\\varepsilon _+^2 (\\varepsilon _-^2)^k$ for every $k \\ge 0$ .", "Then we rewrite the left side of this reaction with $\\varepsilon _-$ terms.", "The result is $1 \\varepsilon _-^{2k} \\varepsilon _+^2&\\rightarrow _\\Phi 1 \\varepsilon _+^2 \\varepsilon _-^{2k}&&\\text{for all $k \\ge 0$.", "}$ We use it to prove the reactions in (REF ) by induction over $k$ .", "For both families of reactions the case $k = 0$ is trivially true, and for $k \\ge 1$ we get $1 (0^3 1)^k&\\rightarrow _\\Phi 1 (0^3 1)^{k-1} \\varepsilon _+^2 \\varepsilon _-^2 \\\\&\\rightarrow _\\Phi 1 \\varepsilon _+^{2(k-1)} \\varepsilon _-^{2(k-1)}\\varepsilon _+^2 \\varepsilon _-^2&&\\text{by induction,} \\\\&\\rightarrow _\\Phi 1 \\varepsilon _+^{2(k-1)} \\varepsilon _+^2\\varepsilon _-^{2(k-1)} \\varepsilon _-^2&&\\text{with~(\\ref {eq:ether-immediate})} \\\\&= \\varepsilon _+^{2k} \\varepsilon _-^{2k}, \\\\1\\varepsilon _-^{2k} \\varepsilon _+^{2\\ell }&\\rightarrow _\\Phi 1\\varepsilon _-^2 \\varepsilon _+^{2\\ell } \\varepsilon _-^{2(k-1)}&&\\text{by induction,} \\\\&\\rightarrow _\\Phi 1\\varepsilon _+^{2\\ell } \\varepsilon _-^2 \\varepsilon _-^{2k-1}&&\\text{with~(\\ref {eq:ether-immediate}).}", "\\\\&= 1\\varepsilon _+^{2\\ell } \\varepsilon _-^{2k}\\,.$ This then proves the theorem.", "The reactions of Theorem REF now serve as an inspiration for the way in which one can express the disturbed ether that arises from a random initial configuration in terms of situations and reactions." ], [ "Random Initial Situations", "We need a method to express with the help of situations the behaviour of random initial configurations.", "First we define what we mean by a random initial configuration.", "Here we assume that the states of a configuration $c\\in \\Sigma ^\\mathbb {Z}$ are chosen at random in such a way that the state of every cell is equal to 1 with probability $p_1$ and that the states of all cells are independent of each other.", "We will always exclude the trivial cases $p_1 = 0$ and $p_1 = 1$ .", "In the language of probability theory [22] we have then performed a random experiment.", "The random choice of a configuration is represented by a random variable $C$ with values in $\\Sigma ^\\mathbb {Z}$ .", "As it is usual in probability theory, this and other random variables are written in capital letters.", "The configuration $c$ mentioned above is then a possible outcome of the experiment.", "The probabilities that define the random experiment will however refer to whole sets of outcomes.", "These sets are called in probability theory “events”, but here they will be called stochastic events, to avoid confusion with the already existing use of the word “event” in the context of cellular processes.", "First we must now specify the probabilities for the outcomes of $C$ and make precise the informal definition given above.", "We can express the random choice of $C$ by saying that for all $x \\in \\mathbb {Z}$ , $P(C(x) = 1) = p_1,$ In a more formal way of speaking, this equation assigns the probability $p_1$ to the set of outcomes, or stochastic event, $\\lbrace \\, c\\in \\Sigma ^\\mathbb {Z}\\colon c(x) = 1 \\,\\rbrace $ .", "Other expressions with random variables are understood in a similar way.", "From (REF ) we can derive the probailities for other stochastic events.", "These are the events that belong to the $\\sigma $ -field [22] that is generated by the sets $\\lbrace \\, c \\in \\Sigma ^\\mathbb {Z}\\colon c(x) = 1 \\,\\rbrace $ .", "We will however not use this $\\sigma $ -field explicitly and work with probabilities in a less formal way.", "For the work with situations we can only use finite pieces of the initial configuration $C$ .", "So we define now the sequence $(U_n)_{n\\ge 0}$ of random intervals.", "Each $U_n$ is a random variable with values in $\\Sigma ^n$ and defined by the relation $U_n(c) = C(-n + 1) \\dots C(1) C(0)\\,.$ With this definition we have a growing sequence of random cellular processes, $\\dots \\supset \\mathrm {pr}([-n] U_n) \\supset \\dots \\supset \\mathrm {pr}([-1] U_{1}) \\supset \\mathrm {pr}([0]U_0) = \\emptyset ,$ that ultimately incorporate all cells of $\\Sigma ^Z$ with indices $\\le 0$ .", "For each $n$ , the closure of $\\mathrm {pr}([-n]U_n)$ then represents a view into the evolution of an element of $C$ .", "The larger $n$ becomes, the larger the window on $C$ becomes.", "All these windows contain only events with non-positive space coordinates, but this does not matter, since the probability distribution for $C$ is invariant under left-to-right shifts.", "Later we will need to do induction proofs over the length of the intervals $U_n$ .", "In them we need to express the sequence $(U_n)_{n\\ge 0}$ in a recursive way.", "To do this we now imagine the construction of the $U_n$ as a stochastic process in which the random interval $U_{n-1}$ is extended by an event that is in state 1 with probability $p_1$ and in state 0 with probability $1 - p_1$ .", "We can then express the values of the $P(U_n)$ in the language of conditional probability with the equations $P(U_n = \\sigma u \\mid U_{n-1} = u) ={\\left\\lbrace \\begin{array}{ll}p_1 & \\text{if $\\sigma = 1$,} \\\\(1 - p_1) & \\text{if $\\sigma = 0$,}\\end{array}\\right.", "}$ valid for all $n \\ge 1$ and $u \\in \\Sigma ^n$ .", "The starting point of this recursion is the trivial case of $n = 0$ , with $P(U_0 = [0]) =1$ ." ], [ "Ether Fragments", "We also need a way to express the fact that the closure of $\\mathrm {pr}(U_n)$ contains fragments of the empty ether.", "To do this we will first consider reactions of the form $U_n \\rightarrow a_+ a_-$ with $a_+ \\in \\mathop {\\mathrm {dom}}\\Phi _+$ and $a_- \\in \\mathop {\\mathrm {dom}}\\Phi _-$ .", "They are a generalisation of the triangle reactions (REF ) for the empty ether.", "Since however $U_n$ has in general not the form $1 (0^31)^k$ , the situation $a_+$ will in general not consist entirely of $\\varepsilon _+$ terms and $a_-$ not only of $\\varepsilon _-$ terms.", "Our goal in this chapter then is to prove that for large $n$ , the closure of $\\mathrm {pr}(U_n)$ nevertheless contains pieces of the ether.", "We will therefore look for reactions of the form (REF ) in which $a_+$ contains factors of the form $1 \\varepsilon _+^2$ .", "The exponent of 2 in $\\varepsilon _+^2$ occurs because all the ether reactions in (REF ) involve only even numbers as exponents for $\\varepsilon _+$ and $\\varepsilon _-$ .", "Figure: Location of the ether fragmentsof () in the situation a + a - a_+ a_-.We will say that an ether fragment occurs at time $t$ if $a_+ = a^{\\prime }_+ 1 \\varepsilon _+^2 a^{\\prime \\prime }_+\\qquad \\text{and}\\qquad \\delta (a^{\\prime }_+)_T = t\\,.$ The situation $a_+ a_-$ serves here as a probe into the closure of $\\mathrm {pr}(U_n)$ .", "We can then use it as a means to express ether formation in such a way that it can be proved with the help of situations and reactions.", "We could also have looked for reactions in which $a_-$ contains factors of the form $\\varepsilon _+^2 1$ .", "The results would be equivalent because Rule 54 is symmetric under exchange of left and right.", "The advantage of the definition we use here is that with it we can express more easily the time at which an ether fragment occurs." ], [ "Reactions and Probability", "Now we will consider arbitrary reactions of the form $U_n \\rightarrow _\\Phi a$ for a given situation $a$ .", "Whether such a reaction is possible depends on the value of $U_n$ .", "We will write the probability of such a reaction as $P(U_n \\rightarrow _\\Phi a)$ ; it is the probability that $U_n$ belongs to the set of situations $u\\in \\Sigma ^n$ for which there is a reaction with result $a$ .", "So we can write $P(U_n \\rightarrow _\\Phi a) =P(\\exists u \\in \\Sigma ^n \\colon u \\rightarrow _\\Phi a)\\,.$ A generalisation of this is the case where a random interval may react to a whole set of possible outcomes.", "When $\\mathcal {A} \\subseteq \\mathcal {S}$ is a set of situations, we will write $P(U_n \\rightarrow _\\Phi \\mathcal {A}) =P(\\exists u \\in \\Sigma ^n, a \\in \\mathcal {A} \\colon u \\rightarrow _\\Phi a)\\,.$ This definition counts the probability that the random situation $U_n$ can react to at least one element of $\\mathcal {A}$ .", "So it is meaningful even if there is more than one possible reaction result for a certain value of $u$ .", "We have therefore always $P(U_n \\rightarrow _\\Phi \\mathcal {S}) = 1$ ." ], [ "Notation for Fragments", "The definitions of the last paragraphs are aimed at a special kind of situations that are needed in the following proof.", "We need to express that a certain situation $f$ , the “fragment”, occurs at one or more specified time steps.", "The set of situations that have in common a certain fragment $f \\in \\mathcal {S}$ , beginning at one of the time steps $t_1$ , ..., $t_n$ , is then written $\\mathcal {F}(f, t_1, \\dots , t_n) =\\lbrace \\, a_+ f a^{\\prime } \\colon a_+ \\in \\mathop {\\mathrm {dom}}\\Phi _+, a^{\\prime } \\in \\mathop {\\mathrm {dom}}\\Phi , \\\\\\delta (a_+)_T \\in \\lbrace t_1, \\dots t_n \\rbrace \\,\\rbrace \\,.$ Among these sets of situations, the most important one is that in which $f$ is an ether fragment; it is called $\\mathcal {E}(t_1, \\dots , t_n) =\\mathcal {F}(1 \\varepsilon _+^2, t_1, \\dots , t_n)\\,.$" ], [ "The Ether Is Inevitable", "The following theorem is a way to express the necessity of ether formation in a weak form.", "It shows that for a random initial configuration, ether fragments can be found arbitrarily far in the future.", "Ether fragments never vanish totally, and we can view that as a sign (but not a proof) that the ether, too, persists.", "Theorem 8.11 (Ether Formation) Let $\\varepsilon > 0$ be a probability.", "The random intervals $(U_k)_{k\\ge 0}$ are defined as in (REF ).", "Then for every $t \\in \\mathbb {N}$ there is an $n \\in \\mathbb {N}$ such that $P(U_n \\rightarrow _\\Phi \\mathcal {E}(t, t + 1, t + 2)) \\ge 1 - \\varepsilon \\,.$ Since $U_n$ is part of the random initial configuration $C$ , the theorem shows that every given sequence of three time steps contains with probability 1 the starting time of an ether fragment.", "Moreover, $C$ consists of infinitely many intervals of length $n$ and these intervals are stochastically independent of each other.", "Therefore the theorem also shows that during every sequence of three steps time, infinitely many ether fragments begin.", "The theorem will now be proved with help of several lemmas.", "Among them, Lemma REF shows how ether fragments arise from the initial configuration, and Lemma REF shows how they propagate to later times.", "Lemma 8.12 (Formation of Ether Fragments) Let $(U_k)_{k \\ge 0}$ be the sequence of random intervals from (REF ).", "Let $n \\in \\mathbb {N}_0$ and let $u \\in \\Sigma ^n$ be an interval.", "Then for every $\\varepsilon > 0$ there is an integer $m > n$ such that $P(U_m \\rightarrow _\\Phi \\mathcal {E}(0) \\mid U_n = u)\\ge 1 - \\varepsilon \\,.$ When extending a random interval $U_k$ by 5 events to $U_{k+5}$ , the probability that $U_{k+5}$ begins with $1 0^3 1$ is $P(U_{k+5} = 1 0^3 1 U_k) = p_1^2 (1 - p_1)^3,$ independent of the value of $U_k$ .", "We will call this probability $p_{10^31}$ .", "Then $p_{10^31} > 0$ because we have required earlier that $p_1 \\notin \\lbrace 0, 1\\rbrace $ .", "Let now $k_0 \\ge 0$ be an integer.", "Then the probability that one of the random intervals $U_{n + 5k}$ with $k \\le k_0$ begins with $10^3 1$ is $(1 - p_{10^31})^{k_0}$ .", "These probabilities are independent of each other because each of them only depends on the states of the cells at the locations $-(n + 5k) + 1$ , ..., $-(n +5k) + 5$ , and those intervals do not overlap.Note that the intervals $U_k$ grow to the left, as described in (REF ).", "Since $1 - p_{10^31} < 1$ , we can find a $k_0$ such that $(1 -p_{10^31})^{k_0} < \\varepsilon $ .", "Now we can set $n_0 = n + 5 k_0$ .", "Then with a probability greater than $1 - \\varepsilon $ there is a $m \\le n_0$ such that $U_m$ begins with $1 0^3 1$ .", "With such an $m$ we also have a reaction $U_m \\rightarrow _\\Phi 1 \\varepsilon _+^2 \\varepsilon _-^2 U_{m-5}$ .", "So we see that as $n$ approaches infinity, there is always an ether fragment $1 \\varepsilon _+^2$ at time 0 that is generated by $U_n$ .", "Next we will prove that these fragments are almost always copied by other reactions to space-time locations at later times.", "For this proof we will need reactions that create a situation with an interval of a given minimal length and at a specified time as a factor.", "Furthermore, initial and final situation of the reaction must be positive slopes.", "This means that the required reaction must be a part of the positive slope reaction system $\\Phi _+$ .", "The following lemma then tells us how we can find reactions that create such an interval from a given situation $a_+ \\in \\mathop {\\mathrm {dom}}\\Phi _+$ .", "When the conditions of the lemma are met, we will say that every $u$ enforces $v$ .", "Lemma 8.13 (Enforcement of Intervals) Let $a_+ \\in \\mathop {\\mathrm {dom}}\\Phi _+$ be chosen arbitrarily.", "Then for every $\\ell > 0$ there is a $k > 0$ such that for all $u \\in \\Sigma ^$ with $\\mathopen \|u\\mathclose \| \\ge k$ there is a reaction of the form $u a_+ \\rightarrow _{\\Phi _+} b_+ v$ with $b_+ \\in \\mathop {\\mathrm {dom}}\\Phi _+$ and $v \\in \\Sigma ^\\ell $ .", "We first consider the case where $\\ell = 1$ and $\\delta (a_+)_T = 1$ .", "Because $a_+$ is achronal we must then have a decomposition $a_+ =u_1 s_+ u_2$ , with $u_1$ , $u_2 \\in \\Sigma ^$ and $s_+$ an element of the set of positive generating slopes for Rule 54.", "They are found in Table REF , and we will refer to them as the set $S_+ =\\lbrace 00 \\oplus , 01 \\oplus _2 1, 10 \\oplus 1, 11 \\oplus _2 00 \\rbrace \\,.$ The worst possible case for the proof occurs when both $u_1$ and $u_2$ are the empty situation.", "So we may assume without loss of generality that $a_+ \\in S_+$ .", "Now, by checking the generating reactions in Table REF , we see that as long as $s_+ \\ne 01 \\oplus _2 1$ , they always have the form $\\sigma s_+\\rightarrow _{\\Phi _+} s^{\\prime }_+ v_1$ , with $\\sigma \\in \\Sigma $ , $v_1 \\in \\Sigma ^$ and $\\mathopen \|v\\mathclose \| \\ge 1$ .", "The two remaining generating reactions, those with $s_+ = 01 \\oplus _2 1$ , have the form $\\sigma s_+ \\rightarrow _{\\Phi _+} s^{\\prime }_+$ , with $s^{\\prime }_+ \\ne s_+$ .", "Now we consider the chain of two generating reaction that starts from $\\sigma _1 \\sigma _2s_+ $ , with $\\sigma _1$ , $\\sigma _2 \\in \\Sigma $ and $v_1$ , $v_2 \\in \\Sigma ^$ .", "$\\sigma _1 \\sigma _2 s_+\\rightarrow _{\\Phi _+} \\sigma _1 s^{\\prime }_+ v_1\\rightarrow _{\\Phi _+} s^{\\prime \\prime }_+ v_2 v_1\\,.$ We have just seen that if $\\mathopen \|v_1\\mathclose \| = 0$ , then $\\mathopen \|v_2\\mathclose \| = 1$ .", "Therefore $\\mathopen \|v_1 v_2\\mathclose \| \\ge 1$ for all $s_+ \\in S_+$ .", "This means that for $\\delta (a_+)_T = 1$ and $\\ell = 1$ we always can set $k = 2$ .", "By induction we see then that for $\\delta (a_+)_T = 1$ and arbitrary $\\ell $ , a value of $k = 2 \\ell $ is enough.", "When we also drop the condition that $\\delta (a_+)_T = 1$ , a value of $k = 2\\ell \\delta (a_+)_T$ is enough for the proposition of the lemma.", "The following two reactions are not derived in a completely formal mode; they can instead be derived from the illustrations.", "Together they will show how an ether fragment causes another ether fragment to occur at a later time.", "Lemma 8.14 (Destruction of an Ether Fragment) For every $\\sigma \\in \\Sigma $ there is a reaction of the form $\\sigma 1 \\varepsilon _+^2 &\\rightarrow _\\Phi a_+ 1^3 a^{\\prime }_+$ with $a_+$ , $a^{\\prime }_+ \\in \\mathop {\\mathrm {dom}}\\Phi _+$ and $\\delta (a_+)_T \\in \\lbrace 1, 2\\rbrace $ .", "Figure: Destruction of ether fragments.Depending on the value of $\\sigma $ , one of the following two reactions can be applied to the left side of the reaction (REF ), $01 \\varepsilon _+^2 \\rightarrow _\\Phi 01 \\oplus _2 1^3 \\varepsilon _+\\qquad \\text{or}\\qquad 11 \\varepsilon _+^2 \\rightarrow _\\Phi 11 \\oplus _2 00 \\oplus 1^3\\,.$ An example for these reactions can be seen in Figure REF .", "Now we can set either $a_+ =01 \\oplus _2$ and $a^{\\prime }_+ = \\varepsilon _+$ or $a^{\\prime }_+ = 11 \\oplus _2 00\\oplus $ and $a^{\\prime }_+ = [0]$ .", "Lemma 8.15 (Creation of an Ether Fragment) For every $j \\in \\mathbb {N}_0$ there is a reaction of the form $0 1^{2j+3} \\rightarrow _\\Phi a_+ 1 \\varepsilon _+^{j+2} a,$ with $a_+ \\in \\mathop {\\mathrm {dom}}\\Phi _+$ , $a \\in \\mathop {\\mathrm {dom}}\\Phi $ and $\\delta (a_+)_T =1$ .", "Figure: Creation of new ether fragments.We use the following family of reactions, $\\lbrace \\, 0 1^{2j +3} \\rightarrow _\\Phi 01 \\oplus _2 1 \\varepsilon _+^{j+2}\\ominus _2 10 (\\ominus 00)^{j+1} \\ominus _2 11\\colon j \\ge 0 \\,\\rbrace \\,.$ It can be derived in a way similar to that of the triangle reactions in Table REF .", "An example for these reactions is shown in Figure REF .", "We can then set $b_+ =01 \\oplus _2$ and $b = \\ominus _2 10 (\\ominus 00)^{j+1} \\ominus _2 11$ .", "These two types of reactions then play a role in the following lemma.", "Its proof uses the fact that the existence of an ether fragment $1\\varepsilon _+^2$ at a given time causes the existence of a fragment of another type, which then causes the existence of another ether fragment at still another time.", "This second type of fragment may be any situation of form $0 1^{2j + 3}$ with $j \\ge 0$ .", "For this kind of fragment we need another set of situations analogous to $\\mathcal {E}(t_1, \\dots , t_n)$ .", "We will therefore write $\\mathcal {E}_1(t_1, \\dots , t_n) =\\bigcup _{j \\ge 0} \\mathcal {F}(0 1^{2j + 3}, t_1, \\dots , t_n)\\,.$ for the set of intermediate fragments that occur in the propagation of ether fragments to later times.", "Lemma 8.16 (Propagation of Ether Fragments) Let $(U_n)_{n \\ge 0}$ be the sequence of random intervals in (REF ).", "Assume that there is a reaction $u\\rightarrow _\\Phi a$ with $u \\in \\Sigma ^n$ and $a \\in \\mathcal {E}(t)$ .", "Then for every $\\varepsilon > 0$ there is a number $m \\ge n$ for which $P(U_m \\rightarrow _\\Phi \\mathcal {E}(t + 2, t + 3) \\mid U_n = u)\\ge 1 - \\varepsilon \\,.$ The number $m$ depends only on $\\varepsilon $ and $t$ , not on $a$ .", "Figure: Decomposition of aa in ().Since $a$ is an element of $\\mathcal {E}(t)$ , it has a representation (Figure REF ) $a = a_+ 1 \\varepsilon _+^2 a^{\\prime }$ with $a_+ \\in \\mathop {\\mathrm {dom}}\\Phi _+$ , $a^{\\prime } \\in \\mathop {\\mathrm {dom}}\\Phi $ and $\\delta (a_+)_T =t$ .", "We now will show that by putting a random interval $v$ at the left of $a$ we will get a situation that reacts to an element of $\\mathcal {E}(t + 2, t + 3)$ , with the probability that this happens growing arbitrarily large as the length of $v$ becomes arbitrarily large.", "Figure: Reaction () puts an eventσ\\sigma to the left of the ether fragment.We know from Lemma REF that we can find a number $m_1$ such that for all $v_1 \\in \\Sigma ^{m_1}$ there is a reaction $v_1 a_+ \\rightarrow _\\Phi a_{1+} \\sigma $ with $a_{1+} \\in \\mathop {\\mathrm {dom}}\\Phi _+$ and $\\sigma \\in \\Sigma $ , independent of the value of $v_1$ .", "The values of $a_{1+}$ and $\\sigma $ depend on the value of $u$ .", "This also means that for every $a_+$ there is a reaction (Figure REF ) $v_1 a_+ 1 \\varepsilon _+^2 a^{\\prime }\\rightarrow _\\Phi a_{1+} \\sigma 1 \\varepsilon _+^2 a^{\\prime }\\,.$ This reaction can now be extended to a reaction that modifies the factor $1 \\varepsilon _+^2$ in the result term.", "In order to find this reaction we must take both possible values for $\\sigma $ into consideration and need to understand the reactions that start at $01\\varepsilon ^2$ and $11 \\varepsilon ^2$ .", "From Lemma REF we know that of all $\\sigma \\in \\Sigma $ there is a reaction $0 \\sigma \\varepsilon _+^2 \\rightarrow _\\Phi x_+ 1^3a^{\\prime }_{2+}$ , with $x_+$ , $a^{\\prime }_{2+} \\in \\mathop {\\mathrm {dom}}\\Phi _+$ and $\\delta (x_+)_T\\in \\lbrace 1, 2 \\rbrace $ .", "One of these two reactions can then always be applied to the result of (REF ).", "The result is a new reaction, $v_1 a_+ 1 \\varepsilon _+^2 \\rightarrow _\\Phi a_{1+} x_+ 1^3a^{\\prime }_{2+}$ .", "Now we introduce a new situation, $a_{2+} = a_{1+} x_+$ and have then found that for every $v_1 \\in \\Sigma ^{m_1}$ there is a reaction $v_1 a_+ 1 \\varepsilon _+^2 a^{\\prime }\\rightarrow _\\Phi a_{2+} 1^3 a^{\\prime }$ with $\\delta (a_{2+})_T \\in \\lbrace t + 1, t + 2 \\rbrace $ .", "For the next step of the proof we need to find a reaction that transforms the result of (REF ) into a situation with $1 0^{2j+3}$ as factor, where $j \\ge 0$ .", "It would be nice if there were a reaction of the form $a_{2+} 1^3 a^{\\prime } \\rightarrow _\\Phi a_{3+} 01^{2j + 3} a^{\\prime \\prime }$ , because then we could apply one the reactions of Lemma REF to the term $0 1^{2j + 3}$ and get another situation that contains an ether fragment.", "However we will prove instead a weaker form of this statement: We will show that there is a number $m_2 \\ge 0$ such that the probability for a reaction $v_2 a_{2+} 1^3 a^{\\prime } \\rightarrow _\\Phi a_{3+} 0 1^{2j + 3} a^{\\prime \\prime }$ , with a random interval $v_2 \\in \\Sigma ^{m_2}$ , is arbitrarily close to 1.", "The interval $v_2$ is then another fragment of the random initial configuration.", "We begin with another application of Lemma REF .", "It shows that by choosing an appropriate minimal length for the interval $v_2$ we can ensure that there is always a reaction $v_2 a_{2+} 1^3 \\rightarrow _\\Phi x_+ w$ , with $x_+ \\in \\mathop {\\mathrm {dom}}\\Phi _+$ and $w \\in \\Sigma ^$ , in which $w$ is arbitrarily long.", "We now multiply $a^{\\prime }$ to the right of this reaction and get $v_2 a_{2+} 1^3 a^{\\prime } \\rightarrow _\\Phi x_+ w a^{\\prime }$ .", "The result of this reaction will now written in a different form, depending on the value of $w$ .", "If $w$ does not consist entirely of events with state 1, we can write it as $w = w^{\\prime } 0 1^\\ell $ .", "Here we must have $\\ell \\ge 3$ , because $w$ is an extension of the interval $1^3$ .", "Now we can set $j = \\left\\lfloor \\frac{\\ell - 3}{2}\\right\\rfloor $ .", "Then we have either $\\ell = 2j + 3$ or $\\ell = 2 j + 4$ .", "In the first case we have $w = w^{\\prime } 01^{2j + 3}$ , and we can set $a_{3+} = x_+ w^{\\prime }$ and $a^{\\prime \\prime } = a^{\\prime }$ ; in the second case we have $w = w^{\\prime } 0 1^{2j + 3} 1$ and can set $a_{3+} =x_+ w^{\\prime }$ and $a^{\\prime \\prime } = 1 a^{\\prime }$ .", "The reaction becomes in both cases $v_2a_{2+} 1^3 a^{\\prime } \\rightarrow _\\Phi a_{3+} 0 1^{2j + 3} a^{\\prime \\prime }$ , as required.", "Figure: Reaction ().", "The interval 01 2j+3 01^{2j+3} occurs later than the ether fragment.When we now multiply both sides of (REF ) from the right with $v_2$ and apply the previous reaction to its result, we get (Figure REF ) $v_2 v_1 a = v_2 v_1 a_+ 1 \\varepsilon _+^2 a^{\\prime }\\rightarrow _\\Phi a_{3+} 0 1^{2j + 3} a^{\\prime \\prime }\\,.$ We have in this reaction $\\delta (a_{3+})_T \\in \\lbrace t + 1, t + 2 \\rbrace $ , because $\\delta (a_{3+})_T = \\delta (a_{2+})_T$ .", "The proof for the existence of this reaction is however only valid if $w$ does not consist entirely of ones.", "Next we must therefore show that the probability that $w$ contains a zero can be made arbitrarily large by choosing $m_2$ large enough.", "For the proof we use the fact that the events of $pr_{x_+}(w)$ in the reaction $v_2a_{2+} 1^3 \\rightarrow _\\Phi x_+ w$ above must occur either at the time step $t + 1$ or $t + 2$ .", "We must therefore show that with a high probability there is an event with state 0 at these time steps.", "For this we use the fact that if the random interval $v_2$ becomes long enough, the probability that it contains a given interval as its factor becomes arbitrarily close to 1.", "(This can be proved in a similar way as Theorem REF .)", "In the current proof we use the factor $1 0^{2t + 3} 1$ because its closure is a triangle process, and it contains zeros at all time steps from 0 to $t + 2$ .", "Therefore, if $m_2$ is large enough, the probability that a random $v_2$ contains the interval $1 0^{2t + 3} 1$ becomes arbitrarily close to 1, and when $v_2$ contains such an interval, there is a cell in state 0 in the time steps $t + 1$ and $t + 2$ .", "Since the interval $v_2$ is at the left of the situation $a_+ 1^3$ , a cell in state 0 must therefore occur in $w$ .", "The result of reaction (REF ) is always an element of $\\mathcal {E}_1(t + 1, t + 2)$ .", "We have therefore shown that for every $\\varepsilon > 0$ there is a number $m = m_2 + m_1 + n$ such that we have the probability $P(U_m \\rightarrow _\\Phi \\mathcal {E}_1(t + 1, t + 2) \\mid U_n = u)\\ge 1 - \\varepsilon \\,.$ As a final step we now prove that if there is a reaction $v\\rightarrow _\\Phi b$ with $v \\in \\Sigma ^m$ and $b \\in \\mathcal {E}_1(t + 1, t+ 2)$ , then there is a reaction from $b$ to an element of $\\mathcal {E}(t + 2, t + 3)$ .", "Figure: Reaction ().", "A new etherfragment is generated from the interval 01 2j+3 0 1^{2j + 3}.For this we do the same with $b$ as we did before with $a$ and write it as $b = b_+ 0 1^{2j + 3} b^{\\prime }$ .", "We have already seen in Lemma REF that there is a reaction $0 1^{2j+ 3} \\rightarrow _\\Phi y_+ 1 \\varepsilon _2^{j + 2} y$ , with $\\delta (y_+)_T =1$ .", "We now set $b_{1+} = b_+ y_+$ .", "Then we have found a reaction (Figure REF ) $b_+ 0 1^{2j + 3} b^{\\prime }\\rightarrow _\\Phi b_{1+} 1 \\varepsilon _+^{j+2} y b^{\\prime }$ with $\\delta (b_{1+})_T \\in \\lbrace t + 2, t + 3 \\rbrace $ , because $\\delta (b_{1+})_T = \\delta (b_+)_T + 1$ .", "Therefore the right side of the reaction (REF ) is always an element of $\\mathcal {E}(t + 2, t + 3)$ .", "Putting all these steps together we see therefore that we have found for every $\\varepsilon > u$ a probability $P(U_m \\rightarrow _\\Phi \\mathcal {E}(t + 2, t + 3) \\mid U_n \\rightarrow _\\Phi a)\\ge 1 - \\varepsilon ,$ as stated in the lemma.", "With the reactions in Lemma REF we see that the ether fragments move to the future, and with Lemma REF we see that ether fragments are generated from random initial configurations.", "Together the lemmas show that ether fragments continue to exist for arbitrarily long times.", "This is now described in more detail in the following proof.", "[Proof of Theorem REF ] The proof is done by induction.", "We can see from Lemma REF that the theorem is true for $t = 0$ : The lemma shows that there is a $m \\ge 0$ such that $P(U_m \\rightarrow _\\Phi \\mathcal {E}(0)) \\ge 1 - \\varepsilon $ .", "Therefore we have $P(U_m \\rightarrow _\\Phi \\mathcal {E}(0, 1, 2)) \\ge P(U_m \\rightarrow _\\Phi \\mathcal {E}(0) \\ge 1 - \\varepsilon )$ .", "In the main part of the induction we assume that the theorem is true for a given time $t \\ge 0$ .", "With this we mean that for every probability $\\varepsilon _1 \\ge 0$ there is a number $n$ such that $P(U_n \\rightarrow _\\Phi \\mathcal {E}(t, t + 1, t + 2)) \\ge 1 - \\varepsilon _1$ .", "We then need to show that for every probability $\\varepsilon > 0$ there is a length $m \\ge n$ such that $P(U_m \\rightarrow _\\Phi \\mathcal {E}(t + 1, t + 2, t + 3))\\ge 1 - \\varepsilon \\,.$ In order to do this we split the reaction $U_m \\rightarrow _\\Phi \\mathcal {E}(t + 1, t + 2, t + 3)$ into two subreactions.", "The first one is derived from the reaction $U_n \\rightarrow _\\Phi \\mathcal {E}(t, t +1, t + 2)$ .", "This reaction exists by the induction assumption, and we can make its probability arbitrarily high by choosing the right $n$ , but its input interval $U_n$ is by definition shorter than $U_m$ .", "Nevertheless, a similar reaction starting from $U_m$ does also exist.", "It will be our first subreaction.", "In the second subreaction, the result of the first subreaction then reacts to an element of $\\mathcal {E}(t + 1, t + 2, t + 3)$ , and we will show that we can make its probability as large as we want by making $m$ large enough.", "To combine these subreactions we need to split the set of possible input intervals for the first reaction, $U_n \\rightarrow _\\Phi \\mathcal {E}(t, t + 1, t + 2)$ , into two disjoint sets.", "For this we introduce the sets $\\mathcal {U}_0$ and $\\mathcal {U}_1 \\subseteq \\Sigma ^n$ , satisfying the requirement that for every reaction $u\\rightarrow _\\Phi a$ with $u \\in \\Sigma ^n$ and $a \\in \\mathcal {E}(t, t + 1,t + 2)$ we have either $u \\in \\mathcal {U}_0$ and $a \\in \\mathcal {E}(t)$ , or $u \\in \\mathcal {U}_1$ and $a \\in \\mathcal {E}(t +1, t + 2)$ .", "If $u$ satisfies both conditions, then we may choose arbitrarily $u \\in \\mathcal {U}_0$ or $u \\in \\mathcal {U}_1$ .", "The same is true if there is no such reaction $u \\rightarrow _\\Phi a$ for a given $u$ .", "Then we can write the probability for the first subreaction as a sum of the probability of two independent stochastic events, namely as $P(U_n \\rightarrow _\\Phi \\mathcal {E}(t, t + 1, t + 2)) =P(U_n \\in \\mathcal {U}_0) + P(U_n \\in \\mathcal {U}_1)\\,.$ The requirements for the reactions from the sets $\\mathcal {U}_0$ and $\\mathcal {U}_1$ are expressed by the (trivial) probabilities $P(U_n &\\rightarrow _\\Phi \\mathcal {E}(t)\\mid U_n \\in \\mathcal {U}_0) = 1, \\\\P(U_n &\\rightarrow _\\Phi \\mathcal {E}(t + 1, t + 2)\\mid U_n \\in \\mathcal {U}_1) = 1\\,.$ The equations stay true if we replace the leftmost $U_n$ in them with a larger random interval $U_m$ , where $m \\ge n$ .", "We will use this now for the second subreaction.", "The results () are the input situations for the second subreaction, so it has two cases as well.", "We need to show that in both cases there is a reaction into the set $\\mathcal {E}(t +1, t + 2, t + 3)$ .", "If $u \\in \\mathcal {U}_0$ , we use Lemma REF .", "It applies to situation $u \\in \\Sigma ^n$ for which there is a reaction $u \\rightarrow _\\Phi a \\in \\mathcal {E}(t)$ .", "This property is of course equivalent to the condition that $u \\in \\mathcal {U}_0$ .", "The lemma then states that for every $\\varepsilon _2 > 0$ there is a number $m \\ge n$ such that $P(U_m\\rightarrow _\\Phi \\mathcal {E}(t + 2, t + 3) \\mid U_n = u) \\ge 1 -\\varepsilon _2$ .", "We now collect the probabilities for all $u \\in \\mathcal {U}_0$ and get the following reaction, $P(U_m \\rightarrow _\\Phi \\mathcal {E}(t + 2, t + 3)\\mid U_n \\in \\mathcal {U}_0) \\ge 1 - \\varepsilon _2\\,.$ If $u \\in \\mathcal {U}_1$ , then we can use a lengthened form of reaction (), $P(U_m \\rightarrow _\\Phi \\mathcal {E}(t + 1, t + 2) \\mid U_n \\in \\mathcal {U}_1) = 1\\,.$ The results of both reactions are elements of $\\mathcal {E}(t + 1, t+ 2, t + 3)$ , as was required.", "Now we can perform the complete induction step.", "The following computation begins by splitting the probability for the reaction $U_m \\rightarrow _\\Phi \\mathcal {E}(t + 1, t + 2, t + 3)$ into two cases, depending on whether the right end of $U_m$ , i. e. the interval $U_n$ , is an element of $\\mathcal {U}_0$ or $\\mathcal {U}_1$ .", "Then the equations () are used to get estimates for these probabilities, and later, at the penultimate step, the two cases are unified again with the help of (REF ).", "$P(U_m \\rightarrow _\\Phi \\mathcal {E}(t + 1, t + 2, t + 3)) \\\\\\begin{aligned}[b]&\\ge P(U_m \\rightarrow _\\Phi \\mathcal {E}(t + 1, t + 2, t + 3)\\mid U_n \\in \\mathcal {U}_0) P(U_n \\in \\mathcal {U}_0) \\\\&\\quad + P(U_m \\rightarrow _\\Phi \\mathcal {E}(t + 1, t + 2, t + 3)\\mid U_n \\in \\mathcal {U}_1) P(U_n \\in \\mathcal {U}_1) \\\\&\\ge P(U_m \\rightarrow _\\Phi \\mathcal {E}(t + 1, t + 2)\\mid U_n \\in \\mathcal {U}_0) P(U_n \\in \\mathcal {U}_0) \\\\&\\quad + P(U_m \\rightarrow _\\Phi \\mathcal {E}(t + 2, t + 3)\\mid U_n \\in \\mathcal {U}_1) P(U_n \\in \\mathcal {U}_1) \\\\&\\ge (1 - \\varepsilon _2) P(U_n \\in \\mathcal {U}_0)+ P(U_n \\in \\mathcal {U}_1) \\\\&\\ge (1 - \\varepsilon _2)(P(U_n \\in \\mathcal {U}_0) + P(U_n \\in \\mathcal {U}_1)) \\\\&= (1 - \\varepsilon _2) P(U_n \\rightarrow _\\Phi \\mathcal {E}(t, t + 1, t + 2)) \\\\&\\ge (1 - \\varepsilon _2) (1 - \\varepsilon _1)\\,.\\end{aligned}$ Therefore the probability for this reaction can be made greater than $1 - \\varepsilon $ by making both $\\varepsilon _1$ and $\\varepsilon _2$ small enough." ], [ "Generalisation to Other Rules", "While the arguments in this chapter were tailored specifically for the use with Rule 54, it was with the hope that they would lead to ideas that are useful for the understanding of ether formation under other rules, e. g. Rule 110.", "To find these new ideas we must generalise the proofs and definitions of this chapter.", "For some of them it is easy to see how to generalise them, but in other places new ideas are needed.", "I will now describe the necessary changes in more detail.", "First we need a characterisation of the ether in the relevant cellular automaton.", "The definition of the ether reaction (Definition REF ) would be generalised to a pair of reactions $e_0 v \\rightarrow _\\Phi e_0 e_+ e_-\\qquad \\text{and}\\qquad e_0 e_- e_+ \\rightarrow _\\Phi e_0 e_+ e_-,$ with $e_0$ , $v \\in \\Sigma ^$ , $e_+ \\in \\mathop {\\mathrm {dom}}\\Phi _+$ and $e_- \\in \\mathop {\\mathrm {dom}}\\Phi _-$ .", "Under Rule 54 we had $e_0 = 1$ , $v = 0^3 1$ , $e_+ =\\varepsilon _+^2$ and $e_- = \\varepsilon _-^2$ .", "This can be done in every one-dimensional cellular automaton with an ether.", "The situation $e_0e_+$ then plays the role of the ether fragment.", "Then we can construct sets of situations $\\mathcal {E}^{\\prime }(t_1, \\dots ,t_n) = \\mathcal {F}(e_0 e_+, t_1, \\dots , t_n)$ , in analogy to the sets $\\mathcal {E}(t_1, \\dots , t_n)$ in (REF ).", "Lemma REF , which proves the generation of an ether fragment from a large enough random interval, can be extended to $\\mathcal {E}^{\\prime }$ : The only requirement for its proof is the existence of a reaction $e_0 v \\rightarrow \\mathcal {E}^{\\prime }(0)$ , but this is the first reaction in (REF ).", "Then, if we have an equivalent to Lemma REF , we can prove ether formation in essentially the same way as here, by showing that if there is an ether fragment at time $t$ , there is always a starting time $t^{\\prime } > t$ at which another ether fragment exists with an arbitrarily high probability.", "Lemma REF , however, is proved with the help of the Lemmas REF and REF , which are highly specific to Rule 54.", "It is not clear whether an equivalent to these lemmas exists for other cellular automata with an ether, and if it exists, how to find it in a systematic way.", "In Rule 54, they were the result of some experimenting and an already well-developed understanding of the way this rule works.", "This means that an expert for, say, Rule 110 could find an equivalent to Lemma REF and REF after a similar amount of experimenting, but there is no recipe for a proof of ether formation if the behaviour of a rule is not yet well understood—even if one has already seen that it has an ether." ], [ "Summary", "In this chapter we have begun to define concepts with which one can express larger structures in a cellular automaton.", "There is a small theory of regular structures like triangles.", "We have seen how one can derive families of reactions.", "Then we turned our view to the ether.", "We first found a way to express the empty ether with situations and reactions.", "The ether reactions motivated the definition of ether fragments, the situations $1\\varepsilon _+^2$ .", "The existence of ether fragments at arbitrary times was defined as a way to express the existence of ether when the initial configuration was chosen at random.", "We had to invent an extension of the calculus of Flexible Time that can handle probabilities.", "This was done in an incomplete way, just enough to get the proofs done.", "Nevertheless a basic principle did appear: in a random reaction the input had to be a random variable, while the result had to be a set of situations.", "With these extensions we expressed how ether fragments were generated in the initial configuration and how an ether fragment that was positioned at a given time $t$ caused the existence of an ether fragment at a later time.", "This then lead to a proof that ether fragments must exist at all time steps when the cellular automaton started from a random initial configuration.", "Finally we considered the question whether the proof that we had done for Rule 54 could be generalised to other transition rules.", "The answer was that it could be only partially, and that filling the gaps would require knowledge about the specific rule involved." ], [ "Conclusions", "This work was about finding a way to speak about cellular automata in terms of “traditional mathematics”, as I have called it in the introduction.", "The main results of this thesis are therefore concepts, not theorems." ], [ "Reaction Systems", "The idea that started my work on cellular automata was that of situations and reaction, first understood only in an intuitive way and for concrete cellular automata.", "There was much choice in the way the situations were defined, and it was resolved by trial and error for a specific transition rule.", "My article [51] was a product of this phase.", "Already then I tried to find definitions that applied to a large number of cellular automata, even if the work was done for one specific transition rule.", "This was so because only when a definition applies to a large number of cases, the methods developed for it can also be used to explore the behaviour of unknown cellular automata.", "In this work I have therefore searched for principles on which I could base the definition of a reaction system that are valid for a large number of transition rules.", "It was still possible to exclude certain rules from consideration if their behaviour was difficult to express with situations.", "One of the first results was the restriction to interval-preserving rules.", "The reason for this decision was that intervals are especially easy to express with situations.", "We have also seen that the closure of an interval has a simple structure and that the possibilities for their left-to-right arrangement are limited.", "Only the restriction to interval-preserving rules made the theoretical understanding of cellular processes and their closures possible.", "The question which kinds of situations to choose was then resolved by finding the concept of separating intervals.", "A separating interval forms a boundary between the cells left of it and the cells right of it.", "If two separating intervals form the left and right ends of an interval process $\\pi $ , then the two intervals already determine which points are determined by $\\pi $ , even if the rest of $\\pi $ is unknown.", "This occurs especially when the events between the separating intervals belong to the closure of a larger process.", "This then occurs in achronal situations: They contain separating intervals at strategic places, so that it is possible to limit the extension of their closure.", "Thanks to this construction we have a guarantee that the closure exists at all.", "The definition of achronal situations and the proof that ordered achronal situations have a closure is another important progress in comparison to [51].", "Separating intervals are also interesting in a more general context.", "They tell about the information transmission in a cellular automaton.", "The set of separating intervals for a transition rule is a better measure for the speed of information transmission than the radius of the transition rule, another a measure for the speed of information transmission.", "The radius does however only measure of the maximal possible speed.", "In contrast to this, the set of separating intervals is an invariant of the cellular automaton that is radius-invariant in the sense of Definition REF ." ], [ "Ether Formation", "The sections about ether formation were intended as a reality check for the formalism of Flexible Time.", "They are a test whether the formalism has been developed far enough to find solutions for a natural-looking question about cellular automata, i. e. a question that did not occur as part of the development of the formalism.", "The question of ether formation was such a problem.", "To become answerable the ether problem had to be reduced to a very simple question.", "We reduced the question of ether formation from the general case to that of Rule 54, and then characterised the ether by the ether fragments $1 \\varepsilon _+^2$ .", "The result was a theorem that only showed that such ether fragments exist at arbitrary times when starting from a random initial configuration, not that they become more common with time.", "The latter fact is clearly visible from computer simulations.", "A more positive result is that the proof of the Ether Formation Theorem REF is valid for all probability distributions in which both cells in state 0 and in state 1 can occur in the initial configuration.", "This is more general than the empirical results, which usually refer to the case that zeros and ones initially occur with equal probability.", "Another positive result of the approach with Flexible Time is that it can express the mechanism with which the ether is created and preserved.", "We have seen that the initial configuration creates with a high probability an ether fragment $1 \\varepsilon _+^2$ , which then causes the creation of an interval $0 1^{2j + 3}$ at a later time, which in turn reacts to another ether fragment at a still later time: All this was expressed with help of the sets $\\mathcal {E}$ and $\\mathcal {E}_1$ .", "The mechanism of Lemma REF then lets us formulate one cause of ether formation under Rule 54.", "This is the connection of structure formation with loss of information about the initial state.", "Structure formation in a dynamical system can always be expressed as a case of information loss: Several initial configurations must evolve to the same, more ordered, later configuration.", "Under Rule 54, information loss occurs during the propagation of ether fragments: In the reactions of Lemma REF , the two situations $01\\varepsilon _+^2$ and $11 \\varepsilon _+^2$ react to a situation that contains the term $1^3$ .", "This idea has a chance to be also the cause of ether formation in other cellular automata." ], [ "The Results in Context", "This work has therefore shown that there is an essentially two-dimensional approach to structure formation in cellular automata and that it can prove nontrivial facts about a cellular automaton.", "Which place does then the formalism of Flexible Time take among the approaches to understand the behaviour of one-dimensional cellular automata, especially among those described in Section ?", "In its two-dimensionality it is similar especially to the work of Ollinger and Richard [48], [53], which was however mainly applied to model a network of colliding particles.", "While this kind of research is also possible with Flexible Time [52], it has not been done in detail.", "In its formalisation of the ether, the approach of this thesis diverges from most other works in that it concentrates directly on fragments of the ether.", "Most other works, not just Ollinger and Richard, describe in great details the particles that move through the ether.", "This kind of approach could also become the basis of a proof of ether formation: an analysis how the particles interact and how they gradually destroy each other.", "(The destruction of gliders does occur under Rule 54 and 110 and was shown experimentally by Boccara et al.", "[4] and by Li and Nordahl [32].)", "The number of particles and their possible interactions can become however quite large, as it does under Rule 110 [37], and it would be tedious to use them all in a formal proof.", "Among the works about the behaviour of random initial configurations, many concentrate on the behaviour of the defects that occur between domains [15], [27].", "It has been shown that under many conditions such a defect performs a random walk.", "In our proof for ether formation we also have a moving “particle”, namely the ether fragment, but its position cannot be directly identified from an evolution diagram.", "Nevertheless it performs a kind of random walk, driven by the states of the cells in the initial configuration.", "With Flexible Time we have therefore another tool with which one can trace the influence of a random initial configuration to later time step.", "(The tool however has not yet been developed very far.)", "There is also some similarity to the “grouping” approach [13], [14], since Flexible Time also groups several cells to a greater entity.", "With Flexible Time there is however much more freedom in the choice of the situations, and as a result this approach does not automatically provide tools to put cellular automata into groups according to their behaviour.", "Unintentionally, Flexible Time may however lead to new ideas for the classification of cellular automata.", "We could now classify them by their separating intervals (Table REF ) or by the pattern of their generating reactions (Table REF ).", "What this classification means is not yet clear, but it must have something to do with the information transmission in the cellular automata." ], [ "Ideas for Further Research", "Finally I list here a small number of ideas for further research.", "Their purpose is to extend the system of Flexible Time and also to apply its ideas to other domains." ], [ "Separating Intervals", "We have defined separating intervals as a purely technical tool.", "It is not yet clear whether they have an intrinsic meaning, except that they are somewhat related to signal transmission.", "A possible starting point for further research is therefore the question whether transition rules with the same set of separating intervals have something in common.", "Which properties of a transition rule can be derived from knowing its separating intervals?", "Another starting point to find out more about separating intervals is the growing body of research about cellular automata with memory [3].", "Does the addition of memory change the separating intervals of cellular automata, and if so, in which way?" ], [ "Explicit Probabilities", "Another idea for later work is the search for good explicit probabilities for ether propagation.", "The reactions only show that the ether fragments survive over time, but they do not give a good estimate about their density.", "A description of the ether propagation that was more detailed will be needed to get better estimates.", "With it, there is a chance to find a proof that the density of ether fragments actually grows over time." ], [ "Generalising the Way to Find the Ether", "Some ideas for this were already outlined in Section .", "The current argument for the ether required many ad hoc constructions.", "An example are the ether fragments, which were only found after studying the evolution of configurations under Rule 54 for a long time.", "There was nothing systematic in their construction.", "Another example is the structure of the proof for Lemma REF .", "All relied on phenomenology.", "Nevertheless the current proof may contain the ideas that can be generalised to a more systematic proof.", "This in turn will of course require a further development of the formalism in order to make it more streamlined and easier to use." ], [ "Analysis of Particle Interactions", "The ether in Rule 54 is the medium in which particles move.", "There has been a large amount of research about the particles and their interactions under Rule 54 [4], [34], [36], Rule 110 [6], [37], [38], [39], [40], [41], [48], [53] and other rules [29], [30].", "My own earlier paper [52] contains the beginnings of an analysis of the particle interactions under Rule 54.", "This line of research was left incomplete because the structure of the reaction system for Rule 54 was not yet clear.", "Now it could be continued, with the hope for a reasonably simple algebra of particle interactions for Rule 54.", "A good knowledge of such particle interactions could make another proof for ether formation possible.", "It has already been noted by Boccara, Nasser and Roger [4] that in the typical evolution of a random initial configuration at a very early time configurations arise that consist of small regions of ether, with particles between them.", "The particles then interact and slowly destroy each other, such that the ether between them grows.", "All this has been found in computer simulations, but not proved.", "A good understanding of particle interactions would therefore allow to understand this process in detail and provide a quantitative estimate for the speed with which the ether grows." ], [ "A Generic Case of Self-Organisation", "Finally an idea for a larger project.", "It is inspired by a paper by Boccara and Roger [5], in which the authors describe a whole class of self-organising rules.", "They are generated from totalistic rules, in which the states of the cells are numbers and the next state of a cell only depends on the sum of the states of the cells in the neighbourhood.", "The authors have found a transformation that transforms an arbitrary totalistic rule into a rule that shows a certain amount of self-organisation.", "In it a rule $\\varphi $ of radius $r$ is transformed into a rule $\\varphi ^{\\prime }$ of radius $n r$ which works on initial configurations in which every block of $n$ cells have the same state.", "It is required that if every cell in the initial configuration of $\\varphi $ is expanded to $n$ cells, then the evolution of this configuration under $\\varphi ^{\\prime }$ corresponds exactly to the evolution of the original configuration under $\\varphi $ .", "If both $\\varphi $ and $\\varphi ^{\\prime }$ are totalistic, then $\\varphi ^{\\prime }$ is uniquely determined by $\\varphi $ .", "Now if the initial configuration is arbitrary, then in the following time steps in the evolution under $\\varphi ^{\\prime }$ the cells begin to organise in blocks of length $n$ , again with defects between the blocks that move randomly and sometimes annihilate.", "The understanding of this kind of pattern formation, in the same or a different way as we have done this here for Rule 54, would lead to the understanding of the behaviour of a whole class of cellular automata, not just one.", "It is therefore very interesting and would also allow the formalism to grow." ] ]	1403.0193
[ [ "The BANANA project. V. Misaligned and precessing stellar rotation axes\n in CV Velorum" ], [ "Abstract As part of the BANANA project (Binaries Are Not Always Neatly Aligned), we have found that the eclipsing binary CV Velorum has misaligned rotation axes.", "Based on our analysis of the Rossiter-McLaughlin effect, we find sky-projected spin-orbit angles of $\\beta_{\\rm p} = -52\\pm6^{\\circ}$ and $\\beta_{\\rm s}= 3\\pm7^{\\circ}$ for the primary and secondary stars (B2.5V $+$ B2.5V, $P=6.9$ d).", "We combine this information with several measurements of changing projected stellar rotation speeds ($v \\sin i_{\\star}$) over the last $30$ years, leading to a model in which the primary star's obliquity is $\\approx65^{\\circ}$, and its spin axis precesses around the total angular momentum vector with a period of about $140$ years.", "The geometry of the secondary star is less clear, although a significant obliquity is also implicated by the observed time variations in the $v \\sin i_{\\star}$.", "By integrating the secular tidal evolution equations backward in time, we find that the system could have evolved from a state of even stronger misalignment similar to DI Herculis, a younger but otherwise comparable binary." ], [ "Introduction", "Stellar obliquities (spin-orbit angles) have been measured in only a handful of binary stars.", "[4].", "The case of DI Herculis, in which both stars have large obliquities [1], has taught us that it is risky to assume that the orbital and spin axes are aligned.", "For one thing, misalignment can influence the observed stellar parameters; for example, the rotation axes may precess, producing time variations in the sky-projected rotation speeds [1], [33].", "Precession would also produce small changes in the orbital inclination, and therefore changes in any eclipse signals.", "Misalignment also influences the rate of apsidal precession, as predicted by [36] for the case of DI Her and confirmed by [1], and as is suspected to be the case for AS Cam [30].", "Furthermore, measurements of obliquities should be helpful in constraining the formation and evolution of binary systems.", "Larger obliquities might indicate that a third star on a wide, inclined orbit gave rise to Kozai cycles in the close pair during which the close pair's orbital eccentricity and inclination oscillate [28], [13], [15], [29].", "A third body on an inclined orbit might also cause orbital precession of the inner orbit around the total angular momentum, again creating large opening angles between the stellar rotation and the orbit of the close pair [13].", "There are also other mechanisms which can lead to apparent misalignments.", "For example [34], [35] suggest that for stars with an outer radiative layer, large angles between the convective core and the outer layer may be created by internal gravity waves.", "The aim of the BANANA project (Binaries Are Not Always Neatly Aligned) is to measure obliquities in close binaries and thereby constrain theories of binary formation and evolution.", "We refer the reader to [4] for a listing of different techniques to measure or constrain obliquities.", "[39], [26], [32], and [45] have also presented new obliquity measurements in some binary star systems.", "This paper is about the CV Vel system, the fifth BANANA system.", "Previous papers have examined the V1143 Cyg, DI Her, NY Cep, and EP Cru systems [2], [1], [4], [3].", "CV Vel was first described by [40].", "A spectroscopic orbit and a light curve were obtained by [16] and [17].", "Two decades later, [5] analyzed the system in detail.", "Together with four Strömgen light curves obtained and analyzed by [11], this has allowed the absolute dimensions of the system to be known with an accuracy of about 1%.", "More recently [43] conducted another study of the system, finding that the stars in the system belong to the class of slowly pulsating B stars [42], [12].", "Table REF summarizes the basic data.", "Table: General data on CV VelorumOne reason why this binary was selected for BANANA was the disagreement in the measured projected stellar rotation speeds measured by [5] and [43].", "[5] found $v \\sin i_{\\star }=28\\pm 3$ km s$^{-1}$ for both stars.", "Employing data obtained nearly 30 years later, [43] found $v \\sin i_{\\rm p} = 19\\pm 1$ km s$^{-1}$ and $v\\sin i_{\\rm s} = 31\\pm 2$ km s$^{-1}$ , indicating a significantly lower $v \\sin i_{\\star }$ for the primary star.", "This could indicate that the stellar rotation axes are misaligned and precessing around the total angular momentum vector, as has been observed for DI Herculis [1].", "This paper is structured as follows.", "In the next section we describe the observations.", "In Section we describe our analysis method and results.", "Then, in Section , we discuss our findings in the framework of tidal evolution.", "We end in Section with a summary of our conclusions." ], [ "Spectroscopic observations", "We observed CV Vel with the FEROS spectrograph [22] on the $2.2$ m telescope at ESO's La Silla observatory.", "We obtained 100 observations on multiple nights between December 2009 and March 2011 with a typical integration time of 5-10 min.", "We obtained 26 spectra during 3 primary eclipses, 30 spectra during 4 secondary eclipses, and another 44 spectra outside of eclipses.", "In addition we observed the system with the CORALIE spectrograph at the Euler Telescope at La Silla.", "Two spectra were obtained in 2010, and another 4 spectra were obtained in spring 2013.", "The CORALIE observations were made near quadrature.", "In all cases, we used the software installed on the observatory computers to reduce the raw 2-d CCD images and to obtain stellar flux density as a function of wavelength.", "The uncertainty in the wavelength solution leads to a velocity uncertainty of a few m s$^{-1}$ , which is negligible for our purposes.", "The resulting spectra have a resolution of $\\approx $$50\\,000$ around 4500 Å, the wavelength region relevant to our analysis.", "We corrected for the radial velocity (RV) of the observatory, performed initial flat fielding with the nightly blaze function, and flagged and omitted bad pixels.", "While this work is mainly based on our new spectra, some parts of our analysis also make use of the spectra obtained by [43].", "Those earlier spectra help to establish the time evolution of the stellar rotation axes over the last few years.", "Figure: Photometry ofCV Vel.", "Top.—Loss of light during primary eclipses,as observed with TRAPPIST.", "The “O-C”(observed -- calculated) subpanel shows the residuals betweenthe data and the best-fitting model.", "The model fits were alsobased on data from Fig.", "1 of , which is notshown here.", "Bottom.—Same, for secondaryeclipses.", "In these plots phase 0 is offset by 0.250.25 from the time ofprimary mid eclipse." ], [ "Photometric observations", "To establish a modern eclipse ephemeris we obtained new photometric data.", "CV Vel was observed with the 0.6m TRAPPIST telescope in the I and z bandpasses [18] in La Sillahttp://www.astro.ulg.ac.be/Sci/Trappist.", "We observed the system during several eclipses from November 2010 to January 2011.", "Since the eclipses last nearly 12 hr, only a portion of an eclipse can be observed in a single night.", "We observed primary eclipses on three different nights, and secondary eclipses on six different nights.", "This gave full coverage of all phases of the secondary eclipse, and coverage of about three-quarters of the primary eclipse (see Figure REF ).", "We also include the Strömgen photometry obtained by [11] in our study.", "Figure: Spectroscopicobservations of CV Vel.", "Grayscale depiction of the spectraobtained outside eclipses and during both eclipses.", "Theobservations are sorted by orbital phase.", "The horizontal dashedlines indicate the begin and end of the primary (red) andsecondary (blue) eclipses.", "The dark bands represent absorptionlines; the darkest band is the Mg ii line.", "One can alsosee the weaker AI iii and S ii lines, at shorterand longer wavelengths, respectively.", "The small vertical blueand red lines indicate the calculated wavelength position of thecentral Mg ii line, as expected from orbital motion.", "Thelines overlap at times of eclipses.", "The discontinuities arisebecause of uneven coverage in orbital phase.To measure the sky-projected obliquities, we take advantage of the Rossiter-McLaughlin (RM) effect, which occurs during eclipses.", "Below we describe our general approach to analyzing the RM effect.", "Section REF describes some factors specific to the case of CV Vel.", "Section REF presents the results." ], [ "Model", "Our approach is similar to that described in Papers I–IV, where it is described in more detail.", "The projected obliquity of stellar rotation axes can be derived from the deformations of stellar absorption lines during eclipses, when parts of the rotating photospheres are blocked from view, as the exact shape of the deformations depend on the geometry of the eclipse.", "We simulate spectra containing light from two stars.", "The simulated spectra are then compared to the observed spectra, and the model parameters are adjusted to provide the best fit.", "Our model includes the orbital motion of both stars, and the broadening of the absorption lines due to rotation, turbulent velocities, and the point-spread function of the spectrograph (PSF).", "For observations made during eclipses, the code only integrates the light from the exposed portions of the stellar disks.", "The resulting master absorption line (which we will call the “kernel”) is then convolved with a line list which we obtain from the Vienna Atomic Line Database [24].", "The lines are shifted in wavelength space according to their orbital radial velocity, and weighed by the relative light contribution from the respective stars.", "The model is specified by a number of parameters.", "Figure: Absorption lines ofCV Vel in 2001/2002.", "Left.—The lines of the primary,in the region spanning the Mg ii line.", "The best-fittingmodel for the secondary lines has been subtracted.", "Thin graylines show all the out-of-eclipse observations.", "The red lineshows our model for the primary lines.", "The lower panel shows thedifference between individual observations and the meanline.", "Right.—The lines of the secondary, aftersubtracting the model of the primary lines.", "The blue line showsour model for the secondary.", "The pulsations of the secondarycause a larger scatter in the residuals.", "These spectra wereobtained in December 2001 and January 2002 with the CORALIE spectrograph by .", "Note that mislabeled the primary as the secondary, andvice versa; here we have labeled the spectra correctly.See also .Figure: Absorption lines of CV Velin 2009/2010.", "Same as Fig.", "butfor our FEROS and CORALIE spectra obtained outside ofeclipses.", "Comparison to Fig.", "revealsthat the primary absorption lines became significantly narrowerbetween 2001/2002 and 2009/2010." ], [ "Model parameters", "The orbit is specified by the eccentricity ($e$ ), argument of periastron ($\\omega $ ), inclination ($i_o$ ), period ($P$ ), and RV semi-amplitudes of the primary and secondary stars ($K_{\\rm p}$ and $K_{\\rm s}$ ).", "The position of the stars on their orbits, and therefore the times of eclipses, are defined by a particular epoch of primary mid-eclipse ($T_{\\rm min,I}$ ).", "In addition, additive velocity offsets ($\\gamma _{\\rm i}$ ) are needed.We use one velocity offset for each star.", "Due to subtle factors specific to each star the $\\gamma _{\\rm i}$ can differ from the barycentric velocity of the system, and they can also differ between the stars.", "Differences in gravitational redshift, line blending, and stellar surface flows could cause such shifts.", "To calculate the duration of eclipses and the loss of light, we need to specify the fractional radius ($r \\equiv R/a$ , where $a$ is the orbital semimajor axis) and quadratic limb darkening parameters ($u_1$ and $u_2$ ) for each star, as well as the light ratio between the two stars ($L_{\\rm s}/L_{\\rm p}$ ) at the wavelength of interest.", "Figure: Observations of CV Vel during primary eclipse.", "(a) Grayscaledepiction of the time dependence of the Mg ii lines ofboth stars, obtained throughout primary eclipses.", "The spectrahave been shifted into the rest frame of the primary.Horizontal dashed lines mark the approximate boundaries of theprimary lines, and vertical dashed lines mark the start and endof the eclipse.", "(b) After subtracting the secondary lines,based on the best-fitting model (See alsoFigure ).", "(c) Afterfurther subtracting a model of the primary lines which does notaccount for the RM effect, but only the light loss duringeclipse.", "This exposes the deformations due to the RM effect.Darkness indicates a deeper absorption line, lightness indicatesa shallower depth than expected in the zero-RM transitmodel.", "Throughout most of the primary eclipse, the blueshiftedside of the absorption line is deeper, indicating that thecompanion is almost exclusively eclipsing the receding half ofthe primary star.", "From this we can conclude that the primaryrotation axis and the orbital axis are misaligned.", "(d) Aftersubtracting a model including the RM effect.Figure: Observations of CV Vel during secondary eclipse.", "Similar toFigure , but forspectra obtained during secondary eclipses.", "In contrast to thedeformations observed during primary eclipse, here the RM effectis antisymmetric in time and covers the full vsini ☆ v \\sin i_{\\star }range, indicating alignment between the rotational and orbitalaxes on the sky.The kernel depends on various broadening mechanisms.", "Assuming uniform rotation, the rotational broadening is specified by $v \\sin i_{\\star }$ .", "Turbulent velocities of the stellar surfaces are described with the micro-macro turbulence model of [19].", "For this model two parameters are required: the Gaussian width of the macroturbulence ($\\zeta _{\\rm i}$ ); and the microturbulence parameter, which is degenerate with the width of the spectrograph PSF.", "We specify the sky-projected spin-orbit angles ($\\beta _{\\rm p}$ and $\\beta _{\\rm s}$ ) using the coordinate system (and sign convention) of [20]." ], [ "Normalization", "To take into account the uncertainties due to imperfect continuum normalization, we add 2 free parameters for each spectrum, to model any residual slope of the continuum as a linear function of wavelength.", "The parameters of the linear function are optimized (in a separate minimization) every time a set of global parameters are evaluated.", "This process is similar to the “Hyperplane Least Squares” method that was described by [6] and used in the context of eclipses in double star systems by [3].", "Figure: Absorptionlines of CV Vel during primary eclipse.", "Each panel shows theMg ii line of the primary star for a particular eclipsephase.", "The solid gray lines indicate the obtained spectra aftersubtraction of the best fitting model for the secondary andshifting the system in the restframe of the primary.", "The centralwavelength of the Mg ii line is indicated by the dashedline at 4481.2284481.228 Å.", "These nine panels show a subset of theobservation presented inFigure and arepresented in a way equivalent to panel (b).", "We also show our bestfitting primary model as red solid line and as black dash-dottedline a model assuming co-aligned stellar and orbital axes,representing the data poorly.", "Each panel has two insetsdepicting the projected rotational velocity of the uncoveredsurface of the primary star, with blue and red indicatingapproaching and receding velocities, respectively.", "The leftinsets show the model with the misaligned stellar axis, and theright insets show the same for the co-aligned model." ], [ "Parameter estimation", "To obtain parameter uncertainties we used a Markov Chain Monte Carlo (MCMC) code.", "Our stepping parameters were as listed above, except that instead of $i_o$ we stepped in $\\cos i_o$ , and for the eccentricity parameters we stepped in $\\sqrt{e} \\cos \\omega $ and $\\sqrt{e} \\sin \\omega $ .", "The chains consisted of $0.5$ million calculations of $\\chi ^2$ .", "The results reported below are the median values of the posterior distribution, and the uncertainty intervals are the values which exclude $15.85$ % of the values at each extreme of the posterior and encompass $68.3$ % of the posterior probability." ], [ "Implementation of the model for CV Vel", "For CV Vel we focused on the Mg ii line at 4481 Å, as this line is relatively deep and broadened mainly by stellar rotation.", "The projected stellar rotation speeds of both stars are small (Table REF ).", "This means that the Mg ii line is well separated from the pressure-broadened He i line at 4471 Å, simplifying our analysis compared to Papers II–IV.", "In addition to the Mg ii line, an Al iii doublet and a S ii line are present in our spectral window from 4476 Å to 4486 Å.", "Thus we include these lines in our model.", "Figure REF shows a grayscale representation of all our observations in this wavelength range.", "[43] reported that the two members of the CV Vel system belong to the class of slowly pulsating B stars [42].", "Using CORALIE spectra from December 2001 and January 2002, they observed pulsations in both stars, with the pulsation amplitude for the primary being larger than for the secondary (see their Figure 5c).", "We reanalyzed their spectra and found that they had mislabeled the primary as the secondary, and vice versa.", "It is the secondary star which showed the larger pulsations in their CORALIE spectra.", "In addition, the values of $v \\sin i_\\star $ quoted by [43] were assigned to the wrong stars [44].", "In fact, their measurement of $v \\sin i_\\star = 31\\pm 2$ km s$^{-1}$ belongs to the primary, and their measurement of $v\\sin i_\\star =19\\pm 1$ km s$^{-1}$ belongs to the secondary.", "See Figure REF .", "Our observations took place about a decade later.", "We also observed large pulsations in the spectra of the secondary star (Figure REF ).", "Here we describe how we dealt with the pulsations while determining the projected obliquIties.", "The pulsation period is a few days.", "The out-of-eclipse observations spanned many months, averaging over many pulsation periods.", "Thus the pulsations likely introduce additional scatter into the derived orbital parameters, but probably do not introduce large systematic biases in the results.", "The situation is different for observations taken during eclipses.", "Over the relatively short timespan of an eclipse, the spectral-line deformation due to pulsation is nearly static or changes coherently, and can introduce biases in the parameters which are extracted from eclipse data.", "This is true not only for the parameters of the pulsating star, but also for the parameters of the companion star, since the light from both stars is modeled simultaneously.", "Given the S/N of our spectra, the pulsations of the primary star are too small to be a concern, but the pulsations of the secondary need to be taken into account.", "The effects of pulsations are most noticeable in the first two moments of the absorption lines: shifts in the wavelength, and changes in line width.", "We therefore decided to allow the first two moments of the secondary lines to vary freely for each observation obtained during a primary or secondary eclipse.", "Each time a trial model is compared to the data the position and width of the lines are adjusted.", "This scheme is similar to the scheme for the normalization, but now focusing on the lines of the secondary measured during eclipses.", "The average shift in velocity is about 2 km s$^{-1}$ and never larger than 3 km s$^{-1}$ .", "The changes in width are always smaller then 2%.", "Table: Results for the CV Vel system.Along with the spectroscopic data, we fitted the photometric data described in Section .", "Because the eclipses last nearly 12 hr the data was obtained during different nights and cover large ranges in airmass.", "We found that, even after performing differential photometry on several comparison stars, the measured flux exhibits significant trends with airmass.", "Therefore, for each nightly time series, we added two parameters describing a linear function of airmass which were optimized upon each calculation of $\\chi ^2$ .", "As mentioned in Section we also fitted the Strömgren uvby photometry from [11].", "To constrain the quadratic limb darkening parameters $u_{1, \\rm i}$ and $u_{2, \\rm i}$ for the relevant bandpasses, we used the 'jktld'http://www.astro.keele.ac.uk/jkt/codes/jktld.html tool to query the predictions of ATLAS atmosphere models [8].", "We queried the models for the spectroscopic region (around 4500 Å), the 'Ic' band, and the Strömgren uvby observations by [11].", "We placed a Gaussian prior on $u_1 + u_2$ with a width of $0.1$ and held the difference $u_1 - u_2$ fixed at the tabulated value.", "As the two known members of the CV Vel system are of similar spectral type we used the same values for $\\zeta $ , $u_1$ , $u_2$ , and the line strengths for both components.", "Similar to the other groups who studied this system, we do not find any sign of an eccentric orbit during our initial trials.", "We therefore decided to set $e\\equiv 0$ , in agreement with the results by [11], who had gathered the most complete eclipse photometry of the system to date.", "We found no sign of a systemic drift in $\\gamma $ over the three years of observations, and therefore we did not include a linear drift term in our model.", "However, this does not translate into a stringent constraint on the presence of a potential third body, because most of our observations took place in 2010/2011.", "As described above, we used the line list from VALD in our model.", "To derive results which can be compared with earlier works, we also ran our model using the rest wavelengths given by [31].", "For the obliquity work we prefer the VALD line list, as it allows us to treat the Mg ii as doublet, which is important because of the relatively slow rotation in CV Vel (Figure REF ).", "Table REF presents the $\\gamma $ values from both runs.", "[43] used different wavelengths for Mg ii which lead to a different values of $\\gamma _{\\rm p}$ and $\\gamma _{\\rm s}$ .", "Adjusting for the difference in the wavelength position we find that the results by [43] are also consistent with the results by [5].", "Figure: Anomalous RVs duringprimary eclipses of CV Vel.", "Left.—Apparent RVof the primary, after subtraction of the best-fitting orbitalmodel.", "The solid line is the best-fitting model for the RM effect.The dashed line indicates the expected signal for a well-aligned system.Right.—Same, but neglecting any correction for pulsationsof the secondary.Figure: Orbit of CV Vel Top.—Apparent RVs for the primary and secondary stars inthe CV Vel system.", "This plot is similar toFigure but now for all RVs taken2009-2013.", "These RVs are shown for illustrative purposes onlyand are not used in the analysis of the system.Bottom.— RVs minus best fitting model from the fit tothe line shapes.", "The secondary star exhibits, due to its largerpulsations (Figure ), a larger RV scatterthan the primary star." ], [ "Results", "The results for the model parameters are given in Table REF .", "Figure REF shows a grayscale representation of the primary spectra in the vicinity of the Mg ii line during the eclipse.", "Figures REF shows the same for the spectra obtained during secondary eclipses.", "Figure REF presents a subset of primary eclipse observations in a more traditional way.", "Concerning the orbital parameters, we find results that are consistent with earlier works.", "The uncertainties in the fractional stellar radii are small, with significant leverage coming from the spectroscopic eclipse data.", "Since we have not fully explored how the pulsations in the absorption lines influence our results for the scaled radii, for the purpose of calculating absolute radii we have taken the conservative approach of using the previously determined fractional radii, which have larger uncertainties from [11].", "We have not tested how the exact timing of the observations in combination with the pulsations might influence the results for the velocity semi-amplitudes and suggest that $0.1$ km s$^{-1}$ is a more realistic uncertainty interval for $K_{\\rm i}$ then the statistical uncertainty of $0.035$ km s$^{-1}$ .", "We use the enlarged uncertainties in calculating the absolute dimensions of the system.", "The results for the macroturbulent width $\\zeta $ and the microturbulent/PSF width are strongly correlated.", "The inferred breakdown between these types of broadening depends on our choice of limb darkening parameters.", "At this point we can only say that any additional broadening beyond rotation is about $8-9$ km s$^{-1}$ .", "Figure: Anomalous RVsduring primary eclipses 2001/2002 in the CV Vel system.Same as Figure , but for the data from2001/2002 obtained by .", "Note the larger scaleof the yy-axis compared to the two panels inFigure .", "This is because the vsini ☆ v \\sin i_\\star of the primary was higher during 2001/2002." ], [ "Projected obliquities and projected rotation speeds", "We find that the sky projection of the primary rotation axis is misaligned against the orbital angular momentum, with $\\beta _{\\rm p}=-52.0\\pm 0.7^{\\circ }$ .", "The projection of the secondary axis appears to be aligned ($\\beta _{\\rm s}=3.7\\pm 1.4^{\\circ }$ ).", "Our method of correcting for pulsations of the secondary turned out to be important, but even with no such corrections the result of a misaligned primary is robust.", "This is illustrated in Figure REF , which shows the anomalous RVs during primary eclipse.", "To create this figure we subtracted our best-fitting model of the secondary spectrum from each of the observed spectra.", "We then measured the RV of the primary star at each epoch, by fitting a Gaussian function to the Mg ii line.", "We then isolated the RM effect by subtracting the orbital RV, taken from the best-fitting orbital model.", "The right panel in Figure REF shows the results for the case when no correction was made for pulsations.", "There is evidently scatter between the results from different nights, but the predominance of the blueshift throughout the transit implies a misaligned system (a formal fit gives $\\beta _{\\rm p}=-37^{\\circ }$ and $\\beta _{\\rm s}=-1^{\\circ }$ ).", "The left panel shows the results for the case in which we have corrected for the time variations in the first two moments of the secondary lines.", "The scatter is much reduced and the fit to the geometric model is much improved.", "Figure REF shows for completeness all RVs obtained.", "As mentioned above the RVs out of eclipse have not been corrected for the influence of pulsations.", "None of the RVs are used in the analysis they are shown here for comparison only.", "We further repeated our analysis on two additional lines, the Si iii line at $4552.6$ Å and the He I line at 6678 Å.", "For Si iii we obtain $\\beta _{\\rm p}=-58^{\\circ }$ and $\\beta _{\\rm s}=-4^{\\circ }$ , and for He I we measure $\\beta _{\\rm p}=-52^{\\circ }$ and $\\beta _{\\rm s}=-1^{\\circ }$ .", "The Si iii line is weaker than the Mg ii line and the He I is pressure broadened, which make the analysis more complex [4].", "We therefore prefer the result from the Mg ii line.", "However we judge that the total spread in the results $6^\\circ $ and $7^\\circ $ are probably closer to the true uncertainty in the projected obliquities, than our formal errors.", "This is because our formal uncertainty intervals rely on measurements taken during 3 and 4 nights, for the primary and secondary, respectively.", "For a better uncertainty estimation measurements obtained during more different nights, or a more carefully handling of the pulsations, would be needed.", "Table: Projected rotation speed measurements in the in CV Velsystem.For the projected rotation speeds we find $v \\sin i_{\\rm p}=21.5\\pm 0.3$ km s$^{-1}$ and $v \\sin i_{\\rm s}=21.1\\pm 0.2$ km s$^{-1}$ .", "Making the same measurement in the Si iii lines one would obtain $v \\sin i_{\\rm p}=20.6$ km s$^{-1}$ and $v \\sin i_{\\rm s}= 20.0$ km s$^{-1}$ .", "For similar reasons as mentioned above for the projected obliquity we suspect that also the formal uncertainties for $v \\sin i_\\star $ are underestimated.", "In what follows we assume that an uncertainty of 2 km s$^{-1}$ is appropriate.", "[43] obtained 4 of their 30 observations during primary eclipses.", "We performed a similar analysis of their spectra, in the same manner as our own data.", "For the projected obliquity of the primary star in 2001/2002 we obtained $\\beta _{\\rm p\\,2002}=-55\\pm 3^{\\circ }$ .", "As this result rests mainly on two observations, obtained nearly at the same eclipse phase (Figure REF ), we judge that the true uncertainty is much larger, probably about $15^\\circ $ .", "For the projected rotation speeds we obtained $v \\sin i_{\\rm p\\,2002}=29.5\\pm 2$ km s$^{-1}$ and $v \\sin i_{\\rm s\\,2002}=19.0\\pm 2$ km s$^{-1}$ , adopting a conservative uncertainty interval as we did for our spectra.", "Figure: Precession of the stellarrotation axes in the CV Vel system.", "The upper panel shows themeasured vsini ☆ v \\sin i_\\star values of the primary (red solidsymbol) and secondary (blue open symbol) stars.", "The linesindicate the time evolution in our best-fittingmodel.", "The lower panel shows the measurements of β\\beta and thepredictions of our model." ], [ "Precession of the Rotation Axes", "Our results for $v \\sin i_{\\star }$ differ from the values found by [5] and from the values found by [43] (see Table REF ).", "Evidently the projected rotation rates are changing on a timescale of decades.", "We are only aware of a few cases in which such changes have been definitively observed, one being the DI Her system [33], [1], [32].", "[43] used the CORALIE spectrograph on the $1.2$ m Swiss telescope for their observations.", "To exclude any systematic effects due to the choice of instrument—as unlikely as it might seem—we also collected a number of spectra with the CORALIE spectrograph, as described in Section , which confirmed the time variation of the $v\\sin i_{\\star }$ of the primary.", "The line width of the secondary appears to have changed between the observations conducted by [5] and [43].", "Assuming that $v$ remained constant over the interval of observations ($\\sim 30$ yr), we interpret these results as variations in $\\sin i_{\\star }$ for both stars.", "This allows us to learn about the precession rates of the stellar rotation axes around the total angular momentum vector of the system.", "Employing the formulas from [33] we can use the $v \\sin i_{\\star }$ values from Table REF together with our measurements of the projected obliquity to obtain values for the stellar obliquities ($\\psi $ ) and rotation velocities of the two stars.", "For this purpose, in addition to the system parameters of CV Vel which are presented in Table REF , we need values for the apsidal motion constant ($k_2$ ) and the radius of gyration ($\\theta $ ) of each star.", "These we obtain from the tables presented by [9].", "We use the model with a mass of $6.3$ M$_{\\odot }$ , close to the mass of the stars in the CV Vel system, and estimate the uncertainty by considering the age interval from $30-50$ Myr.", "The age of CV Vel is estimated to be 40 Myr [43].", "The results for both stars are $k_2=0.005\\pm 0.002$ and $\\theta =0.044\\pm 0.012$ .", "Using these values we carried out a Monte Carlo experiment, in which we draw system parameters by taking the best-fitting values and adding random Gaussian perturbations with a standard deviation equal to the 1$\\sigma $ uncertainty.", "For each draw, we minimize a $\\chi ^2$ function by adjusting $\\psi $ and $v$ for each star, as well as the particular times when the spin and orbital axes are aligned on the sky.", "Furthermore we allow the $k_2$ and $\\theta $ values to vary with a penalty function given by the prior information mentioned above.", "The resulting parameters are presented in Table REF .", "In Figure REF we show the data for $v \\sin i_{\\star }$ and $\\beta $ , as well as our model for their time evolution.", "We obtain $\\psi _{\\rm p}= 67\\pm 4^{\\circ }$ and $\\psi _{\\rm s}= 46\\pm 9^{\\circ }$ for the obliquities, and $v_{\\rm p}=35\\pm 5$ km s$^{-1}$ and $v_{\\rm s}= 28\\pm 4$ km s$^{-1}$ for the rotation speeds.", "The formal uncertainties for $\\psi $ and $v$ should be taken with a grain of salt.", "We did not observe even half a precession period, which makes an estimation of $\\psi $ and $v$ strongly dependent on our assumptions regarding $k_2$ and $\\theta $ .", "We have only a small number of measurements: 3 $v \\sin i_{\\star }$ and one or two $\\beta $ measurement per star, amounting to 9 data points.", "With these we aim to constrain 6 parameters: $v$ , $\\psi $ , and a reference time for each star.", "In this situation we can determine parameter values, but we cannot critically test our underlying assumptions.", "For the secondary in particular we have only little information to constrain $\\psi $ and $v$ .", "The only indication we have for this star that it is not aligned is the change in $v \\sin i_{\\star }$ between 1973 and 2001/2002.", "Clearly, future observations would be helpful to confirm the time variations.", "Measurements of the projected obliquity in only a few years should be able to establish if this star's axis is indeed misaligned (Figure REF ).", "Finally we obtain somewhat different values for $k_2$ and $\\theta $ for the two stars, which have similar masses and the same age.", "This is because for the primary the fast change in $v \\sin i_\\star $ between 2002 and 2010 requires a fast precession timescale.", "Table: Precession of the stellar axes in CV VelThe last point could reflect a shortcoming of our simple model (some missing physics), an underestimation of the errors in the $v \\sin i_\\star $ measurements or the presence of a third body.", "Nevertheless the finding of a large projected misalignment for the primary and the changes in $v \\sin i_{\\star }$ measured for both stars makes it difficult to escape the conclusion that the stars have a large obliquity and precess, even if the precise values are difficult to determine at this point.", "A more detailed precession model and more data on $\\beta $ and $v \\sin i_{\\star }$ , obtained over the next few years, would help in drawing a more complete picture.", "We note that in principle, one can also use the effect of gravity darkening on the eclipse profiles to constrain $\\psi $ , as was done recently by [37], [7] and [32] for the KOI-13 and DI Her systems.", "However as the rotation speed in CV Vel is a factor few slower then in these two systems this would require very precise photometric data.", "We also note that small changes in the orbital inclination of CV Vel are expected, as another consequence of precession.", "This might be detected with precise photometry obtained over many years." ], [ "Time evolution of the spins", "With an age of 40 Myr [43] CV Vel is an order of magnitude older than the even-more misaligned system DI Her ($4.5\\pm 2.5$ Myr, $\\beta _p=72\\pm 4^{\\circ }$ $\\beta _s=-84\\pm 8^{\\circ }$ , [1], [10]).", "In this section we investigate if CV Vel could have evolved from a DI Her-like configuration, through the steady action of tidal dissipation.", "If so, CV Vel might represent a link between young systems with large misalignment, and older systems where tidal interactions have had enough time to attain the equilibrium condition of a circular orbit with aligned and synchronized spins.", "In Paper IV, we found that the EP Cru system (age $57\\pm 5$ Myr) could not have evolved out of a DI Her like system, despite the strong similarities of all the system parameters except the stellar obliquity and age.", "This is because the $v \\sin i_\\star $ values in EP Cru are about 9 times the expected value for the pseudosynchronized state.", "Theories of tidal interactions predict that damping of any significant spin-orbit misalignment should occur on a similar same time scale as synchronization of the rotation [21], [13].", "This is because in these tidal models, a single coefficient describes the coupling between tides and rotation.", "If the stellar rotation frequency is much larger than the synchronized value, then rotation around any axis is damped at about the same rate.", "[25] recently suggested that, for the case of stars with an connective envelope – stars of much lower mass then the stars we study here – dynamical tides can damp different components of the stellar spin on very different timescales.", "Therefore while the rotation speed is reduced, the angle between the overall angular momentum and stellar rotation axis does not change.", "When the stellar rotation around the axis parallel to the orbital angular momentum approaches the synchronized value, then rotation around this axis becomes weakly coupled to the orbit.", "Tidal damping of rotation around any other axis will only cease when the rotation around these axes stops, and the stellar spin is aligned with the orbital axis.", "Therefore, finding a system in an aligned state that is rotating significantly faster than synchronized rotation indicates, according to these tidal theories, that the alignment was primordial.", "In Paper III, we found that NY Cep is also inconsistent with having evolved from a state with large misalignment.", "Figure: Tidal evolution ofCV Vel.", "The red and blue crosses mark the derived rotationspeeds and obliquities of the primary and secondary.", "Red andblue lines show the theoretical obliquity evolution of a systemlike CV Vel.", "Here we used the currently measured values forCV Vel and evolved the system back in time.", "The model includesthe evolutionary changes in stellar radius with time, and adoptsa viscous timescale (t V t_{\\rm V}) of 300 000 yr, about 6 000times larger than what is normally assumed for late typestars.", "A lower value of t V t_{\\rm V} would lead to an overallfaster tidal evolution.", "It will leave the ratio of the alignmentand synchronization timescales unchanged.", "According to thesesimulations and measurements it is conceivable that CV Vel hadlarger (DI Her-like) misalignments when it was younger, and iscurrently undergoing tidal realignment.", "The vertical lineindicates the synchronized rotation speed (V syn V_{\\rm syn}) forthe current orbital configuration and stellar radii ofCV Vel.", "We also show the measured vsini ☆ v \\sin i_\\star andβ ☆ \\beta _\\star for DI Her, NY Cep, and EP Cru, three othersystems from the BANANA survey.", "While the exact value ofV syn V_{\\rm syn} for these systems differ from the value forCV Vel, all these systems do rotate significantly faster thantheir synchronized or pseudosynchronized values.For CV Vel, synchronized rotation would correspond to $v\\approx 30$ km s$^{-1}$ for both stars.", "The slow rotation speeds and misaligned axes suggest that we are observing this system in a state in which tides are currently aligning the axes.", "To illustrate we use the TOPPLE tidal-evolution code [13] with the parameters from Table REF and Table REF and evolve the system backwards in time.", "The results are shown in Figure REF .", "It appears as if the current rotational state of the two stars is consistent with an evolution out of a higher-obliquity state.", "We reiterate, though, that the rotation speed and obliquity of the lower mass star are rather uncertain.", "The results of the tidal evolution do depend on the exact parameters we use for CV Vel, taken from the confidence intervals of our measurements.", "Therefore we can not make strong statements about the exact evolution CV Vel has taken.", "However the qualitative character of the evolution did remain the same in all of our runs.", "CV Vel did evolve out of a state with larger obliquities and faster rotation.", "Under this scenario, we are seeing the system after only about one obliquity-damping timescale, which implies that only a small fraction of an eccentricity damping timescale has elapsed (due to the angular momentum in the orbit being greater than that in the spins).", "This appears to be a counterintuitive result as one would expect that whatever creates high obliquities would also create a high eccentricity, which should still be present, according to our simple simulation.", "Scenarios involving a third body, may account for the misaligned spins despite tidal damping of the eccentricity.", "For example [13] showed that in the triple system SS Lac, with inner and outer orbits non-parallel, the spin orientations of the two inner components could vary on a timescale of just several hundred years.", "As long as the inner and outer orbits remain non-coplanar, the inner orbit will precess around the total angular momentum.", "The orbital precession timescale will most likely be not the same as the precession timescale of the two stars.", "Therefore the angle between the stellar spins and the orbital plane of the inner orbit can remain large even after many obliquity damping timescales.", "The system would settle into a Cassini state, with the oblique spins precessing at the same rate as the inner orbit.", "A pseudo-synchronous spin rate would settle in for the oblique yet circular orbits [27], [14].", "Of course such a scenario remains speculative as long as no third body is searched for and detected.", "The state of the obliquities suggests that DI Her and CV Vel have a history which is qualitatively different from the history of NY Cep, and EP Cru.", "The two later systems had good alignment throughout their main sequence lifetime, while DI Her and CV Vel did at some point acquire a larger misalignment." ], [ "Summary", "We have analyzed spectra and photometry of the CV Vel system, obtained during primary and secondary eclipses as well as outside of eclipses.", "Taking advantage of the Rossiter-McLaughlin effect, we find that the rotation axis of the primary star is tilted by $-52\\pm 6^{\\circ }$ against the orbital angular momentum, as seen on the sky.", "The sky projections of the secondary rotation axis and the orbital axis are well aligned ($3\\pm 7^{\\circ }$ ).", "Furthermore we find that the projected rotation speeds ($v \\sin i_\\star $ ) of both stars are changing on a timescale of decades.", "We interpret these changes as a sign of precession of the stellar rotation axes around the total angular momentum of the system.", "Using the $v \\sin i_\\star $ measurements (ours and literature measurements dating 30 years back) in combination with our projected obliquity measurements, we calculate the rotation speed ($v$ ) as well as the true obliquity ($\\psi $ ) of both stars.", "We find obliquities of $\\psi _{\\rm p}= 64\\pm 4^{\\circ }$ and $\\psi _{\\rm s}= 46\\pm 9^{\\circ }$ and rotation speeds of $v_{\\rm p}=33\\pm 4$ km s$^{-1}$ and $v_{\\rm s}= 28\\pm 4$ km s$^{-1}$ for the two stars.", "While the results for the primary star are relatively solid, the results for the secondary star rely on changes in the measured line width only, and need to be confirmed with future spectroscopic observations.", "Our results for the stellar rotation in CV Vel are consistent with long-term tidal evolution from a state in which the stars had higher rotation speeds as well as higher obliquities, similar to what we found in the younger binary system DI Her.", "In this sense it seems plausible that DI Her and CV Vel are two points on an evolutionary sequence from misaligned to aligned systems.", "Given the simplest tidal theories, the other systems in our sample (NY Cep, and EP Cru) could not have realigned via tides.", "So far it is not clear what causes the difference between these two groups.", "Given recent findings that close binaries are often accompanied by a third body, it is tempting to hypothesize that the influence of a third body is the key factor that is associated with a large misalignment.", "No third body has yet been detected in either the CV Vel nor DI Her systems, nor have these systems been thoroughly searched.", "[23] found a possible pattern in the eclipse timing of DI Her, indicating a third body.", "However [10] found no evidence for a third body, employing a dataset which includes the timings from [23].", "Such a search should be a priority for future work.", "We would like to thank the anonymous referee for timely suggestions, which improved the manuscript.", "We thank Kadri Yakut and Conny Aerts for providing us with a digital version of the [11] photometry, as well as their own CORALIE spectra and comments on the manuscript.", "S.A. acknowledges support during part of this project by a Rubicon fellowship from the Netherlands Organisation for Scientific Research (NWO).", "Work by S.A. and J.N.W.", "was supported by NASA Origins award NNX09AB33G and NSF grant no.", "1108595.", "TRAPPIST is a project funded by the Belgian Fund for Scientific Research (FNRS) with the participation of the Swiss National Science Fundation (SNF).", "MG and EJ are FNRS Research Associates.", "A. H.M.J.", "Triaud received funding from of a fellowship provided by the Swiss National Science Foundation under grant number PBGEP2-14559.", "This research has made use of the following web resources: simbad.u-strasbg.fr, adswww.harvard.edu,arxiv.org, http://arxiv.org" ] ]	1403.0583
[ [ "Enumerating Transformation Semigroups" ], [ "Abstract We describe general methods for enumerating subsemigroups of finite semigroups and techniques to improve the algorithmic efficiency of the calculations.", "As a particular application we use our algorithms to enumerate all transformation semigroups up to degree 4.", "Classification of these semigroups up to conjugacy, isomorphism and anti-isomorphism, by size and rank, provides a solid base for further investigations of transformation semigroups." ], [ "Introduction", "When studying finite structures it can be helpful to generate small examples using a computer [9], [24], [16], [15], [2], [18].", "By investigating these sample objects we can discover patterns, formulate new hypotheses, look for counterexamples, and so on.", "More diverse sample sets make it easier to greater evidence for or against a conjecture.", "Therefore, to maximize the usefulness of these small examples we naturally aim both to enumerate all objects of a certain parameter value (such as size or dimension), and to increase the value of this parameter.", "So far, efforts to enumerate semigroups have focused on the abstract case, enumerating by order (the size of the semigroup).", "The basic idea for enumerating by size is to find all valid multiplication tables (up to isomorphism and anti-isomorphism) of the given size [3], [10], [21], [22], [25], [26], [28], [29], [4].", "Our approach here is to enumerate not by size but, rather, by degree.", "Recall that Cayley's Theorem (for semigroups) says that any finite semigroup $S$ is isomorphic to a semigroup of functions on a finite set; the degree of $S$ is defined to be the minimal size of such a set.", "So we aim to enumerate the valid subtables inside the multiplication table of the full transformation semigroup ${\\mathcal {T}}_n$ , which consists of all transformations of the set $\\lbrace 1,\\ldots ,n\\rbrace $ (i.e., all functions from this set to itself); more specifically, we wish to find all such subtables (up to isomorphism and anti-isomorphism) that are not also (isomorphic to) subtables of ${\\mathcal {T}}_{n-1}$ .", "All finite semigroups will eventually be listed when we enumerate by size or by degree, but the order of the list is different from one method to the other.", "For example, there are 52,989,400,714,478 abstract semigroups of order 9 [3], [4], so one could barely imagine the number of semigroups of order 27, where ${\\mathcal {T}}_3$ would first appear when enumerating by size.", "On the other hand, our results show that there are only 25 different transformation semigroups on 3 points of order 9.", "Metaphorically speaking, enumeration by size and by degree go in completely different directions and proceed at a different rate.", "The complexity of the multiplication of two transformations from ${\\mathcal {T}}_n$ is linear in $n$ .", "However, multiplication in a semigroup defined by a multiplication table has constant complexity.", "Hence, we choose multiplication tables as the main way of representing semigroups.", "This decision has two consequences.", "First, our algorithms fall into the class of semigroup algorithms that fully enumerate the elements.", "This of course restricts us to relatively small semigroups.", "On the other hand, multiplication table algorithms are completely representation independent, so our methods are widely applicable across different types of finite semigroups.", "The article is organised as follows.", "In Section we describe a basic multiplication table method to calculate subsemigroups generated by a subset.", "In Section we present generic search algorithms to enumerate subsemigroups of finite semigroups.", "In Section we discuss techniques for improving the efficiency of the algorithms in Section , based on more specific algebraic results.", "Finally, in Section we apply the developed methods for enumerating transformation semigroups acting on up to 4 points.", "For the computational enumeration we used the Gap computer algebra system [12] and its Semigroups package [23] and developed a new package SubSemi for subsemigroup enumeration [7].", "Instructions for recomputing the results can be found in the package documentation.", "We now take a moment to establish some notation." ], [ "Notation", "Let $S$ be a finite semigroup with $\|S\|=n\\in \\mathbb {N}$ .", "We fix an order on the semigroup elements and for convenience denote them simply by $1,\\ldots ,n$ .", "The multiplication table, or Cayley table of $S$ is an $n\\times n$ matrix $M_S$ with entries from $\\lbrace 1,\\ldots ,n\\rbrace $ , such that $M_{i,j}=k$ if $ij=k$ in $S$ (we denote multiplication in $S$ by juxtaposition).", "The subarray of $M_S$ with rows and columns indexed by a subset $A\\subseteq \\lbrace 1,\\ldots ,n\\rbrace $ is denoted by $M_A$ .", "We denote the set of the entries of a vector $v$ by $\\mathbf {Set}(v)=\\lbrace x\\mid x\\text{ isan entry of } v\\rbrace $ , and similarly for multiplication tables or subarrays, so $\\mathbf {Set}(M_A)=\\lbrace x\\mid x\\text{ is an entry of } M_A\\rbrace $ .", "In particular, $A$ is a subsemigroup of $S$ (denoted $A\\le S$ ) if and only if $\\mathbf {Set}(M_A)\\subseteq A$ .", "The set of all subsemigroups of $S$ is denoted by $\\mathbf {Sub}(S)=\\big \\lbrace T\\mid T\\le S\\big \\rbrace $ .", "We consider the empty set a semigroup, so $\\varnothing \\in \\mathbf {Sub}(S)$ .", "The set of maximal proper subsemigroups of $S$ is denoted by $\\mathbf {Max}(S)$ .", "For $A\\subseteq S$ , $\\langle A\\rangle $ denotes the least subsemigroup of $S$ containing $A$ , the semigroup generated by $A$ .", "If $I$ is an ideal of $S$ then the Rees factor semigroup $S/I$ has elements $(S\\setminus I)\\cup \\lbrace 0\\rbrace $ , where 0 is a new symbol that does not belong to $S$ , and with multiplication $\\cdot $ defined by $s\\cdot t = {\\left\\lbrace \\begin{array}{ll}st &\\text{if $s,t,st\\in S\\setminus I$}\\\\0 &\\text{otherwise.}\\end{array}\\right.", "}$ Note that if $I=\\varnothing $ is the empty ideal, then $S/I\\cong S^0$ is the semigroup obtained from $S$ by adjoining a zero.", "A transformation $t\\in {\\mathcal {T}}_n$ will often be written by simply listing the images of the points, $t=[1t,\\ldots ,nt]$ .", "The group consisting of all permutations of $\\lbrace 1,\\ldots ,n\\rbrace $ is the symmetric group, denoted ${\\mathcal {S}}_n$ , and is the group of units of ${\\mathcal {T}}_n$ ." ], [ "The closure algorithm", "A basic question in computational semigroup theory is: Given a subset $A$ of a semigroup $S$ , what is the subsemigroup $\\langle A\\rangle $ generated by $A$ ?", "We will also write $\\mathbf {Cl}(A)$ for $\\langle A\\rangle $ , and refer to it as the closure of $A$ (in $S$ ).", "Note that on the level of the Cayley table, $\\langle A\\rangle $ is the minimal set $B$ (in the containment order on subsets of $S$ ) such that the subarray $M_B$ contains $M_A$ and all entries from $M_B$ belong to $B$ .", "An algorithm for obtaining $B=\\mathbf {Cl}(A)$ is as follows.", "First we determine the set $\\mathbf {m}(A)=\\mathbf {Set}(M_A)\\setminus A$ of products of elements of $A$ that are missing from $A$ .", "We then recursively define $\\mathbf {Cl}_1(A)=A\\cup \\mathbf {m}(A),\\qquad \\mathbf {Cl}_{i+1}(A)=\\mathbf {Cl}_1(\\mathbf {Cl}_{i}(A))\\quad \\text{for $i\\ge 1$.", "}$ Note that $\\mathbf {Cl}(A)=\\mathbf {Cl}_j(A)$ , where $j$ is minimal such that $\\mathbf {Cl}_j(A)=\\mathbf {Cl}_{j+1}(A)$ , or equivalently $\\mathbf {m}(\\mathbf {Cl}_j(A))=\\varnothing $ .", "For example, consider the symmetric group ${\\mathcal {S}}_3$ , with its elements ordered $1=()$ , $2=(2,3)$ , $3=(1,2)$ , $4=(1,2,3)$ , $5=(1,3,2)$ , $6=(1,3)$ .", "The subarray $M_{\\lbrace 4\\rbrace }$ contains only one entry, 5, which is different from 4, so the subarray is not closed.", "With the above notation, $\\mathbf {m}(\\lbrace 4\\rbrace )=\\lbrace 5\\rbrace $ , $\\mathbf {m}(\\lbrace 4,5\\rbrace )=\\lbrace 1\\rbrace $ and $\\mathbf {m}(\\lbrace 1,4,5\\rbrace )=\\varnothing $ , so $\\mathbf {Cl}_1(\\lbrace 4\\rbrace )=\\lbrace 4,5\\rbrace $ and $\\mathbf {Cl}_2(\\lbrace 4\\rbrace )=\\mathbf {Cl}_3(\\lbrace 4\\rbrace )=\\mathbf {Cl}(\\lbrace 4\\rbrace )=\\lbrace 1,4,5\\rbrace $ , corresponding to the unique subgroup of order 3 in ${\\mathcal {S}}_3$ .", "See Fig.", "REF .", "Figure: The Cayley table of the symmetric group 𝒮 3 {\\mathcal {S}}_3 (left), and calculation of 𝐂𝐥({4})={1,4,5}\\mathbf {Cl}(\\lbrace 4\\rbrace )=\\lbrace 1,4,5\\rbrace (right).", "See text for further explanation.The above recursive definition describes an algorithm for calculating the closure, but not an efficient one.", "We can avoid the full calculation of the missing elements in the recursive steps.", "When extending the subarray $M_A$ by a single element $i$ , if $\\mathbf {m}(A)$ is already calculated, then all new missing elements in $\\mathbf {m}(A\\cup \\lbrace i\\rbrace )$ can only come from the $i$ th row or the $i$ th column.", "So, to calculate the closure we can extend the subarray one-by-one using the elements of $\\mathbf {m}(A)$ and any new missing elements encountered during the recursion.", "This way each table entry is checked only once." ], [ "Basic Search Algorithms for Subsemigroup Enumeration", "One of our primary goals is to enumerate the semigroups of a given degree, and this involves enumerating the subsemigroups of ${\\mathcal {T}}_n$ .", "This is of course a special instance of the following more general problem.", "Problem 3.1 For a semigroup $S$ , find all of its subsemigroups: $\\mathbf {Sub}(S)=\\left\\lbrace T\\mid T\\le S\\right\\rbrace .$ In this section, we discuss a number of algorithmic approaches to this problem.", "Thinking in terms of the multiplication table $M_S$ , we are looking for all subarrays $M_A$ that are also multiplication tables; i.e., they do not contain elements not in $A$ : $\\mathbf {Set}(M_A)\\subseteq A$ ." ], [ "Enumerating by Brute Force", "The obvious brute-force algorithm for constructing $\\mathbf {Sub}(S)$ proceeds by first constructing the powerset $2^S=\\left\\lbrace A\\mid A\\subseteq S\\right\\rbrace $ and then checking each subset for closure.", "For some semigroups, any method essentially reduces to the brute-force algorithm (e.g., left or right zero semigroups, where every subset is a subsemigroup), but it is inefficient in cases where only a fraction of the subsets are closed under multiplication." ], [ "Enumerating by Minimal Generating Sets", "The rank of a semigroup is the least size of a generating set.", "The rank of a subsemigroup can be bigger than the rank of the semigroup itself.", "For example, the full transformation semigroup has rank 3 [11], but its minimal ideal (which is a left zero semigroup) has rank $n$ , while its maximal proper ideal (the semigroup of all singular transformations) has rank ${n\\atopwithdelims ()2}=n(n-1)/2$ for $n\\ge 3$ [13], [19].", "Assuming that we know the maximum rank for subsemigroups of $S$ , we can check all subsets of $S$ with cardinality up to that value to see what subsemigroups they generate.", "The same subsemigroup may be generated by many generating sets but the maximality guarantees that we construct all of $\\mathbf {Sub}(S)$ .", "On each level $k$ , we check $\\binom{\|S\|}{k}$ many generating sets.", "Therefore, the method is only feasible if the maximum value of the ranks of the subsemigroups is known to be small.", "To the knowledge of the authors, the maximal rank of a subsemigroup of ${\\mathcal {T}}_n$ is not currently known, though the maximal rank of an ideal is equal to the largest of the Stirling numbers $S(n,2),\\ldots ,S(n,n-1)$ [20].", "What can we do if we do not know the maximum rank value?", "We can keep going until no new subsemigroup is generated.", "First we check all subsemigroups generated by one element, then those generated by two; we subtract the former set from the latter to obtain the set of rank-2 subsemigroups.", "We then continue up to $m$ where the set of rank-$m$ subsemigroups is empty.", "Unfortunately this last step is wasted, unless rank$(S)=S$ , e.g.", "left zero semigroups, in which case, this is just the brute-force search." ], [ "Enumerating by Minimal Extensions", "A minimal extension of a subsemigroup $T\\le S$ is a subsemigroup $\\langle T\\cup \\lbrace u\\rbrace \\rangle $ , where $u\\in S\\setminus T$ .", "We simply add a new element to $T$ and calculate the closure.", "If we recursively calculate minimal extensions, then we obtain all subsemigroups of $S$ containing $T$ .", "So this represents a solution to a natural generalisation of Problem REF , that of calculating the interval $[T,S]=\\left\\lbrace U\\mid T\\le U\\le S\\right\\rbrace $ in the lattice $\\mathbf {Sub}(S)$ .", "This algorithm is a graph search.", "The nodes are the subsemigroups containing $T$ , and there is a directed edge labelled $u$ from $V$ to $V^{\\prime }$ if $V^{\\prime }=\\langle V\\cup \\lbrace u\\rbrace \\rangle $ .", "In general, there may be many incoming edges to a subsemigroup.", "The efficiency of the algorithm comes from the fact that the search tree is cut when the search encounters a subsemigroup already known, simply by making no further extensions.", "The details are provided in Algorithm REF .", "Depending on how exts, the storage for extensions, behaves under the Store/Retrieve operations we get different search strategies.", "If exts is a stack, then the algorithm performs a depth-first search, and if it is a queue, then a breadth-first search is performed.", "A full subsemigroup enumeration can be done by starting the algorithm with parameters $T=\\varnothing $ , the semigroup is simply $S$ , and $X=S$ .", "This is simply extending the empty set by all elements of $S$ recursively.", "When using the breadth-first search strategy, the generating set is minimal, so Algorithm REF can easily be modified to enumerate minimal generating sets.", "A little consideration shows that this is a more efficient version of the minimal generating sets algorithm (Section REF ), but it does not escape checking generating sets one bigger than the maximal rank.", "[ht] Inputinput Outputoutput subssubs gensgens extsexts StoreStore RetrieveRetrieve MinimalExtensionsSubSemigroupsByMinimalExtensions all $T^{\\prime }\\subseteq S$ such that $T^{\\prime }=\\langle T\\cup Y\\rangle $ for some $Y\\subseteq X$ Name($T$ ,$S$ ,$X$ ) $\\leftarrow \\lbrace T\\rbrace $ $\\leftarrow \\varnothing $ $s\\in $ $(S\\setminus T)\\cap X$ (, $T\\cup \\lbrace s\\rbrace $ ) $\|\|>0$ $T^{\\prime }\\leftarrow \\langle ()\\rangle $ $T^{\\prime }\\notin $ $\\leftarrow $ $\\cup \\ \\lbrace T^{\\prime }\\rbrace $ $s\\in $ $(S\\setminus T^{\\prime })\\cap X$ (, $T^{\\prime }\\cup \\lbrace s\\rbrace $ ) Finding subsemigroups by minimal extensions." ], [ "Enumerating by Maximal Subsemigroups", "Assuming that we have calculated the maximal subsemigroups of $S$ , we can parallelize subsemigroup enumeration by enumerating subsemigroups of the maximal subsemigroups and merging the results, using the obvious fact that $\\mathbf {Sub}(S)=\\lbrace S\\rbrace \\cup \\bigcup _{T\\in \\mathbf {Max}(S)}\\mathbf {Sub}(T).$ A description is given in [14] of the possible maximal subsemigroups of an arbitrary finite semigroup $S$ , i.e.", "a maximal subsemigroup of $S$ is one of the possibilities described in [14].", "Algorithms, based on these properties, are being implemented in the Semigroups package [23], [6].", "The sets of subsemigroups of the maximal subsemigroups do overlap in general, so the same subsemigroup may get enumerated many times, and merging is a nontrivial step.", "Note that recursively iterating the maximal subsemigroups is a variant of the depth-first search algorithm, moving from large subsemigroups to small in contrast to the Minimal Extensions method discussed in Section REF ." ], [ "Advanced Algorithmic Techniques", "Since we are dealing with well-studied algebraic structures, there are many mathematical results we may exploit in order to improve the efficiency of any basic subsemigroup enumeration algorithm.", "In this section, we outline a number of improvements on the algorithms described in the previous section, the most powerful of which involves using an ideal to parallelize the enumeration (see Sections REF and REF )." ], [ "Equivalent Generators", "We define an equivalence relation $\\equiv $ on $S$ by $ s\\equiv t \\Longleftrightarrow \\langle s \\rangle = \\langle t \\rangle .$ For instance, $[ 2, 3, 1, 1 ]\\equiv [ 3, 1, 2, 2 ]$ in ${\\mathcal {T}}_4$ , both transformations generating $\\big \\lbrace [ 2, 3, 1, 1 ], [ 3, 1, 2, 2 ], [ 1, 2, 3, 3 ]\\big \\rbrace $ .", "Note that if the $\\equiv $ -class of $s\\in S$ is nontrivial, then $\\langle s\\rangle $ is a cyclic group.", "Note also that if $s\\equiv t$ , then $\\langle U\\cup \\lbrace s\\rbrace \\rangle =\\langle U\\cup \\lbrace t\\rbrace \\rangle $ for any $U\\subseteq S$ , so when using the Minimal Extensions enumeration method (Section REF ), we only need to calculate extensions with respect to $\\equiv $ -class representatives." ], [ "Exploiting Symmetries", "If we have information about the symmetries $S$ might possess, then we can accelerate any subsemigroup enumeration algorithm.", "For example, if $S$ is a monoid with group of units $G$ , then $T^g=\\left\\lbrace g^{-1}tg \\mid t\\in T \\right\\rbrace $ is a subsemigroup of $S$ for any $T\\in \\mathbf {Sub}(S)$ and $g\\in G$ .", "More generally, if $\\theta $ is an automorphism of $S$ , then $T\\theta =\\left\\lbrace t\\theta \\mid t\\in T \\right\\rbrace \\in \\mathbf {Sub}(S)$ for all $T\\in \\mathbf {Sub}(S)$ .", "There is an algorithm for computing the automorphism group $\\mathbf {Aut}(S)$ of a finite semigroup $S$ [1].", "So, during subsemigroup enumeration, whenever a subsemigroup $T\\le S$ is found, we may quickly find the subsemigroups $T\\theta $ .", "If $H\\le \\mathbf {Aut}(S)$ is a group of automorphisms of $S$ , then we write $\\mathbf {Sub}_H(S)$ for a set of automorphism class representatives of the subsemigroups of $S$ ; so for any $T\\in \\mathbf {Sub}(S)$ , there is a unique subsemigroup $U\\in \\mathbf {Sub}_H(S)$ such that $T=U\\theta $ for some $\\theta \\in H$ .", "Moreover, when extending a subsemigroup $T$ by elements8 from $S\\setminus T$ one by one (Section REF ), we can cut the search tree further.", "By taking the normalizer of $T$ in $S$ , i.e.", "the stabilizer of $T$ under the conjugation, we know that the conjugacy classes of $T\\cup \\lbrace x\\rbrace $ will be of the form $T\\cup \\lbrace x_i\\rbrace $ , where $x_i$ is an element of the orbit of $x$ under conjugation, thus we only need to extend with one element of the orbit.", "Therefore, the search algorithm has to visit only the orbit representatives of $S\\setminus T$ under the normalizer." ], [ "Parallel Enumeration in Ideals and Rees Quotients", "In general, an ideal $I$ of $S$ divides a subsemigroup $T\\le S$ into two parts: a subsemigroup contained in the ideal, $L=T\\cap I$ , and a subset outside the ideal, $U=T\\cap (S\\setminus I)$ .", "We call $L$ and $U$ the lower and upper torso of $T$ with respect to $I$, respectively (see Fig.", "REF ).", "Note that $U$ or $L$ may be empty.", "Figure: If SS has an ideal II, then a subsemigroup T≤ST\\le S is partitioned into two parts by the ideal, the upper torso U=T∩(S∖I)U=T\\cap (S\\setminus I) and the lower torso L=T∩IL=T\\cap I.The upper torso $U$ need not be a subsemigroup of $T$ in general, but $U$ may be turned into a semigroup by adjoining a zero element in an obvious way, giving precisely the Rees quotient of $T$ by the ideal $T\\cap I$ .", "The next lemma shows that subsemigroup enumeration can be done in parallel in $I$ and $S/I$ , though a combination step is still required.", "Lemma 4.1 If $I$ is an ideal of $S$ , then $\\mathbf {Sub}(S)=\\big \\lbrace \\langle (U\\setminus \\lbrace 0\\rbrace )\\cup L \\rangle \\mid U\\in \\mathbf {Sub}(S/I),\\ L\\in \\mathbf {Sub}(I)\\big \\rbrace .$ Let $T\\in \\mathbf {Sub}(S)$ and put $L=T\\cap I$ and $U=T\\cap (S\\setminus I)\\cup \\lbrace 0\\rbrace $ .", "Then $T=(U\\setminus \\lbrace 0\\rbrace )\\cup L=\\langle (U\\setminus \\lbrace 0\\rbrace )\\cup L\\rangle $ .", "This establishes the forward set containment.", "The other is trivial.", "Note that a subsemigroup $U\\le S/I$ need not contain the zero element.", "The method suggested by Lemma REF requires calculating $\|\\mathbf {Sub}(S/I)\|\\cdot \|\\mathbf {Sub}(I)\|$ set unions and subsequent closures.", "However, this inefficient calculation is avoidable by using the Lower Torso Enumeration technique, as we now describe." ], [ "Lower Torso Enumeration", "Suppose we have enumerated $\\mathbf {Sub}(S/I)$ for some ideal $I$ of the semigroup $S$ .", "Then $\\left\\lbrace U\\setminus \\lbrace 0\\rbrace \\mid U\\in \\mathbf {Sub}(S/I) \\right\\rbrace $ is precisely the set of all upper torsos.", "The next task is to find all the matching lower torsos for an upper torso.", "That is, for each upper torso $U$ , we must find all subsemigroups $L\\le I$ such that $U\\cup L\\le S$ .", "The method described in Section REF involved enumerating $\\mathbf {Sub}(S/I)$ and $\\mathbf {Sub}(I)$ and checking what the combinations generated.", "We can do better.", "The idea is that an upper torso $U$ acts on the elements of the ideal $I$ by multiplication, so if we do a minimal extension search (Section REF ) the extensions will often be `large jumps' (see Fig.", "REF ).", "We can use Algorithm REF starting from $U$ and extending only by the elements from the ideal.", "In practice, for the full transformation semigroups, this is a very useful trick.", "The main general advantage of the Lower Torso Enumeration method is that no merge is required.", "Figure: Calculating lower torsos for TT by minimal extensions, first extending with x 1 x_1 then by x 2 x_2, elements from II.", "The idea is that often \|T\|≪\|〈T∪{x 1 }〉\|≪\|〈T∪{x 1 ,x 2 }〉\|\|T\| \\ll \|\\langle T\\cup \\lbrace x_1\\rbrace \\rangle \| \\ll \|\\langle T\\cup \\lbrace x_1,x_2\\rbrace \\rangle \|.", "The jumps in size are due to the upper torso acting on the elements of the ideal." ], [ "Enumerating transformation semigroups of degree 2, 3 and 4", "In order to enumerate all semigroups of degree $n$ , we construct all subsemigroups of the full transformation semigroup ${\\mathcal {T}}_n$ .", "We use the ideal structure to make the enumeration more efficient by making the calculation parallel along the lines discussed in Sections REF and REF .", "Recall that the rank of a transformation $t$ is $\|\\!\\!\\:\\operatorname{im}(t)\|$ .", "The ideal of ${\\mathcal {T}}_n$ containing all elements of rank at most $i$ is denoted by $K_{n,i}$ .", "The ideal structure of ${\\mathcal {T}}_n$ is a linear order of nested ideals: $\\varnothing \\subset K_{n,1}\\subset K_{n,2}\\subset \\ldots \\subset K_{n,n-1}\\subset K_{n,n}={\\mathcal {T}}_n.$ The maximal proper ideal $K_{n,n-1}={\\mathcal {T}}_n\\setminus {\\mathcal {S}}_n$ is also called the singular transformation semigroup of degree $n$ , and consists of all transformations but the permutations.", "It is well known that $\\mathbf {Aut}({\\mathcal {T}}_n)$ is isomorphic to ${\\mathcal {S}}_n$ , with every automorphism induced by conjugating the elements of ${\\mathcal {T}}_n$ by a permutation [17], [27].", "As such, we are primarily interested in calculating $\\mathbf {Sub}_{{\\mathcal {S}}_n}({\\mathcal {T}}_n)$ , a set of conjugacy class representatives of the subsemigroups of ${\\mathcal {T}}_n$ .", "Note, however, that a pair of subsemigroups may be isomorphic but not conjugate.", "A separate backtrack search algorithm is needed to construct semigroup embeddings in general, then in particular we can find a set of isomorphism class representatives [8]." ], [ "Subsemigroups of ${\\mathcal {T}}_2$ , the pen and paper case", "The semigroup ${\\mathcal {T}}_2$ has only four elements and consequently the brute-force search space size is only $2^4=16$ .", "It is an easy exercise to find all of its subsemigroups.", "We order the elements lexicographically, 1=[1,1], 2=[1,2], 3=[2,1], 4=[2,2], and list the closed subarrays: Table: NO_CAPTION, Table: NO_CAPTION, Table: NO_CAPTION, Table: NO_CAPTION, Table: NO_CAPTION, Table: NO_CAPTION, Table: NO_CAPTION, Table: NO_CAPTION, Table: NO_CAPTION, Table: NO_CAPTION.", "Using these we can draw the subsemigroup lattice (see Fig.", "REF ).", "Note that the subsemigroups $\\lbrace 1\\rbrace $ and $\\lbrace 2\\rbrace $ are isomorphic but not conjugate.", "Figure: The subsemigroup lattice of 𝒯 2 {\\mathcal {T}}_2.", "The horizontal levels correspond to classes of subsemigroups of the same size.", "Dark (resp., light) grey blobs indicate nontrivial conjugacy (resp., isomorphism) classes.The obvious subdivision of $\\mathbf {Sub}({\\mathcal {T}}_2)$ is according to the sizes of the subsemigroups (as in Fig.", "REF ).", "It turns out that another way of partitioning the elements will also be important for higher degrees.", "One big chunk of the subsemigroup lattice of ${\\mathcal {T}}_n$ is formed by the subsemigroups of the singular part, $\\mathbf {Sub}(K_{n,n-1})$ , and this has an order-isomorphic copy when we adjoin the identity of ${\\mathcal {T}}_n$ to each subsemigroup of $K_{n,n-1}$ , denoted by $\\mathbf {Sub}(K_{n,n-1})^\\#$ .", "The remaining part is the set of subsemigroups that contain nontrivial permutations.", "Since we have no problems with fully calculating and displaying $\\mathbf {Sub}({\\mathcal {T}}_2)$ , this division has no significance, but can be visualized easily (see Fig.", "REF ).", "Figure: The subsemigroup lattice of 𝒯 2 {\\mathcal {T}}_2.", "The subsemigroups of the singular part are indicated by the lowest light grey blob.", "The middle group is an order-isomorphic copy of the latter, with the identity of 𝒯 2 {\\mathcal {T}}_2 adjoined to each subsemigroup.", "The upper dark grey part consists of the subsemigroups containing nontrivial permutations.", "The size of the dark group appears to get smaller relative to the singular part for higher degrees." ], [ "Subsemigroups of ${\\mathcal {T}}_3$ , the limits of brute-force", "The brute-force search space size for ${\\mathcal {T}}_3$ is $2^{3^3}=\\text{134 217 728}$ , or approximately 134.2 million.", "In contrast, using the Minimal Extension method (Section REF ) together with the Equivalent Generators trick (Section REF ), only 4344 subsets need to be checked to enumerate the 283 conjugacy classes.", "For all the 1299 subsemigroups, 25041 checks were required.", "This demonstrated efficiency of the graph search algorithm highlights the benefits of our approach.", "The frequency distribution of the sizes of subsemigroups of $\\mathbf {Sub}({\\mathcal {T}}_3)$ , as well as conjugacy and isomorphism classes, is as follows: Table: NO_CAPTIONIt is easy to see why ${\\mathcal {T}}_3$ has no subsemigroups of size 25 and 26: the biggest maximal subsemigroup is of order 24, since subsemigroups of order $\\ge 21$ correspond to those of the form $K_{3,2}\\cup H$ where $H\\le {\\mathcal {S}}_3$ .", "On the other hand, we have no such explanation for the missing orders 18, 19 and 20.", "Observe also that isomorphism classes only break up into multiple conjugacy classes for low cardinalities (indicated by grey cells in the table).", "There is only one example of anti-isomorphism in ${\\mathcal {T}}_3$ (and a copy of it with identity included).", "By applying the Minimal Generating sets method (Section REF ), we obtain the following rank value distribution of conjugacy classes of $\\mathbf {Sub}({\\mathcal {T}}_3)$ : Table: NO_CAPTIONIn particular, the maximum rank of a subsemigroup of ${\\mathcal {T}}_3$ is 6.", "For example, the six transformations [1,1,1], [1,2,2], $\\operatorname{id}$ , [1,3,3], [2,2,3], [2,2,1] generate a 16-element semigroup, while replacing the last generator with [1,2,1] yields a semigroup of order 22.", "These are the representatives of the two isomorphism classes." ], [ "Subsemigroups of ${\\mathcal {T}}_4$ , the need for parallelization", "Since $\|{\\mathcal {T}}_4\|=4^4=256$ , the brute-force search space is already enormous; $2^{256}$ is a 78-digit number (by contrast, it is currently estimated that there are approximately $10^{80}$ atoms in the observable universe).", "Therefore, the practical calculation of $\\mathbf {Sub}_{{\\mathcal {S}}_4}({\\mathcal {T}}_4)$ requires the strategy of cutting $K_{4,3}$ into two parts for doing the search for subsemigroups in parallel.", "The exact algorithm for enumerating $\\mathbf {Sub}_{{\\mathcal {S}}_4}({\\mathcal {T}}_4)$ is described in the following six steps.", "Calculate $\\mathbf {Sub}_{{\\mathcal {S}}_4}(K_{4,3}/K_{4,2})$ by the minimal extension algorithm (Section REF ).", "There are 10 002 390 conjugacy classes, slightly more than 10 million.", "In parallel, enumerate all lower torsos for all the upper torsos derived from $\\mathbf {Sub}_{{\\mathcal {S}}_4}(K_{4,3}/K_{4,2})$ with the Lower Torso limited enumeration method (Section REF ).", "This gives $\\mathbf {Sub}_{{\\mathcal {S}}_4}(K_{4,3})$ , with 65 997 018 conjugacy classes.", "The calculation is truly parallel since the upper torsos always differ, so there is no need for merging the elements.", "(The subsemigroups from $\\mathbf {Sub}_{{\\mathcal {S}}_4}(K_{4,2})\\subseteq \\mathbf {Sub}_{{\\mathcal {S}}_4}(K_{4,3})$ are obtained in this step when extending the empty upper torso.)", "To get the order-isomorphic copy of $\\mathbf {Sub}_{{\\mathcal {S}}_4}(K_{4,3})$ , we simply adjoin the identity to all subsemigroups: $\\mathbf {Sub}_{{\\mathcal {S}}_4}(K_{4,3})^\\#=\\left\\lbrace S\\cup \\lbrace \\operatorname{id}\\rbrace \\mid S\\in \\mathbf {Sub}_{{\\mathcal {S}}_4}(K_{4,3}) \\right\\rbrace $ .", "To extend $\\mathbf {Sub}_{{\\mathcal {S}}_4}(K_{4,3})$ to $\\mathbf {Sub}_{{\\mathcal {S}}_4}({\\mathcal {T}}_4)$ , note that ${\\mathcal {T}}_4\\setminus K_{4,3}={\\mathcal {S}}_4$ is a sub(semi)group of ${\\mathcal {T}}_4$ .", "So we enumerate $\\mathbf {Sub}_{{\\mathcal {S}}_4}({\\mathcal {S}}_4)$ instead of $\\mathbf {Sub}_{{\\mathcal {S}}_4}({\\mathcal {T}}_4/K_{4,3})$ , using the Minimal Extensions method.", "These are all closed upper torsos.", "This is a much easier subgroup enumeration problem.", "In parallel, find all lower torsos for all nontrivial subgroups in $\\mathbf {Sub}_{{\\mathcal {S}}_4}({\\mathcal {S}}_4)$ .", "Let $P$ be the set of subsemigroups of ${\\mathcal {T}}_4$ with nontrivial permutations (including the subgroups as well).", "This part corresponds to the dark blob on Fig.", "REF .", "Though the search space is the set of subsets (or in this case the subsemigroups) of $K_{4,3}$ , the search is surprisingly quick.", "This is due to the fact that a subgroup of ${\\mathcal {S}}_4$ acts on the singular part $K_{4,3}$ , making each minimal extension into a huge jump, meaning that adding an extra generator yields a relatively large number of new elements.", "In other words, we take each (conjugacy class representative) subgroup $1\\ne G\\le {\\mathcal {S}}_4$ and look for subsemigroups of $K_{4,3}$ closed under the products with $G$ .", "Even a single nontrivial permutation makes the closure relatively big.", "For instance, there are only 71147 lower torsos in $K_{4,3}$ for $\\mathbb {Z}_2=\\langle (1,2)\\rangle $ .", "The total number of elements in $P$ is 75741.", "Finally, we have $\\mathbf {Sub}({\\mathcal {T}}_4)=\\mathbf {Sub}(K_{4,3})\\cup \\mathbf {Sub}(K_{4,3})^\\# \\cup P$ , the set of subsemigroups of the singular part, its order-isomorphic copy with the identity adjoined to each subsemigroup, and the subsemigroups containing nontrivial permutations.", "The total number of subsemigroups up to conjugacy is $\|\\mathbf {Sub}_{{\\mathcal {S}}_4}({\\mathcal {T}}_4)\|= 132 069 776$ and $\|\\mathbf {Sub}({\\mathcal {T}}_4)\|=3 161 965 550$ .", "Note that the ratio of these two numbers is $\\approx 23.94$ , almost the order of ${\\mathcal {S}}_4$ , while $\\mathbf {Sub}({\\mathcal {S}}_4)/\\mathbf {Sub}_{{\\mathcal {S}}_4}({\\mathcal {S}}_4)=\\frac{30}{11}\\approx 2.72$ .", "We note that if $M$ is an arbitrary finite monoid with group of units $G$ , then there is a similar decomposition $\\mathbf {Sub}(M)=\\mathbf {Sub}(M\\setminus G)\\cup \\mathbf {Sub}(M\\setminus G)^\\# \\cup P$ , where $P$ is defined analogously.", "The size distribution of $\\mathbf {Sub}({\\mathcal {T}}_4)$ shows an interesting pattern (see Fig.", "REF ).", "For subgroups of a group, only the divisors of its order can have nonzero frequency values.", "If we considered the size distribution of all subsets, the maximal (middle) binomial coefficient would define the peak value.", "For $\\mathbf {Sub}({\\mathcal {T}}_4)$ the situation is far more involved.", "The numbers are big and they give the impression of continuous change with several peaks.", "The authors do not currently have an explanation for the shape of the distribution; a systematic study of the size classes is needed.", "Figure: The size distribution of 𝐒𝐮𝐛 𝒮 4 (𝒯 4 )\\mathbf {Sub}_{{\\mathcal {S}}_4}({\\mathcal {T}}_4).", "The maximum is at size 60.", "There are 58 different size values with no subsemigroup (no corresponding dot in the figure)." ], [ "Nilpotency", "Recall that a semigroup $S$ is nilpotent if it has a zero element 0 and $S^k=\\lbrace 0\\rbrace $ for some $k\\in \\mathbb {N}$ .", "It is $k$ -nilpotent if $k$ is the minimal such number.", "(The empty semigroup is not nilpotent as it does not have a zero.)", "To decide $k$ -nilpotency algorithmically, in general we do not need to calculate the power $S^k$ .", "We can take a random $k$ -tuple of semigroup elements, evaluate it as a product and assume that this value is the zero element.", "If we find any other $k$ -tuple evaluating to a different value, then the semigroup is not $k$ -nilpotent.", "The worst case is when $S$ is indeed $k$ -nilpotent, we end up checking all $k$ -tuples.", "It turns out that that there are only 4 nilpotent transformation semigroups on 3 points up to conjugacy.", "The trivial monoid is 1-nilpotent and it can be realized by three different conjugacy classes: by the identity transformation, by a constant map, and by a conjugate of $[1,1,3]$ .", "The only 2-nilpotent conjugacy class has the representative $\\lbrace [1,1,1],[1,1,2]\\rbrace $ .", "There are only 22 nilpotent subsemigroups of ${\\mathcal {T}}_4$ up to conjugacy; 4 of them are 1-nilpotent, 7 are 2-nilpotent and 11 are 3-nilpotent.", "The biggest 3-nilpotent subsemigroup has 6 elements: $\\lbrace [1,1,1,1]$ , $[1,1,1,2]$ , $[1,1,1,3]$ , $[1,1,2,1]$ , $[1,1,2,2]$ , $[1,1,2,3]\\rbrace .$ Note also that the fraction of nilpotent conjugacy classes among all conjugacy classes of subsemigroups of ${\\mathcal {T}}_n$ ($n=1,2,3,4$ ) is 1, $0.28$ , $0.0141$ , $1.66\\times 10^{-7}$ .", "In other words, at least for the first four values of $n$ (which may not be a large enough sample), nilpotent semigroups appear to become exceedingly rare.", "This is in sharp contrast to the situation when enumerating semigroups by order, which yields almost exclusively 3-nilpotent semigroups as the order increases [5].", "It is an intriguing question as to whether the fraction of nilpotents continues to decrease when enumerating by degree.", "It would be very curious indeed if the two methods of “slicing up” the (infinite) set of finite semigroups (by order or by degree) led to the two seemingly contradictory (and ultimately meaningless, as we are simply decomposing a countably infinite set into two infinite subsets) intuitions that “almost all semigroups are 3-nilotent” and “almost all semigroups are non-nilpotent”." ], [ "Summary and Conclusion", "We enumerated and classified all transformation semigroups up to degree 4.", "The methods developed here, with more concentrated effort and computational power, may be able to enumerate $\\mathbf {Sub}({\\mathcal {T}}_5)$ or at least the subsemigroups of some of its ideals/Rees quotients.", "However, a better usage of the results would be to investigate the possibility of a more constructive theory of all transformation semigroups.", "For instance, by studying how many different ways semigroups from $\\mathbf {Sub}({\\mathcal {T}}_n)$ can be embedded into semigroups from $\\mathbf {Sub}({\\mathcal {T}}_{n+1})$ , we might be able to estimate $\|\\mathbf {Sub}({\\mathcal {T}}_{n+1})\|$ , or perhaps construct some recursive formula.", "Table: Number of subsemigroups of full transformation semigroups.Table: Number conjugacy classes that contain bands, commutative, and regular semigroups.", "(The empty semigroup is not counted.)" ], [ "Acknowledgements", "This work was partially supported by the NeCTAR Research Cloud, an initiative of the Australian Government's Super Science scheme and the Education Investment Fund; and by the EU project BIOMICS (contract number CNECT-ICT-318202)." ] ]	1403.0274
[ [ "Natural orbital functional theory and pairing correlation effects in\n electron momentum density" ], [ "Abstract Occupation numbers of natural orbitals capture the physics of strong electron correlations in momentum space.", "A Natural Orbital Density Functional Theory based on the antisymmetrized geminal product provides these occupation numbers and the corresponding electron momentum density.", "A practical implementation of this theory approximates the natural orbitals by the Kohn-Sham orbitals and uses a mean-field approach to estimate pairing amplitudes leading to corrections for the independent particle model.", "The method is applied to weakly doped $\\mbox{La$_2$CuO$_4$}$." ], [ "Natural Orbital Functional Theory and Pairing Correlation Effects in Electron Momentum Density B. Barbiellini Department of Physics, Northeastern University, Boston, MA 02115 USA Occupation numbers of natural orbitals capture the physics of strong electron correlations in momentum space.", "A Natural Orbital Density Functional Theory based on the antisymmetrized geminal product provides these occupation numbers and the corresponding electron momentum density.", "A practical implementation of this theory approximates the natural orbitals by the Kohn-Sham orbitals and uses a mean-field approach to estimate pairing amplitudes leading to corrections for the independent particle model.", "The method is applied to weakly doped La$_2$ CuO$_4$ .", "A key characteristic of an interacting electron system is the electron momentum density (EMD).", "For metallic systems one can also define the Fermi Surface (FS) as the break in the EMD whose presence reveals the existence of quasi-particles and the validity of the Landau-Fermi liquid theory [1].", "FS studies are particular needed in the field of high temperature superconductivity.", "Figure 1 shows the calculated FS of the well known HgBa$_2$ CuO$_4$ [2] while Figure 2 illustrates the evolution of the FS topology with doping of a less known compound studied by Jarlborg et al.", "[3].", "One can notice in Fig.", "2 a topological transition (also called Lifshitz transition), which does not involve any symmetry breaking [4].", "Positron annihilation has been successful for the determination of the FS in many metallic systems, but similar studies of the copper oxide high temperature superconductors have met difficulties since positrons do not probe well the FS contribution of the Cu-O planes [5].", "Another direct probe of FS is the Angular Resolved Photo-Emission Spectroscopy (ARPES) [6].", "However, a concern with ARPES is that most of the information of the interacting electron liquid is based on measurements from a surface sensitive technique that can be applied only to a limited number of materials that cleave such as Bi$_2$ Sr$_2$ CaCu$_2$ O$_{8-\\delta }$ .", "Thus, a risk is that experimental artifacts may be interpreted as fundamental physics.", "At low doping, the FS signal from ARPES breaks up into Fermi arcs [7], which could be part of closed hole pockets [8], [9].", "The formation of small Fermi pockets in other underdoped cuprates also emerges from quantum oscillation (QO) measurements in high magnetic fields [10], [11], [12], [13], [14].", "These FS pieces seen by QOs could be in fact produced by FS reconstructions when some symmetry is broken [15].", "According the theory by Lifshitz and Kosevich [16], [17], a period of QO is linked to an extreme cross section of the FS.", "Nevertheless, QOs in the layered and quasi-two-dimensional (2D) conductors may deviate from the LK theory developed for 3D conventional metals [18], [19].", "Figure: The FS of HgBa 2 _2CuO 4 _4 is shown in the first Brillouin zone.It separatesthe occupied states (yellow grid) from theunoccupied states (black grid).Figure: Evolution of the Fermi surfacein the k z =0k_z = 0-plane in Ba 2 _2CuO 3 _3(where one layer of apical oxygen is missing)as a function of the rigid-band doping for 0, 0.15 and0.30 holes per unit cell.", "The FS evolution is almost identicalin La 2 _2CuO 4 _4.", "Only 1/4 of the first Brillouin zone is shown.The momentum units are 1/a1/a, where aa is the lattice constant.Inelastic x-ray scattering [20], [21] in the deeply inelastic limit, can help to clarify the nature of the FS in copper oxide high temperature superconductors since the corresponding Compton scattering cross-section is well-known to become proportional to the ground state EMD [22].", "A Compton scattering study in single crystals of La$_{2-x}$ Sr$_{x}$ CuO$_{4}$ has directly imaged in momentum space the character of holes doped into this material [23].", "However, improvements in the momentum resolution are still needed to bring Compton scattering into the fold of mainstream probes for the cuprates FS.", "A recent Compton study of overdoped La$_{2-x}$ Sr$_2$ CuO$_4$ [24] shows the difficulty of extracting details of the FS with the present momentum resolution of about $0.15$ a.u.", "Higher momentum resolution can also allow the study of the FS smearing due to the superconducting energy gap opening [25] and to the breakdown of the Landau-Fermi liquid picture [26].", "Surprisingly, Compton scattering experiments even on a simpler material such Li indicate that the EMD of the ground state is not well described by the conventional Landau-Fermi liquid framework since the size of the discontinuity $Z$ at the FS seems to be anomalously small [27], [28], [29], [30].", "Such deviations from the standard metallic picture can be ascribed to the possible existence of significant pairing correlations in the ground state [26], [31], [32].", "The notion of stabilizing the metallic state through the resonant valence bond (RVB) state dates back to the early works of Pauling, who first applied this model to the Li ground state [33].", "Anderson then proposed the RVB wave function as a ground state for the high temperature superconducting materials [34], showing that this hypothesis is able of describing many aspects of the phase diagram of the cuprates [35].", "This paper shows how pairing correlation effects modify the occupation of the natural orbitals [36], [37], which are used to calculate the electron momentum density [31] via a simple generalization of independent particle model (IPM).", "In the IPM, states are either occupied or empty [38].", "The occupation numbers (corresponding to natural orbitals in Bloch states [39], [40]) are calculated through a variational scheme based on the Antisymmetrized Geminal Product (AGP) many-body wave function [26], [31], [41], [42], [44], [43], [45], [46].", "The AGP total energy functional is given by [31], [42], [45] $E_{AGP}=E_{HF} + E_{BCS} + O(1/N)~,$ where $E_{HF}$ is the Hartree Fock energy functional, $E_{BCS}$ is a BCS-type functional and $N$ is the number of electrons in the system.", "Since the Coulomb interaction contained in $E_{BCS}$ is repulsive, energy can be gained only through the exchange part $E_{HF}$ of the Hartree-Fock functional.", "However, energy can also be gained through the term $E_{BCS}$ by the introduction interactions with phonons [47].", "Several authors [48], [49], [50], [51], [52], [53], [54], [55] have considered similar functionals of natural orbitals.", "Nevertheless, some of these functionals violate the $N$ -representability [31] and can become over-correlated.", "This problem is avoided here because the $E_{AGP}$ is $N$ -representable by construction.", "To efficiently extract occupation numbers in a correlated electron gas, we start by approximating the natural orbitals by the Kohn-Sham orbitals [56].", "For the sake of simplicity, we suppose that the eigenvalues are described by a single energy band denoted by ${\\cal E}_{\\vec{k}}$ with $\\mu $ defining the chemical potential.", "The result for the AGP energy functional [31] minimization gives the occupation numbers $n_{\\vec{k}}= {1\\over 2}\\left( 1 -{({\\cal E}_{\\vec{k}} - \\mu )\\over E_{\\vec{k}}}\\right),$ where $E_{\\vec{k}}$ is given by $E_{\\vec{k}} = \\sqrt{({\\cal E}_{\\vec{k}} - \\mu )^2 +\|\\Delta _{\\vec{k}}\|^2}~.$ Two self-consistent equations are also involved, one giving $\\Delta _{\\vec{k}}$ [57] $\\Delta _{\\vec{k}}={1\\over N}\\sum _{\\vec{k}^\\prime } \\frac{J_{\\vec{k}\\vec{k}^\\prime }\\Delta _{\\vec{k}^\\prime }}{2E_{\\vec{k}}},$ and the other determining the chemical potential $\\mu $ $N=\\sum _{\\vec{k}} n_{\\vec{k}}.$ Following Ref.", "[31], one can assume that $J_{\\vec{k}\\vec{k}^\\prime }$ is mostly given by an exchange integral.", "Thus, an approximation for $J_{\\vec{k}\\vec{k}^\\prime }$ is given by [31], [58] $J_{\\vec{k}\\vec{k}^\\prime }=\\delta _{\\vec{k}\\vec{k}^\\prime } I_{\\vec{k}}~,$ with $I_{\\vec{k}}=\\frac{1}{3} \\int d^3{\\vec{r}}\|\\psi _{\\vec{k}}({\\vec{r}})\|^4 \\frac{v_x( {\\vec{r}})}{n({\\vec{r}})}~,$ where $v_x( {\\vec{r}})=2/\\pi (3\\pi ^2n({\\vec{r}})^{1/3})$ is the Kohn-Sham exchange potential [56] and $n({\\vec{r}})$ is the electron density.", "By inserting this approximation in Eq.", "REF , one obtains $\\Delta _{\\vec{k}}^2=\\frac{I_{\\vec{k}}^2-4({\\cal E}_{\\vec{k}} - \\mu )^2}{4}.$ Therefore $\\Delta _{\\vec{k}}$ is different of zero only if $I_{\\vec{k}} > 2({\\cal E}_{\\vec{k}} - \\mu )$ .", "The calculation of the occupation numbers for La$_{2-x}$ Sr$_{x}$ CuO$_{4}$ shown in Fig.", "3 has been performed within an efficient linear muffin-tin orbital (LMTO) band structure method [59].", "In this case, the average $I$ is about $1.67$ eV.", "Therefore, at the Fermi energy $\\Delta \\sim I/2= 0.83$ eV.", "The momentum smearing produced by $\\Delta $ is given by $\\delta k = \\frac{\\Delta }{v_F},$ where $v_F$ is the Fermi velocity.", "By taking $\\hbar v_F\\sim \\pi /a$ (where $a=7.16$ a.u.", "is the lattice constant) we find $\\delta k \\sim 0.07$ a.u.", "This momentum smearing is slightly below the current experimental momentum resolution of $0.15$ a.u.", "available in Compton scattering experiments [24].", "A similar $\\delta k$ can be produced by the antiferromagnetic order [60] when $x \\rightarrow 0$ , but the emergence of ferromagnetic fluctuations for $x \\sim 0.25$ leads to the destruction of both RVB correlations and of the AF order in the over-doped regime [61].", "Figure: Occupation number for La 2-x _{2-x}Sr x _{x}CuO 4 _{4}in the limit x→0x \\rightarrow 0 as a function of theKohn-Sham eigenvalues.The Fermi level is at 0.In conclusion, the AGP method has been used to study the occupation numbers $n_{\\vec{k}}$ of natural orbitals in Bloch states of crystals.", "Values of $n_{\\vec{k}}$ can be extracted from EMD experiments [40] and compared to the present model.", "Strong modification of the occupation numbers due to pairing correlations effects are predicted for La$_{2-x}$ Sr$_{x}$ CuO$_{4}$ with $x \\rightarrow 0$ in an energy window of $0.83$ eV around the Fermi energy.", "This effect produces a smearing of the occupation in momentum space given by $\\delta k \\sim 0.07$ a.u.", "This work is supported by the US Department of Energy, Office of Science, Basic Energy Sciences Contract No.", "DE-FG02-07ER46352.", "It has also benefited from Northeastern University's Advanced Scientific Computation Center (ASCC), theory support at the Advanced Light Source, Berkeley, and the allocation of computer time at NERSC through Grant No.", "DE-AC02-05CH11231." ] ]	1403.0004
[ [ "Coulomb branch Hilbert series and Hall-Littlewood polynomials" ], [ "Abstract There has been a recent progress in understanding the chiral ring of 3d $\\mathcal{N}=4$ superconformal gauge theories by explicitly constructing an exact generating function (Hilbert series) counting BPS operators on the Coulomb branch.", "In this paper we introduce Coulomb branch Hilbert series in the presence of background magnetic charges for flavor symmetries, which are useful for computing the Hilbert series of more general theories through gluing techniques.", "We find a simple formula of the Hilbert series with background magnetic charges for $T_\\rho(G)$ theories in terms of Hall-Littlewood polynomials.", "Here $G$ is a classical group and $\\rho$ is a certain partition related to the dual group of $G$.", "The Hilbert series for vanishing background magnetic charges show that Coulomb branches of $T_\\rho(G)$ theories are complete intersections.", "We also demonstrate that mirror symmetry maps background magnetic charges to baryonic charges." ], [ "Introduction", "Identifying the chiral ring and moduli space on the Coulomb branch of an ${\\cal N}= 4$ supersymmetric gauge theory in $2+1$ dimensions has been a long standing problem.", "On a generic point of the Coulomb branch, the triplet of scalars in the ${\\cal N}= 4$ vector multiplets acquires a vacuum expectation value, and the gauge fields that remain massless are abelian and can be dualized to scalar fields.", "Semiclassically, the Coulomb branch is parametrised by the vacuum expectation values of these four scalars.", "This classical description, however, receives quantum corrections.", "The chiral ring associated with the Coulomb branch, in fact, has a complicated structure involving monopole operators in addition to the classical fields in the Lagrangian.", "In spite of the complicated structure of the chiral ring and quantum corrections on the Coulomb branch, it is still possible to enumerate in an exact way the gauge invariant BPS operators that have a non-zero expectation value along the Coulomb branch [1].", "The idea is that the chiral ring of the quantum Coulomb branch can be described in terms of monopole operators dressed with scalar fields from the vector multiplet.", "The generating function that enumerates such BPS operators according to their quantum numbers is called the Coulomb branch Hilbert series.", "This function can be computed for any $3d$ $\\mathcal {N}=4$ supersymmetric gauge theory that has a Lagrangian description and that are good or ugly in the sense of [2].", "We review the method proposed in [1], henceforth called the monopole formula for Coulomb branch Hilbert series, in section .", "Another way to compute the Coulomb branch Hilbert series of a given theory is to use mirror symmetry as a working assumption (see e.g. [3]).", "Mirror symmetry exchanges the Coulomb branch of the theory in question with the Higgs branch of another theory [4], where the latter does not receive quantum corrections.", "The Higgs branch Hilbert series can be computed in a conventional way from the Lagrangian of the mirror theory using Molien integrals.", "This method has certain limitations, for example, when the Lagrangian of the mirror theory is not available.", "Even when the latter is known, if the theory contains gauge groups of large ranks or large number of hypermultiplets, the computation of the Molien integrals can become very cumbersome in practice.", "One of the aims of this paper (and its companion [5]) is to develop a machinery for efficiently computing Coulomb branch Hilbert series for several classes of $\\mathcal {N}=4$ gauge theories.", "We can obtain the Hilbert series of the theories in question by `gluing' together the Hilbert series of building blocks.", "A similar method has been applied successfully to the computation of Higgs branch Hilbert series.", "The gluing procedure consists in gauging a common global flavor symmetry of the building blocks.", "For the gluing machinery to work, we need to define and compute Coulomb branch Hilbert series in the presence of background magnetic fluxes associated to monopole operators for the global symmetry.", "The gluing is performed by summing over the background monopole fluxes with an appropriate weight, as discussed in detail in section .", "In this paper we discuss the general properties of the Hilbert series with background fluxes and we provide computations for a class of simple theories, the three-dimensional superconformal field theories known as $T_{\\mathbf {\\rho }}(G)$ [2].", "The latter are linear quiver theories with non-decreasing ranks associated with a partition $\\mathbf {\\rho }$ and a flavor symmetry $G$ and were defined in terms of boundary conditions for 4d $\\mathcal {N}=4$ SYM with gauge group $G$ [2].", "We are able to give a closed analytic expression for the Coulomb branch Hilbert series of $T_{\\mathbf {\\rho }}(G)$ .", "These expressions serve as basic building blocks for constructing a large class of more complicated theories.", "In a companion paper [5] we discuss the particularly interesting case of the mirror of Sicilian theories arising from twisted compactification of the $6d$ $(2,0)$ theory on a circle times a Riemann surface with punctures, which can be obtained by gluing copies of $T_{\\mathbf {\\rho }}(G)$ theories [6].", "One of the main results of this paper is an intriguing relation between the Coulomb branch Hilbert series of the $T_{\\mathbf {\\rho }}(G)$ and a class of symmetric functions, as discussed in sections and .", "We conjecture that, given a classical group $G$ and a corresponding partition $\\mathbf {\\rho }$ of the dual group, the Coulomb branch Hilbert series of $T_{\\mathbf {\\rho }}(G)$ with background monopole fluxes for the flavor symmetry $G$ can be written in terms of Hall-Littlewood polynomials (see, e.g.", "[7] and Appendix ).", "We give several pieces of evidence in support of our conjecture and others are given in the companion paper [5].", "Our general formula is (REF ), and for the special case of $G=SU(N)$ the formula is given by (REF ).", "We shall henceforth refer to this form of the Coulomb branch Hilbert series as the Hall-Littlewood formula.Modified Hall-Littlewood polynomials have appeared in the context of Hilbert series of affinized flag varieties in [8].", "It would be interesting to relate that formalism to the one of this paper.", "Hall-Littlewood polynomials have also appeared in the recent literature in the context of the superconformal index of four dimensional ${\\cal N}=2$ Sicilian theories [9].", "As we will see in [5], this is not a coincidence.", "Our conjecture is actually inspired by the results in [9].", "Turning off the background monopole fluxes for the flavor group in (REF ), we obtain a simple expression for the Hilbert series of the Coulomb branch of $T_{\\mathbf {\\rho }}(G)$ for any classical $G$ , formula (REF ), which shows that Coulomb branches of $T_{\\mathbf {\\rho }}(G)$ theories are complete intersections.", "In the rest of the paper we examine the structure of the Coulomb branch Hilbert series of the theory and the physical meaning of the background monopole fluxes.", "In section we study the action of mirror symmetry on the background monopole charges under the flavor symmetry of a theory, in order to shed light on their physical meaning.", "These charges are mapped to baryonic charges on the Higgs branch of the mirror theory.", "Indeed, in many examples we consider, we compute the generating function of the Coulomb branch Hilbert series and match it with the baryonic generating function [10] on the Higgs branch of the mirror theory.", "This relation is given by (REF ).", "We continue and examine the analytic properties of the Coulomb branch Hilbert series of $T_{\\mathbf {\\rho }} (SU(N))$ in section .", "Similarly to the observation of [11] in the context of superconformal indices, we find that the Coulomb branch Hilbert series of $T_{\\rho }(SU(N))$ has a pole whose residue corresponds to that of a new theory $T_{\\mathbf {\\rho }^{\\prime }}(SU(N))$ , where the Young diagram of $\\mathbf {\\rho }^{\\prime }$ can be obtained from that of $\\mathbf {\\rho }$ by moving one box to a different position.", "The analytic structure further substantiates our conjecture that the Hall-Littlewood formula computes the Coulomb branch Hilbert series of $T_{\\rho }(G)$ theories.", "Let us summarize the key results of this paper below.", "The Coulomb branch Hilbert series for theories arising from the `gluing' precedure is given by (REF ).", "The Hall-Littlewood formula for a general $T_{\\mathbf {\\rho }}(G)$ is given by (REF ), and by (REF ) in the special case of $G=SU(N)$ .", "Turning off background fluxes, these formulae reduce to (REF ) and (REF ).", "The relations between the generating function of Coulomb branch Hilbert series and the baryonic generating function of the mirror theory is given by (REF ).", "In the next section we review the monopole formula for the Coulomb branch Hilbert series [1]." ], [ "Note added:", "1.", "After the submission of version 1 of this paper to arXiv, we learnt from the discussions in MathOverflow that there are related works by mathematicians on the Hall-Littlewood formula.", "We would like to acknowledge the contributors in such discussions.", "2.", "One might ask whether there is any relation between the Coulomb branch Hilbert series and the $3d$ superconformal index [12], [13], [14].", "Indeed, there is a recent work [15] showing that, under a particular limit, the superconformal index of a $3d$ ${\\cal N}=4$ theory reduces to the Hilbert series." ], [ "Coulomb branch Hilbert series of a 3d $\\mathcal {N}=4$ gauge theory", "We are interested in the Coulomb branch of three-dimensional $\\mathcal {N}=4$ superconformal field theories which have a Lagrangian ultraviolet description as gauge theories of vector multiplets and hypermultiplets.", "The branch is parameterized by the vacuum expectation value of the triplet of scalars in the $\\mathcal {N}=4$ vector multiplets and by the vacuum expectation value of the dual photons, at a generic point where the gauge group is spontaneously broken to its maximal torus.", "This results in a singular HyperKähler cone of quaternionic dimension equal to the rank of the gauge group $G$ .", "The Coulomb branch is not protected against quantum corrections and the associated chiral ring has a complicated structure involving monopole operators in addition to the classical fields in the Lagrangian." ], [ "The monopole formula for the Coulomb branch Hilbert series", "In [1] a general formula for the Hilbert series of the Coulomb branch of an $\\mathcal {N}=4$ theory was proposed, which we now review.", "We will work in the $\\mathcal {N}=2$ formulation, where the $\\mathcal {N}=4$ vector multiplet decomposes into an $\\mathcal {N}=2$ vector multiplet and a chiral multiplet $\\Phi $ transforming in the adjoint representation of the gauge group.", "The Hilbert series is the generating function of the chiral ring, which enumerates gauge invariant BPS operators which have a non-zero expectation value along the Coulomb branch modulo holomorphic relations.", "The $\\mathcal {N}=2$ vector multiplets are replaced in the description of the chiral ring by monopole operators, which are subject to relations that arise at the quantum level.", "The magnetic charges of the monopoles $m$ are labeled by the weight lattice of the GNO dual gauge group $G^\\vee $ [16].", "The monopoles can be dressed with the scalar components $\\phi $ of the chiral multiplet $\\Phi $ that preserve some supersymmetry.", "As can be seen from the supersymmetry transformations of an $\\mathcal {N}=4$ theory [1], the components $\\phi $ that are BPS live in the Lie algebra of the group $H_m$ which is left unbroken by the monopole flux.", "The residual gauge symmetry which is left in the monopole background consists of a continuous part $H_m$ and of a discrete part corresponding to the Weyl group $W_{G^\\vee }$ of $G^\\vee $ , which acts on both the monopole flux $m$ and on the $\\phi $ .", "Due to the action of the Weyl group, the gauge invariant operators can be labeled by a flux belonging to a Weyl Chamber of the weight lattice $\\Gamma _{G^\\vee }$ .", "They will be dressed by all possible products of $\\phi $ invariant under the action of the residual group $H_m$ .", "The final formula counts all gauge invariant BPS operators according to their dimension and reads: $H_G(t)=\\sum _{\\mathbf {m}\\,\\in \\, \\Gamma _{G^\\vee }/W_{G^\\vee }} t^{\\Delta (\\mathbf {m})} P_G(t;\\mathbf {m}) \\;.$ The sum is over a Weyl Chamber of the weight lattice $\\Gamma _{G^\\vee }$ of the GNO dual group [16].", "$P_G(t;m)$ is a factor which counts the gauge invariants of the residual gauge group $H_{\\mathbf {m}}$ made with the adjoint $\\phi $ , according to their dimension.", "It is given by $P_G(t; \\mathbf {m})=\\prod _{i=1}^r \\frac{1}{1-t^{d_i(\\mathbf {m})}} \\;,$ where $d_i(\\mathbf {m})$ , $i=1,\\dots ,{\\rm rank}\\; H_{\\mathbf {m}}$ are the degrees of the independent Casimir invariants of $H_{\\mathbf {m}}$ , also known as exponents of $H_{\\mathbf {m}}$ .", "$t^{\\Delta (m)}$ is the quantum dimension of the monopole operator, which is given by [17], [2], [18], [19] $\\Delta (\\mathbf {m})=-\\sum _{\\mathbf {\\alpha }\\in \\Delta _+(G)} \|\\alpha (\\mathbf {m})\| + \\frac{1}{2}\\sum _{i=1}^n\\sum _{\\mathbf {\\rho }_i \\in R_i}\|\\mathbf {\\rho }_i(\\mathbf {m})\|\\;,$ where the first sum over positive roots $\\mathbf {\\alpha }\\in \\Delta _+(G)$ of $G$ is the contribution of $\\mathcal {N}=4$ vector multiplets and the second sum over the weights of the matter field representation $R_i$ under the gauge group is the contribution of the $\\mathcal {N}=4$ hypermultiplets $H_i$ , $i=1,\\dots ,n$ .", "Half-hypermultiplets contribute to $\\Delta (m)$ with a factor of $\\frac{1}{4}$ instead of $\\frac{1}{2}$ .", "If the gauge group $G$ is not simply connected there is a nontrivial topological symmetry group under which the monopole operators may be charged, the center $Z(G^\\vee )$ of $G^\\vee $ .", "Let $z$ be a fugacity valued in the topological symmetry group and $J(\\mathbf {m})$ the topological charge of a monopole operator of GNO charges $\\mathbf {m}$ .", "The Hilbert series of the Coulomb branch (REF ) can then be refined to $H_G(t,z)=\\sum _{\\mathbf {m}\\,\\in \\, \\Gamma _{G^\\vee }/W_{G^\\vee }} z^{J(\\mathbf {m})} t^{\\Delta (\\mathbf {m})} P_G(t;\\mathbf {m}) \\;.$ The formula can be applied to `good' or `ugly' theories (according to the classification in [2]) where the dimension of all monopole operators satisfies the unitarity bound $\\Delta \\ge 1/2$ .", "This ensures that the Hilbert series (REF ) is a Taylor series of the form $1+\\mathcal {O}(t^{1/2})$ at $t\\rightarrow 0$ .", "The formula bypasses previous techniques for determining the Coulomb moduli space, which were based on compactification of 4d $\\mathcal {N}=2$ theories, the computation of the quantum corrections to the metric of the moduli space or the use of mirror symmetry.", "The latter method is useful only when the mirror gauge group is sufficiently small.", "We demonstrated the utility of the Coloumb branch formula with many explicit examples in [1].", "On the other hand, even the Coloumb branch formula is difficult to evaluate when the gauge group becomes large.", "The main problems are the number of independent sums, which is equal to the rank of the gauge group, and the presence of absolute values in (REF ).", "So we need to find alternative tools for evaluating the formula.", "A quite efficient way is to look for an analytic formula for the Coulomb branch Hilbert series in the presence of background magnetic fluxes for the flavor symmetry group.", "Such Coulomb branch Hilbert series with background magnetic fluxes can then be used in the gluing technique to derive the Coulomb branch Hilbert series of more general theories." ], [ "The Hilbert series with background magnetic fluxes", "Hilbert series with background monopole fluxes are defined as follows.", "If a theory has gauge group $G$ and a global flavor symmetry $G_F$ acting on the matter fields we can define a Hilbert series as in (REF ) but in the presence of background monopole fluxes for the global symmetry group $G_F$ : $H_{G,G_F}(t,{\\mathbf {m}_F})=\\sum _{\\mathbf {m}\\,\\in \\, \\Gamma _{G^\\vee }/W_{G^\\vee }} t^{\\Delta (\\mathbf {m}, \\mathbf {\\mathbf {m}_F})} P_G(t;\\mathbf {m}) \\;.$ In this formula ${\\mathbf {m}_F}$ is a weight of the dual group $G^\\vee _F$ .", "By using the full global symmetry we can restrict its value to a Weyl chamber of $G^\\vee _F$ and take ${\\mathbf {m}_F}\\in \\Gamma ^_{G^\\vee _F}/W_{G^\\vee _F}$ .", "The sum in (REF ) is only over the magnetic fluxes $\\mathbf {m}$ of the gauge group $G$ .", "The background fluxes ${\\mathbf {m}_F}$ enter explicitly in the dimension formula (REF ) through all the hypermultiplets that are charged under the global symmetry $G_F$ , which acts on the Higgs branch of our theory.", "Note that background fluxes in the context of $3d$ superconformal indices are studied in [20].", "A general way of constructing complicated theories is to start with a collection of theories and gauge some common global symmetry $G_F$ that they share.", "The Hilbert series of the final theory is given by summing over the monopoles of $G_F$ and including the contribution to the dimension formula of the $\\mathcal {N}=4$ dynamical vector multiplets associated with $G_F$ : $H(t)=\\sum _{{\\mathbf {m}_F}\\,\\in \\, \\Gamma _{G^\\vee _F}/W_{G^\\vee _F}} t^{-\\sum _{\\mathbf {\\alpha }_F \\in \\Delta _+(G_F)} \\alpha _F({\\mathbf {m}_F})} P_{G_F}(t;{\\mathbf {m}_F}) \\prod _i H^{(i)}_{G,G_F}(t,{\\mathbf {m}_F})\\; ,$ where $\\alpha _F$ are the positive roots of $G_F$ and the product with the index $i$ runs over the Hilbert series of the $i$ -th theory that is taken into the gluing procedure.", "If we have explicit analytic formulae for the Hilbert series $H^{(i)}_{G,G_F}(t,{\\mathbf {m}_F})$ with background fluxes of the original theories, that resum the RHS of (REF ), the evaluation of $H(t)$ requires to perform a sum without any absolute value, since we can always make $\\alpha _F({\\mathbf {m}_F})$ positive by choosing ${\\mathbf {m}_F}$ in the main Weyl chamber.", "The formulae (REF ) and (REF ) can be immediately generalized to include fugacities for the topological symmetries acting on the Coulomb branch.", "The Hilbert series with background magnetic fluxes are interesting objects per se.", "We may ask what happens to the background fluxes under mirror symmetry.", "As we will see in section , the magnetic fluxes ${\\mathbf {m}_F}$ are mapped to baryonic charges in the mirror theory and the Hilbert series with background magnetic charges are mapped to baryonic generating functions [21], [10], [22] for the mirror theory with certain $U(1)$ gauge factors removed.", "We will provide many explicit examples in Section .", "In the next sections we will provide explicit and general formulae for an interesting class of 3d ${\\cal N}=4$ superconformal theories which may serve as building blocks for the construction of more general theories.", "In a companion paper [5] we will apply the result to the mirrors of M5-brane theories compactified on a circle times a Riemann surface with punctures." ], [ "Hilbert series and Hall-Littlewood polynomials", "In this and the following section we discuss the Coulomb branch Hilbert series of a certain class of ${\\cal N}=4$ supersymmetric gauge theories in three dimensions, the theories called $T_{\\mathbf {\\rho }}(G)$ [2], with $G$ a classical group and $\\rho $ a partition of a certain number discussed in detail below.We adhere to the standard notation where $G$ is a Lie group.", "More precisely, the theory is specified by a choice of the Lie algebra of $G$ .", "Such theories can be naturally realized using brane configurations as in [23].", "In this section we discuss the Coulomb branch Hilbert series of $T_{\\mathbf {\\rho }}(SU(N))$ .", "Other classical groups $G$ are discussed in section REF .", "We shall discuss how the Coulomb branch Hilbert series of $T_{\\mathbf {\\rho }}(SU(N))$ in the presence of background magnetic fluxes for $SU(N)$ can be expressed in terms of certain symmetric functions known as Hall-Littlewood polynomials (see e.g.", "[7]).", "The main formula of this section is the Coulomb branch Hilbert series (REF ) for the basic building block $T_{\\mathbf {\\rho }}(SU(N))$ .", "We summarize key information about Hall-Littlewood polynomials in Appendix ." ], [ "The theory $T_{\\mathbf {\\rho }}(SU(N))$ ", "The quiver diagram for $T_{\\mathbf {\\rho }}(SU(N))$ isWe use standard notations where the links denote hypermultiplets transforming in the fundamental representations of the groups they connect.", "$(G)$ denotes a gauge group and $[G]$ a flavor symmetry.", "$[U(N)]-(U(N_1))-(U(N_2))- \\cdots -(U(N_d)), $ where the partition $\\mathbf {\\rho }$ of $N$ is given by $\\mathbf {\\rho }= (N-N_1, N_{1}-N_{2}, N_{2} - N_{3}, \\ldots , N_{d-1}-N_d, N_d)~,$ with the restriction that $\\mathbf {\\rho }$ is a non-increasing sequence: $N-N_1 \\ge N_{1}-N_{2} \\ge N_{2} - N_{3} \\ge \\cdots \\ge N_{d-1}-N_d \\ge N_d > 0~.", "$ This condition ensures that the $T_{\\mathbf {\\rho }} (SU(N))$ is a `good' theory [2].", "The brane configuration of $T_{\\mathbf {\\rho }}(SU(N))$ is depicted in Figure REF .", "$\\mathbf {\\rho }$ corresponds to a collection of the linking numbers of each NS5-brane.", "The theory associated with the partition $\\mathbf {\\rho }=(1,\\cdots ,1)$ is usually called $T (SU(N))$ without any further specification.", "The $U(1)$ center of the $U(N)$ flavor node in (REF ) is actually gauged, consequently the flavor symmetry is $U(N)/U(1)$ rather than $U(N)$ .", "Figure: The brane configuration of T ρ (SU(N))T_{\\mathbf {\\rho }}(SU(N)).", "The numbers in red indicates the number of D3-branes in each intervals.The labels in blue denote the fugacities x i x_i for NS5-branes and the background fluxes n j n_j for D5-branes.In addition to this flavor symmetry, the theory has a manifest $U(1)^d$ topological symmetry associated to the center of the dual gauge group, with conserved currents $J_i=\\mathop {\\rm Tr}(\\ast F_i)$ , where $i=1,\\dots ,d$ and $F_i$ is the field strength of the $i$ -th gauge group.", "The topological symmetry is enhanced by quantum corrections to a non-abelian global symmetry which is determined as follows [2].", "We refer to $\\rho _i$ as the parts of the partition $\\mathbf {\\rho }$ .", "Let $r_k$ be the number of times that part $k$ appears in the partition $\\mathbf {\\rho }$ .", "The Coulomb branch global symmetry associated with the theory $T_{\\mathbf {\\rho }}(SU(N))$ is $G_{\\mathbf {\\rho }} = S \\left( \\prod _{k} U(r_k) \\right)~, $ where $S$ denotes the removal of the overall $U(1)$ .", "For example, the flavor symmetry associated with $\\mathbf {\\rho }= (5,5,4,4,4,2,1,1)$ is $S(U(2)\\times U(3) \\times U(1) \\times U(2))$ .", "From (REF ), the number of gauge groups $d$ is related to $r_k$ in (REF ) by $d = \\sum _{k} r_k - 1~;$ this is the rank of the global symmetry $G_{\\mathbf {\\rho }}$ acting on the Coulomb branch.", "The maximal torus $U(1)^d$ of $G_{\\mathbf {\\rho }}$ , which is manifest as a topological symmetry in the quiver, enhances to a non-abelian $G_{\\mathbf {\\rho }}$ due to hidden symmetry generators whose associated conserved currents are monopole operators.", "There is also a manifest $SU(N)$ flavor symmetry acting on the Higgs branch.", "In the following we provide two formulae to compute the Coulomb branch Hilbert series of $T_{\\mathbf {\\rho }}(SU(N))$ with background fluxes.", "One is the first principle formula given in (REF ), that we present in (REF ) and refer to as the monopole formula; the other is the formula involving the Hall-Littlewood polynomial, that we present in (REF ) and refer to as the Hall-Littlewood formula.", "We conjecture the equivalence of the two formulae, which we have analytically checked for small values of $N$ and tested perturbatively at very large order in $t$ in many different cases.", "As shown in section , the equivalence of the monopole formula (REF ) with the Hall-Littlewood formula (REF ) for the case of the maximal partition $\\rho = (1,1,\\cdots ,1)$ implies the equivalence of the two formulae for a generic partition $\\rho $ , due to their common analytic structure." ], [ "Monopole formula for the Coulomb branch Hilbert series", "As discussed in section , we can define a Coulomb branch Hilbert series of the theory depending on magnetic background fluxes $n_i$ for the $SU(N)$ flavor symmetry and refined by fugacities $z_i$ for the $U(1)^d$ topological symmetry.", "The monopole formula (REF ) for the Coulomb branch Hilbert series of $T_{\\mathbf {\\rho }} (SU(N))$ with background fluxes reads $& H[T_{\\mathbf {\\rho }} (SU(N))](t; z_0, z_1, \\ldots , z_{d}; n_1, \\ldots , n_N) \\nonumber \\\\&= z_0^{\\sum _{j=1}^N n_j} \\sum _{m_{1,N_1} \\ge m_{2,N_1} \\ge \\ldots \\ge m_{N_1,N_1} > -\\infty }\\cdots \\sum _{m_{1,N_d} \\ge m_{2,N_d} \\ge \\ldots \\ge m_{N_d,N_d} > -\\infty } t^{\\Delta \\left(\\mathbf {n}; \\lbrace m_{i, \\ell }\\rbrace _{\\ell =1}^{N_d} \\right) } \\times \\nonumber \\\\& \\quad \\prod _{k=1}^d z_k^{\\sum _{j=1}^{N_k} m_{j, N_k}} P_{U(N_k)} (t; m_{1,N_k}, \\cdots , m_{N_k,N_k}) ~,$ where the dimension of bare monopole operators is given by (REF ) $\\Delta \\left(\\mathbf {n}; \\lbrace m_{i,k}\\rbrace _{i=1}^k \\right) &= \\frac{1}{2} \\left( \\sum _{i^{\\prime }=1}^{N} \\sum _{i=1}^{N_{1}} \|n_{i^{\\prime }}-m_{i,N_{1}}\| +\\sum _{j=1}^{d-1} \\sum _{i=1}^{N_j} \\sum _{i^{\\prime }=1}^{N_{j+1}} \|m_{i,N_{j}}-m_{i^{\\prime },N_{j+1}}\| \\right) \\nonumber \\\\& \\qquad - \\sum _{j=1}^d \\sum _{1 \\le i <i^{\\prime } \\le N_j} \|m_{i, N_j}-m_{i^{\\prime }, N_{j}} \| ~.$ In (REF ), the integers $(m_{1,N_i},\\cdots , m_{N_i,N_i})$ are the GNO magnetic fluxes for the group $U(N_i)$ and the sum is restricted to the fundamental Weyl chamber by restricting to ordered $N_i$ -tuples $m_{1,N_i}\\ge \\cdots , \\ge m_{N_i,N_i}$ .", "The integers $(n_{1},\\cdots , n_{N})$ are instead the background magnetic fluxes for the flavor symmetry group $SU(N)$ .", "The three sets of sums in the dimension $\\Delta $ take into account the contribution of the fundamental hypermultiplets, the bi-fundamental hypermultiplets and the vector multiplets respectively.", "Notice that the background fluxes $n_i$ enter explicitly in the dimension $\\Delta $ through the contribution of the $N$ fundamental hypermultiplets.", "Finally, the classical factors $ P_{U(N_k)} $ are defined in (REF ) and more details are given in Appendix A of [1].", "In the same paper the reader can find many simple examples of the use of the monopole formula.", "In (REF ), we also use the fugacities $z_k$ to keep track of the topological charges $\\sum _{j=1}^{N_k} m_{j, N_k}$ of $U(1)\\subset U(N_{k})$ gauge group, with $k=1, \\ldots , d$ .", "an extra fugacity $z_0$ which keeps track of the background charge $n_1+\\ldots + n_N$ of $U(1)=Z(U(N))$ .", "The flavor symmetry $G_F$ is in fact $U(N)/U(1)$ rather than $U(N)$ , therefore there is no associated topological symmetry even when $G_F$ is (weakly) gauged: $z_0$ is not a physically independent fugacity.", "We remove the extra topological $U(1)$ by imposing the constraint $z_{0}^N \\prod _{k=1}^{d} z_k^{N_k} = 1 ~.$ With this convenient choice the Hilbert series (REF ) is invariant under a common shift of the fluxes $n_i$ , corresponding to the $U(1)$ which is not part of the flavor symmetry.", "The formula is also invariant under permutations of the $n_i$ , the Weyl group of $SU(N)$ .", "Combining the two invariances, we can always restrict the values of the fluxes to $n_1 \\ge n_2 \\ge \\cdots \\ge n_N \\ge 0$ , which will allow to compare with the Hall-Littlewood formula (REF ).", "Using the shift symmetry we could further set $n_N=0$ ." ], [ "Hall-Littlewood formula for the Coulomb branch Hilbert series", "We claim that the Coulomb branch Hilbert series of this theory (REF ) can be written in terms of HL polynomials asThe variables $x_i$ are usually taken to be independent in the literature on Hall-Littlewood polynomials.", "The monopole formula (REF ) without the constraint (REF ) reproduces the HL formula (REF ) without the constraint (REF ), under the fugacity map (REF ).", "$\\begin{split}&H[{T_\\rho (SU(N))}] (t; x_1, \\ldots , x_{d+1} ; n_1, \\ldots , n_N) \\\\&= t^{\\frac{1}{2} \\delta _{U(N)}(\\mathbf {n})} (1-t)^N K^{U(N)}_{\\mathbf {\\rho }} (\\mathbf {x};t)\\Psi _{U(N)}^{\\mathbf {n}}(\\mathbf {x} t^{\\frac{1}{2}\\mathbf {w}_{\\mathbf {\\rho }}}; t) ~,\\end{split}$ where we explain the notations below: $n_1, \\ldots , n_N$ are the background GNO charges for $U(N)$ group, with $n_1 \\ge n_2 \\ge \\cdots \\ge n_N \\ge 0~.$ The Hall-Littlewood polynomial associated with the group $U(N)$ is given by $\\Psi ^{\\mathbf {n}}_{U(N)} (x_1,\\dots ,x_N;t)=\\sum _{\\sigma \\in S_N}x_{\\sigma (1)}^{n_1} \\dots x_{\\sigma (N)}^{n_N}\\prod _{1 \\le i<j \\le N} \\frac{ 1-t x_{\\sigma (i)}^{-1} x_{\\sigma (j)} }{1-x_{\\sigma (i)}^{-1} x_{\\sigma (j)}}~.$ The notation $\\delta _{U(N)}$ denotes the sum over positive roots of the group $U(N)$ acting on the background charges $n_i$ : $\\delta _{U(N)}(\\mathbf {n}) = \\sum _{1\\le i < j \\le N} (n_i - n_j) = \\sum _{j=1}^{N} (N+1-2j) n_j~.$ Note that this is minus the contribution of the background vector multiplet in the monopole dimension formula.", "The fugacities $x_1, \\ldots , x_{d+1}$ are naturally associated to the NS5-branes in the brane construction as depicted in Figure REF .", "They are related to the fugacities $z_0, \\ldots , z_{d}$ for the manifest topological symmetry group $U(1)^{d+1}/U(1)$ by the fugacity map $z_{0} &=x_1~, \\qquad z_k = x_{k+1} / x_{k}~, \\qquad k=1,\\ldots , d~.$ The symmetry $U(1)^{d+1}/U(1)$ enhances to the non-abelian $G_{\\mathbf {\\rho }}$ due to monopole operators.", "$x_1, \\ldots , x_{d+1}$ are subject to the constraint which fixes the overall $U(1)$ .", "Using the map (REF ), the constraint (REF ) is rewritten as $\\prod _{i=1}^{d+1} x_i^{\\rho _i} =1~.$ $\\mathbf {w}_{{r}}$ denotes the weights of the $SU(2)$ representation of dimension $r$ : $\\mathbf {w}_{{r}} = (r-1, r-3, \\ldots , 3-r, 1-r)~.$ Hence the notation $t^{\\frac{1}{2}\\mathbf {w}_{r}}$ represents the vector $t^{\\frac{1}{2}\\mathbf {w}_{r}} = (t^{\\frac{1}{2}(r-1)}, t^{\\frac{1}{2}(r-3)}, \\ldots , t^{-\\frac{1}{2}(r-3)},t^{-\\frac{1}{2}(r-1)})~.$ In (REF ) and from now on, we abbreviate $\\Psi _{U(N)}^{\\mathbf {n}}(\\mathbf {x} t^{\\frac{1}{2}\\mathbf {w}_{\\mathbf {\\rho }}}; t) := \\Psi _{U(N)}^{(n_1, \\ldots , n_N)}(x_1 t^{\\frac{1}{2}\\mathbf {w}_{\\rho _1}}, x_2 t^{\\frac{1}{2}\\mathbf {w}_{\\rho _2}} , \\ldots , x_{d+1} t^{\\frac{1}{2}\\mathbf {w}_{\\rho _{d+1}}};t)~.$ The reader interested in a graphical illustration of the previous definition of $\\mathbf {x} t^{\\frac{1}{2}\\mathbf {w}_{\\rho }}$ may refer to Fig.", "2 of [9].", "The prefactor $K^{U(N)}_{\\mathbf {\\rho }} (\\mathbf {x};t)$ is given by $K^{U(N)}_{\\mathbf {\\rho }} (\\mathbf {x};t) = \\prod _{i=1}^{\\text{length}({\\mathbf {\\rho }}^T)} \\prod _{j,k=1}^{\\rho ^T_i} \\frac{1}{1-a^i_j \\overline{a}^i_k}~,$ where $\\rho ^T$ denotes the transpose of the partition $\\rho $ and we associate the factors $\\begin{split}a^i_j &= x_j \\;\\; t^{\\frac{1}{2} (\\rho _j-i+1)}~, \\qquad i=1,\\dots ,\\rho _j \\\\{\\overline{a}}^i_k &= x_k^{-1} t^{\\frac{1}{2} (\\rho _k-i+1)}~, \\qquad i=1,\\dots ,\\rho _k \\end{split}$ to each box in the Young tableau.A graphical illustration of $a^i_j$ is given by Fig.", "3 of [9], where $j$ labels the column from left to right and $i$ labels the rows from the bottom to the top of the Young tableau.", "The powers of $t$ inside $a^i_j$ and ${\\overline{a}}^i_k$ are positive by construction.", "For instance: For the full punctureWe often write `puncture' in analogy to the literature on M5 branes on a Riemann surface, but take it to mean `partition' in this paper.", "We use the shorthand notation $(r^s)=(\\underbrace{r,\\cdots ,r}_\\text{$s$ times})$ for partitions.", "$\\mathbf {\\rho }= (1^N)$ , we have $\\mathbf {\\rho }^T = (N)$ and so $K^{U(N)}_{(1^N)} (\\mathbf {x}; t)= \\prod _{1\\le j, k\\le N} \\frac{1}{1- x_j x_k^{-1} t} = \\mathop {\\rm PE}[t \\chi ^{U(N)}_{\\bf Adj} (\\mathbf {x}) ]~,$ where $\\mathop {\\rm PE}$ denotes the plethystic exponential.The plethystic exponential of a multivariate function $f(t_1, .", ".", ".", ", t_n)$ that vanishes at the origin is defined as ${\\rm PE} \\left[ f(t_1, t_2, \\ldots , t_n) \\right] = \\exp \\left( \\sum _{k=1}^\\infty \\frac{1}{k} f(t_1^k, \\cdots , t_n^k) \\right)$ .", "For instance $ \\mathop {\\rm PE}[n t^m]=(1-t^m)^{-n}$ .", "For the simple puncture $\\mathbf {\\rho }=(N-1,1)$ , we have $\\mathbf {\\rho }^T = (2,1^{N-2})$ and so $K^{U(N)}_{(N-1,1)} (\\mathbf {x}; t)= \\mathop {\\rm PE}\\left[t^{N/2} (x_1 x_2^{-1} + x_2 x_1^{-1}) +t+ \\sum _{j=1}^{N-1} t^j \\right]~.$ The representation theoretic explanations for $\\mathbf {x} t^{\\frac{1}{2}\\mathbf {w}_{\\rho }}$ and the prefactor $K^{U(N)}_{\\mathbf {\\rho }} (\\mathbf {x};t)$ are presented in Sec.", "4.1 of [24].", "We summarize this in Sec.", "REF of this paper.", "We have explicitly checked in a large number of examples that the HL formula (REF ) coincides with the monopole formula (REF ).", "For instance, we can consider the case $N=3$ .", "There are two relevant partitions, $(1,1,1)$ and $(2,1)$ .", "The $T_{(1,1,1)}(SU(3))$ theory, also known simply as $T(SU(3))$ , has $SU(3)$ global symmetry acting on the Coulomb branch, as it follows from (REF ).", "The monopole formula (REF ) reads $\\begin{split}& H[T_{(1,1,1)}(SU(3))] (t; x_1, x_2, x_3; n_1, n_2) \\\\& =x_1^{n_1+n_2 }\\sum _{m_{1,1} \\in \\mathbb {Z}} ~\\sum _{m_{2,2}\\ge m_{1,2} > -\\infty } (x_2 x_1^{-1})^{m_{1,2}+m_{2,2}} (x_3 x_2^{-1})^{m_{1,1}} \\times \\\\& \\hspace{56.9055pt} t^{\\frac{1}{2}\\Delta _{(1,1,1)}(m_{1,1}; m_{1,2},m_{2,2};\\mathbf {n})} P_{U(2)}(t;m_{1,2},m_{2,2}) P_{U(1)}(t)~,\\end{split}$ where $m_{i,k}$ , with $i=1, \\ldots , k$ , denote the GNO charges for the $U(k)$ gauge group, $x_1 x_2 x_3=1$ , and $\\Delta _{(1,1,1)}$ is twice the dimension of monopole operators in the $[3]-(2)-(1)$ quiver: $\\Delta _{(1,1,1)} =\\sum _{i=1}^2\|m_{1,1}-m_{i,2}\|+\\sum _{i=1}^2\\sum _{j=1}^3 \|m_{i,2}-n_j\| -2\|m_{1,2}-m_{2,2}\|~, \\quad n_3=0~.$ We used the fugacity map (REF ) and we imposed the constraint (REF ).", "One can check that (REF ) reproduces the Hall-Littlewood formula $\\begin{split}& H[T_{(1,1,1)}(SU(3))] (t; x_2,x_2,x_3; n_1, n_2) \\\\&=t^{ n_1} (1-t)^3 {\\rm PE} \\left[t \\sum _{1 \\le i, j \\le 3} x_i x_j^{-1} \\right]\\Psi ^{(n_1,n_2,0)}_{U(3)}(x_1,x_2,x_3;t) ~.", "\\end{split}$ The $T_{(2,1)}(SU(3))$ theory has Coulomb branch symmetry $U(1)$ and the monopole formula reads $H[T_{(2,1)}(SU(3))] (t; x_1, x_2; n_1, n_2,n_3) = \\frac{x_1^{n_1+n_2}}{1-t} \\sum _{m\\in \\mathbb {Z}} t^{\\frac{1}{2}\\Delta _{(2,1)}(m; \\mathbf {n})}(x_2 x_1^{-1})^m~,$ where $m$ denotes the GNO charge for the $U(1)$ gauge group, $x_1^2 x_2=1$ , and $\\Delta _{(2,1)}$ is twice the dimension of monopole operators in the $[3]-(1)$ quiver: $\\Delta _{(2,1)} = \\sum _{i=1}^3 \|m-n_i\|~, \\quad n_3=0~.$ We again used the fugacity map (REF ) and we imposed the constraint (REF ).", "Again one can check that (REF ) reproduces the Hall-Littlewood formula $\\begin{split}&H[T_{(2,1)}(SU(3))] (t; x_1, x_2; n_1,n_2) \\\\&= t^{n_1} (1-t)^3 {\\rm PE} [2t +t^\\frac{3}{2} ( x_1 x_2^{-1} +x_1^{-1} x_2) +t^2] \\Psi ^{(n_1,n_2,0)}_{U(3)}( x_1 t^{\\frac{1}{2}}, x_1 t^{-\\frac{1}{2}}, x_2; t)~.", "\\end{split}$ We will further demonstrate the HL formula (REF ) in a number of examples in a companion paper [5], where we will successfully compare our formula for the Hilbert series of the Coulomb branch of mirrors of genus 0 3d Sicilian theories with the Hilbert series of the Higgs branch of the Sicilian theories themselves, computed as the Hall-Littlewood limit of the superconformal index of the 4d theories in [9], [25]." ], [ "The Coulomb branch of $T_{\\mathbf {\\rho }}(SU(N))$ is a complete intersection", "Coulomb branches of $T_{\\mathbf {\\rho }}(SU(N))$ theories for various partitions $\\mathbf {\\rho }$ were studied in [3] using mirror symmetry; it was found that many of these algebraic varieties are complete intersections.", "In this section, we provide a direct argument that for any partition $\\mathbf {\\rho }$ , the Coulomb branch of $T_{\\mathbf {\\rho }}(SU(N))$ is a complete intersection.", "Setting $n_1=n_2= \\ldots =n_N=0$ in (REF ) and using the identity $(1-t)^{N} \\Psi _{U(N)}^{(0,\\ldots ,0)}(\\mathbf {x};t) = \\prod _{k=1}^{N} (1-t^k)~, $ we obtain $\\begin{split}H[T_{\\mathbf {\\rho }}(SU(N))](t; \\mathbf {x}; \\mathbf {0}) &= K^{U(N)}_{\\mathbf {\\rho }}(\\mathbf {x}; t) \\prod _{k=1}^{N} (1-t^k) \\\\&= \\mathop {\\rm PE}\\left[ \\sum _{i=1}^{\\text{length}({\\mathbf {\\rho }}^T)} \\sum _{j,k=1}^{\\rho ^T_i} x_j x^{-1}_k t^{\\frac{1}{2}(\\rho _j+\\rho _k)-i+1} -\\sum _{k=1}^N t^k\\right] ~, \\end{split}$ where we have used (REF ) in the second equality.", "It follows from the remark below (REF ) that the powers of $t$ appearing inside the $\\mathop {\\rm PE}$ are strictly positive.", "The form of (REF ) shows that the Coulomb branch, denoted by ${\\rm C}[T_{\\mathbf {\\rho }}(SU(N))]$ , is a complete intersection, i.e.", "it is described by a number $n$ of generators subject to a number $r$ of relations equal to the complex codimension of the variety in the embedding space $\\mathbb {C}^n$ .", "Its complex dimension $n-r$ is given by $\\dim _{\\mathbb {C}} {\\text{\\sc C}}[T_{\\mathbf {\\rho }} (SU(N))] = \\sum _{i=1}^{\\text{length}({\\mathbf {\\rho }}^T)} (\\rho _i^T)^2 -N ~,$ where the positive contribution counts the number of positive terms (representing generators) and the negative contribution counts the number of negative terms (representing relations) inside the $\\mathop {\\rm PE}$ in (REF ).", "This result is in agreement with (2.3) and (2.4) of [26].", "In fact, one can cancel a common factor $\\mathop {\\rm PE}[t-t]=1$ in (REF ), suggesting that a putative generator is eliminated by a relation: we conclude that there are $\\sum _{i=1}^{\\text{length}({\\mathbf {\\rho }}^T)} (\\rho _i^T)^2-1$ generators subject to $N-1$ relations, one per Casimir invariant of $SU(N)$ ." ], [ "Coulomb branch of $T_\\rho (SO(N))$ and {{formula:a63c98a4-c8f9-49df-b355-2f0a39a91b3a}}", "In this section we generalize the results on $T_{\\mathbf {\\rho }}(G^\\vee )$ to other classical groups beyond $G=SU(N)$ , namely $SO(N)$ and $USp(2N)$ .", "One of the key results in this section is the formula (REF ) for the Coulomb branch Hilbert series, involving the Hall-Littlewood polynomial.", "We demonstrate in many examples below that this formula is uniform for all classical gauge groups.", "We conjecture that it can be used also for large classes of `bad' theories, where the monopole formula is not working, as we shall discuss in several examples below and in the Appendix.Recall that a theory is called bad when, using the ultraviolet R-symmetry as in (REF ), there are monopole operators with $\\Delta <\\frac{1}{2}$ .", "When that is the case, one cannot assume that the ultraviolet R-symmetry computes the conformal dimension of BPS operators, because the unitarity bound is violated.", "In the following construction, we will also need the GNO, or Langlands, dual of $G$ , denoted as $G^\\vee $ [16].", "Recall that the Lie algebras $A$ and $D$ are self-dual, while $B$ and $C$ are exchanged by GNO duality.", "As pointed out in [2] and [26], $T_{\\mathbf {\\rho }}(G^\\vee )$ is constructed as a boundary theory of $4d$ ${\\cal N}=4$ super Yang-Mills on a half-space, with the half-BPS boundary condition specified by a homomorphism $\\mathbf {\\rho }: \\mathrm {Lie}(SU(2)) \\rightarrow \\mathrm {Lie}(G)$ .", "The homomorphisms $\\mathbf {\\rho }$ can be classified, up to conjugation, by the nilpotent orbits of the Lie algebra of $G$ .", "This classification puts certain restrictions on the partition $\\mathbf {\\rho }$ which we discuss in section REF .", "The quiver diagrams of the corresponding $T_{\\mathbf {\\rho }}(G^\\vee )$ theories are presented in section REF .", "We compute their Coulomb branch Hilbert series in section REF ." ], [ "$B$ , {{formula:c7597621-72de-4bde-9921-8e2c495ab153}} and {{formula:d3426425-93bd-45cc-adf3-91850a1b238f}} partitions", "The partition $\\mathbf {\\rho }= (\\rho _1, \\rho _2, \\ldots )$ defines a homomorphism $\\mathbf {\\rho }: \\mathrm {Lie}(SU(2)) \\rightarrow \\mathrm {Lie}(G)$ , such that the fundamental representation of $G$ decomposes into a direct sum of irreducible representations of $SU(2)$ of dimensions $\\rho _1, \\rho _2, \\rho _3, \\ldots $ .", "We call $\\rho _i$ parts of the partition $\\mathbf {\\rho }$ .", "Due to the Jacobson-Morozov theorem, such embedding can be classified up to a conjugacy by the nilpotent orbit of $G$ .", "As discussed in [27] and section 2.1 of [26], the possible cases are as follows: For $G=SO(N)$ , the partition $\\mathbf {\\rho }$ of $N$ satisfies the condition that any even part in $\\mathbf {\\rho }$ must appear an even number of times.", "The partition $\\mathbf {\\rho }$ is called a B- or a D-partition if $N$ is odd or even, respectively.", "For instance, the B-partitions for $SO(3)$ are $(3)$ and $(1,1,1)$ ; the D-partitions for $SO(4)$ are $(4)$ , $(3,1)$ , $(2,2)$ and $(1,1,1,1)$ .", "Given a partition $\\mathbf {\\rho }$ satisfying this condition, there is a unique nilpotent orbit associated to it, except for the case when all the parts $\\rho _i$ are even and each even integer appears even times.", "Such a partition is referred to as a very even partition, whose distinct nilpotent orbits are exchanged by the outer automorphism of $SO(N)$ .", "An example of a very even partition is $\\mathbf {\\rho }=(4,4)$ for $G=SO(8)$ .", "For $G=USp(2N)$ , the partition $\\mathbf {\\rho }$ of $2N$ satisfies the condition that any odd part in $\\mathbf {\\rho }$ must appear an even number of times.", "Such a partition is called a C-partition.", "In this case each partition corresponds to a unique nilpotent orbit.", "For instance, the C-partitions $\\mathbf {\\rho }$ for $USp(2)$ are $(2)$ and $(1,1)$ ; for $USp(4)$ , $(4)$ , $( 2,2)$ and $( 1,1,1,1)$ ; for $USp(6)$ , $(6)$ , $(4,2)$ , $(3,3)$ , $(2,2,2)$ , $(2,2,1,1)$ , $(2,1,1,1,1)$ and $(1,1,1,1,1,1)$ .", "Below we present quiver diagrams for $T_{\\mathbf {\\rho }} (G^\\vee )$ for $G=SO(N)$ and $USp(2N)$ .", "Such quivers already appeared as subquivers in the work of [28]." ], [ "Quiver diagrams", "The quiver diagram for $T_{\\mathbf {\\rho }} (SO(2N))$ and $T_{\\mathbf {\\rho }} (SO(2N+1))$ for a given partition $\\mathbf {\\rho }$ is presented in (6.3) and (6.5) of [6].", "The quiver for $T_{\\mathbf {\\rho }} (USp(2N))$ can be easily obtained by generalising that of $T_{\\mathbf {\\rho }} (SO(2N))$ .", "We summarize the necessary information below." ], [ "$T_{\\mathbf {\\rho }}(SO(2N))$ theory.", "In this case $\\mathbf {\\rho }= (\\rho _1, \\ldots , \\rho _\\ell )$ is a $D$ -partition of $2N$ .", "The quiver diagram for $T_{\\mathbf {\\rho }} (SO(2N))$ is $[SO(2N)] - (USp(s_1)) - (O(s_2)) - \\cdots -(O(s_{\\ell -2})) - (USp(s_{\\ell -1}))~,$ where $\\ell $ is even and $s_i = \\left[\\sum _{j=i+1}^\\ell \\rho _{j} \\right]_{+, -} \\qquad \\text{$+$ for $O$ and $-$ for $USp$}~,$ with $[n]_{+(\\text{{\\it resp.", "}}-)}$ the smallest (resp.", "largest) even integer $\\ge n$ (resp.", "$\\le n$ ) and the node $USp(0)$ being removed." ], [ "$T_{\\mathbf {\\rho }}(SO(2N+1))$ theory.", "In this case we consider the Langlands dual of $B_N = SO(2N+1)$ , namely $C_N = USp(2N)$ .", "The partition $\\mathbf {\\rho }= (\\rho _1, \\ldots , \\rho _\\ell )$ is a $C$ -partition of $2N$ .", "The corresponding quiver diagram for $T_{\\mathbf {\\rho }} (SO(2N+1))$ is $[SO(2N+1)] -(USp(s_1)) - (O(s_2)) - \\cdots - (O(s_{[\\ell ]_-}))~,$ where $s_i = \\left[1+ \\sum _{j=i+1}^\\ell \\rho _{j} \\right]_{\\widetilde{+}, -} \\qquad \\text{$\\widetilde{+}$ for $O$ and $-$ for $USp$}~,$ with $[n]_{{\\widetilde{+}}}$ the smallest odd integer $\\ge n$ and $[n]_\\pm $ defined as above." ], [ "$T_{\\mathbf {\\rho }}(USp(2N))$ theory.", "In this case we consider the Langlands dual of $C_N = USp(2N)$ , namely $B_N = SO(2N+1)$ .", "The partition $\\mathbf {\\rho }= (\\rho _1, \\ldots , \\rho _\\ell )$ is a $B$ -partition of $2N+1$ .", "The corresponding quiver diagram for $T_{\\mathbf {\\rho }} (USp(2N))$ is $[USp(2N)] -(O(s_1)) - (USp(s_2)) - \\cdots - (USp(s_{\\ell -1}))~,$ where $\\ell $ is odd and $s_i = \\left[\\sum _{j=i+1}^\\ell \\rho _{j} \\right]_{+, -} \\qquad \\text{$+$ for $O$ and $-$ for $USp$}~,$ with $[n]_\\pm $ defined as above and the node $USp(0)$ being removed.", "In the previous theories some of the gauge groups of type $O$ can be replaced by groups of type $SO$ .", "The distinction between $SO(s)$ and $O(s)$ gauge groups is important.", "Theories with $SO(s)$ gauge groups have typically more BPS gauge invariant operators compared with the same theory with gauge group $O(s)$ and we have different theories according to the choice of $O/SO$ factors." ], [ "Hall-Littlewood formula for $T_{\\mathbf {\\rho }} (G^\\vee )$ with a classical group {{formula:f06ed769-babb-4dae-985c-1db02c5d4b53}}", "In this section we generalize the Hall-Littlewood formula (REF ) to a more general group $G$ .", "We follow closely a similar discussion in section 4.1 of [24]; see also [26] for a comprehensive presentation.", "Several explicit examples are presented in subsequent subsections and in Appendix .", "As discussed earlier, the partition $\\mathbf {\\rho }$ defines a homomorphism $\\mathbf {\\rho }: \\mathrm {Lie}(SU(2)) \\rightarrow \\mathrm {Lie}(G)$ such that $[1,0,\\ldots ,0]_G = \\bigoplus _i [\\rho _i-1]_{SU(2)}~.$ The global symmetry $G_{\\mathbf {\\rho }}$ associated to the puncture and acting on the Coulomb branch is given by the commutant of ${\\mathbf {\\rho }}(\\mathrm {SU}(2))$ in $G$ .", "Explicitly, for a given group $G=U(N)$ , $SO(N)$ or $USp(2N)$ and a puncture $\\mathbf {\\rho }= [\\rho _i]$ with $r_k$ the number of times that part $k$ appears in the partition $\\mathbf {\\rho }$ , we have $G_{\\mathbf {\\rho }} = {\\left\\lbrace \\begin{array}{ll} S \\left( \\prod _{k} U(r_k) \\right) & \\qquad G= U(N)~, \\\\\\prod _{k~\\text{odd}} SO(r_k) \\times \\prod _{k~\\text{even}} USp(r_k) & \\qquad G= SO(2N+1)~\\text{or}~SO(2N)~, \\\\\\prod _{k~\\text{odd}} USp(r_k) \\times \\prod _{k~\\text{even}} SO(r_k) & \\qquad G= USp(2N)~.\\end{array}\\right.", "}$ Let $x_1, x_2, \\ldots $ be fugacities for the global symmetry $G_{\\mathbf {\\rho }}$ , and $r(G)$ the rank of $G$ .", "We conjecture that the Coulomb branch Hilbert series is given by the HL formula $H[T_{\\mathbf {\\rho }}(G^\\vee )](t; \\mathbf {x};n_1,\\ldots , n_{r(G)}) = t^{\\frac{1}{2} \\delta _{G^\\vee }(\\mathbf {n})} (1-t)^{r(G)} K^{G}_{\\mathbf {\\rho }} (\\mathbf {x};t) \\Psi ^{\\mathbf {n}}_{G}(\\mathbf {a}(t, \\mathbf {x});t)~, $ where we explain the notations below.", "The Hall-Littlewood polynomial $\\Psi ^{\\mathbf {n}}_{G}(\\mathbf {a}(t, \\mathbf {x});t)$ associated with a classical group $G$ is given in Appendix .", "The power $\\delta _{G^\\vee }(\\mathbf {n})$ is the sum over positive roots $\\mathbf {\\alpha }\\in \\Delta _+(G^\\vee )$ of the flavor group $G^\\vee $ acting on the background monopole charges $\\mathbf {n}$ : $\\delta _{G^\\vee }(\\mathbf {n}) = \\sum _{\\mathbf {\\alpha }\\in \\Delta _+(G^\\vee )} \\mathbf {\\alpha }(\\mathbf {n})~.$ Explicitly, for classical groups $G$ , these are given by $\\delta _{G^\\vee } (\\mathbf {n}) &= {\\left\\lbrace \\begin{array}{ll}\\sum _{j=1}^N (N+1-2j) n_j & \\qquad G^\\vee =G=U(N), \\\\\\sum _{j=1}^{N} (2N+1-2j)n_j & \\qquad G^\\vee =B_N,~ G=C_N\\\\\\sum _{j=1}^{N} (2N+2-2j)n_j & \\qquad G^\\vee =C_N,~ G=B_N \\\\\\sum _{j=1}^{N-1} (2N-2j)n_j & \\qquad G^\\vee =G=D_N~.\\end{array}\\right.", "}$ The argument $\\mathbf {a}(t, \\mathbf {x})$ , which we shall henceforth abbreviate as $\\mathbf {a}$ , of the HL polynomial is determined by the following decomposition of the fundamental representation of $G$ to $G_{\\mathbf {\\rho }} \\times {\\mathbf {\\rho }} (SU(2))$ : $\\chi ^G_{{\\bf fund}} (\\mathbf {a}) = \\sum _{k} \\chi ^{G_{\\rho _k}}_{{\\bf fund}}( \\mathbf {x}_k) \\chi ^{SU(2)}_{[\\rho _k -1]} (t^{1/2})~,$ where $G_{\\rho _k}$ denotes a subgroup of $G_{\\mathbf {\\rho }}$ corresponding to the part $k$ of the partition $\\mathbf {\\rho }$ that appears $r_k$ times.", "Formula (REF ) determines $\\mathbf {a}$ as a function of $t$ and $\\lbrace \\mathbf {x}_k \\rbrace $ as required.", "Of course, there are many possible choices for $\\mathbf {a}$ ; the choices that are related to each other by outer automorphisms of $G$ are equivalent.", "We provide some examples, such as two inequivalent choices (REF ) for $\\mathbf {\\rho }=(4,4)$ of $G=SO(8)$ , in the subsequent subsections.", "$K^G_{\\mathbf {\\rho }}(\\mathbf {x}; t)$ is a prefactor independent of ${\\mathbf {n}}$ , determined as follows.", "The embedding associated with $\\mathbf {\\rho }$ induces the decomposition of the adjoint representation of $G$ $\\chi ^G_{\\bf Adj} (\\mathbf {a}) = \\sum _{j =0, \\frac{1}{2}, 1, \\frac{3}{2}, \\ldots } \\chi ^{G_{\\mathbf {\\rho }}}_{R_j}(\\mathbf {x}_j) \\chi ^{SU(2)}_{[2j]}(t^{1/2})~, $ where $\\mathbf {a}$ on the left hand side is the same $\\mathbf {a}$ as in (REF ).", "Note that $\\bigoplus _j R_j$ gives the decomposition of the Slodowy slice [26].", "Each component in the slice gives rise to a plethystic exponential, giving $K^G_{\\mathbf {\\rho }}(\\mathbf {x}; t)=\\mathop {\\rm PE}\\left[\\sum _{j =0, \\frac{1}{2}, 1, \\frac{3}{2}, \\ldots } t^{j+1} \\chi ^{G_{\\mathbf {\\rho }}}_{R_j}({\\mathbf {x}}_j )\\right].", "$ We provide a number of examples below; see for example (REF ) for $\\mathbf {\\rho }=(3,3,1,1)$ of $G=SO(8)$ and (REF ) for $\\mathbf {\\rho }=(3,2,2,1)$ of $G=SO(8)$ ." ], [ "The Coulomb branch of $T_{\\mathbf {\\rho }}(G^\\vee )$ is a complete intersection", "In this subsection we generalize the computation in Section REF for any classical group $G$ .", "We will show that formula (REF ) implies that the Coulomb branch of $T_{\\mathbf {\\rho }}(G^\\vee )$ is a complete intersection for any classical group $G$ and any partition $\\mathbf {\\rho }$ .", "Turning off the background monopole fluxes $n_1=n_2 =\\ldots = n_{r(G)} =0$ , the HL polynomials reduce to $\\Psi ^{(0,\\ldots ,0)}_{G} (\\mathbf {x} ;t) = (1-t)^{-r(G)} \\prod _{i=1}^{r(G)} (1-t^{d_i(G)})~,$ where $d_i(G)$ , with $i=1, \\ldots , r(G)$ , are the exponents of $G$ : $d_i(G) ={\\left\\lbrace \\begin{array}{ll}1, 2, 3, \\ldots , N, &\\qquad G= U(N) \\\\2, 4, 6, \\ldots , 2N, & \\qquad G=SO(2N+1), ~USp(2N) \\\\N, 2, 4, 6, \\ldots , 2N-2, & \\qquad G=SO(2N)~.\\end{array}\\right.", "}$ Thus we obtain $\\begin{split}H[T_{\\mathbf {\\rho }}(G^\\vee )](t; \\mathbf {x};\\mathbf {0})&= K^{G}_{\\mathbf {\\rho }} (\\mathbf {x}; t) \\prod _{i=1}^{r(G)} (1-t^{d_i(G)}) \\\\&= \\mathop {\\rm PE}\\left[\\sum _{j =0, \\frac{1}{2}, 1, \\frac{3}{2}, \\ldots } t^{j+1} \\chi ^{G_{\\mathbf {\\rho }}}_{R_j}({\\mathbf {x}}_j )- \\sum _{i=1}^{r(G)} t^{d_i(G)}\\right]~,\\end{split}$ where the representations $R_j$ of the group $G_{\\mathbf {\\rho }}$ are given by (REF ).", "This shows that the Coulomb branch, denoted by ${\\text{\\sc C}}(T_{\\mathbf {\\rho }} (G^\\vee ))$ , of $T_{\\mathbf {\\rho }}(G^\\vee )$ is indeed a complete intersection: there are $\\sum _{j} \\dim _{G_{\\mathbf {\\rho }}}(R_j)$ generators subject to $r(G)$ relations, one per independent Casimir invariant of $G$ .", "The complex dimension of the Coulomb branch is given by $\\dim _{\\mathbb {C}} {\\text{\\sc C}}[T_{\\mathbf {\\rho }} (G^\\vee )] =\\sum _{j} \\dim _{G_{\\mathbf {\\rho }}}(R_j) - r(G)~,$ where $\\dim _{G_{\\mathbf {\\rho }}}(R_j)$ denotes the dimension of representation $R_j$ of the group $G_{\\mathbf {\\rho }}$ .", "According to Theorem 6.1.3 of [27], the first term in (REF ) can be related to the partition as $\\sum _{j} \\dim _{G_{\\mathbf {\\rho }}}(R_j)={\\left\\lbrace \\begin{array}{ll} \\sum _{i} {(\\rho ^T_i)^2} & G= U(N) \\\\ \\frac{1}{2}\\sum _{i} {(\\rho ^T_i)^2} -\\frac{1}{2} \\sum _{i~\\text{odd}} r_i & G=B_N, ~D_N \\\\ \\frac{1}{2}\\sum _{i} {(\\rho ^T_i)^2} +\\frac{1}{2} \\sum _{i~\\text{odd}} r_i & G=C_N~, \\end{array}\\right.", "}$ where $r_k$ is the number of times that $k$ appears in the partition $\\mathbf {\\rho }$ .", "Thus, from (REF ) and (REF ), we have the dimension formula in accordance with (2.3) of [26]: $\\dim _{\\mathbb {C}} {\\text{\\sc C}}[T_{\\mathbf {\\rho }} (G^\\vee )] ={\\left\\lbrace \\begin{array}{ll} \\sum _{i} {(\\rho ^T_i)^2}-N \\qquad & G= U(N) \\\\ \\frac{1}{2}\\sum _{i} {(\\rho ^T_i)^2} -\\frac{1}{2} \\sum _{i~\\text{odd}} r_i -N \\qquad & G=B_N, \\; D_N \\\\ \\frac{1}{2}\\sum _{i} {(\\rho ^T_i)^2} +\\frac{1}{2} \\sum _{i~\\text{odd}} r_i -N \\qquad & G=C_N~.\\end{array}\\right.", "}$" ], [ "$T(SO(N))$ and {{formula:8f76ec3e-c4d5-4f43-a22c-6ed0496b1e62}}", "In this subsection we focus on $\\mathbf {\\rho }=(1,1, \\ldots , 1)$ .", "The case of non maximal punctures is discussed in Appendix .", "From now on, we shall abbreviate $T_{(1,\\ldots ,1)}(G)$ as $T(G)$ .", "The quiver diagrams for $T(SO(N))$ and $T(USp(2N))$ are given in Fig.", "54 of [2]; we summarize them below.With this choice of $SO(N)$ gauge groups for $T(D_N)$ and $T(C_N)$ the Coulomb and Higgs moduli space are precisely the nilpotent cone of the corresponding $D_N$ and $C_N$ group [2].", "$&T(D_N): \\nonumber \\\\& \\qquad (SO(2))-(USp(2))- \\cdots - (SO(2N-2))-(USp(2N-2))-[SO(2N)] \\\\&T(C_N): \\nonumber \\\\& \\qquad (SO(2))-(USp(2))- \\cdots -(USp(2N-2))-(SO(2N))-[USp(2N)] \\\\& T(B_N): \\nonumber \\\\& \\qquad (O(1))-(USp(2))-(O(3))- \\cdots - (O(2N-1))-(USp(2N))-[SO(2N+1)]~.", "$ Note that edges connecting $O$ and $USp$ represent bifundamental half-hypermultiplets.", "Remark also that the quiver in (REF ) is a `bad theory'; for example, the number of flavors under the $USp(2)$ gauge group is 2 (because there are 4 half-hypers charged under this gauge group), which is smaller than $2(1)+1=3$ (see (5.9) of [2]).", "Mirror symmetry exchanges $G$ with the Langlands or GNO dual $G^\\vee $ , therefore $T(D_N)$ is self-dual, whereas $T(C_N)$ and $T(B_N)$ form a mirror pair [2].", "From many examples presented in the following subsections, we deduce that the Hilbert series of Coulomb branch for each of such theories is given by $H[T(G^\\vee )](t;x_1, \\ldots , x_N;n_1,\\ldots , n_N)&= t^{\\frac{1}{2} \\delta _{G^\\vee }(\\mathbf {n})} (1-t)^{r(G)} K^{G} (\\mathbf {x};t) \\Psi ^{\\mathbf {n}}_{G}(\\mathbf {x};t)~.$ where $n_1, \\ldots , n_N$ are the background charges, $\\delta _{G^\\vee }(\\mathbf {n})$ is given by (REF ), and $K^G ( \\mathbf {x};t) &= \\mathop {\\rm PE}\\left[t (\\chi ^G_{\\bf Adj} (\\mathbf {x}) )\\right]~.$ Since the quiver (REF ) for $T(B_N)$ is a bad quiver, we cannot compute the Coulomb branch Hilbert series from the monopole formula, which diverges.", "On the other hand one may expect that the quiver flows to an interacting conformal fixed point in the infrared.", "Since the formula (REF ) involving the Hall-Littlewood polynomial is well defined, we conjecture that it computes the Hilbert series of the Coulomb branch of the infrared SCFT.", "Below we demonstrate this by computing the Coulomb branch Hilbert series of $T(SO(5))$ , which is a bad theory, using the Hall-Littlewood formula and find a matching (REF ) with that of $T(USp(4))$ , which is a good theory.", "When comparing with the monopole formula, we will face the problem of matching fugacities.", "Unlike in the case of unitary groups, in the Coulomb branch of the $T(G^\\vee )$ theories with orthogonal and symplectic groups the Cartan subalgebra of the global symmetry is not fully manifest.", "Some Cartan generators do not correspond to topological symmetry of the Coulomb branch, but arise instead as monopole operators [2].", "It is not clear to us how to introduce fugacities for these generators in the monopole formula.", "Let us demonstrate these formulae in the examples below." ], [ "$T(SO(4))$", "The quiver of $T(SO(4))$ is $(SO(2)) - (USp(2)) - [SO(4)] , $ with a Coulomb branch symmetry $SO(4)$ .", "In addition there is a flavor symmetry $SO(4)$ acting on the Higgs branch.", "The conjectured HL formula for the Coulomb branch Hilbert series of $T(SO(4))$ is $H[T(SO(4)](t; \\mathbf {x}; \\mathbf {n})= t^{n_1} \\mathop {\\rm PE}\\left[t \\left(\\frac{1}{x_1 x_2}+\\frac{x_1}{x_2}+\\frac{x_2}{x_1}+x_1 x_2 \\right) \\right] \\Psi ^{(n_1,n_2)} _{D_2} (\\mathbf {x}; t)~,$ where explicit expression for small values of $n_1$ and $n_2$ are given in Appendix .", "For $n_1=n_2=0$ , we have $\\Psi ^{(0,0)} _{D_2} (\\mathbf {x}; t) = (1+t)^2 = \\frac{\\left(1-t^2\\right)^2}{(1-t)^2}~,$ therefore $\\begin{split}H[T(SO(4)](t; x_1 x_2, x_1 x_2^{-1}; 0,0) &= (1-t^2)^2\\mathop {\\rm PE}\\left[t (x_1^2+x_2^2 +x_1^{-2}+x_2^{-2}+2) \\right] \\\\&=g_{\\mathbb {C}^2/\\mathbb {Z}_2} (t, x_1) g_{\\mathbb {C}^2/\\mathbb {Z}_2} (t, x_2)~,\\end{split}$ in terms of the Hilbert series of $\\mathbb {C}^2/\\mathbb {Z}_2$ $g_{\\mathbb {C}^2/\\mathbb {Z}_2} (t, x) = (1-t^2) \\mathop {\\rm PE}\\left[t( x^2 + 1 +x^{-2}) \\right] ~.$ Hence the Coulomb branch of $T(SO(4))$ is $(\\mathbb {C}^2/\\mathbb {Z}_2)^2$ ; this is identical to the Higgs branch of the same theory, in agreement with the self-mirror property." ], [ "Comparison with monopole formula.", "We can compare the conjectured formula (REF ) with the monopole formula for Coulomb branch Hilbert series $H_{\\rm mon}[T(SO(4)](t; x_1; \\mathbf {n})= \\sum _{m=-\\infty }^\\infty \\sum _{k=0}^\\infty x_1^{2m} t^{\\Delta (m,k,\\mathbf {n})} P_{USp(2)}(t;k) P_{SO(2)}(t;m)~,$ where $m$ is the topological charge for $SO(2)$ gauge group with $x_1$ the corresponding fugacity, and $k$ is the monopole charges in $USp(2)$ gauge group.", "Here $\\begin{split}\\Delta (m,k, \\mathbf {n}) &= \\frac{1}{2} \\left(\|m-k\|+\|m+k\|+ \\sum _{i=1}^2 \\left(\|n_i +k\|+ \|n_i -k\| \\right)\\right) - \|2k\|~, \\\\P_{USp(2)}(t;k) &= P_{SU(2)} (t;k) ={\\left\\lbrace \\begin{array}{ll} \\frac{1}{1-t^2}~, \\qquad & k=0 \\\\ \\frac{1}{1-t}~, \\qquad & k >0~.", "\\end{array}\\right.}", "\\\\P_{SO(2)}(t;m) &= P_{U(1)}(t;m)=\\frac{1}{1-t}~.\\end{split}$ It can be checked that $H[T(SO(4)](t; x_1^2, x_2=1; n_1, n_2) = H_{\\rm mon}[T(SO(4)](t; x_1; n_1,n_2)~.", "$ Note that in the monopole formula (REF ) we can refine only one fugacity $x_1$ of $SO(4)$ , whereas in the conjectured Hall-Littlewood formula (REF ) for the Hilbert series both fugacities $x_1$ and $x_2$ appear.", "This requires some explanation.", "Although the Coulomb branch symmetry is $SO(4)$ , only a $U(1)$ subgroup is manifest.", "The reason is that the only manifest symmetries in the Coulomb branch are the topological symmetries and the quiver $(SO(2))-(USp(2))-[SO(4)]$ has a single abelian factor.", "The remaining $SO(4)$ generators, including the other generator of the Cartan subalgebra, correspond to monopole operators [2].", "In particular the other generator of the Cartan subalgebra is provided by the monopole operator with $m=0$ , $k=1$ along with $n_i=0$ .", "It would be interesting to understand how to include fugacities for the full Cartan subalgebra in the monopole formula for the Coulomb branch Hilbert series.", "We will encounter this phenomenon many times in the following." ], [ "$T(USp(4))$ and {{formula:35bd345b-532b-4066-bf2c-1262768e6b7e}}", "The HL formula for the Coulomb branch of $T(SO(5))$ is given by $H[T(SO(5))](t; \\mathbf {x}; \\mathbf {m}) = t^{\\frac{1}{2}(3m_1+m_2)} \\mathop {\\rm PE}\\left[t (\\chi _{[2,0]}^{C_2} (x_1, x_2)-2) \\right] \\Psi ^{(m_1,m_2)} _{C_2} (x_1,x_2; t),$ where the character of the adjoint representation $[2,0]$ of $USp(4)$ is $\\chi _{[2,0]}^{C_2} (x_1, x_2) =2+\\frac{1}{x_1^2}+x_1^2+\\frac{1}{x_2^2}+\\frac{1}{x_1 x_2}+\\frac{x_1}{x_2}+\\frac{x_2}{x_1}+x_1 x_2+x_2^2~.$ Since the Lie algebra of $USp(4)$ is isomorphic to that of $SO(5)$ , we also have $H[T(USp(4))](t; \\mathbf {y}; \\mathbf {n})= t^{\\frac{1}{2}(4n_1+2n_2)} \\mathop {\\rm PE}\\left[t (\\chi _{[0,2]}^{B_2} (y_1, y_2)-2) \\right] \\Psi ^{(n_1,n_2)} _{B_2} (y_1, y_2; t)~,$ where the character of adjoint representation $[0,2]$ of $SO(5)$ is $\\chi _{[0,2]}^{B_2} (x_1, x_2)=2+\\frac{1}{x_1}+x_1+\\frac{1}{x_2}+\\frac{1}{x_1 x_2}+\\frac{x_1}{x_2}+x_2+\\frac{x_2}{x_1}+x_1 x_2~.$ These two expressions can be matched as follows: $H[T(SO(5))](t; x_1, x_2; n_1+n_2, n_1-n_2)=H[T(USp(4))](t; x_1 x_2, x_1 x_2^{-1}; n_1, n_2)~.$ The matching reflects the translation between representations of $USp(4)$ and $SO(5)$ : $\\begin{split}\\chi ^{C_2}_{(n_1+n_2, n_1-n_2)} (x_1,x_2) &= \\chi ^{B_2}_{(n_1,n_2)} (x_1 x_2, x_1 x_2^{-1})~,\\end{split}$ where $(n_1, n_2)$ denotes the representation of $B_2=SO(5)$ in the standard $e$ -basis,This representation corresponds to Dynkin labels $[n_1-n_2, n_1+n_2]_{SO(5)}$ or $[n_1+n_2, n_1-n_2]_{USp(4)}$ .", "where $n_1 \\ge n_2 \\ge 0$ with $n_1, n_2$ all integers or all half-integers.", "One can explicitly check that (REF ) with $y_2=1$ can also be obtained from the monopole formula (REF ) for the Coulomb branch Hilbert series.", "Again, only one fugacity corresponding to the topological charge of the $SO(2)$ group is manifest." ], [ "Comparison with gluing.", "We can obtain $T(USp(4))$ by gluing $T(SO(4))$ theory with $[SO(4)]-[USp(4)]$ via the $SO(4)$ group.", "We have the Coulomb branch Hilbert series: $\\begin{split}& H_{\\text{glued}}[T(USp(4))] (t; x_1, x_2; n_1, n_2) \\\\&=\\sum _{\\begin{array}{c}k_1 \\ge \|k_2\|\\\\ k_2 \\in \\mathbb {Z}\\end{array}} H[T(SO(4))] (t; x_1, x_2 ; k_1, k_2) \\times \\\\& \\qquad \\qquad t^{-\\delta _{D_2}(k_1,k_2)+\\frac{1}{2}\\delta _{D_2-C_2}(k_1,k_2, n_1,n_2;t)} P_{SO(4)} (t;k_1,k_2)~.\\end{split}$ where $P_{SO(4)} (t;k_1,k_2)$ is given by (A.10) of [1] and $\\begin{split}\\delta _{D_2}(k_1,k_2) &=\\sum _{j=1}^2 \\sum _{i=1}^{j-1} \|k_i-k_j\| +\|k_i+k_j\|~, \\\\\\delta _{D_2-C_2}(k_1,k_2, n_1,n_2;t) &= \\sum _{s=0}^1 \\sum _{i=1}^2 \\sum _{j=1}^2 \|(-1)^s n_i - k_j\|~.\\end{split}$ Then, (REF ) can be matched with (REF ) as follows: $\\begin{split}&H[T(SO(5)](t; x_1,x_2; n_1+n_2, n_1-n_2) \\\\&= H_{\\text{glued}}[T(USp(4))] (t; x_1 x_2, x_1 x_2^{-1}; n_1, n_2)~.\\end{split}$ Comparing (REF ) with (REF ), we see that $H_{\\text{glued}}[T(USp(4))] (t; x_1, x_2; n_1, n_2)= H[T(USp(4))](t; x_1, x_2; n_1, n_2)~.$" ], [ "$T(SO(6))$ and comparison with {{formula:bdd96556-cafa-444d-ab42-47fed1a06c8c}}", "The HL formula for the Coulomb branch of $T(SO(6))$ is given by $H[T(SO(6))(t; \\mathbf {x}; \\mathbf {n})= t^{\\frac{1}{2}(4n_1+2n_2)} \\mathop {\\rm PE}\\left[t \\left(\\chi _{[0,1,1]}^{D_3} ({\\mathbf {x}})-3 \\right) \\right] \\Psi ^{(n_1,n_2,n_3)} _{D_3} (\\mathbf {x}; t)~.$ Since the Lie algebra of $SO(6)$ is isomorphic to that of $SU(4)$ , we expect the matching between the Coulomb branch Hilbert series of $T[SU(4)]$ and that of $T[SO(6)]$ .", "Indeed, the Coulomb branch Hilbert series (REF ) for $G=SU(4)$ and $\\rho =(1^4)$ , $\\begin{split}& H[T(SU(4))](t; \\mathbf {y}; n_1,n_2,n_3,0 ) \\\\&= t^{\\frac{1}{2}(3n_1+n_2-n_3)} \\mathop {\\rm PE}\\left[ t \\left(\\chi ^{SU(4)}_{[1,0,1]} (\\mathbf {y}) -4 \\right) \\right] \\Psi ^{(n_1,n_2,n_3,0)}_{U(4)} (\\mathbf {y};t)~,\\end{split}$ agrees with (REF ) upon a suitable translation.", "Explicitly, for any $a_1, a_2, a_3 \\ge 0$ , $H[T(SO(6)] \\left(t;x_1,x_2,x_3; \\mathbf {m} (\\mathbf {a}) \\right) = H[T(SU(4))](t; y_1,y_2,y_3; \\mathbf {n}(\\mathbf {a}) ) ~,$ with $\\begin{split}\\mathbf {m} (\\mathbf {a}) &= \\left(\\frac{1}{2}a_1+ a_2+ \\frac{1}{2}a_3, \\; \\frac{1}{2}a_1+\\frac{1}{2}a_3, \\; -\\frac{1}{2}a_1+\\frac{1}{2}a_3 \\right)~, \\qquad \\\\\\mathbf {n} (\\mathbf {a}) &= (a_1+a_2+a_3, \\; a_2+a_3, \\; a_3, \\;0)~,\\end{split}$ and the fugacity map $y_1^2=\\frac{x_1}{x_2 x_3}~, \\qquad y_2^2=\\frac{x_2}{x_1 x_3}~, \\qquad y_3^2=\\frac{x_3}{x_1 x_2}~,\\qquad y_4^2 = x_1 x_2 x_3~.$ For the case of $SO(6)$ , (REF ) with $x_2=x_3=1$ can also be reproduced from the monopole formula (REF ) for the Coulomb branch Hilbert series.", "Again, only one fugacity corresponding to topological charge can be made manifest in the latter." ], [ "Background magnetic charges and baryonic charges", "Having computed the Coulomb branch Hilbert series of many theories with background magnetic charges turned on, it is natural to ask which quantities on the Higgs branch are such background charges mapped to under mirror symmetry.", "In this section we show that the answer is the baryonic charges in the mirror of the theory in question.", "So, in a sense, as expected from mirror symmetry which acts as S-duality on brane configurations in type IIB string theory, the magnetic background fluxes are mapped to background electric fluxes.", "The relation between the Hilbert series with background fluxes of a theory and its mirror can be made precise as following.", "If the gauge group $G^M$ of the mirror theory contains a $U(1)$ factor, it also has a $U(1)$ topological symmetry acting on its Coulomb branch.", "In the original theory, this $U(1)$ will be acting on the Higgs branch.", "Ungauging the $U(1)$ gauge symmetry in the mirror is equivalent to gauging the corresponding global $U(1)$ symmetry in the original theory.", "We thus have the following equality ${\\cal G}[G^M/U(1)] (t,b) = \\sum _{m\\in \\mathbb {Z}} {\\cal G}[G^M/U(1),m](t) b^m = \\frac{1}{1-t} \\sum _{m\\in \\mathbb {Z}} H[G,U(1)](t,m) b^m~,$ where we explain the notation as follows.", "${\\cal G}[G^M/U(1)](t,b) $ is the Hilbert series for the Higgs branch of the mirror theory with gauge group $G^M/U(1)$ , graded according to the dimension and the baryonic symmetry corresponding to the ungauged $U(1)$ .", "This function is known as the bayonic generating function of the mirror theory [21], [10], [22].", "${\\cal G}[G^M/U(1)](t,b)$ can be decomposed into sectors of definite baryonic charge by writing the Hilbert series as a formal Laurent series in the baryonic fugacity $b$ .", "We denote such a function in the $m$ -th sector by ${\\cal G}[G^M/U(1),m](t)$ .", "$H[G,U(1)](t,m)$ is the Hilbert series of the Coulomb branch of the original theory with a background magnetic flux for the global $U(1)$ .", "The right hand side of the previous formula is the result of gauging this global $U(1)$ : $b$ is a fugacity for the topological symmetry that arises after gauging.", "Even though the correspondence (REF ) holds separately for each $U(1)$ group, for simplicity we will mostly present examples where we gauge (resp.", "ungauge) the maximal torus of the symmetry group acting on the Coulomb branch (resp.", "Higgs branch of the mirror theory).", "Given a theory $T$ , we use the notations $T_{\\mathbf {B}}$ for the theory obtained by replacing all $U(N_i)$ gauge groups in $T$ by $SU(N_i)$ gauge groups, and $T_{\\mathbf {J}}$ for the theory obtained by gauging the maximal torus of the flavor symmetry.", "$\\mathbf {B}$ and $\\mathbf {J}$ label the baryonic and topological symmetries gained in the two processes.", "If theories $T$ and $T^M$ are mirror theories, so are $T_{\\mathbf {B}}$ and $T^M_{\\mathbf {J}}$ .", "The equivalence of ungauging a $U(1)$ on the Coulomb branch of a theory and gauging a $U(1)$ on the Higgs branch of the mirror theory is straightforward to see at the level of Hilbert series [1].", "The equivalence of gauging a $U(1)$ on the Coulomb branch of a theory and ungauging a $U(1)$ on the Higgs branch of the mirror theory is much less trivial to see at the level of Hilbert series.", "We show it in a number of examples in the rest of the section." ], [ "$T(SU(N))$ theory: {{formula:fd0629a5-e867-4528-b51c-0197389a657d}} quiver", "Let us first consider $T(SU(N))$ theories, which are self-mirror.", "In the following we provide explicit examples of $N=2$ and $N=3$ , from which a general formula for any $N$ can be concluded." ], [ "$T(SU(2))$ theory", "The baryonic generating function of $(1)-[2]$ quiver is defined as the Hilbert series of the Higgs branch of the same theory with the $U(1)$ node ungauged [10], i.e.", "that of $[1]-[2]$ quiver.", "The charge of the hypermultiplets under this ungauged $U(1)$ has an interpretation of the baryonic charge of the $(1)-[2]$ theory.", "Let $b$ be the fugacity associated with the baryonic charges.", "The baryonic generating function is then given by ${\\cal G}[T(SU(2))_B](t; x; b) = \\mathop {\\rm PE}[ (b+b^{-1})(x+x^{-1}) t^{1/2}]~,$ where $x$ is the fugacity for $SU(2)$ .", "After a rescaling $t \\rightarrow t^2$ needed to compare with 4d quantities, the function $(1-t){\\cal G}[T(SU(2))_B](t; x; b) \\nonumber $ is indeed the $F$ -flat Hilbert series [29], [30] of $(1)-[2]$ quiver and the baryonic generating function of $\\mathbb {C}^2/\\mathbb {Z}_2$ ; see (4.6) of [10].", "The baryonic generating function can be related to the Coulomb branch Hilbert series of $T(SU(2))$ with the background flux turned on.", "Indeed, we find that $\\begin{split}{\\cal G}[T(SU(2))_B](t; x; b)&= \\mathop {\\rm PE}[ (b+b^{-1})(x+x^{-1}) t^{1/2}] \\\\&= \\frac{1}{1-t} \\sum _{n=-\\infty }^\\infty H[T(SU(2))] (t; x, x^{-1}; \|n\|, 0) \\; b^{n} \\\\&= H[T(SU(2))_J](t;x;b) ~, \\end{split}$ where $H[T(SU(2))]$ is given by (REF ) with $\\mathbf {\\rho }=(1,1)$ , which extends to the Weyl chamber $n<0$ by replacing $n\\rightarrow -n$ , or equivalently by the monopole formula (REF ): $H[T(SU(2))](t; x_1, x_2; n_1, n_2) = x_1^{n_1+n_2} \\sum _{u=-\\infty }^\\infty t^{\\frac{1}{2}(\|u-n_1\|+\|u-n_2\|)} \\frac{1}{1-t} (x_2 x_1^{-1})^{u}~.$ The summation in (REF ) has an interpretation of generating function for the Coulomb branch Hilbert series with background fluxes.", "Equation (REF ) relates this generating function to the baryonic generating function on the Higgs branch of the theory.", "For reference, we provide the expressions for such a function in the regions $n \\ge 0$ and $n<0$ : $\\sum _{n\\ge 0} H[T(SU(2))] (t; x, x^{-1}; \|n\|, 0) \\; b^{n} &=\\frac{\\left(1+t\\right) -t^{3/2} \\left(x^{-1}+x\\right) b}{\\left(1-tx^{-2}\\right) \\left(1-t x^2\\right) \\left(1-t^{1/2} b x^{-1}\\right) (1-t^{1/2} b x)} ~,\\nonumber \\\\\\sum _{n<0} H[T(SU(2))] (t; x, x^{-1}; \|n\|, 0) \\; b^{n} &= \\frac{t^{1/2} b^{-2} \\left[\\left(x^{-1}+x\\right) b-t^{1/2} \\left(1+t\\right)\\right]}{\\left(1-tx^{-2}\\right) \\left(1-t x^2\\right) \\left(1-t^{1/2} x b^{-1}\\right) \\left(1-t^{1/2} x^{-1} b^{-1}\\right)}~.\\nonumber \\\\$ As anticipated, (REF ) shows that ungauging the $U(1)$ gauge symmetry on the Higgs branch side corresponds to gauging a $U(1)$ global symmetry of the Coulomb branch of the mirror, and the baryonic charges are just the topological charge in the Coulomb branch.", "In this gauging process we introduce the factor $(1-t)^{-1}$ on the right hand side of (REF ).", "The generating function of the Coulomb branch Hilbert series of the quiver $(1)-[2]$ is nothing but the Hilbert series of the Coulomb branch of the theory obtained by gauging the $U(1)$ Cartan subgroup of the flavor symmetry $U(2)/U(1)$ , namely the quiver $(1)-(1)-[1]$ , or equivalently $(1)-(1)-(1)$ with the overall $U(1)$ gauge group factored out." ], [ "$T(SU(3))$ theory", "As before, the baryonic generating function of $(1)-(2)-[3]$ quiver is equal to the Higgs branch Hilbert series of the same theory with the two unitary gauge groups replaced by special unitary groups: $\\begin{split}& {\\cal G}[T(SU(3))_{\\mathbf {B}}](t; x_1, x_2, x_3; b_1, b_2) \\\\&= \\int {\\rm d} \\mu _{SU(2)} ( z) \\frac{1}{\\mathop {\\rm PE}[ t(z^2+1+z^{-2}) ]} \\mathop {\\rm PE}\\Big [(b_1^{-1}b_2 (z+z^{-1}) +b_1 b_2^{-1} (z+z^{-1}) \\\\& \\qquad + b_2(z+z^{-1})\\sum _{i=1}^3 x_i^{-1} +b_2^{-1}(z+z^{-1})\\sum _{i=1}^3 x_i ) t^{1/2} \\Big ]~,\\end{split}$ where $b_1$ and $b_2$ are baryonic fugacities; they can also be viewed as electric fugacities for the $U(1)$ node and for the $U(1)$ center of the $U(2)$ node respectively, $x_1,x_2,x_3$ , with $x_1 x_2 x_3=1$ , are fugacities of the $SU(3)$ flavor symmetry, the Haar measure of $SU(2)$ is $\\int {\\rm d} \\mu _{SU(2)} ( z) = \\frac{1}{2 \\pi i} \\oint _{\|z\|=1} \\frac{1-z^2}{z}~.$" ], [ "Coulomb branch Hilbert series and baryonic generating function", "We are interested in relating (REF ) to the Coulomb branch Hilbert series of $T(SU(3))$ .", "It is convenient to write the latter using the monopole formula (REF ) as follows: $\\begin{split}& H[T(SU(3))](t; x_1, x_2, x_3; n_1,n_2, n_3 =0 ) \\\\&=x_1^{n_1+n_2} \\sum _{m_{1,1}=-\\infty }^\\infty \\sum _{m_{1,2} \\ge m_{2,2} > -\\infty }^\\infty (x_2 x_1^{-1})^{m_{1,2}+m_{2,2}} (x_3 x_2^{-1})^{m_{1,1}} \\times \\\\& \\qquad t^{\\Delta (m_{1,1}; m_{1,2},m_{2,2}; n_1, n_2,0)} P_{U(2)}(m_{1,2},m_{2,2};t)P_{U(1)}(t) ~,\\end{split}$ with $x_1 x_2 x_3=1$ , $P_{U(2)}(\\mathbf {m}; t)$ and $P_{U(1)}(t)$ given by (REF ), and $\\begin{split}&\\Delta (m_{1,1}; m_{1,2},m_{2,2}; n_1, n_2,n_3) \\\\&= \\frac{1}{2}\\sum _{i=1}^2 \|m_{1,1}-m_{i,2}\|+ \\frac{1}{2}\\sum _{i=1}^2 \\sum _{j=1}^3 \|m_{i,2}-n_j\|-\|m_{1,2}-m_{2,2}\|~.\\end{split}$ Indeed we find that (REF ) and (REF ) are related as $\\begin{split}& {\\cal G}[T(SU(3))_{\\mathbf {B}}](t, \\mathbf {x}, \\mathbf {b}) \\\\& = (1-t)^{-2} \\sum _{n_1, n_2 \\in \\mathbb {Z}} H[T(SU(3))](t; x_1^{-1}, x_2^{-1},x_3^{-1};n_1,n_2,0) b_1^{n_1} (b_2^2/b_1)^{n_2}\\\\& = H[T(SU(3))_{\\mathbf {J}}](t; x_1^{-1}, x_2^{-1},x_3^{-1};b_1, b_2^2 b_1^{-1})~.\\end{split}$ This indeed confirms the relation between the generating function of the Coulomb branch Hilbert series and the baryonic generating function, or the Higgs branch Hilbert series of $T(SU(3))_{\\mathbf {B}}$ and the Coulomb branch Hilbert series of $T(SU(3))_{\\mathbf {J}}$ .", "Note that $b_2$ appears only as $b_2^2$ in (REF ) because it is the baryonic fugacity of an $SU(2)$ gauge group normalized in such a way that gauge invariants have charges in $2\\mathbb {Z}$ .", "Alternatively to the monopole formula (REF ), the Coulomb branch Hilbert series is also given by the HL formula (REF ): $H_3(t; x_1,x_2, x_3; n_1,n_2,0) = t^{n_1} (1-t)^3 \\mathop {\\rm PE}\\left[t \\sum _{i,j=1}^3 x_i x_j^{-1} \\right] \\Psi ^{(n_1,n_2,0)}_{U(3)} (\\mathbf {x}; t)~,$ where we emphasize that this formula is valid only if $n_1 \\ge n_2 \\ge 0$ and we take $x_1 x_2 x_3=1$ .", "The relation with (REF ) can therefore be separated into 6 regions, as in Appendix A.2 of [22].", "Explicitly, we obtain ${\\cal G}[T(SU(3))_{\\mathbf {B}}](t; \\mathbf {x}; \\mathbf {b}) &= (1-t)^{-2} \\times \\nonumber \\\\&\\Bigg [ \\sum _{m=0}^\\infty \\sum _{n=0}^\\infty H_3(t; x_1^{-1},x_2^{-1},x_3^{-1}; m+n,n) b_1^m b_2^{2n} \\nonumber \\\\&+ \\sum _{m=1}^\\infty \\sum _{n=1}^\\infty H_3(t; x_1,x_2,x_3; m+n,n) b_1^{-m} b_2^{-2n} \\nonumber \\\\&+ \\sum _{n=1}^\\infty \\sum _{m=n+1}^\\infty H_3(t; x_1^{-1},x_2^{-1},x_3^{-1}; m,n) b_1^{m} b_2^{-2n} \\\\&+ \\sum _{n=0}^\\infty \\sum _{m=n+1}^\\infty H_3(t; x_1,x_2,x_3; m,n) b_1^{-m} b_2^{2n} \\nonumber \\\\&+ \\sum _{m=0}^\\infty \\sum _{\\begin{array}{c}n=m \\\\ n\\ne 0\\end{array}}^\\infty H_3(t; x_1^{-1},x_2^{-1},x_3^{-1}; n,m) b_1^{m} b_2^{-2n} \\nonumber \\\\&+ \\sum _{m=1}^\\infty \\sum _{n=m}^\\infty H_3(t; x_1,x_2,x_3; n,m) b_1^{-m} b_2^{2n} \\Bigg ]~, \\nonumber $ where $x_1x_2x_3=1$ .", "Each summand indicates the baryonic generating function, in terms of the Coulomb branch Hilbert series, in one of the 6 Weyl chambers of $SU(3)$ ." ], [ "$T(SU(N))$ theory", "It is straightforward to derive a similar relation to (REF ) and (REF ) for a general $T(SU(N))$ theory.", "Using the same notation as before, we obtain $& {\\cal G}[T(SU(N))_{\\mathbf {B}}](t, x_1, \\ldots , x_N, b_1, \\ldots , b_{N-1}) = \\frac{1}{(1-t)^{N-1}} \\times \\nonumber \\\\& \\sum _{n_1, \\cdots , n_{N-1} \\in \\mathbb {Z}} H[T(SU(N))](t; x_1^{\\pm }, x_{2}^{\\pm },\\cdots ,x_{N}^{\\pm };n_1,n_2, \\cdots , n_{N-1},0) b_1^{n_1} \\prod _{k=2}^{N-1} (b_{k}^{k} b_{k-1}^{1-k})^{n_{k}}~ = \\nonumber \\\\&~~~= H[T(SU(N))_{\\mathbf {J}}](t; x_1^{\\pm }, x_{2}^{\\pm },\\cdots ,x_{N}^{\\pm };b_1, b_2^2 b_1^{-1}, \\cdots , b_{N-1}^{N-1} b_{N-2}^{-(N-2)}),$ where the power $\\pm $ is $+$ for $N$ even and $-$ for $N$ odd.", "(REF ) shows that the Hilbert series of the Higgs branch of $T(SU(N))_{\\mathbf {B}}$ and the Coulomb branch of $T(SU(N))_{\\mathbf {J}}$ coincide.", "The latter is described by a quiver where the flavor group is replaced by $N$ $U(1)$ nodes joined to the $U(N-1)$ node, with the overall $U(1)$ gauge group factored out.", "The corresponding quiver diagram is depicted in Figure REF .", "Figure: The quiver diagram of T(SU(N)) 𝐉 T(SU(N))_{\\mathbf {J}}.", "In this quiver, the flavor group U(N)U(N) of T(SU(N))T(SU(N)) is replaced by NN U(1)U(1) nodes joined to the U(N-1)U(N-1) node, with the overall U(1)U(1) gauge group factored out." ], [ "$T_{(2,1)}(SU(3))$ theory: {{formula:20006c3f-bae2-46df-98d1-6536234e7d09}} quiver", "The mirror theory of $T_{(2,1)}(SU(3))$ is $\\begin{tikzpicture}[font=\\scriptsize ]\\begin{scope}[auto,every node/.style={draw, minimum size=1cm}, node distance=1cm];\\node [circle] (U1a) at (0, 0) {U(1)};\\node [circle, right=of U1a] (U1b) {U(1)};\\node [rectangle, below=of U1a] (U1f1) {U(1)};\\node [rectangle, below=of U1b] (U1f2) {U(1)};\\end{scope}(U1a) -- (U1b)(U1a)--(U1f1)(U1b)--(U1f2);\\end{tikzpicture}$ The baryonic generating function of the mirror of $T_{(2,1)}(SU(3))$ is given by $\\begin{split}& {\\cal G}[(\\text{mirror $T_{(2,1)}(SU(3))$})_{\\mathbf {B}}] (t; x_1, x_2; b_1, b_2) \\\\& = \\mathop {\\rm PE}\\left[ ( b_1 b_2^{-1}+b_1^{-1} b_2 + b_1 x_1^{-1} +b_1^{-1} x_1+ b_2 x_2^{-1} +b_2^{-1} x_2) t^{1/2} \\right]~.\\end{split}$ On the other hand, the Hilbert series of the Coulomb branch of $T_{(2,1)}(SU(3)): (1)-[3]$ is given by (REF ).", "Formulae (REF ) and (REF ) are related as $\\begin{split}& {\\cal G}[(\\text{mirror $T_{(2,1)}(SU(3))$})_{\\mathbf {B}}] (t; x_1, x_2; b_1, b_2) \\\\& = \\frac{1}{(1 - t)^{2}}\\sum _{n_1,n_2 \\in \\mathbb {Z}} (x_1 x_2)^{-n_2} H[T_{(2,1)}(SU(3))] (t; x_1, x_2; n_1, n_2,0) b_1^{-n_1} b_2^{n_2} \\\\& = \\frac{1}{(1 - t)^{3}}\\sum _{m,n_1,n_2 \\in \\mathbb {Z}} \\left(\\frac{x_2}{x_1}\\right)^{m}\\left(\\frac{x_1}{b_1}\\right)^{n_1} \\left(\\frac{b_2}{x_2}\\right)^{n_2} t^{\\frac{1}{2}\\left(\\sum _{i=1}^2 \|m-n_i\|+\|m\|\\right)} ~.\\end{split}$ This again confirms the relation between the generating function of the Coulomb branch Hilbert series and the baryonic generating function of the mirror theory." ], [ "$T_{(3,1,1)}(USp(4))$ theory: {{formula:07dd2d5f-0e78-49ea-8eee-280c531e7962}} quiver", "We now consider the $T_{(3,1,1)}(USp(4))$ theory, which corresponds to $SO(2)$ gauge theory with 4 flavors of half-hypermultiplets in the two-dimensional vector representation.", "The aim of this section is to present certain subtleties that do not appear in $T_{\\mathbf {\\rho }} (SU(N))$ theory.", "The Coulomb branch Hilbert series of this theory is discussed in appendix REF .", "In the following we shall compute the baryonic generating of the Higgs branch of the mirror theory and establish some relations with such a Coulomb branch Hilbert series." ], [ "The Coulomb branch Hilbert series of $T_{(3,1,1)}(USp(4))$", "The monopole formula for the Coulomb branch Hilbert series of $T_{(3,1,1)}(USp(4))$ is $H[T_{(3,1,1)}(USp(4))](t; z; n_1, n_2) = \\sum _{m=-\\infty }^{\\infty } t^{\\Delta (n_1,n_2; m)} P_{SO(2)}(t) z^{2m}~,$ where $z^2$ keeps track of the topological charge of the gauge group $SO(2)$ , and $P_{SO(2)}(t) = \\frac{1}{1-t}~, \\qquad \\Delta (n_1,n_2; m) = \\frac{1}{2}\\sum _{i=1}^2 \\sum _{s=0}^1 \|(-1)^s n_i +m\|~.$ If we set $n_1=n_2=0$ , the Coulomb branch Hilbert series simplifies to $H[T_{(3,1,1)}(USp(4))](t; z; 0,0) = \\mathop {\\rm PE}\\left[ t + t^2( z^2+ z^{-2}) -t^4\\right] ~.$ Note that this is the Hilbert series of $\\mathbb {C}^2/\\mathbb {Z}_4$ , the Coulomb branch of the $U(1)$ gauge theory with 4 flavors of hypermultiplets of charge 1, which is the same as the $SO(2)$ gauge theory with 4 flavors of half-hypermultiplets in the vector representation of $SO(2)$ .", "Next we compute the baryonic generating function of the mirror of $T_{(3,1,1)}(USp(4))$ .", "The mirror theory of $T_{(3,1,1)}(USp(4))$ theory was discussed in section 6.2 of [28].", "Its quiver is given by $\\begin{split}& \\text{Mirror of $T_{(3,1,1)}(USp(4))$:} \\qquad \\\\& \\qquad [SO(2)]-(USp(2))-(SO(2))-(USp(2))-[SO(2)]~.\\end{split}$ This is a `bad' theory, since the number of flavors of each $USp(2)$ gauge node is 2, which is less than $2(2)+1=5$ .", "In fact, this theory flows to the mirror theory of $U(1)$ gauge theory with 4 flavors, as we shall demonstrate below (REF ).", "Here we are interested in the Higgs branch, which is protected from quantum corrections, so we can use the `bad' quiver to compute it.", "The computation of the $F$ -flat Hilbert series and the baryonic generating function of this mirror theory, obtained by ungauging the $SO(2)$ group, is rather techical, and we relegate it to Appendix REF .", "We present only the end result of the baryonic generating function, which is given by (REF ): $\\begin{split}{\\cal G}(t; x, y; b)&:={\\cal G}[(\\text{Mirror of $T_{(3,1,1)}(USp(4))$})_B](t; x, y; b) \\\\&= \\mathop {\\rm PE}\\left[ \\lbrace 2+(b x^{-1} + b^{-1} x) + (b y^{-1} + b^{-1} y) \\rbrace t - 2t^2\\right] \\\\&= \\mathop {\\rm PE}\\left[ \\lbrace 2+(b_1+b_1^{-1}) + (b_2+b_2^{-1}) \\rbrace t - 2t^2 \\right]~.\\end{split}$ where $x$ and $y$ are the fugacities of the two $SO(2)$ flavor symmetries, and $b$ is the baryonic fugacity corresponding to the ungauged $SO(2)$ in (REF ).", "Two combinations of the baryonic and flavor fugacities appear in this generating function, namely $b_1 := b x^{-1}~, \\qquad b_2 := b y^{-1}~.$ Note that (REF ) is, in fact, the Hilbert series of $(\\mathbb {C}^2/\\mathbb {Z}_2)^2$ , where $b_1$ and $b_2$ are fugacities for the $SU(2)$ isometry associated with each copy of $\\mathbb {C}^2/\\mathbb {Z}_2$ .", "The Higgs branch Hilbert series of quiver (REF ) is given by (REF ): $\\begin{split}\\oint _{\|b\|=1} \\frac{{\\rm d} b}{2 \\pi i b} (1-t) {\\cal G}(t; x, y; b) &= \\mathop {\\rm PE}\\left[ t+t^2( x y^{-1} + y x^{-1}) - t^4\\right] \\\\&= H[T_{(3,1,1)}(USp(4))](t; x^{1/2}y^{-1/2}; 0, 0)~; \\end{split}$ this is equal to the Coulomb branch Hilbert series (REF ) with vanishing background fluxes $n_1=n_2=0$ , as predicted by mirror symmetry.", "Indeed, this indicates that the `bad' theory (REF ) flows to the mirror theory of $U(1)$ gauge theory with 4 flavors in the infra-red.", "We find that ${\\cal G}_{\\mathbb {Z}_2}(t; x=z, y=z^{-1}; b)= (1-t)^{-1} \\sum _{n=-\\infty }^\\infty H[T_{(3,1,1)}(USp(4))](t; z; n, n) b^{2n}~, $ where ${\\cal G}_{\\mathbb {Z}_2}(t; z, z^{-1}; b)$ is a $\\mathbb {Z}_2$ projection of ${\\cal G}(t; z, z^{-1}; b)$ defined as follows: ${\\cal G}_{\\mathbb {Z}_2}(t; z, z^{-1}; b) := \\frac{1}{2} \\left[ {\\cal G}(t; z, z^{-1}; b)+{\\cal G}(t; z, z^{-1}; -b) \\right] ~.$ Note that this $\\mathbb {Z}_2$ projection is not to be identified with the parity of the orthogonal gauge group $O(2)$ , cf.", "(REF ), which does not commute with the Cartan elements of $SO(2)$ .", "It is rather the $\\mathbb {Z}_2$ subgroup of $SO(2)$ that consists of gauge rotations by multiples of $\\pi $ .", "As such, it projects out the odd powers of $b$ in ${\\cal G}(t; z, z^{-1}; b)$ .", "This example displays several subtleties that are not present for $T_{\\mathbf {\\rho }}(SU(N))$ .", "Let us comment and list certain open questions as follows: There are two background fluxes $n_1$ and $n_2$ for the flavor symmetry $USp(4)$ of the theory $T_{(3,1,1)} (USp(4))$ but only one baryonic charge $b$ (and correspondingly only one manifest topological $SO(2)$ symmetry) in the mirror theory.", "This mismatch is very much like the situation discussed in (REF ).", "This is related to the absence of Cartan generators of the symmetry $USp(4)$ on the Coulomb branch of the mirror of $T_{(3,1,1)} (USp(4))$ , where only $SO(2)$ is manifest.", "The remaining Cartan generators of $USp(4)$ correspond to monopole operators.", "The coefficients of odd powers of $b$ in (REF ) (i.e.", "those with odd baryonic charges) cannot be matched with the Coulomb branch Hilbert series (REF ) for any background fluxes.", "Only those of even powers of $b$ can be matched with the Coulomb branch Hilbert series (REF ); this happens when $n_1$ is set to be equal to $n_2$ .", "The $\\mathbb {Z}_2$ projection is there to get rid of the odd powers of $b$ in (REF ) and hence the matching can be done as in (REF ).", "We also observe that by gauging the $U(1)$ 's associated to $n_1$ and $n_2$ and ungauging the $U(1)$ associated to $m$ on the Coulomb branch side, one can reproduce the full ${\\cal G}$ in (REF )." ], [ "Analytic structure of the Coulomb branch Hilbert series", "In this section we examine the analytic structure of the Coulomb branch Hilbert series of $T_{\\mathbf {\\rho }}(SU(N))$ , from the complementary perspectives of the Hall-Littlewood formula and of the monopole formula.", "The study of the analytic structure further substantiates our conjecture that the Hall-Littlewood formula computes the Coulomb branch Hilbert series of $T_{\\mathbf {\\rho }}(SU(N))$ theories.", "The analysis of the HL formula is very similar to that of HL index of 4d Sicilian theories in [11].", "The prefactor $K^{U(N)}_{\\mathbf {\\rho }}(\\mathbf {x}; t)$ in the HL formula (REF ) has a pole corresponding to a particular box corresponding to the rightmost part of the partition $\\mathbf {\\rho }$ .", "Taking the residue of the Hilbert series at this pole isolates the Coulomb branch Hilbert series of a new theory $T_{\\mathbf {\\rho }^{\\prime }}(SU(N))$ , where $\\mathbf {\\rho }^{\\prime }$ is obtained from $\\mathbf {\\rho }$ by moving such a box to a previous column.", "We can identify the same simple pole in the monopole formula for the Coulomb branch Hilbert series with background fluxes of $T_{\\mathbf {\\rho }}(SU(N))$ .", "Computing the residue yields the monopole formula for $T_{\\mathbf {\\rho }^{\\prime }}(SU(N))$ , in complete analogy with the analysis of the HL formula.", "It thus follows that the equivalence of the monopole formula (REF ) with the Hall-Littlewood formula (REF ) in the case the maximal partition $\\rho = (1,1,\\cdots ,1)$ implies the equivalence of the two formulae for a generic partition $\\rho $ .", "Let us demonstrate the idea in the following example.", "The general case is discussed in Section REF and Appendix .", "Another example of the application of the same technique is discussed in Appendix ." ], [ "Obtaining $T_{(2,1)}(SU(3))$ from {{formula:63712e5b-9b4b-4390-aef9-b4dc0ceb3ae2}}", "Let us compare the Hilbert series with background fluxes for the theories $T_{(1,1,1)}(SU(3)): [3]-(2)-(1)$ and $T_{(2,1)}(SU(3)): [3]-(1)$ .", "The latter theory can be obtained from the former by moving one box as follows: ${\\begin{ytableau} ~& ~& (blue!20) \\end{ytableau}} & \\qquad \\longrightarrow \\qquad {\\begin{ytableau} ~& ~\\\\ (blue!20) \\end{ytableau}} \\nonumber \\\\\\qquad \\qquad \\mathbf {\\rho }=(1,1,1) & \\qquad \\qquad \\qquad \\mathbf {\\rho }^\\prime =(2,1)$ The relation between 4d superconformal indices of Sicilian theories including the former or the latter puncture was studied in [11].", "In this section, we perform a similar computation to find the relation between the Coulomb branch Hilbert series for the two $T_{\\mathbf {\\rho }}(SU(3))$ theories.", "From the HL formula (REF ), we obtain the expressions (REF ) and (REF ) for the Coulomb branch Hilbert series of $T_{(1,1,1)}(SU(3))$ and $T_{(2,1)}(SU(3))$ respectively.", "Formula (REF ) has simple poles at $x_i x_j^{-1}\\rightarrow t$ for $i\\ne j$ , due to the plethystic exponential factor in the second line (the HL polynomials are regular).", "In view of the permutation symmetry, we can focus on the pole at $x_1 x_3^{-1}\\rightarrow t$ .", "We set $x_1 = y_1 t^{1/2} z~, \\qquad x_2=y_2~, \\qquad x_3 = y_1 (t^{1/2} z)^{-1}~$ and consider the limit where $z\\rightarrow 1$ .", "Similarly to the discussion in section 2.2 of [11] for the HL limit of superconformal indices, the residue of (REF ) as $z\\rightarrow 1$ in (REF ) reproduces the Hilbert series (REF ) with a certain prefactor that is easily determined: $\\begin{split}& \\underset{z\\rightarrow 1}{\\rm Res} ~ H[T_{(1,1,1)}(SU(3))] (t; y_1 t^{1/2} z, y_2, y_1 t^{-1/2} z^{-1}; n_1, n_2) \\\\&=\\frac{1}{2} \\mathop {\\rm PE}\\left[t^{1/2} (y_1 y_2^{-1}+y_2 y_1^{-1})+t \\right] H[T_{(2,1)}(SU(3))] (t; y_1, y_2; n_1,n_2)~.\\end{split}$" ], [ "Analytic structure of the monopole formula", "The aim of this section is to show that the residue formula (REF ) can also be derived by means of the monopole formula of the Hilbert series (REF ).", "The idea is that when evaluating this expression at the values of $x_i$ corresponding to a pole, one introduces extra powers of $t$ depending on the monopole fluxes which make the series diverge along certain directions in the GNO lattice.", "Taking the residue, some gauge groups effectively disappear, reproducing the quiver corresponding to a different partition." ], [ "Example: pole at $x_1 x_3^{-1}\\rightarrow t$ in (", "The monopole formula (REF ) has a simple pole when the fugacities $\\mathbf {x}$ are as in (REF ) and $z\\rightarrow 1$ , because there is a region of the summation in (REF ) where the series diverges.", "To see this, we consider the region where $m_{1,1}, m_{2,2} \\gg m_{1,2},n_j~.$ Changing summation variable from $m_{1,1}$ to $q = m_{1,1} - m_{2,2}~,$ the power of $t^{1/2}$ in (REF ) becomes $\\begin{split}&\\sum _{j=1}^3 n_j-(m_{1,1}+m_{1,2}+m_{2,2})+ \\Delta _{(1,1,1)}(m_{1,1} = m_{2,2}+q; m_{1,2},m_{2,2}; \\mathbf {n}) \\\\&= \\sum _{j=1}^3 \|m_{1,2}-n_j\| + \|q\| = \\Delta _{(2,1)}(m_{1,2}; \\mathbf {n}) + \|q\| ~, \\end{split}$ where $\\Delta _{(2,1)}(m_{1,2}; \\mathbf {n}) = \\sum _{j=1}^3 \|m_{1,2}-n_j\| ~, \\quad n_3=0~,$ is twice the dimension of monopole operators in the quiver $[3]-(1)$ .", "In this region, the summations in (REF ) diverge when $z\\rightarrow 1$ because $m_{2,2}$ disappears from the summands.", "The singularity depends on the summation over $m_{2,2}$ , which depends only on $z$ : $\\underset{z \\rightarrow 1}{\\rm Res} \\sum _{m_{2,2}=L}^\\infty z^{n_1+n_2-q-m_{1,2}-2m_{2,2}} = \\underset{z \\rightarrow 1}{\\rm Res}~\\frac{z^{2-2L+n_1+n_2-q-m_{1,2}}}{z^2-1} = \\frac{1}{2}~,$ where $L$ is a lower cutoff larger than $m_{1,2}$ and $n_j$ .", "Because $P_{U(2)}(t; m_{1,2}, m_{2,2})$ becomes $1/(1-t)^2$ for $m_{2,2} \\gg m_{1,2}$ , we find that $\\begin{split}& \\underset{z\\rightarrow 1}{\\rm Res} ~ H[T_{(1,1,1)}(SU(3))] (t; y_1 t^{1/2} z, y_2, y_1 t^{-1/2} z^{-1}; n_1, n_2) \\\\&=\\frac{1}{2} \\frac{1}{1-t}\\left[\\frac{1}{1-t} \\sum _{q\\in \\mathbb {Z}} t^{\\frac{1}{2}\|q\|} \\left(\\frac{y_1}{y_2}\\right)^{q} \\right] \\left[\\frac{y_1^{n_1+n_2}}{1-t} \\sum _{m_{1,2}\\in \\mathbb {Z}} \\left( \\frac{y_2}{y_1} \\right)^{m_{1,2}} t^{\\frac{1}{2}\\Delta _{(2,1)}(m_{1,2};\\mathbf {n})}\\right]~.\\end{split}$ The factor in the first square brackets in (REF ) is the Coulomb branch Hilbert series of $U(1)$ with one flavor, which is mirror to a twisted hypermultiplet and evaluates to $\\frac{1}{1-t}\\sum _{q\\in \\mathbb {Z}} t^{\\frac{1}{2}\|q\|} (y_1 y_2^{-1})^{q} = \\frac{1}{(1-t^{1/2} y_1 y_2^{-1})(1-t^{1/2} y_2 y_1^{-1})}~.$ The factor in the second square brackets is the Coulomb branch Hilbert series for the quiver $[3]-(1)$ with background fluxes $\\mathbf {n}$ , corresponding to the partition $\\mathbf {\\rho }^\\prime =(2,1)$ .", "Therefore we conclude from the monopole formula that $\\begin{split}& \\underset{z\\rightarrow 1}{\\rm Res} ~ H[T_{(1,1,1)}(SU(3))] (t; y_1 t^{1/2} z, y_2, y_1 t^{-1/2} z^{-1}; n_1, n_2) \\\\&=\\frac{1}{2} \\mathop {\\rm PE}\\left[t^{1/2} (y_1 y_2^{-1}+y_2 y_1^{-1})+t \\right] H[T_{(2,1)}(SU(3))] (t; y_1,y_2; n_1,n_2)~,\\end{split}$ as deduced previously from the HL formula in (REF )." ], [ "Moving the last box: from partition $\\mathbf {\\rho }=(\\rho _1, \\cdots , \\rho _{d-h},\\rho _{d-h+1},1^{h})$ to {{formula:2145867e-08b8-4ec0-885b-687f75026e42}}", "The previous argument can be generalized to the following theories: $\\begin{split}T_{\\mathbf {\\rho }}(SU(N)): &\\quad {[N]-\\cdots -(h+\\rho _{d-h+1})-(h)-(h-1)-\\cdots -(2)-(1)}~ \\\\T_{\\mathbf {\\rho }^\\prime }(SU(N)): &\\quad {[N]-\\cdots -(h+\\rho _{d-h+1})-(h-1)-\\cdots -(2)-(1)}~\\end{split}$ where the total numbers of gauge groups are respectively $d$ and $d-1$ , and $N = h + \\sum _{k=1}^{d-h+1} \\rho _k~.", "$ This corresponds to moving the last box in a partition ending with 1 as follows: $\\begin{split}{boxsize=15pt}&\\begin{ytableau}(black!20) & [\\ldots ] & (black!20) & (blue!20) & (red!20) &(red!20)& [\\ldots ] & (red!20) & (blue!60) \\\\[\\vdots ] & [\\vdots ] & [\\vdots ] & [\\vdots ] \\\\(black!20) & [\\ldots ] & (black!20)& (blue!20) \\\\[\\vdots ] & [\\vdots ] & [\\vdots ] \\\\(black!20) & [\\ldots ] & (black!20) \\\\[\\vdots ] \\\\(black!20)\\end{ytableau} \\\\& \\qquad \\mathbf {\\rho }=(\\rho _1, \\cdots , \\rho _{d-h+1},1^{h}) \\\\& \\qquad \\qquad \\qquad \\rotatebox {-90}{\\XMLaddatt {origin}{c}\\scalebox {1.5}{\\longrightarrow }}\\\\ \\\\& \\begin{ytableau}(black!20) & [\\ldots ] & (black!20) & (blue!40) & (red!20) &(red!20)& [\\ldots ] & (red!20) \\\\[\\vdots ] & [\\vdots ] & [\\vdots ] & [\\vdots ] \\\\(black!20) & [\\ldots ] & (black!20) & (blue!40) \\\\[\\vdots ] & [\\vdots ] & [\\vdots ] & (blue!40) \\\\(black!20) & [\\ldots ] & (black!20) \\\\[\\vdots ] \\\\(black!20)\\end{ytableau} \\\\& \\quad \\mathbf {\\rho }^\\prime =(\\rho _1, \\cdots , \\rho _{d-h+1}+1,1^{h-1})\\end{split}$ The grey boxes constitute the first spectator block $(\\rho _1, \\cdots , \\rho _{d-h+1})$ , the pink boxes constitute the second spectator block $(1^{h-1})$ of the partition $(\\rho _1, \\cdots , \\rho _{d-h+1},1^{h-1})$ , and the blue boxes belong to the columns involved in the move.", "Note that any partition $\\sigma $ of $N$ can be obtained from $(1^N)$ by an iteration of the previous move.", "Therefore by repeated residue computations we may extract the Coulomb branch Hilbert series of any $T_{\\mathbf {\\sigma }}(SU(N))$ theory from that of $T(SU(N))$ .", "Let us denote the fugacities corresponding to each column of $\\mathbf {\\rho }$ and $\\mathbf {\\rho }^\\prime $ by $\\begin{split}\\mathbf {\\rho }: &\\qquad (x_1, \\ldots , x_{d-h}, {\\color {blue!60} x_{d-h+1}}, {\\color {red} x_{d-h+2}, \\ldots , x_d},{\\color {blue!100!}", "x_{d+1}})~, \\\\\\mathbf {\\rho }^\\prime : &\\qquad (x_1, \\ldots ,x_{d-h}, {\\color {blue!85} y_{d-h+1}}, {\\color {red} x_{d-h+2}, \\ldots , x_d} )~,\\end{split}$ where the fugacities are coloured in accordance with the boxes in (REF ).", "They are subject to the constraints $& \\left( \\sum _{k=1}^{d-h} x_k^{\\rho _k} \\right) (x_{d-h+1}^{\\ell }) \\left( \\prod _{i=1}^{h-1} x_{d-h+1+i} \\right) x_{d+1} = 1~, \\\\& y^{\\ell +1}_{d-h+1} = x_{d-h+1}^\\ell x_{d+1}~, \\qquad \\text{with $\\ell := \\rho _{d-h+1}$}~.", "$ The corresponding brane configurations are depicted in Figure REF .", "Figure: The top and bottom brane configurations correspond to the partitions ρ\\mathbf {\\rho } to ρ ' \\mathbf {\\rho }^\\prime , respectively.", "The notation is as indicated in Figure .", "Here ℓ=ρ d-h+1 \\ell = \\rho _{d-h+1}.Analogously to the analysis of the HL limit of superconformal indices of 4d Sicilian theories in [11], we find that the HL formula (REF ) for the Coulomb branch Hilbert series of $T_{\\mathbf {\\rho }}(SU(N))$ has a simple pole at $x_{d-h+1} x_{d+1}^{-1} = t^{\\frac{1}{2}(1+\\ell )}$ .", "We parametrize $x_{d-h+1}=y_{d-h+1} t^{\\frac{1}{2}}z~, \\qquad x_{d+1}=y_{d-h+1} (t^{\\frac{1}{2}}z)^{-\\ell }~,$ in agreement with (REF ), and compute the residue at the pole $z\\rightarrow 1$ .", "The residue of the prefactor (REF ) is $\\begin{split}& \\underset{z \\rightarrow 1}{\\rm Res} ~ K_{{\\mathbf {\\rho }}} (x_1, \\ldots , x_{d-h},y_{d-h+1} t^{\\frac{1}{2}}z,x_{d-h+2}, \\ldots , x_{d},y_{d-h+1} (t^{\\frac{1}{2}}z)^{-\\ell };t) \\\\&= {\\cal P}_{\\mathbf {\\rho }\\mathbf {\\rho }^\\prime } (\\mathbf {x}; y_{d-h+1};t) K_{{\\mathbf {\\rho }}^\\prime } (x_1, \\ldots , x_{d-h},y_{d-h+1},x_{d-h+2}, \\ldots , x_{d};t) ~,\\end{split}$ where ${\\cal P}_{\\mathbf {\\rho }\\mathbf {\\rho }^\\prime } (\\mathbf {x}; y_{d-h+1};t)= \\frac{1}{1+\\ell } \\mathop {\\rm PE}\\left[t+ t^{\\frac{1}{2}} \\sum _{i=1}^{h-1} \\sum _{s=\\pm 1}\\left(t^{\\frac{1}{2}(\\ell -1)} \\frac{x_{d-h+1+i}}{y_{d-h+1}}\\right)^s \\right]~,$ whereas the HL polynomial with arguments determined by the partition $\\mathbf {\\rho }$ in (REF ) tends to the HL polynomial with arguments determined by $\\mathbf {\\rho }^\\prime $ .", "Therefore we find that $\\begin{split}& \\underset{z \\rightarrow 1}{\\rm Res} ~ H[T_{{\\mathbf {\\rho }}}(SU(N))] (t; x_1, \\ldots , x_{d-h},y_{d-h+1} t^{\\frac{1}{2}}z,x_{d-h+2}, \\ldots , x_{d},y_{d-h+1} (t^{\\frac{1}{2}}z)^{-\\ell }; \\mathbf {n}) \\\\&= {\\cal P}_{\\mathbf {\\rho }\\mathbf {\\rho }^\\prime } (\\mathbf {x}; y_{d-h+1};t) H[T_{{\\mathbf {\\rho }}^\\prime }(SU(N))] (t; x_1, \\ldots , x_{d-h},y_{d-h+1},x_{d-h+2}, \\ldots , x_{d}; \\mathbf {n})~.\\end{split}$ This result can be reproduced from the monopole formula for the Coulomb branch Hilbert series along the lines of section REF .", "The simple pole corresponds to a noncompact flat direction in the Coulomb branch where the D3-branes depicted in purple in Figure REF are sent to infinity.", "The identification of the simple pole and the computation of the residue is tedious but straightforward.", "We leave it to Appendix .", "This agreement of analytic structures is a nontrivial consistency check of the conjectured equivalence of the monopole and the Hall-Littlewood formula for the Coulomb branch Hilbert series.", "It guarantees the equivalence of the two formulae for any $T_{\\mathbf {\\rho }}(SU(N))$ theories once this is established for $T(SU(N))$ ." ], [ "Conclusions", "This paper is a first step towards the computation of the Coulomb branch Hilbert series for wide classes of $\\mathcal {N}=4$ gauge theories.", "The monopole formula introduced in [1] allows to determine the Hilbert series of any good or ugly $\\mathcal {N}=4$ theory as an infinite sum over magnetic fluxes.", "Due to the presence of many sums and absolute values in the dimension formula for the monopoles, it is sometimes difficult to obtain explicit analytic expressions for the Hilbert series, especially when the gauge group becomes large.", "The gluing techniques that we have introduced in section allows to solve this problem in many cases.", "Knowing analytic formulae for the Coulomb branch Hilbert series in the presence of background fluxes for a general class of building blocks, we can determine the Hilbert series of more general theories by gluing.", "The mechanism requires a sum over the background monopole fluxes and involves no absolute values.", "We have given a closed analytic expression for the Coulomb branch Hilbert series with background fluxes for the class of theories $T_{\\mathbf {\\rho }}(G)$ which can serve as building blocks for constructing a wide classes of $\\mathcal {N}=4$ gauge theories.", "In particular, all the mirrors of Sicilian theories, obtained from M5-branes compactified on $S^1$ times a Riemann sphere with punctures $\\lbrace \\mathbf {\\rho }_i \\rbrace $ , can be obtained by gluing the corresponding $T_{\\mathbf {\\rho }}(G)$ theories.", "In the companion paper [5] we will compute the Hilbert series for the Coulomb branch of the mirror of Sicilian theories of type A and D. By mirror symmetry, this is equal to the Higgs branch Hilbert series of the Sicilian theory, which can can be evaluated by the Hall-Littlewood (HL) limit of the superconformal index [9], [25] when the genus of the Riemann surface is zero.", "We will find perfect agreement with the results in [9], [25], which also involve Hall-Littlewood polynomials and were obtained in a completely different manner.", "The agreement with [9] further substantiates our conjecture that the Hall-Littlewood formula computes the Coulomb branch Hilbert series of $T_{\\rho }(G)$ theories.", "In [5] we will also compute the Coulomb branch Hilbert series of mirrors of Sicilian theories with genus greater than one, for which there is no other available method.", "It would be interesting to extend our analysis to cover the more general class of theories $T_{\\mathbf {\\rho }}^{\\mathbf {\\mu }}(G)$ defined in [2], where $\\mathbf {\\mu }$ and $\\mathbf {\\rho }$ are partitions related to $G$ and the dual $G^\\vee $ respectively.", "The Coulomb and Higgs branches of these theories are not generally complete intersections.", "Computing their Coulomb branch Hilbert series with background fluxes would allow us to obtain the Hilbert series of an even wider class of $\\mathcal {N}=4$ gauge theories." ], [ "Acknowledgements", "We thank Francesco Benini, Nick Halmagyi, Yuji Tachikawa and Alessandro Tomasiello for useful discussions, and the following institutes and workshops for hospitality and partial support: the Galileo Galilei Institute for Theoretical Physics and INFN and the Geometry of Strings and Fields workshop (SC), the Simons Center for Geometry and Physics and the 2013 Summer Workshop, and Chulalongkorn University and the 3rd Bangkok Workshop on High Energy Theory (AH and NM), École Polytechnique and the String Theory Groups of the universities of Rome “Tor Vergata” and of Oviedo (NM).", "NM is also grateful to Diego Rodríguez-Gómez, Yolanda Lozano, Raffaele Savelli, Jasmina Selmic, Hagen Triendl, Sarah Maupeu and Mario Pelliccioni for their very kind hospitality.", "We were partially supported by the STFC Consolidated Grant ST/J000353/1 (SC), the EPSRC programme grant EP/K034456/1 (AH), the ERC grant, Short Term Scientific Mission of COST Action MP1210, and World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan (NM), and INFN and the MIUR-FIRB grant RBFR10QS5J “String Theory and Fundamental Interactions” (AZ)." ], [ "Notations and conventions", "Unless stated otherwise, the following notation and conventions are used throughout the paper.", "The plethystic exponential of a multivariate function $f(t_1, \\ldots , t_n)$ that vanishes at the origin, $f(0,\\ldots , 0)=0$ , is defined as $\\mathop {\\rm PE}\\left[ f(t_1, \\ldots , t_n) \\right] = \\exp \\left( \\sum _{k=1}^\\infty \\frac{1}{k} f(t_1^k, \\ldots , t_n^k) \\right)~.$ An irreducible representation of a simple group $G$ can be denoted by its highest weight vector.", "With respect to a basis consisting of the fundamental weights (also known as the $\\omega $ -basis), we write the highest vector as $[a_1, \\ldots , a_r]$ with $r = \\mathrm {rank}~G$ .", "This is the Dynkin label.", "With respect to a basis of the dual Cartan subalgebra (also known as the $e$ -basis or the standard basis), we denote the the highest vector by $(\\lambda _1, \\ldots , \\lambda _r)$ ." ], [ "The Weyl character formulae", "In the following, we present the character formulae that we use throughout the paper.", "Our convention for characters is different from that of LiEhttp://www-math.univ-poitiers.fr/~maavl/LiE/form.html.", "For $U(n)$ , the Dynkin label $[a_1, a_2, \\ldots , a_n]$ is related to $(\\lambda _1, \\lambda _2, \\ldots , \\lambda _n)$ by the formula $\\lambda _i = a_i +\\ldots + a_n~.$ The partition $\\mathbf {\\lambda }$ is subject to $\\lambda _1 \\ge \\lambda _2 \\ge \\ldots \\ge \\lambda _n \\ge 0$ , with all $\\lambda _n$ integers.", "The character is given by the Schur polynomial $\\chi ^{U(n)}_{(\\lambda _1,\\ldots , \\lambda _n)}(\\mathbf {y}) = \\frac{\\mathop {\\rm det}\\left( y_j^{(\\lambda _{i}+n-i)} \\right)_{i,j=1}^n}{\\mathop {\\rm det}\\left( y_j^{(n-i)} \\right)_{i,j=1}^n}~.$ For $A_{n-1}=SU(n)$ , one simply restricts $\\lambda _n=0$ and imposes $y_1 \\cdots y_n =1$ .", "For $B_n =SO(2n+1)$ , the Dynkin label $[a_1, a_2, \\ldots , a_n]$ is related to $(\\lambda _1, \\lambda _2, \\ldots , \\lambda _n)$ by the formula $\\lambda _i &= a_i + a_{i+1} + \\ldots + a_{n-1} + \\frac{1}{2}a_n~, \\quad 1\\le i \\le n-1~, \\nonumber \\\\\\lambda _n &= \\frac{1}{2} a_n~, $ The partition $\\mathbf {\\lambda }$ is subject to $\\lambda _1 \\ge \\lambda _2 \\ge \\ldots \\ge \\lambda _N \\ge 0$ with all $\\lambda _i$ integers or all half-integers.", "The character is given by $\\chi ^{B_n}_{(\\lambda _1,\\ldots , \\lambda _n)}(\\mathbf {y}) = \\frac{\\mathop {\\rm det}\\left( y_j^{(\\lambda _{i}+n-i+\\frac{1}{2})}- y_j^{-(\\lambda _{i}+n-i+\\frac{1}{2})} \\right)_{i,j=1}^n}{\\mathop {\\rm det}\\left( y_j^{(n-i+\\frac{1}{2})}- y_j^{-(n-i+\\frac{1}{2})} \\right)_{i,j=1}^n}~.$ For $C_n =USp(2n)$ , the Dynkin label $[a_1, a_2, \\ldots , a_n]$ is related to $(\\lambda _1, \\lambda _2, \\ldots , \\lambda _n)$ by the formula $\\lambda _i = a_i +\\ldots + a_n~.$ The partition $\\mathbf {\\lambda }$ is subject to $\\lambda _1 \\ge \\lambda _2 \\ge \\ldots \\ge \\lambda _n \\ge 0$ , with all $\\lambda _n$ integers.", "The character is given by $\\chi ^{C_n}_{(\\lambda _1,\\ldots , \\lambda _n)}(\\mathbf {y}) = \\frac{\\mathop {\\rm det}\\left( y_j^{(\\lambda _{i}+n-i+1)}- y_j^{-(\\lambda _{i}+n-i+1)} \\right)_{i,j=1}^n}{\\mathop {\\rm det}\\left( y_j^{(n-i+1)}- y_j^{-(n-i+1)} \\right)_{i,j=1}^n}~.$ For $D_n= SO(2n)$ , the Dynkin label $[a_1, a_2, \\ldots , a_n]$ is related to $(\\lambda _1, \\lambda _2, \\ldots , \\lambda _n)$ by the formula $\\lambda _i &= a_i +\\ldots a_{n-2} +\\frac{1}{2} (a_{n-1} + a_n)~, \\qquad 1\\le i \\le n-2 \\nonumber \\\\\\lambda _{n-1} &=\\frac{1}{2} (a_{n-1} + a_n)~, \\qquad \\lambda _{n} =\\frac{1}{2} (-a_{n-1} + a_n)~.$ The partition $\\mathbf {\\lambda }$ is subject to $\\lambda _1 \\ge \\lambda _2 \\ge \\ldots \\ge \|\\lambda _n\| \\ge 0$ .", "The character is given by $\\chi ^{D_n}_{(\\lambda _1,\\ldots , \\lambda _n)}(\\mathbf {y}) = \\frac{\\mathop {\\rm det}\\left( y_j^{(\\lambda _{i}+n-i)}- y_j^{-(\\lambda _{i}+n-i)} \\right)_{i,j=1}^n}{\\mathop {\\rm det}\\left( y_j^{(n-i)}- y_j^{-(n-i)} \\right)_{i,j=1}^n}~.$" ], [ "Hall-Littlewood polynomials", "The Hall-Littlewood (HL) polynomial associated to a group $G$ and a representation $\\mathbf {\\lambda }$ is a polynomial labelled by the highest weight vector $\\mathbf {\\lambda }= \\sum _{i=1}^r \\lambda _i \\mathbf {e}_i$ , with $\\lbrace \\mathbf {e}_1, \\ldots , \\mathbf {e}_r\\rbrace $ the standard basis of the weight lattice and $r$ the rank of $G$ , defined as (see e.g.", "[7])Note that, for $G=U(N)$ , we define the Hall-Littlewood polynomial without a normalisation factor ${\\cal N}(t)$ in comparison with (5.17) of [9].", "This coincides with $R_\\lambda $ in Section III.1 of [7].", "$\\Psi ^{\\mathbf {\\lambda }}_{G} (x_1, \\ldots , x_r; t) = \\sum _{w \\in W_G} {\\mathbf {x}}^{w(\\mathbf {\\lambda })}\\prod _{\\mathbf {\\alpha }\\in \\Delta _+(G)} \\frac{ 1-t {\\mathbf {x}}^{-w({\\mathbf {\\alpha }})} }{1- {\\mathbf {x}}^{-w({\\mathbf {\\alpha }})} }~,$ where $W_G$ denotes the Weyl group of $G$ and $\\Delta _+(G)$ the set of positive roots of $G$ .", "Explicit details for classical groups $G$ can be listed as follows: For $G=U(N)$ , $W_G=S_N$ is a group of permutations of the elements in $\\lbrace \\mathbf {e}_1,\\ldots , \\mathbf {e}_N\\rbrace $ .", "The partition $\\mathbf {\\lambda }$ is subject to $\\lambda _1 \\ge \\lambda _2 \\ge \\ldots \\ge \\lambda _N \\ge 0$ with $\\lambda _i \\in \\mathbb {Z}$ .", "The representation $\\mathbf {\\lambda }$ corresponds to the Dynkin label $\\mathbf {a} = (\\lambda _1 - \\lambda _2, \\lambda _2-\\lambda _3, \\ldots , \\lambda _{N-1}-\\lambda _{N}, \\lambda _N)~.$ The corresponding HL polynomial is $\\Psi ^{\\mathbf {\\lambda }}_{U(N)} (x_1,\\dots ,x_N;t)=\\sum _{\\sigma \\in S_N}x_{\\sigma (1)}^{\\lambda _1} \\dots x_{\\sigma (N)}^{\\lambda _N}\\prod _{1 \\le i<j \\le N} \\frac{ 1-t x_{\\sigma (i)}^{-1} x_{\\sigma (j)} }{1-x_{\\sigma (i)}^{-1} x_{\\sigma (j)}}\\,,$ where the positive roots are $\\Delta _+ (U(N))= \\lbrace \\mathbf {e}_i - \\mathbf {e}_j \\rbrace _{1 \\le i< j \\le N}~.$ For $G=A_{N-1} = SU(N)$ , we set $x_1 \\cdots x_N=1$ and restrict $\\lambda _N=0$ .", "For $G=B_N = SO(2N+1)$ , $W_G = S_N \\rtimes \\mathbb {Z}_2^N$ under which ${\\mathbf {e}}_i \\rightarrow \\pm {\\mathbf {e}}_{\\sigma (i)}$ , with $\\sigma \\in S_N$ .", "The partition $\\mathbf {\\lambda }$ is subject to $\\lambda _1 \\ge \\lambda _2 \\ge \\ldots \\ge \\lambda _N \\ge 0$ with all $\\lambda _i$ integers or all half-integers.", "The corresponding HL polynomial is $\\Psi ^{\\mathbf {\\lambda }}_{B_N} (x_1,\\dots ,x_N;t) &= \\sum _{s_1,\\ldots , s_N= \\pm 1} \\; \\sum _{\\sigma \\in S_N}\\left( \\prod _{i=1}^N x_{\\sigma (i)}^{s_i \\lambda _i} \\frac{1-t x_{\\sigma (i)}^{-s_i} }{1-x_{\\sigma (i)}^{-s_i} } \\right) \\times \\nonumber \\\\& \\qquad \\left( \\prod _{1 \\le i<j \\le N} \\frac{1-t x_{\\sigma (i)}^{-s_i} x_{\\sigma (j)}^{s_j} }{1-x_{\\sigma (i)}^{-s_i} x_{\\sigma (j)}^{s_j}} \\cdot \\frac{1-t x_{\\sigma (i)}^{-s_i} x_{\\sigma (j)}^{-s_j} }{1-x_{\\sigma (i)}^{-s_i} x_{\\sigma (j)}^{-s_j} } \\right)~,$ where $\\Delta _+(B_N)= \\lbrace \\mathbf {e}_i + \\mathbf {e}_j \\rbrace _{1 \\le i< j \\le N} \\cup \\lbrace \\mathbf {e}_i - \\mathbf {e}_j \\rbrace _{1 \\le i< j \\le N} \\cup \\lbrace \\mathbf {e}_i\\rbrace _{1 \\le i\\le N}~.$ For $G=C_N = USp(2N)$ , $W_G = S_N \\rtimes \\mathbb {Z}_2^N$ under which ${\\mathbf {e}}_i \\rightarrow \\pm {\\mathbf {e}}_{\\sigma (i)}$ , with $\\sigma \\in S_N$ .", "The partition $\\mathbf {\\lambda }$ is subject to $\\lambda _1 \\ge \\lambda _2 \\ge \\ldots \\ge \\lambda _N \\ge 0$ , with all $\\lambda _i$ integers.", "The corresponding HL polynomial is $\\Psi ^{\\mathbf {\\lambda }}_{C_N} (x_1,\\dots ,x_N;t) &= \\sum _{s_1,\\ldots , s_N= \\pm 1} \\; \\sum _{\\sigma \\in S_N}\\left( \\prod _{i=1}^N x_{\\sigma (i)}^{s_i \\lambda _i} \\frac{1-t x_{\\sigma (i)}^{-2s_i} }{1-x_{\\sigma (i)}^{-2s_i} } \\right) \\times \\nonumber \\\\& \\qquad \\left( \\prod _{1 \\le i<j \\le N} \\frac{1-t x_{\\sigma (i)}^{-s_i} x_{\\sigma (j)}^{s_j} }{1-x_{\\sigma (i)}^{-s_i} x_{\\sigma (j)}^{s_j}} \\cdot \\frac{1-t x_{\\sigma (i)}^{-s_i} x_{\\sigma (j)}^{-s_j} }{1-x_{\\sigma (i)}^{-s_i} x_{\\sigma (j)}^{-s_j} } \\right)~,$ where $\\Delta _+ (C_N)= \\lbrace \\mathbf {e}_i + \\mathbf {e}_j \\rbrace _{1 \\le i< j \\le N} \\cup \\lbrace \\mathbf {e}_i - \\mathbf {e}_j \\rbrace _{1 \\le i< j \\le N} \\cup \\lbrace 2\\mathbf {e}_i\\rbrace _{1 \\le i\\le N}~.$ For $G=D_N = SO(2N)$ , $W_G = S_N \\rtimes \\mathbb {Z}_2^{N-1}$ under which ${\\mathbf {e}}_i \\rightarrow (-1)^{b_i} {\\mathbf {e}}_{\\sigma (i)}$ , with $\\sigma \\in S_N$ , $b_i = 0, 1$ and $\\sum _{i=1}^N b_i$ is even.", "The partition $\\mathbf {\\lambda }$ is subject to $\\lambda _1 \\ge \\lambda _2 \\ge \\ldots \\ge \|\\lambda _N\| \\ge 0$ .", "The Dynkin label of the representation $\\mathbf {\\lambda }$ is $\\mathbf {a} =(a_1, \\ldots , a_N)$ , with $\\lambda _i &= a_i +\\ldots a_{N-2} +\\frac{1}{2} (a_{N-1} + a_N)~, \\qquad 1\\le i \\le N-2 \\nonumber \\\\\\lambda _{N-1} &=\\frac{1}{2} (a_{N-1} + a_N)~, \\qquad \\lambda _{N} =\\frac{1}{2} (-a_{N-1} + a_N)~.$ The corresponding HL polynomial is $\\Psi ^{\\mathbf {\\lambda }}_{D_N} (x_1,\\dots ,x_N;t) &= \\sum _{\\begin{array}{c}s_1,\\ldots , s_N= \\pm 1 \\\\ s_1 \\ldots s_N =1\\end{array}} \\; \\sum _{\\sigma \\in S_N}\\left( \\prod _{i=1}^N x_{\\sigma (i)}^{s_i \\lambda _i} \\right) \\times \\nonumber \\\\& \\qquad \\left( \\prod _{1 \\le i<j \\le N} \\frac{1-t x_{\\sigma (i)}^{-s_i} x_{\\sigma (j)}^{s_j} }{1-x_{\\sigma (i)}^{-s_i} x_{\\sigma (j)}^{s_j}} \\cdot \\frac{1-t x_{\\sigma (i)}^{-s_i} x_{\\sigma (j)}^{-s_j} }{1-x_{\\sigma (i)}^{-s_i} x_{\\sigma (j)}^{-s_j} } \\right)~,$ where $\\Delta _+ (D_N)= \\lbrace \\mathbf {e}_i + \\mathbf {e}_j \\rbrace _{1 \\le i< j \\le N} \\cup \\lbrace \\mathbf {e}_i - \\mathbf {e}_j \\rbrace _{1 \\le i< j \\le N} ~.$ We collect explicit expressions for HL polynomials for groups with small ranks below.", "$& \\Psi ^{n_1}_{U(1)} (x_1; t) = x_1^{n_1} \\\\& \\Psi ^{(n_1,n_2)}_{U(2)} (x_1,x_2; t) = \\frac{t x_1^{1+n_2} x_2^{n_1}-x_1^{n_2} x_2^{1+n_1}+x_1^{1+n_1} x_2^{n_2}-t x_1^{n_1} x_2^{1+n_2}}{x_1-x_2}\\\\& \\Psi ^n_{SO(3)}(x;t) = \\frac{x^{-n} \\left[ -1+x^{1+2 n}+x t \\left(1-x^{2 n-1}\\right)\\right]}{x-1} \\\\& \\Psi ^n_{USp(2)}(x;t) = \\frac{x^{-n} \\left[ -1+x^{2+2 n}+x^2 t \\left(1-x^{2 n-2}\\right)\\right]}{x^2-1} = \\Psi ^{(n,0)}_{U(2)} (x,x^{-1}; t) \\\\& \\Psi ^{(n_1,n_2)}_{SO(4)}(x_1,x_2;t) = \\frac{1}{\\left(x_1-x_2\\right) \\left(x_1 x_2-1\\right)} \\Big [-x_1^{1-n_2} x_2^{-n_1}-x_1^{1+n_2} x_2^{2+n_1}+x_1^{-n_1} x_2^{1-n_2}\\nonumber \\\\& \\qquad +x_1^{2+n_1} x_2^{1+n_2} +t \\Big \\lbrace x_1^{2-n_2} x_2^{1-n_1}+x_1^{-n_2} x_2^{1-n_1}+x_1^{n_2} x_2^{1+n_1}+x_1^{2+n_2} x_2^{1+n_1} \\nonumber \\\\& \\qquad -x_1^{1-n_1} x_2^{2-n_2}-x_1^{1-n_1} x_2^{-n_2}-x_1^{1+n_1} x_2^{n_2} \\left(1+x_2^2\\right) \\Big \\rbrace +t^2 \\Big \\lbrace -x_1^{1+n_2} x_2^{n_1}+x_1^{n_1} x_2^{1+n_2} \\nonumber \\\\& \\qquad +x_1 x_2 \\Big (-x_1^{-n_2} x_2^{1-n_1}+x_1^{1-n_1} x_2^{-n_2}\\Big )\\Big \\rbrace \\Big ]~.&$" ], [ "Examples of non-maximal $B$ and {{formula:80a16678-82fe-417a-924b-9995ee01f611}} punctures", "In this appendix, we focus on non-trivial partitions $\\mathbf {\\rho }$ providing many examples for $SO$ and $USp$ groups.", "For groups with low ranks, there are certain isomorphism between their Lie algebras, e.g.", "$SO(5)$ and $USp(4)$ , $SO(6)$ and $SU(4)$ .", "We use such isomorphisms as a tool to check the Hall-Littlewood formula (REF ) in many examples; these checks are similar in spirit to what we performed in section REF .", "Furthermore, in many examples below, we use the Hilbert series as a tool to understand the relationships between the nilpotent orbit and the Higgs/Coulomb branches of the theories.", "Let us summarize some highlights below: In appendix REF we consider the “very even” partition $\\mathbf {\\rho }=(4,4)$ of $SO(8)$ , which corresponds to two distinct nilpotent orbits of $SO(8)$ [27], [31], [26].", "We study how the Coulomb branches of the theories corresponding to these two orbits are related.", "In appendix REF , we study two distinct partitions of $SO(8)$ , namely $\\mathbf {\\rho }_1=(3,3,1,1)$ and $\\mathbf {\\rho }_2=(3,2,2,1)$ .", "Although these two partitions are different, their images under the Spaltenstein map are identical [26].", "Physically, the latter describe the Higgs branches of $T_{\\mathbf {\\rho }_1}(SO(8))$ and $T_{\\mathbf {\\rho }_2}(SO(8))$ as the moduli spaces of the same Hitchin system [31], [26].", "In particular, we show that the Higgs branch Hilbert series of two theories are indeed equal." ], [ "$T_{(3,1,1,1)}(SO(6))$", "The quiver for $T_{(3,1,1,1)}(SO(6))$ is $T_{(3,1,1,1)}(SO(6)): \\qquad [SO(6)]-(USp(2))-(O(2))~.$ The mirror of this theory is given below: $\\begin{tikzpicture}[font=\\scriptsize ]\\begin{scope}[auto,every node/.style={draw, minimum size=1cm}, node distance=1cm];\\node [circle] (SO2) at (0, 0) {SO(2)};\\node [circle, right=of SO2] (USp2) {USp(2)};\\node [circle, right=of USp2] (SO3) {SO(3)};\\node [circle, right=of SO3] (USp2p) {USp(2)};\\node [rectangle, below=of USp2] (SO1) {SO(1)};\\node [rectangle, below=of USp2p] (SO3f) {SO(3)};\\end{scope}(SO2) -- (USp2)(USp2) -- (SO3)(USp2) -- (SO1)(SO3) -- (USp2p)(USp2p) -- (SO3f);\\end{tikzpicture}$ The mirror is obtained using the brane configuration as discussed in [28]." ], [ "Hall-Littlewood formula", "The Coulomb branch Hilbert series of $T_{3,1,1,1} (SO(6))$ is given by $\\begin{split}&H[T_{(3,1,1,1)} (SO(6))] (t; x; n_1,n_2,n_3) \\\\& = t^{2n_1+n_2} (1-t)^3 K^{SO(6)}_{(3,1,1,1)}(x, t) \\Psi ^{(n_1,n_2,n_3)}_D (t, 1, x^2;t)~,\\end{split}$ where $x$ is the fugacity associated with the topological charge of $SO(2)$ gauge group in (REF ) and $K^{SO(6)}_{(3,1,1,1)}(x_1, t) = \\mathop {\\rm PE}\\left[t( \\chi ^{SO(3)}_{[1]}(x)) +t^2 ( \\chi ^{SO(3)}_{[1]}(x)+1) \\right]~,$ with $\\chi ^{SO(3)}_{[1]}(x) = x^2+1+x^{-2}~.$ The argument of the HL polynomial in (REF ) and the factor $K^{SO(6)}_{(3,1,1,1)}$ follow from the decompositions of the fundamental and adjoint representations of $SO(6)$ $\\begin{split}\\chi ^{SO(6)}_{[1,0,0]}(\\mathbf {a}) & = \\sum _{i=1}^3 \\left(a_i + a_i^{-1}\\right) = (t+\\frac{1}{t} + x^2 +\\frac{1}{x^2} +2) = \\chi ^{SU(2)}_{[2]}(t^{\\frac{1}{2}}) + \\chi ^{SO(3)}_{[1]}(x) ~, \\\\\\chi ^{SO(6)}_{[0,1,1]} (\\mathbf {a}) &= \\chi ^{SO(3)}_{[1]}(x) \\chi ^{SU(2)}_{[2]}(t^{\\frac{1}{2}}) + \\chi ^{SU(2)}_{[2]}(t^{\\frac{1}{2}}) + \\chi ^{SO(3)}_{[1]}(x) ~.\\end{split}$ As predicted in (REF ), the Coulomb branch symmetry is enhanced to $SO(3)$ .", "For $n_1=n_2=0$ , we obtain the Hilbert series $\\begin{split}&H[T_{(3,1,1,1)} (SO(6))] (t; x; 0,0,0) \\\\&= (1-t^3)(1-t^4)\\mathop {\\rm PE}\\left[ t \\chi ^{SO(3)}_{[1]}(x) + t^2 \\chi ^{SO(3)}_{[1]}(x) \\right]~,\\end{split}$ which agrees with the Higgs branch Hilbert series of the mirror quiver (REF ).", "We observe that the Hilbert series (REF ) is equal to the Coulomb branch Hilbert series of $T_{(2,2)} (SU(4))$ theory, given by (4.7) of [3].", "As a check of formula (REF ), we compare this to the HL formula for $T_{(2,2)}(SU(4))$ , given by (REF ), with the background charges $n_1,n_2,n_3$ turned on: $\\begin{split}&H[T_{(2,2)}(SU(4))] (t, x_1,x_2; n_1,n_2,n_3,0) \\\\&= t^{\\frac{1}{2}(3n_1+n_2-n_3)} (1-t)^4 K^{U(4)}_{(2,2)}(t; \\mathbf {x}) \\Psi ^{(n_1,n_2,n_3,0)}_{U(4)} (\\mathbf {x}t^{\\frac{1}{2} w_{(2,2)}} ; t)~,\\end{split}$ where $\\mathbf {x} t^{\\frac{1}{2} w_{(2,2)}} &= (x_1 t^{1/2}, x_1 t^{-1/2}, x_2 t^{1/2}, x_2 t^{-1/2}) \\\\K^{U(4)}_{(2,2)}(t; \\mathbf {x}) &= \\mathop {\\rm PE}\\left[ (t+t^2) ( 2+x_1 x_2^{-1}+ x_1^{-1} x_2) \\right]~.$ Note that the prefactor $K^{U(4)}_{(2,2)}(t; \\mathbf {x})$ corresponds to the following decomposition of (REF ): $\\begin{split}\\chi ^{U(4)}_{\\mathbf {\\mathbf {Adj}}} (\\mathbf {a}) &= \\sum _{i,j =1}^4 {a_i}a_j^{-1} \\\\&= ( 2+x_1 x_2^{-1}+ x_1^{-1} x_2) \\left[ \\chi ^{SU(2)}_{[0]}(t^{1/2})+ \\chi ^{SU(2)}_{[2]}(t^{1/2}) \\right]~, \\quad \\mathbf {a} = \\mathbf {x} t^{\\frac{1}{2} w_{(2,2)}}~.\\end{split}$ Indeed, we find that for any $a_1, a_2, a_3 \\ge 0$ , $H[T_{(3,1,1,1)} (SO(6))] \\left(t;x; \\mathbf {m} (\\mathbf {a}) \\right) = H[T_{2,2}(SU(4))](t; x, x^{-1}; \\mathbf {n}(\\mathbf {a}) ) ~,$ with $\\begin{split}\\mathbf {m} (\\mathbf {a}) &= \\left(\\frac{1}{2}a_1+a_2+ \\frac{1}{2}a_3, \\; \\frac{1}{2}a_1+\\frac{1}{2}a_3, \\; -\\frac{1}{2}a_1+\\frac{1}{2}a_3 \\right)~, \\qquad \\\\\\mathbf {n} (\\mathbf {a}) &= (a_1+a_2+a_3, \\; a_2+a_3, \\; a_3, \\;0)~.\\end{split}$" ], [ "$T_{(4,4)}(SO(8))$ and the very even partition {{formula:07e0b743-5195-4445-8ef2-4260d33c8fce}}", "In this appendix, we consider the partition $(4,4)$ of $SO(8)$ .", "This partition is “very even”, therefore it corresponds to two different nilpotent orbits of $SO(8)$ (see, e.g.", "Recipe 5.2.6 of [27] and [31], [26]).", "These two types of puncture $(4,4)$ are related by an outer automorphism of $SO(8)$ that interchanges the two spinor representations $[0,0,0,1]$ and $[0,0,1,0]$ ; we distinguish these punctures by subscripts $I$ and $II$ .", "Even though the distinction of these two types is not apparent in the quiver diagram, the Hall-Littlewood formulae for these two partitions are not equal, even though they can be related." ], [ "Quiver and mirror theory", "For both types of the $(4,4)$ puncture, the quiver diagram of $T_{(4,4)}(SO(8))$ is given by $USp(4)$ gauge theory with 4 flavors, namely $T_{(4,4)}(SO(8)): \\qquad (USp(4))-[SO(8)]~.", "$ This theory is a `bad theory', since the number of flavors is 4, less than $2(2)+1 =5$ .", "The Coulomb branch Hilbert series cannot be computed from the monopole formula, but we expect that the Hall-Littlewood formula gives the correct result.", "The mirror theory of this quiver can be determined using brane configurations as in Figure 13 of [28]; the quiver diagram is given by $\\begin{tikzpicture}[font=\\scriptsize ]\\begin{scope}[auto,every node/.style={draw, minimum size=1cm}, node distance=1cm];\\node [circle] (SO2) at (0, 0) {SO(2)};\\node [circle, right=of SO2] (USp2) {USp(2)};\\node [circle, right=of USp2] (SO4) {SO(4)};\\node [circle, right=of SO4] (USp2p) {USp(2)};\\node [circle, right=of USp2p] (SO2p) {SO(2)};\\node [rectangle, below=of SO4] (USp2f) {USp(2)};\\end{scope}(SO2) -- (USp2)(USp2) -- (SO4)(SO4) -- (USp2p)(USp2p) -- (SO2p)(SO4) -- (USp2f);\\end{tikzpicture}$ The global symmetry group of the $(4,4)$ puncture is $USp(2)$ .", "Two types of punctures corresponds to different embeddings of $USp(2)$ in $SO(8)$ .", "From (REF ), we consider the following decomposition: $\\chi ^{SO(8)}_{[1,0,0,0]}(\\mathbf {a}) = \\sum _{i=1}^4 \\left(a_i + a_i^{-1}\\right) = (t^{3/2}+t^{1/2}+t^{-1/2}+t^{-3/2}) (x+x^{-1})~.$ There are two inequivalent choices of the fugacity maps corresponding to the two types of (4,4) puncture:In this paper we take $a_3 = x^{-1} t^{1/2}$ , differently from Fig.", "8 of [25].", "Our choice is to be consistent with $K_{(4,4)}^{SO(8)}(t; x)$ given in (REF ).", "$\\begin{split}(I): &\\qquad a_1= x t^{3/2}, \\quad a_2 = x t^{1/2}, \\quad a_3 = x^{-1} t^{1/2}, \\quad a_4 = x t^{-3/2}~, \\\\(II): & \\qquad a_1= x t^{3/2}, \\quad a_2 = x t^{1/2}, \\quad a_3 = x^{-1} t^{1/2}, \\quad a_4 = x^{-1} t^{3/2}~.\\end{split}$ For the two types of (4,4) puncture, the HL formula for the Coulomb branch Hilbert series of (REF ) is given by $\\begin{split}H[T_{(4,4)_I}(SO(8))] (t; x; \\mathbf {n}) &= t^{3 n_1+2 n_2 +n_3} (1-t)^4 K_{(4,4)}^{SO(8)} (t; x) \\times \\\\& \\qquad \\Psi ^{(n_1,n_2,n_3,n_4)} _{D} (t^{3/2} x, t^{1/2} x, t^{1/2} x, t^{-3/2} x; t)~,\\end{split} \\\\\\begin{split}H[T_{(4,4)_{II}}(SO(8))] (t; x; \\mathbf {n}) &= t^{3 n_1+2 n_2 +n_3} (1-t)^4 K_{(4,4)}^{SO(8)} (t; x) \\times \\\\& \\qquad \\Psi ^{(n_1,n_2,n_3,n_4)} _{D} (t^{3/2} x, t^{1/2} x, t^{1/2} x, t^{3/2} x^{-1}; t)~,\\end{split}$ where $K_{(4,4)}^{SO(8)}(t,x)= \\mathop {\\rm PE}\\left[ t \\chi _{[2]}(x) + t^2 +t^3 \\chi _{[2]}(x) +t^4 \\right]~.$ Note that (REF ) is consistent with (REF ), namely $\\begin{split}\\chi ^{SO(8)}_{[0,1,0,0]} (\\mathbf {a}) &= \\chi ^{USp(2)}_{[2]}(x) \\chi ^{SU(2)}_{[0]}(t^{\\frac{1}{2}}) + \\chi ^{SU(2)}_{[2]}(t^{\\frac{1}{2}}) + \\chi ^{USp(2)}_{[2]}(x) \\chi ^{SU(2)}_{[4]}(t^{\\frac{1}{2}}) \\\\& \\qquad + \\chi ^{SU(2)}_{[6]}(t^{\\frac{1}{2}})~.\\end{split}$ For $\\mathbf {n} =(0,0,0,0)$ , we obtain the Coulomb branch Hilbert series of quiver (REF ) and the Higgs branch Hilbert series of quiver (REF ): $\\begin{split}& H[T_{(4,4)_I}(SO(8))] (t; x; 0,0,0,0) = H[T_{(4,4)_{II}}(SO(8))] (t; x; 0,0,0,0) \\\\&= \\mathop {\\rm PE}\\left[ t^2 \\chi _{[2]} (x) + t^3 \\chi _{[2]} (x) - t^4 - t^6\\right]~.\\end{split}$ Note that the space is 4 complex dimensional as required.", "For $SO(8)$ , the vector representation $[1,0,0,0]$ corresponds to $\\mathbf {n}=(1,0,0,0)$ , and the two spinor representations $[0,0,1,0]$ and $[0,0,0,1]$ correspond to $\\mathbf {n} = \\frac{1}{2} (1,1,1,-1)$ and $\\frac{1}{2}(1,1,1,1)$ .", "The outer automorphism that relates the two spinor representations exchanges $(4,4)$ punctures of types I and II.", "In general we find that $H[T_{(4,4)_I}(SO(8))] \\left(t; x; n_1,n_2,n_3, n_4\\right) = H[T_{(4,4)_{II}}(SO(8))] \\left(t; x; n_1,n_2,n_3, -n_4 \\right)~.$" ], [ "$T_{(3,3,1,1)}(SO(8))$ and {{formula:e010ffc1-9028-494a-92af-e16af51bc1b2}}", "In this appendix we consider $T_{(3,3,1,1)}(SO(8))$ and $T_{(3,2,2,1)}(SO(8))$ theories, which are mirrors to the second and third example on pp.", "24-25 of [26].", "Their quiver diagrams are given below: $T_{(3,3,1,1)}(SO(8)): & \\qquad (O(2))-(USp(4))-[SO(8)]~, \\\\T_{(3,2,2,1)}(SO(8)): & \\qquad (SO(4))-(USp(4))-[SO(8)] ~.$ Note that the quiver for $T_{(3,3,1,1)}(SO(8))$ is a `good theory', whereas the quiver for $T_{(3,2,2,1)}(SO(8))$ is a `bad theory' since the number of flavors under the $SO(4)$ gauge group is 2, smaller than $4-1=3$ .", "The partitions $\\mathbf {\\rho }= (3,3,1,1)$ and $\\mathbf {\\rho }=(3,2,2,1)$ define the Nahm poles of the above theories.", "The Hitchin pole of each theory can be obtained by a procedure consisting of a transposition and a series of D-collapses [31], [26], whereby the box in the bottom row of the left most column is moved to the next right column.", "The D-collapsing is to be repeated until the number of columns that contain even boxes is even.This condition corresponds to the decomposition of the $2N$ dimensional representation into irreducible representations of $SU(2)$ with dimensions $n_i$ : $2N \\rightarrow n_1+ n_2 + \\ldots +n_k$ .", "In this decomposition, each even $n_i$ appeas even times.", "The resulting D-partition $\\widetilde{\\mathbf {\\rho }}$ defines the Hitchin pole of the theory.", "Below we show step-by-step the procedure to obtain the Hitchin poles for both theories.", "Let us start from $\\mathbf {\\rho }= (3,3,1,1)$ .", "${boxsize=0.7em}{aligntableaux=bottom}\\begin{array}{cccc}{4,2,2} & \\qquad {\\begin{ytableau} ~& ~& ~\\\\ ~& ~& ~\\\\ ~\\\\ (blue!20) \\end{ytableau}} & \\qquad {\\begin{ytableau} ~& ~& ~\\\\ ~& ~& (blue!20)~\\\\ ~ & ~ \\end{ytableau}} & \\qquad {4,2,2}\\\\{\\mathbf {\\rho }}=(3,3,1,1) \\quad &\\xrightarrow{} \\quad (4,2,2) \\quad &\\xrightarrow{} \\quad (3,3,2) \\quad &\\xrightarrow{} \\quad \\widetilde{\\mathbf {\\rho }} = (3,3,1,1) \\\\\\end{array}$ where in each D-collapse the blue box is moved to the next column, according to the rule given in [31].", "Similarly for $\\mathbf {\\rho }= (3,2,2,1)$ we have ${boxsize=0.7em}{aligntableaux=bottom}\\begin{array}{cccc}{4,3,1} & \\qquad {\\begin{ytableau} ~& ~& ~\\\\ ~& ~\\\\~& ~\\\\ (blue!20) \\end{ytableau}} & \\qquad {\\begin{ytableau} ~& ~& ~\\\\ ~& ~& *(blue!20)~\\\\ ~ & ~ \\end{ytableau}} & \\qquad {4,2,2}\\\\{\\mathbf {\\rho }}=(3,2,2,1) \\quad &\\xrightarrow{} \\quad (4,3,1) \\quad &\\xrightarrow{} \\quad (3,3,2) \\quad &\\xrightarrow{} \\quad \\widetilde{\\mathbf {\\rho }} = (3,3,1,1) \\\\\\end{array}$ As a result, the Hitchin poles of the two theories are identical, namely $\\widetilde{\\mathbf {\\rho }} =(3,3,1,1)$ .", "In other words, the Higgs branches of $T_{(3,3,1,1)}(SO(8))$ and $T_{(3,2,2,1)}(SO(8))$ correspond to the moduli space of the same Hitchin system.We thank Yuji Tachikawa for pointing this out to us.", "Indeed it can be shown, using Hilbert series, that the hypermultiplet moduli spaces of the quivers $(O(2))-[USp(4)]~, \\qquad \\qquad (SO(4))-[USp(4)]~ $ are identical.We give details of the computation in Appendix REF .", "Thus, upon gluing with $[USp(4)]-[SO(8)]$ via the $USp(4)$ group, we obtain the same Higgs branch Hilbert series for both $T_{(3,3,1,1)}(SO(8))$ and $T_{(3,2,2,1)}(SO(8))$ .", "Let us now turn to the Coulomb branch Hilbert series of $T_{(3,3,1,1)}(SO(8))$ and $T_{(3,2,2,1)}(SO(8))$ .", "The Coulomb branches of these theories are different; the former is 3 quaternionic dimensional, whereas the latter is 4 quaternionic dimensional.", "The $G_{\\mathbf {\\rho }}$ global symmetries associated with the punctures ${(3,3,1,1)}$ and ${(3,2,2,1)}$ are $SO(2)\\times SO(2)$ and $USp(2)$ respectively.", "The HL formulae for the Hilbert series are given by $ \\begin{split}H[T_{(3,3,1,1)}(SO(8))] (t; x_1, x_2; \\mathbf {n}) &= t^{3 n_1+2 n_2 +n_3} (1-t)^4 K_{(3,3,1,1)}^{SO(8)} (t; x) \\times \\\\& \\qquad \\Psi ^{(n_1,n_2,n_3,n_4)} _{D} ( x_1 t^{-1}, x_1 t, x_1, x_2; t)~,\\end{split}\\\\\\begin{split}H[T_{(3,2,2,1)}(SO(8))] (t; x; \\mathbf {n}) &= t^{3 n_1+2 n_2 +n_3} (1-t)^4 K_{(3,2,2,1)}^{SO(8)} (t; x) \\times \\\\& \\qquad \\Psi ^{(n_1,n_2,n_3,n_4)} _{D} ( t^{1/2} x, t^{-1/2} x,t ,1; t)~,\\end{split}$ where the notations are explained below: $(x_1, x_2)$ are fugacities for $SO(2) \\times SO(2)$ and $x$ is a fugacity for $USp(2)$ .", "They are related to the following embeddings.", "For punctures ${(3,3,1,1)}$ and ${(3,2,2,1)}$ , the decompositions (REF ) are respectively $\\chi ^{SO(8)}_{[1,0,0,0]} (\\mathbf {a}) &=\\sum _{i=1}^4 \\left(a_i+a_i^{-1}\\right) = \\chi ^{SU(2)}_{[2]}(t^{1/2}) (x_1+x_1^{-1}) + (x_2+x_2^{-1})~, \\\\\\chi ^{SO(8)}_{[1,0,0,0]} (\\mathbf {b}) &= \\sum _{i=1}^4 \\left(b_i+b_i^{-1}\\right) = \\chi ^{SU(2)}_{[2]}(t^{1/2}) + \\chi ^{SU(2)}_{[1]}(t^{1/2}) (x+x^{-1}) +1~.$ We pick $\\mathbf {a}= ( x_1 t^{-1}, x_1 t, x_1, x_2)~, \\qquad \\mathbf {b}= ( t^{1/2} x, t^{-1/2} x,t ,1);$ these are the argument of the above Hall-Littlewood polynomials.", "The prefactor $K_{(3,3,1,1)}^{SO(8)}$ is given by $K_{(3,3,1,1)}^{SO(8)} (t; x_1,x_2) &= \\mathop {\\rm PE}\\left[ 2 t + \\lbrace x_1^2+1+x_1^{-2}+(x_1+x_1^{-1})(x_2+x_2^{-1}) \\rbrace t^2 +t^3\\right]~;$ this corresponds to the following decomposition in (REF ): $\\begin{split}\\chi ^{SO(8)}_{[0,1,0,0]} (\\mathbf {a}) &= 2 +\\lbrace x_1^2+1+x_1^{-2}+(x_1+x_1^{-1})(x_2+x_2^{-1}) \\rbrace \\chi ^{SU(2)}_{[2]}(t^{1/2}) \\\\& \\qquad + \\chi ^{SU(2)}_{[4]}(t^{1/2})~.\\end{split}$ The prefactor $ K_{(3,2,2,1)}^{SO(8)}$ is given by $K_{(3,2,2,1)}^{SO(8)} (t; x) &= \\mathop {\\rm PE}\\left[ \\chi ^{USp(2)}_{[2]}(x)t + 2\\chi ^{USp(2)}_{[1]}(x) t^{3/2} + 3t^2 +\\chi ^{USp(2)}_{[1]}(x)t^{5/2} \\right]~;$ this corresponds to the following decomposition in (REF ): $\\begin{split}\\chi ^{SO(8)}_{[0,1,0,0]} (\\mathbf {b}) &= \\chi ^{USp(2)}_{[2]}(x) +2\\chi ^{USp(2)}_{[1]}(x) \\chi ^{SU(2)}_{[1]}(t^{1/2}) +3\\chi ^{SU(2)}_{[2]}(t^{1/2}) \\\\& \\qquad +\\chi ^{USp(2)}_{[1]}(x) \\chi ^{SU(2)}_{[3]}(t^{1/2})~.\\end{split}$ For $\\mathbf {n}=(0,0,0,0)$ , we have $&H[T_{(3,3,1,1)}(SO(8))] (t; x_1, x_2; \\mathbf {0}) \\nonumber \\\\& \\qquad = \\mathop {\\rm PE}\\left[2 t +\\lbrace x_1^2+x_1^{-2}+(x_1+x_1^{-1})(x_2+x_2^{-1}) \\rbrace t^2 +t^3 - 2t^4 -t^6\\right]~, \\\\&H[T_{(3,2,2,1)}(SO(8))] (t; x; \\mathbf {0}) \\nonumber \\\\& \\qquad = \\mathop {\\rm PE}\\left[\\chi ^{USp(2)}_{[2]}(x) t +2 \\chi ^{USp(2)}_{[1]}(x) t^{3/2} +2 t^2+ \\chi ^{USp(2)}_{[1]}(x) t^{5/2} - 2t^4 -t^6 \\right]~.$ These formulae show that the Coulomb branches are complete intersections of complex dimension 6 and 8 respectively.", "Let us compare the results with the prediction of the monopole formula (REF ).", "Since $T_{(3,2,2,1)}(SO(8))$ is a bad theory, the monopole formula diverges in this case.", "For $T_{(3,3,1,1)}(SO(8))$ , only one fugacity corresponding to topological charge of the gauge group $SO(2)$ can be made manifest; this corresponds to the fugacity $x_2$ in (REF ).", "We precisely reproduce (REF ) with $x_1$ set to unity." ], [ "$T_{(3,1,1)}(USp(4))$ and {{formula:13a95260-b548-4159-af43-235f3defa44e}} ", "The quiver diagrams for $T_{(3,1,1)}(USp(4))$ and $T_{(2,2)}(SO(5))$ are $T_{(3,1,1)}(USp(4)): &\\qquad [USp(4)]-(O(2)) \\\\T_{(2,2)}(SO(5)): &\\qquad [SO(5)]-(USp(2))-(O(1))~.$ In this appendix we show that the Hilbert series of the Coulomb branch of these quivers, which are both 1 quaternionic dimensional in agreement with (2.3) of [26], are equal.", "Note that $(3,1,1)$ is a $B$ -partition for $SO(5)$ , and so the global symmetry $G_{\\mathbf {\\rho }}$ associated with this puncture is $SO(2)$ , according to (REF ).", "Similarly, $(2,2)$ is a $C$ -partition for $USp(4)$ , and the corresponding symmetry is therefore also $SO(2)$ .", "Using formula (REF ), we obtain for $T_{(2,2)}(USp(4))$ $H[T_{(3,1,1)}(USp(4))] (t; x; n_1, n_2)= t^{2n_1+n_2} (1-t)^2 K^{SO(5)}_{(3,1,1)} (x; t) \\Psi ^{n_1,n_2}_{SO(5)} (t, x; t)~,$ where the argument $(t, x)$ of the HL polynomial comes from the decomposition (REF ) $\\begin{split}\\chi ^{SO(5)}_{[1,0]}(\\mathbf {a}) &=1+a_1+a_2+a_1^{-1}+a_2^{-1} \\\\&= (t+1+t^{-1})+(x+x^{-1})~, \\qquad \\mathbf {a} = (t, x),\\end{split}$ with $x$ a fugacity of $SO(2)$ , and the prefactor corresponding to the decomposition (REF ) is $K^{SO(5)}_{(3,1,1)} (x; t) = \\mathop {\\rm PE}\\left[ t + t^2(1+x+x^{-1})\\right]~.$ Similarly, using formula (REF ) we obtain for $T_{(2,2,1)}(SO(5))$ , $\\begin{split} &H[T_{(2,2)}(SO(5))] (t; x; n_1, n_2) \\\\& = t^{\\frac{1}{2}(3n_1+n_2)} (1-t)^2 K^{C_2}_{(2,2)} (x; t) \\Psi ^{n_1,n_2}_{C_2} (t^{1/2} x, t^{1/2} x^{-1}; t)~,\\end{split}$ where the argument $(t^{1/2} x, t^{1/2} x^{-1})$ of the HL polynomial comes from the decomposition (REF ): $\\begin{split}\\chi ^{USp(4)}_{[1,0]}(\\mathbf {a}) &=a_1+a_2+a_1^{-1}+a_2^{-1} \\\\&= (t^{1/2}+t^{-1/2})(x+x^{-1})~, \\qquad \\mathbf {a} = (t^{1/2} x, t^{1/2} x^{-1}),\\end{split}$ and the prefactor corresponding to the decomposition (REF ) is $K^{C_2}_{(2,2)} (x; t) = \\mathop {\\rm PE}\\left[ t + t^2(1+x^2+x^{-2})\\right]~.$ The two Hilbert series can be equated as follows: $H[T_{(3,1,1)}(USp(4))] (t; x^2; n_1, n_2) = H[T_{(2,2)}(SO(5))] (t; x; n_1+n_2, n_1-n_2)~.$ For reference we present the result for $n_1=n_2=0$ , $H[T_{(3,1,1)}(USp(4))] (t; x; \\mathbf {0}) = \\mathop {\\rm PE}\\left[ t + t^2 (x+x^{-1}) -t^4\\right]~.$ This is the Hilbert series for $\\mathbb {C}^2/\\widehat{D}_3 = \\mathbb {C}^2/\\mathbb {Z}_4$ , as expected for the Coulomb branch of $U(1)$ gauge theory with 4 flavors.", "Let us compare the results with the prediction of the monopole formula (REF ).", "For $T_{(3,1,1)}(USp(4))$ , the monopole formula gives the same answer as (REF ).", "However, for $T_{(2,2)}(SO(5))$ , it is not possible to refine the Hilbert series with respect to a topological charge, since there is no factor of $U(1)$ in the quiver diagram; the unrefined monopole formula gives the same Hilbert series as (REF ), with $x=1$ ." ], [ "$T_{(2,2,1)}(USp(4))$ and {{formula:4755d7a3-d9c4-4e78-ba59-e47e7624f34b}}", "The quiver diagrams for $T_{(2,2,1)}(USp(4))$ and $T_{(2,1,1)}(SO(5))$ are $T_{(2,2,1)}(USp(4)): &\\qquad [USp(4)]-(O(4)) \\\\T_{(2,1,1)}(SO(5)): &\\qquad [SO(5)]-(USp(2))-(O(3))~.$ Note that both are `bad' theories in the sense of [2], since in $T_{(2,2,1)}(USp(4))$ the global symmetry $USp(4)$ that connects to gauge group $O(4)$ has rank 2, which is smaller than $4-1=3$ , and in $T_{(2,1,1)}(SO(5))$ the global symmetry $USp(2)$ that connects to gauge group $O(3)$ has rank 1, which is smaller than $3-1=2$ .", "Thus the monopole formula (REF ) diverges for both theories.", "In this appendix, we show that the Hilbert series of the Coulomb branch of these quivers, which are both 2 quaternionic dimensional, are equal.", "As discussed in (REF ), the Coulomb branch global symmetry $G_{\\mathbf {\\rho }}$ corresponding to both partitions is $USp(2)$ .", "From formula (REF ), we obtain $\\begin{split}&H[T_{(2,2,1)}(USp(4))] (t; x; n_1, n_2) \\\\&\\qquad = t^{2n_1+n_2} (1-t)^2 K^{B_2}_{(2,2,1)} (x; t) \\Psi ^{n_1,n_2}_{B_2} (t^{\\frac{1}{2}} x, t^{\\frac{1}{2}} x^{-1}; t)~,\\end{split}\\\\\\begin{split}&H[T_{(2,1,1)}(SO(5))] (t; x; n_1, n_2) \\\\& \\qquad = t^{\\frac{1}{2}(3n_1+n_2)} (1-t)^2 K^{C_2}_{(2,1,1)} (x; t) \\Psi ^{n_1,n_2}_{C_2} (t^{1/2}, x; t)~,\\end{split}$ where the prefactors are $\\begin{split}K^{B_2}_{(2,2,1)} (x; t) &= K^{C_2}_{(2,1,1)} (x; t) \\\\&= \\mathop {\\rm PE}\\left[ t \\chi ^{SU(2)}_{[2]}(x) +t^{3/2} \\chi ^{SU(2)}_{[1]}(x) +t^2 \\right]~.\\end{split}$ The two Hilbert series can be equated as follows: $H[T_{(2,2,1)}(USp(4))] (t; x; n_1, n_2) = H[T_{(3,1,1)}(SO(5))] (t; x; n_1+n_2, n_1-n_2)~.$ The result for $n_1=n_2=0$ is $H[T_{(2,2,1)}(USp(4))] (t; x; \\mathbf {0}) = \\mathop {\\rm PE}\\left[ t \\chi ^{SU(2)}_{[2]} (x)+ t^{3/2} \\chi ^{SU(2)}_{[1]} (x) - t^4 \\right]~.$" ], [ "Hilbert series of hypermultiplet moduli spaces", "In this appendix we compute the Higgs branch Hilbert series of the mirror of various examples considered in the paper." ], [ "Hypermultiplet spaces of $(O(2))-[USp(4)]$ and {{formula:0db01c26-b154-46fe-8e35-caf2422a34a6}}", "In appendix REF , we provide certain evidence that the hypermultiplet moduli spaces of $T_{(3,3,1,1)}(SO(8))$ and $T_{(3,2,2,1)}(SO(8))$ are identical.", "In this appendix, we compute Hilbert series for the hypermultiplet moduli spaces of the uncommon part of the two theories, namely $(O(2))-[USp(4)]$ and $(SO(4))-[USp(4)]$ and show that they equal to each other.", "After `gluing'See [32], [3] for more details on the computations of Higgs branch Hilbert series when two or more theories are `glued' together.", "the resulting Hilbert series with that of $[USp(4)]-[SO(8)]$ via the $USp(4)$ group, we then expect the same Hilbert series for $T_{(3,3,1,1)}(SO(8))$ and $T_{(3,2,2,1)}(SO(8))$ as required.", "In the following we work with the variable $\\tau = t^{1/2}~.$" ], [ "The Higgs branch of $(O(2))-[USp(4)]$", "The quiver $(O(2))-[USp(4)]$ is almost identical to the ADHM quiver for 2 $USp(4)$ instantons on $\\mathbb {C}^2$ (see Fig.", "7 of [33]), except that the former has no symmetric hypermultiplet under $O(2)$ gauge group.", "Thus, the Hilbert series can be computed in a similar way as Section 4 of [33]." ], [ "The contribution from the positive parity of $O(2)$ .", "This is the Higgs branch of $(SO(2))-[USp(4)]$ , whose Hilbert series is $g_+(\\tau ; \\mathbf {x}) = \\oint _{\|z\|=1} \\frac{{\\rm d} z}{2 \\pi i z} (1-\\tau ^2) \\mathop {\\rm PE}\\left[ \\chi ^{USp(4)}_{[1,0]} (\\mathbf {x}) (z+z^{-1}) \\tau \\right]~.$ The first few terms in the $USp(4)$ character expansion of $g_+(\\tau ; \\mathbf {x})$ are $\\begin{split}g_+(\\tau ; \\mathbf {x}) &= 1+ (\\chi ^{C_2}_{[2,0]}(\\mathbf {x})+\\chi ^{C_2}_{[0,1]}(\\mathbf {x})) \\tau ^2 \\\\& \\qquad +(\\chi ^{C_2}_{[4,0]}(\\mathbf {x})+\\chi ^{C_2}_{[0,2]}(\\mathbf {x})+\\chi ^{C_2}_{[2,1]}(\\mathbf {x})) \\tau ^4 + \\ldots ~, \\end{split}$ and the unrefined Hilbert series is $g_+(\\tau ; \\lbrace x_i =1 \\rbrace ) = \\frac{1 + 9 \\tau ^2 + 9 \\tau ^4 + \\tau ^6}{(1 - \\tau ^2)^6}~.$ It should be observed that the $USp(4)$ representations appearing in (REF ) embed into $SU(4)$ representations, namely $g_+(\\tau ; \\mathbf {y}) =\\sum _{m=0}^\\infty \\chi ^{SU(4)}_{[m,0,m]} (\\mathbf {y}) \\tau ^{2m}~, $ where a fugacity map between $SU(4)$ and $USp(4)$ isHere we take $\\chi ^{SU(4)}_{[1,0,0]} (\\mathbf {y})= \\sum _{i=1}^4 y_i$ and $\\chi ^{USp(4)}_{[1,0]} (\\mathbf {x})= \\sum _{i=1}^2 (x_i+ x_i^{-1})$ .", "$y_1 = x_1^{-1}~, \\quad y_2= x_1~, \\quad y_3 = x_2^{-1}~, \\quad y_4 = x_2~.$ Note that (REF ) is in fact the Hilbert series of the reduced moduli space of 1 $SU(4)$ instanton on $\\mathbb {C}^2$ , or of the Higgs branch of $U(1)$ gauge theory with 4 flavors." ], [ "The contribution from the negative parity of $O(2)$ .", "This is given by $\\begin{split}g_-(\\tau ; \\mathbf {x}) &= \\frac{1+\\tau ^2}{ \\prod _{i=1}^2 \\mathop {\\rm det}( {\\bf 1}_{2} - \\tau x_i \\sigma _-(z))\\mathop {\\rm det}( {\\bf 1}_{2} - \\tau x_i^{-1} \\sigma _-(z)) } \\\\&= (1+\\tau ^2) \\mathop {\\rm PE}[ \\tau ^2 \\chi ^{USp(4)}_{[1,0]} (x_1^2, x_2^2) ]~,\\end{split}$ where $ {\\bf 1}_{2}$ denotes a two-by-two identity matrix and $\\sigma _-(z) = \\begin{pmatrix} 0 & z \\\\ z^{-1} & 0 \\end{pmatrix}~.", "$" ], [ "Combining $g_+$ and {{formula:23440a37-ddcb-4d3b-bdc4-1014b5aad092}} .", "The Higgs branch Hilbert series of $(O(2))-[USp(4)]$ is therefore $\\begin{split}& g[(O(2))-[USp(4)]](\\tau ; \\mathbf {x})= \\frac{1}{2} \\left(g_+(\\tau ; \\mathbf {x}) + g_-(\\tau ; \\mathbf {x}) \\right) \\\\&\\qquad = 1+ [2,0]_{\\mathbf {x}} \\tau ^2 +( [4,0]_{\\mathbf {x}} +[0,2]_{\\mathbf {x}})t^2 +( [6,0]_{\\mathbf {x}} +[2,2]_{\\mathbf {x}})\\tau ^3 \\\\& \\hspace{42.67912pt} +( [8,0]_{\\mathbf {x}} +[4,2]_{\\mathbf {x}}+[0,4]_{\\mathbf {x}})\\tau ^4+\\ldots ~,\\end{split}$ where we abbreviate $[a_1, a_2]_{\\mathbf {x}} = \\chi ^{USp(4)}_{[a_1,a_2]} (\\mathbf {x})$ .", "The unrefined Hilbert series is $g[(O(2))-[USp(4)]](\\tau ; \\lbrace x_i =1\\rbrace ) = \\frac{1+4 \\tau ^2+4 \\tau ^4+\\tau ^6}{\\left(1-\\tau ^2\\right)^6}~.", "$" ], [ "The Kibble branch of $(SO(4))-[USp(4)]$", "Let us denote the half-hypermultiplets in the theory by $Q_a^i$ , where $a=1,\\ldots , 4$ is an $SO(4)$ gauge index and $i=1, \\ldots , 4$ is a $USp(4)$ flavor index.", "The superpotential is $W= Q_a^i Q_b^j \\varphi ^{ab} J_{ij}$ , where $J$ is a symplectic matrix and $\\varphi $ is an adjoint field under $SO(4)$ .", "Using Macaulay2 [34], we find the space of $F$ -term solutions (i.e.", "the $F$ -flat space) defined by $0 = \\partial _{\\phi ^{ab}} W = Q_a^i Q_b^j J_{ij}$ is a 11 complex dimensional space; not $16-\\frac{1}{2}(4 \\times 3) =10$ complex dimensional if the gauge group $SO(4)$ is completely broken.", "Hence we conclude that at a generic point of the hypermultiplet moduli space (also known as the Kibble branch [35]), the gauge symmetry $SO(4)$ is broken to $SO(2)$ .", "The remaining unbroken symmetry on the Kibble branch is indeed as expected, since $(SO(4))-[USp(4)]$ is a `bad' theory.", "The Hilbert series of the $F$ -flat space can be computed using Macaulay2: $\\begin{split}{\\cal F}^\\flat (\\tau ; \\mathbf {x}; \\mathbf {z}) &= \\mathop {\\rm PE}\\left[ \\tau [1,0]_{\\mathbf {x}} [1,0]_{\\mathbf {z}} \\right] \\times \\\\& \\Big [ 1 -([2,0]_{\\mathbf {z}} + [0,2]_{\\mathbf {z}}) \\tau ^2 +([2,2]_{\\mathbf {z}} +[2,0]_{\\mathbf {z}}+[0,2]_{\\mathbf {z}}+[0,1]_{\\mathbf {x}})\\tau ^4 \\\\& -([1,1]_{\\mathbf {z}}[1,0]_{\\mathbf {x}}) \\tau ^5 -([2, 2]_{\\mathbf {z}} + 1)\\tau ^6+([1,1]_{\\mathbf {z}}[1,0]_{\\mathbf {x}}) \\tau ^7- [0,1]_{\\mathbf {z}} \\tau ^8 \\Big ]~,\\end{split}$ where we abbreviate $[a_1, a_2]_{\\mathbf {x}} = \\chi ^{USp(4)}_{[a_1,a_2]} (\\mathbf {x})$ and $[b_1, b_2]_{\\mathbf {z}} = \\chi ^{SO(4)}_{[b_1,b_2]} (\\mathbf {z})$ .", "The unrefined Hilbert series of the $F$ -flat space is ${\\cal F}^\\flat (\\tau ; \\lbrace x_i=1 \\rbrace ; \\lbrace z_i=1 \\rbrace ) &= \\frac{1 + 5 \\tau + 9 \\tau ^2 + 5 \\tau ^3}{(1 - \\tau )^{11}}~.$ Note that the $F$ -flat space is indeed 11 complex dimensional.", "Integrating over the Haar measure of $SO(4)$ , we obtain the Kibble branch Hilbert series of $(SO(4))-[USp(4)]$ $g[(SO(4))-[USp(4)]](\\tau ; \\mathbf {x}) &= \\int {\\rm d} \\mu _{SO(4)} (\\mathbf {z}) \\; {\\cal F}^\\flat (\\tau ; \\mathbf {x}; \\mathbf {z}) \\nonumber \\\\&= 1+ [2,0]_{\\mathbf {x}} \\tau ^2 +( [4,0]_{\\mathbf {x}} +[0,2]_{\\mathbf {x}})\\tau ^2 +( [6,0]_{\\mathbf {x}} +[2,2]_{\\mathbf {x}})\\tau ^3 \\nonumber \\\\& \\quad +( [8,0]_{\\mathbf {x}} +[4,2]_{\\mathbf {x}}+[0,4]_{\\mathbf {x}})\\tau ^4+\\ldots ~.$ The corresponding unrefined Hilbert series is $g[(SO(4))-[USp(4)]](\\tau ; \\lbrace x=1 \\rbrace ) = \\frac{1+4 \\tau ^2+4 \\tau ^4+ \\tau ^6}{\\left(1-\\tau ^2\\right)^6}~,$ which is is equal to (REF )." ], [ "The Higgs branch of the mirror of $T_{(3,1,1)}(USp(4))$ and the baryonic generating function", "In this section of the appendix, we discuss the computation of the Higgs branch Hilbert series and the baryonic generating function of the mirror theory of $T_{(3,1,1)}(USp(4))$ theory, whose quiver is given by (REF ).", "Let us now compute the baryonic generating on the Higgs branch of this mirror theory.", "Let us denote the chiral fields in the quiver as follows: $\\begin{array}{lll}[SO(2)]-(USp(2)): &\\quad Q^{i_1}_{~a_1} & \\quad \\text{$i_1=1,2$ of $[SO(2)]$}, ~\\text{$a_1=1,2$ of $(USp(2))$,} \\\\(USp(2))-(SO(2)): &\\quad X^{a_2}_{~a_1} & \\quad \\text{$a_2=1,2$ of $(SO(2))$,} \\\\(SO(2))-(USp(2)): &\\quad Y^{a_2}_{~a_3} & \\quad \\text{$a_3=1,2$ of $(USp(2))$,}\\\\(USp(2))-[SO(2)]: &\\quad q^{j_2}_{~b_3} & \\quad \\text{$j_2=1,2$ of $[SO(2)]$}, ~\\text{$b_3=1,2$ of $(USp(2))$}\\end{array}$ The superpotential is given by $\\begin{split}W&= M^{SO(2)}_{i_1 i^{\\prime }_1} \\phi _1^{a_1 b_1} Q^{i_1}_{~a_1} Q^{i^{\\prime }_1}_{b_1} + \\epsilon ^{a_1 a^{\\prime }_1}(\\phi _2)_{a_2 a^{\\prime }_2} X^{a_2}_{~a_1} X^{a^{\\prime }_2}_{~a^{\\prime }_1} -\\phi _1^{a_1 a^{\\prime }_1} M^{SO(2)}_{a_2 a^{\\prime }_2} X^{a_2}_{~a_1} X^{a^{\\prime }_2}_{~a^{\\prime }_1} \\\\& \\quad + \\epsilon ^{a_3 a^{\\prime }_3}(\\phi _2)_{a_2 a^{\\prime }_2} Y^{a_2}_{~a_3} Y^{a^{\\prime }_2}_{~a^{\\prime }_3} -\\phi _3^{a_3 a^{\\prime }_3} M^{SO(2)}_{a_2 a^{\\prime }_2} Y^{a_2}_{~a_3} Y^{a^{\\prime }_2}_{~a^{\\prime }_3}+ M^{SO(2)}_{j_2 j^{\\prime }_2} \\phi _3^{a_3 b_3} Q^{j_2}_{~a_3} Q^{j^{\\prime }_2}_{b_3}~,\\end{split}$ where $M^{SO(2)}$ is a matrix associated with the bilinear form of $SO(2)$ .", "In order for the Lie algebra to contain a nonzero diagonal matrix, which is important for fugacity assignments, we takeIf $M^{SO(2)}$ is taken to be the identity matrix, the Lie algebra would consist of anti-symmetric matrices, and there would be no nonzero diagonal matrix in the Lie algebra.", "$M^{SO(2)} = \\begin{pmatrix} 0& 1 \\\\ 1& 0 \\end{pmatrix}~.$ Since the mirror theory is a `bad' theory, we expect that the space of $F$ -terms solutions (i.e.", "the $F$ -flat space) $\\lbrace \\partial _{\\phi _1} W =0, ~ \\partial _{\\phi _2} W =0,~ \\partial _{\\phi _3} W =0 \\rbrace ~,$ has many branches.", "In order to determine the relevant branch (9 complex dimensional space), we perform the primary decomposition, which can be done using mathematical packages such as STRINGVACUA [36] or SINGULAR [37].", "After such a decomposition, we find using Macaulay2 [34] the Hilbert series of the relevant branch of the $F$ -flat space.", "Since the result is too long to be reported here, we present the first few terms in the series expansion in $\\tau $ : $\\begin{split}& \\mathcal {F}^\\flat (t; x, y; z_1, b, z_2) \\\\&= 1+ \\tau \\Big \\lbrace ( x^{\\pm 1}+ b^{\\pm 1})[1]_{z_1} +( y^{\\pm 1}+ b^{\\pm 1})[1]_{z_2} \\Big \\rbrace \\\\& \\quad + \\tau ^2 \\Big \\lbrace (b^{\\pm 2}+x^{\\pm 2}+b^{\\pm 1}x^{\\pm 1}+1 )[2]_{z_1} + (x \\rightarrow y, \\; z_1 \\rightarrow z_2) \\\\& \\qquad \\quad + (x^{\\pm 1} y^{\\pm 1}+b^{\\pm 1} y^{\\pm 1}+x^{\\pm 1} b^{\\pm 1}+b^{\\pm 2}+2 )[1]_{z_1}[1]_{z_2} \\\\& \\qquad \\quad + (b x^{-1})^{\\pm 1} + (b y^{-1})^{\\pm 1} +1\\Big \\rbrace + \\ldots ~,\\end{split}$ where $x, y$ are the fugacities of the two $SO(2)$ flavor symmetries and $z_1, b, z_2$ are respectively the fugacitites of the gauge groups $USp(2), SO(2), USp(2)$ .", "We have used the shorthand notation $a^{\\pm n} = a^n+a^{-n}$ .", "The unrefined $F$ -flat Hilbert series is $\\mathcal {F}^\\flat (\\tau ; 1, 1; 1, 1, 1) = \\frac{(1+\\tau ) (1+3 \\tau )^2}{(1-\\tau )^9}~.$ Let us next integrate over the two $USp(2)$ gauge groups, but not the $SO(2)$ gauge group: $&\\frac{1}{(2 \\pi i)^2}\\oint _{\|z_1\|=1} \\frac{1-z_1^2}{z_1} \\mathrm {d} z_1 \\oint _{\|z_2\|=1} \\frac{1-z_2^2}{z_2} \\mathrm {d} z_2\\; \\mathcal {F}^\\flat (\\tau ; x, y; z_1, b, z_2) \\nonumber \\\\&= \\mathop {\\rm PE}\\left[ \\lbrace 1+ b (x^{-1}+y^{-1}) + b^{-1} (x+y) \\rbrace \\tau ^2 -2\\tau ^4 \\right]~.$ The baryonic generating function for this theory is obtained by ungauging the $SO(2)$ group; this amounts to multiplying the above function by $(1-\\tau ^2)^{-1}$ to remove the contribution of the $F$ -term from this $SO(2)$ group: $&{\\cal G}[\\text{Mirror of $T_{(3,1,1)}(USp(4))$}/SO(2)](t; x, y; b) \\nonumber \\\\&= \\mathop {\\rm PE}\\left[ \\lbrace 2+(b_1+b_1^{-1}) + (b_2+b_2^{-1}) \\rbrace \\tau ^2 - 2\\tau ^4 \\right]~.$ The Higgs branch Hilbert series is then given by $&\\oint _{\|b\|=1} \\frac{{\\rm d} b}{2 \\pi i b} (1-\\tau ^2){\\cal G}[\\text{mirror of $T_{(3,1,1)}(USp(4))/SO(2)$}](\\tau ; x, y; b) \\nonumber \\\\&= \\mathop {\\rm PE}\\left[ \\tau ^2 + \\tau ^4( x y^{-1} + y x^{-1}) -\\tau ^8 \\right]~.$" ], [ "Derivation of (", "The monopole formula for the Coulomb branch Hilbert series for $T_{\\mathbf {\\rho }} (SU(N))$ reads $\\begin{split}& H[T_{\\mathbf {\\rho }}(SU(N))] (t; \\mathbf {x}; \\mathbf {n}) \\\\& =x_1^{\\sum _{j=1}^N n_j} \\sum _{m_{1,1} \\in \\mathbb {Z}} ~\\sum _{m_{2,2}\\ge m_{1,2} > -\\infty } \\cdots ~\\sum _{m_{N-\\rho _1,N-\\rho _1}\\ge \\cdots \\ge m_{1,N-\\rho _1} > -\\infty } \\\\& \\quad \\left(\\frac{x_2}{x_1}\\right)^{\\sum \\limits _{i=1}^{N-\\rho _1} m_{i,N-\\rho _1}} \\cdots \\left(\\frac{x_{d-h+1}}{ x_{d-h}}\\right)^{\\sum \\limits _{i=1}^{h+\\ell } m_{i,h+\\ell }} \\left(\\frac{x_{d-h+2}}{ x_{d-h+1}}\\right)^{\\sum \\limits _{i=1}^{h} m_{i,h}} \\cdots \\left(\\frac{x_{d+1}}{ x_{d}}\\right)^{m_{1,1}} \\times \\\\& \\quad t^{\\frac{1}{2} \\Delta {(\\rho _1, \\cdots , \\rho _{d-h})} + \\frac{1}{2} \\Delta {(\\ell ,1^h)}} P_{U(N-\\rho _1)}(t;\\lbrace m_{i,N-\\rho _1} \\rbrace ) \\cdots P_{U(2)}(t;\\lbrace m_{i,2} \\rbrace ) P_{U(1)}(t)~,\\end{split}$ where the dimension formula $ \\Delta {(\\rho _1, \\cdots , \\rho _{d-h})} $ corresponds to the quiver tail $[N]-\\cdots -(h+\\ell +\\rho _{d-h})-(h+\\ell )$ including the contribution of the $U(h+\\ell )$ vector multiplet, and $\\Delta {(\\ell ,1^h)}$ corresponds to the tail $[h+\\ell ]-(h)-\\cdots -(2)-(1)$ without the contribution of the $U(h+\\ell )$ vector multiplet.", "Explicitly, the latter is given by $\\begin{split}\\Delta _{(\\ell ,1^h)} &= \\sum _{j=1}^{h-1} \\sum _{i=1}^j \\sum _{i^{\\prime }=1}^{j+1} \| m_{i,j}-m_{i^{\\prime },j+1} \| +\\sum _{i=1}^h \\sum _{i^{\\prime }=1}^{h+\\ell } \| m_{i,h}-m_{i^{\\prime },h+\\ell } \| \\\\& \\quad -2 \\sum _{j=2}^h \\sum _{1\\le i<i^{\\prime }\\le j} (m_{i,j}-m_{i^{\\prime },j})~.\\end{split}$ We will show that the monopole formula (REF ) for the Coulomb branch Hilbert series has a simple pole at $x_{d-h+1}x_{d+1}^{-1} = t^{\\frac{1}{2}(1+\\ell )}$ , and compute the residue at $z\\rightarrow 1$ when the fugacities associated to the partition $\\rho $ satisfy (REF ), to reproduce (REF ).", "The pole is due to the region in which $m_{1,1}, m_{2,2}, \\ldots , m_{h,h}$ are much larger than other $m_{i,j}$ and all $n_i$ .", "In the brane picture at the top of Figure REF , this limit corresponds to considering the $h$ rightmost pairs of adjacent NS5-branes, and moving one D3-brane attached to each such pair far out on the Coulomb branch.", "This limit leaves the brane picture at the bottom of Figure REF at finite distance.", "Setting $m_{h-k,h-k}=m_{h,h} + \\sum _{i=1}^k q_i~, \\qquad k = 1, \\ldots , h-1~,$ we obtain $\\Delta {(\\ell ,1^h)} = \\Delta {(\\ell +1,1^{h-1})} + \\widehat{\\Delta }~,$ where $\\begin{split}\\widehat{\\Delta }&= \\sum _{j=1}^{h-1} \\sum _{i^{\\prime }=1}^j (m_{j,j}-m_{i^{\\prime },j+1}) + \\sum _{j=2}^{h-1} \\sum _{i=1}^{j-1} (m_{j+1,j+1} - m_{i,j}) + \\sum _{i=1}^{h-1} \|m_{i,i}-m_{i+1,i+1}\| \\\\& \\quad + \\sum _{i^{\\prime }=1}^{h+\\ell } (m_{h,h} - m_{i^{\\prime },h+\\ell }) - 2 \\sum _{j=2}^h \\sum _{i=1}^{j-1} (m_{j,j}-m_{i,j}) \\\\&= -\\sum _{i=1}^{h+\\ell } m_{i,h+\\ell } + \\sum _{i=1}^{h-1} \|q_i\| + m_{1,1}+ \\ell m_{h,h} + \\sum _{i=1}^{h-1} m_{i,h} \\\\&= -\\sum _{i=1}^{h+\\ell } m_{i,h+\\ell } + \\sum _{i=1}^{h-1} \|q_i\| +(\\ell +1) m_{h,h}+ \\sum _{i=1}^{h-1}q_i+ \\sum _{i=1}^{h-1} m_{i,h}\\end{split}$ and $\\Delta {(\\ell +1,1^{h-1})}$ corresponds to the tail $[h+\\ell ]-(h-1)-\\cdots -(1)$ without the contribution of the $U(h+\\ell )$ vector multiplet.", "Note that the magnetic charges of the $U(j-1)$ gauge group in the $[h+\\ell ]-(h-1)-\\cdots -(1)$ tail of $T_{\\mathbf {\\rho }^\\prime }(SU(N))$ , with $j=2,\\dots ,h$ , arise from the magnetic charges of the $U(j)$ gauge group in the original $[h+\\ell ]-(h-1)-\\cdots -(1)$ tail of $T_{\\mathbf {\\rho }}(SU(N))$ which are kept finite: $\\text{GNO charges for the $U(j-1)$ group of $T_{\\mathbf {\\rho }^\\prime }(SU(N))$}: \\qquad (m_{i,j})_{i=1}^{j-1}~.$ Next, we focus on the changes in the topological factors in the second line of (REF ).", "The factor that involves the fugacities $x_{d-h+1}$ and $x_{d+1}$ associated to the blue boxes becomes, using (REF ) and (REF ), $\\begin{split}&(x_{d-h+1})^{\\sum _{i=1}^{h+\\ell } m_{i,h+\\ell }-\\sum _{i=1}^{h} m_{i,h}} \\cdot x_{d+1}^{m_{1,1}} = \\\\&= (t^{1/2} z)^{\\sum _{i=1}^{h+\\ell } m_{i,h+\\ell }-\\sum _{i=1}^{h-1} m_{i,h} -(\\ell +1) m_{h,h}- \\ell \\sum _{i=1}^{h-1}q_i} \\times \\\\& \\quad y_{g-h+1}^{\\sum _{i=1}^{h+\\ell } m_{i,h+\\ell }-\\sum _{i=1}^{h-1} m_{i,h} + \\sum _{i=1}^{h-1}q_i}~.\\end{split}$ The factor that involves the fugacities associated to the $h-1$ pink boxes becomes $\\begin{split}& \\prod _{j=1}^{h-1} (x_{d-h+1+j})^{\\sum _{i=1}^{h+1-j} m_{i,h+1-j} - \\sum _{i=1}^{h-j} m_{i,h-j}} = \\\\&= \\left\\lbrace \\prod _{j=1}^{h-2} (x_{d-h+1+j})^{\\sum _{i=1}^{h-j} m_{i,h+1-j} - \\sum _{i=1}^{h-j-1} m_{i,h-j}} \\times x_d^{m_{1,2}} \\right\\rbrace \\prod _{j=1}^{h-1} x_{d-h+1+j}^{-q_j}~,\\end{split}$ where the quantity in the curly bracket $\\lbrace \\cdots \\rbrace $ becomes parts of the new topological factors after the box is moved.", "The topological factors associated to the grey boxes are not affected.", "Finally we consider the fate of the classical factors of the $U(j)$ gauge groups, for $j=1,\\dots ,h$ , in the limit where $m_{1,1}, m_{2,2}, \\ldots , m_{h,h}$ are much larger than other $m_{i,j}$ and all $n_i$ .", "The limit breaks $U(j)\\rightarrow U(j-1)\\times U(1)$ along the Coulomb branch, and $P_{U(j)} (t; \\lbrace m_{i,j}\\rbrace _{i=1}^{j-1}, m_{j,j} ) = P_{U(j-1)} (t; \\lbrace m_{i,j}\\rbrace _{i=1}^{j-1} ) \\times P_{U(1)}(t)~.$ Combining the relevant terms together, we summarize each factor below: The power of $t^{1/2}$ .", "We combine (REF ) and (REF ) and yield $\\Delta {(\\rho _{g-h+1}+1,1^{h-1})} +\\sum _{i=1}^{h-1} \|q_i\|-(\\ell -1)\\sum _{i=1}^{h-1} q_i~.$ The power of $z$ .", "This comes from (REF ), $\\sum _{i=1}^{h+\\ell } m_{i,h+\\ell }-\\sum _{i=1}^{h-1} m_{i,h} -(\\ell +1) m_{h,h}- \\ell \\sum _{i=1}^{h-1}q_i~.$ The power of $y_{d-h+1}$ .", "This comes from (REF ), $\\sum _{i=1}^{h+\\ell } m_{i,h+\\ell }-\\sum _{i=1}^{h-1} m_{i,h} + \\sum _{i=1}^{h-1}q_i~.$ The factors containing $x$ 's in the $(1^{h-1})$ block.", "These come from (REF ), $\\left\\lbrace \\prod _{j=1}^{h-2} (x_{d-h+1+j})^{\\sum _{i=1}^{h-j} m_{i,h+1-j} - \\sum _{i=1}^{h-j-1} m_{i,h-j}} \\times x_d^{m_{1,2}} \\right\\rbrace \\prod _{j=1}^{h-1} x_{d-h+1+j}^{-q_j}~.$ The classical factors.", "These come from (REF ), $\\prod _{j=1}^h P_{U(j)} (t, \\lbrace m_{i,j}\\rbrace _{i=1}^j ) = \\mathop {\\rm PE}[ h t ] ~ \\prod _{j=2}^{h} P_{U(j-1)} (t, \\lbrace m_{i,j}\\rbrace _{i=1}^{j-1} )~.", "$ Finally, we perform the summations over the large magnetic charges $\\lbrace m_{j,j}\\rbrace _{j=1}^h$ , or equivalently over $m_{h,h}$ and $q_1,\\dots ,q_{h-1}$ : The summation over $m_{h,h}$ : the summand depends on $m_{h,h}$ only via $z^{-(\\ell +1)m_{h,h}}$ .", "This is responsible for a simple pole when $z\\rightarrow 1$ , with residue $\\mathop {\\rm Res}_{z \\rightarrow 1} \\sum _{m_{h,h} \\ge L} z^{-(1+\\ell ) m_{h,h}} = \\frac{1}{1+\\ell } ~.", "$ The remaining factors are finite as $z\\rightarrow 1$ , that we set in the following.", "The summations over $q_i$ yield $\\begin{split}&\\prod _{i=1}^{h-1} \\sum _{q_i \\in \\mathbb {Z}} t^{\\frac{1}{2} \|q_i\|} \\left(t^{-\\frac{1}{2}(\\ell -1)} \\frac{y_{d-h+1}}{x_{d-h+1+i}} \\right)^{q_i} \\\\&= \\prod _{i=1}^{h-1} \\mathop {\\rm PE}\\left[-t +t^{\\frac{1}{2}} \\sum _{s = \\pm 1} \\left(t^{-\\frac{1}{2}(\\ell -1)} \\frac{y_{d-h+1}}{x_{d-h+1+i}} \\right)^s \\right] \\\\&= \\mathop {\\rm PE}\\left[ -(h-1)t + t^\\frac{1}{2} \\sum _{i=1}^{h-1} \\sum _{s =\\pm 1} \\left(t^{-\\frac{1}{2}(\\ell -1)} \\frac{y_{d-h+1}}{x_{d-h+1+i}} \\right)^s \\right]~.\\end{split}$ Combining (REF ) and (REF ) with the classical factor $\\mathop {\\rm PE}[h t]$ of $U(1)^h$ in (REF ), we recover ${\\cal P}_{\\mathbf {\\rho }\\mathbf {\\rho }^\\prime }$ in (REF ).", "The remaining factors and summations combine with the factors and summations of the spectator part of the quiver to reconstruct the monopole formula for the Coulomb branch Hilbert series of $T_{\\mathbf {\\rho }^\\prime }(SU(N))$ .", "Hence we have reproduced formula (REF ) from the monopole formulae." ], [ "From $U(N_c)$ gauge theory with {{formula:896db447-7392-4ac9-b8b1-d36aee2e753b}} flavors to {{formula:e032396a-1ead-4993-9609-fcbf0c984246}} flavors", "In this appendix we show how one can obtain the Coulomb branch Hilbert series of $U(N_c)$ gauge theory with $N_f-1$ flavors from that with $N_f$ flavors by means of a residue computation.", "The idea can be explained physically as follows.", "We gauge a $U(1)$ subgroup of the $SU(N_f)$ flavor symmetry, introducing a flat direction from the Coulomb branch of the $U(1)$ gauge group.", "Associated to the flat direction there is a pole in the Coulomb branch Hilbert series, whose residue is related to the Hilbert series of the leftover $U(N_c)$ gauge theory with $N_f-1$ flavors.", "We start from the Hilbert series for $U(N_c)$ gauge theory with $N_f$ flavors, $& H[U(N_c),N_f] (t; z, z_0; n_1, \\ldots , n_{N_f}) \\nonumber \\\\& = z_0^{\\sum _{i=1}^{N_f} n_i} \\sum _{m_1 \\ge \\ldots \\ge m_{N_c} > -\\infty } t^{\\Delta _{N_c,N_f}(\\mathbf {m}, \\mathbf {n})} P_{U(N_c)} (t; m_1, \\ldots , m_{N_c}) z^{m_1+\\ldots +m_{N_c}}~,$ where $\\Delta _{N_c,N_f}(\\mathbf {m}, \\mathbf {n}) &= \\frac{1}{2} \\sum _{i=1}^{N_c} \\sum _{f=1}^{N_f} \|m_i-n_f\| - \\sum _{1 \\le i <j \\le N_c} \|m_i - m_j\|~.$ If we gauge the $U(1)$ subgroup of $SU(N_f)$ associated to the magnetic flux $n_{N_{f}}$ , and let $w$ be the fugacity of the corresponding topological symmetry, the Hilbert series becomes $\\begin{split}& H(t; z_0, z, w; n_1, \\ldots , n_{N_f-1}) \\nonumber \\\\& = \\frac{1}{1-t} ~ z_0^{\\sum _{i=1}^{N_f-1} n_i} \\sum _{n_{N_f} \\in \\mathbb {Z}} \\sum _{m_1 \\ge \\ldots \\ge m_{N_c} > -\\infty } t^{\\Delta (\\mathbf {m}, \\mathbf {n})} P_{U(N_c)} (t; \\mathbf {m}) z^{m_1+\\ldots +m_{N_c}} w^{n_{N_f}}~.\\end{split}$ Now we set $z = t^{1/2} y x~, \\qquad w = t^{-\\frac{1}{2}N_c} x^{-1}$ and consider the $x\\rightarrow 1$ limit.", "Along the lines of the previous subsection, we find that there is a simple pole due to the region where $n_{N_f}\\gg m_i, n_j$ , with $i=1,\\dots ,N_c$ and $j=1,\\dots ,N_f-1$ , and the residue is $\\begin{split}\\mathop {\\rm Res}_{x \\rightarrow 1} & ~H(t;\\;z_0,\\; t^{\\frac{1}{2}} y x, \\; t^{-\\frac{1}{2} N_c} x^{-1}; n_1, \\ldots , n_{N_f-1}) \\nonumber \\\\&= \\frac{1}{1-t} ~ H[U(N_c),N_f-1] (t; y, n_1, \\ldots , n_{N_f-1})~,\\end{split}$ where $H[U(N_c),N_f-1] $ is the Coulomb branch Hilbert series of the $U(N_c)$ gauge theory with $N_f-1$ hypermultiplets." ] ]	1403.0585
[ [ "Power spectrum of the geodesic flow on hyperbolic manifolds" ], [ "Abstract We describe the complex poles of the power spectrum of correlations for the geodesic flow on compact hyperbolic manifolds in terms of eigenvalues of the Laplacian acting on certain natural tensor bundles.", "These poles are a special case of Pollicott-Ruelle resonances, which can be defined for general Anosov flows.", "In our case, resonances are stratified into bands by decay rates.", "The proof also gives an explicit relation between resonant states and eigenstates of the Laplacian." ], [ "Outline and structure", "In this section, we give the ideas of the proof of Theorem REF , first in dimension 2 and then in higher dimensions, and describe the structure of the paper." ], [ "Dimension 2", "We start by using the following criterion (Lemma REF ): $\\lambda \\in \\mathbb {C}$ is a Pollicott–Ruelle resonance if and only if the space $\\operatorname{Res}_X(\\lambda ):=\\lbrace u\\in \\mathcal {D}^{\\prime }(SM)\\mid (X+\\lambda ) u=0,\\ \\operatorname{WF}(u)\\subset E_u^\\rbrace $ is nontrivial.", "Here $\\mathcal {D}^{\\prime }(SM)$ denotes the space of distributions on $M$ (see [33]), $\\operatorname{WF}(u)\\subset T^(SM)$ is the wavefront set of $u$ (see [33]), and $E_u^\\subset T^(SM)$ is the dual unstable foliation described in (REF ).", "It is more convenient to use the condition $\\operatorname{WF}(u)\\subset E_u^$ rather than $u\\in \\mathcal {H}^r$ because this condition is invariant under differential operators of any order.", "The key tools for the proof are the horocyclic vector fields $U_\\pm $ on $SM$ , pictured on Figure REF (a) below.", "To define them, we represent $M=\\Gamma \\backslash \\mathbf {H}^2$ , where $\\mathbf {H}^2=\\lbrace z\\in \\mathbb {C}\\mid \\operatorname{Im}z>0\\rbrace $ is the hyperbolic plane and $\\Gamma \\subset \\operatorname{PSL}(2;\\mathbb {R})$ is a co-compact Fuchsian group of isometries acting by Möbius transformations.", "(See Appendix for the relation of the notation we use in dimension 2, based on the half-plane model of the hyperbolic space, to the notation used elsewhere in the paper which is based on the hyperboloid model.)", "Then $SM$ is covered by $S\\mathbf {H}^2$ , which is isomorphic to the group $G:=\\operatorname{PSL}(2;\\mathbb {R})$ by the map $\\gamma \\in G\\mapsto (\\gamma (i),d\\gamma (i)\\cdot i)$ .", "Consider the left invariant vector fields on $G$ corresponding to the following elements of its Lie algebra: $X=\\begin{pmatrix} {1\\over 2}&0\\\\0&-{1\\over 2}\\end{pmatrix},\\quad U_+=\\begin{pmatrix} 0&1\\\\0&0\\end{pmatrix},\\quad U_-=\\begin{pmatrix} 0&0\\\\1&0\\end{pmatrix},$ then $X,U_\\pm $ descend to vector fields on $SM$ , with $X$ becoming the generator of the geodesic flow.", "We have the commutation relations $[X,U_\\pm ]=\\pm U_\\pm ,\\quad [U_+,U_-]=2X.$ For each $\\lambda $ and $m\\in \\mathbb {N}_0$ , define the spaces $V_m(\\lambda ):=\\lbrace u\\in \\mathcal {D}^{\\prime }(SM)\\mid (X+\\lambda ) u=0,\\ U_-^m u=0,\\ \\operatorname{WF}(u)\\subset E_u^\\rbrace ,$ and put $\\operatorname{Res}^0_X(\\lambda ):=V_1(\\lambda ).$ By (REF ), $U_-^m(\\operatorname{Res}_X(\\lambda ))\\subset \\operatorname{Res}_X(\\lambda +m)$ .", "Since there are no Pollicott–Ruelle resonances in the right half-plane, we conclude that $\\operatorname{Res}_X(\\lambda )=V_m(\\lambda )\\quad \\text{for }m>-\\operatorname{Re}\\lambda .$ We now use the diagram (writing $\\operatorname{Id}=U_\\pm ^0$ , $U_\\pm =U_\\pm ^1$ for uniformity of notation) Figure: NO_CAPTION where $\\iota $ denotes the inclusion maps and unless $\\lambda \\in -1-{1\\over 2}\\mathbb {N}_0$ , we have $V_{m+1}(\\lambda )=V_m(\\lambda )\\oplus U_+^m(\\operatorname{Res}^0_X(\\lambda +m)),$ and $U_+^m$ is one-to-one on $\\operatorname{Res}^0_X(\\lambda +m)$ ; indeed, using (REF ) we calculate $U_-^m U_+^m= m!\\bigg (\\prod _{j=1}^m (2\\lambda +m+j)\\bigg )\\operatorname{Id}\\quad \\text{on }\\operatorname{Res}^0_X(\\lambda +m)$ and the coefficient above is nonzero when $\\lambda \\notin -1-{1\\over 2}\\mathbb {N}_0$ .", "We then see that $\\operatorname{Res}_X(\\lambda )=\\bigoplus _{m\\ge 0} U_+^m(\\operatorname{Res}^0_X(\\lambda +m)).$ It remains to describe the space of resonant states in the first band, $\\operatorname{Res}^0_X(\\lambda )=\\lbrace u\\in \\mathcal {D}^{\\prime }(SM)\\mid (X+\\lambda ) u=0,\\ U_-u=0,\\ \\operatorname{WF}(u)\\subset E_u^\\rbrace .$ We can remove the condition $\\operatorname{WF}(u)\\subset E_u^$ as it follows from the other two, see the remark following Lemma REF .", "We claim that the pushforward map $u\\in \\operatorname{Res}^0_X(\\lambda )\\mapsto f(x):=\\int _{S_x M} u(x,\\xi )\\,dS(\\xi )$ is an isomorphism from $\\operatorname{Res}^0_X(\\lambda )$ onto $\\operatorname{Eig}(-\\lambda (1+\\lambda ))$ , where $\\operatorname{Eig}(\\sigma )=\\lbrace u\\in \\mathcal {C}^\\infty (M)\\mid \\Delta u=\\sigma u\\rbrace $ ; this would finish the proof.", "In other words, the eigenstate of the Laplacian corresponding to $u$ is obtained by integrating $u$ over the fibers of $SM$ .", "To show that (REF ) is an isomorphism, we reduce the elements of $\\operatorname{Res}^0_X(\\lambda )$ to the conformal boundary $\\mathbb {S}^1$ of the ball model $\\mathbb {B}^2$ of the hyperbolic space as follows: $\\operatorname{Res}^0_X(\\lambda )=\\lbrace P(y,B_-(y,\\xi ))^\\lambda w(B_-(y,\\xi ))\\mid w\\in \\operatorname{Bd}(\\lambda )\\rbrace ,$ where $P(y,\\nu )$ is the Poisson kernel: $P(y,\\nu )={1-\|y\|^2\\over \|y-\\nu \|^2}$ , $y\\in \\mathbb {B}^2$ , $\\nu \\in \\mathbb {S}^1$ ; $B_-:S\\mathbb {B}^2\\rightarrow \\mathbb {S}^1$ maps $(y,\\xi )$ to the limiting point of the geodesic $\\varphi _t(y,\\xi )$ as $t\\rightarrow -\\infty $ , see Figure REF (a); and $\\operatorname{Bd}(\\lambda )\\subset \\mathcal {D}^{\\prime }(\\mathbb {S}^1)$ is the space of distributions satisfying certain equivariance property with respect to $\\Gamma $ .", "Here we lifted $\\operatorname{Res}_X^0(\\lambda )$ to distributions on $S\\mathbb {H}^2$ and used the fact that the map $B_-$ is invariant under both $X$ and $U_-$ ; see Lemma REF for details.", "It remains to show that the map $w\\mapsto f$ defined via (REF ) and (REF ) is an isomorphism from $\\operatorname{Bd}(\\lambda )$ to $\\operatorname{Eig}(-\\lambda (1+\\lambda ))$ .", "This map is given by (see Lemma REF ) $f(y)=P_\\lambda ^- w(y):=\\int _{\\mathbb {S}^1}P(y,\\nu )^{1+\\lambda }w(\\nu )\\,dS(\\nu )$ and is the Poisson operator for the (scalar) Laplacian corresponding to the eigenvalue $s(1-s)$ , $s=1+\\lambda $ .", "This Poisson operator is known to be an isomorphism for $\\lambda \\notin -1-\\mathbb {N}$ , see the remark following Theorem REF in Section REF , finishing the proof." ], [ "Higher dimensions", "In higher dimensions, the situation is made considerably more difficult by the fact we can no longer define the vector fields $U_\\pm $ on $SM$ .", "To get around this problem, we remark that in dimension 2, $U_-u$ is the derivative of $u$ along a certain canonical vector in the one-dimensional unstable foliation $E_u\\subset T(SM)$ and similarly $U_+u$ is the derivative along an element of the stable foliation $E_s$ .", "(See Section REF .)", "In dimension $n+1>2$ , the foliations $E_u,E_s$ are $n$ -dimensional and one cannot trivialize them.", "However, each of these foliations is canonically parametrized by the following vector bundle $\\mathcal {E}$ over $SM$ : $\\mathcal {E}(x,\\xi )=\\lbrace \\eta \\in T_x M\\mid \\eta \\perp \\xi \\rbrace ,\\quad (x,\\xi )\\in SM.$ This makes it possible to define horocyclic operators $\\mathcal {U}_\\pm ^m:\\mathcal {D}^{\\prime }(SM)\\rightarrow \\mathcal {D}^{\\prime }(SM;\\otimes ^m_S\\mathcal {E}^),$ where $\\otimes ^m_S$ stands for the $m$ -th symmetric tensor power, and we have the diagram Figure: NO_CAPTION where $\\mathcal {V}_+^m=(-1)^m(\\mathcal {U}_+^m)^$ and we put for a certain extension $\\mathcal {X}$ of $X$ to $\\otimes ^m_S\\mathcal {E}^$ ${\\begin{array}{c}V_m(\\lambda ):=\\lbrace u\\in \\mathcal {D}^{\\prime }(SM)\\mid (X+\\lambda ) u=0,\\ \\mathcal {U}_-^m u=0,\\ \\operatorname{WF}(u)\\subset E_u^\\rbrace ,\\\\\\operatorname{Res}^m_{\\mathcal {X}}(\\lambda ):=\\lbrace v\\in \\mathcal {D}^{\\prime }(SM;\\otimes ^m_S\\mathcal {E}^)\\mid (\\mathcal {X}+\\lambda )v=0,\\ \\mathcal {U}_- v=0,\\ \\operatorname{WF}(v)\\subset E_u^\\rbrace .\\end{array}}$ Similarly to dimension 2, we reduce the problem to understanding the spaces $\\operatorname{Res}^m_{\\mathcal {X}}(\\lambda )$ , and an operator similar to (REF ) maps these spaces to eigenspaces of the Laplacian on divergence-free symmetric tensors.", "However, to make this statement precise, we have to further decompose $\\operatorname{Res}^m_{\\mathcal {X}}(\\lambda )$ into terms coming from traceless tensors of degrees $m,m-2,m-4,\\dots $ , explaining the appearance of the parameter $\\ell $ in the theorem.", "(Here the trace of a symmetric tensor of order $m$ is the result of contracting two of its indices with the metric, yielding a tensor of order $m-2$ .)", "The procedure of reducing elements of $\\operatorname{Res}^m_{\\mathcal {X}}(\\lambda )$ to the conformal boundary $\\mathbb {S}^n$ is also made more difficult since the boundary distributions $w$ are now sections of $\\otimes ^m_S(T^\\mathbb {S}^n)$ .", "A significant part of the paper is dedicated to proving that the higher-dimensional analog of (REF ) on symmetric tensors is indeed an isomorphism between appropriate spaces.", "To show that the Poisson operator $P^-_\\lambda $ is injective, we prove a weak expansion of $f(y)\\in \\mathcal {C}^\\infty (\\mathbb {B}^{n+1})$ in powers of $1-\|y\|$ as $y\\in \\mathbb {B}^{n+1}$ approaches the conformal boundary $\\mathbb {S}^n$ ; since $w$ appears as the coefficient in one of the terms of the expansion, $P^-_\\lambda w=0$ implies $w=0$ .", "To show the surjectivity of $P^-_\\lambda $ , we prove that the lift to $\\mathbb {H}^{n+1}$ of every trace-free divergence-free eigenstate $f$ of the Laplacian admits a weak expansion at the conformal boundary (this requires a fine analysis of the Laplacian and divergence operators on symmetric tensors); putting $w$ to be the coefficient next to one of the terms of this expansion, we can prove that $f=P^-_\\lambda w$ ." ], [ "Structure of the paper", " In Section , we study in detail the geometry of the hyperbolic space $\\mathbb {H}^{n+1}$ , which is the covering space of $M$ ; in Section , we introduce and study the horocyclic operators; in Section , we prove Theorems REF and REF , modulo properties of the Poisson operator; in Sections and , we show the injectivity and the surjectivity of the Poisson operator; Appendix contains several technical lemmas; Appendix shows how the discussion of Section REF fits into the framework of the remainder of the paper; Appendix shows a Weyl law for divergence free symmetric tensors and relates the $m=1$ case to the Hodge Laplacian." ], [ "Geometry of the hyperbolic space", "In this section, we review the structure of the hyperbolic space and its geodesic flow and introduce various objects to be used later, including: the isometry group $G$ of the hyperbolic space and its Lie algebra, including the horocyclic vector fields $U^\\pm _i$ (Section REF ); the stable/unstable foliations $E_s,E_u$ (Section REF ); the conformal compactification of the hyperbolic space, the maps $B_\\pm $ , the coefficients $\\Phi _\\pm $ , and the Poisson kernel (Section REF ); parallel transport to conformal infinity and the maps $\\mathcal {A}_\\pm $ (Section REF )." ], [ "Models of the hyperbolic space", "Consider the Minkowski space $\\mathbb {R}^{1,n+1}$ with the Lorentzian metric $g_M=dx_0^2-\\sum _{j=1}^{n+1}dx_j^2.$ The corresponding scalar product is denoted $\\langle \\cdot ,\\cdot \\rangle _M$ .", "We denote by $e_0,\\dots ,e_{n+1}$ the canonical basis of $\\mathbb {R}^{1,n+1}$ .", "The hyperbolic space of dimension $n+1$ is defined to be one sheet of the two-sheeted hyperboloid $\\mathbb {H}^{n+1}:=\\lbrace x\\in \\mathbb {R}^{1,n+1}\\mid \\langle x,x\\rangle _M=1,\\ x_0>0\\rbrace $ equipped with the Riemannian metric $g_H:=-g_M\|_{T\\mathbb {H}^{n+1}}.$ We denote the unit tangent bundle of $\\mathbb {H}^{n+1}$ by $S\\mathbb {H}^{n+1}:=\\lbrace (x,\\xi )\\mid x\\in \\mathbb {H}^{n+1},\\ \\xi \\in \\mathbb {R}^{1,n+1},\\ \\langle \\xi ,\\xi \\rangle _M=-1,\\ \\langle x,\\xi \\rangle _M=0\\rbrace .$ Another model of the hyperbolic space is the unit ball $\\mathbb {B}^{n+1}=\\lbrace y\\in \\mathbb {R}^{n+1}; \|y\|<1\\rbrace $ , which is identified with $\\mathbb {H}^{n+1}\\subset \\mathbb {R}^{1,n+1}$ via the map (here $x=(x_0,x^{\\prime })\\in \\mathbb {R}\\times \\mathbb {R}^{n+1}$ ) $\\psi : \\mathbb {H}^{n+1} \\rightarrow \\mathbb {B}^{n+1}, \\quad \\psi (x)=\\frac{x^{\\prime }}{x_0+1},\\quad \\psi ^{-1}(y)={1\\over 1-\|y\|^2}(1+\|y\|^2,2y).$ and the metric $g_H$ pulls back to the following metric on $\\mathbb {B}^{n+1}$ : $(\\psi ^{-1})^g_H=\\frac{4\\, dy^2}{(1-\|y\|^2)^2}.$ We will also use the upper half-space model $\\mathbb {U}^{n+1}=\\mathbb {R}_{z_0}^+\\times \\mathbb {R}_{z}^n$ with the metric $(\\psi ^{-1}\\psi _1^{-1})^g_H={dz_0^2+dz^2\\over z_0^2},$ where the diffeomorphism $\\psi _1:\\mathbb {B}^{n+1}\\rightarrow \\mathbb {U}^{n+1}$ is given by (here $y=(y_1,y^{\\prime })\\in \\mathbb {R}\\times \\mathbb {R}^n$ ) $\\psi _1(y_1,y^{\\prime })={(1-\|y\|^2,2y^{\\prime })\\over 1+\|y\|^2-2y_1},\\quad \\psi _1^{-1}(z_0,z)={(z_0^2+\|z\|^2-1,2z)\\over (1+z_0)^2+\|z\|^2}.$" ], [ "Isometry group", "We consider the group $G=\\operatorname{PSO}(1,n+1)\\subset \\operatorname{SL}(n+2;\\mathbb {R})$ of all linear transformations of $\\mathbb {R}^{1,n+1}$ preserving the Minkowski metric, the orientation, and the sign of $x_0$ on timelike vectors.", "For $x\\in \\mathbb {R}^{1,n+1}$ and $\\gamma \\in G$ , denote by $\\gamma \\cdot x$ the result of multiplying $x$ by the matrix $\\gamma $ .", "The group $G$ is exactly the group of orientation preserving isometries of $\\mathbb {H}^{n+1}$ ; under the identification (REF ), it corresponds to the group of direct Möbius transformations of $\\mathbb {R}^{n+1}$ preserving the unit ball.", "The Lie algebra of $G$ is spanned by the matrices $X= E_{0,1}+E_{1,0}, \\quad A_k= E_{0,k}+E_{k,0}, \\quad R_{i,j}=E_{i,j}-E_{j,i}$ for $i,j\\ge 1$ and $k\\ge 2$ , where $E_{i,j}$ is the elementary matrix if $0\\le i,j\\le n+1$ (that is, $E_{i,j}e_k=\\delta _{jk}e_i$ ).", "Denote for $i=1,\\dots , n$ $U_i^+:= -A_{i+1}-R_{1,i+1}, \\quad U^-_i:=-A_{i+1}+R_{1,i+1}$ and observe that $X,U_i^+,U_i^-,R_{i+1,j+1}$ (for $1\\le i<j\\le n$ ) also form a basis.", "Henceforth we identify elements of the Lie algebra of $G$ with left invariant vector fields on $G$ .", "We have the commutator relations (for $1\\le i,j,k\\le n$ and $i\\ne j$ ) ${\\begin{array}{c}[X,U_i^\\pm ]=\\pm U_i^\\pm ,\\quad [U_i^\\pm ,U_j^\\pm ]=0,\\quad [U_i^+,U_i^-]=2X,\\quad [U_i^\\pm ,U_j^\\mp ]=2R_{i+1,j+1},\\\\[R_{i+1,j+1},X]=0,\\quad [R_{i+1,j+1},U_k^\\pm ]=\\delta _{jk}U_i^\\pm -\\delta _{ik}U_j^\\pm .\\end{array}}$ The Lie algebra elements $U_i^\\pm $ are very important in our argument since they generate horocylic flows, see Section REF .", "The flows of $U^1_\\pm $ in the case $n=1$ are shown in Figure REF (a); for $n>1$ , the flows of $U^j_\\pm $ do not descend to $S\\mathbb {H}^{n+1}$ .", "The group $G$ acts on $\\mathbb {H}^{n+1}$ transitively, with the isotropy group of $e_0\\in \\mathbb {H}^{n+1}$ isomorphic to $\\operatorname{SO}(n+1)$ .", "It also acts transitively on the unit tangent bundle $S\\mathbb {H}^{n+1}$ , by the rule $\\gamma .", "(x,\\xi )=(\\gamma \\cdot x,\\gamma \\cdot \\xi )$ , with the isotropy group of $(e_0,e_1)\\in S\\mathbb {H}^{n+1}$ being $H=\\lbrace \\gamma \\in G\\mid \\gamma \\cdot e_0=e_0,\\ \\gamma \\cdot e_1=e_1\\rbrace \\simeq \\operatorname{SO}(n).$ Note that $H$ is the connected Lie subgroup of $G$ with Lie algebra spanned by $R_{i+1,j+1}$ for $1\\le i,j\\le n$ .", "We can then write $S\\mathbb {H}^{n+1}\\simeq G/H$ , where the projection $\\pi _S:G\\rightarrow S\\mathbb {H}^{n+1}$ is given by $\\pi _S(\\gamma )=(\\gamma \\cdot e_0,\\gamma \\cdot e_1)\\in S\\mathbb {H}^{n+1},\\quad \\gamma \\in G.$ Figure: (a) The horocyclic flows exp(±U 1 ± )\\exp (\\pm U^\\pm _1) in dimension n+1=2n+1=2,pulled back to the ball model by the map ψ\\psi from ().", "The thick linesare geodesics and the dashed lines are horocycles.", "(b) The map 𝒜 + \\mathcal {A}_+ and the parallel transport of an element of ℰ\\mathcal {E}along a geodesic." ], [ "Geodesic flow", "The geodesic flow, $\\varphi _t:S\\mathbb {H}^{n+1}\\rightarrow S\\mathbb {H}^{n+1},\\quad t\\in \\mathbb {R},$ is given in the parametrization (REF ) by $\\varphi _t(x,\\xi )=(x\\cosh t+\\xi \\sinh t,x\\sinh t+\\xi \\cosh t).$ We note that, with the projection $\\pi _S:G\\rightarrow S\\mathbb {H}^{n+1}$ defined in (REF ), $\\varphi _t(\\pi _S(\\gamma ))=\\pi _S(\\gamma \\exp (tX)),$ where $X$ is defined in (REF ).", "This means that the generator of the geodesic flow can be obtained by pushing forward the left invariant field on $G$ generated by $X$ by the map $\\pi _S$ (which is possible since $X$ is invariant under right multiplications by elements of the subgroup $H$ defined in (REF )).", "By abuse of notation, we then denote by $X$ also the generator of the geodesic flow on $S\\mathbb {H}^{n+1}$ : $X=\\xi \\cdot \\partial _x+x\\cdot \\partial _\\xi .$ We now provide the stable/unstable decomposition for the geodesic flow, demonstrating that it is hyperbolic (and thus the flow on a compact quotient by a discrete group will be Anosov).", "For $\\rho =(x,\\xi )\\in S\\mathbb {H}^{n+1}$ , the tangent space $T_\\rho (S\\mathbb {H}^{n+1})$ can be written as $T_\\rho (S\\mathbb {H}^{n+1})=\\lbrace (v_x,v_\\xi )\\in (\\mathbb {R}^{1,n+1})^2\\mid \\langle x,v_x\\rangle _M=\\langle \\xi ,v_\\xi \\rangle _M=\\langle x,v_\\xi \\rangle _M+\\langle \\xi ,v_x\\rangle _M=0\\rbrace .$ The differential of the geodesic flow acts by $d\\varphi _t(\\rho )\\cdot (v_x,v_\\xi )=(v_x\\cosh t+v_\\xi \\sinh t,v_x\\sinh t+v_\\xi \\cosh t).$ We have $T_\\rho (S\\mathbb {H}^{n+1})=E^0(\\rho )\\oplus \\widetilde{T}_\\rho (S\\mathbb {H}^{n+1})$ , where $E^0(\\rho ):=\\mathbb {R} X$ is the flow direction and $\\widetilde{T}_\\rho (S\\mathbb {H}^{n+1})=\\lbrace (v_x,v_\\xi )\\in (\\mathbb {R}^{1,n+1})^2\\mid \\langle x,v_x\\rangle _M=\\langle x,v_\\xi \\rangle _M=\\langle \\xi ,v_x\\rangle _M=\\langle \\xi ,v_\\xi \\rangle _M=0\\rbrace ,$ and this splitting is invariant under $d\\varphi _t$ .", "A natural norm on $\\widetilde{T}_\\rho (S\\mathbb {H}^{n+1})$ is given by the formula $\|(v_x,v_\\xi )\|^2:=-\\langle v_x,v_x\\rangle _M-\\langle v_\\xi ,v_\\xi \\rangle _M,$ using the fact that $v_x,v_\\xi $ are Minkowski orthogonal to the timelike vector $x$ and thus must be spacelike or zero.", "Note that this norm is invariant under the action of $G$ .", "We now define the stable/unstable decomposition $\\widetilde{T}_\\rho (S\\mathbb {H}^{n+1})=E_s(\\rho )\\oplus E_u(\\rho )$ , where ${\\begin{array}{c}E_s(\\rho ):=\\lbrace (v,-v)\\mid \\langle x,v\\rangle _M=\\langle \\xi ,v\\rangle _M=0\\rbrace ,\\\\E_u(\\rho ):=\\lbrace (v,v)\\mid \\langle x,v\\rangle _M=\\langle \\xi ,v\\rangle _M=0\\rbrace .\\end{array}}$ Then $T_\\rho (S\\mathbb {H}^{n+1})=E_0(\\rho )\\oplus E_s(\\rho )\\oplus E_u(\\rho )$ , this splitting is invariant under $\\varphi _t$ and under the action of $G$ , and, using the norm from (REF ), $\|d\\varphi _t(\\rho )\\cdot w\|=e^{-t}\|w\|,\\ w\\in E_s(\\rho );\\quad \|d\\varphi _t(\\rho )\\cdot w\|=e^{t}\|w\|,\\ w\\in E_u(\\rho ).$ Finally, we remark that the vector subbundles $E_s$ and $E_u$ are spanned by the left-invariant vector fields $U^+_1,\\dots , U^+_n$ and $U^-_1,\\dots ,U^-_n$ from (REF ) in the sense that $\\pi _S^ E_s=\\operatorname{span}(U^+_1,\\dots ,U^+_n)\\oplus \\mathfrak {h},\\quad \\pi _S^* E_u=\\operatorname{span}(U^-_1,\\dots ,U^-_n)\\oplus \\mathfrak {h}.$ Here $\\pi _S^* E_s:=\\lbrace (\\gamma ,w)\\in TG\\mid (\\pi _S(\\gamma ),d\\pi _S(\\gamma )\\cdot w)\\in E_s\\rbrace $ and $\\pi _S^* E_u$ is defined similarly; $\\mathfrak {h}$ is the left translation of the Lie algebra of $H$ , or equivalently the kernel of $d\\pi _S$ .", "Note that while the individual vector fields $U^\\pm _1,\\dots ,U^\\pm _n$ are not invariant under right multiplications by elements of $H$ in dimensions $n+1>2$ (and thus do not descend to vector fields on $S\\mathbb {H}^{n+1}$ by the map $\\pi _S$ ), their spans are invariant under $H$ by (REF ).", "The dual decomposition, used in the construction of Pollicott–Ruelle resonances, is $T^_\\rho (S\\mathbb {H}^{n+1})=E_0^(\\rho )\\oplus E_s^(\\rho )\\oplus E_u^(\\rho ),$ where $E_0^(\\rho ),E_s^(\\rho ),E_u^(\\rho )$ are dual to $E_0(\\rho ),E_u(\\rho ),E_s(\\rho )$ in the original decomposition (that is, for instance $E_s^(\\rho )$ consists of all covectors annihilating $E_0(\\rho )\\oplus E_s(\\rho )$ ).", "The switching of the roles of $E_s$ and $E_u$ is due to the fact that the flow on the cotangent bundle is $(d\\varphi _t^{-1})^$ ." ], [ "Conformal infinity", "The metric (REF ) in the ball model $\\mathbb {B}^{n+1}$ is conformally compact; namely the metric $(1-\|y\|^2)^2(\\psi ^{-1})^g_H$ continues smoothly to the closure $\\overline{\\mathbb {B}^{n+1}}$ , which we call the conformal compactification of $\\mathbb {H}^{n+1}$ ; note that $\\mathbb {H}^{n+1}$ embeds into the interior of $\\overline{\\mathbb {B}^{n+1}}$ by the map (REF ).", "The boundary $\\mathbb {S}^n=\\partial \\overline{\\mathbb {B}^{n+1}}$ , endowed with the standard metric on the sphere, is called conformal infinity.", "On the hyperboloid model, it is natural to associate to a point at conformal infinity $\\nu \\in \\mathbb {S}^n$ the lightlike ray $\\lbrace (s,s\\nu )\\mid s>0\\rbrace \\subset \\mathbb {R}^{1,n+1}$ ; note that this ray is asymptotic to the curve $\\lbrace (\\sqrt{1+s^2},s\\nu )\\mid s>0\\rbrace \\subset \\mathbb {H}^{n+1}$ , which converges to $\\nu $ in $\\overline{\\mathbb {B}^{n+1}}$ .", "Take $(x,\\xi )\\in S\\mathbb {H}^{n+1}$ .", "Then $\\langle x\\pm \\xi ,x\\pm \\xi \\rangle _M=0$ and $x_0\\pm \\xi _0>0$ , therefore we can write $x\\pm \\xi =\\Phi _\\pm (x,\\xi )(1,B_\\pm (x,\\xi )),$ for some maps $\\Phi _\\pm :S\\mathbb {H}^{n+1}\\rightarrow \\mathbb {R}^+,\\quad B_\\pm :S\\mathbb {H}^{n+1}\\rightarrow \\mathbb {S}^n.$ Then $B_\\pm (x,\\xi )$ is the limit as $t\\rightarrow \\pm \\infty $ of the $x$ -projection of the geodesic $\\varphi _t(x,\\xi )$ in $\\overline{\\mathbb {B}^{n+1}}$ : $B_\\pm (x,\\xi )=\\lim _{t\\rightarrow \\pm \\infty }\\pi (\\varphi _t(x,\\xi )),\\quad \\pi :S\\mathbb {H}^{n+1}\\rightarrow \\mathbb {H}^{n+1}.$ Note that this implies that for $X$ defined in (REF ), $dB_\\pm \\cdot X=0$ since $B_\\pm (\\varphi _s(x,\\xi ))=B_\\pm (x,\\xi )$ for all $s\\in \\mathbb {R}$ .", "Moreover, since $\\Phi _\\pm (\\varphi _t(x,\\xi ))=e^{\\pm t}(x_0+\\xi _0)=e^{\\pm t}\\Phi _\\pm (x,\\xi )$ from (REF ), we find $X\\Phi _\\pm =\\pm \\Phi _\\pm .$ For $(x,\\nu )\\in \\mathbb {H}^{n+1}\\times \\mathbb {S}^n$ (in the hyperboloid model), define the function $P(x,\\nu )=(x_0-x^{\\prime }\\cdot \\nu )^{-1}=(\\langle x,(1,\\nu )\\rangle _M)^{-1}, \\quad \\textrm { if }x=(x_0,x^{\\prime })\\in \\mathbb {H}^{n+1}.$ Note that $P(x,\\nu )>0$ everywhere, and in the Poincaré ball model $\\mathbb {B}^{n+1}$ , we have $P(\\psi ^{-1}(y),\\nu )=\\frac{1-\|y\|^2}{\|y-\\nu \|^2}, \\quad y\\in \\mathbb {B}^{n+1}$ which is the usual Poisson kernel.", "Here $\\psi $ is defined in (REF ).", "For $(x,\\nu )\\in \\mathbb {H}^{n+1}\\times \\mathbb {S}^n$ , there exist unique $\\xi _\\pm \\in S_x\\mathbb {H}^{n+1}$ such that $B_\\pm (x,\\xi _\\pm )=\\nu $ : these are given by $\\xi _\\pm (x,\\nu )=\\mp x\\pm P(x,\\nu )(1,\\nu )$ and the following formula holds $\\Phi _\\pm (x,\\xi _\\pm (x,\\nu ))=P(x,\\nu ).$ Notice that the equation $B_\\pm (x,\\xi _\\pm (x,\\nu ))=\\nu $ implies that $B_\\pm $ are submersions.", "The map $\\nu \\rightarrow \\xi _\\pm (x,\\nu )$ is conformal with the standard choice of metrics on $\\mathbb {S}^n$ and $S_x\\mathbb {H}^{n+1}$ ; in fact, for $\\zeta _1,\\zeta _2\\in T_\\nu \\mathbb {S}^n$ , $\\langle \\partial _\\nu \\xi _\\pm (x,\\nu )\\cdot \\zeta _1,\\partial _\\nu \\xi _\\pm (x,\\nu )\\cdot \\zeta _2\\rangle _{M}=- P(x,\\nu )^{2} \\langle \\zeta _1,\\zeta _2\\rangle _{\\mathbb {R}^{n+1}}.$ Using that $\\langle x+\\xi ,x-\\xi \\rangle _M=2$ , we see that $\\Phi _+(x,\\xi )\\Phi _-(x,\\xi )(1-B_+(x,\\xi )\\cdot B_-(x,\\xi ))=2.$ One can parametrize $S\\mathbb {H}^{n+1}$ by $(\\nu _-,\\nu _+,s)=\\bigg (B_-(x,\\xi ),B_+(x,\\xi ),{1\\over 2}\\log {\\Phi _+(x,\\xi )\\over \\Phi _-(x,\\xi )}\\bigg )\\in (\\mathbb {S}^n\\times \\mathbb {S}^n)_\\Delta \\times \\mathbb {R},$ where $(\\mathbb {S}^n\\times \\mathbb {S}^n)_\\Delta $ is $\\mathbb {S}^n\\times \\mathbb {S}^n$ minus the diagonal.", "In fact, the geodesic $\\gamma (t)=\\varphi _t(x,\\xi )$ goes from $\\nu _-$ to $\\nu _+$ in $\\overline{\\mathbb {B}^{n+1}}$ and $\\gamma (-s)$ is the point of $\\gamma $ closest to $e_0\\in \\mathbb {H}^{n+1}$ (corresponding to $0\\in \\mathbb {B}^{n+1}$ ).", "In the parametrization (REF ), the geodesic flow $\\varphi _t$ is simply $(\\nu _-,\\nu _+,s)\\mapsto (\\nu _-,\\nu _+,s+t).$ We finally remark that the stable/unstable subspaces of the cotangent bundle $E_s^,E_u^\\subset T^(S\\mathbb {H}^{n+1})$ , defined in (REF ), are in fact the conormal bundles of the fibers of the maps $B_\\pm $ : $E_s^(\\rho )=N^(B_+^{-1}(B_+(\\rho ))),\\quad E_u^(\\rho )=N^(B_-^{-1}(B_-(\\rho ))),\\quad \\rho \\in S\\mathbb {H}^{n+1}.$ This is equivalent to saying that the fibers of $B_+$ integrate (i.e.", "are tangent to) the subbundle $E_0\\oplus E_s\\subset T(S\\mathbb {H}^{n+1})$ , while the fibers of $B_-$ integrate the subbundle $E_0\\oplus E_u$ .", "To see the latter statement, for say $B_+$ , it is enough to note that $dB_+\\cdot X=0$ and differentiation along vectors in $E_s$ annihilates the function $x+\\xi $ and thus the map $B_+$ ; therefore, the kernel of $dB_+$ contains $E_0\\oplus E_s$ , and this containment is an equality since the dimensions of both spaces are equal to $n+1$ ." ], [ "Action of $G$ on the conformal infinity", "For $\\gamma \\in G$ and $\\nu \\in \\mathbb {S}^n$ , $\\gamma \\cdot (1,\\nu )$ is a lightlike vector with positive zeroth component.", "We can then define $N_\\gamma (\\nu )>0$ , $L_\\gamma (\\nu )\\in \\mathbb {S}^n$ by $\\gamma \\cdot (1,\\nu )=N_\\gamma (\\nu )(1,L_\\gamma (\\nu )).$ The map $L_\\gamma $ gives the action of $G$ on the conformal infinity $\\mathbb {S}^n=\\partial \\overline{\\mathbb {B}^{n+1}}$ .", "This action is transitive and the isotropy groups of $\\pm e_1\\in \\mathbb {S}^n$ are given by $H_\\pm =\\lbrace \\gamma \\in G\\mid \\exists s>0: \\gamma \\cdot (e_0\\pm e_1)=s(e_0\\pm e_1)\\rbrace .$ The isotropy groups $H_\\pm $ are the connected subgroups of $G$ with the Lie algebras generated by $R_{i+1,j+1}$ for $1\\le i<j\\le n$ , $X$ , and $U^\\pm _i$ for $1\\le i\\le n$ .", "To see that $H_\\pm $ are connected, for $n=1$ we can check directly that every $\\gamma \\in H_\\pm $ can be written as a product $e^{tX}e^{sU^\\pm _1}$ for some $t,s\\in \\mathbb {R}$ , and for $n>1$ we can use the fact that $\\mathbb {S}^n\\simeq G/H_\\pm $ is simply connected and $G$ is connected, and the homotopy long exact sequence of a fibration.", "The differentials of $N_\\gamma $ and $L_\\gamma $ (in $\\nu $ ) can be written as $dN_\\gamma (\\nu )\\cdot \\zeta =\\langle dx_0,\\gamma \\cdot (0,\\zeta )\\rangle ,\\quad (0,dL_\\gamma (\\nu )\\cdot \\zeta )={\\gamma \\cdot (0,\\zeta )-(dN_\\gamma (\\nu )\\cdot \\zeta )(1,L_\\gamma (\\nu ))\\over N_\\gamma (\\nu )},$ here $\\zeta \\in T_\\nu \\mathbb {S}^n$ .", "We see that the map $\\nu \\mapsto L_\\gamma (\\nu )$ is conformal with respect to the standard metric on $\\mathbb {S}^n$ , in fact for $\\zeta _1,\\zeta _2\\in T_\\nu \\mathbb {S}^n$ , $\\langle dL_\\gamma (\\nu )\\cdot \\zeta _1,dL_\\gamma (\\nu )\\cdot \\zeta _2\\rangle _{\\mathbb {R}^{n+1}}=N_\\gamma (\\nu )^{-2} \\langle \\zeta _1,\\zeta _2\\rangle _{\\mathbb {R}^{n+1}}.$ The maps $B_\\pm :S\\mathbb {H}^{n+1}\\rightarrow \\mathbb {S}^n$ are equivariant under the action of $G$ : $B_\\pm (\\gamma .", "(x,\\xi ))=L_\\gamma (B_\\pm (x,\\xi )).$ Moreover, the functions $\\Phi _\\pm (x,\\xi )$ and $P(x,\\nu )$ enjoy the following properties: $\\Phi _\\pm (\\gamma .", "(x,\\xi ))=N_\\gamma (B_\\pm (x,\\xi ))\\Phi _\\pm (x,\\xi ),\\quad P(\\gamma \\cdot x,L_\\gamma (\\nu ))=N_\\gamma (\\nu )P(x,\\nu ).$" ], [ "The bundle $\\mathcal {E}$ and parallel transport to the conformal infinity", "Consider the vector bundle $\\mathcal {E}$ over $S\\mathbb {H}^{n+1}$ defined as follows: $\\mathcal {E}=\\lbrace (x,\\xi ,v)\\in S\\mathbb {H}^{n+1}\\times T_{x}\\mathbb {H}^{n+1}\\mid g_H(\\xi ,v)=0 \\rbrace ,$ i.e.", "the fibers $\\mathcal {E}(x,\\xi )$ consist of all tangent vectors in $T_x\\mathbb {H}^{n+1}$ orthogonal to $\\xi $ ; equivalently, $\\mathcal {E}(x,\\xi )$ consists of all vectors in $\\mathbb {R}^{1,n+1}$ orthogonal to $x$ and $\\xi $ with respect to the Minkowski inner product.", "Note that $G$ naturally acts on $\\mathcal {E}$ , by putting $\\gamma .", "(x,\\xi ,v):=(\\gamma \\cdot x,\\gamma \\cdot \\xi ,\\gamma \\cdot v)$ .", "The bundle $\\mathcal {E}$ is invariant under parallel transport along geodesics.", "Therefore, one can consider the first order differential operator $\\mathcal {X}:\\mathcal {C}^\\infty (S\\mathbb {H}^{n+1};\\mathcal {E})\\rightarrow \\mathcal {C}^\\infty (S\\mathbb {H}^{n+1};\\mathcal {E})$ which is the generator of parallel transport, namely if $\\mathbf {v}$ is a section of $\\mathcal {E}$ and $(x,\\xi )\\in S\\mathbb {H}^{n+1}$ , then $\\mathcal {X}\\mathbf {v}(x,\\xi )$ is the covariant derivative at $t=0$ of the vector field $\\mathbf {v}(t):=\\mathbf {v}(\\varphi _t(x,\\xi ))$ on the geodesic $\\varphi _t(x,\\xi )$ .", "Note that $\\mathcal {E}(\\varphi _t(x,\\xi ))$ is independent of $t$ as a subspace of $\\mathbb {R}^{1,n+1}$ , and under this embedding, $\\mathcal {X}$ just acts as $X$ on each coordinate of $v$ in $\\mathbb {R}^{1,n+1}$ .", "The operator ${1\\over i}\\mathcal {X}$ is a symmetric operator with respect to the standard volume form on $S\\mathbb {H}^{n+1}$ and the inner product on $\\mathcal {E}$ inherited from $T\\mathbb {H}^{n+1}$ .", "We now consider parallel transport of vectors along geodesics going off to infinity.", "Let $(x,\\xi )\\in S\\mathbb {H}^{n+1}$ and $v\\in T_x\\mathbb {H}^{n+1}$ .", "We let $(x(t),\\xi (t))=\\varphi _t(x,\\xi )$ be the corresponding geodesic and $v(t)\\in T_{x(t)} \\mathbb {H}^{n+1}$ be the parallel transport of $v$ along this geodesic.", "We embed $v(t)$ into the unit ball model $\\mathbb {B}^{n+1}$ by defining $w(t)=d\\psi (x(t))\\cdot v(t)\\in \\mathbb {R}^{n+1},$ where $\\psi $ is defined in (REF ).", "Then $w(t)$ converges to 0 as $t\\rightarrow \\pm \\infty $ , but the limits $\\lim _{t\\rightarrow \\pm \\infty }x_0(t) w(t)$ are nonzero for nonzero $v$ ; we call the transformation mapping $v$ to these limits the transport to conformal infinity as $t\\rightarrow \\pm \\infty $ .", "More precisely, if $v=c\\xi +u,\\quad u\\in \\mathcal {E}(x,\\xi ),$ then we calculate $\\lim _{t\\rightarrow \\pm \\infty } x_0(t)w(t)=\\pm cB_\\pm (x,\\xi )+u^{\\prime }-u_0B_\\pm (x,\\xi ),$ where $B_\\pm (x,\\xi )\\in \\mathbb {S}^n$ is defined in Section REF .", "We will in particular use the inverse of the map $\\mathcal {E}(x,\\xi )\\ni u\\mapsto u^{\\prime }-u_0B_\\pm (x,\\xi )\\in T_{B_\\pm (x,\\xi )}\\mathbb {S}^n$ : for $(x,\\xi )\\in S\\mathbb {H}^{n+1}$ and $\\zeta \\in T_{B_\\pm (x,\\xi )}\\mathbb {S}^n$ , define (see Figure REF (b)) $\\mathcal {A}_\\pm (x,\\xi )\\zeta = (0,\\zeta )-\\langle (0,\\zeta ),x\\rangle _M (x\\pm \\xi )=\\pm \\frac{\\partial _\\nu \\xi _\\pm (x,B_\\pm (x,\\xi ))\\cdot \\zeta }{P(x,B_\\pm (x,\\xi ))}\\in \\mathcal {E}(x,\\xi ).$ Here $\\xi _\\pm $ is defined in (REF ).", "Note that by (REF ), $\\mathcal {A}_\\pm $ is an isometry: $\|\\mathcal {A}_\\pm (x,\\xi )\\zeta \|_{g_H}=\|\\zeta \|_{\\mathbb {R}^n},\\quad \\zeta \\in T_{B_\\pm (x,\\xi )}\\mathbb {S}^n.$ Also, $\\mathcal {A}_\\pm $ is equivariant under the action of $G$ : $\\mathcal {A}_\\pm (\\gamma \\cdot x,\\gamma \\cdot \\xi )\\cdot dL_\\gamma (B_\\pm (x,\\xi ))\\cdot \\zeta =N_\\gamma (B_\\pm (x,\\xi ))^{-1}\\,\\gamma \\cdot (\\mathcal {A}_\\pm (x,\\xi )\\zeta ).$ We now write the limits (REF ) in terms of the 0-tangent bundle of Mazzeo–Melrose [40].", "Consider the boundary defining function $\\rho _0:=2(1-\|y\|)/(1+\|y\|)$ on $\\overline{\\mathbb {B}^{n+1}}$ ; note that in the hyperboloid model, with the map $\\psi $ defined in (REF ), $\\rho _0(\\psi (x))=2{\\sqrt{x_0+1}-\\sqrt{x_0-1}\\over \\sqrt{x_0+1}+\\sqrt{x_0-1}}=x_0^{-1}+\\mathcal {O}(x_0^{-2})\\quad \\text{as }x_0\\rightarrow \\infty .$ The hyperbolic metric can be written near the boundary as $g_H=(d\\rho _0^2+h_{\\rho _0})/\\rho _0^2$ with $h_{\\rho _0}$ a smooth family of metrics on $\\mathbb {S}^n$ and $h_0=d\\theta ^2$ is the canonical metric on the sphere (with curvature 1).", "Define the 0-tangent bundle ${^0}T\\overline{\\mathbb {B}^{n+1}}$ to be the smooth bundle over $\\overline{\\mathbb {B}^{n+1}}$ whose smooth sections are the elements of the Lie algebra $\\mathcal {V}_0(\\overline{\\mathbb {B}^{n+1}})$ of smooth vectors fields vanishing at $\\mathbb {S}^n=\\overline{\\mathbb {B}^{n+1}}\\cap \\lbrace \\rho _0=0\\rbrace $ ; near the boundary, this algebra is locally spanned over $\\mathcal {C}^\\infty (\\overline{\\mathbb {B}^{n+1}})$ by the vector fields $\\rho _0 \\partial _{\\rho _0},\\rho _0\\partial _{\\theta _1},\\dots ,\\rho _0\\partial _{\\theta _n}$ if $\\theta _i$ are local coordinates on $\\mathbb {S}^n$ .", "Note that there is a natural map ${^0}T\\overline{\\mathbb {B}^{n+1}}\\rightarrow T\\overline{\\mathbb {B}^{n+1}}$ which is an isomorphism when restricted to the interior $\\mathbb {B}^{n+1}$ .", "We denote by ${^0}T^\\overline{\\mathbb {B}^{n+1}}$ the dual bundle to ${^0}T\\overline{\\mathbb {B}^{n+1}}$ , generated locally near $\\rho _0=0$ by the covectors $d\\rho _0/\\rho _0,d\\theta _1/\\rho _0,\\dots ,d\\theta _n/\\rho _0$ .", "Note that $T^\\overline{\\mathbb {B}^{n+1}}$ naturally embeds into ${^0}T^\\overline{\\mathbb {B}^{n+1}}$ and this embedding is an isomorphism in the interior.", "The metric $g_H$ is a smooth non-degenerate positive definite quadratic form on ${^0}T\\overline{\\mathbb {B}^{n+1}}$ , that is $g_H\\in \\mathcal {C}^\\infty (\\overline{\\mathbb {B}^{n+1}}; \\otimes _S^2({^0}T^\\overline{\\mathbb {B}^{n+1}}))$ , where $\\otimes _S^2$ denotes the space of symmetric 2-tensors.", "We refer the reader to [40] for further details (in particular, for an explanation of why 0-bundles are smooth vector bundles); see also [42] for the similar $b$ -setting.", "We can then interpret (REF ) as follows: for each $(y,\\eta )\\in S\\mathbb {B}^{n+1}$ and each $w\\in T_y\\mathbb {B}^{n+1}$ , the parallel transport $w(t)$ of $w$ along the geodesic $\\varphi _t(y,\\eta )$ (this geodesic extends smoothly to a curve on $\\mathbb {B}^{n+1}$ , as it is part of a line or a circle) has limits as $t\\rightarrow \\pm \\infty $ in the 0-tangent bundle $^0T\\overline{\\mathbb {B}^{n+1}}$ .", "In fact (see [28]), the parallel transport $\\tau (y^{\\prime },y): {^0}T_y{\\mathbb {B}^{n+1}}\\rightarrow {^0}T_{y^{\\prime }}{\\mathbb {B}^{n+1}}$ from $y$ to $y^{\\prime }\\in \\mathbb {B}^{n+1}$ along the geodesic starting at $y$ and ending at $y^{\\prime }$ extends smoothly to the boundary $(y,y^{\\prime })\\in \\overline{\\mathbb {B}^{n+1}}\\times \\overline{\\mathbb {B}^{n+1}}\\setminus {\\rm diag}(\\mathbb {S}^n\\times \\mathbb {S}^n)$ as an endomorphism ${^0}T_y\\overline{\\mathbb {B}^{n+1}}\\rightarrow {^0}T_{y^{\\prime }}\\overline{\\mathbb {B}^{n+1}}$ , where ${\\rm diag}(\\mathbb {S}^n\\times \\mathbb {S}^n)$ denotes the diagonal in the boundary; this parallel transport is an isometry with respect to $g_H$ .", "The same properties hold for parallel transport of covectors in ${^0}T^\\overline{\\mathbb {B}^{n+1}}$ , using the duality provided by the metric $g_H$ .", "An explicit relation to the maps $\\mathcal {A}_\\pm $ is given by the following formula: $\\mathcal {A}_\\pm (x,\\xi )\\cdot \\zeta =d\\psi (x)^{-1}\\cdot \\tau (\\psi (x),B_\\pm (x,\\xi ))\\cdot (\\rho _0\\zeta ),$ where $\\rho _0\\zeta \\in {}^0T_{B_\\pm (x,\\xi )}\\overline{\\mathbb {B}^{n+1}}$ is tangent to the conformal boundary $\\mathbb {S}^n$ ." ], [ "Horocyclic operators", "In this section, we build on the results of Section to construct horocyclic operators $\\mathcal {U}_\\pm :\\mathcal {D}^{\\prime }(S\\mathbb {H}^{n+1};\\otimes ^j \\mathcal {E}^)\\rightarrow \\mathcal {D}^{\\prime }(S\\mathbb {H}^{n+1};\\otimes ^{j+1}\\mathcal {E}^)$ ." ], [ "Symmetric tensors", "In this subsection, we assume that $E$ is a vector space of finite dimension $N$ , equipped with an inner product $g_E$ , and let $E^$ denote the dual space, which has a scalar product induced by $g_E$ (also denoted $g_E$ ).", "(In what follows, we shall take either $E=\\mathcal {E}(x,\\xi )$ or $E=T_x\\mathbb {H}^{n+1}$ for some $(x,\\xi )\\in S\\mathbb {H}^{n+1}$ , and the scalar product $g_E$ in both case is given by the hyperbolic metric $g_H$ on those vector spaces.)", "In this section, we will work with tensor powers of $E^$ , but the constructions apply to tensor powers of $E$ by swapping $E$ with $E^$ .", "We introduce some notation for finite sequences to simplify the calculations below.", "Denote by $A^m$ the space of all sequences $K=k_1\\dots k_m$ with $1\\le k_\\ell \\le N$ .", "For $k_1\\dots k_m\\in A^m$ , $j_1\\dots j_r\\in A^r$ , and a sequence of distinct numbers $1\\le \\ell _1,\\dots ,\\ell _r\\le m$ , denote by $\\lbrace \\ell _1\\rightarrow j_1,\\dots ,\\ell _r\\rightarrow j_r\\rbrace K\\in A^m$ the result of replacing the $\\ell _p$ th element of $K$ by $j_p$ , for all $p$ .", "We can also replace some of $j_p$ by blank space, which means that the corresponding indices are removed from $K$ .", "For $m\\ge 0$ denote by $\\otimes ^m E^$ the $m$ th tensor power of $E^$ and by $\\otimes _S^m E^$ the subset of those tensors which are symmetric, i.e.", "$u\\in \\otimes _S^m E^$ if $u(v_{\\sigma (1)},\\dots v_{\\sigma (m)})=u(v_1,\\dots ,v_m)$ for all $\\sigma \\in \\Pi _m$ and all $v_1,\\dots ,v_m\\in E$ , where $\\Pi _m$ is the permutation group of $\\lbrace 1,\\dots ,m\\rbrace $ .", "There is a natural linear projection $\\mathcal {S}: \\otimes ^m E^\\rightarrow \\otimes ^m_SE^$ defined by $\\mathcal {S}(\\eta ^_{1}\\otimes \\dots \\otimes \\eta ^_{m})=\\frac{1}{m!", "}\\sum _{\\sigma \\in \\Pi _m}\\eta ^_{\\sigma (1)}\\otimes \\dots \\otimes \\eta ^_{\\sigma (m)}, \\quad \\eta ^_{k}\\in E^$ The metric $g_E$ induces a scalar product on $\\otimes ^m E^$ as follows $ \\langle v^_{1}\\otimes \\dots \\otimes v^_{m},w^_{1}\\otimes \\dots \\otimes w^_{m}\\rangle _{g_E}=\\prod _{j=1}^m \\langle v_j^,w_j^\\rangle _{g_E}, \\quad w_{i}^,v_{i}^\\in E^.$ The operator $\\mathcal {S}$ is self-adjoint and thus an orthogonal projection with respect to this scalar product.", "Using the metric $g_E$ , one can decompose the vector space $\\otimes _S^m$ as follows.", "Let $(\\mathbf {e}_i)_{i=1}^{N}$ be an orthonormal basis of $E$ for the metric $g_E$ and $(\\mathbf {e}_i^)$ be the dual basis.", "First of all, introduce the trace map $\\mathcal {T}:\\otimes ^m E^\\rightarrow \\otimes ^{m-2} E^$ contracting the first two indices by the metric: for $v_i\\in E$ , define $\\mathcal {T}(u)(v_1,\\dots ,v_{m-2}):=\\sum _{i=1}^{N}u(\\mathbf {e}_i,\\mathbf {e}_i,v_1,\\dots ,v_{m-2})$ (the result is independent of the choice of the basis).", "For $m< 2$ , we define $\\mathcal {T}$ to be zero on $\\otimes ^m E^$ .", "Note that $\\mathcal {T}$ maps $\\otimes _S^{m+2}E^$ onto $\\otimes ^m_SE^$ .", "Set $\\mathbf {e}^_K:=\\mathbf {e}^_{k_1}\\otimes \\dots \\otimes \\mathbf {e}^_{k_m}\\in \\otimes ^m E^,\\quad K=k_1\\dots k_m\\in {A}^{m}.$ Then $\\mathcal {T}\\Big (\\sum _{K\\in A^{m+2}}f_K \\mathbf {e}^_K\\Big )=\\sum _{K\\in A^m}\\sum _{q\\in A} f_{qqK}\\mathbf {e}_K^.$ The adjoint of $\\mathcal {T}:\\otimes _S^{m+2}E^\\rightarrow \\otimes _S^mE^$ with respect to the scalar product $g_E$ is given by the map $u\\mapsto \\mathcal {S}(g_E\\otimes u)$ .", "To simplify computations, we define a scaled version of it: let $\\mathcal {I}:\\otimes ^m_SE^\\rightarrow \\otimes ^{m+2}_SE^$ be defined by $\\mathcal {I}(u)=\\frac{(m+2)(m+1)}{2}\\mathcal {S}(g_{E}\\otimes u)={(m+2)(m+1)\\over 2} \\mathcal {T}^(u).$ Then $\\mathcal {I}\\Big (\\sum _{K\\in A^m} f_K\\mathbf {e}^_K\\Big )=\\sum _{K\\in A^{m+2}}\\sum _{\\ell ,r=1\\atop \\ell <r}^{m+2}\\delta _{k_\\ell k_r}f_{\\lbrace \\ell \\rightarrow ,r\\rightarrow \\rbrace K}\\mathbf {e}^_K.$ Note that for $u\\in \\otimes ^m_SE^$ , $\\mathcal {T}(\\mathcal {I} u)=(2m+N)u+\\mathcal {I}(\\mathcal {T} u).$ By (REF ) and (REF ), the homomorphism $\\mathcal {T}\\mathcal {I}:\\otimes _S^m E^\\rightarrow \\otimes _S^mE^$ is positive definite and thus an isomorphism.", "Therefore, for $u\\in \\otimes ^m_S E^$ , we can decompose $u=u_1+\\mathcal {I}(u_2)$ , where $u_1\\in \\otimes ^m_S E^$ satisfies $\\mathcal {T}(u_1)=0$ and $u_2=(\\mathcal {T}\\mathcal {I})^{-1}\\mathcal {T}u\\in \\otimes ^{m-2}_S E^$ .", "Iterating this process, we can decompose any $u\\in \\otimes _S^mE^$ into $u=\\sum _{r=0}^{\\lfloor m/2 \\rfloor } \\mathcal {I}^r(u_r),\\quad u_r\\in \\otimes _S^{m-2r}E^,\\quad \\mathcal {T}(u_r)=0,$ with $u_r$ determined uniquely by $u$ .", "Another operation on tensors which will be used is the interior product: if $v\\in E$ and $u\\in \\otimes _S^m E^$ , we denote by $\\iota _v(u)\\in \\otimes _S^{m-1} E^$ the interior product of $u$ by $v$ given by $ \\iota _vu(v_1,\\dots ,v_{m-1}):=u(v,v_1,\\dots ,v_{m-1}).$ If $v^\\in E^$ , we denote $\\iota _{v^}u$ for the tensor $\\iota _vu$ by $g_E(v,\\cdot )=v^$ .", "We conclude this section with a correspondence which will be useful in certain calculations later.", "There is a linear isomorphism between $\\otimes _S^m E^$ and the space $\\operatorname{Pol}^m(E)$ of homogeneous polynomials of degree $m$ on $E$ : to a tensor $u\\in \\otimes _S^m E^$ we associate the function on $E$ given by $x\\rightarrow P_u(x):=u(x,\\dots ,x)$ .", "If we write $x=\\sum _{i=1}^Nx_i\\mathbf {e}_i$ in a given orthonormal basis then $P_{\\mathcal {S}(e^_K)}(x)=\\prod _{j=1}^m x_{k_j},\\quad K=k_1\\dots k_m\\in A^m.$ The flat Laplacian associated to $g_E$ is given by $\\Delta _E=-\\sum _{i=1}^N\\partial _{x_i}^2$ in the coordinates induced by the basis $(\\mathbf {e}_i)$ .", "Then it is direct to see that $\\Delta _E P_u(x)=-m(m-1)P_{\\mathcal {T}(u)}(x),\\quad u\\in \\otimes ^m_SE^.$ which means that the trace corresponds to applying the Laplacian (see [11]).", "In particular, trace-free symmetric tensors of order $m$ correspond to homogeneous harmonic polynomials, and thus restrict to spherical harmonics on the sphere $\|x\|_{g_E}=1$ of $E$ .", "We also have $P_{\\mathcal {I}(u)}(x)={(m+2)(m+1)\\over 2}\\, \|x\|^2 P_u(x),\\quad u\\in \\otimes ^m_SE^.$" ], [ "Horocyclic operators", "We now consider the left-invariant vector fields $X$ , $U_\\pm ^i$ , $R_{i+1,j+1}$ on the isometry group $G$ , identified with the elements of the Lie algebra of $G$ introduced in (REF ), (REF ).", "Recall that $G$ acts on $S\\mathbb {H}^{n+1}$ transitively with the isotropy group $H\\simeq \\operatorname{SO}(n)$ and this action gives rise to the projection $\\pi _S:G\\rightarrow S\\mathbb {H}^{n+1}$ – see (REF ).", "Note that, with the maps $\\Phi _\\pm :S\\mathbb {H}^{n+1}\\rightarrow \\mathbb {R}^+,B_\\pm :S\\mathbb {H}^{n+1}\\rightarrow \\mathbb {S}^n$ defined in (REF ), we have $B_\\pm (\\pi _S(\\gamma ))=L_\\gamma (\\pm e_1),\\quad \\Phi _\\pm (\\pi _S(\\gamma ))=N_\\gamma (\\pm e_1),\\quad \\quad \\gamma \\in G,$ where $N_\\gamma :\\mathbb {S}^n\\rightarrow \\mathbb {R}^+,L_\\gamma :\\mathbb {S}^n\\rightarrow \\mathbb {S}^n$ are defined in (REF ).", "Since $H_\\pm $ , the isotropy group of $\\pm e_1$ under the action $L_\\gamma $ , contains $X,U_i^\\pm $ in its Lie algebra (see (REF ) and Figure REF (a)), we find $d(B_\\pm \\circ \\pi _S)\\cdot U^\\pm _i=0, \\quad d(B_\\pm \\circ \\pi _S)\\cdot X=0.$ We also calculate $d(\\Phi _\\pm \\circ \\pi _S)\\cdot U_i^\\pm =0.$ Define the differential operator on $G$ $U^\\pm _K:=U^\\pm _{k_1}\\dots U^\\pm _{k_m},\\quad K=k_1\\dots k_m\\in A^m.$ Note that the order in which $k_1,\\dots ,k_m$ are listed does not matter by (REF ).", "Moreover, by (REF ) $[R_{i+1,j+1},U^\\pm _K]=\\sum _{\\ell =1}^m (\\delta _{jk_\\ell } U^\\pm _{\\lbrace \\ell \\rightarrow i\\rbrace K}-\\delta _{ik_\\ell }U^\\pm _{\\lbrace \\ell \\rightarrow j\\rbrace K}).$ Since $H$ is generated by the vector fields $R_{i+1,j+1}$ , we see that in dimensions $n+1>2$ the horocyclic vector fields $U^\\pm _i$ , and more generally the operators $U^\\pm _K$ , are not invariant under right multiplication by elements of $H$ and therefore do not descend to differential operators on $S\\mathbb {H}^{n+1}$ – in other words, if $u\\in \\mathcal {D}^{\\prime }(S\\mathbb {H}^{n+1})$ , then $U^\\pm _K (\\pi _S^ u)\\in \\mathcal {D}^{\\prime }(G)$ is not in the image of $\\pi _S^$ .", "However, in this section we will show how to differentiate distributions on $S\\mathbb {H}^{n+1}$ along the horocyclic vector fields, resulting in sections of the vector bundle $\\mathcal {E}$ introduced in Section REF and its tensor powers.", "First of all, we note that by (REF ), the stable and unstable bundles $E_s(x,\\xi )$ and $E_u(x,\\xi )$ are canonically isomorphic to $\\mathcal {E}(x,\\xi )$ by the maps $\\theta _+:\\mathcal {E}(x,\\xi )\\rightarrow E_s(x,\\xi ),\\ \\theta _-:\\mathcal {E}(x,\\xi )\\rightarrow E_u(x,\\xi ),\\quad \\theta _\\pm (v)=(-v,\\pm v).$ For $u\\in \\mathcal {D}^{\\prime }(S\\mathbb {H}^{n+1})$ , we then define the horocyclic derivatives $\\mathcal {U}_\\pm u\\in \\mathcal {D}^{\\prime }(S\\mathbb {H}^{n+1};\\mathcal {E}^)$ by restricting the differential $du\\in \\mathcal {D}^{\\prime }(S\\mathbb {H}^{n+1};T^(S\\mathbb {H}^{n+1}))$ to the stable/unstable foliations and pulling it back by $\\theta _\\pm $ : $\\mathcal {U}_\\pm u(x,\\xi ):=du(x,\\xi )\\circ \\theta _\\pm \\in \\mathcal {E}^(x,\\xi ).$ To relate $\\mathcal {U}_\\pm $ to the vector fields $U^\\pm _i$ on the group $G$ , consider the orthonormal frame $\\mathbf {e}_1^,\\dots ,\\mathbf {e}_n^$ of the bundle $\\pi _S^\\mathcal {E}^$ over $G$ defined by $\\mathbf {e}_j^(\\gamma ):=\\gamma ^{-}(e_{j+1}^)\\in \\mathcal {E}^(\\pi _S(\\gamma )).$ where the $e_j^=dx_j$ form the dual basis to the canonical basis $(e_j)_{j=0,\\dots ,n+1}$ of $\\mathbb {R}^{1,n+1}$ , and $\\gamma ^{-}=(\\gamma ^{-1})^:(\\mathbb {R}^{1,n+1})^\\rightarrow (\\mathbb {R}^{1,n+1})^$ .", "More generally, we can define the orthonormal frame $\\mathbf {e}^_K$ of $\\pi _S^(\\otimes ^m \\mathcal {E}^)$ by $\\mathbf {e}^_K:=\\mathbf {e}_{k_1}^\\otimes \\dots \\otimes \\mathbf {e}_{k_m}^,\\quad K=k_1\\dots k_m\\in A^m.$ We compute for $u\\in \\mathcal {D}^{\\prime }(S\\mathbb {H}^{n+1})$ , $du(\\pi _S(\\gamma ))\\cdot \\theta _\\pm (\\gamma (e_{j+1}))=U_j^\\pm (\\pi _S^u)(\\gamma )$ and thus $\\pi _S^(\\mathcal {U}_\\pm u)=\\sum _{j=1}^n U_j^\\pm (\\pi _S^ u) \\mathbf {e}_j^.$ We next use the formula (REF ) to define $\\mathcal {U}_\\pm $ as an operator $\\mathcal {U}_\\pm :\\mathcal {D}^{\\prime }(S\\mathbb {H}^{n+1};\\otimes ^m \\mathcal {E}^)\\rightarrow \\mathcal {D}^{\\prime }(S\\mathbb {H}^{n+1};\\otimes ^{m+1}\\mathcal {E}^)$ as follows: for $u\\in \\mathcal {D}^{\\prime }(S\\mathbb {H}^{n+1};\\otimes ^m\\mathcal {E}^)$ , define $\\mathcal {U}_\\pm u$ by $\\pi _S^(\\mathcal {U}_\\pm u)=\\sum _{r=1}^n\\sum _{K\\in A^m} (U_r^\\pm u_K)\\mathbf {e}^_{r K},\\quad \\pi _S^u=\\sum _{K\\in A^m}u_K\\mathbf {e}_K^.$ This definition makes sense (that is, the right-hand side of the first formula in (REF ) lies in the image of $\\pi _S^$ ) since a section $f=\\sum _{K\\in A^m}f_K\\mathbf {e}^_K\\in \\mathcal {D}^{\\prime }(G;\\pi _S^(\\otimes ^m\\mathcal {E}^)),\\quad f_K\\in \\mathcal {D}^{\\prime }(G)$ lies in the image of $\\pi _S^$ if and only if $R_{i+1,j+1}f=0$ for $1\\le i<j\\le n$ (the differentiation is well-defined since the fibers of $\\pi _S^(\\otimes ^m\\mathcal {E}^)$ are the same along each integral curve of $R_{i+1,j+1}$ ), and this translates to $R_{i+1,j+1}f_K=\\sum _{\\ell =1}^m (\\delta _{j k_\\ell }f_{\\lbrace \\ell \\rightarrow i\\rbrace K}-\\delta _{i k_\\ell } f_{\\lbrace \\ell \\rightarrow j\\rbrace K}),\\quad 1\\le i<j\\le n,\\ K\\in A^m;$ to show (REF ) for $f_{r K}=U_r^\\pm u_K$ , we use (REF ): ${\\begin{array}{c}R_{i+1,j+1}f_{rK}=[R_{i+1,j+1},U_r^\\pm ] u_K+U_r^\\pm R_{i+1,j+1}u_K\\\\=\\delta _{jr}U_i^\\pm u_K-\\delta _{ir}U_j^\\pm u_K+\\sum _{\\ell =1}^m \\delta _{jk_\\ell } U_r^\\pm u_{\\lbrace \\ell \\rightarrow i\\rbrace K}-\\delta _{ik_\\ell }U_r^\\pm u_{\\lbrace \\ell \\rightarrow j\\rbrace K}.\\end{array}}$ To interpret the operator (REF ) in terms of the stable/unstable foliations in a manner similar to (REF ), consider the connection $\\nabla ^S$ on the bundle $\\mathcal {E}$ over $S\\mathbb {H}^{n+1}$ defined as follows: for $(x,\\xi )\\in S\\mathbb {H}^{n+1}$ , $(v,w)\\in T_{(x,\\xi )}(S\\mathbb {H}^{n+1})$ , and $u\\in \\mathcal {D}^{\\prime }(S\\mathbb {H}^{n+1};\\mathcal {E})$ , let $\\nabla ^S_{(v,w)}u(x,\\xi )$ be the orthogonal projection of $\\nabla ^{\\mathbb {R}^{1,n+1}}_{(v,w)}u(x,\\xi )$ onto $\\mathcal {E}(x,\\xi )\\subset \\mathbb {R}^{1,n+1}$ , where $\\nabla ^{\\mathbb {R}^{1,n+1}}$ is the canonical connection on the trivial bundle $S\\mathbb {H}^{n+1}\\times \\mathbb {R}^{1,n+1}$ over $S\\mathbb {H}^{n+1}$ (corresponding to differentiating the coordinates of $u$ in $\\mathbb {R}^{1,n+1}$ ).", "Then $\\nabla ^S$ naturally induces a connection on $\\otimes ^m\\mathcal {E}^$ , also denoted $\\nabla ^S$ , and we have for $v,v_1,\\dots ,v_m\\in \\mathcal {E}(x,\\xi )$ and $u\\in \\mathcal {D}^{\\prime }(S\\mathbb {H}^{n+1};\\otimes ^m\\mathcal {E}^)$ , $\\mathcal {U}_\\pm u(x,\\xi )(v,v_1,\\dots ,v_m)=(\\nabla ^S_{\\theta _\\pm (v)}u)(v_1,\\dots ,v_m).$ Indeed, if $\\gamma (t)=\\gamma (0)e^{tU^\\pm _j}$ is an integral curve of $U^\\pm _j$ on $G$ , then $\\gamma (t) e_2,\\dots ,\\gamma (t)e_{n+1}$ form a parallel frame of $\\mathcal {E}$ over the curve $(x(t),\\xi (t))=\\pi _S(\\gamma (t))$ with respect to $\\nabla ^S$ , since the covariant derivative of $\\gamma (t)e_k$ in $t$ with respect to $\\nabla ^{\\mathbb {R}^{1,n+1}}$ is simply $\\gamma (t)U^\\pm _j e_k$ ; by (REF ) this is a linear combination of $x(t)=\\gamma (t)e_0$ and $\\xi (t)=\\gamma (t)e_1$ and thus $\\nabla ^S_t (\\gamma (t)e_k)=0$ .", "Note also that the operator $\\mathcal {X}$ defined in (REF ) can be interpreted as the covariant derivative on $\\mathcal {E}$ along the generator $X$ of the geodesic flow by the connection $\\nabla ^S$ .", "One can naturally generalize $\\mathcal {X}$ to a first order differential operator $\\mathcal {X}:\\mathcal {D}^{\\prime }(S\\mathbb {H}^{n+1};\\otimes ^m\\mathcal {E}^)\\rightarrow \\mathcal {D}^{\\prime }(S\\mathbb {H}^{n+1};\\otimes ^m\\mathcal {E}^)$ and ${1\\over i}\\mathcal {X}$ is still symmetric with respect to the natural measure on $S\\mathbb {H}^{n+1}$ and the inner product on $\\otimes ^m\\mathcal {E}^$ induced by the Minkowski metric.", "A characterization of $X$ in terms of the frame $\\mathbf {e}_K^$ is given by $\\pi _S^(\\mathcal {X} u)=\\sum _{K\\in A^m} (Xu_K)\\mathbf {e}_K^,\\quad \\pi _S^u=\\sum _{K\\in A^m} u_K\\mathbf {e}_K^.$ It follows from (REF ) that for $u\\in \\mathcal {D}^{\\prime }(S\\mathbb {H}^{n+1};\\otimes ^m \\mathcal {E}^)$ , $\\mathcal {X}\\mathcal {U}_\\pm u-\\mathcal {U}_\\pm \\mathcal {X}u=\\pm \\mathcal {U}_\\pm u.$ We also observe that, since $[U^\\pm _i,U^\\pm _j]=0$ , for each scalar distribution $u\\in \\mathcal {D}^{\\prime }(S\\mathbb {H}^{n+1})$ and $m\\in \\mathbb {N}$ , we have $\\mathcal {U}_\\pm ^m u\\in \\mathcal {D}^{\\prime }(S\\mathbb {H}^{n+1};\\otimes _S^m\\mathcal {E}^)$ , where $\\otimes ^m_S\\mathcal {E}^\\subset \\otimes ^m\\mathcal {E}^$ denotes the space of all symmetric cotensors of order $m$ .", "Inversion of the operator $\\mathcal {U}_\\pm ^m$ is the topic of the next subsection.", "We conclude with the following lemma describing how the operator $\\mathcal {U}_\\pm ^m$ acts on distributions invariant under the left action of an element of $G$ : Lemma 4.1 Let $\\gamma \\in G$ and $u\\in \\mathcal {D}^{\\prime }(S\\mathbb {H}^{n+1})$ .", "Assume also that $u$ is invariant under left multiplications by $\\gamma $ , namely $u(\\gamma .", "(x,\\xi ))=u(x,\\xi )$ for allStrictly speaking, this statement should be formulated in terms of the pullback of the distribution $u$ by the map $(x,\\xi )\\mapsto \\gamma .", "(x,\\xi )$ .", "$(x,\\xi )\\in S\\mathbb {H}^{n+1}$ .", "Then $v=\\mathcal {U}_\\pm ^m u$ is equivariant under left multiplication by $\\gamma $ in the following sense: $v(\\gamma .", "(x,\\xi ))=\\gamma .v(x,\\xi ),$ where the action of $\\gamma $ on $\\otimes _S^m\\mathcal {E}^$ is naturally induced by its action on $\\mathcal {E}$ (by taking inverse transposes), which in turn comes from the action of $\\gamma $ on $\\mathbb {R}^{1,n+1}$ .", "We have for $\\gamma ^{\\prime }\\in G$ , $\\mathcal {U}_\\pm ^m u(\\pi _S(\\gamma ^{\\prime }))=\\sum _{K\\in A^m}(U_K^\\pm (u\\circ \\pi _S)(\\gamma ^{\\prime }))\\mathbf {e}_K^(\\gamma ^{\\prime }).$ Therefore, since $U^\\pm _j$ are left invariant vector fields on $G$ , $\\mathcal {U}_\\pm ^m u(\\gamma .\\pi _S(\\gamma ^{\\prime }))=\\mathcal {U}_\\pm ^m u(\\pi _S(\\gamma \\gamma ^{\\prime }))=\\sum _{K\\in A^m} (U_K^\\pm (u\\circ \\pi _S)(\\gamma ^{\\prime }))\\mathbf {e}_K^(\\gamma \\gamma ^{\\prime }).$ It remains to note that $\\mathbf {e}_K^(\\gamma \\gamma ^{\\prime })=\\gamma .\\mathbf {e}_K^(\\gamma ^{\\prime })$ ." ], [ "Inverting horocyclic operators", "In this subsection, we will show that distributions $v\\in \\mathcal {D}^{\\prime }(S\\mathbb {H}^{n+1};\\otimes _S^m \\mathcal {E}^)$ satisfying certain conditions are in fact in the image of $\\mathcal {U}_\\pm ^m$ acting on $\\mathcal {D}^{\\prime }(S\\mathbb {H}^{n+1})$ .", "This is an important step in our construction of Pollicott–Ruelle resonances, as it will make it possible to recover a scalar resonant state corresponding to a resonance in the $m$ th band.", "More precisely, we prove Lemma 4.2 Assume that $v\\in \\mathcal {D}^{\\prime }(S\\mathbb {H}^{n+1};\\otimes ^m_S\\mathcal {E}^)$ satisfies $\\mathcal {U}_\\pm v=0$ , and $\\mathcal {X}v=\\pm \\lambda v$ for $\\lambda \\notin {1\\over 2}\\mathbb {Z}$ .", "Then there exists $u\\in \\mathcal {D}^{\\prime }(S\\mathbb {H}^{n+1})$ such that $\\mathcal {U}_\\pm ^m u=v$ and $Xu=\\pm (\\lambda -m) u$ .", "Moreover, if $v$ is equivariant under left multiplication by some $\\gamma \\in G$ in the sense of (REF ), then $u$ is invariant under left multiplication by $\\gamma $ .", "The proof of Lemma REF is modeled on the following well-known formula recovering a homogeneous polynomial of degree $m$ from its coefficients: given constants $a_\\alpha $ for each multiindex $\\alpha $ of length $m$ , we have $\\partial ^\\beta _x \\sum _{\|\\alpha \|=m} {1\\over \\alpha !", "}x^\\alpha a_\\alpha =a_\\beta ,\\quad \|\\beta \|=m.$ The formula recovering $u$ from $v$ in Lemma REF is morally similar to (REF ), with $U^\\pm _j$ taking the role of $\\partial _{x_j}$ , the condition $\\mathcal {U}_\\pm v=0$ corresponding to $a_\\alpha $ being constants, and $U^\\mp _j$ taking the role of the multiplication operators $x_j$ .", "However, the commutation structure of $U^\\pm _j$ , given by (REF ), is more involved than that of $\\partial _{x_j}$ and $x_j$ and in particular it involves the vector field $X$ , explaining the need for the condition $\\mathcal {X}v=\\pm \\lambda v$ (which is satisfied by resonant states).", "To prove Lemma REF , we define the operator $\\mathcal {V}_\\pm :\\mathcal {D}^{\\prime }(S\\mathbb {H}^{n+1};\\otimes ^{m+1} \\mathcal {E}^)\\rightarrow \\mathcal {D}^{\\prime }(S\\mathbb {H}^{n+1};\\otimes ^m\\mathcal {E}^),\\quad \\mathcal {V}_\\pm :=\\mathcal {T}\\mathcal {U}_\\pm ,$ where $\\mathcal {T}$ is defined in Section REF .", "Then by (REF ) $\\pi _S^(\\mathcal {V}_\\pm u)=\\sum _{K\\in A^m}\\sum _{q\\in A}(U^\\pm _q u_{qK})\\mathbf {e}^_K,\\quad u=\\sum _{K\\in A^{m+1}} u_K\\mathbf {e}^_K.$ For later use, we record the following fact: Lemma 4.3 $\\mathcal {U}_\\pm ^=-\\mathcal {V}_\\pm $ , where the adjoint is understood in the formal sense.", "If $u\\in \\mathcal {C}^\\infty _0(S\\mathbb {H}^{n+1};\\otimes ^m \\mathcal {E}^)$ , $v\\in \\mathcal {C}^\\infty (S\\mathbb {H}^{n+1};\\otimes ^{m+1} \\mathcal {E}^)$ and $u_K$ , $v_J$ are the coordinates of $\\pi _S^u$ and $\\pi _S^v$ in the bases $(\\mathbf {e}^_K)_{K\\in A^m}$ and $(\\mathbf {e}^_J)_{J\\in A^{m+1}}$ , then by (REF ), we compute the following pointwise identity on $S\\mathbb {H}^{n+1}$ : $\\langle \\mathcal {U}_\\pm u,\\bar{v}\\rangle +\\langle u,\\overline{\\mathcal {V}_\\pm v}\\rangle =\\mathcal {V}_\\pm w,\\quad w\\in \\mathcal {C}^\\infty _0(S\\mathbb {H}^{n+1};\\mathcal {E}^),\\quad \\pi _S^w=\\sum _{K\\in A^m\\atop q\\in A} u_K\\overline{v_{qK}}\\,\\mathbf {e}_q^.$ It remains to show that for each $w$ , the integral of $\\mathcal {V}_\\pm w$ is equal to zero.", "Since $\\mathcal {V}_\\pm $ is a differential operator of order 1, we must have $\\int _{S\\mathbb {H}^{n+1}}\\mathcal {V}_\\pm w=\\int _{S\\mathbb {H}^{n+1}} \\langle w,\\eta _\\pm \\rangle $ for all $w$ and some $\\eta _\\pm \\in \\mathcal {C}^\\infty (S\\mathbb {H}^{n+1};\\mathcal {E}^)$ independent of $w$ .", "Then $\\eta _\\pm $ is equivariant under the action of the isometry group $G$ and in particular, $\|\\eta _\\pm \|$ is a constant function on $S\\mathbb {H}^{n+1}$ .", "Moreover, using that $\\int Xf=0$ for all $f\\in \\mathcal {C}^\\infty _0(S\\mathbb {H}^{n+1})$ and $\\mathcal {V}_\\pm (\\mathcal {X}w)=(X\\mp 1)\\mathcal {V}_\\pm w$ , we get for all $w\\in \\mathcal {C}^\\infty _0$ , $\\mp \\int _{S\\mathbb {H}^{n+1}} \\langle w,\\eta _\\pm \\rangle =\\int _{S\\mathbb {H}^{n+1}} \\mathcal {V}_\\pm (\\mathcal {X}w)=-\\int _{S\\mathbb {H}^{n+1}} \\langle w,\\mathcal {X}\\eta _\\pm \\rangle .$ This implies that $\\mathcal {X}\\eta _\\pm =\\pm \\eta _\\pm $ and in particular $X\|\\eta _\\pm \|^2=2\\langle \\mathcal {X}\\eta _\\pm ,\\eta _\\pm \\rangle =\\pm 2\|\\eta _\\pm \|^2.$ Since $\|\\eta _\\pm \|^2$ is a constant function, this implies $\\eta _\\pm =0$ , finishing the proof.", "To construct $u$ from $v$ in Lemma REF , we first handle the case when $\\mathcal {T}(v)=0$ ; this condition is automatically satisfied when $m\\le 1$ .", "Lemma 4.4 Assume that $v\\in \\mathcal {D}^{\\prime }(S\\mathbb {H}^{n+1};\\otimes ^m_S\\mathcal {E}^)$ and $\\mathcal {U}_\\pm v=0$ , $\\mathcal {T}(v)=0$ .", "Define $u=\\mathcal {V}_\\mp ^m v\\in \\mathcal {D}^{\\prime }(S\\mathbb {H}^{n+1})$ .", "Then $\\mathcal {U}_\\pm ^m u=2^m m!\\bigg (\\prod _{\\ell =n-1}^{n+m-2} (\\ell \\pm \\mathcal {X})\\bigg )v.$ Assume that $\\pi _S^ v=\\sum _{K\\in A^m} f_K \\mathbf {e}^_K,\\quad f_K\\in \\mathcal {D}^{\\prime }(G).$ Then $\\pi _S^u=\\sum _{K\\in A^m} U^\\mp _K f_K,\\quad \\pi _S^(\\mathcal {U}^m_\\pm u)=\\sum _{K,J\\in A^m} U^\\pm _J U^\\mp _K f_K \\mathbf {e}^_J.$ For $0\\le r<m$ , $J\\in A^{m-1-r}$ , and $p\\in A$ , we have by (REF ) $\\sum _{K\\in A^r\\atop q\\in A} [U^\\pm _p,U^\\mp _q]U^\\mp _K f_{qKJ}=\\pm 2X\\sum _{K\\in A^r}U^\\mp _K f_{pKJ}+2\\sum _{K\\in A^r\\atop q\\in A} R_{p+1,q+1}U^\\mp _K f_{qKJ}.$ To compute the second term on the right-hand side, we commute $R_{p+1,q+1}$ with $U^\\mp _K$ by (REF ) and use (REF ) to get ${\\begin{array}{c}\\sum _{K\\in A^r\\atop q\\in A} R_{p+1,q+1}U^\\mp _K f_{qKJ}=\\sum _{K\\in A^r\\atop q\\in A}\\bigg ( \\sum _{\\ell =1}^{r}(\\delta _{qk_\\ell }U^\\mp _{\\lbrace \\ell \\rightarrow p\\rbrace K} f_{qKJ}-\\delta _{pk_\\ell }U^\\mp _{\\lbrace \\ell \\rightarrow q\\rbrace K} f_{qKJ})\\\\+U^\\mp _K f_{pKJ}-\\delta _{pq}U^\\mp _K f_{qKJ}+\\sum _{\\ell =1}^r (\\delta _{qk_\\ell }U^\\mp _Kf_{q(\\lbrace \\ell \\rightarrow p\\rbrace K)J}-\\delta _{pk_\\ell }U^\\mp _K f_{q(\\lbrace \\ell \\rightarrow q\\rbrace K)J})\\\\+\\sum _{\\ell =1}^{m-1-r} (\\delta _{qj_\\ell }U^\\mp _K f_{qK(\\lbrace \\ell \\rightarrow p\\rbrace J)}-\\delta _{pj_\\ell } U^\\mp _K f_{qK(\\lbrace \\ell \\rightarrow q\\rbrace J)})\\bigg ).\\end{array}}$ Since $v$ is symmetric and $\\mathcal {T}(v)=0$ , the expressions $\\sum _{K\\in A^r,\\,q\\in A}\\delta _{qk_\\ell }U^\\mp _{\\lbrace \\ell \\rightarrow p\\rbrace K}f_{qKJ}$ , $\\sum _{q\\in A}f_{q(\\lbrace \\ell \\rightarrow q\\rbrace K)J}$ , and $\\sum _{q\\in A}f_{qK(\\lbrace \\ell \\rightarrow q\\rbrace J)}$ are zero.", "Further using the symmetry of $v$ , we find $\\sum _{K\\in A^r\\atop q\\in A}R_{p+1,q+1}U^\\mp _K f_{qKJ}=(n+m-r-2)\\sum _{K\\in A^r} U^\\mp _K f_{pKJ}.$ and thus $\\sum _{K\\in A^r\\atop q\\in A}[U^\\pm _p,U^\\mp _q]U^\\mp _K f_{qKJ}=2\\sum _{K\\in A^r} U^\\mp _K (\\pm X+n+m-2r-2) f_{pKJ}.$ Then, using that $\\mathcal {U}_\\pm v=0$ , we find $\\begin{split}\\sum _{K\\in A^{r+1}} U^\\pm _pU^\\mp _K f_{KJ}&=\\sum _{K\\in A^r\\atop q\\in A}\\sum _{\\ell =1}^{r+1} U^\\mp _{k_\\ell \\dots k_r}[U^\\pm _p,U^\\mp _q] U^\\mp _{k_1\\dots k_{\\ell -1}} f_{qKJ}\\\\&= 2\\sum _{K\\in A^r}\\sum _{\\ell =1}^{r+1}U^\\mp _K(\\pm X+n+m-2\\ell )f_{pKJ}\\\\&= 2(r+1)\\sum _{K\\in A^r}U^\\mp _K(\\pm X+n+m-r-2)f_{pKJ}.\\end{split}$ By iterating (REF ) we obtain (using also that $v$ is symmetric) for $J\\in A^m$ , $\\begin{split}U^\\pm _J\\sum _{K\\in A^m}U^\\mp _K f_K&= 2m\\,U^\\pm _{j_1\\dots j_{m-1}}\\sum _{K\\in A^{m-1}}U_K^\\mp (\\pm X+n-1)f_{Kj_m}\\\\&= 4m(m-1)\\,U^\\pm _{j_1\\dots j_{m-2}}\\sum _{K\\in {A}^{m-2}}U_K^\\mp (\\pm X+n)(\\pm X+n-1)f_{Kj_{m-1}j_m}\\\\&= \\dots \\\\&= 2^m m!", "\\prod _{\\ell =n-1}^{n+m-2}(\\pm X+\\ell )f_{J}\\end{split}$ which achieves the proof.", "To handle the case $\\mathcal {T} (v)\\ne 0$ , define also the horocyclic Laplacians $\\Delta _\\pm :=-\\mathcal {T}\\mathcal {U}_\\pm ^2=-\\mathcal {V}_\\pm \\mathcal {U}_\\pm :\\mathcal {D}^{\\prime }(S\\mathbb {H}^{n+1})\\rightarrow \\mathcal {D}^{\\prime }(S\\mathbb {H}^{n+1}),$ so that for $u\\in \\mathcal {D}^{\\prime }(S\\mathbb {H}^{n+1})$ , $\\pi _S^\\Delta _\\pm u=-\\sum _{q=1}^n U^\\pm _q U^\\pm _q (\\pi _S^u).$ Note that, by the commutation relation (REF ), $[X,\\Delta _\\pm ]=\\pm 2\\Delta _\\pm .$ Also, by Lemma REF , $\\Delta _\\pm $ are symmetric operators.", "Lemma 4.5 Assume that $u\\in \\mathcal {D}^{\\prime }(S\\mathbb {H}^{n+1})$ and $\\mathcal {U}_\\pm ^{m+1} u=0$ .", "Then $\\mathcal {U}_\\pm ^{m+2}\\Delta _\\mp u=-4(\\mathcal {X}\\mp m)(2\\mathcal {X}\\pm (n-2))\\mathcal {I}(\\mathcal {U}_\\pm ^m u)-4\\mathcal {I}^2(\\mathcal {T}(\\mathcal {U}_\\pm ^m u)).$ We have $\\pi _S^(\\mathcal {U}_\\pm ^{m+2}\\Delta _\\mp u)=-\\sum _{K\\in A^{m+2}\\atop q\\in A} U^\\pm _K U^\\mp _q U^\\mp _q u\\, \\mathbf {e}^_{K}.$ Using (REF ), we compute for $K\\in A^{m+2}$ and $q\\in A$ , ${\\begin{array}{c}[U^\\pm _K,U^\\mp _q]=\\sum _{\\ell =1}^{m+2} U^\\pm _{k_1\\dots k_{\\ell -1}} [U^\\pm _{k_\\ell }, U^\\mp _q] U^\\pm _{k_{\\ell +1}\\dots k_{m+2}}\\\\=2\\sum _{\\ell =1}^{m+2}\\big ( \\delta _{q k_\\ell }U^\\pm _{\\lbrace \\ell \\rightarrow \\rbrace K}(\\pm X+m-\\ell +2)+U^\\pm _{k_1\\dots k_{\\ell -1}}R_{k_\\ell +1,q+1}U^\\pm _{k_{\\ell +1}\\dots k_{m+2}}\\big )\\\\=2\\sum _{\\ell =1}^{m+2}\\bigg ( U^\\pm _{\\lbrace \\ell \\rightarrow \\rbrace K}\\big (\\delta _{q k_\\ell }(\\pm X+m-\\ell +2)+R_{k_\\ell +1,q+1}\\big )+\\sum _{r=\\ell +1}^{m+2}(\\delta _{qk_r}U^\\pm _{\\lbrace r\\rightarrow \\rbrace K}-\\delta _{k_\\ell k_r}U^\\pm _{\\lbrace \\ell \\rightarrow ,r\\rightarrow q\\rbrace K})\\bigg )\\\\=2\\sum _{\\ell =1}^{m+2}\\bigg (U^\\pm _{\\lbrace \\ell \\rightarrow \\rbrace K}\\big (\\delta _{qk_\\ell }(\\pm X+m+1)+R_{k_\\ell +1,q+1}\\big )-\\sum _{r=\\ell +1}^{m+2}\\delta _{k_\\ell k_r}U^\\pm _{\\lbrace \\ell \\rightarrow ,r\\rightarrow q\\rbrace K}\\bigg ).\\end{array}}$ Since $\\mathcal {U}_\\pm ^{m+1}u=0$ , for $K\\in A^{m+2}$ and $q\\in A$ we have $U^\\pm _Ku=[U_K^\\pm ,U^\\mp _q]u=0$ and thus $U^\\pm _K U^\\mp _q U^\\mp _q u=[[U_K^\\pm ,U_q^\\mp ],U^\\mp _q]u.$ We calculate $\\sum _{q\\in A}[\\delta _{qk_\\ell }(\\pm X+m+1)+R_{k_\\ell +1,q+1},U_q^\\mp ]=(n-2)U_{k_\\ell }^\\mp $ and thus for $K\\in A^{m+2}$ , ${\\begin{array}{c}\\sum _{q\\in A} U^\\pm _K U^\\mp _q U^\\mp _qu=2\\sum _{\\ell =1}^{m+2}\\bigg ([U^\\pm _{\\lbrace \\ell \\rightarrow \\rbrace K},U^\\mp _{k_\\ell }](\\pm X+m+n-1)\\\\-\\sum _{r=\\ell +1}^{m+2}\\delta _{k_\\ell k_r}\\sum _{q\\in A}[U^\\pm _{\\lbrace \\ell \\rightarrow ,r\\rightarrow q\\rbrace K},U^\\mp _q]\\bigg )u.\\end{array}}$ Now, for $K\\in A^{m+2}$ , ${\\begin{array}{c}\\sum _{\\ell =1}^{m+2}[U^\\pm _{\\lbrace \\ell \\rightarrow \\rbrace K},U^\\mp _{k_\\ell }](\\pm X+m+n-1)u=2\\sum _{\\ell ,s=1\\atop \\ell \\ne s}^{m+2}\\bigg (\\delta _{k_\\ell k_s}U^\\pm _{\\lbrace \\ell \\rightarrow ,s\\rightarrow \\rbrace K}(\\pm X+m)\\\\-\\sum _{r=s+1\\atop r\\ne \\ell }^{m+2}\\delta _{k_sk_r}U^\\pm _{\\lbrace s\\rightarrow ,r\\rightarrow \\rbrace K}\\bigg )(\\pm X+m+n-1)u\\\\=2\\sum _{\\ell ,r=1\\atop \\ell <r}^{m+2} \\delta _{k_\\ell k_r}U^\\pm _{\\lbrace \\ell \\rightarrow ,r\\rightarrow \\rbrace K}(\\pm 2X+m)(\\pm X+m+n-1)u.\\end{array}}$ Furthermore, we have for $K\\in A^m$ , ${\\begin{array}{c}\\sum _{q\\in A}[U^\\pm _{qK},U^\\mp _q]u=2U^\\pm _K\\big ((m+n)(\\pm X+m)-m\\big )u-2\\sum _{q\\in A}\\sum _{s,p=1\\atop s<p}^m \\delta _{k_sk_p}U^\\pm _{qq\\lbrace s\\rightarrow ,p\\rightarrow \\rbrace K}u\\end{array}}$ We finally compute ${\\begin{array}{c}\\sum _{q\\in A} U^\\pm _K U^\\mp _q U^\\mp _qu=4\\sum _{\\ell ,r=1\\atop \\ell <r}^{m+2}\\delta _{k_\\ell k_r} U^\\pm _{\\lbrace \\ell \\rightarrow ,r\\rightarrow \\rbrace K}X(2X\\pm (n+2m-2))u\\\\+4\\sum _{q\\in A} \\sum _{\\ell ,r=1\\atop \\ell <r}^{m+2}\\sum _{s,p=1\\atop s<p;\\, \\lbrace s,p\\rbrace \\cap \\lbrace \\ell ,r\\rbrace =\\emptyset }^{m+2}\\delta _{k_\\ell k_r}\\delta _{k_sk_p}U^\\pm _{qq\\lbrace \\ell \\rightarrow ,r\\rightarrow ,s\\rightarrow ,p\\rightarrow \\rbrace K}u,\\end{array}}$ which finishes the proof.", "Arguing by induction using (REF ) and applying Lemma REF to $\\Delta _\\mp ^ru$ , we get Lemma 4.6 Assume that $u\\in \\mathcal {D}^{\\prime }(S\\mathbb {H}^{n+1})$ and $\\mathcal {U}^{m+1}_\\pm u=0$ , $\\mathcal {T}(\\mathcal {U}^m_\\pm u)=0$ .", "Then for each $r\\ge 0$ , $\\mathcal {U}_\\pm ^{m+2r}\\Delta _\\mp ^r u=(-1)^r2^{2r}\\bigg (\\prod _{j=0}^{r-1} (\\mathcal {X}\\mp (m+j))\\bigg )\\bigg (\\prod _{j=1}^r (2\\mathcal {X}\\pm (n-2j))\\bigg )\\mathcal {I}^r (\\mathcal {U}^m_\\pm u).$ Moreover, for $r\\ge 1$ ${\\begin{array}{c}\\mathcal {T}(\\mathcal {U}_\\pm ^{m+2r}\\Delta _\\mp ^ru)=(-1)^r2^{2r}r(n+2m+2r-2)\\\\\\cdot \\bigg (\\prod _{j=0}^{r-1} (\\mathcal {X}\\mp (m+j))\\bigg )\\bigg (\\prod _{j=1}^r (2\\mathcal {X}\\pm (n-2j))\\bigg )\\mathcal {I}^{r-1}(\\mathcal {U}^m_\\pm u).\\end{array}}$ We are now ready to finish the proof of Lemma REF .", "Following (REF ), we decompose $v$ as $v=\\sum _{r=0}^{\\lfloor m/2\\rfloor }\\mathcal {I}^r(v_r)$ with $v_r\\in \\mathcal {D^{\\prime }}(S\\mathbb {H}^{n+1}; \\otimes _S^{m-2r}\\mathcal {E}^)$ and $\\mathcal {T}(v_r)=0$ .", "Since $X$ commutes with $\\mathcal {T}$ and $\\mathcal {I}$ , we find $X v_r=\\pm \\lambda v_r$ .", "Moreover, since $\\mathcal {U}_\\pm v=0$ , we have $\\mathcal {U}_\\pm v_r=0$ .", "Put $u_r:=(-\\Delta _\\mp )^r \\mathcal {V}_\\mp ^{m-2r} v_r\\in \\mathcal {D}^{\\prime }(S\\mathbb {H}^{n+1}).$ By Lemma REF (applied to $v_r$ ) and Lemma REF (applied to $\\mathcal {V}_\\mp ^{m-2r}v_r$ and $m$ replaced by $m-2r$ ), ${\\begin{array}{c}\\mathcal {U}_\\pm ^m u_r=2^{2r}\\bigg (\\prod _{j=0}^{r-1}(\\lambda -(m-2r+j))\\bigg )\\bigg (\\prod _{j=1}^r(2\\lambda +n-2j)\\bigg )\\mathcal {I}^r(\\mathcal {U}^{m-2r}_\\pm \\mathcal {V}_\\mp ^{m-2r}v_r)\\\\=2^m(m-2r)!\\bigg (\\prod _{j=n-1}^{n+m-2r-2}(\\lambda +j)\\bigg )\\bigg (\\prod _{j=m-2r}^{m-r-1}(\\lambda - j)\\bigg )\\bigg (\\prod _{j=1}^r(2\\lambda + n-2j)\\bigg )\\mathcal {I}^r(v_r).\\end{array}}$ Since $\\lambda \\notin {1\\over 2}\\mathbb {Z}$ , we see that $v=\\mathcal {U}_\\pm ^m u$ , where $u$ is a linear combination of $u_0,\\dots ,u_{\\lfloor m/2\\rfloor }$ .", "The relation $Xu=\\pm (\\lambda - m)u$ follows immediately from (REF ) and (REF ).", "Finally, the equivariance property under $G$ follows similarly to Lemma REF ." ], [ "Reduction to the conformal boundary", "We now describe the tensors $v\\in \\mathcal {D}^{\\prime }(S\\mathbb {H}^{n+1};\\otimes ^m_S\\mathcal {E}^)$ that satisfy $\\mathcal {U}_\\pm v=0$ and $Xv=0$ via symmetric tensors on the conformal boundary $\\mathbb {S}^n$ .", "For that we define the operators $\\mathcal {Q}_\\pm :\\mathcal {D}^{\\prime }(\\mathbb {S}^n;\\otimes ^m (T^\\mathbb {S}^n))\\rightarrow \\mathcal {D}^{\\prime }(S\\mathbb {H}^{n+1};\\otimes ^m\\mathcal {E}^)$ by the following formula: if $w\\in \\mathcal {C}^\\infty (\\mathbb {S}^{n};\\otimes ^m(T^\\mathbb {S}^n))$ , we set for $\\eta _i\\in \\mathcal {E}(x,\\xi )$ $\\mathcal {Q}_\\pm w(x,\\xi )(\\eta _1,\\dots ,\\eta _m):=(w\\circ B_\\pm (x,\\xi ))(\\mathcal {A}_\\pm ^{-1}(x,\\xi )\\eta _1,\\dots ,\\mathcal {A}_\\pm ^{-1}(x,\\xi )\\eta _m)$ where $\\mathcal {A}_\\pm (x,\\xi ):T_{B_\\pm (x,\\xi )}\\mathbb {S}^n\\rightarrow \\mathcal {E}(x,\\xi )$ is the parallel transport defined in (REF ), and we see that the operator (REF )Ê extends continuously to $\\mathcal {D}^{\\prime }(\\mathbb {S}^n;\\otimes ^m_S (T^\\mathbb {S}^n))$ since the map $B_\\pm :S\\mathbb {H}^{n+1}\\rightarrow \\mathbb {S}^n$ defined in (REF ) is a submersion, see [33]; the result can be written as $\\mathcal {Q}_\\pm w= (\\otimes ^m (\\mathcal {A}_\\pm ^{-1})^T).w\\circ B_\\pm $ where $T$ means transpose.", "Lemma 4.7 The operator $\\mathcal {Q}_\\pm $ is a linear isomorphism from $\\mathcal {D}^{\\prime }(\\mathbb {S}^n;\\otimes ^m_S(T^\\mathbb {S}^n))$ onto the space $\\lbrace v\\in \\mathcal {D}^{\\prime }(S\\mathbb {H}^{n+1};\\otimes _S^m\\mathcal {E}^)\\mid \\mathcal {U}_\\pm v=0,\\ \\mathcal {X}v=0\\rbrace .$ It is clear that $\\mathcal {Q}_\\pm $ is injective.", "Next, we show that the image of $\\mathcal {Q}_\\pm $ is contained in (REF ).", "For that it suffices to show that for $w\\in \\mathcal {C}^\\infty (\\mathbb {S}^n;\\otimes _S^m(T^\\mathbb {S}^n))$ , we have $\\mathcal {U}_\\pm (\\mathcal {Q}_\\pm w)=0$ and $\\mathcal {X}(\\mathcal {Q}_\\pm w)=0$ .", "We prove the first statement, the second one is established similarly.", "Let $\\gamma \\in G$ , $w_1,\\dots ,w_m\\in \\mathcal {C}^\\infty (\\mathbb {S}^n;T\\mathbb {S}^n)$ , and $w_i^=\\langle w_i, \\cdot \\rangle _{g_{\\mathbb {S}^n}}$ be the duals through the metric.", "Then ${\\begin{array}{c}\\mathcal {Q}_\\pm (w^_1\\otimes \\dots \\otimes w^_m)(\\pi _S(\\gamma ))=\\\\\\sum _{k_1,\\dots ,k_m=1}^n \\Big ( \\prod _{j=1}^m(w^_j\\circ B_\\pm \\circ \\pi _S(\\gamma ))(\\mathcal {A}^{-1}_\\pm (\\pi _S(\\gamma ))\\gamma \\cdot e_{k_j+1})\\Big )\\mathbf {e}_K^(\\gamma )=\\\\(-1)^m\\sum _{k_1,\\dots ,k_m=1}^n \\Big ( \\prod _{j=1}^m\\langle (\\mathcal {A}_\\pm .w_j\\circ B_\\pm )\\circ \\pi _S(\\gamma ),\\gamma \\cdot e_{k_j+1}\\rangle _{M}\\Big )\\mathbf {e}_K^(\\gamma )\\end{array}}$ where we have used (REF ) in the second identity.", "Now we have from (REF ) $\\mathcal {A}_\\pm (\\pi _S(\\gamma ))\\zeta =(0,\\zeta )-\\langle (0,\\zeta ),\\gamma \\cdot e_0\\rangle _M\\gamma (e_0+e_1)$ thus $\\mathcal {Q}_\\pm (w^_1\\otimes \\dots \\otimes w^_m)(\\pi _S(\\gamma ))=\\sum _{k_1,\\dots ,k_m=1}^n \\Big ( \\prod _{j=1}^m\\langle (0,-w_j(B_\\pm (\\pi _S(\\gamma )))),\\gamma \\cdot e_{k_j+1}\\rangle _M\\Big )\\mathbf {e}_K^(\\gamma ).$ Since $d(B_\\pm \\circ \\pi _S)\\cdot U^\\pm _\\ell =0$ by (REF ) and $U^\\pm _\\ell (\\gamma \\cdot e_{k_j+1})=\\gamma \\cdot U^\\pm _\\ell \\cdot e_{k_j+1}$ is a multiple of $\\gamma \\cdot (e_0\\pm e_1)=\\Phi _\\pm (\\pi _S(\\gamma ))(1,B_\\pm (\\pi _S(\\gamma )))$ , we see that $\\mathcal {U}_\\pm (\\mathcal {Q}_\\pm w)=0$ for all $w$ .", "It remains to show that for $v$ in (REF ), we have $v=\\mathcal {Q}_\\pm (w)$ for some $w$ .", "For that, define $\\tilde{v}=(\\otimes ^m\\mathcal {A}_\\pm ^{T})\\cdot v\\in \\mathcal {D}^{\\prime }(S\\mathbb {H}^{n+1};B_\\pm ^(\\otimes _S^m T^\\mathbb {S}^n))$ where $\\mathcal {A}_\\pm ^T$ denotes the tranpose of $\\mathcal {A}_\\pm $ .", "Then $\\mathcal {U}_\\pm v=0$ , $\\mathcal {X}v=0$ imply that $U^\\pm _\\ell (\\pi _S^\\tilde{v})=0$ and $X\\tilde{v}=0$ (where to define differentiation we embed $T^\\mathbb {S}^n$ into $\\mathbb {R}^{n+1}$ ).", "Additionally, $R_{i+1,j+1}(\\pi _S^\\tilde{v})=0$ , therefore $\\pi _S^\\tilde{v}$ is constant on the right cosets of the subgroup $H_\\pm \\subset G$ defined in (REF ).", "Since $(B_\\pm \\circ \\pi _S)^{-1}(B_\\pm \\circ \\pi _S(\\gamma ))=\\gamma H_\\pm $ , we see that $\\tilde{v}$ is the pull-back under $B_\\pm $ of some $w\\in \\mathcal {D}^{\\prime }(\\mathbb {S}^n;\\otimes _S^m T^\\mathbb {S}^n)$ , and it follows that $v=\\mathcal {Q}_\\pm (w)$ .", "In fact, using (REF ) and the expression of $\\xi _\\pm (x,\\nu )$ in (REF ) in terms of Poisson kernel, it is not difficult to show that $\\mathcal {Q}_\\pm (w)$ belongs to a smaller space of tempered distributions: in the ball model, this can be described as the dual space to the Frechet space of smooth sections of $\\otimes ^m ({^0}S\\overline{\\mathbb {B}^{n+1}})$ over $\\overline{\\mathbb {B}^{n+1}}$ which vanish to infinite order at the conformal boundary $\\mathbb {S}^n=\\partial \\overline{\\mathbb {B}^{n+1}}$ .", "We finally give a useful criterion for invariance of $\\mathcal {Q}_\\pm (w)$ under the left action of an element of $G$ : Lemma 4.8 Take $\\gamma \\in G$ and let $w\\in \\mathcal {D}^{\\prime }(\\mathbb {S}^n;\\otimes ^m_S(T^\\mathbb {S}^n))$ .", "Take $s\\in \\mathbb {C}$ and define $v=\\Phi _\\pm ^s \\mathcal {Q}_\\pm (w)$ .", "Then $v$ is equivariant under left multiplication by $\\gamma $ , in the sense of (REF ), if and only if $w$ satisfies the condition $L_\\gamma ^* w(\\nu )=N_\\gamma (\\nu )^{-s-m} w(\\nu ),\\quad \\nu \\in \\mathbb {S}^n.$ Here $L_\\gamma (\\nu )\\in \\mathbb {S}^n$ and $N_\\gamma (\\nu )>0$ are defined in (REF ).", "The lemma follows by a direct calculation from (REF ) and (REF )." ], [ "Pollicott–Ruelle resonances", "In this section, we first recall the results of Butterley–Liverani [10] and Faure–Sjöstrand [18] on the Pollicott–Ruelle resonances for Anosov flows.", "We next state several useful microlocal properties of these resonances and prove Theorem REF , modulo properties of Poisson kernels (Lemma REF and Theorem REF ) which will be proved in Sections and .", "Finally, we prove a pairing formula for resonances and Theorem REF ." ], [ "Definition and properties", "We follow the presentation of [18]; a more recent treatment using different technical tools is also given in [17].", "We refer the reader to these two papers for the necessary notions of microlocal analysis.", "Let $\\mathcal {M}$ be a smooth compact manifold of dimension $2n+1$ and $\\varphi _t=e^{tX}$ be an Anosov flow on $\\mathcal {M}$ , generated by a smooth vector field $X$ .", "(In our case, $\\mathcal {M}=SM$ , $M=\\Gamma \\backslash \\mathbb {H}^{n+1}$ , and $\\varphi _t$ is the geodesic flow – see Section REF .)", "The Anosov property is defined as follows: there exists a continuous splitting $T_y \\mathcal {M}=E_0(y)\\oplus E_u(y)\\oplus E_s(y),\\quad y\\in \\mathcal {M};\\quad E_0(y):=\\mathbb {R}X(y),$ invariant under $d\\varphi _t$ and such that the stable/unstable subbundles $E_s,E_u\\subset T\\mathcal {M}$ satisfy for some fixed smooth norm $\|\\cdot \|$ on the fibers of $T\\mathcal {M}$ and some constants $C$ and $\\theta >0$ , ${\\begin{array}{c}\|d\\varphi _t(y)v\|\\le Ce^{-\\theta t}\|v\|,\\quad v\\in E_s(y);\\\\\|d\\varphi _{-t}(y)v\|\\le Ce^{-\\theta t}\|v\|,\\quad v\\in E_u(y).\\end{array}}$ We make an additional assumption that $\\mathcal {M}$ is equipped with a smooth measure $\\mu $ which is invariant under $\\varphi _t$ , that is, $\\mathcal {L}_X \\mu =0$ .", "We will use the dual decomposition to (REF ), given by $T_y^\\mathcal {M}=E_0^(y)\\oplus E_u^(y)\\oplus E_s^(y),\\quad y\\in \\mathcal {M},$ where $E_0^,E_u^,E_s^$ are dual to $E_0,E_s,E_u$ respectively (note that $E_u,E_s$ are switched places), so for example $E_u^(y)$ consists of covectors annihilating $E_0(y)\\oplus E_u(y)$ .", "Following [18], we now consider for each $r\\ge 0$ an anisotropic Sobolev space $\\mathcal {H}^r(\\mathcal {M}),\\quad \\mathcal {C}^\\infty (\\mathcal {M})\\subset \\mathcal {H}^r(\\mathcal {M})\\subset \\mathcal {D}^{\\prime }(\\mathcal {M}).$ Here we put $u:=-r,s:=r$ in [18].", "Microlocally near $E_u^$ , the space $\\mathcal {H}^r$ is equivalent to the Sobolev space $H^{-r}$ , in the sense that for each pseudodifferential operator $A$ of order 0 whose wavefront set is contained in a small enough conic neighborhood of $E_u^$ , the operator $A$ is bounded $\\mathcal {H}^r\\rightarrow H^{-r}$ and $H^{-r}\\rightarrow \\mathcal {H}^r$ .", "Similarly, microlocally near $E_s^$ , the space $\\mathcal {H}^r$ is equivalent to the Sobolev space $H^r$ .", "We also have $\\mathcal {H}^0=L^2$ .", "The first order differential operator $X$ admits a unique closed unbounded extension from $\\mathcal {C}^\\infty $ to $\\mathcal {H}^r$ , see [18].", "The following theorem, defining Pollicott–Ruelle resonances associated to $\\varphi _t$ , is due to Faure and Sjöstrand [18]; see also [17].", "Theorem 5 Fix $r\\ge 0$ .", "Then the closed unbounded operator $-X:\\mathcal {H}^r(\\mathcal {M})\\rightarrow \\mathcal {H}^r(\\mathcal {M})$ has discrete spectrum in the region $\\lbrace \\operatorname{Re}\\lambda >-r/C_0\\rbrace $ , for some constant $C_0$ independent of $r$ .", "The eigenvalues of $-X$ on $\\mathcal {H}^r$ , called Pollicott–Ruelle resonances, and taken with multiplicities, do not depend on the choice of $r$ as long as they lie in the appropriate region.", "We have the following criterion for Pollicott–Ruelle resonances which does not use the $\\mathcal {H}^r$ spaces explicitly: Lemma 5.1 A number $\\lambda \\in \\mathbb {C}$ is a Pollicott–Ruelle resonance of $X$ if and only the space $\\operatorname{Res}_X(\\lambda ):=\\lbrace u\\in \\mathcal {D}^{\\prime }(\\mathcal {M})\\mid (X+\\lambda )u=0,\\ \\operatorname{WF}(u)\\subset E_u^\\rbrace $ is nontrivial.", "Here $\\operatorname{WF}$ denotes the wavefront set, see for instance [18].", "The elements of $\\operatorname{Res}_X(\\lambda )$ are called resonant states associated to $\\lambda $ and the dimension of this space is called geometric multiplicity of $\\lambda $ .", "Assume first that $\\lambda $ is a Pollicott–Ruelle resonance.", "Take $r>0$ such that $\\operatorname{Re}\\lambda > - r/C_0$ .", "Then $\\lambda $ is an eigenvalue of $-X$ on $\\mathcal {H}^r$ , which implies that there exists nonzero $u\\in \\mathcal {H}^r$ such that $(X+\\lambda ) u=0$ .", "By [18], we have $\\operatorname{WF}(u)\\subset E_u^$ , thus $u$ lies in (REF ).", "Assume now that $u\\in \\mathcal {D}^{\\prime }(\\mathcal {M})$ is a nonzero element of (REF ).", "For large enough $r$ , we have $\\operatorname{Re}\\lambda >-r/C_0$ and $u\\in H^{-r}(\\mathcal {M})$ .", "Since $\\operatorname{WF}(u)\\subset E_u^$ and $\\mathcal {H}^r$ is equivalent to $H^{-r}$ microlocally near $E_u^$ , we have $u\\in \\mathcal {H}^r$ .", "Together with the identity $(X+\\lambda ) u$ , this shows that $\\lambda $ is an eigenvalue of $-X$ on $\\mathcal {H}^r$ and thus a Pollicott–Ruelle resonance.", "For each $\\lambda $ with $\\operatorname{Re}\\lambda >-r/C_0$ , the operator $X+\\lambda :\\mathcal {H}^r\\rightarrow \\mathcal {H}^r$ is Fredholm of index zero on its domain; this follows from the proof of Theorem REF .", "Therefore, $\\dim \\operatorname{Res}_X(\\lambda )$ is equal to the dimension of the kernel of the adjoint operator $X^+\\bar{\\lambda }$ on the $L^2$ dual of $\\mathcal {H}^r$ , which we denote by $\\mathcal {H}^{-r}$ .", "Since ${1\\over i}X$ is symmetric on $L^2$ , we see that $\\operatorname{Res}_X(\\lambda )$ has the same dimension as the following space of coresonant states at $\\lambda $ : $\\operatorname{Res}_{X^}(\\lambda ):=\\lbrace u\\in \\mathcal {D}^{\\prime }(\\mathcal {M})\\mid (X-\\bar{\\lambda }) u=0,\\ \\operatorname{WF}(u)\\subset E_s^\\rbrace .$ The main difference of (REF ) from (REF ) is that the subbundle $E_s^$ is used instead of $E_u^$ ; this can be justified by applying Lemma REF to the vector field $-X$ instead of $X$ , since the roles of the stable/unstable spaces for the corresponding flow $\\varphi _{-t}$ are reversed.", "Note also that for any $\\lambda ,\\lambda ^\\in \\mathbb {C}$ , one can define a pairing $\\langle u,u^\\rangle \\in \\mathbb {C},\\quad u\\in \\operatorname{Res}_X(\\lambda ),\\ u^\\in \\operatorname{Res}_{X^}(\\lambda ^).$ One way of doing that is using the fact that wavefront sets of $u,u^$ intersect only at the zero section, and applying [33].", "An equivalent definition is noting that $u\\in \\mathcal {H}^r$ and $u^\\in \\mathcal {H}^{-r}$ for $r>0$ large enough and using the duality of $\\mathcal {H}^r$ and $\\mathcal {H}^{-r}$ .", "Note that for $\\lambda \\ne \\lambda ^$ , we have $\\langle u,u^\\rangle =0$ ; indeed, $X(u\\overline{u^})=(\\lambda ^-\\lambda )u\\overline{u^}$ integrates to 0.", "The question of computing the product $\\langle u,u^\\rangle $ for $\\lambda =\\lambda ^$ is much more subtle and related to algebraic multiplicities, see Section REF .", "Since ${1\\over i}X$ is self-adjoint on $L^2=\\mathcal {H}^0$ (see [18]), it has no eigenvalues on this space away from the real line; this implies that there are no Pollicott–Ruelle resonances in the right half-plane.", "In other words, we have Lemma 5.2 The spaces $\\operatorname{Res}_X(\\lambda )$ and $\\operatorname{Res}_{X^}(\\lambda )$ are trivial for $\\operatorname{Re}\\lambda >0$ .", "Finally, we note that the results above apply to certain operators on vector bundles.", "More precisely, let $E$ be a smooth vector bundle over $\\mathcal {M}$ and assume that $\\mathcal {X}$ is a first order differential operator on $\\mathcal {D}^{\\prime }(\\mathcal {M};E)$ whose principal part is given by $X$ , namely $\\mathcal {X}(f \\mathbf {u})= f\\, \\mathcal {X}(\\mathbf {u})+(Xf) \\mathcal {X}(\\mathbf {u}),\\quad f\\in \\mathcal {D}^{\\prime }(\\mathcal {M}),\\ \\mathbf {u}\\in \\mathcal {C}^\\infty (\\mathcal {M};E).$ Assume moreover that $E$ is endowed with an inner product $\\langle \\cdot ,\\cdot \\rangle _{E}$ and ${1\\over i}\\mathcal {X}$ is symmetric on $L^2$ with respect to this inner product and the measure $\\mu $ .", "By an easy adaptation of the results of [18] (see [21] and [17]), one can construct anisotropic Sobolev spaces $\\mathcal {H}^r(\\mathcal {M};E)$ and Theorem REF and Lemmas REF , REF apply to $\\mathcal {X}$ on these spaces." ], [ "Proof of the main theorem", "We now concentrate on the case $\\mathcal {M}=SM=\\Gamma \\backslash (S\\mathbb {H}^{n+1}),\\quad M=\\Gamma \\backslash \\mathbb {H}^{n+1},$ with $\\varphi _t$ the geodesic flow.", "Here $\\Gamma \\subset G=\\operatorname{PSO}(1,n+1)$ is a co-compact discrete subgroup with no fixed points, so that $M$ is a compact smooth manifold.", "Henceforth we identify functions on the sphere bundle $SM$ with functions on $S\\mathbb {H}^{n+1}$ invariant under $\\Gamma $ , and similar identifications will be used for other geometric objects.", "It is important to note that the constructions of the previous sections, except those involving the conformal infinity, are invariant under left multiplication by elements of $G$ and thus descend naturally to $SM$ .", "The lift of the geodesic flow on $SM$ is the generator of the geodesic flow on $S\\mathbb {H}^{n+1}$ (see Section REF ); both are denoted $X$ .", "The lifts of the stable/unstable spaces $E_s,E_u$ to $S\\mathbb {H}^{n+1}$ are given in (REF ), and we see that (REF ) holds with $\\theta =1$ .", "The invariant measure $\\mu $ on $SM$ is just the product of the volume measure on $M$ and the standard measure on the fibers of $SM$ induced by the metric.", "Consider the bundle $\\mathcal {E}$ on $SM$ defined in Section REF .", "Then for each $m$ , the operator $\\mathcal {X}:\\mathcal {D}^{\\prime }(SM;\\otimes _S^m\\mathcal {E}^)\\rightarrow \\mathcal {D}^{\\prime }(SM;\\otimes _S^m\\mathcal {E}^)$ defined in (REF ) satisfies (REF ) and ${1\\over i}\\mathcal {X}$ is symmetric.", "The results of Section REF apply both to $X$ and $\\mathcal {X}$ .", "Recall the operator $\\mathcal {U}_-$ introduced in Section REF and its powers, for $m\\ge 0$ , $\\mathcal {U}_-^m:\\mathcal {D}^{\\prime }(SM)\\rightarrow \\mathcal {D}^{\\prime }(SM;\\otimes _S^m\\mathcal {E}^).$ The significance of $\\mathcal {U}_-^m$ for Pollicott–Ruelle resonances is explained by the following Lemma 5.3 Assume that $\\lambda \\in \\mathbb {C}$ is a Pollicott–Ruelle resonance of $X$ and $u\\in \\operatorname{Res}_X(\\lambda )$ is a corresponding resonant state as defined in (REF ).", "Then $\\mathcal {U}_-^m u=0\\quad \\text{for }m>-\\operatorname{Re}\\lambda .$ By (REF ), $(\\mathcal {X}+\\lambda +m) \\mathcal {U}_-^m u=0.$ Note also that $\\operatorname{WF}(\\mathcal {U}_-^mu)\\subset E_u^$ since $\\operatorname{WF}(u)\\subset E_u^$ and $\\mathcal {U}_-^m$ is a differential operator.", "Since $\\lambda +m$ lies in the right half-plane, it remains to apply Lemma REF to $\\mathcal {U}_-^mu$ .", "We can then use the operators $\\mathcal {U}_-^m$ to split the resonance spectrum into bands: Lemma 5.4 Assume that $\\lambda \\in \\mathbb {C}\\setminus {1\\over 2}\\mathbb {Z}$ .", "Then $\\dim \\operatorname{Res}_X(\\lambda )=\\sum _{m\\ge 0}\\dim \\operatorname{Res}_{\\mathcal {X}}^m(\\lambda +m),$ where $\\operatorname{Res}_{\\mathcal {X}}^m(\\lambda ):=\\lbrace v\\in \\mathcal {D}^{\\prime }(SM;\\otimes _S^m\\mathcal {E}^)\\mid (\\mathcal {X}+\\lambda )v=0,\\ \\mathcal {U}_-v=0,\\ \\operatorname{WF}(v)\\subset E_u^\\rbrace .$ The space $\\operatorname{Res}_{\\mathcal {X}}^m(\\lambda )$ is trivial for $\\operatorname{Re}\\lambda >0$ (by Lemma REF ).", "If $\\lambda \\in {1\\over 2}\\mathbb {Z}$ , then we have $\\dim \\operatorname{Res}_X(\\lambda )\\le \\sum _{m\\ge 0}\\dim \\operatorname{Res}_{\\mathcal {X}}^m(\\lambda +m).$ Denote for $m\\ge 1$ , $V_m(\\lambda ):=\\lbrace u\\in \\mathcal {D}^{\\prime }(SM)\\mid (X+\\lambda ) u=0,\\ \\mathcal {U}_-^m u=0,\\ \\operatorname{WF}(u)\\subset E_u^\\rbrace .$ Clearly, $V_m(\\lambda )\\subset V_{m+1}(\\lambda )$ .", "Moreover, by Lemma REF we have $\\operatorname{Res}_X(\\lambda )=V_m(\\lambda )$ for $m$ large enough depending on $\\lambda $ .", "By (REF ), the operator $\\mathcal {U}_-^m$ acts $\\mathcal {U}_-^m:V_{m+1}(\\lambda )\\rightarrow \\operatorname{Res}_{\\mathcal {X}}^m(\\lambda +m),$ and the kernel of (REF ) is exactly $V_m(\\lambda )$ , with the convention that $V_0(\\lambda )=0$ .", "Therefore $\\dim V_{m+1}(\\lambda )\\le \\dim V_m(\\lambda )+\\dim \\operatorname{Res}_{\\mathcal {X}}^m(\\lambda +m)$ and (REF ) follows.", "To show (REF ), it remains to prove that the operator (REF ) is onto; this follows from Lemma REF (which does not enlarge the wavefront set of the resulting distribution since it only employs differential operators in the proof).", "The space $\\operatorname{Res}_{\\mathcal {X}}^m(\\lambda +m)$ is called the space of resonant states at $\\lambda $ associated to $m$ th band; later we see that most of the corresponding Pollicott–Ruelle resonances satisfy $\\operatorname{Re}\\lambda =-n/2-m$ .", "Similarly, we can describe $\\operatorname{Res}_{X^}(\\lambda )$ via the spaces $\\operatorname{Res}_{\\mathcal {X}^}^m(\\lambda +m)$ , where $\\operatorname{Res}_{\\mathcal {X}^}^m(\\lambda ):=\\lbrace v\\in \\mathcal {D}^{\\prime }(SM;\\otimes _S^m\\mathcal {E}^)\\mid (\\mathcal {X}-\\bar{\\lambda }) v=0,\\ \\mathcal {U}_+v=0,\\ \\operatorname{WF}(v)\\subset E_s^\\rbrace ;$ note that here $\\mathcal {U}_+$ is used in place of $\\mathcal {U}_-$ .", "We further decompose $\\operatorname{Res}_{\\mathcal {X}}^m(\\lambda )$ using trace free tensors: Lemma 5.5 Recall the homomorphisms $\\mathcal {T}:\\otimes _S^m\\mathcal {E}^\\rightarrow \\otimes _S^{m-2} \\mathcal {E}^$ , $\\mathcal {I}:\\otimes _S^m\\mathcal {E}^\\rightarrow \\otimes _S^{m-2} \\mathcal {E}^$ defined in Section REF (we put $\\mathcal {T}=0$ for $m=0,1$ ).", "Define the space $\\operatorname{Res}_{\\mathcal {X}}^{m,0}(\\lambda ):=\\lbrace v\\in \\operatorname{Res}_{\\mathcal {X}}^m(\\lambda )\\mid \\mathcal {T}(v)=0\\rbrace .$ Then for all $m\\ge 0$ and $\\lambda $ , $\\dim \\operatorname{Res}_{\\mathcal {X}}^m(\\lambda )=\\sum _{\\ell =0}^{\\lfloor {m\\over 2}\\rfloor } \\dim \\operatorname{Res}_{\\mathcal {X}}^{m-2\\ell ,0}(\\lambda ).$ In fact, $\\operatorname{Res}_{\\mathcal {X}}^{m,0}(\\lambda )=\\bigoplus _{\\ell =0}^{\\lfloor {m\\over 2}\\rfloor } \\mathcal {I}^\\ell (\\operatorname{Res}_{\\mathcal {X}}^{m-2\\ell ,0}(\\lambda )).$ The identity (REF ) follows immediately from (REF ); it is straightforward to see that the defining properties of $\\operatorname{Res}_{\\mathcal {X}}^m(\\lambda )$ are preserved by the canonical tensorial operations involved.", "The identity (REF ) then follows since $\\mathcal {I}$ is one to one by the paragraph following (REF ).", "The elements of $\\operatorname{Res}_{\\mathcal {X}}^{m,0}(\\lambda )$ can be expressed via distributions on the conformal boundary $\\mathbb {S}^n$ : Lemma 5.6 Let $\\mathcal {Q}_-$ be the operator defined in (REF ); recall that it is injective.", "If $\\pi _\\Gamma :S\\mathbb {H}^{n+1}\\rightarrow SM$ is the natural projection map, then $\\pi _\\Gamma ^\\operatorname{Res}_{\\mathcal {X}}^{m,0}(\\lambda )=\\Phi _-^{\\lambda }\\mathcal {Q}_-(\\operatorname{Bd}^{m,0}(\\lambda )),$ where $\\operatorname{Bd}^{m,0}(\\lambda )\\subset \\mathcal {D}^{\\prime }(\\mathbb {S}^n;\\otimes _S^m(T^\\mathbb {S}^n))$ consists of all distributions $w$ such that $\\mathcal {T}(w)=0$ and $L_\\gamma ^* w(\\nu )=N_{\\gamma }(\\nu )^{-\\lambda -m}w(\\nu ),\\quad \\nu \\in \\mathbb {S}^n,\\ \\gamma \\in \\Gamma ,$ where $L_\\gamma ,N_\\gamma $ are defined in (REF ).", "Similarly $\\pi _\\Gamma ^\\operatorname{Res}_{\\mathcal {X}^}^{m,0}(\\lambda )=\\Phi _+^{\\bar{\\lambda }}\\mathcal {Q}_+(\\operatorname{Bd}^{m,0}(\\bar{\\lambda })),\\quad \\operatorname{Bd}^{m,0}(\\bar{\\lambda })=\\overline{\\operatorname{Bd}^{m,0}(\\lambda )}.$ Assume first that $w\\in \\operatorname{Bd}^{m,0}(\\lambda )$ and put $\\tilde{v}=\\Phi _-^{\\lambda }\\mathcal {Q}_-(w)$ .", "Then by Lemma REF and (REF ), $\\tilde{v}$ is invariant under $\\Gamma $ and thus descends to a distribution $v\\in \\mathcal {D}^{\\prime }(SM;\\otimes _S^m\\mathcal {E}^)$ .", "Since $X\\Phi _-^{\\lambda }=-\\lambda \\Phi _-^{\\lambda }$ and $U_j^-(\\Phi _-^{\\lambda }\\circ \\pi _S)=0$ by (REF ) and (REF ), and $\\mathcal {X}$ and $\\mathcal {U}_-$ annihilate the image of $\\mathcal {Q}_-$ by Lemma REF , we have $(\\mathcal {X}+\\lambda ) v=0$ and $\\mathcal {U}_- v=0$ .", "Moreover, by [33] the wavefront set of $\\tilde{v}$ is contained in the conormal bundle to the fibers of the map $B_-$ ; by (REF ), we see that $\\operatorname{WF}(v)\\subset E_u^$ .", "Finally, $\\mathcal {T}(v)=0$ since the map $\\mathcal {A}_-(x,\\xi )$ used in the definition of $\\mathcal {Q}_-$ is an isometry.", "Therefore, $v\\in \\operatorname{Res}_{\\mathcal {X}}^{m,0}(\\lambda )$ and we proved the containment $\\pi _\\Gamma ^\\operatorname{Res}_{\\mathcal {X}}^{m,0}(\\lambda )\\supset \\Phi _-^{\\lambda }\\mathcal {Q}_-(\\operatorname{Bd}^{m,0}(\\lambda ))$ .", "The opposite containment is proved by reversing this argument.", "Remark.", "It follows from the proof of Lemma REF that the condition $\\operatorname{WF}(v)\\subset E_u^$ in (REF ) is unnecessary.", "This could also be seen by applying [34] to the equations $(\\mathcal {X}+\\lambda )v=0$ , $\\mathcal {U}_-v=0$ , since $\\mathcal {X}$ differentiates along the direction $E_0$ , $\\mathcal {U}_-$ differentiates along the direction $E_u$ (see (REF ) and (REF )), and the annihilator of $E_0\\oplus E_u$ (that is, the joint critical set of $\\mathcal {X}+\\lambda ,\\mathcal {U}_-$ ) is exactly $E_u^$ .", "It now remains to relate the space $\\operatorname{Bd}^{m,0}(\\lambda )$ to an eigenspace of the Laplacian on symmetric tensors.", "For that, we introduce the following operator obtained by integrating the corresponding elements of $\\operatorname{Res}^{m,0}_{\\mathcal {X}}(\\lambda )$ along the fibers of $\\mathbb {S}^n$ : Definition 5.7 Take $\\lambda \\in \\mathbb {C}$ .", "The Poisson operators $P^\\pm _\\lambda :\\mathcal {D}^{\\prime }(\\mathbb {S}^n; \\otimes ^m T^\\mathbb {S}^n)\\rightarrow \\mathcal {C}^\\infty (\\mathbb {H}^{n+1}; \\otimes ^m T^\\mathbb {H}^{n+1})$ are defined by the formulas ${\\begin{array}{c}P^-_\\lambda w(x)=\\int _{S_x\\mathbb {H}^{n+1}}\\Phi _-(x,\\xi )^{\\lambda }\\mathcal {Q}_-(w)(x,\\xi )\\,dS(\\xi ),\\\\P^+_\\lambda w(x)=\\int _{S_x\\mathbb {H}^{n+1}}\\Phi _+(x,\\xi )^{\\bar{\\lambda }}\\mathcal {Q}_+(w)(x,\\xi )\\,dS(\\xi ).\\end{array}}$ Here integration of elements of $\\otimes ^m\\mathcal {E}^(x,\\xi )$ is performed by embedding them in $\\otimes ^m T^_x \\mathbb {H}^{n+1}$ using composition with the orthogonal projection $T_x\\mathbb {H}^{n+1}\\rightarrow \\mathcal {E}(x,\\xi )$ .", "The operators $P^\\pm _\\lambda $ are related by the identity $\\overline{P^\\pm _\\lambda w}=P^\\mp _\\lambda \\overline{w}.$ By Lemma REF , $P^-_\\lambda $ maps $\\operatorname{Bd}^{m,0}(\\lambda )$ onto symmetric $\\Gamma $ -equivariant tensors, which can thus be considered as elements of $\\mathcal {C}^\\infty (M;\\otimes _S^m T^M)$ .", "The relation with the Laplacian is given by the following fact, proved in Section REF : Lemma 5.8 For each $\\lambda $ , the image of $\\operatorname{Bd}^{m,0}(\\lambda )$ under $P^-_\\lambda $ is contained in the eigenspace $\\operatorname{Eig}^m(-\\lambda (n+\\lambda )+m)$ , where $\\operatorname{Eig}^m(\\sigma ):=\\lbrace f\\in \\mathcal {C}^\\infty (M;\\otimes _S^m T^M)\\mid \\Delta f= \\sigma f,\\ \\nabla ^ f=0,\\ \\mathcal {T}(f)=0\\rbrace .$ Here the trace $\\mathcal {T}$ was defined in Section REF and the Laplacian $\\Delta $ and the divergence $\\nabla ^$ are introduced in Section REF .", "(A similar result for $P^+_\\lambda $ follows from (REF ).)", "Furthermore, in Sections REF and we show the following crucial Theorem 6 Assume that $\\lambda \\notin \\mathcal {R}_m$ , where $\\mathcal {R}_m=\\left\\lbrace \\begin{array}{ll}-\\tfrac{n}{2}-\\tfrac{1}{2}\\mathbb {N}_0 & \\textrm { if }n>1 \\textrm { or }m=0\\\\-\\tfrac{1}{2}\\mathbb {N}_0 & \\textrm { if }n=1 \\textrm { and }m>0~\\end{array}\\right.$ Then the map $P^-_\\lambda :\\operatorname{Bd}^{m,0}(\\lambda )\\rightarrow \\operatorname{Eig}^m(-\\lambda (n+\\lambda )+m)$ is an isomorphism.", "Remark.", "In Theorem REF , the set of exceptional points where we do not show isomorphism is not optimal but sufficient for our application (we only need $\\mathcal {R}_m\\subset m-{n\\over 2}-{1\\over 2}\\mathbb {N}_0$ ); we expect the exceptional set to be contained in $-n+1-\\mathbb {N}_0$ .", "This result is known for functions, that is for $m=0$ , with the exceptional set being $-n-\\mathbb {N}$ .", "This was proved by Helgason, Minemura in the case of hyperfunctions on $\\mathbb {S}^n$ and by Oshima–Sekiguchi [48] and Schlichtkrull–Van Den Ban [4] for distributions; Grellier–Otal [27] studied the sharp functional spaces on $\\mathbb {S}^n$ of the boundary values of bounded eigenfunctions on $\\mathbb {H}^{n+1}$ .", "The extension to $m>0$ does not seem to be known in the literature and is not trivial, it takes most of Sections and .", "We finally provide the following refinement of Lemma REF , needed to handle the case $\\lambda \\in (-n/2,\\infty )\\cap {1\\over 2}\\mathbb {Z}$ : Lemma 5.9 Assume that $\\lambda \\in -{n\\over 2}+{1\\over 2}\\mathbb {N}$ .", "If $\\lambda \\in - 2\\mathbb {N}$ , then $\\dim \\operatorname{Res}_X(\\lambda )=\\sum _{m\\ge 0\\atop m\\ne -\\lambda }\\dim \\operatorname{Res}^{m}_{\\mathcal {X}}(\\lambda +m).$ If $\\lambda \\notin -2\\mathbb {N}$ , then (REF ) holds.", "We use the proof of Lemma REF .", "We first show that for $m$ odd or $\\lambda \\ne -m$ , $\\mathcal {U}_-^m(V_{m+1}(\\lambda ))=\\operatorname{Res}^{m}_{\\mathcal {X}}(\\lambda +m).$ Using (REF ), it suffices to prove that for $0\\le \\ell \\le {m\\over 2}$ , the space $\\mathcal {I}^\\ell (\\operatorname{Res}^{m-2\\ell ,0}_{\\mathcal {X}}(\\lambda +m))$ is contained in $\\mathcal {U}_-^m(V_{m+1}(\\lambda ))$ .", "This follows from the proof of Lemma REF as long as ${\\begin{array}{c}\\lambda +m\\notin \\mathbb {Z}\\cap \\big ( [2\\ell +2-n-m,1-n]\\cup [m-2\\ell ,m-\\ell -1]\\big ),\\\\\\lambda +m+{n\\over 2}\\notin \\mathbb {Z}\\cap [1,\\ell ];\\end{array}}$ using that $\\lambda >-{n\\over 2}$ , it suffices to prove that $\\lambda \\notin \\mathbb {Z}\\cap [-2\\ell , -\\ell -1].$ On the other hand by Lemma REF , Theorem REF , and Lemma REF , if $\\ell < {m\\over 2}$ and the space $\\operatorname{Res}^{m-2\\ell ,0}_{\\mathcal {X}}(\\lambda +m)$ is nontrivial, then $-\\Big (\\lambda +m+{n\\over 2}\\Big )^2+{n^2\\over 4}+m-2\\ell \\ge m-2\\ell +n-1,$ implying $\\Big \| \\lambda +m+{n\\over 2}\\Big \|\\le \\Big \|{n\\over 2}-1\\Big \|$ and (REF ) follows.", "For the case $\\ell ={m\\over 2}$ , since $\\Delta \\ge 0$ on functions, we have $-\\Big (\\lambda +m+{n\\over 2}\\Big )^2+{n^2\\over 4}\\ge 0,$ which implies that $\\lambda \\le -m$ and thus (REF ) holds unless $\\lambda =-m$ .", "It remains to consider the case when $m=2\\ell $ is even and $\\lambda =-m$ .", "We have $\\operatorname{Res}^m_{\\mathcal {X}}(0)=\\mathcal {I}^{\\ell }(\\operatorname{Res}^{0,0}_{\\mathcal {X}}(0));$ that is, $\\operatorname{Res}^{m-2\\ell ^{\\prime },0}_{\\mathcal {X}}(0)$ is trivial for $\\ell ^{\\prime }<{m\\over 2}$ .", "For $n>1$ , this follows immediately from (REF ), and for $n=1$ , since the bundle $\\mathcal {E}^$ is one-dimensional we get $\\operatorname{Res}^{m^{\\prime },0}_{\\mathcal {X}}(\\lambda )=0$ for $m^{\\prime }\\ge 2$ .", "Now, $\\operatorname{Res}^{0,0}_{\\mathcal {X}}(0)=\\operatorname{Res}^0_{\\mathcal {X}}(0)$ corresponds via Lemma REF and Theorem REF to the kernel of the scalar Laplacian, that is, to the space of constant functions.", "Therefore, $\\operatorname{Res}^{0,0}_{\\mathcal {X}}$ is one-dimensional and it is spanned by the constant function 1 on $SM$ ; it follows that $\\operatorname{Res}^m_{\\mathcal {X}}(0)$ is spanned by $\\mathcal {I}^{\\ell }(1)$ .", "However, by Lemma REF , for each $u\\in \\mathcal {D}^{\\prime }(SM)$ , $\\langle \\mathcal {I}^{\\ell }(1),\\mathcal {U}_-^m u\\rangle _{L^2}=(-1)^m\\langle \\mathcal {V}_-^m\\mathcal {I}^{\\ell }(1),u\\rangle _{L^2}=0.$ Since $\\mathcal {U}_-^m(V_{m+1}(\\lambda ))\\subset \\operatorname{Res}^m_{\\mathcal {X}}(0)$ , we have $\\mathcal {U}_-^m=0$ on $V_{m+1}(\\lambda )$ , which implies that $V_{m+1}(\\lambda )=V_m(\\lambda )$ , finishing the proof.", "To prove Theorem REF , it now suffices to combine Lemmas REF –REF with Theorem REF ." ], [ "Resonance pairing and algebraic multiplicity", "In this section, we prove Theorem REF .", "The key component is a pairing formula which states that the inner product between a resonant and a coresonant state, defined in (REF ), is determined by the inner product between the corresponding eigenstates of the Laplacian.", "The nondegeneracy of the resulting inner product as a bilinear operator on $\\operatorname{Res}_X(\\lambda )\\times \\operatorname{Res}_{X^}(\\lambda )$ for $\\lambda \\notin {1\\over 2}\\mathbb {Z}$ immediately implies the fact that the algebraic and geometric multiplicities of $\\lambda $ coincide (that is, $X+\\lambda $ does not have any nontrivial Jordan cells).", "To state the pairing formula, we first need a decomposition of the space $\\operatorname{Res}_X(\\lambda )$ , which is an effective version of the formulas (REF ) and (REF ).", "Take $m\\ge 0$ , $\\ell \\le \\lfloor m/2\\rfloor $ , $w\\in \\operatorname{Bd}^{m-2\\ell ,0}(\\lambda )$ .", "Let $\\mathcal {I}$ be the operator defined in Section REF .", "Then (REF ) and Lemma REF show that $\\operatorname{Res}_{\\mathcal {X}}^m(\\lambda )=\\bigoplus _{\\ell =0}^{\\lfloor m/2\\rfloor }\\mathcal {I}^\\ell (\\operatorname{Res}_{\\mathcal {X}}^{m-2\\ell ,0}(\\lambda ))=\\bigoplus _{\\ell =0}^{\\lfloor m/2\\rfloor } \\mathcal {I}^\\ell (\\Phi _-^{\\lambda }\\mathcal {Q}_-(\\operatorname{Bd}^{m-2\\ell , 0}(\\lambda ))).$ Next, let $\\mathcal {V}_\\pm ^m:\\mathcal {D}^{\\prime }(SM;\\otimes ^m_S \\mathcal {E}^)\\rightarrow \\mathcal {D}^{\\prime }(SM),\\quad \\Delta _\\pm :\\mathcal {D}^{\\prime }(SM)\\rightarrow \\mathcal {D}^{\\prime }(SM)$ be the operators introduced in Section REF .", "Then the proofs of Lemma REF and Lemma REF show that for $\\lambda \\notin {1\\over 2}\\mathbb {Z}$ , ${\\begin{array}{c}\\operatorname{Res}_X(\\lambda )=\\bigoplus _{m\\ge 0}\\bigoplus _{\\ell =0}^{\\lfloor m/2\\rfloor } V_{m\\ell }(\\lambda ),\\quad \\operatorname{Res}_{X^}(\\lambda )=\\bigoplus _{m\\ge 0}\\bigoplus _{\\ell =0}^{\\lfloor m/2\\rfloor }V_{m\\ell }^(\\lambda );\\\\V_{m\\ell }(\\lambda ):=\\Delta _+^\\ell \\mathcal {V}_+^{m-2\\ell }(\\Phi _-^{\\lambda +m}\\mathcal {Q}_-(\\operatorname{Bd}^{m-2\\ell , 0}(\\lambda +m))),\\\\V_{m\\ell }^(\\lambda ):=\\Delta _-^\\ell \\mathcal {V}_-^{m-2\\ell }(\\Phi _+^{\\bar{\\lambda }+m}\\mathcal {Q}_+(\\overline{\\operatorname{Bd}^{m-2\\ell , 0}(\\lambda +m)})),\\end{array}}$ and the operators in the definitions of $V_{m\\ell }(\\lambda ),V^_{m\\ell }(\\lambda )$ are one-to-one on the corresponding spaces.", "By the proof of Lemma REF , the decomposition (REF ) is also valid for $\\lambda \\in (-n/2,\\infty )\\setminus (-2\\mathbb {N})$ ; for $\\lambda \\in (-n/2,\\infty )\\cap (-2\\mathbb {N})$ , we have ${\\begin{array}{c}\\operatorname{Res}_X(\\lambda )=\\bigoplus _{m\\ge 0\\atop m\\ne -\\lambda }\\bigoplus _{\\ell =0}^{\\lfloor m/2\\rfloor } V_{m\\ell }(\\lambda ),\\quad \\operatorname{Res}_{X^}(\\lambda )=\\bigoplus _{m\\ge 0\\atop m\\ne -\\lambda }\\bigoplus _{\\ell =0}^{\\lfloor m/2\\rfloor }V_{m\\ell }^(\\lambda ).\\end{array}}$ We can now state the pairing formula: Lemma 5.10 Let $\\lambda \\notin -{n\\over 2}-{1\\over 2}\\mathbb {N}_0$ and $u\\in \\operatorname{Res}_X(\\lambda )$ , $u^\\in \\operatorname{Res}_{X^}(\\lambda )$ .", "Let $\\langle u,u^\\rangle _{L^2(SM)}$ be defined by (REF ).", "Then: 1.", "If $u\\in V_{m\\ell }(\\lambda ), u^\\in V^_{m^{\\prime }\\ell ^{\\prime }}(\\lambda )$ , and $(m,\\ell )\\ne (m^{\\prime },\\ell ^{\\prime })$ , then $\\langle u,u^\\rangle _{L^2(SM)}=0$ .", "2.", "If $u\\in V_{m\\ell }(\\lambda )$ , $u^\\in V^_{m\\ell }(\\lambda )$ and $w\\in \\operatorname{Bd}^{m-2\\ell ,0}(\\lambda +m)$ , $w^\\in \\overline{\\operatorname{Bd}^{m-2\\ell ,0}(\\lambda +m)}$ are the elements generating $u,u^$ according to (REF ), then $\\langle u,u^\\rangle _{L^2(SM)}= c_{m\\ell }(\\lambda )\\langle P^-_{\\lambda +m}(w), P^+_{\\lambda +m}(w^)\\rangle _{L^2(M)},$ where ${\\begin{array}{c}c_{m\\ell }(\\lambda )=2^{m+2\\ell -n}\\pi ^{-1-{n\\over 2}}\\ell !", "(m-2\\ell )!\\sin \\Big (\\pi \\Big ({n\\over 2}+\\lambda \\Big )\\Big )\\\\\\cdot {\\Gamma (m+{n\\over 2}-\\ell )\\Gamma (\\lambda +n+2m-2\\ell )\\Gamma (-\\lambda -\\ell )\\Gamma (-\\lambda -m-{n\\over 2}+\\ell +1)\\over \\Gamma (m+{n\\over 2}-2\\ell )\\Gamma (-\\lambda -2\\ell )}.\\end{array}}$ and under the conditions (i) either $\\lambda \\notin -2\\mathbb {N}$ or $m\\ne -\\lambda $ and (ii) $V_{m\\ell }(\\lambda )$ is nontrivial, we have $c_{m\\ell }(\\lambda )\\ne 0$ .", "Remarks.", "(i) The proofs below are rather technical, and it is suggested that the reader start with the case of resonances in the first band, $m=\\ell =0$ , which preserves the essential analytic difficulties of the proof but considerably reduces the amount of calculations needed (in particular, one can go immediately to Lemma REF , and the proof of this lemma for the case $m=\\ell =0$ does not involve the operator $C_\\eta $ ).", "We have $c_{00}(\\lambda )=(4\\pi )^{-n/2}{\\Gamma (n+\\lambda )\\over \\Gamma ({n\\over 2}+\\lambda )}.$ (ii) In the special case of $n=1,m=\\ell =0$ , Lemma REF is a corollary of [2], where the product $uu^\\in \\mathcal {D}^{\\prime }(SM)$ lifts to a Patterson–Sullivan distribution on $S\\mathbb {H}^2$ .", "In general, if $\|\\operatorname{Re}\\lambda \|\\le C$ and $\\operatorname{Im}\\lambda \\rightarrow \\infty $ , then $c_{m\\ell }(\\lambda )$ grows like $\|\\lambda \|^{{n\\over 2}+m}$ .", "Lemma REF immediately gives By Theorem REF , we know that $P^-_\\lambda :\\operatorname{Bd}^{m-2\\ell ,0}(\\lambda +m)\\rightarrow \\operatorname{Eig}^{m-2\\ell }(-(\\lambda +m+n/2)^2+n^2/4+m-2\\ell )$ is an isomorphism.", "Given (REF ), we also get the isomorphism $P^+_\\lambda :\\overline{\\operatorname{Bd}^{m-2\\ell ,0}(\\lambda +m)}\\rightarrow \\operatorname{Eig}^{m-2\\ell }(-(\\lambda +m+n/2)^2+n^2/4+m-2\\ell ).$ Here we used that the target space is invariant under complex conjugation.", "By Lemma REF , the bilinear product $\\operatorname{Res}_X(\\lambda )\\times \\operatorname{Res}_{X^}(\\lambda )\\rightarrow \\mathbb {C},\\quad (u,u^)\\mapsto \\langle u,u^\\rangle _{L^2(SM)}$ is nondenegerate, since the $L^2(M)$ inner product restricted to $\\operatorname{Eig}^{m-2\\ell }(-(\\lambda +m+n/2)^2+n^2/4+m-2\\ell )$ is nondegenerate for all $m,\\ell $ .", "Assume now that $\\tilde{u}\\in \\mathcal {D}^{\\prime }(SM)$ satisfies $(X+\\lambda )^2\\tilde{u}=0$ and $\\tilde{u}\\in \\mathcal {H}^r$ for some $r$ , $\\operatorname{Re}\\lambda >-r/C_0$ ; we need to show that $(X+\\lambda )\\tilde{u}=0$ .", "Put $u:=(X+\\lambda )\\tilde{u}$ .", "Then $u\\in \\operatorname{Res}_X(\\lambda )$ .", "However, $u$ also lies in the image of $X+\\lambda $ on $\\mathcal {H}^r$ , therefore we have $\\langle u,u^\\rangle =0$ for each $u^\\in \\operatorname{Res}_{X^}(\\lambda )$ .", "Since the product (REF ) is nondegenerate, we see that $u=0$ , finishing the proof.", "In the remaining part of this section, we prove Lemma REF .", "Take some $m,m^{\\prime },\\ell ,\\ell ^{\\prime }\\ge 0$ such that $2\\ell \\le m$ , $2\\ell ^{\\prime }\\le m^{\\prime }$ , and consider $u\\in V_{m\\ell }(\\lambda )$ , $u^\\in V_{m^{\\prime }\\ell ^{\\prime }}^(\\lambda )$ given by $u=\\Delta _+^\\ell \\mathcal {V}_+^{m-2\\ell } v,\\quad u^=\\Delta _-^{\\ell ^{\\prime }}\\mathcal {V}_-^{m^{\\prime }-2\\ell ^{\\prime }} v^,$ where for some $w\\in \\operatorname{Bd}^{m-2\\ell ,0}(\\lambda +m)$ and $w^\\in \\overline{\\operatorname{Bd}^{m^{\\prime }-2\\ell ^{\\prime },0}(\\lambda +m^{\\prime })}$ , $v=\\Phi _-^{\\lambda +m}\\mathcal {Q}_-(w)\\in \\mathcal {\\operatorname{Res}}^{m-2\\ell ,0}_{\\mathcal {X}}(\\lambda +m),\\quad v^=\\Phi _+^{\\bar{\\lambda }+m^{\\prime }}\\mathcal {Q}_+(w^)\\in \\operatorname{Res}^{m^{\\prime }-2\\ell ^{\\prime },0}_{\\mathcal {X}^}(\\lambda +m^{\\prime }).$ Using Lemma REF and the fact that $\\Delta _\\pm $ are symmetric, we get $\\langle u,u^\\rangle _{L^2(SM)}=(-1)^{m^{\\prime }}\\langle \\mathcal {U}_-^{m^{\\prime }-2\\ell ^{\\prime }}\\Delta _-^{\\ell ^{\\prime }}\\Delta _+^\\ell \\mathcal {V}_+^{m-2\\ell } v,v^\\rangle _{L^2(SM;\\otimes ^{m^{\\prime }-2\\ell ^{\\prime }}\\mathcal {E}^)}.$ By Lemmas REF and REF , we have $\\mathcal {U}_-^{m+1}\\Delta _+^\\ell \\mathcal {V}_+^{m-2\\ell } v=0$ .", "Therefore, if $m^{\\prime }>m$ , we derive that $\\langle u,u^\\rangle _{L^2(SM)}=0$ ; by swapping $u$ and $u^$ , one can similarly handle the case $m^{\\prime }<m$ .", "We therefore assume that $m=m^{\\prime }$ .", "Then by Lemmas REF and REF (see the proof of Lemma REF ), ${\\begin{array}{c}(-1)^{\\ell +\\ell ^{\\prime }}\\mathcal {U}_-^{m-2\\ell ^{\\prime }}\\Delta _-^{\\ell ^{\\prime }}\\Delta _+^\\ell \\mathcal {V}_+^{m-2\\ell } v=\\mathcal {T}^{\\ell ^{\\prime }}\\mathcal {U}_-^m(-\\Delta _+)^\\ell \\mathcal {V}_+^{m-2\\ell } v\\\\=2^{m+\\ell }(m-2\\ell )!", "{\\Gamma (\\lambda +n+2m-2\\ell -1)\\Gamma (-\\lambda -\\ell )\\Gamma (-\\lambda -m-{n\\over 2}+\\ell +1)\\over \\Gamma (\\lambda +m+n-1)\\Gamma (-\\lambda -2\\ell )\\Gamma (-\\lambda -m-{n\\over 2}+1)}\\mathcal {T}^{\\ell ^{\\prime }}\\mathcal {I}^{\\ell }v.\\end{array}}$ If $\\ell ^{\\prime }>\\ell $ , this implies that $\\langle u,u^\\rangle _{L^2(SM)}=0$ , and the case $\\ell ^{\\prime }<\\ell $ is handled similarly.", "(Recall that $\\mathcal {T}(v)=0$ .)", "We therefore assume that $m=m^{\\prime },\\ell =\\ell ^{\\prime }$ .", "In this case, by (REF ), $\\mathcal {T}^{\\ell }\\mathcal {I}^{\\ell }v=2^\\ell \\ell !", "{\\Gamma (m+{n\\over 2}-\\ell )\\over \\Gamma (m+{n\\over 2}-2\\ell )}v,$ which implies that ${\\begin{array}{c}\\langle u,u^\\rangle _{L^2(SM)}=(-2)^{m+2\\ell }\\ell !", "(m-2\\ell )!", "{\\Gamma (m+{n\\over 2}-\\ell )\\Gamma (\\lambda +n+2m-2\\ell -1)\\over \\Gamma (m+{n\\over 2}-2\\ell )\\Gamma (\\lambda +n+m-1)}\\\\{\\Gamma (-\\lambda -\\ell )\\Gamma (-\\lambda -m-{n\\over 2}+\\ell +1)\\over \\Gamma (-\\lambda -2\\ell )\\Gamma (-\\lambda -m-{n\\over 2}+1)}\\langle v,v^\\rangle _{L^2(SM;\\otimes ^{m-2\\ell }\\mathcal {E}^)}.\\end{array}}$ Note that under assumptions (i) and (ii) of Lemma REF , the coefficient in the formula above is nonzero, see the proof of Lemma REF .", "It then remains to prove the following identity (note that the coefficient there is nonzero for $\\lambda \\notin \\mathbb {Z}$ or $\\operatorname{Re}\\lambda >m-{n\\over 2}$ ): Lemma 5.11 Assume that $v\\in \\operatorname{Res}_{\\mathcal {X}}^{m,0}(\\lambda )$ and $v^\\in \\operatorname{Res}_{\\mathcal {X}^}^{m,0}(\\lambda )$ .", "Define $f(x):=\\int _{S_xM} v(x,\\xi )\\,dS(\\xi ),\\quad f^(x):=\\int _{S_xM} v^(x,\\xi )\\,dS(\\xi ),$ where integration of tensors is understood as in Definition REF .", "If $\\lambda \\notin -({n\\over 2}+\\mathbb {N}_0)$ , then $\\langle f,f^\\rangle _{L^2(M;\\otimes ^m T^M)}=2^n\\pi ^{n\\over 2}{\\Gamma ({n\\over 2}+\\lambda )\\over (n+\\lambda +m-1)\\Gamma (n-1+\\lambda )}\\langle v,v^\\rangle _{L^2(SM;\\otimes ^m\\mathcal {E}^)}.$ We write $\\langle f,f^\\rangle _{L^2(M;\\otimes ^m T^M)}=\\int _{S^2 M} \\langle v(y,\\eta _-),\\overline{v^(y,\\eta _+)}\\rangle _{\\otimes ^m T^_y M}\\,dyd\\eta _-d\\eta _+$ where the bundle $S^2M$ is given by $S^2M=\\lbrace (y,\\eta _-,\\eta _+)\\mid y\\in M,\\ \\eta _\\pm \\in S_yM\\rbrace .$ Define also $S^2_\\Delta M=\\lbrace (y,\\eta _-,\\eta _+)\\in S^2M\\mid \\eta _-+\\eta _+\\ne 0\\rbrace .$ On the other hand $\\langle v,v^\\rangle _{L^2(SM;\\otimes ^m\\mathcal {E}^)}=\\int _{SM} \\langle v(x,\\xi ),\\overline{v^(x,\\xi )}\\rangle _{\\otimes ^m\\mathcal {E}^(x,\\xi )}\\,dxd\\xi .$ The main idea of the proof is to reduce (REF ) to (REF ) by applying the coarea formula to a correctly chosen map $S^2_\\Delta M\\rightarrow SM$ .", "More precisely, consider the following map $\\Psi :\\mathcal {E}\\rightarrow S^2_{\\Delta }\\mathbb {H}^{n+1}$ : for $(x,\\xi )\\in S\\mathbb {H}^{n+1}$ and $\\eta \\in \\mathcal {E}(x,\\xi )$ , define $\\Psi (x,\\xi ,\\eta ):=(y,\\eta _-,\\eta _+)$ , with $\\begin{pmatrix} y\\\\\\eta _-\\\\\\eta _+ \\end{pmatrix}=A\\big (\|\\eta \|^2\\big )\\begin{pmatrix} x\\\\\\xi \\\\\\eta \\end{pmatrix},\\quad A(s)=\\begin{pmatrix}\\sqrt{s+1}&0&1\\\\{s\\over \\sqrt{s+1}}&{1\\over \\sqrt{s+1}}&1\\\\-{s\\over \\sqrt{s+1}}&{1\\over \\sqrt{s+1}}&-1\\end{pmatrix}.$ Figure: (a) The map Ψ:(x,ξ,η)↦(y,η - ,η + )\\Psi :(x,\\xi ,\\eta )\\mapsto (y,\\eta _-,\\eta _+).", "(b) The vectors 𝒜 ± (x,ξ)ζ ± \\mathcal {A}_\\pm (x,\\xi )\\zeta _\\pm (equal in the case drawn)and 𝒜 ± (y,η ± )ζ ± \\mathcal {A}_\\pm (y,\\eta _\\pm )\\zeta _\\pm .Note that, with $\|\\eta \|$ denoting the Riemannian length of $\\eta $ (that is, $\|\\eta \|^2=-\\langle \\eta ,\\eta \\rangle _M$ ), $\\Phi _\\pm (y,\\eta _\\pm )={\\Phi _\\pm (x,\\xi )\\over \\sqrt{1+\|\\eta \|^2}},\\quad B_\\pm (y,\\eta _\\pm )=B_\\pm (x,\\xi ),\\quad \|\\eta _++\\eta _-\|={2\\over \\sqrt{1+\|\\eta \|^2}}.$ Also, $\\det A(s)=-{2\\over s+1},\\quad A(s)^{-1}=\\begin{pmatrix}\\sqrt{s+1}&-{\\sqrt{s+1}\\over 2}&{\\sqrt{s+1}\\over 2}\\\\0&{\\sqrt{s+1}\\over 2}&{\\sqrt{s+1}\\over 2}\\\\-s&{s+1\\over 2}&-{s+1\\over 2}\\end{pmatrix}.$ The map $\\Psi $ is a diffeomorphism; the inverse is given by the formulas $x={2y+\\eta _+-\\eta _-\\over \|\\eta _++\\eta _-\|},\\quad \\xi ={\\eta _++\\eta _-\\over \|\\eta _++\\eta _-\|},\\quad \\eta ={2(\\eta _--\\eta _+)-\|\\eta _+-\\eta _-\|^2 y\\over \|\\eta _++\\eta _-\|^2}.$ The map $\\Psi ^{-1}$ can be visualized as follows (see Figure REF (a)): given $(y,\\eta _-,\\eta _+)$ , the corresponding tangent vector $(x,\\xi )$ is the closest to $y$ point on the geodesic going from $\\nu _-=B_-(y,\\eta _-)$ to $\\nu _+=B_+(y,\\eta _+)$ and the vector $\\eta $ measures both the distance between $x$ and $y$ and the direction of the geodesic from $x$ to $y$ .", "The exceptional set $\\lbrace \\eta _++\\eta _-=0\\rbrace $ corresponds to $\|\\eta \|=\\infty $ .", "A calculation using (REF ) shows that for $\\zeta _\\pm \\in T_{B_\\pm (x,\\xi )} \\mathbb {S}^n$ , $\\mathcal {A}_\\pm (y,\\eta _\\pm )\\zeta _\\pm =\\mathcal {A}_\\pm (x,\\xi )\\zeta _\\pm +{(\\mathcal {A}_\\pm (x,\\xi )\\zeta _\\pm )\\cdot \\eta \\over \\sqrt{1+\|\\eta \|^2}}(x\\pm \\xi ).$ Here $\\cdot $ stands for the Riemannian inner product on $\\mathcal {E}$ which is equal to $-\\langle \\cdot ,\\cdot \\rangle _M$ restricted to $\\mathcal {E}$ .", "Then (see Figure REF (b)) ${\\begin{array}{c}(\\mathcal {A}_+(y,\\eta _+)\\zeta _+)\\cdot (\\mathcal {A}_-(y,\\eta _-)\\zeta _-)=(\\mathcal {A}_+(x,\\xi )\\zeta _+)\\cdot (\\mathcal {A}_-(x,\\xi )\\zeta _-)\\\\-{2\\over 1+\|\\eta \|^2} \\big ((\\mathcal {A}_+(x,\\xi )\\zeta _+)\\cdot \\eta \\big )\\big ((\\mathcal {A}_-(x,\\xi )\\zeta _-)\\cdot \\eta \\big )\\\\=\\big (C_\\eta (\\mathcal {A}_+(x,\\xi )\\zeta _+)\\big )\\cdot (\\mathcal {A}_-(x,\\xi )\\zeta _-),\\end{array}}$ where $C_\\eta :\\mathcal {E}(x,\\xi )\\rightarrow \\mathcal {E}(x,\\xi )$ is given by $C_\\eta (\\tilde{\\eta })=\\tilde{\\eta }-{2\\over 1+\|\\eta \|^2}(\\tilde{\\eta }\\cdot \\eta )\\eta .$ We can similarly define $C_\\eta ^:\\mathcal {E}(x,\\xi )^\\rightarrow \\mathcal {E}(x,\\xi )^$ .", "Then for $\\zeta _\\pm \\in \\otimes ^m T^_{B_\\pm (x,\\xi )}\\mathbb {S}^n$ , ${\\begin{array}{c}\\langle \\otimes ^m(\\mathcal {A}_+^{-1}(y,\\eta _+)^T)\\zeta _+,\\otimes ^m(\\mathcal {A}_-^{-1}(y,\\eta _-)^T)\\zeta _-\\rangle _{\\otimes ^m T^_y\\mathbb {H}^{n+1}}\\\\=\\langle \\otimes ^mC_\\eta ^\\otimes ^m(\\mathcal {A}_+^{-1}(x,\\xi )^T)\\zeta _+,\\otimes ^m(\\mathcal {A}_-^{-1}(x,\\xi )^T)\\zeta _-\\rangle _{\\otimes ^m \\mathcal {E}^(x,\\xi )}.\\end{array}}$ The Jacobian of $\\Psi $ with respect to naturally arising volume forms on $\\mathcal {E}$ and $S^2_\\Delta \\mathbb {H}^{n+1}$ is given by (see Appendix REF for the proof) $J_\\Psi (x,\\xi ,\\eta )=2^n(1+\|\\eta \|^2)^{-n}.$ Now, $\\Psi $ is equivariant under $G$ , therefore it descends to a diffeomorphism $\\Psi :\\mathcal {E}_M\\rightarrow S^2_\\Delta M,\\quad \\mathcal {E}_M:=\\lbrace (x,\\xi ,\\eta )\\mid (x,\\xi )\\in SM,\\ \\eta \\in \\mathcal {E}(x,\\xi )\\rbrace .$ Using Lemma REF and (REF ), we calculate for $(x,\\xi ,\\eta )\\in \\mathcal {E}_M$ and $(y,\\eta _-,\\eta _+)=\\Psi (x,\\xi ,\\eta )$ , $\\langle v(y,\\eta _-),\\overline{v^(y,\\eta _+)}\\rangle _{\\otimes ^mT_y^M}=(1+\|\\eta \|^2)^{-\\lambda }\\langle \\otimes ^mC_\\eta ^ v(x,\\xi ),\\overline{v^(x,\\xi )}\\rangle _{\\otimes ^m\\mathcal {E}^(x,\\xi )}.$ We would now like to plug this expression into (REF ), make the change of variables from $(y,\\eta _-,\\eta _+)$ to $(x,\\xi ,\\eta )$ , and integrate $\\eta $ out, obtaining a multiple of (REF ).", "However, this is not directly possible because (i) the integral in $\\eta $ typically diverges (ii) since the expression integrated in (REF ) is a distribution, one cannot simply replace $S^2M$ by $S^2_\\Delta M$ in the integral.", "We will instead use the asymptotic behavior of both integrals as one approaches the set $\\lbrace \\eta _++\\eta _-=0\\rbrace $ , and Hadamard regularization in $\\eta $ in the $(x,\\xi ,\\eta )$ variables.", "For that, fix $\\chi \\in \\mathcal {C}^\\infty _0(\\mathbb {R})$ such that $\\chi =1$ near 0, and define for $\\varepsilon >0$ , $\\chi _\\varepsilon (y,\\eta _-,\\eta _+)=\\chi \\big (\\varepsilon \\, \|\\eta (y,\\eta _-,\\eta _+)\|\\big ),$ where $\\eta (y,\\eta _-,\\eta _+)$ is the corresponding component of $\\Psi ^{-1}$ ; in fact, we can write $\\chi _\\varepsilon (y,\\eta _-,\\eta _+)=\\chi \\Big (\\varepsilon \\, {\|\\eta _+-\\eta _-\|\\over \|\\eta _++\\eta _-\|}\\Big ).$ Then $\\chi _\\varepsilon \\in \\mathcal {D}^{\\prime }(S^2M)$ .", "In fact, $\\chi _\\varepsilon $ is supported inside $S^2_\\Delta M$ ; by making the change of variables $(y,\\eta _-,\\eta _+)=\\Psi (x,\\xi ,\\eta )$ and using (REF ) and (REF ), we get ${\\begin{array}{c}\\int _{S^2M} \\chi _\\varepsilon (y,\\eta _-,\\eta _+) \\langle v(y,\\eta _-),\\overline{v^(y,\\eta _+)}\\rangle _{\\otimes ^m T_y^M}\\,dyd\\eta _- d\\eta _+\\\\=2^n\\int _{\\mathcal {E}_M} \\chi (\\varepsilon \|\\eta \|)(1+\|\\eta \|^2)^{-\\lambda -n}\\langle \\otimes ^m C_\\eta ^v(x,\\xi ),\\overline{v^(x,\\xi )}\\rangle _{\\otimes ^m \\mathcal {E}^(x,\\xi )}\\,dxd\\xi d\\eta .\\end{array}}$ By Lemma REF , (REF ) has the asymptotic expansion ${\\begin{array}{c}2^n\\pi ^{n\\over 2}{\\Gamma ({n\\over 2}+\\lambda )\\over (n+\\lambda +m-1)\\Gamma (n-1+\\lambda )}\\langle v,v^\\rangle _{L^2(SM;\\otimes ^m_S\\mathcal {E}^)}\\\\+\\sum _{0\\le j\\le -\\operatorname{Re}\\lambda -{n\\over 2}}c_j\\varepsilon ^{n+2\\lambda +2j}+o(1)\\end{array}}$ for some constants $c_j$ .", "It remains to prove the following asymptotic expansion as $\\varepsilon \\rightarrow 0$ : $\\int _{S^2M} (1-\\chi _\\varepsilon (y,\\eta _-,\\eta _+)) \\langle v(y,\\eta _-),\\overline{v^(y,\\eta _+)}\\rangle _{\\otimes ^m T_y^M}\\,dyd\\eta _- d\\eta _+\\sim \\sum _{j=0}^\\infty c^{\\prime }_j \\varepsilon ^{n+2\\lambda +2j}$ where $c^{\\prime }_j$ are some constants.", "Indeed, $\\langle f,f^\\rangle _{L^2(M;\\otimes ^m T^M)}$ is equal to the sum of (REF ) and (REF ); since (REF ) does not have a constant term, $\\langle f,f^\\rangle $ is equal to the constant term in the expansion (REF ).", "To show (REF ), we use the dilation vector field $\\eta \\cdot \\partial _\\eta $ on $\\mathcal {E}$ , which under $\\Psi $ becomes the following vector field on $S^2_\\Delta M$ extending smoothly to $S^2M$ : $L_{(y,\\eta _-,\\eta _+)}=\\bigg ({\\eta _--\\eta _+\\over 2},{\|\\eta _+-\\eta _-\|^2\\over 4}\\,y-{\\eta _+\\over 2}+{\\eta _-\\cdot \\eta _+\\over 2}\\,\\eta _-,-{\|\\eta _+-\\eta _-\|^2\\over 4}\\,y-{\\eta _-\\over 2}+{\\eta _-\\cdot \\eta _+\\over 2}\\,\\eta _+\\bigg ).$ The vector field $L$ is tangent to the submanifold $\\lbrace \\eta _++\\eta _-=0\\rbrace $ , in fact $L(\|\\eta _+-\\eta _-\|^2)=-L(\|\\eta _++\\eta _-\|^2)={\|\\eta _+-\\eta _-\|^2\\cdot \|\\eta _++\\eta _-\|^2\\over 2}.$ We can then compute (following the identity $L\|\\eta \|=\|\\eta \|$ ) $L\\bigg ({\|\\eta _+-\\eta _-\|\\over \|\\eta _++\\eta _-\|}\\bigg )={\|\\eta _+-\\eta _-\|\\over \|\\eta _++\\eta _-\|}\\quad \\text{on }S^2_\\Delta M.$ Using the $(x,\\xi ,\\eta )$ coordinates and (REF ), we can compute the divergence of $L$ with respect to the standard volume form on $S^2M$ : $\\operatorname{Div}L=n(\\eta _+\\cdot \\eta _-).$ Moreover, $B_\\pm (y,\\eta _\\pm )$ are constant along the trajectories of $L$ , and $L(\\Phi _\\pm (y,\\eta _\\pm ))=-{\|\\eta _+-\\eta _-\|^2\\over 4}\\Phi _\\pm (y,\\eta _\\pm ).$ We also use (REF ) to calculate for $\\zeta _\\pm \\in T_{B_\\pm (y,\\eta _\\pm )}\\mathbb {S}^n$ , ${\\begin{array}{c}L\\big ((\\mathcal {A}_+(y,\\eta _+)\\zeta _+)\\cdot (\\mathcal {A}_-(y,\\eta _-)\\zeta _-)\\big )=\\big ((\\mathcal {A}_+(y,\\eta _+)\\zeta _+)\\cdot \\eta _-\\big )\\big ((\\mathcal {A}_-(y,\\eta _-)\\zeta _-)\\cdot \\eta _+\\big ),\\\\L\\big ((\\mathcal {A}_\\pm (y,\\eta _\\pm )\\zeta _\\pm )\\cdot \\eta _\\mp \\big )=(\\eta _+\\cdot \\eta _-)\\big ((\\mathcal {A}_\\pm (y,\\eta _\\pm )\\zeta _\\pm )\\cdot \\eta _\\mp \\big ).\\end{array}}$ Combining these identities and using Lemma REF , we get ${\\begin{array}{c}\\Big ( L+{\\lambda \\over 2}\|\\eta _+-\\eta _-\|^2\\Big )\\langle v(y,\\eta _-),\\overline{v^(y,\\eta _+)}\\rangle _{\\otimes ^m T_y^M}\\\\=m\\langle \\iota _{\\eta _+} v(y,\\eta _-), \\iota _{\\eta _-}\\overline{v^(y,\\eta _+)}\\rangle _{\\otimes ^{m-1} T_y^M}.\\end{array}}$ Integrating by parts, we find ${\\begin{array}{c}\\varepsilon \\partial _\\varepsilon \\int _{S^2 M} (1-\\chi _\\varepsilon (y,\\eta _-,\\eta _+))\\langle v(y,\\eta _-),\\overline{v^(y,\\eta _+)}\\rangle _{\\otimes ^m T_y^M}\\,dyd\\eta _-d\\eta _+\\\\=\\int _{S^2M} L\\big (1-\\chi _\\varepsilon (y,\\eta _-,\\eta _+)\\big )\\,\\langle v(y,\\eta _-),\\overline{v^(y,\\eta _+)}\\rangle _{\\otimes ^m T_y^M}\\,dyd\\eta _-d\\eta _+\\\\=\\int _{S^2M} \\bigg ({\\lambda \\over 2}\|\\eta _+-\\eta _-\|^2-n(\\eta _+\\cdot \\eta _-)\\bigg )(1-\\chi _\\varepsilon (y,\\eta _-,\\eta _+))\\langle v(y,\\eta _-),\\overline{v^(y,\\eta _+)}\\rangle _{\\otimes ^m T_y^M}\\,dyd\\eta _-d\\eta _+\\\\-m\\int _{S^2M} (1-\\chi _\\varepsilon (y,\\eta _-,\\eta _+))\\langle \\iota _{\\eta _+}v(y,\\eta _-),\\iota _{\\eta _-}\\overline{v^(y,\\eta _+)}\\rangle _{\\otimes ^{m-1}T_y^M}\\,dyd\\eta _-d\\eta _+.\\end{array}}$ Arguing similarly, we see that if for integers $0\\le r\\le m$ , $p\\ge 0$ , we put $I_{r,p}(\\varepsilon ):=\\int _{S^2 M} \|\\eta _-+\\eta _+\|^{2p}(1-\\chi _\\varepsilon (y,\\eta _-,\\eta _+))\\langle \\iota _{\\eta _+}^r v(y,\\eta _-),\\iota _{\\eta _-}^r\\overline{v^(y,\\eta _+)}\\rangle _{\\otimes ^{m-r} T_y^M}\\,dyd\\eta _-d\\eta _+$ then $(\\varepsilon \\partial _\\varepsilon -2\\lambda -n-2(r+p))I_{r,p}(\\varepsilon )$ is a finite linear combination of $I_{r^{\\prime },p^{\\prime }}(\\varepsilon )$ , where $r^{\\prime }\\ge r,p^{\\prime }\\ge p$ , and $(r^{\\prime },p^{\\prime })\\ne (r,p)$ .", "For example, the calculation above shows that $(\\varepsilon \\partial _\\varepsilon -2\\lambda -n)I_{0,0}(\\varepsilon )=-{\\lambda +n\\over 2}I_{0,1}(\\varepsilon )-mI_{1,0}(\\varepsilon ).$ Moreover, if $N$ is fixed and $p$ is large enough depending on $N$ , then $I_{r,p}(\\varepsilon )=\\mathcal {O}(\\varepsilon ^N)$ ; to see this, note that $I_{r,p}(\\varepsilon )$ is bounded by some fixed $\\mathcal {C}^\\infty $ -seminorm of $\|\\eta _-+\\eta _+\|^{2p}(1-\\chi _\\varepsilon (y,\\eta _-,\\eta _+))$ .", "It follows that if $N$ is fixed and $\\widetilde{N}$ is large depending on $N$ , then $\\bigg (\\prod _{j=0}^{\\widetilde{N}}(\\varepsilon \\partial _\\varepsilon -2\\lambda -n-2j)\\bigg )I_{0,0}(\\varepsilon )=\\mathcal {O}(\\varepsilon ^N)$ which implies the existence of the decomposition (REF ) and finishes the proof." ], [ "Properties of the Laplacian", "In this section, we introduce the Laplacian and study its basic properties (Section REF ).", "We then give formulas for the Laplacian on symmetric tensors in the half-plane model (Section REF ) which will be the basis for the analysis of the following sections.", "Using these formulas, we study the Poisson kernel and in particular prove Lemma REF and the injectivity of the Poisson kernel (Section REF )." ], [ "Definition and Bochner identity", "The Levi–Civita connection associated to the hyperbolic metric $g_H$ is the operator $\\nabla : \\mathcal {C}^\\infty (\\mathbb {H}^{n+1},T\\mathbb {H}^{n+1})\\rightarrow \\mathcal {C}^\\infty (\\mathbb {H}^{n+1},T^\\mathbb {H}^{n+1}\\otimes T\\mathbb {H}^{n+1})$ which induces a natural covariant derivative, still denoted $\\nabla $ , on sections of $\\otimes ^mT^\\mathbb {H}^{n+1}$ .", "We can work in the ball model $\\mathbb {B}^{n+1}$ and use the 0-tangent structure (see Section REF ) and nabla can be viewed as a differential operator of order 1 $\\nabla : \\mathcal {C}^\\infty (\\overline{\\mathbb {B}^{n+1}}; \\otimes ^m ({^0}T^\\overline{\\mathbb {B}^{n+1}}))\\rightarrow \\mathcal {C}^\\infty (\\overline{\\mathbb {B}^{n+1}},\\otimes ^{m+1} ({^0}T^\\overline{\\mathbb {B}^{n+1}}))$ and we denote by $\\nabla ^$ its adjoint with respect to the $L^2$ scalar product, $\\nabla ^$ is called the divergence: it is given by $\\nabla ^u=-\\mathcal {T}(\\nabla u)$ where $\\mathcal {T}$ denotes the trace, see Section REF .", "Define the rough Laplacian acting on $\\mathcal {C}^\\infty (\\overline{\\mathbb {B}^{n+1}}; \\otimes ^m ({^0}T^\\overline{\\mathbb {B}^{n+1}}))$ by $\\Delta :=\\nabla ^\\nabla $ and this operator maps symmetric tensors to symmetric tensors.", "It also extends to $\\mathcal {D^{\\prime }}(\\mathbb {B}^{n+1}; \\otimes _S^m ({^0}T^\\overline{\\mathbb {B}^{n+1}}))$ by duality.", "The operator $\\Delta $ commutes with $\\mathcal {T}$ and $\\mathcal {I}$ : $\\Delta \\mathcal {T}(u)=\\mathcal {T}(\\Delta u), \\quad \\Delta \\mathcal {I}(u)=\\mathcal {I}(\\Delta u)$ for all $u\\in \\mathcal {D^{\\prime }}(\\mathbb {B}^{n+1}; \\otimes _S^m ({^0}T^\\overline{\\mathbb {B}^{n+1}}))$ .", "There is another natural operator given by $\\Delta _D=D^D$ if $D: \\mathcal {C}^\\infty (\\overline{\\mathbb {B}^{n+1}}; \\otimes _S^{m} ({^0}T^\\overline{\\mathbb {B}^{n+1}}))\\rightarrow \\mathcal {C}^\\infty (\\overline{\\mathbb {B}^{n+1}}; \\otimes _S^{m+1} ({^0}T^\\overline{\\mathbb {B}^{n+1}}))$ is defined by $D:= \\mathcal {S}\\circ \\nabla $ , where $\\mathcal {S}$ is the symmetrization defined by (REF ), and $D^=\\nabla ^$ is the formal adjoint.", "There is a Bochner–Weitzenböck formula relating $\\Delta $ and $\\Delta _D$ , and using that the curvature is constant, we have on trace-free symmetric tensors of order $m$ by [11] $\\Delta _D=\\frac{1}{m+1}( m\\, DD^+\\Delta +m(m+n-1)).$ In particular, since $\|\\mathcal {S}\\nabla u\|^2\\le \|\\nabla u\|^2$ pointwise by the fact that $\\mathcal {S}$ is an orthogonal projection, we see that for $u$ smooth and compactly supported, $\\Vert Du\\Vert _{L^2}^2\\le \\Vert \\nabla u\\Vert ^2_{L^2}$ and thus for $m\\ge 1$ , $u\\in \\mathcal {C}^\\infty _0(\\mathbb {H}^{n+1};\\otimes _S^m(T^\\mathbb {H}^{n+1}))$ , and $\\mathcal {T}u=0$ , $\\langle \\Delta u,u\\rangle _{L^2} \\ge (m+n-1)\\Vert u\\Vert ^2.$ Since the Bochner identity is local, the same inequality clearly descends to co-compact quotients $\\Gamma \\backslash \\mathbb {H}^{n+1}$ (where $\\Delta $ is self-adjoint and has compact resolvent by standard theory of elliptic operators, as its principal part is given by the scalar Laplacian), and this implies Lemma 6.1 The spectrum of $\\Delta $ acting on trace-free symmetric tensors of order $m\\ge 1$ on hyperbolic compact manifolds of dimension $n+1$ is bounded below by $m+n-1$ .", "We finally define $E^{(m)}:=\\otimes _S^{m} ({^0}T^\\overline{\\mathbb {B}^{n+1}})\\cap \\ker \\mathcal {T}$ to be the bundle of trace-free symmetric $m$ -cotensors over the ball model of hyperbolic space." ], [ "Laplacian in the half-plane model", "We now give concrete formulas concerning the Laplacian on symmetric tensors in the half-space model $\\mathbb {U}^{n+1}$ (see (REF )).", "We fix $\\nu \\in \\mathbb {S}^n$ and map $\\mathbb {B}^{n+1}$ to $\\mathbb {U}^{n+1}$ by a composition of a rotation of $\\mathbb {B}^{n+1}$ and the map (REF ); the rotation is chosen so that $\\nu $ is mapped to $0\\in \\overline{\\mathbb {U}^{n+1}}$ and $-\\nu $ is mapped to infinity.", "The 0-cotangent and tangent bundles ${^0}T^\\overline{\\mathbb {B}^{n+1}}$ and ${^0}T\\overline{\\mathbb {B}^{n+1}}$ pull back to the half-space, we denote them ${^0}T^\\mathbb {U}^{n+1}$ and ${^0}T\\mathbb {U}^{n+1}$ .", "The coordinates on $\\mathbb {U}^{n+1}$ are $(z_0,z)\\in \\mathbb {R}^+\\times \\mathbb {R}^n$ and $z=(z_1,\\dots ,z_n)$ .", "We use the following orthonormal bases of ${^0}T\\mathbb {U}^{n+1}$ and ${^0}T^\\mathbb {U}^{n+1}$ : $Z_i=z_0\\partial _{z_i},\\quad Z_i^={dz_i\\over z_0};\\quad 0\\le i\\le n.$ Note that in the compactification $\\overline{\\mathbb {B}^{n+1}}$ this basis is smooth only on $\\overline{\\mathbb {B}^{n+1}}\\setminus \\lbrace -\\nu \\rbrace $ .", "Denote $A:=\\lbrace 1,\\dots ,n\\rbrace $ .", "We can decompose the vector bundle $\\otimes _S^m({^0T}^{\\mathbb {U}^{n+1}})$ into an orthogonal direct sum $\\otimes _S^m({^0T}^{\\mathbb {U}^{n+1}})=\\bigoplus _{k=0}^{m} E^{(m)}_k, \\quad E^{(m)}_k={\\rm span}\\big (\\mathcal {S}((Z_0^)^{\\otimes k}\\otimes Z_I^))_{I\\in {A}^{m-k}}\\big )$ and we let $\\pi _i$ be the orthogonal projection onto $E_i^{(m)}$ .", "Now, each tensor $u\\in \\otimes _S^{m}({^0T}^{\\mathbb {U}^{n+1}})$ can be decomposed as $u=\\sum _{i=0}^{m}u_i$ , with $u_i=\\pi _i(u)\\in E^{(m)}_i$ which we can write as $u=\\sum _{i=0}^m u_i,\\quad u_i= \\mathcal {S}((Z_0^)^{\\otimes i}\\otimes u^{\\prime }_i),\\quad u^{\\prime }_i\\in E^{(m-i)}_0.$ We can therefore identify $E^{(m)}_k$ with $E^{(m-k)}_0$ and view $E^{(m)}$ as a direct sum $E^{(m)}=\\bigoplus _{k=0}^mE^{(m-k)}_0$ .", "The trace-free condition $\\mathcal {T}(u)=0$ is equivalent to the relations $\\mathcal {T}(u^{\\prime }_r)=-\\frac{(r+2)(r+1)}{(m-r)(m-r-1)}u^{\\prime }_{r+2},\\quad 0\\le r\\le m-2.$ and in particular all $u_i$ are determined by $u_0$ and $u_1$ by iterating the trace map $\\mathcal {T}$ .", "The $u_i^{\\prime }$ are related to the elements in the decomposition (REF ) of $u_0$ and $u_1$ viewed as a symmetric $m$ -cotensor on the bundle $(Z_0)^\\perp $ using the metric $z_0^{-2}h=\\sum _{i}Z_i^\\otimes Z_i^$ .", "We see that a nonzero trace-free tensor $u$ on $\\mathbb {U}^{n+1}$ must have a nonzero $u_0$ or $u_1$ component.", "Koszul formula gives us for $i,j\\ge 1$ $\\nabla _{Z_i}Z_j=\\delta _{ij}Z_0, \\quad \\nabla _{Z_0}Z_j=0,\\quad \\nabla _{Z_i}Z_0=-Z_i,\\quad \\nabla _{Z_0}Z_0=0$ which implies $\\nabla Z_0^= -\\sum _{j=1}^nZ_j^\\otimes Z_j^=-\\frac{h}{z_0^2}, \\quad \\nabla Z_j^=Z_j^\\otimes Z_0^.$ We shall use the following notations: if $\\Pi _m$ denotes the set of permutations of $\\lbrace 1,\\dots ,m\\rbrace $ , we write $\\sigma (I):=(i_{\\sigma (1)},\\dots ,i_{\\sigma (m)})$ if $\\sigma \\in \\Pi _m$ .", "If $S=S_1\\otimes \\dots \\otimes S_\\ell $ is a tensor in $\\otimes ^\\ell ({^0}T^\\mathbb {U}^{n+1})$ , we denote by $\\tau _{i\\leftrightarrow j}(S)$ the tensor obtained by permuting $S_i$ with $S_j$ in $S$ , and by $\\rho _{i\\rightarrow V}(S)$ the operation of replacing $S_i$ by $V\\in {^0}T^\\mathbb {U}^{n+1}$ in $S$ .", "The Laplacian and $\\nabla ^$ acting on $E_0^{(m)}$ and $E_1^{(m)}$.", "We start by computing the action of $\\Delta $ on sections of $E^{(m)}_0, E^{(m)}_1$ , and we will later deduce from this computation the action on $E^{(m)}_k$ .", "Let us consider the tensor $Z^_I:=Z^_{i_1}\\otimes \\dots \\otimes Z^_{i_m}\\in E_0^{(m)}$ where $I=(i_1,\\dots ,i_m)\\in A^{m}$ and $Z^_{\\sigma (I)}:= Z^_{i_{\\sigma (1)}}\\otimes \\dots \\otimes Z^_{i_{\\sigma (m)}}.$ The symmetrization of $Z^_I$ is given by $\\mathcal {S}(Z^_I)=\\frac{1}{m!", "}\\sum _{\\sigma \\in \\Pi _m}Z^_{\\sigma (I)}$ and those elements form a basis of the space $E^{(m)}_0$ when $I$ ranges over all combinations of $m$ -uplet in $A=\\lbrace 1,\\dots ,n\\rbrace $ .", "Lemma 6.2 Let $u_0=\\sum _{I\\in {A}^m}f_I\\mathcal {S}(Z^_I)$ with $f_I\\in \\mathcal {C}^\\infty (\\mathbb {U}^{n+1})$ .", "Then one has $ \\begin{split}\\Delta u_0= & \\sum _{I\\in {A}^m}((\\Delta +m)f_I)\\mathcal {S}(Z_I^)+2m\\, \\mathcal {S}(\\nabla ^u_0\\otimes Z_0^) \\\\& +m(m-1)\\mathcal {S}(\\mathcal {T}(u_0)\\otimes Z_0^\\otimes Z_0^)\\end{split}$ while, denoting $d_{z}f_I=\\sum _{i=1}^nZ_i(f_I)Z_i^$ , the divergence is given by $\\begin{split}\\nabla ^u_0= &-(m-1)\\mathcal {S}(\\mathcal {T}(u_0)\\otimes Z_0^)- \\sum _{I\\in {A}^m}\\iota _{d_{z}f_I}\\mathcal {S}(Z_I^).\\end{split}$ Using (REF ), we compute $ \\nabla (f_I\\mathcal {S}(Z_I^))= \\sum _{i=0}^n(Z_if_I)(z) Z_i^\\otimes \\mathcal {S}(Z_I^)+ \\frac{f_I(z)}{m!}", "\\sum _{k=1}^m \\sum _{\\sigma \\in \\Pi _m}\\tau _{1\\leftrightarrow k+1}(Z_0^\\otimes Z^_{\\sigma (I)}).", "$ Then taking the trace of $\\nabla (f_I\\mathcal {S}(Z_I^))$ gives $\\begin{split}\\nabla ^(f_I\\mathcal {S}(Z_I^))=& -\\frac{f_I}{m!}", "\\sum _{k=2}^m \\sum _{\\sigma \\in \\Pi _m}\\delta _{i_{\\sigma (1)},i_{\\sigma (k)}}\\rho _{k-1\\rightarrow Z_0^}(Z^_{i_{\\sigma (2)}}\\otimes \\dots \\otimes Z^_{i_{\\sigma (m)}})\\\\& - \\sum _{i=1}^n(Z_if_I) \\frac{1}{m!}", "\\sum _{\\sigma \\in \\Pi _m}\\delta _{i,i_{\\sigma (1)}}(Z^_{i_{\\sigma (2)}}\\otimes \\dots \\otimes Z^_{i_{\\sigma (m)}})\\end{split}$ We notice that $\\mathcal {S}(\\mathcal {T}(\\mathcal {S}(Z_I^))\\otimes Z_0^)$ is given by $\\mathcal {S}(\\mathcal {T}(\\mathcal {S}(Z_I^))\\otimes Z_0^)=\\frac{1}{m!", "(m-1)} \\sum _{\\sigma \\in \\Pi _m}\\sum _{k=1}^{m-1}\\delta _{i_{\\sigma (1)},i_{\\sigma (2)}}\\tau _{1\\leftrightarrow k}(Z_0^\\otimes Z^_{i_{\\sigma (3)}}\\otimes \\dots \\otimes Z^_{i_{\\sigma (m)}}).$ which implies (REF ).", "Let us now compute $\\nabla ^2(f_I\\mathcal {S}(Z_I^))$ : $\\nabla ^2(f_I\\mathcal {S}(Z_I^)) &= \\sum _{i,j=0}^nZ_jZ_i(f_I) Z_j^\\otimes Z_i^\\otimes \\mathcal {S}(Z_I^)- Z_0(f_I) z_0^{-2} h\\otimes \\mathcal {S}(Z_I^) \\\\& +\\sum _{j=1}^nZ_j(f_I)Z_j^\\otimes Z_0^\\otimes \\mathcal {S}(Z_I^) + \\frac{Z_0(f_I)}{m!}", "\\sum _{\\sigma \\in \\Pi _m}\\sum _{k=1}^m \\tau _{1\\leftrightarrow k+2}(Z_0^\\otimes Z_0^\\otimes Z^_{\\sigma (I)})\\\\&+\\sum _{i=1}^n \\frac{Z_i(f_I)}{m!", "}\\sum _{\\sigma \\in \\Pi _m}\\sum _{k=1}^m \\tau _{1\\leftrightarrow k+2}(Z_0^\\otimes Z_i^\\otimes Z^_{\\sigma (I)})\\\\& + \\sum _{i=1}^n\\frac{Z_i(f_I)}{m!}", "\\sum _{\\sigma \\in \\Pi _m}\\sum _{k=1}^m \\tau _{2\\leftrightarrow k+2}(Z_i^\\otimes Z_0^\\otimes Z^_{\\sigma (I)})[0]\\\\&+ \\frac{Z_0(f_I)}{m!}", "Z_0^\\otimes \\sum _{k=1}^m \\sum _{\\sigma \\in \\Pi _m}\\tau _{1\\leftrightarrow k+1}(Z_0^\\otimes Z^_{\\sigma (I)})\\\\& - \\frac{f_I}{m!}", "\\sum _{j=1}^{n}Z_j^\\otimes \\sum _{\\sigma \\in \\Pi _m}\\sum _{k=1}^m\\tau _{1\\leftrightarrow k+1}(Z_j^\\otimes Z^_{\\sigma (I)})\\\\&+ \\frac{f_I}{m!}", "\\sum _{k=1}^m \\sum _{\\begin{array}{c}\\ell =1\\\\ \\ell \\ne k+1\\end{array}}^{m+1}\\tau _{1\\leftrightarrow \\ell +1}(Z_0^\\otimes \\tau _{1\\leftrightarrow k+1}(Z_0^\\otimes Z^_{\\sigma (I)})).$ We then take the trace: the first line has trace $-(\\Delta f_I) \\mathcal {S}(Z_I^)$ , the second and fifth lines have vanishing trace, the sixth line has trace $-mf_I\\mathcal {S}(Z_I^)$ , the last line has trace $\\frac{2f_I}{m!}", "\\sum _{\\sigma \\in \\Pi _m}\\sum _{1\\le k<\\ell \\le m} \\delta _{i_{\\sigma (k)},i_{\\sigma (\\ell )}}\\rho _{k\\rightarrow Z_0^}\\rho _{\\ell \\rightarrow Z_0^}(Z^_{\\sigma (I)})$ and the sum of the third and fourth lines has trace $2\\sum _{i=1}^n\\frac{Z_i(f_I)}{m!", "}\\sum _{\\sigma \\in \\Pi _m}\\sum _{k=1}^m \\delta _{i,i_{\\sigma (k)}}\\rho _{k\\rightarrow Z_0^}(Z^_{\\sigma (I)}).$ Computing $\\mathcal {S}(\\mathcal {T}(\\mathcal {S}(Z_I^))\\otimes Z_0^\\otimes Z_0^)$ gives ${\\begin{array}{c}\\mathcal {S}\\big (\\mathcal {T}(\\mathcal {S}(Z_I^))\\otimes Z_0^\\otimes Z_0^\\big )= \\\\\\frac{2}{m!m(m-1)}\\sum _{1\\le k<\\ell \\le m}\\sum _{\\sigma \\in \\Pi _m}\\delta _{i_{\\sigma (1)},i_{\\sigma (2)}}\\tau _{1\\leftrightarrow k+2}\\tau _{2\\leftrightarrow \\ell +2}(Z_0^\\otimes Z_0^\\otimes Z^_{i_{\\sigma (3)}}\\otimes \\dots \\otimes Z^_{i_{\\sigma (m)}})\\end{array}}$ therefore the term (REF ) can be simplified to $m(m-1)f_I\\mathcal {S}\\big (\\mathcal {T}(\\mathcal {S}(Z_I^))\\otimes Z_0^\\otimes Z_0^\\big ).$ Similarly to simplify (REF ), we compute ${\\begin{array}{c}\\mathcal {S}\\big (\\nabla ^(f_I\\mathcal {S}(Z_I^))\\otimes Z_0^\\big )= -(m-1)\\mathcal {S}(\\mathcal {T}(f_I\\mathcal {S}(Z_I^))\\otimes Z_0^\\otimes Z_0^)\\\\-\\sum _{i=1}^n(Z_if_I) \\frac{1}{m!m} \\sum _{k=1}^{m}\\sum _{\\sigma \\in \\Pi _m}\\delta _{i,i_{\\sigma (1)}}\\tau _{1\\leftrightarrow k}(Z_0^\\otimes Z^_{i_{\\sigma (2)}}\\otimes \\dots \\otimes Z^_{i_{\\sigma (m)}})\\end{array}}$ so that ${\\begin{array}{c}2\\sum _{i=1}^n\\frac{Z_i(f_I)}{m!", "}\\sum _{\\sigma \\in \\Pi _m}\\sum _{k=1}^m \\delta _{i,i_{\\sigma (k)}}\\rho _{k\\rightarrow Z_0^}(Z^_{\\sigma (I)})\\\\= -2m\\, \\mathcal {S}(\\nabla ^(f_I\\mathcal {S}(Z_I^))\\otimes Z_0^)-2m(m-1)\\mathcal {S}(\\mathcal {T}(f_I\\mathcal {S}(Z_I^))\\otimes Z_0^\\otimes Z_0^).\\end{array}}$ and this achieves the proof of (REF ).Ê A similarly tedious calculation, omitted here, yields Lemma 6.3 Let $u_1=\\mathcal {S}(Z_0^\\otimes u^{\\prime }_1)$ , $u^{\\prime }_1=\\sum _{J\\in {A}^{m-1}}g_J \\mathcal {S}(Z^_J)$ with $g_J\\in \\mathcal {C}^\\infty (\\mathbb {U}^{n+1})$ , then the $E_0^{(m)}\\oplus E_1^{(m)}$ components of the Laplacian of $u_1$ are $ \\begin{split}\\Delta u_1=&\\sum _{J\\in {A}^{m-1}} \\big ((\\Delta +n+3(m-1))g_J\\big )\\mathcal {S}(Z_0^ \\otimes Z_J^)\\\\&+2\\sum _{J\\in {A}^{m-1}} \\mathcal {S}(d_{z}g_J\\otimes Z_J^)+ \\operatorname{Ker}(\\pi _0+\\pi _1)\\end{split}$ and the $E_0^{(m)}\\oplus E_1^{(m)}$ components of divergence of $u_1$ are $\\begin{split}\\nabla ^u_1=&\\,\\frac{1}{m}\\sum _{J\\in {A}^{m-1}}((n+m-1)g_J -Z_0(g_J))\\mathcal {S}(Z_J^)\\\\&-\\frac{(m-1)}{m}\\sum _{J\\in {A}^{m-1}}\\mathcal {S}( Z_0^\\otimes \\iota _{d_{z}g_J}\\mathcal {S}(Z_J^))+\\operatorname{Ker}(\\pi _0+\\pi _1).\\end{split}$ General formulas for Laplacian and divergence.", "Armed with Lemmas REF and REF , we can show the following fact which, together with (REF ), determines completely the Laplacian on trace-free symmetric tensors.", "Lemma 6.4 Assume that $u\\in \\mathcal {D}^{\\prime }(\\mathbb {U}^{n+1};\\otimes _S^mT^\\mathbb {U}^{n+1})$ satisfies $\\mathcal {T}(u)=0$ and is written in the form (REF ).", "Let $u_0=\\sum _{I\\in A^m} f_I\\mathcal {S}(Z_I^),\\quad u_1=\\sum _{J\\in A^{m-1}}g_J\\mathcal {S}(Z_0\\otimes Z_J^).$ Then the projection of $\\Delta u$ onto $E^{(m)}_0\\oplus E^{(m)}_1$ can be written $\\begin{split}\\pi _0(\\Delta u)=& \\sum _{I\\in {A}^m}((\\Delta +m) f_I)\\mathcal {S}(Z^_{I})+2\\sum _{J\\in {A}^{m-1}} \\mathcal {S}(d_{z}g_J\\otimes Z_J^)\\\\&+m(m-1)\\mathcal {S}(z_0^{-2}h\\otimes \\mathcal {T}(u_0)),\\end{split}$ $\\begin{split}\\pi _1(\\Delta u)=& \\sum _{J\\in {A}^{m-1}} ((\\Delta +n+3(m-1))g_J)\\mathcal {S}(Z_0^ \\otimes Z_J^)\\\\&-2m\\sum _{I\\in {A}^m}\\mathcal {S}(Z_0^\\otimes \\iota _{d_{z}f_I}\\mathcal {S}(Z_I^))\\\\&+(m-1)(m-2)\\mathcal {S}(Z_0^\\otimes z_0^{-2}h\\otimes \\mathcal {T}(u^{\\prime }_1)) \\\\& -2m(m-1)\\sum _{I\\in {A}^m}\\mathcal {S}(Z_0^\\otimes d_{z}f_I \\otimes \\mathcal {T}(\\mathcal {S}(Z_I^))).\\end{split}$ First, it is easily seen from (REF ) that $\\Delta u_k$ is a section of $\\bigoplus _{j=k-2}^{k+2}E^{(m)}_{j}$ .", "From Lemmas REF and REF , we have $\\pi _0(\\Delta (u_0+u_1))=\\sum _{I\\in {A}^m}((\\Delta +m) f_I)\\mathcal {S}(Z^_{I})+2\\sum _{J\\in {A}^{m-1}} \\mathcal {S}(d_{z}g_J\\otimes Z_J^).$ Then for $u_2$ , using $\\mathcal {S}((Z_0^)^{\\otimes 2}\\otimes u_2^{\\prime })=\\mathcal {S}(g_H\\otimes u_2^{\\prime })-\\mathcal {S}(z_0^{-2}h\\otimes u_2^{\\prime })$ and $\\Delta \\mathcal {I}=\\mathcal {I}\\Delta $ , $\\pi _0(\\Delta u_2)=\\pi _0(\\mathcal {S}(z_0^{-2}h\\otimes \\Delta u^{\\prime }_2))-\\pi _0(\\Delta (\\mathcal {S}(z_0^{-2}h\\otimes u_2^{\\prime })))$ and writing $u_2^{\\prime }=-\\tfrac{m(m-1)}{2}\\mathcal {T}(u_0)$ by (REF ), we obtain, using (REF ) $\\pi _0(\\Delta u_2)=m(m-1)\\mathcal {S}(z_0^{-2}h\\otimes \\mathcal {T}(u_0))$ We therefore obtain (REF ).", "Now we consider the projection on $E^{(m)}_1$ of the equation $(\\Delta -s)T=0$ .", "We have from (REF ) $\\pi _1(\\Delta u_0)=-2m\\sum _{I\\in {A}^m}\\mathcal {S}(Z_0^\\otimes \\iota _{d_{z}f_I}\\mathcal {S}(Z_I^))$ where $\\iota _{d_{z}f_I}$ means $\\sum _{j=1}^nZ_j(f_I)\\iota _{Z_j}$ .", "Then, from (REF ) $\\begin{split}\\pi _1(\\Delta u_1)=&\\sum _{J\\in {A}^{m-1}} \\big ((\\Delta +n+3(m-1))g_J\\big )\\mathcal {S}(Z_0^ \\otimes Z_J^).\\end{split}$ Using again $\\mathcal {S}((Z_0^)^{\\otimes 2}\\otimes u_2^{\\prime })=\\mathcal {S}(g_H\\otimes u_2^{\\prime })-\\mathcal {S}(z_0^{-2}h \\otimes u_2^{\\prime })$ and $\\Delta \\mathcal {I}=\\mathcal {I}\\Delta $ , (REF ) gives $\\begin{split}\\pi _1(\\Delta u_2)= -2m(m-1)\\sum _{I\\in {A}^m}\\mathcal {S}(Z_0^\\otimes d_{z}f_I \\otimes \\mathcal {T}\\mathcal {S}(Z_I^)).\\end{split}$ Finally, we compute $\\pi _1(\\Delta u_3)$ , using the computation (REF ) we get $\\begin{split}\\pi _1(\\Delta u_3)=&\\pi _1(\\mathcal {S}(z_0^{-2}h\\otimes \\Delta \\mathcal {S}(Z_0^\\otimes u^{\\prime }_3))-\\pi _1(\\Delta \\mathcal {S}(Z_0^\\otimes z_0^{-2}h\\otimes u^{\\prime }_3))\\\\=& (m-1)(m-2)\\mathcal {S}(Z_0^\\otimes z_0^{-2}h\\otimes \\mathcal {T}(u_1^{\\prime })).\\end{split}$ We conclude that $\\pi _1(\\Delta u)$ is given by (REF ).", "Similarly, we also have Lemma 6.5 Let $u$ be as in Lemma REF .", "Then the projection onto $E^{(m-1)}_{0}\\oplus E^{(m-1)}_1$ of the divergence of $u$ is given by $\\pi _0(\\nabla ^u)=-\\sum _{I\\in {A}^{m}}\\iota _{d_{z} f_I}\\mathcal {S}(Z_I^)+\\frac{1}{m}\\sum _{J\\in {A}^{m-1}}((n+m-1)g_J -Z_0(g_J))\\mathcal {S}(Z_J^),$ $\\begin{split}\\pi _1(\\nabla ^u)=& (m-1)\\sum _{I\\in {A}^m}(Z_0f_I-(m+n-1)f_I)\\mathcal {S}\\big (\\mathcal {T}(\\mathcal {S}(Z_I^))\\otimes Z_0^\\big )\\\\& -\\frac{(m-1)}{m}\\sum _{J\\in {A}^{m-1}}\\mathcal {S}( Z_0^\\otimes \\iota _{d_{z}g_J}\\mathcal {S}(Z_J^)).\\end{split}$ The $\\pi _0$ part follows from (REF ) and (REF ).", "For the $\\pi _1$ part, we also use (REF ) and (REF ) but we need to see the contribution from $\\nabla ^u_2$ as well.", "For that, we write as before $u_2^{\\prime }=-\\tfrac{m(m-1)}{2}\\sum _{I\\in {A}^m}f_I\\mathcal {T}(\\mathcal {S}(Z_I^))$ and a direct calculation shows that $\\pi _1(\\nabla ^u_2)=(m-1)\\sum _{I\\in {A}^m}(Z_0f_I-(m+n-2)f_I)\\mathcal {S}(\\mathcal {T}(\\mathcal {S}(Z_I^))\\otimes Z_0^)$ implying the desired result." ], [ "Properties of the Poisson kernel", "In this section, we study the Poisson kernel $P^-_\\lambda $ defined by (REF ).", "Pairing on the sphere.", "We start by proving the following formula: Lemma 6.6 Let $\\lambda \\in \\mathbb {C}$ and $w\\in \\mathcal {D}^{\\prime }(\\mathbb {S}^n;\\otimes _S^m(T^\\mathbb {S}^n))$ .", "Then $P^-_\\lambda w(x)=\\int _{\\mathbb {S}^n} P(x,\\nu )^{n+\\lambda }(\\otimes ^m(\\mathcal {A}^{-1}_-(x,\\xi _-(x,\\nu )))^T) w(\\nu )\\,dS(\\nu )$ where the map $\\xi _-$ is defined in (REF ).", "Making the change of variables $\\xi =\\xi _-(x,\\nu )$ defined in (REF ), and using (REF ) and (REF ), we have ${\\begin{array}{c}P^-_\\lambda w(x)=\\int _{S_x\\mathbb {H}^{n+1}}\\Phi _-(x,\\xi )^{\\lambda } (\\otimes ^m(\\mathcal {A}^{-1}_-(x,\\xi ))^T) w(B_-(x,\\xi ))\\,dS(\\xi )\\\\=\\int _{\\mathbb {S}^n} P(x,\\nu )^{n+\\lambda }(\\otimes ^m(\\mathcal {A}^{-1}_-(x,\\xi _-(x,\\nu )))^T) w(\\nu )\\,dS(\\nu )\\end{array}}$ as required.", "Poisson maps to eigenstates.", "To show that $P^-_\\lambda w(x)$ is an eigenstate of the Laplacian, we use the following Lemma 6.7 Assume that $w\\in \\mathcal {D}^{\\prime }(\\mathbb {S}^n;\\otimes ^m (T^\\mathbb {S}^n))$ is the delta function centered at $e_1=\\partial _{x_1}\\in \\mathbb {S}^n$ with the value $e^_{j_1+1}\\otimes \\dots \\otimes e^_{j_m+1}$ , where $1\\le j_1,\\dots ,j_m\\le n$ .", "Then under the identifications (REF ) and (REF ), we have $P^-_\\lambda w(z_0,z)=z_0^{n+\\lambda }Z_{j_1}^\\otimes \\dots \\otimes Z_{j_m}^.$ We first calculate $P(z,e_1)=z_0.$ It remains to show the following identity in the half-space model $\\mathcal {A}_-^{-T}(z,\\xi _-(z,\\nu ))e^_{j+1}=Z_j^,\\quad 1\\le j\\le n.$ One can verify (REF ) by a direct computation: since $\\mathcal {A}_-$ is an isometry, one can instead calculate the image of $e_{j+1}$ under $\\mathcal {A}_-$ , and then apply to it the differentials of the maps $\\psi $ and $\\psi _1$ defined in (REF ) and (REF ).", "Another way to show (REF ) is to use the interpretation of $\\mathcal {A}_-$ as parallel transport to conformal infinity, see (REF ).", "Note that under the diffeomorphism $\\psi _1:\\mathbb {B}^{n+1}\\rightarrow \\mathbb {U}^{n+1}$ , $\\nu =e_1$ is sent to infinity and geodesics terminating at $\\nu $ , to straight lines parallel to the $z_0$ axis.", "By (REF ), the covector field $Z_j^$ is parallel along these geodesics and orthogonal to their tangent vectors.", "It remains to verify that the limit of the field $\\rho _0 Z_j^$ along these geodesics as $z\\rightarrow \\infty $ , considered as a covector in the ball model, is equal to $e_{j+1}^$ .", "Proof of Lemma REF.", "It suffices to show that for each $\\nu \\in \\mathbb {S}^n$ , if $w$ is a delta function centered at $\\nu $ with value being some symmetric trace-free tensor in $\\otimes _S^m T^_\\nu \\mathbb {S}^n$ , then $\\big (\\Delta +\\lambda (n+\\lambda )-m\\big ) P^-_\\lambda w=0,\\quad \\nabla ^P^-_\\lambda w=0,\\quad \\mathcal {T}(P^-_\\lambda w)=0.$ Since the group of symmetries $G$ of $\\mathbb {H}^{n+1}$ acts transitively on $\\mathbb {S}^n$ , we may assume that $\\nu =\\partial _1$ .", "Applying Lemma REF , we write in the upper half-plane model, $P^-_\\lambda w=z_0^{n+\\lambda } u_0,\\quad u_0\\in E^{(m)}_0,\\quad \\mathcal {T}(u_0)=0.$ It immediately follows that $\\mathcal {T}(P^-_\\lambda w)=0$ .", "To see the other two identities, it suffices to apply Lemma REF together with the formula $\\Delta z_0^{n+\\lambda }=-\\lambda (n+\\lambda ) z_0^{n+\\lambda }.$ Injectivity of Poisson.", "Notice that $P^-_\\lambda $ is an analytic family of operators in $\\lambda $ .", "We define the set $\\mathcal {R}_m=\\left\\lbrace \\begin{array}{ll}-\\tfrac{n}{2}-\\tfrac{1}{2}\\mathbb {N}_0 & \\textrm { if }n>1 \\textrm { or }m=0\\\\-\\tfrac{1}{2}\\mathbb {N}_0 & \\textrm { if }n=1 \\textrm { and }m>0~\\end{array}\\right.$ and we will prove that if $\\lambda \\notin \\mathcal {R}_m$ and $w\\in \\mathcal {D}^{\\prime }(\\mathbb {S}^n;\\otimes _S^mT^\\mathbb {S}^n)$ is trace-free, then $P^-_\\lambda (w)$ has a weak asymptotic expansion at the conformal infinity with the leading term given by a multiple of $w$ , proving injectivity of $P^-_\\lambda $ .", "We shall use the 0-cotangent bundle approach in the ball model and rewrite $\\mathcal {A}^{-1}_\\pm (x,\\xi _\\pm (x,\\nu ))$ as the parallel transport $\\tau (y^{\\prime },y)$ in ${^0}T\\overline{\\mathbb {B}^{n+1}}$ with $\\psi (x)=y$ and $y^{\\prime }=\\nu $ , as explained in (REF ).", "Let $\\rho \\in \\mathcal {C}^\\infty (\\overline{\\mathbb {B}^{n+1}})$ be a smooth boundary defining function which satisfies $\\rho >0$ in $\\mathbb {B}^{n+1}$ , $\|d\\rho \|_{\\rho ^2g_H}=1$ near $\\mathbb {S}^n=\\lbrace \\rho =0\\rbrace $ , where $g_H$ is the hyperbolic metric on the ball.", "We can for example take the function $\\rho =\\rho _0$ defined in (REF ) and smooth it near the center $y=0$ of the ball.", "Such function is called geodesic boundary defining function and induces a diffeomorphism $\\theta : [0,\\epsilon )_t\\times \\mathbb {S}^n\\rightarrow \\overline{\\mathbb {B}^{n+1}}\\cap \\lbrace \\rho < \\epsilon \\rbrace ,\\quad \\theta (t,\\nu ):=\\theta _t(\\nu )$ where $\\theta _t$ is the flow at time $t$ of the gradient $\\nabla ^{\\rho ^2g_H}\\rho $ of $\\rho $ (denoted also $\\partial _\\rho $ ) with respect to the metric $\\rho ^2g_H$ .", "For $\\rho $ given in (REF ), we have for $t$ small $\\theta (t,\\nu )={2-t\\over 2+t}\\,\\nu ,\\quad \\nu \\in \\mathbb {S}^n.$ For a fixed geodesic boundary defining function $\\rho $ , one can identify, over the boundary $\\mathbb {S}^n$ of $\\overline{\\mathbb {B}^{n+1}}$ , the bundle $T^\\mathbb {S}^n$ and $T\\mathbb {S}^n$ with the bundles ${^0}T^\\mathbb {S}^n:={^0}T_{\\mathbb {S}^n}^\\overline{\\mathbb {B}^{n+1}}\\cap \\ker \\iota _{\\rho \\partial _\\rho }$ simply by the isomorphism $v\\mapsto \\rho ^{-1}v$ (and we identify their duals $T\\mathbb {S}^n$ and ${^0}T\\mathbb {S}^n$ as well).", "Similarly, over $\\mathbb {S}^n$ , $E^{(m)}\\cap \\ker \\iota _{\\rho \\partial _\\rho }$ identifies with $\\otimes ^m_ST^\\mathbb {S}^n\\cap \\ker \\mathcal {T}$ by the map $v\\mapsto \\rho ^{-m}v$ .", "We can then view the Poisson operator as an operator $P^-_\\lambda : \\mathcal {D}^{\\prime }(\\mathbb {S}^n; E^{(m)}\\cap \\ker \\iota _{\\rho \\partial _\\rho })\\rightarrow \\mathcal {C}^\\infty (\\mathbb {B}^{n+1}; \\otimes _S^m ( {^0}T^\\overline{\\mathbb {B}^{n+1}})).$ Lemma 6.8 Let $w\\in \\mathcal {D}^{\\prime }(\\mathbb {S}^n; E^{(m)}\\cap \\ker \\iota _{{\\rho _0}\\partial _{\\rho _0}})$ and assume that $\\lambda \\notin \\mathcal {R}_m$ .", "Then $P^-_\\lambda (w)$ has a weak asymptotic expansion at $\\mathbb {S}^n$ as follows: for each $\\nu \\in \\mathbb {S}^n$ , there exists a neighbourhood $V_\\nu \\subset \\overline{\\mathbb {B}^{n+1}}$ of $\\nu $ and a boundary defining function $\\rho =\\rho _\\nu $ such that for any $\\varphi \\in \\mathcal {C}^\\infty (V_\\nu \\cap \\mathbb {S}^n; \\otimes _S^m({^0}T\\mathbb {S}^n))$ , there exist $F_\\pm \\in \\mathcal {C}^\\infty ([0,\\epsilon ))$ such that for $t>0$ small ${\\begin{array}{c}\\int _{\\mathbb {S}^n} \\langle P^-_\\lambda (w)(\\theta (t,\\nu )),\\otimes ^m (\\tau (\\theta (t,\\nu ),\\nu )).\\varphi (\\nu )\\rangle dS_{\\rho }(\\nu )= \\\\\\left\\lbrace \\begin{array}{ll}t^{-\\lambda }F_-(t)+t^{n+\\lambda }F_+(t), &\\lambda \\notin -n/2+\\mathbb {N};\\\\t^{-\\lambda }F_-(t)+t^{n+\\lambda }\\log (t)F_+(t), & \\lambda \\in -n/2+\\mathbb {N}.\\end{array}\\right.\\end{array}}$ using the product collar neighbourhood (REF ) associated to $\\rho $ , and moreover one has $F_-(0)=C \\frac{\\Gamma (\\lambda +\\tfrac{n}{2})}{(\\lambda +n+m-1)\\Gamma (\\lambda +n-1)} \\langle e^{\\lambda f}.w,\\varphi \\rangle $ for some $f\\in \\mathcal {C}^\\infty (\\mathbb {S}^n)$ satisfying $\\rho =\\tfrac{1}{4}e^{f}\\rho _0+\\mathcal {O}(\\rho )$ near $\\rho =0$ and $C\\ne 0$ a constant depending only on $n$ .", "Here $dS_\\rho $ is the Riemannian measure for the metric $(\\rho ^2g_H)\|_{\\mathbb {S}^n}$ and the distributional pairing on $\\mathbb {S}^n$ is with respect to this measure.", "Figure: The covector w(z ' )w(z^{\\prime }), the vector ϕ(z)\\varphi (z), and their parallel transportsto (z 0 ,z)(z_0,z) viewed in the 0-bundles, for the case m=1m=1.First we split $w$ into $w_1+w_2$ where $w_1$ is supported near $\\nu \\in \\mathbb {S}^n$ and $w_2$ is zero near $\\nu $ .", "For the case where $w_2$ has support at positive distance from the support of $\\varphi $ , we have for any geodesic boundary defining function $\\rho $ that $t\\mapsto t^{-n-\\lambda }\\int _{\\mathbb {S}^n} \\langle P^-_\\lambda (w_2)(\\theta (t,\\nu )),\\otimes ^m (\\tau (\\theta (t,\\nu ),\\nu )).\\varphi (\\nu )\\rangle dS_\\rho (\\nu )\\in \\mathcal {C}^\\infty ([0,\\epsilon )),$ this is a direct consequence of Lemma REF and the following smoothness properties ${\\begin{array}{c}(y,\\nu )\\mapsto \\log \\big (P(\\psi ^{-1}(y),\\nu )/\\rho (y)\\big )\\in \\mathcal {C}^\\infty (\\overline{\\mathbb {B}^{n+1}}\\times \\mathbb {S}^n \\setminus {\\rm diag}(\\mathbb {S}^n\\times \\mathbb {S}^n))\\\\\\tau (\\cdot ,\\cdot )\\in \\mathcal {C}^\\infty (\\overline{\\mathbb {B}^{n+1}}\\times \\overline{\\mathbb {B}^{n+1}} \\setminus {\\rm diag}(\\mathbb {S}^n\\times \\mathbb {S}^n));{^0}T^\\overline{\\mathbb {B}^{n+1}}\\otimes {^0}T\\overline{\\mathbb {B}^{n+1}}).\\end{array}}$ This reduces the consideration of the Lemma to the case where $w$ is $w_1$ supported near $\\nu $ , and to simplify we shall keep the notation $w$ instead of $w_1$ .", "We thus consider now $w$ and $\\varphi $ to have support near $\\nu $ .", "For convenience of calculations and as we did before, we work in the half-space model $\\mathbb {R}^+_{z_0}\\times \\mathbb {R}_{z}^n$ by mapping $\\nu $ to $(z_0,z)=(0,0)$ (using the composition of a rotation on the ball model with the map defined in (REF )) and we choose a nighbourhood $V_\\nu $ of $\\nu $ which is mapped to $z_0^2+\|z\|^2<1$ in $\\mathbb {U}^{n+1}$ and choose the geodesic defining function $\\rho =z_0$ (and thus $\\theta (z_0,z)=(z_0,z)$ ).", "(See Figure REF .)", "The geodesic boundary defining function $\\rho _0=\\frac{2(1-\|y\|)}{1+\|y\|}$ in the ball equals $\\rho _0(z_0,z)=4z_0/(1+z_0^2+\|z\|^2)$ in the half-space model.", "The metric $dS_\\rho $ becomes the Euclidean metric $dz$ on $\\mathbb {R}^n$ near 0 and $w$ has compact support in $\\mathbb {R}^n$ .", "By (REF ) and (REF ), the Poisson kernel in these coordinates becomes $\\widetilde{P}(z_0,z;z^{\\prime })=e^{f(z^{\\prime })}P(z_0,z;z^{\\prime })\\textrm { with } P(z_0,z;z^{\\prime }):={z_0\\over z_0^2+\|z-z^{\\prime }\|^2}, \\,\\,f(z^{\\prime })=\\log (1+\|z^{\\prime }\|^2)$ where $z,z^{\\prime }\\in \\mathbb {R}^n$ and $z_0>0$ .", "One has $\\rho =\\tfrac{1}{4}e^f\\rho _0+\\mathcal {O}(\\rho )$ near $\\rho =0$ .", "In the Appendix of [28], the parallel transport $\\tau (z_0,z; 0,z^{\\prime })$ is computed for $z^{\\prime }\\in \\mathbb {R}^n$ is a neighbourhood of 0: in the local orthonormal basis $Z_0=z_0\\partial _{z_0},Z_i=z_0\\partial _{z_i}$ of the bundle ${^0}T\\mathbb {U}^{n+1}$ , near $\\nu $ , the matrix of $\\tau (z_0,z;z^{\\prime }):=\\tau (z_0,z;0,z^{\\prime })$ is given by ${\\begin{array}{c}\\tau _{00}=1-2P(z_0,z;z^{\\prime })\\frac{\|z-z^{\\prime }\|^2}{z_0}, \\quad \\tau _{0i}=-\\tau _{i0}=-2z_0(z_i-z_i^{\\prime })\\frac{P(z_0,z;z^{\\prime })}{z_0}, \\\\\\tau _{ij}=\\delta _{ij}-2P(z_0,z;z^{\\prime })\\frac{(z_i-z_i^{\\prime }).", "(z_j-z_j^{\\prime })}{z_0}.\\end{array}}$ In particular we see that $\\tau (z_0,z;z)$ is the identity matrix in the basis $(Z_i)_i$ and thus $\\tau (\\theta (z_0,z),z)$ as well.", "We denote $(Z_j^)_j$ the dual basis to $(Z_j)_j$ as before.", "Now, we use the correspondence between symmetric tensors and homogeneous polynomials to facilitate computations, as explained in Section REF .", "To $\\mathcal {S}(Z^_I)$ , we associate the polynomial on $\\mathbb {R}^n$ given by $P_I(x)=\\mathcal {S}(Z_I^)\\bigg (\\sum _{i=1}^nx_iZ_I,\\dots ,\\sum _{i=1}^nx_iZ_I\\bigg )=x_I$ where $x_I=\\prod _{k=1}^m x_{i_k}$ if $I=(i_1,\\dots ,i_m)$ .", "We denote by $\\operatorname{Pol}^m(\\mathbb {R}^n)$ the space of homogeneous polynomials of degree $m$ on $\\mathbb {R}^n$ and $\\operatorname{Pol}_0^m(\\mathbb {R}^n)$ those which are harmonic (thus corresponding to trace free symmetric tensors in $E^{(m)}_0$ ).", "Then we can write $w=\\sum _{\\alpha }w_\\alpha p_\\alpha (x)$ for some $w_\\alpha \\in \\mathcal {D^{\\prime }}(\\mathbb {R}^n)$ supported near 0 and $p_\\alpha (x)\\in {\\rm Pol}^m_0(\\mathbb {R}^n)$ .", "Each $p_\\alpha (x)$ composed with the linear map $\\tau (z^{\\prime };z_0,z)\|_{Z_0^\\perp }$ becomes the homogeneous polynomial in $x$ $p_{\\alpha }\\big (x-2 (z-z^{\\prime })\\langle z-z^{\\prime },x\\rangle .", "\\tfrac{P(z_0,z;z^{\\prime })}{z_0}\\big )$ where $\\langle \\cdot ,\\cdot \\rangle $ just denotes the Euclidean scalar product.", "To prove the desired asymptotic expansion, it suffices to take $\\varphi \\in \\mathcal {C}^\\infty _0([0,\\infty )_{z_0}\\times \\mathbb {R}^n)$ and to analyze the following homogeneous polynomial in $x$ as $z_0\\rightarrow 0$ ${\\begin{array}{c}\\int _{\\mathbb {R}^n} \\sum _{\\alpha } \\Big \\langle e^{(n+\\lambda )f}w_\\alpha , \\varphi (z_0,z)P(z_0,z;\\cdot )^{n+\\lambda }p_\\alpha \\big (x-2 (z-\\cdot )\\langle z-\\cdot ,x\\rangle .", "\\tfrac{P(z_0,z;\\cdot )}{z_0}\\big )\\Big \\rangle dz\\end{array}}$ where the bracket $\\langle w_\\alpha , \\cdot \\rangle $ means the distributional pairing coming from pairing with respect to the canonical measure $dS$ on $\\mathbb {S}^n$ , which in $\\mathbb {R}^n$ becomes the measure $4^ne^{-nf}dz$ , and so the $e^{nf}$ in (REF ) cancels out if one works with the Euclidean measure $dz$ , which we do now.", "We remark a convolution kernel in $z$ and thus apply Fourier transform in $z$ (denoted $\\mathcal {F}$ ): denoting $P(z_0;\|z-z^{\\prime }\|)$ for $P(z_0,z;z^{\\prime })$ , the integral (REF ) becomes (up to non-zero multiplicative constant) $I(z_0,x):=\\sum _\\alpha \\Big \\langle \\mathcal {F}^{-1}(e^{\\lambda f}w_\\alpha ), \\mathcal {F}(\\varphi ).\\mathcal {F}_{\\zeta \\rightarrow \\cdot }\\Big (P(z_0;\|\\zeta \|)^{n+\\lambda }p_\\alpha \\big (x-2\\tfrac{\\zeta \\langle \\zeta ,x\\rangle }{z_0} P(z_0;\|\\zeta \|)\\big )\\Big )\\Big \\rangle _{\\mathbb {R}^n}$ We can expand $p_\\alpha \\big (x-2\\frac{\\zeta \\langle \\zeta ,x\\rangle }{z_0} P(z_0;\|\\zeta \|)\\big )$ so that $P(z_0;\|\\zeta \|)^{n+\\lambda }p_\\alpha \\big (x-2\\tfrac{\\zeta \\langle \\zeta ,x\\rangle }{z_0} P(z_0;\|\\zeta \|)\\big )=\\sum _{r=0}^{m}Q_{r,\\alpha }(\\zeta ,x)z_0^{-r}2^rP(z_0;\|\\zeta \|)^{n+\\lambda +r}$ where $Q_{r,\\alpha }(\\zeta )$ is homogeneous of degree $m$ in $x$ and $2r$ in $\\zeta $ .", "Now we have (for some $C\\ne 0$ independent of $\\lambda ,r,\\alpha $ ) ${\\begin{array}{c}\\frac{2^r}{z_0^r}\\mathcal {F}_{\\zeta \\rightarrow \\xi }(P^{n+\\lambda +r}(z_0;\|\\zeta \|)Q_{r,\\alpha }(\\zeta ,x))=\\\\\\frac{C2^{-\\lambda }z_0^{-\\lambda }}{\\Gamma (\\lambda +n+r)}[Q_{r,\\alpha }(i\\partial _\\zeta ,x)(\|\\zeta \|^{\\lambda +\\tfrac{n}{2}+r}K_{\\lambda +\\tfrac{n}{2}+r}(\|\\zeta \|))]_{\\zeta =z_0\\xi }\\\\\\end{array}}$ where $K_\\nu (\\cdot )$ is the modified Bessel function (see [1]) defined by ${\\begin{array}{c}K_\\nu (z):=\\frac{\\pi }{2}\\frac{(I_{-\\nu }(z)-I_{\\nu }(z))}{\\sin (\\nu \\pi )}\\,\\, \\textrm { if } I_{\\nu }(z):= \\sum _{\\ell =0}^\\infty \\frac{1}{\\ell !\\Gamma (\\ell +\\nu +1)}\\Big (\\frac{z}{2}\\Big )^{2\\ell +\\nu }\\end{array}}$ satisfying that $\|K_\\nu (z)\|=\\mathcal {O}(\\frac{e^{-z}}{\\sqrt{z}})$ as $z\\rightarrow \\infty $ , and for $s\\notin \\mathbb {N}_0$ $\\mathcal {F}((1+\|z\|^2)^{-s})(\\xi )= \\frac{2^{-s+1}(2\\pi )^{n/2}}{\\Gamma (s)}\|\\xi \|^{s-n/2}K_{s-n/2}(\|\\xi \|).$ When $\\lambda \\notin (-\\tfrac{n}{2}+ \\mathbb {Z}) \\cup (-n-\\tfrac{1}{2}\\mathbb {N}_0)$ , we have ${\\begin{array}{c}2^{-\\lambda }z_0^{-\\lambda }Q_{r,\\alpha }(i\\partial _{\\zeta },x)(\|\\zeta \|^{\\lambda +\\tfrac{n}{2}+r}K_{\\lambda +\\tfrac{n}{2}+r}(\|\\zeta \|))\|_{\\zeta =z_0\\xi }=\\frac{ 2^{r+\\tfrac{n}{2}}\\pi z_0^{-\\lambda } }{2\\sin (\\pi (\\lambda +\\tfrac{n}{2}+r))}\\\\\\times \\, \\Big ( \\sum _{\\ell =0}^\\infty \\frac{z_0^{2(\\ell -r)}Q_{r,\\alpha }(i\\partial _\\xi ,x)(\|\\tfrac{1}{2}\\xi \|^{2\\ell })}{\\ell !\\Gamma (\\ell -\\lambda -\\tfrac{n}{2}-r+1)}-z_0^{2\\lambda +n}\\sum _{\\ell =0}^\\infty \\frac{z_0^{2\\ell }Q_{r,\\alpha }(i\\partial _\\xi ,x)(\|\\tfrac{1}{2}\\xi \|^{2(\\lambda +r+\\ell )+n})}{\\ell !\\Gamma (\\ell +\\lambda +\\tfrac{n}{2}+r+1)} \\Big ).\\end{array}}$ Here the powers of $\|\\xi \|$ are homogeneous distributions (note that for $\\lambda \\notin \\mathcal {R}_m$ , the exceptional powers $\|\\xi \|^{-n-j}$ , $j\\in \\mathbb {N}_0$ , do not appear) and the pairing of (REF ) with $\\mathcal {F}^{-1}(e^{\\lambda f}w_\\alpha )\\mathcal {F}(\\varphi )$ makes sense since this distribution is Schwartz as $w_\\alpha $ has compact support.", "We deduce from this expansion that for any $w_\\alpha \\in \\mathcal {D}^{\\prime }(\\mathbb {R}^n)$ supported near 0 and $\\varphi \\in \\mathcal {C}^\\infty _0(\\mathbb {R}^n)$ , when $\\lambda \\notin (-\\tfrac{n}{2}+\\mathbb {Z})\\cup \\mathcal {(}-n-\\tfrac{1}{2}\\mathbb {N}_0)$ $I(z_0,x)=z_0^{-\\lambda }F_-(z_0,x)+z_0^{n+\\lambda }F_+(z_0,x)$ for some smooth function $F_\\pm \\in \\mathcal {C}^\\infty ([0,\\epsilon )\\times \\mathbb {R}^n)$ homogeneous of degree $m$ in $x$ .", "We need to analyze $F_-(0,x)$ , which is obtained by computing the term of order 0 in $\\xi $ in the expansion (REF ) (that is, the terms with $\\ell =r$ in the first sum; note that the terms with $\\ell <r$ in this sum are zero): we obtain for some universal constant $C\\ne 0$ $F_-(0,x)=C\\sum _{\\alpha }\\langle e^{\\lambda f}w_\\alpha ,\\varphi \\rangle _{\\mathbb {R}^n} \\sum _{r=0}^m\\frac{(-1)^{r} 2^{-r}\\Gamma (\\lambda +\\tfrac{n}{2})}{r!\\Gamma (\\lambda +n+r)}Q_{r,\\alpha }(i\\partial _\\xi ,x)(\|\\xi \|^{2r})$ where we have used the inversion formula $\\Gamma (1-z)\\Gamma (z)=\\pi /\\sin (\\pi z)$ and $Q_{r,\\alpha }(i\\partial _\\xi ,x)(\|\\xi \|^{2r})$ is constant in $\\xi $ .", "Using Fourier transform, we notice that $Q_{r,\\alpha }(i\\partial _\\xi ,x)(\|\\xi \|^{2r})=\\Delta _\\zeta ^{r}Q_{r,\\alpha }(\\zeta ,x)\|_{\\zeta =0}= \\Delta _\\zeta ^r( p_\\alpha (x- \\zeta \\langle \\zeta ,x\\rangle ))\|_{\\zeta =0}$ We use Lemma REF to deduce that $F_-(0,x)=C\\sum _{\\alpha }\\langle e^{\\lambda f}w_\\alpha ,\\varphi \\rangle _{\\mathbb {R}^n} p_{\\alpha }(x)m!\\frac{\\Gamma (\\lambda +\\tfrac{n}{2})}{\\Gamma (\\lambda +n+m)}\\sum _{r=0}^m\\frac{(-1)^{r}\\Gamma (\\lambda +n+m)}{(m-r)!\\Gamma (\\lambda +n+r)}.$ The sum over $r$ is a non-zero polynomial of order $m$ in $\\lambda $ , and using the binomial formula, we see that its roots are $\\lambda =-n-m+2,\\dots ,-n+1$ , therefore we deduce that $F_-(0,x)=C\\langle e^{\\lambda f}w,\\varphi \\rangle _{\\mathbb {R}^n} \\frac{\\Gamma (\\lambda +\\tfrac{n}{2})}{(\\lambda +n+m-1)\\Gamma (\\lambda +n-1)}.$ We obtain the claimed result except for $\\lambda \\in -\\tfrac{n}{2}+\\mathbb {N}$ by using that the volume measure on $\\mathbb {S}^n$ is $4^{-n}e^{nf}$ .", "Now assume that $\\lambda =-n/2+j$ with $j\\in \\mathbb {N}$ .", "The Bessel function satisfies for $j\\in \\mathbb {N}$ : $\|\\xi \|^{j}K_{j}(\|\\xi \|)=-\\sum _{\\ell =0}^{j-1}\\frac{(-1)^\\ell 2^{j-1-2\\ell }(j-\\ell -1)!", "}{\\ell !", "}\|\\xi \|^{2\\ell }+\|\\xi \|^{2j}(\\log (\|\\xi \|)L_{j}(\|\\xi \|)+H_j(\|\\xi \|))$ for some function $L_{j},H_j\\in \\mathcal {C}^\\infty (\\mathbb {R}^+)\\cap L^2(\\mathbb {R}^+)$ with $L_j(0)\\ne 0$ .", "Then we apply the same arguments as before and this implies the desired statement.", "We obtain as a corollary: Corollary 6.9 For $m\\in \\mathbb {N}_0$ and $\\lambda \\notin \\mathcal {R}_m$ , the operator $P^-_\\lambda :\\mathcal {D}^{\\prime }(\\mathbb {S}^n;\\otimes _S^m(T^\\mathbb {S}^n)\\cap \\ker \\mathcal {T})\\rightarrow \\mathcal {C}^\\infty (\\mathbb {H}^{n+1}; \\otimes _S^m(T^\\mathbb {H}^{n+1}))$ is injective.", "This corollary immediately implies the injectivity part of Theorem REF in Section REF ." ], [ "Expansions of eigenstates of the Laplacian", "In this section, we show the surjectivity of the Poisson operator $P^-_\\lambda $ (see Theorem REF in Section REF ).", "For that, we take an eigenstate $u$ of the Laplacian on $M$ and lift it to $\\mathbb {H}^{n+1}$ .", "The resulting tensor is tempered and thus expected to have a weak asymptotic expansion at the conformal boundary $\\mathbb {S}^n$ ; a precise form of this expansion is obtained by a careful analysis of both the Laplacian and the divergence-free condition.", "We then show that $u=P^-_\\lambda w$ , where $w$ is some constant times the coefficient of $\\rho ^{-\\lambda }$ in the expansion of $u$ (compare with Lemma REF )." ], [ "Indicial calculus and general weak expansion", "Recall the bundle $E^{(m)}$ defined in (REF ).", "The operator $\\Delta $ acting on $\\mathcal {C}^\\infty (\\overline{\\mathbb {B}^{n+1}}; E^{(m)})$ is an elliptic differential operator of order 2 which lies in the 0-calculus of Mazzeo–Melrose [40], which essentially means that it is an elliptic polynomial in elements of the Lie algebra $\\mathcal {V}_0(\\overline{\\mathbb {B}^{n+1}})$ of smooth vector fields vanishing at the boundary of the closed unit ball $\\overline{\\mathbb {B}^{n+1}}$ .", "Let $\\rho \\in \\mathcal {C}^\\infty (\\overline{\\mathbb {B}^{n+1}})$ be a smooth geodesic boundary defining function (see the paragraph preceding (REF )).", "The theory developped by Mazzeo [39] shows that solutions of $\\Delta u=su$ which are in $\\rho ^{-N}L^2(\\mathbb {B}^{n+1};E^{(m)})$ for some $N$ have weak asymptotic expansions at the boundary $\\mathbb {S}^n=\\partial \\overline{\\mathbb {B}^{n+1}}$ where $\\rho $ is any geodesic boundary defining function.", "To make this more precise, we introduce the indicial family of $\\Delta $ : if $\\lambda \\in \\mathbb {C}, \\nu \\in \\mathbb {S}^n$ , then there exists a family $I_{\\lambda ,\\nu }(\\Delta )\\in {\\rm End}(E^{(m)}(\\nu ))$ depending smoothly on $\\nu \\in \\mathbb {S}^n$ and holomorphically on $\\lambda $ so that for all $u\\in \\mathcal {C}^\\infty (\\overline{\\mathbb {B}^{n+1}}; E^{(m)})$ , $t^{-\\lambda }\\Delta (\\rho ^{\\lambda } u)(\\theta (t,\\nu ))=I_{\\lambda ,\\nu }(\\Delta )u(\\theta (0,\\nu ))+ \\mathcal {O}(t)$ near $\\mathbb {S}^n$ , where the remainder is estimated with respect to the metric $g_H$ .", "Notice that $I_{\\lambda ,\\nu }(\\Delta )$ is independent of the choice of boundary defining function $\\rho $ .", "For $\\sigma \\in \\mathbb {C}$ , the indicial set ${\\rm spec}_b(\\Delta -\\sigma ;\\nu )$ at $\\nu \\in \\mathbb {S}^n$ of $\\Delta -\\sigma $ is the set ${\\rm spec}_b(\\Delta -\\sigma ;\\nu ):=\\lbrace \\lambda \\in \\mathbb {C}\\mid I_{\\lambda ,\\nu }(\\Delta )-\\sigma \\operatorname{Id}\\textrm { is not invertible}\\rbrace .$ Then [39] gives the followingThe full power of [39] is not needed for this lemma.", "In fact, it can be proved in a direct way by viewing the equation $(\\Delta -\\sigma )u=0$ as an ordinary differential equation in the variable $\\log \\rho $ .", "The indicial operator gives the constant coefficient principal part and the remaining terms are exponentially decaying; an iterative argument shows the needed asymptotics.", "Lemma 7.1 Fix $\\sigma $ and assume that $\\mathrm {spec}_b(\\Delta -\\sigma ;\\nu )$ is independent of $\\nu \\in \\mathbb {S}^n$ .", "If $u\\in \\rho ^{\\delta }L^2(\\overline{\\mathbb {B}^{n+1}}; E^{(m)})$ with respect to the Euclidean measure for some $\\delta \\in \\mathbb {R}$ , and $(\\Delta -\\sigma )u=0$ , then $u$ has a weak asymptotic expansion at $\\mathbb {S}^n=\\lbrace \\rho =0\\rbrace $ of the form $u =\\sum _{\\lambda \\in {\\rm spec}_b(\\Delta -\\sigma )\\atop {\\rm Re}(\\lambda )>\\delta -1/2}\\sum _{\\ell \\in \\mathbb {N}_0,\\atop {\\rm Re}(\\lambda )+\\ell <\\delta -1/2+N} \\sum _{p=0}^{k_{\\lambda ,\\ell }}\\rho ^{\\lambda +\\ell }(\\log \\rho )^{p}w_{\\lambda ,\\ell ,p}+\\mathcal {O}(\\rho ^{\\delta +N-\\tfrac{1}{2}-\\epsilon })$ for all $N\\in \\mathbb {N}$ and all $\\epsilon >0$ small, where $k_{\\lambda ,\\ell }\\in \\mathbb {N}_0$ , and $w_{\\lambda ,\\ell ,p}$ are in the Sobolev spaces $w_{\\lambda ,\\ell ,p}\\in H^{-{\\rm Re}(\\lambda )-\\ell +\\delta -\\tfrac{1}{2}}(\\mathbb {S}^n; E^{(m)}).$ Here the weak asymptotic means that for any $\\varphi \\in \\mathcal {C}^{\\infty }(\\mathbb {S}^n)$ , as $t\\rightarrow 0$ $\\begin{split}\\int _{\\mathbb {S}^n} u(\\theta (t,\\nu ))\\varphi (\\nu )dS_\\rho (\\nu )= & \\sum _{\\lambda \\in {\\rm spec}_b(\\Delta -\\sigma )\\atop {\\rm Re}(\\lambda )>\\delta -1/2}\\sum _{\\ell \\in \\mathbb {N}_0\\atop {\\rm Re}(\\lambda )+\\ell <\\delta -1/2+N} \\sum _{p=0}^{k_{\\lambda ,\\ell }}t^{\\lambda +\\ell }\\log (t)^{p}\\langle w_{\\lambda ,\\ell ,p},\\varphi \\rangle \\\\& + \\mathcal {O}(t^{\\delta +N-\\tfrac{1}{2}-\\epsilon })\\end{split}$ where $dS_\\rho $ is measure on $\\mathbb {S}^n$ induced by the metric $(\\rho ^2g_H)\|_{\\mathbb {S}^n}$ and the distributional pairing is with respect to this measure.", "Moreover the remainder $\\mathcal {O}(t^{\\delta +N-\\tfrac{1}{2}-\\epsilon })$ is conormal in the sense that it remains an $\\mathcal {O}(t^{\\delta +N-\\tfrac{1}{2}-\\epsilon })$ after applying any finite number of times the operator $t\\partial _t$ , and it depends on some Sobolev norm of $\\varphi $ .", "Remark.", "The existence of the expansion (REF ) proved by Mazzeo in [39] is independent of the choice of $\\rho $ , but the coefficients in the expansion depend on the choice of $\\rho $ .", "Let $\\lambda _0\\in {\\rm spec}_b(\\Delta -\\sigma )$ with ${\\rm Re}(\\lambda _0)>\\delta -1/2$ be an element in the indicial set and assume that $k_{\\lambda _0,0}=0$ , which means that the exponent $\\rho ^{\\lambda _0}$ in the weak expansion $(\\ref {weakas})$ has no log term.", "Assume also that there is no element $\\lambda \\in {\\rm spec}_b(\\Delta -\\sigma )$ with ${\\rm Re}(\\lambda _0)>{\\rm Re}(\\lambda )>\\delta -1/2$ such that $\\lambda \\in \\lambda _0-\\mathbb {N}$ .", "Then it is direct to see from the weak expansion that for a fixed function $\\chi \\in \\mathcal {C}^\\infty (\\mathbb {B}^{n+1})$ equal to 1 near $\\mathbb {S}^n$ and supported close to $\\mathbb {S}^n$ and for each $\\varphi \\in \\mathcal {C}^\\infty (\\mathbb {B}^{n+1})$ , the Mellin transform $h(\\zeta ):= \\int _{\\mathbb {B}^{n+1}}\\rho (y)^\\zeta \\chi (y)\\varphi (y)u(y)\\, {\\rm dvol}_{g_H}(y),\\quad \\operatorname{Re}\\zeta >n+{1\\over 2}-\\delta ,$ (with values in $E^m$ ) has a meromorphic extension to $\\zeta \\in \\mathbb {C}$ with a simple pole at $\\zeta =n-\\lambda _0$ and residue ${\\rm Res}_{\\zeta =n-\\lambda _0}h(\\zeta )=\\langle w_{\\lambda _0,0,0},\\varphi \|_{\\mathbb {S}^n}\\rangle .$ As an application, if $\\rho ^{\\prime }$ is another geodesic boundary defining function, one has $\\rho =e^{f}\\rho ^{\\prime }+\\mathcal {O}(\\rho ^{\\prime })$ for some $f\\in \\mathcal {C}^\\infty (\\mathbb {S}^n)$ and we deduce that if $w^{\\prime }_{\\lambda _0,0,0}$ is the coefficient of $(\\rho ^{\\prime })^{\\lambda _0}$ in the weak expansion of $u$ using $\\rho ^{\\prime }$ , then as distribution on $\\mathbb {S}^n$ $w^{\\prime }_{\\lambda _0,0,0}=e^{\\lambda _0f}w_{\\lambda _0,0,0}$ In particular, under the assumption above for $\\lambda _0$ (this assumption can similarly be seen to be independent of the choice of $\\rho $ ), if one knows the exponents of the asymptotic expansion, then proving that the coefficient of $\\rho ^{\\lambda _0}$ term is nonzero can be done locally near any point of $\\mathbb {S}^n$ and with any choice of geodesic boundary defining function.", "Finally, if $w_{\\lambda _0,0,0}$ is the coefficient of $\\rho _0^{\\lambda _0}$ in the weak expansion with boundary defining function $\\rho _0$ defined in (REF ) and if $\\gamma ^u=u$ for some hyperbolic isometry $\\gamma \\in G$ , we can use that $\\rho _0\\circ \\gamma =N_\\gamma ^{-1}\\cdot \\rho _0+\\mathcal {O}(\\rho _0^2)$ near $\\mathbb {S}^n$ , together with (REF ) to get $L_{\\gamma }^w_{\\lambda _0,0,0}= N_\\gamma ^{\\lambda _0} w_{\\lambda _0,0,0} \\in \\mathcal {D}^{\\prime }(\\mathbb {S}^n; E^{(m)})$ as distributions on $\\mathbb {S}^n$ (with respect to the canonical measure on $\\mathbb {S}^n$ ) with values in $E^{(m)}$ .", "Here $N_\\gamma $ , $L_\\gamma $ are defined in Section REF .", "If we view $w_{\\lambda _0,0,0}$ as a distribution with values in $\\otimes _S^mT^\\mathbb {S}^n$ , the covariance becomes $L_{\\gamma }^w_{\\lambda _0,0,0}= N_\\gamma ^{\\lambda _0-m} w_{\\lambda _0,0,0} \\in \\mathcal {D}^{\\prime }(\\mathbb {S}^n; \\otimes _S^mT^\\mathbb {S}^n).$ Using the calculations of Section REF , we will compute the indicial family of the Laplacian on $E^{(m)}$ : Lemma 7.2 Let $\\Delta $ be the Laplacian on sections of $E^{(m)}$ .", "Then the indicial set $\\mathrm {spec}_b(\\Delta -\\sigma ,\\nu )$ does not depend on $\\nu \\in \\mathbb {S}^n$ and is equal toOur argument in the next section does not actually use the precise indicial roots, as long as they are independent of $\\nu $ and form a discrete set.", "${\\begin{array}{c}\\bigcup _{k=0}^{\\lfloor {m\\over 2}\\rfloor } \\lbrace \\lambda \\mid -\\lambda ^2+n\\lambda +m+2k(2m+n-2k-2)=\\sigma \\rbrace \\\\\\cup \\bigcup _{k=0}^{\\lfloor {m-1\\over 2}\\rfloor } \\lbrace \\lambda \\mid -\\lambda ^2+n\\lambda +n+3(m-1)+2k(n+2m-2k-4)=\\sigma \\rbrace .\\end{array}}$ We consider an isometry mapping the ball model $\\mathbb {B}^{n+1}$ to the half-plane model $\\mathbb {U}^{n+1}$ which also maps $\\nu $ to 0 and do all the calculations in $\\mathbb {U}^{n+1}$ with the geodesic boundary defining function $z_0$ near 0.", "By (REF ), each tensor $u\\in E^{(m)}$ is determined uniquely by its $E^{(m)}_0$ and $E^{(m)}_1$ components, which are denoted $u_0$ and $u_1$ ; therefore, it suffices to understand how the corresponding components of $I_{\\lambda ,\\nu }(\\Delta )u$ are determined by $u_0,u_1$ .", "We can use the geodesic boundary defining function $\\rho =z_0$ ; note that $\\Delta z_0^\\lambda =\\lambda (n-\\lambda )z_0^\\lambda $ for all $\\lambda \\in \\mathbb {C}$ .", "Assume first that $u$ satisfies $u_1=0$ and $u_0$ is constant in the frame $\\mathcal {S}(Z_I^)$ .", "Then by Lemma REF , ${\\begin{array}{c}\\pi _0(z_0^{-\\lambda }\\Delta (z_0^\\lambda u))=R_0u_0=(\\lambda (n-\\lambda )+m) u_0+m(m-1)\\mathcal {S}(z_0^{-2}h\\otimes \\mathcal {T}(u_0)),\\\\\\pi _1(z_0^{-\\lambda }\\Delta (z_0^\\lambda u))=0.\\end{array}}$ Assume now that $u$ satisfies $u_0=0$ and $u_1$ is constant in the frame $\\mathcal {S}(Z_0^\\otimes Z_J^)$ .", "Then by Lemma REF , ${\\begin{array}{c}\\pi _0(z_0^{-\\lambda }\\Delta (z_0^\\lambda u))=0,\\\\\\pi _1(z_0^{-\\lambda }\\Delta (z_0^\\lambda u))=R_1u_1=(\\lambda (n-\\lambda )+n+3(m-1))u_1\\\\+(m-1)(m-2)\\mathcal {S}(Z_0^\\otimes z_0^{-2}h\\otimes \\mathcal {T}(u^{\\prime }_1)).\\end{array}}$ We see that the indicial operator does not intertwine the $u_0$ and $u_1$ components and it remains to understand for which $\\lambda $ the number $s$ is a root of $R_0$ or $R_1$ .", "Next, we consider the decomposition (REF ), where for $u\\in E^{(m)}_0$ , we define $\\mathcal {I}(u)={(m+2)(m+1)\\over 2}\\mathcal {S}(z_0^{-2}h\\otimes u)$ : $u_0=\\sum _{k=0}^{\\lfloor {m\\over 2}\\rfloor } \\mathcal {I}^k(\\otimes u_0^k),\\quad u_1=\\sum _{k=0}^{\\lfloor {m-1\\over 2}\\rfloor } \\mathcal {S}(Z_0^\\otimes \\mathcal {I}^k(u_1^k)),$ where $u_0^k\\in E^{(m-2k)}_0, u_1^k\\in E^{(m-2k-1)}_0$ are trace-free tensors.", "Using (REF ), we calculate ${\\begin{array}{c}R_0 (\\mathcal {I}^k(u_0^k))=(\\lambda (n-\\lambda )+m)\\mathcal {I}^k(u_0^k)+2\\mathcal {I}(\\mathcal {T}(\\mathcal {I}^k(u_0^k)))\\\\=\\big (-\\lambda ^2+n\\lambda +m+2k(2m+n-2k-2)\\big ) \\mathcal {I}^k(u_0^k),\\\\R_1(\\mathcal {S}(Z_0^\\otimes \\mathcal {I}^k(u_1^k)))\\\\=(\\lambda (n-\\lambda )+n+3(m-1))\\mathcal {S}(Z_0^\\otimes \\mathcal {I}^k(u_1^k))+2\\mathcal {S}(Z_0^\\otimes \\mathcal {I}(\\mathcal {T}(\\mathcal {I}^k(u_1^k))))\\\\=\\big (-\\lambda ^2+n\\lambda +n+3(m-1)+2k(n+2m-2k-4)\\big )\\mathcal {S}(Z_0^\\otimes \\mathcal {I}^k(u_1^k)),\\end{array}}$ which finishes the proof of the lemma." ], [ "Weak expansions in the divergence-free case", "By Lemma REF , we now know that solutions of $\\Delta u=\\sigma u$ which are trace-free symmetric tensors of order $m$ in some weighted $L^2$ space have weak asymptotic expansions at the boundary of $\\overline{\\mathbb {B}^{n+1}}$ with exponents obtained from the indicial set of Lemma REF .", "In fact we can be more precise about the exponents which really appear in the weak asymptotic expansion if we ask that $u$ also be divergence-free: Lemma 7.3 Let $u\\in \\rho ^{\\delta }L^2(\\overline{\\mathbb {B}^{n+1}}; E^{(m)})$ be a trace-free symmetric $m$ -cotensor with $\\rho $ a geodesic boundary defining function and $\\delta \\in (-\\infty ,\\tfrac{1}{2})$ , where the measure is the Euclidean Lebesgue measure on the ball.", "Assume that $u$ is a nonzero divergence-free eigentensor for the Laplacian on hyperbolic space: $\\Delta u=\\sigma u,\\quad \\nabla ^u=0$ for some $\\sigma =m+\\frac{n^2}{4}-\\mu ^2$ with ${\\rm Re}(\\mu )\\in [0,\\frac{n+1}{2}-\\delta )$ and $\\mu \\ne 0$ .", "Then the following weak expansion holds: for all $r\\in [0, m]$ , $N>0$ , and $\\epsilon >0$ small $\\begin{split}(\\iota _{\\rho \\partial _\\rho })^ru&= \\sum _{\\ell \\in \\mathbb {N}_0\\atop {\\rm Re}(-\\mu )+\\ell <N-\\epsilon }\\rho ^{\\tfrac{n}{2}-\\mu +r+\\ell } w^r_{-\\mu ,\\ell } \\\\&+\\sum _{\\ell \\in \\mathbb {N}_0\\atop {\\rm Re}(\\mu )+\\ell <N-\\epsilon }\\sum _{p=0}^{k_{\\mu ,\\ell }}\\rho ^{\\tfrac{n}{2}+\\mu +r+\\ell }\\log (\\rho )^{p}w^r_{\\mu ,\\ell ,p}+\\mathcal {O}(\\rho ^{\\tfrac{n}{2}+N+r-\\epsilon })\\end{split}$ with $w^r_{-\\mu ,\\ell }\\in H^{-\\frac{n}{2}+ {\\rm Re}(\\mu )-r-\\ell +\\delta -\\frac{1}{2}}(\\mathbb {S}^n; E^{(m-r)})$ , $w^r_{\\mu ,\\ell ,p}\\in H^{-\\frac{n}{2}- {\\rm Re}(\\mu )-r-\\ell +\\delta -\\frac{1}{2}}(\\mathbb {S}^n; E^{(m-r)})$ .", "Moreover, if $\\mu \\notin {1\\over 2}\\mathbb {N}_0$ , then $k_{\\mu ,\\ell }=0$ .", "Remarks.", "(i) If $u$ is the lift to $\\mathbb {H}^{n+1}$ of an eigentensor on a compact quotient $M=\\Gamma \\backslash \\mathbb {H}^{n+1}$ , then $u\\in L^\\infty (\\mathbb {B}^{n+1};E^{(m)})$ and so for all $\\epsilon >0$ the following regularity holds $w_{-\\mu ,0}\\in H^{-\\frac{n}{2}+ {\\rm Re}(\\mu )-\\epsilon }(\\mathbb {S}^n; E^{(m)}) , \\quad w_{\\mu ,0,0}\\in H^{-\\frac{n}{2}-{\\rm Re}(\\mu )-\\epsilon }(\\mathbb {S}^n; E^{(m)}).$ (ii) The existence of the expansion (REF ) does not depend on the choice of $\\rho $ .", "For $r=0$ , this follows from analysing the Mellin transform of $u$ as in the remark following Lemma REF .", "For $r>0$ , we additionally use that if $\\rho ^{\\prime }$ is another geodesic boundary defining function, then $\\rho \\partial _\\rho -\\rho ^{\\prime }\\partial _{\\rho ^{\\prime }}\\in \\rho \\cdot {}^0T\\overline{\\mathbb {B}^{n+1}}$ (indeed, the dual covector by the metric is $\\rho ^{-1}d\\rho -(\\rho ^{\\prime })^{-1}d\\rho ^{\\prime }$ and we have $\\rho ^{\\prime }=e^f\\rho $ for some smooth function $f$ on $\\overline{\\mathbb {B}^{n+1}}$ ).", "Therefore, $(\\iota _{\\rho ^{\\prime }\\partial _{\\rho ^{\\prime }}})^r u$ is a linear combination of contractions with 0-vector fields of $\\rho ^{r-r^{\\prime }}(\\iota _{\\rho \\partial _\\rho })^{r^{\\prime }} u$ for $0\\le r^{\\prime }\\le r$ , which have the desired asymptotic expansion.", "Moreover, as follows from (REF ), for each $r\\in [0,m]$ , the condition that $w^{r^{\\prime }}_{-\\mu ,0}=0$ for all $r^{\\prime }\\in [0,r]$ also does not depend on the choice of $\\rho $ , and same can be said about $w^{r^{\\prime }}_{\\mu ,0,0}$ when $\\mu \\notin {1\\over 2}\\mathbb {N}_0$ .", "It suffices to describe the weak asymptotic expansion of $u$ near any point $\\nu \\in \\mathbb {S}^n$ .", "For that, we work in the half-space model $\\mathbb {U}^{n+1}$ by sending $-\\nu $ to $\\infty $ and $\\nu $ to 0 as we did before (composing a rotation of the ball model with the map (REF )).", "Since the choice of geodesic boundary defining function does not change the nature of the weak asymptotic expansion (but only the coefficients), we can take the geodesic boundary defining function $\\rho $ to be equal to $\\rho (z_0,z)=z_0$ inside $\|z\|+z_0<1$ (which corresponds to a neighbourhood of $\\nu $ in the ball model).", "Considering the weak asymptotic (REF ) of $u$ near 0 amounts to taking $\\varphi $ supported near $\\nu $ in $\\mathbb {S}^n$ in (REF ): for instance, if we work in the half-space model we shall consider $\\varphi (z)$ supported in $\|z\|<1$ in the boundary of $\\mathbb {U}^{n+1}$ .", "We decompose $u=\\sum _{k=0}^mu_k$ with $u_k\\in \\rho ^{\\delta }L^2(\\mathbb {U}^{n+1}; E^{(m)}_k)$ and we write $u_k=\\mathcal {S}((Z_0^)^{\\otimes k}\\otimes u^{\\prime }_k)$ for some $u^{\\prime }_k\\in \\rho ^{\\delta }L^2(\\mathbb {U}^{n+1}; E^{(m-k)}_0)$ following what we did in (REF ).", "Now, since $u\\in \\rho ^{\\delta }L^2(\\overline{\\mathbb {B}^{n+1}})=\\rho _0^{\\delta }L^2(\\overline{\\mathbb {B}^{n+1}})$ satisfies $\\Delta u=\\sigma u$ , we deduce from the form of the Laplacian near $\\rho =0$ that $u\\in \\rho _0^{\\delta -2k}H^{2k}(\\overline{\\mathbb {B}^{n+1}}; E^{(m)})$ for all $k\\in \\mathbb {N}$ where $H^k$ denotes the Sobolev space of order $k$ associated to the Euclidean Laplacian on the closed unit ball.", "Then by Sobolev embedding one has that for each $t>0$ , $u\|_{z_0=t}$ belongs to $(1+\|z\|)^{N}L^2(\\mathbb {R}_z^n; E^{(m)})$ for some $N\\in \\mathbb {N}$ and we can consider its Fourier transform in $z$ , as a tempered distribution.Unlike Lemma REF , we only use Fourier analysis here for convenience of notation – all the calculations below could be done with differential operators in $z$ instead.", "Then Fourier transforming the equation $(\\pi _0+\\pi _1)(\\Delta u-\\sigma u)=0$ in the $z$ -variable (recall that $\\pi _i$ is the orthogonal projection on $E^{(m)}_i$ ), and writing the Fourier variable $\\xi $ as $\\xi =\\sum _{i=1}^n\\xi _i dz_i=\\sum _{i=1}^nz_0\\xi _i Z_i^$ , with the notations of Lemma REF , we get ${\\begin{array}{c}\\sum _{I\\in {A}^m}((-(Z_0)^2+nZ_0+z_0^2\|\\xi \|^2+m-\\sigma ) \\hat{f}_I)\\mathcal {S}(Z^_{I})+2i\\sum _{J\\in {A}^{m-1}} \\hat{g}_J\\mathcal {S}(\\xi \\otimes Z_J^)\\\\+m(m-1)\\sum _{I}\\hat{f}_I \\mathcal {S}(z_0^{-2}h\\otimes \\mathcal {T}(\\mathcal {S}(Z_I^)))=0.\\end{array}}$ and ${\\begin{array}{c}\\sum _{J\\in A^{m-1}}((-(Z_0)^2+nZ_0+z_0^2\|\\xi \|^2+n+3(m-1)-\\sigma ) \\hat{g}_J)\\mathcal {S}(Z^_J)\\\\-2im\\sum _{I\\in A^m} \\hat{f}_I \\iota _\\xi \\mathcal {S}(Z_I^)-2im(m-1)\\sum _{I\\in A^m}\\hat{f}_I\\mathcal {S}(\\xi \\otimes \\mathcal {T}(\\mathcal {S}(Z_I^)))\\\\+(m-1)(m-2)\\sum _{J\\in A^{m-1}}\\hat{g}_J \\mathcal {S}(z_0^{-2}h\\otimes \\mathcal {T}(\\mathcal {S}(Z_J^)))=0.\\end{array}}$ where hat denotes Fourier transform in $z$ and $\\iota _\\xi $ means $\\sum _{j=1}^nz_0\\xi _j\\iota _{Z_j}$ .", "Similarly we Fourier transform in $z$ the equation $(\\pi _0+\\pi _1)(\\nabla ^u)=0$ using Lemma REF to obtain ${\\begin{array}{c}\\sum _{I\\in {A}^{m}}i\\hat{f}_I\\iota _{\\xi }\\mathcal {S}(Z_I^)=\\frac{1}{m}\\sum _{J\\in {A}^{m-1}}((n+m-1)\\hat{g}_J -Z_0(\\hat{g}_J))\\mathcal {S}(Z_J^),\\\\\\sum _{I\\in {A}^{m}} (Z_0\\hat{f}_I-(n+m-1)\\hat{f}_I)\\mathcal {T}(\\mathcal {S}(Z_I^))=\\frac{1}{m}\\sum _{J\\in {A}^{m-1}}i\\hat{g}_J\\iota _{\\xi }\\mathcal {S}(Z_J^).\\end{array}}$ Now, we use the correspondence between symmetric tensors and homogeneous polynomials to facilitate computations, as explained in Section REF and in the proof of Lemma REF ; that is, to $\\mathcal {S}(Z^_I)$ , we associate the polynomial $x_I$ on $\\mathbb {R}^n$ .", "If $\\xi \\in \\mathbb {R}^n$ is a fixed element and $u\\in {\\rm Pol}^m(\\mathbb {R}^n)$ , we write $\\partial _\\xi u=du.\\xi \\in {\\rm Pol}^{m-1}(\\mathbb {R}^n)$ for the derivative of $u$ in the direction of $\\xi $ and $\\xi ^ u$ for the element $\\langle \\xi ,\\cdot \\rangle _{\\mathbb {R}^n} u\\in {\\rm Pol}^{m+1}(\\mathbb {R}^n)$ .", "The trace map $\\mathcal {T}$ becomes $-\\tfrac{1}{(m(m-1))}\\Delta _x$ .", "We define $\\hat{u}_0 :=\\sum _{I\\in {A}^m}\\hat{f}_Ix_I$ and $\\hat{u}_1=\\sum _{J\\in {A}^{m-1}}\\hat{g}_Jx_J$ .", "The elements $\\hat{f}_I(z_0,\\xi ),\\hat{g}_I(z_0,\\xi )$ belong to the space $\\mathcal {C}^\\infty (\\mathbb {R}^+_{z_0}; {S}^{\\prime }(\\mathbb {R}^n_\\xi ))$ .", "We decompose them as $\\hat{u}_0 =\\sum _{j=0}^{\\lfloor {m\\over 2}\\rfloor }\|x\|^{2j}\\hat{u}_0^{2j},\\quad \\hat{u}_1=\\sum _{j=0}^{\\lfloor {m-1\\over 2}\\rfloor }\|x\|^{2j}\\hat{u}_1^{2j}$ for some $\\hat{u}_i^{2j}\\in {\\rm Pol}_0^{m-i-2j}(\\mathbb {R}^n)$ (harmonic in $x$ , that is trace-free).", "Using the homogeneous polynomial description of $u_0$ , equation (REF ) becomes ${\\begin{array}{c}(-(Z_0)^2+nZ_0+z_0^2\|\\xi \|^2+m-\\sigma ) \\hat{u}_0+2iz_0\\xi ^* \\hat{u}_1-\|x\|^2\\Delta _x\\hat{u}_0=0.\\end{array}}$ First, if $W$ is a harmonic homogeneous polynomial in $x$ of degree $j$ , one has $\\Delta _x(\\xi ^* W)=-2\\partial _{\\xi }W$ and $\\Delta ^2_x(\\xi ^* W)=0$ , thus one can write $\\xi ^* W=\\Big (\\xi ^* W-\\frac{\\partial _{\\xi }W}{n+2(j-1)}\|x\|^2\\Big )+\\frac{\\partial _{\\xi }W}{n+2(j-1)}\|x\|^2$ for the decomposition (REF ) of $\\xi ^* W$ .", "In particular, one can write the decomposition (REF ) of $\\xi ^* \\hat{u}_1$ as $\\xi ^* \\hat{u}_1 =\\sum _{j=0}^{\\lfloor {m-1\\over 2}\\rfloor }\|x\|^{2j}\\Big (\\xi ^* \\hat{u}^{2j}_1-\\frac{\\partial _{\\xi } \\hat{u}^{2j}_1}{n+2(m-2-2j)}\|x\|^2+\\frac{\\partial _{\\xi } \\hat{u}^{2(j-1)}_1}{n+2(m-2j)}\\Big )$ We can write $\\Delta _x \\hat{u}_0=\\sum _{j=0}^{\\lfloor m/2\\rfloor }\\lambda _j \|x\|^{2j-2}\\hat{u}^{2j}_0$ for $\\lambda _j=-2j(n+2(m-j-1))$ .", "Thus (REF ) gives for $j\\le \\lfloor m/2\\rfloor $ ${\\begin{array}{c}(-(Z_0)^2+nZ_0+z_0^2\|\\xi \|^2+m-\\sigma -\\lambda _j) \\hat{u}^{2j}_0\\\\+2iz_0 \\Big (\\xi ^* \\hat{u}^{2j}_1-\\frac{\|x\|^2\\partial _{\\xi } \\hat{u}^{2j}_1}{n+2(m-2-2j)}+\\frac{\\partial _{\\xi } \\hat{u}^{2(j-1)}_1}{n+2(m-2j)}\\Big )=0.\\end{array}}$ Notice that $\\iota _{\\xi }(\\mathcal {S}(Z^_I))$ corresponds to the polynomial $\\frac{z_0}{m}dx_I.\\xi =\\frac{z_0}{m}\\partial _\\xi .x_I$ if $I\\in {A}^m$ .", "From (REF ) we thus have for $c_m:=n+m-1$ $\\begin{split}-iz_0\\partial _\\xi \\hat{u}_0=& (Z_0-c_m)\\hat{u}_1, \\\\-iz_0\\partial _\\xi \\hat{u}_1= & (Z_0-c_m)\\Delta _x\\hat{u}_0.\\end{split}$ Next, (REF ) implies ${\\begin{array}{c}(-(Z_0)^2+nZ_0+z_0^2\|\\xi \|^2+n+3(m-1)-\\sigma )\\hat{u}_1-2iz_0\\partial _\\xi \\hat{u}_0\\\\+2iz_0\\xi ^ \\Delta _x\\hat{u}_0-\|x\|^2\\Delta _x \\hat{u}_1=0.\\end{array}}$ Using (REF ), this can be rewritten as ${\\begin{array}{c}(-(Z_0)^2+(n+2)Z_0+z_0^2\|\\xi \|^2-n+m-1-\\sigma ) \\hat{u}_1\\\\+2iz_0\\xi ^* \\Delta _x\\hat{u}_0-\|x\|^2\\Delta _x\\hat{u}_1=0.\\end{array}}$ We can write $\\Delta _x \\hat{u}_1=\\sum _{j=0}^{[(m-1)/2]}\\lambda ^{\\prime }_j \|x\|^{2j-2}\\hat{u}^{2j}_1$ for $\\lambda ^{\\prime }_j=-2j(n+2(m-j-2))$ .", "We get from (REF ) ${\\begin{array}{c}\\Big (-(Z_0)^2+(n+2)Z_0+z_0^2\|\\xi \|^2-n+m-1-\\sigma -\\lambda ^{\\prime }_j\\Big ) \\hat{u}^{2j}_1\\\\+2iz_0\\Big (\\lambda _{j+1}\\xi ^* \\hat{u}^{2(j+1)}_0-\\frac{\\lambda _{j+1}\\partial _{\\xi } \\hat{u}^{2(j+1)}_0}{n+2(m-3-2j)}\|x\|^2+\\frac{\\lambda _j\\partial _{\\xi } \\hat{u}^{2j}_0}{n+2(m-1-2j)}\\Big )=0.\\end{array}}$ We shall now partially uncouple the system of equations for $\\hat{u}_0^{2j}$ and $\\hat{u}_1^{2j}$ .", "Using (REF ) and applying the decomposition (REF ), we have $\\partial _\\xi (\|x\|^{2j}\\hat{u}_0^{2j})=\|x\|^{2j}\\partial _\\xi \\hat{u}^{2j}_0\\frac{n+2(m-j-1)}{n+2(m-2j-1)}+2j\|x\|^{2j-2}\\Big (\\xi ^* \\hat{u}_0^{2j}-\\frac{\|x\|^2\\partial _{\\xi }\\hat{u}_0^{2j}}{n+2(m-2j-1)}\\Big )$ $\\partial _\\xi (\|x\|^{2j}\\hat{u}_1^{2j})=\|x\|^{2j}\\partial _\\xi \\hat{u}^{2j}_1\\frac{n+2(m-j-2)}{n+2(m-2j-2)}+2j\|x\|^{2j-2}\\Big (\\xi ^* \\hat{u}_1^{2j}-\\frac{\|x\|^2\\partial _{\\xi }\\hat{u}_1^{2j}}{n+2(m-2j-2)}\\Big )$ and from (REF ), this implies that for $j\\ge 0$ ${\\begin{array}{c}(Z_0-c_m)\\hat{u}_1^{2j}=\\\\-iz_0\\Big (\\partial _\\xi \\hat{u}_0^{2j}\\frac{n+2(m-j-1)}{n+2(m-2j-1)}+2(j+1)\\big (\\xi ^* \\hat{u}_0^{2(j+1)}-\\frac{\|x\|^2\\partial _{\\xi }\\hat{u}_0^{2(j+1)}}{n+2(m-2j-3)}\\big )\\Big ),\\end{array}}$ and for $j>0$ ${\\begin{array}{c}(Z_0-c_m)\\hat{u}_0^{2j}=\\\\iz_0\\Big (\\frac{\\partial _\\xi \\hat{u}_1^{2(j-1)}}{2j(n+2(m-2j))}+\\frac{1}{n+2(m-j-1)}\\big (\\xi ^* \\hat{u}_1^{2j}-\\frac{\|x\|^2\\partial _{\\xi }\\hat{u}_1^{2j}}{n+2(m-2j-2)}\\big )\\Big ).\\end{array}}$ Combining with (REF ) and (REF ) we get for $j\\ge 0$ ${\\begin{array}{c}(-(Z_0)^2+(n+4j)Z_0+z_0^2\|\\xi \|^2+m-\\sigma -\\lambda _j -4jc_m) \\hat{u}^{2j}_0\\\\+2iz_0\\frac{n+2(m-2j-1)}{n+2(m-j-1)}\\Big (\\xi ^* \\hat{u}^{2j}_1-\\frac{\|x\|^2\\partial _{\\xi } \\hat{u}^{2j}_1}{n+2(m-2-2j)}\\Big )=0,\\end{array}}$ ${\\begin{array}{c}(-(Z_0)^2+(n+2+4j)Z_0+z_0^2\|\\xi \|^2-n+m-1-\\sigma -\\lambda _j^{\\prime }-4jc_m) \\hat{u}^{2j}_1\\\\+2iz_0(\\lambda _{j+1}+4j(j+1))\\Big (\\xi ^* \\hat{u}^{2(j+1)}_0-\\frac{\|x\|^2\\partial _{\\xi } \\hat{u}^{2(j+1)}_0}{n+2(m-3-2j)}\\Big )=0,\\end{array}}$ ${\\begin{array}{c}(-(Z_0)^2+(n+2-\\tfrac{\\lambda _{j+1}}{j+1})Z_0+z_0^2\|\\xi \|^2-n+m-1-\\sigma +\\tfrac{\\lambda _{j+1}}{j+1}(c_m-j)) \\hat{u}^{2j}_1\\\\+2iz_0\\frac{(n+2(m-j-1))(n+2(m-2j-2))}{n+2(m-2j-1)}\\partial _\\xi \\hat{u}_0^{2j}=0\\end{array}}$ and for $j>0$ ${\\begin{array}{c}(-Z_0^2+(n-\\tfrac{\\lambda _j}{j})Z_0+z_0^2\|\\xi \|^2+m-\\sigma +\\tfrac{\\lambda _j}{j}(c_m-j)) \\hat{u}^{2j}_0\\\\-iz_0\\frac{2(m-1-2j)+n}{j(n+2(m-2j))}\\partial _\\xi \\hat{u}_1^{2(j-1)}=0.\\end{array}}$ To prove the lemma, we will show the following weak asymptotic expansion for $i=0,1$ : ${\\begin{array}{c}\\langle \\hat{u}_i^{2j}(z_0,\\cdot ), \\hat{\\varphi }\\rangle = \\sum _{\\ell \\in \\mathbb {N}_0,\\atop {\\rm Re}(-\\mu )+\\ell <N-\\epsilon }z_0^{\\tfrac{n}{2}-\\mu +2j+i+\\ell }\\langle \\widetilde{w}^{2j}_{i;-\\mu ,\\ell },\\varphi \\rangle \\\\+\\sum _{\\ell \\in \\mathbb {N}_0,\\atop {\\rm Re}(\\mu )+\\ell <N-\\epsilon }\\sum _{p=0}^{k_{\\mu ,\\ell }}z_0^{\\tfrac{n}{2}+\\mu +2j +i+\\ell }\\log (z_0)^{p}\\langle \\widetilde{w}^{2j}_{i;\\mu ,\\ell ,p},\\varphi \\rangle +\\mathcal {O}(z_0^{\\tfrac{n}{2}+2j+i+N-\\epsilon }),\\end{array}}$ where $\\widetilde{w}^{2j}_{i;-\\mu ,\\ell }$ and $\\widetilde{w}^{2j}_{i;\\mu ;\\ell ,p}$ are distributions in some Sobolev spaces in $\\lbrace \|z\|<1\\rbrace \\subset \\mathbb {R}^n$ and for $\\mu \\notin {1\\over 2}\\mathbb {N}_0$ , we have $k_{\\mu ,\\ell }=0$ .", "Define for $0\\le r\\le m$ and $\\varphi \\in \\mathcal {C}_0^\\infty (\\mathbb {R}^n)$ supported in $\\lbrace \|z\|<1\\rbrace $ , $F^r(\\varphi )(z_0):={\\left\\lbrace \\begin{array}{ll}\\langle \\hat{u}_0^{r}(z_0,\\cdot ),\\hat{\\varphi }\\rangle ,&\\text{$r$ is even};\\\\\\langle \\hat{u}_1^{r-1}(z_0,\\cdot ),\\hat{\\varphi }\\rangle ,&\\text{$r$ is odd}.\\\\\\end{array}\\right.", "}$ Since $\\hat{u}_i^{r-i}$ is the Fourier transform in $z$ of iterated traces of $u_i$ , Lemma REF gives that the function $F^r(\\varphi )(z_0)$ satisfies for all $N\\in \\mathbb {N}$ , $\\epsilon >0$ $F^r(\\varphi )(z_0) = \\sum _{\\lambda \\in {\\rm spec}_b(\\Delta -\\sigma )\\atop {\\rm Re}(\\lambda )>\\delta -1/2}\\sum _{\\ell \\in \\mathbb {N}_0,\\atop {\\rm Re}(\\lambda )+\\ell <N-\\epsilon } \\sum _{p=0}^{k_{\\lambda ,\\ell }^r}z_0^{\\lambda +\\ell }\\log (z_0)^{p}\\langle w_{\\lambda ,\\ell ,p}^r,\\varphi \\rangle +\\mathcal {O}(z_0^{N-\\epsilon })$ as $z_0\\rightarrow 0$ , and some $w_{\\lambda ,\\ell ,p}^r$ in some Sobolev space on $\\lbrace \|z\|<1\\rbrace $ .", "We pair (REF ), (REF ) with $\\hat{\\varphi }$ , and it is direct to see that we obtain a differential equation in $z_0$ of the form $P^r(Z_0)F^r(\\varphi )(z_0)=-z_0^2 F^r(\\Delta \\varphi )(z_0)+z_0F^{r+1}(Q^r\\varphi )(z_0)$ for $Z_0=z_0\\partial _{z_0}$ , $P^r(\\lambda ):=-\\lambda ^2+(n+2r)\\lambda -r(n+r)-{n^2\\over 4}+\\mu ^2=-\\Big (\\lambda -{n\\over 2}-r\\Big )^2+\\mu ^2,$ and $Q^r$ some differential operator of order 1 with values in homomorphisms on the space of polynomials in $x$ .", "Here we denote $F^{m+1}=0$ .", "We now show the expansion (REF ) by induction on $r=2j+i=m,m-1,\\dots , 0$ .", "By plugging the expansion (REF ) in the equation (REF ) and using ${\\begin{array}{c}P^r(Z_0)z_0^{\\lambda }\\log (z_0)^p=z_0^\\lambda \\big (P_0^r(\\lambda )(\\log z_0)^p+p\\partial _\\lambda P_0^r(\\lambda )(\\log z_0)^{p-1}\\\\+\\mathcal {O}((\\log z_0)^{p-2})\\big )\\end{array}} $ we see that if for some $p$ , $z_0^{\\lambda }(\\log z_0)^p$ is featured in the asymptotic expansion of $F^r(\\varphi )(z_0)$ , then either $\\lambda \\in n/2+r-\\mu +\\mathbb {N}_0$ , or $\\lambda \\in n/2+r+\\mu +\\mathbb {N}_0$ , or $z_0^{\\lambda -2}(\\log z_0)^p$ is featured in the expansion of $F^r(\\Delta \\varphi )(z_0)$ .", "Moreover, if $p>0$ and $\\lambda \\notin \\lbrace n/2+r\\pm \\mu \\rbrace $ , then either $z_0^{\\lambda }(\\log z_0)^{p^{\\prime }}$ is featured in $F^r(\\varphi )(z_0)$ for some $p^{\\prime }>p$ , or $z_0^{\\lambda -2}(\\log z_0)^p$ is featured in $F^r(\\Delta \\varphi )(z_0)$ , or $z_0^{\\lambda -1}(\\log z_0)^p$ is featured in $F^{r+1}(Q^r\\varphi )(z_0)$ .", "If $p>0$ and $\\lambda = n/2+r\\pm \\mu $ , then (since $\\mu \\ne 0$ and thus $\\partial _\\lambda P_0^r(\\lambda )\\ne 0$ ) either $z_0^{\\lambda }(\\log z_0)^{p^{\\prime }}$ is featured in $F^r(\\varphi )(z_0)$ for some $p^{\\prime }>p$ , or $z_0^{\\lambda -2}(\\log z_0)^{p-1}$ is featured in $F^r(\\Delta \\varphi )(z_0)$ , or $z_0^{\\lambda -1}(\\log z_0)^{p-1}$ is featured in $F^{r+1}(Q^r\\varphi )(z_0)$ , however the latter two cases are only possible when $\\lambda =n/2+r+\\mu $ and $\\mu \\in {1\\over 2}\\mathbb {N}_0$ .", "Together these facts (applied to $\\varphi $ as well as its images under combinations of $\\Delta $ and $Q^r$ ) imply that the weak expansion of $u^{2j}_i$ has the form (REF ).", "The asymptotic expansions (REF ) now follow from (REF ) since $\\rho \\partial _\\rho =Z_0$ for our choice of $\\rho $ and for each $r\\in [0,m]$ , by (REF ) and (REF ) we see that (identifying symmetric tensors with homogeneous polynomials in $(x_0,x)$ ) $(\\iota _{Z_0})^r u(x_0,x)=\\sum _{r^{\\prime }=r}^m \\sum _{s\\ge 0\\atop r^{\\prime }+2s\\le m}c_{m,r,r^{\\prime },s}\\,x_0^{r^{\\prime }-r}\|x\|^{2s} u_{r^{\\prime }-2\\lfloor r^{\\prime }/2\\rfloor }^{2\\lfloor r^{\\prime }/2\\rfloor +2s}(x)$ for some constants $c_{m,r,r^{\\prime },s}$ ; for later use, we also note that $c_{m,r,r,0}\\ne 0$ ." ], [ "Surjectivity of the Poisson operator", "In this section, we prove the surjectivity part of Theorem REF in Section REF (together with the injectivity part established in Corollary REF , this finishes the proof of that theorem).", "The remaining essential component of the proof is showing that unless $u\\equiv 0$ , a certain term in the asymptotic expansion of Lemma REF is nonzero (in particular we will see that $u$ cannot be vanishing to infinite order on $\\mathbb {S}^n$ in the weak sense).", "We start with Lemma 7.4 Take some $u$ satisfying (REF ).", "Assume that for all $r\\in [0,m]$ , the coefficient $w^r_{-\\mu ,0}$ of the weak expansion (REF ) is zero.", "(By Remark (ii) following Lemma REF , this condition is independent of the choice of $\\rho $ .)", "Then $u\\equiv 0$ .", "If $\\mu \\notin {1\\over 2}\\mathbb {N}_0$ , then we can replace $w^r_{-\\mu ,0}$ by $w^r_{\\mu ,0,0}$ in the assumption above.", "We choose some $\\nu \\in \\mathbb {S}^n$ and transform $\\mathbb {B}^{n+1}$ to the half-space model as explained in the proof of Lemma REF , and use the notation of that proof.", "Define the function $f\\in \\mathcal {C}^\\infty (\\mathbb {B}^{n+1})$ in the half-space model as follows: $f={\\left\\lbrace \\begin{array}{ll}z_0^{-m}u^{2m}_0&\\quad \\text{if $m$ is even};\\\\z_0^{-m}u^{2m-1}_1&\\quad \\text{if $m$ is odd}.\\end{array}\\right.", "}$ Here $u^{2j}_0$ , $u^{2j}_1$ are obtained by taking the inverse Fourier transform of $\\hat{u}^{2j}_0,\\hat{u}^{2j}_1$ .", "By (REF ), (REF ) (see also (REF )) we have $(\\Delta _{\\mathbb {H}^{n+1}}-n^2/4+\\mu ^2)f= 0.$ Denote by $\\mathcal {C}_{\\rm temp}^\\infty (\\mathbb {B}^{n+1})$ the set of smooth functions $f$ in $\\mathbb {B}^{n+1}$ which are tempered in the sense that there exists $N\\in \\mathbb {R}$ such that $\\rho _0^{N}f\\in L^2(\\mathbb {B}^{n+1})$ .", "Set $\\lambda :=-n/2+\\mu $ ; it is proved in [4], [48] (see also [27] for a simpler presentation in the case $\|{\\rm Re}(\\lambda )+n/2\|<n/2$ ) that the Poisson operator acting on distributions on hyperbolic space is an isomorphism $P^-_\\lambda : \\mathcal {D}^{\\prime }(\\mathbb {S}^n)\\rightarrow \\ker (\\Delta _{\\mathbb {H}^{n+1}}+\\lambda (n+\\lambda ))\\cap \\mathcal {C}_{\\rm temp}^\\infty (\\mathbb {B}^{n+1})$ for $\\lambda \\notin -n-\\mathbb {N}_0$ , and if ${\\rm Re}(\\lambda )\\ge -n/2$ with $\\lambda \\ne 0$ any element $v\\in \\mathcal {C}_{\\rm temp}^\\infty (\\mathbb {B}^{n+1})$ with $(\\Delta _{\\mathbb {H}^{n+1}}+\\lambda (n+\\lambda ))v=0$ and $v\\lnot \\equiv 0$ satisfies a weak expansion for any $N\\in \\mathbb {N}$ $v=P^-_\\lambda (v_{-\\mu ,\\ell })=\\sum _{\\ell =0}^N\\Big (\\rho _0^{n/2-\\mu +\\ell }v_{-\\mu ,\\ell }+\\sum _{p=1}^{k_{\\mu ,\\ell }}\\rho _0^{n/2+\\mu +\\ell }\\log (\\rho _0)^pv_{\\mu ,\\ell ,p}\\Big )+\\mathcal {O}(\\rho _0^{n/2-\\mu +N})$ with $v_{-\\mu ,0}\\lnot \\equiv 0$ ; moreover $k_{\\mu ,\\ell }=0$ if $\\lambda \\notin -\\tfrac{n}{2}+\\tfrac{1}{2}\\mathbb {N}_0$ , and $v_{\\mu ,0,0}\\ne 0$ for such $\\lambda $ (here $v_{-\\mu ,\\ell },v_{\\mu ,\\ell ,p}$ are distributions on $\\mathbb {S}^n$ as before).The existence of the weak expansion with known coefficients for elements in the image of $P^-_\\lambda $ is directly related to the special case $m=0$ of Lemma REF and the existence of a weak expansion for scalar eigenfunctions of the Laplacian follows from the $m=0$ case of Lemma REF .", "However, neither the surjectivity of the scalar Poisson operator nor the fact that eigenfunctions have nontrivial terms in their weak expansions follows from these statements.", "Next, by (REF ), for some nonzero constant $c$ we have $f=c (z_0^{-1}\\iota _{Z_0})^m u=c\\langle u,\\otimes ^m\\partial _{z_0}\\rangle .$ A calculation using (REF ) shows that in the ball model, using the geodesic boundary defining function $\\rho _0$ from (REF ), $\\partial _{z_0}=-\\bigg ({1-\|y\|^2\\over 2} \\,\\nu +(1+y\\cdot \\nu )\\,y\\bigg )\\partial _y$ is a $\\mathcal {C}^\\infty (\\overline{\\mathbb {B}^{n+1}})$ -linear combination of $\\partial _{\\rho _0}$ and a 0-vector field.", "It follows from the form of the expansion (REF ) and the assumption of this lemma that the coefficient of $\\rho _0^{{n\\over 2}-\\mu }$ of the weak expansion of $f$ is zero.", "(If $\\mu \\notin {1\\over 2}\\mathbb {N}_0$ , then we can also consider instead the coefficient of $\\rho _0^{{n\\over 2}+\\mu }$ .)", "By (REF ) and the surjectivity of the scalar Poisson kernel discussed above, we now see that $f\\equiv 0$ .", "Now, for each fixed $y\\in \\mathbb {B}^{n+1}$ and each $\\eta \\in T_y\\mathbb {B}^{n+1}$ , we can choose $\\nu $ such that $\\eta $ is a multiple of (REF ) at $y$ ; in fact, it suffices to take $\\nu $ so that the geodesic $\\varphi _t(y,\\eta )$ converges to $-\\nu $ as $t\\rightarrow +\\infty $ .", "Therefore, for each $y,\\eta $ , we have $\\langle u,\\otimes ^m \\eta \\rangle =0$ at $y$ .", "Since $u$ is a symmetric tensor, this implies $u\\equiv 0$ .", "We now relax the assumptions of Lemma REF to only include the term with $r=0$ : Lemma 7.5 Take some $u$ satisfying (REF ).", "If $n=1$ and $m>0$ , then we additionally assume that $\\mu \\ne {1\\over 2}$ .", "Assume that the coefficient $w^0_{-\\mu ,0}$ of the weak expansion (REF ) is zero.", "(By Remark (ii) following Lemma REF , this condition is independent of the choice of $\\rho $ .)", "Then $u\\equiv 0$ .", "If $\\mu \\notin {1\\over 2}\\mathbb {N}_0$ , then we can replace $w^0_{-\\mu ,0}$ by $w^0_{\\mu ,0,0}$ in our assumption.", "Assume that $w^0_{\\pm \\mu ,0}=0$ ; here we consider the case of $w^0_{\\mu ,0}:=w^0_{\\mu ,0,0}$ only when $\\mu \\notin {1\\over 2}\\mathbb {N}_0$ .", "By Lemma REF , it suffices to prove that $w^r_{\\pm \\mu ,0}=0$ for $r=0,\\dots ,m$ .", "This is a local statement and we use the half-plane model and the notation of the proof of Lemma REF .", "By (REF ), it then suffices to show that if $\\widetilde{w}^0_{0;\\pm \\mu ,0}=0$ in the expansion (REF ), then $\\widetilde{w}^{2j}_{i;\\pm \\mu ,0}=0$ for all $i,j$ .", "We argue by induction on $r=2j+i=0,\\dots ,m$ .", "Assume first that $i=0$ , $j>0$ , and $\\widetilde{w}^{2(j-1)}_{1;\\pm \\mu ,0}=0$ .", "Then we plug (REF ) into (REF ) and consider the coefficient next to $z_0^{{n\\over 2}\\pm \\mu +2j}$ ; this gives $\\widetilde{w}^{2j}_{0;\\pm \\mu ,0}=0$ if for $\\lambda ={n\\over 2}\\pm \\mu +2j$ , the following constant is nonzero: ${\\begin{array}{c}-\\lambda ^2+\\Big (n-{\\lambda _j\\over j}\\Big )\\lambda +m-\\sigma +{\\lambda _j\\over j}(c_m-j)\\\\=(n+2m-2-4j)(\\pm 2\\mu -n-2m+2+4j).\\end{array}}$ We see immediately that (REF ) is nonzero unless $m=2j$ .", "For the case $m=2j$ , we can use (REF ) directly; taking the coefficient next to $z_0^{{n\\over 2}\\pm \\mu +m}$ , we get $\\widetilde{w}^{2j}_{0;\\pm \\mu ,0}=0$ as long as ${n\\over 2}\\pm \\mu +m\\ne c_m$ , or equivalently $\\pm \\mu \\ne {n\\over 2}-1$ ; the latter inequality is immediately true unless $n=1$ , and it is explicitely excluded by the statement of the present lemma when $n=1$ .", "Similarly, assume that $i=1$ , $0\\le 2j<m$ , and $\\widetilde{w}^{2j}_{0;\\pm \\mu ,0}=0$ .", "Then we plus (REF ) into (REF ) and consider the coefficient next to $z_0^{{n\\over 2}\\pm \\mu +2j+1}$ ; this gives $\\widetilde{w}^{2j}_{1;\\pm \\mu ,0}=0$ if for $\\lambda ={n\\over 2}\\pm \\mu +2j+1$ , the following constant is nonzero: ${\\begin{array}{c}-\\lambda ^2+(n+2-\\tfrac{\\lambda _{j+1}}{j+1})\\lambda -n+m-1-\\sigma +\\tfrac{\\lambda _{j+1}}{j+1}(c_m-j)\\\\=(n+2m-4-4j)(\\pm 2\\mu -n-2m+4+4j).\\end{array}}$ We see immediately that (REF ) is nonzero unless $m=2j+1$ .", "For the case $m=2j+1$ , we can use (REF ) directly; taking the coefficient next to $z_0^{{n\\over 2}\\pm \\mu +m}$ , we get $\\widetilde{w}^{2j}_{1;\\pm \\mu ,0}=0$ as long as ${n\\over 2}\\pm \\mu +m\\ne c_m$ , which we have already established is true.", "We finish the section by the following statement, which immediately implies the surjectivity part of Theorem REF .", "Note that for the lifts of elements of $\\operatorname{Eig}^m(-\\lambda (n+\\lambda )+m)$ , we can take any $\\delta <1/2$ below.", "The condition $\\operatorname{Re}\\lambda <{1\\over 2}-\\delta $ for $m>0$ follows from Lemma REF .", "Corollary 7.6 Let $u\\in \\rho ^{\\delta }L^2(\\overline{\\mathbb {B}^{n+1}}; E^{(m)})$ be a trace-free symmetric $m$ -cotensor with $\\rho $ a geodesic boundary defining function and $\\delta \\in (-\\infty ,\\tfrac{1}{2})$ , where the measure is the Euclidean Lebesgue measure on the ball.", "Assume that $u$ is a nonzero divergence-free eigentensor for the Laplacian on hyperbolic space: $\\Delta u=(-\\lambda (n+\\lambda )+m) u,\\quad \\nabla ^u=0$ with ${\\rm Re}(\\lambda )<\\frac{1}{2}-\\delta $ and $\\lambda \\notin \\mathcal {R}_m$ , with $\\mathcal {R}_m$ defined in (REF ).", "Then there exists $w \\in H^{{\\rm Re}(\\lambda )+\\delta -\\frac{1}{2}}(\\mathbb {S}^n;\\otimes _S^mT^\\mathbb {S}^n)$ such that $u=P^-_\\lambda (w)$ .", "Moreover if $\\gamma ^u=u$ for some $\\gamma \\in G$ , then $L_\\gamma ^w=N_\\gamma ^{-\\lambda -m}w$ .", "For the case ${\\rm Re}(\\lambda )\\ge -n/2$ we set $\\mu =n/2+\\lambda $ and apply Lemma REF : the distribution $w$ will be given by $C(\\lambda )w_{-\\mu ,0}$ for some constant $C(\\lambda )$ to be chosen, and this has the desired covariance with respect to elements of $G$ by using (REF ) from the Remark after Lemma REF .", "To see that $u=P^-_\\lambda (w)$ for a certain $C(\\lambda )$ , it suffices to use the weak expansion in Lemma REF and the identity (REF ) from the Remark following Lemma REF , to deduce that $C(\\lambda )B(\\lambda )w_{-\\mu ,0}$ appears as the leading coefficient of the power $\\rho _0^{-\\lambda }$ in the expansion of $u$ , where $B(\\lambda )$ is a non-zero constant times the factor appearing in (REF ); here $\\rho _0$ is defined in (REF ).", "(The factor $B(\\lambda )$ does not depend on the point $\\nu \\in \\mathbb {S}^n$ since the Poisson operator is equivariant under rotations of $\\overline{\\mathbb {B}^{n+1}}$ .)", "Then choosing $C(\\lambda ):=B(\\lambda )^{-1}$ , we observe that $u$ and $P^-_\\lambda (w)$ both satisfy (REF ) and have the same asymptotic coefficient of $\\rho _0^{-\\lambda }$ in their weak expansion (REF ); thus from Lemma REF we have $u=P^-_\\lambda (w)$ .", "Finally, for ${\\rm Re}(\\lambda )<-n/2$ with $\\lambda \\notin -\\tfrac{n}{2}-\\tfrac{1}{2}\\mathbb {N}_0$ we do the same thing but setting $\\mu :=-n/2-\\lambda $ in Proposition REF ." ], [ "Asymptotic expansions for certain integrals", "In this subsection, we prove the following version of Hadamard regularization: Lemma 1.1 Fix $\\chi \\in \\mathcal {C}^\\infty _0(\\mathbb {R})$ and define for $\\operatorname{Re}\\alpha >0$ , $\\beta \\in \\mathbb {C}$ , and $\\varepsilon >0$ , $F_{\\alpha \\beta }(\\varepsilon ):=\\int _0^\\infty t^{\\alpha -1}(1+t)^{-\\beta }\\chi (\\varepsilon t)\\,dt.$ If $\\alpha -\\beta \\notin \\mathbb {N}_0$ , then $F_{\\alpha \\beta }(\\varepsilon )$ has the following asymptotic expansion as $\\varepsilon \\rightarrow +0$ : $F_{\\alpha \\beta }(\\varepsilon )= {\\Gamma (\\alpha )\\Gamma (\\beta -\\alpha )\\over \\Gamma (\\beta )}\\chi (0)+\\sum _{0\\le j\\le \\operatorname{Re}(\\alpha -\\beta )} c_j \\varepsilon ^{\\beta -\\alpha +j}+o(1),$ for some constants $c_j$ depending on $\\chi $ .", "We use the following identity obtained by integrating by parts: ${\\begin{array}{c}\\varepsilon \\partial _\\varepsilon F_{\\alpha \\beta }(\\varepsilon )=\\int _0^\\infty t^\\alpha (1+t)^{-\\beta }\\partial _t(\\chi (\\varepsilon t))\\,dt\\\\=(\\beta -\\alpha )F_{\\alpha \\beta }(\\varepsilon )-\\beta F_{\\alpha ,\\beta +1}(\\varepsilon ).\\end{array}}$ By using the Taylor expansion of $\\chi $ at zero, we also see that $\\chi (\\varepsilon t)=\\chi (0)+\\mathcal {O}(\\varepsilon t);$ given the following formula obtained by the change of variables $s=(1+t)^{-1}$ and using the beta function, $\\int _0^\\infty t^{\\alpha -1} (1+t)^{-\\beta }\\,dt={\\Gamma (\\alpha )\\Gamma (\\beta -\\alpha )\\over \\Gamma (\\beta )},\\quad \\text{if }\\operatorname{Re}\\beta >\\operatorname{Re}\\alpha >0,$ we see that $F_{\\alpha \\beta }(\\varepsilon )={\\Gamma (\\alpha )\\Gamma (\\beta -\\alpha )\\over \\Gamma (\\beta )}\\chi (0) +\\mathcal {O}(\\varepsilon )\\quad \\text{if }\\operatorname{Re}(\\beta -\\alpha )>1.$ By applying this asymptotic expansion to $F_{\\alpha ,\\beta +M}$ for large integer $M$ and iterating (REF ), we derive the expansion (REF ).", "For the next result, we need the following two calculations (see Section REF for some of the notation used): Lemma 1.2 For each $\\ell \\ge 0$ , $\\int _{\\mathbb {S}^{n-1}} (\\otimes ^{2\\ell }\\eta )\\, dS(\\eta )={2\\pi ^{n-1\\over 2}\\Gamma (\\ell +{1\\over 2})\\over \\Gamma (\\ell +{n\\over 2})}\\mathcal {S}(\\otimes ^\\ell I),$ where $I=\\sum _{j=1}^n \\partial _j\\otimes \\partial _j$ .", "Since both sides are symmetric tensors, it suffices to show that for each $x\\in \\mathbb {R}^n$ , $\\int _{\\mathbb {S}^{n-1}}(x\\cdot \\eta )^{2\\ell }\\,dS(\\eta )={2\\pi ^{n-1\\over 2}\\Gamma (\\ell +{1\\over 2})\\over \\Gamma (\\ell +{n\\over 2})}\|x\|^{2\\ell }.$ Without loss of generality (using homogeneity and rotational invariance), we may assume that $x=\\partial _1$ .", "Then using polar coordinates and Fubini's theorem, we have ${\\Gamma (\\ell +{n\\over 2})\\over 2}\\int _{\\mathbb {S}^{n-1}}\\eta _1^{2\\ell }\\,dS(\\eta )=\\int _{\\mathbb {R}^n} e^{-\|\\eta \|^2}\\eta _1^{2\\ell }\\,d\\eta =\\pi ^{n-1\\over 2}\\Gamma \\Big (\\ell +{1\\over 2}\\Big )$ finishing the proof.", "Lemma 1.3 For each $\\eta \\in \\mathbb {R}^n$ , define the linear map $C_\\eta :\\mathbb {R}^n\\rightarrow \\mathbb {R}^n$ by $C_\\eta (\\tilde{\\eta })=\\tilde{\\eta }-{2\\over 1+\|\\eta \|^2}(\\tilde{\\eta }\\cdot \\eta )\\eta .$ Then for each $A_1,A_2\\in \\otimes ^m_S\\mathbb {R}^n$ with $\\mathcal {T}(A_1)=\\mathcal {T}(A_2)=0$ , and each $r\\ge 0$ , we have $\\int _{\\mathbb {S}^{n-1}} \\langle (\\otimes ^m C_{r\\eta }) A_1,A_2\\rangle \\,dS(\\eta )=2\\pi ^{n\\over 2}\\sum _{\\ell =0}^m {m!\\over (m-\\ell )!\\Gamma ({n\\over 2}+\\ell )}\\bigg (-{r^2\\over 1+r^2}\\bigg )^\\ell \\langle A_1,A_2\\rangle .$ We have $C_{r\\eta }=\\operatorname{Id}-{2r^2\\over 1+r^2}\\, \\eta ^\\otimes \\eta ,$ where $\\eta ^\\in (\\mathbb {R}^n)^$ is the dual to $\\eta $ by the standard metric.", "Then $\\int _{\\mathbb {S}^{n-1}}\\langle (\\otimes ^mC_{r\\eta })A_1,A_2\\rangle \\,dS(\\eta )=\\int _{\\mathbb {S}^{n-1}}\\Big \\langle \\otimes ^m\\Big (I-{2r^2\\over 1+r^2}\\,\\eta \\otimes \\eta \\Big ),\\sigma (A_1\\otimes A_2)\\Big \\rangle \\,dS(\\eta ).$ where $\\sigma $ is the operator defined by $\\sigma (\\eta _1\\otimes \\dots \\otimes \\eta _m\\otimes \\eta ^{\\prime }_1\\otimes \\dots \\otimes \\eta ^{\\prime }_m)=\\eta _1\\otimes \\eta ^{\\prime }_1\\otimes \\dots \\otimes \\eta _m\\otimes \\eta ^{\\prime }_m.$ We use Lemma REF , a binomial expansion, and the fact that $A_j$ are symmetric, to calculate ${\\begin{array}{c}\\int _{\\mathbb {S}^{n-1}}\\Big \\langle \\otimes ^m\\Big (I-{2r^2\\over 1+r^2}\\,\\eta \\otimes \\eta \\Big ),\\sigma (A_1\\otimes A_2)\\Big \\rangle \\,dS(\\eta )\\\\=\\sum _{\\ell =0}^m {m!\\over \\ell !", "(m-\\ell )!", "}\\Big (-{2r^2\\over 1+r^2}\\Big )^\\ell \\int _{\\mathbb {S}^{n-1}}\\langle (\\otimes ^{2\\ell }\\eta )\\otimes (\\otimes ^{m-\\ell }I),\\sigma (A_1\\otimes A_2)\\rangle \\,dS(\\eta )\\\\=2\\pi ^{n-1\\over 2}\\sum _{\\ell =0}^m {m!\\over \\ell !", "(m-\\ell )!", "}\\cdot {\\Gamma (\\ell +{1\\over 2})\\over \\Gamma (\\ell +{n\\over 2})}\\Big (-{2r^2\\over 1+r^2}\\Big )^\\ell \\langle \\mathcal {S}(\\otimes ^\\ell I)\\otimes (\\otimes ^{m-\\ell } I),\\sigma (A_1\\otimes A_2)\\rangle .\\end{array}}$ Since $\\mathcal {T}(A_1)=\\mathcal {T}(A_2)=0$ , we can compute $\\langle \\mathcal {S}(\\otimes ^\\ell I)\\otimes (\\otimes ^{m-\\ell } I),\\sigma (A_1\\otimes A_2)\\rangle ={2^\\ell (\\ell !", ")^2\\over (2\\ell )!", "}\\langle A_1,A_2\\rangle .$ Here $2^\\ell (\\ell !", ")^2/ (2\\ell )!$ is the proportion of all permutations $\\tau $ of $2\\ell $ elements such that for each $j$ , $\\tau (2j-1)+\\tau (2j)$ is odd.", "It remains to calculate ${\\begin{array}{c}\\sum _{\\ell =0}^m {m!\\over \\ell !", "(m-\\ell )!", "}\\cdot {\\Gamma (\\ell +{1\\over 2})\\over \\Gamma (\\ell +{n\\over 2})}\\cdot {2^\\ell (\\ell !", ")^2\\over (2\\ell )!", "}\\,t^\\ell =\\sum _{\\ell =0}^m {\\sqrt{\\pi }\\,m!\\over (m-\\ell )!\\Gamma (\\ell +{n\\over 2})}(t/2)^\\ell .\\end{array}}$ We can now state the following asymptotic formula, used in the proof of Lemma REF : Lemma 1.4 Let $\\chi \\in \\mathcal {C}^\\infty _0(\\mathbb {R})$ be equal to 1 near 0, and take $A_1,A_2\\in \\otimes _S^m \\mathbb {R}^n$ satisfying $\\mathcal {T}(A_1)=\\mathcal {T}(A_2)=0$ .", "Then for $\\lambda \\in \\mathbb {C}$ , $\\lambda \\notin -({n\\over 2}+\\mathbb {N}_0)$ , we have as $\\varepsilon \\rightarrow +0$ , ${\\begin{array}{c}\\int _{\\mathbb {R}^n}\\chi (\\varepsilon \|\\eta \|)(1+\|\\eta \|^2)^{-\\lambda -n}\\langle (\\otimes ^mC_\\eta )A_1,A_2\\rangle \\,d\\eta \\\\=\\pi ^{n\\over 2}{\\Gamma ({n\\over 2}+\\lambda )\\over (n+\\lambda +m-1)\\Gamma (n-1+\\lambda )}\\langle A_1,A_2\\rangle +\\sum _{0\\le j\\le -\\operatorname{Re}\\lambda -{n\\over 2}}c_j\\varepsilon ^{n+2\\lambda +2j}+o(1),\\end{array}}$ for some constants $c_j$ .", "We write, using the change of variables $\\eta =\\sqrt{t}\\theta $ , $\\theta \\in \\mathbb {S}^n$ , and $\\chi (s)=\\tilde{\\chi }(s^2)$ , and by Lemma REF ${\\begin{array}{c}\\int _{\\mathbb {R}^n}\\chi (\\varepsilon \|\\eta \|)(1+\|\\eta \|^2)^{-\\lambda -n}\\langle (\\otimes ^mC_\\eta )A_1,A_2\\rangle \\,d\\eta \\\\={1\\over 2}\\int _0^\\infty \\tilde{\\chi }(\\varepsilon ^2 t)t^{{n\\over 2}-1} (1+t)^{-\\lambda -n} \\int _{\\mathbb {S}^{n-1}}\\langle (\\otimes ^mC_{\\sqrt{t}\\theta })A_1,A_2\\rangle \\,dS(\\theta ) dt\\\\=\\pi ^{n\\over 2}\\sum _{\\ell =0}^m {(-1)^\\ell m!\\over (m-\\ell )!\\Gamma ({n\\over 2}+\\ell )}\\langle A_1,A_2\\rangle \\int _0^\\infty \\tilde{\\chi }(\\varepsilon ^2 t)t^{{n\\over 2}+\\ell -1} (1+t)^{-\\lambda -n-\\ell } \\,dt.\\end{array}}$ We now apply Lemma REF to get the required asymptotic expansion.", "The constant term in the expansion is $\\langle A_1,A_2\\rangle $ times ${\\begin{array}{c}\\pi ^{n\\over 2}\\Gamma \\Big ({n\\over 2}+\\lambda \\Big )\\sum _{\\ell =0}^m {(-1)^\\ell m!\\over (m-\\ell )!\\Gamma (n+\\lambda +\\ell )}\\\\=\\pi ^{n\\over 2}(-1)^m m!\\Gamma \\Big ({n\\over 2}+\\lambda \\Big )\\sum _{\\ell =0}^m {(-1)^\\ell \\over \\ell !\\Gamma (n+\\lambda +m-\\ell )}.\\end{array}}$ We now use the binomial expansion ${(1-t)^{n+\\lambda +m-1}\\over \\Gamma (n+\\lambda +m)}=\\sum _{\\ell =0}^\\infty {(-1)^\\ell \\over \\ell !\\Gamma (n+\\lambda +m-\\ell )}\\,t^\\ell $ and the sum in the last line of (REF ) is the $t^m$ coefficient of ${\\begin{array}{c}(1-t)^{-1}\\cdot {(1-t)^{n+\\lambda +m-1}\\over \\Gamma (n+\\lambda +m)}={(1-t)^{n+\\lambda +m-2}\\over \\Gamma (n+\\lambda +m)}\\\\={1\\over n+\\lambda +m-1}\\sum _{j=0}^\\infty {(-1)^j \\over j!\\Gamma (n+\\lambda +m-j-1)}\\,t^j;\\end{array}}$ this finishes the proof." ], [ "The Jacobian of $\\Psi $", "Here we compute the Jacobian of the map $\\Psi :\\mathcal {E}\\rightarrow S^2_\\Delta \\mathbb {H}^{n+1}$ appearing in the proof of Lemma REF , proving (REF ).", "By the $G$ -equivariance of $\\Psi $ we may assume that $x=\\partial _0,\\xi =\\partial _1,\\eta =\\sqrt{s}\\,\\partial _2$ for some $s\\ge 0$ .", "We then consider the following volume 1 basis of $T_{(x,\\xi ,\\eta )}\\mathcal {E}$ : ${\\begin{array}{c}X_1=(\\partial _1,\\partial _0,0),\\ X_2=(\\partial _2,0,\\sqrt{s}\\,\\partial _0),\\ X_3=(0,\\partial _2,-\\sqrt{s}\\,\\partial _1),\\ X_4=(0,0,\\partial _2);\\\\\\partial _{x_j},\\partial _{\\xi _j},\\partial _{\\eta _j},\\quad 3\\le j\\le n+1.\\end{array}}$ We have $\\Psi (x,\\xi ,\\eta )=(y,\\eta _-,\\eta _+)$ , where $y=(\\sqrt{s+1},0,\\sqrt{s},0,\\dots ,0),\\quad \\eta _\\pm =\\Big (\\mp {s\\over \\sqrt{s+1}},{1\\over \\sqrt{s+1}},\\mp \\sqrt{s},0,\\dots ,0\\Big ).$ Then we can consider the following volume 1 basis for $T_{(y,\\eta _-,\\eta _+)}S^2_\\Delta \\mathbb {H}^{n+1}$ : ${\\begin{array}{c}Y_1=\\Big (\\partial _1,{y\\over \\sqrt{s+1}},{y\\over \\sqrt{s+1}}\\Big ),\\ Y_2=\\Big (\\sqrt{s}\\,\\partial _0+\\sqrt{s+1}\\,\\partial _2,{\\sqrt{s}\\over \\sqrt{s+1}}y,-{\\sqrt{s}\\over \\sqrt{s+1}}y\\Big ),\\\\Y_3={(0,\\sqrt{s}\\,\\partial _0-\\sqrt{s}\\,\\partial _1+\\sqrt{s+1}\\,\\partial _2,0)\\over \\sqrt{s+1}},\\ Y_4={(0,0,\\sqrt{s}\\,\\partial _0+\\sqrt{s}\\,\\partial _1+\\sqrt{s+1}\\,\\partial _2)\\over \\sqrt{s+1}};\\\\\\partial _{y_j},\\partial _{\\nu _{-j}},\\partial _{\\nu _{+j}},\\quad 3\\le j\\le n+1.\\end{array}}$ Then the differential $d\\Psi {(x,\\xi ,\\eta )}$ maps ${\\begin{array}{c}X_1\\mapsto \\sqrt{s+1}\\,Y_1-\\sqrt{s}\\,Y_3-\\sqrt{s}\\,Y_4,\\\\X_2\\mapsto Y_2,\\\\X_3\\mapsto -\\sqrt{s}\\,Y_1+\\sqrt{s+1}\\,Y_3+\\sqrt{s+1}\\,Y_4,\\\\X_4\\mapsto {1\\over \\sqrt{s+1}}\\,Y_2+{1\\over s+1}Y_3-{1\\over s+1}Y_4.\\end{array}}$ Moreover, for $3\\le j\\le n+1$ , $d\\Psi {(x,\\xi ,\\eta )}$ maps linear combinations of $\\partial _{x_j},\\partial _{\\xi _j},\\partial _{\\eta _j}$ to linear combinations of $\\partial _{y_j},\\partial _{\\nu _{-j}},\\partial _{\\nu _{+j}}$ by the matrix $A(s)$ .", "The identity (REF ) now follows by a direct calculation." ], [ "An identity for harmonic polynomials", "We give a technical lemma which is used in the proof of Lemma REF (injectivity of the Poisson kernel).", "Lemma 1.5 Let $P$ be a harmonic homogeneous polynomial of order $m$ in $\\mathbb {R}^n$ , then for $r\\le m$ , we have for all $x\\in \\mathbb {R}^n$ $\\Delta _\\zeta ^rP(x-\\zeta \\langle \\zeta ,x\\rangle )\|_{\\zeta =0}= 2^r\\frac{m!", "r!}{(m-r)!", "}P(x).$ By homogeneity, it suffices to choose $\|x\|=1$ .", "We set $t=\\langle \\zeta ,x\\rangle $ and $u=\\zeta -tx$ and $P(x-\\zeta \\langle \\zeta ,x\\rangle )$ viewed in the $(t,u)$ coordinates is the homogeneous polynomial $(t,u)\\mapsto P((1-t^2)x-tu)$ .", "Now, we write for all $u\\in (\\mathbb {R}x)^\\perp $ and $t>0$ $ P(tx-u)=\\sum _{j=0}^m t^{m-j}P_j(u)$ where $P_j$ is a homogeneous polynomial of degree $j$ in $u\\in (\\mathbb {R}x)^{\\perp }$ , and since the Laplacian $\\Delta _\\zeta $ written in the $t,u$ coordinates is $-\\partial _t^2+\\Delta _u$ , the condition $\\Delta _xP=0$ can be rewritten $\\Delta _uP_j(u)=(m-j+2)(m-j+1)P_{j-2}(u), \\quad \\Delta _uP_1(u)=\\Delta _u P_0=0,$ which gives for all $j$ and $\\ell \\ge 1$ $\\Delta _u^\\ell P_{2\\ell }(u)=m(m-1)\\cdots (m-2\\ell +1)P_0, \\quad \\Delta ^j P_{2\\ell -1}(u)\|_{u=0}=0.$ We write $\\Delta _\\zeta ^r=\\sum _{k=0}^r \\tfrac{r!}{k!(r-k)!", "}(-1)^k\\partial _t^{2k}\\Delta _u^{r-k}$ and using parity and homogeneity considerations, we have $\\begin{split}\\Delta _\\zeta ^rP(x-\\zeta \\langle \\zeta ,x\\rangle )\|_{\\zeta =0} & =\\sum _{k=0}^r\\frac{(-1)^k r!}{k!(r-k)!", "}\\sum _{2j\\le m}[\\partial _t^{2k}((1-t^2)^{m-2j}t^{2j})\\Delta _u^{r-k}P_{2j}(u)]\|_{(t,u)=0}\\\\& = \\sum _{\\max (0,r-m/2)\\le k\\le r}\\frac{(-1)^k r!}{k!(r-k)!", "}(\\partial _t^{2k}((1-t^2)^{m-2(r-k)}t^{2(r-k)}))\|_{t=0}\\,\\Delta _u^{r-k}P_{2(r-k)}\\\\& = P_0\\cdot \\frac{m!", "r!}{(m-r)!}", "\\sum _{r/2\\le k\\le r}\\frac{(-1)^{k+r} (2k)!}{k!(r-k)!", "(2k-r)!}=2^r\\frac{m!", "r!}{(m-r)!", "}P_0\\end{split}$ and $P_0$ is the constant given by $P(x)$ .", "Here we used the identity ${\\begin{array}{c}\\sum _{r/2\\le k\\le r}\\frac{(-1)^{k+r} (2k)!}{k!(r-k)!", "(2k-r)!", "}=\\sum _{0\\le k\\le r/2} (-1)^k{ r!\\over k!(r-k)!", "}\\cdot {(2r-2k)!\\over r!(r-2k)!", "}=2^r\\end{array}}$ which holds since both sides are equal to the $t^r$ coefficient of the product ${\\begin{array}{c}(1-t^2)^r\\cdot (1-t)^{-1-r}={(1+t)^r\\over 1-t},\\\\(1-t)^{-1-r}={1\\over r!", "}d_t^{r}(1-t)^{-1}=\\sum _{j=0}^\\infty {(j+r)!\\over j!r!", "}\\, t^j;\\end{array}}$ the $t^r$ coefficient of $(1+t)^r/(1-t)$ equals the sum of the $t^0,t^1,\\dots , t^r$ coefficients of $(1+t)^r$ , or simply $(1+1)^r=2^r$ ." ], [ "The special case of dimension 2", "We explain how the argument of Section REF fits into the framework of Sections and .", "In dimension 2 it is more standard to use the upper half-plane model $\\mathbf {H}^2:=\\lbrace w\\in \\mathbb {C}\\mid \\operatorname{Im}w>0\\rbrace ,$ which is related to the half-space model of Section REF by the formula $w=-z_1+iz_0$ .", "The group of all isometries of $\\mathbf {H}^2$ is $\\operatorname{PSL}(2;\\mathbb {R})$ , the quotient of $\\operatorname{SL}(2;\\mathbb {R})$ by the group generated by the matrix $-\\operatorname{Id}$ , and the action of $\\operatorname{PSL}(2;\\mathbb {R})$ on $\\mathbf {H}^2$ is by Möbius transformations: $\\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} .", "z= {az+b\\over cz+d},\\quad z\\in \\mathbf {H}^2\\subset \\mathbb {C}.$ Under the identifications (REF ) and (REF ), this action corresponds to the action of $\\operatorname{PSO}(1,2)$ on $\\mathbb {H}^2\\subset \\mathbb {R}^{1,2}$ by the group isomorphism $\\operatorname{PSL}(2;\\mathbb {R})\\rightarrow \\operatorname{PSO}(1,2)$ defined by $\\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} \\mapsto \\begin{pmatrix}{a^2+b^2+c^2+d^2\\over 2} & {a^2-b^2+c^2-d^2\\over 2} & -ab - cd\\\\{a^2+b^2-c^2-d^2\\over 2} & {a^2-b^2-c^2+d^2\\over 2} & cd - ab\\\\-ac - bd & bd - ac & ad + bc\\end{pmatrix}.$ The induced Lie algebra isomorphism maps the vector fields $X,U_-,U_+$ of (REF ) to the fields $X,U^-_1,U^+_1$ of (REF ), (REF ).", "The horocyclic operators $\\mathcal {U}_\\pm :\\mathcal {D}^{\\prime }(S\\mathbb {H}^2)\\rightarrow \\mathcal {D}^{\\prime }(S\\mathbb {H}^2;\\mathcal {E}^)$ of Section REF (and analogously horocyclic operators of higher orders) then take the following form: $\\mathcal {U}_\\pm u=(U_\\pm u)\\eta ^,$ where $\\eta ^$ is the dual to the section $\\eta \\in \\mathcal {C}^\\infty (S\\mathbb {H}^2;\\mathcal {E})$ defined as follows: for $(x,\\xi )\\in S\\mathbb {H}^2$ , $\\eta (x,\\xi )$ is the unique vector in $T_x \\mathbb {H}^2$ such that $(\\xi ,\\eta )$ is a positively oriented orthonormal frame.", "Note also that $\\eta (x,\\xi )=\\pm \\mathcal {A}_\\pm (x,\\xi )\\cdot \\zeta (B_\\pm (x,\\xi ))$ , where $\\mathcal {A}_\\pm (x,\\xi )$ is defined in Section REF and $\\zeta (\\nu )\\in T_\\nu \\mathbb {S}^1$ , $\\nu \\in \\mathbb {S}^1$ , is the result of rotating $\\nu $ counterclockwise by $\\pi /2$ ; therefore, if we use $\\eta $ and $\\zeta $ to trivialize the relevant vector bundles, then the operators $\\mathcal {Q}_\\pm $ of (REF ) are simply the pullback operators by $B_\\pm $ , up to multiplication by $\\pm 1$ ." ], [ "Weyl law", "In this section, we prove the following asymptotic of the counting function for trace free divergence free tensors (see Sections REF and REF for the notation): Proposition 3.1 If $(M,g)$ is a compact Riemannian manifold of dimension $n+1$ and constant sectional curvature $-1$ , and if $\\operatorname{Eig}^m(\\sigma )=\\lbrace u\\in \\mathcal {C}^\\infty (M;\\otimes ^m_S T^M)\\mid \\Delta u = \\sigma u,\\ \\nabla ^u = 0,\\ \\mathcal {T}(u)=0\\rbrace ,$ then the following Weyl law holds as $R\\rightarrow \\infty $ $\\sum _{\\sigma \\le R^2}\\dim \\operatorname{Eig}^m(\\sigma )= c_0(n)(c_1(n,m)-c_1(n,m-2))\\operatorname{Vol}(M)R^{n+1}+\\mathcal {O}(R^n),$ where $c_0(n)={(2\\sqrt{\\pi })^{-n-1}\\over \\Gamma ({n+3\\over 2})}$ and $c_1(n,m)={(m+n-1)!\\over m!(n-1)!", "}$ is the dimension of the space of homogeneous polynomials of order $m$ in $n$ variables.", "(We put $c_1(n,m):=0$ for $m<0$ .)", "Remark.", "The constant $c_2(n,m):=c_1(n,m)-c_1(n,m-2)$ is the dimension of the space of harmonic homogeneous polynomials of order $m$ in $n$ variables.", "We have $c_2(n,0)=1,\\quad c_2(n,1)=n.$ For $m\\ge 2$ , we have $c_2(n,m)>0$ if and only if $n>1$ .", "The proof of Proposition REF uses the following two technical lemmas: Lemma 3.2 Take $u\\in \\mathcal {D}^{\\prime }(M;\\otimes ^m_S T^M)$ .", "Then, denoting $D=\\mathcal {S}\\circ \\nabla $ as in Section REF , $[\\Delta ,\\nabla ^]u&=(2-2m-n)\\nabla ^u-2(m-1)D (\\mathcal {T}(u)),\\\\[\\Delta ,D]u&=(2m+n) Du+2m\\mathcal {S}(g\\otimes \\nabla ^ u).$ We have $\\Delta \\nabla ^u=\\mathcal {T}^2(\\nabla ^3 u),\\quad \\nabla ^\\Delta u=\\mathcal {T}^2(\\tau _{1\\leftrightarrow 3}\\nabla ^3 u).$ where $\\tau _{j\\leftrightarrow k} v$ denotes the result of swapping $j$ th and $k$ th indices in a cotensor $v$ .", "We have $\\operatorname{Id}-\\tau _{1\\leftrightarrow 3}=(\\operatorname{Id}-\\tau _{1\\leftrightarrow 2})+\\tau _{1\\leftrightarrow 2}(\\operatorname{Id}-\\tau _{2\\leftrightarrow 3})+\\tau _{1\\leftrightarrow 2}\\tau _{2\\leftrightarrow 3}(\\operatorname{Id}-\\tau _{1\\leftrightarrow 2}),$ therefore (using that $\\mathcal {T}\\tau _{1\\leftrightarrow 2}=\\mathcal {T}$ ) $[\\Delta ,\\nabla ^]u=\\mathcal {T}^2\\big (\\nabla (\\operatorname{Id}-\\tau _{1\\leftrightarrow 2})\\nabla ^2 u+\\tau _{2\\leftrightarrow 3}(\\operatorname{Id}-\\tau _{1\\leftrightarrow 2})\\nabla ^3 u\\big )$ Since $M$ has sectional curvature $-1$ , we have for any cotensor $v$ of rank $m$ , $(\\operatorname{Id}-\\tau _{1\\leftrightarrow 2})\\nabla ^2 v=\\sum _{\\ell =1}^m(\\tau _{1\\leftrightarrow \\ell +2}-\\tau _{2\\leftrightarrow \\ell +2})(g\\otimes v).$ Then we compute (using that $\\mathcal {T}(\\tau _{2\\leftrightarrow 3}\\tau _{1\\leftrightarrow 3})=\\mathcal {T}(\\tau _{2\\leftrightarrow 3})$ ) ${\\begin{array}{c}[\\Delta ,\\nabla ^]u=\\mathcal {T}^2\\bigg (\\tau _{2\\leftrightarrow 3}-\\operatorname{Id}+\\sum _{\\ell =1}^m ((\\tau _{2\\leftrightarrow \\ell +3}-\\tau _{3\\leftrightarrow \\ell +3})\\tau _{1\\leftrightarrow 3}+\\tau _{2\\leftrightarrow 3}(\\tau _{1\\leftrightarrow \\ell +3}-\\tau _{2\\leftrightarrow \\ell +3}))\\bigg )(g\\otimes \\nabla u).\\end{array}}$ Now, ${\\begin{array}{c}\\mathcal {T}^2(g\\otimes \\nabla u)=\\mathcal {T}^2(\\tau _{2\\leftrightarrow 4}\\tau _{1\\leftrightarrow 3}(g\\otimes \\nabla u))=\\mathcal {T}^2(\\tau _{2\\leftrightarrow 3}\\tau _{1\\leftrightarrow 4}(g\\otimes \\nabla u))=-(n+1)\\nabla ^u,\\\\\\mathcal {T}^2(\\tau _{2\\leftrightarrow 3}(g\\otimes \\nabla u))=\\mathcal {T}^2(\\tau _{3\\leftrightarrow 4}\\tau _{1\\leftrightarrow 3}(g\\otimes \\nabla u))=\\mathcal {T}^2(\\tau _{2\\leftrightarrow 3}\\tau _{2\\leftrightarrow 4}(g\\otimes \\nabla u))=-\\nabla ^u,\\end{array}}$ and since $u$ is symmetric, for $1<\\ell \\le m$ , ${\\begin{array}{c}\\mathcal {T}^2(\\tau _{2\\leftrightarrow \\ell +3}\\tau _{1\\leftrightarrow 3}(g\\otimes \\nabla u))=\\mathcal {T}^2(\\tau _{2\\leftrightarrow 3}\\tau _{1\\leftrightarrow \\ell +3}(g\\otimes \\nabla u))=-\\nabla ^u,\\\\\\mathcal {T}^2(\\tau _{3\\leftrightarrow \\ell +3}\\tau _{1\\leftrightarrow 3}(g\\otimes \\nabla u))=\\mathcal {T}^2(\\tau _{2\\leftrightarrow 3}\\tau _{2\\leftrightarrow \\ell +3}(g\\otimes \\nabla u))=\\tau _{1\\leftrightarrow \\ell -1}\\nabla (\\mathcal {T}(u)).\\end{array}}$ We then compute $[\\Delta ,\\nabla ^]u=(2-2m-n)\\nabla ^u-2\\sum _{\\ell =1}^{m-1}\\tau _{1\\leftrightarrow \\ell }\\nabla (\\mathcal {T}(u)),$ finishing the proof of (REF ).", "The identity () follows from (REF ) by taking the adjoint on the space of symmetric tensors.", "Lemma 3.3 Denote by $\\tilde{\\pi }_m:\\otimes ^m_S T^M\\rightarrow \\otimes ^m_S T^M$ the orthogonal projection onto the space $\\ker \\mathcal {T}$ of trace free tensors.", "Then for each $m$ , the space $F^m:=\\lbrace v\\in C^\\infty (M;\\otimes ^m_S T^M)\\mid \\mathcal {T}(v)=0,\\ \\tilde{\\pi }_{m+1}(Dv)=0\\rbrace $ is finite dimensional.", "The space $F^m$ is contained in the kernel of the operator $P_m:=\\nabla ^* \\tilde{\\pi }_{m+1} D$ acting on trace free sections of $\\otimes ^m_ST^M$ .", "By [11], the operator $P_m$ is elliptic; therefore, its kernel is finite dimensional.", "We now prove Proposition REF .", "For each $m\\ge 0$ and $s\\in \\mathbb {R}$ , denote $W^m(\\sigma ):=\\lbrace u\\in \\mathcal {D}^{\\prime }(M;\\otimes ^m_S T^M)\\mid \\Delta u=\\sigma u,\\ \\mathcal {T}(u)=0\\rbrace .$ The operator $\\Delta $ acting on trace free symmetric tensors is elliptic and in fact, its principal symbol coincides with that of the scalar Laplacian: $p(x,\\xi )=\|\\xi \|_g^2$ .", "It follows that $W^m(\\sigma )$ are finite dimensional and consist of smooth sections.", "By the general argument of Hörmander [34] (see also [13] and [58]; all of these arguments adapt straightforwardly to the case of operators with diagonal principal symbols acting on vector bundles), we have the following Weyl law: $\\sum _{\\sigma \\le R^2} \\dim W^m(\\sigma )=c_0(n)(c_1(n+1,m)-c_1(n+1,m-2))\\operatorname{Vol}(M) R^{n+1}+\\mathcal {O}(R^n);$ here $c_1(n+1,m)-c_1(n+1,m+2)$ is the dimension of the vector bundle on which we consider the operator $\\Delta $ .", "By (REF ), for $m\\ge 1$ the divergence operator acts $\\nabla ^:W^m(\\sigma )\\rightarrow W^{m-1}(\\sigma +2-2m-n).$ This operator is surjective except at finitely many points $\\sigma $ : Lemma 3.4 Let $C_1=\\dim F^{m-1}$ , where $F^{m-1}$ is defined in (REF ).", "Then the number of values $\\sigma $ such that (REF ) is not surjective does not exceed $C_1$ .", "Assume that (REF ) is not surjective for some $\\sigma $ .", "Then there exists nonzero $v\\in W^{m-1}(\\sigma +2-2m-n)$ which is orthogonal to $\\nabla ^(W^m(\\sigma ))$ .", "Since the spaces $W^{m-1}(\\sigma )$ are mutually orthogonal, we see from (REF ) that $v$ is also orthogonal to $\\nabla ^(W^m(\\sigma ))$ for all $\\sigma \\ne \\sigma $ .", "It follows that for each $\\sigma $ and each $u\\in W^m(\\sigma )$ , we have $\\langle Dv,u\\rangle _{L^2}=0$ .", "Since $\\bigoplus _\\sigma W^m(\\sigma )$ is dense in the space of trace free tensors, we see that for each $u\\in C^\\infty (M;\\otimes ^m_ST^M)$ with $\\mathcal {T}(u)=0$ , we have $\\langle Dv, u\\rangle _{L^2}=0$ , which implies that $v\\in F^{m-1}$ .", "It remains to note that $F^{m-1}$ can have a nontrivial intersection with at most $C_1$ of the spaces $W^{m-1}(\\sigma +2-2m-n)$ .", "Since $\\operatorname{Eig}^m(\\sigma )$ is the kernel of (REF ), we have $\\dim \\operatorname{Eig}^m(\\sigma )\\ge \\dim W^m(\\sigma )-\\dim W^{m-1}(\\sigma +2-2m-n),$ and this inequality is an equality if (REF ) is surjective.", "We then see that for some constant $C_2$ independent of $R$ , ${\\begin{array}{c}\\sum _{\\sigma \\le R^2}\\dim W^m(\\sigma )-\\sum _{\\sigma \\le R^2+2-2m-n}\\dim W^{m-1}(\\sigma )\\le \\sum _{\\sigma \\le R^2}\\dim \\operatorname{Eig}^m(\\sigma )\\\\\\le C_2+\\sum _{\\sigma \\le R^2}\\dim W^m(\\sigma )-\\sum _{\\sigma \\le R^2+2-2m-n}\\dim W^{m-1}(\\sigma )\\end{array}}$ and Proposition REF now follows from (REF ) and the identity $c_1(n+1,m)-c_1(n+1,m-1)=c_1(n,m)$ ." ], [ "The case $m=1$", "In this section, we describe space $\\operatorname{Eig}^1(\\sigma )$ in terms of Hodge theory; see for instance [50] for the notation used.", "Note that symmetric cotensors of order 1 are exactly differential 1-forms on $M$ .", "Since the operator $\\nabla :C^\\infty (M)\\rightarrow C^\\infty (M;T^M)$ is equal to the operator $d$ on 0-forms, we have $\\operatorname{Eig}^1(\\sigma )=\\lbrace u\\in \\Omega ^1(M)\\mid \\Delta u=\\sigma u,\\ \\delta u=0\\rbrace .$ Here $\\Delta =\\nabla ^\\nabla $ ; using that $M$ has sectional curvature $-1$ , we write $\\Delta $ in terms of the Hodge Laplacian $\\Delta _\\Omega :=d\\delta +\\delta d$ on 1-forms using the following Weitzenböck formula [50]: $\\Delta u=(\\Delta _\\Omega +n)u,\\quad u\\in \\Omega ^1(M).$ We then see that $\\operatorname{Eig}^1(\\sigma )=\\lbrace u\\in \\Omega ^1(M)\\mid \\Delta _\\Omega u=(\\sigma -n)u,\\ \\delta u=0\\rbrace .$ Finally, let us consider the case $n=1$ .", "The Hodge star operator acts from $\\Omega ^1(M)$ to itself, and we see that for $\\sigma \\ne 1$ , ${\\begin{array}{c}\\operatorname{Eig}^1(\\sigma )=\\lbrace u\\mid u\\in \\Omega ^1(M),\\ \\Delta _\\Omega u =(\\sigma -1)u,\\ du = 0\\rbrace \\\\=\\lbrace (df)\\mid f\\in \\mathcal {C}^\\infty (M),\\ \\Delta f = (\\sigma -1)f \\rbrace .\\end{array}}$ Note that $*(df)$ can be viewed as the Hamiltonian field of $f$ with respect to the naturally induced symplectic form (that is, volume form) on $M$ .", "Acknowledgements.", "We would like to thank Maciej Zworski, Richard Melrose, Steve Zelditch, Rafe Mazzeo, and Kiril Datchev for many useful discussions and suggestions regarding this project, an anonymous referee for many useful suggestions, and Tobias Weich for corrections and references.", "The first author would like to thank MSRI (NSF grant 0932078 000, Fall 2013) where part of this work was done, another part was supported by the NSF grant DMS-1201417, and part of this project was completed during the period SD served as a Clay Research Fellow.", "CG and FF have been supported by “Agence Nationale de la Recherche” under the grant ANR-13-BS01-0007-01." ] ]	1403.0256
[ [ "Leverage effect in energy futures" ], [ "Abstract We propose a comprehensive treatment of the leverage effect, i.e.", "the relationship between returns and volatility of a specific asset, focusing on energy commodities futures, namely Brent and WTI crude oils, natural gas and heating oil.", "After estimating the volatility process without assuming any specific form of its behavior, we find the volatility to be long-term dependent with the Hurst exponent on a verge of stationarity and non-stationarity.", "Bypassing this using by using the detrended cross-correlation and the detrending moving-average cross-correlation coefficients, we find the standard leverage effect for both crude oil.", "For heating oil, the effect is not statistically significant, and for natural gas, we find the inverse leverage effect.", "Finally, we also show that none of the effects between returns and volatility is detected as the long-term cross-correlated one.", "These findings can be further utilized to enhance forecasting models and mainly in the risk management and portfolio diversification." ], [ "Introduction", "The leverage effect is one of the well-established phenomena of the financial economics.", "Historically, [7] discusses a possible relationship between returns and changes in volatility of stocks.", "The argumentation is based on changes in earnings, where decreasing expected earnings of the company push the price down and in turn it decreases the market value of the company which drives the leverage (ratio between debt and equity) up.", "Negative relationship between returns and volatility is thus referred to as `the leverage effect'.", "However, in the modern, high-speed, markets where the market prices of assets are driven by many more forces than simple expected earnings, such an explanation of the effect serves as just a little more than an anecdote.", "The leverage effect can be simply understood as a negative relationship between returns and volatility which are driven by opposite forces.", "When negative news reaches the market, volatility of the corresponding asset usually increases because of an uncertain future development.", "Contrarily, the negative news drives the prices down forming a negative return.", "The leverage effect thus seems as a natural connection of the two characteristics (returns and volatility) of the traded assets.", "The leverage effect is usually tightly connected, and sometimes even interchanged, with a notion of the asymmetric volatility.", "The standard asymmetric volatility is characterized by a lower volatility connected to a bull (growing) market and a higher volatility connected to a bear (declining) market.", "The definition and interconnection between the two effects – the leverage effect and the asymmetric volatility – is thus very close and sometimes hard to distinguish between.", "Nonetheless, most authors agree on several characteristics of the relationship between returns and volatility – returns and volatility are negatively correlated, the correlation is quite weak yet still persists over quite long time (with slowly decaying cross-correlations), and the causality goes from returns to volatility and not vice versa [38], [10], [9], [8].", "Here we analyze the leverage effect in the future contracts of energy commodities, namely WTI and Brent crude oils, natural gas and heating oil.", "We try to provide a coherent treatment of the leverage effect starting from the long-term memory characteristics of volatility and its potential non-stationarity, then moving to the estimation of the correlation between returns and volatility under borderline (non-)stationary and a typical seasonality of futures contracts, and finally checking the slow decay of the cross-correlation function characteristic for long-range cross-correlated processes.", "We find that the leverage effect in its purest form (significant negative correlation between returns and volatility) is found for two out of four studied commodities.", "However, the level of correlation is very low – lower than levels standardly reported for stocks and stock indices.", "Moreover, we show that the cross-correlations are not identified as hyperbolically decaying, i.e.", "there are no long-range cross-correlations between returns and volatility of the studied commodities.", "An important aspect of our analysis stems in not assuming anything about the relationship between returns and volatility which distinguishes our study from the other studies which are majorly built around assuming some kind of asymmetric volatility model, i.e.", "the leverage effect and asymmetric volatility are assumed ex ante to be frequently found ex post.", "The paper is structured as follows.", "In Section 2, we provide a literature review of recent studies on the leverage effect and asymmetric volatility on energy markets.", "Section 3 introduces the most important methodological aspects of our work – volatility estimation, long-term memory and its tests and estimators, estimation of correlations under borderline (non-)stationarity and seasonality, and long-range cross-correlations testing.", "Section 4 presents the analyzed dataset and results.", "Section 5 concludes." ], [ "Literature review", "In this section, we review recent literature on the topic of leverage effect and asymmetric volatility in energy commodities in chronological order.", "[19] examine WTI and Brent crude oil prices with various specifications of the generalized autoregressive conditional heteroskedasticity (GARCH) models for purposes of risk management.", "They find significant two-way spillover effect between both crude oil markets as well as asymmetric leverage effect in the WTI returns but not in the Brent returns.", "Interestingly, the uncovered leverage effect implies that positive shocks have much higher impact on the future dynamics of the series than the negative ones which is opposite to the leverage effect found in stocks and it can be thus treated as an inverse leverage effect.", "[54] study an interrelation between the US dollar exchange rates and crude oil prices with a special focus on spillover effects which they separate into three – mean spillover, volatility spillover and risk spillover.", "Apart from a significant long-term cointegration relationship, the authors find significant volatility asymmetry.", "In a similar way to the previous reference, they find the inverse leverage effect which they attribute mainly the non-renewable property of oil and very different roles and behavior of suppliers and demanders of the commodity.", "[3] examine the relationship between crude oil and stock markets utilizing a two regime Markov switching exponential GARCH model.", "They show that the volatility clustering and the leverage effect can be significantly reduced by allowing for the regime switching.", "Transition between regimes is mainly connected to economic recessions together with stock markets behavior.", "[1] compares predictive powers of GARCH-type and implied volatility models on the WTI future contract.", "Apart from showing that the GARCH-type models outperform the implied volatility models, the author also finds no leverage effect for the WTI contract.", "[14] then focuses on both WTI and Brent crude oil markets and applies GARCH specification.", "The author finds that the WTI volatility is more persistent than the one of the Brent crude oil.", "Even though the leverage effect is found for the Brent market and not for the WTI market, the out-of-sample forecasting exercise provides an evidence that a reduced GARCH model with no asymmetric volatility outperforms the others.", "[51] study both the WTI and Brent futures and compare a wide portfolio of GARCH-type models.", "Focusing on the performance of 1-day, 5-day and 20-day forecasting, they find that no single model is a clear winner in the horse race of testing.", "However, the authors favor the non-linear specifications of GARCh which can control for long-term memory as well as asymmetry.", "Similarly to the previous studies, the results on asymmetry are mixed for the two markets.", "Even though the asymmetry is found for a strong majority of specifications for the Brent market, the WTI shows mixed evidence.", "[13] focus on the relationship between crude oil and biofuels.", "Specifically, they are interested in the dynamics of volatility (using the exponential GARCH model) conditional on various phases of the market with respect to the crude oil prices.", "A significant asymmetric volatility reaction is found only for the soybean futures during the high oil prices.", "Other futures show no significant asymmetry.", "[18] examine the linkage between the crude oil volatility and agricultural commodity markets using the stochastic volatility approach in the Bayesian framework.", "The authors show that speculation, scalping and petroleum investors form important aspects of the volatility formation.", "In the model, they find a weak leverage effect between instantaneous volatility and prices.", "[42] inspects the crude oil dependence structure with various copula functions.", "He shows that the correlation structure is similar during both bear and bull markets and further states that the crude oil market is strongly globalized.", "For the favorited model of the marginals – exponential GARCH – the volatility asymmetry is found for all studied crude oil series.", "The same methodology is then applied in [43] where the relationship between oil price and exchange rates is examined.", "In general, the connection between the oil and exchange rate markets is reported to be very weak.", "The evidence of volatility asymmetry is mixed as well.", "[52] propose a copula-based GARCH model and use it to model dependence between crude oil and the US dollar.", "In their specification, the leverage effect is not significant for either of the studied futures.", "[12] employs a combined regime switching exponential GARCH model with Student-$t$ distributed error terms to model crude oil futures returns.", "The model is able to capture the main stylized facts of the crude oil futures.", "Importantly, the model combines both the regime switching and asymmetric volatility to capture nonlinear dependencies between returns, volatility and higher moments.", "In accordance to other works, no leverage effect is found for the WTI futures.", "[25] analyze the effect of crude oil volatility spillovers on non-energy commodities.", "After controlling for exchange rates, the authors utilize a bivariate exponential GARCH model with time-varying correlation structure.", "They show that the crude oil plays a core role in the commodities structure as its volatility spills over to other, non-energy, markets as well.", "The strength of these spillovers even increases after the 2008 financial crisis.", "Volatility asymmetry is studied as a difference in reaction to bad and good news.", "The authors find the effect to be significant for majority of the studied pairs.", "[37] investigate dynamics of eight energy spot markets on NYMEX.", "The authors combine a mean-reverting and a spike model with GARCH-type time-varying volatility focusing on risk management issues as well as their forecasting performance.", "The leverage effect is found for WTI, heating oil and heating oil-WTI crack spread, and the inverse leverage effect is uncovered for gasoline, natural gas, propane and gasoline-WTI crack spread.", "Copulas are further utilized by [49] who study tail dependence between crude oil and refined petroleum markets.", "Positive dependence is found in both tails so that the markets tend to move together in both bear and bull periods.", "Asymmetry in tail dependence is found between crude and heating oils, and between crude oil and jet fuel.", "Interestingly, the upper tail dependence is stronger than in the lower tail for the pre-crisis period.", "The authors report that the leverage effect, which is found in its standard form, is much stronger for the post-crisis period.", "[45] study the WTI and Brent crude oil with respect to the structural breaks while controlling for potential volatility asymmetry.", "Persistence as well as asymmetry of volatility is reported even after controlling for two structural breaks (Iraqi/Kuwait conflict and the financial crisis of 2008) identified for both oil markets.", "The authors stress that neither of the effects should be studied separately and the constructed models should consider each of structural breaks, volatility persistence and asymmetry.", "And [15] examine crude oil, natural gas, gold and silver markets using various linear and nonlinear GARCH-type specifications.", "The nonlinear specifications are found to fare better in a sense of in-sample and out-of-sample performance as well as risk management issues under the Basel II regulations.", "The direction and significance of the leverage effect are found to be strongly dependent on the model choice." ], [ "Methodology", "Studying leverage effect stems primarily on the analysis of the relationship between returns and volatility of the series.", "As such, this is connected with several issues.", "Firstly, the volatility itself needs to be extracted from the series.", "Secondly, the volatility is standardly considered as a long-term memory process.", "Thirdly, not only is the volatility process long-term dependent but it is usually on the edge of stationarity, i.e.", "its fractional integration parameter $d \\approx 0.5$ and it is thus somewhere between a stationary short-term memory process with $d=0$ and a unit root process with $d=1$ .", "In this section, we introduce methodology and instruments to deal with such specifics." ], [ "Volatility estimation", "In majority of the leverage effect and asymmetric volatility studies covered in the Literature review, the volatility process has been estimated as a part of the complete model under various assumptions and restrictions.", "In turn, the volatility series and its characteristics are strongly dependent on the model choice and specifications.", "For our purposes, the leverage effect emerges from the model only if we assume correlation between the returns and volatility processes.", "However, if the effect is in reality not present, it can simply occur to be significant during the estimation procedure due to the model misspecification.", "In our study, we bypass this issue by estimating the volatility outside the returns model.", "Historically, the volatility and variance series were estimated simply as a squared or absolute returns of the series.", "In a sense, the GARCH-type models are built in the same logic.", "However, these simple measures turn out to be very poor estimators of the true volatility [16].", "Range-based estimators of volatility turn out to be much more efficient and precise than the absolute and squared errors and they stay close to the most efficient realized variance family measuresWe do not opt for the realized variance family measures due to their need of high-frequency data.", "Moreover, our study is a study of the relationship between returns and volatility, not of finding the best measure of volatility.", "The range-based estimators are in turn a very fitting compromise as these need only daily open, close, high and low prices which are freely available for practically all publicly traded assets.. From several possibilities, we select the Garman–Klass estimator (GKE) as a highly efficient estimator of daily variance.", "The estimator is defined as $\\widehat{\\sigma ^2_{GK,t}}=\\frac{(\\log (H_t/L_t))^2}{2}-(2\\log 2-1)(\\log (C_t/O_t))^2$ where $H_t$ and $L_t$ are daily highs and lows, respectively, and $C_t$ and $O_t$ are daily closing and opening prices, respectively [20].", "As the estimator does not take the overnight volatility into consideration, we further work with the open-close returns, i.e.", "$r_t=\\log (C_t)-\\log (O_t)$ ." ], [ "Long-term memory", "Long-term memory (long memory, long-range dependence) is connected to specific features of the series in both time and frequency domains.", "In the time domain, the long-term memory process has asymptotically power-law decaying autocorrelation function $\\rho (k)$ with lag $k$ such that $\\rho (k) \\propto k^{2H-2}$ for $k \\rightarrow +\\infty $ .", "In the frequency domain, the long-term memory process has divergent at origin spectrum $f(\\lambda )$ with frequency $\\lambda $ such that $f(\\lambda )\\propto \\lambda ^{1-2H}$ for $\\lambda \\rightarrow 0+$ .", "In both definitions, the Hurst exponent $H$ plays a crucial role.", "For stationary series, $H$ is standardly bounded between 0 and 1 so that $0 \\le H <1$ .", "No long-term memory is connected to $H=0.5$ , positive long-term autocorrelations are found for $H>0.5$ and negative ones for $H<0.5$ .", "The Hurst exponent is connected to the fractional differencing parameter $d$ in a strict way – $H=d+0.5$ [6].", "Hurst exponent is crucial for our further analysis.", "However, before estimating the exponent itself, we need to test the series for actually being long-range dependent.", "It has been shown that the estimators of Hurst exponent might report values different from 0.5, and thus hinting long-term memory, even if the series are not long-range dependent [46], [47], [48], [35], [5], [27], [28], [55].", "To deal with this matter, we firstly test for the presence of long-range dependence in the series before estimating the Hurst exponent.", "We opt for two tests – modified rescaled range and rescaled variance.", "The modified rescaled range test [36] is an adjusted version of the traditional rescaled range test [24] controlling for short-term memory of the series.", "The testing statistic $V$ is defined as $V_{T}=\\frac{(R/S)_{T}}{\\sqrt{T}}$ where the range $R$ is defined as a difference between the maximum and the minimum of the profile (cumulative demeaned original series), $S$ is the standard deviation of the series and $T$ is the time series length.", "Here $(R/S)_T$ is the rescaled range of the series of length $T$ .", "To control for the potential short-term memory bias (strong short-term memory might be mistaken for the long-term memory), the standard deviation $S$ is used in its heteroskedasticity and autocorrelation consistent (HAC) version.", "For these purposes, we utilize the following specification which is later used in the bivariate setting and the rescaled covariance test as well: $\\widehat{s_{xy,q}}=\\sum _{k=-q}^q{\\left(1-\\frac{\|k\|}{q+1}\\right)\\widehat{\\gamma _{xy}}(k)}$ where $\\widehat{\\gamma _{xy}}(k)$ is a sample cross-covariance at lag $k$ , $q$ is a number of lags taken into consideration and the cross-covariances are weighted with the Barlett-kernel weights.", "For the purposes of the modified rescaled range, we set $S \\equiv \\widehat{s_{xx,q}}$ as the autocovariance function is symmetric.", "We follow the suggestion of [36] and use lag $q$ according to the following formula for the optimal lag: $q^{\\ast }=\\left\\lfloor \\left(\\frac{3T}{2}\\right)^{\\frac{1}{3}}\\left(\\frac{2\\widehat{\|\\rho (1)\|}}{1-\\widehat{\\rho (1)}^2}\\right)^{\\frac{2}{3}}\\right\\rfloor $ where $\\widehat{\\rho (1)}$ is a sample first order autocorrelation and $\\lfloor \\rfloor $ is the lower integer operator.", "Under the null hypothesis of no long-range dependence, the statistic is distributed as $F_V(x)=1+2\\sum _{k=1}^{\\infty }(1-4k^2x^2)e^{-2(kx)^2}.$ As an alternative to the modified rescaled range test, [22] propose the rescaled variance test which simply substitutes the range in Eq.", "REF by variance of the profile.", "The testing statistic $M$ is then defined as $M_T=\\frac{var(X)}{TS^2},$ where $X$ is the profile of the original series and the standard deviation $S$ is defined in the same way as for the modified rescaled range test.", "[22] show that the rescaled variance test has better properties than the modified rescaled range which is further supported by [33] and [34].", "Under the null hypothesis of no long-term memory, the statistic is distributed as $F_M(x)=1+2\\sum ^\\infty _{k=1}{(-1)^ke^{-2k^2\\pi ^2x}}.$ For the estimation of the Hurst exponent itself, we utilize two frequency domain estimators – the local Whittle estimator and the GPH estimator.", "We opt for the frequency domain estimators as these have well defined asymptotic properties and are well suited even for non-stationary or boundary series which turns out to be the case for the analysis we present.", "[44] proposes the local Whittle estimator as a semi-parametric maximum likelihood estimator using the likelihood of [32] while focusing only on a part of spectrum near the origin.", "As an estimator of the spectrum $f(\\lambda )$ , the periodogram $I(\\lambda )$ is utilized.", "For the time series of length $T$ , and setting $m \\le T/2$ and $\\lambda _j=2\\pi j/T$ , the Hurst exponent is estimated as $\\widehat{H}=\\arg \\min R(H),$ where $R(H)=\\log \\left(\\frac{1}{m}\\sum _{j=1}^m{\\lambda _j^{2H-1}I(\\lambda _j)}\\right)-\\frac{2H-1}{m}\\sum _{j=1}^m{\\log \\lambda _j}.$ [21] introduce an estimator based on a full functional specification of the underlying process as the fractional Gaussian noise, which is labeled as the GPH estimator after the authors.", "The assumption of the underlying process is connected to a specific spectral density which is in turn utilized in the regression estimation of the following equation: $\\log I(\\lambda _j) \\propto -(H-0.5)\\log [4\\sin ^2(\\lambda _j/2)].$ Both estimators are consistent and asymptotically normal.", "To avoid bias due to short-term memory, we estimate both the local Whittle and GPH estimators only on parts of the estimated periodogram that are close to the origin (short-term memory is present at high frequencies and thus far from the origin).", "Specifically, we use $m=T^{0.6}$ ." ], [ "Correlation coefficient for non-stationary series", "As the leverage effect can be seen as a correlation between returns and volatility, a need for efficient estimators of correlation between potentially non-stationary series is high.", "Recently, two methods have been proposed in the literature – detrended cross-correlation coefficient [53] and detrending moving-average cross-correlation coefficient [30].", "[53] proposes the detrended cross-correlation coefficient as a combination of the detrended cross-correlation analysis (DCCA) [41] and the detrended fluctuation analysis (DFA) [39], [40], [26].", "The detrended cross-correlation coefficient $\\rho _{DCCA}(s)$ , which measures the correlation even between non-stationary as well as seasonal series, is defined as $\\rho _{DCCA}(s)=\\frac{F^2_{DCCA}(s)}{F_{DFA,x}(s)F_{DFA,y}(s)},$ where $F^2_{DCCA}(s)$ is a detrended covariance between profiles of the two series based on a window of size $s$ , and $F^2_{DFA,x}$ and $F^2_{DFA,y}$ are detrended variances of profiles of the separate series, respectively, for a window size $s$ .", "For more technical details about the methods, please refer to [26], [41] and [31].", "In words, the method is based on calculating the correlation coefficient between series detrended by a linear trend while the detrending is performed in each window of length $s$ .", "[30] introduces the detrending moving-average cross-correlation coefficient as an alternative to the above mentioned coefficient.", "The method connects the detrending moving average (DMA) procedure [50], [2] and detrending moving-average cross-correlation analysis (DMCA) [4], [23].", "The detrending moving-average cross-correlation coefficient $\\rho _{DMCA}(\\lambda )$ is defined as $\\rho _{DMCA}(\\lambda )=\\frac{F_{DMCA}^2(\\lambda )}{F_{x,DMA}(\\lambda )F_{y,DMA}(\\lambda )},$ where $F^2_{DMCA}(\\lambda )$ , $F^2_{DMA,x}(\\lambda )$ and $F^2_{DMA,y}(\\lambda )$ are, similarly to the DCCA-based coefficient, detrended covariance between profiles of the two studied series and detrended variances of the separate series, respectively, with a moving average parameter $\\lambda $ .", "Contrary to the previous DCCA-based method, the DMCA variant does not require box-splitting but estimates the correlation from the profile series detrended simply by the moving average of length $\\lambda $ .", "[11] show that the centered moving average outperforms the backward and forward ones so that we apply the centered one in our analysis.", "For more detailed description of the procedures, please refer to [2], [4] and [30]." ], [ "Rescaled covariance test", "Motivated by the rescaled variance test for the univariate series, [29] proposes the rescaled covariance test which is able to distinguish between long-term and short-term memory between two series.", "In a similar way as for the univariate series, the long-term memory can be generalized to the bivariate setting so that the long-range cross-correlated (cross-persistent) processes are characterized by asymptotically power-law decaying cross-correlation function and divergent at origin cross-spectrum.", "By applying the test to the relationship between returns and volatility, we can comment on possible power-law cross-correlated relationship between the two series which is usually connected to the leverage effect [17].", "The testing statistic for the rescaled covariance test is defined as $M_{xy,T}(q)=q^{\\widehat{H_x}+\\widehat{H_y}-1}\\frac{\\widehat{\\text{Cov}}(X_T,Y_T)}{T\\widehat{s_{xy,q}}},$ where $\\widehat{s_{xy,q}}$ is the HAC-estimator of the covariance of the studied series defined in Eq.", "REF , $\\widehat{\\text{Cov}}(X_T,Y_T)$ is the estimated covariance between profiles of the series, and $\\widehat{H_x}$ and $\\widehat{H_y}$ are estimated Hurst exponents for the separate processes.", "For the estimated Hurst exponents, we use the average of the local Whittle and GPH estimators if the process is found to be long-range dependent.", "Otherwise, we set the corresponding exponent equal to 0.5.", "Critical and $p$ -values for the test are obtained from the moving-block bootstrap methodology.", "For more details, please refer to [29].", "We analyze front futures contracts of Brent crude oil, WTI (West Texas Intermediate) crude oil, heating oil and natural gas between 1.1.2000 and 30.6.2013.", "As we are interested in the leverage effect, we focus on returns and volatility of the future prices.", "In Figs.", "REF and REF , we present returns and volatility based on the Garman-Klass estimator given in Eq.", "REF .", "From the returns charts, we observe that these behave very similarly to the standard financial returns with volatility clustering and non-Gaussian distribution.", "However, we also notice, mainly for the natural gas, that returns undergo certain seasonal pattern which is connected to the rolling of the front and back futures contracts.", "This is dealt with by utilizing detrended cross-correlation and detrending moving-average cross-correlation coefficients which are constructed for such seasonalities.", "Volatility dynamics again reminds of standard volatility of other financial assets with evident persistence, which is dealt with later on.", "Again, the natural gas series stands out with more frequent volatility jumps and more erratic behavior.", "In Tab.", "REF , we summarize standard descriptive statistics and tests.", "All returns series follow quite standard characteristics such as excess volatility, negative skewness (apart from natural gas in this case), non-Gaussian distribution and asymptotic stationarity.", "For each series, we also find significant autocorrelations.", "Later, we test whether these can be treated as the long-term ones or not.", "Apart from the returns and volatility, which we examine in its logarithmic form, we focus on the standardized returns as well.", "Note that the returns standardized by their volatility are usually close to being normally distributed and in general, they are more suitable for statistical analysis.", "From this point onward, we focus solely on the relationship between standardized returns and logarithmic volatility so that if returns and volatility are referred to, we work with the transformed series.", "Standardized returns are all approximately symmetric and do not exceed kurtosis of the normal distribution.", "Moreover, the autocorrelations have been filtered out by standardizing for three out of four series.", "For the volatility, we strongly reject normality of the distribution and we find very strong autocorrelations.", "Moreover, we reject both unit root and stationary behavior of the series.", "This leads us to an inspection of potential long-term memory in the analyzed series.", "In Tab.", "REF , we show results for the modified rescaled range and the rescaled variance tests.", "Optimal lag has been chosen according to Eq.", "REF .", "We find that neither of the returns series are long-range autocorrelated, even though the testing statistics for natural gas are close to the critical levels.", "As expected, long-term memory is identified for all volatility series even after controlling for rather high number of lags (between 15 and 20).", "The results of the long-term memory tests thus give expected results – no long-term memory for the returns and statistically significant long-term memory for the volatility series.", "Based on the previous tests, we take that the returns series are not long-term dependent so that their Hurst exponent is equal to 0.5, which is later used in the rescaled covariance test.", "For the volatility series, we estimate the Hurst exponent $H$ using the local Whittle and GPH estimators.", "The estimates are summarized in Tab.", "REF .", "We observe that both estimators give similar results – the Hurst exponent for volatility for all four studied series is estimated around $H=1$ .", "Based on the reported standard errors, we cannot distinguish whether the Hurst exponents are below or above the unity value.", "Therefore, we cannot easily decide whether the volatility series are stationary long-range dependent or non-stationary long-range dependent but still mean reverting.", "Nevertheless, this does not discredit any of the following instruments and tests.", "As the volatility series are long-term correlated, we need to apply correlation measures which are able to deal with such series.", "[31] shows that the standard correlation coefficient is not able to do so.", "We thus apply the detrended cross-correlation coefficient and detrending moving-average cross-correlation measures which are not only able to work under long-term memory and even non-stationarity but they can also filter out well-defined trends.", "In the case of the studied futures, the rolling period of a trading month is well-established so that we can set $s=\\lambda =20$ and the methods filter the seasonality away.", "Tab.", "REF reports the estimated correlation coefficients between returns and volatility of each studied commodity.", "We find that both crude oils are partially driven by the standard leverage effect connected to negative correlation between returns and volatility.", "For heating oil, the estimated correlation is also negative but not statistically significant$p$ -values are constructed using 10,000 series generated using Fourier randomization, which ensures that the autocorrelation structure remains untouched but the cross-correlations are shuffled away.", "at 1% level.", "Natural gas is then characterized by the inverse leverage effect, i.e.", "the positive correlation between returns and volatility.", "Note that even though some of the correlations are found to be statistically significant, the levels are rather weak compared to standardly reported ones for stocks or stock indices.", "Tab.", "REF then summarizes the results of the rescaled covariance test which test possible long-range cross-correlations.", "We use the same number of lags as for the univariate volatility tests in Tab.", "REF .", "Based on the reported $p$ -values, we find no sings of long-range dependence in the bivariate setting.", "This is tightly connected to rather weak correlations found above.", "Even though the series might be correlated, creating the leverage or the inverse leverage effects, the influence is not strong enough to translate into a long-term connection." ], [ "Conclusion", "In this paper, we propose a comprehensive treatment of the leverage effect, focusing on energy commodities futures, namely Brent and WTI crude oils, natural gas and heating oil.", "After estimating the volatility process without assuming any specific form of its behavior, we find the volatility to be long-term dependent with the Hurst exponent on a verge of stationarity and non-stationarity.", "Bypassing this using by using the detrended cross-correlation and the detrending moving-average cross-correlation coefficients, we find the standard leverage effect for both crude oils.", "For heating oil, the effect is not statistically significant, and for natural gas, we find the inverse leverage effect.", "This points out a need for initial testing for the presence of the leverage effect before constructing any specific models to avoid inefficient estimation or even biased results.", "Finally, we also show that none of the effects between returns and volatility is detected as the long-term cross-correlated one.", "The dynamics of the crude oil futures, as ones of the most traded ones, is thus closer to the one of stocks and stock indices whereas the less popular heating oil and natural gas somewhat deviate from the standard behavior.", "These findings can be further utilized to enhance forecasting models and mainly in the risk management and portfolio diversification." ], [ "Acknowledgements", "The research leading to these results has received funding from the European Union's Seventh Framework Programme (FP7/2007-2013) under grant agreement No.", "FP7-SSH-612955 (FinMaP).", "Support from the Czech Science Foundation under projects No.", "P402/11/0948 and No.", "14-11402P is also gratefully acknowledged." ] ]	1403.0064
[ [ "Exponential decay for the Schr{\\\"o}dinger equation on a dissipative\n waveguide" ], [ "Abstract We prove exponential decay for the solution of the Schr{\\\"o}dinger equation on a dissipative waveguide.", "The absorption is effective everywhere on the boundary but the geometric control condition is not satisfied.", "The proof relies on separation of variables and the Riesz basis property for the eigenfunctions of the transverse operator.", "The case where the absorption index takes negative values is also discussed." ], [ "Introduction", "Let $l> 0$ and $d\\geqslant 2$ .", "Let $\\Omega $ denote the straight waveguide $\\mathbb {R}^{d-1} \\times ]0, l[ \\subset \\mathbb {R}^d$ .", "We consider on $\\Omega $ the Schrödinger equation with dissipative boundary condition $ {\\left\\lbrace \\begin{array}{ll}i \\partial _t u + u̥ = 0 & \\text{on } ]0,+\\infty [ \\times \\Omega ,\\\\\\partial _\\nu u = i a u & \\text{on } ]0,+\\infty [ \\times \\partial \\Omega ,\\\\u(0,\\cdot ) = u_0 & \\text{on } \\Omega .\\end{array}\\right.", "}$ Here $u_0 \\in L^2(\\Omega )$ and $\\partial _\\nu $ denotes the outward normal derivative.", "The absorption index $a$ belongs to $W^{1,\\infty }(\\partial \\Omega )$ .", "In the main result of the paper, $a$ takes positive values, but we also discuss the case where $a$ takes (small) negative values.", "We will first prove well-posedness for this problem.", "Then it is standard computation to check that when $a \\geqslant 0$ the norm of $u(t)$ in $L^2(\\Omega )$ is non-increasing, and that the decay is due to the boundary condition: $\\frac{d}{dt} \\left\\Vert u(t)\\right\\Vert _{L^2(\\Omega )}^2 = - 2 \\int _{\\partial \\Omega } a \\left\|u(t)\\right\|^2 \\leqslant 0.$ Whether this norm goes to 0 for large times, and then the rate of decay, are questions which have been extensively studied in different contexts.", "For the Schrödinger equation as in the present paper, or for the (damped) wave equation which is a closely related problem.", "Many papers deal with the wave equation on compact manifolds, with dissipation in the interior of the domain or at the boundary.", "We know from [21], [25] that a weak assumption on the absorption index $a$ (for instance the dissipation is effective on any open subset of the domain) is enough to ensure that the energy goes to 0 for any initial datum.", "Uniform exponential decay has been obtained in [38], [11] under the now usual geometric control condition.", "Roughly speaking, the assumption is that any (generalized) bicharacteristic (or classical trajectory, or ray of geometric optics) meets the damping region (in the interior of the domain or at the boundary).", "For the free wave equation on a subset of $\\mathbb {R}^d$ , the spatial projections of these bicharacteristics are straight lines, reflected at the boundary according to classical laws of geometric optics.", "This condition is essentially necessary and sufficient (we do not discuss here the subtilities due to the trajectories which meet the boundary tangentially).", "Then the question was to understand what happens when this damping condition fails to hold.", "In [25], [27] it is proved that we have at least a logarithmic decay of the energy if the initial datum belongs to the domain of the infinitesimal generator of the problem.", "This can be optimal, in particular when non controlled trajectories are stable.", "Intermediate rates of decay have been obtained for several examples where the flow is unstable near these trajectories (see for instance [28], [9], [16], [39], [5]).", "The same questions have been investigated for the Schrödinger or wave equation on the Euclidean space.", "Even in the self-adjoint case, where the norm for the solution of the Schrödinger equation or the energy for the solution of the wave equation are conserved, it is interesting to study the local energy decay, which measures the fact that the energy escapes at infinity.", "For the free case and with compactly supported initial conditions, explicit computations show that the energy on a compact vanishes after finite time for the wave in odd dimension $d\\geqslant 3$ , while the decay is of size $t^{-d}$ in even dimension.", "For the Schrödinger equation, the norm of the solution decays like $t^{-\\frac{d}{2}}$ .", "Many authors have proved similar estimates for perturbed problems.", "For instance the wave or Schrödinger equations can be stated outside a compact obstacle of $\\mathbb {R}^d$ (we can also consider a perturbation of the Laplace operator).", "For the wave equation on an exterior domain, we have uniform (exponential) decay for the local energy if and only if the non-trapping assumption holds.", "This means that there is no trapped bicharacteristic.", "See [26], [33], [29].", "See [14] for logarithmic decay without the non-trapping assumption, and [32] for an intermediate situation.", "For the dissipative equations, this non-trapping assumption can be replaced by the same damping condition (geometric control condition) on bounded trajectories as in the compact case (where all the trajectories were bounded).", "This means that the energy of the wave (or at least the contribution of high frequencies) escapes at infinity or is dissipated by the medium.", "This has been used in [1] for a dissipation in the interior of the domain and in [6], [2] for dissipation at the boundary, for the free equations on an exterior domain in both cases.", "See also [36] for the corresponding resolvent estimates, [12] for the damped wave equation with a Laplace-Beltrami operator corresponding to a metric which is a long-range perturbation of the flat one, and [4] where the damping condition is not satisfied but the dissipation is stronger.", "In this paper we consider a domain which is neither bounded nor the complement of a bounded obstacle.", "In particular, compared to the situations mentioned above, the boundary of the waveguide is not compact.", "More precisely, we are going to use the fact that $\\Omega $ is a Euclidean space in some directions and compact with respect to the last coordinate, so that properties of both compact and Euclidean domains will appear in our analysis.", "The main result of this paper is the following: Theorem 1.1 Assume that there exist two constants $a_0,a_1$ such that on $\\partial \\Omega $ we have $ 0 < a_0 \\leqslant a \\leqslant a_1.$ Then there exist $\\gamma > 0$ and $C \\geqslant 0$ such that for all $u_0 \\in L^2(\\Omega )$ the solution $u$ of the problem (REF ) satisfies $\\forall t \\geqslant 0, \\quad \\left\\Vert u(t)\\right\\Vert _{L^2(\\Omega )} \\leqslant C e^{-\\gamma t} \\left\\Vert u_0\\right\\Vert _{L^2(\\Omega )}.$ The same result holds if $a$ vanishes on one side of the boundary and satisfies (REF ) on the other side.", "In this theorem the absorption is effective at least on one side of the boundary, so the damping condition is clearly satisfied on bounded trajectories (which have to meet both sides of $\\partial \\Omega $ ).", "However, it is important to note that we prove exponential decay for the total energy and not only for the local energy.", "This means that all the energy is dissipated by the medium, including the energy going at infinity.", "This suggests that all the classical trajectories, and not only the bounded ones, should be controlled by the dissipation.", "This is not the case here.", "Thus this theorem provides a new example for the already mentioned general idea that given a result for which the non-trapping condition (or the damping condition for dissipative problems) is necessary, we get a close result if there are only a few trajectories which contradict the assumption.", "Here the initial conditions in $T^* \\Omega $ whose corresponding trajectories avoid the boundary form a submanifold of codimension 1 (the frequency vector has no tranversal component).", "Moreover the flow is linearly unstable.", "Compared to the related results on compact or Euclidean domains, our analysis will be based on simpler arguments.", "The key argument is that our problem will inherit the decay property of the transverse problem (that is the Schrödinger equation on $]0,l[$ with the same dissipative boundary condition), for which the geometric control assumption holds.", "In order to prove time-dependent estimates for an evolution equation as in Theorem REF , we often use spectral properties for the generator of the problem (spectral gap for the eigenvalues on a compact manifold, absence of resonances close to a positive energy on a perturbation of the Euclidean space, etc.).", "Here, the problem is governed by the operator $ H_a -$ defined on the domain $ {\\mathcal {D}}(H_a) = \\left\\lbrace H^2(\\Omega ) \\,:\\,\\partial _\\nu i a { on } \\partial \\Omega \\right\\rbrace \\subset L^2(\\Omega ).$ In particular, when $a \\geqslant 0$ the solution $u$ of the problem (REF ) is given by the semi-group $e^{-itH_a}$ generated by this operator.", "When $u_0 \\in {\\mathcal {D}}(H_a)$ this solution belongs to $C^0(\\mathbb {R}_+ , {\\mathcal {D}}(H_a)) \\cap C^1(\\mathbb {R}_+,L^2(\\Omega ))$ .", "The main part of this paper is devoted to the proof of resolvent estimates for the operator $H_a$ .", "More precisely, in order to prove Theorem REF we will use the following result: Theorem 1.2 Let $a$ be as in Theorem REF .", "Then there exist $\\tilde{\\gamma }> 0$ and $C \\geqslant 0$ such that any $z \\in with $ Im(z) - $ is in the resolvent set of $ Ha$ and moreover$$\\left\\Vert (H_a- z)^{-1}\\right\\Vert _{{\\mathcal {L}}(L^2(\\Omega ))} \\leqslant C.$$$ Here ${\\mathcal {L}}(L^2(\\Omega ))$ stands for the set of bounded operators on $L^2(\\Omega )$ .", "When $a$ is constant, our analysis will provide a good description of the spectrum of $H_a$ , and in particular there is a spectral gap as stated in Theorem REF .", "However, for a non-selfadjoint operator the norm of the resolvent can be large far from the spectrum, so the uniform bound of the resolvent on a strip around the real axis is not a consequence of the spectral gap and has to be proved directly.", "In the papers we have mentioned above concerning the damped Schrödinger equation [8], [7], [4], [3], it is proved that this equation satisfies the Kato smoothing effect, that is an estimate of the form $\\int _0^\\infty \\left\\Vert w(x) (1-^\\frac{1}{4} u(t)\\right\\Vert _{L^2}^2 \\, dt \\lesssim \\left\\Vert u_0\\right\\Vert _{L^2}^2,$ for some suitable weight function $w$ .", "In these papers, the absorption is in the interior of the domain and, more important, it is of the form $a(x) (1-^{\\frac{1}{2}} a(x)$ .", "This is stronger than multiplication by $a(x)^2$ for high-frequencies.", "It is not the case here, which is why we have no smoothing effect for our problem.", "However, we can at least recover the same result as in the non-selfadjoint case $a=0$ (see for instance [17]), namely the smoothing property in the unbounded directions: Theorem 1.3 Let $a$ be as in Theorem REF .", "Let $\\delta > \\frac{1}{2}$ .", "Then there exists $C \\geqslant 0$ such that for all $u_0 \\in L^2(\\Omega )$ the solution $u$ of (REF ) satisfies $\\int _0^\\infty \\left\\Vert \\left< x \\right> ^{-\\delta } \\big ( 1- x\\big )^{\\frac{1}{4}} u(t)\\right\\Vert _{L^2(\\Omega )}^2 \\, dt \\leqslant C \\left\\Vert u_0\\right\\Vert _{L^2(\\Omega )}^2,$ where $x = \\sum _{n=1}^{d-1} \\partial _{x_n}^2$ is the partial Laplacian acting on the unbounded directions.", "Here and everywhere below we write $\\left< x \\right>$ for $\\big ( 1 + \\left\|x\\right\|^2 \\big )^{\\frac{1}{2}}$ .", "The proof of Theorem REF relies on the theory of relatively smooth operators in the sense of Kato.", "In order to apply it in our non-selfadjoint setting we will use a self-adjoint dilation of our dissipative operator.", "In the results above, we can relax the assumption that the absorption index is positive everywhere.", "In Theorem REF we consider the case where $a$ is positive on one side of the boundary, say $\\mathbb {R}^{d-1} \\times \\left\\lbrace l \\right\\rbrace $ , and vanishes on the other side $\\mathbb {R}^{d-1} \\times \\left\\lbrace 0 \\right\\rbrace $ .", "What happens if the absorption index is a negative constant on one side is not so clear.", "Let $a_l,a_0\\in \\mathbb {R}$ be such that $ a_l+ a_0> 0$ and consider the problem $ {\\left\\lbrace \\begin{array}{ll}i \\partial _t u + u̥ = 0 & \\text{on } ]0,+\\infty [ \\times \\Omega ,\\\\\\partial _\\nu u = i a_lu & \\text{on } ]0,+\\infty [ \\times \\mathbb {R}^{d-1} \\times \\left\\lbrace l \\right\\rbrace ,\\\\\\partial _\\nu u = i a_0u & \\text{on } ]0,+\\infty [ \\times \\mathbb {R}^{d-1} \\times \\left\\lbrace 0 \\right\\rbrace ,\\\\u(0,\\cdot ) = u_0 & \\text{on } \\Omega .\\end{array}\\right.", "}$ We denote by $H_{a_l,a_0}$ the corresponding operator: $H_{a_l,a_0}= - with domain$${\\mathcal {D}}(H_{a_l,a_0}) = \\left\\lbrace u \\in H^2(\\Omega ) \\,:\\,\\partial _\\nu u = i a_lu \\text{ on } \\mathbb {R}^{d-1} \\times \\left\\lbrace l \\right\\rbrace , \\partial _\\nu u = i a_0u \\text{ on } \\mathbb {R}^{d-1} \\times \\left\\lbrace 0 \\right\\rbrace \\right\\rbrace .$$$ When $a_l$ and $a_0$ have different signs but satisfy (REF ), we will say that the boundary condition is weakly dissipative.", "Estimates like those of Theorems REF and REF are not likely to hold without this assumption.", "For instance, when $a_0= - a_l$ we get a ${\\mathcal {P}}{\\mathcal {T}}$ -symmetric waveguide.", "Such a boundary condition has been studied in [23], [10].", "See also [24].", "In particular it is known that in this case the spectrum is real, so we cannot expect any generalization of our results.", "The case $a_l+a_0< 0$ is dual to the case $a_l+ a_0> 0$ .", "We will see that in this case the norm of the solution grows exponentially (see Remark REF ).", "In the weakly dissipative case we have the following theorem: Theorem 1.4 Let $a_l, a_0\\in \\mathbb {R}$ satisfy Assumption (REF ).", "If $u_0 \\in {\\mathcal {D}}(H_{a_l,a_0})$ then the problem (REF ) has a unique solution $u \\in C^1(]0,+\\infty [,L^2(\\Omega )) \\cap C^0 ([0,+\\infty [,{\\mathcal {D}}(H_{a_l,a_0}))$ .", "Moreover, if $\\left\|a_l\\right\|$ and $\\left\|a_0\\right\|$ are small enough there exist $\\gamma > 0$ and $C \\geqslant 0$ such that for all $u_0$ we have $\\forall t \\geqslant 0, \\quad \\left\\Vert u(t)\\right\\Vert _{L^2(\\Omega )} \\leqslant C e^{-\\gamma t} \\left\\Vert u_0\\right\\Vert _{L^2(\\Omega )}.$ Let us discuss the assumption that the absorption has to be small.", "In [34] it was proved in another context that for high frequencies the properties of a dissipative problem remain valid when the absorption index is positive on average along the corresponding classical flow.", "This is the case here under Assumption (REF ).", "The restriction comes from low frequencies.", "When $a \\ne 0$ , the boundary condition can be rewritten $u = \\frac{1}{ia} \\partial _\\nu u \\quad \\text{on } \\partial \\Omega .$ When $a$ is large compared to the frequency, this is close to a Dirichlet condition.", "That for large dissipation we recover a self-adjoint problem is usually called the overdamping phenomenon.", "This suggests that the problem is now governed by the quantity $1/a$ .", "And when $a_la_0< 0$ , Assumption (REF ) can be rewritten as $\\frac{1}{a_l} + \\frac{1}{a_0} < 0.$ It turns out that for $a_l,a_0$ large enough with $a_la_0< 0$ and (REF ), the contribution of low frequencies is indeed exponentially increasing (see Proposition REF ).", "In this paper we prove all these results on the model case of a straight waveguide with a one-dimensional section.", "The purpose is on one hand to observe all the non-trivial phenomenons mentioned above on a quite simple example.", "On the other hand our analysis is the first step toward the understanding of similar properties for the wave equation on a more general domain.", "On this model case, and with the additional assumption that $a$ is constant, we can rewrite $H_a$ as the sum of the usual Laplacian on the Euclidean space $\\mathbb {R}^{d-1}$ and a Laplace operator on the compact section.", "Since this section is of dimension 1, we can give quite explicitely many spectral properties for this operator.", "In particular, we will see that its eigenfunctions form a Riesz basis, which will give a good description of the spectrum of $H_a$ , first when $a$ is constant and then in the general case.", "The time dependent estimate will follow.", "In Section we prove that if $a \\geqslant 0$ then $H_a$ is maximal dissipative, which gives in particular well-posedness for the problem (REF ).", "In Section we study the transverse operator, that is the Laplace operator on the section $]0,l[$ .", "Spectral properties of $H_a$ are obtained for a constant absorption index in Section , and then Theorems REF and REF are proved in Section .", "Once the spectral properties of $H_a$ are well-understood, the proof of Theorem REF is given in Section .", "Finally, the problem (REF ) where $a$ can take negative values is discussed in Section .", "All along the paper, a general point in $\\Omega $ will be written $(x,y) \\in \\Omega \\simeq \\mathbb {R}^{d-1} \\times ]0,l[$ , with $x \\in \\mathbb {R}^{d-1}$ and $y \\in ]0,l[$ .", "As in Theorem REF , we denote by $x$ the usual laplacian on $\\mathbb {R}^{d-1}$ .", "For Hilbert spaces ${\\mathcal {H}}_1$ and ${\\mathcal {H}}_2$ , ${\\mathcal {L}}({\\mathcal {H}}_1,{\\mathcal {H}}_2)$ is the set of bounded operators from ${\\mathcal {H}}_1$ to ${\\mathcal {H}}_2$ .", "For $\\gamma > 0$ we finally set $\\gamma = \\left\\lbrace z \\in \\mathop {\\rm {Im}}\\nolimits z > -\\gamma \\right\\rbrace , \\quad \\text{and then} \\quad + := 0.$ Acknowledgements: I am grateful to Petr Siegl for stimulating discussions which motivated this paper and helped me through its realization.", "This work is partially supported by the French-Czech BARRANDE Project 26473UL and by the French National Research Project NOSEVOL (ANR 2011 BS01019 01)." ], [ "Operator associated to the dissipative waveguide", "In this section we consider a more general waveguide $\\Omega $ of the form $\\mathbb {R}^{p}\\times \\omega \\subset \\mathbb {R}^d$ where $p \\in \\lbrace 1,\\dots ,d-1\\rbrace $ and $\\omega $ is a smooth open bounded subset of $\\mathbb {R}^{d-p}$ .", "In particular, $\\Omega $ is open in $\\mathbb {R}^d$ .", "Let $a \\in W^{1,\\infty }(\\partial \\Omega )$ .", "Until Proposition REF , we make no assumption on the sign of $a$ .", "We consider on $L^2(\\Omega )$ the operator $H_a$ defined by (REF ) with domain (REF ).", "For all ${\\mathcal {D}}(H_a)$ we have $ \\left< H_a >_{L^2(\\Omega )} = - \\left< , >_{L^2(\\Omega )} = \\left< \\nabla \\nabla >_{L^2(\\Omega )} -i \\left< a >_{L^2(\\partial \\Omega )}.\\right.On the other hand, we consider the quadratic form defined for \\right.\\right.\\right.", "{\\mathcal {D}}(q_a) = H^1(\\Omega ) byq_a( = \\int _\\Omega \\left\|\\nabla \\vert ^2 - i \\int _{\\partial \\Omega } a\\left\|\\vert ^2 .\\right.\\right.We also denote by q_a the corresponding sesquilinear form on {\\mathcal {D}}(q_a)^2.", "That this quadratic form is sectorial and closed follows from the following lemma and traces theorems:$ Lemma 2.1 Let $q_R$ be a non-negative, densely defined, closed form on a Hilbert space ${\\mathcal {H}}$ .", "Let $q_I$ be a symmetric form relatively bounded with respect to $q_R$ .", "Then the form $q_R - i q_I$ is sectorial and closed.", "It is important to note that there is no smallness assumption on the relative bound of $q_I$ with respect to $q_R$ .", "In particular, for $q_a$ we do not need any assumption on the size of $a$ in $L^\\infty (\\partial \\Omega )$ .", "There exists $C >0$ such that for all ${\\mathcal {D}}(q_R)$ we have $\\left\|q_I(\\right\| \\leqslant C \\left( q_R( + \\left\\Vert \\Vert ^2_{\\mathcal {H}}\\right).\\right.$ Let $\\varepsilon _0 = \\frac{1}{2C}$ .", "If we already know that $(q_R -i\\lambda q_I)$ is sectorial and closed for some $\\lambda \\geqslant 0$ , then $(q_R -i(\\lambda +\\varepsilon ) q_I)$ is sectorial and closed for all $\\varepsilon \\in [0,\\varepsilon _0]$ according to Theorem VI.3.4 in [22].", "Now since $q_R$ is sectorial and closed, we can prove by induction on $n \\in \\mathbb {N}$ that $(q_R -i\\lambda q_I)$ is sectorial and closed for all $\\lambda \\in [0,n\\varepsilon _0]$ , and hence for all $\\lambda \\geqslant 0$ .", "This is in particular the case when $\\lambda =1$ .", "We recall the definitions of accretive and dissipative operators (note that the conventions may be different for other authors): Definition 2.2 We say that an operator $T$ on the Hilbert space ${\\mathcal {H}}$ is accretive (respectively dissipative) if $\\forall {\\mathcal {D}}(T), \\quad \\mathop {\\rm {Re}}\\nolimits \\left< T >_{\\mathcal {H}}\\geqslant 0, \\quad \\big (\\text{respectively} \\quad \\mathop {\\rm {Im}}\\nolimits \\left< T >_{\\mathcal {H}}\\leqslant 0\\big ).\\right.\\right.$ Moreover $T$ is said to be maximal accretive (maximal dissipative) if it has no other accretive (dissipative) extension on ${\\mathcal {H}}$ than itself.", "In particuliar $T$ is (maximal) dissipative if and only if $iT$ is (maximal) accretive.", "Let us recall that an accretive operator $T$ is maximal accretive if and only if $(T-z)$ has a bounded inverse on ${\\mathcal {H}}$ for some (and hence any) $z \\in with $ Re(z) < 0$.", "In this case we know from the Hille-Yosida Theorem that $ -T$ generates a contractions semi-group $ t e-tT$.", "Then for all $ u0 D(T)$ the map $ u : t e-tTu0$ belongs to $ C1(R+,H) C0(R+,D(T))$ and solves the problem$${\\left\\lbrace \\begin{array}{ll}u^{\\prime }(t) + T u(t) = 0, \\quad \\forall t > 0,\\\\u(0) = u_0.\\end{array}\\right.", "}$$$ Let us come back to our context.", "According to Lemma REF and the Representation Theorem VI.2.1 in [22], there exists a unique maximal accretive operator $\\hat{H}_a$ on $L^2(\\Omega )$ such that ${\\mathcal {D}}(\\hat{H}_a) \\subset {\\mathcal {D}}(q_a)$ and $\\forall {\\mathcal {D}}(\\hat{H}_a), \\forall \\psi \\in {\\mathcal {D}}(q_a) , \\quad \\left< \\hat{H}_a \\psi \\right>_{L^2(\\Omega )} = q_a(\\psi ).$ Moreover we have ${\\mathcal {D}}(\\hat{H}_a) = \\left\\lbrace u \\in {\\mathcal {D}}(q_a) \\,:\\,\\exists f \\in L^2(\\Omega ) , \\forall \\phi \\in {\\mathcal {D}}(q_a), q_a(u,\\phi ) = \\left< f , \\phi \\right> \\right\\rbrace ,$ and for $u \\in {\\mathcal {D}}(\\hat{H}_a)$ the corresponding $f$ is unique and given by $\\hat{H}_au = f$ .", "Proposition 2.3 We have $\\hat{H}_a= H_a$ .", "In particular $H_a$ is maximal accretive.", "For a one-dimensional section we can essentially follow the proof of Lemma 3.2 in [10].", "This would be enough for our purpose but, for further use, we prove this result in the general setting.", "It is easy to check that ${\\mathcal {D}}(H_a) \\subset {\\mathcal {D}}(\\hat{H}_a)$ and $H_a= \\hat{H}_a$ on ${\\mathcal {D}}(H_a)$ .", "Now let $u \\in {\\mathcal {D}}(\\hat{H}_a)$ .", "By definition there exists $f \\in L^2(\\Omega )$ such that $\\forall \\phi \\in H^1(\\Omega ), \\quad \\int _{\\Omega } \\nabla u \\cdot \\nabla \\overline{\\phi } -i \\int _{\\partial \\Omega } a u \\overline{\\phi }= \\int _{\\Omega } f \\overline{\\phi }.$ Considering $\\phi \\in C_0^\\infty (\\Omega )$ we see that $-u̥ = f$ in the sense of distributions and hence in $L^2(\\Omega )$ .", "This proves that $u \\in H^2_{\\rm {loc}}(\\Omega )$ .", "It remains to prove that $u \\in H^2(\\Omega )$ and that the boundary condition $\\partial _\\nu u = i a u$ holds on $\\partial \\Omega $ .", "Let $j \\in \\lbrace 1,\\dots ,p\\rbrace $ and let $e_j$ be the $j$ -th vector in the canonical basis of $\\mathbb {R}^{p}$ .", "Let $\\delta \\in \\mathbb {R}^$ and $u_\\delta : (x,y) \\mapsto \\frac{1}{\\delta }( u(x + \\delta e_j , y ) - u(x,y)) \\in H^1(\\Omega )$ .", "See for instance [19] for the properties of the difference quotients.", "For all $\\phi \\in H^1(\\Omega )$ we have $q_a(u_\\delta , \\phi ) = -\\int _\\Omega f(x,y) \\overline{\\phi _{-\\delta }} (x,y) \\, dx \\,d\\sigma (y) - i \\int _{\\partial \\Omega } u (x,y) a_{-\\delta }(x,y) \\overline{\\phi }(x-\\delta e_j,y) \\, dx \\, d\\sigma (y),$ where $\\sigma $ is the Lebesgue measure on $\\omega $ .", "Since $a \\in W^{1,\\infty }(\\partial \\Omega )$ there exists $C \\geqslant 0$ such that for all $\\phi \\in H^1(\\Omega )$ and $\\delta > 0$ we have $\\left\|q_a (u_\\delta , \\phi )\\right\| \\leqslant C \\left\\Vert \\phi \\right\\Vert _{H^1(\\Omega )}.$ Applied with $\\phi = u_\\delta $ this gives $\\left\\Vert u_\\delta \\right\\Vert _{\\dot{H}^1(\\Omega )}^2 = \\mathop {\\rm {Re}}\\nolimits q_a (u_\\delta ,u_\\delta ) \\leqslant C \\left\\Vert u_\\delta \\right\\Vert _{H^1(\\Omega )}.$ Since we already know that $u_\\delta \\in L^2(\\Omega )$ uniformly in $\\delta > 0$ , we have $\\left\\Vert u_\\delta \\right\\Vert _{H^1(\\Omega )}^2 \\lesssim 1 + \\left\\Vert u_\\delta \\right\\Vert _{H^1(\\Omega )},$ which implies that $u_\\delta $ is uniformly in $H^1(\\Omega )$ .", "This means that $\\partial _{x_j} u \\in H^1(\\Omega )$ .", "Since this holds for any $j \\in \\lbrace 1,\\dots ,p\\rbrace $ , this proves that all the derivatives of order 2 with at least one derivative in the first $p$ directions belong to $L^2(\\Omega )$ .", "Then we get $-y u = f + x u \\in L^2(\\Omega ).$ According to the Green Formula we have for all $\\phi \\in H^1(\\Omega )$ $\\int _{\\Omega } - y u \\, \\overline{\\phi }\\, dx \\, dy = \\int _{\\Omega } \\nabla _y u \\cdot \\nabla _y \\overline{\\phi }\\, dx \\, dy - \\left< \\partial _\\nu u , \\phi \\right>_{H^{-1/2}(\\partial \\Omega ),H^{1/2}(\\partial \\Omega )}$ (see for instance [20]).", "By density of the trace map, we obtain that $ \\partial _\\nu u = ia u \\quad \\text{on } \\partial \\Omega .$ In particular $\\partial _\\nu u \\in H^{1/2}(\\partial \\Omega )$ .", "Then there exists $v \\in H^2(\\Omega )$ such that $\\partial _\\nu v = \\partial _\\nu u$ (see [20] for a fonction on $\\mathbb {R}^d_+$ ; for a function on $\\Omega $ we follow the same idea as for a fonction on a bounded domain, except that we only use a partition of unity for $\\partial \\omega $ , which allows to cover $\\partial \\Omega $ by a finite number of strips, each of which is diffeomorphic to a strip on $\\mathbb {R}^{p}$ ).", "Let $w = u - v$ .", "We have $w \\in H^1(\\Omega )$ , $y w \\in L^2(\\Omega )$ and ${\\left\\lbrace \\begin{array}{ll}-y w + w = f + x u - y v + w & \\text{on } \\Omega , \\\\\\partial _\\nu w = 0& \\text{on } \\partial \\Omega .\\end{array}\\right.", "}$ Then for almost all $x \\in \\mathbb {R}^{p}$ we have ${\\left\\lbrace \\begin{array}{ll}-y w(x) + w(x) = f(x) + x u(x) - y v(x) + w(x) & \\text{on } \\omega , \\\\\\partial _\\nu w(x) = 0& \\text{on } \\partial \\omega .\\end{array}\\right.", "}$ By elliptic regularity for the Neumann problem (see for instance Theorem 9.26 in [13]) we obtain that $w(x) \\in H^2(\\omega )$ with $\\left\\Vert w(x)\\right\\Vert _{H^2(\\omega )} \\lesssim \\left\\Vert f(x) + x u(x) - y v(x) + w(x)\\right\\Vert _{L^2(\\omega )}.$ After integration over $x \\in \\mathbb {R}^{p}$ , this gives $\\left\\Vert u\\right\\Vert _{H^2_y(\\Omega )} \\lesssim \\left\\Vert f+ x u - y v + w\\right\\Vert _{L^2(\\Omega )} + \\left\\Vert v\\right\\Vert _{H^2(\\Omega )}.$ Since we already know that second derivatives of $u$ involving a derivation in $x$ are in $L^2(\\Omega )$ , this proves that $u \\in H^2(\\Omega )$ and concludes the proof.", "Remark 2.4 We have $H_a^ = H_{-a}$ .", "Now assume that $a$ takes non-negative values.", "According to (REF ), $H_a$ is a dissipative operator.", "Since it is maximal accretive, it is easy to see that it is in fact maximal dissipative: Proposition 2.5 The maximal accretive operator $H_a$ is also maximal dissipative.", "We already know that $H_a$ is dissipative.", "Since it is maximal accretive, any $z \\in with $ Rez < 0$ is in its resolvent set.", "Then it is easy to find $ z$ in the resolvent set of $ Ha$ with $ Imz > 0$.$ Proposition 2.6 If $a > 0$ in an open subset of $\\partial \\Omega $ then $H_a$ has no real eigenvalue.", "Let $u \\in {\\mathcal {D}}(H_a)$ , $\\lambda \\in \\mathbb {R}$ , and assume that $H_au = \\lambda u$ .", "Taking the imaginary part of the equality $q_a (u,u) = \\lambda \\left\\Vert u\\right\\Vert ^2$ gives $\\int _{\\partial \\Omega } a \\left\|u\\right\|^2 = 0.$ This implies that $u = 0$ where $ a \\ne 0$ and $\\partial _\\nu u = ia u = 0$ everywhere on $\\partial \\Omega $ .", "By unique continuation, this implies that $u = 0$ on $\\Omega $ .", "The Transverse Operator Let us come back to the case of a one-dimensional cross-section $\\omega = ]0,l[$ .", "Under the additional assumption that the absorption index $a$ is constant on $\\partial \\Omega $ the operator $H_a$ can be written as $ H_a= -x \\otimes \\operatorname{Id}_{L^2(0,l)} + \\operatorname{Id}_{L^2(\\mathbb {R}^{d-1})} \\otimes T_a,$ where $-x$ is as before the usual flat Laplacian on $\\mathbb {R}^{d-1}$ and $T_a$ is the transverse Laplacian on $]0,l[$ .", "More precisely, we consider on $L^2(0,l)$ the operator $T_a= - \\frac{d^2}{dy^2}$ with domain ${\\mathcal {D}}(T_a) = \\left\\lbrace u \\in H^2(0,l) \\,:\\,u^{\\prime }(0) = -ia u(0), u^{\\prime }(l) = ia u(l) \\right\\rbrace .$ This is the maximal accretive and dissipative operator corresponding to the form $q : u \\in H^1(0,l) \\mapsto \\int _{0}^{l} \\left\|u^{\\prime }(x)\\right\|^2 \\, dx - i a \\left\|u(l)\\right\|^2 - i a \\left\|u(0)\\right\|^2.$ In this section we give the spectral properties of $T_a$ which we need to study the full operator $H_a$ .", "This operator has compact resolvent, and hence its spectrum is given by a sequence of isolated eigenvalues.", "When $a = 0$ , which corresponds to the Neumann problem, we know that the eigenvalues of $T_0$ are the real numbers $n^2 \\nu ^2$ for $n \\in \\mathbb {N}$ , where we have set $\\nu = \\frac{\\pi }{ l}.$ These eigenvalues are algebraically simple.", "Proposition 3.1 There exists a sequence $\\left({\\lambda }_{n}\\right)_{n \\in \\mathbb {N}}$ of continuous functions on $\\mathbb {R}$ such that $\\lambda _n(0) = n \\nu $ and for all $a \\in \\mathbb {R}$ the set of eigenvalues of $T_a$ is $\\left\\lbrace \\lambda _n(a)^2, n\\in \\mathbb {N} \\right\\rbrace $ .", "Moreover: For $(n,a) \\in (\\mathbb {N}\\times \\mathbb {R}) \\setminus \\lbrace (0,0)\\rbrace $ the eigenvalue $\\lambda _n(a)^2$ is algebraically simple and a corresponding eigenvector is given by $ {n}(a) : x \\mapsto A_n(a) \\left( e^{i\\lambda _n(a) x} + \\frac{\\lambda _n(a) + a}{\\lambda _n(a) - a} e^{-i \\lambda _n(a) x} \\right) ,$ where we can choose $A_n(a) \\in \\mathbb {R}_+^$ in such a way that $\\left\\Vert n(a)\\right\\Vert _{L^2(0,l)} = 1$ (when $a = 0$ then 0 is a simple eigenvalue and corresponding eigenvectors are non-zero constant functions).", "For $n \\in \\mathbb {N}$ and $a \\in \\mathbb {R}$ we have $\\lambda _n(-a)= \\overline{\\lambda _n(a)}$ .", "Let $n\\in \\mathbb {N}$ .", "For all $a \\in \\mathbb {R}^$ we have $\\mathop {\\rm {Re}}\\nolimits (\\lambda _n(a)) \\in ] n\\nu ,(n+1)\\nu [$ (when $n=0$ , we have chosen the square root of $\\lambda _0^2(a)$ which has a positive real part).", "For all $n \\in \\mathbb {N}$ there exists $C_n > 0$ such that for $a > 0$ we have $-C_n < \\mathop {\\rm {Im}}\\nolimits (\\lambda _n(a)) < 0$ .", "Let $a > 0$ be fixed.", "We have $\\lambda _n(a) = n \\nu - \\frac{2ia}{n\\nu l} + \\mathop {O}\\limits _{n \\rightarrow +\\infty } \\big (n^{-2} \\big )$ and hence $\\lambda _n(a)^2 = (n\\nu )^2 - \\frac{4ia}{l} + \\mathop {O}\\limits _{n \\rightarrow +\\infty } \\big (n^{-1} \\big ).$ $\\bullet $ It is straightforward computations to check that 0 is an eigenvalue of $T_a$ if and only if $a = 0$ and, if $\\lambda \\in $ , $\\lambda ^2$ is an eigenvalue if and only if $ (a-\\lambda )^2 e^{2i\\lambda l} = ( a+\\lambda )^2.$ If $\\lambda ^2$ is an eigenvalue then the corresponding eigenfunction is of the form $ x \\mapsto A e^{i\\lambda x} + B e^{-i\\lambda x}$ with $ A = \\frac{\\lambda -a}{\\lambda +a} B = \\frac{\\lambda + a}{\\lambda -a} e^{-2i\\lambda l}B.$ Moreover, all these eigenvalues have geometric multiplicity 1.", "Indeed, given $n \\in \\mathbb {N}$ , the space of eigenvectors corresponding to the eigenvalue $\\lambda _n(a)^2$ is strictly included in the space of $H^2$ functions which are solutions of $-u^{\\prime \\prime } - \\lambda _n(a)^2 u = 0$ , and this space is of dimension 2.", "The fact that the eigenvalues of $H_{-a}$ are conjugated to the eigenvalues of $H_{a}$ is a consequence of Remark REF .", "$\\bullet $ Let $a > 0$ and $\\lambda \\in $ be such that $\\lambda ^2$ is an eigenvalue of $T_a$ .", "Assume that $\\mathop {\\rm {Re}}\\nolimits \\lambda \\in \\nu \\mathbb {N}$ .", "Then $\\left( \\frac{a+\\lambda }{a-\\lambda } \\right)^2 = e^{2i\\lambda l} \\in \\mathbb {R}_+$ (note that $\\lambda $ cannot be equal to $a$ in (REF )) and hence $r := \\frac{a+\\lambda }{a-\\lambda } \\in \\mathbb {R}.$ If $r = -1$ then $a = 0$ .", "Otherwise $\\lambda = \\frac{a(1-r)}{1+r} \\in \\mathbb {R}$ .", "In both cases we obtain a contradiction (see Proposition REF ), and hence $\\mathop {\\rm {Re}}\\nolimits \\lambda \\notin \\nu \\mathbb {N}$ .", "This proves that for $a > 0$ the operator $T_a$ has no eigenvalue with real part in $\\nu \\mathbb {N}$ .", "$\\bullet $ Now let $R > 0$ .", "We prove that if $C_R \\geqslant 0$ is chosen large enough and if $a \\in \\mathbb {R}$ and $\\lambda \\in $ are such that $\\lambda ^2$ is an eigenvalue of $T_a$ , then $ \\left\|\\mathop {\\rm {Re}}\\nolimits \\lambda \\right\| \\leqslant R \\quad \\Rightarrow \\quad \\left\|\\mathop {\\rm {Im}}\\nolimits \\lambda \\right\| \\leqslant C_R.$ Assume by contradiction that this is not the case.", "Then for all $m \\in \\mathbb {N}$ we can find $x_m \\in [-R,R]$ and $y_m \\in \\mathbb {R}$ with $\\left\|y_m\\right\| \\geqslant m$ such that $(x_m+ i y_m)^2$ is an eigenvalue of $T_{a_m}$ for some $a_m \\in \\mathbb {R}$ .", "We have $e^{-2y_m l} = \\left\|\\frac{a_m + x_m + iy_m}{a_m - x_m - iy_m}\\right\|^2 = \\frac{(a_m + x_m)^2 + y_m^2}{(a_m - x_m)^2 + y_m^2} \\xrightarrow[m \\rightarrow \\infty ]{}1,$ which gives a contradiction.", "$\\bullet $ The family of operators $T_a$ for $a \\in \\mathbb {R}$ is an analytic family of operators of type B in the sense of Kato [22].", "We already know that the spectrum of $T_0$ is given by $\\left\\lbrace (n\\nu )^2, n \\in \\mathbb {N} \\right\\rbrace $ .", "Then for all $n \\in \\mathbb {N}$ there exists an analytic function $\\lambda _n^2$ such that, at least for small $a$ , $\\lambda _n^2(a)$ is in the spectrum of $T_a$ (and then we define $\\lambda _n$ as the square root of $\\lambda _n^2$ with positive real part).", "See Theorem VII.1.7 in [22].", "$\\bullet $ Let $n \\in \\mathbb {N}^$ .", "We write $\\lambda _n(a) = n\\nu + \\beta a + \\gamma a^2 + O_{a \\rightarrow 0} (a^3)$ .", "We have $e^{2i \\lambda _n(a) l}= 1 + 2il\\beta a + 2il\\gamma a^2 - 2l^2 \\beta ^2 a^2 + O(a^3),$ and on the other hand: $\\left( \\frac{\\lambda _n(a) + a}{\\lambda _n(a) - a} \\right)^2= 1 + \\frac{4a}{n\\nu } - \\frac{4(\\beta -2) a^2}{n^2\\nu ^2} + O\\big (a^3\\big ).$ Since $\\lambda _n(a)$ solves (REF ) for any $a > 0$ we obtain $\\beta = \\frac{2}{iln\\nu } = -\\frac{2i}{\\pi n}$ and $\\mathop {\\rm {Re}}\\nolimits (\\gamma )= \\frac{4 l}{n^3\\pi ^3} .$ Since $\\mathop {\\rm {Re}}\\nolimits (\\beta ) = 0$ and $\\mathop {\\rm {Re}}\\nolimits (\\gamma ) > 0$ we have $\\mathop {\\rm {Re}}\\nolimits \\big (\\lambda _n(a)\\big ) \\in \\left] n\\nu , (n+1)\\nu \\right[$ for $a > 0$ small enough.", "The functions $a \\mapsto \\mathop {\\rm {Re}}\\nolimits \\big (\\lambda _n(a)\\big )$ are continuous and cannot reach $\\nu \\mathbb {N}$ unless $a=0$ , so this remains true for any $a > 0$ such that $\\lambda _n(a)$ is defined.", "Similarly $\\mathop {\\rm {Re}}\\nolimits \\big (\\lambda _0(a)\\big ) \\in ]0, \\nu [$ for all $a > 0$ .", "In particular the curves $a \\mapsto \\lambda _n(a)$ for $n \\in \\mathbb {N}$ never meet.", "Moreover we know from (REF ) that $\\lambda _n(a)$ remains in a bounded set of $, so the curves $ a n(a)$ are defined for all $ a R$ and for all $ a R$ the eigenvalues of $ Ta$ are exactly given by $ n(a)2$ for $ n N$.$ $\\bullet $ It remains to prove that the asymptotic expansion of $\\lambda _n(a)$ for $n$ fixed and $a$ small is also valid for $a$ fixed and $n$ large.", "Let $a>0$ be fixed.", "Derivating (REF ) and using the fact that $\\left\|\\lambda _n(s)\\right\| \\geqslant n \\nu $ for all $s \\in \\mathbb {R}$ we obtain that $\\sup _{s \\in [0,a]} \\left\|\\lambda ^{\\prime }_n(s)\\right\| = O \\big (n^{-1}\\big ).$ This means that $\\lambda _n(a) = n\\nu + O(n^{-1})$ .", "Then we obtain the asymptotic expansion of $\\lambda _n(a)$ for large $n$ as before, using again (REF ).", "This gives the last statement of the proposition and concludes the proof.", "Now that we have proved what we need concerning the spectrum of the operator $T_a$ , we study the corresponding sequence of eigenfunctions.", "In the self-adjoint case $a = 0$ , we know that the eigenfunctions $n(0)$ form an orthonormal basis.", "Of course this is no longer the case for the non-selfadjoint operator $T_a$ with $a \\ne 0$ .", "However we can prove that in this case we have a Riesz basis.", "We recall that the sequence $\\left({_{n}_{n \\in \\mathbb {N}} of vectors in the Hilbert space {\\mathcal {H}} is said to be a Riesz basis if there exists a bounded operator \\Theta \\in {\\mathcal {L}}({\\mathcal {H}}) with bounded inverse and an orthonormal basis \\left({e}_{n}\\right)_{n \\in \\mathbb {N}} of {\\mathcal {H}} such that n = \\Theta e_n for all n \\in \\mathbb {N}.In this case any f \\in {\\mathcal {H}} can be written as \\sum _{n\\in \\mathbb {N}} f_n n with \\left({f}_{n}\\right)_{n \\in \\mathbb {N}} \\in l^2(, and there exists C \\geqslant 1 such that for all f = \\sum _{n\\in \\mathbb {N}} f_n n\\in {\\mathcal {H}} we haveC^{-1}\\sum _{n\\in \\mathbb {N}} \\left\|f_n\\right\|^2 \\leqslant \\left\\Vert \\sum _{n\\in \\mathbb {N}} f_n n\\right\\Vert _{\\mathcal {H}}^2 \\leqslant C \\sum _{n\\in \\mathbb {N}} \\left\|f_n\\right\|^2.In these estimates we can take C = \\max \\left( \\left\\Vert \\Theta \\right\\Vert ^2 , \\left\\Vert \\Theta ^{-1}\\right\\Vert ^2 \\right).", "}Let \\right.$ (n)n N$ be a Riesz basis of $ H$ with $$ and $ (en)n N$ as above.", "If we set $ * n = (-1)* en$ for all $ n N$ then $ (n)nN$ is also a Riesz basis, called the dual basis of $ (n)n N$.", "In particular for all $ j,k N$ we have\\begin{equation} \\left< j , _k \\right>_{\\mathcal {H}}= \\delta _{j,k}.\\end{equation}We refer to \\cite {christensen} for more details about Riesz bases.", "Now we want to prove that for any $ a R$ the sequence of eigenfunctions for the operator $ Ta$ is a Riesz basis of $ L2(0,l)$.$ Proposition 3.2 For all $a \\in \\mathbb {R}$ the sequence $(n(a))_{n\\in \\mathbb {N}}$ defined in (REF ) is a Riesz basis of $L^2(0,l)$ .", "Moreover for all $R > 0$ there exists $C \\geqslant 0$ such that for $a \\in [-R,R]$ and $\\left({c}_{n}\\right)_{n \\in \\mathbb {N}} \\in l^2($ we have $C^{-1}\\sum _{n = 0}^{\\infty } \\left\|c_n\\right\|^2 \\leqslant \\left\\Vert \\sum _{n = 0}^{\\infty } c_n n(a)\\right\\Vert ^2 \\leqslant C \\sum _{n = 0}^{\\infty } \\left\|c_n\\right\|^2.$ If $C$ was chosen large enough and if $\\left({c}_{n}\\right)_{n \\in \\mathbb {N}} \\in l^2($ is such that $\\sum _{n=0}^\\infty \\left\|\\lambda _n(a) c_n\\right\|^2 < \\infty $ we also have $C^{-1}\\sum _{n = 0}^{\\infty } \\left\|\\lambda _n(a) c_n\\right\|^2 \\leqslant \\left\\Vert \\sum _{n = 0}^{\\infty } c_n _n(a)\\right\\Vert ^2 \\leqslant C \\sum _{n = 0}^{\\infty } \\left\|\\lambda _n(a) c_n\\right\|^2.$ For similar results we refer to [30] (see also Lemma XIX.3.10 in [18]).", "For the proof we need the following lemma: Lemma 3.3 Let $R > 0$ .", "Then there exists $C \\geqslant 0$ such that for $a \\in [-R,R]$ and $j,k \\in \\mathbb {N}$ with $j < k$ we have $\\left< j(a) , k(a) \\right>_{L^2(0,l)} \\leqslant \\frac{C}{\\left< j \\right> (k-j)} \\quad \\text{and} \\quad \\left< _j(a) , _k(a) \\right>_{L^2(0,l)} \\leqslant C \\frac{k}{k-j}.$ Let $e_n(a,x) = e^{i \\lambda _n (a) x}$ and $\\tilde{e}_n(a,x) = e^{-i \\lambda _n (a) x}$ .", "According to Proposition REF we have $\\lambda _n(a) = n\\nu + O(n^{-1})$ (here and below all the estimates are uniform in $a \\in [-R,R]$ ), so $\\begin{aligned}\\left\\Vert e_n(a)\\right\\Vert ^2_{L^2(0,l)}= \\frac{e^{-2l\\mathop {\\rm {Im}}\\nolimits (\\lambda _n(a))} - 1 }{-2 \\mathop {\\rm {Im}}\\nolimits (\\lambda _n(a))} = l+ O(n^{-1}).\\end{aligned}$ Similarly $\\left\\Vert \\tilde{e}_n(a)\\right\\Vert ^2_{L^2(0,l)} = l+ O(n^{-1})$ , and $\\left< e_n(a) , \\tilde{e}_n(a) \\right> =O(n^{-1})$ .", "Moreover, with (REF ) we see that $A_n(a) = \\frac{1}{\\sqrt{2l}} + O(n^{-1}).$ Now let $j,k \\in \\mathbb {N}$ with $j < k$ .", "We omit the dependence in $a$ for $j$ , $e_j$ , $\\tilde{e}_j$ and $\\lambda _j$ .", "We have $\\left< e_j , e_k \\right> = \\frac{e^{i(\\lambda _j - \\overline{\\lambda _k}) l} -1}{\\lambda _j - \\overline{\\lambda _k}}\\quad \\text{and} \\quad \\left< \\tilde{e}_j , \\tilde{e}_k \\right> = \\frac{e^{-i(\\lambda _j - \\overline{\\lambda _k}) l} -1}{-(\\lambda _j - \\overline{\\lambda _k})} .$ Since $\\lambda _j - \\overline{\\lambda _k} = (j-k) \\nu + O(1/j)$ we have $\\left\|\\left< e_j , e_k \\right>\\right\| + \\left\|\\left< \\tilde{e}_j , \\tilde{e}_k \\right>\\right\| \\lesssim \\frac{1}{k-j}\\quad \\text{and} \\quad \\left\|\\left< e_j , e_k \\right> + \\left< \\tilde{e}_j , \\tilde{e}_k \\right>\\right\| \\lesssim \\frac{1}{\\left< j \\right> (k-j)} .$ Similarly $\\left\|\\left< \\tilde{e}_j , e_k \\right>\\right\| + \\left\|\\left< e_j , \\tilde{e}_k \\right>\\right\| \\lesssim \\frac{1}{k+j}\\quad \\text{and} \\quad \\left\|\\left< \\tilde{e}_j , e_k \\right> + \\left< e_j , \\tilde{e}_k \\right>\\right\| \\lesssim \\frac{1}{\\left< j \\right> (k+j)} .$ And finally $\\left\|\\left< j , k \\right>\\right\|& = \\left\|A_j\\overline{A_k}\\left< e_j + \\frac{\\lambda _j - a }{\\lambda _j + a } \\tilde{e}_j , e_k + \\frac{\\lambda _k - a }{\\lambda _k + a } \\tilde{e}_k \\right>\\right\|\\\\& \\lesssim \\left\| \\left< e_j , e_k \\right> +\\left< e_j , \\tilde{e}_k \\right>+\\left< \\tilde{e}_j , e_k \\right>+\\left< \\tilde{e}_j , \\tilde{e}_k \\right> \\right\|\\\\& \\quad + \\left< j \\right> ^{-1}\\left( \\left\|\\left< e_j , e_k \\right>\\right\| + \\left\|\\left< e_j , \\tilde{e}_k \\right>\\right\| + \\left\|\\left< \\tilde{e}_j , e_k \\right>\\right\| + \\left\|\\left< \\tilde{e}_j , \\tilde{e}_k \\right>\\right\| \\right)\\\\& \\lesssim \\frac{1}{\\left< j \\right> (k-j)}.$ The second estimate is proved similarly, using again that $\\left\|\\lambda _n(a)\\right\| = n\\nu + O(n^{-1})$ for large $n$ .", "$\\bullet $ Let $a \\in \\mathbb {R}$ and $\\left({c}_{n}\\right)_{n \\in \\mathbb {N}} \\in l^2($ .", "Let $C\\geqslant 0$ be given by Lemma REF .", "For $N,p \\in \\mathbb {N}$ we have $\\left\\Vert \\sum _{n = N}^{N+p} c_n n(a)\\right\\Vert ^2 - \\sum _{n=N }^{N+p} \\left\|c_n\\right\|^2&= \\sum _{j=N}^{N+p} \\sum _{k=j+1}^{N+p} 2 \\mathop {\\rm {Re}}\\nolimits \\left( c_j \\overline{c_k} \\left< j , k \\right> \\right) \\\\& \\leqslant 2C \\sum _{j=0}^\\infty \\frac{\\left\|c_j\\right\|}{\\left< j \\right>} \\sum _{k=1}^\\infty \\frac{\\left\|c_{k+j}\\right\|}{k} \\\\& \\lesssim \\left\\Vert c\\right\\Vert _{l^2(}^2.$ This proves that the series $\\sum _{n = 0}^\\infty c_n n(a)$ converges in $L^2(0,l)$ and $\\left\\Vert \\sum _{n = 0}^\\infty c_n n(a)\\right\\Vert _{L^2(0,l)}^2 \\lesssim \\sum _{n=0}^\\infty \\left\|c_n\\right\|^2.$ $\\bullet $ Let $n,m \\in \\mathbb {N}$ be such that $n\\ne m$ .", "In $L^2(0,l)$ we have $\\lambda _n(a)^2 \\left< n(a) , m(-a) \\right>& = \\left< T_an(a) , m(-a) \\right> = \\left< n(a) , T_a^* m(-a) \\right>\\\\& = {\\lambda _m(a)^2} \\left< n(a) , m(-a) \\right>,$ and hence $\\left< n(a) , m(-a) \\right> = 0$ .", "On the other hand, with (REF ) we can check that for $n$ large enough we have $\\left< n(a) , n(-a) \\right> \\ne 0$ .", "Now let $\\left({c}_{n}\\right)_{n \\in \\mathbb {N}} \\in l^2($ be such that $\\sum _{n=0}^\\infty c_n n(a) = 0$ .", "Taking the inner product with $m(-a)$ we see that $c_m = 0$ for $m > N$ if $N$ is chosen large enough.", "Then for all $k \\in \\lbrace 0,\\dots ,N\\rbrace $ we have $0 = T_a^k \\sum _{n=0}^N c_n n(a) = \\sum _{n=0}^N \\lambda _n(a)^{2k} c_n n(a).$ Since the eigenvalues $\\lambda _n(a)^2$ for $n\\in \\lbrace 0,\\dots ,N\\rbrace $ are pairwise disjoint, this proves that for all $n \\in \\lbrace 0,\\dots ,N\\rbrace $ we have $c_n n(a) = 0$ , and hence $c_n=0$ .", "Finally the map $(c_n) \\in l^2( \\mapsto \\sum c_nn(a) \\in L^2(0,l)$ is one-to-one.", "$\\bullet $ As above we can check that for $N\\in \\mathbb {N}$ we have $\\left\\Vert \\sum _{n = N}^{\\infty } c_n n(a)\\right\\Vert ^2 - \\sum _{n=N}^{\\infty } \\left\|c_n\\right\|^2\\lesssim \\sum _{n=N}^{\\infty } \\left\|c_n\\right\|^2 \\sqrt{\\sum _{n=N}^\\infty \\frac{1}{\\left< n \\right>^2}}.$ This is less that $\\frac{1}{2} \\sum _{n=N}^{\\infty } \\left\|c_n\\right\|^2$ if $N$ is chosen large enough.", "Let such an integer $N$ be fixed.", "Now assume by contradiction that the sequences $\\left({a}_{m}\\right)_{m \\in \\mathbb {N}} \\in [-R,R]^\\mathbb {N}$ and $(c^m)_{m\\in \\mathbb {N}}$ in $l^2(^\\mathbb {N}$ are such that $\\left\\Vert c^m\\right\\Vert _{l^2} = 1$ for all $m \\in \\mathbb {N}$ and $\\left\\Vert \\sum _{n=0}^\\infty c_n^m n(a_m)\\right\\Vert ^2 \\xrightarrow[m \\rightarrow \\infty ]{}0.$ For $m \\in \\mathbb {N}$ we set $f_m = \\sum _{n=0}^{N-1} c_n^m n(a_m)$ and $g_m = \\sum _{n=N}^{\\infty } c_n^m n(a_m)$ .", "After extracting a subsequence if necessary we can assume that $a_m$ converges to some $a \\in [-R,R]$ and $f_m$ converges to some $f \\in L^2(0,l)$ .", "Let $P_m$ (respectively $P$ ) denote the orthogonal projection on $\\operatorname{span}(j(a_m))_{j\\geqslant N}$ (respectively on $\\operatorname{span}(n(a))_{n\\geqslant N}$ ).", "We have $\\left\\Vert f_m + g_m\\right\\Vert ^2 = \\left\\Vert f_m\\right\\Vert ^2 + 2 \\mathop {\\rm {Re}}\\nolimits \\left< P_m f_m , g_m \\right> + \\left\\Vert g_m\\right\\Vert ^2 \\geqslant \\left\\Vert f_m\\right\\Vert ^2 - \\left\\Vert P_m f_m\\right\\Vert ^2 \\xrightarrow[m \\rightarrow \\infty ]{}\\left\\Vert f\\right\\Vert ^2 - \\left\\Vert P f\\right\\Vert ^2.$ This gives $f = P f$ , and hence $f = 0$ .", "This gives a contradiction with $0 = \\lim _{m \\rightarrow \\infty } \\left\\Vert g_m\\right\\Vert ^2 \\geqslant \\lim _{m\\rightarrow \\infty } \\frac{1}{2} \\sum _{n=N}^\\infty \\left\|c_n^m\\right\|^2 = \\frac{1}{2}$ and proves the first inequality of the proposition.", "$\\bullet $ It remains to prove that the sequence $(n(a))$ is complete.", "We know that the family $\\big ( n(0) \\big )_{n\\in \\mathbb {N}}$ is an orthonormal basis of $L^2(0,l)$ .", "Since $n(a) = n(0) + O \\big ( n^{-1}\\big )$ in $L^2(0,l)$ , this follows from a perturbation argument (see Theorem V.2.20 in [22]).", "This concludes the proof.", "For $a \\in \\mathbb {R}$ we denote by $(n^(a))_{n\\in \\mathbb {N}}$ the dual basis of $(n(a))_{n\\in \\mathbb {N}}$ .", "We have $n^(a) = n(-a) = \\overline{n(a)}$ .", "Separation of variables - Spectrum of the model operator In this section we use the results on the transversal operator $T_a$ to prove spectral properties for the full operator $H_a$ when $a$ is constant on $\\partial \\Omega $ .", "Most of the results of this section are inspired by the ${\\mathcal {P}}{\\mathcal {T}}$ -symmetric analogs (see [10]).", "Let $a > 0$ be fixed.", "We set $\\mathfrak {S}_a = \\bigcup _{n \\in \\mathbb {N}} \\lbrace \\lambda _n(a)^2\\rbrace + \\mathbb {R}_+ = \\left\\lbrace \\lambda _n(a)^2 + r, n \\in \\mathbb {N}, r \\in \\mathbb {R}_+ \\right\\rbrace \\subset $ Proposition 4.1 We have $\\mathfrak {S} _a \\subset \\sigma (H_a)$ .", "Let $n \\in \\mathbb {N}$ , $r \\geqslant 0$ and $z = \\lambda _n(a)^2 + r \\in \\mathfrak {S}_a$ .", "Let $\\left({f}_{m}\\right)_{m \\in \\mathbb {N}}$ be a sequence in $H^2(\\mathbb {R}^{d-1})$ such that $\\left\\Vert f_m\\right\\Vert _{L^2(\\mathbb {R}^{d-1})} = 1$ for all $m \\in \\mathbb {N}$ and $\\left\\Vert (-x -r)f_m\\right\\Vert _{L^2(\\mathbb {R}^{d-1})} \\rightarrow 0$ as $m \\rightarrow \\infty $ .", "For $m \\in \\mathbb {N}$ and $(x,y) \\in \\mathbb {R}^{d-1} \\times ]0,l[$ we set $u_m(x,y) = f_m (x) n(a;y)$ .", "Then $u_m \\in {\\mathcal {D}}(H_a)$ and $\\left\\Vert u_m\\right\\Vert _{L^2(\\Omega )} = 1$ for all $m \\in \\mathbb {N}$ .", "Moreover, according to (REF ) we have $\\left\\Vert (H_a- z)u_m\\right\\Vert _{L^2(\\Omega )} = \\left\\Vert (-x - r) f_m\\right\\Vert _{L^2(\\mathbb {R}^{d-1})} \\xrightarrow[m \\rightarrow \\infty ]{}0.$ This implies that $z \\in \\sigma (H_a)$ .", "For $u \\in L^2(\\Omega )$ , $n\\in \\mathbb {N}$ and $x \\in \\mathbb {R}^{d-1}$ we set $u_n (x) = \\left< u(x, \\cdot ) , _n(a) \\right>_{L^2(0,l)} \\in .", "This gives a sequence of functions $ un$ defined almost everywhere on $ Rd-1$.$ Proposition 4.2 Let $u \\in L^2(\\Omega )$ .", "Then $u_n \\in L^2(\\mathbb {R}^{d-1})$ for all $n \\in \\mathbb {N}$ and on $L^2(\\Omega )$ we have $u= \\sum _{n\\in \\mathbb {N}} u_n \\otimes n(a).$ For $N \\in \\mathbb {N}$ we set $v_N = \\sum _{n = 0}^N u_n \\otimes n(a)$ .", "For almost all $x \\in \\mathbb {R}^{d-1}$ , $v_N(x)$ defines a function in $L^2(0,l)$ which goes to $u(x)$ as $N \\rightarrow \\infty $ .", "According to Proposition REF we have $\\left\\Vert v_N(x)\\right\\Vert ^2_{L^2(0,l)} \\lesssim \\sum _{n=0}^N \\left\|u_n(x)\\right\|^2 \\lesssim \\sum _{n=0}^\\infty \\left\|u_n(x)\\right\|^2 \\lesssim \\left\\Vert u(x)\\right\\Vert ^2_{L^2(0,l)},$ where all the estimates are uniform in $N$ .", "Since the map $x \\mapsto \\left\\Vert u(x)\\right\\Vert ^2_{L^2(0,l)}$ belongs to $L^1(\\mathbb {R}^{d-1})$ , we can apply the dominated convergence theorem to conclude that $u-v_N \\rightarrow 0$ in $L^2(\\Omega )$ .", "Proposition 4.3 Let $u \\in {\\mathcal {D}}(H_a)$ .", "Then $u_n \\in H^2(\\mathbb {R}^{d-1})$ for all $n \\in \\mathbb {N}$ and in $L^2(\\Omega )$ we have $H_au = \\sum _{n\\in \\mathbb {N}} \\big ( - x + \\lambda _n(a)^2 \\big ) u_n \\otimes n(a).$ Let $z \\in \\mathfrak {S}_a$ .", "Then $z$ belongs to the resolvent set of $H_a$ and for all $u \\in L^2 (\\Omega )$ we have $(H_a-z)^{-1}u = \\sum _{n\\in \\mathbb {N}} (-x + \\lambda _n(a)^2-z)^{-1}u_n \\otimes n(a).$ In particular there exists $C \\geqslant 0$ such that for all $z \\in \\mathfrak {S}_a$ and $u \\in L^2(\\Omega )$ we have $\\left\\Vert (H_a-z)^{-1}u\\right\\Vert _{L^2(\\Omega )} \\leqslant \\frac{C}{d(z,\\mathfrak {S}_a)} \\left\\Vert u\\right\\Vert _{L^2(\\Omega )}.$ $\\bullet $ Let $z \\in \\mathfrak {S}_a$ and $u \\in L^2(\\Omega )$ .", "Let $u_n \\in L^2(\\mathbb {R}^{d-1})$ , $n\\in \\mathbb {N}$ , be as above.", "For $n \\in \\mathbb {N}$ we set $\\tilde{R}_n(u)= (-x + \\lambda _n(a)^2 - z)^{-1}u_n \\in H^2 (\\mathbb {R}^{d-1})$ and $R_n(u)= \\tilde{R}_n(u)\\otimes n(a) \\in {\\mathcal {D}}(H_a)$ .", "Using the standard spectral properties of the self-adjoint operator $-x$ , we see that on $L^2(\\mathbb {R}^{d-1})$ we have $ \\left\\Vert \\tilde{R}_n(u)\\right\\Vert \\lesssim \\frac{\\left\\Vert u_n\\right\\Vert }{d(z, \\lambda _n(a)^2 +\\mathbb {R}_+ )} \\lesssim _z \\frac{\\left\\Vert u_n\\right\\Vert }{\\left< n \\right> ^{2}} \\quad \\text{and} \\quad \\left\\Vert \\partial _x \\tilde{R}_n(u)\\right\\Vert \\lesssim _z \\frac{\\left\\Vert u_n\\right\\Vert }{\\left< n \\right> }.$ The first estimate is uniform in $z$ but not the others.", "With Proposition REF we obtain for $N,p \\in \\mathbb {N}$ $\\left\\Vert \\sum _{n=N}^{N+p} R_n(u)\\right\\Vert ^2_{H^1(\\Omega )} \\lesssim _z \\sum _{n=N}^{N+p} \\left( \\left\\Vert \\tilde{R}_n(u)\\right\\Vert ^2_{H^1(\\mathbb {R}^{d-1})} + \\left< n \\right>^2 \\left\\Vert \\tilde{R}_n(u)\\right\\Vert ^2_{L^2(\\mathbb {R}^{d-1})} \\right) \\xrightarrow[N \\rightarrow \\infty ]{}0.$ This proves that the series $\\sum R_n(u)$ converges to some $R(u) \\in H^1(\\Omega )$ .", "Moreover, with the first inequality of (REF ) and Proposition REF again, we see that $ \\left\\Vert R(u)\\right\\Vert _{L^2(\\Omega )} \\lesssim \\frac{\\left\\Vert u\\right\\Vert _{L^2(\\Omega )}}{d (z ,\\mathfrak {S}_a)},$ uniformly in $z$ .", "It remains to see that $\\forall \\phi \\in H^1(\\Omega ), \\quad q_a(R(u),\\phi ) - z \\left< R(u) , \\phi \\right>= \\left< u , \\phi \\right>,$ which comes from the fact that this is true with $R(u)$ and $u$ replaced by $R_n(u)$ and $u_n \\otimes n(a)$ respectively.", "This proves that $R(u) \\in {\\mathcal {D}}(H_a)$ and $(H_a-z) R(u) = u$ .", "If $\\mathop {\\rm {Re}}\\nolimits (z) < 0$ , we already know that $(H_a-z)$ has a bounded inverse on $L^2(\\Omega )$ , and hence we have $(H_a-z)^{-1}= R$ .", "This proves the second statement of the proposition when $\\mathop {\\rm {Re}}\\nolimits (z) < 0$ .", "$\\bullet $ Let $ u \\in {\\mathcal {D}}(H_a)$ and $v = (H_a+1) u \\in L^2(\\Omega )$ .", "For $n \\in \\mathbb {N}$ and almost all $x \\in \\mathbb {R}^{d-1}$ we set $v_n(x) = \\left< v(x,\\cdot ) , n^(a) \\right>_{L^2(0,l)}$ .", "According to (ii) applied with $z = -1$ we have $u = (H_a+1)^{-1}v = \\sum _{n\\in \\mathbb {N}} \\big (-x + \\lambda _n(a)^2 +1\\big )^{-1}v_n \\otimes n(a).$ By uniqueness for the decomposition of $u(x,\\cdot )$ with respect to the Riesz basis $(n(a))_{n\\in \\mathbb {N}}$ , we have for all $n \\in \\mathbb {N}$ $u_n = \\big (-x + \\lambda _n(a)^2 +1\\big )^{-1}v_n.$ This proves that $u_n \\in H^2(\\mathbb {R}^{d-1})$ .", "Then $H_au = v - u& = \\sum _{n \\in \\mathbb {N}} \\left( 1 - \\big (-x + \\lambda _n(a)^2 +1\\big )^{-1}\\right) v_n \\otimes n(a)\\\\& = \\sum _{n \\in \\mathbb {N}} \\big (-x + \\lambda _n(a)^2 \\big ) u_n \\otimes n(a).$ This proves the first statement of the proposition.", "$\\bullet $ It remains to finish the proof of (ii).", "Let $z \\in \\mathfrak {S}_a$ and $w \\in {\\mathcal {D}}(H_a)$ .", "With (i) we see that $R\\big ((H_a-z)w\\big ) = w$ .", "Since we already know that $(H_a-z)R(u) = u$ for all $u \\in L^2(\\Omega )$ , this proves that $R$ is a bounded inverse for $(H_a-z)$ on $L^2(\\Omega )$ .", "The estimate on $(H_a-z)^{-1}$ follows from (REF ), and the proposition is proved.", "As a first application of this proposition, we can check that the operator $H_a$ has no eigenvalue, as is the case for $-x$ : Corollary 4.4 The operator $H_a$ has no eigenvalue.", "Let $z \\in and $ u D(Ha)$ be such that $ (Ha-z) u =0$.", "Then according to the first item of Proposition \\ref {prop-dec-res} we have$$\\sum _{n \\in \\mathbb {N}} \\left( - \\lambda _n(a)^2 - z \\right) u_n \\otimes n(a) = 0.$$This implies that for all $ n N$ we have$$\\left( - \\lambda _n(a)^2 - z \\right) u_n = 0$$in $ L2(Rd-1)$, and hence $ un = 0$ since the operator $ -x$ has no eigenvalue.", "Finally $ u = 0$, and the proposition is proved.$ However, the main point in Proposition REF is the second: Corollary 4.5 Theorem REF holds when $a > 0$ is constant.", "According to the last statement of Proposition REF and the fact that $\\mathop {\\rm {Im}}\\nolimits (\\lambda _n(a)^2) < 0$ for all $n \\in \\mathbb {N}$ , there exists $\\tilde{\\gamma }> 0$ such that for all $z \\in \\mathfrak {S}_a$ we have $\\mathop {\\rm {Im}}\\nolimits (z) \\leqslant - \\tilde{\\gamma }$ .", "With Proposition REF , the conclusion follows.", "Remark 4.6 To simplify the proof we have only considered the case where $a$ is equal to the same constant on both sides of the boundary.", "However we can similarly consider the case where $a$ is equal to some positive constant $a_0$ on $\\mathbb {R}^{d-1} \\times \\left\\lbrace 0 \\right\\rbrace $ and to another positive constant $a_l$ on $\\mathbb {R}^{d-1} \\times \\left\\lbrace l \\right\\rbrace $ .", "One of these two constants may even be zero.", "We refer to Section for this kind of computations.", "Non-constant absorption index In this section we prove Theorems REF and REF for a non-constant absorption index $a$ .", "For $b \\in W^{1,\\infty }(\\partial \\Omega )$ we denote by $\\Theta _b \\in {\\mathcal {L}}(H^1(\\Omega ) , H^{-1}(\\Omega ))$ the operator such that for all $\\psi \\in H^1(\\Omega )$ $\\left< \\Theta _b \\psi \\right>_{H^{-1}(\\Omega ),H^1(\\Omega )} = \\int _{\\partial \\Omega } b\\psi .$ We denote by $\\theta _b$ the corresponding quadratic form on $H^1(\\Omega )$ .", "We also denote by $\\tilde{H}_{a}$ the operator in ${\\mathcal {L}}\\big (H^1(\\Omega ),H^{-1}(\\Omega ) \\big )$ such that $\\left< \\tilde{H}_{a} \\psi \\right>_{H^{-1},H^1} = q_a(\\psi )$ for all $\\psi \\in H^1(\\Omega )$ .", "Let $z \\in +$ .", "According to the Lax-Milgram Theorem, $(1+i)(\\tilde{H}_{a}-z)$ is an isomorphism from $H^1(\\Omega )$ to $H^{-1}(\\Omega )$ .", "Moreover, for $f \\in L^2(\\Omega ) \\subset H^{-1}(\\Omega )$ we have $(\\tilde{H}_{a}-z)^{-1}f = (H_a-z)^{-1}f.$ The following proposition relies on a suitable version of the so-called quadratic estimates: Proposition 5.1 Let $a_0 > 0$ be as in the statement of Theorem REF .", "Assume that (REF ) holds everywhere on $\\partial \\Omega $ .", "Let $B \\in {\\mathcal {L}}\\big (H^1(\\Omega ),L^2(\\Omega ) \\big )$ .", "Then there exists $C \\geqslant 0$ such that for all $z \\in +$ we have $\\left\\Vert B (\\tilde{H}_{a}-z)^{-1}B^\\right\\Vert _{{\\mathcal {L}}(L^2(\\Omega ))} \\leqslant C \\left\\Vert B (\\tilde{H}_{a_0}-z)^{-1}B^\\right\\Vert _{{\\mathcal {L}}(L^2(\\Omega ))}.$ For $z \\in +$ the resolvent identity applied to $\\tilde{H}_{a}= \\tilde{H}_{a_0}+ \\Theta _{a-a_0}$ gives $ B(\\tilde{H}_{a}-z)^{-1}B^* = B (\\tilde{H}_{a_0}-z)^{-1}B^* - B (\\tilde{H}_{a_0}-z)^{-1}\\Theta _{a-a_0} (\\tilde{H}_{a}-z)^{-1}B^.$ Let $\\psi \\in L^2(\\Omega )$ .", "Since $\\Theta _{a-a_0}$ is associated to a non-negative quadratic form on $H^1(\\Omega )$ , the Cauchy-Schwarz inequality gives ${\\left< B(\\tilde{H}_{a_0}-z)^{-1}\\Theta _{a-a_0} (\\tilde{H}_{a}-z)^{-1}B^ \\psi \\right>_{L^2}}\\\\&& = \\left< \\Theta _{a-a_0} (\\tilde{H}_{a}-z)^{-1}B^* (\\tilde{H}_{a_0}^* - \\overline{z})^{-1}B^* \\psi \\right> _{H^{-1},H^1}\\\\&& \\leqslant \\theta _{a-a_0} \\big ((\\tilde{H}_{a}-z)^{-1}B^* )^{\\frac{1}{2}} \\times \\theta _{a-a_0} \\big ((\\tilde{H}_{a_0}^* -\\overline{z})^{-1}B^* \\psi \\big ) ^{\\frac{1}{2}}.$ The first factor is estimated as follows: ${\\theta _{a-a_0} \\big ((\\tilde{H}_{a}-z)^{-1}B^* )\\leqslant \\theta _{a} \\big ((\\tilde{H}_{a}-z)^{-1}B^* )}\\\\&& \\leqslant \\frac{1}{2i} \\left< 2i (\\Theta _{a} + \\mathop {\\rm {Im}}\\nolimits z) (\\tilde{H}_{a}-z)^{-1}B^* (\\tilde{H}_{a}- z)^{-1}B^* > _{H^{-1},H^1}\\\\&& \\leqslant \\frac{1}{2i} \\left< B \\big ((\\tilde{H}_{a}-z)^{-1}-(\\tilde{H}_{a}^* - \\overline{z})^{-1}\\big ) B^* > _{L^2}\\\\&& \\leqslant \\left\\Vert B(\\tilde{H}_{a}-z)^{-1}B^\\right\\Vert _{{\\mathcal {L}}(L^2)} \\left\\Vert \\Vert _{L^2}^2.\\right.We can proceed similarly for the other factor, using the fact that \\right.\\right.a-a_0 \\leqslant \\alpha a_0 for some \\alpha \\geqslant 0.", "Thus there exists C \\geqslant 0 such that{\\begin{@align}{1}{-1}\\left\\Vert B (\\tilde{H}_{a_0}-z)^{-1}\\Theta _{a-a_0} (\\tilde{H}_{a}-z)^{-1}B^* \\right\\Vert & \\leqslant C \\left\\Vert B (\\tilde{H}_{a_0}-z)^{-1}B^\\right\\Vert ^{\\frac{1}{2}} \\left\\Vert B (\\tilde{H}_{a}-z)^{-1}B^\\right\\Vert ^{\\frac{1}{2}}\\\\& \\leqslant \\frac{C^2}{2} \\left\\Vert B (\\tilde{H}_{a_0}-z)^{-1}B^\\right\\Vert + \\frac{1}{2} \\left\\Vert B (\\tilde{H}_{a}-z)^{-1}B^\\right\\Vert .\\end{@align}}With (\\ref {eq-res-identity}), the conclusion follows.$ Now we can finish the proof of Theorem REF : According to Corollary REF and Proposition REF applied with $B = \\operatorname{Id}_{L^2(\\Omega )}$ (note that we can simply replace $\\tilde{H}_{a}$ by $H_a$ when $B \\in {\\mathcal {L}}(L^2(\\Omega ))$ ), there exists $C > 0$ such that $\\left\\Vert (H_a-z)^{-1}\\right\\Vert _{{\\mathcal {L}}(L^2(\\Omega ))} \\leqslant C$ for all $z \\in +$ .", "Then it only remains to choose $\\tilde{\\gamma }\\in \\big ] 0, \\frac{1}{C} \\big [$ to conclude.", "The second statement, concerning the case where $a$ vanishes on one side of the boundary, is proved similarly.", "Let us now turn to the proof of Theorem REF .", "We first prove another resolvent estimate in which we see the smoothing effect in weighted spaces: Proposition 5.2 Let $\\delta > \\frac{1}{2}$ .", "Then there exists $C \\geqslant 0$ such that for all $z \\in \\mathbb {C}_+$ we have $\\left\\Vert \\left< x \\right> ^{-\\delta } (1- \\Delta _x)^{\\frac{1}{4}} (H_a-z)^{-1} (1- \\Delta _x)^{\\frac{1}{4}} \\left< x \\right> ^{-\\delta }\\right\\Vert _{\\mathcal {L} (L^2(\\Omega ))} \\leqslant C.$ It is known for the free laplacian that for $\\left\| \\mathop {\\rm {Re}} (\\zeta ) \\right\| \\gg 1$ and $\\mathop {\\rm {Im}}(\\zeta ) > 0$ we have $\\left\\Vert \\left< x \\right>^{-\\delta } (-\\Delta _x -\\zeta )^{-1} \\left< x \\right>^{-\\delta }\\right\\Vert _{\\mathcal {L}(L^2(\\mathbb {R}^{d-1}))} \\lesssim \\left< \\zeta \\right> ^{-\\frac{1}{2}}$ and hence $\\left\\Vert \\left< x \\right>^{-\\delta } (1- x)^{\\frac{1}{4}}(-\\Delta _x -\\zeta )^{-1} (1- \\Delta _x)^{\\frac{1}{4}}\\left< x \\right>^{-\\delta } \\right\\Vert _{\\mathcal {L} (L^2(\\mathbb {R}^{d-1}))} \\lesssim 1.$ Thus if $a$ is constant we have $\\left\\Vert \\left< x \\right>^{-\\delta } (1- x)^{\\frac{1}{4}}(-\\Delta _x + \\lambda _n(a)^2 -z)^{-1} (1- \\Delta _x)^{\\frac{1}{4}}\\left< x \\right>^{-\\delta }\\right\\Vert _{\\mathcal {L}(L^2(\\mathbb {R}^{d-1})) } \\lesssim 1,$ uniformly in $z \\in +$ and $n \\in \\mathbb {N}$ .", "In this case we obtain the result using the separation of variables as in Section .", "Then we conclude with Proposition REF applied with $B = \\left< x \\right> ^{-\\delta } (1-\\Delta _x)^{\\frac{1}{4}}$ .", "In fact, we first obtain an estimate on the resolvent $(\\tilde{H}_a -z)^{-1}$ , but this proves that the operator $\\left< x \\right> ^{-\\delta } (1- \\Delta _x)^{\\frac{1}{4}} (H_a-z)^{-1} (1- \\Delta _x)^{\\frac{1}{4}} \\left< x \\right> ^{-\\delta }$ extends to a bounded operator on $L^2(\\Omega )$ , and then the same estimate holds for the corresponding closure.", "With the second estimate of Proposition REF , we can apply the theory of relatively smooth operators (see §XIII.7 in [37]).", "However, since $H_a$ is not self-adjoint but only maximal dissipative, we have to use a self-adjoint dilation (see [31]) of $H_a$ , as is done in the proof of [36] (see also Proposition 2.24 in [35]).", "Time decay for the Schrödinger equation In this section we prove Theorem REF .", "Let $u_0 \\in {\\mathcal {D}}(H_a)$ and let $u$ be the solution of the problem (REF ).", "We know that $\\left\\Vert u(t)\\right\\Vert _{L^2(\\Omega )} \\leqslant \\left\\Vert u_0\\right\\Vert _{L^2(\\Omega )}$ for all $t \\geqslant 0$ , so the result only concerns large times.", "Let $\\tilde{\\gamma }> 0$ be given by Theorem REF and $\\gamma = \\tilde{\\gamma }/ 3$ .", "Let $C^\\infty (\\mathbb {R})$ be equal to 0 on $]-\\infty ,1]$ and equal to 1 on $[2,+\\infty [$ .", "For $t \\in \\mathbb {R}$ we set $u_t) = t) u(t),$ and for $z \\in +$ : $v(z) = \\int _\\mathbb {R}e^{itz} u_t) \\, dt.$ The map $t \\mapsto e^{-\\gamma t}u_t)$ belongs to $L^1(\\mathbb {R}) \\cap L^2(\\mathbb {R}) \\cap C^1(\\mathbb {R})$ and its derivative is in $L^1(\\mathbb {R})$ so $\\mapsto v(+i\\gamma )$ is bounded and decays at least like $\\left< \\right>^{-1}$ .", "In particular it is in $L^2(\\mathbb {R})$ .", "For $R > 0$ we set $u_R(t) = \\frac{1}{2\\pi } \\int _{-R}^R e^{-it(+i\\gamma )} v(+i\\gamma ) \\, d.$ Then $\\left\\Vert e^{-t\\gamma } (u_ u_R)\\right\\Vert _{L^2(\\mathbb {R}_t,L^2(\\Omega ))} \\xrightarrow[R \\rightarrow +\\infty ]{} 0.$ Since $u_ is continuous, Theorem \\ref {th-energy-decay} will be proved if we can show that there exists $ C 0$ which does not depend on $ u0$ and such that for all $ t 0$ we have\\begin{equation} \\limsup _{R \\rightarrow \\infty } \\left\\Vert u_R(t)\\right\\Vert _{L^2(\\Omega )} \\leqslant C e^{-\\gamma t} \\left\\Vert u_0\\right\\Vert _{L^2(\\Omega )}.\\end{equation}For $ z we set $\\theta (z) = -i \\int _\\mathbb {R}e^{itz} (t) u(t) \\, dt = -i \\int _1^2 e^{itz} (t) u(t) \\, dt.$ Let $z \\in +$ .", "We multiply (REF ) by $t) e^{itz}$ and integrate over $t \\in \\mathbb {R}$ .", "After partial integration we obtain $ v(z) = (H_a-z)^{-1}\\theta (z).$ Then $v$ extends to a holomorphic function on ${3\\gamma }$ , taking this equality as a definition.", "According to the Cauchy Theorem we have in $L^2(\\mathbb {R}_t)$ $ \\begin{aligned}\\lim _{R \\rightarrow \\infty } u_R(t)= \\frac{1}{2\\pi } \\lim _{R \\rightarrow \\infty } \\int _{-R}^R e^{-it(-2i\\gamma )} v(-2i\\gamma )\\,d= {e^{-2\\gamma t}} \\lim _{R \\rightarrow \\infty } \\widetilde{u_R}(t),\\end{aligned}$ where for $t \\in \\mathbb {R}$ we have set $\\widetilde{u_R}(t) = \\int _{-R}^R e^{-it}v(-2i\\gamma ) \\, d.$ According to Plancherel's equality and Theorem REF we have uniformly in $R >0$ : $\\int _\\mathbb {R}\\left\\Vert \\widetilde{u_R} (t)\\right\\Vert ^2_{L^2(\\Omega )} \\, dt& = \\int _{-R}^R \\left\\Vert \\big (H_a-(-2i\\gamma )\\big )^{-1}\\theta (-2i\\gamma )\\right\\Vert _{L^2(\\Omega )}^2 \\, d\\\\& \\lesssim \\int _\\mathbb {R}\\left\\Vert \\theta (-2i\\gamma )\\right\\Vert _{L^2(\\Omega )}^2 \\, d\\\\& \\lesssim \\int _\\mathbb {R}e^{2\\gamma t}\\left\|(t)\\right\| \\left\\Vert u(t)\\right\\Vert _{L^2(\\Omega )}^2 \\, dt \\\\& \\lesssim \\left\\Vert u_0\\right\\Vert ^2_{L^2(\\Omega )}.$ In particular there exists $C \\geqslant 0$ such that for $u_0 \\in {\\mathcal {D}}(H_a)$ and $R > 0$ we can find $T(u_0,R) \\in [0,1]$ which satisfies $\\left\\Vert \\widetilde{u_R} (T(u_0,R))\\right\\Vert _{L^2(\\Omega )} \\leqslant C \\left\\Vert u_0\\right\\Vert _{L^2(\\Omega )}.$ Let $R > 0$ .", "Then $\\widetilde{u_R} \\in C^1(\\mathbb {R})$ and for $t \\geqslant 1$ we have $\\widetilde{u_R} (t) = e^{-i(t-T(u_0,R))H_a} \\widetilde{u_R} (T(u_0,R)) + \\int _{T(u_0,R)} ^t \\frac{\\partial }{\\partial s} \\left( e^{-i(t-s)H_a} \\widetilde{u_R}(s) \\right) \\,ds,$ where $\\frac{\\partial }{\\partial s} \\left( e^{-i(t-s)H_a} \\widetilde{u_R}(s) \\right)& = \\frac{\\partial }{\\partial s}\\int _{-R}^R e^{-is} e^{-i(t-s)H_a} \\big (H_a-(-2i\\gamma )\\big )^{-1}\\theta (-2i\\gamma ) \\, d\\\\& = i \\int _{-R}^R e^{-i(t-s)H_a} e^{-is} (H_a-) \\big ( H_a- (-2i\\gamma ) \\big ) ^{-1}\\theta (-2i\\gamma ) \\, d\\\\& = 2\\gamma e^{-i(t-s)H_a} \\widetilde{u_R} (s) + i e^{-i(t-s)H_a} \\int _{-R}^R e^{-is} \\theta (-2i\\gamma ) \\, d.$ This proves that the map $s \\mapsto \\frac{\\partial }{\\partial s} \\left( e^{-i(t-s)H_a} \\widetilde{u_R}(s) \\right)$ belongs to $L^2([0,t],L^2(\\Omega ))$ uniformly in $t$ and $R>0$ , and its $L^2([0,t],L^2(\\Omega ))$ norm is controlled by the norm of $u_0$ in $L^2(\\Omega )$ .", "We finally obtain $C \\geqslant 0$ such that for all $t \\in \\mathbb {R}$ and $R > 0$ we have $\\left\\Vert \\widetilde{u_R} (t)\\right\\Vert _{L^2(\\Omega )} \\leqslant C \\left< t \\right>^{\\frac{1}{2}} \\left\\Vert u_0\\right\\Vert _{L^2(\\Omega )}.$ With (REF ) this proves () and concludes the proof of Theorem REF .", "The case of a weakly dissipative boundary condition In this section we prove Theorem REF about the problem (REF ).", "The absorption index $a$ now takes the value $a_l$ on $\\mathbb {R}^{d-1} \\times \\left\\lbrace l \\right\\rbrace $ and $a_0$ on $\\mathbb {R}^{d-1} \\times \\left\\lbrace 0 \\right\\rbrace $ .", "The proof follows the same lines as in the dissipative case, except that well-posedness of the problem is not an easy consequence of the general dissipative theory.", "We will use the separation of variables as in Section instead.", "Once we have a decomposition as in Proposition REF for the initial datum, we can propagate each term by means of the unitary group generated by $-x$ and define the solution of (REF ) as a series of solutions on $\\mathbb {R}^{d-1}$ .", "Let us first look at the transverse problem.", "The transverse operator on $L^2(0,l)$ corresponding to the problem (REF ) is now given by $T_{a_l,a_0}= - \\frac{d^2}{dy^2}$ with domain ${\\mathcal {D}}(T_{a_l,a_0}) = \\left\\lbrace u \\in H^2(0,l) \\,:\\,u^{\\prime }(0) = -ia_0u(0), u^{\\prime }(l) = ia_lu(l) \\right\\rbrace .$ As already mentioned in Remark REF , we can reproduce exactly the same analysis as in Section if $a_l> 0$ and $a_0>0$ (or if one of them vanishes).", "In particular, there is no restriction on the sizes of these coefficients.", "The results we give here to handle the weakly dissipative case are also valid in this situation.", "The strategy will be the same as in Section , so we will only emphasize the differences.", "We first remark that 0 is an eigenvalue of $T_{a_l,a_0}$ if and only if $a_l=a_0=0$ .", "Otherwise, $\\lambda ^2 \\in $ is an eigenvalue of $T_{a_l,a_0}$ if and only if $ (\\lambda - a_l)(\\lambda - a_0) e^{2i\\lambda l} = (\\lambda + a_l) (\\lambda + a_0).$ We recover (REF ) when $a_l= a_0$ .", "Lemma 7.1 Let $a_l,a_0\\in \\mathbb {R}$ and $\\lambda \\in $ be such that $a_l+a_0\\ne 0$ and $\\lambda ^2$ is an eigenvalue of $T_{a_l,a_0}$ .", "Then $\\mathop {\\rm {Re}}\\nolimits (\\lambda ) \\notin \\nu \\mathbb {N}$ .", "We recall from [23] that if $a_l+a_0= 0$ (${\\mathcal {P}}{\\mathcal {T}}$ -symmectric case) then $n^2 \\nu ^2 \\in \\sigma (T_{a_l,a_0})$ for all $n \\in \\mathbb {N}^$ (see also Figure REF for $a_l+a_0> 0$ small).", "$\\bullet $ We assume by contradiction that $\\mathop {\\rm {Re}}\\nolimits (\\lambda ) \\in \\nu \\mathbb {N}$ .", "According to (REF ) we have $\\frac{(\\lambda + a_l)(\\lambda +a_0)}{(\\lambda -a_l)(\\lambda -a_0)} = e^{2il\\lambda } = e^{-2 l\\mathop {\\rm {Im}}\\nolimits (\\lambda )} \\in \\mathbb {R}_+^.$ After multiplication by $\\left\|\\lambda - a_l\\right\|^2 \\left\|\\lambda - a_0\\right\|^2 \\in \\mathbb {R}_+^$ we obtain $\\big ( \\left\|\\lambda \\right\|^2 - 2 i a_l\\mathop {\\rm {Im}}\\nolimits (\\lambda ) - a_l^2 \\big ) \\big ( \\left\|\\lambda \\right\|^2 - 2 i a_0\\mathop {\\rm {Im}}\\nolimits (\\lambda ) - a_0^2 \\big ) \\in \\mathbb {R}_+^.$ Taking the real and imaginary parts gives $ \\left\|\\lambda \\right\|^4 - (a_l^2 + a_0^2) \\left\|\\lambda \\right\|^2 + a_l^2 a_0^2 - 4 a_la_0\\mathop {\\rm {Im}}\\nolimits (\\lambda )^2 > 0$ and $ 2 \\mathop {\\rm {Im}}\\nolimits (\\lambda ) (a_l+a_0) \\big (\\left\|\\lambda \\right\|^2 - a_la_0\\big ) = 0.$ $\\bullet $ Assume that $a_la_0\\geqslant 0$ .", "In this case $\\mathop {\\rm {Im}}\\nolimits (\\lambda ) \\ne 0$ (for the same reason as in the proof of Proposition REF ), so (REF ) implies $\\left\|\\lambda \\right\|^2 = a_la_0$ .", "Then (REF ) reads $- a_la_0(a_l- a_0)^2 - 4a_la_0\\mathop {\\rm {Im}}\\nolimits (\\lambda )^2 > 0,$ which gives a contradiction.", "$\\bullet $ Now assume that $a_la_0< 0$ .", "Then (REF ) implies $\\mathop {\\rm {Im}}\\nolimits (\\lambda ) = 0$ and hence $e^{2il\\lambda } = 1$ .", "From (REF ) we now obtain $(\\lambda -a_l) (\\lambda -a_0) = (\\lambda +a_l) (\\lambda +a_0),$ which is impossible since $\\lambda (a_l+ a_0) \\ne 0$ .", "This concludes the proof.", "Proposition 7.2 There exists $\\rho > 0$ such that if $\\left\|a_l\\right\| + \\left\|a_0\\right\| \\leqslant \\rho $ and $a_l+ a_0> 0$ then the spectrum of $T_{a_l,a_0}$ is given by a sequence $(\\lambda _n(a_l,a_0)^2)_{n\\in \\mathbb {N}}$ of algebraically simple eigenvalues such that $\\sup _{n\\in \\mathbb {N}} \\, \\mathop {\\rm {Im}}\\nolimits \\left(\\lambda _n(a_l,a_0)^2\\right) < 0.$ Moreover, any sequence of normalized eigenfunctions corresponding to these eigenvalues forms a Riesz basis.", "Figure: λ n (a l ,a 0 )\\lambda _n(a_l,a_0) for n∈{0,⋯,30}n \\in \\lbrace 0,\\dots ,30\\rbrace and l=πl= \\pi .As in the proof of Proposition REF we can see that for any $R > 0$ there exists $C_R \\geqslant 0$ such that if $\\lambda ^2 \\in $ is an eigenvalue of $T_{a_l,a_0}$ we have $ \\left\|\\mathop {\\rm {Re}}\\nolimits \\lambda \\right\| \\leqslant R \\quad \\Rightarrow \\quad \\left\|\\mathop {\\rm {Im}}\\nolimits \\lambda \\right\| \\leqslant C_R.$ The operator $T_{a_l,a_0}$ depends analytically on the parameters $a_l$ and $a_0$ , and we know that when $a_l= a_0= 0$ the eigenvalues $n^2\\nu ^2$ for $n \\in \\mathbb {N}$ are algebraically simple.", "With the restrictions given by (REF ) and Lemma REF , we obtain as in Section a sequence of maps $(a_l,a_0) \\mapsto \\lambda _n(a_l,a_0)$ such that the eigenvalues of $T_{a_l,a_0}$ are $\\lambda _n(a_l,a_0)^2$ for $n \\in \\mathbb {N}$ .", "Let $n \\in \\mathbb {N}^$ .", "We have $ \\lambda _n(a_l,a_0)= n\\nu - \\frac{i}{n\\pi } (a_l+ a_0) + \\gamma (a_l+a_0)^2 + O \\left( \\left\|a_l\\right\|^3 , \\left\|a_0\\right\|^3 \\right),$ with $\\mathop {\\rm {Re}}\\nolimits (\\gamma ) = l/(n\\pi )^3 > 0$ .", "As in the dissipative case, we obtain that for any $a_l,a_0$ with $a_l+ a_0> 0$ these eigenvalues $\\lambda _n(a_l,a_0)^2$ are simple.", "If moreover $a_l$ and $a_0$ are small enough, the eigenvalue $\\lambda _n(a_l,a_0)^2$ is close to $(n\\nu )^2$ and away from the real axis uniformly in $n \\in \\mathbb {N}^$ (the first two terms in (REF ) are also the first two terms of the asymptotic expansion for large $n$ and fixed $a_l$ and $a_0$ ).", "It remains to check that we also have $\\mathop {\\rm {Im}}\\nolimits (\\lambda _0(a_l,a_0)^2) < 0$ .", "For small $a_l,a_0$ we denote by ${a_l,a_0}(0)$ a normalized eigenvector corresponding to the eigenvalue $\\lambda _0(a_l,a_0)^2$ and depending analytically on $a_l$ and $a_0$ .", "For all $\\psi \\in H^1(0,l)$ we have $\\left< {a_l,a_0}^{\\prime } , \\psi ^{\\prime } \\right>_{L^2(0,l)} - i a_l{{a_l,a_0}(l)} \\overline{\\psi (l)} - i a_0{{a_l,a_0}(0)} \\overline{\\psi (0)} = \\lambda _0(a_l,a_0)^2 \\left< {a_l,a_0} , \\psi \\right>_{L^2(0,l)}$ We apply this with $\\psi = {a_l,a_0}$ , take the derivatives with respect to $a_l$ and $a_0$ at point $(a_l,a_0) = (0,0)$ , and use the facts that ${0,0}$ is constant and $\\lambda _{0}(0,0) = 0$ .", "We obtain $\\nabla _{a_l,a_0} \\big (\\lambda _0^2 \\big ) = -\\frac{i}{l}\\big ( 1 , 1 \\big ).$ This proves that $\\mathop {\\rm {Im}}\\nolimits \\big (\\lambda _0(a_l,a_0)^2\\big )<0$ if $a_l$ and $a_0$ are small enough with $a_l+ a_0> 0$ .", "The Riesz basis property relies as before on the fact that $\\left\|\\lambda _n(a_l,a_0)- n\\nu \\right\| = O(n^{-1}).$ For this point we can follow what is done in Section for the dissipative case.", "For $n \\in \\mathbb {N}$ and $a_l,a_0\\in \\mathbb {R}$ with $a_l+ a_0> 0$ we consider a normalized eigenvector $n({a_l,a_0})\\in L^2(0,l)$ corresponding to the eigenvalue $\\lambda _n(a_l,a_0)^2$ of $T_{a_l,a_0}$ .", "We denote by $(n^(a_l,a_0))_{n\\in \\mathbb {N}}$ the dual basis.", "Proposition 7.3 Let $u_0 \\in {\\mathcal {D}}(H_{a_l,a_0})$ .", "Then the problem (REF ) has a unique solution $u \\in C^1 (\\mathbb {R},L^2(\\Omega )) \\cap C^0(\\mathbb {R},{\\mathcal {D}}(H_{a_l,a_0}))$ .", "Moreover if we write $u_0 = \\sum _{n\\in \\mathbb {N}} u_{0,n} \\otimes n({a_l,a_0})$ where $u_{0,n} \\in L^2(\\mathbb {R}^{d-1})$ , then $u$ is given by $u(t) = \\sum _{n\\in \\mathbb {N}} \\left( e^{-it \\left(-x+\\lambda _n(a_l,a_0)^2 \\right)} u_{0,n}\\right) \\otimes n({a_l,a_0}).$ $\\bullet $ Assume that $u \\in C^0(\\mathbb {R}_+, {\\mathcal {D}}(H_{a_l,a_0})) \\cap C^1(\\mathbb {R}_+^,L^2(\\Omega ))$ is a solution of (REF ).", "Let $t \\in \\mathbb {R}_+^$ .", "For all $n \\in \\mathbb {N}$ and almost all $x \\in \\mathbb {R}^{d-1}$ we can define $u_n(t,x) = \\left< u(t;x,\\cdot ) , _n(a_l,a_0) \\right>_{L^2(0,l)},$ so that in $L^2(\\Omega )$ we have $u(t) = \\sum _{n\\in \\mathbb {N}} u_n(t) \\otimes n({a_l,a_0}).$ According to Proposition REF (which can be proved similarly in this context) we have $u_n(t) \\in H^2(\\mathbb {R}^{d-1})$ for all $t \\in \\mathbb {R}_+^$ and $n \\in \\mathbb {N}$ , and for $s \\in \\mathbb {R}^$ we have $i\\frac{u(t+s)-u(t)}{s} - H_{a_l,a_0}u(t) \\\\ = \\sum _{n\\in \\mathbb {N}} \\left( i\\frac{u_n(t+s)-u_n(t)}{s} - \\left(-x +\\lambda _n(a_l,a_0)^2 \\right) u_n(t) \\right) \\otimes n({a_l,a_0}).$ Let $n\\in \\mathbb {N}$ .", "According to Proposition REF we have $\\left\\Vert i\\frac{u_n(t+s)-u_n(t)}{s} - \\left(-x +\\lambda _n(a_l,a_0)^2 \\right) u_n(t)\\right\\Vert _{L^2(\\mathbb {R}^{d-1})}\\\\\\lesssim \\left\\Vert i\\frac{u(t+s)-u(t)}{s} - H_{a_l,a_0}u(t)\\right\\Vert _{L^2(\\Omega )}\\xrightarrow[s \\rightarrow 0]{} 0.$ This proves that $u_n$ is differentiable and for all $t > 0$ $i u_n^{\\prime }(t) = \\left( - x + \\lambda _n(a_l,a_0)^2 \\right) u_n(t).$ Then for all $t > 0$ $u_n(t) = e^{-it\\left( -x + \\lambda _n(a_l,a_0)^2 \\right)} u_n(0) = e^{-it\\left( -x + \\lambda _n(a_l,a_0)^2 \\right)} u_{0,n}.$ $\\bullet $ Conversely, let us prove that the function $u$ defined by the statement of the proposition is indeed a solution of (REF ).", "Let $t \\in \\mathbb {R}$ .", "According to Proposition REF , $u_{0,n}$ and hence $e^{-it \\left(-x+\\lambda _n(a_l,a_0)^2 \\right)} u_{0,n}$ belong to $H^2(\\mathbb {R}^{d-1})$ for all $n \\in \\mathbb {N}$ .", "Therefore $u(t) \\in {\\mathcal {D}}(H_{a_l,a_0})$ .", "Then for all $s \\in \\mathbb {R}^$ we have ${ \\sum _{n\\in \\mathbb {N}}\\left\\Vert \\left( i\\frac{e^{-is \\left(-x + \\lambda _n(a_l,a_0)^2 \\right)} - 1}{s} -\\left(- x + \\lambda _n(a_l,a_0)^2\\right) \\right) \\, e^{-it(-x + \\lambda _n(a_l,a_0)^2)} u_{0,n}\\right\\Vert ^2 }\\\\&& \\lesssim _t \\sum _{n\\in \\mathbb {N}}\\left\\Vert \\frac{1}{s} \\int _0^s \\big ( e^{-i\\theta (-x + \\lambda _n(a_l,a_0)^2)} - 1 \\big ) \\left(- x + \\lambda _n(a_l,a_0)^2\\right) u_{0,n} \\, d\\theta \\right\\Vert ^2_{L^2(\\mathbb {R}^{d-1})}\\\\&& \\lesssim _t \\sum _{n\\in \\mathbb {N}}\\left\\Vert \\left(- x + \\lambda _n(a_l,a_0)^2\\right) u_{0,n} \\right\\Vert ^2_{L^2(\\mathbb {R}^{d-1})}\\\\&& \\lesssim _t \\left\\Vert H_{a_l,a_0}u_0\\right\\Vert _{L^2(\\Omega )}^2$ This series of functions converges uniformly in $s$ so we can take the limit, which proves that for any $t \\in \\mathbb {R}$ $\\left\\Vert i\\frac{u(t+s)-u(t)}{s} -H_{a_l,a_0}u(t)\\right\\Vert ^2 \\xrightarrow[s \\rightarrow 0]{} 0.$ This proves that $u$ is differentiable and $i u^{\\prime }(t) + H_{a_l,a_0}u(t) = 0$ , so $u$ is indeed a solution of (REF ).", "Now we can prove Theorem REF : According to Proposition REF we have existence and uniqueness for the solution $u$ of the problem REF .", "Then with Proposition REF we have $\\left\\Vert u(t)\\right\\Vert _{L^2(\\Omega )}^2& \\lesssim \\sum _{n \\in \\mathbb {N}} \\left\\Vert e^{-it \\left(-x + \\lambda _n(a_l,a_0)^2 \\right)} u_{0,n}\\right\\Vert _{L^2(\\mathbb {R}^{d-1})}^2 \\\\& \\lesssim \\sum _{n \\in \\mathbb {N}} e^{t \\mathop {\\rm {Im}}\\nolimits \\left(\\lambda _n(a_l,a_0)^2\\right)}\\left\\Vert u_{0,n}\\right\\Vert _{L^2(\\mathbb {R}^{d-1})}^2 .$ Proposition REF gives $\\gamma _{a_l,a_0} > 0$ such that $\\left\\Vert u(t)\\right\\Vert _{L^2(\\Omega )}^2 \\lesssim e^{-t \\gamma _{a_l,a_0}} \\sum _{n \\in \\mathbb {N}} \\left\\Vert u_{0,n}\\right\\Vert _{L^2(\\mathbb {R}^{d-1})}^2 \\lesssim e^{-t \\gamma _{a_l,a_0}} \\left\\Vert u_0\\right\\Vert _{L^2(\\Omega )}^2,$ which concludes the proof.", "In the end of this section we show that the smallness assumption on $\\left\|a_l\\right\|+ \\left\|a_0\\right\|$ is necessary in Theorem REF .", "More precisely, if $\\left\|a_l\\right\|$ and $\\left\|a_0\\right\|$ are too large, then the transverse operator $T_{a_l,a_0}$ has eigenvalues with positive imaginary parts and hence the solution of the Schrödinger equation grows exponentially.", "Proposition 7.4 Let $a_l, a_0\\in \\mathbb {R}$ with $a_l+ a_0> 0$ and $a_la_0< 0$ .", "Let $n \\in \\mathbb {N}$ .", "If $s > 0$ is large enough, we have $\\mathop {\\rm {Im}}\\nolimits \\big (\\lambda _n(sa_l,sa_0)^2\\big ) > 0$ .", "We know that the curves $s \\mapsto \\lambda _n(sa_l,sa_0)$ for $n \\in \\mathbb {N}$ are defined for all $s \\in \\mathbb {R}$ and remain bounded.", "Moreover we have chosen the square root $\\lambda _n(sa_l,sa_0)$ of $\\lambda _n(sa_l,sa_0)^2$ which has a non-negative real part, so the imaginary parts of $\\lambda _n(sa_l,sa_0)$ and $\\lambda _n(sa_l,sa_0)^2$ have the same signs.", "Assume that $a_la_0< 0$ , and let $n \\in \\mathbb {N}^$ be fixed.", "We have $ \\frac{(\\lambda _n(sa_l,sa_0)+s a_l)(\\lambda _n(sa_l,sa_0)+sa_0)}{(\\lambda _n(sa_l,sa_0)-sa_l)(\\lambda _n(sa_l,sa_0)-sa_0)} = 1 + \\frac{2\\lambda _n(sa_l,sa_0)}{s} \\left( \\frac{1}{a_l} + \\frac{1}{a_0} \\right) + \\mathop {O}\\limits _{s \\rightarrow +\\infty } \\big ( s^{-2} \\big ).$ Since $\\mathop {\\rm {Re}}\\nolimits (\\lambda _n(sa_l,sa_0)) > n\\nu $ and $\\mathop {\\rm {Im}}\\nolimits (\\lambda _n(sa_l,sa_0))$ is bounded, this quantity is of norm less than 1 when $s > 0$ is large enough, so $e^{-2l\\mathop {\\rm {Im}}\\nolimits (\\lambda _n(sa_l,sa_0))} = \\left\|e^{2il\\lambda _n(sa_l,sa_0)}\\right\| < 1,$ and hence $\\mathop {\\rm {Im}}\\nolimits (\\lambda _n(sa_l,sa_0)) > 0$ .", "When $n = 0$ , the same holds if we can prove that $\\lambda _0(sa_l,sa_0)$ does not go to 0 for large $s$ .", "Indeed, in this case the only possibility to have $e^{2il\\lambda _0(sa_l,sa_0)} \\rightarrow 1$ is that $\\lambda _0(sa_l,sa_0)$ goes to $\\nu $ , and then $\\mathop {\\rm {Re}}\\nolimits (\\lambda _0(sa_l,sa_0))$ is bounded by below by a positive constant, and we can conclude as before.", "So assume by contradiction that $\\lambda _0(sa_l,sa_0)$ goes to 0 as $s$ goes to $+\\infty $ .", "Then we have $e^{2il\\lambda _0(sa_l,sa_0)} = 1 + 2il\\lambda _0(sa_l,sa_0)+ \\mathop {O}\\limits _{s \\rightarrow +\\infty } \\big ( \\left\|\\lambda _0(sa_l,sa_0))\\right\|^2 \\big ),$ which gives a contradiction with (REF ), where the rest $O(s^{-2})$ has to be replaced by $O\\big ( \\lambda _0^2 s^{-2}\\big )$ .", "This concludes the proof.", "Remark 7.5 We remark from (REF ) (see also Figure REF ) that given $a_l,a_0\\in \\mathbb {R}$ such that $a_l+a_0> 0$ we always have $\\mathop {\\rm {Im}}\\nolimits (\\lambda _n(a_l,a_0)) < 0$ if $n$ is large enough.", "By duality, this means that there is always an eigenvalue with positive imaginary part when $a_l+ a_0< 0$ , and hence the norm of the solution of (REF ) is always exponentially increasing in this case.", "In this section we consider a more general waveguide $\\Omega $ of the form $\\mathbb {R}^{p}\\times \\omega \\subset \\mathbb {R}^d$ where $p \\in \\lbrace 1,\\dots ,d-1\\rbrace $ and $\\omega $ is a smooth open bounded subset of $\\mathbb {R}^{d-p}$ .", "In particular, $\\Omega $ is open in $\\mathbb {R}^d$ .", "Let $a \\in W^{1,\\infty }(\\partial \\Omega )$ .", "Until Proposition REF , we make no assumption on the sign of $a$ .", "We consider on $L^2(\\Omega )$ the operator $H_a$ defined by (REF ) with domain (REF ).", "For all ${\\mathcal {D}}(H_a)$ we have $ \\left< H_a >_{L^2(\\Omega )} = - \\left< , >_{L^2(\\Omega )} = \\left< \\nabla \\nabla >_{L^2(\\Omega )} -i \\left< a >_{L^2(\\partial \\Omega )}.\\right.On the other hand, we consider the quadratic form defined for \\right.\\right.\\right.", "{\\mathcal {D}}(q_a) = H^1(\\Omega ) byq_a( = \\int _\\Omega \\left\|\\nabla \\vert ^2 - i \\int _{\\partial \\Omega } a\\left\|\\vert ^2 .\\right.\\right.We also denote by q_a the corresponding sesquilinear form on {\\mathcal {D}}(q_a)^2.", "That this quadratic form is sectorial and closed follows from the following lemma and traces theorems:$ Lemma 2.1 Let $q_R$ be a non-negative, densely defined, closed form on a Hilbert space ${\\mathcal {H}}$ .", "Let $q_I$ be a symmetric form relatively bounded with respect to $q_R$ .", "Then the form $q_R - i q_I$ is sectorial and closed.", "It is important to note that there is no smallness assumption on the relative bound of $q_I$ with respect to $q_R$ .", "In particular, for $q_a$ we do not need any assumption on the size of $a$ in $L^\\infty (\\partial \\Omega )$ .", "There exists $C >0$ such that for all ${\\mathcal {D}}(q_R)$ we have $\\left\|q_I(\\right\| \\leqslant C \\left( q_R( + \\left\\Vert \\Vert ^2_{\\mathcal {H}}\\right).\\right.$ Let $\\varepsilon _0 = \\frac{1}{2C}$ .", "If we already know that $(q_R -i\\lambda q_I)$ is sectorial and closed for some $\\lambda \\geqslant 0$ , then $(q_R -i(\\lambda +\\varepsilon ) q_I)$ is sectorial and closed for all $\\varepsilon \\in [0,\\varepsilon _0]$ according to Theorem VI.3.4 in [22].", "Now since $q_R$ is sectorial and closed, we can prove by induction on $n \\in \\mathbb {N}$ that $(q_R -i\\lambda q_I)$ is sectorial and closed for all $\\lambda \\in [0,n\\varepsilon _0]$ , and hence for all $\\lambda \\geqslant 0$ .", "This is in particular the case when $\\lambda =1$ .", "We recall the definitions of accretive and dissipative operators (note that the conventions may be different for other authors): Definition 2.2 We say that an operator $T$ on the Hilbert space ${\\mathcal {H}}$ is accretive (respectively dissipative) if $\\forall {\\mathcal {D}}(T), \\quad \\mathop {\\rm {Re}}\\nolimits \\left< T >_{\\mathcal {H}}\\geqslant 0, \\quad \\big (\\text{respectively} \\quad \\mathop {\\rm {Im}}\\nolimits \\left< T >_{\\mathcal {H}}\\leqslant 0\\big ).\\right.\\right.$ Moreover $T$ is said to be maximal accretive (maximal dissipative) if it has no other accretive (dissipative) extension on ${\\mathcal {H}}$ than itself.", "In particuliar $T$ is (maximal) dissipative if and only if $iT$ is (maximal) accretive.", "Let us recall that an accretive operator $T$ is maximal accretive if and only if $(T-z)$ has a bounded inverse on ${\\mathcal {H}}$ for some (and hence any) $z \\in with $ Re(z) < 0$.", "In this case we know from the Hille-Yosida Theorem that $ -T$ generates a contractions semi-group $ t e-tT$.", "Then for all $ u0 D(T)$ the map $ u : t e-tTu0$ belongs to $ C1(R+,H) C0(R+,D(T))$ and solves the problem$${\\left\\lbrace \\begin{array}{ll}u^{\\prime }(t) + T u(t) = 0, \\quad \\forall t > 0,\\\\u(0) = u_0.\\end{array}\\right.", "}$$$ Let us come back to our context.", "According to Lemma REF and the Representation Theorem VI.2.1 in [22], there exists a unique maximal accretive operator $\\hat{H}_a$ on $L^2(\\Omega )$ such that ${\\mathcal {D}}(\\hat{H}_a) \\subset {\\mathcal {D}}(q_a)$ and $\\forall {\\mathcal {D}}(\\hat{H}_a), \\forall \\psi \\in {\\mathcal {D}}(q_a) , \\quad \\left< \\hat{H}_a \\psi \\right>_{L^2(\\Omega )} = q_a(\\psi ).$ Moreover we have ${\\mathcal {D}}(\\hat{H}_a) = \\left\\lbrace u \\in {\\mathcal {D}}(q_a) \\,:\\,\\exists f \\in L^2(\\Omega ) , \\forall \\phi \\in {\\mathcal {D}}(q_a), q_a(u,\\phi ) = \\left< f , \\phi \\right> \\right\\rbrace ,$ and for $u \\in {\\mathcal {D}}(\\hat{H}_a)$ the corresponding $f$ is unique and given by $\\hat{H}_au = f$ .", "Proposition 2.3 We have $\\hat{H}_a= H_a$ .", "In particular $H_a$ is maximal accretive.", "For a one-dimensional section we can essentially follow the proof of Lemma 3.2 in [10].", "This would be enough for our purpose but, for further use, we prove this result in the general setting.", "It is easy to check that ${\\mathcal {D}}(H_a) \\subset {\\mathcal {D}}(\\hat{H}_a)$ and $H_a= \\hat{H}_a$ on ${\\mathcal {D}}(H_a)$ .", "Now let $u \\in {\\mathcal {D}}(\\hat{H}_a)$ .", "By definition there exists $f \\in L^2(\\Omega )$ such that $\\forall \\phi \\in H^1(\\Omega ), \\quad \\int _{\\Omega } \\nabla u \\cdot \\nabla \\overline{\\phi } -i \\int _{\\partial \\Omega } a u \\overline{\\phi }= \\int _{\\Omega } f \\overline{\\phi }.$ Considering $\\phi \\in C_0^\\infty (\\Omega )$ we see that $-u̥ = f$ in the sense of distributions and hence in $L^2(\\Omega )$ .", "This proves that $u \\in H^2_{\\rm {loc}}(\\Omega )$ .", "It remains to prove that $u \\in H^2(\\Omega )$ and that the boundary condition $\\partial _\\nu u = i a u$ holds on $\\partial \\Omega $ .", "Let $j \\in \\lbrace 1,\\dots ,p\\rbrace $ and let $e_j$ be the $j$ -th vector in the canonical basis of $\\mathbb {R}^{p}$ .", "Let $\\delta \\in \\mathbb {R}^$ and $u_\\delta : (x,y) \\mapsto \\frac{1}{\\delta }( u(x + \\delta e_j , y ) - u(x,y)) \\in H^1(\\Omega )$ .", "See for instance [19] for the properties of the difference quotients.", "For all $\\phi \\in H^1(\\Omega )$ we have $q_a(u_\\delta , \\phi ) = -\\int _\\Omega f(x,y) \\overline{\\phi _{-\\delta }} (x,y) \\, dx \\,d\\sigma (y) - i \\int _{\\partial \\Omega } u (x,y) a_{-\\delta }(x,y) \\overline{\\phi }(x-\\delta e_j,y) \\, dx \\, d\\sigma (y),$ where $\\sigma $ is the Lebesgue measure on $\\omega $ .", "Since $a \\in W^{1,\\infty }(\\partial \\Omega )$ there exists $C \\geqslant 0$ such that for all $\\phi \\in H^1(\\Omega )$ and $\\delta > 0$ we have $\\left\|q_a (u_\\delta , \\phi )\\right\| \\leqslant C \\left\\Vert \\phi \\right\\Vert _{H^1(\\Omega )}.$ Applied with $\\phi = u_\\delta $ this gives $\\left\\Vert u_\\delta \\right\\Vert _{\\dot{H}^1(\\Omega )}^2 = \\mathop {\\rm {Re}}\\nolimits q_a (u_\\delta ,u_\\delta ) \\leqslant C \\left\\Vert u_\\delta \\right\\Vert _{H^1(\\Omega )}.$ Since we already know that $u_\\delta \\in L^2(\\Omega )$ uniformly in $\\delta > 0$ , we have $\\left\\Vert u_\\delta \\right\\Vert _{H^1(\\Omega )}^2 \\lesssim 1 + \\left\\Vert u_\\delta \\right\\Vert _{H^1(\\Omega )},$ which implies that $u_\\delta $ is uniformly in $H^1(\\Omega )$ .", "This means that $\\partial _{x_j} u \\in H^1(\\Omega )$ .", "Since this holds for any $j \\in \\lbrace 1,\\dots ,p\\rbrace $ , this proves that all the derivatives of order 2 with at least one derivative in the first $p$ directions belong to $L^2(\\Omega )$ .", "Then we get $-y u = f + x u \\in L^2(\\Omega ).$ According to the Green Formula we have for all $\\phi \\in H^1(\\Omega )$ $\\int _{\\Omega } - y u \\, \\overline{\\phi }\\, dx \\, dy = \\int _{\\Omega } \\nabla _y u \\cdot \\nabla _y \\overline{\\phi }\\, dx \\, dy - \\left< \\partial _\\nu u , \\phi \\right>_{H^{-1/2}(\\partial \\Omega ),H^{1/2}(\\partial \\Omega )}$ (see for instance [20]).", "By density of the trace map, we obtain that $ \\partial _\\nu u = ia u \\quad \\text{on } \\partial \\Omega .$ In particular $\\partial _\\nu u \\in H^{1/2}(\\partial \\Omega )$ .", "Then there exists $v \\in H^2(\\Omega )$ such that $\\partial _\\nu v = \\partial _\\nu u$ (see [20] for a fonction on $\\mathbb {R}^d_+$ ; for a function on $\\Omega $ we follow the same idea as for a fonction on a bounded domain, except that we only use a partition of unity for $\\partial \\omega $ , which allows to cover $\\partial \\Omega $ by a finite number of strips, each of which is diffeomorphic to a strip on $\\mathbb {R}^{p}$ ).", "Let $w = u - v$ .", "We have $w \\in H^1(\\Omega )$ , $y w \\in L^2(\\Omega )$ and ${\\left\\lbrace \\begin{array}{ll}-y w + w = f + x u - y v + w & \\text{on } \\Omega , \\\\\\partial _\\nu w = 0& \\text{on } \\partial \\Omega .\\end{array}\\right.", "}$ Then for almost all $x \\in \\mathbb {R}^{p}$ we have ${\\left\\lbrace \\begin{array}{ll}-y w(x) + w(x) = f(x) + x u(x) - y v(x) + w(x) & \\text{on } \\omega , \\\\\\partial _\\nu w(x) = 0& \\text{on } \\partial \\omega .\\end{array}\\right.", "}$ By elliptic regularity for the Neumann problem (see for instance Theorem 9.26 in [13]) we obtain that $w(x) \\in H^2(\\omega )$ with $\\left\\Vert w(x)\\right\\Vert _{H^2(\\omega )} \\lesssim \\left\\Vert f(x) + x u(x) - y v(x) + w(x)\\right\\Vert _{L^2(\\omega )}.$ After integration over $x \\in \\mathbb {R}^{p}$ , this gives $\\left\\Vert u\\right\\Vert _{H^2_y(\\Omega )} \\lesssim \\left\\Vert f+ x u - y v + w\\right\\Vert _{L^2(\\Omega )} + \\left\\Vert v\\right\\Vert _{H^2(\\Omega )}.$ Since we already know that second derivatives of $u$ involving a derivation in $x$ are in $L^2(\\Omega )$ , this proves that $u \\in H^2(\\Omega )$ and concludes the proof.", "Remark 2.4 We have $H_a^ = H_{-a}$ .", "Now assume that $a$ takes non-negative values.", "According to (REF ), $H_a$ is a dissipative operator.", "Since it is maximal accretive, it is easy to see that it is in fact maximal dissipative: Proposition 2.5 The maximal accretive operator $H_a$ is also maximal dissipative.", "We already know that $H_a$ is dissipative.", "Since it is maximal accretive, any $z \\in with $ Rez < 0$ is in its resolvent set.", "Then it is easy to find $ z$ in the resolvent set of $ Ha$ with $ Imz > 0$.$ Proposition 2.6 If $a > 0$ in an open subset of $\\partial \\Omega $ then $H_a$ has no real eigenvalue.", "Let $u \\in {\\mathcal {D}}(H_a)$ , $\\lambda \\in \\mathbb {R}$ , and assume that $H_au = \\lambda u$ .", "Taking the imaginary part of the equality $q_a (u,u) = \\lambda \\left\\Vert u\\right\\Vert ^2$ gives $\\int _{\\partial \\Omega } a \\left\|u\\right\|^2 = 0.$ This implies that $u = 0$ where $ a \\ne 0$ and $\\partial _\\nu u = ia u = 0$ everywhere on $\\partial \\Omega $ .", "By unique continuation, this implies that $u = 0$ on $\\Omega $ .", "The Transverse Operator Let us come back to the case of a one-dimensional cross-section $\\omega = ]0,l[$ .", "Under the additional assumption that the absorption index $a$ is constant on $\\partial \\Omega $ the operator $H_a$ can be written as $ H_a= -x \\otimes \\operatorname{Id}_{L^2(0,l)} + \\operatorname{Id}_{L^2(\\mathbb {R}^{d-1})} \\otimes T_a,$ where $-x$ is as before the usual flat Laplacian on $\\mathbb {R}^{d-1}$ and $T_a$ is the transverse Laplacian on $]0,l[$ .", "More precisely, we consider on $L^2(0,l)$ the operator $T_a= - \\frac{d^2}{dy^2}$ with domain ${\\mathcal {D}}(T_a) = \\left\\lbrace u \\in H^2(0,l) \\,:\\,u^{\\prime }(0) = -ia u(0), u^{\\prime }(l) = ia u(l) \\right\\rbrace .$ This is the maximal accretive and dissipative operator corresponding to the form $q : u \\in H^1(0,l) \\mapsto \\int _{0}^{l} \\left\|u^{\\prime }(x)\\right\|^2 \\, dx - i a \\left\|u(l)\\right\|^2 - i a \\left\|u(0)\\right\|^2.$ In this section we give the spectral properties of $T_a$ which we need to study the full operator $H_a$ .", "This operator has compact resolvent, and hence its spectrum is given by a sequence of isolated eigenvalues.", "When $a = 0$ , which corresponds to the Neumann problem, we know that the eigenvalues of $T_0$ are the real numbers $n^2 \\nu ^2$ for $n \\in \\mathbb {N}$ , where we have set $\\nu = \\frac{\\pi }{ l}.$ These eigenvalues are algebraically simple.", "Proposition 3.1 There exists a sequence $\\left({\\lambda }_{n}\\right)_{n \\in \\mathbb {N}}$ of continuous functions on $\\mathbb {R}$ such that $\\lambda _n(0) = n \\nu $ and for all $a \\in \\mathbb {R}$ the set of eigenvalues of $T_a$ is $\\left\\lbrace \\lambda _n(a)^2, n\\in \\mathbb {N} \\right\\rbrace $ .", "Moreover: For $(n,a) \\in (\\mathbb {N}\\times \\mathbb {R}) \\setminus \\lbrace (0,0)\\rbrace $ the eigenvalue $\\lambda _n(a)^2$ is algebraically simple and a corresponding eigenvector is given by $ {n}(a) : x \\mapsto A_n(a) \\left( e^{i\\lambda _n(a) x} + \\frac{\\lambda _n(a) + a}{\\lambda _n(a) - a} e^{-i \\lambda _n(a) x} \\right) ,$ where we can choose $A_n(a) \\in \\mathbb {R}_+^$ in such a way that $\\left\\Vert n(a)\\right\\Vert _{L^2(0,l)} = 1$ (when $a = 0$ then 0 is a simple eigenvalue and corresponding eigenvectors are non-zero constant functions).", "For $n \\in \\mathbb {N}$ and $a \\in \\mathbb {R}$ we have $\\lambda _n(-a)= \\overline{\\lambda _n(a)}$ .", "Let $n\\in \\mathbb {N}$ .", "For all $a \\in \\mathbb {R}^$ we have $\\mathop {\\rm {Re}}\\nolimits (\\lambda _n(a)) \\in ] n\\nu ,(n+1)\\nu [$ (when $n=0$ , we have chosen the square root of $\\lambda _0^2(a)$ which has a positive real part).", "For all $n \\in \\mathbb {N}$ there exists $C_n > 0$ such that for $a > 0$ we have $-C_n < \\mathop {\\rm {Im}}\\nolimits (\\lambda _n(a)) < 0$ .", "Let $a > 0$ be fixed.", "We have $\\lambda _n(a) = n \\nu - \\frac{2ia}{n\\nu l} + \\mathop {O}\\limits _{n \\rightarrow +\\infty } \\big (n^{-2} \\big )$ and hence $\\lambda _n(a)^2 = (n\\nu )^2 - \\frac{4ia}{l} + \\mathop {O}\\limits _{n \\rightarrow +\\infty } \\big (n^{-1} \\big ).$ $\\bullet $ It is straightforward computations to check that 0 is an eigenvalue of $T_a$ if and only if $a = 0$ and, if $\\lambda \\in $ , $\\lambda ^2$ is an eigenvalue if and only if $ (a-\\lambda )^2 e^{2i\\lambda l} = ( a+\\lambda )^2.$ If $\\lambda ^2$ is an eigenvalue then the corresponding eigenfunction is of the form $ x \\mapsto A e^{i\\lambda x} + B e^{-i\\lambda x}$ with $ A = \\frac{\\lambda -a}{\\lambda +a} B = \\frac{\\lambda + a}{\\lambda -a} e^{-2i\\lambda l}B.$ Moreover, all these eigenvalues have geometric multiplicity 1.", "Indeed, given $n \\in \\mathbb {N}$ , the space of eigenvectors corresponding to the eigenvalue $\\lambda _n(a)^2$ is strictly included in the space of $H^2$ functions which are solutions of $-u^{\\prime \\prime } - \\lambda _n(a)^2 u = 0$ , and this space is of dimension 2.", "The fact that the eigenvalues of $H_{-a}$ are conjugated to the eigenvalues of $H_{a}$ is a consequence of Remark REF .", "$\\bullet $ Let $a > 0$ and $\\lambda \\in $ be such that $\\lambda ^2$ is an eigenvalue of $T_a$ .", "Assume that $\\mathop {\\rm {Re}}\\nolimits \\lambda \\in \\nu \\mathbb {N}$ .", "Then $\\left( \\frac{a+\\lambda }{a-\\lambda } \\right)^2 = e^{2i\\lambda l} \\in \\mathbb {R}_+$ (note that $\\lambda $ cannot be equal to $a$ in (REF )) and hence $r := \\frac{a+\\lambda }{a-\\lambda } \\in \\mathbb {R}.$ If $r = -1$ then $a = 0$ .", "Otherwise $\\lambda = \\frac{a(1-r)}{1+r} \\in \\mathbb {R}$ .", "In both cases we obtain a contradiction (see Proposition REF ), and hence $\\mathop {\\rm {Re}}\\nolimits \\lambda \\notin \\nu \\mathbb {N}$ .", "This proves that for $a > 0$ the operator $T_a$ has no eigenvalue with real part in $\\nu \\mathbb {N}$ .", "$\\bullet $ Now let $R > 0$ .", "We prove that if $C_R \\geqslant 0$ is chosen large enough and if $a \\in \\mathbb {R}$ and $\\lambda \\in $ are such that $\\lambda ^2$ is an eigenvalue of $T_a$ , then $ \\left\|\\mathop {\\rm {Re}}\\nolimits \\lambda \\right\| \\leqslant R \\quad \\Rightarrow \\quad \\left\|\\mathop {\\rm {Im}}\\nolimits \\lambda \\right\| \\leqslant C_R.$ Assume by contradiction that this is not the case.", "Then for all $m \\in \\mathbb {N}$ we can find $x_m \\in [-R,R]$ and $y_m \\in \\mathbb {R}$ with $\\left\|y_m\\right\| \\geqslant m$ such that $(x_m+ i y_m)^2$ is an eigenvalue of $T_{a_m}$ for some $a_m \\in \\mathbb {R}$ .", "We have $e^{-2y_m l} = \\left\|\\frac{a_m + x_m + iy_m}{a_m - x_m - iy_m}\\right\|^2 = \\frac{(a_m + x_m)^2 + y_m^2}{(a_m - x_m)^2 + y_m^2} \\xrightarrow[m \\rightarrow \\infty ]{}1,$ which gives a contradiction.", "$\\bullet $ The family of operators $T_a$ for $a \\in \\mathbb {R}$ is an analytic family of operators of type B in the sense of Kato [22].", "We already know that the spectrum of $T_0$ is given by $\\left\\lbrace (n\\nu )^2, n \\in \\mathbb {N} \\right\\rbrace $ .", "Then for all $n \\in \\mathbb {N}$ there exists an analytic function $\\lambda _n^2$ such that, at least for small $a$ , $\\lambda _n^2(a)$ is in the spectrum of $T_a$ (and then we define $\\lambda _n$ as the square root of $\\lambda _n^2$ with positive real part).", "See Theorem VII.1.7 in [22].", "$\\bullet $ Let $n \\in \\mathbb {N}^$ .", "We write $\\lambda _n(a) = n\\nu + \\beta a + \\gamma a^2 + O_{a \\rightarrow 0} (a^3)$ .", "We have $e^{2i \\lambda _n(a) l}= 1 + 2il\\beta a + 2il\\gamma a^2 - 2l^2 \\beta ^2 a^2 + O(a^3),$ and on the other hand: $\\left( \\frac{\\lambda _n(a) + a}{\\lambda _n(a) - a} \\right)^2= 1 + \\frac{4a}{n\\nu } - \\frac{4(\\beta -2) a^2}{n^2\\nu ^2} + O\\big (a^3\\big ).$ Since $\\lambda _n(a)$ solves (REF ) for any $a > 0$ we obtain $\\beta = \\frac{2}{iln\\nu } = -\\frac{2i}{\\pi n}$ and $\\mathop {\\rm {Re}}\\nolimits (\\gamma )= \\frac{4 l}{n^3\\pi ^3} .$ Since $\\mathop {\\rm {Re}}\\nolimits (\\beta ) = 0$ and $\\mathop {\\rm {Re}}\\nolimits (\\gamma ) > 0$ we have $\\mathop {\\rm {Re}}\\nolimits \\big (\\lambda _n(a)\\big ) \\in \\left] n\\nu , (n+1)\\nu \\right[$ for $a > 0$ small enough.", "The functions $a \\mapsto \\mathop {\\rm {Re}}\\nolimits \\big (\\lambda _n(a)\\big )$ are continuous and cannot reach $\\nu \\mathbb {N}$ unless $a=0$ , so this remains true for any $a > 0$ such that $\\lambda _n(a)$ is defined.", "Similarly $\\mathop {\\rm {Re}}\\nolimits \\big (\\lambda _0(a)\\big ) \\in ]0, \\nu [$ for all $a > 0$ .", "In particular the curves $a \\mapsto \\lambda _n(a)$ for $n \\in \\mathbb {N}$ never meet.", "Moreover we know from (REF ) that $\\lambda _n(a)$ remains in a bounded set of $, so the curves $ a n(a)$ are defined for all $ a R$ and for all $ a R$ the eigenvalues of $ Ta$ are exactly given by $ n(a)2$ for $ n N$.$ $\\bullet $ It remains to prove that the asymptotic expansion of $\\lambda _n(a)$ for $n$ fixed and $a$ small is also valid for $a$ fixed and $n$ large.", "Let $a>0$ be fixed.", "Derivating (REF ) and using the fact that $\\left\|\\lambda _n(s)\\right\| \\geqslant n \\nu $ for all $s \\in \\mathbb {R}$ we obtain that $\\sup _{s \\in [0,a]} \\left\|\\lambda ^{\\prime }_n(s)\\right\| = O \\big (n^{-1}\\big ).$ This means that $\\lambda _n(a) = n\\nu + O(n^{-1})$ .", "Then we obtain the asymptotic expansion of $\\lambda _n(a)$ for large $n$ as before, using again (REF ).", "This gives the last statement of the proposition and concludes the proof.", "Now that we have proved what we need concerning the spectrum of the operator $T_a$ , we study the corresponding sequence of eigenfunctions.", "In the self-adjoint case $a = 0$ , we know that the eigenfunctions $n(0)$ form an orthonormal basis.", "Of course this is no longer the case for the non-selfadjoint operator $T_a$ with $a \\ne 0$ .", "However we can prove that in this case we have a Riesz basis.", "We recall that the sequence $\\left({_{n}_{n \\in \\mathbb {N}} of vectors in the Hilbert space {\\mathcal {H}} is said to be a Riesz basis if there exists a bounded operator \\Theta \\in {\\mathcal {L}}({\\mathcal {H}}) with bounded inverse and an orthonormal basis \\left({e}_{n}\\right)_{n \\in \\mathbb {N}} of {\\mathcal {H}} such that n = \\Theta e_n for all n \\in \\mathbb {N}.In this case any f \\in {\\mathcal {H}} can be written as \\sum _{n\\in \\mathbb {N}} f_n n with \\left({f}_{n}\\right)_{n \\in \\mathbb {N}} \\in l^2(, and there exists C \\geqslant 1 such that for all f = \\sum _{n\\in \\mathbb {N}} f_n n\\in {\\mathcal {H}} we haveC^{-1}\\sum _{n\\in \\mathbb {N}} \\left\|f_n\\right\|^2 \\leqslant \\left\\Vert \\sum _{n\\in \\mathbb {N}} f_n n\\right\\Vert _{\\mathcal {H}}^2 \\leqslant C \\sum _{n\\in \\mathbb {N}} \\left\|f_n\\right\|^2.In these estimates we can take C = \\max \\left( \\left\\Vert \\Theta \\right\\Vert ^2 , \\left\\Vert \\Theta ^{-1}\\right\\Vert ^2 \\right).", "}Let \\right.$ (n)n N$ be a Riesz basis of $ H$ with $$ and $ (en)n N$ as above.", "If we set $ * n = (-1)* en$ for all $ n N$ then $ (n)nN$ is also a Riesz basis, called the dual basis of $ (n)n N$.", "In particular for all $ j,k N$ we have\\begin{equation} \\left< j , _k \\right>_{\\mathcal {H}}= \\delta _{j,k}.\\end{equation}We refer to \\cite {christensen} for more details about Riesz bases.", "Now we want to prove that for any $ a R$ the sequence of eigenfunctions for the operator $ Ta$ is a Riesz basis of $ L2(0,l)$.$ Proposition 3.2 For all $a \\in \\mathbb {R}$ the sequence $(n(a))_{n\\in \\mathbb {N}}$ defined in (REF ) is a Riesz basis of $L^2(0,l)$ .", "Moreover for all $R > 0$ there exists $C \\geqslant 0$ such that for $a \\in [-R,R]$ and $\\left({c}_{n}\\right)_{n \\in \\mathbb {N}} \\in l^2($ we have $C^{-1}\\sum _{n = 0}^{\\infty } \\left\|c_n\\right\|^2 \\leqslant \\left\\Vert \\sum _{n = 0}^{\\infty } c_n n(a)\\right\\Vert ^2 \\leqslant C \\sum _{n = 0}^{\\infty } \\left\|c_n\\right\|^2.$ If $C$ was chosen large enough and if $\\left({c}_{n}\\right)_{n \\in \\mathbb {N}} \\in l^2($ is such that $\\sum _{n=0}^\\infty \\left\|\\lambda _n(a) c_n\\right\|^2 < \\infty $ we also have $C^{-1}\\sum _{n = 0}^{\\infty } \\left\|\\lambda _n(a) c_n\\right\|^2 \\leqslant \\left\\Vert \\sum _{n = 0}^{\\infty } c_n _n(a)\\right\\Vert ^2 \\leqslant C \\sum _{n = 0}^{\\infty } \\left\|\\lambda _n(a) c_n\\right\|^2.$ For similar results we refer to [30] (see also Lemma XIX.3.10 in [18]).", "For the proof we need the following lemma: Lemma 3.3 Let $R > 0$ .", "Then there exists $C \\geqslant 0$ such that for $a \\in [-R,R]$ and $j,k \\in \\mathbb {N}$ with $j < k$ we have $\\left< j(a) , k(a) \\right>_{L^2(0,l)} \\leqslant \\frac{C}{\\left< j \\right> (k-j)} \\quad \\text{and} \\quad \\left< _j(a) , _k(a) \\right>_{L^2(0,l)} \\leqslant C \\frac{k}{k-j}.$ Let $e_n(a,x) = e^{i \\lambda _n (a) x}$ and $\\tilde{e}_n(a,x) = e^{-i \\lambda _n (a) x}$ .", "According to Proposition REF we have $\\lambda _n(a) = n\\nu + O(n^{-1})$ (here and below all the estimates are uniform in $a \\in [-R,R]$ ), so $\\begin{aligned}\\left\\Vert e_n(a)\\right\\Vert ^2_{L^2(0,l)}= \\frac{e^{-2l\\mathop {\\rm {Im}}\\nolimits (\\lambda _n(a))} - 1 }{-2 \\mathop {\\rm {Im}}\\nolimits (\\lambda _n(a))} = l+ O(n^{-1}).\\end{aligned}$ Similarly $\\left\\Vert \\tilde{e}_n(a)\\right\\Vert ^2_{L^2(0,l)} = l+ O(n^{-1})$ , and $\\left< e_n(a) , \\tilde{e}_n(a) \\right> =O(n^{-1})$ .", "Moreover, with (REF ) we see that $A_n(a) = \\frac{1}{\\sqrt{2l}} + O(n^{-1}).$ Now let $j,k \\in \\mathbb {N}$ with $j < k$ .", "We omit the dependence in $a$ for $j$ , $e_j$ , $\\tilde{e}_j$ and $\\lambda _j$ .", "We have $\\left< e_j , e_k \\right> = \\frac{e^{i(\\lambda _j - \\overline{\\lambda _k}) l} -1}{\\lambda _j - \\overline{\\lambda _k}}\\quad \\text{and} \\quad \\left< \\tilde{e}_j , \\tilde{e}_k \\right> = \\frac{e^{-i(\\lambda _j - \\overline{\\lambda _k}) l} -1}{-(\\lambda _j - \\overline{\\lambda _k})} .$ Since $\\lambda _j - \\overline{\\lambda _k} = (j-k) \\nu + O(1/j)$ we have $\\left\|\\left< e_j , e_k \\right>\\right\| + \\left\|\\left< \\tilde{e}_j , \\tilde{e}_k \\right>\\right\| \\lesssim \\frac{1}{k-j}\\quad \\text{and} \\quad \\left\|\\left< e_j , e_k \\right> + \\left< \\tilde{e}_j , \\tilde{e}_k \\right>\\right\| \\lesssim \\frac{1}{\\left< j \\right> (k-j)} .$ Similarly $\\left\|\\left< \\tilde{e}_j , e_k \\right>\\right\| + \\left\|\\left< e_j , \\tilde{e}_k \\right>\\right\| \\lesssim \\frac{1}{k+j}\\quad \\text{and} \\quad \\left\|\\left< \\tilde{e}_j , e_k \\right> + \\left< e_j , \\tilde{e}_k \\right>\\right\| \\lesssim \\frac{1}{\\left< j \\right> (k+j)} .$ And finally $\\left\|\\left< j , k \\right>\\right\|& = \\left\|A_j\\overline{A_k}\\left< e_j + \\frac{\\lambda _j - a }{\\lambda _j + a } \\tilde{e}_j , e_k + \\frac{\\lambda _k - a }{\\lambda _k + a } \\tilde{e}_k \\right>\\right\|\\\\& \\lesssim \\left\| \\left< e_j , e_k \\right> +\\left< e_j , \\tilde{e}_k \\right>+\\left< \\tilde{e}_j , e_k \\right>+\\left< \\tilde{e}_j , \\tilde{e}_k \\right> \\right\|\\\\& \\quad + \\left< j \\right> ^{-1}\\left( \\left\|\\left< e_j , e_k \\right>\\right\| + \\left\|\\left< e_j , \\tilde{e}_k \\right>\\right\| + \\left\|\\left< \\tilde{e}_j , e_k \\right>\\right\| + \\left\|\\left< \\tilde{e}_j , \\tilde{e}_k \\right>\\right\| \\right)\\\\& \\lesssim \\frac{1}{\\left< j \\right> (k-j)}.$ The second estimate is proved similarly, using again that $\\left\|\\lambda _n(a)\\right\| = n\\nu + O(n^{-1})$ for large $n$ .", "$\\bullet $ Let $a \\in \\mathbb {R}$ and $\\left({c}_{n}\\right)_{n \\in \\mathbb {N}} \\in l^2($ .", "Let $C\\geqslant 0$ be given by Lemma REF .", "For $N,p \\in \\mathbb {N}$ we have $\\left\\Vert \\sum _{n = N}^{N+p} c_n n(a)\\right\\Vert ^2 - \\sum _{n=N }^{N+p} \\left\|c_n\\right\|^2&= \\sum _{j=N}^{N+p} \\sum _{k=j+1}^{N+p} 2 \\mathop {\\rm {Re}}\\nolimits \\left( c_j \\overline{c_k} \\left< j , k \\right> \\right) \\\\& \\leqslant 2C \\sum _{j=0}^\\infty \\frac{\\left\|c_j\\right\|}{\\left< j \\right>} \\sum _{k=1}^\\infty \\frac{\\left\|c_{k+j}\\right\|}{k} \\\\& \\lesssim \\left\\Vert c\\right\\Vert _{l^2(}^2.$ This proves that the series $\\sum _{n = 0}^\\infty c_n n(a)$ converges in $L^2(0,l)$ and $\\left\\Vert \\sum _{n = 0}^\\infty c_n n(a)\\right\\Vert _{L^2(0,l)}^2 \\lesssim \\sum _{n=0}^\\infty \\left\|c_n\\right\|^2.$ $\\bullet $ Let $n,m \\in \\mathbb {N}$ be such that $n\\ne m$ .", "In $L^2(0,l)$ we have $\\lambda _n(a)^2 \\left< n(a) , m(-a) \\right>& = \\left< T_an(a) , m(-a) \\right> = \\left< n(a) , T_a^* m(-a) \\right>\\\\& = {\\lambda _m(a)^2} \\left< n(a) , m(-a) \\right>,$ and hence $\\left< n(a) , m(-a) \\right> = 0$ .", "On the other hand, with (REF ) we can check that for $n$ large enough we have $\\left< n(a) , n(-a) \\right> \\ne 0$ .", "Now let $\\left({c}_{n}\\right)_{n \\in \\mathbb {N}} \\in l^2($ be such that $\\sum _{n=0}^\\infty c_n n(a) = 0$ .", "Taking the inner product with $m(-a)$ we see that $c_m = 0$ for $m > N$ if $N$ is chosen large enough.", "Then for all $k \\in \\lbrace 0,\\dots ,N\\rbrace $ we have $0 = T_a^k \\sum _{n=0}^N c_n n(a) = \\sum _{n=0}^N \\lambda _n(a)^{2k} c_n n(a).$ Since the eigenvalues $\\lambda _n(a)^2$ for $n\\in \\lbrace 0,\\dots ,N\\rbrace $ are pairwise disjoint, this proves that for all $n \\in \\lbrace 0,\\dots ,N\\rbrace $ we have $c_n n(a) = 0$ , and hence $c_n=0$ .", "Finally the map $(c_n) \\in l^2( \\mapsto \\sum c_nn(a) \\in L^2(0,l)$ is one-to-one.", "$\\bullet $ As above we can check that for $N\\in \\mathbb {N}$ we have $\\left\\Vert \\sum _{n = N}^{\\infty } c_n n(a)\\right\\Vert ^2 - \\sum _{n=N}^{\\infty } \\left\|c_n\\right\|^2\\lesssim \\sum _{n=N}^{\\infty } \\left\|c_n\\right\|^2 \\sqrt{\\sum _{n=N}^\\infty \\frac{1}{\\left< n \\right>^2}}.$ This is less that $\\frac{1}{2} \\sum _{n=N}^{\\infty } \\left\|c_n\\right\|^2$ if $N$ is chosen large enough.", "Let such an integer $N$ be fixed.", "Now assume by contradiction that the sequences $\\left({a}_{m}\\right)_{m \\in \\mathbb {N}} \\in [-R,R]^\\mathbb {N}$ and $(c^m)_{m\\in \\mathbb {N}}$ in $l^2(^\\mathbb {N}$ are such that $\\left\\Vert c^m\\right\\Vert _{l^2} = 1$ for all $m \\in \\mathbb {N}$ and $\\left\\Vert \\sum _{n=0}^\\infty c_n^m n(a_m)\\right\\Vert ^2 \\xrightarrow[m \\rightarrow \\infty ]{}0.$ For $m \\in \\mathbb {N}$ we set $f_m = \\sum _{n=0}^{N-1} c_n^m n(a_m)$ and $g_m = \\sum _{n=N}^{\\infty } c_n^m n(a_m)$ .", "After extracting a subsequence if necessary we can assume that $a_m$ converges to some $a \\in [-R,R]$ and $f_m$ converges to some $f \\in L^2(0,l)$ .", "Let $P_m$ (respectively $P$ ) denote the orthogonal projection on $\\operatorname{span}(j(a_m))_{j\\geqslant N}$ (respectively on $\\operatorname{span}(n(a))_{n\\geqslant N}$ ).", "We have $\\left\\Vert f_m + g_m\\right\\Vert ^2 = \\left\\Vert f_m\\right\\Vert ^2 + 2 \\mathop {\\rm {Re}}\\nolimits \\left< P_m f_m , g_m \\right> + \\left\\Vert g_m\\right\\Vert ^2 \\geqslant \\left\\Vert f_m\\right\\Vert ^2 - \\left\\Vert P_m f_m\\right\\Vert ^2 \\xrightarrow[m \\rightarrow \\infty ]{}\\left\\Vert f\\right\\Vert ^2 - \\left\\Vert P f\\right\\Vert ^2.$ This gives $f = P f$ , and hence $f = 0$ .", "This gives a contradiction with $0 = \\lim _{m \\rightarrow \\infty } \\left\\Vert g_m\\right\\Vert ^2 \\geqslant \\lim _{m\\rightarrow \\infty } \\frac{1}{2} \\sum _{n=N}^\\infty \\left\|c_n^m\\right\|^2 = \\frac{1}{2}$ and proves the first inequality of the proposition.", "$\\bullet $ It remains to prove that the sequence $(n(a))$ is complete.", "We know that the family $\\big ( n(0) \\big )_{n\\in \\mathbb {N}}$ is an orthonormal basis of $L^2(0,l)$ .", "Since $n(a) = n(0) + O \\big ( n^{-1}\\big )$ in $L^2(0,l)$ , this follows from a perturbation argument (see Theorem V.2.20 in [22]).", "This concludes the proof.", "For $a \\in \\mathbb {R}$ we denote by $(n^(a))_{n\\in \\mathbb {N}}$ the dual basis of $(n(a))_{n\\in \\mathbb {N}}$ .", "We have $n^(a) = n(-a) = \\overline{n(a)}$ .", "Separation of variables - Spectrum of the model operator In this section we use the results on the transversal operator $T_a$ to prove spectral properties for the full operator $H_a$ when $a$ is constant on $\\partial \\Omega $ .", "Most of the results of this section are inspired by the ${\\mathcal {P}}{\\mathcal {T}}$ -symmetric analogs (see [10]).", "Let $a > 0$ be fixed.", "We set $\\mathfrak {S}_a = \\bigcup _{n \\in \\mathbb {N}} \\lbrace \\lambda _n(a)^2\\rbrace + \\mathbb {R}_+ = \\left\\lbrace \\lambda _n(a)^2 + r, n \\in \\mathbb {N}, r \\in \\mathbb {R}_+ \\right\\rbrace \\subset $ Proposition 4.1 We have $\\mathfrak {S} _a \\subset \\sigma (H_a)$ .", "Let $n \\in \\mathbb {N}$ , $r \\geqslant 0$ and $z = \\lambda _n(a)^2 + r \\in \\mathfrak {S}_a$ .", "Let $\\left({f}_{m}\\right)_{m \\in \\mathbb {N}}$ be a sequence in $H^2(\\mathbb {R}^{d-1})$ such that $\\left\\Vert f_m\\right\\Vert _{L^2(\\mathbb {R}^{d-1})} = 1$ for all $m \\in \\mathbb {N}$ and $\\left\\Vert (-x -r)f_m\\right\\Vert _{L^2(\\mathbb {R}^{d-1})} \\rightarrow 0$ as $m \\rightarrow \\infty $ .", "For $m \\in \\mathbb {N}$ and $(x,y) \\in \\mathbb {R}^{d-1} \\times ]0,l[$ we set $u_m(x,y) = f_m (x) n(a;y)$ .", "Then $u_m \\in {\\mathcal {D}}(H_a)$ and $\\left\\Vert u_m\\right\\Vert _{L^2(\\Omega )} = 1$ for all $m \\in \\mathbb {N}$ .", "Moreover, according to (REF ) we have $\\left\\Vert (H_a- z)u_m\\right\\Vert _{L^2(\\Omega )} = \\left\\Vert (-x - r) f_m\\right\\Vert _{L^2(\\mathbb {R}^{d-1})} \\xrightarrow[m \\rightarrow \\infty ]{}0.$ This implies that $z \\in \\sigma (H_a)$ .", "For $u \\in L^2(\\Omega )$ , $n\\in \\mathbb {N}$ and $x \\in \\mathbb {R}^{d-1}$ we set $u_n (x) = \\left< u(x, \\cdot ) , _n(a) \\right>_{L^2(0,l)} \\in .", "This gives a sequence of functions $ un$ defined almost everywhere on $ Rd-1$.$ Proposition 4.2 Let $u \\in L^2(\\Omega )$ .", "Then $u_n \\in L^2(\\mathbb {R}^{d-1})$ for all $n \\in \\mathbb {N}$ and on $L^2(\\Omega )$ we have $u= \\sum _{n\\in \\mathbb {N}} u_n \\otimes n(a).$ For $N \\in \\mathbb {N}$ we set $v_N = \\sum _{n = 0}^N u_n \\otimes n(a)$ .", "For almost all $x \\in \\mathbb {R}^{d-1}$ , $v_N(x)$ defines a function in $L^2(0,l)$ which goes to $u(x)$ as $N \\rightarrow \\infty $ .", "According to Proposition REF we have $\\left\\Vert v_N(x)\\right\\Vert ^2_{L^2(0,l)} \\lesssim \\sum _{n=0}^N \\left\|u_n(x)\\right\|^2 \\lesssim \\sum _{n=0}^\\infty \\left\|u_n(x)\\right\|^2 \\lesssim \\left\\Vert u(x)\\right\\Vert ^2_{L^2(0,l)},$ where all the estimates are uniform in $N$ .", "Since the map $x \\mapsto \\left\\Vert u(x)\\right\\Vert ^2_{L^2(0,l)}$ belongs to $L^1(\\mathbb {R}^{d-1})$ , we can apply the dominated convergence theorem to conclude that $u-v_N \\rightarrow 0$ in $L^2(\\Omega )$ .", "Proposition 4.3 Let $u \\in {\\mathcal {D}}(H_a)$ .", "Then $u_n \\in H^2(\\mathbb {R}^{d-1})$ for all $n \\in \\mathbb {N}$ and in $L^2(\\Omega )$ we have $H_au = \\sum _{n\\in \\mathbb {N}} \\big ( - x + \\lambda _n(a)^2 \\big ) u_n \\otimes n(a).$ Let $z \\in \\mathfrak {S}_a$ .", "Then $z$ belongs to the resolvent set of $H_a$ and for all $u \\in L^2 (\\Omega )$ we have $(H_a-z)^{-1}u = \\sum _{n\\in \\mathbb {N}} (-x + \\lambda _n(a)^2-z)^{-1}u_n \\otimes n(a).$ In particular there exists $C \\geqslant 0$ such that for all $z \\in \\mathfrak {S}_a$ and $u \\in L^2(\\Omega )$ we have $\\left\\Vert (H_a-z)^{-1}u\\right\\Vert _{L^2(\\Omega )} \\leqslant \\frac{C}{d(z,\\mathfrak {S}_a)} \\left\\Vert u\\right\\Vert _{L^2(\\Omega )}.$ $\\bullet $ Let $z \\in \\mathfrak {S}_a$ and $u \\in L^2(\\Omega )$ .", "Let $u_n \\in L^2(\\mathbb {R}^{d-1})$ , $n\\in \\mathbb {N}$ , be as above.", "For $n \\in \\mathbb {N}$ we set $\\tilde{R}_n(u)= (-x + \\lambda _n(a)^2 - z)^{-1}u_n \\in H^2 (\\mathbb {R}^{d-1})$ and $R_n(u)= \\tilde{R}_n(u)\\otimes n(a) \\in {\\mathcal {D}}(H_a)$ .", "Using the standard spectral properties of the self-adjoint operator $-x$ , we see that on $L^2(\\mathbb {R}^{d-1})$ we have $ \\left\\Vert \\tilde{R}_n(u)\\right\\Vert \\lesssim \\frac{\\left\\Vert u_n\\right\\Vert }{d(z, \\lambda _n(a)^2 +\\mathbb {R}_+ )} \\lesssim _z \\frac{\\left\\Vert u_n\\right\\Vert }{\\left< n \\right> ^{2}} \\quad \\text{and} \\quad \\left\\Vert \\partial _x \\tilde{R}_n(u)\\right\\Vert \\lesssim _z \\frac{\\left\\Vert u_n\\right\\Vert }{\\left< n \\right> }.$ The first estimate is uniform in $z$ but not the others.", "With Proposition REF we obtain for $N,p \\in \\mathbb {N}$ $\\left\\Vert \\sum _{n=N}^{N+p} R_n(u)\\right\\Vert ^2_{H^1(\\Omega )} \\lesssim _z \\sum _{n=N}^{N+p} \\left( \\left\\Vert \\tilde{R}_n(u)\\right\\Vert ^2_{H^1(\\mathbb {R}^{d-1})} + \\left< n \\right>^2 \\left\\Vert \\tilde{R}_n(u)\\right\\Vert ^2_{L^2(\\mathbb {R}^{d-1})} \\right) \\xrightarrow[N \\rightarrow \\infty ]{}0.$ This proves that the series $\\sum R_n(u)$ converges to some $R(u) \\in H^1(\\Omega )$ .", "Moreover, with the first inequality of (REF ) and Proposition REF again, we see that $ \\left\\Vert R(u)\\right\\Vert _{L^2(\\Omega )} \\lesssim \\frac{\\left\\Vert u\\right\\Vert _{L^2(\\Omega )}}{d (z ,\\mathfrak {S}_a)},$ uniformly in $z$ .", "It remains to see that $\\forall \\phi \\in H^1(\\Omega ), \\quad q_a(R(u),\\phi ) - z \\left< R(u) , \\phi \\right>= \\left< u , \\phi \\right>,$ which comes from the fact that this is true with $R(u)$ and $u$ replaced by $R_n(u)$ and $u_n \\otimes n(a)$ respectively.", "This proves that $R(u) \\in {\\mathcal {D}}(H_a)$ and $(H_a-z) R(u) = u$ .", "If $\\mathop {\\rm {Re}}\\nolimits (z) < 0$ , we already know that $(H_a-z)$ has a bounded inverse on $L^2(\\Omega )$ , and hence we have $(H_a-z)^{-1}= R$ .", "This proves the second statement of the proposition when $\\mathop {\\rm {Re}}\\nolimits (z) < 0$ .", "$\\bullet $ Let $ u \\in {\\mathcal {D}}(H_a)$ and $v = (H_a+1) u \\in L^2(\\Omega )$ .", "For $n \\in \\mathbb {N}$ and almost all $x \\in \\mathbb {R}^{d-1}$ we set $v_n(x) = \\left< v(x,\\cdot ) , n^(a) \\right>_{L^2(0,l)}$ .", "According to (ii) applied with $z = -1$ we have $u = (H_a+1)^{-1}v = \\sum _{n\\in \\mathbb {N}} \\big (-x + \\lambda _n(a)^2 +1\\big )^{-1}v_n \\otimes n(a).$ By uniqueness for the decomposition of $u(x,\\cdot )$ with respect to the Riesz basis $(n(a))_{n\\in \\mathbb {N}}$ , we have for all $n \\in \\mathbb {N}$ $u_n = \\big (-x + \\lambda _n(a)^2 +1\\big )^{-1}v_n.$ This proves that $u_n \\in H^2(\\mathbb {R}^{d-1})$ .", "Then $H_au = v - u& = \\sum _{n \\in \\mathbb {N}} \\left( 1 - \\big (-x + \\lambda _n(a)^2 +1\\big )^{-1}\\right) v_n \\otimes n(a)\\\\& = \\sum _{n \\in \\mathbb {N}} \\big (-x + \\lambda _n(a)^2 \\big ) u_n \\otimes n(a).$ This proves the first statement of the proposition.", "$\\bullet $ It remains to finish the proof of (ii).", "Let $z \\in \\mathfrak {S}_a$ and $w \\in {\\mathcal {D}}(H_a)$ .", "With (i) we see that $R\\big ((H_a-z)w\\big ) = w$ .", "Since we already know that $(H_a-z)R(u) = u$ for all $u \\in L^2(\\Omega )$ , this proves that $R$ is a bounded inverse for $(H_a-z)$ on $L^2(\\Omega )$ .", "The estimate on $(H_a-z)^{-1}$ follows from (REF ), and the proposition is proved.", "As a first application of this proposition, we can check that the operator $H_a$ has no eigenvalue, as is the case for $-x$ : Corollary 4.4 The operator $H_a$ has no eigenvalue.", "Let $z \\in and $ u D(Ha)$ be such that $ (Ha-z) u =0$.", "Then according to the first item of Proposition \\ref {prop-dec-res} we have$$\\sum _{n \\in \\mathbb {N}} \\left( - \\lambda _n(a)^2 - z \\right) u_n \\otimes n(a) = 0.$$This implies that for all $ n N$ we have$$\\left( - \\lambda _n(a)^2 - z \\right) u_n = 0$$in $ L2(Rd-1)$, and hence $ un = 0$ since the operator $ -x$ has no eigenvalue.", "Finally $ u = 0$, and the proposition is proved.$ However, the main point in Proposition REF is the second: Corollary 4.5 Theorem REF holds when $a > 0$ is constant.", "According to the last statement of Proposition REF and the fact that $\\mathop {\\rm {Im}}\\nolimits (\\lambda _n(a)^2) < 0$ for all $n \\in \\mathbb {N}$ , there exists $\\tilde{\\gamma }> 0$ such that for all $z \\in \\mathfrak {S}_a$ we have $\\mathop {\\rm {Im}}\\nolimits (z) \\leqslant - \\tilde{\\gamma }$ .", "With Proposition REF , the conclusion follows.", "Remark 4.6 To simplify the proof we have only considered the case where $a$ is equal to the same constant on both sides of the boundary.", "However we can similarly consider the case where $a$ is equal to some positive constant $a_0$ on $\\mathbb {R}^{d-1} \\times \\left\\lbrace 0 \\right\\rbrace $ and to another positive constant $a_l$ on $\\mathbb {R}^{d-1} \\times \\left\\lbrace l \\right\\rbrace $ .", "One of these two constants may even be zero.", "We refer to Section for this kind of computations.", "Non-constant absorption index In this section we prove Theorems REF and REF for a non-constant absorption index $a$ .", "For $b \\in W^{1,\\infty }(\\partial \\Omega )$ we denote by $\\Theta _b \\in {\\mathcal {L}}(H^1(\\Omega ) , H^{-1}(\\Omega ))$ the operator such that for all $\\psi \\in H^1(\\Omega )$ $\\left< \\Theta _b \\psi \\right>_{H^{-1}(\\Omega ),H^1(\\Omega )} = \\int _{\\partial \\Omega } b\\psi .$ We denote by $\\theta _b$ the corresponding quadratic form on $H^1(\\Omega )$ .", "We also denote by $\\tilde{H}_{a}$ the operator in ${\\mathcal {L}}\\big (H^1(\\Omega ),H^{-1}(\\Omega ) \\big )$ such that $\\left< \\tilde{H}_{a} \\psi \\right>_{H^{-1},H^1} = q_a(\\psi )$ for all $\\psi \\in H^1(\\Omega )$ .", "Let $z \\in +$ .", "According to the Lax-Milgram Theorem, $(1+i)(\\tilde{H}_{a}-z)$ is an isomorphism from $H^1(\\Omega )$ to $H^{-1}(\\Omega )$ .", "Moreover, for $f \\in L^2(\\Omega ) \\subset H^{-1}(\\Omega )$ we have $(\\tilde{H}_{a}-z)^{-1}f = (H_a-z)^{-1}f.$ The following proposition relies on a suitable version of the so-called quadratic estimates: Proposition 5.1 Let $a_0 > 0$ be as in the statement of Theorem REF .", "Assume that (REF ) holds everywhere on $\\partial \\Omega $ .", "Let $B \\in {\\mathcal {L}}\\big (H^1(\\Omega ),L^2(\\Omega ) \\big )$ .", "Then there exists $C \\geqslant 0$ such that for all $z \\in +$ we have $\\left\\Vert B (\\tilde{H}_{a}-z)^{-1}B^\\right\\Vert _{{\\mathcal {L}}(L^2(\\Omega ))} \\leqslant C \\left\\Vert B (\\tilde{H}_{a_0}-z)^{-1}B^\\right\\Vert _{{\\mathcal {L}}(L^2(\\Omega ))}.$ For $z \\in +$ the resolvent identity applied to $\\tilde{H}_{a}= \\tilde{H}_{a_0}+ \\Theta _{a-a_0}$ gives $ B(\\tilde{H}_{a}-z)^{-1}B^* = B (\\tilde{H}_{a_0}-z)^{-1}B^* - B (\\tilde{H}_{a_0}-z)^{-1}\\Theta _{a-a_0} (\\tilde{H}_{a}-z)^{-1}B^.$ Let $\\psi \\in L^2(\\Omega )$ .", "Since $\\Theta _{a-a_0}$ is associated to a non-negative quadratic form on $H^1(\\Omega )$ , the Cauchy-Schwarz inequality gives ${\\left< B(\\tilde{H}_{a_0}-z)^{-1}\\Theta _{a-a_0} (\\tilde{H}_{a}-z)^{-1}B^ \\psi \\right>_{L^2}}\\\\&& = \\left< \\Theta _{a-a_0} (\\tilde{H}_{a}-z)^{-1}B^* (\\tilde{H}_{a_0}^* - \\overline{z})^{-1}B^* \\psi \\right> _{H^{-1},H^1}\\\\&& \\leqslant \\theta _{a-a_0} \\big ((\\tilde{H}_{a}-z)^{-1}B^* )^{\\frac{1}{2}} \\times \\theta _{a-a_0} \\big ((\\tilde{H}_{a_0}^* -\\overline{z})^{-1}B^* \\psi \\big ) ^{\\frac{1}{2}}.$ The first factor is estimated as follows: ${\\theta _{a-a_0} \\big ((\\tilde{H}_{a}-z)^{-1}B^* )\\leqslant \\theta _{a} \\big ((\\tilde{H}_{a}-z)^{-1}B^* )}\\\\&& \\leqslant \\frac{1}{2i} \\left< 2i (\\Theta _{a} + \\mathop {\\rm {Im}}\\nolimits z) (\\tilde{H}_{a}-z)^{-1}B^* (\\tilde{H}_{a}- z)^{-1}B^* > _{H^{-1},H^1}\\\\&& \\leqslant \\frac{1}{2i} \\left< B \\big ((\\tilde{H}_{a}-z)^{-1}-(\\tilde{H}_{a}^* - \\overline{z})^{-1}\\big ) B^* > _{L^2}\\\\&& \\leqslant \\left\\Vert B(\\tilde{H}_{a}-z)^{-1}B^\\right\\Vert _{{\\mathcal {L}}(L^2)} \\left\\Vert \\Vert _{L^2}^2.\\right.We can proceed similarly for the other factor, using the fact that \\right.\\right.a-a_0 \\leqslant \\alpha a_0 for some \\alpha \\geqslant 0.", "Thus there exists C \\geqslant 0 such that{\\begin{@align}{1}{-1}\\left\\Vert B (\\tilde{H}_{a_0}-z)^{-1}\\Theta _{a-a_0} (\\tilde{H}_{a}-z)^{-1}B^* \\right\\Vert & \\leqslant C \\left\\Vert B (\\tilde{H}_{a_0}-z)^{-1}B^\\right\\Vert ^{\\frac{1}{2}} \\left\\Vert B (\\tilde{H}_{a}-z)^{-1}B^\\right\\Vert ^{\\frac{1}{2}}\\\\& \\leqslant \\frac{C^2}{2} \\left\\Vert B (\\tilde{H}_{a_0}-z)^{-1}B^\\right\\Vert + \\frac{1}{2} \\left\\Vert B (\\tilde{H}_{a}-z)^{-1}B^\\right\\Vert .\\end{@align}}With (\\ref {eq-res-identity}), the conclusion follows.$ Now we can finish the proof of Theorem REF : According to Corollary REF and Proposition REF applied with $B = \\operatorname{Id}_{L^2(\\Omega )}$ (note that we can simply replace $\\tilde{H}_{a}$ by $H_a$ when $B \\in {\\mathcal {L}}(L^2(\\Omega ))$ ), there exists $C > 0$ such that $\\left\\Vert (H_a-z)^{-1}\\right\\Vert _{{\\mathcal {L}}(L^2(\\Omega ))} \\leqslant C$ for all $z \\in +$ .", "Then it only remains to choose $\\tilde{\\gamma }\\in \\big ] 0, \\frac{1}{C} \\big [$ to conclude.", "The second statement, concerning the case where $a$ vanishes on one side of the boundary, is proved similarly.", "Let us now turn to the proof of Theorem REF .", "We first prove another resolvent estimate in which we see the smoothing effect in weighted spaces: Proposition 5.2 Let $\\delta > \\frac{1}{2}$ .", "Then there exists $C \\geqslant 0$ such that for all $z \\in \\mathbb {C}_+$ we have $\\left\\Vert \\left< x \\right> ^{-\\delta } (1- \\Delta _x)^{\\frac{1}{4}} (H_a-z)^{-1} (1- \\Delta _x)^{\\frac{1}{4}} \\left< x \\right> ^{-\\delta }\\right\\Vert _{\\mathcal {L} (L^2(\\Omega ))} \\leqslant C.$ It is known for the free laplacian that for $\\left\| \\mathop {\\rm {Re}} (\\zeta ) \\right\| \\gg 1$ and $\\mathop {\\rm {Im}}(\\zeta ) > 0$ we have $\\left\\Vert \\left< x \\right>^{-\\delta } (-\\Delta _x -\\zeta )^{-1} \\left< x \\right>^{-\\delta }\\right\\Vert _{\\mathcal {L}(L^2(\\mathbb {R}^{d-1}))} \\lesssim \\left< \\zeta \\right> ^{-\\frac{1}{2}}$ and hence $\\left\\Vert \\left< x \\right>^{-\\delta } (1- x)^{\\frac{1}{4}}(-\\Delta _x -\\zeta )^{-1} (1- \\Delta _x)^{\\frac{1}{4}}\\left< x \\right>^{-\\delta } \\right\\Vert _{\\mathcal {L} (L^2(\\mathbb {R}^{d-1}))} \\lesssim 1.$ Thus if $a$ is constant we have $\\left\\Vert \\left< x \\right>^{-\\delta } (1- x)^{\\frac{1}{4}}(-\\Delta _x + \\lambda _n(a)^2 -z)^{-1} (1- \\Delta _x)^{\\frac{1}{4}}\\left< x \\right>^{-\\delta }\\right\\Vert _{\\mathcal {L}(L^2(\\mathbb {R}^{d-1})) } \\lesssim 1,$ uniformly in $z \\in +$ and $n \\in \\mathbb {N}$ .", "In this case we obtain the result using the separation of variables as in Section .", "Then we conclude with Proposition REF applied with $B = \\left< x \\right> ^{-\\delta } (1-\\Delta _x)^{\\frac{1}{4}}$ .", "In fact, we first obtain an estimate on the resolvent $(\\tilde{H}_a -z)^{-1}$ , but this proves that the operator $\\left< x \\right> ^{-\\delta } (1- \\Delta _x)^{\\frac{1}{4}} (H_a-z)^{-1} (1- \\Delta _x)^{\\frac{1}{4}} \\left< x \\right> ^{-\\delta }$ extends to a bounded operator on $L^2(\\Omega )$ , and then the same estimate holds for the corresponding closure.", "With the second estimate of Proposition REF , we can apply the theory of relatively smooth operators (see §XIII.7 in [37]).", "However, since $H_a$ is not self-adjoint but only maximal dissipative, we have to use a self-adjoint dilation (see [31]) of $H_a$ , as is done in the proof of [36] (see also Proposition 2.24 in [35]).", "Time decay for the Schrödinger equation In this section we prove Theorem REF .", "Let $u_0 \\in {\\mathcal {D}}(H_a)$ and let $u$ be the solution of the problem (REF ).", "We know that $\\left\\Vert u(t)\\right\\Vert _{L^2(\\Omega )} \\leqslant \\left\\Vert u_0\\right\\Vert _{L^2(\\Omega )}$ for all $t \\geqslant 0$ , so the result only concerns large times.", "Let $\\tilde{\\gamma }> 0$ be given by Theorem REF and $\\gamma = \\tilde{\\gamma }/ 3$ .", "Let $C^\\infty (\\mathbb {R})$ be equal to 0 on $]-\\infty ,1]$ and equal to 1 on $[2,+\\infty [$ .", "For $t \\in \\mathbb {R}$ we set $u_t) = t) u(t),$ and for $z \\in +$ : $v(z) = \\int _\\mathbb {R}e^{itz} u_t) \\, dt.$ The map $t \\mapsto e^{-\\gamma t}u_t)$ belongs to $L^1(\\mathbb {R}) \\cap L^2(\\mathbb {R}) \\cap C^1(\\mathbb {R})$ and its derivative is in $L^1(\\mathbb {R})$ so $\\mapsto v(+i\\gamma )$ is bounded and decays at least like $\\left< \\right>^{-1}$ .", "In particular it is in $L^2(\\mathbb {R})$ .", "For $R > 0$ we set $u_R(t) = \\frac{1}{2\\pi } \\int _{-R}^R e^{-it(+i\\gamma )} v(+i\\gamma ) \\, d.$ Then $\\left\\Vert e^{-t\\gamma } (u_ u_R)\\right\\Vert _{L^2(\\mathbb {R}_t,L^2(\\Omega ))} \\xrightarrow[R \\rightarrow +\\infty ]{} 0.$ Since $u_ is continuous, Theorem \\ref {th-energy-decay} will be proved if we can show that there exists $ C 0$ which does not depend on $ u0$ and such that for all $ t 0$ we have\\begin{equation} \\limsup _{R \\rightarrow \\infty } \\left\\Vert u_R(t)\\right\\Vert _{L^2(\\Omega )} \\leqslant C e^{-\\gamma t} \\left\\Vert u_0\\right\\Vert _{L^2(\\Omega )}.\\end{equation}For $ z we set $\\theta (z) = -i \\int _\\mathbb {R}e^{itz} (t) u(t) \\, dt = -i \\int _1^2 e^{itz} (t) u(t) \\, dt.$ Let $z \\in +$ .", "We multiply (REF ) by $t) e^{itz}$ and integrate over $t \\in \\mathbb {R}$ .", "After partial integration we obtain $ v(z) = (H_a-z)^{-1}\\theta (z).$ Then $v$ extends to a holomorphic function on ${3\\gamma }$ , taking this equality as a definition.", "According to the Cauchy Theorem we have in $L^2(\\mathbb {R}_t)$ $ \\begin{aligned}\\lim _{R \\rightarrow \\infty } u_R(t)= \\frac{1}{2\\pi } \\lim _{R \\rightarrow \\infty } \\int _{-R}^R e^{-it(-2i\\gamma )} v(-2i\\gamma )\\,d= {e^{-2\\gamma t}} \\lim _{R \\rightarrow \\infty } \\widetilde{u_R}(t),\\end{aligned}$ where for $t \\in \\mathbb {R}$ we have set $\\widetilde{u_R}(t) = \\int _{-R}^R e^{-it}v(-2i\\gamma ) \\, d.$ According to Plancherel's equality and Theorem REF we have uniformly in $R >0$ : $\\int _\\mathbb {R}\\left\\Vert \\widetilde{u_R} (t)\\right\\Vert ^2_{L^2(\\Omega )} \\, dt& = \\int _{-R}^R \\left\\Vert \\big (H_a-(-2i\\gamma )\\big )^{-1}\\theta (-2i\\gamma )\\right\\Vert _{L^2(\\Omega )}^2 \\, d\\\\& \\lesssim \\int _\\mathbb {R}\\left\\Vert \\theta (-2i\\gamma )\\right\\Vert _{L^2(\\Omega )}^2 \\, d\\\\& \\lesssim \\int _\\mathbb {R}e^{2\\gamma t}\\left\|(t)\\right\| \\left\\Vert u(t)\\right\\Vert _{L^2(\\Omega )}^2 \\, dt \\\\& \\lesssim \\left\\Vert u_0\\right\\Vert ^2_{L^2(\\Omega )}.$ In particular there exists $C \\geqslant 0$ such that for $u_0 \\in {\\mathcal {D}}(H_a)$ and $R > 0$ we can find $T(u_0,R) \\in [0,1]$ which satisfies $\\left\\Vert \\widetilde{u_R} (T(u_0,R))\\right\\Vert _{L^2(\\Omega )} \\leqslant C \\left\\Vert u_0\\right\\Vert _{L^2(\\Omega )}.$ Let $R > 0$ .", "Then $\\widetilde{u_R} \\in C^1(\\mathbb {R})$ and for $t \\geqslant 1$ we have $\\widetilde{u_R} (t) = e^{-i(t-T(u_0,R))H_a} \\widetilde{u_R} (T(u_0,R)) + \\int _{T(u_0,R)} ^t \\frac{\\partial }{\\partial s} \\left( e^{-i(t-s)H_a} \\widetilde{u_R}(s) \\right) \\,ds,$ where $\\frac{\\partial }{\\partial s} \\left( e^{-i(t-s)H_a} \\widetilde{u_R}(s) \\right)& = \\frac{\\partial }{\\partial s}\\int _{-R}^R e^{-is} e^{-i(t-s)H_a} \\big (H_a-(-2i\\gamma )\\big )^{-1}\\theta (-2i\\gamma ) \\, d\\\\& = i \\int _{-R}^R e^{-i(t-s)H_a} e^{-is} (H_a-) \\big ( H_a- (-2i\\gamma ) \\big ) ^{-1}\\theta (-2i\\gamma ) \\, d\\\\& = 2\\gamma e^{-i(t-s)H_a} \\widetilde{u_R} (s) + i e^{-i(t-s)H_a} \\int _{-R}^R e^{-is} \\theta (-2i\\gamma ) \\, d.$ This proves that the map $s \\mapsto \\frac{\\partial }{\\partial s} \\left( e^{-i(t-s)H_a} \\widetilde{u_R}(s) \\right)$ belongs to $L^2([0,t],L^2(\\Omega ))$ uniformly in $t$ and $R>0$ , and its $L^2([0,t],L^2(\\Omega ))$ norm is controlled by the norm of $u_0$ in $L^2(\\Omega )$ .", "We finally obtain $C \\geqslant 0$ such that for all $t \\in \\mathbb {R}$ and $R > 0$ we have $\\left\\Vert \\widetilde{u_R} (t)\\right\\Vert _{L^2(\\Omega )} \\leqslant C \\left< t \\right>^{\\frac{1}{2}} \\left\\Vert u_0\\right\\Vert _{L^2(\\Omega )}.$ With (REF ) this proves () and concludes the proof of Theorem REF .", "The case of a weakly dissipative boundary condition In this section we prove Theorem REF about the problem (REF ).", "The absorption index $a$ now takes the value $a_l$ on $\\mathbb {R}^{d-1} \\times \\left\\lbrace l \\right\\rbrace $ and $a_0$ on $\\mathbb {R}^{d-1} \\times \\left\\lbrace 0 \\right\\rbrace $ .", "The proof follows the same lines as in the dissipative case, except that well-posedness of the problem is not an easy consequence of the general dissipative theory.", "We will use the separation of variables as in Section instead.", "Once we have a decomposition as in Proposition REF for the initial datum, we can propagate each term by means of the unitary group generated by $-x$ and define the solution of (REF ) as a series of solutions on $\\mathbb {R}^{d-1}$ .", "Let us first look at the transverse problem.", "The transverse operator on $L^2(0,l)$ corresponding to the problem (REF ) is now given by $T_{a_l,a_0}= - \\frac{d^2}{dy^2}$ with domain ${\\mathcal {D}}(T_{a_l,a_0}) = \\left\\lbrace u \\in H^2(0,l) \\,:\\,u^{\\prime }(0) = -ia_0u(0), u^{\\prime }(l) = ia_lu(l) \\right\\rbrace .$ As already mentioned in Remark REF , we can reproduce exactly the same analysis as in Section if $a_l> 0$ and $a_0>0$ (or if one of them vanishes).", "In particular, there is no restriction on the sizes of these coefficients.", "The results we give here to handle the weakly dissipative case are also valid in this situation.", "The strategy will be the same as in Section , so we will only emphasize the differences.", "We first remark that 0 is an eigenvalue of $T_{a_l,a_0}$ if and only if $a_l=a_0=0$ .", "Otherwise, $\\lambda ^2 \\in $ is an eigenvalue of $T_{a_l,a_0}$ if and only if $ (\\lambda - a_l)(\\lambda - a_0) e^{2i\\lambda l} = (\\lambda + a_l) (\\lambda + a_0).$ We recover (REF ) when $a_l= a_0$ .", "Lemma 7.1 Let $a_l,a_0\\in \\mathbb {R}$ and $\\lambda \\in $ be such that $a_l+a_0\\ne 0$ and $\\lambda ^2$ is an eigenvalue of $T_{a_l,a_0}$ .", "Then $\\mathop {\\rm {Re}}\\nolimits (\\lambda ) \\notin \\nu \\mathbb {N}$ .", "We recall from [23] that if $a_l+a_0= 0$ (${\\mathcal {P}}{\\mathcal {T}}$ -symmectric case) then $n^2 \\nu ^2 \\in \\sigma (T_{a_l,a_0})$ for all $n \\in \\mathbb {N}^$ (see also Figure REF for $a_l+a_0> 0$ small).", "$\\bullet $ We assume by contradiction that $\\mathop {\\rm {Re}}\\nolimits (\\lambda ) \\in \\nu \\mathbb {N}$ .", "According to (REF ) we have $\\frac{(\\lambda + a_l)(\\lambda +a_0)}{(\\lambda -a_l)(\\lambda -a_0)} = e^{2il\\lambda } = e^{-2 l\\mathop {\\rm {Im}}\\nolimits (\\lambda )} \\in \\mathbb {R}_+^.$ After multiplication by $\\left\|\\lambda - a_l\\right\|^2 \\left\|\\lambda - a_0\\right\|^2 \\in \\mathbb {R}_+^$ we obtain $\\big ( \\left\|\\lambda \\right\|^2 - 2 i a_l\\mathop {\\rm {Im}}\\nolimits (\\lambda ) - a_l^2 \\big ) \\big ( \\left\|\\lambda \\right\|^2 - 2 i a_0\\mathop {\\rm {Im}}\\nolimits (\\lambda ) - a_0^2 \\big ) \\in \\mathbb {R}_+^.$ Taking the real and imaginary parts gives $ \\left\|\\lambda \\right\|^4 - (a_l^2 + a_0^2) \\left\|\\lambda \\right\|^2 + a_l^2 a_0^2 - 4 a_la_0\\mathop {\\rm {Im}}\\nolimits (\\lambda )^2 > 0$ and $ 2 \\mathop {\\rm {Im}}\\nolimits (\\lambda ) (a_l+a_0) \\big (\\left\|\\lambda \\right\|^2 - a_la_0\\big ) = 0.$ $\\bullet $ Assume that $a_la_0\\geqslant 0$ .", "In this case $\\mathop {\\rm {Im}}\\nolimits (\\lambda ) \\ne 0$ (for the same reason as in the proof of Proposition REF ), so (REF ) implies $\\left\|\\lambda \\right\|^2 = a_la_0$ .", "Then (REF ) reads $- a_la_0(a_l- a_0)^2 - 4a_la_0\\mathop {\\rm {Im}}\\nolimits (\\lambda )^2 > 0,$ which gives a contradiction.", "$\\bullet $ Now assume that $a_la_0< 0$ .", "Then (REF ) implies $\\mathop {\\rm {Im}}\\nolimits (\\lambda ) = 0$ and hence $e^{2il\\lambda } = 1$ .", "From (REF ) we now obtain $(\\lambda -a_l) (\\lambda -a_0) = (\\lambda +a_l) (\\lambda +a_0),$ which is impossible since $\\lambda (a_l+ a_0) \\ne 0$ .", "This concludes the proof.", "Proposition 7.2 There exists $\\rho > 0$ such that if $\\left\|a_l\\right\| + \\left\|a_0\\right\| \\leqslant \\rho $ and $a_l+ a_0> 0$ then the spectrum of $T_{a_l,a_0}$ is given by a sequence $(\\lambda _n(a_l,a_0)^2)_{n\\in \\mathbb {N}}$ of algebraically simple eigenvalues such that $\\sup _{n\\in \\mathbb {N}} \\, \\mathop {\\rm {Im}}\\nolimits \\left(\\lambda _n(a_l,a_0)^2\\right) < 0.$ Moreover, any sequence of normalized eigenfunctions corresponding to these eigenvalues forms a Riesz basis.", "Figure: λ n (a l ,a 0 )\\lambda _n(a_l,a_0) for n∈{0,⋯,30}n \\in \\lbrace 0,\\dots ,30\\rbrace and l=πl= \\pi .As in the proof of Proposition REF we can see that for any $R > 0$ there exists $C_R \\geqslant 0$ such that if $\\lambda ^2 \\in $ is an eigenvalue of $T_{a_l,a_0}$ we have $ \\left\|\\mathop {\\rm {Re}}\\nolimits \\lambda \\right\| \\leqslant R \\quad \\Rightarrow \\quad \\left\|\\mathop {\\rm {Im}}\\nolimits \\lambda \\right\| \\leqslant C_R.$ The operator $T_{a_l,a_0}$ depends analytically on the parameters $a_l$ and $a_0$ , and we know that when $a_l= a_0= 0$ the eigenvalues $n^2\\nu ^2$ for $n \\in \\mathbb {N}$ are algebraically simple.", "With the restrictions given by (REF ) and Lemma REF , we obtain as in Section a sequence of maps $(a_l,a_0) \\mapsto \\lambda _n(a_l,a_0)$ such that the eigenvalues of $T_{a_l,a_0}$ are $\\lambda _n(a_l,a_0)^2$ for $n \\in \\mathbb {N}$ .", "Let $n \\in \\mathbb {N}^$ .", "We have $ \\lambda _n(a_l,a_0)= n\\nu - \\frac{i}{n\\pi } (a_l+ a_0) + \\gamma (a_l+a_0)^2 + O \\left( \\left\|a_l\\right\|^3 , \\left\|a_0\\right\|^3 \\right),$ with $\\mathop {\\rm {Re}}\\nolimits (\\gamma ) = l/(n\\pi )^3 > 0$ .", "As in the dissipative case, we obtain that for any $a_l,a_0$ with $a_l+ a_0> 0$ these eigenvalues $\\lambda _n(a_l,a_0)^2$ are simple.", "If moreover $a_l$ and $a_0$ are small enough, the eigenvalue $\\lambda _n(a_l,a_0)^2$ is close to $(n\\nu )^2$ and away from the real axis uniformly in $n \\in \\mathbb {N}^$ (the first two terms in (REF ) are also the first two terms of the asymptotic expansion for large $n$ and fixed $a_l$ and $a_0$ ).", "It remains to check that we also have $\\mathop {\\rm {Im}}\\nolimits (\\lambda _0(a_l,a_0)^2) < 0$ .", "For small $a_l,a_0$ we denote by ${a_l,a_0}(0)$ a normalized eigenvector corresponding to the eigenvalue $\\lambda _0(a_l,a_0)^2$ and depending analytically on $a_l$ and $a_0$ .", "For all $\\psi \\in H^1(0,l)$ we have $\\left< {a_l,a_0}^{\\prime } , \\psi ^{\\prime } \\right>_{L^2(0,l)} - i a_l{{a_l,a_0}(l)} \\overline{\\psi (l)} - i a_0{{a_l,a_0}(0)} \\overline{\\psi (0)} = \\lambda _0(a_l,a_0)^2 \\left< {a_l,a_0} , \\psi \\right>_{L^2(0,l)}$ We apply this with $\\psi = {a_l,a_0}$ , take the derivatives with respect to $a_l$ and $a_0$ at point $(a_l,a_0) = (0,0)$ , and use the facts that ${0,0}$ is constant and $\\lambda _{0}(0,0) = 0$ .", "We obtain $\\nabla _{a_l,a_0} \\big (\\lambda _0^2 \\big ) = -\\frac{i}{l}\\big ( 1 , 1 \\big ).$ This proves that $\\mathop {\\rm {Im}}\\nolimits \\big (\\lambda _0(a_l,a_0)^2\\big )<0$ if $a_l$ and $a_0$ are small enough with $a_l+ a_0> 0$ .", "The Riesz basis property relies as before on the fact that $\\left\|\\lambda _n(a_l,a_0)- n\\nu \\right\| = O(n^{-1}).$ For this point we can follow what is done in Section for the dissipative case.", "For $n \\in \\mathbb {N}$ and $a_l,a_0\\in \\mathbb {R}$ with $a_l+ a_0> 0$ we consider a normalized eigenvector $n({a_l,a_0})\\in L^2(0,l)$ corresponding to the eigenvalue $\\lambda _n(a_l,a_0)^2$ of $T_{a_l,a_0}$ .", "We denote by $(n^(a_l,a_0))_{n\\in \\mathbb {N}}$ the dual basis.", "Proposition 7.3 Let $u_0 \\in {\\mathcal {D}}(H_{a_l,a_0})$ .", "Then the problem (REF ) has a unique solution $u \\in C^1 (\\mathbb {R},L^2(\\Omega )) \\cap C^0(\\mathbb {R},{\\mathcal {D}}(H_{a_l,a_0}))$ .", "Moreover if we write $u_0 = \\sum _{n\\in \\mathbb {N}} u_{0,n} \\otimes n({a_l,a_0})$ where $u_{0,n} \\in L^2(\\mathbb {R}^{d-1})$ , then $u$ is given by $u(t) = \\sum _{n\\in \\mathbb {N}} \\left( e^{-it \\left(-x+\\lambda _n(a_l,a_0)^2 \\right)} u_{0,n}\\right) \\otimes n({a_l,a_0}).$ $\\bullet $ Assume that $u \\in C^0(\\mathbb {R}_+, {\\mathcal {D}}(H_{a_l,a_0})) \\cap C^1(\\mathbb {R}_+^,L^2(\\Omega ))$ is a solution of (REF ).", "Let $t \\in \\mathbb {R}_+^$ .", "For all $n \\in \\mathbb {N}$ and almost all $x \\in \\mathbb {R}^{d-1}$ we can define $u_n(t,x) = \\left< u(t;x,\\cdot ) , _n(a_l,a_0) \\right>_{L^2(0,l)},$ so that in $L^2(\\Omega )$ we have $u(t) = \\sum _{n\\in \\mathbb {N}} u_n(t) \\otimes n({a_l,a_0}).$ According to Proposition REF (which can be proved similarly in this context) we have $u_n(t) \\in H^2(\\mathbb {R}^{d-1})$ for all $t \\in \\mathbb {R}_+^$ and $n \\in \\mathbb {N}$ , and for $s \\in \\mathbb {R}^$ we have $i\\frac{u(t+s)-u(t)}{s} - H_{a_l,a_0}u(t) \\\\ = \\sum _{n\\in \\mathbb {N}} \\left( i\\frac{u_n(t+s)-u_n(t)}{s} - \\left(-x +\\lambda _n(a_l,a_0)^2 \\right) u_n(t) \\right) \\otimes n({a_l,a_0}).$ Let $n\\in \\mathbb {N}$ .", "According to Proposition REF we have $\\left\\Vert i\\frac{u_n(t+s)-u_n(t)}{s} - \\left(-x +\\lambda _n(a_l,a_0)^2 \\right) u_n(t)\\right\\Vert _{L^2(\\mathbb {R}^{d-1})}\\\\\\lesssim \\left\\Vert i\\frac{u(t+s)-u(t)}{s} - H_{a_l,a_0}u(t)\\right\\Vert _{L^2(\\Omega )}\\xrightarrow[s \\rightarrow 0]{} 0.$ This proves that $u_n$ is differentiable and for all $t > 0$ $i u_n^{\\prime }(t) = \\left( - x + \\lambda _n(a_l,a_0)^2 \\right) u_n(t).$ Then for all $t > 0$ $u_n(t) = e^{-it\\left( -x + \\lambda _n(a_l,a_0)^2 \\right)} u_n(0) = e^{-it\\left( -x + \\lambda _n(a_l,a_0)^2 \\right)} u_{0,n}.$ $\\bullet $ Conversely, let us prove that the function $u$ defined by the statement of the proposition is indeed a solution of (REF ).", "Let $t \\in \\mathbb {R}$ .", "According to Proposition REF , $u_{0,n}$ and hence $e^{-it \\left(-x+\\lambda _n(a_l,a_0)^2 \\right)} u_{0,n}$ belong to $H^2(\\mathbb {R}^{d-1})$ for all $n \\in \\mathbb {N}$ .", "Therefore $u(t) \\in {\\mathcal {D}}(H_{a_l,a_0})$ .", "Then for all $s \\in \\mathbb {R}^$ we have ${ \\sum _{n\\in \\mathbb {N}}\\left\\Vert \\left( i\\frac{e^{-is \\left(-x + \\lambda _n(a_l,a_0)^2 \\right)} - 1}{s} -\\left(- x + \\lambda _n(a_l,a_0)^2\\right) \\right) \\, e^{-it(-x + \\lambda _n(a_l,a_0)^2)} u_{0,n}\\right\\Vert ^2 }\\\\&& \\lesssim _t \\sum _{n\\in \\mathbb {N}}\\left\\Vert \\frac{1}{s} \\int _0^s \\big ( e^{-i\\theta (-x + \\lambda _n(a_l,a_0)^2)} - 1 \\big ) \\left(- x + \\lambda _n(a_l,a_0)^2\\right) u_{0,n} \\, d\\theta \\right\\Vert ^2_{L^2(\\mathbb {R}^{d-1})}\\\\&& \\lesssim _t \\sum _{n\\in \\mathbb {N}}\\left\\Vert \\left(- x + \\lambda _n(a_l,a_0)^2\\right) u_{0,n} \\right\\Vert ^2_{L^2(\\mathbb {R}^{d-1})}\\\\&& \\lesssim _t \\left\\Vert H_{a_l,a_0}u_0\\right\\Vert _{L^2(\\Omega )}^2$ This series of functions converges uniformly in $s$ so we can take the limit, which proves that for any $t \\in \\mathbb {R}$ $\\left\\Vert i\\frac{u(t+s)-u(t)}{s} -H_{a_l,a_0}u(t)\\right\\Vert ^2 \\xrightarrow[s \\rightarrow 0]{} 0.$ This proves that $u$ is differentiable and $i u^{\\prime }(t) + H_{a_l,a_0}u(t) = 0$ , so $u$ is indeed a solution of (REF ).", "Now we can prove Theorem REF : According to Proposition REF we have existence and uniqueness for the solution $u$ of the problem REF .", "Then with Proposition REF we have $\\left\\Vert u(t)\\right\\Vert _{L^2(\\Omega )}^2& \\lesssim \\sum _{n \\in \\mathbb {N}} \\left\\Vert e^{-it \\left(-x + \\lambda _n(a_l,a_0)^2 \\right)} u_{0,n}\\right\\Vert _{L^2(\\mathbb {R}^{d-1})}^2 \\\\& \\lesssim \\sum _{n \\in \\mathbb {N}} e^{t \\mathop {\\rm {Im}}\\nolimits \\left(\\lambda _n(a_l,a_0)^2\\right)}\\left\\Vert u_{0,n}\\right\\Vert _{L^2(\\mathbb {R}^{d-1})}^2 .$ Proposition REF gives $\\gamma _{a_l,a_0} > 0$ such that $\\left\\Vert u(t)\\right\\Vert _{L^2(\\Omega )}^2 \\lesssim e^{-t \\gamma _{a_l,a_0}} \\sum _{n \\in \\mathbb {N}} \\left\\Vert u_{0,n}\\right\\Vert _{L^2(\\mathbb {R}^{d-1})}^2 \\lesssim e^{-t \\gamma _{a_l,a_0}} \\left\\Vert u_0\\right\\Vert _{L^2(\\Omega )}^2,$ which concludes the proof.", "In the end of this section we show that the smallness assumption on $\\left\|a_l\\right\|+ \\left\|a_0\\right\|$ is necessary in Theorem REF .", "More precisely, if $\\left\|a_l\\right\|$ and $\\left\|a_0\\right\|$ are too large, then the transverse operator $T_{a_l,a_0}$ has eigenvalues with positive imaginary parts and hence the solution of the Schrödinger equation grows exponentially.", "Proposition 7.4 Let $a_l, a_0\\in \\mathbb {R}$ with $a_l+ a_0> 0$ and $a_la_0< 0$ .", "Let $n \\in \\mathbb {N}$ .", "If $s > 0$ is large enough, we have $\\mathop {\\rm {Im}}\\nolimits \\big (\\lambda _n(sa_l,sa_0)^2\\big ) > 0$ .", "We know that the curves $s \\mapsto \\lambda _n(sa_l,sa_0)$ for $n \\in \\mathbb {N}$ are defined for all $s \\in \\mathbb {R}$ and remain bounded.", "Moreover we have chosen the square root $\\lambda _n(sa_l,sa_0)$ of $\\lambda _n(sa_l,sa_0)^2$ which has a non-negative real part, so the imaginary parts of $\\lambda _n(sa_l,sa_0)$ and $\\lambda _n(sa_l,sa_0)^2$ have the same signs.", "Assume that $a_la_0< 0$ , and let $n \\in \\mathbb {N}^$ be fixed.", "We have $ \\frac{(\\lambda _n(sa_l,sa_0)+s a_l)(\\lambda _n(sa_l,sa_0)+sa_0)}{(\\lambda _n(sa_l,sa_0)-sa_l)(\\lambda _n(sa_l,sa_0)-sa_0)} = 1 + \\frac{2\\lambda _n(sa_l,sa_0)}{s} \\left( \\frac{1}{a_l} + \\frac{1}{a_0} \\right) + \\mathop {O}\\limits _{s \\rightarrow +\\infty } \\big ( s^{-2} \\big ).$ Since $\\mathop {\\rm {Re}}\\nolimits (\\lambda _n(sa_l,sa_0)) > n\\nu $ and $\\mathop {\\rm {Im}}\\nolimits (\\lambda _n(sa_l,sa_0))$ is bounded, this quantity is of norm less than 1 when $s > 0$ is large enough, so $e^{-2l\\mathop {\\rm {Im}}\\nolimits (\\lambda _n(sa_l,sa_0))} = \\left\|e^{2il\\lambda _n(sa_l,sa_0)}\\right\| < 1,$ and hence $\\mathop {\\rm {Im}}\\nolimits (\\lambda _n(sa_l,sa_0)) > 0$ .", "When $n = 0$ , the same holds if we can prove that $\\lambda _0(sa_l,sa_0)$ does not go to 0 for large $s$ .", "Indeed, in this case the only possibility to have $e^{2il\\lambda _0(sa_l,sa_0)} \\rightarrow 1$ is that $\\lambda _0(sa_l,sa_0)$ goes to $\\nu $ , and then $\\mathop {\\rm {Re}}\\nolimits (\\lambda _0(sa_l,sa_0))$ is bounded by below by a positive constant, and we can conclude as before.", "So assume by contradiction that $\\lambda _0(sa_l,sa_0)$ goes to 0 as $s$ goes to $+\\infty $ .", "Then we have $e^{2il\\lambda _0(sa_l,sa_0)} = 1 + 2il\\lambda _0(sa_l,sa_0)+ \\mathop {O}\\limits _{s \\rightarrow +\\infty } \\big ( \\left\|\\lambda _0(sa_l,sa_0))\\right\|^2 \\big ),$ which gives a contradiction with (REF ), where the rest $O(s^{-2})$ has to be replaced by $O\\big ( \\lambda _0^2 s^{-2}\\big )$ .", "This concludes the proof.", "Remark 7.5 We remark from (REF ) (see also Figure REF ) that given $a_l,a_0\\in \\mathbb {R}$ such that $a_l+a_0> 0$ we always have $\\mathop {\\rm {Im}}\\nolimits (\\lambda _n(a_l,a_0)) < 0$ if $n$ is large enough.", "By duality, this means that there is always an eigenvalue with positive imaginary part when $a_l+ a_0< 0$ , and hence the norm of the solution of (REF ) is always exponentially increasing in this case." ], [ "The Transverse Operator", "Let us come back to the case of a one-dimensional cross-section $\\omega = ]0,l[$ .", "Under the additional assumption that the absorption index $a$ is constant on $\\partial \\Omega $ the operator $H_a$ can be written as $ H_a= -x \\otimes \\operatorname{Id}_{L^2(0,l)} + \\operatorname{Id}_{L^2(\\mathbb {R}^{d-1})} \\otimes T_a,$ where $-x$ is as before the usual flat Laplacian on $\\mathbb {R}^{d-1}$ and $T_a$ is the transverse Laplacian on $]0,l[$ .", "More precisely, we consider on $L^2(0,l)$ the operator $T_a= - \\frac{d^2}{dy^2}$ with domain ${\\mathcal {D}}(T_a) = \\left\\lbrace u \\in H^2(0,l) \\,:\\,u^{\\prime }(0) = -ia u(0), u^{\\prime }(l) = ia u(l) \\right\\rbrace .$ This is the maximal accretive and dissipative operator corresponding to the form $q : u \\in H^1(0,l) \\mapsto \\int _{0}^{l} \\left\|u^{\\prime }(x)\\right\|^2 \\, dx - i a \\left\|u(l)\\right\|^2 - i a \\left\|u(0)\\right\|^2.$ In this section we give the spectral properties of $T_a$ which we need to study the full operator $H_a$ .", "This operator has compact resolvent, and hence its spectrum is given by a sequence of isolated eigenvalues.", "When $a = 0$ , which corresponds to the Neumann problem, we know that the eigenvalues of $T_0$ are the real numbers $n^2 \\nu ^2$ for $n \\in \\mathbb {N}$ , where we have set $\\nu = \\frac{\\pi }{ l}.$ These eigenvalues are algebraically simple.", "Proposition 3.1 There exists a sequence $\\left({\\lambda }_{n}\\right)_{n \\in \\mathbb {N}}$ of continuous functions on $\\mathbb {R}$ such that $\\lambda _n(0) = n \\nu $ and for all $a \\in \\mathbb {R}$ the set of eigenvalues of $T_a$ is $\\left\\lbrace \\lambda _n(a)^2, n\\in \\mathbb {N} \\right\\rbrace $ .", "Moreover: For $(n,a) \\in (\\mathbb {N}\\times \\mathbb {R}) \\setminus \\lbrace (0,0)\\rbrace $ the eigenvalue $\\lambda _n(a)^2$ is algebraically simple and a corresponding eigenvector is given by $ {n}(a) : x \\mapsto A_n(a) \\left( e^{i\\lambda _n(a) x} + \\frac{\\lambda _n(a) + a}{\\lambda _n(a) - a} e^{-i \\lambda _n(a) x} \\right) ,$ where we can choose $A_n(a) \\in \\mathbb {R}_+^$ in such a way that $\\left\\Vert n(a)\\right\\Vert _{L^2(0,l)} = 1$ (when $a = 0$ then 0 is a simple eigenvalue and corresponding eigenvectors are non-zero constant functions).", "For $n \\in \\mathbb {N}$ and $a \\in \\mathbb {R}$ we have $\\lambda _n(-a)= \\overline{\\lambda _n(a)}$ .", "Let $n\\in \\mathbb {N}$ .", "For all $a \\in \\mathbb {R}^$ we have $\\mathop {\\rm {Re}}\\nolimits (\\lambda _n(a)) \\in ] n\\nu ,(n+1)\\nu [$ (when $n=0$ , we have chosen the square root of $\\lambda _0^2(a)$ which has a positive real part).", "For all $n \\in \\mathbb {N}$ there exists $C_n > 0$ such that for $a > 0$ we have $-C_n < \\mathop {\\rm {Im}}\\nolimits (\\lambda _n(a)) < 0$ .", "Let $a > 0$ be fixed.", "We have $\\lambda _n(a) = n \\nu - \\frac{2ia}{n\\nu l} + \\mathop {O}\\limits _{n \\rightarrow +\\infty } \\big (n^{-2} \\big )$ and hence $\\lambda _n(a)^2 = (n\\nu )^2 - \\frac{4ia}{l} + \\mathop {O}\\limits _{n \\rightarrow +\\infty } \\big (n^{-1} \\big ).$ $\\bullet $ It is straightforward computations to check that 0 is an eigenvalue of $T_a$ if and only if $a = 0$ and, if $\\lambda \\in $ , $\\lambda ^2$ is an eigenvalue if and only if $ (a-\\lambda )^2 e^{2i\\lambda l} = ( a+\\lambda )^2.$ If $\\lambda ^2$ is an eigenvalue then the corresponding eigenfunction is of the form $ x \\mapsto A e^{i\\lambda x} + B e^{-i\\lambda x}$ with $ A = \\frac{\\lambda -a}{\\lambda +a} B = \\frac{\\lambda + a}{\\lambda -a} e^{-2i\\lambda l}B.$ Moreover, all these eigenvalues have geometric multiplicity 1.", "Indeed, given $n \\in \\mathbb {N}$ , the space of eigenvectors corresponding to the eigenvalue $\\lambda _n(a)^2$ is strictly included in the space of $H^2$ functions which are solutions of $-u^{\\prime \\prime } - \\lambda _n(a)^2 u = 0$ , and this space is of dimension 2.", "The fact that the eigenvalues of $H_{-a}$ are conjugated to the eigenvalues of $H_{a}$ is a consequence of Remark REF .", "$\\bullet $ Let $a > 0$ and $\\lambda \\in $ be such that $\\lambda ^2$ is an eigenvalue of $T_a$ .", "Assume that $\\mathop {\\rm {Re}}\\nolimits \\lambda \\in \\nu \\mathbb {N}$ .", "Then $\\left( \\frac{a+\\lambda }{a-\\lambda } \\right)^2 = e^{2i\\lambda l} \\in \\mathbb {R}_+$ (note that $\\lambda $ cannot be equal to $a$ in (REF )) and hence $r := \\frac{a+\\lambda }{a-\\lambda } \\in \\mathbb {R}.$ If $r = -1$ then $a = 0$ .", "Otherwise $\\lambda = \\frac{a(1-r)}{1+r} \\in \\mathbb {R}$ .", "In both cases we obtain a contradiction (see Proposition REF ), and hence $\\mathop {\\rm {Re}}\\nolimits \\lambda \\notin \\nu \\mathbb {N}$ .", "This proves that for $a > 0$ the operator $T_a$ has no eigenvalue with real part in $\\nu \\mathbb {N}$ .", "$\\bullet $ Now let $R > 0$ .", "We prove that if $C_R \\geqslant 0$ is chosen large enough and if $a \\in \\mathbb {R}$ and $\\lambda \\in $ are such that $\\lambda ^2$ is an eigenvalue of $T_a$ , then $ \\left\|\\mathop {\\rm {Re}}\\nolimits \\lambda \\right\| \\leqslant R \\quad \\Rightarrow \\quad \\left\|\\mathop {\\rm {Im}}\\nolimits \\lambda \\right\| \\leqslant C_R.$ Assume by contradiction that this is not the case.", "Then for all $m \\in \\mathbb {N}$ we can find $x_m \\in [-R,R]$ and $y_m \\in \\mathbb {R}$ with $\\left\|y_m\\right\| \\geqslant m$ such that $(x_m+ i y_m)^2$ is an eigenvalue of $T_{a_m}$ for some $a_m \\in \\mathbb {R}$ .", "We have $e^{-2y_m l} = \\left\|\\frac{a_m + x_m + iy_m}{a_m - x_m - iy_m}\\right\|^2 = \\frac{(a_m + x_m)^2 + y_m^2}{(a_m - x_m)^2 + y_m^2} \\xrightarrow[m \\rightarrow \\infty ]{}1,$ which gives a contradiction.", "$\\bullet $ The family of operators $T_a$ for $a \\in \\mathbb {R}$ is an analytic family of operators of type B in the sense of Kato [22].", "We already know that the spectrum of $T_0$ is given by $\\left\\lbrace (n\\nu )^2, n \\in \\mathbb {N} \\right\\rbrace $ .", "Then for all $n \\in \\mathbb {N}$ there exists an analytic function $\\lambda _n^2$ such that, at least for small $a$ , $\\lambda _n^2(a)$ is in the spectrum of $T_a$ (and then we define $\\lambda _n$ as the square root of $\\lambda _n^2$ with positive real part).", "See Theorem VII.1.7 in [22].", "$\\bullet $ Let $n \\in \\mathbb {N}^$ .", "We write $\\lambda _n(a) = n\\nu + \\beta a + \\gamma a^2 + O_{a \\rightarrow 0} (a^3)$ .", "We have $e^{2i \\lambda _n(a) l}= 1 + 2il\\beta a + 2il\\gamma a^2 - 2l^2 \\beta ^2 a^2 + O(a^3),$ and on the other hand: $\\left( \\frac{\\lambda _n(a) + a}{\\lambda _n(a) - a} \\right)^2= 1 + \\frac{4a}{n\\nu } - \\frac{4(\\beta -2) a^2}{n^2\\nu ^2} + O\\big (a^3\\big ).$ Since $\\lambda _n(a)$ solves (REF ) for any $a > 0$ we obtain $\\beta = \\frac{2}{iln\\nu } = -\\frac{2i}{\\pi n}$ and $\\mathop {\\rm {Re}}\\nolimits (\\gamma )= \\frac{4 l}{n^3\\pi ^3} .$ Since $\\mathop {\\rm {Re}}\\nolimits (\\beta ) = 0$ and $\\mathop {\\rm {Re}}\\nolimits (\\gamma ) > 0$ we have $\\mathop {\\rm {Re}}\\nolimits \\big (\\lambda _n(a)\\big ) \\in \\left] n\\nu , (n+1)\\nu \\right[$ for $a > 0$ small enough.", "The functions $a \\mapsto \\mathop {\\rm {Re}}\\nolimits \\big (\\lambda _n(a)\\big )$ are continuous and cannot reach $\\nu \\mathbb {N}$ unless $a=0$ , so this remains true for any $a > 0$ such that $\\lambda _n(a)$ is defined.", "Similarly $\\mathop {\\rm {Re}}\\nolimits \\big (\\lambda _0(a)\\big ) \\in ]0, \\nu [$ for all $a > 0$ .", "In particular the curves $a \\mapsto \\lambda _n(a)$ for $n \\in \\mathbb {N}$ never meet.", "Moreover we know from (REF ) that $\\lambda _n(a)$ remains in a bounded set of $, so the curves $ a n(a)$ are defined for all $ a R$ and for all $ a R$ the eigenvalues of $ Ta$ are exactly given by $ n(a)2$ for $ n N$.$ $\\bullet $ It remains to prove that the asymptotic expansion of $\\lambda _n(a)$ for $n$ fixed and $a$ small is also valid for $a$ fixed and $n$ large.", "Let $a>0$ be fixed.", "Derivating (REF ) and using the fact that $\\left\|\\lambda _n(s)\\right\| \\geqslant n \\nu $ for all $s \\in \\mathbb {R}$ we obtain that $\\sup _{s \\in [0,a]} \\left\|\\lambda ^{\\prime }_n(s)\\right\| = O \\big (n^{-1}\\big ).$ This means that $\\lambda _n(a) = n\\nu + O(n^{-1})$ .", "Then we obtain the asymptotic expansion of $\\lambda _n(a)$ for large $n$ as before, using again (REF ).", "This gives the last statement of the proposition and concludes the proof.", "Now that we have proved what we need concerning the spectrum of the operator $T_a$ , we study the corresponding sequence of eigenfunctions.", "In the self-adjoint case $a = 0$ , we know that the eigenfunctions $n(0)$ form an orthonormal basis.", "Of course this is no longer the case for the non-selfadjoint operator $T_a$ with $a \\ne 0$ .", "However we can prove that in this case we have a Riesz basis.", "We recall that the sequence $\\left({_{n}_{n \\in \\mathbb {N}} of vectors in the Hilbert space {\\mathcal {H}} is said to be a Riesz basis if there exists a bounded operator \\Theta \\in {\\mathcal {L}}({\\mathcal {H}}) with bounded inverse and an orthonormal basis \\left({e}_{n}\\right)_{n \\in \\mathbb {N}} of {\\mathcal {H}} such that n = \\Theta e_n for all n \\in \\mathbb {N}.In this case any f \\in {\\mathcal {H}} can be written as \\sum _{n\\in \\mathbb {N}} f_n n with \\left({f}_{n}\\right)_{n \\in \\mathbb {N}} \\in l^2(, and there exists C \\geqslant 1 such that for all f = \\sum _{n\\in \\mathbb {N}} f_n n\\in {\\mathcal {H}} we haveC^{-1}\\sum _{n\\in \\mathbb {N}} \\left\|f_n\\right\|^2 \\leqslant \\left\\Vert \\sum _{n\\in \\mathbb {N}} f_n n\\right\\Vert _{\\mathcal {H}}^2 \\leqslant C \\sum _{n\\in \\mathbb {N}} \\left\|f_n\\right\|^2.In these estimates we can take C = \\max \\left( \\left\\Vert \\Theta \\right\\Vert ^2 , \\left\\Vert \\Theta ^{-1}\\right\\Vert ^2 \\right).", "}Let \\right.$ (n)n N$ be a Riesz basis of $ H$ with $$ and $ (en)n N$ as above.", "If we set $ * n = (-1)* en$ for all $ n N$ then $ (n)nN$ is also a Riesz basis, called the dual basis of $ (n)n N$.", "In particular for all $ j,k N$ we have\\begin{equation} \\left< j , _k \\right>_{\\mathcal {H}}= \\delta _{j,k}.\\end{equation}We refer to \\cite {christensen} for more details about Riesz bases.", "Now we want to prove that for any $ a R$ the sequence of eigenfunctions for the operator $ Ta$ is a Riesz basis of $ L2(0,l)$.$ Proposition 3.2 For all $a \\in \\mathbb {R}$ the sequence $(n(a))_{n\\in \\mathbb {N}}$ defined in (REF ) is a Riesz basis of $L^2(0,l)$ .", "Moreover for all $R > 0$ there exists $C \\geqslant 0$ such that for $a \\in [-R,R]$ and $\\left({c}_{n}\\right)_{n \\in \\mathbb {N}} \\in l^2($ we have $C^{-1}\\sum _{n = 0}^{\\infty } \\left\|c_n\\right\|^2 \\leqslant \\left\\Vert \\sum _{n = 0}^{\\infty } c_n n(a)\\right\\Vert ^2 \\leqslant C \\sum _{n = 0}^{\\infty } \\left\|c_n\\right\|^2.$ If $C$ was chosen large enough and if $\\left({c}_{n}\\right)_{n \\in \\mathbb {N}} \\in l^2($ is such that $\\sum _{n=0}^\\infty \\left\|\\lambda _n(a) c_n\\right\|^2 < \\infty $ we also have $C^{-1}\\sum _{n = 0}^{\\infty } \\left\|\\lambda _n(a) c_n\\right\|^2 \\leqslant \\left\\Vert \\sum _{n = 0}^{\\infty } c_n _n(a)\\right\\Vert ^2 \\leqslant C \\sum _{n = 0}^{\\infty } \\left\|\\lambda _n(a) c_n\\right\|^2.$ For similar results we refer to [30] (see also Lemma XIX.3.10 in [18]).", "For the proof we need the following lemma: Lemma 3.3 Let $R > 0$ .", "Then there exists $C \\geqslant 0$ such that for $a \\in [-R,R]$ and $j,k \\in \\mathbb {N}$ with $j < k$ we have $\\left< j(a) , k(a) \\right>_{L^2(0,l)} \\leqslant \\frac{C}{\\left< j \\right> (k-j)} \\quad \\text{and} \\quad \\left< _j(a) , _k(a) \\right>_{L^2(0,l)} \\leqslant C \\frac{k}{k-j}.$ Let $e_n(a,x) = e^{i \\lambda _n (a) x}$ and $\\tilde{e}_n(a,x) = e^{-i \\lambda _n (a) x}$ .", "According to Proposition REF we have $\\lambda _n(a) = n\\nu + O(n^{-1})$ (here and below all the estimates are uniform in $a \\in [-R,R]$ ), so $\\begin{aligned}\\left\\Vert e_n(a)\\right\\Vert ^2_{L^2(0,l)}= \\frac{e^{-2l\\mathop {\\rm {Im}}\\nolimits (\\lambda _n(a))} - 1 }{-2 \\mathop {\\rm {Im}}\\nolimits (\\lambda _n(a))} = l+ O(n^{-1}).\\end{aligned}$ Similarly $\\left\\Vert \\tilde{e}_n(a)\\right\\Vert ^2_{L^2(0,l)} = l+ O(n^{-1})$ , and $\\left< e_n(a) , \\tilde{e}_n(a) \\right> =O(n^{-1})$ .", "Moreover, with (REF ) we see that $A_n(a) = \\frac{1}{\\sqrt{2l}} + O(n^{-1}).$ Now let $j,k \\in \\mathbb {N}$ with $j < k$ .", "We omit the dependence in $a$ for $j$ , $e_j$ , $\\tilde{e}_j$ and $\\lambda _j$ .", "We have $\\left< e_j , e_k \\right> = \\frac{e^{i(\\lambda _j - \\overline{\\lambda _k}) l} -1}{\\lambda _j - \\overline{\\lambda _k}}\\quad \\text{and} \\quad \\left< \\tilde{e}_j , \\tilde{e}_k \\right> = \\frac{e^{-i(\\lambda _j - \\overline{\\lambda _k}) l} -1}{-(\\lambda _j - \\overline{\\lambda _k})} .$ Since $\\lambda _j - \\overline{\\lambda _k} = (j-k) \\nu + O(1/j)$ we have $\\left\|\\left< e_j , e_k \\right>\\right\| + \\left\|\\left< \\tilde{e}_j , \\tilde{e}_k \\right>\\right\| \\lesssim \\frac{1}{k-j}\\quad \\text{and} \\quad \\left\|\\left< e_j , e_k \\right> + \\left< \\tilde{e}_j , \\tilde{e}_k \\right>\\right\| \\lesssim \\frac{1}{\\left< j \\right> (k-j)} .$ Similarly $\\left\|\\left< \\tilde{e}_j , e_k \\right>\\right\| + \\left\|\\left< e_j , \\tilde{e}_k \\right>\\right\| \\lesssim \\frac{1}{k+j}\\quad \\text{and} \\quad \\left\|\\left< \\tilde{e}_j , e_k \\right> + \\left< e_j , \\tilde{e}_k \\right>\\right\| \\lesssim \\frac{1}{\\left< j \\right> (k+j)} .$ And finally $\\left\|\\left< j , k \\right>\\right\|& = \\left\|A_j\\overline{A_k}\\left< e_j + \\frac{\\lambda _j - a }{\\lambda _j + a } \\tilde{e}_j , e_k + \\frac{\\lambda _k - a }{\\lambda _k + a } \\tilde{e}_k \\right>\\right\|\\\\& \\lesssim \\left\| \\left< e_j , e_k \\right> +\\left< e_j , \\tilde{e}_k \\right>+\\left< \\tilde{e}_j , e_k \\right>+\\left< \\tilde{e}_j , \\tilde{e}_k \\right> \\right\|\\\\& \\quad + \\left< j \\right> ^{-1}\\left( \\left\|\\left< e_j , e_k \\right>\\right\| + \\left\|\\left< e_j , \\tilde{e}_k \\right>\\right\| + \\left\|\\left< \\tilde{e}_j , e_k \\right>\\right\| + \\left\|\\left< \\tilde{e}_j , \\tilde{e}_k \\right>\\right\| \\right)\\\\& \\lesssim \\frac{1}{\\left< j \\right> (k-j)}.$ The second estimate is proved similarly, using again that $\\left\|\\lambda _n(a)\\right\| = n\\nu + O(n^{-1})$ for large $n$ .", "$\\bullet $ Let $a \\in \\mathbb {R}$ and $\\left({c}_{n}\\right)_{n \\in \\mathbb {N}} \\in l^2($ .", "Let $C\\geqslant 0$ be given by Lemma REF .", "For $N,p \\in \\mathbb {N}$ we have $\\left\\Vert \\sum _{n = N}^{N+p} c_n n(a)\\right\\Vert ^2 - \\sum _{n=N }^{N+p} \\left\|c_n\\right\|^2&= \\sum _{j=N}^{N+p} \\sum _{k=j+1}^{N+p} 2 \\mathop {\\rm {Re}}\\nolimits \\left( c_j \\overline{c_k} \\left< j , k \\right> \\right) \\\\& \\leqslant 2C \\sum _{j=0}^\\infty \\frac{\\left\|c_j\\right\|}{\\left< j \\right>} \\sum _{k=1}^\\infty \\frac{\\left\|c_{k+j}\\right\|}{k} \\\\& \\lesssim \\left\\Vert c\\right\\Vert _{l^2(}^2.$ This proves that the series $\\sum _{n = 0}^\\infty c_n n(a)$ converges in $L^2(0,l)$ and $\\left\\Vert \\sum _{n = 0}^\\infty c_n n(a)\\right\\Vert _{L^2(0,l)}^2 \\lesssim \\sum _{n=0}^\\infty \\left\|c_n\\right\|^2.$ $\\bullet $ Let $n,m \\in \\mathbb {N}$ be such that $n\\ne m$ .", "In $L^2(0,l)$ we have $\\lambda _n(a)^2 \\left< n(a) , m(-a) \\right>& = \\left< T_an(a) , m(-a) \\right> = \\left< n(a) , T_a^* m(-a) \\right>\\\\& = {\\lambda _m(a)^2} \\left< n(a) , m(-a) \\right>,$ and hence $\\left< n(a) , m(-a) \\right> = 0$ .", "On the other hand, with (REF ) we can check that for $n$ large enough we have $\\left< n(a) , n(-a) \\right> \\ne 0$ .", "Now let $\\left({c}_{n}\\right)_{n \\in \\mathbb {N}} \\in l^2($ be such that $\\sum _{n=0}^\\infty c_n n(a) = 0$ .", "Taking the inner product with $m(-a)$ we see that $c_m = 0$ for $m > N$ if $N$ is chosen large enough.", "Then for all $k \\in \\lbrace 0,\\dots ,N\\rbrace $ we have $0 = T_a^k \\sum _{n=0}^N c_n n(a) = \\sum _{n=0}^N \\lambda _n(a)^{2k} c_n n(a).$ Since the eigenvalues $\\lambda _n(a)^2$ for $n\\in \\lbrace 0,\\dots ,N\\rbrace $ are pairwise disjoint, this proves that for all $n \\in \\lbrace 0,\\dots ,N\\rbrace $ we have $c_n n(a) = 0$ , and hence $c_n=0$ .", "Finally the map $(c_n) \\in l^2( \\mapsto \\sum c_nn(a) \\in L^2(0,l)$ is one-to-one.", "$\\bullet $ As above we can check that for $N\\in \\mathbb {N}$ we have $\\left\\Vert \\sum _{n = N}^{\\infty } c_n n(a)\\right\\Vert ^2 - \\sum _{n=N}^{\\infty } \\left\|c_n\\right\|^2\\lesssim \\sum _{n=N}^{\\infty } \\left\|c_n\\right\|^2 \\sqrt{\\sum _{n=N}^\\infty \\frac{1}{\\left< n \\right>^2}}.$ This is less that $\\frac{1}{2} \\sum _{n=N}^{\\infty } \\left\|c_n\\right\|^2$ if $N$ is chosen large enough.", "Let such an integer $N$ be fixed.", "Now assume by contradiction that the sequences $\\left({a}_{m}\\right)_{m \\in \\mathbb {N}} \\in [-R,R]^\\mathbb {N}$ and $(c^m)_{m\\in \\mathbb {N}}$ in $l^2(^\\mathbb {N}$ are such that $\\left\\Vert c^m\\right\\Vert _{l^2} = 1$ for all $m \\in \\mathbb {N}$ and $\\left\\Vert \\sum _{n=0}^\\infty c_n^m n(a_m)\\right\\Vert ^2 \\xrightarrow[m \\rightarrow \\infty ]{}0.$ For $m \\in \\mathbb {N}$ we set $f_m = \\sum _{n=0}^{N-1} c_n^m n(a_m)$ and $g_m = \\sum _{n=N}^{\\infty } c_n^m n(a_m)$ .", "After extracting a subsequence if necessary we can assume that $a_m$ converges to some $a \\in [-R,R]$ and $f_m$ converges to some $f \\in L^2(0,l)$ .", "Let $P_m$ (respectively $P$ ) denote the orthogonal projection on $\\operatorname{span}(j(a_m))_{j\\geqslant N}$ (respectively on $\\operatorname{span}(n(a))_{n\\geqslant N}$ ).", "We have $\\left\\Vert f_m + g_m\\right\\Vert ^2 = \\left\\Vert f_m\\right\\Vert ^2 + 2 \\mathop {\\rm {Re}}\\nolimits \\left< P_m f_m , g_m \\right> + \\left\\Vert g_m\\right\\Vert ^2 \\geqslant \\left\\Vert f_m\\right\\Vert ^2 - \\left\\Vert P_m f_m\\right\\Vert ^2 \\xrightarrow[m \\rightarrow \\infty ]{}\\left\\Vert f\\right\\Vert ^2 - \\left\\Vert P f\\right\\Vert ^2.$ This gives $f = P f$ , and hence $f = 0$ .", "This gives a contradiction with $0 = \\lim _{m \\rightarrow \\infty } \\left\\Vert g_m\\right\\Vert ^2 \\geqslant \\lim _{m\\rightarrow \\infty } \\frac{1}{2} \\sum _{n=N}^\\infty \\left\|c_n^m\\right\|^2 = \\frac{1}{2}$ and proves the first inequality of the proposition.", "$\\bullet $ It remains to prove that the sequence $(n(a))$ is complete.", "We know that the family $\\big ( n(0) \\big )_{n\\in \\mathbb {N}}$ is an orthonormal basis of $L^2(0,l)$ .", "Since $n(a) = n(0) + O \\big ( n^{-1}\\big )$ in $L^2(0,l)$ , this follows from a perturbation argument (see Theorem V.2.20 in [22]).", "This concludes the proof.", "For $a \\in \\mathbb {R}$ we denote by $(n^(a))_{n\\in \\mathbb {N}}$ the dual basis of $(n(a))_{n\\in \\mathbb {N}}$ .", "We have $n^(a) = n(-a) = \\overline{n(a)}$ ." ], [ "Separation of variables - Spectrum of the model operator", "In this section we use the results on the transversal operator $T_a$ to prove spectral properties for the full operator $H_a$ when $a$ is constant on $\\partial \\Omega $ .", "Most of the results of this section are inspired by the ${\\mathcal {P}}{\\mathcal {T}}$ -symmetric analogs (see [10]).", "Let $a > 0$ be fixed.", "We set $\\mathfrak {S}_a = \\bigcup _{n \\in \\mathbb {N}} \\lbrace \\lambda _n(a)^2\\rbrace + \\mathbb {R}_+ = \\left\\lbrace \\lambda _n(a)^2 + r, n \\in \\mathbb {N}, r \\in \\mathbb {R}_+ \\right\\rbrace \\subset $ Proposition 4.1 We have $\\mathfrak {S} _a \\subset \\sigma (H_a)$ .", "Let $n \\in \\mathbb {N}$ , $r \\geqslant 0$ and $z = \\lambda _n(a)^2 + r \\in \\mathfrak {S}_a$ .", "Let $\\left({f}_{m}\\right)_{m \\in \\mathbb {N}}$ be a sequence in $H^2(\\mathbb {R}^{d-1})$ such that $\\left\\Vert f_m\\right\\Vert _{L^2(\\mathbb {R}^{d-1})} = 1$ for all $m \\in \\mathbb {N}$ and $\\left\\Vert (-x -r)f_m\\right\\Vert _{L^2(\\mathbb {R}^{d-1})} \\rightarrow 0$ as $m \\rightarrow \\infty $ .", "For $m \\in \\mathbb {N}$ and $(x,y) \\in \\mathbb {R}^{d-1} \\times ]0,l[$ we set $u_m(x,y) = f_m (x) n(a;y)$ .", "Then $u_m \\in {\\mathcal {D}}(H_a)$ and $\\left\\Vert u_m\\right\\Vert _{L^2(\\Omega )} = 1$ for all $m \\in \\mathbb {N}$ .", "Moreover, according to (REF ) we have $\\left\\Vert (H_a- z)u_m\\right\\Vert _{L^2(\\Omega )} = \\left\\Vert (-x - r) f_m\\right\\Vert _{L^2(\\mathbb {R}^{d-1})} \\xrightarrow[m \\rightarrow \\infty ]{}0.$ This implies that $z \\in \\sigma (H_a)$ .", "For $u \\in L^2(\\Omega )$ , $n\\in \\mathbb {N}$ and $x \\in \\mathbb {R}^{d-1}$ we set $u_n (x) = \\left< u(x, \\cdot ) , _n(a) \\right>_{L^2(0,l)} \\in .", "This gives a sequence of functions $ un$ defined almost everywhere on $ Rd-1$.$ Proposition 4.2 Let $u \\in L^2(\\Omega )$ .", "Then $u_n \\in L^2(\\mathbb {R}^{d-1})$ for all $n \\in \\mathbb {N}$ and on $L^2(\\Omega )$ we have $u= \\sum _{n\\in \\mathbb {N}} u_n \\otimes n(a).$ For $N \\in \\mathbb {N}$ we set $v_N = \\sum _{n = 0}^N u_n \\otimes n(a)$ .", "For almost all $x \\in \\mathbb {R}^{d-1}$ , $v_N(x)$ defines a function in $L^2(0,l)$ which goes to $u(x)$ as $N \\rightarrow \\infty $ .", "According to Proposition REF we have $\\left\\Vert v_N(x)\\right\\Vert ^2_{L^2(0,l)} \\lesssim \\sum _{n=0}^N \\left\|u_n(x)\\right\|^2 \\lesssim \\sum _{n=0}^\\infty \\left\|u_n(x)\\right\|^2 \\lesssim \\left\\Vert u(x)\\right\\Vert ^2_{L^2(0,l)},$ where all the estimates are uniform in $N$ .", "Since the map $x \\mapsto \\left\\Vert u(x)\\right\\Vert ^2_{L^2(0,l)}$ belongs to $L^1(\\mathbb {R}^{d-1})$ , we can apply the dominated convergence theorem to conclude that $u-v_N \\rightarrow 0$ in $L^2(\\Omega )$ .", "Proposition 4.3 Let $u \\in {\\mathcal {D}}(H_a)$ .", "Then $u_n \\in H^2(\\mathbb {R}^{d-1})$ for all $n \\in \\mathbb {N}$ and in $L^2(\\Omega )$ we have $H_au = \\sum _{n\\in \\mathbb {N}} \\big ( - x + \\lambda _n(a)^2 \\big ) u_n \\otimes n(a).$ Let $z \\in \\mathfrak {S}_a$ .", "Then $z$ belongs to the resolvent set of $H_a$ and for all $u \\in L^2 (\\Omega )$ we have $(H_a-z)^{-1}u = \\sum _{n\\in \\mathbb {N}} (-x + \\lambda _n(a)^2-z)^{-1}u_n \\otimes n(a).$ In particular there exists $C \\geqslant 0$ such that for all $z \\in \\mathfrak {S}_a$ and $u \\in L^2(\\Omega )$ we have $\\left\\Vert (H_a-z)^{-1}u\\right\\Vert _{L^2(\\Omega )} \\leqslant \\frac{C}{d(z,\\mathfrak {S}_a)} \\left\\Vert u\\right\\Vert _{L^2(\\Omega )}.$ $\\bullet $ Let $z \\in \\mathfrak {S}_a$ and $u \\in L^2(\\Omega )$ .", "Let $u_n \\in L^2(\\mathbb {R}^{d-1})$ , $n\\in \\mathbb {N}$ , be as above.", "For $n \\in \\mathbb {N}$ we set $\\tilde{R}_n(u)= (-x + \\lambda _n(a)^2 - z)^{-1}u_n \\in H^2 (\\mathbb {R}^{d-1})$ and $R_n(u)= \\tilde{R}_n(u)\\otimes n(a) \\in {\\mathcal {D}}(H_a)$ .", "Using the standard spectral properties of the self-adjoint operator $-x$ , we see that on $L^2(\\mathbb {R}^{d-1})$ we have $ \\left\\Vert \\tilde{R}_n(u)\\right\\Vert \\lesssim \\frac{\\left\\Vert u_n\\right\\Vert }{d(z, \\lambda _n(a)^2 +\\mathbb {R}_+ )} \\lesssim _z \\frac{\\left\\Vert u_n\\right\\Vert }{\\left< n \\right> ^{2}} \\quad \\text{and} \\quad \\left\\Vert \\partial _x \\tilde{R}_n(u)\\right\\Vert \\lesssim _z \\frac{\\left\\Vert u_n\\right\\Vert }{\\left< n \\right> }.$ The first estimate is uniform in $z$ but not the others.", "With Proposition REF we obtain for $N,p \\in \\mathbb {N}$ $\\left\\Vert \\sum _{n=N}^{N+p} R_n(u)\\right\\Vert ^2_{H^1(\\Omega )} \\lesssim _z \\sum _{n=N}^{N+p} \\left( \\left\\Vert \\tilde{R}_n(u)\\right\\Vert ^2_{H^1(\\mathbb {R}^{d-1})} + \\left< n \\right>^2 \\left\\Vert \\tilde{R}_n(u)\\right\\Vert ^2_{L^2(\\mathbb {R}^{d-1})} \\right) \\xrightarrow[N \\rightarrow \\infty ]{}0.$ This proves that the series $\\sum R_n(u)$ converges to some $R(u) \\in H^1(\\Omega )$ .", "Moreover, with the first inequality of (REF ) and Proposition REF again, we see that $ \\left\\Vert R(u)\\right\\Vert _{L^2(\\Omega )} \\lesssim \\frac{\\left\\Vert u\\right\\Vert _{L^2(\\Omega )}}{d (z ,\\mathfrak {S}_a)},$ uniformly in $z$ .", "It remains to see that $\\forall \\phi \\in H^1(\\Omega ), \\quad q_a(R(u),\\phi ) - z \\left< R(u) , \\phi \\right>= \\left< u , \\phi \\right>,$ which comes from the fact that this is true with $R(u)$ and $u$ replaced by $R_n(u)$ and $u_n \\otimes n(a)$ respectively.", "This proves that $R(u) \\in {\\mathcal {D}}(H_a)$ and $(H_a-z) R(u) = u$ .", "If $\\mathop {\\rm {Re}}\\nolimits (z) < 0$ , we already know that $(H_a-z)$ has a bounded inverse on $L^2(\\Omega )$ , and hence we have $(H_a-z)^{-1}= R$ .", "This proves the second statement of the proposition when $\\mathop {\\rm {Re}}\\nolimits (z) < 0$ .", "$\\bullet $ Let $ u \\in {\\mathcal {D}}(H_a)$ and $v = (H_a+1) u \\in L^2(\\Omega )$ .", "For $n \\in \\mathbb {N}$ and almost all $x \\in \\mathbb {R}^{d-1}$ we set $v_n(x) = \\left< v(x,\\cdot ) , n^(a) \\right>_{L^2(0,l)}$ .", "According to (ii) applied with $z = -1$ we have $u = (H_a+1)^{-1}v = \\sum _{n\\in \\mathbb {N}} \\big (-x + \\lambda _n(a)^2 +1\\big )^{-1}v_n \\otimes n(a).$ By uniqueness for the decomposition of $u(x,\\cdot )$ with respect to the Riesz basis $(n(a))_{n\\in \\mathbb {N}}$ , we have for all $n \\in \\mathbb {N}$ $u_n = \\big (-x + \\lambda _n(a)^2 +1\\big )^{-1}v_n.$ This proves that $u_n \\in H^2(\\mathbb {R}^{d-1})$ .", "Then $H_au = v - u& = \\sum _{n \\in \\mathbb {N}} \\left( 1 - \\big (-x + \\lambda _n(a)^2 +1\\big )^{-1}\\right) v_n \\otimes n(a)\\\\& = \\sum _{n \\in \\mathbb {N}} \\big (-x + \\lambda _n(a)^2 \\big ) u_n \\otimes n(a).$ This proves the first statement of the proposition.", "$\\bullet $ It remains to finish the proof of (ii).", "Let $z \\in \\mathfrak {S}_a$ and $w \\in {\\mathcal {D}}(H_a)$ .", "With (i) we see that $R\\big ((H_a-z)w\\big ) = w$ .", "Since we already know that $(H_a-z)R(u) = u$ for all $u \\in L^2(\\Omega )$ , this proves that $R$ is a bounded inverse for $(H_a-z)$ on $L^2(\\Omega )$ .", "The estimate on $(H_a-z)^{-1}$ follows from (REF ), and the proposition is proved.", "As a first application of this proposition, we can check that the operator $H_a$ has no eigenvalue, as is the case for $-x$ : Corollary 4.4 The operator $H_a$ has no eigenvalue.", "Let $z \\in and $ u D(Ha)$ be such that $ (Ha-z) u =0$.", "Then according to the first item of Proposition \\ref {prop-dec-res} we have$$\\sum _{n \\in \\mathbb {N}} \\left( - \\lambda _n(a)^2 - z \\right) u_n \\otimes n(a) = 0.$$This implies that for all $ n N$ we have$$\\left( - \\lambda _n(a)^2 - z \\right) u_n = 0$$in $ L2(Rd-1)$, and hence $ un = 0$ since the operator $ -x$ has no eigenvalue.", "Finally $ u = 0$, and the proposition is proved.$ However, the main point in Proposition REF is the second: Corollary 4.5 Theorem REF holds when $a > 0$ is constant.", "According to the last statement of Proposition REF and the fact that $\\mathop {\\rm {Im}}\\nolimits (\\lambda _n(a)^2) < 0$ for all $n \\in \\mathbb {N}$ , there exists $\\tilde{\\gamma }> 0$ such that for all $z \\in \\mathfrak {S}_a$ we have $\\mathop {\\rm {Im}}\\nolimits (z) \\leqslant - \\tilde{\\gamma }$ .", "With Proposition REF , the conclusion follows.", "Remark 4.6 To simplify the proof we have only considered the case where $a$ is equal to the same constant on both sides of the boundary.", "However we can similarly consider the case where $a$ is equal to some positive constant $a_0$ on $\\mathbb {R}^{d-1} \\times \\left\\lbrace 0 \\right\\rbrace $ and to another positive constant $a_l$ on $\\mathbb {R}^{d-1} \\times \\left\\lbrace l \\right\\rbrace $ .", "One of these two constants may even be zero.", "We refer to Section for this kind of computations.", "Non-constant absorption index In this section we prove Theorems REF and REF for a non-constant absorption index $a$ .", "For $b \\in W^{1,\\infty }(\\partial \\Omega )$ we denote by $\\Theta _b \\in {\\mathcal {L}}(H^1(\\Omega ) , H^{-1}(\\Omega ))$ the operator such that for all $\\psi \\in H^1(\\Omega )$ $\\left< \\Theta _b \\psi \\right>_{H^{-1}(\\Omega ),H^1(\\Omega )} = \\int _{\\partial \\Omega } b\\psi .$ We denote by $\\theta _b$ the corresponding quadratic form on $H^1(\\Omega )$ .", "We also denote by $\\tilde{H}_{a}$ the operator in ${\\mathcal {L}}\\big (H^1(\\Omega ),H^{-1}(\\Omega ) \\big )$ such that $\\left< \\tilde{H}_{a} \\psi \\right>_{H^{-1},H^1} = q_a(\\psi )$ for all $\\psi \\in H^1(\\Omega )$ .", "Let $z \\in +$ .", "According to the Lax-Milgram Theorem, $(1+i)(\\tilde{H}_{a}-z)$ is an isomorphism from $H^1(\\Omega )$ to $H^{-1}(\\Omega )$ .", "Moreover, for $f \\in L^2(\\Omega ) \\subset H^{-1}(\\Omega )$ we have $(\\tilde{H}_{a}-z)^{-1}f = (H_a-z)^{-1}f.$ The following proposition relies on a suitable version of the so-called quadratic estimates: Proposition 5.1 Let $a_0 > 0$ be as in the statement of Theorem REF .", "Assume that (REF ) holds everywhere on $\\partial \\Omega $ .", "Let $B \\in {\\mathcal {L}}\\big (H^1(\\Omega ),L^2(\\Omega ) \\big )$ .", "Then there exists $C \\geqslant 0$ such that for all $z \\in +$ we have $\\left\\Vert B (\\tilde{H}_{a}-z)^{-1}B^\\right\\Vert _{{\\mathcal {L}}(L^2(\\Omega ))} \\leqslant C \\left\\Vert B (\\tilde{H}_{a_0}-z)^{-1}B^\\right\\Vert _{{\\mathcal {L}}(L^2(\\Omega ))}.$ For $z \\in +$ the resolvent identity applied to $\\tilde{H}_{a}= \\tilde{H}_{a_0}+ \\Theta _{a-a_0}$ gives $ B(\\tilde{H}_{a}-z)^{-1}B^* = B (\\tilde{H}_{a_0}-z)^{-1}B^* - B (\\tilde{H}_{a_0}-z)^{-1}\\Theta _{a-a_0} (\\tilde{H}_{a}-z)^{-1}B^.$ Let $\\psi \\in L^2(\\Omega )$ .", "Since $\\Theta _{a-a_0}$ is associated to a non-negative quadratic form on $H^1(\\Omega )$ , the Cauchy-Schwarz inequality gives ${\\left< B(\\tilde{H}_{a_0}-z)^{-1}\\Theta _{a-a_0} (\\tilde{H}_{a}-z)^{-1}B^ \\psi \\right>_{L^2}}\\\\&& = \\left< \\Theta _{a-a_0} (\\tilde{H}_{a}-z)^{-1}B^* (\\tilde{H}_{a_0}^* - \\overline{z})^{-1}B^* \\psi \\right> _{H^{-1},H^1}\\\\&& \\leqslant \\theta _{a-a_0} \\big ((\\tilde{H}_{a}-z)^{-1}B^* )^{\\frac{1}{2}} \\times \\theta _{a-a_0} \\big ((\\tilde{H}_{a_0}^* -\\overline{z})^{-1}B^* \\psi \\big ) ^{\\frac{1}{2}}.$ The first factor is estimated as follows: ${\\theta _{a-a_0} \\big ((\\tilde{H}_{a}-z)^{-1}B^* )\\leqslant \\theta _{a} \\big ((\\tilde{H}_{a}-z)^{-1}B^* )}\\\\&& \\leqslant \\frac{1}{2i} \\left< 2i (\\Theta _{a} + \\mathop {\\rm {Im}}\\nolimits z) (\\tilde{H}_{a}-z)^{-1}B^* (\\tilde{H}_{a}- z)^{-1}B^* > _{H^{-1},H^1}\\\\&& \\leqslant \\frac{1}{2i} \\left< B \\big ((\\tilde{H}_{a}-z)^{-1}-(\\tilde{H}_{a}^* - \\overline{z})^{-1}\\big ) B^* > _{L^2}\\\\&& \\leqslant \\left\\Vert B(\\tilde{H}_{a}-z)^{-1}B^\\right\\Vert _{{\\mathcal {L}}(L^2)} \\left\\Vert \\Vert _{L^2}^2.\\right.We can proceed similarly for the other factor, using the fact that \\right.\\right.a-a_0 \\leqslant \\alpha a_0 for some \\alpha \\geqslant 0.", "Thus there exists C \\geqslant 0 such that{\\begin{@align}{1}{-1}\\left\\Vert B (\\tilde{H}_{a_0}-z)^{-1}\\Theta _{a-a_0} (\\tilde{H}_{a}-z)^{-1}B^* \\right\\Vert & \\leqslant C \\left\\Vert B (\\tilde{H}_{a_0}-z)^{-1}B^\\right\\Vert ^{\\frac{1}{2}} \\left\\Vert B (\\tilde{H}_{a}-z)^{-1}B^\\right\\Vert ^{\\frac{1}{2}}\\\\& \\leqslant \\frac{C^2}{2} \\left\\Vert B (\\tilde{H}_{a_0}-z)^{-1}B^\\right\\Vert + \\frac{1}{2} \\left\\Vert B (\\tilde{H}_{a}-z)^{-1}B^\\right\\Vert .\\end{@align}}With (\\ref {eq-res-identity}), the conclusion follows.$ Now we can finish the proof of Theorem REF : According to Corollary REF and Proposition REF applied with $B = \\operatorname{Id}_{L^2(\\Omega )}$ (note that we can simply replace $\\tilde{H}_{a}$ by $H_a$ when $B \\in {\\mathcal {L}}(L^2(\\Omega ))$ ), there exists $C > 0$ such that $\\left\\Vert (H_a-z)^{-1}\\right\\Vert _{{\\mathcal {L}}(L^2(\\Omega ))} \\leqslant C$ for all $z \\in +$ .", "Then it only remains to choose $\\tilde{\\gamma }\\in \\big ] 0, \\frac{1}{C} \\big [$ to conclude.", "The second statement, concerning the case where $a$ vanishes on one side of the boundary, is proved similarly.", "Let us now turn to the proof of Theorem REF .", "We first prove another resolvent estimate in which we see the smoothing effect in weighted spaces: Proposition 5.2 Let $\\delta > \\frac{1}{2}$ .", "Then there exists $C \\geqslant 0$ such that for all $z \\in \\mathbb {C}_+$ we have $\\left\\Vert \\left< x \\right> ^{-\\delta } (1- \\Delta _x)^{\\frac{1}{4}} (H_a-z)^{-1} (1- \\Delta _x)^{\\frac{1}{4}} \\left< x \\right> ^{-\\delta }\\right\\Vert _{\\mathcal {L} (L^2(\\Omega ))} \\leqslant C.$ It is known for the free laplacian that for $\\left\| \\mathop {\\rm {Re}} (\\zeta ) \\right\| \\gg 1$ and $\\mathop {\\rm {Im}}(\\zeta ) > 0$ we have $\\left\\Vert \\left< x \\right>^{-\\delta } (-\\Delta _x -\\zeta )^{-1} \\left< x \\right>^{-\\delta }\\right\\Vert _{\\mathcal {L}(L^2(\\mathbb {R}^{d-1}))} \\lesssim \\left< \\zeta \\right> ^{-\\frac{1}{2}}$ and hence $\\left\\Vert \\left< x \\right>^{-\\delta } (1- x)^{\\frac{1}{4}}(-\\Delta _x -\\zeta )^{-1} (1- \\Delta _x)^{\\frac{1}{4}}\\left< x \\right>^{-\\delta } \\right\\Vert _{\\mathcal {L} (L^2(\\mathbb {R}^{d-1}))} \\lesssim 1.$ Thus if $a$ is constant we have $\\left\\Vert \\left< x \\right>^{-\\delta } (1- x)^{\\frac{1}{4}}(-\\Delta _x + \\lambda _n(a)^2 -z)^{-1} (1- \\Delta _x)^{\\frac{1}{4}}\\left< x \\right>^{-\\delta }\\right\\Vert _{\\mathcal {L}(L^2(\\mathbb {R}^{d-1})) } \\lesssim 1,$ uniformly in $z \\in +$ and $n \\in \\mathbb {N}$ .", "In this case we obtain the result using the separation of variables as in Section .", "Then we conclude with Proposition REF applied with $B = \\left< x \\right> ^{-\\delta } (1-\\Delta _x)^{\\frac{1}{4}}$ .", "In fact, we first obtain an estimate on the resolvent $(\\tilde{H}_a -z)^{-1}$ , but this proves that the operator $\\left< x \\right> ^{-\\delta } (1- \\Delta _x)^{\\frac{1}{4}} (H_a-z)^{-1} (1- \\Delta _x)^{\\frac{1}{4}} \\left< x \\right> ^{-\\delta }$ extends to a bounded operator on $L^2(\\Omega )$ , and then the same estimate holds for the corresponding closure.", "With the second estimate of Proposition REF , we can apply the theory of relatively smooth operators (see §XIII.7 in [37]).", "However, since $H_a$ is not self-adjoint but only maximal dissipative, we have to use a self-adjoint dilation (see [31]) of $H_a$ , as is done in the proof of [36] (see also Proposition 2.24 in [35]).", "Time decay for the Schrödinger equation In this section we prove Theorem REF .", "Let $u_0 \\in {\\mathcal {D}}(H_a)$ and let $u$ be the solution of the problem (REF ).", "We know that $\\left\\Vert u(t)\\right\\Vert _{L^2(\\Omega )} \\leqslant \\left\\Vert u_0\\right\\Vert _{L^2(\\Omega )}$ for all $t \\geqslant 0$ , so the result only concerns large times.", "Let $\\tilde{\\gamma }> 0$ be given by Theorem REF and $\\gamma = \\tilde{\\gamma }/ 3$ .", "Let $C^\\infty (\\mathbb {R})$ be equal to 0 on $]-\\infty ,1]$ and equal to 1 on $[2,+\\infty [$ .", "For $t \\in \\mathbb {R}$ we set $u_t) = t) u(t),$ and for $z \\in +$ : $v(z) = \\int _\\mathbb {R}e^{itz} u_t) \\, dt.$ The map $t \\mapsto e^{-\\gamma t}u_t)$ belongs to $L^1(\\mathbb {R}) \\cap L^2(\\mathbb {R}) \\cap C^1(\\mathbb {R})$ and its derivative is in $L^1(\\mathbb {R})$ so $\\mapsto v(+i\\gamma )$ is bounded and decays at least like $\\left< \\right>^{-1}$ .", "In particular it is in $L^2(\\mathbb {R})$ .", "For $R > 0$ we set $u_R(t) = \\frac{1}{2\\pi } \\int _{-R}^R e^{-it(+i\\gamma )} v(+i\\gamma ) \\, d.$ Then $\\left\\Vert e^{-t\\gamma } (u_ u_R)\\right\\Vert _{L^2(\\mathbb {R}_t,L^2(\\Omega ))} \\xrightarrow[R \\rightarrow +\\infty ]{} 0.$ Since $u_ is continuous, Theorem \\ref {th-energy-decay} will be proved if we can show that there exists $ C 0$ which does not depend on $ u0$ and such that for all $ t 0$ we have\\begin{equation} \\limsup _{R \\rightarrow \\infty } \\left\\Vert u_R(t)\\right\\Vert _{L^2(\\Omega )} \\leqslant C e^{-\\gamma t} \\left\\Vert u_0\\right\\Vert _{L^2(\\Omega )}.\\end{equation}For $ z we set $\\theta (z) = -i \\int _\\mathbb {R}e^{itz} (t) u(t) \\, dt = -i \\int _1^2 e^{itz} (t) u(t) \\, dt.$ Let $z \\in +$ .", "We multiply (REF ) by $t) e^{itz}$ and integrate over $t \\in \\mathbb {R}$ .", "After partial integration we obtain $ v(z) = (H_a-z)^{-1}\\theta (z).$ Then $v$ extends to a holomorphic function on ${3\\gamma }$ , taking this equality as a definition.", "According to the Cauchy Theorem we have in $L^2(\\mathbb {R}_t)$ $ \\begin{aligned}\\lim _{R \\rightarrow \\infty } u_R(t)= \\frac{1}{2\\pi } \\lim _{R \\rightarrow \\infty } \\int _{-R}^R e^{-it(-2i\\gamma )} v(-2i\\gamma )\\,d= {e^{-2\\gamma t}} \\lim _{R \\rightarrow \\infty } \\widetilde{u_R}(t),\\end{aligned}$ where for $t \\in \\mathbb {R}$ we have set $\\widetilde{u_R}(t) = \\int _{-R}^R e^{-it}v(-2i\\gamma ) \\, d.$ According to Plancherel's equality and Theorem REF we have uniformly in $R >0$ : $\\int _\\mathbb {R}\\left\\Vert \\widetilde{u_R} (t)\\right\\Vert ^2_{L^2(\\Omega )} \\, dt& = \\int _{-R}^R \\left\\Vert \\big (H_a-(-2i\\gamma )\\big )^{-1}\\theta (-2i\\gamma )\\right\\Vert _{L^2(\\Omega )}^2 \\, d\\\\& \\lesssim \\int _\\mathbb {R}\\left\\Vert \\theta (-2i\\gamma )\\right\\Vert _{L^2(\\Omega )}^2 \\, d\\\\& \\lesssim \\int _\\mathbb {R}e^{2\\gamma t}\\left\|(t)\\right\| \\left\\Vert u(t)\\right\\Vert _{L^2(\\Omega )}^2 \\, dt \\\\& \\lesssim \\left\\Vert u_0\\right\\Vert ^2_{L^2(\\Omega )}.$ In particular there exists $C \\geqslant 0$ such that for $u_0 \\in {\\mathcal {D}}(H_a)$ and $R > 0$ we can find $T(u_0,R) \\in [0,1]$ which satisfies $\\left\\Vert \\widetilde{u_R} (T(u_0,R))\\right\\Vert _{L^2(\\Omega )} \\leqslant C \\left\\Vert u_0\\right\\Vert _{L^2(\\Omega )}.$ Let $R > 0$ .", "Then $\\widetilde{u_R} \\in C^1(\\mathbb {R})$ and for $t \\geqslant 1$ we have $\\widetilde{u_R} (t) = e^{-i(t-T(u_0,R))H_a} \\widetilde{u_R} (T(u_0,R)) + \\int _{T(u_0,R)} ^t \\frac{\\partial }{\\partial s} \\left( e^{-i(t-s)H_a} \\widetilde{u_R}(s) \\right) \\,ds,$ where $\\frac{\\partial }{\\partial s} \\left( e^{-i(t-s)H_a} \\widetilde{u_R}(s) \\right)& = \\frac{\\partial }{\\partial s}\\int _{-R}^R e^{-is} e^{-i(t-s)H_a} \\big (H_a-(-2i\\gamma )\\big )^{-1}\\theta (-2i\\gamma ) \\, d\\\\& = i \\int _{-R}^R e^{-i(t-s)H_a} e^{-is} (H_a-) \\big ( H_a- (-2i\\gamma ) \\big ) ^{-1}\\theta (-2i\\gamma ) \\, d\\\\& = 2\\gamma e^{-i(t-s)H_a} \\widetilde{u_R} (s) + i e^{-i(t-s)H_a} \\int _{-R}^R e^{-is} \\theta (-2i\\gamma ) \\, d.$ This proves that the map $s \\mapsto \\frac{\\partial }{\\partial s} \\left( e^{-i(t-s)H_a} \\widetilde{u_R}(s) \\right)$ belongs to $L^2([0,t],L^2(\\Omega ))$ uniformly in $t$ and $R>0$ , and its $L^2([0,t],L^2(\\Omega ))$ norm is controlled by the norm of $u_0$ in $L^2(\\Omega )$ .", "We finally obtain $C \\geqslant 0$ such that for all $t \\in \\mathbb {R}$ and $R > 0$ we have $\\left\\Vert \\widetilde{u_R} (t)\\right\\Vert _{L^2(\\Omega )} \\leqslant C \\left< t \\right>^{\\frac{1}{2}} \\left\\Vert u_0\\right\\Vert _{L^2(\\Omega )}.$ With (REF ) this proves () and concludes the proof of Theorem REF .", "The case of a weakly dissipative boundary condition In this section we prove Theorem REF about the problem (REF ).", "The absorption index $a$ now takes the value $a_l$ on $\\mathbb {R}^{d-1} \\times \\left\\lbrace l \\right\\rbrace $ and $a_0$ on $\\mathbb {R}^{d-1} \\times \\left\\lbrace 0 \\right\\rbrace $ .", "The proof follows the same lines as in the dissipative case, except that well-posedness of the problem is not an easy consequence of the general dissipative theory.", "We will use the separation of variables as in Section instead.", "Once we have a decomposition as in Proposition REF for the initial datum, we can propagate each term by means of the unitary group generated by $-x$ and define the solution of (REF ) as a series of solutions on $\\mathbb {R}^{d-1}$ .", "Let us first look at the transverse problem.", "The transverse operator on $L^2(0,l)$ corresponding to the problem (REF ) is now given by $T_{a_l,a_0}= - \\frac{d^2}{dy^2}$ with domain ${\\mathcal {D}}(T_{a_l,a_0}) = \\left\\lbrace u \\in H^2(0,l) \\,:\\,u^{\\prime }(0) = -ia_0u(0), u^{\\prime }(l) = ia_lu(l) \\right\\rbrace .$ As already mentioned in Remark REF , we can reproduce exactly the same analysis as in Section if $a_l> 0$ and $a_0>0$ (or if one of them vanishes).", "In particular, there is no restriction on the sizes of these coefficients.", "The results we give here to handle the weakly dissipative case are also valid in this situation.", "The strategy will be the same as in Section , so we will only emphasize the differences.", "We first remark that 0 is an eigenvalue of $T_{a_l,a_0}$ if and only if $a_l=a_0=0$ .", "Otherwise, $\\lambda ^2 \\in $ is an eigenvalue of $T_{a_l,a_0}$ if and only if $ (\\lambda - a_l)(\\lambda - a_0) e^{2i\\lambda l} = (\\lambda + a_l) (\\lambda + a_0).$ We recover (REF ) when $a_l= a_0$ .", "Lemma 7.1 Let $a_l,a_0\\in \\mathbb {R}$ and $\\lambda \\in $ be such that $a_l+a_0\\ne 0$ and $\\lambda ^2$ is an eigenvalue of $T_{a_l,a_0}$ .", "Then $\\mathop {\\rm {Re}}\\nolimits (\\lambda ) \\notin \\nu \\mathbb {N}$ .", "We recall from [23] that if $a_l+a_0= 0$ (${\\mathcal {P}}{\\mathcal {T}}$ -symmectric case) then $n^2 \\nu ^2 \\in \\sigma (T_{a_l,a_0})$ for all $n \\in \\mathbb {N}^$ (see also Figure REF for $a_l+a_0> 0$ small).", "$\\bullet $ We assume by contradiction that $\\mathop {\\rm {Re}}\\nolimits (\\lambda ) \\in \\nu \\mathbb {N}$ .", "According to (REF ) we have $\\frac{(\\lambda + a_l)(\\lambda +a_0)}{(\\lambda -a_l)(\\lambda -a_0)} = e^{2il\\lambda } = e^{-2 l\\mathop {\\rm {Im}}\\nolimits (\\lambda )} \\in \\mathbb {R}_+^.$ After multiplication by $\\left\|\\lambda - a_l\\right\|^2 \\left\|\\lambda - a_0\\right\|^2 \\in \\mathbb {R}_+^$ we obtain $\\big ( \\left\|\\lambda \\right\|^2 - 2 i a_l\\mathop {\\rm {Im}}\\nolimits (\\lambda ) - a_l^2 \\big ) \\big ( \\left\|\\lambda \\right\|^2 - 2 i a_0\\mathop {\\rm {Im}}\\nolimits (\\lambda ) - a_0^2 \\big ) \\in \\mathbb {R}_+^.$ Taking the real and imaginary parts gives $ \\left\|\\lambda \\right\|^4 - (a_l^2 + a_0^2) \\left\|\\lambda \\right\|^2 + a_l^2 a_0^2 - 4 a_la_0\\mathop {\\rm {Im}}\\nolimits (\\lambda )^2 > 0$ and $ 2 \\mathop {\\rm {Im}}\\nolimits (\\lambda ) (a_l+a_0) \\big (\\left\|\\lambda \\right\|^2 - a_la_0\\big ) = 0.$ $\\bullet $ Assume that $a_la_0\\geqslant 0$ .", "In this case $\\mathop {\\rm {Im}}\\nolimits (\\lambda ) \\ne 0$ (for the same reason as in the proof of Proposition REF ), so (REF ) implies $\\left\|\\lambda \\right\|^2 = a_la_0$ .", "Then (REF ) reads $- a_la_0(a_l- a_0)^2 - 4a_la_0\\mathop {\\rm {Im}}\\nolimits (\\lambda )^2 > 0,$ which gives a contradiction.", "$\\bullet $ Now assume that $a_la_0< 0$ .", "Then (REF ) implies $\\mathop {\\rm {Im}}\\nolimits (\\lambda ) = 0$ and hence $e^{2il\\lambda } = 1$ .", "From (REF ) we now obtain $(\\lambda -a_l) (\\lambda -a_0) = (\\lambda +a_l) (\\lambda +a_0),$ which is impossible since $\\lambda (a_l+ a_0) \\ne 0$ .", "This concludes the proof.", "Proposition 7.2 There exists $\\rho > 0$ such that if $\\left\|a_l\\right\| + \\left\|a_0\\right\| \\leqslant \\rho $ and $a_l+ a_0> 0$ then the spectrum of $T_{a_l,a_0}$ is given by a sequence $(\\lambda _n(a_l,a_0)^2)_{n\\in \\mathbb {N}}$ of algebraically simple eigenvalues such that $\\sup _{n\\in \\mathbb {N}} \\, \\mathop {\\rm {Im}}\\nolimits \\left(\\lambda _n(a_l,a_0)^2\\right) < 0.$ Moreover, any sequence of normalized eigenfunctions corresponding to these eigenvalues forms a Riesz basis.", "Figure: λ n (a l ,a 0 )\\lambda _n(a_l,a_0) for n∈{0,⋯,30}n \\in \\lbrace 0,\\dots ,30\\rbrace and l=πl= \\pi .As in the proof of Proposition REF we can see that for any $R > 0$ there exists $C_R \\geqslant 0$ such that if $\\lambda ^2 \\in $ is an eigenvalue of $T_{a_l,a_0}$ we have $ \\left\|\\mathop {\\rm {Re}}\\nolimits \\lambda \\right\| \\leqslant R \\quad \\Rightarrow \\quad \\left\|\\mathop {\\rm {Im}}\\nolimits \\lambda \\right\| \\leqslant C_R.$ The operator $T_{a_l,a_0}$ depends analytically on the parameters $a_l$ and $a_0$ , and we know that when $a_l= a_0= 0$ the eigenvalues $n^2\\nu ^2$ for $n \\in \\mathbb {N}$ are algebraically simple.", "With the restrictions given by (REF ) and Lemma REF , we obtain as in Section a sequence of maps $(a_l,a_0) \\mapsto \\lambda _n(a_l,a_0)$ such that the eigenvalues of $T_{a_l,a_0}$ are $\\lambda _n(a_l,a_0)^2$ for $n \\in \\mathbb {N}$ .", "Let $n \\in \\mathbb {N}^$ .", "We have $ \\lambda _n(a_l,a_0)= n\\nu - \\frac{i}{n\\pi } (a_l+ a_0) + \\gamma (a_l+a_0)^2 + O \\left( \\left\|a_l\\right\|^3 , \\left\|a_0\\right\|^3 \\right),$ with $\\mathop {\\rm {Re}}\\nolimits (\\gamma ) = l/(n\\pi )^3 > 0$ .", "As in the dissipative case, we obtain that for any $a_l,a_0$ with $a_l+ a_0> 0$ these eigenvalues $\\lambda _n(a_l,a_0)^2$ are simple.", "If moreover $a_l$ and $a_0$ are small enough, the eigenvalue $\\lambda _n(a_l,a_0)^2$ is close to $(n\\nu )^2$ and away from the real axis uniformly in $n \\in \\mathbb {N}^$ (the first two terms in (REF ) are also the first two terms of the asymptotic expansion for large $n$ and fixed $a_l$ and $a_0$ ).", "It remains to check that we also have $\\mathop {\\rm {Im}}\\nolimits (\\lambda _0(a_l,a_0)^2) < 0$ .", "For small $a_l,a_0$ we denote by ${a_l,a_0}(0)$ a normalized eigenvector corresponding to the eigenvalue $\\lambda _0(a_l,a_0)^2$ and depending analytically on $a_l$ and $a_0$ .", "For all $\\psi \\in H^1(0,l)$ we have $\\left< {a_l,a_0}^{\\prime } , \\psi ^{\\prime } \\right>_{L^2(0,l)} - i a_l{{a_l,a_0}(l)} \\overline{\\psi (l)} - i a_0{{a_l,a_0}(0)} \\overline{\\psi (0)} = \\lambda _0(a_l,a_0)^2 \\left< {a_l,a_0} , \\psi \\right>_{L^2(0,l)}$ We apply this with $\\psi = {a_l,a_0}$ , take the derivatives with respect to $a_l$ and $a_0$ at point $(a_l,a_0) = (0,0)$ , and use the facts that ${0,0}$ is constant and $\\lambda _{0}(0,0) = 0$ .", "We obtain $\\nabla _{a_l,a_0} \\big (\\lambda _0^2 \\big ) = -\\frac{i}{l}\\big ( 1 , 1 \\big ).$ This proves that $\\mathop {\\rm {Im}}\\nolimits \\big (\\lambda _0(a_l,a_0)^2\\big )<0$ if $a_l$ and $a_0$ are small enough with $a_l+ a_0> 0$ .", "The Riesz basis property relies as before on the fact that $\\left\|\\lambda _n(a_l,a_0)- n\\nu \\right\| = O(n^{-1}).$ For this point we can follow what is done in Section for the dissipative case.", "For $n \\in \\mathbb {N}$ and $a_l,a_0\\in \\mathbb {R}$ with $a_l+ a_0> 0$ we consider a normalized eigenvector $n({a_l,a_0})\\in L^2(0,l)$ corresponding to the eigenvalue $\\lambda _n(a_l,a_0)^2$ of $T_{a_l,a_0}$ .", "We denote by $(n^(a_l,a_0))_{n\\in \\mathbb {N}}$ the dual basis.", "Proposition 7.3 Let $u_0 \\in {\\mathcal {D}}(H_{a_l,a_0})$ .", "Then the problem (REF ) has a unique solution $u \\in C^1 (\\mathbb {R},L^2(\\Omega )) \\cap C^0(\\mathbb {R},{\\mathcal {D}}(H_{a_l,a_0}))$ .", "Moreover if we write $u_0 = \\sum _{n\\in \\mathbb {N}} u_{0,n} \\otimes n({a_l,a_0})$ where $u_{0,n} \\in L^2(\\mathbb {R}^{d-1})$ , then $u$ is given by $u(t) = \\sum _{n\\in \\mathbb {N}} \\left( e^{-it \\left(-x+\\lambda _n(a_l,a_0)^2 \\right)} u_{0,n}\\right) \\otimes n({a_l,a_0}).$ $\\bullet $ Assume that $u \\in C^0(\\mathbb {R}_+, {\\mathcal {D}}(H_{a_l,a_0})) \\cap C^1(\\mathbb {R}_+^,L^2(\\Omega ))$ is a solution of (REF ).", "Let $t \\in \\mathbb {R}_+^$ .", "For all $n \\in \\mathbb {N}$ and almost all $x \\in \\mathbb {R}^{d-1}$ we can define $u_n(t,x) = \\left< u(t;x,\\cdot ) , _n(a_l,a_0) \\right>_{L^2(0,l)},$ so that in $L^2(\\Omega )$ we have $u(t) = \\sum _{n\\in \\mathbb {N}} u_n(t) \\otimes n({a_l,a_0}).$ According to Proposition REF (which can be proved similarly in this context) we have $u_n(t) \\in H^2(\\mathbb {R}^{d-1})$ for all $t \\in \\mathbb {R}_+^$ and $n \\in \\mathbb {N}$ , and for $s \\in \\mathbb {R}^$ we have $i\\frac{u(t+s)-u(t)}{s} - H_{a_l,a_0}u(t) \\\\ = \\sum _{n\\in \\mathbb {N}} \\left( i\\frac{u_n(t+s)-u_n(t)}{s} - \\left(-x +\\lambda _n(a_l,a_0)^2 \\right) u_n(t) \\right) \\otimes n({a_l,a_0}).$ Let $n\\in \\mathbb {N}$ .", "According to Proposition REF we have $\\left\\Vert i\\frac{u_n(t+s)-u_n(t)}{s} - \\left(-x +\\lambda _n(a_l,a_0)^2 \\right) u_n(t)\\right\\Vert _{L^2(\\mathbb {R}^{d-1})}\\\\\\lesssim \\left\\Vert i\\frac{u(t+s)-u(t)}{s} - H_{a_l,a_0}u(t)\\right\\Vert _{L^2(\\Omega )}\\xrightarrow[s \\rightarrow 0]{} 0.$ This proves that $u_n$ is differentiable and for all $t > 0$ $i u_n^{\\prime }(t) = \\left( - x + \\lambda _n(a_l,a_0)^2 \\right) u_n(t).$ Then for all $t > 0$ $u_n(t) = e^{-it\\left( -x + \\lambda _n(a_l,a_0)^2 \\right)} u_n(0) = e^{-it\\left( -x + \\lambda _n(a_l,a_0)^2 \\right)} u_{0,n}.$ $\\bullet $ Conversely, let us prove that the function $u$ defined by the statement of the proposition is indeed a solution of (REF ).", "Let $t \\in \\mathbb {R}$ .", "According to Proposition REF , $u_{0,n}$ and hence $e^{-it \\left(-x+\\lambda _n(a_l,a_0)^2 \\right)} u_{0,n}$ belong to $H^2(\\mathbb {R}^{d-1})$ for all $n \\in \\mathbb {N}$ .", "Therefore $u(t) \\in {\\mathcal {D}}(H_{a_l,a_0})$ .", "Then for all $s \\in \\mathbb {R}^$ we have ${ \\sum _{n\\in \\mathbb {N}}\\left\\Vert \\left( i\\frac{e^{-is \\left(-x + \\lambda _n(a_l,a_0)^2 \\right)} - 1}{s} -\\left(- x + \\lambda _n(a_l,a_0)^2\\right) \\right) \\, e^{-it(-x + \\lambda _n(a_l,a_0)^2)} u_{0,n}\\right\\Vert ^2 }\\\\&& \\lesssim _t \\sum _{n\\in \\mathbb {N}}\\left\\Vert \\frac{1}{s} \\int _0^s \\big ( e^{-i\\theta (-x + \\lambda _n(a_l,a_0)^2)} - 1 \\big ) \\left(- x + \\lambda _n(a_l,a_0)^2\\right) u_{0,n} \\, d\\theta \\right\\Vert ^2_{L^2(\\mathbb {R}^{d-1})}\\\\&& \\lesssim _t \\sum _{n\\in \\mathbb {N}}\\left\\Vert \\left(- x + \\lambda _n(a_l,a_0)^2\\right) u_{0,n} \\right\\Vert ^2_{L^2(\\mathbb {R}^{d-1})}\\\\&& \\lesssim _t \\left\\Vert H_{a_l,a_0}u_0\\right\\Vert _{L^2(\\Omega )}^2$ This series of functions converges uniformly in $s$ so we can take the limit, which proves that for any $t \\in \\mathbb {R}$ $\\left\\Vert i\\frac{u(t+s)-u(t)}{s} -H_{a_l,a_0}u(t)\\right\\Vert ^2 \\xrightarrow[s \\rightarrow 0]{} 0.$ This proves that $u$ is differentiable and $i u^{\\prime }(t) + H_{a_l,a_0}u(t) = 0$ , so $u$ is indeed a solution of (REF ).", "Now we can prove Theorem REF : According to Proposition REF we have existence and uniqueness for the solution $u$ of the problem REF .", "Then with Proposition REF we have $\\left\\Vert u(t)\\right\\Vert _{L^2(\\Omega )}^2& \\lesssim \\sum _{n \\in \\mathbb {N}} \\left\\Vert e^{-it \\left(-x + \\lambda _n(a_l,a_0)^2 \\right)} u_{0,n}\\right\\Vert _{L^2(\\mathbb {R}^{d-1})}^2 \\\\& \\lesssim \\sum _{n \\in \\mathbb {N}} e^{t \\mathop {\\rm {Im}}\\nolimits \\left(\\lambda _n(a_l,a_0)^2\\right)}\\left\\Vert u_{0,n}\\right\\Vert _{L^2(\\mathbb {R}^{d-1})}^2 .$ Proposition REF gives $\\gamma _{a_l,a_0} > 0$ such that $\\left\\Vert u(t)\\right\\Vert _{L^2(\\Omega )}^2 \\lesssim e^{-t \\gamma _{a_l,a_0}} \\sum _{n \\in \\mathbb {N}} \\left\\Vert u_{0,n}\\right\\Vert _{L^2(\\mathbb {R}^{d-1})}^2 \\lesssim e^{-t \\gamma _{a_l,a_0}} \\left\\Vert u_0\\right\\Vert _{L^2(\\Omega )}^2,$ which concludes the proof.", "In the end of this section we show that the smallness assumption on $\\left\|a_l\\right\|+ \\left\|a_0\\right\|$ is necessary in Theorem REF .", "More precisely, if $\\left\|a_l\\right\|$ and $\\left\|a_0\\right\|$ are too large, then the transverse operator $T_{a_l,a_0}$ has eigenvalues with positive imaginary parts and hence the solution of the Schrödinger equation grows exponentially.", "Proposition 7.4 Let $a_l, a_0\\in \\mathbb {R}$ with $a_l+ a_0> 0$ and $a_la_0< 0$ .", "Let $n \\in \\mathbb {N}$ .", "If $s > 0$ is large enough, we have $\\mathop {\\rm {Im}}\\nolimits \\big (\\lambda _n(sa_l,sa_0)^2\\big ) > 0$ .", "We know that the curves $s \\mapsto \\lambda _n(sa_l,sa_0)$ for $n \\in \\mathbb {N}$ are defined for all $s \\in \\mathbb {R}$ and remain bounded.", "Moreover we have chosen the square root $\\lambda _n(sa_l,sa_0)$ of $\\lambda _n(sa_l,sa_0)^2$ which has a non-negative real part, so the imaginary parts of $\\lambda _n(sa_l,sa_0)$ and $\\lambda _n(sa_l,sa_0)^2$ have the same signs.", "Assume that $a_la_0< 0$ , and let $n \\in \\mathbb {N}^$ be fixed.", "We have $ \\frac{(\\lambda _n(sa_l,sa_0)+s a_l)(\\lambda _n(sa_l,sa_0)+sa_0)}{(\\lambda _n(sa_l,sa_0)-sa_l)(\\lambda _n(sa_l,sa_0)-sa_0)} = 1 + \\frac{2\\lambda _n(sa_l,sa_0)}{s} \\left( \\frac{1}{a_l} + \\frac{1}{a_0} \\right) + \\mathop {O}\\limits _{s \\rightarrow +\\infty } \\big ( s^{-2} \\big ).$ Since $\\mathop {\\rm {Re}}\\nolimits (\\lambda _n(sa_l,sa_0)) > n\\nu $ and $\\mathop {\\rm {Im}}\\nolimits (\\lambda _n(sa_l,sa_0))$ is bounded, this quantity is of norm less than 1 when $s > 0$ is large enough, so $e^{-2l\\mathop {\\rm {Im}}\\nolimits (\\lambda _n(sa_l,sa_0))} = \\left\|e^{2il\\lambda _n(sa_l,sa_0)}\\right\| < 1,$ and hence $\\mathop {\\rm {Im}}\\nolimits (\\lambda _n(sa_l,sa_0)) > 0$ .", "When $n = 0$ , the same holds if we can prove that $\\lambda _0(sa_l,sa_0)$ does not go to 0 for large $s$ .", "Indeed, in this case the only possibility to have $e^{2il\\lambda _0(sa_l,sa_0)} \\rightarrow 1$ is that $\\lambda _0(sa_l,sa_0)$ goes to $\\nu $ , and then $\\mathop {\\rm {Re}}\\nolimits (\\lambda _0(sa_l,sa_0))$ is bounded by below by a positive constant, and we can conclude as before.", "So assume by contradiction that $\\lambda _0(sa_l,sa_0)$ goes to 0 as $s$ goes to $+\\infty $ .", "Then we have $e^{2il\\lambda _0(sa_l,sa_0)} = 1 + 2il\\lambda _0(sa_l,sa_0)+ \\mathop {O}\\limits _{s \\rightarrow +\\infty } \\big ( \\left\|\\lambda _0(sa_l,sa_0))\\right\|^2 \\big ),$ which gives a contradiction with (REF ), where the rest $O(s^{-2})$ has to be replaced by $O\\big ( \\lambda _0^2 s^{-2}\\big )$ .", "This concludes the proof.", "Remark 7.5 We remark from (REF ) (see also Figure REF ) that given $a_l,a_0\\in \\mathbb {R}$ such that $a_l+a_0> 0$ we always have $\\mathop {\\rm {Im}}\\nolimits (\\lambda _n(a_l,a_0)) < 0$ if $n$ is large enough.", "By duality, this means that there is always an eigenvalue with positive imaginary part when $a_l+ a_0< 0$ , and hence the norm of the solution of (REF ) is always exponentially increasing in this case." ], [ "Non-constant absorption index", "In this section we prove Theorems REF and REF for a non-constant absorption index $a$ .", "For $b \\in W^{1,\\infty }(\\partial \\Omega )$ we denote by $\\Theta _b \\in {\\mathcal {L}}(H^1(\\Omega ) , H^{-1}(\\Omega ))$ the operator such that for all $\\psi \\in H^1(\\Omega )$ $\\left< \\Theta _b \\psi \\right>_{H^{-1}(\\Omega ),H^1(\\Omega )} = \\int _{\\partial \\Omega } b\\psi .$ We denote by $\\theta _b$ the corresponding quadratic form on $H^1(\\Omega )$ .", "We also denote by $\\tilde{H}_{a}$ the operator in ${\\mathcal {L}}\\big (H^1(\\Omega ),H^{-1}(\\Omega ) \\big )$ such that $\\left< \\tilde{H}_{a} \\psi \\right>_{H^{-1},H^1} = q_a(\\psi )$ for all $\\psi \\in H^1(\\Omega )$ .", "Let $z \\in +$ .", "According to the Lax-Milgram Theorem, $(1+i)(\\tilde{H}_{a}-z)$ is an isomorphism from $H^1(\\Omega )$ to $H^{-1}(\\Omega )$ .", "Moreover, for $f \\in L^2(\\Omega ) \\subset H^{-1}(\\Omega )$ we have $(\\tilde{H}_{a}-z)^{-1}f = (H_a-z)^{-1}f.$ The following proposition relies on a suitable version of the so-called quadratic estimates: Proposition 5.1 Let $a_0 > 0$ be as in the statement of Theorem REF .", "Assume that (REF ) holds everywhere on $\\partial \\Omega $ .", "Let $B \\in {\\mathcal {L}}\\big (H^1(\\Omega ),L^2(\\Omega ) \\big )$ .", "Then there exists $C \\geqslant 0$ such that for all $z \\in +$ we have $\\left\\Vert B (\\tilde{H}_{a}-z)^{-1}B^\\right\\Vert _{{\\mathcal {L}}(L^2(\\Omega ))} \\leqslant C \\left\\Vert B (\\tilde{H}_{a_0}-z)^{-1}B^\\right\\Vert _{{\\mathcal {L}}(L^2(\\Omega ))}.$ For $z \\in +$ the resolvent identity applied to $\\tilde{H}_{a}= \\tilde{H}_{a_0}+ \\Theta _{a-a_0}$ gives $ B(\\tilde{H}_{a}-z)^{-1}B^* = B (\\tilde{H}_{a_0}-z)^{-1}B^* - B (\\tilde{H}_{a_0}-z)^{-1}\\Theta _{a-a_0} (\\tilde{H}_{a}-z)^{-1}B^.$ Let $\\psi \\in L^2(\\Omega )$ .", "Since $\\Theta _{a-a_0}$ is associated to a non-negative quadratic form on $H^1(\\Omega )$ , the Cauchy-Schwarz inequality gives ${\\left< B(\\tilde{H}_{a_0}-z)^{-1}\\Theta _{a-a_0} (\\tilde{H}_{a}-z)^{-1}B^ \\psi \\right>_{L^2}}\\\\&& = \\left< \\Theta _{a-a_0} (\\tilde{H}_{a}-z)^{-1}B^* (\\tilde{H}_{a_0}^* - \\overline{z})^{-1}B^* \\psi \\right> _{H^{-1},H^1}\\\\&& \\leqslant \\theta _{a-a_0} \\big ((\\tilde{H}_{a}-z)^{-1}B^* )^{\\frac{1}{2}} \\times \\theta _{a-a_0} \\big ((\\tilde{H}_{a_0}^* -\\overline{z})^{-1}B^* \\psi \\big ) ^{\\frac{1}{2}}.$ The first factor is estimated as follows: ${\\theta _{a-a_0} \\big ((\\tilde{H}_{a}-z)^{-1}B^* )\\leqslant \\theta _{a} \\big ((\\tilde{H}_{a}-z)^{-1}B^* )}\\\\&& \\leqslant \\frac{1}{2i} \\left< 2i (\\Theta _{a} + \\mathop {\\rm {Im}}\\nolimits z) (\\tilde{H}_{a}-z)^{-1}B^* (\\tilde{H}_{a}- z)^{-1}B^* > _{H^{-1},H^1}\\\\&& \\leqslant \\frac{1}{2i} \\left< B \\big ((\\tilde{H}_{a}-z)^{-1}-(\\tilde{H}_{a}^* - \\overline{z})^{-1}\\big ) B^* > _{L^2}\\\\&& \\leqslant \\left\\Vert B(\\tilde{H}_{a}-z)^{-1}B^\\right\\Vert _{{\\mathcal {L}}(L^2)} \\left\\Vert \\Vert _{L^2}^2.\\right.We can proceed similarly for the other factor, using the fact that \\right.\\right.a-a_0 \\leqslant \\alpha a_0 for some \\alpha \\geqslant 0.", "Thus there exists C \\geqslant 0 such that{\\begin{@align}{1}{-1}\\left\\Vert B (\\tilde{H}_{a_0}-z)^{-1}\\Theta _{a-a_0} (\\tilde{H}_{a}-z)^{-1}B^* \\right\\Vert & \\leqslant C \\left\\Vert B (\\tilde{H}_{a_0}-z)^{-1}B^\\right\\Vert ^{\\frac{1}{2}} \\left\\Vert B (\\tilde{H}_{a}-z)^{-1}B^\\right\\Vert ^{\\frac{1}{2}}\\\\& \\leqslant \\frac{C^2}{2} \\left\\Vert B (\\tilde{H}_{a_0}-z)^{-1}B^\\right\\Vert + \\frac{1}{2} \\left\\Vert B (\\tilde{H}_{a}-z)^{-1}B^\\right\\Vert .\\end{@align}}With (\\ref {eq-res-identity}), the conclusion follows.$ Now we can finish the proof of Theorem REF : According to Corollary REF and Proposition REF applied with $B = \\operatorname{Id}_{L^2(\\Omega )}$ (note that we can simply replace $\\tilde{H}_{a}$ by $H_a$ when $B \\in {\\mathcal {L}}(L^2(\\Omega ))$ ), there exists $C > 0$ such that $\\left\\Vert (H_a-z)^{-1}\\right\\Vert _{{\\mathcal {L}}(L^2(\\Omega ))} \\leqslant C$ for all $z \\in +$ .", "Then it only remains to choose $\\tilde{\\gamma }\\in \\big ] 0, \\frac{1}{C} \\big [$ to conclude.", "The second statement, concerning the case where $a$ vanishes on one side of the boundary, is proved similarly.", "Let us now turn to the proof of Theorem REF .", "We first prove another resolvent estimate in which we see the smoothing effect in weighted spaces: Proposition 5.2 Let $\\delta > \\frac{1}{2}$ .", "Then there exists $C \\geqslant 0$ such that for all $z \\in \\mathbb {C}_+$ we have $\\left\\Vert \\left< x \\right> ^{-\\delta } (1- \\Delta _x)^{\\frac{1}{4}} (H_a-z)^{-1} (1- \\Delta _x)^{\\frac{1}{4}} \\left< x \\right> ^{-\\delta }\\right\\Vert _{\\mathcal {L} (L^2(\\Omega ))} \\leqslant C.$ It is known for the free laplacian that for $\\left\| \\mathop {\\rm {Re}} (\\zeta ) \\right\| \\gg 1$ and $\\mathop {\\rm {Im}}(\\zeta ) > 0$ we have $\\left\\Vert \\left< x \\right>^{-\\delta } (-\\Delta _x -\\zeta )^{-1} \\left< x \\right>^{-\\delta }\\right\\Vert _{\\mathcal {L}(L^2(\\mathbb {R}^{d-1}))} \\lesssim \\left< \\zeta \\right> ^{-\\frac{1}{2}}$ and hence $\\left\\Vert \\left< x \\right>^{-\\delta } (1- x)^{\\frac{1}{4}}(-\\Delta _x -\\zeta )^{-1} (1- \\Delta _x)^{\\frac{1}{4}}\\left< x \\right>^{-\\delta } \\right\\Vert _{\\mathcal {L} (L^2(\\mathbb {R}^{d-1}))} \\lesssim 1.$ Thus if $a$ is constant we have $\\left\\Vert \\left< x \\right>^{-\\delta } (1- x)^{\\frac{1}{4}}(-\\Delta _x + \\lambda _n(a)^2 -z)^{-1} (1- \\Delta _x)^{\\frac{1}{4}}\\left< x \\right>^{-\\delta }\\right\\Vert _{\\mathcal {L}(L^2(\\mathbb {R}^{d-1})) } \\lesssim 1,$ uniformly in $z \\in +$ and $n \\in \\mathbb {N}$ .", "In this case we obtain the result using the separation of variables as in Section .", "Then we conclude with Proposition REF applied with $B = \\left< x \\right> ^{-\\delta } (1-\\Delta _x)^{\\frac{1}{4}}$ .", "In fact, we first obtain an estimate on the resolvent $(\\tilde{H}_a -z)^{-1}$ , but this proves that the operator $\\left< x \\right> ^{-\\delta } (1- \\Delta _x)^{\\frac{1}{4}} (H_a-z)^{-1} (1- \\Delta _x)^{\\frac{1}{4}} \\left< x \\right> ^{-\\delta }$ extends to a bounded operator on $L^2(\\Omega )$ , and then the same estimate holds for the corresponding closure.", "With the second estimate of Proposition REF , we can apply the theory of relatively smooth operators (see §XIII.7 in [37]).", "However, since $H_a$ is not self-adjoint but only maximal dissipative, we have to use a self-adjoint dilation (see [31]) of $H_a$ , as is done in the proof of [36] (see also Proposition 2.24 in [35]).", "Time decay for the Schrödinger equation In this section we prove Theorem REF .", "Let $u_0 \\in {\\mathcal {D}}(H_a)$ and let $u$ be the solution of the problem (REF ).", "We know that $\\left\\Vert u(t)\\right\\Vert _{L^2(\\Omega )} \\leqslant \\left\\Vert u_0\\right\\Vert _{L^2(\\Omega )}$ for all $t \\geqslant 0$ , so the result only concerns large times.", "Let $\\tilde{\\gamma }> 0$ be given by Theorem REF and $\\gamma = \\tilde{\\gamma }/ 3$ .", "Let $C^\\infty (\\mathbb {R})$ be equal to 0 on $]-\\infty ,1]$ and equal to 1 on $[2,+\\infty [$ .", "For $t \\in \\mathbb {R}$ we set $u_t) = t) u(t),$ and for $z \\in +$ : $v(z) = \\int _\\mathbb {R}e^{itz} u_t) \\, dt.$ The map $t \\mapsto e^{-\\gamma t}u_t)$ belongs to $L^1(\\mathbb {R}) \\cap L^2(\\mathbb {R}) \\cap C^1(\\mathbb {R})$ and its derivative is in $L^1(\\mathbb {R})$ so $\\mapsto v(+i\\gamma )$ is bounded and decays at least like $\\left< \\right>^{-1}$ .", "In particular it is in $L^2(\\mathbb {R})$ .", "For $R > 0$ we set $u_R(t) = \\frac{1}{2\\pi } \\int _{-R}^R e^{-it(+i\\gamma )} v(+i\\gamma ) \\, d.$ Then $\\left\\Vert e^{-t\\gamma } (u_ u_R)\\right\\Vert _{L^2(\\mathbb {R}_t,L^2(\\Omega ))} \\xrightarrow[R \\rightarrow +\\infty ]{} 0.$ Since $u_ is continuous, Theorem \\ref {th-energy-decay} will be proved if we can show that there exists $ C 0$ which does not depend on $ u0$ and such that for all $ t 0$ we have\\begin{equation} \\limsup _{R \\rightarrow \\infty } \\left\\Vert u_R(t)\\right\\Vert _{L^2(\\Omega )} \\leqslant C e^{-\\gamma t} \\left\\Vert u_0\\right\\Vert _{L^2(\\Omega )}.\\end{equation}For $ z we set $\\theta (z) = -i \\int _\\mathbb {R}e^{itz} (t) u(t) \\, dt = -i \\int _1^2 e^{itz} (t) u(t) \\, dt.$ Let $z \\in +$ .", "We multiply (REF ) by $t) e^{itz}$ and integrate over $t \\in \\mathbb {R}$ .", "After partial integration we obtain $ v(z) = (H_a-z)^{-1}\\theta (z).$ Then $v$ extends to a holomorphic function on ${3\\gamma }$ , taking this equality as a definition.", "According to the Cauchy Theorem we have in $L^2(\\mathbb {R}_t)$ $ \\begin{aligned}\\lim _{R \\rightarrow \\infty } u_R(t)= \\frac{1}{2\\pi } \\lim _{R \\rightarrow \\infty } \\int _{-R}^R e^{-it(-2i\\gamma )} v(-2i\\gamma )\\,d= {e^{-2\\gamma t}} \\lim _{R \\rightarrow \\infty } \\widetilde{u_R}(t),\\end{aligned}$ where for $t \\in \\mathbb {R}$ we have set $\\widetilde{u_R}(t) = \\int _{-R}^R e^{-it}v(-2i\\gamma ) \\, d.$ According to Plancherel's equality and Theorem REF we have uniformly in $R >0$ : $\\int _\\mathbb {R}\\left\\Vert \\widetilde{u_R} (t)\\right\\Vert ^2_{L^2(\\Omega )} \\, dt& = \\int _{-R}^R \\left\\Vert \\big (H_a-(-2i\\gamma )\\big )^{-1}\\theta (-2i\\gamma )\\right\\Vert _{L^2(\\Omega )}^2 \\, d\\\\& \\lesssim \\int _\\mathbb {R}\\left\\Vert \\theta (-2i\\gamma )\\right\\Vert _{L^2(\\Omega )}^2 \\, d\\\\& \\lesssim \\int _\\mathbb {R}e^{2\\gamma t}\\left\|(t)\\right\| \\left\\Vert u(t)\\right\\Vert _{L^2(\\Omega )}^2 \\, dt \\\\& \\lesssim \\left\\Vert u_0\\right\\Vert ^2_{L^2(\\Omega )}.$ In particular there exists $C \\geqslant 0$ such that for $u_0 \\in {\\mathcal {D}}(H_a)$ and $R > 0$ we can find $T(u_0,R) \\in [0,1]$ which satisfies $\\left\\Vert \\widetilde{u_R} (T(u_0,R))\\right\\Vert _{L^2(\\Omega )} \\leqslant C \\left\\Vert u_0\\right\\Vert _{L^2(\\Omega )}.$ Let $R > 0$ .", "Then $\\widetilde{u_R} \\in C^1(\\mathbb {R})$ and for $t \\geqslant 1$ we have $\\widetilde{u_R} (t) = e^{-i(t-T(u_0,R))H_a} \\widetilde{u_R} (T(u_0,R)) + \\int _{T(u_0,R)} ^t \\frac{\\partial }{\\partial s} \\left( e^{-i(t-s)H_a} \\widetilde{u_R}(s) \\right) \\,ds,$ where $\\frac{\\partial }{\\partial s} \\left( e^{-i(t-s)H_a} \\widetilde{u_R}(s) \\right)& = \\frac{\\partial }{\\partial s}\\int _{-R}^R e^{-is} e^{-i(t-s)H_a} \\big (H_a-(-2i\\gamma )\\big )^{-1}\\theta (-2i\\gamma ) \\, d\\\\& = i \\int _{-R}^R e^{-i(t-s)H_a} e^{-is} (H_a-) \\big ( H_a- (-2i\\gamma ) \\big ) ^{-1}\\theta (-2i\\gamma ) \\, d\\\\& = 2\\gamma e^{-i(t-s)H_a} \\widetilde{u_R} (s) + i e^{-i(t-s)H_a} \\int _{-R}^R e^{-is} \\theta (-2i\\gamma ) \\, d.$ This proves that the map $s \\mapsto \\frac{\\partial }{\\partial s} \\left( e^{-i(t-s)H_a} \\widetilde{u_R}(s) \\right)$ belongs to $L^2([0,t],L^2(\\Omega ))$ uniformly in $t$ and $R>0$ , and its $L^2([0,t],L^2(\\Omega ))$ norm is controlled by the norm of $u_0$ in $L^2(\\Omega )$ .", "We finally obtain $C \\geqslant 0$ such that for all $t \\in \\mathbb {R}$ and $R > 0$ we have $\\left\\Vert \\widetilde{u_R} (t)\\right\\Vert _{L^2(\\Omega )} \\leqslant C \\left< t \\right>^{\\frac{1}{2}} \\left\\Vert u_0\\right\\Vert _{L^2(\\Omega )}.$ With (REF ) this proves () and concludes the proof of Theorem REF .", "The case of a weakly dissipative boundary condition In this section we prove Theorem REF about the problem (REF ).", "The absorption index $a$ now takes the value $a_l$ on $\\mathbb {R}^{d-1} \\times \\left\\lbrace l \\right\\rbrace $ and $a_0$ on $\\mathbb {R}^{d-1} \\times \\left\\lbrace 0 \\right\\rbrace $ .", "The proof follows the same lines as in the dissipative case, except that well-posedness of the problem is not an easy consequence of the general dissipative theory.", "We will use the separation of variables as in Section instead.", "Once we have a decomposition as in Proposition REF for the initial datum, we can propagate each term by means of the unitary group generated by $-x$ and define the solution of (REF ) as a series of solutions on $\\mathbb {R}^{d-1}$ .", "Let us first look at the transverse problem.", "The transverse operator on $L^2(0,l)$ corresponding to the problem (REF ) is now given by $T_{a_l,a_0}= - \\frac{d^2}{dy^2}$ with domain ${\\mathcal {D}}(T_{a_l,a_0}) = \\left\\lbrace u \\in H^2(0,l) \\,:\\,u^{\\prime }(0) = -ia_0u(0), u^{\\prime }(l) = ia_lu(l) \\right\\rbrace .$ As already mentioned in Remark REF , we can reproduce exactly the same analysis as in Section if $a_l> 0$ and $a_0>0$ (or if one of them vanishes).", "In particular, there is no restriction on the sizes of these coefficients.", "The results we give here to handle the weakly dissipative case are also valid in this situation.", "The strategy will be the same as in Section , so we will only emphasize the differences.", "We first remark that 0 is an eigenvalue of $T_{a_l,a_0}$ if and only if $a_l=a_0=0$ .", "Otherwise, $\\lambda ^2 \\in $ is an eigenvalue of $T_{a_l,a_0}$ if and only if $ (\\lambda - a_l)(\\lambda - a_0) e^{2i\\lambda l} = (\\lambda + a_l) (\\lambda + a_0).$ We recover (REF ) when $a_l= a_0$ .", "Lemma 7.1 Let $a_l,a_0\\in \\mathbb {R}$ and $\\lambda \\in $ be such that $a_l+a_0\\ne 0$ and $\\lambda ^2$ is an eigenvalue of $T_{a_l,a_0}$ .", "Then $\\mathop {\\rm {Re}}\\nolimits (\\lambda ) \\notin \\nu \\mathbb {N}$ .", "We recall from [23] that if $a_l+a_0= 0$ (${\\mathcal {P}}{\\mathcal {T}}$ -symmectric case) then $n^2 \\nu ^2 \\in \\sigma (T_{a_l,a_0})$ for all $n \\in \\mathbb {N}^$ (see also Figure REF for $a_l+a_0> 0$ small).", "$\\bullet $ We assume by contradiction that $\\mathop {\\rm {Re}}\\nolimits (\\lambda ) \\in \\nu \\mathbb {N}$ .", "According to (REF ) we have $\\frac{(\\lambda + a_l)(\\lambda +a_0)}{(\\lambda -a_l)(\\lambda -a_0)} = e^{2il\\lambda } = e^{-2 l\\mathop {\\rm {Im}}\\nolimits (\\lambda )} \\in \\mathbb {R}_+^.$ After multiplication by $\\left\|\\lambda - a_l\\right\|^2 \\left\|\\lambda - a_0\\right\|^2 \\in \\mathbb {R}_+^$ we obtain $\\big ( \\left\|\\lambda \\right\|^2 - 2 i a_l\\mathop {\\rm {Im}}\\nolimits (\\lambda ) - a_l^2 \\big ) \\big ( \\left\|\\lambda \\right\|^2 - 2 i a_0\\mathop {\\rm {Im}}\\nolimits (\\lambda ) - a_0^2 \\big ) \\in \\mathbb {R}_+^.$ Taking the real and imaginary parts gives $ \\left\|\\lambda \\right\|^4 - (a_l^2 + a_0^2) \\left\|\\lambda \\right\|^2 + a_l^2 a_0^2 - 4 a_la_0\\mathop {\\rm {Im}}\\nolimits (\\lambda )^2 > 0$ and $ 2 \\mathop {\\rm {Im}}\\nolimits (\\lambda ) (a_l+a_0) \\big (\\left\|\\lambda \\right\|^2 - a_la_0\\big ) = 0.$ $\\bullet $ Assume that $a_la_0\\geqslant 0$ .", "In this case $\\mathop {\\rm {Im}}\\nolimits (\\lambda ) \\ne 0$ (for the same reason as in the proof of Proposition REF ), so (REF ) implies $\\left\|\\lambda \\right\|^2 = a_la_0$ .", "Then (REF ) reads $- a_la_0(a_l- a_0)^2 - 4a_la_0\\mathop {\\rm {Im}}\\nolimits (\\lambda )^2 > 0,$ which gives a contradiction.", "$\\bullet $ Now assume that $a_la_0< 0$ .", "Then (REF ) implies $\\mathop {\\rm {Im}}\\nolimits (\\lambda ) = 0$ and hence $e^{2il\\lambda } = 1$ .", "From (REF ) we now obtain $(\\lambda -a_l) (\\lambda -a_0) = (\\lambda +a_l) (\\lambda +a_0),$ which is impossible since $\\lambda (a_l+ a_0) \\ne 0$ .", "This concludes the proof.", "Proposition 7.2 There exists $\\rho > 0$ such that if $\\left\|a_l\\right\| + \\left\|a_0\\right\| \\leqslant \\rho $ and $a_l+ a_0> 0$ then the spectrum of $T_{a_l,a_0}$ is given by a sequence $(\\lambda _n(a_l,a_0)^2)_{n\\in \\mathbb {N}}$ of algebraically simple eigenvalues such that $\\sup _{n\\in \\mathbb {N}} \\, \\mathop {\\rm {Im}}\\nolimits \\left(\\lambda _n(a_l,a_0)^2\\right) < 0.$ Moreover, any sequence of normalized eigenfunctions corresponding to these eigenvalues forms a Riesz basis.", "Figure: λ n (a l ,a 0 )\\lambda _n(a_l,a_0) for n∈{0,⋯,30}n \\in \\lbrace 0,\\dots ,30\\rbrace and l=πl= \\pi .As in the proof of Proposition REF we can see that for any $R > 0$ there exists $C_R \\geqslant 0$ such that if $\\lambda ^2 \\in $ is an eigenvalue of $T_{a_l,a_0}$ we have $ \\left\|\\mathop {\\rm {Re}}\\nolimits \\lambda \\right\| \\leqslant R \\quad \\Rightarrow \\quad \\left\|\\mathop {\\rm {Im}}\\nolimits \\lambda \\right\| \\leqslant C_R.$ The operator $T_{a_l,a_0}$ depends analytically on the parameters $a_l$ and $a_0$ , and we know that when $a_l= a_0= 0$ the eigenvalues $n^2\\nu ^2$ for $n \\in \\mathbb {N}$ are algebraically simple.", "With the restrictions given by (REF ) and Lemma REF , we obtain as in Section a sequence of maps $(a_l,a_0) \\mapsto \\lambda _n(a_l,a_0)$ such that the eigenvalues of $T_{a_l,a_0}$ are $\\lambda _n(a_l,a_0)^2$ for $n \\in \\mathbb {N}$ .", "Let $n \\in \\mathbb {N}^$ .", "We have $ \\lambda _n(a_l,a_0)= n\\nu - \\frac{i}{n\\pi } (a_l+ a_0) + \\gamma (a_l+a_0)^2 + O \\left( \\left\|a_l\\right\|^3 , \\left\|a_0\\right\|^3 \\right),$ with $\\mathop {\\rm {Re}}\\nolimits (\\gamma ) = l/(n\\pi )^3 > 0$ .", "As in the dissipative case, we obtain that for any $a_l,a_0$ with $a_l+ a_0> 0$ these eigenvalues $\\lambda _n(a_l,a_0)^2$ are simple.", "If moreover $a_l$ and $a_0$ are small enough, the eigenvalue $\\lambda _n(a_l,a_0)^2$ is close to $(n\\nu )^2$ and away from the real axis uniformly in $n \\in \\mathbb {N}^$ (the first two terms in (REF ) are also the first two terms of the asymptotic expansion for large $n$ and fixed $a_l$ and $a_0$ ).", "It remains to check that we also have $\\mathop {\\rm {Im}}\\nolimits (\\lambda _0(a_l,a_0)^2) < 0$ .", "For small $a_l,a_0$ we denote by ${a_l,a_0}(0)$ a normalized eigenvector corresponding to the eigenvalue $\\lambda _0(a_l,a_0)^2$ and depending analytically on $a_l$ and $a_0$ .", "For all $\\psi \\in H^1(0,l)$ we have $\\left< {a_l,a_0}^{\\prime } , \\psi ^{\\prime } \\right>_{L^2(0,l)} - i a_l{{a_l,a_0}(l)} \\overline{\\psi (l)} - i a_0{{a_l,a_0}(0)} \\overline{\\psi (0)} = \\lambda _0(a_l,a_0)^2 \\left< {a_l,a_0} , \\psi \\right>_{L^2(0,l)}$ We apply this with $\\psi = {a_l,a_0}$ , take the derivatives with respect to $a_l$ and $a_0$ at point $(a_l,a_0) = (0,0)$ , and use the facts that ${0,0}$ is constant and $\\lambda _{0}(0,0) = 0$ .", "We obtain $\\nabla _{a_l,a_0} \\big (\\lambda _0^2 \\big ) = -\\frac{i}{l}\\big ( 1 , 1 \\big ).$ This proves that $\\mathop {\\rm {Im}}\\nolimits \\big (\\lambda _0(a_l,a_0)^2\\big )<0$ if $a_l$ and $a_0$ are small enough with $a_l+ a_0> 0$ .", "The Riesz basis property relies as before on the fact that $\\left\|\\lambda _n(a_l,a_0)- n\\nu \\right\| = O(n^{-1}).$ For this point we can follow what is done in Section for the dissipative case.", "For $n \\in \\mathbb {N}$ and $a_l,a_0\\in \\mathbb {R}$ with $a_l+ a_0> 0$ we consider a normalized eigenvector $n({a_l,a_0})\\in L^2(0,l)$ corresponding to the eigenvalue $\\lambda _n(a_l,a_0)^2$ of $T_{a_l,a_0}$ .", "We denote by $(n^(a_l,a_0))_{n\\in \\mathbb {N}}$ the dual basis.", "Proposition 7.3 Let $u_0 \\in {\\mathcal {D}}(H_{a_l,a_0})$ .", "Then the problem (REF ) has a unique solution $u \\in C^1 (\\mathbb {R},L^2(\\Omega )) \\cap C^0(\\mathbb {R},{\\mathcal {D}}(H_{a_l,a_0}))$ .", "Moreover if we write $u_0 = \\sum _{n\\in \\mathbb {N}} u_{0,n} \\otimes n({a_l,a_0})$ where $u_{0,n} \\in L^2(\\mathbb {R}^{d-1})$ , then $u$ is given by $u(t) = \\sum _{n\\in \\mathbb {N}} \\left( e^{-it \\left(-x+\\lambda _n(a_l,a_0)^2 \\right)} u_{0,n}\\right) \\otimes n({a_l,a_0}).$ $\\bullet $ Assume that $u \\in C^0(\\mathbb {R}_+, {\\mathcal {D}}(H_{a_l,a_0})) \\cap C^1(\\mathbb {R}_+^,L^2(\\Omega ))$ is a solution of (REF ).", "Let $t \\in \\mathbb {R}_+^$ .", "For all $n \\in \\mathbb {N}$ and almost all $x \\in \\mathbb {R}^{d-1}$ we can define $u_n(t,x) = \\left< u(t;x,\\cdot ) , _n(a_l,a_0) \\right>_{L^2(0,l)},$ so that in $L^2(\\Omega )$ we have $u(t) = \\sum _{n\\in \\mathbb {N}} u_n(t) \\otimes n({a_l,a_0}).$ According to Proposition REF (which can be proved similarly in this context) we have $u_n(t) \\in H^2(\\mathbb {R}^{d-1})$ for all $t \\in \\mathbb {R}_+^$ and $n \\in \\mathbb {N}$ , and for $s \\in \\mathbb {R}^$ we have $i\\frac{u(t+s)-u(t)}{s} - H_{a_l,a_0}u(t) \\\\ = \\sum _{n\\in \\mathbb {N}} \\left( i\\frac{u_n(t+s)-u_n(t)}{s} - \\left(-x +\\lambda _n(a_l,a_0)^2 \\right) u_n(t) \\right) \\otimes n({a_l,a_0}).$ Let $n\\in \\mathbb {N}$ .", "According to Proposition REF we have $\\left\\Vert i\\frac{u_n(t+s)-u_n(t)}{s} - \\left(-x +\\lambda _n(a_l,a_0)^2 \\right) u_n(t)\\right\\Vert _{L^2(\\mathbb {R}^{d-1})}\\\\\\lesssim \\left\\Vert i\\frac{u(t+s)-u(t)}{s} - H_{a_l,a_0}u(t)\\right\\Vert _{L^2(\\Omega )}\\xrightarrow[s \\rightarrow 0]{} 0.$ This proves that $u_n$ is differentiable and for all $t > 0$ $i u_n^{\\prime }(t) = \\left( - x + \\lambda _n(a_l,a_0)^2 \\right) u_n(t).$ Then for all $t > 0$ $u_n(t) = e^{-it\\left( -x + \\lambda _n(a_l,a_0)^2 \\right)} u_n(0) = e^{-it\\left( -x + \\lambda _n(a_l,a_0)^2 \\right)} u_{0,n}.$ $\\bullet $ Conversely, let us prove that the function $u$ defined by the statement of the proposition is indeed a solution of (REF ).", "Let $t \\in \\mathbb {R}$ .", "According to Proposition REF , $u_{0,n}$ and hence $e^{-it \\left(-x+\\lambda _n(a_l,a_0)^2 \\right)} u_{0,n}$ belong to $H^2(\\mathbb {R}^{d-1})$ for all $n \\in \\mathbb {N}$ .", "Therefore $u(t) \\in {\\mathcal {D}}(H_{a_l,a_0})$ .", "Then for all $s \\in \\mathbb {R}^$ we have ${ \\sum _{n\\in \\mathbb {N}}\\left\\Vert \\left( i\\frac{e^{-is \\left(-x + \\lambda _n(a_l,a_0)^2 \\right)} - 1}{s} -\\left(- x + \\lambda _n(a_l,a_0)^2\\right) \\right) \\, e^{-it(-x + \\lambda _n(a_l,a_0)^2)} u_{0,n}\\right\\Vert ^2 }\\\\&& \\lesssim _t \\sum _{n\\in \\mathbb {N}}\\left\\Vert \\frac{1}{s} \\int _0^s \\big ( e^{-i\\theta (-x + \\lambda _n(a_l,a_0)^2)} - 1 \\big ) \\left(- x + \\lambda _n(a_l,a_0)^2\\right) u_{0,n} \\, d\\theta \\right\\Vert ^2_{L^2(\\mathbb {R}^{d-1})}\\\\&& \\lesssim _t \\sum _{n\\in \\mathbb {N}}\\left\\Vert \\left(- x + \\lambda _n(a_l,a_0)^2\\right) u_{0,n} \\right\\Vert ^2_{L^2(\\mathbb {R}^{d-1})}\\\\&& \\lesssim _t \\left\\Vert H_{a_l,a_0}u_0\\right\\Vert _{L^2(\\Omega )}^2$ This series of functions converges uniformly in $s$ so we can take the limit, which proves that for any $t \\in \\mathbb {R}$ $\\left\\Vert i\\frac{u(t+s)-u(t)}{s} -H_{a_l,a_0}u(t)\\right\\Vert ^2 \\xrightarrow[s \\rightarrow 0]{} 0.$ This proves that $u$ is differentiable and $i u^{\\prime }(t) + H_{a_l,a_0}u(t) = 0$ , so $u$ is indeed a solution of (REF ).", "Now we can prove Theorem REF : According to Proposition REF we have existence and uniqueness for the solution $u$ of the problem REF .", "Then with Proposition REF we have $\\left\\Vert u(t)\\right\\Vert _{L^2(\\Omega )}^2& \\lesssim \\sum _{n \\in \\mathbb {N}} \\left\\Vert e^{-it \\left(-x + \\lambda _n(a_l,a_0)^2 \\right)} u_{0,n}\\right\\Vert _{L^2(\\mathbb {R}^{d-1})}^2 \\\\& \\lesssim \\sum _{n \\in \\mathbb {N}} e^{t \\mathop {\\rm {Im}}\\nolimits \\left(\\lambda _n(a_l,a_0)^2\\right)}\\left\\Vert u_{0,n}\\right\\Vert _{L^2(\\mathbb {R}^{d-1})}^2 .$ Proposition REF gives $\\gamma _{a_l,a_0} > 0$ such that $\\left\\Vert u(t)\\right\\Vert _{L^2(\\Omega )}^2 \\lesssim e^{-t \\gamma _{a_l,a_0}} \\sum _{n \\in \\mathbb {N}} \\left\\Vert u_{0,n}\\right\\Vert _{L^2(\\mathbb {R}^{d-1})}^2 \\lesssim e^{-t \\gamma _{a_l,a_0}} \\left\\Vert u_0\\right\\Vert _{L^2(\\Omega )}^2,$ which concludes the proof.", "In the end of this section we show that the smallness assumption on $\\left\|a_l\\right\|+ \\left\|a_0\\right\|$ is necessary in Theorem REF .", "More precisely, if $\\left\|a_l\\right\|$ and $\\left\|a_0\\right\|$ are too large, then the transverse operator $T_{a_l,a_0}$ has eigenvalues with positive imaginary parts and hence the solution of the Schrödinger equation grows exponentially.", "Proposition 7.4 Let $a_l, a_0\\in \\mathbb {R}$ with $a_l+ a_0> 0$ and $a_la_0< 0$ .", "Let $n \\in \\mathbb {N}$ .", "If $s > 0$ is large enough, we have $\\mathop {\\rm {Im}}\\nolimits \\big (\\lambda _n(sa_l,sa_0)^2\\big ) > 0$ .", "We know that the curves $s \\mapsto \\lambda _n(sa_l,sa_0)$ for $n \\in \\mathbb {N}$ are defined for all $s \\in \\mathbb {R}$ and remain bounded.", "Moreover we have chosen the square root $\\lambda _n(sa_l,sa_0)$ of $\\lambda _n(sa_l,sa_0)^2$ which has a non-negative real part, so the imaginary parts of $\\lambda _n(sa_l,sa_0)$ and $\\lambda _n(sa_l,sa_0)^2$ have the same signs.", "Assume that $a_la_0< 0$ , and let $n \\in \\mathbb {N}^$ be fixed.", "We have $ \\frac{(\\lambda _n(sa_l,sa_0)+s a_l)(\\lambda _n(sa_l,sa_0)+sa_0)}{(\\lambda _n(sa_l,sa_0)-sa_l)(\\lambda _n(sa_l,sa_0)-sa_0)} = 1 + \\frac{2\\lambda _n(sa_l,sa_0)}{s} \\left( \\frac{1}{a_l} + \\frac{1}{a_0} \\right) + \\mathop {O}\\limits _{s \\rightarrow +\\infty } \\big ( s^{-2} \\big ).$ Since $\\mathop {\\rm {Re}}\\nolimits (\\lambda _n(sa_l,sa_0)) > n\\nu $ and $\\mathop {\\rm {Im}}\\nolimits (\\lambda _n(sa_l,sa_0))$ is bounded, this quantity is of norm less than 1 when $s > 0$ is large enough, so $e^{-2l\\mathop {\\rm {Im}}\\nolimits (\\lambda _n(sa_l,sa_0))} = \\left\|e^{2il\\lambda _n(sa_l,sa_0)}\\right\| < 1,$ and hence $\\mathop {\\rm {Im}}\\nolimits (\\lambda _n(sa_l,sa_0)) > 0$ .", "When $n = 0$ , the same holds if we can prove that $\\lambda _0(sa_l,sa_0)$ does not go to 0 for large $s$ .", "Indeed, in this case the only possibility to have $e^{2il\\lambda _0(sa_l,sa_0)} \\rightarrow 1$ is that $\\lambda _0(sa_l,sa_0)$ goes to $\\nu $ , and then $\\mathop {\\rm {Re}}\\nolimits (\\lambda _0(sa_l,sa_0))$ is bounded by below by a positive constant, and we can conclude as before.", "So assume by contradiction that $\\lambda _0(sa_l,sa_0)$ goes to 0 as $s$ goes to $+\\infty $ .", "Then we have $e^{2il\\lambda _0(sa_l,sa_0)} = 1 + 2il\\lambda _0(sa_l,sa_0)+ \\mathop {O}\\limits _{s \\rightarrow +\\infty } \\big ( \\left\|\\lambda _0(sa_l,sa_0))\\right\|^2 \\big ),$ which gives a contradiction with (REF ), where the rest $O(s^{-2})$ has to be replaced by $O\\big ( \\lambda _0^2 s^{-2}\\big )$ .", "This concludes the proof.", "Remark 7.5 We remark from (REF ) (see also Figure REF ) that given $a_l,a_0\\in \\mathbb {R}$ such that $a_l+a_0> 0$ we always have $\\mathop {\\rm {Im}}\\nolimits (\\lambda _n(a_l,a_0)) < 0$ if $n$ is large enough.", "By duality, this means that there is always an eigenvalue with positive imaginary part when $a_l+ a_0< 0$ , and hence the norm of the solution of (REF ) is always exponentially increasing in this case." ], [ "Time decay for the Schrödinger equation", "In this section we prove Theorem REF .", "Let $u_0 \\in {\\mathcal {D}}(H_a)$ and let $u$ be the solution of the problem (REF ).", "We know that $\\left\\Vert u(t)\\right\\Vert _{L^2(\\Omega )} \\leqslant \\left\\Vert u_0\\right\\Vert _{L^2(\\Omega )}$ for all $t \\geqslant 0$ , so the result only concerns large times.", "Let $\\tilde{\\gamma }> 0$ be given by Theorem REF and $\\gamma = \\tilde{\\gamma }/ 3$ .", "Let $C^\\infty (\\mathbb {R})$ be equal to 0 on $]-\\infty ,1]$ and equal to 1 on $[2,+\\infty [$ .", "For $t \\in \\mathbb {R}$ we set $u_t) = t) u(t),$ and for $z \\in +$ : $v(z) = \\int _\\mathbb {R}e^{itz} u_t) \\, dt.$ The map $t \\mapsto e^{-\\gamma t}u_t)$ belongs to $L^1(\\mathbb {R}) \\cap L^2(\\mathbb {R}) \\cap C^1(\\mathbb {R})$ and its derivative is in $L^1(\\mathbb {R})$ so $\\mapsto v(+i\\gamma )$ is bounded and decays at least like $\\left< \\right>^{-1}$ .", "In particular it is in $L^2(\\mathbb {R})$ .", "For $R > 0$ we set $u_R(t) = \\frac{1}{2\\pi } \\int _{-R}^R e^{-it(+i\\gamma )} v(+i\\gamma ) \\, d.$ Then $\\left\\Vert e^{-t\\gamma } (u_ u_R)\\right\\Vert _{L^2(\\mathbb {R}_t,L^2(\\Omega ))} \\xrightarrow[R \\rightarrow +\\infty ]{} 0.$ Since $u_ is continuous, Theorem \\ref {th-energy-decay} will be proved if we can show that there exists $ C 0$ which does not depend on $ u0$ and such that for all $ t 0$ we have\\begin{equation} \\limsup _{R \\rightarrow \\infty } \\left\\Vert u_R(t)\\right\\Vert _{L^2(\\Omega )} \\leqslant C e^{-\\gamma t} \\left\\Vert u_0\\right\\Vert _{L^2(\\Omega )}.\\end{equation}For $ z we set $\\theta (z) = -i \\int _\\mathbb {R}e^{itz} (t) u(t) \\, dt = -i \\int _1^2 e^{itz} (t) u(t) \\, dt.$ Let $z \\in +$ .", "We multiply (REF ) by $t) e^{itz}$ and integrate over $t \\in \\mathbb {R}$ .", "After partial integration we obtain $ v(z) = (H_a-z)^{-1}\\theta (z).$ Then $v$ extends to a holomorphic function on ${3\\gamma }$ , taking this equality as a definition.", "According to the Cauchy Theorem we have in $L^2(\\mathbb {R}_t)$ $ \\begin{aligned}\\lim _{R \\rightarrow \\infty } u_R(t)= \\frac{1}{2\\pi } \\lim _{R \\rightarrow \\infty } \\int _{-R}^R e^{-it(-2i\\gamma )} v(-2i\\gamma )\\,d= {e^{-2\\gamma t}} \\lim _{R \\rightarrow \\infty } \\widetilde{u_R}(t),\\end{aligned}$ where for $t \\in \\mathbb {R}$ we have set $\\widetilde{u_R}(t) = \\int _{-R}^R e^{-it}v(-2i\\gamma ) \\, d.$ According to Plancherel's equality and Theorem REF we have uniformly in $R >0$ : $\\int _\\mathbb {R}\\left\\Vert \\widetilde{u_R} (t)\\right\\Vert ^2_{L^2(\\Omega )} \\, dt& = \\int _{-R}^R \\left\\Vert \\big (H_a-(-2i\\gamma )\\big )^{-1}\\theta (-2i\\gamma )\\right\\Vert _{L^2(\\Omega )}^2 \\, d\\\\& \\lesssim \\int _\\mathbb {R}\\left\\Vert \\theta (-2i\\gamma )\\right\\Vert _{L^2(\\Omega )}^2 \\, d\\\\& \\lesssim \\int _\\mathbb {R}e^{2\\gamma t}\\left\|(t)\\right\| \\left\\Vert u(t)\\right\\Vert _{L^2(\\Omega )}^2 \\, dt \\\\& \\lesssim \\left\\Vert u_0\\right\\Vert ^2_{L^2(\\Omega )}.$ In particular there exists $C \\geqslant 0$ such that for $u_0 \\in {\\mathcal {D}}(H_a)$ and $R > 0$ we can find $T(u_0,R) \\in [0,1]$ which satisfies $\\left\\Vert \\widetilde{u_R} (T(u_0,R))\\right\\Vert _{L^2(\\Omega )} \\leqslant C \\left\\Vert u_0\\right\\Vert _{L^2(\\Omega )}.$ Let $R > 0$ .", "Then $\\widetilde{u_R} \\in C^1(\\mathbb {R})$ and for $t \\geqslant 1$ we have $\\widetilde{u_R} (t) = e^{-i(t-T(u_0,R))H_a} \\widetilde{u_R} (T(u_0,R)) + \\int _{T(u_0,R)} ^t \\frac{\\partial }{\\partial s} \\left( e^{-i(t-s)H_a} \\widetilde{u_R}(s) \\right) \\,ds,$ where $\\frac{\\partial }{\\partial s} \\left( e^{-i(t-s)H_a} \\widetilde{u_R}(s) \\right)& = \\frac{\\partial }{\\partial s}\\int _{-R}^R e^{-is} e^{-i(t-s)H_a} \\big (H_a-(-2i\\gamma )\\big )^{-1}\\theta (-2i\\gamma ) \\, d\\\\& = i \\int _{-R}^R e^{-i(t-s)H_a} e^{-is} (H_a-) \\big ( H_a- (-2i\\gamma ) \\big ) ^{-1}\\theta (-2i\\gamma ) \\, d\\\\& = 2\\gamma e^{-i(t-s)H_a} \\widetilde{u_R} (s) + i e^{-i(t-s)H_a} \\int _{-R}^R e^{-is} \\theta (-2i\\gamma ) \\, d.$ This proves that the map $s \\mapsto \\frac{\\partial }{\\partial s} \\left( e^{-i(t-s)H_a} \\widetilde{u_R}(s) \\right)$ belongs to $L^2([0,t],L^2(\\Omega ))$ uniformly in $t$ and $R>0$ , and its $L^2([0,t],L^2(\\Omega ))$ norm is controlled by the norm of $u_0$ in $L^2(\\Omega )$ .", "We finally obtain $C \\geqslant 0$ such that for all $t \\in \\mathbb {R}$ and $R > 0$ we have $\\left\\Vert \\widetilde{u_R} (t)\\right\\Vert _{L^2(\\Omega )} \\leqslant C \\left< t \\right>^{\\frac{1}{2}} \\left\\Vert u_0\\right\\Vert _{L^2(\\Omega )}.$ With (REF ) this proves () and concludes the proof of Theorem REF ." ], [ "The case of a weakly dissipative boundary condition", "In this section we prove Theorem REF about the problem (REF ).", "The absorption index $a$ now takes the value $a_l$ on $\\mathbb {R}^{d-1} \\times \\left\\lbrace l \\right\\rbrace $ and $a_0$ on $\\mathbb {R}^{d-1} \\times \\left\\lbrace 0 \\right\\rbrace $ .", "The proof follows the same lines as in the dissipative case, except that well-posedness of the problem is not an easy consequence of the general dissipative theory.", "We will use the separation of variables as in Section instead.", "Once we have a decomposition as in Proposition REF for the initial datum, we can propagate each term by means of the unitary group generated by $-x$ and define the solution of (REF ) as a series of solutions on $\\mathbb {R}^{d-1}$ .", "Let us first look at the transverse problem.", "The transverse operator on $L^2(0,l)$ corresponding to the problem (REF ) is now given by $T_{a_l,a_0}= - \\frac{d^2}{dy^2}$ with domain ${\\mathcal {D}}(T_{a_l,a_0}) = \\left\\lbrace u \\in H^2(0,l) \\,:\\,u^{\\prime }(0) = -ia_0u(0), u^{\\prime }(l) = ia_lu(l) \\right\\rbrace .$ As already mentioned in Remark REF , we can reproduce exactly the same analysis as in Section if $a_l> 0$ and $a_0>0$ (or if one of them vanishes).", "In particular, there is no restriction on the sizes of these coefficients.", "The results we give here to handle the weakly dissipative case are also valid in this situation.", "The strategy will be the same as in Section , so we will only emphasize the differences.", "We first remark that 0 is an eigenvalue of $T_{a_l,a_0}$ if and only if $a_l=a_0=0$ .", "Otherwise, $\\lambda ^2 \\in $ is an eigenvalue of $T_{a_l,a_0}$ if and only if $ (\\lambda - a_l)(\\lambda - a_0) e^{2i\\lambda l} = (\\lambda + a_l) (\\lambda + a_0).$ We recover (REF ) when $a_l= a_0$ .", "Lemma 7.1 Let $a_l,a_0\\in \\mathbb {R}$ and $\\lambda \\in $ be such that $a_l+a_0\\ne 0$ and $\\lambda ^2$ is an eigenvalue of $T_{a_l,a_0}$ .", "Then $\\mathop {\\rm {Re}}\\nolimits (\\lambda ) \\notin \\nu \\mathbb {N}$ .", "We recall from [23] that if $a_l+a_0= 0$ (${\\mathcal {P}}{\\mathcal {T}}$ -symmectric case) then $n^2 \\nu ^2 \\in \\sigma (T_{a_l,a_0})$ for all $n \\in \\mathbb {N}^$ (see also Figure REF for $a_l+a_0> 0$ small).", "$\\bullet $ We assume by contradiction that $\\mathop {\\rm {Re}}\\nolimits (\\lambda ) \\in \\nu \\mathbb {N}$ .", "According to (REF ) we have $\\frac{(\\lambda + a_l)(\\lambda +a_0)}{(\\lambda -a_l)(\\lambda -a_0)} = e^{2il\\lambda } = e^{-2 l\\mathop {\\rm {Im}}\\nolimits (\\lambda )} \\in \\mathbb {R}_+^.$ After multiplication by $\\left\|\\lambda - a_l\\right\|^2 \\left\|\\lambda - a_0\\right\|^2 \\in \\mathbb {R}_+^$ we obtain $\\big ( \\left\|\\lambda \\right\|^2 - 2 i a_l\\mathop {\\rm {Im}}\\nolimits (\\lambda ) - a_l^2 \\big ) \\big ( \\left\|\\lambda \\right\|^2 - 2 i a_0\\mathop {\\rm {Im}}\\nolimits (\\lambda ) - a_0^2 \\big ) \\in \\mathbb {R}_+^.$ Taking the real and imaginary parts gives $ \\left\|\\lambda \\right\|^4 - (a_l^2 + a_0^2) \\left\|\\lambda \\right\|^2 + a_l^2 a_0^2 - 4 a_la_0\\mathop {\\rm {Im}}\\nolimits (\\lambda )^2 > 0$ and $ 2 \\mathop {\\rm {Im}}\\nolimits (\\lambda ) (a_l+a_0) \\big (\\left\|\\lambda \\right\|^2 - a_la_0\\big ) = 0.$ $\\bullet $ Assume that $a_la_0\\geqslant 0$ .", "In this case $\\mathop {\\rm {Im}}\\nolimits (\\lambda ) \\ne 0$ (for the same reason as in the proof of Proposition REF ), so (REF ) implies $\\left\|\\lambda \\right\|^2 = a_la_0$ .", "Then (REF ) reads $- a_la_0(a_l- a_0)^2 - 4a_la_0\\mathop {\\rm {Im}}\\nolimits (\\lambda )^2 > 0,$ which gives a contradiction.", "$\\bullet $ Now assume that $a_la_0< 0$ .", "Then (REF ) implies $\\mathop {\\rm {Im}}\\nolimits (\\lambda ) = 0$ and hence $e^{2il\\lambda } = 1$ .", "From (REF ) we now obtain $(\\lambda -a_l) (\\lambda -a_0) = (\\lambda +a_l) (\\lambda +a_0),$ which is impossible since $\\lambda (a_l+ a_0) \\ne 0$ .", "This concludes the proof.", "Proposition 7.2 There exists $\\rho > 0$ such that if $\\left\|a_l\\right\| + \\left\|a_0\\right\| \\leqslant \\rho $ and $a_l+ a_0> 0$ then the spectrum of $T_{a_l,a_0}$ is given by a sequence $(\\lambda _n(a_l,a_0)^2)_{n\\in \\mathbb {N}}$ of algebraically simple eigenvalues such that $\\sup _{n\\in \\mathbb {N}} \\, \\mathop {\\rm {Im}}\\nolimits \\left(\\lambda _n(a_l,a_0)^2\\right) < 0.$ Moreover, any sequence of normalized eigenfunctions corresponding to these eigenvalues forms a Riesz basis.", "Figure: λ n (a l ,a 0 )\\lambda _n(a_l,a_0) for n∈{0,⋯,30}n \\in \\lbrace 0,\\dots ,30\\rbrace and l=πl= \\pi .As in the proof of Proposition REF we can see that for any $R > 0$ there exists $C_R \\geqslant 0$ such that if $\\lambda ^2 \\in $ is an eigenvalue of $T_{a_l,a_0}$ we have $ \\left\|\\mathop {\\rm {Re}}\\nolimits \\lambda \\right\| \\leqslant R \\quad \\Rightarrow \\quad \\left\|\\mathop {\\rm {Im}}\\nolimits \\lambda \\right\| \\leqslant C_R.$ The operator $T_{a_l,a_0}$ depends analytically on the parameters $a_l$ and $a_0$ , and we know that when $a_l= a_0= 0$ the eigenvalues $n^2\\nu ^2$ for $n \\in \\mathbb {N}$ are algebraically simple.", "With the restrictions given by (REF ) and Lemma REF , we obtain as in Section a sequence of maps $(a_l,a_0) \\mapsto \\lambda _n(a_l,a_0)$ such that the eigenvalues of $T_{a_l,a_0}$ are $\\lambda _n(a_l,a_0)^2$ for $n \\in \\mathbb {N}$ .", "Let $n \\in \\mathbb {N}^$ .", "We have $ \\lambda _n(a_l,a_0)= n\\nu - \\frac{i}{n\\pi } (a_l+ a_0) + \\gamma (a_l+a_0)^2 + O \\left( \\left\|a_l\\right\|^3 , \\left\|a_0\\right\|^3 \\right),$ with $\\mathop {\\rm {Re}}\\nolimits (\\gamma ) = l/(n\\pi )^3 > 0$ .", "As in the dissipative case, we obtain that for any $a_l,a_0$ with $a_l+ a_0> 0$ these eigenvalues $\\lambda _n(a_l,a_0)^2$ are simple.", "If moreover $a_l$ and $a_0$ are small enough, the eigenvalue $\\lambda _n(a_l,a_0)^2$ is close to $(n\\nu )^2$ and away from the real axis uniformly in $n \\in \\mathbb {N}^$ (the first two terms in (REF ) are also the first two terms of the asymptotic expansion for large $n$ and fixed $a_l$ and $a_0$ ).", "It remains to check that we also have $\\mathop {\\rm {Im}}\\nolimits (\\lambda _0(a_l,a_0)^2) < 0$ .", "For small $a_l,a_0$ we denote by ${a_l,a_0}(0)$ a normalized eigenvector corresponding to the eigenvalue $\\lambda _0(a_l,a_0)^2$ and depending analytically on $a_l$ and $a_0$ .", "For all $\\psi \\in H^1(0,l)$ we have $\\left< {a_l,a_0}^{\\prime } , \\psi ^{\\prime } \\right>_{L^2(0,l)} - i a_l{{a_l,a_0}(l)} \\overline{\\psi (l)} - i a_0{{a_l,a_0}(0)} \\overline{\\psi (0)} = \\lambda _0(a_l,a_0)^2 \\left< {a_l,a_0} , \\psi \\right>_{L^2(0,l)}$ We apply this with $\\psi = {a_l,a_0}$ , take the derivatives with respect to $a_l$ and $a_0$ at point $(a_l,a_0) = (0,0)$ , and use the facts that ${0,0}$ is constant and $\\lambda _{0}(0,0) = 0$ .", "We obtain $\\nabla _{a_l,a_0} \\big (\\lambda _0^2 \\big ) = -\\frac{i}{l}\\big ( 1 , 1 \\big ).$ This proves that $\\mathop {\\rm {Im}}\\nolimits \\big (\\lambda _0(a_l,a_0)^2\\big )<0$ if $a_l$ and $a_0$ are small enough with $a_l+ a_0> 0$ .", "The Riesz basis property relies as before on the fact that $\\left\|\\lambda _n(a_l,a_0)- n\\nu \\right\| = O(n^{-1}).$ For this point we can follow what is done in Section for the dissipative case.", "For $n \\in \\mathbb {N}$ and $a_l,a_0\\in \\mathbb {R}$ with $a_l+ a_0> 0$ we consider a normalized eigenvector $n({a_l,a_0})\\in L^2(0,l)$ corresponding to the eigenvalue $\\lambda _n(a_l,a_0)^2$ of $T_{a_l,a_0}$ .", "We denote by $(n^(a_l,a_0))_{n\\in \\mathbb {N}}$ the dual basis.", "Proposition 7.3 Let $u_0 \\in {\\mathcal {D}}(H_{a_l,a_0})$ .", "Then the problem (REF ) has a unique solution $u \\in C^1 (\\mathbb {R},L^2(\\Omega )) \\cap C^0(\\mathbb {R},{\\mathcal {D}}(H_{a_l,a_0}))$ .", "Moreover if we write $u_0 = \\sum _{n\\in \\mathbb {N}} u_{0,n} \\otimes n({a_l,a_0})$ where $u_{0,n} \\in L^2(\\mathbb {R}^{d-1})$ , then $u$ is given by $u(t) = \\sum _{n\\in \\mathbb {N}} \\left( e^{-it \\left(-x+\\lambda _n(a_l,a_0)^2 \\right)} u_{0,n}\\right) \\otimes n({a_l,a_0}).$ $\\bullet $ Assume that $u \\in C^0(\\mathbb {R}_+, {\\mathcal {D}}(H_{a_l,a_0})) \\cap C^1(\\mathbb {R}_+^,L^2(\\Omega ))$ is a solution of (REF ).", "Let $t \\in \\mathbb {R}_+^$ .", "For all $n \\in \\mathbb {N}$ and almost all $x \\in \\mathbb {R}^{d-1}$ we can define $u_n(t,x) = \\left< u(t;x,\\cdot ) , _n(a_l,a_0) \\right>_{L^2(0,l)},$ so that in $L^2(\\Omega )$ we have $u(t) = \\sum _{n\\in \\mathbb {N}} u_n(t) \\otimes n({a_l,a_0}).$ According to Proposition REF (which can be proved similarly in this context) we have $u_n(t) \\in H^2(\\mathbb {R}^{d-1})$ for all $t \\in \\mathbb {R}_+^$ and $n \\in \\mathbb {N}$ , and for $s \\in \\mathbb {R}^$ we have $i\\frac{u(t+s)-u(t)}{s} - H_{a_l,a_0}u(t) \\\\ = \\sum _{n\\in \\mathbb {N}} \\left( i\\frac{u_n(t+s)-u_n(t)}{s} - \\left(-x +\\lambda _n(a_l,a_0)^2 \\right) u_n(t) \\right) \\otimes n({a_l,a_0}).$ Let $n\\in \\mathbb {N}$ .", "According to Proposition REF we have $\\left\\Vert i\\frac{u_n(t+s)-u_n(t)}{s} - \\left(-x +\\lambda _n(a_l,a_0)^2 \\right) u_n(t)\\right\\Vert _{L^2(\\mathbb {R}^{d-1})}\\\\\\lesssim \\left\\Vert i\\frac{u(t+s)-u(t)}{s} - H_{a_l,a_0}u(t)\\right\\Vert _{L^2(\\Omega )}\\xrightarrow[s \\rightarrow 0]{} 0.$ This proves that $u_n$ is differentiable and for all $t > 0$ $i u_n^{\\prime }(t) = \\left( - x + \\lambda _n(a_l,a_0)^2 \\right) u_n(t).$ Then for all $t > 0$ $u_n(t) = e^{-it\\left( -x + \\lambda _n(a_l,a_0)^2 \\right)} u_n(0) = e^{-it\\left( -x + \\lambda _n(a_l,a_0)^2 \\right)} u_{0,n}.$ $\\bullet $ Conversely, let us prove that the function $u$ defined by the statement of the proposition is indeed a solution of (REF ).", "Let $t \\in \\mathbb {R}$ .", "According to Proposition REF , $u_{0,n}$ and hence $e^{-it \\left(-x+\\lambda _n(a_l,a_0)^2 \\right)} u_{0,n}$ belong to $H^2(\\mathbb {R}^{d-1})$ for all $n \\in \\mathbb {N}$ .", "Therefore $u(t) \\in {\\mathcal {D}}(H_{a_l,a_0})$ .", "Then for all $s \\in \\mathbb {R}^$ we have ${ \\sum _{n\\in \\mathbb {N}}\\left\\Vert \\left( i\\frac{e^{-is \\left(-x + \\lambda _n(a_l,a_0)^2 \\right)} - 1}{s} -\\left(- x + \\lambda _n(a_l,a_0)^2\\right) \\right) \\, e^{-it(-x + \\lambda _n(a_l,a_0)^2)} u_{0,n}\\right\\Vert ^2 }\\\\&& \\lesssim _t \\sum _{n\\in \\mathbb {N}}\\left\\Vert \\frac{1}{s} \\int _0^s \\big ( e^{-i\\theta (-x + \\lambda _n(a_l,a_0)^2)} - 1 \\big ) \\left(- x + \\lambda _n(a_l,a_0)^2\\right) u_{0,n} \\, d\\theta \\right\\Vert ^2_{L^2(\\mathbb {R}^{d-1})}\\\\&& \\lesssim _t \\sum _{n\\in \\mathbb {N}}\\left\\Vert \\left(- x + \\lambda _n(a_l,a_0)^2\\right) u_{0,n} \\right\\Vert ^2_{L^2(\\mathbb {R}^{d-1})}\\\\&& \\lesssim _t \\left\\Vert H_{a_l,a_0}u_0\\right\\Vert _{L^2(\\Omega )}^2$ This series of functions converges uniformly in $s$ so we can take the limit, which proves that for any $t \\in \\mathbb {R}$ $\\left\\Vert i\\frac{u(t+s)-u(t)}{s} -H_{a_l,a_0}u(t)\\right\\Vert ^2 \\xrightarrow[s \\rightarrow 0]{} 0.$ This proves that $u$ is differentiable and $i u^{\\prime }(t) + H_{a_l,a_0}u(t) = 0$ , so $u$ is indeed a solution of (REF ).", "Now we can prove Theorem REF : According to Proposition REF we have existence and uniqueness for the solution $u$ of the problem REF .", "Then with Proposition REF we have $\\left\\Vert u(t)\\right\\Vert _{L^2(\\Omega )}^2& \\lesssim \\sum _{n \\in \\mathbb {N}} \\left\\Vert e^{-it \\left(-x + \\lambda _n(a_l,a_0)^2 \\right)} u_{0,n}\\right\\Vert _{L^2(\\mathbb {R}^{d-1})}^2 \\\\& \\lesssim \\sum _{n \\in \\mathbb {N}} e^{t \\mathop {\\rm {Im}}\\nolimits \\left(\\lambda _n(a_l,a_0)^2\\right)}\\left\\Vert u_{0,n}\\right\\Vert _{L^2(\\mathbb {R}^{d-1})}^2 .$ Proposition REF gives $\\gamma _{a_l,a_0} > 0$ such that $\\left\\Vert u(t)\\right\\Vert _{L^2(\\Omega )}^2 \\lesssim e^{-t \\gamma _{a_l,a_0}} \\sum _{n \\in \\mathbb {N}} \\left\\Vert u_{0,n}\\right\\Vert _{L^2(\\mathbb {R}^{d-1})}^2 \\lesssim e^{-t \\gamma _{a_l,a_0}} \\left\\Vert u_0\\right\\Vert _{L^2(\\Omega )}^2,$ which concludes the proof.", "In the end of this section we show that the smallness assumption on $\\left\|a_l\\right\|+ \\left\|a_0\\right\|$ is necessary in Theorem REF .", "More precisely, if $\\left\|a_l\\right\|$ and $\\left\|a_0\\right\|$ are too large, then the transverse operator $T_{a_l,a_0}$ has eigenvalues with positive imaginary parts and hence the solution of the Schrödinger equation grows exponentially.", "Proposition 7.4 Let $a_l, a_0\\in \\mathbb {R}$ with $a_l+ a_0> 0$ and $a_la_0< 0$ .", "Let $n \\in \\mathbb {N}$ .", "If $s > 0$ is large enough, we have $\\mathop {\\rm {Im}}\\nolimits \\big (\\lambda _n(sa_l,sa_0)^2\\big ) > 0$ .", "We know that the curves $s \\mapsto \\lambda _n(sa_l,sa_0)$ for $n \\in \\mathbb {N}$ are defined for all $s \\in \\mathbb {R}$ and remain bounded.", "Moreover we have chosen the square root $\\lambda _n(sa_l,sa_0)$ of $\\lambda _n(sa_l,sa_0)^2$ which has a non-negative real part, so the imaginary parts of $\\lambda _n(sa_l,sa_0)$ and $\\lambda _n(sa_l,sa_0)^2$ have the same signs.", "Assume that $a_la_0< 0$ , and let $n \\in \\mathbb {N}^*$ be fixed.", "We have $ \\frac{(\\lambda _n(sa_l,sa_0)+s a_l)(\\lambda _n(sa_l,sa_0)+sa_0)}{(\\lambda _n(sa_l,sa_0)-sa_l)(\\lambda _n(sa_l,sa_0)-sa_0)} = 1 + \\frac{2\\lambda _n(sa_l,sa_0)}{s} \\left( \\frac{1}{a_l} + \\frac{1}{a_0} \\right) + \\mathop {O}\\limits _{s \\rightarrow +\\infty } \\big ( s^{-2} \\big ).$ Since $\\mathop {\\rm {Re}}\\nolimits (\\lambda _n(sa_l,sa_0)) > n\\nu $ and $\\mathop {\\rm {Im}}\\nolimits (\\lambda _n(sa_l,sa_0))$ is bounded, this quantity is of norm less than 1 when $s > 0$ is large enough, so $e^{-2l\\mathop {\\rm {Im}}\\nolimits (\\lambda _n(sa_l,sa_0))} = \\left\|e^{2il\\lambda _n(sa_l,sa_0)}\\right\| < 1,$ and hence $\\mathop {\\rm {Im}}\\nolimits (\\lambda _n(sa_l,sa_0)) > 0$ .", "When $n = 0$ , the same holds if we can prove that $\\lambda _0(sa_l,sa_0)$ does not go to 0 for large $s$ .", "Indeed, in this case the only possibility to have $e^{2il\\lambda _0(sa_l,sa_0)} \\rightarrow 1$ is that $\\lambda _0(sa_l,sa_0)$ goes to $\\nu $ , and then $\\mathop {\\rm {Re}}\\nolimits (\\lambda _0(sa_l,sa_0))$ is bounded by below by a positive constant, and we can conclude as before.", "So assume by contradiction that $\\lambda _0(sa_l,sa_0)$ goes to 0 as $s$ goes to $+\\infty $ .", "Then we have $e^{2il\\lambda _0(sa_l,sa_0)} = 1 + 2il\\lambda _0(sa_l,sa_0)+ \\mathop {O}\\limits _{s \\rightarrow +\\infty } \\big ( \\left\|\\lambda _0(sa_l,sa_0))\\right\|^2 \\big ),$ which gives a contradiction with (REF ), where the rest $O(s^{-2})$ has to be replaced by $O\\big ( \\lambda _0^2 s^{-2}\\big )$ .", "This concludes the proof.", "Remark 7.5 We remark from (REF ) (see also Figure REF ) that given $a_l,a_0\\in \\mathbb {R}$ such that $a_l+a_0> 0$ we always have $\\mathop {\\rm {Im}}\\nolimits (\\lambda _n(a_l,a_0)) < 0$ if $n$ is large enough.", "By duality, this means that there is always an eigenvalue with positive imaginary part when $a_l+ a_0< 0$ , and hence the norm of the solution of (REF ) is always exponentially increasing in this case." ] ]	1403.0424
[ [ "Planetary Transit Candidates in the CSTAR Field: Analysis of the 2008\n Data" ], [ "Abstract The Chinese Small Telescope ARray (CSTAR) is a group of four identical, fully automated, static 14.5 cm telescopes.", "CSTAR is located at Dome A, Antarctica and covers 20 square degree of sky around the South Celestial Pole.", "The installation is designed to provide high-cadence photometry for the purpose of monitoring the quality of the astronomical observing conditions at Dome A and detecting transiting exoplanets.", "CSTAR has been operational since 2008, and has taken a rich and high-precision photometric data set of 10,690 stars.", "In the first observing season, we obtained 291,911 qualified science frames with 20-second integrations in the i-band.", "Photometric precision reaches about 4 mmag at 20-second cadence at i=7.5, and is about 20 mmag at i=12.", "Using robust detection methods, ten promising exoplanet candidates were found.", "Four of these were found to be giants using spectroscopic follow-up.", "All of these transit candidates are presented here along with the discussion of their detailed properties as well as the follow-up observations." ], [ "Introduction", "The detection and study of exoplanets is one of the most exciting and fastest growing fields in astrophysics.", "At the present time, several different detection methods have yielded success.", "Two of most productive methods among them have been the radial velocity method and the transit method.", "Even though among the confirmed exoplanets, the radial velocity method has been more productive, the transit method also has its own advantages.", "The spectroscopic radial velocity method measures the doppler velocity signatures of individual stars at multiple epochs, which is a very time consuming procedure.", "The photometric transit method can yield the light curves of thousands of stars simultaneously.", "More importantly, the photometric transit method provides information on planetary radius and the inclination of the planetary orbit relative to the line of sight, not possible from radial velocity detections.", "In addition, a wide array of studies are possible for transiting exoplanets, which cannot be done with non-transiting systems, e.g.", "the study of planetary atmospheres [37], temperature, surface brightness [40], [41], and the misalignment between the planetary orbit and the stellar spin [54].", "Ideally, to search for transit exoplanet requires high-quality, wide-field, long-baseline continuous time-series photometry.", "This kind of monitoring can be achieved effectively by the ambitious space-based programs such as CoRoT [2] and Kepler [6] or complicated longitude-distributed network programs such as HATNet [3] and HATSouth [4].", "However, the circumpolar locations offer a potentially comparable alternative.", "The circumpolar locations provide favorable conditions for a wide and diverse range of astronomical observations, including photometric transiting detections.", "Thanks to the extremely cold, calm atmosphere and thin turbulent surface boundary layer, as well as the absence of light and air pollution, we can obtain high quality photometric images in circumpolar locations [9], [43], [44], [45].", "Furthermore, the long polar nights offer an opportunity to obtain continuous photometric monitoring.", "As shown by a series of previous thorough and meticulous studies (cf.", "Pont & Bouchy 2005; Crouzet et al.", "2010; Daban et al.", "2010; Law et al.", "2013), it greatly increases the detectability of transiting exoplanets, particularly those with periods in excess of a few days.", "Additionally, decreased high-altitude turbulence will result in reduced scintillation noise that will lead to superior photometric precision [23].", "The significant photometric advantages of the polar regions have been proven and utilized by the observing facilities at different polar sites such as two AWCam [25] at Canadian High Arctic, SPOT [47] at the South Pole, small-IRAIT [49], ASTEP-South [13] and ASTEP-400 [16] at Dome C. Dome A, located in the deep interior of Antarctica, with the surface elevation $4,093\\,\\rm {m}$ , is the highest astronomical site on the continent and is also one of the coldest places on Earth.", "In a study that considered the weather, the boundary layer, airglow, aurorae, precipitable water vapor, surface temperature, thermal sky emission, and the free atmosphere, [36] concluded that Dome A might be the best astronomical site on Earth.", "In order to take the advantage of these remarkable observing conditions at the Dome A, the Chinese Small Telescope ARray (CSTAR) was established at Dome A in 2008 January.", "CSTAR undertook both site testing and science research tasks.", "In 2008, 291,911 qualified i-band photometric images were acquired.", "Based on these data, the first version of photometric catalog has been released by [58], and updated three times [52], [53], [29] to correct for various systematic errors.", "The resulting CSTAR photometric precision typically reaches $\\sim 4\\,\\rm {mmag}$ at 20-second cadence at $i=7.5$ , and is $\\sim 20\\,\\rm {mmag}$ at $i=12$ (see Figure REF ), which is sufficient for the detection of giant transiting exoplanets around F, G, K, dwarf stars.", "In this paper, we present ten exoplanet candidates to come from 10,690 high precision light curves selected from the CSTAR data of 2008 [53].", "From all these candidates four were found to be giants using spectroscopic follow-up.", "Since this is the first effort to find exoplanets from these data, we describe the CSTAR instrument, observations, previous data reductions and the methods used for the transit searching in detail, as well as the procedures used to eliminate the false positives.", "The layout of the paper is as follows.", "A brief description of the CSTAR instrument, observations and previous data reduction, as well as the photometric precision of the light curves, is presented in Section 2.", "In Section 3, we detail the techniques we used for transit detection and the robust procedures of data validation.", "The spectroscopic and radial velocity follow-up are briefly described in Section 4.", "We report the exoplanet candidates along with the detailed properties for each system in Section 5.", "Lastly, the work is summarized and prospects for future work are discussed in Section 6.", "CSTAR, as a part of PLATeau Observatory (PLATO) [26], [56], is the first photometric instrument to enter operation at Dome A.", "Full details of CSTAR instrument can be found in [57] and [59].", "Here we summarize the features relevant to this work.", "The CSTAR facility consists of four static, co-aligned Schmidt-Cassegrain telescopes on a fixed mount with the same $4^\\circ .5 \\times 4^\\circ .5$ Field of View (FOV) around the South Celestial Pole, each telescope housing a different filter in SDSS bands: r, g, i and open.", "Each telescope gives a $145\\,\\rm {mm}$ entrance pupil diameter (effective aperture of $100\\,\\rm { mm}$ ) and is coupled to a $1\\rm K \\times 1 \\rm K$ Andor DV 435 frame transfer CCD array which yields the plate-scale of $15\\,\\rm {arcsec}\\,\\rm {pixel}^{-1}$ ." ], [ "Observations", "CSTAR was successfully shipped and deployed at Dome A in 2008 January and operated for the following four years.", "This work is based on the data obtained in 2008.", "In the 2008 observing season (2008 March 4 to August 8), intermittent problems with the CSTAR computers and hard disks [56] prevent us from obtaining useful data in the g, r and open band.", "Fortunately the i band data were not affected, and observations were carried out for $1728\\,\\rm {hours}$ (291,911 qualified frames with 20-second exposure times) during the Antarctic polar nights, with only a few short interruptions due to cloudy weather [60] or temporary instrument problems [56].", "These observations provide well-sampled light curves with a baseline of more than one hundred days.", "Additional details of the CSTAR observations in 2008 are presented in [58]." ], [ "Previous Data Reductions", "Reduction of the CSTAR data aim to produce millimagnitude photometric precision for the bright stars.", "A custom reduction pipeline was developed which is able to achieve this goal and is described in more detail in [58] and [52], [53] as well as [29].", "Here we will only briefly review the main factors to be considered when reducing the wide-field data from CSTAR.", "After preliminary reductions, aperture photometry was performed on the sources that were detected in the all calibrated images.", "Using 48 brightest local calibrators, the instrumental magnitudes were calibrated to the $i$ magnitudes of the stars in the USNO-B 1.0 catalog [30], which were derived from the UNSO-B 1.0 magnitudes according to the transformation between USNO-B 1.0 magnitude and SDSS $i$ magnitude given by [30].", "Finally, the first version of CSTAR catalog, detailed in [58], was released.", "For transit searching, the photometric data were further refined by applying corrections for additional systematic errors, as briefly reviewed below.", "Poor weather will lead to spatial variations in extinction across the large CSTAR FOV ($4.5^{\\circ } \\times 4.5^{\\circ }$ ).", "This spatially uneven extinction can be modelled and corrected by comparing each frame to a master (median) frame.", "The more detailed procedures has been described in [52].", "The residual of the flat-field correction results in spatially dependent errors, which show up as daily variations when the stars are centered on the different pixels in different exposure frames during their diurnal motion around the south celestial pole on the static CSTAR optical system.", "This kind of diurnal effect can be effectively corrected by specific differential photometry: comparing the target object to a bright reference star in the nearby diurnal path.", "For more details, see [53].", "Since CSTAR is a static telescope and fixed to point at the South Celestial Pole, star images move clockwise on the CCD due to diurnal motion.", "Ghost images, located in symmetrical position of the CCD, move counterclockwise.", "For that reason, ghost images move and contaminate the photometry of stars.", "The significant contamination arising from the ghost images, detailed in [29], was also studied and corrected.", "The resulting light curves typically achieve a photometric precision of $\\sim 4\\,\\rm {mmag}$ at 20-second cadence for the brightest non-saturated stars ($i=7.5$ ), rising to $\\sim 20\\,\\rm {mmag}$ at $i=12$ .", "The distribution of RMS values as a function of i magnitude is shown in Figure REF .", "Each of points represents a 20-second sampled light curve with one-day observations.", "The abrupt upturn in variability at $i < 7.5$ signifies the onset of saturation, and our photometry is complete to a limiting magnitude of $i = 14$ .", "For that reason, we use the $i$ -band time-series data on the 10,690 point sources, restricted to $7.5 < i <14$ in our study, to detect transit events." ], [ "Transiting Searching Algorithm", "To search for planetary transits in the light curves, the BLS algorithm [24] is applied to the data.", "The search is limited within $1.05-30.0\\,\\rm {days}$ periods range, with 4500 period steps, 1,500 phase bins, and fractional transit length from $q_{\\rm {min}}=0.01$ to $q_{\\rm {max}}=0.1$ .", "The BLS spectra of CSTAR light curves generally display an increasing background power towards the lower frequency.", "This is caused by slight long-term systematic trends of the light curves [3].", "To remove this effects from the BLS spectra, a fourth-order polynomial is fitted and then subtracted from the spectra.", "For the most significant residual peak which do not lie at a known alias, fit statistics and parameters of the box-fitting transit model are obtained and then used to provide a ranked list of the best candidates." ], [ "Candidate Selection Criteria", "The systematic errors and true astrophysical variabilities, such as low-mass star, “blended stellar binaries\", and “grazing stellar binaries\", can mimic the true transit signals and result in a high false-positive rate.", "For this reason, it is imperative to distinguish false-positive signals from the true exoplanet candidates.", "This section describes the procedures of candidate inspection based on the techniques used in previous successful transit surveys, such as WASP [31], HATNet [3], HATSouth [4], CoRoT [2], Optical Gravitational Lensing Experiment (OGLE) [50], Kepler [6], XO [28], and Trans-Atlantic Exoplanet Survey (TrES) [1]." ], [ "Stage 1: Pre Filter", "As described in section 2.3, a total of 10,690 stars with sufficiently high precision were selected from the CSTAR data set for transit searching.", "They are processed by the detection algorithm, yielding an output of fit statistics and parameters of the box-fitting transit model.", "The large number of stars make visual inspection of every light curve infeasible.", "So we require that a number of conditions should be satisfied before subjecting the candidates to visual inspection.", "To avoid missing any interesting candidates before visual inspection, the initial selection criteria are deliberately set relatively low.", "The thresholds for rejection are: Photometric transit depth greater than 10 percent.", "The fractional change in brightness of transit depth is essentially determined by the square of the ratio of the planet radius to the host star radius.", "Giant transiting planets typically have depths on the order of one percent.", "We set a relatively loose depth criteria (10 percent) to avoid loss of interesting objects.", "Although a $\\rm {R=2\\,R_J}$ planet will block out a quarter of the light of late-type stars (e.g.", "M0 V star), as [22] pointed out, these kinds of detections from bright, wide-field surveys would be extremely rare.", "Frequencies with empty phases.", "The incomplete phase coverage leads to aliasing and can often cause false-positive detection.", "We use a simple model to exclude frequencies with poor phase coverage.", "The folded light curve is split with the expected transit width.", "A frequency is considered systematic if the number of empty intervals is larger than 2.", "Period $<$ $1.05\\,day$ or periods at a known alias The BLS algorithm, similar to other pattern matching methods, suffers from aliasing effects originating from nearly periodic sampling [24].", "Therefore, it creates false frequency peaks at period associated with one sidereal day and uniform 20-second sampling interval.", "The BLS spectra clearly display such peaks, as well as some other commonly occurring frequencies associated with the remaining systematic errors.", "For that reason stars exhibiting these periodicities are excluded in order to minimize the number of aliases.", "We have also elected not to search for transits with periods less than $1.05\\,\\rm {day}$ , due to the large number of false frequency peaks in that region.", "Even these relatively low selection criteria remove more than 85 percent of the initial detections.", "Only 1,583 candidates pass and these are then visually inspected as set out in the next section." ], [ "Stage 2: Visual Inspection", "Our visual inspection procedure is based upon that used for the successful WASP program as described in [11], [27] and [46].", "During the visual inspection of the folded light curves in conjunction with the corresponding BLS spectra, surviving candidates are require to have: Plausible transit shape.", "Since the transit depth has been limited in the stage 1, transit shape becomes the first important aspect in this stage.", "A visible transit dip is a basic requirement for a candidate to be called “transit candidate\".", "Flat out-of-transit light curve.", "The light curve before and after transit should be flat.", "Candidates are removed if they show the clear evidence of variability out of transit, including the secondary eclipse, ellipsoidal variation as well as realistic variability of other form.", "Smoothly phase coverage.", "Although candidates were systematically removed in stage 1 if their frequencies are associated with gaps in the folded light curves, some with uneven distribution of data points in the folded light curves, which may not be effectively identified in the stage 1, were deselected from further consideration by visual inspection.", "This step is also used to discard light curves of poor quality.", "Credible measured period.", "BLS spectra together with the folded light curves are inspected to confirm whether the clear period peaks are arising from secure transit signals or other variabilities.", "As the visual inspection process is somewhat subjective, it was carried out independently by the two authors (Songhu Wang and Ming Yang).", "After a comparison of the analysis, this examination reduced the 1,583 candidates down to 208 transit-like candidates, which required further investigation." ], [ "Stage 3: Statistical Filter", "The main purpose of this stage is to facilitate the further identification of the true planetary candidates from false-positive transit detections caused by systematic trends or true astrophysical variability.", "Candidates are passed forward if: Signal-to-red noise $(S_{\\rm {r}}) \\ge 7.0$.", "Contrary to the white-noise (uncorrelated-noise) assumptions, the errors on ground-based millimagnitude photometry are usually red (correlated) [33].", "In the CSTAR data, the uncertainty of the mean decreases more slowly than $n^{1/2}$ , suggesting that red-noise is present.", "This can mimic transit signal with a time-scale similar to the duration of the true close-in planetary transit.", "So, $S_{\\rm {r}}$ , a simple and robust statistical parameter to assess the significance level of detected transit in the presence of red noise, is calculated for each light curve by $S_{\\rm r} = \\frac{d \\sqrt{N_{\\rm tr}}}{\\sigma _{\\rm r}},$ where $d$ is the best-fitting transit depth, $N_{\\rm tr}$ is the number of transits observed, $\\sigma _{\\rm r}$ is the uncertainty of transit depth binned on the expected transit duration in the presence of red noise.", "The simplest method of assessing the level of red-noise ($\\sigma _{\\rm r}$ ) present in the data is to compute a sliding average of the out-of-transit data over the $n$ data points contained in a transit-length interval.", "This method is proposed by [33] and has been successfully applied to the SuperWASP candidates [10], [11], [22], [27], [46].", "The typical level of $\\sigma _{\\rm r}$ in the CSTAR data is of $2.1\\,{\\rm mmag}$ .", "It is slightly lower when compared to $3\\,\\rm {mmag}$ for the OGLE [33] and SuperWASP [39].", "For that reason, although there is no confirmed planetary transit and no simulation was performed for the $S_{\\rm r}$ threshold in the CSTAR survey, to attempt to detect more transiting planets, it is reasonable to set our $S_{\\rm r}$ threshold to the lower boundary of the typical range (7-9) of that given by [33] based on the detailed simulation with $S_{\\rm r}=3\\,\\rm {mmag}$ .", "This threshold is also consistent with that used for the SuperWASP candidates [10].", "The transit to antitransit ratio $(\\Delta \\chi ^2/\\Delta \\chi ^2_-) \\ge 1.5$.", "The systematic variations and the stellar intrinsic variables with timescale similar to the planetary transit can give rise to false-positive transit detections.", "A light curve with a genuine transit will result in only a strong transit (dimming) detection and not a strong antitransit (brightening) detection.", "On the contrary, one could expect the strong correlated measurements caused by the systematics or the stellar intrinsic variables should produce both significant transit and antitransit detections.", "Consequently, $\\Delta \\chi ^2/\\Delta \\chi ^2_-$ , measuring the ratio of improvements of best-fit transit to the improvements of the best-fit antitransit, is calculated for each light curve.", "This provides an estimate to which a detection has the expected properties of a credible transit signal rather than the properties of the systematics or intrinsic stellar sinusoidal variability [8].", "Signal to noise of the ellipsoidal variation $(S_{\\rm {ellip}})<5.0$ .", "Blended systems, gazing eclipsing binaries and eclipsing systems with a planet-sized star (e.g.", "brown dwarf) are the most common astrophysical imposters that mimic a transiting planet signal.", "It can be very difficult to distinguish these systems from genuine transiting planets using the properties of the transit event itself (e.g.", "shape, depth, etc).", "Nevertheless, evidence of ellipsoidal variability, due to tidal distortions and gravity brightening, can be used to remove from the remaining candidates which have massive, and therefore not planetary companions.", "The method, proposed by [38], was successfully applied to the OGLE [50] and the WASP [31] candidates.", "No statistical differences between odd and even transits.", "A blended or grazing eclipsing binary system can produce a shallow dip similar to an exoplanet transit.", "A true exoplanet would ideally lead to the evenly spaced transits with the same depths.", "In contrast, the depths of primary and secondary eclipses of a blended or grazing eclipsing binaries are generally different due to the difference in size and temperature of the two components.", "In addition, the primary and secondary eclipse are usually unevenly spaced in the time series since the orbit of binaries is generally eccentric [55].", "We use the significant level of the consistency in transit depth ($P_\\delta $ ) and epoch ($P_t$ ), as detailed in [55], to assess whether the odd and even transits are drawn from the same population.", "The smaller this statistic, the more likely the event is an astrophysical false positive.", "The significant level ($P_\\delta $ or $P_t$ ) of 0.05 or less denotes the transit signal is unlikely to be caused by a transiting planet.", "No aperture blends.", "Blended eclipsing binary systems are some of the most common imposters identified as the transiting planets in wide-field transit surveys such as CSTAR.", "The large plate-scale of CSTAR makes it likely that there will be more than one bright object within a single CSTAR pixel ($15\\, \\rm {arcsec}$ ) or the applied photometric aperture ($\\rm {radius}=45\\, \\rm {arcsec}$ ) of the CSTAR photometry.", "This can lead to a dilution of depth of a stellar eclipsing binary, making it appear similar to a transiting exoplanet.", "If the angular separation of the blend is less than or comparable to the pixel scale of CSTAR, we cannot eliminate the false positive arising from blended eclipsing in this step, however, imposters arising from the wider blends can be eliminated here: The candidates are eliminated if the center of a brighter object is present within 45-arcsec aperture.", "In addition, for some candidates, aperture photometry is subject to contamination by nearby bright objects (just outside the photometric aperture).", "The detected transit-like shallow dip could be due to the nearby object with a deep eclipse.", "These spurious candidates are rejected by comparing their light curves to those of nearby objects.", "We note that to avoid missing some interesting systems, some candidates with parameters just outside these thresholds have also been carried forward to the next stage.", "We find just ten candidates of the initial 208 candidates pass through these statistical filters." ], [ "Stage 4: Additional System Information", "The ten candidates which pass through the third stage are analyzed in the following manner: Stellar information.", "To estimate the radius of the transiting candidate, the radius of the host star must be determined.", "The color indices, derived from Tycho-2 $B-V$ [21], are used to estimate the spectral type and radius of the host stars based on the data from [12], assuming the host stars to be main sequence.", "Using the besancon model [34] we estimate that $40\\%$ of the stars in our FOV between $i=7.5-12$ are giants, for which the detected transit signals would then due to other stars, not planets.", "Taking [7] as a guide, the 2MASS $J-K$ colors [15] can act as a rough indicator of the luminosity class of the target.", "Candidates with $J-K>0.5$ are flagged as potentially giants.", "Refined transit parameters.", "The remaining transit light curves are modelled using the jktebop code [42].", "The refined parameters of these system, such as period, epoch, particularly the planetary radius ($R_{\\rm p}$ ), are obtained from these modeling results together with the derived host stellar radius ($R_$ ).", "Although gas giant planets, brown dwarfs and white dwarfs can all have similar radii, we regard CSTAR candidates with estimated radii less than $2\\,\\rm {R_{Jup}}$ as realistic candidates.", "The ratio of the theoretical duration and the observed duration ($\\eta $ ).", "For each candidate, we provide the ratio of the theoretical duration and the observed duration ($\\eta $ ), which is introduced for the OGLE candidates by [48] and then has been successfully applied in the WASP candidates.", "$\\eta $ of strong exoplanet candidate is expected to close to 1.", "The analysis set out in this section was only to provide additional information to remaining system but we did not use it to cull any candidates.", "In this section we describe the follow-up spectroscopy that we have undertaken to help identify two common sources of false positives in transit surveys: eclipses around giant host stars and eclipsing binaries." ], [ "Spectral Typing Follow-Up", "If a candidate host star is a giant, then its large stellar radius means that the transit event see in the discovery data cannot be due to a transiting exoplanet.", "We therefore spectral typed each of the 10 candidates to check for giant hosts.", "On the night of 2013 September 9 we took a single spectrum of each candidate with the Wide Field Spectrograph [17] on the Australian National University (ANU) $2.3\\,\\rm {m}$ telescope.", "Spectra we taken using the B3000 grating which results in a resolution of R=3000 and a wavelength range of 3500 to 6000 Å. Spectra were reduced and flux calibrated in accordance with the methodology set out in [5].", "The spectra were compared to a grid of template spectra from the MARCS models [19].", "The candidates CSTAR J021535.71-871122.5, CSTAR J014026.01-873057.1, CSTAR J203905.43-872328.2 and CSTAR J231620.78-871626.8 all showed $\\log {g}$$<3.1$ , indicating that they are giants and can be ruled out as candidates.", "The six remaining candidates are dwarfs and we therefore continued with multiple epoch RV measurements for these candidates to check for high-amplitude RV variations indicative of eclipsing binaries." ], [ "Radial Velocity Follow-Up", "For the six dwarf candidates we obtained multi-epoch radial velocity measurements using WiFeS with the R7000 grating.", "Details on the technique for obtaining radial velocity measurements on WiFeS are set out in [5].", "On nights spanning 2013 September 20-25 we took between 3 and 5 RV measurements for each six candidates spanning a range of phases for each candidate.", "None of the candidates showed any RV variation beyond the intrinsic measurement scatter of $2\\,\\rm {km\\,s^{-1}}$ , indicating that none of these unblended eclipsing binaries.", "All six therefore remain as good candidates for future high resolution radial velocity follow-up and/or photometric follow-up." ], [ "Result and Discussion", "This section we present the ten CSTAR candidates in detail and discuss the follow-up observations we have made for each candidate." ], [ "Result of transit search", "The candidate selection process result in ten promising exoplanet candidates, four of them were found to be giants using spectroscopic follow-up.", "Med-resolution radial velocity showed none of the remaining six candidates have radial velocity variation great than $2\\,\\rm {km\\,s^{-1}}$ .", "All of these candidates are listed in Table , along with the detailed information of them.", "The candidate ID is of the form `CSTAR J$hhmmss.ss-ddmmss.s$ ', with the position coordinates based on Tycho (J2000.0) position [20].", "In Figure REF we plot the theoretical curves of transit depth produced by planets of 0.5, 1.0 and 1.5 Jupiter radii as a function of host star radius assuming central crossing transit ($i=90$ ).", "All of candidates are shown as open circles.", "Those with giant host stars are over-plotted as crosses.", "It can be seen all the six remaining candidates have reasonable planetary radii between 0.5 and $1.6\\,\\rm {R_J}$ ." ], [ "Discussion of Candidates", "In this section we provide a detailed description of each of ten candidates.", "In addition, and for completeness, we also discuss the system `CSTAR J183056.78-884317.0', an eclipsing binaries with a light curve that is similar to a transiting exoplanet light curve and which has been previously identified by other groups.", "The details are summarized in Table .", "The binned phase-folded light curves of these candidates along with their respective BLS periodograms are shown in Figures 5 to 14.", "CSTAR J183056.78-884317.0 As shown in Figure REF , this system exhibits a classic, flat-bottomed transit signature in the binned folded light curve of this bright ($i=9.84$ ) star and there is a strong periodic peak at $9.93\\,\\rm {d}$ from 13 detected transits.", "However, a relatively marked ellipsoidal variation ($S/N_{\\rm {ellip}}=5.87$ ) together with a long duration ($\\sim 10\\,\\rm {h}$ ) and high value of $\\eta $ (2.03), suggest that it more likely to be an eclipsing binary.", "This object is also identified by the ASTEP team [13] and another CSTAR analysis team [51].", "To verify our analysis results, the spectroscopic observations are applied to the object using both low-resolution Wide Field Spectrograph [17] and higher-resolution echelle on the ANU $2.3\\,\\rm m$ telescope.", "The results from five observations are presented in Figure REF and show a radial velocity semi-amplitude of $K=12\\,\\rm {km\\,s^{-1}}$ , indicating that the candidate is an eclipsing binary.", "The ASTEP identification of this candidate is detailed in [14].", "CSTAR J001238.65-871811.0 This candidate has 24 transits with two percent depth and has the longest period ($5.37\\,\\rm d$ ) of the ten candidates.", "The companion radius of $0.96\\, \\rm {R_J}$ is supported by a slightly low but acceptable value of $\\eta $ (0.65).", "As all parameters of this candidate easily pass the transit-sift threshold, it is worth high-priority follow-up, although there is a relatively large scatter in the light curve and periodogram (Figure REF ).", "CSTAR J014026.01-873057.1 As show in Figure REF , the object displays a relatively shallow (0.9 percent) transit in an otherwise flat, if noisy, folded light curve with a well-defined period of $4.16\\, \\rm d$ .", "The Tycho-2 color $(B-V=1.5)$ suggests a M4 primary with $0.71\\,\\rm {R_\\odot }$ , leading to a rather small planetary radius of $0.52\\, \\rm {R_J}$ and a reasonable $\\eta =0.71$ if it was a dwarf.", "However, the very red color of the host star ($J-K=0.67$ ) suggested it was more likely to be a giant [7] and this was confirmed by our spectroscopic follow-up which gave $\\rm {log}(g)=0.6$ .", "CSTAR J021535.71-871122.5 Although there is some scatter in the light curve over the transit (Figure REF ), there is a strong peak in the periodogram.", "The observed short period ($1.438\\,\\rm d$ ) may place this candidate a very hot Jupiter.", "The exceptional high $\\bigtriangleup x^2$ /$\\bigtriangleup x^2_{-}$ (2.69) and $S_{\\rm {r}}$ (12.10) together with low $S/N_{\\rm {ellip}}$ (0.48) plus well agreed odd- and even-transits make this seem to be a strong candidate.", "However, the infrared color of the host star ($J-K=0.80$ ) suggests this object may be a giant and this was confirmed by our spectroscopic follow-up ($\\rm {log}(g)=3.3$ ).", "CSTAR J022810.02-871521.3 The object displays a transit with strong period ($2.586\\,\\rm d$ ) in an otherwise flat, if noisy, folded light curve (Figure REF ).", "The F-type primary star implies a $1.55\\,\\rm {R_{Jup}}$ companion (the largest companion of the ten candidates) and an acceptable $\\eta $ (0.61).", "These factors together with the high $\\bigtriangleup x^2$ /$\\bigtriangleup x^2_{-}$ (2.63) and low $S/N_{\\rm {ellip}}$ (0.65) make this target a good candidates.", "CSTAR J075108.62-871131.3 This candidate displays a clear transit-like dip in the folded light curve (Figure REF ) and well meet all of the selection criteria.", "The low $S/N_{\\rm {ellip}}$ (0.75) plus the high $S_{\\rm {r}}$ (8.6) as well as $\\eta \\sim 1$ make this a strong candidate.", "Although the very red color of the host star ($J-K=0.95$ ) suggests it may have been a giant, our spectroscopic follow-up ($\\rm {log}(g)=4.5$ ) suggests it more likely to be a dwarf.", "CSTAR J110005.67-871200.4 As shown in Figure REF , the transit in this candidate is obvious and there is a strong peak ($3.23\\, \\rm d$ ) in the periodogram.", "The high $S_{\\rm {r}}$ (10.6) and $\\bigtriangleup x^2$ /$\\bigtriangleup x^2_{-}$ (2.02) indicate the transit is not due to systematics.", "The $S/N_{\\rm {ellip}}$ is low at 1.2 and the light curve is flat outside of transit.", "The estimate of the host radius and transit depth indicate a companion with moderate radius ($1.34\\,\\rm {R_{Jup}}$ ) and an acceptable, if a bit low, $\\eta $ (0.55).", "The combination of these factors makes this candidate a high-priority target.", "CSTAR J113310.22-865758.3 This candidate displays a prototypical transit of one and half percent depth over an otherwise flat, if a little bit noisy, folded light curve (Figure REF ).", "The strong peak ($1.65\\,\\rm d$ ) in the periodogram together with low ellipsoidal variation ($S/N_{\\rm {ellip}}=2.17$ ) as well as a reasonable $\\eta =1.03$ indicated this brightest candidate ($i=9.97$ ) a good exoplanet candidate.", "CSTAR J132821.71-870903.3 The object clearly shows a `U'-shaped dip in an otherwise flat light curve (Figure REF ).", "This candidate has a relatively long period of $4.27\\, \\rm d$ .", "We derive a reasonable radius ($1.26\\,\\rm {R_{Jup}}$ ) of the companion for its G0 spectral type.", "However, an acceptable, but relatively low $\\eta $ (0.53) together with a slightly difference between odd-and even transit depth make this object a lower priority candidate.", "CSTAR J203905.43-872328.2 This object displays a very shallow ($\\sim 0.007\\, \\rm mag$ ) but clear flat-bottom dip with a flat out of transit light curve (Figure REF ) which shows no signs of ellipsoidal variation ($S/N_{\\rm {ellip}}=0.53$ ).", "There is a strong peak ($2.22\\,\\rm d$ ) in periodogram.", "The predicted relatively small companion radius of $0.64\\,\\rm {R_{Jup}}$ is slightly tempered by $\\eta =1.15$ .", "The relatively red 2MASS $J-K$ color (0.68) suggests a possible giant host star and it was confirmed by our spectroscopic follow-up which gave $\\rm {log}(g)=1.5$ .", "CSTAR J231620.78-871626.8 While noisy, this folded light curve (Figure REF ) exhibits a shallow transit.", "The strongest peak in the periodogram corresponds to $1.41\\,\\rm d$ which is the shortest companion of the final candidates.", "The derived radius ($0.69\\,\\rm {R_{Jup}}$ ) of companion are relatively small but the calculated transit duration is close to the observed one ($\\eta =0.94$ ).", "However the relatively red color ($J-K=0.81$ ) suggested this object may be a giant and this was confirmed by our spectroscopic follow-up ($\\rm {log}(g)=1.5$ ).", "We also note that the relatively low $S_{\\rm {r}}\\,(6.7)$ together with a slightly difference between odd- and even-transit depth indicated this candidate may have been a false positive." ], [ "Discussion of Further Follow-up Observations", "The transit method has proven to be an excellent way of finding exoplanets, however final confirmation and determination of the planetary mass and radius requires high precision photometry and radial velocity follow up.", "Such observations of the candidates in our list are being performed by our colleagues at Australia now." ], [ "Conclusion", "In 2008, more than 100 days of observations for a $20\\,\\rm {deg}^2$ field centered at the South Celestial Pole with the Antarctic CSTAR telescope provided high-precision, long-baseline light curves of 10,690 stars with a cadence of 20 seconds.", "From this data set we found ten bright exoplanet candidates with short period.", "Subsequent spectral follow-up showed that four of these were giants, leaving six candidates.", "Med-resolution radial velocity showed none of the six candidates have radial velocity variation great than $2\\,\\rm {km\\,s^{-1}}$ .", "These detections have enriched the relatively limited optical astronomy fruit in Antarctica and indirectly reflects the favorable quality of Dome A for continuous photometric observations.", "However, the real strength of CSTAR will be realized when the 2008 data are combined with the multi-color observations of following years.", "We expect to find many more candidates, especially those with longer periods and small radii, as a result of longer baseline along with higher signal to noise ratio.", "The photometric data, including all of the CSTAR catalog and the light curves, are a valuable data set for the study of variable stars as well as hunting for transit exoplanets.", "We thank the anonymous referee for the suggestions to improve the manuscript.", "This research is supported by the National Basic Research Program of China (No.", "2013CB834900, 2014CB845704, 2013CB834902, 2014CB845702), the National Natural Science Foundation of China under grant Nos.", "11333002, 11073032, 11003010, 10925313, 11373033, 11373035, 11203034, 11203031, the Strategic Priority Research Program-The Emergence of Cosmological Structures of the Chinese Academy of Sciences (Grant No.", "XDB09000000), 985 project of Nanjing University and Superiority Discipline Construction Project of Jiangsu Province, fund of Astronomy of the National Nature Science Foundation of China and the Chinese Academy of Science under Grants U1231113, the Natural Science Foundation for the Youth of Jiangsu Province (NO.", "BK20130547).", "cccccccccccccccccc 0pt Summary of CSTAR exoplanet transit candidates CSTAR ID Epoch i Period Duration Depth $R_$ $R_p$ $B-V$ $J-K$ $\\rm {T_{eff}}$ log(g) $S_p$ $\\bigtriangleup x^2$ /$\\bigtriangleup x^2_{-}$ $S/N_{\\rm {ellip}}$ $S_{\\rm {r}}$ $\\eta $ $P_{\\delta }\\mid P_t$ CSTAR J+ (2454500.0 +) (mag) (d) (h) (mag) $(\\rm {R_\\odot })$ $(\\rm {R_{Jup}})$ (mag) (mag) K 183056.78-884317.0 53.69665 9.84 9.924 10.004 0.021 1.214 1.531 0.48 0.31 — — F5 4.23 5.87 22.32 2.03 $0.42\\mid 0.38$ 001238.65-871811.0 48.80221 10.59 5.371 2.269 0.021 0.959 1.356 0.69 0.43 5900 4.9 G5 3.53 0.28 8.78 0.65 $0.66\\mid 0.74$ 014026.01-873057.1 46.69858 10.26 4.164 1.847 0.009 0.714 0.519 1.54 0.67 4800 0.6 Giant 1.48 0.26 10.37 0.71 $ 0.15\\mid 0.44$ 021535.71-871122.5 46.50898 10.69 1.438 1.360 0.018 0.740 0.862 1.65 0.80 4600 3.3 Giant 2.69 0.45 12.10 0.71 $ 0.48\\mid 0.23 $ 022810.02-871521.3 50.90359 10.62 2.586 2.048 0.021 1.274 1.547 0.44 0.36 6100 3.5 F5 2.63 0.65 7.11 0.61 $ 0.64\\mid 0.11 $ 075108.62-871131.3 47.59870 10.41 2.630 2.298 0.016 0.693 0.742 1.24 0.95 4800 4.5 K7 1.52 0.75 8.60 1.02 $0.17\\mid 0.42$ 110005.67-871200.4 47.11239 10.84 3.228 1.633 0.025 0.969 1.335 0.68 0.33 6300 3.9 G5 2.02 1.19 10.60 0.55 $ 0.07\\mid 0.62 $ 113310.22-865758.3 47.14206 9.97 1.652 2.045 0.016 0.727 0.794 1.06 0.60 4900 5.0 K4 1.63 1.72 6.96 1.03 $0.45\\mid 0.40$ 132821.71-870903.3 46.53672 10.41 4.273 1.797 0.018 1.068 1.255 0.59 0.41 6000 4.5 G0 1.62 2.17 7.05 0.53 $0.01\\mid 0.20$ 203905.43-872328.2 47.21003 10.35 2.216 2.691 0.007 0.872 0.636 0.79 0.68 4800 1.5 Giant 1.64 0.53 7.68 1.15 $0.22\\mid 0.91$ 231620.78-871626.8 46.99121 10.76 1.408 1.676 0.009 0.693 0.569 1.39 0.81 4300 2.4 Giant 2.86 0.36 6.68 0.94 $0.02\\mid 0.82$" ] ]	1403.0086
[ [ "Non-linear matter bispectrum in general relativity" ], [ "Abstract We show that the relativistic effects are negligibly small in the non-linear density and velocity bispectra.", "Although the non-linearities of Einstein equation introduce additional non-linear terms to the Newtonian fluid equations, the corrections to the bispectrum only show up on super-horizon scales.", "We show this with the next-to-leading order non-linear bispectrum for a pressureless fluid in a flat Friedmann-Robertson-Walker background, by calculating the density and velocity fields up to fourth order.", "We work in the comoving gauge, where the dynamics is identical to the Newtonian up to second order.", "We also discuss the leading order matter bispectrum in various gauges, and show yet another relativistic effect near horizon scales that the matter bispectrum strongly depends on the gauge choice." ], [ "Introduction", "Recent advances in cosmology has been greatly spurred by precise cosmological observations.", "The accurate measurements of the temperature anisotropies and polarizations in the cosmic microwave background (CMB) by the Wilkinson Microwave Anisotropy Probe (WMAP) have opened the era of precision cosmology [1], and with the most recent PLANCK data we can constrain the cosmological parameters with less than $\\mathcal {O}(1)$ percent error [2].", "With the planned experiments such as PIXIE [3], PRISM [4], and LiteBIRD [5] to mention a few, it is guaranteed that we continue our success in the CMB observations and that we can constrain the cosmological parameters further and can obtain more information on the early universe as well.", "Large scale structure (LSS) of the universe is yet another powerful cosmological probe, and its importance has ever been increasing with galaxy surveys such as SDSS [6], WiggleZ [7] and VIPERS [8].", "The LSS observations can provide the measurement of geometrical distances, growth of structures, and shape of primordial correlation functions.", "These lower redshift information combined with the CMB data can break down the degeneracies among cosmological parameters that yields better constraints than CMB alone [2].", "Furthermore, the full three-dimensional information with a huge redshift coverage available for the LSS observations naturally yields measurement of properties of dark energy, neutrino properties as well as physics of the early universe.", "A number of future observations such as HETDEX [10], MS-DESI [9], LSST [11] and Euclid [12] are proposed to observe LSS with improved accuracy in near future.", "Provided that unprecedentedly accurate data will be soon available in both CMB and LSS, our theoretical endeavour should also meet the observational precision.", "This introduces, however, a number of interesting and important questions to be addressed, especially for LSS: Non-linearity: With increasing observational accuracy, we can probe the signal beyond the two-point correlation function in CMB and LSS.", "The higher-order correlation functions are the signature of non-linearities.", "Searching for the primordial non-Gaussianity [13] is a prime example.", "The current best constraint from PLANCK is consistent with that the primordial fluctuations follow the Gaussian statistics with the local non-linearity parameter $f_{\\rm NL}= 2.7 \\pm 5.8$ at $2\\sigma $ confidence level.", "Non-linearity is more prominent in LSS: gravitational instability amplifies the density fluctuations to form non-linear structures such as galaxies and clusters of galaxies.", "As a result, the non-linearities deviates the matter power spectrum from the linear theory predictions [14], [15], and generates large higher-order correlation functions such as bispectrum and trispectrum.", "Accurate modeling of non-linearities is, therefore, the key requirement of exploiting the LSS data at the accuracy level similar to the CMB.", "Relevance of general relativity: Most studies on LSS in the past have been done in the context of the Newtonian gravity [16], which works fine in the small scale, sub-horizon limit.", "In order to achieve robust measurements of dark energy properties, for example from Baryon Acoustic Oscillations (See [17] for a recent review), planned future LSS surveys will probe larger and larger volume, and access the scales comparable to the horizon.", "Modeling the LSS observables on those large scales demands that we work in the fully general relativistic context.", "The first question that must be addressed is whether the purely relativistic effects are large enough to be detected or not.", "Furthermore, attempted modifications to general relativity (to explain the recent cosmic acceleration) mostly show up on such very large scales.", "Thus LSS is a perfect playground to test modified theories of gravity.", "Gauge: As we should resort to general relativity, at least in principle, to study LSS properly, it is crucial to clarify which `gauge' we are using to interpret the data from LSS surveys.", "Different gauges are mathematically equivalent, but it does not mean that physical clarity is also equally shared.", "In particular, in the small scale limit the `density contrasts' $\\delta \\equiv T^0{}_0/\\overline{T}^0{}_0 -1$ in almost all popular gauges are equivalent to the Newtonian density contrast [18], but equivalence does not hold on large enough scales close to the horizon.", "Of course, by properly choosing the gauge that we interpret the data, the gauge ambiguity on large scales disappears to yield the gauge invariant expression for the observable such as the galaxy power spectrum [19].", "Bearing these in mind, we are encouraged to go beyond the two-point correlation function or power spectrum, and study the higher-order correlation functions arisen from the non-linearity in general relativity.", "In this article, we study the next-to-leading order non-linearities in the matter bispectrum in the comoving gauge.", "The non-linear matter power spectrum in the same gauge was computed in [15].", "In the comoving gauge, the physical interpretation of the relativistic variables is transparent and the set of dynamical equations becomes particularly simple.", "Furthermore, the equations governing the dynamics of the density and velocity fields exactly coincide with the usual Newtonian hydrodynamic equations up to second order [20].", "Therefore, the leading order matter bispectrum, which results from correlating one second order density contrast to two linear order ones, in the full relativistic calculation must be the same as that of the Newtonian calculation, and the purely relativistic contributions appear from the third order.", "To obtain the self-consistent next-to-leading order non-linearities, we calculate the density contrast to the fourth-order.", "We compute the one-loop matter density and velocity bispectra, and confirm that the purely relativistic corrections are subdominant on cosmologically relevant scales.", "Going beyond the comoving gauge, we also calculate the leading order matter bispectrum from various other gauges to demonstrate the wild gauge dependence of the density and velocity bispectra.", "As in the case for the galaxy power spectrum, such a gauge dependence should go away when one calculate the `observable' quantities in each gauge.", "This article is organized as follows.", "In Section we present the perturbation equations of a pressureless matter in the comoving gauge.", "In Section we give the fourth order solutions of the perturbation equations in terms of kernels, and compute the matter bispectrum including one-loop corrections.", "In Section we show the total bispectrum in particular configurations of interest.", "In Section we show gauge dependence of the leading bispectrum in general relativity for large scale study.", "We conclude in Section ." ], [ "Setup and equations", "First we present the setup of the background around which we will introduce density contrast $\\delta $ and the peculiar velocity $\\mathbf {v}$ .", "We consider a flat Friedmann-Robertson-Walker universe as a background.", "Furthermore, to simplify the analysis, we consider the Einstein-de Sitter universe, i.e.", "a flat Friedmann model dominated by a pressureless matter.", "This is a good enough approximation of our universe at high redshifts.", "We find the Arnowitt-Deser-Misner formulation of 3+1 decomposition [21] particularly convenient for tracing the dynamical degrees of freedom in the system.", "The four-dimensional line element is given by $ds^2 = -N^2dt^2 + \\gamma _{ij} \\left( N^idt + dx^i \\right) \\left( N^jdt + dx^j \\right) \\, ,$ where $N$ , $N^i$ and $\\gamma _{ij}$ are, respectively, the lapse, shift and spatial metric.", "We use the Roman indices to indicate the spatial dimensions, which are raised and lowered by the spatial metric.", "We only consider scalar perturbations, because the vector and tensor contributions may be negligibly small on scales where the relativistic effects are important.", "To fix the coordinate system, we choose the comoving gauge, which is defined by $T^0{}_i = 0 \\, .$ This completely fixes the temporal gauge degree of freedom even at non-linear order [20].", "The spatial gauge degree of freedom can be fixed by taking only a trace component of perturbation in the spatial metric, $\\gamma _{ij} = & a^2(1+2\\varphi )\\delta _{ij} \\, .$ The comoving gauge condition (REF ) gives rise to a particularly simple form of the energy-momentum tensor $T_{\\mu \\nu }$ .", "We can write $T^\\mu {}_\\nu $ in the perfect fluid form $T^\\mu {}_\\nu = (\\rho +p)u^\\mu u_\\nu + p\\delta ^\\mu {}_\\nu \\, ,$ from which we can see that the comoving gauge condition demands $u_i=0$ .", "Then, for a pressureless matter, the energy and momentum densities and the spatial energy-momentum tensor which appear in the equations we are to solve are $\\mathcal {E}& \\equiv N^2T^{00} = \\rho \\, ,\\\\\\mathcal {J}_i & \\equiv NT^0{}_i = 0 \\, ,\\\\\\mathcal {S}_{ij} & \\equiv T_{ij} = 0 \\, .$ This will lead to a great simplification of the equations.", "Having the setup, we can now write the dynamical equations.", "The relevant equations are [22] $R^{(3)} + \\frac{2}{3}K^2 - \\overline{K}^i{}_j\\overline{K}^j{}_i & = 2\\mathcal {E}\\, ,\\\\\\overline{K}^j{}_{i\|j} - \\frac{2}{3}K_{\|i} & = \\mathcal {J}_i \\, ,\\\\\\mathcal {E}_{,0} - N^i\\mathcal {E}_{,i} & = NK\\left( \\mathcal {E}+ \\frac{\\mathcal {S}}{3} \\right) + N\\overline{K}^i{}_j\\overline{S}^j{}_i + \\frac{1}{N}\\left( N^2\\mathcal {J}^i \\right)_{\|i} \\, ,\\\\\\mathcal {J}_{i,0} - N^j\\mathcal {J}_{i,j} - N^j{}_{,i}\\mathcal {J}_j & = NK\\mathcal {J}_i - \\left( \\mathcal {E}\\delta ^j{}_i + \\mathcal {S}^j{}_i \\right) N_{\|j} - N\\mathcal {S}^j{}_{i\|j} \\, ,\\\\K_{,0} - N^iK_{,i} & = -N^{\|i}{}_{\|i} + N \\left( R^{(3)} + K^2 + \\frac{1}{2}\\mathcal {S}- \\frac{3}{2}\\mathcal {E}\\right) \\, ,$ which are, respectively, the energy and momentum constraints, energy and momentum conservations, and the trace part of the evolution equation.", "Here, $R^{(3)}$ is the 3-curvature scalar constructed from $\\gamma _{ij}$ , $K$ is the trace of the extrinsic curvature tensor $K_{ij}\\equiv (N_{i\|j} + N_{j\|i} - \\dot{\\gamma }_{ij})/N$ , an overbar denotes the traceless part, and a vertical bar denotes a covariant derivative with respect to $\\gamma _{ij}$ .", "Now, applying our gauge conditions in the Einstein-de Sitter universe, from the momentum conservation we can see that $N_{,i} = 0$ , i.e.", "the lapse function is homogeneous.", "Further, we can identify the perturbation variables as $\\rho = \\rho _0 + \\delta \\rho (t,{\\mathbf {x}})$ and $K = 3H-\\theta (t,{\\mathbf {x}})$ with $\\theta (t,{\\mathbf {x}}) = \\nabla \\cdot {\\mathbf {v}}(t,{\\mathbf {x}})/a$ , so that their equations in the comoving gauge exactly coincide with the Newtonian continuity and Euler equations respectively [20].", "Then, from the energy and momentum constraint equations we can write respectively $\\varphi $ and $N^i$ in terms of $\\delta \\equiv \\delta \\rho /\\rho _0$ and ${\\mathbf {v}}$ .", "We arrive at the relativistic version of the continuity and Euler equations, which are up to fourth order: $\\dot{\\delta }+ \\frac{1}{a}\\nabla \\cdot \\mathbf {v}= &-\\frac{1}{a}\\nabla \\cdot (\\delta \\mathbf {v})\\nonumber \\\\& - \\frac{1}{a} \\left[ -2\\varphi _1\\mathbf {v} + \\nabla \\left( \\Delta ^{-1}X_2 \\right) \\right]\\cdot (\\nabla \\delta )\\nonumber \\\\& - \\frac{1}{a} \\left\\lbrace -2\\varphi _2\\mathbf {v} + \\nabla \\left( \\Delta ^{-1}X_3 \\right) - 2\\varphi _1\\left[ -2\\varphi _1\\mathbf {v} + \\nabla \\left( \\Delta ^{-1}X_2 \\right) \\right] \\right\\rbrace \\cdot (\\nabla \\delta ) \\, ,$ $&-\\frac{1}{a}\\nabla \\cdot \\left( \\dot{\\mathbf {v}} + H\\mathbf {v} \\right) - \\frac{\\rho _0}{2}\\delta \\nonumber \\\\= & \\frac{1}{a^2} \\nabla \\cdot \\left[ (\\mathbf {v}\\cdot \\nabla )\\mathbf {v} \\right]\\nonumber \\\\& + \\frac{1}{a^2} \\bigg ( \\Delta \\left[ (\\mathbf {v}\\cdot \\nabla )\\Delta ^{-1}X_2 \\right] - (\\mathbf {v}\\cdot \\nabla )X_2 - \\frac{2}{3}X_2(\\nabla \\cdot \\mathbf {v}) + \\frac{2}{3}\\varphi _1(\\mathbf {v}\\cdot \\nabla )(\\nabla \\cdot \\mathbf {v}) - 4\\nabla \\cdot \\left\\lbrace \\varphi _1 \\left[ (\\mathbf {v}\\cdot \\nabla )\\mathbf {v} - \\frac{1}{3}(\\nabla \\cdot \\mathbf {v})\\mathbf {v} \\right] \\right\\rbrace \\bigg )\\nonumber \\\\& + \\frac{1}{a^2} \\bigg ( \\Delta \\left[ (\\mathbf {v}\\cdot \\nabla )\\Delta ^{-1}X_3 \\right] - (\\mathbf {v}\\cdot \\nabla )X_3 - \\frac{2}{3}X_3(\\nabla \\cdot \\mathbf {v}) + \\frac{2}{3}\\varphi _2(\\mathbf {v}\\cdot \\nabla )(\\nabla \\cdot \\mathbf {v}) - 4\\nabla \\cdot \\left\\lbrace \\varphi _2 \\left[ (\\mathbf {v}\\cdot \\nabla )\\mathbf {v} - \\frac{1}{3}(\\nabla \\cdot \\mathbf {v})\\mathbf {v} \\right] \\right\\rbrace \\nonumber \\\\& \\hspace{28.45274pt} - 4 \\nabla \\cdot \\left[ \\varphi _1 \\left\\lbrace \\left[\\nabla \\left(\\Delta ^{-1}X_2\\right)\\right]\\cdot \\nabla - \\frac{1}{3}X_2 \\right\\rbrace \\mathbf {v} + \\varphi _1 \\left\\lbrace \\left(\\mathbf {v}\\cdot \\nabla \\right) - \\frac{1}{3}\\left(\\nabla \\cdot \\mathbf {v}\\right) \\right\\rbrace \\left[\\nabla (\\Delta ^{-1}X_2)\\right] \\right]\\nonumber \\\\& \\hspace{28.45274pt} + \\frac{2}{3}\\varphi _1 \\left[ \\nabla (\\nabla \\cdot \\mathbf {v})\\cdot \\nabla \\left( \\Delta ^{-1}X_2 \\right) + 4(\\mathbf {v}\\cdot \\nabla )X_2 \\right] + 12 \\nabla \\cdot \\left\\lbrace \\varphi _1^2 \\left[ (\\mathbf {v}\\cdot \\nabla )\\mathbf {v} - \\frac{1}{3}(\\nabla \\cdot \\mathbf {v})\\mathbf {v} \\right] \\right\\rbrace - 4\\varphi _1^2(\\mathbf {v}\\cdot \\nabla )(\\nabla \\cdot \\mathbf {v})\\nonumber \\\\& \\hspace{28.45274pt} + 2 \\left(\\nabla \\varphi _1\\right)\\cdot \\left(\\nabla \\varphi _1\\right)\\mathbf {v}\\cdot \\mathbf {v} + \\frac{2}{3}\\left( \\nabla \\varphi _1\\cdot \\mathbf {v} \\right)^2 + \\nabla \\cdot \\left\\lbrace \\left[ \\nabla \\left( \\Delta ^{-1}X_2 \\right)\\cdot \\nabla \\right] \\nabla \\left( \\Delta ^{-1}X_2 \\right) - X_2 \\nabla \\left(\\Delta ^{-1}X_2\\right) \\right\\rbrace + \\frac{2}{3}X_2^2 \\bigg ) \\, ,$ where $\\Delta \\equiv \\delta ^{ij}\\partial _i\\partial _j$ and $\\Delta ^{-1}$ are spatial Laplacian and inverse Laplacian operators respectively, and $-\\frac{\\Delta }{a^2}\\varphi _1 = & \\frac{\\rho _0}{2}\\delta - \\frac{H}{a}\\nabla \\cdot \\mathbf {v} \\, ,\\\\-\\frac{\\Delta }{a^2}\\varphi _2 = & \\frac{1}{4a^2} \\left\\lbrace \\nabla \\cdot \\left[ (\\mathbf {v}\\cdot \\nabla )\\mathbf {v} \\right] - (\\mathbf {v}\\cdot \\nabla )(\\nabla \\cdot \\mathbf {v}) - (\\nabla \\cdot \\mathbf {v})^2 \\right\\rbrace - \\frac{1}{2a^2} \\left[ 3\\left(\\nabla \\varphi _1\\right)\\cdot \\left(\\nabla \\varphi _1\\right) + 8\\varphi _1\\Delta \\varphi _1 \\right] \\, ,\\\\X_2 = & 2\\varphi _1\\nabla \\cdot \\mathbf {v} - (\\mathbf {v}\\cdot \\nabla )\\varphi _1 + \\frac{3}{2}\\Delta ^{-1}\\nabla \\cdot \\left[ \\Delta \\varphi _1\\mathbf {v} + (\\mathbf {v}\\cdot \\nabla ) \\left(\\nabla \\varphi _1\\right) \\right] \\, ,\\\\X_3 = & 2\\varphi _1X_2 + 2\\varphi _2(\\nabla \\cdot \\mathbf {v}) - \\left(\\nabla \\varphi _1\\right) \\cdot \\left[\\nabla \\left(\\Delta ^{-1}X_2\\right)\\right] - (\\mathbf {v}\\cdot \\nabla )\\varphi _2- 4\\varphi _1^2(\\nabla \\cdot \\mathbf {v}) + 4\\varphi _1(\\mathbf {v}\\cdot \\nabla )\\varphi _1\\nonumber \\\\& + \\frac{3}{2}\\Delta ^{-1}\\nabla \\cdot \\left[ \\Delta \\varphi _1\\nabla \\left(\\Delta ^{-1}X_2\\right) + \\Delta \\varphi _2\\mathbf {v} + \\nabla \\left( \\Delta ^{-1}X_2 \\right) \\cdot \\nabla \\left( \\nabla \\varphi _1 \\right) + (\\mathbf {v}\\cdot \\nabla )\\nabla \\varphi _2 \\right]\\nonumber \\\\& - \\frac{3}{2}\\Delta ^{-1}\\nabla \\cdot \\left\\lbrace \\left( \\nabla \\varphi _1 \\right) \\cdot \\left( \\nabla \\varphi _1 \\right)\\mathbf {v} + 3(\\mathbf {v}\\cdot \\nabla )\\varphi _1\\nabla \\varphi _1 + 4\\varphi _1 \\left[ (\\mathbf {v}\\cdot \\nabla )\\nabla \\varphi _1 + \\Delta \\varphi _1\\mathbf {v} \\right] \\right\\rbrace \\, .$ Note that if $\\varphi _1=\\varphi _2=0$ , we recover the Newtonian continuity and Euler equations as can be read from (REF ) and (REF ), respectively.", "Thus, relativistic contributions are originated from $\\varphi _1$ and $\\varphi _2$ ." ], [ "Solutions", "We can find the non-linear solutions of (REF ) and (REF ) perturbatively as follows.", "First the order linear solutions is the same as the standard ones for the linear perturbation theory, $\\delta _1({\\mathbf {k}},t) = & D(t)\\delta _1({\\mathbf {k}},t_0) \\, ,\\\\\\theta _1({\\mathbf {k}},t) = & -aHD(t)\\delta _1({\\mathbf {k}},t_0) \\, ,$ where $D(t)$ is the linear growth factor which is normalized to unity at the present time $t=t_0$ , and $f\\equiv d\\log D/d\\log a$ is the logarithmic derivative of the linear growth factor.", "Note that $D(t) = a(t)$ in the Einstein de-Sitter universe that we are considering here.", "With these linear solutions for density and velocity, we perturbatively expand the full non-linear solutions using momentum dependent symmetric kernels as $\\delta ({\\mathbf {k}},t) = & \\sum _{n=1}^\\infty \\delta _n = \\sum _{n=1}^\\infty D^n(t) \\int \\frac{d^3q_1\\cdots d^3q_n}{(2\\pi )^{3(n-1)}} \\delta ^{(3)}({\\mathbf {k}}-{\\mathbf {q}}_{12\\cdots n}) F_n^{(s)}({\\mathbf {q}}_1, \\cdots {\\mathbf {q}}_n) \\delta _1({\\mathbf {q}}_1)\\cdots \\delta _1({\\mathbf {q}}_n) \\, ,\\\\\\theta ({\\mathbf {k}},t) = & \\sum _{n=1}^\\infty \\theta _n = -aH\\sum _{n=1}^\\infty D^n(t) \\int \\frac{d^3q_1\\cdots d^3q_n}{(2\\pi )^{3(n-1)}} \\delta ^{(3)}({\\mathbf {k}}-{\\mathbf {q}}_{12\\cdots n}) G_n^{(s)}({\\mathbf {q}}_1, \\cdots {\\mathbf {q}}_n) \\delta _1({\\mathbf {q}}_1)\\cdots \\delta _1({\\mathbf {q}}_n) \\, ,$ where $F_1({\\mathbf {k}}) = G_1({\\mathbf {k}}) = 1$ and ${\\mathbf {q}}_{12\\cdots n} \\equiv \\sum _{i=1}^n{\\mathbf {q}}_i$ .", "Note that we only consider the fastest growing mode at each order in perturbations.", "With this ansatz, (REF ) and (REF ) become simply differential equations of $F_n$ and $G_n$ .", "Because the Newtonian hydrodynamical equations are closed at second order and the relativistic equations coincide with the Newtonian ones up to second order, the purely relativistic solutions appears from third order.", "Note that, in the comoving gauge, purely relativistic terms explicitly include the comoving horizon scale $k_H \\equiv aH$ .", "The second order kernels are the same as standard perturbation theory [16], and the third order kernels are presented in (12) and (13) of [15].", "For completeness, we present the equations and solutions for the fourth order kernels in Appendix ." ], [ "Tree bispectrum", "The matter bispectrum is defined as $\\left\\langle \\delta ({\\mathbf {k}}_1,t)\\delta ({\\mathbf {k}}_2,t)\\delta ({\\mathbf {k}}_3,t) \\right\\rangle \\equiv (2\\pi )^3 \\delta ^{(3)}(\\textbf {\\em k}_{123}) B({\\mathbf {k}}_1,{\\mathbf {k}}_2,{\\mathbf {k}}_3,t) \\, ,$ and the velocity bispectrum is defined in the same way for $\\theta (\\textbf {\\em k},t)$ .", "Assuming that the linear density perturbation $\\delta _1$ follows the Gaussian statistics, any higher order correlation functions beyond the linear power spectrum $P_{11}$ , defined by $\\left\\langle \\delta _1({\\mathbf {k}}_1,t)\\delta _1({\\mathbf {k}}_2,t) \\right\\rangle \\equiv (2\\pi )^3 \\delta ^{(3)}({\\mathbf {k}}_1+{\\mathbf {k}}_2) P_{11}(k_1,t) \\, ,$ can be written in terms of $P_{11}$ .", "Note that from (), we can see that the linear power spectrum of the velocity perturbation is simply $P_{11}$ multiplied by $k_H^2 \\equiv (aH)^2$ .", "With Gaussian $\\delta _1$ , the next-to-leading order bispectrum is given by $& \\left\\langle \\delta ({\\mathbf {k}}_1)\\delta ({\\mathbf {k}}_2)\\delta ({\\mathbf {k}}_3) \\right\\rangle \\nonumber \\\\= & \\Big [ \\left\\langle \\delta _1({\\mathbf {k}}_1)\\delta _1({\\mathbf {k}}_2)\\delta _2({\\mathbf {k}}_3) \\right\\rangle + \\text{(2 cyclic)} \\Big ]\\nonumber \\\\& + \\left\\langle \\delta _2({\\mathbf {k}}_1)\\delta _2({\\mathbf {k}}_2)\\delta _2({\\mathbf {k}}_3) \\right\\rangle + \\Big [ \\left\\langle \\delta _1({\\mathbf {k}}_1)\\delta _1({\\mathbf {k}}_2)\\delta _4({\\mathbf {k}}_3) \\right\\rangle + \\text{(2 cyclic)} \\Big ] + \\Big [ \\left\\langle \\delta _1({\\mathbf {k}}_1)\\delta _2({\\mathbf {k}}_2)\\delta _3({\\mathbf {k}}_3) \\right\\rangle + \\text{(5 cyclic)} \\Big ]\\nonumber \\\\\\equiv & (2\\pi )^3 \\delta ^{(3)}({\\mathbf {k}}_{123}) \\bigg \\lbrace B^{(0)}({\\mathbf {k}}_1,{\\mathbf {k}}_2,{\\mathbf {k}}_3) + \\Big [ B^{(1)}_{222}({\\mathbf {k}}_1,{\\mathbf {k}}_2,{\\mathbf {k}}_3) + B^{(1)}_{114}({\\mathbf {k}}_1,{\\mathbf {k}}_2,{\\mathbf {k}}_3) + B^{(1)}_{123}({\\mathbf {k}}_1,{\\mathbf {k}}_2,{\\mathbf {k}}_3) \\Big ] \\bigg \\rbrace \\, ,$ where we have suppressed the time dependence notation.", "The leading bispectrum $B^{(0)}$ does not contain any internal momentum integration, and is thus usually dubbed as the “tree-level” bispectrum.", "Meanwhile, the leading corrections $B^{(1)}$ all contain one internal momentum integration and are frequently called as “one-loop” corrections.", "In the following, we present matter bispectrum only.", "The velocity bispectrum is obtained in essentially the same way by replacing the kernel $G_i$ and supplying the additional factor $-k_H^3$ .", "We can straightforwardly compute the tree level bispectrum $B^{(0)}$ .", "We first consider $\\left\\langle \\delta _1({\\mathbf {k}}_1)\\delta _1({\\mathbf {k}}_2)\\delta _2({\\mathbf {k}}_3) \\right\\rangle $ .", "This reads $\\left\\langle \\delta _1({\\mathbf {k}}_1)\\delta _1({\\mathbf {k}}_2)\\delta _2({\\mathbf {k}}_3) \\right\\rangle = \\int \\frac{d^3q_1d^3q_2}{(2\\pi )^3} \\delta ^{(3)}({\\mathbf {k}}_3-{\\mathbf {q}}_{12})F_2^{(s)}({\\mathbf {q}}_1,{\\mathbf {q}}_2) \\Big \\langle \\delta _1({\\mathbf {k}}_1)\\delta _1({\\mathbf {k}}_2)\\Big [\\delta _1({\\mathbf {q}}_1)\\delta _1({\\mathbf {q}}_2)\\Big ] \\Big \\rangle \\, .$ Then, we can immediately find $\\left\\langle \\delta _1({\\mathbf {k}}_1)\\delta _1({\\mathbf {k}}_2)\\delta _2({\\mathbf {k}}_3) \\right\\rangle = (2\\pi )^3 \\delta ^{(3)}({\\mathbf {k}}_{123}) 2F_2^{(s)}(-{\\mathbf {k}}_1,-{\\mathbf {k}}_2)P_{11}(k_1)P_{11}(k_2) \\, ,$ and the tree bispectrum is thus $B^{(0)}({\\mathbf {k}}_1,{\\mathbf {k}}_2,{\\mathbf {k}}_3) = 2F_2^{(s)}(-{\\mathbf {k}}_1,-{\\mathbf {k}}_2)P_{11}(k_1)P_{11}(k_2) + \\text{(2 cyclic)} \\, .$" ], [ "$B^{(1)}_{222}$", "Next we consider the first one-loop correction term, $B^{(1)}_{222}({\\mathbf {k}}_1,{\\mathbf {k}}_2,{\\mathbf {k}}_3)$ .", "From the full expression $\\left\\langle \\delta _2({\\mathbf {k}}_1)\\delta _2({\\mathbf {k}}_2)\\delta _2({\\mathbf {k}}_3) \\right\\rangle =& \\int \\frac{d^3q_1\\cdots d^3q_2}{(2\\pi )^{3\\cdot 3}} \\delta ^{(3)}({\\mathbf {k}}_1-{\\mathbf {q}}_{12})\\delta ^{(3)}({\\mathbf {k}}_2-{\\mathbf {q}}_{34})\\delta ^{(3)}({\\mathbf {k}}_3-{\\mathbf {q}}_{56})\\nonumber \\\\& \\times F_2^{(s)}({\\mathbf {q}}_1,{\\mathbf {q}}_2)F_2^{(s)}({\\mathbf {q}}_3,{\\mathbf {q}}_4)F_2^{(s)}({\\mathbf {q}}_5,{\\mathbf {q}}_6)\\Big \\langle \\Big [\\delta _1({\\mathbf {q}}_1)\\delta _1({\\mathbf {q}}_2)\\Big ] \\Big [\\delta _1({\\mathbf {q}}_3)\\delta _1({\\mathbf {q}}_4)\\Big ] \\Big [\\delta _1({\\mathbf {q}}_5)\\delta _1({\\mathbf {q}}_6)\\Big ] \\Big \\rangle \\, ,$ we can find $B^{(1)}_{222}({\\mathbf {k}}_1,{\\mathbf {k}}_2,{\\mathbf {k}}_3) =& 8 \\int \\frac{d^3q}{(2\\pi )^3}F_2^{(s)}({\\mathbf {q}},{\\mathbf {k}}_1-{\\mathbf {q}}) F_2^{(s)}(-{\\mathbf {q}},{\\mathbf {k}}_2+{\\mathbf {q}}) F_2^{(s)}(-{\\mathbf {k}}_1+{\\mathbf {q}},-{\\mathbf {k}}_2-{\\mathbf {q}})\\nonumber \\\\&\\times P_{11}(q)P_{11}(\|{\\mathbf {k}}_1-{\\mathbf {q}}\|)P_{11}(\|{\\mathbf {k}}_2+{\\mathbf {q}}\|) \\, .$" ], [ "$B^{(1)}_{114}$", "For the next term $B^{(1)}_{114}({\\mathbf {k}}_1,{\\mathbf {k}}_2,{\\mathbf {k}}_3)$ , we can proceed in the same manner.", "Let us consider $\\left\\langle \\delta _1({\\mathbf {k}}_1)\\delta _1({\\mathbf {k}}_2)\\delta _4({\\mathbf {k}}_3) \\right\\rangle = &\\int \\frac{d^3q_1\\cdots d^3q_4}{(2\\pi )^{3\\cdot 3}} \\delta ^{(3)}({\\mathbf {k}}_3-{\\mathbf {q}}_{1234})\\nonumber \\\\&\\times F_4^{(s)}({\\mathbf {q}}_1,{\\mathbf {q}}_2,{\\mathbf {q}}_3,{\\mathbf {q}}_4)\\Big \\langle \\delta _1({\\mathbf {k}}_1)\\delta _1({\\mathbf {k}}_2)\\Big [\\delta _1({\\mathbf {q}}_1)\\delta _1({\\mathbf {q}}_2)\\delta _1({\\mathbf {q}}_3)\\delta _1({\\mathbf {q}}_4)\\Big ] \\Big \\rangle \\, .$ Then after straightforward calculations we find $B^{(1)}_{114}({\\mathbf {k}}_1,{\\mathbf {k}}_2,{\\mathbf {k}}_3) = 12 \\int \\frac{d^3q}{(2\\pi )^3}F_4^{(s)}(-{\\mathbf {k}}_1,-{\\mathbf {k}}_2,{\\mathbf {q}},-{\\mathbf {q}}) P_{11}(k_1)P_{11}(k_2)P_{11}(q) + \\text{(2 cyclic)} \\, .$" ], [ "$B^{(1)}_{123}$", "Finally, we consider the last contribution $B^{(1)}_{123}({\\mathbf {k}}_1,{\\mathbf {k}}_2,{\\mathbf {k}}_3)$ with $\\left\\langle \\delta _1({\\mathbf {k}}_1)\\delta _2({\\mathbf {k}}_2)\\delta _3({\\mathbf {k}}_3) \\right\\rangle = & \\int \\frac{d^3q_1\\cdots d^3q_5}{(2\\pi )^{3\\cdot 3}}\\delta ^{(3)}({\\mathbf {k}}_2-{\\mathbf {q}}_{12})\\delta ^{(3)}({\\mathbf {k}}_3-{\\mathbf {q}}_{345})\\nonumber \\\\& \\times F_2^{(s)}({\\mathbf {q}}_1,{\\mathbf {q}}_2)F_3^{(s)}({\\mathbf {q}}_3,{\\mathbf {q}}_4,{\\mathbf {q}}_5)\\Big \\langle \\delta _1({\\mathbf {k}}_1) \\Big [\\delta _1({\\mathbf {q}}_1)\\delta _1({\\mathbf {q}}_2)\\Big ]\\Big [\\delta _1({\\mathbf {q}}_3)\\delta _1({\\mathbf {q}}_4)\\delta _1({\\mathbf {q}}_5)\\Big ] \\Big \\rangle \\, .$ There are two different ways of correlating the six $\\delta _1$ 's.", "Let us call them ($a$ ) and ($b$ ).", "First, ($a$ ) is that the two propagators from $\\delta _2$ vertex are connected to both $\\delta _1$ and $\\delta _3$ vertices, and the remaining two propagators within $\\delta _3$ are inter-connected and form a loop.", "And ($b$ ) is that the two propagators from $\\delta _2$ vertex are both connected to $\\delta _3$ vertex, and the remaining one propagator of $\\delta _3$ is connected to $\\delta _1$ vertex.", "That is, in terms of momentum shown in (REF ), ($a$ ) corresponds to the correlations that one of $\\lbrace {\\mathbf {q}}_1,{\\mathbf {q}}_2\\rbrace $ is correlated to ${\\mathbf {k}}_1$ , and the remaining one is correlated to $\\lbrace {\\mathbf {q}}_3,{\\mathbf {q}}_4,{\\mathbf {q}}_5\\rbrace $ .", "For ($b$ ), two of $\\lbrace {\\mathbf {q}}_3,{\\mathbf {q}}_4,{\\mathbf {q}}_5\\rbrace $ are correlated to $\\lbrace {\\mathbf {q}}_1,{\\mathbf {q}}_2\\rbrace $ , and the remaining one is correlated to ${\\mathbf {k}}_1$ .", "There are six non-zero contributions for each of ($a$ ) and ($b$ ), and we work out to find $B^{(1)}_{123}({\\mathbf {k}}_1,{\\mathbf {k}}_2,{\\mathbf {k}}_3) = & B^{(1)}_{123a}({\\mathbf {k}}_1,{\\mathbf {k}}_2,{\\mathbf {k}}_3) + B^{(1)}_{123b}({\\mathbf {k}}_1,{\\mathbf {k}}_2,{\\mathbf {k}}_3) \\, ,\\\\B^{(1)}_{123a}({\\mathbf {k}}_1,{\\mathbf {k}}_2,{\\mathbf {k}}_3) = & 6F_2^{(s)}(-{\\mathbf {k}}_1,-{\\mathbf {k}}_3) \\int \\frac{d^3q}{(2\\pi )^3} F_3^{(s)}({\\mathbf {k}}_3,{\\mathbf {q}},-{\\mathbf {q}}) P_{11}(k_1)P_{11}(k_3)P_{11}(q) + \\text{(5 cyclic)} \\, ,\\\\B^{(1)}_{123b}({\\mathbf {k}}_1,{\\mathbf {k}}_2,{\\mathbf {k}}_3) = & 6 \\int \\frac{d^3q}{(2\\pi )^3} F_2^{(s)}({\\mathbf {q}},{\\mathbf {k}}_2-{\\mathbf {q}}) F_3^{(s)}(-{\\mathbf {k}}_1,-{\\mathbf {k}}_2+{\\mathbf {q}},-{\\mathbf {q}}) P_{11}(k_1)P_{11}(q)P_{11}(\|{\\mathbf {k}}_2-{\\mathbf {q}}\|)+ \\text{(5 cyclic)} \\, .$ One can find the diagramatic representation of the one-loop bispectrum in [23]." ], [ "Results", "To highlight the general relativistic effect at one-loop level, we find it sufficient to show some special triangular configurations.", "We set the three momenta $\\textbf {\\em k}_1$ , $\\textbf {\\em k}_2$ and $\\textbf {\\em k}_3$ in such a way that $\|\\textbf {\\em k}_1\| = \|\\textbf {\\em k}_2\| = k$ and $\|\\textbf {\\em k}_3\| = k/\\alpha $ , and vary $\\alpha $ for different configurations of interest.", "For example, $\\alpha = 1/2$ , $\\alpha = 1$ and $\\alpha \\gg 1$ correspond to the folded, equilateral and squeezed configurations, respectively.", "We implement the integration in the one-loop calculation by setting $\\textbf {\\em k}_1$ , $\\textbf {\\em k}_2$ and $\\textbf {\\em k}_3$ on the $xz$ plane with $\\textbf {\\em k}_1$ being aligned along the $z$ axis.", "To perform the integration over $\\textbf {\\em q}$ , we introduce the magnitude of $\\textbf {\\em q}$ and the cosine between $\\textbf {\\em q}$ and $\\textbf {\\em k}_1$ as $q = rk$ and $\\textbf {\\em k}_1\\cdot \\textbf {\\em q} = k^2r\\mu $ with $0 \\le r \\le \\infty $ and $-1 \\le \\mu \\le 1$ .", "Then, each vector including the internal momentum $\\textbf {\\em q}$ is given respectively by $\\textbf {\\em k}_1 & = (0,0,k) \\, ,\\\\\\textbf {\\em k}_2 & = \\left( \\frac{\\sqrt{4\\alpha ^2-1}}{2\\alpha ^2}k, 0, \\frac{1-2\\alpha ^2}{2\\alpha ^2}k \\right) \\, ,\\\\\\textbf {\\em k}_3 & = -\\textbf {\\em k}_1-\\textbf {\\em k}_3 \\, ,\\\\\\textbf {\\em q} & = \\left( kr\\sqrt{1-\\mu ^2}\\cos \\phi , kr\\sqrt{1-\\mu ^2}\\sin \\phi , kr\\mu \\right) \\, .$ We calculate the linear matter power spectrum $P_{11}(k)$ from CAMB code with cosmological parameters given in Table 1 of [24].", "We find that setting the radial integral range for $r$ from $r_\\text{min}=10^{-2}k_H/k$ to $r_\\text{max}=10^6k_H/k$ is sufficient to guarantee the convergence.", "All results we show hereafter are for $z=0$ .", "In Figure REF we show the matter bispectrum $B_{\\rm total}$ up to one-loop corrections as well as individual component: leading order $B_{\\rm tree}$ , Newtonian one-loop $B_{1-{\\rm loop}}^{\\rm NT}$ , relativistic one-loop $B_{1-{\\rm loop}}^{\\rm GR}$ and their sum $B_{1-{\\rm loop}}^{\\rm NT + GR}$ .", "For each curve, dashes lines show the absolute value of the negative quantity.", "The Newtonian one-loop corrections are appreciable on sub-horizon scales, $k \\gtrsim 0.1h$ Mpc$^{-1}$ , and dominates the tree contribution for large $k$ indicating the strong non-linearities due to gravitational instability.", "They change sign at around $k \\sim 0.1h$ Mpc$^{-1}$ , and on smaller (larger) scales the Newtonian corrections are negative (positive).", "The general relativistic one-loop corrections are strongly suppressed on small scales, but we note that on very large scales ($k\\rightarrow 0$ limit) they approach a constant value.", "While sub-dominant on all scales in the equilateral and folded configurations, the relativistic corrections give rise to the notable changes to the matter bispectrum on large scales for more squeezed triangles.", "In the tightly squeezed limit ($\\alpha =100$ ) they even dominate the tree contribution and make the total bispectrum negative, i.e.", "anti-correlated.", "This peculiar behavior is mainly coming from the components that carry $k_H^4$ factor in the fourth order kernel $F_4$ .", "Figure: In each panel, we present the matter density bispectrum in the (clockwise from top left) equilateral, folded, tightly (α=100\\alpha =100) and slightly (α=10\\alpha =10) squeezed configuration at z=0z=0.", "Solid (dashed) lines indicate that the corresponding contributions have positive (negative) values.", "The vertical dotted line denotes the Hubble horizon scale k H k_H.We estimate the behavior of the large scale plateau as the following.", "For simplicity, let us abbreviate the radial integration with the $k_H^4$ factor of $F_4$ in $B_{114}^{(1)}(k,k,k/\\alpha )$ as $\\int dr P_{11}(kr)f(r)$ .", "The variable $r \\equiv q/k$ is very large on large scale since $k\\rightarrow 0$ .", "With this, we can understand the asymptotic behavior of $f(r)$ on large scale ($r\\gg 1$ ) and in the squeezed limit ($\\alpha \\gg 1$ ) as $f(r) = -\\frac{75k_H^4}{28\\pi ^2k}\\alpha ^2 P_{11}(k)P_{11}\\left(\\frac{k}{\\alpha }\\right) + \\mathcal {O}(r^{-2})\\, .$ Note that $f(r)$ is to leading order independent of $r$ in the large scale limit.", "Using the fact $P_{11}(k)\\propto k^{n_\\mathcal {R}}$ for small $k$ with $n_\\mathcal {R}\\sim 0.96$ being the spectral index of the primordial perturbation, we can simply write the squeezed bispectrum on very large scales as $B_{114}^{(1)}(k,k,k/\\alpha ) \\propto -\\frac{75k_H^4}{28\\pi ^2}\\alpha k^{2(n_\\mathcal {R}-1)} \\, .$ Therefore, the matter bispectrum in the squeezed configuration is proportional to $\\alpha $ and is nearly independent of $k$ .", "We show in Figure REF the velocity bispectrum.", "The non-linear velocity bispectrum shows the similar features as the density bispectrum in Figure REF .", "Especially, the plateau on large scales in the squeezed configuration can be estimated in a similar way: writing the relevant component from $G_4$ schematically as $k_H^3\\int dr P_{11}(kr)g(r)$ , in the large scale limit we can find $g(r) = f(r)/3$ .", "Note, however, that the magnitude is much smaller than the matter bispectrum, due to the suppression by a factor of $k_H^3$ .", "As we see from both figures, the relativistic corrections to the density and velocity bispectra are very well-regulated, as the general relativistic signature shows only with a small amplitude on smaller scales and is noticible only for the large scale where the smallest mode is beyond the horizon scale, $k_H=aH$ , shown as a vertical dashed line in the figures.", "Figure: Velocity bispectrum shown in the same manner as Figure ." ], [ "Tree bispectrum in other gauges", "As we have seen above, the matter and velocity bispectra are well-regulated on all scales, and the relativistic effects are noticeable only on large scales beyond the horizon scale.", "This is because the comoving gauge is privileged in such a way that we have the same results as the Newtonian calculation up to second order.", "However, in other gauges this is not guaranteed and we in general expect deviations from each other, especially on large scales.", "To illustrate this point, we show in a few popular gauges the tree level bispectrum for which we need second order perturbation $\\delta _2$ , or equivalently, second order kernel $F_2$ .", "Note that, while we calculate the second order solutions with various gauge choices, the second order kernels $F_2$ we present here are still defined in Eq.", "(REF ) with the linear density contrast in the comoving gauge that we are referring as $\\delta _1$ throughout this paper.", "Comoving gauge: This is the main gauge we work with in this article.", "As we have worked out in the previous section, $\\delta _1$ and $\\delta _2$ are identical to the Newtonian density perturbations in the Eulerian coordinate.", "Thus there is no relativistic contributions at tree level.", "Synchronous gauge: Synchronous gauge takes no perturbation in the 00 and $0i$ components of the metric, $ds^2 = -dt^2 + a^2(1+2\\varphi )\\delta _{ij}dx^idx^j \\, ,$ so that the time coordinate agrees with proper time.", "In this gauge $\\delta _1$ is the same as that in the comoving gauge, but the second order kernel is found to be $F_2^\\text{(sg)}(\\textbf {\\em q}_1,\\textbf {\\em q}_2) = \\frac{5}{7}+ \\frac{2}{7}\\frac{(\\textbf {\\em q}_1\\cdot \\textbf {\\em q}_2)^2}{q_1^2q_2^2} \\, .$ Thus although there is no divergence on large scales, the tree bispectrum does not match that in the comoving gauge everywhere.", "This is because the density field in the synchronous gauge can be interpreted as the Newtonian density perturbation in the Lagrangian point of view following the moving volume elements.", "Zero shear gauge: In this gauge the metric is written as $ds^2 = -(1+2\\Phi )dt^2 + a^2(1-2\\Psi )\\delta _{ij}dx^idx^j \\, .$ We find the linear perturbation is given by $\\delta ^\\text{(zsg)}_1(\\textbf {\\em k}) = \\left( 1+\\frac{3k_H^2}{k^2} \\right) \\delta _1(\\textbf {\\em k}) \\, .$ Thus, while we recover the same result on sub-horizon scales as in the comoving gauge, deviation becomes prominent as we approach the horizon scale and eventually we face divergence on super-horizon scales.", "We can find the second order kernel as $F_2&^\\text{(zsg)}(\\textbf {\\em q}_1,\\textbf {\\em q}_2) = F_2^\\text{(cg)}(\\textbf {\\em q}_1,\\textbf {\\em q}_2)\\nonumber \\\\& + \\left( \\frac{k_H}{k} \\right)^2 \\left[ \\frac{9}{7} + \\frac{q_{12}^2}{q_1^2} + \\frac{q_{12}^2}{q_2^2} + \\frac{12 \\left(\\textbf {\\em q}_1\\cdot \\textbf {\\em q}_2\\right)^2}{7 q_1^2 q_2^2} + \\frac{3\\textbf {\\em q}_1\\cdot \\textbf {\\em q}_2}{2 q_1^2} + \\frac{3\\textbf {\\em q}_1\\cdot \\textbf {\\em q}_2}{2 q_2^2} \\right]\\nonumber \\\\& + \\left(\\frac{k_H}{k}\\right)^4 \\left[ \\frac{105}{4} - \\frac{15 \\left(\\textbf {\\em q}_1\\cdot \\textbf {\\em q}_2\\right)^2}{4 q_1^2 q_2^2} + \\frac{75 \\textbf {\\em q}_1\\cdot \\textbf {\\em q}_2}{4 q_1^2} + \\frac{75 \\textbf {\\em q}_1\\cdot \\textbf {\\em q}_2}{4 q_2^2} + \\frac{15 q_2^2}{2 q_1^2} + \\frac{15 q_1^2}{2 q_2^2} + \\frac{9q_{12}^4}{2q_1^2q_2^2} + \\frac{3q_{12}^2\\textbf {\\em q}_1\\cdot \\textbf {\\em q}_2}{2q_1^2q_2^2} \\right] \\, .$ This also matches the comoving gauge kernel on small scales but diverges in the limit $k\\rightarrow 0$ .", "The relativistic effect of gauge dependence in this gauge is characterized by the second and the third terms in the kernel with a factor of $k_H$ .", "Uniform curvature gauge: This gauge is also called as the flat gauge.", "In this gauge the spatial metric is set to be unperturbed, $ds^2 = -(1+2A)dt^2 - 2B_i dx^i dt + a^2\\delta _{ij}dx^idx^j \\, .$ The linear perturbation in this gauge is $\\delta ^\\text{(ucg)}_1(\\textbf {\\em k}) = \\left( 1+\\frac{15k_H^2}{2k^2} \\right) \\delta _1(\\textbf {\\em k}) \\, ,$ thus as in the zero shear gauge we find divergence on large scales.", "The second order kernel is given by $F_2^\\text{(ucg)}(\\textbf {\\em q}_1,\\textbf {\\em q}_2) = & F_2^\\text{(cg)}(\\textbf {\\em q}_1,\\textbf {\\em q}_2)\\nonumber \\\\& + \\left( \\frac{k_H}{k} \\right)^2 \\left[ \\frac{15}{4} + \\frac{5q_{12}^2}{2q_1^2} + \\frac{5q_{12}^2}{2q_2^2} + \\frac{15 \\left(\\textbf {\\em q}_1\\cdot \\textbf {\\em q}_2\\right)^2}{4 q_1^2 q_2^2} + \\frac{15 \\textbf {\\em q}_1\\cdot \\textbf {\\em q}_2}{4 q_1^2}+\\frac{15 \\textbf {\\em q}_1\\cdot \\textbf {\\em q}_2}{4 q_2^2} \\right]\\nonumber \\\\& + \\left( \\frac{k_H}{k} \\right)^4\\left(\\frac{225q_{12}^2 \\textbf {\\em q}_1\\cdot \\textbf {\\em q}_2}{8 q_1^2 q_2^2}+\\frac{75q_{12}^2}{2 q_1^2}+\\frac{75q_{12}^2}{2 q_2^2}+\\frac{75q_{12}^4}{8q_1^2q_2^2}\\right) \\, .$ Likewise, we recover the comoving gauge result on small scales $k\\rightarrow \\infty $ .", "We compare the tree level bispectra in all the aforementioned gauges in Figure REF .", "On small scales, the tree-level bispectra from all gauges except for the synchronous gauge converge to the Newtonian tree level bispectrum: the bispectrum in the synchronous gauge does not converge to the Newtonian (Eulerian) bispectrum, because the coordinate system is the similar to the Lagrangian fluid view.", "On larger scales ($k_3 \\lesssim k_H$ ), we start to see the gauge dependence of the matter density fields and the tree level bispectra from all four gauges are different from each other.", "Figure: We show the tree bispectrum in the comoving (CG), synchronous (SG), zero shear (ZSG) and uniform curvature (UCG) gauges.", "From top left clockwise, the bispectra are projected onto the equilateral, folded, tightly (α=100\\alpha =100) and slightly (α=10\\alpha =10) squeezed configurations at z=0z=0." ], [ "Conclusions", "We have studied how the non-linearities in general relativistic affects the non-linear density and velocity bispectra.", "Using the full general relativistic formalism, we calculate one-loop bispectra of density and velocity fields in a flat, matter dominated universe.", "We have assumed that the initial density perturbation is perfectly Gaussian, so that the matter bispectrum comes solely from the non-linear dynamics.", "As we work in the comoving gauge, where the relativistic density and velocity fields coincide with those in the Newtonian theory, the pure relativistic corrections appear from third order.", "We have computed the non-linear bispectrum in the equilateral, folded and squeezed triangular configurations and have shown that the relativistic effects are appreciable only on the scale larger than the horizon.", "On small scales, the Newtonian one-loop corrections dominate the relativistic ones and even the tree contributions at $k\\gtrsim 0.2h$ Mpc${}^{-1}$ , indicating the non-linear evolution of the bispectrum due to gravitational instability.", "The general relativistic corrections appear to be dominant over the Newtonian ones when the longest wavemode is near comoving horizon scale.", "That is the reason why we see the domination of the relativistic corrections in the squeezed configurations on very large scales.", "We then have demonstrated the gauge dependence of the matter bispectrum by explicitly computing the tree-level matter bispectrum in four different gauges: comoving, synchronous, zero shear and uniform curvature gauges.", "Except for the synchronous gauge, whose time coordinate is comoving with the observer and thus the meaning of the density contrast differs from the other gauges even on the small scales, the matter bispectrum computed in the other gauges are the same on small scales.", "On large scales, the gauge dependence begin to show up more prominently and the matter bisepctra from all four gauges diverge from each other.", "This result is, again, consistent with the one-loop result in the comoving gauge that the general relativistic effects are only important near the horizon scales.", "The gauge dependence that we see near the horizon scale is the outcome of that the matter bispectrum itself is not a direct observable.", "When considering the observable quantities such as the bispectrum of weak-lensing shear and convergence field, or the galaxy bispectrum including all the relevant effects such as galaxy bias, light reflection, etc, the gauge dependence must disappear.", "In the case of the galaxy power spectrum, [19] have shown that the observed galaxy power spectrum written in terms of the observed coordinate system is indeed gauge independent.", "Likewise, we surmise that calculating the bispectrum of observed quantities should resolve this gauge dependent ambiguities.", "Such a calculation requires extending the previous work done for the galaxy power spectrum to second order." ], [ "Acknowledgements", "SGB appreciates Jai-chan Hwang for suggesting helpful comments and supporting this research.", "We thank the Topical Research Program “Theories and practices in large scale structure formation”, supported by the Asia Pacific Center for Theoretical Physics, while this work was under progress.", "JG acknowledges the Max-Planck-Gesellschaft, the Korea Ministry of Education, Science and Technology, Gyeongsangbuk-Do and Pohang City for the support of the Independent Junior Research Group at the Asia Pacific Center for Theoretical Physics.", "JG is also supported by a Starting Grant through the Basic Science Research Program of the National Research Foundation of Korea (2013R1A1A1006701).", "DJ is supported by DoE SC-0008108 and NASA NNX12AE86G, and acknowledges support from the John Templeton Foundation." ], [ "Fourth order solutions", "Substituting (REF ) and () into (REF ) and (REF ), in Fourier space we obtain the differential equations for the fourth order kernels $F_4$ and $G_4$ as $& \\frac{1}{H}\\frac{dF_4}{dt} + 4F_4 - G_4\\nonumber \\\\=& \\frac{{\\mathbf {k}}\\cdot {\\mathbf {q}}_{234}}{q_{234}^2}G_3({\\mathbf {q}}_2,{\\mathbf {q}}_3,{\\mathbf {q}}_4) + \\frac{{\\mathbf {k}}\\cdot {\\mathbf {q}}_{34}}{q_{34}^2}F_2({\\mathbf {q}}_1,{\\mathbf {q}}_2)G_2({\\mathbf {q}}_3,{\\mathbf {q}}_4) + \\frac{{\\mathbf {k}}\\cdot {\\mathbf {q}}_1}{q_1^2}F_3({\\mathbf {q}}_2,{\\mathbf {q}}_3,{\\mathbf {q}}_4)\\nonumber \\\\& - (aH)^2 \\bigg \\lbrace \\frac{5}{2} \\bigg [ \\frac{{\\mathbf {q}}_{12}\\cdot {\\mathbf {q}}_{34}}{q_{12}^2} \\bigg ( -\\frac{2}{q_2^2} + \\frac{{\\mathbf {q}}_1\\cdot {\\mathbf {q}}_2}{q_1^2q_2^2} - \\frac{3}{2}\\frac{{\\mathbf {q}}_{12}\\cdot {\\mathbf {q}}_1}{q_{12}^2q_1^2} - \\frac{3}{2}\\frac{{\\mathbf {q}}_{12}\\cdot {\\mathbf {q}}_2}{q_{12}^2}\\frac{{\\mathbf {q}}_1\\cdot {\\mathbf {q}}_2}{q_1^2q_2^2} \\bigg ) + 2\\frac{{\\mathbf {q}}_1\\cdot {\\mathbf {q}}_{34}}{q_1^2q_2^2} \\bigg ] F_2({\\mathbf {q}}_3,{\\mathbf {q}}_4)\\nonumber \\\\& \\hspace{17.07182pt} + \\bigg [ \\frac{{\\mathbf {q}}_{123}\\cdot {\\mathbf {q}}_4}{q_{123}^2} \\bigg ( -\\frac{2}{q_{23}^2} + \\frac{{\\mathbf {q}}_1\\cdot {\\mathbf {q}}_{23}}{q_1^2q_{23}^2} - \\frac{3}{2}\\frac{{\\mathbf {q}}_{123}\\cdot {\\mathbf {q}}_1}{q_{123}^2q_1^2} - \\frac{3}{2}\\frac{{\\mathbf {q}}_{123}\\cdot {\\mathbf {q}}_{23}}{q_{123}^2}\\frac{{\\mathbf {q}}_1\\cdot {\\mathbf {q}}_{23}}{q_1^2q_{23}^2} \\bigg ) + 2\\frac{{\\mathbf {q}}_1\\cdot {\\mathbf {q}}_4}{q_1^2q_{23}^2} \\bigg ] \\bigg [ \\frac{3}{2}F_2({\\mathbf {q}}_2,{\\mathbf {q}}_3) + G_2({\\mathbf {q}}_2,{\\mathbf {q}}_3) \\bigg ]\\nonumber \\\\& \\hspace{17.07182pt} + \\frac{5}{2} \\bigg [ \\frac{{\\mathbf {q}}_{123}\\cdot {\\mathbf {q}}_4}{q_{123}^2} \\bigg ( -\\frac{2}{q_1^2} + \\frac{{\\mathbf {q}}_1\\cdot {\\mathbf {q}}_{23}}{q_1^2q_{23}^2} - \\frac{3}{2}\\frac{{\\mathbf {q}}_{123}\\cdot {\\mathbf {q}}_{23}}{q_{123}^2q_{23}^2} - \\frac{3}{2}\\frac{{\\mathbf {q}}_{123}\\cdot {\\mathbf {q}}_1}{q_{123}^2}\\frac{{\\mathbf {q}}_1\\cdot {\\mathbf {q}}_{23}}{q_1^2q_{23}^2} \\bigg ) + 2\\frac{{\\mathbf {q}}_{23}\\cdot {\\mathbf {q}}_4}{q_1^2q_{23}^2} \\bigg ] G_2({\\mathbf {q}}_2,{\\mathbf {q}}_3) \\bigg \\rbrace \\nonumber \\\\& + \\frac{{\\mathbf {q}}_1\\cdot {\\mathbf {q}}_{234}}{q_{234}^2} \\bigg \\lbrace \\frac{25}{4}(aH)^4 \\bigg ( \\frac{2}{q_4^2} - \\frac{{\\mathbf {q}}_{23}\\cdot {\\mathbf {q}}_4}{q_{23}^2q_4^2} + \\frac{3}{2}\\frac{{\\mathbf {q}}_{234}\\cdot {\\mathbf {q}}_{23}}{q_{234}^2q_{23}^2} + \\frac{3}{2}\\frac{{\\mathbf {q}}_{234}\\cdot {\\mathbf {q}}_4}{q_{234}^2}\\frac{{\\mathbf {q}}_{23}\\cdot {\\mathbf {q}}_4}{q_{23}^2q_4^2} \\bigg )& \\hspace{93.89418pt} \\times \\bigg ( \\frac{2}{q_2^2} - \\frac{{\\mathbf {q}}_2\\cdot {\\mathbf {q}}_3}{q_2^2q_3^2} + \\frac{3}{2}\\frac{{\\mathbf {q}}_{23}\\cdot {\\mathbf {q}}_2}{q_{23}^2q_2^2} + \\frac{3}{2}\\frac{{\\mathbf {q}}_{23}\\cdot {\\mathbf {q}}_3}{q_{23}^2}\\frac{{\\mathbf {q}}_2\\cdot {\\mathbf {q}}_3}{q_2^2q_3^2} \\bigg )\\nonumber \\\\& \\hspace{17.07182pt} + \\frac{(aH)^2}{2q_{34}^2} \\bigg ( -2 + \\frac{{\\mathbf {q}}_2\\cdot {\\mathbf {q}}_{34}}{q_2^2} - \\frac{3}{2}\\frac{{\\mathbf {q}}_{234}\\cdot {\\mathbf {q}}_2}{q_{234}^2}\\frac{q_{34}^2}{q_2^2} - \\frac{3}{2}\\frac{{\\mathbf {q}}_{234}\\cdot {\\mathbf {q}}_{34}}{q_{234}^2}\\frac{{\\mathbf {q}}_2\\cdot {\\mathbf {q}}_{34}}{q_{34}^2} \\bigg )\\nonumber \\\\& \\hspace{51.21504pt} \\times \\bigg \\lbrace \\frac{1}{2} \\bigg [ 1 + \\frac{{\\mathbf {q}}_3\\cdot {\\mathbf {q}}_4}{q_3^2} \\bigg ( 1 - \\frac{{\\mathbf {q}}_{34}\\cdot {\\mathbf {q}}_4}{q_4^2} \\bigg ) \\bigg ] - \\frac{25}{4} \\frac{(aH)^2}{q_3^2q_4^2} \\left( 3{\\mathbf {q}}_3\\cdot {\\mathbf {q}}_4 + 8q_4^2 \\right) \\bigg \\rbrace \\nonumber \\\\& \\hspace{17.07182pt} + \\frac{25}{4}\\frac{(aH)^4}{q_3^2q_4^2} \\bigg [ -4 + 4\\frac{{\\mathbf {q}}_2\\cdot {\\mathbf {q}}_3}{q_2^2} - \\frac{3}{2} \\bigg ( \\frac{{\\mathbf {q}}_{234}\\cdot {\\mathbf {q}}_2}{q_{234}^2}\\frac{{\\mathbf {q}}_3\\cdot {\\mathbf {q}}_4}{q_2^2} + 3\\frac{{\\mathbf {q}}_{234}\\cdot {\\mathbf {q}}_4}{q_{234}^2}\\frac{{\\mathbf {q}}_2\\cdot {\\mathbf {q}}_3}{q_2^2}+ 4\\frac{{\\mathbf {q}}_{234}\\cdot {\\mathbf {q}}_3}{q_{234}^2}\\frac{{\\mathbf {q}}_2\\cdot {\\mathbf {q}}_3}{q_2^2} + 4\\frac{{\\mathbf {q}}_{234}\\cdot {\\mathbf {q}}_2}{q_{234}^2}\\frac{q_3^2}{q_2^2} \\bigg ) \\bigg ] \\bigg \\rbrace \\nonumber \\\\& + 2 \\bigg \\lbrace \\frac{{\\mathbf {q}}_1\\cdot {\\mathbf {q}}_2}{2q_2^2q_{34}^2}\\bigg ( \\frac{(aH)^2}{2} \\bigg [ 1 + \\frac{{\\mathbf {q}}_3\\cdot {\\mathbf {q}}_4}{q_3^2} \\bigg ( 1 - \\frac{{\\mathbf {q}}_{34}\\cdot {\\mathbf {q}}_4}{q_4^2} \\bigg ) \\bigg ] - \\frac{25}{4}\\frac{(aH)^4}{q_3^2q_4^2} \\left( 3{\\mathbf {q}}_3\\cdot {\\mathbf {q}}_4 + 8q_4^2 \\right) \\bigg )\\nonumber \\\\& \\hspace{17.07182pt} + \\frac{25}{4}\\frac{(aH)^4}{q_2^2}\\frac{{\\mathbf {q}}_1\\cdot {\\mathbf {q}}_{34}}{q_{34}^2} \\bigg [ -\\frac{2}{q_4^2} + \\frac{{\\mathbf {q}}_3\\cdot {\\mathbf {q}}_4}{q_3^2q_4^2} - \\frac{3}{2}\\frac{{\\mathbf {q}}_{34}\\cdot {\\mathbf {q}}_3}{q_{34}^2q_3^2} - \\frac{3}{2}\\frac{{\\mathbf {q}}_{34}\\cdot {\\mathbf {q}}_4}{q_{34}^2}\\frac{{\\mathbf {q}}_3\\cdot {\\mathbf {q}}_4}{q_3^2q_4^2} \\bigg ] \\bigg \\rbrace - 25\\frac{{\\mathbf {q}}_1\\cdot {\\mathbf {q}}_2}{q_2^2}\\frac{(aH)^4}{q_3^2q_4^2} \\, ,$ and $& \\frac{1}{H}\\frac{dG_4}{dt} + \\frac{3}{2} \\left( 3G_4 - F_4 \\right)\\nonumber \\\\= & \\frac{k^2{\\mathbf {q}}_1\\cdot {\\mathbf {q}}_{234}}{q_1^2q_{234}^2}G_3({\\mathbf {q}}_2,{\\mathbf {q}}_3,{\\mathbf {q}}_4) + \\frac{k^2{\\mathbf {q}}_{12}\\cdot {\\mathbf {q}}_{34}}{2q_{12}^2q_{34}^2}G_2({\\mathbf {q}}_1,{\\mathbf {q}}_2)G_2({\\mathbf {q}}_3,{\\mathbf {q}}_4)\\nonumber \\\\& + (aH)^2 \\bigg ( \\bigg \\lbrace \\bigg [ \\frac{2}{3} + \\frac{{\\mathbf {q}}_1\\cdot {\\mathbf {q}}_{234}}{q_1^2} \\bigg ( 1 - \\frac{k^2}{q_{234}^2} \\bigg ) \\bigg ] \\bigg ( -\\frac{2}{q_2^2} + \\frac{{\\mathbf {q}}_2\\cdot {\\mathbf {q}}_{34}}{q_2^2q_{34}^2} - \\frac{3}{2}\\frac{{\\mathbf {q}}_{234}\\cdot {\\mathbf {q}}_{34}}{q_{234}^2q_{34}^2} - \\frac{3}{2}\\frac{{\\mathbf {q}}_{234}\\cdot {\\mathbf {q}}_2}{q_{234}^2}\\frac{{\\mathbf {q}}_2\\cdot {\\mathbf {q}}_{34}}{q_2^2q_{34}^2} \\bigg )\\nonumber \\\\& \\hspace{17.07182pt} + \\frac{1}{q_2^2} \\bigg [ \\frac{2}{3}\\frac{{\\mathbf {q}}_1\\cdot {\\mathbf {q}}_{34}}{q_{34}^2} - 4 \\bigg ( \\frac{{\\mathbf {q}}_1\\cdot {\\mathbf {q}}_{34}}{q_{34}^2} - \\frac{1}{3} \\bigg ) \\frac{{\\mathbf {k}}\\cdot {\\mathbf {q}}_1}{q_1^2} \\bigg ] \\bigg \\rbrace \\frac{5}{2}G_2({\\mathbf {q}}_3,{\\mathbf {q}}_4)\\nonumber \\\\& \\hspace{42.67912pt} + \\bigg \\lbrace \\bigg [ \\frac{2}{3} + \\frac{{\\mathbf {q}}_{12}\\cdot {\\mathbf {q}}_{34}}{q_{34}^2} \\bigg ( 1 - \\frac{k^2}{q_{12}^2} \\bigg ) \\bigg ] \\bigg ( -\\frac{2}{q_2^2} + \\frac{{\\mathbf {q}}_1\\cdot {\\mathbf {q}}_2}{q_1^2q_2^2} - \\frac{3}{2}\\frac{{\\mathbf {q}}_{12}\\cdot {\\mathbf {q}}_1}{q_{12}^2q_1^2} - \\frac{3}{2}\\frac{{\\mathbf {q}}_{12}\\cdot {\\mathbf {q}}_2}{q_{12}^2}\\frac{{\\mathbf {q}}_1\\cdot {\\mathbf {q}}_2}{q_1^2q_2^2} \\bigg )\\nonumber \\\\& \\hspace{56.9055pt} + \\frac{1}{q_2^2} \\bigg [ \\frac{2}{3}\\frac{{\\mathbf {q}}_1\\cdot {\\mathbf {q}}_{34}}{q_1^2} - 4 \\bigg ( \\frac{{\\mathbf {q}}_1\\cdot {\\mathbf {q}}_{34}}{q_1^2} - \\frac{1}{3} \\bigg ) \\frac{{\\mathbf {k}}\\cdot {\\mathbf {q}}_{34}}{q_{34}^2} \\bigg ] \\bigg \\rbrace \\frac{5}{2}G_2({\\mathbf {q}}_3,{\\mathbf {q}}_4)\\nonumber \\\\& \\hspace{42.67912pt} + \\bigg \\lbrace \\bigg [ \\frac{2}{3} + \\frac{{\\mathbf {q}}_1\\cdot {\\mathbf {q}}_{234}}{q_1^2} \\bigg ( 1 - \\frac{k^2}{q_{234}^2} \\bigg ) \\bigg ] \\bigg ( -\\frac{2}{q_{34}^2} + \\frac{{\\mathbf {q}}_2\\cdot {\\mathbf {q}}_{34}}{q_2^2q_{34}^2} - \\frac{3}{2}\\frac{{\\mathbf {q}}_2\\cdot {\\mathbf {q}}_{234}}{q_2^2q_{234}^2} - \\frac{3}{2}\\frac{{\\mathbf {q}}_{234}\\cdot {\\mathbf {q}}_{34}}{q_{234}^2}\\frac{{\\mathbf {q}}_2\\cdot {\\mathbf {q}}_{34}}{q_2^2q_{34}^2} \\bigg )\\nonumber \\\\& \\hspace{56.9055pt} + \\frac{1}{q_{34}^2} \\bigg [ \\frac{2}{3}\\frac{{\\mathbf {q}}_1\\cdot {\\mathbf {q}}_2}{q_2^2} - 4 \\bigg ( \\frac{{\\mathbf {q}}_1\\cdot {\\mathbf {q}}_2}{q_2^2} - \\frac{1}{3} \\bigg ) \\frac{{\\mathbf {k}}\\cdot {\\mathbf {q}}_1}{q_1^2} \\bigg ] \\bigg \\rbrace \\bigg [ \\frac{3}{2}F_2({\\mathbf {q}}_3,{\\mathbf {q}}_4) + G_2({\\mathbf {q}}_3,{\\mathbf {q}}_4) \\bigg ] \\bigg )\\nonumber \\\\& + \\bigg ( -\\frac{2}{3} - \\frac{{\\mathbf {q}}_1\\cdot {\\mathbf {q}}_{234}}{q_1^2} + \\frac{k^2}{q_{234}^2}\\frac{{\\mathbf {q}}_1\\cdot {\\mathbf {q}}_{234}}{q_1^2} \\bigg )\\nonumber \\\\& \\hspace{8.5359pt} \\times \\bigg \\lbrace \\frac{25}{4} (aH)^4 \\bigg ( \\frac{2}{q_4^2} - \\frac{{\\mathbf {q}}_{23}\\cdot {\\mathbf {q}}_4}{q_{23}^2q_4^2} + \\frac{3}{2}\\frac{{\\mathbf {q}}_{234}\\cdot {\\mathbf {q}}_{23}}{q_{234}^2q_{23}^2} + \\frac{3}{2}\\frac{{\\mathbf {q}}_{234}\\cdot {\\mathbf {q}}_4}{q_{234}^2}\\frac{{\\mathbf {q}}_{23}\\cdot {\\mathbf {q}}_4}{q_{23}^2q_4^2} \\bigg )\\nonumber \\\\& \\hspace{62.59596pt}\\times \\bigg ( \\frac{2}{q_3^2} - \\frac{{\\mathbf {q}}_2\\cdot {\\mathbf {q}}_3}{q_2^2q_3^2} + \\frac{3}{2}\\frac{{\\mathbf {q}}_{23}\\cdot {\\mathbf {q}}_2}{q_{23}^2q_2^2} + \\frac{3}{2}\\frac{{\\mathbf {q}}_{23}\\cdot {\\mathbf {q}}_3}{q_{23}^2}\\frac{{\\mathbf {q}}_2\\cdot {\\mathbf {q}}_3}{q_2^2q_3^2} \\bigg )\\nonumber \\\\& \\hspace{22.76228pt} + \\frac{(aH)^2}{2q_{34}^2} \\bigg ( -2 + \\frac{{\\mathbf {q}}_2\\cdot {\\mathbf {q}}_{34}}{q_2^2} - \\frac{3}{2}\\frac{{\\mathbf {q}}_{234}\\cdot {\\mathbf {q}}_2}{q_{234}^2}\\frac{q_{34}^2}{q_2^2} - \\frac{3}{2}\\frac{{\\mathbf {q}}_{234}\\cdot {\\mathbf {q}}_{34}}{q_{234}^2}\\frac{{\\mathbf {q}}_2\\cdot {\\mathbf {q}}_{34}}{q_{34}^2} \\bigg )\\nonumber \\\\& \\hspace{65.44142pt} \\times \\bigg \\lbrace \\frac{1}{2} \\bigg [ 1 + \\frac{{\\mathbf {q}}_3\\cdot {\\mathbf {q}}_4}{q_3^2} \\bigg ( 1 - \\frac{{\\mathbf {q}}_{34}\\cdot {\\mathbf {q}}_4}{q_4^2} \\bigg ) \\bigg ] - \\frac{25}{4}\\frac{(aH)^2}{q_3^2q_4^2} \\left( 3{\\mathbf {q}}_3\\cdot {\\mathbf {q}}_4 + 8q_4^2 \\right) \\bigg \\rbrace \\nonumber \\\\& \\hspace{22.76228pt} + \\frac{25}{4}\\frac{(aH)^4}{q_3^2q_4^2} \\bigg [ -4 + 4\\frac{{\\mathbf {q}}_2\\cdot {\\mathbf {q}}_3}{q_2^2} - \\frac{3}{2} \\bigg ( \\frac{{\\mathbf {q}}_{234}\\cdot {\\mathbf {q}}_2}{q_{234}^2}\\frac{{\\mathbf {q}}_3\\cdot {\\mathbf {q}}_4}{q_2^2} + 3\\frac{{\\mathbf {q}}_{234}\\cdot {\\mathbf {q}}_4}{q_{234}^2}\\frac{{\\mathbf {q}}_2\\cdot {\\mathbf {q}}_3}{q_2^2} + 4\\frac{{\\mathbf {q}}_{234}\\cdot {\\mathbf {q}}_3}{q_{234}^2}\\frac{{\\mathbf {q}}_2\\cdot {\\mathbf {q}}_3}{q_2^2} + 4\\frac{{\\mathbf {q}}_{234}\\cdot {\\mathbf {q}}_2}{q_{234}^2}\\frac{q_3^2}{q_2^2} \\bigg ) \\bigg ] \\bigg \\rbrace \\nonumber \\\\&+ \\frac{(aH)^2}{2q_{34}^2} \\bigg ( -\\frac{2}{3}\\frac{{\\mathbf {q}}_1\\cdot {\\mathbf {q}}_2}{q_1^2} + 4\\frac{{\\mathbf {q}}_1\\cdot {\\mathbf {q}}_2}{q_1^2}\\frac{{\\mathbf {k}}\\cdot {\\mathbf {q}}_2}{q_2^2} - \\frac{4}{3}\\frac{{\\mathbf {k}}\\cdot {\\mathbf {q}}_2}{q_2^2} \\bigg )\\nonumber \\\\&\\hspace{56.9055pt}\\times \\bigg \\lbrace \\frac{1}{2} \\bigg [ 1 + \\frac{{\\mathbf {q}}_3\\cdot {\\mathbf {q}}_4}{q_3^2} \\bigg ( 1 - \\frac{{\\mathbf {q}}_{34}\\cdot {\\mathbf {q}}_4}{q_4^2} \\bigg ) \\bigg ] - \\frac{25}{4}\\frac{(aH)^2}{q_3^2q_4^2} \\left( 3{\\mathbf {q}}_3\\cdot {\\mathbf {q}}_4 + 8q_4^2 \\right) \\bigg \\rbrace \\nonumber \\\\& + \\frac{25}{4}\\frac{(aH)^4}{q_1^2} \\bigg ( -\\frac{2}{q_4^2} + \\frac{{\\mathbf {q}}_3\\cdot {\\mathbf {q}}_4}{q_3^2q_4^2} - \\frac{3}{2}\\frac{{\\mathbf {q}}_{34}\\cdot {\\mathbf {q}}_3}{q_{34}^2q_3^2} - \\frac{3}{2}\\frac{{\\mathbf {q}}_{34}\\cdot {\\mathbf {q}}_4}{q_{34}^2}\\frac{{\\mathbf {q}}_3\\cdot {\\mathbf {q}}_4}{q_3^2q_4^2} \\bigg )\\nonumber \\\\& \\hspace{56.9055pt} \\times \\bigg ( 4\\frac{{\\mathbf {k}}\\cdot {\\mathbf {q}}_2}{q_2^2}\\frac{{\\mathbf {q}}_2\\cdot {\\mathbf {q}}_{34}}{q_{34}^2} - \\frac{4}{3}\\frac{{\\mathbf {k}}\\cdot {\\mathbf {q}}_2}{q_2^2} + 4\\frac{{\\mathbf {k}}\\cdot {\\mathbf {q}}_{34}}{q_{34}^2}\\frac{{\\mathbf {q}}_2\\cdot {\\mathbf {q}}_{34}}{q_2^2} - \\frac{4}{3}\\frac{{\\mathbf {k}}\\cdot {\\mathbf {q}}_{34}}{q_{34}^2} - \\frac{2}{3}\\frac{{\\mathbf {q}}_2\\cdot {\\mathbf {q}}_{34}}{q_{34}^2} - \\frac{8}{3}\\frac{{\\mathbf {q}}_2\\cdot {\\mathbf {q}}_{34}}{{q_2}^2} \\bigg )\\nonumber \\\\& + \\frac{25}{4}\\frac{(aH)^4}{q_1^2q_2^2} \\bigg ( 12\\frac{{\\mathbf {q}}_3\\cdot {\\mathbf {q}}_4}{q_3^2}\\frac{{\\mathbf {k}}\\cdot {\\mathbf {q}}_4}{q_4^2} - 4\\frac{{\\mathbf {k}}\\cdot {\\mathbf {q}}_4}{q_4^2} - 4\\frac{{\\mathbf {q}}_3\\cdot {\\mathbf {q}}_4}{q_3^2} + 2\\frac{{\\mathbf {q}}_1\\cdot {\\mathbf {q}}_2}{q_3^2}\\frac{{\\mathbf {q}}_3\\cdot {\\mathbf {q}}_4}{q_4^2} + \\frac{2}{3}\\frac{{\\mathbf {q}}_1\\cdot {\\mathbf {q}}_3}{q_3^2}\\frac{{\\mathbf {q}}_2\\cdot {\\mathbf {q}}_4}{q_4^2} \\bigg )\\nonumber \\\\& + \\frac{25}{4}(aH)^4 \\bigg ( \\frac{2}{3} - \\frac{{\\mathbf {k}}\\cdot {\\mathbf {q}}_{34}}{q_{34}^2} + \\frac{{\\mathbf {k}}\\cdot {\\mathbf {q}}_{34}}{q_{34}^2}\\frac{{\\mathbf {q}}_{12}\\cdot {\\mathbf {q}}_{34}}{q_{12}^2} \\bigg )\\nonumber \\\\& \\hspace{56.9055pt} \\times \\bigg ( \\frac{2}{q_2^2} - \\frac{{\\mathbf {q}}_1\\cdot {\\mathbf {q}}_2}{q_1^2q_2^2} + \\frac{3}{2}\\frac{{\\mathbf {q}}_{12}\\cdot {\\mathbf {q}}_1}{q_{12}^2q_1^2} + \\frac{3}{2}\\frac{{\\mathbf {q}}_{12}\\cdot {\\mathbf {q}}_2}{q_{12}^2}\\frac{{\\mathbf {q}}_1\\cdot {\\mathbf {q}}_2}{q_1^2q_2^2} \\bigg ) \\bigg ( \\frac{2}{q_4^2} - \\frac{{\\mathbf {q}}_3\\cdot {\\mathbf {q}}_4}{q_3^2q_4^2} + \\frac{3}{2}\\frac{{\\mathbf {q}}_{34}\\cdot {\\mathbf {q}}_3}{q_{34}^2q_3^2} + \\frac{3}{2}\\frac{{\\mathbf {q}}_{34}\\cdot {\\mathbf {q}}_4}{q_{34}^2}\\frac{{\\mathbf {q}}_3\\cdot {\\mathbf {q}}_4}{q_3^2q_4^2} \\bigg ) \\, .$ We can obtain readily the solutions of these equations.", "We can divide the kernels into the Newtonian and general relativistic parts.", "The Newtonian kernels are time independent, and can be calculated algebraically in terms of lower order kernels as $F_4^\\text{(N)} = & \\frac{1}{33} \\bigg [ \\frac{9{\\mathbf {k}}\\cdot {\\mathbf {q}}_1}{q_1^2}F_3^\\text{(N)}({\\mathbf {q}}_2,{\\mathbf {q}}_3,{\\mathbf {q}}_4) + \\frac{9{\\mathbf {k}}\\cdot {\\mathbf {q}}_{234}}{q_{234}^2}G_3^\\text{(N)}({\\mathbf {q}}_2,{\\mathbf {q}}_3,{\\mathbf {q}}_4) + \\frac{9{\\mathbf {k}}\\cdot {\\mathbf {q}}_{34}}{q_{34}^2}F_2({\\mathbf {q}}_1,{\\mathbf {q}}_2)G_2({\\mathbf {q}}_3,{\\mathbf {q}}_4)\\nonumber \\\\& \\hspace{14.22636pt} + \\frac{2k^2{\\mathbf {q}}_1\\cdot {\\mathbf {q}}_{234}}{q_1^2q_{234}^2}G_3^\\text{(N)}({\\mathbf {q}}_2,{\\mathbf {q}}_3,{\\mathbf {q}}_4) + \\frac{2k^2{\\mathbf {q}}_{12}\\cdot {\\mathbf {q}}_{34}}{2q_{12}^2q_{34}^2}G_2({\\mathbf {q}}_1,{\\mathbf {q}}_2)G_2({\\mathbf {q}}_3,{\\mathbf {q}}_4) \\bigg ] \\, ,\\\\G_4^\\text{(N)} = & \\frac{1}{33} \\bigg [ \\frac{3{\\mathbf {k}}\\cdot {\\mathbf {q}}_1}{q_1^2}F_3^\\text{(N)}({\\mathbf {q}}_2,{\\mathbf {q}}_3,{\\mathbf {q}}_4) + \\frac{3{\\mathbf {k}}\\cdot {\\mathbf {q}}_{234}}{q_{234}^2}G_3^\\text{(N)}({\\mathbf {q}}_2,{\\mathbf {q}}_3,{\\mathbf {q}}_4) + \\frac{3{\\mathbf {k}}\\cdot {\\mathbf {q}}_{34}}{q_{34}^2}F_2({\\mathbf {q}}_1,{\\mathbf {q}}_2)G_2({\\mathbf {q}}_3,{\\mathbf {q}}_4)\\nonumber \\\\& \\hspace{14.22636pt} + \\frac{8k^2{\\mathbf {q}}_1\\cdot {\\mathbf {q}}_{234}}{q_1^2q_{234}^2}G_3^\\text{(N)}({\\mathbf {q}}_2,{\\mathbf {q}}_3,{\\mathbf {q}}_4) + \\frac{8k^2{\\mathbf {q}}_{12}\\cdot {\\mathbf {q}}_{34}}{2q_{12}^2q_{34}^2}G_2({\\mathbf {q}}_1,{\\mathbf {q}}_2)G_2({\\mathbf {q}}_3,{\\mathbf {q}}_4) \\bigg ] \\, .$ Meanwhile, the general relativistic kernels are time dependent.", "Introducing the horizon scale wavenumber $k_H \\equiv aH$ , we can find $F_4^\\text{(GR)} & = \\frac{7A_1+2B_1}{18} k_H^2 + \\frac{5A_2+2B_2}{7}k_H^4 \\, ,\\\\G_4^\\text{(GR)} & = \\frac{A_1+2B_1}{6}k_H^2 + \\frac{3A_2+4B_2}{7}k_H^4 \\, ,$ where $& A_1 = \\frac{{\\mathbf {k}}\\cdot {\\mathbf {q}}_1}{q_1^2}k_H^{-2}F_3^\\text{(GR)}({\\mathbf {q}}_2,{\\mathbf {q}}_3,{\\mathbf {q}}_4) + \\frac{{\\mathbf {k}}\\cdot {\\mathbf {q}}_{234}}{q_{234}^2}k_H^{-2}G_3^\\text{(GR)}({\\mathbf {q}}_2,{\\mathbf {q}}_3,{\\mathbf {q}}_4)\\nonumber \\\\& - \\frac{5}{2} \\bigg [ \\frac{{\\mathbf {q}}_{12}\\cdot {\\mathbf {q}}_{34}}{q_{12}^2} \\bigg ( -\\frac{2}{q_2^2} + \\frac{{\\mathbf {q}}_1\\cdot {\\mathbf {q}}_2}{q_1^2q_2^2} - \\frac{3}{2}\\frac{{\\mathbf {q}}_{12}\\cdot {\\mathbf {q}}_1}{q_{12}^2q_1^2} - \\frac{3}{2}\\frac{{\\mathbf {q}}_{12}\\cdot {\\mathbf {q}}_2}{q_{12}^2}\\frac{{\\mathbf {q}}_1\\cdot {\\mathbf {q}}_2}{q_1^2q_2^2} \\bigg ) + 2\\frac{{\\mathbf {q}}_1\\cdot {\\mathbf {q}}_{34}}{q_1^2q_2^2} \\bigg ] F_2({\\mathbf {q}}_3,{\\mathbf {q}}_4)\\nonumber \\\\& - \\bigg [ \\frac{{\\mathbf {q}}_{123}\\cdot {\\mathbf {q}}_4}{q_{123}^2} \\bigg ( -\\frac{2}{q_{23}^2} + \\frac{{\\mathbf {q}}_1\\cdot {\\mathbf {q}}_{23}}{q_1^2q_{23}^2} - \\frac{3}{2}\\frac{{\\mathbf {q}}_{123}\\cdot {\\mathbf {q}}_1}{q_{123}^2q_1^2} - \\frac{3}{2}\\frac{{\\mathbf {q}}_{123}\\cdot {\\mathbf {q}}_{23}}{q_{123}^2}\\frac{{\\mathbf {q}}_1\\cdot {\\mathbf {q}}_{23}}{q_1^2q_{23}^2} \\bigg ) + 2\\frac{{\\mathbf {q}}_1\\cdot {\\mathbf {q}}_4}{q_1^2q_{23}^2} \\bigg ] \\bigg [ \\frac{3}{2}F_2({\\mathbf {q}}_2,{\\mathbf {q}}_3) + G_2({\\mathbf {q}}_2,{\\mathbf {q}}_3) \\bigg ]\\nonumber \\\\& - \\frac{5}{2} \\bigg [ \\frac{{\\mathbf {q}}_{123}\\cdot {\\mathbf {q}}_4}{q_{123}^2} \\bigg ( -\\frac{2}{q_1^2} + \\frac{{\\mathbf {q}}_1\\cdot {\\mathbf {q}}_{23}}{q_1^2q_{23}^2} - \\frac{3}{2}\\frac{{\\mathbf {q}}_{123}\\cdot {\\mathbf {q}}_{23}}{q_{123}^2q_{23}^2} - \\frac{3}{2}\\frac{{\\mathbf {q}}_{123}\\cdot {\\mathbf {q}}_1}{q_{123}^2}\\frac{{\\mathbf {q}}_1\\cdot {\\mathbf {q}}_{23}}{q_1^2q_{23}^2} \\bigg ) + 2\\frac{{\\mathbf {q}}_{23}\\cdot {\\mathbf {q}}_4}{q_1^2q_{23}^2} \\bigg ] G_2({\\mathbf {q}}_2,{\\mathbf {q}}_3)\\nonumber \\\\& + \\frac{{\\mathbf {q}}_1\\cdot {\\mathbf {q}}_{234}}{q_{234}^2} \\frac{1}{4q_{34}^2} \\bigg ( -2 + \\frac{{\\mathbf {q}}_2\\cdot {\\mathbf {q}}_{34}}{q_2^2} - \\frac{3}{2}\\frac{{\\mathbf {q}}_{234}\\cdot {\\mathbf {q}}_2}{q_{234}^2}\\frac{q_{34}^2}{q_2^2} - \\frac{3}{2}\\frac{{\\mathbf {q}}_{234}\\cdot {\\mathbf {q}}_{34}}{q_{234}^2}\\frac{{\\mathbf {q}}_2\\cdot {\\mathbf {q}}_{34}}{q_{34}^2} \\bigg ) \\bigg [ 1 + \\frac{{\\mathbf {q}}_3\\cdot {\\mathbf {q}}_4}{q_3^2} \\bigg ( 1 - \\frac{{\\mathbf {q}}_{34}\\cdot {\\mathbf {q}}_4}{q_4^2} \\bigg ) \\bigg ]\\nonumber \\\\& + \\frac{{\\mathbf {q}}_1\\cdot {\\mathbf {q}}_2}{2q_2^2q_{34}^2} \\bigg [ 1 + \\frac{{\\mathbf {q}}_3\\cdot {\\mathbf {q}}_4}{q_3^2} \\bigg ( 1 - \\frac{{\\mathbf {q}}_{34}\\cdot {\\mathbf {q}}_4}{q_4^2} \\bigg ) \\bigg ] \\, ,\\\\& B_1 = \\frac{k^2{\\mathbf {q}}_1\\cdot {\\mathbf {q}}_{234}}{q_1^2q_{234}^2}k_H^{-2}G_3^\\text{(GR)}({\\mathbf {q}}_2,{\\mathbf {q}}_3,{\\mathbf {q}}_4)\\nonumber \\\\& + \\frac{5}{2}\\bigg \\lbrace \\bigg [ \\frac{2}{3} + \\frac{{\\mathbf {q}}_1\\cdot {\\mathbf {q}}_{234}}{q_1^2} \\bigg ( 1 - \\frac{k^2}{q_{234}^2} \\bigg ) \\bigg ] \\bigg ( -\\frac{2}{q_2^2} + \\frac{{\\mathbf {q}}_2\\cdot {\\mathbf {q}}_{34}}{q_2^2q_{34}^2} - \\frac{3}{2}\\frac{{\\mathbf {q}}_{234}\\cdot {\\mathbf {q}}_{34}}{q_{234}^2q_{34}^2} - \\frac{3}{2}\\frac{{\\mathbf {q}}_{234}\\cdot {\\mathbf {q}}_2}{q_{234}^2}\\frac{{\\mathbf {q}}_2\\cdot {\\mathbf {q}}_{34}}{q_2^2q_{34}^2} \\bigg )\\nonumber \\\\& \\hspace{22.76228pt} + \\frac{1}{q_2^2} \\bigg [ \\frac{2}{3}\\frac{{\\mathbf {q}}_1\\cdot {\\mathbf {q}}_{34}}{q_{34}^2} - 4 \\bigg ( \\frac{{\\mathbf {q}}_1\\cdot {\\mathbf {q}}_{34}}{q_{34}^2} - \\frac{1}{3} \\bigg ) \\frac{{\\mathbf {k}}\\cdot {\\mathbf {q}}_1}{q_1^2} \\bigg ] \\bigg \\rbrace G_2({\\mathbf {q}}_3,{\\mathbf {q}}_4)\\nonumber \\\\& + \\frac{5}{2} \\bigg \\lbrace \\bigg [ \\frac{2}{3} + \\frac{{\\mathbf {q}}_{12}\\cdot {\\mathbf {q}}_{34}}{q_{34}^2} \\bigg ( 1 - \\frac{k^2}{q_{12}^2} \\bigg ) \\bigg ] \\bigg ( -\\frac{2}{q_2^2} + \\frac{{\\mathbf {q}}_1\\cdot {\\mathbf {q}}_2}{q_1^2q_2^2} - \\frac{3}{2}\\frac{{\\mathbf {q}}_{12}\\cdot {\\mathbf {q}}_1}{q_{12}^2q_1^2} - \\frac{3}{2}\\frac{{\\mathbf {q}}_{12}\\cdot {\\mathbf {q}}_2}{q_{12}^2}\\frac{{\\mathbf {q}}_1\\cdot {\\mathbf {q}}_2}{q_1^2q_2^2} \\bigg )\\nonumber \\\\& \\hspace{22.76228pt} + \\frac{1}{q_2^2} \\bigg [ \\frac{2}{3}\\frac{{\\mathbf {q}}_1\\cdot {\\mathbf {q}}_{34}}{q_1^2} - 4 \\bigg ( \\frac{{\\mathbf {q}}_1\\cdot {\\mathbf {q}}_{34}}{q_1^2} - \\frac{1}{3} \\bigg ) \\frac{{\\mathbf {k}}\\cdot {\\mathbf {q}}_{34}}{q_{34}^2} \\bigg ] \\bigg \\rbrace G_2({\\mathbf {q}}_3,{\\mathbf {q}}_4)\\nonumber \\\\& + \\bigg \\lbrace \\bigg [ \\frac{2}{3} + \\frac{{\\mathbf {q}}_1\\cdot {\\mathbf {q}}_{234}}{q_1^2} \\bigg ( 1 - \\frac{k^2}{q_{234}^2} \\bigg ) \\bigg ] \\bigg ( -\\frac{2}{q_{34}^2} + \\frac{{\\mathbf {q}}_2\\cdot {\\mathbf {q}}_{34}}{q_2^2q_{34}^2} - \\frac{3}{2}\\frac{{\\mathbf {q}}_2\\cdot {\\mathbf {q}}_{234}}{q_2^2q_{234}^2} - \\frac{3}{2}\\frac{{\\mathbf {q}}_{234}\\cdot {\\mathbf {q}}_{34}}{q_{234}^2}\\frac{{\\mathbf {q}}_2\\cdot {\\mathbf {q}}_{34}}{q_2^2q_{34}^2} \\bigg )\\nonumber \\\\& \\hspace{22.76228pt} + \\frac{1}{q_{34}^2} \\bigg [ \\frac{2}{3}\\frac{{\\mathbf {q}}_1\\cdot {\\mathbf {q}}_2}{q_2^2} - 4 \\bigg ( \\frac{{\\mathbf {q}}_1\\cdot {\\mathbf {q}}_2}{q_2^2} - \\frac{1}{3} \\bigg ) \\frac{{\\mathbf {k}}\\cdot {\\mathbf {q}}_1}{q_1^2} \\bigg ] \\bigg \\rbrace \\bigg [ \\frac{3}{2}F_2({\\mathbf {q}}_3,{\\mathbf {q}}_4) + G_2({\\mathbf {q}}_3,{\\mathbf {q}}_4) \\bigg ]\\nonumber \\\\& + \\bigg ( -\\frac{2}{3} - \\frac{{\\mathbf {q}}_1\\cdot {\\mathbf {q}}_{234}}{q_1^2} + \\frac{k^2}{q_{234}^2}\\frac{{\\mathbf {q}}_1\\cdot {\\mathbf {q}}_{234}}{q_1^2} \\bigg )\\nonumber \\\\& \\hspace{14.22636pt} \\times \\frac{1}{4q_{34}^2} \\bigg ( -2 + \\frac{{\\mathbf {q}}_2\\cdot {\\mathbf {q}}_{34}}{q_2^2} - \\frac{3}{2}\\frac{{\\mathbf {q}}_{234}\\cdot {\\mathbf {q}}_2}{q_{234}^2}\\frac{q_{34}^2}{q_2^2} - \\frac{3}{2}\\frac{{\\mathbf {q}}_{234}\\cdot {\\mathbf {q}}_{34}}{q_{234}^2}\\frac{{\\mathbf {q}}_2\\cdot {\\mathbf {q}}_{34}}{q_{34}^2} \\bigg ) \\bigg [ 1 + \\frac{{\\mathbf {q}}_3\\cdot {\\mathbf {q}}_4}{q_3^2} \\bigg ( 1 - \\frac{{\\mathbf {q}}_{34}\\cdot {\\mathbf {q}}_4}{q_4^2} \\bigg ) \\bigg ]\\nonumber \\\\& + \\frac{1}{4q_{34}^2} \\bigg ( -\\frac{2}{3}\\frac{{\\mathbf {q}}_1\\cdot {\\mathbf {q}}_2}{q_1^2} + 4\\frac{{\\mathbf {q}}_1\\cdot {\\mathbf {q}}_2}{q_1^2}\\frac{{\\mathbf {k}}\\cdot {\\mathbf {q}}_2}{q_2^2} - \\frac{4}{3}\\frac{{\\mathbf {k}}\\cdot {\\mathbf {q}}_2}{q_2^2} \\bigg ) \\bigg [ 1 + \\frac{{\\mathbf {q}}_3\\cdot {\\mathbf {q}}_4}{q_3^2} \\bigg ( 1 - \\frac{{\\mathbf {q}}_{34}\\cdot {\\mathbf {q}}_4}{q_4^2} \\bigg ) \\bigg ] \\, ,$ and $& A_2 = \\frac{25}{4} \\frac{{\\mathbf {q}}_1\\cdot {\\mathbf {q}}_{234}}{q_{234}^2} \\bigg ( \\frac{2}{q_4^2} - \\frac{{\\mathbf {q}}_{23}\\cdot {\\mathbf {q}}_4}{q_{23}^2q_4^2} + \\frac{3}{2}\\frac{{\\mathbf {q}}_{234}\\cdot {\\mathbf {q}}_{23}}{q_{234}^2q_{23}^2} + \\frac{3}{2}\\frac{{\\mathbf {q}}_{234}\\cdot {\\mathbf {q}}_4}{q_{234}^2}\\frac{{\\mathbf {q}}_{23}\\cdot {\\mathbf {q}}_4}{q_{23}^2q_4^2} \\bigg ) \\bigg ( \\frac{2}{q_2^2} - \\frac{{\\mathbf {q}}_2\\cdot {\\mathbf {q}}_3}{q_2^2q_3^2} + \\frac{3}{2}\\frac{{\\mathbf {q}}_{23}\\cdot {\\mathbf {q}}_2}{q_{23}^2q_2^2} + \\frac{3}{2}\\frac{{\\mathbf {q}}_{23}\\cdot {\\mathbf {q}}_3}{q_{23}^2}\\frac{{\\mathbf {q}}_2\\cdot {\\mathbf {q}}_3}{q_2^2q_3^2} \\bigg )\\nonumber \\\\& - \\frac{25}{8} \\frac{{\\mathbf {q}}_1\\cdot {\\mathbf {q}}_{234}}{q_{34}^2q_{234}^2} \\bigg ( -2 + \\frac{{\\mathbf {q}}_2\\cdot {\\mathbf {q}}_{34}}{q_2^2} - \\frac{3}{2}\\frac{{\\mathbf {q}}_{234}\\cdot {\\mathbf {q}}_2}{q_{234}^2}\\frac{q_{34}^2}{q_2^2} - \\frac{3}{2}\\frac{{\\mathbf {q}}_{234}\\cdot {\\mathbf {q}}_{34}}{q_{234}^2}\\frac{{\\mathbf {q}}_2\\cdot {\\mathbf {q}}_{34}}{q_{34}^2} \\bigg ) \\frac{3{\\mathbf {q}}_3\\cdot {\\mathbf {q}}_4 + 8q_4^2}{q_3^2q_4^2}\\nonumber \\\\& + \\frac{25}{4} \\frac{{\\mathbf {q}}_1\\cdot {\\mathbf {q}}_{234}}{q_{234}^2} \\frac{1}{q_3^2q_4^2} \\bigg [ -4 + 4\\frac{{\\mathbf {q}}_2\\cdot {\\mathbf {q}}_3}{q_2^2} - \\frac{3}{2} \\bigg ( \\frac{{\\mathbf {q}}_{234}\\cdot {\\mathbf {q}}_2}{q_{234}^2}\\frac{{\\mathbf {q}}_3\\cdot {\\mathbf {q}}_4}{q_2^2} + 3\\frac{{\\mathbf {q}}_{234}\\cdot {\\mathbf {q}}_4}{q_{234}^2}\\frac{{\\mathbf {q}}_2\\cdot {\\mathbf {q}}_3}{q_2^2}+ 4\\frac{{\\mathbf {q}}_{234}\\cdot {\\mathbf {q}}_3}{q_{234}^2}\\frac{{\\mathbf {q}}_2\\cdot {\\mathbf {q}}_3}{q_2^2} + 4\\frac{{\\mathbf {q}}_{234}\\cdot {\\mathbf {q}}_2}{q_{234}^2}\\frac{q_3^2}{q_2^2} \\bigg ) \\bigg ]\\nonumber \\\\& - \\frac{25}{4} \\frac{{\\mathbf {q}}_1\\cdot {\\mathbf {q}}_2}{q_2^2q_{34}^2} \\frac{3{\\mathbf {q}}_3\\cdot {\\mathbf {q}}_4 + 8q_4^2}{q_3^2q_4^2} + \\frac{25}{4}\\frac{{\\mathbf {q}}_1\\cdot {\\mathbf {q}}_{34}}{q_2^2q_{34}^2} \\bigg [ -\\frac{2}{q_4^2} + \\frac{{\\mathbf {q}}_3\\cdot {\\mathbf {q}}_4}{q_3^2q_4^2} - \\frac{3}{2}\\frac{{\\mathbf {q}}_{34}\\cdot {\\mathbf {q}}_3}{q_{34}^2q_3^2} - \\frac{3}{2}\\frac{{\\mathbf {q}}_{34}\\cdot {\\mathbf {q}}_4}{q_{34}^2}\\frac{{\\mathbf {q}}_3\\cdot {\\mathbf {q}}_4}{q_3^2q_4^2} \\bigg ] - \\frac{25}{q_3^2q_4^2}\\frac{{\\mathbf {q}}_1\\cdot {\\mathbf {q}}_2}{q_2^2} \\, ,\\\\& B_2 = \\bigg ( -\\frac{2}{3} - \\frac{{\\mathbf {q}}_1\\cdot {\\mathbf {q}}_{234}}{q_1^2} + \\frac{k^2}{q_{234}^2}\\frac{{\\mathbf {q}}_1\\cdot {\\mathbf {q}}_{234}}{q_1^2} \\bigg )\\nonumber \\\\& \\hspace{8.5359pt} \\times \\bigg \\lbrace \\frac{25}{4} \\bigg ( \\frac{2}{q_4^2} - \\frac{{\\mathbf {q}}_{23}\\cdot {\\mathbf {q}}_4}{q_{23}^2q_4^2} + \\frac{3}{2}\\frac{{\\mathbf {q}}_{234}\\cdot {\\mathbf {q}}_{23}}{q_{234}^2q_{23}^2} + \\frac{3}{2}\\frac{{\\mathbf {q}}_{234}\\cdot {\\mathbf {q}}_4}{q_{234}^2}\\frac{{\\mathbf {q}}_{23}\\cdot {\\mathbf {q}}_4}{q_{23}^2q_4^2} \\bigg ) \\bigg ( \\frac{2}{q_3^2} - \\frac{{\\mathbf {q}}_2\\cdot {\\mathbf {q}}_3}{q_2^2q_3^2} + \\frac{3}{2}\\frac{{\\mathbf {q}}_{23}\\cdot {\\mathbf {q}}_2}{q_{23}^2q_2^2} + \\frac{3}{2}\\frac{{\\mathbf {q}}_{23}\\cdot {\\mathbf {q}}_3}{q_{23}^2}\\frac{{\\mathbf {q}}_2\\cdot {\\mathbf {q}}_3}{q_2^2q_3^2} \\bigg )\\nonumber \\\\& \\hspace{22.76228pt} - \\frac{25}{8q_{34}^2} \\bigg ( -2 + \\frac{{\\mathbf {q}}_2\\cdot {\\mathbf {q}}_{34}}{q_2^2} - \\frac{3}{2}\\frac{{\\mathbf {q}}_{234}\\cdot {\\mathbf {q}}_2}{q_{234}^2}\\frac{q_{34}^2}{q_2^2} - \\frac{3}{2}\\frac{{\\mathbf {q}}_{234}\\cdot {\\mathbf {q}}_{34}}{q_{234}^2}\\frac{{\\mathbf {q}}_2\\cdot {\\mathbf {q}}_{34}}{q_{34}^2} \\bigg ) \\frac{3{\\mathbf {q}}_3\\cdot {\\mathbf {q}}_4 + 8q_4^2}{q_3^2q_4^2}\\nonumber \\\\& \\hspace{22.76228pt} + \\frac{25}{4q_3^2q_4^2} \\bigg [ -4 + 4\\frac{{\\mathbf {q}}_2\\cdot {\\mathbf {q}}_3}{q_2^2} - \\frac{3}{2} \\bigg ( \\frac{{\\mathbf {q}}_{234}\\cdot {\\mathbf {q}}_2}{q_{234}^2}\\frac{{\\mathbf {q}}_3\\cdot {\\mathbf {q}}_4}{q_2^2} + 3\\frac{{\\mathbf {q}}_{234}\\cdot {\\mathbf {q}}_4}{q_{234}^2}\\frac{{\\mathbf {q}}_2\\cdot {\\mathbf {q}}_3}{q_2^2} + 4\\frac{{\\mathbf {q}}_{234}\\cdot {\\mathbf {q}}_3}{q_{234}^2}\\frac{{\\mathbf {q}}_2\\cdot {\\mathbf {q}}_3}{q_2^2} + 4\\frac{{\\mathbf {q}}_{234}\\cdot {\\mathbf {q}}_2}{q_{234}^2}\\frac{q_3^2}{q_2^2} \\bigg ) \\bigg ] \\bigg \\rbrace \\nonumber \\\\& - \\frac{25}{8q_{34}^2} \\bigg ( -\\frac{2}{3}\\frac{{\\mathbf {q}}_1\\cdot {\\mathbf {q}}_2}{q_1^2} + 4\\frac{{\\mathbf {q}}_1\\cdot {\\mathbf {q}}_2}{q_1^2}\\frac{{\\mathbf {k}}\\cdot {\\mathbf {q}}_2}{q_2^2} - \\frac{4}{3}\\frac{{\\mathbf {k}}\\cdot {\\mathbf {q}}_2}{q_2^2} \\bigg ) \\frac{3{\\mathbf {q}}_3\\cdot {\\mathbf {q}}_4 + 8q_4^2}{q_3^2q_4^2}\\nonumber \\\\& + \\frac{25}{4q_1^2} \\bigg ( -\\frac{2}{q_4^2} + \\frac{{\\mathbf {q}}_3\\cdot {\\mathbf {q}}_4}{q_3^2q_4^2} - \\frac{3}{2}\\frac{{\\mathbf {q}}_{34}\\cdot {\\mathbf {q}}_3}{q_{34}^2q_3^2} - \\frac{3}{2}\\frac{{\\mathbf {q}}_{34}\\cdot {\\mathbf {q}}_4}{q_{34}^2}\\frac{{\\mathbf {q}}_3\\cdot {\\mathbf {q}}_4}{q_3^2q_4^2} \\bigg )\\nonumber \\\\& \\hspace{28.45274pt} \\times \\bigg ( 4\\frac{{\\mathbf {k}}\\cdot {\\mathbf {q}}_2}{q_2^2}\\frac{{\\mathbf {q}}_2\\cdot {\\mathbf {q}}_{34}}{q_{34}^2} - \\frac{4}{3}\\frac{{\\mathbf {k}}\\cdot {\\mathbf {q}}_2}{q_2^2} + 4\\frac{{\\mathbf {k}}\\cdot {\\mathbf {q}}_{34}}{q_{34}^2}\\frac{{\\mathbf {q}}_2\\cdot {\\mathbf {q}}_{34}}{q_2^2} - \\frac{4}{3}\\frac{{\\mathbf {k}}\\cdot {\\mathbf {q}}_{34}}{q_{34}^2} - \\frac{2}{3}\\frac{{\\mathbf {q}}_2\\cdot {\\mathbf {q}}_{34}}{q_{34}^2} - \\frac{8}{3}\\frac{{\\mathbf {q}}_2\\cdot {\\mathbf {q}}_{34}}{{q_2}^2} \\bigg )\\nonumber \\\\& + \\frac{25}{4q_1^2q_2^2} \\bigg ( 12\\frac{{\\mathbf {q}}_3\\cdot {\\mathbf {q}}_4}{q_3^2}\\frac{{\\mathbf {k}}\\cdot {\\mathbf {q}}_4}{q_4^2} - 4\\frac{{\\mathbf {k}}\\cdot {\\mathbf {q}}_4}{q_4^2} - 4\\frac{{\\mathbf {q}}_3\\cdot {\\mathbf {q}}_4}{q_3^2} + 2\\frac{{\\mathbf {q}}_1\\cdot {\\mathbf {q}}_2}{q_3^2}\\frac{{\\mathbf {q}}_3\\cdot {\\mathbf {q}}_4}{q_4^2} + \\frac{2}{3}\\frac{{\\mathbf {q}}_1\\cdot {\\mathbf {q}}_3}{q_3^2}\\frac{{\\mathbf {q}}_2\\cdot {\\mathbf {q}}_4}{q_4^2} \\bigg )\\nonumber \\\\& + \\frac{25}{4} \\bigg ( \\frac{2}{3} - \\frac{{\\mathbf {k}}\\cdot {\\mathbf {q}}_{34}}{q_{34}^2} + \\frac{{\\mathbf {k}}\\cdot {\\mathbf {q}}_{34}}{q_{34}^2}\\frac{{\\mathbf {q}}_{12}\\cdot {\\mathbf {q}}_{34}}{q_{12}^2} \\bigg )\\nonumber \\\\& \\hspace{28.45274pt} \\times \\bigg ( \\frac{2}{q_2^2} - \\frac{{\\mathbf {q}}_1\\cdot {\\mathbf {q}}_2}{q_1^2q_2^2} + \\frac{3}{2}\\frac{{\\mathbf {q}}_{12}\\cdot {\\mathbf {q}}_1}{q_{12}^2q_1^2} + \\frac{3}{2}\\frac{{\\mathbf {q}}_{12}\\cdot {\\mathbf {q}}_2}{q_{12}^2}\\frac{{\\mathbf {q}}_1\\cdot {\\mathbf {q}}_2}{q_1^2q_2^2} \\bigg ) \\bigg ( \\frac{2}{q_4^2} - \\frac{{\\mathbf {q}}_3\\cdot {\\mathbf {q}}_4}{q_3^2q_4^2} + \\frac{3}{2}\\frac{{\\mathbf {q}}_{34}\\cdot {\\mathbf {q}}_3}{q_{34}^2q_3^2} + \\frac{3}{2}\\frac{{\\mathbf {q}}_{34}\\cdot {\\mathbf {q}}_4}{q_{34}^2}\\frac{{\\mathbf {q}}_3\\cdot {\\mathbf {q}}_4}{q_3^2q_4^2} \\bigg ) \\, .$ The fully symmetric kernels can be obtained by taking into account possible permutations, $F_4^{(s)}({\\mathbf {q}}_1,{\\mathbf {q}}_2,{\\mathbf {q}}_3,{\\mathbf {q}}_4) = & \\frac{1}{4!}", "\\Big [ F_4({\\mathbf {q}}_1,{\\mathbf {q}}_2,{\\mathbf {q}}_3,{\\mathbf {q}}_4) + \\text{(23 cyclic)} \\Big ]\\, ,\\\\G_4^{(s)}({\\mathbf {q}}_1,{\\mathbf {q}}_2,{\\mathbf {q}}_3,{\\mathbf {q}}_4) = & \\frac{1}{4!}", "\\Big [ G_4({\\mathbf {q}}_1,{\\mathbf {q}}_2,{\\mathbf {q}}_3,{\\mathbf {q}}_4) + \\text{(23 cyclic)} \\Big ] \\, .$ Note that the equations and solutions up to third order can be found in [15]." ] ]	1403.0438
[ [ "Comment on \"Spin-Gradient-Driven Light Amplification in a Quantum\n Plasma\"" ], [ "Abstract A comment on the Letter by S. Braun, F. A. Asenjo and S. M. Mahajan, Phys.", "Rev.", "Lett., 109, 175003 (2012).", "We show that recent arguments for light amplification driven by inhomogeneous quantum spin fields in low temperature electron plasmas in metals are invalid.", "In essence, a neglect of Pauli `blocking' led the authors to over-estimate the effects of intrinsic spin." ], [ "Comment on “Spin-Gradient-Driven Light Amplification in a Quantum Plasma” Govind S. Krishnaswami$^{1}$ , Rajaram Nityananda$^{2}$ , Abhijit Sen$^{3}$ , Anantanarayanan Thyagaraja$^{4}$ $^{1}$ Chennai Mathematical Institute, SIPCOT IT Park, Siruseri 603103, India $^{2}$ National Centre for Radio Astrophysics, TIFR, Pune 411007, India $^{3}$ Institute for Plasma Research, Bhat, Gandhinagar 382428, India $^{4}$ Astrophysics Group, University of Bristol, Bristol, BS8 1TL, United Kingdom Email: [email protected], [email protected], [email protected], [email protected] 52.35.-g, 52.35.Hr; 52.35.We; 67.10.-j; 67.30.hj A comment on the Letter by S. Braun, F. A. Asenjo and S. M. Mahajan, Phys.", "Rev.", "Lett., 109, 175003 (2012).", "We show that recent arguments for light amplification driven by inhomogeneous quantum spin fields in low temperature electron plasmas in metals are invalid.", "In essence, a neglect of Pauli `blocking' led the authors to over-estimate the effects of intrinsic spin.", "Journal Reference: Phys.", "Rev.", "Lett.", "112, 129501 (2014).", "Submitted 17 Aug, 2013, accepted 19 Feb, 2014.", "Recently, Braun, Asenjo, and Mahajan [1] made a startling prediction: an electromagnetic (EM) wave above the cut off frequency, incident on a metal at low temperatures ($\\sim 30$ K), will amplify in the presence of a conduction electron spin field ${\\bf S}$ with large gradients.", "This prediction is based on spin quantum hydrodynamic (SQHD) equations derived in [2] and the `quantum spin vorticity' formalism of [3].", "The authors treat the metal as a free electron plasma with a uniform neutralizing ion background.", "However, their equations are not valid for conduction electron densities ($n \\simeq 10^{29}/$ m$^3$ ) in metals at low temperatures, due essentially to a neglect of Pauli blocking in the derivations of [2], [3].", "For such $n$ , the Fermi temperature $T_{\\rm F} = ({\\hbar ^2/2 m_e}) (3\\pi ^2 n)^{2/3}$ is a few eV.", "Hence, even at room temperature, the electron gas is highly degenerate.", "When $T \\ll T_{\\rm F}$ the (thermal) de Broglie wavelength exceeds the interelectron distance, the electron gas obeys Fermi-Dirac (FD) statistics.", "Unfortunately, the authors' equations (based on [2]) assume a factorized form for the $N$ -electron wave function, ignoring the antisymmetrization required by the Pauli exclusion principle.", "Consequently, the electron spin magnetization current is greatly overestimated.", "They use the formula ${\\bf j}_{\\rm sp} = \\nabla \\times [2n\\mu _{B}{\\bf S}]$ [Eqs.", "(2), (3) in [1]] where $\\mu _B = e \\hbar /2m_e$ .", "However, in a degenerate electron gas, the spin magnetization ought, by Pauli blocking, to be of order $2(n_+ - n_-) \\mu _B {\\bf b}$ where ${\\bf b}$ is the unit vector in the local magnetic field (${\\bf B}_0$ ) direction.", "Here, $n_\\pm = \\frac{n}{2} \\left[ 1 \\pm \\frac{3 \\mu _B B_0}{2 T_F} \\right]$ [4], [5] are the number densities of electrons with spins parallel and antiparallel to ${\\bf B}_0$ and ${(n_+ - n_-)}/{n} = {\\cal O}(T/T_F)$ .", "At temperatures $T \\ll T_F$ , only electrons close to the Fermi level [estimated by $(T/T_F)n$ ] contribute to the magnetization current, as well as any electron dynamics.", "Thus, $\\bf S$ must have a size $\\simeq [n_+ - n_-]{\\bf b}/n $ , invalidating the authors' assumption [Eq.", "(1) of [1]] that $\\bf S$ is a unit vector.", "This assumption leads to a huge $j_{\\rm sp} \\simeq 2n({\\mu _B}/{L_{S}}) \\simeq 10^{6}{\\rm A.m}^{-2}$ , where, $L_S = 1$ m is a macroscopic gradient length-scale.", "Any smaller $L_S$ will make $j_{\\rm sp}$ even larger.", "This corresponds to a near `saturation magnetic induction' of 1 Tesla, a value more typical of spin-polarized core electrons in a ferromagnet than conduction electrons in a metal.", "A saturation magnetization is problematic, coming at the cost of Fermi energy, which corresponds to creating a highly excited state.", "Thus, when $T \\ll T_F$ , Pauli blocking ensures that quantum spin effects are suppressed, as is well known [5] from the smallness of Pauli paramagnetism and Landau diamagnetism.", "Fermi liquid theory and quantum Boltzmann equations are required in this regime[4].", "On the other hand, when $T \\gg T_F$ and Maxwell-Boltzmann statistics apply, quantum spin effects are negligible, being a small perturbation [of ${\\cal O}(\\mu _B B/T) \\ll 1$ ] to the standard Vlasov kinetics (as also stated in [2]).", "At high temperatures, Larmor moments $\\mu _L = T/B$ dominate over spin moments ($\\mu _L \\gg \\mu _B$ ).", "Also, at any $T$ , the quantum spin force $\\mu _B \\nabla B$ is a very small perturbation to the classical orbit theory based on the Lorentz force.", "Furthermore, Coulomb collisions are non-negligible at high densities.", "The authors require a low $T \\simeq 30$ K to prevent collisions from damping the EM wave moving into their inhomogeneous spin density-dominated medium.", "This low collision rate is due to Pauli blocking[4], [5].", "It is inconsistent to use Pauli blocking to suppress collisions, but ignore its effect on the spin magnetization.", "Owing to these omissions and contradictions, the authors' equations are valid neither at high nor low $T$ .", "Moreover, they do not consider the equation of state of the electron gas nor the basic equilibrium state [involving significant spin gradients], around which they linearize their equations [(6)-(8) of [1]] for spin and EM fields.", "Discussion of the conditions needed [in principle] to create such a medium, and a Poynting theorem describing the pumping of the EM wave by the spin gradients, would lend credence to their predictions.", "For other critiques of hydrodynamical treatments of quantum plasmas (“QHD”) see [6], [7].", "This Comment (see [8] for a more detailed discussion and references) focuses on the erroneous treatment of electrons in [1].", "Addendum to Comment Date: March 2, 2014.", "In [9], the authors of [1] offer a Reply to our Comment.", "They accept the validity of our criticisms.", "In particular, they agree that (1) the average spin field $\\bf S$ is not a unit vector and (2) that Pauli blocking will greatly reduce the spin magnetization current in low temperature metallic plasmas, and thereby reduce any instability.", "They go on to state that at low temperatures, the light wave growth rate as well as electron-electron collision/damping frequency will both be brought down by the factor $\\alpha = T/T_F$ , yielding $\\Gamma _{\\rm PB} = \\alpha \\Gamma _{old} \\quad \\text{and} \\quad \\nu _{ee} \\sim \\frac{k_B T^2}{\\hbar T_F}.$ Thus they estimate the ratio of collisional damping to growth rate as $\\frac{\\nu _{ee}}{\\Gamma _{\\rm PB}} \\sim \\frac{k_B T}{\\hbar \\Gamma _{old}}.$ They assert that, in principle, this ratio could be less than unity for sufficiently low temperatures, thereby implying amplification of the light wave.", "It is true that in an `ideal metal', the electron collision rate $\\nu _{ee}$ will scale like $T^{2}$ [cf.", "[4], also Eq.", "(17.66) in [5]].", "In reality, at very low temperatures [when, $T \\ll T_{\\rm Debye}$ ], the `residual resistance' of a metal due to impurity or lattice defect scattering leads to a temperature-independent collision rate $\\nu _0$ .", "If such a temperature-independent collision frequency is used, then the authors' estimate for the ratio of collision rate to growth rate $\\frac{\\nu _0}{\\Gamma _{\\rm PB}} \\sim \\frac{T_F \\nu _0}{T \\Gamma _{\\rm old}}$ would be more than unity for sufficiently low temperatures, implying that collisions prevent any light amplification.", "Even if we accept $\\nu _{ee} = k_B T^2/\\hbar T_F$ as a reasonable low temperature collision rate, as well as their method of estimating the effect of Pauli blocking [i.e., that $\\Gamma _{\\rm PB} = \\frac{T}{T_F} \\Gamma _{\\rm old}$ ], we find for solid state plasmas, that the ratio of collision to growth rate $\\nu _{ee}/\\Gamma _{\\rm PB}$ is less than unity only for very low temperatures $T \\lesssim 0.025 K$ .", "Collisional damping of the wave will overwhelm the claimed effect for any higher temperature.", "What is more, even at such a low temperature, the EM wave would have to travel at least $c/\\Gamma _{PB} \\sim 30$ km in their medium to be significantly amplified.", "To obtain these estimates for solid state plasmas considered by the authors, we take $n_e \\approx 10^{29}$ m$^{-3}$ , corresponding to a Fermi temperature $k_B T_F \\approx 1$ eV $\\approx 10^4$ K and a plasma frequency $\\omega _{pe} \\approx 1.6 \\times 10^{16}$ s$^{-1}$ .", "Now from Fig.", "1 of their Letter [1], the maximum value of $\\Gamma _{\\rm old}$ is $2.5 \\times 10^{-7} \\times \\omega _{pe} \\approx 4 \\times 10^{9}$ s$^{-1}$ .", "Using their formulas $\\nu _{ee} \\sim \\frac{k_B T^2}{\\hbar T_F}$ and $\\frac{\\nu _{ee}}{\\Gamma _{PB}} \\sim \\frac{k_B T}{\\hbar \\Gamma _{\\rm old}}$ , we find that $k_B T \\lesssim 2.5 \\times 10^{-6}$ eV $\\approx 0.025$ K for the growth rate to exceed the collision rate.", "At $T = 0.025$ K, $\\Gamma _{PB} \\approx 10^4$ s$^{-1}$ and so the wave must travel $c/\\Gamma _{PB} \\approx 30$ km in the medium to be amplified significantly.", "Thus, even assuming SQHD to be corrected as suggested by the authors, the effects predicted are negligible [using their own numbers and formulae] and are far smaller than many other neglected effects such as collisionless damping, impurity scattering, etc.", "However, there is a more basic problem.", "The authors estimate the effect of FD statistics on the growth rate to be given by simply multiplying the old (uncorrected) growth rate $\\Gamma _{\\rm old}$ by $(\\frac{T}{T_{F}})$ .", "This is unacceptable, since the growth rate has not been derived ab initio using correct SQHD equations when $T \\ll T_{F}$ .", "The correct equations will have drastically reduced spin forces in the electron momentum equation and much lower spin magnetization currents in Maxwell's equations.", "It is to be checked afresh using correct equations, whether the mode will grow at all, for the stated conditions [especially the `WKB' condition $kL_{S} \\gg 1$ , where $k$ is the wave number and $L_{S}$ the gradient length-scale of the thermally averaged spin field].", "As mentioned in our Comment, it is readily shown from a consideration of the single-particle electron Hamiltonian in classical physics, that an added `spin magnetic moment force' [$\\simeq \|\\mu _{\\rm Bohr}\\nabla B\|$ ] borrowed from relativistic quantum mechanics, is tiny compared to the Lorentz forces arising from self-consistent electromagnetic fields.", "The spin-dependent dipole force can be shown from perturbation theory, to modify standard results of wave propagation in plasmas by minuscule effects.", "When large numbers of electrons are considered, there is the further crucial point that the spins can be oriented [in quantum theory] along or anti-parallel to the local magnetic field, and there will be very large cancellations of an already small effect by quantum and thermal averaging.", "The authors offer no clear argument to evade these standard conclusions." ] ]	1403.0228
[ [ "Equilibrium time-correlation functions for one-dimensional hard-point\n systems" ], [ "Abstract As recently proposed, the long-time behavior of equilibrium time-correlation functions for one-dimensional systems are expected to be captured by a nonlinear extension of fluctuating hydrodynamics.", "We outline the predictions from the theory aimed at the comparison with molecular dynamics.", "We report on numerical simulations of a fluid with a hard-shoulder potential and of a hard-point gas with alternating masses.", "These models have in common that the collision time is zero and their dynamics amounts to iterating collision by collision.", "The theory is well confirmed, with the twist that the non-universal coefficients are still changing at longest accessible times." ], [ "Introduction", "As very well understood, in thermal equilibrium one-dimensional classical fluids show no phase transitions and have rapidly decaying static correlations, provided the interaction potential is sufficiently short-ranged [1].", "On the other hand, as discovered in the early 1970ies, time correlations have anomalous decay.", "In particular, total current-current correlations decay non-integrably and the Green-Kubo definition of transport coefficients yields divergent expressions [2], [3].", "At the time only fairly qualitative predictions were available.", "But over the last 15 years there has been a considerable spectrum of molecular dynamics (MD) simulations, which do provide quantitative information [4], [5].", "Currently the conventional system size is of the order of $10^4$ particles and the maximal simulation time is such that the right and left going sound modes first collide in a ring geometry.", "Most efforts have been directed towards the numerical value of the dynamical exponents and the issue of universality.", "Very recently, in addition to exponents, universal scaling functions have been proposed on the basis of nonlinear fluctuating hydrodynamics [6], [7].", "The main goal of our contribution is to compare these theoretical predictions with MD simulations.", "For such purpose we consider hard-point systems, mainly because there is then no need to simulate differential equations.", "The dynamics proceeds from collision to collision with free motion in-between.", "Such models have been studied extensively before [8], [9], [10], [11], [12], [13], [14], [15].", "While there are important hints in the literature, the available simulations are not specific enough to test the theory.", "Therefore we decided to redo three of the most common models.", "(1) a hard-point gas of particles with equal mass and interaction between neighbors through a “shoulder potential\", (2) a hard-point gas of particles with alternating masses, and (3) the same as (2) but with the hard-point potential replaced by an infinitely high square-well potential.", "In other words, when neighboring particles reach a maximal distance, say $a$ , then there is an inward collision.", "These models have in common that they are particular instances of anharmonic chains, as characterized by having an interaction only between particles of adjacent label.", "For hard-point systems, in addition, the spatial order coincides with the label order.", "The equilibrium measure of a generic anharmonic chain is of product form.", "Therefore no equilibration step is required.", "The true equilibrium distribution is swiftly produced by a random number generator.", "We believe that this is of advantage as compared to the more conventional dynamical equilibration, which has always the risk of systematic errors, even though the necessarily limited numerical tests indicate an equilibrated system.", "Nonlinear fluctuating hydrodynamics makes the implicit assumptions that there are no further local conservation laws beyond the three standard ones and that upon fixing their values the dynamics is sufficiently well mixing in time.", "These assumptions cannot be checked easily.", "Counterexamples are completely integrable chains, as the Toda chain.", "For the shoulder potential the scattering induced through the potential step seems to suffice.", "Models (2) and (3) from above become integrable in case of equal masses.", "Presumably any other mass ratio destroys their integrability.", "Based on the experience from MD simulations a mass ratio around 3 is sufficiently well mixing.", "As an outline: In the following two sections we introduce the hard-point systems under study and review the theory.", "More details are recorded in [16].", "In Section 4 we report our MD results and in Section 5 we arrive at conclusions and compare with other MD data available." ], [ "Monoatomic chains.", "The hamiltonian of a one-dimensional fluid is of the form $H_\\mathrm {f}=\\sum ^N_{j=1} \\tfrac{1}{2}p^2_j +\\tfrac{1}{2}\\sum ^N_{i\\ne j=1}V(q_i-q_j)\\,.$ Here $q_j$ is the position and $p_j$ the momentum of the $j$ -th particle.", "Momentum equals velocity, since we use units for which the mass equals 1.", "$V$ is the interaction potential.", "We now choose specifically the shoulder potential $V_\\mathrm {sh}(x) = \\infty \\mathrm {\\hspace{4.0pt}for\\hspace{4.0pt}} \|x\|\\le \\tfrac{1}{2}\\,,\\quad V_\\mathrm {sh}(x) = 1 \\mathrm {\\hspace{4.0pt}for\\hspace{4.0pt}} \\tfrac{1}{2}< \|x\| < 1\\,,\\quad V_\\mathrm {sh}(x) = 0 \\mathrm {\\hspace{4.0pt}for\\hspace{4.0pt}} 1\\le \|x\|\\,.$ If one initially imposes $q_j +\\tfrac{1}{2}\\le q_{j + 1}$ , then this order is preserved in time and the interaction is only between neighboring particles.", "We introduce the $j$ -th stretch $r_j = q_{j+1} - q_j \\,.$ Then $\\dot{r}_j = p_{j+1} - p_j\\,,\\quad \\dot{p}_j = V^{\\prime }_\\mathrm {sh}(r_j)- V^{\\prime }_\\mathrm {sh}(r_{j-1})\\,.$ Without loss of generality, the potential height is chosen to be 1 and the hard core size to be $\\tfrac{1}{2}$ .", "The width of the potential step could be any value between 0 and $\\tfrac{1}{2}$ .", "We study here only the maximal width.", "It is of advantage to view $(r_j,p_j)_{j = 1,...,N}$ as a one-dimensional field theory with two components.", "Periodic boundary conditions, $r_{j+N} = r_j$ , $p_{j+N} = p_j$ , are imposed throughout.", "For hard-point particles $r_j\\ge 0$ and $p_j \\in \\mathbb {R}$ .", "The somewhat singular force in (REF ) translates into the following collision rules.", "Between collisions one has free motion with $\\dot{p}_j = 0$ .", "There are two types of collisions, at $r_j = \\tfrac{1}{2}$ and at $r_j = 1$ .", "(i) $r_j = \\frac{1}{2}$.", "If $p_{j+1} - p_j <0$ , then there is a point collision as $\\begin{split}p_j^{\\prime } &= p_{j+1}\\,, \\\\p_{j+1}^{\\prime } &= p_{j}\\,,\\end{split}$ where $^{\\prime }$ denotes the momentum after the collision.", "If $p_{j+1} - p_j > 0$ , particles separate under free motion.", "(ii) $r_j = 1$ .", "There is scattering at the potential step depending on whether particles approach or recede from each other.", "In the latter case, i.e., $p_{j+1} - p_{j} > 0$ , the collision rule reads $\\begin{split}p_j^{\\prime } &= \\tfrac{1}{2}\\Big (p_j + p_{j+1} - \\sqrt{(p_{j+1} - p_j)^2 + 4}\\Big )\\,, \\\\p_{j+1}^{\\prime } &= \\tfrac{1}{2}\\Big (p_j + p_{j+1} + \\sqrt{(p_{j+1} - p_j)^2 + 4}\\Big ) \\,.\\end{split}$ For approaching particles with large momentum difference $p_{j} - p_{j+1} > 2$ , the momentum transfer is sufficient to enter the shoulder plateau and the collision rule reads $\\begin{split}p_j^{\\prime } &= \\tfrac{1}{2}\\Big (p_j + p_{j+1} + \\sqrt{(p_{j+1} - p_j)^2 - 4}\\Big )\\,, \\\\p_{j+1}^{\\prime } &= \\tfrac{1}{2}\\Big (p_j + p_{j+1} - \\sqrt{(p_{j+1} - p_j)^2 - 4}\\Big ) \\,.\\end{split}$ If the incoming momentum transfer is too small, then the particles are specularly reflected, i.e., if $0 < p_{j} - p_{j+1} < 2$ , then $\\begin{split}p_j^{\\prime } &= p_{j+1}\\,, \\\\p_{j+1}^{\\prime } &= p_{j}\\,.\\end{split}$ An anharmonic chain, in general, still evolves according to (REF ), but with $V_\\mathrm {sh}$ replaced by some potential $V$ and generically without constraints on $r_j$ .", "$V$ is assumed to be bounded from below and to have at least a one-sided linearly increasing bound at infinity.", "Thermal equilibrium is described by the canonical Gibbs measure at zero average momentum.", "It is given by a product measure, i.e., the $(r_j,p_j)_{j=1,...,N}$ are independent.", "At a single site, the momentum $p_j$ has a Maxwellian distribution with mean zero and variance $1/2\\beta $ , while the probability density of the stretch $r_j$ is given by $Z^{-1} \\mathrm {e}^{-\\beta (V(y)+py)} \\,,\\quad Z(p,\\beta )=\\int _\\mathbb {R}dy \\mathrm {e}^{-\\beta (V(y)+py)}\\,.$ $p$ controls the stretch and $\\beta $ the energy.", "Averages with respect to (REF ) are denoted by $\\langle \\cdot \\rangle _{p,\\beta }$ .", "Note that $p=-\\langle V^{\\prime }(y)\\rangle _{p,\\beta }$ and, as average force on a specified particle, $p$ is identified with the thermodynamic pressure.", "In our simulations the average is always with respect to this canonical equilibrium measure.", "In the literature one finds an effort to impose zero momentum strictly and not only on average.", "But for the sizes and time spans under investigation, this makes hardly any difference.", "For the purpose of nonlinear fluctuating hydrodynamics we also record the Euler equations of an anharmonic chain, see [16] for more details.", "These are evolution equations of the conserved fields on a macroscopic scale.", "From (REF ) we infer that $r_j$ and $p_j$ are locally conserved.", "As for any mechanical system also the local energy is conserved.", "The energy at site $j$ is $e_j=\\tfrac{1}{2}p^2_j + V(r_j)\\,.$ Then $\\dot{e}_j= p_{j+1} V^{\\prime }(r_j)-p_j V^{\\prime }(r_{j-1})\\,,$ hence the local energy current is $-p_j V^{\\prime }(r_{j-1})$ .", "We collect the conserved fields as the 3-vector $\\vec{g} = (g_1,g_2,g_3)$ , $\\vec{g}(j,t) = \\big (r_j(t),p_j(t),e_j(t)\\big ) \\,,$ $\\vec{g}(j,0) = \\vec{g}(j)$ .", "Then $\\frac{d}{dt}\\vec{g}(j,t) + \\vec{\\mathcal {J}}(j+1,t) - \\vec{\\mathcal {J}}(j,t)=0 \\,,$ where the local current functions are given by $\\vec{\\mathcal {J}}(j) = \\big ( -p_j,-V^{\\prime }(r_{j-1}), - p_jV^{\\prime }(r_{j-1})\\big )\\,.$ Once the conserved fields are identified, the Euler equations follow from the assumption of local equilibrium.", "More precisely, we introduce the microcanonical parameters $\\ell $ , $\\mathsf {e}$ through $\\ell =\\langle r_j\\rangle _{p,\\beta }\\,,\\quad \\mathsf {e}=\\langle e_j\\rangle _{p,\\beta }=\\frac{1}{2\\beta }+\\langle V(r_j)\\rangle _{p,\\beta }\\,.$ (REF ) defines $(p,\\beta ) \\mapsto (\\ell (p,\\beta ),\\mathsf {e}(p,\\beta ))$ , thereby the inverse map $(\\ell , \\mathsf {e}) \\mapsto (p(\\ell ,\\mathsf {e}),$ $ \\beta (\\ell ,\\mathsf {e}))$ , and thus accomplishes the switch between the microcanonical variables $\\ell , \\mathsf {e}$ and the canonical variables $p, \\beta $ .", "Next let us choose an initial state, for which $p,\\beta $ , and mean velocity are slowly varying on the scale of the lattice.", "This induces a slow variation of stretch $\\ell $ , velocity $\\mathsf {u}$ , and total energy $\\mathfrak {e} =\\frac{1}{2} \\mathsf {u}^2+ \\mathsf {e}$ .", "Then by averaging the fields in a local equilibrium state, the microscopic conservation laws (REF ) turn into the Euler equations of an anharmonic chain as $\\partial _t\\ell (x,t) +\\partial _x \\mathsf {j}_\\ell (x,t) =0\\,,\\quad \\partial _t \\mathsf {u}(x,t) +\\partial _x \\mathsf {j}_\\mathsf {u}(x,t) =0\\,,\\quad \\partial _t \\mathfrak {e}(x,t) +\\partial _x \\mathsf {j}_\\mathfrak {e}(x,t) =0\\,,$ where the hydrodynamic currents are given by $\\langle \\vec{\\mathcal {J}}(j)\\rangle _{\\ell ,\\mathsf {u},\\mathfrak {e}} = \\big (-\\mathsf {u},p(\\ell ,\\mathfrak {e}-\\tfrac{1}{2}\\mathsf {u}^2), \\mathsf {u} p(\\ell ,\\mathfrak {e}-\\tfrac{1}{2}\\mathsf {u}^2)\\big )= \\vec{\\mathsf {j}}$ with $p(\\ell ,\\mathsf {e})$ defined implicitly through (REF ).", "By construction the slow variation refers to the particle label $j$ .", "Hence “$x$ \" in (REF ) stands for its continuum approximation.", "Returning to the hard-point system with shoulder potential, one obtains $&&Z(p,\\beta ) = \\frac{1}{p\\beta }\\mathrm {e}^{-p\\beta } \\big (1 + \\mathrm {e}^{-\\beta }( \\mathrm {e}^{p\\beta /2} -1)\\big )\\,,\\nonumber \\\\&& \\ell = -\\frac{1}{\\beta }\\partial _p \\log Z(p,\\beta )\\,,\\quad \\mathsf {e} = \\frac{1}{2\\beta } + \\frac{1}{Z(p,\\beta )}\\frac{1}{p\\beta }\\mathrm {e}^{-\\beta -p\\beta }( \\mathrm {e}^{p\\beta /2} -1)\\,.\\medskip $" ], [ "Biatomic chains.", "We reintroduce the mass $m_j$ of the $j$ -th particle and also a site-dependent interaction potential $V_j$ .", "Then the equations of motion for the chain become $\\dot{r}_j = \\frac{1}{m_{j+1}}p_{j+1} - \\frac{1}{m_j}p_j\\,,\\quad \\dot{p}_j = V^{\\prime }_j(r_j)- V^{\\prime }_{j-1}(r_{j-1})\\,.$ For a biatomic chain $m_j$ and $V_j$ have period 2 and hence the unit cell consists of two adjacent particles.", "We normalize by $m_0$ and set $\\kappa = m_1/m_0$ .", "Then $m_j = 1$ for even $j$ and $m_j = \\kappa $ for odd $j$ .", "We consider the particular case, in which particles interact through the square-well potential $V_\\mathrm {sw}(x) = 0 \\mathrm {\\hspace{4.0pt}for\\hspace{4.0pt}} 0 < \|x\| < a\\,,\\quad V_\\mathrm {sw}(x) = \\infty \\mathrm {\\hspace{4.0pt}otherwise\\hspace{0.0pt}}\\,.$ Then between collisions there is free motion with $\\dot{p}_j = 0$ .", "For $r_j = 0$ the incoming momenta are defined by $p_j - p_{j+1} > 0$ and for $r_j = a$ by $p_j - p_{j+1} < 0$ .", "In either case the collision rule reads $&&\\hspace{0.0pt}p_j^{\\prime } = p_j + 2\\,\\frac{m_j p_{j+1} - m_{j+1}p_j}{m_j + m_{j+1}}\\,,\\nonumber \\\\&&\\hspace{0.0pt}p_{j+1}^{\\prime } = p_{j+1} - 2\\, \\frac{m_j p_{j+1} - m_{j+1}p_j}{m_j + m_{j+1}} \\,.$ Note that the transformation (REF ) depends only on the mass ratio $\\kappa $ .", "Since there is zero potential energy, $\\mathsf {e} = \\frac{1}{2\\beta }$ and the pressure factorizes as $a\\beta p = h(\\ell /a)\\,,$ where $h$ is the inverse function to $y \\mapsto y^{-1} - (\\mathrm {e}^y -1)^{-1}$ .", "Clearly, length can be normalized such that $a=1$ .", "This is then our model of hard-point particles with alternating masses and square-well potential.", "For the hard-point gas with merely alternating masses, we take the limit $a \\rightarrow \\infty $ which amounts to delete the option $r_j = a$ in the collision rules (REF ).", "The pressure simplifies to $\\beta p = \\frac{1}{\\ell }\\,.$ For the hydrodynamic equations one has to take into account that momentum and energy transfer in a collision depend on the masses.", "Hence the currents (REF ) are modified to $\\big (-\\frac{1}{\\bar{m}}\\mathsf {u},p,\\frac{1}{\\bar{m}}\\mathsf {u}p\\big )\\,, \\quad p = p(\\ell ,\\mathfrak {e} - \\frac{1}{2\\bar{m}}\\mathsf {u}^2)\\,,$ where $\\bar{m}$ stands for the average mass, $\\bar{m} =(m_0 + m_1)/2$ ." ], [ "Nonlinear fluctuating hydrodynamics", "A standing issue of statistical mechanics is to understand the long-time behavior of dynamical correlations for the chain in thermal equilibrium.", "The modes with the longest life time will come from the locally conserved fields $\\vec{g}(j,t) = \\big (r_j(t),p_j(t),e_j(t)\\big ) $ .", "Hence one studies their correlations defined through $S_{\\alpha \\alpha ^{\\prime }}(j,t)=\\langle g_{\\alpha }(j,t) g_{\\alpha ^{\\prime }}(0,0)\\rangle _{p,\\beta } - \\langle g_{\\alpha }(j,t)\\rangle _{p,\\beta } \\langle g_{\\alpha ^{\\prime }}(0)\\rangle _{p,\\beta }\\,,$ $\\alpha ,\\alpha ^{\\prime }=1,2,3$ .", "Note that by space-time stationarity $\\langle g_{\\alpha }(j,t)\\rangle _{p,\\beta } = \\langle g_{\\alpha }(0)\\rangle _{p,\\beta }$ and $S_{\\alpha \\alpha ^{\\prime }}(j,t)= S_{\\alpha ^{\\prime }\\alpha }(-j,-t)\\,.$ Also at time $t=0$ , $S(j,0)=\\delta _{j0} C$ which defines the static susceptibility matrix $C$ .", "For the theory it is convenient to study directly the infinite one-dimensional lattice $\\mathbb {Z}$ , on which the correlations can spread forever.", "But MD is on a ring with $N$ sites and the dynamics is run only up to time $t_{\\mathrm {max}}$ , the first time when the two sound modes collide, i.e., $2c\\, t_{\\mathrm {max}}=N$ with $c$ the speed of sound.", "In higher spatial dimensions the long-time properties of the correlation functions (REF ) are well captured by (linear) fluctuating hydrodynamics.", "This is a Gaussian fluctuation theory for the hydrodynamic fields.", "The drift part of the corresponding Langevin equations is obtained by linearizing the Navier-Stokes equations around equilibrium and consists of the Euler flow term, linear in $\\partial _x$ , and the dissipative transport terms, quadratic in $\\partial _x$ .", "The noisy part is obtained by adding random currents with space-time white noise statistics to the systematic currents.", "These random currents model all the left out degrees of freedom from the exact conservation laws.", "The strength of the random currents is determined by the fluctuation dissipation theorem.", "Fluctuating hydrodynamics predicts diffusive broadening of the peaks.", "Such a behavior holds in dimension $d > 2$ and is well-known to break down in one dimension.", "The minimal proposal in [6], [7] is to generalize to a nonlinear version, for which the Euler currents are kept up to second order in the deviation from equilibrium, while the dissipative part and the noise are taken from the linear theory.", "We give here a brief review with more details provided in [16].", "We fix the equilibrium parameters $p,\\beta $ and denote the small deviations from equilibrium by $\\vec{u}(x,t) = (u_1(x,t),u_2(x,t),u_3(x,t))$ .", "When dissipation and noise are added, $\\vec{u}(x,t)$ becomes a random field with zero average.", "By construction, the fluctuation field is governed by the Langevin equations $\\partial _t \\vec{u}(x,t) +\\partial _x \\big (A \\vec{u}(x,t) + \\tfrac{1}{2} \\langle u(x,t),\\vec{H}u(x,t)\\rangle -\\partial _x \\tilde{D} \\vec{u}(x,t) +\\tilde{B}\\vec{\\xi }(x,t)\\,\\big )=0\\,.$ Here the Euler currents have been expanded relative to the reference background $\\vec{u}_0 = (\\ell ,0,\\mathsf {e})$ up to second order as $\\mathsf {j}_\\alpha (\\vec{u}_0 + \\vec{u}) = \\mathsf {j}_\\alpha (\\vec{u}_0) + \\sum _{\\beta =1}^{3}\\partial _{u_\\beta } \\mathsf {j}_\\alpha (\\vec{u}_0)u_\\beta +\\tfrac{1}{2} \\sum _{\\beta ,\\beta ^{\\prime }=1}^{3}\\partial _{u_\\beta } \\partial _{u_{\\beta ^{\\prime }}} \\mathsf {j}_\\alpha (\\vec{u}_0)u_\\beta u_{\\beta ^{\\prime }}\\,.$ This defines the $3\\times 3$ linearization matrix $A$ and the three-vector of the Hessian matrices $\\vec{H}$ of second derivatives, $A_{\\alpha \\beta }= \\partial _{u_\\beta } \\mathsf {j}_\\alpha \\,, \\quad H^\\alpha _{\\beta \\beta ^{\\prime }} =\\partial _{u_\\beta } \\partial _{u_{\\beta ^{\\prime }}} \\mathsf {j}_\\alpha \\,.$ $\\vec{\\xi }$ is Gaussian white noise with mean 0 and covariance $\\langle \\xi _\\alpha (x,t) \\xi _{\\alpha ^{\\prime }}(x^{\\prime },t^{\\prime })\\rangle = \\delta _{\\alpha \\alpha ^{\\prime }} \\delta (x-x^{\\prime }) \\delta (t-t^{\\prime })\\,,$ where $\\tilde{B}\\tilde{B}\\mathrm {^T}$ is the noise strength matrix with $^\\mathrm {T}$ denoting transpose.", "The susceptibility matrix $C$ and the diffusion matrix $\\tilde{D}$ satisfy the fluctuation-dissipation relation $\\tilde{D}C+C\\tilde{D}=\\tilde{B}\\tilde{B}^\\mathrm {T}\\,.$ If one had set $\\vec{H} = 0$ in (REF ), then this Langevin equation would agree with fluctuating hydrodynamics specialized to one dimension.", "In principle, one could include higher orders in the expansion.", "By power counting they are subdominant.", "Of course, if quadratic coefficients vanish, one should study the effect of cubic terms.", "Most likely, logarithmic corrections could result.", "But other features will be more important than such fine details.", "We consider the stationary, mean zero solution to (REF ), again denoted by $\\vec{u}(x,t)$ .", "Then the claim is that for long times and large spatial scales $\\langle u_\\alpha (x,t)u_\\beta (0,0) \\rangle \\simeq S_{\\alpha \\beta }(j,t)$ with $x$ the continuum approximation for $j$ .", "For a single component Eq.", "(REF ) is the stochastic Burgers equation, equivalently in its space integrated version, the one-dimensional Kadar-Parisi-Zhang equation [17].", "Multi-component KPZ type equations have been proposed before [18], [19], however with degenerate, i.e., vanishing velocities.", "We refer to [20] for pointing out the importance of distinct mode velocities.", "The linearization $A$ has the eigenvalues $-c,0,c$ corresponding to the left and right going sound peaks and the heat peak.", "In Eq.", "(REF ), this linear term dominates all other terms.", "To better understand its role one has to make a linear transformation in component space, denoted by $R$ , such that $A$ becomes diagonal.", "In addition, as a convenient normalization, the transformed susceptibility matrix is required to be the unit matrix.", "Both conditions lead to $RAR^{-1}= \\mathrm {diag}(-c,0,c)\\,,\\quad RCR\\mathrm {^T}=1\\,,$ which determine $R$ up to an overall sign.", "We set $\\vec{\\phi } = R\\vec{u}$ and call $\\vec{\\phi } = (\\phi _{-1},\\phi _0,\\phi _1)$ the normal modes.", "The transformed Langevin equations read $\\partial _t \\phi _\\alpha + \\partial _x \\big (c_\\alpha \\phi _\\alpha + \\langle \\vec{\\phi }, G^{\\alpha }\\vec{\\phi }\\rangle -\\partial _x(D\\phi )_\\alpha +(B\\xi )_\\alpha \\big )=0\\,,$ $\\alpha = -1,0,1$ , where $ D = R \\tilde{D}R^{-1}$ and $B = R\\tilde{B}$ with noise strength $ B B^{\\mathrm {T}} = 2 D$ .", "The velocity of the $\\alpha $ -th normal mode is $c_\\alpha $ , $c_\\sigma =\\sigma c$ , $c_0=0$ , $\\sigma =\\pm 1$ .", "The inner product $\\langle \\cdot ,\\cdot \\rangle $ is in component space and the $G^\\alpha $ matrix of coefficients stands for $G^\\alpha = \\tfrac{1}{2}\\sum ^3_{\\alpha ^{\\prime }=1} R_{\\alpha \\alpha ^{\\prime }} (R^{-1})^{\\mathrm {T}} H^{\\alpha ^{\\prime }}R^{-1}\\,.$ As before, we have to consider the stationary process $\\vec{\\phi }(x,t)$ with mean zero, $\\langle \\vec{\\phi }(x,t)\\rangle = 0$ , satisfying Eq.", "(REF ).", "The $\\vec{\\phi }$ -$\\vec{\\phi }$ correlations are defined by $S^{\\sharp \\phi }_{\\alpha \\alpha ^{\\prime }}(x,t) = \\langle \\phi _{\\alpha }(x,t) \\phi _{\\alpha ^{\\prime }}(0,0) \\rangle \\,,$ where the superscript $^\\sharp $ reminds of normal mode and $^\\phi $ of the underlying stochastic process.", "The central claim is that, as $3\\times 3$ matrices, $RS(j,t)R^\\mathrm {T} = S^\\sharp (j,t) \\simeq S^{\\sharp \\phi }(x,t)$ on a mesoscopic scale.", "Eq.", "(REF ) is a stochastic non-linear field theory and its two-point correlation cannot be readily computed.", "We summarize the main findings up to now.", "Diagonality.", "By construction $S_{\\alpha \\alpha ^{\\prime }}^{\\sharp \\phi }(x,0) = \\delta _{\\alpha \\alpha ^{\\prime }}\\delta (x) \\,.$ Using space-time stationarity and the conservation laws, one deduces the sum rule $\\int dx\\, S_{\\alpha \\alpha ^{\\prime }}^{\\sharp \\phi }(x,t) = \\int dx\\, S_{\\alpha \\alpha ^{\\prime }}^{\\sharp \\phi }(x,0) = \\delta _{\\alpha \\alpha ^{\\prime }} \\,,$ but there is no reason for $S^{\\sharp \\phi }(x,t)$ to remain pointwise diagonal at later times.", "But the distinct velocities of the modes enforce such a behavior and, in the one-loop mode coupling approximation to Eq.", "(REF ), the off-diagonal matrix elements are very small after some transient time.", "This is also seen in MD simulations and leads to $S_{\\alpha \\alpha ^{\\prime }}^{\\sharp \\phi }(x,t) \\simeq \\delta _{\\alpha \\alpha ^{\\prime }}f_\\alpha (x,t)\\,.$ By (REF ), (REF ), the diagonal terms satisfy $f_\\alpha (x,0) = \\delta (x)\\,,\\quad \\int _{\\mathbb {R}} dx\\, f_\\alpha (x,t) = 1\\,.$ For the physical fields $S_{\\alpha \\alpha } (j,t) \\simeq \\sum _{\\alpha ^{\\prime }=1}^{3}\|(R^{-1})_{\\alpha \\alpha ^{\\prime }}\|^2f_{\\alpha ^{\\prime }}(x,t)\\,.$ Hence the Landau-Placzek ratios can be read off from the $R$ matrix.", "Generically, $S_{\\alpha \\alpha } (j,t)$ has three peaks located at 0 (heat peak) and at $\\pm ct$ (sound peaks).", "But for special parameter values some of the Landau-Plazcek ratios may vanish and less peaks are visible.", "KPZ scaling, sound peaks.", "Since the three modes have distinct propagation velocities, one expects Eq.", "(REF ) to decouple into three independent equations, each of which then has the structure of the noisy Burgers equation.", "Thus, if $G^\\sigma _{\\sigma \\sigma } \\ne 0$ , one will have the KPZ scaling, $f_\\sigma (x,t)\\cong (\\lambda _\\mathrm {s} t)^{-2/3} f_{\\mathrm {KPZ}} \\big ((\\lambda _\\mathrm {s} t)^{-2/3}(x-\\sigma ct)\\big )\\,.$ $f_{\\mathrm {KPZ}}$ is the exact scaling function for the two-point correlation of the noisy Burgers equation, see Appendix .", "According to KPZ scaling theory, the non-universal coefficient reads $\\lambda _\\mathrm {s} = \|G^\\sigma _{\\sigma \\sigma }\| \\, a_\\mathrm {s}\\,,\\quad a_\\mathrm {s} = 2\\sqrt{2}\\,,$ where we divided into the material parameter $G^\\sigma _{\\sigma \\sigma }$ and the universal pure number $a_\\mathrm {s}$ .", "Of course, $a_\\mathrm {s}$ depends on the convention for $f_{\\mathrm {KPZ}}$ .", "Lévy scaling, heat mode.", "For anharmonic chains $G_{00}^0 = 0$ , always.", "Thus the leading KPZ scaling (REF ) degenerates and one has to study the interaction between the modes.", "So far, this goal has been accomplished only on the level of mode-coupling, which leads to the prediction $f_0(x,t) = (\\lambda _\\mathrm {h} t)^{-3/5}f_{\\mathrm {L},5/3}((\\lambda _\\mathrm {h} t)^{-3/5} x)$ with $f_{\\mathrm {L},\\alpha }$ the symmetric $\\alpha $ -stable distribution, also known as $\\alpha $ -Lévy distribution, see Appendix .", "As a result of previous numerical simulations [10], [11], [13], and also confirmed here, $f_{\\mathrm {L},5/3}$ seems to be the exact scaling function.", "If so, one can use the scaling properties of non-linear fluctuating hydrodynamics to deduce that $\\lambda _\\mathrm {h} = c^{-1/3}\\lambda ^{-2/3}_\\mathrm {s} (G^0_{\\sigma \\sigma })^2 \\, a_\\mathrm {h}\\,.$ As before, $a_\\mathrm {h}$ is a pure number, not depending on the particular model.", "To determine $a_\\mathrm {h}$ one would have to rely on the exact solution of some model in the same universality class.", "According to mode-coupling theory, $a_\\mathrm {h} = 4 \\int _0^\\infty \\!\\!ds \\, s^{-2/3}\\cos s \\int _{\\mathbb {R}} dx f_{\\mathrm {KPZ}}(x)^2 = 2 \\sqrt{3}\\, \\Gamma \\big (\\tfrac{1}{3}\\big ) \\int _{\\mathbb {R}} dx f_{\\mathrm {KPZ}}(x)^2 \\simeq 3.617\\,.$ Physically one expects that there are no correlations propagating beyond the sound cone, which is confirmed in our simulations.", "Thus the Levy peak is cut off at the location of the sound modes.", "Even potential, zero pressure.", "In principle, also $G^\\sigma _{\\sigma \\sigma }$ could vanish implying that the prediction based on the noisy Burgers equation becomes invalid.", "One generic case for this to happen is $p=0$ and a potential symmetric relative to some reference point.", "An example is $V_\\mathrm {sw}$ with reference point $a/2$ , implying the non-KPZ value $\\ell = a/2$ .", "Mode-coupling theory predicts the sound peaks to be diffusive, $f_\\sigma (x,t) = (\\lambda _\\mathrm {s} t)^{-1/2}f_{\\mathrm {G}}((\\lambda _\\mathrm {s} t)^{-1/2}(x - \\sigma ct))$ and the heat peak to be $\\tfrac{3}{2}$ -Lévy, $f_0(x,t) = (\\lambda _\\mathrm {h} t)^{-2/3}f_{\\mathrm {L},3/2}((\\lambda _\\mathrm {h} t)^{-2/3}x)\\,.$ Based on a recent exact solution for models in the same universality class [21], [22], and also confirmed by our simulations, the scalings (REF ), (REF ) are expected to be the true asymptotic behavior.", "From the self-similarity of non-linear fluctuating hydrodynamics one then deduces $\\lambda _\\mathrm {h} = c^{-1/2}\\lambda ^{-1/2}_\\mathrm {s} (G^0_{11})^2 a_\\mathrm {h}\\,.$ The exact solution implies $a_\\mathrm {h} = 4 \\int _0^\\infty \\!\\!ds s^{-1/2} \\cos s \\int _{\\mathbb {R}} dx f_{\\mathrm {G}}(x)^2 = \\sqrt{2}\\,,$ which happens to agree with the mode-coupling computation.", "Remark.", "Eq.", "(4.11) of [16] should read $\\exp [-\|2\\pi k\|^{5/3}\\lambda _\\mathrm {h}t ]$ and consequently $\\lambda _\\mathrm {h}$ of Eq.", "(4.12) has to be multiplied by $(2\\pi )^{-5/3}$ .", "On the same footing, in Eq.", "(4.18) it should read $\\exp [-\|2\\pi k\|^{3/2}\\lambda _\\mathrm {h}t ] $ and consequently $\\lambda _\\mathrm {h}$ of Eq.", "(4.19) has to be multiplied by $(2\\pi )^{-3/2}$ .", "For the hard-point systems under study, the free energy and the Euler currents have been provided already and this allows for the computation of the non-universal constants, at least in principle.", "However, for equal masses with square shoulder potential, while (REF ) looks still simple, to compute, say, $G$ as a function of $p,\\beta $ turns out to be cumbersome.", "Therefore we rely on a Mathematica code, which computes all coefficients numerically.", "For alternating masses with square-well potential, since the pressure factorizes, all coefficients are expressed in terms of $h$ and its derivatives.", "To have at least one explicit example, we provide the details in Appendix .", "Note that nonlinear fluctuating hydrodynamics makes predictions in essence independent of the specific value of the mass ratio $\\kappa $ .", "So we could set $\\kappa = 1$ .", "But for the mechanical system this amounts to a mere relabeling.", "Thus in our derivations implicitly we have assumed that the dynamics is sufficiently chaotic and that the system has no other conservation laws than the three listed already." ], [ "Molecular dynamics simulations", "Nonlinear fluctuating hydrodynamics is based on several assumptions.", "To find out about the accuracy of the theory one has to rely on MD simulations, which have been carried out for all three models, in each case for a single choice of parameters.", "The lattice size is always $N=4096$ .", "For given initial conditions the dynamics is obtained by iterating collision after collision.", "As an example, for the shoulder potential at our choice of parameters there are approximately 1200 collisions per particle up to the maximal time $t_{\\max } = 1024$ .", "In our implementation we use an “event table” consisting of 8192 ($= 2^{13}$ ) time slots, each of which covers the interval $1/8192$ .", "At the beginning of the simulation, pairwise collision events are determined from the positions and momenta of neighboring particles.", "Each anticipated collision event is stored in the time slot covering the event timing modulo 1.", "A time slot can store more than one event, but the time interval of a slot is chosen such that there is typically only one event per slot.", "Conceptually, the event table resembles a hash table, with the event time serving as index.", "During the actual simulation, the time slots are cyclically traversed one after another: we pick the (closest in time) event from the current time slot and update the positions and momenta of the particle pair associated with the event to the time point immediately after the collision.", "Since momenta have changed, the predictions for the neighboring particles have to be revised, and associated collision events are moved to the time slot covering the newly predicted collision time.", "The phase space functions defining $S(j,t)$ are averaged over all lattice sites and recorded for all $j$ and at times $t = 256$ , 512, 1024.", "We use fast Fourier transformation to accelerate this step.", "A simulation for the shoulder potential at our parameters takes approximately $1.5\\,\\mathrm {s}$ on a commodity laptop computer.", "The scheme is repeated $10^7$ times with initial conditions sampled by means of a random number generator from the i.i.d.", "distribution defined at and above Eq.", "(REF ).", "In the last step we perform the linear transformation $RS(j,t)R^\\mathrm {T} = S^{\\sharp }(j,t)$ .", "In fact, as striking qualitative prediction, this matrix should be diagonal in good approximation.", "Indeed, the off-diagonal matrix elements have size less than $4\\%$ of the diagonal entries, and in the figures below we only show the diagonal entries $S^{\\sharp }_{\\alpha \\alpha }(j,t)$ , $\\alpha = \\pm 1,0$ .", "By symmetry $S^{\\sharp }_{\\alpha \\alpha }(j,t)= S^{\\sharp }_{-\\alpha -\\alpha }(-j,t)$ and only one sound peak needs to be plotted.", "Having obtained the numerical peak, $f^\\mathrm {num}_\\alpha $ , one has to compare with the theoretical prediction $f^\\mathrm {th}_\\alpha $ .", "Since by construction the area under each peak equals 1 and since the peaks turn out to be positive, it is natural to use the $L^1$ -norm as a numerical value for the distance between $f^\\mathrm {num}_\\alpha $ and $f^\\mathrm {th}_\\alpha $ .", "As only free parameter we adopt the linear scale and minimize the expression $\\sum _{j=1}^{N} \\big \| f^\\mathrm {num}_\\alpha (j,t) -(\\lambda t)^{-\\gamma _{\\alpha }} f^\\mathrm {th}_\\alpha ((\\lambda t)^{-\\gamma _{\\alpha }}(j -c_\\alpha t))\\big \|$ with respect to $\\lambda > 0$ for fixed $t$ .", "Here $c_\\alpha $ is the velocity of mode $\\alpha $ and $\\gamma _{\\alpha }$ is the theoretical scaling exponent.", "We record the minimal $L^1$ -distance and the respective value of $\\lambda $ .", "Obviously, there is some level of arbitrariness in our choice.", "We discuss the MD results for each model separately.", "The transformation matrix $R$ and the nonlinear couplings $G$ are listed in Appendix .", "Our conventions for the scaling functions can be found in Appendix ." ], [ "Shoulder potential.", "This is an equal mass chain with potential (REF ).", "The parameters are $p = 1.2$ and $\\beta = 2$ , yielding the sound speed $c = 1.743$ and the average stretch $\\langle r_j \\rangle = 1.246$ .", "In Fig.", "REF a the three peaks are superimposed.", "For the correlations of the physical fields each peak comes with a weight, see Eq.", "(REF ).", "For the stretch correlations the weights are $0.082 : 0.065 : 0.082$ , for the momentum correlations $0.25 : 0 : 0.25$ , and for the energy correlations $0.119 : 0.07 : 0.119$ .", "In Fig.", "REF b,c the scaled heat and sound peaks are compared with the theoretical predictions.", "We note that the deviation from the theoretical shape is fairly small, but the non-universal $\\lambda $ -coefficients are still dropping in time.", "The prediction for $a_\\mathrm {s}$ is based on decoupling, which is expected to be exact.", "The theoretical value is $a_\\mathrm {s} = 2\\sqrt{2} \\simeq 2.828$ , to be compared with the $t = 1024$ molecular dynamics value of 3.936, indicating that the simulation has not yet reached the asymptotic regime.", "The theoretical $\\tfrac{5}{3}$ -Lévy distribution of the heat peak is based on mode-coupling.", "From this perspective, it is not even sure that the true scaling function is given by $\\tfrac{5}{3}$ -Lévy.", "But our simulations, and also earlier results [10], [13], support a symmetric stable distribution with exponent $\\tfrac{5}{3}$ .", "Figure: (Color online) MD simulation of an equal mass chain with shoulder potential as defined in Eq.", "() and parameters N=4096N = 4096, p=1.2p = 1.2, β=2\\beta = 2, at t=1024t = 1024.", "(a) Diagonal matrix entries, S αα ♯ (j,t)S^{\\sharp }_{\\alpha \\alpha }(j,t), of the two-point correlations.", "The gray vertical lines show the sound speed predicted from theory.", "The tails of the sound peaks reappear on the opposite side due to periodic boundary conditions.", "(b) Rescaled heat and (c) right sound peak.", "The theoretical scaling exponents are used and λ\\lambda is fitted numerically to minimize the L 1 L^1-distance between simulation and prediction.", "The dashed orange curve is the predicted 5 3\\tfrac{5}{3}-Lévy distribution f L,5/3 f_{\\mathrm {L},5/3} and the dashed red curve shows f KPZ f_{\\mathrm {KPZ}}.We record the still drifting non-universal coefficients in Table REF , together with the $L^1$ distance defined in Eq.", "(REF ).", "For ease of comparison we provide the universal coefficients $a_\\mathrm {s}$ , $a_\\mathrm {h}$ .", "The theory value for $a_\\mathrm {s}$ is exact, whereas $a_\\mathrm {h}$ employs mode-coupling theory.", "In Table REF the theory value for $\\lambda _\\mathrm {h}$ is based on the exact value of $\\lambda _\\mathrm {s}$ .", "One could argue that instead the measured value of $\\lambda _\\mathrm {s}$ at the same time should be used.", "This will make the comparison slightly less favorable.", "It is remarkable that the empirical and theoretical values are in reasonable agreement, despite the system not yet having reached the asymptotic regime.", "To have a quantitative test, one would have to simulate for longer times and, consequently, with larger lattices.", "Table: Numerically fitted non-universal coefficients from the shoulder potential simulation of Fig.", ", and the corresponding L 1 L^1-distance to the theoretically predicted stable distribution f L,5/3 f_{\\mathrm {L},5/3} for the heat peak and f KPZ f_{\\mathrm {KPZ}} for the sound peak.Given the good fit in Fig.", "REF , more details are provided by plotting the difference between the simulation data and the theoretical fit at optimal $\\lambda $ , see Fig.", "REF with a logarithmic plot provided in Fig.", "REF .", "Even for this difference, the change from the earliest time, $t =256$ , to the latest one, $t = 1024$ , is not particularly pronounced.", "Figure: (Color online).", "Difference between, respectively, the heat and right sound peaks obtained from the MD simulation with shoulder potential and the theoretical prediction at optimal λ\\lambda , as listed in Table .", "The notches around \|x\|≃12\\vert x \\vert \\simeq 12 in (a) are due to feedback from the sound modes.We also simulated the dynamics with an attractive potential, for which (REF ) is modified such that $V_\\mathrm {sh}^{-}(x) = -1$ for $\\frac{1}{2} < \|x\| < 1$ .", "The parameters are fixed as $\\beta = \\frac{2}{5}$ and $p = \\frac{3}{2}$ , with a corresponding sound velocity $c = 1.745$ .", "The coupling matrix $G^{0}$ hardly changes, while $G^{1}$ is roughly doubled.", "This leads to broader sound peaks and thus a stronger interaction between the peaks.", "The heat peak has the same error bars as in case of the repulsive potential.", "At the longest time the sound peaks still have a slight asymmetry, increasing the $L^1$ distance by a factor of 3.", "The attractive potential, at the given parameters, seems to have a considerably slower convergence.", "Indicative are $a_{\\mathrm {h}} = 32.447$ , $a_{\\mathrm {s}} = 12.413$ , both at $t = 1024$ and corresponding to $\\lambda _{\\mathrm {h}} = 7.209$ , $\\lambda _{\\mathrm {s}} = 9.449$ , which deviate even further from the theoretical values." ], [ "Hard-point gas with alternating masses.", "For biatomic chains, the unit cell consists of two adjacent particles.", "To allow for direct comparison with monoatomic chains, we average the two-point correlations according to $\\tilde{S}_{\\alpha \\alpha ^{\\prime }}(j,t) = \\tfrac{1}{4} \\big ( 2\\, S_{\\alpha \\alpha ^{\\prime }}(j,t) + S_{\\alpha \\alpha ^{\\prime }}(j-1,t) + S_{\\alpha \\alpha ^{\\prime }}(j+1,t) \\big ).$ Omitting such average, the two-point correlations would have a pronounced period of 2.", "The parameters of the hard-point gas are alternating masses $m_0 = 1$ , $m_1 = 3$ and $p = 2$ , $\\beta = 1/2$ , yielding a sound speed of $c_{\\bar{m}} = \\sqrt{3}$ .", "The peak structure is comparable to Fig.", "REF .", "For the stretch correlations the weights are $\\frac{1}{6}:\\frac{2}{3}:\\frac{1}{6}$ , for the momentum correlations $2:0:2$ , and for the energy correlations $\\frac{2}{3}:\\frac{2}{3}:\\frac{2}{3}$ .", "The difference between the rescaled sound and heat peaks and the theoretical prediction is displayed in Fig.", "REF .", "We record the still drifting non-universal coefficients in Table REF together with the prediction for the universal coefficients.", "Note that, despite different material parameters, the accuracy is comparable to the chain with shoulder potential.", "Figure: (Color online).", "Difference between, respectively, the heat and right sound peaks obtained from the MD simulation with alternating masses m 0 =1m_0 = 1, m 1 =3m_1 = 3, and the theoretical prediction at optimal λ\\lambda .", "Each λ\\lambda is fitted numerically to minimize the L 1 L^1-distance, see Table .", "Note the notches at \|x\|≃16\\vert x \\vert \\simeq 16 in (a) resulting from feedback of the sound modes.Table: Numerically fitted non-universal coefficients for the hard-point gas with alternating masses m 0 =1m_0 = 1, m 1 =3m_1 = 3, and the corresponding L 1 L^1-distance to the 5 3\\tfrac{5}{3}-Lévy distributionf L,5/3 f_{\\mathrm {L},5/3} for the heat peak and KPZ scaling function f KPZ f_{\\mathrm {KPZ}} for the sound peak." ], [ "Square-well potential with alternating masses at zero pressure.", "The universality classes of nonlinear fluctuating hydrodynamics depend on the vanishing of some of the leading couplings $G^\\alpha _{\\alpha ^{\\prime }\\alpha ^{\\prime }}$ .", "For anharmonic chains, $G^0_{00} =0$ always.", "The make some other leading coefficient vanish is not so easily achieved, except for a symmetric potential at zero pressure.", "A specific example is the square-well potential at zero pressure.", "While the overall appearance looks similar, one can test a universality class different from the previous two examples.", "The square-well potential is defined in Eq.", "(REF ).", "We use the maximal distance $a = 1$ , alternating masses $m_0 = 1$ , $m_1 = 3$ , and $p = 0$ , $\\beta = 2$ , yielding a sound speed of $c_{\\bar{m}} = \\sqrt{3}$ .", "The peak structure is comparable to Fig.", "REF .", "For the stretch correlations the weights are $\\frac{1}{24}:0:\\frac{1}{24}$ , for the momentum correlations $\\frac{1}{2}:0:\\frac{1}{2}$ , and for the energy correlations $0:\\frac{1}{8}:0$ .", "We record the still drifting non-universal coefficients in Table REF .", "For the theoretical prediction, as input for (REF ) we use the measured value of $\\lambda _{\\mathrm {s}}$ at $t = 1024$ .", "$\\lambda _{\\mathrm {s}}$ is a regular transport coefficient not covered by our version of fluctuating hydrodynamics.", "Table: Numerically fitted non-universal coefficients for the simulation with alternating masses m 0 =1m_0 = 1, m 1 =3m_1 = 3 and square-well potential, and the corresponding L 1 L^1-distance to the 3 2\\tfrac{3}{2}-Lévy distributionf L,3/2 f_{\\mathrm {L},3/2} for the heat peak and Gaussian f G f_{\\mathrm {G}} for the sound peak.", "The theory value uses the current numerical value of λ s \\lambda _{\\mathrm {s}}.Figure: (Color online).", "Difference between the, respectively, rescaled heat and right sound peaks obtainedfrom the MD simulation with maximal distance a=1a = 1 and alternating masses, and the theoretical predictionat optimal λ\\lambda .", "Note that the exponents and asymptotic functions are different from the previous cases.", "The fitted λ\\lambda values are provided in Table .", "In the top row, the feedback from the sound modes to the heat mode is clearly visible.The difference between the rescaled sound and heat peaks and the theoretical prediction is displayed in Fig.", "REF .", "It is also instructive to have a logarithmic plot of the simulation data.", "In Fig.", "REF , for each of the three models, we only show the longest time.", "The fit is for optimal $\\lambda $ .", "The asymmetry of the sound peak is still visible with a slightly slower decay towards the heat peak.", "Figure: (Color online).", "Logarithmic plot of the heat and right sound peaks for all three models, at t=1024t = 1024.", "The dashed orange curve in (a) and (b) is the f L,5/3 f_{\\mathrm {L},5/3} and the dashed brown curve in (c) the f L,3/2 f_{\\mathrm {L},3/2}.", "The dashed red curve in (d) and (e) shows f KPZ f_{\\mathrm {KPZ}}, and the magenta dashed curve in (f) isthe Gaussian f G f_{\\mathrm {G}}." ], [ "Conclusions", "For a few anharmonic chains with hard collisions, our MD simulations support the predictions from nonlinear fluctuating hydrodynamics.", "Two of the models have sound peaks satisfying KPZ scaling and a heat peak which scales as the symmetric $\\tfrac{5}{3}$ -Lévy distribution.", "The square-well potential chain is anomalous at the fine-tuned parameters $p=0$ , $\\ell =a/2$ , in having two diffusive sound peaks and a symmetric $\\tfrac{3}{2}$ -Lévy heat peak.", "Of course, it would be of interest to expand the evidence by investigating FPU chains and possibly one-dimensional classical fluids.", "The, to us, most surprising discovery is the precision at which the peaks fit the predicted scaling functions.", "In particular, we add to the evidence that the stable law with exponent $\\tfrac{5}{3}$ is indeed the exact scaling function.", "The peaks attain their theoretical shape already for fairly short times.", "On the other side, the non-universal coefficients $\\lambda _{\\mathrm {s}}$ and $\\lambda _{\\mathrm {h}}$ are still slowly drifting on the appropriate self-similar scale.", "On the sizes and times accessible by the simulation, the $\\lambda $ coefficients have not reached a limiting value.", "The deviation of $\\lambda _{\\mathrm {s}}$ , $\\lambda _{\\mathrm {h}}$ from their theoretical value is significant, but one could believe that eventually the predicted asymptotics will be reached.", "Our observation, if true in more generality, would shed some light on earlier discrepancies in determining scaling exponents.", "It is like averaging over systems with the same scaling exponent but distinct non-universal parameters.", "In the same spirit we point out that in Figs.", "REF to REF the sound peaks show a slight asymmetry and the heat peak has bumps resulting from the interaction with the sound mode.", "We conjecture that these are transient effects which will disappear for longer times and correspondingly larger system size.", "In the literature there are contributions which point to similar conclusions.", "We mention the early measurement of the total energy-energy current correlations with a decay as $t^{-2/3}$ [8].", "Also, on a purely phenomenological basis, the $\\tfrac{5}{3}$ -Lévy distribution has been reported before [10], [11], [13], although at a closer look not exactly the same quantity as here is monitored.", "Here we focus on quantities predicted by our theory and, in addition, implement several numerical innovations.", "(i) We average over $10^7$ initial conditions which are drawn from the exact equilibrium distribution.", "There is no equilibration time step.", "(ii) We employ the field theory version of anharmonic chains and measure the locally conserved fields in this representation.", "We use normal mode coordinates, so to unambiguously separate the three peaks, and restrict our simulation time up to the first collision between the sound peaks.", "(iii) It is easier to fit theoretical predictions than to measure accurately scaling exponents.", "In our context, one has six correlation functions depending on space-time.", "Nonlinear fluctuating hydrodynamics suggests to use the finest resolution for either spatial lattice or Fourier modes and only a few time points.", "In other MD simulations the converse is pursued, namely fine time, respectively frequency, resolution and only a few smallest wave numbers.", "With such data the peak structure is well resolved in frequency space, but the translation back to $(x,t)$ -space cannot be readily achieved.", "While writing, there are further MD simulations on the way.", "H. van Beijeren and H. Posch proposed the shoulder potential for which they run extensive MD simulations.", "Accurate scaling plots are reported, but the non-universal coefficients still deviate from their theoretical value [15].", "A. Dhar et. al.", "[25] simulate FPU chains with 8192 particles and up to $t = 1600$ .", "The interaction potential is of the form $V(x) = \\tfrac{1}{2} x^2 + \\tfrac{1}{3}\\mathsf {a} x^3 + \\tfrac{1}{4}x^4$ .", "The simulated parameter sets include the asymmetric case $\\mathsf {a} = 2$ , $p = 1$ , $\\beta = 2$ and the case of even potential at zero pressure, $\\mathsf {a} = 0$ , $\\beta = 1$ , $p = 0$ .", "Such simulations are challenging, since one has to solve the differential equations of motion.", "The results indicate that the heat peak scales as the $\\tfrac{5}{3}$ -Lévy distribution with wave-like small perturbations receding outwards.", "On the other hand the sound peaks are still slightly asymmetric.", "The decay outside the sound cone is well approximated by $f_{\\mathrm {KPZ}}$ , while the opposite shoulder still exhibits slow tails resulting from the interaction with the heat peak.", "S. Lepri [26] simulates the FPU chain with $\\mathsf {a} = 2$ , $\\ell = 1$ , and $\\mathsf {e} = 0.1$ , corresponding to the pressure $p = -0.0077$ , alternatively $\\mathsf {e} = 0.5$ , corresponding to the pressure $p = -0.026$ .", "Lepri works in frequency and wave number space.", "The sound peak is highly resolved.", "For the lowest wave numbers, the scaling plot fits well with the Fourier transformed KPZ scaling function.", "However the measured non-universal $\\lambda _\\mathrm {s}$ is off by a factor of roughly 3 in each of the two cases.", "More detailed results are reported in [27].", "Motivated by quantum fluids, M. Kulkarni and A. Lamacraft simulate the nonlinear Schrödinger equation on a lattice.", "Only the two sound peaks are observed and an effective hydrodynamic model works with number and momentum as only conserved fields.", "They report on the sound peak in frequency space for a few lowest wave numbers [28], in spirit similar to [26].", "A fit to the corresponding KPZ scaling function turns out to be fairly precise [29].", "An interesting variant is studied by G. Stoltz based on [30].", "His random field is specified by $\\lbrace y_j,j\\in \\mathbb {Z}\\rbrace $ with $y_j \\in \\mathbb {R}$ .", "The deterministic part of the evolution is governed by $\\frac{d}{dt} y_j = V^{\\prime }(y_{j+1}) - V^{\\prime }(y_{j-1})\\,.$ In addition there are random exchanges $y_j,y_{j+1}$ to $y_{j+1},y_j$ independently at each bond with rate 1.", "The conserved fields are $y_j$ and $V(y_j)$ .", "The dynamics is non-reversible.", "The invariant measures are identical to the $\\lbrace r_j\\rbrace $ -part of the anharmonic chain.", "The canonical parameters are $p,\\beta $ , as before, conjugate to the stretch $\\ell $ and internal potential energy $\\mathsf {e}$ , $\\ell = \\langle y_j\\rangle _{p,\\beta }\\,,\\quad \\mathsf {e} = \\langle V(y_j) \\rangle _{p,\\beta }\\,.$ There are no momenta.", "The Euler equations have only two components and read $\\partial _t \\ell + 2\\partial _x p =0\\,,\\quad \\partial _t \\mathsf {e} - \\partial _x p^2 =0$ with $p = p(\\ell ,\\mathsf {e})$ .", "Following the standard route one obtains the mode velocities $c_1 =0$ , $c_2 = 2(-p\\partial _\\mathsf {e} p +\\partial _\\ell p)$ and the $G$ -couplings $G^1_{11} =0$ , $G^1_{12} =0$ , $G^1_{22} \\ne 0$ , while $G^2_{\\alpha \\alpha ^{\\prime }}$ is generically different from 0.", "Thus the peak with label 2 is expected to have KPZ scaling, in analogy to our sound peaks.", "The peak with label 1 will be $\\tfrac{5}{3}$ -Lévy.", "However, since there is no symmetrically located third peak, it will be the asymmetric $\\tfrac{5}{3}$ -Lévy distribution at maximally allowed asymmetry, see Appendix D in [16].", "MD simulations using the exponential potential $V_{\\mathrm {exp}}(x) = \\mathrm {e}^{-x} +x$ confirm such predictions [31].", "If $V_{\\mathrm {exp}}$ is replaced by the harmonic potential $V_{\\mathrm {ha}}(x) = x^2$ , then one switches to a different universality class, which is the two mode version of our square-well potential at zero pressure and $\\ell = a/2$ .", "Nonlinear fluctuating hydrodynamics predicts a diffusive peak and a $\\frac{3}{2}$ -Lévy peak at maximal asymmetry.", "Mathematically rigorous proofs have been posted recently [21], [22].", "Nonlinear fluctuating hydrodynamics is fairly insensitive to the underlying dynamics and only relies on having uniform, current carrying steady states.", "Thus it applies to quantum fluids, but also to nonreversible stochastic particle systems, as lattice gases with several locally conserved components.", "The latter systems are accessible through Monte-Carlo simulations.", "In [32] the AHR model [33] is studied.", "The steady state is computed via matrix product ansatz.", "Hence all coefficients are known analytically.", "For the normal modes one finds $c_1 \\ne c_2$ and also $G^{1}_{11} \\ne 0$ , $G^{2}_{22} \\ne 0$ .", "However the subleading coefficients vanish, $G^{1}_{22} = 0 = G^{2}_{11}$ .", "In Monte Carlo simulations one observes a rapid relaxation to $f_{\\mathrm {KPZ}}$ for each mode.", "The non-universal $\\lambda $ coefficient is relaxed and attains precisely the value as deduced from the theory.", "A coupled two-lane TASEP is studied in [34], which besides two KPZ peaks allows one to also realize the cases of a KPZ peak with a $\\tfrac{5}{3}$ -Lévy peak and, more exotically, of a KPZ peak with a diffusive peak." ], [ "Acknowledgments.", "We are grateful to H. van Beijeren and H. Posch for sharing with us their insights on simulating a fluid with hard-shoulder potential.", "We greatly profited from discussions with S. Denisov, A. Dhar, P. Ferrari, D. Huse, J. Krug, M. Kulkarni, S. Lepri, R. Livi, A. Politi, T. Sasamoto, G. Schütz, and G. Stoltz.", "Computing resources of the Leibniz-Rechenzentrum are thankfully acknowledged." ], [ "Square-well potential", "The square-well potential serves as an example, for which the transformation to normal modes and the second order expansion are still reasonably explicit.", "We choose the spatial unit such that $a = 1$ .", "Then the thermodynamic potentials are given by $&&\\hspace{-10.0pt} p(\\ell ,\\mathsf {e}) = 2 \\mathsf {e} h(\\ell )\\,,\\quad 2\\mathsf {e}\\beta = 1\\,,\\nonumber \\\\&&\\hspace{-10.0pt} \\partial _\\ell p = 2 \\mathsf {e} h^{\\prime }\\,,\\hspace{3.0pt} \\partial _\\mathsf {e} = 2h\\,,\\quad \\partial _\\ell ^2 p = 2 \\mathsf {e} h^{\\prime }{^{\\prime }}\\,,\\hspace{3.0pt}\\partial _\\ell \\partial _\\mathsf {e} p = 2h^{\\prime }\\,,\\hspace{3.0pt}\\partial _\\mathsf {e} ^2p = 0\\,.$ For the function $h$ we use only that $h^{\\prime } <0$ .", "The concrete $h$ is given below (REF ).", "For the hard-point gas, $h = 1/\\ell $ .", "The sound speed is $c_{\\bar{m}} = c / \\sqrt{\\bar{m}}\\,,\\quad c^2 = 2\\mathsf {e}(-h^{\\prime } +2 h^2)\\,.$ The susceptibility is $C=\\begin{pmatrix} (-h^{\\prime })^{-1} & 0 & 0 \\\\0 & 2\\mathsf {e} \\bar{m}& 0 \\\\0 & 0 & 2\\mathsf {e}^2\\end{pmatrix}\\,,$ and the linearized Euler equations are governed by $A=\\begin{pmatrix} 0 & -\\bar{m}^{-1} & 0 \\\\2 \\mathsf {e} h^{\\prime }& 0 & 2h \\\\0 & 2 \\mathsf {e}h \\bar{m}^{-1}& 0\\end{pmatrix}\\,.$ From the eigenvectors of $A$ one obtains the transformation matrix $R$ as $R =\\frac{1}{2c\\sqrt{\\mathsf {e}}}\\begin{pmatrix}2\\mathsf {e}h^{\\prime }&- c_{\\bar{m}}&2h\\\\4 \\mathsf {e}h \\sqrt{-h^{\\prime }}&0& 2 \\sqrt{-h^{\\prime }} \\\\2\\mathsf {e}h^{\\prime }&c_{\\bar{m}} &2h\\\\\\end{pmatrix}\\,,$ $R^{-1}= \\frac{\\sqrt{\\mathsf {e}}}{c}\\begin{pmatrix}-1&2h/\\sqrt{-h^{\\prime }}&-1\\\\- c_{\\bar{m}} \\bar{m}&0& c_{\\bar{m}} \\bar{m}\\\\2\\mathsf {e}h&2\\mathsf {e}\\sqrt{-h^{\\prime }}&2\\mathsf {e}h\\\\\\end{pmatrix}\\,.$ Next we compute the $G$ matrices for the nonlinear coupling constants.", "Firstly, by direct differentiation of $p$ , $H^\\ell =0\\,,\\quad H^\\mathsf {u}=2\\begin{pmatrix} \\mathsf {e} h^{\\prime }{^{\\prime }}& 0 & h^{\\prime }\\\\0 & -h\\bar{m}^{-1} & 0 \\\\h^{\\prime } & 0 & 0\\end{pmatrix}\\,,\\quad H^\\mathsf {e}= \\frac{2}{\\bar{m}}\\begin{pmatrix} 0 &\\mathsf {e}h^{\\prime } & 0 \\\\\\mathsf {e}h^{\\prime }& 0 & h \\\\0 & h & 0\\end{pmatrix}$ and transformed as $(R^{-1})^{\\mathrm {T}}H^\\mathsf {u} R^{-1}= \\frac{2\\mathsf {e}^2}{c^2}\\begin{pmatrix} a_3 & a_1& a_4 \\\\a_1 & a_2 & a_1 \\\\a_4 & a_1 & a_3\\end{pmatrix},\\quad (R^{-1})^{\\mathrm {T}}H^\\mathsf {e} R^{-1} = 2\\mathsf {e}c\\bar{m}^{-1/2}\\begin{pmatrix} -1 & 0& 0 \\\\0 & 0 & 0 \\\\0 & 0 & 1\\end{pmatrix}\\,,$ where $\\begin{split}a_1 &= 2(-h^{\\prime })^{-1/2}\\big (-hh^{\\prime }{^{\\prime }} + h^{\\prime 2} +2 h^2h^{\\prime }\\big )\\,,\\quad a_2 = 4(-h^{\\prime })^{-1}\\big ( h^2h^{\\prime }{^{\\prime }} - 2h h^{\\prime 2}\\big )\\,,\\\\a_3 &= h^{\\prime }{^{\\prime }} - 2 h h^{\\prime } -4h^3\\,, \\qquad a_4 = h^{\\prime }{^{\\prime }} -6hh^{\\prime } +4h^3\\,.\\end{split}$ One still has to apply $R$ to $(R^{-1})^{\\mathrm {T}}\\vec{H} R^{-1}$ .", "The final result reads $G^0 &= \\sqrt{-h^{\\prime }\\,\\mathsf {e}/\\bar{m}}\\begin{pmatrix} -1 & 0& 0 \\\\0 & 0 & 0 \\\\0 & 0 & 1\\end{pmatrix}\\,,\\\\G^\\sigma &= \\frac{1}{2} \\sqrt{\\mathsf {e}/\\bar{m}} \\left(\\sigma \\frac{1}{2(-h^{\\prime } + 2h^2)}\\begin{pmatrix} a_3 & a_1& a_4 \\\\a_1 & a_2 & a_1 \\\\a_4 & a_1 & a_3\\end{pmatrix}+2h\\begin{pmatrix} -1 & 0& 0 \\\\0 & 0 & 0 \\\\0 & 0 & 1\\end{pmatrix}\\right)\\,.$ At $2\\ell = a$ , $p = 0$ the sound speed simplifies to $c_{\\bar{m}} = 2 \\sqrt{6\\,\\mathsf {e}/\\bar{m}}$ and the coupling matrices to $G^{0} = 2 \\sqrt{3\\,\\mathsf {e}/\\bar{m}}\\begin{pmatrix}-1 & 0 & 0 \\\\0 & 0 & 0 \\\\0 & 0 & 1 \\\\\\end{pmatrix}, \\quad G^{\\sigma } = \\sigma \\sqrt{3\\,\\mathsf {e}/\\bar{m}}\\begin{pmatrix}0 & 1 & 0 \\\\1 & 0 & 1 \\\\0 & 1 & 0 \\\\\\end{pmatrix}\\,.$ For the hard-point gas, $a = \\infty $ , hence $p(\\ell ) = 1/\\ell $ , and one finds $a_1 = -6\\ell ^{-3}$ , $a_2 = 0$ , $a_3 =0$ , $a_4 = 12\\ell ^{-3}$ ." ], [ "Speed of sound, $R$ matrix, and {{formula:4380958c-a10b-4b09-b13e-eea9bd83c467}} couplings", "For each model, at our parameters, we record $c$ , $R$ , and $G$ .", "One has the relation $G^{-1} = - (G^{1})^{\\mathcal {T}}$ , where ${}^\\mathcal {T}$ stands for transpose relative to the anti-diagonal.", "Thus only $G^1$ is listed.", "The entries are rounded to four digits for visual clarity." ], [ "Shoulder potential.", "Our parameters $p = 1.2$ , $\\beta = 2$ imply $ c \\simeq 1.743$ and $R =\\begin{pmatrix}-0.8067 & -1 & 0.7800 \\\\2.1031 & 0 & 1.7526 \\\\-0.8067 & 1 & 0.7800 \\\\\\end{pmatrix}\\,, \\quad R^{-1} =\\begin{pmatrix}-0.2869 & 0.2554 & -0.2869 \\\\-0.5 & 0 & 0.5 \\\\0.3443 & 0.2641 & 0.3443 \\\\\\end{pmatrix}\\,,$ as well as $G^{1} =\\begin{pmatrix}-0.3131 & -0.0123 & 0.3664 \\\\-0.0123 & 0.2014 & -0.0123 \\\\0.3664 & -0.0123 & 0.3664 \\\\\\end{pmatrix}\\,,\\quad G^{0} =\\begin{pmatrix}-0.7635 & 0 & 0 \\\\0 & 0 & 0 \\\\0 & 0 & 0.7635 \\\\\\end{pmatrix}.", "\\\\$" ], [ "Hard-point gas with alternating masses.", "The general expression for $R$ reads $R = \\frac{1}{\\sqrt{6}} \\begin{pmatrix}-\\beta p & -\\sqrt{3 \\beta / \\bar{m}} & 2 \\beta \\\\2 \\beta p & 0 & 2 \\beta \\\\-\\beta p & \\sqrt{3 \\beta / \\bar{m}} & 2 \\beta \\\\\\end{pmatrix}.$ The $G$ matrices only depend on the sound speed, and the general formula is $G^{1} =\\frac{c_{\\bar{m}}}{2 \\sqrt{6}}\\begin{pmatrix}-2 & -1 & 2 \\\\-1 & 0 & -1 \\\\2 & -1 & 2 \\\\\\end{pmatrix}\\,, \\quad G^{0} =\\frac{c_{\\bar{m}}}{\\sqrt{6}}\\begin{pmatrix}-1 & 0 & 0 \\\\0 & 0 & 0 \\\\0 & 0 & 1 \\\\\\end{pmatrix}, \\\\$ with sound speed $c_{\\bar{m}} = c / \\sqrt{\\bar{m}}$ and $c = \\sqrt{3\\beta }\\,p$ .", "Specifically for $m_0 = 1$ , $m_1 = 3$ , $p = 2$ , $\\beta = 1/2$ , one obtains $c_{\\bar{m}} = \\sqrt{3} \\simeq 1.732$ and $R =\\begin{pmatrix}-0.4082 & -0.3536 & 0.4082 \\\\0.8165 & 0 & 0.4082 \\\\-0.4082 & 0.3536 & 0.4082 \\\\\\end{pmatrix}\\,, \\quad R^{-1} =\\begin{pmatrix}-0.4082 & 0.8165 & -0.4082 \\\\-1 & 0 & 1 \\\\0.8165 & 0.8165 & 0.8165 \\\\\\end{pmatrix},$ $G^{1} =\\begin{pmatrix}-0.7071 & -0.3536 & 0.7071 \\\\-0.3536 & 0 & -0.3536 \\\\0.7071 & -0.3536 & 0.7071 \\\\\\end{pmatrix}\\,, \\quad G^{0} =\\begin{pmatrix}-0.7071 & 0 & 0 \\\\0 & 0 & 0 \\\\0 & 0 & 0.7071 \\\\\\end{pmatrix}\\,.$" ], [ "Square-well potential, $a = 1$ and {{formula:d7b857ce-ec36-4999-a307-7c75c28c39f6}} .", "The general formula for $R$ is provided in Eq.", "(REF ) and for $G$ in Eq.", "(REF ).", "Inserting $\\beta = 2$ and alternating masses $m_0 = 1$ , $m_1 = 3$ results in $c_{\\bar{m}} = \\sqrt{3}$ and $R =\\begin{pmatrix}-2.4495 & -0.7071 & 0 \\\\0 & 0 & 2.8284 \\\\-2.4495 & 0.7071 & 0 \\\\\\end{pmatrix}\\,, \\quad R^{-1} =\\begin{pmatrix}-0.2041 & 0 & -0.2041 \\\\-0.7071 & 0 & 0.7071 \\\\0 & 0.3536 & 0 \\\\\\end{pmatrix}\\,,$ $G^{1} =\\begin{pmatrix}0 & 0.6124 & 0 \\\\0.6124 & 0 & 0.6124 \\\\0 & 0.6124 & 0 \\\\\\end{pmatrix}, \\quad G^{0} =\\begin{pmatrix}-1.2247 & 0 & 0 \\\\0 & 0 & 0 \\\\0 & 0 & 1.2247 \\\\\\end{pmatrix}.$" ], [ "Scaling functions", "The non-universal $\\lambda $ coefficients are defined relative to a conventional choice of the scaling functions, which we list for convenience.", "The Gaussian of unit variance is defined as $f_\\mathrm {G}(x) = (2\\pi )^{-1/2}\\,\\mathrm {e}^{-x^2/2}\\,.$ The symmetric Lévy distribution with index $\\alpha $ , $0 < \\alpha <2$ , is given by $f_{\\mathrm {L},\\alpha }(x) = \\frac{1}{2\\pi }\\int _{\\mathbb {R}}dk \\mathrm {e}^{\\mathrm {i}kx}\\mathrm {e}^{-\|k\|^\\alpha }\\,.$ $f_{\\mathrm {L},\\alpha }(x) \\simeq \|x\|^{-\\alpha -1}$ for large $\|x\|$ .", "The KPZ scaling function $f_{\\mathrm {KPZ}}$ is tabulated in [23], denoted there by $f$ .", "It holds $f_{\\mathrm {KPZ}}\\ge 0,\\hspace{6.0pt} \\int _{\\mathbb {R}} dxf_{\\mathrm {KPZ}}(x)=1,\\hspace{6.0pt} f_{\\mathrm {KPZ}}(x)=f_{\\mathrm {KPZ}}(-x),\\hspace{6.0pt} \\int _{\\mathbb {R}} dxf_{\\mathrm {KPZ}}(x)x^2=0.510523\\ldots \\,.$ $f_{\\mathrm {KPZ}}$ looks like a Gaussian with suppressed tails, more precisely a large $\|x\|$ decay as $\\exp [-0.295\|x\|^{3}]$ [24].", "The natural definition of $f_{\\mathrm {KPZ}}$ involves the Fredholm determinant of the Airy kernel, which then implies our particular value of the variance." ] ]	1403.0213
[ [ "Affine permutations and rational slope parking functions" ], [ "Abstract We introduce a new approach to the enumeration of rational slope parking functions with respect to the area and a generalized dinv statistics, and relate the combinatorics of parking functions to that of affine permutations.", "We relate our construction to two previously known combinatorial constructions: Haglund's bijection exchanging the pairs of statistics (area,dinv) and (bounce,area) on Dyck paths, and Pak-Stanley labeling of the regions of k-Shi hyperplane arrangements by k-parking functions.", "Essentially, our approach can be viewed as a generalization and a unification of these two constructions.", "We also relate our combinatorial constructions to representation theory.", "We derive new formulas for the Poincare polynomials of certain affine Springer fibers and describe a connection to the theory of finite dimensional representations of DAHA and nonsymmetric Macdonald polynomials." ], [ "Introduction", "Parking functions are ubiquitous in the modern combinatorics.", "There is a natural action of the symmetric group on parking functions, and the orbits are labeled by the non-decreasing parking functions which correspond naturally to the Dyck paths.", "This provides a link between parking functions and various combinatorial objects counted by Catalan numbers.", "In a series of papers Garsia, Haglund, Haiman, et al.", "[18], [19], related the combinatorics of Catalan numbers and parking functions to the space of diagonal harmonics.", "There are also deep connections to the geometry of the Hilbert scheme.", "Since the works of Pak and Stanley [29], Athanasiadis and Linusson [5] , it became clear that parking functions are tightly related to the combinatorics of the affine symmetric group.", "In particular, they provided two different bijections between the parking functions and the regions of Shi hyperplane arrangement.", "It has been remarked in [2], [11], [27] that the inverses of the affine permutations labeling the minimal alcoves in Shi regions belong to a certain simplex $D_{n}^{n+1}$ , which is isometric to the $(n+1)$ -dilated fundamental alcove.", "As a result, the alcoves in $D_{n}^{n+1}$ can be labeled by parking functions in two different ways.", "In this paper we develop a “rational slope” generalization of this correspondence.", "A function $f:\\lbrace 1,\\ldots , n\\rbrace \\rightarrow \\mathbb {Z}_{\\ge 0}$ is called an $m/n$ -parking function if the Young diagram with row lengths equal to $f(1),\\dots , f(n)$ put in the decreasing order, fits under the diagonal in an $n\\times m$ rectangle.", "Recall that a bijection ${\\omega }:\\mathbb {Z}\\rightarrow \\mathbb {Z}$ is called an affine permutation if ${\\omega }(x+n)={\\omega }(x)+n$ for all $x$ and $\\sum \\limits _{i=1}^n {\\omega }(i)=\\frac{n(n+1)}{2}$ .", "Given a positive integer $m$ , we call an affine permutation $m$ -stable, if the inequality ${\\omega }(x+m)> {\\omega }(x)$ holds for all $x$ .", "All constructions in the present paper are based on the following basic observation (see Section REF for details).", "Proposition 1.1 If $m$ and $n$ are coprime then $m$ -stable affine permutations label the alcoves in a certain simplex $D_n^m$ which is isometric to the $m$ -dilated fundamental alcove.", "In particular, the number of $m$ -stable affine permutations equals $m^{n-1}$ .", "The simplex $D_n^{m}$ (first defined in [10], [27]) plays the central role in our study.", "We show that the alcoves in it naturally label various algebraic and geometric objects such as cells in certain affine Springer fibres and nonsymmetric Macdonald polynomials at $q^m=t^n$ .", "We provide a clear combinatorial dictionary that allows one to pass from one description to another.", "We define two maps $\\operatorname{\\mathcal {A}}, \\operatorname{\\mathcal {PS}}$ between the $m$ -stable affine permutations and $m/n$ -parking functions and prove the following results about them.", "Theorem 1.2 Maps $\\operatorname{\\mathcal {A}}$ and $\\operatorname{\\mathcal {PS}}$ satisfy the following properties: The map $\\operatorname{\\mathcal {A}}$ is a bijection for all $m$ and $n$ .", "The map $\\operatorname{\\mathcal {PS}}$ is a bijection for $m=kn\\pm 1$ .", "For $m=kn+1$ , it is equivalent to the Pak-Stanley labeling of Shi regions.", "The map $\\operatorname{\\mathcal {PS}}\\circ \\operatorname{\\mathcal {A}}^{-1}$ generalizes the bijection $\\zeta $ constructed by Haglund in [18].", "More concretely, if one takes $m=n+1$ and restricts the maps $\\operatorname{\\mathcal {A}}$ and $\\operatorname{\\mathcal {PS}}$ to minimal length right coset representatives of $S_n\\backslash \\widetilde{S}_n,$ then $\\operatorname{\\mathcal {PS}}\\circ \\operatorname{\\mathcal {A}}^{-1}$ specializes to Haglund's $\\zeta .$ Remark 1.3 For $m=n+1$ the bijection $\\operatorname{\\mathcal {A}}$ is similar to the Athanasiadis-Linusson [5] labeling of Shi regions, but actually differs from it.", "Conjecture 1.4 The map $\\operatorname{\\mathcal {PS}}$ is bijective for all relatively prime $m$ and $n$ .", "The map $\\operatorname{\\mathcal {PS}}$ has an important geometric meaning.", "In [24] Lusztig and Smelt considered a certain Springer fibre $\\mathcal {F}_{m/n}$ in the affine flag variety and proved that it can be paved by $m^{n-1}$ affine cells.", "In [14], [15] a related subvariety of the affine Grassmannian has been studied under the name of Jacobi factor, and a bijection between its cells and the Dyck paths in $m\\times n$ rectangle has been constructed.", "In [20] Hikita generalized this combinatorial analysis and constructed a bijection between the cells in the affine Springer fiber and $m/n$ -parking functions (in slightly different terminology).", "He gave a quite involved combinatorial formula for the dimension of a cell.", "We reformulate his result in terms of the map $\\operatorname{\\mathcal {PS}}$ .", "Theorem 1.5 The affine Springer fiber $\\mathcal {F}_{m/n}$ admits a paving by affine cells $\\Sigma _{\\omega }$ naturally labeled by the $m$ -stable affine permutations $\\omega $ .", "The dimension of such a cell equals $\\dim \\Sigma _{\\omega }=\\sum _{i=1}^{n} \\operatorname{\\mathcal {PS}}_{\\omega }(i).$ Corollary 1.6 If the map $\\operatorname{\\mathcal {PS}}$ is a bijection (in particular, if $m=kn\\pm 1$ ), then the Poincaré polynomial of $\\mathcal {F}_{m/n}$ is given by the following simple formula: $\\sum _{k=0}^{\\infty }t^{k}\\dim \\operatorname{H}^{k}\\left(\\mathcal {F}_{m/n}\\right)=\\sum _{f\\in \\operatorname{\\mathcal {PF}}_{m/n}}t^{2\\sum _{i}f(i)}.$ It had been proven by Varagnolo, Vasserot and Yun [32], [33] that the cohomology of affine Springer fibers $\\mathcal {F}_{m/n}$ carry the action of double affine Hecke algebra (DAHA).", "In fact, all finite-dimensional DAHA representations can be constructed this way.", "On the other hand, Cherednik, the third named author and Suzuki [7], [30] gave a combinatorial description of DAHA representations in terms of periodic standard Young tableaux and nonsymmetric Macdonald polynomials.", "Theorem 1.7 There is a basis (of nonsymmetric Macdonald polynomials) in the finite-dimensional DAHA representation naturally labeled by the alcoves of the $m$ -dilated fundamental simplex.", "By Proposition REF , these alcoves can be identified with the $m$ -stable permutations $\\omega $ .", "The weight of such a nonsymmetric Macdonald polynomial can be explicitly computed in terms of the parking function $\\operatorname{\\mathcal {A}}_\\omega $ .", "The maps $\\operatorname{\\mathcal {A}}$ and $\\operatorname{\\mathcal {PS}}$ can be used to define two statistics on $m$ -stable permutations (or, equivalently, on $m/n$ -parking functions): $\\operatorname{area}(\\omega ):=\\frac{(m-1)(n-1)}{2}-\\sum \\operatorname{\\mathcal {A}}_{\\omega }(i),\\ \\operatorname{dinv}(\\omega ):=\\frac{(m-1)(n-1)}{2}-\\sum \\operatorname{\\mathcal {PS}}_{\\omega }(i).$ For the case $m=n+1$ Armstrong showed in [2] (in slightly different terms) that $\\operatorname{area}$ and $\\operatorname{dinv}$ statistic agrees with the statistics defined in [19] as a part of “Shuffle Conjecture”.", "Conjecture 1.8 The combinatorial Hilbert series $H_{m/n}(q,t):=\\sum _{\\omega }q^{\\operatorname{area}(\\omega )}t^{\\operatorname{dinv}(\\omega )}$ is symmetric in $q$ and $t$ for all $m$ and $n$ : $H_{m/n}(q,t)=H_{m/n}(t,q).$ To support this conjecture, let us remark that the “weak symmetry” $H_{m/n}(q,1)=H_{m/n}(1,q)$ would follow from the bijectivity of the map $\\operatorname{\\mathcal {PS}}.$ Indeed, $H_{m/n}(q,1)=\\sum _{\\omega }q^{\\frac{(m-1)(n-1)}{2}-\\sum \\operatorname{\\mathcal {A}}_{\\omega }(i)}=\\sum _{f\\in \\operatorname{\\mathcal {PF}}_{m/n}}q^{\\frac{(m-1)(n-1)}{2}-\\sum f(i)}=\\sum _{\\omega }q^{\\frac{(m-1)(n-1)}{2}-\\sum \\operatorname{\\mathcal {PS}}_{\\omega }(i)}=H_{m/n}(1,q).$ The second equation follows from the bijectivity of the map $\\operatorname{\\mathcal {A}}$ , and the third one follows from the bijectivity of the map $\\operatorname{\\mathcal {PS}}$ .", "In particular, the “weak symmetry” holds for $m=kn\\pm 1.$ Surprisingly enough, we found a version of the map $\\operatorname{\\mathcal {PS}}$ for the finite symmetric group $S_n$ .", "A permutation $\\omega \\in S_n$ is called $m$ -stable, if $\\omega (i+m)>\\omega (i)$ for all $i\\le n-m.$ It is easy to see that the number of $m$ -stable permutations is given by a certain multinomial coefficient.", "We define $\\operatorname{\\mathcal {PS}}_{\\omega }({\\omega }(i))$ as the number of inversions of height at most $m$ in $\\omega $ , containing $i$ as the right end.", "Theorem 1.9 The restriction of the map $\\operatorname{\\mathcal {PS}}$ to the finite symmetric group $S_n$ is injective for all $m$ and $n$ .", "For example, for $m=2$ the map $\\operatorname{\\mathcal {PS}}$ provides a bijection between the set of 2-stable permutations in $S_n$ and the set of length $n$ Dyck paths with free right end.", "We also discuss a relation of this finite version of our construction to the theory of Springer fibers.", "The rest of the paper is organized as follows.", "In Section we introduce and review the main ingredients of our construction: rational slope parking functions, affine permutations, and Sommers regions.", "In Section we construct the maps $\\operatorname{\\mathcal {A}}$ and $\\operatorname{\\mathcal {PS}}$ from the set of $m$ -stable affine permutations to the set of rational slope parking functions and prove that $\\operatorname{\\mathcal {A}}$ is a bijection.", "We also discuss the statistics arising from our construction and introduce the combinatorial Hilbert polynomial.", "In Section we study the case $m=kn\\pm 1$ and its relation to the theory of extended Shi arrangements and Pak-Stanley labeling.", "In Section we discuss the specializations of the maps $\\operatorname{\\mathcal {A}}$ and $\\operatorname{\\mathcal {PS}}$ to minimal length coset representatives and their relation to Haglund's bijection $\\zeta .$ In Section we relate our construction to the theory of finite dimensional representations of Cherednik's DAHA and nonsymmetric Macdonald polynomials.", "In Section we discuss a version of the map $\\operatorname{\\mathcal {PS}}$ for the finite symmetric group and prove its injectivity.", "In Section we discuss how our constructions are related to the theory of Springer fibers.", "Finally, we consider some examples for $m\\ne kn\\pm 1$ in Section .", "It is worth to mention that the combinatorial structure of the dilated fundamental alcove has been recently investigated in [31], where the alcoves in it were labeled by certain sequences of numbers (but not parking functions).", "We plan to investigate the connections of our work to [31] in the future." ], [ "Acknowledgements", "The authors would like to thank American Institute of Mathematics (AIM) for hospitality, and D. Armstrong, F. Bergeron, S. Fishel, I. Pak, R. Stanley, V. Reiner, B. Rhoades, A. Varchenko, G. Warrington and N. Williams for useful discussions and suggestions.", "The work of E. G. was partially supported by the grants RFBR-13-01-00755, NSh-4850.2012.1.", "The work of M. V. was partially supported by the grant NSA MSP H98230-12-1-0232." ], [ "Tools and definitions", "We start with a brief review of the definitions and basic results involving parking functions, affine permutations, and hyperplane arrangements, which will play the key role in our constructions." ], [ "Parking Functions", "Definition 2.1 A function $f:\\lbrace 1,\\dots , n\\rbrace \\rightarrow \\mathbb {Z}_{\\ge 0}$ is called an $m/n$ -parking function if the Young diagram with row lengths equal to $f(1),\\dots , f(n)$ put in the decreasing order, bottom to top, fits under the diagonal in an $n\\times m$ rectangle.", "The set of such functions is denoted by $\\operatorname{\\mathcal {PF}}_{m/n}.$ We will often use the notation $f=\\llparenthesis f(1)f(2)\\ldots f(n) \\rrparenthesis $ for parking functions.", "Example 2.2 Consider the function $f:\\lbrace 1,2,3,4\\rbrace \\rightarrow \\mathbb {Z}_{\\ge 0}$ given by $f(1)=2,$ $f(2)=0,$ $f(3)=4,$ and $f(4)=0$ (i.e.", "$f=\\llparenthesis 2040 \\rrparenthesis $ ).", "The corresponding Young diagram fits under the diagonal in a $4\\times 7$ rectangle, but it does not fit under the diagonal in a $4\\times 5$ rectangle.", "Therefore, $f\\in PF_{7/4}$ but $f\\notin PF_{5/4}$ (see Figure REF ).", "Figure: The labeled diagram for the parking function f=⦇2040⦈.f=\\llparenthesis 2040 \\rrparenthesis .Equivalently, a function $f:\\lbrace 1,\\ldots ,n\\rbrace \\rightarrow \\mathbb {Z}_{\\ge 0}$ belongs to $\\operatorname{\\mathcal {PF}}_{m/n}$ if and only if it satisfies one of the following two equivalent conditions: $\\forall \\ell \\in \\lbrace 0,\\ldots ,m-1\\rbrace ,\\ \\sharp \\lbrace k\\in \\lbrace 1,\\dots , n\\rbrace \\mid f(k)< \\ell \\rbrace \\ge \\frac{\\ell n}{m},$ or $\\forall i\\in \\lbrace 0,\\ldots ,n-1\\rbrace ,\\ \\sharp \\lbrace k\\in \\lbrace 1,\\dots , n\\rbrace \\mid f(k)\\le \\frac{im}{n}\\rbrace \\ge i+1.$ Let $P:\\operatorname{\\mathcal {PF}}_{m/n}\\rightarrow Y_{m,n}$ denote the natural map from the set of parking functions to the set $Y_{m,n}$ of Young diagrams that fit under diagonal in an $n\\times m$ rectangle.", "To recover a parking function $f\\in \\operatorname{\\mathcal {PF}}_{m/n}$ from the corresponding Young diagram $P(f)$ one needs some extra information.", "Lengths of the rows of $P(f)$ correspond to the values of $f,$ but one needs also to assign the preimages to them.", "That is, one should label the rows of $P(f)$ by integers $1,2,\\ldots ,n.$ Note that if $P(f)$ has two rows of the same length, then the order of the corresponding labels does not matter.", "One should choose one of the possible orders.", "We choose the decreasing order (read from bottom to top).", "Definition 2.3 Let $\\widehat{Y}_{m,n}$ denote the set of couples $(D,\\tau )$ of a Young diagram $D\\in Y_{m,n}$ and a (finite) permutation $\\tau \\in S_n,$ such that if $k$ th and $(k+1)$ th rows of $D$ have the same length, than $\\tau (k+1)<\\tau (k).$ We will refer to $\\tau $ as the row-labeling of $D.$ Note that $\\tau \\in S_n$ is the permutation of maximal length such that $f \\circ \\tau $ is non-increasing.", "Example 2.4 In Example REF , one has $\\tau = [3,1,4,2]$ , so $f\\circ \\tau =\\llparenthesis 2040 \\rrparenthesis \\circ [3,1,4,2] = \\llparenthesis 4200 \\rrparenthesis $ .", "We get the following lemma: Lemma 2.5 The set of $m/n$ -parking functions $\\operatorname{\\mathcal {PF}}_{m/n}$ is in bijection with the set of labeled Young diagrams $\\widehat{Y}_{m,n}.$ Remark 2.6 Note that for $m=n+1$ the set $\\operatorname{\\mathcal {PF}}_{m/n}$ is exactly the set of classical parking functions $\\operatorname{\\mathcal {PF}}$ , and for $m=kn+1$ it is the set of $k$ -parking functions $\\operatorname{\\mathcal {PF}}_k$ (e.g.", "[18]).", "From now on we will assume that $m$ and $n$ are coprime, so there are no lattice points in the diagonal of $n\\times m$ rectangle.", "By abuse of notation, we will call a non-decreasing parking function increasing.", "The number of increasing parking functions equals to the generalized Catalan number $\\sharp Y_{m,n}= \\frac{1}{n+m}\\binom{n+m}{n}$ .", "The number of all parking functions equals $m^{n-1}$ ." ], [ "Affine Permutations", "Definition 2.7 The affine symmetric group $\\widetilde{S}_n$ is generated by elements $s_1,\\ldots ,s_{n-1},s_0$ subject to the relations $s_i^2=1,$ $s_is_j=s_js_i$ for $i-j\\lnot \\equiv \\pm 1$ mod $n,$ $s_is_js_i=s_js_is_j$ for $i-j\\equiv \\pm 1$ mod $n$ (if $n > 2$ ).", "Let ${\\bf \\overline{x}}=\\left( \\begin{array}{l} x_1 \\\\ \\vdots \\\\ x_n \\end{array} \\right),\\ \\ V:=\\lbrace {\\bf \\overline{x}}\\in {\\mathbb {R}}^n \\mid x_1+\\ldots +x_n=0\\rbrace \\subset \\mathbb {R}^n$ and let $H_{ij}^k$ be the hyperplane $\\lbrace {\\bf \\overline{x}}\\in V \\mid x_i-x_j=k\\rbrace \\subset V.$ The hyperplane arrangement $\\widetilde{B}_n=\\lbrace H_{ij}^k:0<i<j\\le n,k\\in \\mathbb {Z}\\rbrace $ is called the affine braid arrangement.", "The connected components of the complement to the affine braid arrangement are called alcoves.", "The group $\\widetilde{S}_n$ acts on $V$ with the generators $s_i$ acting by reflections in hyperplanes $H_{i,i+1}^0$ for $i>0,$ and $s_0$ acting by reflection in the hyperplane $H_{1,n}^1.$ The action is free and transitive on the set of alcoves, so that the map $\\omega \\mapsto \\omega ({\\rm A}_0),$ where ${\\rm A}_0:=\\lbrace {\\bf \\overline{x}}\\in V \\mid x_1>x_2>\\ldots >x_n>x_1-1\\rbrace $ is the fundamental alcove, gives a bijection between the group $\\widetilde{S}_n$ and the set of alcoves.", "Observe $H_{i, j}^k = H_{j,i}^{-k}$ , so we may always take $i<j$ .", "It is convenient to extend our notation to allow subscripts in ${\\mathbb {Z}}$ via $H_{i+tn, j+tn}^k = H_{i,j}^k$ and $H_{i, j}^k = H_{i,j-n}^{k-1}$ .", "In this way, we can uniquely write each hyperplane in $\\widetilde{B}_n$ as $H_{i, {\\ell }}^0$ with $1 \\le i \\le n$ , $i < {\\ell }$ , ${\\ell }\\in {\\mathbb {Z}}$ .", "Then we can define the height of the hyperplane $H_{i, {\\ell }}^0$ to be ${\\ell }-i$ .", "Observe, in this manner, the reflecting hyperplane of $s_0$ is $H_{1,n}^1 = H_{1,0}^0 = H_{0,1}^0$ of height 1.", "Note that with this notation, the action of the group $\\widetilde{S}_n$ on the hyperplanes $H_{i,j}^k$ is given by ${\\omega }(H_{i,j}^k)=H_{{\\omega }(i),{\\omega }(j)}^k.$ There is another way to think about the affine symmetric group: Definition 2.8 A bijection $\\omega :\\mathbb {Z}\\rightarrow \\mathbb {Z}$ is called an affine $S_n$ -permutation, if $\\omega (x+n)=\\omega (x)+n$ for all $x,$ and $\\sum _{i=1}^{n}\\omega (i)=\\frac{n(n+1)}{2}.$ In this presentation the operation is composition and the generators $s_1,\\ldots , s_{n-1},s_0$ are given by $s_i(x)=x+1$ for $x\\equiv i$ mod $n,$ $s_i(x)=x-1$ for $x\\equiv i+1$ mod $n,$ $s_i(x)=x$ otherwise.", "It is convenient to use list or window notation for $\\omega \\in \\widetilde{S}_n$ as the list $[{\\omega }(1), {\\omega }(2), \\cdots , {\\omega }(n)]$ .", "Since ${\\omega }(x+n)={\\omega }(x)+n,$ this determines ${\\omega }$ .", "The bijection between $\\widetilde{S}_n$ and the set of alcoves can be made more explicit in the following way.", "Lemma 2.9 Every alcove ${\\rm A}$ contains exactly one point $(x_1,\\ldots ,x_n)^T\\in {\\rm A}$ in its interior such that the numbers $\\frac{n+1}{2}-nx_1,\\ldots ,\\frac{n+1}{2}-nx_n$ are all integers.", "Moreover, if ${\\bf \\overline{x}}\\in \\omega ({\\rm A}_0)$ is such a point, then in the window notation one has $\\omega ^{-1}=[\\frac{n+1}{2}-nx_1,\\ldots ,\\frac{n+1}{2}-nx_n].$ These points are called centroids of alcoves.", "Window notation for the identity permutation is $\\mathrm {id}=[1,2,\\ldots ,n].$ By (REF ), the corresponding point is $\\frac{1}{2n}(n-1,n-3,\\ldots ,1-n).$ Note that it belongs to the fundamental alcove ${\\rm A}_0=\\lbrace {\\bf \\overline{x}}\\in V \\mid x_1>x_2>\\ldots >x_n>x_1-1\\rbrace .$ Moreover, it is the unique point ${\\bf \\overline{x}}\\in {\\rm A}_0$ such that the numbers $\\frac{n+1}{2}-nx_i$ are all integers.", "Indeed, let $a_i=\\frac{n+1}{2}-nx_i$ for all $1\\le i\\le n.$ Since $x_1>x_2>\\ldots >x_n>x_1-1$ we get $a_1<a_2<\\ldots <a_n<a_1+n.$ Moreover, since $x_1+\\ldots +x_n=0,$ we have $a_1+\\ldots +a_n=\\frac{n(n+1)}{2}.$ There is a unique collection of integers satisfying these conditions: $a_1=1,a_2=2,\\ldots ,a_n=n.$ Since $\\widetilde{S}_n$ acts freely and transitively on the set of alcoves, all we need to prove is that (REF ) is preserved under the action of the generators $s_0,\\ldots ,s_n.$ Indeed, for $1\\le i\\le n$ we have $(s_i\\omega )^{-1}=\\omega ^{-1}s_i=[\\omega ^{-1}(s_i(1)),\\ldots ,\\omega ^{-1}(s_i(n))]$ $=[\\omega ^{-1}(1),\\ldots ,\\omega ^{-1}(i+1),\\omega ^{-1}(i),\\ldots ,\\omega ^{-1}(n)],$ and for $i=0$ we have $(s_0\\omega )^{-1}=[\\omega ^{-1}(n)-n,\\omega ^{-1}(2),\\ldots ,\\omega ^{-1}(n-1),\\omega ^{-1}(1)+n].$ On the other side, generators $s_1,\\ldots ,s_n\\in S_n$ simply permute the coordinates of points in $V,$ while $s_0$ acts by sending $(x_1,\\ldots ,x_n)$ to $(x_n+1,x_2,\\ldots ,x_{n-1},x_1-1).$ Therefore, Equation REF is preserved by the action of the group $\\widetilde{S}_n.$ The minimal length left coset representatives ${\\omega }\\in \\widetilde{S}_n/S_n$ , also known as affine Grassmannian permutations, satisfy ${\\omega }(1) < {\\omega }(2) < \\cdots < {\\omega }(n)$ , so that their window notation is an increasing list of integers (summing to $\\frac{ n(n+1)}{2}$ and with distinct remainders $\\mod {n}$ ).", "Their inverses ${\\omega }^{-1}$ are the minimal length right coset representatives and satisfy that the centroid of the alcove ${\\omega }^{-1}({\\rm A}_0)$ are precisely those whose coordinates are decreasing.", "That is to say, ${\\omega }^{-1}({\\rm A}_0)$ is in the dominant chamber $\\lbrace {\\bf \\overline{x}}\\in V\\mid x_1 > x_2 > \\cdots > x_n \\rbrace $ .", "By a slight abuse of notation, we will refer to the set of minimal length left (right) coset representatives as $\\widetilde{S}_n/S_n$ (respectively $S_n\\backslash \\widetilde{S}_n$ )." ], [ "Sommers region", "The notions of an inversion and the length of a permutation generalizes from the symmetric group $S_n$ to the affine symmetric group $\\widetilde{S}_n.$ However, the set $\\lbrace (i,j)\\in {\\mathbb {Z}}^2\\mid i<j, {\\omega }(i)>{\\omega }(j)\\rbrace $ is infinite for all ${\\omega }\\in \\widetilde{S}_n$ except identity.", "That is why it makes more sense to consider inversions up to shifts by multiples of $n:$ Definition 2.10 Let ${\\omega }$ be an affine permutation.", "The set of its inversions is defined as $\\operatorname{{Inv}}({\\omega }) : =\\lbrace (i,j) \\in {\\mathbb {Z}}\\times {\\mathbb {Z}}\\mid 1\\le i\\le n,\\ i < j, {\\omega }(i) > {\\omega }(j) \\rbrace .$ The length of a permutation ${\\omega }$ is then defined as $\\ell ({\\omega }) = \\sharp \\operatorname{{Inv}}({\\omega }).$ We shall say the height of an inversion $(i,j)$ is $j-i$ .", "We will also use the notation $\\operatorname{{\\overline{Inv}}}({\\omega }) : =\\lbrace (i,j) \\in {\\mathbb {Z}}\\times {\\mathbb {Z}}\\mid \\ i < j, {\\omega }(i) > {\\omega }(j) \\rbrace $ for unnormalized inversions.", "Remark 2.11 If $(i,j)\\in \\operatorname{{Inv}}({\\omega }),$ then obviously $i+kn<j+kn$ and ${\\omega }(i+kn)>{\\omega }(j+kn)$ for any integer $k.$ Essentially, these couples of integers represent the same inversion of ${\\omega }.$ The condition $1\\le i\\le n$ allows us to count each inversion exactly once.", "Alternatively, one could also require $1\\le j\\le n,$ $1\\le {\\omega }(i)\\le n,$ or $1\\le {\\omega }(j)\\le n.$ Example 2.12 Consider ${\\omega }= [-3,2,3,8] \\in \\widetilde{S}_{4}/S_4$ , whose inverse is ${\\omega }^{-1} =[5,2,3,0]$ .", "The centroid of ${\\omega }^{-1}({\\rm A}_0)$ is $\\frac{1}{8} (11,1,-1,-11)^T$ .", "Note ${\\omega }$ is translation by the vector $\\mu = (-1,0,0,1)^T$ , as ${\\omega }= [1 -1(4), 2+0(4), 3+0(4), 4+1(4)]$ and likewise ${\\omega }^{-1}$ is translation by $-\\mu $ .", "One can see the centroid above is the centroid of the fundamental alcove translated by $-\\mu $ .", "In terms of Coxeter generators, ${\\omega }= s_1 s_2 s_3 s_2 s_1 s_0$ and ${\\omega }^{-1} = s_0 s_1 s_2 s_3 s_2 s_1$ .", "Note $\\operatorname{{Inv}}({\\omega }) = \\lbrace (4,5), (4,6), (4,7), (4,9), (3,5), (2,5) \\rbrace $ and $\\ell ({\\omega }) = 6$ which is also its Coxeter length.", "The inversions are of height $1, 2, 3, 5, 2, 3$ respectively.", "Additionally, $\\operatorname{{Inv}}({\\omega }^{-1}) =\\lbrace (1,2), (1,3), (1,4), (1,8), (2,4), (3,4) \\rbrace $ .", "Geometrically, ${\\omega }$ has an inversion of height $m$ if and only if the alcove ${\\omega }^{-1}({\\rm A}_0)$ is separated from ${\\rm A}_0$ by a (corresponding) hyperplane of height $m$ .", "More precisely, that hyperplane is $H_{i, i+m}^0$ if the inversion is $(i,i+m)$ .", "The following definition will play the key role in our constructions: Definition 2.13 An affine permutation $\\omega \\in \\widetilde{S}_n$ is called $m$ -stable if for all $x$ the inequality $\\omega (x+m)>\\omega (x)$ holds, i.e.", "${\\omega }$ has no inversions of height $m$ .", "The set of all $m$ -stable affine permutations is denoted by $\\widetilde{S}_n^m$ .", "Definition 2.14 An affine permutation $\\omega \\in \\widetilde{S}_n$ is called $m$ -restricted if $\\omega ^{-1}\\in \\widetilde{S}_n^m$ .", "We will denote the set of $m$ -restricted permutations by ${}^m\\widetilde{S}_n$ .", "Note ${\\omega }\\in {}^m\\widetilde{S}_n$ if and only if for all $i < j$ , ${\\omega }(i) - {\\omega }(j) \\ne m$ .", "Lemma REF implies an important corollary for the set $\\widetilde{S}_n^m$ : Lemma 2.15 Let $m=kn+r,$ where $0<r<n.$ The set of alcoves $\\lbrace \\omega ({\\rm A}_0):\\omega \\in {}^m\\widetilde{S}_n\\rbrace $ coincides with the set of alcoves that fit inside the region $D_n^m\\subset V$ defined by the inequalities: $x_i - x_{i+r} \\ge -k$ for $1\\le i\\le n-r,$ $x_{i+r-n}-x_i \\le k+1$ for $n-r+1\\le i\\le n.$ Remark that $D_n^m$ is precisely the region cut out by the hyperplanes of height $m$ , as $H_{i, i+r}^{-k} = H_{i, i+r + kn}^0 = H_{i, i+m}^0$ and likewise $H_{ i+r-n, i}^{k+1} = H_{ i+r -n + (k+1)n, i}^0 = H_{i+m, i}^0= H_{i, i+m}^0$ .", "This means that alcove ${\\omega }^{-1}({\\rm A}_0)$ is inside $D_{n}^{m}$ if and only if the permutation ${\\omega }$ has no inversions of height $m$ .", "The region $D_n^m$ was considered by Sommers in [27], therefore we call it the Sommers region.", "It is known that $D_{n}^{m}$ is isometric to the $m$ th dilation of the fundamental alcove.", "This was proven for all types by Fan [10] and Sommers [27], based on an earlier unpublished observation of Lusztig.", "It is worth emphasizing that in type $A$ the construction of the isometry is very clear.", "Lemma 2.16 Let $c=\\frac{(m-1)(n+1)}{2}$ The affine permutation ${\\omega }_m:=[m-c,2m-c,\\ldots ,nm-c]$ induces an isometry between $D_{n}^{m}$ and the simplex $mD_{n}^{1} = m {\\rm A}_0$ in the dominant region cut out by the hyperplane $x_1-x_n=m$ : ${\\omega }_m\\left(mD_{n}^{1}\\right)=D_{n}^{m}.$ By the proof of Lemma REF , the region $D_n^m$ is cut out by the hyperplanes $H_{i,i+m}^{0}$ for all integer $i$ .", "Since $m$ and $n$ are coprime, one can equivalently say that it is cut out by the hyperplanes $H_{m-c,2m-c}^{0}, H_{2m-c,3m-c}^{0}, \\ldots , H_{nm-c,(n+1)m-c}^{0}.$ On the other hand, the hyperplane $x_1-x_n=m$ can be written as $H_{n,mn+1}^0$ , so the simplex $mD_{n}^{1}$ is cut out by the hyperplanes $H_{1,2}^{0},H_{2,3}^{0},\\ldots , H_{n-1,n}^{0},H_{n,mn+1}^0$ .", "Remark that ${\\omega }_m(i)=mi-c$ for $1\\le i\\le n$ , and ${\\omega }_m(mn+1)={\\omega }_m(1)+mn=m-c+mn=m(n+1)-c.$ Therefore ${\\omega }_m(H_{i,i+1}^{0})=H_{mi-c,m(i+1)-c}^0$ for $1\\le i<n$ , and ${\\omega }_m(H_{n,mn+1}^{0})=H_{mn-c,m(n+1)-c}^{0} \\, ,$ hence ${\\omega }_m(mD_{n}^{1})=D_{n}^{m}.$ Observe that, since $m$ and $n$ are coprime, the image of the origin under this isometry will be the unique vertex of $D_n^m$ with all integer entries (in other words, in the root lattice).", "Example 2.17 For $n=3$ and $m=2$ we have ${\\omega }_2=[024]$ .", "The dilated fundamental alcove is bounded by the hyperplanes $H_{2,3}^{0},H_{1,2}^{0}$ and $H_{1,3}^2$ , the Sommers region $D_{3}^{2}$ is bounded by the hyperplanes $H_{1,3}^0, H_{2,4}^{0}$ and $H_{3,5}^{0}$ .", "Note that $ {\\omega }_2(H_{2,3}^{0})=H_{2,4}^{0},\\ {\\omega }_2(H_{1,2}^{0})=H_{0,2}^{0}=H_{3,5}^{0},\\ {\\omega }_2(H_{1,3}^2)=H_{0,4}^{2}=H_{1,3}^{0}.", "$ Similarly, for $m=4$ we have ${\\omega }_4=[-226]$ .", "The dilated fundamental alcove is bounded by the hyperplanes $H_{2,3}^{0},H_{1,2}^{0}$ and $H_{1,3}^4$ , the Sommers region $D_{3}^{4}$ is bounded by the hyperplanes $H_{1,5}^{0}, H_{2,6}^{0}$ and $H_{3,7}^{0}$ .", "Note that ${\\omega }_4(H_{2,3}^{0})=H_{2,6}^{0},\\ {\\omega }_4(H_{1,2}^{0})=H_{-2,2}^{0}=H_{1,5}^{0},\\ {\\omega }_4(H_{1,3}^4)=H_{-2,6}^{4}=H_{3,7}^{0}.", "$ All these hyperplanes are shown in Figure REF and Figure REF .", "Figure: Dilated fundamental alcove (left) and Sommers regions (right) for m=2m=2 ,ω 2 =[024]{\\omega }_2=[024]Figure: Dilated fundamental alcoves (left) and Sommers regions (right) for m=4m=4,ω 4 =[-226] {\\omega }_4=[-226]" ], [ "Bijection $\\operatorname{\\mathcal {A}}:\\widetilde{S}_n^m\\rightarrow \\operatorname{\\mathcal {PF}}_{m/n}$", "We define the map $\\operatorname{\\mathcal {A}}:\\widetilde{S}_n^m\\rightarrow \\operatorname{\\mathcal {PF}}_{m/n}$ by the following procedure.", "Given ${\\omega }\\in \\widetilde{S}_n^m$ , consider the set $\\Delta _{\\omega }:=\\lbrace i\\in \\mathbb {Z}: \\omega (i)>0\\rbrace \\subset \\mathbb {Z}$ and let $M_{{\\omega }}$ be its minimal element.", "Note that the set $\\Delta _{\\omega }$ is invariant under addition of $m$ and $n.$ Indeed, if $i\\in \\Delta _{{\\omega }}$ then ${\\omega }(i+m)>{\\omega }(i)>0$ and ${\\omega }(i+n)={\\omega }(i)+n>n>0.$ Therefore $i+m\\in \\Delta _{\\omega }$ and $i+n\\in \\Delta _{\\omega }.$ Consider the integer lattice $(\\mathbb {Z})^2.$ We prefer to think about it as of the set of square boxes, rather than the set of integer points.", "Consider the rectangle $R_{m,n}:=\\lbrace (x,y)\\in (\\mathbb {Z})^2 \\mid 0\\le x<m, 0\\le y<n\\rbrace .$ Let us label the boxes of the lattice according to the linear function $l(x,y):=(mn-m-n)+M_{{\\omega }}-nx-my.$ The function $l(x,y)$ is chosen in such a way that a box is labeled by $M_{\\omega }$ if and only if its NE corner touches the line containing the NW-SE diagonal of the rectangle $R_{m,n},$ so $l(x,y)\\ge M_{{\\omega }}$ if and only if the box $(x,y)$ is below this line.", "The Young diagram $D_{{\\omega }}$ defined by $D_{{\\omega }}:=\\lbrace (x,y)\\in R_{m,n}\\mid l(x,y) \\in \\Delta _{{\\omega }}\\rbrace .$ If $(x,y)\\in D_{{\\omega }},$ then $\\omega (l(x,y))>0$ , hence $\\omega (l(x-1,y))=\\omega (l(x,y)+n)>0,$ and $\\omega (l(x,y-1))=\\omega (l(x,y)+m)>\\omega (l(x,y))>0.$ Therefore, if $x-1\\ge 0,$ then $(x-1,y)\\in D_{{\\omega }},$ and if $y-1\\ge 0,$ then $(x,y-1)\\in D_{{\\omega }}$ .", "We conclude that $D_{{\\omega }}\\subset R_{m,n}$ is indeed a Young diagram with the SW corner box $(0,0).$ Note also that $D_{\\omega }$ fits under the NW-SE diagonal of $R_{m,n}.$ Therefore, $D_w\\in Y_{m,n}.$ Observe that the boxes in the $i^{\\mathrm {th}}$ row of the diagram correspond to coordinates with $y = i-1$ .", "The row-labeling $\\tau _{{\\omega }}$ is given by $\\tau _{{\\omega }}(i)={\\omega }(a_i),$ where $a_i$ is the label on the rightmost box of the $i$ th row of $D_{{\\omega }}$ (if a row has length 0 we take the label on the box $(-1,i-1),$ just outside the rectangle in the same row).", "Note that if $i$ th and $(i+1)$ th rows have the same length, then $a_{i+1}=a_i-m$ and $\\tau _{\\omega }(i+1)={\\omega }(a_{i+1})={\\omega }(a_i-m)<{\\omega }(a_i)=\\tau _{\\omega }(i).$ Therefore, $(D_{{\\omega }},\\tau _{{\\omega }})\\in \\widehat{Y}_{m,n}$ .", "We define $\\operatorname{\\mathcal {A}}({\\omega })\\in \\operatorname{\\mathcal {PF}}_{m/n}$ to be the parking function corresponding to $(D_{\\omega },\\tau _{\\omega }).$ Example 3.1 Let $n=4$ , $m=7$ .", "Consider the affine permutation ${\\omega }= [0,6,3,1] = s_1 s_0 s_2 s_3 s_2:$ $\\begin{array}{ccccccccccccccc}x &\\ldots &-3&-2&-1&0&1&2&3&4&5&6&7&8&\\ldots \\\\\\omega (x) &\\ldots &-4&2&-1&-3&0&6&3&1&4&10&7&5&\\ldots ,\\\\\\end{array}$ The inversion set is $\\operatorname{{Inv}}({\\omega }) = \\lbrace (2,3), (2,4), (2,5), (2,8), (3,4) \\rbrace $ .", "Note that there are no inversions of height $7,$ so $\\omega $ is 7-stable.", "Equivalently, ${\\omega }^{-1} = [4,-2,3,5]$ is 7-restricted.", "The set $\\Delta _{{\\omega }}=\\lbrace -2,2,3,4,\\ldots \\rbrace $ is invariant under the addition of 4 and $7,$ and $M_{{\\omega }}=-2.$ The diagram $D_{{\\omega }}$ is shown in Figure REF .", "Note that the labels $3,4,5,$ and $-2$ on the rightmost boxes of the rows of $D_{{\\omega }}$ are the 4-generators of the set $\\Delta _{{\\omega }},$ i.e.", "they are the smallest numbers in $\\Delta _{{\\omega }}$ in the corresponding congruence classes $\\mod {4}.$ It follows then that the corresponding values ${\\omega }(3),{\\omega }(4),{\\omega }(5),$ and ${\\omega }(-2)$ are a permutation of $1,2,3,4.$ Indeed, read bottom to top, $({\\omega }(3),{\\omega }(4),{\\omega }(5),{\\omega }(-2))=(3,1,4,2).$ This defines the row-labeling $\\tau _{\\omega }:=[3,1,4,2].$ Note that the last (top) two rows of the diagram have the same length $0.$ Therefore, the difference between the corresponding labels is $5-(-2)=7.$ The 7-stability condition then implies that $\\tau (3)={\\omega }(5)>{\\omega }(-2)=\\tau (4),$ which is exactly the required monotonicity condition on the labeling.", "Using the bijection from Lemma REF , one obtains the parking function $\\operatorname{\\mathcal {A}}_{{\\omega }}=\\llparenthesis 2040 \\rrparenthesis .$ Figure: The labeled diagram corresponding to the permutation ω=[0,6,3,1].", "{\\omega }= [0,6,3,1].Equivalently, one can start directly from ${\\omega }^{-1} = [4,-2,3,5] \\in {}^m\\widetilde{S}_n$ and form the same labeled rectangle, noting $M_{\\omega }= \\min \\lbrace {{\\omega }^{-1}}(i) \\mid 1 \\le i \\le n \\rbrace $ .", "Then $\\operatorname{\\mathcal {A}}_{\\omega }(i) = 1 +x$ where $x$ is the $x$ -coordinate of the box labeled ${{\\omega }^{-1}}(i)$ .", "Alternatively, one can define the map $\\operatorname{\\mathcal {A}}$ in a more compact, but less pictorial way: Definition 3.2 Let $\\omega \\in \\widetilde{S}_n^m.$ We define the corresponding parking function $\\operatorname{\\mathcal {A}}_{\\omega }$ as follows.", "Let $M_{\\omega }:=\\min \\lbrace i\\in \\mathbb {Z}: \\omega (i)>0\\rbrace .$ Given $\\alpha \\in \\lbrace 1,\\ldots ,n\\rbrace ,$ there is a unique way to express $\\omega ^{-1}(\\alpha )-M_{{\\omega }}$ as a linear combination $rm-kn$ with the condition $r\\in \\lbrace 0,\\ldots ,n-1\\rbrace .$ Note that one automatically gets $k\\ge 0.$ Indeed, otherwise $\\alpha =\\omega (M_{{\\omega }}+rm-kn)\\ge \\omega (M_{{\\omega }})-kn>-kn\\ge n,$ which contradicts the assumption $\\alpha \\in \\lbrace 1,\\ldots ,n\\rbrace .$ We set $\\operatorname{\\mathcal {A}}_{\\omega }(\\alpha ):=k.$ Lemma 3.3 The two above definitions of the map $\\operatorname{\\mathcal {A}}$ are equivalent.", "Let $(x,i-1)$ be the rightmost box in the $i$ th row of the diagram $D_{\\omega }.$ Let $\\alpha :=\\omega (l(x,i-1))=\\omega (M_{{\\omega }}+(mn-m-n)-xn-(i-1)m)=\\tau _{\\omega }(i).$ We need to check that $\\operatorname{\\mathcal {A}}_{\\omega }(\\alpha )$ is equal to the length of the $i$ th row, which is $x+1.$ Indeed, $\\omega ^{-1}(\\alpha )=M_{{\\omega }}+(mn-m-n)-xn-(i-1)m,$ or $\\omega ^{-1}(\\alpha )-M_{{\\omega }}=(n-i)m-(x+1)n,$ with $n-i\\in \\lbrace 0,\\ldots n-1\\rbrace .$ Therefore, $\\operatorname{\\mathcal {A}}_{\\omega }(\\alpha )=x+1.$ Theorem 3.4 The map $\\operatorname{\\mathcal {A}}:\\widetilde{S}_n^m\\rightarrow \\operatorname{\\mathcal {PF}}_{m/n}$ is a bijection.", "Injectivity of the map $\\operatorname{\\mathcal {A}}:\\widetilde{S}_n^m\\rightarrow \\operatorname{\\mathcal {PF}}_{m/n}$ is immediate from the construction.", "Indeed, the diagram $D_{\\omega }$ completely determines the set $\\Delta _{\\omega },$ while the row-labeling $\\tau _{\\omega }$ determines the values of $\\omega $ on the $n$ -generators of $\\Delta _{\\omega },$ which suffices to determine $\\omega .$ This gives an injective map $\\phi :\\operatorname{\\mathcal {PF}}_{m/n}\\rightarrow \\widetilde{S}_n,$ such that $\\phi \\circ \\operatorname{\\mathcal {A}}=id_{\\widetilde{S}_n^m}.$ To prove that $\\operatorname{\\mathcal {A}}$ is also surjective, it suffices to show that $\\phi (f)$ is $m$ -stable for any $f\\in \\operatorname{\\mathcal {PF}}_{m/n}.$ Indeed, let $f\\in \\operatorname{\\mathcal {PF}}_{m/n}$ be a parking function, $(D_f,\\tau _f)$ be the corresponding Young diagram with row labeling, and $\\omega :=\\phi (f).$ Suppose that $(l,l+m)$ is an inversion of height $m,$ i.e.", "$\\omega (l)>\\omega (l+m).$ By shifting $l$ by a multiple of $n$ if necessary, one can assume that $\\omega (l)\\in \\lbrace 1,\\ldots ,n\\rbrace ,$ so that $l$ labels the rightmost box of one of the rows of $D_f.$ Suppose that $l$ labels the box $(x,y)$ with $y>0,$ i.e.", "it is not in the first row.", "Then $l+m=l(x,y-1),$ which is the label on the box just below the box $(x,y).$ Since $D_f$ is a Young diagram, $(x,y-1)\\in D_f.$ Suppose that $(z,y-1)\\in D_f$ is the rightmost box in the $y$ th row.", "Then $\\omega (l+m)=\\omega (l(z,y-1)+(z-x)n)=\\tau _f(y)+(z-x)n.$ If $z>x,$ we get $\\omega (l+m)>n>\\omega (l).$ Contradiction.", "If $z=x,$ then the $y$ th and $(y+1)$ th rows are of the same length and, by the condition on the row labeling $\\omega (l+m)=\\tau _f(y)>\\tau _f(y+1)=\\omega (l).$ Contradiction.", "Suppose now that $l$ labels the rightmost box in the first row: $l=l(x,0)=M+mn-m-n-xn$ .", "Then $l+m=M+(m-1-x)n$ and $\\omega (l+m)=\\omega (M)+(m-1-x)n>(m-1-x)n\\ge n,$ because $M$ labels the rightmost box in the $n$ th (top) row and, therefore, $\\omega (M)=\\tau _f(n)>0,$ and $m-1-x \\ge 1,$ because the first row of the diagram $D_f$ has length $x+1,$ which has to be less than $m.$ We conclude that $\\omega (l+m)>n\\ge \\omega (l).$ Contradiction.", "Therefore, $\\omega \\in \\widetilde{S}_n^m$ and $\\operatorname{\\mathcal {A}}:\\widetilde{S}_n^m\\rightarrow \\operatorname{\\mathcal {PF}}_{m/n}$ is a bijection.", "We call $\\operatorname{\\mathcal {A}}$ the Anderson map, since if we restrict the domain to minimal length right coset representatives (which correspond to partitions called $(m,n)$ -cores), and then project to increasing parking functions by sorting, the map agrees with one constructed by Anderson [1].", "Remark 3.5 For ${\\omega }_m$ as in Lemma REF we have $\\operatorname{\\mathcal {A}}_{{\\omega }_m} = \\llparenthesis 00\\cdots 0 \\rrparenthesis $ .", "Example 3.6 Consider the case $n=5, m=3$ .", "Let $\\mathrm {id}= [1,2,3,4,5]$ , $s_1 = [2,1,3,4,5]$ , $s_1 s_2 = [2,3,1,4,5]$ which are all 3-restricted.", "The images of their inverses under the Anderson map are then $f = \\operatorname{\\mathcal {A}}_{\\mathrm {id}} = \\llparenthesis 01201 \\rrparenthesis $ , $\\operatorname{\\mathcal {A}}_{s_1} = \\llparenthesis 10201 \\rrparenthesis = f \\circ s_1$ , and $\\operatorname{\\mathcal {A}}_{s_2 s_1} = \\llparenthesis 12001 \\rrparenthesis = f \\circ s_1s_2$ .", "Indeed the 3-restricted permutations in the shuffle $14 \\,$$$ 25 $\\scriptscriptstyle \\cup {\\hspace{-2.22214pt}}\\cup $ 3$ correspond in a similar manner to the entire finite $ S5$ orbit of$ f$.The precise statement is in the following proposition.$ Proposition 3.7 Let ${\\omega }\\in \\widetilde{S}_n^m$ , $f = \\operatorname{\\mathcal {A}}_{\\omega }\\in \\operatorname{\\mathcal {PF}}_{m/n}$ , and write $u = {{\\omega }^{-1}}\\in {}^m\\widetilde{S}_n$ .", "Let $H = \\lbrace h \\in S_n\\mid f \\circ h = f\\rbrace $ .", "Let $\\mathcal {V} = \\lbrace v \\in S_n\\mid uv \\in {}^m\\widetilde{S}_n\\rbrace $ .", "If $v \\in \\mathcal {V}$ then $\\operatorname{\\mathcal {A}}_{v^{-1}{\\omega }} = \\operatorname{\\mathcal {A}}_{{\\omega }} \\circ v.$ In particular $\\mathcal {V}$ are a complete set of coset representatives for $S_n/H$ .", "We first consider the case $v = s_i$ .", "Note $ u s_i = [u(1), \\cdots , u(i+1), u(i), \\cdots , u(n)]$ , so in particular as $u, us_i \\in {}^m\\widetilde{S}_n$ , $\| u(i) = u(i+1)\| \\ne m$ ; and in fact this difference cannot be a multiple of $m$ .", "We observed at the end of Example REF that $\\operatorname{\\mathcal {A}}_{\\omega }(k) = 1+x$ where $(x,y)$ are the coordinates of the box labeled $u(k)$ .", "Multiplication by $s_i \\in S_n$ does not change $M_{\\omega }$ nor the function that labels the boxes of the rectangle.", "Hence $\\operatorname{\\mathcal {A}}_{s_i{\\omega }}(k) = 1+x$ where $(x,y)$ are the coordinates of the box labeled $us_i(k)$ .", "In other words $\\operatorname{\\mathcal {A}}_{s_i {\\omega }}(k) ={\\left\\lbrace \\begin{array}{ll}\\operatorname{\\mathcal {A}}_{\\omega }(k) & k \\ne i, i+1 \\\\\\operatorname{\\mathcal {A}}_{\\omega }(i+1) & k = i \\\\\\operatorname{\\mathcal {A}}_{\\omega }(i) & k = i+1\\end{array}\\right.", "}\\qquad = \\operatorname{\\mathcal {A}}_{\\omega }\\circ s_i = f \\circ s_i.$ Further, note that $\\operatorname{\\mathcal {A}}_{\\omega }\\circ s_i \\ne \\operatorname{\\mathcal {A}}_{\\omega }$ , otherwise the boxes labeled $u(i), u(i+1)$ would be in the same column and hence differ by a multiple of $m$ .", "Conversely, given $1\\le i \\le n$ with $f \\circ s_i \\ne f$ , we see $u, u s_i \\in {}^m\\widetilde{S}_n$ .", "Hence we may assume $f$ is chosen so that $H$ is a standard parabolic subgroup, in which case the set $\\mathcal {V}$ will correspond to minimal length coset representatives.", "Indeed, repeating the above argument, we see that for any $v \\in S_n$ with $uv \\in \\widetilde{S}_n^m$ and $\\ell (uv) = \\ell (u) + \\ell (v)$ that $\\operatorname{\\mathcal {A}}_{v^{-1}w} = \\operatorname{\\mathcal {A}}_{\\omega }\\circ v$ ." ], [ "Map $\\operatorname{\\mathcal {PS}}:\\widetilde{S}_n^m\\rightarrow \\operatorname{\\mathcal {PF}}_{m/n}$", "Definition 3.8 Let $\\omega \\in \\widetilde{S}_n^m$ .", "Then the map $\\operatorname{\\mathcal {PS}}_{\\omega }:\\lbrace 1,\\ldots ,n\\rbrace \\rightarrow \\mathbb {Z}$ is given by: $\\operatorname{\\mathcal {PS}}_{\\omega }(\\alpha ):=\\sharp \\left\\lbrace \\beta \\mid \\beta >\\alpha ,0 < \\omega ^{-1}(\\alpha ) - \\omega ^{-1}(\\beta )<m \\right\\rbrace $ $=\\sharp \\left\\lbrace i \\mid \\omega (i) >\\alpha ,\\omega ^{-1}(\\alpha )-m < i<\\omega ^{-1}(\\alpha ) \\right\\rbrace .$ In other words, $\\operatorname{\\mathcal {PS}}_{{\\omega }}(\\alpha )$ is equal to the number of inversions $(i,j)\\in \\operatorname{{Inv}}({\\omega })$ of height less than $m$ and such that ${\\omega }(j)\\equiv \\alpha $ mod $n.$ Definition 3.9 Let $\\operatorname{\\mathcal {SP}}: {}^m\\widetilde{S}_n\\rightarrow \\operatorname{\\mathcal {PF}}_{m/n}$ be defined by ${\\omega }\\mapsto \\operatorname{\\mathcal {PS}}_{{\\omega }^{-1}}$ .", "Observe $\\operatorname{\\mathcal {SP}}_u(i) = \\sharp \\lbrace j > i \\mid 0 < u(i) - u(j) < m \\rbrace $ .", "Example 3.10 Using the same permutation as in Example REF , one gets $\\operatorname{\\mathcal {PS}}_{\\omega }(1)=3 = \\sharp \\lbrace (2,4), (3,4), (-2,4) \\rbrace ,$ $\\operatorname{\\mathcal {PS}}_{\\omega }(2)=0,$ $\\operatorname{\\mathcal {PS}}_{\\omega }(3)=1 = \\sharp \\lbrace (2,3)\\rbrace ,$ and $\\operatorname{\\mathcal {PS}}_{\\omega }(4)=1 = \\sharp \\lbrace (2,5)\\rbrace ,$ so $\\operatorname{\\mathcal {PS}}_{\\omega } = \\llparenthesis 3011 \\rrparenthesis $ .", "Likewise, ${{\\omega }^{-1}}= [4, -2, 3, 5]$ and $\\operatorname{\\mathcal {SP}}_{{\\omega }^{-1}}(1) = 3 =\\sharp \\lbrace (1,2), (1,3), (1,6) \\rbrace \\subseteq \\operatorname{{Inv}}({{\\omega }^{-1}})$ , $ 1 = \\sharp \\lbrace (3,6)\\rbrace ,$ and $ 1 = \\sharp \\lbrace (4,6)\\rbrace .$ Example 3.11 Consider $(n,m) = (5,3)$ and $u = [0,3,6,2,4]$ .", "Then $\\operatorname{{Inv}}(u) = \\lbrace (2,4), (3,4), (3,5), (3,6) \\rbrace $ and so $\\operatorname{\\mathcal {SP}}_u = \\llparenthesis 01200 \\rrparenthesis $ .", "Note $u(3) - u(4) = 4 > m$ so this inversion does not contribute to $\\operatorname{\\mathcal {SP}}_u(3).$ Let us prove that $\\operatorname{\\mathcal {PS}}_{\\omega }$ is indeed an $m/n$ -parking function.", "We will need the following definition and lemmas.", "Definition 3.12 A subset $K\\subset \\mathbb {Z}$ is called $\\pm n$ -invariant if for all $x\\in K$ one has $x+n\\in K$ and $x-n\\in K$ .", "Lemma 3.13 Let $K$ be an $\\pm n$ -invariant set, and $\\sharp \\left(K\\cap [1,n]\\right)=k$ .", "Then there exists $i\\in \\mathbb {Z}$ such that $\\sharp \\left(K\\cap [i-m+1,i]\\right)\\le \\frac{km}{n}.$ Consider an interval $I$ in $\\mathbb {Z}$ of length $mn$ .", "On one hand, it is covered by $m$ intervals of length $n$ , containing $k$ points of $K$ each, hence $\\sharp (K\\cap I)=km$ .", "On the other hand, it is covered by $n$ intervals of length $m$ , hence one of these intervals should contain at most $\\frac{km}{n}$ points of $K$ .", "Lemma 3.14 Let $\\omega \\in \\widetilde{S}_n^m$ , let $K$ be an $\\pm n$ -invariant set, and $\\sharp \\left(K\\cap [1,n]\\right)=k$ .", "There exists $l\\in \\mathbb {Z}\\setminus K$ such that the following conditions hold: a) If $j<l$ and $\\omega (j)>\\omega (l)$ then $j\\in K$ b) $\\sharp \\lbrace j\\in K: l-m<j<l, \\omega (j)>\\omega (l)\\rbrace \\le \\frac{km}{n}.$ By Lemma REF there exists $i\\in \\mathbb {Z}$ such that $\\sharp \\left(K\\cap [i-m+1,i]\\right)\\le \\frac{km}{n}$ .", "Since $\\omega (x+n)=\\omega (x)+n$ , the set of values of $\\omega $ on the half line $(-\\infty ,i]$ is bounded from above.", "Let us choose $l\\le i$ such that: $\\omega (l)=\\max \\lbrace \\omega (x):x\\in (-\\infty ,i]\\setminus K\\rbrace ,$ and prove that this $l$ satisfies (a) and (b).", "If $j<l$ and $j\\notin K$ then by (REF ) one has $\\omega (j)<\\omega (l)$ , hence (a) holds.", "To prove (b), define $J_m(l,K):=\\lbrace j\\in K: l-m<j<l, \\omega (j)>\\omega (l)\\rbrace .$ Given $j\\in J_m(l,K)$ , there exists a unique $\\alpha (j)\\in \\mathbb {Z}_{\\ge 0}$ such that $i-m< j+\\alpha (j)m\\le i$ , and for different $j$ the numbers $j+\\alpha (j)m$ are all different.", "Since $\\omega $ is $m$ -stable, we have $\\omega (l)<\\omega (j)<\\omega (j+m)<\\ldots <\\omega (j+\\alpha (j)m).$ By (REF ) we conclude that $j+\\alpha (j)m\\in K$ .", "Therefore we constructed an injective map from $J_m(l,K)$ to $K\\cap [i-m+1,i]$ , and $\\sharp J_m(l,K)\\le \\sharp \\left(K\\cap [i-m+1,i]\\right)\\le \\frac{km}{n}.$ Theorem 3.15 For any $m$ -stable affine permutation $\\omega ,$ the function $\\operatorname{\\mathcal {PS}}_{\\omega }$ is an $m/n$ -parking function.", "Thus one gets a map $\\operatorname{\\mathcal {PS}}:\\widetilde{S}_n^m\\rightarrow \\operatorname{\\mathcal {PF}}_{m/n}.$ Let us construct a chain of $\\pm n$ -invariant subsets $\\emptyset =K_0\\subset K_1\\subset \\ldots K_n={\\mathbb {Z}},$ with $\\sharp \\lbrace K_i\\cap [1,n]\\rbrace =i$ for all $i,$ by the following inductive procedure.", "Given $K_i$ for some $i,$ we use Lemma REF to find an integer $l_{i+1}$ satisfying Lemma REF (a,b).", "Since $K_i$ was $\\pm n$ -invariant and $l_{i+1}\\notin K_i$ , the sets $K_i$ and $l_{i+1}+n\\mathbb {Z}$ do not intersect, hence we can set $K_{i+1}:=K_i\\sqcup (l_{i+1}+n\\mathbb {Z}).$ By shifting the number $l_{i+1}$ by a multiple of $n$ if necessary, we can assume that $\\omega (l_{i+1})\\in \\lbrace 1,\\ldots ,n\\rbrace $ for all $i\\in \\lbrace 0,\\ldots ,n-1\\rbrace .$ Note that $\\lbrace {\\omega }(l_1),{\\omega }(l_2),\\ldots ,{\\omega }(l_n)\\rbrace =\\lbrace 1,\\ldots ,n\\rbrace .$ Let us estimate $\\operatorname{\\mathcal {PS}}_{\\omega }(\\omega (l_{i+1})).$ If $l_{i+1}-m< j<l_{i+1}$ and $\\omega (j)>\\omega (l_{i+1})$ , then by Lemma REF (a) we have $j\\in K_i$ and by Lemma REF (b) the number of such $j$ is at most $\\frac{im}{n}$ .", "Therefore, $\\sharp \\lbrace \\alpha : \\operatorname{\\mathcal {PS}}_{\\omega }(\\alpha )\\le \\frac{im}{n}\\rbrace \\ge i+1,$ because for any $k\\in \\lbrace 0,1,\\ldots ,i\\rbrace $ one has $\\operatorname{\\mathcal {PS}}_{\\omega }(\\omega (l_{k+1}))\\le \\frac{km}{n}\\le \\frac{im}{n}.$ Therefore, $\\operatorname{\\mathcal {PS}}(\\omega )$ is an $m/n$ -parking function.", "Conjecture 3.16 The map $\\operatorname{\\mathcal {PS}}:\\omega \\mapsto \\operatorname{\\mathcal {PS}}_{\\omega }$ is a bijection between $\\widetilde{S}_n^m$ and $\\operatorname{\\mathcal {PF}}_{m/n}.$ In the special cases $m = kn \\pm 1,$ we prove that $\\operatorname{\\mathcal {PS}}$ is a bijection in the next Section.", "It is convenient to extend the domains of the functions $\\operatorname{\\mathcal {PS}}_{\\omega }$ and $\\operatorname{\\mathcal {SP}}_{\\omega }$ to all integers by using exactly the same formula.", "Note that in this case $\\operatorname{\\mathcal {PS}}_{{\\omega }}(\\alpha +n)=\\operatorname{\\mathcal {PS}}_{{\\omega }}(\\alpha ).$ We have the following results, which should be considered as steps towards Conjecture REF : Proposition 3.17 Let ${\\omega }\\in {}^m\\widetilde{S}_n$ and let $1 \\le i \\le n$ , $i < j$ .", "${\\omega }(i) < {\\omega }(i+1) \\iff \\operatorname{\\mathcal {SP}}_{\\omega }(i) \\le \\operatorname{\\mathcal {SP}}_{\\omega }(i+1)$ $(i,j)\\in \\operatorname{{Inv}}({\\omega }) \\Rightarrow \\operatorname{\\mathcal {SP}}_{\\omega }(i) > \\operatorname{\\mathcal {SP}}_{\\omega }(j)$ We first show that if $(i,j) \\in \\operatorname{{Inv}}({\\omega })$ , then ${\\omega }$ has a unique inversion $(i,J)$ with $j \\le J, \\, {\\omega }(j) \\equiv {\\omega }(J) \\mod {m}, \\text{ and } 0 < {\\omega }(i) - {\\omega }(J) < m.$ Since ${\\omega }\\in {}^m\\widetilde{S}_n$ , in the list ${\\omega }(1), {\\omega }(2), \\cdots $ , we have that ${\\omega }(j)$ occurs to the left of ${\\omega }(j) + rm$ for all $r \\ge 1$ .", "Hence, we can pick $r \\ge 0$ such that $rm < {\\omega }(i) - {\\omega }(j) < (r+1)m$ , i.e.", "$0 < {\\omega }(i) - ({\\omega }(j) + rm) < m$ , and set $J = {\\omega }^{-1}({\\omega }(j) + rm)$ .", "Now suppose ${\\omega }(i) < {\\omega }(i+1)$ .", "Let $(i,j) \\in \\operatorname{{Inv}}({\\omega })$ with $0 < {\\omega }(i) - {\\omega }(j) < m$ .", "Then since ${\\omega }(i+1) > {\\omega }(i) > {\\omega }(j)$ we also have $(i+1,j) \\in \\operatorname{{Inv}}({\\omega })$ .", "Observe $i+1 < j$ as $(i,i+1) \\operatorname{{Inv}}({\\omega })$ .", "We pick $(i+1, J) \\in \\operatorname{{Inv}}({\\omega })$ as in (REF ) above.", "The map $(i,j) \\mapsto (i+1, J)$ is clearly an injection, yielding $\\operatorname{\\mathcal {SP}}_{\\omega }(i) \\le \\operatorname{\\mathcal {SP}}_{\\omega }(i+1)$ .", "For ease of exposition, we recall Remark REF which lets us equate an inversion $(i,j)$ with $(i+tn, j+tn)$ .", "Next if $i<j$ with ${\\omega }(i) > {\\omega }(j)$ , suppose we have $(j,k) \\in \\operatorname{{Inv}}({\\omega })$ with $0 < {\\omega }(j) - {\\omega }(k) < m$ .", "Then $(i,k) \\in \\operatorname{{Inv}}({\\omega })$ too.", "We can pick $K \\ge k$ according to (REF ) yielding $(i,K) \\in \\operatorname{{Inv}}({\\omega })$ with $0 < {\\omega }(i) - {\\omega }(K) < m$ and ${\\omega }(k) \\equiv {\\omega }(K) \\mod {m}$ .", "Again the map $(j,k) \\mapsto (i,K)$ is an injection, yielding $\\operatorname{\\mathcal {SP}}_{\\omega }(j) \\le \\operatorname{\\mathcal {SP}}_{\\omega }(i)$ .", "Further there is an extra inversion of the form $(i,J) \\in Inv({\\omega })$ , showing $\\operatorname{\\mathcal {SP}}_{\\omega }(i) > \\operatorname{\\mathcal {SP}}_{\\omega }(j)$ .", "The case $j=i+1$ gives the converse of (REF ).", "Note that if $(n, n+1) \\in \\operatorname{{Inv}}({\\omega })$ then the above proposition implies $\\operatorname{\\mathcal {SP}}_{\\omega }(n) \\le \\operatorname{\\mathcal {SP}}_{\\omega }(1)$ , as we have $\\operatorname{\\mathcal {SP}}_{\\omega }(1) = \\operatorname{\\mathcal {SP}}_{\\omega }(n+1)$ by our convention.", "As a consequence of this, we have the following corollary.", "Corollary 3.18 Let $1 \\le i, j \\le n$ .", "$\\operatorname{\\mathcal {SP}}_{\\omega }(i) = \\operatorname{\\mathcal {SP}}_w(j) \\Rightarrow \|{\\omega }(i) - {\\omega }(j) \| < n$ Without loss of generality $ i< j$ .", "By Proposition REF item (REF ), ${\\omega }(i) < {\\omega }(j)$ .", "If also ${\\omega }(i) + n < {\\omega }(j)$ then as $j < i+n$ , $(j, i+n) \\in \\operatorname{{Inv}}({\\omega })$ so by the proposition $\\operatorname{\\mathcal {SP}}_{\\omega }(j) > \\operatorname{\\mathcal {SP}}_{\\omega }(i+n) = \\operatorname{\\mathcal {SP}}_{\\omega }(i)$ which is a contradiction.", "Proposition 3.19 Let ${\\omega }\\in {}^m\\widetilde{S}_n$ .", "If $0 < {\\omega }(i) - {\\omega }(i+1) < m,$ then ${\\left\\lbrace \\begin{array}{ll}\\operatorname{\\mathcal {SP}}_{{\\omega }s_i}(i) = \\operatorname{\\mathcal {SP}}_{\\omega }(i+1), \\\\\\operatorname{\\mathcal {SP}}_{{\\omega }s_i}(i+1) = \\operatorname{\\mathcal {SP}}_{\\omega }(i) -1, \\\\\\operatorname{\\mathcal {SP}}_{{\\omega }s_i}(j) = \\operatorname{\\mathcal {SP}}_{\\omega }(j)\\ \\mbox{for}\\ j \\lnot \\equiv i, i+1\\mod {n} \\end{array}\\right.", "}$ If $m < {\\omega }(i) - {\\omega }(i+1),$ then ${\\left\\lbrace \\begin{array}{ll}\\operatorname{\\mathcal {SP}}_{{\\omega }s_i}(i) = \\operatorname{\\mathcal {SP}}_{\\omega }(i+1), \\\\\\operatorname{\\mathcal {SP}}_{{\\omega }s_i}(i+1) = \\operatorname{\\mathcal {SP}}_{\\omega }(i), \\\\\\operatorname{\\mathcal {SP}}_{{\\omega }s_i}(j) = \\operatorname{\\mathcal {SP}}_{\\omega }(j)\\ \\mbox{for}\\ j \\lnot \\equiv i, i+1\\mod {n} \\end{array}\\right.", "}$ Write $u = w s_i$ .", "Since $0 < {\\omega }(i) - {\\omega }(i+1)$ , we have $\\operatorname{{Inv}}(u) =s_i(\\operatorname{{Inv}}({\\omega })) \\setminus \\lbrace (i,i+1)\\rbrace $ .", "In other words, $(i,j) \\in \\operatorname{{Inv}}(u)$ iff $(i+1,j) \\in \\operatorname{{Inv}}({\\omega })$ .", "Since $u(i)-u(j) = w(i+1)-w(j)$ this yields $\\operatorname{\\mathcal {SP}}_{u}(i) = \\operatorname{\\mathcal {SP}}_{\\omega }(i+1)$ Similarly, for $k \\ne i, i+1$ , $(k,j) \\in \\operatorname{{Inv}}(u)$ iff $(k,j) \\in \\operatorname{{Inv}}({\\omega })$ , yielding $\\operatorname{\\mathcal {SP}}_{u}(k) = \\operatorname{\\mathcal {SP}}_{\\omega }(k)$ .", "Finally $(i+1,j) \\in \\operatorname{{Inv}}(u)$ iff $j \\ne i+1$ and $(i,j) \\in \\operatorname{{Inv}}({\\omega })$ .", "Again $u(i+1)-u(j) = w(i)-w(j)$ .", "Hence in the case ${\\omega }(i) - {\\omega }(i+1) < m$ , so that $(i,i+1) \\in \\operatorname{{Inv}}({\\omega })$ contributes to $\\operatorname{\\mathcal {SP}}_{\\omega }(i)$ , we see $\\operatorname{\\mathcal {SP}}_{\\omega }(i) = \\operatorname{\\mathcal {SP}}_u(i+1) +1$ .", "When $m < {\\omega }(i) - {\\omega }(i+1)$ this inversion does not contribute, so $\\operatorname{\\mathcal {SP}}_{\\omega }(i) = \\operatorname{\\mathcal {SP}}_u(i+1)$ .", "As a corollary to this proposition, we see that $\\operatorname{\\mathcal {SP}}$ is injective on $\\lbrace {\\omega }\\in {}^m\\widetilde{S}_n\\mid (i,j) \\in \\operatorname{{Inv}}({\\omega }) \\Rightarrow {\\omega }(i)-{\\omega }(j) < m \\rbrace $ .", "Another interpretation of Theorem REF is that $\\operatorname{\\mathcal {SP}}$ not only respects descents but also respects (weakly) increasing subsequences.", "Example 3.20 Let $(n,m) = (3,4)$ .", "Consider these three affine permutations $y = [1,5,0], \\, {\\omega }= y s_2 = [1,0,5], $ and $ {\\omega }s_1 = [0,1,5]\\in {}^{4}\\widetilde{S}_{3}$ .", "The corresponding parking functions are $\\operatorname{\\mathcal {SP}}_y = \\llparenthesis 120 \\rrparenthesis , \\operatorname{\\mathcal {SP}}_w = \\operatorname{\\mathcal {SP}}_{ys_2} = \\llparenthesis 102 \\rrparenthesis $ (note $5-0 > 4$ so their second and third values have swapped), and $\\operatorname{\\mathcal {SP}}_{{\\omega }s_1} = \\llparenthesis 002 \\rrparenthesis $ (note $1-0 < 4$ ).", "Remark 3.21 The maps $\\operatorname{\\mathcal {PS}}$ and $\\operatorname{\\mathcal {SP}}$ preserve a kind of cyclic symmetry, as follows.", "(Compare this to Proposition REF for the map $\\operatorname{\\mathcal {A}}$ .)", "Let the shift operator $\\pi : {\\mathbb {Z}}\\rightarrow {\\mathbb {Z}}$ be defined by $\\pi (i) = i+1.$ Clearly $\\pi (i + tn) = \\pi (i) + tn$ , but $\\pi \\widetilde{S}_n$ as $\\sum _{i=1}^n \\pi (i) = \\frac{n(n+3)}{2}$ .", "(In other contexts, $\\pi $ lives in the extended affine symmetric group $P \\rtimes S_n\\supsetneqq Q \\rtimes S_n\\simeq \\widetilde{S}_n$ .", "It corresponds to the generator of $P/Q$ where $P$ and $Q$ are the weight and root lattices of type $A$ , respectively.)", "The conjugation map $\\widetilde{S}_n\\rightarrow \\widetilde{S}_n$ , ${\\omega }\\mapsto \\pi {\\omega }\\pi ^{-1}$ interacts nicely with the maps $\\operatorname{\\mathcal {SP}}$ and $\\operatorname{\\mathcal {PS}}$ .", "In window notation, conjugation by $\\pi $ corresponds to sliding the “window\" one unit to the left, but then renormalizing so the sum of the entries is still $\\frac{n(n+1)}{2}$ , i.e.", "$\\pi {\\omega }\\pi ^{-1} = [{\\omega }(0) +1, {\\omega }(1)+1, \\cdots , {\\omega }(n-1) +1];$ equivalently $\\pi {\\omega }\\pi ^{-1}(i) = {\\omega }(i+1) +1$ .", "It is clear that $\\operatorname{{\\overline{Inv}}}(\\pi {\\omega }\\pi ^{-1}) = \\lbrace (i+1,j+1) \\mid (i,j) \\in \\operatorname{{\\overline{Inv}}}({\\omega }) \\rbrace $ , and so conjugation by $\\pi $ preserves heights of inversions.", "In particular, it preserves both sets ${}^m\\widetilde{S}_n$ and $\\widetilde{S}_n^m$ .", "It is also clear from the definition of $\\operatorname{\\mathcal {SP}}$ that $&\\operatorname{\\mathcal {SP}}_{\\pi u \\pi ^{-1}}(i+1) = \\operatorname{\\mathcal {SP}}_u(i)& & \\text{for } u \\in {}^m\\widetilde{S}_n\\text{ \\, and hence} \\\\&\\operatorname{\\mathcal {PS}}_{\\pi {\\omega }\\pi ^{-1}} (i+1) = \\operatorname{\\mathcal {PS}}_{\\omega }(i)& & {\\omega }\\in {}^m\\widetilde{S}_n.$ Consider Example REF , for which $u=[4,-2,3,5]$ and $SP_u = \\llparenthesis 3011 \\rrparenthesis $ .", "We get $\\pi u \\pi ^{-1} = [2,5,-1,4]$ (for which $\\operatorname{{Inv}}(\\pi u \\pi ^{-1})=\\lbrace (1,3), (2,3), (2,4), (2,7), (4,7)\\rbrace $ ) and $\\operatorname{\\mathcal {SP}}_{\\pi u \\pi ^{-1}} = \\llparenthesis 1301 \\rrparenthesis $ ." ], [ "Two statistics", "Our work was partially motivated by some open questions posed by Armstrong in [2].", "He managed to describe $\\operatorname{area}$ and $\\operatorname{dinv}$ statistics on parking functions appearing in “Shuffle Conjecture” of [19] in terms of the Shi arrangement.", "We present two natural generalizations of these statistics to the rational case.", "Both were introduced in a different form in [20], but they can be best written in terms of maps $\\operatorname{\\mathcal {A}}$ and $\\operatorname{\\mathcal {PS}}$ .", "Definition 3.22 Let $\\omega \\in \\widetilde{S}_n^m$ be an affine permutation labeling an alcove ${{\\omega }^{-1}}({\\rm A}_0)\\in D_{n}^{m}$ .", "We define: $\\operatorname{area}(\\omega ):=\\frac{(m-1)(n-1)}{2}-\\sum \\operatorname{\\mathcal {A}}_{\\omega }(i),\\ \\operatorname{dinv}(\\omega ):=\\frac{(m-1)(n-1)}{2}-\\sum \\operatorname{\\mathcal {PS}}_{\\omega }(i).$ Proposition 3.23 The statistics $\\operatorname{area}(\\omega )$ can be computed as follows.", "Recall that $\\Delta _{{\\omega }}:=\\lbrace i\\in \\mathbb {Z}: \\omega (i)>0\\rbrace $ , then $\\operatorname{area}(\\omega )=\\sharp \\left([\\min \\Delta _{{\\omega }},+\\infty )\\setminus \\Delta _{{\\omega }}\\right).$ Indeed, there are $\\frac{(m-1)(n-1)}{2}$ boxes in the rectangle $R_{m,n}$ below the diagonal.", "The ones labeled by the elements of the set $\\left([\\min \\Delta _{{\\omega }},+\\infty )\\setminus \\Delta _{{\\omega }}\\right)$ are in 1-to-1 correspondence with the boxes outside of the diagram of $\\operatorname{\\mathcal {A}}(\\omega )$ , so their number equals $\\frac{(m-1)(n-1)}{2}-\\sum \\operatorname{\\mathcal {A}}_i(\\omega )=\\operatorname{area}(\\omega )$ .", "One can also check that the statistic $\\operatorname{area}$ agrees with the statistics $\\operatorname{ish}^{-1}$ of [2].", "Example 3.24 For the fundamental alcove, $\\operatorname{\\mathcal {PS}}_\\mathrm {id}(i)=0$ , so $\\operatorname{dinv}(\\mathrm {id})=\\frac{(m-1)(n-1)}{2}.$ On the other hand, $\\Delta _{\\mathrm {id}}=\\lbrace 1,2,3,\\ldots \\rbrace ,$ so by Proposition REF $\\operatorname{area}(\\mathrm {id})=0$ .", "Example 3.25 Consider the permutation ${\\omega }_m=[m-c,2m-c\\ldots ,nm-c]\\in {}^m\\widetilde{S}_n$ .", "Here the constant $c$ is uniquely determined by the condition $\\sum \\limits _{i=1}^n {\\omega }_m(i)=\\frac{n(n+1)}{2}.$ In fact, $c=\\frac{(n+1)(m-1)}{2}.$ Let us compute $\\operatorname{\\mathcal {SP}}_{{\\omega }_m}=\\operatorname{\\mathcal {PS}}_{{\\omega }_m^{-1}}.$ Since the entries in the window notation for ${\\omega }_m$ are increasing, i.e.", "${\\omega }_m\\in \\widetilde{S}_n/S_n$ , if $(k,t) \\in \\operatorname{{Inv}}({\\omega }_m)$ this forces $t = i +jn$ for some $1 \\le i < k$ and $j \\ge 1$ .", "Since it is an inversion, we have $km-c > im-c+jn$ .", "To contribute to $\\operatorname{\\mathcal {SP}}_{{\\omega }_m}(k)$ we must have $0 < (km-c) - (im-c+jn) < m&\\iff & km> im+jn > (k-1)m\\\\&\\iff & (k-1-i)m < nj < (k-i)m\\\\&\\iff & \\frac{(k-1-i)m}{n} < j < \\frac{(k-i)m}{n}$ Hence $\\operatorname{\\mathcal {SP}}_{{\\omega }_m}(k) =\\sharp \\lbrace j, i \\mid j \\ge 1, 1 \\le i < k, \\frac{(k-1-i)m}{n} < j < \\frac{(k-i)m}{n} \\rbrace $ Since we run over all $1 \\le i < k$ this is just $= \\sharp \\lbrace j \\mid j \\ge 1, j < \\frac{(k-1)m}{n} \\rbrace = \\lfloor \\frac{m(k-1)}{n} \\rfloor .$ By Proposition REF , $\\operatorname{\\mathcal {SP}}_{{\\omega }_m}$ is weakly increasing.", "The corresponding diagram is the maximal diagram that fits under the diagonal in an $m\\times n$ rectangle.", "The area of such a diagram is $\\frac{(n-1)(m-1)}{2},$ therefore $\\operatorname{dinv}({\\omega }_m^{-1})=\\frac{(m-1)(n-1)}{2}-\\sum \\operatorname{\\mathcal {PS}}_{{\\omega }_m^{-1}}(i)=0.$ One can also check that $\\operatorname{\\mathcal {A}}({\\omega }_m^{-1})=0$ , so $\\operatorname{area}({\\omega }_m^{-1})=\\frac{(m-1)(n-1)}{2}$ .", "Definition 3.26 We define the combinatorial Hilbert series as the bigraded generating function: $H_{m/n}(q,t)=\\sum _{\\omega \\in \\widetilde{S}_n^m}q^{\\operatorname{area}(\\omega )}t^{\\operatorname{dinv}(\\omega )}.$ It is clear that $H_{m/n}(1,1)=m^{n-1}$ , since there are $m^{n-1}$ permutations in $\\widetilde{S}_n^m$ .", "Conjecture 3.27 (cf.", "[20]) The combinatorial Hilbert series is symmetric in $q$ and $t$ : $H_{m/n}(q,t)=H_{m/n}(t,q).$ This conjecture is a special case of “Rational Shuffle Conjecture” [16].", "A more general conjecture also implies this identity $H_{m/n}(q,q^{-1})=q^{-\\frac{(m-1)(n-1)}{2}}(1+q+\\ldots +q^{m-1})^{n-1}.$ Both conjectures are open for general $m$ and $n$ .", "Example 3.28 For $n=5$ and $m=2$ , we have (see Example REF below for details): $H_{2/5}(q,t)=5+4(q+t)+(q^2+qt+t^2),$ and the above properties hold: $H_{2/5}(1,1)=16=2^4,$ $H_{2/5}(q,t)=H_{2/5}(t,q),$ $H_{2/5}(q,q^{-1})=q^{-2}+4q^{-1}+6+4q+q^2=q^{-2}(1+q)^4.$" ], [ "Extended Shi Arrangements and Pak-Stanley Labeling.", "Recall the set of $k$ -parking functions $\\operatorname{\\mathcal {PF}}_k : = \\operatorname{\\mathcal {PF}}_{(kn+1)/n}.$ Recall the hyperplanes $H_{ij}^k=\\lbrace {\\bf \\overline{x}}\\in V\\mid x_i-x_j=k\\rbrace $ and the affine braid arrangement $\\widetilde{B}_n=\\lbrace H_{ij}^k\\mid 1\\le i,j\\le n, k\\in {\\mathbb {Z}}\\rbrace $ .", "The extended Shi arrangement, or $k$ -Shi arrangement [25], [29], is defined as a subarrangement of the affine braid arrangement: Definition 4.1 The hyperplane arrangement $\\operatorname{Sh}^k_n:=\\left\\lbrace H_{ij}^{\\ell }:1\\le i<j\\le n,\\ -k<{\\ell }\\le k \\right\\rbrace $ is called the $k$ -Shi arrangement.", "The connected components of the complement to $\\operatorname{Sh}^k_n$ are called $k$ -Shi regions.", "The set of $k$ -Shi regions is denoted $\\operatorname{Reg}^k_n.$ One can use the notations introduced in Section REF to rewrite the definition of the $k$ -Shi arrangement as follows: $\\operatorname{Sh}^k_n=\\left\\lbrace H_{ij}^{\\ell }:1\\le i<j\\le n,\\ -k<{\\ell }\\le k \\right\\rbrace $ $=\\left\\lbrace H_{ij}^{\\ell }:1\\le i<j\\le n,\\ -k<{\\ell }< 0 \\right\\rbrace \\sqcup \\left\\lbrace H_{ij}^{\\ell }:1\\le i<j\\le n,\\ 0\\le {\\ell }\\le k \\right\\rbrace $ $=\\left\\lbrace H_{i,j-n\\ell }^0:1\\le i<j\\le n,\\ -k<{\\ell }< 0 \\right\\rbrace \\sqcup \\left\\lbrace H_{j,i+\\ell n}^0:1\\le i<j\\le n,\\ 0\\le {\\ell }\\le k \\right\\rbrace $ $=\\left\\lbrace H_{ij}^0:1\\le i\\le n, i<j<i+kn, ji \\mod {n}\\right\\rbrace .$ In other words, the $k$ -Shi arrangement consists of all hyperplanes of height less than $kn$ in the affine braid arrangement.", "The hyperplane $H_{ij}^{\\ell }$ divides $V$ into two half-spaces.", "Let $H_{ij}^{{\\ell },\\prec }$ denote the half-space that contains ${\\rm A}_0$ and $ H_{ij}^{{\\ell },\\succ }$ denote the complementary half-space.", "Note that $H_{ij}^{\\ell }$ separates ${\\omega }({\\rm A}_0)$ from ${\\rm A}_0$ iff ${\\omega }({\\rm A}_0) \\subseteq H_{ij}^{{\\ell },\\succ }$ iff $(i, j-{\\ell }n)$ or $(j, i+{\\ell }n) \\in \\operatorname{{Inv}}({\\omega }^{-1})$ (when taking the convention $i,j \\in \\lbrace 1, \\ldots , n\\rbrace $ ).", "Definition 4.2 The Pak-Stanley labeling is the map $\\lambda :\\operatorname{Reg}^k_n\\rightarrow \\operatorname{\\mathcal {PF}}_k$ , $R \\mapsto \\lambda _R$ defined by the formula $\\lambda _R(a)=\\sharp \\lbrace H_{ij}^{\\ell }\\in \\operatorname{Sh}^k_n\\mid R \\subseteq H_{ij}^{{\\ell },\\succ }, {\\ell }>0, i=a\\rbrace +\\sharp \\lbrace H_{ij}^{\\ell }\\in \\operatorname{Sh}^k_n\\mid R \\subseteq H_{ij}^{{\\ell },\\succ }, {\\ell }\\le 0, j=a\\rbrace .$ In other words, one labels the fundamental alcove ${\\rm A}_0$ by the parking function $f=\\llparenthesis 0\\ldots 0 \\rrparenthesis ,$ and then as one crosses the hyperplane $H_{ij}^{\\ell }$ in the positive direction (i.e.", "getting further away from ${\\rm A}_0$ ), one adds 1 to $f(j)$ if ${\\ell }\\le 0$ and adds 1 to $f(i)$ if ${\\ell }>0.$ Remark 4.3 One can rewrite this definition as follows: $\\lambda _R(a)=\\sharp \\lbrace H_{ij}^0\\in \\operatorname{Sh}^k_n\\mid R \\subseteq H_{ij}^{0,\\succ }, a=i<j\\rbrace =\\sharp \\lbrace H_{a \\, a+t}^0 \\mid R \\subseteq H_{a\\, a+t}^{0,\\succ }, 0 < t < kn, t \\lnot \\equiv 0\\mod {n}\\rbrace .$ We illustrate the Pak-Stanley labeling for $n=3,$ $k=1$ ($m=4$ ) in Figure REF .", "Figure: Pak-Stanley labeling for 1-Shi arrangement for n=3.n=3.Theorem 4.4 ([29]) The map $\\lambda :\\operatorname{Reg}^k_n\\rightarrow \\operatorname{\\mathcal {PF}}_k$ is a bijection." ], [ "Relation Between Sommers Regions and Extended Shi Arrangements for $m=kn\\pm 1$", "Consider the case $m=kn+1.$ One can show that each region of an extended Shi arrangement contains a unique minimal alcove (i.e.", "an alcove with the least number of hyperplanes $H_{ij}^k$ separating it from the fundamental alcove ${\\rm A}_0$ ).", "Theorem 4.5 ([11]) An alcove $\\omega ({\\rm A}_0)$ is the minimal alcove of a $k$ -Shi region if and only if $\\omega ^{-1}({\\rm A}_0)\\subset D_n^{kn+1}.$ Example 4.6 We illustrate this theorem in Figure REF , where the minimal alcoves of the 1-Shi region are matched with the alcoves in the Sommers region $D_3^4.$ On the left we have the minimal alcoves $\\omega ({\\rm A}_0)$ labeled by the $m$ -stable permutations $\\omega \\in \\widetilde{S}_n^m$ for $m=4, n=3$ .", "On the right we have alcoves $\\omega ^{-1}({\\rm A}_0)$ that fit inside $D_3^4,$ labeled by the $m$ -restricted permutations $\\omega ^{-1}\\in {}^m\\widetilde{S}_n.$ Note that $[-226]=[420]^{-1},$ $[150]=[1-16]^{-1},$ and $[4-13]=[-253]^{-1}.$ Figure: Minimal alcoves for Sh 3 1 \\operatorname{Sh}_3^{1} and Sommers region D 3 4 D_{3}^{4}.Theorem REF and Lemma REF imply a bijection $\\operatorname{alc}:\\widetilde{S}_n^{kn+1}\\rightarrow \\operatorname{Reg}^k_n.$ Theorem 4.7 One has $\\lambda \\circ \\operatorname{alc}=\\operatorname{\\mathcal {PS}}$ in this case.", "In particular, $\\operatorname{\\mathcal {PS}}$ is a bijection for $m=kn+1.$ As it was mentioned in Section REF , an affine permutation $\\omega $ has an inversion $(i,i+h)$ if and only if $\\omega ^{-1}({\\rm A}_0)$ is separated from ${\\rm A}_0$ by the hyperplane $H_{i,i+h}^0$ or, equivalently, if and only if $\\omega ^{-1}({\\rm A}_0)\\subset H_{i,i+h}^{0,\\succ }.$ Given a region $R$ , for any affine permutation $\\omega $ such that $\\omega ^{-1}({\\rm A}_0)\\subset R,$ the number of inversions of the form $(a,a+h)$ of height $h<kn$ is equal to $\\lambda _R(a).$ If $\\omega ^{-1}\\in \\widetilde{S}_n^{kn+1}$ then the alcove $\\omega ^{-1}({\\rm A}_0)$ is the minimal alcove in the region $R$ and $\\operatorname{alc}(\\omega ^{-1})=R.$ By definition, $\\operatorname{\\mathcal {PS}}_{\\omega ^{-1}}(a)$ is equal to the number of inversions $(\\alpha ,\\beta )$ of $\\omega ^{-1},$ such that $\\beta -\\alpha <kn+1$ and $\\omega ^{-1}(\\beta )=a,$ which is the same as the number of inversions $(a,a+h)$ of $\\omega ,$ such that $\\omega (a)-\\omega (a+h)<kn+1.$ Note that $\\omega (a)-\\omega (a+h)$ cannot be equal to $kn,$ so $\\operatorname{\\mathcal {PS}}_{\\omega ^{-1}}(a)$ is, in fact, equal to the number of inversions $(a,a+h),$ such that $\\omega (a)-\\omega (a+h)<kn.$ To match it with $\\lambda _R(a),$ one has to prove the following equation for any $a\\in \\lbrace 1,\\ldots ,n\\rbrace $ and any ${\\omega }$ : $\\sharp \\left\\lbrace (a,a+h)\\in \\operatorname{{Inv}}(\\omega )\\mid h<kn\\right\\rbrace =\\sharp \\left\\lbrace (a,a+h)\\in \\operatorname{{Inv}}(\\omega )\\mid \\omega (a)-\\omega (a+h)<kn\\right\\rbrace .$ Given $r\\in \\lbrace 1,\\dots ,n-1\\rbrace ,$ define $\\gamma _{a,{\\omega }}(r):=\\sharp \\left\\lbrace (a,a+h)\\in \\operatorname{{Inv}}(\\omega )\\mid h<kn, h\\equiv r \\operatorname{mod}n\\right\\rbrace $ and $\\gamma ^{\\prime }_{a,{\\omega }}(r):=\\sharp \\left\\lbrace (a,a+h)\\in \\operatorname{{Inv}}(\\omega )\\mid \\omega (a)-\\omega (a+h)<kn, h\\equiv r \\operatorname{mod}n\\right\\rbrace .$ Let $h_{\\max }^r$ be the maximal number such that $(a,h_{\\max }^r)\\in \\operatorname{{Inv}}(\\omega )$ and $h_{\\max }^r\\equiv r\\ \\operatorname{mod}n.$ It is not hard to see that $\\gamma _{a,{\\omega }}(r)=\\gamma ^{\\prime }_{a,{\\omega }}(r)=\\min \\left(k,\\left\\lfloor \\frac{h_{\\max }^r}{n}\\right\\rfloor \\right).$ Indeed, the total number of inversions $(a,a+h)$ such that $h\\equiv r\\ \\operatorname{mod}n$ equals $\\lfloor \\frac{h_{\\max }^r}{n}\\rfloor $ .", "If it is less than or equal to $k$ then all of them satisfy both $h<kn$ and $\\omega (a)-\\omega (a+h)<kn.$ In turn, if it is greater than $k,$ then the inversions $(a,a+h)$ for $h=r,r+n,\\ldots ,r+(k-1)n$ satisfy the condition $h<kn,$ while the inversions $(a,a+h)$ for $h=h_{\\max }^r,h_{\\max }^r-n,\\ldots ,h_{\\max }^r-(k-1)n$ satisfy the condition $\\omega (a)-\\omega (a+h)<kn.$ Finally, the sum of identities $\\gamma _{a,{\\omega }}(r)=\\gamma ^{\\prime }_{a,{\\omega }}(r)$ for all $r$ is equivalent to (REF ).", "Example 4.8 When one applies the map $\\operatorname{\\mathcal {PS}}$ to the affine permutations in the left part of Figure REF one gets the Pak-Stanley labeling shown in Figure REF .", "The case $m=kn-1$ is treated similarly.", "The main difference is that instead of the set of all $k$ -Shi regions $\\operatorname{Reg}^k_n$ one should consider the set of bounded $k$ -Shi regions $\\widehat{\\operatorname{Reg}^k_n}.$ One can show that every bounded $k$ -Shi region contains exactly one maximal alcove.", "Theorem 4.9 ([12]) An alcove $\\omega ({\\rm A}_0)$ is the maximal alcove of a bounded $k$ -Shi region if and only if $\\omega ^{-1}({\\rm A}_0)\\subset D_n^{kn-1}.$ As above, we use Lemma REF and Theorem REF to obtain the bijection $\\widehat{\\operatorname{alc}}:\\widetilde{S}_n^{kn-1}\\rightarrow \\widehat{\\operatorname{Reg}^k_n}.$ We prove the following theorem: Theorem 4.10 The image of the subset $\\widehat{\\operatorname{Reg}^k_n}\\subset \\operatorname{Reg}^k_n$ under the Pak-Stanley labeling is exactly $\\operatorname{\\mathcal {PF}}_{(kn-1)/n}\\subset \\operatorname{\\mathcal {PF}}_{(kn+1)/n}.$ Furthermore, one gets $\\lambda \\circ \\widehat{\\operatorname{alc}}=\\operatorname{\\mathcal {PS}}$ in this case.", "In particular, $\\operatorname{\\mathcal {PS}}$ is a bijection for $m=kn-1.$ It is sufficient to prove the formula $\\lambda \\circ \\widehat{\\operatorname{alc}}=\\operatorname{\\mathcal {PS}}.$ Indeed, this would imply that the restriction of the Pak-Stanley labeling to the subset $\\widehat{\\operatorname{Reg}^k_n}\\subset \\operatorname{Reg}^k_n$ is an injective (and, therefore, bijective) map from $\\widehat{\\operatorname{Reg}^k_n}$ to $\\operatorname{\\mathcal {PF}}_{(kn-1)/n}.$ If $\\omega ^{-1}\\in \\widetilde{S}_n^{kn-1}$ then the alcove $\\omega ^{-1}({\\rm A}_0)$ is the maximal alcove of a bounded region $R$ and $\\widehat{\\operatorname{alc}}(\\omega ^{-1})=R.$ Similarly to Theorem REF , we get that $\\operatorname{\\mathcal {PS}}_{\\omega ^{-1}}(a)$ is equal to the number of inversions $(a,a+h)$ of $\\omega $ such that $\\omega (a)-\\omega (a+h)<kn-1.$ Since $\\omega \\in {}^{kn-1}\\widetilde{S}_n,$ one has $\\omega (a)-\\omega (a+h)\\ne kn-1$ for any $h>0.$ Therefore, $\\operatorname{\\mathcal {PS}}_{\\omega ^{-1}}(a)$ is equal to the number of inversions $(a,a+h)$ in $\\omega $ such that $\\omega (a)-\\omega (a+h)<kn.$ In the proof of Theorem REF we have shown that this number is equal to $\\lambda _R(a).$" ], [ "Minimal Length Representatives and the Zeta Map.", "Definition 5.1 Let $\\operatorname{{Mod_{m,n}}}$ be the set of subsets $\\Delta \\subset \\mathbb {Z}_{\\ge 0},$ such that $\\Delta +m\\subset \\Delta ,$ $\\Delta +n\\subset \\Delta ,$ and $\\min (\\Delta )=0.$ A number $a$ is called an $n$ -generator of $\\Delta $ , if $a\\in \\Delta $ and $a-n\\notin \\Delta $ .", "Every $\\Delta \\in \\operatorname{{Mod_{m,n}}}$ has exactly $n$ distinct $n$ -generators.", "In [14], [15] such subsets were called 0-normalized semimodules over the semigroup generated by $m$ and $n$.", "We will simply call them $m,n$ -invariant subsets.", "There is a natural map $R:\\widetilde{S}_n^m\\rightarrow \\operatorname{{Mod_{m,n}}}$ given by $\\omega \\mapsto \\Delta _{{\\omega }}-\\min (\\Delta _{{\\omega }})$ (here, as before, $\\Delta _{\\omega }=\\lbrace i\\in \\mathbb {Z}: \\omega (i)>0\\rbrace $ ).", "Let $\\Omega _n^m$ be the set of $m$ -stable minimal length right coset representatives of $S_n\\backslash \\widetilde{S}_n.$ In other words, $\\Omega _n^m:=\\lbrace \\omega \\in \\widetilde{S}_n^m\\mid \\omega ^{-1}(1)<\\ldots <\\omega ^{-1}(n)\\rbrace .$ One can check that the restriction $R\|_{\\Omega _n^m}:\\Omega _n^m\\rightarrow \\operatorname{{Mod_{m,n}}}$ is a bijection.", "Indeed, the integers $\\omega ^{-1}(1),\\ldots ,\\omega ^{-1}(n)$ are the $n$ -generators of $\\Delta _{{\\omega }},$ and since ${\\omega }\\in \\Omega _n^m$ we have $\\omega ^{-1}(1)<\\ldots <\\omega ^{-1}(n),$ so one can uniquely recover ${\\omega }$ from $\\Delta _{\\omega }.$ Let $\\hat{R}:=R\|_{\\Omega _n^m}:\\Omega _n^m\\rightarrow \\operatorname{{Mod_{m,n}}}$ denote the restriction.", "Recall that $Y_{m,n}$ is the set of Young diagrams that fit under diagonal in an $n\\times m$ rectangle and $P:\\operatorname{\\mathcal {PF}}_{m/n}\\rightarrow Y_{m,n}$ is the natural map.", "In [14], [15] the first two named authors constructed two maps $D:\\operatorname{{Mod_{m,n}}}\\rightarrow Y_{m,n}$ and $G:\\operatorname{{Mod_{m,n}}}\\rightarrow Y_{m,n},$ proved that $D$ is a bijection, and related the two maps to the theory of $q,t$ -Catalan numbers in the following way.", "In the case $m=n+1$ one gets $c_n(q,t)=\\sum \\limits _{\\Delta \\in \\operatorname{{Mod_{n+1,n}}}} q^{\\delta -\|D(\\Delta )\|}t^{\\delta -\|G(\\Delta )\|},$ where $\\delta =\\frac{n(n-1)}{2}$ and $c_n(q,t)$ is the Garsia-Haiman $q,t$ -Catalan polynomial.", "It is known that these polynomials are symmetric $c_n(q,t)=c_n(t,q),$ although the proof is highly non-combinatorial and uses the machinery of Hilbert schemes, developed by Haiman.", "Finding a combinatorial proof of the symmetry of the $q,t$ -Catalan polynomials remains an open problem.", "The above consideration motivates the rational slope generalization of the $q,t$ -Catalan numbers: $c_{m,n}(q,t)=\\sum \\limits _{\\Delta \\in \\operatorname{{Mod_{m,n}}}} q^{\\delta -\|D(\\Delta )\|}t^{\\delta -\|G(\\Delta )\|},$ where $\\delta =\\frac{(m-1)(n-1)}{2}$ is the total number of boxes below the diagonal in an $n\\times m$ rectangle.", "The symmetry of these polynomials remains an open problem beyond the classical case $m=n+1$ and the cases $\\min (m,n)\\le 4$ (see [15] for $\\min (m,n)\\le 3$ and [22] for $\\min (m,n)=4$ ).", "It was also shown in [14] that the composition $G\\circ D^{-1}:Y_{m,n}\\rightarrow Y_{m,n}$ generalizes Haglund's zeta map exchanging the pairs of statistics $(\\operatorname{area},\\operatorname{dinv})$ and $(\\operatorname{bounce}, \\operatorname{area})$ on Dyck paths.", "It was conjectured that the map $G,$ and therefore, the generalized Haglund's zeta, are also bijections.", "This would imply a weaker symmetry property $c_{m,n}(q,1)=c_{m,n}(1,q).$ In [15] the bijectivity of $G$ was proved for $m=kn\\pm 1.$ For more details on this work we refer the reader to [14], [15].", "Let $\\star $ denote the involution on $\\widetilde{S}_n$ : ${\\omega }^{\\star }(x)=1-{\\omega }(1-x)$ .", "Lemma 5.2 The map $\\star $ preserves the set $\\widetilde{S}_n^m$ and the set $\\Omega _n^m$ .", "The map $\\overline{\\star }:(i,j)\\mapsto (1-j,1-i)$ provides a height preserving bijection from $\\lbrace (i,j)\\mid i<j,\\ {\\omega }(i)>{\\omega }(j)\\rbrace $ to $\\lbrace (i,j)\\mid i<j,\\ {\\omega }^\\star (i)>{\\omega }^\\star (j)\\rbrace .$ Let us check that ${\\omega }^\\star $ is an affine permutation: ${\\omega }^{\\star }(x+n)=1-{\\omega }(1-x-n)=1-{\\omega }(1-x)+n,$ $\\sum _{i=1}^{n}{\\omega }^{\\star }(i)=n-\\sum _{i=1-n}^{0}{\\omega }(i)=n-\\sum _{i=1}^{n}({\\omega }(i)-n)=n+n^2-\\frac{n(n+1)}{2}=\\frac{n(n+1)}{2}.$ If ${\\omega }(1)<\\ldots <{\\omega }(n)$ then ${\\omega }(1-n)<\\cdots < {\\omega }(0)$ , so ${\\omega }^{\\star }(1)<\\ldots <{\\omega }^{\\star }(n)$ .", "Let $(i,j)$ be such that $i<j$ and ${\\omega }(i)>{\\omega }(j).$ Then $1-j<1-i,$ and ${\\omega }^\\star (1-j)=1-{\\omega }(1-(1-j))=1-{\\omega }(j)>1-{\\omega }(i)={\\omega }^\\star (1-i).$ Note also that $\\overline{\\star }$ squares to identity.", "Therefore, since $\\star $ is an involution, $\\overline{\\star }$ is a bijection between $\\lbrace (i,j)\\mid i<j,\\ {\\omega }(i)>{\\omega }(j)\\rbrace $ and $\\lbrace (i,j)\\mid i<j,\\ {\\omega }^\\star (i)>{\\omega }^\\star (j)\\rbrace $ .", "Furthermore, since $\\overline{\\star }$ preserves height, $\\star $ preserves the set $\\widetilde{S}_n^m.$ The following Theorem shows that our constructions are direct generalizations of those of [14], [15]: Theorem 5.3 One has the following identities: $P\\circ \\operatorname{\\mathcal {A}}\\circ \\hat{R}^{-1}=D,$ $P\\circ \\operatorname{\\mathcal {PS}}\\circ \\star \\circ \\hat{R}^{-1}=G.$ The first statement follows from the definition of $\\operatorname{\\mathcal {A}}$ and Lemma REF .", "For the second statement, we need to recall the definition of the map $G$ .", "Given an $m,n$ -invariant subset $\\Delta \\in \\operatorname{{Mod_{m,n}}}$ , let $u_1<\\ldots <u_n$ be its $n$ -generators.", "The map $G$ was defined in [14], [15] by the formula $G_{\\Delta }(\\alpha )=\\sharp \\left([u_\\alpha ,u_\\alpha +m]\\setminus \\Delta \\right).$ Given a minimal coset representative $\\omega \\in \\Omega _n^m$ , we can consider an $m,n$ -invariant subset $R(\\omega )=\\Delta _{\\omega }-\\min (\\Delta _{{\\omega }})\\in \\operatorname{{Mod_{m,n}}}$ .", "Its $n$ -generators are $u_\\alpha =\\omega ^{-1}(\\alpha )-\\min (\\Delta _{{\\omega }})$ , and by (REF ) we have $u_1<\\ldots <u_n$ .", "For every $x\\in [u_\\alpha ,u_\\alpha +m]\\setminus R(\\omega )$ , define $x^{\\prime }:=x+\\min (\\Delta _{{\\omega }})$ , then all such $x^{\\prime }$ (and hence $x$ ) are defined by the inequalities $\\omega ^{-1}(\\alpha )<x^{\\prime }<\\omega ^{-1}(\\alpha )+m,\\ \\omega (x^{\\prime })\\le 0.$ Note that by (REF ) the inequality $\\omega (x^{\\prime })<0$ can be replaced by $\\omega (x^{\\prime })<\\alpha $ .", "Indeed, we have ${\\omega }^{-1}(1)<{\\omega }^{-1}(2)<\\ldots <{\\omega }^{-1}(n),$ and, therefore, ${\\omega }(x^{\\prime })\\notin \\lbrace 1,\\ldots ,\\alpha -1\\rbrace $ for $\\omega ^{-1}(\\alpha )<x^{\\prime }<\\omega ^{-1}(\\alpha )+m.$ Therefore the set $[u_\\alpha ,u_\\alpha +m]\\setminus R(\\omega )$ is in bijection with the set $\\lbrace (i,j)\\mid i<j<i+m,\\ {\\omega }(i)>{\\omega }(j),\\ {\\omega }(i)=\\alpha \\rbrace .$ In turn, the map $\\overline{\\star }$ bijectively maps this set to the set $\\lbrace (1-j,1-i)\\mid (1-j)<1-i<(1-j)+m,\\ {\\omega }^\\star (1-j)>{\\omega }^\\star (1-i),\\ {\\omega }^\\star (1-i)=1-{\\omega }(i)=1-\\alpha \\rbrace ,$ or, after a shift by $n$ and a change of variables, $\\lbrace (i,j)\\mid i<j<i+m,\\ {\\omega }^\\star (i)>{\\omega }^\\star (j),\\ {\\omega }^\\star (j)=n+1-\\alpha \\rbrace .$ Therefore, according to the definition of the map $\\operatorname{\\mathcal {PS}},$ we get $G_{R(\\omega )}(\\alpha )=\\operatorname{\\mathcal {PS}}_{\\omega ^{\\star }}(n+1-\\alpha ),$ and thus $G(R({\\omega }))=P(\\operatorname{\\mathcal {PS}}({\\omega }^\\star )).$ The involution $\\star $ could have been avoided in Theorem REF by adjusting the definition of the map $\\operatorname{\\mathcal {PS}}.$ However, in that case one would have to use $\\star $ to match the map $\\operatorname{\\mathcal {PS}}$ for $m=kn+1$ with the Pak-Stanley labeling (see Section ).", "The composition $\\operatorname{\\mathcal {PS}}\\circ \\operatorname{\\mathcal {A}}^{-1}:\\operatorname{\\mathcal {PF}}_{m/n}\\rightarrow \\operatorname{\\mathcal {PF}}_{m/n}$ should be thought of as a rational slope parking function generalization of the Haglund zeta map $\\zeta $ .", "Note that its bijectivity remains conjectural beyond cases $m=kn\\pm 1,$ which follows immediately from Theorems REF and REF ." ], [ "Finite-dimensional representations of DAHA", "It turns out that the map $A$ is tightly related to the representation theory of double affine Hecke algebras (DAHA).", "This theory is quite elaborate and far beyond the scope of this paper, so we refer the reader to Cherednik's book [6] for all details.", "Here we just list the necessary facts about finite-dimensional representations of DAHA.", "Let $\\mathbf {H}_n$ denote the DAHA of type $A_{n-1}$ .", "It contains the finite Hecke algebra generated by the elements $T_i$ , $1 \\le i < n$ as well as two commutative subalgebras $X_i/X_j$ , $1 \\le i \\ne j \\le n$ , and $Y_1^{\\pm 1},\\ldots ,Y_n^{\\pm 1}$ subject to commutation relations between $X^{\\prime }s$ and $Y^{\\prime }s$ that depend on two parameters $q$ and $t$ .", "(Alternatively, one can take generators $T_i$ , $0 \\le i < n$ , $\\pi $ and $X_i/X_j$ , $1 \\le i \\ne j \\le n$ .)", "$\\mathbf {H}_n$ admits a (degree 0 Laurent) polynomial representation $V=X_i/X_j]_{1 \\le i \\ne j \\le n}$ , where $X_i/X_j$ act as multiplication operators, and $Y_i$ act as certain difference operators.", "We can also obtain $V$ by inducing a 1-dimensional representation of the subalgebra generated by the $T_i$ and $Y_i^{\\pm 1}$ up to $\\mathbf {H}_n$ .", "The product $Y_1 Y_2 \\cdots Y_n $ (or equivalently $\\pi ^n$ ) acts as a constant on this representation.", "This constant agrees with the scalar by which the product acts on the initial 1-dimensional representation.", "We usually take this constant to be 1, or indeed impose the relation $Y_1 Y_2 \\cdots Y_n = 1$ in $\\mathbf {H}_n$ .", "However, to match the combinatorics developed in this paper, it is convenient to choose that scalar to be $q^{\\frac{n+1}{2}}$ .", "There exists a basis of $V$ consisting of nonsymmetric Macdonald polynomials $E_{\\sigma }(X_i)$ labeled by minimal length right coset representatives $\\sigma \\in S_n\\backslash \\widetilde{S}_n$ such that $Y_i$ are diagonal in this basis: $Y_i(E_{\\sigma })=a_i(\\sigma )E_{\\sigma }.$ The weights $a_i(\\sigma )$ are directly related to the combinatorial content of this paper and can be described as follows.", "Corresponding to the fundamental alcove $\\sigma = \\mathrm {id}$ we have $E_{\\mathrm {id}}=1$ and its weight equals to: $a(\\mathrm {id})=(a_1(\\mathrm {id}),\\ldots ,a_n(\\mathrm {id}))=q^{\\frac{n+1}{2n}}(t^{\\frac{1-n}{2}}, t^{\\frac{3-n}{2}},\\ldots ,t^{\\frac{n-1}{2}}).$ As we cross the walls (from $\\sigma {\\rm A}_0$ to $\\sigma s_i {\\rm A}_0$ ), the weights are transformed as follows: $s_i(a_1,\\ldots ,a_n)={\\left\\lbrace \\begin{array}{ll}(a_1,\\ldots , a_{i+1},a_i,\\ldots ,a_n),& \\text{if}\\ i\\ne 0\\cr (a_n/q,a_2,\\ldots ,a_{n-1},qa_{n}),& \\text{if}\\ i=0.\\end{array}\\right.", "}$ One can check that (REF ) defines an action of the affine symmetric group on the set of sequences of Laurent monomials in $q$ and $t$ .", "If the parameters $q$ and $t$ are connected by the relation $q^m=t^n$ for coprime $m$ and $n$ , the polynomial representation $V$ becomes reducible, and admits a finite-dimensional quotient $L_{m/n}$ of dimension $m^{n-1}$ .", "The basis of $L_{m/n}$ is again given by the nonsymmetric Macdonald polynomials $E_{\\sigma }$ , but now the permutation $\\sigma $ should have $\\sigma {\\rm A}_0$ belong to the (dominant) region bounded by the hyperplane $x_1-x_n=m$ .", "In other words, we can cross a wall if and only if the ratio of the corresponding weights is not equal to $t^{\\pm 1}$ .", "(See [7] for a discussion on these finite-dimensional quotients.", "See [8] for the formula for the intertwiners that take $E_\\sigma $ to $E_{\\sigma s_i}$ .", "See [9] for the nonsymmetric Macdonald evaluation formula that describes the $E_\\sigma $ in the radical of the polynomial representation.)" ], [ "From DAHA weights to Sommers region", "For the finite-dimensional representation $L_{m/n}$ we have $q^m=t^n$ , so $t=q^{m/n}$ .", "This means that every monomial $q^{x}t^{y}$ can be written as $q^{\\frac{nx+my}{n}}$ , so we can rewrite the DAHA weights as $a(\\sigma )=(a_1,\\ldots ,a_n)=(q^{b_1(\\sigma )/n},\\ldots ,q^{b_n(\\sigma )/n}).$ It turns out that “evaluated weights” $b_i$ are tightly related to the labeling of the region $D_{n}^{m}$ by affine permutations.", "Let $c=\\frac{(m-1)(n+1)}{2}$ .", "Consider the affine permutation ${\\omega }_m=[m-c,2m-c,\\ldots ,nm-c]$ .", "By Lemma REF , ${\\omega }_m$ identifies the dilated fundamental alcove with the simplex $D_{n}^{m}$ .", "Recall ${\\omega }\\in {}^m\\widetilde{S}_n\\iff {\\omega }_m^{-1}\\omega ({\\rm A}_0) \\subseteq m D_n^1 = m {\\rm A}_0$ .", "Theorem 6.1 Under this identification, for every ${\\omega }\\in {}^m\\widetilde{S}_n$ one has: $b({\\omega }_m^{-1}\\omega )=\\omega ,$ by which we mean for $\\sigma = {\\omega }_m^{-1}\\omega $ that $(b_1(\\sigma ),\\ldots ,b_n(\\sigma )) = ({\\omega }(1), \\ldots , {\\omega }(n))$ .", "In the weight picture we start from the fundamental alcove at $\\sigma = \\mathrm {id}$ , where we have weights $a(\\mathrm {id}) =q^{\\frac{n+1}{2n}}(t^{\\frac{1-n}{2}}, t^{\\frac{3-n}{2}},\\ldots ,t^{\\frac{n-1}{2}})=(q^{(m-c)/n}, q^{(2m-c)/n},\\ldots ,q^{(nm-c)/n}),\\\\b(\\mathrm {id}) =b({\\omega }_m^{-1} {\\omega }_m) =(m-c,2m-c,\\ldots ,nm-c)=({\\omega }_m(1), \\ldots , {\\omega }_m(n)).$ By Lemma REF , ${\\rm A}_0\\subseteq mD_n^1$ corresponds to the alcove ${\\omega }_m({\\rm A}_0) \\subseteq D_{n}^{m}$ , which we label by ${\\omega }_m=[m-c,\\ldots ,nm-c] \\in {}^m\\widetilde{S}_n$ .", "Therefore the desired identity holds for $\\sigma = \\mathrm {id}$ and can be extended to any $\\sigma $ with $\\sigma {\\rm A}_0\\subseteq m {\\rm A}_0= m D_n^1$ (equivalently ${\\omega }_m\\sigma \\in {}^m\\widetilde{S}_n$ ) by rules (REF ).", "Figure: DAHA weights for L 4/3 L_{4/3} above.", "When one evaluates at t=q 4/3 t=q^{4/3}, the weightsbecome (q u(1) 3 ,q u(2) 3 ,q u(3) 3 )(q^\\frac{u(1)}{3}, q^\\frac{u(2)}{3}, q^\\frac{u(3)}{3}) for the matchingalcove ω m uA 0 {\\omega }_mu {\\rm A}_0 which is labeled by u∈ 4 S ˜ 3 u \\in {}^{4}\\widetilde{S}_{3} inFigure .Note for the fundamental alcove,a( id )=q 2 3 (1 t,1,t)=q 2 3 (q -4 3 ,q 0 ,q 4 3 )=(q -2 3 ,q 2 3 ,q 6 3 )a(\\mathrm {id}) =q^{\\frac{2}{3}}( \\frac{1}{t}, 1, t)= q^{\\frac{2}{3}}(q^{-\\frac{4}{3}}, q^0, q^{\\frac{4}{3}})= (q^{-\\frac{2}{3}}, q^{\\frac{2}{3}}, q^{\\frac{6}{3}}) and u=[-2,2,6]=ω m id u=[-2,2,6] = {\\omega }_m\\mathrm {id}.Compare this toFigure , where the alcoveA 0 {\\rm A}_0 in the left figure matchesthe alcove labeled [-2,2,6][-2,2,6] in right figure." ], [ "From DAHA weights to parking functions", "Instead of direct evaluation of DAHA weights as powers of $q^{1/n}$ , one can instead draw monomials $q^{x}t^{y}$ on the $(x,y)$ -plane.", "This point of view was used in much wider generality in [30], where the weights were interpreted in terms of periodic skew standard Young tableaux.", "Here we focus on finite-dimensional representations and relate this picture to parking function diagrams.", "Let $a=(a_1,\\ldots ,a_n)$ be a DAHA weight.", "We define a function $T_{a}:\\mathbb {Z}_{\\ge 0}^2\\rightarrow \\mathbb {Z}_{\\ge 0}$ labeling the square lattice by the following rule.", "For every $i$ , let us present $a_i=(q^{\\frac{n+1}{2n}} t^{\\frac{-1-n}{2}}) q^{x_i}t^{y_i}$ and define $T_a(x_i,y_i)=i$ .", "Under this renormalization, $\\lbrace y_1, \\ldots , y_n\\rbrace = \\lbrace 1, \\ldots , n\\rbrace $ .", "Hence we obtain $n$ squares labeled $1,\\ldots , n$ in the rows $1,\\ldots n$ in some order.", "We can extend this labeling to the whole plane by the following two-periodic construction.", "First, one can identify $q^m$ with $t^n$ and write $T_{a}(x+m,y-n)=T_{a}(x,y)$ .", "Secondly, recall that the $b_i$ that correspond to $a$ can be naturally extended to an affine permutation using the quasi-periodic condition $b_{i+n}=b_i+n$ .", "This means that one can define $a_i$ for all integer $i$ by the rule $a_{i+n}=qa_i$ , and $T_{a}(x+1,y)=T_{a}(x,y)+n$ .", "Hence the fillings in the boxes of $T_a$ increase across rows automatically; that is, $T_a$ is row-standard.", "The more interesting question is when is $T_a$ column standard, which in this context means fillings increase up columns.", "Lemma 6.2 The weight $a$ appears in the finite-dimensional representation $L_{m/n}$ if and only if $T_{a}$ is a standard Young tableau (SYT), that is, $T_{a}(x+1,y)>T_{a}(x,y)$ and $T_{a}(x,y+1)>T_{a}(x,y)$ .", "Indeed, in terms of $b_i$ this means that $b_{i}+m$ appears after $b_i$ , which is precisely equivalent to $m$ -stability.", "Corollary 6.3 There is a natural bijection between the alcoves in the Sommers region and surjective maps $T:\\mathbb {Z}_{\\ge 0}^2\\rightarrow \\mathbb {Z}_{\\ge 0}$ satisfying the following conditions: $T(x+1,y)=T(x,y)+n,\\ T(x+m,y-n)=T(x,y),\\ T(x,y+1)>T(x,y).$ Lemma 6.4 There exists a unique up to shift $n\\times m$ rectangle such that all squares labeled by positive numbers are located above the NW-SE diagonal.", "The corresponding parking function diagram coincides with the Anderson-type labeling up to a central symmetry.", "If $(b_1,\\ldots ,b_n)$ corresponds to the weight $a$ , then its corresponding $n$ -invariant subset has n-generators $b_i$ , and contains all fillings in squares to the right of labeled ones, including the periodic shift by $(m, -n)$ .", "There exists a unique line with slope $m/n$ which is tangent to the resulting infinite set of squares, and the tangency points define the $n\\times m$ rectangle.", "Now the statement follows from the definition of the map $\\operatorname{\\mathcal {A}}$ .", "Example 6.5 Consider the weight $(a_1,a_2,a_3)=q^{\\frac{2}{3}}(\\frac{t}{q^2},\\frac{q}{t},q)= (q^{0}, q^{\\frac{1}{3}}, q^{\\frac{5}{3}}) $ for $t=q^{4/3}$ .", "We have $(b_1,b_2,b_3)=(0,1,5)$ and $\\omega =[015]$ .", "The corresponding $(3,4)$ -invariant subset is $\\Delta _{\\omega }=\\omega ({\\mathbb {Z}}_{>0})=\\lbrace 0,1,3,4,5,\\ldots \\rbrace $ , and the parking function diagram is shown in Figure REF on the right.", "On the left side of Figure REF is a piece of $T_a$ , showing rows with $1\\le y \\le 4$ and columns with $-2\\le x \\le 4$ .", "Rewriting $a = q^{\\frac{2}{3}} t^{-2}(q^{-2} t^3, q^1 t^1, q^1 t^2)$ , we see we put the filling 1 in square $(-2,3)$ , 2 in square $(1,1)$ , and 3 in square $(1,2)$ .", "The periodicity conditions fill in the rest of the squares of $T_a$ .", "The unique NW-SE line has been drawn, and the corresponding rectangle it determines is rotated by 180 to obtain a Young diagram below the diagonal.", "To get to the weight $a$ from the trivial weight, we need to apply ${\\omega }_m^{-1} {\\omega }= [-2,2,6]^{-1} \\circ [0,1,5] = [-3,4,5]$ .", "Observe $[-3,4,5] = [-2(3)+3, 1(3) + 1, 1(3) +2]$ .", "From this we could also read off that the fillings 1,2,3 belong in squares $(-2,3)$ , $(1,1)$ , $(1,2)$ respectively.", "(We remind the reader of Figure REF , where the alcove labeled $[-3,4,5]$ in the left figure, matches the alcove labeled $[0,1,5]$ in right figure.)", "Note, if we had instead normalized in the more standard way so that $Y_1 Y_2 Y_3 =1$ and the fundamental alcove had weight $a^{\\prime } =(\\frac{1}{t}, 1, t)$ , then we would have had a shift by $2 = 3 \\frac{2}{3}$ yielding $(b_1^{\\prime },b_2^{\\prime },b_3^{\\prime })=(-2,-1,3)= (0-2,1-2,5-2)$ but still $\\omega =[015]$ and we would draw $T_{a^{\\prime }}$ as above.", "Figure: Periodic SYT on (x,y)(x,y)-plane (left); parking function diagram (right)" ], [ "Injectivity of $\\operatorname{\\mathcal {PS}}$ for the finite symmetric group", "In this Section we prove an analogue of Conjecture REF for the finite group $S_n.$ Definition 7.1 Let $S_n^m$ denote the intersection $S_n\\cap \\widetilde{S}_n^m.$ In other words, ${\\omega }\\in S_n^m$ if for all $1\\le x\\le n-m$ the inequality $\\omega (x+m)>\\omega (x)$ holds and $\\lbrace 1, \\ldots , n\\rbrace = \\lbrace {\\omega }(1),\\ldots , {\\omega }(n)\\rbrace $ .", "We call such permutations finite $m$ -stable.", "Proposition 7.2 The number of finite $m$ -stable permutations equals $\\sharp S_n^m=\\frac{n!", "}{\\prod _{i=1}^{m}n_i!", "},\\ \\mbox{\\rm where}\\ n_i={\\left\\lbrace \\begin{array}{ll}\\left\\lfloor \\frac{n-i}{m}\\right\\rfloor +1& \\mbox{\\rm if}\\ i\\le n,\\\\0& \\mbox{\\rm if}\\ i>n.\\\\\\end{array}\\right.", "}$ The set $X=\\lbrace 1,\\ldots ,n\\rbrace $ can be split into $m$ disjoint subsets $X_i:=\\lbrace x\\in X: x\\equiv i \\operatorname{mod}m\\rbrace $ of cardinality $n_i$ .", "A permutation $\\omega $ is finite $m$ -stable if and only if it increases on each $X_i$ , hence it is uniquely determined by an ordered partition $\\lbrace 1,\\ldots ,n\\rbrace =\\omega (X_1)\\sqcup \\ldots \\sqcup \\omega (X_m).$ Example 7.3 For $n=5, m=3$ , $X_1 = \\lbrace 1,4\\rbrace , X_2 = \\lbrace 2,5\\rbrace , X_3 = \\lbrace 3\\rbrace $ .", "Observe ${\\omega }\\in S_n^m$ iff ${{\\omega }^{-1}}$ occurs in the shuffle $14 \\,$$$ 25 $\\scriptscriptstyle \\cup {\\hspace{-2.22214pt}}\\cup $ 3$,which are precisely the $ m$-restricted permutations in $ S5$.$ Definition 7.4 Given a permutation $\\omega \\in S_n^m$ , let us define $\\operatorname{\\mathcal {PS}}_{\\omega }(\\alpha )$ as the number of inversions $(x,y)$ of $\\omega $ such that $x<y<x+m,\\ \\omega (x)>\\omega (y)=\\alpha $ (the height of such inversion is less than $m$ ).", "Define $\\operatorname{\\mathcal {PS}}_{\\omega }:=(\\operatorname{\\mathcal {PS}}_{\\omega }(1),\\ldots ,\\operatorname{\\mathcal {PS}}_{\\omega }(n)).$ In other words, this is just the restriction $\\operatorname{\\mathcal {PS}}\\mid _{S_n}$ .", "Hence by Theorem REF the integer sequences in the image of $\\operatorname{\\mathcal {PS}}$ are $m/n$ -parking functions.", "Observe that if ${\\omega }\\in S_n$ then $\\operatorname{\\mathcal {PS}}_{\\omega }(n) = 0$ .", "Theorem 7.5 The map $\\operatorname{\\mathcal {PS}}$ from the set $S_n^m$ to $\\operatorname{\\mathcal {PF}}_{m/n}$ is injective.", "We provide two proofs of this Theorem, as they are somewhat different and might both be useful for the future attempts to proof Conjecture REF in the affine case.", "Given a parking function $\\llparenthesis \\operatorname{\\mathcal {PS}}_{\\omega }(1)\\cdots \\operatorname{\\mathcal {PS}}_{\\omega }(n) \\rrparenthesis $ in the image, we need to reconstruct $\\omega $ or, equivalently, $\\omega ^{-1} \\in S_n$ .", "We will first reconstruct the number $x_1={\\omega }^{-1}(1),$ then $x_{2}={\\omega }^{-1}(2),$ and so on, all the way up to $x_n={\\omega }^{-1}(n).$ Note that ${{\\omega }^{-1}}= [x_1, x_2, \\ldots , x_n]$ and $\\operatorname{\\mathcal {PS}}_{\\omega }(i) = \\operatorname{\\mathcal {SP}}_{{{\\omega }^{-1}}}(i) =\\sharp \\lbrace j \\mid i < j \\le n, 0 < x_i - x_j < m \\rbrace $ .", "We have used that since ${\\omega }\\in S_n$ , for all $(i,j) \\in \\operatorname{{Inv}}({\\omega })$ , $1 \\le i < j \\le n$ .", "Also since ${\\omega }\\in S_n$ , for all $j \\ge 1$ we have ${\\omega }(j) \\ge 1$ .", "For the first step, note that $x_1< m+1$ , since otherwise $x_1-m$ and $x_1$ will form an inversion of ${\\omega }$ of height $m$ , as $x_1 -m \\ge 1$ so it occurs to the right of $x_1$ in ${{\\omega }^{-1}}$ .", "For every $1 \\le y<x_1$ , there is an inversion $(y,x_1)$ of height less than $m$ and there are no other inversions of the form $(-,x_1)$ , hence $x_1=\\operatorname{\\mathcal {PS}}_{\\omega }(1)+1.$ On the next step we recover $x_2.$ Note that for every $y<x_2,$ there is an inversion $(y,x_2),$ unless $y=x_1.$ It follows that $x_2$ is either equal to $x_1+m$ or $x_2< m+1.$ It is not hard to see, that all these possible values of $x_2$ correspond to different values of $\\operatorname{\\mathcal {PS}}_{{\\omega }}(2).$ Therefore, knowing $\\operatorname{\\mathcal {PS}}_{{\\omega }}(2),$ one can recover $x_2.$ Let us show that one can proceed in that manner inductively all the way to $x_n.$ Suppose that one has already reconstructed $x_i=\\omega ^{-1}(i)$ for all $i<k$ .", "Define the set $Y_{k-1}=\\lbrace x_1\\ldots ,x_{k-1}\\rbrace \\sqcup \\lbrace l\\in \\mathbb {Z}:l<1\\rbrace =\\lbrace y: 1\\le y \\le n, \\omega (y)<k\\rbrace \\sqcup \\lbrace l\\in \\mathbb {Z}:l<1\\rbrace .$ Let us use the notation $I(y):=(y-m,y]\\cap \\mathbb {Z}$ for any $y\\in \\mathbb {Z}_{\\le n}.$ Consider the function $\\varphi _k(y):=\\sharp \\left(I(y)\\setminus Y_{k-1}\\right)-1$ defined on the domain $y\\in \\mathbb {Z}_{\\le n}\\setminus Y_{k-1}$ .", "Given $\\operatorname{\\mathcal {PS}}_{\\omega }(k)=\\varphi _k(x_k),$ we need to reconstruct $x_k.$ Let us prove that the function $\\varphi _k(y)$ is non-decreasing.", "Indeed, let $y>y^{\\prime }$ and $y,y^{\\prime }\\in \\mathbb {Z}_{\\le n}\\setminus Y_{k-1}.$ Let $z\\in I(y)=(y-m,y]\\cap \\mathbb {Z}$ and $z^{\\prime }\\in I(y^{\\prime })=(y^{\\prime }-m,y^{\\prime }]\\cap \\mathbb {Z}$ be such that $z-z^{\\prime }\\equiv 0$ mod $m.$ It follows that if $z\\in Y_{k-1},$ then also $z^{\\prime }\\in Y_{k-1}.$ Otherwise, if $z^{\\prime }Y_{k-1}$ then $1 \\le z^{\\prime }$ and ${\\omega }(z^{\\prime }) \\ge k$ .", "Further $y > y^{\\prime }$ implies $z > z^{\\prime } \\ge 1$ and $z\\in Y_{k-1}$ gives ${\\omega }(z) < k \\le {\\omega }(z^{\\prime })$ .", "Therefore, $(z^{\\prime },z)$ is an inversion of ${\\omega }$ of height divisible by $m,$ which implies that ${\\omega }$ is not $m$ -stable.", "Contradiction.", "We conclude, that $\\varphi _k(y)=\\sharp \\left(I(y)\\setminus Y_{k-1}\\right)-1\\ge \\sharp \\left(I(y^{\\prime })\\setminus Y_{k-1}\\right)-1=\\varphi _k(y^{\\prime }).$ Finally, we remark that $x_k\\notin Y_{k-1}$ but $x_k-m\\in Y_{k-1},$ since otherwise $x_k-m\\ge 1$ and $\\omega (x_k-m)>k,$ and that produces an inversion of height $m$ .", "Therefore, one check there is a strict inequality $\\varphi _k(y)<\\varphi _k(x_k)$ for any $y<x_k$ with $y \\in \\mathbb {Z}_{\\le n}\\setminus Y_{k-1}.$ Thus, $x_k=\\min \\lbrace y\\in \\mathbb {Z}\\setminus Y_{k-1}\|\\varphi _k(y)=\\operatorname{\\mathcal {PS}}_{\\omega }(k)\\rbrace $ .", "In particular, this set is non-empty.", "We illustrate this proof on an example in Figure REF .", "Define the function $g(\\alpha ,i)$ by the following formula: $g(\\alpha ,i)=\\sharp \\lbrace j\\in (i-m,i]\\cap \\lbrace 1,\\dots ,n\\rbrace : \\omega (j)> \\alpha \\rbrace .$ By definition of $\\operatorname{\\mathcal {PS}}_{\\omega },$ one immediately gets $g(\\omega (i),i)=\\operatorname{\\mathcal {PS}}_{\\omega }(\\omega (i)).$ Lemma 7.6 The function $g(\\alpha ,i)$ is non-decreasing in $i$ for any fixed $\\alpha .$ Indeed, suppose that $g(\\alpha ,i)<g(\\alpha ,i-1).$ The interval $(i-m,i]$ is obtained from the interval $(i-1-m,i-1]$ by dropping $i-m$ and adding $i.$ Therefore, one should have $\\omega (i-m)>\\alpha $ and $\\omega (i)\\le \\alpha $ to get $g(\\alpha ,i)<g(\\alpha ,i-1).$ But that implies $\\omega (i-m)>\\omega (i),$ producing an inversion of height $m.$ Contradiction.", "We will need the following corollary: Corollary 7.7 For any $i\\in \\lbrace 1,\\dots ,n\\rbrace ,$ $\\omega (i)$ is the minimal integer $\\alpha $ , such that $\\alpha \\ne \\omega (j)$ for any $j<i,$ and $\\operatorname{\\mathcal {PS}}_{\\omega }(\\alpha )=\\sharp \\lbrace j\\in (i-m,i)\\cap \\lbrace 1,\\dots ,n\\rbrace : \\omega (j)>\\alpha \\rbrace .$ Fix $i$ .", "Let $\\alpha $ satisfy the above conditions.", "Notice such an $\\alpha $ must exist since ${\\omega }(i)$ satisfies these conditions as $\\operatorname{\\mathcal {PS}}_{\\omega }(\\omega (i))=g(\\omega (i),i)=\\sharp \\left\\lbrace j\\in (i-m,i): \\omega (j)>\\omega (i)\\right\\rbrace .$ By minimality, $\\alpha \\le {\\omega }(i)$ .", "If $\\alpha \\ne {\\omega }(i)$ then we must have $\\alpha < {\\omega }(i)$ , yielding $i \\in \\lbrace j\\in (i-m,i]\\cap \\lbrace 1,\\dots ,n\\rbrace : \\omega (j)> \\alpha \\rbrace $ .", "However $i \\left\\lbrace j\\in (i-m,i) \\cap \\lbrace 1,\\dots ,n\\rbrace :\\omega (j) > \\alpha \\right\\rbrace $ whose cardinality is $ \\operatorname{\\mathcal {PS}}_{\\omega }(\\alpha )$ by assumption.", "Hence $g(\\alpha ,i) = \\operatorname{\\mathcal {PS}}_{\\omega }(\\alpha )+1.$ If it were the case that $\\alpha = {\\omega }(k)$ for some $k>i$ , then since $g(\\alpha ,-)$ is non-decreasing, we get $\\operatorname{\\mathcal {PS}}_w(\\alpha ) = \\operatorname{\\mathcal {PS}}_w(w(k)) = g(w(k),k) =g(\\alpha ,k) \\ge g(\\alpha ,i) > \\operatorname{\\mathcal {PS}}_{\\omega }(\\alpha )$ .", "Contradiction.", "On the other hand, $\\alpha $ was chosen so $\\alpha \\ne \\omega (j)$ for any $j<i$ .", "Hence it must be that $\\alpha = {\\omega }(i).$ Now we can complete the proof of Theorem REF and reconstruct $\\omega $ starting from $\\omega (1),$ then $\\omega (2),$ and so on, using Corollary REF .", "Indeed, if we already reconstructed $\\omega (1),\\omega (2),\\dots ,\\omega (i-1),$ then we can compute $\\sharp \\lbrace j\\in (i-m,i)\\cap \\lbrace 1,\\dots ,n\\rbrace : \\omega (j)>\\alpha \\rbrace $ for all $\\alpha \\in \\lbrace 1,\\ldots ,n\\rbrace .$ Then $\\omega (i)$ is the smallest number $\\alpha \\in \\lbrace 1,\\ldots ,n\\rbrace $ such that $\\alpha \\ne \\omega (j)$ for $j<i,$ and $\\operatorname{\\mathcal {PS}}_{\\omega }(\\alpha )=\\sharp \\lbrace j\\in (i-m,i)\\cap \\lbrace 1,\\dots ,n\\rbrace : \\omega (j)>\\alpha \\rbrace .$ We illustrate this proof on example on Figure REF .", "Figure: Suppose that n=7,n=7, m=3,m=3, and error ω =⦇0010210⦈\\operatorname{\\mathcal {PS}}_{\\omega }=\\llparenthesis 0010210 \\rrparenthesis .", "Let us reconstructω -1 {{\\omega }^{-1}} using the first proof of Theorem .", "We record on every step the numbers that we have already reconstructed and the values of the function ϕ k \\varphi _k for all other numbers.Figure: As in Figure , n=7,n=7, m=3,m=3, and error ω =⦇0010210⦈\\operatorname{\\mathcal {PS}}_{\\omega }=\\llparenthesis 0010210 \\rrparenthesis .", "This time we reconstruct ω{\\omega } using the second proof of Theorem .", "We record on every step the numbers that we have already reconstructed and the difference error ω (α)-♯{j∈{k-m+1,...,k-1}:ω(j)>α}\\operatorname{\\mathcal {PS}}_{\\omega }(\\alpha )-\\sharp \\lbrace j\\in \\lbrace k-m+1,\\ldots , k-1\\rbrace : \\omega (j)>\\alpha \\rbrace for all α∈{1,...,n}∖{ω(1),ω(2),⋯,ω(k-1)},\\alpha \\in \\lbrace 1,\\ldots ,n\\rbrace \\setminus \\lbrace \\omega (1),\\omega (2),\\dots ,\\omega (k-1)\\rbrace , so that on each step we choosethe position of the leftmost 0 in the second column.We do not know how to describe the image $\\operatorname{\\mathcal {PS}}(S_n^m)$ for general $m$ .", "As an example, let us consider the case $m=2$ for which we do have a complete description.", "Let us recall that $S_{n}^{2}$ is the set of finite permutations $\\omega $ of $n$ elements with no inversions of height 2, that is, $\\omega (i+2)>\\omega (i)$ for all $x$ .", "We define the map $\\operatorname{inv}^{(2)}$ from the set $S_{n}^{2}$ to the set of sequences of 0's and 1's as $\\operatorname{inv}^{(2)}_{\\omega }(\\alpha ):=\\chi \\left(\\omega (\\omega ^{-1}(\\alpha )-1)>\\alpha \\right)= {\\left\\lbrace \\begin{array}{ll}1 & \\text{ if } \\omega (\\omega ^{-1}(\\alpha )-1)>\\alpha \\\\0 & \\text{ else.}\\end{array}\\right.", "}$ Lemma 7.8 The image of $\\operatorname{inv}^{2}$ consists of all $n$ -element sequences $f$ of 0's and 1's, such that for every $\\alpha \\in \\lbrace 1,\\ldots ,n\\rbrace $ at least half of the subsequence $(f_\\alpha ,\\ldots f_n)$ are 0's.", "The image of $\\operatorname{inv}^{2}$ agrees with that of $\\operatorname{\\mathcal {PS}}\|_{{S}_{n}^{2}}$ .", "Let $\\omega $ be a permutation in $S_n^{2}$ and let $f=\\operatorname{inv}^{(2)}(\\omega )$ .", "For every $\\alpha $ such that $f_\\alpha =1$ one can find $\\beta =\\omega (\\omega ^{-1}(\\alpha )-1)>\\alpha $ such that $f_\\beta =0$ (otherwise $\\omega $ would have an inversion of height 2).", "In other words, if we consider ${\\omega }^{-1} = [x_1,\\ldots , x_n]$ , $f(\\alpha ) = 1$ iff $x_\\alpha = i$ and $i-1$ occurs to the right, i.e.", "$i-1 = x_\\beta $ with $\\beta > \\alpha $ .", "Which occurs iff $\\operatorname{\\mathcal {PS}}_{\\omega }(\\alpha )=1$ .", "And in this case $i-2$ cannot be to the right of $i-1$ as that would place it to the right of $i$ , i.e.", "$f(\\beta ) = \\operatorname{\\mathcal {PS}}_{\\omega }(\\beta )=0$ .", "Note that the correspondence $\\alpha \\mapsto \\beta =\\omega (\\omega ^{-1}(\\alpha )-1)$ from 1's to 0's in the sequence $f$ is injective and increasing.", "Therefore, for every $\\alpha \\in \\lbrace 1,\\ldots ,n\\rbrace $ at least half of the subsequence $(f_\\alpha ,\\ldots f_n)$ are 0's.", "Since we know that $\\operatorname{inv}^{(2)}$ is injective, the lemma now follows from the comparison of the cardinalities of the two sets.", "The sequences appearing in Lemma REF have a clear combinatorial meaning.", "Let us read the sequence $s$ backwards and replace 0's with a vector $(1,1)$ and 1's with a vector $(1,-1)$ .", "We get a lattice path in $\\mathbb {Z}^2$ which never goes below the horizontal axis.", "Such a path may be called a Dyck path with open right end, and Lemma REF establishes a bijection between the set of such paths of length $n$ and the set of finite 2-stable permutations." ], [ "Algorithm to construct $\\operatorname{\\mathcal {SP}}^{-1}$ in the affine case", "Here we present a conjectural algorithm that inverts $\\operatorname{\\mathcal {SP}}$ .", "While we have not yet shown the algorithm terminates, which in this case means it eventually becomes $n$ -periodic, we have checked it on several examples.", "Given $f \\in \\operatorname{\\mathcal {PF}}_{m/n}$ , extend $f$ to ${\\mathbb {N}}$ by $f(i+tn) = f(i)$ .", "Construct an injective function $U: {\\mathbb {N}}\\rightarrow {\\mathbb {N}}$ as follows.", "Informally, we will think of $U$ as the bottom row in the following table.", "$\\begin{array}{r\|ccc}i & 1&2& \\cdots \\\\\\hline f(i) & f(1) & f(2) & \\cdots \\\\\\hline U(i)& U(1) &U(2) & \\cdots \\end{array}$ Since $U$ is manifestly injective, it will make sense to talk about $U^{-1}$ .", "We will insert the numbers $\\alpha \\in {\\mathbb {N}}$ into the table as follows.", "Place $\\alpha = 1$ under the leftmost 0.", "In other words, let $i = \\min \\lbrace j \\in {\\mathbb {N}}\| f(j) = 0\\rbrace $ and then set $U(i) = 1$ .", "As there always exists some $1 \\le j \\le n$ such that $f(j) = 0$ , this is always possible.", "Assume $\\lbrace 1, 2, \\ldots , \\alpha - 1 \\rbrace $ have already been placed.", "Place $\\alpha $ in the leftmost empty position $i$ (i.e.", "$U(i) = \\alpha $ , with $i \\lbrace U^{-1}(\\beta ) \| 1 \\le \\beta < \\alpha \\rbrace $ for $i$ minimal) such that these two conditions hold.", "($\\mathtt {I}$ ) $\\alpha $ is to the right of $\\alpha - tm$ for $1 \\le t < \\alpha /m$ , $t \\in {\\mathbb {N}}$ .", "More precisely, $i > U^{-1}(\\alpha - tm)$ .", "($\\mathtt {II}$ ) If $U(i) =\\alpha $ , then $f(i) = \\sharp \\lbrace \\beta \|\\beta \\in (\\alpha -m, \\alpha ), U^{-1}(\\beta ) > i \\rbrace $ .", "In other words, we build $U$ so that $f(i) =\\sharp \\lbrace j \| j > i, 0 < U(i)-U(j) < m \\rbrace $ counts the number of $m$ -restricted inversions.", "Note that placing $\\alpha $ is always possible, since a valid (non-minimal) position for $\\alpha $ is under a 0 of $f$ such that it and all spots to the right of it are as yet unoccupied.", "Conjecture 7.9 For the $U$ constructed above, $\\exists N$ such that for all $i \\ge N$ , $t \\in {\\mathbb {N}}$ $U(i+tn) = U(i) + tn$ , so in particular $U(N+j)$ for $1 \\le j \\le n$ have all been assigned values Given $U$ constructed from $f \\in \\operatorname{\\mathcal {PF}}_{m/n}$ as in the algorithm and satisfying the conditions of the conjecture, we construct ${\\omega }\\in {}^m\\widetilde{S}_n$ as follows: Pick $t$ so $1+tn \\ge N$ .", "By the periodicity of $U$ and that $U$ has no “gaps\" after $N$ , $\\lbrace U(i+tn) \\bmod n \| 1\\le i\\le n \\rbrace = \\lbrace 1, 2, \\ldots , n\\rbrace $ .", "Hence $b := \\sum _{i=1}^n U(i+tn) \\equiv \\frac{n(n+1)}{2} \\bmod n$ .", "Let $k$ be such that $b - \\frac{n(n+1)}{2} = kn$ .", "Now set $ {\\omega }(i) = U(i+tn) -k.$ This forces $\\sum _{i=1}^n {\\omega }(i) = \\frac{n(n+1)}{2}$ , and so we see ${\\omega }\\in \\widetilde{S}_n$ .", "By construction, ($\\mathtt {I}$ ) and ($\\mathtt {II}$ ) imply $w \\in {}^m\\widetilde{S}_n$ and $\\operatorname{\\mathcal {SP}}_{\\omega }= f$ .", "We illustrate the algorithm to construct $U$ and ${\\omega }$ on the following example.", "Example 7.10 Let $n=5, m=3$ .", "Let $f = \\llparenthesis 11002 \\rrparenthesis \\in \\operatorname{\\mathcal {PF}}_{3/5}.$ Figure: Algorithm to construct UU fromf=⦇11002⦈∈error 3/5 .f = \\llparenthesis 11002 \\rrparenthesis \\in \\operatorname{\\mathcal {PF}}_{3/5}.Refer to Figure REF for a demonstration of how $U$ is constructed.", "Note that $U(7) \\ne 8$ since that would place 8 before 5, violating being 3-restricted.", "In the above we can in fact take $N=5$ .", "Observe $\\lbrace U(6), U(7), U(8), U(9), U(10) \\rbrace = \\lbrace 6,9,5,8, 12 \\rbrace $ yielding $b = 40$ and $k = 5$ .", "Hence we set ${\\omega }= [1,4,0,3,7]$ .", "Now one can easily verify ${\\omega }\\in {}^{3}\\widetilde{S}_{5}$ and $\\operatorname{\\mathcal {SP}}_{\\omega }= \\llparenthesis 11002 \\rrparenthesis $ .", "In practice, we have found $U$ to be surjective as well; in other words there are no “gaps\" even before $N$ .", "Further, when $f = \\operatorname{\\mathcal {SP}}_u$ for some finite permutation $u \\in S_n$ , we can take $N=1$ ." ], [ "Springer fibers for the symmetric group", "Let $V$ be a finite-dimensional vector space and let $N$ be a nilpotent transformation of $V$ .", "Let $\\operatorname{Fl}(V)$ denote the space of complete flags in $V$ .", "A classical object in the representation theory is the Springer fiber ([28], [26]) defined as $X_N:=\\lbrace {\\bf F}=\\lbrace V=V_1\\supset V_2\\supset \\ldots \\supset V_{n}\\rbrace \\in \\operatorname{Fl}(V): N(V_i)\\subset V_i\\rbrace .$ It is known that $X_N$ admits an affine paving with combinatorics completely determined by the conjugacy class of $N$ (see e.g.", "[26] and references therein).", "We will be interested in a particular case of this construction.", "Let us fix a basis $(e_1,\\ldots ,e_n)$ in the space $V$ , consider the operator of shift by $m$ : $N(e_i):={\\left\\lbrace \\begin{array}{ll}e_{i+m},& i+m\\le n\\\\0,& \\text{otherwise}\\\\\\end{array}\\right.", "}$ The following theorem describes the structure of the affine cells in the variety $X_{N}$ .", "Theorem 8.1 The variety $X_N$ admits an affine paving, where the cells $\\Sigma _{\\omega }$ are parametrized by the finite $m$ -restricted permutations $\\omega \\in S_n^m$ .", "The dimension of $\\Sigma _{\\omega }$ is given by the number of inversions in $\\omega ^{-1}$ of height less than $m$ .", "The cells are essentially given by the intersections of Schubert cells in $\\operatorname{Fl}(V)$ with the subvariety $X_N$ .", "For the sake of completeness, let us recall their construction.", "Given a permutation $\\omega \\in S_n$ , we can define a stratum $\\Sigma _{\\omega }$ in $\\operatorname{Fl}(V)$ consisting of the following flags: ${\\bf F}=\\lbrace V_1\\supset V_2\\supset \\ldots \\supset V_n\\rbrace ,\\ V_i=\\operatorname{span}\\lbrace v^{\\omega (i)},\\ldots ,v^{\\omega (n)} \\rbrace ,$ where $v^\\alpha =e_\\alpha +\\sum _{\\beta >\\alpha }\\lambda ^\\alpha _\\beta e_\\beta .$ Note that the position of $v^\\alpha $ in the basis equals $\\omega ^{-1}(\\alpha )$ .", "After a triangular change of variables, we can assume that $\\lambda ^\\alpha _\\beta =0$ for $\\beta >\\alpha $ with $\\omega ^{-1}(\\beta )>\\omega ^{-1}(\\alpha ).$ Therefore one can write $v^\\alpha =e_\\alpha +\\sum _{\\beta >\\alpha ,\\omega ^{-1}(\\beta )<\\omega ^{-1}(\\alpha )}\\lambda ^\\alpha _\\beta e_\\beta =e_\\alpha +\\sum _{(\\alpha ,\\beta )\\in \\operatorname{{Inv}}({\\omega }^{-1})}\\lambda ^\\alpha _\\beta e_\\beta .$ The parameters $\\lambda ^\\alpha _\\beta $ are uniquely defined by the flag ${\\bf F}$ .", "They serve as coordinates on the affine space $\\Sigma _{\\omega }$ , whose dimension is equal to the length of $\\omega $ , i.e.", "$=\\sharp \\operatorname{{Inv}}({\\omega }) = \\sharp \\operatorname{{Inv}}({{\\omega }^{-1}})$ .", "Let us study the intersection $\\Sigma _{\\omega }^N:=\\Sigma _{\\omega }\\cap X_N.$ Since $Nv^\\alpha $ starts with $e_{\\alpha +m}$ , the vector $v^{\\alpha +m}$ should go after $v^\\alpha $ in the basis, so one needs $\\omega ^{-1}(\\alpha +m)>\\omega ^{-1}(\\alpha )$ .", "Therefore $\\Sigma ^{N}_{\\omega }$ is non-empty if and only if $\\omega ^{-1}$ is $m$ -stable.", "A flag ${\\bf F}$ is $N$ -invariant, if $N(v^\\alpha )$ belongs to $\\operatorname{span}\\lbrace v^\\beta : \\omega ^{-1}(\\beta )>\\omega ^{-1}(\\alpha ) \\rbrace $ for all $\\alpha $ .", "If $\\beta >\\alpha +m\\ \\text{and}\\ \\omega ^{-1}(\\beta )>\\omega ^{-1}(\\alpha ),$ then the coefficient in $v^{\\alpha +m}-N(v^\\alpha )$ at $e_\\beta $ can be eliminated by subtracting an appropriate multiple of $v^\\beta $ .", "Once all these coefficients are eliminated, the remaining coefficients in $v^{\\alpha +m}-N(v^\\alpha ),$ labeled by $\\beta >\\alpha +m$ such that $\\omega ^{-1}(\\beta )<\\omega ^{-1}(\\alpha )$ will vanish automatically.", "Therefore $\\Sigma ^N_{\\omega }$ is cut out in $\\Sigma _{\\omega }$ by the equations: $\\lambda ^{\\alpha +m}_\\beta =\\lambda ^\\alpha _{\\beta -m}+\\phi (\\lambda )\\ \\text{if}\\ \\beta >\\alpha +m,\\omega ^{-1}(\\beta )<\\omega ^{-1}(\\alpha ),$ where $\\phi (\\lambda )$ are certain explicit polynomials in $\\lambda ^\\mu _\\nu $ with $\\nu -\\mu <\\beta -\\alpha -m,$ with no linear terms.", "It is clear that such equations are labeled by the inversions $(\\alpha ,\\beta )$ in $\\omega ^{-1}$ of height bigger than $m$ .", "Note also that the linear parts of these equations are linearly independent.", "Therefore the number of free parameters on $\\Sigma ^N_{\\omega }$ equals to the number of inversions of ${{\\omega }^{-1}}$ of height less than $m$ .", "Example 8.2 Consider a 2-stable permutation $\\omega =[2, 1, 4, 3]=\\omega ^{-1}$ .", "The basis (REF ) has the form: $v^{\\omega (1)}=e_2,\\ v^{\\omega (2)}=e_1+\\lambda ^1_2 e_2,\\ v^{\\omega (3)}=e_4,\\ v^{\\omega (4)}=e_3+\\lambda ^3_4 e_4.$ There are two free parameters, so $\\dim \\Sigma ^{N}_{\\omega }=2$ .", "Note that although $\\lambda ^3_4\\ne \\lambda ^1_2$ , $N(v^{\\omega (2)})=e_3+\\lambda ^1_2 e_4\\in \\operatorname{span}\\lbrace v^{\\omega (3)},v^{\\omega (4)} \\rbrace .$ Example 8.3 Consider a 3-stable permutation $\\omega ^{-1}=[1, 5, 3, 2, 6, 4, 7],$ so $\\omega =[1, 4, 3, 6, 2, 5, 7]$ is 3-restricted.", "The basis (REF ) has a form: $v^{\\omega (1)}=e_1,\\ v^{\\omega (2)}=e_4,\\ v^{\\omega (3)}=e_{3}+\\lambda ^3_4 e_4,\\ v^{\\omega (4)}=e_6,\\ v^{\\omega (5)}=e_2+\\lambda ^2_3 e_3+\\lambda ^2_4 e_4+\\lambda ^2_6 e_6,$ $v^{\\omega (6)}=e_5+\\lambda ^5_6 e_6,\\ v^{\\omega (7)}=e_7.$ Since $N(v^{{\\omega }(5)})-v^{{\\omega }(6)}=(\\lambda ^2_3-\\lambda ^5_6)e_6+\\lambda ^2_4 e_7\\in V_5,$ and $V_5=\\operatorname{span}\\lbrace v^{{\\omega }(5)},v^{{\\omega }(6)},v^{{\\omega }(7)} \\rbrace =\\operatorname{span}\\lbrace e_2+\\lambda ^2_3 e_3+\\lambda ^2_4 e_4+\\lambda ^2_6 e_6,e_5+\\lambda ^5_6 e_6,e_7 \\rbrace ,$ the coefficient of $e_6$ must vanish and so we get the relation $\\lambda ^2_3=\\lambda ^5_6.$ Therefore $\\dim \\Sigma _{\\omega }^N=4.$" ], [ "Springer fibers for the affine symmetric group", "We recall the basic definitions of the type $A$ affine Springer fibres, and refer the reader e. g. to [13], [21], [24] for more details.", "Let us choose an indeterminate $\\varepsilon $ and consider the field $K=(\\varepsilon ))$ of Laurent power series and the ring $\\mathcal {O}=[\\varepsilon ]]$ of power series in $\\varepsilon $ .", "Let $V=n((\\varepsilon ))$ be a $K$ -vector space of dimension $n$ .", "Definition 8.4 The affine Grassmannian $\\mathcal {G}_n$ for the group $GL_n$ is the moduli space of $\\mathcal {O}$ -submodules $M\\subset V$ such that the following three conditions are satisfied: (a) $M$ is $\\mathcal {O}$ -invariant.", "(b) There exists $N$ such that $\\varepsilon ^{-N}n[[\\varepsilon ]]\\supset M\\supset \\varepsilon ^{N}n[[\\varepsilon ]].$ (c) Let $N$ be an integer satisfying the above condition.", "Then the following normalization condition is satisfied: $\\dim _{ \\varepsilon ^{-N}n[[\\varepsilon ]]/M=\\dim _{ M/ \\varepsilon ^{N}n[[\\varepsilon ]].", "}}The{\\em affine complete flag variety}$ Fn$ for the group $ GLn$ is the moduli space of collections $ {M0...Mn}$, where each $ Mi$ satisfies (a) and (b),$ Mi/Mi+1=1,$\\ $ Mn=M0,$ and $ M0Gn,$ i.e.", "$ M0$ also satisfies the normalization condition (c).$ Definition 8.5 Let $T$ be an endomorphism of $V$ .", "It is called nil-elliptic if $\\lim _{k\\rightarrow \\infty }T^{k}=0$ and the characteristic polynomial of $T$ is irreducible over $K$ .", "Given a nil-elliptic operator $T$ , one can extend its action to $\\mathcal {G}_n$ and to $\\mathcal {F}_n$ and define the affine Springer fibers as the corresponding fixed point sets.", "Remark 8.6 The condition $\\lim _{k\\rightarrow \\infty }T^{k}=0$ means that for any $N\\in {\\mathbb {N}}$ there exists $k\\in {\\mathbb {N}}$ such that $T^k(n[[\\varepsilon ]])\\subset \\varepsilon ^Nn[[\\varepsilon ]].$ In [24] Lusztig and Smelt studied the structure of the affine Springer fibers for a particular choice of $T$ .", "Given a $-basis $ {e1,..., en}$ in $ n$,one can consider it as a $ K$-basis of$ V=n(())$.", "Consider the operator $ N$ definedby the equations $ N(ei)=ei+1, N(en)=e1.$ The following theorem is the main result of \\cite {LS91}.$ Theorem 8.7 ([24]) Consider the nil-elliptic operator $T:=N^m$ , where $m$ is coprime to $n$ .", "Then the corresponding affine Springer fiber $\\mathcal {F}_{m/n}\\subset \\mathcal {F}_n$ admits an affine paving by $m^{n-1}$ affine cells.", "It turns out that the affine paving of this affine Springer fiber is tightly related to the combinatorics of the simplex $D_{n}^{m}$ .", "This was implicitly stated in [17], [20], [24], but we would like to make this correspondence precise and explicit.", "Theorem 8.8 There is a natural bijection between the affine cells in $\\mathcal {F}_{m/n}$ and the affine permutations in ${}^m\\widetilde{S}_n$ .", "The dimension of the cell $\\Sigma _{\\omega }$ labeled by the affine permutation $\\omega $ is equal to $\\sum _{i=1}^{n}\\operatorname{\\mathcal {SP}}_{\\omega }(i)$ .", "Let us introduce an auxiliary variable $z=\\varepsilon ^{1/n}$ .", "We can identify the vector space $V=n((\\varepsilon ))$ with the space $\\operatorname{span}_1,z,\\ldots ,z^{n-1} \\rbrace ((z^n))\\simeq (z))$ of Laurent power series in $z$ by sending the basis $\\lbrace e_1,\\ldots ,e_n\\rbrace $ to $\\lbrace 1,z,\\ldots ,z^{n-1}\\rbrace .$ Note that under this identification, $n[[\\varepsilon ]]$ is mapped to $[z]]$ .", "By construction, $N$ coincides with the multiplication operator by $z$ and hence $T=N^m$ coincides with the multiplication operator by $z^m$ .", "Therefore $\\mathcal {F}_{m/n}$ consists of flags $\\lbrace M_0\\supset \\ldots \\supset M_n\\rbrace $ of $[z^n,z^m]]$ -modules, such that $\\dim _{ M_i/M_{i+1}=1, M_n=z^nM_0 and M_0\\in \\mathcal {G}_n.", "Let us extend the notation M_i to arbitrary i\\in {\\mathbb {Z}} by setting M_{i+kn}:=z^{kn}M_i.", "As a result, we get an infinite flag \\lbrace \\ldots \\supset M_0\\supset \\ldots \\supset M_n\\supset \\ldots \\rbrace of [z^n,z^m]]-modules satisfying the same conditions as above and M_{i+n}=z^nM_i.", "}For $ f(z)V=(z)),$ let $ Ord(f)$ denote the order of $ f(z)$ in $ z,$ i.e.", "the smallest degree of $ z,$ such that the corresponding coefficient in $ f(z)$ does not vanish.", "Given a subset $ MV$, define $$\\operatorname{Ord}(M)=\\lbrace \\operatorname{Ord}(f)\\ :\\ f\\in M, f\\ne 0\\rbrace .$$ We will need the following lemma, whose proof is standard and left to the reader:$ Lemma 8.9 Let $L\\subset M\\subset (z))$ be two $[z^m,z^n]]$ -submodules in $z^{-N}[z]]$ for some large $N\\in {\\mathbb {N}}.$ Then $\\sharp \\left(\\operatorname{Ord}(M)\\setminus \\operatorname{Ord}(L)\\right)=\\dim _M̏/L.$ Given a flag $\\lbrace \\ldots \\supset M_0\\supset \\ldots \\supset M_n\\supset \\ldots \\rbrace $ as above, set $ \\operatorname{Ord}(M_i)\\setminus \\operatorname{Ord}(M_{i+1}) = \\lbrace {\\omega }(i)\\rbrace .$ Note that one automatically gets $\\operatorname{Ord}(M_i)=\\lbrace {\\omega }(i),{\\omega }(i+1),\\ldots \\rbrace ,$ because $\\bigcap \\limits _i\\operatorname{Ord}(M_i)=\\emptyset .$ Recall the notation $\\operatorname{{Inv}}({\\omega }) : =\\lbrace (i,j) \\in {\\mathbb {N}}\\times {\\mathbb {N}}\\mid 1\\le i\\le n,\\ i < j, {\\omega }(i) > {\\omega }(j) \\rbrace $ and $\\operatorname{{\\overline{Inv}}}({\\omega }) : =\\lbrace (i,j) \\in {\\mathbb {N}}\\times {\\mathbb {N}}\\mid i < j, {\\omega }(i) > {\\omega }(j) \\rbrace $ for the inversion sets of ${\\omega }.$ For each $i$ there exists a unique $f_i(z)\\in M_i$ such that $f_i=z^{{\\omega }(i)}+\\sum _{(j,i)\\in \\operatorname{{\\overline{Inv}}}({\\omega })}\\lambda ^{{\\omega }(i)}_{{\\omega }(j)} z^{{\\omega }(j)}.$ Indeed, take any function $f\\in M_i$ such that $\\operatorname{Ord}(f)=\\lbrace {\\omega }(i)\\rbrace $ and use functions from $M_{i+1}$ to eliminate coefficients at $z^{{\\omega }(j)}$ for $j>i$ and ${\\omega }(j)>{\\omega }(i).$ The resulting function is unique up to a scalar, because otherwise $\\dim _M̏_i/M_{i+1}$ would be at least $2.$ It follows that $f_{i+n}=z^nf_i.$ We claim that ${\\omega }$ is an affine permutation and, moreover, $\\omega \\in {}^m\\widetilde{S}_n$ .", "Indeed, since $f_{i+n}=z^nf_i$ we get ${\\omega }(i+n)={\\omega }(i)+n,$ and since $z^mf_i\\in M_i$ we get that ${\\omega }(i)+m\\in \\lbrace {\\omega }(i),{\\omega }(i+1),\\ldots \\rbrace ,$ and, therefore, for any $j<i,$ ${\\omega }(j)-{\\omega }(i)\\ne m.$ Finally, we need to check the normalization condition $\\sum \\limits _{i=1}^n {\\omega }(i)=\\frac{n(n+1)}{2},$ which follows form the normalization condition on $M_0\\in \\mathcal {G}_n.$ Indeed, it is not hard to see that for all $L\\in \\mathcal {G}_n$ the sum of elements of $\\operatorname{Ord}(L)\\setminus \\operatorname{Ord}(t^nL)$ should be the same.", "In particular, for $L=[z]]$ we have $\\operatorname{Ord}([z]])\\setminus \\operatorname{Ord}(z^n[z]])=\\lbrace 0,1,\\ldots ,n-1\\rbrace ,$ and their sum is $\\frac{n(n-1)}{2}.$ Therefore, since $\\operatorname{Ord}(M_0)\\setminus \\operatorname{Ord}(z^nM_0)=\\lbrace {\\omega }(0),\\ldots ,{\\omega }(n-1)\\rbrace ,$ we get $\\sum \\limits _{i-0}^{n-1} {\\omega }(i)=\\frac{n(n-1)}{2},$ which equivalent to the required condition.", "The above gives us a map $\\nu :\\mathcal {F}_{m/n}\\rightarrow {}^m\\widetilde{S}_n.$ Let us prove that the fibers $\\Sigma _{{\\omega }}:=\\nu ^{-1}({\\omega })$ of this map are affine cells and compute their dimensions.", "This is very similar to the computation in the finite case (see Theorem REF ).", "Let us set $f^{{\\omega }(i)}:=f_i.$ The expansions (REF ) can be rewritten as $f^\\alpha =z^\\alpha +\\sum _{(\\alpha ,\\beta )\\in \\operatorname{{\\overline{Inv}}}({\\omega }^{-1})}\\lambda ^{\\alpha }_{\\beta } z^\\beta .$ Since $f_{i+n}=z^nf_i,$ one gets $\\lambda ^{\\alpha +n}_{\\beta +n}=\\lambda ^\\alpha _\\beta .$ Let us also extend the notation by setting $\\lambda ^{\\alpha }_{\\beta }=0$ whenever $(\\alpha ,\\beta )\\notin \\operatorname{{\\overline{Inv}}}({\\omega }^{-1})$ , so that one can write $f^\\alpha =z^\\alpha +\\sum _{\\beta >\\alpha } \\lambda ^\\alpha _{\\beta }z^{\\beta }.$ Let us say that the coefficient $\\lambda ^{\\alpha }_{\\beta }$ is of height $\\beta -\\alpha .$ As before, let $\\alpha ={\\omega }(i).$ The condition $z^m f^\\alpha \\in M_i$ implies the following relations on the coefficients.", "The function $z^mf^\\alpha -f^{\\alpha +m}=\\sum _{\\beta >\\alpha } \\lambda ^\\alpha _\\beta z^{\\beta +m}-\\sum _{\\beta >\\alpha +m} \\lambda ^{\\alpha +m}_{\\beta }z^\\beta =\\sum _{\\beta >\\alpha +m}(\\lambda ^\\alpha _{\\beta -m}-\\lambda ^{\\alpha +m}_\\beta )z^\\beta $ should belong to $M_i$ .", "Take $\\beta >\\alpha +m$ and let $j={\\omega }^{-1}(\\beta ).$ If $j>i,$ then the term of degree $\\beta $ can be eliminated by subtracting $f^\\beta =f_j\\in M_i$ with an appropriate coefficient.", "If $j<i,$ then ${{\\omega }^{-1}}(\\beta -m) < {{\\omega }^{-1}}(\\beta ) < {{\\omega }^{-1}}(\\alpha ) < {{\\omega }^{-1}}(\\alpha +m)$ .", "Hence $(\\alpha ,\\beta -m)\\in \\operatorname{{\\overline{Inv}}}({\\omega }^{-1})$ and $(\\alpha +m,\\beta )\\in \\operatorname{{\\overline{Inv}}}({\\omega }^{-1}),$ so the coefficients $\\lambda ^\\alpha _{\\beta -m}$ and $\\lambda ^{\\alpha +m}_\\beta $ are both not forced to be zero, i.e.", "they are parameters on $\\Sigma _{\\omega }$ .", "The term $(\\lambda ^\\alpha _{\\beta -m}-\\lambda ^{\\alpha +m}_\\beta )z^\\beta $ has to vanish automatically after we eliminated all lower order terms.", "As we eliminate terms of degree $\\gamma $ such that $\\alpha +m<\\gamma <\\beta ,$ the coefficient at $z^\\beta $ changes, but the added terms can only depend on coefficients of smaller height.", "More precisely, all additional terms are non-linear, and the total height of each monomial is always $\\beta -\\alpha -m.$ In the end, we get that $\\lambda ^\\alpha _{\\beta -m}-\\lambda ^{\\alpha +m}_\\beta $ should be equal to zero modulo the coefficients of smaller height.", "This means that for each $(\\alpha ,\\beta )\\in \\operatorname{{\\overline{Inv}}}({\\omega }^{-1})$ of height $\\beta -\\alpha >m$ there is an equation that allows one to express $\\lambda ^\\alpha _{\\beta -m}$ in terms of $\\lambda ^{\\alpha +m}_\\beta $ and higher order terms in coefficients with lower height.", "A priori, the linear parts of these equations can be dependent if for all $0\\le q\\le n$ one has $(\\alpha +qm,\\beta +qm)\\in \\operatorname{{\\overline{Inv}}}({\\omega }^{-1}).$ However, since $m$ and $n$ are relatively prime, this would mean that $\\omega ^{-1}(\\gamma )>\\omega ^{-1}(\\gamma +\\beta -\\alpha )$ for all $\\gamma \\in {\\mathbb {Z}},$ which is impossible.", "Therefore, one can resolve the relations on the coefficients with respect to $\\lambda ^\\alpha _\\beta $ such that $(\\alpha ,\\beta )\\in \\operatorname{{\\overline{Inv}}}({\\omega }^{-1})$ and $(\\alpha ,\\beta +m)\\in \\operatorname{{\\overline{Inv}}}({\\omega }^{-1}).$ So, the coordinates on $\\Sigma _{{\\omega }}$ correspond to the inversions $(\\alpha ,\\beta )\\in \\operatorname{{Inv}}({\\omega }^{-1}),$ such that $(\\alpha ,\\beta +m)\\notin \\operatorname{{Inv}}({\\omega }^{-1})$ .", "Since $\\lambda ^\\alpha _\\beta =\\lambda ^{\\alpha +n}_{\\beta +n},$ one should count inversions in $\\operatorname{{Inv}}({\\omega }^{-1})$ only.", "It is not hard to see that such inversions are in bijection with inversions of height less than $m.$ Indeed, the required map is $(\\alpha ,\\beta )\\mapsto (\\alpha ,\\beta -km),$ where $k$ is the maximal integer such that $\\beta -km>\\alpha .$ Alternatively, one can also notice that the relations are in bijection with inversions of height greater than $m.$ Indeed, the relation $\\lambda ^\\alpha _\\beta \\equiv \\lambda ^{\\alpha +m}_{\\beta +m}$ (modulo lower height terms) corresponds to the inversion $(\\alpha ,\\beta +m)\\in \\operatorname{{\\overline{Inv}}}({\\omega }^{-1})$ of height greater than $m.$ Therefore, the dimension of $\\Sigma _{{\\omega }}$ is the total number of inversions minus the number of inversion of height greater than $m.$ Since there are no inversions of height $m,$ the dimension is equal to the number of inversions of height less than $m.$ Since $\\sum _{i=1}^{n}\\operatorname{\\mathcal {PS}}_{\\omega ^{-1}}(i)$ is exactly the total number of inversions of height less than $m,$ we conclude that $\\dim (\\Sigma _{\\omega })=\\sum _{i=1}^{n}\\operatorname{\\mathcal {SP}}_{\\omega }(i)=\\sum _{i=1}^{n}\\operatorname{\\mathcal {PS}}_{\\omega ^{-1}}(i)=\\frac{(m-1)(n-1)}{2}-\\operatorname{dinv}({\\omega }^{-1}).$ For a more abstract proof see e.g.", "[13] and [20].", "Remark 8.10 Similar reasoning shows that the Grassmannian version of the affine Springer fiber $\\mathcal {G}_{m/n}\\subset \\mathcal {G}_n$ parametrizes appropriately normalized $[z^n,z^m]]$ -submodules in $(z)).$ This affine Springer fiber was studied e.g.", "in [14], [15] under the name of Jacobi factor of the plane curve singularity $\\lbrace x^m=y^n\\rbrace $ .", "The cells in it are parametrized by the subsets in $\\mathbb {Z}_{\\ge 0}$ which are invariant under addition of $m$ and $n$ , and can be matched to the lattice points in $D_{n}^{m}$ .", "Note the lattice points in turn correspond to the minimal length left coset representatives ${}^m\\widetilde{S}_n\\cap \\widetilde{S}_n/S_n$ .", "Corollary 8.11 If the map $\\operatorname{\\mathcal {PS}}$ is a bijection then the Poincaré polynomial of $\\mathcal {F}_{m/n}$ is given by the following formula: $\\sum _{k=0}^{\\infty }t^{k}\\dim \\operatorname{H}^{k}\\left(\\mathcal {F}_{m/n}\\right)=\\sum _{f\\in \\operatorname{\\mathcal {PF}}_{m/n}}t^{2\\sum _{i}f(i)}.$ Since the variety $\\mathcal {F}_{m/n}$ can be paved by the even-dimensional cells, it has no odd cohomology and $(2k)$ -th Betti number is equal to the number of cells of complex dimension $k$ .", "Therefore by Theorem REF : $\\sum _{k}t^{k}\\dim \\operatorname{H}^{k}\\left(\\mathcal {F}_{m/n}\\right)=\\sum _{{\\omega }\\in {}^m\\widetilde{S}_n}t^{2\\dim \\Sigma _{{\\omega }}}=\\sum _{{\\omega }\\in \\widetilde{S}_n^m}t^{2\\sum _i \\operatorname{\\mathcal {PS}}_{{\\omega }}(i)}=\\sum _{f\\in \\operatorname{\\mathcal {PF}}_{m/n}}t^{2\\sum _i f(i)}.$ Equation $(\\ref {Poinc})$ was conjectured in [24] for all coprime $m$ and $n$ .", "Some examples for $m \\ne kn\\pm 1.$ In this section we discuss some examples for which $m \\ne kn\\pm 1$ .", "Example 9.1 There are $81 = 3^4$ $3/5$ -parking functions.", "The $7= \\frac{1}{5+3} \\binom{5+3}{5} $ increasing parking functions are $\\llparenthesis 00000 \\rrparenthesis ,\\llparenthesis 00001 \\rrparenthesis , \\llparenthesis 00002 \\rrparenthesis , \\llparenthesis 00011 \\rrparenthesis , \\llparenthesis 00012 \\rrparenthesis , \\llparenthesis 00111 \\rrparenthesis ,\\llparenthesis 00112 \\rrparenthesis $ .", "Grouping them into the $S_{5}$ -orbits $\\lbrace f \\circ {\\omega }\\mid {\\omega }\\in S_{5} \\rbrace $ yields $81 = 1 + 5+5+10+20+10+30$ .", "There are 7 vectors in ${\\mathbb {Z}}^5 \\cap V \\cap D_5^3$ .", "Their transposes are: $(0,0,0,0,0), (1,0,0,0,-1), (0,1,0,0,-1),(1,0,0,-1,0),(0,0,1,-1,0), (0,1,-1,0,0), (1,-1,0,1,-1)$ The 30 parking functions in the $S_{5}$ -orbit of $\\llparenthesis 00112 \\rrparenthesis $ correspond under the map $\\operatorname{\\mathcal {A}}$ to the 30 permutations in $S_{5} \\cap {}^{3}\\widetilde{S}_{5}$ which are those in the support of the shuffle $14 \\,$$$ 25 $\\scriptscriptstyle \\cup {\\hspace{-2.22214pt}}\\cup $ 3$(that is to say, the intersection of the Sommers region with the orbit ofthe identity permutation).", "\\\\On the other hand, the parking function $ 00000 = A(m)$corresponds under $ A$ to the affine permutation$ m= [-3,0,3,6,9] 3S5$.", "Anything else in its right $ S5$-orbitlies outside the Sommers region.$ Despite the fact many of the above theorems and constructions use $\\widetilde{S}_n^m$ , it is more uniform to study the set $\\lbrace u {\\rm A}_0\\mid u \\in {}^m\\widetilde{S}_n\\rbrace $ and $\\operatorname{\\mathcal {SP}}$ than $\\lbrace {\\omega }{\\rm A}_0\\mid {\\omega }\\in \\widetilde{S}_n^m\\rbrace $ and $\\operatorname{\\mathcal {PS}}$ .", "One reason is that while the Sommers region can always be defined for $\\gcd (m,n)=1$ , a hyperplane arrangement that is the correct analogue of the Shi arrangement cannot.", "Consider the following example.", "Example 9.2 .", "In the case $(n,m) = (5,3)$ it is impossible to find a set of hyperplanes that separate the alcoves $\\lbrace {\\omega }{\\rm A}_0\\mid {\\omega }\\in \\widetilde{S}_n^m\\rbrace $ and has $\\frac{1}{5+3} \\binom{5+3}{5} = 7$ dominant regions, i.e.", "that there are exactly 7 dominant regions with a unique ${\\omega }{\\rm A}_0$ in each.", "In other words, the notion of $\\operatorname{Reg}^k_n$ does not extend well when $m \\ne kn \\pm 1$ .", "Indeed, 7 dominant regions corresponding to 3-restricted affine permutations ${\\omega }\\in {}^{3}\\widetilde{S}_{5} \\cap \\widetilde{S}_{5}/S_{5}$ are shown in bold in Figure REF .", "Each permutation drawn corresponds to the dominant alcove ${{\\omega }^{-1}}{\\rm A}_0$ .", "Hence the hyperplanes crossed (by the pictured ${\\omega }$ ) correspond exactly to $\\operatorname{{Inv}}({\\omega })$ .", "The hyperplanes $H_{4,6}^0,H_{5,6}^0$ and $H_{5,7}^0$ separate $[02346]$ from other 3-restricted permutations.", "To separate $[-21457]$ from $[-11258]$ , one must add either $H_{3,6}^0$ or $H_{5,8}^0$ to the arrangement, but this would leave either of the non-3-restricted permutations $[-22456]$ or $[01248]$ in a region with no 3-restricted permutations.", "Therefore any extension of the classical braid arrangement for $S_5$ would either have a region with two 3-restricted permutations or a region with none of them.", "Note there are more hyperplanes ($H_{4,7}^0, H_{5,11}^0$ ) that ${\\omega }_m= [-3,0,3,6,9]$ has crossed that we did not draw on the picture.", "Figure: 7 permutations ω∈ 3 S ˜ 5 {\\omega }\\in {}^{3}\\widetilde{S}_{5} labeling ω -1 A 0 {\\omega }^{-1} {\\rm A}_0in the dominant coneExample 9.3 We list all affine permutations in $\\widetilde{S}_{5}^{2}$ together with their images under the maps $\\operatorname{\\mathcal {A}}$ and $\\operatorname{\\mathcal {PS}}$ in Figure REF .", "Here ${\\omega }$ is a 2-stable affine permutation (that is, ${\\omega }(i+2)>{\\omega }(i)$ ), and ${\\omega }^{-1}$ is 2-restricted .", "Note that for $m=2$ one has $\\operatorname{\\mathcal {A}}_{{\\omega }}(k)={\\omega }^{-1}(k)-M_{{\\omega }}\\ \\operatorname{mod}\\ 2$ , where, as above, $M_{{\\omega }}=\\min \\lbrace k : \\omega (k)>0\\rbrace $ .", "The combinatorial Hilbert series has a form: $H_{2/5}(q,t)=5+4(q+t)+(q^2+qt+t^2).$ In particular, it is symmetric in $q$ and $t$ and thus answers a question posed in [2].", "The special vertex of $D_{2/5}$ corresponding to the fundamental alcove by Lemma REF is described by the affine permutation ${\\omega }_2=[-1,1,3,5,7]$ .", "Figure: Affine permutations in S ˜ 5 2 \\widetilde{S}_{5}^{2}, their inverses in 2 S ˜ 5 {}^{2}\\widetilde{S}_{5}; maps error\\operatorname{\\mathcal {A}} and error\\operatorname{\\mathcal {PS}} to error 2/5 \\operatorname{\\mathcal {PF}}_{2/5};area\\operatorname{area} and dinv\\operatorname{dinv} statistics" ], [ "Extended Shi Arrangements and Pak-Stanley Labeling.", "Recall the set of $k$ -parking functions $\\operatorname{\\mathcal {PF}}_k : = \\operatorname{\\mathcal {PF}}_{(kn+1)/n}.$ Recall the hyperplanes $H_{ij}^k=\\lbrace {\\bf \\overline{x}}\\in V\\mid x_i-x_j=k\\rbrace $ and the affine braid arrangement $\\widetilde{B}_n=\\lbrace H_{ij}^k\\mid 1\\le i,j\\le n, k\\in {\\mathbb {Z}}\\rbrace $ .", "The extended Shi arrangement, or $k$ -Shi arrangement [25], [29], is defined as a subarrangement of the affine braid arrangement: Definition 4.1 The hyperplane arrangement $\\operatorname{Sh}^k_n:=\\left\\lbrace H_{ij}^{\\ell }:1\\le i<j\\le n,\\ -k<{\\ell }\\le k \\right\\rbrace $ is called the $k$ -Shi arrangement.", "The connected components of the complement to $\\operatorname{Sh}^k_n$ are called $k$ -Shi regions.", "The set of $k$ -Shi regions is denoted $\\operatorname{Reg}^k_n.$ One can use the notations introduced in Section REF to rewrite the definition of the $k$ -Shi arrangement as follows: $\\operatorname{Sh}^k_n=\\left\\lbrace H_{ij}^{\\ell }:1\\le i<j\\le n,\\ -k<{\\ell }\\le k \\right\\rbrace $ $=\\left\\lbrace H_{ij}^{\\ell }:1\\le i<j\\le n,\\ -k<{\\ell }< 0 \\right\\rbrace \\sqcup \\left\\lbrace H_{ij}^{\\ell }:1\\le i<j\\le n,\\ 0\\le {\\ell }\\le k \\right\\rbrace $ $=\\left\\lbrace H_{i,j-n\\ell }^0:1\\le i<j\\le n,\\ -k<{\\ell }< 0 \\right\\rbrace \\sqcup \\left\\lbrace H_{j,i+\\ell n}^0:1\\le i<j\\le n,\\ 0\\le {\\ell }\\le k \\right\\rbrace $ $=\\left\\lbrace H_{ij}^0:1\\le i\\le n, i<j<i+kn, ji \\mod {n}\\right\\rbrace .$ In other words, the $k$ -Shi arrangement consists of all hyperplanes of height less than $kn$ in the affine braid arrangement.", "The hyperplane $H_{ij}^{\\ell }$ divides $V$ into two half-spaces.", "Let $H_{ij}^{{\\ell },\\prec }$ denote the half-space that contains ${\\rm A}_0$ and $ H_{ij}^{{\\ell },\\succ }$ denote the complementary half-space.", "Note that $H_{ij}^{\\ell }$ separates ${\\omega }({\\rm A}_0)$ from ${\\rm A}_0$ iff ${\\omega }({\\rm A}_0) \\subseteq H_{ij}^{{\\ell },\\succ }$ iff $(i, j-{\\ell }n)$ or $(j, i+{\\ell }n) \\in \\operatorname{{Inv}}({\\omega }^{-1})$ (when taking the convention $i,j \\in \\lbrace 1, \\ldots , n\\rbrace $ ).", "Definition 4.2 The Pak-Stanley labeling is the map $\\lambda :\\operatorname{Reg}^k_n\\rightarrow \\operatorname{\\mathcal {PF}}_k$ , $R \\mapsto \\lambda _R$ defined by the formula $\\lambda _R(a)=\\sharp \\lbrace H_{ij}^{\\ell }\\in \\operatorname{Sh}^k_n\\mid R \\subseteq H_{ij}^{{\\ell },\\succ }, {\\ell }>0, i=a\\rbrace +\\sharp \\lbrace H_{ij}^{\\ell }\\in \\operatorname{Sh}^k_n\\mid R \\subseteq H_{ij}^{{\\ell },\\succ }, {\\ell }\\le 0, j=a\\rbrace .$ In other words, one labels the fundamental alcove ${\\rm A}_0$ by the parking function $f=\\llparenthesis 0\\ldots 0 \\rrparenthesis ,$ and then as one crosses the hyperplane $H_{ij}^{\\ell }$ in the positive direction (i.e.", "getting further away from ${\\rm A}_0$ ), one adds 1 to $f(j)$ if ${\\ell }\\le 0$ and adds 1 to $f(i)$ if ${\\ell }>0.$ Remark 4.3 One can rewrite this definition as follows: $\\lambda _R(a)=\\sharp \\lbrace H_{ij}^0\\in \\operatorname{Sh}^k_n\\mid R \\subseteq H_{ij}^{0,\\succ }, a=i<j\\rbrace =\\sharp \\lbrace H_{a \\, a+t}^0 \\mid R \\subseteq H_{a\\, a+t}^{0,\\succ }, 0 < t < kn, t \\lnot \\equiv 0\\mod {n}\\rbrace .$ We illustrate the Pak-Stanley labeling for $n=3,$ $k=1$ ($m=4$ ) in Figure REF .", "Figure: Pak-Stanley labeling for 1-Shi arrangement for n=3.n=3.Theorem 4.4 ([29]) The map $\\lambda :\\operatorname{Reg}^k_n\\rightarrow \\operatorname{\\mathcal {PF}}_k$ is a bijection." ], [ "Relation Between Sommers Regions and Extended Shi Arrangements for $m=kn\\pm 1$", "Consider the case $m=kn+1.$ One can show that each region of an extended Shi arrangement contains a unique minimal alcove (i.e.", "an alcove with the least number of hyperplanes $H_{ij}^k$ separating it from the fundamental alcove ${\\rm A}_0$ ).", "Theorem 4.5 ([11]) An alcove $\\omega ({\\rm A}_0)$ is the minimal alcove of a $k$ -Shi region if and only if $\\omega ^{-1}({\\rm A}_0)\\subset D_n^{kn+1}.$ Example 4.6 We illustrate this theorem in Figure REF , where the minimal alcoves of the 1-Shi region are matched with the alcoves in the Sommers region $D_3^4.$ On the left we have the minimal alcoves $\\omega ({\\rm A}_0)$ labeled by the $m$ -stable permutations $\\omega \\in \\widetilde{S}_n^m$ for $m=4, n=3$ .", "On the right we have alcoves $\\omega ^{-1}({\\rm A}_0)$ that fit inside $D_3^4,$ labeled by the $m$ -restricted permutations $\\omega ^{-1}\\in {}^m\\widetilde{S}_n.$ Note that $[-226]=[420]^{-1},$ $[150]=[1-16]^{-1},$ and $[4-13]=[-253]^{-1}.$ Figure: Minimal alcoves for Sh 3 1 \\operatorname{Sh}_3^{1} and Sommers region D 3 4 D_{3}^{4}.Theorem REF and Lemma REF imply a bijection $\\operatorname{alc}:\\widetilde{S}_n^{kn+1}\\rightarrow \\operatorname{Reg}^k_n.$ Theorem 4.7 One has $\\lambda \\circ \\operatorname{alc}=\\operatorname{\\mathcal {PS}}$ in this case.", "In particular, $\\operatorname{\\mathcal {PS}}$ is a bijection for $m=kn+1.$ As it was mentioned in Section REF , an affine permutation $\\omega $ has an inversion $(i,i+h)$ if and only if $\\omega ^{-1}({\\rm A}_0)$ is separated from ${\\rm A}_0$ by the hyperplane $H_{i,i+h}^0$ or, equivalently, if and only if $\\omega ^{-1}({\\rm A}_0)\\subset H_{i,i+h}^{0,\\succ }.$ Given a region $R$ , for any affine permutation $\\omega $ such that $\\omega ^{-1}({\\rm A}_0)\\subset R,$ the number of inversions of the form $(a,a+h)$ of height $h<kn$ is equal to $\\lambda _R(a).$ If $\\omega ^{-1}\\in \\widetilde{S}_n^{kn+1}$ then the alcove $\\omega ^{-1}({\\rm A}_0)$ is the minimal alcove in the region $R$ and $\\operatorname{alc}(\\omega ^{-1})=R.$ By definition, $\\operatorname{\\mathcal {PS}}_{\\omega ^{-1}}(a)$ is equal to the number of inversions $(\\alpha ,\\beta )$ of $\\omega ^{-1},$ such that $\\beta -\\alpha <kn+1$ and $\\omega ^{-1}(\\beta )=a,$ which is the same as the number of inversions $(a,a+h)$ of $\\omega ,$ such that $\\omega (a)-\\omega (a+h)<kn+1.$ Note that $\\omega (a)-\\omega (a+h)$ cannot be equal to $kn,$ so $\\operatorname{\\mathcal {PS}}_{\\omega ^{-1}}(a)$ is, in fact, equal to the number of inversions $(a,a+h),$ such that $\\omega (a)-\\omega (a+h)<kn.$ To match it with $\\lambda _R(a),$ one has to prove the following equation for any $a\\in \\lbrace 1,\\ldots ,n\\rbrace $ and any ${\\omega }$ : $\\sharp \\left\\lbrace (a,a+h)\\in \\operatorname{{Inv}}(\\omega )\\mid h<kn\\right\\rbrace =\\sharp \\left\\lbrace (a,a+h)\\in \\operatorname{{Inv}}(\\omega )\\mid \\omega (a)-\\omega (a+h)<kn\\right\\rbrace .$ Given $r\\in \\lbrace 1,\\dots ,n-1\\rbrace ,$ define $\\gamma _{a,{\\omega }}(r):=\\sharp \\left\\lbrace (a,a+h)\\in \\operatorname{{Inv}}(\\omega )\\mid h<kn, h\\equiv r \\operatorname{mod}n\\right\\rbrace $ and $\\gamma ^{\\prime }_{a,{\\omega }}(r):=\\sharp \\left\\lbrace (a,a+h)\\in \\operatorname{{Inv}}(\\omega )\\mid \\omega (a)-\\omega (a+h)<kn, h\\equiv r \\operatorname{mod}n\\right\\rbrace .$ Let $h_{\\max }^r$ be the maximal number such that $(a,h_{\\max }^r)\\in \\operatorname{{Inv}}(\\omega )$ and $h_{\\max }^r\\equiv r\\ \\operatorname{mod}n.$ It is not hard to see that $\\gamma _{a,{\\omega }}(r)=\\gamma ^{\\prime }_{a,{\\omega }}(r)=\\min \\left(k,\\left\\lfloor \\frac{h_{\\max }^r}{n}\\right\\rfloor \\right).$ Indeed, the total number of inversions $(a,a+h)$ such that $h\\equiv r\\ \\operatorname{mod}n$ equals $\\lfloor \\frac{h_{\\max }^r}{n}\\rfloor $ .", "If it is less than or equal to $k$ then all of them satisfy both $h<kn$ and $\\omega (a)-\\omega (a+h)<kn.$ In turn, if it is greater than $k,$ then the inversions $(a,a+h)$ for $h=r,r+n,\\ldots ,r+(k-1)n$ satisfy the condition $h<kn,$ while the inversions $(a,a+h)$ for $h=h_{\\max }^r,h_{\\max }^r-n,\\ldots ,h_{\\max }^r-(k-1)n$ satisfy the condition $\\omega (a)-\\omega (a+h)<kn.$ Finally, the sum of identities $\\gamma _{a,{\\omega }}(r)=\\gamma ^{\\prime }_{a,{\\omega }}(r)$ for all $r$ is equivalent to (REF ).", "Example 4.8 When one applies the map $\\operatorname{\\mathcal {PS}}$ to the affine permutations in the left part of Figure REF one gets the Pak-Stanley labeling shown in Figure REF .", "The case $m=kn-1$ is treated similarly.", "The main difference is that instead of the set of all $k$ -Shi regions $\\operatorname{Reg}^k_n$ one should consider the set of bounded $k$ -Shi regions $\\widehat{\\operatorname{Reg}^k_n}.$ One can show that every bounded $k$ -Shi region contains exactly one maximal alcove.", "Theorem 4.9 ([12]) An alcove $\\omega ({\\rm A}_0)$ is the maximal alcove of a bounded $k$ -Shi region if and only if $\\omega ^{-1}({\\rm A}_0)\\subset D_n^{kn-1}.$ As above, we use Lemma REF and Theorem REF to obtain the bijection $\\widehat{\\operatorname{alc}}:\\widetilde{S}_n^{kn-1}\\rightarrow \\widehat{\\operatorname{Reg}^k_n}.$ We prove the following theorem: Theorem 4.10 The image of the subset $\\widehat{\\operatorname{Reg}^k_n}\\subset \\operatorname{Reg}^k_n$ under the Pak-Stanley labeling is exactly $\\operatorname{\\mathcal {PF}}_{(kn-1)/n}\\subset \\operatorname{\\mathcal {PF}}_{(kn+1)/n}.$ Furthermore, one gets $\\lambda \\circ \\widehat{\\operatorname{alc}}=\\operatorname{\\mathcal {PS}}$ in this case.", "In particular, $\\operatorname{\\mathcal {PS}}$ is a bijection for $m=kn-1.$ It is sufficient to prove the formula $\\lambda \\circ \\widehat{\\operatorname{alc}}=\\operatorname{\\mathcal {PS}}.$ Indeed, this would imply that the restriction of the Pak-Stanley labeling to the subset $\\widehat{\\operatorname{Reg}^k_n}\\subset \\operatorname{Reg}^k_n$ is an injective (and, therefore, bijective) map from $\\widehat{\\operatorname{Reg}^k_n}$ to $\\operatorname{\\mathcal {PF}}_{(kn-1)/n}.$ If $\\omega ^{-1}\\in \\widetilde{S}_n^{kn-1}$ then the alcove $\\omega ^{-1}({\\rm A}_0)$ is the maximal alcove of a bounded region $R$ and $\\widehat{\\operatorname{alc}}(\\omega ^{-1})=R.$ Similarly to Theorem REF , we get that $\\operatorname{\\mathcal {PS}}_{\\omega ^{-1}}(a)$ is equal to the number of inversions $(a,a+h)$ of $\\omega $ such that $\\omega (a)-\\omega (a+h)<kn-1.$ Since $\\omega \\in {}^{kn-1}\\widetilde{S}_n,$ one has $\\omega (a)-\\omega (a+h)\\ne kn-1$ for any $h>0.$ Therefore, $\\operatorname{\\mathcal {PS}}_{\\omega ^{-1}}(a)$ is equal to the number of inversions $(a,a+h)$ in $\\omega $ such that $\\omega (a)-\\omega (a+h)<kn.$ In the proof of Theorem REF we have shown that this number is equal to $\\lambda _R(a).$" ], [ "Minimal Length Representatives and the Zeta Map.", "Definition 5.1 Let $\\operatorname{{Mod_{m,n}}}$ be the set of subsets $\\Delta \\subset \\mathbb {Z}_{\\ge 0},$ such that $\\Delta +m\\subset \\Delta ,$ $\\Delta +n\\subset \\Delta ,$ and $\\min (\\Delta )=0.$ A number $a$ is called an $n$ -generator of $\\Delta $ , if $a\\in \\Delta $ and $a-n\\notin \\Delta $ .", "Every $\\Delta \\in \\operatorname{{Mod_{m,n}}}$ has exactly $n$ distinct $n$ -generators.", "In [14], [15] such subsets were called 0-normalized semimodules over the semigroup generated by $m$ and $n$.", "We will simply call them $m,n$ -invariant subsets.", "There is a natural map $R:\\widetilde{S}_n^m\\rightarrow \\operatorname{{Mod_{m,n}}}$ given by $\\omega \\mapsto \\Delta _{{\\omega }}-\\min (\\Delta _{{\\omega }})$ (here, as before, $\\Delta _{\\omega }=\\lbrace i\\in \\mathbb {Z}: \\omega (i)>0\\rbrace $ ).", "Let $\\Omega _n^m$ be the set of $m$ -stable minimal length right coset representatives of $S_n\\backslash \\widetilde{S}_n.$ In other words, $\\Omega _n^m:=\\lbrace \\omega \\in \\widetilde{S}_n^m\\mid \\omega ^{-1}(1)<\\ldots <\\omega ^{-1}(n)\\rbrace .$ One can check that the restriction $R\|_{\\Omega _n^m}:\\Omega _n^m\\rightarrow \\operatorname{{Mod_{m,n}}}$ is a bijection.", "Indeed, the integers $\\omega ^{-1}(1),\\ldots ,\\omega ^{-1}(n)$ are the $n$ -generators of $\\Delta _{{\\omega }},$ and since ${\\omega }\\in \\Omega _n^m$ we have $\\omega ^{-1}(1)<\\ldots <\\omega ^{-1}(n),$ so one can uniquely recover ${\\omega }$ from $\\Delta _{\\omega }.$ Let $\\hat{R}:=R\|_{\\Omega _n^m}:\\Omega _n^m\\rightarrow \\operatorname{{Mod_{m,n}}}$ denote the restriction.", "Recall that $Y_{m,n}$ is the set of Young diagrams that fit under diagonal in an $n\\times m$ rectangle and $P:\\operatorname{\\mathcal {PF}}_{m/n}\\rightarrow Y_{m,n}$ is the natural map.", "In [14], [15] the first two named authors constructed two maps $D:\\operatorname{{Mod_{m,n}}}\\rightarrow Y_{m,n}$ and $G:\\operatorname{{Mod_{m,n}}}\\rightarrow Y_{m,n},$ proved that $D$ is a bijection, and related the two maps to the theory of $q,t$ -Catalan numbers in the following way.", "In the case $m=n+1$ one gets $c_n(q,t)=\\sum \\limits _{\\Delta \\in \\operatorname{{Mod_{n+1,n}}}} q^{\\delta -\|D(\\Delta )\|}t^{\\delta -\|G(\\Delta )\|},$ where $\\delta =\\frac{n(n-1)}{2}$ and $c_n(q,t)$ is the Garsia-Haiman $q,t$ -Catalan polynomial.", "It is known that these polynomials are symmetric $c_n(q,t)=c_n(t,q),$ although the proof is highly non-combinatorial and uses the machinery of Hilbert schemes, developed by Haiman.", "Finding a combinatorial proof of the symmetry of the $q,t$ -Catalan polynomials remains an open problem.", "The above consideration motivates the rational slope generalization of the $q,t$ -Catalan numbers: $c_{m,n}(q,t)=\\sum \\limits _{\\Delta \\in \\operatorname{{Mod_{m,n}}}} q^{\\delta -\|D(\\Delta )\|}t^{\\delta -\|G(\\Delta )\|},$ where $\\delta =\\frac{(m-1)(n-1)}{2}$ is the total number of boxes below the diagonal in an $n\\times m$ rectangle.", "The symmetry of these polynomials remains an open problem beyond the classical case $m=n+1$ and the cases $\\min (m,n)\\le 4$ (see [15] for $\\min (m,n)\\le 3$ and [22] for $\\min (m,n)=4$ ).", "It was also shown in [14] that the composition $G\\circ D^{-1}:Y_{m,n}\\rightarrow Y_{m,n}$ generalizes Haglund's zeta map exchanging the pairs of statistics $(\\operatorname{area},\\operatorname{dinv})$ and $(\\operatorname{bounce}, \\operatorname{area})$ on Dyck paths.", "It was conjectured that the map $G,$ and therefore, the generalized Haglund's zeta, are also bijections.", "This would imply a weaker symmetry property $c_{m,n}(q,1)=c_{m,n}(1,q).$ In [15] the bijectivity of $G$ was proved for $m=kn\\pm 1.$ For more details on this work we refer the reader to [14], [15].", "Let $\\star $ denote the involution on $\\widetilde{S}_n$ : ${\\omega }^{\\star }(x)=1-{\\omega }(1-x)$ .", "Lemma 5.2 The map $\\star $ preserves the set $\\widetilde{S}_n^m$ and the set $\\Omega _n^m$ .", "The map $\\overline{\\star }:(i,j)\\mapsto (1-j,1-i)$ provides a height preserving bijection from $\\lbrace (i,j)\\mid i<j,\\ {\\omega }(i)>{\\omega }(j)\\rbrace $ to $\\lbrace (i,j)\\mid i<j,\\ {\\omega }^\\star (i)>{\\omega }^\\star (j)\\rbrace .$ Let us check that ${\\omega }^\\star $ is an affine permutation: ${\\omega }^{\\star }(x+n)=1-{\\omega }(1-x-n)=1-{\\omega }(1-x)+n,$ $\\sum _{i=1}^{n}{\\omega }^{\\star }(i)=n-\\sum _{i=1-n}^{0}{\\omega }(i)=n-\\sum _{i=1}^{n}({\\omega }(i)-n)=n+n^2-\\frac{n(n+1)}{2}=\\frac{n(n+1)}{2}.$ If ${\\omega }(1)<\\ldots <{\\omega }(n)$ then ${\\omega }(1-n)<\\cdots < {\\omega }(0)$ , so ${\\omega }^{\\star }(1)<\\ldots <{\\omega }^{\\star }(n)$ .", "Let $(i,j)$ be such that $i<j$ and ${\\omega }(i)>{\\omega }(j).$ Then $1-j<1-i,$ and ${\\omega }^\\star (1-j)=1-{\\omega }(1-(1-j))=1-{\\omega }(j)>1-{\\omega }(i)={\\omega }^\\star (1-i).$ Note also that $\\overline{\\star }$ squares to identity.", "Therefore, since $\\star $ is an involution, $\\overline{\\star }$ is a bijection between $\\lbrace (i,j)\\mid i<j,\\ {\\omega }(i)>{\\omega }(j)\\rbrace $ and $\\lbrace (i,j)\\mid i<j,\\ {\\omega }^\\star (i)>{\\omega }^\\star (j)\\rbrace $ .", "Furthermore, since $\\overline{\\star }$ preserves height, $\\star $ preserves the set $\\widetilde{S}_n^m.$ The following Theorem shows that our constructions are direct generalizations of those of [14], [15]: Theorem 5.3 One has the following identities: $P\\circ \\operatorname{\\mathcal {A}}\\circ \\hat{R}^{-1}=D,$ $P\\circ \\operatorname{\\mathcal {PS}}\\circ \\star \\circ \\hat{R}^{-1}=G.$ The first statement follows from the definition of $\\operatorname{\\mathcal {A}}$ and Lemma REF .", "For the second statement, we need to recall the definition of the map $G$ .", "Given an $m,n$ -invariant subset $\\Delta \\in \\operatorname{{Mod_{m,n}}}$ , let $u_1<\\ldots <u_n$ be its $n$ -generators.", "The map $G$ was defined in [14], [15] by the formula $G_{\\Delta }(\\alpha )=\\sharp \\left([u_\\alpha ,u_\\alpha +m]\\setminus \\Delta \\right).$ Given a minimal coset representative $\\omega \\in \\Omega _n^m$ , we can consider an $m,n$ -invariant subset $R(\\omega )=\\Delta _{\\omega }-\\min (\\Delta _{{\\omega }})\\in \\operatorname{{Mod_{m,n}}}$ .", "Its $n$ -generators are $u_\\alpha =\\omega ^{-1}(\\alpha )-\\min (\\Delta _{{\\omega }})$ , and by (REF ) we have $u_1<\\ldots <u_n$ .", "For every $x\\in [u_\\alpha ,u_\\alpha +m]\\setminus R(\\omega )$ , define $x^{\\prime }:=x+\\min (\\Delta _{{\\omega }})$ , then all such $x^{\\prime }$ (and hence $x$ ) are defined by the inequalities $\\omega ^{-1}(\\alpha )<x^{\\prime }<\\omega ^{-1}(\\alpha )+m,\\ \\omega (x^{\\prime })\\le 0.$ Note that by (REF ) the inequality $\\omega (x^{\\prime })<0$ can be replaced by $\\omega (x^{\\prime })<\\alpha $ .", "Indeed, we have ${\\omega }^{-1}(1)<{\\omega }^{-1}(2)<\\ldots <{\\omega }^{-1}(n),$ and, therefore, ${\\omega }(x^{\\prime })\\notin \\lbrace 1,\\ldots ,\\alpha -1\\rbrace $ for $\\omega ^{-1}(\\alpha )<x^{\\prime }<\\omega ^{-1}(\\alpha )+m.$ Therefore the set $[u_\\alpha ,u_\\alpha +m]\\setminus R(\\omega )$ is in bijection with the set $\\lbrace (i,j)\\mid i<j<i+m,\\ {\\omega }(i)>{\\omega }(j),\\ {\\omega }(i)=\\alpha \\rbrace .$ In turn, the map $\\overline{\\star }$ bijectively maps this set to the set $\\lbrace (1-j,1-i)\\mid (1-j)<1-i<(1-j)+m,\\ {\\omega }^\\star (1-j)>{\\omega }^\\star (1-i),\\ {\\omega }^\\star (1-i)=1-{\\omega }(i)=1-\\alpha \\rbrace ,$ or, after a shift by $n$ and a change of variables, $\\lbrace (i,j)\\mid i<j<i+m,\\ {\\omega }^\\star (i)>{\\omega }^\\star (j),\\ {\\omega }^\\star (j)=n+1-\\alpha \\rbrace .$ Therefore, according to the definition of the map $\\operatorname{\\mathcal {PS}},$ we get $G_{R(\\omega )}(\\alpha )=\\operatorname{\\mathcal {PS}}_{\\omega ^{\\star }}(n+1-\\alpha ),$ and thus $G(R({\\omega }))=P(\\operatorname{\\mathcal {PS}}({\\omega }^\\star )).$ The involution $\\star $ could have been avoided in Theorem REF by adjusting the definition of the map $\\operatorname{\\mathcal {PS}}.$ However, in that case one would have to use $\\star $ to match the map $\\operatorname{\\mathcal {PS}}$ for $m=kn+1$ with the Pak-Stanley labeling (see Section ).", "The composition $\\operatorname{\\mathcal {PS}}\\circ \\operatorname{\\mathcal {A}}^{-1}:\\operatorname{\\mathcal {PF}}_{m/n}\\rightarrow \\operatorname{\\mathcal {PF}}_{m/n}$ should be thought of as a rational slope parking function generalization of the Haglund zeta map $\\zeta $ .", "Note that its bijectivity remains conjectural beyond cases $m=kn\\pm 1,$ which follows immediately from Theorems REF and REF ." ], [ "Finite-dimensional representations of DAHA", "It turns out that the map $A$ is tightly related to the representation theory of double affine Hecke algebras (DAHA).", "This theory is quite elaborate and far beyond the scope of this paper, so we refer the reader to Cherednik's book [6] for all details.", "Here we just list the necessary facts about finite-dimensional representations of DAHA.", "Let $\\mathbf {H}_n$ denote the DAHA of type $A_{n-1}$ .", "It contains the finite Hecke algebra generated by the elements $T_i$ , $1 \\le i < n$ as well as two commutative subalgebras $X_i/X_j$ , $1 \\le i \\ne j \\le n$ , and $Y_1^{\\pm 1},\\ldots ,Y_n^{\\pm 1}$ subject to commutation relations between $X^{\\prime }s$ and $Y^{\\prime }s$ that depend on two parameters $q$ and $t$ .", "(Alternatively, one can take generators $T_i$ , $0 \\le i < n$ , $\\pi $ and $X_i/X_j$ , $1 \\le i \\ne j \\le n$ .)", "$\\mathbf {H}_n$ admits a (degree 0 Laurent) polynomial representation $V=X_i/X_j]_{1 \\le i \\ne j \\le n}$ , where $X_i/X_j$ act as multiplication operators, and $Y_i$ act as certain difference operators.", "We can also obtain $V$ by inducing a 1-dimensional representation of the subalgebra generated by the $T_i$ and $Y_i^{\\pm 1}$ up to $\\mathbf {H}_n$ .", "The product $Y_1 Y_2 \\cdots Y_n $ (or equivalently $\\pi ^n$ ) acts as a constant on this representation.", "This constant agrees with the scalar by which the product acts on the initial 1-dimensional representation.", "We usually take this constant to be 1, or indeed impose the relation $Y_1 Y_2 \\cdots Y_n = 1$ in $\\mathbf {H}_n$ .", "However, to match the combinatorics developed in this paper, it is convenient to choose that scalar to be $q^{\\frac{n+1}{2}}$ .", "There exists a basis of $V$ consisting of nonsymmetric Macdonald polynomials $E_{\\sigma }(X_i)$ labeled by minimal length right coset representatives $\\sigma \\in S_n\\backslash \\widetilde{S}_n$ such that $Y_i$ are diagonal in this basis: $Y_i(E_{\\sigma })=a_i(\\sigma )E_{\\sigma }.$ The weights $a_i(\\sigma )$ are directly related to the combinatorial content of this paper and can be described as follows.", "Corresponding to the fundamental alcove $\\sigma = \\mathrm {id}$ we have $E_{\\mathrm {id}}=1$ and its weight equals to: $a(\\mathrm {id})=(a_1(\\mathrm {id}),\\ldots ,a_n(\\mathrm {id}))=q^{\\frac{n+1}{2n}}(t^{\\frac{1-n}{2}}, t^{\\frac{3-n}{2}},\\ldots ,t^{\\frac{n-1}{2}}).$ As we cross the walls (from $\\sigma {\\rm A}_0$ to $\\sigma s_i {\\rm A}_0$ ), the weights are transformed as follows: $s_i(a_1,\\ldots ,a_n)={\\left\\lbrace \\begin{array}{ll}(a_1,\\ldots , a_{i+1},a_i,\\ldots ,a_n),& \\text{if}\\ i\\ne 0\\cr (a_n/q,a_2,\\ldots ,a_{n-1},qa_{n}),& \\text{if}\\ i=0.\\end{array}\\right.", "}$ One can check that (REF ) defines an action of the affine symmetric group on the set of sequences of Laurent monomials in $q$ and $t$ .", "If the parameters $q$ and $t$ are connected by the relation $q^m=t^n$ for coprime $m$ and $n$ , the polynomial representation $V$ becomes reducible, and admits a finite-dimensional quotient $L_{m/n}$ of dimension $m^{n-1}$ .", "The basis of $L_{m/n}$ is again given by the nonsymmetric Macdonald polynomials $E_{\\sigma }$ , but now the permutation $\\sigma $ should have $\\sigma {\\rm A}_0$ belong to the (dominant) region bounded by the hyperplane $x_1-x_n=m$ .", "In other words, we can cross a wall if and only if the ratio of the corresponding weights is not equal to $t^{\\pm 1}$ .", "(See [7] for a discussion on these finite-dimensional quotients.", "See [8] for the formula for the intertwiners that take $E_\\sigma $ to $E_{\\sigma s_i}$ .", "See [9] for the nonsymmetric Macdonald evaluation formula that describes the $E_\\sigma $ in the radical of the polynomial representation.)" ], [ "From DAHA weights to Sommers region", "For the finite-dimensional representation $L_{m/n}$ we have $q^m=t^n$ , so $t=q^{m/n}$ .", "This means that every monomial $q^{x}t^{y}$ can be written as $q^{\\frac{nx+my}{n}}$ , so we can rewrite the DAHA weights as $a(\\sigma )=(a_1,\\ldots ,a_n)=(q^{b_1(\\sigma )/n},\\ldots ,q^{b_n(\\sigma )/n}).$ It turns out that “evaluated weights” $b_i$ are tightly related to the labeling of the region $D_{n}^{m}$ by affine permutations.", "Let $c=\\frac{(m-1)(n+1)}{2}$ .", "Consider the affine permutation ${\\omega }_m=[m-c,2m-c,\\ldots ,nm-c]$ .", "By Lemma REF , ${\\omega }_m$ identifies the dilated fundamental alcove with the simplex $D_{n}^{m}$ .", "Recall ${\\omega }\\in {}^m\\widetilde{S}_n\\iff {\\omega }_m^{-1}\\omega ({\\rm A}_0) \\subseteq m D_n^1 = m {\\rm A}_0$ .", "Theorem 6.1 Under this identification, for every ${\\omega }\\in {}^m\\widetilde{S}_n$ one has: $b({\\omega }_m^{-1}\\omega )=\\omega ,$ by which we mean for $\\sigma = {\\omega }_m^{-1}\\omega $ that $(b_1(\\sigma ),\\ldots ,b_n(\\sigma )) = ({\\omega }(1), \\ldots , {\\omega }(n))$ .", "In the weight picture we start from the fundamental alcove at $\\sigma = \\mathrm {id}$ , where we have weights $a(\\mathrm {id}) =q^{\\frac{n+1}{2n}}(t^{\\frac{1-n}{2}}, t^{\\frac{3-n}{2}},\\ldots ,t^{\\frac{n-1}{2}})=(q^{(m-c)/n}, q^{(2m-c)/n},\\ldots ,q^{(nm-c)/n}),\\\\b(\\mathrm {id}) =b({\\omega }_m^{-1} {\\omega }_m) =(m-c,2m-c,\\ldots ,nm-c)=({\\omega }_m(1), \\ldots , {\\omega }_m(n)).$ By Lemma REF , ${\\rm A}_0\\subseteq mD_n^1$ corresponds to the alcove ${\\omega }_m({\\rm A}_0) \\subseteq D_{n}^{m}$ , which we label by ${\\omega }_m=[m-c,\\ldots ,nm-c] \\in {}^m\\widetilde{S}_n$ .", "Therefore the desired identity holds for $\\sigma = \\mathrm {id}$ and can be extended to any $\\sigma $ with $\\sigma {\\rm A}_0\\subseteq m {\\rm A}_0= m D_n^1$ (equivalently ${\\omega }_m\\sigma \\in {}^m\\widetilde{S}_n$ ) by rules (REF ).", "Figure: DAHA weights for L 4/3 L_{4/3} above.", "When one evaluates at t=q 4/3 t=q^{4/3}, the weightsbecome (q u(1) 3 ,q u(2) 3 ,q u(3) 3 )(q^\\frac{u(1)}{3}, q^\\frac{u(2)}{3}, q^\\frac{u(3)}{3}) for the matchingalcove ω m uA 0 {\\omega }_mu {\\rm A}_0 which is labeled by u∈ 4 S ˜ 3 u \\in {}^{4}\\widetilde{S}_{3} inFigure .Note for the fundamental alcove,a( id )=q 2 3 (1 t,1,t)=q 2 3 (q -4 3 ,q 0 ,q 4 3 )=(q -2 3 ,q 2 3 ,q 6 3 )a(\\mathrm {id}) =q^{\\frac{2}{3}}( \\frac{1}{t}, 1, t)= q^{\\frac{2}{3}}(q^{-\\frac{4}{3}}, q^0, q^{\\frac{4}{3}})= (q^{-\\frac{2}{3}}, q^{\\frac{2}{3}}, q^{\\frac{6}{3}}) and u=[-2,2,6]=ω m id u=[-2,2,6] = {\\omega }_m\\mathrm {id}.Compare this toFigure , where the alcoveA 0 {\\rm A}_0 in the left figure matchesthe alcove labeled [-2,2,6][-2,2,6] in right figure." ], [ "From DAHA weights to parking functions", "Instead of direct evaluation of DAHA weights as powers of $q^{1/n}$ , one can instead draw monomials $q^{x}t^{y}$ on the $(x,y)$ -plane.", "This point of view was used in much wider generality in [30], where the weights were interpreted in terms of periodic skew standard Young tableaux.", "Here we focus on finite-dimensional representations and relate this picture to parking function diagrams.", "Let $a=(a_1,\\ldots ,a_n)$ be a DAHA weight.", "We define a function $T_{a}:\\mathbb {Z}_{\\ge 0}^2\\rightarrow \\mathbb {Z}_{\\ge 0}$ labeling the square lattice by the following rule.", "For every $i$ , let us present $a_i=(q^{\\frac{n+1}{2n}} t^{\\frac{-1-n}{2}}) q^{x_i}t^{y_i}$ and define $T_a(x_i,y_i)=i$ .", "Under this renormalization, $\\lbrace y_1, \\ldots , y_n\\rbrace = \\lbrace 1, \\ldots , n\\rbrace $ .", "Hence we obtain $n$ squares labeled $1,\\ldots , n$ in the rows $1,\\ldots n$ in some order.", "We can extend this labeling to the whole plane by the following two-periodic construction.", "First, one can identify $q^m$ with $t^n$ and write $T_{a}(x+m,y-n)=T_{a}(x,y)$ .", "Secondly, recall that the $b_i$ that correspond to $a$ can be naturally extended to an affine permutation using the quasi-periodic condition $b_{i+n}=b_i+n$ .", "This means that one can define $a_i$ for all integer $i$ by the rule $a_{i+n}=qa_i$ , and $T_{a}(x+1,y)=T_{a}(x,y)+n$ .", "Hence the fillings in the boxes of $T_a$ increase across rows automatically; that is, $T_a$ is row-standard.", "The more interesting question is when is $T_a$ column standard, which in this context means fillings increase up columns.", "Lemma 6.2 The weight $a$ appears in the finite-dimensional representation $L_{m/n}$ if and only if $T_{a}$ is a standard Young tableau (SYT), that is, $T_{a}(x+1,y)>T_{a}(x,y)$ and $T_{a}(x,y+1)>T_{a}(x,y)$ .", "Indeed, in terms of $b_i$ this means that $b_{i}+m$ appears after $b_i$ , which is precisely equivalent to $m$ -stability.", "Corollary 6.3 There is a natural bijection between the alcoves in the Sommers region and surjective maps $T:\\mathbb {Z}_{\\ge 0}^2\\rightarrow \\mathbb {Z}_{\\ge 0}$ satisfying the following conditions: $T(x+1,y)=T(x,y)+n,\\ T(x+m,y-n)=T(x,y),\\ T(x,y+1)>T(x,y).$ Lemma 6.4 There exists a unique up to shift $n\\times m$ rectangle such that all squares labeled by positive numbers are located above the NW-SE diagonal.", "The corresponding parking function diagram coincides with the Anderson-type labeling up to a central symmetry.", "If $(b_1,\\ldots ,b_n)$ corresponds to the weight $a$ , then its corresponding $n$ -invariant subset has n-generators $b_i$ , and contains all fillings in squares to the right of labeled ones, including the periodic shift by $(m, -n)$ .", "There exists a unique line with slope $m/n$ which is tangent to the resulting infinite set of squares, and the tangency points define the $n\\times m$ rectangle.", "Now the statement follows from the definition of the map $\\operatorname{\\mathcal {A}}$ .", "Example 6.5 Consider the weight $(a_1,a_2,a_3)=q^{\\frac{2}{3}}(\\frac{t}{q^2},\\frac{q}{t},q)= (q^{0}, q^{\\frac{1}{3}}, q^{\\frac{5}{3}}) $ for $t=q^{4/3}$ .", "We have $(b_1,b_2,b_3)=(0,1,5)$ and $\\omega =[015]$ .", "The corresponding $(3,4)$ -invariant subset is $\\Delta _{\\omega }=\\omega ({\\mathbb {Z}}_{>0})=\\lbrace 0,1,3,4,5,\\ldots \\rbrace $ , and the parking function diagram is shown in Figure REF on the right.", "On the left side of Figure REF is a piece of $T_a$ , showing rows with $1\\le y \\le 4$ and columns with $-2\\le x \\le 4$ .", "Rewriting $a = q^{\\frac{2}{3}} t^{-2}(q^{-2} t^3, q^1 t^1, q^1 t^2)$ , we see we put the filling 1 in square $(-2,3)$ , 2 in square $(1,1)$ , and 3 in square $(1,2)$ .", "The periodicity conditions fill in the rest of the squares of $T_a$ .", "The unique NW-SE line has been drawn, and the corresponding rectangle it determines is rotated by 180 to obtain a Young diagram below the diagonal.", "To get to the weight $a$ from the trivial weight, we need to apply ${\\omega }_m^{-1} {\\omega }= [-2,2,6]^{-1} \\circ [0,1,5] = [-3,4,5]$ .", "Observe $[-3,4,5] = [-2(3)+3, 1(3) + 1, 1(3) +2]$ .", "From this we could also read off that the fillings 1,2,3 belong in squares $(-2,3)$ , $(1,1)$ , $(1,2)$ respectively.", "(We remind the reader of Figure REF , where the alcove labeled $[-3,4,5]$ in the left figure, matches the alcove labeled $[0,1,5]$ in right figure.)", "Note, if we had instead normalized in the more standard way so that $Y_1 Y_2 Y_3 =1$ and the fundamental alcove had weight $a^{\\prime } =(\\frac{1}{t}, 1, t)$ , then we would have had a shift by $2 = 3 \\frac{2}{3}$ yielding $(b_1^{\\prime },b_2^{\\prime },b_3^{\\prime })=(-2,-1,3)= (0-2,1-2,5-2)$ but still $\\omega =[015]$ and we would draw $T_{a^{\\prime }}$ as above.", "Figure: Periodic SYT on (x,y)(x,y)-plane (left); parking function diagram (right)" ], [ "Injectivity of $\\operatorname{\\mathcal {PS}}$ for the finite symmetric group", "In this Section we prove an analogue of Conjecture REF for the finite group $S_n.$ Definition 7.1 Let $S_n^m$ denote the intersection $S_n\\cap \\widetilde{S}_n^m.$ In other words, ${\\omega }\\in S_n^m$ if for all $1\\le x\\le n-m$ the inequality $\\omega (x+m)>\\omega (x)$ holds and $\\lbrace 1, \\ldots , n\\rbrace = \\lbrace {\\omega }(1),\\ldots , {\\omega }(n)\\rbrace $ .", "We call such permutations finite $m$ -stable.", "Proposition 7.2 The number of finite $m$ -stable permutations equals $\\sharp S_n^m=\\frac{n!", "}{\\prod _{i=1}^{m}n_i!", "},\\ \\mbox{\\rm where}\\ n_i={\\left\\lbrace \\begin{array}{ll}\\left\\lfloor \\frac{n-i}{m}\\right\\rfloor +1& \\mbox{\\rm if}\\ i\\le n,\\\\0& \\mbox{\\rm if}\\ i>n.\\\\\\end{array}\\right.", "}$ The set $X=\\lbrace 1,\\ldots ,n\\rbrace $ can be split into $m$ disjoint subsets $X_i:=\\lbrace x\\in X: x\\equiv i \\operatorname{mod}m\\rbrace $ of cardinality $n_i$ .", "A permutation $\\omega $ is finite $m$ -stable if and only if it increases on each $X_i$ , hence it is uniquely determined by an ordered partition $\\lbrace 1,\\ldots ,n\\rbrace =\\omega (X_1)\\sqcup \\ldots \\sqcup \\omega (X_m).$ Example 7.3 For $n=5, m=3$ , $X_1 = \\lbrace 1,4\\rbrace , X_2 = \\lbrace 2,5\\rbrace , X_3 = \\lbrace 3\\rbrace $ .", "Observe ${\\omega }\\in S_n^m$ iff ${{\\omega }^{-1}}$ occurs in the shuffle $14 \\,$$$ 25 $\\scriptscriptstyle \\cup {\\hspace{-2.22214pt}}\\cup $ 3$,which are precisely the $ m$-restricted permutations in $ S5$.$ Definition 7.4 Given a permutation $\\omega \\in S_n^m$ , let us define $\\operatorname{\\mathcal {PS}}_{\\omega }(\\alpha )$ as the number of inversions $(x,y)$ of $\\omega $ such that $x<y<x+m,\\ \\omega (x)>\\omega (y)=\\alpha $ (the height of such inversion is less than $m$ ).", "Define $\\operatorname{\\mathcal {PS}}_{\\omega }:=(\\operatorname{\\mathcal {PS}}_{\\omega }(1),\\ldots ,\\operatorname{\\mathcal {PS}}_{\\omega }(n)).$ In other words, this is just the restriction $\\operatorname{\\mathcal {PS}}\\mid _{S_n}$ .", "Hence by Theorem REF the integer sequences in the image of $\\operatorname{\\mathcal {PS}}$ are $m/n$ -parking functions.", "Observe that if ${\\omega }\\in S_n$ then $\\operatorname{\\mathcal {PS}}_{\\omega }(n) = 0$ .", "Theorem 7.5 The map $\\operatorname{\\mathcal {PS}}$ from the set $S_n^m$ to $\\operatorname{\\mathcal {PF}}_{m/n}$ is injective.", "We provide two proofs of this Theorem, as they are somewhat different and might both be useful for the future attempts to proof Conjecture REF in the affine case.", "Given a parking function $\\llparenthesis \\operatorname{\\mathcal {PS}}_{\\omega }(1)\\cdots \\operatorname{\\mathcal {PS}}_{\\omega }(n) \\rrparenthesis $ in the image, we need to reconstruct $\\omega $ or, equivalently, $\\omega ^{-1} \\in S_n$ .", "We will first reconstruct the number $x_1={\\omega }^{-1}(1),$ then $x_{2}={\\omega }^{-1}(2),$ and so on, all the way up to $x_n={\\omega }^{-1}(n).$ Note that ${{\\omega }^{-1}}= [x_1, x_2, \\ldots , x_n]$ and $\\operatorname{\\mathcal {PS}}_{\\omega }(i) = \\operatorname{\\mathcal {SP}}_{{{\\omega }^{-1}}}(i) =\\sharp \\lbrace j \\mid i < j \\le n, 0 < x_i - x_j < m \\rbrace $ .", "We have used that since ${\\omega }\\in S_n$ , for all $(i,j) \\in \\operatorname{{Inv}}({\\omega })$ , $1 \\le i < j \\le n$ .", "Also since ${\\omega }\\in S_n$ , for all $j \\ge 1$ we have ${\\omega }(j) \\ge 1$ .", "For the first step, note that $x_1< m+1$ , since otherwise $x_1-m$ and $x_1$ will form an inversion of ${\\omega }$ of height $m$ , as $x_1 -m \\ge 1$ so it occurs to the right of $x_1$ in ${{\\omega }^{-1}}$ .", "For every $1 \\le y<x_1$ , there is an inversion $(y,x_1)$ of height less than $m$ and there are no other inversions of the form $(-,x_1)$ , hence $x_1=\\operatorname{\\mathcal {PS}}_{\\omega }(1)+1.$ On the next step we recover $x_2.$ Note that for every $y<x_2,$ there is an inversion $(y,x_2),$ unless $y=x_1.$ It follows that $x_2$ is either equal to $x_1+m$ or $x_2< m+1.$ It is not hard to see, that all these possible values of $x_2$ correspond to different values of $\\operatorname{\\mathcal {PS}}_{{\\omega }}(2).$ Therefore, knowing $\\operatorname{\\mathcal {PS}}_{{\\omega }}(2),$ one can recover $x_2.$ Let us show that one can proceed in that manner inductively all the way to $x_n.$ Suppose that one has already reconstructed $x_i=\\omega ^{-1}(i)$ for all $i<k$ .", "Define the set $Y_{k-1}=\\lbrace x_1\\ldots ,x_{k-1}\\rbrace \\sqcup \\lbrace l\\in \\mathbb {Z}:l<1\\rbrace =\\lbrace y: 1\\le y \\le n, \\omega (y)<k\\rbrace \\sqcup \\lbrace l\\in \\mathbb {Z}:l<1\\rbrace .$ Let us use the notation $I(y):=(y-m,y]\\cap \\mathbb {Z}$ for any $y\\in \\mathbb {Z}_{\\le n}.$ Consider the function $\\varphi _k(y):=\\sharp \\left(I(y)\\setminus Y_{k-1}\\right)-1$ defined on the domain $y\\in \\mathbb {Z}_{\\le n}\\setminus Y_{k-1}$ .", "Given $\\operatorname{\\mathcal {PS}}_{\\omega }(k)=\\varphi _k(x_k),$ we need to reconstruct $x_k.$ Let us prove that the function $\\varphi _k(y)$ is non-decreasing.", "Indeed, let $y>y^{\\prime }$ and $y,y^{\\prime }\\in \\mathbb {Z}_{\\le n}\\setminus Y_{k-1}.$ Let $z\\in I(y)=(y-m,y]\\cap \\mathbb {Z}$ and $z^{\\prime }\\in I(y^{\\prime })=(y^{\\prime }-m,y^{\\prime }]\\cap \\mathbb {Z}$ be such that $z-z^{\\prime }\\equiv 0$ mod $m.$ It follows that if $z\\in Y_{k-1},$ then also $z^{\\prime }\\in Y_{k-1}.$ Otherwise, if $z^{\\prime }Y_{k-1}$ then $1 \\le z^{\\prime }$ and ${\\omega }(z^{\\prime }) \\ge k$ .", "Further $y > y^{\\prime }$ implies $z > z^{\\prime } \\ge 1$ and $z\\in Y_{k-1}$ gives ${\\omega }(z) < k \\le {\\omega }(z^{\\prime })$ .", "Therefore, $(z^{\\prime },z)$ is an inversion of ${\\omega }$ of height divisible by $m,$ which implies that ${\\omega }$ is not $m$ -stable.", "Contradiction.", "We conclude, that $\\varphi _k(y)=\\sharp \\left(I(y)\\setminus Y_{k-1}\\right)-1\\ge \\sharp \\left(I(y^{\\prime })\\setminus Y_{k-1}\\right)-1=\\varphi _k(y^{\\prime }).$ Finally, we remark that $x_k\\notin Y_{k-1}$ but $x_k-m\\in Y_{k-1},$ since otherwise $x_k-m\\ge 1$ and $\\omega (x_k-m)>k,$ and that produces an inversion of height $m$ .", "Therefore, one check there is a strict inequality $\\varphi _k(y)<\\varphi _k(x_k)$ for any $y<x_k$ with $y \\in \\mathbb {Z}_{\\le n}\\setminus Y_{k-1}.$ Thus, $x_k=\\min \\lbrace y\\in \\mathbb {Z}\\setminus Y_{k-1}\|\\varphi _k(y)=\\operatorname{\\mathcal {PS}}_{\\omega }(k)\\rbrace $ .", "In particular, this set is non-empty.", "We illustrate this proof on an example in Figure REF .", "Define the function $g(\\alpha ,i)$ by the following formula: $g(\\alpha ,i)=\\sharp \\lbrace j\\in (i-m,i]\\cap \\lbrace 1,\\dots ,n\\rbrace : \\omega (j)> \\alpha \\rbrace .$ By definition of $\\operatorname{\\mathcal {PS}}_{\\omega },$ one immediately gets $g(\\omega (i),i)=\\operatorname{\\mathcal {PS}}_{\\omega }(\\omega (i)).$ Lemma 7.6 The function $g(\\alpha ,i)$ is non-decreasing in $i$ for any fixed $\\alpha .$ Indeed, suppose that $g(\\alpha ,i)<g(\\alpha ,i-1).$ The interval $(i-m,i]$ is obtained from the interval $(i-1-m,i-1]$ by dropping $i-m$ and adding $i.$ Therefore, one should have $\\omega (i-m)>\\alpha $ and $\\omega (i)\\le \\alpha $ to get $g(\\alpha ,i)<g(\\alpha ,i-1).$ But that implies $\\omega (i-m)>\\omega (i),$ producing an inversion of height $m.$ Contradiction.", "We will need the following corollary: Corollary 7.7 For any $i\\in \\lbrace 1,\\dots ,n\\rbrace ,$ $\\omega (i)$ is the minimal integer $\\alpha $ , such that $\\alpha \\ne \\omega (j)$ for any $j<i,$ and $\\operatorname{\\mathcal {PS}}_{\\omega }(\\alpha )=\\sharp \\lbrace j\\in (i-m,i)\\cap \\lbrace 1,\\dots ,n\\rbrace : \\omega (j)>\\alpha \\rbrace .$ Fix $i$ .", "Let $\\alpha $ satisfy the above conditions.", "Notice such an $\\alpha $ must exist since ${\\omega }(i)$ satisfies these conditions as $\\operatorname{\\mathcal {PS}}_{\\omega }(\\omega (i))=g(\\omega (i),i)=\\sharp \\left\\lbrace j\\in (i-m,i): \\omega (j)>\\omega (i)\\right\\rbrace .$ By minimality, $\\alpha \\le {\\omega }(i)$ .", "If $\\alpha \\ne {\\omega }(i)$ then we must have $\\alpha < {\\omega }(i)$ , yielding $i \\in \\lbrace j\\in (i-m,i]\\cap \\lbrace 1,\\dots ,n\\rbrace : \\omega (j)> \\alpha \\rbrace $ .", "However $i \\left\\lbrace j\\in (i-m,i) \\cap \\lbrace 1,\\dots ,n\\rbrace :\\omega (j) > \\alpha \\right\\rbrace $ whose cardinality is $ \\operatorname{\\mathcal {PS}}_{\\omega }(\\alpha )$ by assumption.", "Hence $g(\\alpha ,i) = \\operatorname{\\mathcal {PS}}_{\\omega }(\\alpha )+1.$ If it were the case that $\\alpha = {\\omega }(k)$ for some $k>i$ , then since $g(\\alpha ,-)$ is non-decreasing, we get $\\operatorname{\\mathcal {PS}}_w(\\alpha ) = \\operatorname{\\mathcal {PS}}_w(w(k)) = g(w(k),k) =g(\\alpha ,k) \\ge g(\\alpha ,i) > \\operatorname{\\mathcal {PS}}_{\\omega }(\\alpha )$ .", "Contradiction.", "On the other hand, $\\alpha $ was chosen so $\\alpha \\ne \\omega (j)$ for any $j<i$ .", "Hence it must be that $\\alpha = {\\omega }(i).$ Now we can complete the proof of Theorem REF and reconstruct $\\omega $ starting from $\\omega (1),$ then $\\omega (2),$ and so on, using Corollary REF .", "Indeed, if we already reconstructed $\\omega (1),\\omega (2),\\dots ,\\omega (i-1),$ then we can compute $\\sharp \\lbrace j\\in (i-m,i)\\cap \\lbrace 1,\\dots ,n\\rbrace : \\omega (j)>\\alpha \\rbrace $ for all $\\alpha \\in \\lbrace 1,\\ldots ,n\\rbrace .$ Then $\\omega (i)$ is the smallest number $\\alpha \\in \\lbrace 1,\\ldots ,n\\rbrace $ such that $\\alpha \\ne \\omega (j)$ for $j<i,$ and $\\operatorname{\\mathcal {PS}}_{\\omega }(\\alpha )=\\sharp \\lbrace j\\in (i-m,i)\\cap \\lbrace 1,\\dots ,n\\rbrace : \\omega (j)>\\alpha \\rbrace .$ We illustrate this proof on example on Figure REF .", "Figure: Suppose that n=7,n=7, m=3,m=3, and error ω =⦇0010210⦈\\operatorname{\\mathcal {PS}}_{\\omega }=\\llparenthesis 0010210 \\rrparenthesis .", "Let us reconstructω -1 {{\\omega }^{-1}} using the first proof of Theorem .", "We record on every step the numbers that we have already reconstructed and the values of the function ϕ k \\varphi _k for all other numbers.Figure: As in Figure , n=7,n=7, m=3,m=3, and error ω =⦇0010210⦈\\operatorname{\\mathcal {PS}}_{\\omega }=\\llparenthesis 0010210 \\rrparenthesis .", "This time we reconstruct ω{\\omega } using the second proof of Theorem .", "We record on every step the numbers that we have already reconstructed and the difference error ω (α)-♯{j∈{k-m+1,...,k-1}:ω(j)>α}\\operatorname{\\mathcal {PS}}_{\\omega }(\\alpha )-\\sharp \\lbrace j\\in \\lbrace k-m+1,\\ldots , k-1\\rbrace : \\omega (j)>\\alpha \\rbrace for all α∈{1,...,n}∖{ω(1),ω(2),⋯,ω(k-1)},\\alpha \\in \\lbrace 1,\\ldots ,n\\rbrace \\setminus \\lbrace \\omega (1),\\omega (2),\\dots ,\\omega (k-1)\\rbrace , so that on each step we choosethe position of the leftmost 0 in the second column.We do not know how to describe the image $\\operatorname{\\mathcal {PS}}(S_n^m)$ for general $m$ .", "As an example, let us consider the case $m=2$ for which we do have a complete description.", "Let us recall that $S_{n}^{2}$ is the set of finite permutations $\\omega $ of $n$ elements with no inversions of height 2, that is, $\\omega (i+2)>\\omega (i)$ for all $x$ .", "We define the map $\\operatorname{inv}^{(2)}$ from the set $S_{n}^{2}$ to the set of sequences of 0's and 1's as $\\operatorname{inv}^{(2)}_{\\omega }(\\alpha ):=\\chi \\left(\\omega (\\omega ^{-1}(\\alpha )-1)>\\alpha \\right)= {\\left\\lbrace \\begin{array}{ll}1 & \\text{ if } \\omega (\\omega ^{-1}(\\alpha )-1)>\\alpha \\\\0 & \\text{ else.}\\end{array}\\right.", "}$ Lemma 7.8 The image of $\\operatorname{inv}^{2}$ consists of all $n$ -element sequences $f$ of 0's and 1's, such that for every $\\alpha \\in \\lbrace 1,\\ldots ,n\\rbrace $ at least half of the subsequence $(f_\\alpha ,\\ldots f_n)$ are 0's.", "The image of $\\operatorname{inv}^{2}$ agrees with that of $\\operatorname{\\mathcal {PS}}\|_{{S}_{n}^{2}}$ .", "Let $\\omega $ be a permutation in $S_n^{2}$ and let $f=\\operatorname{inv}^{(2)}(\\omega )$ .", "For every $\\alpha $ such that $f_\\alpha =1$ one can find $\\beta =\\omega (\\omega ^{-1}(\\alpha )-1)>\\alpha $ such that $f_\\beta =0$ (otherwise $\\omega $ would have an inversion of height 2).", "In other words, if we consider ${\\omega }^{-1} = [x_1,\\ldots , x_n]$ , $f(\\alpha ) = 1$ iff $x_\\alpha = i$ and $i-1$ occurs to the right, i.e.", "$i-1 = x_\\beta $ with $\\beta > \\alpha $ .", "Which occurs iff $\\operatorname{\\mathcal {PS}}_{\\omega }(\\alpha )=1$ .", "And in this case $i-2$ cannot be to the right of $i-1$ as that would place it to the right of $i$ , i.e.", "$f(\\beta ) = \\operatorname{\\mathcal {PS}}_{\\omega }(\\beta )=0$ .", "Note that the correspondence $\\alpha \\mapsto \\beta =\\omega (\\omega ^{-1}(\\alpha )-1)$ from 1's to 0's in the sequence $f$ is injective and increasing.", "Therefore, for every $\\alpha \\in \\lbrace 1,\\ldots ,n\\rbrace $ at least half of the subsequence $(f_\\alpha ,\\ldots f_n)$ are 0's.", "Since we know that $\\operatorname{inv}^{(2)}$ is injective, the lemma now follows from the comparison of the cardinalities of the two sets.", "The sequences appearing in Lemma REF have a clear combinatorial meaning.", "Let us read the sequence $s$ backwards and replace 0's with a vector $(1,1)$ and 1's with a vector $(1,-1)$ .", "We get a lattice path in $\\mathbb {Z}^2$ which never goes below the horizontal axis.", "Such a path may be called a Dyck path with open right end, and Lemma REF establishes a bijection between the set of such paths of length $n$ and the set of finite 2-stable permutations." ], [ "Algorithm to construct $\\operatorname{\\mathcal {SP}}^{-1}$ in the affine case", "Here we present a conjectural algorithm that inverts $\\operatorname{\\mathcal {SP}}$ .", "While we have not yet shown the algorithm terminates, which in this case means it eventually becomes $n$ -periodic, we have checked it on several examples.", "Given $f \\in \\operatorname{\\mathcal {PF}}_{m/n}$ , extend $f$ to ${\\mathbb {N}}$ by $f(i+tn) = f(i)$ .", "Construct an injective function $U: {\\mathbb {N}}\\rightarrow {\\mathbb {N}}$ as follows.", "Informally, we will think of $U$ as the bottom row in the following table.", "$\\begin{array}{r\|ccc}i & 1&2& \\cdots \\\\\\hline f(i) & f(1) & f(2) & \\cdots \\\\\\hline U(i)& U(1) &U(2) & \\cdots \\end{array}$ Since $U$ is manifestly injective, it will make sense to talk about $U^{-1}$ .", "We will insert the numbers $\\alpha \\in {\\mathbb {N}}$ into the table as follows.", "Place $\\alpha = 1$ under the leftmost 0.", "In other words, let $i = \\min \\lbrace j \\in {\\mathbb {N}}\| f(j) = 0\\rbrace $ and then set $U(i) = 1$ .", "As there always exists some $1 \\le j \\le n$ such that $f(j) = 0$ , this is always possible.", "Assume $\\lbrace 1, 2, \\ldots , \\alpha - 1 \\rbrace $ have already been placed.", "Place $\\alpha $ in the leftmost empty position $i$ (i.e.", "$U(i) = \\alpha $ , with $i \\lbrace U^{-1}(\\beta ) \| 1 \\le \\beta < \\alpha \\rbrace $ for $i$ minimal) such that these two conditions hold.", "($\\mathtt {I}$ ) $\\alpha $ is to the right of $\\alpha - tm$ for $1 \\le t < \\alpha /m$ , $t \\in {\\mathbb {N}}$ .", "More precisely, $i > U^{-1}(\\alpha - tm)$ .", "($\\mathtt {II}$ ) If $U(i) =\\alpha $ , then $f(i) = \\sharp \\lbrace \\beta \|\\beta \\in (\\alpha -m, \\alpha ), U^{-1}(\\beta ) > i \\rbrace $ .", "In other words, we build $U$ so that $f(i) =\\sharp \\lbrace j \| j > i, 0 < U(i)-U(j) < m \\rbrace $ counts the number of $m$ -restricted inversions.", "Note that placing $\\alpha $ is always possible, since a valid (non-minimal) position for $\\alpha $ is under a 0 of $f$ such that it and all spots to the right of it are as yet unoccupied.", "Conjecture 7.9 For the $U$ constructed above, $\\exists N$ such that for all $i \\ge N$ , $t \\in {\\mathbb {N}}$ $U(i+tn) = U(i) + tn$ , so in particular $U(N+j)$ for $1 \\le j \\le n$ have all been assigned values Given $U$ constructed from $f \\in \\operatorname{\\mathcal {PF}}_{m/n}$ as in the algorithm and satisfying the conditions of the conjecture, we construct ${\\omega }\\in {}^m\\widetilde{S}_n$ as follows: Pick $t$ so $1+tn \\ge N$ .", "By the periodicity of $U$ and that $U$ has no “gaps\" after $N$ , $\\lbrace U(i+tn) \\bmod n \| 1\\le i\\le n \\rbrace = \\lbrace 1, 2, \\ldots , n\\rbrace $ .", "Hence $b := \\sum _{i=1}^n U(i+tn) \\equiv \\frac{n(n+1)}{2} \\bmod n$ .", "Let $k$ be such that $b - \\frac{n(n+1)}{2} = kn$ .", "Now set $ {\\omega }(i) = U(i+tn) -k.$ This forces $\\sum _{i=1}^n {\\omega }(i) = \\frac{n(n+1)}{2}$ , and so we see ${\\omega }\\in \\widetilde{S}_n$ .", "By construction, ($\\mathtt {I}$ ) and ($\\mathtt {II}$ ) imply $w \\in {}^m\\widetilde{S}_n$ and $\\operatorname{\\mathcal {SP}}_{\\omega }= f$ .", "We illustrate the algorithm to construct $U$ and ${\\omega }$ on the following example.", "Example 7.10 Let $n=5, m=3$ .", "Let $f = \\llparenthesis 11002 \\rrparenthesis \\in \\operatorname{\\mathcal {PF}}_{3/5}.$ Figure: Algorithm to construct UU fromf=⦇11002⦈∈error 3/5 .f = \\llparenthesis 11002 \\rrparenthesis \\in \\operatorname{\\mathcal {PF}}_{3/5}.Refer to Figure REF for a demonstration of how $U$ is constructed.", "Note that $U(7) \\ne 8$ since that would place 8 before 5, violating being 3-restricted.", "In the above we can in fact take $N=5$ .", "Observe $\\lbrace U(6), U(7), U(8), U(9), U(10) \\rbrace = \\lbrace 6,9,5,8, 12 \\rbrace $ yielding $b = 40$ and $k = 5$ .", "Hence we set ${\\omega }= [1,4,0,3,7]$ .", "Now one can easily verify ${\\omega }\\in {}^{3}\\widetilde{S}_{5}$ and $\\operatorname{\\mathcal {SP}}_{\\omega }= \\llparenthesis 11002 \\rrparenthesis $ .", "In practice, we have found $U$ to be surjective as well; in other words there are no “gaps\" even before $N$ .", "Further, when $f = \\operatorname{\\mathcal {SP}}_u$ for some finite permutation $u \\in S_n$ , we can take $N=1$ ." ], [ "Springer fibers for the symmetric group", "Let $V$ be a finite-dimensional vector space and let $N$ be a nilpotent transformation of $V$ .", "Let $\\operatorname{Fl}(V)$ denote the space of complete flags in $V$ .", "A classical object in the representation theory is the Springer fiber ([28], [26]) defined as $X_N:=\\lbrace {\\bf F}=\\lbrace V=V_1\\supset V_2\\supset \\ldots \\supset V_{n}\\rbrace \\in \\operatorname{Fl}(V): N(V_i)\\subset V_i\\rbrace .$ It is known that $X_N$ admits an affine paving with combinatorics completely determined by the conjugacy class of $N$ (see e.g.", "[26] and references therein).", "We will be interested in a particular case of this construction.", "Let us fix a basis $(e_1,\\ldots ,e_n)$ in the space $V$ , consider the operator of shift by $m$ : $N(e_i):={\\left\\lbrace \\begin{array}{ll}e_{i+m},& i+m\\le n\\\\0,& \\text{otherwise}\\\\\\end{array}\\right.", "}$ The following theorem describes the structure of the affine cells in the variety $X_{N}$ .", "Theorem 8.1 The variety $X_N$ admits an affine paving, where the cells $\\Sigma _{\\omega }$ are parametrized by the finite $m$ -restricted permutations $\\omega \\in S_n^m$ .", "The dimension of $\\Sigma _{\\omega }$ is given by the number of inversions in $\\omega ^{-1}$ of height less than $m$ .", "The cells are essentially given by the intersections of Schubert cells in $\\operatorname{Fl}(V)$ with the subvariety $X_N$ .", "For the sake of completeness, let us recall their construction.", "Given a permutation $\\omega \\in S_n$ , we can define a stratum $\\Sigma _{\\omega }$ in $\\operatorname{Fl}(V)$ consisting of the following flags: ${\\bf F}=\\lbrace V_1\\supset V_2\\supset \\ldots \\supset V_n\\rbrace ,\\ V_i=\\operatorname{span}\\lbrace v^{\\omega (i)},\\ldots ,v^{\\omega (n)} \\rbrace ,$ where $v^\\alpha =e_\\alpha +\\sum _{\\beta >\\alpha }\\lambda ^\\alpha _\\beta e_\\beta .$ Note that the position of $v^\\alpha $ in the basis equals $\\omega ^{-1}(\\alpha )$ .", "After a triangular change of variables, we can assume that $\\lambda ^\\alpha _\\beta =0$ for $\\beta >\\alpha $ with $\\omega ^{-1}(\\beta )>\\omega ^{-1}(\\alpha ).$ Therefore one can write $v^\\alpha =e_\\alpha +\\sum _{\\beta >\\alpha ,\\omega ^{-1}(\\beta )<\\omega ^{-1}(\\alpha )}\\lambda ^\\alpha _\\beta e_\\beta =e_\\alpha +\\sum _{(\\alpha ,\\beta )\\in \\operatorname{{Inv}}({\\omega }^{-1})}\\lambda ^\\alpha _\\beta e_\\beta .$ The parameters $\\lambda ^\\alpha _\\beta $ are uniquely defined by the flag ${\\bf F}$ .", "They serve as coordinates on the affine space $\\Sigma _{\\omega }$ , whose dimension is equal to the length of $\\omega $ , i.e.", "$=\\sharp \\operatorname{{Inv}}({\\omega }) = \\sharp \\operatorname{{Inv}}({{\\omega }^{-1}})$ .", "Let us study the intersection $\\Sigma _{\\omega }^N:=\\Sigma _{\\omega }\\cap X_N.$ Since $Nv^\\alpha $ starts with $e_{\\alpha +m}$ , the vector $v^{\\alpha +m}$ should go after $v^\\alpha $ in the basis, so one needs $\\omega ^{-1}(\\alpha +m)>\\omega ^{-1}(\\alpha )$ .", "Therefore $\\Sigma ^{N}_{\\omega }$ is non-empty if and only if $\\omega ^{-1}$ is $m$ -stable.", "A flag ${\\bf F}$ is $N$ -invariant, if $N(v^\\alpha )$ belongs to $\\operatorname{span}\\lbrace v^\\beta : \\omega ^{-1}(\\beta )>\\omega ^{-1}(\\alpha ) \\rbrace $ for all $\\alpha $ .", "If $\\beta >\\alpha +m\\ \\text{and}\\ \\omega ^{-1}(\\beta )>\\omega ^{-1}(\\alpha ),$ then the coefficient in $v^{\\alpha +m}-N(v^\\alpha )$ at $e_\\beta $ can be eliminated by subtracting an appropriate multiple of $v^\\beta $ .", "Once all these coefficients are eliminated, the remaining coefficients in $v^{\\alpha +m}-N(v^\\alpha ),$ labeled by $\\beta >\\alpha +m$ such that $\\omega ^{-1}(\\beta )<\\omega ^{-1}(\\alpha )$ will vanish automatically.", "Therefore $\\Sigma ^N_{\\omega }$ is cut out in $\\Sigma _{\\omega }$ by the equations: $\\lambda ^{\\alpha +m}_\\beta =\\lambda ^\\alpha _{\\beta -m}+\\phi (\\lambda )\\ \\text{if}\\ \\beta >\\alpha +m,\\omega ^{-1}(\\beta )<\\omega ^{-1}(\\alpha ),$ where $\\phi (\\lambda )$ are certain explicit polynomials in $\\lambda ^\\mu _\\nu $ with $\\nu -\\mu <\\beta -\\alpha -m,$ with no linear terms.", "It is clear that such equations are labeled by the inversions $(\\alpha ,\\beta )$ in $\\omega ^{-1}$ of height bigger than $m$ .", "Note also that the linear parts of these equations are linearly independent.", "Therefore the number of free parameters on $\\Sigma ^N_{\\omega }$ equals to the number of inversions of ${{\\omega }^{-1}}$ of height less than $m$ .", "Example 8.2 Consider a 2-stable permutation $\\omega =[2, 1, 4, 3]=\\omega ^{-1}$ .", "The basis (REF ) has the form: $v^{\\omega (1)}=e_2,\\ v^{\\omega (2)}=e_1+\\lambda ^1_2 e_2,\\ v^{\\omega (3)}=e_4,\\ v^{\\omega (4)}=e_3+\\lambda ^3_4 e_4.$ There are two free parameters, so $\\dim \\Sigma ^{N}_{\\omega }=2$ .", "Note that although $\\lambda ^3_4\\ne \\lambda ^1_2$ , $N(v^{\\omega (2)})=e_3+\\lambda ^1_2 e_4\\in \\operatorname{span}\\lbrace v^{\\omega (3)},v^{\\omega (4)} \\rbrace .$ Example 8.3 Consider a 3-stable permutation $\\omega ^{-1}=[1, 5, 3, 2, 6, 4, 7],$ so $\\omega =[1, 4, 3, 6, 2, 5, 7]$ is 3-restricted.", "The basis (REF ) has a form: $v^{\\omega (1)}=e_1,\\ v^{\\omega (2)}=e_4,\\ v^{\\omega (3)}=e_{3}+\\lambda ^3_4 e_4,\\ v^{\\omega (4)}=e_6,\\ v^{\\omega (5)}=e_2+\\lambda ^2_3 e_3+\\lambda ^2_4 e_4+\\lambda ^2_6 e_6,$ $v^{\\omega (6)}=e_5+\\lambda ^5_6 e_6,\\ v^{\\omega (7)}=e_7.$ Since $N(v^{{\\omega }(5)})-v^{{\\omega }(6)}=(\\lambda ^2_3-\\lambda ^5_6)e_6+\\lambda ^2_4 e_7\\in V_5,$ and $V_5=\\operatorname{span}\\lbrace v^{{\\omega }(5)},v^{{\\omega }(6)},v^{{\\omega }(7)} \\rbrace =\\operatorname{span}\\lbrace e_2+\\lambda ^2_3 e_3+\\lambda ^2_4 e_4+\\lambda ^2_6 e_6,e_5+\\lambda ^5_6 e_6,e_7 \\rbrace ,$ the coefficient of $e_6$ must vanish and so we get the relation $\\lambda ^2_3=\\lambda ^5_6.$ Therefore $\\dim \\Sigma _{\\omega }^N=4.$" ], [ "Springer fibers for the affine symmetric group", "We recall the basic definitions of the type $A$ affine Springer fibres, and refer the reader e. g. to [13], [21], [24] for more details.", "Let us choose an indeterminate $\\varepsilon $ and consider the field $K=(\\varepsilon ))$ of Laurent power series and the ring $\\mathcal {O}=[\\varepsilon ]]$ of power series in $\\varepsilon $ .", "Let $V=n((\\varepsilon ))$ be a $K$ -vector space of dimension $n$ .", "Definition 8.4 The affine Grassmannian $\\mathcal {G}_n$ for the group $GL_n$ is the moduli space of $\\mathcal {O}$ -submodules $M\\subset V$ such that the following three conditions are satisfied: (a) $M$ is $\\mathcal {O}$ -invariant.", "(b) There exists $N$ such that $\\varepsilon ^{-N}n[[\\varepsilon ]]\\supset M\\supset \\varepsilon ^{N}n[[\\varepsilon ]].$ (c) Let $N$ be an integer satisfying the above condition.", "Then the following normalization condition is satisfied: $\\dim _{ \\varepsilon ^{-N}n[[\\varepsilon ]]/M=\\dim _{ M/ \\varepsilon ^{N}n[[\\varepsilon ]].", "}}The{\\em affine complete flag variety}$ Fn$ for the group $ GLn$ is the moduli space of collections $ {M0...Mn}$, where each $ Mi$ satisfies (a) and (b),$ Mi/Mi+1=1,$\\ $ Mn=M0,$ and $ M0Gn,$ i.e.", "$ M0$ also satisfies the normalization condition (c).$ Definition 8.5 Let $T$ be an endomorphism of $V$ .", "It is called nil-elliptic if $\\lim _{k\\rightarrow \\infty }T^{k}=0$ and the characteristic polynomial of $T$ is irreducible over $K$ .", "Given a nil-elliptic operator $T$ , one can extend its action to $\\mathcal {G}_n$ and to $\\mathcal {F}_n$ and define the affine Springer fibers as the corresponding fixed point sets.", "Remark 8.6 The condition $\\lim _{k\\rightarrow \\infty }T^{k}=0$ means that for any $N\\in {\\mathbb {N}}$ there exists $k\\in {\\mathbb {N}}$ such that $T^k(n[[\\varepsilon ]])\\subset \\varepsilon ^Nn[[\\varepsilon ]].$ In [24] Lusztig and Smelt studied the structure of the affine Springer fibers for a particular choice of $T$ .", "Given a $-basis $ {e1,..., en}$ in $ n$,one can consider it as a $ K$-basis of$ V=n(())$.", "Consider the operator $ N$ definedby the equations $ N(ei)=ei+1, N(en)=e1.$ The following theorem is the main result of \\cite {LS91}.$ Theorem 8.7 ([24]) Consider the nil-elliptic operator $T:=N^m$ , where $m$ is coprime to $n$ .", "Then the corresponding affine Springer fiber $\\mathcal {F}_{m/n}\\subset \\mathcal {F}_n$ admits an affine paving by $m^{n-1}$ affine cells.", "It turns out that the affine paving of this affine Springer fiber is tightly related to the combinatorics of the simplex $D_{n}^{m}$ .", "This was implicitly stated in [17], [20], [24], but we would like to make this correspondence precise and explicit.", "Theorem 8.8 There is a natural bijection between the affine cells in $\\mathcal {F}_{m/n}$ and the affine permutations in ${}^m\\widetilde{S}_n$ .", "The dimension of the cell $\\Sigma _{\\omega }$ labeled by the affine permutation $\\omega $ is equal to $\\sum _{i=1}^{n}\\operatorname{\\mathcal {SP}}_{\\omega }(i)$ .", "Let us introduce an auxiliary variable $z=\\varepsilon ^{1/n}$ .", "We can identify the vector space $V=n((\\varepsilon ))$ with the space $\\operatorname{span}_1,z,\\ldots ,z^{n-1} \\rbrace ((z^n))\\simeq (z))$ of Laurent power series in $z$ by sending the basis $\\lbrace e_1,\\ldots ,e_n\\rbrace $ to $\\lbrace 1,z,\\ldots ,z^{n-1}\\rbrace .$ Note that under this identification, $n[[\\varepsilon ]]$ is mapped to $[z]]$ .", "By construction, $N$ coincides with the multiplication operator by $z$ and hence $T=N^m$ coincides with the multiplication operator by $z^m$ .", "Therefore $\\mathcal {F}_{m/n}$ consists of flags $\\lbrace M_0\\supset \\ldots \\supset M_n\\rbrace $ of $[z^n,z^m]]$ -modules, such that $\\dim _{ M_i/M_{i+1}=1, M_n=z^nM_0 and M_0\\in \\mathcal {G}_n.", "Let us extend the notation M_i to arbitrary i\\in {\\mathbb {Z}} by setting M_{i+kn}:=z^{kn}M_i.", "As a result, we get an infinite flag \\lbrace \\ldots \\supset M_0\\supset \\ldots \\supset M_n\\supset \\ldots \\rbrace of [z^n,z^m]]-modules satisfying the same conditions as above and M_{i+n}=z^nM_i.", "}For $ f(z)V=(z)),$ let $ Ord(f)$ denote the order of $ f(z)$ in $ z,$ i.e.", "the smallest degree of $ z,$ such that the corresponding coefficient in $ f(z)$ does not vanish.", "Given a subset $ MV$, define $$\\operatorname{Ord}(M)=\\lbrace \\operatorname{Ord}(f)\\ :\\ f\\in M, f\\ne 0\\rbrace .$$ We will need the following lemma, whose proof is standard and left to the reader:$ Lemma 8.9 Let $L\\subset M\\subset (z))$ be two $[z^m,z^n]]$ -submodules in $z^{-N}[z]]$ for some large $N\\in {\\mathbb {N}}.$ Then $\\sharp \\left(\\operatorname{Ord}(M)\\setminus \\operatorname{Ord}(L)\\right)=\\dim _M̏/L.$ Given a flag $\\lbrace \\ldots \\supset M_0\\supset \\ldots \\supset M_n\\supset \\ldots \\rbrace $ as above, set $ \\operatorname{Ord}(M_i)\\setminus \\operatorname{Ord}(M_{i+1}) = \\lbrace {\\omega }(i)\\rbrace .$ Note that one automatically gets $\\operatorname{Ord}(M_i)=\\lbrace {\\omega }(i),{\\omega }(i+1),\\ldots \\rbrace ,$ because $\\bigcap \\limits _i\\operatorname{Ord}(M_i)=\\emptyset .$ Recall the notation $\\operatorname{{Inv}}({\\omega }) : =\\lbrace (i,j) \\in {\\mathbb {N}}\\times {\\mathbb {N}}\\mid 1\\le i\\le n,\\ i < j, {\\omega }(i) > {\\omega }(j) \\rbrace $ and $\\operatorname{{\\overline{Inv}}}({\\omega }) : =\\lbrace (i,j) \\in {\\mathbb {N}}\\times {\\mathbb {N}}\\mid i < j, {\\omega }(i) > {\\omega }(j) \\rbrace $ for the inversion sets of ${\\omega }.$ For each $i$ there exists a unique $f_i(z)\\in M_i$ such that $f_i=z^{{\\omega }(i)}+\\sum _{(j,i)\\in \\operatorname{{\\overline{Inv}}}({\\omega })}\\lambda ^{{\\omega }(i)}_{{\\omega }(j)} z^{{\\omega }(j)}.$ Indeed, take any function $f\\in M_i$ such that $\\operatorname{Ord}(f)=\\lbrace {\\omega }(i)\\rbrace $ and use functions from $M_{i+1}$ to eliminate coefficients at $z^{{\\omega }(j)}$ for $j>i$ and ${\\omega }(j)>{\\omega }(i).$ The resulting function is unique up to a scalar, because otherwise $\\dim _M̏_i/M_{i+1}$ would be at least $2.$ It follows that $f_{i+n}=z^nf_i.$ We claim that ${\\omega }$ is an affine permutation and, moreover, $\\omega \\in {}^m\\widetilde{S}_n$ .", "Indeed, since $f_{i+n}=z^nf_i$ we get ${\\omega }(i+n)={\\omega }(i)+n,$ and since $z^mf_i\\in M_i$ we get that ${\\omega }(i)+m\\in \\lbrace {\\omega }(i),{\\omega }(i+1),\\ldots \\rbrace ,$ and, therefore, for any $j<i,$ ${\\omega }(j)-{\\omega }(i)\\ne m.$ Finally, we need to check the normalization condition $\\sum \\limits _{i=1}^n {\\omega }(i)=\\frac{n(n+1)}{2},$ which follows form the normalization condition on $M_0\\in \\mathcal {G}_n.$ Indeed, it is not hard to see that for all $L\\in \\mathcal {G}_n$ the sum of elements of $\\operatorname{Ord}(L)\\setminus \\operatorname{Ord}(t^nL)$ should be the same.", "In particular, for $L=[z]]$ we have $\\operatorname{Ord}([z]])\\setminus \\operatorname{Ord}(z^n[z]])=\\lbrace 0,1,\\ldots ,n-1\\rbrace ,$ and their sum is $\\frac{n(n-1)}{2}.$ Therefore, since $\\operatorname{Ord}(M_0)\\setminus \\operatorname{Ord}(z^nM_0)=\\lbrace {\\omega }(0),\\ldots ,{\\omega }(n-1)\\rbrace ,$ we get $\\sum \\limits _{i-0}^{n-1} {\\omega }(i)=\\frac{n(n-1)}{2},$ which equivalent to the required condition.", "The above gives us a map $\\nu :\\mathcal {F}_{m/n}\\rightarrow {}^m\\widetilde{S}_n.$ Let us prove that the fibers $\\Sigma _{{\\omega }}:=\\nu ^{-1}({\\omega })$ of this map are affine cells and compute their dimensions.", "This is very similar to the computation in the finite case (see Theorem REF ).", "Let us set $f^{{\\omega }(i)}:=f_i.$ The expansions (REF ) can be rewritten as $f^\\alpha =z^\\alpha +\\sum _{(\\alpha ,\\beta )\\in \\operatorname{{\\overline{Inv}}}({\\omega }^{-1})}\\lambda ^{\\alpha }_{\\beta } z^\\beta .$ Since $f_{i+n}=z^nf_i,$ one gets $\\lambda ^{\\alpha +n}_{\\beta +n}=\\lambda ^\\alpha _\\beta .$ Let us also extend the notation by setting $\\lambda ^{\\alpha }_{\\beta }=0$ whenever $(\\alpha ,\\beta )\\notin \\operatorname{{\\overline{Inv}}}({\\omega }^{-1})$ , so that one can write $f^\\alpha =z^\\alpha +\\sum _{\\beta >\\alpha } \\lambda ^\\alpha _{\\beta }z^{\\beta }.$ Let us say that the coefficient $\\lambda ^{\\alpha }_{\\beta }$ is of height $\\beta -\\alpha .$ As before, let $\\alpha ={\\omega }(i).$ The condition $z^m f^\\alpha \\in M_i$ implies the following relations on the coefficients.", "The function $z^mf^\\alpha -f^{\\alpha +m}=\\sum _{\\beta >\\alpha } \\lambda ^\\alpha _\\beta z^{\\beta +m}-\\sum _{\\beta >\\alpha +m} \\lambda ^{\\alpha +m}_{\\beta }z^\\beta =\\sum _{\\beta >\\alpha +m}(\\lambda ^\\alpha _{\\beta -m}-\\lambda ^{\\alpha +m}_\\beta )z^\\beta $ should belong to $M_i$ .", "Take $\\beta >\\alpha +m$ and let $j={\\omega }^{-1}(\\beta ).$ If $j>i,$ then the term of degree $\\beta $ can be eliminated by subtracting $f^\\beta =f_j\\in M_i$ with an appropriate coefficient.", "If $j<i,$ then ${{\\omega }^{-1}}(\\beta -m) < {{\\omega }^{-1}}(\\beta ) < {{\\omega }^{-1}}(\\alpha ) < {{\\omega }^{-1}}(\\alpha +m)$ .", "Hence $(\\alpha ,\\beta -m)\\in \\operatorname{{\\overline{Inv}}}({\\omega }^{-1})$ and $(\\alpha +m,\\beta )\\in \\operatorname{{\\overline{Inv}}}({\\omega }^{-1}),$ so the coefficients $\\lambda ^\\alpha _{\\beta -m}$ and $\\lambda ^{\\alpha +m}_\\beta $ are both not forced to be zero, i.e.", "they are parameters on $\\Sigma _{\\omega }$ .", "The term $(\\lambda ^\\alpha _{\\beta -m}-\\lambda ^{\\alpha +m}_\\beta )z^\\beta $ has to vanish automatically after we eliminated all lower order terms.", "As we eliminate terms of degree $\\gamma $ such that $\\alpha +m<\\gamma <\\beta ,$ the coefficient at $z^\\beta $ changes, but the added terms can only depend on coefficients of smaller height.", "More precisely, all additional terms are non-linear, and the total height of each monomial is always $\\beta -\\alpha -m.$ In the end, we get that $\\lambda ^\\alpha _{\\beta -m}-\\lambda ^{\\alpha +m}_\\beta $ should be equal to zero modulo the coefficients of smaller height.", "This means that for each $(\\alpha ,\\beta )\\in \\operatorname{{\\overline{Inv}}}({\\omega }^{-1})$ of height $\\beta -\\alpha >m$ there is an equation that allows one to express $\\lambda ^\\alpha _{\\beta -m}$ in terms of $\\lambda ^{\\alpha +m}_\\beta $ and higher order terms in coefficients with lower height.", "A priori, the linear parts of these equations can be dependent if for all $0\\le q\\le n$ one has $(\\alpha +qm,\\beta +qm)\\in \\operatorname{{\\overline{Inv}}}({\\omega }^{-1}).$ However, since $m$ and $n$ are relatively prime, this would mean that $\\omega ^{-1}(\\gamma )>\\omega ^{-1}(\\gamma +\\beta -\\alpha )$ for all $\\gamma \\in {\\mathbb {Z}},$ which is impossible.", "Therefore, one can resolve the relations on the coefficients with respect to $\\lambda ^\\alpha _\\beta $ such that $(\\alpha ,\\beta )\\in \\operatorname{{\\overline{Inv}}}({\\omega }^{-1})$ and $(\\alpha ,\\beta +m)\\in \\operatorname{{\\overline{Inv}}}({\\omega }^{-1}).$ So, the coordinates on $\\Sigma _{{\\omega }}$ correspond to the inversions $(\\alpha ,\\beta )\\in \\operatorname{{Inv}}({\\omega }^{-1}),$ such that $(\\alpha ,\\beta +m)\\notin \\operatorname{{Inv}}({\\omega }^{-1})$ .", "Since $\\lambda ^\\alpha _\\beta =\\lambda ^{\\alpha +n}_{\\beta +n},$ one should count inversions in $\\operatorname{{Inv}}({\\omega }^{-1})$ only.", "It is not hard to see that such inversions are in bijection with inversions of height less than $m.$ Indeed, the required map is $(\\alpha ,\\beta )\\mapsto (\\alpha ,\\beta -km),$ where $k$ is the maximal integer such that $\\beta -km>\\alpha .$ Alternatively, one can also notice that the relations are in bijection with inversions of height greater than $m.$ Indeed, the relation $\\lambda ^\\alpha _\\beta \\equiv \\lambda ^{\\alpha +m}_{\\beta +m}$ (modulo lower height terms) corresponds to the inversion $(\\alpha ,\\beta +m)\\in \\operatorname{{\\overline{Inv}}}({\\omega }^{-1})$ of height greater than $m.$ Therefore, the dimension of $\\Sigma _{{\\omega }}$ is the total number of inversions minus the number of inversion of height greater than $m.$ Since there are no inversions of height $m,$ the dimension is equal to the number of inversions of height less than $m.$ Since $\\sum _{i=1}^{n}\\operatorname{\\mathcal {PS}}_{\\omega ^{-1}}(i)$ is exactly the total number of inversions of height less than $m,$ we conclude that $\\dim (\\Sigma _{\\omega })=\\sum _{i=1}^{n}\\operatorname{\\mathcal {SP}}_{\\omega }(i)=\\sum _{i=1}^{n}\\operatorname{\\mathcal {PS}}_{\\omega ^{-1}}(i)=\\frac{(m-1)(n-1)}{2}-\\operatorname{dinv}({\\omega }^{-1}).$ For a more abstract proof see e.g.", "[13] and [20].", "Remark 8.10 Similar reasoning shows that the Grassmannian version of the affine Springer fiber $\\mathcal {G}_{m/n}\\subset \\mathcal {G}_n$ parametrizes appropriately normalized $[z^n,z^m]]$ -submodules in $(z)).$ This affine Springer fiber was studied e.g.", "in [14], [15] under the name of Jacobi factor of the plane curve singularity $\\lbrace x^m=y^n\\rbrace $ .", "The cells in it are parametrized by the subsets in $\\mathbb {Z}_{\\ge 0}$ which are invariant under addition of $m$ and $n$ , and can be matched to the lattice points in $D_{n}^{m}$ .", "Note the lattice points in turn correspond to the minimal length left coset representatives ${}^m\\widetilde{S}_n\\cap \\widetilde{S}_n/S_n$ .", "Corollary 8.11 If the map $\\operatorname{\\mathcal {PS}}$ is a bijection then the Poincaré polynomial of $\\mathcal {F}_{m/n}$ is given by the following formula: $\\sum _{k=0}^{\\infty }t^{k}\\dim \\operatorname{H}^{k}\\left(\\mathcal {F}_{m/n}\\right)=\\sum _{f\\in \\operatorname{\\mathcal {PF}}_{m/n}}t^{2\\sum _{i}f(i)}.$ Since the variety $\\mathcal {F}_{m/n}$ can be paved by the even-dimensional cells, it has no odd cohomology and $(2k)$ -th Betti number is equal to the number of cells of complex dimension $k$ .", "Therefore by Theorem REF : $\\sum _{k}t^{k}\\dim \\operatorname{H}^{k}\\left(\\mathcal {F}_{m/n}\\right)=\\sum _{{\\omega }\\in {}^m\\widetilde{S}_n}t^{2\\dim \\Sigma _{{\\omega }}}=\\sum _{{\\omega }\\in \\widetilde{S}_n^m}t^{2\\sum _i \\operatorname{\\mathcal {PS}}_{{\\omega }}(i)}=\\sum _{f\\in \\operatorname{\\mathcal {PF}}_{m/n}}t^{2\\sum _i f(i)}.$ Equation $(\\ref {Poinc})$ was conjectured in [24] for all coprime $m$ and $n$ .", "Some examples for $m \\ne kn\\pm 1.$ In this section we discuss some examples for which $m \\ne kn\\pm 1$ .", "Example 9.1 There are $81 = 3^4$ $3/5$ -parking functions.", "The $7= \\frac{1}{5+3} \\binom{5+3}{5} $ increasing parking functions are $\\llparenthesis 00000 \\rrparenthesis ,\\llparenthesis 00001 \\rrparenthesis , \\llparenthesis 00002 \\rrparenthesis , \\llparenthesis 00011 \\rrparenthesis , \\llparenthesis 00012 \\rrparenthesis , \\llparenthesis 00111 \\rrparenthesis ,\\llparenthesis 00112 \\rrparenthesis $ .", "Grouping them into the $S_{5}$ -orbits $\\lbrace f \\circ {\\omega }\\mid {\\omega }\\in S_{5} \\rbrace $ yields $81 = 1 + 5+5+10+20+10+30$ .", "There are 7 vectors in ${\\mathbb {Z}}^5 \\cap V \\cap D_5^3$ .", "Their transposes are: $(0,0,0,0,0), (1,0,0,0,-1), (0,1,0,0,-1),(1,0,0,-1,0),(0,0,1,-1,0), (0,1,-1,0,0), (1,-1,0,1,-1)$ The 30 parking functions in the $S_{5}$ -orbit of $\\llparenthesis 00112 \\rrparenthesis $ correspond under the map $\\operatorname{\\mathcal {A}}$ to the 30 permutations in $S_{5} \\cap {}^{3}\\widetilde{S}_{5}$ which are those in the support of the shuffle $14 \\,$$$ 25 $\\scriptscriptstyle \\cup {\\hspace{-2.22214pt}}\\cup $ 3$(that is to say, the intersection of the Sommers region with the orbit ofthe identity permutation).", "\\\\On the other hand, the parking function $ 00000 = A(m)$corresponds under $ A$ to the affine permutation$ m= [-3,0,3,6,9] 3S5$.", "Anything else in its right $ S5$-orbitlies outside the Sommers region.$ Despite the fact many of the above theorems and constructions use $\\widetilde{S}_n^m$ , it is more uniform to study the set $\\lbrace u {\\rm A}_0\\mid u \\in {}^m\\widetilde{S}_n\\rbrace $ and $\\operatorname{\\mathcal {SP}}$ than $\\lbrace {\\omega }{\\rm A}_0\\mid {\\omega }\\in \\widetilde{S}_n^m\\rbrace $ and $\\operatorname{\\mathcal {PS}}$ .", "One reason is that while the Sommers region can always be defined for $\\gcd (m,n)=1$ , a hyperplane arrangement that is the correct analogue of the Shi arrangement cannot.", "Consider the following example.", "Example 9.2 .", "In the case $(n,m) = (5,3)$ it is impossible to find a set of hyperplanes that separate the alcoves $\\lbrace {\\omega }{\\rm A}_0\\mid {\\omega }\\in \\widetilde{S}_n^m\\rbrace $ and has $\\frac{1}{5+3} \\binom{5+3}{5} = 7$ dominant regions, i.e.", "that there are exactly 7 dominant regions with a unique ${\\omega }{\\rm A}_0$ in each.", "In other words, the notion of $\\operatorname{Reg}^k_n$ does not extend well when $m \\ne kn \\pm 1$ .", "Indeed, 7 dominant regions corresponding to 3-restricted affine permutations ${\\omega }\\in {}^{3}\\widetilde{S}_{5} \\cap \\widetilde{S}_{5}/S_{5}$ are shown in bold in Figure REF .", "Each permutation drawn corresponds to the dominant alcove ${{\\omega }^{-1}}{\\rm A}_0$ .", "Hence the hyperplanes crossed (by the pictured ${\\omega }$ ) correspond exactly to $\\operatorname{{Inv}}({\\omega })$ .", "The hyperplanes $H_{4,6}^0,H_{5,6}^0$ and $H_{5,7}^0$ separate $[02346]$ from other 3-restricted permutations.", "To separate $[-21457]$ from $[-11258]$ , one must add either $H_{3,6}^0$ or $H_{5,8}^0$ to the arrangement, but this would leave either of the non-3-restricted permutations $[-22456]$ or $[01248]$ in a region with no 3-restricted permutations.", "Therefore any extension of the classical braid arrangement for $S_5$ would either have a region with two 3-restricted permutations or a region with none of them.", "Note there are more hyperplanes ($H_{4,7}^0, H_{5,11}^0$ ) that ${\\omega }_m= [-3,0,3,6,9]$ has crossed that we did not draw on the picture.", "Figure: 7 permutations ω∈ 3 S ˜ 5 {\\omega }\\in {}^{3}\\widetilde{S}_{5} labeling ω -1 A 0 {\\omega }^{-1} {\\rm A}_0in the dominant coneExample 9.3 We list all affine permutations in $\\widetilde{S}_{5}^{2}$ together with their images under the maps $\\operatorname{\\mathcal {A}}$ and $\\operatorname{\\mathcal {PS}}$ in Figure REF .", "Here ${\\omega }$ is a 2-stable affine permutation (that is, ${\\omega }(i+2)>{\\omega }(i)$ ), and ${\\omega }^{-1}$ is 2-restricted .", "Note that for $m=2$ one has $\\operatorname{\\mathcal {A}}_{{\\omega }}(k)={\\omega }^{-1}(k)-M_{{\\omega }}\\ \\operatorname{mod}\\ 2$ , where, as above, $M_{{\\omega }}=\\min \\lbrace k : \\omega (k)>0\\rbrace $ .", "The combinatorial Hilbert series has a form: $H_{2/5}(q,t)=5+4(q+t)+(q^2+qt+t^2).$ In particular, it is symmetric in $q$ and $t$ and thus answers a question posed in [2].", "The special vertex of $D_{2/5}$ corresponding to the fundamental alcove by Lemma REF is described by the affine permutation ${\\omega }_2=[-1,1,3,5,7]$ .", "Figure: Affine permutations in S ˜ 5 2 \\widetilde{S}_{5}^{2}, their inverses in 2 S ˜ 5 {}^{2}\\widetilde{S}_{5}; maps error\\operatorname{\\mathcal {A}} and error\\operatorname{\\mathcal {PS}} to error 2/5 \\operatorname{\\mathcal {PF}}_{2/5};area\\operatorname{area} and dinv\\operatorname{dinv} statistics" ], [ "Springer fibers for the symmetric group", "Let $V$ be a finite-dimensional vector space and let $N$ be a nilpotent transformation of $V$ .", "Let $\\operatorname{Fl}(V)$ denote the space of complete flags in $V$ .", "A classical object in the representation theory is the Springer fiber ([28], [26]) defined as $X_N:=\\lbrace {\\bf F}=\\lbrace V=V_1\\supset V_2\\supset \\ldots \\supset V_{n}\\rbrace \\in \\operatorname{Fl}(V): N(V_i)\\subset V_i\\rbrace .$ It is known that $X_N$ admits an affine paving with combinatorics completely determined by the conjugacy class of $N$ (see e.g.", "[26] and references therein).", "We will be interested in a particular case of this construction.", "Let us fix a basis $(e_1,\\ldots ,e_n)$ in the space $V$ , consider the operator of shift by $m$ : $N(e_i):={\\left\\lbrace \\begin{array}{ll}e_{i+m},& i+m\\le n\\\\0,& \\text{otherwise}\\\\\\end{array}\\right.", "}$ The following theorem describes the structure of the affine cells in the variety $X_{N}$ .", "Theorem 8.1 The variety $X_N$ admits an affine paving, where the cells $\\Sigma _{\\omega }$ are parametrized by the finite $m$ -restricted permutations $\\omega \\in S_n^m$ .", "The dimension of $\\Sigma _{\\omega }$ is given by the number of inversions in $\\omega ^{-1}$ of height less than $m$ .", "The cells are essentially given by the intersections of Schubert cells in $\\operatorname{Fl}(V)$ with the subvariety $X_N$ .", "For the sake of completeness, let us recall their construction.", "Given a permutation $\\omega \\in S_n$ , we can define a stratum $\\Sigma _{\\omega }$ in $\\operatorname{Fl}(V)$ consisting of the following flags: ${\\bf F}=\\lbrace V_1\\supset V_2\\supset \\ldots \\supset V_n\\rbrace ,\\ V_i=\\operatorname{span}\\lbrace v^{\\omega (i)},\\ldots ,v^{\\omega (n)} \\rbrace ,$ where $v^\\alpha =e_\\alpha +\\sum _{\\beta >\\alpha }\\lambda ^\\alpha _\\beta e_\\beta .$ Note that the position of $v^\\alpha $ in the basis equals $\\omega ^{-1}(\\alpha )$ .", "After a triangular change of variables, we can assume that $\\lambda ^\\alpha _\\beta =0$ for $\\beta >\\alpha $ with $\\omega ^{-1}(\\beta )>\\omega ^{-1}(\\alpha ).$ Therefore one can write $v^\\alpha =e_\\alpha +\\sum _{\\beta >\\alpha ,\\omega ^{-1}(\\beta )<\\omega ^{-1}(\\alpha )}\\lambda ^\\alpha _\\beta e_\\beta =e_\\alpha +\\sum _{(\\alpha ,\\beta )\\in \\operatorname{{Inv}}({\\omega }^{-1})}\\lambda ^\\alpha _\\beta e_\\beta .$ The parameters $\\lambda ^\\alpha _\\beta $ are uniquely defined by the flag ${\\bf F}$ .", "They serve as coordinates on the affine space $\\Sigma _{\\omega }$ , whose dimension is equal to the length of $\\omega $ , i.e.", "$=\\sharp \\operatorname{{Inv}}({\\omega }) = \\sharp \\operatorname{{Inv}}({{\\omega }^{-1}})$ .", "Let us study the intersection $\\Sigma _{\\omega }^N:=\\Sigma _{\\omega }\\cap X_N.$ Since $Nv^\\alpha $ starts with $e_{\\alpha +m}$ , the vector $v^{\\alpha +m}$ should go after $v^\\alpha $ in the basis, so one needs $\\omega ^{-1}(\\alpha +m)>\\omega ^{-1}(\\alpha )$ .", "Therefore $\\Sigma ^{N}_{\\omega }$ is non-empty if and only if $\\omega ^{-1}$ is $m$ -stable.", "A flag ${\\bf F}$ is $N$ -invariant, if $N(v^\\alpha )$ belongs to $\\operatorname{span}\\lbrace v^\\beta : \\omega ^{-1}(\\beta )>\\omega ^{-1}(\\alpha ) \\rbrace $ for all $\\alpha $ .", "If $\\beta >\\alpha +m\\ \\text{and}\\ \\omega ^{-1}(\\beta )>\\omega ^{-1}(\\alpha ),$ then the coefficient in $v^{\\alpha +m}-N(v^\\alpha )$ at $e_\\beta $ can be eliminated by subtracting an appropriate multiple of $v^\\beta $ .", "Once all these coefficients are eliminated, the remaining coefficients in $v^{\\alpha +m}-N(v^\\alpha ),$ labeled by $\\beta >\\alpha +m$ such that $\\omega ^{-1}(\\beta )<\\omega ^{-1}(\\alpha )$ will vanish automatically.", "Therefore $\\Sigma ^N_{\\omega }$ is cut out in $\\Sigma _{\\omega }$ by the equations: $\\lambda ^{\\alpha +m}_\\beta =\\lambda ^\\alpha _{\\beta -m}+\\phi (\\lambda )\\ \\text{if}\\ \\beta >\\alpha +m,\\omega ^{-1}(\\beta )<\\omega ^{-1}(\\alpha ),$ where $\\phi (\\lambda )$ are certain explicit polynomials in $\\lambda ^\\mu _\\nu $ with $\\nu -\\mu <\\beta -\\alpha -m,$ with no linear terms.", "It is clear that such equations are labeled by the inversions $(\\alpha ,\\beta )$ in $\\omega ^{-1}$ of height bigger than $m$ .", "Note also that the linear parts of these equations are linearly independent.", "Therefore the number of free parameters on $\\Sigma ^N_{\\omega }$ equals to the number of inversions of ${{\\omega }^{-1}}$ of height less than $m$ .", "Example 8.2 Consider a 2-stable permutation $\\omega =[2, 1, 4, 3]=\\omega ^{-1}$ .", "The basis (REF ) has the form: $v^{\\omega (1)}=e_2,\\ v^{\\omega (2)}=e_1+\\lambda ^1_2 e_2,\\ v^{\\omega (3)}=e_4,\\ v^{\\omega (4)}=e_3+\\lambda ^3_4 e_4.$ There are two free parameters, so $\\dim \\Sigma ^{N}_{\\omega }=2$ .", "Note that although $\\lambda ^3_4\\ne \\lambda ^1_2$ , $N(v^{\\omega (2)})=e_3+\\lambda ^1_2 e_4\\in \\operatorname{span}\\lbrace v^{\\omega (3)},v^{\\omega (4)} \\rbrace .$ Example 8.3 Consider a 3-stable permutation $\\omega ^{-1}=[1, 5, 3, 2, 6, 4, 7],$ so $\\omega =[1, 4, 3, 6, 2, 5, 7]$ is 3-restricted.", "The basis (REF ) has a form: $v^{\\omega (1)}=e_1,\\ v^{\\omega (2)}=e_4,\\ v^{\\omega (3)}=e_{3}+\\lambda ^3_4 e_4,\\ v^{\\omega (4)}=e_6,\\ v^{\\omega (5)}=e_2+\\lambda ^2_3 e_3+\\lambda ^2_4 e_4+\\lambda ^2_6 e_6,$ $v^{\\omega (6)}=e_5+\\lambda ^5_6 e_6,\\ v^{\\omega (7)}=e_7.$ Since $N(v^{{\\omega }(5)})-v^{{\\omega }(6)}=(\\lambda ^2_3-\\lambda ^5_6)e_6+\\lambda ^2_4 e_7\\in V_5,$ and $V_5=\\operatorname{span}\\lbrace v^{{\\omega }(5)},v^{{\\omega }(6)},v^{{\\omega }(7)} \\rbrace =\\operatorname{span}\\lbrace e_2+\\lambda ^2_3 e_3+\\lambda ^2_4 e_4+\\lambda ^2_6 e_6,e_5+\\lambda ^5_6 e_6,e_7 \\rbrace ,$ the coefficient of $e_6$ must vanish and so we get the relation $\\lambda ^2_3=\\lambda ^5_6.$ Therefore $\\dim \\Sigma _{\\omega }^N=4.$" ], [ "Springer fibers for the affine symmetric group", "We recall the basic definitions of the type $A$ affine Springer fibres, and refer the reader e. g. to [13], [21], [24] for more details.", "Let us choose an indeterminate $\\varepsilon $ and consider the field $K=(\\varepsilon ))$ of Laurent power series and the ring $\\mathcal {O}=[\\varepsilon ]]$ of power series in $\\varepsilon $ .", "Let $V=n((\\varepsilon ))$ be a $K$ -vector space of dimension $n$ .", "Definition 8.4 The affine Grassmannian $\\mathcal {G}_n$ for the group $GL_n$ is the moduli space of $\\mathcal {O}$ -submodules $M\\subset V$ such that the following three conditions are satisfied: (a) $M$ is $\\mathcal {O}$ -invariant.", "(b) There exists $N$ such that $\\varepsilon ^{-N}n[[\\varepsilon ]]\\supset M\\supset \\varepsilon ^{N}n[[\\varepsilon ]].$ (c) Let $N$ be an integer satisfying the above condition.", "Then the following normalization condition is satisfied: $\\dim _{ \\varepsilon ^{-N}n[[\\varepsilon ]]/M=\\dim _{ M/ \\varepsilon ^{N}n[[\\varepsilon ]].", "}}The{\\em affine complete flag variety}$ Fn$ for the group $ GLn$ is the moduli space of collections $ {M0...Mn}$, where each $ Mi$ satisfies (a) and (b),$ Mi/Mi+1=1,$\\ $ Mn=M0,$ and $ M0Gn,$ i.e.", "$ M0$ also satisfies the normalization condition (c).$ Definition 8.5 Let $T$ be an endomorphism of $V$ .", "It is called nil-elliptic if $\\lim _{k\\rightarrow \\infty }T^{k}=0$ and the characteristic polynomial of $T$ is irreducible over $K$ .", "Given a nil-elliptic operator $T$ , one can extend its action to $\\mathcal {G}_n$ and to $\\mathcal {F}_n$ and define the affine Springer fibers as the corresponding fixed point sets.", "Remark 8.6 The condition $\\lim _{k\\rightarrow \\infty }T^{k}=0$ means that for any $N\\in {\\mathbb {N}}$ there exists $k\\in {\\mathbb {N}}$ such that $T^k(n[[\\varepsilon ]])\\subset \\varepsilon ^Nn[[\\varepsilon ]].$ In [24] Lusztig and Smelt studied the structure of the affine Springer fibers for a particular choice of $T$ .", "Given a $-basis $ {e1,..., en}$ in $ n$,one can consider it as a $ K$-basis of$ V=n(())$.", "Consider the operator $ N$ definedby the equations $ N(ei)=ei+1, N(en)=e1.$ The following theorem is the main result of \\cite {LS91}.$ Theorem 8.7 ([24]) Consider the nil-elliptic operator $T:=N^m$ , where $m$ is coprime to $n$ .", "Then the corresponding affine Springer fiber $\\mathcal {F}_{m/n}\\subset \\mathcal {F}_n$ admits an affine paving by $m^{n-1}$ affine cells.", "It turns out that the affine paving of this affine Springer fiber is tightly related to the combinatorics of the simplex $D_{n}^{m}$ .", "This was implicitly stated in [17], [20], [24], but we would like to make this correspondence precise and explicit.", "Theorem 8.8 There is a natural bijection between the affine cells in $\\mathcal {F}_{m/n}$ and the affine permutations in ${}^m\\widetilde{S}_n$ .", "The dimension of the cell $\\Sigma _{\\omega }$ labeled by the affine permutation $\\omega $ is equal to $\\sum _{i=1}^{n}\\operatorname{\\mathcal {SP}}_{\\omega }(i)$ .", "Let us introduce an auxiliary variable $z=\\varepsilon ^{1/n}$ .", "We can identify the vector space $V=n((\\varepsilon ))$ with the space $\\operatorname{span}_1,z,\\ldots ,z^{n-1} \\rbrace ((z^n))\\simeq (z))$ of Laurent power series in $z$ by sending the basis $\\lbrace e_1,\\ldots ,e_n\\rbrace $ to $\\lbrace 1,z,\\ldots ,z^{n-1}\\rbrace .$ Note that under this identification, $n[[\\varepsilon ]]$ is mapped to $[z]]$ .", "By construction, $N$ coincides with the multiplication operator by $z$ and hence $T=N^m$ coincides with the multiplication operator by $z^m$ .", "Therefore $\\mathcal {F}_{m/n}$ consists of flags $\\lbrace M_0\\supset \\ldots \\supset M_n\\rbrace $ of $[z^n,z^m]]$ -modules, such that $\\dim _{ M_i/M_{i+1}=1, M_n=z^nM_0 and M_0\\in \\mathcal {G}_n.", "Let us extend the notation M_i to arbitrary i\\in {\\mathbb {Z}} by setting M_{i+kn}:=z^{kn}M_i.", "As a result, we get an infinite flag \\lbrace \\ldots \\supset M_0\\supset \\ldots \\supset M_n\\supset \\ldots \\rbrace of [z^n,z^m]]-modules satisfying the same conditions as above and M_{i+n}=z^nM_i.", "}For $ f(z)V=(z)),$ let $ Ord(f)$ denote the order of $ f(z)$ in $ z,$ i.e.", "the smallest degree of $ z,$ such that the corresponding coefficient in $ f(z)$ does not vanish.", "Given a subset $ MV$, define $$\\operatorname{Ord}(M)=\\lbrace \\operatorname{Ord}(f)\\ :\\ f\\in M, f\\ne 0\\rbrace .$$ We will need the following lemma, whose proof is standard and left to the reader:$ Lemma 8.9 Let $L\\subset M\\subset (z))$ be two $[z^m,z^n]]$ -submodules in $z^{-N}[z]]$ for some large $N\\in {\\mathbb {N}}.$ Then $\\sharp \\left(\\operatorname{Ord}(M)\\setminus \\operatorname{Ord}(L)\\right)=\\dim _M̏/L.$ Given a flag $\\lbrace \\ldots \\supset M_0\\supset \\ldots \\supset M_n\\supset \\ldots \\rbrace $ as above, set $ \\operatorname{Ord}(M_i)\\setminus \\operatorname{Ord}(M_{i+1}) = \\lbrace {\\omega }(i)\\rbrace .$ Note that one automatically gets $\\operatorname{Ord}(M_i)=\\lbrace {\\omega }(i),{\\omega }(i+1),\\ldots \\rbrace ,$ because $\\bigcap \\limits _i\\operatorname{Ord}(M_i)=\\emptyset .$ Recall the notation $\\operatorname{{Inv}}({\\omega }) : =\\lbrace (i,j) \\in {\\mathbb {N}}\\times {\\mathbb {N}}\\mid 1\\le i\\le n,\\ i < j, {\\omega }(i) > {\\omega }(j) \\rbrace $ and $\\operatorname{{\\overline{Inv}}}({\\omega }) : =\\lbrace (i,j) \\in {\\mathbb {N}}\\times {\\mathbb {N}}\\mid i < j, {\\omega }(i) > {\\omega }(j) \\rbrace $ for the inversion sets of ${\\omega }.$ For each $i$ there exists a unique $f_i(z)\\in M_i$ such that $f_i=z^{{\\omega }(i)}+\\sum _{(j,i)\\in \\operatorname{{\\overline{Inv}}}({\\omega })}\\lambda ^{{\\omega }(i)}_{{\\omega }(j)} z^{{\\omega }(j)}.$ Indeed, take any function $f\\in M_i$ such that $\\operatorname{Ord}(f)=\\lbrace {\\omega }(i)\\rbrace $ and use functions from $M_{i+1}$ to eliminate coefficients at $z^{{\\omega }(j)}$ for $j>i$ and ${\\omega }(j)>{\\omega }(i).$ The resulting function is unique up to a scalar, because otherwise $\\dim _M̏_i/M_{i+1}$ would be at least $2.$ It follows that $f_{i+n}=z^nf_i.$ We claim that ${\\omega }$ is an affine permutation and, moreover, $\\omega \\in {}^m\\widetilde{S}_n$ .", "Indeed, since $f_{i+n}=z^nf_i$ we get ${\\omega }(i+n)={\\omega }(i)+n,$ and since $z^mf_i\\in M_i$ we get that ${\\omega }(i)+m\\in \\lbrace {\\omega }(i),{\\omega }(i+1),\\ldots \\rbrace ,$ and, therefore, for any $j<i,$ ${\\omega }(j)-{\\omega }(i)\\ne m.$ Finally, we need to check the normalization condition $\\sum \\limits _{i=1}^n {\\omega }(i)=\\frac{n(n+1)}{2},$ which follows form the normalization condition on $M_0\\in \\mathcal {G}_n.$ Indeed, it is not hard to see that for all $L\\in \\mathcal {G}_n$ the sum of elements of $\\operatorname{Ord}(L)\\setminus \\operatorname{Ord}(t^nL)$ should be the same.", "In particular, for $L=[z]]$ we have $\\operatorname{Ord}([z]])\\setminus \\operatorname{Ord}(z^n[z]])=\\lbrace 0,1,\\ldots ,n-1\\rbrace ,$ and their sum is $\\frac{n(n-1)}{2}.$ Therefore, since $\\operatorname{Ord}(M_0)\\setminus \\operatorname{Ord}(z^nM_0)=\\lbrace {\\omega }(0),\\ldots ,{\\omega }(n-1)\\rbrace ,$ we get $\\sum \\limits _{i-0}^{n-1} {\\omega }(i)=\\frac{n(n-1)}{2},$ which equivalent to the required condition.", "The above gives us a map $\\nu :\\mathcal {F}_{m/n}\\rightarrow {}^m\\widetilde{S}_n.$ Let us prove that the fibers $\\Sigma _{{\\omega }}:=\\nu ^{-1}({\\omega })$ of this map are affine cells and compute their dimensions.", "This is very similar to the computation in the finite case (see Theorem REF ).", "Let us set $f^{{\\omega }(i)}:=f_i.$ The expansions (REF ) can be rewritten as $f^\\alpha =z^\\alpha +\\sum _{(\\alpha ,\\beta )\\in \\operatorname{{\\overline{Inv}}}({\\omega }^{-1})}\\lambda ^{\\alpha }_{\\beta } z^\\beta .$ Since $f_{i+n}=z^nf_i,$ one gets $\\lambda ^{\\alpha +n}_{\\beta +n}=\\lambda ^\\alpha _\\beta .$ Let us also extend the notation by setting $\\lambda ^{\\alpha }_{\\beta }=0$ whenever $(\\alpha ,\\beta )\\notin \\operatorname{{\\overline{Inv}}}({\\omega }^{-1})$ , so that one can write $f^\\alpha =z^\\alpha +\\sum _{\\beta >\\alpha } \\lambda ^\\alpha _{\\beta }z^{\\beta }.$ Let us say that the coefficient $\\lambda ^{\\alpha }_{\\beta }$ is of height $\\beta -\\alpha .$ As before, let $\\alpha ={\\omega }(i).$ The condition $z^m f^\\alpha \\in M_i$ implies the following relations on the coefficients.", "The function $z^mf^\\alpha -f^{\\alpha +m}=\\sum _{\\beta >\\alpha } \\lambda ^\\alpha _\\beta z^{\\beta +m}-\\sum _{\\beta >\\alpha +m} \\lambda ^{\\alpha +m}_{\\beta }z^\\beta =\\sum _{\\beta >\\alpha +m}(\\lambda ^\\alpha _{\\beta -m}-\\lambda ^{\\alpha +m}_\\beta )z^\\beta $ should belong to $M_i$ .", "Take $\\beta >\\alpha +m$ and let $j={\\omega }^{-1}(\\beta ).$ If $j>i,$ then the term of degree $\\beta $ can be eliminated by subtracting $f^\\beta =f_j\\in M_i$ with an appropriate coefficient.", "If $j<i,$ then ${{\\omega }^{-1}}(\\beta -m) < {{\\omega }^{-1}}(\\beta ) < {{\\omega }^{-1}}(\\alpha ) < {{\\omega }^{-1}}(\\alpha +m)$ .", "Hence $(\\alpha ,\\beta -m)\\in \\operatorname{{\\overline{Inv}}}({\\omega }^{-1})$ and $(\\alpha +m,\\beta )\\in \\operatorname{{\\overline{Inv}}}({\\omega }^{-1}),$ so the coefficients $\\lambda ^\\alpha _{\\beta -m}$ and $\\lambda ^{\\alpha +m}_\\beta $ are both not forced to be zero, i.e.", "they are parameters on $\\Sigma _{\\omega }$ .", "The term $(\\lambda ^\\alpha _{\\beta -m}-\\lambda ^{\\alpha +m}_\\beta )z^\\beta $ has to vanish automatically after we eliminated all lower order terms.", "As we eliminate terms of degree $\\gamma $ such that $\\alpha +m<\\gamma <\\beta ,$ the coefficient at $z^\\beta $ changes, but the added terms can only depend on coefficients of smaller height.", "More precisely, all additional terms are non-linear, and the total height of each monomial is always $\\beta -\\alpha -m.$ In the end, we get that $\\lambda ^\\alpha _{\\beta -m}-\\lambda ^{\\alpha +m}_\\beta $ should be equal to zero modulo the coefficients of smaller height.", "This means that for each $(\\alpha ,\\beta )\\in \\operatorname{{\\overline{Inv}}}({\\omega }^{-1})$ of height $\\beta -\\alpha >m$ there is an equation that allows one to express $\\lambda ^\\alpha _{\\beta -m}$ in terms of $\\lambda ^{\\alpha +m}_\\beta $ and higher order terms in coefficients with lower height.", "A priori, the linear parts of these equations can be dependent if for all $0\\le q\\le n$ one has $(\\alpha +qm,\\beta +qm)\\in \\operatorname{{\\overline{Inv}}}({\\omega }^{-1}).$ However, since $m$ and $n$ are relatively prime, this would mean that $\\omega ^{-1}(\\gamma )>\\omega ^{-1}(\\gamma +\\beta -\\alpha )$ for all $\\gamma \\in {\\mathbb {Z}},$ which is impossible.", "Therefore, one can resolve the relations on the coefficients with respect to $\\lambda ^\\alpha _\\beta $ such that $(\\alpha ,\\beta )\\in \\operatorname{{\\overline{Inv}}}({\\omega }^{-1})$ and $(\\alpha ,\\beta +m)\\in \\operatorname{{\\overline{Inv}}}({\\omega }^{-1}).$ So, the coordinates on $\\Sigma _{{\\omega }}$ correspond to the inversions $(\\alpha ,\\beta )\\in \\operatorname{{Inv}}({\\omega }^{-1}),$ such that $(\\alpha ,\\beta +m)\\notin \\operatorname{{Inv}}({\\omega }^{-1})$ .", "Since $\\lambda ^\\alpha _\\beta =\\lambda ^{\\alpha +n}_{\\beta +n},$ one should count inversions in $\\operatorname{{Inv}}({\\omega }^{-1})$ only.", "It is not hard to see that such inversions are in bijection with inversions of height less than $m.$ Indeed, the required map is $(\\alpha ,\\beta )\\mapsto (\\alpha ,\\beta -km),$ where $k$ is the maximal integer such that $\\beta -km>\\alpha .$ Alternatively, one can also notice that the relations are in bijection with inversions of height greater than $m.$ Indeed, the relation $\\lambda ^\\alpha _\\beta \\equiv \\lambda ^{\\alpha +m}_{\\beta +m}$ (modulo lower height terms) corresponds to the inversion $(\\alpha ,\\beta +m)\\in \\operatorname{{\\overline{Inv}}}({\\omega }^{-1})$ of height greater than $m.$ Therefore, the dimension of $\\Sigma _{{\\omega }}$ is the total number of inversions minus the number of inversion of height greater than $m.$ Since there are no inversions of height $m,$ the dimension is equal to the number of inversions of height less than $m.$ Since $\\sum _{i=1}^{n}\\operatorname{\\mathcal {PS}}_{\\omega ^{-1}}(i)$ is exactly the total number of inversions of height less than $m,$ we conclude that $\\dim (\\Sigma _{\\omega })=\\sum _{i=1}^{n}\\operatorname{\\mathcal {SP}}_{\\omega }(i)=\\sum _{i=1}^{n}\\operatorname{\\mathcal {PS}}_{\\omega ^{-1}}(i)=\\frac{(m-1)(n-1)}{2}-\\operatorname{dinv}({\\omega }^{-1}).$ For a more abstract proof see e.g.", "[13] and [20].", "Remark 8.10 Similar reasoning shows that the Grassmannian version of the affine Springer fiber $\\mathcal {G}_{m/n}\\subset \\mathcal {G}_n$ parametrizes appropriately normalized $[z^n,z^m]]$ -submodules in $(z)).$ This affine Springer fiber was studied e.g.", "in [14], [15] under the name of Jacobi factor of the plane curve singularity $\\lbrace x^m=y^n\\rbrace $ .", "The cells in it are parametrized by the subsets in $\\mathbb {Z}_{\\ge 0}$ which are invariant under addition of $m$ and $n$ , and can be matched to the lattice points in $D_{n}^{m}$ .", "Note the lattice points in turn correspond to the minimal length left coset representatives ${}^m\\widetilde{S}_n\\cap \\widetilde{S}_n/S_n$ .", "Corollary 8.11 If the map $\\operatorname{\\mathcal {PS}}$ is a bijection then the Poincaré polynomial of $\\mathcal {F}_{m/n}$ is given by the following formula: $\\sum _{k=0}^{\\infty }t^{k}\\dim \\operatorname{H}^{k}\\left(\\mathcal {F}_{m/n}\\right)=\\sum _{f\\in \\operatorname{\\mathcal {PF}}_{m/n}}t^{2\\sum _{i}f(i)}.$ Since the variety $\\mathcal {F}_{m/n}$ can be paved by the even-dimensional cells, it has no odd cohomology and $(2k)$ -th Betti number is equal to the number of cells of complex dimension $k$ .", "Therefore by Theorem REF : $\\sum _{k}t^{k}\\dim \\operatorname{H}^{k}\\left(\\mathcal {F}_{m/n}\\right)=\\sum _{{\\omega }\\in {}^m\\widetilde{S}_n}t^{2\\dim \\Sigma _{{\\omega }}}=\\sum _{{\\omega }\\in \\widetilde{S}_n^m}t^{2\\sum _i \\operatorname{\\mathcal {PS}}_{{\\omega }}(i)}=\\sum _{f\\in \\operatorname{\\mathcal {PF}}_{m/n}}t^{2\\sum _i f(i)}.$ Equation $(\\ref {Poinc})$ was conjectured in [24] for all coprime $m$ and $n$ .", "Some examples for $m \\ne kn\\pm 1.$ In this section we discuss some examples for which $m \\ne kn\\pm 1$ .", "Example 9.1 There are $81 = 3^4$ $3/5$ -parking functions.", "The $7= \\frac{1}{5+3} \\binom{5+3}{5} $ increasing parking functions are $\\llparenthesis 00000 \\rrparenthesis ,\\llparenthesis 00001 \\rrparenthesis , \\llparenthesis 00002 \\rrparenthesis , \\llparenthesis 00011 \\rrparenthesis , \\llparenthesis 00012 \\rrparenthesis , \\llparenthesis 00111 \\rrparenthesis ,\\llparenthesis 00112 \\rrparenthesis $ .", "Grouping them into the $S_{5}$ -orbits $\\lbrace f \\circ {\\omega }\\mid {\\omega }\\in S_{5} \\rbrace $ yields $81 = 1 + 5+5+10+20+10+30$ .", "There are 7 vectors in ${\\mathbb {Z}}^5 \\cap V \\cap D_5^3$ .", "Their transposes are: $(0,0,0,0,0), (1,0,0,0,-1), (0,1,0,0,-1),(1,0,0,-1,0),(0,0,1,-1,0), (0,1,-1,0,0), (1,-1,0,1,-1)$ The 30 parking functions in the $S_{5}$ -orbit of $\\llparenthesis 00112 \\rrparenthesis $ correspond under the map $\\operatorname{\\mathcal {A}}$ to the 30 permutations in $S_{5} \\cap {}^{3}\\widetilde{S}_{5}$ which are those in the support of the shuffle $14 \\,$$$ 25 $\\scriptscriptstyle \\cup {\\hspace{-2.22214pt}}\\cup $ 3$(that is to say, the intersection of the Sommers region with the orbit ofthe identity permutation).", "\\\\On the other hand, the parking function $ 00000 = A(m)$corresponds under $ A$ to the affine permutation$ m= [-3,0,3,6,9] 3S5$.", "Anything else in its right $ S5$-orbitlies outside the Sommers region.$ Despite the fact many of the above theorems and constructions use $\\widetilde{S}_n^m$ , it is more uniform to study the set $\\lbrace u {\\rm A}_0\\mid u \\in {}^m\\widetilde{S}_n\\rbrace $ and $\\operatorname{\\mathcal {SP}}$ than $\\lbrace {\\omega }{\\rm A}_0\\mid {\\omega }\\in \\widetilde{S}_n^m\\rbrace $ and $\\operatorname{\\mathcal {PS}}$ .", "One reason is that while the Sommers region can always be defined for $\\gcd (m,n)=1$ , a hyperplane arrangement that is the correct analogue of the Shi arrangement cannot.", "Consider the following example.", "Example 9.2 .", "In the case $(n,m) = (5,3)$ it is impossible to find a set of hyperplanes that separate the alcoves $\\lbrace {\\omega }{\\rm A}_0\\mid {\\omega }\\in \\widetilde{S}_n^m\\rbrace $ and has $\\frac{1}{5+3} \\binom{5+3}{5} = 7$ dominant regions, i.e.", "that there are exactly 7 dominant regions with a unique ${\\omega }{\\rm A}_0$ in each.", "In other words, the notion of $\\operatorname{Reg}^k_n$ does not extend well when $m \\ne kn \\pm 1$ .", "Indeed, 7 dominant regions corresponding to 3-restricted affine permutations ${\\omega }\\in {}^{3}\\widetilde{S}_{5} \\cap \\widetilde{S}_{5}/S_{5}$ are shown in bold in Figure REF .", "Each permutation drawn corresponds to the dominant alcove ${{\\omega }^{-1}}{\\rm A}_0$ .", "Hence the hyperplanes crossed (by the pictured ${\\omega }$ ) correspond exactly to $\\operatorname{{Inv}}({\\omega })$ .", "The hyperplanes $H_{4,6}^0,H_{5,6}^0$ and $H_{5,7}^0$ separate $[02346]$ from other 3-restricted permutations.", "To separate $[-21457]$ from $[-11258]$ , one must add either $H_{3,6}^0$ or $H_{5,8}^0$ to the arrangement, but this would leave either of the non-3-restricted permutations $[-22456]$ or $[01248]$ in a region with no 3-restricted permutations.", "Therefore any extension of the classical braid arrangement for $S_5$ would either have a region with two 3-restricted permutations or a region with none of them.", "Note there are more hyperplanes ($H_{4,7}^0, H_{5,11}^0$ ) that ${\\omega }_m= [-3,0,3,6,9]$ has crossed that we did not draw on the picture.", "Figure: 7 permutations ω∈ 3 S ˜ 5 {\\omega }\\in {}^{3}\\widetilde{S}_{5} labeling ω -1 A 0 {\\omega }^{-1} {\\rm A}_0in the dominant coneExample 9.3 We list all affine permutations in $\\widetilde{S}_{5}^{2}$ together with their images under the maps $\\operatorname{\\mathcal {A}}$ and $\\operatorname{\\mathcal {PS}}$ in Figure REF .", "Here ${\\omega }$ is a 2-stable affine permutation (that is, ${\\omega }(i+2)>{\\omega }(i)$ ), and ${\\omega }^{-1}$ is 2-restricted .", "Note that for $m=2$ one has $\\operatorname{\\mathcal {A}}_{{\\omega }}(k)={\\omega }^{-1}(k)-M_{{\\omega }}\\ \\operatorname{mod}\\ 2$ , where, as above, $M_{{\\omega }}=\\min \\lbrace k : \\omega (k)>0\\rbrace $ .", "The combinatorial Hilbert series has a form: $H_{2/5}(q,t)=5+4(q+t)+(q^2+qt+t^2).$ In particular, it is symmetric in $q$ and $t$ and thus answers a question posed in [2].", "The special vertex of $D_{2/5}$ corresponding to the fundamental alcove by Lemma REF is described by the affine permutation ${\\omega }_2=[-1,1,3,5,7]$ .", "Figure: Affine permutations in S ˜ 5 2 \\widetilde{S}_{5}^{2}, their inverses in 2 S ˜ 5 {}^{2}\\widetilde{S}_{5}; maps error\\operatorname{\\mathcal {A}} and error\\operatorname{\\mathcal {PS}} to error 2/5 \\operatorname{\\mathcal {PF}}_{2/5};area\\operatorname{area} and dinv\\operatorname{dinv} statistics" ], [ "Some examples for $m \\ne kn\\pm 1.$", "In this section we discuss some examples for which $m \\ne kn\\pm 1$ .", "Example 9.1 There are $81 = 3^4$ $3/5$ -parking functions.", "The $7= \\frac{1}{5+3} \\binom{5+3}{5} $ increasing parking functions are $\\llparenthesis 00000 \\rrparenthesis ,\\llparenthesis 00001 \\rrparenthesis , \\llparenthesis 00002 \\rrparenthesis , \\llparenthesis 00011 \\rrparenthesis , \\llparenthesis 00012 \\rrparenthesis , \\llparenthesis 00111 \\rrparenthesis ,\\llparenthesis 00112 \\rrparenthesis $ .", "Grouping them into the $S_{5}$ -orbits $\\lbrace f \\circ {\\omega }\\mid {\\omega }\\in S_{5} \\rbrace $ yields $81 = 1 + 5+5+10+20+10+30$ .", "There are 7 vectors in ${\\mathbb {Z}}^5 \\cap V \\cap D_5^3$ .", "Their transposes are: $(0,0,0,0,0), (1,0,0,0,-1), (0,1,0,0,-1),(1,0,0,-1,0),(0,0,1,-1,0), (0,1,-1,0,0), (1,-1,0,1,-1)$ The 30 parking functions in the $S_{5}$ -orbit of $\\llparenthesis 00112 \\rrparenthesis $ correspond under the map $\\operatorname{\\mathcal {A}}$ to the 30 permutations in $S_{5} \\cap {}^{3}\\widetilde{S}_{5}$ which are those in the support of the shuffle $14 \\,$$$ 25 $\\scriptscriptstyle \\cup {\\hspace{-2.22214pt}}\\cup $ 3$(that is to say, the intersection of the Sommers region with the orbit ofthe identity permutation).", "\\\\On the other hand, the parking function $ 00000 = A(m)$corresponds under $ A$ to the affine permutation$ m= [-3,0,3,6,9] 3S5$.", "Anything else in its right $ S5$-orbitlies outside the Sommers region.$ Despite the fact many of the above theorems and constructions use $\\widetilde{S}_n^m$ , it is more uniform to study the set $\\lbrace u {\\rm A}_0\\mid u \\in {}^m\\widetilde{S}_n\\rbrace $ and $\\operatorname{\\mathcal {SP}}$ than $\\lbrace {\\omega }{\\rm A}_0\\mid {\\omega }\\in \\widetilde{S}_n^m\\rbrace $ and $\\operatorname{\\mathcal {PS}}$ .", "One reason is that while the Sommers region can always be defined for $\\gcd (m,n)=1$ , a hyperplane arrangement that is the correct analogue of the Shi arrangement cannot.", "Consider the following example.", "Example 9.2 .", "In the case $(n,m) = (5,3)$ it is impossible to find a set of hyperplanes that separate the alcoves $\\lbrace {\\omega }{\\rm A}_0\\mid {\\omega }\\in \\widetilde{S}_n^m\\rbrace $ and has $\\frac{1}{5+3} \\binom{5+3}{5} = 7$ dominant regions, i.e.", "that there are exactly 7 dominant regions with a unique ${\\omega }{\\rm A}_0$ in each.", "In other words, the notion of $\\operatorname{Reg}^k_n$ does not extend well when $m \\ne kn \\pm 1$ .", "Indeed, 7 dominant regions corresponding to 3-restricted affine permutations ${\\omega }\\in {}^{3}\\widetilde{S}_{5} \\cap \\widetilde{S}_{5}/S_{5}$ are shown in bold in Figure REF .", "Each permutation drawn corresponds to the dominant alcove ${{\\omega }^{-1}}{\\rm A}_0$ .", "Hence the hyperplanes crossed (by the pictured ${\\omega }$ ) correspond exactly to $\\operatorname{{Inv}}({\\omega })$ .", "The hyperplanes $H_{4,6}^0,H_{5,6}^0$ and $H_{5,7}^0$ separate $[02346]$ from other 3-restricted permutations.", "To separate $[-21457]$ from $[-11258]$ , one must add either $H_{3,6}^0$ or $H_{5,8}^0$ to the arrangement, but this would leave either of the non-3-restricted permutations $[-22456]$ or $[01248]$ in a region with no 3-restricted permutations.", "Therefore any extension of the classical braid arrangement for $S_5$ would either have a region with two 3-restricted permutations or a region with none of them.", "Note there are more hyperplanes ($H_{4,7}^0, H_{5,11}^0$ ) that ${\\omega }_m= [-3,0,3,6,9]$ has crossed that we did not draw on the picture.", "Figure: 7 permutations ω∈ 3 S ˜ 5 {\\omega }\\in {}^{3}\\widetilde{S}_{5} labeling ω -1 A 0 {\\omega }^{-1} {\\rm A}_0in the dominant coneExample 9.3 We list all affine permutations in $\\widetilde{S}_{5}^{2}$ together with their images under the maps $\\operatorname{\\mathcal {A}}$ and $\\operatorname{\\mathcal {PS}}$ in Figure REF .", "Here ${\\omega }$ is a 2-stable affine permutation (that is, ${\\omega }(i+2)>{\\omega }(i)$ ), and ${\\omega }^{-1}$ is 2-restricted .", "Note that for $m=2$ one has $\\operatorname{\\mathcal {A}}_{{\\omega }}(k)={\\omega }^{-1}(k)-M_{{\\omega }}\\ \\operatorname{mod}\\ 2$ , where, as above, $M_{{\\omega }}=\\min \\lbrace k : \\omega (k)>0\\rbrace $ .", "The combinatorial Hilbert series has a form: $H_{2/5}(q,t)=5+4(q+t)+(q^2+qt+t^2).$ In particular, it is symmetric in $q$ and $t$ and thus answers a question posed in [2].", "The special vertex of $D_{2/5}$ corresponding to the fundamental alcove by Lemma REF is described by the affine permutation ${\\omega }_2=[-1,1,3,5,7]$ .", "Figure: Affine permutations in S ˜ 5 2 \\widetilde{S}_{5}^{2}, their inverses in 2 S ˜ 5 {}^{2}\\widetilde{S}_{5}; maps error\\operatorname{\\mathcal {A}} and error\\operatorname{\\mathcal {PS}} to error 2/5 \\operatorname{\\mathcal {PF}}_{2/5};area\\operatorname{area} and dinv\\operatorname{dinv} statistics" ] ]	1403.0303
[ [ "Optoelectronic down-conversion by four-wave mixing in a highly nonlinear\n fiber for millimeter-wave and THz phase-locking" ], [ "Abstract Optoelectronic down-conversion of a THz optical beatnote to a RF intermediate frequency is performed with a standard Mach-Zehnder modulator followed by a zero dispersion-slope fiber.", "The two interleaved optical spectra obtained by four-wave mixing are shown to contain more than 75 harmonics enabling to conveniently recover the phase noise of a beatnote at 770 GHz at around 500 MHz.", "This four-wave mixing down-conversion technique is implemented in a two-frequency solid-state laser in order to directly phase-lock its 168 GHz beatnote to a 10 MHz local oscillator." ], [ "Antoine Rolland, Lucien Pouget, Marc Brunel and Mehdi Alouini [email protected] [Second University] Institut de Physique de Rennes, Département d'Optique et Photonique, UMR CNRS-Université de Rennes 1 6251, Campus de Beaulieu, Rennes cedex 35042, FRANCE [An achemso demo] Optoelectronic down-conversion by four-wave mixing in a highly nonlinear fiber for millimeter-wave and THz phase-locking Optoelectronic down-conversion of a THz optical beatnote to a RF intermediate frequency is performed with a standard Mach-Zehnder modulator followed by a zero dispersion-slope fiber.", "The two interleaved optical spectra obtained by four-wave mixing are shown to contain more than 75 harmonics enabling to conveniently recover the phase noise of a beatnote at 770 GHz at around 500 MHz.", "This four-wave mixing down-conversion technique is implemented in a two-frequency solid-state laser in order to directly phase-lock its 168 GHz beatnote to a 10 MHz local oscillator.", "Ultra-high resolution spectroscopy [1], THz metrology [2], or THz radio-astronomy [3], [4] are applications that need ultra-stable continuous-wave (cw) sources.", "While heterodyning two continuous optical waves in a photo-detector is a well-established approach to generate cw THz signals, stability usually relies on phase-locked loops (PLL) [5], [6], [7], [8].", "When the heterodyne beat is at THz frequencies, the major issue is to down-convert the THz beat to RF frequencies in order to implement standard PLLs.", "In this perspective, an important experimental step up was made recently by using a nonlinear electro-optic modulator (EOM) to bridge the gap between widely spaced optical frequencies.", "The opto-electronic phase locked-loop principle (OEPLL) principle was successfully implemented to stabilize two-frequency laser beat signals at 100 GHz [9], [10], up to 250 GHz [11] in an improved set-up, and also to stabilize microcombs at 140 GHz mode spacing [12].", "In all cases, the frequency spacing is limited to the number of harmonics that the EOM can generate, typically 10.", "Besides, four-wave mixing in optical fibers is an optical nonlinear effect which has been studied extensively for applications such as telecommunication when several optical carriers propagate simultaneously in the fiber [13].", "In particular, the benefit of using this effect has been shown for the generation of optical frequency combs [14].", "Indeed, a dispersion management in the fiber allows to generate very wide comb [15].", "Moreover, the use of a zero-dispersion-slope highly nonlinear fiber offer an important potentiality to generate wide and flat optical frequency combs [16], [17], [18], [19].", "Thus, One can wonder if this nonlinear effect could be advantageously implemented to achieve an optoelectronic down-conversion from the THz to the RF band for the purpose of detecting an intermediate frequency which give access to the THz phase noise.", "In this letter we report on a new solution including a highly non-linear fiber (HNLF) to bridge the wide frequency gap between the THz and GHz domains.", "Actually, by sending a two-frequency beam into the HNLF, one expects to generate two interleaved frequency combs which should give access to an intermediate frequency signal carrying the THz phase noise.", "The principle of the experimental setup is depicted in Fig.REF (a).", "It involves two independent lasers or a single laser providing two optical frequencies, $\\nu _e$ and $\\nu _o$ , whose difference is in the THz range.", "The two beams are combined and then focused into a polarization-maintaining (PM) fiber.", "Contrary to our previous work in which a dedicated nonlinear electrooptic modulator (EOM) was required [9], we now use a standard EOM.", "This EOM is driven at $f_{RF}$ leading to the generation of two sidebands around $\\nu _e$ and $\\nu _o$ .", "In order to optimize the four-wave mixing (FWM) efficiency in the HNLF, an erbium-doped fiber amplifier (EDFA) follows the EOM.", "The optical power sent into the 100 m-long HNLF fiber can thus be adjusted from 20 dBm to 28 dBm.", "Note that the HNLF dispersion slope is chosen close to 0 ps/(nm$^{2}$ .km), which is required to achieve optimal phase matching of the generated frequency combs [16].", "Finally, we adjust the bias voltage of the modulator, as well as the RF power in order to maximize the efficiency of the frequency comb generation.", "Figure: (a) Optoelectronic down-conversion experimental setup.", "DFL, Dual-Frequency Laser; EOM, ElectroOptic Modulator; EDFA, Erbium Doped Fiber Amplifier; HNLF, Highly NonLinear Fiber; IF, Intermediate Frequency; OSA, Optical Spectrum Analyzer; ESA, Electrical Spectrum Analyzer.", "(b) Two-axis-propagation dual-frequency laser, see text for details.In our experiment, the two optical frequencies are provided by a two-propagation-axis laser schematized in Fig.REF (b).", "The active medium is a 1.5-mm-long phosphate glass doped Er/Yb.", "The resonator is closed by the input mirror M1 directly coated on the external face of the active medium and by a 5-cm radius of curvature mirror.", "Due to an anti-reflection coated 10-mm-long YVO4 crystal (cut at 45$^\\circ $ of its optical axis) inserted into the cavity, two orthogonally polarized eigenmodes (labeled respectively o and e) are separated by 1 mm in the active medium while superimposed at the output coupler.", "The active medium is pumped at 980 nm using a laser diode.", "In order to efficiently pump the two eigenmodes, the pump beam is split into two parallel 400-mW 100 $\\mu $ m-diameter beams separated by 1 mm.", "To this aim, a second YVO$_4$ crystal (not shown in the figure) is inserted between the pump focusing lens and the laser input mirror.", "To ensure single mode oscillation of each eigenpolarization and to adjust independently their wavelengths, we insert two 40 µm-thick silica etalons (E$_e$ and E$_o$ ) coated on both sides for 30 % reflection at 1550 nm.", "Such a cavity architecture leads to the simultaneous oscillation of two tunable wavelengths which are linearly cross-polarized.", "By tilting the etalons, we are able to tune independently the two wavelengths over the whole erbium gain bandwidth.", "It leads to a beatnote adjustable between a few MHz and 2 THz [20], [21].", "Here we keep E$_o$ perpendicular to the ordinary propagation axis while E$_e$ is tilted to sweep the frequency difference $\\Delta \\nu =\\nu _e-\\nu _o$ .", "Furthermore, a LiTaO$_3$ electrooptic crystal is inserted on the ordinary path of the cavity offering a continuous tunability of the beatnote.", "This laser is consequently turned into a THz voltage controlled oscillator [9].", "The TEM$_{00}$ laser output beam is sent through a polarizer oriented at 45$^\\circ $ in order to make the two modes beating before entering the EOM.", "Experimental results are presented in Fig.", "REF .", "In order to evaluate the electrical spectrum of the generated combs, we first send through the EOM one optical frequency and we turn $f_{RF}$ to a low frequency (1 GHz).", "Fig.", "REF (a) displays the 15 first harmonics of $f_{RF}$ observed with a 16 GHz bandwidth photodiode after propagation along the HNLF.", "The frequency comb has a quite flat amplitude, which greatly improves previous achievements using a nonlinear EOM or a phase modulator [9], [10].", "Although $f_{RF}$ is low, the entire comb cannot be displayed because of the limited cut-off frequency of our photodiode.", "To observe higher harmonics and then evaluate the comb span, we now set the frequency modulation $f_{RF}$ at 10 GHz and monitor the comb in the optical domain.", "The generated comb spans 1 THz leading to measurable lines up the 75th harmonics, as shown on the optical spectrum of Fig.", "REF (b) .", "Figure: (a) Electrical spectrum, with f RF f_{RF} = 1 GHz.", "(b) Optical spectrum (RBW 0.05 nm).", "The inset shows the optical spectrum beyond 700 GHz from the carrier.An accurate spectral analysis of the comb lines is performed by measuring the RF phase noise of the different harmonics numbered N. To this aim, we turn again to a modulation frequency $f_{RF}$ of 1 GHz.", "For illustration purpose, two phase noise spectra for (1) $N$ = 10 (10 GHz) and (2) $N$ = 20 (20 GHz) are reported in Fig.", "REF (a).", "The expected phase noise degradation of $20\\log (N)$ [22] is confirmed experimentally.", "One can thus conclude that the 100m-long fiber does not bring additional noise degradation at least for the first 20th harmonics generated by FWM.", "Let us remind here that our goal is to down convert a THz beatnote in the GHz range.", "We now send to the modulator the two optical frequencies at the same time.", "Setting the frequency difference of the laser to $\\Delta \\nu =\\nu _e-\\nu _o=$ 770 GHz, we were able to measure an intermediate frequency between the interleaved combs at around 500 MHz (see Fig.", "REF (b)).", "A dynamic range as high as 70 dB is then obtained on this intermediate frequency with a 30 kHz measurement bandwidth.", "This high dynamic range makes it possible to evaluate the spectral purity of the THz beatnote without any electrical component operating in the THz range.", "Indeed, the highest cut-off frequency in the experiment is that of the EOM, i.e., 10 GHz.", "Although demonstrated with a dual frequency laser, this FWM-assisted down-conversion technique is obviously well suited for two independent solid-state or semiconductor lasers.", "Moreover, the technique demonstrated here involves only widely spread and commercially available components in contrast to ref [9].", "Figure: (a) SSB Phase noise spectra.", "(1) N = 10 (10GHz) and (2) N = 20 (20GHz).", "(b) Intermediate frequency for a frequency difference Δν=ν e -ν o =\\Delta \\nu =\\nu _e-\\nu _o=770 GHz (RBW 30 kHz).Now, one can wonder if we can use this approach in order to actively stabilize the optical beatnote $\\Delta \\nu $ through the generated intermediate frequency.", "To this aim, we implement the experimental servo-loop depicted in Fig.", "REF .", "For the purpose of phase locking , it is convenient to down convert the laser beatnote directly to $f_i$ = 10 MHz, i.e., at the frequency of the quartz oscillator that will be used as frequency reference.", "This is obtained by adjusting precisely the modulation frequency $f_{RF}$ .", "We then electrically mix $f_i$ with the 10 MHz reference quartz oscillator of the RF synthesizer leading to a DC voltage proportional to the phase error.", "Through a loop filter (100 kHz bandwidth), we finally apply the error signal on the LiTaO$_3$ crystal located inside the laser cavity.", "The laser frequency difference $\\Delta \\nu $ is set at around 170 GHz.", "We have chosen this value for OEPLL demonstration purpose, instead of 770 GHz as before, because our laser was actually optimized in terms of optical power and long-term stability for optimal operation around $\\Delta \\nu =$ 100 GHz.", "To make the intermediate frequency $f_i$ close to 10 MHz, the EOM frequency $f_{RF}$ has to be adjusted to 10.5525 GHz.", "The value of the frequency difference being given by $\\Delta \\nu =f_i+2Nf_{RF}$ , where N is the locking harmonic order, its exact value is actually $\\Delta \\nu =$ 168.85 GHz with $N=$ 8.", "By closing the loop we stabilize the intermediate frequency and, consequently, the frequency difference $\\Delta \\nu $ .", "This result is reported in Fig.", "REF .", "Fig.", "REF (a) displays the electrical spectrum of the intermediate frequency $f_i$ when the loop is closed.", "The full width at half maximum is measured to be lower than 1 Hz limited by the resolution of our electrical spectrum analyzer.", "This leads to an in-loop relative instability $\\delta \\nu $ /$\\Delta \\nu $ at the 10$^{-11}$ level.", "As already mentioned, the signal-to-noise ratio is greatly improved as compared to the results presented in Ref.", "[9], 80 dB here.", "It is worthwhile to notice that the spectral purity of the stabilized beatnote is governed by the spectral purity of the RF synthesizer delivering $f_{RF}$ and the 10 MHz reference.", "To evaluate precisely the noise closer to the carrier, we measure now the single sideband (SSB) phase noise of the intermediate frequency (see Fig.", "REF (b)).", "The absolute phase noise is found to be $-$ 110 dBc/Hz at 10 kHz from the carrier.", "However, one can notice a resonant noise at 550 kHz whose origin is still not identified.", "Indeed, this frequency corresponds neither to the relaxation oscillation frequency of the laser (90 kHz) nor to the free spectral range of the optelectronic loop which is estimated at around 180 kHz.", "Further studies are actually devoted to figure out the origin of this excess noise.", "Figure: Complete experimental setup for the FWM-based OEPLL.", "Att, optical attenuator; LF, Loop Filter.Figure: (a)Electrical spectrum of the stabilized intermediate frequency f i f_i for a frequency difference Δν=\\Delta \\nu =168.85 GHz with a 1 Hz resolution bandwidth and a 80 Hz span.", "(b) Absolute SSB phase noise of the intermediate frequencyIn conclusion, we propose and demonstrate an optoelectronic down-conversion of a THz beat note to RF frequencies using four-wave mixing in a highly nonlinear fiber.", "This principle is illustrated using a dual frequency laser whose frequency difference is set at 770 GHz.", "Two interleaved combs are then generated around each optical carrier using a standard Mach-Zehnder modulator at 10 GHz followed by a highly nonlinear zero dispersion slope fiber.", "These combs are shown to contain more than 75 harmonics offering an intermediate frequency in the MHz range with an SNR of 70 dB.", "Moreover we show that the four wave mixing mechanism does not bring any additional degradation of the measured phase noise, at least up to the 20th harmonic.", "Hence, the intermediate frequency gives access to the phase noise of the THz beatnote using only commercially available optoelectronic components operating at room temperature.", "Although demonstrated using a dual frequency laser, the proposed down conversion technique could apply to a couple of detuned single mode lasers including semiconductor lasers.", "In a second part, this four-wave mixing down-conversion technique is advantageously implemented in an optoelectronic phase locked loop in order to directly phase-lock the laser beat note at 168 GHz to a 10 MHz local oscillator.", "The beat note linewidth is then reduced down to 1 Hz corresponding to an in-loop relative instability of 10$^{-11}$ .", "In these conditions, the phase noise level is measured to be $-$ 110 dBc/Hz at 10 kHz from the carrier and is shown to be limited by the RF synthesizer phase noise.", "This second part of the work is a new step for the generation of ultra- high spectral purity THz waves.", "Further studies include a detailed analysis and understanding of the resonant phase noise we observed at 500 kHz from the carrier and which is still unexplained.", "Moreover, out-of-loop measurements will be undertaken in order to fully characterize the stability of the THz beatnote.", "According to the significantly high down-conversion SNR offered by this four wave mixing approach, the phase locking of two independent lasers is now envisaged using a loop filter with increased bandwidth.", "The authors are very grateful to Goulc'hen Loas, Ludovic Frein, Cyril Hamel, Steve Bouhier and Anthony Carré for their help.", "This work is partially funded by Région Bretagne, Rennes Métropole, FEDER, and DGA." ] ]	1403.0508
[ [ "Representing, reasoning and answering questions about biological\n pathways - various applications" ], [ "Abstract Biological organisms are composed of numerous interconnected biochemical processes.", "Diseases occur when normal functionality of these processes is disrupted.", "Thus, understanding these biochemical processes and their interrelationships is a primary task in biomedical research and a prerequisite for diagnosing diseases, and drug development.", "Scientists studying these processes have identified various pathways responsible for drug metabolism, and signal transduction, etc.", "Newer techniques and speed improvements have resulted in deeper knowledge about these pathways, resulting in refined models that tend to be large and complex, making it difficult for a person to remember all aspects of it.", "Thus, computer models are needed to analyze them.", "We want to build such a system that allows modeling of biological systems and pathways in such a way that we can answer questions about them.", "Many existing models focus on structural and/or factoid questions, using surface-level knowledge that does not require understanding the underlying model.", "We believe these are not the kind of questions that a biologist may ask someone to test their understanding of the biological processes.", "We want our system to answer the kind of questions a biologist may ask.", "Such questions appear in early college level text books.", "Thus the main goal of our thesis is to develop a system that allows us to encode knowledge about biological pathways and answer such questions about them demonstrating understanding of the pathway.", "To that end, we develop a language that will allow posing such questions and illustrate the utility of our framework with various applications in the biological domain.", "We use some existing tools with modifications to accomplish our goal.", "Finally, we apply our system to real world applications by extracting pathway knowledge from text and answering questions related to drug development." ], [ "Introduction", "Biological organisms are composed of cells that contain numerous interconnected and interacting biochemical processes occurring simultaneously.", "Disruptions in the normal functionality of these processes causes diseases, which appear as symptoms (of these diseases).", "As a result understanding these processes is a fundamental activity in the biological domain and is prerequisite for activities such as disease diagnosis and drug discovery.", "One aspect of understanding the biological systems is the identification of pathways responsible for drug metabolism, diseases, and signal transduction, etc.", "The availability of high throughput approaches like micro-arrays, improvements in algorithms, and hardware that have come online during the last decade has resulted in significant refinement in these pathways.", "The pathways have become much larger in size and complexity to the degree that it is not reasonable for one person to fully retain all aspects of the pathway.", "As a result, computer based models of pathways are needed that allow the biologists to ask questions against them and compare them with real-world knowledge.", "The model should be such that it has an understanding of the pathway.", "Such a system would be considered intelligent and would assist the biologists in expanding the breadth of their search for new drugs and diagnoses.", "Source knowledge for these pathways comes from volumes of research papers published every year.", "Though there are a number of curated pathway resources available, they significantly lag behind the current state of the research in biology.", "As a result, we need a way to extract this pathway information from published text." ], [ "Choosing the right questions", "A large body of research exists on computer modeling of biological processes and it continues to be an active area of research.", "However, many such models focus on surface properties, like structure; or factoid questions.", "Though important, we feel these systems do not test the understanding of the underlying system being modeled.", "We want to go beyond this surface level information and answer questions requiring deeper reasoning.", "We want our system to answer questions that a biology teacher expects his / her students to answer after reading the required text.", "So, we turned to college level biological text books for the questions that we feel are more indicative of such understanding.", "Following questions from [64] illustrate the kind of questions we are interested in answering: “What would happen to the rate of glycolysis if DHAP were removed from the process of glycolysis as quickly as it was produced?” “A muscle cell had used up its supply of oxygen and ATP.", "Explain what affect would this have on the rate of cellular respiration and glycolysis?” These questions and others like it were the subject of a recent deep knowledge representation challengehttps://sites.google.com/site/2nddeepkrchallenge/.", "In this thesis, we focus on questions that require reasoning over simulations." ], [ "Choosing the right tools", "Data about biological systems can be qualitative or quantitative in nature.", "The fully quantitative data about reaction dynamics is based on ordinary differential equations (ODEs) of reaction kinetics, which are often lacking [16].", "Qualitative data is more prevalent.", "It is less precise, but tends to capture general relationships between various components of a biological pathway.", "Adding quantitative information to a qualitative model provides the next step in refinement of the biological pathways [36], providing better coverage of biological systems and processes.", "We want to use this qualitative+quantitative data for our modeling.", "To simulate and reason with the pathways, we need tools that can model a biological pathway that contains qualitative+quantitative information, simulate the pathway and reason with the results." ], [ "Need for a pathway a specification and a query language", "Pathway information comes in various formats, such as cartoon drawings, formal graphical representations like Kohn's Maps [43], curated databases of pathways [40], [41], [19] and free text.", "The depth of this knowledge as well as its taxonomy varies with the source.", "Thus, a common specification language is needed.", "Such a language must be easy to understand and must have a well defined semantics.", "Queries are normally specified in natural language, which is vague.", "So, a more precise query language is needed.", "One could ask queries in one of the existing formal languages [27], but that will be burdensome for a user to become fluent.", "As a result, we need a language that has a simple English-like syntax, but a well defined semantics, so that it does not have the vagaries of the natural language." ], [ "Text extraction", "Knowledge about biological pathways is spread over collections of published papers as nuggets of information, such as relationships between proteins; between proteins and drugs; genetic variation; and association of population groups with genetic variation; to name a few.", "Published research may also contain contradictory information, e.g.", "an earlier conjecture that was proven to be untrue in later research, or knowledge with limited amount of certainty.", "To extract these nuggets and to assemble them into a coherent pathway requires background knowledge, similar to other technical fields.", "Portions of this knowledge are published in books and online repositories.", "Thus, we need a method of text extraction that allows one to extract nuggets of information, consult available databases and produce knowledge about pathway that is self-consistent." ], [ "Overview", "In this thesis, we propose to build a system, called BioPathQA, to answer deeper reasoning questions using existing tools with modifications.", "To that end, we develop a language to specify pathways and queries.", "Our system is designed to answer reasoning questions requiring simulation.", "We demonstrate the applicability of our system with applications to drug development on knowledge obtained from text extraction.", "To implement an answering system that can answer simulation based reasoning questions, we first looked for available tools that could help in this task and we found Petri Nets as providing the right level of formalism for our application.", "Petri Nets [61] are a popular representation formalism used to model biological systems and to simulate them.", "They have been used to model and analyze the dynamic behavior as well as structural properties of biological systems.", "However, such analysis is usually limited to invariant determination, liveness, boundedness and reachability.", "To our knowledge they have not been used to answer questions similar to the aforementioned.", "In order to represent deeper reasoning questions, we have to make extensions to the Petri Net model as the basic model lacks sufficient richness.", "For example, we may want to change the firing semantics to limit the state space or maximize parallel activity.", "Although numerous Petri Net modeling, simulation and analysis systems exist [39], [67], [44], [8], [58], [48], we found certain limitations in the default implementation of these systems as well that prevented us from using them as is.", "For example, the Colored Petri Net implementation CPNtools http://cpntools.org does not allow inhibitor arcs (we use to model protein inhibition); Cell Illustrator [58] is closed source and does not support colored tokens (we use to model locations); Snoopy [67] supports a large number of extensions, but it is unclear how one exports the simulation results for further reasoning; and most did not allow exploring all possible state evolutions of a pathway or using different firing semantics.", "To make these extensions in an easy manner we use Answer Set Programming (ASP) [54] as the language to represent and simulate Petri Nets.", "It allows a simple encoding of the Petri Net and can be easily extended to incorporate extensions Certain commercial tools, like Cell Illustrator (http://www.cellillustrator.com) do allow exporting their model into a high level language, but we believe that a declarative language is more suited to succinctly describe the problem..", "In addition, ASP allows powerful reasoning capability and the possibility of implementing additional constructs not supported by Petri Nets directly, such as the ability to filter trajectories.", "Petri Net to ASP translation has been studied before [6], [37].", "However, these implementations have been limited to specific classes of Petri Nets and have different focus.", "For example, the Simple Logic Petri Nets [6] do not allow numerical accumulation of the same tokens from multiple transitions to a single place and the Binary Petri Nets [37] do not allow more than one tokens at any place." ], [ "Specific contributions", "The research contribution of this thesis can be divided into four major parts.", "The first part gives a general encoding of Petri Nets in ASP, which allows easy extension by making local changes.", "The second part shows how the ASP encoding of Petri Nets can be used to answer simulation based reasoning questions.", "The third part describes the high-level language for pathway and query specification; and the system that we have developed to answer deep reasoning questions.", "The fourth part shows how knowledge is extracted from text of research papers, cleaned and assembled into a pathway to answer simulation based reasoning questions using our system." ], [ "General ASP encoding of Petri Net for simulation", "Although previous work on encoding Petri Nets in ASP exists, it is limited to specific classes of Petri Nets.", "We present an encoding of a basic Petri Net in ASP to show it is done in an intuitive yet concise way.", "The default execution semantics of a Petri Net is the set-semantics, which allows a subset of transitions to fire simultaneously when ready.", "This can result in far too many combinations of transition firing arrangements.", "A simpler approach is to use the so called interleaved execution semantics, in which at most one transition fires when ready.", "This too can generate many firing arrangements.", "Biological systems are highly parallel in nature, as a result it is beneficial to model maximum parallel activity.", "So, we introduce a new firing semantics, called the maximal firing set semantics by extending the set semantics.", "In this semantics, a maximal subset of non-conflicting enabled transitions fire simultaneously when ready.", "Then, we extend the basic ASP encoding to include Petri Net extensions like reset-arcs (to model immediate consumption of any amount of substrate), inhibit-arcs (to model gene/protein inhibition), read-arcs (to model additional pre-conditions of a reaction, such as different start vs. maintenance quantity of a reactant), colored-tokens (to model quantities of different types of substances at the same location), priority-transitions (to select between alternate metabolic paths), and timed-transitions (to model slow reactions) that allow modeling of various concepts in biological systems.", "We show how ASP allows us to make these extensions with small amount of local changes.", "This component is one of the major focuses of our research.", "It is described in Chapter and is the basis for implementation of our system to model pathways and answer questions about them." ], [ "Answering simulation based reasoning questions", "We use the encoding developed in Chapter to questions in [64] that were a part of the Second Deep Knowledge Representation Challenge https://sites.google.com/site/2nddeepkrchallenge/.", "These questions are focused on the mechanism of cellular respiration and test the understanding of the student studying the material; and appear in two main forms: [(i)] inquiry about change of rate of a process due to a change in the modeled system, and explanation of a change due to a change in the modeled system.", "We built Petri Net models for the situations specified in the questions, encoded them in ASP and simulated them over a period of time.", "For change of rate questions, we computed the rate for both nominal and modified cases and observed that they matched the responses provided with the challenge questions.", "For the explanation of change questions, we collected the summary of firing transitions as well as substance quantities produced at various times.", "This information formed the basis of our answer.", "We compared our results with the answers provided with the challenge questions.", "A novel aspect of our approach is that we apply the initial conditions and interventions mentioned in the questions as modifications to the pathway representation.", "These interventions can be considered as a generalized form of actions.", "For certain questions, additional domain knowledge outside the source material was required.", "We filled this gap in knowledge as necessary.", "We also kept the models to a subset of the pathway for performance as well as to reduce clutter in the output that can bury the results with unnecessary details.", "This component of our research is described in Chapter .", "BioPathQA: a system and a language to represent pathways and query them We combined the techniques learned from Chapter , action languages, and biological modeling languages to build a question answering system that takes a pathway and a query as input.", "Both the pathway specification language and the query language have strict formal semantics, which allow them to be free of the vagaries of natural language, the language of the research papers as well as the query statements.", "Guarded-Arc Petri Net Since the biological pathways are constructed of biochemical reactions, they are effected by environmental changes.", "Mutations within the cell can also result in conditional change in behavior of certain processes.", "As a result, we needed actions with conditional effects.", "Our Petri Net model wasn't rich enough to model conditional actions, so we extended the Petri Nets with conditional arcs.", "We call this extension, the Guarded-Arc Petri Net, where a guard is a condition on an arc, which must be true for that arc to be traversed.", "With this extension, a Petri Net transition can have different outcomes for different markings.", "Our model is similar to the model in [39] in many aspects.", "This component of our research is described in Chapter .", "Text Extraction to Answer Questions about Real World Applications To apply our system to real world applications, we have to extract pathway knowledge from published papers, which are published in natural language text.", "For this, we use a system called the Parse Tree Query Language (PTQL) [75] to nuggets of information from the abstracts published on PubMed http://www.ncbi.nlm.nih.gov/pubmed.", "Sentences are parsed using the Link Grammar [69] or Stanford Parser [20]; with various object-classes identified within the sentence.", "Unlike Information Retrieval (IR) approaches that tend to treat documents as unstructured bags-of-words, PTQL treats words (or word-groups) as sentence elements with syntactic as well as dependency relationships between them.", "PTQL queries combine lexical, syntactic and semantic features of sentence elements.", "Thus with PTQL, one can ask for the first-noun of a noun-phrase that is the direct-object of a verb-phrase for some specific verb string.", "To accomplish its task, PTQL performs a number of pre-processing steps on its input useful for text extraction and leverages on various existing databases.", "These include sentence splitting, tokenization, part-of-speech (POS) tagging, named entity recognition, entity-mention normalization, cross-linking with concepts from external databases, such as Gene Ontology [14] and UniProt [18].", "We extract gene-gene, gene-drug, and gene-disease relationships using PTQL, assemble them into a pathway specification and reason with the extracted knowledge to determine possible drug interactions.", "Facts and relationships extracted using PTQL are further subject to filtering to remove inconsistent information.", "A pathway specification is then constructed from the extracted facts, which can be queried using the query specification language.", "We illustrate the use of our deep reasoning system by an example from the drug-drug interaction domain.", "This component is described in Chapter .", "Summary The main contributions of this thesis can be summarized as follows: Generalized Petri Net encoding in ASP, including a new maximal firing set semantics (Chapter ) An easy to extend encoding is developed, that allows adding extensions using local changes A new Petri Net firing semantics, the so called maximal firing set semantics is defined, which ensures maximum possible parallel activity at any given point Answering simulation based deep reasoning questions using our ASP encoding (Chapter ) It is shown, how deep reasoning questions requiring simulation based reasoning can be answered.", "Developed a system called BioPathQA and a language to specify biological pathways and answer deep reasoning questions about it (Chapter ) A pathway specification language is developed, combining concepts from Petri Nets, Action Languages, and Biological Pathways A query specification language is developed, which is english like, with well defined semantics, avoiding the vagaries of Natural Language A description of our implementation using ASP and Python is given; and an execution trace is shown Performed text extraction to extract pathway knowledge (Chapter ) It is shown pathway knowledge is extracted and used to answer questions in the drug-drug interaction domain Petri Net Encoding in ASP for Biological Domain Introduction Petri Net [61] is a graphical modeling language with formal semantics used for description of distributed systems.", "It is named after Carl Adam Petri, who formally defined Petri Nets in his PhD thesis in the 1960's [11].", "Petri nets have been widely used to model a wide range of systems, from distributed systems to biological pathways.", "The main advantages of Petri Net representation include its simplicity and the ability to model concurrent and asynchronous systems and processes.", "A variety of Petri Net extensions have been proposed in the literature, e.g.", "inhibitor arcs, reset transitions, timed transitions, stochastic transitions, prioritized transitions, colored petri nets, logic petri nets, hierarchical petri nets, hybrid petri nets and functional petri nets to a name a few [6], [57], [34].", "Our interest in Petri Nets is for representing biological pathways and simulating them in order to answer simulation based reasoning questions.", "We show how Petri nets can be represented in ASP.", "We also demonstrate how various extensions of basic Petri nets can be easily expressed and implemented by making small changes to the initial encoding.", "During this process we will relate the extensions to their use in the biological domain.", "Later chapters will show how this representation and simulation is used to answer biologically relevant questions.", "The rest of this chapter is organized as follows: We present some background material on Answer Set Programming (ASP) and Petri Nets.", "Following that, we present the Answer Set encoding of a basic Petri Net.", "After that we will introduce various Petri Nets extensions and the relevant ASP code changes to implement such extensions.", "Background Answer Set Programming Answer Set Programming (ASP) is a declarative logic programming language that is based on the Stable Model Semantics [25].", "It has been applied to a problems ranging from spacecrafts, work flows, natural language processing and biological systems modeling.", "Although ASP language is quite general, we limit ourselves to language and extensions relevant to our work.", "Definition 1 (Term) A term is a term in the propositional logic sense.", "Definition 2 (Literal) A literal is an atom in the propositional logic sense.", "A literal prefixed with $\\mathbf {not}$ is referred to as a negation-as-failure literal or a naf-literal, with $\\mathbf {not}$ representing negation-as-failure.", "We will refer to propositional atoms as basic atoms to differentiate them from other atoms, such as the aggregate atoms defined below.", "Definition 3 (Aggregate Atom) A sum aggregate atom is of the form: $L \\; [ B_0=w_0,\\dots ,B_m=w_m ] \\; U$ where, $B_i$ are basic atoms, $w_i$ are positive integer weight terms, $L,U$ are integer terms specifying the lower and upper limits of aggregate weights.", "The lower and upper limits are assumed to be $-\\infty $ and $\\infty $ , if not specified.", "A count aggregate atom is a special case of the sum aggregate atom in which all weights are 1, i.e.", "$L \\; [ B_0=1,\\dots ,B_m=1] \\; U$ and it is represented by: $L \\; \\lbrace B_0,\\dots ,B_m \\rbrace \\; U$ A choice atom is a special case of the count aggregate atom (REF ) in which $n=m$ .", "Definition 4 (ASP Program) An ASP program $\\Pi $ is a finite set of rules of the following form: $A_0 \\leftarrow A_1,\\dots ,A_m,\\mathbf {not~} B_{1},\\dots ,\\mathbf {not~} B_n, C_1,\\dots ,C_k.$ where each $A_0$ is either a basic atom or a choice atom, $A_i$ and $B_i$ are basic atoms, $C_i$ are aggregate atoms and $\\mathbf {not}$ is negation-as-failure.", "In rule (REF ), $\\lbrace A_0 \\rbrace $ is called the head of the rule, and $\\lbrace A_1,\\dots ,$ $A_m,$ $\\mathbf {not~} B_{1},\\dots ,$ $\\mathbf {not~} B_n, $ $C_1,\\dots ,$ $C_k\\rbrace $ is called its tail.", "A rule in which $A_0$ is a choice atom is called a choice rule.", "A rule without a head is called a constraint.", "A rule with a basic atom as its head and empty tail is called a fact in which case the “$\\leftarrow $ ” is dropped.", "Let $R$ be an ASP rule of the form (REF ) and let $pos(R) = \\lbrace A_1,\\dots ,A_m \\rbrace $ represent the positive atoms, $neg(R) = \\lbrace B_1,\\dots ,B_n \\rbrace $ the negation-as-failure atoms, and $agg(R) = \\lbrace C_1,\\dots ,C_k \\rbrace $ represent the aggregate atoms in the body of a rule $R$ .", "Let $lit(A)$ represent the set of basic literals in atom $A$ , i.e.", "$lit(A)=\\lbrace A\\rbrace $ if $A$ is a basic atom; $lit(A)=\\lbrace B_0,\\dots ,B_n\\rbrace $ if $A$ is an aggregate atom.", "Let $C$ be an aggregate atom of the form (REF ) and let $pos(C) = \\lbrace B_0,\\dots ,B_m \\rbrace $ be the sets of basic positive literals such that $lit(C) = pos(C)$ .", "Let $lit(R) = lit(head(R)) \\cup pos(R) \\cup neg(R) \\cup \\bigcup _{C \\in agg(R)}{lit(C)} $ for a rule $R \\in \\Pi $ and $lit(\\Pi ) = \\bigcup _{R \\in \\Pi }{lit(R)}$ be the set of basic literals in ASP program $\\Pi $ .", "Definition 5 (Aggregate Atom Satisfaction) A ground aggregate atom $C$ of the form (REF ) is satisfied by a set of basic ground atoms $S$ , if $L \\le \\sum _{0 \\le i \\le m, B_i \\in S}{w_i} \\le U$ and we write $S \\models C$ .", "Given a set of basic ground literals $S$ and a basic ground atom $A$ , we say $S \\models A$ if $A \\in S$ , $S \\models \\mathbf {not~} A$ if $A \\notin S$ .", "For a rule $R$ of the form (REF ) $S \\models body(R)$ if $\\forall A \\in \\lbrace A_1,\\dots ,A_m\\rbrace , S \\models A$ , $\\forall B \\in \\lbrace B_{1},\\dots ,B_n \\rbrace , S \\models \\mathbf {not~} B$ , and $\\forall C \\in \\lbrace C_1,\\dots ,C_k \\rbrace , S \\models C$ ; $S \\models head(R)$ if $S \\models A_0$ .", "Definition 6 (Rule Satisfaction) A ground rule $R \\in \\Pi $ is satisfied by a set of basic ground atoms $S$ , iff, $S \\models body(R)$ implies $S \\models head(R)$ .", "A constraint rule $R \\in \\Pi $ is satisfied by set $S$ if $S \\lnot \\models body(R)$ .", "We define reduct of an ASP program by treating aggregate atoms in a similar way as negation-as-failure literals, since our code does not contain recursion through aggregation (which can yield non-intuitive answer-sets [70]).", "Definition 7 (Reduct) Let $S$ be a set of ground basic atoms, the reduct of ground ASP program $\\Pi $ w.r.t.", "$S$ , written $\\Pi ^S$ is the set of rules: $\\lbrace p \\leftarrow A_1,\\dots ,A_m.", "\\; \| \\; A_0 \\leftarrow A_1, \\dots , $ $A_m, $ $\\mathbf {not~} B_{1}, \\dots , $ $\\mathbf {not~} B_n, $ $C_1,\\dots ,$ $C_k.", "\\in \\Pi , p \\in lit(A_0) \\cap S, \\lbrace A_1,\\dots ,A_m\\rbrace \\subseteq S, \\lbrace B_1,\\dots ,B_n \\rbrace \\cap S = \\emptyset , $ $\\nexists C \\in \\lbrace C_1,\\dots ,C_k \\rbrace , S \\lnot \\models C \\rbrace $ .", "Intuitively, this definition of reduct removes all rules which contain a naf-literal or an aggregate atom in their bodies that does not hold in $S$ , and it removes aggregate atoms as well as naf-literals from the body of the remaining rules.", "Heads of choice-rules are split into multiple rules containing at most one atom in their heads.", "The resulting reduct is a program that does not contain any aggregate atoms or negative literals.", "The rules of such a program are monotonic, such that if it satisfied by a set $S$ of atoms, it is also satisfied by any superset of $S$ .", "A deductive closure of such a (positive) monotonic program is defined as the unique smallest set of atoms $S$ such that whenever all body atoms of a rule hold in $S$ , the head also holds in $S$ .", "The deductive closure can be iteratively computed by starting with an empty set and adding heads of rules for which the bodies are satisfied, until a fix point is reached, where no additional rules can be satisfied.", "(adopted from [4]) Definition 8 (Answer Set) A set of basic ground atoms $S$ is an answer set of a ground ASP program $\\Pi $ , iff $S$ is equal to the deductive closure of $\\Pi ^S$ and $S$ satisfies each rule of $\\Pi $ .", "(adopted from [4]) Clingo Specific Syntactic Elements The ASP code in this thesis is in the syntax of ASP solver called clingo [24].", "The “$\\leftarrow $ ” in ASP rules is replaced by the symbol “:-”.", "Though the semantics of ASP are defined on ground programs, Clingo allows variables and other constructs for compact representation.", "We intuitively describe specific syntactic elements and their meanings below: Comments: Text following “%” to the end of the line is treated as a comment.", "Interval: Atoms defined over an contiguous range of integer values can be compactly written as intervals, e.g.", "$p(1\\;..\\;5)$ represents atoms $p(1), p(2), p(3), p(4), p(5)$ .", "Pooling: Symbol “;” allows for pooling alternative terms.", "For example, an atom $p(\\dots ,X,\\dots )$ and $p(\\dots ,Y,\\dots )$ can be pooled together into a single atom as $p(\\dots ,X;Y,\\dots )$ .", "Aggregate assignment atom: An aggregate assignment atom $Q=\\#sum[A_0=w_0,$ $\\dots ,$ $A_m=w_m,$ $\\mathbf {not~} A_{m+1}=w_{m+1},\\dots ,$ $\\mathbf {not~} A_n=w_n]$ assigns the sum $\\sum _{A_i \\in S, 0 \\le i \\le m}{w_i} + $ $\\sum _{A_j \\notin S, m+1 \\le n}{w_j}$ to $Q$ w.r.t.", "a consistent set of basic ground atoms $S$ .", "Condition: Conditions allow instantiating variables to collections of terms within aggregates, e.g.", "$\\lbrace p(X) : q(X) \\rbrace $ instantiates $p(X)$ for only those $X$ that $q(X)$ satisfies.", "For example, if we have $p(1..5)$ but only $q(3;5)$ , then $\\lbrace p(X) : q(X) \\rbrace $ is expanded to $\\lbrace p(3), p(5) \\rbrace $ .", "Grounding Grounding makes a program variable free by replacing variables with the possible values they can take.", "Clingo uses the grounder Gringo for “smart” grounding, which results in substantial reduction in the size of the program.", "Details of this grounding are implementation specific.", "We present the intuitive process of grounding below.", "A set of ground terms is constructed, where a ground term is a term that contains no variables.", "The variables are split into two categories: local and global.", "Local variables are the ones that appear only within an aggregate atom (minus the limits) and nowhere else in a rule.", "Such variables are considered local writ.", "the aggregate atom.", "All other variables are considered global.", "First the global variables are eliminated in the rules as follows: Each rule $r$ containing an aggregate assignment atom of the form (REF ) is replaced with set of rules $r^{\\prime }$ in which the aggregate assignment atom is is replaced with an aggregate atom with lower and upper bounds of $Q$ for all possible substitutions of $Q$ .", "This is generalized to multiple aggregate assignment atoms by repeating this step for each such atom, where output of previous iteration forms the input of the next iteration.", "Each rule $r^{\\prime }$ , is replaced with the set of all rules $r^{\\prime \\prime }$ obtained by all possible substitutions of ground terms for global variables in $r$ .", "Then the local variables are eliminated in the rules by expanding conditions, such that $p(\\dots ,X,\\dots ) : d(X)$ are replaced by $p(\\dots ,d_1,\\dots ), \\dots , p(\\dots ,d_k,\\dots )$ for the extent $\\lbrace d_1,\\dots ,d_k\\rbrace $ of $d(X)$ .", "This is generalized to multiple conditions in the obvious way.", "Following the convention of the Clingo system, Variables in rules presented in this thesis start with capital letters while lower-case text and numbers are constants.", "Italicized text represents a constant term from a definition in context.", "A recent work [35] gives the semantics of Gringo with ASP Core 2 syntax [13] using Infintary Propositional Formulas, which translate Gringo to propositional formulas with infinitely long conjunctions and disjunctions.", "Their approach removes the safety requirement, but the subset of Gringo presented does appear to cover assignments.", "Although their approach provides a way to improve our ASP encoding by removing the requirement of specifying the maximum number of tokens or running simulations until a condition holds, our simpler (limited) semantics is sufficient for the limited syntax and semantics we use.", "Multiset A multiset $A$ over a domain set $D$ is a pair $\\langle D,m \\rangle $ , where $m: D \\rightarrow {N}$ is a function giving the multiplicity of $d \\in D$ in $A$ .", "Given two multsets $A = \\langle D,m_A \\rangle , B = \\langle D,m_B \\rangle $ , $A \\odot B$ if $\\forall d \\in D: m_A(d) \\odot m_B(d)$ , where $\\odot \\in \\lbrace <,>,\\le ,\\ge ,=\\rbrace $ , and $A \\ne B$ if $\\exists d \\in D : m_A(d) \\ne m_B(d)$ .", "Multiset sum/difference is defined in the usual way.", "We use the short-hands $d \\in A$ to represent $m_A(d) > 0$ , $A = \\emptyset $ to represent $\\forall d \\in D, m(d) = 0$ , $A \\otimes n$ to represent $\\forall d \\in D, m(d) \\otimes n$ , where $n \\in {N}$ , $\\otimes \\in \\lbrace <,>,\\le ,\\ge ,=,\\ne \\rbrace $ .", "We use the notation $d/n \\in A$ to represent that $d$ appears $n$ -times in $A$ ; we drop $A$ when clear from context.", "The reader is referred to [71] for details.", "Petri Net A Petri Net is a graph of a finite set of nodes and directed arcs, where nodes are split between places and transitions, and each arc either connects a place to a transition or a transition to a place.", "Each place has a number of tokens (called the its marking) Standard convention is to use dots in place nodes to represent the marking of the place.", "We use numbers for compact representation..", "Collective marking of all places in a Petri Net is called its marking (or state).", "Arc labels represent arc weights.", "When missing, arc-weight is assumed as one, and place marking is assumed as zero.", "Figure: Petri Net graph (of sub-section of glycolysis pathway) showing places as circles, transitions as boxes and arcs as directed arrows.", "Places have token count (or marking) written above them, assumed 0 when missing.", "Arcs labels represent arc-weights, assumed 1 when missing.The set of place nodes on incoming and outgoing arcs of a transition are called its pre-set (input place set or input-set) and post-set (output place set or output-set), respectively.", "A transition $t$ is enabled when each of its pre-set place $p$ has at least the number of tokens equal to the arc-weight from $p$ to $t$ .", "An enabled transition may fire, consuming tokens equal to arc-weight from place $p$ to transition $t$ from each pre-set place $p$ , producing tokens equal to arc-weight from transition $t$ to place $p$ to each post-set place $p$ .", "Multiple transitions may fire as long as they consume no more than the available tokens, with the assumption that tokens cannot be shared.", "Fig.", "REF shows a representation of a portion of the glycolysis pathway as given in [64].", "In this figure, places represent reactants and products, transitions represent reactions, and arc weights represent reactant quantity consumed or the product quantity produced by the reaction.", "When unspecified, arc-weight is assumed to be 1 and place-marking is assumed to be 0.", "Definition 9 (Petri Net) A Petri Net is a tuple $PN=(P,T,E,W)$ , where, $P=\\lbrace p_1, \\dots , p_n\\rbrace $ is a finite set of places; $T=\\lbrace t_1, \\dots , t_m\\rbrace $ is a finite set of transitions, $P \\cap T = \\emptyset $ ; $E^+ \\subseteq T \\times P$ is a set of arcs from transitions to places; $E^- \\subseteq P \\times T$ is a set of arcs from places to transitions; $E= E^+ \\cup E^- $ ; and $W: E \\rightarrow {N}\\setminus \\lbrace 0\\rbrace $ is the arc-weight function Definition 10 (Marking) A marking $M=(M(p_1),\\dots ,M(p_{n}))$ is the token assignment of each place node $p_i \\in P$ of $PN$ , where $M(p_i) \\in {N}$ .", "Initial token assignment $M_0: P \\rightarrow {N}$ is called the initial marking.", "Marking at step $k$ is written as $M_k$ .", "Definition 11 (Pre-set & post-set of a transition) Pre-set / input-set of a transition $t \\in T$ of $PN$ is $\\bullet t = \\lbrace p \\in P : (p,t) \\in E^- \\rbrace $ , while the post-set / output-set is $t \\bullet = \\lbrace p \\in P : (t,p) \\in E^+ \\rbrace $ Definition 12 (Enabled Transition) A transition $t \\in T$ of $PN$ is enabled with respect to marking $M$ , $enabled_M(t)$ , if $\\forall p \\in \\bullet t, W(p,t) \\le M(p)$ .", "An enabled transition may fire.", "Definition 13 (Transition Execution) A transition execution is the simulation of change of marking from $M_k$ to $M_{k+1}$ due to firing of a transition $t \\in T$ of $PN$ .", "$M_{k+1}$ is computed as follows: $ \\forall p_i \\in \\bullet t, M_{k+1}(p_i) = M_{k}(p_i) - W(p_i,t) $ $ \\forall p_j \\in t \\bullet , M_{k+1}(p_j) = M_{k}(p_j)+ W(t,p_j) $ Petri Nets allow simultaneous firing of a set of enabled transitions w.r.t.", "a marking as long as they do not conflict.", "Definition 14 (Conflicting Transitions) Given $PN$ with marking $M$ .", "A set of enabled transitions $T_e = \\lbrace t \\in T : enabled_M(t) \\rbrace $ of $PN$ conflict if their simultaneous firing will consume more tokens than are available at an input place: $\\exists p \\in P : M(p) < \\displaystyle \\sum _{\\begin{array}{c}t \\in T_e \\wedge p \\in \\bullet t\\end{array}}{W(p,t)}$ Definition 15 (Firing Set) A firing set is a set $T_k=\\lbrace t_{1},\\dots ,t_{m}\\rbrace \\subseteq T$ of simultaneously firing transitions that are enabled and do not conflict w.r.t.", "to the current marking $M_k$ of $PN$ .", "Definition 16 (Firing Set Execution) Execution of a firing set $T_k$ of $PN$ on a marking $M_{k}$ computes the new marking $M_{k+1}$ as follows: $\\forall p \\in P, M_{k+1}(p) = M_k(p)- \\sum _{\\begin{array}{c}t \\in T_k \\wedge p \\in \\bullet t\\end{array}} W(p,t)+ \\sum _{\\begin{array}{c}t \\in T_k \\wedge p \\in t \\bullet \\end{array}} W(t,p)$ where $\\sum _{\\begin{array}{c}t \\in T_k \\wedge p \\in \\bullet t\\end{array}} W(p,t)$ is the total consumption from place $p$ and $\\sum _{\\begin{array}{c}t \\in T_k \\wedge p \\in t \\bullet \\end{array}} W(t,p)$ is the total production at place $p$ .", "Definition 17 (Execution Sequence) An execution sequence $X = M_0, T_0, M_1, T_1, \\dots , $ $M_k, T_k, M_{k+1}$ of $PN$ is the simulation of a firing set sequence $\\sigma = T_1,T_2,\\dots ,T_k$ w.r.t.", "an initial marking $M_0$ , producing the final marking $M_{k+1}$ .", "$M_{k+1}$ is the transitive closure of firing set executions, where subsequent marking become the initial marking for the next firing set.", "For an execution sequence $X = M_0, T_0, M_1, T_1, \\dots , M_k, T_k, M_{k+1}$ , the firing of $T_0$ with respect to marking $M_0$ produces the marking $M_1$ which becomes the initial marking for $T_1$ , which produces $M_2$ and so on.", "Translating Basic Petri Net Into ASP In this section we present ASP encoding of simple Petri Nets.", "We describe, how a given Petri Net $PN$ , and an initial marking $M_0$ are encoded into ASP for a simulation length $k$ .", "Following sections will show how Petri Net extensions can be easily added to it.", "We represent a Petri Net with the following facts: f1: Facts place($p_i$ ).", "where $p_i \\in P$ is a place.", "f2: Facts trans($t_j$ ).", "where $t_j \\in T$ is a transition.", "f3: Facts ptarc($p_i,t_j,W(p_i,t_j)$ ).", "where $(p_i,t_j) \\in E^-$ with weight $W(p_i,t_j)$ .", "f4: Facts tparc($t_i,p_j,W(t_i,p_j)$ ).", "where $(t_i,p_j) \\in E^+$ with weight $W(t_i,p_j)$ .", "Petri Net execution simulation proceeds in discrete time-steps, these time steps are encoded by the following facts: f5: Facts time($ts_i$ ) where $0 \\le ts_i \\le k$ .", "The initial marking (or initial state) of the Petri Net is represented by the following facts: i1: Facts holds($p_i,M_0(p_i),0$ ) for every place $p_i \\in P$ with initial marking $M_0(p_i)$ .", "ASP requires all variables in rule bodies be domain restricted.", "So, we add the following facts to capture the possible token quantities produced during the simulation Note that $ntok$ can be arbitrarily chosen to be larger than the maximum expected token quantity produced during the simulation.", ": f6: Facts num($n$ )., where $0 \\le n \\le ntok$ A transition $t_i$ is enabled if each of its input places $p_j \\in \\bullet t_i$ has at least arc-weight $W(p_j, t_i)$ tokens.", "Conversely, $t_i$ is not enabled if $\\exists p_j \\in \\bullet t_i : M(p_j) < W(p_j,t_i)$ , and is only enabled when no such place $p_j$ exists.", "These are captured in $e\\ref {e:ne:ptarc}$ and $e\\ref {e:enabled}$ respectively: e1: notenabled(T,TS):-ptarc(P,T,N),holds(P,Q,TS),Q<N, place(P), trans(T), time(TS),num(N),num(Q).", "e2: enabled(T,TS) :- trans(T), time(TS), not notenabled(T, TS).", "Rule $e\\ref {e:ne:ptarc}$ encodes notenabled(T,TS) which captures the existence of an input place $P$ of transition $T$ that violates the minimum token requirement $N$ at time-step $TS$ .", "Where, the predicate holds(P,Q,TS) encodes the marking $Q$ of place $P$ at $TS$ .", "Rule $e\\ref {e:enabled}$ encodes enabled(T,TS) which captures that transition $T$ is enabled at $TS$ since there is no input place $P$ of transition $T$ that violates the minimum input token requirement at $TS$ .", "In biological context, $e\\ref {e:enabled}$ captures the conditions when a reaction (represented by $T$ ) is ready to proceed.", "A subset of enabled transitions may fire simultaneously at a given time-step.", "This is encoded as: a1: {fires(T,TS)} :- enabled(T,TS), trans(T), time(TS).", "Rule $a\\ref {a:fires}$ encodes fires(T,TS), which captures the firing of transition $T$ at $TS$ .", "The rule is encoded with a count atom as its head, which makes it a choice rule.", "This rule either picks the enabled transition $T$ for firing at $TS$ or not, effectively enumerating a subset of enabled transitions to fire.", "Whether this set can fire or not in an answer set is subject to conflict checking, which is done by rules $a\\ref {a:overc:place},a\\ref {a:overc:gen},a\\ref {a:overc:elim}$ shown later.", "In biological context, the selected transition-set models simultaneously occurring reactions and the conflict models limited reactant supply that cannot be shared.", "Such a conflict can lead to multiple choices in parallel reaction evolutions and different outcomes.", "The next set of rules captures the consumption and production of tokens due to the firing of individual transitions in a firing-set as well as their aggregate effect, which computes the marking for the next time step: r1: add(P,Q,T,TS) :- fires(T,TS), tparc(T,P,Q), time(TS).", "r2: del(P,Q,T,TS) :- fires(T,TS), ptarc(P,T,Q), time(TS).", "r3: tot_incr(P,QQ,TS) :- QQ=#sum[add(P,Q,T,TS)=Q:num(Q):trans(T)], time(TS), num(QQ), place(P).", "r4: tot_decr(P,QQ,TS) :- QQ=#sum[del(P,Q,T,TS)=Q:num(Q):trans(T)], time(TS), num(QQ), place(P).", "r5: holds(P,Q,TS+1) :-holds(P,Q1,TS),tot_incr(P,Q2,TS),time(TS+1), tot_decr(P,Q3,TS),Q=Q1+Q2-Q3,place(P),num(Q;Q1;Q2;Q3),time(TS).", "Rule $r\\ref {r:add}$ encodes add(P,Q,T,TS) and captures the addition of $Q$ tokens to place $P$ due to firing of transition $T$ at time-step $TS$ .", "Rule $r\\ref {r:del}$ encodes del(P,Q,T,TS) and captures the deletion of $Q$ tokens from place $P$ due to firing of transition $T$ at $TS$ .", "Rules $r\\ref {r:totincr}$ and $r\\ref {r:totdecr}$ aggregate all add's and del's for place $P$ due to $r\\ref {r:add}$ and $r\\ref {r:del}$ at time-step $TS$ , respectively, by using the QQ=#sum[] construct to sum the $Q$ values into $QQ$ .", "Rule $r\\ref {r:nextstate}$ which encodes holds(P,Q,TS+1) uses these aggregate adds and removes and updates $P$ 's marking for the next time-step $TS+1$ .", "In biological context, these rules capture the effect of a reaction on reactant and product quantities available in the next simulation step.", "To prevent overconsumption at a place following rules are added: a2: consumesmore(P,TS) :- holds(P,Q,TS), tot_decr(P,Q1,TS), Q1 > Q. a3: consumesmore :- consumesmore(P,TS).", "a4: :- consumesmore.", "Rule $a\\ref {a:overc:place}$ encodes consumesmore(P,TS) which captures overconsumption of tokens at input place $P$ at time $TS$ due to the firing set selected by $a\\ref {a:fires}$ .", "Overconsumption (and hence conflict) occurs when tokens $Q1$ consumed by the firing set are greater than the tokens $Q$ available at $P$ .", "Rule $a\\ref {a:overc:gen}$ generalizes this notion of overconsumption and constraint $a\\ref {a:overc:elim}$ eliminates answers where overconsumption is possible.", "Definition 18 Given a Petri Net $PN$ , its initial marking $M_0$ and its encoding $\\Pi (PN,$ $M_0,$ $k,$ $ntok)$ for $k$ -steps and maximum $ntok$ tokens at any place.", "We say that there is a 1-1 correspondence between the answer sets of $\\Pi (PN,M_0,k,ntok)$ and the execution sequences of $PN$ iff for each answer set $A$ of $\\Pi (PN,M_0,k,ntok)$ , there is a corresponding execution sequence $X=M_0,T_0,M_1,\\dots ,M_k,T_k,M_{k+1}$ of $PN$ and for each execution sequence $X$ of $PN$ there is an answer-set $A$ of $\\Pi (PN,M_0,k,ntok)$ such that $\\lbrace fires(t,ts) : t \\in T_{ts}, 0 \\le ts \\le k \\rbrace = \\lbrace fires(t,ts) : fires(t,ts) \\in A \\rbrace $ $\\begin{split}\\lbrace holds(p,q,ts) &: p \\in P, q=M_{ts}(p), 0 \\le ts \\le k+1 \\rbrace \\\\&= \\lbrace holds(p,q,ts) : holds(p,q,ts) \\in A\\rbrace \\end{split}$ Proposition 1 There is a 1-1 correspondence between the answer sets of $\\Pi ^0(PN,$ $M_0,$ $k,$ $ntok)$ and the execution sequences of $PN$ .", "An example execution Next we look at an example execution of the Petri Net shown in Figure REF .", "The Petri Net and its initial marking are encoded as follows{holds(p1,0,0),...,holds(pN,0,0)}, {num(0),...,num(60)}, {time(0),...,time(5)} have been written as holds(p1;...;pN,0,0), num(0..60), time(0..5), respectively, to save space.", ": num(0..60).time(0..5).place(f16bp;dhap;g3p;bpg13).", "trans(t3;t4;t5a;t5b;t6).tparc(t3,f16bp,1).ptarc(f16bp,t4,1).", "tparc(t4,dhap,1).tparc(t4,g3p,1).ptarc(dhap,t5a,1).", "tparc(t5a,g3p,1).ptarc(g3p,t5b,1).tparc(t5b,dhap,1).", "ptarc(g3p,t6,1).tparc(t6,bpg13,2).holds(f16bp;dhap;g3p;bgp13,0,0).", "we get thousands of answer-sets, for example{fires(t1,ts1),...,fires(tN,ts1)} have been written as fires(t1;...;tN;ts1) to save space.", ": holds(bpg13,0,0) holds(dhap,0,0) holds(f16bp,0,0) holds(g3p,0,0) holds(bpg13,0,1) holds(dhap,0,1) holds(f16bp,1,1) holds(g3p,0,1) holds(bpg13,0,2) holds(dhap,1,2) holds(f16bp,1,2) holds(g3p,1,2) holds(bpg13,0,3) holds(dhap,2,3) holds(f16bp,1,3) holds(g3p,2,3) holds(bpg13,2,4) holds(dhap,3,4) holds(f16bp,1,4) holds(g3p,2,4) holds(bpg13,4,5) holds(dhap,4,5) holds(f16bp,1,5) holds(g3p,2,5) fires(t3,0) fires(t3;t4,1) fires(t3;t4;t5a;t5b,2) fires(t3;t4;t5a;t5b;t6,3) fires(t3;t4;t5a;t5b;t6,4) fires(t3;t4;t5a;t5b;t6,5) Changing Firing Semantics The ASP code above implements the set firing semantics.", "It can produce a large number of answer-sets, since any subset of a firing set will also be fired as a firing set.", "For our biological system modeling, it is often beneficial to simulate only the maximum activity at any given time-step.", "We accomplish this by defining the maximal firing set semantics, which requires that a maximal subset of non-conflicting transitions fires at a single time stepSuch a semantics reduces the reachable markings.", "See [12] for the analysis of its computational power.. Our semantics is different from the firing multiplier approach used by [46], in which a transition can fire as many times as allowed by the tokens available in its source places.", "Their approach requires an exponential time firing algorithm in the number of transitions.", "Our maximal firing set semantics is implemented by adding the following rules to the encoding in Section REF : a5: could_not_have(T,TS) :- enabled(T,TS), not fires(T,TS), ptarc(S,T,Q), holds(S,QQ,TS), tot_decr(S,QQQ,TS), Q > QQ - QQQ.", "a6: :- not could_not_have(T,TS), enabled(T,TS), not fires(T,TS), trans(T), time(TS).", "Rule $a\\ref {a:maxfire:cnh}$ encodes could_not_have(T,TS) which means that an enabled transition $T$ that did not fire at time $TS$ , could not have fired because its firing would have resulted in overconsumption.", "Rule $a\\ref {a:maxfire:elim}$ eliminates any answer-sets in which an enabled transition did not fire, that could not have caused overconsumption.", "Intuitively, these two rules guarantee that the only reason for an enabled transition to not fire is conflict avoidance (due to overconsumption).", "With this firing semantics, the number of answer-sets produced for Petri Net in Figure REF reduces to 2.", "Proposition 2 There is 1-1 correspondence between the answer sets of $\\Pi ^1(PN,$ $M_0,$ $k,$ $ntok)$ and the execution sequences of $PN$ .", "Other firing semantics can be encoded with similar ease.", "For example, if interleaved firing semantics is desired, replace rules $a\\ref {a:maxfire:cnh},a\\ref {a:maxfire:elim}$ with the following: aREF ': more_than_one_fires :- fires(T1,TS), fires(T2, TS), T1 != T2, time(TS).", "aREF ': :- more_than_one_fires.", "We now look at Petri Net extensions and show how they can be easily encoded in ASP.", "Extension - Reset Arcs Definition 19 (Reset Arc) A Reset Arc in a Petri Net $PN^R$ is an arc from place $p$ to transition $t$ that consumes all tokens from its input place $p$ upon firing of $t$ .", "A Reset Petri Net is a tuple $PN^R = (P,T,E,W,R)$ where, $P, T, E, W$ are the same as for PN; and $R: T \\rightarrow 2^P$ defines reset arcs Figure: Petri Net of Fig extended with a reset arc from dhapdhap to trtr shown with double arrowhead.Figure REF shows an extended version of the Petri Net in Figure REF with a reset arc from $dhap$ to $tr$ (shown with double arrowhead).", "In biological context it models the removal of all quantity of compound $dhap$ .", "Petri Net execution semantics with reset arcs is modified for conflict detection and execution as follows: Definition 20 (Reset Transition) A transition $t \\in T$ of $PN^R$ is called a reset-transition if it has a reset arc incident on it, i.e.", "$R(t) \\ne \\emptyset $ .", "Definition 21 (Firing Set) A firing set is a set $T_k=\\lbrace t_{1},\\dots ,t_{m}\\rbrace \\subseteq T$ of simultaneously firing transitions that are enabled and do not conflict w.r.t.", "to the current marking $M_k$ of $PN^R$ .", "$T_k$ is not a firing set if there is an enabled reset-transition that is not in $T_k$ , i.e.", "$\\exists t : enabled_{M_k}(t), R(t) \\ne \\emptyset , t \\notin T_k$ .", "The reset arc is involved here because we use a modified execution semantics of reset arcs compared to the standard definition [3].", "Even though both capture similar operation, our definition allows us to model elimination of all quantity of a substance as soon as it is produced, even in a maximal firing set semantics.", "Our semantics considers reset arc's token consumption in contention with other arcs, while the standard definition does not..", "Definition 22 (Transition Execution in $PN^R$ ) A transition execution is the simulation of change of marking from $M_k$ to $M_{k+1}$ due to firing of a transition $t \\in T$ of $PN^R$ .", "$M_{k+1}$ is computed as follows: $ \\forall p_i \\in \\bullet t, M_{k+1}(p_i) = M_{k}(p_i) - W(p_i,t) $ $ \\forall p_j \\in t \\bullet , M_{k+1}(p_j) = M_{k}(p_j) + W(t,p_j) $ $ \\forall p_r \\in R(t), M_{k+1}(p_r) = M_{k}(p_r) - M_{k}(p_r) $ Definition 23 (Conflicting Transitions in $PN^R$ ) A set of enabled transitions conflict in $PN^R$ w.r.t.", "$M_k$ if firing them simultaneously will consume more tokens than are available at any one of their common input-places.", "$T_e = \\lbrace t \\in T : enabled_{M_k}(t) \\rbrace $ conflict if: $\\exists p \\in P : M_k(p) < (\\displaystyle \\sum _{t \\in T_e \\wedge (p,t) \\in E^-}{W(p,t)} + \\displaystyle \\sum _{t \\in T_e \\wedge p \\in R(t)}{M_k(p)})$ Definition 24 (Firing Set Execution in $PN^R$ ) Execution of a transition set $T_i$ in $PN^R$ has the following effect: $\\forall p \\in P \\setminus R(T_i), M_{k+1}(p) = M_k(p) - \\sum _{\\begin{array}{c}t \\in T_i \\wedge p \\in \\bullet t\\end{array}} W(p,t) + \\sum _{\\begin{array}{c}t \\in T_i \\wedge p \\in t \\bullet \\end{array}} W(t,p)$ $\\forall p \\in R(T_i), M_{k+1}(p) = \\sum _{t \\in T_i \\wedge p \\in t \\bullet } W(t,p)$ where $R(T_i)=\\displaystyle \\bigcup _{\\begin{array}{c}t \\in T_i\\end{array}} R(t)$ and represents the places emptied by $T_i$ due to reset arcs Our definition of conflicting transitions allows at most one transition with a reset arc from a place to fire, any more create a conflict.", "Thus, the new marking computation is equivalent to $\\forall p \\in P, M_{k+1}(p) = M_k(p) - (\\sum _{\\begin{array}{c}t \\in T_k \\wedge p \\in \\bullet t\\end{array}}{W(p,t)} + \\sum _{\\begin{array}{c}t \\in T_k \\wedge p \\in R(t)\\end{array}}{M_k(p)})+ \\sum _{\\begin{array}{c}t \\in T_k \\wedge p \\in t \\bullet \\end{array}} W(t,p)$.", "Since a reset arc from $p$ to $t$ , $p \\in R(t)$ consumes current marking dependent tokens, we extend ptarc to include time-step and replace $f\\ref {f:ptarc},f\\ref {f:tparc},e\\ref {e:ne:ptarc},r\\ref {r:add},r2,a\\ref {a:maxfire:cnh}$ with $f\\ref {f:r:ptarc},f\\ref {f:r:tparc},e\\ref {e:r:ne:ptarc},r\\ref {r:r:add},r\\ref {r:r:del},a\\ref {a:r:maxfire:cnh}$ , respectively in the Section REF encoding and add rule $f\\ref {f:rptarc}$ for each reset arc: f7: Rules ptarc($p_i,t_j,W(p_i,t_j),ts_k$ ):-time($ts_k$ ).", "for each non-reset arc $(p_i,t_j) \\in E^-$ f8: Rules tparc($t_i,p_j,W(t_i,p_j),ts_k$ ):-time($ts_k$ ).", "for each non-reset arc $(t_i,p_j) \\in E^+$ e3: notenabled(T,TS) :- ptarc(P,T,N,TS), holds(P,Q,TS), Q < N, place(P), trans(T), time(TS), num(N), num(Q).", "r6: add(P,Q,T,TS) :- fires(T,TS), tparc(T,P,Q,TS), time(TS).", "r7: del(P,Q,T,TS) :- fires(T,TS), ptarc(P,T,Q,TS), time(TS).", "f9: Rules ptarc($p_i,t_j,X,ts_k$ ) :- holds($p_i,X,ts_k$ ), num($X$ ), $X>0$ .", "for each reset arc between $p_i$ and $t_j$ using $X=M_k(p_i)$ as arc-weight at time step $ts_k$ .", "f10: Rules :- enabled($t_j,ts_k$ ),not fires($t_j,ts_k$ ), time($ts_k$ ).", "for each transition $t_j$ with an incoming reset arc, i.e.", "$R(t_j) \\ne \\emptyset $ .", "a7: could_not_have(T,TS) :- enabled(T,TS), not fires(T,TS), ptarc(S,T,Q,TS), holds(S,QQ,TS), tot_decr(S,QQQ,TS), Q>QQ-QQQ.", "Rule $f\\ref {f:rptarc}$ encodes place-transition arc with marking dependent weight to capture the notion of a reset arc, while rule $f\\ref {f:rptarc:elim}$ ensures that the reset-transition (i.e.", "the transition on which the reset arc terminates) always fires when enabled.", "Proposition 3 There is 1-1 correspondence between the answer sets of $\\Pi ^2(PN^R,M_0,$ $k,ntok)$ and the execution sequences of $PN^R$ .", "The execution semantics of our definition are slightly different from the standard definition in [3], even though both capture similar operations.", "Our implementation considers token consumption by reset arc in contention with other token consuming arcs from the same place, while the standard definition considers token consumption as a side effect, not in contention with other arcs.", "We chose our definition to allow modeling of biological process that removes all available quantity of a substance in a maximal firing set.", "Consider Figure REF , if $dhap$ has 1 or more tokens, our semantics would only permit either $t5a$ or $tr$ to fire in a single time-step, while the standard semantics can allow both $t5a$ and $tr$ to fire simultaneously, such that the reset arc removes left over tokens after $(dhap,t5a)$ consumes one token.", "We could have, instead, extended our encoding to include self-modifying nets [77], but our modified-definition provides a simpler solution.", "Standard semantics, however, can be easily encoded by replacing $r\\ref {r:nextstate}$ by $r\\ref {r:nextstate}a^{\\prime }, r\\ref {r:nextstate}b^{\\prime }$ ; replacing $f\\ref {f:rptarc},f\\ref {f:rptarc:elim}$ with $f\\ref {f:rptarc}^{\\prime }$ ; and adding $a\\ref {a:reset:std}$ as follows: fREF ': rptarc($p_i$ ,$t_j$ ).", "- for each reset arc between $p_i \\in R(t_j)$ and $t_j$ .", "a8: reset(P,TS) :- rptarc(P,T), place(P), trans(T), fires(T,TS), time(TS).", "rREF a': holds(P,Q,TS+1) :- holds(P,Q1,TS), tot_incr(P,Q2,TS), tot_decr(P,Q3,TS), Q=Q1+Q2-Q3, place(P), num(Q;Q1;Q2;Q3), time(TS), time(TS+1), not reset(P,TS).", "rREF b': holds(P,Q,TS+1) :- tot_incr(P,Q,TS), place(P), num(Q), time(TS), time(TS+1), reset(P,TS).", "where, the fact $f\\ref {f:rptarc}^{\\prime }$ encodes the reset arc; rule $a\\ref {a:reset:std}$ encodes if place $P$ will be reset at time $TS$ due to firing of transition $T$ that has a reset arc on it from $P$ to $T$ ; rule $r\\ref {r:nextstate}a^{\\prime }$ computes marking at $TS+1$ when place $P$ is not being reset; and rule $r\\ref {r:nextstate}b^{\\prime }$ computes marking at $TS+1$ when $P$ is being reset.", "Extension - Inhibitor Arcs Definition 25 (Inhibitor Arc) An inhibitor arc [61] is a place-transition arc that inhibits its transition from firing as long as the place has any tokens in it.", "An inhibitor arc does not consume any tokens from its input place.", "A Petri Net with reset and inhibitor arcs is a tuple $PN^{RI}=(P,T,E,W,R,I)$ , where, $P, T, E, W, R$ are the same as for $PN^R$ ; and $I: T \\rightarrow 2^P$ defines inhibitor arcs.", "Figure: Petri Net showing feedback inhibition arc from atpatp to gly1gly1 with a bullet arrowhead.", "Inhibitor arc weight is assumed 1 when not specified.Figure REF shows a Petri Net with inhibition arc from $atp$ to $gly1$ with a bulleted arrowhead.", "It models biological feedback regulation in simplistic terms, where excess $atp$ downstream causes the upstream $atp$ production by glycolysis $gly$ to be inhibited until the excess quantity is consumed [64].", "Petri Net execution semantics with inhibit arcs is modified for determining enabled transitions as follows: Definition 26 (Enabled Transition in $PN^{RI}$ ) A transition $t$ is enabled with respect to marking $M$ , $enabled_M(t)$ , if all its input places $p$ have at least the number of tokens as the arc-weight $W(p,t)$ and all $p \\in I(t)$ have zero tokens, i.e.", "$(\\forall p \\in \\bullet t, W(p,t) \\le M(p)) \\wedge (\\forall p \\in I(t), M(p) = 0)$ We add inhibitor arcs to our encoding in Section REF as follows: f11: Rules iptarc($p_i,t_j,1,ts_k$ ):-time($ts_k$ ).", "for each inhibitor arc between $p_i \\in I(t_j)$ and $t_j$ .", "e4: notenabled(T,TS) :- iptarc(P,T,N,TS), holds(P,Q,TS), place(P), trans(T), time(TS), num(N), num(Q), Q >= N. The new rule $e\\ref {e:ne:iptarc}$ encodes another reason for a transition to be disabled (or not enabled).", "An inhibitor arc from $p$ to $t$ with arc weight $N$ will cause its target transition $t$ to not enable when the number of tokens at its source place $p$ is greater than or equal to $N$ , where $N$ is always 1 per rule $f\\ref {f:iptarc}$ .", "Proposition 4 There is 1-1 correspondence between the answer sets of $\\Pi ^3(PN^{RI},M_0,k,ntok)$ and the execution sequences of $PN$ .", "Extension - Read Arcs Definition 27 (Read Arc) A read arc (a test arc or a query arc) [17] is an arc from place to transition, which enables its transition only when its source place has at least the number of tokens as its arc weight.", "It does not consume any tokens from its input place.", "A Petri Net with reset, inhibitor and read arcs is a tuple $PN^{RIQ}=(P,T,W,R,I,Q,QW)$ , where, $P,T,E,W,R,I$ are the same as for $PN^{RI}$ ; $Q \\subseteq P \\times T $ defines read arcs; and $QW: Q \\rightarrow {N} \\setminus \\lbrace 0\\rbrace $ defines read arc weight.", "Figure: Petri Net with read arc from h_ish\\_is to synsyn shown with arrowhead on both ends.", "The transition synsyn will not fire unless there are at least 25 tokens in h_ish\\_is, but when it executes, it only consumes 3 tokens.Figure REF shows a Petri Net with read arc from $h\\_is$ to $syn$ shown with arrowhead on both ends.", "It models the ATP synthase $syn$ activation requiring a higher concentration of $H+$ ions $h\\_is$ in the intermembrane space This is an oversimplified model of $syn$ (ATP synthase) activation, since the actual model requires an $H+$ concentration differential across membrane..", "The reaction itself consumes a lower quantity of $H+$ ions represented by the regular place-transition arc [64], [7].", "Petri Net execution semantics with read arcs is modified for determining enabled transitions as follows: Definition 28 (Enabled Transition in $PN^{RIQ}$ ) A transition $t$ is enabled with respect to marking $M$ , $enabled_M(t)$ , if all its input places $p$ have at least the number of tokens as the arc-weight $W(p,t)$ , all $p_i \\in I(t)$ have zero tokens and all $p_q : (p_q,t) \\in Q$ have at least the number of tokens as the arc-weight $W(p,t)$ , i.e.", "$(\\forall p \\in \\bullet t, W(p,t) \\le M(p)) \\wedge (\\forall p \\in I(t), M(p) = 0) \\wedge (\\forall (p,t) \\in Q, M(p) \\ge QW(p,t))$ We add read arcs to our encoding of Section REF as follows: f12: Rules tptarc($p_i,t_j,QW(p_i,t_j),ts_k$ ):-time($ts_k$ ).", "for each read arc $(p_i,t_j) \\in Q$ .", "e5: notenabled(T,TS):-tptarc(P,T,N,TS),holds(P,Q,TS), place(P),trans(T), time(TS), num(N), num(Q), Q < N. The new rule $f\\ref {f:tptarc}$ captures the read arc and its arc-weight; and the new rule $e\\ref {e:ne:tptarc}$ encodes another reason for a transition to not be enabled.", "A read arc from $p$ to $t$ with arc weight $N$ will cause its target transition $t$ to not enable when the number of tokens at its source place $p$ is less than the arc weight $N$ .", "Proposition 5 There is a 1-1 correspondence between the answer sets of $\\Pi ^4(PN^{RIQ},M_0,$ $k,ntok)$ and the execution sequences of $PN^{RIQ}$ .", "Extension - Colored Tokens Higher level Petri Nets extend the notion of tokens to typed (or colored) tokens.", "This allows a more compact representation of complicated networks [60].", "Definition 29 (Petri Net with Colored Tokens) A Petri Net with Colored Tokens (with reset, inhibit and read arcs) is a tuple $PN^C=(P,T,E,C,W,R,I,Q,QW)$ , where $P,T,E,R,I,Q$ are the same as for basic Petri Nets, $C=\\lbrace c_1,\\dots ,c_l\\rbrace $ is a finite set of colors (or types), and arc weights $W : E \\rightarrow \\langle C,m \\rangle $ , $QW : Q \\rightarrow \\langle C,m \\rangle $ are specified as multi-sets of colored tokens over color set $C$ .", "The state (or marking) of place nodes $M(p_i) = \\langle C,m \\rangle , p_i \\in P$ is specified as a multiset of colored tokens over set $C$ .", "We will now update some definitions related to Petri Nets to include colored tokens.", "Definition 30 (Marking) A marking $M=(M(p_1),\\dots ,M(p_n))$ assigns a colored multi-set of tokens over the domain of colors $C$ to each place $\\lbrace p_1,\\dots ,p_n\\rbrace \\in P$ of $PN^C$ .", "The initial marking is the initial token assignment of place nodes and is represented by $M_0$ .", "The marking at time-step $k$ is written as $M_k$ .", "Definition 31 (Pre-set and post-set of a transition) The pre-set (or input-set) of a transition $t$ is $\\bullet t = \\lbrace p \\in P \| (p,t) \\in E^- \\rbrace $ , while the post-set (or output-set) is $t \\bullet = \\lbrace p \\in P \| (t,p) \\in E^+ \\rbrace $ .", "Definition 32 (Enabled Transition) A transition $t$ is enabled with respect to marking $M$ , $enabled_M(t)$ , if each of its input places $p$ has at least the number of colored-tokens as the arc-weight $W(p,t)$In the following text, for simplicity, we will use $W(p,t)$ to mean $W(\\langle p,t \\rangle )$ .", "We use similar simpler notation for $QW$ ., each of its inhibiting places $p_i \\in I(t)$ have zero tokens and each of its read places $p_q : (p_q,t) \\in Q$ have at least the number of colored-tokens as the read-arc-weight $QW(p_q,t)$ , i.e.", "$(\\forall p \\in \\bullet t, W(p,t) \\le M(p)) \\wedge (\\forall p \\in I(t), M(p) = \\emptyset ) \\wedge (\\forall (p,t) \\in Q, M(p) \\ge QW(p,t))$ for a given $t$ .This is equivalent to $\\forall c \\in C, (\\forall p \\in \\bullet t, m_{W(p,t)}(c) \\le m_{M(p)}(c)) \\wedge (\\forall p \\in I(t), m_{M(p)}(c) = 0) \\wedge (\\forall (p,t) \\in Q, m_{M(p)}(c) \\ge m_{QW(p,t)}(c))$ .", "Definition 33 (Transition Execution) Execution of a transition $t$ of $PN^C$ on a marking $M_k$ computes a new marking $M_{k+1}$ as: $\\forall p \\in \\bullet t M_{k+1}(p) = M_k(p) - W(p,t)$ $\\forall p \\in t\\bullet M_{k+1}(p) = M_k(p) + W(t,p)$ $\\forall p \\in R(t) M_{k+1}(p) = M_k(p) - M_k(p)$ Any number of enabled transitions may fire simultaneously as long as they don't conflict.", "A transition when fired consumed tokens from its pre-set places equivalent to the (place,transition) arc-weight.", "Definition 34 (Conflicting Transitions) A set of transitions $T_c \\subseteq \\lbrace t : $ $enabled_{M_k}(t) \\rbrace $ is in conflict in $PN^C$ with respect to $M_k$ if firing them will consume more tokens than are available at one of their common input places, i.e., $\\exists p \\in P : M_k(p) < (\\sum _{t \\in T_c \\wedge p \\in \\bullet t}{W(p,t)} + \\sum _{\\begin{array}{c}t \\in T_c \\wedge p \\in R(t)\\end{array}}{M_k(p)})$ Definition 35 (Firing Set) A firing set is a set $T_k=\\lbrace t_{k_1},\\dots ,t_{k_n}\\rbrace \\subseteq T$ of simultaneously firing transitions of $PN^C$ that are enabled and do not conflict w.r.t.", "to the current marking $M_k$ of $PN$ .", "A set $T_k$ is not a firing set if there is an enabled reset-transition that is not in $T_k$ , i.e.", "$\\exists t \\in enabled_{M_k}, R(t) \\ne \\emptyset , t \\notin T_k$ .", "See footnote REF Definition 36 (Firing Set Execution) Execution of a firing set $T_k$ of $PN^C$ on a marking $M_k$ computes a new marking $M_{k+1}$ as: $\\forall p \\in P \\setminus R(T_k), M_{k+1}(p) = M_k(p)- \\sum _{\\begin{array}{c}t \\in T_k \\wedge p \\in \\bullet t\\end{array}} W(p,t)+ \\sum _{\\begin{array}{c}t \\in T_k \\wedge p \\in t \\bullet \\end{array}} W(t,p)$ $\\forall p \\in R(T_k), M_{k+1}(p) = \\sum _{\\begin{array}{c}t \\in T_k \\wedge p \\in t \\bullet \\end{array}} W(t,p)$ where $R(T_k)=\\bigcup _{t \\in T_k} R(t)$ See footnote REF Definition 37 (Execution Sequence) An execution sequence $X = M_0, T_0, M_1, $ $T_1, \\dots , $ $M_k, $ $T_k, M_{k+1}$ of $PN$ is the simulation of a firing set sequence $\\sigma = T_1,T_2,\\dots ,T_k$ w.r.t.", "an initial marking $M_0$ , producing the final marking $M_{k+1}$ .", "$M_{k+1}$ is the transitive closure of firing set executions, where subsequent marking become the initial marking for the next firing set.", "For an execution sequence $X = M_0, T_0, M_1, T_1, \\dots , $ $M_k, T_k, M_{k+1}$ , the firing of $T_0$ with respect to marking $M_0$ produces the marking $M_1$ which becomes the initial marking for $T_1$ , which produces $M_2$ and so on.", "Figure: Petri Net with tokens of colors {e,h,h2o,nadh,nadp,o2}\\lbrace e,h,h2o,nadh,nadp,o2\\rbrace .", "Circles represent places, and rectangles represent transitions.", "Arc weights such as “nadh/2,h/2nadh/2,h/2”, “h/2,h2o/1h/2,h2o/1” specify the number of tokens consumed and produced during the execution of their respective transitions, where “nadh/2,h/2nadh/2,h/2” means 2 tokens of color nadhnadh and 2 tokens of hh.", "Similar notation is used to specify marking on places, when not present, the place is assumed to be empty of tokens.If the Figure REF Petri Net has the marking: $M_0(mm)=[nadh/2,h/6]$ , $M_0(q)=[e/2]$ , $M_0(cytc)=[e/2]$ , $M_0(is)=[o2/1]$ , then transitions $t1,t3,t4$ are enabled.", "However, either $\\lbrace t1,t3\\rbrace $ or $\\lbrace t4\\rbrace $ can fire simultaneously in a single firing at time 0 due to limited $h$ tokens in $mm$ .", "$t4$ is said to be in conflict with $t1,t3$ .", "Translating Petri Nets with Colored Tokens to ASP In order to represent the Petri Net $PN^C$ with colored tokens, initial marking $M_0$ , and simulation length $k$ , we modify our encoding in Section REF to add a new color parameter to all rules and facts containing token counts in them.", "We keep rules $f\\ref {f:c:place}, f\\ref {f:c:trans}, f\\ref {f:c:time}, f\\ref {f:c:num}, f\\ref {f:rptarc:elim}$ remain as they were for basic Petri Nets.", "We add a new rule $f\\ref {f:c:col}$ for possible set of token colors and replace rules $f\\ref {f:r:ptarc}, f\\ref {f:r:tparc}, f\\ref {f:rptarc}, f\\ref {f:iptarc}, f\\ref {f:tptarc}, i\\ref {i:holds}$ with $f\\ref {f:c:ptarc}, f\\ref {f:c:tparc}, f\\ref {f:c:rptarc}, f\\ref {f:c:iptarc}, f\\ref {f:c:tptarc}, i\\ref {i:c:init}$ to add the color parameter as follows: f13: Facts col($c_k$ ) where $c_k \\in C$ is a color.", "f14: Rules ptarc($p_i,t_j,n_c,c,ts_k$ ) :- time($ts_k$ ).", "for each $(p_i,t_j) \\in E^-$ , $c \\in C$ , $n_c=m_{W(p_i,t_j)}(c) : n_c > 0$ .The time parameter $ts_k$ allows us to capture reset arcs, which consume tokens equal to the current (time-step based) marking of their source nodes.", "f15: Rules tparc($t_i,p_j,n_c,c,ts_k$ ) :- time($ts_k$ ).", "for each $(t_i,p_j) \\in E^+$ , $c \\in C$ , $n_c=m_{W(t_i,p_j)}(c) : n_c > 0$ .", "f16: Rules ptarc($p_i,t_j,n_c,c,ts_k$ ) :- holds($p_i,n_c,c,ts_k$ ), num($n_c$ ), $n_c$ >0, time($ts_k$ ).", "for each $(p_i,t_j):$ $p_i \\in R(t_j)$ , $c \\in C$ , $n_c=m_{M_k(p_i)}(c)$ .", "f17: Rules iptarc($p_i,t_j,1,c,ts_k$ ) :- time($ts_k$ ).", "for each $(p_i,t_j): p_i \\in I(t_j)$ , $c \\in C$ .", "f18: Rules tptarc($p_i,t_j,n_c,c,ts_k$ ) :- time($ts_k$ ).", "for each $(p_i,t_j) \\in Q$ , $c \\in C$ , $n_c=m_{QW(p_i,t_j)}(c) : n_c > 0 $ .", "i2: Facts holds($p_i,n_c,c,0$ ).", "for each place $p_i \\in P, c \\in C, n_c=m_{M_0(p_i)}(c)$ .", "Next, we encode Petri Net's execution behavior, which proceeds in discrete time steps.", "Rules $e\\ref {e:r:ne:ptarc}, e\\ref {e:ne:iptarc}, e\\ref {e:ne:tptarc}, e\\ref {e:enabled}$ are replaced by $e\\ref {e:c:ne:ptarc}, e\\ref {e:c:ne:iptarc}, e\\ref {e:c:ne:tptarc}, e\\ref {e:c:enabled}$ .", "For a transition $t_i$ to be enabled, it must satisfy the following conditions: [(i)] $\\nexists p_j \\in \\bullet t_i : M(p_j) < W(p_j,t_i)$ , $\\nexists p_j \\in I(t_i) : M(p_j) > 0$ , and $\\nexists (p_j,t_i) \\in Q : M(p_j) < QW(p_j,t_i)$ .", "These three conditions are encoded as $e\\ref {e:c:ne:ptarc},e\\ref {e:c:ne:iptarc},e\\ref {e:c:ne:tptarc}$ , respectively and we encode the absence of any of these conditions for a transition as $e\\ref {e:c:enabled}$ : e6: notenabled(T,TS) :- ptarc(P,T, N,C,TS), holds(P,Q,C,TS), place(P), trans(T), time(TS), num(N), num(Q), col(C), Q<N.", "e7: notenabled(T,TS) :- iptarc(P,T,N,C,TS), holds(P,Q,C,TS), place(P), trans(T), time(TS), num(N), num(Q), col(C), Q>=N.", "e8: notenabled(T,TS) :- tptarc(P,T,N,C,TS), holds(P,Q,C,TS), place(P), trans(T), time(TS), num(N), num(Q), col(C), Q<N.", "e9: enabled(T,TS) :- trans(T), time(TS), not notenabled(T,TS).", "Rule $e\\ref {e:c:ne:ptarc}$ captures the existence of an input place $P$ with insufficient number of tokens for transition $T$ to fire.", "Rule $e\\ref {e:c:ne:iptarc}$ captures existence of a non-empty source place $P$ of an inhibitor arc to $T$ preventing $T$ from firing.", "Rule $e\\ref {e:c:ne:tptarc}$ captures existence of a source place $P$ with less than arc-weight tokens required by the read arc to transition $T$ for $T$ to be enabled.", "The, holds(P,Q,C,TS) predicate captures the marking of place $P$ at time $TS$ as $Q$ tokens of color $C$ .", "Rule $e\\ref {e:c:enabled}$ captures enabling of transition $T$ when no reason for it to be not enabled is determined by $e\\ref {e:c:ne:ptarc},e\\ref {e:c:ne:iptarc},e\\ref {e:c:ne:tptarc}$ .", "In a biological context, this enabling is equivalent to a reaction's pre-conditions being satisfied.", "A reaction can proceed when its input substances are available in the required quantities, it is not inhibited, and any required activation quantity of activating substances is available.", "Any subset of enabled transitions can fire simultaneously at a given time-step.", "We select a subset of fireable transitions using the choice rule $a\\ref {a:c:fires}$ The choice rule $a\\ref {a:c:fires}$ either picks an enabled transition $T$ for firing at time $TS$ or not.", "The combined effect over all transitions is to pick a subset of enabled transitions to fire.", "Rule $f\\ref {f:c:rptarc:elim}$ ensures that enabled reset-transitions will be a part of this firing set.", "Whether these transitions are in conflict are checked by later rules $a\\ref {a:c:overc:place},a\\ref {a:c:overc:gen},a\\ref {a:c:overc:elim}$ .", "In a biological context, the multiple firing models parallel processes occurring simultaneously.", "The marking is updated according to the firing set using rules $r\\ref {r:c:add}, r\\ref {r:c:del}, r\\ref {r:c:totincr}, r\\ref {r:c:totdecr}, r\\ref {r:c:nextstate}$ which replaced $r\\ref {r:r:add}, r\\ref {r:r:del}, r\\ref {r:totincr}, r\\ref {r:totdecr}, r\\ref {r:nextstate}$ as follows: r8: add(P,Q,T,C,TS) :- fires(T,TS), tparc(T,P,Q,C,TS), time(TS).", "r9: del(P,Q,T,C,TS) :- fires(T,TS), ptarc(P,T,Q,C,TS), time(TS).", "r10: tot_incr(P,QQ,C,TS) :- col(C), QQ = #sum[add(P,Q,T,C,TS) = Q : num(Q) : trans(T)], time(TS), num(QQ), place(P).", "r11: tot_decr(P,QQ,C,TS) :- col(C), QQ = #sum[del(P,Q,T,C,TS) = Q : num(Q) : trans(T)], time(TS), num(QQ), place(P).", "r12: holds(P,Q,C,TS+1):-place(P),num(Q;Q1;Q2;Q3),time(TS),time(TS+1),col(C), holds(P,Q1,C,TS), tot_incr(P,Q2,C,TS), tot_decr(P,Q3,C,TS), Q=Q1+Q2-Q3.", "Rules $r\\ref {r:c:add}$ and $r\\ref {r:c:del}$ capture that $Q$ tokens of color $C$ will be added or removed to/from place $P$ due to firing of transition $T$ at the respective time-step $TS$ .", "Rules $r\\ref {r:c:totincr}$ and $r\\ref {r:c:totdecr}$ aggregate these tokens for each $C$ for each place $P$ (using aggregate assignment QQ = #sum[...]) at the respective time-step $TS$ .", "Rule $r\\ref {r:c:nextstate}$ uses the aggregates to compute the next marking of $P$ for color $C$ at the time-step ($TS+1$ ) by subtracting removed tokens and adding added tokens to the current marking.", "In a biological context, this captures the effect of a process / reaction, which consumes its inputs and produces outputs for the downstream processes.", "We capture token overconsumption using the rules $a\\ref {a:c:overc:place}, a\\ref {a:c:overc:gen}, a\\ref {a:c:overc:elim}$ of which $a\\ref {a:c:overc:place}$ is a colored replacement for $a\\ref {a:overc:place}$ and is encoded as follows: a9: consumesmore(P,TS) :- holds(P,Q,C,TS), tot_decr(P,Q1,C,TS), Q1 > Q.", "Rule $a\\ref {a:c:overc:place}$ determines whether firing set selected by $a\\ref {a:c:fires}$ will cause overconsumption of tokens at $P$ at time $TS$ by comparing available tokens to aggregate tokens removed as determined by $r\\ref {r:c:totdecr}$ .", "Rule $a\\ref {a:c:overc:gen}$ generalizes the notion of overconsumption, while rule $a\\ref {a:c:overc:elim}$ eliminates answer with such overconsumption.", "In a biological context, conflict (through overconsumption) models the limitation of input substances, which dictate which downstream processes can occur simultaneously.", "We remove rules $a\\ref {a:maxfire:cnh}, a\\ref {a:maxfire:elim}$ from previous encoding to get the set firing semantics.", "Now, we extend the definition (REF ) of $\\text{1-1}$ correspondence between the execution sequence of Petri Net and the answer-sets of its ASP encoding to Petri Nets with colored tokens as follows.", "Definition 38 Given a Petri Net $PN$ with colored tokens, its initial marking $M_0$ and its encoding $\\Pi (PN,M_0,k,ntok)$ for $k$ -steps and maximum $ntok$ tokens at any place.", "We say that there is a 1-1 correspondence between the answer sets of $\\Pi (PN,M_0,k,ntok)$ and the execution sequences of $PN$ iff for each answer set $A$ of $\\Pi (PN,M_0,k,ntok)$ , there is a corresponding execution sequence $X=M_0,T_0,M_1,\\dots ,M_k,T_k,M_{k+1}$ of $PN$ and for each execution sequence $X$ of $PN$ there is an answer-set $A$ of $\\Pi (PN,M_0,$ $k,ntok)$ such that $\\lbrace fires(t,ts) : t \\in T_{ts}, 0 \\le ts \\le k \\rbrace = \\lbrace fires(t,ts) : fires(t,ts) \\in A \\rbrace $ $\\begin{split}\\lbrace holds(p,q,c,ts) &: p \\in P, c/q=M_{ts}(p), 0 \\le ts \\le k+1 \\rbrace \\\\&= \\lbrace holds(p,q,c,ts) : holds(p,q,c,ts) \\in A\\rbrace \\end{split}$ Proposition 6 There is 1-1 correspondence between the answer sets of $\\Pi ^5(PN^C,M_0,$ $k,ntok)$ and the execution sequences of $PN$ .", "To add maximal firing semantics, we add $a\\ref {a:c:maxfire:elim}$ as it is and replace $a\\ref {a:maxfire:cnh}$ with $a\\ref {a:c:maxfire:cnh}$ as follows: a10: could_not_have(T,TS):-enabled(T,TS),not fires(T,TS), ptarc(S,T,Q,C,TS), holds(S,QQ,C,TS), tot_decr(S,QQQ,C,TS), Q > QQ - QQQ.", "Rule $a\\ref {a:c:maxfire:cnh}$ captures the fact that transition $T$ , though enabled, could not have fired at $TS$ , as its firing would have caused overconsumption.", "Rule $a\\ref {a:c:maxfire:elim}$ eliminates any answers where an enabled transition could have fired without causing overconsumption but did not.", "This modification reduces the number of answers produced for the Petri Net in Figure REF to 4.", "We can encode other firing semantics with similar easeFor example, if interleaved semantics is desired, rules $a\\ref {a:c:maxfire:cnh},a\\ref {a:c:maxfire:elim}$ can changed to capture and eliminate answer-sets in which more than one transition fires in a firing set as: aREF ': more_than_one_fires :- fires(T1,TS),fires(T2,TS),T1!=T2,time(TS).", "aREF ': :-more_than_one_fires.", ".", "We now look at how additional extensions can be easily encoded by making small code changes.", "Extension - Priority Transitions Priority transitions enable ordering of Petri Net transitions, favoring high priority transitions over lower priority ones [9].", "In a biological context, this is used to model primary (or dominant) vs. secondary pathways / processes in a biological system.", "This prioritization may be due to an intervention (such as prioritizing elimination of a metabolite over recycling it).", "Definition 39 (Priority Colored Petri Net) A Priority Colored Petri Net with reset, inhibit, and read arcs is a tuple $PN^{pri} = (P,T,E,C,W,R,I,Q,QW,Z)$ , where: $P,T,E,C,W,R,I,Q,QW$ are the same as for $PN^C$ , and $Z : T \\rightarrow {N}$ is a priority function that assigns priorities to transitions.", "Lower number signifies higher priority.", "Definition 40 (Enabled Transition) A transition $t_i$ is enabled in $PN^{pri}$ w.r.t.", "a marking $M$ (prenabled$_{M}(t)$ ) if it would be enabled in $PN^C$ w.r.t.", "$M$ and there isn't another transition $t_j$ that would be enabled in $PN^C$ (with respect to M) s.t.", "$Z(t_j) < Z(t_i)$ .", "Definition 41 (Firing Set) A firing set is a set $T_k=\\lbrace t_{k_1},\\dots ,t_{k_n}\\rbrace \\subseteq T$ of simultaneously firing transitions of $PN^{pri}$ that are priority enabled and do not conflict w.r.t.", "to the current marking $M_k$ of $PN$ .", "A set $T_k$ is not a firing set if there is an priority enabled reset-transition that is not in $T_k$ , i.e.", "$\\exists t : prenabled_{M_k}(t), R(t) \\ne \\emptyset , t \\notin T_k$ .", "See footnote REF We add the following facts and rules to encode transition priority and enabled priority transitions: f19: Facts transpr($t_i$ ,$pr_i$ ) where $pr_i = Z(t_i)$ is $t_i^{\\prime }s$ priority.", "a11: notprenabled(T,TS) :- enabled(T,TS), transpr(T,P), enabled(TT,TS), transpr(TT,PP), PP < P. a12: prenabled(T,TS) :- enabled(T,TS), not notprenabled(T,TS).", "Rule $a\\ref {a:c:prne}$ captures that an enabled transition $T$ is not priority-enabled, if there is another enabled transition with higher priority at $TS$ .", "Rule $a\\ref {a:c:prenabled}$ captures that transition $T$ is priority-enabled at $TS$ since there is no enabled transition with higher priority.", "We replace rules $a\\ref {a:c:fires},f\\ref {f:c:rptarc:elim},a\\ref {a:c:maxfire:cnh},a\\ref {a:c:maxfire:elim}$ with $a\\ref {a:c:prfires},f\\ref {f:c:pr:rptarc:elim}, a\\ref {a:c:prmaxfire:cnh},a\\ref {a:c:prmaxfire:elim}$ respectively to propagate priority as follows: a13: {fires(T,TS)} :- prenabled(T,TS), trans(T), time(TS).", "f20: Rules :- prenabled($t_j,ts_k$ ),not fires($t_j,ts_k$ ), time($ts_k$ ).", "for each transition $t_j$ with an incoming reset arc.", "a14: could_not_have(T,TS) :- prenabled(T,TS), not fires(T,TS), ptarc(S,T,Q,C,TS), holds(S,QQ,C,TS), tot_decr(S,QQQ,C,TS), Q > QQ - QQQ.", "a15: :- not could_not_have(T,TS), time(TS), prenabled(T,TS), not fires(T,TS), trans(T).", "Rules $a\\ref {a:c:prfires},f\\ref {f:c:rptarc:elim},a\\ref {a:c:prmaxfire:cnh},a\\ref {a:c:prmaxfire:elim}$ perform the same function as $a\\ref {a:c:fires},f\\ref {f:c:pr:rptarc:elim},a\\ref {a:c:maxfire:cnh},a\\ref {a:c:maxfire:elim}$ , except that they consider only priority-enabled transitions as compared all enabled transitions.", "Proposition 7 There is 1-1 correspondence between the answer sets of $\\Pi ^6(PN^{pri},M_0,$ $k,ntok)$ and the execution sequences of $PN^{pri}$ .", "Extension - Timed Transitions Biological processes vary in time required for them to complete.", "Timed transitions [63] model this variation of duration.", "The timed transitions can be reentrant or non-reentrantA reentrant transition is like a vehicle assembly line, which accepts new parts while working on multiple vehicles at various stages of completion; whereas a non-reentrant transition only accepts new input when the current processing is finished.. We extend our encoding to allow reentrant timed transitions.", "Figure: An extended version of the Petri Net model from Fig. .", "The new transitions tq,tcytctq,tcytc have a duration of 2 each (shown in square brackets (“[ ]”) next to the transition).", "When missing, transition duration is assumed to be 1.Definition 42 (Priority Colored Petri Net with Timed Transitions) A Priority Colored Petri Net with Timed Transitions, reset, inhibit, and query arcs is a tuple $PN^D=(P,T,E,C,W,R,I,Q,QW,Z,D)$ , where $P,T,E,C,W,R,I,Q,QW,Z$ are the same as for $PN^{pri}$ , and $D : T \\rightarrow {N} \\setminus \\lbrace 0\\rbrace $ is a duration function that assigns positive integer durations to transitions.", "Figure REF shows an extended version of Petri Net model of the Electron Transport Chain [64] shown in Figure REF .", "The new transitions $tq$ and $tcytc$ (shown in dotted outline) are timed transitions modeling the speed of the small carrier molecules, Coenzyme Q ($q$ ) and Cytochrome C ($cytc$ ) as an effect of membrane fluidity.", "Higher numbers for transition duration represent slower movement of the carrier molecules due to lower fluidity.", "Definition 43 (Transition Execution) A transition $t$ in $PN^D$ consumes tokens from its input places and reset places immediately, while it produces tokens in its output places at the end of transition duration $D(t)$ , as follows: $\\forall p \\in \\bullet t, M_{k+1}(p) = M_k(p) - W(p,t)$ $\\forall p \\in t\\bullet , M_{k+D(t)}(p) = M_{k+D(t)-1}(p) + W(p,t)$ $\\forall p \\in R(t), M_{k+1}(p) = M_k(p) - M_k(p)$ Execution in $PN^D$ changes, since the token update from $M_k$ to $M_{k+1}$ can involve transitions that started at some time $l$ before time $k$ , but finish at $k+1$ .", "Definition 44 (Firing Set Execution) New marking due to firing set execution is computed as follows: $\\forall p \\in P \\setminus R(T_k), M_{k+1}(p) = M_k(p)- \\sum _{\\begin{array}{c}t \\in T_k, p \\in \\bullet t\\end{array}} W(p,t)+ \\sum _{\\begin{array}{c}t \\in T_l, p \\in t \\bullet : 0 \\le l \\le k, l+D(t) = k+1\\end{array}} W(t,p)$ $\\forall p \\in R(T_k), M_{k+1}(p) = \\sum _{\\begin{array}{c}t \\in T_l , p \\in t \\bullet : l \\le k, l+D(t) = k+1 \\end{array}} W(t,p)$ where $R(T_i)=\\displaystyle \\cup _{\\begin{array}{c}t \\in T_i\\end{array}} R(t)$ .", "A timed transition $t$ produces its output $D(t)$ time units after being fired.", "We replace $f\\ref {f:c:tparc}$ with $f\\ref {f:c:dur:tparc}$ adding transition duration and replace rule $r\\ref {r:c:add}$ with $r\\ref {r:c:dur:add}$ that produces tokens at the end of transition duration: f21: Rules tparc($t_i,p_j,n_c,c,ts_k,D(t_i)$ ):-time($ts_k$ ).", "for each $(t_i,p_j) \\in E^+$ , $c \\in C$ , $n_c=m_{W(t_i,p_j)}(c) : n_c > 0$ .", "r13: add(P,Q,T,C,TS):-fires(T,TS0),time(TS0;TS), tparc(T,P,Q,C,TS0,D), TS=TS0+D-1.", "Proposition 8 There is 1-1 correspondence between the answer sets of $\\Pi ^7(PN^D,M_0,$ $k,ntok)$ and the execution sequences of $PN^D$ .", "Above implementation of timed-transition is reentrant, however, we can easily make these timed transitions non-reentrant by adding rule $e\\ref {e:c:ne:dur}$ that disallows a transition from being enabled if it is already in progress: e10: notenabled(T,TS):-fires(T,TS0), num(N), TS>TS0, tparc(T,P,N,C,TS0,D), col(C), time(TS0), time(TS), TS<(TS0+D).", "Other Extensions Other Petri Net extensions can be implemented with similar ease.", "For example, Guard Conditions on transitions can be trivially implemented as a notenabled/2 rules.", "Self Modifying Petri Nets [77], which allow marking-dependent arc-weights can be implemented in a similar manner as the Reset Arc extension in section REF .", "Object Petri Nets [78], in which each token is a Petri Net itself can be implemented (using token reference semantics) by adding an additional “network-id” parameter to our encoding, where “id=0” is reserved for system net and higher numbers are used for token nets.", "Transition coordination between system & token nets is enforced through constraints on transition labels, where transition labels are added as additional facts about transitions.", "Related Work Petri Nets have been previously encoded in ASP, but the previous implementations have been limited to restricted classes of Petri Nets.", "For example, 1-safe Petri Net to ASP translation has been presented in [37], which is limited to binary Petri Nets.", "Translation of Logic Petri Nets to ASP has been presented in [6], but their model cannot handle numerical aggregation of tokens from multiple input transitions to the same place.", "Our work focused on problems in the biological domain and is more generalized.", "We can represent reset arcs, inhibition arcs, priority arcs as well as durative transitions.", "Conclusion We have presented an encoding of basic Petri Nets in ASP and showed how it can be easily extended to include extension to model various biological constructs.", "Portions of this work were published in [2] and [1].", "In the next chapter we will use Petri Nets and their ASP encoding to model biological pathways to answer questions about them.", "Answering Questions using Petri Nets and ASP Introduction In this chapter we use various Petri Net extensions presented in Chapter and their ASP encoding to answer question from [64] that were a part of the Second Deep Knowledge Representation Challengehttps://sites.google.com/site/2nddeepkrchallenge/.", "Definition 45 (Rate) Rate of product P is defined as the quantity of P produced per unit-time.", "Rate of an action A is defined as the number of time A occurs per unit-time.", "Comparing Altered Trajectories due to Reset Intervention Question 1 At one point in the process of glycolysis, both dihydroxyacetone phosphate (DHAP) and glyceraldehyde 3-phosphate (G3P) are produced.", "Isomerase catalyzes the reversible conversion between these two isomers.", "The conversion of DHAP to G3P never reaches equilibrium and G3P is used in the next step of glycolysis.", "What would happen to the rate of glycolysis if DHAP were removed from the process of glycolysis as quickly as it was produced?", "Provided Answer: “Glycolysis is likely to stop, or at least slow it down.", "The conversion of the two isomers is reversible, and the removal of DHAP will cause the reaction to shift in that direction so more G3P is converted to DHAP.", "If less (or no) G3P were available, the conversion of G3P into DHAP would slow down (or be unable to occur).” Solution 1 The process of glycolysis is shown in Fig 9.9 of Campbell's book.", "Glycolysis splits Glucose into Pyruvate.", "In the process it produces ATP and NADH.", "Any one of these can be used to gauge the glycolysis rate, since they will be produced in proportion to the input Glucose.", "The amount of pyruvate produced is the best choice since it is the direct end product of glycolysis.", "The ratio of the quantity of pyruvate produced over a representative span of time gives us the glycolysis rate.", "We assume a steady supply of Glucose is available and also assume that sufficient quantity of various enzymes used in glycolysis is available, since the question does not place any restriction on these substances.", "We narrow our focus to a subsection from Fructose 1,6-bisphosphate (F16BP) to 1,3-Bisphosphoglycerate (BPG13) as shown in Figure REF since that is the part the question is concerned with.", "We can ignore the linear chain up-stream of F16BP as well as the linear chain down-stream of BPG13 since the amount of F16BP available will be equal to Glucose and the amount of BPG13 will be equal to the amount of Pyruvate given our steady supply assumption.", "Figure: Petri Net graph relevant to question .", "“f16bp” is the compound Fructose 1,6-biphosphate, “bpg13” is 1,3-Bisphosphoglycerate.", "Transition trtr shown in dotted lines is added to model the elimination of dhapdhap as soon as it is produced.We fulfill the steady supply requirement of Glucose by a source transition-node $t3$ .", "We fulfill sufficient enzyme supply by a fixed quantity for each enzyme such that this quantity is in excess of what can be consumed during our simulation interval.", "Where the simulation interval is the number of time-steps over which we will measure the rate of glycolysis.", "We model the elimination of DHAP as soon as it is produced with a reset arc, shown with a dotted style in Figures REF .", "Such an arc removes all tokens from its source place when it fires.", "Since we have added it as an unconditional arc, it is always enabled for firing.", "We encode both situations in ASP with the maximal firing set policy.", "Both situations (without and with reset arc) are encoded in ASP and run for 10 steps.", "At the end of those 10 steps the amount of BPG13 is compared to determine the difference in the rate of glycolysis.", "In normal situation (without $(dhap,tr)$ reset arc), unique quantities of “bpg13” from all (2) answer-sets after 10 steps were as follows: holds(bpg13,14,10) holds(bpg13,16,10) with reset arc $tr$ , unique quantities of “bpg13” from all (512) answer-sets after 10 steps were as follows: holds(bpg13,0,10) holds(bpg13,10,10) holds(bpg13,12,10) holds(bpg13,14,10) holds(bpg13,16,10) holds(bpg13,2,10) holds(bpg13,4,10) holds(bpg13,6,10) holds(bpg13,8,10) Figure: Amount of “bpg13” produced in unique answer-sets produced by a 10 step simulation.", "The graph shows two situations, without the (dhap,tr)(dhap,tr) reset arc (normal situation) and with the reset arc (abnormal situation).", "The purpose of this graph is to depict the variation in the amounts of glycolysis produced in various answer sets.Note that the rate of glycolysis is generally lower when DHAP is immediately consumed.", "It is as low as zero essentially stopping glycolysis.", "The range of values are due to the choice between G3P being converted to DHAP or BPG13.", "If more G3P is converted to DHAP, then less BPG13 is produced and vice versa.", "Also, note that if G3P is not converted to BPG13, no NADH or ATP is produced either due to the liner chain from G3P to Pyruvate.", "The unique quantities of BPG13 are shown in a graphical format in Figure REF , while a trend of average quantity of BPG13 produced is shown in Figure REF .", "Figure: Average amount of “bpg13” produced during the 10-step simulation at various time steps.", "The average is over all answer-sets.", "The graph shows two situations, without the (dhap,tr)(dhap,tr) reset arc (normal situation) and with the reset arc (abnormal situation).", "The divergence in “bpg13” production is clearly shown.We created a minimal model of the Petri Net in Figure REF by removing enzymes and reactants that were not relevant to the question and did not contribute to the estimation of glycolysis.", "This is shown in Figure REF .", "Figure: Minimal version of the Petri Net graph in Figure .", "All reactants that do not contribute to the estimation of the rate of glycolysis have been removed.Simulating it for 10 steps with the same initial marking as the Petri Net in Figure REF produced the same results as for Figure REF .", "Determining Conditions Leading to an Observation Question 2 When and how does the body switch to B oxidation versus glycolysis as the major way of burning fuel?", "Provided Answer: “The relative volumes of the raw materials for B oxidation and glycolysis indicate which of these two processes will occur.", "Glycolysis uses the raw material glucose, and B oxidation uses Acyl CoA from fatty acids.", "When the blood sugar level decreases below its homeostatic level, then B oxidation will occur with available fatty acids.", "If no fatty acids are immediately available, glucagon and other hormones regulate release of stored sugar and fat, or even catabolism of proteins and nucleic acids, to be used as energy sources.” Solution 2 The answer provided requires background knowledge about the mechanism that regulates which source of energy will be used.", "This information is not presented in Chapter 9 of Campbell's book, which is the source material of this exercise.", "However, we can model it based on background information combined with Figure 9.19 of Campbell's book.", "Our model is presented in Figure REFWe can extend this model by adding expressions to inhibition arcs that compare available substances..", "Figure: Petri Net graph relevant to question .", "“fats” are fats, “prot” are proteins, “fac” are fatty acids, “sug” are sugars, “amin” are amino acids and “acoa” is ACoA.", "Transition “box” is the beta oxidation, “t5” is glycolysis, “t1” is fat digestion into fatty acids, and “t9” is protein deamination.We can test the model by simulating it for a time period and testing whether beta oxidation (box) is started when sugar (sug) is finished.", "We do not need a steady supply of sugar in this case, just enough to be consumed in a few time steps to capture the switch over.", "Fats and proteins may or may not modeled as a steady supply, since their bioavailability is dependent upon a number of external factors.", "We assume a steady supply of both and model it with large enough initial quantity that will last beyond the simulation period.", "We translate the petri net model into ASP and run it for 10 iterations.", "Following are the results: holds(acoa,0,0) holds(amin,0,0) holds(fac,0,0) holds(fats,5,0) holds(prot,3,0) holds(sug,4,0) fires(t1,0) fires(t5,0) fires(t9,0) holds(acoa,1,1) holds(amin,1,1) holds(fac,1,1) holds(fats,4,1) holds(prot,3,1) holds(sug,3,1) fires(t5,1) holds(acoa,2,2) holds(amin,1,2) holds(fac,1,2) holds(fats,4,2) holds(prot,3,2) holds(sug,2,2) fires(t5,2) holds(acoa,3,3) holds(amin,1,3) holds(fac,1,3) holds(fats,4,3) holds(prot,3,3) holds(sug,1,3) fires(t5,3) holds(acoa,4,4) holds(amin,1,4) holds(fac,1,4) holds(fats,4,4) holds(prot,3,4) holds(sug,0,4) fires(box,4) holds(acoa,5,5) holds(amin,1,5) holds(fac,0,5) holds(fats,4,5) holds(prot,3,5) holds(sug,0,5) fires(t1,5) fires(t9,5) holds(acoa,5,6) holds(amin,2,6) holds(fac,1,6) holds(fats,3,6) holds(prot,3,6) holds(sug,0,6) fires(box,6)c holds(acoa,6,7) holds(amin,2,7) holds(fac,0,7) holds(fats,3,7) holds(prot,3,7) holds(sug,0,7) fires(t1,7) fires(t9,7) holds(acoa,6,8) holds(amin,3,8) holds(fac,1,8) holds(fats,2,8) holds(prot,3,8) holds(sug,0,8) fires(box,8) holds(acoa,7,9) holds(amin,3,9) holds(fac,0,9) holds(fats,2,9) holds(prot,3,9) holds(sug,0,9) fires(t1,9) fires(t9,9) holds(acoa,7,10) holds(amin,4,10) holds(fac,1,10) holds(fats,1,10) holds(prot,3,10) holds(sug,0,10) fires(box,10) We can see that by time-step 4, the sugar supply is depleted and beta oxidation starts occurring.", "Comparing Altered Trajectories due to Accumulation Intervention Question 3 ATP is accumulating in the cell.", "What affect would this have on the rate of glycolysis?", "Explain.", "Provided Answer: “ATP and AMP regulate the activity of phosphofructokinase.", "When there is an abundance of AMP in the cell, this indicates that the rate of ATP consumption is high.", "The cell is in need for more ATP.", "If ATP is accumulating in the cell, this indicates that the cell's demand for ATP had decreased.", "The cell can decrease its production of ATP.", "Therefore, the rate of glycolysis will decrease.” Solution 3 Control of cellular respiration is summarized in Fig 9.20 of Campbell's book.", "We can gauge the rate of glycolysis by the amount of Pyruvate produced, which is the end product of glycolysis.", "We assume a steady supply of glucose is available.", "Its availability is not impacted by any of the feedback mechanism depicted in Fig 9.20 of Campbell's book or restricted by the question.", "We can ignore the respiration steps after glycolysis, since they are directly dependent upon the end product of glycolysis, i.e.", "Pyruvate.", "These steps only reinforce the negative effect of ATP.", "The Citrate feed-back shown in Campbell's Fig 9.20 is also not relevant to the question, so we can assume a constant level of it and leave it out of the picture.", "Another simplification that we do is to treat the inhibition of Phosphofructokinase (PFK) by ATP as the inhibition of glycolysis itself.", "This is justified, since PFK is on a linear path from Glucose to Fructose 1,6-bisphosphate (F16BP), and all downstream product quantities are directly dependent upon the amount of F16BP (as shown in Campbell's Fig 9.9), given steady supply of substances involved in glycolysis.", "Our assumption also applies to ATP consumed in Fig 9.9.", "Our simplified picture is shown in Figure REF as a Petri Net.", "Figure: Petri Net graph relevant to question .", "“glu” is Glucose, “pyr” is Pyruvate.", "Transitions “gly1” represents glycolysis and “cw1” is cellular work that consumes ATP and produces AMP.", "Transition “gly1” is inhibited only when the number of atpatp tokens is greater than 4.We model cellular work that recycles ATP to AMP (see p/181 of Campbell's book) by the $cw1$ transition, shown in dotted style.", "In normal circumstances, this arc does not let ATP to collect.", "If we reduce the arc-weights incident on $cw1$ to 1, we get the situation where less work is being done and some ATP will collect, as a result glycolysis will pause and resume.", "If we remove $cw1$ (representing no cellular work), ATP will start accumulating and glycolysis will stop.", "We use an arbitrary arc-weight of 4 on the inhibition arc $(atp,gly1)$ to model an elevated level of ATP beyond normal that would cause inhibition An alternate modeling would be compare the number of tokens on the $amp$ node and the $atp$ node and set a level-threshold that inhibits $gly1$ .", "Such technique is common in colored-peri nets.. We encode all three situations in ASP with maximal firing set policy.", "We run them for 10 steps and compare the quantity of pyruvate produced to determine the difference in the rate of glycolysis.", "In normal situation when cellular work is being performed ($cw1$ arc is present), unique quantities of “pyr” after 10 step are as follows: holds(pyr,20,10) when the cellular work is reduced, i.e.", "($(atp,cw1)$ , $(cw1,amp)$ arc weights changed to 1), unique quantities of “pyr” after 10 steps are as follows: holds(pyr,14,10) with no cellular work ($cw1$ arc removed), unique quantities of “pyr” after 10 steps are as follows: holds(pyr,6,10) The results show the rate of glycolysis reducing as the cellular work decreases to the point where it stops once ATP reaches the inhibition threshold.", "Higher numbers of ATP produced in later steps of cellular respiration will reinforce this inhibition even more quickly.", "Trend of answers from various runs is shown in Figure REF .", "Figure: Amount of pyruvate produced from various lengths of runs.", "Comparing Altered Trajectories due to Initial Value Intervention Question 4 A muscle cell had used up its supply of oxygen and ATP.", "Explain what affect would this have on the rate of cellular respiration and glycolysis?", "Provided Answer: “Oxygen is needed for cellular respiration to occur.", "Therefore, cellular respiration would stop.", "The cell would generate ATP by glycolysis only.", "Decrease in the concentration of ATP in the cell would stimulate an increased rate of glycolysis in order to produce more ATP.” Solution 4 Figure 9.18 of Campbell's book gives the general idea of what happens when oxygen is not present.", "Figure 9.20 of Campbell's book shows the control of glycolysis by ATP.", "To formulate the answer, we need pieces from both.", "ATP inhibits Phosphofructokinase (Fig 9.20 of Campbell), which is an enzyme used in glycolysis.", "No ATP means that enzyme is no longer inhibited and glycolysis can proceed at full throttle.", "Pyruvate either goes through aerobic respiration when oxygen is present or it goes through fermentation when oxygen is absent (Fig 9.18 of Campbell).", "We can monitor the rate of glycolysis and cellular respiration by observing these operations occurring (by looking at corresponding transition firing) over a simulation time period.", "Our simplified Petri Net model is shown in Figure REF .", "We ignore the details of processes following glycolysis, except that these steps produce additional ATP.", "We do not need an exact number of ATP produced as long as we keep it higher than the ATP produced by glycolysis.", "Higher numbers will just have a higher negative feed-back (or inhibition) effect on glycolysis.", "We ignore citrate's inhibition of glycolysis since that is not relevant to the question and since it gets recycled by the citric acid cycle (see Fig 9.12 of Campbell).", "We also ignore AMP, since it is not relevant to the question, by assuming sufficient supply to maintain glycolysis.", "We also assume continuous cellular work consuming ATP, without that ATP will accumulate almost immediately and stop glycolysis.", "We assume a steady supply of glucose is available to carry out glycolysis and fulfill this requirement by having a quantity in excess of the consumption during our simulation interval.", "All other substances participating in glycolysis are assumed to be available in a steady supply so that glycolysis can continue.", "Figure: Petri Net graph relevant to question .", "“glu” is Glucose, “pyr” is Pyruvate, “atp” is ATP, “eth” is ethenol or other products of fermentation, and “o2” is Oxygen.", "Transitions “gly1” represents glycolysis, “res1” is respiration in presence of oxygen, “fer1” is fermentation when no oxygen is present, and “cw1” is cellular work that consumes ATP.", "Transition “gly1” is inhibited only when the number of atpatp tokens is greater than 4.We then consider two scenarios, one where oxygen is present and where oxygen is absent and determine the change in rate of glycolysis and respiration by counting the firings of their respective transitions.", "We encode both situations in ASP with maximal firing set policy.", "Both situations are executed for 10 steps.", "At the end of those steps the firing count of “gly1” and “res1” is computed and compared to determine the difference in the rates of glycolysis and respiration respectively.", "In the normal situation (when oxygen is present), we get the following answer sets: fires(gly1,0) fires(cw1,1) fires(gly1,1) fires(res1,1) fires(cw1,2) fires(res1,2) fires(cw1,3) fires(cw1,4) fires(cw1,5) fires(gly1,5) fires(cw1,6) fires(gly1,6) fires(res1,6) fires(cw1,7) fires(res1,7) fires(cw1,8) fires(cw1,9) fires(cw1,10) while in the abnormal situation (when oxygen is absent), we get the following firings: fires(gly1,0) fires(cw1,1) fires(fer1,1) fires(gly1,1) fires(cw1,2) fires(fer1,2) fires(gly1,2) fires(cw1,3) fires(fer1,3) fires(gly1,3) fires(cw1,4) fires(fer1,4) fires(gly1,4) fires(cw1,5) fires(fer1,5) fires(gly1,5) fires(cw1,6) fires(fer1,6) fires(gly1,6) fires(cw1,7) fires(fer1,7) fires(gly1,7) fires(cw1,8) fires(fer1,8) fires(gly1,8) fires(cw1,9) fires(fer1,9) fires(gly1,9) fires(cw1,10) fires(fer1,10) fires(gly1,10) Note that the number of firings of glycolysis for normal situation is lower when oxygen is present and higher when oxygen is absent.", "While, the number of firings is zero when no oxygen is present.", "Thus, respiration stops when no oxygen is present and the need of ATP by cellular work is fulfilled by a higher amount of glycolysis.", "Trend from various runs is shown in Figure REF .", "Figure: Firing counts of glycolysis (gly1) and respiration (res1) for different simulation lengths for the petri net in Figure Comparing Altered Trajectories due to Inhibition Intervention Question 5 The final protein complex in the electron transport chain of the mitochondria is non-functional.", "Explain the effect of this on pH of the intermembrane space of the mitochondria.", "Provided Answer: “The H+ ion gradient would gradually decrease and the pH would gradually increase.", "The other proteins in the chain are still able to produce the H+ ion gradient.", "However, a non-functional, final protein in the electron transport chain would mean that oxygen is not shuttling electrons away from the electron transport chain.", "This would cause a backup in the chain, and the other proteins in the electron transport chain would no longer be able to accept electrons and pump H+ ions into the intermembrane space.", "A concentration decrease in the H+ ions means an increase in the pH.” Solution 5 The electron transport chain is shown in Fig 9.15 (1) of Campbell's book.", "In order to explain the effect on pH, we will show the change in the execution of the electron transport chain with both a functioning and non-functioning final protein.", "Since pH depends upon the concentration of H+ ions, we will quantify the difference its quantity in the intermembrane space in both scenarios as well.", "We assume that a steady input of NADH, FADH2, H+ and O2 are available in the mitochondrial matrix.", "We also assume an electron carrying capacity of 2 for both ubiquinone (Q)http://www.benbest.com/nutrceut/CoEnzymeQ.html and cytochrome c (Cyt c).", "This carrying capacity is background information not provided in Campbell's Chapter 9.", "As with previous questions, we fulfill the steady supply requirement of substances by having input quantities in excess of what would be consumed during our simulation interval.", "Figure: Petri Net graph relevant to question .", "“is” is the intermembrane space, “mm” is mitochondrial matrix, “q” is ubiquinone and “cytc” is cytochrome c. The inhibition arcs (q,t1)(q,t1), (q,t2)(q,t2) and (cytc,t3)(cytc,t3) capture the electron carrying capacities of qq and cytccytc.", "Over capacity will cause backup in electron transport chain.", "Tokens are colored, e.g.", "“h:1, nadh:1” specify one token of hh and nadhnadh each.", "Token types are “h” for H+, “nadh” for NADH, “nadp” for NADP, “fadh2” for FADH2, “fad” for FAD, “e” for electrons, “o2” for oxygen and “h2o” for water.", "We remove t4t4 to model non-functioning protein complex IVIV.We model this problem as a colored petri net shown in Figure REF .", "The normal situation is made up of the entire graph.", "The abnormal situation (with non-functional final protein complex) is modeled by removing transition $t4$ from the graphAlternatively, we can model a non-functioning transition by attaching an inhibition arc to it with one token at its source place.", "We encode both situations in ASP with maximal firing set policy.", "Both are run for 10 steps.", "The amount of $h$ (H+) is compared in the $is$ (intermembrane space) to determine change in pH and the firing sequence is compared to explain the effect.", "In normal situation (entire graph), we get the following: fires(t1,0) fires(t2,0) fires(t3,1) fires(t1,2) fires(t2,2) fires(t3,2) fires(t4,2) fires(t3,3) fires(t4,3) fires(t6,3) fires(t1,4) fires(t2,4) fires(t3,4) fires(t4,4) fires(t6,4) fires(t3,5) fires(t4,5) fires(t6,5) fires(t1,6) fires(t2,6) fires(t3,6) fires(t4,6) fires(t6,6) fires(t3,7) fires(t4,7) fires(t6,7) fires(t1,8) fires(t2,8) fires(t3,8) fires(t4,8) fires(t6,8) fires(t3,9) fires(t4,9) fires(t6,9) fires(t1,10) fires(t2,10) fires(t3,10) fires(t4,10) fires(t6,10) holds(is,15,h,10) with $t4$ removed, we get the following: fires(t1,0) fires(t2,0) fires(t3,1) fires(t1,2) fires(t2,2) fires(t3,2) fires(t6,3) fires(t6,4) holds(is,2,h,10) Note that the amount of H+ ($h$ ) produced in the intermembrane space ($is$ ) is much smaller when the final protein complex is non-functional ($t4$ removed).", "Lower H+ translates to higher pH.", "Thus, the pH of intermembrane space will increase as a result of nonfunctional final protein.", "Also, note that the firing of $t3$ , $t1$ and $t2$ responsible for shuttling electrons also stop very quickly when $t4$ no longer removes the electrons ($e$ ) from Cyt c ($cytc$ ) to produce $H_2O$ .", "This is because $cytc$ and $q$ are at their capacity on electrons that they can carry and stop the electron transport chain by inhibiting transitions $t3$ , $t2$ and $t1$ .", "Trend for various runs is shown in Figure REF .", "Figure: Simulation of Petri Net in Figure .", "In a complete model of the biological system, there will be a mechanism that keeps the quantity of H+ in check in the intermembrane space and will plateau at some point.", "Comparing Altered Trajectories due to Gradient Equilization Intervention Question 6 Exposure to a toxin caused the membranes to become permeable to ions.", "In a mitochondrion, how would this affect the pH in the intermembrane space and also ATP production?", "Provided Answer: “The pH of the intermembrane space would decrease as H+ ions diffuse through the membrane, and the H+ ion gradient is lost.", "The H+ gradient is essential in ATP production b/c facilitated diffusion of H+ through ATP synthase drives ATP synthesis.", "Decreasing the pH would lead to a decrease in the rate of diffusion through ATP synthase and therefore a decrease in the production of ATP.” Solution 6 Oxidative phosphorylation is shown in Fig 9.15 of Campbell's book.", "In order to explain the effect on pH in the intermembrane space and the ATP production we will show the change in the amount of H+ ions in the intermembrane space as well as the amount of ATP produced when the inner mitochondrial membrane is impermeable and permeable.", "Note that the concentration of H+ determines the pH.", "we have chosen to simplify the diagram by not having FADH2 in the picture.", "Its removal does not change the response, since it provides an alternate input mechanism to electron transport chain.", "We will assume that a steady input of NADH, H+, O2, ADP and P is available in the mitochondrial matrix.", "We also assume an electron carrying capacity of 2 for both ubiquinone (Q) and cytochrome c (Cyt c).", "We fulfill the steady supply requirement of substances by having input quantities in excess of what would be consumed during our simulation interval.", "We model this problem as a colored petri net shown in Figure REF .", "Transition $t6,t7$ shown in dotted style are added to model the abnormal situationIf reverse permeability is also desired additional arcs may be added from mm to is.", "They capture the diffusion of H+ ions back from Intermembrane Space to the Mitochondrial matrix.", "One or both may be enabled to capture degrees of permeability.", "we have added a condition on the firing of $t5$ (ATP Synthase activation) to enforce gradient to pump ATP Synthase.", "Figure: Petri Net graph relevant to question .", "“is” is the intermembrane space, “mm” is mitochondrial matrix, “q” is ubiquinone and “cytc” is cytochrome c. Tokens are colored, e.g.", "“h:1, nadh:1” specify one token of hh and nadhnadh each.", "Token types are “h” for H+, “nadh” for NADH, “e” for electrons, “o2” for oxygen, “h2o” for water, “atp” for ATP and “adp” for ADP.", "We add t6,t7t6,t7 to model cross domain diffusion from intermembrane space to mitochondrial matrix.", "One or both of t6,t7t6,t7 may be enabled at a time to control the degree of permeability.", "The text above “t5” is an additional condition which must be satisfied for “t5” to be enabled.We encode both situations in ASP with maximal firing set policy.", "Both are run for 10 steps each and the amount of $h$ and $atp$ is compared to determine the effect of pH and ATP production.", "We capture the gradient requirement as the following ASP codeWe can alternatively model this by having a threshold arc from “is” to “t5” if only a minimum trigger quantity is required in the intermembrane space.", ": notenabled(T,TS) :- T==t5, C==h, trans(T), col(C), holds(is,Qis,C,TS), holds(mm,Qmm,C,TS), Qmm+3 > Qis, num(Qis;Qmm), time(TS).", "In the normal situation, we get the following $h$ token distribution after 10 steps: holds(is,11,h,10) holds(mm,1,h,10) holds(mm,6,atp,10) we change the permeability to 1 ($t6$ enabled), we get the following token distribution instead: holds(is,10,h,10) holds(mm,2,h,10) holds(mm,5,atp,10) we change the permeability to 2 ($t6,t7$ enabled), the distribution changes as follows: holds(is,8,h,10) holds(mm,4,h,10) holds(mm,2,atp,10) Note that as the permeability increases, the amount of H+ ($h$ ) in intermembrane space ($is$ ) decreases and so does the amount of ATP ($h$ ) in mitochondrial matrix.", "Thus, an increase in permeability will increase the pH.", "If the permeability increases even beyond 2, no ATP will be produced from ADP due to insufficient H+ gradient.", "Trend from various runs is shown in Figure REF .", "Figure: Quantities of H+ and ATP at various run lengths and permeabilities for the Petri Net model in Figure .", "Comparing Altered Trajectories due to Delay Intervention Question 7 Membranes must be fluid to function properly.", "How would decreased fluidity of the membrane affect the efficiency of the electron transport chain?", "Provided Answer: “Some of the components of the electron transport chain are mobile electron carriers, which means they must be able to move within the membrane.", "If fluidity decreases, these movable components would be encumbered and move more slowly.", "This would cause decreased efficiency of the electron transport chain.” Solution 7 The answer deals requires background knowledge about fluidity and how it relates to mobile carriers not presented in the source chapter.", "From background knowledge we find that the higher the fluidity, higher the mobility.", "The electron transport chain is presented in Fig 9.15 of Campbell's book.", "From background knowledge, we know that the efficiency of the electron transport chain is measured by the amount of ATP produced per NADH/FADH2.", "The ATP production happens due to the gradient of H+ ions across the mitochondrial membrane.", "The higher the number of H+ ions in the intermembrane space, the higher would be the gradient and the resulting efficiency.", "So we measure the efficiency of the chain by the amount of H+ transported to intermembrane space, assuming all other (fixed) molecules behave normally.", "This is a valid assumption since H+ transported from mitochondrial matrix is directly proportional to the amount of electrons shuttled through the non-mobile complexes and there is a linear chain from the electron carrier to oxygen.", "We model this chain using a Petri Net with durative transitions shown in Figure REF .", "Higher the duration of transitions, lower the fluidity of the membrane.", "We assume that a steady supply of NADH and H+ is available in the mitochondrial membrane.", "We fulfill this requirement by having quantities in excess of what will be consumed during the simulation.", "We ignore FADH2 from the diagram, since it is just an alternate path to the electron chain.", "Using it by itself will produce a lower number of H+ transporter to intermembrane space, but it will not change the result.", "We compare the amount of H+ transported into the intermembrane space to gauge the efficiency of the electron transport chain.", "More efficient the chain is, more H+ will it transport.", "We model three scenarios: normal fluidity, low fluidity with transitions $t3$ and $t4$ having an execution time of 2 and an lower fluidity with transitions $t3,t4$ having execution time of 4.", "Figure: Petri Net graph relevant to question .", "“is” is the intermembrane space, “mm” is mitochondrial matrix, “q” is ubiquinone and “cytc” is cytochrome c. Tokens are colored, e.g.", "“h:1, nadh:1” specify one token of hh and nadhnadh each.", "Token types are “h” for H+, “nadh” for NADH, “e” for electrons.", "Numbers in square brackets below the transition represent transition durations with default of one time unit, if the number is missing.We encode these cases in ASP with maximal firing set semantics and simulate them for 10 time steps.", "For the normal fluidity we get: holds(is,27,h,10) for low fluidity we get: holds(is,24,h,10) for lower fluidity we get: holds(is,18,h,10) Note that as the fluidity decreases, so does the amount of H+ transported to intermembrane space, pointing to lower efficiency of electron transport chain.", "Trend of various runs is shown in Figure REF .", "Figure: Quantities of H+ produced in the intermembrane space at various run lengths and fluidities for the Petri Net model in Figure .", "Comparing Altered Trajectories due to Priority and Read Interventions Question 8 Phosphofructokinase (PFK) is allosterically regulated by ATP.", "Considering the result of glycolysis, is the allosteric regulation of PFK likely to increase or decrease the rate of activity for this enzyme?", "Provided Answer: “Considering that one of the end products of glycolysis is ATP, PFK is inhibited when ATP is abundant and bound to the enzyme.", "The inhibition decreases ATP production along this pathway.” Solution 8 Regulation of Phosphofructokinase (PFK) is presented in Figure 9.20 of Campbell's book.", "We ignore substances upstream of Fructose 6-phosphate (F6P) by assuming they are available in abundance.", "We also ignore AMP by assuming normal supply of it.", "We also ignore any output of glycolysis other than ATP production since the downstream processes ultimately produce additional ATP.", "Citric acid is also ignored since it is not relevant to the question at hand.", "We monitor the rate of activity of PFK by the number of times it gets used for glycolysis.", "We model this problem as a Petri Net shown in Figure REF .", "Allosteric regulation of PFK is modeled by a compound “pfkatp” which represents PFK's binding with ATP to form a compound.", "Details of allosteric regulation are not provided in the same chapter, they are background knowledge from external sources.", "Higher than normal quantity of ATP is modeled by a threshold arc (shown with arrow-heads at both ends) with an arbitrary threshold value of 4.", "This number can be increased as necessary.", "The output of glycolysis and down stream processes “t3” has been set to 2 to run the simulation in a reasonable amount of time.", "It can be made larger as necessary.", "The allosteric regulation transition “t4” has also been given a higher priority than glycolysis transition “t3”.", "This way, ATP in excess will cause PFK to be converted to PFK+ATP compound, reducing action of PFK.", "We assume that F6P is available in sufficient quantity and so is PFK.", "This requirement is fulfilled by having more quantity than can be consumed in the simulation duration.", "We model both the normal situation including transition $t4$ shown in dotted style and the abnormal situation where $t4$ is removed.", "Figure: Petri Net graph relevant to question .", "“pfk” is phosphofructokinase, “f6p” is fructose 6-phosphate, “atp” is ATP and “pfkatp” is the pfk bound with atp for allosteric regulation.", "Transition “t3” represents enzymic action of pfk, “t4” represents the binding of pfk with atp.", "The double arrowed arc represents a threshold arc, which enables “t4” when there are at least 4 tokens available at “atp”.", "Numbers above transitions in angular brackets represent arc priorities.We encode both situations in ASP with maximal firing set policy and run them for 10 time steps.", "At the end of the run we compare the firing count of transition $t3$ for both cases.", "For the normal case (with $t4$ ), we get the following results: holds(atp,0,c,0) holds(f6p,20,c,0) holds(pfk,20,c,0) holds(pfkatp,0,c,0) fires(t3,0) holds(atp,2,c,1) holds(f6p,19,c,1) holds(pfk,19,c,1) holds(pfkatp,0,c,1) fires(t3,1) holds(atp,4,c,2) holds(f6p,18,c,2) holds(pfk,18,c,2) holds(pfkatp,0,c,2) fires(t4,2) holds(atp,3,c,3) holds(f6p,18,c,3) holds(pfk,17,c,3) holds(pfkatp,1,c,3) fires(t3,3) holds(atp,5,c,4) holds(f6p,17,c,4) holds(pfk,16,c,4) holds(pfkatp,1,c,4) fires(t4,4) holds(atp,4,c,5) holds(f6p,17,c,5) holds(pfk,15,c,5) holds(pfkatp,2,c,5) fires(t4,5) holds(atp,3,c,6) holds(f6p,17,c,6) holds(pfk,14,c,6) holds(pfkatp,3,c,6) fires(t3,6) holds(atp,5,c,7) holds(f6p,16,c,7) holds(pfk,13,c,7) holds(pfkatp,3,c,7) fires(t4,7) holds(atp,4,c,8) holds(f6p,16,c,8) holds(pfk,12,c,8) holds(pfkatp,4,c,8) fires(t4,8) holds(atp,3,c,9) holds(f6p,16,c,9) holds(pfk,11,c,9) holds(pfkatp,5,c,9) fires(t3,9) holds(atp,5,c,10) holds(f6p,15,c,10) holds(pfk,10,c,10) holds(pfkatp,5,c,10) fires(t4,10) Note that $t3$ fires only when the ATP falls below our set threshold, above it PFK is converted to PFK+ATP compound via $t4$ .", "For the abnormal case (without $t4$ ) we get the following results: holds(atp,0,c,0) holds(f6p,20,c,0) holds(pfk,20,c,0) holds(pfkatp,0,c,0) fires(t3,0) holds(atp,2,c,1) holds(f6p,19,c,1) holds(pfk,19,c,1) holds(pfkatp,0,c,1) fires(t3,1) holds(atp,4,c,2) holds(f6p,18,c,2) holds(pfk,18,c,2) holds(pfkatp,0,c,2) fires(t3,2) holds(atp,6,c,3) holds(f6p,17,c,3) holds(pfk,17,c,3) holds(pfkatp,0,c,3) fires(t3,3) holds(atp,8,c,4) holds(f6p,16,c,4) holds(pfk,16,c,4) holds(pfkatp,0,c,4) fires(t3,4) holds(atp,10,c,5) holds(f6p,15,c,5) holds(pfk,15,c,5) holds(pfkatp,0,c,5) fires(t3,5) holds(atp,12,c,6) holds(f6p,14,c,6) holds(pfk,14,c,6) holds(pfkatp,0,c,6) fires(t3,6) holds(atp,14,c,7) holds(f6p,13,c,7) holds(pfk,13,c,7) holds(pfkatp,0,c,7) fires(t3,7) holds(atp,16,c,8) holds(f6p,12,c,8) holds(pfk,12,c,8) holds(pfkatp,0,c,8) fires(t3,8) holds(atp,18,c,9) holds(f6p,11,c,9) holds(pfk,11,c,9) holds(pfkatp,0,c,9) fires(t3,9) holds(atp,20,c,10) holds(f6p,10,c,10) holds(pfk,10,c,10) holds(pfkatp,0,c,10) fires(t3,10) Note that when ATP is not abundant, transition $t3$ fires continuously, which represents the enzymic activity that converts F6P to downstream substances.", "Trend of various runs is shown in Figure REF .", "Figure: Petri Net model in Figure .", "Comparing Altered Trajectories due to Automatic Conversion Intervention Question 9 How does the oxidation of NADH affect the rate of glycolysis?", "Provided Answer: “NADH must be oxidized back to NAD+ in order to be used in glycolysis.", "Without this molecule, glycolysis cannot occur.” Solution 9 Cellular respiration is summarized in Fig 9.6 of Campbell's book.", "NAD+ is reduced to NADH during glycolysis (see Campbell's Fig 9.9) during the process of converting Glyceraldehyde 3-phosphate (G3P) to 1,3-Bisphosphoglycerate (BPG13).", "NADH is oxidized back to NAD+ during oxidative phosphorylation by the electron transport chain (see Campbell's Fig 9.15).", "We can gauge the rate of glycolysis by the amount of Pyruvate produced, which is the end product of glycolysis.", "We simplify our model by abstracting glycolysis as a black-box that takes Glucose and NAD+ as input and produces NADH and Pyruvate as output, since there is a linear chain from Glucose to Pyruvate that depends upon the availability of NAD+.", "We also abstract oxidative phosphorylation as a black-box which takes NADH as input and produces NAD+ as output.", "None of the other inner workings of oxidative phosphorylation play a role in answering the question assuming they are functioning normally.", "We also ignore the pyruvate oxidation and citric acid cycle stages of cellular respiration since their end products only provide additional raw material for oxidative phosphorylation and do not add value to answering the question.", "We assume a steady supply of Glucose and all other substances used in glycolysis but a limited supply of NAD+, since it can be recycled from NADH and we want to model its impact.", "We fulfill the steady supply requirement of Glucose with sufficient initial quantity in excess of what will be consumed during our simulation interval.", "We also ensure that we have sufficient initial quantity of NAD+ to maintain glycolysis as long as it can be recycled.", "Figure: Petri Net graph relevant to question .", "“glu” represents glucose, “gly1” represents glycolysis, “pyr” represents pyruvate, “ox1” represents oxidative phosphorylation, “nadh” represents NADH and “nadp” represents NAD+.", "“ox1” is removed to model stoppage of oxidation of NADH to NAD+.Figure REF is a Petri Net representation of our simplified model.", "Normal situation is modeled by the entire graph, where NADH is recycled back to NAD+, while the abnormal situation is modeled by the graph with the transition $ox1$ (shown in dotted style) removed.", "We encode both situations in ASP with the maximal firing set policy.", "Both situations are run for 5 steps and the amount of pyruvate is compared to determine the difference in the rate of glycolysis.", "In normal situation (with $ox1$ transition), unique quantities of pyruvate ($pyr$ ) are as follows: fires(gly1,0) fires(gly1,1) fires(gly1,2) fires(gly1,3) fires(gly1,4) holds(pyr,10,5) while in abnormal situation (without $ox1$ transition), unique quantities of pyruvate are as follows: fires(gly1,0) fires(gly1,1) fires(gly1,2) holds(pyr,6,5) Note that the rate of glycolysis is lower when NADH is not recycled back to NAD+, as the glycolysis stops after the initial quantity of 6 NAD+ is consumed.", "Also, the $gly1$ transition does not fire after time-step 2, indicating glycolysis has stopped.", "Trend of various runs is shown in Figure REF .", "Figure: Amount of “pyr” produced by runs of various lengths of Petri Net in Figure .", "It shows results for both normal situation where “nadh” is recycled to “nadp” as well as the abnormal situation where this recycling is stopped.", "Comparing Altered Trajectories due to Initial Value Intervention Question 10 During intense exercise, can a muscle cell use fat as a concentrated source of chemical energy?", "Explain.", "Provided Answer: “When oxygen is present, the fatty acid chains containing most of the energy of a fat are oxidized and fed into the citric acid cycle and the electron transport chain.", "During intense exercise, however, oxygen is scarce in muscle cells, so ATP must be generated by glycolysis alone.", "A very small part of the fat molecule, the glycerol backbone, can be oxidized via glycolysis, but the amount of energy released by this portion is insignificant compared to that released by the fatty acid chains.", "(This is why moderate exercise, staying below 70% maximum heart rate, is better for burning fat because enough oxygen remains available to the muscles.", ")” Solution 10 The process of fat consumption in glycolysis and citric acid cycle is summarized in Fig 9.19 of Campbell's book.", "Fats are digested into glycerol and fatty acids.", "Glycerol gets fed into glycolysis after being converted into Gyceraldehyde 3-phosphate (G3P), while fatty acids get fed into citric acid cycle after being broken down through beta oxidation and converted into Acetyl CoA.", "Campbell's Fig 9.18 identify a junction in catabolism where aerobic respiration or fermentation take place depending upon whether oxygen is present or not.", "Energy produced at various steps is in terms of ATP produced.", "In order to explain whether fat can be used as a concentrated source of chemical energy or not, we have to show the different ways of ATP production and when they kick in.", "We combine the various pieces of information collected from Fig 9.19, second paragraph on second column of p/180, Fig 9.15, Fig 9.16 and Fig 9.18 of Campbell's book into Figure REF .", "We model two situations when oxygen is not available in the muscle cells (at the start of a intense exercise) and when oxygen is available in the muscle cells (after the exercise intensity is plateaued).", "We then compare and contrast them on the amount of ATP produced and the reasons for the firing sequences.", "Figure: Petri Net graph relevant to question .", "“fats” are fats, “dig” is digestion of fats, “gly” is glycerol, “fac” is fatty acid, “g3p” is Glyceraldehyde 3-phosphate, “pyr” is pyruvate, “o2” is oxygen, “nadh” is NADH, “acoa” is Acyl CoA, “atp” is ATP, “op1” is oxidative phosphorylation, “cac1” is citric acid cycle, “fer1” is fermentation, “ox1” is oxidation of pyruvate to Acyl CoA and “box1” is beta oxidation.Figure REF is a petri net representation of our simplified model.", "Our edge labels have lower numbers on them than the yield in Fig 9.16 of Campbell's book but they still capture the difference in volume that would be produced due to oxidative phosphorylation vs. glycolysis.", "Using exact amounts will only increase the difference of ATP production due to the two mechanisms.", "We encode both situations (when oxygen is present and when it is not) in ASP with maximal firing set policy.", "We run them for 10 steps.", "The firing sequence and the resulting yield of ATP explain what the possible use of fat as a source of chemical energy.", "At he start of intense exercise, when oxygen is in short supply: holds(acoa,0,0) holds(atp,0,0) holds(fac,0,0) holds(fats,5,0) holds(g3p,0,0) holds(gly,0,0) holds(nadh,0,0) holds(o2,0,0) holds(pyr,0,0) fires(dig,0) holds(acoa,0,1) holds(atp,0,1) holds(fac,1,1) holds(fats,4,1) holds(g3p,0,1) holds(gly,1,1) holds(nadh,0,1) holds(o2,0,1) holds(pyr,0,1) fires(box1,1) fires(dig,1) fires(t2,1) holds(acoa,1,2) holds(atp,0,2) holds(fac,1,2) holds(fats,3,2) holds(g3p,1,2) holds(gly,1,2) holds(nadh,1,2) holds(o2,0,2) holds(pyr,0,2) fires(box1,2) fires(cac1,2) fires(dig,2) fires(gly6,2) fires(t2,2) holds(acoa,1,3) holds(atp,2,3) holds(fac,1,3) holds(fats,2,3) holds(g3p,1,3) holds(gly,1,3) holds(nadh,3,3) holds(o2,0,3) holds(pyr,1,3) fires(box1,3) fires(cac1,3) fires(dig,3) fires(fer1,3) fires(gly6,3) fires(t2,3) holds(acoa,1,4) holds(atp,4,4) holds(fac,1,4) holds(fats,1,4) holds(g3p,1,4) holds(gly,1,4) holds(nadh,5,4) holds(o2,0,4) holds(pyr,1,4) fires(box1,4) fires(cac1,4) fires(dig,4) fires(fer1,4) fires(gly6,4) fires(t2,4) holds(acoa,1,5) holds(atp,6,5) holds(fac,1,5) holds(fats,0,5) holds(g3p,1,5) holds(gly,1,5) holds(nadh,7,5) holds(o2,0,5) holds(pyr,1,5) fires(box1,5) fires(cac1,5) fires(fer1,5) fires(gly6,5) fires(t2,5) holds(acoa,1,6) holds(atp,8,6) holds(fac,0,6) holds(fats,0,6) holds(g3p,1,6) holds(gly,0,6) holds(nadh,9,6) holds(o2,0,6) holds(pyr,1,6) fires(cac1,6) fires(fer1,6) fires(gly6,6) holds(acoa,0,7) holds(atp,10,7) holds(fac,0,7) holds(fats,0,7) holds(g3p,0,7) holds(gly,0,7) holds(nadh,10,7) holds(o2,0,7) holds(pyr,1,7) fires(fer1,7) holds(acoa,0,8) holds(atp,10,8) holds(fac,0,8) holds(fats,0,8) holds(g3p,0,8) holds(gly,0,8) holds(nadh,10,8) holds(o2,0,8) holds(pyr,0,8) holds(acoa,0,9) holds(atp,10,9) holds(fac,0,9) holds(fats,0,9) holds(g3p,0,9) holds(gly,0,9) holds(nadh,10,9) holds(o2,0,9) holds(pyr,0,9) holds(acoa,0,10) holds(atp,10,10) holds(fac,0,10) holds(fats,0,10) holds(g3p,0,10) holds(gly,0,10) holds(nadh,10,10) holds(o2,0,10) holds(pyr,0,10) when the exercise intensity has plateaued and oxygen is no longer in short supply: holds(acoa,0,0) holds(atp,0,0) holds(fac,0,0) holds(fats,5,0) holds(g3p,0,0) holds(gly,0,0) holds(nadh,0,0) holds(o2,10,0) holds(pyr,0,0) fires(dig,0) holds(acoa,0,1) holds(atp,0,1) holds(fac,1,1) holds(fats,4,1) holds(g3p,0,1) holds(gly,1,1) holds(nadh,0,1) holds(o2,10,1) holds(pyr,0,1) fires(box1,1) fires(dig,1) fires(t2,1) holds(acoa,1,2) holds(atp,0,2) holds(fac,1,2) holds(fats,3,2) holds(g3p,1,2) holds(gly,1,2) holds(nadh,1,2) holds(o2,10,2) holds(pyr,0,2) fires(box1,2) fires(cac1,2) fires(dig,2) fires(gly6,2) fires(op1,2) fires(t2,2) holds(acoa,1,3) holds(atp,5,3) holds(fac,1,3) holds(fats,2,3) holds(g3p,1,3) holds(gly,1,3) holds(nadh,2,3) holds(o2,9,3) holds(pyr,1,3) fires(box1,3) fires(cac1,3) fires(dig,3) fires(gly6,3) fires(op1,3) fires(ox1,3) fires(t2,3) holds(acoa,2,4) holds(atp,10,4) holds(fac,1,4) holds(fats,1,4) holds(g3p,1,4) holds(gly,1,4) holds(nadh,4,4) holds(o2,8,4) holds(pyr,1,4) fires(box1,4) fires(cac1,4) fires(dig,4) fires(gly6,4) fires(op1,4) fires(ox1,4) fires(t2,4) holds(acoa,3,5) holds(atp,15,5) holds(fac,1,5) holds(fats,0,5) holds(g3p,1,5) holds(gly,1,5) holds(nadh,6,5) holds(o2,7,5) holds(pyr,1,5) fires(box1,5) fires(cac1,5) fires(gly6,5) fires(op1,5) fires(ox1,5) fires(t2,5) holds(acoa,4,6) holds(atp,20,6) holds(fac,0,6) holds(fats,0,6) holds(g3p,1,6) holds(gly,0,6) holds(nadh,8,6) holds(o2,6,6) holds(pyr,1,6) fires(cac1,6) fires(gly6,6) fires(op1,6) fires(ox1,6) holds(acoa,4,7) holds(atp,25,7) holds(fac,0,7) holds(fats,0,7) holds(g3p,0,7) holds(gly,0,7) holds(nadh,9,7) holds(o2,5,7) holds(pyr,1,7) fires(cac1,7) fires(op1,7) fires(ox1,7) holds(acoa,4,8) holds(atp,29,8) holds(fac,0,8) holds(fats,0,8) holds(g3p,0,8) holds(gly,0,8) holds(nadh,10,8) holds(o2,4,8) holds(pyr,0,8) fires(cac1,8) fires(op1,8) holds(acoa,3,9) holds(atp,33,9) holds(fac,0,9) holds(fats,0,9) holds(g3p,0,9) holds(gly,0,9) holds(nadh,10,9) holds(o2,3,9) holds(pyr,0,9) fires(cac1,9) fires(op1,9) holds(acoa,2,10) holds(atp,37,10) holds(fac,0,10) holds(fats,0,10) holds(g3p,0,10) holds(gly,0,10) holds(nadh,10,10) holds(o2,2,10) holds(pyr,0,10) fires(cac1,10) fires(op1,10) We see that more ATP is produced when oxygen is available.", "Most ATP (energy) is produced by the oxidative phosphorylation which requires oxygen.", "When oxygen is not available, small amount of energy is produced due to glycolysis of glycerol ($gly$ ).", "With oxygen a lot more energy is produced, most of it due to fatty acids ($fac$ ).", "Trend of various runs is shown in Figure REF .", "Figure: Amount of “atp” produced by runs of various lengths of Petri Net in Figure .", "Two situations are shown: when oxygen is in short supply and when it is abundant.", "Conclusion In this chapter we presented how to model biological systems as Petri Nets, translated them into ASP, reasoned with them and answered questions about them.", "We used diagrams from Campbell's book, background knowledge and assumptions to facilitate our modeling work.", "However, source knowledge for real world applications comes from published papers, magazines and books.", "This means that we have to do text extraction.", "In one of the following chapters we look at some of the real applications that we have worked on in the past in collaboration with other researchers to develop models using text extraction.", "But first, we look at how we use the concept of answering questions using Petri Nets to build a question answering system.", "We will extend the Petri Nets even more for this.", "BioPathQA - A System for Modeling, Simulating, and Querying Biological PathwaysThe BioPathQA System Introduction In this chapter we combine the methods from Chapter , notions from action languages, and ASP to build a system BioPathQA and a language to specify pathways and query them.", "We show how various biological pathways are encoded in BioPathQA and how it computes answers of queries against them.", "Description of BioPathQA Our system has the following components: [(i)] a pathway specification language a query language to specify the deep reasoning question, an ASP program that encodes the pathway model and its extensions for simulation.", "Knowledge about biological pathways comes in many different forms, such as cartoon diagrams, maps with well defined syntax and semantics (e.g.", "Kohn's maps [43]), and biological publications.", "Similar to other technical domains, some amount of domain knowledge is also required.", "Users want to collect information from disparate sources and encode it in a pathway specification.", "We have developed a language to allow users to describe their pathway.", "This description includes describing the substances and actions that make up the pathway, the initial state of the substances, and how the state of the pathway changes due to the actions.", "An evolution of a pathway's state from the initial state, through a set of actions is called a trajectory.", "Being a specification language targeted at biological systems, multiple actions autonomously execute in parallel as soon as their preconditions are satisfied.", "The amount of parallelism is dictated by any resource conflicts between the actions.", "When that occurs, only one sub-set of the possible actions can execute, leading to multiple outcomes from that point on.", "Questions are usually provided in natural language, which is vague.", "To avoid the vagaries of natural language, we developed a language with syntax close to natural language but with a well defined formal semantics.", "The query language allows a user to make changes to the pathway through interventions, and restrict its trajectories through observations and query on aggregate values in a trajectory, across a set of trajectories and even over two sets of trajectories.", "This allows the user to compare a base case of a pathway specification with an alternate case modified due to interventions and observations.", "This new feature is a major contribution of our research.", "Inspiration for our high level language comes from action languages and query languages such as [26].", "While action languages generally describe transition systems [27], our language describes trajectories.", "In addition, our language is geared towards modeling natural systems, in which actions occur autonomously [66] when their pre-conditions are satisfied; and we do not allow the quantities to become negative (as the quantities represent amounts of physical entities).", "Next we describe the syntax of our pathway specification language and the query language.", "Following that we will describe the syntax of our language and how we encode it in ASP.", "Syntax of Pathway Specification Language (BioPathQA-PL) The alphabet of pathway specification language $\\mathcal {P}$ consists of disjoint nonempty domain-dependent sets $A$ , $F$ , $L$ representing actions, fluents, and locations, respectively; a fixed set $S$ of firing styles; a fixed set $K$ of keywords providing syntactic sugar (shown in bold face in pathway specification language below); a fixed set of punctuations $\\lbrace `,^{\\prime } \\rbrace $ ; and a fixed set of special constants $\\lbrace `1^{\\prime },`^{\\prime },`max^{\\prime }\\rbrace $ ; and integers.", "Each fluent $f \\in F$ has a domain $dom(f)$ which is either integer or binary and specifies the values $f$ can take.", "A fluent is either simple, such as $f$ or locational, such as $f[l]$ , where $l \\in L$ .", "A state $s$ is an interpretation of $F$ that maps fluents to their values.", "We write $s(f)=v$ to represent “$f$ has the value $v$ in state $s$ ”.", "States are indexed, such that consecutive states $s_i$ and $s_{i+1}$ represent an evolution over one time step from $i$ to $i+1$ due to firing of an action set $T_i$ in $s_i$ .", "We illustrate the role of various symbols in the alphabet with examples from the biological domain.", "Consider the following example of a hypothetical pathway specification: $&\\mathbf {domain~of~} sug \\mathbf {~is~} integer, fac \\mathbf {~is~} integer, acoa \\mathbf {~is~} integer, h2o \\mathbf {~is~} integer\\\\&gly \\mathbf {~may~execute~causing~} sug \\mathbf {~change~value~by~} -1, acoa \\mathbf {~change~value~by~} +1\\\\&box \\mathbf {~may~execute~causing~} fac \\mathbf {~change~value~by~} -1, acoa \\mathbf {~change~value~by~} +1 \\\\&\\mathbf {~~~~~~~~~~~if~} h2o \\mathbf {~has~value~} 1 \\mathbf {~or~higher~}\\\\&\\mathbf {inhibit~} box \\mathbf {~if~} sug \\mathbf {~has~value~} 1 \\mathbf {~or~higher~}\\\\&\\mathbf {initially~} sug \\mathbf {~has~value~} 3, fac \\mathbf {~has~value~} 4, acoa \\mathbf {~has~value~} 0 &$ It describes two processes glycolysis and beta-oxidation represented by actions `$gly$ ' and `$box$ ' in lines () and ()-() respectively.", "Substances used by the pathway, i.e.", "sugar, fatty-acids, acetyl-CoA, and water are represented by numeric fluents `$sug$ ',`$fac$ ',`$acoa$ ', and `$h2o$ ' respectively in line (REF ).", "When glycolysis occurs, it consumes 1 unit of sugar and produces 1 unit of acetyl-CoA (line ().", "When beta-oxidation occurs, it consumes 1 unit of fatty-acids and produces 1 unit of acetyl-CoA (line ()).", "The inputs of glycolysis implicitly impose a requirement that glycolysis can only occur when at least 1 unit of sugar is available.", "Similarly, the input of beta-oxidation implicitly a requirement that beta-oxidation can only occur when at least 1 unit of fatty-acids is available.", "Beta oxidation has an additional condition imposed on it in line () that there must be at least 1 unit of water available.", "We call this a guard condition on beta-oxidation.", "Line () explictly inhibits beta-oxidation when there is any sugar available; and line () sets up the initial conditions of the pathway, i.e.", "Initially 3 units of each sugar, 4 units of fatty-acids are available and no acetyl-CoA is available.", "The words `$\\lbrace domain,$ $is,$ $may,$ $execute,$ $causing,$ $change,$ $value,$ $by,$ $has,$ $or,$ $higher,$ $inhibit,$ $if,$ $initially\\rbrace $ ' are keywords.", "When locations are involved, locational fluents take place of simple fluents and our representation changes to include locations.", "For example: $&gly \\mathbf {~may~execute~causing~} \\nonumber \\\\&~~~~sug \\mathbf {~atloc~} mm \\mathbf {~change~value~by~} -1, \\nonumber \\\\&~~~~acoa \\mathbf {~atloc~} mm \\mathbf {~change~value~by~} +1$ represents glycolysis taking 1 unit of sugar from mitochondrial matrix (represented by `$mm$ ') and produces acetyl-CoA in the mitochondrial matrix.", "Here `$atloc$ ' is an additional keyword.", "A pathway is composed of a collection of different types of statements and clauses.", "We first introduce their syntax, following that we will give intuitive definitions, and following that we will show how they are combined together to construct a pathway specification.", "Definition 46 (Fluent domain declaration statement) A fluent domain declaration statement declares the values a fluent can take.", "It has the form: $\\mathbf {domain~of~} f \\mathbf {~is~} `integer^{\\prime }\|`binary^{\\prime } \\\\\\mathbf {domain~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} `integer^{\\prime }\|`binary^{\\prime }$ for simple fluent “$f$ ”, and locational fluent “$f[l]$ ”.", "Multiple domain statements are compactly written as: $\\mathbf {domain~of~} f_1 \\mathbf {~is~} `integer^{\\prime }\|`binary^{\\prime } , \\dots , f_n \\mathbf {~is~} `integer^{\\prime }\|`binary^{\\prime } \\\\\\mathbf {domain~of~} f_1 \\mathbf {~atloc~} l_1 \\mathbf {~is~} `integer^{\\prime }\|`binary^{\\prime } , \\dots , f_n \\mathbf {~atloc~} l_n \\mathbf {~is~} `integer^{\\prime }\|`binary^{\\prime }$ Binary domain is usually used for representing substances in a signaling pathway, while a metabolic pathways take positive numeric values.", "Since the domain is for a physical entity, we do not allow negative values for fluents.", "Definition 47 (Guard condition) A guard condition takes one of the following forms: $&f \\mathbf {~has~value~} w \\mathbf {~or~higher~} \\\\&f \\mathbf {~atloc~} l \\mathbf {~has~value~} w \\mathbf {~or~higher~}\\\\&f \\mathbf {~has~value~lower~than~} w \\\\&f \\mathbf {~atloc~} l \\mathbf {~has~value~lower~than~} w \\\\&f \\mathbf {~has~value~equal~to~} w \\\\&f \\mathbf {~atloc~} l \\mathbf {~has~value~equal~to~} w \\\\&f_1 \\mathbf {~has~value~higher~than~} f_2\\\\&f_1 \\mathbf {~atloc~} l_1 \\mathbf {~has~value~higher~than~} f_2 \\mathbf {~atloc~} l_2$ where, each $f$ in (REF ), (), (), () is a simple fluent, each $f[l]$ in (), (), (), () is a locational fluent with location $l$ , and each $w \\in \\mathbb {N}^+ \\cup \\lbrace 0 \\rbrace $ .", "Definition 48 (Effect clause) An effect clause can take one of the following forms: $&f \\mathbf {~change~value~by~} e\\\\&f \\mathbf {~atloc~} l \\mathbf {~change~value~by~} e$ where $a$ is an action, $f$ in (REF ) is a simple fluent, $f[l]$ in () is a locational fluent with location $l$ , $e \\in \\mathbb {N}^+ \\cup \\lbrace \\rbrace $ for integer fluents or $e \\in \\lbrace 1,-1,\\rbrace $ for binary fluents.", "Definition 49 (May-execute statement) A may-execute statement captures the conditions for firing an action $a$ and its impact.", "It is of the form: $a \\mathbf {~may~execute~causing~} & \\mathit {effect}_1, \\dots , \\mathit {effect}_m \\nonumber \\\\\\mathbf {~if~} &guard\\_cond_1, \\dots , guard\\_cond_n$ where $\\mathit {effect}_i$ is an effect clause; and $\\mathit {guard\\_cond}_j$ is a guard condition clause, $m > 0$ , and $n \\ge 0$ .", "If $n = 0$ , the effect statement is unconditional (guarded by $\\top $ ) and the $\\mathbf {if}$ is dropped.", "A single may-execute statement must not have $\\mathit {effect}_i, \\mathit {effect}_j$ with $e_i < 0, e_j < 0$ for the same fluent; or $e_i > 0, e_j > 0$ for the same fluent.", "Definition 50 (Must-execute statement) An must-execute statement captures the impact of firing of an action $a$ that must fire when enabled (as long as it is not inhibited).", "It is an expression of the form: $a \\mathbf {~normally~must~execute~causing~} & \\mathit {effect}_1, \\dots , \\mathit {effect}_m \\nonumber \\\\\\mathbf {~if~} &guard\\_cond_{1}, \\dots , guard\\_cond_n$ where $\\mathit {effect}_i$ , and $\\mathit {guard\\_cond}_j$ are as in (REF ) above.", "Definition 51 (Inhibit statement) An inhibit statement captures the conditions that inhibit an action from occurring.", "It is an expression of the form: $\\mathbf {inhibit~} a \\mathbf {~if~} guard\\_cond_1, \\dots , guard\\_cond_n$ where $a$ is an action, $guard\\_cond_i$ is a guard condition clause, and $n \\ge 0$ .", "if $n = 0$ , the inhibition of action `$a$ ' is unconditional `$\\mathbf {if}$ ' is dropped.", "Definition 52 (Initial condition statement) An initial condition statement captures the initial state of pathway.", "It is of the form: $\\mathbf {initially~} & f \\mathbf {~has~value~} w\\\\\\mathbf {initially~} & f \\mathbf {~atloc~} l \\mathbf {~has~value~} w$ where each $f$ in (REF ) is a simple fluent, $f[l]$ in () is a locational fluent with location $l$ , and each $w$ is a non-negative integer.", "Multiple initial condition statements are written compactly as: $\\mathbf {initially~} & f_1\\mathbf {~has~value~} w_1, \\dots , f_n \\mathbf {~has~value~} w_n\\\\\\mathbf {initially~} & f_1 \\mathbf {~atloc~} l_1 \\mathbf {~has~value~} w_1, \\dots , f_n \\mathbf {~atloc~} l_n \\mathbf {~has~value~} w_n$ Definition 53 (Duration Statement) A duration statement represents the duration of an action that takes longer than a single time unit to execute.", "It is of the form: $a \\mathbf {~executes~in~} d \\mathbf {~time~units}$ where $d$ is a positive integer representing the action duration.", "Definition 54 (Stimulate Statement) A stimulate statement changes the rate of an action.", "It is an expression of the form: $\\mathbf {normally~stimulate~} a \\mathbf {~by~factor~} n \\mathbf {~if~} guard\\_cond_1,\\dots ,guard\\_cond_n$ where $guard\\_cond_i$ is a condition, $n > 0$ .", "When $n=0$ , the stimulation is unconditional and $\\mathbf {~if~}$ is dropped.", "A stimulation causes the $\\mathit {effect}$ in may-cause and must-fire multiplied by $n$ .", "Actions execute automatically when fireable, subject to the available fluent quantities.", "Definition 55 (Firing Style Statement) A firing style statement specifies how many actions execute simultaneously (or action parallelism).", "It is of the form: $\\mathbf {firing~style~} S$ where, $S$ is either “1”, “$$ ”, or “$max$ ” for serial execution, interleaved execution, and maximum parallelism.", "We will now give the intuitive meaning of these statements and put them into context w.r.t.", "the biological domain.", "Though our description below uses simple fluents only, it applies to locational fluents in a obvious manner.", "The reason for having locational fluents at all is that they allow a more natural pathway specification when substance locations are involved instead of devising one's own encoding scheme.", "For example, in a mitochondria, hydrogen ions (H+) appear in multiple locations (intermembrane space and mitochondrial matrix), with each location carrying its distinct quantity separate from other locations.", "Intuitively, a may-execute statement (REF ) represents an action $a$ that may fire if all conditions `$\\mathit {guard\\_cond}_{1},\\dots ,\\mathit {guard\\_cond}_n$ ' hold in the current state.", "When it executes, it impacts the state as specified in $\\mathit {effect}$ s. In biological context, action $a$ represents a process, such as a reaction, $\\mathit {effect}$ s represent the inputs / ingredients of the reaction, and $guard\\_cond$ represent additional preconditions necessary for the reaction to proceed.", "Condition (REF ) holds in a state $s$ if $s(f) \\ge w$ .", "It could represent an initiation concentration $w$ of a substance $f$ which is higher than the quantity consumed by the reaction $a$ .", "Condition () holds in a state $s$ if $s(f) < w$ .", "Condition () holds in a state $s$ if $s(f) = w$ .", "Condition () holds in a state $s$ if $s(f_1) > s(f_2)$ capturing a situation where a substance gradient is required for a biological process to occur.", "An example of one such process is the synthesis of ATP by ATP Synthase, which requires a H+ (Hydrogen ion) gradient across the inner mitochondrial matrix [64].", "Intuitively, the effect clause (REF ) of an action describes the impact of an action on a fluent.", "When an action $a$ fires in a state $s$ , the value of $f$ changes according to the effect clause for $f$ .", "The value of $f$ increases by $e$ if $e > 0$ , decreases by $e$ if $e < 0$ , or decreases by $s(f)$ if $e = `^{\\prime }$ (where $`^{\\prime }$ can be interpreted as $-s(f)$ ).", "For a reaction $a$ , a fluent with $e < 0$ represents an ingredient consumed in quantity $\|e\|$ by the reaction; a fluent with $e > 0$ represents a product of the reaction in quantity $e$ ; a fluent with $e = `^{\\prime }$ represents consuming all quantity of the substance due to the reaction.", "Since the fluents represent physical substances, their quantities cannot become negative.", "As a result, any action that may cause a fluent quantity to go below zero is disallowed.", "Intuitively, a must-execute statement (REF ) is similar to a may-exec, except that when enabled, it preferentially fires over other actions as long as there isn't an inhibit proposition that will cause the action to become inactive.", "It captures the effect of an action that must happen whenever enabled.", "Intuitively, an inhibit statement (REF ) specifies the conditions that inhibits an action.", "In a biological context, it defines inhibition of reactions, e.g., through biological feedback control.", "Though we could have added these conditions to may-exec, it is more intuitive to keep them separate as inhibition conditions are usually discovered separately in a biological domain.", "Including them as part of may-fire would constitute a surgery of existing knowledge bases.", "Intuitively, an initial condition statement (REF ) specifies the initial values of fluents.", "The collection of such propositions defines the initial state $s_0$ of the pathway.", "In a biological context, this defines the initial distribution of substances in the biological system.", "Intuitively, an action duration statement (REF ) represents action durations, especially when an action takes longer to execute.", "When an action $a$ with duration $d$ fires in state $s_k$ , it immediately decreases the values of fluents with $e < 0$ and $e = $ upon execution, however, it does not increase the value of fluents with $e > 0$ until time the end of its execution in state $s_{k+d}$ .", "In a biological context the action duration captures a reaction's duration.", "A reaction consumes its ingredients immediately on firing, processes them for duration $d$ and generates its products at the end of this duration.", "Intuitively, a stimulate statement (REF ) represents a change in the rate of an action $a$ .", "The stimulation causes the action to change its rate of consumption of its ingredients and production of its products by a factor $n$ .", "In biological context, this stimulation can be a result of an enzyme or a stimulant's availability, causing a reaction that normally proceeds slowly to occur faster.", "Intuitively, a firing style statement (REF ) specifies the parallelism of actions.", "When it is “1”, at most one action may fire, when it is “$max$ ”, the maximum allowable actions must fire, and when it is “$$ ”, any subset of fireable actions may fire simultaneously.", "In a biological domain the firing style allows one to model serial operations, parallel operations and maximally parallel operations.", "The maximum parallelism is also useful in quickly discovering changes that occur in a biological system.", "Definition 56 (Pathway Specification) A pathway specification is composed of one or more may-execute, must-execute, effect, inhibit, stimulate, initially, priority, duration statements, and one firing style statement.", "When a duration statement is not specified for an action, it is assumed to be 1.", "Any fluents for which an initial quantity is not specified are assumed to have a value of zero.", "A pathway specification is consistent if [(i)] there is at most one firing style, priority, duration statement for each action $a$ ; the $guard\\_cond_1,\\dots ,guard\\_cond_n$ from a may-execute or must-execute are disjoint from any other may-execute or must-execute Note that `$f1 \\text{ has value } 5 \\text{ or higher }$ ' overlaps with `$f1 \\text{ has value } 7 \\text{ or higher}$ ' and the two conditions are not considered disjoint.", "; locational and non-locational fluents may not be intermixed; domain of fluents, effects, conditions and numeric values are consistent, i.e., effects and conditions on binary fluents must be binary; and the pathway specification does not cause it to violate fluent domains by producing non-binary values for binary fluents.", "Each pathway specification $\\mathbf {D}$ represents a collection of trajectories of the form: $\\sigma = s_0, T_0, s_1, \\dots , s_{k-1}, T_{k-1}, s_k$ .", "Each trajectory encodes an evolution of the pathway starting from an initial state $s_0$ , where $s_i$ 's are states, and $T_i$ 's are sets of actions that fired in state $s_i$ .", "Intuitively, a trajectory starts from the initial state $s_0$ .", "Each $s_i,s_{i+1}$ pair represents the state evolution in one time step due to the action set $T_i$ .", "An action set $T_i$ is only executable in state $s_i$ , if the sum of changes to fluents due to $e_i < 0$ and $e_i = $ will not result in any of the fluents going negative.", "Changes to fluents due to $e_i > 0$ for the action set $T_i$ occur over subsequent time-steps depending upon the durations of actions involved.", "Thus, the state $s_i(f_i)$ is the sum of $e_i > 0$ for actions of duration $d$ that occurred $d$ time steps before (current time step) $i$ , i.e.", "$a \\in T_{i-d}$ , where the default duration $d$ of an action is 1 if none specified.", "Next we describe the semantics of the pathway specification language, which describes how these trajectories are generated.", "Semantics of Pathway Specification Language (BioPathQA-PL) The semantics of the pathway specification language are defined in terms of the trajectories of the domain description $\\mathbf {D}$ .", "Since our pathway specification language is inspired by Petri Nets, we use Petri Nets execution semantics to define its trajectories.", "However, some constructs in our pathway language specification are not directly representable in standard Petri Nets, as a result, we will have to extend them.", "Let an arc-guard be a conjunction of guard conditions of the form (REF )-(), such that it is wholly constructed of either locational or non-locational fluents, but not both.", "We introduce a new type of Guarded-arc Petri Net in which each arc has an arc-guard expression associated with it.", "Arcs with the same arc-guard are traversed when a transition connected to them fires and the arc-guard is found to hold in the current state.", "The arc-guards of arcs connected to the same transition form an exclusive set, such that only arcs corresponding to one guard expression may fire (for one transition).", "This setup can lead to different outcomes of an actionArcs for different guard expressions emanating / terminating at a place can further be combined into a single conditional arc with conditional arc-weights.", "If none of the condition applies then the arc is assumed to be missing..", "The transitions in this new type of Petri Net can have the following inscriptions on them: Propositional formula, specifying the executability conditions of the transition.", "Arc-durations, represented as “$dur(n)$ ”, where $n \\in \\mathbb {N}^+$ A must-execute inscription, “$must\\text{-}execute(guard)$ ”, requires that when the $guard$ holds in a state where this transition is enabled, it must fire, unless explicitly inhibited.", "The $guard$ has the same form as an $arc\\text{-}guard$ A stimulation inscription, “$stimulate(n,guard)$ ”, applies a multiplication factor $n \\in \\mathbb {N}^+$ to the input and output quantities consumed and produced by the transition, when $guard$ hold in the current state, where $guard$ has the same form as an $arc\\text{-}guard$ .", "Certain aspects of our nets are similar to CPNs [39].", "However, the CPNs do not allow our semantics of the reset arcs, or must-fire guards.", "Guarded-Arc Petri Net Figure: Example of a guarded-arc Petri Net.Figure REF shows an example of a guarded-arc Petri Net.", "There are two arc-guard expressions $f1<5$ and $f1>5$ .", "When $f1<5$ , $t1$ consumes one token from place $f1$ and produces one token in place $f2$ .", "When $f1>5$ , $t1$ 's consumption and production of the same tokens doubles.", "The transition $t1$ implicitly gets the guards for each arc represented the or-ed conditions $(f1<5) \\vee (f1>5))$ .", "The arc also has two conditions inhibiting it, they are represented by the and-ed conditions $\\lnot (f1>7) \\wedge \\lnot (f1<3)$ , where `$\\lnot $ ' represents logical not.", "Transition $t1$ is stimulated by factor 3 when $f1>5$ and it has a duration of 10 time units.", "A transition cannot fire even though one of its arc-guards is enabled, unless the token requirements on the arc itself are also fulfilled, e.g.", "if $f1$ has value 0 in the current state, even though $f1 < 5$ guard is satisfied, the transition cannot execute, because the input arc $(f1,t1)$ for this guard needs 1 token.", "Definition 57 (Guard) A condition is of the form: $(f < v), (f \\le v), (f > v), (f \\ge v), (f = v)$ , where $f$ is a fluent and $v$ either a fluent or a numeric constant.", "Then, a guard is a propositional formula of conditions, with each condition treated as a proposition, subject to the restriction that all fluents in all conditions in a guard are either locational or simple, but not both.", "Definition 58 (Interpretation of a Guard) An interpretation of a guard $G$ is a possible assignment of a value to each fleuent $f \\in G$ from the domain of $f$ .", "Definition 59 (Guard Satisfaction) A guard $G$ with simple fluents is satisfied w.r.t.", "a state $s$ , written $s \\models G$ iff $G$ has an interpretation in which each of its fluents $f$ has the value $s(f)$ and $G$ is true.", "A guard $G$ with locational fluents is satisfied w.r.r.", "a state $s$ , written $s \\models G$ iff $G$ has an interpretation in which each of its fluents $f[l]$ has the value $m_{s(l)}(f)$ and $G$ is true, where $m_X(f)$ is the multiplicity of $f$ in $X$ .", "Definition 60 (Guarded-Arc Petri Net) A Guarded-Arc Petri Net is a tuple $PN^G=(P,T,G,E,R,W,D,B,TG,MF,L)$ , where: $P & \\text{ is a finite set of places}\\\\T & \\text{ is a finite set of transitions}\\\\G & \\text{ is a set of guards as defined in definition~(\\ref {def:guard})}\\\\TG &: T \\rightarrow G \\text{ are the transition guards }\\\\E &\\subseteq (T \\times P \\times G) \\cup (P \\times T \\times G) \\text{ are the guarded arcs }\\\\R &\\subseteq P \\times T \\times G \\text{ are the guarded reset arcs }\\\\W &: E \\rightarrow \\mathbb {N}^+ \\text{ are arc weights }\\\\D &: T \\rightarrow \\mathbb {N}^+ \\text{ are the transition durations}\\\\B &: T \\rightarrow G \\times \\mathbb {N}^+ \\text{ transition stimulation or boost }\\\\MF &: T \\rightarrow 2^{G} \\text{ must-fire guards for a transition}\\\\L &: P \\rightarrow \\mathbb {N}^+ \\text{ specifies maximum number to tokens for each place }$ subject to constraints: $P \\cap T = \\emptyset $ $R \\cap E = \\emptyset $ Let $t_1 \\in T$ and $t_2 \\in T$ be any two distinct transitions, then $g1 \\in MF(t_1)$ and $g2 \\in MF(t_2)$ must not have an interpretation that make both $g1$ and $g2$ true.", "Let $t \\in T$ be a transition, and $gg_t = \\lbrace g: (t,p,g) \\in E \\rbrace \\cup \\lbrace g: (p,t,g) \\in E \\rbrace \\cup \\lbrace g: (p,t,g) \\in R \\rbrace $ be the set of arc-guards for normal and reset arcs connected to it, then $g_1 \\in gg_t, g_2 \\in gg_t$ must not have an interpretation that makes both $g_1$ and $g_2$ true.", "Let $t \\in T$ be a transition, and $gg_t = \\lbrace g: (g,n) \\in B(t) \\rbrace $ be its stimulation arc-guards, then $g_1 \\in gg_t, g_2 \\in gg_t$ must not have an interpretation that makes both $g_1$ and $g_2$ true.", "Let $t \\in T$ be a transition, and $(g,n) \\in B(t) : n>1$ , then there must not exist a place $p \\in P : (p,t,g) \\in E, L(p) = 1$ .", "Intuitively, stimulation of binary inputs is not supported.", "We will make a simplifying assumption that all places are readable by using their place names.", "Execution of the $PN^G$ occurs in discrete time steps.", "Definition 61 (Marking (or State)) Marking (or State) of a Guarded-Arc Petri Net $PN^G$ is the token assignment of each place $p_i \\in P$ .", "Initial marking $M_0 : P \\rightarrow \\mathbb {N}^0$ , while the token assignment at step $k$ is written as $M_k$ .", "Next we define the execution semantics of $PN^G$ .", "First we introduce some terminology that will be used below.", "Let $s_0 = M_0$ represent the the initial marking (or state), $s_k = M_k$ represent the marking (or state) at time step $k$ , $s_k(p)$ represent the marking of place $p$ at time step $k$ , such that $s_k = [s_k(p_0), \\dots , $ $s_k(p_n)]$ , where $P=\\lbrace p_0, \\dots , p_n\\rbrace $ $T_k$ be the firing-set that fired in step $k$ , $b_k(t)$ be the stimulation value applied to a transition $t$ w.r.t.", "step $k$ $en_k$ be the set of enabled transitions in state $s_k$ , $mf_k$ be the set of must-execute transitions in state $s_k$ , $consume_k(p,\\lbrace t_1,\\dots ,t_n\\rbrace )$ be the sum of tokens that will be consumed from place $p$ if transitions $t_1,\\dots ,t_n$ fired in state $s_k$ , $overc_k(\\lbrace t_1,\\dots ,t_n\\rbrace )$ be the set of places that will have over-consumption of tokens if transitions $t_1,\\dots ,t_n$ were to fire simultaneously in state $s_k$ , $sel_k(fs)$ be the set of possible firing-set choices in state $s_k$ using $\\mathit {fs}$ firing style $produce_k(p)$ be the total production of tokens in place $p$ (in state $s_k$ ) due to actions terminating in state $s_k$ , $s_{k+1}$ be the next state computed from current state $s_k$ due to firing of transition-set $T_k$ $min(a,b)$ gives the minimum of numbers $a,b$ , such that $min(a,b) = a \\text{ if } a < b \\text{ or } b \\text{ otherwise }$ .", "Then, the execution semantics of the guarded-arc Petri Net starting from state $s_0$ using firing-style $\\mathit {fs}$ is given as follows: $b_k(t) &={\\left\\lbrace \\begin{array}{ll}n &\\text{ if } (g,n) \\in B(t), s_k \\models g\\\\1 &\\text{ otherwise }\\end{array}\\right.", "}\\nonumber \\\\en_k &= \\lbrace t : t \\in T, s_k \\models TG(t), \\forall (p,t,g) \\in E, \\nonumber \\\\&~~~~~~~~(s_k \\models g, s_k(p) \\ge W(p,t,g) * b_k(t)) \\rbrace \\nonumber \\\\mf_k &= \\lbrace t : t \\in en_k, \\exists g \\in MF(t), s_k \\models g \\rbrace \\nonumber \\\\consume_k(p,\\lbrace t_1,\\dots ,t_n\\rbrace ) &= \\sum _{i=1,\\dots ,n}{W(p,t_i,g) * b_k(t) : (p,t_i,g) \\in E, s_k \\models g} \\nonumber \\\\&+ \\sum _{i=1,\\dots ,n}{s_k(p) : (p,t_i,g) \\in R, s_k \\models g} \\nonumber \\\\overc_k(\\lbrace t_1,\\dots ,t_n\\rbrace ) &= \\lbrace p : p \\in P, s_k(p) < consume_k(p,\\lbrace t_1,\\dots ,t_n\\rbrace ) \\rbrace \\nonumber \\\\sel_k(1) &={\\left\\lbrace \\begin{array}{ll}mf_k & \\text{ if } \|mf_k\| = 1\\\\\\lbrace \\lbrace t \\rbrace : t \\in en_k \\rbrace & \\text{ if } \|mf_k\| < 1\\end{array}\\right.", "}\\nonumber \\\\sel_k() &= \\lbrace ss : ss \\in 2^{en_k}, mf_k \\subseteq ss, overc_k(ss) = \\emptyset \\rbrace \\nonumber \\\\sel_k(max) &= \\lbrace ss : ss \\in 2^{en_k}, mf_k \\subseteq ss, overc_k(p,ss) = \\emptyset , \\nonumber \\\\&~~~~~~(\\nexists ss^{\\prime } \\in 2^{en_k} : ss \\subset ss^{\\prime }, mf_k \\subseteq ss^{\\prime }, overc_k(ss^{\\prime }) = \\emptyset ) \\rbrace \\nonumber \\\\T_k &= T^{\\prime }_k : T^{\\prime }_k \\in sel_k(\\mathit {fs}), (\\nexists t \\in en_k \\setminus T^{\\prime }_k, t \\text{ is a reset transition } ) \\nonumber \\\\produce_k(p) &= \\sum _{j=0,\\dots ,k}{W(t_i,p,g) b_j(t_i) : (t_i,p,g) \\in E, t_i \\in T_j, D(t_i)+j = k+1}\\nonumber \\\\s_{k+1}(p) &= min(s_k(p) - consume_k(p,T_k) + produce_k(p), L(p))$ Definition 62 (Trajectory) $\\sigma = s_0, T_0, s_1, \\dots , s_{k}, T_{k}, s_{k+1}$ is a trajectory of $PN^G$ iff given $s_0 = M_0$ , each $T_i$ is a possible firing-set in $s_i$ whose firing produces $s_{i+1}$ per $PN^G$ 's execution semantics in (REF ).", "Construction of Guarded-Arc Petri Net from a Pathway Specification Now we describe how to construct such a Petri Net from a given pathway specification $\\mathbf {D}$ with locational fluents.", "We proceed as follows: The set of transitions $T = \\lbrace a : a \\text{ is an action in } \\mathbf {D}\\rbrace $ .", "The set of places $P = \\lbrace f : f \\text{ is a fluent in } \\mathbf {D}\\rbrace $ .", "The limit relation for places $L(f) ={\\left\\lbrace \\begin{array}{ll}1 & \\text{ if } `\\mathbf {domain~of~} f \\mathbf {~is~} binary^{\\prime } \\in \\mathbf {D}\\\\\\infty & \\text{ otherwise }\\\\\\end{array}\\right.", "}$ .", "An arc-guard expression $guard\\_cond_1,\\dots ,guard\\_cond_n$ is translated into the conjunction $(guard_1,\\dots ,guard_n)$ , where $guard_i$ is obtained from $guard\\_cond_i$ as follows: A guard condition (REF ) is translated to $f \\ge w$ A guard condition () is translated to $f < w$ A guard condition () is translated to $f = w$ A guard condition () is translated to $f_1 > f_2$ A may-execute statement (REF ) is translated into guarded arcs as follows: Let guard $G$ be the translation of arc-guard conditions $guard\\_cond_1,\\dots ,$ $guard\\_cond_n$ specified in the may-execute proposition.", "The effect clause (REF ) are translated into arcs as follows: An effect clause with $e < 0$ is translated into an input arc $(f,a,G)$ , with arc-weight $W(f,a,G) = \|e\|$ .", "An effect clause with $e = `^{\\prime }$ is translated into a reset set $(f,a,G)$ with arc-weight $W(f,a,G) = $ .", "An effect clause with $e > 0$ is translated into an output arc $(a,f,G)$ , with arc-weight $W(a,f,G) = e$ .", "A must-execute statement (REF ) is translated into guarded arcs in the same way as may execute.", "In addition, it adds an arc-inscription $must\\text{-}exec(G)$ , where $G$ is the translation of the arc-guard.", "An inhibit statement (REF ) is translated into $IG = (guard_1,\\dots ,$ $guard_n)$ , where $(guard_1,\\dots ,$ $guard_n)$ is the translation of $(guard\\_cond\\_1,\\dots ,$ $guard\\_cond_n)$ An initial condition statement () sets the initial marking of a specific place $p$ to $w$ , i.e.", "$M_0(p) = w$ .", "An duration statement (REF ) adds a $dur(d)$ inscription to transition $a$ .", "A stimulate statement (REF ) adds a $stimulate(n,G)$ inscription to transition $a$ , where $G$ is the translation of the stimulate guard, a conjunction of $guard\\_cond_1,$ $\\dots ,guard\\_cond_n$ .", "A guard $(G_1 \\vee \\dots \\vee G_n) \\wedge (\\lnot IG_1 \\wedge \\dots \\wedge \\lnot IG_m)$ is added to each transition $a$ , where $G_i, 1 \\le i \\le n$ is a guard for a may-execute or a must-execute statement and $IG_i, 1 \\le i \\le m$ is a guard for an inhibit statement.", "A firing style statement (REF ) does not visibly appear on a Petri Net diagram, but it specifies the transition firing regime the Petri Net follows.", "Example: Consider the following pathway specification: $\\begin{array}{llll}\\mathbf {domain~of~} &f_1 \\mathbf {~is~} integer, &f_2 \\mathbf {~is~} integer\\\\t_1 \\mathbf {~may~execute~}\\\\\\;\\;\\;\\;\\;\\;\\;\\;\\;\\mathbf {causing~} & f_1 \\mathbf {~change~value~by~} -1, & f_2 \\mathbf {~change~value~by~} +1\\\\\\;\\;\\;\\;\\;\\;\\;\\;\\;\\mathbf {if~} & f_1 \\mathbf {~has~value~lower~than~} 5\\\\t_1 \\mathbf {~may~execute~}\\\\\\;\\;\\;\\;\\;\\;\\;\\;\\;\\mathbf {causing~} & f_1 \\mathbf {~change~value~by~} -2, & f_2 \\mathbf {~change~value~by~} +2\\\\\\;\\;\\;\\;\\;\\;\\;\\;\\;\\mathbf {if~} & f_1 \\mathbf {~has~value~higher~than~} 5\\\\\\mathbf {duration~of~} & t1 \\mathbf {~is~} 10\\\\\\mathbf {inhibit~} t1 \\mathbf {~if~} & f_1 \\mathbf {~has~value~higher~than~} 7\\\\\\mathbf {inhibit~} t1 \\mathbf {~if~} & f_1 \\mathbf {~has~value~lower~than~} 3\\\\\\mathbf {normally~stimulate~} t1 & \\mathbf {~by~factor~} 3 \\\\\\;\\;\\;\\;\\;\\;\\;\\;\\;\\mathbf {~if~} & f_2 \\mathbf {~has~value~higher~than~} 5\\\\\\mathbf {initially~}& f_1 \\mathbf {~has~value~} 0, & f_2 \\mathbf {~has~value~} 0,\\\\\\mathbf {firing~style~} & max\\end{array}$ This pathway specification is encoded as the Petri Net in figure REF .", "Guarded-Arc Petri Net with Colored Tokens Next we extend the Guarded-arc Petri Nets to add Colored tokens.", "We will use this extension to model pathways with locational fluents.", "Figure: Example of a guarded-arc Petri Net with colored tokens.Figure REF shows an example of a guarded-arc Petri Net with colored tokens.", "There are two arc-guard expressions $f3[p1]<5$ and $f3[p1]>5$ .", "When $f3[p1]<5$ , $t1$ consumes one token of color $f1$ from place $p1$ and produces one token of color $f2$ in place $p2$ .", "When $f3[p1]>5$ , $t1$ 's consumption and production of the same colored tokens doubles.", "The transition $t1$ implicitly gets the guards for each arc represented the or-ed conditions $((f3[p1]<5) \\vee (f3[p1]>5))$ .", "The arc also has two conditions inhibiting it, they are represented by the and-ed conditions $\\lnot (f3[p1]>7) \\wedge \\lnot (f3[p1]<3)$ , where `$\\lnot $ ' represents logical not.", "Transition $t1$ is stimulated by factor 3 when $f3[p1]>5$ and it has a duration of 10 time units.", "Definition 63 (Guarded-Arc Petri Net with Colored Tokens) A Guarded-Arc Petri Net with Colored Tokens is a tuple $PN^{GC}=(P,T,C,G,E,R,W,D,B,TG,$ $MF,L)$ , such that: $P &: \\text{finite set of places}\\\\T &: \\text{finite set of transitions}\\\\C &: \\text{finite set of colors}\\\\G &: \\text{set of guards as defined in definition~(\\ref {def:guard}) with locational fluents}\\\\TG &: T \\rightarrow G \\text{ are the transition guards }\\\\E &\\subseteq (T \\times P \\times G) \\cup (P \\times T \\times G) \\text{ are the guarded arcs }\\\\R &\\subseteq P \\times T \\times G \\text{ are the guarded reset arcs }\\\\W &: E \\rightarrow \\langle C,m \\rangle \\text{ are arc weights; each arc weight is a multiset over } C\\\\D &: T \\rightarrow \\mathbb {N}^+ \\text{ are the transition durations}\\\\B &: T \\rightarrow G \\times \\mathbb {N}^+ \\text{ transition stimulation or boost }\\\\MF &: T \\rightarrow 2^{G} \\text{ must-fire guards for a transition}\\\\L &: P \\times C \\rightarrow \\mathbb {N}^+ \\text{ specifies maximum number of tokens for each color in each place}\\\\$ subject to constraints: $P \\cap T = \\emptyset $ $R \\cap E = \\emptyset $ Let $t_1 \\in T$ and $t_2 \\in T$ be any two distinct transitions, then $g1 \\in MF(t_1)$ and $g2 \\in MF(t_2)$ must not have an interpretation that make both $g1$ and $g2$ true.", "Let $t \\in T$ be a transition, and $gg_t = \\lbrace g: (t,p,g) \\in E \\rbrace \\cup \\lbrace g: (p,t,g) \\in E \\rbrace \\cup \\lbrace g: (p,t,g) \\in R \\rbrace $ be the set of arc-guards for normal and reset arcs connected to it, then $g_1 \\in gg_t, g_2 \\in gg_t$ must not have an interpretation that makes both $g_1$ and $g_2$ true.", "Let $t \\in T$ be a transition, and $gg_t = \\lbrace g: (g,n) \\in B(t) \\rbrace $ be its stimulation guards, then $g_1 \\in gg_t, g_2 \\in gg_t$ must not have an interpretation that makes both $g_1$ and $g_2$ true.", "Let $t \\in T$ be a transition, and $(g,n) \\in B(t) : n>1$ , then there must not exist a place $p \\in P$ and a color $c \\in C$ such that $(p,t,g) \\in E, L(p,c) = 1$ .", "Intuitively, stimulation of binary inputs is not supported.", "Definition 64 (Marking (or State)) Marking (or State) of a Guarded-Arc Petri Net with Colored Tokens $PN^{GC}$ is the colored token assignment of each place $p_i \\in P$ .", "Initial marking is written as $M_0 : P \\rightarrow \\langle C,m \\rangle $ , while the token assignment at step $k$ is written as $M_k$ .", "We make a simplifying assumption that all places are readable by using their place name.", "Next we define the execution semantics of the guarded-arc Petri Net.", "First we introduce some terminology that will be used below.", "Let $s_0 = M_0$ represent the the initial marking (or state), $s_k = M_k$ represent the marking (or state) at time step $k$ , $s_k(p)$ represent the marking of place $p$ at time step $k$ , such that $s_k = [s_k(p_0),$ $\\dots ,s_k(p_n)]$ , where $P=\\lbrace p_0,\\dots ,p_n\\rbrace $ .", "$T_k$ be the firing-set that fired in state $s_k$ , $b_k(t)$ be the stimulation value applied to transition $t$ w.r.t.", "state $s_k$ , $en_k$ be the set of enabled transitions in state $s_k$ , $mf_k$ be the set of must-fire transitions in state $s_k$ , $consume_k(p,\\lbrace t_1,\\dots ,t_n\\rbrace )$ be the sum of colored tokens that will be consumed from place $p$ if transitions $t_1,\\dots ,t_n$ fired in state $s_k$ , $overc_k(\\lbrace t_1,\\dots ,t_n\\rbrace )$ be the set of places that will have over-consumption of tokens if transitions $t_1,\\dots ,t_n$ were to fire simultaneously in state $s_k$ , $sel_k(fs)$ be the set of possible firing-sets in state $s_k$ using $\\mathit {fs}$ firing style, $produce_k(p)$ be the total production of tokens in place $p$ in state $s_k$ due to actions terminating in state $s_k$ , $s_{k+1}$ be the next state computed from current state $s_k$ and $T_k$ , $m_X(c)$ represents the multiplicity of $c \\in C$ in multiset $X=\\langle C,m \\rangle $ , $c/n$ represents repetition of an element $c$ of a multi-set $n$ -times, multiplication of multiset $X= \\langle C,m \\rangle $ with a number $n$ be defined in terms of multiplication of element multiplicities by $n$ , i.e.", "$\\forall c \\in C, m_X(c)n$ , and $min(a,b)$ gives the minimum of numbers $a,b$ , such that $min(a,b) = a \\text{ if } a < b \\text{ or } b \\text{ otherwise }$ .", "Then, the execution semantics of the guarded-arc Petri Net starting from state $s_0$ using firing-style $\\mathit {fs}$ is given as follows: $b_k(t) &={\\left\\lbrace \\begin{array}{ll}n & \\text{ if } (g,n) \\in B(t), s_k \\models g\\\\1 &\\text{ otherwise }\\end{array}\\right.", "}\\nonumber \\\\en_k &= \\lbrace t \\in T, s_k \\models TG(t), \\forall (p,t,g) \\in E, \\nonumber \\\\&~~~~~~(s_k \\models g, s_k(p) \\ge W(p,t,g) b_k(t)) \\rbrace \\nonumber \\\\mf_k &= \\lbrace t \\in en_k, \\exists g \\in MF(t), s_k \\models g \\rbrace \\nonumber \\\\consume_k(p,\\lbrace t_1,\\dots ,t_n\\rbrace ) &= \\sum _{i=1,\\dots ,n}{W(p,t_i,g) * b_k(t) : (p,t_i,g) \\in E, s_k \\models g} \\nonumber \\\\&+ \\sum _{i=1,\\dots ,n}{s_k(p) : (p,t_i,g) \\in R, s_k \\models g} \\nonumber \\\\overc_k(\\lbrace t_1,\\dots ,t_n\\rbrace ) &= \\lbrace p \\in P : \\exists c \\in C, m_{s_k(p)}(c) < m_{consume_k(p,\\lbrace t_1,\\dots ,t_n\\rbrace )}(c) \\rbrace \\nonumber \\\\sel_k(1) &={\\left\\lbrace \\begin{array}{ll}mf_k & \\text{ if } \|mf_k\| = 1\\\\\\lbrace \\lbrace t \\rbrace : t \\in en_k \\rbrace & \\text{ if } \|mf_k\| < 1\\end{array}\\right.", "}\\nonumber \\\\sel_k() &={\\left\\lbrace \\begin{array}{ll}ss \\in 2^{en_k} : mf_k \\subseteq ss, overc_k(ss) = \\emptyset \\end{array}\\right.", "}\\nonumber \\\\sel_k(max) &={\\left\\lbrace \\begin{array}{ll}ss \\in 2^{en_k} : mf_k \\subseteq ss, overc_k(p,ss) = \\emptyset , \\\\(\\nexists ss^{\\prime } \\in 2^{en_k} : ss \\subset ss^{\\prime }, mf_k \\subseteq ss^{\\prime }, overc_k(ss^{\\prime }) = \\emptyset )\\end{array}\\right.", "}\\nonumber \\\\T_k &= T^{\\prime }_k : T^{\\prime }_k \\in sel_k(\\mathit {fs}), (\\nexists t \\in en_k \\setminus T^{\\prime }_k, t \\text{ is a reset transition } ) \\nonumber \\\\produce_k(p) &= \\sum _{j=0,\\dots ,k}{W(t_i,p,g) b_j(t_i) : (t_i,p,g) \\in E, t_i \\in T_j), D(t_i)+j = k+1}\\nonumber \\\\s_{k+1} &= [ c/n : c \\in C, \\nonumber \\\\&~~~~n=min(m_{s_k(p)}(c) \\nonumber \\\\&~~~~~~~~~~~~ - m_{consume_k(p,T_k)}(c) \\nonumber \\\\&~~~~~~~~~~~~ + m_{produce_k(p)}(c), L(p,c)) ]$ Definition 65 (Trajectory) $\\sigma = s_0, T_0, s_1, \\dots , s_{k}, T_{k}, s_{k+1}$ is a trajectory of $PN^{GC}$ iff given $s_0=M_0$ , each $T_i$ is a possible firing-set in $s_i$ whose firing produces $s_{i+1}$ per $PN^{GC}$ 's execution semantics in (REF ).", "Construction of Guarded-Arc Petri Net with Colored Tokens from a Pathway Specification with Locational Fluents Now we describe how to construct such a Petri Net from a given pathway specification $\\mathbf {D}$ with locational fluents.", "We proceed as follows: The set of transitions $T = \\lbrace a : a \\text{ is an action in } \\mathbf {D}\\rbrace $ .", "The set of colors $C = \\lbrace f : f[l] \\text{ is a fluent in } \\mathbf {D}\\rbrace $ .", "The set of places $P = \\lbrace l : f[l] \\text{ is a fluent in } \\mathbf {D}\\rbrace $ .", "The limit relation for each colored token in a place $L(f,c) ={\\left\\lbrace \\begin{array}{ll}1 & \\text{ if } `\\mathbf {domain~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} binary^{\\prime } \\in \\mathbf {D}\\\\\\infty & \\text{ otherwise }\\\\\\end{array}\\right.", "}$ .", "A guard expression $guard\\_cond_1,\\dots ,guard\\_cond_n$ is translated into the conjunction $(guard_1,\\dots ,$ $guard_n)$ , where $guard_i$ is obtained from $guard\\_cond_i$ as follows: A guard condition () is translated to $f[l] < w$ A guard condition () is translated to $f[l] = w$ A guard condition () is translated to $f[l] \\ge w$ A guard condition () is translated to $f_1[l_1] > f_2[l_2]$ A may-execute statement (REF ) is translated into guarded arcs as follows: Let guard $G$ be the translation of guard conditions $guard\\_cond_1,$ $\\dots ,guard\\_cond_n$ specified in the may-execute proposition.", "The effect clauses of the form () are grouped into input, reset and output effect sets for an action as follows: The clauses with $e < 0$ for the same place $l$ are grouped together into an input set of $a$ requiring input from place $l$ .", "The clauses with $e = `^{\\prime }$ for the same place $l$ are grouped together into a reset set of $a$ requiring input from place $l$ .", "The clauses with $e > 0$ for the same place $l$ are grouped together into an output set of $a$ to place $l$ .", "A group of input effect clauses $\\mathit {effect}_1,\\dots ,\\mathit {effect}_m, m > 0$ of the form () is translated into an input arc $(l,a,G)$ , with arc-weight $W(l,a,G) = w^+$ , where $w^+$ is the multi-set union of $f_i/\|e_i\|$ in $\\mathit {effect}_i, 1 \\le i \\le m$ .", "A group of output effect clauses $\\mathit {effect}_1,\\dots ,\\mathit {effect}_m, m > 0$ of the form () is translated into an output arc $(a,l,G)$ , with arc-weight $W(a,l,G) = w^-$ , where $w^-$ is the multi-set union of $f_i/e_i$ in $\\mathit {effect}_i, 1 \\le i \\le m$ .", "A group of reset effect clauses $\\mathit {effect}_1,\\dots ,\\mathit {effect}_m, m > 0$ of the form () is translated into a reset arc $(l,a,G)$ with arc-weight $W(l,a,G) = $ .", "A must-execute statement (REF ) is translated into guarded arcs in the same way as may execute.", "In addition, it adds an arc-inscription $must\\text{-}exec(G)$ , where $G$ is the guard, which is the translation of $guard\\_cond_1,\\dots ,guard\\_cond_n$ .", "An inhibit statement (REF ) is translated into $IG = (guard_1,\\dots ,guard_n)$ , where $(guard_1,\\dots ,guard_n)$ is the translation of $(guard\\_cond\\_1,\\dots ,guard\\_cond_n)$ An initial condition statement () sets the initial marking of a specific place $l$ for a specific color $f$ to $w$ , i.e.", "$m_{(M_0(l))}(f) = w$ .", "An duration statement (REF ) adds a $dur(d)$ inscription to transition $a$ .", "A stimulate statement (REF ) adds a $stimulate(n,G)$ inscription to transition $a$ , where guard $G$ is the translation of its guard expression $guard\\_cond_1,$ $\\dots ,guard\\_cond_n$ .", "A guard $(G_1 \\vee \\dots \\vee G_n) \\wedge (\\lnot IG_1 \\wedge \\dots \\wedge \\lnot IG_m)$ is added to each transition $a$ , where $G_i, 1 \\le i \\le n$ is the guard for a may-execute or a must-execute proposition and $IG_i, 1 \\le i \\le m$ is a guard for an inhibit proposition.", "A firing style statement (REF ) does not visibly show on a Petri Net, but it specifies the transition firing regime the Petri Net follows.", "Example Consider the following pathway specification: $&\\mathbf {domain~of~} \\nonumber \\\\&~~~~f_1 \\mathbf {~atloc~} l_1 \\mathbf {~is~} integer, f_2 \\mathbf {~atloc~} l_1 \\mathbf {~is~} integer, \\nonumber \\\\&~~~~f_3 \\mathbf {~atloc~} l_1 \\mathbf {~is~} integer, f_1 \\mathbf {~atloc~} l_2 \\mathbf {~is~} integer, \\nonumber \\\\&~~~~f_2 \\mathbf {~atloc~} l_2 \\mathbf {~is~} integer, f_3 \\mathbf {~atloc~} l_2 \\mathbf {~is~} integer \\nonumber \\\\&t_1 \\mathbf {~may~execute~causing} \\nonumber \\\\&~~~~~~~~f_1 \\mathbf {~atloc~} l_1 \\mathbf {~change~value~by~} -1, \\nonumber \\\\&~~~~~~~~f_2 \\mathbf {~atloc~} l_2 \\mathbf {~change~value~by~} +1 \\nonumber \\\\&~~~~\\mathbf {if~} f_3 \\mathbf {~atloc~} l_1 \\mathbf {~has~value~lower~than~} 5 \\nonumber \\\\&t_1 \\mathbf {~may~execute~causing} \\nonumber \\\\&~~~~~~~~f_1 \\mathbf {~atloc~} l_1 \\mathbf {~change~value~by~} -2, \\nonumber \\\\&~~~~~~~~f_2 \\mathbf {~atloc~} l_2 \\mathbf {~change~value~by~} +2 \\nonumber \\\\&~~~~\\mathbf {if~} f_3 \\mathbf {~atloc~} l_1 \\mathbf {~has~value~higher~than~} 5 \\nonumber \\\\&\\mathbf {duration~of~} t1 \\mathbf {~is~} 10 \\nonumber \\\\&\\mathbf {inhibit~} t1 \\mathbf {~if~} f_3 \\mathbf {~atloc~} l_1 \\mathbf {~has~value~higher~than~} 7 \\nonumber \\\\&\\mathbf {inhibit~} t1 \\mathbf {~if~} f_3 \\mathbf {~atloc~} l_1 \\mathbf {~has~value~lower~than~} 3 \\nonumber \\\\&\\mathbf {normally~stimulate~} t1 \\mathbf {~by~factor~} 3 \\nonumber \\\\&~~~~\\mathbf {~if~} f_2 \\mathbf {~atloc~} l_2 \\mathbf {~has~value~higher~than~} 5 \\nonumber \\\\&\\mathbf {initially~} \\nonumber \\\\&~~~~f_1 \\mathbf {~atloc~} l_1 \\mathbf {~has~value~} 0, f_1 \\mathbf {~atloc~} l_2 \\mathbf {~has~value~} 0, \\nonumber \\\\&~~~~f_2 \\mathbf {~atloc~} l_1 \\mathbf {~has~value~} 0, f_2 \\mathbf {~atloc~} l_2 \\mathbf {~has~value~} 0, \\nonumber \\\\&~~~~f_3 \\mathbf {~atloc~} l_1 \\mathbf {~has~value~} 0, f_3 \\mathbf {~atloc~} l_2 \\mathbf {~has~value~} 0 \\nonumber \\\\&\\mathbf {firing~style~} max$ Syntax of Query Language (BioPathQA-QL) The alphabet of query language $\\mathcal {Q}$ consists of the same sets $A,F,L$ from $\\mathcal {P}$ representing actions, fluents, and locations, respectively; a fixed set of reserved keywords $K$ shown in bold in syntax below; a fixed set $\\lbrace `:^{\\prime }, `;^{\\prime }, `,^{\\prime } , `^{\\prime \\prime }\\rbrace $ of punctuations; a fixed set of $\\lbrace `<^{\\prime },`>^{\\prime },`=^{\\prime }\\rbrace $ of directions; and constants.", "Our query language asks questions about biological entities and processes in a biological pathway described through the pathway specification language.", "This is our domain description.", "A query statement is composed of a query description (the quantity, observation, or condition being sought by the question), interventions (changes to the pathway), observations (about states and actions of the pathway), and initial setup conditions.", "The query statement is evaluated against the trajectories of the pathway, generated by simulating the pathway.", "These trajectories are modified by the initial setup and interventions.", "The resulting trajectories are then filtered to retain only those which satisfy the observations specified in the query statement.", "A query statement can take various forms: The simplest queries do not modify the pathway and check if a specific observation is true on a trajectory or not.", "An observation can be a point observation or an interval observation depending upon whether they can be evaluated w.r.t.", "a point or an interval on a trajectory.", "More complex queries modify the pathway in various ways and ask for comparison of an observation before and after such modification.", "Following query statements about the rate of production of `$bpg13$ ' illustrate the kind of queries that can be asked from our system about the specified glycolysis pathway as given in [64].", "Determine if `$n$ ' is a possible rate of production of substance `$bpg13$ ': $\\mathbf {rate~} & \\mathbf {of~production~of~} ^{\\prime }bpg13^{\\prime } \\mathbf {~is~} n \\nonumber \\\\&\\mathbf {when~observed~between~time~step~} 0 \\mathbf {~and~} \\mathbf {~time~step~} k ; &$ Determine if `$n$ ' is a possible rate of production of substance `$bpg13$ ' in a pathway when it is being supplied with a limited supply of an upstream substance `$f16bp$ ': $\\mathbf {rate}&\\mathbf {~of~production~of~} ^{\\prime }bpg13^{\\prime } \\mathbf {~is~} n \\nonumber \\\\&\\mathbf {when~observed~between~time~step~} 0 \\mathbf {~and~} \\mathbf {~time~step~} k ; \\nonumber \\\\&\\mathbf {using~initial~setup:~} \\mathbf {set~value~of~} `f16bp^{\\prime } \\mathbf {~to~} 5 ; &$ Determine if `$n$ ' is a possible rate of production of substance `$bpg13$ ' in a pathway when it is being supplied with a steady state supply of an upstream substance `$f16bp$ ' at the rate of 5 units per time-step: $\\mathbf {rate}&\\mathbf {~of~production~of~} ^{\\prime }bpg13^{\\prime } \\mathbf {~is~} n \\nonumber \\\\&\\mathbf {when~observed~between~time~step~} 0 \\mathbf {~and~} \\mathbf {~time~step~} k ; \\nonumber \\\\&\\mathbf {using~initial~setup:~} \\mathbf {continuously~supply~} `f16bp^{\\prime } \\mathbf {~in~quantity~} 5 ; &$ Determine if `$n$ ' is a possible rate of production of substance `$bpg13$ ' in a pathway when it is being supplied with a steady state supply of an upstream substance `$f16bp$ ' at the rate of 5 units per time-step and the pathway is modified to remove all quantity of the substance '$dhap$ ' as soon as it is produced: $\\mathbf {rate}&\\mathbf {~of~production~of~} ^{\\prime }bpg13^{\\prime } \\mathbf {~is~} n \\nonumber \\\\&\\mathbf {when~observed~between~time~step~} 0 \\mathbf {~and~} \\mathbf {~time~step~} k ; \\nonumber \\\\&\\mathbf {due~to~interventions:~} \\mathbf {remove~} `dhap^{\\prime } \\mathbf {~as~soon~as~produced} ;\\nonumber \\\\&\\mathbf {using~initial~setup:~} \\mathbf {continuously~supply~} `f16bp^{\\prime } \\mathbf {~in~quantity~} 5 ; &$ Determine if `$n$ ' is a possible rate of production of substance `$bpg13$ ' in a pathway when it is being supplied with a steady state supply of an upstream substance `$f16bp$ ' at the rate of 5 units per time-step and the pathway is modified to remove all quantity of the substance '$dhap$ ' as soon as it is produced and a non-functional pathway process / reaction named `$t5b$ ': $\\mathbf {rate}&\\mathbf {~of~production~of~} ^{\\prime }bpg13^{\\prime } \\mathbf {~is~} n \\nonumber \\\\&\\mathbf {when~observed~between~time~step~} 0 \\mathbf {~and~} \\mathbf {~time~step~} k ; \\nonumber \\\\&\\mathbf {due~to~interventions:~} \\mathbf {remove~} `dhap^{\\prime } \\mathbf {~as~soon~as~produced} ;\\nonumber \\\\&\\mathbf {due~to~observations:~} `t5b^{\\prime } \\mathbf {~does~not~occur} ; \\nonumber \\\\&\\mathbf {using~initial~setup:~} \\mathbf {continuously~supply~} `f16bp^{\\prime } \\mathbf {~in~quantity~} 5 ; &$ Determine if `$n$ ' is the average rate of production of substance `$bpg13$ ' in a pathway when it is being supplied with a steady state supply of an upstream substance `$f16bp$ ' at the rate of 5 units per time-step and the pathway is modified to remove all quantity of the substance '$dhap$ ' as soon as it is produced and a non-functional pathway process / reaction named `$t5b$ ': $\\mathbf {average}&\\mathbf {~rate~of~production~of~} ^{\\prime }bpg13^{\\prime } \\mathbf {~is~} n \\nonumber \\\\&\\mathbf {when~observed~between~time~step~} 0 \\mathbf {~and~} \\mathbf {~time~step~} k ; \\nonumber \\\\&\\mathbf {due~to~interventions:~} \\mathbf {remove~} `dhap^{\\prime } \\mathbf {~as~soon~as~produced} ;\\nonumber \\\\&\\mathbf {due~to~observations:~} `t5b^{\\prime } \\mathbf {~does~not~occur} ; \\nonumber \\\\&\\mathbf {using~initial~setup:~} \\mathbf {continuously~supply~} `f16bp^{\\prime } \\mathbf {~in~quantity~} 5 ; &$ Determine if `$d$ ' is the direction of change in the average rate of production of substance `$bpg13$ ' with a steady state supply of an upstream pathway input when compared with a pathway with the same steady state supply of an upstream pathway input, but in which the substance `$dhap$ ' is removed from the pathway as soon as it is produced and pathway process / reaction called `$t5b$ ' is non-functional: $\\mathbf {dir}&\\mathbf {ection~of~change~in~average~rate~of~production~of~} ^{\\prime }bpg13^{\\prime } \\mathbf {~is~} d \\nonumber \\\\&\\mathbf {when~observed~between~time~step~} 0 \\mathbf {~and~} \\mathbf {~time~step~} k ; \\nonumber \\\\&\\mathbf {comparing~nominal~pathway~with~modified~pathway~obtained~} \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\mathbf {due~to~interventions:~} \\mathbf {remove~} `dhap^{\\prime } \\mathbf {~as~soon~as~produced~} ;\\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\mathbf {due~to~observations:~} `t5b^{\\prime } \\mathbf {~does~not~occur} ; \\nonumber \\\\&\\mathbf {using~initial~setup:~} \\mathbf {continuously~supply~} `f16bp^{\\prime } \\mathbf {~in~quantity~} 5 ; &$ Queries can also be about actions, as illustrated in the following examples.", "Determine if action `$t5b$ ' ever occurs when there is a continuous supply of `$f16bp$ ' is available and `$t5a$ ' is disabled: $`t5b^{\\prime } &\\mathbf {~occurs~} ;\\nonumber \\\\&\\mathbf {due~to~interventions:~} \\mathbf {disable~} `t5a^{\\prime } ; \\nonumber \\\\&\\mathbf {using~initial~setup:~} \\mathbf {continuously~produce~} `f16bp^{\\prime } \\mathbf {~in~quantity~} 5 ; &$ Determine if glycolysis ($`gly^{\\prime }$ ) gets replaced with beta-oxidation ($`box^{\\prime }$ ) when sugar ($`sug^{\\prime }$ ) is exhausted but fatty acids ($`fac^{\\prime }$ ) are available, when starting with a fixed initial supply of sugar and fatty acids in quantity 5: $&`gly^{\\prime } \\mathbf {~switches~to~} `box^{\\prime } \\mathbf {~when~} \\nonumber \\\\&~~~~\\mathbf {value~of~} ^{\\prime }sug^{\\prime } \\mathbf {~is~} 0, \\nonumber \\\\&~~~~\\mathbf {value~of~} ^{\\prime }fac^{\\prime } \\mathbf {~is~higher~than~} 0 \\nonumber \\\\&~~~~\\mathbf {in~all~trajectories} ; \\nonumber \\\\&\\mathbf {due~to~observations:} \\nonumber \\\\&~~~~`gly^{\\prime } \\mathbf {~switches~to~} `box^{\\prime } \\nonumber \\\\&\\mathbf {using~initial~setup:~} \\nonumber \\\\&~~~~\\mathbf {set~value~of~} `sug^{\\prime } \\mathbf {~to~} 5, \\nonumber \\\\&~~~~\\mathbf {set~value~of~} `fac^{\\prime } \\mathbf {~to~} 5 ; &$ Next we define various syntactic elements of a query statement, give their intuitive meaning, and how these components fit together to form a query statement.", "We will define the formal semantics in a later section.", "Note that some of the single-trajectory queries can be represented as LTL formulas.", "However, we have chosen to keep the current representation as it is more intuitive for our biological domain.", "In the following description, $f_i$ 's are fluents, $l_i$ 's are locations, $n$ 's are numbers, $q$ 's are positive integer numbers, $d$ is one of the directions from $\\lbrace <,>,=\\rbrace $ .", "Definition 66 (Point) A point is a time-step on the trajectory.", "It has the form: $&\\mathbf {time~step~} ts$ Definition 67 (Interval) An interval is a sub-sequence of time-steps on a trajectory.", "It has the form: $&\\langle point \\rangle \\mathbf {~and~} \\langle point \\rangle $ Definition 68 (Aggregate Operator (aggop)) An aggregate operator computes an aggregate quantity over a sequence of values.", "It can be one of the following: $& \\mathbf {minimum}\\\\& \\mathbf {maximum}\\\\& \\mathbf {average}$ Definition 69 (Quantitative Interval Formula) A quantitative interval formula is a formula that is evaluated w.r.t.", "an interval on a trajectory for some quantity $n$ .", "$&\\mathbf {rate~of~production~of~} f \\mathbf {~is~} n\\\\&\\mathbf {rate~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n\\\\& \\mathbf {rate~of~firing~of~} a \\mathbf {~is~} n\\\\&\\mathbf {total~production~of~} f \\mathbf {~is~} n\\\\&\\mathbf {total~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n$ Intuitively, the rate of production of a fluent $f$ in interval $s_i,\\dots ,s_j$ on a trajectory $s_0,T_0,\\dots ,T_{k-1},s_k$ is $n=(s_j(f)-s_i(f))/(j-i)$ ; rate of firing of an action $a$ in interval $s_i,\\dots ,s_j$ is $n=\|\\lbrace T_l : a \\in T_l, i \\le l \\le j-1\\rbrace \|/(j-i)$ ; and total production of a fluent $f$ in interval $s_i,\\dots ,s_j$ is $n=s_j(f)-s_i(f)$ .", "If the given $n$ equals the computed $n$ , then the formula holds.", "The same intuition extends to locational fluents, except that fluent $f$ is replaced by $f[l]$ , e.g.", "rate of production of fluent $f$ at location $l$ in interval $s_i,\\dots ,s_j$ on a trajectory is $n=(s_j(f[l])-s_i(f[l]))/(j-i)$ .", "In biological context, the actions represent reactions and fluents substances used in these reactions.", "The rate and total production formulas are used in aggregate observations to determine if reactions are slowing down or speeding up during various portions of a simulation.", "Definition 70 (Quantitative Point Formula) A quantitative point formula is a formula that is evaluated w.r.t.", "a point on a trajectory for some quantity $n$ .", "$&\\mathbf {value~of~} f \\mathbf {~is~higher~than~} n\\\\&\\mathbf {value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~higher~than~} n\\\\&\\mathbf {value~of~} f \\mathbf {~is~lower~than~} n\\\\&\\mathbf {value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~lower~than~} n\\\\&\\mathbf {value~of~} f \\mathbf {~is~} n\\\\&\\mathbf {value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n$ Definition 71 (Qualitative Interval Formula) A qualitative interval formula is a formula that is evaluated w.r.t.", "an interval on a trajectory.", "$& f \\mathbf {~is~accumulating~}\\\\& f \\mathbf {~is~accumulating~} \\mathbf {~atloc~} l \\\\& f \\mathbf {~is~decreasing~}\\\\& f \\mathbf {~is~decreasing~} \\mathbf {~atloc~} l$ Intuitively, a fluent $f$ is accumulating in interval $s_i,\\dots ,s_j$ on a trajectory if $f$ 's value monotonically increases during the interval.", "A fluent $f$ is decreasing in interval $s_i,\\dots ,s_j$ on a trajectory if $f$ 's value monotonically decreases during the interval.", "The same intuition extends to locational fluents by replacing $f$ with $f[l]$ .", "Definition 72 (Qualitative Point Formula) A qualitative point formula is a formula that is evaluated w.r.t.", "a point on a trajectory.", "$& a \\mathbf {~occurs}\\\\& a \\mathbf {~does~not~occur}\\\\& a1 \\mathbf {~switches~to~} a2\\\\$ Intuitively, an action occurs at a point $i$ on the trajectory if $a \\in T_i$ ; an action does not occur at point $i$ if $a \\notin T_i$ ; an action $a1$ switches to $a2$ at point $i$ if $a1 \\in T_{i-1}$ , $a2 \\notin T_{i-1}$ , $a1 \\notin T_i$ , $a2 \\in T_i$ .", "Definition 73 (Quantitative All Interval Formula) A quantitative all interval formula is a formula that is evaluated w.r.t.", "an interval on a set of trajectories $\\sigma _1,\\dots ,\\sigma _m$ and corresponding quantities $r_1,\\dots ,r_m$ .", "$&\\mathbf {~rates~of~production~of~} f \\mathbf {~are~} [r_1,\\dots ,r_m]\\\\&\\mathbf {~rates~of~production~of~} f \\mathbf {~altoc~} l \\mathbf {~are~} [r_1,\\dots ,r_m]\\\\&\\mathbf {~rates~of~firing~of~} f \\mathbf {~are~} [r_1,\\dots ,r_m]\\\\&\\mathbf {~totals~of~production~of~} f \\mathbf {~are~} [r_1,\\dots ,r_m]\\\\&\\mathbf {~totals~of~production~of~} f \\mathbf {~altoc~} l \\mathbf {~are~} [r_1,\\dots ,r_m]$ Intuitively, a quantitative all interval formula holds on some interval $[i,j]$ over a set of trajectories $\\sigma _1,\\dots ,\\sigma _m$ for values $[r_1,\\dots ,r_m]$ if for each $r_x$ the corresponding quantitative interval formula holds in interval $[i,j]$ in trajectory $\\sigma _x$ .", "For example, $\\mathbf {rates~of~production~of~} f \\mathbf {~are~} [r_1,\\dots ,r_m]$ in interval $[i,j]$ over a set of trajectories $\\sigma _1,\\dots ,\\sigma _m$ if for each $x \\in \\lbrace 1 \\dots m\\rbrace $ , $\\textbf {rate~of~production~of~} f \\mathbf {~is~} r_x$ in interval $[i,j]$ in trajectory $\\sigma _x$ .", "Definition 74 (Quantitative All Point Formula) A quantitative all point formula is a formula that is evaluated w.r.t.", "a point on a set of trajectories $\\sigma _1,\\dots ,\\sigma _m$ and corresponding quantities $r_1,\\dots ,r_m$ .", "$&\\mathbf {values~of~} f \\mathbf {~are~} [r_1,\\dots ,r_m]\\\\&\\mathbf {values~of~} f \\mathbf {~atloc~} l \\mathbf {~are~} [r_1,\\dots ,r_m]$ Intuitively, a quantitative all point formula holds at some point $i$ over a set of trajectories $\\sigma _1,\\dots ,\\sigma _m$ for values $[r_1,\\dots ,r_m]$ if for each $r_x$ the corresponding quantitative point formula holds at point $i$ in trajectory $\\sigma _x$ .", "For example, $\\mathbf {values~of~} f \\mathbf {~are~} [r_1,\\dots ,r_m]$ at point $i$ over a set of trajectories $\\sigma _1,\\dots ,\\sigma _m$ if for each $x \\in \\lbrace 1\\dots m\\rbrace $ , $\\textbf {value~of~} f \\mathbf {~is~} r_x$ at point $i$ in trajectory $\\sigma _x$ .", "Definition 75 (Quantitative Aggregate Interval Formula) A quantitative aggregate interval formula is a formula that is evaluated w.r.t.", "an interval on a set of trajectories $\\sigma _1,\\dots ,\\sigma _m$ and an aggregate value $r$ , where $r$ is the aggregate of $[r_1,\\dots ,r_m]$ using $aggop$ .", "$&\\langle aggop \\rangle \\mathbf {~rate~of~production~of~} f \\mathbf {~is~} n\\\\&\\langle aggop \\rangle \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n\\\\&\\langle aggop \\rangle \\mathbf {~rate~of~firing~of~} a \\mathbf {~is~} n\\\\&\\langle aggop \\rangle \\mathbf {~total~production~of~} f \\mathbf {~is~} n\\\\&\\langle aggop \\rangle \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n$ Intuitively, a quantitative aggregate interval formula holds on some interval $[i,j]$ over a set of trajectories $\\sigma _1,\\dots ,\\sigma _m$ for a value $r$ if there exist $[r_1,\\dots ,r_m]$ whose aggregate value per $aggop$ is $r$ , such that for each $r_x$ the quantitative interval formula (corresponding to the quantitative aggregate interval formual) holds in interval $[i,j]$ in trajectory $\\sigma _x$ .", "For example, $average \\mathbf {~rate~of~}$ $\\mathbf {production~of~} f \\mathbf {~is~} r$ in interval $[i,j]$ over a set of trajectories $\\sigma _1,\\dots ,\\sigma _m$ if $r=(r_1+\\dots +r_m)/m$ and for each $x \\in \\lbrace 1\\dots m\\rbrace $ , $\\textbf {rate~of~production~of~} f \\mathbf {~is~} r_x$ in interval $[i,j]$ in trajectory $\\sigma _x$ .", "Definition 76 (Quantitative Aggregate Point Formula) A quantitative aggregate point formula is a formula that is evaluated w.r.t.", "a point on a set of trajectories $\\sigma _1,\\dots ,\\sigma _m$ and an aggregate value $r$ , where $r$ is the aggregate of $[r_1,\\dots ,r_m]$ using $aggop$ .", "$&\\langle aggop \\rangle \\mathbf {~value~of~} f \\mathbf {~is~} r\\\\&\\langle aggop \\rangle \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} r$ Intuitively, a quantitative aggregate point formula holds at some point $i$ over a set of trajectories $\\sigma _1,\\dots ,\\sigma _m$ for a value $r$ if there exist $[r_1,\\dots ,r_m]$ whose aggregate value per $aggop$ is $r$ , such that for each $r_x$ the quantitative point formula (corresponding to the quantitative aggregate point formual) holds at point $i$ in trajectory $\\sigma _x$ .", "For example, $average \\mathbf {~value~of~} f \\mathbf {~is~} r$ at point $i$ over a set of trajectories $\\sigma _1,\\dots ,\\sigma _m$ if $r=(r_1+\\dots +r_m)/m$ and for each $x \\in \\lbrace 1\\dots m\\rbrace $ , $\\textbf {value~of~} f \\mathbf {~is~} r_x$ at point $i$ in trajectory $\\sigma _x$ .", "Definition 77 (Quantitative Comparative Aggregate Interval Formula) A quantitative comparative aggregate interval formula is a formula that is evaluated w.r.t.", "an interval over two sets of trajectories and a direction `$d$ ' such that `$d$ ' relates the two sets of trajectories.", "$&\\mathbf {direction~of~change~in~} \\langle aggop \\rangle \\mathbf {~rate~of~production~of~} f \\mathbf {~is~} d\\\\&\\mathbf {direction~of~change~in~} \\langle aggop \\rangle \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} d\\\\&\\mathbf {direction~of~change~in~} \\langle aggop \\rangle \\mathbf {~rate~of~firing~of~} a \\mathbf {~is~} d\\\\&\\mathbf {direction~of~change~in~} \\langle aggop \\rangle \\mathbf {~total~production~of~} f \\mathbf {~is~} d\\\\&\\mathbf {direction~of~change~in~} \\langle aggop \\rangle \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} d$ Intuitively, a comparative quantitative aggregate interval formula compares two quantitative interval formulas over using the direction $d$ over a given interval.", "Definition 78 (Quantitative Comparative Aggregate Point Formula) A quantitative comparative aggregate point formula is a formula that is evaluated w.r.t.", "a point over two sets of trajectories and a direction `$d$ ' such that `$d$ ' relates the two sets of trajectories.", "$&\\mathbf {direction~of~change~in~} \\langle aggop \\rangle \\mathbf {~value~of~} f \\mathbf {~is~} d\\\\&\\mathbf {direction~of~change~in~} \\langle aggop \\rangle \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} d$ Intuitively, a comparative quantitative aggregate point formula compares two quantitative point formulas over using the direction $d$ at a given point.", "Definition 79 (Simple Interval Formula) A simple interval formula takes the following forms: $&\\langle \\text{quantitative interval formula} \\rangle \\\\&\\langle \\text{qualitative interval formula} \\rangle $ Definition 80 (Simple Point Formula) A simple interval formula takes the following forms: $&\\langle \\text{quantitative point formula} \\rangle \\\\&\\langle \\text{qualitative point formula} \\rangle $ Definition 81 (Internal Observation Description) An internal observation description takes the following form: $&\\langle \\text{simple point formula} \\rangle \\\\&\\langle \\text{simple point formula} \\rangle \\mathbf {~at~} \\langle point \\rangle \\\\&\\langle \\text{simple interval formula} \\rangle \\\\&\\langle \\text{simple interval formula} \\rangle \\mathbf {~when~observed~between~} \\langle interval \\rangle $ Definition 82 (Simple Point Formula Cascade) A simple point formula cascade takes the following form: $&\\langle \\text{simple point formula} \\rangle _0 \\nonumber \\\\&~~~~~~~~\\mathbf {~after~} \\langle \\text{simple point formula} \\rangle _{1,1}, \\dots , \\langle \\text{simple point formula} \\rangle _{1,n_1} \\nonumber \\\\&~~~~~~~~\\vdots \\nonumber \\\\&~~~~~~~~\\mathbf {~after~} \\langle \\text{simple point formula} \\rangle _{u,1}, \\dots , \\langle \\text{simple point formula} \\rangle _{u,n_u} \\\\&\\langle \\text{simple point formula} \\rangle _0 \\nonumber \\\\&~~~~~~~~\\mathbf {~when~} \\langle \\text{simple point formula} \\rangle _{1,1}, \\dots , \\langle \\text{simple point formula} \\rangle _{1,n_1}\\\\&\\langle \\text{simple point formula} \\rangle _0 \\nonumber \\\\&~~~~~~~~\\mathbf {~when~} \\langle \\text{cond} \\rangle $ where $u \\ge 1$ and `$\\text{cond}$ ' is a conjunction of $\\text{simple point formula}$ s that is true in the same point as the $\\text{simple point formula}$ .", "Intuitively, the simple point formula cascade (REF ) holds if a given sequence of point formulas hold in order in a trajectory.", "Intuitively, simple point formula cascade () holds if a given point formula occurs at the same point as a set of simple point formulas in a trajectory.", "Note that these formulas and many other of our single trajectory formulas can be replaced by an LTL [56] formula, but we have kept this syntax as it is more relevant to the question answering needs in the biological domain.", "Definition 83 (Query Description) A query description specifies a non-comparative observation that can be made either on a trajectory or a set of trajectories.", "$& \\langle \\text{quantitative aggregate interval formula} \\rangle \\mathbf {~when~observed~between~} \\langle \\text{interval} \\rangle \\\\& \\langle \\text{quantitative aggregate point formula} \\rangle \\mathbf {~when~observed~at~} \\langle \\text{point} \\rangle \\\\& \\langle \\text{quantitative all interval formula} \\rangle \\mathbf {~when~observed~between~} \\langle \\text{interval} \\rangle \\\\& \\langle \\text{quantitative all point formula} \\rangle \\mathbf {~when~observed~at~} \\langle \\text{point} \\rangle \\\\& \\langle \\text{internal observation description} \\rangle \\\\& \\langle \\text{internal observation description} \\rangle \\mathbf {~in~all~trajectories}\\\\& \\langle \\text{simple point formula cascade} \\rangle \\\\& \\langle \\text{simple point formula cascade} \\rangle \\mathbf {~in~all~trajectories}$ The single trajectory observations are can be represented using LTL formulas, but we have chosen to keep them in this form for ease of use by users from the biological domain.", "Definition 84 (Comparative Query Description) A comparative query description specifies a comparative observation that can be made w.r.t.", "two sets of trajectories.", "$& \\langle \\text{quantitative comparative aggregate interval formula} \\rangle \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\;\\mathbf {~when~observed~between~} \\langle \\text{interval} \\rangle \\\\& \\langle \\text{quantitative comparative aggregate point formula} \\rangle \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\;\\mathbf {~when~observed~at~} \\langle \\text{point} \\rangle $ Definition 85 (Intervention) Interventions define modifications to domain descriptions.", "$&\\mathbf {remove~} f_1 \\mathbf {~as~soon~as~produced} \\\\&\\mathbf {remove~} f_1 \\mathbf {~atloc~} l_1 \\mathbf {~as~soon~as~produced} \\\\&\\mathbf {disable~} a_2 \\\\&\\mathbf {continuously~transform~} f_1\\mathbf {~in~quantity~} q_1 \\mathbf {~to~} f_2 \\\\&\\mathbf {continuously~transform~} f_1\\mathbf {~atloc~} l_1 \\mathbf {~in~quantity~} q_1 \\mathbf {~to~} f_2 \\mathbf {~atloc~} l_2 \\\\&\\mathbf {make~} f_3 \\mathbf {~inhibit~} a_3 \\\\&\\mathbf {make~} f_3 \\mathbf {~atloc~} l_3 \\mathbf {~inhibit~} a_3 \\\\&\\mathbf {continuously~supply~} f_4 \\mathbf {~in~quantity~} q_4 \\\\&\\mathbf {contiunously~supply~} f_4 \\mathbf {~atloc~} l_4 \\mathbf {~in~quantity~} q_4 \\\\&\\mathbf {continuously~transfer~} f_1 \\mathbf {~in~quantity~} q_1 \\mathbf {~across~} l_1,l_2 \\mathbf {~to~lower~gradient~} \\\\&\\mathbf {add~delay~of~} q_1 \\mathbf {~time~units~in~availability~of~} f_1 \\\\&\\mathbf {add~delay~of~} q_1 \\mathbf {~time~units~in~availability~of~} f_1 \\mathbf {~atloc~} l_1 \\\\&\\mathbf {set~value~of~} f_4 \\mathbf {~to~} q_4 \\\\&\\mathbf {set~value~of~} f_4 \\mathbf {~atloc~} l_4 \\mathbf {~to~} q_4$ Intuitively, intervention (REF ) modifies the pathway such that all quantity of $f_1$ is removed as soon as it is produced; intervention () modifies the pathway such that all quantity of $f_1[l_1]$ is removed as soon as it is produced; intervention () disables the action $a_2$ ; intervention () modifies the pathway such that $f_1$ gets converted to $f_2$ at the rate of $q_1$ units per time-unit; intervention () modifies the pathway such that $f_1[l_1]$ gets converted to $f_2[l_2]$ at the rate of $q_1$ units per time-unit; intervention () modifies the pathway such that action $a_3$ is now inhibited each time there is 1 or more units of $f_3$ and sets value of $f_3$ to 1 to initially inhibit $a_3$ ; intervention () modifies the pathway such that action $a_3$ is now inhibited each time there is 1 or more units of $f_3[l_3]$ and sets value of $f_3[l_3]$ to 1 to initially inhibit $a_3$ ; intervention () modifies the pathway to continuously supply $f_4$ at the rate of $q_4$ units per time-unit; intervention () modifies the pathway to continuously supply $f_4[l_4]$ at the rate of $q_4$ units per time-unit; intervention () modifies the pathway to transfer $f_1[l_1]$ to $f_1[l_2]$ in quantity $q_1$ or back depending upon whether $f_1[l_1]$ is higher than $f_1[l_2]$ or lower; intervention () modifies the pathway to add delay of $q_1$ time units between when $f_1$ is produced to when it is made available to next action; intervention () modifies the pathway to add delay of $q_1$ time units between when $f_1[l_1]$ is produced to when it is made available to next action; intervention () modifies the pathway to set the initial value of $f_4$ to $q_4$ ; and intervention () modifies the pathway to set the initial value of $f_4[l_4]$ to $q_4$ .", "Definition 86 (Initial Condition) An initial condition is one of the intervention (), (), (), () as given in definition REF .", "Intuitively, initial conditions are interventions that setup fixed or continuous supply of substances participating in a pathway.", "Definition 87 (Query Statement) A query statement can be of the following forms: $& \\langle \\text{query description} \\rangle ; \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\mathbf {due~to~interventions:} \\; \\langle intervention \\rangle _1, \\dots , \\langle intervention \\rangle _{N1}; \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\mathbf {due~to~observations:} \\; \\langle \\text{internal observation} \\rangle _1, \\dots , \\langle \\text{internal observation} \\rangle _{N2}; \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\mathbf {using~initial~setup:} \\; \\langle \\text{initial condition} \\rangle _1, \\dots , \\langle \\text{initial condition} \\rangle _{N3};\\\\\\nonumber \\\\& \\langle \\text{comparative query description} \\rangle ; \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\;\\mathbf {comparing~nominal~pathway~with~modified~pathway~obtained~} \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\mathbf {due~to~interventions:} \\; \\langle intervention \\rangle _1, \\dots , \\langle intervention \\rangle _{N1} ; \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\mathbf {due~to~observations:} \\; \\langle \\text{internal observation} \\rangle _1, \\dots , \\langle \\text{internal observation} \\rangle _{N2}; \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\mathbf {using~initial~setup:} \\; \\langle \\text{initial condition} \\rangle _1, \\dots , \\langle \\text{initial condition} \\rangle _{N3}; &$ where interventions, observations, and initial setup are optional.", "Intuitively, a query statement asks whether a query description holds in a pathway, perhaps after modifying it with initial setup, interventions and observations.", "Intuitively, a comparative query statement asks whether a comparative query description holds with a nominal pathway is compared against a modified pathway, where both pathways have the same initial setup, but only the modified pathway has been modified with interventions and observations.", "Semantics of the Query Language (BioPathQA-QL) In this section we give the semantics of our pathway specification language and the query language.", "The semantics of the query language is in terms of the trajectories of a domain description $\\mathbf {D}$ that satisfy a query $\\mathbf {Q}$ .", "We will present the semantics using LTL-style formulas.", "First, we informally define the semantics of the query language as follows.", "Let $\\mathbf {Q}$ be a query statement of the form (REF ) with a query description $U$ , interventions $V_1,\\dots ,V_{\|V\|}$ , internal observations $O_1,\\dots ,O_{\|O\|}$ , and initial setup conditions $I_1,\\dots ,I_{\|I\|}$ .", "We construct a modified domain description $\\mathbf {D_1}$ by applying $I_1,\\dots ,I_{\|I\|}$ and$V_1,\\dots ,V_{\|V\|}$ to $\\mathbf {D}$ .", "We filter the trajectories of $\\mathbf {D_1}$ to retain only those trajectories that satisfy the observations $O_1,\\dots ,O_{\|O\|}$ .", "Then we determine if $U$ holds on any of the retained trajectories.", "If it does, then we say that $\\mathbf {D}$ satisfies $\\mathbf {Q}$ .", "Let $\\mathbf {Q}$ be a comparative query statement of the form () with quantitative comparative aggregate query description $U$ , interventions $V_1,\\dots ,V_{\|V\|}$ , internal observations $O_1,\\dots ,O_{\|O\|}$ , and initial conditions $I_1,\\dots ,I_{\|I\|}$ .", "Then we evaluate $\\mathbf {Q}$ by deriving two sub-query statements.", "$\\mathbf {Q_0}$ is constructed by removing the interventions $V_1,\\dots ,V_{\|V\|}$ and observations $O_1,\\dots ,O_{\|O\|}$ from $\\mathbf {Q}$ and replacing the quantitative comparative aggregate query description $U$ with the corresponding quantitative aggregate query description $U^{\\prime }$ , $\\mathbf {Q_1}$ is constructed by replacing the quantitative comparative aggregate query description $U$ with the corresponding quantitative aggregate query description $U^{\\prime }$ .", "Then $\\mathbf {D}$ satisfies $\\mathbf {Q}$ iff we can find $d \\in \\lbrace <,>,= \\rbrace $ s.t.", "$n \\; d \\; n^{\\prime }$ , where $\\mathbf {D}$ satisfies $\\mathbf {Q_0}$ for some value $n$ and $\\mathbf {D}$ satisfies $\\mathbf {Q_1}$ for some value $n^{\\prime }$ .", "An Illustrative Example In this section, we illustrate with an example how we intuitively evaluate a comparative query statement.", "In the later sections, we will give the formal semantics of query satisfaction.", "Consider the following simple pathway specification: $\\begin{array}{llll}&\\mathbf {domain~of~} & f_1 \\mathbf {~is~} integer, & f_2 \\mathbf {~is~} integer \\\\&t_1 \\mathbf {~may~fire~causing~} & f_1 \\mathbf {~change~value~by~} -1, & f_2 \\mathbf {~change~value~by~} +1\\\\& \\mathbf {initially~} & f_1 \\mathbf {~has~value~} 0, & f_2 \\mathbf {~has~value~} 0,\\\\&\\mathbf {firing~style~} & max\\end{array}$ Let the following specify a query statement $\\mathbf {Q}$ : $\\mathbf {dir}&\\mathbf {ection~of~change~in~} average \\mathbf {~rate~of~production~of~} f_2 \\mathbf {~is~} d\\\\&\\mathbf {when~observed~between~time~step~} 0 \\mathbf {~and~time~step~} k;\\\\&\\mathbf {comparing~nominal~pathway~with~modified~pathway~obtained~}\\\\&\\;\\;\\;\\;\\;\\;\\mathbf {due~to~interventions:~} \\mathbf {remove~} f_2 \\mathbf {~as~soon~as~produced} ;\\\\&\\mathbf {using~initial~setup:~} \\mathbf {continuously~supply~} f_1 \\mathbf {~in~quantity~} 1;$ that we want to evaluate against $\\mathbf {D}$ using a simulation length $k$ with maximum $ntok$ tokens at any place to determine `$d$ ' that satisfies it.", "We construct the baseline query $\\mathbf {Q_0}$ by removing interventions and observations, and replacing the comparative aggregate quantitative query description with the corresponding aggregate quantitative query description as follows: $average &\\mathbf {~rate~of~production~of~} f_2 \\mathbf {~is~} n\\\\&\\mathbf {when~observed~between~time~step~} 0 \\mathbf {~and~time~step~} k ;\\\\&\\mathbf {using~initial~setup:~} \\mathbf {continuously~supply~} f_1 \\mathbf {~in~quantity~} 1;\\\\$ We construct the alternate query $\\mathbf {Q_1}$ by replacing the comparative aggregate quantitative query description with the corresponding aggregate quantitative query description as follows: $average&\\mathbf {~rate~of~production~of~} f_2 \\mathbf {~is~} n^{\\prime }\\\\&\\mathbf {when~observed~between~time~step~} 0 \\mathbf {~and~time~step~} k ;\\\\&\\mathbf {due~to~interventions:~} \\mathbf {remove~} f_2 \\mathbf {~as~soon~as~produced} ;\\\\&\\mathbf {using~initial~setup:~} \\mathbf {continuously~supply~} f_1 \\mathbf {~in~quantity~} 1;\\\\$ We build a modified domain description $\\mathbf {D_0}$ as $\\mathbf {D} \\diamond (\\mathbf {continuously~supply~} f_1 $ $\\mathbf {~in~quantity~} 1)$ based on initial conditions in $\\mathbf {Q_0}$ .", "We get: $\\begin{array}{llll}&\\mathbf {domain~of~} & f_1 \\mathbf {~is~} integer, & f_2 \\mathbf {~is~} integer \\\\&t_1 \\mathbf {~may~fire~causing~} & f_1 \\mathbf {~change~value~by~} -1, & f_2 \\mathbf {~change~value~by~} +1\\\\&t_{f_1} \\mathbf {~may~fire~causing~} & f_1 \\mathbf {~change~value~by~} +1\\\\& \\mathbf {initially~} & f_1 \\mathbf {~has~value~} 0, & f_2 \\mathbf {~has~value~} 0,\\\\&\\mathbf {firing~style~} & max\\end{array}$ We evaluate $\\mathbf {Q_0}$ against $\\mathbf {D_0}$ .", "It results in $m_0$ trajectories with rate of productions $n_j=(s_k(f_2)-s_0(f_2))/k$ on trajectory $\\tau _j=s_0,\\dots ,s_k, 1 \\le j \\le m_0$ using time interval $[0,k]$ .", "The rate of productions are averaged to produce $n=(n_1+\\dots +n_{m_0})/m_0$ .", "Next, we construct the alternate domain description $\\mathbf {D_1}$ as $\\mathbf {D_0} \\diamond (\\mathbf {remove~} f_2 \\mathbf {~as~soon~} $ $\\mathbf {as~produced~})$ based on initial conditions and interventions in $\\mathbf {Q_1}$ .", "We get: $\\begin{array}{llll}&\\mathbf {domain~of~} & f_1 \\mathbf {~is~} integer, & f_2 \\mathbf {~is~} integer \\\\&t_1 \\mathbf {~may~fire~causing~} & f_1 \\mathbf {~change~value~by~} -1, & f_2 \\mathbf {~change~value~by~} +1\\\\&t_{f_1} \\mathbf {~may~fire~causing~} & f_1 \\mathbf {~change~value~by~} +1\\\\&t_{f_2} \\mathbf {~may~fire~causing~} & f_2 \\mathbf {~change~value~by~} \\\\& \\mathbf {initially~} & f_1 \\mathbf {~has~value~} 0, & f_2 \\mathbf {~has~value~} 0\\\\&\\mathbf {firing~style~} & max\\end{array}$ We evaluate $\\mathbf {Q_1}$ against $\\mathbf {D_1}$ .", "Since there are no observations, no filtering is required.", "This results in $m_1$ trajectories, each with rate of production $n^{\\prime }_j=(s_k(f_2)-s_0(f_2))/k$ on trajectory $\\tau ^{\\prime }_j = s_0,\\dots ,s_k, 1 \\le j \\le m_1$ using time interval $[0,k]$ .", "The rate of productions are averaged to produce $n^{\\prime }=(n^{\\prime }_1+\\dots +n^{\\prime }_{m_1})/m_1$ .", "Due to the simple nature of our domain description, it has only one trajectory for each of the two domains.", "As a result, for any $k > 1$ , $n^{\\prime } < n$ .", "Thus, $\\mathbf {D}$ satisfies $\\mathbf {Q}$ iff $d = ``<^{\\prime \\prime }$ .", "We will now define the semantics of how a domain description $\\mathbf {D}$ is modified according to the interventions and initial conditions, the semantics of conditions imposed by the internal observations.", "We will then formally define how $\\mathbf {Q}$ is entailed in $\\mathbf {D}$ .", "Domain Transformation due to Interventions and Initial Conditions An intervention $I$ modifies a given domain description $\\mathbf {D}$ , potentially resulting in a different set of trajectories than $\\mathbf {D}$ .", "We define a binary operator $\\diamond $ that transforms $\\mathbf {D}$ by applying an intervention $I$ as a set of edits to $\\mathbf {D}$ using the pathway specification language.", "The trajectories of the modified domain description $\\mathbf {D^{\\prime }} = \\mathbf {D} \\diamond I$ are given by the semantics of the pathway specification language.", "Below, we give the intuitive impact and edits required by each of the interventions.", "Domain modification by intervention (REF ) $\\mathbf {D^{\\prime }} = \\mathbf {D} \\diamond (\\mathbf {remove~} f_1 \\mathbf {~as~soon~}$ $\\mathbf {as~produced})$ modifies the pathway by removing all existing quantity of $f_1$ at each time step: $\\mathbf {D^{\\prime }} &= \\mathbf {D}+ \\left\\lbrace \\begin{array}{l}tr \\mathbf {~may~execute~causing~} f_1 \\mathbf {~change~value~by~} \\end{array}\\right\\rbrace $ Domain modification by intervention () $\\mathbf {D^{\\prime }} = \\mathbf {D} \\diamond (\\mathbf {remove~} f_1 \\mathbf {~atloc~} l_1)$ modifies the pathway by removing all existing quantity of $f_1$ at each time step: $\\mathbf {D^{\\prime }} &= \\mathbf {D}+ \\left\\lbrace \\begin{array}{l}tr \\mathbf {~may~execute~causing~} f_1 \\mathbf {~atloc~} l_1 \\mathbf {~change~value~by~} \\end{array}\\right\\rbrace $ Domain modification by intervention () $\\mathbf {D^{\\prime }} = \\mathbf {D} \\diamond (\\mathbf {disable~} a_2)$ modifies the pathway such that its trajectories have $a_2 \\notin T_i$ , where $i \\ge 0$ .", "$\\mathbf {D^{\\prime }} &= \\mathbf {D}+ \\left\\lbrace \\begin{array}{ll}\\mathbf {inhibit~} a_2\\end{array}\\right\\rbrace $ Domain modification by intervention () $\\mathbf {D^{\\prime }} = \\mathbf {D} \\diamond (\\mathbf {continuously~transform~} f_1 $ $\\mathbf {~in~quantity~} q_1 $ $\\mathbf {~to~} f_2)$ where $s_{i+1}(f_1)$ decreases, and $s_{i+1}(f_2)$ increases by $q_1$ at each time step $i \\ge 0$ , when $s_i(f_1) \\ge q_1$ .", "$\\mathbf {D^{\\prime }} = \\mathbf {D}&+ \\left\\lbrace \\begin{array}{ll}a_{f_{1,2}} \\mathbf {~may~execute~causing~} & f_1 \\mathbf {~change~value~by~} -q_1, \\\\ & f_2 \\mathbf {~change~value~by~} +q_1\\end{array}\\right\\rbrace $ Domain modification by intervention () $\\mathbf {D^{\\prime }} = \\mathbf {D} \\diamond (\\mathbf {continuously~transform~} $ $f_1 \\mathbf {~atloc~} l_1$ $ \\mathbf {~in~quantity~} q_1 $ $\\mathbf {~to~} f_2 \\mathbf {~atloc~} l_2)$ where $s_{i+1}(f_1[l_1])$ decreases, and $s_{i+1}(f_2[l_2])$ increases by $q_1$ at each time step $i \\ge 0$ , when $s_i(f_1[l_1]) \\ge q_1$ .", "$\\mathbf {D^{\\prime }} = \\mathbf {D}&+ \\left\\lbrace \\begin{array}{ll}a_{f_{1,2}} \\mathbf {~may~execute~causing~} & f_1 \\mathbf {~atloc~} l_1 \\mathbf {~change~value~by~} -q_1, \\\\ & f_2 \\mathbf {~atloc~} l_2 \\mathbf {~change~value~by~} +q_1\\end{array}\\right\\rbrace $ Domain modification by intervention () $\\mathbf {D^{\\prime }} = \\mathbf {D} \\diamond (\\mathbf {make~} f_3 \\mathbf {~inhibit~} a_3 )$ modifies the pathway such that it has $a_3$ inhibited due to $f_3$ .", "$\\mathbf {D^{\\prime }} = \\mathbf {D}&- \\left\\lbrace \\begin{array}{l}\\mathbf {initially~} f_3 \\mathbf {~has~value~} q \\in \\mathbf {D}\\end{array}\\right\\rbrace \\\\&+ \\left\\lbrace \\begin{array}{ll}\\mathbf {inhibit~} a_3 \\mathbf {~if~} & f_3 \\mathbf {~has~value~} 1 \\mathbf {~or~higher}\\\\\\mathbf {initially~} & f_3 \\mathbf {~has~value~} 1\\end{array}\\right\\rbrace $ Domain modification by intervention () $\\mathbf {D^{\\prime }} = \\mathbf {D} \\diamond (\\mathbf {make~} f_3 \\mathbf {~atloc~} l_3 \\mathbf {~inhibit~} a_3)$ modifies the pathway such that it has $a_3$ inhibited due to $f_3[l_3]$ .", "$\\mathbf {D^{\\prime }} = \\mathbf {D}&- \\left\\lbrace \\begin{array}{l}\\mathbf {initially~} f_3 \\mathbf {~atloc~} l_3 \\mathbf {~has~value~} q \\in \\mathbf {D}\\end{array}\\right\\rbrace \\\\&+ \\left\\lbrace \\begin{array}{ll}\\mathbf {inhibit~} a_3 \\mathbf {~if~} & f_3 \\mathbf {~atloc~} l_3 \\mathbf {~has~value~} 1 \\mathbf {~or~higher}, \\\\\\mathbf {initially~} & f_3 \\mathbf {~atloc~} l_3 \\mathbf {~has~value~} 1\\end{array}\\right\\rbrace $ Domain modification by intervention () $\\mathbf {D^{\\prime }} = \\mathbf {D} \\diamond (\\mathbf {~continuously~supply~} f_4 $ $\\mathbf {~in~quantity~} q_4)$ modifies the pathway such that a quantity $q_4$ of substance $f_4$ is supplied at each time step.", "$\\mathbf {D^{\\prime }} = \\mathbf {D}&+ \\left\\lbrace \\begin{array}{ll}t_{f_4} \\mathbf {~may~execute~causing~} & f_4 \\mathbf {~change~value~by~} +q_4\\end{array}\\right\\rbrace $ Domain modification by intervention () $\\mathbf {D^{\\prime }} = \\mathbf {D} \\diamond (\\mathbf {~continuously~supply~} f_4 $ $\\mathbf {~atloc~} l_4 $ $\\mathbf {~in~quantity~} q_4 )$ modifies the pathway such that a quantity $q_4$ of substance $f_4$ at location $l_4$ is supplied at each time step.", "$\\mathbf {D^{\\prime }} = \\mathbf {D}&+ \\left\\lbrace \\begin{array}{ll}t_{f_4} \\mathbf {~may~execute~causing~} & f_4 \\mathbf {~atloc~} l_4 \\mathbf {~change~value~by~} +q_4\\end{array}\\right\\rbrace $ Domain modification by intervention () $\\mathbf {D^{\\prime }} = \\mathbf {D} \\diamond (\\mathbf {continuously~transfer~} f_1 $ $\\mathbf {~in~quantity~} q_1 $ $\\mathbf {~across~} l_1, l_2 $ $\\mathbf {~to~lower~gradient})$ modifies the pathway such that substance represented by $f_1$ is transferred from location $l_1$ to $l_2$ or $l_2$ to $l_1$ depending upon whether it is at a higher quantity at $l_1$ or $l_2$ .", "$\\mathbf {D^{\\prime }} = \\mathbf {D}&+ \\left\\lbrace \\begin{array}{ll}t_{f_1} \\mathbf {~may~execute~causing~} & f_1 \\mathbf {~atloc~} l_1 \\mathbf {~change~value~by~} -q_1, \\\\ & f_1 \\mathbf {~atloc~} l_2 \\mathbf {~change~value~by~} +q_1 \\\\\\mathbf {~~~~~~~if}& f_1 \\mathbf {~atloc~} l_1 \\mathbf {~has~higher~value~than~} f_1 \\mathbf {~atloc~} l_2, \\\\t^{\\prime }_{f_1} \\mathbf {~may~execute~causing~} & f_1 \\mathbf {~atloc~} l_2 \\mathbf {~change~value~by~} -q_1, \\\\ & f_1 \\mathbf {~atloc~} l_1 \\mathbf {~change~value~by~} +q_1\\\\\\mathbf {~~~~~~~if}& f_1 \\mathbf {~atloc~} l_2 \\mathbf {~has~higher~value~than~} f_1 \\mathbf {~atloc~} l_1, \\\\\\end{array}\\right\\rbrace $ Domain modification by intervention () $\\mathbf {D^{\\prime }} = \\mathbf {D} \\diamond (\\mathbf {add~delay~of~} q_1 $ $\\mathbf {~time~units~} $ $\\mathbf {in~availability~of~} f_1)$ modifies the pathway such that $f_1$ 's arrival is delayed by $q_1$ time units.", "We create additional cases for all actions that produce $f_1$ , such that it goes through an additional delay action.", "$\\mathbf {D^{\\prime }} = \\mathbf {D}& - \\left\\lbrace \\begin{array}{ll}a \\mathbf {~may~execute~causing~} & f_1 \\mathbf {~change~value~by~} +w_1, \\\\ & \\mathit {effect}_1,\\dots ,\\mathit {effect}_n \\\\ & \\mathbf {~if~} cond_1,\\dots ,cond_m \\in \\mathbf {D}\\\\\\end{array}\\right\\rbrace \\\\& + \\left\\lbrace \\begin{array}{ll}a \\mathbf {~may~execute~causing~} & f^{\\prime }_1 \\mathbf {~change~value~by~} +w_1, \\\\ & \\mathit {effect}_1,\\dots ,\\mathit {effect}_n, \\\\ & \\mathbf {~if~} cond_1,\\dots ,cond_m\\\\a_{f_1} \\mathbf {~may~execute~causing~} & f^{\\prime }_1 \\mathbf {~change~value~by~} -w_1, \\\\ & f_1 \\mathbf {~change~value~by~} +w_1\\\\a_{f_1} \\mathbf {~executes~} & \\mathbf {~in~} q_1 \\mathbf {~time~units}\\\\\\end{array}\\right\\rbrace $ Domain modification by intervention () $\\mathbf {D^{\\prime }} = \\mathbf {D} \\diamond (\\mathbf {add~delay~of} q_1 $ $\\mathbf {~time~units~} $ $\\mathbf {in~availability~of~} f_1 \\mathbf {~atloc~} l_1)$ modifies trajectories such that $f_1[l_1]$ 's arrival is delayed by $q_1$ time units.", "We create additional cases for all actions that produce $f_1 \\mathbf {~atloc~} l_1$ , such that it goes through an additional delay action.", "$\\mathbf {D^{\\prime }} = \\mathbf {D}&- \\left\\lbrace \\begin{array}{ll}a \\mathbf {~may~execute~causing~} & f_1 \\mathbf {~atloc~} l_1 \\mathbf {~change~value~by~} +w_1, \\\\ & \\mathit {effect}_1,\\dots ,\\mathit {effect}_n \\\\ & \\mathbf {~if~} cond_1,\\dots ,cond_m \\in \\mathbf {D}\\end{array}\\right\\rbrace \\\\&+ \\left\\lbrace \\begin{array}{ll}a \\mathbf {~may~execute~causing~} & f_1 \\mathbf {~atloc~} l^{\\prime }_1 \\mathbf {~change~value~by~} +w_1, \\\\ & \\mathit {effect}_1,\\dots ,\\mathit {effect}_n \\\\ & \\mathbf {~if~} cond_1,\\dots ,cond_n\\\\a_{f_1} \\mathbf {~may~execute~causing~} & f_1 \\mathbf {~atloc~} l^{\\prime }_1 \\mathbf {~change~value~by~} -w_1,\\\\ & f_1 \\mathbf {~atloc~} l_1 \\mathbf {~change~value~by~} +w_1\\\\a_{f_1} \\mathbf {~executes~} & \\mathbf {~in~} q_1 \\mathbf {~time~units}\\\\\\end{array}\\right\\rbrace $ Domain modification by intervention () $\\mathbf {D^{\\prime }} = \\mathbf {D} \\diamond (\\mathbf {~set~value~of~} f_4 $ $\\mathbf {~to~} q_4)$ modifies the pathway such that its trajectories have $s_0(f_4) = q_4$ .", "$\\mathbf {D^{\\prime }} = \\mathbf {D}& - \\left\\lbrace \\begin{array}{l}\\mathbf {initially~} f_4 \\mathbf {~has~value~} n \\in \\mathbf {D}\\end{array}\\right\\rbrace \\\\& + \\left\\lbrace \\begin{array}{l}\\mathbf {initially~} f_4 \\mathbf {~has~value~} q_4\\end{array}\\right\\rbrace $ Domain modification by intervention () $\\mathbf {D^{\\prime }} = \\mathbf {D} \\diamond (\\mathbf {~set~value~of~} f_4 $ $\\mathbf {~atloc~} l_4 $ $\\mathbf {~to~} q_4 )$ modifies the pathway such that its trajectories have $s_0({f_4}_{l_4}) = q_4$ .", "$\\mathbf {D^{\\prime }} = \\mathbf {D}& - \\left\\lbrace \\begin{array}{l}\\mathbf {initially~} f_4 \\mathbf {~atloc~} l_4 \\mathbf {~has~value~} n \\in \\mathbf {D}\\end{array}\\right\\rbrace \\\\& + \\left\\lbrace \\begin{array}{l}\\mathbf {initially~} f_4 \\mathbf {~atloc~} l_4 \\mathbf {~has~value~} q_4\\end{array}\\right\\rbrace $ Formula Semantics We will now define the semantics of some common formulas that we will use in the following sections.", "First we introduce the LTL-style formulas that we will be using to define the syntax.", "A formula $\\langle s_i,\\sigma \\rangle \\models F$ represents that $F$ holds at point $i$ .", "A formula $\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace \\models F$ represents that $F$ holds at point $i$ on a set of trajectories $\\sigma _1,\\dots ,\\sigma _m$ .", "A formula $\\big \\lbrace \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\big \\rbrace \\models F$ represents that $F$ holds at point $i$ on two sets of trajectories $\\sigma _1,\\dots ,\\sigma _m$ and $\\lbrace \\bar{\\sigma }_1,\\dots ,\\bar{\\sigma }_{\\bar{m}} \\rbrace $ .", "A formula $(\\langle s_i,\\sigma \\rangle ,j) \\models F$ represents that $F$ holds in the interval $[i,j]$ on trajectory $\\sigma $ .", "A formula $(\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models F$ represents that $F$ holds in the interval $[i,j]$ on a set of trajectories $\\sigma _1,\\dots ,\\sigma _m$ .", "A formula $(\\big \\lbrace \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\big \\rbrace , j) \\models F$ represents that $F$ holds in the interval $[i,j]$ over two sets of trajectories $\\lbrace \\sigma _1,\\dots , \\sigma _m \\rbrace $ and $\\lbrace \\bar{\\sigma }_1,\\dots ,\\bar{\\sigma }_{\\bar{m}} \\rbrace $ .", "Given a domain description $\\mathbf {D}$ with simple fluents represented by a Guarded-Arc Petri Net as defined in definition REF .", "Let $\\sigma = s_0,T_0,s_1,\\dots ,T_{k-1},s_{k}$ be its trajectory as defined in (REF ), and $s_i$ be a state on that trajectory.", "Let actions $T_i$ firing in state $s_i$ be observable in $s_i$ such that $T_i \\subseteq s_i$ .", "First we define how interval formulas are satisfied on a trajectory $\\sigma $ , starting state $s_i$ and an ending point $j$ : $(\\langle s_i,\\sigma \\rangle , j) \\models & \\mathbf {~rate~of~production~of~} f \\mathbf {~is~} n \\nonumber \\\\&\\text{ if } n=(s_j(f)-s_i(f))/(j-i)\\\\(\\langle s_i,\\sigma \\rangle , j) \\models & \\mathbf {~rate~of~firing~of~} a \\mathbf {~is~} n \\nonumber \\\\&\\text{ if } n=\\sum _{i\\le k \\le j, \\langle s_k,\\sigma \\rangle \\models a \\mathbf {~occurs~}}{1}/(j-i)\\\\(\\langle s_i,\\sigma \\rangle , j) \\models & \\mathbf {~total~production~of~} f \\mathbf {~is~} n \\nonumber \\\\&\\text{ if } n=(s_j(f)-s_i(f))\\\\(\\langle s_i,\\sigma \\rangle , j) \\models & f \\mathbf {~is~accumulating~} \\nonumber \\\\&\\text{ if } (\\nexists k, i \\le k \\le j : s_{k+1}(f) < s_k(f)) \\text{ and } s_j(f) > s_i(f)\\\\(\\langle s_i,\\sigma \\rangle , j) \\models & f \\mathbf {~is~decreasing~} \\nonumber \\\\&\\text{ if } (\\nexists k, i \\le k \\le j : s_{k+1}(f) > s_k(f)) \\text{ and } s_j(f) < s_i(f)$ Next we define how a point formula is satisfied on a trajectory $\\sigma $ , in a state $s_i$ : $\\langle s_i,\\sigma \\rangle \\models & \\mathbf {~value~of~} f \\mathbf {~is~higher~than~} n \\nonumber \\\\&\\text{ if } s_i(f) > n\\\\\\langle s_i,\\sigma \\rangle \\models & \\mathbf {~value~of~} f \\mathbf {~is~lower~than~} n \\nonumber \\\\&\\text{ if } s_i(f) < n\\\\\\langle s_i,\\sigma \\rangle \\models & \\mathbf {~value~of~} f \\mathbf {~is~} n \\nonumber \\\\&\\text{ if } s_i(f) = n\\\\\\langle s_i,\\sigma \\rangle \\models & a \\mathbf {~occurs~} \\nonumber \\\\&\\text{ if } a \\in s_i\\\\\\langle s_i,\\sigma \\rangle \\models & a \\mathbf {~does~not~occur~} \\nonumber \\\\&\\text{ if } a \\notin s_i\\\\\\langle s_i,\\sigma \\rangle \\models & a_1 \\mathbf {~switches~to~} a_2 \\nonumber \\\\&\\text{ if } a_1 \\in s_{i-1} \\text{ and } a_2 \\notin s_{i-1} \\text{ and } a_1 \\notin s_i \\text{ and } a_2 \\in s_i$ Next we define how a quantitative all interval formula is satisfied on a set of trajectories $\\sigma _1,\\dots ,\\sigma _m$ with starting states $s^1_i,\\dots ,s^m_i$ and end point $j$ : $(\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models & \\mathbf {~rates~of~production~of~} f \\mathbf {~are~} [r_1,\\dots ,r_m] \\nonumber \\\\&\\text{ if } (\\langle s^1_i,\\sigma _1 \\rangle , j) \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~is~} r_1\\nonumber \\\\&\\vdots \\nonumber \\\\&\\text{ and } (\\langle s^m_i,\\sigma _m \\rangle , j) \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~is~} r_m \\\\(\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models & \\mathbf {~rates~of~firing~of~} a \\mathbf {~are~} [r_1,\\dots ,r_m] \\nonumber \\\\&\\text{ if } (\\langle s^1_i,\\sigma _1 \\rangle , j) \\models \\mathbf {~rate~of~firing~of~} a \\mathbf {~is~} r_1\\nonumber \\\\&\\vdots \\nonumber \\\\&\\text{ and } (\\langle s^m_i,\\sigma _m \\rangle , j) \\models \\mathbf {~rate~of~firing~of~} a \\mathbf {~is~} r_m \\\\(\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models & \\mathbf {~totals~of~production~of~} f \\mathbf {~are~} [r_1,\\dots ,r_m] \\nonumber \\\\&\\text{ if } (\\langle s^1_i,\\sigma _1 \\rangle , j) \\models \\mathbf {~total~production~of~} f \\mathbf {~is~} r_1\\nonumber \\\\&\\vdots \\nonumber \\\\&\\text{ and } (\\langle s^m_i,\\sigma _m \\rangle , j) \\models \\mathbf {~total~production~of~} f \\mathbf {~is~} r_m$ Next we define how a quantitative all point formula is satisfied on a set of trajectories $\\sigma _1,\\dots ,\\sigma _m$ in states $s^1_i,\\dots ,s^m_i$ : $\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace \\models & \\mathbf {~values~of~} f \\mathbf {~are~} [r_1,\\dots ,r_m] \\nonumber \\\\&\\text{ if } \\langle s^1_i,\\sigma _1 \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~is~} r_1\\nonumber \\\\&\\vdots \\nonumber \\\\&\\text{ and } \\langle s^m_i,\\sigma _m \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~is~} r_m$ Next we define how a quantitative aggregate interval formula is satisfied on a set of trajectories $\\sigma _1,\\dots ,\\sigma _m$ with starting states $s^1_i,\\dots ,s^m_i$ and end point $j$ : $(\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models & \\; average \\mathbf {~rate~of~production~of~} f \\mathbf {~is~} r \\nonumber \\\\&\\text{ if } \\exists [r_1,\\dots ,r_m] : \\nonumber \\\\&~~~~~~(\\langle s^1_i,\\sigma _1 \\rangle , j) \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~is~} r_1\\nonumber \\\\&~~~~~~\\vdots \\nonumber \\\\&~~~~~~(\\langle s^m_i,\\sigma _m \\rangle , j) \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~is~} r_m \\nonumber \\\\&\\text{ and } r=(r_1+\\dots +r_m)/m\\\\(\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models & \\; minimum \\mathbf {~rate~of~production~of~} f \\mathbf {~is~} r \\nonumber \\\\&\\text{ if } \\exists [r_1,\\dots ,r_m] : \\nonumber \\\\&~~~~~~(\\langle s^1_i,\\sigma _1 \\rangle , j) \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~is~} r_1\\nonumber \\\\&~~~~~~\\vdots \\nonumber \\\\&~~~~~~(\\langle s^m_i,\\sigma _m \\rangle , j) \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~is~} r_m \\nonumber \\\\&\\text{ and } \\exists k, 1 \\le k \\le m : r=r_k \\text{ and } \\forall x, 1 \\le x \\le m, r_k \\le r_x\\\\(\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models & \\; maximum \\mathbf {~rate~of~production~of~} f \\mathbf {~is~} r \\nonumber \\\\&\\text{ if } \\exists [r_1,\\dots ,r_m] : \\nonumber \\\\&~~~~~~(\\langle s^1_i,\\sigma _1 \\rangle , j) \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~is~} r_1\\nonumber \\\\&~~~~~~\\vdots \\nonumber \\\\&~~~~~~(\\langle s^m_i,\\sigma _m \\rangle , j) \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~is~} r_m \\nonumber \\\\&\\text{ and } \\exists k, 1 \\le k \\le m : r=r_k \\text{ and } \\forall x, 1 \\le x \\le m, r_k \\ge r_x\\\\\\vspace{20.0pt}\\nonumber \\\\(\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models & \\; average \\mathbf {~rate~of~firing~of~} f \\mathbf {~is~} r \\nonumber \\\\&\\text{ iff } \\exists [r_1,\\dots ,r_m] : \\nonumber \\\\&~~~~~~(\\langle s^1_i,\\sigma _1 \\rangle , j) \\models \\mathbf {~rate~of~firing~of~} f \\mathbf {~is~} r_1\\nonumber \\\\&~~~~~~\\vdots \\nonumber \\\\&~~~~~~(\\langle s^m_i,\\sigma _m \\rangle , j) \\models \\mathbf {~rate~of~firing~of~} f \\mathbf {~is~} r_m \\nonumber \\\\&\\text{ and } r=(r_1+\\dots +r_m)/m\\\\(\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models & \\; minimum \\mathbf {~rate~of~firing~of~} f \\mathbf {~is~} r \\nonumber \\\\&\\text{ iff } \\exists [r_1,\\dots ,r_m] : \\nonumber \\\\&~~~~~~(\\langle s^1_i,\\sigma _1 \\rangle , j) \\models \\mathbf {~rate~of~firing~of~} f \\mathbf {~is~} r_1\\nonumber \\\\&~~~~~~\\vdots \\nonumber \\\\&~~~~~~(\\langle s^m_i,\\sigma _m \\rangle , j) \\models \\mathbf {~rate~of~firing~of~} f \\mathbf {~is~} r_m \\nonumber \\\\&\\text{ and } \\exists k, 1 \\le k \\le m : r=r_k \\text{ and } \\forall x, 1 \\le x \\le m, r_k \\le r_x\\\\(\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models & \\; maximum \\mathbf {~rate~of~firing~of~} f \\mathbf {~is~} r \\nonumber \\\\&\\text{ iff } \\exists [r_1,\\dots ,r_m] : \\nonumber \\\\&~~~~~~(\\langle s^1_i,\\sigma _1 \\rangle , j) \\models \\mathbf {~rate~of~firing~of~} f \\mathbf {~is~} r_1\\nonumber \\\\&~~~~~~\\vdots \\nonumber \\\\&~~~~~~(\\langle s^m_i,\\sigma _m \\rangle , j) \\models \\mathbf {~rate~of~firing~of~} f \\mathbf {~is~} r_m \\nonumber \\\\&\\text{ and } \\exists k, 1 \\le k \\le m : r=r_k \\text{ and } \\forall x, 1 \\le x \\le m, r_k \\ge r_x\\\\\\vspace{20.0pt}\\nonumber \\\\(\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models & \\; average \\mathbf {~total~production~of~} f \\mathbf {~is~} r \\nonumber \\\\&\\text{ if } \\exists [r_1,\\dots ,r_m] : \\nonumber \\\\&~~~~~~(\\langle s^1_i,\\sigma _1 \\rangle , j) \\models \\mathbf {~total~production~of~} f \\mathbf {~is~} r_1\\nonumber \\\\&~~~~~~\\vdots \\nonumber \\\\&~~~~~~(\\langle s^m_i,\\sigma _m \\rangle , j) \\models \\mathbf {~total~production~of~} f \\mathbf {~is~} r_m \\nonumber \\\\&\\text{ and } r=(r_1+\\dots +r_m)/m\\\\(\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models & \\; minimum \\mathbf {~total~production~of~} f \\mathbf {~is~} r \\nonumber \\\\&\\text{ if } \\exists [r_1,\\dots ,r_m] : \\nonumber \\\\&~~~~~~(\\langle s^1_i,\\sigma _1 \\rangle , j) \\models \\mathbf {~total~production~of~} f \\mathbf {~is~} r_1\\nonumber \\\\&~~~~~~\\vdots \\nonumber \\\\&~~~~~~(\\langle s^m_i,\\sigma _m \\rangle , j) \\models \\mathbf {~total~production~of~} f \\mathbf {~is~} r_m \\nonumber \\\\&\\text{ and } \\exists k, 1 \\le k \\le m : r=r_k \\text{ and } \\forall x, 1 \\le x \\le m, r_k \\le r_x\\\\(\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models & \\; maximum \\mathbf {~total~production~of~} f \\mathbf {~is~} r \\nonumber \\\\&\\text{ if } \\exists [r_1,\\dots ,r_m] : \\nonumber \\\\&~~~~~~(\\langle s^1_i,\\sigma _1 \\rangle , j) \\models \\mathbf {~total~production~of~} f \\mathbf {~is~} r_1\\nonumber \\\\&~~~~~~\\vdots \\nonumber \\\\&~~~~~~(\\langle s^m_i,\\sigma _m \\rangle , j) \\models \\mathbf {~total~production~of~} f \\mathbf {~is~} r_m \\nonumber \\\\&\\text{ and } \\exists k, 1 \\le k \\le m : r=r_k \\text{ and } \\forall x, 1 \\le x \\le m, r_k \\ge r_x$ Next we define how a quantitative aggregate point formula is satisfied on a set of trajectories $\\sigma _1,\\dots ,\\sigma _m$ in states $s^1_i,\\dots ,s^m_i$ : $\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace \\models & \\; average \\mathbf {~value~of~} f \\mathbf {~is~} r \\nonumber \\\\&\\text{ if } \\exists [r_1,\\dots ,r_m] : \\nonumber \\\\&~~~~~~\\langle s^1_i,\\sigma _1 \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~is~} r_1\\nonumber \\\\&~~~~~~\\vdots \\nonumber \\\\&~~~~~~\\langle s^m_i,\\sigma _m \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~is~} r_m \\nonumber \\\\&\\text{ and } r=(r_1+\\dots +r_m)/m\\\\\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace \\models & \\; minimum \\mathbf {~value~of~} f \\mathbf {~is~} r \\nonumber \\\\&\\text{ if } \\exists [r_1,\\dots ,r_m] : \\nonumber \\\\&~~~~~~\\langle s^1_i,\\sigma _1 \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~is~} r_1\\nonumber \\\\&~~~~~~\\vdots \\nonumber \\\\&~~~~~~\\langle s^m_i,\\sigma _m \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~is~} r_m \\nonumber \\\\&\\text{ and } \\exists k, 1 \\le k \\le m : r=r_k \\text{ and } \\forall x, 1 \\le x \\le m, r_k \\le r_x\\\\\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace \\models & \\; maximum \\mathbf {~value~of~} f \\mathbf {~is~} r \\nonumber \\\\&\\text{ if } \\exists [r_1,\\dots ,r_m] : \\nonumber \\\\&~~~~~~\\langle s^1_i,\\sigma _1 \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~is~} r_1\\nonumber \\\\&~~~~~~\\vdots \\nonumber \\\\&~~~~~~\\langle s^m_i,\\sigma _m \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~is~} r_m \\nonumber \\\\&\\text{ and } \\exists k, 1 \\le k \\le m : r=r_k \\text{ and } \\forall x, 1 \\le x \\le m, r_k \\ge r_x$ Next we define how a comparative quantitative aggregate interval formula is satisfied on two sets of trajectories $\\sigma _1,\\dots ,\\sigma _m$ , and $\\bar{\\sigma }_1,\\dots ,\\bar{\\sigma }_{\\bar{m}}$ a starting point $i$ and an ending point $j$ : $&\\Big (\\Big \\lbrace \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace , j \\Big ) \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\models \\; \\mathbf {~direction~of~change~in~} average \\mathbf {~rate~of~production~of~} f \\mathbf {~is~} d \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ if } \\exists \\; n_1 : (\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models \\; average \\mathbf {~rate~of~production~of~} f \\text{~is~} n_1 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } \\exists \\; n_2 : (\\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace , j) \\models \\; average \\mathbf {~rate~of~production~of~} f \\text{~is~} n_2 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } n_2 \\; d \\; n_1\\\\&\\Big (\\Big \\lbrace \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace , j \\Big ) \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\models \\; \\mathbf {~direction~of~change~in~} minimum \\mathbf {~rate~of~production~of~} f \\mathbf {~is~} d \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ if } \\exists \\; n_1 : (\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models \\; minimum \\mathbf {~rate~of~production~of~} f \\text{~is~} n_1 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } \\exists \\; n_2 : (\\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace , j) \\models \\; minimum \\mathbf {~rate~of~production~of~} f \\text{~is~} n_2 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } n_2 \\; d \\; n_1\\\\&\\Big (\\Big \\lbrace \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace , j \\Big ) \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\models \\; \\mathbf {~direction~of~change~in~} maximum \\mathbf {~rate~of~production~of~} f \\mathbf {~is~} d \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ if } \\exists \\; n_1 : (\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models \\; maximum \\mathbf {~rate~of~production~of~} f \\text{~is~} n_1 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } \\exists \\; n_2 : (\\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace , j) \\models \\; maximum \\mathbf {~rate~of~production~of~} f \\text{~is~} n_2 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } n_2 \\; d \\; n_1\\\\\\vspace{20.0pt}\\nonumber \\\\&\\Big (\\Big \\lbrace \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace , j \\Big ) \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\models \\; \\mathbf {~direction~of~change~in~} average \\mathbf {~rate~of~firing~of~} f \\mathbf {~is~} d \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ if } \\exists \\; n_1 : (\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models \\; average \\mathbf {~rate~of~firing~of~} f \\text{~is~} n_1 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } \\exists \\; n_2 : (\\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace , j) \\models \\; average \\mathbf {~rate~of~firing~of~} f \\text{~is~} n_2 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } n_2 \\; d \\; n_1\\\\&\\Big (\\Big \\lbrace \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace , j \\Big ) \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\models \\; \\mathbf {~direction~of~change~in~} minimum \\mathbf {~rate~of~firing~of~} f \\mathbf {~is~} d \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ if } \\exists \\; n_1 : (\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models \\; minimum \\mathbf {~rate~of~firing~of~} f \\text{~is~} n_1 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } \\exists \\; n_2 : (\\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace , j) \\models \\; minimum \\mathbf {~rate~of~firing~of~} f \\text{~is~} n_2 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } n_2 \\; d \\; n_1\\\\&\\Big (\\Big \\lbrace \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace , j \\Big ) \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\models \\; \\mathbf {~direction~of~change~in~} maximum \\mathbf {~rate~of~firing~of~} f \\mathbf {~is~} d \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ if } \\exists \\; n_1 : (\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models \\; maximum \\mathbf {~rate~of~firing~of~} f \\text{~is~} n_1 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } \\exists \\; n_2 : (\\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace , j) \\models \\; maximum \\mathbf {~rate~of~firing~of~} f \\text{~is~} n_2 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } n_2 \\; d \\; n_1\\\\\\vspace{20.0pt}\\nonumber \\\\&\\Big (\\Big \\lbrace \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace , j \\Big ) \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\models \\; \\mathbf {~direction~of~change~in~} average \\mathbf {~total~production~of~} f \\mathbf {~is~} d \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ if } \\exists \\; n_1 : (\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models \\; average \\mathbf {~total~production~of~} f \\text{~is~} n_1 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } \\exists \\; n_2 : (\\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace , j) \\models \\; average \\mathbf {~total~production~of~} f \\text{~is~} n_2 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } n_2 \\; d \\; n_1\\\\&\\Big (\\Big \\lbrace \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace , j \\Big ) \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\models \\; \\mathbf {~direction~of~change~in~} minimum \\mathbf {~total~production~of~} f \\mathbf {~is~} d \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ if } \\exists \\; n_1 : (\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models \\; minimum \\mathbf {~total~production~of~} f \\text{~is~} n_1 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } \\exists \\; n_2 : (\\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace , j) \\models \\; minimum \\mathbf {~total~production~of~} f \\text{~is~} n_2 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } n_2 \\; d \\; n_1\\\\&\\Big (\\Big \\lbrace \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace , j \\Big ) \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\models \\; \\mathbf {~direction~of~change~in~} maximum \\mathbf {~total~production~of~} f \\mathbf {~is~} d \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ if } \\exists \\; n_1 : (\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models \\; maximum \\mathbf {~total~production~of~} f \\text{~is~} n_1 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } \\exists \\; n_2 : (\\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace , j) \\models \\; maximum \\mathbf {~total~production~of~} f \\text{~is~} n_2 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } n_2 \\; d \\; n_1$ Next we define how a comparative quantitative aggregate point formula is satisfied on two sets of trajectories $\\sigma _1,\\dots ,\\sigma _m$ , and $\\bar{\\sigma }_1,\\dots ,\\bar{\\sigma }_{\\bar{m}}$ at point $i$ : $&\\Big \\lbrace \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\models \\; \\mathbf {~direction~of~change~in~} average \\mathbf {~value~of~} f \\mathbf {~is~} d \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ if } \\exists \\; n_1 : \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace \\models \\; average \\mathbf {~value~of~} f \\text{~is~} n_1 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } \\exists \\; n_2 : \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\models \\; average \\mathbf {~value~of~} f \\text{~is~} n_2 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } n_2 \\; d \\; n_1\\\\&\\Big \\lbrace \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\models \\; \\mathbf {~direction~of~change~in~} minimum \\mathbf {~value~of~} f \\mathbf {~is~} d \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ if } \\exists \\; n_1 : \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace \\models \\; minimum \\mathbf {~value~of~} f \\text{~is~} n_1 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } \\exists \\; n_2 : \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\models \\; minimum \\mathbf {~value~of~} f \\text{~is~} n_2 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } n_2 \\; d \\; n_1\\\\&\\Big \\lbrace \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\models \\; \\mathbf {~direction~of~change~in~} maximum \\mathbf {~value~of~} f \\mathbf {~is~} d \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ if } \\exists \\; n_1 : \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace \\models \\; maximum \\mathbf {~value~of~} f \\text{~is~} n_1 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } \\exists \\; n_2 : \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\models \\; maximum \\mathbf {~value~of~} f \\text{~is~} n_2 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } n_2 \\; d \\; n_1$ Given a domain description $\\mathbf {D}$ with simple fluents represented by a Guarded-Arc Petri Net with Colored tokens as defined in definition REF .", "Let $\\sigma = s_0,T_0,s_1,\\dots ,T_{k-1},s_k$ be its trajectory as defined in definition REF , and $s_i$ be a state on that trajectory.", "Let actions $T_i$ firing in state $s_i$ be observable in $s_i$ such that $T_i \\subseteq s_i$ .", "We define observation semantics using LTL below.", "We will use $s_i(f[l])$ to represent $m_{s_i(l)}(f)$ (multiplicity / value of $f$ in location $l$ ) in state $s_i$ .", "First we define how interval formulas are satisfied on a trajectory $\\sigma $ , starting state $s_i$ and an ending point $j$ : $(\\langle s_i,\\sigma \\rangle , j) \\models & \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n \\nonumber \\\\&\\text{ if } n=(s_j(f[l])-s_i(f[l]))/(j-i)\\\\(\\langle s_i,\\sigma \\rangle , j) \\models & \\mathbf {~rate~of~firing~of~} a \\mathbf {~is~} n \\nonumber \\\\&\\text{ if } n=\\sum _{i\\le k \\le j, \\langle s_k,\\sigma \\rangle \\models a \\mathbf {~occurs~}}{1}/(j-i)\\\\(\\langle s_i,\\sigma \\rangle , j) \\models & \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n \\nonumber \\\\&\\text{ if } n=(s_j(f[l])-s_i(f[l]))\\\\(\\langle s_i,\\sigma \\rangle , j) \\models & f \\mathbf {~is~accumulating~} \\mathbf {~atloc~} l \\nonumber \\\\&\\text{ if } (\\nexists k, i \\le k \\le j : s_{k+1}(f[l]) < s_k(f[l])) \\text{ and } s_j(f[l]) > s_i(f[l])\\\\(\\langle s_i,\\sigma \\rangle , j) \\models & f \\mathbf {~is~decreasing~} \\mathbf {~atloc~} l \\nonumber \\\\&\\text{ if } (\\nexists k, i \\le k \\le j : s_{k+1}(f[l]) > s_k(f[l])) \\text{ and } s_j(f[l]) < s_i(f[l])$ Next we define how a point formula is satisfied on a trajectory $\\sigma $ , in a state $s_i$ : $\\langle s_i,\\sigma \\rangle \\models & \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~higher~than~} n \\nonumber \\\\&\\text{ if } s_i(f[l]) > n\\\\\\langle s_i,\\sigma \\rangle \\models & \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~lower~than~} n \\nonumber \\\\&\\text{ if } s_i(f[l]) < n\\\\\\langle s_i,\\sigma \\rangle \\models & \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n \\nonumber \\\\&\\text{ if } s_i(f[l]) = n\\\\\\langle s_i,\\sigma \\rangle \\models & a \\mathbf {~occurs~} \\nonumber \\\\&\\text{ if } a \\in s_i\\\\\\langle s_i,\\sigma \\rangle \\models & a \\mathbf {~does~not~occur~} \\nonumber \\\\&\\text{ if } a \\notin s_i\\\\\\langle s_i,\\sigma \\rangle \\models & a_1 \\mathbf {~switches~to~} a_2 \\nonumber \\\\&\\text{ if } a_1 \\in s_{i-1} \\text{ and } a_2 \\notin s_{i-1} \\text{ and } a_1 \\notin s_i \\text{ and } a_2 \\in s_i$ Next we define how a quantitative all interval formula is satisfied on a set of trajectories $\\sigma _1,\\dots ,\\sigma _m$ with starting states $s^1_i,\\dots ,s^m_i$ and end point $j$ : $(\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models & \\mathbf {~rates~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~are~} [r_1,\\dots ,r_m] \\nonumber \\\\&\\text{ if } (\\langle s^1_i,\\sigma _1 \\rangle , j) \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} r_1\\nonumber \\\\&\\vdots \\nonumber \\\\&\\text{ and } (\\langle s^m_i,\\sigma _m \\rangle , j) \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} r_m \\\\(\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models & \\mathbf {~rates~of~firing~of~} a \\mathbf {~are~} [r_1,\\dots ,r_m] \\nonumber \\\\&\\text{ if } (\\langle s^1_i,\\sigma _1 \\rangle , j) \\models \\mathbf {~rate~of~firing~of~} a \\mathbf {~is~} r_1\\nonumber \\\\&\\vdots \\nonumber \\\\&\\text{ and } (\\langle s^m_i,\\sigma _m \\rangle , j) \\models \\mathbf {~rate~of~firing~of~} a \\mathbf {~is~} r_m \\\\(\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models & \\mathbf {~totals~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~are~} [r_1,\\dots ,r_m] \\nonumber \\\\&\\text{ if } (\\langle s^1_i,\\sigma _1 \\rangle , j) \\models \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} r_1\\nonumber \\\\&\\vdots \\nonumber \\\\&\\text{ and } (\\langle s^m_i,\\sigma _m \\rangle , j) \\models \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} r_m$ Next we define how a quantitative all point formula is satisfied w.r.t.", "a set of trajectories $\\sigma _1,\\dots ,\\sigma _m$ in states $s^1_i,\\dots ,s^m_i$ : $\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace \\models & \\mathbf {~values~of~} f \\mathbf {~atloc~} l\\mathbf {~are~} [r_1,\\dots ,r_m] \\nonumber \\\\&\\text{ if } \\langle s^1_i,\\sigma _1 \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} r_1\\nonumber \\\\&\\vdots \\nonumber \\\\&\\text{ and } \\langle s^m_i,\\sigma _m \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} r_m$ Next we define how a quantitative aggregate interval formula is satisfied w.r.t.", "a set of trajectories $\\sigma _1,\\dots ,\\sigma _m$ with starting states $s^1_i,\\dots ,s^m_i$ and end point $j$ : $(\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models & \\; average \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} r \\nonumber \\\\&\\text{ if } \\exists [r_1,\\dots ,r_m] : \\nonumber \\\\&~~~~~~(\\langle s^1_i,\\sigma _1 \\rangle , j) \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} r_1\\nonumber \\\\&~~~~~~\\vdots \\nonumber \\\\&~~~~~~(\\langle s^m_i,\\sigma _m \\rangle , j) \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} r_m \\nonumber \\\\&\\text{ and } r=(r_1+\\dots +r_m)/m\\\\(\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models & \\; minimum \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} r \\nonumber \\\\&\\text{ if } \\exists [r_1,\\dots ,r_m] : \\nonumber \\\\&~~~~~~(\\langle s^1_i,\\sigma _1 \\rangle , j) \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} r_1\\nonumber \\\\&~~~~~~\\vdots \\nonumber \\\\&~~~~~~(\\langle s^m_i,\\sigma _m \\rangle , j) \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} r_m \\nonumber \\\\&\\text{ and } \\exists k, 1 \\le k \\le m : r=r_k \\text{ and } \\forall x, 1 \\le x \\le m, r_k \\le r_x\\\\(\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models & \\; maximum \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} r \\nonumber \\\\&\\text{ if } \\exists [r_1,\\dots ,r_m] : \\nonumber \\\\&~~~~~~(\\langle s^1_i,\\sigma _1 \\rangle , j) \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} r_1\\nonumber \\\\&~~~~~~\\vdots \\nonumber \\\\&~~~~~~(\\langle s^m_i,\\sigma _m \\rangle , j) \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} r_m \\nonumber \\\\&\\text{ and } \\exists k, 1 \\le k \\le m : r=r_k \\text{ and } \\forall x, 1 \\le x \\le m, r_k \\ge r_x\\\\\\vspace{20.0pt}\\nonumber \\\\(\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models & \\; average \\mathbf {~rate~of~firing~of~} f \\mathbf {~is~} r \\nonumber \\\\&\\text{ iff } \\exists [r_1,\\dots ,r_m] : \\nonumber \\\\&~~~~~~(\\langle s^1_i,\\sigma _1 \\rangle , j) \\models \\mathbf {~rate~of~firing~of~} f \\mathbf {~is~} r_1\\nonumber \\\\&~~~~~~\\vdots \\nonumber \\\\&~~~~~~(\\langle s^m_i,\\sigma _m \\rangle , j) \\models \\mathbf {~rate~of~firing~of~} f \\mathbf {~is~} r_m \\nonumber \\\\&\\text{ and } r=(r_1+\\dots +r_m)/m\\\\(\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models & \\; minimum \\mathbf {~rate~of~firing~of~} f \\mathbf {~is~} r \\nonumber \\\\&\\text{ iff } \\exists [r_1,\\dots ,r_m] : \\nonumber \\\\&~~~~~~(\\langle s^1_i,\\sigma _1 \\rangle , j) \\models \\mathbf {~rate~of~firing~of~} f \\mathbf {~is~} r_1\\nonumber \\\\&~~~~~~\\vdots \\nonumber \\\\&~~~~~~(\\langle s^m_i,\\sigma _m \\rangle , j) \\models \\mathbf {~rate~of~firing~of~} f \\mathbf {~is~} r_m \\nonumber \\\\&\\text{ and } \\exists k, 1 \\le k \\le m : r=r_k \\text{ and } \\forall x, 1 \\le x \\le m, r_k \\le r_x\\\\(\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models & \\; maximum \\mathbf {~rate~of~firing~of~} f \\mathbf {~is~} r \\nonumber \\\\&\\text{ iff } \\exists [r_1,\\dots ,r_m] : \\nonumber \\\\&~~~~~~(\\langle s^1_i,\\sigma _1 \\rangle , j) \\models \\mathbf {~rate~of~firing~of~} f \\mathbf {~is~} r_1\\nonumber \\\\&~~~~~~\\vdots \\nonumber \\\\&~~~~~~(\\langle s^m_i,\\sigma _m \\rangle , j) \\models \\mathbf {~rate~of~firing~of~} f \\mathbf {~is~} r_m \\nonumber \\\\&\\text{ and } \\exists k, 1 \\le k \\le m : r=r_k \\text{ and } \\forall x, 1 \\le x \\le m, r_k \\ge r_x\\\\\\vspace{20.0pt}\\nonumber \\\\(\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models & \\; average \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} r \\nonumber \\\\&\\text{ if } \\exists [r_1,\\dots ,r_m] : \\nonumber \\\\&~~~~~~(\\langle s^1_i,\\sigma _1 \\rangle , j) \\models \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} r_1\\nonumber \\\\&~~~~~~\\vdots \\nonumber \\\\&~~~~~~(\\langle s^m_i,\\sigma _m \\rangle , j) \\models \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} r_m \\nonumber \\\\&\\text{ and } r=(r_1+\\dots +r_m)/m\\\\(\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models & \\; minimum \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} r \\nonumber \\\\&\\text{ if } \\exists [r_1,\\dots ,r_m] : \\nonumber \\\\&~~~~~~(\\langle s^1_i,\\sigma _1 \\rangle , j) \\models \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} r_1\\nonumber \\\\&~~~~~~\\vdots \\nonumber \\\\&~~~~~~(\\langle s^m_i,\\sigma _m \\rangle , j) \\models \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} r_m \\nonumber \\\\&\\text{ and } \\exists k, 1 \\le k \\le m : r=r_k \\text{ and } \\forall x, 1 \\le x \\le m, r_k \\le r_x\\\\(\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models & \\; maximum \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} r \\nonumber \\\\&\\text{ if } \\exists [r_1,\\dots ,r_m] : \\nonumber \\\\&~~~~~~(\\langle s^1_i,\\sigma _1 \\rangle , j) \\models \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} r_1\\nonumber \\\\&~~~~~~\\vdots \\nonumber \\\\&~~~~~~(\\langle s^m_i,\\sigma _m \\rangle , j) \\models \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} r_m \\nonumber \\\\&\\text{ and } \\exists k, 1 \\le k \\le m : r=r_k \\text{ and } \\forall x, 1 \\le x \\le m, r_k \\ge r_x$ Next we define how a quantitative aggregate point formula is satisfied w.r.t.", "a set of trajectories $\\sigma _1,\\dots ,\\sigma _m$ in states $s^1_i,\\dots ,s^m_i$ : $\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace \\models & \\; average \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} r \\nonumber \\\\&\\text{ if } \\exists [r_1,\\dots ,r_m] : \\nonumber \\\\&~~~~~~\\langle s^1_i,\\sigma _1 \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} r_1\\nonumber \\\\&~~~~~~\\vdots \\nonumber \\\\&~~~~~~\\langle s^m_i,\\sigma _m \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} r_m \\nonumber \\\\&\\text{ and } r=(r_1+\\dots +r_m)/m\\\\\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace \\models & \\; minimum \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} r \\nonumber \\\\&\\text{ if } \\exists [r_1,\\dots ,r_m] : \\nonumber \\\\&~~~~~~\\langle s^1_i,\\sigma _1 \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} r_1\\nonumber \\\\&~~~~~~\\vdots \\nonumber \\\\&~~~~~~\\langle s^m_i,\\sigma _m \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} r_m \\nonumber \\\\&\\text{ and } \\exists k, 1 \\le k \\le m : r=r_k \\text{ and } \\forall x, 1 \\le x \\le m, r_k \\le r_x\\\\\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace \\models & \\; maximum \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} r \\nonumber \\\\&\\text{ if } \\exists [r_1,\\dots ,r_m] : \\nonumber \\\\&~~~~~~\\langle s^1_i,\\sigma _1 \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} r_1\\nonumber \\\\&~~~~~~\\vdots \\nonumber \\\\&~~~~~~\\langle s^m_i,\\sigma _m \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} r_m \\nonumber \\\\&\\text{ and } \\exists k, 1 \\le k \\le m : r=r_k \\text{ and } \\forall x, 1 \\le x \\le m, r_k \\ge r_x$ Next we define how a comparative quantitative aggregate interval formula is satisfied w.r.t.", "two sets of trajectories $\\sigma _1,\\dots ,\\sigma _m$ , and $\\bar{\\sigma }_1,\\dots ,\\bar{\\sigma }_{\\bar{m}}$ a starting point $i$ and an ending point $j$ : $&\\Big (\\Big \\lbrace \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace , j \\Big ) \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\models \\; \\mathbf {~direction~of~change~in~} average \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} d \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ if } \\exists \\; n_1 : (\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models \\; average \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\text{~is~} n_1 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } \\exists \\; n_2 : (\\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace , j) \\models \\; average \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\text{~is~} n_2 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } n_2 \\; d \\; n_1\\\\&\\Big (\\Big \\lbrace \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace , j \\Big ) \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\models \\; \\mathbf {~direction~of~change~in~} minimum \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} d \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ if } \\exists \\; n_1 : (\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models \\; minimum \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\text{~is~} n_1 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } \\exists \\; n_2 : (\\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace , j) \\models \\; minimum \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\text{~is~} n_2 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } n_2 \\; d \\; n_1\\\\&\\Big (\\Big \\lbrace \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace , j \\Big ) \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\models \\; \\mathbf {~direction~of~change~in~} maximum \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} d \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ if } \\exists \\; n_1 : (\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models \\; maximum \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\text{~is~} n_1 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } \\exists \\; n_2 : (\\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace , j) \\models \\; maximum \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\text{~is~} n_2 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } n_2 \\; d \\; n_1\\\\\\vspace{20.0pt}\\nonumber \\\\&\\Big (\\Big \\lbrace \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace , j \\Big ) \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\models \\; \\mathbf {~direction~of~change~in~} average \\mathbf {~rate~of~firing~of~} f \\mathbf {~is~} d \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ if } \\exists \\; n_1 : (\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models \\; average \\mathbf {~rate~of~firing~of~} f \\text{~is~} n_1 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } \\exists \\; n_2 : (\\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace , j) \\models \\; average \\mathbf {~rate~of~firing~of~} f \\text{~is~} n_2 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } n_2 \\; d \\; n_1\\\\&\\Big (\\Big \\lbrace \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace , j \\Big ) \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\models \\; \\mathbf {~direction~of~change~in~} minimum \\mathbf {~rate~of~firing~of~} f \\mathbf {~is~} d \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ if } \\exists \\; n_1 : (\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models \\; minimum \\mathbf {~rate~of~firing~of~} f \\text{~is~} n_1 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } \\exists \\; n_2 : (\\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace , j) \\models \\; minimum \\mathbf {~rate~of~firing~of~} f \\text{~is~} n_2 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } n_2 \\; d \\; n_1\\\\&\\Big (\\Big \\lbrace \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace , j \\Big ) \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\models \\; \\mathbf {~direction~of~change~in~} maximum \\mathbf {~rate~of~firing~of~} f \\mathbf {~is~} d \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ if } \\exists \\; n_1 : (\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models \\; maximum \\mathbf {~rate~of~firing~of~} f \\text{~is~} n_1 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } \\exists \\; n_2 : (\\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace , j) \\models \\; maximum \\mathbf {~rate~of~firing~of~} f \\text{~is~} n_2 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } n_2 \\; d \\; n_1\\\\\\vspace{20.0pt}\\nonumber \\\\&\\Big (\\Big \\lbrace \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace , j \\Big ) \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\models \\; \\mathbf {~direction~of~change~in~} average \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} d \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ if } \\exists \\; n_1 : (\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models \\; average \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\text{~is~} n_1 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } \\exists \\; n_2 : (\\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace , j) \\models \\; average \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\text{~is~} n_2 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } n_2 \\; d \\; n_1\\\\&\\Big (\\Big \\lbrace \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace , j \\Big ) \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\models \\; \\mathbf {~direction~of~change~in~} minimum \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} d \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ if } \\exists \\; n_1 : (\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models \\; minimum \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\text{~is~} n_1 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } \\exists \\; n_2 : (\\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace , j) \\models \\; minimum \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\text{~is~} n_2 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } n_2 \\; d \\; n_1\\\\&\\Big (\\Big \\lbrace \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace , j \\Big ) \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\models \\; \\mathbf {~direction~of~change~in~} maximum \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} d \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ if } \\exists \\; n_1 : (\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models \\; maximum \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\text{~is~} n_1 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } \\exists \\; n_2 : (\\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace , j) \\models \\; maximum \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\text{~is~} n_2 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } n_2 \\; d \\; n_1$ Next we define how a comparative quantitative aggregate point formula is satisfied w.r.t.", "two sets of trajectories $\\sigma _1,\\dots ,\\sigma _m$ , and $\\bar{\\sigma }_1,\\dots ,\\bar{\\sigma }_{\\bar{m}}$ at point $i$ : $&\\Big \\lbrace \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\models \\; \\mathbf {~direction~of~change~in~} average \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} d \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ if } \\exists \\; n_1 : \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace \\models \\; average \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\text{~is~} n_1 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } \\exists \\; n_2 : \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\models \\; average \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\text{~is~} n_2 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } n_2 \\; d \\; n_1\\\\&\\Big \\lbrace \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\models \\; \\mathbf {~direction~of~change~in~} minimum \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} d \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ if } \\exists \\; n_1 : \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace \\models \\; minimum \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\text{~is~} n_1 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } \\exists \\; n_2 : \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\models \\; minimum \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\text{~is~} n_2 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } n_2 \\; d \\; n_1\\\\&\\Big \\lbrace \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\models \\; \\mathbf {~direction~of~change~in~} maximum \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} d \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ if } \\exists \\; n_1 : \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace \\models \\; maximum \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\text{~is~} n_1 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } \\exists \\; n_2 : \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\models \\; maximum \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\text{~is~} n_2 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } n_2 \\; d \\; n_1$ Trajectory Filtering due to Internal Observations The trajectories produced by the Guarded-Arc Petri Net execution are filtered to retain only the trajectories that satisfy all internal observations in a query statement.", "Let $\\sigma = s_0,\\dots ,s_k$ be a trajectory as given in definition REF .", "Then $\\sigma $ satisfies an observation: $&\\mathbf {~rate~of~production~of~} f \\mathbf {~is~} n \\nonumber \\\\&~~~~~~~~\\text{ if } (\\langle s_0,\\sigma \\rangle , k) \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~is~} n \\\\&\\mathbf {~rate~of~production~of~} f \\mathbf {~is~} n \\nonumber \\\\&~~~~~~~~\\text{ if } (\\langle s_0,\\sigma \\rangle , k) \\models \\mathbf {~rate~of~firing~of~} a \\mathbf {~is~} n \\\\&\\mathbf {~total~production~of~} f \\mathbf {~is~} n \\nonumber \\\\&~~~~~~~~\\text{ if } (\\langle s_0,\\sigma \\rangle , k) \\models \\mathbf {~total~production~of~} f \\mathbf {~is~} n \\\\&f \\mathbf {~is~accumulating~} \\nonumber \\\\&~~~~~~~~\\text{ if } (\\langle s_0,\\sigma \\rangle , k) \\models f \\mathbf {~is~accumulating~} \\\\&f \\mathbf {~is~decreasing~} \\nonumber \\\\&~~~~~~~~\\text{ if } (\\langle s_0,\\sigma \\rangle , k) \\models f \\mathbf {~is~decreasing~} \\\\\\vspace{20.0pt}\\nonumber \\\\&\\mathbf {~rate~of~production~of~} f \\mathbf {~is~} n \\nonumber \\\\&~~~~~~~~\\mathbf {when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&~~~~~~~~\\text{ if } (\\langle s_i,\\sigma \\rangle , j) \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~is~} n \\\\&\\mathbf {~rate~of~production~of~} f \\mathbf {~is~} n \\nonumber \\\\&~~~~~~~~\\mathbf {when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&~~~~~~~~\\text{ if } (\\langle s_i,\\sigma \\rangle , j) \\models \\mathbf {~rate~of~firing~of~} a \\mathbf {~is~} n \\\\&\\mathbf {~total~production~of~} f \\mathbf {~is~} n \\nonumber \\\\&~~~~~~~~\\mathbf {when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&~~~~~~~~\\text{ if } (\\langle s_i,\\sigma \\rangle , j) \\models \\mathbf {~total~production~of~} f \\mathbf {~is~} n \\\\&f \\mathbf {~is~accumulating~} \\nonumber \\\\&~~~~~~~~\\mathbf {when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&~~~~~~~~\\text{ if } (\\langle s_i,\\sigma \\rangle , j) \\models f \\mathbf {~is~accumulating~} \\\\&f \\mathbf {~is~decreasing~} \\nonumber \\\\&~~~~~~~~\\mathbf {when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&~~~~~~~~\\text{ if } (\\langle s_i,\\sigma \\rangle , j) \\models f \\mathbf {~is~decreasing~} \\\\\\vspace{20.0pt}\\nonumber \\\\&f \\mathbf {~is~higher~than~} n \\nonumber \\\\&~~~~~~~~\\text{ if } \\exists i, 0 \\le i \\le k: \\langle s_i,\\sigma \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~is~higher~than~} n \\\\&f \\mathbf {~is~lower~than~} n \\nonumber \\\\&~~~~~~~~\\text{ if } \\exists i, 0 \\le i \\le k: \\langle s_i,\\sigma \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~is~lower~than~} n \\\\&f \\mathbf {~is~} n \\nonumber \\\\&~~~~~~~~\\text{ if } \\exists i, 0 \\le i \\le k: \\langle s_i,\\sigma \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~is~} n \\\\&a \\mathbf {~occurs~} \\nonumber \\\\&~~~~~~~~\\text{ if } \\exists i, 0 \\le i \\le k: \\langle s_i,\\sigma \\rangle \\models a \\mathbf {~occurs~} \\\\&a \\mathbf {~does~not~occur~} \\nonumber \\\\&~~~~~~~~\\text{ if } \\exists i, 0 \\le i \\le k: \\langle s_i,\\sigma \\rangle \\models a \\mathbf {~does~not~occur~} \\\\&a_1 \\mathbf {~switches~to~} a_2 \\nonumber \\\\&~~~~~~~~\\text{ if } \\exists i, 0 \\le i \\le k: \\langle s_i,\\sigma \\rangle \\models a_1 \\mathbf {~switches~to~} a_2\\vspace{20.0pt}\\nonumber \\\\&f \\mathbf {~is~higher~than~} n \\nonumber \\\\&~~~~~~~~\\mathbf {at~time~step~} i \\nonumber \\\\&~~~~~~~~\\text{ if } \\langle s_i,\\sigma \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~is~higher~than~} n \\\\&f \\mathbf {~is~lower~than~} n \\nonumber \\\\&~~~~~~~~\\mathbf {at~time~step~} i \\nonumber \\\\&~~~~~~~~\\text{ if } \\langle s_i,\\sigma \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~is~lower~than~} n \\\\&f \\mathbf {~is~} n \\nonumber \\\\&~~~~~~~~\\mathbf {at~time~step~} i \\nonumber \\\\&~~~~~~~~\\text{ if } \\langle s_i,\\sigma \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~is~} n \\\\&a \\mathbf {~occurs~} \\nonumber \\\\&~~~~~~~~\\mathbf {at~time~step~} i \\nonumber \\\\&~~~~~~~~\\text{ if } \\langle s_i,\\sigma \\rangle \\models a \\mathbf {~occurs~} \\\\&a \\mathbf {~does~not~occur~} \\nonumber \\\\&~~~~~~~~\\mathbf {at~time~step~} i \\nonumber \\\\&~~~~~~~~\\text{ if } \\langle s_i,\\sigma \\rangle \\models a \\mathbf {~does~not~occur~} \\\\&a_1 \\mathbf {~switches~to~} a_2 \\nonumber \\\\&~~~~~~~~\\mathbf {at~time~step~} i \\nonumber \\\\&~~~~~~~~\\text{ if } \\langle s_i,\\sigma \\rangle \\models a_1 \\mathbf {~switches~to~} a_2$ Let $\\sigma = s_0,\\dots ,s_k$ be a trajectory of the form (REF ).", "Then $\\sigma $ satisfies an observation: $&\\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n \\nonumber \\\\&~~~~~~~~\\text{ if } (\\langle s_0,\\sigma \\rangle , k) \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n \\\\&\\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n \\nonumber \\\\&~~~~~~~~\\text{ if } (\\langle s_0,\\sigma \\rangle , k) \\models \\mathbf {~rate~of~firing~of~} a \\mathbf {~is~} n \\\\&\\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n \\nonumber \\\\&~~~~~~~~\\text{ if } (\\langle s_0,\\sigma \\rangle , k) \\models \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n \\\\&f \\mathbf {~is~accumulating~} \\mathbf {~atloc~} l \\nonumber \\\\&~~~~~~~~\\text{ if } (\\langle s_0,\\sigma \\rangle , k) \\models f \\mathbf {~is~accumulating~} \\mathbf {~atloc~} l \\\\&f \\mathbf {~is~decreasing~} \\mathbf {~atloc~} l \\nonumber \\\\&~~~~~~~~\\text{ if } (\\langle s_0,\\sigma \\rangle , k) \\models f \\mathbf {~is~decreasing~} \\mathbf {~atloc~} l \\\\\\vspace{20.0pt}\\nonumber \\\\&\\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n \\nonumber \\\\&~~~~~~~~\\mathbf {when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&~~~~~~~~\\text{ if } (\\langle s_i,\\sigma \\rangle , j) \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n \\\\&\\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n \\nonumber \\\\&~~~~~~~~\\mathbf {when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&~~~~~~~~\\text{ if } (\\langle s_i,\\sigma \\rangle , j) \\models \\mathbf {~rate~of~firing~of~} a \\mathbf {~is~} n \\\\&\\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n \\nonumber \\\\&~~~~~~~~\\mathbf {when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&~~~~~~~~\\text{ if } (\\langle s_i,\\sigma \\rangle , j) \\models \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n \\\\&f \\mathbf {~is~accumulating~} \\mathbf {~atloc~} l \\nonumber \\\\&~~~~~~~~\\mathbf {when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&~~~~~~~~\\text{ if } (\\langle s_i,\\sigma \\rangle , j) \\models f \\mathbf {~is~accumulating~} \\mathbf {~atloc~} l \\\\&f \\mathbf {~is~decreasing~} \\mathbf {~atloc~} l \\nonumber \\\\&~~~~~~~~\\mathbf {when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&~~~~~~~~\\text{ if } (\\langle s_i,\\sigma \\rangle , j) \\models f \\mathbf {~is~decreasing~} \\mathbf {~atloc~} l \\\\\\vspace{20.0pt}\\nonumber \\\\&f \\mathbf {~atloc~} l \\mathbf {~is~higher~than~} n \\nonumber \\\\&~~~~~~~~\\text{ if } \\exists i, 0 \\le i \\le k: \\langle s_i,\\sigma \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~higher~than~} n \\\\&f \\mathbf {~atloc~} l \\mathbf {~is~lower~than~} n \\nonumber \\\\&~~~~~~~~\\text{ if } \\exists i, 0 \\le i \\le k: \\langle s_i,\\sigma \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~lower~than~} n \\\\&f \\mathbf {~atloc~} l \\mathbf {~is~} n \\nonumber \\\\&~~~~~~~~\\text{ if } \\exists i, 0 \\le i \\le k: \\langle s_i,\\sigma \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n \\\\&a \\mathbf {~occurs~} \\nonumber \\\\&~~~~~~~~\\text{ if } \\exists i, 0 \\le i \\le k: \\langle s_i,\\sigma \\rangle \\models a \\mathbf {~occurs~} \\\\&a \\mathbf {~does~not~occur~} \\nonumber \\\\&~~~~~~~~\\text{ if } \\exists i, 0 \\le i \\le k: \\langle s_i,\\sigma \\rangle \\models a \\mathbf {~does~not~occur~} \\\\&a_1 \\mathbf {~switches~to~} a_2 \\nonumber \\\\&~~~~~~~~\\text{ if } \\exists i, 0 \\le i \\le k: \\langle s_i,\\sigma \\rangle \\models a_1 \\mathbf {~switches~to~} a_2\\vspace{20.0pt}\\nonumber \\\\&f \\mathbf {~atloc~} l \\mathbf {~is~higher~than~} n \\nonumber \\\\&~~~~~~~~\\mathbf {at~time~step~} i \\nonumber \\\\&~~~~~~~~\\text{ if } \\langle s_i,\\sigma \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~higher~than~} n \\\\&f \\mathbf {~atloc~} l \\mathbf {~is~lower~than~} n \\nonumber \\\\&~~~~~~~~\\mathbf {at~time~step~} i \\nonumber \\\\&~~~~~~~~\\text{ if } \\langle s_i,\\sigma \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~lower~than~} n \\\\&f \\mathbf {~atloc~} l \\mathbf {~is~} n \\nonumber \\\\&~~~~~~~~\\mathbf {at~time~step~} i \\nonumber \\\\&~~~~~~~~\\text{ if } \\langle s_i,\\sigma \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n \\\\&a \\mathbf {~occurs~} \\nonumber \\\\&~~~~~~~~\\mathbf {at~time~step~} i \\nonumber \\\\&~~~~~~~~\\text{ if } \\langle s_i,\\sigma \\rangle \\models a \\mathbf {~occurs~} \\\\&a \\mathbf {~does~not~occur~} \\nonumber \\\\&~~~~~~~~\\mathbf {at~time~step~} i \\nonumber \\\\&~~~~~~~~\\text{ if } \\langle s_i,\\sigma \\rangle \\models a \\mathbf {~does~not~occur~} \\\\&a_1 \\mathbf {~switches~to~} a_2 \\nonumber \\\\&~~~~~~~~\\mathbf {at~time~step~} i \\nonumber \\\\&~~~~~~~~\\text{ if } \\langle s_i,\\sigma \\rangle \\models a_1 \\mathbf {~switches~to~} a_2$ A trajectory $\\sigma $ is kept for further processing w.r.t.", "a set of internal observations $\\langle \\text{internal observation} \\rangle _1,\\dots ,\\langle \\text{internal observation} \\rangle _n$ if $\\sigma \\models \\langle \\text{internal observation} \\rangle _i$ , $1 \\le i \\le n$ .", "Query Description Satisfaction Now, we define query statement semantics using LTL syntax.", "Let $\\mathbf {D}$ be a domain description with simple fluents and $\\sigma = s_0,\\dots ,s_k$ be its trajectory of length $k$ as defined in (REF ).", "Let $\\sigma _1,\\dots ,\\sigma _m$ represent the set of trajectories of $\\mathbf {D}$ filtered by observations as necessary, with each trajectory has the form $\\sigma _i = s^i_0,\\dots ,s^i_k$ $1\\le i \\le m$ .", "Let $\\mathbf {\\bar{D}}$ be a modified domain description and $\\bar{\\sigma }_1,\\dots ,\\bar{\\sigma }_{\\bar{m}}$ be its trajectories of the form $\\bar{\\sigma }_i = \\bar{s}^i_0,\\dots ,\\bar{s}^i_k$ .", "Then we define query satisfaction using the formula satisfaction in section REF as follows.", "Two sets of trajectories $\\sigma _1,\\dots ,\\sigma _m$ and $\\bar{\\sigma }_1,\\dots ,\\bar{\\sigma }_{\\bar{m}}$ satisfy a comparative query description based on formula satisfaction of section REF as follows: $&\\Big \\lbrace \\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_0,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_0,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\models \\; \\mathbf {~direction~of~change~in~} aggop \\mathbf {~rate~of~production~of~} f \\mathbf {~is~} d \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\mathbf {~when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ if } \\Big (\\Big \\lbrace \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace , j \\Big ) \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\models \\mathbf {~direction~of~change~in~} aggop \\mathbf {~rate~of~production~of~} f \\mathbf {~is~} d \\\\&\\Big \\lbrace \\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_0,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_0,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\models \\; \\mathbf {~direction~of~change~in~} aggop \\mathbf {~rate~of~firing~of~} f \\mathbf {~is~} d \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\mathbf {~when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ if } \\Big (\\Big \\lbrace \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace , j\\Big ) \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\models \\mathbf {~direction~of~change~in~} aggop \\mathbf {~rate~of~firing~of~} f \\mathbf {~is~} d \\\\&\\Big \\lbrace \\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_0,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_0,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\models \\; \\mathbf {~direction~of~change~in~} aggop \\mathbf {~total~production~of~} f \\mathbf {~is~} d \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\mathbf {~when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ if } \\Big (\\Big \\lbrace \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace , j\\Big ) \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\models \\mathbf {~direction~of~change~in~} aggop \\mathbf {~total~production~of~} f \\mathbf {~is~} d \\\\&\\Big \\lbrace \\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_0,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_0,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\models \\; \\mathbf {~direction~of~change~in~} aggop \\mathbf {~value~of~} f \\mathbf {~is~} d \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\mathbf {~when~observed~at~time~step~} i \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ if } \\Big \\lbrace \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\models \\mathbf {~direction~of~change~in~} aggop \\mathbf {~value~of~} f \\mathbf {~is~} d$ A set of trajectories $\\sigma _1,\\dots ,\\sigma _m$ , each of length $k$ satisfies a quantitative or a qualitative interval query description based on formula satisfaction of section REF as follows: $&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~is~} n \\nonumber \\\\&~~~~\\text{ if } \\exists x, 1 \\le x \\le m : (\\langle s^x_0,\\sigma _x \\rangle , k) \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~is~} n \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~rate~of~firing~of~} a \\mathbf {~is~} n \\nonumber \\\\&~~~~\\text{ if } \\exists x, 1 \\le x \\le m : (\\langle s^x_0,\\sigma _x \\rangle , k) \\models \\mathbf {~rate~of~firing~of~} a \\mathbf {~is~} n \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~total~production~of~} f \\mathbf {~is~} n \\nonumber \\\\&~~~~\\text{ if } \\exists x, 1 \\le x \\le m : (\\langle s^x_0,\\sigma _x \\rangle , k) \\models \\mathbf {~total~production~of~} f \\mathbf {~is~} n \\\\\\vspace{20.0pt}\\nonumber \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~is~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&~~~~\\text{ if } \\exists x, 1 \\le x \\le m : (\\langle s^x_i,\\sigma _x \\rangle , j) \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~is~} n \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~rate~of~firing~of~} a \\mathbf {~is~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&~~~~\\text{ if } \\exists x, 1 \\le x \\le m : (\\langle s^x_i,\\sigma _x \\rangle , j) \\models \\mathbf {~rate~of~firing~of~} a \\mathbf {~is~} n \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~total~production~of~} f \\mathbf {~is~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&~~~~\\text{ if } \\exists x, 1 \\le x \\le m : (\\langle s^x_i,\\sigma _x \\rangle , j) \\models \\mathbf {~total~production~of~} f \\mathbf {~is~} n \\\\\\vspace{20.0pt}\\nonumber \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models f \\mathbf {~is~accumulating~} \\nonumber \\\\&~~~~\\text{ if } \\exists x, 1 \\le x \\le m : (\\langle s^x_0,\\sigma _x \\rangle , k) \\models f \\mathbf {~is~accumulating~} \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models f \\mathbf {~is~decreasing~} \\nonumber \\\\&~~~~\\text{ if } \\exists x, 1 \\le x \\le m : (\\langle s^x_0,\\sigma _x \\rangle , k) \\models f \\mathbf {~is~decreasing~} \\\\\\vspace{20.0pt}\\nonumber \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models f \\mathbf {~is~accumulating~} \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&~~~~\\text{ if } \\exists x, 1 \\le x \\le m : (\\langle s^x_i,\\sigma _x \\rangle , j) \\models f \\mathbf {~is~accumulating~} \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models f \\mathbf {~is~decreasing~} \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&~~~~\\text{ if } \\exists x, 1 \\le x \\le m : (\\langle s^x_i,\\sigma _x \\rangle , j) \\models f \\mathbf {~is~decreasing~} \\\\\\vspace{20.0pt}\\nonumber \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~is~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~in~all~trajectories~} \\nonumber \\\\&~~~~\\text{ if } \\forall x, 1 \\le x \\le m : (\\langle s^x_0,\\sigma _x \\rangle , k) \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~is~} n \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~rate~of~firing~of~} a \\mathbf {~is~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~in~all~trajectories~} \\nonumber \\\\&~~~~\\text{ if } \\forall x, 1 \\le x \\le m : (\\langle s^x_0,\\sigma _x \\rangle , k) \\models \\mathbf {~rate~of~firing~of~} a \\mathbf {~is~} n \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~total~production~of~} f \\mathbf {~is~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~in~all~trajectories~} \\nonumber \\\\&~~~~\\text{ if } \\forall x, 1 \\le x \\le m : (\\langle s^x_0,\\sigma _x \\rangle , k) \\models \\mathbf {~total~production~of~} f \\mathbf {~is~} n \\\\\\vspace{20.0pt}\\nonumber \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~is~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~in~all~trajectories~} \\nonumber \\\\&~~~~\\text{ if } \\forall x, 1 \\le x \\le m : (\\langle s^x_i,\\sigma _x \\rangle , j) \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~is~} n \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~rate~of~firing~of~} a \\mathbf {~is~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~in~all~trajectories~} \\nonumber \\\\&~~~~\\text{ if } \\forall x, 1 \\le x \\le m : (\\langle s^x_i,\\sigma _x \\rangle , j) \\models \\mathbf {~rate~of~firing~of~} a \\mathbf {~is~} n \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~total~production~of~} f \\mathbf {~is~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~in~all~trajectories~} \\nonumber \\\\&~~~~\\text{ if } \\forall x, 1 \\le x \\le m : (\\langle s^x_i,\\sigma _x \\rangle , j) \\models \\mathbf {~total~production~of~} f \\mathbf {~is~} n \\\\\\vspace{20.0pt}\\nonumber \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models f \\mathbf {~is~accumulating~} \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~in~all~trajectories~} \\nonumber \\\\&~~~~\\text{ if } \\forall x, 1 \\le x \\le m : (\\langle s^x_0,\\sigma _x \\rangle , k) \\models f \\mathbf {~is~accumulating~} \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models f \\mathbf {~is~decreasing~} \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~in~all~trajectories~} \\nonumber \\\\&~~~~\\text{ if } \\forall x, 1 \\le x \\le m : (\\langle s^x_0,\\sigma _x \\rangle , k) \\models f \\mathbf {~is~decreasing~} \\\\\\vspace{20.0pt}\\nonumber \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models f \\mathbf {~is~accumulating~} \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~in~all~trajectories~} \\nonumber \\\\&~~~~\\text{ if } \\forall x, 1 \\le x \\le m : (\\langle s^x_i,\\sigma _x \\rangle , j) \\models f \\mathbf {~is~accumulating~} \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models f \\mathbf {~is~decreasing~} \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~in~all~trajectories~} \\nonumber \\\\&~~~~\\text{ if } \\forall x, 1 \\le x \\le m : (\\langle s^x_i,\\sigma _x \\rangle , j) \\models f \\mathbf {~is~decreasing~}$ A set of trajectories $\\sigma _1,\\dots ,\\sigma _m$ , each of length $k$ satisfies a quantitative or a qualitative point query description based on formula satisfaction of section REF as follows: $&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~value~of~} f \\mathbf {~is~higher~than~} n \\nonumber \\\\&~~~~\\text{ if } \\exists x \\exists i, 1 \\le x \\le m, 0 \\le i \\le k : \\langle s^x_i,\\sigma _x \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~is~higher~than~} n \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~value~of~} f \\mathbf {~is~lower~than~} n \\nonumber \\\\&~~~~\\text{ if } \\exists x \\exists i, 1 \\le x \\le m, 0 \\le i \\le k : \\langle s^x_i,\\sigma _x \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~is~lower~than~} n \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~value~of~} f \\mathbf {~is~} n \\nonumber \\\\&~~~~\\text{ if } \\exists x \\exists i, 1 \\le x \\le m, 0 \\le i \\le k : \\langle s^x_i,\\sigma _x \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~is~} n \\\\\\vspace{20.0pt}\\nonumber \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~value~of~} f \\mathbf {~is~higher~than~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~at~time~step~} i \\nonumber \\\\&~~~~\\text{ if } \\exists x, 1 \\le x \\le m : \\langle s^x_i,\\sigma _x \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~is~higher~than~} n \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~value~of~} f \\mathbf {~is~lower~than~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~at~time~step~} i \\nonumber \\\\&~~~~\\text{ if } \\exists x, 1 \\le x \\le m : \\langle s^x_i,\\sigma _x \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~is~lower~than~} n \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~value~of~} f \\mathbf {~is~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~at~time~step~} i \\nonumber \\\\&~~~~\\text{ if } \\exists x, 1 \\le x \\le m : \\langle s^x_i,\\sigma _x \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~is~} n \\\\\\vspace{20.0pt}\\nonumber \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~value~of~} f \\mathbf {~is~higher~than~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~in~all~trajectories~} \\nonumber \\\\&~~~~\\text{ if } \\forall x \\exists i, 1 \\le x \\le m, 0 \\le i \\le k : \\langle s^x_i,\\sigma _x \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~is~higher~than~} n \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~value~of~} f \\mathbf {~is~lower~than~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~in~all~trajectories~} \\nonumber \\\\&~~~~\\text{ if } \\forall x \\exists i, 1 \\le x \\le m, 0 \\le i \\le k : \\langle s^x_i,\\sigma _x \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~is~lower~than~} n \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~value~of~} f \\mathbf {~is~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~in~all~trajectories~} \\nonumber \\\\&~~~~\\text{ if } \\forall x \\exists i, 1 \\le x \\le m, 0 \\le i \\le k : \\langle s^x_i,\\sigma _x \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~is~} n \\\\\\vspace{20.0pt}\\nonumber \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~value~of~} f \\mathbf {~is~higher~than~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~at~time~step~} i \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~in~all~trajectories~} \\nonumber \\\\&~~~~\\text{ if } \\forall x, 1 \\le x \\le m : \\langle s^x_i,\\sigma _x \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~is~higher~than~} n \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~value~of~} f \\mathbf {~is~lower~than~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~at~time~step~} i \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~in~all~trajectories~} \\nonumber \\\\&~~~~\\text{ if } \\forall x, 1 \\le x \\le m : \\langle s^x_i,\\sigma _x \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~is~lower~than~} n \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~value~of~} f \\mathbf {~is~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~at~time~step~} i \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~in~all~trajectories~} \\nonumber \\\\&~~~~\\text{ if } \\forall x, 1 \\le x \\le m : \\langle s^x_i,\\sigma _x \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~is~} n$ Now, we turn our attention to domain descriptions with locational fluents.", "Let $\\mathbf {D}$ be a domain description and $\\sigma = s_0,\\dots ,s_k$ be its trajectory of length $k$ as defined in (REF ).", "Let $\\sigma _1,\\dots ,\\sigma _m$ represent the set of trajectories of $\\mathbf {D}$ filtered by observations as necessary, with each trajectory has the form $\\sigma _i = s^i_0,\\dots ,s^i_k$ $1\\le i \\le m$ .", "Let $\\mathbf {\\bar{D}}$ be a modified domain description and $\\bar{\\sigma }_1,\\dots ,\\bar{\\sigma }_{\\bar{m}}$ be its trajectories of the form $\\bar{\\sigma }_i = \\bar{s}^i_0,\\dots ,\\bar{s}^i_k$ .", "Then we define query satisfaction using the formula satisfaction in section REF as follows.", "Two sets of trajectories $\\sigma _1,\\dots ,\\sigma _m$ and $\\bar{\\sigma }_1,\\dots ,\\bar{\\sigma }_{\\bar{m}}$ satisfy a comparative query description based on formula satisfaction of section REF as follows: $&\\Big \\lbrace \\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_0,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_0,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\models \\; \\mathbf {~direction~of~change~in~} \\langle \\text{aggop} \\rangle \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} d \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\mathbf {~when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ if } \\Big (\\Big \\lbrace \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace , j \\Big ) \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\models \\mathbf {~direction~of~change~in~} \\langle \\text{aggop} \\rangle \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} d \\\\&\\Big \\lbrace \\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_0,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_0,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\models \\; \\mathbf {~direction~of~change~in~} \\langle \\text{aggop} \\rangle \\mathbf {~rate~of~firing~of~} f \\mathbf {~is~} d \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\mathbf {~when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ if } \\Big (\\Big \\lbrace \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace , j\\Big ) \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\models \\mathbf {~direction~of~change~in~} \\langle \\text{aggop} \\rangle \\mathbf {~rate~of~firing~of~} f \\mathbf {~is~} d \\\\&\\Big \\lbrace \\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_0,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_0,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\models \\; \\mathbf {~direction~of~change~in~} \\langle \\text{aggop} \\rangle \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} d \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\mathbf {~when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ if } \\Big (\\Big \\lbrace \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace , j\\Big ) \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\models \\mathbf {~direction~of~change~in~} \\langle \\text{aggop} \\rangle \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} d \\\\&\\Big \\lbrace \\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_0,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_0,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\models \\; \\mathbf {~direction~of~change~in~} \\langle \\text{aggop} \\rangle \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} d \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\mathbf {~when~observed~at~time~step~} i \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ if } \\Big \\lbrace \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\models \\mathbf {~direction~of~change~in~} \\langle \\text{aggop} \\rangle \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} d$ A set of trajectories $\\sigma _1,\\dots ,\\sigma _m$ , each of length $k$ satisfies a quantitative or a qualitative interval query description based on formula satisfaction of section REF as follows: $&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n \\nonumber \\\\&~~~~\\text{ if } \\exists x, 1 \\le x \\le m : (\\langle s^x_0,\\sigma _x \\rangle , k) \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~rate~of~firing~of~} a \\mathbf {~is~} n \\nonumber \\\\&~~~~\\text{ if } \\exists x, 1 \\le x \\le m : (\\langle s^x_0,\\sigma _x \\rangle , k) \\models \\mathbf {~rate~of~firing~of~} a \\mathbf {~is~} n \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n \\nonumber \\\\&~~~~\\text{ if } \\exists x, 1 \\le x \\le m : (\\langle s^x_0,\\sigma _x \\rangle , k) \\models \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n \\\\\\vspace{20.0pt}\\nonumber \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&~~~~\\text{ if } \\exists x, 1 \\le x \\le m : (\\langle s^x_i,\\sigma _x \\rangle , j) \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~rate~of~firing~of~} a \\mathbf {~is~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&~~~~\\text{ if } \\exists x, 1 \\le x \\le m : (\\langle s^x_i,\\sigma _x \\rangle , j) \\models \\mathbf {~rate~of~firing~of~} a \\mathbf {~is~} n \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&~~~~\\text{ if } \\exists x, 1 \\le x \\le m : (\\langle s^x_i,\\sigma _x \\rangle , j) \\models \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n \\\\\\vspace{20.0pt}\\nonumber \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models f \\mathbf {~is~accumulating~} \\mathbf {~atloc~} l \\nonumber \\\\&~~~~\\text{ if } \\exists x, 1 \\le x \\le m : (\\langle s^x_0,\\sigma _x \\rangle , k) \\models f \\mathbf {~is~accumulating~} \\mathbf {~atloc~} l \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models f \\mathbf {~is~decreasing~} \\mathbf {~atloc~} l \\nonumber \\\\&~~~~\\text{ if } \\exists x, 1 \\le x \\le m : (\\langle s^x_0,\\sigma _x \\rangle , k) \\models f \\mathbf {~is~decreasing~} \\mathbf {~atloc~} l \\\\\\vspace{20.0pt}\\nonumber \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models f \\mathbf {~is~accumulating~} \\mathbf {~atloc~} l \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&~~~~\\text{ if } \\exists x, 1 \\le x \\le m : (\\langle s^x_i,\\sigma _x \\rangle , j) \\models f \\mathbf {~is~accumulating~} \\mathbf {~atloc~} l \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models f \\mathbf {~is~decreasing~} \\mathbf {~atloc~} l \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&~~~~\\text{ if } \\exists x, 1 \\le x \\le m : (\\langle s^x_i,\\sigma _x \\rangle , j) \\models f \\mathbf {~is~decreasing~} \\mathbf {~atloc~} l \\\\\\vspace{30.0pt} \\nonumber \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~in~all~trajectories~} \\nonumber \\\\&~~~~\\text{ if } \\forall x, 1 \\le x \\le m : (\\langle s^x_0,\\sigma _x \\rangle , k) \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~rate~of~firing~of~} a \\mathbf {~is~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~in~all~trajectories~} \\nonumber \\\\&~~~~\\text{ if } \\forall x, 1 \\le x \\le m : (\\langle s^x_0,\\sigma _x \\rangle , k) \\models \\mathbf {~rate~of~firing~of~} a \\mathbf {~is~} n \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~in~all~trajectories~} \\nonumber \\\\&~~~~\\text{ if } \\forall x, 1 \\le x \\le m : (\\langle s^x_0,\\sigma _x \\rangle , k) \\models \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n \\\\\\vspace{20.0pt}\\nonumber \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~in~all~trajectories~} \\nonumber \\\\&~~~~\\text{ if } \\forall x, 1 \\le x \\le m : (\\langle s^x_i,\\sigma _x \\rangle , j) \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~rate~of~firing~of~} a \\mathbf {~is~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~in~all~trajectories~} \\nonumber \\\\&~~~~\\text{ if } \\forall x, 1 \\le x \\le m : (\\langle s^x_i,\\sigma _x \\rangle , j) \\models \\mathbf {~rate~of~firing~of~} a \\mathbf {~is~} n \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~in~all~trajectories~} \\nonumber \\\\&~~~~\\text{ if } \\forall x, 1 \\le x \\le m : (\\langle s^x_i,\\sigma _x \\rangle , j) \\models \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n \\text{ till } j\\\\\\vspace{20.0pt}\\nonumber \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models f \\mathbf {~is~accumulating~} \\mathbf {~atloc~} l \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~in~all~trajectories~} \\nonumber \\\\&~~~~\\text{ if } \\forall x, 1 \\le x \\le m : (\\langle s^x_0,\\sigma _x \\rangle , k) \\models f \\mathbf {~is~accumulating~} \\mathbf {~atloc~} l \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models f \\mathbf {~is~decreasing~} \\mathbf {~atloc~} l \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~in~all~trajectories~} \\nonumber \\\\&~~~~\\text{ if } \\forall x, 1 \\le x \\le m : (\\langle s^x_0,\\sigma _x \\rangle , k) \\models f \\mathbf {~is~decreasing~} \\mathbf {~atloc~} l \\\\\\vspace{20.0pt}\\nonumber \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models f \\mathbf {~is~accumulating~} \\mathbf {~atloc~} l \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~in~all~trajectories~} \\nonumber \\\\&~~~~\\text{ if } \\forall x, 1 \\le x \\le m : (\\langle s^x_i,\\sigma _x \\rangle , j) \\models f \\mathbf {~is~accumulating~} \\mathbf {~atloc~} l \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models f \\mathbf {~is~decreasing~} \\mathbf {~atloc~} l \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~in~all~trajectories~} \\nonumber \\\\&~~~~\\text{ if } \\forall x, 1 \\le x \\le m : (\\langle s^x_i,\\sigma _x \\rangle , j) \\models f \\mathbf {~is~decreasing~} \\mathbf {~atloc~} l$ A set of trajectories $\\sigma _1,\\dots ,\\sigma _m$ , each of length $k$ satisfies a quantitative or a qualitative point query description based on formula satisfaction of section REF as follows: $&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l\\mathbf {~is~higher~than~} n \\nonumber \\\\&~~~~\\text{ if } \\exists x \\exists i, 1 \\le x \\le m, 0 \\le i \\le k : \\langle s^x_i,\\sigma _x \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~higher~than~} n \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l\\mathbf {~is~lower~than~} n \\nonumber \\\\&~~~~\\text{ if } \\exists x \\exists i, 1 \\le x \\le m, 0 \\le i \\le k : \\langle s^x_i,\\sigma _x \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~lower~than~} n \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l\\mathbf {~is~} n \\nonumber \\\\&~~~~\\text{ if } \\exists x \\exists i, 1 \\le x \\le m, 0 \\le i \\le k : \\langle s^x_i,\\sigma _x \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n\\\\\\vspace{20.0pt}\\nonumber \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l\\mathbf {~is~higher~than~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~at~time~step~} i \\nonumber \\\\&~~~~\\text{ if } \\exists x, 1 \\le x \\le m : \\langle s^x_i,\\sigma _x \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~higher~than~} n \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l\\mathbf {~is~lower~than~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~at~time~step~} i \\nonumber \\\\&~~~~\\text{ if } \\exists x, 1 \\le x \\le m : \\langle s^x_i,\\sigma _x \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~lower~than~} n \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l\\mathbf {~is~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~at~time~step~} i \\nonumber \\\\&~~~~\\text{ if } \\exists x, 1 \\le x \\le m : \\langle s^x_i,\\sigma _x \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n\\\\\\vspace{20.0pt}\\nonumber \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l\\mathbf {~is~higher~than~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~in~all~trajectories~} \\nonumber \\\\&~~~~\\text{ if } \\forall x \\exists i, 1 \\le x \\le m, 0 \\le i \\le k : \\langle s^x_i,\\sigma _x \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~higher~than~} n \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~lower~than~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~in~all~trajectories~} \\nonumber \\\\&~~~~\\text{ if } \\forall x \\exists i, 1 \\le x \\le m, 0 \\le i \\le k : \\langle s^x_i,\\sigma _x \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~lower~than~} n \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l\\mathbf {~is~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~in~all~trajectories~} \\nonumber \\\\&~~~~\\text{ if } \\forall x \\exists i, 1 \\le x \\le m, 0 \\le i \\le k : \\langle s^x_i,\\sigma _x \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n\\\\\\vspace{20.0pt}\\nonumber \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~higher~than~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~at~time~step~} i \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~in~all~trajectories~} \\nonumber \\\\&~~~~\\text{ if } \\forall x, 1 \\le x \\le m : \\langle s^x_i,\\sigma _x \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~higher~than~} n \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l\\mathbf {~is~lower~than~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~at~time~step~} i \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~in~all~trajectories~} \\nonumber \\\\&~~~~\\text{ if } \\forall x, 1 \\le x \\le m : \\langle s^x_i,\\sigma _x \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~lower~than~} n \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l\\mathbf {~is~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~at~time~step~} i \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~in~all~trajectories~} \\nonumber \\\\&~~~~\\text{ if } \\forall x, 1 \\le x \\le m : \\langle s^x_i,\\sigma _x \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n$ Next, we generically define the satisfaction of a simple point formula cascade query w.r.t.", "a set of trajectories $\\sigma _1,\\dots ,\\sigma _m$ .", "The trajectories will either be as defined in definitions (REF ) or (REF ) for simple point formula cascade query statement made up of simple fluents or locational fluents, respectively.", "A set of trajectories $\\sigma _1,\\dots ,\\sigma _m$ , each of length $k$ satisfies a simple point formula cascade based on formula satisfaction of section REF as follows: $&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\langle \\text{simple point formula} \\rangle _0 \\nonumber \\\\&~~~~~~~~\\mathbf {~after~} \\langle \\text{simple point formula} \\rangle _{1,1}, \\dots , \\langle \\text{simple point formula} \\rangle _{1,n_1} \\nonumber \\\\&~~~~~~~~\\vdots \\nonumber \\\\&~~~~~~~~\\mathbf {~after~} \\langle \\text{simple point formula} \\rangle _{u,1}, \\dots , \\langle \\text{simple point formula} \\rangle _{u,n_u} \\nonumber \\\\&~~~~\\text{ if } \\exists x \\exists i_0 \\exists i_1 \\dots \\exists i_u, 1 \\le x \\le m, i_1 < i_0 \\le k, \\dots , i_u < i_{u-1} \\le k, 0 \\le i_u \\le k: \\nonumber \\\\&~~~~~~~~\\langle s^x_{i_0},\\sigma _x \\rangle \\models \\langle \\text{simple point formula} \\rangle _0 \\text{ and } \\nonumber \\\\&~~~~~~~~\\langle s^x_{i_1},\\sigma _x \\rangle \\models \\langle \\text{simple point formula} \\rangle _{1,1} \\text{ and } \\nonumber \\hdots \\text{ and } \\langle s^x_{i_1},\\sigma _x \\rangle \\models \\langle \\text{simple point formula} \\rangle _{1,n_1} \\text{ and } \\nonumber \\\\&~~~~~~~~~~~~\\vdots \\nonumber \\\\&~~~~~~~~\\langle s^x_{i_u},\\sigma _x \\rangle \\models \\langle \\text{simple point formula} \\rangle _{u,1} \\text{ and } \\nonumber \\hdots \\text{ and } \\langle s^x_{i_u},\\sigma _x \\rangle \\models \\langle \\text{simple point formula} \\rangle _{u,n_u} \\\\\\\\\\vspace{10.0pt}\\nonumber \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\langle \\text{simple point formula} \\rangle _0 \\nonumber \\\\&~~~~~~~~\\mathbf {~after~} \\langle \\text{simple point formula} \\rangle _{1,1}, \\dots , \\langle \\text{simple point formula} \\rangle _{1,n_1} \\nonumber \\\\&~~~~~~~~\\vdots \\nonumber \\\\&~~~~~~~~\\mathbf {~after~} \\langle \\text{simple point formula} \\rangle _{u,1}, \\dots , \\langle \\text{simple point formula} \\rangle _{u,n_u} \\nonumber \\\\&~~~~~~~~\\mathbf {~in~all~trajectories} \\nonumber \\\\&~~~~\\text{ if } \\forall x, 1 \\le x \\le m, \\exists i_0 \\exists i_1 \\dots \\exists i_u, i_1 < i_0 \\le k, \\dots , i_u < i_{u-1} \\le k, 0 \\le i_u \\le k: \\nonumber \\\\&~~~~~~~~\\langle s^x_{i_0},\\sigma _x \\rangle \\models \\langle \\text{simple point formula} \\rangle _0 \\text{ and } \\nonumber \\\\&~~~~~~~~\\langle s^x_{i_1},\\sigma _x \\rangle \\models \\langle \\text{simple point formula} \\rangle _{1,1} \\text{ and } \\nonumber \\hdots \\text{ and } \\langle s^x_{i_1},\\sigma _x \\rangle \\models \\langle \\text{simple point formula} \\rangle _{1,n_1} \\text{ and } \\nonumber \\\\&~~~~~~~~~~~~\\vdots \\nonumber \\\\&~~~~~~~~\\langle s^x_{i_u},\\sigma _x \\rangle \\models \\langle \\text{simple point formula} \\rangle _{u,1} \\text{ and } \\nonumber \\hdots \\text{ and } \\langle s^x_{i_u},\\sigma _x \\rangle \\models \\langle \\text{simple point formula} \\rangle _{u,n_u} \\\\\\\\\\vspace{10.0pt}\\nonumber \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\langle \\text{simple point formula} \\rangle _0 \\nonumber \\\\&~~~~~~~~\\mathbf {~when~} \\langle \\text{simple point formula} \\rangle _{1,1}, \\dots , \\langle \\text{simple point formula} \\rangle _{1,n_1} \\nonumber \\\\&~~~~\\text{ if } \\exists x \\exists i, 1 \\le x \\le m, 0 \\le i \\le k: \\nonumber \\\\&~~~~~~~~\\langle s^x_{i_0},\\sigma _x \\rangle \\models \\langle \\text{simple point formula} \\rangle _0 \\text{ and } \\nonumber \\\\&~~~~~~~~\\langle s^x_{i_1},\\sigma _x \\rangle \\models \\langle \\text{simple point formula} \\rangle _{1,1} \\text{ and } \\nonumber \\hdots \\text{ and } \\langle s^x_{i_1},\\sigma _x \\rangle \\models \\langle \\text{simple point formula} \\rangle _{1,n_1} \\\\\\\\\\vspace{10.0pt}\\nonumber \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\langle \\text{simple point formula} \\rangle _0 \\nonumber \\\\&~~~~~~~~\\mathbf {~when~} \\langle \\text{simple point formula} \\rangle _{1,1}, \\dots , \\langle \\text{simple point formula} \\rangle _{1,n_1} \\nonumber \\\\&~~~~~~~~\\mathbf {~in~all~trajectories} \\nonumber \\\\&~~~~\\text{ if } \\forall x, 1 \\le x \\le m, \\exists i, 0 \\le i \\le k: \\nonumber \\\\&~~~~~~~~\\langle s^x_{i_0},\\sigma _x \\rangle \\models \\langle \\text{simple point formula} \\rangle _0 \\text{ and } \\nonumber \\\\&~~~~~~~~\\langle s^x_{i_1},\\sigma _x \\rangle \\models \\langle \\text{simple point formula} \\rangle _{1,1} \\text{ and } \\nonumber \\hdots \\text{ and } \\langle s^x_{i_1},\\sigma _x \\rangle \\models \\langle \\text{simple point formula} \\rangle _{1,n_1} \\\\$ Query Statement Satisfaction Let $\\mathbf {D}$ as defined in section REF be a domain description and $\\mathbf {Q}$ be a query statement (REF ) as defined in section REF with query description $U$ , interventions $V_1,\\dots ,V_{\|V\|}$ , internal observations $O_1,\\dots ,O_{\|O\|}$ , and initial conditions $I_1,\\dots ,I_{\|I\|}$ .", "Let $\\mathbf {D_1} \\equiv \\mathbf {D} \\diamond I_1 \\diamond \\dots \\diamond I_{\|I\|} \\diamond V_1 \\diamond \\dots \\diamond V_{\|V\|}$ be the modified domain description constructed by applying the initial conditions and interventions from $\\mathbf {Q}$ as defined in section REF .", "Let $\\sigma _1,\\dots ,\\sigma _m$ be the trajectories of $\\mathbf {D_1}$ that satisfy $O_1,\\dots ,O_{\|O\|}$ as given in section REF .", "Then, $\\mathbf {D}$ satisfies $\\mathbf {Q}$ if $\\lbrace \\sigma _1,\\dots ,\\sigma _m\\rbrace \\models U$ as defined in section REF .", "Let $\\mathbf {D}$ as defined in section REF be a domain description and $\\mathbf {Q}$ be a query statement () as defined in section REF with query description $U$ , interventions $V_1,\\dots ,V_{\|V\|}$ , internal observations $O_1,\\dots ,O_{\|O\|}$ , and initial conditions $I_1,\\dots ,I_{\|I\|}$ .", "Let $\\mathbf {D_)} \\equiv \\mathbf {D} \\diamond I_1 \\diamond \\dots \\diamond I_{\|I\|} $ be the nominal domain description constructed by applying the initial conditions from $\\mathbf {Q}$ as defined in section REF .", "Let $\\sigma _1,\\dots ,\\sigma _m$ be the trajectories of $\\mathbf {D_0}$ .", "Let $\\mathbf {D_1} \\equiv \\mathbf {D} \\diamond I_1 \\diamond \\dots \\diamond I_{\|I\|} \\diamond V_1 \\diamond \\dots \\diamond V_{\|V\|}$ be the modified domain description constructed by applying the initial conditions and interventions from $\\mathbf {Q}$ as defined in section REF .", "Let $\\bar{\\sigma }_1,\\dots ,\\bar{\\sigma }_{\\bar{m}}$ be the trajectories of $\\mathbf {D_1}$ that satisfy $O_1,\\dots ,O_{\|O\|}$ as given in section REF .", "Then, $\\mathbf {D}$ satisfies $\\mathbf {Q}$ if $\\Big \\lbrace \\lbrace \\sigma _1,\\dots ,\\sigma _m\\rbrace , \\lbrace \\bar{\\sigma }_1,\\dots ,\\bar{\\sigma }_{\\bar{m}}\\rbrace \\Big \\rbrace \\models U$ as defined in section REF .", "Example Encodings In this section we give some examples of how we will encode queries and pathways related to these queries.", "We will also show how the pathway is modified to answer questions Some of the same pathways appear in previous chapters, they have been updated here with additional background knowledge..", "Question 11 At one point in the process of glycolysis, both DHAP and G3P are produced.", "Isomerase catalyzes the reversible conversion between these two isomers.", "The conversion of DHAP to G3P never reaches equilibrium and G3P is used in the next step of glycolysis.", "What would happen to the rate of glycolysis if DHAP were removed from the process of glycolysis as quickly as it was produced?", "Figure: Petri Net for question The question is asking for the direction of change in the rate of glycolysis when the nominal glycolysis pathway is compared against a modified pathway in which dhap is removed as soon as it is produced.", "Since this rate can vary with the trajectory followed by the world evolution, we consider the average change in rate.", "From the domain knowledge [64] we know that the rate of $glycolysis$ can be measured by the rate of $pyruvate$ (the end product of glycolysis) and that the rate of $pyruvate$ is equal to the rate of $bpg13$ (due to linear chain from $bpg13$ to $pyruvate$ ).", "Thus, we can monitor the rate of $bpg13$ instead to determine the rate of glycolysis.", "To ensure that our pathway is not starved due to source ingredients, we add a continuous supply of $f16bp$ in quantity 1 to the pathway.", "Then, the following pathway specification encodes the domain description $\\mathbf {D}$ for question REF and produces the PN in Fig.", "REF minus the $tr,t3$ transitions: $&\\begin{array}{llll}&\\mathbf {domain~of~} & f16bp \\mathbf {~is~} integer, & dhap \\mathbf {~is~} integer, \\\\&& g3p \\mathbf {~is~} integer, & bpg13 \\mathbf {~is~} integer\\nonumber \\\\&t4 \\mathbf {~may~execute~causing~} & f16bp \\mathbf {~change~value~by~} -1, & dhap \\mathbf {~change~value~by~} +1,\\\\ && g3p \\mathbf {~change~value~by~} +1\\nonumber \\\\&t5a \\mathbf {~may~execute~causing~} & dhap \\mathbf {~change~value~by~} -1, & g3p \\mathbf {~change~value~by~} +1\\nonumber \\\\&t5b \\mathbf {~may~execute~causing~} & g3p \\mathbf {~change~value~by~} -1, & dhap \\mathbf {~change~value~by~} +1\\nonumber \\\\&t6 \\mathbf {~may~execute~causing~} & g3p \\mathbf {~change~value~by~} -1, & bpg13 \\mathbf {~change~value~by~} +2\\nonumber \\\\& \\mathbf {initially~} & f16bp \\mathbf {~has~value~} 0, & dhap \\mathbf {~has~value~} 0,\\\\ && g3p \\mathbf {~has~value~} 0, & bpg13 \\mathbf {~has~value~} 0\\nonumber \\\\&\\mathbf {firing~style~} & max\\end{array}\\\\$ And the following query $\\mathbf {Q}$ for a simulation of length $k$ encodes the question: $&\\mathbf {direction~of~change~in~} average \\mathbf {~rate~of~production~of~} bpg13 \\mathbf {~is~} d\\nonumber \\\\&~~~~~~\\mathbf {when~observed~between~time~step~} 0 \\mathbf {~and~time~step~} k;\\nonumber \\\\&~~~~~~\\mathbf {comparing~nominal~pathway~with~modified~pathway~obtained~}\\nonumber \\\\&~~~~~~~~~~~~\\mathbf {due~to~interventions:} \\mathbf {~remove~} dhap \\mathbf {~as~soon~as~produced};\\nonumber \\\\&~~~~~~\\mathbf {using~initial~setup:~} \\mathbf {continuously~supply~} f16bp \\mathbf {~in~quantity~} 1;$ Since this is a comparative quantitative query statement, it is decomposed into two sub-queries, $\\mathbf {Q_0}$ capturing the nominal case of average rate of production w.r.t.", "given initial conditions: $\\begin{array}{lll}average \\mathbf {~rate~of~production~of~} bpg13 \\mathbf {~is~} n_{avg}\\\\~~~~~~~~~~~~\\mathbf {when~observed~between~time~step~} 0 \\mathbf {~and~time~step~} k; \\\\~~~~~~\\mathbf {using~initial~setup:~} \\mathbf {continuously~supply~} f16bp \\mathbf {~in~quantity~} 1;\\end{array}$ and $\\mathbf {Q_1}$ capturing the modified case in which the pathway is subject to interventions and observations w.r.t.", "initial conditions: $\\begin{array}{lll}average \\mathbf {~rate~of~production~of~} bpg13 \\mathbf {~is~} n^{\\prime }_{avg}\\\\~~~~~~~~~~~~\\mathbf {when~observed~between~time~step~} 0 \\mathbf {~and~time~step~} k; \\\\~~~~~~\\mathbf {due~to~interventions:} \\mathbf {~remove~} dhap \\mathbf {~as~soon~as~produced} ;\\\\~~~~~~\\mathbf {using~initial~setup:~} \\mathbf {continuously~supply~} f16bp \\mathbf {~in~quantity~} 1;\\end{array}$ Then the task is to determine $d$ , such that $\\mathbf {D} \\models \\mathbf {Q_0}$ for some value of $n_{avg}$ , $\\mathbf {D} \\models \\mathbf {Q_1}$ for some value of $n^{\\prime }_{avg}$ , and $n^{\\prime }_{avg} \\; d \\; n_{avg}$ .", "To answer the sub-queries $\\mathbf {Q_0}$ and $\\mathbf {Q_1}$ we build modified domain descriptions $\\mathbf {D_0}$ and $\\mathbf {D_1}$ , where, $\\mathbf {D_0} \\equiv \\mathbf {D} \\diamond (\\mathbf {continuously~supply~} f16bp $ $\\mathbf {~in~quantity~} 1)$ is the nominal domain description $\\mathbf {D}$ modified according to $\\mathbf {Q_0}$ to include the initial conditions; and $\\mathbf {D_1} \\equiv \\mathbf {D_0} \\diamond (\\mathbf {remove~} dhap $ $\\mathbf {~as~soon~as~}$ $\\mathbf {produced})$ is the modified domain description $\\mathbf {D}$ modified according to $\\mathbf {Q_1}$ to include the initial conditions as well as the interventions.", "The $\\diamond $ operator modifies the domain description to its left by adding, removing or modifying pathway specification language statements to add the interventions and initial conditions description to its right.", "Thus, $\\mathbf {D_0} &= \\mathbf {D} + \\left\\lbrace \\begin{array}{llll}tf_{f16bp} \\mathbf {~may~execute~causing~} &f16bp \\mathbf {~change~value~by~} +1\\\\\\end{array}\\right.\\\\\\mathbf {D_1} &= \\mathbf {D_0} + \\left\\lbrace \\begin{array}{llll}tr \\mathbf {~may~execute~causing~} &dhap \\mathbf {~change~value~by~} \\\\\\end{array}\\right.$ Performing a simulation of $k=5$ steps with $ntok=20$ max tokens, we find that the average rate of $bpg13$ production decreases from $n_{avg}=0.83$ to $n^{\\prime }_{avg}=0.5$ .", "Thus, $\\mathbf {D} \\models \\mathbf {Q}$ iff $d = ^{\\prime }<^{\\prime }$ .", "Alternatively, we say that the rate of glycolysis decreases when DHAP is removed as quickly as it is produced.", "Question 12 When and how does the body switch to B-oxidation versus glycolysis as the major way of burning fuel?", "Figure: Petri Net for question The following pathway specification encodes the domain description $\\mathbf {D}$ for question REF and produces the PN in Fig.", "REF : $\\begin{array}{llll}&\\mathbf {domain~of~} & gly \\mathbf {~is~} integer, & sug \\mathbf {~is~} integer, \\\\ && fac \\mathbf {~is~} integer, & acoa \\mathbf {~is~} integer \\\\&gly \\mathbf {~may~execute~causing~} & sug \\mathbf {~change~value~by~} -1, & acoa \\mathbf {~change~value~by~} +1\\\\&box \\mathbf {~may~execute~causing~} & fac \\mathbf {~change~value~by~} -1, & acoa \\mathbf {~change~value~by~} +1\\\\&\\mathbf {inhibit~} box \\mathbf {~if~} & sug \\mathbf {~has~value~} 1 \\mathbf {~or~higher}\\\\&\\mathbf {initially~} & sug \\mathbf {~has~value~} 3, & fac \\mathbf {~has~value~} 3\\\\ && acoa \\mathbf {~has~value~} 0 \\\\&t1 \\mathbf {~may~execute~causing~} & sug \\mathbf {~change~value~by~} +1\\\\&t2 \\mathbf {~may~execute~causing~} & fac \\mathbf {~change~value~by~} +1\\\\& \\mathbf {firing~style~} & \\end{array}$ where, $fac$ represents fatty acids, $sug$ represents sugar, $acoa$ represents acetyl coenzyme-A, $gly$ represents the process of glycolysis, and $box$ represents the process of beta oxidation.", "The question is asking for the general conditions when glycolysis switches to beta-oxidation, which is some property “$p$ ” that holds after which the switch occurs.", "The query $\\mathbf {Q}$ is encoded as: $&gly \\mathbf {~switches~to~} box \\mathbf {~when~} p; \\nonumber \\\\&~~~~\\mathbf {due~to~observations:} \\nonumber \\\\&~~~~~~~~gly \\mathbf {~switches~to~} box \\nonumber \\\\&~~~~\\mathbf {~in~all~trajectories}$ where condition `$p$ ' is a conjunction of simple point formulas.", "Then the task is to determine a minimal such conjunction of formulas that is satisfied in the state where $`gly^{\\prime } \\mathbf {~switches~to~} `box^{\\prime }$ holds over all trajectories.Note that this could be an LTL formula that must hold in all trajectories, but we did not add it here to keep the language simple.", "Since there is no change in initial conditions of the pathway and there are no interventions, the modified domain description $\\mathbf {D_1} \\equiv \\mathbf {D}$ .", "Intuitively, $p$ is the property that holds over fluents of the transitional state $s_j$ in which the switch takes palce, such that $gly \\in T_{j-1}, box \\notin T_{j-1}, gly \\notin T_j, box \\in T_j$ and the minimal set of firings leading up to it.", "The only trajectories to consider are the ones in which the observation is true.", "Thus the condition $p$ is determined as the intersection of sets of fluent based conditions that were true at the time of the switch, such as: $\\lbrace &sug \\mathbf {~has~value~} s_j(sug), sug \\mathbf {~has~value~higher~than~} 0, \\dots \\\\&sug \\mathbf {~has~value~higher~than~} s_j(sug)-1,sug \\mathbf {~has~value~lower~than~} s_j(sug)+1,\\\\&fac \\mathbf {~has~value~} s_j(fac), fac \\mathbf {~has~value~higher~than~} 0, \\dots \\\\&fac \\mathbf {~has~value~higher~than~} s_j(fac)-1, fac \\mathbf {~has~value~lower~than~} s_j(fac)+1, \\\\&acoa \\mathbf {~has~value~} s_j(acoa), acoa \\mathbf {~has~value~higher~than~} 0, \\dots \\\\&acoa \\mathbf {~has~value~higher~than~} s_j(acoa)-1, acoa \\mathbf {~has~value~lower~than~} s_j(acoa)+1\\rbrace $ Simulating it for $k=5$ steps with $ntok=20$ max tokens, we find the condition $p = acoa \\text{ has value greater than } 0, $ $sug \\text{ has value } 0, $ $sug \\text{ has value lower than } 1, $ $fac \\text{ has value higher than } 0 \\rbrace $ .", "Thus, the state when this switch occurs must sugar ($sug$ ) depleted and available supply of fatty acids ($fac$ ).", "Question 13 The final protein complex in the electron transport chain of the mitochondria is non-functional.", "Explain the effect of this on pH of the intermembrane space of the mitochondria.", "Figure: Petri Net for question The question is asking for the direction of change in the pH of the intermembrane space when the nominal case is compared against a modified pathway in which the complex 4 ($t4$ ) is defective.", "Since pH is defined as $-log_{10}(H+)$ , we monitor the total production of $H+$ ions to determine the change in pH value.", "However, since different world evolutions can follow different trajectories, we consider the average production of H+.", "Furthermore, we model the defective $t4$ as being unable to carry out its reaction, by disabling/inhibiting it.", "Then the following pathway specification encodes the domain description $\\mathbf {D}$ for question REF and produces the PN in Fig.", "REF without the $ft4$ place node.", "In the pathway description below, we have included only one domain declaration for all fluents as an example and left the rest out to save space.", "Assume integer domain declaration for all fluents.", "$&\\mathbf {domain~of~} \\\\&~~~~ nadh \\mathbf {~atloc~} mm \\mathbf {~is~} integer \\\\&t1 \\mathbf {~may~execute~causing~} \\\\&~~~~ nadh \\mathbf {~atloc~} mm \\mathbf {~change~value~by~} -2, h \\mathbf {~atloc~} mm \\mathbf {~change~value~by~} -2,\\\\&~~~~ e \\mathbf {~atloc~} q \\mathbf {~change~value~by~} +2, h \\mathbf {~atloc~} is \\mathbf {~change~value~by~} +2, \\\\&~~~~ nadp \\mathbf {~atloc~} mm \\mathbf {~change~value~by~} +2\\\\&t2 \\mathbf {~may~execute~causing~} \\\\&~~~~ fadh2 \\mathbf {~atloc~} mm \\mathbf {~change~value~by~} -2, e \\mathbf {~atloc~} q \\mathbf {~change~value~by~} +2,\\\\&~~~~ fad \\mathbf {~atloc~} mm \\mathbf {~change~value~by~} +2\\\\&t3 \\mathbf {~may~execute~causing~} \\\\&~~~~ e \\mathbf {~atloc~} q \\mathbf {~change~value~by~} -2, h \\mathbf {~atloc~} mm \\mathbf {~change~value~by~} -2,\\\\&~~~~ e \\mathbf {~atloc~} cytc \\mathbf {~change~value~by~} +2, h \\mathbf {~atloc~} is \\mathbf {~change~value~by~} +2\\\\&t4 \\mathbf {~may~execute~causing~} \\\\&~~~~ o2 \\mathbf {~atloc~} mm \\mathbf {~change~value~by~} -1, e \\mathbf {~atloc~} cytc \\mathbf {~change~value~by~} -2,\\\\&~~~~ h \\mathbf {~atloc~} mm \\mathbf {~change~value~by~} -6, h2o \\mathbf {~atloc~} mm \\mathbf {~change~value~by~} +2,\\\\&~~~~ h \\mathbf {~atloc~} is \\mathbf {~change~value~by~} +2\\\\&t10 \\mathbf {~may~execute~causing~} \\\\&~~~~ nadh \\mathbf {~atloc~} mm \\mathbf {~change~value~by~} +2, \\\\&~~~~ h \\mathbf {~atloc~} \\mathbf {~change~value~by~} +4, o2 \\mathbf {~atloc~} mm \\mathbf {~change~value~by~} +1\\\\&\\mathbf {initially~} \\\\&~~~~ fadh2 \\mathbf {~atloc~} mm \\mathbf {~has~value~} 0, e \\mathbf {~atloc~} mm \\mathbf {~has~value~} 0,\\\\&~~~~ fad \\mathbf {~atloc~} mm \\mathbf {~has~value~} 0, o2 \\mathbf {~atloc~} mm \\mathbf {~has~value~} 0,\\\\&~~~~ h2o \\mathbf {~atloc~} mm \\mathbf {~has~value~} 0, atp \\mathbf {~atloc~} mm \\mathbf {~has~value~} 0\\\\&\\mathbf {initially~} \\\\&~~~~ fadh2 \\mathbf {~atloc~} is \\mathbf {~has~value~} 0, e \\mathbf {~atloc~} is \\mathbf {~has~value~} 0,\\\\&~~~~ fad \\mathbf {~atloc~} is \\mathbf {~has~value~} 0, o2 \\mathbf {~atloc~} is \\mathbf {~has~value~} 0,\\\\&~~~~ h2o \\mathbf {~atloc~} is \\mathbf {~has~value~} 0, atp \\mathbf {~atloc~} is \\mathbf {~has~value~} 0\\\\&\\mathbf {initially~} \\\\&~~~~ fadh2 \\mathbf {~atloc~} q \\mathbf {~has~value~} 0, e \\mathbf {~atloc~} q \\mathbf {~has~value~} 0,\\\\&~~~~ fad \\mathbf {~atloc~} q \\mathbf {~has~value~} 0, o2 \\mathbf {~atloc~} q \\mathbf {~has~value~} 0,\\\\&~~~~ h2o \\mathbf {~atloc~} q \\mathbf {~has~value~} 0, atp \\mathbf {~atloc~} q \\mathbf {~has~value~} 0\\\\&\\mathbf {initially~} \\\\&~~~~ fadh2 \\mathbf {~atloc~} cytc \\mathbf {~has~value~} 0, e \\mathbf {~atloc~} cytc \\mathbf {~has~value~} 0,\\\\&~~~~ fad \\mathbf {~atloc~} cytc \\mathbf {~has~value~} 0, o2 \\mathbf {~atloc~} cytc \\mathbf {~has~value~} 0,\\\\&~~~~ h2o \\mathbf {~atloc~} cytc \\mathbf {~has~value~} 0, atp \\mathbf {~atloc~} cytc \\mathbf {~has~value~} 0\\\\&~~~~ \\mathbf {firing~style~} max\\\\$ where $mm$ represents the mitochondrial matrix, $is$ represents the intermembrane space, $t1-t4$ represent the reaction of the four complexes making up the electron transport chain, $h$ is the $H+$ ion, $nadh$ is $NADH$ , $fadh2$ is $FADH_2$ , $fad$ is $FAD$ , $e$ is electrons, $o2$ is oxygen $O_2$ , $atp$ is $ATP$ , $h2o$ is water $H_2O$ , and $t10$ is a source transition that supply a continuous supply of source ingredients for the chain to function, such as $nadh$ , $h$ , $o2$ .", "As a result, the query $\\mathbf {Q}$ asked by the question is encoded as follows: $\\begin{array}{l}\\mathbf {direction~of~change~in~} average \\mathbf {~total~production~of~} h \\mathbf {~atloc~} is \\mathbf {~is~} d\\\\~~~~~~~~\\mathbf {when~observed~between~time~step~} 0 \\mathbf {~and~time~step~} k ;\\\\~~~~~~~~\\mathbf {comparing~nominal~pathway~with~modified~pathway~obtained}\\\\~~~~~~~~\\mathbf {due~to~intervention~} t4 \\mathbf {~disabled} ;\\end{array}$ Since this is a comparative quantitative query statement, it is decomposed into two queries, $\\mathbf {Q_0}$ capturing the nominal case of average production w.r.t.", "given initial conditions: $\\begin{array}{l}average \\mathbf {~total~production~of~} h \\mathbf {~atloc~} is \\mathbf {~is~} n_{avg}\\\\~~~~~~~~\\mathbf {when~observed~between~time~step~} 0 \\mathbf {~and~time~step~} k ;\\end{array}$ and $\\mathbf {Q_1}$ the modified case w.r.t.", "initial conditions, modified to include interventions and subject to observations: $\\begin{array}{l}average \\mathbf {~total~production~of~} h \\mathbf {~atloc~} is \\mathbf {~is~} n^{\\prime }_{avg}\\\\~~~~~~~~\\mathbf {when~observed~between~time~step~} 0 \\mathbf {~and~time~step~} k ;\\\\~~~~~~~~\\mathbf {due~to~intervention~} t4 \\mathbf {~disabled} ;\\end{array}$ The the task is to determine $d$ , such that $\\mathbf {D} \\models \\mathbf {Q_0}$ for some value of $n_{avg}$ , $\\mathbf {D} \\models \\mathbf {Q_1}$ for some value of $n^{\\prime }_{avg}$ , and $n^{\\prime }_{avg} \\; d \\; n_{avg}$ .", "To answer the sub-queries $\\mathbf {Q_0}$ and $\\mathbf {Q_1}$ we build nominal description $\\mathbf {D_0}$ and modified pathway $\\mathbf {D_1}$ , where $\\mathbf {D_0} \\equiv \\mathbf {D}$ since there are no initial conditions; and $\\mathbf {D_1} \\equiv \\mathbf {D_0} \\diamond (t4 \\mathbf {~disabled})$ is the domain description $\\mathbf {D}$ modified according to $\\mathbf {Q_1}$ to modified to include the initial conditions as well as interventions.", "The $\\diamond $ operator modifies the domain description to its left by adding, removing or modifying pathway specification language statements to add the intervention and initial conditions to its right.", "Thus, $\\mathbf {D_1} = \\mathbf {D_0} + \\left\\lbrace \\begin{array}{llll}\\mathbf {inhibit~} t4\\\\\\end{array}\\right.$ Performing a simulation of $k=5$ steps with $ntok=20$ max tokens, we find that the average total production of H+ in the intermembrane space ($h$ at location $is$ ) reduces from 16 to 14.", "Lower quantity of H+ translates to a higer numeric value of $-log_{10}$ , as a result the pH increases.", "Question 14 Membranes must be fluid to function properly.", "How would decreased fluidity of the membrane affect the efficiency of the electron transport chain?", "Figure: Petri Net with colored tokens alternate for question From background knowledge, we know that, “Establishing the H+ gradient is a major function of the electron transport chain” [64], we measure the efficiency in terms of H+ ions moved to the intermembrane space ($is$ ) over time.", "Thus, we interpret the question is asking for the direction of change in the production of H+ moved to the intermembrane space when the nominal case is compared against a modified pathway with decreased fluidity of membrane.", "Additional background knowledge from [64] tells us that the decreased fluidity reduces the speed of mobile carriers such as $q$ and $cytc$ .", "Fluidity can span a range of values, but we will consider one such value $v$ per query.", "The following pathway specification encodes the domain description $\\mathbf {D}$ for question REF and produces the PN in Fig.", "REF minus the places $q\\_3,cytc\\_4$ and transitions $tq,tcytc$ .", "In the pathway description below, we have included only one domain declaration for all fluents as an example and left the rest out to save space.", "Assume integer domain declaration for all fluents.", "$&\\mathbf {domain~of~} \\\\&~~~~nadh \\mathbf {~atloc~} mm \\mathbf {~is~} integer \\\\&t1 \\mathbf {~may~execute~causing~} \\\\&~~~~ nadh \\mathbf {~atloc~} mm \\mathbf {~change~value~by~} -2, h \\mathbf {~atloc~} mm \\mathbf {~change~value~by~} -2,\\\\&~~~~ e \\mathbf {~atloc~} q \\mathbf {~change~value~by~} +2, h \\mathbf {~atloc~} is \\mathbf {~change~value~by~} +2,\\\\&~~~~ nadp \\mathbf {~atloc~} mm \\mathbf {~change~value~by~} +2\\\\&t3 \\mathbf {~may~execute~causing~} \\\\&~~~~ e \\mathbf {~atloc~} q \\mathbf {~change~value~by~} -2, h \\mathbf {~atloc~} mm \\mathbf {~change~value~by~} -2,\\\\&~~~~ e \\mathbf {~atloc~} cytc \\mathbf {~change~value~by~} +2, h \\mathbf {~atloc~} is \\mathbf {~change~value~by~} +2\\\\&t4 \\mathbf {~may~execute~causing~} \\\\&~~~~ o2 \\mathbf {~atloc~} mm \\mathbf {~change~value~by~} -1, e \\mathbf {~atloc~} cytc \\mathbf {~change~value~by~} -2, \\\\&~~~~ h \\mathbf {~atloc~} mm \\mathbf {~change~value~by~} -2, h2o \\mathbf {~atloc~} mm \\mathbf {~change~value~by~} +2,\\\\&~~~~ h \\mathbf {~atloc~} is \\mathbf {~change~value~by~} +2\\\\&t10 \\mathbf {~may~execute~causing~} \\\\&~~~~ nadh \\mathbf {~atloc~} mm \\mathbf {~change~value~by~} +2, h \\mathbf {~atloc~} mm \\mathbf {~change~value~by~} +4,\\\\&~~~~ o2 \\mathbf {~atloc~} mm \\mathbf {~change~value~by~} +1\\\\& \\mathbf {initially~} \\\\&~~~~ nadh \\mathbf {~atloc~} mm \\mathbf {~has~value~} 0, h \\mathbf {~atloc~} mm \\mathbf {~has~value~} 0,\\\\&~~~~ nadp \\mathbf {~atloc~} mm \\mathbf {~has~value~} 0, o2 \\mathbf {~atloc~} mm \\mathbf {~has~value~} 0\\\\&~~~~ h2o \\mathbf {~atloc~} mm \\mathbf {~has~value~} 0 \\\\& \\mathbf {initially~} \\\\&~~~~ nadh \\mathbf {~atloc~} is \\mathbf {~has~value~} 0, h \\mathbf {~atloc~} is \\mathbf {~has~value~} 0,\\\\&~~~~ nadp \\mathbf {~atloc~} is \\mathbf {~has~value~} 0, o2 \\mathbf {~atloc~} is \\mathbf {~has~value~} 0\\\\&~~~~ h2o \\mathbf {~atloc~} is \\mathbf {~has~value~} 0 \\\\& \\mathbf {initially~} \\\\&~~~~ nadh \\mathbf {~atloc~} q \\mathbf {~has~value~} 0, h \\mathbf {~atloc~} q \\mathbf {~has~value~} 0,\\\\&~~~~ nadp \\mathbf {~atloc~} q \\mathbf {~has~value~} 0, o2 \\mathbf {~atloc~} q \\mathbf {~has~value~} 0\\\\&~~~~ h2o \\mathbf {~atloc~} q \\mathbf {~has~value~} 0 \\\\& \\mathbf {initially~} \\\\&~~~~ nadh \\mathbf {~atloc~} cytc \\mathbf {~has~value~} 0, h \\mathbf {~atloc~} cytc \\mathbf {~has~value~} 0,\\\\&~~~~ nadp \\mathbf {~atloc~} cytc \\mathbf {~has~value~} 0, o2 \\mathbf {~atloc~} cytc \\mathbf {~has~value~} 0\\\\&~~~~ h2o \\mathbf {~atloc~} cytc \\mathbf {~has~value~} 0 \\\\&\\mathbf {firing~style~} max\\\\$ The query $\\mathbf {Q}$ asked by the question is encoded as follows: $\\begin{array}{l}\\mathbf {direction~of~change~in~} average \\mathbf {~total~production~of~} h \\mathbf {~atloc~} is \\mathbf {~is~} d\\\\~~~~~~\\mathbf {when~observed~between~time~step~} 0 \\mathbf {~and~time~step~} k;\\\\~~~~~~\\mathbf {comparing~nominal~pathway~with~modified~pathway~obtained~}\\\\~~~~~~~~~~~~\\mathbf {due~to~interventions:} \\\\~~~~~~~~~~~~~~~~~~\\mathbf {add~delay~of~} v \\mathbf {~time~units~in~availability~of~} e \\mathbf {~atloc~} q,\\\\~~~~~~~~~~~~~~~~~~\\mathbf {add~delay~of~} v \\mathbf {~time~units~in~availability~of~} e \\mathbf {~atloc~} cytc;\\end{array}$ Since this is a comparative quantitative query statement, we decompose it into two queries, $\\mathbf {Q_0}$ capturing the nominal case of average rate of production w.r.t.", "given initial conditions: $\\begin{array}{l}average \\mathbf {~total~production~of~} h \\mathbf {~atloc~} is \\mathbf {~is~} d\\\\~~~~~~~~~~~~\\mathbf {when~observed~between~time~step~} 0 \\mathbf {~and~time~step~} k;\\\\\\end{array}$ and $\\mathbf {Q_1}$ the modified case w.r.t.", "initial conditions, modified to include interventions and subject to observations: $\\begin{array}{l}average \\mathbf {~total~production~of~} h \\mathbf {~atloc~} is \\mathbf {~is~} d\\\\~~~~~~~~~~~~\\mathbf {when~observed~between~time~step~} 0 \\mathbf {~and~time~step~} k;\\\\~~~~~~\\mathbf {due~to~interventions:} \\\\~~~~~~~~~~~~\\mathbf {add~delay~of~} v \\mathbf {~time~units~in~availability~of~} e \\mathbf {~atloc~} q,\\\\~~~~~~~~~~~~\\mathbf {add~delay~of~} v \\mathbf {~time~units~in~availability~of~} e \\mathbf {~atloc~} cytc;\\end{array}$ Then the task is to determine $d$ , such that $\\mathbf {D} \\models \\mathbf {Q_0}$ for some value of $n_{avg}$ , $\\mathbf {D} \\models \\mathbf {Q_1}$ for some value of $n^{\\prime }_{avg}$ , and $n^{\\prime }_{avg} \\; d \\; n_{avg}$ .", "To answer the sub-queries $\\mathbf {Q_0}$ and $\\mathbf {Q_1}$ we build modified domain descriptions $\\mathbf {D_0}$ and $\\mathbf {D_1}$ , where, $\\mathbf {D_0} \\equiv \\mathbf {D}$ is the nominal domain description $\\mathbf {D}$ modified according to $\\mathbf {Q_0}$ to include the initial conditions; and $\\mathbf {D_1} \\equiv \\mathbf {D_0} \\diamond (\\mathbf {add~delay~of~} v \\mathbf {~time~units~in~availability~of~} $ $e \\mathbf {~atloc~} q) \\diamond (\\mathbf {add~delay~of~} v \\mathbf {~time~units~in~availability~of~} $ $e \\mathbf {~atloc~} cytc)$ is the modified domain description $\\mathbf {D}$ modified according to $\\mathbf {Q_1}$ to include the initial conditions as well as the interventions.", "The $\\diamond $ operator modifies the domain description to its left by adding, removing or modifying pathway specification language statements to add the interventions and initial conditions description to its right.", "Thus, $\\mathbf {D_1} = \\mathbf {D} &-\\left\\lbrace \\begin{array}{llll}t1 \\mathbf {~may~execute~causing~} & nadh \\mathbf {~atloc~} mm \\mathbf {~change~value~by~} -2, \\\\& h \\mathbf {~atloc~} mm \\mathbf {~change~value~by~} -2,\\\\& e \\mathbf {~atloc~} q \\mathbf {~change~value~by~} +2, \\\\& h \\mathbf {~atloc~} is \\mathbf {~change~value~by~} +2,\\\\& nadp \\mathbf {~atloc~} mm \\mathbf {~change~value~by~} +2\\\\t3 \\mathbf {~may~execute~causing~} & e \\mathbf {~atloc~} q \\mathbf {~change~value~by~} -2, \\\\& h \\mathbf {~atloc~} mm \\mathbf {~change~value~by~} -2,\\\\& e \\mathbf {~atloc~} cytc \\mathbf {~change~value~by~} +2, \\\\& h \\mathbf {~atloc~} is \\mathbf {~change~value~by~} +2\\\\\\end{array}\\right\\rbrace \\\\& +\\left\\lbrace \\begin{array}{llll}t1 \\mathbf {~may~execute~causing~} & nadh \\mathbf {~atloc~} mm \\mathbf {~change~value~by~} -2, \\\\& h \\mathbf {~atloc~} mm \\mathbf {~change~value~by~} -2,\\\\& e \\mathbf {~atloc~} q\\_3 \\mathbf {~change~value~by~} +2, \\\\& h \\mathbf {~atloc~} is \\mathbf {~change~value~by~} +2,\\\\& nadp \\mathbf {~atloc~} mm \\mathbf {~change~value~by~} +2\\\\t3 \\mathbf {~may~execute~causing~} & e \\mathbf {~atloc~} q \\mathbf {~change~value~by~} -2, \\\\& h \\mathbf {~atloc~} mm \\mathbf {~change~value~by~} -2,\\\\& e \\mathbf {~atloc~} cytc\\_4 \\mathbf {~change~value~by~} +2, \\\\& h \\mathbf {~atloc~} is \\mathbf {~change~value~by~} +2\\\\tq \\mathbf {~may~fire~causing~} & e \\mathbf {~atloc~} q\\_3 \\mathbf {~change~value~by~} -2, \\\\& e \\mathbf {~atloc~} q \\mathbf {~change~value~by~} +2\\\\tcytc \\mathbf {~may~fire~causing~} & e \\mathbf {~atloc~} cytc\\_4 \\mathbf {~change~value~by~} -2, \\\\& e \\mathbf {~atloc~} cytc \\mathbf {~change~value~by~} +2\\\\tq \\mathbf {~executes~} & \\mathbf {~in~} 4 \\mathbf {~time~units}\\\\tcytc \\mathbf {~executes~} & \\mathbf {~in~} 4 \\mathbf {~time~units}\\\\\\end{array}\\right\\rbrace \\\\$ Performing a simulation of k = 5 steps with ntok = 20 max tokens with a fluidity based delay of 2, we find that the average total production of H+ in the intermembrane space (h at location is) reduces from 16 to 10.", "Lower quantity of H+ going into the intermembrane space means lower efficiency, where we define the efficiency as the total amount of $H+$ ions transferred to the intermembrane space over the simulation run.", "Example Encoding with Conditional Actions Next, we illustrate how conditional actions would be encoded in our high-level language with an example.", "Consider the pathway from question REF .", "Say, the reaction step $t4$ has developed a fault, in which it has two modes of operation, in the first mode, when $f16bp$ has less than 3 units available, the reaction proceeds normally, but when $f16bp$ is available in 3 units or higher, the reaction continues to produce $g3p$ but not $dhap$ directly.", "$dhap$ can still be produced by subsequent step from $g3p$ .", "The modified pathway is given in our pathway specification language below: $\\begin{array}{llll}&t3 \\mathbf {~may~execute~causing~} & f16bp \\mathbf {~change~value~by~} +1\\\\&t4 \\mathbf {~may~execute~causing~} & f16bp \\mathbf {~change~value~by~} -1, & dhap \\mathbf {~change~value~by~} +1,\\\\ && g3p \\mathbf {~change~value~by~} +1\\\\&~~~~~~~~~~~~~~~~\\mathbf {if~}& g3p \\mathbf {~has~value~lower~than~} 3\\\\&t4 \\mathbf {~may~execute~causing~} & f16bp \\mathbf {~change~value~by~} -1, & dhap \\mathbf {~change~value~by~} +1\\\\&~~~~~~~~~~~~~~~~\\mathbf {~if}& g3p \\mathbf {~has~value~} 3 \\mathbf {~or~higher}\\\\&t5a \\mathbf {~may~execute~causing~} & dhap \\mathbf {~change~value~by~} -1, & g3p \\mathbf {~change~value~by~} +1\\\\&t5b \\mathbf {~may~execute~causing~} & g3p \\mathbf {~change~value~by~} -1, & dhap \\mathbf {~change~value~by~} +1\\\\&t6 \\mathbf {~may~execute~causing~} & g3p \\mathbf {~change~value~by~} -1, & bpg13 \\mathbf {~change~value~by~} +2\\\\& \\mathbf {initially~} & f16bp \\mathbf {~has~value~} 0, & dhap \\mathbf {~has~value~} 0,\\\\ && g3p \\mathbf {~has~value~} 0, & bpg13 \\mathbf {~has~value~} 0\\\\&\\mathbf {firing~style~} & max\\end{array}$ We ask the same question $\\mathbf {Q}$ : $&\\mathbf {direction~of~change~in~} average \\mathbf {~rate~of~production~of~} bpg13 \\mathbf {~is~} d\\\\&~~~~~~\\mathbf {when~observed~between~time~step~} 0 \\mathbf {~and~time~step~} k; \\\\&~~~~~~\\mathbf {comparing~nominal~pathway~with~modified~pathway~obtained~}\\\\&~~~~~~~~~~~~\\mathbf {due~to~interventions:~} \\mathbf {remove~} dhap \\mathbf {~as~soon~as~produced};\\\\&~~~~~~~~~~~~\\mathbf {using~initial~setup:~} \\mathbf {continuously~supply~} f16bp \\mathbf {~in~quantity~} 1;$ Since this is a comparative quantitative query statement, we decompose it into two queries, $\\mathbf {Q_0}$ capturing the nominal case of average rate of production w.r.t.", "given initial conditions: $\\begin{array}{l}average \\mathbf {~rate~of~production~of~} bpg13 \\mathbf {~is~} n\\\\~~~~~~~~~~~~\\mathbf {when~observed~between~time~step~} 0 \\mathbf {~and~time~step~} k;\\\\~~~~~~\\mathbf {using~initial~setup:~} \\mathbf {continuously~supply~} f16bp \\mathbf {~in~quantity~} 1;\\end{array}$ and $\\mathbf {Q_1}$ the modified case w.r.t.", "initial conditions, modified to include interventions and subject to observations: $\\begin{array}{l}average \\mathbf {~rate~of~production~of~} bpg13 \\mathbf {~is~} n\\\\~~~~~~~~~~~~\\mathbf {when~observed~between~time~step~} 0 \\mathbf {~and~time~step~} k;\\\\~~~~~~\\mathbf {due~to~interventions:~} \\mathbf {remove~} dhap \\mathbf {~as~soon~as~produced};\\\\~~~~~~\\mathbf {using~initial~setup:~} \\mathbf {continuously~supply~} f16bp \\mathbf {~in~quantity~} 1;\\end{array}$ Then the task is to determine $d$ , such that $\\mathbf {D} \\models \\mathbf {Q_0}$ for some value of $n_{avg}$ , $\\mathbf {D} \\models \\mathbf {Q_1}$ for some value of $n^{\\prime }_{avg}$ , and $n^{\\prime }_{avg} \\; d \\; n_{avg}$ .", "To answer the sub-queries $\\mathbf {Q_0}$ and $\\mathbf {Q_1}$ we build modified domain descriptions $\\mathbf {D_0}$ and $\\mathbf {D_1}$ , where, $\\mathbf {D_0} \\equiv \\mathbf {D} \\diamond (\\mathbf {continuously~supply~} f16bp $ $\\mathbf {~in~quantity~} 1)$ is the nominal domain description $\\mathbf {D}$ modified according to $\\mathbf {Q_0}$ to include the initial conditions; and $\\mathbf {D_1} \\equiv \\mathbf {D_0} \\diamond (\\mathbf {remove~} dhap \\mathbf {~as~soon~as~produced})$ is the modified domain description $\\mathbf {D}$ modified according to $\\mathbf {Q_1}$ to include the initial conditions as well as the interventions.", "The $\\diamond $ operator modifies the domain description to its left by adding, removing or modifying pathway specification language statements to add the interventions and initial conditions description to its right.", "Thus, $\\mathbf {D_0} &= \\mathbf {D} + \\left\\lbrace \\begin{array}{llll}tf_{f16bp} \\mathbf {~may~execute~causing~} &f16bp \\mathbf {~change~value~by~} +1\\\\\\end{array}\\right.\\\\\\mathbf {D_1} &= \\mathbf {D_0} + \\left\\lbrace \\begin{array}{llll}tr \\mathbf {~may~execute~causing~} &dhap \\mathbf {~change~value~by~} \\\\\\end{array}\\right.$ ASP Program Next we briefly outline how the pathway specification and the query statement components are encoded in ASP (using Clingo syntax).", "In the following section, we will illustrate the process using an example.", "As evident from the previous sections, we need to simulate non-comparative queries only.", "Any comparative queries are translated into non-comparative sub-queries, each of which is simulated and their results compared to evaluate the comparative query.", "The ASP program is a concatenation of the translation of a pathway specification (domain description which includes the firing style) and internal observations.", "Any initial setup conditions and interventions are pre-applied to the pathway specification using intervention semantics in section REF before it is translated to ASP using the translation in chapter as our basis.", "The encoded pathway specification has the semantics defined in section REF .", "Internal observations in the `due to observations:' portion of query statement are translated into ASP constraints using the internal observation semantics defined in section REF and added to the encoding of the pathway specification.", "The program if simulated for a specified simulation length $k$ produces all trajectories of the pathway for the specified firing style, filtered by the internal observations.", "The query description specified in the query statement is then evaluated w.r.t.", "these trajectories.", "Although this part can be done in ASP, we have currently implemented it outside ASP in our implementation for ease of using floating point math.", "Next, we describe an implementation of our high level language and illustrate the construction of an ASP program, its simulation, and query statement evaluation.", "Implementation We have developed an implementation Implementation available at: https://sites.google.com/site/deepqa2014/ of a subset of our high level (Pathway and Query specification) language in Python.", "We use the Clingo ASP implementation for our simulation.", "In this section we describe various components of this implementation.", "An architectural overview of our implementation is shown in figure REF .", "Figure: BioPathQA Implementation System ArchitectureThe Pathway Specification Language (BioPathQA-PL) Parser component is responsible for parsing the Pathway Specification Language (BioPathQA-PL).", "It use PLY (Python Lex-Yacc)http://www.dabeaz.com/ply to parse a given pathway specification using grammar based on section REF .", "On a successful parse, a Guarded-Arc Petri Net pathway model based on section REF is constructed for the pathway specification.", "The Query Language Parser component is responsible for parsing the Query Specification Language (BioPathQA-QL).", "It uses PLY to parse a given query statement using grammar based on section REF .", "On a successful parse, an internal representation of the query statement is constructed.", "Elements of this internal representation include objects representing the query description, the list of interventions, the list of internal observations, and the list of initial setup conditions.", "Each intervention and initial setup condition object has logic in it to modify a given pathway per the intervention semantics described in section REF .", "The Query Statement Model component is also responsible for generating basic queries for aggregate queries and implementing interventions in the Petri Net Pathway Model.", "The Dictionary of fluents, locations, and actions is consulted by the ASP code generator to standardize symbol names in the ASP code produced for the pathway specification and the internal observations.", "The ASP Translator component is responsible for translating the Guarded-Arc Petri Net model into ASP facts and rules; and the driver needed to simulate the model using the firing semantics specified in the pathway model.", "The code generated is based on the ASP translation of Petri Nets and its various extensions given in chapter .", "To reduce the ASP code and its complexity, the translator limits the output model to the extensions used in the Petri Net model to be translated.", "Thus, the colored tokens extension code is not produced unless colored tokens have been used.", "Similarly, guarded-arcs code is not produced if no arc-guards are used in the model.", "The ASP Translator component is also responsible for translating internal observations from the Query Statement into ASP constraints to filter Petri Net trajectories based on the observation semantics in section REF .", "Following examples illustrate our encoding scheme.", "The observation `$a_1 \\mathbf {~switches~to~} a_2$ ' is encoded as a constraint using the following rules: obs_1_occurred(TS+1) :- time(TS;TS+1), trans(a1;a2), fires(a1,TS), not fires(a2,TS), not fires(a1,TS+1), fires(a2,TS+1).", "obs_1_occurred :- obs_1_occurred(TS), time(TS).", "obs_1_had_occurred(TSS) :- obs_1_occurred(TS), TS<=TSS, time(TSS;TS).", ":- not obs_1_occurred.", "The observation `$a_1 \\mathbf {~occurs~at~time~step~} 5$ ' is encoded as a constraint using the following rules: obs_2_occurred(TS) :- fires(a1,TS), trans(a1), time(TS), TS=5.", "obs_2_occurred :- obs_2_occurred(TS), time(TS).", "obs_2_had_occurred(TSS) :- obs_2_occurred(TS), TS<=TSS, time(TSS;TS).", ":- not obs_2_occurred.", "The observation `$s_1 \\mathbf {~is~decreasing~atloc~} l_1 \\mathbf {~when~observed~between~time~step~} 0 $ $\\mathbf {~and~time~step~} 5$ ' is encoded as a constraint using the following rules: obs_3_violated(TS) :- place(l1), col(s1), holds(l1,Q1,s1,TS), holds(l1,Q2,s1,TS+1), num(Q1;Q2), Q2 > Q1, time(TS;TS+1), TS=0, TS+1=5.", "obs_3_violated :- obs_3_violated(TS), time(TS).", "obs_3_occurred(TS+1) :- not obs_3_violated, holds(l1,Q1,s1,TS), holds(l1,Q2,s1,TS+1), time(TS;TS+1), num(Q1;Q2), Q2<Q1, TS=0, TS+1=5.", "obs_3_occurred :- obs_3_occurred(TS), time(TS).", "obs_3_had_occurred(TSS) :- obs_3_occurred(TS), TS<=TSS, time(TSS;TS).", ":- obs_3_occurred.", "In addition, the translator is also responsible for any rules needed to ease post-processing of the query description.", "For example, for qualitative queries, a generic predicate tgt_obs_occurred(TS) is generated that is true when the given qualitative description holds in an answer-set at time step $TS$ .", "The output of the translator is an ASP program, which when simulated using Clingo produces the (possibly) filtered trajectories of the pathway.", "The Post Processor component is responsible for parsing the ASP answer sets, identifying the correct atoms from it, extracting quantities from atom-bodies as necessary, organizing them into a matrix form, and aggregating them as needed.", "Figure: BioPathQA Graphical User InterfaceThe User Interface component is responsible for coordinating the processing of query statement.", "It presents the user with a graphical user interface shown in figure REF .", "The user types a Pathway Specification (in BioPathQA-PL syntax), a Query Specification (in BioPathQA-QL syntax), and simulation parameters.", "On pressing “Execute Query”, the user interface component processes the query as prints results in the bottom box.", "Query evaluation differs by the type of query.", "We describe the query evaluation methodology used below.", "For non-comparative quantitative queries: Pathway specification is parsed into a Guarded-Arc Petri Net model.", "Query statement is parsed into an internal form.", "Initial conditions from the query are applied to the pathway model.", "Interventions are applied to the pathway model.", "Modified pathway model is translated to ASP.", "Internal observations are added to the ASP code as ASP constraints.", "Answer sets of the ASP code are computed using Clingo.", "Relevant atoms are extracted: fires/2 predicate for firing rate, holds/3 (or holds/4 – colored tokens) predicate for fluent quantity or rate formulas.", "Fluent value or firing-count values are extracted and organized as matrices with rows representing answer-sets and columns representing time-steps.", "Within answer-set interval or point value sub-select is done and the values converted to rates or totals as needed.", "If aggregation, such as average, minimum, or maximum is desired, it is performed over rows of values from the last step.", "If a value was specified in the query, it is compared against the computed aggregate for boolean result.", "If a value was not specified, the computed value is returned as the value satisfying the query statement.", "For queries over all trajectories, the same value must hold over all trajectories, otherwise, only one match is required to satisfy the query.", "For non-comparative qualitative queries: Follow steps (REF )-(REF ) of non-comparative quantitative queries.", "Add rules for the query description for post-processing.", "Answer sets of the ASP code are computed using Clingo.", "Relevant atoms are extracted: tgt_obs_occurred/1 identifying the time step when the observation within the query description is satisfied.", "Truth value of the query observation is determined, including determining the truth value over all trajectories.", "For comparative quantitative queries: Query statement is decomposed into two non-comparative quantitative sub-query statements as illustrated in section REF : A nominal sub-query which has the same initial conditions as the comparative query, but none of its interventions or observations A modified sub-query which has the same initial conditions, interventions, and observations as the comparative query both sub-query statements have the same query description, which is the non-aggregate form of the comparative query description.", "Thus, a comparative average rate query is translated to non-comparative average rate sub-queries.", "Each sub-query statements is evaluated using steps (REF )-(REF ) from the non-comparative quantitative query processing.", "A direction of change is computed by comparing the computed aggregate value for the modified query statement to the nominal query statement.", "If the comparative quantitative query has a direction specified, it is the compared against the computed value for a boolean result.", "If the comparative quantitative query did not have a direction specified, the computed value is returned as the value satisfying the query statement.", "For explanation queries with query description with formula of the form (), it is expected that the number of answer-sets will be quite large.", "So, we avoid generating all answer-sets before processing them, instead we process them in-line as they are generated.", "It is a bit slower, but leads to a smaller memory foot print.", "Follow steps (REF )-(REF ) of non-comparative quantitative queries.", "Add rules for the query description for post-processing.", "Compute answer sets of the ASP code using Clingo.", "Extract relevant atoms: extract tgt_obs_occurred/1 identifying the time step when the query description is satisfied extract holds/3 (or holds/4 – for colored tokens) at the same time-step as tgt_obs_occurred/1 to construct fluent-based conditions Construct fluent-based conditions as explanation of the query observation.", "If the query is over all trajectories, fluent-based conditions for each trajectory are intersected across trajectories to determine the minimum set of conditions explaining the query observation.", "Next we illustrate query processing through an execution trace of question (REF ).", "The following shows the encoding of the base case domain, which includes the pathway specification from (REF ) with initial setup conditions from the query statement (REF ) applied: #const nts=1.", "time(0..nts).", "#const ntok=1.", "num(0..ntok).", "place(bpg13).", "place(dhap).", "place(f16bp).", "place(g3p).", "trans(src_f16bp_1).", "trans(t3).", "trans(t4).", "trans(t5a).", "trans(t5b).", "trans(t6).", "tparc(src_f16bp_1,f16bp,1,TS) :- time(TS).", "tparc(t3,f16bp,1,TS) :- time(TS).", "ptarc(f16bp,t4,1,TS) :- time(TS).", "tparc(t4,g3p,1,TS) :- time(TS).", "tparc(t4,dhap,1,TS) :- time(TS).", "ptarc(dhap,t5a,1,TS) :- time(TS).", "tparc(t5a,g3p,1,TS) :- time(TS).", "ptarc(g3p,t5b,1,TS) :- time(TS).", "tparc(t5b,dhap,1,TS) :- time(TS).", "ptarc(g3p,t6,1,TS) :- time(TS).", "tparc(t6,bpg13,2,TS) :- time(TS).", "holds(bpg13,0,0).", "holds(dhap,0,0).", "holds(f16bp,0,0).", "holds(g3p,0,0).", "could_not_have(T,TS) :- enabled(T,TS), not fires(T,TS), ptarc(P,T,Q,TS), holds(P,QQ,TS), tot_decr(P,QQQ,TS), Q > QQ - QQQ.", ":- not could_not_have(T,TS), time(TS), enabled(T,TS), not fires(T,TS), trans(T).", "min(A,B,A) :- A<=B, num(A;B).", "min(A,B,B) :- B<=A, num(A;B).", "#hide min/3.", "holdspos(P):- holds(P,N,0), place(P), num(N), N > 0. holds(P,0,0) :- place(P), not holdspos(P).", "notenabled(T,TS) :- ptarc(P,T,N,TS), holds(P,Q,TS), Q < N, place(P), trans(T), time(TS), num(N), num(Q).", "enabled(T,TS) :- trans(T), time(TS), not notenabled(T, TS).", "{ fires(T,TS) } :- enabled(T,TS), trans(T), time(TS).", "add(P,Q,T,TS) :- fires(T,TS), tparc(T,P,Q,TS), time(TS).", "del(P,Q,T,TS) :- fires(T,TS), ptarc(P,T,Q,TS), time(TS).", "tot_incr(P,QQ,TS) :- QQ = #sum[add(P,Q,T,TS) = Q : num(Q) : trans(T)], time(TS), num(QQ), place(P).", "tot_decr(P,QQ,TS) :- QQ = #sum[del(P,Q,T,TS) = Q : num(Q) : trans(T)], time(TS), num(QQ), place(P).", "holds(P,Q,TS+1) :- holds(P,Q1,TS), tot_incr(P,Q2,TS), tot_decr(P,Q3,TS), Q=Q1+Q2-Q3, place(P), num(Q;Q1;Q2;Q3), time(TS), time(TS+1).", "consumesmore(P,TS) :- holds(P,Q,TS), tot_decr(P,Q1,TS), Q1 > Q. consumesmore :- consumesmore(P,TS).", ":- consumesmore.", "The following shows encoding of the alternate case domain, which consists of the pathway specification from (REF ) with initial setup conditions and interventions applied; and any internal observations from the query statement (REF ) added: #const nts=1.", "time(0..nts).", "#const ntok=1.", "num(0..ntok).", "place(bpg13).", "place(dhap).", "place(f16bp).", "place(g3p).", "trans(reset_dhap_1).", "trans(src_f16bp_1).", "trans(t3).", "trans(t4).", "trans(t5a).", "trans(t5b).", "trans(t6).", "ptarc(dhap,reset_dhap_1,Q,TS) :- holds(dhap,Q,TS), Q>0, time(TS).", ":- enabled(reset_dhap_1,TS), not fires(reset_dhap_1,TS), time(TS).", "tparc(src_f16bp_1,f16bp,1,TS) :- time(TS).", "tparc(t3,f16bp,1,TS) :- time(TS).", "ptarc(f16bp,t4,1,TS) :- time(TS).", "tparc(t4,g3p,1,TS) :- time(TS).", "tparc(t4,dhap,1,TS) :- time(TS).", "ptarc(dhap,t5a,1,TS) :- time(TS).", "tparc(t5a,g3p,1,TS) :- time(TS).", "ptarc(g3p,t5b,1,TS) :- time(TS).", "tparc(t5b,dhap,1,TS) :- time(TS).", "ptarc(g3p,t6,1,TS) :- time(TS).", "tparc(t6,bpg13,2,TS) :- time(TS).", "holds(bpg13,0,0).", "holds(dhap,0,0).", "holds(f16bp,0,0).", "holds(g3p,0,0).", "could_not_have(T,TS) :- enabled(T,TS), not fires(T,TS), ptarc(P,T,Q,TS), holds(P,QQ,TS), tot_decr(P,QQQ,TS), Q > QQ - QQQ.", ":- not could_not_have(T,TS), time(TS), enabled(T,TS), not fires(T,TS), trans(T).", "min(A,B,A) :- A<=B, num(A;B).", "min(A,B,B) :- B<=A, num(A;B).", "#hide min/3.", "holdspos(P):- holds(P,N,0), place(P), num(N), N > 0. holds(P,0,0) :- place(P), not holdspos(P).", "notenabled(T,TS) :- ptarc(P,T,N,TS), holds(P,Q,TS), Q < N, place(P), trans(T), time(TS), num(N), num(Q).", "enabled(T,TS) :- trans(T), time(TS), not notenabled(T, TS).", "{ fires(T,TS) } :- enabled(T,TS), trans(T), time(TS).", "add(P,Q,T,TS) :- fires(T,TS), tparc(T,P,Q,TS), time(TS).", "del(P,Q,T,TS) :- fires(T,TS), ptarc(P,T,Q,TS), time(TS).", "tot_incr(P,QQ,TS) :- QQ = #sum[add(P,Q,T,TS) = Q : num(Q) : trans(T)], time(TS), num(QQ), place(P).", "tot_decr(P,QQ,TS) :- QQ = #sum[del(P,Q,T,TS) = Q : num(Q) : trans(T)], time(TS), num(QQ), place(P).", "holds(P,Q,TS+1) :- holds(P,Q1,TS), tot_incr(P,Q2,TS), tot_decr(P,Q3,TS), Q=Q1+Q2-Q3, place(P), num(Q;Q1;Q2;Q3), time(TS), time(TS+1).", "consumesmore(P,TS) :- holds(P,Q,TS), tot_decr(P,Q1,TS), Q1 > Q. consumesmore :- consumesmore(P,TS).", ":- consumesmore.", "Both programs are simulated for 5 time-steps and 20 max tokens using the following Clingo command: clingo 0 -cntok=20 -cnts=5 program.lp Answer sets of the base case are as follows: Answer: 1 holds(bpg13,0,0) holds(dhap,0,0) holds(f16bp,0,0) holds(g3p,0,0) fires(src_f16bp_1,0) fires(t3,0) holds(bpg13,0,1) holds(dhap,0,1) holds(f16bp,2,1) holds(g3p,0,1) fires(src_f16bp_1,1) fires(t3,1) fires(t4,1) holds(bpg13,0,2) holds(dhap,1,2) holds(f16bp,3,2) holds(g3p,1,2) fires(src_f16bp_1,2) fires(t3,2) fires(t4,2) fires(t5a,2) fires(t5b,2) holds(bpg13,0,3) holds(dhap,2,3) holds(f16bp,4,3) holds(g3p,2,3) fires(src_f16bp_1,3) fires(t3,3) fires(t4,3) fires(t5a,3) fires(t5b,3) fires(t6,3) holds(bpg13,2,4) holds(dhap,3,4) holds(f16bp,5,4) holds(g3p,2,4) fires(src_f16bp_1,4) fires(t3,4) fires(t4,4) fires(t5a,4) fires(t5b,4) fires(t6,4) holds(bpg13,4,5) holds(dhap,4,5) holds(f16bp,6,5) holds(g3p,2,5) fires(src_f16bp_1,5) fires(t3,5) fires(t4,5) fires(t5a,5) fires(t5b,5) fires(t6,5) Answer: 2 holds(bpg13,0,0) holds(dhap,0,0) holds(f16bp,0,0) holds(g3p,0,0) fires(src_f16bp_1,0) fires(t3,0) holds(bpg13,0,1) holds(dhap,0,1) holds(f16bp,2,1) holds(g3p,0,1) fires(src_f16bp_1,1) fires(t3,1) fires(t4,1) holds(bpg13,0,2) holds(dhap,1,2) holds(f16bp,3,2) holds(g3p,1,2) fires(src_f16bp_1,2) fires(t3,2) fires(t4,2) fires(t5a,2) fires(t6,2) holds(bpg13,2,3) holds(dhap,1,3) holds(f16bp,4,3) holds(g3p,2,3) fires(src_f16bp_1,3) fires(t3,3) fires(t4,3) fires(t5a,3) fires(t5b,3) fires(t6,3) holds(bpg13,4,4) holds(dhap,2,4) holds(f16bp,5,4) holds(g3p,2,4) fires(src_f16bp_1,4) fires(t3,4) fires(t4,4) fires(t5a,4) fires(t5b,4) fires(t6,4) holds(bpg13,6,5) holds(dhap,3,5) holds(f16bp,6,5) holds(g3p,2,5) fires(src_f16bp_1,5) fires(t3,5) fires(t4,5) fires(t5a,5) fires(t5b,5) fires(t6,5) Answer sets of the alternate case are as follows: Answer: 1 holds(bpg13,0,0) holds(dhap,0,0) holds(f16bp,0,0) holds(g3p,0,0) fires(reset_dhap_1,0) fires(src_f16bp_1,0) fires(t3,0) holds(bpg13,0,1) holds(dhap,0,1) holds(f16bp,2,1) holds(g3p,0,1) fires(reset_dhap_1,1) fires(src_f16bp_1,1) fires(t3,1) fires(t4,1) holds(bpg13,0,2) holds(dhap,1,2) holds(f16bp,3,2) holds(g3p,1,2) fires(reset_dhap_1,2) fires(src_f16bp_1,2) fires(t3,2) fires(t4,2) fires(t6,2) holds(bpg13,2,3) holds(dhap,1,3) holds(f16bp,4,3) holds(g3p,1,3) fires(reset_dhap_1,3) fires(src_f16bp_1,3) fires(t3,3) fires(t4,3) fires(t5b,3) holds(bpg13,2,4) holds(dhap,2,4) holds(f16bp,5,4) holds(g3p,1,4) fires(reset_dhap_1,4) fires(src_f16bp_1,4) fires(t3,4) fires(t4,4) fires(t5b,4) holds(bpg13,2,5) holds(dhap,2,5) holds(f16bp,6,5) holds(g3p,1,5) fires(reset_dhap_1,5) fires(src_f16bp_1,5) fires(t3,5) fires(t4,5) fires(t5b,5) Answer: 2 holds(bpg13,0,0) holds(dhap,0,0) holds(f16bp,0,0) holds(g3p,0,0) fires(reset_dhap_1,0) fires(src_f16bp_1,0) fires(t3,0) holds(bpg13,0,1) holds(dhap,0,1) holds(f16bp,2,1) holds(g3p,0,1) fires(reset_dhap_1,1) fires(src_f16bp_1,1) fires(t3,1) fires(t4,1) holds(bpg13,0,2) holds(dhap,1,2) holds(f16bp,3,2) holds(g3p,1,2) fires(reset_dhap_1,2) fires(src_f16bp_1,2) fires(t3,2) fires(t4,2) fires(t6,2) holds(bpg13,2,3) holds(dhap,1,3) holds(f16bp,4,3) holds(g3p,1,3) fires(reset_dhap_1,3) fires(src_f16bp_1,3) fires(t3,3) fires(t4,3) fires(t5b,3) holds(bpg13,2,4) holds(dhap,2,4) holds(f16bp,5,4) holds(g3p,1,4) fires(reset_dhap_1,4) fires(src_f16bp_1,4) fires(t3,4) fires(t4,4) fires(t5b,4) holds(bpg13,2,5) holds(dhap,2,5) holds(f16bp,6,5) holds(g3p,1,5) fires(reset_dhap_1,5) fires(src_f16bp_1,5) fires(t3,5) fires(t4,5) fires(t6,5) Answer: 3 holds(bpg13,0,0) holds(dhap,0,0) holds(f16bp,0,0) holds(g3p,0,0) fires(reset_dhap_1,0) fires(src_f16bp_1,0) fires(t3,0) holds(bpg13,0,1) holds(dhap,0,1) holds(f16bp,2,1) holds(g3p,0,1) fires(reset_dhap_1,1) fires(src_f16bp_1,1) fires(t3,1) fires(t4,1) holds(bpg13,0,2) holds(dhap,1,2) holds(f16bp,3,2) holds(g3p,1,2) fires(reset_dhap_1,2) fires(src_f16bp_1,2) fires(t3,2) fires(t4,2) fires(t6,2) holds(bpg13,2,3) holds(dhap,1,3) holds(f16bp,4,3) holds(g3p,1,3) fires(reset_dhap_1,3) fires(src_f16bp_1,3) fires(t3,3) fires(t4,3) fires(t5b,3) holds(bpg13,2,4) holds(dhap,2,4) holds(f16bp,5,4) holds(g3p,1,4) fires(reset_dhap_1,4) fires(src_f16bp_1,4) fires(t3,4) fires(t4,4) fires(t6,4) holds(bpg13,4,5) holds(dhap,1,5) holds(f16bp,6,5) holds(g3p,1,5) fires(reset_dhap_1,5) fires(src_f16bp_1,5) fires(t3,5) fires(t4,5) fires(t5b,5) Answer: 4 holds(bpg13,0,0) holds(dhap,0,0) holds(f16bp,0,0) holds(g3p,0,0) fires(reset_dhap_1,0) fires(src_f16bp_1,0) fires(t3,0) holds(bpg13,0,1) holds(dhap,0,1) holds(f16bp,2,1) holds(g3p,0,1) fires(reset_dhap_1,1) fires(src_f16bp_1,1) fires(t3,1) fires(t4,1) holds(bpg13,0,2) holds(dhap,1,2) holds(f16bp,3,2) holds(g3p,1,2) fires(reset_dhap_1,2) fires(src_f16bp_1,2) fires(t3,2) fires(t4,2) fires(t6,2) holds(bpg13,2,3) holds(dhap,1,3) holds(f16bp,4,3) holds(g3p,1,3) fires(reset_dhap_1,3) fires(src_f16bp_1,3) fires(t3,3) fires(t4,3) fires(t5b,3) holds(bpg13,2,4) holds(dhap,2,4) holds(f16bp,5,4) holds(g3p,1,4) fires(reset_dhap_1,4) fires(src_f16bp_1,4) fires(t3,4) fires(t4,4) fires(t6,4) holds(bpg13,4,5) holds(dhap,1,5) holds(f16bp,6,5) holds(g3p,1,5) fires(reset_dhap_1,5) fires(src_f16bp_1,5) fires(t3,5) fires(t4,5) fires(t6,5) Answer: 5 holds(bpg13,0,0) holds(dhap,0,0) holds(f16bp,0,0) holds(g3p,0,0) fires(reset_dhap_1,0) fires(src_f16bp_1,0) fires(t3,0) holds(bpg13,0,1) holds(dhap,0,1) holds(f16bp,2,1) holds(g3p,0,1) fires(reset_dhap_1,1) fires(src_f16bp_1,1) fires(t3,1) fires(t4,1) holds(bpg13,0,2) holds(dhap,1,2) holds(f16bp,3,2) holds(g3p,1,2) fires(reset_dhap_1,2) fires(src_f16bp_1,2) fires(t3,2) fires(t4,2) fires(t5b,2) holds(bpg13,0,3) holds(dhap,2,3) holds(f16bp,4,3) holds(g3p,1,3) fires(reset_dhap_1,3) fires(src_f16bp_1,3) fires(t3,3) fires(t4,3) fires(t5b,3) holds(bpg13,0,4) holds(dhap,2,4) holds(f16bp,5,4) holds(g3p,1,4) fires(reset_dhap_1,4) fires(src_f16bp_1,4) fires(t3,4) fires(t4,4) fires(t5b,4) holds(bpg13,0,5) holds(dhap,2,5) holds(f16bp,6,5) holds(g3p,1,5) fires(reset_dhap_1,5) fires(src_f16bp_1,5) fires(t3,5) fires(t4,5) fires(t5b,5) Answer: 6 holds(bpg13,0,0) holds(dhap,0,0) holds(f16bp,0,0) holds(g3p,0,0) fires(reset_dhap_1,0) fires(src_f16bp_1,0) fires(t3,0) holds(bpg13,0,1) holds(dhap,0,1) holds(f16bp,2,1) holds(g3p,0,1) fires(reset_dhap_1,1) fires(src_f16bp_1,1) fires(t3,1) fires(t4,1) holds(bpg13,0,2) holds(dhap,1,2) holds(f16bp,3,2) holds(g3p,1,2) fires(reset_dhap_1,2) fires(src_f16bp_1,2) fires(t3,2) fires(t4,2) fires(t5b,2) holds(bpg13,0,3) holds(dhap,2,3) holds(f16bp,4,3) holds(g3p,1,3) fires(reset_dhap_1,3) fires(src_f16bp_1,3) fires(t3,3) fires(t4,3) fires(t5b,3) holds(bpg13,0,4) holds(dhap,2,4) holds(f16bp,5,4) holds(g3p,1,4) fires(reset_dhap_1,4) fires(src_f16bp_1,4) fires(t3,4) fires(t4,4) fires(t5b,4) holds(bpg13,0,5) holds(dhap,2,5) holds(f16bp,6,5) holds(g3p,1,5) fires(reset_dhap_1,5) fires(src_f16bp_1,5) fires(t3,5) fires(t4,5) fires(t6,5) Answer: 7 holds(bpg13,0,0) holds(dhap,0,0) holds(f16bp,0,0) holds(g3p,0,0) fires(reset_dhap_1,0) fires(src_f16bp_1,0) fires(t3,0) holds(bpg13,0,1) holds(dhap,0,1) holds(f16bp,2,1) holds(g3p,0,1) fires(reset_dhap_1,1) fires(src_f16bp_1,1) fires(t3,1) fires(t4,1) holds(bpg13,0,2) holds(dhap,1,2) holds(f16bp,3,2) holds(g3p,1,2) fires(reset_dhap_1,2) fires(src_f16bp_1,2) fires(t3,2) fires(t4,2) fires(t5b,2) holds(bpg13,0,3) holds(dhap,2,3) holds(f16bp,4,3) holds(g3p,1,3) fires(reset_dhap_1,3) fires(src_f16bp_1,3) fires(t3,3) fires(t4,3) fires(t5b,3) holds(bpg13,0,4) holds(dhap,2,4) holds(f16bp,5,4) holds(g3p,1,4) fires(reset_dhap_1,4) fires(src_f16bp_1,4) fires(t3,4) fires(t4,4) fires(t6,4) holds(bpg13,2,5) holds(dhap,1,5) holds(f16bp,6,5) holds(g3p,1,5) fires(reset_dhap_1,5) fires(src_f16bp_1,5) fires(t3,5) fires(t4,5) fires(t5b,5) Answer: 8 holds(bpg13,0,0) holds(dhap,0,0) holds(f16bp,0,0) holds(g3p,0,0) fires(reset_dhap_1,0) fires(src_f16bp_1,0) fires(t3,0) holds(bpg13,0,1) holds(dhap,0,1) holds(f16bp,2,1) holds(g3p,0,1) fires(reset_dhap_1,1) fires(src_f16bp_1,1) fires(t3,1) fires(t4,1) holds(bpg13,0,2) holds(dhap,1,2) holds(f16bp,3,2) holds(g3p,1,2) fires(reset_dhap_1,2) fires(src_f16bp_1,2) fires(t3,2) fires(t4,2) fires(t5b,2) holds(bpg13,0,3) holds(dhap,2,3) holds(f16bp,4,3) holds(g3p,1,3) fires(reset_dhap_1,3) fires(src_f16bp_1,3) fires(t3,3) fires(t4,3) fires(t5b,3) holds(bpg13,0,4) holds(dhap,2,4) holds(f16bp,5,4) holds(g3p,1,4) fires(reset_dhap_1,4) fires(src_f16bp_1,4) fires(t3,4) fires(t4,4) fires(t6,4) holds(bpg13,2,5) holds(dhap,1,5) holds(f16bp,6,5) holds(g3p,1,5) fires(reset_dhap_1,5) fires(src_f16bp_1,5) fires(t3,5) fires(t4,5) fires(t6,5) Answer: 9 holds(bpg13,0,0) holds(dhap,0,0) holds(f16bp,0,0) holds(g3p,0,0) fires(reset_dhap_1,0) fires(src_f16bp_1,0) fires(t3,0) holds(bpg13,0,1) holds(dhap,0,1) holds(f16bp,2,1) holds(g3p,0,1) fires(reset_dhap_1,1) fires(src_f16bp_1,1) fires(t3,1) fires(t4,1) holds(bpg13,0,2) holds(dhap,1,2) holds(f16bp,3,2) holds(g3p,1,2) fires(reset_dhap_1,2) fires(src_f16bp_1,2) fires(t3,2) fires(t4,2) fires(t5b,2) holds(bpg13,0,3) holds(dhap,2,3) holds(f16bp,4,3) holds(g3p,1,3) fires(reset_dhap_1,3) fires(src_f16bp_1,3) fires(t3,3) fires(t4,3) fires(t6,3) holds(bpg13,2,4) holds(dhap,1,4) holds(f16bp,5,4) holds(g3p,1,4) fires(reset_dhap_1,4) fires(src_f16bp_1,4) fires(t3,4) fires(t4,4) fires(t5b,4) holds(bpg13,2,5) holds(dhap,2,5) holds(f16bp,6,5) holds(g3p,1,5) fires(reset_dhap_1,5) fires(src_f16bp_1,5) fires(t3,5) fires(t4,5) fires(t5b,5) Answer: 10 holds(bpg13,0,0) holds(dhap,0,0) holds(f16bp,0,0) holds(g3p,0,0) fires(reset_dhap_1,0) fires(src_f16bp_1,0) fires(t3,0) holds(bpg13,0,1) holds(dhap,0,1) holds(f16bp,2,1) holds(g3p,0,1) fires(reset_dhap_1,1) fires(src_f16bp_1,1) fires(t3,1) fires(t4,1) holds(bpg13,0,2) holds(dhap,1,2) holds(f16bp,3,2) holds(g3p,1,2) fires(reset_dhap_1,2) fires(src_f16bp_1,2) fires(t3,2) fires(t4,2) fires(t5b,2) holds(bpg13,0,3) holds(dhap,2,3) holds(f16bp,4,3) holds(g3p,1,3) fires(reset_dhap_1,3) fires(src_f16bp_1,3) fires(t3,3) fires(t4,3) fires(t6,3) holds(bpg13,2,4) holds(dhap,1,4) holds(f16bp,5,4) holds(g3p,1,4) fires(reset_dhap_1,4) fires(src_f16bp_1,4) fires(t3,4) fires(t4,4) fires(t5b,4) holds(bpg13,2,5) holds(dhap,2,5) holds(f16bp,6,5) holds(g3p,1,5) fires(reset_dhap_1,5) fires(src_f16bp_1,5) fires(t3,5) fires(t4,5) fires(t6,5) Answer: 11 holds(bpg13,0,0) holds(dhap,0,0) holds(f16bp,0,0) holds(g3p,0,0) fires(reset_dhap_1,0) fires(src_f16bp_1,0) fires(t3,0) holds(bpg13,0,1) holds(dhap,0,1) holds(f16bp,2,1) holds(g3p,0,1) fires(reset_dhap_1,1) fires(src_f16bp_1,1) fires(t3,1) fires(t4,1) holds(bpg13,0,2) holds(dhap,1,2) holds(f16bp,3,2) holds(g3p,1,2) fires(reset_dhap_1,2) fires(src_f16bp_1,2) fires(t3,2) fires(t4,2) fires(t5b,2) holds(bpg13,0,3) holds(dhap,2,3) holds(f16bp,4,3) holds(g3p,1,3) fires(reset_dhap_1,3) fires(src_f16bp_1,3) fires(t3,3) fires(t4,3) fires(t6,3) holds(bpg13,2,4) holds(dhap,1,4) holds(f16bp,5,4) holds(g3p,1,4) fires(reset_dhap_1,4) fires(src_f16bp_1,4) fires(t3,4) fires(t4,4) fires(t6,4) holds(bpg13,4,5) holds(dhap,1,5) holds(f16bp,6,5) holds(g3p,1,5) fires(reset_dhap_1,5) fires(src_f16bp_1,5) fires(t3,5) fires(t4,5) fires(t5b,5) Answer: 12 holds(bpg13,0,0) holds(dhap,0,0) holds(f16bp,0,0) holds(g3p,0,0) fires(reset_dhap_1,0) fires(src_f16bp_1,0) fires(t3,0) holds(bpg13,0,1) holds(dhap,0,1) holds(f16bp,2,1) holds(g3p,0,1) fires(reset_dhap_1,1) fires(src_f16bp_1,1) fires(t3,1) fires(t4,1) holds(bpg13,0,2) holds(dhap,1,2) holds(f16bp,3,2) holds(g3p,1,2) fires(reset_dhap_1,2) fires(src_f16bp_1,2) fires(t3,2) fires(t4,2) fires(t5b,2) holds(bpg13,0,3) holds(dhap,2,3) holds(f16bp,4,3) holds(g3p,1,3) fires(reset_dhap_1,3) fires(src_f16bp_1,3) fires(t3,3) fires(t4,3) fires(t6,3) holds(bpg13,2,4) holds(dhap,1,4) holds(f16bp,5,4) holds(g3p,1,4) fires(reset_dhap_1,4) fires(src_f16bp_1,4) fires(t3,4) fires(t4,4) fires(t6,4) holds(bpg13,4,5) holds(dhap,1,5) holds(f16bp,6,5) holds(g3p,1,5) fires(reset_dhap_1,5) fires(src_f16bp_1,5) fires(t3,5) fires(t4,5) fires(t6,5) Answer: 13 holds(bpg13,0,0) holds(dhap,0,0) holds(f16bp,0,0) holds(g3p,0,0) fires(reset_dhap_1,0) fires(src_f16bp_1,0) fires(t3,0) holds(bpg13,0,1) holds(dhap,0,1) holds(f16bp,2,1) holds(g3p,0,1) fires(reset_dhap_1,1) fires(src_f16bp_1,1) fires(t3,1) fires(t4,1) holds(bpg13,0,2) holds(dhap,1,2) holds(f16bp,3,2) holds(g3p,1,2) fires(reset_dhap_1,2) fires(src_f16bp_1,2) fires(t3,2) fires(t4,2) fires(t6,2) holds(bpg13,2,3) holds(dhap,1,3) holds(f16bp,4,3) holds(g3p,1,3) fires(reset_dhap_1,3) fires(src_f16bp_1,3) fires(t3,3) fires(t4,3) fires(t6,3) holds(bpg13,4,4) holds(dhap,1,4) holds(f16bp,5,4) holds(g3p,1,4) fires(reset_dhap_1,4) fires(src_f16bp_1,4) fires(t3,4) fires(t4,4) fires(t5b,4) holds(bpg13,4,5) holds(dhap,2,5) holds(f16bp,6,5) holds(g3p,1,5) fires(reset_dhap_1,5) fires(src_f16bp_1,5) fires(t3,5) fires(t4,5) fires(t6,5) Answer: 14 holds(bpg13,0,0) holds(dhap,0,0) holds(f16bp,0,0) holds(g3p,0,0) fires(reset_dhap_1,0) fires(src_f16bp_1,0) fires(t3,0) holds(bpg13,0,1) holds(dhap,0,1) holds(f16bp,2,1) holds(g3p,0,1) fires(reset_dhap_1,1) fires(src_f16bp_1,1) fires(t3,1) fires(t4,1) holds(bpg13,0,2) holds(dhap,1,2) holds(f16bp,3,2) holds(g3p,1,2) fires(reset_dhap_1,2) fires(src_f16bp_1,2) fires(t3,2) fires(t4,2) fires(t6,2) holds(bpg13,2,3) holds(dhap,1,3) holds(f16bp,4,3) holds(g3p,1,3) fires(reset_dhap_1,3) fires(src_f16bp_1,3) fires(t3,3) fires(t4,3) fires(t6,3) holds(bpg13,4,4) holds(dhap,1,4) holds(f16bp,5,4) holds(g3p,1,4) fires(reset_dhap_1,4) fires(src_f16bp_1,4) fires(t3,4) fires(t4,4) fires(t5b,4) holds(bpg13,4,5) holds(dhap,2,5) holds(f16bp,6,5) holds(g3p,1,5) fires(reset_dhap_1,5) fires(src_f16bp_1,5) fires(t3,5) fires(t4,5) fires(t5b,5) Answer: 15 holds(bpg13,0,0) holds(dhap,0,0) holds(f16bp,0,0) holds(g3p,0,0) fires(reset_dhap_1,0) fires(src_f16bp_1,0) fires(t3,0) holds(bpg13,0,1) holds(dhap,0,1) holds(f16bp,2,1) holds(g3p,0,1) fires(reset_dhap_1,1) fires(src_f16bp_1,1) fires(t3,1) fires(t4,1) holds(bpg13,0,2) holds(dhap,1,2) holds(f16bp,3,2) holds(g3p,1,2) fires(reset_dhap_1,2) fires(src_f16bp_1,2) fires(t3,2) fires(t4,2) fires(t6,2) holds(bpg13,2,3) holds(dhap,1,3) holds(f16bp,4,3) holds(g3p,1,3) fires(reset_dhap_1,3) fires(src_f16bp_1,3) fires(t3,3) fires(t4,3) fires(t6,3) holds(bpg13,4,4) holds(dhap,1,4) holds(f16bp,5,4) holds(g3p,1,4) fires(reset_dhap_1,4) fires(src_f16bp_1,4) fires(t3,4) fires(t4,4) fires(t6,4) holds(bpg13,6,5) holds(dhap,1,5) holds(f16bp,6,5) holds(g3p,1,5) fires(reset_dhap_1,5) fires(src_f16bp_1,5) fires(t3,5) fires(t4,5) fires(t5b,5) Answer: 16 holds(bpg13,0,0) holds(dhap,0,0) holds(f16bp,0,0) holds(g3p,0,0) fires(reset_dhap_1,0) fires(src_f16bp_1,0) fires(t3,0) holds(bpg13,0,1) holds(dhap,0,1) holds(f16bp,2,1) holds(g3p,0,1) fires(reset_dhap_1,1) fires(src_f16bp_1,1) fires(t3,1) fires(t4,1) holds(bpg13,0,2) holds(dhap,1,2) holds(f16bp,3,2) holds(g3p,1,2) fires(reset_dhap_1,2) fires(src_f16bp_1,2) fires(t3,2) fires(t4,2) fires(t6,2) holds(bpg13,2,3) holds(dhap,1,3) holds(f16bp,4,3) holds(g3p,1,3) fires(reset_dhap_1,3) fires(src_f16bp_1,3) fires(t3,3) fires(t4,3) fires(t6,3) holds(bpg13,4,4) holds(dhap,1,4) holds(f16bp,5,4) holds(g3p,1,4) fires(reset_dhap_1,4) fires(src_f16bp_1,4) fires(t3,4) fires(t4,4) fires(t6,4) holds(bpg13,6,5) holds(dhap,1,5) holds(f16bp,6,5) holds(g3p,1,5) fires(reset_dhap_1,5) fires(src_f16bp_1,5) fires(t3,5) fires(t4,5) fires(t6,5) Atoms selected for $bpg13$ quantity extraction for the nominal case: [holds(bpg13,0,0),holds(bpg13,0,1),holds(bpg13,0,2), \tholds(bpg13,0,3),holds(bpg13,2,4),holds(bpg13,4,5)], [holds(bpg13,0,0),holds(bpg13,0,1),holds(bpg13,0,2), \tholds(bpg13,2,3),holds(bpg13,4,4),holds(bpg13,6,5)] Atoms selected for $bpg13$ quantity extraction for the modified case: [holds(bpg13,0,0),holds(bpg13,0,1),holds(bpg13,0,2), \tholds(bpg13,2,3),holds(bpg13,2,4),holds(bpg13,2,5)], [holds(bpg13,0,0),holds(bpg13,0,1),holds(bpg13,0,2), \tholds(bpg13,2,3),holds(bpg13,2,4),holds(bpg13,2,5)], [holds(bpg13,0,0),holds(bpg13,0,1),holds(bpg13,0,2), \tholds(bpg13,2,3),holds(bpg13,2,4),holds(bpg13,4,5)], [holds(bpg13,0,0),holds(bpg13,0,1),holds(bpg13,0,2), \tholds(bpg13,2,3),holds(bpg13,2,4),holds(bpg13,4,5)], [holds(bpg13,0,0),holds(bpg13,0,1),holds(bpg13,0,2), \tholds(bpg13,0,3),holds(bpg13,0,4),holds(bpg13,0,5)], [holds(bpg13,0,0),holds(bpg13,0,1),holds(bpg13,0,2), \tholds(bpg13,0,3),holds(bpg13,0,4),holds(bpg13,0,5)], [holds(bpg13,0,0),holds(bpg13,0,1),holds(bpg13,0,2), \tholds(bpg13,0,3),holds(bpg13,0,4),holds(bpg13,2,5)], [holds(bpg13,0,0),holds(bpg13,0,1),holds(bpg13,0,2), \tholds(bpg13,0,3),holds(bpg13,0,4),holds(bpg13,2,5)], [holds(bpg13,0,0),holds(bpg13,0,1),holds(bpg13,0,2), \tholds(bpg13,0,3),holds(bpg13,2,4),holds(bpg13,2,5)], [holds(bpg13,0,0),holds(bpg13,0,1),holds(bpg13,0,2), \tholds(bpg13,0,3),holds(bpg13,2,4),holds(bpg13,2,5)], [holds(bpg13,0,0),holds(bpg13,0,1),holds(bpg13,0,2), \tholds(bpg13,0,3),holds(bpg13,2,4),holds(bpg13,4,5)], [holds(bpg13,0,0),holds(bpg13,0,1),holds(bpg13,0,2), \tholds(bpg13,0,3),holds(bpg13,2,4),holds(bpg13,4,5)], [holds(bpg13,0,0),holds(bpg13,0,1),holds(bpg13,0,2), \tholds(bpg13,2,3),holds(bpg13,4,4),holds(bpg13,4,5)], [holds(bpg13,0,0),holds(bpg13,0,1),holds(bpg13,0,2), \tholds(bpg13,2,3),holds(bpg13,4,4),holds(bpg13,4,5)], [holds(bpg13,0,0),holds(bpg13,0,1),holds(bpg13,0,2), \tholds(bpg13,2,3),holds(bpg13,4,4),holds(bpg13,6,5)], [holds(bpg13,0,0),holds(bpg13,0,1),holds(bpg13,0,2), \tholds(bpg13,2,3),holds(bpg13,4,4),holds(bpg13,6,5)] Progression from raw matrix of $bpg13$ quantities in various answer-sets (rows) at various simulation steps (columns) to rates and finally to the average aggregate for the nominal case: $\\begin{bmatrix}0 & 0 & 0 & 0 & 2 & 4\\\\0 & 0 & 0 & 2 & 4 & 6\\\\\\end{bmatrix}=rate=>\\begin{bmatrix}0.8\\\\1.2\\\\\\end{bmatrix}=average=>1.0$ Progression from raw matrix of $bpg13$ quantities in various answer-sets (rows) at various simulation steps (columns) to rates and finally to the average aggregate for the modified case: $\\begin{bmatrix}0 & 0 & 0 & 2 & 2 & 2\\\\0 & 0 & 0 & 2 & 2 & 2\\\\0 & 0 & 0 & 2 & 2 & 4\\\\0 & 0 & 0 & 2 & 2 & 4\\\\0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 2\\\\0 & 0 & 0 & 0 & 0 & 2\\\\0 & 0 & 0 & 0 & 2 & 2\\\\0 & 0 & 0 & 0 & 2 & 2\\\\0 & 0 & 0 & 0 & 2 & 4\\\\0 & 0 & 0 & 0 & 2 & 4\\\\0 & 0 & 0 & 2 & 4 & 4\\\\0 & 0 & 0 & 2 & 4 & 4\\\\0 & 0 & 0 & 2 & 4 & 6\\\\0 & 0 & 0 & 2 & 4 & 6\\\\\\end{bmatrix}=rate=>\\begin{bmatrix}0.4\\\\0.4\\\\0.8 \\\\0.8 \\\\0.", "\\\\0.", "\\\\0.4 \\\\0.4 \\\\0.4 \\\\0.4 \\\\0.8 \\\\0.8 \\\\0.8 \\\\0.8 \\\\1.2 \\\\1.2\\\\\\end{bmatrix}=average=>0.6$ We find that $d=^{\\prime }<^{\\prime }$ comparing the modified case rate of $0.6$ to the nominal case rate of $1.0$ .", "Since the direction $d$ was an unknown in the query statement, our system generates produces the full query specification with $d$ replaced by $^{\\prime }<^{\\prime }$ as follows: direction of change in average rate of production of 'bpg13' is '<' (0.6<1) when observed between time step 0 and time step 5 comparing nominal pathway with modified pathway obtained; due to interventions: remove 'dhap' as soon as produced; using initial setup: continuously supply 'f16bp' in quantity 1; Evaluation Methodolgy A direct comparison against other tools is not possible, since most other programs explore one state evolution, while we explore all possible state evolutions.", "In addition ASP has to ground the program completely, irrespective of whether we are computing one answer or all.", "So, to evaluate our system, we compare our results for the questions from the 2nd Deep KR Challenge against the answers they have provided.", "Our results in essence match the responses given for the questions.", "Related Work In this section, we relate our high level language with other high level action languages.", "Comparison with $\\pi $ -Calculus $\\pi $ -calculus is a formalism that is used to model biological systems and pathways by modeling biological systems as mobile communication systems.", "We use the biological model described by [65] for comparison against our system.", "In their model they represent molecules and their domains as computational processes, interacting elements of molecules as communication channels (two molecules interact if they fit together as in a lock-and-key mechanism), and reactions as communication through channel transmission.", "$\\pi $ -calculus models have the ability of changing their structure during simulation.", "Our system on the other hand only allows modification of the pathway at the start of simulation.", "Regular $\\pi $ -calculus models appear qualitative in nature.", "However, stochastic extensions allow representation of quantitative data [62].", "In contrast, the focus of our system is on the quantitative+qualitative representation using numeric fluents.", "It is unclear how one can easily implement maximal-parallelism of our system in $\\pi $ -calculus, where a maximum number of simultaneous actions occur such that they do not cause a conflict.", "Where, a set of actions is said to be in conflict if their simultaneous execution will cause a fluent to become negative.", "Comparison with Action Language $\\mathcal {A}$ Action language $\\mathcal {A}$ [26] is a formalism that has been used to model biological systems and pathways.", "First we give a brief overview of $\\mathcal {A}$ in an intuitive manner.", "Assume two sets of disjoint symbols containing fluent names and action names, then a fluent expression is either a fluent name $F$ or $\\lnot F$ .", "A domain description is composed of propositions of the following form: value proposition: $F \\mathbf {~after~} A_1;\\dots ;A_m$ , where $(m \\ge 0)$ , $F$ is a fluent and $A_1,\\dots ,A_m$ are fluents.", "effect propostion: $A \\mathbf {~causes~} F \\mathbf {~if~} P_1,\\dots ,P_n$ , where $(n \\ge 0)$ , $A$ is an action, $F,P_1,\\dots ,P_n$ are fluent expressions.", "$P_1,\\dots ,P_n$ are called preconditions of $A$ and the effect proposition describes the effect on $F$ .", "We relate it to our work: Fluents are boolean.", "We support numeric valued fluents, with binary fluents.", "Fluents are non-inertial, but inertia can be added.", "Our fluents are always intertial.", "Action description specifies the effect of an action.", "Our domain description specifies `natural'-actions, which execute automatically when their pre-conditions are satisfied, subject to certain conditions.", "As a result our domain description represents trajectories.", "No built in support for aggregates exists.", "We support a selected set of aggregates, on a single trajectory and over multiple trajectories.", "Value propositions in $\\mathcal {A}$ are representable as observations in our query language.", "Comparison with Action Language $\\mathcal {B}$ Action language $\\mathcal {B}$ extends $\\mathcal {A}$ by adding static causal laws, which allows one to specify indirect effects or ramifications of an action [28].", "We relate it to our work below: Inertia is built into the semantics of $\\mathcal {B}$ [28].", "Our language also has intertia built in.", "$\\mathcal {B}$ supports static causal laws that allow defining a fluent in terms of other fluents.", "We do not support static causal laws.", "Comparison with Action Language $\\mathcal {C}$ Action language $\\mathcal {C}$ is based on the theory of causal explanation, i.e.", "a formula is true if there is a cause for it to be true [30].", "It has been previously used to represent and reason about biological pathways [23].", "We relate it to our work below: $\\mathcal {C}$ supports boolean fluents only.", "We support numeric valued fluents, and binary fluents.", "$\\mathcal {C}$ allows both inertial and non-inertial fluents.", "While our fluents are always inertial.", "$\\mathcal {C}$ support static causal laws (or ramifications), that allow defining a fluent in terms of other fluents.", "We do not support them.", "$\\mathcal {C}$ describes causal relationships between fluents and actions.", "Our language on the other hand describes trajectories.", "Comparison with Action Language $\\mathcal {C+}$ First, we give a brief overview of $\\mathcal {C+}$ [29].", "Intuitively, atoms $\\mathcal {C+}$ are of the form $c=v$ , where $c$ is either a fluent or an action constant, $v$ belongs to the domain of $c$ , and fluents and actions form a disjoint set.", "A formula is a propositional combination of atoms.", "A fluent formula is a formula in which all constants are fluent constants; and an action formula is a formula with one action constant but no fluent constants.", "An action description in $\\mathcal {C+}$ is composed of causal laws of the following forms: static law: $\\mathbf {caused~} F \\mathbf {~if~} G$ , where $F$ and $G$ are fluent formulas action dynamic law: $\\mathbf {caused~} F \\mathbf {~if~} G$ , where $F$ is an action formula and $G$ is a formula fluent dynamic law: $\\mathbf {caused~} F \\mathbf {~if~} G \\mathbf {~after~} H$ , where $F$ and $G$ are fluent formulas and $H$ is a formula Concise forms of these laws exist, e.g.", "`$\\mathbf {intertial~} f \\equiv \\mathbf {caused~} f=v \\mathbf {~if~} f=v \\mathbf {~after~} f=v, \\forall v \\in \\text{ domain of } f$ ' that allow a more intuitive program declaration.", "We now relate it to our work.", "Some of the main differences include: Fluents are multi-valued, fluent values can be integer, boolean or other types.", "We support integer and binary valued fluents only.", "Actions are multi-valued.", "We do not support multi-valued actions.", "Both inertial and non-inertial fluents are supported.", "In comparison we allow inertial fluents only.", "Static causal laws are supported that allow changing the value of a fluent based on other fluents (ramifications).", "We do not allow static causal laws.", "Effect of parallel actions on numeric fluents is not additive.", "However, the additive fluents extension [51] adds the capability of additive fluents through new rules.", "The extended language, however, imposes certain restrictions on additive fluents and also restricts the domain of additive actions to boolean actions only.", "Our fluents are always additive.", "Supports defaults.", "We do not have the same notion as defaults, but allow initial values for fluents in our domain description.", "Action's occurrence and its effect are defined in separate statements.", "In our case, the action's occurrence and effect are generally combined in one statement.", "Although parallel actions are supported, it is unclear how one can concisely describe the condition implicit in our system that simultaneously occurring actions may not conflict.", "Two actions conflict if their simultaneous execution will cause a fluent to become negative.", "Exogenous actions seem the closest match to our may execute actions.", "However, our actions are `natural', in that they execute automatically when their pre-conditions are satisfied, they are not explicitly inhibited, and they do not conflict.", "Actions conflict when their simultaneous execution will cause one of the fluents to become negative.", "The exogenous-style character of our actions holds when the firing style is `$$ '.", "When the firing style changes, the execution semantics change as well.", "Consider the following two may execute statements in our language: $a_1 \\mathbf {~may~execute~causing~} f_1 \\mathbf {~change~value~by~} -5 \\mathbf {~if~} f_2 \\mathbf {~has~value~} 3 \\mathbf {~or~higher} \\\\a_2 \\mathbf {~may~execute~causing~} f_1 \\mathbf {~change~value~by~} -3 \\mathbf {~if~} f_2 \\mathbf {~has~value~} 2 \\mathbf {~or~higher}$ and two states: (i) $f_1=10, f_2=5$ , (ii) $f_1=6,f_2=5$ .", "In state (i) both $a_1,a_2$ can occur simultaneously (at one point) resulting in firing-choices $\\big \\lbrace \\lbrace a_1,a_2 \\rbrace ,\\lbrace a_1\\rbrace ,\\lbrace a_2\\rbrace ,\\lbrace \\rbrace ,\\big \\rbrace $ ; whereas, in state (ii) only one of $a_1$ or $a_2$ can occur at one point resulting in the firing-choices: $\\big \\lbrace \\lbrace a_1\\rbrace ,\\lbrace a_2\\rbrace ,\\lbrace \\rbrace ,\\big \\rbrace $ because of a conflict due to the limited amount of $f_1$ .", "These firing choices apply for firing style `', which allows any combination of fireable actions to occur.", "If the firing style is set to `max', the maximum set of non-conflicting actions may fire, and the firing choices for state (i) change to $\\big \\lbrace \\lbrace a_1,a_2 \\rbrace \\big \\rbrace $ and the firing choices for state (ii) change to $\\big \\lbrace \\lbrace a_1\\rbrace ,\\lbrace a_2\\rbrace \\big \\rbrace $ .", "If the firing style is set to `1', at most one action may fire at one point, and the firing choices for both state (i) and state (ii) reduce to $\\big \\lbrace \\lbrace a_1\\rbrace ,\\lbrace a_2\\rbrace ,\\lbrace \\rbrace \\big \\rbrace $ .", "So, the case with `' firing style can be represented in $\\mathcal {C+}$ with exogenous actions $a_1,a_2$ ; the case with `1' firing style can be represented in $\\mathcal {C+}$ with exogenous actions $a_1,a_2$ and a constraint requiring that both $a_1,a_2$ do not occur simultaneously; while the case with `max' firing style can be represented by exogenous actions $a_1,a_2$ with additional action dynamic laws.", "They will still be subject to the conflict checking.", "Action dynamic laws can be used to force actions similar to our must execute actions.", "Specification of initial values of fluents seem possible through the query language.", "The default statement comes close, but it does not have the same notion as setting a fluent to a value once.", "We support specifying initial values both in the domain description as well as the query.", "There does not appear built-in support for aggregation of fluent values within the same answer set, such as sum, count, rate, minimum, maximum etc.", "Although some of it could be implemented using the additive fluents extension.", "We support a set of aggregates, such as total, and rate.", "Additional aggregates can be easily added.", "We support queries over aggregates (such as minimum, maximum, average) of single-trajectory aggregates (such as total, and rate etc.)", "over a set of trajectories.", "We also support comparative queries over two sets of trajectories.", "Our queries allow modification of the domain description as part of query evaluation.", "Comparison with $\\mathcal {BC}$ Action language $\\mathcal {BC}$ combines features of $\\mathcal {B}$ and $\\mathcal {C+}$ [52].", "First we give a brief overview of $\\mathcal {BC}$ .", "Intuitively, $\\mathcal {BC}$ has actions and valued fluents.", "A valued fluent, called an atom, is of the form `$f=v$ ', where $f$ is a fluent constant and $v \\in domain(f)$ .", "A fluent can be regular or statically determined.", "An action description in $\\mathcal {BC}$ is composed of static and dynamic laws of the following form: static law: $A_0 \\mathbf {~if~} A_1,\\dots ,A_m \\mathbf {~ifcons~} A_{m+1},\\dots ,A_n$ , where $(n \\ge m \\ge 0)$ , and each $A_i$ is an atom.", "dynamic law: $A_0 \\mathbf {~after~} A_1,\\dots ,A_m \\mathbf {~ifcons~} A_{m+1},\\dots ,A_n$ , where $(n \\ge m \\ge 0)$ , $A_0$ is a regular fluent atom, each of $A_1,\\dots ,A_m$ is an atom or an action constant, and $A_{m+1},\\dots ,A_n$ are atoms.", "Concise forms of these laws exist that allow a more intuitive program declaration.", "Now we relate it to our work.", "Some of the main differences include: Fluents are multi-valued, fluent values can which can be integer, boolean, or other types.", "We only support integer and binary fluents.", "Static causal laws are allowed.", "We do not support static causal laws.", "Similar to $\\mathcal {C+}$ numeric fluent accumulation is not supported.", "It is supported in our system.", "It is unclear how aggregate queries within a trajectory can be concisely represented.", "Aggregate queries such as rate are supported in our system.", "It does not seem that queries over multiple trajectories or sets of trajectories are supported.", "Such queries are supported in our system.", "Comparison with ASPMT ASPMT combines ASP with Satisfiability Modulo Theories.", "We relate the work in [53] where $\\mathcal {C+}$ is extended using ASPMT with our work.", "It adds support for real valued fluents to $\\mathcal {C+}$ including additive fluents.", "Thus, it allows reasoning with continuous and discrete processes simultaneously.", "Our language does not support real numbers directly.", "Several systems also exist to model and reason with biological pathway.", "For example: Comparison with BioSigNet-RR BioSigNet-RR [5] is a system for representing and reasoning with signaling networks.", "We relate it to our work in an intuitive manner.", "Fluents are boolean, so qualitative queries are possible.", "We support both integer and binary fluents, so quantiative queries are also possible.", "Indirect effects (or ramifications) are supported.", "We do not support these.", "Action effects are captured separately in `$\\mathbf {~causes~}$ statement' from action triggering statements `$\\mathbf {~triggers~}$ ' and `$\\mathbf {~n\\_triggers~}$ '.", "We capture both components in a `$\\mathbf {~may~execute~causing~}$ ' or `$\\mathbf {~normally~must~execute~causing~}$ ' statement.", "Their action triggering semantics have some similarity to our actions.", "Just like their actions get triggered when the action's pre-conditions are satisfied, our actions are also triggered when their pre-conditions are satisfied.", "However, the triggering semantics are different, e.g.", "their triggers statement causes an action to occur even if it is disabled, we do not have an equivalent for it; and their n_triggers is similar in semantics to normally must execute causing statement.", "It is not clear how loops in biological systems can be modeled in their system.", "Loops are possible in our by virtue of the Petri Net semantics.", "Their queries can have time-points and their precedence relations as part of the query.", "Though our queries allow the specification of some time points for interval queries, time-points are not supported in a similar way.", "However, we do support certain types of observation relative queries.", "The intervention in their planning queries has similarities to interventions in our system.", "However, it appears that our intervention descriptions are higher level.", "Conclusion In this chapter we presented the BioPathQA system and the languages to represent and query biological pathways.", "We also presented a new type of Petri Net, the so called Guarded-Arc Petri Net that is used as a model behind our pathway specification language, which shares certain aspects with CPNs [39], but our semantics for reset arcs is different, and we allow must-fire actions that prioritize actions for firing over other actions.", "We also showed how the system can be applied to questions from college level text books that require deeper reasoning and cannot be answered by using surface level knowledge.", "Although our system is developed with respect to the biological domain, it can be applied to non-biological domain as well.", "Some of the features of our language include: natural-actions that automatically fire when their prerequisite conditions are met (subject to certain restrictions); an automatic default constraint that ensures fluents do not go negative, since they model natural systems substances; a more natural representation of locations; and control of the level of parallelism to a certain degree.", "Our query language also allows interventions similar to Pearl's surgeries [59], which are more general than actions.", "Next we want to apply BioPathQA to a real world application by doing text extraction.", "Knowledge for real world is extracted from research papers.", "In the next chapter we show how such text extraction is done for pathway construction and drug development.", "We will then show how we can apply BioPathQA to the extracted knowledge to answer questions about the extracted knowledge.", "Text Extraction for Real World Applications In the previous chapter we looked at the BioPathQA system and how it answers simulation based reasoning questions about biological pathways, including questions that require comparison of alternate scenarios through simulation.", "These so called `what-if' questions arise in biological activities such as drug development, drug interaction, and personalized medicine.", "We will now put our system and language in context of such activities.", "Cutting-edge knowledge about pathways for activities such as drug development, drug interaction, and personalized medicine comes in the form of natural language research papers, thousands of which are published each year.", "To use this knowledge with our system, we need to perform extraction.", "In this chapter we describe techniques we use for such knowledge extraction for discovering drug interactions.", "We illustrate with an example extraction how we organize the extracted knowledge into a pathway specification and give examples of relevant what-if questions that a researcher performing may ask in the drug development domain.", "Introduction Thousands of research papers are published each year about biological systems and pathways over a broad spectrum of activities, including interactions between dugs and diseases, the associated pathways, and genetic variation.", "Thus, one has to perform text extraction to extract relationships between the biochemical processes, their involvement in diseases, and their interaction with drugs.", "For personalized medicine, one is also interested in how these interrelationships change in presence of genetic variation.", "In short, we are looking for relationships between various components of the biochemical processes and their internal and external stimuli.", "Many approaches exist for extracting relationships from a document.", "Most rely on some form of co-occurrence, relative distance, or order of words in a single document.", "Some use shallow parsing as well.", "Although these techniques tend to have a higher recall, they focus on extracting explicit relationships, which are relationships that are fully captured in a sentence or a document.", "These techniques also do not capture implicit relationships that may be spread across multiple documents.", "are spread across multiple documents relating to different species.", "Additional issues arise from the level of detail from in older vs. newer texts and seemingly contradictory information due to various levels of confidence in the techniques used.", "Many do not handle negative statements.", "We primarily use a system called PTQL [75] to extract these relationships, which allows combining the syntactic structure (parse tree), semantic elements, and word order in a relationship query.", "The sentences are pre-processed by using named-entity recognition, and entity normalization to allow querying on classes of entity types, such as drugs, and diseases; and also to allow cross-linking relationships across documents when they refer to the same entity with a different name.", "Queries that use such semantic association between words/phrases are likely to produce higher precision results.", "Source knowledge for extraction primarily comes from thousands of biological abstracts published each year in PubMed http://www.ncbi.nlm.nih.gov/pubmed.", "Next we briefly describe how we extract relationships about drug interactions.", "Following that we briefly describe how we extract association of drugs, and diseases with genetic variation.", "We conclude this chapter with an illustrative example of how the drug interaction relationships are used with our system to answer questions about drug interactions and how genetic variation could be utilized in our system.", "Extracting Relationships about Drug Interactions We summarize the extraction of relationships for our work on drug-drug interactions from [73].", "Studying drug-drug interactions are a major activity in drug development.", "Drug interactions occur due to the interactions between the biological processes / pathways that are responsible metabolizing and transporting drugs.", "Metabolic processes remove a drug from the system within a certain time period.", "For a drug to remain effective, it must be maintained within its therapeutic window for the period of treatment, requiring taking the drug periodically.", "Outside the therapeutic window, a drug can become toxic if a quantity greater than the therapeutic window is retained; or it can become ineffective if a quantity less than the therapeutic window is retained.", "Since liver enzymes metabolize most drugs, it is the location where most metabolic-interaction takes place.", "Induction or inhibition of these enzymes can affect the bioavailability of a drug through transcriptional regulation, either directly or indirectly.", "For example, if drug $A$ inhibits enzyme $E$ , which metabolizes drug $B$ , then the bioavailability of drug $B$ will be higher than normal, rendering it toxic.", "On the other hand, if drug $A$ induces enzyme $E$ , which metabolizes drug $B$ , then drug $B$ 's bioavailability will be lesser than normal, rendering it ineffective.", "Inhibition of enzymes is a common form of drug-drug interactions [10].", "In direct inhibition, a drug $A$ inhibit enzyme $E$ , which is responsible for metabolism of drug $B$ .", "Drug $A$ , leads to a decrease in the level of enzyme $E$ , which in turn can increase bioavailability of drug $B$ potentially leading to toxicity.", "Alternatively, insufficient metabolism of drug $B$ can lead to smaller amount of drug $B$ 's metabolites being produced, leading to therapeutic failure.", "An example of one such direct inhibition is the interaction between CYP2D6 inhibitor quinidine and CYP2D6 substrates (i.e.", "substances metabolized by CYP2D6), such as Codeine.", "The inhibition of CYP2D6 increases the bioavailability of drugs metabolized by CYP2D6 leading to adverse side effects.", "Another form of drug interactions is through induction of enzymes [10].", "In direct induction, a drug $A$ induces enzyme $E$ , which is responsible for metabolism of drug $B$ .", "An example of such direct induction is between phenobarbital, a CYP2C9 inducer and warfarin (a CYP2C9 substrate).", "Phenobarbital leads to increased metabolism of warfarin, decreasing warfarinÕs bioavailability.", "Direct interaction due to induction though possible is not as common as indirect interaction through transcription factors, which regulate the drug metabolizing enzymes.", "In such an interaction, drug $A$ activates a transcription factor $TF$ , which regulates and induces enzyme $E$ , where enzyme $E$ metabolizes drug $B$ .", "Transcription factors are referred to as regulators of xenobiotic-metabolizing enzymes.", "Examples of such regulators include aryl hydrocarbon receptor AhR, pregnane X receptor PXR and constitutive androstane receptor CAR.", "Drug interactions can also occur due to the induction or inhibition of transporters.", "Transporters are mainly responsible for cellular uptake or efflux (elimination) of drugs.", "They play an important part in drug disposition, by transporting drugs into the liver cells, for example.", "Transporter-based drug interactions, however, are not as well studies as metabolism-based interactions [10].", "Method Extraction of drug-drug interactions from the text can either be explicit or implicit.", "Explicit extraction refers to extraction of drug-drug interaction mentioned within a single sentence, while implicit extraction requires extraction of bio-properties of drug transport, metabolism and elimination that can lead to drug-drug interaction.", "This type of indirect extraction combines background information about biological processes, identification of protein families and the interactions that are involved in drug metabolism.", "Our approach is to extract both explicit and implicit drug interactions as summarized in Fig REF and it builds upon the work done in [74].", "Figure: This figure from outlines the effects of drug A on drug B through (a) direct induction/inhibition of enzymes; (b) indirect induction/inhibition of transportation factors that regulate the drug-metabolizing enzymes.Explicit Drug Interaction Extraction Explicit extraction mainly extracts drug-drug interactions directly mentioned in PubMed (or Medline) abstracts.", "For example, the following sentences each have a metabolic interaction mentioned within the same sentence: Ciprofloxacin strongly inhibits clozapine metabolism.", "(PMID: 19067475) Enantioselective induction of cyclophosphamide metabolism by phenytoin.", "which can be extracted by using the following PTQL query using the underlined keywords from above sentences: //S{//?", "[Tag=`Drug'](d1) => //?", "[Value IN {`induce',`induces',`inhibit',`inhibits'}](v) => //?[Tag=`Drug'](d2) => //?", "[Value=`metabolism'](w)} ::: [d1 v d2 w] 5 : d1.value, v.value, d2.value.", "This PTQL query specifies that a drug (denoted by d1) must be followed by one of the keywords from $\\lbrace `induce^{\\prime },`inhibit^{\\prime }, `inhibits^{\\prime }\\rbrace $ (denoted by v), which in turn must be followed by another drug (denoted by d2) followed the keyword $`metabolism^{\\prime }$ (denoted by w); all found within a proximity of 5 words of each other.", "The query produces tripes of $\\langle d1, v, d2 \\rangle $ values as output.", "Thus the results will produce triples $\\langle d1, induces, d2 \\rangle $ and $\\langle d1, inhibits, d2 \\rangle $ which mean that the drug d1 increases the effect of d2 (i.e.", "$\\langle d1, increases, d2 \\rangle $ ) and decreases the effect of d2 (i.e.", "$\\langle d1, decreases, d2 \\rangle $ ) respectively.", "For example, the sentence S1 above matches this PTQL query and the query will produce the triplet $\\langle \\text{ciprofloxacin}, \\text{increases}, \\text{clozapine} \\rangle $ .", "Implicit Drug Interaction Extraction Implicit extraction mainly extracts drug-drug interactions not yet published, but which can be inferred from published articles and properties of drug metabolism.", "The metabolic properties themselves have their origin in various publications.", "The metabolic interactions extracted from published articles and the background knowledge of properties of drug metabolism are reasoned with in an automated fashion to infer drug interactions.", "The following table outlines the kinds of interactions extracted from the text: Table: This table from shows the triplets representing various properties relevant to the extraction of implicit drug interactions and their description.which require multiple PTQL queries for extraction.", "As an example, the following PTQL query is used to extract $\\langle protein, metabolizes, drug \\rangle $ triplets: //S{/?", "[Tag=`Drug'](d1) => //VP{//?", "[Value IN {`metabolized',`metabolised'}](rel) => //?", "[Tag=`GENE'](g1)}} ::: g1.value, rel.value, d1.value which specifies that the extracted triplets must have a drug (denoted by d1) followed by a verb phrase (denoted by VP) with the verb in $\\lbrace `metabolized^{\\prime },`metabolised^{\\prime } \\rbrace $ , followed by a gene (denoted by g1).", "Table REF shows examples of extracted triplets.", "Table: This table from shows the triplets representing various properties relevant to the extraction of implicit drug interactions and their description.", "Data Cleaning The protein-protein and protein-drug relationships extracted from the parse tree database need an extra step of refinement to ensure that they correspond to the known properties of drug metabolism.", "For instance, for a protein to metabolize a drug, the protein must be an enzyme.", "Similarly, for a protein to regulate an enzyme, the protein must be a transcription factor.", "Thus, the $\\langle protein, metabolizes, drug \\rangle $ facts get refined to $\\langle enzyme, metabolizes, drug \\rangle $ and $\\langle protein, regulates, protein \\rangle $ gets refined to $\\langle transcription factor, regulates, enzyme \\rangle $ respectively.", "Classification of proteins is done using UniProt, the Gene Ontology (GO) and Entrez Gene summary by applying rules such as: A protein p is an enzyme if it belongs to a known enzyme family, such as CYP, UGT or SULT gene families; or is annotated under UniProt with the hydrolase, ligase, lyase or transferase keywords; or is listed under the “metabolic process” GO-term; or its Entrez Gene summary mentions key phrases like “drug metabolism” or roots for “enzyme” or “catalyzes”.", "A protein p is considered as a transcription factor if it is annotated with keywords transcription, transcription-regulator or activator under UniProt; or it is listed under the “transcription factor activity” category in GO; or its Entrez Gene summary contains the phrase “transcription factor”.", "Additional rules are applied to remove conflicting information, such as, favoring negative extractions (such as `$P$ does not metabolize $D$ ') over positive extractions (such as `$P$ metabolizes $D$ ').", "For details, see [73].", "Results The correctness of extracted interactions was determined by manually compiling a gold standard for each type of interaction using co-occurrence queries.", "For example, for $\\langle protein, metabolizes, drug \\rangle $ relations, we examined sentences that contain co-occurrence of protein, drug and one of the keywords “metabolized”, “metabolize”, “metabolises”, “metabolise”, “substrate” etc.", "Table REF summarizes the performance of our extraction approach.", "Table: Performance of interactions extracted from Medline abstracts.", "TP represents true-positives, while FN represents false-negatives Extracting Knowledge About Genetic Variants We summarize the relevant portion of our work on associating genetic variants with drugs, diseases and adverse reactions as presented in [33].", "Incorrect drug dosage is the leading cause of adverse drug reactions.", "Doctors prescribe the same amount of medicine to a patient for most drugs using the average drug response, even though a particular person's drug response may be higher or lower than the average case.", "A large part of the difference in drug response can be attributed to single nucleotide polymorphisms (SNPs).", "For example, the enzyme CYP2D6 has 70 known allelic variations, 50 of which are non-functional [31].", "Patients with poor metabolizer variations may retain higher concentration of drug for typical dosage, while patients with rapid metabolizers may not even reach therapeutic level of drug or toxic level of drug metabolites [68].", "Thus, it is important to consider the individual's genetic composition for dosage determination, especially for narrow therapeutic index drugs.", "Scientists studying these variations have grouped metabolizers into categories of poor (PM), intermediate (IM), rapid (RM) and ultra-rapid metabolizers (UM) and found that for some drugs, only 20% of usual dosage was required for PM and up to 140% for UM [38].", "Information about SNPs, their frequency in various population groups, their effect on genes (enzymic activity) and related data is stored in research papers, clinical studies and surveys.", "However, it is spread-out among them.", "Various databases collect this information in different forms.", "PharmGKB collects information such information and how it related to drug response [76].", "However, it is a small percentage of the total number of articles on PharmGKB, due to time consuming manual curation of data.", "Our work focuses on automatically extracting genetic variations and their impact on drug responses from PubMed abstracts to catch up with the current state of research in the biological domain, producing a repository of knowledge to assist in personalized medicine.", "Our approach leverages on as many existing tools as possible.", "Methods Next, we describe the methods used in our extraction, including: named entity recognition, entity normalization, and relation extraction.", "Named Entity Recognition We want to identify entities including genes (also proteins and enzymes), drugs, diseases, ADRs (adverse drug reactions), SNPs, RefSNPs (rs-numbers), names of alleles, populations and frequencies.", "For genes, we use BANNER [49] trained on BioCreative II GM training set [45].", "For genotypes (genetic variations including SNPs) we used a combination of MutationFinder [15] and custom components.", "Custom components were targeted mostly on non-SNPs (“c.76_78delACT”, 11MV324KF”) and insertions/deletions (“1707 del T”, “c.76_77insG”), RefSNPs (rs-numbers) and names of alleles/haplotypes (“CYP2D64”, “T allele”, “null allele”).", "For diseases (and ADRs), we used BANNER trained on a corpus of 3000 sentences with disease annotations [50].", "An additional 200 random sentences containing no disease were added from BioCreative II GM data to offset the low percentage (10%) of sentences without disease in the 3000 sentence corpus.", "In addition to BANNER, we used a dictionary extracted from UMLS.", "This dictionary consisted of 162k terms for 54k concepts from the six categories “Disease or Syndrome”, ”Neoplastic Process”,“Congenital Abnormality”,“Mental or Behavioral Dysfunction”,“Experimental Model of Disease” and “Acquired Abnormality”.", "The list was filtered to remove unspecific as well as spurious disease names such as “symptoms”, “disorder”, .... A dictionary for adverse drug reactions originated from SIDER Side Effect Resource [47], which provides a mapping of ADR terminology to UMLS CUIs.", "It consisted of 1.6k UMLS concepts with 6.5k terms.", "For drugs, we used a dictionary based on DrugBank[80] containing about 29k different drug names including both generic as well as brand names.", "We used the cross-linking information from DrugBank to collect additional synonyms and IDs from PharmGKB.", "We cross linked to Compound and Substance IDs from PubChem to provide hyperlinks to additional information.", "For population, we collected a dictionary of terms referring to countries, regions, regions inhabitants and their ethnicities from WikiPedia, e.g.", "“Caucasian”, “Italian”, “North African”, .... We filtered out irrelevant phrases like “Chinese [hamster]”.", "For frequencies, we extract all numbers and percentages as well as ranges from sentences that contain the word “allele”, “variant”, “mutation”, or “population”.", "The output is filtered in this case as well to remove false positives referring to p-values, odd ratios, confidence intervals and common trigger words.", "Entity Normalization Genes, diseases and drugs can appear with many different names in the text.", "For example, “CYP2D6” can appear as “Cytochrome p450 2D6” or “P450 IID6” among others, but they all refer to the same enzyme (EntrezGene ID 1565).", "We use GNAT on recognized genes [32], but limit them to human, mouse and rat genes.", "The gene name recognized by BANNER is filtered by GNAT to remove non-useful modifiers and looked up against EntrezGene restricted to human, mouse and rat genes to find candidate IDs for each gene name.", "Ambiguity (multiple matches) is resolved by matching the text surrounding the gene mention with gene's annotation from a set of resources, like EntrezGene, UniProt.", "Drugs and diseases/ADRs are resolved to their official IDs from DrugBank or UMLS.", "If none is found, we choose an ID for it based on its name.", "Genetic variants Genetic variations are converted to HGVScite [21] recommended format.", "Alleles were converted to the star notation (e.g.", "“CYP2D61”) and the genotype (“TT allele”) or fixed terms such as “null allele” and “variant allele”.", "Populations mentions are mapped to controlled vocabulary to remove orthographic and lexical variations.", "Relation Extraction Twelve type of relations were extracted between the detected entities as given in Table REF .", "Different methods were applied to detect different relations depending upon relation type, sentence structure and whether another method was able to extraction a relation beforehand.", "Gene-drug, gene-disease, drug-disease were extracted using sentence based co-occurrence (possibly refined by using relation-specific keywords) due to its good precision yield of this method for these relations.", "For other relations additional extraction methods were implemented.", "These include: High-confidence co-occurrence that includes keywords These co-occurences have the relation keyword in them.", "This method is applied to gene-drug, gene-disease, drug-ADR, drug-disease and mutation-disease associations.", "It uses keywords from PolySearch [79] as well as our own.", "Co-occurrence without keywords Such co-occurrences do not require any relationship keyword.", "This method is used for allele-population and variant-population relationships.", "This method can misidentify negative relationships.", "High-confidence relationships, if not found with a keyword drop down to this method for a lower confidence result.", "1:n co-occurrence Relationships where one entity has one instance in a given sentence and the other occurs one or more times.", "Single instance entity may have more than one occurrence.", "This method is useful in identifying gene mutations, where a gene is mentioned in a sentence along with a number of its mutations.", "The gene itself may be repeated.", "Enumerations with matching counts Captures entities in sentences where a list of entities is followed by an equal number of counts.", "This method is useful in capturing alleles and their associated frequencies, e.g.", "“The frequencies of CYP1B11, 2, 3, and 4 alleles were 0.087, 0.293, 0.444, and 0.175, respectively.” Least common ancestor (LCA) sub-tree Assigns associations based on distance in parse tree.", "We used Stanford parser [42] to get grammatical structure of a sentence as a dependency tree.", "This allows relating verb to its subject and noun to its modifiers.", "This method picks the closest pair in the lowest common ancestors (dependency) sub-tree of the entities.", "Maximum distance in terms of edges connecting the entity nodes was set to 10, which was determined empirically to provide the best balance between precision and recall.", "This method associates frequencies with alleles in the sentence “The allele frequencies were 18.3% (-24T) and 21.2% (1249A)”.", "m:n co-occurrence This method builds associations between all pairs of entities.", "Low confidence co-occurrence This acts as the catch-all case if none of the above methods work.", "Table: Unique binary relations identified between detected entities from .These methods were applied in order to each sentence, stoping at the first method that extracted the desired relationship.", "Order of these methods was determined empirically based of their precision.", "The order of the method used determines our confidence in the result.", "If none of the higher confidence methods are successful, a co-occurrence based method is used for extraction with low confidence.", "Abstract-level co-occurrence are also extracted to provide hits on potential relations.", "They appear in the database only when they appear in more than a pre-set threshold number of abstracts.", "Results Performance was evaluated by evaluating the precision and recall of individual components and coverage of existing results.", "Precision and recall were tested by processing 3500 PubMed abstracts found via PharmGKB relations and manually checking the 2500 predictions.", "Coverage was tested against DrugBank and PharmGKB.", "Extracted relations went through manual evaluation for correctness.", "Each extraction was also assigned a confidence value based on the confidence in the method of extraction used.", "We got a coverage of 91% of data in DrugBank and 94% in PharmGKB.", "Taking into false positive rates for genes, drugs and gene-drug relations, SNPshot has more than 10,000 new relations.", "Applying BioPathQA to Drug-Drug Interaction Discovery Now we use our BioPathQA system from chapter to answer questions about drug-drug interaction using knowledge extracted from research publications using the approach in sections REF ,REF .", "We supplement the extracted knowledge with domain knowledge as needed.", "Let the extracted facts be as follows: The drug $gefitinib$ is metabolized by $CYP3A4$ .", "The drug $phenytoin$ induces $CYP3A4$ .", "Following additional facts have been provided about a new drug currently in development: A new drug being developed $test\\_drug$ is a CYP3A4 inhibitor We show the pathway specification based on the above facts and background knowledge, then elaborate on each component: $&\\mathbf {domain~of~} gefitinib \\mathbf {~is~} integer, cyp3a4 \\mathbf {~is~} integer, \\nonumber \\\\&~~~~~~~~~~~~~~~~~~~~ phenytoin \\mathbf {~is~} integer, test\\_drug \\mathbf {~is~} integer\\\\&t1 \\mathbf {~may~execute~causing~} \\nonumber \\\\&~~~~~~~~~~~~~~~~~~~~ gefitinib \\mathbf {~change~value~by~} -1, \\nonumber \\\\&~~~~~~~~~~~~~~~~~~~~ cyp3a4 \\mathbf {~change~value~by~} -1, \\nonumber \\\\&~~~~~~~~~~~~~~~~~~~~ cyp3a4 \\mathbf {~change~value~by~} +1\\\\&\\mathbf {normally~stimulate~} t1 \\mathbf {~by~factor~} 2 \\nonumber \\\\&~~~~~~~~~~~~~~~~~~~~ \\mathbf {~if~} phenytoin \\mathbf {~has~value~} 1 \\mathbf {~or~higher~} \\\\&\\mathbf {inhibit~} t1 \\mathbf {~if~} test\\_drug \\mathbf {~has~value~} 1 \\mathbf {~or~higher~} \\\\&\\mathbf {initially~} gefitinib \\mathbf {~has~value~} 20, cyp3a4 \\mathbf {~has~value~} 60, \\nonumber \\\\&~~~~~~~~~~~~~~~~~~~~ phenytoin \\mathbf {~has~value~} 0, test\\_drug \\mathbf {~has~value~} 0 \\\\&\\mathbf {firing~style~} max$ Line REF declares the domain of the fluents as integer numbers.", "Line represents the activity of enzyme $cyp3a4$ as the action $t1$ .", "Due to the enzymic action $t1$ , one unit of $gefitinib$ is metabolized, and thus converted to various metabolites (not shown here).", "The enzymic action uses one unit of $cyp3a4$ as catalyst, which is used in the reaction and released afterwards.", "Line represents the knowledge that $phenytoin$ induces the activity of $cyp3a4$ .", "From background knowledge we find out that the stimulation in the activity can be as high as 2-times [55].", "Line represents the knowledge that there is a new drug $test\\_drug$ being tested that is known to inhibit the activity of $cyp3a4$ .", "Line specifies the initial distribution of the drugs and enzymes in the pathway.", "Assuming the patient has been given some fixed dose, say 20 units, of the medicine $gefitinib$ .", "It also specifies there is a large 60 units quantity of $cyp3a4$ available to ensure reactions do not slow down due to unavailability of enzyme availability.", "Additionaly, the drug $phenytoin$ is absent from the system and a new drug $test\\_drug$ to be tested is not in the system either.", "This gives us our pathway specification.", "Now we consider two application scenarios for drug development.", "Drug Administration A patient is taking 20 units of $gefitinib$ , and is being prescribed additional drugs to be co-administered.", "The drug administrator wants to know if there will be an interaction with $gefitinib$ if 5 units of $phenytoin$ are co-administered.", "If there is an interaction, what will be the bioavailability of $gefitinib$ so that its dosage could be appropriately adjusted.", "The first question is asking whether giving the patient 5-units of $phenytoin$ in addition to the existing $gefitinib$ dose will cause a drug-interaction.", "It is encoded as the following query statement $\\mathbf {Q}$ for a $k$ -step simulation: $&\\mathbf {direction~of~change~in~} average \\mathbf {~value~of~} gefitinib \\mathbf {~is~} d\\\\&~~~~~~\\mathbf {when~observed~at~time~step~} k; \\\\&~~~~~~\\mathbf {comparing~nominal~pathway~with~modified~pathway~obtained~}\\\\&~~~~~~~~~~~~\\mathbf {due~to~interventions:~} \\mathbf {set~value~of~} phenytoin \\mathbf {~to~} 5;$ If the direction of change is “$=$ ” then there was no drug-interaction.", "Otherwise, an interaction was noticed.", "For a simulation of length $k=5$ , we find 15 units of $gefitinib$ remained at the end of simulation in the nominal case when no $phenytoin$ is administered.", "The amount drops to 10 units of $gefitinib$ when $phenytoin$ is co-administered.", "The change in direction is “$<$ ”.", "Thus there is an interaction.", "The second question is asking about the bioavailability of the drug $gefitinib$ after some after giving $phenytoin$ in 5 units.", "If this bioavailability falls below the efficacy level of the drug, then the drug would not treat the disease effectively.", "It is encoded as the following query statement $\\mathbf {Q}$ for a $k$ -step simulation: $&average \\mathbf {~value~of~} gefitinib \\mathbf {~is~} n\\\\&~~~~\\mathbf {when~observed~at~time~step~} k; \\\\&~~~~\\mathbf {due~to~interventions:~} \\\\&~~~~~~~~\\mathbf {set~value~of~} phenytoin \\mathbf {~to~} 5;$ For a simulation of length $k=5$ , we find 10 units of $gefitinib$ remain.", "A drug administrator (such as a pharmacist) can adjust the drug accordingly.", "Drug Development A drug manufacturer is developing a new drug $test\\_drug$ that is known to inhibit CYP3A4 that will be co-administered with drugs $gefitinib$ and $phenytoin$ .", "He wants to determine the bioavailability of $gefitinib$ over time to determine the risk of toxicity.", "The question is asking about the bioavailability of the drug $gefitinib$ after 10 time units after giving $phenytoin$ in 5 units and the new drug $test\\_drug$ in 5 units.", "If this bioavailability remains high, there is chance for toxicity due to the drug at the subsequent dosage intervals.", "It is encoded as the following query statement $\\mathbf {Q}$ for a $k$ -step simulation: $&average \\mathbf {~value~of~} gefitinib \\mathbf {~is~} n\\\\&~~~~\\mathbf {when~observed~at~time~step~} k; \\\\&~~~~\\mathbf {due~to~interventions:~} \\\\&~~~~~~~~\\mathbf {set~value~of~} phenytoin \\mathbf {~to~} 5,\\\\&~~~~~~~~\\mathbf {set~value~of~} test\\_drug \\mathbf {~to~} 5;$ For a simulation of length $k=5$ , we find all 20 units of $gefitinib$ remain.", "This could lead to toxicity by building high concentration of $gefitinib$ in the body.", "Drug Administration in Presence of Genetic Variation A drug administrator wants to establish the dosage of $morphine$ for a person based on its genetic profile using its bioavailability.", "Consider the following facts extracted about a simplified morphine pathway: $codeine$ is metabolized by $CYP2D6$ producing $morphine$ $CYP2D6$ has three allelic variations “1” – (EM) effective metabolizer (normal case) “2” – (UM) ultra rapid metabolizer “9” – (PM) poor metabolizer For simplicity, assume UM allele doubles the metabolic rate, while PM allele halves the metabolic rate of CYP2D6.", "Then, the resulting pathway is given by: $&\\mathbf {domain~of~} cyp2d6\\_allele \\mathbf {~is~} integer, cyp2d6 \\mathbf {~is~} integer\\\\&\\mathbf {domain~of~} codeine \\mathbf {~is~} integer, morphine \\mathbf {~is~} integer\\\\&cyp2d6\\_action \\mathbf {~may~execute~causing}\\\\&~~~~codeine \\mathbf {~change~value~by~} -2, morphine \\mathbf {~change~value~by~} +2\\\\&~~~~\\mathbf {if~} cyp2d6\\_allele \\mathbf {~has~value~equal~to~} 1\\\\&cyp2d6\\_action \\mathbf {~may~execute~causing}\\\\&~~~~codeine \\mathbf {~change~value~by~} -4, morphine \\mathbf {~change~value~by~} +4\\\\&~~~~\\mathbf {if~} cyp2d6\\_allele \\mathbf {~has~value~equal~to~} 2\\\\&cyp2d6\\_action \\mathbf {~may~execute~causing~}\\\\&~~~~codeine \\mathbf {~change~value~by~} -1, morphine \\mathbf {~change~value~by~} +1\\\\&~~~~\\mathbf {if~} cyp2d6\\_allele \\mathbf {~has~value~equal~to~} 9\\\\&\\mathbf {initially~}\\\\&~~~~codeine \\mathbf {~has~value~} 0, morphine \\mathbf {~has~value~} 0,\\\\&~~~~cyp2d6 \\mathbf {~has~value~} 20, cyp2d6\\_allele \\mathbf {~has~value~} 1\\\\&\\mathbf {firing~style~} max\\\\$ Then, the bioavailability of $morphine$ can be determined by the following query: $&average \\mathbf {~value~of~} morphine \\mathbf {~is~} n\\\\&~~~~\\mathbf {when~observed~at~time~step~} k; \\\\&~~~~\\mathbf {due~to~interventions:~} \\\\&~~~~~~~~\\mathbf {set~value~of~} codeine \\mathbf {~to~} 20,\\\\&~~~~~~~~\\mathbf {set~value~of~} cyp2d6 \\mathbf {~to~} 20,\\\\&~~~~~~~~\\mathbf {set~value~of~} cyp2d6\\_allele \\mathbf {~to~} 9;$ Simulation for 5 time steps reveal that the average bioavailability of $morphine$ after 5 time-steps is 5 for PM (down from 10 for EM).", "Although this is a toy example, it is easy to see the potential of capturing known genetic variations in the pathway and setting the complete genetic profile of a person in the intervention part of the query.", "Conclusion In this chapter we presented how we extract biological pathway knowledge from text, including knowledge about drug-gene interactions and their relationship to genetic variation.", "We showed how the information extracted is used to build pathway specification and illustrated how biologically relevant questions can be answered about drug-drug interaction using the BioPathQA system developed in chapter .", "Next we look at the future directions in which the research work done in this thesis can be extended.", "Conclusion and Future Work The field of knowledge representation and reasoning (KR) is currently one of the most active research areas.", "It represents the next step in the evolution of systems that know how to organize knowledge, and have the ability to intelligently respond to questions about this knowledge.", "Such questions could be about static knowledge or the dynamic processes.", "Biological systems are uniquely positioned as role models for this next evolutionary step due to their precise vocabulary and mechanical nature.", "As a result, a number of recent research challenges in the KR field are focused on it.", "The biological field itself needs systems that can intelligently answer questions about such biological processes and systems in an automated fashion, given the large number of research papers published each year.", "Curating these publications is time consuming and expensive, as a result the state of over all knowledge about biological systems lags behind the cutting edge research.", "An important class of questions asked about biological systems are the so called “what-if” questions that compare alternate scenarios of a biological pathway.", "To answer such questions, one has to perform simulation on a nominal pathway against a pathway modified due to the interventions specified for the alternate scenario.", "Often, this means creating two pathways (for nominal and alternate cases) and simulate them separately.", "This opens up the possibility that the two pathways can become out of synchronization.", "A better approach is to allow the user to specify the needed interventions in the query statement itself.", "In addition, to understand the full spread of possible outcomes, given the parallel nature of biological pathways, one must consider all possible pathway evolutions, otherwise, some outcomes may remain hidden.", "If a system is to be used by biologists, it must have a simple interface, lowering the barrier of entry.", "Since biological pathway knowledge can arrive from different sources, including books, published articles, and lab experiments, a common input format is desired.", "Such a format allows specification of pathways due to automatic extraction, as well as any changes / additions due to locally available information.", "A comprehensive end-to-end system that accomplish all the goals would take a natural language query along with any additional specific knowledge about the pathway as input, extract the relevant portion of the relevant pathway from published material (and background knowledge), simulate it based on the query, and generate the results in a visual format.", "Each of these tasks comes with its own challenges, some of which have been addressed in this thesis.", "In this thesis, we have developed a system and a high level language to specify a biological pathway and answer simulation based reasoning questions about it.", "The high level language uses controlled-English vocabulary to make it more natural for a researcher to use directly.", "The high level language has two components: a pathway specification language, and a query specification language.", "The pathway specification language allows the user to specify a pathway in a source independent form, thus locally obtained knowledge (e.g.", "from lab) can be combined with automatically extracted knowledge.", "We believe that our pathway specification language is easy for a person to understand and encode, lowering the bar to using our system.", "Our pathway specification language allows conditional actions, enabling the encoding of alternate action outcomes due to genetic variation.", "An important aspect of our pathway specification language is that it specifies trajectories, which includes specifying the initial configuration of substances, as well as state evolution style, such as maximal firing of actions, or serialized actions etc.", "Our query specification language provides a bridge between natural language questions and their formal representation.", "It is English-like but with precise semantics.", "A main feature of our query language is its support for comparative queries over alternate scenarios, which is not currently supported by any of the query languages (associated with action languages) we have reviewed.", "Our specification of alternate scenarios uses interventions (a general form of actions), that allow the user to modify the pathway as part of the query processing.", "We believe our query language is easier for a biologist to understand without requiring formal training.", "To model the pathways, we use Petri Nets, which have been used in the past to model and simulate biological pathways.", "Petri Nets have a simple visual representation, which closely matches biological pathways; and they inherently support parallelism.", "We extended the Petri Nets to add features that we needed to suit our domain, e.g., reset arcs that remove all quantity of a substance as soon as it is produced, and conditional arcs that specify the conditional outcome of an action.", "For simulation, we use ASP, which allowed us straight forward way to implement Petri Nets.", "It also gave us the ability to add extensions to the Petri Net by making local edits, implement different firing semantics, filter trajectories based on observations, and reason with the results.", "One of the major advantage of using Petri Net based simulation is the ability to generate all possible state evolutions, enabling us to process queries that determine the conditions when a certain observation becomes true.", "Our post-processing step is done in Python, which allows strong text processing capabilities using regular expressions, as well as libraries to easy process large matrices of numbers for summarization of results.", "Now we present additional challenges that need to be addressed.", "Pathway Extraction In Chapter we described how we extract facts for drug-drug interaction and gene variation.", "This work needs to be extended to include some of the newer databases that have come online recently.", "This may provide us with enzyme reaction rates, and substance quantities used in various reactions.", "The relation extraction for pathways must also be cognizant of any genetic variation mentioned in the text.", "Since the knowledge about the pathway appears in relationships at varying degree of detail, a process needs to be devised to assemble the pathway from the same level to granularity together, while also maintaining pathways at different levels of granularities.", "Since pathway extraction is a time consuming task, it would be best to create a catalog of the pathways.", "The cataloged pathways could be manually edited by the user as necessary.", "Storing pathways in this way means that would have to be updated periodically, requiring merging of new knowledge into existing pathways.", "Manual edits would have to be identified, such that the updated pathway does not overwrite them without the user's knowledge.", "Pathway Selection Questions presented in biological texts do not explicitly mention the relevant pathway to use for answering the question.", "One way to address this issue is to maintain a catalog of pre-defined pathways with keywords associated with them.", "Such keywords can include names of the substances, names of the processes, and other relevant meta-data about the pathway.", "The catalog can be searched to find the closest match to the query being asked.", "An additional aspect in proper pathway selection is to use the proper abstraction level.", "If our catalog contains a pathway at different abstraction levels, the coarsest pathway that contains the processes and substances in the query should be selected.", "Any higher fidelity will increase the processing time and generate too much irrelevant data.", "Alternatively, the catalog could contain the pathway in a hierarchical form, allowing extraction of all elements of a pathway at the same depth.", "A common way to hierarchically organize the pathway related to our system is to have hierarchical actions, which is the approach taken by hierarchical Petri nets.", "Lastly, the question may only ask about a small subsection of a much larger pathway.", "For better performance, it is beneficial to extract the smallest biological pathway network model that can answer the question.", "Pathway Modeling In Chapter , we presented our modeling of biological questions using Petri Nets and their extensions encoded in ASP.", "We came across concepts like allosteric regulation, inhibition of inhibition, and inhibition of activation that we currently do not model.", "In allosteric regulation, an enzyme is not fully enabled or disabled, the enzyme's shape changes, making it more or less active.", "The level of its activity depends upon concentrations of activators and inhibitors.", "In inhibition of inhibition, the inhibition of a reaction is itself inhibited by another inhibition; while in inhibition of activation (or stimulation), a substance inhibits the stimulation produced by a substance.", "Both of these appear to be actions on actions, something that Petri Nets do not allow.", "An alternate coding for these would have to be devised.", "As more detailed information about pathways becomes available, the reactions and processes that we have in current pathways may get replaced with more detailed sub-pathways themselves.", "However, such refinement may not come at the same time for separate legs of the pathway.", "Just replacing the coarse transition with a refined transition may not be sufficient due to relative timing constraints.", "Hence, a hierarchical Petri Net model may need to be implemented (see , ).", "Pathway Simulation In Chapter we presented our approach to encode Petri Nets and their extensions.", "We used a discrete solver called clingo for our ASP encoding.", "As the number of simulation length increases in size or larger quantities are involved, the solver slows down significantly.", "This is due to an increased grounding cost of the program.", "Incremental solving (using iclingo) does not help, since the program size still increases, and the increments merely delays the slow down but does not stop it.", "Systems such as constraint logic solvers (such as ) could be used for discrete cases.", "Alternatively, a system developed on the ASPMT [53] approach could be used, since it can represent longer runs, larger quantities, and real number quantities.", "Extend High Level Language In Chapter we described the BioPathQA system, the pathway specification and the query specification high level languages.", "As we enhance the modeling of the biological pathways, we will need to improve or extend the system as well as the high level language.", "We give a few examples of such extensions.", "Our pathway specification language currently does not support continuous quantities (real numbers).", "Extending to real numbers will improve the coverage of the pathways that can be modeled.", "In addition, derived quantities (fluents) can be added, e.g.", "pH could be defined as a formula that is read-only in nature.", "Certain observations and queries can be easily specified using a language such as LTL, especially for questions requiring conditions leading to an action or a state.", "As a result, it may be useful to add LTL formulas to the query language.", "We did not take this approach because it would have presented an additional non-English style syntax for the biologists.", "Our substance production / consumption rates and amounts are currently tied to the fluents.", "In certain situations it is desirable to analyze the quantity of a substance produced / consumed by a specific action, e.g.", "one is interested in finding the amount of H+ ions produced by a multi-protein complex IV only.", "Interventions (that are a part of the query statement) presented in this thesis are applied at the start of the simulation.", "Eliminating this restriction would allow applying surgeries to the pathway mid execution.", "Thus, instead of specifying the steady state conditions in the query statement, one could apply the intervention when a certain steady state is reached.", "Result Formatting and Visualization In Chapter we described our system that answers questions specified in our high level language.", "At the end of its process, it outputs the final result.", "This output method can be enhanced by allowing to look at the progression of results in addition to the final result.", "This provides the biologist with the whole spread of possible outcomes.", "An example of such a spread is shown in Fig.", "fig:q1:result for question REF .", "A graphical representation of the simulation progression is also beneficial in enhancing the confidence of the biologist.", "Indeed many existing tools do this.", "A similar effect can be achieved by parsing and showing the relevant portion of the answer set.", "Summary In Chapter we introduced the thesis topic and summarized specific research contributions In Chapter we introduced the foundational material of this thesis including Petri Nets and ASP.", "We showed how ASP could be used to encode basic Petri Nets.", "We also showed how ASP's elaboration tolerance and declarative syntax allows us to encode various Petri Net extensions with small localized changes.", "We also introduced a new firing semantics, the so called maximal firing set semantics to simulate a Petri Net with maximum parallel activity.", "In Chapter we showed how the Petri Net extensions and the ASP encoding can be used to answer simulation based deep reasoning questions.", "This and the work in Chapter was published in [1], [2].", "In Chapter we developed a system called BioPathQA to allow users to specify a pathway and answer queries against it.", "We also developed a pathway specification language and a query language for this system in order to avoid the vagaries of natural language.", "We introduced a new type of Guarded-arc Petri Nets to model conditional actions as a model for pathway simulation.", "We also described our implementation developed around a subset of the pathway specification language.", "In Chapter we briefly described how text extraction is done to extract real world knowledge about pathways and drug interactions.", "We then used the extracted knowledge to answer question using BioPathQA.", "The text extraction work was published in [73], [72], [33].", "Proofs of Various Propositions Assumption: The definitions in this section assume the programs $\\Pi $ do not have recursion through aggregate atoms.", "Our ASP translation ensures this due to the construction of programs $\\Pi $ .", "First we extend some definitions and properties related to ASP, such that they apply to rules with aggregate atoms.", "We will refer to the non-aggregate atoms as basic atoms.", "Recall the definitions of an ASP program given in section REF .", "Proposition 9 (Forced Atom Proposition) Let $S$ be an answer set of a ground ASP program $\\Pi $ as defined in definition REF .", "For any ground instance of a rule R in $\\Pi $ of the form $A_0 \\text{:-} A_1,\\dots ,$ $A_m,\\mathbf {not~} B_{1},\\dots ,$ $\\mathbf {not~} B_n, C_1,\\dots ,$ $C_k.$ if $\\forall A_i, 1 \\le i \\le m, S \\models A_i$ , and $\\forall B_j, 1 \\le j \\le n, S \\lnot \\models B_j$ , $\\forall C_l, 1 \\le l \\le k, S \\models C_l$ then $S \\models A_0$ .", "Proof: Let $S$ be an answer set of a ground ASP program $\\Pi $ , $R \\in \\Pi $ be a ground rule such that $\\forall B_j, 1 \\le j \\le n, S \\lnot \\models B_j$ ; and $\\forall C_l, 1 \\le l \\le k, S \\models C_l$ .", "Then, the reduct $R^S \\equiv \\lbrace p_1 \\text{:-} A_1,\\dots ,A_m.", "; \\dots ; p_h \\text{:-} A_1,\\dots ,A_m.", "\\; \| \\; \\lbrace p_1,\\dots ,p_h \\rbrace = S \\cap lit(A_0) \\rbrace $ .", "Since $S$ is an answer set of $\\Pi $ , it is a model of $\\Pi ^S$ .", "As a result, whenever, $\\forall A_i, 1 \\le i \\le m, S \\models A_i$ , $\\lbrace p_1,\\dots ,p_h \\rbrace \\subseteq S$ and $S \\models A_0$ .", "Proposition 10 (Supporting Rule Proposition) If $S$ is an answer set of a ground ASP program $\\Pi $ as defined in definition REF then $S$ is supported by $\\Pi $ .", "That is, if $S \\models A_0$ , then there exists a ground instance of a rule R in $\\Pi $ of the type $A_0 \\text{:-} A_1,\\dots ,$ $A_m,$ $\\mathbf {not~} B_{1},\\dots ,$ $\\mathbf {not~} B_n, $ $C_1,\\dots ,C_k.$ such that $\\forall A_i, 1 \\le i \\le m, S \\models A_i$ , $\\forall B_j, 1 \\le j \\le n, S \\lnot \\models B_j$ , and $\\forall C_l, 1\\le l \\le k, S \\models C_l$ .", "Proof: For $S$ to be an answer set of $\\Pi $ , it must be the deductive closure of reduct $\\Pi ^S$ .", "The deductive closure $S$ of $\\Pi ^S$ is iteratively built by starting from an empty set $S$ , and adding head atoms of rules $R_h^S \\equiv p_h \\text{:-} A_1,\\dots ,A_m., R_h^S \\in \\Pi ^S$ , whenever, $S \\models A_i, 1 \\le i \\le m$ , where, $R_h^S$ is a rule in the reduct of ground rule $R \\in \\Pi $ with $p_h \\in lit(A_0) \\cap S$ .", "Thus, there is a rule $R \\equiv A_0 \\text{:-} A_1,\\dots ,$ $A_m,$ $\\mathbf {not~} B_{1},\\dots ,$ $\\mathbf {not~}B_n, $ $C_1,\\dots ,$ $C_k.$ , $R \\in \\Pi $ , such that $\\forall C_l, 1 \\le l \\le k$ and $\\forall B_j, 1 \\le j \\le n, S \\lnot \\models B_j$ .", "Nothing else belongs in $S$ .", "Next, we extend the splitting set theorem to include aggregate atoms.", "Definition 88 (Splitting Set) A Splitting Set for a program $\\Pi $ is any set $U$ of literals such that, for every rule $R \\in \\Pi $ , if $U \\models head(R)$ then $lit(R) \\subset U$ .", "The set $U$ splits $\\Pi $ into upper and lower parts.", "The set of rules $R \\in \\Pi $ s.t.", "$lit(R) \\subset U$ is called the bottom of $\\Pi $ w.r.t.", "$U$ , denoted by $bot_U(\\Pi )$ .", "The rest of the rules, i.e.", "$\\Pi \\setminus bot_U(\\Pi )$ is called the top of $\\Pi $ w.r.t.", "$U$ , denoted by $top_U(\\Pi )$ .", "Proposition 11 Let $U$ be a splitting set of $\\Pi $ with answer set $S$ and let $X = S \\cap U$ and $Y = S \\setminus U$ .", "Then, the reduct of $\\Pi $ w.r.t.", "$S$ , i.e.", "$\\Pi ^S$ is equal to $bot_U(\\Pi )^X \\cup (\\Pi \\setminus bot_U(\\Pi ))^{X \\cup Y}$ .", "Proof: We can rewrite $\\Pi $ as $bot_U(\\Pi ) \\cup (\\Pi \\setminus bot_U(\\Pi ))$ using the definition of splitting set.", "Then the reduct of $\\Pi $ w.r.t.", "$S$ can be written in terms of $X$ and $Y$ , since $S = X \\cup Y$ .", "$\\Pi ^S =$ $\\Pi ^{X \\cup Y} =$ $(bot_U(\\Pi ) \\cup (\\Pi \\setminus bot_U(\\Pi )))^{X \\cup Y} =$ $bot_U(\\Pi )^{X \\cup Y} \\cup $ $(\\Pi \\setminus bot_U(\\Pi ))^{X \\cup Y}$ .", "Since $lit(bot_U(\\Pi )) \\subseteq U$ and $Y \\cap U = \\emptyset $ , the reduct of $bot_U(\\Pi )^{X \\cup Y} = bot_U(\\Pi )^X$ .", "Thus, $\\Pi ^{X \\cup Y} = bot_U(\\Pi )^X \\cup (\\Pi \\setminus bot_U(\\Pi ))^{X \\cup Y}$ .", "Proposition 12 Let $S$ be an answer set of a program $\\Pi $ , then $S \\subseteq lit(head(\\Pi ))$ .", "Proof: If $S$ is an answer set of a program $\\Pi $ then $S$ is produced by the deductive closure of $\\Pi ^S$ (the reduct of $\\Pi $ w.r.t $S$ ).", "By definition of the deductive closure, nothing can be in $S$ unless it is the head of some rule supported by $S$ .", "Splitting allow computing the answer set of a program $\\Pi $ in layers.", "Answer sets of the bottom layer are first used to partially evaluate the top layer, and then answer sets of the top layer are computed.", "Next, we define how a program is partially evaluated.", "Intuitively, the partial evaluation of an aggregate atom $c$ given splitting set $U$ w.r.t.", "a set of literals $X$ removes all literals that are part of the splitting set $U$ from $c$ and updates $c$ 's lower and upper bounds based on the literals in $X$ , which usually come from $bot_U$ of a program.", "The set $X$ represents our knowledge about the positive literals, while the set $U \\setminus X$ represents our knowledge about naf-literals at this stage.", "We can remove all literals in $U$ from $c$ , since the literals in $U$ will not appear in the head of any rule in $top_U$ .", "Definition 89 (Partial Evaluation of Aggregate Atom) The partial evaluation of an aggregate atom $c = l \\; [ B_0=w_0,\\dots , B_m=w_m ] \\; u$ , given splitting set $U$ w.r.t.", "a set of literals $X$ , written $eval_U(c,X)$ is a new aggregate atom $c^{\\prime }$ constructed from $c$ as follows: $pos(c^{\\prime }) = pos(c) \\setminus U$ $d=\\sum _{B_i \\in pos(c) \\cap U \\cap X}{w_i} $ $l^{\\prime } = l-d$ , $u^{\\prime } = u-d$ are the lower and upper limits of $c^{\\prime }$ Next, we define how a program is partially evaluated given a splitting set $U$ w.r.t.", "a set of literals $X$ that form the answer-set of the lower layer.", "Intuitively, a partial evaluation deletes all rules from the partial evaluation for which the body of the rule is determined to be not supported by $U$ w.r.t.", "$X$ .", "This includes rules which have an aggregate atom $c$ in their body s.t.", "$lit(c) \\subseteq U$ , but $X \\lnot \\models c$ Note that we can fully evaluate an aggregate atom $c$ w.r.t.", "answer-set $X$ if $lit(c) \\subseteq U$ ..", "In the remaining rules, the positive and negative literals that overlap with $U$ are deleted, and so are the aggregate atoms that have $lit(c) \\subseteq U$ (since such a $c$ can be fully evaluated w.r.t.", "$X$ ).", "Each remaining aggregate atom is updated by removing atoms that belong to $U$ Since the atoms in $U$ will not appear in the head of any atoms in $top_U$ and hence will not form a basis in future evaluations of $c$ ., and updating its limits based on the answer-set $X$ The limit update requires knowledge of the current answer-set to update limit values..", "The head atom is not modified, since $eval_U(...)$ is performed on $\\Pi \\setminus bot_U(\\Pi )$ , which already removes all rules with heads atoms that intersect $U$ .", "Definition 90 (Partial Evaluation) The partial evaluation of $\\Pi $ , given splitting set $U$ w.r.t.", "a set of literals $X$ is the program $eval_U(\\Pi ,X)$ composed of rules $R^{\\prime }$ for each $R \\in \\Pi $ that satisfies all the following conditions: $pos(R) \\cap U \\subseteq X,$ $((neg(R) \\cap U) \\cap X) = \\emptyset , \\text{ and }$ if there is a $c \\in agg(R)$ s.t.", "$lit(c) \\subseteq U$ , then $X \\models c$ A new rule $R^{\\prime }$ is constructed from a rule $R$ as follows: $head(R^{\\prime }) = head(R)$ , $pos(R^{\\prime }) = pos(R) \\setminus U$ , $neg(R^{\\prime }) = neg(R) \\setminus U$ , $agg(R^{\\prime }) = \\lbrace eval_U(c,X) : c \\in agg(R), lit(c) \\lnot \\subseteq U \\rbrace $ Proposition 13 Let $U$ be a splitting set for $\\Pi $ , $X$ be an answer set of $bot_U(\\Pi )$ , and $Y$ be an answer set of $eval_U(\\Pi \\setminus bot_U(\\Pi ),X)$ .", "Then, $X \\subseteq lit(\\Pi ) \\cap U$ and $Y \\subseteq lit(\\Pi ) \\setminus U$ .", "Proof: By proposition REF , $X \\subseteq lit(head(bot_U(\\Pi )))$ , and $Y \\subseteq lit(head(eval_U(\\Pi \\setminus bot_U(\\Pi ),X)))$ .", "In addition, $lit(head(bot_U(\\Pi ))) \\subseteq lit(bot_U(\\Pi ))$ and $lit(bot_U(\\Pi )) \\subseteq lit(\\Pi ) \\cap U$ by definition of $bot_U(\\Pi )$ .", "Then $X \\subseteq lit(\\Pi ) \\cap U$ , and $Y \\subseteq lit(\\Pi ) \\setminus U$ .", "Definition 91 (Solution) [4] Let $U$ be a splitting set for a program $\\Pi $ .", "A solution to $\\Pi $ w.r.t.", "$U$ is a pair $\\langle X,Y \\rangle $ of literals such that: $X$ is an answer set for $bot_U(\\Pi )$ $Y$ is an answer set for $eval_U(top_U(\\Pi ),X)$ ; and $X \\cup Y$ is consistent.", "Proposition 14 (Splitting Theorem) [4] Let $U$ be a splitting set for a program $\\Pi $ .", "A set $S$ of literals is a consistent answer set for $\\Pi $ iff $S = X \\cup Y$ for some solution $\\langle X,Y \\rangle $ of $\\Pi $ w.r.t.", "$U$ .", "Lemma 1 Let $U$ be a splitting set of $\\Pi $ , $C$ be an aggregate atom in $\\Pi $ , and $X$ and $Y$ be subsets of $lit(\\Pi )$ s.t.", "$X \\subseteq U$ , and $Y \\cap U = \\emptyset $ .", "Then, $X \\cup Y \\models C$ iff $Y \\models eval_U(C,X)$ .", "Proof: Let $C^{\\prime } = eval_U(C,X)$ , then by definition of partial evaluation of aggregate atom, $pos(C^{\\prime }) = pos(C) \\setminus U$ , with lower limit $l^{\\prime } = l-d$ , and upper limit $u^{\\prime } = u-d$ , computed from $l,u$ , the lower and upper limits of $C$ , where $d=\\displaystyle \\sum _{B_i \\in pos(C) \\cap U \\cap X}{w_i}$ $Y \\models C^{\\prime }$ iff $l^{\\prime } \\le \\left( \\displaystyle \\sum _{B^{\\prime }_i \\in pos(C^{\\prime }) \\cap Y}{w^{\\prime }_i} \\right) \\le u^{\\prime }$ – by definition of aggregate atom satisfaction.", "then $Y \\models C$ iff $l \\le \\left( \\displaystyle \\sum _{B_i \\in pos(C) \\cap U \\cap X}{w_i} +\\displaystyle \\sum _{B^{\\prime }_i \\in (pos(C) \\setminus U) \\cap Y}{w^{\\prime }_i} \\right) \\le u$ however, $(pos(C) \\cap U) \\cap X$ and $(pos(C) \\setminus U) \\cap Y$ combined represent $pos(C) \\cap (X \\cup Y)$ – since $pos(C) \\cap (X \\cup Y) &= ((pos(C) \\cap U) \\cup (pos(C) \\setminus U)) \\cap (X \\cup Y) \\\\&= [((pos(C) \\cap U) \\cup (pos(C) \\setminus U)) \\cap X] \\\\&~~~~~~\\cup [((pos(C) \\cap U) \\cup (pos(C) \\setminus U)) \\cap Y]\\\\&= [(pos(C) \\cap U) \\cap X) \\cup ((pos(C) \\setminus U) \\cap X)]\\\\&~~~~~~\\cup [(pos(C) \\cap U) \\cap Y) \\cup ((pos(C) \\setminus U) \\cap Y)]\\\\&= [((pos(C) \\cap U) \\cap X) \\cup \\emptyset ] \\cup [\\emptyset \\cup ((pos(C) \\setminus U) \\cap Y)]\\\\&= ((pos(C) \\cap U) \\cap X) \\cup ((pos(C) \\setminus U) \\cap Y)$ where $X \\subseteq U \\text{ and } Y \\cap U = \\emptyset $ thus, $Y \\models C$ iff $l \\le \\left( \\displaystyle \\sum _{B_i \\in pos(C) \\cap (X \\cup Y)}{w_i} \\right) \\le u$ which is the same as $X \\cup Y \\models C$ Lemma 2 Let $U$ be a splitting set for $\\Pi $ , and $X, Y$ be subsets of $lit(\\Pi )$ s.t.", "$X \\subseteq U$ and $Y \\cap U = \\emptyset $ .", "Then the body of a rule $R^{\\prime } \\in eval_U(\\Pi ,X)$ is satisfied by $Y$ iff the body of the rule $R \\in \\Pi $ it was constructed from is satisfied by $X \\cup Y$ .", "Proof: $Y$ satisfies $body(R^{\\prime })$ iff $pos(R^{\\prime }) \\subseteq Y$ , $neg(R^{\\prime }) \\cap Y = \\emptyset $ , $Y \\models C^{\\prime }$ for each $C^{\\prime } \\in agg(R^{\\prime })$ – by definition of rule satisfaction iff $(pos(R) \\cap U) \\subseteq X$ , $(pos(R) \\setminus U) \\subseteq Y$ , $(neg(R) \\cap U) \\cap X) = \\emptyset $ , $(neg(R) \\setminus U) \\cap Y) = \\emptyset $ , $X$ satisfies $C$ for all $C \\in agg(C)$ in which $lit(C) \\subseteq U$ , and $Y$ satisfies $eval_U(C,X)$ for all $ C \\in agg(C)$ in which $lit(C) \\lnot \\subseteq U$ – using definition of partial evaluation iff $pos(R) \\subseteq X \\cup Y$ , $neg(R) \\cap (X \\cup Y) = \\emptyset $ , $X \\cup Y \\models C$ – using $(A \\cap U) \\cup (A \\setminus U) = A$ $A \\cap (X \\cup Y) = ((A \\cap U) \\cup (A \\setminus U)) \\cap (X \\cup Y) = ((A \\cap U) \\cap (X \\cup Y)) \\cup ((A \\setminus U) \\cap (X \\cup Y)) = (A \\cap U) \\cap X) \\cup ((A \\setminus U) \\cap Y)$ – given $X \\subseteq U$ and $Y \\cap U = \\emptyset $ .", "and lemma REF Proof of Splitting Theorem: Let $U$ be a splitting set of $\\Pi $ , then a consistent set of literals $S$ is an answer set of $\\Pi $ iff it can be written as $S = X \\cup Y$ , where $X$ is an answer set of $bot_U(\\Pi )$ ; and $Y$ is an answer set of $eval_U(\\Pi \\setminus bot_U(\\Pi ),Y)$ .", "($\\Leftarrow $ ) Let $X$ is an answer set of $bot_U(\\Pi )$ ; and $Y$ is an answer set of $eval_U(\\Pi \\setminus bot_U(\\Pi ),X)$ ; we show that $X \\cup Y$ is an answer set of $\\Pi $ .", "By definition of $bot_U(\\Pi )$ , $lit(bot_U(\\Pi )) \\subseteq U$ .", "In addition, by proposition REF , $Y \\cap U = \\emptyset $ .", "Then, $\\Pi ^{X \\cup Y} = (bot_U(\\Pi ) \\cup (\\Pi \\setminus bot_U(\\Pi )))^{X \\cup Y} = bot_U(\\Pi )^{X \\cup Y} \\cup (\\Pi \\setminus bot_U(\\Pi ))^{X \\cup Y} = bot_U(\\Pi )^X \\cup (\\Pi \\setminus bot_U(\\Pi ))^{X \\cup Y}$ .", "Let $r$ be a rule in $\\Pi ^{X \\cup Y}$ , s.t.", "$X \\cup Y \\models body(r)$ then we show that $X \\cup Y \\models head(r)$ .", "The rule $r$ either belongs to $bot_U(\\Pi )^X$ or $(\\Pi \\setminus bot_U(\\Pi ))^{X \\cup Y}$ .", "Case 1: say $r \\in bot_U(\\Pi )^X$ be a rule whose body is satisfied by $X \\cup Y$ then there is a rule $R \\in bot_U(\\Pi )$ s.t.", "$r \\in R^X$ then $X \\models body(R)$ – since $X \\cup Y \\models body(r)$ ; $lit(bot_U(\\Pi )) \\subseteq U$ and $Y \\cap U = \\emptyset $ we already have $X \\models head(R)$ – since $X$ is an answer set of $bot_U(\\Pi )$ ; given then $X \\cup Y \\models head(R)$ – because $lit(R) \\subseteq U$ and $Y \\cap U = \\emptyset $ consequently, $X \\cup Y \\models head(r)$ Case 2: say $r \\in (\\Pi \\setminus bot_U(\\Pi ))^{X \\cup Y}$ be a rule whose body is satisfied by $X \\cup Y$ then there is a rule $R \\in (\\Pi \\setminus bot_U(\\Pi ))$ s.t.", "$r \\in R^{X \\cup Y}$ then $lit(head(R)) \\cap U = \\emptyset $ – otherwise, $R$ would have belonged to $bot_U(\\Pi )$ , by definition of splitting set then $head(r) \\in Y$ – since $X \\subseteq U$ in addition, $pos(R) \\subseteq X \\cup Y$ , $neg(R) \\cap (X \\cup Y) = \\emptyset $ , $X \\cup Y \\models C$ for each $C \\in agg(R)$ – using definition of reduct then $pos(R) \\cap U \\subseteq X$ or $pos(R) \\setminus U \\subseteq Y$ ; $(neg(R) \\cap U) \\cap X = \\emptyset $ and $(neg(R) \\setminus U) \\cap Y = \\emptyset $ ; and for each $C \\in agg(R)$ , either $X \\models C$ if $lit(C) \\subseteq U$ , or $Y \\models eval_U(C,X)$ if $lit(C) \\lnot \\subseteq U$ – by rearranging, lemma REF , $X \\subseteq U$ , $Y \\cap U = \\emptyset $ , and definition of partial evaluation of an aggregate atom note that $pos(R) \\cap U \\subseteq X$ , $(neg(R) \\cap U) \\cap X = \\emptyset $ , and for each $C \\in agg(R)$ , s.t.", "$lit(C) \\subseteq U$ , $X \\models C$ , represent conditions satisfied by each rule that become part of a partial evaluation – using definition of partial evaluation and $pos(R) \\setminus U$ , $neg(R) \\setminus U$ , and for each $C \\in agg(R)$ , $eval_U(C,X)$ are the modifications made to the rule during partial evaluation given splitting set $U$ w.r.t.", "$X$ – using definition of partial evaluation and $pos(R) \\setminus U \\subseteq Y$ , $(neg(R) \\setminus U) \\cap Y = \\emptyset $ , and for each $C \\in agg(R)$ , $Y \\models eval_U(C,X)$ if $lit(C) \\lnot \\subseteq U$ represent conditions satisfied by rules that become part of the reduct w.r.t $Y$ – using definition of partial evaluation and reduct then $r$ is a rule in reduct $eval_U(\\Pi \\setminus bot_U(\\Pi ),X)^Y$ – using (REF ), (REF ) above in addition, given that $Y$ satisfies $eval_U(\\Pi \\setminus bot_U(\\Pi ),X)$ , and $head(r) \\cap U = \\emptyset $ , we have $X \\cup Y \\models head(r)$ Next we show that $X \\cup Y$ satisfies all rules of $\\Pi $ .", "Say, $R$ is a rule in $\\Pi $ not satisfied by $X \\cup Y$ .", "Then, either it belongs to $bot_U(\\Pi )$ or $(\\Pi \\setminus bot_U(\\Pi ))$ .", "If it belongs to $bot_U(\\Pi )$ , it must not be satisfied by $X$ , since $lit(bot_U(\\Pi )) \\subseteq U$ and $Y \\cap U = \\emptyset $ .", "However, the contrary is given to be true.", "On the other hand if it belongs to $(\\Pi \\setminus bot_U(\\Pi ))$ , then $X \\cup Y$ satisfies $body(R)$ but not $head(R)$ .", "That would mean that its $head(R)$ is not satisfied by $Y$ , since $head(R) \\cap U = \\emptyset $ by definition of splitting set.", "However, from lemma REF we know that if $body(R)$ is satisfied by $X \\cup Y$ , $body(R^{\\prime })$ is satisfied by $Y$ for $R^{\\prime } \\in eval_U(\\Pi \\setminus bot_U(\\Pi ),X)$ .", "We also know that $Y$ satisfies all rules in $eval_U(\\Pi \\setminus bot_U(\\Pi ),X)$ .", "So, $R^{\\prime }$ must be satisfied by $Y$ contradicting our assumption.", "Thus, all rules of $\\Pi $ are satisfied by $X \\cup Y$ and $X \\cup Y$ is an answer set of $\\Pi $ .", "($\\Rightarrow $ ) Let $S$ be a consistent answer set of $\\Pi $ , we show that $S = X \\cup Y$ for sets $X$ and $Y$ s.t.", "$X$ is an answer set of $bot_U(\\Pi )$ and $Y$ is an answer set of $eval_U(\\Pi \\setminus bot_U(\\Pi ),X)$ .", "We take $X=S \\cap U$ , $Y=S \\setminus U$ , then $S=X \\cup Y$ .", "Case 1: We show that $X$ is answer set of $bot_U(\\Pi )$ $\\Pi $ can be split into $bot_U(\\Pi ) \\cup (\\Pi \\setminus bot_U(\\Pi ))$ – by definition of splitting then $X \\cup Y$ satisfies $bot_U(\\Pi )$ – $X \\cup Y$ is an answer set of $\\Pi $ ; given however $lit(bot_U(\\Pi )) \\subseteq U$ , $Y \\cap U = \\emptyset $ – by definition of splitting then $X$ satisfies $bot_U(\\Pi )$ – since elements of $Y$ do not appear in the rules of $bot_U(\\Pi )$ then $X$ is an answer set of $bot_U(\\Pi )$ Case 2: We show that $Y$ is answer set of $eval_U(\\Pi \\setminus bot_U(\\Pi ),X)$ let $r$ be a rule in $eval_U(\\Pi \\setminus bot_U(\\Pi ),X)^Y$ , s.t.", "its body is satisfied by $Y$ then $r \\in R^Y$ for an $R \\in eval_U(\\Pi \\setminus bot_U(\\Pi ),X)$ s.t.", "[(i)] (3) $pos(R) \\subseteq Y$ (4) $neg(R) \\cap Y = \\emptyset $ (5) $Y \\models C$ for all $C \\in agg(R)$ (6) $head(R) \\cap Y \\ne \\emptyset $ – using definition of reduct each $R$ is constructed from $R^{\\prime } \\in \\Pi $ that satisfies all the following conditions [(i)] (8) $pos(R^{\\prime }) \\subseteq U \\cap X$ (9) $(neg(R^{\\prime }) \\cap U) \\cap X = \\emptyset $ (10) if there is a $C^{\\prime } \\in agg(R^{\\prime })$ s.t.", "$lit(C^{\\prime }) \\subseteq U$ , then $X \\models C^{\\prime }$ ; and each $C \\in agg(R)$ is a partial evaluation of $C^{\\prime } \\in agg(R^{\\prime })$ s.t.", "$C = eval_U(C^{\\prime },X)$ – using definition of partial evaluation then the $body(R^{\\prime })$ satisfies all the following conditions: $pos(R^{\\prime }) \\subseteq X \\cup Y$ – since $X \\subseteq U$ , $X \\cap Y = \\emptyset $ $neg(R^{\\prime }) \\cap (X \\cup Y) = \\emptyset $ – since $X \\subseteq U$ , $X \\cap Y = \\emptyset $ $X \\cup Y \\models C^{\\prime }$ for each $C^{\\prime } \\in agg(R^{\\prime })$ – since [(i)] (d) each $C^{\\prime } \\in agg(R^{\\prime })$ with $lit(C^{\\prime }) \\subseteq U$ satisfied by $X$ is also satisfied by $X \\cup Y$ as $lit(Y) \\cap lit(C^{\\prime }) = \\emptyset $ ; and (e) each $C^{\\prime } \\in agg(R^{\\prime })$ with $lit(C^{\\prime }) \\lnot \\subseteq U$ is satisfied by $X \\cup Y$ – using partial evaluation, reduct construction, and $X \\cap Y = \\emptyset $ then $X \\cup Y$ satisfies $body(R^{\\prime })$ – from previous line in addition, $lit(head(R^{\\prime })) \\cap U = \\emptyset $ , otherwise, $R^{\\prime }$ would have belonged to $bot_U(\\Pi )$ by definition of splitting set then $R^{\\prime }$ is a rule in $\\Pi \\setminus bot_U(\\Pi )$ – from the last two lines we know that $X \\cup Y$ satisfies every rule in $(\\Pi \\setminus bot_U(\\Pi ))$ – given; and that elements of $U$ do not appear in the head of rules in $(\\Pi \\setminus bot_U(\\Pi ))$ – from definition of splitting; then $Y$ must satisfy the head of these rules then $Y$ satisfies $head(R^{\\prime })$ – from (REF ) Next we show that $Y$ satisfies all rules of $eval_U(\\Pi \\setminus bot_U(\\Pi ),X)$ .", "Let $R^{\\prime }$ be a rule in $eval_U(\\Pi \\setminus bot_U(\\Pi ),X)$ such that $body(R^{\\prime })$ is satisfied by $Y$ but not $head(R^{\\prime })$ .", "Since $head(R^{\\prime }) \\cap Y = \\emptyset $ , $head(R^{\\prime })$ is not satisfied by $X \\cup Y$ either.", "Then, there is an $R \\in (\\Pi \\setminus bot_U(\\Pi ))$ such that $X \\cup Y$ satisfies $body(R)$ but not $head(R)$ , which contradicts given.", "Thus, $Y$ satisfies all rules of $eval_U(\\Pi \\setminus bot_U(\\Pi ),X)$ .", "Then $Y$ is an answer set of $eval_U(\\Pi \\setminus bot_U(\\Pi ),X)$ Definition 92 (Splitting Sequence) [4] A splitting sequence for a program $\\Pi $ is a monotone, continuous sequence ${\\langle U_{\\alpha } \\rangle }_{\\alpha < \\mu }$ of splitting sets of $\\Pi $ such that $\\bigcup _{\\alpha < \\mu }{U_{\\mu }} = lit(\\Pi )$ .", "Definition 93 (Solution) [4] Let $U={\\langle U_{\\alpha } \\rangle }_{\\alpha < \\mu }$ be a splitting sequence for a program $\\Pi $ .", "A solution to $\\Pi $ w.r.t $U$ is a sequence ${\\langle X_{\\alpha } \\rangle }_{\\alpha < \\mu }$ of sets of literals such that: $X_0$ is an answer set for $bot_{U_0}(\\Pi )$ for any ordinal $\\alpha + 1 < \\mu $ , $X_{\\alpha +1}$ is an answer set of the program: $eval_{U_{\\alpha }}(bot_{U_{\\alpha +1}}(\\Pi ) \\setminus bot_{U_{\\alpha }}(\\Pi ), \\bigcup _{\\nu \\le \\alpha }{X_{\\nu }})$ for any limit ordinal $\\alpha < \\mu , X_{\\alpha } = \\emptyset $ , and $\\bigcup _{\\alpha \\le \\mu }(X_{\\alpha })$ is consistent Proposition 15 (Splitting Sequence Theorem) [4] Let $U={\\langle U_{\\alpha } \\rangle }_{\\alpha < \\mu }$ be a splitting sequence for a program $\\Pi $ .", "A set $S$ of literals is a consistent answer set for $\\Pi $ iff $S=\\bigcup _{\\alpha < \\mu }{X_{\\alpha }}$ for some solution ${\\langle X_{\\alpha } \\rangle }_{\\alpha < \\mu }$ to $\\Pi $ w.r.t $U$ .", "Proof: Let $U = \\langle U_\\alpha \\rangle _{\\alpha < \\mu }$ be a splitting sequence of $\\Pi $ , then a consistent set of literals $S=\\bigcup _{\\alpha < \\mu }{X_{\\alpha }}$ is an answer set of $\\Pi ^S$ iff $X_0$ is an answer set of $bot_{U_0}(\\Pi )$ and for any ordinal $\\alpha + 1 < \\mu $ , $X_{\\alpha +1}$ is an answer set of $eval_{U_{\\alpha }}(bot_{U_{\\alpha +1}}(\\Pi ) \\setminus bot_{U_{\\alpha }}(\\Pi ), \\bigcup _{\\nu \\le \\alpha }X_{\\nu })$ .", "Note that every literal in $bot_{U_0}(\\Pi )$ belongs to $lit(\\Pi ) \\cap U_0$ , and every literal occurring in $eval_{U_{\\alpha }}(bot_{U_{\\alpha +1}}(\\Pi ) \\setminus bot_{U_{\\alpha }}(\\Pi ), \\bigcup _{\\nu \\le \\alpha }X_{\\nu })$ , $(\\alpha + 1 < \\mu )$ belongs to $lit(\\Pi ) \\cap (U_{\\alpha +1} \\setminus U_{\\alpha })$ .", "In addition, $X_0$ , and all $X_{\\alpha +1}$ are pairwise disjoint.", "We prove the theorem by induction over the splitting sequence.", "Base case: $\\alpha = 1$ .", "The splitting sequence is $U_0 \\subseteq U_1$ .", "Then the sub-program $\\Pi _1 = bot_{U_1}(\\Pi )$ contains all literals in $U_1$ ; and $U_0$ splits $\\Pi _1$ into $bot_{U_0}(\\Pi _1)$ and $bot_{U_1}(\\Pi _1) \\setminus bot_{U_0}(\\Pi _1)$ .", "Then, $S_1 = X_0 \\cup X_1$ is a consistent answer set of $\\Pi _1$ iff $X_0 = S_1 \\cap U_0$ is an answer set of $bot_{U_0}(\\Pi _1)$ and $X_1 = S_1 \\setminus U_0$ is an answer set of $eval_{U_0}(\\Pi _1 \\setminus bot_{U_0}(\\Pi _1),X_1)$ – by the splitting theorem Since $bot_{U_0}(\\Pi _1) = bot_{U_0}(\\Pi )$ and $bot_{U_1}(\\Pi _1) \\setminus bot_{U_0}(\\Pi _1) = bot_{U_1}(\\Pi ) \\setminus bot_{U_0}(\\Pi )$ ; $S_1 = X_0 \\cup X_1$ is an answer set for $\\Pi \\setminus bot_{U_1}(\\Pi )$ .", "Induction: Assume theorem holds for $\\alpha = k$ , show theorem holds for $\\alpha = k+1$ .", "The inductive assumption holds for the splitting sequence $U_0 \\subseteq \\dots \\subseteq U_k$ .", "Then the sub-program $\\Pi _k = bot_{U_k}(\\Pi )$ contains all literals in $U_k$ and $S_k = X_0 \\cup \\dots \\cup X_k$ is a consistent answer set of $\\Pi ^{S_k}$ iff $X_0$ is an answer set for $bot_{U_0}(\\Pi _k)$ and for any $\\alpha \\le k$ , $X_{\\alpha +1}$ is answer set of $eval_{U_{\\alpha }}(bot_{U_{\\alpha +1}}(\\Pi ) \\setminus bot_{U_{\\alpha }}(\\Pi ), X_0 \\cup \\dots \\cup X_{\\alpha })$ We show that the theorem holds for $\\alpha = k+1$ .", "The splitting sequence is $U_0 \\subseteq U_{k+1}$ .", "Then the sub-program $\\Pi _{k+1} = bot_{U_{k+1}}(\\Pi )$ contains all literals $U_{k+1}$ .", "We have $U_k$ split $\\Pi _{k+1}$ into $bot_{U_k}(\\Pi _{k+1})$ and $bot_{U_{k+1}}(\\Pi _{k+1}) \\setminus bot_{U_k}(\\Pi _{k+1})$ .", "Then, $S_{k+1} = X_{0:k} \\cup X_{k+1}$ is a consistent answer set of $\\Pi _{k+1}$ iff $X_{0:k} = S_{k+1} \\cap U_k$ is an answer set of $bot_{U_k}(\\Pi _{k+1})$ and $X_{k+1} = S_{k+1} \\setminus U_k$ is an answer set of $eval_{U_k}(\\Pi _{k+1} \\setminus bot_{U_k}(\\Pi _{k+1},X_{k+1})$ – by the splitting theorem Since $bot_{U_k}(\\Pi _{k+1}) = bot_{U_k}(\\Pi )$ and $bot_{U_{k+1}}(\\Pi _{k+1}) \\setminus bot_{U_k}(\\Pi _{k+1}) = bot_{U_{k+1}}(\\Pi ) \\setminus bot_{U_k}(\\Pi )$ ; $S_{k+1} = X_{0:k} \\cup X_1$ is an answer set for $\\Pi \\setminus bot_{U_{k+1}}(\\Pi )$ .", "From the inductive assumption we know that $X_0 \\cup \\dots \\cup X_k$ is a consistent answer set of $bot_{U_k}(\\Pi )$ , $X_0$ is the answer set of $bot_{U_0}(\\Pi )$ , and for each $0 \\le \\alpha \\le k$ , $X_{\\alpha +1}$ is answer set of $eval_{U_{\\alpha }}(bot_{U_{\\alpha +1}}(\\Pi ) \\setminus bot_{U_{\\alpha }}(\\Pi ), X_0 \\cup \\dots \\cup X_{\\alpha })$ .", "Thus, $X_{0:k} = X_0 \\cup \\dots \\cup X_k$ .", "Combining above with the inductive assumption, we get $S_{k+1} = X_0 \\cup \\dots \\cup X_{k+1}$ is a consistent answer set of $\\Pi ^{S_{k+1}}$ iff $X_0$ is an answer set for $bot_{U_0}(\\Pi _{k+1})$ and for any $\\alpha \\le k+1$ , $X_{\\alpha +1}$ is answer set of $eval_{U_{\\alpha }}(bot_{U_{\\alpha +1}}(\\Pi ) \\setminus bot_{U_{\\alpha }}(\\Pi ), X_0 \\cup \\dots \\cup X_{\\alpha })$ .", "In addition, for some $\\alpha < \\mu $ , where $\\mu $ is the length of the splitting sequence $U = \\langle U_{\\alpha } \\rangle _{\\alpha < \\mu }$ of $\\Pi $ , $bot_{U_{\\alpha }}(\\Pi )$ will be the entire $\\Pi $ , i.e.", "$lit(\\Pi ) = U_{\\alpha }$ .", "Then the set $S$ of literals is a consistent answer set of $\\Pi $ iff $S=\\bigcup _{\\alpha < \\mu }(X_{\\alpha })$ for some solution $\\langle X_{\\alpha } \\rangle _{\\alpha < \\mu }$ to $\\Pi $ w.r.t $U$ .", "Proof of Proposition REF Let $PN=(P,T,E,W)$ be a Petri Net, $M_0$ be its initial marking and let $\\Pi ^0(PN,M_0,$ $k,ntok)$ be the ASP encoding of $PN$ and $M_0$ over a simulation length $k$ , with maximum $ntok$ tokens on any place node, as defined in section REF .", "Then $X=M_0,T_0,M_1,\\dots ,M_k,T_k,M_{k+1}$ is an execution sequence of a $PN$ (w.r.t.", "$M_0$ ) iff there is an answer set $A$ of $\\Pi ^0(PN,M_0,k,ntok)$ such that: $\\lbrace fires(t,ts) : t \\in T_{ts}, 0 \\le ts \\le k \\rbrace = \\lbrace fires(t,ts) : fires(t,ts) \\in A \\rbrace $ $\\begin{split}\\lbrace holds(p,q,ts) &: p \\in P, q = M_{ts}(p), 0 \\le ts \\le k+1 \\rbrace \\\\&= \\lbrace holds(p,q,ts) : holds(p,q,ts) \\in A \\rbrace \\end{split}$ We prove this by showing that: Given an execution sequence $X$ , we create a set $A$ such that it satisfies (REF ) and (REF ) and show that $A$ is an answer set of $\\Pi ^0$ Given an answer set $A$ of $\\Pi ^0$ , we create an execution sequence $X$ such that (REF ) and (REF ) are satisfied First we show (REF ): Given $PN$ and its execution sequence $X$ , we create a set $A$ as a union of the following sets: $A_1=\\lbrace num(n) : 0 \\le n \\le ntok \\rbrace $ $A_2=\\lbrace time(ts) : 0 \\le ts \\le k\\rbrace $ $A_3=\\lbrace place(p) : p \\in P \\rbrace $ $A_4=\\lbrace trans(t) : t \\in T \\rbrace $ $A_5=\\lbrace ptarc(p,t,n) : (p,t) \\in E^-, n=W(p,t) \\rbrace $ , where $E^- \\subseteq E$ $A_6=\\lbrace tparc(t,p,n) : (t,p) \\in E^+, n=W(t,p) \\rbrace $ , where $E^+ \\subseteq E$ $A_7=\\lbrace holds(p,q,0) : p \\in P, q=M_{0}(p) \\rbrace $ $A_8=\\lbrace notenabled(t,ts) : t \\in T, 0 \\le ts \\le k, \\exists p \\in \\bullet t, M_{ts}(p) < W(p,t) \\rbrace $ per definition REF (enabled transition) $A_9=\\lbrace enabled(t,ts) : t \\in T, 0 \\le ts \\le k, \\forall p \\in \\bullet t, W(p,t) \\le M_{ts}(p) \\rbrace $ per definition REF (enabled transition) $A_{10}=\\lbrace fires(t,ts) : t \\in T_{ts}, 0 \\le ts \\le k \\rbrace $ from definition REF (firing set) , only an enabled transition may fire $A_{11}=\\lbrace add(p,q,t,ts) : t \\in T_{ts}, p \\in t \\bullet , q=W(t,p), 0 \\le ts \\le k \\rbrace $ per definition REF (transition execution) $A_{12}=\\lbrace del(p,q,t,ts) : t \\in T_{ts}, p \\in \\bullet t, q=W(p,t), 0 \\le ts \\le k \\rbrace $ per definition REF (transition execution) $A_{13}=\\lbrace tot\\_incr(p,q,ts) : p \\in P, q=\\sum _{t \\in T_{ts}, p \\in t \\bullet }{W(t,p)}, 0 \\le ts \\le k \\rbrace $ per definition REF (firing set execution) $A_{14}=\\lbrace tot\\_decr(p,q,ts): p \\in P, q=\\sum _{t \\in T_{ts}, p \\in \\bullet t}{W(p,t)}, 0 \\le ts \\le k \\rbrace $ per definition REF (firing set execution) $A_{15}=\\lbrace consumesmore(p,ts) : p \\in P, q=M_{ts}(p), q1=\\sum _{t \\in T_{ts}, p \\in \\bullet t}{W(p,t)}, q1 > q, 0 \\le ts \\le k \\rbrace $ per definition REF (conflicting transitions) for enabled transition set $T_{ts}$ $A_{16}=\\lbrace consumesmore : \\exists p \\in P : q=M_{ts}(p), q1=\\sum _{t \\in T_{ts}, p \\in \\bullet t}{W(p,t)}, q1 > q, 0 \\le ts \\le k \\rbrace $ per definition REF (conflicting transitions) $A_{17}=\\lbrace holds(p,q,ts+1) : p \\in P, q=M_{ts+1}(p), 0 \\le ts < k\\rbrace $ , where $M_{ts+1}(p) = M_{ts}(p) - \\sum _{\\begin{array}{c}t \\in T_{ts}, p \\in \\bullet t\\end{array}}{W(p,t)} + \\sum _{\\begin{array}{c}t \\in T_{ts}, p \\in t \\bullet \\end{array}}{W(t,p)}$ according to definition REF (firing set execution) We show that $A$ satisfies (REF ) and (REF ), and $A$ is an answer set of $\\Pi ^0$ .", "$A$ satisfies (REF ) and (REF ) by its construction (given above).", "We show $A$ is an answer set of $\\Pi ^0$ by splitting.", "We split $lit(\\Pi ^0)$ (literals of $\\Pi ^0$ ) into a sequence of $6(k+1)+2$ sets: $U_0= head(f\\ref {f:place}) \\cup head(f\\ref {f:trans}) \\cup head(f\\ref {f:ptarc}) \\cup head(f\\ref {f:tparc}) \\cup head(f\\ref {f:time}) \\cup head(f\\ref {f:num}) \\cup head(i\\ref {i:holds}) = \\lbrace place(p) : p \\in P\\rbrace \\cup \\lbrace trans(t) : t \\in T\\rbrace \\cup \\lbrace ptarc(p,t,n) : (p,t) \\in E^-, n=W(p,t) \\rbrace \\cup \\lbrace tparc(t,p,n) : (t,p) \\in E^+, n=W(t,p) \\rbrace \\cup $ $\\lbrace time(0), \\dots , $ $time(k)\\rbrace \\cup \\lbrace num(0), \\dots , num(ntok)\\rbrace \\cup \\lbrace holds(p,q,0) : p \\in P, q=M_0(p) \\rbrace $ $U_{6k+1}=U_{6k+0} \\cup head(e\\ref {e:ne:ptarc})^{ts=k} = U_{6k+0} \\cup \\lbrace notenabled(t,k) : t \\in T \\rbrace $ $U_{6k+2}=U_{6k+1} \\cup head(e\\ref {e:enabled})^{ts=k} = U_{6k+1} \\cup \\lbrace enabled(t,k) : t \\in T \\rbrace $ $U_{6k+3}=U_{6k+2} \\cup head(a\\ref {a:fires})^{ts=k} = U_{6k+2} \\cup \\lbrace fires(t,k) : t \\in T \\rbrace $ $U_{6k+4}=U_{6k+3} \\cup head(r\\ref {r:add})^{ts=k} \\cup head(r\\ref {r:del})^{ts=k} = U_{6k+3} \\cup \\lbrace add(p,q,t,k) : p \\in P, t \\in T, q=W(t,p) \\rbrace \\cup \\lbrace del(p,q,t,k) : p \\in P, t \\in T, q=W(p,t) \\rbrace $ $U_{6k+5}=U_{6k+4} \\cup head(r\\ref {r:totincr})^{ts=k} \\cup head(r\\ref {r:totdecr})^{ts=k} = U_{6k+4} \\cup \\lbrace tot\\_incr(p,q,k) : p \\in P, 0 \\le q \\le ntok \\rbrace \\cup \\lbrace tot\\_decr(p,q,k) : p \\in P, 0 \\le q \\le ntok \\rbrace $ $U_{6k+6}=U_{6k+5} \\cup head(r\\ref {r:nextstate})^{ts=k} \\cup head(a\\ref {a:overc:place})^{ts=k} = U_{6k+5} \\cup \\lbrace holds(p,q,k+1) : p \\in P, 0 \\le q \\le ntok \\rbrace \\cup \\lbrace consumesmore(p,k) : p \\in P\\rbrace $ $U_{6k+7}=U_{6k+6} \\cup head(a\\ref {a:overc:gen}) = U_{7k+6} \\cup \\lbrace consumesmore \\rbrace $ where $head(r_i)^{ts=k}$ are head atoms of ground rule $r_i$ in which $ts=k$ .", "We write $A_i^{ts=k} = \\lbrace a(\\dots ,ts) : a(\\dots ,ts) \\in A_i, ts=k \\rbrace $ as short hand for all atoms in $A_i$ with $ts=k$ .", "$U_{\\alpha }, 0 \\le \\alpha \\le 6k+7$ form a splitting sequence, since each $U_i$ is a splitting set of $\\Pi ^0$ , and $\\langle U_{\\alpha }\\rangle _{\\alpha < \\mu }$ is a monotone continuous sequence, where $U_0 \\subseteq U_1 \\dots \\subseteq U_{6k+7}$ and $\\bigcup _{\\alpha < \\mu }{U_{\\alpha }} = lit(\\Pi ^0)$ .", "We compute the answer set of $\\Pi ^0$ using the splitting sets as follows: $bot_{U_0}(\\Pi ^0) = f\\ref {f:place} \\cup f\\ref {f:trans} \\cup f\\ref {f:ptarc} \\cup f\\ref {f:tparc} \\cup f\\ref {f:time} \\cup i\\ref {i:holds} \\cup f\\ref {f:num}$ and $X_0 = A_1 \\cup \\dots \\cup A_7$ ($= U_0$ ) is its answer set – using forced atom proposition $eval_{U_0}(bot_{U_1}(\\Pi ^0) \\setminus bot_{U_0}(\\Pi ^0), X_0) = \\lbrace notenabled(t,0) \\text{:-} .", "\| \\lbrace trans(t), $ $ ptarc(p,t,n), $ $ holds(p,q,0) \\rbrace \\subseteq X_0, \\text{~where~} q < n \\rbrace $ .", "Its answer set $X_1=A_8^{ts=0}$ – using forced atom proposition and construction of $A_8$ .", "where, $q=M_0(p)$ , $n=W(p,t)$ , and for an arc $(p,t) \\in E^-$ – by construction of $i\\ref {i:holds}$ and $f\\ref {f:ptarc}$ in $\\Pi ^0$ , and in an arc $(p,t) \\in E^-$ , $p \\in \\bullet t$ (by definition REF of preset) thus, $notenabled(t,0) \\in X_1$ represents $\\exists p \\in \\bullet t : M_0(p) < W(p,t)$ .", "$eval_{U_1}(bot_{U_2}(\\Pi ^0) \\setminus bot_{U_1}(\\Pi ^0), X_0 \\cup X_1) = \\lbrace enabled(t,0) \\text{:-}.", "\| trans(t) \\in X_0 \\cup X_1, $ $notenabled(t,0) \\notin X_0 \\cup X_1 \\rbrace $ .", "Its answer set is $X_2 = A_9^{ts=0}$ – using forced atom proposition and construction of $A_9$ .", "since an $enabled(t,0) \\in X_2$ if $\\nexists ~notenabled(t,0) \\in X_0 \\cup X_1$ , which is equivalent to $\\nexists p \\in \\bullet t : M_0(p) < W(p,t) \\equiv \\forall p \\in \\bullet t: M_0(p) \\ge W(p,t)$ .", "$eval_{U_2}(bot_{U_3}(\\Pi ^0) \\setminus bot_{U_2}(\\Pi ^0), X_0 \\cup X_1 \\cup X_2) = \\lbrace \\lbrace fires(t,0)\\rbrace \\text{:-}.", "\| enabled(t,0) \\\\ \\text{~holds in~} X_0 \\cup X_1 \\cup X_2 \\rbrace $ .", "It has multiple answer sets $X_{3.1}, \\dots , X_{3.n}$ , corresponding to elements of power set of $fires(t,0)$ atoms in $eval_{U_2}(...)$ – using supported rule proposition.", "Since we are showing that the union of answer sets of $\\Pi ^0$ determined using splitting is equal to $A$ , we only consider the set that matches the $fires(t,0)$ elements in $A$ and call it $X_3$ , ignoring the rest.", "Thus, $X_3 = A_{10}^{ts=0}$ , representing $T_0$ .", "$eval_{U_3}(bot_{U_4}(\\Pi ^0) \\setminus bot_{U_3}(\\Pi ^0), X_0 \\cup \\dots \\cup X_3) = \\lbrace add(p,n,t,0) \\text{:-}.", "\| \\lbrace fires(t,0), \\\\ tparc(t,p,n) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_3 \\rbrace \\cup \\lbrace del(p,n,t,0) \\text{:-}.", "\| \\lbrace fires(t,0), ptarc(p,t,n) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_3 \\rbrace $ .", "It's answer set is $X_4 = A_{11}^{ts=0} \\cup A_{12}^{ts=0}$ – using forced atom proposition and definitions of $A_{11}$ and $A_{12}$ .", "where each $add$ atom is equivalent to $n=W(t,p) : p \\in t \\bullet $ , and each $del$ atom is equivalent to $n=W(p,t) : p \\in \\bullet t$ , representing the effect of transitions in $T_0$ – by construction $eval_{U_4}(bot_{U_5}(\\Pi ^0) \\setminus bot_{U_4}(\\Pi ^0), X_0 \\cup \\dots \\cup X_4) = \\lbrace tot\\_incr(p,qq,0) \\text{:-}.", "\| \\\\ qq=\\sum _{add(p,q,t,0) \\in X_0 \\cup \\dots \\cup X_4}{q} \\rbrace \\cup $ $\\lbrace tot\\_decr(p,qq,0) \\text{:-}.", "\| qq=\\sum _{del(p,q,t,0) \\in X_0 \\cup \\dots \\cup X_4}{q} \\rbrace $ .", "It's answer set is $X_5 = A_{13}^{ts=0} \\cup A_{14}^{ts=0}$ – using forced atom proposition and definitions of $A_{13}$ and $A_{14}$ , ad definition REF (semantics of aggregate assignment atom).", "where each $tot\\_incr(p,qq,0)$ , $qq=\\sum _{add(p,q,t,0) \\in X_0 \\cup \\dots X_4}{q} \\\\ \\equiv $ $qq=\\sum _{t \\in X_3, p \\in t \\bullet }{W(p,t)}$ , and each $tot\\_decr(p,qq,0)$ , $qq=\\sum _{del(p,q,t,0) \\in X_0 \\cup \\dots X_4}{q} \\\\ \\equiv $ $qq=\\sum _{t \\in X_3, p \\in \\bullet t}{W(t,p)}$ , represent the net effect of transitions in $T_0$ – by construction $eval_{U_5}(bot_{U_6}(\\Pi ^0) \\setminus bot_{U_5}(\\Pi ^0), X_0 \\cup \\dots \\cup X_5) = \\lbrace consumesmore(p,0) \\text{:-}.", "\| \\\\ \\lbrace holds(p,q,0), $ $tot\\_decr(p,q1,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_5, $ $q1 > q \\rbrace \\cup $ $\\lbrace holds(p,q,1) \\text{:-}.", "\| $ $ \\lbrace holds(p,q1,0), $ $tot\\_incr(p,q2,0), $ $tot\\_decr(p,q3,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_5, $ $q=q1+q2-q3 \\rbrace $ .", "It's answer set is $X_6 = A_{15}^{ts=0} \\cup A_{17}^{ts=0}$ – using forced atom proposition.", "where, $consumesmore(p,0)$ represents $\\exists p : q=M_0(p), \\\\ q1=\\sum _{t \\in T_0, p \\in \\bullet t}{W(p,t)}, q1 > q$ – indicating place $p$ will be overconsumed if $T_0$ is fired as defined in definition REF (conflicting transitions) and $holds(p,q,1)$ represents $q=M_1(p)$ – by construction $ \\vdots $ $eval_{U_{6k+0}}(bot_{U_{6k+1}}(\\Pi ^0) \\setminus bot_{U_{6k+0}}(\\Pi ^0), X_0 \\cup \\dots \\cup X_{6k+0}) = \\\\ \\lbrace notenabled(t,k) \\text{:-} .", "\| \\lbrace trans(t), ptarc(p,t,n), holds(p,q,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{6k+0}, \\\\ \\text{~where~} q < n \\rbrace $ .", "Its answer set $X_{6k+1}=A_8^{ts=k}$ – using forced atom proposition and construction of $A_8$ .", "where, $q=M_k(p)$ , and $n=W(p,t)$ for an arc $(p,t) \\in E^-$ – by construction of $holds$ and $ptarc$ predicates in $\\Pi ^0$ , and in an arc $(p,t) \\in E^-$ , $p \\in \\bullet t$ (by definition REF of preset) thus, $notenabled(t,k) \\in X_{6k+1}$ represents $\\exists p \\in \\bullet t : M_k(p) < W(p,t)$ .", "$eval_{U_{6k+1}}(bot_{U_{6k+2}}(\\Pi ^0) \\setminus bot_{U_{6k+1}}(\\Pi ^0), X_0 \\cup \\dots \\cup X_{6k+1}) = \\lbrace enabled(t,k) \\text{:-}.", "\| \\\\ trans(t) \\in $ $X_0 \\cup \\dots \\cup X_{6k+1}, notenabled(t,k) \\notin X_0 \\cup \\dots \\cup X_{6k+1} \\rbrace $ .", "Its answer set is $X_{6k+2} = A_9^{ts=k}$ – using forced atom proposition and construction of $A_9$ .", "since an $enabled(t,k) \\in X_{6k+2}$ if $\\nexists ~notenabled(t,k) \\in X_0 \\cup \\dots \\cup X_{6k+1}$ , which is equivalent to $\\nexists p \\in \\bullet t : M_k(p) < W(p,t) \\equiv \\forall p \\in \\bullet t: M_k(p) \\ge W(p,t)$ .", "$eval_{U_{6k+2}}(bot_{U_{6k+3}}(\\Pi ^0) \\setminus bot_{U_{6k+2}}(\\Pi ^0), X_0 \\cup \\dots \\cup X_{6k+2}) = \\\\ \\lbrace \\lbrace fires(t,k)\\rbrace \\text{:-}.", "\| $ $enabled(t,k) \\text{~holds in~} $ $X_0 \\cup \\dots \\cup X_{6k+2} \\rbrace $ .", "It has multiple answer sets $X_{{6k+3}.1}, \\dots , X_{{6k+3}.n}$ , corresponding to elements of power set of $fires(t,k)$ atoms in $eval_{U_{6k+2}}(...)$ – using supported rule proposition.", "Since we are showing that the union of answer sets of $\\Pi ^0$ determined using splitting is equal to $A$ , we only consider the set that matches the $fires(t,k)$ elements in $A$ and call it $X_{6k+3}$ , ignoring the reset.", "Thus, $X_{6k+3} = A_{10}^{ts=k}$ , representing $T_k$ .", "$eval_{U_{6k+3}}(bot_{U_{6k+4}}(\\Pi ^0) \\setminus bot_{U_{6k+3}}(\\Pi ^0), X_0 \\cup \\dots \\cup X_{6k+3}) = $ $ \\lbrace add(p,n,t,k) \\text{:-}.", "\| \\\\ \\lbrace fires(t,k), tparc(t,p,n) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{6k+3} \\rbrace \\cup $ $ \\lbrace del(p,n,t,k) \\text{:-}.", "\| $ $\\lbrace fires(t,k), $ $ptarc(p,t,n) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{6k+3} \\rbrace $ .", "It's answer set is $X_{6k+4} = A_{11}^{ts=k} \\cup A_{12}^{ts=k}$ – using forced atom proposition and definitions of $A_{11}$ and $A_{12}$ .", "where, each $add$ atom is equivalent to $n=W(t,p) : p \\in t \\bullet $ , and each $del$ atom is equivalent to $n=W(p,t) : p \\in \\bullet t$ representing the effect of transitions in $T_k$ $eval_{U_{6k+4}}(bot_{U_{6k+5}}(\\Pi ^0) \\setminus bot_{U_{6k+4}}(\\Pi ^0), X_0 \\cup \\dots \\cup X_{6k+4}) = $ $\\lbrace tot\\_incr(p,qq,k) \\text{:-}.", "\| $ $ qq=\\sum _{add(p,q,t,k) \\in X_0 \\cup \\dots \\cup X_{6k+4}}{q} \\rbrace \\cup $ $\\lbrace tot\\_decr(p,qq,k) \\text{:-}.", "\| \\\\ qq=\\sum _{del(p,q,t,k) \\in X_0 \\cup \\dots \\cup X_{6k+4}}{q} \\rbrace $ .", "It's answer set is $X_5 = A_{13}^{ts=k} \\cup A_{14}^{ts=k}$ – using forced atom proposition and definitions of $A_{13}$ and $A_{14}$ .", "where, each $tot\\_incr(p,qq,k)$ , $qq=\\sum _{add(p,q,t,k) \\in X_0 \\cup \\dots X_{7k+4}}{q}$ $\\equiv qq=\\sum _{t \\in X_{6k+3}, p \\in t \\bullet }{W(p,t)}$ , and each $tot\\_decr(p,qq,k)$ , $qq=\\sum _{del(p,q,t,k) \\in X_0 \\cup \\dots X_{7k+4}}{q}$ $\\equiv qq=\\sum _{t \\in X_{6k+3}, p \\in \\bullet t}{W(t,p)}$ , represent the net effect of transitions in $T_k$ $eval_{U_{6k+5}}(bot_{U_{6k+6}}(\\Pi ^0) \\setminus bot_{U_{6k+5}}(\\Pi ^0), X_0 \\cup \\dots \\cup X_{6k+5}) = $ $ \\lbrace consumesmore(p,k) \\text{:-}.", "\| $ $ \\lbrace holds(p,q,k), $ $ tot\\_decr(p,q1,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{6k+5} , q1 > q \\rbrace \\cup \\lbrace holds(p,q,k+1) \\text{:-}., \| \\lbrace holds(p,q1,k), tot\\_incr(p,q2,k), \\\\ tot\\_decr(p,q3,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{6k+5}, q=q1+q2-q3 \\rbrace $ .", "It's answer set is $X_{6k+6} = A_{15}^{ts=k} \\cup A_{17}^{ts=k}$ – using forced atom proposition.", "where, $holds(p,q,k+1)$ represents the marking of place $p$ in the next time step due to firing $T_k$ , and, $consumesmore(p,k)$ represents $\\exists p : q=M_{k}(p), q1=\\sum _{t \\in T_{k}, p \\in \\bullet t}{W(p,t)}, $ $q1 > q$ indicating place $p$ that will be overconsumed if $T_k$ is fired as defined in definition REF (conflicting transitions) $eval_{U_{6k+6}}(bot_{U_{6k+7}}(\\Pi ^0) \\setminus bot_{U_{6k+6}}(\\Pi ^0), X_0 \\cup \\dots \\cup X_{6k+6}) = \\lbrace consumesmore \\text{:-}.", "\| $ $\\lbrace consumesmore(p,0),$ $ \\dots , $ $consumesmore(p,k)\\rbrace \\cap (X_0 \\cup \\dots \\cup X_{6k+7}) \\ne \\emptyset \\rbrace $ .", "It's answer set is $X_{6k+7} = A_{16}^{ts=k}$ – using forced atom proposition $X_{6k+7}$ will be empty since none of $consumesmore(p,0),\\dots , \\\\ consumesmore(p,k)$ hold in $X_0 \\cup \\dots \\cup X_{6k+6}$ due to the construction of $A$ and encoding of $a\\ref {a:overc:place}$ , and it is not eliminated by the constraint $a\\ref {a:overc:elim}$ .", "The set $X=X_{0} \\cup \\dots \\cup X_{6k+7}$ is the answer set of $\\Pi ^0$ by the splitting sequence theorem REF .", "Each $X_i, 0 \\le i \\le 6k+7$ matches a distinct partition of $A$ , and $X = A$ , thus $A$ is an answer set of $\\Pi ^0$ .", "Next we show (REF ): Given $\\Pi ^0$ be the encoding of a Petri Net $PN(P,T,E,W)$ with initial marking $M_0$ , and $A$ be an answer set of $\\Pi ^0$ that satisfies (REF ) and (REF ), then we can construct $X=M_0,T_0,\\dots ,M_k,T_k,M_{k+1}$ from $A$ , such that it is an execution sequence of $PN$ .", "We construct the $X$ as follows: $M_i = (M_i(p_0), \\dots , M_i(p_n))$ , where $\\lbrace holds(p_0,M_i(p_0),i), \\dots holds(p_n,M_i(p_n),i) \\rbrace \\\\ \\subseteq A$ , for $0 \\le i \\le k+1$ $T_i = \\lbrace t : fires(t,i) \\in A\\rbrace $ , for $0 \\le i \\le k$ and show that $X$ is indeed an execution sequence of $PN$ .", "We show this by induction over $k$ (i.e.", "given $M_k$ , $T_k$ is a valid firing set and its firing produces marking $M_{k+1}$ ).", "Base case: Let $k=0$ , show [(1)] $T_0$ is a valid firing set for $M_0$ , and $T_0$ 's firing in $M_0$ producing marking $M_1$ .", "We show $T_0$ is a valid firing set w.r.t.", "$M_0$ .", "Let $\\lbrace fires(t_0,0), \\dots , fires(t_x,0) \\rbrace $ be the set of all $fires(\\dots ,0)$ atoms in $A$ , Then for each $fires(t_i,0) \\in A$ $enabled(t_i,0) \\in A$ – from rule $a\\ref {a:fires}$ and supported rule proposition Then $notenabled(t_i,0) \\notin A$ – from rule $e\\ref {e:enabled}$ and supported rule proposition Then $body(e\\ref {e:ne:ptarc})$ must not hold in $A$ – from rule $e\\ref {e:ne:ptarc}$ and forced atom proposition Then $q \\lnot < n_i \\equiv q \\ge n_i$ in $e\\ref {e:ne:ptarc}$ for all $\\lbrace holds(p,q,0), ptarc(p,t_i,n_i)\\rbrace \\subseteq A$ – from $e\\ref {e:ne:ptarc}$ , forced atom proposition, and the following: $holds(p,q,0) \\in A$ represents $q=M_0(p)$ – rule $i\\ref {i:holds}$ construction $ptarc(p,t_i,n_i) \\in A$ represents $n_i=W(p,t_i)$ – rule $f\\ref {f:ptarc} $ construction Then $\\forall p \\in \\bullet t_i, M_0(p) > W(p,t_i)$ – from definition REF of preset $\\bullet t_i$ in PN Then $t_i$ is enabled and can fire in $PN$ , as a result it can belong to $T_0$ – from definition REF of enabled transition And $consumesmore \\notin A$ , since $A$ is an answer set of $\\Pi ^0$ – from rule $a\\ref {a:overc:elim}$ and supported rule proposition Then $\\nexists consumesmore(p,0) \\in A$ – from rule $a\\ref {a:overc:gen}$ and supported rule proposition Then $\\nexists \\lbrace holds(p,q,0), tot\\_decr(p,q1,0) \\rbrace \\subseteq A : q1>q$ in $body(a\\ref {a:overc:place})$ – from $a\\ref {a:overc:place}$ and forced atom proposition Then $\\nexists p : \\sum _{t_i \\in \\lbrace t_0,\\dots ,t_x\\rbrace , p \\in \\bullet t_i}{W(p,t_i)} > M_0(p)$ – from the following $holds(p,q,0)$ represents $q=M_0(p)$ – from rule $i\\ref {i:holds}$ construction, given $tot\\_decr(p,q1,0) \\in A$ if $\\lbrace del(p,q1_0,t_0,0), \\dots del(p,q1_x,t_x,0) \\rbrace \\subseteq A$ , where $q1=q1_0+\\dots +q1_x$ – from $r\\ref {r:totdecr}$ and forced atom proposition $del(p,q1_i,t_i,0) \\in A$ if $\\lbrace fires(t_i,0), ptarc(p,t_i,q1_i) \\rbrace \\subseteq A$ – from $r\\ref {r:del}$ and supported rule proposition $del(p,q1_i,t_i,0) \\in A$ represents removal of $q1_i = W(p,t_i)$ tokens from $p \\in \\bullet t_i$ – from $r\\ref {r:del}$ , supported rule proposition, and definition REF of transition execution in PN Then the set of transitions in $T_0$ do not conflict – by the definition REF of conflicting transitions Then $\\lbrace t_0, \\dots , t_x\\rbrace = T_0$ – using 1(a),1(b) above, and definition of firing set We show $M_1$ is produced by firing $T_0$ in $M_0$ .", "Let $holds(p,q,1) \\in A$ Then $\\lbrace holds(p,q1,0), tot\\_incr(p,q2,0), tot\\_decr(p,q3,0) \\rbrace \\subseteq A : q=q1+q2-q3$ – from rule $r\\ref {r:nextstate}$ and supported rule proposition Then $holds(p,q1,0) \\in A$ represents $q1=M_0(p)$ – given, rule $i\\ref {i:holds}$ construction; and $\\lbrace add(p,q_0,t_0,0), \\dots , $ $add(p,q_j,t_j,0)\\rbrace \\subseteq A : q_0 + \\dots + q_j = q2$ ; and $\\lbrace del(p,q_0,t_0,0), \\dots , $ $del(p,q_l,t_l,0)\\rbrace \\subseteq A : q_0 + \\dots + q_l = q3$ – rules $r\\ref {r:totincr},r\\ref {r:totdecr}$ and supported rule proposition, respectively Then $\\lbrace fires(t_0,0), \\dots , fires(t_j,0) \\rbrace \\subseteq A$ and $\\lbrace fires(t_0,0), \\dots , fires(t_l,0) \\rbrace $ $\\subseteq A$ – rules $r\\ref {r:add},r\\ref {r:del}$ and supported rule proposition; and the following $tparc(t_y,p,q_y) \\in A, 0 \\le y \\le j$ represents $q_y=W(t_y,p)$ – given $ptarc(p,t_z,q_z) \\in A, 0 \\le z \\le l$ represents $q_z=W(p,t_z)$ – given Then $\\lbrace fires(t_0,0), \\dots , fires(t_j,0) \\rbrace \\cup \\lbrace fires(t_0,0), \\dots , fires(t_l,0) \\rbrace \\subseteq A $ $= \\lbrace fires(t_0,0), \\dots , fires(t_x,0) \\rbrace \\subseteq A$ – set union of subsets Then for each $fires(t_x,0) \\in A$ we have $t_x \\in T_0$ – already shown in item REF above Then $q = M_0(p) + \\sum _{t_x \\in T_0 \\wedge p \\in t_x \\bullet }{W(t_x,p)} - \\sum _{t_x \\in T_0 \\wedge p \\in \\bullet t_x}{W(p,t_x)}$ – from (REF ) above and the following Each $add(p,q_j,t_j,0) \\in A$ represents $q_j=W(t_j,p)$ for $p \\in t_j \\bullet $ – rule $r\\ref {r:add}$ encoding, and definition REF of transition execution in $PN$ Each $del(p,t_y,q_y,0) \\in A$ represents $q_y=W(p,t_y)$ for $p \\in \\bullet t_y$ – from rule $r\\ref {r:del}$ encoding, and definition REF of transition execution in $PN$ Each $tot\\_incr(p,q2,0) \\in A$ represents $q2=\\sum _{t_x \\in T_0 \\wedge p \\in t_x \\bullet }{W(t_x,p)}$ – aggregate assignment atom semantics in rule $r\\ref {r:totincr}$ Each $tot\\_decr(p,q3,0) \\in A$ represents $q3=\\sum _{t_x \\in T_0 \\wedge p \\in \\bullet t_x}{W(p,t_x)}$ – aggregate assignment atom semantics in rule $r\\ref {r:totdecr}$ Then, $M_1(p) = q$ – since $holds(p,q,1) \\in A$ encodes $q=M_1(p)$ – by construction Inductive Step: Assume $M_k$ is a valid marking in $X$ for $PN$ , show [(1)] $T_k$ is a valid firing set for $M_k$ , and firing $T_k$ in $M_k$ produces marking $M_{k+1}$ .", "We show that $T_k$ is a valid firing set for $M_k$ .", "Let $\\lbrace fires(t_0,k),\\dots ,fires(t_x,k) \\rbrace $ be the set of all $fires(\\dots ,k)$ atoms in $A$ , Then for each $fires(t_i,k) \\in A$ $enabled(t_i,k) \\in A$ – from rule $a\\ref {a:fires}$ and supported rule proposition Then $notenabled(t_i,k) \\notin A$ – from rule $e\\ref {e:enabled}$ and supported rule proposition Then body of $e\\ref {e:ne:ptarc}$ must hold in $A$ – from rule $e\\ref {e:ne:ptarc}$ and forced proposition Then $q \\lnot < n_i \\equiv q \\ge n_i$ in $e\\ref {e:ne:ptarc}$ for all $\\lbrace holds(p,q,k), ptarc(p,t_i,n_i) \\rbrace \\subseteq A$ – from $e\\ref {e:ne:ptarc}$ using forced atom proposition, and the following $holds(p,q,k) \\in A$ represents $q=M_k(p)$ – construction, inductive assumption $ptarc(p,t,n_i) \\in A$ represents $n_i=W(p,t)$ – rule $f\\ref {f:tparc}$ construction Then $\\forall p \\in \\bullet t_i, M_k(p) > W(p,t_i)$ – from definition REF of preset $\\bullet t_i$ in PN Then $t_i$ is enabled and can fire in $PN$ , as a result it can belong to $T_k$ – from definition REF of enabled transition And $consumesmore \\notin A$ , since $A$ is an answer set of $\\Pi ^0$ – from rule $a\\ref {a:overc:elim}$ and supported rule proposition Then $\\nexists consumesmore(p,k) \\in A$ – from rule $a\\ref {a:overc:gen}$ and forced atom proposition Then $\\nexists \\lbrace holds(p,q,k), tot\\_decr(p,q1,k) \\rbrace \\subseteq A : q1 > q$ in $body(a\\ref {a:overc:place})$ – from $a\\ref {a:overc:place}$ and forced atom proposition Then $\\nexists p : \\sum _{t_i \\in \\lbrace t_0,\\dots ,t_x \\rbrace , p \\in \\bullet t_i}{W(p,t_i)} > M_k(p)$ – from the following $holds(p,q,k) \\in A$ represents $q=M_k(p)$ – by construction, and the inductive assumption $tot\\_decr(p,q1,k) \\in A$ if $\\lbrace del(p,q1_0,t_0,k), \\dots , del(p,q1_x,t_x,k) \\rbrace \\subseteq A$ , where $q1=q1_0+\\dots +q1_x$ – from $r\\ref {r:totdecr}$ and forced atom proposition $del(p,q1_i,t_i,k) \\in A$ if $\\lbrace fires(t_i,k), ptarc(p,t_i,q1_i) \\rbrace \\subseteq A$ – from $r\\ref {r:del}$ and supported rule proposition $del(p,q1_i,t_i,k)$ represents removal of $q1_i = W(p,t_i)$ tokens from $p \\in \\bullet t_i$ – from construction rule $r\\ref {r:del}$ , supported rule proposition, and definition REF of transition execution in $PN$ Then the set of transitions $T_k$ does conflict – by the definition REF of conflicting transitions Then $\\lbrace t_0,\\dots ,t_x\\rbrace = T_k$ – using 1(a),1(b) above We show $M_{k+1}$ is produced by firing $T_k$ in $M_k$ .", "Let $holds(p,q,k+1) \\in A$ Then $\\lbrace holds(p,q1,k), tot\\_incr(p,q2,k), tot\\_decr(p,q3,k) \\rbrace \\in A : q=q2+q2-q3$ – from rule $r\\ref {r:nextstate}$ and supported rule proposition Then $holds(p,q1,k) \\in A$ represents $q1=M_k(p)$ – inductive assumption and construction; and $\\lbrace add(p,q2_0,t_0,k), \\dots , add(p,q2_j,t_j,k) \\rbrace \\subseteq A : q2_0+\\dots +q2_j=q2$ and $\\lbrace del(p,q3_0,t_0,k), \\dots , del(p,q3_l,t_l,k) \\rbrace \\subseteq A : q3_0+\\dots +q3_l=q3$ – from rules $r\\ref {r:totincr},r\\ref {r:totdecr}$ using supported rule proposition, respectively Then $\\lbrace fires(t_0,k), \\dots , fires(t_j,k) \\rbrace \\subseteq A$ and $\\lbrace fires(t_0,k), \\dots , fires(t_l,k) \\rbrace $ $\\subseteq A$ – from rules $r\\ref {r:add} ,r\\ref {r:del} $ using supported rule proposition, respectively Then $\\lbrace fires(t_0,k), \\dots , fires(t_j,k) \\rbrace \\cup \\lbrace fires(t_0,k), \\dots , fires(t_l,k) \\rbrace $ $= \\lbrace fires(t_0,k), \\dots , fires(t_x,k) \\rbrace \\subseteq A$ – subset union property Then for each $fires(t_x,k) \\in A$ we have $t_x \\in T_k$ - already shown in item REF above Then $q = M_k(p) + \\sum _{t_x \\in T_k \\wedge p \\in t_x \\bullet }{W(t_x,p)} - \\sum _{t_x \\in T_k \\wedge p \\in \\bullet t_x}{W(p,t_x}$ – from (REF ) above and the following Each $add(p,q_j,t_j,0) \\in A$ represents $q_j = W(t_j,p)$ for $p \\in t_j \\bullet $ – encoding of $r\\ref {r:add}$ and definition REF of transition execution in PN Each $del(p,t_y,q_y,0) \\in A$ represents $q_y = W(p,t_y)$ for $p \\in \\bullet t_y$ – encoding of $r\\ref {r:del}$ and definition REF of transition execution in PN Each $tot\\_incr(p,q2,0) \\in A$ represents $q2 = \\sum _{t_x \\in T_k \\wedge p \\in t_x \\bullet }{W(t_x,p)}$ – aggregate assignment atom semantics in rule $r\\ref {r:totincr}$ Each $tot\\_decr(p,q3,0) \\in A$ represents $q3 = \\sum _{t_x \\in T_k \\wedge p \\in \\bullet t_x}{W(p,t_x)}$ – aggregate assignment atom semantics in rule $r\\ref {r:totdecr}$ Then $M_{k+1}(p) = q$ – since $holds(p,q,k+1) \\in A$ encodes $q=M_{k+1}(p)$ by construction As a result, for any $n > k$ , $T_n$ is a valid firing set w.r.t.", "$M_n$ and its firing produces marking $M_{n+1}$ .", "Conclusion: Since both REF and REF hold, $X=M_0,T_0,M_1,\\dots ,M_k,T_k,M_{k+1}$ is an execution sequence of $PN(P,T,E,W)$ (w.r.t.", "$M_0$ ) iff there is an answer set $A$ of $\\Pi ^0(PN,M_0,k,ntok)$ such that (REF ) and (REF ) hold.", "Proof of Proposition REF Let $PN=(P,T,E,W)$ be a Petri Net, $M_0$ be its initial marking and let $\\Pi ^1(PN,M_0,$ $k,ntok)$ be the ASP encoding of $PN$ and $M_0$ over a simulation length $k$ , with maximum $ntok$ tokens on any place node, as defined in section REF .", "Then $X=M_0,T_0,M_1,\\dots ,M_k,T_k,M_{k+1}$ is an execution sequence of $PN$ (w.r.t.", "$M_0$ ) iff there is an answer set $A$ of $\\Pi ^1(PN,M_0,k,ntok)$ such that: $\\lbrace fires(t,ts) : t \\in T_{ts}, 0 \\le ts \\le k\\rbrace = \\lbrace fires(t,ts) : fires(t,ts) \\in A \\rbrace $ $\\begin{split}\\lbrace holds(p,q,ts) &: p \\in P, q = M_{ts}(p), 0 \\le ts \\le k+1 \\rbrace \\\\&= \\lbrace holds(p,q,ts) : holds(p,q,ts) \\in A \\rbrace \\end{split}$ We prove this by showing that: Given an execution sequence $X$ , we create a set $A$ such that it satisfies (REF ) and (REF ) and show that $A$ is an answer set of $\\Pi ^1$ Given an answer set $A$ of $\\Pi ^1$ , we create an execution sequence $X$ such that (REF ) and (REF ) are satisfied.", "First we show (REF ): Given a $PN$ and an execution sequence $X$ of $PN$ , we create a set $A$ as a union of the following sets: $A_1=\\lbrace num(n) : 0 \\le n \\le ntok \\rbrace $ $A_2=\\lbrace time(ts) : 0 \\le ts \\le k\\rbrace $ $A_3=\\lbrace place(p) : p \\in P \\rbrace $ $A_4=\\lbrace trans(t) : t \\in T \\rbrace $ $A_5=\\lbrace ptarc(p,t,n) : (p,t) \\in E^-, n=W(p,t) \\rbrace $ , where $E^- \\subseteq E$ $A_6=\\lbrace tparc(t,p,n) : (t,p) \\in E^+, n=W(t,p) \\rbrace $ , where $E^+ \\subseteq E$ $A_7=\\lbrace holds(p,q,0) : p \\in P, q=M_{0}(p) \\rbrace $ $A_8=\\lbrace notenabled(t,ts) : t \\in T, 0 \\le ts \\le k, \\exists p \\in \\bullet t, M_{ts}(p) < W(p,t) \\rbrace $ per definition REF (enabled transition) $A_9=\\lbrace enabled(t,ts) : t \\in T, 0 \\le ts \\le k, \\forall p \\in \\bullet t, W(p,t) \\le M_{ts}(p) \\rbrace $ per definition REF (enabled transition) $A_{10}=\\lbrace fires(t,ts) : t \\in T_{ts}, 0 \\le ts \\le k \\rbrace $ per definition REF (firing set), only an enabled transition may fire $A_{11}=\\lbrace add(p,q,t,ts) : t \\in T_{ts}, p \\in t \\bullet , q=W(t,p), 0 \\le ts \\le k \\rbrace $ per definition REF (transition execution) $A_{12}=\\lbrace del(p,q,t,ts) : t \\in T_{ts}, p \\in \\bullet t, q=W(p,t), 0 \\le ts \\le k \\rbrace $ per definition REF (transition execution) $A_{13}=\\lbrace tot\\_incr(p,q,ts) : p \\in P, q=\\sum _{t \\in T_{ts}, p \\in t \\bullet }{W(t,p)}, 0 \\le ts \\le k \\rbrace $ per definition REF (firing set execution) $A_{14}=\\lbrace tot\\_decr(p,q,ts): p \\in P, q=\\sum _{t \\in T_{ts}, p \\in \\bullet t}{W(p,t)}, 0 \\le ts \\le k \\rbrace $ per definition REF (firing set execution) $A_{15}=\\lbrace consumesmore(p,ts) : p \\in P, q=M_{ts}(p), q1=\\sum _{t \\in T_{ts}, p \\in \\bullet t}{W(p,t)}, q1 > q, 0 \\le ts \\le k \\rbrace $ per definition REF (conflicting transitions) for enabled transition set $T_{ts}$ $A_{16}=\\lbrace consumesmore : \\exists p \\in P : q=M_{ts}(p), q1=\\sum _{t \\in T_{ts}, p \\in \\bullet t}{W(p,t)}, q1 > q, 0 \\le ts \\le k \\rbrace $ per definition REF (conflicting transitions) $A_{17}=\\lbrace could\\_not\\_have(t,ts) : t \\in T, (\\forall p \\in \\bullet t, W(p,t) \\le M_{ts}(p)), t \\notin T_{ts}, (\\exists p \\in \\bullet t : W(p,t) > M_{ts}(p) - \\sum _{t^{\\prime } \\in T_{ts}, p \\in \\bullet t^{\\prime }}{W(p,t^{\\prime })}), 0 \\le ts \\le k \\rbrace $ per the maximal firing set semantics $A_{18}=\\lbrace holds(p,q,ts+1) : p \\in P, q=M_{ts+1}(p), 0 \\le ts < k\\rbrace $ , where $M_{ts+1}(p) = M_{ts}(p) - \\sum _{\\begin{array}{c}t \\in T_{ts}, p \\in \\bullet t\\end{array}}{W(p,t)} + \\sum _{\\begin{array}{c}t \\in T_{ts}, p \\in t \\bullet \\end{array}}{W(t,p)}$ according to definition REF (firing set execution) We show that $A$ satisfies (REF ) and (REF ), and $A$ is an answer set of $\\Pi ^1$ .", "$A$ satisfies (REF ) and (REF ) by its construction above.", "We show $A$ is an answer set of $\\Pi ^1$ by splitting.", "We split $lit(\\Pi ^1)$ into a sequence of $6k+8$ sets: $U_0= head(f\\ref {f:place}) \\cup head(f\\ref {f:trans}) \\cup head(f\\ref {f:ptarc}) \\cup head(f\\ref {f:tparc}) \\cup head(f\\ref {f:time}) \\cup head(f\\ref {f:num}) $ $\\cup head(i\\ref {i:holds}) = \\lbrace place(p) : p \\in P\\rbrace \\cup \\lbrace trans(t) : t \\in T\\rbrace \\cup \\lbrace ptarc(p,t,n) : (p,t) \\in E^-, n=W(p,t) \\rbrace \\cup \\lbrace tparc(t,p,n) : (t,p) \\in E^+, n=W(t,p) \\rbrace \\cup $ $ \\lbrace time(0), \\dots , time(k)\\rbrace \\cup \\lbrace num(0), \\dots , num(ntok)\\rbrace \\cup \\lbrace holds(p,q,0) : p \\in P, q=M_0(p) \\rbrace $ $U_{6k+1}=U_{6k+0} \\cup head(e\\ref {e:ne:ptarc})^{ts=k} = U_{6k+0} \\cup \\lbrace notenabled(t,k) : t \\in T \\rbrace $ $U_{6k+2}=U_{6k+1} \\cup head(e\\ref {e:enabled})^{ts=k} = U_{6k+1} \\cup \\lbrace enabled(t,k) : t \\in T \\rbrace $ $U_{6k+3}=U_{6k+2} \\cup head(a\\ref {a:fires})^{ts=k} = U_{6k+2} \\cup \\lbrace fires(t,k) : t \\in T \\rbrace $ $U_{6k+4}=U_{6k+3} \\cup head(r\\ref {r:add})^{ts=k} \\cup head(r\\ref {r:del})^{ts=k} = U_{6k+3} \\cup \\lbrace add(p,q,t,k) : p \\in P, t \\in T, q=W(t,p) \\rbrace \\cup \\lbrace del(p,q,t,k) : p \\in P, t \\in T, q=W(p,t) \\rbrace $ $U_{6k+5}=U_{6k+4} \\cup head(r\\ref {r:totincr})^{ts=k} \\cup head(r\\ref {r:totdecr})^{ts=k} = U_{6k+4} \\cup \\lbrace tot\\_incr(p,q,k) : p \\in P, 0 \\le q \\le ntok \\rbrace \\cup \\lbrace tot\\_decr(p,q,k) : p \\in P, 0 \\le q \\le ntok \\rbrace $ $U_{6k+6}=U_{6k+5} \\cup head(r\\ref {r:nextstate})^{ts=k} \\cup head(a\\ref {a:overc:place})^{ts=k} \\cup head(a\\ref {a:maxfire:cnh})^{ts=k} $ $= U_{6k+5} \\\\ \\cup \\lbrace consumesmore(p,k) : p \\in P\\rbrace \\cup \\lbrace holds(p,q,k+1) : p \\in P, 0 \\le q \\le ntok \\rbrace \\cup $ $\\lbrace could\\_not\\_have(t,k) : t \\in T \\rbrace $ $U_{6k+7}=U_{6k+6} \\cup head(a\\ref {a:overc:gen}) = U_{6k+6} \\cup \\lbrace consumesmore \\rbrace $ where $head(r_i)^{ts=k}$ are head atoms of ground rule $r_i$ in which $ts=k$ .", "We write $A_i^{ts=k} = \\lbrace a(\\dots ,ts) : a(\\dots ,ts) \\in A_i, ts=k \\rbrace $ as short hand for all atoms in $A_i$ with $ts=k$ .", "$U_{\\alpha }, 0 \\le \\alpha \\le 6k+7$ form a splitting sequence, since each $U_i$ is a splitting set of $\\Pi ^1$ , and $\\langle U_{\\alpha }\\rangle _{\\alpha < \\mu }$ is a monotone continuous sequence, where $U_0 \\subseteq U_1 \\dots \\subseteq U_{6k+7}$ and $\\bigcup _{\\alpha < \\mu }{U_{\\alpha }} = lit(\\Pi ^1)$ .", "We compute the answer set of $\\Pi ^1$ using the splitting sets as follows: $bot_{U_0}(\\Pi ^1) = f\\ref {f:place} \\cup f\\ref {f:trans} \\cup f\\ref {f:ptarc} \\cup f\\ref {f:tparc} \\cup f\\ref {f:time} \\cup i\\ref {i:holds} \\cup f\\ref {f:num}$ and $X_0 = A_1 \\cup \\dots \\cup A_7$ ($= U_0$ ) is its answer set – using forced atom proposition $eval_{U_0}(bot_{U_1}(\\Pi ^1) \\setminus bot_{U_0}(\\Pi ^1), X_0) = \\lbrace notenabled(t,0) \\text{:-} .", "\| \\lbrace trans(t), \\\\ ptarc(p,t,n), holds(p,q,0) \\rbrace \\subseteq X_0, \\text{~where~} q < n \\rbrace $ .", "Its answer set $X_1=A_8^{ts=0}$ – using forced atom proposition and construction of $A_8$ .", "where, $q=M_0(p)$ , and $n=W(p,t)$ for an arc $(p,t) \\in E^-$ – by construction of $i\\ref {i:holds}$ and $f\\ref {f:ptarc}$ in $\\Pi ^1$ , and in an arc $(p,t) \\in E^-$ , $p \\in \\bullet t$ (by definition REF of preset) thus, $notenabled(t,0) \\in X_1$ represents $\\exists p \\in \\bullet t : M_0(p) < W(p,t)$ .", "$eval_{U_1}(bot_{U_2}(\\Pi ^1) \\setminus bot_{U_1}(\\Pi ^1), X_0 \\cup X_1) = \\lbrace enabled(t,0) \\text{:-}.", "\| trans(t) \\in X_0 \\cup X_1, $ $notenabled(t,0) \\notin X_0 \\cup X_1 \\rbrace $ .", "Its answer set is $X_2 = A_9^{ts=0}$ – using forced atom proposition and construction of $A_9$ .", "since an $enabled(t,0) \\in X_2$ if $\\nexists ~notenabled(t,0) \\in X_0 \\cup X_1$ , which is equivalent to $\\nexists p \\in \\bullet t : M_0(p) < W(p,t) \\equiv \\forall p \\in \\bullet t: M_0(p) \\ge W(p,t)$ .", "$eval_{U_2}(bot_{U_3}(\\Pi ^1) \\setminus bot_{U_2}(\\Pi ^1), X_0 \\cup X_1 \\cup X_2) = \\lbrace \\lbrace fires(t,0)\\rbrace \\text{:-}.", "\| enabled(t,0) \\\\ \\text{~holds in~} X_0 \\cup X_1 \\cup X_2 \\rbrace $ .", "It has multiple answer sets $X_{3.1}, \\dots , X_{3.n}$ , corresponding to elements of power set of $fires(t,0)$ atoms in $eval_{U_2}(...)$ – using supported rule proposition.", "Since we are showing that the union of answer sets of $\\Pi ^1$ determined using splitting is equal to $A$ , we only consider the set that matches the $fires(t,0)$ elements in $A$ and call it $X_3$ , ignoring the rest.", "Thus, $X_3 = A_{10}^{ts=0}$ , representing $T_0$ .", "$eval_{U_3}(bot_{U_4}(\\Pi ^1) \\setminus bot_{U_3}(\\Pi ^1), X_0 \\cup \\dots \\cup X_3) = \\lbrace add(p,n,t,0) \\text{:-}.", "\| \\lbrace fires(t,0), \\\\ tparc(t,p,n) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_3 \\rbrace \\cup \\lbrace del(p,n,t,0) \\text{:-}.", "\| \\lbrace fires(t,0), ptarc(p,t,n) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_3 \\rbrace $ .", "It's answer set is $X_4 = A_{11}^{ts=0} \\cup A_{12}^{ts=0}$ – using forced atom proposition and definitions of $A_{11}$ and $A_{12}$ .", "where, each $add$ atom encodes $n=W(t,p) : p \\in t \\bullet $ , and each $del$ atom encodes $n=W(p,t) : p \\in \\bullet t$ representing the effect of transitions in $T_0$ – by construction $eval_{U_4}(bot_{U_5}(\\Pi ^1) \\setminus bot_{U_4}(\\Pi ^1), X_0 \\cup \\dots \\cup X_4) = \\lbrace tot\\_incr(p,qq,0) \\text{:-}.", "\| $ $qq=\\sum _{add(p,q,t,0) \\in X_0 \\cup \\dots \\cup X_4}{q} \\rbrace \\cup \\lbrace tot\\_decr(p,qq,0) \\text{:-}.", "\| qq=\\sum _{del(p,q,t,0) \\in X_0 \\cup \\dots \\cup X_4}{q} \\rbrace $ .", "It's answer set is $X_5 = A_{13}^{ts=0} \\cup A_{14}^{ts=0}$ – using forced atom proposition and definitions of $A_{13}$ , $A_{14}$ , and semantics of aggregate assignment atom where, each $tot\\_incr(p,qq,0)$ , $qq=\\sum _{add(p,q,t,0) \\in X_0 \\cup \\dots X_4}{q}$ $\\equiv qq=\\sum _{t \\in X_3, p \\in t \\bullet }{W(p,t)}$ , and, each $tot\\_decr(p,qq,0)$ , $qq=\\sum _{del(p,q,t,0) \\in X_0 \\cup \\dots X_4}{q}$ $\\equiv qq=\\sum _{t \\in X_3, p \\in \\bullet t}{W(t,p)}$ , represent the net effect of actions in $T_0$ – by construction $eval_{U_5}(bot_{U_6}(\\Pi ^1) \\setminus bot_{U_5}(\\Pi ^1), X_0 \\cup \\dots \\cup X_5) = \\lbrace consumesmore(p,0) \\text{:-}.", "\| \\\\ \\lbrace holds(p,q,0), tot\\_decr(p,q1,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_5 : q1 > q \\rbrace \\cup \\lbrace holds(p,q,1) \\text{:-}., \| \\\\ \\lbrace holds(p,q1,0), tot\\_incr(p,q2,0), tot\\_decr(p,q3,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_5, q=q1+q2-q3 \\rbrace \\cup \\lbrace could\\_not\\_have(t,0) \\text{:-}.", "\| \\lbrace enabled(t,0), ptarc(s,t,q), holds(s,qq,0), \\\\ tot\\_decr(s,qqq,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_5, fires(t,0) \\notin (X_0 \\cup \\dots \\cup X_5), q > qq-qqq \\rbrace $ .", "It's answer set is $X_6 = A_{15}^{ts=0} \\cup A_{17}^{ts=0} \\cup A_{18}^{ts=0}$ – using forced atom proposition and definitions of $A_{15}, A_{17}, A_{18}$ .", "where, $consumesmore(p,0)$ represents $\\exists p : q=M_0(p), q1=\\sum _{t \\in T_0, p \\in \\bullet t}{W(p,t)}, $ $q1 > q$ indicating place $p$ will be overconsumed if $T_0$ is fired, as defined in definition REF (conflicting transitions) and, $holds(p,q,1)$ encodes $q=M_1(p)$ – by construction and $could\\_not\\_have(t,0)$ represents an enabled transition $t$ in $T_0$ that could not fire due to insufficient tokens $X_6$ does not contain $could\\_not\\_have(t,0)$ , when $enabled(t,0) \\in X_0 \\cup \\dots \\cup X_5$ and $fires(t,0) \\notin X_0 \\cup \\dots \\cup X_5$ due to construction of $A$ , encoding of $a\\ref {a:maxfire:cnh}$ and its body atoms.", "As a result it is not eliminated by the constraint $a\\ref {a:maxfire:elim}$ $ \\vdots $ $eval_{U_{6k+0}}(bot_{U_{6k+1}}(\\Pi ^1) \\setminus bot_{U_{6k+0}}(\\Pi ^1), X_0 \\cup \\dots \\cup X_{6k+0}) = \\\\ \\lbrace notenabled(t,k) \\text{:-} .", "\| \\lbrace trans(t), ptarc(p,t,n), holds(p,q,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{6k+0}, \\\\ \\text{~where~} q < n \\rbrace $ .", "Its answer set $X_{6k+1}=A_8^{ts=k}$ – using forced atom proposition and construction of $A_8$ .", "where, $q=M_k(p)$ , and $n=W(p,t)$ for an arc $(p,t) \\in E^-$ – by construction of $holds$ and $ptarc$ predicates in $\\Pi ^1$ , and in an arc $(p,t) \\in E^-$ , $p \\in \\bullet t$ (by definition REF of preset) thus, $notenabled(t,k) \\in X_{6k+1}$ represents $\\exists p \\in \\bullet t : M_k(p) < W(p,t)$ .", "$eval_{U_{6k+1}}(bot_{U_{6k+2}}(\\Pi ^1) \\setminus bot_{U_{6k+1}}(\\Pi ^1), X_0 \\cup \\dots \\cup X_{6k+1}) = \\lbrace enabled(t,k) \\text{:-}.", "\| \\\\ trans(t) \\in X_0 \\cup \\dots \\cup X_{6k+1}, notenabled(t,k) \\notin X_0 \\cup \\dots \\cup X_{6k+1} \\rbrace $ .", "Its answer set is $X_{6k+2} = A_9^{ts=k}$ – using forced atom proposition and construction of $A_9$ .", "since an $enabled(t,k) \\in X_{6k+2}$ if $\\nexists ~notenabled(t,k) \\in X_0 \\cup \\dots \\cup X_{6k+1}$ , which is equivalent to $\\nexists p \\in \\bullet t : M_k(p) < W(p,t) \\equiv \\forall p \\in \\bullet t: M_k(p) \\ge W(p,t)$ .", "$eval_{U_{6k+2}}(bot_{U_{6k+3}}(\\Pi ^1) \\setminus bot_{U_{6k+2}}(\\Pi ^1), X_0 \\cup \\dots \\cup X_{6k+2}) = \\\\ \\lbrace \\lbrace fires(t,k)\\rbrace \\text{:-}.", "\| enabled(t,k) \\text{~holds in~} X_0 \\cup \\dots \\cup X_{6k+2} \\rbrace $ .", "It has multiple answer sets $X_{{6k+3}.1}, \\dots , X_{{6k+3}.n}$ , corresponding to elements of power set of $fires(t,k)$ atoms in $eval_{U_{6k+2}}(...)$ – using supported rule proposition.", "Since we are showing that the union of answer sets of $\\Pi ^1$ determined using splitting is equal to $A$ , we only consider the set that matches the $fires(t,k)$ elements in $A$ and call it $X_{6k+3}$ , ignoring the reset.", "Thus, $X_{6k+3} = A_{10}^{ts=k}$ , representing $T_k$ .", "$eval_{U_{6k+3}}(bot_{U_{6k+4}}(\\Pi ^1) \\setminus bot_{U_{6k+3}}(\\Pi ^1), X_0 \\cup \\dots \\cup X_{6k+3}) = \\\\ \\lbrace add(p,n,t,k) \\text{:-}.", "\| \\lbrace fires(t,k), tparc(t,p,n) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{6k+3} \\rbrace \\cup \\\\ \\lbrace del(p,n,t,k) \\text{:-}.", "\| \\lbrace fires(t,k), ptarc(p,t,n) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{6k+3} \\rbrace $ .", "It's answer set is $X_{6k+4} = A_{11}^{ts=k} \\cup A_{12}^{ts=k}$ – using forced atom proposition and definitions of $A_{11}$ and $A_{12}$ .", "where, each $add$ atom is equivalent to $n=W(t,p) : p \\in t \\bullet $ , and, each $del$ atom is equivalent to $n=W(p,t) : p \\in \\bullet t$ , representing the effect of transitions in $T_k$ $eval_{U_{6k+4}}(bot_{U_{6k+5}}(\\Pi ^1) \\setminus bot_{U_{6k+4}}(\\Pi ^1), X_0 \\cup \\dots \\cup X_{6k+4}) = \\lbrace tot\\_incr(p,qq,k) \\text{:-}.", "\| \\\\ qq=\\sum _{add(p,q,t,k) \\in X_0 \\cup \\dots \\cup X_{6k+4}}{q} \\rbrace \\cup \\lbrace tot\\_decr(p,qq,k) \\text{:-}.", "\| \\\\ qq=\\sum _{del(p,q,t,k) \\in X_0 \\cup \\dots \\cup X_{6k+4}}{q} \\rbrace $ .", "It's answer set is $X_{6k+5} = A_{13}^{ts=k} \\cup A_{14}^{ts=k}$ – using forced atom proposition and definitions of $A_{13}$ and $A_{14}$ .", "where, each $tot\\_incr(p,qq,k)$ , $qq=\\sum _{add(p,q,t,k) \\in X_0 \\cup \\dots X_{6k+4}}{q}$ $\\equiv \\\\ qq=\\sum _{t \\in X_{6k+3}, p \\in t \\bullet }{W(p,t)}$ , and, each $tot\\_decr(p,qq,k)$ , $qq=\\sum _{del(p,q,t,k) \\in X_0 \\cup \\dots X_{6k+4}}{q}$ $\\equiv \\\\ qq=\\sum _{t \\in X_{6k+3}, p \\in \\bullet t}{W(t,p)}$ , represent the net effect of transitions in $T_k$ $eval_{U_{6k+5}}(bot_{U_{6k+6}}(\\Pi ^1) \\setminus bot_{U_{6k+5}}(\\Pi ^1), X_0 \\cup \\dots \\cup X_{6k+5}) = \\\\ \\lbrace consumesmore(p,k) \\text{:-}.", "\| \\lbrace holds(p,q,k), tot\\_decr(p,q1,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{6k+5} : q1 > q \\rbrace \\cup \\lbrace holds(p,q,k+1) \\text{:-}., \| \\lbrace holds(p,q1,k), tot\\_incr(p,q2,k), \\\\ tot\\_decr(p,q3,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{6k+5} : q=q1+q2-q3 \\rbrace \\cup \\lbrace could\\_not\\_have(t,k) \\text{:-} \\\\ \\lbrace enabled(t,k), ptarc(s,t,q), holds(s,qq,k), tot\\_decr(s,qqq,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{6k+5}, $ $fires(t,k) \\notin (X_0 \\cup \\dots \\cup X_{6k+5}), q > qq-qqq \\rbrace $ .", "It's answer set is $X_{6k+6} = A_{15}^{ts=k} \\cup A_{17}^{ts=k} \\cup A_{18}^{ts=k}$ – using forced atom proposition.", "where, $consumesmore(p,k)$ represents $\\exists p : q=M_{k}(p), \\\\ q1=\\sum _{t \\in T_{k}, p \\in \\bullet t}{W(p,t)}, q1 > q$ $holds(p,q,k+1)$ represents $q=M_{k+1}(p)$ indicating place $p$ that will be over consumed if $T_k$ is fired, as defined in definition REF (conflicting transitions), $holds(p,q,k+1)$ represents $q=M_{k+1}(p)$ – by construction and $could\\_not\\_have(t,k)$ represents an enabled transition $t$ in $T_k$ that could not fire due to insufficient tokens $X_{6k+6}$ does not contain $could\\_not\\_have(t,k)$ , when $enabled(t,k) \\in X_0 \\cup \\dots \\cup X_{6k+5}$ and $fires(t,k) \\notin X_0 \\cup \\dots \\cup X_{6k+5}$ due to construction of $A$ , encoding of $a\\ref {a:maxfire:cnh}$ and its body atoms.", "As a result it is note eliminated by the constraint $a\\ref {a:maxfire:elim}$ $eval_{U_{6k+6}}(bot_{U_{6k+7}}(\\Pi ^1) \\setminus bot_{U_{6k+6}}(\\Pi ^1), X_0 \\cup \\dots \\cup X_{6k+6}) = \\lbrace consumesmore \\text{:-}.", "\| $ $\\lbrace consumesmore(p,0),$ $\\dots , $ $consumesmore(p,k) \\rbrace \\cap $ $(X_0 \\cup \\dots \\cup X_{6k+6}) \\ne \\emptyset \\rbrace $ .", "It's answer set is $X_{6k+7} = A_{16}$ – using forced atom proposition $X_{6k+7}$ will be empty since none of $consumesmore(p,0),\\dots , \\\\ consumesmore(p,k)$ hold in $X_0 \\cup \\dots \\cup X_{6k+6}$ due to the construction of $A$ , encoding of $a\\ref {a:overc:place}$ and its body atoms.", "As a result, it is not eliminated by the constraint $a\\ref {a:overc:elim}$ The set $X = X_0 \\cup \\dots \\cup X_{6k+7}$ is the answer set of $\\Pi ^0$ by the splitting sequence theorem REF .", "Each $X_i, 0 \\le i \\le 6k+7$ matches a distinct portion of $A$ , and $X = A$ , thus $A$ is an answer set of $\\Pi ^1$ .", "Next we show (REF ): Given $\\Pi ^1$ be the encoding of a Petri Net $PN(P,T,E,W)$ with initial marking $M_0$ , and $A$ be an answer set of $\\Pi ^1$ that satisfies (REF ) and (REF ), then we can construct $X=M_0,T_0,\\dots ,M_k,T_k,M_{k+1}$ from $A$ , such that it is an execution sequence of $PN$ .", "We construct the $X$ as follows: $M_i = (M_i(p_0), \\dots , M_i(p_n))$ , where $\\lbrace holds(p_0,M_i(p_0),i), \\dots holds(p_n,M_i(p_n),i) \\rbrace \\\\ \\subseteq A$ , for $0 \\le i \\le k+1$ $T_i = \\lbrace t : fires(t,i) \\in A\\rbrace $ , for $0 \\le i \\le k$ and show that $X$ is indeed an execution sequence of $PN$ .", "We show this by induction over $k$ (i.e.", "given $M_k$ , $T_k$ is a valid firing set and its firing produces marking $M_{k+1}$ ).", "Base case: Let $k=0$ , and $M_0$ is a valid marking in $X$ for $PN$ , show [(1)] $T_0$ is a valid firing set for $M_0$ , and $T_0$ 's firing w.r.t.", "marking $M_0$ produces $M_1$ .", "We show $T_0$ is a valid firing set for $M_0$ .", "Let $\\lbrace fires(t_0,0), \\dots , fires(t_x,0) \\rbrace $ be the set of all $fires(\\dots ,0)$ atoms in $A$ , Then for each $fires(t_i,0) \\in A$ $enabled(t_i,0) \\in A$ – from rule $a\\ref {a:fires}$ and supported rule proposition Then $notenabled(t_i,0) \\notin A$ – from rule $e\\ref {e:enabled}$ and supported rule proposition Then $body(e\\ref {e:ne:ptarc})$ must not hold in $A$ – from rule $e\\ref {e:ne:ptarc}$ and forced atom proposition Then $q \\lnot < n_i \\equiv q \\ge n_i$ in $e\\ref {e:ne:ptarc}$ for all $\\lbrace holds(p,q,0), ptarc(p,t_i,n_i)\\rbrace \\subseteq A$ – from $e\\ref {e:ne:ptarc}$ , forced atom proposition, and the following $holds(p,q,0) \\in A$ represents $q=M_0(p)$ – rule $i\\ref {i:holds}$ construction $ptarc(p,t_i,n_i) \\in A$ represents $n_i=W(p,t_i)$ – rule $f\\ref {f:ptarc} $ construction Then $\\forall p \\in \\bullet t_i, M_0(p) > W(p,t_i)$ – from definition REF of preset $\\bullet t_i$ in PN Then $t_i$ is enabled and can fire in $PN$ , as a result it can belong to $T_0$ – from definition REF of enabled transition And $consumesmore \\notin A$ , since $A$ is an answer set of $\\Pi ^1$ – from rule $a\\ref {a:overc:elim}$ and supported rule proposition Then $\\nexists consumesmore(p,0) \\in A$ – from rule $a\\ref {a:overc:gen}$ and supported rule proposition Then $\\nexists \\lbrace holds(p,q,0), tot\\_decr(p,q1,0) \\rbrace \\subseteq A : q1>q$ in $body(a\\ref {a:overc:place})$ – from $a\\ref {a:overc:place}$ and forced atom proposition Then $\\nexists p : \\sum _{t_i \\in \\lbrace t_0,\\dots ,t_x\\rbrace , p \\in \\bullet t_i}{W(p,t_i)} > M_0(p)$ – from the following $holds(p,q,0)$ represents $q=M_0(p)$ – from rule $i\\ref {i:holds}$ construction, given $tot\\_decr(p,q1,0) \\in A$ if $\\lbrace del(p,q1_0,t_0,0), \\dots , del(p,q1_x,t_x,0) \\rbrace \\subseteq A$ , where $q1 = q1_0+\\dots +q1_x$ – from $r\\ref {r:totdecr}$ and forced atom proposition $del(p,q1_i,t_i,0) \\in A$ if $\\lbrace fires(t_i,0), ptarc(p,t_i,q1_i) \\rbrace \\subseteq A$ – from $r\\ref {r:del}$ and supported rule proposition $del(p,q1_i,t_i,0)$ represents removal of $q1_i = W(p,t_i)$ tokens from $p \\in \\bullet t_i$ – from rule $r\\ref {r:del}$ , supported rule proposition, and definition REF of transition execution in $PN$ Then the set of transitions in $T_0$ do not conflict – by the definition REF of conflicting transitions And for each $enabled(t_j,0) \\in A$ and $fires(t_j,0) \\notin A$ , $could\\_not\\_have(t_j,0) \\in A$ , since $A$ is an answer set of $\\Pi ^1$ - from rule $a\\ref {a:maxfire:elim}$ and supported rule proposition Then $\\lbrace enabled(t_j,0), holds(s,qq,0), ptarc(s,t_j,q,0), \\\\ tot\\_decr(s,qqq,0) \\rbrace \\subseteq A$ , such that $q > qq - qqq$ and $fires(t_j,0) \\notin A$ - from rule $a\\ref {a:maxfire:cnh}$ and supported rule proposition Then for an $s \\in \\bullet t_j$ , $W(s,t_j) > M_0(s) - \\sum _{t_i \\in T_0, s \\in \\bullet t_i}{W(s,t_i)}$ - from the following: $ptarc(s,t_i,q)$ represents $q=W(s,t_i)$ – from rule $f\\ref {f:r:ptarc}$ construction $holds(s,qq,0)$ represents $qq=M_0(s)$ – from $i\\ref {i:holds}$ construction $tot\\_decr(s,qqq,0) \\in A$ if $\\lbrace del(s,qqq_0,t_0,0), \\dots , del(s,qqq_x,t_x,0) \\rbrace \\subseteq A$ – from rule $r\\ref {r:totdecr}$ construction and supported rule proposition $del(s,qqq_i,t_i,0) \\in A$ if $\\lbrace fires(t_i,0), ptarc(s,t_i,qqq_i) \\rbrace \\subseteq A$ – from rule $r\\ref {r:r:del} $ and supported rule proposition $del(s,qqq_i,t_i,0)$ represents $qqq_i = W(s,t_i) : t_i \\in T_0, (s,t_i) \\in E^-$ – from rule $f\\ref {f:r:ptarc}$ construction $tot\\_decr(q,qqq,0)$ represents $\\sum _{t_i \\in T_0, s \\in \\bullet t_i}{W(s,t_i)}$ – from (C,D,E) above Then firing $T_0 \\cup \\lbrace t_j \\rbrace $ would have required more tokens than are present at its source place $s \\in \\bullet t_j$ .", "Thus, $T_0$ is a maximal set of transitions that can simultaneously fire.", "Then $\\lbrace t_0, \\dots , t_x\\rbrace = T_0$ – using 1(a),1(b) above; and using 1(c) it is a maximal firing set We show $M_1$ is produced by firing $T_0$ in $M_0$ .", "Let $holds(p,q,1) \\in A$ Then $\\lbrace holds(p,q1,0), tot\\_incr(p,q2,0), tot\\_decr(p,q3,0) \\rbrace \\subseteq A : q=q1+q2-q3$ – from rule $r\\ref {r:nextstate}$ and supported rule proposition Then, $holds(p,q1,0) \\in A$ represents $q1=M_0(p)$ – given, rule $i\\ref {i:holds}$ construction; and $\\lbrace add(p,q2_0,t_0,0), \\dots , $ $add(p,q2_j,t_j,0)\\rbrace \\subseteq A : q2_0 + \\dots + q2_j = q2$ and $\\lbrace del(p,q3_0,t_0,0), \\dots , $ $del(p,q3_l,t_l,0)\\rbrace \\subseteq A : q3_0 + \\dots + q3_l = q3$ – rules $r\\ref {r:totincr},r\\ref {r:totdecr}$ using supported rule proposition Then $\\lbrace fires(t_0,0), \\dots , fires(t_j,0) \\rbrace \\subseteq A$ and $\\lbrace fires(t_0,0), \\dots , fires(t_l,0) \\rbrace \\\\ \\subseteq A$ – rules $r\\ref {r:add},r\\ref {r:del}$ and supported rule proposition, respectively Then $\\lbrace fires(t_0,0), \\dots , $ $fires(t_j,0) \\rbrace \\cup \\lbrace fires(t_0,0), \\dots , $ $fires(t_l,0) \\rbrace \\subseteq A = \\lbrace fires(t_0,0), \\dots , $ $fires(t_x,0) \\rbrace \\subseteq A$ – set union of subsets Then for each $fires(t_x,0) \\in A$ we have $t_x \\in T_0$ – already shown in item REF above Then $q = M_0(p) + \\sum _{t_x \\in T_0 \\wedge p \\in t_x \\bullet }{W(t_x,p)} - \\sum _{t_x \\in T_0 \\wedge p \\in \\bullet t_x}{W(p,t_x)}$ – from (REF ) above and the following Each $add(p,q_j,t_j,0) \\in A$ represents $q_j=W(t_j,p)$ for $p \\in t_j \\bullet $ – rule $r\\ref {r:add}$ encoding, and definition REF of transition execution in $PN$ Each $del(p,t_y,q_y,0) \\in A$ represents $q_y=W(p,t_y)$ for $p \\in \\bullet t_y$ – from rule $r\\ref {r:del}$ encoding, and definition REF of transition execution in $PN$ Each $tot\\_incr(p,q2,0) \\in A$ represents $q2=\\sum _{t_x \\in T_0 \\wedge p \\in t_x \\bullet }{W(t_x,p)}$ – aggregate assignment atom semantics in rule $r\\ref {r:totincr}$ Each $tot\\_decr(p,q3,0) \\in A$ represents $q3=\\sum _{t_x \\in T_0 \\wedge p \\in \\bullet t_x}{W(p,t_x)}$ – aggregate assignment atom semantics in rule $r\\ref {r:totdecr}$ Then, $M_1(p) = q$ – since $holds(p,q,1) \\in A$ encodes $q=M_1(p)$ from construction Inductive Step: Let $k > 0$ , and $M_k$ is a valid marking in $X$ for $PN$ , show [(1)] $T_k$ is a valid firing set for $M_k$ , and $T_k$ 's firing in $M_k$ produces marking $M_{k+1}$ .", "We show $T_k$ is a valid firing set.", "Let $\\lbrace fires(t_0,k),\\dots ,fires(t_x,k) \\rbrace $ be the set of all $fires(\\dots ,k)$ atoms in $A$ , Then for each $fires(t_i,k) \\in A$ $enabled(t_i,k) \\in A$ – from rule $a\\ref {a:fires}$ and supported rule proposition Then $notenabled(t_i,k) \\notin A$ – from rule $e\\ref {e:enabled}$ and supported rule proposition Then $body(e\\ref {e:ne:ptarc})$ must hold in $A$ – from rule $e\\ref {e:ne:ptarc}$ and forced atom proposition Then $q \\lnot < n_i \\equiv q \\ge n_i$ in $e\\ref {e:ne:ptarc}$ for all $\\lbrace holds(p,q,k), ptarc(p,t_i,n) \\rbrace \\subseteq A$ – from $e\\ref {e:ne:ptarc}$ , forced atom proposition, and the following $holds(p,q,k) \\in A$ represents $q=M_k(p)$ – construction, inductive assumption $ptarc(p,t,n_i) \\in A$ represents $n_i=W(p,t_i)$ – rule $f\\ref {f:ptarc}$ construction Then $\\forall p \\in \\bullet t_i$ , $M_k(p) \\ge W(p,t_i)$ – from definition REF of preset $\\bullet t_i$ in $PN$ Then $t_i$ is enabled and can fire in $PN$ , as a result it can belong to $T_k$ – from definition REF of enabled transition And $consumesmore \\notin A$ , since $A$ is an answer set of $\\Pi ^1$ – from rule $a\\ref {a:overc:elim}$ and supported rule proposition Then $\\nexists consumesmore(p,k) \\in A$ – from rule $a\\ref {a:overc:gen}$ and forced atom proposition Then $\\nexists \\lbrace holds(p,q,k), tot\\_decr(p,q1,k) \\rbrace \\subseteq A : q1 > q$ in $body(a\\ref {a:overc:place})$ – from $a\\ref {a:overc:place}$ and forced atom proposition Then $\\nexists p : \\sum _{t_i \\in \\lbrace t_0,\\dots ,t_x \\rbrace , p \\in \\bullet t_i}{W(p,t_i)} > M_k(p)$ – from the following $holds(p,q,k) \\in A$ represents $q=M_k(p)$ – by construction of $\\Pi ^1$ , and the inductive assumption about $M_k(p)$ $tot\\_decr(p,q1,k) \\in A$ if $\\lbrace del(p,q1_0,t_0,k), \\dots , del(p,q1_x,t_x,k) \\rbrace \\subseteq A$ , where $q1=q1_0+\\dots +q1_x$ – from $r\\ref {r:totdecr}$ and forced atom proposition $del(p,q1_i,t_i,k) \\in A$ if $\\lbrace fires(t_i,k), ptarc(p,t_i,q1_i) \\rbrace \\subseteq A$ – from $r\\ref {r:del}$ and supported rule proposition $del(p,q1_i,t_i,k)$ represents removal of $q1_i = W(p,t_i)$ tokens from $p \\in \\bullet t_i$ – from construction rule $r\\ref {r:del}$ , supported rule proposition, and definition REF of transition execution in $PN$ Then the set of transitions $T_k$ do not conf – by the definition REF of conflicting transitions And for each $enabled(t_j,k) \\in A$ and $fires(t_j,k) \\notin A$ , $could\\_not\\_have(t_j,k) \\in A$ , since $A$ is an answer set of $\\Pi ^1$ - from rule $a\\ref {a:maxfire:elim}$ and supported rule proposition Then $\\lbrace enabled(t_j,k), holds(s,qq,k), ptarc(s,t_j,q,k), \\\\ tot\\_decr(s,qqq,k) \\rbrace \\subseteq A$ , such that $q > qq - qqq$ and $fires(t_j,0) \\notin A$ - from rule $a\\ref {a:maxfire:cnh}$ and supported rule proposition Then for an $s \\in \\bullet t_j$ , $W(s,t_j) > M_k(s) - \\sum _{t_i \\in T_k, s \\in \\bullet t_i}{W(s,t_i)}$ - from the following: $ptarc(s,t_i,q)$ represents $q=W(s,t_i)$ – from rule $f\\ref {f:r:ptarc}$ construction $holds(s,qq,k)$ represents $qq=M_k(s)$ – from $i\\ref {i:holds}$ construction $tot\\_decr(s,qqq,k) \\in A$ if $\\lbrace del(s,qqq_0,t_0,k), \\dots , del(s,qqq_x,t_x,k) \\rbrace \\subseteq A$ – from rule $r\\ref {r:totdecr}$ construction and supported rule proposition $del(s,qqq_i,t_i,k) \\in A$ if $\\lbrace fires(t_i,k), ptarc(s,t_i,qqq_i) \\rbrace \\subseteq A$ – from rule $r\\ref {r:r:del} $ and supported rule proposition $del(s,qqq_i,t_i,k)$ represents $qqq_i = W(s,t_i) : t_i \\in T_k, (s,t_i) \\in E^-$ – from rule $f\\ref {f:r:ptarc}$ construction $tot\\_decr(q,qqq,k)$ represents $\\sum _{t_i \\in T_k, s \\in \\bullet t_i}{W(s,t_i)}$ – from (C,D,E) above Then firing $T_k \\cup \\lbrace t_j \\rbrace $ would have required more tokens than are present at its source place $s \\in \\bullet t_j$ .", "Thus, $T_k$ is a maximal set of transitions that can simultaneously fire.", "Then $\\lbrace t_0,\\dots ,t_x \\rbrace = T_k$ – using 1(a),1(b) above; and using 1(c) it is a maximal firing set We show $M_{k+1}$ is produced by firing $T_k$ in $M_k$ .", "Let $holds(p,q,k+1) \\in A$ Then $\\lbrace holds(p,q1,k), tot\\_incr(p,q2,k), tot\\_decr(p,q3,k) \\rbrace \\in A : q=q2+q2-q3$ – from rule $r\\ref {r:nextstate}$ and supported rule proposition $holds(p,q1,k) \\in A$ represents $q1=M_k(p)$ – construction, inductive assumption; and $\\lbrace add(p,q2_0,t_0,k), \\dots , $ $add(p,q2_j,t_j,k) \\rbrace \\subseteq A : q2_0+\\dots +q2_j=q2$ and $\\lbrace del(p,q3_0,t_0,k), \\dots , $ $del(p,q3_l,t_l,k) \\rbrace \\subseteq A : q3_0+\\dots +q3_l=q3$ – from rules $r\\ref {r:totincr},r\\ref {r:totdecr}$ using supported rule proposition, respectively Then $\\lbrace fires(t_0,k), \\dots , fires(t_j,k) \\rbrace \\subseteq A$ and $\\lbrace fires(t_0,k), \\dots , fires(t_l,k) \\rbrace \\\\ \\subseteq A$ – from rules $r\\ref {r:add} ,r\\ref {r:del} $ using supported rule proposition, respectively Then $\\lbrace fires(t_0,k), \\dots , $ $fires(t_j,k) \\rbrace \\cup \\lbrace fires(t_0,k), \\dots , $ $fires(t_l,k) \\rbrace = \\lbrace fires(t_0,k), \\dots , $ $fires(t_x,k) \\rbrace \\subseteq A$ – subset union property Then for each $fires(t_x,k) \\in A$ we have $t_x \\in T_k$ - already shown in item (REF ) above Then $q = M_k(p) + \\sum _{t_x \\in T_k \\wedge p \\in t_x \\bullet }{W(t_x,p)} - \\sum _{t_x \\in T_k \\wedge p \\in \\bullet t_x}{W(p,t_x)}$ – from (REF ) above and the following Each $add(p,q_j,t_j,k) \\in A$ represents $q_j = W(t_j,p)$ for $p \\in t_j \\bullet $ – rule $r\\ref {r:add}$ encoding and definition REF of transition execution in PN Each $del(p,t_y,q_y,k) \\in A$ represents $q_y = W(p,t_y)$ for $p \\in \\bullet t_y$ – rule $r\\ref {r:del}$ encoding and definition REF of transition execution in PN Each $tot\\_incr(p,q2,k) \\in A$ represents $q2 = \\sum _{t_x \\in T_k \\wedge p \\in t_x \\bullet }{W(t_x,p)}$ – aggregate assignment atom semantics in rule $r\\ref {r:totincr}$ Each $tot\\_decr(p,q3,k) \\in A$ represents $q3 = \\sum _{t_x \\in T_k \\wedge p \\in \\bullet t_x}{W(p,t_x)}$ – aggregate assignment atom semantics in rule $r\\ref {r:totdecr}$ Then $M_{k+1}(p) = q$ – since $holds(p,q,k+1) \\in A$ encodes $q=M_{k+1}(p)$ – from construction As a result, for any $n > k$ , $T_n$ will be a valid firing set w.r.t.", "$M_n$ and its firing produces marking $M_{n+1}$ .", "Conclusion: Since both REF and REF hold, $X=M_0,T_0,M_1,\\dots ,M_k,T_k,M_{k+1}$ is an execution sequence of $PN(P,T,E,W)$ (w.r.t.", "$M_0$ ) iff there is an answer set $A$ of $\\Pi ^1(PN,M_0,k,ntok)$ such that (REF ) and (REF ) hold.", "Proof of Proposition REF Let $PN=(P,T,E,W,R)$ be a Petri Net, $M_0$ be its initial marking and let $\\Pi ^2(PN,M_0,k,ntok)$ by the ASP encoding of $PN$ and $M_0$ over a simulation length $k$ , with maximum $ntok$ tokens on any place node, as defined in section REF .", "Then $X=M_0,T_0,M_1,\\dots ,M_k,T_k,M_{k+1}$ is an execution sequence of $PN$ (w.r.t $M_0$ ) iff there is an answer set $A$ of $\\Pi ^2(PN,M_0,k,ntok)$ such that: $\\lbrace fires(t,ts) : t \\in T_{ts}, 0 \\le ts \\le k\\rbrace = \\lbrace fires(t,ts) : fires(t,ts) \\in A \\rbrace $ $\\begin{split}\\lbrace holds(p,q,ts) &: p \\in P, q = M_{ts}(p), 0 \\le ts \\le k+1 \\rbrace \\\\&= \\lbrace holds(p,q,ts) : holds(p,q,ts) \\in A \\rbrace \\end{split}$ We prove this by showing that: Given an execution sequence $X$ , we create a set $A$ such that it satisfies (REF ) and (REF ) and show that $A$ is an answer set of $\\Pi ^2$ Given an answer set $A$ of $\\Pi ^2$ , we create an execution sequence $X$ such that (REF ) and (REF ) are satisfied.", "First we show (REF ): Given $PN$ and an execution sequence $X$ of $PN$ , we create a set $A$ as a union of the following sets: $A_1=\\lbrace num(n) : 0 \\le n \\le ntok \\rbrace $ $A_2=\\lbrace time(ts) : 0 \\le ts \\le k\\rbrace $ $A_3=\\lbrace place(p) : p \\in P \\rbrace $ $A_4=\\lbrace trans(t) : t \\in T \\rbrace $ $A_5=\\lbrace ptarc(p,t,n,ts) : (p,t) \\in E^-, n=W(p,t), 0 \\le ts \\le k\\rbrace $ , where $E^- \\subseteq E$ $A_6=\\lbrace tparc(t,p,n,ts) : (t,p) \\in E^+, n=W(t,p), 0 \\le ts \\le k\\rbrace $ , where $E^+ \\subseteq E$ $A_7=\\lbrace holds(p,q,0) : p \\in P, q=M_{0}(p) \\rbrace $ $A_8=\\lbrace notenabled(t,ts) : t \\in T, 0 \\le ts \\le k, \\exists p \\in \\bullet t, M_{ts}(p) < W(p,t) \\rbrace $ per definition REF (enabled transition) $A_9=\\lbrace enabled(t,ts) : t \\in T, 0 \\le ts \\le k, \\forall p \\in \\bullet t, W(p,t) \\le M_{ts}(p) \\rbrace $ per definition REF (enabled transition) $A_{10}=\\lbrace fires(t,ts) : t \\in T_{ts}, 0 \\le ts \\le k \\rbrace $ per definition REF (firing set), only an enabled transition may fire $A_{11}=\\lbrace add(p,q,t,ts) : t \\in T_{ts}, p \\in t \\bullet , q=W(t,p), 0 \\le ts \\le k \\rbrace $ per definition REF (transition execution) $A_{12}=\\lbrace del(p,q,t,ts) : t \\in T_{ts}, p \\in \\bullet t, q=W(p,t), 0 \\le ts \\le k \\rbrace \\cup \\lbrace del(p,q,t,ts) : t \\in T_{ts}, p \\in R(t), q=M_{ts}(p), 0 \\le ts \\le k \\rbrace $ per definition REF (transition execution) $A_{13}=\\lbrace tot\\_incr(p,q,ts) : p \\in P, q=\\sum _{t \\in T_{ts}, p \\in t \\bullet }{W(t,p)}, 0 \\le ts \\le k \\rbrace $ per definition REF (execution) $A_{14}=\\lbrace tot\\_decr(p,q,ts): p \\in P, q=\\sum _{t \\in T_{ts}, p \\in \\bullet t}{W(p,t)}+\\sum _{t \\in T_{ts}, p \\in R(t)}{M_{ts}(p)}, \\\\ 0 \\le ts \\le k \\rbrace $ per definition REF (execution) $A_{15}=\\lbrace consumesmore(p,ts) : p \\in P, q=M_{ts}(p), q1=\\sum _{t \\in T_{ts}, p \\in \\bullet t}{W(p,t)}+\\sum _{t \\in T_{ts}, p \\in R(t)}{M_{ts}(p)}, q1 > q, 0 \\le ts \\le k \\rbrace $ per definition REF (conflicting transitions) $A_{16}=\\lbrace consumesmore : \\exists p \\in P : q=M_{ts}(p), q1=\\sum _{t \\in T_{ts}, p \\in \\bullet t}{W(p,t)}+\\sum _{t \\in T_{ts}, p \\in R(t)}{M_{ts}(p)}, q1 > q, 0 \\le ts \\le k \\rbrace $ per definition REF (conflicting transitions) $A_{17}=\\lbrace could\\_not\\_have(t,ts) : t \\in T, (\\forall p \\in \\bullet t, W(p,t) \\le M_{ts}(p)), t \\notin T_{ts}, (\\exists p \\in \\bullet t \\cup R(t) : q > M_{ts}(p) - (\\sum _{t^{\\prime } \\in T_{ts}, p \\in \\bullet t^{\\prime }}{W(p,t^{\\prime })} + \\sum _{t^{\\prime } \\in T_{ts}, p \\in R(t^{\\prime })}{M_{ts}(p)}), q=W(p,t) \\text{~if~} p \\in \\bullet t \\text{~or~} M_{ts}(p) \\text{~otherwise~}), 0 \\le ts \\le k \\rbrace $ per the maximal firing set semantics $A_{18}=\\lbrace holds(p,q,ts+1) : p \\in P, q=M_{ts+1}(p), 0 \\le ts < k\\rbrace $ , where $M_{ts+1}(p) = M_{ts}(p) - (\\sum _{\\begin{array}{c}t \\in T_{ts}, p \\in \\bullet t\\end{array}}{W(p,t)} + \\sum _{\\begin{array}{c}t \\in T_{ts}, p \\in R(t)\\end{array}}{M_{ts}(p)}) + \\sum _{\\begin{array}{c}t \\in T_{ts}, p \\in t \\bullet \\end{array}}{W(t,p)}$ according to definition REF (firing set execution) $A_{19}=\\lbrace ptarc(p,t,n,ts) : p \\in R(t), n=M_{ts}(p), n > 0, 0 \\le ts \\le k\\rbrace $ We show that $A$ satisfies (REF ) and (REF ), and $A$ is an answer set of $\\Pi ^1$ .", "$A$ satisfies (REF ) and (REF ) by its construction above.", "We show $A$ is an answer set of $\\Pi ^1$ by splitting.", "We split $lit(\\Pi ^1)$ into a sequence of $7(k+1)+2$ sets: $U_0= head(f\\ref {f:place}) \\cup head(f\\ref {f:trans}) \\cup head(f\\ref {f:time}) \\cup head(f\\ref {f:num}) \\cup head(i\\ref {i:holds}) = \\lbrace place(p) : p \\in P\\rbrace \\cup \\lbrace trans(t) : t \\in T\\rbrace \\cup \\lbrace time(0), \\dots , time(k)\\rbrace \\cup \\lbrace num(0), \\dots , num(ntok)\\rbrace \\cup \\lbrace holds(p,q,0) : p \\in P, q=M_0(p) \\rbrace $ $U_{7k+1}=U_{7k+0} \\cup head(f\\ref {f:r:ptarc})^{ts=k} \\cup head(f\\ref {f:r:tparc})^{ts=k} \\cup head(f\\ref {f:rptarc})^{ts=k} = U_{7k+0} \\cup \\\\ \\lbrace ptarc(p,t,n,k) : (p,t) \\in E^-, n=W(p,t) \\rbrace \\cup \\lbrace tparc(t,p,n,k) : (t,p) \\in E^+, n=W(t,p) \\rbrace \\cup \\lbrace ptarc(p,t,n,k) : p \\in R(t), n=M_{k}(p), n > 0 \\rbrace $ $U_{7k+2}=U_{7k+1} \\cup head(e\\ref {e:r:ne:ptarc} )^{ts=k} = U_{7k+1} \\cup \\lbrace notenabled(t,k) : t \\in T \\rbrace $ $U_{7k+3}=U_{7k+2} \\cup head(e\\ref {e:enabled})^{ts=k} = U_{7k+2} \\cup \\lbrace enabled(t,k) : t \\in T \\rbrace $ $U_{7k+4}=U_{7k+3} \\cup head(a\\ref {a:fires})^{ts=k} = U_{7k+3} \\cup \\lbrace fires(t,k) : t \\in T \\rbrace $ $U_{7k+5}=U_{7k+4} \\cup head(r\\ref {r:r:add} )^{ts=k} \\cup head(r\\ref {r:r:del} )^{ts=k} = U_{7k+4} \\cup \\lbrace add(p,q,t,k) : p \\in P, t \\in T, q=W(t,p) \\rbrace \\cup \\lbrace del(p,q,t,k) : p \\in P, t \\in T, q=W(p,t) \\rbrace \\cup \\lbrace del(p,q,t,k) : p \\in P, t \\in T, q=M_{k}(p) \\rbrace $ $U_{7k+6}=U_{7k+5} \\cup head(r\\ref {r:totincr})^{ts=k} \\cup head(r\\ref {r:totdecr})^{ts=k} = U_{7k+5} \\cup \\lbrace tot\\_incr(p,q,k) : p \\in P, 0 \\le q \\le ntok \\rbrace \\cup \\lbrace tot\\_decr(p,q,k) : p \\in P, 0 \\le q \\le ntok \\rbrace $ $U_{7k+7}=U_{7k+6} \\cup head(r\\ref {r:nextstate})^{ts=k} \\cup head(a\\ref {a:overc:place})^{ts=k} \\cup head(a\\ref {a:r:maxfire:cnh})^{ts=k} $ $= U_{7k+6} \\cup \\\\ \\lbrace consumesmore(p,k) : p \\in P\\rbrace \\cup \\lbrace holds(p,q,k+1) : p \\in P, 0 \\le q \\le ntok \\rbrace \\cup \\lbrace could\\_not\\_have(t,k) : t \\in T \\rbrace $ $U_{7k+8}=U_{7k+7} \\cup head(a\\ref {a:overc:gen}) = U_{7k+7} \\cup \\lbrace consumesmore \\rbrace $ where $head(r_i)^{ts=k}$ are head atoms of ground rule $r_i$ in which $ts=k$ .", "We write $A_i^{ts=k} = \\lbrace a(\\dots ,ts) : a(\\dots ,ts) \\in A_i, ts=k \\rbrace $ as short hand for all atoms in $A_i$ with $ts=k$ .", "$U_{\\alpha }, 0 \\le \\alpha \\le 7k+8$ form a splitting sequence, since each $U_i$ is a splitting set of $\\Pi ^1$ , and $\\langle U_{\\alpha }\\rangle _{\\alpha < \\mu }$ is a monotone continuous sequence, where $U_0 \\subseteq U_1 \\dots \\subseteq U_{7k+8}$ and $\\bigcup _{\\alpha < \\mu }{U_{\\alpha }} = lit(\\Pi ^1)$ .", "We compute the answer set of $\\Pi ^2$ using the splitting sets as follows: $bot_{U_0}(\\Pi ^2) = f\\ref {f:place} \\cup f\\ref {f:trans} \\cup f\\ref {f:time} \\cup i\\ref {i:holds} \\cup f\\ref {f:num}$ and $X_0 = A_1 \\cup \\dots \\cup A_4 \\cup A_7$ ($= U_0$ ) is its answer set – using forced atom proposition $eval_{U_0}(bot_{U_1}(\\Pi ^2) \\setminus bot_{U_0}(\\Pi ^2), X_0) = \\lbrace ptarc(p,t,q,0) \\text{:-}.", "\| q=W(p,t) \\rbrace \\cup \\\\ \\lbrace tparc(t,p,q,0) \\text{:-}.", "\| $ $q=W(t,p) \\rbrace \\cup \\lbrace ptarc(p,t,q,0) \\text{:-}.", "\| $ $q=M_0(p) \\rbrace $ .", "Its answer set $X_1=A_5^{ts=0} \\cup A_6^{ts=0} \\cup A_{19}^{ts=0}$ – using forced atom proposition and construction of $A_5, A_6, A_{19}$ .", "$eval_{U_1}(bot_{U_2}(\\Pi ^2) \\setminus bot_{U_1}(\\Pi ^2), X_0 \\cup X_1) = \\lbrace notenabled(t,0) \\text{:-} .", "\| \\lbrace trans(t), \\\\ ptarc(p,t,n,0), holds(p,q,0) \\rbrace \\subseteq X_0 \\cup X_1, \\text{~where~} q < n \\rbrace $ .", "Its answer set $X_2=A_8^{ts=0}$ – using forced atom proposition and construction of $A_8$ .", "where, $q=M_0(p)$ , and $n=W(p,t)$ for an arc $(p,t) \\in E^-$ – by construction of $i\\ref {i:holds}$ and $f\\ref {f:r:ptarc}$ in $\\Pi ^2$ , and in an arc $(p,t) \\in E^-$ , $p \\in \\bullet t$ (by definition REF of preset) thus, $notenabled(t,0) \\in X_1$ represents $\\exists p \\in \\bullet t : M_0(p) < W(p,t)$ .", "$eval_{U_2}(bot_{U_3}(\\Pi ^2) \\setminus bot_{U_2}(\\Pi ^2), X_0 \\cup \\dots \\cup X_2) = \\lbrace enabled(t,0) \\text{:-}.", "\| trans(t) \\in X_0 \\cup \\dots \\cup X_2, notenabled(t,0) \\notin X_0 \\cup \\dots \\cup X_2 \\rbrace $ .", "Its answer set is $X_3 = A_9^{ts=0}$ – using forced atom proposition and construction of $A_9$ .", "since an $enabled(t,0) \\in X_3$ if $\\nexists ~notenabled(t,0) \\in X_0 \\cup \\dots \\cup X_2$ ; which is equivalent to $\\nexists p \\in \\bullet t : M_0(p) < W(p,t) \\equiv \\forall p \\in \\bullet t: M_0(p) \\ge W(p,t)$ .", "$eval_{U_3}(bot_{U_4}(\\Pi ^2) \\setminus bot_{U_3}(\\Pi ^2), X_0 \\cup \\dots \\cup X_3) = \\lbrace \\lbrace fires(t,0)\\rbrace \\text{:-}.", "\| enabled(t,0) \\\\ \\text{~holds in~} X_0 \\cup \\dots \\cup X_3 \\rbrace $ .", "It has multiple answer sets $X_{4.1}, \\dots , X_{4.n}$ , corresponding to elements of power set of $fires(t,0)$ atoms in $eval_{U_3}(...)$ – using supported rule proposition.", "Since we are showing that the union of answer sets of $\\Pi ^2$ determined using splitting is equal to $A$ , we only consider the set that matches the $fires(t,0)$ elements in $A$ and call it $X_4$ , ignoring the rest.", "Thus, $X_4 = A_{10}^{ts=0}$ , representing $T_0$ .", "in addition, for every $t$ such that $enabled(t,0) \\in X_0 \\cup \\dots \\cup X_3, R(t) \\ne \\emptyset $ ; $fires(t,0) \\in X_4$ – per definition REF (firing set); requiring that a reset transition is fired when enabled thus, the firing set $T_0$ will not be eliminated by the constraint $f\\ref {f:c:rptarc:elim}$ $eval_{U_4}(bot_{U_5}(\\Pi ^2) \\setminus bot_{U_4}(\\Pi ^2), X_0 \\cup \\dots \\cup X_4) = \\lbrace add(p,n,t,0) \\text{:-}.", "\| \\lbrace fires(t,0), \\\\ tparc(t,p,n,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_4 \\rbrace \\cup \\lbrace del(p,n,t,0) \\text{:-}.", "\| \\lbrace fires(t,0), ptarc(p,t,n,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_4 \\rbrace $ .", "It's answer set is $X_5 = A_{11}^{ts=0} \\cup A_{12}^{ts=0}$ – using forced atom proposition and definitions of $A_{11}$ and $A_{12}$ .", "where, each $add$ atom is equivalent to $n=W(t,p) : p \\in t \\bullet $ , and each $del$ atom is equivalent to $n=W(p,t) : p \\in \\bullet t$ ; or $n=M_k(p) : p \\in R(t)$ , representing the effect of transitions in $T_0$ – by construction $eval_{U_5}(bot_{U_6}(\\Pi ^2) \\setminus bot_{U_5}(\\Pi ^2), X_0 \\cup \\dots \\cup X_5) = \\lbrace tot\\_incr(p,qq,0) \\text{:-}.", "\| $ $qq=\\sum _{add(p,q,t,0) \\in X_0 \\cup \\dots \\cup X_5}{q} \\rbrace \\cup \\lbrace tot\\_decr(p,qq,0) \\text{:-}.", "\| qq=\\sum _{del(p,q,t,0) \\in X_0 \\cup \\dots \\cup X_5}{q} \\rbrace $ .", "It's answer set is $X_6 = A_{13}^{ts=0} \\cup A_{14}^{ts=0}$ – using forced atom proposition, definitions of $A_{13}$ , $A_{14}$ , and definition REF (semantics of aggregate assignment atom).", "where, each for $tot\\_incr(p,qq,0)$ , $qq=\\sum _{add(p,q,t,0) \\in X_0 \\cup \\dots X_5}{q}$ $\\equiv qq=\\sum _{t \\in X_4, p \\in t \\bullet }{W(p,t)}$ , and each $tot\\_decr(p,qq,0)$ , $qq=\\sum _{del(p,q,t,0) \\in X_0 \\cup \\dots X_5}{q}$ $\\equiv qq=\\sum _{t \\in X_4, p \\in \\bullet t}{W(t,p)} + \\sum _{t \\in X_4, p \\in R(t)}{M_{k}(p)}$ , represent the net effect of transitions in $T_0$ – by construction $eval_{U_6}(bot_{U_7}(\\Pi ^2) \\setminus bot_{U_6}(\\Pi ^2), X_0 \\cup \\dots \\cup X_6) = \\lbrace consumesmore(p,0) \\text{:-}.", "\| \\\\ \\lbrace holds(p,q,0), tot\\_decr(p,q1,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_6, q1 > q \\rbrace \\cup \\lbrace holds(p,q,1) \\text{:-}., \| \\\\ \\lbrace holds(p,q1,0), tot\\_incr(p,q2,0), tot\\_decr(p,q3,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_6, q=q1+q2-q3 \\rbrace \\cup \\lbrace could\\_not\\_have(t,0) \\text{:-}.", "\| \\lbrace enabled(t,0), ptarc(s,t,q), holds(s,qq,0), \\\\ tot\\_decr(s,qqq,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_6, fires(t,0) \\notin (X_0 \\cup \\dots \\cup X_6), q > qq-qqq \\rbrace $ .", "It's answer set is $X_7 = A_{15}^{ts=0} \\cup A_{17}^{ts=0} \\cup A_{18}^{ts=0}$ – using forced atom proposition and definitions of $A_{15}, A_{17}, A_{18}$ .", "where, $consumesmore(p,0)$ represents $\\exists p : q=M_0(p), q1= \\\\ \\sum _{t \\in T_0, p \\in \\bullet t}{W(p,t)}+$ $\\sum _{t \\in T_0, p \\in R(t)}{M_k(p)}, $ $q1 > q$ , indicating place $p$ will be overconsumed if $T_0$ is fired – as defined in definition REF (conflicting transitions) and, $holds(p,q,1)$ represents $q=M_1(p)$ – by construction of $\\Pi ^2$ and $could\\_not\\_have(t,0)$ represents an enabled transition $t$ in $T_0$ that could not fire due to insufficient tokens $X_7$ does not contain $could\\_not\\_have(t,0)$ , when $enabled(t,0) \\in X_0 \\cup \\dots \\cup X_6$ and $fires(t,0) \\notin X_0 \\cup \\dots \\cup X_6$ due to construction of $A$ , encoding of $a\\ref {a:r:maxfire:cnh}$ and its body atoms.", "As a result it is not eliminated by the constraint $a\\ref {a:maxfire:elim}$ $ \\vdots $ $eval_{U_{7k+0}}(bot_{U_{7k+1}}(\\Pi ^2) \\setminus bot_{U_{7k+0}}(\\Pi ^2), X_0 \\cup \\dots \\cup X_{7k+0}) = \\lbrace ptarc(p,t,q,k) \\text{:-}.", "\| q=W(p,t) \\rbrace \\cup \\lbrace tparc(t,p,q,k) \\text{:-}.", "\| q=W(t,p) \\rbrace \\cup \\lbrace ptarc(p,t,q,k) \\text{:-}.", "\| q=M_0(p) \\rbrace $ .", "Its answer set $X_{7k+1}=A_5^{ts=k} \\cup A_6^{ts=k} \\cup A_{19}^{ts=k}$ – using forced atom proposition and construction of $A_5, A_6, A_{19}$ .", "$eval_{U_{7k+1}}(bot_{U_{7k+2}}(\\Pi ^2) \\setminus bot_{U_{7k+1}}(\\Pi ^2), X_0 \\cup X_{7k+1}) = \\lbrace notenabled(t,k) \\text{:-} .", "\| \\lbrace trans(t), \\\\ ptarc(p,t,n,k), holds(p,q,k) \\rbrace \\subseteq X_0 \\cup X_{7k+1}, \\text{~where~} q < n \\rbrace $ .", "Its answer set $X_{7k+2}=A_8^{ts=k}$ – using forced atom proposition and construction of $A_8$ .", "where, $q=M_0(p)$ , and $n=W(p,t)$ for an arc $(p,t) \\in E^-$ – by construction of $holds$ and $ptarc$ predicates in $\\Pi ^2$ , and in an arc $(p,t) \\in E^-$ , $p \\in \\bullet t$ (by definition REF of preset) thus, $notenabled(t,k) \\in X_{7k+1}$ represents $\\exists p \\in \\bullet t : M_0(p) < W(p,t)$ .", "$eval_{U_{7k+2}}(bot_{U_{7k+3}}(\\Pi ^2) \\setminus bot_{U_{7k+2}}(\\Pi ^2), X_0 \\cup \\dots \\cup X_{7k+2}) = \\lbrace enabled(t,k) \\text{:-}.", "\| trans(t) \\in X_0 \\cup \\dots \\cup X_{7k+2} , notenabled(t,k) \\notin X_0 \\cup \\dots \\cup X_{7k+2} \\rbrace $ .", "Its answer set is $X_{7k+3} = A_9^{ts=k}$ – using forced atom proposition and construction of $A_9$ .", "since an $enabled(t,k) \\in X_{7k+3}$ if $\\nexists ~notenabled(t,k) \\in X_0 \\cup \\dots \\cup X_{7k+2}$ ; which is equivalent to $\\nexists p \\in \\bullet t : M_0(p) < W(p,t) \\equiv \\forall p \\in \\bullet t: M_0(p) \\ge W(p,t)$ .", "$eval_{U_{7k+3}}(bot_{U_{7k+4}}(\\Pi ^2) \\setminus bot_{U_{7k+3}}(\\Pi ^2), X_0 \\cup \\dots \\cup X_{7k+3}) = \\lbrace \\lbrace fires(t,k)\\rbrace \\text{:-}.", "\| \\\\ enabled(t,k) \\\\ \\text{~holds in~} X_0 \\cup \\dots \\cup X_{7k+3} \\rbrace $ .", "It has multiple answer sets $X_{7k+4.1}, \\dots , X_{1k+4.n}$ , corresponding to elements of power set of $fires(t,k)$ atoms in $eval_{U_{7k+3}}(...)$ – using supported rule proposition.", "Since we are showing that the union of answer sets of $\\Pi ^2$ determined using splitting is equal to $A$ , we only consider the set that matches the $fires(t,k)$ elements in $A$ and call it $X_{7k+4}$ , ignoring the rest.", "Thus, $X_{7k+4} = A_{10}^{ts=k}$ , representing $T_k$ .", "in addition, for every $t$ such that $enabled(t,k) \\in X_0 \\cup \\dots \\cup X_{7k+3}, R(t) \\ne \\emptyset $ ; $fires(t,k) \\in X_{7k+4}$ – per definition REF (firing set); requiring that a reset transition is fired when enabled thus, the firing set $T_k$ will not be eliminated by the constraint $f\\ref {f:c:rptarc:elim}$ $eval_{U_{7k+4}}(bot_{U_{7k+5}}(\\Pi ^2) \\setminus bot_{U_{7k+4}}(\\Pi ^2), X_0 \\cup \\dots \\cup X_{7k+4}) = \\\\ \\lbrace add(p,n,t,k) \\text{:-}.", "\| \\lbrace fires(t,k), tparc(t,p,n,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{7k+4} \\rbrace \\cup \\\\ \\lbrace del(p,n,t,k) \\text{:-}.", "\| \\lbrace fires(t,k), ptarc(p,t,n,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{7k+4} \\rbrace $ .", "It's answer set is $X_{7k+5} = A_{11}^{ts=k} \\cup A_{12}^{ts=k}$ – using forced atom proposition and definitions of $A_{11}$ and $A_{12}$ .", "where, each $add$ atom is equivalent to $n=W(t,p) : p \\in \\bullet t$ , and each $del$ atom is equivalent to $n=W(p,t) : p \\in t \\bullet $ ; or $n=M_k(p) : p \\in R(t)$ , representing the effect of transitions in $T_k$ $eval_{U_{7k+5}}(bot_{U_{7k+6}}(\\Pi ^2) \\setminus bot_{U_{7k+5}}(\\Pi ^2), X_0 \\cup \\dots \\cup X_{7k+5}) = \\\\ \\lbrace tot\\_incr(p,qq,k) \\text{:-}.", "\| qq=\\sum _{add(p,q,t,k) \\in X_0 \\cup \\dots \\cup X_{7k+5}}{q} \\rbrace \\cup \\\\ \\lbrace tot\\_decr(p,qq,k) \\text{:-}.", "\| qq=\\sum _{del(p,q,t,k) \\in X_0 \\cup \\dots \\cup X_{7k+5}}{q} \\rbrace $ .", "It's answer set is $X_{7k+6} = A_{13}^{ts=k} \\cup A_{14}^{ts=k}$ – using forced atom proposition and definitions of $A_{13}$ and $A_{14}$ .", "where, each $tot\\_incr(p,qq,k)$ , $qq=\\sum _{add(p,q,t,k) \\in X_0 \\cup \\dots X_{7k+5}}{q}$ $\\equiv qq=\\sum _{t \\in X_{7k+4}, p \\in t \\bullet }{W(p,t)}$ , and each $tot\\_decr(p,qq,k)$ , $qq=\\sum _{del(p,q,t,k) \\in X_0 \\cup \\dots X_{7k+5}}{q}$ $\\equiv qq=\\sum _{t \\in X_{7k+4}, p \\in \\bullet t}{W(t,p)} + \\sum _{t \\in X_{7k+4}, p \\in R(t)}{M_{k}(p)}$ , represent the net effect of transitions in $T_k$ $eval_{U_{7k+6}}(bot_{U_{7k+7}}(\\Pi ^2) \\setminus bot_{U_{7k+6}}(\\Pi ^2), X_0 \\cup \\dots \\cup X_{7k+6}) = \\lbrace consumesmore(p,k) \\text{:-}.", "\| \\\\ \\lbrace holds(p,q,k), tot\\_decr(p,q1,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{7k+6}, q1 > q \\rbrace \\cup \\lbrace holds(p,q,k+1) \\text{:-}., \| \\\\ \\lbrace holds(p,q1,k), tot\\_incr(p,q2,k), tot\\_decr(p,q3,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{7k+6}, q=q1+q2-q3 \\rbrace \\cup \\lbrace could\\_not\\_have(t,k) \\text{:-}.", "\| \\lbrace enabled(t,k), ptarc(s,t,q), holds(s,qq,k), \\\\ tot\\_decr(s,qqq,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{7k+6}, fires(t,k) \\notin (X_0 \\cup \\dots \\cup X_{7k+6}), q > qq-qqq \\rbrace $ .", "It's answer set is $X_{7k+7} = A_{15}^{ts=k} \\cup A_{17}^{ts=k} \\cup A_{18}^{ts=k}$ – using forced atom proposition and definitions of $A_{15}, A_{17}, A_{18}$ .", "where, $consumesmore(p,k)$ represents $\\exists p : q=M_0(p), q1= \\\\ \\sum _{t \\in T_0, p \\in \\bullet t}{W(p,t)}+\\sum _{t \\in T_0, p \\in R(t)}{M_k(p)}, q1 > q$ , indicating place $p$ that will be over consumed if $T_k$ is fired, as defined in definition REF (conflicting transitions) $holds(p,q,k+1)$ represents $q=M_{k+1}(p)$ – by construction of $\\Pi ^2$ , and $could\\_not\\_have(t,k)$ represents enabled transition $t$ in $T_k$ that could not be fired due to insufficient tokens $X_{7k+7}$ does not contain $could\\_not\\_have(t,k)$ , when $enabled(t,k) \\in X_0 \\cup \\dots \\cup X_{7k+6}$ and $fires(t,k) \\notin X_0 \\cup \\dots \\cup X_{7k+6}$ due to the construction of $A$ , encoding of $a\\ref {a:r:maxfire:cnh}$ and its body atoms.", "As a result it is not eliminated by the constraint $a\\ref {a:maxfire:elim}$ $eval_{U_{7k+7}}(bot_{U_{7k+8}}(\\Pi ^2) \\setminus bot_{U_{7k+7}}(\\Pi ^2), X_0 \\cup \\dots \\cup X_{7k+7}) = \\lbrace consumesmore \\text{:-}.", "\| $ $ \\lbrace consumesmore(p,0),\\dots ,$ $consumesmore(p,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{7k+7} \\rbrace $ .", "It's answer set is $X_{7k+8} = A_{16}$ – using forced atom proposition $X_{7k+8}$ will be empty since none of $consumesmore(p,0),\\dots , \\\\ consumesmore(p,k)$ hold in $X_0 \\cup \\dots \\cup X_{7k+7}$ due to the construction of $A$ , encoding of $a\\ref {a:overc:place}$ and its body atoms.", "As a result, it is not eliminated by the constraint $a\\ref {a:overc:elim}$ The set $X = X_0 \\cup \\dots \\cup X_{7k+8}$ is the answer set of $\\Pi ^2$ by the splitting sequence theorem REF .", "Each $X_i, 0 \\le i \\le 7k+8$ matches a distinct portion of $A$ , and $X = A$ , thus $A$ is an answer set of $\\Pi ^2$ .", "Next we show (REF ): Given $\\Pi ^2$ be the encoding of a Petri Net $PN(P,T,E,W,R)$ with initial marking $M_0$ , and $A$ be an answer set of $\\Pi ^2$ that satisfies (REF ) and (REF ), then we can construct $X=M_0,T_0,\\dots ,M_k,T_k,M_{k+1}$ from $A$ , such that it is an execution sequence of $PN$ .", "We construct the $X$ as follows: $M_i = (M_i(p_0), \\dots , M_i(p_n))$ , where $\\lbrace holds(p_0,M_i(p_0),i), \\dots holds(p_n,M_i(p_n),i) \\rbrace \\\\ \\subseteq A$ , for $0 \\le i \\le k+1$ $T_i = \\lbrace t : fires(t,i) \\in A\\rbrace $ , for $0 \\le i \\le k$ and show that $X$ is indeed an execution sequence of $PN$ .", "We show this by induction over $k$ (i.e.", "given $M_k$ , $T_k$ is a valid firing set and its firing produces marking $M_{k+1}$ ).", "Base case: Let $k=0$ , and $M_0$ is a valid marking in $X$ for $PN$ , show [(1)] $T_0$ is a valid firing set for $M_0$ , and $T_0$ 's firing in $M_0$ produces marking $M_1$ .", "We show $T_0$ is a valid firing set for $M_0$ .", "Let $\\lbrace fires(t_0,0), \\dots , fires(t_x,0) \\rbrace $ be the set of all $fires(\\dots ,0)$ atoms in $A$ , Then for each $fires(t_i,0) \\in A$ $enabled(t_i,0) \\in A$ – from rule $a\\ref {a:fires}$ and supported rule proposition Then $notenabled(t_i,0) \\notin A$ – from rule $e\\ref {e:enabled}$ and supported rule proposition Then $body(e\\ref {e:r:ne:ptarc})$ must not hold in $A$ – from rule $e\\ref {e:r:ne:ptarc}$ and forced atom proposition Then $q \\lnot < n_i \\equiv q \\ge n_i$ in $e\\ref {e:r:ne:ptarc}$ for all $\\lbrace holds(p,q,0), ptarc(p,t_i,n_i,0)\\rbrace \\subseteq A$ – from $e\\ref {e:r:ne:ptarc}$ , forced atom proposition, and the following $holds(p,q,0) \\in A$ represents $q=M_0(p)$ – rule $i\\ref {i:holds}$ construction $ptarc(p,t_i,n_i,0) \\in A$ represents $n_i=W(p,t_i)$ – rule $f\\ref {f:r:ptarc}$ construction; or it represents $n_i = M_0(p)$ – rule $f\\ref {f:rptarc}$ construction; the construction of $f\\ref {f:rptarc}$ ensures that $notenabled(t,0)$ is never true due to the reset arc Then $\\forall p \\in \\bullet t_i, M_0(p) \\ge W(p,t_i)$ – from definition REF of preset $\\bullet t_i$ in PN Then $t_i$ is enabled and can fire in $PN$ , as a result it can belong to $T_0$ – from definition REF of enabled transition And $consumesmore \\notin A$ , since $A$ is an answer set of $\\Pi ^2$ – from rule $a\\ref {a:overc:elim}$ and supported rule proposition Then $\\nexists consumesmore(p,0) \\in A$ – from rule $a\\ref {a:overc:gen}$ and supported rule proposition Then $\\nexists \\lbrace holds(p,q,0), tot\\_decr(p,q1,0) \\rbrace \\subseteq A : q1>q$ in $body(a\\ref {a:overc:place})$ – from $a\\ref {a:overc:place}$ and forced atom proposition Then $\\nexists p : (\\sum _{t_i \\in \\lbrace t_0,\\dots ,t_x\\rbrace , p \\in \\bullet t_i}{W(p,t_i)}+\\sum _{t_i \\in \\lbrace t_0,\\dots ,t_x\\rbrace , p \\in R(t_i)}{M_0(p)}) > M_0(p)$ – from the following $holds(p,q,0)$ represents $q=M_0(p)$ – from rule $i\\ref {i:holds}$ construction, given $tot\\_decr(p,q1,0) \\in A$ if $\\lbrace del(p,q1_0,t_0,0), \\dots , del(p,q1_x,t_x,0) \\rbrace \\subseteq A$ , where $q1 = q1_0+\\dots +q1_x$ – from $r\\ref {r:totdecr}$ and forced atom proposition $del(p,q1_i,t_i,0) \\in A$ if $\\lbrace fires(t_i,0), ptarc(p,t_i,q1_i,0) \\rbrace \\subseteq A$ – from $r\\ref {r:r:del} $ and supported rule proposition $del(p,q1_i,t_i,0)$ either represents removal of $q1_i = W(p,t_i)$ tokens from $p \\in \\bullet t_i$ ; or it represents removal of $q1_i = M_0(p)$ tokens from $p \\in R(t_i)$ – from rule $r\\ref {r:r:del} $ , supported rule proposition, and definition REF of transition execution in $PN$ Then the set of transitions in $T_0$ do not conflict – by the definition REF of conflicting transitions And for each $enabled(t_j,0) \\in A$ and $fires(t_j,0) \\notin A$ , $could\\_not\\_have(t_j,0) \\in A$ , since $A$ is an answer set of $\\Pi ^2$ - from rule $a\\ref {a:maxfire:elim}$ and supported rule proposition Then $\\lbrace enabled(t_j,0), holds(s,qq,0), ptarc(s,t_j,q,0), \\\\ tot\\_decr(s,qqq,0) \\rbrace \\subseteq A$ , such that $q > qq - qqq$ and $fires(t_j,0) \\notin A$ - from rule $a\\ref {a:r:maxfire:cnh}$ and supported rule proposition Then for an $s \\in \\bullet t_j \\cup R(t_j)$ , $q > M_0(s) - (\\sum _{t_i \\in T_0, s \\in \\bullet t_i}{W(s,t_i)} + \\sum _{t_i \\in T_0, s \\in R(t_i)}{M_0(s))}$ , where $q=W(s,t_j) \\text{~if~} s \\in \\bullet t_j, \\text{~or~} M_0(s) \\text{~otherwise}$ - from the following: $ptarc(s,t_i,q,0)$ represents $q=W(s,t_i)$ if $(s,t_i) \\in E^-$ or $q=M_0(s)$ if $s \\in R(t_i)$ – from rule $f\\ref {f:r:ptarc},f\\ref {f:rptarc}$ construction $holds(s,qq,0)$ represents $qq=M_0(s)$ – from $i\\ref {i:holds}$ construction $tot\\_decr(s,qqq,0) \\in A$ if $\\lbrace del(s,qqq_0,t_0,0), \\dots , del(s,qqq_x,t_x,0) \\rbrace \\subseteq A$ – from rule $r\\ref {r:totdecr}$ construction and supported rule proposition $del(s,qqq_i,t_i,0) \\in A$ if $\\lbrace fires(t_i,0), ptarc(s,t_i,qqq_i,0) \\rbrace \\subseteq A$ – from rule $r\\ref {r:r:del} $ and supported rule proposition $del(s,qqq_i,t_i,0)$ represents $qqq_i = W(s,t_i) : t_i \\in T_0, (s,t_i) \\in E^-$ , or $qqq_i = M_0(t_i) : t_i \\in T_0, s \\in R(t_i)$ – from rule $f\\ref {f:r:ptarc},f\\ref {f:rptarc}$ construction $tot\\_decr(q,qqq,0)$ represents $\\sum _{t_i \\in T_0, s \\in \\bullet t_i}{W(s,t_i)} + \\\\ \\sum _{t_i \\in T_0, s \\in R(t_i)}{M_0(s)}$ – from (C,D,E) above Then firing $T_0 \\cup \\lbrace t_j \\rbrace $ would have required more tokens than are present at its source place $s \\in \\bullet t_j \\cup R(t_j)$ .", "Thus, $T_0$ is a maximal set of transitions that can simultaneously fire.", "And for each reset transition $t_r$ with $enabled(t_r,0) \\in A$ , $fires(t_r,0) \\in A$ , since $A$ is an answer set of $\\Pi ^2$ - from rule $f\\ref {f:c:rptarc:elim}$ and supported rule proposition Then, the firing set $T_0$ satisfies the reset-transition requirement of definition REF (firing set) Then $\\lbrace t_0, \\dots , t_x\\rbrace = T_0$ – using 1(a),1(b),1(d) above; and using 1(c) it is a maximal firing set We show $M_1$ is produced by firing $T_0$ in $M_0$ .", "Let $holds(p,q,1) \\in A$ Then $\\lbrace holds(p,q1,0), tot\\_incr(p,q2,0), tot\\_decr(p,q3,0) \\rbrace \\subseteq A : q=q1+q2-q3$ – from rule $r\\ref {r:nextstate}$ and supported rule proposition $holds(p,q1,0) \\in A$ represents $q1=M_0(p)$ – given, rule $i\\ref {i:holds}$ construction; and $\\lbrace add(p,q2_0,t_0,0), \\dots , $ $add(p,q2_j,t_j,0)\\rbrace \\subseteq A : q2_0 + \\dots + q2_j = q2$ and $\\lbrace del(p,q3_0,t_0,0), \\dots , $ $del(p,q3_l,t_l,0)\\rbrace \\subseteq A : q3_0 + \\dots + q3_l = q3$ – rules $r\\ref {r:totincr},r\\ref {r:totdecr}$ using supported rule proposition Then $\\lbrace fires(t_0,0), \\dots , fires(t_j,0) \\rbrace \\subseteq A$ and $\\lbrace fires(t_0,0), \\dots , fires(t_l,0) \\rbrace \\\\ \\subseteq A$ – rules $r\\ref {r:r:add},r\\ref {r:r:del}$ and supported rule proposition, respectively Then $\\lbrace fires(t_0,0), \\dots , fires(t_j,0) \\rbrace \\cup \\lbrace fires(t_0,0), \\dots , fires(t_l,0) \\rbrace \\subseteq A = \\lbrace fires(t_0,0), \\dots , fires(t_x,0) \\rbrace \\subseteq A$ – set union of subsets Then for each $fires(t_x,0) \\in A$ we have $t_x \\in T_0$ – already shown in item REF above Then $q = M_0(p) + \\sum _{t_x \\in T_0 \\wedge p \\in t_x \\bullet }{W(t_x,p)} - (\\sum _{t_x \\in T_0 \\wedge p \\in \\bullet t_x}{W(p,t_x)} + \\\\ \\sum _{t_x \\in T_0 \\wedge p \\in R(t_x)}{M_0(p)})$ – from (REF ),(REF ) (REF ) above and the following Each $add(p,q_j,t_j,0) \\in A$ represents $q_j=W(t_j,p)$ for $p \\in t_j \\bullet $ – rule $r\\ref {r:r:add} $ encoding, and definition REF of postset in $PN$ Each $del(p,t_y,q_y,0) \\in A$ represents either $q_y=W(p,t_y)$ for $p \\in \\bullet t_y$ , or $q_y=M_0(p)$ for $p \\in R(t_y)$ – from rule $r\\ref {r:r:del} ,f\\ref {f:r:ptarc} $ encoding and definition REF of preset in $PN$ ; or from rule $r\\ref {r:r:del} ,f\\ref {f:rptarc}$ encoding and definition of reset arc in $PN$ Each $tot\\_incr(p,q2,0) \\in A$ represents $q2=\\sum _{t_x \\in T_0 \\wedge p \\in t_x \\bullet }{W(t_x,p)}$ – aggregate assignment atom semantics in rule $r\\ref {r:totincr}$ Each $tot\\_decr(p,q3,0) \\in A$ represents $q3=\\sum _{t_x \\in T_0 \\wedge p \\in \\bullet t_x}{W(p,t_x)} + $ $\\sum _{t_x \\in T_0 \\wedge p \\in R(t_x)}{M_0(p)}$ – aggregate assignment atom semantics in rule $r\\ref {r:totdecr}$ Then $M_1(p) = q$ – since $holds(p,q,1) \\in A$ encodes $q=M_1(p)$ – from construction Inductive Step: Let $k > 0$ , and $M_k$ is a valid marking in $X$ for $PN$ , show [(1)] $T_k$ is a valid firing set in $M_k$ , and firing $T_k$ in $M_k$ produces marking $M_{k+1}$ .", "We show that $T_k$ is a valid firing set in $M_k$ .", "Let $\\lbrace fires(t_0,k), \\dots , fires(t_x,k) \\rbrace $ be the set of all $fires(\\dots ,k)$ atoms in $A$ , Then for each $fires(t_i,k) \\in A$ $enabled(t_i,k) \\in A$ – from rule $a\\ref {a:fires}$ and supported rule proposition Then $notenabled(t_i,k) \\notin A$ – from rule $e\\ref {e:enabled}$ and supported rule proposition Then $body(e\\ref {e:r:ne:ptarc})$ must not hold in $A$ – from rule $e\\ref {e:r:ne:ptarc}$ and forced atom proposition Then $q \\lnot < n_i \\equiv q \\ge n_i$ in $e\\ref {e:r:ne:ptarc}$ for all $\\lbrace holds(p,q,k), ptarc(p,t_i,n_i,k)\\rbrace \\subseteq A$ – from $e\\ref {e:r:ne:ptarc}$ , forced atom proposition, and the following $holds(p,q,k) \\in A$ represents $q=M_k(p)$ – construction, inductive assumption $ptarc(p,t_i,n_i,k) \\in A$ represents $n_i=W(p,t_i)$ – rule $f\\ref {f:r:ptarc}$ construction; or it represents $n_i=M_k(p)$ – rule $f\\ref {f:rptarc}$ construction; the construction of $f\\ref {f:rptarc}$ ensures that $notenabled(t,0)$ is never true due to the reset arc Then $\\forall p \\in \\bullet t_i, M_k(p) \\ge W(p,t_i)$ – from definition REF of preset $\\bullet t_i$ in PN Then $t_i$ is enabled and can fire in $PN$ , as a result it can belong to $T_k$ – from definition REF of enabled transition And $consumesmore \\notin A$ , since $A$ is an answer set of $\\Pi ^2$ – from rule $a\\ref {a:overc:elim}$ and supported rule proposition Then $\\nexists consumesmore(p,k) \\in A$ – from rule $a\\ref {a:overc:gen}$ and supported rule proposition Then $\\nexists \\lbrace holds(p,q,k), tot\\_decr(p,q1,k) \\rbrace \\subseteq A : q1>q$ in $body(a\\ref {a:overc:place})$ – from $a\\ref {a:overc:place}$ and forced atom proposition Then $\\nexists p : (\\sum _{t_i \\in \\lbrace t_0,\\dots ,t_x\\rbrace , p \\in \\bullet t_i}{W(p,t_i)}+\\sum _{t_i \\in \\lbrace t_0,\\dots ,t_x\\rbrace , p \\in R(t_i)}{M_k(p)}) > M_k(p)$ – from the following $holds(p,q,k)$ represents $q=M_k(p)$ – inductive assumption, given $tot\\_decr(p,q1,k) \\in A$ if $\\lbrace del(p,q1_0,t_0,k), \\dots , del(p,q1_x,t_x,k) \\rbrace \\subseteq A$ , where $q1 = q1_0+\\dots +q1_x$ – from $r\\ref {r:totdecr}$ and forced atom proposition $del(p,q1_i,t_i,k) \\in A$ if $\\lbrace fires(t_i,k), ptarc(p,t_i,q1_i,k) \\rbrace \\subseteq A$ – from $r\\ref {r:r:del} $ and supported rule proposition $del(p,q1_i,t_i,k)$ either represents removal of $q1_i = W(p,t_i)$ tokens from $p \\in \\bullet t_i$ ; or it represents removal of $q1_i = M_k(p)$ tokens from $p \\in R(t_i)$ – from rule $r\\ref {r:r:del} $ , supported rule proposition, and definition REF of transition execution in $PN$ Then the set of transitions in $T_k$ do not conflict – by the definition REF of conflicting transitions And for each $enabled(t_j,k) \\in A$ and $fires(t_j,k) \\notin A$ , $could\\_not\\_have(t_j,k) \\in A$ , since $A$ is an answer set of $\\Pi ^2$ - from rule $a\\ref {a:maxfire:elim}$ and supported rule proposition Then $\\lbrace enabled(t_j,k), holds(s,qq,k), ptarc(s,t_j,q,k), \\\\ tot\\_decr(s,qqq,k) \\rbrace \\subseteq A$ , such that $q > qq - qqq$ and $fires(t_j,k) \\notin A$ - from rule $a\\ref {a:r:maxfire:cnh}$ and supported rule proposition Then for an $s \\in \\bullet t_j \\cup R(t_j)$ , $q > M_k(s) - (\\sum _{t_i \\in T_k, s \\in \\bullet t_i}{W(s,t_i)} + \\sum _{t_i \\in T_k, s \\in R(t_i)}{M_k(s))}$ , where $q=W(s,t_j) \\text{~if~} s \\in \\bullet t_j, \\text{~or~} M_k(s) \\text{~otherwise}$ - from the following: $ptarc(s,t_i,q,k)$ represents $q=W(s,t_i)$ if $(s,t_i) \\in E^-$ or $q=M_k(s)$ if $s \\in R(t_i)$ – from rule $f\\ref {f:r:ptarc},f\\ref {f:rptarc}$ construction $holds(s,qq,k)$ represents $qq=M_k(s)$ – construction $tot\\_decr(s,qqq,k) \\in A$ if $\\lbrace del(s,qqq_0,t_0,k), \\dots , del(s,qqq_x,t_x,k) \\rbrace \\subseteq A$ – from rule $r\\ref {r:totdecr}$ construction and supported rule proposition $del(s,qqq_i,t_i,k) \\in A$ if $\\lbrace fires(t_i,k), ptarc(s,t_i,qqq_i,k) \\rbrace \\subseteq A$ – from rule $r\\ref {r:r:del} $ and supported rule proposition $del(s,qqq_i,t_i,k)$ represents $qqq_i = W(s,t_i) : t_i \\in T_k, (s,t_i) \\in E^-$ , or $qqq_i = M_k(t_i) : t_i \\in T_k, s \\in R(t_i)$ – from rule $f\\ref {f:r:ptarc},f\\ref {f:rptarc}$ construction $tot\\_decr(q,qqq,k)$ represents $\\sum _{t_i \\in T_k, s \\in \\bullet t_i}{W(s,t_i)} + \\\\ \\sum _{t_i \\in T_k, s \\in R(t_i)}{M_k(s)}$ – from (C,D,E) above Then firing $T_k \\cup \\lbrace t_j \\rbrace $ would have required more tokens than are present at its source place $s \\in \\bullet t_j \\cup R(t_j)$ .", "Thus, $T_k$ is a maximal set of transitions that can simultaneously fire.", "And for each reset transition $t_r$ with $enabled(t_r,k) \\in A$ , $fires(t_r,k) \\in A$ , since $A$ is an answer set of $\\Pi ^2$ - from rule $f\\ref {f:c:rptarc:elim}$ and supported rule proposition Then the firing set $T_k$ satisfies the reset transition requirement of definition REF (firing set) Then $\\lbrace t_0, \\dots , t_x\\rbrace = T_k$ – using 1(a),1(b), 1(d) above; and using 1(c) it is a maximal firing set We show that $M_{k+1}$ is produced by firing $T_k$ in $M_k$ .", "Let $holds(p,q,k+1) \\in A$ Then $\\lbrace holds(p,q1,k), tot\\_incr(p,q2,k), tot\\_decr(p,q3,k) \\rbrace \\subseteq A : q=q1+q2-q3$ – from rule $r\\ref {r:nextstate}$ and supported rule proposition $holds(p,q1,k) \\in A$ represents $q1=M_k(p)$ – construction, inductive assumption; and $\\lbrace add(p,q2_0,t_0,k), \\dots , $ $add(p,q2_j,t_j,k)\\rbrace \\subseteq A : q2_0 + \\dots + q2_j = q2$ and $\\lbrace del(p,q3_0,t_0,k), \\dots , $ $del(p,q3_l,t_l,k)\\rbrace \\subseteq A : q3_0 + \\dots + q3_l = q3$ – rules $r\\ref {r:totincr},r\\ref {r:totdecr}$ using supported rule proposition Then $\\lbrace fires(t_0,k), \\dots , $ $fires(t_j,k) \\rbrace \\subseteq A$ and $\\lbrace fires(t_0,k), \\dots , $ $fires(t_l,k) \\rbrace \\\\ \\subseteq A$ – rules $r\\ref {r:r:add},r\\ref {r:r:del}$ and supported rule proposition, respectively Then $\\lbrace fires(t_0,k), \\dots , fires(t_j,k) \\rbrace \\cup \\lbrace fires(t_0,k), \\dots , $ $fires(t_l,k) \\rbrace \\subseteq $ $A = \\lbrace fires(t_0,k), \\dots , $ $ fires(t_x,k) \\rbrace \\subseteq A$ – set union of subsets Then for each $fires(t_x,k) \\in A$ we have $t_x \\in T_k$ – already shown in item REF above Then $q = M_k(p) + \\sum _{t_x \\in T_0 \\wedge p \\in t_x \\bullet }{W(t_x,p)} - (\\sum _{t_x \\in T_k \\wedge p \\in \\bullet t_x}{W(p,t_x)} + \\\\ \\sum _{t_x \\in T_k \\wedge p \\in R(t_x)}{M_k(p)})$ – from (REF ) above and the following Each $add(p,q_j,t_j,k) \\in A$ represents $q_j=W(t_j,p)$ for $p \\in t_j \\bullet $ – rule $r\\ref {r:r:add} $ encoding, and definition REF of transition execution in $PN$ Each $del(p,t_y,q_y,k) \\in A$ represents either $q_y=W(p,t_y)$ for $p \\in \\bullet t_y$ , or $q_y=M_k(p)$ for $p \\in R(t_y)$ – from rule $r\\ref {r:r:del} ,f\\ref {f:r:ptarc} $ encoding and definition REF of transition execution in $PN$ ; or from rule $r\\ref {r:r:del} ,f\\ref {f:rptarc}$ encoding and definition of reset arc in $PN$ Each $tot\\_incr(p,q2,k) \\in A$ represents $q2=\\sum _{t_x \\in T_k \\wedge p \\in t_x \\bullet }{W(t_x,p)}$ – aggregate assignment atom semantics in rule $r\\ref {r:totincr}$ Each $tot\\_decr(p,q3,0) \\in A$ represents $q3=\\sum _{t_x \\in T_k \\wedge p \\in \\bullet t_x}{W(p,t_x)} + $ $\\sum _{t_x \\in T_k \\wedge p \\in R(t_x)}{M_0(p)}$ – aggregate assignment atom semantics in rule $r\\ref {r:totdecr}$ Then, $M_{k+1}(p) = q$ – since $holds(p,q,k+1) \\in A$ encodes $q=M_{k+1}(p)$ – from construction As a result, for any $n > k$ , $T_n$ is a valid firing set w.r.t.", "$M_n$ and its firing produces marking $M_{n+1}$ .", "Conclusion: Since both (REF ) and (REF ) hold, $X=M_0,T_0,M_1,\\dots ,M_k,T_{k+1}$ is an execution sequence of $PN(P,T,E,W,R)$ (w.r.t $M_0$ ) iff there is an answer set $A$ of $\\Pi ^2(PN,M_0,k,ntok)$ such that (REF ) and (REF ) hold.", "Poof of Proposition REF Let $PN=(P,T,E,W,R,I)$ be a Petri Net, $M_0$ be its initial marking and let $\\Pi ^3(PN,M_0,k,ntok)$ be the ASP encoding of $PN$ and $M_0$ over a simulation length $k$ , with maximum $ntok$ tokens on any place node, as defined in section REF .", "Then $X=M_0,T_0,M_1,\\dots ,M_k,T_k,M_{k+1}$ is an execution sequence of $PN$ (w.r.t.", "$M_0$ ) iff there is an answer set $A$ of $\\Pi ^3(PN,M_0,k,ntok)$ such that: $\\lbrace fires(t,ts) : t \\in T_{ts}, 0 \\le ts \\le k\\rbrace = \\lbrace fires(t,ts) : fires(t,ts) \\in A \\rbrace $ $\\begin{split}\\lbrace holds(p,q,ts) &: p \\in P, q = M_{ts}(p), 0 \\le ts \\le k+1 \\rbrace \\\\&= \\lbrace holds(p,q,ts) : holds(p,q,ts) \\in A \\rbrace \\end{split}$ We prove this by showing that: Given an execution sequence $X$ , we create a set $A$ such that it satisfies (REF ) and (REF ) and show that $A$ is an answer set of $\\Pi ^3$ Given an answer set $A$ of $\\Pi ^3$ , we create an execution sequence $X$ such that (REF ) and (REF ) are satisfied.", "First we show (REF ): Given $PN$ and an execution sequence $X$ of $PN$ , we create a set $A$ as a union of the following sets: $A_1=\\lbrace num(n) : 0 \\le n \\le ntok \\rbrace $ $A_2=\\lbrace time(ts) : 0 \\le ts \\le k\\rbrace $ $A_3=\\lbrace place(p) : p \\in P \\rbrace $ $A_4=\\lbrace trans(t) : t \\in T \\rbrace $ $A_5=\\lbrace ptarc(p,t,n,ts) : (p,t) \\in E^-, n=W(p,t), 0 \\le ts \\le k \\rbrace $ , where $E^- \\subseteq E$ $A_6=\\lbrace tparc(t,p,n,ts) : (t,p) \\in E^+, n=W(t,p), 0 \\le ts \\le k \\rbrace $ , where $E^+ \\subseteq E$ $A_7=\\lbrace holds(p,q,0) : p \\in P, q=M_{0}(p) \\rbrace $ $A_8=\\lbrace notenabled(t,ts) : t \\in T, 0 \\le ts \\le k, (\\exists p \\in \\bullet t, M_{ts}(p) < W(p,t)) \\vee (\\exists p \\in I(t), M_{ts}(p) \\ne 0) \\rbrace $ per definition REF (enabled transition) $A_9=\\lbrace enabled(t,ts) : t \\in T, 0 \\le ts \\le k, (\\forall p \\in \\bullet t, W(p,t) \\le M_{ts}(p)) \\wedge (\\forall p \\in I(t), M_{ts}(p) = 0) \\rbrace $ per definition REF (enabled transition) $A_{10}=\\lbrace fires(t,ts) : t \\in T_{ts}, 0 \\le ts \\le k \\rbrace $ per definition REF (firing set), only an enabled transition may fire $A_{11}=\\lbrace add(p,q,t,ts) : t \\in T_{ts}, p \\in t \\bullet , q=W(t,p), 0 \\le ts \\le k \\rbrace $ per definition REF (transition execution) $A_{12}=\\lbrace del(p,q,t,ts) : t \\in T_{ts}, p \\in \\bullet t, q=W(p,t), 0 \\le ts \\le k \\rbrace \\cup \\lbrace del(p,q,t,ts) : t \\in T_{ts}, p \\in R(t), q=M_{ts}(p), 0 \\le ts \\le k \\rbrace $ per definition REF (transition execution) $A_{13}=\\lbrace tot\\_incr(p,q,ts) : p \\in P, q=\\sum _{t \\in T_{ts}, p \\in t \\bullet }{W(t,p)}, 0 \\le ts \\le k \\rbrace $ per definition REF (firing set execution) $A_{14}=\\lbrace tot\\_decr(p,q,ts): p \\in P, q=\\sum _{t \\in T_{ts}, p \\in \\bullet t}{W(p,t)}+\\sum _{t \\in T_{ts}, p \\in R(t) }{M_{ts}(p)}, 0 \\le ts \\le k \\rbrace $ per definition REF (firing set execution) $A_{15}=\\lbrace consumesmore(p,ts) : p \\in P, q=M_{ts}(p), q1=\\sum _{t \\in T_{ts}, p \\in \\bullet t}{W(p,t)} + \\sum _{t \\in T_{ts}, p \\in R(t)}{M_{ts}(p)}, q1 > q, 0 \\le ts \\le k \\rbrace $ per definition REF (conflicting transitions) $A_{16}=\\lbrace consumesmore : \\exists p \\in P : q=M_{ts}(p), q1=\\sum _{t \\in T_{ts}, p \\in \\bullet t}{W(p,t)} + \\sum _{t \\in T_{ts}, p \\in R(t)}(M_{ts}(p)), q1 > q, 0 \\le ts \\le k \\rbrace $ per definition REF (conflicting transitions) $A_{17}=\\lbrace could\\_not\\_have(t,ts) : t \\in T, (\\forall p \\in \\bullet t, W(p,t) \\le M_{ts}(p)), t \\notin T_{ts}, (\\exists p \\in \\bullet t : W(p,t) > M_{ts}(p) - (\\sum _{t^{\\prime } \\in T_{ts}, p \\in \\bullet t^{\\prime }}{W(p,t^{\\prime })} + \\sum _{t^{\\prime } \\in T_{ts}, p \\in R(t^{\\prime })} M_{ts}(p)), 0 \\le ts \\le k \\rbrace $ per the maximal firing set semantics $A_{18}=\\lbrace holds(p,q,ts+1) : p \\in P, q=M_{ts+1}(p), 0 \\le ts < k\\rbrace $ , where $M_{ts+1}(p) = M_{ts}(p) - (\\sum _{\\begin{array}{c}t \\in T_{ts}, p \\in \\bullet t\\end{array}}{W(p,t)} + $ $\\sum _{t \\in T_{ts}, p \\in R(t)}M_{ts}(p)) + \\\\ \\sum _{\\begin{array}{c}t \\in T_{ts}, p \\in t \\bullet \\end{array}}{W(t,p)}$ according to definition REF (firing set execution) $A_{19}=\\lbrace ptarc(p,t,n,ts) : p \\in R(t), n = M_{ts}(p), n > 0, 0 \\le ts \\le k \\rbrace $ $A_{20}=\\lbrace iptarc(p,t,1,ts) : p \\in P, 0 \\le ts < k \\rbrace $ We show that $A$ satisfies (REF ) and (REF ), and $A$ is an answer set of $\\Pi ^3$ .", "$A$ satisfies (REF ) and (REF ) by its construction above.", "We show $A$ is an answer set of $\\Pi ^3$ by splitting.", "We split $lit(\\Pi ^3)$ into a sequence of $7k+9$ sets: $U_0= head(f\\ref {f:place}) \\cup head(f\\ref {f:trans}) \\cup head(f\\ref {f:time}) \\cup head(f\\ref {f:num}) \\cup head(i\\ref {i:holds}) = \\lbrace place(p) : p \\in P\\rbrace \\cup \\lbrace trans(t) : t \\in T\\rbrace \\cup \\lbrace time(0), \\dots , time(k)\\rbrace \\cup \\lbrace num(0), \\dots , num(ntok)\\rbrace \\cup \\lbrace holds(p,q,0) : p \\in P, q=M_0(p) \\rbrace $ $U_{7k+1}=U_{7k+0} \\cup head(f\\ref {f:r:ptarc})^{ts=k} \\cup head(f\\ref {f:r:tparc})^{ts=k} \\cup head(f\\ref {f:rptarc})^{ts=k} \\cup head(f\\ref {f:iptarc})^{ts=k} = U_{7k+0} \\cup \\lbrace ptarc(p,t,n,k) : (p,t) \\in E^-, n=W(p,t) \\rbrace \\cup \\lbrace tparc(t,p,n,k) : (t,p) \\in E^+, n=W(t,p) \\rbrace \\cup \\lbrace ptarc(p,t,n,k) : p \\in R(t), n=M_{k}(p), n > 0 \\rbrace \\cup \\\\ \\lbrace iptarc(p,t,1,k) : p \\in I(t) \\rbrace $ $U_{7k+2}=U_{7k+1} \\cup head(e\\ref {e:r:ne:ptarc} )^{ts=k} \\cup head(e\\ref {e:ne:iptarc})^{ts=k} = U_{7k+1} \\cup \\lbrace notenabled(t,k) : t \\in T \\rbrace $ $U_{7k+3}=U_{7k+2} \\cup head(e\\ref {e:enabled})^{ts=k} = U_{7k+2} \\cup \\lbrace enabled(t,k) : t \\in T \\rbrace $ $U_{7k+4}=U_{7k+3} \\cup head(a\\ref {a:fires})^{ts=k} = U_{7k+3} \\cup \\lbrace fires(t,k) : t \\in T \\rbrace $ $U_{7k+5}=U_{7k+4} \\cup head(r\\ref {r:r:add} )^{ts=k} \\cup head(r\\ref {r:r:del} )^{ts=k} = U_{7k+4} \\cup \\lbrace add(p,q,t,k) : p \\in P, t \\in T, q=W(t,p) \\rbrace \\cup \\lbrace del(p,q,t,k) : p \\in P, t \\in T, q=W(p,t) \\rbrace \\cup \\lbrace del(p,q,t,k) : p \\in P, t \\in T, q=M_{k}(p) \\rbrace $ $U_{7k+6}=U_{7k+5} \\cup head(r\\ref {r:totincr})^{ts=k} \\cup head(r\\ref {r:totdecr})^{ts=k} = U_{7k+5} \\cup \\lbrace tot\\_incr(p,q,k) : p \\in P, 0 \\le q \\le ntok \\rbrace \\cup \\lbrace tot\\_decr(p,q,k) : p \\in P, 0 \\le q \\le ntok \\rbrace $ $U_{7k+7}=U_{7k+6} \\cup head(r\\ref {r:nextstate})^{ts=k} \\cup head(a\\ref {a:overc:place})^{ts=k} \\cup head(a\\ref {a:r:maxfire:cnh})^{ts=k} = $ $U_{7k+6} \\cup \\\\ \\lbrace consumesmore(p,k) : p \\in P\\rbrace \\cup \\lbrace holds(p,q,k+1) : p \\in P, 0 \\le q \\le ntok \\rbrace \\cup \\lbrace could\\_not\\_have(t,k) : t \\in T \\rbrace $ $U_{7k+8}=U_{7k+7} \\cup head(a\\ref {a:overc:gen})^{ts=k} = U_{7k+7} \\cup \\lbrace consumesmore \\rbrace $ where $head(r_i)^{ts=k}$ are head atoms of ground rule $r_i$ in which $ts=k$ .", "We write $A_i^{ts=k} = \\lbrace a(\\dots ,ts) : a(\\dots ,ts) \\in A_i, ts=k \\rbrace $ as short hand for all atoms in $A_i$ with $ts=k$ .", "$U_{\\alpha }, 0 \\le \\alpha \\le 7k+8$ form a splitting sequence, since each $U_i$ is a splitting set of $\\Pi ^1$ , and $\\langle U_{\\alpha }\\rangle _{\\alpha < \\mu }$ is a monotone continuous sequence, where $U_0 \\subseteq U_1 \\dots \\subseteq U_{7k+8}$ and $\\bigcup _{\\alpha < \\mu }{U_{\\alpha }} = lit(\\Pi ^1)$ .", "We compute the answer set of $\\Pi ^3$ using the splitting sets as follows: $bot_{U_0}(\\Pi ^3) = f\\ref {f:place} \\cup f\\ref {f:trans} \\cup f\\ref {f:time} \\cup i\\ref {i:holds} \\cup f\\ref {f:num}$ and $X_0 = A_1 \\cup \\dots \\cup A_4 \\cup A_7$ ($= U_0$ ) is its answer set – using forced atom proposition $eval_{U_0}(bot_{U_1}(\\Pi ^3) \\setminus bot_{U_0}(\\Pi ^3), X_0) = \\lbrace ptarc(p,t,q,0) \\text{:-}.", "\| q=W(p,t) \\rbrace \\cup \\\\ \\lbrace tparc(t,p,q,0) \\text{:-}.", "\| q=W(t,p) \\rbrace \\cup \\lbrace ptarc(p,t,q,0) \\text{:-}.", "\| q=M_0(p) \\rbrace \\cup \\\\ \\lbrace iptarc(p,t,1,0) \\text{:-}.", "\\rbrace $ .", "Its answer set $X_1=A_5^{ts=0} \\cup A_6^{ts=0} \\cup A_{19}^{ts=0} \\cup A_{20}^{ts=0}$ – using forced atom proposition and construction of $A_5, A_6, A_{19}, A_{20}$ .", "$eval_{U_1}(bot_{U_2}(\\Pi ^3) \\setminus bot_{U_1}(\\Pi ^3), X_0 \\cup X_1) = \\lbrace notenabled(t,0) \\text{:-} .", "\| (\\lbrace trans(t), \\\\ ptarc(p,t,n,0), holds(p,q,0) \\rbrace \\subseteq X_0 \\cup X_1, \\text{~where~} q < n) \\text{~or~} \\lbrace notenabled(t,0) \\text{:-} .", "\| \\\\ (\\lbrace trans(t), iptarc(p,t,n2,0), holds(p,q,0) \\rbrace \\subseteq X_0 \\cup X_1, \\text{~where~} q \\ge n2 \\rbrace \\rbrace $ .", "Its answer set $X_2=A_8^{ts=0}$ – using forced atom proposition and construction of $A_8$ .", "where, $q=M_0(p)$ , and $n=W(p,t)$ for an arc $(p,t) \\in E^-$ – by construction $i\\ref {i:holds}$ and $f\\ref {f:r:ptarc}$ in $\\Pi ^3$ , and in an arc $(p,t) \\in E^-$ , $p \\in \\bullet t$ (by definition REF of preset) $n2=1$ – by construction of $iptarc$ predicates in $\\Pi ^3$ , meaning $q \\ge n2 \\equiv q \\ge 1 \\equiv q > 0$ , thus, $notenabled(t,0) \\in X_1$ represents $(\\exists p \\in \\bullet t : M_0(p) < W(p,t)) \\vee (\\exists p \\in I(t) : M_0(p) > 0)$ .", "$eval_{U_2}(bot_{U_3}(\\Pi ^3) \\setminus bot_{U_2}(\\Pi ^3), X_0 \\cup \\dots \\cup X_2) = \\lbrace enabled(t,0) \\text{:-}.", "\| trans(t) \\in X_0 \\cup \\dots \\cup X_2, notenabled(t,0) \\notin X_0 \\cup \\dots \\cup X_2 \\rbrace $ .", "Its answer set is $X_3 = A_9^{ts=0}$ – using forced atom proposition and construction of $A_9$ .", "since an $enabled(t,0) \\in X_3$ if $\\nexists ~notenabled(t,0) \\in X_0 \\cup \\dots \\cup X_2$ ; which is equivalent to $(\\nexists p \\in \\bullet t : M_0(p) < W(p,t)) \\wedge (\\nexists p \\in I(t) : M_0(p) > 0) \\equiv (\\forall p \\in \\bullet t: M_0(p) \\ge W(p,t)) \\wedge (\\forall p \\in I(t) : M_0(p) = 0)$ .", "$eval_{U_3}(bot_{U_4}(\\Pi ^3) \\setminus bot_{U_3}(\\Pi ^3), X_0 \\cup \\dots \\cup X_3) = \\lbrace \\lbrace fires(t,0)\\rbrace \\text{:-}.", "\| enabled(t,0) \\\\ \\text{~holds in~} X_0 \\cup \\dots \\cup X_3 \\rbrace $ .", "It has multiple answer sets $X_{4.1}, \\dots , X_{4.n}$ , corresponding to elements of power set of $fires(t,0)$ atoms in $eval_{U_3}(...)$ – using supported rule proposition.", "Since we are showing that the union of answer sets of $\\Pi ^3$ determined using splitting is equal to $A$ , we only consider the set that matches the $fires(t,0)$ elements in $A$ and call it $X_4$ , ignoring the rest.", "Thus, $X_4 = A_{10}^{ts=0}$ , representing $T_0$ .", "in addition, for every $t$ such that $enabled(t,0) \\in X_0 \\cup \\dots \\cup X_3, R(t) \\ne \\emptyset $ ; $fires(t,0) \\in X_4$ – per definition REF (firing set); requiring that a reset transition is fired when enabled thus, the firing set $T_0$ will not be eliminated by the constraint $f\\ref {f:c:rptarc:elim}$ $eval_{U_4}(bot_{U_5}(\\Pi ^3) \\setminus bot_{U_4}(\\Pi ^3), X_0 \\cup \\dots \\cup X_4) = \\lbrace add(p,n,t,0) \\text{:-}.", "\| \\lbrace fires(t,0), \\\\ tparc(t,p,n,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_4 \\rbrace \\cup \\lbrace del(p,n,t,0) \\text{:-}.", "\| \\lbrace fires(t,0), ptarc(p,t,n,0) \\rbrace \\\\ \\subseteq X_0 \\cup \\dots \\cup X_4 \\rbrace $ .", "It's answer set is $X_5 = A_{11}^{ts=0} \\cup A_{12}^{ts=0}$ – using forced atom proposition and definitions of $A_{11}$ and $A_{12}$ .", "where, each $add$ atom is equivalent to $n=W(t,p) : p \\in t \\bullet $ , and each $del$ atom is equivalent to $n=W(p,t) : p \\in \\bullet t$ ; or $n=M_k(p) : p \\in R(t)$ , representing the effect of transitions in $T_0$ .", "$eval_{U_5}(bot_{U_6}(\\Pi ^3) \\setminus bot_{U_5}(\\Pi ^3), X_0 \\cup \\dots \\cup X_5) = \\lbrace tot\\_incr(p,qq,0) \\text{:-}.", "\| $ $qq=\\sum _{add(p,q,t,0) \\in X_0 \\cup \\dots \\cup X_5}{q} \\rbrace \\cup \\lbrace tot\\_decr(p,qq,0) \\text{:-}.", "\| qq=\\sum _{del(p,q,t,0) \\in X_0 \\cup \\dots \\cup X_5}{q} \\rbrace $ .", "It's answer set is $X_6 = A_{13}^{ts=0} \\cup A_{14}^{ts=0}$ – using forced atom proposition and definitions of $A_{13}$ and $A_{14}$ .", "where, each $tot\\_incr(p,qq,0)$ , $qq=\\sum _{add(p,q,t,0) \\in X_0 \\cup \\dots X_5}{q}$ $\\equiv qq=\\sum _{t \\in X_4, p \\in t \\bullet }{W(p,t)}$ , and each $tot\\_decr(p,qq,0)$ , $qq=\\sum _{del(p,q,t,0) \\in X_0 \\cup \\dots X_5}{q}$ $\\equiv qq=\\sum _{t \\in X_4, p \\in \\bullet t}{W(t,p)} + \\sum _{t \\in X_4, p \\in R(t)}{M_{k}(p)}$ , represent the net effect of transitions in $T_0$ .", "$eval_{U_6}(bot_{U_7}(\\Pi ^3) \\setminus bot_{U_6}(\\Pi ^3), X_0 \\cup \\dots \\cup X_6) = \\lbrace consumesmore(p,0) \\text{:-}.", "\| \\\\ \\lbrace holds(p,q,0), tot\\_decr(p,q1,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_6, q1 > q \\rbrace \\cup \\lbrace holds(p,q,1) \\text{:-}., \| \\\\ \\lbrace holds(p,q1,0), tot\\_incr(p,q2,0), tot\\_decr(p,q3,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_6, q=q1+q2-q3 \\rbrace \\cup \\lbrace could\\_not\\_have(t,0) \\text{:-}.", "\| \\lbrace enabled(t,0), ptarc(s,t,q), holds(s,qq,0), \\\\ tot\\_decr(s,qqq,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_6, fires(t,0) \\notin (X_0 \\cup \\dots \\cup X_6), q > qq-qqq \\rbrace $ .", "It's answer set is $X_7 = A_{15}^{ts=0} \\cup A_{17}^{ts=0} \\cup A_{18}^{ts=0}$ – using forced atom proposition and definitions of $A_{15}, A_{17}, A_{18}, A_9$ .", "where, $consumesmore(p,0)$ represents $\\exists p : q=M_0(p), q1= \\\\ \\sum _{t \\in T_0, p \\in \\bullet t}{W(p,t)}+\\sum _{t \\in T_0, p \\in R(t)}{M_k(p)}, q1 > q$ , indicating place $p$ will be over consumed if $T_0$ is fired, as defined in definition REF (conflicting transitions) $holds(p,q,1)$ represents $q=M_1(p)$ – by construction of $\\Pi ^3$ and $could\\_not\\_have(t,0)$ represents enabled transition $t \\in T_0$ that could not fire due to insufficient tokens $X_7$ does not contain $could\\_not\\_have(t,0)$ , when $enabled(t,0) \\in X_0 \\cup \\dots \\cup X_5$ and $fires(t,0) \\notin X_0 \\cup \\dots \\cup X_6$ due to construction of $A$ , encoding of $a\\ref {a:r:maxfire:cnh}$ and its body atoms.", "As a result it is not eliminated by the constraint $a\\ref {a:maxfire:elim}$ $ \\vdots $ $eval_{U_{7k+0}}(bot_{U_{7k+1}}(\\Pi ^3) \\setminus bot_{U_{7k+0}}(\\Pi ^3), X_0 \\cup \\dots \\cup X_{7k+0}) = \\lbrace ptarc(p,t,q,k) \\text{:-}.", "\| q=W(p,t) \\rbrace \\cup \\lbrace tparc(t,p,q,k) \\text{:-}.", "\| q=W(t,p) \\rbrace \\cup \\lbrace ptarc(p,t,q,k) \\text{:-}.", "\| q=M_k(p) \\rbrace \\cup \\lbrace iptarc(p,t,1,k) \\text{:-}.", "\\rbrace $ .", "Its answer set $X_{7k+1}=A_5^{ts=k} \\cup A_6^{ts=k} \\cup A_{19}^{ts=k} \\cup A_{20}^{ts=k}$ – using forced atom proposition and construction of $A_5, A_6, A_{19}, A_{20}$ .", "$eval_{U_{7k+1}}(bot_{U_{7k+2}}(\\Pi ^3) \\setminus bot_{U_{7k+1}}(\\Pi ^3), X_0 \\cup \\dots \\cup X_{7k+1}) = \\lbrace notenabled(t,k) \\text{:-} .", "\| \\\\ (\\lbrace trans(t), ptarc(p,t,n,k), holds(p,q,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{7k+1}, \\text{~where~} q < n) \\rbrace \\text{~or~} $ $\\lbrace notenabled(t,k) \\text{:-} .", "\| (\\lbrace trans(t), iptarc(p,t,n2,k), holds(p,q,k) \\rbrace \\subseteq \\\\ X_0 \\cup \\dots \\cup X_{7k+1}, \\text{~where~} q \\ge n2 \\rbrace \\rbrace $ .", "Its answer set $X_{7k+2}=A_8^{ts=k}$ – using forced atom proposition and construction of $A_8$ .", "where, $q=M_k(p)$ , and $n=W(p,t)$ for an arc $(p,t) \\in E^-$ – by construction of $holds$ and $ptarc$ predicates in $\\Pi ^3$ , and in an arc $(p,t) \\in E^-$ , $p \\in \\bullet t$ (by definition REF of preset) $n2=1$ – by construction of $iptarc$ predicates in $\\Pi ^3$ , meaning $q \\ge n2 \\equiv q \\ge 1 \\equiv q > 0$ , thus, $notenabled(t,k) \\in X_{7k+1}$ represents $(\\exists p \\in \\bullet t : M_k(p) < W(p,t)) \\vee (\\exists p \\in I(t) : M_k(p) > k)$ .", "$eval_{U_{7k+2}}(bot_{U_{7k+3}}(\\Pi ^3) \\setminus bot_{U_{7k+2}}(\\Pi ^3), X_0 \\cup \\dots \\cup X_{7k+2}) = \\lbrace enabled(t,k) \\text{:-}.", "\| \\\\ trans(t) \\in X_0 \\cup \\dots \\cup X_{7k+2} \\wedge notenabled(t,k) \\notin X_0 \\cup \\dots \\cup X_{7k+2} \\rbrace $ .", "Its answer set is $X_{7k+3} = A_9^{ts=k}$ – using forced atom proposition and construction of $A_9$ .", "since an $enabled(t,k) \\in X_{7k+3}$ if $\\nexists ~notenabled(t,k) \\in X_0 \\cup \\dots \\cup X_{7k+2}$ ; which is equivalent to $(\\nexists p \\in \\bullet t : M_k(p) < W(p,t)) \\wedge (\\nexists p \\in I(t) : M_k(p) > k) \\equiv (\\forall p \\in \\bullet t: M_k(p) \\ge W(p,t)) \\wedge (\\forall p \\in I(t) : M_k(p) = k)$ .", "$eval_{U_{7k+3}}(bot_{U_{7k+4}}(\\Pi ^3) \\setminus bot_{U_{7k+3}}(\\Pi ^3), X_0 \\cup \\dots \\cup X_{7k+3}) = \\lbrace \\lbrace fires(t,k)\\rbrace \\text{:-}.", "\| \\\\ enabled(t,k) \\\\ \\text{~holds in~} X_0 \\cup \\dots \\cup X_{7k+3} \\rbrace $ .", "It has multiple answer sets $X_{7k+4.1}, \\dots , X_{1k+4.n}$ , corresponding to elements of power set of $fires(t,k)$ atoms in $eval_{U_{7k+3}}(...)$ – using supported rule proposition.", "Since we are showing that the union of answer sets of $\\Pi ^3$ determined using splitting is equal to $A$ , we only consider the set that matches the $fires(t,k)$ elements in $A$ and call it $X_{7k+4}$ , ignoring the rest.", "Thus, $X_{7k+4} = A_{10}^{ts=k}$ , representing $T_k$ .", "in addition, for every $t$ such that $enabled(t,k) \\in X_0 \\cup \\dots \\cup X_{7k+3}, R(t) \\ne \\emptyset $ ; $fires(t,k) \\in X_{7k+4}$ – per definition REF (firing set); requiring that a reset transition is fired when enabled thus, the firing set $T_k$ will not be eliminated by the constraint $f\\ref {f:c:rptarc:elim}$ $eval_{U_{7k+4}}(bot_{U_{7k+5}}(\\Pi ^3) \\setminus bot_{U_{7k+4}}(\\Pi ^3), X_0 \\cup \\dots \\cup X_{7k+4}) = \\lbrace add(p,n,t,k) \\text{:-}.", "\| \\\\ \\lbrace fires(t,k), tparc(t,p,n,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{7k+4} \\rbrace \\cup \\lbrace del(p,n,t,k) \\text{:-}.", "\| \\lbrace fires(t,k), \\\\ ptarc(p,t,n,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{7k+4} \\rbrace $ .", "It's answer set is $X_{7k+5} = A_{11}^{ts=k} \\cup A_{12}^{ts=k}$ – using forced atom proposition and definitions of $A_{11}$ and $A_{12}$ .", "where, each $add$ atom is equivalent to $n=W(t,p) : p \\in t \\bullet $ , and each $del$ atom is equivalent to $n=W(p,t) : p \\in \\bullet t$ ; or $n=M_k(p) : p \\in R(t)$ , representing the effect of transitions in $T_k$ $eval_{U_{7k+5}}(bot_{U_{7k+6}}(\\Pi ^3) \\setminus bot_{U_{7k+5}}(\\Pi ^3), X_0 \\cup \\dots \\cup X_{7k+5}) = \\\\ \\lbrace tot\\_incr(p,qq,k) \\text{:-}.", "\| $ $qq=\\sum _{add(p,q,t,k) \\in X_0 \\cup \\dots \\cup X_{7k+5}}{q} \\rbrace \\cup $ $\\lbrace tot\\_decr(p,qq,k) \\text{:-}.", "\| $ $qq=\\sum _{del(p,q,t,k) \\in X_0 \\cup \\dots \\cup X_{7k+5}}{q} \\rbrace $ .", "It's answer set is $X_{7k+6} = A_{13}^{ts=k} \\cup A_{14}^{ts=k}$ – using forced atom proposition and definitions of $A_{13}$ and $A_{14}$ .", "where, each $tot\\_incr(p,qq,k)$ , $qq=\\sum _{add(p,q,t,k) \\in X_0 \\cup \\dots X_{7k+5}}{q}$ $\\equiv qq=\\sum _{t \\in X_{7k+4}, p \\in t \\bullet }{W(p,t)}$ , and each $tot\\_decr(p,qq,k)$ , $qq=\\sum _{del(p,q,t,k) \\in X_0 \\cup \\dots X_{7k+5}}{q}$ $\\equiv qq=\\sum _{t \\in X_{7k+4}, p \\in \\bullet t}{W(t,p)} + \\sum _{t \\in X_{7k+4}, p \\in R(t)}{M_{k}(p)}$ , representing the net effect of transitions in $T_k$ $eval_{U_{7k+6}}(bot_{U_{7k+7}}(\\Pi ^3) \\setminus bot_{U_{7k+6}}(\\Pi ^3), X_0 \\cup \\dots \\cup X_{7k+6}) = \\lbrace consumesmore(p,k) \\text{:-}.", "\| \\\\ \\lbrace holds(p,q,k), tot\\_decr(p,q1,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{7k+6}, q1 > q \\rbrace \\cup \\lbrace holds(p,q,1) \\text{:-}., \| \\\\ \\lbrace holds(p,q1,k), tot\\_incr(p,q2,k), tot\\_decr(p,q3,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{7k+6}, q=q1+q2-q3 \\rbrace \\cup \\lbrace could\\_not\\_have(t,k) \\text{:-}.", "\| \\lbrace enabled(t,k), ptarc(s,t,q), holds(s,qq,k), \\\\ tot\\_decr(s,qqq,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{7k+6}, fires(t,k) \\notin (X_0 \\cup \\dots \\cup X_{7k+6}), q > qq-qqq \\rbrace $ .", "It's answer set is $X_{7k+7} = A_{15}^{ts=k} \\cup A_{17}^{ts=k} \\cup A_{18}^{ts=k}$ – using forced atom proposition and definitions of $A_{15}, A_{17}, A_{18}, A_9$ .", "where, $consumesmore(p,k)$ represents $\\exists p : q=M_k(p), \\\\ q1=\\sum _{t \\in T_k, p \\in \\bullet t}{W(p,t)}+\\sum _{t \\in T_k, p \\in R(t)}{M_k(p)}, q1 > q$ , indicating place $p$ that will be over consumed if $T_k$ is fired, as defined in definition REF (conflicting transitions), $holds(p,q,k+1)$ represents $q=M_{k+1}(p)$ – by construction of $\\Pi ^3$ and $could\\_not\\_have(t,k)$ represents enabled transition $t$ in $T_k$ that could not be fired due to insufficient tokens $X_{7k+7}$ does not contain $could\\_not\\_have(t,k)$ , when $enabled(t,k) \\in X_0 \\cup \\dots \\cup X_{7k+6}$ and $fires(t,k) \\notin X_0 \\cup \\dots \\cup X_{7k+6}$ due to construction of $A$ , encoding of $a\\ref {a:r:maxfire:cnh}$ and its body atoms.", "As a result it is not eliminated by the constraint $a\\ref {a:maxfire:elim}$ $eval_{U_{7k+7}}(bot_{U_{7k+8}}(\\Pi ^3) \\setminus bot_{U_{7k+7}}(\\Pi ^3), X_0 \\cup \\dots \\cup X_{7k+7}) = \\lbrace consumesmore \\text{:-}.", "\| \\\\ \\lbrace consumesmore(p,0),\\dots ,$ $consumesmore(p,k) \\rbrace \\cap (X_0 \\cup \\dots \\cup X_{7k+7}) \\ne \\emptyset \\rbrace $ .", "It's answer set is $X_{7k+8} = A_{16}$ – using forced atom proposition $X_{7k+8}$ will be empty since none of $consumesmore(p,0),\\dots , \\\\ consumesmore(p,k)$ hold in $X_0 \\cup \\dots \\cup X_{7k+7}$ due to the construction of $A$ , encoding of $a\\ref {a:overc:place}$ and its body atoms.", "As a result, it is not eliminated by the constraint $a\\ref {a:overc:elim}$ The set $X = X_0 \\cup \\dots \\cup X_{7k+8}$ is the answer set of $\\Pi ^3$ by the splitting sequence theorem REF .", "Each $X_i, 0 \\le i \\le 7k+8$ matches a distinct portion of $A$ , and $X = A$ , thus $A$ is an answer set of $\\Pi ^3$ .", "Next we show (REF ): Given $\\Pi ^3$ be the encoding of a Petri Net $PN(P,T,E,W,R,I)$ with initial marking $M_0$ , and $A$ be an answer set of $\\Pi ^3$ that satisfies (REF ) and (REF ), then we can construct $X=M_0,T_0,\\dots ,M_k,T_k,M_{k+1}$ from $A$ , such that it is an execution sequence of $PN$ .", "We construct the $X$ as follows: $M_i = (M_i(p_0), \\dots , M_i(p_n))$ , where $\\lbrace holds(p_0,M_i(p_0),i), \\dots holds(p_n,M_i(p_n),i) \\rbrace \\\\ \\subseteq A$ , for $0 \\le i \\le k+1$ $T_i = \\lbrace t : fires(t,i) \\in A\\rbrace $ , for $0 \\le i \\le k$ and show that $X$ is indeed an execution sequence of $PN$ .", "We show this by induction over $k$ (i.e.", "given $M_k$ , $T_k$ is a valid firing set and its firing produces marking $M_{k+1}$ ).", "Base case: Let $k=0$ , and $M_0$ is a valid marking in $X$ for $PN$ , show [(1)] $T_0$ is a valid firing set for $M_0$ , and $M_1$ is $T_0$ 's target marking w.r.t.", "$M_0$ .", "We show $T_0$ is a valid firing set for $M_0$ .", "Let $\\lbrace fires(t_0,0), \\dots , fires(t_x,0) \\rbrace $ be the set of all $fires(\\dots ,0)$ atoms in $A$ , Then for each $fires(t_i,0) \\in A$ $enabled(t_i,0) \\in A$ – from rule $a\\ref {a:fires}$ and supported rule proposition Then $notenabled(t_i,0) \\notin A$ – from rule $e\\ref {e:enabled}$ and supported rule proposition Then $body(e\\ref {e:r:ne:ptarc} )$ must not hold in $A$ and $body(e\\ref {e:ne:iptarc} )$ must not hold in $A$ – from rules $e\\ref {e:r:ne:ptarc} ,e\\ref {e:ne:iptarc} $ and forced atom proposition Then $q \\lnot < n_i \\equiv q \\ge n_i$ in $e\\ref {e:r:ne:ptarc}$ for all $\\lbrace holds(p,q,0), ptarc(p,t_i,n_i,0)\\rbrace \\subseteq A$ – from $e\\ref {e:r:ne:ptarc}$ , forced atom proposition, and given facts ($holds(p,q,0) \\in A, ptarc(p,t_i,n_i,0) \\in A$ ) And $q \\lnot \\ge n_i \\equiv q < n_i$ in $e\\ref {e:ne:iptarc} $ for all $\\lbrace holds(p,q,0), iptarc(p,t_i,n_i,0) \\rbrace \\subseteq A, n_i=1$ ; $q > n_i \\equiv q = 0$ – from $e\\ref {e:ne:iptarc} $ , forced atom proposition, given facts ($holds(p,q,0) \\in A, iptarc(p,t_i,1,0) \\in A$ ), and $q$ is a positive integer Then $(\\forall p \\in \\bullet t_i, M_0(p) \\ge W(p,t_i)) \\wedge (\\forall p \\in I(t_i), M_0(p) = 0)$ – from $holds(p,q,0) \\in A$ represents $q=M_0(p)$ – rule $i\\ref {i:holds}$ construction $ptarc(p,t_i,n_i,0) \\in A$ represents $n_i=W(p,t_i)$ – rule $f\\ref {f:r:ptarc}$ construction; or it represents $n_i = M_0(p)$ – rule $f\\ref {f:rptarc}$ construction; the construction of $f\\ref {f:rptarc}$ ensures that $notenabled(t,0)$ is never true due to the reset arc definition REF of preset $\\bullet t_i$ in PN Then $t_i$ is enabled and can fire in $PN$ , as a result it can belong to $T_0$ – from definition REF of enabled transition And $consumesmore \\notin A$ , since $A$ is an answer set of $\\Pi ^3$ – from rule $a\\ref {a:overc:elim}$ and supported rule proposition Then $\\nexists consumesmore(p,0) \\in A$ – from rule $a\\ref {a:overc:gen}$ and supported rule proposition Then $\\nexists \\lbrace holds(p,q,0), tot\\_decr(p,q1,0) \\rbrace \\subseteq A : q1>q$ in $body(a\\ref {a:overc:place})$ – from $a\\ref {a:overc:place}$ and forced atom proposition Then $\\nexists p : (\\sum _{t_i \\in \\lbrace t_0,\\dots ,t_x\\rbrace , p \\in \\bullet t_i}{W(p,t_i)}+\\sum _{t_i \\in \\lbrace t_0,\\dots ,t_x\\rbrace , p \\in R(t_i)}{M_0(p)}) > M_0(p)$ – from the following $holds(p,q,0)$ represents $q=M_0(p)$ – from rule $i\\ref {i:holds}$ encoding, given $tot\\_decr(p,q1,0) \\in A$ if $\\lbrace del(p,q1_0,t_0,0), \\dots , del(p,q1_x,t_x,0) \\rbrace \\subseteq A$ , where $q1 = q1_0+\\dots +q1_x$ – from $r\\ref {r:totdecr}$ and forced atom proposition $del(p,q1_i,t_i,0) \\in A$ if $\\lbrace fires(t_i,0), ptarc(p,t_i,q1_i,0) \\rbrace \\subseteq A$ – from $r\\ref {r:r:del} $ and supported rule proposition $del(p,q1_i,t_i,0)$ either represents removal of $q1_i = W(p,t_i)$ tokens from $p \\in \\bullet t_i$ ; or it represents removal of $q1_i = M_0(p)$ tokens from $p \\in R(t_i)$ – from rule $r\\ref {r:r:del} $ , supported rule proposition, and definition REF of transition execution in $PN$ Then the set of transitions in $T_0$ do not conflict – by the definition REF of conflicting transitions And for each $enabled(t_j,0) \\in A$ and $fires(t_j,0) \\notin A$ , $could\\_not\\_have(t_j,0) \\in A$ , since $A$ is an answer set of $\\Pi ^3$ - from rule $a\\ref {a:maxfire:elim}$ and supported rule proposition Then $\\lbrace enabled(t_j,0), holds(s,qq,0), ptarc(s,t_j,q,0), \\\\ tot\\_decr(s,qqq,0) \\rbrace \\subseteq A$ , such that $q > qq - qqq$ and $fires(t_j,0) \\notin A$ - from rule $a\\ref {a:r:maxfire:cnh}$ and supported rule proposition Then for an $s \\in \\bullet t_j \\cup R(t_j)$ , $q > M_0(s) - (\\sum _{t_i \\in T_0, s \\in \\bullet t_i}{W(s,t_i)} + \\sum _{t_i \\in T_0, s \\in R(t_i)}{M_0(s))}$ , where $q=W(s,t_j) \\text{~if~} s \\in \\bullet t_j, \\text{~or~} M_0(s) \\text{~otherwise}$ - from the following: $ptarc(s,t_i,q,0)$ represents $q=W(s,t_i)$ if $(s,t_i) \\in E^-$ or $q=M_0(s)$ if $s \\in R(t_i)$ – from rule $f\\ref {f:r:ptarc},f\\ref {f:rptarc}$ construction $holds(s,qq,0)$ represents $qq=M_0(s)$ – from $i\\ref {i:holds}$ construction $tot\\_decr(s,qqq,0) \\in A$ if $\\lbrace del(s,qqq_0,t_0,0), \\dots , del(s,qqq_x,t_x,0) \\rbrace \\subseteq A$ – from rule $r\\ref {r:totdecr}$ construction and supported rule proposition $del(s,qqq_i,t_i,0) \\in A$ if $\\lbrace fires(t_i,0), ptarc(s,t_i,qqq_i,0) \\rbrace \\subseteq A$ – from rule $r\\ref {r:r:del} $ and supported rule proposition $del(s,qqq_i,t_i,0)$ represents $qqq_i = W(s,t_i) : t_i \\in T_0, (s,t_i) \\in E^-$ , or $qqq_i = M_0(t_i) : t_i \\in T_0, s \\in R(t_i)$ – from rule $f\\ref {f:r:ptarc},f\\ref {f:rptarc}$ construction $tot\\_decr(q,qqq,0)$ represents $\\sum _{t_i \\in T_0, s \\in \\bullet t_i}{W(s,t_i)} + \\\\ \\sum _{t_i \\in T_0, s \\in R(t_i)}{M_0(s)}$ – from (C,D,E) above Then firing $T_0 \\cup \\lbrace t_j \\rbrace $ would have required more tokens than are present at its source place $s \\in \\bullet t_j \\cup R(t_j)$ .", "Thus, $T_0$ is a maximal set of transitions that can simultaneously fire.", "And for each reset transition $t_r$ with $enabled(t_r,0) \\in A$ , $fires(t_r,0) \\in A$ , since $A$ is an answer set of $\\Pi ^2$ - from rule $f\\ref {f:c:rptarc:elim}$ and supported rule proposition Then, the firing set $T_0$ satisfies the reset-transition requirement of definition REF (firing set) Then $\\lbrace t_0, \\dots , t_x\\rbrace = T_0$ – using 1(a),1(b),1(d) above; and using 1(c) it is a maximal firing set We show $M_1$ is produced by firing $T_0$ in $M_0$ .", "Let $holds(p,q,1) \\in A$ Then $\\lbrace holds(p,q1,0), tot\\_incr(p,q2,0), tot\\_decr(p,q3,0) \\rbrace \\subseteq A : q=q1+q2-q3$ – from rule $r\\ref {r:nextstate}$ and supported rule proposition Then $holds(p,q1,0) \\in A$ represents $q1=M_0(p)$ – given, rule $i\\ref {i:holds}$ construction; and $\\lbrace add(p,q2_0,t_0,0), \\dots , $ $add(p,q2_j,t_j,0)\\rbrace \\subseteq A : q2_0 + \\dots + q2_j = q2$ and $\\lbrace del(p,q3_0,t_0,0), \\dots , $ $del(p,q3_l,t_l,0)\\rbrace \\subseteq A : q3_0 + \\dots + q3_l = q3$ – rules $r\\ref {r:totincr},r\\ref {r:totdecr}$ and supported rule proposition, respectively Then $\\lbrace fires(t_0,0), \\dots , fires(t_j,0) \\rbrace \\subseteq A$ and $\\lbrace fires(t_0,0), \\dots , fires(t_l,0) \\rbrace \\\\ \\subseteq A$ – rules $r\\ref {r:r:add},r\\ref {r:r:del}$ and supported rule proposition, respectively Then $\\lbrace fires(t_0,0), \\dots , $ $fires(t_j,0) \\rbrace \\cup \\lbrace fires(t_0,0), \\dots , $ $fires(t_l,0) \\rbrace \\subseteq A = \\lbrace fires(t_0,0), \\dots , $ $fires(t_x,0) \\rbrace \\subseteq A$ – set union of subsets Then for each $fires(t_x,0) \\in A$ we have $t_x \\in T_0$ – already shown in item REF above Then $q = M_0(p) + \\sum _{t_x \\in T_0 \\wedge p \\in t_x \\bullet }{W(t_x,p)} - (\\sum _{t_x \\in T_0 \\wedge p \\in \\bullet t_x}{W(p,t_x)} + \\\\ \\sum _{t_x \\in T_0 \\wedge p \\in R(t_x)}{M_0(p)})$ – from (REF ) above and the following Each $add(p,q_j,t_j,0) \\in A$ represents $q_j=W(t_j,p)$ for $p \\in t_j \\bullet $ – rule $r\\ref {r:r:add} $ encoding, and definition REF of transition execution in $PN$ Each $del(p,t_y,q_y,0) \\in A$ represents either $q_y=W(p,t_y)$ for $p \\in \\bullet t_y$ , or $q_y=M_0(p)$ for $p \\in R(t_y)$ – from rule $r\\ref {r:r:del} ,f\\ref {f:r:ptarc} $ encoding and definition REF of transition execution in $PN$ ; or from rule $r\\ref {r:r:del} ,f\\ref {f:rptarc}$ encoding and definition of reset arc in $PN$ Each $tot\\_incr(p,q2,0) \\in A$ represents $q2=\\sum _{t_x \\in T_0 \\wedge p \\in t_x \\bullet }{W(t_x,p)}$ – aggregate assignment atom semantics in rule $r\\ref {r:totincr}$ Each $tot\\_decr(p,q3,0) \\in A$ represents $q3=\\sum _{t_x \\in T_0 \\wedge p \\in \\bullet t_x}{W(p,t_x)} + $ $\\sum _{t_x \\in T_0 \\wedge p \\in R(t_x)}{M_0(p)}$ – aggregate assignment atom semantics in rule $r\\ref {r:totdecr}$ Then, $M_1(p) = q$ – since $holds(p,q,1) \\in A$ encodes $q=M_1(p)$ – from construction Inductive Step: Let $k > 0$ , and $M_k$ is a valid marking in $X$ for $PN$ , show [(1)] $T_k$ is a valid firing set for $M_k$ , and firing $T_k$ in $M_k$ produces marking $M_{k+1}$ .", "We show that $T_k$ is a valid firing set in $M_k$ .", "Let $\\lbrace fires(t_0,k), \\dots , fires(t_x,k) \\rbrace $ be the set of all $fires(\\dots ,k)$ atoms in $A$ , Then for each $fires(t_i,k) \\in A$ $enabled(t_i,k) \\in A$ – from rule $a\\ref {a:fires}$ and supported rule proposition Then $notenabled(t_i,k) \\notin A$ – from rule $e\\ref {e:enabled}$ and supported rule proposition Then $body(e\\ref {e:r:ne:ptarc} )$ must not hold in $A$ and $body(e\\ref {e:ne:iptarc} )$ must not hold in $A$ – from rule $e\\ref {e:r:ne:ptarc} ,e\\ref {e:ne:iptarc} $ and forced atom proposition Then $q \\lnot < n_i \\equiv q \\ge n_i$ in $e\\ref {e:r:ne:ptarc}$ for all $\\lbrace holds(p,q,k), ptarc(p,t_i,n_i,k)\\rbrace \\subseteq A$ – from $e\\ref {e:r:ne:ptarc}$ , forced atom proposition, and given facts ($holds(p,q,k) \\in A, ptarc(p,t_i,n_i,k) \\in A$ ) And $q \\lnot \\ge n_i \\equiv q < n_i$ in $e\\ref {e:ne:iptarc} $ for all $\\lbrace holds(p,q,k), iptarc(p,t_i,n_i,k) \\rbrace \\subseteq A, n_i=1; q > n_i \\equiv q = 0$ – from $e\\ref {e:ne:iptarc} $ , forced atom proposition, given facts $(holds(p,q,k) \\in A, iptarc(p,t_i,1,k) \\in A)$ , and $q$ is a positive integer Then $(\\forall p \\in \\bullet t_i, M_k(p) \\ge W(p,t_i)) \\wedge (\\forall p \\in I(t_i), M_k(p) = 0)$ – from $holds(p,q,k) \\in A$ represents $q=M_k(p)$ – inductive assumption, given $ptarc(p,t_i,n_i,k) \\in A$ represents $n_i=W(p,t_i)$ – rule $f\\ref {f:r:ptarc}$ construction; or it represents $n_i=M_k(p)$ – rule $f\\ref {f:rptarc}$ construction; the construction of $f\\ref {f:rptarc}$ ensures that $notenabled(t,0)$ is never true due to the reset arc definition REF of preset $\\bullet t_i$ in PN Then $t_i$ is enabled and can fire in $PN$ , as a result it can belong to $T_k$ – from definition REF of enabled transition And $consumesmore \\notin A$ , since $A$ is an answer set of $\\Pi ^3$ – from rule $a\\ref {a:overc:elim}$ and supported rule proposition Then $\\nexists consumesmore(p,k) \\in A$ – from rule $a\\ref {a:overc:gen}$ and supported rule proposition Then $\\nexists \\lbrace holds(p,q,k), tot\\_decr(p,q1,k) \\rbrace \\subseteq A : q1>q$ in $body(a\\ref {a:overc:place})$ – from $a\\ref {a:overc:place}$ and forced atom proposition Then $\\nexists p : (\\sum _{t_i \\in \\lbrace t_0,\\dots ,t_x\\rbrace , p \\in \\bullet t_i}{W(p,t_i)}+\\sum _{t_i \\in \\lbrace t_0,\\dots ,t_x\\rbrace , p \\in R(t_i)}{M_k(p)}) > M_k(p)$ – from the following $holds(p,q,k)$ represents $q=M_k(p)$ – inductive assumption, construction $tot\\_decr(p,q1,k) \\in A$ if $\\lbrace del(p,q1_0,t_0,k), \\dots , del(p,q1_x,t_x,k) \\rbrace \\subseteq A$ , where $q1 = q1_0+\\dots +q1_x$ – from $r\\ref {r:totdecr}$ and forced atom proposition $del(p,q1_i,t_i,k) \\in A$ if $\\lbrace fires(t_i,k), ptarc(p,t_i,q1_i,k) \\rbrace \\subseteq A$ – from $r\\ref {r:r:del} $ and supported rule proposition $del(p,q1_i,t_i,k)$ either represents removal of $q1_i = W(p,t_i)$ tokens from $p \\in \\bullet t_i$ ; or it represents removal of $q1_i = M_k(p)$ tokens from $p \\in R(t_i)$ – from rule $r\\ref {r:r:del} $ , supported rule proposition, and definition REF of transition execution in $PN$ Then $T_k$ does not contain conflicting transitions – by the definition REF of conflicting transitions And for each $enabled(t_j,k) \\in A$ and $fires(t_j,k) \\notin A$ , $could\\_not\\_have(t_j,k) \\in A$ , since $A$ is an answer set of $\\Pi ^3$ - from rule $a\\ref {a:maxfire:elim}$ and supported rule proposition Then $\\lbrace enabled(t_j,k), holds(s,qq,k), ptarc(s,t_j,q,k), \\\\ tot\\_decr(s,qqq,k) \\rbrace \\subseteq A$ , such that $q > qq - qqq$ and $fires(t_j,k) \\notin A$ - from rule $a\\ref {a:r:maxfire:cnh}$ and supported rule proposition Then for an $s \\in \\bullet t_j \\cup R(t_j)$ , $q > M_k(s) - (\\sum _{t_i \\in T_k, s \\in \\bullet t_i}{W(s,t_i)} + \\sum _{t_i \\in T_k, s \\in R(t_i)}{M_k(s))}$ , where $q=W(s,t_j) \\text{~if~} s \\in \\bullet t_j, \\text{~or~} M_k(s) \\text{~otherwise}$ - from the following: $ptarc(s,t_i,q,k)$ represents $q=W(s,t_i)$ if $(s,t_i) \\in E^-$ or $q=M_k(s)$ if $s \\in R(t_i)$ – from rule $f\\ref {f:r:ptarc},f\\ref {f:rptarc}$ construction $holds(s,qq,k)$ represents $qq=M_k(s)$ – construction $tot\\_decr(s,qqq,k) \\in A$ if $\\lbrace del(s,qqq_0,t_0,k), \\dots , del(s,qqq_x,t_x,k) \\rbrace \\\\ \\subseteq A$ – from rule $r\\ref {r:totdecr}$ construction and supported rule proposition $del(s,qqq_i,t_i,k) \\in A$ if $\\lbrace fires(t_i,k), ptarc(s,t_i,qqq_i,k) \\rbrace \\subseteq A$ – from rule $r\\ref {r:r:del} $ and supported rule proposition $del(s,qqq_i,t_i,k)$ represents $qqq_i = W(s,t_i) : t_i \\in T_k, (s,t_i) \\in E^-$ , or $qqq_i = M_k(t_i) : t_i \\in T_k, s \\in R(t_i)$ – from rule $f\\ref {f:r:ptarc},f\\ref {f:rptarc}$ construction $tot\\_decr(q,qqq,k)$ represents $\\sum _{t_i \\in T_k, s \\in \\bullet t_i}{W(s,t_i)} + \\\\ \\sum _{t_i \\in T_k, s \\in R(t_i)}{M_k(s)}$ – from (C,D,E) above Then firing $T_k \\cup \\lbrace t_j \\rbrace $ would have required more tokens than are present at its source place $s \\in \\bullet t_j \\cup R(t_j)$ .", "Thus, $T_k$ is a maximal set of transitions that can simultaneously fire.", "And for each reset transition $t_r$ with $enabled(t_r,k) \\in A$ , $fires(t_r,k) \\in A$ , since $A$ is an answer set of $\\Pi ^2$ - from rule $f\\ref {f:c:rptarc:elim}$ and supported rule proposition Then the firing set $T_k$ satisfies the reset transition requirement of definition REF (firing set) Then $\\lbrace t_0, \\dots , t_x\\rbrace = T_k$ – using 1(a),1(b), 1(d) above; and using 1(c) it is a maximal firing set We show that $M_{k+1}$ is produced by firing $T_k$ in $M_k$ .", "Let $holds(p,q,k+1) \\in A$ Then $\\lbrace holds(p,q1,k), tot\\_incr(p,q2,k), tot\\_decr(p,q3,k) \\rbrace \\subseteq A : q=q1+q2-q3$ – from rule $r\\ref {r:nextstate}$ and supported rule proposition Then, $holds(p,q1,k) \\in A$ represents $q1=M_k(p)$ – inductive assumption, construction ; and $\\lbrace add(p,q2_0,t_0,k), \\dots , $ $add(p,q2_j,t_j,k)\\rbrace \\subseteq A : q2_0 + \\dots + q2_j = q2$ and $\\lbrace del(p,q3_0,t_0,k), \\dots , $ $del(p,q3_l,t_l,k)\\rbrace \\subseteq A : q3_0 + \\dots + q3_l = q3$ – rules $r\\ref {r:totincr},r\\ref {r:totdecr}$ and supported rule proposition, respectively Then $\\lbrace fires(t_0,k), \\dots , fires(t_j,k) \\rbrace \\subseteq A$ and $\\lbrace fires(t_0,k), \\dots , fires(t_l,k) \\rbrace \\\\ \\subseteq A$ – rules $r\\ref {r:r:add},r\\ref {r:r:del}$ and supported rule proposition, respectively Then $\\lbrace fires(t_0,k), \\dots , $ $fires(t_j,k) \\rbrace \\cup \\lbrace fires(t_0,k), \\dots , $ $fires(t_l,k) \\rbrace \\subseteq A = $ $\\lbrace fires(t_0,k), \\dots , $ $fires(t_x,k) \\rbrace \\subseteq A$ – set union of subsets Then for each $fires(t_x,k) \\in A$ we have $t_x \\in T_k$ – already shown in item REF above Then $q = M_k(p) + \\sum _{t_x \\in T_0 \\wedge p \\in t_x \\bullet }{W(t_x,p)} - (\\sum _{t_x \\in T_k \\wedge p \\in \\bullet t_x}{W(p,t_x)} + \\\\ \\sum _{t_x \\in T_k \\wedge p \\in R(t_x)}{M_k(p)})$ – from (REF ) above and the following Each $add(p,q_j,t_j,k) \\in A$ represents $q_j=W(t_j,p)$ for $p \\in t_j \\bullet $ – rule $r\\ref {r:r:add} $ encoding, and definition REF of transition execution in $PN$ Each $del(p,t_y,q_y,k) \\in A$ represents either $q_y=W(p,t_y)$ for $p \\in \\bullet t_y$ , or $q_y=M_k(p)$ for $p \\in R(t_y)$ – from rule $r\\ref {r:r:del} ,f\\ref {f:r:ptarc} $ encoding and definition REF of transition execution in $PN$ ; or from rule $r\\ref {r:r:del} ,f\\ref {f:rptarc}$ encoding and definition of reset arc in $PN$ Each $tot\\_incr(p,q2,k) \\in A$ represents $q2=\\sum _{t_x \\in T_k \\wedge p \\in t_x \\bullet }{W(t_x,p)}$ – aggregate assignment atom semantics in rule $r\\ref {r:totincr}$ Each $tot\\_decr(p,q3,0) \\in A$ represents $q3=\\sum _{t_x \\in T_k \\wedge p \\in \\bullet t_x}{W(p,t_x)} + $ $\\sum _{t_x \\in T_k \\wedge p \\in R(t_x)}{M_k(p)}$ – aggregate assignment atom semantics in rule $r\\ref {r:totdecr}$ Then, $M_{k+1}(p) = q$ – since $holds(p,q,k+1) \\in A$ encodes $q=M_{k+1}(p)$ – from construction As a result, for any $n > k$ , $T_n$ will be a valid firing set for $M_n$ and $M_{n+1}$ will be its target marking.", "Conclusion: Since both (REF ) and (REF ) hold, $X=M_0,T_0,M_1,\\dots ,M_k,T_{k+1}$ is an execution sequence of $PN(P,T,E,W,R)$ (w.r.t $M_0$ ) iff there is an answer set $A$ of $\\Pi ^3(PN,M_0,k,ntok)$ such that (REF ) and (REF ) hold.", "Proof of Proposition REF Let $PN=(P,T,E,W,R,I,Q,QW)$ be a Petri Net, $M_0$ be its initial marking and let $\\Pi ^4(PN,M_0,k,ntok)$ be the ASP encoding of $PN$ and $M_0$ over a simulation length $k$ , with maximum $ntok$ tokens on any place node, as defined in section REF .", "Then $X=M_0,T_0,M_1,\\dots ,M_k,T_k,M_{k+1}$ is an execution sequence of $PN$ (w.r.t.", "$M_0$ ) iff there is an answer set $A$ of $\\Pi ^4(PN,M_0,k,ntok)$ such that: $\\lbrace fires(t,ts) : t \\in T_{ts}, 0 \\le ts \\le k\\rbrace = \\lbrace fires(t,ts) : fires(t,ts) \\in A \\rbrace $ $\\begin{split}\\lbrace holds(p,q,ts) &: p \\in P, q = M_{ts}(p), 0 \\le ts \\le k+1 \\rbrace \\\\&= \\lbrace holds(p,q,ts) : holds(p,q,ts) \\in A \\rbrace \\end{split}$ We prove this by showing that: Given an execution sequence $X$ , we create a set $A$ such that it satisfies (REF ) and (REF ) and show that $A$ is an answer set of $\\Pi ^4$ Given an answer set $A$ of $\\Pi ^4$ , we create an execution sequence $X$ such that (REF ) and (REF ) are satisfied.", "First we show (REF ): Given $PN$ and an execution sequence $X$ of $PN$ , we create a set $A$ as a union of the following sets: $A_1=\\lbrace num(n) : 0 \\le n \\le ntok \\rbrace $ $A_2=\\lbrace time(ts) : 0 \\le ts \\le k\\rbrace $ $A_3=\\lbrace place(p) : p \\in P \\rbrace $ $A_4=\\lbrace trans(t) : t \\in T \\rbrace $ $A_5=\\lbrace ptarc(p,t,n,ts) : (p,t) \\in E^-, n=W(p,t), 0 \\le ts \\le k \\rbrace $ , where $E^- \\subseteq E$ $A_6=\\lbrace tparc(t,p,n,ts) : (t,p) \\in E^+, n=W(t,p), 0 \\le ts \\le k \\rbrace $ , where $E^+ \\subseteq E$ $A_7=\\lbrace holds(p,q,0) : p \\in P, q=M_{0}(p) \\rbrace $ $A_8=\\lbrace notenabled(t,ts) : t \\in T, 0 \\le ts \\le k, (\\exists p \\in \\bullet t, M_{ts}(p) < W(p,t)) \\vee (\\exists p \\in I(t), M_{ts}(p) \\ne 0) \\vee (\\exists (p,t) \\in Q, M_{ts}(p) < QW(p,t)) \\rbrace $ per definition REF (enabled transition) $A_9=\\lbrace enabled(t,ts) : t \\in T, 0 \\le ts \\le k, (\\forall p \\in \\bullet t, W(p,t) \\le M_{ts}(p)) \\wedge (\\forall p \\in I(t), M_{ts}(p) = 0) \\wedge (\\forall (p,t) \\in Q, M_{ts}(p) \\ge QW(p,t)) \\rbrace $ per definition REF (enabled transition) $A_{10}=\\lbrace fires(t,ts) : t \\in T_{ts}, 0 \\le ts \\le k \\rbrace $ per definition REF (enabled transitions), only an enabled transition may fire $A_{11}=\\lbrace add(p,q,t,ts) : t \\in T_{ts}, p \\in t \\bullet , q=W(t,p), 0 \\le ts \\le k \\rbrace $ per definition REF (transition execution) $A_{12}=\\lbrace del(p,q,t,ts) : t \\in T_{ts}, p \\in \\bullet t, q=W(p,t), 0 \\le ts \\le k \\rbrace \\cup \\lbrace del(p,q,t,ts) : t \\in T_{ts}, p \\in R(t), q=M_{ts}(p), 0 \\le ts \\le k \\rbrace $ per definition REF (transition execution) $A_{13}=\\lbrace tot\\_incr(p,q,ts) : p \\in P, q=\\sum _{t \\in T_{ts}, p \\in t \\bullet }{W(t,p)}, 0 \\le ts \\le k \\rbrace $ per definition REF (firing set execution) $A_{14}=\\lbrace tot\\_decr(p,q,ts): p \\in P, q=\\sum _{t \\in T_{ts}, p \\in \\bullet t}{W(p,t)}+\\sum _{t \\in T_{ts}, p \\in R(t) }{M_{ts}(p)}, 0 \\le ts \\le k \\rbrace $ per definition REF (firing set execution) $A_{15}=\\lbrace consumesmore(p,ts) : p \\in P, q=M_{ts}(p), q1=\\sum _{t \\in T_{ts}, p \\in \\bullet t}{W(p,t)} + \\sum _{t \\in T_{ts}, p \\in R(t)}{M_{ts}(p)}, q1 > q, 0 \\le ts \\le k \\rbrace $ per definition REF (conflicting transitions) $A_{16}=\\lbrace consumesmore : \\exists p \\in P : q=M_{ts}(p), q1=\\sum _{t \\in T_{ts}, p \\in \\bullet t}{W(p,t)} + \\sum _{t \\in T_{ts}, p \\in R(t)}(M_{ts}(p)), q1 > q, 0 \\le ts \\le k \\rbrace $ per definition REF (conflicting transitions) $A_{17}=\\lbrace could\\_not\\_have(t,ts) : t \\in T, (\\forall p \\in \\bullet t, W(p,t) \\le M_{ts}(p)), t \\notin T_{ts}, (\\exists p \\in \\bullet t : W(p,t) > M_{ts}(p) - (\\sum _{t^{\\prime } \\in T_{ts}, p \\in \\bullet t^{\\prime }}{W(p,t^{\\prime })} + \\sum _{t^{\\prime } \\in T_{ts}, p \\in R(t^{\\prime })} M_{ts}(p)), 0 \\le ts \\le k \\rbrace $ per the maximal firing set semantics $A_{18}=\\lbrace holds(p,q,ts+1) : p \\in P, q=M_{ts+1}(p), 0 \\le ts < k\\rbrace $ , where $M_{ts+1}(p) = M_{ts}(p) - (\\sum _{\\begin{array}{c}t \\in T_{ts}, p \\in \\bullet t\\end{array}}{W(p,t)} + \\\\ \\sum _{t \\in T_{ts}, p \\in R(t)}M_{ts}(p)) + $ $\\sum _{\\begin{array}{c}t \\in T_{ts}, p \\in t \\bullet \\end{array}}{W(t,p)}$ according to definition REF (firing set execution) $A_{19}=\\lbrace ptarc(p,t,n,ts) : p \\in R(t), n = M_{ts}(p), n > 0, 0 \\le ts \\le k \\rbrace $ $A_{20}=\\lbrace iptarc(p,t,1,ts) : p \\in P, 0 \\le ts < k \\rbrace $ $A_{21}=\\lbrace tptarc(p,t,n,ts) : (p,t) \\in Q, n=QW(p,t), 0 \\le ts \\le k \\rbrace $ We show that $A$ satisfies (REF ) and (REF ), and $A$ is an answer set of $\\Pi ^4$ .", "$A$ satisfies (REF ) and (REF ) by its construction above.", "We show $A$ is an answer set of $\\Pi ^4$ by splitting.", "We split $lit(\\Pi ^4)$ into a sequence of $7k+9$ sets: $U_0= head(f\\ref {f:place}) \\cup head(f\\ref {f:trans}) \\cup head(f\\ref {f:time}) \\cup head(f\\ref {f:num}) \\cup head(i\\ref {i:holds}) = \\lbrace place(p) : p \\in P\\rbrace \\cup \\lbrace trans(t) : t \\in T\\rbrace \\cup \\lbrace time(0), \\dots , time(k)\\rbrace \\cup \\lbrace num(0), \\dots , num(ntok)\\rbrace \\cup \\lbrace holds(p,q,0) : p \\in P, q=M_0(p) \\rbrace $ $U_{7k+1}=U_{7k+0} \\cup head(f\\ref {f:r:ptarc})^{ts=k} \\cup head(f\\ref {f:r:tparc})^{ts=k} \\cup head(f\\ref {f:rptarc})^{ts=k} \\cup head(f\\ref {f:iptarc})^{ts=k} \\cup head(f\\ref {f:tptarc})^{ts=k} = U_{7k+0} \\cup \\lbrace ptarc(p,t,n,k) : (p,t) \\in E^-, n=W(p,t) \\rbrace \\cup \\\\ \\lbrace tparc(t,p,n,k) : (t,p) \\in E^+, n=W(t,p) \\rbrace \\cup $ $\\lbrace ptarc(p,t,n,k) : p \\in R(t), n=M_{k}(p), n > 0 \\rbrace \\cup $ $\\lbrace iptarc(p,t,1,k) : p \\in I(t) \\rbrace \\cup $ $\\lbrace tptarc(p,t,n,k) : (p,t) \\in Q, n=QW(p,t) \\rbrace $ $U_{7k+2}=U_{7k+1} \\cup head(e\\ref {e:r:ne:ptarc} )^{ts=k} = U_{7k+1} \\cup \\lbrace notenabled(t,k) : t \\in T \\rbrace $ $U_{7k+3}=U_{7k+2} \\cup head(e\\ref {e:enabled})^{ts=k} = U_{7k+2} \\cup \\lbrace enabled(t,k) : t \\in T \\rbrace $ $U_{7k+4}=U_{7k+3} \\cup head(a\\ref {a:fires})^{ts=k} = U_{7k+3} \\cup \\lbrace fires(t,k) : t \\in T \\rbrace $ $U_{7k+5}=U_{7k+4} \\cup head(r\\ref {r:r:add} )^{ts=k} \\cup head(r\\ref {r:r:del} )^{ts=k} = U_{7k+4} \\cup \\lbrace add(p,q,t,k) : p \\in P, t \\in T, q=W(t,p) \\rbrace \\cup \\lbrace del(p,q,t,k) : p \\in P, t \\in T, q=W(p,t) \\rbrace \\cup \\lbrace del(p,q,t,k) : p \\in P, t \\in T, q=M_{k}(p) \\rbrace $ $U_{7k+6}=U_{7k+5} \\cup head(r\\ref {r:totincr})^{ts=k} \\cup head(r\\ref {r:totdecr})^{ts=k} = U_{7k+5} \\cup \\lbrace tot\\_incr(p,q,k) : p \\in P, 0 \\le q \\le ntok \\rbrace \\cup \\lbrace tot\\_decr(p,q,k) : p \\in P, 0 \\le q \\le ntok \\rbrace $ $U_{7k+7}=U_{7k+6} \\cup head(r\\ref {r:nextstate})^{ts=k} \\cup head(a\\ref {a:overc:place})^{ts=k} \\cup head(a\\ref {a:r:maxfire:cnh})^{ts=k} = U_{7k+6} \\cup \\\\ \\lbrace consumesmore(p,k) : p \\in P\\rbrace \\cup \\lbrace holds(p,q,k+1) : p \\in P, 0 \\le q \\le ntok \\rbrace \\cup \\lbrace could\\_not\\_have(t,k) : t \\in T \\rbrace $ $U_{7k+8}=U_{7k+7} \\cup head(a\\ref {a:overc:gen}) = U_{7k+7} \\cup \\lbrace consumesmore \\rbrace $ where $head(r_i)^{ts=k}$ are head atoms of ground rule $r_i$ in which $ts=k$ .", "We write $A_i^{ts=k} = \\lbrace a(\\dots ,ts) : a(\\dots ,ts) \\in A_i, ts=k \\rbrace $ as short hand for all atoms in $A_i$ with $ts=k$ .", "$U_{\\alpha }, 0 \\le \\alpha \\le 7k+8$ form a splitting sequence, since each $U_i$ is a splitting set of $\\Pi ^4$ , and $\\langle U_{\\alpha }\\rangle _{\\alpha < \\mu }$ is a monotone continuous sequence, where $U_0 \\subseteq U_1 \\dots \\subseteq U_{8(k+1)}$ and $\\bigcup _{\\alpha < \\mu }{U_{\\alpha }} = lit(\\Pi ^4)$ .", "We compute the answer set of $\\Pi ^4$ using the splitting sets as follows: $bot_{U_0}(\\Pi ^4) = f\\ref {f:place} \\cup f\\ref {f:trans} \\cup f\\ref {f:time} \\cup i\\ref {i:holds} \\cup f\\ref {f:num}$ and $X_0 = A_1 \\cup \\dots \\cup A_4 \\cup A_7$ ($= U_0$ ) is its answer set – using forced atom proposition $eval_{U_0}(bot_{U_1}(\\Pi ^4) \\setminus bot_{U_0}(\\Pi ^4), X_0) = \\lbrace ptarc(p,t,q,0) \\text{:-}.", "\| q=W(p,t) \\rbrace \\cup \\\\ \\lbrace tparc(t,p,q,0) \\text{:-}.", "\| q=W(t,p) \\rbrace \\cup \\lbrace ptarc(p,t,q,0) \\text{:-}.", "\| q=M_0(p) \\rbrace \\cup \\\\ \\lbrace iptarc(p,t,1,0) \\text{:-}.", "\\rbrace \\cup \\lbrace tptarc(p,t,q,0) \\text{:-}.", "\| q = QW(p,t) \\rbrace $ .", "Its answer set $X_1=A_5^{ts=0} \\cup A_6^{ts=0} \\cup A_{19}^{ts=0} \\cup A_{20}^{ts=0} \\cup A_{21}^{ts=0}$ – using forced atom proposition and construction of $A_5, A_6, A_{19}, A_{20}, A_{21}$ .", "$eval_{U_1}(bot_{U_2}(\\Pi ^4) \\setminus bot_{U_1}(\\Pi ^4), X_0 \\cup X_1) = \\lbrace notenabled(t,0) \\text{:-} .", "\| (\\lbrace trans(t), \\\\ ptarc(p,t,n,0), holds(p,q,0) \\rbrace \\subseteq X_0 \\cup X_1, \\text{~where~} q < n) \\text{~or~} (\\lbrace notenabled(t,0) \\text{:-} .", "\| \\\\ (\\lbrace trans(t), iptarc(p,t,n2,0), holds(p,q,0) \\rbrace \\subseteq X_0 \\cup X_1, \\text{~where~} q \\ge n2 \\rbrace ) \\text{~or~} \\\\ (\\lbrace trans(t), tptarc(p,t,n3,0), holds(p,q,0) \\rbrace \\subseteq X_0 \\cup X_1, \\text{~where~} q < n3) \\rbrace $ .", "Its answer set $X_2=A_8^{ts=0}$ – using forced atom proposition and construction of $A_8$ .", "where, $q=M_0(p)$ , and $n=W(p,t)$ for an arc $(p,t) \\in E^-$ – by construction of $i\\ref {i:holds}$ and $f\\ref {f:r:ptarc}$ in $\\Pi ^4$ , and in an arc $(p,t) \\in E^-$ , $p \\in \\bullet t$ (by definition REF of preset) $n2=1$ – by construction of $iptarc$ predicates in $\\Pi ^4$ , meaning $q \\ge n2 \\equiv q \\ge 1 \\equiv q > 0$ , $tptarc(p,t,n3,0)$ represents $n3=QW(p,t)$ , where $(p,t) \\in Q$ thus, $notenabled(t,0) \\in X_1$ represents $(\\exists p \\in \\bullet t : M_0(p) < W(p,t)) \\vee (\\exists p \\in I(t) : M_0(p) > 0) \\vee (\\exists (p,t) \\in Q : M_{ts}(p) < QW(p,t))$ .", "$eval_{U_2}(bot_{U_3}(\\Pi ^4) \\setminus bot_{U_2}(\\Pi ^4), X_0 \\cup \\dots \\cup X_2) = \\lbrace enabled(t,0) \\text{:-}.", "\| trans(t) \\in X_0 \\cup \\dots \\cup X_2, notenabled(t,0) \\notin X_0 \\cup \\dots \\cup X_2 \\rbrace $ .", "Its answer set is $X_3 = A_9^{ts=0}$ – using forced atom proposition and construction of $A_9$ .", "since an $enabled(t,0) \\in X_3$ if $\\nexists ~notenabled(t,0) \\in X_0 \\cup \\dots \\cup X_2$ ; which is equivalent to $(\\nexists p \\in \\bullet t : M_0(p) < W(p,t)) \\wedge (\\nexists p \\in I(t) : M_0(p) > 0) \\wedge (\\nexists (p,t) \\in Q : M_0(p) < QW(p,t) ) \\equiv (\\forall p \\in \\bullet t: M_0(p) \\ge W(p,t)) \\wedge (\\forall p \\in I(t) : M_0(p) = 0)$ .", "$eval_{U_3}(bot_{U_4}(\\Pi ^4) \\setminus bot_{U_3}(\\Pi ^4), X_0 \\cup \\dots \\cup X_3) = \\lbrace \\lbrace fires(t,0)\\rbrace \\text{:-}.", "\| enabled(t,0) \\\\ \\text{~holds in~} X_0 \\cup \\dots \\cup X_3 \\rbrace $ .", "It has multiple answer sets $X_{4.1}, \\dots , X_{4.n}$ , corresponding to elements of power set of $fires(t,0)$ atoms in $eval_{U_3}(...)$ – using supported rule proposition.", "Since we are showing that the union of answer sets of $\\Pi ^4$ determined using splitting is equal to $A$ , we only consider the set that matches the $fires(t,0)$ elements in $A$ and call it $X_4$ , ignoring the rest.", "Thus, $X_4 = A_{10}^{ts=0}$ , representing $T_0$ .", "in addition, for every $t$ such that $enabled(t,0) \\in X_0 \\cup \\dots \\cup X_3, R(t) \\ne \\emptyset $ ; $fires(t,0) \\in X_4$ – per definition REF (firing set); requiring that a reset transition is fired when enabled thus, the firing set $T_0$ will not be eliminated by the constraint $f\\ref {f:c:rptarc:elim}$ $eval_{U_4}(bot_{U_5}(\\Pi ^4) \\setminus bot_{U_4}(\\Pi ^4), X_0 \\cup \\dots \\cup X_4) = \\lbrace add(p,n,t,0) \\text{:-}.", "\| \\lbrace fires(t,0), \\\\ tparc(t,p,n,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_4 \\rbrace \\cup \\lbrace del(p,n,t,0) \\text{:-}.", "\| \\lbrace fires(t,0), ptarc(p,t,n,0) \\rbrace \\\\ \\subseteq X_0 \\cup \\dots \\cup X_4 \\rbrace $ .", "It's answer set is $X_5 = A_{11}^{ts=0} \\cup A_{12}^{ts=0}$ – using forced atom proposition and definitions of $A_{11}$ and $A_{12}$ .", "where, each $add$ atom is equivalent to $n=W(t,p),p \\in t \\bullet $ , and each $del$ atom is equivalent to $n=W(p,t), p \\in \\bullet t$ ; or $n=M_k(p), p \\in R(t)$ , representing the effect of transitions in $T_0$ – by construction $eval_{U_5}(bot_{U_6}(\\Pi ^4) \\setminus bot_{U_5}(\\Pi ^4), X_0 \\cup \\dots \\cup X_5) = \\lbrace tot\\_incr(p,qq,0) \\text{:-}.", "\| $ $qq=\\sum _{add(p,q,t,0) \\in X_0 \\cup \\dots \\cup X_5}{q} \\rbrace \\cup \\lbrace tot\\_decr(p,qq,0) \\text{:-}.", "\| qq=\\sum _{del(p,q,t,0) \\in X_0 \\cup \\dots \\cup X_5}{q} \\rbrace $ .", "It's answer set is $X_6 = A_{13}^{ts=0} \\cup A_{14}^{ts=0}$ – using forced atom proposition and definitions of $A_{13}$ and $A_{14}$ .", "where, each $tot\\_incr(p,qq,0)$ , $qq=\\sum _{add(p,q,t,0) \\in X_0 \\cup \\dots X_5}{q}$ $\\equiv qq=\\sum _{t \\in X_4, p \\in t \\bullet }{W(p,t)}$ , and each $tot\\_decr(p,qq,0)$ , $qq=\\sum _{del(p,q,t,0) \\in X_0 \\cup \\dots X_5}{q}$ $\\equiv qq=\\sum _{t \\in X_4, p \\in \\bullet t}{W(t,p)} + \\sum _{t \\in X_4, p \\in R(t)}{M_{k}(p)}$ , represent the net effect of transitions in $T_0$ – by construction $eval_{U_6}(bot_{U_7}(\\Pi ^4) \\setminus bot_{U_6}(\\Pi ^4), X_0 \\cup \\dots \\cup X_6) = \\lbrace consumesmore(p,0) \\text{:-}.", "\| \\\\ \\lbrace holds(p,q,0), tot\\_decr(p,q1,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_6, q1 > q \\rbrace \\cup \\lbrace holds(p,q,1) \\text{:-}., \| \\\\ \\lbrace holds(p,q1,0), tot\\_incr(p,q2,0), tot\\_decr(p,q3,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_6, q=q1+q2-q3 \\rbrace \\cup \\lbrace could\\_not\\_have(t,0) \\text{:-}.", "\| \\lbrace enabled(t,0), ptarc(s,t,q), holds(s,qq,0), \\\\ tot\\_decr(s,qqq,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_6, fires(t,0) \\notin (X_0 \\cup \\dots \\cup X_6), q > qq-qqq \\rbrace $ .", "It's answer set is $X_7 = A_{15}^{ts=0} \\cup A_{17}^{ts=0} \\cup A_{18}^{ts=0}$ – using forced atom proposition and definitions of $A_{15}, A_{17}, A_{18}, A_9$ .", "where, $consumesmore(p,0)$ represents $\\exists p : q=M_0(p), q1= \\\\ \\sum _{t \\in T_0, p \\in \\bullet t}{W(p,t)}+\\sum _{t \\in T_0, p \\in R(t)}{M_0(p)}, q1 > q$ , indicating place $p$ will be over consumed if $T_0$ is fired, as defined in definition REF (conflicting transitions), $holds(p,q,1)$ represents $q=M_1(p)$ – by construction of $\\Pi ^4$ , and $could\\_not\\_have(t,0)$ represents enabled transition $t \\in T_0$ that could not fire due to insufficient tokens $X_7$ does not contain $could\\_not\\_have(t,0)$ , when $enabled(t,0) \\in X_0 \\cup \\dots \\cup X_6$ and $fires(t,0) \\notin X_0 \\cup \\dots \\cup X_6$ due to construction of $A$ , encoding of $a\\ref {a:r:maxfire:cnh}$ and its body atoms.", "As a result it is not eliminated by the constraint $a\\ref {a:maxfire:elim}$ $ \\vdots $ $eval_{U_{7k+0}}(bot_{U_{7k+1}}(\\Pi ^4) \\setminus bot_{U_{7k+0}}(\\Pi ^4), X_0 \\cup \\dots \\cup X_{7k+0}) = \\lbrace ptarc(p,t,q,k) \\text{:-}.", "\| q=W(p,t) \\rbrace \\cup \\lbrace tparc(t,p,q,k) \\text{:-}.", "\| q=W(t,p) \\rbrace \\cup \\lbrace ptarc(p,t,q,k) \\text{:-}.", "\| q=M_k(p) \\rbrace \\cup \\lbrace iptarc(p,t,1,k) \\text{:-}.", "\\rbrace $ .", "Its answer set $X_{7k+1}=A_5^{ts=k} \\cup A_6^{ts=k} \\cup A_{19}^{ts=k} \\cup A_{20}^{ts=k}$ – using forced atom proposition and construction of $A_5, A_6, A_{19}, A_{20}$ .", "$eval_{U_{7k+1}}(bot_{U_{7k+2}}(\\Pi ^4) \\setminus bot_{U_{7k+1}}(\\Pi ^4), X_0 \\cup \\dots \\cup X_{7k+1}) = \\lbrace notenabled(t,k) \\text{:-} .", "\| $ $(\\lbrace trans(t), ptarc(p,t,n,k), holds(p,q,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{7k+1}, \\text{~where~} q < n) \\text{~or~} \\\\ \\lbrace notenabled(t,k) \\text{:-} .", "\| (\\lbrace trans(t), iptarc(p,t,n2,k), holds(p,q,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{7k+1}, \\\\ \\text{~where~} q \\ge n2 \\rbrace \\rbrace $ .", "Its answer set $X_{7k+2}=A_8^{ts=k}$ – using forced atom proposition and construction of $A_8$ .", "where, $q=M_k(p)$ , and $n=W(p,t)$ for an arc $(p,t) \\in E^-$ – by construction of $holds$ and $ptarc$ predicates in $\\Pi ^4$ , and in an arc $(p,t) \\in E^-$ , $p \\in \\bullet t$ (by definition REF of preset) $n2=1$ – by construction of $iptarc$ predicates in $\\Pi ^4$ , meaning $q \\ge n2 \\equiv q \\ge 1 \\equiv q > 0$ , thus, $notenabled(t,k) \\in X_{7k+1}$ represents $(\\exists p \\in \\bullet t : M_k(p) < W(p,t)) \\vee (\\exists p \\in I(t) : M_k(p) > k)$ .", "$eval_{U_{7k+2}}(bot_{U_{7k+3}}(\\Pi ^4) \\setminus bot_{U_{7k+2}}(\\Pi ^4), X_0 \\cup \\dots \\cup X_{7k+2}) = \\lbrace enabled(t,k) \\text{:-}.", "\| \\\\ trans(t) \\in X_0 \\cup \\dots \\cup X_{7k+2} \\wedge notenabled(t,k) \\notin X_0 \\cup \\dots \\cup X_{7k+2} \\rbrace $ .", "Its answer set is $X_{7k+3} = A_9^{ts=k}$ – using forced atom proposition and construction of $A_9$ .", "since an $enabled(t,k) \\in X_{7k+3}$ if $\\nexists ~notenabled(t,k) \\in X_0 \\cup \\dots \\cup X_{7k+2}$ ; which is equivalent to $(\\nexists p \\in \\bullet t : M_k(p) < W(p,t)) \\wedge (\\nexists p \\in I(t) : M_k(p) > k) \\equiv (\\forall p \\in \\bullet t: M_k(p) \\ge W(p,t)) \\wedge (\\forall p \\in I(t) : M_k(p) = k)$ .", "$eval_{U_{7k+3}}(bot_{U_{7k+4}}(\\Pi ^4) \\setminus bot_{U_{7k+3}}(\\Pi ^4), X_0 \\cup \\dots \\cup X_{7k+3}) = \\lbrace \\lbrace fires(t,k)\\rbrace \\text{:-}.", "\| \\\\ enabled(t,k) \\\\ \\text{~holds in~} X_0 \\cup \\dots \\cup X_{7k+3} \\rbrace $ .", "It has multiple answer sets $X_{7k+4.1}, \\dots , X_{7k+4.n}$ , corresponding to elements of power set of $fires(t,k)$ atoms in $eval_{U_{7k+3}}(...)$ – using supported rule proposition.", "Since we are showing that the union of answer sets of $\\Pi ^4$ determined using splitting is equal to $A$ , we only consider the set that matches the $fires(t,k)$ elements in $A$ and call it $X_{7k+4}$ , ignoring the rest.", "Thus, $X_{7k+4} = A_{10}^{ts=k}$ , representing $T_k$ .", "in addition, for every $t$ such that $enabled(t,k) \\in X_0 \\cup \\dots \\cup X_{7k+3}, R(t) \\ne \\emptyset $ ; $fires(t,k) \\in X_{7k+4}$ – per definition REF (firing set); requiring that a reset transition is fired when enabled thus, the firing set $T_k$ will not be eliminated by the constraint $f\\ref {f:c:rptarc:elim}$ $eval_{U_{7k+4}}(bot_{U_{7k+5}}(\\Pi ^4) \\setminus bot_{U_{7k+4}}(\\Pi ^4), X_0 \\cup \\dots \\cup X_{7k+4}) = \\lbrace add(p,n,t,k) \\text{:-}.", "\| \\\\ \\lbrace fires(t,k), tparc(t,p,n,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{7k+4} \\rbrace \\cup \\lbrace del(p,n,t,k) \\text{:-}.", "\| \\lbrace fires(t,k), \\\\ ptarc(p,t,n,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{7k+4} \\rbrace $ .", "It's answer set is $X_{7k+5} = A_{11}^{ts=k} \\cup A_{12}^{ts=k}$ – using forced atom proposition and definitions of $A_{11}$ and $A_{12}$ .", "where, each $add$ atom is equivalent to $n=W(t,p) : p \\in t \\bullet $ , and each $del$ atom is equivalent to $n=W(p,t) : p \\in \\bullet t$ ; or $n=M_k(p) : p \\in R(t)$ , representing the effect of transitions in $T_k$ $eval_{U_{7k+5}}(bot_{U_{7k+6}}(\\Pi ^4) \\setminus bot_{U_{7k+5}}(\\Pi ^4), X_0 \\cup \\dots \\cup X_{7k+5}) = \\\\ \\lbrace tot\\_incr(p,qq,k) \\text{:-}.", "\| qq=\\sum _{add(p,q,t,k) \\in X_0 \\cup \\dots \\cup X_{7k+5}}{q} \\rbrace \\cup \\\\ \\lbrace tot\\_decr(p,qq,k) \\text{:-}.", "\| qq=\\sum _{del(p,q,t,k) \\in X_0 \\cup \\dots \\cup X_{7k+5}}{q} \\rbrace $ .", "It's answer set is $X_{7k+6} = A_{13}^{ts=k} \\cup A_{14}^{ts=k}$ – using forced atom proposition and definitions of $A_{13}$ and $A_{14}$ .", "where, each $tot\\_incr(p,qq,k)$ , $qq=\\sum _{add(p,q,t,k) \\in X_0 \\cup \\dots X_{7k+5}}{q}$ $\\equiv qq=\\sum _{t \\in X_{7k+4}, p \\in t \\bullet }{W(p,t)}$ , and each $tot\\_decr(p,qq,k)$ , $qq=\\sum _{del(p,q,t,k) \\in X_0 \\cup \\dots X_{7k+5}}{q}$ $\\equiv qq=\\sum _{t \\in X_{7k+4}, p \\in \\bullet t}{W(t,p)} + \\sum _{t \\in X_{7k+4}, p \\in R(t)}{M_{k}(p)}$ , represent the net effect of transition in $T_k$ $eval_{U_{7k+6}}(bot_{U_{7k+7}}(\\Pi ^4) \\setminus bot_{U_{7k+6}}(\\Pi ^4), X_0 \\cup \\dots \\cup X_{7k+6}) = \\lbrace consumesmore(p,k) \\text{:-}.", "\| \\\\ \\lbrace holds(p,q,k), tot\\_decr(p,q1,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{7k+6}, q1 > q \\rbrace \\cup \\lbrace holds(p,q,1) \\text{:-}., \| \\\\ \\lbrace holds(p,q1,k), tot\\_incr(p,q2,k), tot\\_decr(p,q3,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{7k+6}, q=q1+q2-q3 \\rbrace \\cup \\lbrace could\\_not\\_have(t,k) \\text{:-}.", "\| \\lbrace enabled(t,k), ptarc(s,t,q), holds(s,qq,k), \\\\ tot\\_decr(s,qqq,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{7k+6}, fires(t,k) \\notin (X_0 \\cup \\dots \\cup X_{7k+6}), q > qq-qqq \\rbrace $ .", "It's answer set is $X_{7k+7} = A_{15}^{ts=k} \\cup A_{17}^{ts=k} \\cup A_{18}^{ts=k}$ – using forced atom proposition and definitions of $A_{15}, A_{17}, A_{18}, A_9$ .", "where, $consumesmore(p,k)$ represents $\\exists p : q=M_k(p), q1= \\\\ \\sum _{t \\in T_k, p \\in \\bullet t}{W(p,t)}+\\sum _{t \\in T_k, p \\in R(t)}{M_k(p)}, q1 > q$ , indicating place $p$ that will be over consumed if $T_k$ is fired, as defined in definition REF (conflicting transitions), $holds(p,q,k+1)$ represents $q=M_{k+1}(p)$ – by construction of $\\Pi ^4$ , and $could\\_not\\_have(t,k)$ represents enabled transition $t$ in $T_k$ that could not be fired due to insufficient tokens $X_{7k+7}$ does not contain $could\\_not\\_have(t,k)$ , when $enabled(t,k) \\in X_0 \\cup \\dots \\cup X_{7k+6}$ and $fires(t,k) \\notin X_0 \\cup \\dots \\cup X_{7k+6}$ due to construction of $A$ , encoding of $a\\ref {a:r:maxfire:cnh}$ and its body atoms.", "As a result it is not eliminated by the constraint $a\\ref {a:maxfire:elim}$ $eval_{U_{7k+7}}(bot_{U_{7k+8}}(\\Pi ^4) \\setminus bot_{U_{7k+7}}(\\Pi ^4), X_0 \\cup \\dots \\cup X_{7k+7}) = \\lbrace consumesmore \\text{:-}.", "\| \\\\ \\lbrace consumesmore(p,0),\\dots , $ $consumesmore(p,k) \\rbrace \\cap (X_0 \\cup \\dots \\cup X_{7k+7}) \\ne \\emptyset \\rbrace $ .", "It's answer set is $X_{7k+8} = A_{16}$ – using forced atom proposition $X_{7k+8}$ will be empty since none of $consumesmore(p,0),\\dots , \\\\ consumesmore(p,k)$ hold in $X_0 \\cup \\dots \\cup X_{7k+8}$ due to the construction of $A$ , encoding of $a\\ref {a:overc:place}$ and its body atoms.", "As a result, it is not eliminated by the constraint $a\\ref {a:overc:elim}$ The set $X = X_0 \\cup \\dots \\cup X_{7k+8}$ is the answer set of $\\Pi ^4$ by the splitting sequence theorem REF .", "Each $X_i, 0 \\le i \\le 7k+8$ matches a distinct portion of $A$ , and $X = A$ , thus $A$ is an answer set of $\\Pi ^4$ .", "Next we show (REF ): Given $\\Pi ^4$ be the encoding of a Petri Net $PN(P,T,E,W,R,I)$ with initial marking $M_0$ , and $A$ be an answer set of $\\Pi ^4$ that satisfies (REF ) and (REF ), then we can construct $X=M_0,T_0,\\dots ,M_k,T_k,M_{k+1}$ from $A$ , such that it is an execution sequence of $PN$ .", "We construct the $X$ as follows: $M_i = (M_i(p_0), \\dots , M_i(p_n))$ , where $\\lbrace holds(p_0,M_i(p_0),i), \\dots holds(p_n,M_i(p_n),i) \\rbrace \\\\ \\subseteq A$ , for $0 \\le i \\le k+1$ $T_i = \\lbrace t : fires(t,i) \\in A\\rbrace $ , for $0 \\le i \\le k$ and show that $X$ is indeed an execution sequence of $PN$ .", "We show this by induction over $k$ (i.e.", "given $M_k$ , $T_k$ is a valid firing set and its firing produces marking $M_{k+1}$ ).", "Base case: Let $k=0$ , and $M_0$ is a valid marking in $X$ for $PN$ , show [(1)] $T_0$ is a valid firing set for $M_0$ , and firing of $T_0$ in $M_0$ produces marking $M_1$ .", "We show $T_0$ is a valid firing set for $M_0$ .", "Let $\\lbrace fires(t_0,0), \\dots , fires(t_x,0) \\rbrace $ be the set of all $fires(\\dots ,0)$ atoms in $A$ , Then for each $fires(t_i,0) \\in A$ $enabled(t_i,0) \\in A$ – from rule $a\\ref {a:fires}$ and supported rule proposition Then $notenabled(t_i,0) \\notin A$ – from rule $e\\ref {e:enabled}$ and supported rule proposition Then either of $body(e\\ref {e:r:ne:ptarc} )$ , $body(e\\ref {e:ne:iptarc} )$ , or $body(e\\ref {e:ne:tptarc} )$ must not hold in $A$ – from rules $e\\ref {e:r:ne:ptarc} ,e\\ref {e:ne:iptarc} ,e\\ref {e:ne:tptarc} $ and forced atom proposition Then $q \\lnot < n_i \\equiv q \\ge n_i$ in $e\\ref {e:r:ne:ptarc} $ for all $\\lbrace holds(p,q,0), ptarc(p,t_i,n_i,0)\\rbrace \\subseteq A$ – from $e\\ref {e:r:ne:ptarc} $ , forced atom proposition, and given facts ($holds(p,q,0) \\in A, ptarc(p,t_i,n_i,0) \\in A$ ) And $q \\lnot \\ge n_i \\equiv q < n_i$ in $e\\ref {e:ne:iptarc} $ for all $\\lbrace holds(p,q,0), iptarc(p,t_i,n_i,0) \\rbrace \\subseteq A, n_i=1$ ; $q > n_i \\equiv q = 0$ – from $e\\ref {e:ne:iptarc} $ , forced atom proposition, given facts ($holds(p,q,0) \\in A, iptarc(p,t_i,1,0) \\in A$ ), and $q$ is a positive integer And $q \\lnot < n_i \\equiv q \\ge n_i$ in $e\\ref {e:ne:tptarc} $ for all $\\lbrace holds(p,q,0), tptarc(p,t_i,n_i,0) \\rbrace \\subseteq A$ – from $e\\ref {e:ne:tptarc} $ , forced atom proposition, and given facts Then $(\\forall p \\in \\bullet t_i, M_0(p) \\ge W(p,t_i)) \\wedge (\\forall p \\in I(t_i), M_0(p) = 0) \\wedge (\\forall (p,t_i) \\in Q, M_0(p) \\ge QW(p,t_i))$ – from the following $holds(p,q,0) \\in A$ represents $q=M_0(p)$ – rule $i\\ref {i:holds}$ construction $ptarc(p,t_i,n_i,0) \\in A$ represents $n_i=W(p,t_i)$ – rule $f\\ref {f:r:ptarc}$ construction; or it represents $n_i = M_0(p)$ – rule $f\\ref {f:rptarc}$ construction; the construction of $f\\ref {f:rptarc}$ ensures that $notenabled(t,0)$ is never true due to the reset arc definition REF of preset $\\bullet t_i$ in $PN$ definition REF of enabled transition in $PN$ Then $t_i$ is enabled and can fire in $PN$ , as a result it can belong to $T_0$ – from definition REF of enabled transition And $consumesmore \\notin A$ , since $A$ is an answer set of $\\Pi ^4$ – from rule $a\\ref {a:overc:elim}$ and supported rule proposition Then $\\nexists consumesmore(p,0) \\in A$ – from rule $a\\ref {a:overc:gen}$ and supported rule proposition Then $\\nexists \\lbrace holds(p,q,0), tot\\_decr(p,q1,0) \\rbrace \\subseteq A : q1>q$ in $body(a\\ref {a:overc:place})$ – from $a\\ref {a:overc:place}$ and forced atom proposition Then $\\nexists p : (\\sum _{t_i \\in \\lbrace t_0,\\dots ,t_x\\rbrace , p \\in \\bullet t_i}{W(p,t_i)}+\\sum _{t_i \\in \\lbrace t_0,\\dots ,t_x\\rbrace , p \\in R(t_i)}{M_0(p)}) > M_0(p)$ – from the following $holds(p,q,0)$ represents $q=M_0(p)$ – from rule $i\\ref {i:holds}$ construction, given $tot\\_decr(p,q1,0) \\in A$ if $\\lbrace del(p,q1_0,t_0,0), \\dots , del(p,q1_x,t_x,0) \\rbrace \\subseteq A$ , where $q1 = q1_0+\\dots +q1_x$ – from $r\\ref {r:totdecr}$ and forced atom proposition $del(p,q1_i,t_i,0) \\in A$ if $\\lbrace fires(t_i,0), ptarc(p,t_i,q1_i,0) \\rbrace \\subseteq A$ – from $r\\ref {r:r:del} $ and supported rule proposition $del(p,q1_i,t_i,0)$ represents removal of $q1_i = W(p,t_i)$ tokens from $p \\in \\bullet t_i$ ; or it represents removal of $q1_i = M_0(p)$ tokens from $p \\in R(t_i)$ – from rule $r\\ref {r:r:del} $ , supported rule proposition, and definition REF of transition execution in $PN$ Then the set of transitions in $T_0$ do not conflict – by the definition REF of conflicting transitions And for each $enabled(t_j,0) \\in A$ and $fires(t_j,0) \\notin A$ , $could\\_not\\_have(t_j,0) \\in A$ , since $A$ is an answer set of $\\Pi ^4$ - from rule $a\\ref {a:maxfire:elim}$ and supported rule proposition Then $\\lbrace enabled(t_j,0), holds(s,qq,0), ptarc(s,t_j,q,0), \\\\ tot\\_decr(s,qqq,0) \\rbrace \\subseteq A$ , such that $q > qq - qqq$ and $fires(t_j,0) \\notin A$ - from rule $a\\ref {a:r:maxfire:cnh}$ and supported rule proposition Then for an $s \\in \\bullet t_j \\cup R(t_j)$ , $q > M_0(s) - (\\sum _{t_i \\in T_0, s \\in \\bullet t_i}{W(s,t_i)} + \\sum _{t_i \\in T_0, s \\in R(t_i)}{M_0(s))}$ , where $q=W(s,t_j) \\text{~if~} s \\in \\bullet t_j, \\text{~or~} M_0(s) \\text{~otherwise}$ - from the following: $ptarc(s,t_i,q,0)$ represents $q=W(s,t_i)$ if $(s,t_i) \\in E^-$ or $q=M_0(s)$ if $s \\in R(t_i)$ – from rule $f\\ref {f:r:ptarc},f\\ref {f:rptarc}$ construction $holds(s,qq,0)$ represents $qq=M_0(s)$ – from $i\\ref {i:holds}$ construction $tot\\_decr(s,qqq,0) \\in A$ if $\\lbrace del(s,qqq_0,t_0,0), \\dots , del(s,qqq_x,t_x,0) \\rbrace \\subseteq A$ – from rule $r\\ref {r:totdecr}$ construction and supported rule proposition $del(s,qqq_i,t_i,0) \\in A$ if $\\lbrace fires(t_i,0), ptarc(s,t_i,qqq_i,0) \\rbrace \\subseteq A$ – from rule $r\\ref {r:r:del} $ and supported rule proposition $del(s,qqq_i,t_i,0)$ represents $qqq_i = W(s,t_i) : t_i \\in T_0, (s,t_i) \\in E^-$ , or $qqq_i = M_0(t_i) : t_i \\in T_0, s \\in R(t_i)$ – from rule $f\\ref {f:r:ptarc},f\\ref {f:rptarc}$ construction $tot\\_decr(q,qqq,0)$ represents $\\sum _{t_i \\in T_0, s \\in \\bullet t_i}{W(s,t_i)} + \\\\ \\sum _{t_i \\in T_0, s \\in R(t_i)}{M_0(s)}$ – from (C,D,E) above Then firing $T_0 \\cup \\lbrace t_j \\rbrace $ would have required more tokens than are present at its source place $s \\in \\bullet t_j \\cup R(t_j)$ .", "Thus, $T_0$ is a maximal set of transitions that can simultaneously fire.", "And for each reset transition $t_r$ with $enabled(t_r,0) \\in A$ , $fires(t_r,0) \\in A$ , since $A$ is an answer set of $\\Pi ^2$ - from rule $f\\ref {f:c:rptarc:elim}$ and supported rule proposition Then, the firing set $T_0$ satisfies the reset-transition requirement of definition REF (firing set) Then $\\lbrace t_0, \\dots , t_x\\rbrace = T_0$ – using 1(a),1(b),1(d) above; and using 1(c) it is a maximal firing set We show $M_1$ is produced by firing $T_0$ in $M_0$ .", "Let $holds(p,q,1) \\in A$ Then $\\lbrace holds(p,q1,0), tot\\_incr(p,q2,0), tot\\_decr(p,q3,0) \\rbrace \\subseteq A : q=q1+q2-q3$ – from rule $r\\ref {r:nextstate}$ and supported rule proposition Then $holds(p,q1,0) \\in A$ represents $q1=M_0(p)$ – given, rule $i\\ref {i:holds}$ construction; Then $\\lbrace add(p,q2_0,t_0,0), \\dots , add(p,q2_j,t_j,0)\\rbrace \\subseteq A : q2_0 + \\dots + q2_j = q2$ and $\\lbrace del(p,q3_0,t_0,0), \\dots , del(p,q3_l,t_l,0)\\rbrace \\subseteq A : q3_0 + \\dots + q3_l = q3$ – rules $r\\ref {r:totincr},r\\ref {r:totdecr}$ and supported rule proposition, respectively Then $\\lbrace fires(t_0,0), \\dots , fires(t_j,0) \\rbrace \\subseteq A$ and $\\lbrace fires(t_0,0), \\dots , fires(t_l,0) \\rbrace \\\\ \\subseteq A$ – rules $r\\ref {r:r:add},r\\ref {r:r:del}$ and supported rule proposition, respectively Then $\\lbrace fires(t_0,0), \\dots , fires(t_j,0) \\rbrace \\cup \\lbrace fires(t_0,0), \\dots , fires(t_l,0) \\rbrace \\subseteq A = \\lbrace fires(t_0,0), \\dots , $ $fires(t_x,0) \\rbrace \\subseteq A$ – set union of subsets Then for each $fires(t_x,0) \\in A$ we have $t_x \\in T_0$ – already shown in item REF above Then $q = M_0(p) + \\sum _{t_x \\in T_0 \\wedge p \\in t_x \\bullet }{W(t_x,p)} - (\\sum _{t_x \\in T_0 \\wedge p \\in \\bullet t_x}{W(p,t_x)} + \\\\ \\sum _{t_x \\in T_0 \\wedge p \\in R(t_x)}{M_0(p)})$ – from (REF ) above and the following Each $add(p,q_j,t_j,0) \\in A$ represents $q_j=W(t_j,p)$ for $p \\in t_j \\bullet $ – rule $r\\ref {r:r:add} $ encoding, and definition REF of transition execution in $PN$ Each $del(p,t_y,q_y,0) \\in A$ represents either $q_y=W(p,t_y)$ for $p \\in \\bullet t_y$ , or $q_y=M_0(p)$ for $p \\in R(t_y)$ – from rule $r\\ref {r:r:del} ,f\\ref {f:r:ptarc} $ encoding and definition REF of transition execution in $PN$ ; or from rule $r\\ref {r:r:del} ,f\\ref {f:rptarc}$ encoding and definition of reset arc in $PN$ Each $tot\\_incr(p,q2,0) \\in A$ represents $q2=\\sum _{t_x \\in T_0 \\wedge p \\in t_x \\bullet }{W(t_x,p)}$ – aggregate assignment atom semantics in rule $r\\ref {r:totincr}$ Each $tot\\_decr(p,q3,0) \\in A$ represents $q3=\\sum _{t_x \\in T_0 \\wedge p \\in \\bullet t_x}{W(p,t_x)} + \\sum _{t_x \\in T_0 \\wedge p \\in R(t_x)}{M_0(p)}$ – aggregate assignment atom semantics in rule $r\\ref {r:totdecr}$ Then, $M_1(p) = q$ – since $holds(p,q,1) \\in A$ encodes $q=M_1(p)$ – from construction Inductive Step: Let $k > 0$ , and $M_k$ is a valid marking in $X$ for $PN$ , show [(1)] $T_k$ is a valid firing set for $M_k$ , and firing $T_k$ in $M_k$ produces marking $M_{k+1}$ .", "We show that $T_k$ is a valid firing set in $M_k$ .", "Let $\\lbrace fires(t_0,k), \\dots , fires(t_x,k) \\rbrace $ be the set of all $fires(\\dots ,k)$ atoms in $A$ , Then for each $fires(t_i,k) \\in A$ $enabled(t_i,k) \\in A$ – from rule $a\\ref {a:fires}$ and supported rule proposition Then $notenabled(t_i,k) \\notin A$ – from rule $e\\ref {e:enabled}$ and supported rule proposition Then either of $body(e\\ref {e:r:ne:ptarc} )$ , $body(e\\ref {e:ne:iptarc} )$ , or $body(e\\ref {e:ne:tptarc} )$ must not hold in $A$ – from rule $e\\ref {e:r:ne:ptarc} ,e\\ref {e:ne:iptarc} ,e\\ref {e:ne:tptarc} $ and forced atom proposition Then $q \\lnot < n_i \\equiv q \\ge n_i$ in $e\\ref {e:r:ne:ptarc}$ for all $\\lbrace holds(p,q,k), ptarc(p,t_i,n_i,k)\\rbrace \\subseteq A$ – from $e\\ref {e:r:ne:ptarc}$ , forced atom proposition, and given facts ($holds(p,q,k) \\in A, ptarc(p,t_i,n_i,k) \\in A$ ) And $q \\lnot \\ge n_i \\equiv q < n_i$ in $e\\ref {e:ne:iptarc} $ for all $\\lbrace holds(p,q,k), iptarc(p,t_i,n_i,k) \\rbrace \\subseteq A, n_i=1; q > n_i \\equiv q = 0$ – from $e\\ref {e:ne:iptarc} $ , forced atom proposition, given facts $(holds(p,q,k) \\in A, iptarc(p,t_i,1,k) \\in A)$ , and $q$ is a positive integer And $q \\lnot < n_i \\equiv q \\ge n_i$ in $e\\ref {e:ne:tptarc} $ for all $\\lbrace holds(p,q,k), tptarc(p,t_i,n_i,k) \\rbrace \\subseteq A$ – from $e\\ref {e:ne:tptarc} $ , forced atom proposition, and given facts Then $(\\forall p \\in \\bullet t_i, M_k(p) \\ge W(p,t_i)) \\wedge (\\forall p \\in I(t_i), M_k(p) = 0) \\wedge (\\forall (p,t_i) \\in Q, M_k(p) \\ge QW(p,t_i))$ – from $holds(p,q,k) \\in A$ represents $q=M_k(p)$ – inductive assumption, given $ptarc(p,t_i,n_i,k) \\in A$ represents $n_i=W(p,t_i)$ – rule $f\\ref {f:r:ptarc}$ construction; or it represents $n_i=M_k(p)$ – rule $f\\ref {f:rptarc}$ construction; the construction of $f\\ref {f:rptarc}$ ensures that $notenabled(t,k)$ is never true due to the reset arc definition REF of preset $\\bullet t_i$ in $PN$ definition REF of enabled transition in $PN$ Then $t_i$ is enabled and can fire in $PN$ , as a result it can belong to $T_k$ – from definition REF of enabled transition And $consumesmore \\notin A$ , since $A$ is an answer set of $\\Pi ^4$ – from rule $a\\ref {a:overc:elim}$ and supported rule proposition Then $\\nexists consumesmore(p,k) \\in A$ – from rule $a\\ref {a:overc:gen}$ and supported rule proposition Then $\\nexists \\lbrace holds(p,q,k), tot\\_decr(p,q1,k) \\rbrace \\subseteq A : q1>q$ in $body(e\\ref {e:r:ne:ptarc})$ – from $a\\ref {a:overc:place}$ and forced atom proposition Then $\\nexists p : (\\sum _{t_i \\in \\lbrace t_0,\\dots ,t_x\\rbrace , p \\in \\bullet t_i}{W(p,t_i)}+\\sum _{t_i \\in \\lbrace t_0,\\dots ,t_x\\rbrace , p \\in R(t_i)}{M_k(p)}) > M_k(p)$ – from the following $holds(p,q,k)$ represents $q=M_k(p)$ – inductive assumption, given $tot\\_decr(p,q1,k) \\in A$ if $\\lbrace del(p,q1_0,t_0,k), \\dots , del(p,q1_x,t_x,k) \\rbrace \\subseteq A$ , where $q1 = q1_0+\\dots +q1_x$ – from $r\\ref {r:totdecr}$ and forced atom proposition $del(p,q1_i,t_i,k) \\in A$ if $\\lbrace fires(t_i,k), ptarc(p,t_i,q1_i,k) \\rbrace \\subseteq A$ – from $r\\ref {r:r:del} $ and supported rule proposition $del(p,q1_i,t_i,k)$ either represents removal of $q1_i = W(p,t_i)$ tokens from $p \\in \\bullet t_i$ ; or it represents removal of $q1_i = M_k(p)$ tokens from $p \\in R(t_i)$ – from rule $r\\ref {r:r:del} $ , supported rule proposition, and definition REF of transition execution in $PN$ Then $T_k$ does not contain conflicting transitions – by the definition REF of conflicting transitions And for each $enabled(t_j,k) \\in A$ and $fires(t_j,k) \\notin A$ , $could\\_not\\_have(t_j,k) \\in A$ , since $A$ is an answer set of $\\Pi ^4$ - from rule $a\\ref {a:maxfire:elim}$ and supported rule proposition Then $\\lbrace enabled(t_j,k), holds(s,qq,k), ptarc(s,t_j,q,k), \\\\ tot\\_decr(s,qqq,k) \\rbrace \\subseteq A$ , such that $q > qq - qqq$ and $fires(t_j,k) \\notin A$ - from rule $a\\ref {a:r:maxfire:cnh}$ and supported rule proposition Then for an $s \\in \\bullet t_j \\cup R(t_j)$ , $q > M_k(s) - (\\sum _{t_i \\in T_k, s \\in \\bullet t_i}{W(s,t_i)} + \\sum _{t_i \\in T_k, s \\in R(t_i)}{M_k(s))}$ , where $q=W(s,t_j) \\text{~if~} s \\in \\bullet t_j, \\text{~or~} M_k(s) \\text{~otherwise}$ - from the following: $ptarc(s,t_i,q,k)$ represents $q=W(s,t_i)$ if $(s,t_i) \\in E^-$ or $q=M_k(s)$ if $s \\in R(t_i)$ – from rule $f\\ref {f:r:ptarc},f\\ref {f:rptarc}$ construction $holds(s,qq,k)$ represents $qq=M_k(s)$ – construction $tot\\_decr(s,qqq,k) \\in A$ if $\\lbrace del(s,qqq_0,t_0,k), \\dots , del(s,qqq_x,t_x,k) \\rbrace \\\\ \\subseteq A$ – from rule $r\\ref {r:totdecr}$ construction and supported rule proposition $del(s,qqq_i,t_i,k) \\in A$ if $\\lbrace fires(t_i,k), ptarc(s,t_i,qqq_i,k) \\rbrace \\subseteq A$ – from rule $r\\ref {r:r:del} $ and supported rule proposition $del(s,qqq_i,t_i,k)$ represents $qqq_i = W(s,t_i) : t_i \\in T_k, (s,t_i) \\in E^-$ , or $qqq_i = M_k(t_i) : t_i \\in T_k, s \\in R(t_i)$ – from rule $f\\ref {f:r:ptarc},f\\ref {f:rptarc}$ construction $tot\\_decr(q,qqq,k)$ represents $\\sum _{t_i \\in T_k, s \\in \\bullet t_i}{W(s,t_i)} + \\\\ \\sum _{t_i \\in T_k, s \\in R(t_i)}{M_k(s)}$ – from (C,D,E) above Then firing $T_k \\cup \\lbrace t_j \\rbrace $ would have required more tokens than are present at its source place $s \\in \\bullet t_j \\cup R(t_j)$ .", "Thus, $T_k$ is a maximal set of transitions that can simultaneously fire.", "And for each reset transition $t_r$ with $enabled(t_r,k) \\in A$ , $fires(t_r,k) \\in A$ , since $A$ is an answer set of $\\Pi ^2$ - from rule $f\\ref {f:c:rptarc:elim}$ and supported rule proposition Then the firing set $T_k$ satisfies the reset transition requirement of definition REF (firing set) Then $\\lbrace t_0, \\dots , t_x\\rbrace = T_k$ – using 1(a),1(b), 1(d) above; and using 1(c) it is a maximal firing set We show that $M_{k+1}$ is produced by firing $T_k$ in $M_k$ .", "Let $holds(p,q,k+1) \\in A$ Then $\\lbrace holds(p,q1,k), tot\\_incr(p,q2,k), tot\\_decr(p,q3,k) \\rbrace \\subseteq A : q=q1+q2-q3$ – from rule $r\\ref {r:nextstate}$ and supported rule proposition Then $holds(p,q1,k) \\in A$ represents $q1=M_k(p)$ – construction, inductive assumption; and $\\lbrace add(p,q2_0,t_0,k), \\dots , add(p,q2_j,t_j,k)\\rbrace \\subseteq A : q2_0 + \\dots + q2_j = q2$ and $\\lbrace del(p,q3_0,t_0,k), \\dots , del(p,q3_l,t_l,k)\\rbrace \\subseteq A : q3_0 + \\dots + q3_l = q3$ – rules $r\\ref {r:totincr},r\\ref {r:totdecr}$ and supported rule proposition, respectively Then $\\lbrace fires(t_0,k), \\dots , fires(t_j,k) \\rbrace \\subseteq A$ and $\\lbrace fires(t_0,k), \\dots , fires(t_l,k) \\rbrace \\\\ \\subseteq A$ – rules $r\\ref {r:r:add},r\\ref {r:r:del}$ and supported rule proposition, respectively Then $\\lbrace fires(t_0,k), \\dots , $ $fires(t_j,k) \\rbrace \\cup \\lbrace fires(t_0,k), \\dots , $ $fires(t_l,k) \\rbrace \\subseteq A = \\lbrace fires(t_0,k), \\dots , $ $fires(t_x,k) \\rbrace \\subseteq A$ – set union of subsets Then for each $fires(t_x,k) \\in A$ we have $t_x \\in T_k$ – already shown in item REF above Then $q = M_k(p) + \\sum _{t_x \\in T_0 \\wedge p \\in t_x \\bullet }{W(t_x,p)} - (\\sum _{t_x \\in T_k \\wedge p \\in \\bullet t_x}{W(p,t_x)} + \\\\ \\sum _{t_x \\in T_k \\wedge p \\in R(t_x)}{M_k(p)})$ – from (REF ) above and the following Each $add(p,q_j,t_j,k) \\in A$ represents $q_j=W(t_j,p)$ for $p \\in t_j \\bullet $ – rule $r\\ref {r:r:add} $ encoding, and definition REF of transition execution in $PN$ Each $del(p,t_y,q_y,k) \\in A$ represents either $q_y=W(p,t_y)$ for $p \\in \\bullet t_y$ , or $q_y=M_k(p)$ for $p \\in R(t_y)$ – from rule $r\\ref {r:r:del} ,f\\ref {f:r:ptarc} $ encoding and definition REF of transition execution in $PN$ ; or from rule $r\\ref {r:r:del} ,f\\ref {f:rptarc}$ encoding and definition of reset arc in $PN$ Each $tot\\_incr(p,q2,k) \\in A$ represents $q2=\\sum _{t_x \\in T_k \\wedge p \\in t_x \\bullet }{W(t_x,p)}$ – aggregate assignment atom semantics in rule $r\\ref {r:totincr}$ Each $tot\\_decr(p,q3,0) \\in A$ represents $q3=\\sum _{t_x \\in T_k \\wedge p \\in \\bullet t_x}{W(p,t_x)} + \\sum _{t_x \\in T_k \\wedge p \\in R(t_x)}{M_0(p)}$ – aggregate assignment atom semantics in rule $r\\ref {r:totdecr}$ Then, $M_{k+1}(p) = q$ – since $holds(p,q,k+1) \\in A$ encodes $q=M_{k+1}(p)$ – from construction As a result, for any $n > k$ , $T_n$ will be a valid firing set for $M_n$ and $M_{n+1}$ will be its target marking.", "Conclusion: Since both (REF ) and (REF ) hold, $X=M_0,T_0,M_1,\\dots ,M_k,T_{k+1}$ is an execution sequence of $PN(P,T,E,W,R)$ (w.r.t $M_0$ ) iff there is an answer set $A$ of $\\Pi ^4(PN,M_0,k,ntok)$ such that (REF ) and (REF ) hold.", "Proof of Proposition REF Let $PN=(P,T,E,C,W,R,I,Q,QW)$ be a Petri Net, $M_0$ be its initial marking and let $\\Pi ^5(PN,M_0,k,ntok)$ be the ASP encoding of $PN$ and $M_0$ over a simulation length $k$ , with maximum $ntok$ tokens on any place node, as defined in section REF .", "Then $X=M_0,T_k,M_1,\\dots ,M_k,T_k,M_{k+1}$ is an execution sequence of $PN$ (w.r.t.", "$M_0$ ) iff there is an answer set $A$ of $\\Pi ^5(PN,M_0,k,ntok)$ such that: $\\lbrace fires(t,ts) : t \\in T_{ts}, 0 \\le ts \\le k\\rbrace = \\lbrace fires(t,ts) : fires(t,ts) \\in A \\rbrace $ $\\begin{split}\\lbrace holds(p,q,c,ts) &: p \\in P, c/q = M_{ts}(p), 0 \\le ts \\le k+1 \\rbrace \\\\&= \\lbrace holds(p,q,c,ts) : holds(p,q,c,ts) \\in A \\rbrace \\end{split}$ We prove this by showing that: Given an execution sequence $X$ , we create a set $A$ such that it satisfies (REF ) and (REF ) and show that $A$ is an answer set of $\\Pi ^5$ Given an answer set $A$ of $\\Pi ^5$ , we create an execution sequence $X$ such that (REF ) and (REF ) are satisfied.", "First we show (REF ): Given $PN$ and an execution sequence $X$ of $PN$ , we create a set $A$ as a union of the following sets: $A_1=\\lbrace num(n) : 0 \\le n \\le ntok \\rbrace $ $A_2=\\lbrace time(ts) : 0 \\le ts \\le k\\rbrace $ $A_3=\\lbrace place(p) : p \\in P \\rbrace $ $A_4=\\lbrace trans(t) : t \\in T \\rbrace $ $A_5=\\lbrace color(c) : c \\in C \\rbrace $ $A_6=\\lbrace ptarc(p,t,n_c,c,ts) : (p,t) \\in E^-, c \\in C, n_c=m_{W(p,t)}(c), n_c > 0, 0 \\le ts \\le k \\rbrace $ , where $E^- \\subseteq E$ $A_7=\\lbrace tparc(t,p,n_c,c,ts) : (t,p) \\in E^+, c \\in C, n_c=m_{W(t,p)}(c), n_c > 0, 0 \\le ts \\le k \\rbrace $ , where $E^+ \\subseteq E$ $A_8=\\lbrace holds(p,q_c,c,0) : p \\in P, c \\in C, q_c=m_{M_{0}(p)}(c) \\rbrace $ $A_9=\\lbrace ptarc(p,t,n_c,c,ts) : p \\in R(t), c \\in C, n_c = m_{M_{ts}(p)}, n_c > 0, 0 \\le ts \\le k \\rbrace $ $A_{10}=\\lbrace iptarc(p,t,1,c,ts) : p \\in I(t), c \\in C, 0 \\le ts < k \\rbrace $ $A_{11}=\\lbrace tptarc(p,t,n_c,c,ts) : (p,t) \\in Q, c \\in C, n_c=m_{QW(p,t)}(c), n_c > 0, 0 \\le ts \\le k \\rbrace $ $A_{12}=\\lbrace notenabled(t,ts) : t \\in T, 0 \\le ts \\le k, \\exists c \\in C, (\\exists p \\in \\bullet t, m_{M_{ts}(p)}(c) < m_{W(p,t)}(c)) \\vee (\\exists p \\in I(t), m_{M_{ts}(p)}(c) > 0) \\vee (\\exists (p,t) \\in Q, m_{M_{ts}(p)}(c) < m_{QW(p,t)}(c)) \\rbrace $ per definition REF (enabled transition) $A_{13}=\\lbrace enabled(t,ts) : t \\in T, 0 \\le ts \\le k, \\forall c \\in C, (\\forall p \\in \\bullet t, m_{W(p,t)}(c) \\le m_{M_{ts}(p)}(c)) \\wedge (\\forall p \\in I(t), m_{M_{ts}(p)}(c) = 0) \\wedge (\\forall (p,t) \\in Q, m_{M_{ts}(p)}(c) \\ge m_{QW(p,t)}(c)) \\rbrace $ per definition REF (enabled transition) $A_{14}=\\lbrace fires(t,ts) : t \\in T_{ts}, 0 \\le ts \\le k \\rbrace $ per definition REF (firing set), only an enabled transition may fire $A_{15}=\\lbrace add(p,q_c,t,c,ts) : t \\in T_{ts}, p \\in t \\bullet , c \\in C, q_c=m_{W(t,p)}(c), 0 \\le ts \\le k \\rbrace $ per definition REF (transition execution) $A_{16}=\\lbrace del(p,q_c,t,c,ts) : t \\in T_{ts}, p \\in \\bullet t, c \\in C, q_c=m_{W(p,t)}(c), 0 \\le ts \\le k \\rbrace \\cup \\lbrace del(p,q_c,t,c,ts) : t \\in T_{ts}, p \\in R(t), c \\in C, q_c=m_{M_{ts}(p)}(c), 0 \\le ts \\le k \\rbrace $ per definition REF (transition execution) $A_{17}=\\lbrace tot\\_incr(p,q_c,c,ts) : p \\in P, c \\in C, q_c=\\sum _{t \\in T_{ts}, p \\in t \\bullet }{m_{W(t,p)}(c)}, 0 \\le ts \\le k \\rbrace $ per definition REF (firing set execution) $A_{18}=\\lbrace tot\\_decr(p,q_c,c,ts): p \\in P, c \\in C, q_c=\\sum _{t \\in T_{ts}, p \\in \\bullet t}{m_{W(p,t)}(c)}+ \\\\ \\sum _{t \\in T_{ts}, p \\in R(t) }{m_{M_{ts}(p)}(c)}, 0 \\le ts \\le k \\rbrace $ per definition REF (firing set execution) $A_{19}=\\lbrace consumesmore(p,ts) : p \\in P, c \\in C, q_c=m_{M_{ts}(p)}(c), \\\\ q1_c=\\sum _{t \\in T_{ts}, p \\in \\bullet t}{m_{W(p,t)}(c)} + \\sum _{t \\in T_{ts}, p \\in R(t)}{m_{M_{ts}(p)}(c)}, q1 > q, 0 \\le ts \\le k \\rbrace $ per definition REF (conflicting transitions) $A_{20}=\\lbrace consumesmore : \\exists p \\in P, c \\in C, q_c=m_{M_{ts}(p)}(c), \\\\q1_c=\\sum _{t \\in T_{ts}, p \\in \\bullet t}{m_{W(p,t)}(c)} + \\sum _{t \\in T_{ts}, p \\in R(t)}(m_{M_{ts}(p)}(c)), q1 > q, 0 \\le ts \\le k \\rbrace $ per definition REF (conflicting transitions) per the maximal firing set semantics $A_{21}=\\lbrace holds(p,q_c,c,ts+1) : p \\in P, c \\in C, q_c=m_{M_{ts+1}(p)}(c), 0 \\le ts < k\\rbrace $ , where $M_{ts+1}(p) = M_{ts}(p) - (\\sum _{\\begin{array}{c}t \\in T_{ts}, p \\in \\bullet t\\end{array}}{W(p,t)} + \\\\ \\sum _{t \\in T_{ts}, p \\in R(t)}M_{ts}(p)) + \\sum _{\\begin{array}{c}t \\in T_{ts}, p \\in t \\bullet \\end{array}}{W(t,p)}$ according to definition REF (firing set execution) We show that $A$ satisfies (REF ) and (REF ), and $A$ is an answer set of $\\Pi ^5$ .", "$A$ satisfies (REF ) and (REF ) by its construction above.", "We show $A$ is an answer set of $\\Pi ^5$ by splitting.", "We split $lit(\\Pi ^5)$ into a sequence of $7k+9$ sets: $U_0= head(f\\ref {f:c:place}) \\cup head(f\\ref {f:c:trans}) \\cup head(f\\ref {f:c:col}) \\cup head(f\\ref {f:c:time}) \\cup head(f\\ref {f:c:num}) \\cup head(i\\ref {i:c:init}) = \\lbrace place(p) : p \\in P\\rbrace \\cup \\lbrace trans(t) : t \\in T\\rbrace \\cup \\lbrace col(c) : c \\in C \\rbrace \\cup \\lbrace time(0), \\dots , time(k)\\rbrace \\cup \\lbrace num(0), \\dots , num(ntok)\\rbrace \\cup \\lbrace holds(p,q_c,c,0) : p \\in P, c \\in C, q_c=m_{M_0(p)}(c) \\rbrace $ $U_{7k+1}=U_{7k+0} \\cup head(f\\ref {f:c:ptarc})^{ts=k} \\cup head(f\\ref {f:c:tparc})^{ts=k} \\cup head(f\\ref {f:c:rptarc})^{ts=k} \\cup head(f\\ref {f:c:iptarc})^{ts=k} \\cup head(f\\ref {f:c:tptarc})^{ts=k} = U_{7k+0} \\cup \\lbrace ptarc(p,t,n_c,c,k) : (p,t) \\in E^-, c \\in C, n_c=m_{W(p,t)}(c) \\rbrace \\cup \\lbrace tparc(t,p,n_c,c,k) : (t,p) \\in E^+, c \\in C, n_c=m_{W(t,p)}(c) \\rbrace \\cup \\lbrace ptarc(p,t,n_c,c,k) : p \\in R(t), c \\in C, n_c=m_{M_{k}(p)}(c), n > 0 \\rbrace \\cup \\lbrace iptarc(p,t,1,c,k) : p \\in I(t), c \\in C \\rbrace \\cup \\lbrace tptarc(p,t,n_c,c,k) : (p,t) \\in Q, c \\in C, n_c=m_{QW(p,t)}(c) \\rbrace $ $U_{7k+2}=U_{7k+1} \\cup head(e\\ref {e:c:ne:ptarc})^{ts=k} \\cup head(e\\ref {e:c:ne:iptarc})^{ts=k} \\cup head(e\\ref {e:c:ne:tptarc})^{ts=k} = U_{7k+1} \\cup \\\\ \\lbrace notenabled(t,k) : t \\in T \\rbrace $ $U_{7k+3}=U_{7k+2} \\cup head(e\\ref {e:c:enabled})^{ts=k} = U_{7k+2} \\cup \\lbrace enabled(t,k) : t \\in T \\rbrace $ $U_{7k+4}=U_{7k+3} \\cup head(a\\ref {a:c:fires})^{ts=k} = U_{7k+3} \\cup \\lbrace fires(t,k) : t \\in T \\rbrace $ $U_{7k+5}=U_{7k+4} \\cup head(r\\ref {r:c:add})^{ts=k} \\cup head(r\\ref {r:c:del})^{ts=k} = U_{7k+4} \\cup \\lbrace add(p,q_c,t,c,k) : p \\in P, t \\in T, c \\in C, q_c=m_{W(t,p)}(c) \\rbrace \\cup \\lbrace del(p,q_c,t,c,k) : p \\in P, t \\in T, c \\in C, q_c=m_{W(p,t)}(c) \\rbrace \\cup \\lbrace del(p,q_c,t,c,k) : p \\in P, t \\in T, c \\in C, q_c=m_{M_{k}(p)}(c) \\rbrace $ $U_{7k+6}=U_{7k+5} \\cup head(r\\ref {r:c:totincr})^{ts=k} \\cup head(r\\ref {r:c:totdecr})^{ts=k} = U_{7k+5} \\cup \\lbrace tot\\_incr(p,q_c,c,k) : p \\in P, c \\in C, 0 \\le q_c \\le ntok \\rbrace \\cup \\lbrace tot\\_decr(p,q_c,c,k) : p \\in P, c \\in C, 0 \\le q_c \\le ntok \\rbrace $ $U_{7k+7}=U_{7k+6} \\cup head(a\\ref {a:c:overc:place})^{ts=k} \\cup head(r\\ref {r:c:nextstate})^{ts=k} = U_{7k+6} \\cup \\lbrace consumesmore(p,k) : p \\in P\\rbrace \\cup \\lbrace holds(p,q,k+1) : p \\in P, 0 \\le q \\le ntok \\rbrace $ $U_{7k+8}=U_{7k+7} \\cup head(a\\ref {a:c:overc:gen}) = U_{7k+7} \\cup \\lbrace consumesmore \\rbrace $ where $head(r_i)^{ts=k}$ are head atoms of ground rule $r_i$ in which $ts=k$ .", "We write $A_i^{ts=k} = \\lbrace a(\\dots ,ts) : a(\\dots ,ts) \\in A_i, ts=k \\rbrace $ as short hand for all atoms in $A_i$ with $ts=k$ .", "$U_{\\alpha }, 0 \\le \\alpha \\le 7k+8$ form a splitting sequence, since each $U_i$ is a splitting set of $\\Pi ^5$ , and $\\langle U_{\\alpha }\\rangle _{\\alpha < \\mu }$ is a monotone continuous sequence, where $U_0 \\subseteq U_1 \\dots \\subseteq U_{7k+8}$ and $\\bigcup _{\\alpha < \\mu }{U_{\\alpha }} = lit(\\Pi ^5)$ .", "We compute the answer set of $\\Pi ^5$ using the splitting sets as follows: $bot_{U_0}(\\Pi ^5) = f\\ref {f:place} \\cup f\\ref {f:trans} \\cup f\\ref {f:c:ptarc} \\cup f9 \\cup f\\ref {f:c:tptarc} \\cup i\\ref {i:c:init}$ and $X_0 = A_1 \\cup \\dots \\cup A_5 \\cup A_8$ ($= U_0$ ) is its answer set – using forced atom proposition $eval_{U_0}(bot_{U_1}(\\Pi ^5) \\setminus bot_{U_0}(\\Pi ^5), X_0) = \\lbrace ptarc(p,t,q_c,c,0) \\text{:-}.", "\| c \\in C, q_c=m_{W(p,t)}(c), q_c > 0 \\rbrace \\cup \\lbrace tparc(t,p,q_c,c,0) \\text{:-}.", "\| c \\in C, q_c=m_{W(t,p)}(c), q_c > 0 \\rbrace \\cup \\lbrace ptarc(p,t,q_c,c,0) \\text{:-}.", "\| c \\in C, q_c=m_{M_0(p)}(c), q_c > 0 \\rbrace \\cup \\lbrace iptarc(p,t,1,c,0) \\text{:-}.", "\| c \\in C \\rbrace \\cup \\lbrace tptarc(p,t,q_c,c,0) \\text{:-}.", "\| c \\in C, q_c = m_{QW(p,t)}(c), q_c > 0 \\rbrace $ .", "Its answer set $X_1=A_6^{ts=0} \\cup A_7^{ts=0} \\cup A_9^{ts=0} \\cup A_{10}^{ts=0} \\cup A_{11}^{ts=0}$ – using forced atom proposition and construction of $A_6, A_7, A_9, A_{10}, A_{11}$ .", "$eval_{U_1}(bot_{U_2}(\\Pi ^5) \\setminus bot_{U_1}(\\Pi ^5), X_0 \\cup X_1) = \\lbrace notenabled(t,0) \\text{:-} .", "\| (\\lbrace trans(t), \\\\ ptarc(p,t,n_c,c,0), holds(p,q_c,c,0) \\rbrace \\subseteq X_0 \\cup X_1, \\text{~where~} q_c < n_c) \\text{~or~} \\\\ (\\lbrace notenabled(t,0) \\text{:-} .", "\| (\\lbrace trans(t), iptarc(p,t,n2_c,c,0), holds(p,q_c,c,0) \\rbrace \\subseteq X_0 \\cup X_1, \\\\ \\text{~where~} q_c \\ge n2_c \\rbrace ) \\text{~or~} $ $(\\lbrace trans(t), tptarc(p,t,n3_c,c,0), holds(p,q_c,c,0) \\rbrace \\subseteq X_0 \\cup X_1, \\text{~where~} q_c < n3_c) \\rbrace $ .", "Its answer set $X_2=A_{12}^{ts=0}$ – using forced atom proposition and construction of $A_{12}$ .", "where, $q_c=m_{M_0(p)}(c)$ , and $n_c=m_{W(p,t)}(c)$ for an arc $(p,t) \\in E^-$ – by construction of $i\\ref {i:c:init}$ and $f\\ref {f:c:ptarc}$ in $\\Pi ^5$ , and in an arc $(p,t) \\in E^-$ , $p \\in \\bullet t$ (by definition REF of preset) $n2_c=1$ – by construction of $iptarc$ predicates in $\\Pi ^5$ , meaning $q_c \\ge n2_c \\equiv q_c \\ge 1 \\equiv q_c > 0$ , $tptarc(p,t,n3_c,c,0)$ represents $n3_c=m_{QW(p,t)}(c)$ , where $(p,t) \\in Q$ thus, $notenabled(t,0) \\in X_1$ represents $\\exists c \\in C, (\\exists p \\in \\bullet t : m_{M_0(p)}(c) < m_{W(p,t)}(c)) \\vee (\\exists p \\in I(t) : m_{M_0(p)}(c) > 0) \\vee (\\exists (p,t) \\in Q : m_{M_0(p)}(c) < m_{QW(p,t)}(c))$ .", "$eval_{U_2}(bot_{U_3}(\\Pi ^5) \\setminus bot_{U_2}(\\Pi ^5), X_0 \\cup \\dots \\cup X_2) = \\lbrace enabled(t,0) \\text{:-}.", "\| trans(t) \\in X_0 \\cup \\dots \\cup X_2, notenabled(t,0) \\notin X_0 \\cup \\dots \\cup X_2 \\rbrace $ .", "Its answer set is $X_3 = A_{13}^{ts=0}$ – using forced atom proposition and construction of $A_{13}$ .", "where, an $enabled(t,0) \\in X_3$ if $\\nexists ~notenabled(t,0) \\in X_0 \\cup \\dots \\cup X_2$ ; which is equivalent to $\\forall c \\in C, (\\nexists p \\in \\bullet t : m_{M_0(p)}(c) < m_{W(p,t)}(c)) \\wedge (\\nexists p \\in I(t) : m_{M_0(p)}(c) > 0) \\wedge (\\nexists (p,t) \\in Q : m_{M_0(p)}(c) < m_{QW(p,t)}(c) ) \\equiv \\forall c \\in C, (\\forall p \\in \\bullet t: m_{M_0(p)}(c) \\ge m_{W(p,t)}(c)) \\wedge (\\forall p \\in I(t) : m_{M_0(p)}(c) = 0)$ .", "$eval_{U_3}(bot_{U_4}(\\Pi ^5) \\setminus bot_{U_3}(\\Pi ^5), X_0 \\cup \\dots \\cup X_3) = \\lbrace \\lbrace fires(t,0)\\rbrace \\text{:-}.", "\| enabled(t,0) \\\\ \\text{~holds in~} X_0 \\cup \\dots \\cup X_3 \\rbrace $ .", "It has multiple answer sets $X_{4.1}, \\dots , X_{4.n}$ , corresponding to elements of power set of $fires(t,0)$ atoms in $eval_{U_3}(...)$ – using supported rule proposition.", "Since we are showing that the union of answer sets of $\\Pi ^5$ determined using splitting is equal to $A$ , we only consider the set that matches the $fires(t,0)$ elements in $A$ and call it $X_4$ , ignoring the rest.", "Thus, $X_4 = A_{14}^{ts=0}$ , representing $T_0$ .", "in addition, for every $t$ such that $enabled(t,0) \\in X_0 \\cup \\dots \\cup X_3, R(t) \\ne \\emptyset $ ; $fires(t,0) \\in X_4$ – per definition REF (firing set); requiring that a reset transition is fired when enabled thus, the firing set $T_0$ will not be eliminated by the constraint $f\\ref {f:c:rptarc:elim}$ $eval_{U_4}(bot_{U_5}(\\Pi ^5) \\setminus bot_{U_4}(\\Pi ^5), X_0 \\cup \\dots \\cup X_4) = \\lbrace add(p,n_c,t,c,0) \\text{:-}.", "\| \\lbrace fires(t,0), \\\\ tparc(t,p,n_c,c,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_4 \\rbrace \\cup \\lbrace del(p,n_c,t,c,0) \\text{:-}.", "\| \\lbrace fires(t,0), \\\\ ptarc(p,t,n_c,c,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_4 \\rbrace $ .", "It's answer set is $X_5 = A_{15}^{ts=0} \\cup A_{16}^{ts=0}$ – using forced atom proposition and definitions of $A_{15}$ and $A_{16}$ .", "where, each $add$ atom is equivalent to $n_c=m_{W(t,p)}(c),c \\in C, p \\in t \\bullet $ , and each $del$ atom is equivalent to $n_c=m_{W(p,t)}(c), c \\in C, p \\in \\bullet t$ ; or $n_c=m_{M_0(p)}(c), c \\in C, p \\in R(t)$ , representing the effect of transitions in $T_0$ $eval_{U_5}(bot_{U_6}(\\Pi ^5) \\setminus bot_{U_5}(\\Pi ^5), X_0 \\cup \\dots \\cup X_5) = \\lbrace tot\\_incr(p,qq_c,c,0) \\text{:-}.", "\| qq_c=\\sum _{add(p,q_c,t,c,0) \\in X_0 \\cup \\dots \\cup X_5}{q_c} \\rbrace \\cup \\lbrace tot\\_decr(p,qq_c,c,0) \\text{:-}.", "\| qq_c= \\\\ \\sum _{del(p,q_c,t,c,0) \\in X_0 \\cup \\dots \\cup X_5}{q_c} \\rbrace $ .", "It's answer set is $X_6 = A_{17}^{ts=0} \\cup A_{18}^{ts=0}$ – using forced atom proposition and definitions of $A_{17}$ and $A_{18}$ .", "where, each $tot\\_incr(p,qq_c,c,0)$ , $qq_c=\\sum _{add(p,q_c,t,c,0) \\in X_0 \\cup \\dots X_5}{q_c}$ $\\equiv qq_c=\\sum _{t \\in X_4, p \\in t \\bullet }{m_{W(p,t)}(c)}$ , and each $tot\\_decr(p,qq_c,c,0)$ , $qq_c=\\sum _{del(p,q_c,t,c,0) \\in X_0 \\cup \\dots X_5}{q_c}$ $\\equiv qq=\\sum _{t \\in X_4, p \\in \\bullet t}{m_{W(t,p)}(c)} + \\sum _{t \\in X_4, p \\in R(t)}{m_{M_0(p)}(c)}$ , represent the net effect of transitions in $T_0$ $eval_{U_6}(bot_{U_7}(\\Pi ^5) \\setminus bot_{U_6}(\\Pi ^5), X_0 \\cup \\dots \\cup X_6) = \\lbrace consumesmore(p,0) \\text{:-}.", "\| \\\\ \\lbrace holds(p,q_c,c,0), tot\\_decr(p,q1_c,c,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_6, q1_c > q_c \\rbrace \\cup \\\\ \\lbrace holds(p,q_c,c,1) \\text{:-}., \| \\lbrace holds(p,q1_c,c,0), $ $tot\\_incr(p,q2_c,c,0), $ $tot\\_decr(p,q3_c,c,0) \\rbrace \\\\ \\subseteq X_0 \\cup \\dots \\cup X_6, q_c=q1_c+q2_c-q3_c \\rbrace $ .", "It's answer set is $X_7 = A_{19}^{ts=0} \\cup A_{21}^{ts=0}$ – using forced atom proposition and definitions of $A_{19}, A_{21}$ .", "where, $consumesmore(p,0)$ represents $\\exists p : q_c=m_{M_0(p)}(c), q1_c= \\\\ \\sum _{t \\in T_0, p \\in \\bullet t}{m_{W(p,t)}(c)}+\\sum _{t \\in T_0, p \\in R(t)}{m_{M_0(p)}(c)}, q1_c > q_c, c \\in C$ , indicating place $p$ will be over consumed if $T_0$ is fired, as defined in definition REF (conflicting transitions), $holds(p,q_c,c,1)$ represents $q_c=m_{M_1(p)}(c)$ – by construction of $\\Pi ^5$ $ \\vdots $ $eval_{U_{7k+0}}(bot_{U_{7k+1}}(\\Pi ^5) \\setminus bot_{U_{7k+0}}(\\Pi ^5), X_0 \\cup \\dots \\cup X_{7k+0}) = \\lbrace ptarc(p,t,q_c,c,k) \\text{:-}.", "\| c \\in C, q_c=m{W(p,t)}(c), q_c > 0 \\rbrace \\cup \\lbrace tparc(t,p,q_c,c,k) \\text{:-}.", "\| c \\in C, q_c=m_{W(t,p)}(c), q_c > 0 \\rbrace \\cup $ $\\lbrace ptarc(p,t,q_c,c,k) \\text{:-}.", "\| $ $c \\in C, q_c=m_{M_k(p)}(c), q_c > 0 \\rbrace \\cup \\lbrace iptarc(p,t,1,c,k) \\text{:-}.", "\| c \\in C \\rbrace \\cup \\lbrace tptarc(p,t,q_c,c,k) \\text{:-}.", "\| c \\in C, q_c = m_{QW(p,t)}(c), q_c > 0 \\rbrace $ .", "Its answer set $X_{7k+1}=A_6^{ts=k} \\cup A_7^{ts=k} \\cup A_9^{ts=k} \\cup A_{10}^{ts=k} \\cup A_{11}^{ts=k}$ – using forced atom proposition and construction of $A_6, A_7, A_9, A_{10}, A_{11}$ .", "$eval_{U_{7k+1}}(bot_{U_{7k+2}}(\\Pi ^5) \\setminus bot_{U_{7k+1}}(\\Pi ^5), X_0 \\cup \\dots \\cup X_{7k+1}) = \\lbrace notenabled(t,k) \\text{:-} .", "\| \\\\ (\\lbrace trans(t), ptarc(p,t,n_c,c,k), holds(p,q_c,c,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{7k+1}, \\text{~where~} q_c < n_c) \\text{~or~} $ $ (\\lbrace notenabled(t,k) \\text{:-} .", "\| (\\lbrace trans(t), iptarc(p,t,n2_c,c,k), holds(p,q_c,c,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{7k+1}, \\text{~where~} q_c \\ge n2_c \\rbrace ) \\text{~or~} (\\lbrace trans(t), tptarc(p,t,n3_c,c,k), \\\\ holds(p,q_c,c,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{7k+1}, \\text{~where~} q_c < n3_c) \\rbrace $ .", "Its answer set $X_{7k+2}=A_{12}^{ts=k}$ – using forced atom proposition and construction of $A_{12}$ .", "since $q_c=m_{M_k(p)}(c)$ , and $n_c=m_{W(p,t)}(c)$ for an arc $(p,t) \\in E^-$ – by construction of $holds$ and $ptarc$ predicates in $\\Pi ^5$ , and in an arc $(p,t) \\in E^-$ , $p \\in \\bullet t$ (by definition REF of preset) $n2_c=1$ – by construction of $iptarc$ predicates in $\\Pi ^5$ , meaning $q_c \\ge n2_c \\equiv q_c \\ge 1 \\equiv q_c > 0$ , $tptarc(p,t,n3_c,c,k)$ represents $n3_c=m_{QW(p,t)}(c)$ , where $(p,t) \\in Q$ thus, $notenabled(t,k) \\in X_{8k+1}$ represents $\\exists c \\in C (\\exists p \\in \\bullet t : m_{M_k(p)}(c) < m_{W(p,t)}(c)) \\vee (\\exists p \\in I(t) : m_{M_k(p)}(c) > 0) \\vee (\\exists (p,t) \\in Q : m_{M_{ts}(p)}(c) < m_{QW(p,t)}(c))$ .", "$eval_{U_{7k+2}}(bot_{U_{7k+3}}(\\Pi ^5) \\setminus bot_{U_{7k+2}}(\\Pi ^5), X_0 \\cup \\dots \\cup X_{7k+2}) = \\lbrace enabled(t,k) \\text{:-}.", "\| trans(t) \\in X_0 \\cup \\dots \\cup X_{7k+2} \\wedge notenabled(t,k) \\notin X_0 \\cup \\dots \\cup X_{7k+2} \\rbrace $ .", "Its answer set is $X_{7k+3} = A_{13}^{ts=k}$ – using forced atom proposition and construction of $A_{13}$ .", "since an $enabled(t,k) \\in X_{7k+3}$ if $\\nexists ~notenabled(t,k) \\in X_0 \\cup \\dots \\cup X_{7k+2}$ ; which is equivalent to $\\forall c \\in C, (\\nexists p \\in \\bullet t : m_{M_k(p)}(c) < m_{W(p,t)}(c)) \\wedge (\\nexists p \\in I(t) : m_{M_k(p)}(c) > 0) \\wedge (\\nexists (p,t) \\in Q : m_{M_k(p)}(c) < m_{QW(p,t)}(c) ) \\equiv \\forall c \\in C, (\\forall p \\in \\bullet t: m_{M_k(p)}(c) \\ge m_{W(p,t)}(c)) \\wedge (\\forall p \\in I(t) : m_{M_k(p)}(c) = 0)$ .", "$eval_{U_{7k+3}}(bot_{U_{7k+4}}(\\Pi ^5) \\setminus bot_{U_{7k+3}}(\\Pi ^5), X_0 \\cup \\dots \\cup X_{7k+3}) = \\lbrace \\lbrace fires(t,k)\\rbrace \\text{:-}.", "\| \\\\ enabled(t,k) \\text{~holds in~} X_0 \\cup \\dots \\cup X_{7k+3} \\rbrace $ .", "It has multiple answer sets $X_{7k+}{4.1}, \\dots , X_{7k+}{4.n}$ , corresponding to elements of power set of $fires(t,k)$ atoms in $eval_{U_3}(...)$ – using supported rule proposition.", "Since we are showing that the union of answer sets of $\\Pi ^5$ determined using splitting is equal to $A$ , we only consider the set that matches the $fires(t,k)$ elements in $A$ and call it $X_{7k+4}$ , ignoring the rest.", "Thus, $X_{7k+4} = A_{14}^{ts=k}$ , representing $T_k$ .", "in addition, for every $t$ such that $enabled(t,k) \\in X_0 \\cup \\dots \\cup X_{7k+3}, R(t) \\ne \\emptyset $ ; $fires(t,k) \\in X_{7k+4}$ – per definition REF (firing set); requiring that a reset transition is fired when enabled thus, the firing set $T_k$ will not be eliminated by the constraint $f\\ref {f:c:rptarc:elim}$ $eval_{U_{7k+4}}(bot_{U_{7k+5}}(\\Pi ^5) \\setminus bot_{U_{7k+4}}(\\Pi ^5), X_0 \\cup \\dots \\cup X_{7k+4}) = \\lbrace add(p,n_c,t,c,k) \\text{:-}.", "\| $ $ \\lbrace fires(t,k), \\\\ tparc(t,p,n_c,c,k) \\rbrace \\subseteq $ $X_0 \\cup \\dots \\cup X_{7k+4} \\rbrace \\cup $ $\\lbrace del(p,n_c,t,c,k) \\text{:-}.", "\| $ $\\lbrace fires(t,k), \\\\ ptarc(p,t,n_c,c,k) \\rbrace \\subseteq $ $X_0 \\cup \\dots \\cup X_{7k+4} \\rbrace $ .", "It's answer set is $X_{7k+5} = A_{15}^{ts=k} \\cup A_{16}^{ts=k}$ – using forced atom proposition and definitions of $A_{15}$ and $A_{16}$ .", "where each $add$ atom is equivalent to $n_c=m_{W(t,p)}(c),c \\in C, p \\in t \\bullet $ , and each $del$ atom is equivalent to $n_c=m_{W(p,t)}(c), c \\in C, p \\in \\bullet t$ ; or $n_c=m_{M_k(p)}(c), c \\in C, p \\in R(t)$ , representing the effect of transitions in $T_k$ $eval_{U_{7k+5}}(bot_{U_{7k+6}}(\\Pi ^5) \\setminus bot_{U_{7k+5}}(\\Pi ^5), X_0 \\cup \\dots \\cup X_{7k+5}) = \\lbrace tot\\_incr(p,qq_c,c,k) \\text{:-}.", "\| $ $qq=\\sum _{add(p,q_c,t,c,k) \\in X_0 \\cup \\dots \\cup X_{7k+5}}{q_c} \\rbrace \\cup $ $ \\lbrace tot\\_decr(p,qq_c,c,k) \\text{:-}.", "\| \\\\ qq_c=\\sum _{del(p,q_c,t,c,k) \\in X_0 \\cup \\dots \\cup X_{7k+5}}{q_c} \\rbrace $ .", "It's answer set is $X_{7k+6} = A_{17}^{ts=k} \\cup A_{18}^{ts=k}$ – using forced atom proposition and definitions of $A_{17}$ and $A_{18}$ .", "where, each $tot\\_incr(p,qq_c,c,k)$ , $qq_c=\\sum _{add(p,q_c,t,c,k) \\in X_0 \\cup \\dots X_{8k+5}}{q_c}$ $\\equiv qq_c=\\sum _{t \\in X_{7k+4}, p \\in t \\bullet }{m_{W(p,t)}(c)}$ , and each $tot\\_decr(p,qq_c,c,k)$ , $qq_c=\\sum _{del(p,q_c,t,c,k) \\in X_0 \\cup \\dots X_{8k+5}}{q_c}$ $\\equiv qq=\\sum _{t \\in X_{7k+4}, p \\in \\bullet t}{m_{W(t,p)}(c)} + \\sum _{t \\in X_{7k+4}, p \\in R(t)}{m_{M_{k}(p)}(c)}$ , represent the net effect of transitions in $T_k$ $eval_{U_{7k+6}}(bot_{U_{7k+7}}(\\Pi ^5) \\setminus bot_{U_{7k+6}}(\\Pi ^5), X_0 \\cup \\dots \\cup X_{7k+6}) = \\lbrace consumesmore(p,k) \\text{:-}.", "\| \\\\ \\lbrace holds(p,q_c,c,k), tot\\_decr(p,q1_c,c,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{7k+6}, q1 > q \\rbrace \\cup \\\\ \\lbrace holds(p,q_c,c,k+1) \\text{:-}.", "\| \\lbrace holds(p,q1_c,c,k), tot\\_incr(p,q2_c,c,k), \\\\ tot\\_decr(p,q3_c,c,k) \\rbrace \\subseteq $ $X_0 \\cup \\dots \\cup X_{7k+6}, q_c=q1_c+q2_c-q3_c \\rbrace $ .", "It's answer set is $X_{7k+7} = A_{19}^{ts=k} \\cup A_{21}^{ts=k}$ – using forced atom proposition and definitions of $A_{19}, A_{21}$ .", "where, $consumesmore(p,k)$ represents $\\exists p : q_c=m_{M_k(p)}(c), q1_c= \\\\ \\sum _{t \\in T_k, p \\in \\bullet t}{m_{W(p,t)}(c)}+\\sum _{t \\in T_k, p \\in R(t)}{m_{M_k(p)}(c)}, q1_c > q_c, c \\in C$ , indicating place $p$ that will be over consumed if $T_k$ is fired, as defined in definition REF (conflicting transitions), and $holds(p,q_c,c,k+1)$ represents $q_c=m_{M_{k+1}(p)}(c)$ – by construction of $\\Pi ^5$ $eval_{U_{7k+7}}(bot_{U_{7k+8}}(\\Pi ^5) \\setminus bot_{U_{7k+7}}(\\Pi ^5), X_0 \\cup \\dots \\cup X_{7k+7}) = \\lbrace consumesmore \\text{:-}.", "\| \\\\ \\lbrace consumesmore(p,0),\\dots ,$ $consumesmore(p,k) \\rbrace \\cap (X_0 \\cup \\dots \\cup X_{7k+7}) \\ne \\emptyset \\rbrace $ .", "It's answer set is $X_{7k+8} = A_{20}^{ts=k}$ – using forced atom proposition and the definition of $A_{20}$ $X_{7k+7}$ will be empty since none of $consumesmore(p,0),\\dots , \\\\ consumesmore(p,k)$ hold in $X_0 \\cup \\dots \\cup X_{7k+7}$ due to the construction of $A$ , encoding of $a\\ref {a:c:overc:place}$ and its body atoms.", "As a result, it is not eliminated by the constraint $a\\ref {a:c:overc:elim}$ The set $X = X_0 \\cup \\dots \\cup X_{7k+8}$ is the answer set of $\\Pi ^5$ by the splitting sequence theorem REF .", "Each $X_i, 0 \\le i \\le 7k+8$ matches a distinct portion of $A$ , and $X = A$ , thus $A$ is an answer set of $\\Pi ^5$ .", "Next we show (REF ): Given $\\Pi ^5$ be the encoding of a Petri Net $PN(P,T,E,C,W,R,I,$ $Q,WQ)$ with initial marking $M_0$ , and $A$ be an answer set of $\\Pi ^5$ that satisfies (REF ) and (REF ), then we can construct $X=M_0,T_k,\\dots ,M_k,T_k,M_{k+1}$ from $A$ , such that it is an execution sequence of $PN$ .", "We construct the $X$ as follows: $M_i = (M_i(p_0), \\dots , M_i(p_n))$ , where $\\lbrace holds(p_0,m_{M_i(p_0)}(c),c,i), \\dots \\\\ holds(p_n,m_{M_i(p_n)}(c),c,i) \\rbrace \\subseteq A$ , for $c \\in C, 0 \\le i \\le k+1$ $T_i = \\lbrace t : fires(t,i) \\in A\\rbrace $ , for $0 \\le i \\le k$ and show that $X$ is indeed an execution sequence of $PN$ .", "We show this by induction over $k$ (i.e.", "given $M_k$ , $T_k$ is a valid firing set and its firing produces marking $M_{k+1}$ ).", "Base case: Let $k=0$ , and $M_0$ is a valid marking in $X$ for $PN$ , show [(1)] $T_0$ is a valid firing set for $M_0$ , and firing $T_0$ in $M_0$ results in marking $M_1$ .", "We show $T_0$ is a valid firing set for $M_0$ .", "Let $\\lbrace fires(t_0,0), \\dots , fires(t_x,0) \\rbrace $ be the set of all $fires(\\dots ,0)$ atoms in $A$ , Then for each $fires(t_i,0) \\in A$ $enabled(t_i,0) \\in A$ – from rule $a\\ref {a:c:fires}$ and supported rule proposition Then $notenabled(t_i,0) \\notin A$ – from rule $e\\ref {e:c:enabled}$ and supported rule proposition Then either of $body(e\\ref {e:c:ne:ptarc})$ , $body(e\\ref {e:c:ne:iptarc})$ , or $body(e\\ref {e:c:ne:tptarc})$ must not hold in $A$ – from rules $e\\ref {e:c:ne:ptarc},e\\ref {e:c:ne:iptarc},e\\ref {e:c:ne:tptarc}$ and forced atom proposition Then $q_c \\lnot < {n_i}_c \\equiv q_c \\ge {n_i}_c$ in $e\\ref {e:c:ne:ptarc}$ for all $\\lbrace holds(p,q_c,c,0), \\\\ ptarc(p,t_i,{n_i}_c,c,0)\\rbrace \\subseteq A$ – from $e\\ref {e:c:ne:ptarc}$ , forced atom proposition, and given facts ($holds(p,q_c,c,0) \\in A, ptarc(p,t_i,{n_i}_c,0) \\in A$ ) And $q_c \\lnot \\ge {n_i}_c \\equiv q_c < {n_i}_c$ in $e\\ref {e:c:ne:iptarc}$ for all $\\lbrace holds(p,q_c,c,0), \\\\ iptarc(p,t_i,{n_i}_c,c,0) \\rbrace \\subseteq A, {n_i}_c=1$ ; $q_c > {n_i}_c \\equiv q_c = 0$ – from $e\\ref {e:c:ne:iptarc}$ , forced atom proposition, given facts ($holds(p,q_c,c,0) \\in A, iptarc(p,t_i,1,c,0) \\in A$ ), and $q_c$ is a positive integer And $q_c \\lnot < {n_i}_c \\equiv q_c \\ge {n_i}_c$ in $e\\ref {e:c:ne:tptarc}$ for all $\\lbrace holds(p,q_c,c,0), \\\\ tptarc(p,t_i,{n_i}_c,c,0) \\rbrace \\subseteq A$ – from $e\\ref {e:c:ne:tptarc}$ , forced atom proposition, and given facts Then $\\forall c \\in C, (\\forall p \\in \\bullet t_i, m_{M_0(p)}(c) \\ge m_{W(p,t_i)}(c)) \\wedge (\\forall p \\in I(t_i), m_{M_0(p)}(c) = 0) \\wedge (\\forall (p,t_i) \\in Q, m_{M_0(p)}(c) \\ge m_{QW(p,t_i)}(c))$ – from the following $holds(p,q_c,c,0) \\in A$ represents $q_c=m_{M_0(p)}(c)$ – rule $i\\ref {i:c:init}$ construction $ptarc(p,t_i,{n_i}_c,0) \\in A$ represents ${n_i}_c=m_{W(p,t_i)}(c)$ – rule $f\\ref {f:c:ptarc}$ construction; or it represents ${n_i}_c = m_{M_0(p)}(c)$ – rule $f\\ref {f:c:rptarc}$ construction; the construction of $f\\ref {f:c:rptarc}$ ensures that $notenabled(t,0)$ is never true for a reset arc definition REF of preset $\\bullet t_i$ in $PN$ definition REF of enabled transition in $PN$ Then $t_i$ is enabled and can fire in $PN$ , as a result it can belong to $T_0$ – from definition REF of enabled transition And $consumesmore \\notin A$ , since $A$ is an answer set of $\\Pi ^5$ – from rule $a\\ref {a:c:overc:gen}$ and supported rule proposition Then $\\nexists consumesmore(p,0) \\in A$ – from rule $a\\ref {a:c:overc:place}$ and supported rule proposition Then $\\nexists \\lbrace holds(p,q_c,c,0), tot\\_decr(p,q1_c,c,0) \\rbrace \\subseteq A, q1_c>q_c$ in $body(a\\ref {a:c:overc:place})$ – from $a\\ref {a:c:overc:place}$ and forced atom proposition Then $\\nexists c \\in C \\nexists p \\in P, (\\sum _{t_i \\in \\lbrace t_0,\\dots ,t_x\\rbrace , p \\in \\bullet t_i}{m_{W(p,t_i)}(c)}+ \\\\ \\sum _{t_i \\in \\lbrace t_0,\\dots ,t_x\\rbrace , p \\in R(t_i)}{m_{M_0(p)}(c)}) > m_{M_0(p)}(c)$ – from the following $holds(p,q_c,c,0)$ represents $q_c=m_{M_0(p)}(c)$ – from rule $i\\ref {i:c:init}$ construction, given $tot\\_decr(p,q1_c,c,0) \\in A$ if $\\lbrace del(p,{q1_0}_c,t_0,c,0), \\dots , \\\\ del(p,{q1_x}_c,t_x,c,0) \\rbrace \\subseteq A$ , where $q1_c = {q1_0}_c+\\dots +{q1_x}_c$ – from $r\\ref {r:c:totdecr}$ and forced atom proposition $del(p,{q1_i}_c,t_i,c,0) \\in A$ if $\\lbrace fires(t_i,0), ptarc(p,t_i,{q1_i}_c,c,0) \\rbrace \\subseteq A$ – from $r\\ref {r:c:del}$ and supported rule proposition $del(p,{q1_i}_c,t_i,c,0)$ either represents removal of ${q1_i}_c = m_{W(p,t_i)}(c)$ tokens from $p \\in \\bullet t_i$ ; or it represents removal of ${q1_i}_c = m_{M_0(p)}(c)$ tokens from $p \\in R(t_i)$ – from rules $r\\ref {r:c:del},f\\ref {f:c:ptarc},f\\ref {f:c:rptarc}$ , supported rule proposition, and definition REF of transition execution in $PN$ Then the set of transitions in $T_0$ do not conflict – by the definition REF of conflicting transitions And for each reset transition $t_r$ with $enabled(t_r,0) \\in A$ , $fires(t_r,0) \\in A$ , since $A$ is an answer set of $\\Pi ^5$ - from rule $f\\ref {f:c:rptarc:elim}$ and supported rule proposition Then, the firing set $T_0$ satisfies the reset-transition requirement of definition REF (firing set) Then $\\lbrace t_0, \\dots , t_x\\rbrace = T_0$ – using 1(a),1(b),1(d) above; and using 1(c) it is a maximal firing set We show $M_1$ is produced by firing $T_0$ in $M_0$ .", "Let $holds(p,q_c,c,1) \\in A$ Then $\\lbrace holds(p,q1_c,c,0), tot\\_incr(p,q2_c,c,0), tot\\_decr(p,q3_c,c,0) \\rbrace \\subseteq A : q_c=q1_c+q2_c-q3_c$ – from rule $r\\ref {r:c:nextstate}$ and supported rule proposition Then, $holds(p,q1_c,c,0) \\in A$ represents $q1_c=m_{M_0(p)}(c)$ – given, rule $i\\ref {i:c:init}$ construction; and $\\lbrace add(p,{q2_0}_c,t_0,c,0), \\dots , $ $add(p,{q2_j}_c,t_j,c,0)\\rbrace \\subseteq A : {q2_0}_c + \\dots + {q2_j}_c = q2_c$ and $\\lbrace del(p,{q3_0}_c,t_0,c,0), \\dots , $ $del(p,{q3_l}_c,t_l,c,0)\\rbrace \\subseteq A : {q3_0}_c + \\dots + {q3_l}_c = q3_c$ – rules $r\\ref {r:c:totincr},r\\ref {r:c:totdecr}$ and supported rule proposition, respectively Then $\\lbrace fires(t_0,0), \\dots , fires(t_j,0) \\rbrace \\subseteq A$ and $\\lbrace fires(t_0,0), \\dots , fires(t_l,0) \\rbrace $ $\\subseteq A$ – rules $r\\ref {r:c:add},r\\ref {r:c:del}$ and supported rule proposition, respectively Then $\\lbrace fires(t_0,0), \\dots , $ $fires(t_j,0) \\rbrace \\cup $ $\\lbrace fires(t_0,0), \\dots , $ $fires(t_l,0) \\rbrace \\subseteq $ $A = \\lbrace fires(t_0,0), \\dots , $ $fires(t_x,0) \\rbrace \\subseteq A$ – set union of subsets Then for each $fires(t_x,0) \\in A$ we have $t_x \\in T_0$ – already shown in item REF above Then $q_c = m_{M_0(p)}(c) + \\sum _{t_x \\in T_0 \\wedge p \\in t_x \\bullet }{m_{W(t_x,p)}(c)} - (\\sum _{t_x \\in T_0 \\wedge p \\in \\bullet t_x}{m_{W(p,t_x)}(c)} + \\sum _{t_x \\in T_0 \\wedge p \\in R(t_x)}{m_{M_0(p)}(c)})$ – from (REF ) above and the following Each $add(p,{q_j}_c,t_j,c,0) \\in A$ represents ${q_j}_c=m_{W(t_j,p)}(c)$ for $p \\in t_j \\bullet $ – rule $r\\ref {r:c:add},f\\ref {f:c:tparc}$ encoding, and definition REF of transition execution in $PN$ Each $del(p,t_y,{q_y}_c,c,0) \\in A$ represents either ${q_y}_c=m_{W(p,t_y)}(c)$ for $p \\in \\bullet t_y$ , or ${q_y}_c=m_{M_0(p)}(c)$ for $p \\in R(t_y)$ – from rule $r\\ref {r:c:del},f\\ref {f:c:ptarc}$ encoding and definition REF of transition execution in $PN$ ; or from rule $r\\ref {r:c:del},f\\ref {f:c:rptarc}$ encoding and definition of reset arc in $PN$ Each $tot\\_incr(p,q2_c,c,0) \\in A$ represents $q2_c=\\sum _{t_x \\in T_0 \\wedge p \\in t_x \\bullet }{m_{W(t_x,p)}(c)}$ – aggregate assignment atom semantics in rule $r\\ref {r:c:totincr}$ Each $tot\\_decr(p,q3_c,c,0) \\in A$ represents $q3_c=\\sum _{t_x \\in T_0 \\wedge p \\in \\bullet t_x}{m_{W(p,t_x)}(c)} + \\sum _{t_x \\in T_0 \\wedge p \\in R(t_x)}{m_{M_0(p)}(c)}$ – aggregate assignment atom semantics in rule $r\\ref {r:c:totdecr}$ Then, $m_{M_1(p)}(c) = q_c$ – since $holds(p,q_c,c,1) \\in A$ encodes $q_c=m_{M_1(p)}(c)$ – from construction Inductive Step: Let $k > 0$ , and $M_k$ is a valid marking in $X$ for $PN$ , show [(1)] $T_k$ is a valid firing set for $M_k$ , and firing $T_k$ in $M_k$ produces marking $M_{k+1}$ .", "We show that $T_k$ is a valid firing set in $M_k$ .", "Let $\\lbrace fires(t_0,k), \\dots , fires(t_x,k) \\rbrace $ be the set of all $fires(\\dots ,k)$ atoms in $A$ , Then for each $fires(t_i,k) \\in A$ $enabled(t_i,k) \\in A$ – from rule $a\\ref {a:c:fires}$ and supported rule proposition Then $notenabled(t_i,k) \\notin A$ – from rule $e\\ref {e:c:enabled}$ and supported rule proposition Then either of $body(e\\ref {e:c:ne:ptarc})$ , $body(e\\ref {e:c:ne:iptarc})$ , or $body(e\\ref {e:c:ne:tptarc})$ must not hold in $A$ – from rules $e\\ref {e:c:ne:ptarc},e\\ref {e:c:ne:iptarc},e\\ref {e:c:ne:tptarc}$ and forced atom proposition Then $q_c \\lnot < {n_i}_c \\equiv q_c \\ge {n_i}_c$ in $e\\ref {e:c:ne:ptarc}$ for all $\\lbrace holds(p,q_c,c,k), \\\\ ptarc(p,t_i,{n_i}_c,c,k)\\rbrace \\subseteq A$ – from $e\\ref {e:c:ne:ptarc}$ , forced atom proposition, given facts, and the inductive assumption And $q_c \\lnot \\ge {n_i}_c \\equiv q_c < {n_i}_c$ in $e\\ref {e:c:ne:iptarc}$ for all $\\lbrace holds(p,q_c,c,k), \\\\ iptarc(p,t_i,{n_i}_c,c,k) \\rbrace \\subseteq A, {n_i}_c=1$ ; $q_c > {n_i}_c \\equiv q_c = 0$ – from $e\\ref {e:c:ne:iptarc}$ , forced atom proposition, given facts ($holds(p,q_c,c,k) \\in A, iptarc(p,t_i,1,c,k) \\in A$ ), and $q_c$ is a positive integer And $q_c \\lnot < {n_i}_c \\equiv q_c \\ge {n_i}_c$ in $e\\ref {e:c:ne:tptarc}$ for all $\\lbrace holds(p,q_c,c,k), \\\\ tptarc(p,t_i,{n_i}_c,c,k) \\rbrace \\subseteq A$ – from $e\\ref {e:c:ne:tptarc}$ , forced atom proposition, and inductive assumption Then $(\\forall p \\in \\bullet t_i, m_{M_k(p)}(c) \\ge m_{W(p,t_i)}(c)) \\wedge (\\forall p \\in I(t_i), m_{M_k(p)}(c) = 0) \\wedge (\\forall (p,t_i) \\in Q, m_{M_k(p)}(c) \\ge m_{QW(p,t_i)}(c))$ – from $holds(p,q_c,c,k) \\in A$ represents $q_c=m_{M_k(p)}(c)$ – construction of $\\Pi ^5$ $ptarc(p,t_i,{n_i}_c,k) \\in A$ represents ${n_i}_c=m_{W(p,t_i)}(c)$ – rule $f\\ref {f:c:ptarc}$ construction; or it represents ${n_i}_c = m_{M_k(p)}(c)$ – rule $f\\ref {f:c:rptarc}$ construction; the construction of $f\\ref {f:c:rptarc}$ ensures that $notenabled(t,k)$ is never true for a reset arc definition REF of preset $\\bullet t_i$ in $PN$ definition REF of enabled transition in $PN$ Then $t_i$ is enabled and can fire in $PN$ , as a result it can belong to $T_k$ – from definition REF of enabled transition And $consumesmore \\notin A$ , since $A$ is an answer set of $\\Pi ^5$ – from rule $a\\ref {a:c:overc:gen}$ and supported rule proposition Then $\\nexists consumesmore(p,k) \\in A$ – from rule $a\\ref {a:c:overc:place}$ and supported rule proposition Then $\\nexists \\lbrace holds(p,q_c,c,k), tot\\_decr(p,q1_c,c,k) \\rbrace \\subseteq A, q1_c>q_c$ in $body(a\\ref {a:c:overc:place})$ – from $a\\ref {a:c:overc:place}$ and forced atom proposition Then $\\nexists c \\in C, \\nexists p \\in P, (\\sum _{t_i \\in \\lbrace t_0,\\dots ,t_x\\rbrace , p \\in \\bullet t_i}{m_{W(p,t_i)}(c)}+ \\\\ \\sum _{t_i \\in \\lbrace t_0,\\dots ,t_x\\rbrace , p \\in R(t_i)}{m_{M_k(p)}(c)}) > m_{M_k(p)}(c)$ – from the following $holds(p,q_c,c,k)$ represents $q_c=m_{M_k(p)}(c)$ – construction of $\\Pi ^5$ , inductive assumption $tot\\_decr(p,q1_c,c,k) \\in A$ if $\\lbrace del(p,{q1_0}_c,t_0,c,k), \\dots , \\\\ del(p,{q1_x}_c,t_x,c,k) \\rbrace \\subseteq A$ , where $q1_c = {q1_0}_c+\\dots +{q1_x}_c$ – from $r\\ref {r:c:totdecr}$ and forced atom proposition $del(p,{q1_i}_c,t_i,c,k) \\in A$ if $\\lbrace fires(t_i,k), ptarc(p,t_i,{q1_i}_c,c,k) \\rbrace \\subseteq A$ – from $r\\ref {r:c:del}$ and supported rule proposition $del(p,{q1_i}_c,t_i,c,k)$ either represents removal of ${q1_i}_c = m_{W(p,t_i)}(c)$ tokens from $p \\in \\bullet t_i$ ; or it represents removal of ${q1_i}_c = m_{M_k(p)}(c)$ tokens from $p \\in R(t_i)$ – from rules $r\\ref {r:c:del},f\\ref {f:c:ptarc},f\\ref {f:c:rptarc}$ , supported rule proposition, and definition REF of transition execution in $PN$ Then the set of transitions in $T_k$ do not conflict – by the definition REF of conflicting transitions And for each reset transition $t_r$ with $enabled(t_r,k) \\in A$ , $fires(t_r,k) \\in A$ , since $A$ is an answer set of $\\Pi ^5$ - from rule $f\\ref {f:c:rptarc:elim}$ and supported rule proposition Then the firing set $T_k$ satisfies the reset transition requirement of definition REF (firing set) Then $\\lbrace t_0, \\dots , t_x\\rbrace = T_k$ – using 1(a),1(b), 1(d) above; and using 1(c) it is a maximal firing set We show that $M_{k+1}$ is produced by firing $T_k$ in $M_k$ .", "Let $holds(p,q_c,c,k+1) \\in A$ Then $\\lbrace holds(p,q1_c,c,k), tot\\_incr(p,q2_c,c,k), tot\\_decr(p,q3_c,c,k) \\rbrace \\subseteq A : q_c=q1_c+q2_c-q3_c$ – from rule $r\\ref {r:c:nextstate}$ and supported rule proposition Then, $holds(p,q1_c,c,k) \\in A$ represents $q1_c=m_{M_k(p)}(c)$ – construction, inductive assumption; and $\\lbrace add(p,{q2_0}_c,t_0,c,k), \\dots , add(p,{q2_j}_c,t_j,c,k)\\rbrace \\subseteq A : {q2_0}_c + \\dots + {q2_j}_c = q2_c$ and $\\lbrace del(p,{q3_0}_c,t_0,c,k), \\dots , del(p,{q3_l}_c,t_l,c,k)\\rbrace \\subseteq A : {q3_0}_c + \\dots + {q3_l}_c = q3_c$ – rules $r\\ref {r:c:totincr},r\\ref {r:c:totdecr}$ and supported rule proposition, respectively Then $\\lbrace fires(t_0,k), \\dots , fires(t_j,k) \\rbrace \\subseteq A$ and $\\lbrace fires(t_0,k), \\dots , fires(t_l,k) \\rbrace \\\\ \\subseteq A$ – rules $r\\ref {r:c:add},r\\ref {r:c:del}$ and supported rule proposition, respectively Then $\\lbrace fires(t_0,k), \\dots , $ $fires(t_j,k) \\rbrace \\cup \\lbrace fires(t_0,k), \\dots , $ $fires(t_l,k) \\rbrace \\subseteq $ $A = \\lbrace fires(t_0,k), \\dots , $ $fires(t_x,k) \\rbrace \\subseteq A$ – set union of subsets Then for each $fires(t_x,k) \\in A$ we have $t_x \\in T_k$ – already shown in item REF above Then $q_c = m_{M_k(p)}(c) + \\sum _{t_x \\in T_k \\wedge p \\in t_x \\bullet }{m_{W(t_x,p)}(c)} - (\\sum _{t_x \\in T_k \\wedge p \\in \\bullet t_x}{m_{W(p,t_x)}(c)} + \\sum _{t_x \\in T_k \\wedge p \\in R(t_x)}{m_{M_k(p)}(c)})$ – from (REF ) above and the following Each $add(p,{q_j}_c,t_j,c,k) \\in A$ represents ${q_j}_c=m_{W(t_j,p)}(c)$ for $p \\in t_j \\bullet $ – rule $r\\ref {r:c:add},f\\ref {f:c:tparc}$ encoding, and definition REF of transition execution in $PN$ Each $del(p,t_y,{q_y}_c,c,k) \\in A$ represents either ${q_y}_c=m_{W(p,t_y)}(c)$ for $p \\in \\bullet t_y$ , or ${q_y}_c=m_{M_k(p)}(c)$ for $p \\in R(t_y)$ – from rule $r\\ref {r:c:del},f\\ref {f:c:ptarc}$ encoding and definition REF of transition execution in $PN$ ; or from rule $r\\ref {r:c:del},f\\ref {f:c:rptarc}$ encoding and definition of reset arc in $PN$ Each $tot\\_incr(p,q2_c,c,k) \\in A$ represents $q2_c=\\sum _{t_x \\in T_k \\wedge p \\in t_x \\bullet }{m_{W(t_x,p)}(c)}$ – aggregate assignment atom semantics in rule $r\\ref {r:c:totincr}$ Each $tot\\_decr(p,q3_c,c,k) \\in A$ represents $q3_c=\\sum _{t_x \\in T_k \\wedge p \\in \\bullet t_x}{m_{W(p,t_x)}(c)} + \\sum _{t_x \\in T_k \\wedge p \\in R(t_x)}{m_{M_k(p)}(c)}$ – aggregate assignment atom semantics in rule $r\\ref {r:c:totdecr}$ Then, $m_{M_{k+1}(p)}(c) = q_c$ – since $holds(p,q_c,c,k+1) \\in A$ encodes $q_c=m_{M_{k+1}(p)}(c)$ – from construction As a result, for any $n > k$ , $T_n$ will be a valid firing set for $M_n$ and $M_{n+1}$ will be its target marking.", "Conclusion: Since both (REF ) and (REF ) hold, $X=M_0,T_k,M_1,\\dots ,M_k,T_{k+1}$ is an execution sequence of $PN(P,T,E,C,W,R,I,Q,QW)$ (w.r.t $M_0$ ) iff there is an answer set $A$ of $\\Pi ^5(PN,M_0,k,ntok)$ such that (REF ) and (REF ) hold.", "Proof of Proposition REF Let $PN=(P,T,E,C,W,R,I,Q,QW,Z)$ be a Petri Net, $M_0$ be its initial marking and let $\\Pi ^6(PN,M_0,k,ntok)$ be the ASP encoding of $PN$ and $M_0$ over a simulation length $k$ , with maximum $ntok$ tokens on any place node, as defined in section REF .", "Then $X=M_0,T_k,M_1,\\dots ,M_k,T_k,M_{k+1}$ is an execution sequence of $PN$ (w.r.t.", "$M_0$ ) iff there is an answer set $A$ of $\\Pi ^6(PN,M_0,k,ntok)$ such that: $\\lbrace fires(t,ts) : t \\in T_{ts}, 0 \\le ts \\le k\\rbrace = \\lbrace fires(t,ts) : fires(t,ts) \\in A \\rbrace $ $\\begin{split}\\lbrace holds(p,q,c,ts) &: p \\in P, c/q = M_{ts}(p), 0 \\le ts \\le k+1 \\rbrace \\\\&= \\lbrace holds(p,q,c,ts) : holds(p,q,c,ts) \\in A \\rbrace \\end{split}$ We prove this by showing that: Given an execution sequence $X$ , we create a set $A$ such that it satisfies (REF ) and (REF ) and show that $A$ is an answer set of $\\Pi ^6$ Given an answer set $A$ of $\\Pi ^6$ , we create an execution sequence $X$ such that (REF ) and (REF ) are satisfied.", "First we show (REF ): We create a set $A$ as a union of the following sets: $A_1=\\lbrace num(n) : 0 \\le n \\le ntok \\rbrace $ $A_2=\\lbrace time(ts) : 0 \\le ts \\le k\\rbrace $ $A_3=\\lbrace place(p) : p \\in P \\rbrace $ $A_4=\\lbrace trans(t) : t \\in T \\rbrace $ $A_5=\\lbrace color(c) : c \\in C \\rbrace $ $A_6=\\lbrace ptarc(p,t,n_c,c,ts) : (p,t) \\in E^-, c \\in C, n_c=m_{W(p,t)}(c), n_c > 0, 0 \\le ts \\le k \\rbrace $ , where $E^- \\subseteq E$ $A_7=\\lbrace tparc(t,p,n_c,c,ts) : (t,p) \\in E^+, c \\in C, n_c=m_{W(t,p)}(c), n_c > 0, 0 \\le ts \\le k \\rbrace $ , where $E^+ \\subseteq E$ $A_8=\\lbrace holds(p,q_c,c,0) : p \\in P, c \\in C, q_c=m_{M_{0}(p)}(c) \\rbrace $ $A_9=\\lbrace ptarc(p,t,n_c,c,ts) : p \\in R(t), c \\in C, n_c = m_{M_{ts}(p)}, n_c > 0, 0 \\le ts \\le k \\rbrace $ $A_{10}=\\lbrace iptarc(p,t,1,c,ts) : p \\in I(t), c \\in C, 0 \\le ts < k \\rbrace $ $A_{11}=\\lbrace tptarc(p,t,n_c,c,ts) : (p,t) \\in Q, c \\in C, n_c=m_{QW(p,t)}(c), n_c > 0, 0 \\le ts \\le k \\rbrace $ $A_{12}=\\lbrace notenabled(t,ts) : t \\in T, 0 \\le ts \\le k, \\exists c \\in C, (\\exists p \\in \\bullet t, m_{M_{ts}(p)}(c) < m_{W(p,t)}(c)) \\vee (\\exists p \\in I(t), m_{M_{ts}(p)}(c) > 0) \\vee (\\exists (p,t) \\in Q, m_{M_{ts}(p)}(c) < m_{QW(p,t)}(c)) \\rbrace $ per definition REF (enabled transition) $A_{13}=\\lbrace enabled(t,ts) : t \\in T, 0 \\le ts \\le k, \\forall c \\in C, (\\forall p \\in \\bullet t, m_{W(p,t)}(c) \\le m_{M_{ts}(p)}(c)) \\wedge (\\forall p \\in I(t), m_{M_{ts}(p)}(c) = 0) \\wedge (\\forall (p,t) \\in Q, m_{M_{ts}(p)}(c) \\ge m_{QW(p,t)}(c)) \\rbrace $ per definition REF (enabled transition) $A_{14}=\\lbrace fires(t,ts) : t \\in T_{ts}, 0 \\le ts \\le k \\rbrace $ per definition REF (firing set), only an enabled transition may fire $A_{15}=\\lbrace add(p,q_c,t,c,ts) : t \\in T_{ts}, p \\in t \\bullet , c \\in C, q_c=m_{W(t,p)}(c), 0 \\le ts \\le k \\rbrace $ per definition REF (transition execution) $A_{16}=\\lbrace del(p,q_c,t,c,ts) : t \\in T_{ts}, p \\in \\bullet t, c \\in C, q_c=m_{W(p,t)}(c), 0 \\le ts \\le k \\rbrace \\cup \\lbrace del(p,q_c,t,c,ts) : t \\in T_{ts}, p \\in R(t), c \\in C, q_c=m_{M_{ts}(p)}(c), 0 \\le ts \\le k \\rbrace $ per definition REF (transition execution) $A_{17}=\\lbrace tot\\_incr(p,q_c,c,ts) : p \\in P, c \\in C, q_c=\\sum _{t \\in T_{ts}, p \\in t \\bullet }{m_{W(t,p)}(c)}, 0 \\le ts \\le k \\rbrace $ per definition REF (firing set execution) $A_{18}=\\lbrace tot\\_decr(p,q_c,c,ts): p \\in P, c \\in C, q_c=\\sum _{t \\in T_{ts}, p \\in \\bullet t}{m_{W(p,t)}(c)}+ \\\\ \\sum _{t \\in T_{ts}, p \\in R(t) }{m_{M_{ts}(p)}(c)}, 0 \\le ts \\le k \\rbrace $ per definition REF (firing set execution) $A_{19}=\\lbrace consumesmore(p,ts) : p \\in P, c \\in C, q_c=m_{M_{ts}(p)}(c), \\\\ q1_c=\\sum _{t \\in T_{ts}, p \\in \\bullet t}{m_{W(p,t)}(c)} + \\sum _{t \\in T_{ts}, p \\in R(t)}{m_{M_{ts}(p)}(c)}, q1_c > q_c, 0 \\le ts \\le k \\rbrace $ per definition REF (conflicting transitions) $A_{20}=\\lbrace consumesmore : \\exists p \\in P, c \\in C, q_c=m_{M_{ts}(p)}(c), \\\\ q1_c=\\sum _{t \\in T_{ts}, p \\in \\bullet t}{m_{W(p,t)}(c)} + \\sum _{t \\in T_{ts}, p \\in R(t)}(m_{M_{ts}(p)}(c)), q1_c > q_c, 0 \\le ts \\le k \\rbrace $ per definition REF (conflicting transitions) $A_{21}=\\lbrace holds(p,q_c,c,ts+1) : p \\in P, c \\in C, q_c=m_{M_{ts+1}(p)}(c), 0 \\le ts < k\\rbrace $ , where $M_{ts+1}(p) = M_j(p) - (\\sum _{\\begin{array}{c}t \\in T_{ts}, p \\in \\bullet t\\end{array}}{W(p,t)} + \\sum _{t \\in T_{ts}, p \\in R(t)}M_{ts}(p)) + \\\\ \\sum _{\\begin{array}{c}t \\in T_{ts}, p \\in t \\bullet \\end{array}}{W(t,p)}$ according to definition REF (firing set execution) $A_{22}=\\lbrace transpr(t,pr) : t \\in T, pr=Z(t) \\rbrace $ $A_{23}=\\lbrace notprenabled(t,ts) : t \\in T, enabled(t,ts) \\in A_{13}, (\\exists tt \\in T, enabled(tt,ts) \\in A_{13}, Z(tt) < Z(t)), 0 \\le ts \\le k \\rbrace \\\\ = \\lbrace notprenabled(t,ts) : t \\in T, (\\forall p \\in \\bullet t, W(p,t) \\le M_{ts}(p)), (\\forall p \\in I(t), M_{ts}(p) = 0), (\\forall (p,t) \\in Q, M_0(p) \\ge WQ(p,t)), \\exists tt \\in T, (\\forall pp \\in \\bullet tt, W(pp,tt) \\le M_{ts}(pp)), \\\\ (\\forall pp \\in I(tt), M_{ts}(pp) = 0), (\\forall (pp,tt) \\in Q, M_{ts}(pp) \\ge QW(pp,tt)), Z(tt) < Z(t), 0 \\le ts \\le k \\rbrace $ $A_{24}=\\lbrace prenabled(t,ts) : t \\in T, enabled(t,ts) \\in A_{13}, (\\nexists tt \\in T: enabled(tt,ts) \\in A_{13}, Z(tt) < Z(t)), 0 \\le ts \\le k \\rbrace = \\lbrace prenabled(t,ts) : t \\in T, (\\forall p \\in \\bullet t, W(p,t) \\le M_{ts}(p)), (\\forall p \\in I(t), M_{ts}(p) = 0), (\\forall (p,t) \\in Q, M_0(p) \\ge WQ(p,t)), \\nexists tt \\in T, (\\forall pp \\in \\bullet tt, W(pp,tt) \\le M_{ts}(pp)), (\\forall pp \\in I(tt), M_{ts}(pp) = 0), (\\forall (pp,tt) \\in Q, M_{ts}(pp) \\ge QW(pp,tt)), Z(tt) < Z(t), 0 \\le ts \\le k \\rbrace $ $A_{25}=\\lbrace could\\_not\\_have(t,ts): t \\in T, prenabled(t,ts) \\in A_{24}, fires(t,ts) \\notin A_{14}, (\\exists p \\in \\bullet t \\cup R(t): q > M_{ts}(p) - (\\sum _{t^{\\prime } \\in T_{ts}, p \\in \\bullet t^{\\prime }}{W(p,t^{\\prime })} + \\sum _{t^{\\prime } \\in T_{ts}, p \\in R(t^{\\prime })}{M_{ts}(p)}), q=W(p,t) \\text{~if~} (p,t) \\in E^- \\text{~or~} R(t) \\text{~otherwise}), 0 \\le ts \\le k\\rbrace =\\lbrace could\\_not\\_have(t,ts) : t \\in T, (\\forall p \\in \\bullet t, W(p,t) \\le M_{ts}(p)), (\\forall p \\in I(t), M_{ts}(p) = 0), (\\forall (p,t) \\in Q, M_0(p) \\ge QW(p,t)), (\\nexists tt \\in T, $ $(\\forall pp \\in \\bullet tt, W(pp,tt) \\le M_{ts}(pp)), (\\forall pp \\in I(tt), M_{ts}(pp) = 0), (\\forall (pp,tt) \\in Q, M_{ts}(pp) \\ge QW(pp,tt)), Z(tt) < Z(t)), t \\notin T_{ts}, (\\exists p \\in \\bullet t \\cup R(t): q > M_{ts}(p) - (\\sum _{t^{\\prime } \\in T_{ts}, p \\in \\bullet t^{\\prime }}{W(p,t^{\\prime })} + \\sum _{t^{\\prime } \\in T_{ts}, p \\in R(t^{\\prime })}{M_{ts}(p)}), q=W(p,t) \\text{~if~} \\\\ (p,t) \\in E^- \\text{~or~} R(t) \\text{~otherwise}), 0 \\le ts \\le k \\rbrace $ per the maximal firing set semantics We show that $A$ satisfies (REF ) and (REF ), and $A$ is an answer set of $\\Pi ^6$ .", "$A$ satisfies (REF ) and (REF ) by its construction above.", "We show $A$ is an answer set of $\\Pi ^6$ by splitting.", "We split $lit(\\Pi ^6)$ into a sequence of $9k+11$ sets: $U_0= head(f\\ref {f:c:place}) \\cup head(f\\ref {f:c:trans}) \\cup head(f\\ref {f:c:col}) \\cup head(f\\ref {f:c:time}) \\cup head(f\\ref {f:c:num}) \\cup head(i\\ref {i:c:init}) \\cup head(f\\ref {f:c:pr}) = \\lbrace place(p) : p \\in P\\rbrace \\cup \\lbrace trans(t) : t \\in T\\rbrace \\cup \\lbrace col(c) : c \\in C \\rbrace \\cup \\lbrace time(0), \\dots , time(k)\\rbrace \\cup \\lbrace num(0), \\dots , num(ntok)\\rbrace \\cup \\lbrace holds(p,q_c,c,0) : p \\in P, c \\in C, q_c=m_{M_0(p)}(c) \\rbrace \\cup \\lbrace transpr(t,pr) : t \\in T, pr=Z(t) \\rbrace $ $U_{9k+1}=U_{9k+0} \\cup head(f\\ref {f:c:ptarc})^{ts=k} \\cup head(f\\ref {f:c:tparc})^{ts=k} \\cup head(f\\ref {f:c:rptarc})^{ts=k} \\cup head(f\\ref {f:c:iptarc})^{ts=k} \\cup head(f\\ref {f:c:tptarc})^{ts=k} = U_{10k+0} \\cup \\lbrace ptarc(p,t,n_c,c,k) : (p,t) \\in E^-, c \\in C, n_c=m_{W(p,t)}(c) \\rbrace \\\\ \\cup \\lbrace tparc(t,p,n_c,c,k) : (t,p) \\in E^+, c \\in C, n_c=m_{W(t,p)}(c) \\rbrace \\cup \\lbrace ptarc(p,t,n_c,c,k) : p \\in R(t), c \\in C, n_c=m_{M_{k}(p)}(c), n > 0 \\rbrace \\cup \\lbrace iptarc(p,t,1,c,k) : p \\in I(t), c \\in C \\rbrace \\cup \\lbrace tptarc(p,t,n_c,c,k) : (p,t) \\in Q, c \\in C, n_c=m_{QW(p,t)}(c) \\rbrace $ $U_{9k+2}=U_{9k+1} \\cup head(e\\ref {e:c:ne:ptarc})^{ts=k} \\cup head(e\\ref {e:c:ne:iptarc})^{ts=k} \\cup head(e\\ref {e:c:ne:tptarc})^{ts=k} = U_{9k+1} \\\\ \\cup \\lbrace notenabled(t,k) : t \\in T \\rbrace $ $U_{9k+3}=U_{9k+2} \\cup head(e\\ref {e:c:enabled})^{ts=k} = U_{9k+2} \\cup \\lbrace enabled(t,k) : t \\in T \\rbrace $ $U_{9k+4}=U_{9k+3} \\cup head(a\\ref {a:c:prne})^{ts=k} = U_{9k+3} \\cup \\lbrace notprenabled(t,k) : t \\in T \\rbrace $ $U_{9k+5}=U_{9k+4} \\cup head(a\\ref {a:c:prenabled})^{ts=k} = U_{9k+4} \\cup \\lbrace prenabled(t,k) : t \\in T \\rbrace $ $U_{9k+6}=U_{9k+5} \\cup head(a\\ref {a:c:prfires})^{ts=k} = U_{9k+5} \\cup \\lbrace fires(t,k) : t \\in T \\rbrace $ $U_{9k+7}=U_{9k+6} \\cup head(r\\ref {r:c:add})^{ts=k} \\cup head(r\\ref {r:c:del})^{ts=k} = U_{9k+6} \\cup \\lbrace add(p,q_c,t,c,k) : p \\in P, t \\in T, c \\in C, q_c=m_{W(t,p)}(c) \\rbrace \\cup \\lbrace del(p,q_c,t,c,k) : p \\in P, t \\in T, c \\in C, q_c=m_{W(p,t)}(c) \\rbrace \\cup \\lbrace del(p,q_c,t,c,k) : p \\in P, t \\in T, c \\in C, q_c=m_{M_{k}(p)}(c) \\rbrace $ $U_{9k+8}=U_{9k+7} \\cup head(r\\ref {r:c:totincr})^{ts=k} \\cup head(r\\ref {r:c:totdecr})^{ts=k} = U_{9k+7} \\cup \\lbrace tot\\_incr(p,q_c,c,k) : p \\in P, c \\in C, 0 \\le q_c \\le ntok \\rbrace \\cup \\lbrace tot\\_decr(p,q_c,c,k) : p \\in P, c \\in C, 0 \\le q_c \\le ntok \\rbrace $ $U_{9k+9}=U_{9k+8} \\cup head(a\\ref {a:c:overc:place})^{ts=k} \\cup head(r\\ref {r:c:nextstate})^{ts=k} \\cup head(a\\ref {a:c:prmaxfire:cnh})^{ts=k} = U_{9k+8} \\cup \\\\ \\lbrace consumesmore(p,k) : p \\in P\\rbrace \\cup \\lbrace holds(p,q,k+1) : p \\in P, 0 \\le q \\le ntok \\rbrace \\cup \\lbrace could\\_not\\_have(t,k) : t \\in T \\rbrace $ $U_{9k+10}=U_{9k+9} \\cup head(a\\ref {a:c:overc:gen})^{ts=k} = U_{9k+9} \\cup \\lbrace consumesmore \\rbrace $ where $head(r_i)^{ts=k}$ are head atoms of ground rule $r_i$ in which $ts=k$ .", "We write $A_i^{ts=k} = \\lbrace a(\\dots ,ts) : a(\\dots ,ts) \\in A_i, ts=k \\rbrace $ as short hand for all atoms in $A_i$ with $ts=k$ .", "$U_{\\alpha }, 0 \\le \\alpha \\le 9k+10$ form a splitting sequence, since each $U_i$ is a splitting set of $\\Pi ^6$ , and $\\langle U_{\\alpha }\\rangle _{\\alpha < \\mu }$ is a monotone continuous sequence, where $U_0 \\subseteq U_1 \\dots \\subseteq U_{9k+10}$ and $\\bigcup _{\\alpha < \\mu }{U_{\\alpha }} = lit(\\Pi ^6)$ .", "We compute the answer set of $\\Pi ^6$ using the splitting sets as follows: $bot_{U_0}(\\Pi ^6) = f\\ref {f:c:place} \\cup f\\ref {f:c:trans} \\cup f\\ref {f:c:col} \\cup f\\ref {f:c:time} \\cup f\\ref {f:c:num} \\cup i\\ref {i:c:init} \\cup f\\ref {f:c:pr}$ and $X_0 = A_1 \\cup \\dots \\cup A_5 \\cup A_8$ ($= U_0$ ) is its answer set – using forced atom proposition $eval_{U_0}(bot_{U_1}(\\Pi ^6) \\setminus bot_{U_0}(\\Pi ^6), X_0) = \\lbrace ptarc(p,t,q_c,c,0) \\text{:-}.", "\| c \\in C, $ $q_c=m_{W(p,t)}(c), q_c > 0 \\rbrace \\cup \\lbrace tparc(t,p,q_c,c,0) \\text{:-}.", "\| c \\in C, q_c=m_{W(t,p)}(c), q_c > 0 \\rbrace \\cup \\lbrace ptarc(p,t,q_c,c,0) \\text{:-}.", "\| c \\in C, $ $q_c=m_{M_0(p)}(c), q_c > 0 \\rbrace \\cup \\lbrace iptarc(p,t,1,c,0) \\text{:-}.", "\| c \\in C \\rbrace \\cup \\lbrace tptarc(p,t,q_c,c,0) \\text{:-}.", "\| c \\in C, q_c = m_{QW(p,t)}(c), q_c > 0 \\rbrace $ .", "Its answer set $X_1=A_6^{ts=0} \\cup A_7^{ts=0} \\cup A_9^{ts=0} \\cup A_{10}^{ts=0} \\cup A_{11}^{ts=0}$ – using forced atom proposition and construction of $A_6, A_7, A_9, A_{10}, A_{11}$ .", "$eval_{U_1}(bot_{U_2}(\\Pi ^6) \\setminus bot_{U_1}(\\Pi ^6), X_0 \\cup X_1) = \\lbrace notenabled(t,0) \\text{:-} .", "\| (\\lbrace trans(t), \\\\ ptarc(p,t,n_c,c,0), holds(p,q_c,c,0) \\rbrace \\subseteq X_0 \\cup X_1, \\text{~where~} q_c < n_c) \\text{~or~} \\\\ (\\lbrace notenabled(t,0) \\text{:-} .", "\| (\\lbrace trans(t), iptarc(p,t,n2_c,c,0), holds(p,q_c,c,0) \\rbrace \\subseteq X_0 \\cup X_1, \\\\ \\text{~where~} q_c \\ge n2_c \\rbrace ) \\text{~or~} (\\lbrace trans(t), tptarc(p,t,n3_c,c,0), $ $holds(p,q_c,c,0) \\rbrace \\subseteq X_0 \\cup X_1, \\text{~where~} $ $q_c < n3_c) \\rbrace $ .", "Its answer set $X_2=A_{12}^{ts=0}$ – using forced atom proposition and construction of $A_{12}$ .", "where, $q_c=m_{M_0(p)}(c)$ , and $n_c=m_{W(p,t)}(c)$ for an arc $(p,t) \\in E^-$ – by construction of $i\\ref {i:c:init}$ and $f\\ref {f:c:ptarc}$ in $\\Pi ^6$ , and in an arc $(p,t) \\in E^-$ , $p \\in \\bullet t$ (by definition REF of preset) $n2_c=1$ – by construction of $iptarc$ predicates in $\\Pi ^6$ , meaning $q_c \\ge n2_c \\equiv q_c \\ge 1 \\equiv q_c > 0$ , $tptarc(p,t,n3_c,c,0)$ represents $n3_c=m_{QW(p,t)}(c)$ , where $(p,t) \\in Q$ thus, $notenabled(t,0) \\in X_1$ represents $\\exists c \\in C, (\\exists p \\in \\bullet t : m_{M_0(p)}(c) < m_{W(p,t)}(c)) \\vee (\\exists p \\in I(t) : m_{M_0(p)}(c) > 0) \\vee (\\exists (p,t) \\in Q : m_{M_0(p)}(c) < m_{QW(p,t)}(c))$ .", "$eval_{U_2}(bot_{U_3}(\\Pi ^6) \\setminus bot_{U_2}(\\Pi ^6), X_0 \\cup \\dots \\cup X_2) = \\lbrace enabled(t,0) \\text{:-}.", "\| trans(t) \\in X_0 \\cup \\dots \\cup X_2, notenabled(t,0) \\notin X_0 \\cup \\dots \\cup X_2 \\rbrace $ .", "Its answer set is $X_3 = A_{13}^{ts=0}$ – using forced atom proposition and construction of $A_{13}$ .", "since an $enabled(t,0) \\in X_3$ if $\\nexists ~notenabled(t,0) \\in X_0 \\cup \\dots \\cup X_2$ ; which is equivalent to $\\nexists t, \\forall c \\in C, (\\nexists p \\in \\bullet t, m_{M_0(p)}(c) < m_{W(p,t)}(c)), (\\nexists p \\in I(t), $ $m_{M_0(p)}(c) > 0), (\\nexists (p,t) \\in Q : m_{M_0(p)}(c) < m_{QW(p,t)}(c) ), \\forall c \\in C, (\\forall p \\in \\bullet t: m_{M_0(p)}(c) \\ge m_{W(p,t)}(c)), (\\forall p \\in I(t) : m_{M_0(p)}(c) = 0)$ .", "$eval_{U_3}(bot_{U_4}(\\Pi ^6) \\setminus bot_{U_3}(\\Pi ^6), X_0 \\cup \\dots \\cup X_3) = \\lbrace notprenabled(t,0) \\text{:-}.", "\| \\\\ \\lbrace enabled(t,0), transpr(t,p), enabled(tt,0), transpr(tt,pp) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_3, pp < p \\rbrace $ .", "Its answer set is $X_4 = A_{23}^{ts=k}$ – using forced atom proposition and construction of $A_{23}$ .", "$enabled(t,0)$ represents $\\exists t \\in T, \\forall c \\in C, (\\forall p \\in \\bullet t, m_{M_0(p)}(c) \\ge m_{W(p,t)}(c)), $ $(\\forall p \\in I(t), m_{M_0(p)}(c) = 0), (\\forall (p,t) \\in Q, m_{M_0(p)}(c) \\ge m_{QW(p,t)}(c))$ $enabled(tt,0)$ represents $\\exists tt \\in T, \\forall c \\in C, (\\forall pp \\in \\bullet tt, m_{M_0(pp)}(c) \\ge \\\\ m_{W(pp,tt)}(c)) \\wedge (\\forall pp \\in I(tt), m_{M_0(pp)}(c) = 0), (\\forall (pp,tt) \\in Q, m_{M_0(pp)}(c) \\ge m_{QW(pp,tt)}(c))$ $transpr(t,p)$ represents $p=Z(t)$ – by construction $transpr(tt,pp)$ represents $pp=Z(tt)$ – by construction thus, $notprenabled(t,0)$ represents $\\forall c \\in C, (\\forall p \\in \\bullet t, m_{M_0(p)}(c) \\ge \\\\ m_{W(p,t)}(c)) \\wedge (\\forall p \\in I(t), m_{M_0(p)}(c) = 0), \\exists tt \\in T, (\\forall pp \\in \\bullet tt, m_{M_0(pp)}(c) \\ge m_{W(pp,tt)}(c)) \\wedge (\\forall pp \\in I(tt), m_{M_0(pp)}(c) = 0), Z(tt) < Z(t)$ which is equivalent to $(\\forall p \\in \\bullet t: M_0(p) \\ge W(p,t)) \\wedge (\\forall p \\in I(t), M_0(p) = 0), \\exists tt \\in T, (\\forall pp \\in \\bullet tt, M_0(pp) \\ge W(pp,tt)), (\\forall pp \\in I(tt), M_0(pp) = 0), Z(tt) < Z(t)$ – assuming multiset domain $C$ for all operations $eval_{U_4}(bot_{U_5}(\\Pi ^6) \\setminus bot_{U_4}(\\Pi ^6), X_0 \\cup \\dots \\cup X_4) = \\lbrace prenabled(t,0) \\text{:-}.", "\| enabled(t,0) \\in X_0 \\cup \\dots \\cup X_4, notprenabled(t,0) \\notin X_0 \\cup \\dots \\cup X_4 \\rbrace $ .", "Its answer set is $X_5 = A_{24}^{ts=k}$ – using forced atom proposition and construction of $A_{24}$ $enabled(t,0)$ represents $\\forall c \\in C, (\\forall p \\in \\bullet t, m_{M_0(p)}(c) \\ge m_{W(p,t)}(c)), (\\forall p \\in I(t), m_{M_0(p)}(c) = 0), (\\forall (p,t) \\in Q, m_{M_0(p)}(c) \\ge m_{QW(p,t)}(c)) \\equiv (\\forall p \\in \\bullet t, \\\\ M_0(p) \\ge W(p,t)), (\\forall p \\in I(t), M_0(p) = 0), (\\forall (p,t) \\in Q, M_0(p) \\ge QW(p,t))$ – from REF above and assuming multiset domain $C$ for all operations $notprenabled(t,0)$ represents $(\\forall p \\in \\bullet t, M_0(p) \\ge W(p,t)), (\\forall p \\in I(t), \\\\ M_0(p) = 0), (\\forall (p,t) \\in Q, M_0(p) \\ge QW(p,t)), \\exists tt \\in T, (\\forall pp \\in \\bullet tt, M_0(pp) \\ge W(pp,tt)), (\\forall pp \\in I(tt), M_0(pp) = 0), (\\forall (pp,tt) \\in Q, M_0(pp) \\ge W(pp,tt)), \\\\ Z(tt) < Z(t)$ – from REF above and assuming multiset domain $C$ for all operations then, $prenabled(t,0)$ represents $(\\forall p \\in \\bullet t, M_0(p) \\ge W(p,t)), (\\forall p \\in I(t), \\\\ M_0(p) = 0), (\\forall (p,t) \\in Q, M_0(p) \\ge QW(p,t)), \\nexists tt \\in T, ((\\forall pp \\in \\bullet tt, M_0(pp) \\ge W(pp,tt)), (\\forall pp \\in I(tt), M_0(pp) = 0), (\\forall (pp,tt) \\in Q, M_0(pp) \\ge W(pp,tt)), \\\\ Z(tt) < Z(t))$ – from (a), (b) and $enabled(t,0) \\in X_0 \\cup \\dots \\cup X_4$ $eval_{U_5}(bot_{U_6}(\\Pi ^6) \\setminus bot_{U_5}(\\Pi ^6), X_0 \\cup \\dots \\cup X_5) = \\lbrace \\lbrace fires(t,0)\\rbrace \\text{:-}.", "\| prenabled(t,0) \\\\ \\text{~holds in~} X_0 \\cup \\dots \\cup X_5 \\rbrace $ .", "It has multiple answer sets $X_{6.1}, \\dots , X_{6.n}$ , corresponding to elements of power set of $fires(t,0)$ atoms in $eval_{U_5}(...)$ – using supported rule proposition.", "Since we are showing that the union of answer sets of $\\Pi ^6$ determined using splitting is equal to $A$ , we only consider the set that matches the $fires(t,0)$ elements in $A$ and call it $X_6$ , ignoring the rest.", "Thus, $X_6 = A_{14}^{ts=0}$ , representing $T_0$ .", "in addition, for every $t$ such that $prenabled(t,0) \\in X_0 \\cup \\dots \\cup X_5, R(t) \\ne \\emptyset $ ; $fires(t,0) \\in X_6$ – per definition REF (firing set); requiring that a reset transition is fired when enabled thus, the firing set $T_0$ will not be eliminated by the constraint $f\\ref {f:c:pr:rptarc:elim}$ $eval_{U_6}(bot_{U_7}(\\Pi ^6) \\setminus bot_{U_6}(\\Pi ^6), X_0 \\cup \\dots \\cup X_6) = \\lbrace add(p,n_c,t,c,0) \\text{:-}.", "\| \\lbrace fires(t,0), \\\\ tparc(t,p,n_c,c,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_6 \\rbrace \\cup \\lbrace del(p,n_c,t,c,0) \\text{:-}.", "\| \\lbrace fires(t,0), \\\\ ptarc(p,t,n_c,c,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_6 \\rbrace $ .", "It's answer set is $X_7 = A_{15}^{ts=0} \\cup A_{16}^{ts=0}$ – using forced atom proposition and definitions of $A_{15}$ and $A_{16}$ .", "where, each $add$ atom is equivalent to $n_c=m_{W(t,p)}(c),c \\in C, p \\in t \\bullet $ , and each $del$ atom is equivalent to $n_c=m_{W(p,t)}(c), c \\in C, p \\in \\bullet t$ ; or $n_c=m_{M_0(p)}(c), c \\in C, p \\in R(t)$ , representing the effect of transitions in $T_0$ – by construction $eval_{U_7}(bot_{U_8}(\\Pi ^6) \\setminus bot_{U_7}(\\Pi ^6), X_0 \\cup \\dots \\cup X_7) = \\lbrace tot\\_incr(p,qq_c,c,0) \\text{:-}.", "\| qq_c=\\sum _{add(p,q_c,t,c,0) \\in X_0 \\cup \\dots \\cup X_7}{q_c} \\rbrace \\cup \\lbrace tot\\_decr(p,qq_c,c,0) \\text{:-}.", "\| qq_c=\\sum _{del(p,q_c,t,c,0) \\in X_0 \\cup \\dots \\cup X_7}{q_c} \\rbrace $ .", "It's answer set is $X_8 = A_{17}^{ts=0} \\cup A_{18}^{ts=0}$ – using forced atom proposition and definitions of $A_{17}$ and $A_{18}$ .", "where, each $tot\\_incr(p,qq_c,c,0)$ , $qq_c=\\sum _{add(p,q_c,t,c,0) \\in X_0 \\cup \\dots X_7}{q_c}$ $\\equiv qq_c=\\sum _{t \\in X_6, p \\in t \\bullet }{m_{W(p,t)}(c)}$ , where, each $tot\\_decr(p,qq_c,c,0)$ , $qq_c=\\sum _{del(p,q_c,t,c,0) \\in X_0 \\cup \\dots X_7}{q_c}$ $\\equiv qq=\\sum _{t \\in X_6, p \\in \\bullet t}{m_{W(t,p)}(c)} + \\sum _{t \\in X_6, p \\in R(t)}{m_{M_0(p)}(c)}$ , represent the net effect of transitions in $T_0$ $eval_{U_8}(bot_{U_9}(\\Pi ^6) \\setminus bot_{U_8}(\\Pi ^6), X_0 \\cup \\dots \\cup X_8) = \\lbrace consumesmore(p,0) \\text{:-}.", "\| \\\\ \\lbrace holds(p,q_c,c,0), tot\\_decr(p,q1_c,c,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_8, q1_c > q_c \\rbrace \\cup \\\\ \\lbrace holds(p,q_c,c,1) \\text{:-}.", "\| \\lbrace holds(p,q1_c,c,0), tot\\_incr(p,q2_c,c,0), tot\\_decr(p,q3_c,c,0) \\rbrace \\\\ \\subseteq X_0 \\cup \\dots \\cup X_6, q_c=q1_c+q2_c-q3_c \\rbrace \\cup \\lbrace could\\_not\\_have(t,0) \\text{:-}.", "\| \\lbrace prenabled(t,0), \\\\ ptarc(s,t,q,c,0), holds(s,qq,c,0), tot\\_decr(s,qqq,c,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_8, \\\\ fires(t,0) \\notin X_0 \\cup \\dots \\cup X_8, q > qq - qqq \\rbrace $ .", "It's answer set is $X_9 = A_{19}^{ts=0} \\cup A_{21}^{ts=0} \\cup A_{25}^{ts=0}$ – using forced atom proposition and definitions of $A_{19}, A_{21}, A_{25}$ .", "where, $consumesmore(p,0)$ represents $\\exists p : q_c=m_{M_0(p)}(c), q1_c= \\\\ \\sum _{t \\in T_0, p \\in \\bullet t}{m_{W(p,t)}(c)}+\\sum _{t \\in T_0, p \\in R(t)}{m_{M_0(p)}(c)}, q1_c > q_c, c \\in C$ , indicating place $p$ will be over consumed if $T_0$ is fired, as defined in definition REF (conflicting transitions), $holds(p,q_c,c,1)$ if $q_c=m_{M_0(p)}(c)+\\sum _{t \\in T_0, p \\in t \\bullet }{m_{W(t,p)}(c)}- \\\\ (\\sum _{t \\in T_0, p \\in \\bullet t}{m_{W(p,t)}(c)}+ \\sum _{t \\in T_0, p \\in R(t)}{m_{M_0(p)}(c)})$ represented by $q_c=m_{M_1(p)}(c)$ for some $c \\in C$ – by construction of $\\Pi ^6$ $could\\_not\\_have(t,0)$ if $(\\forall p \\in \\bullet t, W(p,t) \\le M_0(p)), (\\forall p \\in I(t), M_0(p) = 0), (\\forall (p,t) \\in Q, \\\\ M_0(p) \\ge WQ(p,t)), \\nexists tt \\in T, (\\forall pp \\in \\bullet tt, W(pp,tt) \\le M_{ts}(pp)), (\\forall pp \\in I(tt), M_0(pp) = 0), (\\forall (pp,tt) \\in Q, M_0(pp) \\ge QW(pp,tt)), Z(tt) < Z(t)$ , and $q_c > m_{M_0(s)}(c) - (\\sum _{t^{\\prime } \\in T_0, s \\in \\bullet t^{\\prime }}{m_{W(s,t^{\\prime })}(c)}+ \\sum _{t^{\\prime } \\in T_0, s \\in R(t)}{m_{M_0(s)}(c)}), \\\\ q_c = m_{W(s,t)}(c) \\text{~if~} s \\in \\bullet t \\text{~or~} m_{M_0(s)}(c) \\text{~otherwise}$ for some $c \\in C$ , which becomes $q > M_0(s) - (\\sum _{t^{\\prime } \\in T_0, s \\in \\bullet t^{\\prime }}{W(s,t^{\\prime })}+ \\sum _{t^{\\prime } \\in T_0, s \\in R(t)}{M_0(s)}), q = W(s,t) \\text{~if~} s \\in \\bullet t \\text{~or~} M_0(s) \\text{~otherwise}$ for all $c \\in C$ (i), (ii) above combined match the definition of $A_{25}$ $X_9$ does not contain $could\\_not\\_have(t,0)$ , when $prenabled(t,0) \\in X_0 \\cup \\dots \\cup X_8$ and $fires(t,0) \\notin X_0 \\cup \\dots \\cup X_8$ due to construction of $A$ , encoding of $a\\ref {a:c:overc:place}$ and its body atoms.", "As a result, it is not eliminated by the constraint $a\\ref {a:c:maxfire:elim}$ $ \\vdots $ $eval_{U_{9k+0}}(bot_{U_{9k+1}}(\\Pi ^6) \\setminus bot_{U_{9k+0}}(\\Pi ^6), X_0 \\cup \\dots \\cup X_{9k+0}) = \\lbrace ptarc(p,t,q_c,c,k) \\text{:-}.", "\| c \\in C, q_c=m_{W(p,t)}(c), q_c > 0 \\rbrace \\cup \\lbrace tparc(t,p,q_c,c,k) \\text{:-}.", "\| c \\in C, q_c=m_{W(t,p)}(c), q_c > 0 \\rbrace \\cup \\lbrace ptarc(p,t,q_c,c,k) \\text{:-}.", "\| c \\in C, q_c=m_{M_0(p)}(c), q_c > 0 \\rbrace \\cup \\lbrace iptarc(p,t,1,c,k) \\text{:-}.", "\| c \\in C \\rbrace \\cup \\lbrace tptarc(p,t,q_c,c,k) \\text{:-}.", "\| c \\in C, q_c = m_{QW(p,t)}(c), q_c > 0 \\rbrace $ .", "Its answer set $X_{9k+1}=A_6^{ts=k} \\cup A_7^{ts=k} \\cup A_9^{ts=k} \\cup A_{10}^{ts=k} \\cup A_{11}^{ts=k}$ – using forced atom proposition and construction of $A_6, A_7, A_9, A_{10}, A_{11}$ .", "$eval_{U_{9k+1}}(bot_{U_{9k+2}}(\\Pi ^6) \\setminus bot_{U_{9k+1}}(\\Pi ^6), X_0 \\cup \\dots \\cup X_{9k+1}) = $ $\\lbrace notenabled(t,k) \\text{:-} .", "\| $ $(\\lbrace trans(t), $ $ptarc(p,t,n_c,c,k), $ $holds(p,q_c,c,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{10k+1}, \\text{~where~} $ $q_c < n_c) \\text{~or~} $ $(\\lbrace notenabled(t,k) \\text{:-} .", "\| $ $(\\lbrace trans(t), $ $iptarc(p,t,n2_c,c,k), $ $holds(p,q_c,c,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{10k+1}, \\text{~where~} $ $q_c \\ge n2_c \\rbrace ) \\text{~or~} $ $(\\lbrace trans(t), $ $tptarc(p,t,n3_c,c,k), \\\\ holds(p,q_c,c,k) \\rbrace \\subseteq X_0 \\cup X_{9k+1}, \\text{~where~} q_c < n3_c) \\rbrace $ .", "Its answer set $X_{9k+2}=A_{12}^{ts=k}$ – using forced atom proposition and construction of $A_{12}$ .", "where, $q_c=m_{M_0(p)}(c)$ , and $n_c=m_{W(p,t)}(c)$ for an arc $(p,t) \\in E^-$ – by construction of $i\\ref {i:c:init}$ and $f\\ref {f:c:ptarc}$ predicates in $\\Pi ^6$ , and in an arc $(p,t) \\in E^-$ , $p \\in \\bullet t$ (by definition REF of preset) $n2_c=1$ – by construction of $iptarc$ predicates in $\\Pi ^6$ , meaning $q_c \\ge n2_c \\equiv q_c \\ge 1 \\equiv q_c > 0$ , $tptarc(p,t,n3_c,c,k)$ represents $n3_c=m_{QW(p,t)}(c)$ , where $(p,t) \\in Q$ thus, $notenabled(t,k) \\in X_{9k+1}$ represents $\\exists c \\in C, (\\exists p \\in \\bullet t : m_{M_0(p)}(c) < m_{W(p,t)}(c)) \\vee (\\exists p \\in I(t) : m_{M_0(p)}(c) > k) \\vee (\\exists (p,t) \\in Q : m_{M_0(p)}(c) < m_{QW(p,t)}(c))$ .", "$eval_{U_{9k+2}}(bot_{U_{9k+3}}(\\Pi ^6) \\setminus bot_{U_{9k+2}}(\\Pi ^6), X_0 \\cup \\dots \\cup X_{9k+2}) = \\lbrace enabled(t,k) \\text{:-}.", "\| trans(t) \\in X_0 \\cup \\dots \\cup X_{9k+2}, notenabled(t,k) \\notin X_0 \\cup \\dots \\cup X_{9k+2} \\rbrace $ .", "Its answer set is $X_{9k+3} = A_{13}^{ts=k}$ – using forced atom proposition and construction of $A_{13}$ .", "since an $enabled(t,k) \\in X_{9k+3}$ if $\\nexists ~notenabled(t,k) \\in X_0 \\cup \\dots \\cup X_{9k+2}$ ; which is equivalent to $\\nexists t, \\forall c \\in C, (\\nexists p \\in \\bullet t, m_{M_0(p)}(c) < m_{W(p,t)}(c)), (\\nexists p \\in I(t), m_{M_0(p)}(c) > k), (\\nexists (p,t) \\in Q : m_{M_0(p)}(c) < m_{QW(p,t)}(c) ), \\forall c \\in C, (\\forall p \\in \\bullet t: m_{M_0(p)}(c) \\ge m_{W(p,t)}(c)), (\\forall p \\in I(t) : m_{M_0(p)}(c) = k)$ .", "$eval_{U_{9k+3}}(bot_{U_{9k+4}}(\\Pi ^6) \\setminus bot_{U_{9k+3}}(\\Pi ^6), X_0 \\cup \\dots \\cup X_{9k+3}) = \\lbrace notprenabled(t,k) \\text{:-}.", "\| \\\\ \\lbrace enabled(t,k), transpr(t,p), enabled(tt,k), transpr(tt,pp) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{9k+3}, \\\\ pp < p \\rbrace $ .", "Its answer set is $X_{9k+4} = A_{23}^{ts=k}$ – using forced atom proposition and construction of $A_{23}$ .", "$enabled(t,k)$ represents $\\exists t \\in T, \\forall c \\in C, (\\forall p \\in \\bullet t, m_{M_0(p)}(c) \\ge m_{W(p,t)}(c)), \\\\ (\\forall p \\in I(t), m_{M_0(p)}(c) = k), (\\forall (p,t) \\in Q, m_{M_0(p)}(c) \\ge m_{QW(p,t)}(c))$ $enabled(tt,k)$ represents $\\exists tt \\in T, \\forall c \\in C, (\\forall pp \\in \\bullet tt, m_{M_0(pp)}(c) \\ge \\\\ m_{W(pp,tt)}(c)) \\wedge (\\forall pp \\in I(tt), m_{M_0(pp)}(c) = k), (\\forall (pp,tt) \\in Q, m_{M_0(pp)}(c) \\ge m_{QW(pp,tt)}(c))$ $transpr(t,p)$ represents $p=Z(t)$ – by construction $transpr(tt,pp)$ represents $pp=Z(tt)$ – by construction thus, $notprenabled(t,k)$ represents $\\forall c \\in C, (\\forall p \\in \\bullet t, m_{M_0(p)}(c) \\ge m_{W(p,t)}(c)) \\\\ \\wedge (\\forall p \\in I(t), m_{M_0(p)}(c) = k), \\exists tt \\in T, (\\forall pp \\in \\bullet tt, m_{M_0(pp)}(c) \\ge m_{W(pp,tt)}(c)) \\wedge (\\forall pp \\in I(tt), m_{M_0(pp)}(c) = k), Z(tt) < Z(t)$ which is equivalent to $(\\forall p \\in \\bullet t: M_0(p) \\ge W(p,t)) \\wedge (\\forall p \\in I(t), M_0(p) = k), \\exists tt \\in T, (\\forall pp \\in \\bullet tt, M_0(pp) \\ge W(pp,tt)), (\\forall pp \\in I(tt), M_0(pp) = k), Z(tt) < Z(t)$ – assuming multiset domain $C$ for all operations $eval_{U_{9k+4}}(bot_{U_{9k+5}}(\\Pi ^6) \\setminus bot_{U_{9k+4}}(\\Pi ^6), X_0 \\cup \\dots \\cup X_{9k+4}) = \\lbrace prenabled(t,k) \\text{:-}.", "\| $ $enabled(t,k) \\in X_0 \\cup \\dots \\cup X_{9k+4}, notprenabled(t,k) \\notin X_0 \\cup \\dots \\cup X_{9k+4} \\rbrace $ .", "Its answer set is $X_{9k+5} = A_{24}^{ts=k}$ – using forced atom proposition and construction of $A_{24}$ $enabled(t,k)$ represents $\\forall c \\in C, (\\forall p \\in \\bullet t, m_{M_0(p)}(c) \\ge m_{W(p,t)}(c)), $ $(\\forall p \\in I(t), m_{M_0(p)}(c) = k), (\\forall (p,t) \\in Q, m_{M_0(p)}(c) \\ge m_{QW(p,t)}(c)) \\equiv (\\forall p \\in \\bullet t, M_0(p) \\ge W(p,t)), $ $(\\forall p \\in I(t), M_0(p) = k), (\\forall (p,t) \\in Q, M_0(p) \\ge QW(p,t))$ – from REF above and assuming multiset domain $C$ for all operations $notprenabled(t,k)$ represents $(\\forall p \\in \\bullet t, M_0(p) \\ge W(p,t)), (\\forall p \\in I(t), \\\\ M_0(p) = k), (\\forall (p,t) \\in Q, M_0(p) \\ge QW(p,t)), \\exists tt \\in T, $ $(\\forall pp \\in \\bullet tt, M_0(pp) \\ge W(pp,tt)), $ $(\\forall pp \\in I(tt), M_0(pp) = k), (\\forall (pp,tt) \\in Q, M_0(pp) \\ge W(pp,tt)), \\\\ Z(tt) < Z(t)$ – from REF above and assuming multiset domain $C$ for all operations then, $prenabled(t,k)$ represents $(\\forall p \\in \\bullet t, M_0(p) \\ge W(p,t)), (\\forall p \\in I(t), \\\\ M_0(p) = k), (\\forall (p,t) \\in Q, M_0(p) \\ge QW(p,t)), \\nexists tt \\in T, $ $((\\forall pp \\in \\bullet tt, M_0(pp) \\ge W(pp,tt)), $ $(\\forall pp \\in I(tt), M_0(pp) = k), (\\forall (pp,tt) \\in Q, M_0(pp) \\ge W(pp,tt)), \\\\ Z(tt) < Z(t))$ – from (a), (b) and $enabled(t,k) \\in X_0 \\cup \\dots \\cup X_{9k+4}$ $eval_{U_{9k+5}}(bot_{U_{9k+6}}(\\Pi ^6) \\setminus bot_{U_{9k+5}}(\\Pi ^6), X_0 \\cup \\dots \\cup X_{9k+5}) = \\lbrace \\lbrace fires(t,k)\\rbrace \\text{:-}.", "\| \\\\ prenabled(t,k) \\text{~holds in~} X_0 \\cup \\dots \\cup X_{9k+5} \\rbrace $ .", "It has multiple answer sets $X_{9k+6.1}, \\dots , \\\\ X_{9k+6.n}$ , corresponding to elements of power set of $fires(t,k)$ atoms in $eval_{U_{9k+5}}(...)$ – using supported rule proposition.", "Since we are showing that the union of answer sets of $\\Pi ^6$ determined using splitting is equal to $A$ , we only consider the set that matches the $fires(t,k)$ elements in $A$ and call it $X_{9k+6}$ , ignoring the rest.", "Thus, $X_{9k+6} = A_{14}^{ts=k}$ , representing $T_k$ .", "in addition, for every $t$ such that $prenabled(t,k) \\in X_0 \\cup \\dots \\cup X_{9k+5}, R(t) \\ne \\emptyset $ ; $fires(t,k) \\in X_{9k+6}$ – per definition REF (firing set); requiring that a reset transition is fired when enabled thus, the firing set $T_k$ will not be eliminated by the constraint $f\\ref {f:c:pr:rptarc:elim}$ $eval_{U_{9k+6}}(bot_{U_{9k+7}}(\\Pi ^6) \\setminus bot_{U_{9k+6}}(\\Pi ^6), X_0 \\cup \\dots \\cup X_{9k+6}) = \\lbrace add(p,n_c,t,c,k) \\text{:-}.", "\| \\\\ \\lbrace fires(t,k), tparc(t,p,n_c,c,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{9k+6} \\rbrace \\cup \\lbrace del(p,n_c,t,c,k) \\text{:-}.", "\| \\\\ \\lbrace fires(t,k), ptarc(p,t,n_c,c,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{9k+6} \\rbrace $ .", "It's answer set is $X_{9k+7} = A_{15}^{ts=k} \\cup A_{16}^{ts=k}$ – using forced atom proposition and definitions of $A_{15}$ and $A_{16}$ .", "where, each $add$ atom is equivalent to $n_c=m_{W(t,p)}(c),c \\in C, p \\in t \\bullet $ , and each $del$ atom is equivalent to $n_c=m_{W(p,t)}(c), c \\in C, p \\in \\bullet t$ ; or $n_c=m_{M_0(p)}(c), c \\in C, p \\in R(t)$ , representing the effect of transitions in $T_k$ .", "$eval_{U_{9k+7}}(bot_{U_{9k+8}}(\\Pi ^6) \\setminus bot_{U_{9k+7}}(\\Pi ^6), X_0 \\cup \\dots \\cup X_{9k+7}) = \\lbrace tot\\_incr(p,qq_c,c,k) \\text{:-}.", "\| $ $qq_c=\\sum _{add(p,q_c,t,c,k) \\in X_0 \\cup \\dots \\cup X_{10k+7}}{q_c} \\rbrace \\cup \\lbrace tot\\_decr(p,qq_c,c,k) \\text{:-}.", "\| qq_c= \\\\ \\sum _{del(p,q_c,t,c,k) \\in X_0 \\cup \\dots \\cup X_{9k+7}}{q_c} \\rbrace $ .", "It's answer set is $X_{10k+8} = A_{17}^{ts=k} \\cup A_{18}^{ts=k}$ – using forced atom proposition and definitions of $A_{17}$ and $A_{18}$ .", "where, each $tot\\_incr(p,qq_c,c,k)$ , $qq_c=\\sum _{add(p,q_c,t,c,k) \\in X_0 \\cup \\dots X_{10k+7}}{q_c}$ $\\equiv qq_c=\\sum _{t \\in X_{9k+6}, p \\in t \\bullet }{m_{W(p,t)}(c)}$ , and each $tot\\_decr(p,qq_c,c,k)$ , $qq_c=\\sum _{del(p,q_c,t,c,k) \\in X_0 \\cup \\dots X_{10k+7}}{q_c}$ $\\equiv qq=\\sum _{t \\in X_{10k+6}, p \\in \\bullet t}{m_{W(t,p)}(c)} + \\sum _{t \\in X_{9k+6}, p \\in R(t)}{m_{M_0(p)}(c)}$ , represent the net effect of transitions in $T_k$ $eval_{U_{9k+8}}(bot_{U_{9k+9}}(\\Pi ^6) \\setminus bot_{U_{9k+8}}(\\Pi ^6), X_0 \\cup \\dots \\cup X_{9k+8}) = \\lbrace consumesmore(p,k) \\text{:-}.", "\| \\\\ \\lbrace holds(p,q_c,c,k), tot\\_decr(p,q1_c,c,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{9k+8}, q1_c > q_c \\rbrace \\cup \\\\ \\lbrace holds(p,q_c,c,1) \\text{:-}.", "\| \\lbrace holds(p,q1_c,c,k), tot\\_incr(p,q2_c,c,k), tot\\_decr(p,q3_c,c,k) \\rbrace \\\\ \\subseteq $ $X_0 \\cup \\dots \\cup X_{9k+6}, q_c=q1_c+q2_c-q3_c \\rbrace \\cup \\lbrace could\\_not\\_have(t,k) \\text{:-}.", "\| \\lbrace prenabled(t,k), \\\\ ptarc(s,t,q,c,k), holds(s,qq,c,k), tot\\_decr(s,qqq,c,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{9k+8}, \\\\ fires(t,k) \\notin X_0 \\cup \\dots \\cup X_{9k+8}, q > qq - qqq \\rbrace $ .", "It's answer set is $X_{10k+9} = A_{19}^{ts=k} \\cup A_{21}^{ts=k} \\cup A_{25}^{ts=k}$ – using forced atom proposition and definitions of $A_{19}, A_{21}, A_{25}$ .", "where, $consumesmore(p,k)$ represents $\\exists p : q_c=m_{M_0(p)}(c), \\\\ q1_c=$ $\\sum _{t \\in T_k, p \\in \\bullet t}{m_{W(p,t)}(c)}+\\sum _{t \\in T_k, p \\in R(t)}{m_{M_0(p)}(c)}, q1_c > q_c, c \\in C$ , indicating place $p$ that will be over consumed if $T_k$ is fired, as defined in definition REF (conflicting transition) $holds(p,q_c,c,k+1)$ if $q_c=m_{M_0(p)}(c)+\\sum _{t \\in T_k, p \\in t \\bullet }{m_{W(t,p)}(c)}- \\\\ (\\sum _{t \\in T_k, p \\in \\bullet t}{m_{W(p,t)}(c)}+ \\sum _{t \\in T_k, p \\in R(t)}{m_{M_0(p)}(c)})$ represented by $q_c=m_{M_1(p)}(c)$ for some $c \\in C$ – by construction of $\\Pi ^6$ $could\\_not\\_have(t,k)$ if $(\\forall p \\in \\bullet t, W(p,t) \\le M_0(p)), (\\forall p \\in I(t), M_0(p) = k), (\\forall (p,t) \\in Q, $ $M_0(p) \\ge WQ(p,t)), \\nexists tt \\in T, (\\forall pp \\in \\bullet tt, W(pp,tt) \\le M_{ts}(pp)), (\\forall pp \\in I(tt), M_0(pp) = k), (\\forall (pp,tt) \\in Q, $ $M_0(pp) \\ge QW(pp,tt)), Z(tt) < Z(t)$ , and $q_c > m_{M_0(s)}(c) - (\\sum _{t^{\\prime } \\in T_k, s \\in \\bullet t^{\\prime }}{m_{W(s,t^{\\prime })}(c)}+ \\sum _{t^{\\prime } \\in T_k, s \\in R(t)}{m_{M_0(s)}(c)}), $ $q_c = m_{W(s,t)}(c) \\text{~if~} s \\in \\bullet t \\text{~or~} m_{M_0(s)}(c) \\text{~otherwise}$ for some $c \\in C$ , which becomes $q > M_0(s) - (\\sum _{t^{\\prime } \\in T_k, s \\in \\bullet t^{\\prime }}{W(s,t^{\\prime })}+ \\sum _{t^{\\prime } \\in T_k, s \\in R(t)}{M_0(s)}), q = W(s,t) \\text{~if~} s \\in \\bullet t \\text{~or~} M_0(s) \\text{~otherwise}$ for all $c \\in C$ (i), (ii) above combined match the definition of $A_{25}$ $X_{9k+9}$ does not contain $could\\_not\\_have(t,k)$ , when $prenabled(t,k) \\in X_0 \\cup \\dots \\cup X_{9k+6}$ and $fires(t,k) \\notin X_0 \\cup \\dots \\cup X_{9k+5}$ due to construction of $A$ , encoding of $a\\ref {a:c:maxfire:cnh}$ and its body atoms.", "As a result it is not eliminated by the constraint $a\\ref {a:c:maxfire:elim}$ $eval_{U_{9k+9}}(bot_{U_{9k+10}}(\\Pi ^6) \\setminus bot_{U_{9k+9}}(\\Pi ^6), X_0 \\cup \\dots \\cup X_{9k+9}) = \\lbrace consumesmore \\text{:-}.", "\| \\\\ \\lbrace consumesmore(p,k) \\rbrace \\subseteq A \\rbrace $ .", "It's answer set is $X_{9k+10} = A_{20}$ – using forced atom proposition and the definition of $A_{20}$ $X_{9k+10}$ will be empty since none of $consumesmore(p,0),\\dots , \\\\ consumesmore(p,k)$ hold in $X_0 \\cup \\dots \\cup X_{9k+9}$ due to the construction of $A$ , encoding of $a\\ref {a:c:overc:place}$ and its body atoms.", "As a result, it is not eliminated by the constraint $a\\ref {a:c:overc:elim}$ The set $X = X_0 \\cup \\dots \\cup X_{9k+10}$ is the answer set of $\\Pi ^6$ by the splitting sequence theorem REF .", "Each $X_i, 0 \\le i \\le 9k+10$ matches a distinct portion of $A$ , and $X = A$ , thus $A$ is an answer set of $\\Pi ^6$ .", "Next we show (REF ): Given $\\Pi ^6$ be the encoding of a Petri Net $PN(P,T,E,C,W,R,I,$ $Q,QW,Z)$ with initial marking $M_0$ , and $A$ be an answer set of $\\Pi ^6$ that satisfies (REF ) and (REF ), then we can construct $X=M_0,T_k,\\dots ,M_k,T_k,M_{k+1}$ from $A$ , such that it is an execution sequence of $PN$ .", "We construct the $X$ as follows: $M_i = (M_i(p_0), \\dots , M_i(p_n))$ , where $\\lbrace holds(p_0,m_{M_i(p_0)}(c),c,i), \\dots ,\\\\ holds(p_n,m_{M_i(p_n)}(c),c,i) \\rbrace \\subseteq A$ , for $c \\in C, 0 \\le i \\le k+1$ $T_i = \\lbrace t : fires(t,i) \\in A\\rbrace $ , for $0 \\le i \\le k$ and show that $X$ is indeed an execution sequence of $PN$ .", "We show this by induction over $k$ (i.e.", "given $M_k$ , $T_k$ is a valid firing set and its firing produces marking $M_{k+1}$ ).", "Base case: Let $k=0$ , and $M_0$ is a valid marking in $X$ for $PN$ , show [(1)] $T_0$ is a valid firing set for $M_0$ , and firing $T_0$ in $M_0$ produces marking $M_1$ .", "We show $T_0$ is a valid firing set for $M_0$ .", "Let $\\lbrace fires(t_0,0), \\dots , fires(t_x,0) \\rbrace $ be the set of all $fires(\\dots ,0)$ atoms in $A$ , Then for each $fires(t_i,0) \\in A$ $prenabled(t_i,0) \\in A$ – from rule $a\\ref {a:c:prfires}$ and supported rule proposition Then $enabled(t_i,0) \\in A$ – from rule $a\\ref {a:c:prenabled}$ and supported rule proposition And $notprenabled(t_i,0) \\notin A$ – from rule $a\\ref {a:c:prenabled}$ and supported rule proposition For $enabled(t_i,0) \\in A$ $notenabled(t_i,0) \\notin A$ – from rule $e\\ref {e:c:enabled}$ and supported rule proposition Then either of $body(e\\ref {e:c:ne:ptarc})$ , $body(e\\ref {e:c:ne:iptarc})$ , or $body(e\\ref {e:c:ne:tptarc})$ must not hold in $A$ for $t_i$ – from rules $e\\ref {e:c:ne:ptarc},e\\ref {e:c:ne:iptarc},e\\ref {e:c:ne:tptarc}$ and forced atom proposition Then $q_c \\lnot < {n_i}_c \\equiv q_c \\ge {n_i}_c$ in $e\\ref {e:c:ne:ptarc}$ for all $\\lbrace holds(p,q_c,c,0), \\\\ ptarc(p,t_i,{n_i}_c,c,0)\\rbrace \\subseteq A$ – from $e:\\ref {e:c:ne:ptarc}$ , forced atom proposition, and given facts ($holds(p,q_c,c,0) \\in A, ptarc(p,t_i,{n_i}_c,0) \\in A$ ) And $q_c \\lnot \\ge {n_i}_c \\equiv q_c < {n_i}_c$ in $e\\ref {e:c:ne:iptarc}$ for all $\\lbrace holds(p,q_c,c,0), \\\\ iptarc(p,t_i,{n_i}_c,c,0) \\rbrace \\subseteq A, {n_i}_c=1$ ; $q_c > {n_i}_c \\equiv q_c = 0$ – from $e\\ref {e:c:ne:iptarc}$ , forced atom proposition, given facts ($holds(p,q_c,c,0) \\in A, \\\\ iptarc(p,t_i,1,c,0) \\in A$ ), and $q_c$ is a positive integer And $q_c \\lnot < {n_i}_c \\equiv q_c \\ge {n_i}_c$ in $e\\ref {e:c:ne:tptarc}$ for all $\\lbrace holds(p,q_c,c,0), \\\\ tptarc(p,t_i,{n_i}_c,c,0) \\rbrace \\subseteq A$ – from $e\\ref {e:c:ne:tptarc}$ , forced atom proposition, and given facts Then $\\forall c \\in C, (\\forall p \\in \\bullet t_i, m_{M_0(p)}(c) \\ge m_{W(p,t_i)}(c)) \\wedge (\\forall p \\in I(t_i), \\\\ m_{M_0(p)}(c) = 0) \\wedge (\\forall (p,t_i) \\in Q, m_{M_0(p)}(c) \\ge m_{QW(p,t_i)}(c))$ – from $i\\ref {i:c:init},f\\ref {f:c:ptarc},f\\ref {f:c:iptarc},f\\ref {f:c:tptarc}$ construction, definition REF of preset $\\bullet t_i$ in $PN$ , definition REF of enabled transition in $PN$ , and that the construction of reset arcs by $f\\ref {f:c:rptarc}$ ensures $notenabled(t,0)$ is never true for a reset arc, where $holds(p,q_c,c,0) \\in A$ represents $q_c=m_{M_0(p)}(c)$ , $ptarc(p,t_i,{n_i}_c,0) \\in A$ represents ${n_i}_c=m_{W(p,t_i)}(c)$ , ${n_i}_c = m_{M_0(p)}(c)$ .", "Which is equivalent to $(\\forall p \\in \\bullet t_i, M_0(p) \\ge W(p,t_i)) \\wedge (\\forall p \\in I(t_i), M_0(p) = 0) \\wedge (\\forall (p,t_i) \\in Q, M_0(p) \\ge QW(p,t_i))$ – assuming multiset domain $C$ For $notprenabled(t_i,0) \\notin A$ Either $(\\nexists enabled(tt,0) \\in A : pp < p_i)$ or $(\\forall enabled(tt,0) \\in A : pp \\lnot < p_i)$ where $pp = Z(tt), p_i = Z(t_i)$ – from rule $a\\ref {a:c:prne},f\\ref {f:c:pr}$ and forced atom proposition This matches the definition of an enabled priority transition Then $t_i$ is enabled and can fire in $PN$ , as a result it can belong to $T_0$ – from definition REF of enabled transition And $consumesmore \\notin A$ , since $A$ is an answer set of $\\Pi ^6$ – from rule $a\\ref {a:c:overc:gen}$ and supported rule proposition Then $\\nexists consumesmore(p,0) \\in A$ – from rule $a\\ref {a:c:overc:place}$ and supported rule proposition Then $\\nexists \\lbrace holds(p,q_c,c,0), tot\\_decr(p,q1_c,c,0) \\rbrace \\subseteq A, q1_c>q_c$ in $body(a\\ref {a:c:overc:place})$ – from $a\\ref {a:c:overc:place}$ and forced atom proposition Then $\\nexists c \\in C \\nexists p \\in P, (\\sum _{t_i \\in \\lbrace t_0,\\dots ,t_x\\rbrace , p \\in \\bullet t_i}{m_{W(p,t_i)}(c)}+ \\\\ \\sum _{t_i \\in \\lbrace t_0,\\dots ,t_x\\rbrace , p \\in R(t_i)}{m_{M_0(p)}(c)}) > m_{M_0(p)}(c)$ – from the following $holds(p,q_c,c,0)$ represents $q_c=m_{M_0(p)}(c)$ – from rule $i\\ref {i:c:init}$ encoding, given $tot\\_decr(p,q1_c,c,0) \\in A$ if $\\lbrace del(p,{q1_0}_c,t_0,c,0), \\dots , \\\\ del(p,{q1_x}_c,t_x,c,0) \\rbrace \\subseteq A$ , where $q1_c = {q1_0}_c+\\dots +{q1_x}_c$ – from $r\\ref {r:c:totdecr}$ and forced atom proposition $del(p,{q1_i}_c,t_i,c,0) \\in A$ if $\\lbrace fires(t_i,0), ptarc(p,t_i,{q1_i}_c,c,0) \\rbrace \\subseteq A$ – from $r\\ref {r:c:del}$ and supported rule proposition $del(p,{q1_i}_c,t_i,c,0)$ either represents removal of ${q1_i}_c = m_{W(p,t_i)}(c)$ tokens from $p \\in \\bullet t_i$ ; or it represents removal of ${q1_i}_c = m_{M_0(p)}(c)$ tokens from $p \\in R(t_i)$ – from rules $r\\ref {r:c:del},f\\ref {f:c:ptarc},f\\ref {f:c:rptarc}$ , supported rule proposition, and definition REF of transition execution in $PN$ Then the set of transitions in $T_0$ do not conflict – by the definition REF of conflicting transitions And for each $prenabled(t_j,0) \\in A$ and $fires(t_j,0) \\notin A$ , $could\\_not\\_have(t_j,0) \\in A$ , since $A$ is an answer set of $\\Pi ^6$ - from rule $a\\ref {a:c:prmaxfire:elim}$ and supported rule proposition Then $\\lbrace prenabled(t_j,0), holds(s,qq_c,c,0), ptarc(s,t_j,q_c,c,0), \\\\ tot\\_decr(s,qqq_c,c,0) \\rbrace \\subseteq A$ , such that $q_c > qq_c - qqq_c$ and $fires(t_j,0) \\notin A$ - from rule $a\\ref {a:c:prmaxfire:cnh}$ and supported rule proposition Then for an $s \\in \\bullet t_j \\cup R(t_j)$ , $q_c > m_{M_0(s)}(c) - (\\sum _{t_i \\in T_0, s \\in \\bullet t_i}{m_{W(s,t_i)}(c)} + \\sum _{t_i \\in T_0, s \\in R(t_i)}{m_{M_0(s)}(c))}$ , where $q_c=m_{W(s,t_j)}(c) \\text{~if~} s \\in \\bullet t_j, \\text{~or~} m_{M_0(s)}(c)$ $ \\text{~otherwise}$ – from the following $ptarc(s,t_i,q_c,c,0)$ represents $q_c=m_{W(s,t_i)}(c)$ if $(s,t_i) \\in E^-$ or $q_c=m_{M_0(s)}(c)$ if $s \\in R(t_i)$ – from rule $f\\ref {f:c:ptarc},f\\ref {f:c:rptarc}$ construction $holds(s,qq_c,c,0)$ represents $qq_c=m_{M_0(s)}(c)$ – from $i\\ref {i:c:init}$ construction $tot\\_decr(s,qqq_c,c,0) \\in A$ if $\\lbrace del(s,{qqq_0}_c,t_0,c,0), \\dots , \\\\ del(s,{qqq_x}_c,t_x,c,0) \\rbrace \\subseteq A$ – from rule $r\\ref {r:c:totdecr}$ construction and supported rule proposition $del(s,{qqq_i}_c,t_i,c,0) \\in A$ if $\\lbrace fires(t_i,0), ptarc(s,t_i,{qqq_i}_c,c,0) \\rbrace \\subseteq A$ – from rule $r\\ref {r:c:del}$ and supported rule proposition $del(s,{qqq_i}_c,t_i,c,0)$ either represents ${qqq_i}_c = m_{W(s,t_i)}(c) : t_i \\in T_0, (s,t_i) \\in E^-$ , or ${qqq_i}_c = m_{M_0(t_i)}(c) : t_i \\in T_0, s \\in R(t_i)$ – from rule $f\\ref {f:c:ptarc},f\\ref {f:c:rptarc}$ construction $tot\\_decr(q,qqq_c,c,0)$ represents $\\sum _{t_i \\in T_0, s \\in \\bullet t_i}{m_{W(s,t_i)}(c)} + \\\\ \\sum _{t_i \\in T_0, s \\in R(t_i)}{m_{M_0(s)}(c)}$ – from (C,D,E) above Then firing $T_0 \\cup \\lbrace t_j \\rbrace $ would have required more tokens than are present at its source place $s \\in \\bullet t_j \\cup R(t_j)$ .", "Thus, $T_0$ is a maximal set of transitions that can simultaneously fire.", "And for each reset transition $t_r$ with $prenabled(t_r,0) \\in A$ , $fires(t_r,0) \\in A$ , since $A$ is an answer set of $\\Pi ^6$ - from rule $f\\ref {f:c:pr:rptarc:elim}$ and supported rule proposition Then, the firing set $T_0$ satisfies the reset-transition requirement of definition REF (firing set) Then $\\lbrace t_0, \\dots , t_x\\rbrace = T_0$ – using 1(a),1(b),1(d) above; and using 1(c) it is a maximal firing set We show $M_1$ is produced by firing $T_0$ in $M_0$ .", "Let $holds(p,q_c,c,1) \\in A$ Then $\\lbrace holds(p,q1_c,c,0), tot\\_incr(p,q2_c,c,0), tot\\_decr(p,q3_c,c,0) \\rbrace \\subseteq A : q_c=q1_c+q2_c-q3_c$ – from rule $r\\ref {r:c:nextstate}$ and supported rule proposition Then, $holds(p,q1_c,c,0) \\in A$ represents $q1_c=m_{M_0(p)}(c)$ – given, construction; and $\\lbrace add(p,{q2_0}_c,t_0,c,0), \\dots , $ $add(p,{q2_j}_c,t_j,c,0)\\rbrace \\subseteq A : {q2_0}_c + \\dots + {q2_j}_c = q2_c$ and $\\lbrace del(p,{q3_0}_c,t_0,c,0), \\dots , $ $del(p,{q3_l}_c,t_l,c,0)\\rbrace \\subseteq A : {q3_0}_c + \\dots + {q3_l}_c = q3_c$ – rules $r\\ref {r:c:totincr},r\\ref {r:c:totdecr}$ and supported rule proposition, respectively Then $\\lbrace fires(t_0,0), \\dots , fires(t_j,0) \\rbrace \\subseteq A$ and $\\lbrace fires(t_0,0), \\dots , fires(t_l,0) \\rbrace \\subseteq A$ – rules $r\\ref {r:c:add},r\\ref {r:c:del}$ and supported rule proposition, respectively Then $\\lbrace fires(t_0,0), \\dots , $ $fires(t_j,0) \\rbrace \\cup $ $\\lbrace fires(t_0,0), \\dots , $ $fires(t_l,0) \\rbrace \\subseteq $ $A = \\lbrace fires(t_0,0), \\dots , $ $fires(t_x,0) \\rbrace \\subseteq A$ – set union of subsets Then for each $fires(t_x,0) \\in A$ we have $t_x \\in T_0$ – already shown in item REF above Then $q_c = m_{M_0(p)}(c) + \\sum _{t_x \\in T_0 \\wedge p \\in t_x \\bullet }{m_{W(t_x,p)}(c)} - (\\sum _{t_x \\in T_0 \\wedge p \\in \\bullet t_x}{m_{W(p,t_x)}(c)} + \\sum _{t_x \\in T_0 \\wedge p \\in R(t_x)}{m_{M_0(p)}(c)})$ – from (REF ) above and the following Each $add(p,{q_j}_c,t_j,c,0) \\in A$ represents ${q_j}_c=m_{W(t_j,p)}(c)$ for $p \\in t_j \\bullet $ – rule $r\\ref {r:c:add},f\\ref {f:c:tparc}$ encoding, and definition REF of transition execution in $PN$ Each $del(p,t_y,{q_y}_c,c,0) \\in A$ represents either ${q_y}_c=m_{W(p,t_y)}(c)$ for $p \\in \\bullet t_y$ , or ${q_y}_c=m_{M_0(p)}(c)$ for $p \\in R(t_y)$ – from rule $r\\ref {r:c:del},f\\ref {f:c:ptarc}$ encoding and definition REF of transition execution in $PN$ ; or from rule $r\\ref {r:c:del},f\\ref {f:c:rptarc}$ encoding and definition of reset arc in $PN$ Each $tot\\_incr(p,q2_c,c,0) \\in A$ represents $q2_c=\\sum _{t_x \\in T_0 \\wedge p \\in t_x \\bullet }{m_{W(t_x,p)}(c)}$ – aggregate assignment atom semantics in rule $r\\ref {r:c:totincr}$ Each $tot\\_decr(p,q3_c,c,0) \\in A$ represents $q3_c=\\sum _{t_x \\in T_0 \\wedge p \\in \\bullet t_x}{m_{W(p,t_x)}(c)} + $ $\\sum _{t_x \\in T_0 \\wedge p \\in R(t_x)}{m_{M_0(p)}(c)}$ – aggregate assignment atom semantics in rule $r\\ref {r:c:totdecr}$ Then, $m_{M_1(p)}(c) = q_c$ – since $holds(p,q_c,c,1) \\in A$ encodes $q_c=m_{M_1(p)}(c)$ – from construction Inductive Step: Let $k > 0$ , and $M_k$ is a valid marking in $X$ for $PN$ , show [(1)] $T_k$ is a valid firing set for $M_k$ , and firing $T_k$ in $M_k$ produces marking $M_{k+1}$ .", "We show that $T_k$ is a valid firing set in $M_k$ .", "Let $\\lbrace fires(T_k,k), \\dots , fires(t_x,k) \\rbrace $ be the set of all $fires(\\dots ,k)$ atoms in $A$ , Then for each $fires(t_i,k) \\in A$ $prenabled(t_i,k) \\in A$ – from rule $a\\ref {a:c:prfires}$ and supported rule proposition Then $enabled(t_i,k) \\in A$ – from rule $a\\ref {a:c:prenabled}$ and supported rule proposition And $notprenabled(t_i,k) \\notin A$ – from rule $a\\ref {a:c:prenabled}$ and supported rule proposition For $enabled(t_i,k) \\in A$ $notenabled(t_i,k) \\notin A$ – from rule $e\\ref {e:c:enabled}$ and supported rule proposition Then either of $body(e\\ref {e:c:ne:ptarc})$ , $body(e\\ref {e:c:ne:iptarc})$ , or $body(e\\ref {e:c:ne:tptarc})$ must not hold in $A$ for $t_i$ – from rules $e\\ref {e:c:ne:ptarc},e\\ref {e:c:ne:iptarc},e\\ref {e:c:ne:tptarc}$ and forced atom proposition Then $q_c \\lnot < {n_i}_c \\equiv q_c \\ge {n_i}_c$ in $e\\ref {e:c:ne:ptarc}$ for all $\\lbrace holds(p,q_c,c,k), \\\\ ptarc(p,t_i,{n_i}_c,c,k)\\rbrace \\subseteq A$ – from $e\\ref {e:c:ne:ptarc}$ , forced atom proposition, and given facts ($holds(p,q_c,c,k) \\in A, ptarc(p,t_i,{n_i}_c,k) \\in A$ ) And $q_c \\lnot \\ge {n_i}_c \\equiv q_c < {n_i}_c$ in $e\\ref {e:c:ne:iptarc}$ for all $\\lbrace holds(p,q_c,c,k), \\\\ iptarc(p,t_i,{n_i}_c,c,k) \\rbrace \\subseteq A, {n_i}_c=1$ ; $q_c > {n_i}_c \\equiv q_c = 0$ – from $e\\ref {e:c:ne:iptarc}$ , forced atom proposition, given facts ($holds(p,q_c,c,k) \\in A, \\\\ iptarc(p,t_i,1,c,k) \\in A$ ), and $q_c$ is a positive integer And $q_c \\lnot < {n_i}_c \\equiv q_c \\ge {n_i}_c$ in $e\\ref {e:c:ne:tptarc}$ for all $\\lbrace holds(p,q_c,c,k), \\\\ tptarc(p,t_i,{n_i}_c,c,k) \\rbrace \\subseteq A$ – from $e\\ref {e:c:ne:tptarc}$ , forced atom proposition, and given facts Then $\\forall c \\in C, (\\forall p \\in \\bullet t_i, m_{M_k(p)}(c) \\ge m_{W(p,t_i)}(c)) \\wedge (\\forall p \\in I(t_i), $ $m_{M_k(p)}(c) = k) \\wedge (\\forall (p,t_i) \\in Q, m_{M_k(p)}(c) \\ge m_{QW(p,t_i)}(c))$ – from $f\\ref {f:c:ptarc},f\\ref {f:c:iptarc},f\\ref {f:c:tptarc}$ construction, inductive assumption, definition REF of preset $\\bullet t_i$ in $PN$ , definition REF of enabled transition in $PN$ , and that the construction of reset arcs by $f\\ref {f:c:rptarc}$ ensures $notenabled(t,k)$ is never true for a reset arc, where $holds(p,q_c,c,k) \\in A$ represents $q_c=m_{M_k(p)}(c)$ , $ptarc(p,t_i,{n_i}_c,k) \\in A$ represents ${n_i}_c=m_{W(p,t_i)}(c)$ , ${n_i}_c = m_{M_k(p)}(c)$ .", "Which is equivalent to $(\\forall p \\in \\bullet t_i, M_k(p) \\ge W(p,t_i)) \\wedge (\\forall p \\in I(t_i), M_k(p) = k) \\wedge (\\forall (p,t_i) \\in Q, M_k(p) \\ge QW(p,t_i))$ – assuming multiset domain $C$ For $notprenabled(t_i,k) \\notin A$ Either $(\\nexists enabled(tt,k) \\in A : pp < p_i)$ or $(\\forall enabled(tt,k) \\in A : pp \\lnot < p_i)$ where $pp = Z(tt), p_i = Z(t_i)$ – from rule $a\\ref {a:c:prne}, f\\ref {f:c:pr}$ and forced atom proposition This matches the definition of an enabled priority transition Then $t_i$ is enabled and can fire in $PN$ , as a result it can belong to $T_k$ – from definition REF of enabled transition And $consumesmore \\notin A$ , since $A$ is an answer set of $\\Pi ^6$ – from rule $a\\ref {a:c:overc:elim}$ and supported rule proposition Then $\\nexists consumesmore(p,k) \\in A$ – from rule $a\\ref {a:c:overc:elim}$ and supported rule proposition Then $\\nexists \\lbrace holds(p,q_c,c,k), tot\\_decr(p,q1_c,c,k) \\rbrace \\subseteq A, q1_c>q_c$ in $body(a\\ref {a:c:overc:place})$ – from $a\\ref {a:c:overc:place}$ and forced atom proposition Then $\\nexists c \\in C \\nexists p \\in P, (\\sum _{t_i \\in \\lbrace T_k,\\dots ,t_x\\rbrace , p \\in \\bullet t_i}{m_{W(p,t_i)}(c)}+ \\\\ \\sum _{t_i \\in \\lbrace T_k,\\dots ,t_x\\rbrace , p \\in R(t_i)}{m_{M_k(p)}(c)}) > m_{M_k(p)}(c)$ – from the following $holds(p,q_c,c,k)$ represents $q_c=m_{M_k(p)}(c)$ – from inductive assumption and construction, given $tot\\_decr(p,q1_c,c,k) \\in A$ if $\\lbrace del(p,{q1_0}_c,T_k,c,k), \\dots , \\\\ del(p,{q1_x}_c,t_x,c,k) \\rbrace \\subseteq A$ , where $q1_c = {q1_0}_c+\\dots +{q1_x}_c$ – from $r\\ref {r:c:totdecr}$ and forced atom proposition $del(p,{q1_i}_c,t_i,c,k) \\in A$ if $\\lbrace fires(t_i,k), ptarc(p,t_i,{q1_i}_c,c,k) \\rbrace \\subseteq A$ – from $r\\ref {r:c:del}$ and supported rule proposition $del(p,{q1_i}_c,t_i,c,k)$ either represents removal of ${q1_i}_c = m_{W(p,t_i)}(c)$ tokens from $p \\in \\bullet t_i$ ; or it represents removal of ${q1_i}_c = m_{M_k(p)}(c)$ tokens from $p \\in R(t_i)$ – from rules $r\\ref {r:c:del},f\\ref {f:c:ptarc},f\\ref {f:c:rptarc}$ , supported rule proposition, and definition REF of transition execution in $PN$ Then the set of transitions in $T_k$ do not conflict – by the definition REF of conflicting transitions And for each $prenabled(t_j,k) \\in A$ and $fires(t_j,k) \\notin A$ , $could\\_not\\_have(t_j,k) \\in A$ , since $A$ is an answer set of $\\Pi ^6$ - from rule $a\\ref {a:c:prmaxfire:elim}$ and supported rule proposition Then $\\lbrace prenabled(t_j,k), holds(s,qq_c,c,k), ptarc(s,t_j,q_c,c,k), \\\\ tot\\_decr(s,qqq_c,c,k) \\rbrace \\subseteq A$ , such that $q_c > qq_c - qqq_c$ and $fires(t_j,k) \\notin A$ - from rule $a\\ref {a:c:prmaxfire:cnh}$ and supported rule proposition Then for an $s \\in \\bullet t_j \\cup R(t_j)$ , $q_c > m_{M_k(s)}(c) - (\\sum _{t_i \\in T_k, s \\in \\bullet t_i}{m_{W(s,t_i)}(c)} + \\sum _{t_i \\in T_k, s \\in R(t_i)}{m_{M_k(s)}(c))}$ , where $q_c=m_{W(s,t_j)}(c) \\text{~if~} s \\in \\bullet t_j, \\text{~or~} m_{M_k(s)}(c) $ $\\text{~otherwise}$ .", "$ptarc(s,t_i,q_c,c,k)$ represents $q_c=m_{W(s,t_i)}(c)$ if $(s,t_i) \\in E^-$ or $q_c=m_{M_k(s)}(c)$ if $s \\in R(t_i)$ – from rule $f\\ref {f:c:ptarc},f\\ref {f:c:rptarc}$ construction $holds(s,qq_c,c,k)$ represents $qq_c=m_{M_k(s)}(c)$ – from inductive assumption and construction $tot\\_decr(s,qqq_c,c,k) \\in A$ if $\\lbrace del(s,{qqq_0}_c,T_k,c,k), \\dots , \\\\ del(s,{qqq_x}_c,t_x,c,k) \\rbrace \\subseteq A$ – from rule $r\\ref {r:c:totdecr}$ construction and supported rule proposition $del(s,{qqq_i}_c,t_i,c,k) \\in A$ if $\\lbrace fires(t_i,k), ptarc(s,t_i,{qqq_i}_c,c,k) \\rbrace \\subseteq A$ – from rule $r\\ref {r:c:del}$ and supported rule proposition $del(s,{qqq_i}_c,t_i,c,k)$ either represents ${qqq_i}_c = m_{W(s,t_i)}(c) : t_i \\in T_k, (s,t_i) \\in E^-$ , or ${qqq_i}_c = m_{M_k(t_i)}(c) : t_i \\in T_k, s \\in R(t_i)$ – from rule $f\\ref {f:c:ptarc},f\\ref {f:c:rptarc}$ construction $tot\\_decr(q,qqq_c,c,k)$ represents $\\sum _{t_i \\in T_k, s \\in \\bullet t_i}{m_{W(s,t_i)}(c)} + \\\\ \\sum _{t_i \\in T_k, s \\in R(t_i)}{m_{M_k(s)}(c)}$ – from (C,D,E) above Then firing $T_k \\cup \\lbrace t_j \\rbrace $ would have required more tokens than are present at its source place $s \\in \\bullet t_j \\cup R(t_j)$ .", "Thus, $T_k$ is a maximal set of transitions that can simultaneously fire.", "And for each reset transition $t_r$ with $prenabled(t_r,k) \\in A$ , $fires(t_r,k) \\in A$ , since $A$ is an answer set of $\\Pi ^6$ - from rule $f\\ref {f:c:pr:rptarc:elim}$ and supported rule proposition Then the firing set $T_k$ satisfies the reset transition requirement of definition REF (firing set) Then $\\lbrace t_0, \\dots , t_x\\rbrace = T_k$ – using 1(a),1(b), 1(d) above; and using 1(c) it is a maximal firing set We show that $M_{k+1}$ is produced by firing $T_k$ in $M_k$ .", "Let $holds(p,q_c,c,k+1) \\in A$ Then $\\lbrace holds(p,q1_c,c,k), tot\\_incr(p,q2_c,c,k), tot\\_decr(p,q3_c,c,k) \\rbrace \\subseteq A : q_c=q1_c+q2_c-q3_c$ – from rule $r\\ref {r:c:nextstate}$ and supported rule proposition Then, $holds(p,q1_c,c,k) \\in A$ represents $q1_c=m_{M_k(p)}(c)$ – inductive assumption; and $\\lbrace add(p,{q2_0}_c,T_k,c,k), \\dots , $ $add(p,{q2_j}_c,t_j,c,k)\\rbrace \\subseteq A : {q2_0}_c + \\dots + {q2_j}_c = q2_c$ and $\\lbrace del(p,{q3_0}_c,T_k,c,k), \\dots , $ $del(p,{q3_l}_c,t_l,c,k)\\rbrace \\subseteq A : {q3_0}_c + \\dots + {q3_l}_c = q3_c$ – rules $r\\ref {r:c:totincr},r\\ref {r:c:totdecr}$ and supported rule proposition, respectively Then $\\lbrace fires(T_k,k), \\dots , fires(t_j,k) \\rbrace \\subseteq A$ and $\\lbrace fires(T_k,k), \\dots , \\\\ fires(t_l,k) \\rbrace \\subseteq A$ – rules $r\\ref {r:c:add},r\\ref {r:c:del}$ and supported rule proposition, respectively Then $\\lbrace fires(T_k,k), \\dots , $ $fires(t_j,k) \\rbrace \\cup $ $\\lbrace fires(T_k,k), \\dots , $ $fires(t_l,k) \\rbrace \\subseteq $ $A = \\lbrace fires(T_k,k), \\dots , fires(t_x,k) \\rbrace \\subseteq A$ – set union of subsets Then for each $fires(t_x,k) \\in A$ we have $t_x \\in T_k$ – already shown in item REF above Then $q_c = m_{M_k(p)}(c) + \\sum _{t_x \\in T_k \\wedge p \\in t_x \\bullet }{m_{W(t_x,p)}(c)} - (\\sum _{t_x \\in T_k \\wedge p \\in \\bullet t_x}{m_{W(p,t_x)}(c)} + \\sum _{t_x \\in T_k \\wedge p \\in R(t_x)}{m_{M_k(p)}(c)})$ – from (REF ) above and the following Each $add(p,{q_j}_c,t_j,c,k) \\in A$ represents ${q_j}_c=m_{W(t_j,p)}(c)$ for $p \\in t_j \\bullet $ – rule $r\\ref {r:c:add},f\\ref {f:c:tparc}$ encoding, and definition REF of transition execution in $PN$ Each $del(p,t_y,{q_y}_c,c,k) \\in A$ represents either ${q_y}_c=m_{W(p,t_y)}(c)$ for $p \\in \\bullet t_y$ , or ${q_y}_c=m_{M_k(p)}(c)$ for $p \\in R(t_y)$ – from rule $r\\ref {r:c:del},f\\ref {f:c:ptarc}$ encoding and definition REF of transition execution in $PN$ ; or from rule $r\\ref {r:c:del},f\\ref {f:c:rptarc}$ encoding and definition of reset arc in $PN$ Each $tot\\_incr(p,q2_c,c,k) \\in A$ represents $q2_c=\\sum _{t_x \\in T_k \\wedge p \\in t_x \\bullet }{m_{W(t_x,p)}(c)}$ – aggregate assignment atom semantics in rule $r\\ref {r:c:totincr}$ Each $tot\\_decr(p,q3_c,c,k) \\in A$ represents $q3_c=\\sum _{t_x \\in T_k \\wedge p \\in \\bullet t_x}{m_{W(p,t_x)}(c)} + \\sum _{t_x \\in T_k \\wedge p \\in R(t_x)}{m_{M_k(p)}(c)}$ – aggregate assignment atom semantics in rule $r\\ref {r:c:totdecr}$ Then, $m_{M_{k+1}(p)}(c) = q_c$ – since $holds(p,q_c,c,k+1) \\in A$ encodes $q_c=m_{M_{k+1}(p)}(c)$ – from construction As a result, for any $n > k$ , $T_n$ will be a valid firing set for $M_n$ and $M_{n+1}$ will be its target marking.", "Conclusion: Since both (REF ) and (REF ) hold, $X=M_0,T_k,M_1,\\dots ,M_k,T_{k+1}$ is an execution sequence of $PN(P,T,E,C,W,R,I,Q,QW,Z)$ (w.r.t $M_0$ ) iff there is an answer set $A$ of $\\Pi ^6(PN,M_0,k,ntok)$ such that (REF ) and (REF ) hold.", "Proof of Proposition REF Let $PN=(P,T,E,C,W,R,I,Q,QW,Z,D)$ be a Petri Net, $M_0$ be its initial marking and let $\\Pi ^7(PN,M_0,k,ntok)$ be the ASP encoding of $PN$ and $M_0$ over a simulation length $k$ , with maximum $ntok$ tokens on any place node, as defined in section REF .", "Then $X=M_0,T_k,M_1,\\dots ,M_k,T_k,M_{k+1}$ is an execution sequence of $PN$ (w.r.t.", "$M_0$ ) iff there is an answer set $A$ of $\\Pi ^7(PN,M_0,k,ntok)$ such that: $\\lbrace fires(t,ts) : t \\in T_{ts}, 0 \\le ts \\le k\\rbrace = \\lbrace fires(t,ts) : fires(t,ts) \\in A \\rbrace $ $\\begin{split}\\lbrace holds(p,q,c,ts) &: p \\in P, c/q = M_{ts}(p), 0 \\le ts \\le k+1 \\rbrace \\\\&= \\lbrace holds(p,q,c,ts) : holds(p,q,c,ts) \\in A \\rbrace \\end{split}$ We prove this by showing that: Given an execution sequence $X$ , we create a set $A$ such that it satisfies (REF ) and (REF ) and show that $A$ is an answer set of $\\Pi ^7$ Given an answer set $A$ of $\\Pi ^7$ , we create an execution sequence $X$ such that (REF ) and (REF ) are satisfied.", "First we show (REF ): We create a set $A$ as a union of the following sets: $A_1=\\lbrace num(n) : 0 \\le n \\le ntok \\rbrace $ $A_2=\\lbrace time(ts) : 0 \\le ts \\le k\\rbrace $ $A_3=\\lbrace place(p) : p \\in P \\rbrace $ $A_4=\\lbrace trans(t) : t \\in T \\rbrace $ $A_5=\\lbrace color(c) : c \\in C \\rbrace $ $A_6=\\lbrace ptarc(p,t,n_c,c,ts) : (p,t) \\in E^-, c \\in C, n_c=m_{W(p,t)}(c), n_c > 0, 0 \\le ts \\le k \\rbrace $ , where $E^- \\subseteq E$ $A_7=\\lbrace tparc(t,p,n_c,c,ts,d) : (t,p) \\in E^+, c \\in C, n_c=m_{W(t,p)}(c), n_c > 0, d=D(t), 0 \\le ts \\le k \\rbrace $ , where $E^+ \\subseteq E$ $A_8=\\lbrace holds(p,q_c,c,0) : p \\in P, c \\in C, q_c=m_{M_{0}(p)}(c) \\rbrace $ $A_9=\\lbrace ptarc(p,t,n_c,c,ts) : p \\in R(t), c \\in C, n_c = m_{M_{ts}(p)}, n_c > 0, 0 \\le ts \\le k \\rbrace $ $A_{10}=\\lbrace iptarc(p,t,1,c,ts) : p \\in I(t), c \\in C, 0 \\le ts < k \\rbrace $ $A_{11}=\\lbrace tptarc(p,t,n_c,c,ts) : (p,t) \\in Q, c \\in C, n_c=m_{QW(p,t)}(c), n_c > 0, 0 \\le ts \\le k \\rbrace $ $A_{12}=\\lbrace notenabled(t,ts) : t \\in T, 0 \\le ts \\le k, \\exists c \\in C, (\\exists p \\in \\bullet t, m_{M_{ts}(p)}(c) < m_{W(p,t)}(c)) \\vee (\\exists p \\in I(t), m_{M_{ts}(p)}(c) > 0) \\vee (\\exists (p,t) \\in Q, m_{M_{ts}(p)}(c) < m_{QW(p,t)}(c)) \\rbrace $ per definition REF (enabled transition) $A_{13}=\\lbrace enabled(t,ts) : t \\in T, 0 \\le ts \\le k, \\forall c \\in C, (\\forall p \\in \\bullet t, m_{W(p,t)}(c) \\le m_{M_{ts}(p)}(c)) \\wedge (\\forall p \\in I(t), m_{M_{ts}(p)}(c) = 0) \\wedge (\\forall (p,t) \\in Q, m_{M_{ts}(p)}(c) \\ge m_{QW(p,t)}(c)) \\rbrace $ per definition REF (enabled transition) $A_{14}=\\lbrace fires(t,ts) : t \\in T_{ts}, 0 \\le ts \\le k \\rbrace $ per definition REF (enabled transitions), only an enabled transition may fire $A_{15}=\\lbrace add(p,q_c,t,c,ts+d-1) : t \\in T_{ts}, p \\in t \\bullet , c \\in C, q_c=m_{W(t,p)}(c), \\\\d = D(t), 0 \\le ts \\le k \\rbrace $ per definition REF (transition execution) $A_{16}=\\lbrace del(p,q_c,t,c,ts) : t \\in T_{ts}, p \\in \\bullet t, c \\in C, q_c=m_{W(p,t)}(c), 0 \\le ts \\le k \\rbrace \\cup \\lbrace del(p,q_c,t,c,ts) : t \\in T_{ts}, p \\in R(t), c \\in C, q_c=m_{M_{ts}(p)}(c), 0 \\le ts \\le k \\rbrace $ per definition REF (transition execution) $A_{17}=\\lbrace tot\\_incr(p,q_c,c,ts) : p \\in P, c \\in C, \\\\ q_c=\\sum _{t \\in T_{l}, p \\in t \\bullet , l \\le ts, l+D(t)=ts+1}{m_{W(t,p)}(c)}, 0 \\le ts \\le k \\rbrace $ per definition REF (firing set execution) $A_{18}=\\lbrace tot\\_decr(p,q_c,c,ts): p \\in P, c \\in C, q_c=\\sum _{t \\in T_{ts}, p \\in \\bullet t}{m_{W(p,t)}(c)}+ \\\\ \\sum _{t \\in T_{ts}, p \\in R(t) }{m_{M_{ts}(p)}(c)}, 0 \\le ts \\le k \\rbrace $ per definition REF (firing set execution) $A_{19}=\\lbrace consumesmore(p,ts) : p \\in P, c \\in C, q_c=m_{M_{ts}(p)}(c), \\\\ q1_c=\\sum _{t \\in T_{ts}, p \\in \\bullet t}{m_{W(p,t)}(c)} + \\sum _{t \\in T_{ts}, p \\in R(t)}{m_{M_{ts}(p)}(c)}, q1_c > q_c, 0 \\le ts \\le k \\rbrace $ per definition REF (conflicting transitions) $A_{20}=\\lbrace consumesmore : \\exists p \\in P, c \\in C, q_c=m_{M_{ts}(p)}(c), \\\\ q1_c=\\sum _{t \\in T_{ts}, p \\in \\bullet t}{m_{W(p,t)}(c)} + \\sum _{t \\in T_{ts}, p \\in R(t)}(m_{M_{ts}(p)}(c)), q1_c > q_c, 0 \\le ts \\le k \\rbrace $ per definition REF (conflicting transitions) $A_{21}=\\lbrace holds(p,q_c,c,ts+1) : p \\in P, c \\in C, q_c=m_{M_{ts+1}(p)}(c), 0 \\le ts < k\\rbrace $ , where $M_{ts+1}(p) = M_{ts}(p) - (\\sum _{\\begin{array}{c}t \\in T_{ts}, p \\in \\bullet t\\end{array}}{W(p,t)} + \\sum _{t \\in T_{ts}, p \\in R(t)}M_{ts}(p)) + \\\\ \\sum _{\\begin{array}{c}t \\in T_l, p \\in t \\bullet , l \\le ts, l+D(t)-1=ts\\end{array}}{W(t,p)}$ according to definition REF (firing set execution) $A_{22}=\\lbrace transpr(t,pr) : t \\in T, pr=Z(t) \\rbrace $ $A_{23}=\\lbrace notprenabled(t,ts) : t \\in T, enabled(t,ts) \\in A_{13}, (\\exists tt \\in T, enabled(tt,ts) \\in A_{13}, Z(tt) < Z(t)), 0 \\le ts \\le k \\rbrace \\\\ = \\lbrace notprenabled(t,ts) : t \\in T, (\\forall p \\in \\bullet t, W(p,t) \\le M_{ts}(p)), (\\forall p \\in I(t), M_{ts}(p) = 0), (\\forall (p,t) \\in Q, M_0(p) \\ge WQ(p,t)), \\exists tt \\in T, $ $(\\forall pp \\in \\bullet tt, W(pp,tt) \\le M_{ts}(pp)), $ $(\\forall pp \\in I(tt), M_{ts}(pp) = 0), (\\forall (pp,tt) \\in Q, M_{ts}(pp) \\ge QW(pp,tt)), Z(tt) < Z(t), 0 \\le ts \\le k \\rbrace $ $A_{24}=\\lbrace prenabled(t,ts) : t \\in T, enabled(t,ts) \\in A_{13}, (\\nexists tt \\in T: enabled(tt,ts) \\in A_{13}, Z(tt) < Z(t)), 0 \\le ts \\le k \\rbrace \\\\ = \\lbrace prenabled(t,ts) : t \\in T, (\\forall p \\in \\bullet t, W(p,t) \\le M_{ts}(p)), (\\forall p \\in I(t), M_{ts}(p) = 0), (\\forall (p,t) \\in Q, M_0(p) \\ge WQ(p,t)), \\nexists tt \\in T, $ $(\\forall pp \\in \\bullet tt, W(pp,tt) \\le M_{ts}(pp)), $ $(\\forall pp \\in I(tt), M_{ts}(pp) = 0), (\\forall (pp,tt) \\in Q, M_{ts}(pp) \\ge QW(pp,tt)), Z(tt) < Z(t), 0 \\le ts \\le k \\rbrace $ $A_{25}=\\lbrace could\\_not\\_have(t,ts): t \\in T, prenabled(t,ts) \\in A_{24}, fires(t,ts) \\notin A_{14}, (\\exists p \\in \\bullet t \\cup R(t): q > M_{ts}(p) - (\\sum _{t^{\\prime } \\in T_{ts}, p \\in \\bullet t^{\\prime }}{W(p,t^{\\prime })} + \\sum _{t^{\\prime } \\in T_{ts}, p \\in R(t^{\\prime })}{M_{ts}(p)}), q=W(p,t) \\text{~if~} (p,t) \\in E^- \\text{~or~} R(t) \\text{~otherwise}), 0 \\le ts \\le k\\rbrace \\\\ =\\lbrace could\\_not\\_have(t,ts) : t \\in T, (\\forall p \\in \\bullet t, W(p,t) \\le M_{ts}(p)), (\\forall p \\in I(t), M_{ts}(p) = 0), (\\forall (p,t) \\in Q, M_0(p) \\ge QW(p,t)), (\\nexists tt \\in T, $ $(\\forall pp \\in \\bullet tt, W(pp,tt) \\le M_{ts}(pp)), $ $(\\forall pp \\in I(tt), M_{ts}(pp) = 0), (\\forall (pp,tt) \\in Q, M_{ts}(pp) \\ge QW(pp,tt)), Z(tt) < Z(t)), t \\notin T_{ts}, (\\exists p \\in \\bullet t \\cup R(t): q > M_{ts}(p) - (\\sum _{t^{\\prime } \\in T_{ts}, p \\in \\bullet t^{\\prime }}{W(p,t^{\\prime })} + \\\\ \\sum _{t^{\\prime } \\in T_{ts}, p \\in R(t^{\\prime })}{M_{ts}(p)}), q=W(p,t) \\text{~if~} (p,t) \\in E^- \\text{~or~} R(t) \\text{~otherwise}), 0 \\le ts \\le k \\rbrace $ per the maximal firing set semantics We show that $A$ satisfies (REF ) and (REF ), and $A$ is an answer set of $\\Pi ^7$ .", "$A$ satisfies (REF ) and (REF ) by its construction above.", "We show $A$ is an answer set of $\\Pi ^7$ by splitting.", "We split $lit(\\Pi ^7)$ into a sequence of $9k+11$ sets: $U_0= head(f\\ref {f:c:place}) \\cup head(f\\ref {f:c:trans}) \\cup head(f\\ref {f:c:col}) \\cup head(f\\ref {f:c:time}) \\cup head(f\\ref {f:c:num}) \\cup head(i\\ref {i:c:init}) \\cup head(f\\ref {f:c:pr}) = \\lbrace place(p) : p \\in P\\rbrace \\cup \\lbrace trans(t) : t \\in T\\rbrace \\cup \\lbrace col(c) : c \\in C \\rbrace \\cup \\lbrace time(0), \\dots , time(k)\\rbrace \\cup \\lbrace num(0), \\dots , num(ntok)\\rbrace \\cup \\lbrace holds(p,q_c,c,0) : p \\in P, c \\in C, q_c=m_{M_0(p)}(c) \\rbrace \\cup \\lbrace transpr(t,pr) : t \\in T, pr=Z(t) \\rbrace $ $U_{9k+1}=U_{9k+0} \\cup head(f\\ref {f:c:ptarc})^{ts=k} \\cup head(f\\ref {f:c:tparc})^{ts=k} \\cup head(f\\ref {f:c:rptarc})^{ts=k} \\cup head(f\\ref {f:c:iptarc})^{ts=k} \\cup head(f\\ref {f:c:tptarc})^{ts=k} = U_{9k+0} \\cup $ $\\lbrace ptarc(p,t,n_c,c,k) : (p,t) \\in E^-, c \\in C, n_c=m_{W(p,t)}(c) \\rbrace \\\\ \\cup \\lbrace tparc(t,p,n_c,c,k,d) : (t,p) \\in E^+, c \\in C, n_c=m_{W(t,p)}(c), d=D(t) \\rbrace \\cup $ $\\lbrace ptarc(p,t,n_c,c,k) : p \\in R(t), c \\in C, n_c=m_{M_{k}(p)}(c), n > 0 \\rbrace \\cup \\lbrace iptarc(p,t,1,c,k) : \\\\ p \\in I(t), c \\in C \\rbrace \\cup $ $\\lbrace tptarc(p,t,n_c,c,k) : (p,t) \\in Q, c \\in C, n_c=m_{QW(p,t)}(c) \\rbrace $ $U_{9k+2}=U_{9k+1} \\cup head(e\\ref {e:c:ne:ptarc})^{ts=k} \\cup head(e\\ref {e:c:ne:iptarc})^{ts=k} \\cup head(e\\ref {e:c:ne:tptarc})^{ts=k} = U_{10k+1} \\cup \\\\ \\lbrace notenabled(t,k) : t \\in T \\rbrace $ $U_{9k+3}=U_{9k+2} \\cup head(e\\ref {e:c:enabled})^{ts=k} = U_{9k+2} \\cup \\lbrace enabled(t,k) : t \\in T \\rbrace $ $U_{9k+4}=U_{9k+3} \\cup head(a\\ref {a:c:prne})^{ts=k} = U_{9k+3} \\cup \\lbrace notprenabled(t,k) : t \\in T \\rbrace $ $U_{9k+5}=U_{9k+4} \\cup head(a\\ref {a:c:prenabled})^{ts=k} = U_{9k+4} \\cup \\lbrace prenabled(t,k) : t \\in T \\rbrace $ $U_{9k+6}=U_{9k+5} \\cup head(a\\ref {a:c:prfires})^{ts=k} = U_{9k+5} \\cup \\lbrace fires(t,k) : t \\in T \\rbrace $ $U_{9k+7}=U_{9k+6} \\cup head(r\\ref {r:c:dur:add})^{ts=k} \\cup head(r\\ref {r:c:del})^{ts=k} = U_{9k+6} \\cup \\lbrace add(p,q_c,t,c,k) : p \\in P, t \\in T, c \\in C, q_c=m_{W(t,p)}(c) \\rbrace \\cup \\lbrace del(p,q_c,t,c,k) : p \\in P, t \\in T, c \\in C, q_c=m_{W(p,t)}(c) \\rbrace \\cup \\lbrace del(p,q_c,t,c,k) : p \\in P, t \\in T, c \\in C, q_c=m_{M_{k}(p)}(c) \\rbrace $ $U_{9k+8}=U_{9k+7} \\cup head(r\\ref {r:c:totincr})^{ts=k} \\cup head(r\\ref {r:c:totdecr})^{ts=k} = U_{9k+7} \\cup \\lbrace tot\\_incr(p,q_c,c,k) : p \\in P, c \\in C, 0 \\le q_c \\le ntok \\rbrace \\cup \\lbrace tot\\_decr(p,q_c,c,k) : p \\in P, c \\in C, 0 \\le q_c \\le ntok \\rbrace $ $U_{9k+9}=U_{9k+8} \\cup head(a\\ref {a:c:overc:place})^{ts=k} \\cup head(r\\ref {r:c:nextstate})^{ts=k} \\cup head(a\\ref {a:c:prmaxfire:cnh})^{ts=k} = U_{9k+8} \\cup \\\\ \\lbrace consumesmore(p,k) : p \\in P\\rbrace \\cup \\lbrace holds(p,q,k+1) : p \\in P, 0 \\le q \\le ntok \\rbrace \\cup \\lbrace could\\_not\\_have(t,k) : t \\in T \\rbrace $ $U_{9k+10}=U_{9k+9} \\cup head(a\\ref {a:c:overc:gen}) = U_{9k+9} \\cup \\lbrace consumesmore \\rbrace $ where $head(r_i)^{ts=k}$ are head atoms of ground rule $r_i$ in which $ts=k$ .", "We write $A_i^{ts=k} = \\lbrace a(\\dots ,ts) : a(\\dots ,ts) \\in A_i, ts=k \\rbrace $ as short hand for all atoms in $A_i$ with $ts=k$ .", "$U_{\\alpha }, 0 \\le \\alpha \\le 9k+10$ form a splitting sequence, since each $U_i$ is a splitting set of $\\Pi ^7$ , and $\\langle U_{\\alpha }\\rangle _{\\alpha < \\mu }$ is a monotone continuous sequence, where $U_0 \\subseteq U_1 \\dots \\subseteq U_{9k+10}$ and $\\bigcup _{\\alpha < \\mu }{U_{\\alpha }} = lit(\\Pi ^7)$ .", "We compute the answer set of $\\Pi ^7$ using the splitting sets as follows: $bot_{U_0}(\\Pi ^7) = f\\ref {f:c:place} \\cup f\\ref {f:c:trans} \\cup f\\ref {f:c:col} \\cup f\\ref {f:c:time} \\cup f\\ref {f:c:num} \\cup i\\ref {i:c:init} \\cup f\\ref {f:c:pr}$ and $X_0 = A_1 \\cup \\dots \\cup A_5 \\cup A_8$ ($= U_0$ ) is its answer set – using forced atom proposition $eval_{U_0}(bot_{U_1}(\\Pi ^7) \\setminus bot_{U_0}(\\Pi ^7), X_0) = \\lbrace ptarc(p,t,q_c,c,0) \\text{:-}.", "\| c \\in C, q_c=m_{W(p,t)}(c), q_c > 0 \\rbrace \\cup $ $\\lbrace tparc(t,p,q_c,c,0,d) \\text{:-}.", "\| c \\in C, q_c=m_{W(t,p)}(c), q_c > 0, d=D(t) \\rbrace \\cup \\\\ \\lbrace ptarc(p,t,q_c,c,0) \\text{:-}.", "\| c \\in C, q_c=m_{M_0(p)}(c), q_c > 0 \\rbrace \\cup $ $\\lbrace iptarc(p,t,1,c,0) \\text{:-}.", "\| c \\in C \\rbrace \\cup $ $\\lbrace tptarc(p,t,q_c,c,0) \\text{:-}.", "\| c \\in C, q_c = m_{QW(p,t)}(c), q_c > 0 \\rbrace $ .", "Its answer set $X_1=A_6^{ts=0} \\cup A_7^{ts=0} \\cup A_9^{ts=0} \\cup A_{10}^{ts=0} \\cup A_{11}^{ts=0}$ – using forced atom proposition and construction of $A_6, A_7, A_9, A_{10}, A_{11}$ .", "$eval_{U_1}(bot_{U_2}(\\Pi ^7) \\setminus bot_{U_1}(\\Pi ^7), X_0 \\cup X_1) = \\lbrace notenabled(t,0) \\text{:-} .", "\| (\\lbrace trans(t), \\\\ ptarc(p,t,n_c,c,0), $ $holds(p,q_c,c,0) \\rbrace \\subseteq X_0 \\cup X_1, \\text{~where~} q_c < n_c) \\text{~or~} \\\\ (\\lbrace notenabled(t,0) \\text{:-} .", "\| $ $(\\lbrace trans(t), $ $iptarc(p,t,n2_c,c,0), $ $ holds(p,q_c,c,0) \\rbrace \\subseteq X_0 \\cup X_1, \\text{~where~} q_c \\ge n2_c \\rbrace ) \\text{~or~} $ $(\\lbrace trans(t), $ $tptarc(p,t,n3_c,c,0), $ $holds(p,q_c,c,0) \\rbrace \\subseteq X_0 \\cup X_1, \\text{~where~} q_c < n3_c) \\rbrace $ .", "Its answer set $X_2=A_{12}^{ts=0}$ – using forced atom proposition and construction of $A_{12}$ .", "where, $q_c=m_{M_0(p)}(c)$ , and $n_c=m_{W(p,t)}(c)$ for an arc $(p,t) \\in E^-$ – by construction of $i\\ref {i:c:init}$ and $f\\ref {f:c:ptarc}$ in $\\Pi ^7$ , and in an arc $(p,t) \\in E^-$ , $p \\in \\bullet t$ (by definition REF of preset) $n2_c=1$ – by construction of $iptarc$ predicates in $\\Pi ^7$ , meaning $q_c \\ge n2_c \\equiv q_c \\ge 1 \\equiv q_c > 0$ , $tptarc(p,t,n3_c,c,0)$ represents $n3_c=m_{QW(p,t)}(c)$ , where $(p,t) \\in Q$ thus, $notenabled(t,0) \\in X_1$ means $\\exists c \\in C, (\\exists p \\in \\bullet t : m_{M_0(p)}(c) < m_{W(p,t)}(c)) \\\\ \\vee (\\exists p \\in I(t) : m_{M_0(p)}(c) > 0) \\vee (\\exists (p,t) \\in Q : m_{M_0(p)}(c) < m_{QW(p,t)}(c))$ .", "$eval_{U_2}(bot_{U_3}(\\Pi ^7) \\setminus bot_{U_2}(\\Pi ^7), X_0 \\cup \\dots \\cup X_2) = \\lbrace enabled(t,0) \\text{:-}.", "\| trans(t) \\in X_0 \\cup \\dots \\cup X_2, notenabled(t,0) \\notin X_0 \\cup \\dots \\cup X_2 \\rbrace $ .", "Its answer set is $X_3 = A_{13}^{ts=0}$ – using forced atom proposition and construction of $A_{13}$ .", "since an $enabled(t,0) \\in X_3$ if $\\nexists ~notenabled(t,0) \\in X_0 \\cup \\dots \\cup X_2$ ; which is equivalent to $\\nexists t, \\forall c \\in C, (\\nexists p \\in \\bullet t, m_{M_0(p)}(c) < m_{W(p,t)}(c)), (\\nexists p \\in I(t), \\\\ m_{M_0(p)}(c) > 0), (\\nexists (p,t) \\in Q : m_{M_0(p)}(c) < m_{QW(p,t)}(c) ), \\forall c \\in C, (\\forall p \\in \\bullet t: m_{M_0(p)}(c) \\ge m_{W(p,t)}(c)), (\\forall p \\in I(t) : m_{M_0(p)}(c) = 0)$ .", "$eval_{U_3}(bot_{U_4}(\\Pi ^7) \\setminus bot_{U_3}(\\Pi ^7), X_0 \\cup \\dots \\cup X_3) = \\lbrace notprenabled(t,0) \\text{:-}.", "\| \\\\ \\lbrace enabled(t,0), transpr(t,p), enabled(tt,0), transpr(tt,pp) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_3, pp < p \\rbrace $ .", "Its answer set is $X_4 = A_{23}^{ts=k}$ – using forced atom proposition and construction of $A_{23}$ .", "$enabled(t,0)$ represents $\\exists t \\in T, \\forall c \\in C, (\\forall p \\in \\bullet t, m_{M_0(p)}(c) \\ge m_{W(p,t)}(c)), $ $(\\forall p \\in I(t), m_{M_0(p)}(c) = 0), (\\forall (p,t) \\in Q, m_{M_0(p)}(c) \\ge m_{QW(p,t)}(c))$ $enabled(tt,0)$ represents $\\exists tt \\in T, \\forall c \\in C, $ $(\\forall pp \\in \\bullet tt, m_{M_0(pp)}(c) \\ge \\\\ m_{W(pp,tt)}(c)) \\wedge $ $(\\forall pp \\in I(tt), m_{M_0(pp)}(c) = 0), (\\forall (pp,tt) \\in Q, m_{M_0(pp)}(c) \\ge m_{QW(pp,tt)}(c))$ $transpr(t,p)$ represents $p=Z(t)$ – by construction $transpr(tt,pp)$ represents $pp=Z(tt)$ – by construction thus, $notprenabled(t,0)$ represents $\\forall c \\in C, $ $(\\forall p \\in \\bullet t, m_{M_0(p)}(c) \\ge m_{W(p,t)}(c)), $ $(\\forall p \\in I(t), m_{M_0(p)}(c) = 0), (\\forall (p,t) \\in Q, m_{M_0(p)}(c) \\ge m_{QW(p,t)}(c)), \\exists tt \\in T, (\\forall pp \\in \\bullet tt, m_{M_0(pp)}(c) \\ge m_{W(pp,tt)}(c)), $ $(\\forall pp \\in I(tt), m_{M_0(pp)}(c) = 0), \\\\ (\\forall (pp,tt) \\in Q, m_{M_0(pp)}(c) \\ge m_{QW(pp,tt)}(c)), Z(tt) < Z(t)$ which is equivalent to $(\\forall p \\in \\bullet t: M_0(p) \\ge W(p,t)), $ $(\\forall p \\in I(t), M_0(p) = 0), (\\forall (p,t) \\in Q, M_0(p) \\ge QW(p,t)), \\exists tt \\in T, $ $(\\forall pp \\in \\bullet tt, M_0(pp) \\ge \\\\ W(pp,tt)), $ $(\\forall pp \\in I(tt), M_0(pp) = 0), (\\forall (pp,tt) \\in Q, M_0 \\ge QW(pp,tt)), \\\\ Z(tt) < Z(t)$ – assuming multiset domain $C$ for all operations $eval_{U_4}(bot_{U_5}(\\Pi ^7) \\setminus bot_{U_4}(\\Pi ^7), X_0 \\cup \\dots \\cup X_4) = \\lbrace prenabled(t,0) \\text{:-}.", "\| enabled(t,0) \\in X_0 \\cup \\dots \\cup X_4, notprenabled(t,0) \\notin X_0 \\cup \\dots \\cup X_4 \\rbrace $ .", "Its answer set is $X_5 = A_{24}^{ts=k}$ – using forced atom proposition and construction of $A_{24}$ $enabled(t,0)$ represents $\\forall c \\in C, (\\forall p \\in \\bullet t, m_{M_0(p)}(c) \\ge m_{W(p,t)}(c)), (\\forall p \\in I(t), m_{M_0(p)}(c) = 0), (\\forall (p,t) \\in Q, m_{M_0(p)}(c) \\ge m_{QW(p,t)}(c)) \\equiv $ $(\\forall p \\in \\bullet t, \\\\ M_0(p) \\ge W(p,t)), $ $(\\forall p \\in I(t), M_0(p) = 0), (\\forall (p,t) \\in Q, M_0(p) \\ge QW(p,t))$ – from REF above and assuming multiset domain $C$ for all operations $notprenabled(t,0)$ represents $(\\forall p \\in \\bullet t, M_0(p) \\ge W(p,t)), $ $(\\forall p \\in I(t), \\\\ M_0(p) = 0), (\\forall (p,t) \\in Q, M_0(p) \\ge QW(p,t)), \\exists tt \\in T, $ $(\\forall pp \\in \\bullet tt, M_0(pp) \\ge W(pp,tt)), $ $(\\forall pp \\in I(tt), M_0(pp) = 0), (\\forall (pp,tt) \\in Q, M_0(pp) \\ge QW(pp,tt)), \\\\ Z(tt) < Z(t)$ – from REF above and assuming multiset domain $C$ for all operations then, $prenabled(t,0)$ represents $(\\forall p \\in \\bullet t, M_0(p) \\ge W(p,t)), $ $(\\forall p \\in I(t), \\\\ M_0(p) = 0), (\\forall (p,t) \\in Q, M_0(p) \\ge QW(p,t)), \\nexists tt \\in T, $ $((\\forall pp \\in \\bullet tt, \\\\ M_0(pp) \\ge W(pp,tt)), $ $(\\forall pp \\in I(tt), M_0(pp) = 0), (\\forall (pp,tt) \\in Q, M_0(pp) \\ge QW(pp,tt)), Z(tt) < Z(t))$ – from (a), (b) and $enabled(t,0) \\in X_0 \\cup \\dots \\cup X_4$ $eval_{U_5}(bot_{U_6}(\\Pi ^7) \\setminus bot_{U_5}(\\Pi ^7), X_0 \\cup \\dots \\cup X_5) = \\lbrace \\lbrace fires(t,0)\\rbrace \\text{:-}.", "\| prenabled(t,0) \\\\ \\text{~holds in~} X_0 \\cup \\dots \\cup X_5 \\rbrace $ .", "It has multiple answer sets $X_{6.1}, \\dots , X_{6.n}$ , corresponding to elements of power set of $fires(t,0)$ atoms in $eval_{U_5}(...)$ – using supported rule proposition.", "Since we are showing that the union of answer sets of $\\Pi ^7$ determined using splitting is equal to $A$ , we only consider the set that matches the $fires(t,0)$ elements in $A$ and call it $X_6$ , ignoring the rest.", "Thus, $X_6 = A_{14}^{ts=0}$ , representing $T_0$ .", "in addition, for every $t$ such that $prenabled(t,0) \\in X_0 \\cup \\dots \\cup X_5, R(t) \\ne \\emptyset $ ; $fires(t,0) \\in X_6$ – per definition REF (firing set); requiring that a reset transition is fired when enabled thus, the firing set $T_0$ will not be eliminated by the constraint $f\\ref {f:c:pr:rptarc:elim}$ $eval_{U_6}(bot_{U_7}(\\Pi ^7) \\setminus bot_{U_6}(\\Pi ^7), X_0 \\cup \\dots \\cup X_6) = \\lbrace add(p,n_c,t,c,0) \\text{:-}.", "\| $ $\\lbrace fires(t,0-d+1), $ $tparc(t,p,n_c,c,0,d) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_6 \\rbrace \\cup \\lbrace del(p,n_c,t,c,0) \\text{:-}.", "\| $ $\\lbrace fires(t,0), $ $ptarc(p,t,n_c,c,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_6 \\rbrace $ .", "It's answer set is $X_7 = A_{15}^{ts=0} \\cup A_{16}^{ts=0}$ – using forced atom proposition and definitions of $A_{15}$ and $A_{16}$ .", "where, each $add$ atom is equivalent to $n_c=m_{W(t,p)}(c),c \\in C, p \\in t \\bullet $ , and each $del$ atom is equivalent to $n_c=m_{W(p,t)}(c), c \\in C, p \\in \\bullet t$ ; or $n_c=m_{M_0(p)}(c), c \\in C, p \\in R(t)$ , representing the effect of transitions in $T_0$ $eval_{U_7}(bot_{U_8}(\\Pi ^7) \\setminus bot_{U_7}(\\Pi ^7), X_0 \\cup \\dots \\cup X_7) = \\\\ \\lbrace tot\\_incr(p,qq_c,c,0) \\text{:-}.", "\| qq_c=\\sum _{add(p,q_c,t,c,0) \\in X_0 \\cup \\dots \\cup X_7}{q_c} \\rbrace \\cup \\\\ \\lbrace tot\\_decr(p,qq_c,c,0) \\text{:-}.", "\| qq_c=\\sum _{del(p,q_c,t,c,0) \\in X_0 \\cup \\dots \\cup X_7}{q_c} \\rbrace $ .", "It's answer set is $X_8 = A_{17}^{ts=0} \\cup A_{18}^{ts=0}$ – using forced atom proposition and definitions of $A_{17}$ and $A_{18}$ .", "where, each $tot\\_incr(p,qq_c,c,0)$ , $qq_c=\\sum _{add(p,q_c,t,c,0) \\in X_0 \\cup \\dots X_7}{q_c}$ $\\equiv qq_c=\\sum _{t \\in X_6, p \\in t \\bullet , 0+D(t)-1=0}{m_{W(p,t)}(c)}$ , $tot\\_decr(p,qq_c,c,0)$ , $qq_c=\\sum _{del(p,q_c,t,c,0) \\in X_0 \\cup \\dots X_7}{q_c}$ $\\equiv qq=\\sum _{t \\in X_6, p \\in \\bullet t}{m_{W(t,p)}(c)} + \\sum _{t \\in X_6, p \\in R(t)}{m_{M_0(p)}(c)}$ , represent the net effect of transitions in $T_0$ $eval_{U_8}(bot_{U_9}(\\Pi ^7) \\setminus bot_{U_8}(\\Pi ^7), X_0 \\cup \\dots \\cup X_8) = \\lbrace consumesmore(p,0) \\text{:-}.", "\| \\\\ \\lbrace holds(p,q_c,c,0), tot\\_decr(p,q1_c,c,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_8, q1_c > q_c \\rbrace \\cup $ $\\lbrace holds(p,q_c,c,1) \\text{:-}.", "\| \\lbrace holds(p,q1_c,c,0), tot\\_incr(p,q2_c,c,0), \\\\ tot\\_decr(p,q3_c,c,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_6, q_c=q1_c+q2_c-q3_c \\rbrace \\cup \\lbrace could\\_not\\_have(t,0) \\text{:-}.", "\| \\\\ \\lbrace prenabled(t,0), ptarc(s,t,q,c,0), holds(s,qq,c,0), tot\\_decr(s,qqq,c,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_8, fires(t,0) \\notin X_0 \\cup \\dots \\cup X_8, q > qq - qqq \\rbrace $ .", "It's answer set is $X_9 = A_{19}^{ts=0} \\cup A_{21}^{ts=0} \\cup A_{25}^{ts=0}$ – using forced atom proposition and definitions of $A_{19}, A_{21}, A_{25}$ .", "where $consumesmore(p,0)$ represents $\\exists p : q_c=m_{M_0(p)}(c), q1_c= \\\\ \\sum _{t \\in T_0, p \\in \\bullet t}{m_{W(p,t)}(c)}+\\sum _{t \\in T_0, p \\in R(t)}{m_{M_0(p)}(c)}, q1_c > q_c, c \\in C$ , indicating place $p$ will be over consumed if $T_0$ is fired, as defined in definition REF (conflicting transitions), $holds(p,q_c,c,1)$ if $q_c=m_{M_0(p)}(c)+\\sum _{t \\in T_0, p \\in t \\bullet , 0+D(t)-1=0}{m_{W(t,p)}(c)}- \\\\ (\\sum _{t \\in T_0, p \\in \\bullet t}{m_{W(p,t)}(c)}+ $ $\\sum _{t \\in T_0, p \\in R(t)}{m_{M_0(p)}(c)})$ represented by $q_c=m_{M_1(p)}(c)$ for some $c \\in C$ – by construction of $\\Pi ^7$ and $consumesmore(p,0)$ if $\\sum _{t \\in T_0, p \\in \\bullet t}{m_{W(p,t)}(c)}+ $ $\\sum _{t \\in T_0, p \\in R(t)}{m_{M_0(p)}(c)} > m_{M_0(p)}(c)$ for any $c \\in C$ $could\\_not\\_have(t,0)$ if $(\\forall p \\in \\bullet t, W(p,t) \\le M_0(p)), (\\forall p \\in I(t), M_0(p) = 0), (\\forall (p,t) \\in Q, \\\\ M_0(p) \\ge WQ(p,t)), \\nexists tt \\in T, (\\forall pp \\in \\bullet tt, W(pp,tt) \\le M_{ts}(pp)), (\\forall pp \\in I(tt), M_0(pp) = 0), (\\forall (pp,tt) \\in Q, M_0(pp) \\ge QW(pp,tt)), Z(tt) < Z(t)$ , and $q_c > m_{M_0(s)}(c) - (\\sum _{t^{\\prime } \\in T_0, s \\in \\bullet t^{\\prime }}{m_{W(s,t^{\\prime })}(c)}+ \\sum _{t^{\\prime } \\in T_0, s \\in R(t)}{m_{M_0(s)}(c)}), \\\\ q_c = m_{W(s,t)}(c) \\text{~if~} s \\in \\bullet t \\text{~or~} m_{M_0(s)}(c) \\text{~otherwise}$ for some $c \\in C$ , which becomes $q > M_0(s) - (\\sum _{t^{\\prime } \\in T_0, s \\in \\bullet t^{\\prime }}{W(s,t^{\\prime })}+ \\sum _{t^{\\prime } \\in T_0, s \\in R(t)}{M_0(s)}), q = W(s,t) \\text{~if~} s \\in \\bullet t \\text{~or~} M_0(s) \\text{~otherwise}$ for all $c \\in C$ (i), (ii) above combined match the definition of $A_{25}$ $X_9$ does not contain $could\\_not\\_have(t,0)$ , when $prenabled(t,0) \\in X_0 \\cup \\dots \\cup X_6$ and $fires(t,0) \\notin X_0 \\cup \\dots \\cup X_5$ due to construction of $A$ , encoding of $a\\ref {a:c:maxfire:cnh}$ and its body atoms.", "As a result, it is not eliminated by the constraint $a\\ref {a:c:prmaxfire:elim}$ $ \\vdots $ $eval_{U_{9k+0}}(bot_{U_{9k+1}}(\\Pi ^7) \\setminus bot_{U_{9k+0}}(\\Pi ^7), X_0 \\cup \\dots \\cup X_{9k+0}) = \\lbrace ptarc(p,t,q_c,c,k) \\text{:-}.", "\| c \\in C, q_c=m_{W(p,t)}(c), q_c > 0 \\rbrace \\cup \\lbrace tparc(t,p,q_c,c,k,d) \\text{:-}.", "\| c \\in C, q_c=m_{W(t,p)}(c), q_c > 0, d=D(t) \\rbrace \\cup $ $\\lbrace ptarc(p,t,q_c,c,k) \\text{:-}.", "\| c \\in C, q_c=m_{M_k(p)}(c), q_c > 0 \\rbrace \\cup \\\\ \\lbrace iptarc(p,t,1,c,k) \\text{:-}.", "\| c \\in C \\rbrace \\cup $ $\\lbrace tptarc(p,t,q_c,c,k) \\text{:-}.", "\| c \\in C, q_c = m_{QW(p,t)}(c), q_c > 0 \\rbrace $ .", "Its answer set $X_{9k+1}=A_6^{ts=k} \\cup A_7^{ts=k} \\cup A_9^{ts=k} \\cup A_{10}^{ts=k} \\cup A_{11}^{ts=k}$ – using forced atom proposition and construction of $A_6, A_7, A_9, A_{10}, A_{11}$ .", "$eval_{U_{9k+1}}(bot_{U_{10k+2}}(\\Pi ^7) \\setminus bot_{U_{9k+1}}(\\Pi ^7), X_0 \\cup \\dots \\cup X_{9k+1}) = \\lbrace notenabled(t,k) \\text{:-} .", "\| \\\\ (\\lbrace trans(t), ptarc(p,t,n_c,c,k), holds(p,q_c,c,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{9k+1}, \\text{~where~} q_c < n_c) \\text{~or~} (\\lbrace notenabled(t,k) \\text{:-} .", "\| (\\lbrace trans(t), iptarc(p,t,n2_c,c,k), holds(p,q_c,c,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{9k+1}, $ $\\text{~where~} q_c \\ge n2_c \\rbrace ) \\text{~or~} $ $(\\lbrace trans(t), tptarc(p,t,n3_c,c,k), \\\\ holds(p,q_c,c,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{9k+1}, $ $\\text{~where~} q_c < n3_c) \\rbrace $ .", "Its answer set $X_{9k+2}=A_{12}^{ts=k}$ – using forced atom proposition and construction of $A_{12}$ .", "where, $q_c=m_{M_k(p)}(c)$ , and $n_c=m_{W(p,t)}(c)$ for an arc $(p,t) \\in E^-$ – by construction of $i\\ref {i:c:init}$ and $f\\ref {f:c:ptarc}$ predicates in $\\Pi ^7$ , and in an arc $(p,t) \\in E^-$ , $p \\in \\bullet t$ (by definition REF of preset) $n2_c=1$ – by construction of $iptarc$ predicates in $\\Pi ^7$ , meaning $q_c \\ge n2_c \\equiv q_c \\ge 1 \\equiv q_c > 0$ , $tptarc(p,t,n3_c,c,k)$ represents $n3_c=m_{QW(p,t)}(c)$ , where $(p,t) \\in Q$ thus, $notenabled(t,k) \\in X_{9k+1}$ represents $\\exists c \\in C, (\\exists p \\in \\bullet t : m_{M_k(p)}(c) < m_{W(p,t)}(c)) \\vee (\\exists p \\in I(t) : m_{M_k(p)}(c) > 0) \\vee (\\exists (p,t) \\in Q : m_{M_k(p)}(c) < m_{QW(p,t)}(c))$ .", "$eval_{U_{9k+2}}(bot_{U_{9k+3}}(\\Pi ^7) \\setminus bot_{U_{9k+2}}(\\Pi ^7), X_0 \\cup \\dots \\cup X_{9k+2}) = \\lbrace enabled(t,k) \\text{:-}.", "\| trans(t) \\in X_0 \\cup \\dots \\cup X_{9k+2}, notenabled(t,k) \\notin X_0 \\cup \\dots \\cup X_{9k+2} \\rbrace $ .", "Its answer set is $X_{9k+3} = A_{13}^{ts=k}$ – using forced atom proposition and construction of $A_{13}$ .", "Since an $enabled(t,k) \\in X_{9k+3}$ if $\\nexists ~notenabled(t,k) \\in X_0 \\cup \\dots \\cup X_{9k+2}$ ; which is equivalent to $\\nexists t, \\forall c \\in C, (\\nexists p \\in \\bullet t, m_{M_k(p)}(c) < m_{W(p,t)}(c)), (\\nexists p \\in I(t), m_{M_k(p)}(c) > 0), (\\nexists (p,t) \\in Q : m_{M_k(p)}(c) < m_{QW(p,t)}(c) ), \\forall c \\in C, (\\forall p \\in \\bullet t: m_{M_k(p)}(c) \\ge m_{W(p,t)}(c)), (\\forall p \\in I(t) : m_{M_k(p)}(c) = 0)$ .", "$eval_{U_{9k+3}}(bot_{U_{9k+4}}(\\Pi ^7) \\setminus bot_{U_{9k+3}}(\\Pi ^7), X_0 \\cup \\dots \\cup X_{9k+3}) = \\lbrace notprenabled(t,k) \\text{:-}.", "\| \\\\ \\lbrace enabled(t,k), transpr(t,p), enabled(tt,k), transpr(tt,pp) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{9k+3}, \\\\ pp < p \\rbrace $ .", "Its answer set is $X_{9k+4} = A_{23}^{ts=k}$ – using forced atom proposition and construction of $A_{23}$ .", "$enabled(t,k)$ represents $\\exists t \\in T, \\forall c \\in C, (\\forall p \\in \\bullet t, m_{M_k(p)}(c) \\ge m_{W(p,t)}(c)), $ $(\\forall p \\in I(t), m_{M_k(p)}(c) = 0), (\\forall (p,t) \\in Q, m_{M_k(p)}(c) \\ge m_{QW(p,t)}(c))$ $enabled(tt,k)$ represents $\\exists tt \\in T, \\forall c \\in C, (\\forall pp \\in \\bullet tt, m_{M_k(pp)}(c) \\ge \\\\ m_{W(pp,tt)}(c)) \\wedge (\\forall pp \\in I(tt), m_{M_k(pp)}(c) = 0), (\\forall (pp,tt) \\in Q, m_{M_k(pp)}(c) \\ge m_{QW(pp,tt)}(c))$ $transpr(t,p)$ represents $p=Z(t)$ – by construction $transpr(tt,pp)$ represents $pp=Z(tt)$ – by construction thus, $notprenabled(t,k)$ represents $\\forall c \\in C, (\\forall p \\in \\bullet t, m_{M_k(p)}(c) \\ge \\\\ m_{W(p,t)}(c)), $ $(\\forall p \\in I(t), m_{M_k(p)}(c) = 0), (\\forall (p,t) \\in Q, m_{M_k(p)}(c) \\ge \\\\ m_{QW(p,t)}(c)), \\exists tt \\in T, $ $(\\forall pp \\in \\bullet tt, m_{M_k(pp)}(c) \\ge m_{W(pp,tt)}(c)), $ $(\\forall pp \\in I(tt), \\\\ m_{M_k(pp)}(c) = 0), (\\forall (pp,tt) \\in Q, m_{M_k(pp)}(c) \\ge m_{QW(pp,tt)}(c)), Z(tt) < Z(t)$ which is equivalent to $(\\forall p \\in \\bullet t: M_k(p) \\ge W(p,t)), $ $(\\forall p \\in I(t), \\\\ M_k(p) = 0), (\\forall (p,t) \\in Q, M_k(p) \\ge QW(p,t)), \\exists tt \\in T, $ $(\\forall pp \\in \\bullet tt, \\\\ M_k(pp) \\ge W(pp,tt)), $ $(\\forall pp \\in I(tt), M_k(pp) = 0), (\\forall (pp,tt) \\in Q, M_k(pp) \\ge QW(pp,tt)), Z(tt) < Z(t)$ – assuming multiset domain $C$ for all operations $eval_{U_{9k+4}}(bot_{U_{9k+5}}(\\Pi ^7) \\setminus bot_{U_{9k+4}}(\\Pi ^7), X_0 \\cup \\dots \\cup X_{9k+4}) = \\lbrace prenabled(t,k) \\text{:-}.", "\| \\\\ enabled(t,k) \\in X_0 \\cup \\dots \\cup X_{9k+4}, notprenabled(t,k) \\notin X_0 \\cup \\dots \\cup X_{9k+4} \\rbrace $ .", "Its answer set is $X_{9k+5} = A_{24}^{ts=k}$ – using forced atom proposition and construction of $A_{24}$ $enabled(t,k)$ represents $\\forall c \\in C, (\\forall p \\in \\bullet t, m_{M_k(p)}(c) \\ge m_{W(p,t)}(c)), $ $(\\forall p \\in I(t), m_{M_k(p)}(c) = 0), (\\forall (p,t) \\in Q, m_{M_k(p)}(c) \\ge m_{QW(p,t)}(c)) \\equiv (\\forall p \\in \\bullet t, M_k(p) \\ge W(p,t)), $ $(\\forall p \\in I(t), M_k(p) = 0), (\\forall (p,t) \\in Q, M_k(p) \\ge QW(p,t))$ – from REF above and assuming multiset domain $C$ for all operations $notprenabled(t,k)$ represents $(\\forall p \\in \\bullet t, M_k(p) \\ge W(p,t)), (\\forall p \\in I(t), \\\\ M_k(p) = 0), (\\forall (p,t) \\in Q, M_k(p) \\ge QW(p,t)), \\exists tt \\in T, $ $(\\forall pp \\in \\bullet tt, M_k(pp) \\ge W(pp,tt)), (\\forall pp \\in I(tt), M_k(pp) = 0), (\\forall (pp,tt) \\in Q, M_k(pp) \\ge QW(pp,tt)), \\\\ Z(tt) < Z(t)$ – from REF above and assuming multiset domain $C$ for all operations then, $prenabled(t,k)$ represents $(\\forall p \\in \\bullet t, M_k(p) \\ge W(p,t)), (\\forall p \\in I(t), \\\\ M_k(p) = 0), (\\forall (p,t) \\in Q, M_k(p) \\ge QW(p,t)), \\nexists tt \\in T, $ $((\\forall pp \\in \\bullet tt, M_k(pp) \\ge W(pp,tt)), $ $(\\forall pp \\in I(tt), M_k(pp) = 0), (\\forall (pp,tt) \\in Q, \\\\ M_k(pp) \\ge QW(pp,tt)), Z(tt) < Z(t))$ – from (a), (b) and $enabled(t,k) \\in X_0 \\cup \\dots \\cup X_{9k+4}$ $eval_{U_{9k+5}}(bot_{U_{9k+6}}(\\Pi ^7) \\setminus bot_{U_{9k+5}}(\\Pi ^7), X_0 \\cup \\dots \\cup X_{9k+5}) = \\lbrace \\lbrace fires(t,k)\\rbrace \\text{:-}.", "\| \\\\ prenabled(t,k) \\text{~holds in~} X_0 \\cup \\dots \\cup X_{9k+5} \\rbrace $ .", "It has multiple answer sets $X_{9k+6.1}, \\dots , X_{9k+6.n}$ , corresponding to elements of power set of $fires(t,k)$ atoms in $eval_{U_{9k+5}}(...)$ – using supported rule proposition.", "Since we are showing that the union of answer sets of $\\Pi ^7$ determined using splitting is equal to $A$ , we only consider the set that matches the $fires(t,k)$ elements in $A$ and call it $X_{9k+6}$ , ignoring the rest.", "Thus, $X_{9k+6} = A_{14}^{ts=k}$ , representing $T_k$ .", "in addition, for every $t$ such that $prenabled(t,k) \\in X_0 \\cup \\dots \\cup X_{9k+5}, R(t) \\ne \\emptyset $ ; $fires(t,k) \\in X_{9k+6}$ – per definition REF (firing set); requiring that a reset transition is fired when enabled thus, the firing set $T_k$ will not be eliminated by the constraint $f\\ref {f:c:pr:rptarc:elim}$ $eval_{U_{9k+6}}(bot_{U_{9k+7}}(\\Pi ^7) \\setminus bot_{U_{9k+6}}(\\Pi ^7), X_0 \\cup \\dots \\cup X_{9k+6}) = \\lbrace add(p,n_c,t,c,k) \\text{:-}.", "\| $ $\\lbrace fires(t,k-d+1), tparc(t,p,n_c,c,0,d) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{9k+6} \\rbrace \\cup \\\\ \\lbrace del(p,n_c,t,c,k) \\text{:-}.", "\| \\lbrace fires(t,k), ptarc(p,t,n_c,c,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{9k+6} \\rbrace $ .", "It's answer set is $X_{9k+7} = A_{15}^{ts=k} \\cup A_{16}^{ts=k}$ – using forced atom proposition and definitions of $A_{15}$ and $A_{16}$ .", "where, each $add$ atom is equivalent to $n_c=m_{W(t,p)}(c),c \\in C, p \\in t \\bullet $ , and each $del$ atom is equivalent to $n_c=m_{W(p,t)}(c), c \\in C, p \\in \\bullet t$ ; or $n_c=m_{M_k(p)}(c), c \\in C, p \\in R(t)$ , representing the effect of transitions in $T_k$ $eval_{U_{9k+7}}(bot_{U_{9k+8}}(\\Pi ^7) \\setminus bot_{U_{9k+7}}(\\Pi ^7), X_0 \\cup \\dots \\cup X_{9k+7}) = \\lbrace tot\\_incr(p,qq_c,c,k) \\text{:-}.", "\| $ $ qq_c=\\sum _{add(p,q_c,t,c,k) \\in X_0 \\cup \\dots \\cup X_{9k+7}}{q_c} \\rbrace \\cup \\lbrace tot\\_decr(p,qq_c,c,k) \\text{:-}.", "\| $ $ qq_c=\\sum _{del(p,q_c,t,c,k) \\in X_0 \\cup \\dots \\cup X_{9k+7}}{q_c} \\rbrace $ .", "It's answer set is $X_{9k+8} = A_{17}^{ts=k} \\cup A_{18}^{ts=k}$ – using forced atom proposition and definitions of $A_{17}$ and $A_{18}$ .", "where, each $tot\\_incr(p,qq_c,c,k)$ , $qq_c=\\sum _{add(p,q_c,t,c,k) \\in X_0 \\cup \\dots X_{9k+7}}{q_c}$ $\\equiv qq_c=\\sum _{t \\in X_{9k+6}, p \\in t \\bullet , 0 \\le l \\le k, l+D(t)-1=k }{m_{W(p,t)}(c)}$ , and each $tot\\_decr(p,qq_c,c,k)$ , $qq_c=\\sum _{del(p,q_c,t,c,k) \\in X_0 \\cup \\dots X_{9k+7}}{q_c}$ $\\equiv qq=\\sum _{t \\in X_{9k+6}, p \\in \\bullet t}{m_{W(t,p)}(c)} + \\sum _{t \\in X_{9k+6}, p \\in R(t)}{m_{M_k(p)}(c)}$ , represent the net effect of transitions in $T_k$ $eval_{U_{9k+8}}(bot_{U_{9k+9}}(\\Pi ^7) \\setminus bot_{U_{9k+8}}(\\Pi ^7), X_0 \\cup \\dots \\cup X_{9k+8}) = $ $\\lbrace consumesmore(p,k) \\text{:-}.", "\| \\\\ \\lbrace holds(p,q_c,c,k), tot\\_decr(p,q1_c,c,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{9k+8}, q1_c > q_c \\rbrace \\cup \\\\ \\lbrace holds(p,q_c,c,k+1) \\text{:-}., \| \\lbrace holds(p,q1_c,c,k), tot\\_incr(p,q2_c,c,k), \\\\ tot\\_decr(p,q3_c,c,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{9k+6}, q_c=q1_c+q2_c-q3_c \\rbrace \\cup \\\\ \\lbrace could\\_not\\_have(t,k) \\text{:-}.", "\| \\lbrace prenabled(t,k), \\\\ ptarc(s,t,q,c,k), holds(s,qq,c,k), tot\\_decr(s,qqq,c,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{9k+8}, \\\\ fires(t,k) \\notin X_0 \\cup \\dots \\cup X_{10k+8}, q > qq - qqq \\rbrace $ .", "It's answer set is $X_{9k+9} = A_{19}^{ts=k} \\cup A_{21}^{ts=k} \\cup A_{25}^{ts=k}$ – using forced atom proposition and definitions of $A_{19}, A_{21}, A_{25}$ .", "where, $consumesmore(p,k)$ represents $\\exists p : q_c=m_{M_k(p)}(c), \\\\ q1_c=\\sum _{t \\in T_k, p \\in \\bullet t}{m_{W(p,t)}(c)}+\\sum _{t \\in T_k, p \\in R(t)}{m_{M_k(p)}(c)}, q1_c > q_c, c \\in C$ , indicating place $p$ that will be over consumed if $T_k$ is fired, as defined in definition REF (conflicting transitions), $holds(p,q_c,c,k+1)$ if $q_c=m_{M_k(p)}(c)+\\sum _{t \\in T_l, p \\in t \\bullet , 0 \\le l \\le k, l+D(t)-1=k}{m_{W(t,p)}(c)}-(\\sum _{t \\in T_k, p \\in \\bullet t}{m_{W(p,t)}(c)}+ \\sum _{t \\in T_k, p \\in R(t)}{m_{M_k(p)}(c)})$ represented by $q_c=m_{M_1(p)}(c)$ for some $c \\in C$ – by construction of $\\Pi ^7$ , and $could\\_not\\_have(t,k)$ if $(\\forall p \\in \\bullet t, W(p,t) \\le M_k(p)), (\\forall p \\in I(t), M_k(p) = 0), (\\forall (p,t) \\in Q, \\\\ M_k(p) \\ge WQ(p,t)), \\nexists tt \\in T, (\\forall pp \\in \\bullet tt, W(pp,tt) \\le M_{ts}(pp)), (\\forall pp \\in I(tt), M_k(pp) = 0), (\\forall (pp,tt) \\in Q, M_k(pp) \\ge QW(pp,tt)), Z(tt) < Z(t)$ , and $q_c > m_{M_k(s)}(c) - (\\sum _{t^{\\prime } \\in T_k, s \\in \\bullet t^{\\prime }}{m_{W(s,t^{\\prime })}(c)}+ \\sum _{t^{\\prime } \\in T_k, s \\in R(t)}{m_{M_k(s)}(c)}), \\\\ q_c = m_{W(s,t)}(c) \\text{~if~} s \\in \\bullet t \\text{~or~} m_{M_k(s)}(c) \\text{~otherwise}$ for some $c \\in C$ , which becomes $q > M_k(s) - (\\sum _{t^{\\prime } \\in T_k, s \\in \\bullet t^{\\prime }}{W(s,t^{\\prime })}+ \\sum _{t^{\\prime } \\in T_k, s \\in R(t)}{M_k(s)}), q = W(s,t) \\text{~if~} s \\in \\bullet t \\text{~or~} M_k(s) \\text{~otherwise}$ for all $c \\in C$ (i), (ii) above combined match the definition of $A_{25}$ $X_{9k+9}$ does not contain $could\\_not\\_have(t,k)$ , when $prenabled(t,k) \\in X_0 \\cup \\dots \\cup X_{9k+6}$ and $fires(t,k) \\notin X_0 \\cup \\dots \\cup X_{9k+5}$ due to construction of $A$ , encoding of $a\\ref {a:c:maxfire:cnh}$ and its body atoms.", "As a result it is not eliminated by the constraint $a\\ref {a:c:prmaxfire:elim}$ $eval_{U_{9k+9}}(bot_{U_{9k+10}}(\\Pi ^7) \\setminus bot_{U_{9k+9}}(\\Pi ^7), X_0 \\cup \\dots \\cup X_{9k+9}) = \\lbrace consumesmore \\text{:-}.", "\| \\\\ \\lbrace consumesmore(p,0),\\dots ,$ $consumesmore(p,k) \\rbrace \\cap (X_0 \\cup \\dots \\cup X_{9k+9}) \\ne \\emptyset \\rbrace $ .", "It's answer set is $X_{9k+10} = A_{20}$ – using forced atom proposition and the definition of $A_{20}$ $X_{9k+10}$ will be empty since none of $consumesmore(p,0),\\dots , \\\\ consumesmore(p,k)$ hold in $X_0 \\cup \\dots \\cup X_{9k+10}$ due to the construction of $A$ , encoding of $a\\ref {a:overc:place}$ and its body atoms.", "As a result, it is not eliminated by the constraint $a\\ref {a:overc:elim}$ The set $X = X_0 \\cup \\dots \\cup X_{9k+10}$ is the answer set of $\\Pi ^7$ by the splitting sequence theorem REF .", "Each $X_i, 0 \\le i \\le 9k+10$ matches a distinct portion of $A$ , and $X = A$ , thus $A$ is an answer set of $\\Pi ^7$ .", "Next we show (REF ): Given $\\Pi ^7$ be the encoding of a Petri Net $PN(P,T,E,C,W,R,I,$ $Q,QW,Z,D)$ with initial marking $M_0$ , and $A$ be an answer set of $\\Pi ^7$ that satisfies (REF ) and (REF ), then we can construct $X=M_0,T_k,\\dots ,M_k,T_k,M_{k+1}$ from $A$ , such that it is an execution sequence of $PN$ .", "We construct the $X$ as follows: $M_i = (M_i(p_0), \\dots , M_i(p_n))$ , where $\\lbrace holds(p_0,m_{M_i(p_0)}(c),c,i), \\dots ,\\\\ holds(p_n,m_{M_i(p_n)}(c),c,i) \\rbrace \\subseteq A$ , for $c \\in C, 0 \\le i \\le k+1$ $T_i = \\lbrace t : fires(t,i) \\in A\\rbrace $ , for $0 \\le i \\le k$ and show that $X$ is indeed an execution sequence of $PN$ .", "We show this by induction over $k$ (i.e.", "given $M_k$ , $T_k$ is a valid firing set and its firing produces marking $M_{k+1}$ ).", "Base case: Let $k=0$ , and $M_0$ is a valid marking in $X$ for $PN$ , show [(1)] $T_0$ is a valid firing set for $M_0$ , and firing $T_0$ in $M_0$ produces marking $M_1$ .", "We show $T_0$ is a valid firing set for $M_0$ .", "Let $\\lbrace fires(t_0,0), \\dots , fires(t_x,0) \\rbrace $ be the set of all $fires(\\dots ,0)$ atoms in $A$ , Then for each $fires(t_i,0) \\in A$ $prenabled(t_i,0) \\in A$ – from rule $a\\ref {a:c:prfires}$ and supported rule proposition Then $enabled(t_i,0) \\in A$ – from rule $a\\ref {a:c:prenabled}$ and supported rule proposition And $notprenabled(t_i,0) \\notin A$ – from rule $a\\ref {a:c:prenabled}$ and supported rule proposition For $enabled(t_i,0) \\in A$ $notenabled(t_i,0) \\notin A$ – from rule $e\\ref {e:c:enabled}$ and supported rule proposition Then either of $body(e\\ref {e:c:ne:ptarc})$ , $body(e\\ref {e:c:ne:iptarc})$ , or $body(e\\ref {e:c:ne:tptarc})$ must not hold in $A$ for $t_i$ – from rules $body(e\\ref {e:c:ne:ptarc}), body(e\\ref {e:c:ne:iptarc}), body(e\\ref {e:c:ne:tptarc})$ and forced atom proposition Then $q_c \\lnot < {n_i}_c \\equiv q_c \\ge {n_i}_c$ in $e\\ref {e:c:ne:ptarc}$ for all $\\lbrace holds(p,q_c,c,0), \\\\ ptarc(p,t_i,{n_i}_c,c,0)\\rbrace \\subseteq A$ – from $e\\ref {e:c:ne:ptarc}$ , forced atom proposition, and given facts ($holds(p,q_c,c,0) \\in A, ptarc(p,t_i,{n_i}_c,0) \\in A$ ) And $q_c \\lnot \\ge {n_i}_c \\equiv q_c < {n_i}_c$ in $e\\ref {e:c:ne:iptarc}$ for all $\\lbrace holds(p,q_c,c,0), \\\\ iptarc(p,t_i,{n_i}_c,c,0) \\rbrace \\subseteq A, {n_i}_c=1$ ; $q_c > {n_i}_c \\equiv q_c = 0$ – from $e\\ref {e:c:ne:iptarc}$ , forced atom proposition, given facts ($holds(p,q_c,c,0) \\in A, \\\\ iptarc(p,t_i,1,c,0) \\in A$ ), and $q_c$ is a positive integer And $q_c \\lnot < {n_i}_c \\equiv q_c \\ge {n_i}_c$ in $e\\ref {e:c:ne:tptarc}$ for all $\\lbrace holds(p,q_c,c,0), \\\\ tptarc(p,t_i,{n_i}_c,c,0) \\rbrace \\subseteq A$ – from $e\\ref {e:c:ne:tptarc}$ , forced atom proposition, and given facts Then $\\forall c \\in C, (\\forall p \\in \\bullet t_i, m_{M_0(p)}(c) \\ge m_{W(p,t_i)}(c)) \\wedge (\\forall p \\in I(t_i), \\\\ m_{M_0(p)}(c) = 0) \\wedge (\\forall (p,t_i) \\in Q, m_{M_0(p)}(c) \\ge m_{QW(p,t_i)}(c))$ – from $i\\ref {i:c:init},f\\ref {f:c:ptarc},f\\ref {f:c:rptarc}$ construction, definition REF of preset $\\bullet t_i$ in $PN$ , definition REF of enabled transition in $PN$ , and that the construction of reset arcs by $f\\ref {f:c:rptarc}$ ensures $notenabled(t,0)$ is never true for a reset arc, where $holds(p,q_c,c,0) \\in A$ represents $q_c=m_{M_0(p)}(c)$ , $ptarc(p,t_i,{n_i}_c,0) \\in A$ represents ${n_i}_c=m_{W(p,t_i)}(c)$ , ${n_i}_c = m_{M_0(p)}(c)$ .", "Which is equivalent to $(\\forall p \\in \\bullet t_i, M_0(p) \\ge W(p,t_i)) \\wedge (\\forall p \\in I(t_i), M_0(p) = 0) \\wedge (\\forall (p,t_i) \\in Q, M_0(p) \\ge QW(p,t_i))$ – assuming multiset domain $C$ For $notprenabled(t_i,0) \\notin A$ Either $(\\nexists enabled(tt,0) \\in A : pp < p_i)$ or $(\\forall enabled(tt,0) \\in A : pp \\lnot < p_i)$ where $pp = Z(tt), p_i = Z(t_i)$ – from rule $a\\ref {a:c:prne}, f\\ref {f:c:pr}$ and forced atom proposition This matches the definition of an enabled priority transition Then $t_i$ is enabled and can fire in $PN$ , as a result it can belong to $T_0$ – from definition REF of enabled transition And $consumesmore \\notin A$ , since $A$ is an answer set of $\\Pi ^7$ – from rule $a\\ref {a:c:overc:elim}$ and supported rule proposition Then $\\nexists consumesmore(p,0) \\in A$ – from rule $a\\ref {a:c:overc:gen}$ and supported rule proposition Then $\\nexists \\lbrace holds(p,q_c,c,0), tot\\_decr(p,q1_c,c,0) \\rbrace \\subseteq A, q1_c>q_c$ in $body(a\\ref {a:c:overc:place})$ – from $a\\ref {a:c:overc:place}$ and forced atom proposition Then $\\nexists c \\in C \\nexists p \\in P, (\\sum _{t_i \\in \\lbrace t_0,\\dots ,t_x\\rbrace , p \\in \\bullet t_i}{m_{W(p,t_i)}(c)}+ \\\\ \\sum _{t_i \\in \\lbrace t_0,\\dots ,t_x\\rbrace , p \\in R(t_i)}{m_{M_0(p)}(c)}) > m_{M_0(p)}(c)$ – from the following $holds(p,q_c,c,0)$ represents $q_c=m_{M_0(p)}(c)$ – from rule $i\\ref {i:c:init}$ encoding, given $tot\\_decr(p,q1_c,c,0) \\in A$ if $\\lbrace del(p,{q1_0}_c,t_0,c,0), \\dots , \\\\ del(p,{q1_x}_c,t_x,c,0) \\rbrace \\subseteq A$ , where $q1_c = {q1_0}_c+\\dots +{q1_x}_c$ – from $r\\ref {r:c:totdecr}$ and forced atom proposition $del(p,{q1_i}_c,t_i,c,0) \\in A$ if $\\lbrace fires(t_i,0), ptarc(p,t_i,{q1_i}_c,c,0) \\rbrace \\subseteq A$ – from $r\\ref {r:c:del}$ and supported rule proposition $del(p,{q1_i}_c,t_i,c,0)$ represents removal of ${q1_i}_c = m_{W(p,t_i)}(c)$ tokens from $p \\in \\bullet t_i$ ; or it represents removal of ${q1_i}_c = m_{M_0(p)}(c)$ tokens from $p \\in R(t_i)$ – from rules $r\\ref {r:c:del},f\\ref {f:c:ptarc},f\\ref {f:c:rptarc}$ , supported rule proposition, and definition REF of preset in $PN$ Then the set of transitions in $T_0$ do not conflict – by the definition REF of conflicting transitions And for each $prenabled(t_j,0) \\in A$ and $fires(t_j,0) \\notin A$ , $could\\_not\\_have(t_j,0) \\in A$ , since $A$ is an answer set of $\\Pi ^7$ - from rule $a\\ref {a:c:prmaxfire:elim}$ and supported rule proposition Then $\\lbrace prenabled(t_j,0), holds(s,qq_c,c,0), ptarc(s,t_j,q_c,c,0), \\\\ tot\\_decr(s,qqq_c,c,0) \\rbrace \\subseteq A$ , such that $q_c > qq_c - qqq_c$ and $fires(t_j,0) \\notin A$ - from rule $a\\ref {a:c:prmaxfire:cnh}$ and supported rule proposition Then for an $s \\in \\bullet t_j \\cup R(t_j)$ , $q_c > m_{M_0(s)}(c) - (\\sum _{t_i \\in T_0, s \\in \\bullet t_i}{m_{W(s,t_i)}(c)} + \\sum _{t_i \\in T_0, s \\in R(t_i)}{m_{M_0(s)}(c))}$ , where $q_c=m_{W(s,t_j)}(c) \\text{~if~} s \\in \\bullet t_j, \\text{~or~} m_{M_0(s)}(c)$ $ \\text{~otherwise}$ .", "$ptarc(s,t_i,q_c,c,0)$ represents $q_c=m_{W(s,t_i)}(c)$ if $(s,t_i) \\in E^-$ or $q_c=m_{M_0(s)}(c)$ if $s \\in R(t_i)$ – from rule $f\\ref {f:c:ptarc},f\\ref {f:c:rptarc}$ construction $holds(s,qq_c,c,0)$ represents $qq_c=m_{M_0(s)}(c)$ – from $i\\ref {i:c:init}$ construction $tot\\_decr(s,qqq_c,c,0) \\in A$ if $\\lbrace del(s,{qqq_0}_c,t_0,c,0), \\dots , \\\\ del(s,{qqq_x}_c,t_x,c,0) \\rbrace \\subseteq A$ – from rule $r\\ref {r:c:totdecr}$ construction and supported rule proposition $del(s,{qqq_i}_c,t_i,c,0) \\in A$ if $\\lbrace fires(t_i,0), ptarc(s,t_i,{qqq_i}_c,c,0) \\rbrace \\subseteq A$ – from rule $r\\ref {r:c:del}$ and supported rule proposition $del(s,{qqq_i}_c,t_i,c,0)$ either represents ${qqq_i}_c = m_{W(s,t_i)}(c) : t_i \\in T_0, (s,t_i) \\in E^-$ , or ${qqq_i}_c = m_{M_0(t_i)}(c) : t_i \\in T_0, s \\in R(t_i)$ – from rule $f\\ref {f:c:ptarc},f\\ref {f:c:rptarc}$ construction $tot\\_decr(q,qqq_c,c,0)$ represents $\\sum _{t_i \\in T_0, s \\in \\bullet t_i}{m_{W(s,t_i)}(c)} + \\\\ \\sum _{t_i \\in T_0, s \\in R(t_i)}{m_{M_0(s)}(c)}$ – from (C,D,E) above Then firing $T_0 \\cup \\lbrace t_j \\rbrace $ would have required more tokens than are present at its source place $s \\in \\bullet t_j \\cup R(t_j)$ .", "Thus, $T_0$ is a maximal set of transitions that can simultaneously fire.", "And for each reset transition $t_r$ with $prenabled(t_r,0) \\in A$ , $fires(t_r,0) \\in A$ , since $A$ is an answer set of $\\Pi ^7$ - from rule $f\\ref {f:c:pr:rptarc:elim}$ and supported rule proposition Then, the firing set $T_0$ satisfies the reset-transition requirement of definition REF (firing set) Then $\\lbrace t_0, \\dots , t_x\\rbrace = T_0$ – using 1(a),1(b),1(d) above; and using 1(c) it is a maximal firing set Let $holds(p,q_c,c,1) \\in A$ Then $\\lbrace holds(p,q1_c,c,0), tot\\_incr(p,q2_c,c,0), tot\\_decr(p,q3_c,c,0) \\rbrace \\subseteq A : q_c=q1_c+q2_c-q3_c$ – from rule $r\\ref {r:c:nextstate}$ and supported rule proposition Then, $holds(p,q1_c,c,0) \\in A$ represents $q1_c=m_{M_0(p)}(c)$ – given ; and $\\lbrace add(p,{q2_0}_c,t_0,c,0), \\dots , $ $add(p,{q2_j}_c,t_j,c,0)\\rbrace \\subseteq A : {q2_0}_c + \\dots + {q2_j}_c = q2_c$ and $\\lbrace del(p,{q3_0}_c,t_0,c,0), \\dots , $ $del(p,{q3_l}_c,t_l,c,0)\\rbrace \\subseteq A : {q3_0}_c + \\dots + {q3_l}_c = q3_c$ – rules $r\\ref {r:c:totincr},r\\ref {r:c:totdecr}$ and supported rule proposition, respectively Then $\\lbrace fires(t_0,0), \\dots , fires(t_j,0) \\rbrace \\subseteq A$ and $\\lbrace fires(t_0,0), \\dots , fires(t_l,0) \\rbrace \\\\ \\subseteq A$ – rules $r\\ref {r:c:dur:add},r\\ref {r:c:del}$ and supported rule proposition, respectively Then $\\lbrace fires(t_0,0), \\dots , $ $fires(t_j,0) \\rbrace \\cup \\lbrace fires(t_0,0), \\dots , $ $fires(t_l,0) \\rbrace \\subseteq $ $A = \\lbrace fires(t_0,0), \\dots , $ $fires(t_x,0) \\rbrace \\subseteq A$ – set union of subsets Then for each $fires(t_x,0) \\in A$ we have $t_x \\in T_0$ – already shown in item REF above Then $q_c = m_{M_0(p)}(c) + \\sum _{t_x \\in T_0, p \\in t_x \\bullet , 0+D(t_x)-1=0}{m_{W(t_x,p)}(c)} - \\\\ (\\sum _{t_x \\in T_0 \\wedge p \\in \\bullet t_x}{m_{W(p,t_x)}(c)} + $ $\\sum _{t_x \\in T_0 \\wedge p \\in R(t_x)}{m_{M_0(p)}(c)})$ – from (REF ) above and the following Each $add(p,{q_j}_c,t_j,c,0) \\in A$ represents ${q_j}_c=m_{W(t_j,p)}(c)$ for $p \\in t_j \\bullet $ – rule $r\\ref {r:c:dur:add},f\\ref {f:c:dur:tparc}$ encoding, and definition REF of transition execution in $PN$ Each $del(p,t_y,{q_y}_c,c,0) \\in A$ represents either ${q_y}_c=m_{W(p,t_y)}(c)$ for $p \\in \\bullet t_y$ , or ${q_y}_c=m_{M_0(p)}(c)$ for $p \\in R(t_y)$ – from rule $r\\ref {r:c:del},f\\ref {f:c:ptarc}$ encoding and definition REF of transition execution in $PN$ ; or from rule $r\\ref {r:c:del},f\\ref {f:c:rptarc}$ encoding and definition of reset arc in $PN$ Each $tot\\_incr(p,q2_c,c,0) \\in A$ represents $q2_c=\\sum _{t_x \\in T_0 \\wedge p \\in t_x \\bullet , 0+D(t_x)-1=0}{m_{W(t_x,p)}(c)}$ – aggregate assignment atom semantics in rule $r\\ref {r:c:totincr}$ Each $tot\\_decr(p,q3_c,c,0) \\in A$ represents $q3_c=\\sum _{t_x \\in T_0 \\wedge p \\in \\bullet t_x}{m_{W(p,t_x)}(c)} \\\\ + \\sum _{t_x \\in T_0 \\wedge p \\in R(t_x)}{m_{M_0(p)}(c)}$ – aggregate assignment atom semantics in rule $r\\ref {r:c:totdecr}$ Then, $m_{M_1(p)}(c) = q_c$ – since $holds(p,q_c,c,1) \\in A$ encodes $q_c=m_{M_1(p)}(c)$ – from construction Inductive Step: Let $k > 0$ , and $M_k$ is a valid marking in $X$ for $PN$ , show [(1)] $T_k$ is a valid firing set for $M_k$ , and firing $T_k$ in $M_k$ produces marking $M_{k+1}$ .", "We show that $T_k$ is a valid firing set in $M_k$ .", "Let $\\lbrace fires(t_0,k), \\dots , fires(t_x,k) \\rbrace $ be the set of all $fires(\\dots ,k)$ atoms in $A$ , We show that $T_k$ is a valid firing set in $M_k$ .", "Then for each $fires(t_i,k) \\in A$ $prenabled(t_i,k) \\in A$ – from rule $a\\ref {a:c:prfires}$ and supported rule proposition Then $enabled(t_i,k) \\in A$ – from rule $a\\ref {a:c:prenabled}$ and supported rule proposition And $notprenabled(t_i,k) \\notin A$ – from rule $a\\ref {a:c:prenabled}$ and supported rule proposition For $enabled(t_i,k) \\in A$ $notenabled(t_i,k) \\notin A$ – from rule $e\\ref {e:c:enabled}$ and supported rule proposition Then either of $body(e\\ref {e:c:ne:ptarc})$ , $body(e\\ref {e:c:ne:iptarc})$ , or $body(e\\ref {e:c:ne:tptarc})$ must not hold in $A$ for $t_i$ – from rules $body(e\\ref {e:c:ne:ptarc}), body(e\\ref {e:c:ne:iptarc}), body(e\\ref {e:c:ne:tptarc})$ and forced atom proposition Then $q_c \\lnot < {n_i}_c \\equiv q_c \\ge {n_i}_c$ in $e\\ref {e:c:ne:ptarc}$ for all $\\lbrace holds(p,q_c,c,k), \\\\ ptarc(p,t_i,{n_i}_c,c,k)\\rbrace \\subseteq A$ – from $e\\ref {e:c:ne:ptarc}$ , forced atom proposition, and given facts ($holds(p,q_c,c,k) \\in A, ptarc(p,t_i,{n_i}_c,k) \\in A$ ) And $q_c \\lnot \\ge {n_i}_c \\equiv q_c < {n_i}_c$ in $e\\ref {e:c:ne:iptarc}$ for all $\\lbrace holds(p,q_c,c,k), \\\\ iptarc(p,t_i,{n_i}_c,c,k) \\rbrace \\subseteq A, {n_i}_c=1$ ; $q_c > {n_i}_c \\equiv q_c = 0$ – from $e\\ref {e:c:ne:iptarc}$ , forced atom proposition, given facts ($holds(p,q_c,c,k) \\in A, \\\\ iptarc(p,t_i,1,c,k) \\in A$ ), and $q_c$ is a positive integer And $q_c \\lnot < {n_i}_c \\equiv q_c \\ge {n_i}_c$ in $e\\ref {e:c:ne:tptarc}$ for all $\\lbrace holds(p,q_c,c,k), \\\\ tptarc(p,t_i,{n_i}_c,c,k) \\rbrace \\subseteq A$ – from $e\\ref {e:c:ne:tptarc}$ , forced atom proposition, and given facts Then $\\forall c \\in C, (\\forall p \\in \\bullet t_i, m_{M_k(p)}(c) \\ge m_{W(p,t_i)}(c)) \\wedge (\\forall p \\in I(t_i), \\\\ m_{M_k(p)}(c) = 0) \\wedge (\\forall (p,t_i) \\in Q, m_{M_k(p)}(c) \\ge m_{QW(p,t_i)}(c))$ – from the inductive assumption, $f\\ref {f:c:ptarc},f\\ref {f:c:rptarc}$ construction, definition REF of preset $\\bullet t_i$ in $PN$ , definition REF of enabled transition in $PN$ , and that the construction of reset arcs by $f\\ref {f:c:rptarc}$ ensures $notenabled(t,k)$ is never true for a reset arc, where $holds(p,q_c,c,k) \\in A$ represents $q_c=m_{M_k(p)}(c)$ , $ptarc(p,t_i,{n_i}_c,k) \\in A$ represents ${n_i}_c=m_{W(p,t_i)}(c)$ , ${n_i}_c = m_{M_k(p)}(c)$ .", "Which is equivalent to $(\\forall p \\in \\bullet t_i, M_k(p) \\ge W(p,t_i)) \\wedge (\\forall p \\in I(t_i), M_k(p) = 0) \\wedge (\\forall (p,t_i) \\in Q, M_k(p) \\ge QW(p,t_i))$ – assuming multiset domain $C$ For $notprenabled(t_i,k) \\notin A$ Either $(\\nexists enabled(tt,k) \\in A : pp < p_i)$ or $(\\forall enabled(tt,k) \\in A : pp \\lnot < p_i)$ where $pp = Z(tt), p_i = Z(t_i)$ – from rule $a\\ref {a:c:prne}, f\\ref {f:c:pr}$ and forced atom proposition This matches the definition of an enabled priority transition Then $t_i$ is enabled and can fire in $PN$ , as a result it can belong to $T_k$ – from definition REF of enabled transition And $consumesmore \\notin A$ , since $A$ is an answer set of $\\Pi ^7$ – from rule $a\\ref {a:c:overc:elim}$ and supported rule proposition Then $\\nexists consumesmore(p,k) \\in A$ – from rule $a\\ref {a:c:overc:gen}$ and supported rule proposition Then $\\nexists \\lbrace holds(p,q_c,c,k), tot\\_decr(p,q1_c,c,k) \\rbrace \\subseteq A, q1_c>q_c$ in $body(a\\ref {a:c:overc:place})$ – from $a\\ref {a:c:overc:place}$ and forced atom proposition Then $\\nexists c \\in C \\nexists p \\in P, (\\sum _{t_i \\in \\lbrace t_0,\\dots ,t_x\\rbrace , p \\in \\bullet t_i}{m_{W(p,t_i)}(c)}+ \\\\ \\sum _{t_i \\in \\lbrace t_0,\\dots ,t_x\\rbrace , p \\in R(t_i)}{m_{M_k(p)}(c)}) > m_{M_k(p)}(c)$ – from the following $holds(p,q_c,c,k)$ represents $q_c=m_{M_k(p)}(c)$ – from rule $PN$ encoding, given $tot\\_decr(p,q1_c,c,k) \\in A$ if $\\lbrace del(p,{q1_0}_c,t_0,c,k), \\dots , \\\\ del(p,{q1_x}_c,t_x,c,k) \\rbrace \\subseteq A$ , where $q1_c = {q1_0}_c+\\dots +{q1_x}_c$ – from $r\\ref {r:c:totdecr}$ and forced atom proposition $del(p,{q1_i}_c,t_i,c,k) \\in A$ if $\\lbrace fires(t_i,k), ptarc(p,t_i,{q1_i}_c,c,k) \\rbrace \\subseteq A$ – from $r\\ref {r:c:del}$ and supported rule proposition $del(p,{q1_i}_c,t_i,c,k)$ either represents removal of ${q1_i}_c = m_{W(p,t_i)}(c)$ tokens from $p \\in \\bullet t_i$ ; or it represents removal of ${q1_i}_c = m_{M_k(p)}(c)$ tokens from $p \\in R(t_i)$ – from rules $r\\ref {r:c:del},f\\ref {f:c:ptarc},f\\ref {f:c:rptarc}$ , supported rule proposition, and definition REF of transition execution in $PN$ Then the set of transitions in $T_k$ do not conflict – by the definition REF of conflicting transitions And for each $prenabled(t_j,k) \\in A$ and $fires(t_j,k) \\notin A$ , $could\\_not\\_have(t_j,k) \\in A$ , since $A$ is an answer set of $\\Pi ^7$ - from rule $a\\ref {a:c:prmaxfire:elim}$ and supported rule proposition Then $\\lbrace prenabled(t_j,k), holds(s,qq_c,c,k), ptarc(s,t_j,q_c,c,k), \\\\ tot\\_decr(s,qqq_c,c,k) \\rbrace \\subseteq A$ , such that $q_c > qq_c - qqq_c$ and $fires(t_j,k) \\notin A$ - from rule $a\\ref {a:c:prmaxfire:cnh}$ and supported rule proposition Then for an $s \\in \\bullet t_j \\cup R(t_j)$ , $q_c > m_{M_k(s)}(c) - (\\sum _{t_i \\in T_k, s \\in \\bullet t_i}{m_{W(s,t_i)}(c)} + \\sum _{t_i \\in T_k, s \\in R(t_i)}{m_{M_k(s)}(c))}$ , where $q_c=m_{W(s,t_j)}(c) \\text{~if~} s \\in \\bullet t_j, \\text{~or~} m_{M_k(s)}(c) $ $\\text{~otherwise}$ .", "$ptarc(s,t_i,q_c,c,k)$ represents $q_c=m_{W(s,t_i)}(c)$ if $(s,t_i) \\in E^-$ or $q_c=m_{M_k(s)}(c)$ if $s \\in R(t_i)$ – from rule $f\\ref {f:c:ptarc},f\\ref {f:c:rptarc}$ construction $holds(s,qq_c,c,k)$ represents $qq_c=m_{M_k(s)}(c)$ – from $i\\ref {i:c:init}$ construction $tot\\_decr(s,qqq_c,c,k) \\in A$ if $\\lbrace del(s,{qqq_0}_c,t_0,c,k), \\dots , \\\\ del(s,{qqq_x}_c,t_x,c,k) \\rbrace \\subseteq A$ – from rule $r\\ref {r:c:totdecr}$ construction and supported rule proposition $del(s,{qqq_i}_c,t_i,c,k) \\in A$ if $\\lbrace fires(t_i,k), ptarc(s,t_i,{qqq_i}_c,c,k) \\rbrace \\subseteq A$ – from rule $r\\ref {r:c:del}$ and supported rule proposition $del(s,{qqq_i}_c,t_i,c,k)$ represents ${qqq_i}_c = m_{W(s,t_i)}(c) : t_i \\in T_k, (s,t_i) \\in E^-$ , or ${qqq_i}_c = m_{M_k(t_i)}(c) : t_i \\in T_k, s \\in R(t_i)$ – from rule $f\\ref {f:c:ptarc},f\\ref {f:c:rptarc}$ construction $tot\\_decr(q,qqq_c,c,k)$ represents $\\sum _{t_i \\in T_k, s \\in \\bullet t_i}{m_{W(s,t_i)}(c)} + \\\\ \\sum _{t_i \\in T_k, s \\in R(t_i)}{m_{M_k(s)}(c)}$ – from (C,D,E) above Then firing $T_k \\cup \\lbrace t_j \\rbrace $ would have required more tokens than are present at its source place $s \\in \\bullet t_j \\cup R(t_j)$ .", "Thus, $T_k$ is a maximal set of transitions that can simultaneously fire.", "And for each reset transition $t_r$ with $enabled(t_r,k) \\in A$ , $fires(t_r,k) \\in A$ , since $A$ is an answer set of $\\Pi ^7$ - from rule $f\\ref {f:c:pr:rptarc:elim}$ and supported rule proposition Then the firing set $T_k$ satisfies the reset transition requirement of definition REF (firing set) Then $\\lbrace t_0, \\dots , t_x\\rbrace = T_k$ – using 1(a),1(b), 1(d) above; and using 1(c) it is a maximal firing set We show that $M_{k+1}$ is produced by firing $T_k$ in $M_k$ .", "Let $holds(p,q_c,c,k+1) \\in A$ Then $\\lbrace holds(p,q1_c,c,k), tot\\_incr(p,q2_c,c,k), tot\\_decr(p,q3_c,c,k) \\rbrace \\subseteq A : q_c=q1_c+q2_c-q3_c$ – from rule $r\\ref {r:c:nextstate}$ and supported rule proposition Then $holds(p,q1_c,c,k) \\in A$ represents $q1_c=m_{M_k(p)}(c)$ – inductive assumption; and $\\lbrace add(p,{q2_0}_c,t_0,c,k), \\dots , $ $add(p,{q2_j}_c,t_j,c,k)\\rbrace \\subseteq A : {q2_0}_c + \\dots + {q2_j}_c = q2_c$ and $\\lbrace del(p,{q3_0}_c,t_0,c,k), \\dots , $ $del(p,{q3_l}_c,t_l,c,k)\\rbrace \\subseteq A : {q3_0}_c + \\dots + {q3_l}_c = q3_c$ – rules $r\\ref {r:c:totincr},r\\ref {r:c:totdecr}$ and supported rule proposition, respectively Then $\\lbrace fires(t_0,k), \\dots , fires(t_j,k) \\rbrace \\subseteq A$ and $\\lbrace fires(t_0,k), \\dots , fires(t_l,k) \\rbrace \\\\ \\subseteq A$ – rules $r\\ref {r:c:dur:add},r\\ref {r:c:del}$ and supported rule proposition, respectively Then $\\lbrace fires(t_0,k), \\dots , fires(t_j,k) \\rbrace \\cup \\lbrace fires(t_0,k), \\dots , fires(t_l,k) \\rbrace \\subseteq A = \\lbrace fires(t_0,k), \\dots , fires(t_x,k) \\rbrace \\subseteq A$ – set union of subsets Then for each $fires(t_x,k) \\in A$ we have $t_x \\in T_k$ – already shown in item REF above Then $q_c = m_{M_k(p)}(c) + \\sum _{t_x \\in T_l, p \\in t_x \\bullet , 0 \\le l \\le k, l+D(t_x)-1=k}{m_{W(t_x,p)}(c)} - \\\\ (\\sum _{t_x \\in T_k \\wedge p \\in \\bullet t_x}{m_{W(p,t_x)}(c)} + \\sum _{t_x \\in T_k \\wedge p \\in R(t_x)}{m_{M_k(p)}(c)})$ – from (REF ) above and the following Each $add(p,{q_j}_c,t_j,c,k) \\in A$ represents ${q_j}_c=m_{W(t_j,p)}(c)$ for $p \\in t_j \\bullet $ – rule $r\\ref {r:c:dur:add},f\\ref {f:c:dur:tparc}$ encoding, and definition REF of postset in $PN$ Each $del(p,t_y,{q_y}_c,c,k) \\in A$ represents either ${q_y}_c=m_{W(p,t_y)}(c)$ for $p \\in \\bullet t_y$ , or ${q_y}_c=m_{M_k(p)}(c)$ for $p \\in R(t_y)$ – from rule $r\\ref {r:c:del},f\\ref {f:c:ptarc}$ encoding and definition REF of preset in $PN$ ; or from rule $r\\ref {r:c:del},f\\ref {f:c:rptarc}$ encoding and definition of reset arc in $PN$ Each $tot\\_incr(p,q2_c,c,k) \\in A$ represents $q2_c=\\sum _{t_x \\in T_l, p \\in t_x \\bullet , 0 \\le l \\le k, l+D(t_x)-1=k}{m_{W(t_x,p)}(c)}$ – aggregate assignment atom semantics in rule $r\\ref {r:c:totincr}$ Each $tot\\_decr(p,q3_c,c,k) \\in A$ represents $q3_c=\\sum _{t_x \\in T_k \\wedge p \\in \\bullet t_x}{m_{W(p,t_x)}(c)} \\\\ + \\sum _{t_x \\in T_k \\wedge p \\in R(t_x)}{m_{M_k(p)}(c)}$ – aggregate assignment atom semantics in rule $r\\ref {r:c:totdecr}$ Then, $m_{M_{k+1}(p)}(c) = q_c$ – since $holds(p,q_c,c,k+1) \\in A$ encodes $q_c=m_{M_{k+1}(p)}(c)$ – from construction As a result, for any $n > k$ , $T_n$ will be a valid firing set for $M_n$ and $M_{n+1}$ will be its target marking.", "Conclusion: Since both (REF ) and (REF ) hold, $X=M_0,T_k,M_1,\\dots ,M_k,T_{k+1}$ is an execution sequence of $PN(P,T,E,C,W,R,I,Q,QW,Z,D)$ (w.r.t $M_0$ ) iff there is an answer set $A$ of $\\Pi ^7(PN,M_0,k,ntok)$ such that (REF ) and (REF ) hold.", "Complete Set of Queries Used for Drug-Drug InteractionDrug-Drug Interaction Queries Drug Activates Gene 1 //S{/NP{//?[Value='activation'](kw1)=>//?[Value='of'](kw2)=>//?[Tag='GENE'](kw0)=>//?[Value='by'](kw4)=>//?", "[Tag='DRUG'](kw3)}} ::: distinct sent.cid, kw3.value, kw1.value, kw0.value, sent.value 2 //NP{/NP{/?[Tag='DRUG'](kw3)=>/?[Value='activation'](kw1)}=>/PP{/?[Value='of'](kw2)=>//?", "[Tag='GENE'](kw0)}} ::: distinct sent.cid, kw3.value, kw1.value, kw0.value, sent.value 3 //NP{/?[Value='activation'](kw1)=>/PP{/?[Value='of'](kw2)=>//?[Tag='GENE'](kw0)=>//?[Value='by'](kw4)=>//?", "[Tag='DRUG'](kw3)}} ::: distinct sent.cid, kw3.value, kw1.value, kw0.value, sent.value 4 //S{/NP{//?[Tag='GENE'](kw2)}=>/VP{//?[Value='activation'](kw1)=>//?", "[Tag='DRUG'](kw0)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 5 //S{/NP{//?[Tag='DRUG'](kw0)}=>/VP{//?[Value='activation'](kw1)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 6 //NP{/NP{/?[Tag='DRUG'](kw0)=>/?[Value='activation'](kw1)}=>/PP{//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 7 //S{/NP{//?[Value='activation'](kw1)=>//?[Tag='GENE'](kw2)}=>/VP{//?", "[Tag='DRUG'](kw0)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 8 //S{/NP{//?[Value='activation'](kw1)=>//?[Value='of'](kw2)=>//?[Tag='GENE'](kw0)=>//?[Value='by'](kw4)=>//?", "[Tag='DRUG'](kw3)}} ::: distinct sent.cid, kw3.value, kw1.value, kw0.value, sent.value 9 //NP{/NP{/?[Tag='DRUG'](kw3)=>/?[Value='activation'](kw1)}=>/PP{/?[Value='of'](kw2)=>//?", "[Tag='GENE'](kw0)}} ::: distinct sent.cid, kw3.value, kw1.value, kw0.value, sent.value 10 //NP{/?[Value='activation'](kw1)=>/PP{/?[Value='of'](kw2)=>//?[Tag='GENE'](kw0)=>//?[Value='by'](kw4)=>//?", "[Tag='DRUG'](kw3)}} ::: distinct sent.cid, kw3.value, kw1.value, kw0.value, sent.value Gene Induces Gene 1 //S{/NP{//?[Tag='GENE'](kw0)}=>/VP{//?[Value='induced'](kw1)=>//?[Value='by'](kw3)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 2 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{//?[Value='induced'](kw1)=>//?[Value='by'](kw3)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 3 //S{/NP{//?[Tag='GENE'](kw0)}=>/VP{//?[Value IN {'increase','increased'}](kw1)=>//?[Tag='GENE'](kw2)=>//?", "[Value='activity'](kw3)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 4 //S{/NP{//?[Tag='GENE'](kw0)=>//?[Value='activity'](kw3)}=>/VP{//?[Value='increase'](kw1)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 5 //S{/NP{//?[Tag='GENE'](kw0)}=>/VP{//?[Value IN {'stimulates','stimulate', 'stimulated'}](kw1)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 6 //S{/NP{//?[Tag='GENE'](kw0)}=>/VP{/?[Value IN {'stimulates','stimulate'}](kw1)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 7 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/?[Value='stimulates'](kw1)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 8 //S{/NP{//?[Tag='GENE'](kw0)}=>/VP{/?[Value='activated'](kw1)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 9 //NP{/NP{/?[Value='induction'](kw1)}=>/PP{/?[Value='of'](kw3)=>/NP{//?[Tag='GENE'](kw0)=>//?[Value='by'](kw4)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 10 //NP{/NP{/?[Value='stimulation'](kw1)}=>/PP{/?[Value='of'](kw3)=>/NP{//?[Tag='GENE'](kw0)=>//?[Value='by'](kw4)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 11 //NP{/NP{/?[Value='activation'](kw1)}=>/PP{/?[Value IN {'of'}](kw3)=>/NP{//?[Tag='GENE'](kw0)=>//?[Value='by'](kw4)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 12 //S{/NP{//?[Tag='GENE'](kw0)}=>/VP{//?[Value='inducible'](kw1)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 13 //S{/NP{//?[Tag='GENE'](kw0)}=>/VP{//?[Value='induced'](kw1)=>//?[Value='by'](kw3)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 14 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{//?[Value='induced'](kw1)=>//?[Value='by'](kw3)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 15 //S{/NP{//?[Tag='GENE'](kw0)}=>/VP{//?[Value IN {'increase','increased'}](kw1)=>//?[Tag='GENE'](kw2)=>//?", "[Value='activity'](kw3)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 16 //S{/NP{//?[Tag='GENE'](kw0)=>//?[Value='activity'](kw3)}=>/VP{//?[Value='increase'](kw1)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 17 //S{/NP{//?[Tag='GENE'](kw0)}=>/VP{//?[Value='stimulated'](kw1)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 18 //S{/NP{//?[Tag='GENE'](kw0)}=>/VP{/?[Value IN {'stimulates','stimulate'}](kw1)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 19 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/?[Value='stimulates'](kw1)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 20 //S{/NP{//?[Tag='GENE'](kw0)}=>/VP{/?[Value='activated'](kw1)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 21 //VP{/?[Value='activated'](kw1)=>/PP{//?[Tag='GENE'](kw0)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 22 //NP{/NP{/?[Value='induction'](kw1)}=>/PP{/?[Value='of'](kw3)=>/NP{//?[Tag='GENE'](kw0)=>//?[Value='by'](kw4)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 23 //NP{/NP{/?[Value='stimulation'](kw1)}=>/PP{/?[Value='of'](kw3)=>/NP{//?[Tag='GENE'](kw0)=>//?[Value='by'](kw4)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 24 //NP{/NP{/?[Value='activation'](kw1)}=>/PP{/?[Value IN {'of'}](kw3)=>/NP{//?[Tag='GENE'](kw0)=>//?[Value='by'](kw4)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 25 //S{/NP{//?[Tag='GENE'](kw0)}=>/VP{//?[Value='inducible'](kw1)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value Gene Inhibits Gene 1 //S{/NP{//?[Tag='GENE'](kw0)}=>/VP{//?[Value='induced'](kw1)=>//?[Value='by'](kw3)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 2 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{//?[Value='induced'](kw1)=>//?[Value='by'](kw3)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 3 //S{/NP{//?[Tag='GENE'](kw0)}=>/VP{//?[Value IN {'increase','increased'}](kw1)=>//?[Tag='GENE'](kw2)=>//?", "[Value='activity'](kw3)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 4 //S{/NP{//?[Tag='GENE'](kw0)=>//?[Value='activity'](kw3)}=>/VP{//?[Value='increase'](kw1)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 5 //S{/NP{//?[Tag='GENE'](kw0)}=>/VP{//?[Value IN {'stimulates','stimulate', 'stimulated'}](kw1)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 6 //S{/NP{//?[Tag='GENE'](kw0)}=>/VP{/?[Value IN {'stimulates','stimulate'}](kw1)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 7 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/?[Value='stimulates'](kw1)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 8 //S{/NP{//?[Tag='GENE'](kw0)}=>/VP{/?[Value='activated'](kw1)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 9 //NP{/NP{/?[Value='induction'](kw1)}=>/PP{/?[Value='of'](kw3)=>/NP{//?[Tag='GENE'](kw0)=>//?[Value='by'](kw4)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 10 //NP{/NP{/?[Value='stimulation'](kw1)}=>/PP{/?[Value='of'](kw3)=>/NP{//?[Tag='GENE'](kw0)=>//?[Value='by'](kw4)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 11 //NP{/NP{/?[Value='activation'](kw1)}=>/PP{/?[Value IN {'of'}](kw3)=>/NP{//?[Tag='GENE'](kw0)=>//?[Value='by'](kw4)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 12 //S{/NP{//?[Tag='GENE'](kw0)}=>/VP{//?[Value='inducible'](kw1)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value Drug Changes Gene Expression/Activity 1 //S{/?[Tag='DRUG'](kw2)=>/?[Value IN {'increased','increase','increases'}](kw1)=>/?[Value IN {'levels','level'}](kw3)=>/?", "[Tag='GENE'](kw0)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 2 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/?[Value IN {'increased','increases'}](kw1)=>/NP{//?[Tag='GENE'](kw0)=>//?", "[Value IN {'expression','level','activity','activities','levels'}](kw3)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 3 //S{/?[Tag='DRUG'](kw2)=>/VP{/NP{//?[Value IN {'increases','increase'}](kw1)=>//?[Tag='GENE'](kw0)=>//?", "[Value IN {'levels','activity'}](kw3)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 4 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value IN {'increased','increases'}](kw1)=>/NP{//?[Tag='GENE'](kw0)=>//?", "[Value IN {'activity','activities','levels'}](kw3)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 5 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/?[Value='increased'](kw1)=>/NP{//?[Value IN {'activity','levels','expression'}](kw3)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 6 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{//?[Value IN {'increased','increases','increase'}](kw1)=>//?[Tag='GENE'](kw0)=>//?", "[Value IN {'activity','levels'}](kw3)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 7 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{/?[Value='increased'](kw1)=>//?[Tag='GENE'](kw0)=>//?", "[Value IN {'expression','activity'}](kw3)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 8 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value IN {'increased','increases'}](kw1)=>/NP{//?[Value IN {'expression','activity','activities','level','levels'}](kw3)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 9 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value IN {'decreased','decreases'}](kw1)=>/NP{//?[Tag='GENE'](kw0)=>//?", "[Value IN {'activity','expression'}](kw3)}}} ::: distinct sent.cid, kw2.value, kw1,value, kw0.value, sent.value 10 //S{/?[Tag='DRUG'](kw2)=>/?[Value IN {'increased','increase','increases'}](kw1)=>/?[Value IN {'levels','level'}](kw3)=>/?", "[Tag='GENE'](kw0)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 11 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/?[Value IN {'increased','increases'}](kw1)=>/NP{//?[Tag='GENE'](kw0)=>//?", "[Value IN {'expression','level','activity','activities','levels'}](kw3)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 12 //S{/?[Tag='DRUG'](kw2)=>/VP{/NP{//?[Value IN {'increases','increase'}](kw1)=>//?[Tag='GENE'](kw0)=>//?", "[Value IN {'levels','activity'}](kw3)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 13 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value IN {'increased','increases'}](kw1)=>/NP{//?[Tag='GENE'](kw0)=>//?", "[Value IN {'activity','activities','levels'}](kw3)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 14 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/?[Value='increased'](kw1)=>/NP{//?[Value IN {'activity','levels','expression'}](kw3)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 15 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{//?[Value IN {'increased','increases','increase'}](kw1)=>//?[Tag='GENE'](kw0)=>//?", "[Value IN {'activity','levels'}](kw3)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 16 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{/?[Value='increased'](kw1)=>//?[Tag='GENE'](kw0)=>//?", "[Value IN {'expression','activity'}](kw3)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 17 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value IN {'increased','increases'}](kw1)=>/NP{//?[Value IN {'expression','activity','activities','level','levels'}](kw3)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 18 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value IN {'decreased','decreases'}](kw1)=>/NP{//?[Tag='GENE'](kw0)=>//?", "[Value IN {'activity','expression'}](kw3)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value Drug Induces/Stimulates Gene 1 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value IN {'stimulated','induced'}](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 2 //VP{/?[Value IN {'stimulated','induced'}](kw1)=>/PP{/NP{//?[Tag='DRUG'](kw2)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 3 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/?[Value IN {'stimulated','induced'}](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 4 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/?[Value IN {'stimulated','induced'}](kw1)=>/PP{//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 5 //S{/?[Tag='DRUG'](kw2)=>/VP{/NP{//?[Tag='GENE'](kw0)=>//?", "[Value IN {'stimulated','induced'}](kw1)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 6 //S{/NP{/PP{//?[Tag='DRUG'](kw2)}}=>/VP{/?[Value IN {'stimulated','induced'}](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 7 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/VP{/?[Value IN {'stimulated','induced'}](kw1)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 8 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/VP{//?[Value='induced'](kw1)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 9 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value IN {'stimulated','induced'}](kw1)=>/NP{/?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 10 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{//?[Value IN {'stimulate','induce'}](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 11 //NP{/NP{/?[Value IN {'induction','stimulation'}](kw1)}=>/PP{/NP{//?[Tag='GENE'](kw0)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 12 //S{/NP{/?[Value IN {'stimulation','induction'}](kw1)=>/PP{//?[Tag='GENE'](kw0)}}=>/VP{/VP{//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 13 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/NP{//?[Value IN {'stimulation','induction'}](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 14 //NP{/NP{/NP{/?[Value='induction'](kw1)}=>/PP{//?[Tag='GENE'](kw0)}}=>/PP{/NP{/?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 15 //NP{/NP{/?[Tag='GENE'](kw0)=>/?[Value IN {'stimulation','induction'}](kw1)}=>/PP{/NP{/?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 16 //NP{/?[Value IN {'stimulation','induction'}](kw1)=>/PP{/NP{//?[Tag='GENE'](kw0)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 17 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/?[Value IN {'stimulates','induces'}](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 18 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value IN {'stimulates','induces'}](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 19 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value IN {'stimulated','induced'}](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 20 //VP{/?[Value IN {'stimulated','induced'}](kw1)=>/PP{/NP{//?[Tag='DRUG'](kw2)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 21 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/?[Value IN {'stimulated','induced'}](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 22 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/?[Value IN {'stimulated','induced'}](kw1)=>/PP{//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 23 //S{/?[Tag='DRUG'](kw2)=>/VP{/NP{//?[Tag='GENE'](kw0)=>//?", "[Value IN {'stimulated','induced'}](kw1)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 24 //S{/NP{/PP{//?[Tag='DRUG'](kw2)}}=>/VP{/?[Value IN {'stimulated','induced'}](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 25 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/VP{/?[Value IN {'stimulated','induced'}](kw1)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 26 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/VP{//?[Value='induced'](kw1)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 27 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value IN {'stimulated','induced'}](kw1)=>/NP{/?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 28 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{//?[Value IN {'stimulate','induce'}](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 29 //NP{/NP{/?[Value IN {'induction','stimulation'}](kw1)}=>/PP{/NP{//?[Tag='GENE'](kw0)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 30 //S{/NP{/?[Value IN {'stimulation','induction'}](kw1)=>/PP{//?[Tag='GENE'](kw0)}}=>/VP{/VP{//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 31 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/NP{//?[Value IN {'stimulation','induction'}](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 32 //NP{/NP{/NP{/?[Value='induction'](kw1)}=>/PP{//?[Tag='GENE'](kw0)}}=>/PP{/NP{/?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 33 //NP{/NP{/?[Tag='GENE'](kw0)=>/?[Value IN {'stimulation','induction'}](kw1)}=>/PP{/NP{/?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 34 //NP{/?[Value IN {'stimulation','induction'}](kw1)=>/PP{/NP{//?[Tag='GENE'](kw0)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 35 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/?[Value IN {'stimulates','induces'}](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 36 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value IN {'stimulates','induces'}](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value Drug Inhibits Gene 1 //NP{/NP{/NP{/?[Value='inhibitory'](kw1)}=>/PP{//?[Tag='DRUG'](kw2)}}=>/PP{/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 2 //S{/?[Tag='DRUG'](kw2)=>/VP{/NP{/?[Tag='GENE'](kw0)=>/?", "[Value='inhibitor'](kw1)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 3 //NP{/?[Tag='DRUG'](kw2)=>/NP{/?[Tag='GENE'](kw0)=>/?", "[Value='inhibitor'](kw1)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 4 //NP{/NP{/?[Tag='GENE'](kw0)=>/?[Value='inhibitor'](kw1)}=>/?", "[Tag='DRUG'](kw2)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 5 //NP{/NP{/NP{/?[Tag='GENE'](kw0)=>/?[Tag='DRUG'](kw2)}}=>/?", "[Value='inhibitor'](kw1)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 6 //NP{/?[Tag='GENE'](kw0)=>/?[Value='inhibitor'](kw1)=>/?", "[Tag='DRUG'](kw2)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 7 //NP{/NP{/?[Value='inhibitor'](kw1)}=>/PP{/NP{//?[Tag='GENE'](kw0)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 8 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{//?[Tag='GENE'](kw0)=>//?", "[Value='inhibitor'](kw1)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 9 //S{/?[Tag='DRUG'](kw2)=>/VP{/NP{//?[Value='inhibitor'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 10 //NP{/?[Tag='DRUG'](kw2)=>/NP{/NP{/?[Tag='GENE'](kw0)}}=>/?", "[Value='inhibitor'](kw1)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 11 //S{/?[Tag='DRUG'](kw2)=>/?[Value='inhibitor'](kw1)=>/?", "[Tag='GENE'](kw0)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 12 //NP{/NP{/?[Tag='DRUG'](kw2)=>/NP{/?[Value='inhibitor'](kw1)}=>/PP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 13 //S{/?[Tag='GENE'](kw0)=>/?[Value='inhibitor'](kw1)=>/?", "[Tag='DRUG'](kw2)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 14 //NP{/?[Tag='DRUG'](kw2)=>/NP{/NP{/?[Value='inhibitor'](kw1)}=>/PP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 15 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{//?[Value='inhibitor'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 16 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/NP{//?[Value='inhibitor'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 17 //NP{/?[Tag='DRUG'](kw2)=>/NP{/NP{/?[Tag='GENE'](kw0)=>/?", "[Value='inhibitor'](kw1)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 18 //NP{/NP{/PP{//?[Tag='DRUG'](kw2)=>//?[Tag='GENE'](kw0)}=>/NP{/?", "[Value='inhibitor'](kw1)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 19 //NP{/NP{/?[Tag='GENE'](kw0)=>/?[Value='inhibitor'](kw1)=>/?", "[Tag='DRUG'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 20 //S{/?[Tag='DRUG'](kw2)=>/?[Tag='GENE'](kw0)=>/?", "[Value='inhibitor'](kw1)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 21 //NP{/NP{/?[Tag='DRUG'](kw2)=>/?[Tag='GENE'](kw0)=>/?", "[Value='inhibitor'](kw1)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 22 //NP{/NP{/?[Tag='DRUG'](kw2)=>/NP{/?[Tag='GENE'](kw0)=>/?", "[Value='inhibitor'](kw1)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 23 //NP{/NP{/PP{//?[Tag='DRUG'](kw2)}=>/?[Tag='GENE'](kw0)=>/?", "[Value='inhibitor'](kw1)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 24 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value='inhibited'](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 25 //S{/NP{/NP{/?[Tag='DRUG'](kw2)}}=>/VP{/?[Value='inhibited'](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 26 //S{/?[Tag='DRUG'](kw2)=>/?[Value='inhibited'](kw1)=>/?", "[Tag='GENE'](kw0)} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 27 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/?[Value='inhibited'](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 28 //S{/NP{/NP{//?[Tag='DRUG'](kw2)}}=>/VP{/?[Value='inhibited'](kw1)=>/NP{/?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 29 //S{/NP{/PP{//?[Tag='GENE'](kw0)}}=>/VP{/VP{/?[Value='inhibited'](kw1)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 30 //S{/NP{/NP{//?[Tag='DRUG'](kw2)}}=>/VP{/?[Value='inhibited'](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 31 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{/?[Value='inhibited'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 32 //S{/NP{/PP{//?[Tag='DRUG'](kw2)}}=>/VP{/?[Value='inhibited'](kw1)=>/NP{/?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 33 //S{/NP{/NP{/?[Tag='GENE'](kw0)}}=>/VP{/VP{/?[Value='inhibited'](kw1)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 34 //S{/NP{/NP{/?[Tag='GENE'](kw0)}}=>/VP{/?[Value='inhibited'](kw1)=>/PP{//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 35 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value='inhibited'](kw1)=>/PP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 36 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/VP{/?[Value='inhibited'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 37 //S{/NP{/NP{/?[Tag='DRUG'](kw2)}}=>/VP{/?[Value='inhibited'](kw1)=>/NP{/?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 38 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/VP{/?[Value='inhibited'](kw1)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 39 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/?[Value='inhibited'](kw1)=>/NP{/?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 40 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/VP{//?[Value='inhibited'](kw1)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 41 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value='inhibited'](kw1)=>/NP{/?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 42 //S{/NP{/NP{/?[Tag='GENE'](kw0)=>/?[Tag='DRUG'](kw2)}}=>/VP{/?", "[Value='inhibit'](kw1)}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 43 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/VP{//?[Value='inhibit'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 44 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{//?[Value='inhibit'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 45 //NP{/NP{/?[Tag='GENE'](kw0)=>/?[Value='inhibition'](kw1)}=>/PP{/NP{/?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 46 //S{/?[Tag='DRUG'](kw2)=>/VP{/PP{//?[Value='inhibition'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 47 //S{/NP{/PP{//?[Tag='DRUG'](kw2)}}=>/VP{/NP{//?[Value='inhibition'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 48 //NP{/?[Value='inhibition'](kw1)=>/PP{/NP{//?[Tag='GENE'](kw0)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 49 //S{/?[Value='inhibition'](kw1)=>/PP{/NP{//?[Tag='GENE'](kw0)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 50 //NP{/NP{/?[Tag='DRUG'](kw2)=>/?[Value='inhibition'](kw1)}=>/PP{/NP{/?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 51 //NP{/NP{/?[Value='inhibition'](kw1)}=>/PP{/NP{/?[Tag='GENE'](kw0)=>/?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 52 //S{/NP{/?[Value='inhibition'](kw1)=>/PP{//?[Tag='GENE'](kw0)}}=>/VP{/PP{//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 53 //S{/NP{/NP{/?[Value='inhibition'](kw1)}=>/PP{//?[Tag='GENE'](kw0)}}=>/VP{/NP{//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 54 //S{/?[Value IN {'inhibition','Inhibition'}](kw1)=>/?[Tag='GENE'](kw0)=>/?", "[Tag='DRUG'](kw2)} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 55 //NP{/NP{/NP{/?[Value='inhibition'](kw1)}=>/PP{//?[Tag='GENE'](kw0)}}=>/PP{/NP{//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 56 //NP{/?[Tag='DRUG'](kw2)=>/NP{/?[Value='inhibition'](kw1)}=>/PP{/NP{/?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 57 //NP{/NP{/?[Value='inhibition'](kw1)}=>/PP{/NP{//?[Tag='GENE'](kw0)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 58 //S{/?[Tag='DRUG'](kw2)=>/VP{/NP{//?[Value='inhibition'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 59 //S{/NP{/NP{/?[Value='inhibition'](kw1)}=>/PP{//?[Tag='DRUG'](kw2)}}=>/VP{/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 60 //NP{/NP{/?[Value='inhibition'](kw1)=>/PP{//?[Tag='GENE'](kw0)}}=>/PP{/NP{//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 61 //NP{/NP{/?[Tag='DRUG'](kw2)=>/?[Value='inhibition'](kw1)}=>/PP{/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 62 //NP{/NP{/?[Value='inhibition'](kw1)=>/PP{//?[Tag='GENE'](kw0)}}=>/PP{/NP{/?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 63 //NP{/NP{/NP{/?[Value='inhibition'](kw1)}=>/PP{//?[Tag='GENE'](kw0)}}=>/PP{/NP{/?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 64 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/?[Value='inhibits'](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 65 /S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value='inhibits'](kw1)=>/NP{/?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 66 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{/?[Value='inhibits'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 67 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value='inhibits'](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 68 //NP{/NP{/NP{/?[Value='inhibitory'](kw1)}=>/PP{//?[Tag='DRUG'](kw2)}}=>/PP{/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 69 //S{/?[Tag='DRUG'](kw2)=>/VP{/NP{/?[Tag='GENE'](kw0)=>/?", "[Value='inhibitor'](kw1)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 70 //NP{/?[Tag='DRUG'](kw2)=>/NP{/?[Tag='GENE'](kw0)=>/?", "[Value='inhibitor'](kw1)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 71 //NP{/NP{/?[Tag='GENE'](kw0)=>/?[Value='inhibitor'](kw1)}=>/?", "[Tag='DRUG'](kw2)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 72 //NP{/NP{/NP{/?[Tag='GENE'](kw0)=>/?[Tag='DRUG'](kw2)}}=>/?", "[Value='inhibitor'](kw1)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 73 //NP{/?[Tag='GENE'](kw0)=>/?[Value='inhibitor'](kw1)=>/?", "[Tag='DRUG'](kw2)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 74 //NP{/NP{/?[Value='inhibitor'](kw1)}=>/PP{/NP{//?[Tag='GENE'](kw0)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 75 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{//?[Tag='GENE'](kw0)=>//?", "[Value='inhibitor'](kw1)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 76 //S{/?[Tag='DRUG'](kw2)=>/VP{/NP{//?[Value='inhibitor'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 77 //NP{/?[Tag='DRUG'](kw2)=>/NP{/NP{/?[Tag='GENE'](kw0)}}=>/?", "[Value='inhibitor'](kw1)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 78 //S{/?[Tag='DRUG'](kw2)=>/?[Value='inhibitor'](kw1)=>/?", "[Tag='GENE'](kw0)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 79 //NP{/NP{/?[Tag='DRUG'](kw2)=>/NP{/?[Value='inhibitor'](kw1)}=>/PP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 80 //S{/?[Tag='GENE'](kw0)=>/?[Value='inhibitor'](kw1)=>/?", "[Tag='DRUG'](kw2)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 81 //NP{/?[Tag='DRUG'](kw2)=>/NP{/NP{/?[Value='inhibitor'](kw1)}=>/PP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 82 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{//?[Value='inhibitor'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 83 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/NP{//?[Value='inhibitor'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 84 //NP{/?[Tag='DRUG'](kw2)=>/NP{/NP{/?[Tag='GENE'](kw0)=>/?", "[Value='inhibitor'](kw1)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 85 //NP{/NP{/PP{//?[Tag='DRUG'](kw2)=>//?[Tag='GENE'](kw0)}=>/NP{/?", "[Value='inhibitor'](kw1)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 86 //NP{/NP{/?[Tag='GENE'](kw0)=>/?[Value='inhibitor'](kw1)=>/?", "[Tag='DRUG'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 87 //S{/?[Tag='DRUG'](kw2)=>/?[Tag='GENE'](kw0)=>/?", "[Value='inhibitor'](kw1)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 88 //NP{/NP{/?[Tag='DRUG'](kw2)=>/?[Tag='GENE'](kw0)=>/?", "[Value='inhibitor'](kw1)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 89 //NP{/NP{/?[Tag='DRUG'](kw2)=>/NP{/?[Tag='GENE'](kw0)=>/?", "[Value='inhibitor'](kw1)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 90 //NP{/NP{/PP{//?[Tag='DRUG'](kw2)}=>/?[Tag='GENE'](kw0)=>/?", "[Value='inhibitor'](kw1)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 91 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value='inhibited'](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 92 //S{/NP{/NP{/?[Tag='DRUG'](kw2)}}=>/VP{/?[Value='inhibited'](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 93 //S{/?[Tag='DRUG'](kw2)=>/?[Value='inhibited'](kw1)=>/?", "[Tag='GENE'](kw0)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 94 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/?[Value='inhibited'](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 95 //S{/NP{/NP{//?[Tag='DRUG'](kw2)}}=>/VP{/?[Value='inhibited'](kw1)=>/NP{/?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 96 //S{/NP{/PP{//?[Tag='GENE'](kw0)}}=>/VP{/VP{/?[Value='inhibited'](kw1)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 97 //S{/NP{/NP{//?[Tag='DRUG'](kw2)}}=>/VP{/?[Value='inhibited'](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 98 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{/?[Value='inhibited'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 99 //S{/NP{/PP{//?[Tag='DRUG'](kw2)}}=>/VP{/?[Value='inhibited'](kw1)=>/NP{/?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 100 //S{/NP{/NP{/?[Tag='GENE'](kw0)}}=>/VP{/VP{/?[Value='inhibited'](kw1)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 101 //S{/NP{/NP{/?[Tag='GENE'](kw0)}}=>/VP{/?[Value='inhibited'](kw1)=>/PP{//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 102 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value='inhibited'](kw1)=>/PP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 103 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/VP{/?[Value='inhibited'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 104 //S{/NP{/NP{/?[Tag='DRUG'](kw2)}}=>/VP{/?[Value='inhibited'](kw1)=>/NP{/?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 105 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/VP{/?[Value='inhibited'](kw1)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 106 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/?[Value='inhibited'](kw1)=>/NP{/?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 107 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/VP{//?[Value='inhibited'](kw1)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 108 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value='inhibited'](kw1)=>/NP{/?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 109 //S{/NP{/NP{/?[Tag='GENE'](kw0)=>/?[Tag='DRUG'](kw2)}}=>/VP{/?", "[Value='inhibit'](kw1)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 110 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/VP{//?[Value='inhibit'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 111 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{//?[Value='inhibit'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 112 //NP{/NP{/?[Tag='GENE'](kw0)=>/?[Value='inhibition'](kw1)}=>/PP{/NP{/?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 113 //S{/?[Tag='DRUG'](kw2)=>/VP{/PP{//?[Value='inhibition'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 114 //S{/NP{/PP{//?[Tag='DRUG'](kw2)}}=>/VP{/NP{//?[Value='inhibition'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 115 //NP{/?[Value='inhibition'](kw1)=>/PP{/NP{//?[Tag='GENE'](kw0)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 116 //S{/?[Value='inhibition'](kw1)=>/PP{/NP{//?[Tag='GENE'](kw0)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 117 //NP{/NP{/?[Tag='DRUG'](kw2)=>/?[Value='inhibition'](kw1)}=>/PP{/NP{/?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 118 //NP{/NP{/?[Value='inhibition'](kw1)}=>/PP{/NP{/?[Tag='GENE'](kw0)=>/?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 119 //S{/NP{/?[Value='inhibition'](kw1)=>/PP{//?[Tag='GENE'](kw0)}}=>/VP{/PP{//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 120 //S{/NP{/NP{/?[Value='inhibition'](kw1)}=>/PP{//?[Tag='GENE'](kw0)}}=>/VP{/NP{//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 121 //S{/?[Value IN {'inhibition','Inhibition'}](kw1)=>/?[Tag='GENE'](kw0)=>/?", "[Tag='DRUG'](kw2)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 122 //NP{/NP{/NP{/?[Value='inhibition'](kw1)}=>/PP{//?[Tag='GENE'](kw0)}}=>/PP{/NP{//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 123 //NP{/?[Tag='DRUG'](kw2)=>/NP{/?[Value='inhibition'](kw1)}=>/PP{/NP{/?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 124 //NP{/NP{/?[Value='inhibition'](kw1)}=>/PP{/NP{//?[Tag='GENE'](kw0)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 125 //S{/?[Tag='DRUG'](kw2)=>/VP{/NP{//?[Value='inhibition'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 126 //S{/NP{/NP{/?[Value='inhibition'](kw1)}=>/PP{//?[Tag='DRUG'](kw2)}}=>/VP{/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 127 //NP{/NP{/?[Value='inhibition'](kw1)=>/PP{//?[Tag='GENE'](kw0)}}=>/PP{/NP{//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 128 //NP{/NP{/?[Tag='DRUG'](kw2)=>/?[Value='inhibition'](kw1)}=>/PP{/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 129 //NP{/NP{/?[Value='inhibition'](kw1)=>/PP{//?[Tag='GENE'](kw0)}}=>/PP{/NP{/?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 130 //NP{/NP{/NP{/?[Value='inhibition'](kw1)}=>/PP{//?[Tag='GENE'](kw0)}}=>/PP{/NP{/?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 131 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/?[Value='inhibits'](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 132 /S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value='inhibits'](kw1)=>/NP{/?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 133 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{/?[Value='inhibits'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 134 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value='inhibits'](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value Gene Metabolized Drug 1 //S{/NP{/NP{/?[Tag='DRUG'](kw2)}}=>/VP{/VP{/?[Value IN {'metabolised','metabolized'}](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 2 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{//?[Value IN {'metabolised','metabolized'}](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 3 //S{/NP{/SBAR{//?[Tag='DRUG'](kw2)}}=>/VP{/?[Value IN {'metabolised','metabolized'}](kw1)=>/PP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 4 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/VP{/?[Value IN {'metabolised','metabolized'}](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 5 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/VP{//?[Value IN {'metabolized','metabolised'}](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 6 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{/?[Value IN {'metabolised','metabolized'}](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 7 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value='metabolized'](kw1)=>/PP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 8 //S{/?[Tag='DRUG'](kw2)=>/NP{/?[Value='metabolism'](kw1)}=>/VP{/VP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 9 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/NP{//?[Tag='DRUG'](kw2)=>//?", "[Value='metabolism'](kw1)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 10 //NP{/NP{/?[Tag='GENE'](kw0)=>/?[Value='metabolism'](kw1)}=>/PP{/NP{/?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 11 //NP{/?[Value='metabolism'](kw1)=>/PP{/NP{//?[Tag='DRUG'](kw2)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 12 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/VP{//?[Value='metabolism'](kw1)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 13 //S{/NP{/NP{/?[Tag='GENE'](kw0)}}=>/VP{/NP{//?[Value='metabolism'](kw1)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 14 //NP{/NP{/PP{//?[Tag='GENE'](kw0)}}=>/PP{/NP{/?[Tag='DRUG'](kw2)=>/?", "[Value='metabolism'](kw1)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 15 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/PP{//?[Value='metabolism'](kw1)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 16 //NP{/NP{/?[Tag='DRUG'](kw2)=>/?[Value='metabolism'](kw1)}=>/PP{/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 17 //NP{/NP{/?[Tag='GENE'](kw0)=>/?[Value='metabolism'](kw1)}=>/PP{/NP{//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 18 //S{/NP{/NP{/?[Value='metabolism'](kw1)}=>/PP{//?[Tag='DRUG'](kw2)}}=>/VP{/VP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 19 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/PP{//?[Tag='DRUG'](kw2)=>//?", "[Value='metabolism'](kw1)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 20 //NP{/NP{/PP{//?[Tag='GENE'](kw0)}}=>/PP{/NP{//?[Value='metabolism'](kw1)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 21 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/NP{//?[Value='substrate'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 22 //NP{/NP{/?[Tag='DRUG'](kw2)=>/NP{/?[Value='substrate'](kw1)}=>/PP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 23 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{//?[Value='substrate'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 24 //NP{/?[Tag='GENE'](kw0)=>/?[Value='substrate'](kw1)=>/?", "[Tag='DRUG'](kw2)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 25 //NP{/?[Tag='DRUG'](kw2)=>/NP{/NP{/?[Value='substrate'](kw1)}=>/PP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 26 //S{/NP{/NP{/?[Tag='DRUG'](kw2)}}=>/VP{/VP{/?[Value IN {'metabolised','metabolized'}](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 27 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{//?[Value IN {'metabolised','metabolized'}](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 28 //S{/NP{/SBAR{//?[Tag='DRUG'](kw2)}}=>/VP{/?[Value IN {'metabolised','metabolized'}](kw1)=>/PP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 29 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/VP{/?[Value IN {'metabolised','metabolized'}](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 30 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/VP{//?[Value IN {'metabolized','metabolised'}](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 31 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{/?[Value IN {'metabolised','metabolized'}](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 32 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value='metabolized'](kw1)=>/PP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 33 //S{/?[Tag='DRUG'](kw2)=>/NP{/?[Value='metabolism'](kw1)}=>/VP{/VP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 34 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/NP{//?[Tag='DRUG'](kw2)=>//?", "[Value='metabolism'](kw1)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 35 //NP{/NP{/?[Tag='GENE'](kw0)=>/?[Value='metabolism'](kw1)}=>/PP{/NP{/?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 36 //NP{/?[Value='metabolism'](kw1)=>/PP{/NP{//?[Tag='DRUG'](kw2)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 37 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/VP{//?[Value='metabolism'](kw1)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 38 //S{/NP{/NP{/?[Tag='GENE'](kw0)}}=>/VP{/NP{//?[Value='metabolism'](kw1)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 39 //NP{/NP{/PP{//?[Tag='GENE'](kw0)}}=>/PP{/NP{/?[Tag='DRUG'](kw2)=>/?", "[Value='metabolism'](kw1)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 40 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/PP{//?[Value='metabolism'](kw1)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 41 //NP{/NP{/?[Tag='DRUG'](kw2)=>/?[Value='metabolism'](kw1)}=>/PP{/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 42 //NP{/NP{/?[Tag='GENE'](kw0)=>/?[Value='metabolism'](kw1)}=>/PP{/NP{//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 43 //S{/NP{/NP{/?[Value='metabolism'](kw1)}=>/PP{//?[Tag='DRUG'](kw2)}}=>/VP{/VP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 44 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/PP{//?[Tag='DRUG'](kw2)=>//?", "[Value='metabolism'](kw1)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 45 //NP{/NP{/PP{//?[Tag='GENE'](kw0)}}=>/PP{/NP{//?[Value='metabolism'](kw1)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 46 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/NP{//?[Value='substrate'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 47 //NP{/NP{/?[Tag='DRUG'](kw2)=>/NP{/?[Value='substrate'](kw1)}=>/PP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 48 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{//?[Value='substrate'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 49 //NP{/?[Tag='GENE'](kw0)=>/?[Value='substrate'](kw1)=>/?", "[Tag='DRUG'](kw2)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 50 //NP{/?[Tag='DRUG'](kw2)=>/NP{/NP{/?[Value='substrate'](kw1)}=>/PP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value Gene Regulates Gene 1 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/VP{//?[Value IN {'regulated', 'upregulated', 'downregulated', 'up-regulated', 'down-regulated'}](kw1)=>//?[Value='by'](kw3)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 2 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/VP{/?[Value IN {'regulated', 'upregulated', 'downregulated', 'up-regulated', 'down-regulated'}](kw1)=>//?[Value='by'](kw3)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 3 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/?[Value IN {'regulated','down-regulated'}](kw1)=>/NP{//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 4 //NP{/NP{/?[Value IN {'regulation', 'upregulation', 'downregulation', 'up-regulation', 'down-regulation'}](kw1)}=>/PP{/NP{//?[Tag='GENE'](kw0)=>//?[Value='by'](kw3)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 5 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/?[Value IN {'regulates', 'upregulates', 'downregulates', 'up-regulates', 'down-regulates'}](kw1)=>/NP{//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 6 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/VP{//?[Value='in'](kw3)=>//?[Value='regulating'](kw1)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value Gene Regulate Gene (Xenobiotic Metabolism) 1 //S{//?[Tag='GENE' AND Canonical LIKE 'CYP\\%'](kw0)<=>//?", "[Tag='GENE' AND Canonical IN {'AhR', 'CASR', 'CAR', 'PXR', 'NR1I2', 'NR1I3'}](kw1)} ::: distinct sent.cid, kw0.value, kw1.value, sent.value 2 //S{//?[Tag='GENE' AND Value LIKE 'cytochrome\\%'](kw0)<=>//?", "[Tag='GENE' AND Canonical IN {'AhR', 'CASR', 'CAR', 'PXR', 'NR1I2', 'NR1I3'}](kw1)} ::: distinct sent.cid, kw0.value, kw1.value, sent.value Negative Drug Induces/Metabolizes/Inhibits Gene 1 //S{/?[Tag='DRUG'](kw0)=>/VP{/VP{/?[Value IN {'induced', 'inhibited', 'metabolized', 'metabolised'}](kw1)=>//?[Tag='GENE'](kw2)=>//?[Value='not'](kw3)=>//?", "[Tag='GENE'](kw4)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw4.value, kw3.value, sent.value 2 //S{/?[Tag='DRUG'](kw0)=>/?[Value='not'](kw3)=>/?[Tag='DRUG'](kw4)=>/?[Value='inhibited'](kw1)=>/?", "[Tag='GENE'](kw2)} ::: distinct sent.cid, kw4.value, kw1.value, kw2.value, kw3.value, sent.value 3 //S{/SBAR{/S{//?[Tag='DRUG'](kw0)}}=>/S{/S{//?[Value='metabolized'](kw1)=>//?[Tag='GENE'](kw2)=>//?[Value='not'](kw3)=>//?", "[Tag='GENE'](kw4)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw4.value, kw3.value, sent.value 4 //S{/NP{/PP{//?[Tag='GENE'](kw2)}}=>/VP{/VP{/?[Value IN {'induced', 'inhibited', 'metabolized', 'metabolised'}](kw1)=>//?[Tag='DRUG'](kw0)=>//?[Value='not'](kw3)=>//?", "[Tag='DRUG'](kw4)}}} ::: distinct sent.cid, kw4.value, kw1.value, kw2.value, kw3.value, sent.value 5 //S{/NP{/PP{//?[Tag='DRUG'](kw0)}}=>/VP{/VP{/?[Value IN {'induced', 'inhibited', 'metabolized', 'metabolised'}](kw1)=>//?[Tag='GENE'](kw2)=>//?[Value IN {'not','no'}](kw3)=>//?", "[Tag='GENE'](kw4)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw4.value, kw3.value, sent.value 6 //S{/?[Tag='DRUG'](kw0)=>/VP{/?[Value='not'](kw3)=>/VP{//?[Value IN {'induced', 'inhibited', 'metabolized', 'metabolised'}](kw1)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 7 //S{/NP{/?[Tag='GENE'](kw2)}=>/VP{/VP{/?[Value IN {'induced', 'inhibited', 'metabolized', 'metabolised'}](kw1)=>//?[Tag='DRUG'](kw0)=>//?[Value='not'](kw3)=>//?", "[Tag='DRUG'](kw4)}}} ::: distinct sent.cid, kw4.value, kw1.value, kw2.value, kw3.value, sent.value 8 //S{/NP{/NP{/?[Value='not'](kw3)=>/?[Tag='DRUG'](kw0)}}=>/VP{/?[Value IN {'inhibited','induced'}](kw1)=>/NP{//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 9 //S{/NP{/NP{/?[Value IN {'no','not'}](kw3)=>/?[Tag='DRUG'](kw0)}}=>/VP{/?[Value IN {'inhibited','induced'}](kw1)=>/NP{/?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 10 //S{/?[Tag='DRUG'](kw0)=>/S{/S{//?[Value='not'](kw3)=>//?[Value IN {'induce', 'inhibit'}](kw1)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 11 //S{/?[Tag='DRUG'](kw0)=>/?[Tag='GENE'](kw2)=>/?[Value IN {'not'}](kw3)=>/?", "[Value IN {'inhibit','induce'}](kw1)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 12 //S{/?[Tag='DRUG'](kw0)=>/VP{/?[Value='not'](kw3)=>/VP{/?[Value IN {'inhibit','induce'}](kw1)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 13 //S{/NP{/?[Tag='DRUG'](kw0)}=>/VP{/VP{/?[Value='not'](kw3)=>//?[Value='inhibit'](kw1)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 14 //S{/NP{/NP{/?[Tag='GENE'](kw2)}}=>/VP{/?[Value='not'](kw3)=>/VP{/?[Value='metabolize'](kw1)=>//?", "[Tag='DRUG'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 15 //S{/NP{/NP{//?[Tag='DRUG'](kw0)}}=>/VP{/?[Value='not'](kw3)=>/VP{/?[Value IN {'inhibit','induce'}](kw1)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 16 //S{/NP{/?[Tag='DRUG'](kw0)}=>/VP{/?[Value='not'](kw3)=>/VP{/?[Value='inhibit'](kw1)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 17 //S{/NP{/?[Tag='DRUG'](kw0)}=>/VP{/?[Value IN {'induces', 'inhibits'}](kw1)=>/NP{//?[Tag='GENE'](kw2)=>//?[Value='not'](kw3)=>//?", "[Tag='GENE'](kw4)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw4.value, kw3.value, sent.value 18 //S{/?[Tag='DRUG'](kw0)=>/VP{/?[Value='inhibits'](kw1)=>/NP{//?[Tag='GENE'](kw2)=>//?[Value='not'](kw3)=>//?", "[Tag='GENE'](kw4)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw4.value, kw3.value, sent.value 19 //S{/?[Tag='DRUG'](kw0)=>/VP{/NP{//?[Value='no'](kw3)=>//?[Value IN {'induction','metabolism','inhibition'}](kw1)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 20 //S{/NP{/NP{//?[Tag='DRUG'](kw0)=>//?[Tag='GENE'](kw2)=>//?[Value IN {'induction','metabolism','inhibition'}](kw1)}}=>/VP{/?", "[Value='not'](kw3)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 21 //S{/?[Tag='DRUG'](kw0)=>/VP{/?[Value='not'](kw3)=>/VP{//?[Value IN {'induction','metabolism','inhibition'}](kw1)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 22 //S{/NP{/PP{//?[Tag='DRUG'](kw0)}}=>/VP{/?[Value='not'](kw3)=>/VP{//?[Value IN {'induction','metabolism','inhibition'}](kw1)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value Negative Drug Induces Gene 1 //S{//?[Tag='GENE'](kw0)=>//?[Value IN {'induced','increased','stimulated'}](kw1)=>//?[Value='by'](kw3)=>//?[Tag='DRUG'](kw2)=>//?[Value='not'](kw4)=>//?", "[Tag='DRUG'](kw5)} ::: distinct sent.cid, kw5.value, kw1.value, kw0.value, kw4.value, sent.value 2 //S{//?[Tag='GENE'](kw0)=>//?[Value='not'](kw4)=>//?[Tag='GENE'](kw5)=>//?[Value IN {'induced','increased','stimulated'}](kw1)=>//?[Value='by'](kw3)=>//?", "[Tag='DRUG'](kw2)} ::: distinct sent.cid, kw2.value, kw1.value, kw5.value, kw4.value, sent.value 3 //S{//?[Tag='DRUG'](kw0)=>//?[Value='not'](kw4)=>//?[Value IN {'induce','induced','increase','increased','stimulate','stimulated'}](kw1)=>//?", "[Tag='GENE'](kw2)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw4.value, sent.value 4 //S{//?[Tag='DRUG'](kw0)=>//?[Value='not'](kw4)=>//?[Tag='DRUG'](kw5)=>//?[Value IN {'induces','increases','stimulates','induced','increased','stimulated'}](kw1)=>//?", "[Tag='GENE'](kw2)} ::: distinct sent.cid, kw5.value, kw1.value, kw2.value, kw4.value, sent.value 5 //S{//?[Value='no'](kw5)=>//?[value IN {'induction','stimulation'}](kw1)=>//?[Value IN {'of'}](kw2)=>//?[Tag='GENE'](kw0)=>//?[value IN {'by'}](kw3)=>//?", "[Tag='DRUG'](kw4)} ::: distinct sent.cid, kw4.value, kw1.value, kw0.value, kw5.value, sent.value Negative Drug Inhibits Gene 1 //S{//?[Tag='GENE'](kw0)=>//?[Value IN {'inhibited','decreased'}](kw1)=>//?[Value='by'](kw3)=>//?[Tag='DRUG'](kw2)=>//?[Value='not'](kw4)=>//?", "[Tag='DRUG'](kw5)} ::: distinct sent.cid, kw5.value, kw1.value, kw0.value, kw4.value, sent.value 2 //S{//?[Tag='GENE'](kw0)=>//?[Value='not'](kw4)=>//?[Tag='GENE'](kw5)=>//?[Value IN {'inhibited','decreased'}](kw1)=>//?[Value='by'](kw3)=>//?", "[Tag='DRUG'](kw2)} ::: distinct sent.cid, kw2.value, kw1.value, kw5.value, kw4.value, sent.value 3 //S{//?[Tag='DRUG'](kw0)=>//?[Value='not'](kw4)=>//?[Value IN {'inhibit','inhibited','decrease','decreased'}](kw1)=>//?", "[Tag='GENE'](kw2)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw4.value, sent.value 4 //S{//?[Tag='DRUG'](kw0)=>//?[Value='not'](kw4)=>//?[Tag='DRUG'](kw5)=>//?[Value IN {'inhibits','decreases','inhibited','decreased'}](kw1)=>//?", "[Tag='GENE'](kw2)} ::: distinct sent.cid, kw5.value, kw1.value, kw2.value, kw4.value, sent.value 5 //S{//?[Value='no'](kw5)=>//?[value IN {'inhibition'}](kw1)=>//?[Value IN {'of'}](kw2)=>//?[Tag='GENE'](kw0)=>//?[value IN {'by'}](kw3)=>//?", "[Tag='DRUG'](kw4)} ::: distinct sent.cid, kw4.value, kw1.value, kw0.value, kw5.value, sent.value Negative Gene Metabolizes Drug 1 //S{//?[Tag='DRUG'](kw0)=>//?[Value IN {'metabolized','metabolised'}](kw1)=>//?[Value='by'](kw3)=>//?[Tag='GENE'](kw2)=>//?[Value='not'](kw4)=>//?", "[Tag='GENE'](kw5)} ::: distinct sent.cid, kw0.value, kw1.value, kw5.value, kw4.value, sent.value 2 //S{//?[Tag='DRUG'](kw0)=>//?[Value='not'](kw4)=>//?[Tag='DRUG'](kw5)=>//?[Value IN {'metabolized','metabolised'}](kw1)=>//?[Value='by'](kw3)=>//?", "[Tag='GENE'](kw2)} ::: distinct sent.cid, kw5.value, kw1.value, kw2.value, kw4.value, sent.value 3 //S{//?[Tag='GENE'](kw0)=>//?[Value='not'](kw4)=>//?[Value IN {'metabolize','metabolise'}](kw1)=>//?", "[Tag='DRUG'](kw2)} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, kw4.value, sent.value 4 //S{//?[Tag='GENE'](kw0)=>//?[Value='not'](kw4)=>//?[Tag='GENE'](kw5)=>//?[Value IN {'metabolize','metabolise','metabolizes','metabolises'}](kw1)=>//?", "[Tag='DRUG'](kw2)} ::: distinct sent.cid, kw2.value, kw1.value, kw5.value, kw4.value, sent.value Negative Gene Downregulates Gene 1 //S{//?[Tag='GENE'](kw0)=>//?[Value IN {'suppressed','downregulated','inhibited','down-regulated','repressed','disrupted'}](kw1)=>//?[Value='by'](kw3)=>//?[Tag='GENE'](kw2)=>//?[Value='not'](kw4)=>//?", "[Tag='GENE'](kw5)} ::: distinct sent.cid, kw5.value, kw1.value, kw0.value, kw4.value, sent.value 2 //S{//?[Tag='GENE'](kw0)=>//?[Value='not'](kw4)=>//?[Tag='GENE'](kw5)=>//?[Value IN {'suppressed','downregulated','inhibited','down-regulated','repressed','disrupted'}](kw1)=>//?[Value='by'](kw3)=>//?", "[Tag='GENE'](kw2)} ::: distinct sent.cid, kw2.value, kw1.value, kw5.value, kw4.value, sent.value 3 //S{//?[Tag='GENE'](kw0)=>//?[Value='not'](kw4)=>//?[Value IN {'suppressed','suppress','downregulated','downregulate','inhibited','inhibit','down-regulated','down-regulate','repressed','repress','disrupted','disrupt'}](kw1)=>//?", "[Tag='GENE'](kw2)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw4.value, sent.value 4 //S{//?[Tag='GENE'](kw0)=>//?[Value='not'](kw4)=>//?[Tag='GENE'](kw5)=>//?[Value IN {'suppresses','downregulates','inhibits','down-regulates','represses','disrupts','suppressed','downregulated','inhibited','down-regulated','repressed','disrupted'}](kw1)=>//?", "[Tag='GENE'](kw2)} ::: distinct sent.cid, kw5.value, kw1.value, kw2.value, kw4.value, sent.value 5 //S{//?[Value='no'](kw5)=>//?[value IN {'inhibition','downregulation','down-regulation'}](kw1)=>//?[Value IN {'of'}](kw2)=>//?[Tag='GENE'](kw0)=>//?[value IN {'on'}](kw3)=>//?", "[Tag='GENE'](kw4)} ::: distinct sent.cid, kw0.value, kw1.value, kw4.value, kw5.value, sent.value Negative Gene Upregulates Gene 1 //S{//?[Tag='GENE'](kw0)=>//?[Value IN {'activated','induced','stimulated','regulated','upregulated','up-regulated'}](kw1)=>//?[Value='by'](kw3)=>//?[Tag='GENE'](kw2)=>//?[Value='not'](kw4)=>//?", "[Tag='GENE'](kw5)} ::: distinct sent.cid, kw5.value, kw1.value, kw0.value, kw4.value, sent.value 2 //S{//?[Tag='GENE'](kw0)=>//?[Value='not'](kw4)=>//?[Tag='GENE'](kw5)=>//?", "[Value IN {'activated','induced','stimulated','regulated','upregulated','up-regulated'}](kw1) ::: distinct sent.cid, kw2.value, kw1.value, kw5.value, kw4.value, sent.value 3 //S{//?[Tag='GENE'](kw0)=>//?[Value='not'](kw4)=>//?[Value IN {'activated','induced','stimulated','regulated','upregulated','up-regulated','activate','induce','stimulate','regulate','upregulate','up-regulate'}](kw1)=>//?", "[Tag='GENE'](kw2)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw4.value, sent.value 4 //S{//?[Tag='GENE'](kw0)=>//?[Value='not'](kw4)=>//?[Tag='GENE'](kw5)=>//?[Value IN {'activates','induces','stimulates','regulates','upregulates','up-regulates','activate','induce','stimulate','regulate','upregulate','up-regulate'}](kw1)=>//?", "[Tag='GENE'](kw2)} ::: distinct sent.cid, kw5.value, kw1.value, kw2.value, kw4.value, sent.value 5 //S{//?[Value='no'](kw5)=>//?[value IN {'induction','activation','stimulation','regulation','upregulation','up-regulation'}](kw1)=>//?[Value IN {'of'}](kw2)=>//?[Tag='GENE'](kw0)=>//?[value IN {'by'}](kw3)=>//?", "[Tag='GENE'](kw4)} ::: distinct sent.cid, kw4.value, kw1.value, kw0.value, kw5.value, sent.value Drug Gene Co-Occurrence 1 //S{//?[Tag='DRUG'](kw0)<=>//?", "[Tag='GENE'](kw1)} ::: distinct sent.cid, kw0.value, kw1.value, kw0.type, kw1.type, sent.value" ], [ "Introduction", "Petri Net [61] is a graphical modeling language with formal semantics used for description of distributed systems.", "It is named after Carl Adam Petri, who formally defined Petri Nets in his PhD thesis in the 1960's [11].", "Petri nets have been widely used to model a wide range of systems, from distributed systems to biological pathways.", "The main advantages of Petri Net representation include its simplicity and the ability to model concurrent and asynchronous systems and processes.", "A variety of Petri Net extensions have been proposed in the literature, e.g.", "inhibitor arcs, reset transitions, timed transitions, stochastic transitions, prioritized transitions, colored petri nets, logic petri nets, hierarchical petri nets, hybrid petri nets and functional petri nets to a name a few [6], [57], [34].", "Our interest in Petri Nets is for representing biological pathways and simulating them in order to answer simulation based reasoning questions.", "We show how Petri nets can be represented in ASP.", "We also demonstrate how various extensions of basic Petri nets can be easily expressed and implemented by making small changes to the initial encoding.", "During this process we will relate the extensions to their use in the biological domain.", "Later chapters will show how this representation and simulation is used to answer biologically relevant questions.", "The rest of this chapter is organized as follows: We present some background material on Answer Set Programming (ASP) and Petri Nets.", "Following that, we present the Answer Set encoding of a basic Petri Net.", "After that we will introduce various Petri Nets extensions and the relevant ASP code changes to implement such extensions." ], [ "Answer Set Programming", "Answer Set Programming (ASP) is a declarative logic programming language that is based on the Stable Model Semantics [25].", "It has been applied to a problems ranging from spacecrafts, work flows, natural language processing and biological systems modeling.", "Although ASP language is quite general, we limit ourselves to language and extensions relevant to our work.", "Definition 1 (Term) A term is a term in the propositional logic sense.", "Definition 2 (Literal) A literal is an atom in the propositional logic sense.", "A literal prefixed with $\\mathbf {not}$ is referred to as a negation-as-failure literal or a naf-literal, with $\\mathbf {not}$ representing negation-as-failure.", "We will refer to propositional atoms as basic atoms to differentiate them from other atoms, such as the aggregate atoms defined below.", "Definition 3 (Aggregate Atom) A sum aggregate atom is of the form: $L \\; [ B_0=w_0,\\dots ,B_m=w_m ] \\; U$ where, $B_i$ are basic atoms, $w_i$ are positive integer weight terms, $L,U$ are integer terms specifying the lower and upper limits of aggregate weights.", "The lower and upper limits are assumed to be $-\\infty $ and $\\infty $ , if not specified.", "A count aggregate atom is a special case of the sum aggregate atom in which all weights are 1, i.e.", "$L \\; [ B_0=1,\\dots ,B_m=1] \\; U$ and it is represented by: $L \\; \\lbrace B_0,\\dots ,B_m \\rbrace \\; U$ A choice atom is a special case of the count aggregate atom (REF ) in which $n=m$ .", "Definition 4 (ASP Program) An ASP program $\\Pi $ is a finite set of rules of the following form: $A_0 \\leftarrow A_1,\\dots ,A_m,\\mathbf {not~} B_{1},\\dots ,\\mathbf {not~} B_n, C_1,\\dots ,C_k.$ where each $A_0$ is either a basic atom or a choice atom, $A_i$ and $B_i$ are basic atoms, $C_i$ are aggregate atoms and $\\mathbf {not}$ is negation-as-failure.", "In rule (REF ), $\\lbrace A_0 \\rbrace $ is called the head of the rule, and $\\lbrace A_1,\\dots ,$ $A_m,$ $\\mathbf {not~} B_{1},\\dots ,$ $\\mathbf {not~} B_n, $ $C_1,\\dots ,$ $C_k\\rbrace $ is called its tail.", "A rule in which $A_0$ is a choice atom is called a choice rule.", "A rule without a head is called a constraint.", "A rule with a basic atom as its head and empty tail is called a fact in which case the “$\\leftarrow $ ” is dropped.", "Let $R$ be an ASP rule of the form (REF ) and let $pos(R) = \\lbrace A_1,\\dots ,A_m \\rbrace $ represent the positive atoms, $neg(R) = \\lbrace B_1,\\dots ,B_n \\rbrace $ the negation-as-failure atoms, and $agg(R) = \\lbrace C_1,\\dots ,C_k \\rbrace $ represent the aggregate atoms in the body of a rule $R$ .", "Let $lit(A)$ represent the set of basic literals in atom $A$ , i.e.", "$lit(A)=\\lbrace A\\rbrace $ if $A$ is a basic atom; $lit(A)=\\lbrace B_0,\\dots ,B_n\\rbrace $ if $A$ is an aggregate atom.", "Let $C$ be an aggregate atom of the form (REF ) and let $pos(C) = \\lbrace B_0,\\dots ,B_m \\rbrace $ be the sets of basic positive literals such that $lit(C) = pos(C)$ .", "Let $lit(R) = lit(head(R)) \\cup pos(R) \\cup neg(R) \\cup \\bigcup _{C \\in agg(R)}{lit(C)} $ for a rule $R \\in \\Pi $ and $lit(\\Pi ) = \\bigcup _{R \\in \\Pi }{lit(R)}$ be the set of basic literals in ASP program $\\Pi $ .", "Definition 5 (Aggregate Atom Satisfaction) A ground aggregate atom $C$ of the form (REF ) is satisfied by a set of basic ground atoms $S$ , if $L \\le \\sum _{0 \\le i \\le m, B_i \\in S}{w_i} \\le U$ and we write $S \\models C$ .", "Given a set of basic ground literals $S$ and a basic ground atom $A$ , we say $S \\models A$ if $A \\in S$ , $S \\models \\mathbf {not~} A$ if $A \\notin S$ .", "For a rule $R$ of the form (REF ) $S \\models body(R)$ if $\\forall A \\in \\lbrace A_1,\\dots ,A_m\\rbrace , S \\models A$ , $\\forall B \\in \\lbrace B_{1},\\dots ,B_n \\rbrace , S \\models \\mathbf {not~} B$ , and $\\forall C \\in \\lbrace C_1,\\dots ,C_k \\rbrace , S \\models C$ ; $S \\models head(R)$ if $S \\models A_0$ .", "Definition 6 (Rule Satisfaction) A ground rule $R \\in \\Pi $ is satisfied by a set of basic ground atoms $S$ , iff, $S \\models body(R)$ implies $S \\models head(R)$ .", "A constraint rule $R \\in \\Pi $ is satisfied by set $S$ if $S \\lnot \\models body(R)$ .", "We define reduct of an ASP program by treating aggregate atoms in a similar way as negation-as-failure literals, since our code does not contain recursion through aggregation (which can yield non-intuitive answer-sets [70]).", "Definition 7 (Reduct) Let $S$ be a set of ground basic atoms, the reduct of ground ASP program $\\Pi $ w.r.t.", "$S$ , written $\\Pi ^S$ is the set of rules: $\\lbrace p \\leftarrow A_1,\\dots ,A_m.", "\\; \| \\; A_0 \\leftarrow A_1, \\dots , $ $A_m, $ $\\mathbf {not~} B_{1}, \\dots , $ $\\mathbf {not~} B_n, $ $C_1,\\dots ,$ $C_k.", "\\in \\Pi , p \\in lit(A_0) \\cap S, \\lbrace A_1,\\dots ,A_m\\rbrace \\subseteq S, \\lbrace B_1,\\dots ,B_n \\rbrace \\cap S = \\emptyset , $ $\\nexists C \\in \\lbrace C_1,\\dots ,C_k \\rbrace , S \\lnot \\models C \\rbrace $ .", "Intuitively, this definition of reduct removes all rules which contain a naf-literal or an aggregate atom in their bodies that does not hold in $S$ , and it removes aggregate atoms as well as naf-literals from the body of the remaining rules.", "Heads of choice-rules are split into multiple rules containing at most one atom in their heads.", "The resulting reduct is a program that does not contain any aggregate atoms or negative literals.", "The rules of such a program are monotonic, such that if it satisfied by a set $S$ of atoms, it is also satisfied by any superset of $S$ .", "A deductive closure of such a (positive) monotonic program is defined as the unique smallest set of atoms $S$ such that whenever all body atoms of a rule hold in $S$ , the head also holds in $S$ .", "The deductive closure can be iteratively computed by starting with an empty set and adding heads of rules for which the bodies are satisfied, until a fix point is reached, where no additional rules can be satisfied.", "(adopted from [4]) Definition 8 (Answer Set) A set of basic ground atoms $S$ is an answer set of a ground ASP program $\\Pi $ , iff $S$ is equal to the deductive closure of $\\Pi ^S$ and $S$ satisfies each rule of $\\Pi $ .", "(adopted from [4])" ], [ "Clingo Specific Syntactic Elements", "The ASP code in this thesis is in the syntax of ASP solver called clingo [24].", "The “$\\leftarrow $ ” in ASP rules is replaced by the symbol “:-”.", "Though the semantics of ASP are defined on ground programs, Clingo allows variables and other constructs for compact representation.", "We intuitively describe specific syntactic elements and their meanings below: Comments: Text following “%” to the end of the line is treated as a comment.", "Interval: Atoms defined over an contiguous range of integer values can be compactly written as intervals, e.g.", "$p(1\\;..\\;5)$ represents atoms $p(1), p(2), p(3), p(4), p(5)$ .", "Pooling: Symbol “;” allows for pooling alternative terms.", "For example, an atom $p(\\dots ,X,\\dots )$ and $p(\\dots ,Y,\\dots )$ can be pooled together into a single atom as $p(\\dots ,X;Y,\\dots )$ .", "Aggregate assignment atom: An aggregate assignment atom $Q=\\#sum[A_0=w_0,$ $\\dots ,$ $A_m=w_m,$ $\\mathbf {not~} A_{m+1}=w_{m+1},\\dots ,$ $\\mathbf {not~} A_n=w_n]$ assigns the sum $\\sum _{A_i \\in S, 0 \\le i \\le m}{w_i} + $ $\\sum _{A_j \\notin S, m+1 \\le n}{w_j}$ to $Q$ w.r.t.", "a consistent set of basic ground atoms $S$ .", "Condition: Conditions allow instantiating variables to collections of terms within aggregates, e.g.", "$\\lbrace p(X) : q(X) \\rbrace $ instantiates $p(X)$ for only those $X$ that $q(X)$ satisfies.", "For example, if we have $p(1..5)$ but only $q(3;5)$ , then $\\lbrace p(X) : q(X) \\rbrace $ is expanded to $\\lbrace p(3), p(5) \\rbrace $ ." ], [ "Grounding", "Grounding makes a program variable free by replacing variables with the possible values they can take.", "Clingo uses the grounder Gringo for “smart” grounding, which results in substantial reduction in the size of the program.", "Details of this grounding are implementation specific.", "We present the intuitive process of grounding below.", "A set of ground terms is constructed, where a ground term is a term that contains no variables.", "The variables are split into two categories: local and global.", "Local variables are the ones that appear only within an aggregate atom (minus the limits) and nowhere else in a rule.", "Such variables are considered local writ.", "the aggregate atom.", "All other variables are considered global.", "First the global variables are eliminated in the rules as follows: Each rule $r$ containing an aggregate assignment atom of the form (REF ) is replaced with set of rules $r^{\\prime }$ in which the aggregate assignment atom is is replaced with an aggregate atom with lower and upper bounds of $Q$ for all possible substitutions of $Q$ .", "This is generalized to multiple aggregate assignment atoms by repeating this step for each such atom, where output of previous iteration forms the input of the next iteration.", "Each rule $r^{\\prime }$ , is replaced with the set of all rules $r^{\\prime \\prime }$ obtained by all possible substitutions of ground terms for global variables in $r$ .", "Then the local variables are eliminated in the rules by expanding conditions, such that $p(\\dots ,X,\\dots ) : d(X)$ are replaced by $p(\\dots ,d_1,\\dots ), \\dots , p(\\dots ,d_k,\\dots )$ for the extent $\\lbrace d_1,\\dots ,d_k\\rbrace $ of $d(X)$ .", "This is generalized to multiple conditions in the obvious way.", "Following the convention of the Clingo system, Variables in rules presented in this thesis start with capital letters while lower-case text and numbers are constants.", "Italicized text represents a constant term from a definition in context.", "A recent work [35] gives the semantics of Gringo with ASP Core 2 syntax [13] using Infintary Propositional Formulas, which translate Gringo to propositional formulas with infinitely long conjunctions and disjunctions.", "Their approach removes the safety requirement, but the subset of Gringo presented does appear to cover assignments.", "Although their approach provides a way to improve our ASP encoding by removing the requirement of specifying the maximum number of tokens or running simulations until a condition holds, our simpler (limited) semantics is sufficient for the limited syntax and semantics we use." ], [ "Multiset", "A multiset $A$ over a domain set $D$ is a pair $\\langle D,m \\rangle $ , where $m: D \\rightarrow {N}$ is a function giving the multiplicity of $d \\in D$ in $A$ .", "Given two multsets $A = \\langle D,m_A \\rangle , B = \\langle D,m_B \\rangle $ , $A \\odot B$ if $\\forall d \\in D: m_A(d) \\odot m_B(d)$ , where $\\odot \\in \\lbrace <,>,\\le ,\\ge ,=\\rbrace $ , and $A \\ne B$ if $\\exists d \\in D : m_A(d) \\ne m_B(d)$ .", "Multiset sum/difference is defined in the usual way.", "We use the short-hands $d \\in A$ to represent $m_A(d) > 0$ , $A = \\emptyset $ to represent $\\forall d \\in D, m(d) = 0$ , $A \\otimes n$ to represent $\\forall d \\in D, m(d) \\otimes n$ , where $n \\in {N}$ , $\\otimes \\in \\lbrace <,>,\\le ,\\ge ,=,\\ne \\rbrace $ .", "We use the notation $d/n \\in A$ to represent that $d$ appears $n$ -times in $A$ ; we drop $A$ when clear from context.", "The reader is referred to [71] for details." ], [ "Petri Net", "A Petri Net is a graph of a finite set of nodes and directed arcs, where nodes are split between places and transitions, and each arc either connects a place to a transition or a transition to a place.", "Each place has a number of tokens (called the its marking) Standard convention is to use dots in place nodes to represent the marking of the place.", "We use numbers for compact representation..", "Collective marking of all places in a Petri Net is called its marking (or state).", "Arc labels represent arc weights.", "When missing, arc-weight is assumed as one, and place marking is assumed as zero.", "Figure: Petri Net graph (of sub-section of glycolysis pathway) showing places as circles, transitions as boxes and arcs as directed arrows.", "Places have token count (or marking) written above them, assumed 0 when missing.", "Arcs labels represent arc-weights, assumed 1 when missing.The set of place nodes on incoming and outgoing arcs of a transition are called its pre-set (input place set or input-set) and post-set (output place set or output-set), respectively.", "A transition $t$ is enabled when each of its pre-set place $p$ has at least the number of tokens equal to the arc-weight from $p$ to $t$ .", "An enabled transition may fire, consuming tokens equal to arc-weight from place $p$ to transition $t$ from each pre-set place $p$ , producing tokens equal to arc-weight from transition $t$ to place $p$ to each post-set place $p$ .", "Multiple transitions may fire as long as they consume no more than the available tokens, with the assumption that tokens cannot be shared.", "Fig.", "REF shows a representation of a portion of the glycolysis pathway as given in [64].", "In this figure, places represent reactants and products, transitions represent reactions, and arc weights represent reactant quantity consumed or the product quantity produced by the reaction.", "When unspecified, arc-weight is assumed to be 1 and place-marking is assumed to be 0.", "Definition 9 (Petri Net) A Petri Net is a tuple $PN=(P,T,E,W)$ , where, $P=\\lbrace p_1, \\dots , p_n\\rbrace $ is a finite set of places; $T=\\lbrace t_1, \\dots , t_m\\rbrace $ is a finite set of transitions, $P \\cap T = \\emptyset $ ; $E^+ \\subseteq T \\times P$ is a set of arcs from transitions to places; $E^- \\subseteq P \\times T$ is a set of arcs from places to transitions; $E= E^+ \\cup E^- $ ; and $W: E \\rightarrow {N}\\setminus \\lbrace 0\\rbrace $ is the arc-weight function Definition 10 (Marking) A marking $M=(M(p_1),\\dots ,M(p_{n}))$ is the token assignment of each place node $p_i \\in P$ of $PN$ , where $M(p_i) \\in {N}$ .", "Initial token assignment $M_0: P \\rightarrow {N}$ is called the initial marking.", "Marking at step $k$ is written as $M_k$ .", "Definition 11 (Pre-set & post-set of a transition) Pre-set / input-set of a transition $t \\in T$ of $PN$ is $\\bullet t = \\lbrace p \\in P : (p,t) \\in E^- \\rbrace $ , while the post-set / output-set is $t \\bullet = \\lbrace p \\in P : (t,p) \\in E^+ \\rbrace $ Definition 12 (Enabled Transition) A transition $t \\in T$ of $PN$ is enabled with respect to marking $M$ , $enabled_M(t)$ , if $\\forall p \\in \\bullet t, W(p,t) \\le M(p)$ .", "An enabled transition may fire.", "Definition 13 (Transition Execution) A transition execution is the simulation of change of marking from $M_k$ to $M_{k+1}$ due to firing of a transition $t \\in T$ of $PN$ .", "$M_{k+1}$ is computed as follows: $ \\forall p_i \\in \\bullet t, M_{k+1}(p_i) = M_{k}(p_i) - W(p_i,t) $ $ \\forall p_j \\in t \\bullet , M_{k+1}(p_j) = M_{k}(p_j)+ W(t,p_j) $ Petri Nets allow simultaneous firing of a set of enabled transitions w.r.t.", "a marking as long as they do not conflict.", "Definition 14 (Conflicting Transitions) Given $PN$ with marking $M$ .", "A set of enabled transitions $T_e = \\lbrace t \\in T : enabled_M(t) \\rbrace $ of $PN$ conflict if their simultaneous firing will consume more tokens than are available at an input place: $\\exists p \\in P : M(p) < \\displaystyle \\sum _{\\begin{array}{c}t \\in T_e \\wedge p \\in \\bullet t\\end{array}}{W(p,t)}$ Definition 15 (Firing Set) A firing set is a set $T_k=\\lbrace t_{1},\\dots ,t_{m}\\rbrace \\subseteq T$ of simultaneously firing transitions that are enabled and do not conflict w.r.t.", "to the current marking $M_k$ of $PN$ .", "Definition 16 (Firing Set Execution) Execution of a firing set $T_k$ of $PN$ on a marking $M_{k}$ computes the new marking $M_{k+1}$ as follows: $\\forall p \\in P, M_{k+1}(p) = M_k(p)- \\sum _{\\begin{array}{c}t \\in T_k \\wedge p \\in \\bullet t\\end{array}} W(p,t)+ \\sum _{\\begin{array}{c}t \\in T_k \\wedge p \\in t \\bullet \\end{array}} W(t,p)$ where $\\sum _{\\begin{array}{c}t \\in T_k \\wedge p \\in \\bullet t\\end{array}} W(p,t)$ is the total consumption from place $p$ and $\\sum _{\\begin{array}{c}t \\in T_k \\wedge p \\in t \\bullet \\end{array}} W(t,p)$ is the total production at place $p$ .", "Definition 17 (Execution Sequence) An execution sequence $X = M_0, T_0, M_1, T_1, \\dots , $ $M_k, T_k, M_{k+1}$ of $PN$ is the simulation of a firing set sequence $\\sigma = T_1,T_2,\\dots ,T_k$ w.r.t.", "an initial marking $M_0$ , producing the final marking $M_{k+1}$ .", "$M_{k+1}$ is the transitive closure of firing set executions, where subsequent marking become the initial marking for the next firing set.", "For an execution sequence $X = M_0, T_0, M_1, T_1, \\dots , M_k, T_k, M_{k+1}$ , the firing of $T_0$ with respect to marking $M_0$ produces the marking $M_1$ which becomes the initial marking for $T_1$ , which produces $M_2$ and so on." ], [ "Translating Basic Petri Net Into ASP", "In this section we present ASP encoding of simple Petri Nets.", "We describe, how a given Petri Net $PN$ , and an initial marking $M_0$ are encoded into ASP for a simulation length $k$ .", "Following sections will show how Petri Net extensions can be easily added to it.", "We represent a Petri Net with the following facts: f1: Facts place($p_i$ ).", "where $p_i \\in P$ is a place.", "f2: Facts trans($t_j$ ).", "where $t_j \\in T$ is a transition.", "f3: Facts ptarc($p_i,t_j,W(p_i,t_j)$ ).", "where $(p_i,t_j) \\in E^-$ with weight $W(p_i,t_j)$ .", "f4: Facts tparc($t_i,p_j,W(t_i,p_j)$ ).", "where $(t_i,p_j) \\in E^+$ with weight $W(t_i,p_j)$ .", "Petri Net execution simulation proceeds in discrete time-steps, these time steps are encoded by the following facts: f5: Facts time($ts_i$ ) where $0 \\le ts_i \\le k$ .", "The initial marking (or initial state) of the Petri Net is represented by the following facts: i1: Facts holds($p_i,M_0(p_i),0$ ) for every place $p_i \\in P$ with initial marking $M_0(p_i)$ .", "ASP requires all variables in rule bodies be domain restricted.", "So, we add the following facts to capture the possible token quantities produced during the simulation Note that $ntok$ can be arbitrarily chosen to be larger than the maximum expected token quantity produced during the simulation.", ": f6: Facts num($n$ )., where $0 \\le n \\le ntok$ A transition $t_i$ is enabled if each of its input places $p_j \\in \\bullet t_i$ has at least arc-weight $W(p_j, t_i)$ tokens.", "Conversely, $t_i$ is not enabled if $\\exists p_j \\in \\bullet t_i : M(p_j) < W(p_j,t_i)$ , and is only enabled when no such place $p_j$ exists.", "These are captured in $e\\ref {e:ne:ptarc}$ and $e\\ref {e:enabled}$ respectively: e1: notenabled(T,TS):-ptarc(P,T,N),holds(P,Q,TS),Q<N, place(P), trans(T), time(TS),num(N),num(Q).", "e2: enabled(T,TS) :- trans(T), time(TS), not notenabled(T, TS).", "Rule $e\\ref {e:ne:ptarc}$ encodes notenabled(T,TS) which captures the existence of an input place $P$ of transition $T$ that violates the minimum token requirement $N$ at time-step $TS$ .", "Where, the predicate holds(P,Q,TS) encodes the marking $Q$ of place $P$ at $TS$ .", "Rule $e\\ref {e:enabled}$ encodes enabled(T,TS) which captures that transition $T$ is enabled at $TS$ since there is no input place $P$ of transition $T$ that violates the minimum input token requirement at $TS$ .", "In biological context, $e\\ref {e:enabled}$ captures the conditions when a reaction (represented by $T$ ) is ready to proceed.", "A subset of enabled transitions may fire simultaneously at a given time-step.", "This is encoded as: a1: {fires(T,TS)} :- enabled(T,TS), trans(T), time(TS).", "Rule $a\\ref {a:fires}$ encodes fires(T,TS), which captures the firing of transition $T$ at $TS$ .", "The rule is encoded with a count atom as its head, which makes it a choice rule.", "This rule either picks the enabled transition $T$ for firing at $TS$ or not, effectively enumerating a subset of enabled transitions to fire.", "Whether this set can fire or not in an answer set is subject to conflict checking, which is done by rules $a\\ref {a:overc:place},a\\ref {a:overc:gen},a\\ref {a:overc:elim}$ shown later.", "In biological context, the selected transition-set models simultaneously occurring reactions and the conflict models limited reactant supply that cannot be shared.", "Such a conflict can lead to multiple choices in parallel reaction evolutions and different outcomes.", "The next set of rules captures the consumption and production of tokens due to the firing of individual transitions in a firing-set as well as their aggregate effect, which computes the marking for the next time step: r1: add(P,Q,T,TS) :- fires(T,TS), tparc(T,P,Q), time(TS).", "r2: del(P,Q,T,TS) :- fires(T,TS), ptarc(P,T,Q), time(TS).", "r3: tot_incr(P,QQ,TS) :- QQ=#sum[add(P,Q,T,TS)=Q:num(Q):trans(T)], time(TS), num(QQ), place(P).", "r4: tot_decr(P,QQ,TS) :- QQ=#sum[del(P,Q,T,TS)=Q:num(Q):trans(T)], time(TS), num(QQ), place(P).", "r5: holds(P,Q,TS+1) :-holds(P,Q1,TS),tot_incr(P,Q2,TS),time(TS+1), tot_decr(P,Q3,TS),Q=Q1+Q2-Q3,place(P),num(Q;Q1;Q2;Q3),time(TS).", "Rule $r\\ref {r:add}$ encodes add(P,Q,T,TS) and captures the addition of $Q$ tokens to place $P$ due to firing of transition $T$ at time-step $TS$ .", "Rule $r\\ref {r:del}$ encodes del(P,Q,T,TS) and captures the deletion of $Q$ tokens from place $P$ due to firing of transition $T$ at $TS$ .", "Rules $r\\ref {r:totincr}$ and $r\\ref {r:totdecr}$ aggregate all add's and del's for place $P$ due to $r\\ref {r:add}$ and $r\\ref {r:del}$ at time-step $TS$ , respectively, by using the QQ=#sum[] construct to sum the $Q$ values into $QQ$ .", "Rule $r\\ref {r:nextstate}$ which encodes holds(P,Q,TS+1) uses these aggregate adds and removes and updates $P$ 's marking for the next time-step $TS+1$ .", "In biological context, these rules capture the effect of a reaction on reactant and product quantities available in the next simulation step.", "To prevent overconsumption at a place following rules are added: a2: consumesmore(P,TS) :- holds(P,Q,TS), tot_decr(P,Q1,TS), Q1 > Q. a3: consumesmore :- consumesmore(P,TS).", "a4: :- consumesmore.", "Rule $a\\ref {a:overc:place}$ encodes consumesmore(P,TS) which captures overconsumption of tokens at input place $P$ at time $TS$ due to the firing set selected by $a\\ref {a:fires}$ .", "Overconsumption (and hence conflict) occurs when tokens $Q1$ consumed by the firing set are greater than the tokens $Q$ available at $P$ .", "Rule $a\\ref {a:overc:gen}$ generalizes this notion of overconsumption and constraint $a\\ref {a:overc:elim}$ eliminates answers where overconsumption is possible.", "Definition 18 Given a Petri Net $PN$ , its initial marking $M_0$ and its encoding $\\Pi (PN,$ $M_0,$ $k,$ $ntok)$ for $k$ -steps and maximum $ntok$ tokens at any place.", "We say that there is a 1-1 correspondence between the answer sets of $\\Pi (PN,M_0,k,ntok)$ and the execution sequences of $PN$ iff for each answer set $A$ of $\\Pi (PN,M_0,k,ntok)$ , there is a corresponding execution sequence $X=M_0,T_0,M_1,\\dots ,M_k,T_k,M_{k+1}$ of $PN$ and for each execution sequence $X$ of $PN$ there is an answer-set $A$ of $\\Pi (PN,M_0,k,ntok)$ such that $\\lbrace fires(t,ts) : t \\in T_{ts}, 0 \\le ts \\le k \\rbrace = \\lbrace fires(t,ts) : fires(t,ts) \\in A \\rbrace $ $\\begin{split}\\lbrace holds(p,q,ts) &: p \\in P, q=M_{ts}(p), 0 \\le ts \\le k+1 \\rbrace \\\\&= \\lbrace holds(p,q,ts) : holds(p,q,ts) \\in A\\rbrace \\end{split}$ Proposition 1 There is a 1-1 correspondence between the answer sets of $\\Pi ^0(PN,$ $M_0,$ $k,$ $ntok)$ and the execution sequences of $PN$ ." ], [ "An example execution", "Next we look at an example execution of the Petri Net shown in Figure REF .", "The Petri Net and its initial marking are encoded as follows{holds(p1,0,0),...,holds(pN,0,0)}, {num(0),...,num(60)}, {time(0),...,time(5)} have been written as holds(p1;...;pN,0,0), num(0..60), time(0..5), respectively, to save space.", ": num(0..60).time(0..5).place(f16bp;dhap;g3p;bpg13).", "trans(t3;t4;t5a;t5b;t6).tparc(t3,f16bp,1).ptarc(f16bp,t4,1).", "tparc(t4,dhap,1).tparc(t4,g3p,1).ptarc(dhap,t5a,1).", "tparc(t5a,g3p,1).ptarc(g3p,t5b,1).tparc(t5b,dhap,1).", "ptarc(g3p,t6,1).tparc(t6,bpg13,2).holds(f16bp;dhap;g3p;bgp13,0,0).", "we get thousands of answer-sets, for example{fires(t1,ts1),...,fires(tN,ts1)} have been written as fires(t1;...;tN;ts1) to save space.", ": holds(bpg13,0,0) holds(dhap,0,0) holds(f16bp,0,0) holds(g3p,0,0) holds(bpg13,0,1) holds(dhap,0,1) holds(f16bp,1,1) holds(g3p,0,1) holds(bpg13,0,2) holds(dhap,1,2) holds(f16bp,1,2) holds(g3p,1,2) holds(bpg13,0,3) holds(dhap,2,3) holds(f16bp,1,3) holds(g3p,2,3) holds(bpg13,2,4) holds(dhap,3,4) holds(f16bp,1,4) holds(g3p,2,4) holds(bpg13,4,5) holds(dhap,4,5) holds(f16bp,1,5) holds(g3p,2,5) fires(t3,0) fires(t3;t4,1) fires(t3;t4;t5a;t5b,2) fires(t3;t4;t5a;t5b;t6,3) fires(t3;t4;t5a;t5b;t6,4) fires(t3;t4;t5a;t5b;t6,5)" ], [ "Changing Firing Semantics", "The ASP code above implements the set firing semantics.", "It can produce a large number of answer-sets, since any subset of a firing set will also be fired as a firing set.", "For our biological system modeling, it is often beneficial to simulate only the maximum activity at any given time-step.", "We accomplish this by defining the maximal firing set semantics, which requires that a maximal subset of non-conflicting transitions fires at a single time stepSuch a semantics reduces the reachable markings.", "See [12] for the analysis of its computational power.. Our semantics is different from the firing multiplier approach used by [46], in which a transition can fire as many times as allowed by the tokens available in its source places.", "Their approach requires an exponential time firing algorithm in the number of transitions.", "Our maximal firing set semantics is implemented by adding the following rules to the encoding in Section REF : a5: could_not_have(T,TS) :- enabled(T,TS), not fires(T,TS), ptarc(S,T,Q), holds(S,QQ,TS), tot_decr(S,QQQ,TS), Q > QQ - QQQ.", "a6: :- not could_not_have(T,TS), enabled(T,TS), not fires(T,TS), trans(T), time(TS).", "Rule $a\\ref {a:maxfire:cnh}$ encodes could_not_have(T,TS) which means that an enabled transition $T$ that did not fire at time $TS$ , could not have fired because its firing would have resulted in overconsumption.", "Rule $a\\ref {a:maxfire:elim}$ eliminates any answer-sets in which an enabled transition did not fire, that could not have caused overconsumption.", "Intuitively, these two rules guarantee that the only reason for an enabled transition to not fire is conflict avoidance (due to overconsumption).", "With this firing semantics, the number of answer-sets produced for Petri Net in Figure REF reduces to 2.", "Proposition 2 There is 1-1 correspondence between the answer sets of $\\Pi ^1(PN,$ $M_0,$ $k,$ $ntok)$ and the execution sequences of $PN$ .", "Other firing semantics can be encoded with similar ease.", "For example, if interleaved firing semantics is desired, replace rules $a\\ref {a:maxfire:cnh},a\\ref {a:maxfire:elim}$ with the following: aREF ': more_than_one_fires :- fires(T1,TS), fires(T2, TS), T1 != T2, time(TS).", "aREF ': :- more_than_one_fires.", "We now look at Petri Net extensions and show how they can be easily encoded in ASP." ], [ "Extension - Reset Arcs", "Definition 19 (Reset Arc) A Reset Arc in a Petri Net $PN^R$ is an arc from place $p$ to transition $t$ that consumes all tokens from its input place $p$ upon firing of $t$ .", "A Reset Petri Net is a tuple $PN^R = (P,T,E,W,R)$ where, $P, T, E, W$ are the same as for PN; and $R: T \\rightarrow 2^P$ defines reset arcs Figure: Petri Net of Fig extended with a reset arc from dhapdhap to trtr shown with double arrowhead.Figure REF shows an extended version of the Petri Net in Figure REF with a reset arc from $dhap$ to $tr$ (shown with double arrowhead).", "In biological context it models the removal of all quantity of compound $dhap$ .", "Petri Net execution semantics with reset arcs is modified for conflict detection and execution as follows: Definition 20 (Reset Transition) A transition $t \\in T$ of $PN^R$ is called a reset-transition if it has a reset arc incident on it, i.e.", "$R(t) \\ne \\emptyset $ .", "Definition 21 (Firing Set) A firing set is a set $T_k=\\lbrace t_{1},\\dots ,t_{m}\\rbrace \\subseteq T$ of simultaneously firing transitions that are enabled and do not conflict w.r.t.", "to the current marking $M_k$ of $PN^R$ .", "$T_k$ is not a firing set if there is an enabled reset-transition that is not in $T_k$ , i.e.", "$\\exists t : enabled_{M_k}(t), R(t) \\ne \\emptyset , t \\notin T_k$ .", "The reset arc is involved here because we use a modified execution semantics of reset arcs compared to the standard definition [3].", "Even though both capture similar operation, our definition allows us to model elimination of all quantity of a substance as soon as it is produced, even in a maximal firing set semantics.", "Our semantics considers reset arc's token consumption in contention with other arcs, while the standard definition does not..", "Definition 22 (Transition Execution in $PN^R$ ) A transition execution is the simulation of change of marking from $M_k$ to $M_{k+1}$ due to firing of a transition $t \\in T$ of $PN^R$ .", "$M_{k+1}$ is computed as follows: $ \\forall p_i \\in \\bullet t, M_{k+1}(p_i) = M_{k}(p_i) - W(p_i,t) $ $ \\forall p_j \\in t \\bullet , M_{k+1}(p_j) = M_{k}(p_j) + W(t,p_j) $ $ \\forall p_r \\in R(t), M_{k+1}(p_r) = M_{k}(p_r) - M_{k}(p_r) $ Definition 23 (Conflicting Transitions in $PN^R$ ) A set of enabled transitions conflict in $PN^R$ w.r.t.", "$M_k$ if firing them simultaneously will consume more tokens than are available at any one of their common input-places.", "$T_e = \\lbrace t \\in T : enabled_{M_k}(t) \\rbrace $ conflict if: $\\exists p \\in P : M_k(p) < (\\displaystyle \\sum _{t \\in T_e \\wedge (p,t) \\in E^-}{W(p,t)} + \\displaystyle \\sum _{t \\in T_e \\wedge p \\in R(t)}{M_k(p)})$ Definition 24 (Firing Set Execution in $PN^R$ ) Execution of a transition set $T_i$ in $PN^R$ has the following effect: $\\forall p \\in P \\setminus R(T_i), M_{k+1}(p) = M_k(p) - \\sum _{\\begin{array}{c}t \\in T_i \\wedge p \\in \\bullet t\\end{array}} W(p,t) + \\sum _{\\begin{array}{c}t \\in T_i \\wedge p \\in t \\bullet \\end{array}} W(t,p)$ $\\forall p \\in R(T_i), M_{k+1}(p) = \\sum _{t \\in T_i \\wedge p \\in t \\bullet } W(t,p)$ where $R(T_i)=\\displaystyle \\bigcup _{\\begin{array}{c}t \\in T_i\\end{array}} R(t)$ and represents the places emptied by $T_i$ due to reset arcs Our definition of conflicting transitions allows at most one transition with a reset arc from a place to fire, any more create a conflict.", "Thus, the new marking computation is equivalent to $\\forall p \\in P, M_{k+1}(p) = M_k(p) - (\\sum _{\\begin{array}{c}t \\in T_k \\wedge p \\in \\bullet t\\end{array}}{W(p,t)} + \\sum _{\\begin{array}{c}t \\in T_k \\wedge p \\in R(t)\\end{array}}{M_k(p)})+ \\sum _{\\begin{array}{c}t \\in T_k \\wedge p \\in t \\bullet \\end{array}} W(t,p)$.", "Since a reset arc from $p$ to $t$ , $p \\in R(t)$ consumes current marking dependent tokens, we extend ptarc to include time-step and replace $f\\ref {f:ptarc},f\\ref {f:tparc},e\\ref {e:ne:ptarc},r\\ref {r:add},r2,a\\ref {a:maxfire:cnh}$ with $f\\ref {f:r:ptarc},f\\ref {f:r:tparc},e\\ref {e:r:ne:ptarc},r\\ref {r:r:add},r\\ref {r:r:del},a\\ref {a:r:maxfire:cnh}$ , respectively in the Section REF encoding and add rule $f\\ref {f:rptarc}$ for each reset arc: f7: Rules ptarc($p_i,t_j,W(p_i,t_j),ts_k$ ):-time($ts_k$ ).", "for each non-reset arc $(p_i,t_j) \\in E^-$ f8: Rules tparc($t_i,p_j,W(t_i,p_j),ts_k$ ):-time($ts_k$ ).", "for each non-reset arc $(t_i,p_j) \\in E^+$ e3: notenabled(T,TS) :- ptarc(P,T,N,TS), holds(P,Q,TS), Q < N, place(P), trans(T), time(TS), num(N), num(Q).", "r6: add(P,Q,T,TS) :- fires(T,TS), tparc(T,P,Q,TS), time(TS).", "r7: del(P,Q,T,TS) :- fires(T,TS), ptarc(P,T,Q,TS), time(TS).", "f9: Rules ptarc($p_i,t_j,X,ts_k$ ) :- holds($p_i,X,ts_k$ ), num($X$ ), $X>0$ .", "for each reset arc between $p_i$ and $t_j$ using $X=M_k(p_i)$ as arc-weight at time step $ts_k$ .", "f10: Rules :- enabled($t_j,ts_k$ ),not fires($t_j,ts_k$ ), time($ts_k$ ).", "for each transition $t_j$ with an incoming reset arc, i.e.", "$R(t_j) \\ne \\emptyset $ .", "a7: could_not_have(T,TS) :- enabled(T,TS), not fires(T,TS), ptarc(S,T,Q,TS), holds(S,QQ,TS), tot_decr(S,QQQ,TS), Q>QQ-QQQ.", "Rule $f\\ref {f:rptarc}$ encodes place-transition arc with marking dependent weight to capture the notion of a reset arc, while rule $f\\ref {f:rptarc:elim}$ ensures that the reset-transition (i.e.", "the transition on which the reset arc terminates) always fires when enabled.", "Proposition 3 There is 1-1 correspondence between the answer sets of $\\Pi ^2(PN^R,M_0,$ $k,ntok)$ and the execution sequences of $PN^R$ .", "The execution semantics of our definition are slightly different from the standard definition in [3], even though both capture similar operations.", "Our implementation considers token consumption by reset arc in contention with other token consuming arcs from the same place, while the standard definition considers token consumption as a side effect, not in contention with other arcs.", "We chose our definition to allow modeling of biological process that removes all available quantity of a substance in a maximal firing set.", "Consider Figure REF , if $dhap$ has 1 or more tokens, our semantics would only permit either $t5a$ or $tr$ to fire in a single time-step, while the standard semantics can allow both $t5a$ and $tr$ to fire simultaneously, such that the reset arc removes left over tokens after $(dhap,t5a)$ consumes one token.", "We could have, instead, extended our encoding to include self-modifying nets [77], but our modified-definition provides a simpler solution.", "Standard semantics, however, can be easily encoded by replacing $r\\ref {r:nextstate}$ by $r\\ref {r:nextstate}a^{\\prime }, r\\ref {r:nextstate}b^{\\prime }$ ; replacing $f\\ref {f:rptarc},f\\ref {f:rptarc:elim}$ with $f\\ref {f:rptarc}^{\\prime }$ ; and adding $a\\ref {a:reset:std}$ as follows: fREF ': rptarc($p_i$ ,$t_j$ ).", "- for each reset arc between $p_i \\in R(t_j)$ and $t_j$ .", "a8: reset(P,TS) :- rptarc(P,T), place(P), trans(T), fires(T,TS), time(TS).", "rREF a': holds(P,Q,TS+1) :- holds(P,Q1,TS), tot_incr(P,Q2,TS), tot_decr(P,Q3,TS), Q=Q1+Q2-Q3, place(P), num(Q;Q1;Q2;Q3), time(TS), time(TS+1), not reset(P,TS).", "rREF b': holds(P,Q,TS+1) :- tot_incr(P,Q,TS), place(P), num(Q), time(TS), time(TS+1), reset(P,TS).", "where, the fact $f\\ref {f:rptarc}^{\\prime }$ encodes the reset arc; rule $a\\ref {a:reset:std}$ encodes if place $P$ will be reset at time $TS$ due to firing of transition $T$ that has a reset arc on it from $P$ to $T$ ; rule $r\\ref {r:nextstate}a^{\\prime }$ computes marking at $TS+1$ when place $P$ is not being reset; and rule $r\\ref {r:nextstate}b^{\\prime }$ computes marking at $TS+1$ when $P$ is being reset." ], [ "Extension - Inhibitor Arcs", "Definition 25 (Inhibitor Arc) An inhibitor arc [61] is a place-transition arc that inhibits its transition from firing as long as the place has any tokens in it.", "An inhibitor arc does not consume any tokens from its input place.", "A Petri Net with reset and inhibitor arcs is a tuple $PN^{RI}=(P,T,E,W,R,I)$ , where, $P, T, E, W, R$ are the same as for $PN^R$ ; and $I: T \\rightarrow 2^P$ defines inhibitor arcs.", "Figure: Petri Net showing feedback inhibition arc from atpatp to gly1gly1 with a bullet arrowhead.", "Inhibitor arc weight is assumed 1 when not specified.Figure REF shows a Petri Net with inhibition arc from $atp$ to $gly1$ with a bulleted arrowhead.", "It models biological feedback regulation in simplistic terms, where excess $atp$ downstream causes the upstream $atp$ production by glycolysis $gly$ to be inhibited until the excess quantity is consumed [64].", "Petri Net execution semantics with inhibit arcs is modified for determining enabled transitions as follows: Definition 26 (Enabled Transition in $PN^{RI}$ ) A transition $t$ is enabled with respect to marking $M$ , $enabled_M(t)$ , if all its input places $p$ have at least the number of tokens as the arc-weight $W(p,t)$ and all $p \\in I(t)$ have zero tokens, i.e.", "$(\\forall p \\in \\bullet t, W(p,t) \\le M(p)) \\wedge (\\forall p \\in I(t), M(p) = 0)$ We add inhibitor arcs to our encoding in Section REF as follows: f11: Rules iptarc($p_i,t_j,1,ts_k$ ):-time($ts_k$ ).", "for each inhibitor arc between $p_i \\in I(t_j)$ and $t_j$ .", "e4: notenabled(T,TS) :- iptarc(P,T,N,TS), holds(P,Q,TS), place(P), trans(T), time(TS), num(N), num(Q), Q >= N. The new rule $e\\ref {e:ne:iptarc}$ encodes another reason for a transition to be disabled (or not enabled).", "An inhibitor arc from $p$ to $t$ with arc weight $N$ will cause its target transition $t$ to not enable when the number of tokens at its source place $p$ is greater than or equal to $N$ , where $N$ is always 1 per rule $f\\ref {f:iptarc}$ .", "Proposition 4 There is 1-1 correspondence between the answer sets of $\\Pi ^3(PN^{RI},M_0,k,ntok)$ and the execution sequences of $PN$ ." ], [ "Extension - Read Arcs", "Definition 27 (Read Arc) A read arc (a test arc or a query arc) [17] is an arc from place to transition, which enables its transition only when its source place has at least the number of tokens as its arc weight.", "It does not consume any tokens from its input place.", "A Petri Net with reset, inhibitor and read arcs is a tuple $PN^{RIQ}=(P,T,W,R,I,Q,QW)$ , where, $P,T,E,W,R,I$ are the same as for $PN^{RI}$ ; $Q \\subseteq P \\times T $ defines read arcs; and $QW: Q \\rightarrow {N} \\setminus \\lbrace 0\\rbrace $ defines read arc weight.", "Figure: Petri Net with read arc from h_ish\\_is to synsyn shown with arrowhead on both ends.", "The transition synsyn will not fire unless there are at least 25 tokens in h_ish\\_is, but when it executes, it only consumes 3 tokens.Figure REF shows a Petri Net with read arc from $h\\_is$ to $syn$ shown with arrowhead on both ends.", "It models the ATP synthase $syn$ activation requiring a higher concentration of $H+$ ions $h\\_is$ in the intermembrane space This is an oversimplified model of $syn$ (ATP synthase) activation, since the actual model requires an $H+$ concentration differential across membrane..", "The reaction itself consumes a lower quantity of $H+$ ions represented by the regular place-transition arc [64], [7].", "Petri Net execution semantics with read arcs is modified for determining enabled transitions as follows: Definition 28 (Enabled Transition in $PN^{RIQ}$ ) A transition $t$ is enabled with respect to marking $M$ , $enabled_M(t)$ , if all its input places $p$ have at least the number of tokens as the arc-weight $W(p,t)$ , all $p_i \\in I(t)$ have zero tokens and all $p_q : (p_q,t) \\in Q$ have at least the number of tokens as the arc-weight $W(p,t)$ , i.e.", "$(\\forall p \\in \\bullet t, W(p,t) \\le M(p)) \\wedge (\\forall p \\in I(t), M(p) = 0) \\wedge (\\forall (p,t) \\in Q, M(p) \\ge QW(p,t))$ We add read arcs to our encoding of Section REF as follows: f12: Rules tptarc($p_i,t_j,QW(p_i,t_j),ts_k$ ):-time($ts_k$ ).", "for each read arc $(p_i,t_j) \\in Q$ .", "e5: notenabled(T,TS):-tptarc(P,T,N,TS),holds(P,Q,TS), place(P),trans(T), time(TS), num(N), num(Q), Q < N. The new rule $f\\ref {f:tptarc}$ captures the read arc and its arc-weight; and the new rule $e\\ref {e:ne:tptarc}$ encodes another reason for a transition to not be enabled.", "A read arc from $p$ to $t$ with arc weight $N$ will cause its target transition $t$ to not enable when the number of tokens at its source place $p$ is less than the arc weight $N$ .", "Proposition 5 There is a 1-1 correspondence between the answer sets of $\\Pi ^4(PN^{RIQ},M_0,$ $k,ntok)$ and the execution sequences of $PN^{RIQ}$ ." ], [ "Extension - Colored Tokens", "Higher level Petri Nets extend the notion of tokens to typed (or colored) tokens.", "This allows a more compact representation of complicated networks [60].", "Definition 29 (Petri Net with Colored Tokens) A Petri Net with Colored Tokens (with reset, inhibit and read arcs) is a tuple $PN^C=(P,T,E,C,W,R,I,Q,QW)$ , where $P,T,E,R,I,Q$ are the same as for basic Petri Nets, $C=\\lbrace c_1,\\dots ,c_l\\rbrace $ is a finite set of colors (or types), and arc weights $W : E \\rightarrow \\langle C,m \\rangle $ , $QW : Q \\rightarrow \\langle C,m \\rangle $ are specified as multi-sets of colored tokens over color set $C$ .", "The state (or marking) of place nodes $M(p_i) = \\langle C,m \\rangle , p_i \\in P$ is specified as a multiset of colored tokens over set $C$ .", "We will now update some definitions related to Petri Nets to include colored tokens.", "Definition 30 (Marking) A marking $M=(M(p_1),\\dots ,M(p_n))$ assigns a colored multi-set of tokens over the domain of colors $C$ to each place $\\lbrace p_1,\\dots ,p_n\\rbrace \\in P$ of $PN^C$ .", "The initial marking is the initial token assignment of place nodes and is represented by $M_0$ .", "The marking at time-step $k$ is written as $M_k$ .", "Definition 31 (Pre-set and post-set of a transition) The pre-set (or input-set) of a transition $t$ is $\\bullet t = \\lbrace p \\in P \| (p,t) \\in E^- \\rbrace $ , while the post-set (or output-set) is $t \\bullet = \\lbrace p \\in P \| (t,p) \\in E^+ \\rbrace $ .", "Definition 32 (Enabled Transition) A transition $t$ is enabled with respect to marking $M$ , $enabled_M(t)$ , if each of its input places $p$ has at least the number of colored-tokens as the arc-weight $W(p,t)$In the following text, for simplicity, we will use $W(p,t)$ to mean $W(\\langle p,t \\rangle )$ .", "We use similar simpler notation for $QW$ ., each of its inhibiting places $p_i \\in I(t)$ have zero tokens and each of its read places $p_q : (p_q,t) \\in Q$ have at least the number of colored-tokens as the read-arc-weight $QW(p_q,t)$ , i.e.", "$(\\forall p \\in \\bullet t, W(p,t) \\le M(p)) \\wedge (\\forall p \\in I(t), M(p) = \\emptyset ) \\wedge (\\forall (p,t) \\in Q, M(p) \\ge QW(p,t))$ for a given $t$ .This is equivalent to $\\forall c \\in C, (\\forall p \\in \\bullet t, m_{W(p,t)}(c) \\le m_{M(p)}(c)) \\wedge (\\forall p \\in I(t), m_{M(p)}(c) = 0) \\wedge (\\forall (p,t) \\in Q, m_{M(p)}(c) \\ge m_{QW(p,t)}(c))$ .", "Definition 33 (Transition Execution) Execution of a transition $t$ of $PN^C$ on a marking $M_k$ computes a new marking $M_{k+1}$ as: $\\forall p \\in \\bullet t M_{k+1}(p) = M_k(p) - W(p,t)$ $\\forall p \\in t\\bullet M_{k+1}(p) = M_k(p) + W(t,p)$ $\\forall p \\in R(t) M_{k+1}(p) = M_k(p) - M_k(p)$ Any number of enabled transitions may fire simultaneously as long as they don't conflict.", "A transition when fired consumed tokens from its pre-set places equivalent to the (place,transition) arc-weight.", "Definition 34 (Conflicting Transitions) A set of transitions $T_c \\subseteq \\lbrace t : $ $enabled_{M_k}(t) \\rbrace $ is in conflict in $PN^C$ with respect to $M_k$ if firing them will consume more tokens than are available at one of their common input places, i.e., $\\exists p \\in P : M_k(p) < (\\sum _{t \\in T_c \\wedge p \\in \\bullet t}{W(p,t)} + \\sum _{\\begin{array}{c}t \\in T_c \\wedge p \\in R(t)\\end{array}}{M_k(p)})$ Definition 35 (Firing Set) A firing set is a set $T_k=\\lbrace t_{k_1},\\dots ,t_{k_n}\\rbrace \\subseteq T$ of simultaneously firing transitions of $PN^C$ that are enabled and do not conflict w.r.t.", "to the current marking $M_k$ of $PN$ .", "A set $T_k$ is not a firing set if there is an enabled reset-transition that is not in $T_k$ , i.e.", "$\\exists t \\in enabled_{M_k}, R(t) \\ne \\emptyset , t \\notin T_k$ .", "See footnote REF Definition 36 (Firing Set Execution) Execution of a firing set $T_k$ of $PN^C$ on a marking $M_k$ computes a new marking $M_{k+1}$ as: $\\forall p \\in P \\setminus R(T_k), M_{k+1}(p) = M_k(p)- \\sum _{\\begin{array}{c}t \\in T_k \\wedge p \\in \\bullet t\\end{array}} W(p,t)+ \\sum _{\\begin{array}{c}t \\in T_k \\wedge p \\in t \\bullet \\end{array}} W(t,p)$ $\\forall p \\in R(T_k), M_{k+1}(p) = \\sum _{\\begin{array}{c}t \\in T_k \\wedge p \\in t \\bullet \\end{array}} W(t,p)$ where $R(T_k)=\\bigcup _{t \\in T_k} R(t)$ See footnote REF Definition 37 (Execution Sequence) An execution sequence $X = M_0, T_0, M_1, $ $T_1, \\dots , $ $M_k, $ $T_k, M_{k+1}$ of $PN$ is the simulation of a firing set sequence $\\sigma = T_1,T_2,\\dots ,T_k$ w.r.t.", "an initial marking $M_0$ , producing the final marking $M_{k+1}$ .", "$M_{k+1}$ is the transitive closure of firing set executions, where subsequent marking become the initial marking for the next firing set.", "For an execution sequence $X = M_0, T_0, M_1, T_1, \\dots , $ $M_k, T_k, M_{k+1}$ , the firing of $T_0$ with respect to marking $M_0$ produces the marking $M_1$ which becomes the initial marking for $T_1$ , which produces $M_2$ and so on.", "Figure: Petri Net with tokens of colors {e,h,h2o,nadh,nadp,o2}\\lbrace e,h,h2o,nadh,nadp,o2\\rbrace .", "Circles represent places, and rectangles represent transitions.", "Arc weights such as “nadh/2,h/2nadh/2,h/2”, “h/2,h2o/1h/2,h2o/1” specify the number of tokens consumed and produced during the execution of their respective transitions, where “nadh/2,h/2nadh/2,h/2” means 2 tokens of color nadhnadh and 2 tokens of hh.", "Similar notation is used to specify marking on places, when not present, the place is assumed to be empty of tokens.If the Figure REF Petri Net has the marking: $M_0(mm)=[nadh/2,h/6]$ , $M_0(q)=[e/2]$ , $M_0(cytc)=[e/2]$ , $M_0(is)=[o2/1]$ , then transitions $t1,t3,t4$ are enabled.", "However, either $\\lbrace t1,t3\\rbrace $ or $\\lbrace t4\\rbrace $ can fire simultaneously in a single firing at time 0 due to limited $h$ tokens in $mm$ .", "$t4$ is said to be in conflict with $t1,t3$ ." ], [ "Translating Petri Nets with Colored Tokens to ASP", "In order to represent the Petri Net $PN^C$ with colored tokens, initial marking $M_0$ , and simulation length $k$ , we modify our encoding in Section REF to add a new color parameter to all rules and facts containing token counts in them.", "We keep rules $f\\ref {f:c:place}, f\\ref {f:c:trans}, f\\ref {f:c:time}, f\\ref {f:c:num}, f\\ref {f:rptarc:elim}$ remain as they were for basic Petri Nets.", "We add a new rule $f\\ref {f:c:col}$ for possible set of token colors and replace rules $f\\ref {f:r:ptarc}, f\\ref {f:r:tparc}, f\\ref {f:rptarc}, f\\ref {f:iptarc}, f\\ref {f:tptarc}, i\\ref {i:holds}$ with $f\\ref {f:c:ptarc}, f\\ref {f:c:tparc}, f\\ref {f:c:rptarc}, f\\ref {f:c:iptarc}, f\\ref {f:c:tptarc}, i\\ref {i:c:init}$ to add the color parameter as follows: f13: Facts col($c_k$ ) where $c_k \\in C$ is a color.", "f14: Rules ptarc($p_i,t_j,n_c,c,ts_k$ ) :- time($ts_k$ ).", "for each $(p_i,t_j) \\in E^-$ , $c \\in C$ , $n_c=m_{W(p_i,t_j)}(c) : n_c > 0$ .The time parameter $ts_k$ allows us to capture reset arcs, which consume tokens equal to the current (time-step based) marking of their source nodes.", "f15: Rules tparc($t_i,p_j,n_c,c,ts_k$ ) :- time($ts_k$ ).", "for each $(t_i,p_j) \\in E^+$ , $c \\in C$ , $n_c=m_{W(t_i,p_j)}(c) : n_c > 0$ .", "f16: Rules ptarc($p_i,t_j,n_c,c,ts_k$ ) :- holds($p_i,n_c,c,ts_k$ ), num($n_c$ ), $n_c$ >0, time($ts_k$ ).", "for each $(p_i,t_j):$ $p_i \\in R(t_j)$ , $c \\in C$ , $n_c=m_{M_k(p_i)}(c)$ .", "f17: Rules iptarc($p_i,t_j,1,c,ts_k$ ) :- time($ts_k$ ).", "for each $(p_i,t_j): p_i \\in I(t_j)$ , $c \\in C$ .", "f18: Rules tptarc($p_i,t_j,n_c,c,ts_k$ ) :- time($ts_k$ ).", "for each $(p_i,t_j) \\in Q$ , $c \\in C$ , $n_c=m_{QW(p_i,t_j)}(c) : n_c > 0 $ .", "i2: Facts holds($p_i,n_c,c,0$ ).", "for each place $p_i \\in P, c \\in C, n_c=m_{M_0(p_i)}(c)$ .", "Next, we encode Petri Net's execution behavior, which proceeds in discrete time steps.", "Rules $e\\ref {e:r:ne:ptarc}, e\\ref {e:ne:iptarc}, e\\ref {e:ne:tptarc}, e\\ref {e:enabled}$ are replaced by $e\\ref {e:c:ne:ptarc}, e\\ref {e:c:ne:iptarc}, e\\ref {e:c:ne:tptarc}, e\\ref {e:c:enabled}$ .", "For a transition $t_i$ to be enabled, it must satisfy the following conditions: [(i)] $\\nexists p_j \\in \\bullet t_i : M(p_j) < W(p_j,t_i)$ , $\\nexists p_j \\in I(t_i) : M(p_j) > 0$ , and $\\nexists (p_j,t_i) \\in Q : M(p_j) < QW(p_j,t_i)$ .", "These three conditions are encoded as $e\\ref {e:c:ne:ptarc},e\\ref {e:c:ne:iptarc},e\\ref {e:c:ne:tptarc}$ , respectively and we encode the absence of any of these conditions for a transition as $e\\ref {e:c:enabled}$ : e6: notenabled(T,TS) :- ptarc(P,T, N,C,TS), holds(P,Q,C,TS), place(P), trans(T), time(TS), num(N), num(Q), col(C), Q<N.", "e7: notenabled(T,TS) :- iptarc(P,T,N,C,TS), holds(P,Q,C,TS), place(P), trans(T), time(TS), num(N), num(Q), col(C), Q>=N.", "e8: notenabled(T,TS) :- tptarc(P,T,N,C,TS), holds(P,Q,C,TS), place(P), trans(T), time(TS), num(N), num(Q), col(C), Q<N.", "e9: enabled(T,TS) :- trans(T), time(TS), not notenabled(T,TS).", "Rule $e\\ref {e:c:ne:ptarc}$ captures the existence of an input place $P$ with insufficient number of tokens for transition $T$ to fire.", "Rule $e\\ref {e:c:ne:iptarc}$ captures existence of a non-empty source place $P$ of an inhibitor arc to $T$ preventing $T$ from firing.", "Rule $e\\ref {e:c:ne:tptarc}$ captures existence of a source place $P$ with less than arc-weight tokens required by the read arc to transition $T$ for $T$ to be enabled.", "The, holds(P,Q,C,TS) predicate captures the marking of place $P$ at time $TS$ as $Q$ tokens of color $C$ .", "Rule $e\\ref {e:c:enabled}$ captures enabling of transition $T$ when no reason for it to be not enabled is determined by $e\\ref {e:c:ne:ptarc},e\\ref {e:c:ne:iptarc},e\\ref {e:c:ne:tptarc}$ .", "In a biological context, this enabling is equivalent to a reaction's pre-conditions being satisfied.", "A reaction can proceed when its input substances are available in the required quantities, it is not inhibited, and any required activation quantity of activating substances is available.", "Any subset of enabled transitions can fire simultaneously at a given time-step.", "We select a subset of fireable transitions using the choice rule $a\\ref {a:c:fires}$ The choice rule $a\\ref {a:c:fires}$ either picks an enabled transition $T$ for firing at time $TS$ or not.", "The combined effect over all transitions is to pick a subset of enabled transitions to fire.", "Rule $f\\ref {f:c:rptarc:elim}$ ensures that enabled reset-transitions will be a part of this firing set.", "Whether these transitions are in conflict are checked by later rules $a\\ref {a:c:overc:place},a\\ref {a:c:overc:gen},a\\ref {a:c:overc:elim}$ .", "In a biological context, the multiple firing models parallel processes occurring simultaneously.", "The marking is updated according to the firing set using rules $r\\ref {r:c:add}, r\\ref {r:c:del}, r\\ref {r:c:totincr}, r\\ref {r:c:totdecr}, r\\ref {r:c:nextstate}$ which replaced $r\\ref {r:r:add}, r\\ref {r:r:del}, r\\ref {r:totincr}, r\\ref {r:totdecr}, r\\ref {r:nextstate}$ as follows: r8: add(P,Q,T,C,TS) :- fires(T,TS), tparc(T,P,Q,C,TS), time(TS).", "r9: del(P,Q,T,C,TS) :- fires(T,TS), ptarc(P,T,Q,C,TS), time(TS).", "r10: tot_incr(P,QQ,C,TS) :- col(C), QQ = #sum[add(P,Q,T,C,TS) = Q : num(Q) : trans(T)], time(TS), num(QQ), place(P).", "r11: tot_decr(P,QQ,C,TS) :- col(C), QQ = #sum[del(P,Q,T,C,TS) = Q : num(Q) : trans(T)], time(TS), num(QQ), place(P).", "r12: holds(P,Q,C,TS+1):-place(P),num(Q;Q1;Q2;Q3),time(TS),time(TS+1),col(C), holds(P,Q1,C,TS), tot_incr(P,Q2,C,TS), tot_decr(P,Q3,C,TS), Q=Q1+Q2-Q3.", "Rules $r\\ref {r:c:add}$ and $r\\ref {r:c:del}$ capture that $Q$ tokens of color $C$ will be added or removed to/from place $P$ due to firing of transition $T$ at the respective time-step $TS$ .", "Rules $r\\ref {r:c:totincr}$ and $r\\ref {r:c:totdecr}$ aggregate these tokens for each $C$ for each place $P$ (using aggregate assignment QQ = #sum[...]) at the respective time-step $TS$ .", "Rule $r\\ref {r:c:nextstate}$ uses the aggregates to compute the next marking of $P$ for color $C$ at the time-step ($TS+1$ ) by subtracting removed tokens and adding added tokens to the current marking.", "In a biological context, this captures the effect of a process / reaction, which consumes its inputs and produces outputs for the downstream processes.", "We capture token overconsumption using the rules $a\\ref {a:c:overc:place}, a\\ref {a:c:overc:gen}, a\\ref {a:c:overc:elim}$ of which $a\\ref {a:c:overc:place}$ is a colored replacement for $a\\ref {a:overc:place}$ and is encoded as follows: a9: consumesmore(P,TS) :- holds(P,Q,C,TS), tot_decr(P,Q1,C,TS), Q1 > Q.", "Rule $a\\ref {a:c:overc:place}$ determines whether firing set selected by $a\\ref {a:c:fires}$ will cause overconsumption of tokens at $P$ at time $TS$ by comparing available tokens to aggregate tokens removed as determined by $r\\ref {r:c:totdecr}$ .", "Rule $a\\ref {a:c:overc:gen}$ generalizes the notion of overconsumption, while rule $a\\ref {a:c:overc:elim}$ eliminates answer with such overconsumption.", "In a biological context, conflict (through overconsumption) models the limitation of input substances, which dictate which downstream processes can occur simultaneously.", "We remove rules $a\\ref {a:maxfire:cnh}, a\\ref {a:maxfire:elim}$ from previous encoding to get the set firing semantics.", "Now, we extend the definition (REF ) of $\\text{1-1}$ correspondence between the execution sequence of Petri Net and the answer-sets of its ASP encoding to Petri Nets with colored tokens as follows.", "Definition 38 Given a Petri Net $PN$ with colored tokens, its initial marking $M_0$ and its encoding $\\Pi (PN,M_0,k,ntok)$ for $k$ -steps and maximum $ntok$ tokens at any place.", "We say that there is a 1-1 correspondence between the answer sets of $\\Pi (PN,M_0,k,ntok)$ and the execution sequences of $PN$ iff for each answer set $A$ of $\\Pi (PN,M_0,k,ntok)$ , there is a corresponding execution sequence $X=M_0,T_0,M_1,\\dots ,M_k,T_k,M_{k+1}$ of $PN$ and for each execution sequence $X$ of $PN$ there is an answer-set $A$ of $\\Pi (PN,M_0,$ $k,ntok)$ such that $\\lbrace fires(t,ts) : t \\in T_{ts}, 0 \\le ts \\le k \\rbrace = \\lbrace fires(t,ts) : fires(t,ts) \\in A \\rbrace $ $\\begin{split}\\lbrace holds(p,q,c,ts) &: p \\in P, c/q=M_{ts}(p), 0 \\le ts \\le k+1 \\rbrace \\\\&= \\lbrace holds(p,q,c,ts) : holds(p,q,c,ts) \\in A\\rbrace \\end{split}$ Proposition 6 There is 1-1 correspondence between the answer sets of $\\Pi ^5(PN^C,M_0,$ $k,ntok)$ and the execution sequences of $PN$ .", "To add maximal firing semantics, we add $a\\ref {a:c:maxfire:elim}$ as it is and replace $a\\ref {a:maxfire:cnh}$ with $a\\ref {a:c:maxfire:cnh}$ as follows: a10: could_not_have(T,TS):-enabled(T,TS),not fires(T,TS), ptarc(S,T,Q,C,TS), holds(S,QQ,C,TS), tot_decr(S,QQQ,C,TS), Q > QQ - QQQ.", "Rule $a\\ref {a:c:maxfire:cnh}$ captures the fact that transition $T$ , though enabled, could not have fired at $TS$ , as its firing would have caused overconsumption.", "Rule $a\\ref {a:c:maxfire:elim}$ eliminates any answers where an enabled transition could have fired without causing overconsumption but did not.", "This modification reduces the number of answers produced for the Petri Net in Figure REF to 4.", "We can encode other firing semantics with similar easeFor example, if interleaved semantics is desired, rules $a\\ref {a:c:maxfire:cnh},a\\ref {a:c:maxfire:elim}$ can changed to capture and eliminate answer-sets in which more than one transition fires in a firing set as: aREF ': more_than_one_fires :- fires(T1,TS),fires(T2,TS),T1!=T2,time(TS).", "aREF ': :-more_than_one_fires.", ".", "We now look at how additional extensions can be easily encoded by making small code changes.", "Extension - Priority Transitions Priority transitions enable ordering of Petri Net transitions, favoring high priority transitions over lower priority ones [9].", "In a biological context, this is used to model primary (or dominant) vs. secondary pathways / processes in a biological system.", "This prioritization may be due to an intervention (such as prioritizing elimination of a metabolite over recycling it).", "Definition 39 (Priority Colored Petri Net) A Priority Colored Petri Net with reset, inhibit, and read arcs is a tuple $PN^{pri} = (P,T,E,C,W,R,I,Q,QW,Z)$ , where: $P,T,E,C,W,R,I,Q,QW$ are the same as for $PN^C$ , and $Z : T \\rightarrow {N}$ is a priority function that assigns priorities to transitions.", "Lower number signifies higher priority.", "Definition 40 (Enabled Transition) A transition $t_i$ is enabled in $PN^{pri}$ w.r.t.", "a marking $M$ (prenabled$_{M}(t)$ ) if it would be enabled in $PN^C$ w.r.t.", "$M$ and there isn't another transition $t_j$ that would be enabled in $PN^C$ (with respect to M) s.t.", "$Z(t_j) < Z(t_i)$ .", "Definition 41 (Firing Set) A firing set is a set $T_k=\\lbrace t_{k_1},\\dots ,t_{k_n}\\rbrace \\subseteq T$ of simultaneously firing transitions of $PN^{pri}$ that are priority enabled and do not conflict w.r.t.", "to the current marking $M_k$ of $PN$ .", "A set $T_k$ is not a firing set if there is an priority enabled reset-transition that is not in $T_k$ , i.e.", "$\\exists t : prenabled_{M_k}(t), R(t) \\ne \\emptyset , t \\notin T_k$ .", "See footnote REF We add the following facts and rules to encode transition priority and enabled priority transitions: f19: Facts transpr($t_i$ ,$pr_i$ ) where $pr_i = Z(t_i)$ is $t_i^{\\prime }s$ priority.", "a11: notprenabled(T,TS) :- enabled(T,TS), transpr(T,P), enabled(TT,TS), transpr(TT,PP), PP < P. a12: prenabled(T,TS) :- enabled(T,TS), not notprenabled(T,TS).", "Rule $a\\ref {a:c:prne}$ captures that an enabled transition $T$ is not priority-enabled, if there is another enabled transition with higher priority at $TS$ .", "Rule $a\\ref {a:c:prenabled}$ captures that transition $T$ is priority-enabled at $TS$ since there is no enabled transition with higher priority.", "We replace rules $a\\ref {a:c:fires},f\\ref {f:c:rptarc:elim},a\\ref {a:c:maxfire:cnh},a\\ref {a:c:maxfire:elim}$ with $a\\ref {a:c:prfires},f\\ref {f:c:pr:rptarc:elim}, a\\ref {a:c:prmaxfire:cnh},a\\ref {a:c:prmaxfire:elim}$ respectively to propagate priority as follows: a13: {fires(T,TS)} :- prenabled(T,TS), trans(T), time(TS).", "f20: Rules :- prenabled($t_j,ts_k$ ),not fires($t_j,ts_k$ ), time($ts_k$ ).", "for each transition $t_j$ with an incoming reset arc.", "a14: could_not_have(T,TS) :- prenabled(T,TS), not fires(T,TS), ptarc(S,T,Q,C,TS), holds(S,QQ,C,TS), tot_decr(S,QQQ,C,TS), Q > QQ - QQQ.", "a15: :- not could_not_have(T,TS), time(TS), prenabled(T,TS), not fires(T,TS), trans(T).", "Rules $a\\ref {a:c:prfires},f\\ref {f:c:rptarc:elim},a\\ref {a:c:prmaxfire:cnh},a\\ref {a:c:prmaxfire:elim}$ perform the same function as $a\\ref {a:c:fires},f\\ref {f:c:pr:rptarc:elim},a\\ref {a:c:maxfire:cnh},a\\ref {a:c:maxfire:elim}$ , except that they consider only priority-enabled transitions as compared all enabled transitions.", "Proposition 7 There is 1-1 correspondence between the answer sets of $\\Pi ^6(PN^{pri},M_0,$ $k,ntok)$ and the execution sequences of $PN^{pri}$ .", "Extension - Timed Transitions Biological processes vary in time required for them to complete.", "Timed transitions [63] model this variation of duration.", "The timed transitions can be reentrant or non-reentrantA reentrant transition is like a vehicle assembly line, which accepts new parts while working on multiple vehicles at various stages of completion; whereas a non-reentrant transition only accepts new input when the current processing is finished.. We extend our encoding to allow reentrant timed transitions.", "Figure: An extended version of the Petri Net model from Fig. .", "The new transitions tq,tcytctq,tcytc have a duration of 2 each (shown in square brackets (“[ ]”) next to the transition).", "When missing, transition duration is assumed to be 1.Definition 42 (Priority Colored Petri Net with Timed Transitions) A Priority Colored Petri Net with Timed Transitions, reset, inhibit, and query arcs is a tuple $PN^D=(P,T,E,C,W,R,I,Q,QW,Z,D)$ , where $P,T,E,C,W,R,I,Q,QW,Z$ are the same as for $PN^{pri}$ , and $D : T \\rightarrow {N} \\setminus \\lbrace 0\\rbrace $ is a duration function that assigns positive integer durations to transitions.", "Figure REF shows an extended version of Petri Net model of the Electron Transport Chain [64] shown in Figure REF .", "The new transitions $tq$ and $tcytc$ (shown in dotted outline) are timed transitions modeling the speed of the small carrier molecules, Coenzyme Q ($q$ ) and Cytochrome C ($cytc$ ) as an effect of membrane fluidity.", "Higher numbers for transition duration represent slower movement of the carrier molecules due to lower fluidity.", "Definition 43 (Transition Execution) A transition $t$ in $PN^D$ consumes tokens from its input places and reset places immediately, while it produces tokens in its output places at the end of transition duration $D(t)$ , as follows: $\\forall p \\in \\bullet t, M_{k+1}(p) = M_k(p) - W(p,t)$ $\\forall p \\in t\\bullet , M_{k+D(t)}(p) = M_{k+D(t)-1}(p) + W(p,t)$ $\\forall p \\in R(t), M_{k+1}(p) = M_k(p) - M_k(p)$ Execution in $PN^D$ changes, since the token update from $M_k$ to $M_{k+1}$ can involve transitions that started at some time $l$ before time $k$ , but finish at $k+1$ .", "Definition 44 (Firing Set Execution) New marking due to firing set execution is computed as follows: $\\forall p \\in P \\setminus R(T_k), M_{k+1}(p) = M_k(p)- \\sum _{\\begin{array}{c}t \\in T_k, p \\in \\bullet t\\end{array}} W(p,t)+ \\sum _{\\begin{array}{c}t \\in T_l, p \\in t \\bullet : 0 \\le l \\le k, l+D(t) = k+1\\end{array}} W(t,p)$ $\\forall p \\in R(T_k), M_{k+1}(p) = \\sum _{\\begin{array}{c}t \\in T_l , p \\in t \\bullet : l \\le k, l+D(t) = k+1 \\end{array}} W(t,p)$ where $R(T_i)=\\displaystyle \\cup _{\\begin{array}{c}t \\in T_i\\end{array}} R(t)$ .", "A timed transition $t$ produces its output $D(t)$ time units after being fired.", "We replace $f\\ref {f:c:tparc}$ with $f\\ref {f:c:dur:tparc}$ adding transition duration and replace rule $r\\ref {r:c:add}$ with $r\\ref {r:c:dur:add}$ that produces tokens at the end of transition duration: f21: Rules tparc($t_i,p_j,n_c,c,ts_k,D(t_i)$ ):-time($ts_k$ ).", "for each $(t_i,p_j) \\in E^+$ , $c \\in C$ , $n_c=m_{W(t_i,p_j)}(c) : n_c > 0$ .", "r13: add(P,Q,T,C,TS):-fires(T,TS0),time(TS0;TS), tparc(T,P,Q,C,TS0,D), TS=TS0+D-1.", "Proposition 8 There is 1-1 correspondence between the answer sets of $\\Pi ^7(PN^D,M_0,$ $k,ntok)$ and the execution sequences of $PN^D$ .", "Above implementation of timed-transition is reentrant, however, we can easily make these timed transitions non-reentrant by adding rule $e\\ref {e:c:ne:dur}$ that disallows a transition from being enabled if it is already in progress: e10: notenabled(T,TS):-fires(T,TS0), num(N), TS>TS0, tparc(T,P,N,C,TS0,D), col(C), time(TS0), time(TS), TS<(TS0+D).", "Other Extensions Other Petri Net extensions can be implemented with similar ease.", "For example, Guard Conditions on transitions can be trivially implemented as a notenabled/2 rules.", "Self Modifying Petri Nets [77], which allow marking-dependent arc-weights can be implemented in a similar manner as the Reset Arc extension in section REF .", "Object Petri Nets [78], in which each token is a Petri Net itself can be implemented (using token reference semantics) by adding an additional “network-id” parameter to our encoding, where “id=0” is reserved for system net and higher numbers are used for token nets.", "Transition coordination between system & token nets is enforced through constraints on transition labels, where transition labels are added as additional facts about transitions.", "Related Work Petri Nets have been previously encoded in ASP, but the previous implementations have been limited to restricted classes of Petri Nets.", "For example, 1-safe Petri Net to ASP translation has been presented in [37], which is limited to binary Petri Nets.", "Translation of Logic Petri Nets to ASP has been presented in [6], but their model cannot handle numerical aggregation of tokens from multiple input transitions to the same place.", "Our work focused on problems in the biological domain and is more generalized.", "We can represent reset arcs, inhibition arcs, priority arcs as well as durative transitions.", "Conclusion We have presented an encoding of basic Petri Nets in ASP and showed how it can be easily extended to include extension to model various biological constructs.", "Portions of this work were published in [2] and [1].", "In the next chapter we will use Petri Nets and their ASP encoding to model biological pathways to answer questions about them.", "Answering Questions using Petri Nets and ASP Introduction In this chapter we use various Petri Net extensions presented in Chapter and their ASP encoding to answer question from [64] that were a part of the Second Deep Knowledge Representation Challengehttps://sites.google.com/site/2nddeepkrchallenge/.", "Definition 45 (Rate) Rate of product P is defined as the quantity of P produced per unit-time.", "Rate of an action A is defined as the number of time A occurs per unit-time.", "Comparing Altered Trajectories due to Reset Intervention Question 1 At one point in the process of glycolysis, both dihydroxyacetone phosphate (DHAP) and glyceraldehyde 3-phosphate (G3P) are produced.", "Isomerase catalyzes the reversible conversion between these two isomers.", "The conversion of DHAP to G3P never reaches equilibrium and G3P is used in the next step of glycolysis.", "What would happen to the rate of glycolysis if DHAP were removed from the process of glycolysis as quickly as it was produced?", "Provided Answer: “Glycolysis is likely to stop, or at least slow it down.", "The conversion of the two isomers is reversible, and the removal of DHAP will cause the reaction to shift in that direction so more G3P is converted to DHAP.", "If less (or no) G3P were available, the conversion of G3P into DHAP would slow down (or be unable to occur).” Solution 1 The process of glycolysis is shown in Fig 9.9 of Campbell's book.", "Glycolysis splits Glucose into Pyruvate.", "In the process it produces ATP and NADH.", "Any one of these can be used to gauge the glycolysis rate, since they will be produced in proportion to the input Glucose.", "The amount of pyruvate produced is the best choice since it is the direct end product of glycolysis.", "The ratio of the quantity of pyruvate produced over a representative span of time gives us the glycolysis rate.", "We assume a steady supply of Glucose is available and also assume that sufficient quantity of various enzymes used in glycolysis is available, since the question does not place any restriction on these substances.", "We narrow our focus to a subsection from Fructose 1,6-bisphosphate (F16BP) to 1,3-Bisphosphoglycerate (BPG13) as shown in Figure REF since that is the part the question is concerned with.", "We can ignore the linear chain up-stream of F16BP as well as the linear chain down-stream of BPG13 since the amount of F16BP available will be equal to Glucose and the amount of BPG13 will be equal to the amount of Pyruvate given our steady supply assumption.", "Figure: Petri Net graph relevant to question .", "“f16bp” is the compound Fructose 1,6-biphosphate, “bpg13” is 1,3-Bisphosphoglycerate.", "Transition trtr shown in dotted lines is added to model the elimination of dhapdhap as soon as it is produced.We fulfill the steady supply requirement of Glucose by a source transition-node $t3$ .", "We fulfill sufficient enzyme supply by a fixed quantity for each enzyme such that this quantity is in excess of what can be consumed during our simulation interval.", "Where the simulation interval is the number of time-steps over which we will measure the rate of glycolysis.", "We model the elimination of DHAP as soon as it is produced with a reset arc, shown with a dotted style in Figures REF .", "Such an arc removes all tokens from its source place when it fires.", "Since we have added it as an unconditional arc, it is always enabled for firing.", "We encode both situations in ASP with the maximal firing set policy.", "Both situations (without and with reset arc) are encoded in ASP and run for 10 steps.", "At the end of those 10 steps the amount of BPG13 is compared to determine the difference in the rate of glycolysis.", "In normal situation (without $(dhap,tr)$ reset arc), unique quantities of “bpg13” from all (2) answer-sets after 10 steps were as follows: holds(bpg13,14,10) holds(bpg13,16,10) with reset arc $tr$ , unique quantities of “bpg13” from all (512) answer-sets after 10 steps were as follows: holds(bpg13,0,10) holds(bpg13,10,10) holds(bpg13,12,10) holds(bpg13,14,10) holds(bpg13,16,10) holds(bpg13,2,10) holds(bpg13,4,10) holds(bpg13,6,10) holds(bpg13,8,10) Figure: Amount of “bpg13” produced in unique answer-sets produced by a 10 step simulation.", "The graph shows two situations, without the (dhap,tr)(dhap,tr) reset arc (normal situation) and with the reset arc (abnormal situation).", "The purpose of this graph is to depict the variation in the amounts of glycolysis produced in various answer sets.Note that the rate of glycolysis is generally lower when DHAP is immediately consumed.", "It is as low as zero essentially stopping glycolysis.", "The range of values are due to the choice between G3P being converted to DHAP or BPG13.", "If more G3P is converted to DHAP, then less BPG13 is produced and vice versa.", "Also, note that if G3P is not converted to BPG13, no NADH or ATP is produced either due to the liner chain from G3P to Pyruvate.", "The unique quantities of BPG13 are shown in a graphical format in Figure REF , while a trend of average quantity of BPG13 produced is shown in Figure REF .", "Figure: Average amount of “bpg13” produced during the 10-step simulation at various time steps.", "The average is over all answer-sets.", "The graph shows two situations, without the (dhap,tr)(dhap,tr) reset arc (normal situation) and with the reset arc (abnormal situation).", "The divergence in “bpg13” production is clearly shown.We created a minimal model of the Petri Net in Figure REF by removing enzymes and reactants that were not relevant to the question and did not contribute to the estimation of glycolysis.", "This is shown in Figure REF .", "Figure: Minimal version of the Petri Net graph in Figure .", "All reactants that do not contribute to the estimation of the rate of glycolysis have been removed.Simulating it for 10 steps with the same initial marking as the Petri Net in Figure REF produced the same results as for Figure REF .", "Determining Conditions Leading to an Observation Question 2 When and how does the body switch to B oxidation versus glycolysis as the major way of burning fuel?", "Provided Answer: “The relative volumes of the raw materials for B oxidation and glycolysis indicate which of these two processes will occur.", "Glycolysis uses the raw material glucose, and B oxidation uses Acyl CoA from fatty acids.", "When the blood sugar level decreases below its homeostatic level, then B oxidation will occur with available fatty acids.", "If no fatty acids are immediately available, glucagon and other hormones regulate release of stored sugar and fat, or even catabolism of proteins and nucleic acids, to be used as energy sources.” Solution 2 The answer provided requires background knowledge about the mechanism that regulates which source of energy will be used.", "This information is not presented in Chapter 9 of Campbell's book, which is the source material of this exercise.", "However, we can model it based on background information combined with Figure 9.19 of Campbell's book.", "Our model is presented in Figure REFWe can extend this model by adding expressions to inhibition arcs that compare available substances..", "Figure: Petri Net graph relevant to question .", "“fats” are fats, “prot” are proteins, “fac” are fatty acids, “sug” are sugars, “amin” are amino acids and “acoa” is ACoA.", "Transition “box” is the beta oxidation, “t5” is glycolysis, “t1” is fat digestion into fatty acids, and “t9” is protein deamination.We can test the model by simulating it for a time period and testing whether beta oxidation (box) is started when sugar (sug) is finished.", "We do not need a steady supply of sugar in this case, just enough to be consumed in a few time steps to capture the switch over.", "Fats and proteins may or may not modeled as a steady supply, since their bioavailability is dependent upon a number of external factors.", "We assume a steady supply of both and model it with large enough initial quantity that will last beyond the simulation period.", "We translate the petri net model into ASP and run it for 10 iterations.", "Following are the results: holds(acoa,0,0) holds(amin,0,0) holds(fac,0,0) holds(fats,5,0) holds(prot,3,0) holds(sug,4,0) fires(t1,0) fires(t5,0) fires(t9,0) holds(acoa,1,1) holds(amin,1,1) holds(fac,1,1) holds(fats,4,1) holds(prot,3,1) holds(sug,3,1) fires(t5,1) holds(acoa,2,2) holds(amin,1,2) holds(fac,1,2) holds(fats,4,2) holds(prot,3,2) holds(sug,2,2) fires(t5,2) holds(acoa,3,3) holds(amin,1,3) holds(fac,1,3) holds(fats,4,3) holds(prot,3,3) holds(sug,1,3) fires(t5,3) holds(acoa,4,4) holds(amin,1,4) holds(fac,1,4) holds(fats,4,4) holds(prot,3,4) holds(sug,0,4) fires(box,4) holds(acoa,5,5) holds(amin,1,5) holds(fac,0,5) holds(fats,4,5) holds(prot,3,5) holds(sug,0,5) fires(t1,5) fires(t9,5) holds(acoa,5,6) holds(amin,2,6) holds(fac,1,6) holds(fats,3,6) holds(prot,3,6) holds(sug,0,6) fires(box,6)c holds(acoa,6,7) holds(amin,2,7) holds(fac,0,7) holds(fats,3,7) holds(prot,3,7) holds(sug,0,7) fires(t1,7) fires(t9,7) holds(acoa,6,8) holds(amin,3,8) holds(fac,1,8) holds(fats,2,8) holds(prot,3,8) holds(sug,0,8) fires(box,8) holds(acoa,7,9) holds(amin,3,9) holds(fac,0,9) holds(fats,2,9) holds(prot,3,9) holds(sug,0,9) fires(t1,9) fires(t9,9) holds(acoa,7,10) holds(amin,4,10) holds(fac,1,10) holds(fats,1,10) holds(prot,3,10) holds(sug,0,10) fires(box,10) We can see that by time-step 4, the sugar supply is depleted and beta oxidation starts occurring.", "Comparing Altered Trajectories due to Accumulation Intervention Question 3 ATP is accumulating in the cell.", "What affect would this have on the rate of glycolysis?", "Explain.", "Provided Answer: “ATP and AMP regulate the activity of phosphofructokinase.", "When there is an abundance of AMP in the cell, this indicates that the rate of ATP consumption is high.", "The cell is in need for more ATP.", "If ATP is accumulating in the cell, this indicates that the cell's demand for ATP had decreased.", "The cell can decrease its production of ATP.", "Therefore, the rate of glycolysis will decrease.” Solution 3 Control of cellular respiration is summarized in Fig 9.20 of Campbell's book.", "We can gauge the rate of glycolysis by the amount of Pyruvate produced, which is the end product of glycolysis.", "We assume a steady supply of glucose is available.", "Its availability is not impacted by any of the feedback mechanism depicted in Fig 9.20 of Campbell's book or restricted by the question.", "We can ignore the respiration steps after glycolysis, since they are directly dependent upon the end product of glycolysis, i.e.", "Pyruvate.", "These steps only reinforce the negative effect of ATP.", "The Citrate feed-back shown in Campbell's Fig 9.20 is also not relevant to the question, so we can assume a constant level of it and leave it out of the picture.", "Another simplification that we do is to treat the inhibition of Phosphofructokinase (PFK) by ATP as the inhibition of glycolysis itself.", "This is justified, since PFK is on a linear path from Glucose to Fructose 1,6-bisphosphate (F16BP), and all downstream product quantities are directly dependent upon the amount of F16BP (as shown in Campbell's Fig 9.9), given steady supply of substances involved in glycolysis.", "Our assumption also applies to ATP consumed in Fig 9.9.", "Our simplified picture is shown in Figure REF as a Petri Net.", "Figure: Petri Net graph relevant to question .", "“glu” is Glucose, “pyr” is Pyruvate.", "Transitions “gly1” represents glycolysis and “cw1” is cellular work that consumes ATP and produces AMP.", "Transition “gly1” is inhibited only when the number of atpatp tokens is greater than 4.We model cellular work that recycles ATP to AMP (see p/181 of Campbell's book) by the $cw1$ transition, shown in dotted style.", "In normal circumstances, this arc does not let ATP to collect.", "If we reduce the arc-weights incident on $cw1$ to 1, we get the situation where less work is being done and some ATP will collect, as a result glycolysis will pause and resume.", "If we remove $cw1$ (representing no cellular work), ATP will start accumulating and glycolysis will stop.", "We use an arbitrary arc-weight of 4 on the inhibition arc $(atp,gly1)$ to model an elevated level of ATP beyond normal that would cause inhibition An alternate modeling would be compare the number of tokens on the $amp$ node and the $atp$ node and set a level-threshold that inhibits $gly1$ .", "Such technique is common in colored-peri nets.. We encode all three situations in ASP with maximal firing set policy.", "We run them for 10 steps and compare the quantity of pyruvate produced to determine the difference in the rate of glycolysis.", "In normal situation when cellular work is being performed ($cw1$ arc is present), unique quantities of “pyr” after 10 step are as follows: holds(pyr,20,10) when the cellular work is reduced, i.e.", "($(atp,cw1)$ , $(cw1,amp)$ arc weights changed to 1), unique quantities of “pyr” after 10 steps are as follows: holds(pyr,14,10) with no cellular work ($cw1$ arc removed), unique quantities of “pyr” after 10 steps are as follows: holds(pyr,6,10) The results show the rate of glycolysis reducing as the cellular work decreases to the point where it stops once ATP reaches the inhibition threshold.", "Higher numbers of ATP produced in later steps of cellular respiration will reinforce this inhibition even more quickly.", "Trend of answers from various runs is shown in Figure REF .", "Figure: Amount of pyruvate produced from various lengths of runs.", "Comparing Altered Trajectories due to Initial Value Intervention Question 4 A muscle cell had used up its supply of oxygen and ATP.", "Explain what affect would this have on the rate of cellular respiration and glycolysis?", "Provided Answer: “Oxygen is needed for cellular respiration to occur.", "Therefore, cellular respiration would stop.", "The cell would generate ATP by glycolysis only.", "Decrease in the concentration of ATP in the cell would stimulate an increased rate of glycolysis in order to produce more ATP.” Solution 4 Figure 9.18 of Campbell's book gives the general idea of what happens when oxygen is not present.", "Figure 9.20 of Campbell's book shows the control of glycolysis by ATP.", "To formulate the answer, we need pieces from both.", "ATP inhibits Phosphofructokinase (Fig 9.20 of Campbell), which is an enzyme used in glycolysis.", "No ATP means that enzyme is no longer inhibited and glycolysis can proceed at full throttle.", "Pyruvate either goes through aerobic respiration when oxygen is present or it goes through fermentation when oxygen is absent (Fig 9.18 of Campbell).", "We can monitor the rate of glycolysis and cellular respiration by observing these operations occurring (by looking at corresponding transition firing) over a simulation time period.", "Our simplified Petri Net model is shown in Figure REF .", "We ignore the details of processes following glycolysis, except that these steps produce additional ATP.", "We do not need an exact number of ATP produced as long as we keep it higher than the ATP produced by glycolysis.", "Higher numbers will just have a higher negative feed-back (or inhibition) effect on glycolysis.", "We ignore citrate's inhibition of glycolysis since that is not relevant to the question and since it gets recycled by the citric acid cycle (see Fig 9.12 of Campbell).", "We also ignore AMP, since it is not relevant to the question, by assuming sufficient supply to maintain glycolysis.", "We also assume continuous cellular work consuming ATP, without that ATP will accumulate almost immediately and stop glycolysis.", "We assume a steady supply of glucose is available to carry out glycolysis and fulfill this requirement by having a quantity in excess of the consumption during our simulation interval.", "All other substances participating in glycolysis are assumed to be available in a steady supply so that glycolysis can continue.", "Figure: Petri Net graph relevant to question .", "“glu” is Glucose, “pyr” is Pyruvate, “atp” is ATP, “eth” is ethenol or other products of fermentation, and “o2” is Oxygen.", "Transitions “gly1” represents glycolysis, “res1” is respiration in presence of oxygen, “fer1” is fermentation when no oxygen is present, and “cw1” is cellular work that consumes ATP.", "Transition “gly1” is inhibited only when the number of atpatp tokens is greater than 4.We then consider two scenarios, one where oxygen is present and where oxygen is absent and determine the change in rate of glycolysis and respiration by counting the firings of their respective transitions.", "We encode both situations in ASP with maximal firing set policy.", "Both situations are executed for 10 steps.", "At the end of those steps the firing count of “gly1” and “res1” is computed and compared to determine the difference in the rates of glycolysis and respiration respectively.", "In the normal situation (when oxygen is present), we get the following answer sets: fires(gly1,0) fires(cw1,1) fires(gly1,1) fires(res1,1) fires(cw1,2) fires(res1,2) fires(cw1,3) fires(cw1,4) fires(cw1,5) fires(gly1,5) fires(cw1,6) fires(gly1,6) fires(res1,6) fires(cw1,7) fires(res1,7) fires(cw1,8) fires(cw1,9) fires(cw1,10) while in the abnormal situation (when oxygen is absent), we get the following firings: fires(gly1,0) fires(cw1,1) fires(fer1,1) fires(gly1,1) fires(cw1,2) fires(fer1,2) fires(gly1,2) fires(cw1,3) fires(fer1,3) fires(gly1,3) fires(cw1,4) fires(fer1,4) fires(gly1,4) fires(cw1,5) fires(fer1,5) fires(gly1,5) fires(cw1,6) fires(fer1,6) fires(gly1,6) fires(cw1,7) fires(fer1,7) fires(gly1,7) fires(cw1,8) fires(fer1,8) fires(gly1,8) fires(cw1,9) fires(fer1,9) fires(gly1,9) fires(cw1,10) fires(fer1,10) fires(gly1,10) Note that the number of firings of glycolysis for normal situation is lower when oxygen is present and higher when oxygen is absent.", "While, the number of firings is zero when no oxygen is present.", "Thus, respiration stops when no oxygen is present and the need of ATP by cellular work is fulfilled by a higher amount of glycolysis.", "Trend from various runs is shown in Figure REF .", "Figure: Firing counts of glycolysis (gly1) and respiration (res1) for different simulation lengths for the petri net in Figure Comparing Altered Trajectories due to Inhibition Intervention Question 5 The final protein complex in the electron transport chain of the mitochondria is non-functional.", "Explain the effect of this on pH of the intermembrane space of the mitochondria.", "Provided Answer: “The H+ ion gradient would gradually decrease and the pH would gradually increase.", "The other proteins in the chain are still able to produce the H+ ion gradient.", "However, a non-functional, final protein in the electron transport chain would mean that oxygen is not shuttling electrons away from the electron transport chain.", "This would cause a backup in the chain, and the other proteins in the electron transport chain would no longer be able to accept electrons and pump H+ ions into the intermembrane space.", "A concentration decrease in the H+ ions means an increase in the pH.” Solution 5 The electron transport chain is shown in Fig 9.15 (1) of Campbell's book.", "In order to explain the effect on pH, we will show the change in the execution of the electron transport chain with both a functioning and non-functioning final protein.", "Since pH depends upon the concentration of H+ ions, we will quantify the difference its quantity in the intermembrane space in both scenarios as well.", "We assume that a steady input of NADH, FADH2, H+ and O2 are available in the mitochondrial matrix.", "We also assume an electron carrying capacity of 2 for both ubiquinone (Q)http://www.benbest.com/nutrceut/CoEnzymeQ.html and cytochrome c (Cyt c).", "This carrying capacity is background information not provided in Campbell's Chapter 9.", "As with previous questions, we fulfill the steady supply requirement of substances by having input quantities in excess of what would be consumed during our simulation interval.", "Figure: Petri Net graph relevant to question .", "“is” is the intermembrane space, “mm” is mitochondrial matrix, “q” is ubiquinone and “cytc” is cytochrome c. The inhibition arcs (q,t1)(q,t1), (q,t2)(q,t2) and (cytc,t3)(cytc,t3) capture the electron carrying capacities of qq and cytccytc.", "Over capacity will cause backup in electron transport chain.", "Tokens are colored, e.g.", "“h:1, nadh:1” specify one token of hh and nadhnadh each.", "Token types are “h” for H+, “nadh” for NADH, “nadp” for NADP, “fadh2” for FADH2, “fad” for FAD, “e” for electrons, “o2” for oxygen and “h2o” for water.", "We remove t4t4 to model non-functioning protein complex IVIV.We model this problem as a colored petri net shown in Figure REF .", "The normal situation is made up of the entire graph.", "The abnormal situation (with non-functional final protein complex) is modeled by removing transition $t4$ from the graphAlternatively, we can model a non-functioning transition by attaching an inhibition arc to it with one token at its source place.", "We encode both situations in ASP with maximal firing set policy.", "Both are run for 10 steps.", "The amount of $h$ (H+) is compared in the $is$ (intermembrane space) to determine change in pH and the firing sequence is compared to explain the effect.", "In normal situation (entire graph), we get the following: fires(t1,0) fires(t2,0) fires(t3,1) fires(t1,2) fires(t2,2) fires(t3,2) fires(t4,2) fires(t3,3) fires(t4,3) fires(t6,3) fires(t1,4) fires(t2,4) fires(t3,4) fires(t4,4) fires(t6,4) fires(t3,5) fires(t4,5) fires(t6,5) fires(t1,6) fires(t2,6) fires(t3,6) fires(t4,6) fires(t6,6) fires(t3,7) fires(t4,7) fires(t6,7) fires(t1,8) fires(t2,8) fires(t3,8) fires(t4,8) fires(t6,8) fires(t3,9) fires(t4,9) fires(t6,9) fires(t1,10) fires(t2,10) fires(t3,10) fires(t4,10) fires(t6,10) holds(is,15,h,10) with $t4$ removed, we get the following: fires(t1,0) fires(t2,0) fires(t3,1) fires(t1,2) fires(t2,2) fires(t3,2) fires(t6,3) fires(t6,4) holds(is,2,h,10) Note that the amount of H+ ($h$ ) produced in the intermembrane space ($is$ ) is much smaller when the final protein complex is non-functional ($t4$ removed).", "Lower H+ translates to higher pH.", "Thus, the pH of intermembrane space will increase as a result of nonfunctional final protein.", "Also, note that the firing of $t3$ , $t1$ and $t2$ responsible for shuttling electrons also stop very quickly when $t4$ no longer removes the electrons ($e$ ) from Cyt c ($cytc$ ) to produce $H_2O$ .", "This is because $cytc$ and $q$ are at their capacity on electrons that they can carry and stop the electron transport chain by inhibiting transitions $t3$ , $t2$ and $t1$ .", "Trend for various runs is shown in Figure REF .", "Figure: Simulation of Petri Net in Figure .", "In a complete model of the biological system, there will be a mechanism that keeps the quantity of H+ in check in the intermembrane space and will plateau at some point.", "Comparing Altered Trajectories due to Gradient Equilization Intervention Question 6 Exposure to a toxin caused the membranes to become permeable to ions.", "In a mitochondrion, how would this affect the pH in the intermembrane space and also ATP production?", "Provided Answer: “The pH of the intermembrane space would decrease as H+ ions diffuse through the membrane, and the H+ ion gradient is lost.", "The H+ gradient is essential in ATP production b/c facilitated diffusion of H+ through ATP synthase drives ATP synthesis.", "Decreasing the pH would lead to a decrease in the rate of diffusion through ATP synthase and therefore a decrease in the production of ATP.” Solution 6 Oxidative phosphorylation is shown in Fig 9.15 of Campbell's book.", "In order to explain the effect on pH in the intermembrane space and the ATP production we will show the change in the amount of H+ ions in the intermembrane space as well as the amount of ATP produced when the inner mitochondrial membrane is impermeable and permeable.", "Note that the concentration of H+ determines the pH.", "we have chosen to simplify the diagram by not having FADH2 in the picture.", "Its removal does not change the response, since it provides an alternate input mechanism to electron transport chain.", "We will assume that a steady input of NADH, H+, O2, ADP and P is available in the mitochondrial matrix.", "We also assume an electron carrying capacity of 2 for both ubiquinone (Q) and cytochrome c (Cyt c).", "We fulfill the steady supply requirement of substances by having input quantities in excess of what would be consumed during our simulation interval.", "We model this problem as a colored petri net shown in Figure REF .", "Transition $t6,t7$ shown in dotted style are added to model the abnormal situationIf reverse permeability is also desired additional arcs may be added from mm to is.", "They capture the diffusion of H+ ions back from Intermembrane Space to the Mitochondrial matrix.", "One or both may be enabled to capture degrees of permeability.", "we have added a condition on the firing of $t5$ (ATP Synthase activation) to enforce gradient to pump ATP Synthase.", "Figure: Petri Net graph relevant to question .", "“is” is the intermembrane space, “mm” is mitochondrial matrix, “q” is ubiquinone and “cytc” is cytochrome c. Tokens are colored, e.g.", "“h:1, nadh:1” specify one token of hh and nadhnadh each.", "Token types are “h” for H+, “nadh” for NADH, “e” for electrons, “o2” for oxygen, “h2o” for water, “atp” for ATP and “adp” for ADP.", "We add t6,t7t6,t7 to model cross domain diffusion from intermembrane space to mitochondrial matrix.", "One or both of t6,t7t6,t7 may be enabled at a time to control the degree of permeability.", "The text above “t5” is an additional condition which must be satisfied for “t5” to be enabled.We encode both situations in ASP with maximal firing set policy.", "Both are run for 10 steps each and the amount of $h$ and $atp$ is compared to determine the effect of pH and ATP production.", "We capture the gradient requirement as the following ASP codeWe can alternatively model this by having a threshold arc from “is” to “t5” if only a minimum trigger quantity is required in the intermembrane space.", ": notenabled(T,TS) :- T==t5, C==h, trans(T), col(C), holds(is,Qis,C,TS), holds(mm,Qmm,C,TS), Qmm+3 > Qis, num(Qis;Qmm), time(TS).", "In the normal situation, we get the following $h$ token distribution after 10 steps: holds(is,11,h,10) holds(mm,1,h,10) holds(mm,6,atp,10) we change the permeability to 1 ($t6$ enabled), we get the following token distribution instead: holds(is,10,h,10) holds(mm,2,h,10) holds(mm,5,atp,10) we change the permeability to 2 ($t6,t7$ enabled), the distribution changes as follows: holds(is,8,h,10) holds(mm,4,h,10) holds(mm,2,atp,10) Note that as the permeability increases, the amount of H+ ($h$ ) in intermembrane space ($is$ ) decreases and so does the amount of ATP ($h$ ) in mitochondrial matrix.", "Thus, an increase in permeability will increase the pH.", "If the permeability increases even beyond 2, no ATP will be produced from ADP due to insufficient H+ gradient.", "Trend from various runs is shown in Figure REF .", "Figure: Quantities of H+ and ATP at various run lengths and permeabilities for the Petri Net model in Figure .", "Comparing Altered Trajectories due to Delay Intervention Question 7 Membranes must be fluid to function properly.", "How would decreased fluidity of the membrane affect the efficiency of the electron transport chain?", "Provided Answer: “Some of the components of the electron transport chain are mobile electron carriers, which means they must be able to move within the membrane.", "If fluidity decreases, these movable components would be encumbered and move more slowly.", "This would cause decreased efficiency of the electron transport chain.” Solution 7 The answer deals requires background knowledge about fluidity and how it relates to mobile carriers not presented in the source chapter.", "From background knowledge we find that the higher the fluidity, higher the mobility.", "The electron transport chain is presented in Fig 9.15 of Campbell's book.", "From background knowledge, we know that the efficiency of the electron transport chain is measured by the amount of ATP produced per NADH/FADH2.", "The ATP production happens due to the gradient of H+ ions across the mitochondrial membrane.", "The higher the number of H+ ions in the intermembrane space, the higher would be the gradient and the resulting efficiency.", "So we measure the efficiency of the chain by the amount of H+ transported to intermembrane space, assuming all other (fixed) molecules behave normally.", "This is a valid assumption since H+ transported from mitochondrial matrix is directly proportional to the amount of electrons shuttled through the non-mobile complexes and there is a linear chain from the electron carrier to oxygen.", "We model this chain using a Petri Net with durative transitions shown in Figure REF .", "Higher the duration of transitions, lower the fluidity of the membrane.", "We assume that a steady supply of NADH and H+ is available in the mitochondrial membrane.", "We fulfill this requirement by having quantities in excess of what will be consumed during the simulation.", "We ignore FADH2 from the diagram, since it is just an alternate path to the electron chain.", "Using it by itself will produce a lower number of H+ transporter to intermembrane space, but it will not change the result.", "We compare the amount of H+ transported into the intermembrane space to gauge the efficiency of the electron transport chain.", "More efficient the chain is, more H+ will it transport.", "We model three scenarios: normal fluidity, low fluidity with transitions $t3$ and $t4$ having an execution time of 2 and an lower fluidity with transitions $t3,t4$ having execution time of 4.", "Figure: Petri Net graph relevant to question .", "“is” is the intermembrane space, “mm” is mitochondrial matrix, “q” is ubiquinone and “cytc” is cytochrome c. Tokens are colored, e.g.", "“h:1, nadh:1” specify one token of hh and nadhnadh each.", "Token types are “h” for H+, “nadh” for NADH, “e” for electrons.", "Numbers in square brackets below the transition represent transition durations with default of one time unit, if the number is missing.We encode these cases in ASP with maximal firing set semantics and simulate them for 10 time steps.", "For the normal fluidity we get: holds(is,27,h,10) for low fluidity we get: holds(is,24,h,10) for lower fluidity we get: holds(is,18,h,10) Note that as the fluidity decreases, so does the amount of H+ transported to intermembrane space, pointing to lower efficiency of electron transport chain.", "Trend of various runs is shown in Figure REF .", "Figure: Quantities of H+ produced in the intermembrane space at various run lengths and fluidities for the Petri Net model in Figure .", "Comparing Altered Trajectories due to Priority and Read Interventions Question 8 Phosphofructokinase (PFK) is allosterically regulated by ATP.", "Considering the result of glycolysis, is the allosteric regulation of PFK likely to increase or decrease the rate of activity for this enzyme?", "Provided Answer: “Considering that one of the end products of glycolysis is ATP, PFK is inhibited when ATP is abundant and bound to the enzyme.", "The inhibition decreases ATP production along this pathway.” Solution 8 Regulation of Phosphofructokinase (PFK) is presented in Figure 9.20 of Campbell's book.", "We ignore substances upstream of Fructose 6-phosphate (F6P) by assuming they are available in abundance.", "We also ignore AMP by assuming normal supply of it.", "We also ignore any output of glycolysis other than ATP production since the downstream processes ultimately produce additional ATP.", "Citric acid is also ignored since it is not relevant to the question at hand.", "We monitor the rate of activity of PFK by the number of times it gets used for glycolysis.", "We model this problem as a Petri Net shown in Figure REF .", "Allosteric regulation of PFK is modeled by a compound “pfkatp” which represents PFK's binding with ATP to form a compound.", "Details of allosteric regulation are not provided in the same chapter, they are background knowledge from external sources.", "Higher than normal quantity of ATP is modeled by a threshold arc (shown with arrow-heads at both ends) with an arbitrary threshold value of 4.", "This number can be increased as necessary.", "The output of glycolysis and down stream processes “t3” has been set to 2 to run the simulation in a reasonable amount of time.", "It can be made larger as necessary.", "The allosteric regulation transition “t4” has also been given a higher priority than glycolysis transition “t3”.", "This way, ATP in excess will cause PFK to be converted to PFK+ATP compound, reducing action of PFK.", "We assume that F6P is available in sufficient quantity and so is PFK.", "This requirement is fulfilled by having more quantity than can be consumed in the simulation duration.", "We model both the normal situation including transition $t4$ shown in dotted style and the abnormal situation where $t4$ is removed.", "Figure: Petri Net graph relevant to question .", "“pfk” is phosphofructokinase, “f6p” is fructose 6-phosphate, “atp” is ATP and “pfkatp” is the pfk bound with atp for allosteric regulation.", "Transition “t3” represents enzymic action of pfk, “t4” represents the binding of pfk with atp.", "The double arrowed arc represents a threshold arc, which enables “t4” when there are at least 4 tokens available at “atp”.", "Numbers above transitions in angular brackets represent arc priorities.We encode both situations in ASP with maximal firing set policy and run them for 10 time steps.", "At the end of the run we compare the firing count of transition $t3$ for both cases.", "For the normal case (with $t4$ ), we get the following results: holds(atp,0,c,0) holds(f6p,20,c,0) holds(pfk,20,c,0) holds(pfkatp,0,c,0) fires(t3,0) holds(atp,2,c,1) holds(f6p,19,c,1) holds(pfk,19,c,1) holds(pfkatp,0,c,1) fires(t3,1) holds(atp,4,c,2) holds(f6p,18,c,2) holds(pfk,18,c,2) holds(pfkatp,0,c,2) fires(t4,2) holds(atp,3,c,3) holds(f6p,18,c,3) holds(pfk,17,c,3) holds(pfkatp,1,c,3) fires(t3,3) holds(atp,5,c,4) holds(f6p,17,c,4) holds(pfk,16,c,4) holds(pfkatp,1,c,4) fires(t4,4) holds(atp,4,c,5) holds(f6p,17,c,5) holds(pfk,15,c,5) holds(pfkatp,2,c,5) fires(t4,5) holds(atp,3,c,6) holds(f6p,17,c,6) holds(pfk,14,c,6) holds(pfkatp,3,c,6) fires(t3,6) holds(atp,5,c,7) holds(f6p,16,c,7) holds(pfk,13,c,7) holds(pfkatp,3,c,7) fires(t4,7) holds(atp,4,c,8) holds(f6p,16,c,8) holds(pfk,12,c,8) holds(pfkatp,4,c,8) fires(t4,8) holds(atp,3,c,9) holds(f6p,16,c,9) holds(pfk,11,c,9) holds(pfkatp,5,c,9) fires(t3,9) holds(atp,5,c,10) holds(f6p,15,c,10) holds(pfk,10,c,10) holds(pfkatp,5,c,10) fires(t4,10) Note that $t3$ fires only when the ATP falls below our set threshold, above it PFK is converted to PFK+ATP compound via $t4$ .", "For the abnormal case (without $t4$ ) we get the following results: holds(atp,0,c,0) holds(f6p,20,c,0) holds(pfk,20,c,0) holds(pfkatp,0,c,0) fires(t3,0) holds(atp,2,c,1) holds(f6p,19,c,1) holds(pfk,19,c,1) holds(pfkatp,0,c,1) fires(t3,1) holds(atp,4,c,2) holds(f6p,18,c,2) holds(pfk,18,c,2) holds(pfkatp,0,c,2) fires(t3,2) holds(atp,6,c,3) holds(f6p,17,c,3) holds(pfk,17,c,3) holds(pfkatp,0,c,3) fires(t3,3) holds(atp,8,c,4) holds(f6p,16,c,4) holds(pfk,16,c,4) holds(pfkatp,0,c,4) fires(t3,4) holds(atp,10,c,5) holds(f6p,15,c,5) holds(pfk,15,c,5) holds(pfkatp,0,c,5) fires(t3,5) holds(atp,12,c,6) holds(f6p,14,c,6) holds(pfk,14,c,6) holds(pfkatp,0,c,6) fires(t3,6) holds(atp,14,c,7) holds(f6p,13,c,7) holds(pfk,13,c,7) holds(pfkatp,0,c,7) fires(t3,7) holds(atp,16,c,8) holds(f6p,12,c,8) holds(pfk,12,c,8) holds(pfkatp,0,c,8) fires(t3,8) holds(atp,18,c,9) holds(f6p,11,c,9) holds(pfk,11,c,9) holds(pfkatp,0,c,9) fires(t3,9) holds(atp,20,c,10) holds(f6p,10,c,10) holds(pfk,10,c,10) holds(pfkatp,0,c,10) fires(t3,10) Note that when ATP is not abundant, transition $t3$ fires continuously, which represents the enzymic activity that converts F6P to downstream substances.", "Trend of various runs is shown in Figure REF .", "Figure: Petri Net model in Figure .", "Comparing Altered Trajectories due to Automatic Conversion Intervention Question 9 How does the oxidation of NADH affect the rate of glycolysis?", "Provided Answer: “NADH must be oxidized back to NAD+ in order to be used in glycolysis.", "Without this molecule, glycolysis cannot occur.” Solution 9 Cellular respiration is summarized in Fig 9.6 of Campbell's book.", "NAD+ is reduced to NADH during glycolysis (see Campbell's Fig 9.9) during the process of converting Glyceraldehyde 3-phosphate (G3P) to 1,3-Bisphosphoglycerate (BPG13).", "NADH is oxidized back to NAD+ during oxidative phosphorylation by the electron transport chain (see Campbell's Fig 9.15).", "We can gauge the rate of glycolysis by the amount of Pyruvate produced, which is the end product of glycolysis.", "We simplify our model by abstracting glycolysis as a black-box that takes Glucose and NAD+ as input and produces NADH and Pyruvate as output, since there is a linear chain from Glucose to Pyruvate that depends upon the availability of NAD+.", "We also abstract oxidative phosphorylation as a black-box which takes NADH as input and produces NAD+ as output.", "None of the other inner workings of oxidative phosphorylation play a role in answering the question assuming they are functioning normally.", "We also ignore the pyruvate oxidation and citric acid cycle stages of cellular respiration since their end products only provide additional raw material for oxidative phosphorylation and do not add value to answering the question.", "We assume a steady supply of Glucose and all other substances used in glycolysis but a limited supply of NAD+, since it can be recycled from NADH and we want to model its impact.", "We fulfill the steady supply requirement of Glucose with sufficient initial quantity in excess of what will be consumed during our simulation interval.", "We also ensure that we have sufficient initial quantity of NAD+ to maintain glycolysis as long as it can be recycled.", "Figure: Petri Net graph relevant to question .", "“glu” represents glucose, “gly1” represents glycolysis, “pyr” represents pyruvate, “ox1” represents oxidative phosphorylation, “nadh” represents NADH and “nadp” represents NAD+.", "“ox1” is removed to model stoppage of oxidation of NADH to NAD+.Figure REF is a Petri Net representation of our simplified model.", "Normal situation is modeled by the entire graph, where NADH is recycled back to NAD+, while the abnormal situation is modeled by the graph with the transition $ox1$ (shown in dotted style) removed.", "We encode both situations in ASP with the maximal firing set policy.", "Both situations are run for 5 steps and the amount of pyruvate is compared to determine the difference in the rate of glycolysis.", "In normal situation (with $ox1$ transition), unique quantities of pyruvate ($pyr$ ) are as follows: fires(gly1,0) fires(gly1,1) fires(gly1,2) fires(gly1,3) fires(gly1,4) holds(pyr,10,5) while in abnormal situation (without $ox1$ transition), unique quantities of pyruvate are as follows: fires(gly1,0) fires(gly1,1) fires(gly1,2) holds(pyr,6,5) Note that the rate of glycolysis is lower when NADH is not recycled back to NAD+, as the glycolysis stops after the initial quantity of 6 NAD+ is consumed.", "Also, the $gly1$ transition does not fire after time-step 2, indicating glycolysis has stopped.", "Trend of various runs is shown in Figure REF .", "Figure: Amount of “pyr” produced by runs of various lengths of Petri Net in Figure .", "It shows results for both normal situation where “nadh” is recycled to “nadp” as well as the abnormal situation where this recycling is stopped.", "Comparing Altered Trajectories due to Initial Value Intervention Question 10 During intense exercise, can a muscle cell use fat as a concentrated source of chemical energy?", "Explain.", "Provided Answer: “When oxygen is present, the fatty acid chains containing most of the energy of a fat are oxidized and fed into the citric acid cycle and the electron transport chain.", "During intense exercise, however, oxygen is scarce in muscle cells, so ATP must be generated by glycolysis alone.", "A very small part of the fat molecule, the glycerol backbone, can be oxidized via glycolysis, but the amount of energy released by this portion is insignificant compared to that released by the fatty acid chains.", "(This is why moderate exercise, staying below 70% maximum heart rate, is better for burning fat because enough oxygen remains available to the muscles.", ")” Solution 10 The process of fat consumption in glycolysis and citric acid cycle is summarized in Fig 9.19 of Campbell's book.", "Fats are digested into glycerol and fatty acids.", "Glycerol gets fed into glycolysis after being converted into Gyceraldehyde 3-phosphate (G3P), while fatty acids get fed into citric acid cycle after being broken down through beta oxidation and converted into Acetyl CoA.", "Campbell's Fig 9.18 identify a junction in catabolism where aerobic respiration or fermentation take place depending upon whether oxygen is present or not.", "Energy produced at various steps is in terms of ATP produced.", "In order to explain whether fat can be used as a concentrated source of chemical energy or not, we have to show the different ways of ATP production and when they kick in.", "We combine the various pieces of information collected from Fig 9.19, second paragraph on second column of p/180, Fig 9.15, Fig 9.16 and Fig 9.18 of Campbell's book into Figure REF .", "We model two situations when oxygen is not available in the muscle cells (at the start of a intense exercise) and when oxygen is available in the muscle cells (after the exercise intensity is plateaued).", "We then compare and contrast them on the amount of ATP produced and the reasons for the firing sequences.", "Figure: Petri Net graph relevant to question .", "“fats” are fats, “dig” is digestion of fats, “gly” is glycerol, “fac” is fatty acid, “g3p” is Glyceraldehyde 3-phosphate, “pyr” is pyruvate, “o2” is oxygen, “nadh” is NADH, “acoa” is Acyl CoA, “atp” is ATP, “op1” is oxidative phosphorylation, “cac1” is citric acid cycle, “fer1” is fermentation, “ox1” is oxidation of pyruvate to Acyl CoA and “box1” is beta oxidation.Figure REF is a petri net representation of our simplified model.", "Our edge labels have lower numbers on them than the yield in Fig 9.16 of Campbell's book but they still capture the difference in volume that would be produced due to oxidative phosphorylation vs. glycolysis.", "Using exact amounts will only increase the difference of ATP production due to the two mechanisms.", "We encode both situations (when oxygen is present and when it is not) in ASP with maximal firing set policy.", "We run them for 10 steps.", "The firing sequence and the resulting yield of ATP explain what the possible use of fat as a source of chemical energy.", "At he start of intense exercise, when oxygen is in short supply: holds(acoa,0,0) holds(atp,0,0) holds(fac,0,0) holds(fats,5,0) holds(g3p,0,0) holds(gly,0,0) holds(nadh,0,0) holds(o2,0,0) holds(pyr,0,0) fires(dig,0) holds(acoa,0,1) holds(atp,0,1) holds(fac,1,1) holds(fats,4,1) holds(g3p,0,1) holds(gly,1,1) holds(nadh,0,1) holds(o2,0,1) holds(pyr,0,1) fires(box1,1) fires(dig,1) fires(t2,1) holds(acoa,1,2) holds(atp,0,2) holds(fac,1,2) holds(fats,3,2) holds(g3p,1,2) holds(gly,1,2) holds(nadh,1,2) holds(o2,0,2) holds(pyr,0,2) fires(box1,2) fires(cac1,2) fires(dig,2) fires(gly6,2) fires(t2,2) holds(acoa,1,3) holds(atp,2,3) holds(fac,1,3) holds(fats,2,3) holds(g3p,1,3) holds(gly,1,3) holds(nadh,3,3) holds(o2,0,3) holds(pyr,1,3) fires(box1,3) fires(cac1,3) fires(dig,3) fires(fer1,3) fires(gly6,3) fires(t2,3) holds(acoa,1,4) holds(atp,4,4) holds(fac,1,4) holds(fats,1,4) holds(g3p,1,4) holds(gly,1,4) holds(nadh,5,4) holds(o2,0,4) holds(pyr,1,4) fires(box1,4) fires(cac1,4) fires(dig,4) fires(fer1,4) fires(gly6,4) fires(t2,4) holds(acoa,1,5) holds(atp,6,5) holds(fac,1,5) holds(fats,0,5) holds(g3p,1,5) holds(gly,1,5) holds(nadh,7,5) holds(o2,0,5) holds(pyr,1,5) fires(box1,5) fires(cac1,5) fires(fer1,5) fires(gly6,5) fires(t2,5) holds(acoa,1,6) holds(atp,8,6) holds(fac,0,6) holds(fats,0,6) holds(g3p,1,6) holds(gly,0,6) holds(nadh,9,6) holds(o2,0,6) holds(pyr,1,6) fires(cac1,6) fires(fer1,6) fires(gly6,6) holds(acoa,0,7) holds(atp,10,7) holds(fac,0,7) holds(fats,0,7) holds(g3p,0,7) holds(gly,0,7) holds(nadh,10,7) holds(o2,0,7) holds(pyr,1,7) fires(fer1,7) holds(acoa,0,8) holds(atp,10,8) holds(fac,0,8) holds(fats,0,8) holds(g3p,0,8) holds(gly,0,8) holds(nadh,10,8) holds(o2,0,8) holds(pyr,0,8) holds(acoa,0,9) holds(atp,10,9) holds(fac,0,9) holds(fats,0,9) holds(g3p,0,9) holds(gly,0,9) holds(nadh,10,9) holds(o2,0,9) holds(pyr,0,9) holds(acoa,0,10) holds(atp,10,10) holds(fac,0,10) holds(fats,0,10) holds(g3p,0,10) holds(gly,0,10) holds(nadh,10,10) holds(o2,0,10) holds(pyr,0,10) when the exercise intensity has plateaued and oxygen is no longer in short supply: holds(acoa,0,0) holds(atp,0,0) holds(fac,0,0) holds(fats,5,0) holds(g3p,0,0) holds(gly,0,0) holds(nadh,0,0) holds(o2,10,0) holds(pyr,0,0) fires(dig,0) holds(acoa,0,1) holds(atp,0,1) holds(fac,1,1) holds(fats,4,1) holds(g3p,0,1) holds(gly,1,1) holds(nadh,0,1) holds(o2,10,1) holds(pyr,0,1) fires(box1,1) fires(dig,1) fires(t2,1) holds(acoa,1,2) holds(atp,0,2) holds(fac,1,2) holds(fats,3,2) holds(g3p,1,2) holds(gly,1,2) holds(nadh,1,2) holds(o2,10,2) holds(pyr,0,2) fires(box1,2) fires(cac1,2) fires(dig,2) fires(gly6,2) fires(op1,2) fires(t2,2) holds(acoa,1,3) holds(atp,5,3) holds(fac,1,3) holds(fats,2,3) holds(g3p,1,3) holds(gly,1,3) holds(nadh,2,3) holds(o2,9,3) holds(pyr,1,3) fires(box1,3) fires(cac1,3) fires(dig,3) fires(gly6,3) fires(op1,3) fires(ox1,3) fires(t2,3) holds(acoa,2,4) holds(atp,10,4) holds(fac,1,4) holds(fats,1,4) holds(g3p,1,4) holds(gly,1,4) holds(nadh,4,4) holds(o2,8,4) holds(pyr,1,4) fires(box1,4) fires(cac1,4) fires(dig,4) fires(gly6,4) fires(op1,4) fires(ox1,4) fires(t2,4) holds(acoa,3,5) holds(atp,15,5) holds(fac,1,5) holds(fats,0,5) holds(g3p,1,5) holds(gly,1,5) holds(nadh,6,5) holds(o2,7,5) holds(pyr,1,5) fires(box1,5) fires(cac1,5) fires(gly6,5) fires(op1,5) fires(ox1,5) fires(t2,5) holds(acoa,4,6) holds(atp,20,6) holds(fac,0,6) holds(fats,0,6) holds(g3p,1,6) holds(gly,0,6) holds(nadh,8,6) holds(o2,6,6) holds(pyr,1,6) fires(cac1,6) fires(gly6,6) fires(op1,6) fires(ox1,6) holds(acoa,4,7) holds(atp,25,7) holds(fac,0,7) holds(fats,0,7) holds(g3p,0,7) holds(gly,0,7) holds(nadh,9,7) holds(o2,5,7) holds(pyr,1,7) fires(cac1,7) fires(op1,7) fires(ox1,7) holds(acoa,4,8) holds(atp,29,8) holds(fac,0,8) holds(fats,0,8) holds(g3p,0,8) holds(gly,0,8) holds(nadh,10,8) holds(o2,4,8) holds(pyr,0,8) fires(cac1,8) fires(op1,8) holds(acoa,3,9) holds(atp,33,9) holds(fac,0,9) holds(fats,0,9) holds(g3p,0,9) holds(gly,0,9) holds(nadh,10,9) holds(o2,3,9) holds(pyr,0,9) fires(cac1,9) fires(op1,9) holds(acoa,2,10) holds(atp,37,10) holds(fac,0,10) holds(fats,0,10) holds(g3p,0,10) holds(gly,0,10) holds(nadh,10,10) holds(o2,2,10) holds(pyr,0,10) fires(cac1,10) fires(op1,10) We see that more ATP is produced when oxygen is available.", "Most ATP (energy) is produced by the oxidative phosphorylation which requires oxygen.", "When oxygen is not available, small amount of energy is produced due to glycolysis of glycerol ($gly$ ).", "With oxygen a lot more energy is produced, most of it due to fatty acids ($fac$ ).", "Trend of various runs is shown in Figure REF .", "Figure: Amount of “atp” produced by runs of various lengths of Petri Net in Figure .", "Two situations are shown: when oxygen is in short supply and when it is abundant.", "Conclusion In this chapter we presented how to model biological systems as Petri Nets, translated them into ASP, reasoned with them and answered questions about them.", "We used diagrams from Campbell's book, background knowledge and assumptions to facilitate our modeling work.", "However, source knowledge for real world applications comes from published papers, magazines and books.", "This means that we have to do text extraction.", "In one of the following chapters we look at some of the real applications that we have worked on in the past in collaboration with other researchers to develop models using text extraction.", "But first, we look at how we use the concept of answering questions using Petri Nets to build a question answering system.", "We will extend the Petri Nets even more for this.", "BioPathQA - A System for Modeling, Simulating, and Querying Biological PathwaysThe BioPathQA System Introduction In this chapter we combine the methods from Chapter , notions from action languages, and ASP to build a system BioPathQA and a language to specify pathways and query them.", "We show how various biological pathways are encoded in BioPathQA and how it computes answers of queries against them.", "Description of BioPathQA Our system has the following components: [(i)] a pathway specification language a query language to specify the deep reasoning question, an ASP program that encodes the pathway model and its extensions for simulation.", "Knowledge about biological pathways comes in many different forms, such as cartoon diagrams, maps with well defined syntax and semantics (e.g.", "Kohn's maps [43]), and biological publications.", "Similar to other technical domains, some amount of domain knowledge is also required.", "Users want to collect information from disparate sources and encode it in a pathway specification.", "We have developed a language to allow users to describe their pathway.", "This description includes describing the substances and actions that make up the pathway, the initial state of the substances, and how the state of the pathway changes due to the actions.", "An evolution of a pathway's state from the initial state, through a set of actions is called a trajectory.", "Being a specification language targeted at biological systems, multiple actions autonomously execute in parallel as soon as their preconditions are satisfied.", "The amount of parallelism is dictated by any resource conflicts between the actions.", "When that occurs, only one sub-set of the possible actions can execute, leading to multiple outcomes from that point on.", "Questions are usually provided in natural language, which is vague.", "To avoid the vagaries of natural language, we developed a language with syntax close to natural language but with a well defined formal semantics.", "The query language allows a user to make changes to the pathway through interventions, and restrict its trajectories through observations and query on aggregate values in a trajectory, across a set of trajectories and even over two sets of trajectories.", "This allows the user to compare a base case of a pathway specification with an alternate case modified due to interventions and observations.", "This new feature is a major contribution of our research.", "Inspiration for our high level language comes from action languages and query languages such as [26].", "While action languages generally describe transition systems [27], our language describes trajectories.", "In addition, our language is geared towards modeling natural systems, in which actions occur autonomously [66] when their pre-conditions are satisfied; and we do not allow the quantities to become negative (as the quantities represent amounts of physical entities).", "Next we describe the syntax of our pathway specification language and the query language.", "Following that we will describe the syntax of our language and how we encode it in ASP.", "Syntax of Pathway Specification Language (BioPathQA-PL) The alphabet of pathway specification language $\\mathcal {P}$ consists of disjoint nonempty domain-dependent sets $A$ , $F$ , $L$ representing actions, fluents, and locations, respectively; a fixed set $S$ of firing styles; a fixed set $K$ of keywords providing syntactic sugar (shown in bold face in pathway specification language below); a fixed set of punctuations $\\lbrace `,^{\\prime } \\rbrace $ ; and a fixed set of special constants $\\lbrace `1^{\\prime },`^{\\prime },`max^{\\prime }\\rbrace $ ; and integers.", "Each fluent $f \\in F$ has a domain $dom(f)$ which is either integer or binary and specifies the values $f$ can take.", "A fluent is either simple, such as $f$ or locational, such as $f[l]$ , where $l \\in L$ .", "A state $s$ is an interpretation of $F$ that maps fluents to their values.", "We write $s(f)=v$ to represent “$f$ has the value $v$ in state $s$ ”.", "States are indexed, such that consecutive states $s_i$ and $s_{i+1}$ represent an evolution over one time step from $i$ to $i+1$ due to firing of an action set $T_i$ in $s_i$ .", "We illustrate the role of various symbols in the alphabet with examples from the biological domain.", "Consider the following example of a hypothetical pathway specification: $&\\mathbf {domain~of~} sug \\mathbf {~is~} integer, fac \\mathbf {~is~} integer, acoa \\mathbf {~is~} integer, h2o \\mathbf {~is~} integer\\\\&gly \\mathbf {~may~execute~causing~} sug \\mathbf {~change~value~by~} -1, acoa \\mathbf {~change~value~by~} +1\\\\&box \\mathbf {~may~execute~causing~} fac \\mathbf {~change~value~by~} -1, acoa \\mathbf {~change~value~by~} +1 \\\\&\\mathbf {~~~~~~~~~~~if~} h2o \\mathbf {~has~value~} 1 \\mathbf {~or~higher~}\\\\&\\mathbf {inhibit~} box \\mathbf {~if~} sug \\mathbf {~has~value~} 1 \\mathbf {~or~higher~}\\\\&\\mathbf {initially~} sug \\mathbf {~has~value~} 3, fac \\mathbf {~has~value~} 4, acoa \\mathbf {~has~value~} 0 &$ It describes two processes glycolysis and beta-oxidation represented by actions `$gly$ ' and `$box$ ' in lines () and ()-() respectively.", "Substances used by the pathway, i.e.", "sugar, fatty-acids, acetyl-CoA, and water are represented by numeric fluents `$sug$ ',`$fac$ ',`$acoa$ ', and `$h2o$ ' respectively in line (REF ).", "When glycolysis occurs, it consumes 1 unit of sugar and produces 1 unit of acetyl-CoA (line ().", "When beta-oxidation occurs, it consumes 1 unit of fatty-acids and produces 1 unit of acetyl-CoA (line ()).", "The inputs of glycolysis implicitly impose a requirement that glycolysis can only occur when at least 1 unit of sugar is available.", "Similarly, the input of beta-oxidation implicitly a requirement that beta-oxidation can only occur when at least 1 unit of fatty-acids is available.", "Beta oxidation has an additional condition imposed on it in line () that there must be at least 1 unit of water available.", "We call this a guard condition on beta-oxidation.", "Line () explictly inhibits beta-oxidation when there is any sugar available; and line () sets up the initial conditions of the pathway, i.e.", "Initially 3 units of each sugar, 4 units of fatty-acids are available and no acetyl-CoA is available.", "The words `$\\lbrace domain,$ $is,$ $may,$ $execute,$ $causing,$ $change,$ $value,$ $by,$ $has,$ $or,$ $higher,$ $inhibit,$ $if,$ $initially\\rbrace $ ' are keywords.", "When locations are involved, locational fluents take place of simple fluents and our representation changes to include locations.", "For example: $&gly \\mathbf {~may~execute~causing~} \\nonumber \\\\&~~~~sug \\mathbf {~atloc~} mm \\mathbf {~change~value~by~} -1, \\nonumber \\\\&~~~~acoa \\mathbf {~atloc~} mm \\mathbf {~change~value~by~} +1$ represents glycolysis taking 1 unit of sugar from mitochondrial matrix (represented by `$mm$ ') and produces acetyl-CoA in the mitochondrial matrix.", "Here `$atloc$ ' is an additional keyword.", "A pathway is composed of a collection of different types of statements and clauses.", "We first introduce their syntax, following that we will give intuitive definitions, and following that we will show how they are combined together to construct a pathway specification.", "Definition 46 (Fluent domain declaration statement) A fluent domain declaration statement declares the values a fluent can take.", "It has the form: $\\mathbf {domain~of~} f \\mathbf {~is~} `integer^{\\prime }\|`binary^{\\prime } \\\\\\mathbf {domain~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} `integer^{\\prime }\|`binary^{\\prime }$ for simple fluent “$f$ ”, and locational fluent “$f[l]$ ”.", "Multiple domain statements are compactly written as: $\\mathbf {domain~of~} f_1 \\mathbf {~is~} `integer^{\\prime }\|`binary^{\\prime } , \\dots , f_n \\mathbf {~is~} `integer^{\\prime }\|`binary^{\\prime } \\\\\\mathbf {domain~of~} f_1 \\mathbf {~atloc~} l_1 \\mathbf {~is~} `integer^{\\prime }\|`binary^{\\prime } , \\dots , f_n \\mathbf {~atloc~} l_n \\mathbf {~is~} `integer^{\\prime }\|`binary^{\\prime }$ Binary domain is usually used for representing substances in a signaling pathway, while a metabolic pathways take positive numeric values.", "Since the domain is for a physical entity, we do not allow negative values for fluents.", "Definition 47 (Guard condition) A guard condition takes one of the following forms: $&f \\mathbf {~has~value~} w \\mathbf {~or~higher~} \\\\&f \\mathbf {~atloc~} l \\mathbf {~has~value~} w \\mathbf {~or~higher~}\\\\&f \\mathbf {~has~value~lower~than~} w \\\\&f \\mathbf {~atloc~} l \\mathbf {~has~value~lower~than~} w \\\\&f \\mathbf {~has~value~equal~to~} w \\\\&f \\mathbf {~atloc~} l \\mathbf {~has~value~equal~to~} w \\\\&f_1 \\mathbf {~has~value~higher~than~} f_2\\\\&f_1 \\mathbf {~atloc~} l_1 \\mathbf {~has~value~higher~than~} f_2 \\mathbf {~atloc~} l_2$ where, each $f$ in (REF ), (), (), () is a simple fluent, each $f[l]$ in (), (), (), () is a locational fluent with location $l$ , and each $w \\in \\mathbb {N}^+ \\cup \\lbrace 0 \\rbrace $ .", "Definition 48 (Effect clause) An effect clause can take one of the following forms: $&f \\mathbf {~change~value~by~} e\\\\&f \\mathbf {~atloc~} l \\mathbf {~change~value~by~} e$ where $a$ is an action, $f$ in (REF ) is a simple fluent, $f[l]$ in () is a locational fluent with location $l$ , $e \\in \\mathbb {N}^+ \\cup \\lbrace \\rbrace $ for integer fluents or $e \\in \\lbrace 1,-1,\\rbrace $ for binary fluents.", "Definition 49 (May-execute statement) A may-execute statement captures the conditions for firing an action $a$ and its impact.", "It is of the form: $a \\mathbf {~may~execute~causing~} & \\mathit {effect}_1, \\dots , \\mathit {effect}_m \\nonumber \\\\\\mathbf {~if~} &guard\\_cond_1, \\dots , guard\\_cond_n$ where $\\mathit {effect}_i$ is an effect clause; and $\\mathit {guard\\_cond}_j$ is a guard condition clause, $m > 0$ , and $n \\ge 0$ .", "If $n = 0$ , the effect statement is unconditional (guarded by $\\top $ ) and the $\\mathbf {if}$ is dropped.", "A single may-execute statement must not have $\\mathit {effect}_i, \\mathit {effect}_j$ with $e_i < 0, e_j < 0$ for the same fluent; or $e_i > 0, e_j > 0$ for the same fluent.", "Definition 50 (Must-execute statement) An must-execute statement captures the impact of firing of an action $a$ that must fire when enabled (as long as it is not inhibited).", "It is an expression of the form: $a \\mathbf {~normally~must~execute~causing~} & \\mathit {effect}_1, \\dots , \\mathit {effect}_m \\nonumber \\\\\\mathbf {~if~} &guard\\_cond_{1}, \\dots , guard\\_cond_n$ where $\\mathit {effect}_i$ , and $\\mathit {guard\\_cond}_j$ are as in (REF ) above.", "Definition 51 (Inhibit statement) An inhibit statement captures the conditions that inhibit an action from occurring.", "It is an expression of the form: $\\mathbf {inhibit~} a \\mathbf {~if~} guard\\_cond_1, \\dots , guard\\_cond_n$ where $a$ is an action, $guard\\_cond_i$ is a guard condition clause, and $n \\ge 0$ .", "if $n = 0$ , the inhibition of action `$a$ ' is unconditional `$\\mathbf {if}$ ' is dropped.", "Definition 52 (Initial condition statement) An initial condition statement captures the initial state of pathway.", "It is of the form: $\\mathbf {initially~} & f \\mathbf {~has~value~} w\\\\\\mathbf {initially~} & f \\mathbf {~atloc~} l \\mathbf {~has~value~} w$ where each $f$ in (REF ) is a simple fluent, $f[l]$ in () is a locational fluent with location $l$ , and each $w$ is a non-negative integer.", "Multiple initial condition statements are written compactly as: $\\mathbf {initially~} & f_1\\mathbf {~has~value~} w_1, \\dots , f_n \\mathbf {~has~value~} w_n\\\\\\mathbf {initially~} & f_1 \\mathbf {~atloc~} l_1 \\mathbf {~has~value~} w_1, \\dots , f_n \\mathbf {~atloc~} l_n \\mathbf {~has~value~} w_n$ Definition 53 (Duration Statement) A duration statement represents the duration of an action that takes longer than a single time unit to execute.", "It is of the form: $a \\mathbf {~executes~in~} d \\mathbf {~time~units}$ where $d$ is a positive integer representing the action duration.", "Definition 54 (Stimulate Statement) A stimulate statement changes the rate of an action.", "It is an expression of the form: $\\mathbf {normally~stimulate~} a \\mathbf {~by~factor~} n \\mathbf {~if~} guard\\_cond_1,\\dots ,guard\\_cond_n$ where $guard\\_cond_i$ is a condition, $n > 0$ .", "When $n=0$ , the stimulation is unconditional and $\\mathbf {~if~}$ is dropped.", "A stimulation causes the $\\mathit {effect}$ in may-cause and must-fire multiplied by $n$ .", "Actions execute automatically when fireable, subject to the available fluent quantities.", "Definition 55 (Firing Style Statement) A firing style statement specifies how many actions execute simultaneously (or action parallelism).", "It is of the form: $\\mathbf {firing~style~} S$ where, $S$ is either “1”, “$$ ”, or “$max$ ” for serial execution, interleaved execution, and maximum parallelism.", "We will now give the intuitive meaning of these statements and put them into context w.r.t.", "the biological domain.", "Though our description below uses simple fluents only, it applies to locational fluents in a obvious manner.", "The reason for having locational fluents at all is that they allow a more natural pathway specification when substance locations are involved instead of devising one's own encoding scheme.", "For example, in a mitochondria, hydrogen ions (H+) appear in multiple locations (intermembrane space and mitochondrial matrix), with each location carrying its distinct quantity separate from other locations.", "Intuitively, a may-execute statement (REF ) represents an action $a$ that may fire if all conditions `$\\mathit {guard\\_cond}_{1},\\dots ,\\mathit {guard\\_cond}_n$ ' hold in the current state.", "When it executes, it impacts the state as specified in $\\mathit {effect}$ s. In biological context, action $a$ represents a process, such as a reaction, $\\mathit {effect}$ s represent the inputs / ingredients of the reaction, and $guard\\_cond$ represent additional preconditions necessary for the reaction to proceed.", "Condition (REF ) holds in a state $s$ if $s(f) \\ge w$ .", "It could represent an initiation concentration $w$ of a substance $f$ which is higher than the quantity consumed by the reaction $a$ .", "Condition () holds in a state $s$ if $s(f) < w$ .", "Condition () holds in a state $s$ if $s(f) = w$ .", "Condition () holds in a state $s$ if $s(f_1) > s(f_2)$ capturing a situation where a substance gradient is required for a biological process to occur.", "An example of one such process is the synthesis of ATP by ATP Synthase, which requires a H+ (Hydrogen ion) gradient across the inner mitochondrial matrix [64].", "Intuitively, the effect clause (REF ) of an action describes the impact of an action on a fluent.", "When an action $a$ fires in a state $s$ , the value of $f$ changes according to the effect clause for $f$ .", "The value of $f$ increases by $e$ if $e > 0$ , decreases by $e$ if $e < 0$ , or decreases by $s(f)$ if $e = `^{\\prime }$ (where $`^{\\prime }$ can be interpreted as $-s(f)$ ).", "For a reaction $a$ , a fluent with $e < 0$ represents an ingredient consumed in quantity $\|e\|$ by the reaction; a fluent with $e > 0$ represents a product of the reaction in quantity $e$ ; a fluent with $e = `^{\\prime }$ represents consuming all quantity of the substance due to the reaction.", "Since the fluents represent physical substances, their quantities cannot become negative.", "As a result, any action that may cause a fluent quantity to go below zero is disallowed.", "Intuitively, a must-execute statement (REF ) is similar to a may-exec, except that when enabled, it preferentially fires over other actions as long as there isn't an inhibit proposition that will cause the action to become inactive.", "It captures the effect of an action that must happen whenever enabled.", "Intuitively, an inhibit statement (REF ) specifies the conditions that inhibits an action.", "In a biological context, it defines inhibition of reactions, e.g., through biological feedback control.", "Though we could have added these conditions to may-exec, it is more intuitive to keep them separate as inhibition conditions are usually discovered separately in a biological domain.", "Including them as part of may-fire would constitute a surgery of existing knowledge bases.", "Intuitively, an initial condition statement (REF ) specifies the initial values of fluents.", "The collection of such propositions defines the initial state $s_0$ of the pathway.", "In a biological context, this defines the initial distribution of substances in the biological system.", "Intuitively, an action duration statement (REF ) represents action durations, especially when an action takes longer to execute.", "When an action $a$ with duration $d$ fires in state $s_k$ , it immediately decreases the values of fluents with $e < 0$ and $e = $ upon execution, however, it does not increase the value of fluents with $e > 0$ until time the end of its execution in state $s_{k+d}$ .", "In a biological context the action duration captures a reaction's duration.", "A reaction consumes its ingredients immediately on firing, processes them for duration $d$ and generates its products at the end of this duration.", "Intuitively, a stimulate statement (REF ) represents a change in the rate of an action $a$ .", "The stimulation causes the action to change its rate of consumption of its ingredients and production of its products by a factor $n$ .", "In biological context, this stimulation can be a result of an enzyme or a stimulant's availability, causing a reaction that normally proceeds slowly to occur faster.", "Intuitively, a firing style statement (REF ) specifies the parallelism of actions.", "When it is “1”, at most one action may fire, when it is “$max$ ”, the maximum allowable actions must fire, and when it is “$$ ”, any subset of fireable actions may fire simultaneously.", "In a biological domain the firing style allows one to model serial operations, parallel operations and maximally parallel operations.", "The maximum parallelism is also useful in quickly discovering changes that occur in a biological system.", "Definition 56 (Pathway Specification) A pathway specification is composed of one or more may-execute, must-execute, effect, inhibit, stimulate, initially, priority, duration statements, and one firing style statement.", "When a duration statement is not specified for an action, it is assumed to be 1.", "Any fluents for which an initial quantity is not specified are assumed to have a value of zero.", "A pathway specification is consistent if [(i)] there is at most one firing style, priority, duration statement for each action $a$ ; the $guard\\_cond_1,\\dots ,guard\\_cond_n$ from a may-execute or must-execute are disjoint from any other may-execute or must-execute Note that `$f1 \\text{ has value } 5 \\text{ or higher }$ ' overlaps with `$f1 \\text{ has value } 7 \\text{ or higher}$ ' and the two conditions are not considered disjoint.", "; locational and non-locational fluents may not be intermixed; domain of fluents, effects, conditions and numeric values are consistent, i.e., effects and conditions on binary fluents must be binary; and the pathway specification does not cause it to violate fluent domains by producing non-binary values for binary fluents.", "Each pathway specification $\\mathbf {D}$ represents a collection of trajectories of the form: $\\sigma = s_0, T_0, s_1, \\dots , s_{k-1}, T_{k-1}, s_k$ .", "Each trajectory encodes an evolution of the pathway starting from an initial state $s_0$ , where $s_i$ 's are states, and $T_i$ 's are sets of actions that fired in state $s_i$ .", "Intuitively, a trajectory starts from the initial state $s_0$ .", "Each $s_i,s_{i+1}$ pair represents the state evolution in one time step due to the action set $T_i$ .", "An action set $T_i$ is only executable in state $s_i$ , if the sum of changes to fluents due to $e_i < 0$ and $e_i = $ will not result in any of the fluents going negative.", "Changes to fluents due to $e_i > 0$ for the action set $T_i$ occur over subsequent time-steps depending upon the durations of actions involved.", "Thus, the state $s_i(f_i)$ is the sum of $e_i > 0$ for actions of duration $d$ that occurred $d$ time steps before (current time step) $i$ , i.e.", "$a \\in T_{i-d}$ , where the default duration $d$ of an action is 1 if none specified.", "Next we describe the semantics of the pathway specification language, which describes how these trajectories are generated.", "Semantics of Pathway Specification Language (BioPathQA-PL) The semantics of the pathway specification language are defined in terms of the trajectories of the domain description $\\mathbf {D}$ .", "Since our pathway specification language is inspired by Petri Nets, we use Petri Nets execution semantics to define its trajectories.", "However, some constructs in our pathway language specification are not directly representable in standard Petri Nets, as a result, we will have to extend them.", "Let an arc-guard be a conjunction of guard conditions of the form (REF )-(), such that it is wholly constructed of either locational or non-locational fluents, but not both.", "We introduce a new type of Guarded-arc Petri Net in which each arc has an arc-guard expression associated with it.", "Arcs with the same arc-guard are traversed when a transition connected to them fires and the arc-guard is found to hold in the current state.", "The arc-guards of arcs connected to the same transition form an exclusive set, such that only arcs corresponding to one guard expression may fire (for one transition).", "This setup can lead to different outcomes of an actionArcs for different guard expressions emanating / terminating at a place can further be combined into a single conditional arc with conditional arc-weights.", "If none of the condition applies then the arc is assumed to be missing..", "The transitions in this new type of Petri Net can have the following inscriptions on them: Propositional formula, specifying the executability conditions of the transition.", "Arc-durations, represented as “$dur(n)$ ”, where $n \\in \\mathbb {N}^+$ A must-execute inscription, “$must\\text{-}execute(guard)$ ”, requires that when the $guard$ holds in a state where this transition is enabled, it must fire, unless explicitly inhibited.", "The $guard$ has the same form as an $arc\\text{-}guard$ A stimulation inscription, “$stimulate(n,guard)$ ”, applies a multiplication factor $n \\in \\mathbb {N}^+$ to the input and output quantities consumed and produced by the transition, when $guard$ hold in the current state, where $guard$ has the same form as an $arc\\text{-}guard$ .", "Certain aspects of our nets are similar to CPNs [39].", "However, the CPNs do not allow our semantics of the reset arcs, or must-fire guards.", "Guarded-Arc Petri Net Figure: Example of a guarded-arc Petri Net.Figure REF shows an example of a guarded-arc Petri Net.", "There are two arc-guard expressions $f1<5$ and $f1>5$ .", "When $f1<5$ , $t1$ consumes one token from place $f1$ and produces one token in place $f2$ .", "When $f1>5$ , $t1$ 's consumption and production of the same tokens doubles.", "The transition $t1$ implicitly gets the guards for each arc represented the or-ed conditions $(f1<5) \\vee (f1>5))$ .", "The arc also has two conditions inhibiting it, they are represented by the and-ed conditions $\\lnot (f1>7) \\wedge \\lnot (f1<3)$ , where `$\\lnot $ ' represents logical not.", "Transition $t1$ is stimulated by factor 3 when $f1>5$ and it has a duration of 10 time units.", "A transition cannot fire even though one of its arc-guards is enabled, unless the token requirements on the arc itself are also fulfilled, e.g.", "if $f1$ has value 0 in the current state, even though $f1 < 5$ guard is satisfied, the transition cannot execute, because the input arc $(f1,t1)$ for this guard needs 1 token.", "Definition 57 (Guard) A condition is of the form: $(f < v), (f \\le v), (f > v), (f \\ge v), (f = v)$ , where $f$ is a fluent and $v$ either a fluent or a numeric constant.", "Then, a guard is a propositional formula of conditions, with each condition treated as a proposition, subject to the restriction that all fluents in all conditions in a guard are either locational or simple, but not both.", "Definition 58 (Interpretation of a Guard) An interpretation of a guard $G$ is a possible assignment of a value to each fleuent $f \\in G$ from the domain of $f$ .", "Definition 59 (Guard Satisfaction) A guard $G$ with simple fluents is satisfied w.r.t.", "a state $s$ , written $s \\models G$ iff $G$ has an interpretation in which each of its fluents $f$ has the value $s(f)$ and $G$ is true.", "A guard $G$ with locational fluents is satisfied w.r.r.", "a state $s$ , written $s \\models G$ iff $G$ has an interpretation in which each of its fluents $f[l]$ has the value $m_{s(l)}(f)$ and $G$ is true, where $m_X(f)$ is the multiplicity of $f$ in $X$ .", "Definition 60 (Guarded-Arc Petri Net) A Guarded-Arc Petri Net is a tuple $PN^G=(P,T,G,E,R,W,D,B,TG,MF,L)$ , where: $P & \\text{ is a finite set of places}\\\\T & \\text{ is a finite set of transitions}\\\\G & \\text{ is a set of guards as defined in definition~(\\ref {def:guard})}\\\\TG &: T \\rightarrow G \\text{ are the transition guards }\\\\E &\\subseteq (T \\times P \\times G) \\cup (P \\times T \\times G) \\text{ are the guarded arcs }\\\\R &\\subseteq P \\times T \\times G \\text{ are the guarded reset arcs }\\\\W &: E \\rightarrow \\mathbb {N}^+ \\text{ are arc weights }\\\\D &: T \\rightarrow \\mathbb {N}^+ \\text{ are the transition durations}\\\\B &: T \\rightarrow G \\times \\mathbb {N}^+ \\text{ transition stimulation or boost }\\\\MF &: T \\rightarrow 2^{G} \\text{ must-fire guards for a transition}\\\\L &: P \\rightarrow \\mathbb {N}^+ \\text{ specifies maximum number to tokens for each place }$ subject to constraints: $P \\cap T = \\emptyset $ $R \\cap E = \\emptyset $ Let $t_1 \\in T$ and $t_2 \\in T$ be any two distinct transitions, then $g1 \\in MF(t_1)$ and $g2 \\in MF(t_2)$ must not have an interpretation that make both $g1$ and $g2$ true.", "Let $t \\in T$ be a transition, and $gg_t = \\lbrace g: (t,p,g) \\in E \\rbrace \\cup \\lbrace g: (p,t,g) \\in E \\rbrace \\cup \\lbrace g: (p,t,g) \\in R \\rbrace $ be the set of arc-guards for normal and reset arcs connected to it, then $g_1 \\in gg_t, g_2 \\in gg_t$ must not have an interpretation that makes both $g_1$ and $g_2$ true.", "Let $t \\in T$ be a transition, and $gg_t = \\lbrace g: (g,n) \\in B(t) \\rbrace $ be its stimulation arc-guards, then $g_1 \\in gg_t, g_2 \\in gg_t$ must not have an interpretation that makes both $g_1$ and $g_2$ true.", "Let $t \\in T$ be a transition, and $(g,n) \\in B(t) : n>1$ , then there must not exist a place $p \\in P : (p,t,g) \\in E, L(p) = 1$ .", "Intuitively, stimulation of binary inputs is not supported.", "We will make a simplifying assumption that all places are readable by using their place names.", "Execution of the $PN^G$ occurs in discrete time steps.", "Definition 61 (Marking (or State)) Marking (or State) of a Guarded-Arc Petri Net $PN^G$ is the token assignment of each place $p_i \\in P$ .", "Initial marking $M_0 : P \\rightarrow \\mathbb {N}^0$ , while the token assignment at step $k$ is written as $M_k$ .", "Next we define the execution semantics of $PN^G$ .", "First we introduce some terminology that will be used below.", "Let $s_0 = M_0$ represent the the initial marking (or state), $s_k = M_k$ represent the marking (or state) at time step $k$ , $s_k(p)$ represent the marking of place $p$ at time step $k$ , such that $s_k = [s_k(p_0), \\dots , $ $s_k(p_n)]$ , where $P=\\lbrace p_0, \\dots , p_n\\rbrace $ $T_k$ be the firing-set that fired in step $k$ , $b_k(t)$ be the stimulation value applied to a transition $t$ w.r.t.", "step $k$ $en_k$ be the set of enabled transitions in state $s_k$ , $mf_k$ be the set of must-execute transitions in state $s_k$ , $consume_k(p,\\lbrace t_1,\\dots ,t_n\\rbrace )$ be the sum of tokens that will be consumed from place $p$ if transitions $t_1,\\dots ,t_n$ fired in state $s_k$ , $overc_k(\\lbrace t_1,\\dots ,t_n\\rbrace )$ be the set of places that will have over-consumption of tokens if transitions $t_1,\\dots ,t_n$ were to fire simultaneously in state $s_k$ , $sel_k(fs)$ be the set of possible firing-set choices in state $s_k$ using $\\mathit {fs}$ firing style $produce_k(p)$ be the total production of tokens in place $p$ (in state $s_k$ ) due to actions terminating in state $s_k$ , $s_{k+1}$ be the next state computed from current state $s_k$ due to firing of transition-set $T_k$ $min(a,b)$ gives the minimum of numbers $a,b$ , such that $min(a,b) = a \\text{ if } a < b \\text{ or } b \\text{ otherwise }$ .", "Then, the execution semantics of the guarded-arc Petri Net starting from state $s_0$ using firing-style $\\mathit {fs}$ is given as follows: $b_k(t) &={\\left\\lbrace \\begin{array}{ll}n &\\text{ if } (g,n) \\in B(t), s_k \\models g\\\\1 &\\text{ otherwise }\\end{array}\\right.", "}\\nonumber \\\\en_k &= \\lbrace t : t \\in T, s_k \\models TG(t), \\forall (p,t,g) \\in E, \\nonumber \\\\&~~~~~~~~(s_k \\models g, s_k(p) \\ge W(p,t,g) * b_k(t)) \\rbrace \\nonumber \\\\mf_k &= \\lbrace t : t \\in en_k, \\exists g \\in MF(t), s_k \\models g \\rbrace \\nonumber \\\\consume_k(p,\\lbrace t_1,\\dots ,t_n\\rbrace ) &= \\sum _{i=1,\\dots ,n}{W(p,t_i,g) * b_k(t) : (p,t_i,g) \\in E, s_k \\models g} \\nonumber \\\\&+ \\sum _{i=1,\\dots ,n}{s_k(p) : (p,t_i,g) \\in R, s_k \\models g} \\nonumber \\\\overc_k(\\lbrace t_1,\\dots ,t_n\\rbrace ) &= \\lbrace p : p \\in P, s_k(p) < consume_k(p,\\lbrace t_1,\\dots ,t_n\\rbrace ) \\rbrace \\nonumber \\\\sel_k(1) &={\\left\\lbrace \\begin{array}{ll}mf_k & \\text{ if } \|mf_k\| = 1\\\\\\lbrace \\lbrace t \\rbrace : t \\in en_k \\rbrace & \\text{ if } \|mf_k\| < 1\\end{array}\\right.", "}\\nonumber \\\\sel_k() &= \\lbrace ss : ss \\in 2^{en_k}, mf_k \\subseteq ss, overc_k(ss) = \\emptyset \\rbrace \\nonumber \\\\sel_k(max) &= \\lbrace ss : ss \\in 2^{en_k}, mf_k \\subseteq ss, overc_k(p,ss) = \\emptyset , \\nonumber \\\\&~~~~~~(\\nexists ss^{\\prime } \\in 2^{en_k} : ss \\subset ss^{\\prime }, mf_k \\subseteq ss^{\\prime }, overc_k(ss^{\\prime }) = \\emptyset ) \\rbrace \\nonumber \\\\T_k &= T^{\\prime }_k : T^{\\prime }_k \\in sel_k(\\mathit {fs}), (\\nexists t \\in en_k \\setminus T^{\\prime }_k, t \\text{ is a reset transition } ) \\nonumber \\\\produce_k(p) &= \\sum _{j=0,\\dots ,k}{W(t_i,p,g) b_j(t_i) : (t_i,p,g) \\in E, t_i \\in T_j, D(t_i)+j = k+1}\\nonumber \\\\s_{k+1}(p) &= min(s_k(p) - consume_k(p,T_k) + produce_k(p), L(p))$ Definition 62 (Trajectory) $\\sigma = s_0, T_0, s_1, \\dots , s_{k}, T_{k}, s_{k+1}$ is a trajectory of $PN^G$ iff given $s_0 = M_0$ , each $T_i$ is a possible firing-set in $s_i$ whose firing produces $s_{i+1}$ per $PN^G$ 's execution semantics in (REF ).", "Construction of Guarded-Arc Petri Net from a Pathway Specification Now we describe how to construct such a Petri Net from a given pathway specification $\\mathbf {D}$ with locational fluents.", "We proceed as follows: The set of transitions $T = \\lbrace a : a \\text{ is an action in } \\mathbf {D}\\rbrace $ .", "The set of places $P = \\lbrace f : f \\text{ is a fluent in } \\mathbf {D}\\rbrace $ .", "The limit relation for places $L(f) ={\\left\\lbrace \\begin{array}{ll}1 & \\text{ if } `\\mathbf {domain~of~} f \\mathbf {~is~} binary^{\\prime } \\in \\mathbf {D}\\\\\\infty & \\text{ otherwise }\\\\\\end{array}\\right.", "}$ .", "An arc-guard expression $guard\\_cond_1,\\dots ,guard\\_cond_n$ is translated into the conjunction $(guard_1,\\dots ,guard_n)$ , where $guard_i$ is obtained from $guard\\_cond_i$ as follows: A guard condition (REF ) is translated to $f \\ge w$ A guard condition () is translated to $f < w$ A guard condition () is translated to $f = w$ A guard condition () is translated to $f_1 > f_2$ A may-execute statement (REF ) is translated into guarded arcs as follows: Let guard $G$ be the translation of arc-guard conditions $guard\\_cond_1,\\dots ,$ $guard\\_cond_n$ specified in the may-execute proposition.", "The effect clause (REF ) are translated into arcs as follows: An effect clause with $e < 0$ is translated into an input arc $(f,a,G)$ , with arc-weight $W(f,a,G) = \|e\|$ .", "An effect clause with $e = `^{\\prime }$ is translated into a reset set $(f,a,G)$ with arc-weight $W(f,a,G) = $ .", "An effect clause with $e > 0$ is translated into an output arc $(a,f,G)$ , with arc-weight $W(a,f,G) = e$ .", "A must-execute statement (REF ) is translated into guarded arcs in the same way as may execute.", "In addition, it adds an arc-inscription $must\\text{-}exec(G)$ , where $G$ is the translation of the arc-guard.", "An inhibit statement (REF ) is translated into $IG = (guard_1,\\dots ,$ $guard_n)$ , where $(guard_1,\\dots ,$ $guard_n)$ is the translation of $(guard\\_cond\\_1,\\dots ,$ $guard\\_cond_n)$ An initial condition statement () sets the initial marking of a specific place $p$ to $w$ , i.e.", "$M_0(p) = w$ .", "An duration statement (REF ) adds a $dur(d)$ inscription to transition $a$ .", "A stimulate statement (REF ) adds a $stimulate(n,G)$ inscription to transition $a$ , where $G$ is the translation of the stimulate guard, a conjunction of $guard\\_cond_1,$ $\\dots ,guard\\_cond_n$ .", "A guard $(G_1 \\vee \\dots \\vee G_n) \\wedge (\\lnot IG_1 \\wedge \\dots \\wedge \\lnot IG_m)$ is added to each transition $a$ , where $G_i, 1 \\le i \\le n$ is a guard for a may-execute or a must-execute statement and $IG_i, 1 \\le i \\le m$ is a guard for an inhibit statement.", "A firing style statement (REF ) does not visibly appear on a Petri Net diagram, but it specifies the transition firing regime the Petri Net follows.", "Example: Consider the following pathway specification: $\\begin{array}{llll}\\mathbf {domain~of~} &f_1 \\mathbf {~is~} integer, &f_2 \\mathbf {~is~} integer\\\\t_1 \\mathbf {~may~execute~}\\\\\\;\\;\\;\\;\\;\\;\\;\\;\\;\\mathbf {causing~} & f_1 \\mathbf {~change~value~by~} -1, & f_2 \\mathbf {~change~value~by~} +1\\\\\\;\\;\\;\\;\\;\\;\\;\\;\\;\\mathbf {if~} & f_1 \\mathbf {~has~value~lower~than~} 5\\\\t_1 \\mathbf {~may~execute~}\\\\\\;\\;\\;\\;\\;\\;\\;\\;\\;\\mathbf {causing~} & f_1 \\mathbf {~change~value~by~} -2, & f_2 \\mathbf {~change~value~by~} +2\\\\\\;\\;\\;\\;\\;\\;\\;\\;\\;\\mathbf {if~} & f_1 \\mathbf {~has~value~higher~than~} 5\\\\\\mathbf {duration~of~} & t1 \\mathbf {~is~} 10\\\\\\mathbf {inhibit~} t1 \\mathbf {~if~} & f_1 \\mathbf {~has~value~higher~than~} 7\\\\\\mathbf {inhibit~} t1 \\mathbf {~if~} & f_1 \\mathbf {~has~value~lower~than~} 3\\\\\\mathbf {normally~stimulate~} t1 & \\mathbf {~by~factor~} 3 \\\\\\;\\;\\;\\;\\;\\;\\;\\;\\;\\mathbf {~if~} & f_2 \\mathbf {~has~value~higher~than~} 5\\\\\\mathbf {initially~}& f_1 \\mathbf {~has~value~} 0, & f_2 \\mathbf {~has~value~} 0,\\\\\\mathbf {firing~style~} & max\\end{array}$ This pathway specification is encoded as the Petri Net in figure REF .", "Guarded-Arc Petri Net with Colored Tokens Next we extend the Guarded-arc Petri Nets to add Colored tokens.", "We will use this extension to model pathways with locational fluents.", "Figure: Example of a guarded-arc Petri Net with colored tokens.Figure REF shows an example of a guarded-arc Petri Net with colored tokens.", "There are two arc-guard expressions $f3[p1]<5$ and $f3[p1]>5$ .", "When $f3[p1]<5$ , $t1$ consumes one token of color $f1$ from place $p1$ and produces one token of color $f2$ in place $p2$ .", "When $f3[p1]>5$ , $t1$ 's consumption and production of the same colored tokens doubles.", "The transition $t1$ implicitly gets the guards for each arc represented the or-ed conditions $((f3[p1]<5) \\vee (f3[p1]>5))$ .", "The arc also has two conditions inhibiting it, they are represented by the and-ed conditions $\\lnot (f3[p1]>7) \\wedge \\lnot (f3[p1]<3)$ , where `$\\lnot $ ' represents logical not.", "Transition $t1$ is stimulated by factor 3 when $f3[p1]>5$ and it has a duration of 10 time units.", "Definition 63 (Guarded-Arc Petri Net with Colored Tokens) A Guarded-Arc Petri Net with Colored Tokens is a tuple $PN^{GC}=(P,T,C,G,E,R,W,D,B,TG,$ $MF,L)$ , such that: $P &: \\text{finite set of places}\\\\T &: \\text{finite set of transitions}\\\\C &: \\text{finite set of colors}\\\\G &: \\text{set of guards as defined in definition~(\\ref {def:guard}) with locational fluents}\\\\TG &: T \\rightarrow G \\text{ are the transition guards }\\\\E &\\subseteq (T \\times P \\times G) \\cup (P \\times T \\times G) \\text{ are the guarded arcs }\\\\R &\\subseteq P \\times T \\times G \\text{ are the guarded reset arcs }\\\\W &: E \\rightarrow \\langle C,m \\rangle \\text{ are arc weights; each arc weight is a multiset over } C\\\\D &: T \\rightarrow \\mathbb {N}^+ \\text{ are the transition durations}\\\\B &: T \\rightarrow G \\times \\mathbb {N}^+ \\text{ transition stimulation or boost }\\\\MF &: T \\rightarrow 2^{G} \\text{ must-fire guards for a transition}\\\\L &: P \\times C \\rightarrow \\mathbb {N}^+ \\text{ specifies maximum number of tokens for each color in each place}\\\\$ subject to constraints: $P \\cap T = \\emptyset $ $R \\cap E = \\emptyset $ Let $t_1 \\in T$ and $t_2 \\in T$ be any two distinct transitions, then $g1 \\in MF(t_1)$ and $g2 \\in MF(t_2)$ must not have an interpretation that make both $g1$ and $g2$ true.", "Let $t \\in T$ be a transition, and $gg_t = \\lbrace g: (t,p,g) \\in E \\rbrace \\cup \\lbrace g: (p,t,g) \\in E \\rbrace \\cup \\lbrace g: (p,t,g) \\in R \\rbrace $ be the set of arc-guards for normal and reset arcs connected to it, then $g_1 \\in gg_t, g_2 \\in gg_t$ must not have an interpretation that makes both $g_1$ and $g_2$ true.", "Let $t \\in T$ be a transition, and $gg_t = \\lbrace g: (g,n) \\in B(t) \\rbrace $ be its stimulation guards, then $g_1 \\in gg_t, g_2 \\in gg_t$ must not have an interpretation that makes both $g_1$ and $g_2$ true.", "Let $t \\in T$ be a transition, and $(g,n) \\in B(t) : n>1$ , then there must not exist a place $p \\in P$ and a color $c \\in C$ such that $(p,t,g) \\in E, L(p,c) = 1$ .", "Intuitively, stimulation of binary inputs is not supported.", "Definition 64 (Marking (or State)) Marking (or State) of a Guarded-Arc Petri Net with Colored Tokens $PN^{GC}$ is the colored token assignment of each place $p_i \\in P$ .", "Initial marking is written as $M_0 : P \\rightarrow \\langle C,m \\rangle $ , while the token assignment at step $k$ is written as $M_k$ .", "We make a simplifying assumption that all places are readable by using their place name.", "Next we define the execution semantics of the guarded-arc Petri Net.", "First we introduce some terminology that will be used below.", "Let $s_0 = M_0$ represent the the initial marking (or state), $s_k = M_k$ represent the marking (or state) at time step $k$ , $s_k(p)$ represent the marking of place $p$ at time step $k$ , such that $s_k = [s_k(p_0),$ $\\dots ,s_k(p_n)]$ , where $P=\\lbrace p_0,\\dots ,p_n\\rbrace $ .", "$T_k$ be the firing-set that fired in state $s_k$ , $b_k(t)$ be the stimulation value applied to transition $t$ w.r.t.", "state $s_k$ , $en_k$ be the set of enabled transitions in state $s_k$ , $mf_k$ be the set of must-fire transitions in state $s_k$ , $consume_k(p,\\lbrace t_1,\\dots ,t_n\\rbrace )$ be the sum of colored tokens that will be consumed from place $p$ if transitions $t_1,\\dots ,t_n$ fired in state $s_k$ , $overc_k(\\lbrace t_1,\\dots ,t_n\\rbrace )$ be the set of places that will have over-consumption of tokens if transitions $t_1,\\dots ,t_n$ were to fire simultaneously in state $s_k$ , $sel_k(fs)$ be the set of possible firing-sets in state $s_k$ using $\\mathit {fs}$ firing style, $produce_k(p)$ be the total production of tokens in place $p$ in state $s_k$ due to actions terminating in state $s_k$ , $s_{k+1}$ be the next state computed from current state $s_k$ and $T_k$ , $m_X(c)$ represents the multiplicity of $c \\in C$ in multiset $X=\\langle C,m \\rangle $ , $c/n$ represents repetition of an element $c$ of a multi-set $n$ -times, multiplication of multiset $X= \\langle C,m \\rangle $ with a number $n$ be defined in terms of multiplication of element multiplicities by $n$ , i.e.", "$\\forall c \\in C, m_X(c)n$ , and $min(a,b)$ gives the minimum of numbers $a,b$ , such that $min(a,b) = a \\text{ if } a < b \\text{ or } b \\text{ otherwise }$ .", "Then, the execution semantics of the guarded-arc Petri Net starting from state $s_0$ using firing-style $\\mathit {fs}$ is given as follows: $b_k(t) &={\\left\\lbrace \\begin{array}{ll}n & \\text{ if } (g,n) \\in B(t), s_k \\models g\\\\1 &\\text{ otherwise }\\end{array}\\right.", "}\\nonumber \\\\en_k &= \\lbrace t \\in T, s_k \\models TG(t), \\forall (p,t,g) \\in E, \\nonumber \\\\&~~~~~~(s_k \\models g, s_k(p) \\ge W(p,t,g) b_k(t)) \\rbrace \\nonumber \\\\mf_k &= \\lbrace t \\in en_k, \\exists g \\in MF(t), s_k \\models g \\rbrace \\nonumber \\\\consume_k(p,\\lbrace t_1,\\dots ,t_n\\rbrace ) &= \\sum _{i=1,\\dots ,n}{W(p,t_i,g) * b_k(t) : (p,t_i,g) \\in E, s_k \\models g} \\nonumber \\\\&+ \\sum _{i=1,\\dots ,n}{s_k(p) : (p,t_i,g) \\in R, s_k \\models g} \\nonumber \\\\overc_k(\\lbrace t_1,\\dots ,t_n\\rbrace ) &= \\lbrace p \\in P : \\exists c \\in C, m_{s_k(p)}(c) < m_{consume_k(p,\\lbrace t_1,\\dots ,t_n\\rbrace )}(c) \\rbrace \\nonumber \\\\sel_k(1) &={\\left\\lbrace \\begin{array}{ll}mf_k & \\text{ if } \|mf_k\| = 1\\\\\\lbrace \\lbrace t \\rbrace : t \\in en_k \\rbrace & \\text{ if } \|mf_k\| < 1\\end{array}\\right.", "}\\nonumber \\\\sel_k() &={\\left\\lbrace \\begin{array}{ll}ss \\in 2^{en_k} : mf_k \\subseteq ss, overc_k(ss) = \\emptyset \\end{array}\\right.", "}\\nonumber \\\\sel_k(max) &={\\left\\lbrace \\begin{array}{ll}ss \\in 2^{en_k} : mf_k \\subseteq ss, overc_k(p,ss) = \\emptyset , \\\\(\\nexists ss^{\\prime } \\in 2^{en_k} : ss \\subset ss^{\\prime }, mf_k \\subseteq ss^{\\prime }, overc_k(ss^{\\prime }) = \\emptyset )\\end{array}\\right.", "}\\nonumber \\\\T_k &= T^{\\prime }_k : T^{\\prime }_k \\in sel_k(\\mathit {fs}), (\\nexists t \\in en_k \\setminus T^{\\prime }_k, t \\text{ is a reset transition } ) \\nonumber \\\\produce_k(p) &= \\sum _{j=0,\\dots ,k}{W(t_i,p,g) b_j(t_i) : (t_i,p,g) \\in E, t_i \\in T_j), D(t_i)+j = k+1}\\nonumber \\\\s_{k+1} &= [ c/n : c \\in C, \\nonumber \\\\&~~~~n=min(m_{s_k(p)}(c) \\nonumber \\\\&~~~~~~~~~~~~ - m_{consume_k(p,T_k)}(c) \\nonumber \\\\&~~~~~~~~~~~~ + m_{produce_k(p)}(c), L(p,c)) ]$ Definition 65 (Trajectory) $\\sigma = s_0, T_0, s_1, \\dots , s_{k}, T_{k}, s_{k+1}$ is a trajectory of $PN^{GC}$ iff given $s_0=M_0$ , each $T_i$ is a possible firing-set in $s_i$ whose firing produces $s_{i+1}$ per $PN^{GC}$ 's execution semantics in (REF ).", "Construction of Guarded-Arc Petri Net with Colored Tokens from a Pathway Specification with Locational Fluents Now we describe how to construct such a Petri Net from a given pathway specification $\\mathbf {D}$ with locational fluents.", "We proceed as follows: The set of transitions $T = \\lbrace a : a \\text{ is an action in } \\mathbf {D}\\rbrace $ .", "The set of colors $C = \\lbrace f : f[l] \\text{ is a fluent in } \\mathbf {D}\\rbrace $ .", "The set of places $P = \\lbrace l : f[l] \\text{ is a fluent in } \\mathbf {D}\\rbrace $ .", "The limit relation for each colored token in a place $L(f,c) ={\\left\\lbrace \\begin{array}{ll}1 & \\text{ if } `\\mathbf {domain~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} binary^{\\prime } \\in \\mathbf {D}\\\\\\infty & \\text{ otherwise }\\\\\\end{array}\\right.", "}$ .", "A guard expression $guard\\_cond_1,\\dots ,guard\\_cond_n$ is translated into the conjunction $(guard_1,\\dots ,$ $guard_n)$ , where $guard_i$ is obtained from $guard\\_cond_i$ as follows: A guard condition () is translated to $f[l] < w$ A guard condition () is translated to $f[l] = w$ A guard condition () is translated to $f[l] \\ge w$ A guard condition () is translated to $f_1[l_1] > f_2[l_2]$ A may-execute statement (REF ) is translated into guarded arcs as follows: Let guard $G$ be the translation of guard conditions $guard\\_cond_1,$ $\\dots ,guard\\_cond_n$ specified in the may-execute proposition.", "The effect clauses of the form () are grouped into input, reset and output effect sets for an action as follows: The clauses with $e < 0$ for the same place $l$ are grouped together into an input set of $a$ requiring input from place $l$ .", "The clauses with $e = `^{\\prime }$ for the same place $l$ are grouped together into a reset set of $a$ requiring input from place $l$ .", "The clauses with $e > 0$ for the same place $l$ are grouped together into an output set of $a$ to place $l$ .", "A group of input effect clauses $\\mathit {effect}_1,\\dots ,\\mathit {effect}_m, m > 0$ of the form () is translated into an input arc $(l,a,G)$ , with arc-weight $W(l,a,G) = w^+$ , where $w^+$ is the multi-set union of $f_i/\|e_i\|$ in $\\mathit {effect}_i, 1 \\le i \\le m$ .", "A group of output effect clauses $\\mathit {effect}_1,\\dots ,\\mathit {effect}_m, m > 0$ of the form () is translated into an output arc $(a,l,G)$ , with arc-weight $W(a,l,G) = w^-$ , where $w^-$ is the multi-set union of $f_i/e_i$ in $\\mathit {effect}_i, 1 \\le i \\le m$ .", "A group of reset effect clauses $\\mathit {effect}_1,\\dots ,\\mathit {effect}_m, m > 0$ of the form () is translated into a reset arc $(l,a,G)$ with arc-weight $W(l,a,G) = $ .", "A must-execute statement (REF ) is translated into guarded arcs in the same way as may execute.", "In addition, it adds an arc-inscription $must\\text{-}exec(G)$ , where $G$ is the guard, which is the translation of $guard\\_cond_1,\\dots ,guard\\_cond_n$ .", "An inhibit statement (REF ) is translated into $IG = (guard_1,\\dots ,guard_n)$ , where $(guard_1,\\dots ,guard_n)$ is the translation of $(guard\\_cond\\_1,\\dots ,guard\\_cond_n)$ An initial condition statement () sets the initial marking of a specific place $l$ for a specific color $f$ to $w$ , i.e.", "$m_{(M_0(l))}(f) = w$ .", "An duration statement (REF ) adds a $dur(d)$ inscription to transition $a$ .", "A stimulate statement (REF ) adds a $stimulate(n,G)$ inscription to transition $a$ , where guard $G$ is the translation of its guard expression $guard\\_cond_1,$ $\\dots ,guard\\_cond_n$ .", "A guard $(G_1 \\vee \\dots \\vee G_n) \\wedge (\\lnot IG_1 \\wedge \\dots \\wedge \\lnot IG_m)$ is added to each transition $a$ , where $G_i, 1 \\le i \\le n$ is the guard for a may-execute or a must-execute proposition and $IG_i, 1 \\le i \\le m$ is a guard for an inhibit proposition.", "A firing style statement (REF ) does not visibly show on a Petri Net, but it specifies the transition firing regime the Petri Net follows.", "Example Consider the following pathway specification: $&\\mathbf {domain~of~} \\nonumber \\\\&~~~~f_1 \\mathbf {~atloc~} l_1 \\mathbf {~is~} integer, f_2 \\mathbf {~atloc~} l_1 \\mathbf {~is~} integer, \\nonumber \\\\&~~~~f_3 \\mathbf {~atloc~} l_1 \\mathbf {~is~} integer, f_1 \\mathbf {~atloc~} l_2 \\mathbf {~is~} integer, \\nonumber \\\\&~~~~f_2 \\mathbf {~atloc~} l_2 \\mathbf {~is~} integer, f_3 \\mathbf {~atloc~} l_2 \\mathbf {~is~} integer \\nonumber \\\\&t_1 \\mathbf {~may~execute~causing} \\nonumber \\\\&~~~~~~~~f_1 \\mathbf {~atloc~} l_1 \\mathbf {~change~value~by~} -1, \\nonumber \\\\&~~~~~~~~f_2 \\mathbf {~atloc~} l_2 \\mathbf {~change~value~by~} +1 \\nonumber \\\\&~~~~\\mathbf {if~} f_3 \\mathbf {~atloc~} l_1 \\mathbf {~has~value~lower~than~} 5 \\nonumber \\\\&t_1 \\mathbf {~may~execute~causing} \\nonumber \\\\&~~~~~~~~f_1 \\mathbf {~atloc~} l_1 \\mathbf {~change~value~by~} -2, \\nonumber \\\\&~~~~~~~~f_2 \\mathbf {~atloc~} l_2 \\mathbf {~change~value~by~} +2 \\nonumber \\\\&~~~~\\mathbf {if~} f_3 \\mathbf {~atloc~} l_1 \\mathbf {~has~value~higher~than~} 5 \\nonumber \\\\&\\mathbf {duration~of~} t1 \\mathbf {~is~} 10 \\nonumber \\\\&\\mathbf {inhibit~} t1 \\mathbf {~if~} f_3 \\mathbf {~atloc~} l_1 \\mathbf {~has~value~higher~than~} 7 \\nonumber \\\\&\\mathbf {inhibit~} t1 \\mathbf {~if~} f_3 \\mathbf {~atloc~} l_1 \\mathbf {~has~value~lower~than~} 3 \\nonumber \\\\&\\mathbf {normally~stimulate~} t1 \\mathbf {~by~factor~} 3 \\nonumber \\\\&~~~~\\mathbf {~if~} f_2 \\mathbf {~atloc~} l_2 \\mathbf {~has~value~higher~than~} 5 \\nonumber \\\\&\\mathbf {initially~} \\nonumber \\\\&~~~~f_1 \\mathbf {~atloc~} l_1 \\mathbf {~has~value~} 0, f_1 \\mathbf {~atloc~} l_2 \\mathbf {~has~value~} 0, \\nonumber \\\\&~~~~f_2 \\mathbf {~atloc~} l_1 \\mathbf {~has~value~} 0, f_2 \\mathbf {~atloc~} l_2 \\mathbf {~has~value~} 0, \\nonumber \\\\&~~~~f_3 \\mathbf {~atloc~} l_1 \\mathbf {~has~value~} 0, f_3 \\mathbf {~atloc~} l_2 \\mathbf {~has~value~} 0 \\nonumber \\\\&\\mathbf {firing~style~} max$ Syntax of Query Language (BioPathQA-QL) The alphabet of query language $\\mathcal {Q}$ consists of the same sets $A,F,L$ from $\\mathcal {P}$ representing actions, fluents, and locations, respectively; a fixed set of reserved keywords $K$ shown in bold in syntax below; a fixed set $\\lbrace `:^{\\prime }, `;^{\\prime }, `,^{\\prime } , `^{\\prime \\prime }\\rbrace $ of punctuations; a fixed set of $\\lbrace `<^{\\prime },`>^{\\prime },`=^{\\prime }\\rbrace $ of directions; and constants.", "Our query language asks questions about biological entities and processes in a biological pathway described through the pathway specification language.", "This is our domain description.", "A query statement is composed of a query description (the quantity, observation, or condition being sought by the question), interventions (changes to the pathway), observations (about states and actions of the pathway), and initial setup conditions.", "The query statement is evaluated against the trajectories of the pathway, generated by simulating the pathway.", "These trajectories are modified by the initial setup and interventions.", "The resulting trajectories are then filtered to retain only those which satisfy the observations specified in the query statement.", "A query statement can take various forms: The simplest queries do not modify the pathway and check if a specific observation is true on a trajectory or not.", "An observation can be a point observation or an interval observation depending upon whether they can be evaluated w.r.t.", "a point or an interval on a trajectory.", "More complex queries modify the pathway in various ways and ask for comparison of an observation before and after such modification.", "Following query statements about the rate of production of `$bpg13$ ' illustrate the kind of queries that can be asked from our system about the specified glycolysis pathway as given in [64].", "Determine if `$n$ ' is a possible rate of production of substance `$bpg13$ ': $\\mathbf {rate~} & \\mathbf {of~production~of~} ^{\\prime }bpg13^{\\prime } \\mathbf {~is~} n \\nonumber \\\\&\\mathbf {when~observed~between~time~step~} 0 \\mathbf {~and~} \\mathbf {~time~step~} k ; &$ Determine if `$n$ ' is a possible rate of production of substance `$bpg13$ ' in a pathway when it is being supplied with a limited supply of an upstream substance `$f16bp$ ': $\\mathbf {rate}&\\mathbf {~of~production~of~} ^{\\prime }bpg13^{\\prime } \\mathbf {~is~} n \\nonumber \\\\&\\mathbf {when~observed~between~time~step~} 0 \\mathbf {~and~} \\mathbf {~time~step~} k ; \\nonumber \\\\&\\mathbf {using~initial~setup:~} \\mathbf {set~value~of~} `f16bp^{\\prime } \\mathbf {~to~} 5 ; &$ Determine if `$n$ ' is a possible rate of production of substance `$bpg13$ ' in a pathway when it is being supplied with a steady state supply of an upstream substance `$f16bp$ ' at the rate of 5 units per time-step: $\\mathbf {rate}&\\mathbf {~of~production~of~} ^{\\prime }bpg13^{\\prime } \\mathbf {~is~} n \\nonumber \\\\&\\mathbf {when~observed~between~time~step~} 0 \\mathbf {~and~} \\mathbf {~time~step~} k ; \\nonumber \\\\&\\mathbf {using~initial~setup:~} \\mathbf {continuously~supply~} `f16bp^{\\prime } \\mathbf {~in~quantity~} 5 ; &$ Determine if `$n$ ' is a possible rate of production of substance `$bpg13$ ' in a pathway when it is being supplied with a steady state supply of an upstream substance `$f16bp$ ' at the rate of 5 units per time-step and the pathway is modified to remove all quantity of the substance '$dhap$ ' as soon as it is produced: $\\mathbf {rate}&\\mathbf {~of~production~of~} ^{\\prime }bpg13^{\\prime } \\mathbf {~is~} n \\nonumber \\\\&\\mathbf {when~observed~between~time~step~} 0 \\mathbf {~and~} \\mathbf {~time~step~} k ; \\nonumber \\\\&\\mathbf {due~to~interventions:~} \\mathbf {remove~} `dhap^{\\prime } \\mathbf {~as~soon~as~produced} ;\\nonumber \\\\&\\mathbf {using~initial~setup:~} \\mathbf {continuously~supply~} `f16bp^{\\prime } \\mathbf {~in~quantity~} 5 ; &$ Determine if `$n$ ' is a possible rate of production of substance `$bpg13$ ' in a pathway when it is being supplied with a steady state supply of an upstream substance `$f16bp$ ' at the rate of 5 units per time-step and the pathway is modified to remove all quantity of the substance '$dhap$ ' as soon as it is produced and a non-functional pathway process / reaction named `$t5b$ ': $\\mathbf {rate}&\\mathbf {~of~production~of~} ^{\\prime }bpg13^{\\prime } \\mathbf {~is~} n \\nonumber \\\\&\\mathbf {when~observed~between~time~step~} 0 \\mathbf {~and~} \\mathbf {~time~step~} k ; \\nonumber \\\\&\\mathbf {due~to~interventions:~} \\mathbf {remove~} `dhap^{\\prime } \\mathbf {~as~soon~as~produced} ;\\nonumber \\\\&\\mathbf {due~to~observations:~} `t5b^{\\prime } \\mathbf {~does~not~occur} ; \\nonumber \\\\&\\mathbf {using~initial~setup:~} \\mathbf {continuously~supply~} `f16bp^{\\prime } \\mathbf {~in~quantity~} 5 ; &$ Determine if `$n$ ' is the average rate of production of substance `$bpg13$ ' in a pathway when it is being supplied with a steady state supply of an upstream substance `$f16bp$ ' at the rate of 5 units per time-step and the pathway is modified to remove all quantity of the substance '$dhap$ ' as soon as it is produced and a non-functional pathway process / reaction named `$t5b$ ': $\\mathbf {average}&\\mathbf {~rate~of~production~of~} ^{\\prime }bpg13^{\\prime } \\mathbf {~is~} n \\nonumber \\\\&\\mathbf {when~observed~between~time~step~} 0 \\mathbf {~and~} \\mathbf {~time~step~} k ; \\nonumber \\\\&\\mathbf {due~to~interventions:~} \\mathbf {remove~} `dhap^{\\prime } \\mathbf {~as~soon~as~produced} ;\\nonumber \\\\&\\mathbf {due~to~observations:~} `t5b^{\\prime } \\mathbf {~does~not~occur} ; \\nonumber \\\\&\\mathbf {using~initial~setup:~} \\mathbf {continuously~supply~} `f16bp^{\\prime } \\mathbf {~in~quantity~} 5 ; &$ Determine if `$d$ ' is the direction of change in the average rate of production of substance `$bpg13$ ' with a steady state supply of an upstream pathway input when compared with a pathway with the same steady state supply of an upstream pathway input, but in which the substance `$dhap$ ' is removed from the pathway as soon as it is produced and pathway process / reaction called `$t5b$ ' is non-functional: $\\mathbf {dir}&\\mathbf {ection~of~change~in~average~rate~of~production~of~} ^{\\prime }bpg13^{\\prime } \\mathbf {~is~} d \\nonumber \\\\&\\mathbf {when~observed~between~time~step~} 0 \\mathbf {~and~} \\mathbf {~time~step~} k ; \\nonumber \\\\&\\mathbf {comparing~nominal~pathway~with~modified~pathway~obtained~} \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\mathbf {due~to~interventions:~} \\mathbf {remove~} `dhap^{\\prime } \\mathbf {~as~soon~as~produced~} ;\\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\mathbf {due~to~observations:~} `t5b^{\\prime } \\mathbf {~does~not~occur} ; \\nonumber \\\\&\\mathbf {using~initial~setup:~} \\mathbf {continuously~supply~} `f16bp^{\\prime } \\mathbf {~in~quantity~} 5 ; &$ Queries can also be about actions, as illustrated in the following examples.", "Determine if action `$t5b$ ' ever occurs when there is a continuous supply of `$f16bp$ ' is available and `$t5a$ ' is disabled: $`t5b^{\\prime } &\\mathbf {~occurs~} ;\\nonumber \\\\&\\mathbf {due~to~interventions:~} \\mathbf {disable~} `t5a^{\\prime } ; \\nonumber \\\\&\\mathbf {using~initial~setup:~} \\mathbf {continuously~produce~} `f16bp^{\\prime } \\mathbf {~in~quantity~} 5 ; &$ Determine if glycolysis ($`gly^{\\prime }$ ) gets replaced with beta-oxidation ($`box^{\\prime }$ ) when sugar ($`sug^{\\prime }$ ) is exhausted but fatty acids ($`fac^{\\prime }$ ) are available, when starting with a fixed initial supply of sugar and fatty acids in quantity 5: $&`gly^{\\prime } \\mathbf {~switches~to~} `box^{\\prime } \\mathbf {~when~} \\nonumber \\\\&~~~~\\mathbf {value~of~} ^{\\prime }sug^{\\prime } \\mathbf {~is~} 0, \\nonumber \\\\&~~~~\\mathbf {value~of~} ^{\\prime }fac^{\\prime } \\mathbf {~is~higher~than~} 0 \\nonumber \\\\&~~~~\\mathbf {in~all~trajectories} ; \\nonumber \\\\&\\mathbf {due~to~observations:} \\nonumber \\\\&~~~~`gly^{\\prime } \\mathbf {~switches~to~} `box^{\\prime } \\nonumber \\\\&\\mathbf {using~initial~setup:~} \\nonumber \\\\&~~~~\\mathbf {set~value~of~} `sug^{\\prime } \\mathbf {~to~} 5, \\nonumber \\\\&~~~~\\mathbf {set~value~of~} `fac^{\\prime } \\mathbf {~to~} 5 ; &$ Next we define various syntactic elements of a query statement, give their intuitive meaning, and how these components fit together to form a query statement.", "We will define the formal semantics in a later section.", "Note that some of the single-trajectory queries can be represented as LTL formulas.", "However, we have chosen to keep the current representation as it is more intuitive for our biological domain.", "In the following description, $f_i$ 's are fluents, $l_i$ 's are locations, $n$ 's are numbers, $q$ 's are positive integer numbers, $d$ is one of the directions from $\\lbrace <,>,=\\rbrace $ .", "Definition 66 (Point) A point is a time-step on the trajectory.", "It has the form: $&\\mathbf {time~step~} ts$ Definition 67 (Interval) An interval is a sub-sequence of time-steps on a trajectory.", "It has the form: $&\\langle point \\rangle \\mathbf {~and~} \\langle point \\rangle $ Definition 68 (Aggregate Operator (aggop)) An aggregate operator computes an aggregate quantity over a sequence of values.", "It can be one of the following: $& \\mathbf {minimum}\\\\& \\mathbf {maximum}\\\\& \\mathbf {average}$ Definition 69 (Quantitative Interval Formula) A quantitative interval formula is a formula that is evaluated w.r.t.", "an interval on a trajectory for some quantity $n$ .", "$&\\mathbf {rate~of~production~of~} f \\mathbf {~is~} n\\\\&\\mathbf {rate~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n\\\\& \\mathbf {rate~of~firing~of~} a \\mathbf {~is~} n\\\\&\\mathbf {total~production~of~} f \\mathbf {~is~} n\\\\&\\mathbf {total~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n$ Intuitively, the rate of production of a fluent $f$ in interval $s_i,\\dots ,s_j$ on a trajectory $s_0,T_0,\\dots ,T_{k-1},s_k$ is $n=(s_j(f)-s_i(f))/(j-i)$ ; rate of firing of an action $a$ in interval $s_i,\\dots ,s_j$ is $n=\|\\lbrace T_l : a \\in T_l, i \\le l \\le j-1\\rbrace \|/(j-i)$ ; and total production of a fluent $f$ in interval $s_i,\\dots ,s_j$ is $n=s_j(f)-s_i(f)$ .", "If the given $n$ equals the computed $n$ , then the formula holds.", "The same intuition extends to locational fluents, except that fluent $f$ is replaced by $f[l]$ , e.g.", "rate of production of fluent $f$ at location $l$ in interval $s_i,\\dots ,s_j$ on a trajectory is $n=(s_j(f[l])-s_i(f[l]))/(j-i)$ .", "In biological context, the actions represent reactions and fluents substances used in these reactions.", "The rate and total production formulas are used in aggregate observations to determine if reactions are slowing down or speeding up during various portions of a simulation.", "Definition 70 (Quantitative Point Formula) A quantitative point formula is a formula that is evaluated w.r.t.", "a point on a trajectory for some quantity $n$ .", "$&\\mathbf {value~of~} f \\mathbf {~is~higher~than~} n\\\\&\\mathbf {value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~higher~than~} n\\\\&\\mathbf {value~of~} f \\mathbf {~is~lower~than~} n\\\\&\\mathbf {value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~lower~than~} n\\\\&\\mathbf {value~of~} f \\mathbf {~is~} n\\\\&\\mathbf {value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n$ Definition 71 (Qualitative Interval Formula) A qualitative interval formula is a formula that is evaluated w.r.t.", "an interval on a trajectory.", "$& f \\mathbf {~is~accumulating~}\\\\& f \\mathbf {~is~accumulating~} \\mathbf {~atloc~} l \\\\& f \\mathbf {~is~decreasing~}\\\\& f \\mathbf {~is~decreasing~} \\mathbf {~atloc~} l$ Intuitively, a fluent $f$ is accumulating in interval $s_i,\\dots ,s_j$ on a trajectory if $f$ 's value monotonically increases during the interval.", "A fluent $f$ is decreasing in interval $s_i,\\dots ,s_j$ on a trajectory if $f$ 's value monotonically decreases during the interval.", "The same intuition extends to locational fluents by replacing $f$ with $f[l]$ .", "Definition 72 (Qualitative Point Formula) A qualitative point formula is a formula that is evaluated w.r.t.", "a point on a trajectory.", "$& a \\mathbf {~occurs}\\\\& a \\mathbf {~does~not~occur}\\\\& a1 \\mathbf {~switches~to~} a2\\\\$ Intuitively, an action occurs at a point $i$ on the trajectory if $a \\in T_i$ ; an action does not occur at point $i$ if $a \\notin T_i$ ; an action $a1$ switches to $a2$ at point $i$ if $a1 \\in T_{i-1}$ , $a2 \\notin T_{i-1}$ , $a1 \\notin T_i$ , $a2 \\in T_i$ .", "Definition 73 (Quantitative All Interval Formula) A quantitative all interval formula is a formula that is evaluated w.r.t.", "an interval on a set of trajectories $\\sigma _1,\\dots ,\\sigma _m$ and corresponding quantities $r_1,\\dots ,r_m$ .", "$&\\mathbf {~rates~of~production~of~} f \\mathbf {~are~} [r_1,\\dots ,r_m]\\\\&\\mathbf {~rates~of~production~of~} f \\mathbf {~altoc~} l \\mathbf {~are~} [r_1,\\dots ,r_m]\\\\&\\mathbf {~rates~of~firing~of~} f \\mathbf {~are~} [r_1,\\dots ,r_m]\\\\&\\mathbf {~totals~of~production~of~} f \\mathbf {~are~} [r_1,\\dots ,r_m]\\\\&\\mathbf {~totals~of~production~of~} f \\mathbf {~altoc~} l \\mathbf {~are~} [r_1,\\dots ,r_m]$ Intuitively, a quantitative all interval formula holds on some interval $[i,j]$ over a set of trajectories $\\sigma _1,\\dots ,\\sigma _m$ for values $[r_1,\\dots ,r_m]$ if for each $r_x$ the corresponding quantitative interval formula holds in interval $[i,j]$ in trajectory $\\sigma _x$ .", "For example, $\\mathbf {rates~of~production~of~} f \\mathbf {~are~} [r_1,\\dots ,r_m]$ in interval $[i,j]$ over a set of trajectories $\\sigma _1,\\dots ,\\sigma _m$ if for each $x \\in \\lbrace 1 \\dots m\\rbrace $ , $\\textbf {rate~of~production~of~} f \\mathbf {~is~} r_x$ in interval $[i,j]$ in trajectory $\\sigma _x$ .", "Definition 74 (Quantitative All Point Formula) A quantitative all point formula is a formula that is evaluated w.r.t.", "a point on a set of trajectories $\\sigma _1,\\dots ,\\sigma _m$ and corresponding quantities $r_1,\\dots ,r_m$ .", "$&\\mathbf {values~of~} f \\mathbf {~are~} [r_1,\\dots ,r_m]\\\\&\\mathbf {values~of~} f \\mathbf {~atloc~} l \\mathbf {~are~} [r_1,\\dots ,r_m]$ Intuitively, a quantitative all point formula holds at some point $i$ over a set of trajectories $\\sigma _1,\\dots ,\\sigma _m$ for values $[r_1,\\dots ,r_m]$ if for each $r_x$ the corresponding quantitative point formula holds at point $i$ in trajectory $\\sigma _x$ .", "For example, $\\mathbf {values~of~} f \\mathbf {~are~} [r_1,\\dots ,r_m]$ at point $i$ over a set of trajectories $\\sigma _1,\\dots ,\\sigma _m$ if for each $x \\in \\lbrace 1\\dots m\\rbrace $ , $\\textbf {value~of~} f \\mathbf {~is~} r_x$ at point $i$ in trajectory $\\sigma _x$ .", "Definition 75 (Quantitative Aggregate Interval Formula) A quantitative aggregate interval formula is a formula that is evaluated w.r.t.", "an interval on a set of trajectories $\\sigma _1,\\dots ,\\sigma _m$ and an aggregate value $r$ , where $r$ is the aggregate of $[r_1,\\dots ,r_m]$ using $aggop$ .", "$&\\langle aggop \\rangle \\mathbf {~rate~of~production~of~} f \\mathbf {~is~} n\\\\&\\langle aggop \\rangle \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n\\\\&\\langle aggop \\rangle \\mathbf {~rate~of~firing~of~} a \\mathbf {~is~} n\\\\&\\langle aggop \\rangle \\mathbf {~total~production~of~} f \\mathbf {~is~} n\\\\&\\langle aggop \\rangle \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n$ Intuitively, a quantitative aggregate interval formula holds on some interval $[i,j]$ over a set of trajectories $\\sigma _1,\\dots ,\\sigma _m$ for a value $r$ if there exist $[r_1,\\dots ,r_m]$ whose aggregate value per $aggop$ is $r$ , such that for each $r_x$ the quantitative interval formula (corresponding to the quantitative aggregate interval formual) holds in interval $[i,j]$ in trajectory $\\sigma _x$ .", "For example, $average \\mathbf {~rate~of~}$ $\\mathbf {production~of~} f \\mathbf {~is~} r$ in interval $[i,j]$ over a set of trajectories $\\sigma _1,\\dots ,\\sigma _m$ if $r=(r_1+\\dots +r_m)/m$ and for each $x \\in \\lbrace 1\\dots m\\rbrace $ , $\\textbf {rate~of~production~of~} f \\mathbf {~is~} r_x$ in interval $[i,j]$ in trajectory $\\sigma _x$ .", "Definition 76 (Quantitative Aggregate Point Formula) A quantitative aggregate point formula is a formula that is evaluated w.r.t.", "a point on a set of trajectories $\\sigma _1,\\dots ,\\sigma _m$ and an aggregate value $r$ , where $r$ is the aggregate of $[r_1,\\dots ,r_m]$ using $aggop$ .", "$&\\langle aggop \\rangle \\mathbf {~value~of~} f \\mathbf {~is~} r\\\\&\\langle aggop \\rangle \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} r$ Intuitively, a quantitative aggregate point formula holds at some point $i$ over a set of trajectories $\\sigma _1,\\dots ,\\sigma _m$ for a value $r$ if there exist $[r_1,\\dots ,r_m]$ whose aggregate value per $aggop$ is $r$ , such that for each $r_x$ the quantitative point formula (corresponding to the quantitative aggregate point formual) holds at point $i$ in trajectory $\\sigma _x$ .", "For example, $average \\mathbf {~value~of~} f \\mathbf {~is~} r$ at point $i$ over a set of trajectories $\\sigma _1,\\dots ,\\sigma _m$ if $r=(r_1+\\dots +r_m)/m$ and for each $x \\in \\lbrace 1\\dots m\\rbrace $ , $\\textbf {value~of~} f \\mathbf {~is~} r_x$ at point $i$ in trajectory $\\sigma _x$ .", "Definition 77 (Quantitative Comparative Aggregate Interval Formula) A quantitative comparative aggregate interval formula is a formula that is evaluated w.r.t.", "an interval over two sets of trajectories and a direction `$d$ ' such that `$d$ ' relates the two sets of trajectories.", "$&\\mathbf {direction~of~change~in~} \\langle aggop \\rangle \\mathbf {~rate~of~production~of~} f \\mathbf {~is~} d\\\\&\\mathbf {direction~of~change~in~} \\langle aggop \\rangle \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} d\\\\&\\mathbf {direction~of~change~in~} \\langle aggop \\rangle \\mathbf {~rate~of~firing~of~} a \\mathbf {~is~} d\\\\&\\mathbf {direction~of~change~in~} \\langle aggop \\rangle \\mathbf {~total~production~of~} f \\mathbf {~is~} d\\\\&\\mathbf {direction~of~change~in~} \\langle aggop \\rangle \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} d$ Intuitively, a comparative quantitative aggregate interval formula compares two quantitative interval formulas over using the direction $d$ over a given interval.", "Definition 78 (Quantitative Comparative Aggregate Point Formula) A quantitative comparative aggregate point formula is a formula that is evaluated w.r.t.", "a point over two sets of trajectories and a direction `$d$ ' such that `$d$ ' relates the two sets of trajectories.", "$&\\mathbf {direction~of~change~in~} \\langle aggop \\rangle \\mathbf {~value~of~} f \\mathbf {~is~} d\\\\&\\mathbf {direction~of~change~in~} \\langle aggop \\rangle \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} d$ Intuitively, a comparative quantitative aggregate point formula compares two quantitative point formulas over using the direction $d$ at a given point.", "Definition 79 (Simple Interval Formula) A simple interval formula takes the following forms: $&\\langle \\text{quantitative interval formula} \\rangle \\\\&\\langle \\text{qualitative interval formula} \\rangle $ Definition 80 (Simple Point Formula) A simple interval formula takes the following forms: $&\\langle \\text{quantitative point formula} \\rangle \\\\&\\langle \\text{qualitative point formula} \\rangle $ Definition 81 (Internal Observation Description) An internal observation description takes the following form: $&\\langle \\text{simple point formula} \\rangle \\\\&\\langle \\text{simple point formula} \\rangle \\mathbf {~at~} \\langle point \\rangle \\\\&\\langle \\text{simple interval formula} \\rangle \\\\&\\langle \\text{simple interval formula} \\rangle \\mathbf {~when~observed~between~} \\langle interval \\rangle $ Definition 82 (Simple Point Formula Cascade) A simple point formula cascade takes the following form: $&\\langle \\text{simple point formula} \\rangle _0 \\nonumber \\\\&~~~~~~~~\\mathbf {~after~} \\langle \\text{simple point formula} \\rangle _{1,1}, \\dots , \\langle \\text{simple point formula} \\rangle _{1,n_1} \\nonumber \\\\&~~~~~~~~\\vdots \\nonumber \\\\&~~~~~~~~\\mathbf {~after~} \\langle \\text{simple point formula} \\rangle _{u,1}, \\dots , \\langle \\text{simple point formula} \\rangle _{u,n_u} \\\\&\\langle \\text{simple point formula} \\rangle _0 \\nonumber \\\\&~~~~~~~~\\mathbf {~when~} \\langle \\text{simple point formula} \\rangle _{1,1}, \\dots , \\langle \\text{simple point formula} \\rangle _{1,n_1}\\\\&\\langle \\text{simple point formula} \\rangle _0 \\nonumber \\\\&~~~~~~~~\\mathbf {~when~} \\langle \\text{cond} \\rangle $ where $u \\ge 1$ and `$\\text{cond}$ ' is a conjunction of $\\text{simple point formula}$ s that is true in the same point as the $\\text{simple point formula}$ .", "Intuitively, the simple point formula cascade (REF ) holds if a given sequence of point formulas hold in order in a trajectory.", "Intuitively, simple point formula cascade () holds if a given point formula occurs at the same point as a set of simple point formulas in a trajectory.", "Note that these formulas and many other of our single trajectory formulas can be replaced by an LTL [56] formula, but we have kept this syntax as it is more relevant to the question answering needs in the biological domain.", "Definition 83 (Query Description) A query description specifies a non-comparative observation that can be made either on a trajectory or a set of trajectories.", "$& \\langle \\text{quantitative aggregate interval formula} \\rangle \\mathbf {~when~observed~between~} \\langle \\text{interval} \\rangle \\\\& \\langle \\text{quantitative aggregate point formula} \\rangle \\mathbf {~when~observed~at~} \\langle \\text{point} \\rangle \\\\& \\langle \\text{quantitative all interval formula} \\rangle \\mathbf {~when~observed~between~} \\langle \\text{interval} \\rangle \\\\& \\langle \\text{quantitative all point formula} \\rangle \\mathbf {~when~observed~at~} \\langle \\text{point} \\rangle \\\\& \\langle \\text{internal observation description} \\rangle \\\\& \\langle \\text{internal observation description} \\rangle \\mathbf {~in~all~trajectories}\\\\& \\langle \\text{simple point formula cascade} \\rangle \\\\& \\langle \\text{simple point formula cascade} \\rangle \\mathbf {~in~all~trajectories}$ The single trajectory observations are can be represented using LTL formulas, but we have chosen to keep them in this form for ease of use by users from the biological domain.", "Definition 84 (Comparative Query Description) A comparative query description specifies a comparative observation that can be made w.r.t.", "two sets of trajectories.", "$& \\langle \\text{quantitative comparative aggregate interval formula} \\rangle \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\;\\mathbf {~when~observed~between~} \\langle \\text{interval} \\rangle \\\\& \\langle \\text{quantitative comparative aggregate point formula} \\rangle \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\;\\mathbf {~when~observed~at~} \\langle \\text{point} \\rangle $ Definition 85 (Intervention) Interventions define modifications to domain descriptions.", "$&\\mathbf {remove~} f_1 \\mathbf {~as~soon~as~produced} \\\\&\\mathbf {remove~} f_1 \\mathbf {~atloc~} l_1 \\mathbf {~as~soon~as~produced} \\\\&\\mathbf {disable~} a_2 \\\\&\\mathbf {continuously~transform~} f_1\\mathbf {~in~quantity~} q_1 \\mathbf {~to~} f_2 \\\\&\\mathbf {continuously~transform~} f_1\\mathbf {~atloc~} l_1 \\mathbf {~in~quantity~} q_1 \\mathbf {~to~} f_2 \\mathbf {~atloc~} l_2 \\\\&\\mathbf {make~} f_3 \\mathbf {~inhibit~} a_3 \\\\&\\mathbf {make~} f_3 \\mathbf {~atloc~} l_3 \\mathbf {~inhibit~} a_3 \\\\&\\mathbf {continuously~supply~} f_4 \\mathbf {~in~quantity~} q_4 \\\\&\\mathbf {contiunously~supply~} f_4 \\mathbf {~atloc~} l_4 \\mathbf {~in~quantity~} q_4 \\\\&\\mathbf {continuously~transfer~} f_1 \\mathbf {~in~quantity~} q_1 \\mathbf {~across~} l_1,l_2 \\mathbf {~to~lower~gradient~} \\\\&\\mathbf {add~delay~of~} q_1 \\mathbf {~time~units~in~availability~of~} f_1 \\\\&\\mathbf {add~delay~of~} q_1 \\mathbf {~time~units~in~availability~of~} f_1 \\mathbf {~atloc~} l_1 \\\\&\\mathbf {set~value~of~} f_4 \\mathbf {~to~} q_4 \\\\&\\mathbf {set~value~of~} f_4 \\mathbf {~atloc~} l_4 \\mathbf {~to~} q_4$ Intuitively, intervention (REF ) modifies the pathway such that all quantity of $f_1$ is removed as soon as it is produced; intervention () modifies the pathway such that all quantity of $f_1[l_1]$ is removed as soon as it is produced; intervention () disables the action $a_2$ ; intervention () modifies the pathway such that $f_1$ gets converted to $f_2$ at the rate of $q_1$ units per time-unit; intervention () modifies the pathway such that $f_1[l_1]$ gets converted to $f_2[l_2]$ at the rate of $q_1$ units per time-unit; intervention () modifies the pathway such that action $a_3$ is now inhibited each time there is 1 or more units of $f_3$ and sets value of $f_3$ to 1 to initially inhibit $a_3$ ; intervention () modifies the pathway such that action $a_3$ is now inhibited each time there is 1 or more units of $f_3[l_3]$ and sets value of $f_3[l_3]$ to 1 to initially inhibit $a_3$ ; intervention () modifies the pathway to continuously supply $f_4$ at the rate of $q_4$ units per time-unit; intervention () modifies the pathway to continuously supply $f_4[l_4]$ at the rate of $q_4$ units per time-unit; intervention () modifies the pathway to transfer $f_1[l_1]$ to $f_1[l_2]$ in quantity $q_1$ or back depending upon whether $f_1[l_1]$ is higher than $f_1[l_2]$ or lower; intervention () modifies the pathway to add delay of $q_1$ time units between when $f_1$ is produced to when it is made available to next action; intervention () modifies the pathway to add delay of $q_1$ time units between when $f_1[l_1]$ is produced to when it is made available to next action; intervention () modifies the pathway to set the initial value of $f_4$ to $q_4$ ; and intervention () modifies the pathway to set the initial value of $f_4[l_4]$ to $q_4$ .", "Definition 86 (Initial Condition) An initial condition is one of the intervention (), (), (), () as given in definition REF .", "Intuitively, initial conditions are interventions that setup fixed or continuous supply of substances participating in a pathway.", "Definition 87 (Query Statement) A query statement can be of the following forms: $& \\langle \\text{query description} \\rangle ; \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\mathbf {due~to~interventions:} \\; \\langle intervention \\rangle _1, \\dots , \\langle intervention \\rangle _{N1}; \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\mathbf {due~to~observations:} \\; \\langle \\text{internal observation} \\rangle _1, \\dots , \\langle \\text{internal observation} \\rangle _{N2}; \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\mathbf {using~initial~setup:} \\; \\langle \\text{initial condition} \\rangle _1, \\dots , \\langle \\text{initial condition} \\rangle _{N3};\\\\\\nonumber \\\\& \\langle \\text{comparative query description} \\rangle ; \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\;\\mathbf {comparing~nominal~pathway~with~modified~pathway~obtained~} \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\mathbf {due~to~interventions:} \\; \\langle intervention \\rangle _1, \\dots , \\langle intervention \\rangle _{N1} ; \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\mathbf {due~to~observations:} \\; \\langle \\text{internal observation} \\rangle _1, \\dots , \\langle \\text{internal observation} \\rangle _{N2}; \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\mathbf {using~initial~setup:} \\; \\langle \\text{initial condition} \\rangle _1, \\dots , \\langle \\text{initial condition} \\rangle _{N3}; &$ where interventions, observations, and initial setup are optional.", "Intuitively, a query statement asks whether a query description holds in a pathway, perhaps after modifying it with initial setup, interventions and observations.", "Intuitively, a comparative query statement asks whether a comparative query description holds with a nominal pathway is compared against a modified pathway, where both pathways have the same initial setup, but only the modified pathway has been modified with interventions and observations.", "Semantics of the Query Language (BioPathQA-QL) In this section we give the semantics of our pathway specification language and the query language.", "The semantics of the query language is in terms of the trajectories of a domain description $\\mathbf {D}$ that satisfy a query $\\mathbf {Q}$ .", "We will present the semantics using LTL-style formulas.", "First, we informally define the semantics of the query language as follows.", "Let $\\mathbf {Q}$ be a query statement of the form (REF ) with a query description $U$ , interventions $V_1,\\dots ,V_{\|V\|}$ , internal observations $O_1,\\dots ,O_{\|O\|}$ , and initial setup conditions $I_1,\\dots ,I_{\|I\|}$ .", "We construct a modified domain description $\\mathbf {D_1}$ by applying $I_1,\\dots ,I_{\|I\|}$ and$V_1,\\dots ,V_{\|V\|}$ to $\\mathbf {D}$ .", "We filter the trajectories of $\\mathbf {D_1}$ to retain only those trajectories that satisfy the observations $O_1,\\dots ,O_{\|O\|}$ .", "Then we determine if $U$ holds on any of the retained trajectories.", "If it does, then we say that $\\mathbf {D}$ satisfies $\\mathbf {Q}$ .", "Let $\\mathbf {Q}$ be a comparative query statement of the form () with quantitative comparative aggregate query description $U$ , interventions $V_1,\\dots ,V_{\|V\|}$ , internal observations $O_1,\\dots ,O_{\|O\|}$ , and initial conditions $I_1,\\dots ,I_{\|I\|}$ .", "Then we evaluate $\\mathbf {Q}$ by deriving two sub-query statements.", "$\\mathbf {Q_0}$ is constructed by removing the interventions $V_1,\\dots ,V_{\|V\|}$ and observations $O_1,\\dots ,O_{\|O\|}$ from $\\mathbf {Q}$ and replacing the quantitative comparative aggregate query description $U$ with the corresponding quantitative aggregate query description $U^{\\prime }$ , $\\mathbf {Q_1}$ is constructed by replacing the quantitative comparative aggregate query description $U$ with the corresponding quantitative aggregate query description $U^{\\prime }$ .", "Then $\\mathbf {D}$ satisfies $\\mathbf {Q}$ iff we can find $d \\in \\lbrace <,>,= \\rbrace $ s.t.", "$n \\; d \\; n^{\\prime }$ , where $\\mathbf {D}$ satisfies $\\mathbf {Q_0}$ for some value $n$ and $\\mathbf {D}$ satisfies $\\mathbf {Q_1}$ for some value $n^{\\prime }$ .", "An Illustrative Example In this section, we illustrate with an example how we intuitively evaluate a comparative query statement.", "In the later sections, we will give the formal semantics of query satisfaction.", "Consider the following simple pathway specification: $\\begin{array}{llll}&\\mathbf {domain~of~} & f_1 \\mathbf {~is~} integer, & f_2 \\mathbf {~is~} integer \\\\&t_1 \\mathbf {~may~fire~causing~} & f_1 \\mathbf {~change~value~by~} -1, & f_2 \\mathbf {~change~value~by~} +1\\\\& \\mathbf {initially~} & f_1 \\mathbf {~has~value~} 0, & f_2 \\mathbf {~has~value~} 0,\\\\&\\mathbf {firing~style~} & max\\end{array}$ Let the following specify a query statement $\\mathbf {Q}$ : $\\mathbf {dir}&\\mathbf {ection~of~change~in~} average \\mathbf {~rate~of~production~of~} f_2 \\mathbf {~is~} d\\\\&\\mathbf {when~observed~between~time~step~} 0 \\mathbf {~and~time~step~} k;\\\\&\\mathbf {comparing~nominal~pathway~with~modified~pathway~obtained~}\\\\&\\;\\;\\;\\;\\;\\;\\mathbf {due~to~interventions:~} \\mathbf {remove~} f_2 \\mathbf {~as~soon~as~produced} ;\\\\&\\mathbf {using~initial~setup:~} \\mathbf {continuously~supply~} f_1 \\mathbf {~in~quantity~} 1;$ that we want to evaluate against $\\mathbf {D}$ using a simulation length $k$ with maximum $ntok$ tokens at any place to determine `$d$ ' that satisfies it.", "We construct the baseline query $\\mathbf {Q_0}$ by removing interventions and observations, and replacing the comparative aggregate quantitative query description with the corresponding aggregate quantitative query description as follows: $average &\\mathbf {~rate~of~production~of~} f_2 \\mathbf {~is~} n\\\\&\\mathbf {when~observed~between~time~step~} 0 \\mathbf {~and~time~step~} k ;\\\\&\\mathbf {using~initial~setup:~} \\mathbf {continuously~supply~} f_1 \\mathbf {~in~quantity~} 1;\\\\$ We construct the alternate query $\\mathbf {Q_1}$ by replacing the comparative aggregate quantitative query description with the corresponding aggregate quantitative query description as follows: $average&\\mathbf {~rate~of~production~of~} f_2 \\mathbf {~is~} n^{\\prime }\\\\&\\mathbf {when~observed~between~time~step~} 0 \\mathbf {~and~time~step~} k ;\\\\&\\mathbf {due~to~interventions:~} \\mathbf {remove~} f_2 \\mathbf {~as~soon~as~produced} ;\\\\&\\mathbf {using~initial~setup:~} \\mathbf {continuously~supply~} f_1 \\mathbf {~in~quantity~} 1;\\\\$ We build a modified domain description $\\mathbf {D_0}$ as $\\mathbf {D} \\diamond (\\mathbf {continuously~supply~} f_1 $ $\\mathbf {~in~quantity~} 1)$ based on initial conditions in $\\mathbf {Q_0}$ .", "We get: $\\begin{array}{llll}&\\mathbf {domain~of~} & f_1 \\mathbf {~is~} integer, & f_2 \\mathbf {~is~} integer \\\\&t_1 \\mathbf {~may~fire~causing~} & f_1 \\mathbf {~change~value~by~} -1, & f_2 \\mathbf {~change~value~by~} +1\\\\&t_{f_1} \\mathbf {~may~fire~causing~} & f_1 \\mathbf {~change~value~by~} +1\\\\& \\mathbf {initially~} & f_1 \\mathbf {~has~value~} 0, & f_2 \\mathbf {~has~value~} 0,\\\\&\\mathbf {firing~style~} & max\\end{array}$ We evaluate $\\mathbf {Q_0}$ against $\\mathbf {D_0}$ .", "It results in $m_0$ trajectories with rate of productions $n_j=(s_k(f_2)-s_0(f_2))/k$ on trajectory $\\tau _j=s_0,\\dots ,s_k, 1 \\le j \\le m_0$ using time interval $[0,k]$ .", "The rate of productions are averaged to produce $n=(n_1+\\dots +n_{m_0})/m_0$ .", "Next, we construct the alternate domain description $\\mathbf {D_1}$ as $\\mathbf {D_0} \\diamond (\\mathbf {remove~} f_2 \\mathbf {~as~soon~} $ $\\mathbf {as~produced~})$ based on initial conditions and interventions in $\\mathbf {Q_1}$ .", "We get: $\\begin{array}{llll}&\\mathbf {domain~of~} & f_1 \\mathbf {~is~} integer, & f_2 \\mathbf {~is~} integer \\\\&t_1 \\mathbf {~may~fire~causing~} & f_1 \\mathbf {~change~value~by~} -1, & f_2 \\mathbf {~change~value~by~} +1\\\\&t_{f_1} \\mathbf {~may~fire~causing~} & f_1 \\mathbf {~change~value~by~} +1\\\\&t_{f_2} \\mathbf {~may~fire~causing~} & f_2 \\mathbf {~change~value~by~} \\\\& \\mathbf {initially~} & f_1 \\mathbf {~has~value~} 0, & f_2 \\mathbf {~has~value~} 0\\\\&\\mathbf {firing~style~} & max\\end{array}$ We evaluate $\\mathbf {Q_1}$ against $\\mathbf {D_1}$ .", "Since there are no observations, no filtering is required.", "This results in $m_1$ trajectories, each with rate of production $n^{\\prime }_j=(s_k(f_2)-s_0(f_2))/k$ on trajectory $\\tau ^{\\prime }_j = s_0,\\dots ,s_k, 1 \\le j \\le m_1$ using time interval $[0,k]$ .", "The rate of productions are averaged to produce $n^{\\prime }=(n^{\\prime }_1+\\dots +n^{\\prime }_{m_1})/m_1$ .", "Due to the simple nature of our domain description, it has only one trajectory for each of the two domains.", "As a result, for any $k > 1$ , $n^{\\prime } < n$ .", "Thus, $\\mathbf {D}$ satisfies $\\mathbf {Q}$ iff $d = ``<^{\\prime \\prime }$ .", "We will now define the semantics of how a domain description $\\mathbf {D}$ is modified according to the interventions and initial conditions, the semantics of conditions imposed by the internal observations.", "We will then formally define how $\\mathbf {Q}$ is entailed in $\\mathbf {D}$ .", "Domain Transformation due to Interventions and Initial Conditions An intervention $I$ modifies a given domain description $\\mathbf {D}$ , potentially resulting in a different set of trajectories than $\\mathbf {D}$ .", "We define a binary operator $\\diamond $ that transforms $\\mathbf {D}$ by applying an intervention $I$ as a set of edits to $\\mathbf {D}$ using the pathway specification language.", "The trajectories of the modified domain description $\\mathbf {D^{\\prime }} = \\mathbf {D} \\diamond I$ are given by the semantics of the pathway specification language.", "Below, we give the intuitive impact and edits required by each of the interventions.", "Domain modification by intervention (REF ) $\\mathbf {D^{\\prime }} = \\mathbf {D} \\diamond (\\mathbf {remove~} f_1 \\mathbf {~as~soon~}$ $\\mathbf {as~produced})$ modifies the pathway by removing all existing quantity of $f_1$ at each time step: $\\mathbf {D^{\\prime }} &= \\mathbf {D}+ \\left\\lbrace \\begin{array}{l}tr \\mathbf {~may~execute~causing~} f_1 \\mathbf {~change~value~by~} \\end{array}\\right\\rbrace $ Domain modification by intervention () $\\mathbf {D^{\\prime }} = \\mathbf {D} \\diamond (\\mathbf {remove~} f_1 \\mathbf {~atloc~} l_1)$ modifies the pathway by removing all existing quantity of $f_1$ at each time step: $\\mathbf {D^{\\prime }} &= \\mathbf {D}+ \\left\\lbrace \\begin{array}{l}tr \\mathbf {~may~execute~causing~} f_1 \\mathbf {~atloc~} l_1 \\mathbf {~change~value~by~} \\end{array}\\right\\rbrace $ Domain modification by intervention () $\\mathbf {D^{\\prime }} = \\mathbf {D} \\diamond (\\mathbf {disable~} a_2)$ modifies the pathway such that its trajectories have $a_2 \\notin T_i$ , where $i \\ge 0$ .", "$\\mathbf {D^{\\prime }} &= \\mathbf {D}+ \\left\\lbrace \\begin{array}{ll}\\mathbf {inhibit~} a_2\\end{array}\\right\\rbrace $ Domain modification by intervention () $\\mathbf {D^{\\prime }} = \\mathbf {D} \\diamond (\\mathbf {continuously~transform~} f_1 $ $\\mathbf {~in~quantity~} q_1 $ $\\mathbf {~to~} f_2)$ where $s_{i+1}(f_1)$ decreases, and $s_{i+1}(f_2)$ increases by $q_1$ at each time step $i \\ge 0$ , when $s_i(f_1) \\ge q_1$ .", "$\\mathbf {D^{\\prime }} = \\mathbf {D}&+ \\left\\lbrace \\begin{array}{ll}a_{f_{1,2}} \\mathbf {~may~execute~causing~} & f_1 \\mathbf {~change~value~by~} -q_1, \\\\ & f_2 \\mathbf {~change~value~by~} +q_1\\end{array}\\right\\rbrace $ Domain modification by intervention () $\\mathbf {D^{\\prime }} = \\mathbf {D} \\diamond (\\mathbf {continuously~transform~} $ $f_1 \\mathbf {~atloc~} l_1$ $ \\mathbf {~in~quantity~} q_1 $ $\\mathbf {~to~} f_2 \\mathbf {~atloc~} l_2)$ where $s_{i+1}(f_1[l_1])$ decreases, and $s_{i+1}(f_2[l_2])$ increases by $q_1$ at each time step $i \\ge 0$ , when $s_i(f_1[l_1]) \\ge q_1$ .", "$\\mathbf {D^{\\prime }} = \\mathbf {D}&+ \\left\\lbrace \\begin{array}{ll}a_{f_{1,2}} \\mathbf {~may~execute~causing~} & f_1 \\mathbf {~atloc~} l_1 \\mathbf {~change~value~by~} -q_1, \\\\ & f_2 \\mathbf {~atloc~} l_2 \\mathbf {~change~value~by~} +q_1\\end{array}\\right\\rbrace $ Domain modification by intervention () $\\mathbf {D^{\\prime }} = \\mathbf {D} \\diamond (\\mathbf {make~} f_3 \\mathbf {~inhibit~} a_3 )$ modifies the pathway such that it has $a_3$ inhibited due to $f_3$ .", "$\\mathbf {D^{\\prime }} = \\mathbf {D}&- \\left\\lbrace \\begin{array}{l}\\mathbf {initially~} f_3 \\mathbf {~has~value~} q \\in \\mathbf {D}\\end{array}\\right\\rbrace \\\\&+ \\left\\lbrace \\begin{array}{ll}\\mathbf {inhibit~} a_3 \\mathbf {~if~} & f_3 \\mathbf {~has~value~} 1 \\mathbf {~or~higher}\\\\\\mathbf {initially~} & f_3 \\mathbf {~has~value~} 1\\end{array}\\right\\rbrace $ Domain modification by intervention () $\\mathbf {D^{\\prime }} = \\mathbf {D} \\diamond (\\mathbf {make~} f_3 \\mathbf {~atloc~} l_3 \\mathbf {~inhibit~} a_3)$ modifies the pathway such that it has $a_3$ inhibited due to $f_3[l_3]$ .", "$\\mathbf {D^{\\prime }} = \\mathbf {D}&- \\left\\lbrace \\begin{array}{l}\\mathbf {initially~} f_3 \\mathbf {~atloc~} l_3 \\mathbf {~has~value~} q \\in \\mathbf {D}\\end{array}\\right\\rbrace \\\\&+ \\left\\lbrace \\begin{array}{ll}\\mathbf {inhibit~} a_3 \\mathbf {~if~} & f_3 \\mathbf {~atloc~} l_3 \\mathbf {~has~value~} 1 \\mathbf {~or~higher}, \\\\\\mathbf {initially~} & f_3 \\mathbf {~atloc~} l_3 \\mathbf {~has~value~} 1\\end{array}\\right\\rbrace $ Domain modification by intervention () $\\mathbf {D^{\\prime }} = \\mathbf {D} \\diamond (\\mathbf {~continuously~supply~} f_4 $ $\\mathbf {~in~quantity~} q_4)$ modifies the pathway such that a quantity $q_4$ of substance $f_4$ is supplied at each time step.", "$\\mathbf {D^{\\prime }} = \\mathbf {D}&+ \\left\\lbrace \\begin{array}{ll}t_{f_4} \\mathbf {~may~execute~causing~} & f_4 \\mathbf {~change~value~by~} +q_4\\end{array}\\right\\rbrace $ Domain modification by intervention () $\\mathbf {D^{\\prime }} = \\mathbf {D} \\diamond (\\mathbf {~continuously~supply~} f_4 $ $\\mathbf {~atloc~} l_4 $ $\\mathbf {~in~quantity~} q_4 )$ modifies the pathway such that a quantity $q_4$ of substance $f_4$ at location $l_4$ is supplied at each time step.", "$\\mathbf {D^{\\prime }} = \\mathbf {D}&+ \\left\\lbrace \\begin{array}{ll}t_{f_4} \\mathbf {~may~execute~causing~} & f_4 \\mathbf {~atloc~} l_4 \\mathbf {~change~value~by~} +q_4\\end{array}\\right\\rbrace $ Domain modification by intervention () $\\mathbf {D^{\\prime }} = \\mathbf {D} \\diamond (\\mathbf {continuously~transfer~} f_1 $ $\\mathbf {~in~quantity~} q_1 $ $\\mathbf {~across~} l_1, l_2 $ $\\mathbf {~to~lower~gradient})$ modifies the pathway such that substance represented by $f_1$ is transferred from location $l_1$ to $l_2$ or $l_2$ to $l_1$ depending upon whether it is at a higher quantity at $l_1$ or $l_2$ .", "$\\mathbf {D^{\\prime }} = \\mathbf {D}&+ \\left\\lbrace \\begin{array}{ll}t_{f_1} \\mathbf {~may~execute~causing~} & f_1 \\mathbf {~atloc~} l_1 \\mathbf {~change~value~by~} -q_1, \\\\ & f_1 \\mathbf {~atloc~} l_2 \\mathbf {~change~value~by~} +q_1 \\\\\\mathbf {~~~~~~~if}& f_1 \\mathbf {~atloc~} l_1 \\mathbf {~has~higher~value~than~} f_1 \\mathbf {~atloc~} l_2, \\\\t^{\\prime }_{f_1} \\mathbf {~may~execute~causing~} & f_1 \\mathbf {~atloc~} l_2 \\mathbf {~change~value~by~} -q_1, \\\\ & f_1 \\mathbf {~atloc~} l_1 \\mathbf {~change~value~by~} +q_1\\\\\\mathbf {~~~~~~~if}& f_1 \\mathbf {~atloc~} l_2 \\mathbf {~has~higher~value~than~} f_1 \\mathbf {~atloc~} l_1, \\\\\\end{array}\\right\\rbrace $ Domain modification by intervention () $\\mathbf {D^{\\prime }} = \\mathbf {D} \\diamond (\\mathbf {add~delay~of~} q_1 $ $\\mathbf {~time~units~} $ $\\mathbf {in~availability~of~} f_1)$ modifies the pathway such that $f_1$ 's arrival is delayed by $q_1$ time units.", "We create additional cases for all actions that produce $f_1$ , such that it goes through an additional delay action.", "$\\mathbf {D^{\\prime }} = \\mathbf {D}& - \\left\\lbrace \\begin{array}{ll}a \\mathbf {~may~execute~causing~} & f_1 \\mathbf {~change~value~by~} +w_1, \\\\ & \\mathit {effect}_1,\\dots ,\\mathit {effect}_n \\\\ & \\mathbf {~if~} cond_1,\\dots ,cond_m \\in \\mathbf {D}\\\\\\end{array}\\right\\rbrace \\\\& + \\left\\lbrace \\begin{array}{ll}a \\mathbf {~may~execute~causing~} & f^{\\prime }_1 \\mathbf {~change~value~by~} +w_1, \\\\ & \\mathit {effect}_1,\\dots ,\\mathit {effect}_n, \\\\ & \\mathbf {~if~} cond_1,\\dots ,cond_m\\\\a_{f_1} \\mathbf {~may~execute~causing~} & f^{\\prime }_1 \\mathbf {~change~value~by~} -w_1, \\\\ & f_1 \\mathbf {~change~value~by~} +w_1\\\\a_{f_1} \\mathbf {~executes~} & \\mathbf {~in~} q_1 \\mathbf {~time~units}\\\\\\end{array}\\right\\rbrace $ Domain modification by intervention () $\\mathbf {D^{\\prime }} = \\mathbf {D} \\diamond (\\mathbf {add~delay~of} q_1 $ $\\mathbf {~time~units~} $ $\\mathbf {in~availability~of~} f_1 \\mathbf {~atloc~} l_1)$ modifies trajectories such that $f_1[l_1]$ 's arrival is delayed by $q_1$ time units.", "We create additional cases for all actions that produce $f_1 \\mathbf {~atloc~} l_1$ , such that it goes through an additional delay action.", "$\\mathbf {D^{\\prime }} = \\mathbf {D}&- \\left\\lbrace \\begin{array}{ll}a \\mathbf {~may~execute~causing~} & f_1 \\mathbf {~atloc~} l_1 \\mathbf {~change~value~by~} +w_1, \\\\ & \\mathit {effect}_1,\\dots ,\\mathit {effect}_n \\\\ & \\mathbf {~if~} cond_1,\\dots ,cond_m \\in \\mathbf {D}\\end{array}\\right\\rbrace \\\\&+ \\left\\lbrace \\begin{array}{ll}a \\mathbf {~may~execute~causing~} & f_1 \\mathbf {~atloc~} l^{\\prime }_1 \\mathbf {~change~value~by~} +w_1, \\\\ & \\mathit {effect}_1,\\dots ,\\mathit {effect}_n \\\\ & \\mathbf {~if~} cond_1,\\dots ,cond_n\\\\a_{f_1} \\mathbf {~may~execute~causing~} & f_1 \\mathbf {~atloc~} l^{\\prime }_1 \\mathbf {~change~value~by~} -w_1,\\\\ & f_1 \\mathbf {~atloc~} l_1 \\mathbf {~change~value~by~} +w_1\\\\a_{f_1} \\mathbf {~executes~} & \\mathbf {~in~} q_1 \\mathbf {~time~units}\\\\\\end{array}\\right\\rbrace $ Domain modification by intervention () $\\mathbf {D^{\\prime }} = \\mathbf {D} \\diamond (\\mathbf {~set~value~of~} f_4 $ $\\mathbf {~to~} q_4)$ modifies the pathway such that its trajectories have $s_0(f_4) = q_4$ .", "$\\mathbf {D^{\\prime }} = \\mathbf {D}& - \\left\\lbrace \\begin{array}{l}\\mathbf {initially~} f_4 \\mathbf {~has~value~} n \\in \\mathbf {D}\\end{array}\\right\\rbrace \\\\& + \\left\\lbrace \\begin{array}{l}\\mathbf {initially~} f_4 \\mathbf {~has~value~} q_4\\end{array}\\right\\rbrace $ Domain modification by intervention () $\\mathbf {D^{\\prime }} = \\mathbf {D} \\diamond (\\mathbf {~set~value~of~} f_4 $ $\\mathbf {~atloc~} l_4 $ $\\mathbf {~to~} q_4 )$ modifies the pathway such that its trajectories have $s_0({f_4}_{l_4}) = q_4$ .", "$\\mathbf {D^{\\prime }} = \\mathbf {D}& - \\left\\lbrace \\begin{array}{l}\\mathbf {initially~} f_4 \\mathbf {~atloc~} l_4 \\mathbf {~has~value~} n \\in \\mathbf {D}\\end{array}\\right\\rbrace \\\\& + \\left\\lbrace \\begin{array}{l}\\mathbf {initially~} f_4 \\mathbf {~atloc~} l_4 \\mathbf {~has~value~} q_4\\end{array}\\right\\rbrace $ Formula Semantics We will now define the semantics of some common formulas that we will use in the following sections.", "First we introduce the LTL-style formulas that we will be using to define the syntax.", "A formula $\\langle s_i,\\sigma \\rangle \\models F$ represents that $F$ holds at point $i$ .", "A formula $\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace \\models F$ represents that $F$ holds at point $i$ on a set of trajectories $\\sigma _1,\\dots ,\\sigma _m$ .", "A formula $\\big \\lbrace \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\big \\rbrace \\models F$ represents that $F$ holds at point $i$ on two sets of trajectories $\\sigma _1,\\dots ,\\sigma _m$ and $\\lbrace \\bar{\\sigma }_1,\\dots ,\\bar{\\sigma }_{\\bar{m}} \\rbrace $ .", "A formula $(\\langle s_i,\\sigma \\rangle ,j) \\models F$ represents that $F$ holds in the interval $[i,j]$ on trajectory $\\sigma $ .", "A formula $(\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models F$ represents that $F$ holds in the interval $[i,j]$ on a set of trajectories $\\sigma _1,\\dots ,\\sigma _m$ .", "A formula $(\\big \\lbrace \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\big \\rbrace , j) \\models F$ represents that $F$ holds in the interval $[i,j]$ over two sets of trajectories $\\lbrace \\sigma _1,\\dots , \\sigma _m \\rbrace $ and $\\lbrace \\bar{\\sigma }_1,\\dots ,\\bar{\\sigma }_{\\bar{m}} \\rbrace $ .", "Given a domain description $\\mathbf {D}$ with simple fluents represented by a Guarded-Arc Petri Net as defined in definition REF .", "Let $\\sigma = s_0,T_0,s_1,\\dots ,T_{k-1},s_{k}$ be its trajectory as defined in (REF ), and $s_i$ be a state on that trajectory.", "Let actions $T_i$ firing in state $s_i$ be observable in $s_i$ such that $T_i \\subseteq s_i$ .", "First we define how interval formulas are satisfied on a trajectory $\\sigma $ , starting state $s_i$ and an ending point $j$ : $(\\langle s_i,\\sigma \\rangle , j) \\models & \\mathbf {~rate~of~production~of~} f \\mathbf {~is~} n \\nonumber \\\\&\\text{ if } n=(s_j(f)-s_i(f))/(j-i)\\\\(\\langle s_i,\\sigma \\rangle , j) \\models & \\mathbf {~rate~of~firing~of~} a \\mathbf {~is~} n \\nonumber \\\\&\\text{ if } n=\\sum _{i\\le k \\le j, \\langle s_k,\\sigma \\rangle \\models a \\mathbf {~occurs~}}{1}/(j-i)\\\\(\\langle s_i,\\sigma \\rangle , j) \\models & \\mathbf {~total~production~of~} f \\mathbf {~is~} n \\nonumber \\\\&\\text{ if } n=(s_j(f)-s_i(f))\\\\(\\langle s_i,\\sigma \\rangle , j) \\models & f \\mathbf {~is~accumulating~} \\nonumber \\\\&\\text{ if } (\\nexists k, i \\le k \\le j : s_{k+1}(f) < s_k(f)) \\text{ and } s_j(f) > s_i(f)\\\\(\\langle s_i,\\sigma \\rangle , j) \\models & f \\mathbf {~is~decreasing~} \\nonumber \\\\&\\text{ if } (\\nexists k, i \\le k \\le j : s_{k+1}(f) > s_k(f)) \\text{ and } s_j(f) < s_i(f)$ Next we define how a point formula is satisfied on a trajectory $\\sigma $ , in a state $s_i$ : $\\langle s_i,\\sigma \\rangle \\models & \\mathbf {~value~of~} f \\mathbf {~is~higher~than~} n \\nonumber \\\\&\\text{ if } s_i(f) > n\\\\\\langle s_i,\\sigma \\rangle \\models & \\mathbf {~value~of~} f \\mathbf {~is~lower~than~} n \\nonumber \\\\&\\text{ if } s_i(f) < n\\\\\\langle s_i,\\sigma \\rangle \\models & \\mathbf {~value~of~} f \\mathbf {~is~} n \\nonumber \\\\&\\text{ if } s_i(f) = n\\\\\\langle s_i,\\sigma \\rangle \\models & a \\mathbf {~occurs~} \\nonumber \\\\&\\text{ if } a \\in s_i\\\\\\langle s_i,\\sigma \\rangle \\models & a \\mathbf {~does~not~occur~} \\nonumber \\\\&\\text{ if } a \\notin s_i\\\\\\langle s_i,\\sigma \\rangle \\models & a_1 \\mathbf {~switches~to~} a_2 \\nonumber \\\\&\\text{ if } a_1 \\in s_{i-1} \\text{ and } a_2 \\notin s_{i-1} \\text{ and } a_1 \\notin s_i \\text{ and } a_2 \\in s_i$ Next we define how a quantitative all interval formula is satisfied on a set of trajectories $\\sigma _1,\\dots ,\\sigma _m$ with starting states $s^1_i,\\dots ,s^m_i$ and end point $j$ : $(\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models & \\mathbf {~rates~of~production~of~} f \\mathbf {~are~} [r_1,\\dots ,r_m] \\nonumber \\\\&\\text{ if } (\\langle s^1_i,\\sigma _1 \\rangle , j) \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~is~} r_1\\nonumber \\\\&\\vdots \\nonumber \\\\&\\text{ and } (\\langle s^m_i,\\sigma _m \\rangle , j) \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~is~} r_m \\\\(\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models & \\mathbf {~rates~of~firing~of~} a \\mathbf {~are~} [r_1,\\dots ,r_m] \\nonumber \\\\&\\text{ if } (\\langle s^1_i,\\sigma _1 \\rangle , j) \\models \\mathbf {~rate~of~firing~of~} a \\mathbf {~is~} r_1\\nonumber \\\\&\\vdots \\nonumber \\\\&\\text{ and } (\\langle s^m_i,\\sigma _m \\rangle , j) \\models \\mathbf {~rate~of~firing~of~} a \\mathbf {~is~} r_m \\\\(\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models & \\mathbf {~totals~of~production~of~} f \\mathbf {~are~} [r_1,\\dots ,r_m] \\nonumber \\\\&\\text{ if } (\\langle s^1_i,\\sigma _1 \\rangle , j) \\models \\mathbf {~total~production~of~} f \\mathbf {~is~} r_1\\nonumber \\\\&\\vdots \\nonumber \\\\&\\text{ and } (\\langle s^m_i,\\sigma _m \\rangle , j) \\models \\mathbf {~total~production~of~} f \\mathbf {~is~} r_m$ Next we define how a quantitative all point formula is satisfied on a set of trajectories $\\sigma _1,\\dots ,\\sigma _m$ in states $s^1_i,\\dots ,s^m_i$ : $\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace \\models & \\mathbf {~values~of~} f \\mathbf {~are~} [r_1,\\dots ,r_m] \\nonumber \\\\&\\text{ if } \\langle s^1_i,\\sigma _1 \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~is~} r_1\\nonumber \\\\&\\vdots \\nonumber \\\\&\\text{ and } \\langle s^m_i,\\sigma _m \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~is~} r_m$ Next we define how a quantitative aggregate interval formula is satisfied on a set of trajectories $\\sigma _1,\\dots ,\\sigma _m$ with starting states $s^1_i,\\dots ,s^m_i$ and end point $j$ : $(\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models & \\; average \\mathbf {~rate~of~production~of~} f \\mathbf {~is~} r \\nonumber \\\\&\\text{ if } \\exists [r_1,\\dots ,r_m] : \\nonumber \\\\&~~~~~~(\\langle s^1_i,\\sigma _1 \\rangle , j) \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~is~} r_1\\nonumber \\\\&~~~~~~\\vdots \\nonumber \\\\&~~~~~~(\\langle s^m_i,\\sigma _m \\rangle , j) \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~is~} r_m \\nonumber \\\\&\\text{ and } r=(r_1+\\dots +r_m)/m\\\\(\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models & \\; minimum \\mathbf {~rate~of~production~of~} f \\mathbf {~is~} r \\nonumber \\\\&\\text{ if } \\exists [r_1,\\dots ,r_m] : \\nonumber \\\\&~~~~~~(\\langle s^1_i,\\sigma _1 \\rangle , j) \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~is~} r_1\\nonumber \\\\&~~~~~~\\vdots \\nonumber \\\\&~~~~~~(\\langle s^m_i,\\sigma _m \\rangle , j) \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~is~} r_m \\nonumber \\\\&\\text{ and } \\exists k, 1 \\le k \\le m : r=r_k \\text{ and } \\forall x, 1 \\le x \\le m, r_k \\le r_x\\\\(\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models & \\; maximum \\mathbf {~rate~of~production~of~} f \\mathbf {~is~} r \\nonumber \\\\&\\text{ if } \\exists [r_1,\\dots ,r_m] : \\nonumber \\\\&~~~~~~(\\langle s^1_i,\\sigma _1 \\rangle , j) \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~is~} r_1\\nonumber \\\\&~~~~~~\\vdots \\nonumber \\\\&~~~~~~(\\langle s^m_i,\\sigma _m \\rangle , j) \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~is~} r_m \\nonumber \\\\&\\text{ and } \\exists k, 1 \\le k \\le m : r=r_k \\text{ and } \\forall x, 1 \\le x \\le m, r_k \\ge r_x\\\\\\vspace{20.0pt}\\nonumber \\\\(\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models & \\; average \\mathbf {~rate~of~firing~of~} f \\mathbf {~is~} r \\nonumber \\\\&\\text{ iff } \\exists [r_1,\\dots ,r_m] : \\nonumber \\\\&~~~~~~(\\langle s^1_i,\\sigma _1 \\rangle , j) \\models \\mathbf {~rate~of~firing~of~} f \\mathbf {~is~} r_1\\nonumber \\\\&~~~~~~\\vdots \\nonumber \\\\&~~~~~~(\\langle s^m_i,\\sigma _m \\rangle , j) \\models \\mathbf {~rate~of~firing~of~} f \\mathbf {~is~} r_m \\nonumber \\\\&\\text{ and } r=(r_1+\\dots +r_m)/m\\\\(\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models & \\; minimum \\mathbf {~rate~of~firing~of~} f \\mathbf {~is~} r \\nonumber \\\\&\\text{ iff } \\exists [r_1,\\dots ,r_m] : \\nonumber \\\\&~~~~~~(\\langle s^1_i,\\sigma _1 \\rangle , j) \\models \\mathbf {~rate~of~firing~of~} f \\mathbf {~is~} r_1\\nonumber \\\\&~~~~~~\\vdots \\nonumber \\\\&~~~~~~(\\langle s^m_i,\\sigma _m \\rangle , j) \\models \\mathbf {~rate~of~firing~of~} f \\mathbf {~is~} r_m \\nonumber \\\\&\\text{ and } \\exists k, 1 \\le k \\le m : r=r_k \\text{ and } \\forall x, 1 \\le x \\le m, r_k \\le r_x\\\\(\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models & \\; maximum \\mathbf {~rate~of~firing~of~} f \\mathbf {~is~} r \\nonumber \\\\&\\text{ iff } \\exists [r_1,\\dots ,r_m] : \\nonumber \\\\&~~~~~~(\\langle s^1_i,\\sigma _1 \\rangle , j) \\models \\mathbf {~rate~of~firing~of~} f \\mathbf {~is~} r_1\\nonumber \\\\&~~~~~~\\vdots \\nonumber \\\\&~~~~~~(\\langle s^m_i,\\sigma _m \\rangle , j) \\models \\mathbf {~rate~of~firing~of~} f \\mathbf {~is~} r_m \\nonumber \\\\&\\text{ and } \\exists k, 1 \\le k \\le m : r=r_k \\text{ and } \\forall x, 1 \\le x \\le m, r_k \\ge r_x\\\\\\vspace{20.0pt}\\nonumber \\\\(\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models & \\; average \\mathbf {~total~production~of~} f \\mathbf {~is~} r \\nonumber \\\\&\\text{ if } \\exists [r_1,\\dots ,r_m] : \\nonumber \\\\&~~~~~~(\\langle s^1_i,\\sigma _1 \\rangle , j) \\models \\mathbf {~total~production~of~} f \\mathbf {~is~} r_1\\nonumber \\\\&~~~~~~\\vdots \\nonumber \\\\&~~~~~~(\\langle s^m_i,\\sigma _m \\rangle , j) \\models \\mathbf {~total~production~of~} f \\mathbf {~is~} r_m \\nonumber \\\\&\\text{ and } r=(r_1+\\dots +r_m)/m\\\\(\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models & \\; minimum \\mathbf {~total~production~of~} f \\mathbf {~is~} r \\nonumber \\\\&\\text{ if } \\exists [r_1,\\dots ,r_m] : \\nonumber \\\\&~~~~~~(\\langle s^1_i,\\sigma _1 \\rangle , j) \\models \\mathbf {~total~production~of~} f \\mathbf {~is~} r_1\\nonumber \\\\&~~~~~~\\vdots \\nonumber \\\\&~~~~~~(\\langle s^m_i,\\sigma _m \\rangle , j) \\models \\mathbf {~total~production~of~} f \\mathbf {~is~} r_m \\nonumber \\\\&\\text{ and } \\exists k, 1 \\le k \\le m : r=r_k \\text{ and } \\forall x, 1 \\le x \\le m, r_k \\le r_x\\\\(\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models & \\; maximum \\mathbf {~total~production~of~} f \\mathbf {~is~} r \\nonumber \\\\&\\text{ if } \\exists [r_1,\\dots ,r_m] : \\nonumber \\\\&~~~~~~(\\langle s^1_i,\\sigma _1 \\rangle , j) \\models \\mathbf {~total~production~of~} f \\mathbf {~is~} r_1\\nonumber \\\\&~~~~~~\\vdots \\nonumber \\\\&~~~~~~(\\langle s^m_i,\\sigma _m \\rangle , j) \\models \\mathbf {~total~production~of~} f \\mathbf {~is~} r_m \\nonumber \\\\&\\text{ and } \\exists k, 1 \\le k \\le m : r=r_k \\text{ and } \\forall x, 1 \\le x \\le m, r_k \\ge r_x$ Next we define how a quantitative aggregate point formula is satisfied on a set of trajectories $\\sigma _1,\\dots ,\\sigma _m$ in states $s^1_i,\\dots ,s^m_i$ : $\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace \\models & \\; average \\mathbf {~value~of~} f \\mathbf {~is~} r \\nonumber \\\\&\\text{ if } \\exists [r_1,\\dots ,r_m] : \\nonumber \\\\&~~~~~~\\langle s^1_i,\\sigma _1 \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~is~} r_1\\nonumber \\\\&~~~~~~\\vdots \\nonumber \\\\&~~~~~~\\langle s^m_i,\\sigma _m \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~is~} r_m \\nonumber \\\\&\\text{ and } r=(r_1+\\dots +r_m)/m\\\\\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace \\models & \\; minimum \\mathbf {~value~of~} f \\mathbf {~is~} r \\nonumber \\\\&\\text{ if } \\exists [r_1,\\dots ,r_m] : \\nonumber \\\\&~~~~~~\\langle s^1_i,\\sigma _1 \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~is~} r_1\\nonumber \\\\&~~~~~~\\vdots \\nonumber \\\\&~~~~~~\\langle s^m_i,\\sigma _m \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~is~} r_m \\nonumber \\\\&\\text{ and } \\exists k, 1 \\le k \\le m : r=r_k \\text{ and } \\forall x, 1 \\le x \\le m, r_k \\le r_x\\\\\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace \\models & \\; maximum \\mathbf {~value~of~} f \\mathbf {~is~} r \\nonumber \\\\&\\text{ if } \\exists [r_1,\\dots ,r_m] : \\nonumber \\\\&~~~~~~\\langle s^1_i,\\sigma _1 \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~is~} r_1\\nonumber \\\\&~~~~~~\\vdots \\nonumber \\\\&~~~~~~\\langle s^m_i,\\sigma _m \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~is~} r_m \\nonumber \\\\&\\text{ and } \\exists k, 1 \\le k \\le m : r=r_k \\text{ and } \\forall x, 1 \\le x \\le m, r_k \\ge r_x$ Next we define how a comparative quantitative aggregate interval formula is satisfied on two sets of trajectories $\\sigma _1,\\dots ,\\sigma _m$ , and $\\bar{\\sigma }_1,\\dots ,\\bar{\\sigma }_{\\bar{m}}$ a starting point $i$ and an ending point $j$ : $&\\Big (\\Big \\lbrace \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace , j \\Big ) \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\models \\; \\mathbf {~direction~of~change~in~} average \\mathbf {~rate~of~production~of~} f \\mathbf {~is~} d \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ if } \\exists \\; n_1 : (\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models \\; average \\mathbf {~rate~of~production~of~} f \\text{~is~} n_1 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } \\exists \\; n_2 : (\\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace , j) \\models \\; average \\mathbf {~rate~of~production~of~} f \\text{~is~} n_2 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } n_2 \\; d \\; n_1\\\\&\\Big (\\Big \\lbrace \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace , j \\Big ) \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\models \\; \\mathbf {~direction~of~change~in~} minimum \\mathbf {~rate~of~production~of~} f \\mathbf {~is~} d \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ if } \\exists \\; n_1 : (\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models \\; minimum \\mathbf {~rate~of~production~of~} f \\text{~is~} n_1 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } \\exists \\; n_2 : (\\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace , j) \\models \\; minimum \\mathbf {~rate~of~production~of~} f \\text{~is~} n_2 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } n_2 \\; d \\; n_1\\\\&\\Big (\\Big \\lbrace \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace , j \\Big ) \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\models \\; \\mathbf {~direction~of~change~in~} maximum \\mathbf {~rate~of~production~of~} f \\mathbf {~is~} d \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ if } \\exists \\; n_1 : (\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models \\; maximum \\mathbf {~rate~of~production~of~} f \\text{~is~} n_1 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } \\exists \\; n_2 : (\\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace , j) \\models \\; maximum \\mathbf {~rate~of~production~of~} f \\text{~is~} n_2 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } n_2 \\; d \\; n_1\\\\\\vspace{20.0pt}\\nonumber \\\\&\\Big (\\Big \\lbrace \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace , j \\Big ) \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\models \\; \\mathbf {~direction~of~change~in~} average \\mathbf {~rate~of~firing~of~} f \\mathbf {~is~} d \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ if } \\exists \\; n_1 : (\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models \\; average \\mathbf {~rate~of~firing~of~} f \\text{~is~} n_1 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } \\exists \\; n_2 : (\\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace , j) \\models \\; average \\mathbf {~rate~of~firing~of~} f \\text{~is~} n_2 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } n_2 \\; d \\; n_1\\\\&\\Big (\\Big \\lbrace \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace , j \\Big ) \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\models \\; \\mathbf {~direction~of~change~in~} minimum \\mathbf {~rate~of~firing~of~} f \\mathbf {~is~} d \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ if } \\exists \\; n_1 : (\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models \\; minimum \\mathbf {~rate~of~firing~of~} f \\text{~is~} n_1 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } \\exists \\; n_2 : (\\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace , j) \\models \\; minimum \\mathbf {~rate~of~firing~of~} f \\text{~is~} n_2 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } n_2 \\; d \\; n_1\\\\&\\Big (\\Big \\lbrace \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace , j \\Big ) \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\models \\; \\mathbf {~direction~of~change~in~} maximum \\mathbf {~rate~of~firing~of~} f \\mathbf {~is~} d \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ if } \\exists \\; n_1 : (\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models \\; maximum \\mathbf {~rate~of~firing~of~} f \\text{~is~} n_1 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } \\exists \\; n_2 : (\\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace , j) \\models \\; maximum \\mathbf {~rate~of~firing~of~} f \\text{~is~} n_2 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } n_2 \\; d \\; n_1\\\\\\vspace{20.0pt}\\nonumber \\\\&\\Big (\\Big \\lbrace \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace , j \\Big ) \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\models \\; \\mathbf {~direction~of~change~in~} average \\mathbf {~total~production~of~} f \\mathbf {~is~} d \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ if } \\exists \\; n_1 : (\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models \\; average \\mathbf {~total~production~of~} f \\text{~is~} n_1 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } \\exists \\; n_2 : (\\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace , j) \\models \\; average \\mathbf {~total~production~of~} f \\text{~is~} n_2 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } n_2 \\; d \\; n_1\\\\&\\Big (\\Big \\lbrace \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace , j \\Big ) \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\models \\; \\mathbf {~direction~of~change~in~} minimum \\mathbf {~total~production~of~} f \\mathbf {~is~} d \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ if } \\exists \\; n_1 : (\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models \\; minimum \\mathbf {~total~production~of~} f \\text{~is~} n_1 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } \\exists \\; n_2 : (\\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace , j) \\models \\; minimum \\mathbf {~total~production~of~} f \\text{~is~} n_2 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } n_2 \\; d \\; n_1\\\\&\\Big (\\Big \\lbrace \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace , j \\Big ) \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\models \\; \\mathbf {~direction~of~change~in~} maximum \\mathbf {~total~production~of~} f \\mathbf {~is~} d \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ if } \\exists \\; n_1 : (\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models \\; maximum \\mathbf {~total~production~of~} f \\text{~is~} n_1 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } \\exists \\; n_2 : (\\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace , j) \\models \\; maximum \\mathbf {~total~production~of~} f \\text{~is~} n_2 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } n_2 \\; d \\; n_1$ Next we define how a comparative quantitative aggregate point formula is satisfied on two sets of trajectories $\\sigma _1,\\dots ,\\sigma _m$ , and $\\bar{\\sigma }_1,\\dots ,\\bar{\\sigma }_{\\bar{m}}$ at point $i$ : $&\\Big \\lbrace \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\models \\; \\mathbf {~direction~of~change~in~} average \\mathbf {~value~of~} f \\mathbf {~is~} d \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ if } \\exists \\; n_1 : \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace \\models \\; average \\mathbf {~value~of~} f \\text{~is~} n_1 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } \\exists \\; n_2 : \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\models \\; average \\mathbf {~value~of~} f \\text{~is~} n_2 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } n_2 \\; d \\; n_1\\\\&\\Big \\lbrace \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\models \\; \\mathbf {~direction~of~change~in~} minimum \\mathbf {~value~of~} f \\mathbf {~is~} d \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ if } \\exists \\; n_1 : \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace \\models \\; minimum \\mathbf {~value~of~} f \\text{~is~} n_1 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } \\exists \\; n_2 : \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\models \\; minimum \\mathbf {~value~of~} f \\text{~is~} n_2 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } n_2 \\; d \\; n_1\\\\&\\Big \\lbrace \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\models \\; \\mathbf {~direction~of~change~in~} maximum \\mathbf {~value~of~} f \\mathbf {~is~} d \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ if } \\exists \\; n_1 : \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace \\models \\; maximum \\mathbf {~value~of~} f \\text{~is~} n_1 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } \\exists \\; n_2 : \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\models \\; maximum \\mathbf {~value~of~} f \\text{~is~} n_2 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } n_2 \\; d \\; n_1$ Given a domain description $\\mathbf {D}$ with simple fluents represented by a Guarded-Arc Petri Net with Colored tokens as defined in definition REF .", "Let $\\sigma = s_0,T_0,s_1,\\dots ,T_{k-1},s_k$ be its trajectory as defined in definition REF , and $s_i$ be a state on that trajectory.", "Let actions $T_i$ firing in state $s_i$ be observable in $s_i$ such that $T_i \\subseteq s_i$ .", "We define observation semantics using LTL below.", "We will use $s_i(f[l])$ to represent $m_{s_i(l)}(f)$ (multiplicity / value of $f$ in location $l$ ) in state $s_i$ .", "First we define how interval formulas are satisfied on a trajectory $\\sigma $ , starting state $s_i$ and an ending point $j$ : $(\\langle s_i,\\sigma \\rangle , j) \\models & \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n \\nonumber \\\\&\\text{ if } n=(s_j(f[l])-s_i(f[l]))/(j-i)\\\\(\\langle s_i,\\sigma \\rangle , j) \\models & \\mathbf {~rate~of~firing~of~} a \\mathbf {~is~} n \\nonumber \\\\&\\text{ if } n=\\sum _{i\\le k \\le j, \\langle s_k,\\sigma \\rangle \\models a \\mathbf {~occurs~}}{1}/(j-i)\\\\(\\langle s_i,\\sigma \\rangle , j) \\models & \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n \\nonumber \\\\&\\text{ if } n=(s_j(f[l])-s_i(f[l]))\\\\(\\langle s_i,\\sigma \\rangle , j) \\models & f \\mathbf {~is~accumulating~} \\mathbf {~atloc~} l \\nonumber \\\\&\\text{ if } (\\nexists k, i \\le k \\le j : s_{k+1}(f[l]) < s_k(f[l])) \\text{ and } s_j(f[l]) > s_i(f[l])\\\\(\\langle s_i,\\sigma \\rangle , j) \\models & f \\mathbf {~is~decreasing~} \\mathbf {~atloc~} l \\nonumber \\\\&\\text{ if } (\\nexists k, i \\le k \\le j : s_{k+1}(f[l]) > s_k(f[l])) \\text{ and } s_j(f[l]) < s_i(f[l])$ Next we define how a point formula is satisfied on a trajectory $\\sigma $ , in a state $s_i$ : $\\langle s_i,\\sigma \\rangle \\models & \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~higher~than~} n \\nonumber \\\\&\\text{ if } s_i(f[l]) > n\\\\\\langle s_i,\\sigma \\rangle \\models & \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~lower~than~} n \\nonumber \\\\&\\text{ if } s_i(f[l]) < n\\\\\\langle s_i,\\sigma \\rangle \\models & \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n \\nonumber \\\\&\\text{ if } s_i(f[l]) = n\\\\\\langle s_i,\\sigma \\rangle \\models & a \\mathbf {~occurs~} \\nonumber \\\\&\\text{ if } a \\in s_i\\\\\\langle s_i,\\sigma \\rangle \\models & a \\mathbf {~does~not~occur~} \\nonumber \\\\&\\text{ if } a \\notin s_i\\\\\\langle s_i,\\sigma \\rangle \\models & a_1 \\mathbf {~switches~to~} a_2 \\nonumber \\\\&\\text{ if } a_1 \\in s_{i-1} \\text{ and } a_2 \\notin s_{i-1} \\text{ and } a_1 \\notin s_i \\text{ and } a_2 \\in s_i$ Next we define how a quantitative all interval formula is satisfied on a set of trajectories $\\sigma _1,\\dots ,\\sigma _m$ with starting states $s^1_i,\\dots ,s^m_i$ and end point $j$ : $(\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models & \\mathbf {~rates~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~are~} [r_1,\\dots ,r_m] \\nonumber \\\\&\\text{ if } (\\langle s^1_i,\\sigma _1 \\rangle , j) \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} r_1\\nonumber \\\\&\\vdots \\nonumber \\\\&\\text{ and } (\\langle s^m_i,\\sigma _m \\rangle , j) \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} r_m \\\\(\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models & \\mathbf {~rates~of~firing~of~} a \\mathbf {~are~} [r_1,\\dots ,r_m] \\nonumber \\\\&\\text{ if } (\\langle s^1_i,\\sigma _1 \\rangle , j) \\models \\mathbf {~rate~of~firing~of~} a \\mathbf {~is~} r_1\\nonumber \\\\&\\vdots \\nonumber \\\\&\\text{ and } (\\langle s^m_i,\\sigma _m \\rangle , j) \\models \\mathbf {~rate~of~firing~of~} a \\mathbf {~is~} r_m \\\\(\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models & \\mathbf {~totals~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~are~} [r_1,\\dots ,r_m] \\nonumber \\\\&\\text{ if } (\\langle s^1_i,\\sigma _1 \\rangle , j) \\models \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} r_1\\nonumber \\\\&\\vdots \\nonumber \\\\&\\text{ and } (\\langle s^m_i,\\sigma _m \\rangle , j) \\models \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} r_m$ Next we define how a quantitative all point formula is satisfied w.r.t.", "a set of trajectories $\\sigma _1,\\dots ,\\sigma _m$ in states $s^1_i,\\dots ,s^m_i$ : $\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace \\models & \\mathbf {~values~of~} f \\mathbf {~atloc~} l\\mathbf {~are~} [r_1,\\dots ,r_m] \\nonumber \\\\&\\text{ if } \\langle s^1_i,\\sigma _1 \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} r_1\\nonumber \\\\&\\vdots \\nonumber \\\\&\\text{ and } \\langle s^m_i,\\sigma _m \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} r_m$ Next we define how a quantitative aggregate interval formula is satisfied w.r.t.", "a set of trajectories $\\sigma _1,\\dots ,\\sigma _m$ with starting states $s^1_i,\\dots ,s^m_i$ and end point $j$ : $(\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models & \\; average \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} r \\nonumber \\\\&\\text{ if } \\exists [r_1,\\dots ,r_m] : \\nonumber \\\\&~~~~~~(\\langle s^1_i,\\sigma _1 \\rangle , j) \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} r_1\\nonumber \\\\&~~~~~~\\vdots \\nonumber \\\\&~~~~~~(\\langle s^m_i,\\sigma _m \\rangle , j) \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} r_m \\nonumber \\\\&\\text{ and } r=(r_1+\\dots +r_m)/m\\\\(\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models & \\; minimum \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} r \\nonumber \\\\&\\text{ if } \\exists [r_1,\\dots ,r_m] : \\nonumber \\\\&~~~~~~(\\langle s^1_i,\\sigma _1 \\rangle , j) \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} r_1\\nonumber \\\\&~~~~~~\\vdots \\nonumber \\\\&~~~~~~(\\langle s^m_i,\\sigma _m \\rangle , j) \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} r_m \\nonumber \\\\&\\text{ and } \\exists k, 1 \\le k \\le m : r=r_k \\text{ and } \\forall x, 1 \\le x \\le m, r_k \\le r_x\\\\(\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models & \\; maximum \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} r \\nonumber \\\\&\\text{ if } \\exists [r_1,\\dots ,r_m] : \\nonumber \\\\&~~~~~~(\\langle s^1_i,\\sigma _1 \\rangle , j) \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} r_1\\nonumber \\\\&~~~~~~\\vdots \\nonumber \\\\&~~~~~~(\\langle s^m_i,\\sigma _m \\rangle , j) \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} r_m \\nonumber \\\\&\\text{ and } \\exists k, 1 \\le k \\le m : r=r_k \\text{ and } \\forall x, 1 \\le x \\le m, r_k \\ge r_x\\\\\\vspace{20.0pt}\\nonumber \\\\(\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models & \\; average \\mathbf {~rate~of~firing~of~} f \\mathbf {~is~} r \\nonumber \\\\&\\text{ iff } \\exists [r_1,\\dots ,r_m] : \\nonumber \\\\&~~~~~~(\\langle s^1_i,\\sigma _1 \\rangle , j) \\models \\mathbf {~rate~of~firing~of~} f \\mathbf {~is~} r_1\\nonumber \\\\&~~~~~~\\vdots \\nonumber \\\\&~~~~~~(\\langle s^m_i,\\sigma _m \\rangle , j) \\models \\mathbf {~rate~of~firing~of~} f \\mathbf {~is~} r_m \\nonumber \\\\&\\text{ and } r=(r_1+\\dots +r_m)/m\\\\(\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models & \\; minimum \\mathbf {~rate~of~firing~of~} f \\mathbf {~is~} r \\nonumber \\\\&\\text{ iff } \\exists [r_1,\\dots ,r_m] : \\nonumber \\\\&~~~~~~(\\langle s^1_i,\\sigma _1 \\rangle , j) \\models \\mathbf {~rate~of~firing~of~} f \\mathbf {~is~} r_1\\nonumber \\\\&~~~~~~\\vdots \\nonumber \\\\&~~~~~~(\\langle s^m_i,\\sigma _m \\rangle , j) \\models \\mathbf {~rate~of~firing~of~} f \\mathbf {~is~} r_m \\nonumber \\\\&\\text{ and } \\exists k, 1 \\le k \\le m : r=r_k \\text{ and } \\forall x, 1 \\le x \\le m, r_k \\le r_x\\\\(\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models & \\; maximum \\mathbf {~rate~of~firing~of~} f \\mathbf {~is~} r \\nonumber \\\\&\\text{ iff } \\exists [r_1,\\dots ,r_m] : \\nonumber \\\\&~~~~~~(\\langle s^1_i,\\sigma _1 \\rangle , j) \\models \\mathbf {~rate~of~firing~of~} f \\mathbf {~is~} r_1\\nonumber \\\\&~~~~~~\\vdots \\nonumber \\\\&~~~~~~(\\langle s^m_i,\\sigma _m \\rangle , j) \\models \\mathbf {~rate~of~firing~of~} f \\mathbf {~is~} r_m \\nonumber \\\\&\\text{ and } \\exists k, 1 \\le k \\le m : r=r_k \\text{ and } \\forall x, 1 \\le x \\le m, r_k \\ge r_x\\\\\\vspace{20.0pt}\\nonumber \\\\(\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models & \\; average \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} r \\nonumber \\\\&\\text{ if } \\exists [r_1,\\dots ,r_m] : \\nonumber \\\\&~~~~~~(\\langle s^1_i,\\sigma _1 \\rangle , j) \\models \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} r_1\\nonumber \\\\&~~~~~~\\vdots \\nonumber \\\\&~~~~~~(\\langle s^m_i,\\sigma _m \\rangle , j) \\models \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} r_m \\nonumber \\\\&\\text{ and } r=(r_1+\\dots +r_m)/m\\\\(\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models & \\; minimum \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} r \\nonumber \\\\&\\text{ if } \\exists [r_1,\\dots ,r_m] : \\nonumber \\\\&~~~~~~(\\langle s^1_i,\\sigma _1 \\rangle , j) \\models \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} r_1\\nonumber \\\\&~~~~~~\\vdots \\nonumber \\\\&~~~~~~(\\langle s^m_i,\\sigma _m \\rangle , j) \\models \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} r_m \\nonumber \\\\&\\text{ and } \\exists k, 1 \\le k \\le m : r=r_k \\text{ and } \\forall x, 1 \\le x \\le m, r_k \\le r_x\\\\(\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models & \\; maximum \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} r \\nonumber \\\\&\\text{ if } \\exists [r_1,\\dots ,r_m] : \\nonumber \\\\&~~~~~~(\\langle s^1_i,\\sigma _1 \\rangle , j) \\models \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} r_1\\nonumber \\\\&~~~~~~\\vdots \\nonumber \\\\&~~~~~~(\\langle s^m_i,\\sigma _m \\rangle , j) \\models \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} r_m \\nonumber \\\\&\\text{ and } \\exists k, 1 \\le k \\le m : r=r_k \\text{ and } \\forall x, 1 \\le x \\le m, r_k \\ge r_x$ Next we define how a quantitative aggregate point formula is satisfied w.r.t.", "a set of trajectories $\\sigma _1,\\dots ,\\sigma _m$ in states $s^1_i,\\dots ,s^m_i$ : $\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace \\models & \\; average \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} r \\nonumber \\\\&\\text{ if } \\exists [r_1,\\dots ,r_m] : \\nonumber \\\\&~~~~~~\\langle s^1_i,\\sigma _1 \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} r_1\\nonumber \\\\&~~~~~~\\vdots \\nonumber \\\\&~~~~~~\\langle s^m_i,\\sigma _m \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} r_m \\nonumber \\\\&\\text{ and } r=(r_1+\\dots +r_m)/m\\\\\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace \\models & \\; minimum \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} r \\nonumber \\\\&\\text{ if } \\exists [r_1,\\dots ,r_m] : \\nonumber \\\\&~~~~~~\\langle s^1_i,\\sigma _1 \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} r_1\\nonumber \\\\&~~~~~~\\vdots \\nonumber \\\\&~~~~~~\\langle s^m_i,\\sigma _m \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} r_m \\nonumber \\\\&\\text{ and } \\exists k, 1 \\le k \\le m : r=r_k \\text{ and } \\forall x, 1 \\le x \\le m, r_k \\le r_x\\\\\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace \\models & \\; maximum \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} r \\nonumber \\\\&\\text{ if } \\exists [r_1,\\dots ,r_m] : \\nonumber \\\\&~~~~~~\\langle s^1_i,\\sigma _1 \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} r_1\\nonumber \\\\&~~~~~~\\vdots \\nonumber \\\\&~~~~~~\\langle s^m_i,\\sigma _m \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} r_m \\nonumber \\\\&\\text{ and } \\exists k, 1 \\le k \\le m : r=r_k \\text{ and } \\forall x, 1 \\le x \\le m, r_k \\ge r_x$ Next we define how a comparative quantitative aggregate interval formula is satisfied w.r.t.", "two sets of trajectories $\\sigma _1,\\dots ,\\sigma _m$ , and $\\bar{\\sigma }_1,\\dots ,\\bar{\\sigma }_{\\bar{m}}$ a starting point $i$ and an ending point $j$ : $&\\Big (\\Big \\lbrace \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace , j \\Big ) \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\models \\; \\mathbf {~direction~of~change~in~} average \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} d \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ if } \\exists \\; n_1 : (\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models \\; average \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\text{~is~} n_1 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } \\exists \\; n_2 : (\\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace , j) \\models \\; average \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\text{~is~} n_2 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } n_2 \\; d \\; n_1\\\\&\\Big (\\Big \\lbrace \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace , j \\Big ) \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\models \\; \\mathbf {~direction~of~change~in~} minimum \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} d \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ if } \\exists \\; n_1 : (\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models \\; minimum \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\text{~is~} n_1 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } \\exists \\; n_2 : (\\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace , j) \\models \\; minimum \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\text{~is~} n_2 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } n_2 \\; d \\; n_1\\\\&\\Big (\\Big \\lbrace \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace , j \\Big ) \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\models \\; \\mathbf {~direction~of~change~in~} maximum \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} d \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ if } \\exists \\; n_1 : (\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models \\; maximum \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\text{~is~} n_1 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } \\exists \\; n_2 : (\\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace , j) \\models \\; maximum \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\text{~is~} n_2 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } n_2 \\; d \\; n_1\\\\\\vspace{20.0pt}\\nonumber \\\\&\\Big (\\Big \\lbrace \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace , j \\Big ) \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\models \\; \\mathbf {~direction~of~change~in~} average \\mathbf {~rate~of~firing~of~} f \\mathbf {~is~} d \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ if } \\exists \\; n_1 : (\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models \\; average \\mathbf {~rate~of~firing~of~} f \\text{~is~} n_1 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } \\exists \\; n_2 : (\\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace , j) \\models \\; average \\mathbf {~rate~of~firing~of~} f \\text{~is~} n_2 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } n_2 \\; d \\; n_1\\\\&\\Big (\\Big \\lbrace \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace , j \\Big ) \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\models \\; \\mathbf {~direction~of~change~in~} minimum \\mathbf {~rate~of~firing~of~} f \\mathbf {~is~} d \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ if } \\exists \\; n_1 : (\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models \\; minimum \\mathbf {~rate~of~firing~of~} f \\text{~is~} n_1 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } \\exists \\; n_2 : (\\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace , j) \\models \\; minimum \\mathbf {~rate~of~firing~of~} f \\text{~is~} n_2 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } n_2 \\; d \\; n_1\\\\&\\Big (\\Big \\lbrace \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace , j \\Big ) \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\models \\; \\mathbf {~direction~of~change~in~} maximum \\mathbf {~rate~of~firing~of~} f \\mathbf {~is~} d \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ if } \\exists \\; n_1 : (\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models \\; maximum \\mathbf {~rate~of~firing~of~} f \\text{~is~} n_1 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } \\exists \\; n_2 : (\\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace , j) \\models \\; maximum \\mathbf {~rate~of~firing~of~} f \\text{~is~} n_2 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } n_2 \\; d \\; n_1\\\\\\vspace{20.0pt}\\nonumber \\\\&\\Big (\\Big \\lbrace \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace , j \\Big ) \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\models \\; \\mathbf {~direction~of~change~in~} average \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} d \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ if } \\exists \\; n_1 : (\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models \\; average \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\text{~is~} n_1 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } \\exists \\; n_2 : (\\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace , j) \\models \\; average \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\text{~is~} n_2 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } n_2 \\; d \\; n_1\\\\&\\Big (\\Big \\lbrace \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace , j \\Big ) \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\models \\; \\mathbf {~direction~of~change~in~} minimum \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} d \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ if } \\exists \\; n_1 : (\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models \\; minimum \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\text{~is~} n_1 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } \\exists \\; n_2 : (\\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace , j) \\models \\; minimum \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\text{~is~} n_2 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } n_2 \\; d \\; n_1\\\\&\\Big (\\Big \\lbrace \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace , j \\Big ) \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\models \\; \\mathbf {~direction~of~change~in~} maximum \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} d \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ if } \\exists \\; n_1 : (\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models \\; maximum \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\text{~is~} n_1 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } \\exists \\; n_2 : (\\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace , j) \\models \\; maximum \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\text{~is~} n_2 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } n_2 \\; d \\; n_1$ Next we define how a comparative quantitative aggregate point formula is satisfied w.r.t.", "two sets of trajectories $\\sigma _1,\\dots ,\\sigma _m$ , and $\\bar{\\sigma }_1,\\dots ,\\bar{\\sigma }_{\\bar{m}}$ at point $i$ : $&\\Big \\lbrace \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\models \\; \\mathbf {~direction~of~change~in~} average \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} d \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ if } \\exists \\; n_1 : \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace \\models \\; average \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\text{~is~} n_1 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } \\exists \\; n_2 : \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\models \\; average \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\text{~is~} n_2 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } n_2 \\; d \\; n_1\\\\&\\Big \\lbrace \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\models \\; \\mathbf {~direction~of~change~in~} minimum \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} d \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ if } \\exists \\; n_1 : \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace \\models \\; minimum \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\text{~is~} n_1 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } \\exists \\; n_2 : \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\models \\; minimum \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\text{~is~} n_2 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } n_2 \\; d \\; n_1\\\\&\\Big \\lbrace \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\models \\; \\mathbf {~direction~of~change~in~} maximum \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} d \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ if } \\exists \\; n_1 : \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace \\models \\; maximum \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\text{~is~} n_1 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } \\exists \\; n_2 : \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\models \\; maximum \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\text{~is~} n_2 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } n_2 \\; d \\; n_1$ Trajectory Filtering due to Internal Observations The trajectories produced by the Guarded-Arc Petri Net execution are filtered to retain only the trajectories that satisfy all internal observations in a query statement.", "Let $\\sigma = s_0,\\dots ,s_k$ be a trajectory as given in definition REF .", "Then $\\sigma $ satisfies an observation: $&\\mathbf {~rate~of~production~of~} f \\mathbf {~is~} n \\nonumber \\\\&~~~~~~~~\\text{ if } (\\langle s_0,\\sigma \\rangle , k) \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~is~} n \\\\&\\mathbf {~rate~of~production~of~} f \\mathbf {~is~} n \\nonumber \\\\&~~~~~~~~\\text{ if } (\\langle s_0,\\sigma \\rangle , k) \\models \\mathbf {~rate~of~firing~of~} a \\mathbf {~is~} n \\\\&\\mathbf {~total~production~of~} f \\mathbf {~is~} n \\nonumber \\\\&~~~~~~~~\\text{ if } (\\langle s_0,\\sigma \\rangle , k) \\models \\mathbf {~total~production~of~} f \\mathbf {~is~} n \\\\&f \\mathbf {~is~accumulating~} \\nonumber \\\\&~~~~~~~~\\text{ if } (\\langle s_0,\\sigma \\rangle , k) \\models f \\mathbf {~is~accumulating~} \\\\&f \\mathbf {~is~decreasing~} \\nonumber \\\\&~~~~~~~~\\text{ if } (\\langle s_0,\\sigma \\rangle , k) \\models f \\mathbf {~is~decreasing~} \\\\\\vspace{20.0pt}\\nonumber \\\\&\\mathbf {~rate~of~production~of~} f \\mathbf {~is~} n \\nonumber \\\\&~~~~~~~~\\mathbf {when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&~~~~~~~~\\text{ if } (\\langle s_i,\\sigma \\rangle , j) \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~is~} n \\\\&\\mathbf {~rate~of~production~of~} f \\mathbf {~is~} n \\nonumber \\\\&~~~~~~~~\\mathbf {when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&~~~~~~~~\\text{ if } (\\langle s_i,\\sigma \\rangle , j) \\models \\mathbf {~rate~of~firing~of~} a \\mathbf {~is~} n \\\\&\\mathbf {~total~production~of~} f \\mathbf {~is~} n \\nonumber \\\\&~~~~~~~~\\mathbf {when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&~~~~~~~~\\text{ if } (\\langle s_i,\\sigma \\rangle , j) \\models \\mathbf {~total~production~of~} f \\mathbf {~is~} n \\\\&f \\mathbf {~is~accumulating~} \\nonumber \\\\&~~~~~~~~\\mathbf {when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&~~~~~~~~\\text{ if } (\\langle s_i,\\sigma \\rangle , j) \\models f \\mathbf {~is~accumulating~} \\\\&f \\mathbf {~is~decreasing~} \\nonumber \\\\&~~~~~~~~\\mathbf {when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&~~~~~~~~\\text{ if } (\\langle s_i,\\sigma \\rangle , j) \\models f \\mathbf {~is~decreasing~} \\\\\\vspace{20.0pt}\\nonumber \\\\&f \\mathbf {~is~higher~than~} n \\nonumber \\\\&~~~~~~~~\\text{ if } \\exists i, 0 \\le i \\le k: \\langle s_i,\\sigma \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~is~higher~than~} n \\\\&f \\mathbf {~is~lower~than~} n \\nonumber \\\\&~~~~~~~~\\text{ if } \\exists i, 0 \\le i \\le k: \\langle s_i,\\sigma \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~is~lower~than~} n \\\\&f \\mathbf {~is~} n \\nonumber \\\\&~~~~~~~~\\text{ if } \\exists i, 0 \\le i \\le k: \\langle s_i,\\sigma \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~is~} n \\\\&a \\mathbf {~occurs~} \\nonumber \\\\&~~~~~~~~\\text{ if } \\exists i, 0 \\le i \\le k: \\langle s_i,\\sigma \\rangle \\models a \\mathbf {~occurs~} \\\\&a \\mathbf {~does~not~occur~} \\nonumber \\\\&~~~~~~~~\\text{ if } \\exists i, 0 \\le i \\le k: \\langle s_i,\\sigma \\rangle \\models a \\mathbf {~does~not~occur~} \\\\&a_1 \\mathbf {~switches~to~} a_2 \\nonumber \\\\&~~~~~~~~\\text{ if } \\exists i, 0 \\le i \\le k: \\langle s_i,\\sigma \\rangle \\models a_1 \\mathbf {~switches~to~} a_2\\vspace{20.0pt}\\nonumber \\\\&f \\mathbf {~is~higher~than~} n \\nonumber \\\\&~~~~~~~~\\mathbf {at~time~step~} i \\nonumber \\\\&~~~~~~~~\\text{ if } \\langle s_i,\\sigma \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~is~higher~than~} n \\\\&f \\mathbf {~is~lower~than~} n \\nonumber \\\\&~~~~~~~~\\mathbf {at~time~step~} i \\nonumber \\\\&~~~~~~~~\\text{ if } \\langle s_i,\\sigma \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~is~lower~than~} n \\\\&f \\mathbf {~is~} n \\nonumber \\\\&~~~~~~~~\\mathbf {at~time~step~} i \\nonumber \\\\&~~~~~~~~\\text{ if } \\langle s_i,\\sigma \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~is~} n \\\\&a \\mathbf {~occurs~} \\nonumber \\\\&~~~~~~~~\\mathbf {at~time~step~} i \\nonumber \\\\&~~~~~~~~\\text{ if } \\langle s_i,\\sigma \\rangle \\models a \\mathbf {~occurs~} \\\\&a \\mathbf {~does~not~occur~} \\nonumber \\\\&~~~~~~~~\\mathbf {at~time~step~} i \\nonumber \\\\&~~~~~~~~\\text{ if } \\langle s_i,\\sigma \\rangle \\models a \\mathbf {~does~not~occur~} \\\\&a_1 \\mathbf {~switches~to~} a_2 \\nonumber \\\\&~~~~~~~~\\mathbf {at~time~step~} i \\nonumber \\\\&~~~~~~~~\\text{ if } \\langle s_i,\\sigma \\rangle \\models a_1 \\mathbf {~switches~to~} a_2$ Let $\\sigma = s_0,\\dots ,s_k$ be a trajectory of the form (REF ).", "Then $\\sigma $ satisfies an observation: $&\\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n \\nonumber \\\\&~~~~~~~~\\text{ if } (\\langle s_0,\\sigma \\rangle , k) \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n \\\\&\\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n \\nonumber \\\\&~~~~~~~~\\text{ if } (\\langle s_0,\\sigma \\rangle , k) \\models \\mathbf {~rate~of~firing~of~} a \\mathbf {~is~} n \\\\&\\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n \\nonumber \\\\&~~~~~~~~\\text{ if } (\\langle s_0,\\sigma \\rangle , k) \\models \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n \\\\&f \\mathbf {~is~accumulating~} \\mathbf {~atloc~} l \\nonumber \\\\&~~~~~~~~\\text{ if } (\\langle s_0,\\sigma \\rangle , k) \\models f \\mathbf {~is~accumulating~} \\mathbf {~atloc~} l \\\\&f \\mathbf {~is~decreasing~} \\mathbf {~atloc~} l \\nonumber \\\\&~~~~~~~~\\text{ if } (\\langle s_0,\\sigma \\rangle , k) \\models f \\mathbf {~is~decreasing~} \\mathbf {~atloc~} l \\\\\\vspace{20.0pt}\\nonumber \\\\&\\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n \\nonumber \\\\&~~~~~~~~\\mathbf {when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&~~~~~~~~\\text{ if } (\\langle s_i,\\sigma \\rangle , j) \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n \\\\&\\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n \\nonumber \\\\&~~~~~~~~\\mathbf {when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&~~~~~~~~\\text{ if } (\\langle s_i,\\sigma \\rangle , j) \\models \\mathbf {~rate~of~firing~of~} a \\mathbf {~is~} n \\\\&\\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n \\nonumber \\\\&~~~~~~~~\\mathbf {when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&~~~~~~~~\\text{ if } (\\langle s_i,\\sigma \\rangle , j) \\models \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n \\\\&f \\mathbf {~is~accumulating~} \\mathbf {~atloc~} l \\nonumber \\\\&~~~~~~~~\\mathbf {when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&~~~~~~~~\\text{ if } (\\langle s_i,\\sigma \\rangle , j) \\models f \\mathbf {~is~accumulating~} \\mathbf {~atloc~} l \\\\&f \\mathbf {~is~decreasing~} \\mathbf {~atloc~} l \\nonumber \\\\&~~~~~~~~\\mathbf {when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&~~~~~~~~\\text{ if } (\\langle s_i,\\sigma \\rangle , j) \\models f \\mathbf {~is~decreasing~} \\mathbf {~atloc~} l \\\\\\vspace{20.0pt}\\nonumber \\\\&f \\mathbf {~atloc~} l \\mathbf {~is~higher~than~} n \\nonumber \\\\&~~~~~~~~\\text{ if } \\exists i, 0 \\le i \\le k: \\langle s_i,\\sigma \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~higher~than~} n \\\\&f \\mathbf {~atloc~} l \\mathbf {~is~lower~than~} n \\nonumber \\\\&~~~~~~~~\\text{ if } \\exists i, 0 \\le i \\le k: \\langle s_i,\\sigma \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~lower~than~} n \\\\&f \\mathbf {~atloc~} l \\mathbf {~is~} n \\nonumber \\\\&~~~~~~~~\\text{ if } \\exists i, 0 \\le i \\le k: \\langle s_i,\\sigma \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n \\\\&a \\mathbf {~occurs~} \\nonumber \\\\&~~~~~~~~\\text{ if } \\exists i, 0 \\le i \\le k: \\langle s_i,\\sigma \\rangle \\models a \\mathbf {~occurs~} \\\\&a \\mathbf {~does~not~occur~} \\nonumber \\\\&~~~~~~~~\\text{ if } \\exists i, 0 \\le i \\le k: \\langle s_i,\\sigma \\rangle \\models a \\mathbf {~does~not~occur~} \\\\&a_1 \\mathbf {~switches~to~} a_2 \\nonumber \\\\&~~~~~~~~\\text{ if } \\exists i, 0 \\le i \\le k: \\langle s_i,\\sigma \\rangle \\models a_1 \\mathbf {~switches~to~} a_2\\vspace{20.0pt}\\nonumber \\\\&f \\mathbf {~atloc~} l \\mathbf {~is~higher~than~} n \\nonumber \\\\&~~~~~~~~\\mathbf {at~time~step~} i \\nonumber \\\\&~~~~~~~~\\text{ if } \\langle s_i,\\sigma \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~higher~than~} n \\\\&f \\mathbf {~atloc~} l \\mathbf {~is~lower~than~} n \\nonumber \\\\&~~~~~~~~\\mathbf {at~time~step~} i \\nonumber \\\\&~~~~~~~~\\text{ if } \\langle s_i,\\sigma \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~lower~than~} n \\\\&f \\mathbf {~atloc~} l \\mathbf {~is~} n \\nonumber \\\\&~~~~~~~~\\mathbf {at~time~step~} i \\nonumber \\\\&~~~~~~~~\\text{ if } \\langle s_i,\\sigma \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n \\\\&a \\mathbf {~occurs~} \\nonumber \\\\&~~~~~~~~\\mathbf {at~time~step~} i \\nonumber \\\\&~~~~~~~~\\text{ if } \\langle s_i,\\sigma \\rangle \\models a \\mathbf {~occurs~} \\\\&a \\mathbf {~does~not~occur~} \\nonumber \\\\&~~~~~~~~\\mathbf {at~time~step~} i \\nonumber \\\\&~~~~~~~~\\text{ if } \\langle s_i,\\sigma \\rangle \\models a \\mathbf {~does~not~occur~} \\\\&a_1 \\mathbf {~switches~to~} a_2 \\nonumber \\\\&~~~~~~~~\\mathbf {at~time~step~} i \\nonumber \\\\&~~~~~~~~\\text{ if } \\langle s_i,\\sigma \\rangle \\models a_1 \\mathbf {~switches~to~} a_2$ A trajectory $\\sigma $ is kept for further processing w.r.t.", "a set of internal observations $\\langle \\text{internal observation} \\rangle _1,\\dots ,\\langle \\text{internal observation} \\rangle _n$ if $\\sigma \\models \\langle \\text{internal observation} \\rangle _i$ , $1 \\le i \\le n$ .", "Query Description Satisfaction Now, we define query statement semantics using LTL syntax.", "Let $\\mathbf {D}$ be a domain description with simple fluents and $\\sigma = s_0,\\dots ,s_k$ be its trajectory of length $k$ as defined in (REF ).", "Let $\\sigma _1,\\dots ,\\sigma _m$ represent the set of trajectories of $\\mathbf {D}$ filtered by observations as necessary, with each trajectory has the form $\\sigma _i = s^i_0,\\dots ,s^i_k$ $1\\le i \\le m$ .", "Let $\\mathbf {\\bar{D}}$ be a modified domain description and $\\bar{\\sigma }_1,\\dots ,\\bar{\\sigma }_{\\bar{m}}$ be its trajectories of the form $\\bar{\\sigma }_i = \\bar{s}^i_0,\\dots ,\\bar{s}^i_k$ .", "Then we define query satisfaction using the formula satisfaction in section REF as follows.", "Two sets of trajectories $\\sigma _1,\\dots ,\\sigma _m$ and $\\bar{\\sigma }_1,\\dots ,\\bar{\\sigma }_{\\bar{m}}$ satisfy a comparative query description based on formula satisfaction of section REF as follows: $&\\Big \\lbrace \\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_0,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_0,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\models \\; \\mathbf {~direction~of~change~in~} aggop \\mathbf {~rate~of~production~of~} f \\mathbf {~is~} d \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\mathbf {~when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ if } \\Big (\\Big \\lbrace \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace , j \\Big ) \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\models \\mathbf {~direction~of~change~in~} aggop \\mathbf {~rate~of~production~of~} f \\mathbf {~is~} d \\\\&\\Big \\lbrace \\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_0,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_0,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\models \\; \\mathbf {~direction~of~change~in~} aggop \\mathbf {~rate~of~firing~of~} f \\mathbf {~is~} d \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\mathbf {~when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ if } \\Big (\\Big \\lbrace \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace , j\\Big ) \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\models \\mathbf {~direction~of~change~in~} aggop \\mathbf {~rate~of~firing~of~} f \\mathbf {~is~} d \\\\&\\Big \\lbrace \\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_0,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_0,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\models \\; \\mathbf {~direction~of~change~in~} aggop \\mathbf {~total~production~of~} f \\mathbf {~is~} d \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\mathbf {~when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ if } \\Big (\\Big \\lbrace \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace , j\\Big ) \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\models \\mathbf {~direction~of~change~in~} aggop \\mathbf {~total~production~of~} f \\mathbf {~is~} d \\\\&\\Big \\lbrace \\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_0,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_0,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\models \\; \\mathbf {~direction~of~change~in~} aggop \\mathbf {~value~of~} f \\mathbf {~is~} d \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\mathbf {~when~observed~at~time~step~} i \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ if } \\Big \\lbrace \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\models \\mathbf {~direction~of~change~in~} aggop \\mathbf {~value~of~} f \\mathbf {~is~} d$ A set of trajectories $\\sigma _1,\\dots ,\\sigma _m$ , each of length $k$ satisfies a quantitative or a qualitative interval query description based on formula satisfaction of section REF as follows: $&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~is~} n \\nonumber \\\\&~~~~\\text{ if } \\exists x, 1 \\le x \\le m : (\\langle s^x_0,\\sigma _x \\rangle , k) \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~is~} n \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~rate~of~firing~of~} a \\mathbf {~is~} n \\nonumber \\\\&~~~~\\text{ if } \\exists x, 1 \\le x \\le m : (\\langle s^x_0,\\sigma _x \\rangle , k) \\models \\mathbf {~rate~of~firing~of~} a \\mathbf {~is~} n \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~total~production~of~} f \\mathbf {~is~} n \\nonumber \\\\&~~~~\\text{ if } \\exists x, 1 \\le x \\le m : (\\langle s^x_0,\\sigma _x \\rangle , k) \\models \\mathbf {~total~production~of~} f \\mathbf {~is~} n \\\\\\vspace{20.0pt}\\nonumber \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~is~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&~~~~\\text{ if } \\exists x, 1 \\le x \\le m : (\\langle s^x_i,\\sigma _x \\rangle , j) \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~is~} n \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~rate~of~firing~of~} a \\mathbf {~is~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&~~~~\\text{ if } \\exists x, 1 \\le x \\le m : (\\langle s^x_i,\\sigma _x \\rangle , j) \\models \\mathbf {~rate~of~firing~of~} a \\mathbf {~is~} n \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~total~production~of~} f \\mathbf {~is~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&~~~~\\text{ if } \\exists x, 1 \\le x \\le m : (\\langle s^x_i,\\sigma _x \\rangle , j) \\models \\mathbf {~total~production~of~} f \\mathbf {~is~} n \\\\\\vspace{20.0pt}\\nonumber \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models f \\mathbf {~is~accumulating~} \\nonumber \\\\&~~~~\\text{ if } \\exists x, 1 \\le x \\le m : (\\langle s^x_0,\\sigma _x \\rangle , k) \\models f \\mathbf {~is~accumulating~} \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models f \\mathbf {~is~decreasing~} \\nonumber \\\\&~~~~\\text{ if } \\exists x, 1 \\le x \\le m : (\\langle s^x_0,\\sigma _x \\rangle , k) \\models f \\mathbf {~is~decreasing~} \\\\\\vspace{20.0pt}\\nonumber \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models f \\mathbf {~is~accumulating~} \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&~~~~\\text{ if } \\exists x, 1 \\le x \\le m : (\\langle s^x_i,\\sigma _x \\rangle , j) \\models f \\mathbf {~is~accumulating~} \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models f \\mathbf {~is~decreasing~} \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&~~~~\\text{ if } \\exists x, 1 \\le x \\le m : (\\langle s^x_i,\\sigma _x \\rangle , j) \\models f \\mathbf {~is~decreasing~} \\\\\\vspace{20.0pt}\\nonumber \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~is~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~in~all~trajectories~} \\nonumber \\\\&~~~~\\text{ if } \\forall x, 1 \\le x \\le m : (\\langle s^x_0,\\sigma _x \\rangle , k) \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~is~} n \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~rate~of~firing~of~} a \\mathbf {~is~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~in~all~trajectories~} \\nonumber \\\\&~~~~\\text{ if } \\forall x, 1 \\le x \\le m : (\\langle s^x_0,\\sigma _x \\rangle , k) \\models \\mathbf {~rate~of~firing~of~} a \\mathbf {~is~} n \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~total~production~of~} f \\mathbf {~is~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~in~all~trajectories~} \\nonumber \\\\&~~~~\\text{ if } \\forall x, 1 \\le x \\le m : (\\langle s^x_0,\\sigma _x \\rangle , k) \\models \\mathbf {~total~production~of~} f \\mathbf {~is~} n \\\\\\vspace{20.0pt}\\nonumber \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~is~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~in~all~trajectories~} \\nonumber \\\\&~~~~\\text{ if } \\forall x, 1 \\le x \\le m : (\\langle s^x_i,\\sigma _x \\rangle , j) \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~is~} n \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~rate~of~firing~of~} a \\mathbf {~is~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~in~all~trajectories~} \\nonumber \\\\&~~~~\\text{ if } \\forall x, 1 \\le x \\le m : (\\langle s^x_i,\\sigma _x \\rangle , j) \\models \\mathbf {~rate~of~firing~of~} a \\mathbf {~is~} n \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~total~production~of~} f \\mathbf {~is~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~in~all~trajectories~} \\nonumber \\\\&~~~~\\text{ if } \\forall x, 1 \\le x \\le m : (\\langle s^x_i,\\sigma _x \\rangle , j) \\models \\mathbf {~total~production~of~} f \\mathbf {~is~} n \\\\\\vspace{20.0pt}\\nonumber \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models f \\mathbf {~is~accumulating~} \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~in~all~trajectories~} \\nonumber \\\\&~~~~\\text{ if } \\forall x, 1 \\le x \\le m : (\\langle s^x_0,\\sigma _x \\rangle , k) \\models f \\mathbf {~is~accumulating~} \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models f \\mathbf {~is~decreasing~} \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~in~all~trajectories~} \\nonumber \\\\&~~~~\\text{ if } \\forall x, 1 \\le x \\le m : (\\langle s^x_0,\\sigma _x \\rangle , k) \\models f \\mathbf {~is~decreasing~} \\\\\\vspace{20.0pt}\\nonumber \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models f \\mathbf {~is~accumulating~} \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~in~all~trajectories~} \\nonumber \\\\&~~~~\\text{ if } \\forall x, 1 \\le x \\le m : (\\langle s^x_i,\\sigma _x \\rangle , j) \\models f \\mathbf {~is~accumulating~} \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models f \\mathbf {~is~decreasing~} \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~in~all~trajectories~} \\nonumber \\\\&~~~~\\text{ if } \\forall x, 1 \\le x \\le m : (\\langle s^x_i,\\sigma _x \\rangle , j) \\models f \\mathbf {~is~decreasing~}$ A set of trajectories $\\sigma _1,\\dots ,\\sigma _m$ , each of length $k$ satisfies a quantitative or a qualitative point query description based on formula satisfaction of section REF as follows: $&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~value~of~} f \\mathbf {~is~higher~than~} n \\nonumber \\\\&~~~~\\text{ if } \\exists x \\exists i, 1 \\le x \\le m, 0 \\le i \\le k : \\langle s^x_i,\\sigma _x \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~is~higher~than~} n \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~value~of~} f \\mathbf {~is~lower~than~} n \\nonumber \\\\&~~~~\\text{ if } \\exists x \\exists i, 1 \\le x \\le m, 0 \\le i \\le k : \\langle s^x_i,\\sigma _x \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~is~lower~than~} n \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~value~of~} f \\mathbf {~is~} n \\nonumber \\\\&~~~~\\text{ if } \\exists x \\exists i, 1 \\le x \\le m, 0 \\le i \\le k : \\langle s^x_i,\\sigma _x \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~is~} n \\\\\\vspace{20.0pt}\\nonumber \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~value~of~} f \\mathbf {~is~higher~than~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~at~time~step~} i \\nonumber \\\\&~~~~\\text{ if } \\exists x, 1 \\le x \\le m : \\langle s^x_i,\\sigma _x \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~is~higher~than~} n \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~value~of~} f \\mathbf {~is~lower~than~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~at~time~step~} i \\nonumber \\\\&~~~~\\text{ if } \\exists x, 1 \\le x \\le m : \\langle s^x_i,\\sigma _x \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~is~lower~than~} n \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~value~of~} f \\mathbf {~is~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~at~time~step~} i \\nonumber \\\\&~~~~\\text{ if } \\exists x, 1 \\le x \\le m : \\langle s^x_i,\\sigma _x \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~is~} n \\\\\\vspace{20.0pt}\\nonumber \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~value~of~} f \\mathbf {~is~higher~than~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~in~all~trajectories~} \\nonumber \\\\&~~~~\\text{ if } \\forall x \\exists i, 1 \\le x \\le m, 0 \\le i \\le k : \\langle s^x_i,\\sigma _x \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~is~higher~than~} n \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~value~of~} f \\mathbf {~is~lower~than~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~in~all~trajectories~} \\nonumber \\\\&~~~~\\text{ if } \\forall x \\exists i, 1 \\le x \\le m, 0 \\le i \\le k : \\langle s^x_i,\\sigma _x \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~is~lower~than~} n \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~value~of~} f \\mathbf {~is~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~in~all~trajectories~} \\nonumber \\\\&~~~~\\text{ if } \\forall x \\exists i, 1 \\le x \\le m, 0 \\le i \\le k : \\langle s^x_i,\\sigma _x \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~is~} n \\\\\\vspace{20.0pt}\\nonumber \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~value~of~} f \\mathbf {~is~higher~than~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~at~time~step~} i \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~in~all~trajectories~} \\nonumber \\\\&~~~~\\text{ if } \\forall x, 1 \\le x \\le m : \\langle s^x_i,\\sigma _x \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~is~higher~than~} n \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~value~of~} f \\mathbf {~is~lower~than~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~at~time~step~} i \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~in~all~trajectories~} \\nonumber \\\\&~~~~\\text{ if } \\forall x, 1 \\le x \\le m : \\langle s^x_i,\\sigma _x \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~is~lower~than~} n \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~value~of~} f \\mathbf {~is~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~at~time~step~} i \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~in~all~trajectories~} \\nonumber \\\\&~~~~\\text{ if } \\forall x, 1 \\le x \\le m : \\langle s^x_i,\\sigma _x \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~is~} n$ Now, we turn our attention to domain descriptions with locational fluents.", "Let $\\mathbf {D}$ be a domain description and $\\sigma = s_0,\\dots ,s_k$ be its trajectory of length $k$ as defined in (REF ).", "Let $\\sigma _1,\\dots ,\\sigma _m$ represent the set of trajectories of $\\mathbf {D}$ filtered by observations as necessary, with each trajectory has the form $\\sigma _i = s^i_0,\\dots ,s^i_k$ $1\\le i \\le m$ .", "Let $\\mathbf {\\bar{D}}$ be a modified domain description and $\\bar{\\sigma }_1,\\dots ,\\bar{\\sigma }_{\\bar{m}}$ be its trajectories of the form $\\bar{\\sigma }_i = \\bar{s}^i_0,\\dots ,\\bar{s}^i_k$ .", "Then we define query satisfaction using the formula satisfaction in section REF as follows.", "Two sets of trajectories $\\sigma _1,\\dots ,\\sigma _m$ and $\\bar{\\sigma }_1,\\dots ,\\bar{\\sigma }_{\\bar{m}}$ satisfy a comparative query description based on formula satisfaction of section REF as follows: $&\\Big \\lbrace \\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_0,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_0,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\models \\; \\mathbf {~direction~of~change~in~} \\langle \\text{aggop} \\rangle \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} d \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\mathbf {~when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ if } \\Big (\\Big \\lbrace \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace , j \\Big ) \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\models \\mathbf {~direction~of~change~in~} \\langle \\text{aggop} \\rangle \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} d \\\\&\\Big \\lbrace \\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_0,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_0,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\models \\; \\mathbf {~direction~of~change~in~} \\langle \\text{aggop} \\rangle \\mathbf {~rate~of~firing~of~} f \\mathbf {~is~} d \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\mathbf {~when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ if } \\Big (\\Big \\lbrace \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace , j\\Big ) \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\models \\mathbf {~direction~of~change~in~} \\langle \\text{aggop} \\rangle \\mathbf {~rate~of~firing~of~} f \\mathbf {~is~} d \\\\&\\Big \\lbrace \\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_0,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_0,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\models \\; \\mathbf {~direction~of~change~in~} \\langle \\text{aggop} \\rangle \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} d \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\mathbf {~when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ if } \\Big (\\Big \\lbrace \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace , j\\Big ) \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\models \\mathbf {~direction~of~change~in~} \\langle \\text{aggop} \\rangle \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} d \\\\&\\Big \\lbrace \\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_0,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_0,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\models \\; \\mathbf {~direction~of~change~in~} \\langle \\text{aggop} \\rangle \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} d \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\mathbf {~when~observed~at~time~step~} i \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ if } \\Big \\lbrace \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\models \\mathbf {~direction~of~change~in~} \\langle \\text{aggop} \\rangle \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} d$ A set of trajectories $\\sigma _1,\\dots ,\\sigma _m$ , each of length $k$ satisfies a quantitative or a qualitative interval query description based on formula satisfaction of section REF as follows: $&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n \\nonumber \\\\&~~~~\\text{ if } \\exists x, 1 \\le x \\le m : (\\langle s^x_0,\\sigma _x \\rangle , k) \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~rate~of~firing~of~} a \\mathbf {~is~} n \\nonumber \\\\&~~~~\\text{ if } \\exists x, 1 \\le x \\le m : (\\langle s^x_0,\\sigma _x \\rangle , k) \\models \\mathbf {~rate~of~firing~of~} a \\mathbf {~is~} n \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n \\nonumber \\\\&~~~~\\text{ if } \\exists x, 1 \\le x \\le m : (\\langle s^x_0,\\sigma _x \\rangle , k) \\models \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n \\\\\\vspace{20.0pt}\\nonumber \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&~~~~\\text{ if } \\exists x, 1 \\le x \\le m : (\\langle s^x_i,\\sigma _x \\rangle , j) \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~rate~of~firing~of~} a \\mathbf {~is~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&~~~~\\text{ if } \\exists x, 1 \\le x \\le m : (\\langle s^x_i,\\sigma _x \\rangle , j) \\models \\mathbf {~rate~of~firing~of~} a \\mathbf {~is~} n \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&~~~~\\text{ if } \\exists x, 1 \\le x \\le m : (\\langle s^x_i,\\sigma _x \\rangle , j) \\models \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n \\\\\\vspace{20.0pt}\\nonumber \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models f \\mathbf {~is~accumulating~} \\mathbf {~atloc~} l \\nonumber \\\\&~~~~\\text{ if } \\exists x, 1 \\le x \\le m : (\\langle s^x_0,\\sigma _x \\rangle , k) \\models f \\mathbf {~is~accumulating~} \\mathbf {~atloc~} l \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models f \\mathbf {~is~decreasing~} \\mathbf {~atloc~} l \\nonumber \\\\&~~~~\\text{ if } \\exists x, 1 \\le x \\le m : (\\langle s^x_0,\\sigma _x \\rangle , k) \\models f \\mathbf {~is~decreasing~} \\mathbf {~atloc~} l \\\\\\vspace{20.0pt}\\nonumber \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models f \\mathbf {~is~accumulating~} \\mathbf {~atloc~} l \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&~~~~\\text{ if } \\exists x, 1 \\le x \\le m : (\\langle s^x_i,\\sigma _x \\rangle , j) \\models f \\mathbf {~is~accumulating~} \\mathbf {~atloc~} l \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models f \\mathbf {~is~decreasing~} \\mathbf {~atloc~} l \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&~~~~\\text{ if } \\exists x, 1 \\le x \\le m : (\\langle s^x_i,\\sigma _x \\rangle , j) \\models f \\mathbf {~is~decreasing~} \\mathbf {~atloc~} l \\\\\\vspace{30.0pt} \\nonumber \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~in~all~trajectories~} \\nonumber \\\\&~~~~\\text{ if } \\forall x, 1 \\le x \\le m : (\\langle s^x_0,\\sigma _x \\rangle , k) \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~rate~of~firing~of~} a \\mathbf {~is~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~in~all~trajectories~} \\nonumber \\\\&~~~~\\text{ if } \\forall x, 1 \\le x \\le m : (\\langle s^x_0,\\sigma _x \\rangle , k) \\models \\mathbf {~rate~of~firing~of~} a \\mathbf {~is~} n \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~in~all~trajectories~} \\nonumber \\\\&~~~~\\text{ if } \\forall x, 1 \\le x \\le m : (\\langle s^x_0,\\sigma _x \\rangle , k) \\models \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n \\\\\\vspace{20.0pt}\\nonumber \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~in~all~trajectories~} \\nonumber \\\\&~~~~\\text{ if } \\forall x, 1 \\le x \\le m : (\\langle s^x_i,\\sigma _x \\rangle , j) \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~rate~of~firing~of~} a \\mathbf {~is~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~in~all~trajectories~} \\nonumber \\\\&~~~~\\text{ if } \\forall x, 1 \\le x \\le m : (\\langle s^x_i,\\sigma _x \\rangle , j) \\models \\mathbf {~rate~of~firing~of~} a \\mathbf {~is~} n \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~in~all~trajectories~} \\nonumber \\\\&~~~~\\text{ if } \\forall x, 1 \\le x \\le m : (\\langle s^x_i,\\sigma _x \\rangle , j) \\models \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n \\text{ till } j\\\\\\vspace{20.0pt}\\nonumber \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models f \\mathbf {~is~accumulating~} \\mathbf {~atloc~} l \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~in~all~trajectories~} \\nonumber \\\\&~~~~\\text{ if } \\forall x, 1 \\le x \\le m : (\\langle s^x_0,\\sigma _x \\rangle , k) \\models f \\mathbf {~is~accumulating~} \\mathbf {~atloc~} l \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models f \\mathbf {~is~decreasing~} \\mathbf {~atloc~} l \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~in~all~trajectories~} \\nonumber \\\\&~~~~\\text{ if } \\forall x, 1 \\le x \\le m : (\\langle s^x_0,\\sigma _x \\rangle , k) \\models f \\mathbf {~is~decreasing~} \\mathbf {~atloc~} l \\\\\\vspace{20.0pt}\\nonumber \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models f \\mathbf {~is~accumulating~} \\mathbf {~atloc~} l \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~in~all~trajectories~} \\nonumber \\\\&~~~~\\text{ if } \\forall x, 1 \\le x \\le m : (\\langle s^x_i,\\sigma _x \\rangle , j) \\models f \\mathbf {~is~accumulating~} \\mathbf {~atloc~} l \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models f \\mathbf {~is~decreasing~} \\mathbf {~atloc~} l \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~in~all~trajectories~} \\nonumber \\\\&~~~~\\text{ if } \\forall x, 1 \\le x \\le m : (\\langle s^x_i,\\sigma _x \\rangle , j) \\models f \\mathbf {~is~decreasing~} \\mathbf {~atloc~} l$ A set of trajectories $\\sigma _1,\\dots ,\\sigma _m$ , each of length $k$ satisfies a quantitative or a qualitative point query description based on formula satisfaction of section REF as follows: $&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l\\mathbf {~is~higher~than~} n \\nonumber \\\\&~~~~\\text{ if } \\exists x \\exists i, 1 \\le x \\le m, 0 \\le i \\le k : \\langle s^x_i,\\sigma _x \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~higher~than~} n \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l\\mathbf {~is~lower~than~} n \\nonumber \\\\&~~~~\\text{ if } \\exists x \\exists i, 1 \\le x \\le m, 0 \\le i \\le k : \\langle s^x_i,\\sigma _x \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~lower~than~} n \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l\\mathbf {~is~} n \\nonumber \\\\&~~~~\\text{ if } \\exists x \\exists i, 1 \\le x \\le m, 0 \\le i \\le k : \\langle s^x_i,\\sigma _x \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n\\\\\\vspace{20.0pt}\\nonumber \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l\\mathbf {~is~higher~than~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~at~time~step~} i \\nonumber \\\\&~~~~\\text{ if } \\exists x, 1 \\le x \\le m : \\langle s^x_i,\\sigma _x \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~higher~than~} n \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l\\mathbf {~is~lower~than~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~at~time~step~} i \\nonumber \\\\&~~~~\\text{ if } \\exists x, 1 \\le x \\le m : \\langle s^x_i,\\sigma _x \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~lower~than~} n \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l\\mathbf {~is~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~at~time~step~} i \\nonumber \\\\&~~~~\\text{ if } \\exists x, 1 \\le x \\le m : \\langle s^x_i,\\sigma _x \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n\\\\\\vspace{20.0pt}\\nonumber \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l\\mathbf {~is~higher~than~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~in~all~trajectories~} \\nonumber \\\\&~~~~\\text{ if } \\forall x \\exists i, 1 \\le x \\le m, 0 \\le i \\le k : \\langle s^x_i,\\sigma _x \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~higher~than~} n \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~lower~than~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~in~all~trajectories~} \\nonumber \\\\&~~~~\\text{ if } \\forall x \\exists i, 1 \\le x \\le m, 0 \\le i \\le k : \\langle s^x_i,\\sigma _x \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~lower~than~} n \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l\\mathbf {~is~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~in~all~trajectories~} \\nonumber \\\\&~~~~\\text{ if } \\forall x \\exists i, 1 \\le x \\le m, 0 \\le i \\le k : \\langle s^x_i,\\sigma _x \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n\\\\\\vspace{20.0pt}\\nonumber \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~higher~than~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~at~time~step~} i \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~in~all~trajectories~} \\nonumber \\\\&~~~~\\text{ if } \\forall x, 1 \\le x \\le m : \\langle s^x_i,\\sigma _x \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~higher~than~} n \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l\\mathbf {~is~lower~than~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~at~time~step~} i \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~in~all~trajectories~} \\nonumber \\\\&~~~~\\text{ if } \\forall x, 1 \\le x \\le m : \\langle s^x_i,\\sigma _x \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~lower~than~} n \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l\\mathbf {~is~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~at~time~step~} i \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~in~all~trajectories~} \\nonumber \\\\&~~~~\\text{ if } \\forall x, 1 \\le x \\le m : \\langle s^x_i,\\sigma _x \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n$ Next, we generically define the satisfaction of a simple point formula cascade query w.r.t.", "a set of trajectories $\\sigma _1,\\dots ,\\sigma _m$ .", "The trajectories will either be as defined in definitions (REF ) or (REF ) for simple point formula cascade query statement made up of simple fluents or locational fluents, respectively.", "A set of trajectories $\\sigma _1,\\dots ,\\sigma _m$ , each of length $k$ satisfies a simple point formula cascade based on formula satisfaction of section REF as follows: $&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\langle \\text{simple point formula} \\rangle _0 \\nonumber \\\\&~~~~~~~~\\mathbf {~after~} \\langle \\text{simple point formula} \\rangle _{1,1}, \\dots , \\langle \\text{simple point formula} \\rangle _{1,n_1} \\nonumber \\\\&~~~~~~~~\\vdots \\nonumber \\\\&~~~~~~~~\\mathbf {~after~} \\langle \\text{simple point formula} \\rangle _{u,1}, \\dots , \\langle \\text{simple point formula} \\rangle _{u,n_u} \\nonumber \\\\&~~~~\\text{ if } \\exists x \\exists i_0 \\exists i_1 \\dots \\exists i_u, 1 \\le x \\le m, i_1 < i_0 \\le k, \\dots , i_u < i_{u-1} \\le k, 0 \\le i_u \\le k: \\nonumber \\\\&~~~~~~~~\\langle s^x_{i_0},\\sigma _x \\rangle \\models \\langle \\text{simple point formula} \\rangle _0 \\text{ and } \\nonumber \\\\&~~~~~~~~\\langle s^x_{i_1},\\sigma _x \\rangle \\models \\langle \\text{simple point formula} \\rangle _{1,1} \\text{ and } \\nonumber \\hdots \\text{ and } \\langle s^x_{i_1},\\sigma _x \\rangle \\models \\langle \\text{simple point formula} \\rangle _{1,n_1} \\text{ and } \\nonumber \\\\&~~~~~~~~~~~~\\vdots \\nonumber \\\\&~~~~~~~~\\langle s^x_{i_u},\\sigma _x \\rangle \\models \\langle \\text{simple point formula} \\rangle _{u,1} \\text{ and } \\nonumber \\hdots \\text{ and } \\langle s^x_{i_u},\\sigma _x \\rangle \\models \\langle \\text{simple point formula} \\rangle _{u,n_u} \\\\\\\\\\vspace{10.0pt}\\nonumber \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\langle \\text{simple point formula} \\rangle _0 \\nonumber \\\\&~~~~~~~~\\mathbf {~after~} \\langle \\text{simple point formula} \\rangle _{1,1}, \\dots , \\langle \\text{simple point formula} \\rangle _{1,n_1} \\nonumber \\\\&~~~~~~~~\\vdots \\nonumber \\\\&~~~~~~~~\\mathbf {~after~} \\langle \\text{simple point formula} \\rangle _{u,1}, \\dots , \\langle \\text{simple point formula} \\rangle _{u,n_u} \\nonumber \\\\&~~~~~~~~\\mathbf {~in~all~trajectories} \\nonumber \\\\&~~~~\\text{ if } \\forall x, 1 \\le x \\le m, \\exists i_0 \\exists i_1 \\dots \\exists i_u, i_1 < i_0 \\le k, \\dots , i_u < i_{u-1} \\le k, 0 \\le i_u \\le k: \\nonumber \\\\&~~~~~~~~\\langle s^x_{i_0},\\sigma _x \\rangle \\models \\langle \\text{simple point formula} \\rangle _0 \\text{ and } \\nonumber \\\\&~~~~~~~~\\langle s^x_{i_1},\\sigma _x \\rangle \\models \\langle \\text{simple point formula} \\rangle _{1,1} \\text{ and } \\nonumber \\hdots \\text{ and } \\langle s^x_{i_1},\\sigma _x \\rangle \\models \\langle \\text{simple point formula} \\rangle _{1,n_1} \\text{ and } \\nonumber \\\\&~~~~~~~~~~~~\\vdots \\nonumber \\\\&~~~~~~~~\\langle s^x_{i_u},\\sigma _x \\rangle \\models \\langle \\text{simple point formula} \\rangle _{u,1} \\text{ and } \\nonumber \\hdots \\text{ and } \\langle s^x_{i_u},\\sigma _x \\rangle \\models \\langle \\text{simple point formula} \\rangle _{u,n_u} \\\\\\\\\\vspace{10.0pt}\\nonumber \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\langle \\text{simple point formula} \\rangle _0 \\nonumber \\\\&~~~~~~~~\\mathbf {~when~} \\langle \\text{simple point formula} \\rangle _{1,1}, \\dots , \\langle \\text{simple point formula} \\rangle _{1,n_1} \\nonumber \\\\&~~~~\\text{ if } \\exists x \\exists i, 1 \\le x \\le m, 0 \\le i \\le k: \\nonumber \\\\&~~~~~~~~\\langle s^x_{i_0},\\sigma _x \\rangle \\models \\langle \\text{simple point formula} \\rangle _0 \\text{ and } \\nonumber \\\\&~~~~~~~~\\langle s^x_{i_1},\\sigma _x \\rangle \\models \\langle \\text{simple point formula} \\rangle _{1,1} \\text{ and } \\nonumber \\hdots \\text{ and } \\langle s^x_{i_1},\\sigma _x \\rangle \\models \\langle \\text{simple point formula} \\rangle _{1,n_1} \\\\\\\\\\vspace{10.0pt}\\nonumber \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\langle \\text{simple point formula} \\rangle _0 \\nonumber \\\\&~~~~~~~~\\mathbf {~when~} \\langle \\text{simple point formula} \\rangle _{1,1}, \\dots , \\langle \\text{simple point formula} \\rangle _{1,n_1} \\nonumber \\\\&~~~~~~~~\\mathbf {~in~all~trajectories} \\nonumber \\\\&~~~~\\text{ if } \\forall x, 1 \\le x \\le m, \\exists i, 0 \\le i \\le k: \\nonumber \\\\&~~~~~~~~\\langle s^x_{i_0},\\sigma _x \\rangle \\models \\langle \\text{simple point formula} \\rangle _0 \\text{ and } \\nonumber \\\\&~~~~~~~~\\langle s^x_{i_1},\\sigma _x \\rangle \\models \\langle \\text{simple point formula} \\rangle _{1,1} \\text{ and } \\nonumber \\hdots \\text{ and } \\langle s^x_{i_1},\\sigma _x \\rangle \\models \\langle \\text{simple point formula} \\rangle _{1,n_1} \\\\$ Query Statement Satisfaction Let $\\mathbf {D}$ as defined in section REF be a domain description and $\\mathbf {Q}$ be a query statement (REF ) as defined in section REF with query description $U$ , interventions $V_1,\\dots ,V_{\|V\|}$ , internal observations $O_1,\\dots ,O_{\|O\|}$ , and initial conditions $I_1,\\dots ,I_{\|I\|}$ .", "Let $\\mathbf {D_1} \\equiv \\mathbf {D} \\diamond I_1 \\diamond \\dots \\diamond I_{\|I\|} \\diamond V_1 \\diamond \\dots \\diamond V_{\|V\|}$ be the modified domain description constructed by applying the initial conditions and interventions from $\\mathbf {Q}$ as defined in section REF .", "Let $\\sigma _1,\\dots ,\\sigma _m$ be the trajectories of $\\mathbf {D_1}$ that satisfy $O_1,\\dots ,O_{\|O\|}$ as given in section REF .", "Then, $\\mathbf {D}$ satisfies $\\mathbf {Q}$ if $\\lbrace \\sigma _1,\\dots ,\\sigma _m\\rbrace \\models U$ as defined in section REF .", "Let $\\mathbf {D}$ as defined in section REF be a domain description and $\\mathbf {Q}$ be a query statement () as defined in section REF with query description $U$ , interventions $V_1,\\dots ,V_{\|V\|}$ , internal observations $O_1,\\dots ,O_{\|O\|}$ , and initial conditions $I_1,\\dots ,I_{\|I\|}$ .", "Let $\\mathbf {D_)} \\equiv \\mathbf {D} \\diamond I_1 \\diamond \\dots \\diamond I_{\|I\|} $ be the nominal domain description constructed by applying the initial conditions from $\\mathbf {Q}$ as defined in section REF .", "Let $\\sigma _1,\\dots ,\\sigma _m$ be the trajectories of $\\mathbf {D_0}$ .", "Let $\\mathbf {D_1} \\equiv \\mathbf {D} \\diamond I_1 \\diamond \\dots \\diamond I_{\|I\|} \\diamond V_1 \\diamond \\dots \\diamond V_{\|V\|}$ be the modified domain description constructed by applying the initial conditions and interventions from $\\mathbf {Q}$ as defined in section REF .", "Let $\\bar{\\sigma }_1,\\dots ,\\bar{\\sigma }_{\\bar{m}}$ be the trajectories of $\\mathbf {D_1}$ that satisfy $O_1,\\dots ,O_{\|O\|}$ as given in section REF .", "Then, $\\mathbf {D}$ satisfies $\\mathbf {Q}$ if $\\Big \\lbrace \\lbrace \\sigma _1,\\dots ,\\sigma _m\\rbrace , \\lbrace \\bar{\\sigma }_1,\\dots ,\\bar{\\sigma }_{\\bar{m}}\\rbrace \\Big \\rbrace \\models U$ as defined in section REF .", "Example Encodings In this section we give some examples of how we will encode queries and pathways related to these queries.", "We will also show how the pathway is modified to answer questions Some of the same pathways appear in previous chapters, they have been updated here with additional background knowledge..", "Question 11 At one point in the process of glycolysis, both DHAP and G3P are produced.", "Isomerase catalyzes the reversible conversion between these two isomers.", "The conversion of DHAP to G3P never reaches equilibrium and G3P is used in the next step of glycolysis.", "What would happen to the rate of glycolysis if DHAP were removed from the process of glycolysis as quickly as it was produced?", "Figure: Petri Net for question The question is asking for the direction of change in the rate of glycolysis when the nominal glycolysis pathway is compared against a modified pathway in which dhap is removed as soon as it is produced.", "Since this rate can vary with the trajectory followed by the world evolution, we consider the average change in rate.", "From the domain knowledge [64] we know that the rate of $glycolysis$ can be measured by the rate of $pyruvate$ (the end product of glycolysis) and that the rate of $pyruvate$ is equal to the rate of $bpg13$ (due to linear chain from $bpg13$ to $pyruvate$ ).", "Thus, we can monitor the rate of $bpg13$ instead to determine the rate of glycolysis.", "To ensure that our pathway is not starved due to source ingredients, we add a continuous supply of $f16bp$ in quantity 1 to the pathway.", "Then, the following pathway specification encodes the domain description $\\mathbf {D}$ for question REF and produces the PN in Fig.", "REF minus the $tr,t3$ transitions: $&\\begin{array}{llll}&\\mathbf {domain~of~} & f16bp \\mathbf {~is~} integer, & dhap \\mathbf {~is~} integer, \\\\&& g3p \\mathbf {~is~} integer, & bpg13 \\mathbf {~is~} integer\\nonumber \\\\&t4 \\mathbf {~may~execute~causing~} & f16bp \\mathbf {~change~value~by~} -1, & dhap \\mathbf {~change~value~by~} +1,\\\\ && g3p \\mathbf {~change~value~by~} +1\\nonumber \\\\&t5a \\mathbf {~may~execute~causing~} & dhap \\mathbf {~change~value~by~} -1, & g3p \\mathbf {~change~value~by~} +1\\nonumber \\\\&t5b \\mathbf {~may~execute~causing~} & g3p \\mathbf {~change~value~by~} -1, & dhap \\mathbf {~change~value~by~} +1\\nonumber \\\\&t6 \\mathbf {~may~execute~causing~} & g3p \\mathbf {~change~value~by~} -1, & bpg13 \\mathbf {~change~value~by~} +2\\nonumber \\\\& \\mathbf {initially~} & f16bp \\mathbf {~has~value~} 0, & dhap \\mathbf {~has~value~} 0,\\\\ && g3p \\mathbf {~has~value~} 0, & bpg13 \\mathbf {~has~value~} 0\\nonumber \\\\&\\mathbf {firing~style~} & max\\end{array}\\\\$ And the following query $\\mathbf {Q}$ for a simulation of length $k$ encodes the question: $&\\mathbf {direction~of~change~in~} average \\mathbf {~rate~of~production~of~} bpg13 \\mathbf {~is~} d\\nonumber \\\\&~~~~~~\\mathbf {when~observed~between~time~step~} 0 \\mathbf {~and~time~step~} k;\\nonumber \\\\&~~~~~~\\mathbf {comparing~nominal~pathway~with~modified~pathway~obtained~}\\nonumber \\\\&~~~~~~~~~~~~\\mathbf {due~to~interventions:} \\mathbf {~remove~} dhap \\mathbf {~as~soon~as~produced};\\nonumber \\\\&~~~~~~\\mathbf {using~initial~setup:~} \\mathbf {continuously~supply~} f16bp \\mathbf {~in~quantity~} 1;$ Since this is a comparative quantitative query statement, it is decomposed into two sub-queries, $\\mathbf {Q_0}$ capturing the nominal case of average rate of production w.r.t.", "given initial conditions: $\\begin{array}{lll}average \\mathbf {~rate~of~production~of~} bpg13 \\mathbf {~is~} n_{avg}\\\\~~~~~~~~~~~~\\mathbf {when~observed~between~time~step~} 0 \\mathbf {~and~time~step~} k; \\\\~~~~~~\\mathbf {using~initial~setup:~} \\mathbf {continuously~supply~} f16bp \\mathbf {~in~quantity~} 1;\\end{array}$ and $\\mathbf {Q_1}$ capturing the modified case in which the pathway is subject to interventions and observations w.r.t.", "initial conditions: $\\begin{array}{lll}average \\mathbf {~rate~of~production~of~} bpg13 \\mathbf {~is~} n^{\\prime }_{avg}\\\\~~~~~~~~~~~~\\mathbf {when~observed~between~time~step~} 0 \\mathbf {~and~time~step~} k; \\\\~~~~~~\\mathbf {due~to~interventions:} \\mathbf {~remove~} dhap \\mathbf {~as~soon~as~produced} ;\\\\~~~~~~\\mathbf {using~initial~setup:~} \\mathbf {continuously~supply~} f16bp \\mathbf {~in~quantity~} 1;\\end{array}$ Then the task is to determine $d$ , such that $\\mathbf {D} \\models \\mathbf {Q_0}$ for some value of $n_{avg}$ , $\\mathbf {D} \\models \\mathbf {Q_1}$ for some value of $n^{\\prime }_{avg}$ , and $n^{\\prime }_{avg} \\; d \\; n_{avg}$ .", "To answer the sub-queries $\\mathbf {Q_0}$ and $\\mathbf {Q_1}$ we build modified domain descriptions $\\mathbf {D_0}$ and $\\mathbf {D_1}$ , where, $\\mathbf {D_0} \\equiv \\mathbf {D} \\diamond (\\mathbf {continuously~supply~} f16bp $ $\\mathbf {~in~quantity~} 1)$ is the nominal domain description $\\mathbf {D}$ modified according to $\\mathbf {Q_0}$ to include the initial conditions; and $\\mathbf {D_1} \\equiv \\mathbf {D_0} \\diamond (\\mathbf {remove~} dhap $ $\\mathbf {~as~soon~as~}$ $\\mathbf {produced})$ is the modified domain description $\\mathbf {D}$ modified according to $\\mathbf {Q_1}$ to include the initial conditions as well as the interventions.", "The $\\diamond $ operator modifies the domain description to its left by adding, removing or modifying pathway specification language statements to add the interventions and initial conditions description to its right.", "Thus, $\\mathbf {D_0} &= \\mathbf {D} + \\left\\lbrace \\begin{array}{llll}tf_{f16bp} \\mathbf {~may~execute~causing~} &f16bp \\mathbf {~change~value~by~} +1\\\\\\end{array}\\right.\\\\\\mathbf {D_1} &= \\mathbf {D_0} + \\left\\lbrace \\begin{array}{llll}tr \\mathbf {~may~execute~causing~} &dhap \\mathbf {~change~value~by~} \\\\\\end{array}\\right.$ Performing a simulation of $k=5$ steps with $ntok=20$ max tokens, we find that the average rate of $bpg13$ production decreases from $n_{avg}=0.83$ to $n^{\\prime }_{avg}=0.5$ .", "Thus, $\\mathbf {D} \\models \\mathbf {Q}$ iff $d = ^{\\prime }<^{\\prime }$ .", "Alternatively, we say that the rate of glycolysis decreases when DHAP is removed as quickly as it is produced.", "Question 12 When and how does the body switch to B-oxidation versus glycolysis as the major way of burning fuel?", "Figure: Petri Net for question The following pathway specification encodes the domain description $\\mathbf {D}$ for question REF and produces the PN in Fig.", "REF : $\\begin{array}{llll}&\\mathbf {domain~of~} & gly \\mathbf {~is~} integer, & sug \\mathbf {~is~} integer, \\\\ && fac \\mathbf {~is~} integer, & acoa \\mathbf {~is~} integer \\\\&gly \\mathbf {~may~execute~causing~} & sug \\mathbf {~change~value~by~} -1, & acoa \\mathbf {~change~value~by~} +1\\\\&box \\mathbf {~may~execute~causing~} & fac \\mathbf {~change~value~by~} -1, & acoa \\mathbf {~change~value~by~} +1\\\\&\\mathbf {inhibit~} box \\mathbf {~if~} & sug \\mathbf {~has~value~} 1 \\mathbf {~or~higher}\\\\&\\mathbf {initially~} & sug \\mathbf {~has~value~} 3, & fac \\mathbf {~has~value~} 3\\\\ && acoa \\mathbf {~has~value~} 0 \\\\&t1 \\mathbf {~may~execute~causing~} & sug \\mathbf {~change~value~by~} +1\\\\&t2 \\mathbf {~may~execute~causing~} & fac \\mathbf {~change~value~by~} +1\\\\& \\mathbf {firing~style~} & \\end{array}$ where, $fac$ represents fatty acids, $sug$ represents sugar, $acoa$ represents acetyl coenzyme-A, $gly$ represents the process of glycolysis, and $box$ represents the process of beta oxidation.", "The question is asking for the general conditions when glycolysis switches to beta-oxidation, which is some property “$p$ ” that holds after which the switch occurs.", "The query $\\mathbf {Q}$ is encoded as: $&gly \\mathbf {~switches~to~} box \\mathbf {~when~} p; \\nonumber \\\\&~~~~\\mathbf {due~to~observations:} \\nonumber \\\\&~~~~~~~~gly \\mathbf {~switches~to~} box \\nonumber \\\\&~~~~\\mathbf {~in~all~trajectories}$ where condition `$p$ ' is a conjunction of simple point formulas.", "Then the task is to determine a minimal such conjunction of formulas that is satisfied in the state where $`gly^{\\prime } \\mathbf {~switches~to~} `box^{\\prime }$ holds over all trajectories.Note that this could be an LTL formula that must hold in all trajectories, but we did not add it here to keep the language simple.", "Since there is no change in initial conditions of the pathway and there are no interventions, the modified domain description $\\mathbf {D_1} \\equiv \\mathbf {D}$ .", "Intuitively, $p$ is the property that holds over fluents of the transitional state $s_j$ in which the switch takes palce, such that $gly \\in T_{j-1}, box \\notin T_{j-1}, gly \\notin T_j, box \\in T_j$ and the minimal set of firings leading up to it.", "The only trajectories to consider are the ones in which the observation is true.", "Thus the condition $p$ is determined as the intersection of sets of fluent based conditions that were true at the time of the switch, such as: $\\lbrace &sug \\mathbf {~has~value~} s_j(sug), sug \\mathbf {~has~value~higher~than~} 0, \\dots \\\\&sug \\mathbf {~has~value~higher~than~} s_j(sug)-1,sug \\mathbf {~has~value~lower~than~} s_j(sug)+1,\\\\&fac \\mathbf {~has~value~} s_j(fac), fac \\mathbf {~has~value~higher~than~} 0, \\dots \\\\&fac \\mathbf {~has~value~higher~than~} s_j(fac)-1, fac \\mathbf {~has~value~lower~than~} s_j(fac)+1, \\\\&acoa \\mathbf {~has~value~} s_j(acoa), acoa \\mathbf {~has~value~higher~than~} 0, \\dots \\\\&acoa \\mathbf {~has~value~higher~than~} s_j(acoa)-1, acoa \\mathbf {~has~value~lower~than~} s_j(acoa)+1\\rbrace $ Simulating it for $k=5$ steps with $ntok=20$ max tokens, we find the condition $p = acoa \\text{ has value greater than } 0, $ $sug \\text{ has value } 0, $ $sug \\text{ has value lower than } 1, $ $fac \\text{ has value higher than } 0 \\rbrace $ .", "Thus, the state when this switch occurs must sugar ($sug$ ) depleted and available supply of fatty acids ($fac$ ).", "Question 13 The final protein complex in the electron transport chain of the mitochondria is non-functional.", "Explain the effect of this on pH of the intermembrane space of the mitochondria.", "Figure: Petri Net for question The question is asking for the direction of change in the pH of the intermembrane space when the nominal case is compared against a modified pathway in which the complex 4 ($t4$ ) is defective.", "Since pH is defined as $-log_{10}(H+)$ , we monitor the total production of $H+$ ions to determine the change in pH value.", "However, since different world evolutions can follow different trajectories, we consider the average production of H+.", "Furthermore, we model the defective $t4$ as being unable to carry out its reaction, by disabling/inhibiting it.", "Then the following pathway specification encodes the domain description $\\mathbf {D}$ for question REF and produces the PN in Fig.", "REF without the $ft4$ place node.", "In the pathway description below, we have included only one domain declaration for all fluents as an example and left the rest out to save space.", "Assume integer domain declaration for all fluents.", "$&\\mathbf {domain~of~} \\\\&~~~~ nadh \\mathbf {~atloc~} mm \\mathbf {~is~} integer \\\\&t1 \\mathbf {~may~execute~causing~} \\\\&~~~~ nadh \\mathbf {~atloc~} mm \\mathbf {~change~value~by~} -2, h \\mathbf {~atloc~} mm \\mathbf {~change~value~by~} -2,\\\\&~~~~ e \\mathbf {~atloc~} q \\mathbf {~change~value~by~} +2, h \\mathbf {~atloc~} is \\mathbf {~change~value~by~} +2, \\\\&~~~~ nadp \\mathbf {~atloc~} mm \\mathbf {~change~value~by~} +2\\\\&t2 \\mathbf {~may~execute~causing~} \\\\&~~~~ fadh2 \\mathbf {~atloc~} mm \\mathbf {~change~value~by~} -2, e \\mathbf {~atloc~} q \\mathbf {~change~value~by~} +2,\\\\&~~~~ fad \\mathbf {~atloc~} mm \\mathbf {~change~value~by~} +2\\\\&t3 \\mathbf {~may~execute~causing~} \\\\&~~~~ e \\mathbf {~atloc~} q \\mathbf {~change~value~by~} -2, h \\mathbf {~atloc~} mm \\mathbf {~change~value~by~} -2,\\\\&~~~~ e \\mathbf {~atloc~} cytc \\mathbf {~change~value~by~} +2, h \\mathbf {~atloc~} is \\mathbf {~change~value~by~} +2\\\\&t4 \\mathbf {~may~execute~causing~} \\\\&~~~~ o2 \\mathbf {~atloc~} mm \\mathbf {~change~value~by~} -1, e \\mathbf {~atloc~} cytc \\mathbf {~change~value~by~} -2,\\\\&~~~~ h \\mathbf {~atloc~} mm \\mathbf {~change~value~by~} -6, h2o \\mathbf {~atloc~} mm \\mathbf {~change~value~by~} +2,\\\\&~~~~ h \\mathbf {~atloc~} is \\mathbf {~change~value~by~} +2\\\\&t10 \\mathbf {~may~execute~causing~} \\\\&~~~~ nadh \\mathbf {~atloc~} mm \\mathbf {~change~value~by~} +2, \\\\&~~~~ h \\mathbf {~atloc~} \\mathbf {~change~value~by~} +4, o2 \\mathbf {~atloc~} mm \\mathbf {~change~value~by~} +1\\\\&\\mathbf {initially~} \\\\&~~~~ fadh2 \\mathbf {~atloc~} mm \\mathbf {~has~value~} 0, e \\mathbf {~atloc~} mm \\mathbf {~has~value~} 0,\\\\&~~~~ fad \\mathbf {~atloc~} mm \\mathbf {~has~value~} 0, o2 \\mathbf {~atloc~} mm \\mathbf {~has~value~} 0,\\\\&~~~~ h2o \\mathbf {~atloc~} mm \\mathbf {~has~value~} 0, atp \\mathbf {~atloc~} mm \\mathbf {~has~value~} 0\\\\&\\mathbf {initially~} \\\\&~~~~ fadh2 \\mathbf {~atloc~} is \\mathbf {~has~value~} 0, e \\mathbf {~atloc~} is \\mathbf {~has~value~} 0,\\\\&~~~~ fad \\mathbf {~atloc~} is \\mathbf {~has~value~} 0, o2 \\mathbf {~atloc~} is \\mathbf {~has~value~} 0,\\\\&~~~~ h2o \\mathbf {~atloc~} is \\mathbf {~has~value~} 0, atp \\mathbf {~atloc~} is \\mathbf {~has~value~} 0\\\\&\\mathbf {initially~} \\\\&~~~~ fadh2 \\mathbf {~atloc~} q \\mathbf {~has~value~} 0, e \\mathbf {~atloc~} q \\mathbf {~has~value~} 0,\\\\&~~~~ fad \\mathbf {~atloc~} q \\mathbf {~has~value~} 0, o2 \\mathbf {~atloc~} q \\mathbf {~has~value~} 0,\\\\&~~~~ h2o \\mathbf {~atloc~} q \\mathbf {~has~value~} 0, atp \\mathbf {~atloc~} q \\mathbf {~has~value~} 0\\\\&\\mathbf {initially~} \\\\&~~~~ fadh2 \\mathbf {~atloc~} cytc \\mathbf {~has~value~} 0, e \\mathbf {~atloc~} cytc \\mathbf {~has~value~} 0,\\\\&~~~~ fad \\mathbf {~atloc~} cytc \\mathbf {~has~value~} 0, o2 \\mathbf {~atloc~} cytc \\mathbf {~has~value~} 0,\\\\&~~~~ h2o \\mathbf {~atloc~} cytc \\mathbf {~has~value~} 0, atp \\mathbf {~atloc~} cytc \\mathbf {~has~value~} 0\\\\&~~~~ \\mathbf {firing~style~} max\\\\$ where $mm$ represents the mitochondrial matrix, $is$ represents the intermembrane space, $t1-t4$ represent the reaction of the four complexes making up the electron transport chain, $h$ is the $H+$ ion, $nadh$ is $NADH$ , $fadh2$ is $FADH_2$ , $fad$ is $FAD$ , $e$ is electrons, $o2$ is oxygen $O_2$ , $atp$ is $ATP$ , $h2o$ is water $H_2O$ , and $t10$ is a source transition that supply a continuous supply of source ingredients for the chain to function, such as $nadh$ , $h$ , $o2$ .", "As a result, the query $\\mathbf {Q}$ asked by the question is encoded as follows: $\\begin{array}{l}\\mathbf {direction~of~change~in~} average \\mathbf {~total~production~of~} h \\mathbf {~atloc~} is \\mathbf {~is~} d\\\\~~~~~~~~\\mathbf {when~observed~between~time~step~} 0 \\mathbf {~and~time~step~} k ;\\\\~~~~~~~~\\mathbf {comparing~nominal~pathway~with~modified~pathway~obtained}\\\\~~~~~~~~\\mathbf {due~to~intervention~} t4 \\mathbf {~disabled} ;\\end{array}$ Since this is a comparative quantitative query statement, it is decomposed into two queries, $\\mathbf {Q_0}$ capturing the nominal case of average production w.r.t.", "given initial conditions: $\\begin{array}{l}average \\mathbf {~total~production~of~} h \\mathbf {~atloc~} is \\mathbf {~is~} n_{avg}\\\\~~~~~~~~\\mathbf {when~observed~between~time~step~} 0 \\mathbf {~and~time~step~} k ;\\end{array}$ and $\\mathbf {Q_1}$ the modified case w.r.t.", "initial conditions, modified to include interventions and subject to observations: $\\begin{array}{l}average \\mathbf {~total~production~of~} h \\mathbf {~atloc~} is \\mathbf {~is~} n^{\\prime }_{avg}\\\\~~~~~~~~\\mathbf {when~observed~between~time~step~} 0 \\mathbf {~and~time~step~} k ;\\\\~~~~~~~~\\mathbf {due~to~intervention~} t4 \\mathbf {~disabled} ;\\end{array}$ The the task is to determine $d$ , such that $\\mathbf {D} \\models \\mathbf {Q_0}$ for some value of $n_{avg}$ , $\\mathbf {D} \\models \\mathbf {Q_1}$ for some value of $n^{\\prime }_{avg}$ , and $n^{\\prime }_{avg} \\; d \\; n_{avg}$ .", "To answer the sub-queries $\\mathbf {Q_0}$ and $\\mathbf {Q_1}$ we build nominal description $\\mathbf {D_0}$ and modified pathway $\\mathbf {D_1}$ , where $\\mathbf {D_0} \\equiv \\mathbf {D}$ since there are no initial conditions; and $\\mathbf {D_1} \\equiv \\mathbf {D_0} \\diamond (t4 \\mathbf {~disabled})$ is the domain description $\\mathbf {D}$ modified according to $\\mathbf {Q_1}$ to modified to include the initial conditions as well as interventions.", "The $\\diamond $ operator modifies the domain description to its left by adding, removing or modifying pathway specification language statements to add the intervention and initial conditions to its right.", "Thus, $\\mathbf {D_1} = \\mathbf {D_0} + \\left\\lbrace \\begin{array}{llll}\\mathbf {inhibit~} t4\\\\\\end{array}\\right.$ Performing a simulation of $k=5$ steps with $ntok=20$ max tokens, we find that the average total production of H+ in the intermembrane space ($h$ at location $is$ ) reduces from 16 to 14.", "Lower quantity of H+ translates to a higer numeric value of $-log_{10}$ , as a result the pH increases.", "Question 14 Membranes must be fluid to function properly.", "How would decreased fluidity of the membrane affect the efficiency of the electron transport chain?", "Figure: Petri Net with colored tokens alternate for question From background knowledge, we know that, “Establishing the H+ gradient is a major function of the electron transport chain” [64], we measure the efficiency in terms of H+ ions moved to the intermembrane space ($is$ ) over time.", "Thus, we interpret the question is asking for the direction of change in the production of H+ moved to the intermembrane space when the nominal case is compared against a modified pathway with decreased fluidity of membrane.", "Additional background knowledge from [64] tells us that the decreased fluidity reduces the speed of mobile carriers such as $q$ and $cytc$ .", "Fluidity can span a range of values, but we will consider one such value $v$ per query.", "The following pathway specification encodes the domain description $\\mathbf {D}$ for question REF and produces the PN in Fig.", "REF minus the places $q\\_3,cytc\\_4$ and transitions $tq,tcytc$ .", "In the pathway description below, we have included only one domain declaration for all fluents as an example and left the rest out to save space.", "Assume integer domain declaration for all fluents.", "$&\\mathbf {domain~of~} \\\\&~~~~nadh \\mathbf {~atloc~} mm \\mathbf {~is~} integer \\\\&t1 \\mathbf {~may~execute~causing~} \\\\&~~~~ nadh \\mathbf {~atloc~} mm \\mathbf {~change~value~by~} -2, h \\mathbf {~atloc~} mm \\mathbf {~change~value~by~} -2,\\\\&~~~~ e \\mathbf {~atloc~} q \\mathbf {~change~value~by~} +2, h \\mathbf {~atloc~} is \\mathbf {~change~value~by~} +2,\\\\&~~~~ nadp \\mathbf {~atloc~} mm \\mathbf {~change~value~by~} +2\\\\&t3 \\mathbf {~may~execute~causing~} \\\\&~~~~ e \\mathbf {~atloc~} q \\mathbf {~change~value~by~} -2, h \\mathbf {~atloc~} mm \\mathbf {~change~value~by~} -2,\\\\&~~~~ e \\mathbf {~atloc~} cytc \\mathbf {~change~value~by~} +2, h \\mathbf {~atloc~} is \\mathbf {~change~value~by~} +2\\\\&t4 \\mathbf {~may~execute~causing~} \\\\&~~~~ o2 \\mathbf {~atloc~} mm \\mathbf {~change~value~by~} -1, e \\mathbf {~atloc~} cytc \\mathbf {~change~value~by~} -2, \\\\&~~~~ h \\mathbf {~atloc~} mm \\mathbf {~change~value~by~} -2, h2o \\mathbf {~atloc~} mm \\mathbf {~change~value~by~} +2,\\\\&~~~~ h \\mathbf {~atloc~} is \\mathbf {~change~value~by~} +2\\\\&t10 \\mathbf {~may~execute~causing~} \\\\&~~~~ nadh \\mathbf {~atloc~} mm \\mathbf {~change~value~by~} +2, h \\mathbf {~atloc~} mm \\mathbf {~change~value~by~} +4,\\\\&~~~~ o2 \\mathbf {~atloc~} mm \\mathbf {~change~value~by~} +1\\\\& \\mathbf {initially~} \\\\&~~~~ nadh \\mathbf {~atloc~} mm \\mathbf {~has~value~} 0, h \\mathbf {~atloc~} mm \\mathbf {~has~value~} 0,\\\\&~~~~ nadp \\mathbf {~atloc~} mm \\mathbf {~has~value~} 0, o2 \\mathbf {~atloc~} mm \\mathbf {~has~value~} 0\\\\&~~~~ h2o \\mathbf {~atloc~} mm \\mathbf {~has~value~} 0 \\\\& \\mathbf {initially~} \\\\&~~~~ nadh \\mathbf {~atloc~} is \\mathbf {~has~value~} 0, h \\mathbf {~atloc~} is \\mathbf {~has~value~} 0,\\\\&~~~~ nadp \\mathbf {~atloc~} is \\mathbf {~has~value~} 0, o2 \\mathbf {~atloc~} is \\mathbf {~has~value~} 0\\\\&~~~~ h2o \\mathbf {~atloc~} is \\mathbf {~has~value~} 0 \\\\& \\mathbf {initially~} \\\\&~~~~ nadh \\mathbf {~atloc~} q \\mathbf {~has~value~} 0, h \\mathbf {~atloc~} q \\mathbf {~has~value~} 0,\\\\&~~~~ nadp \\mathbf {~atloc~} q \\mathbf {~has~value~} 0, o2 \\mathbf {~atloc~} q \\mathbf {~has~value~} 0\\\\&~~~~ h2o \\mathbf {~atloc~} q \\mathbf {~has~value~} 0 \\\\& \\mathbf {initially~} \\\\&~~~~ nadh \\mathbf {~atloc~} cytc \\mathbf {~has~value~} 0, h \\mathbf {~atloc~} cytc \\mathbf {~has~value~} 0,\\\\&~~~~ nadp \\mathbf {~atloc~} cytc \\mathbf {~has~value~} 0, o2 \\mathbf {~atloc~} cytc \\mathbf {~has~value~} 0\\\\&~~~~ h2o \\mathbf {~atloc~} cytc \\mathbf {~has~value~} 0 \\\\&\\mathbf {firing~style~} max\\\\$ The query $\\mathbf {Q}$ asked by the question is encoded as follows: $\\begin{array}{l}\\mathbf {direction~of~change~in~} average \\mathbf {~total~production~of~} h \\mathbf {~atloc~} is \\mathbf {~is~} d\\\\~~~~~~\\mathbf {when~observed~between~time~step~} 0 \\mathbf {~and~time~step~} k;\\\\~~~~~~\\mathbf {comparing~nominal~pathway~with~modified~pathway~obtained~}\\\\~~~~~~~~~~~~\\mathbf {due~to~interventions:} \\\\~~~~~~~~~~~~~~~~~~\\mathbf {add~delay~of~} v \\mathbf {~time~units~in~availability~of~} e \\mathbf {~atloc~} q,\\\\~~~~~~~~~~~~~~~~~~\\mathbf {add~delay~of~} v \\mathbf {~time~units~in~availability~of~} e \\mathbf {~atloc~} cytc;\\end{array}$ Since this is a comparative quantitative query statement, we decompose it into two queries, $\\mathbf {Q_0}$ capturing the nominal case of average rate of production w.r.t.", "given initial conditions: $\\begin{array}{l}average \\mathbf {~total~production~of~} h \\mathbf {~atloc~} is \\mathbf {~is~} d\\\\~~~~~~~~~~~~\\mathbf {when~observed~between~time~step~} 0 \\mathbf {~and~time~step~} k;\\\\\\end{array}$ and $\\mathbf {Q_1}$ the modified case w.r.t.", "initial conditions, modified to include interventions and subject to observations: $\\begin{array}{l}average \\mathbf {~total~production~of~} h \\mathbf {~atloc~} is \\mathbf {~is~} d\\\\~~~~~~~~~~~~\\mathbf {when~observed~between~time~step~} 0 \\mathbf {~and~time~step~} k;\\\\~~~~~~\\mathbf {due~to~interventions:} \\\\~~~~~~~~~~~~\\mathbf {add~delay~of~} v \\mathbf {~time~units~in~availability~of~} e \\mathbf {~atloc~} q,\\\\~~~~~~~~~~~~\\mathbf {add~delay~of~} v \\mathbf {~time~units~in~availability~of~} e \\mathbf {~atloc~} cytc;\\end{array}$ Then the task is to determine $d$ , such that $\\mathbf {D} \\models \\mathbf {Q_0}$ for some value of $n_{avg}$ , $\\mathbf {D} \\models \\mathbf {Q_1}$ for some value of $n^{\\prime }_{avg}$ , and $n^{\\prime }_{avg} \\; d \\; n_{avg}$ .", "To answer the sub-queries $\\mathbf {Q_0}$ and $\\mathbf {Q_1}$ we build modified domain descriptions $\\mathbf {D_0}$ and $\\mathbf {D_1}$ , where, $\\mathbf {D_0} \\equiv \\mathbf {D}$ is the nominal domain description $\\mathbf {D}$ modified according to $\\mathbf {Q_0}$ to include the initial conditions; and $\\mathbf {D_1} \\equiv \\mathbf {D_0} \\diamond (\\mathbf {add~delay~of~} v \\mathbf {~time~units~in~availability~of~} $ $e \\mathbf {~atloc~} q) \\diamond (\\mathbf {add~delay~of~} v \\mathbf {~time~units~in~availability~of~} $ $e \\mathbf {~atloc~} cytc)$ is the modified domain description $\\mathbf {D}$ modified according to $\\mathbf {Q_1}$ to include the initial conditions as well as the interventions.", "The $\\diamond $ operator modifies the domain description to its left by adding, removing or modifying pathway specification language statements to add the interventions and initial conditions description to its right.", "Thus, $\\mathbf {D_1} = \\mathbf {D} &-\\left\\lbrace \\begin{array}{llll}t1 \\mathbf {~may~execute~causing~} & nadh \\mathbf {~atloc~} mm \\mathbf {~change~value~by~} -2, \\\\& h \\mathbf {~atloc~} mm \\mathbf {~change~value~by~} -2,\\\\& e \\mathbf {~atloc~} q \\mathbf {~change~value~by~} +2, \\\\& h \\mathbf {~atloc~} is \\mathbf {~change~value~by~} +2,\\\\& nadp \\mathbf {~atloc~} mm \\mathbf {~change~value~by~} +2\\\\t3 \\mathbf {~may~execute~causing~} & e \\mathbf {~atloc~} q \\mathbf {~change~value~by~} -2, \\\\& h \\mathbf {~atloc~} mm \\mathbf {~change~value~by~} -2,\\\\& e \\mathbf {~atloc~} cytc \\mathbf {~change~value~by~} +2, \\\\& h \\mathbf {~atloc~} is \\mathbf {~change~value~by~} +2\\\\\\end{array}\\right\\rbrace \\\\& +\\left\\lbrace \\begin{array}{llll}t1 \\mathbf {~may~execute~causing~} & nadh \\mathbf {~atloc~} mm \\mathbf {~change~value~by~} -2, \\\\& h \\mathbf {~atloc~} mm \\mathbf {~change~value~by~} -2,\\\\& e \\mathbf {~atloc~} q\\_3 \\mathbf {~change~value~by~} +2, \\\\& h \\mathbf {~atloc~} is \\mathbf {~change~value~by~} +2,\\\\& nadp \\mathbf {~atloc~} mm \\mathbf {~change~value~by~} +2\\\\t3 \\mathbf {~may~execute~causing~} & e \\mathbf {~atloc~} q \\mathbf {~change~value~by~} -2, \\\\& h \\mathbf {~atloc~} mm \\mathbf {~change~value~by~} -2,\\\\& e \\mathbf {~atloc~} cytc\\_4 \\mathbf {~change~value~by~} +2, \\\\& h \\mathbf {~atloc~} is \\mathbf {~change~value~by~} +2\\\\tq \\mathbf {~may~fire~causing~} & e \\mathbf {~atloc~} q\\_3 \\mathbf {~change~value~by~} -2, \\\\& e \\mathbf {~atloc~} q \\mathbf {~change~value~by~} +2\\\\tcytc \\mathbf {~may~fire~causing~} & e \\mathbf {~atloc~} cytc\\_4 \\mathbf {~change~value~by~} -2, \\\\& e \\mathbf {~atloc~} cytc \\mathbf {~change~value~by~} +2\\\\tq \\mathbf {~executes~} & \\mathbf {~in~} 4 \\mathbf {~time~units}\\\\tcytc \\mathbf {~executes~} & \\mathbf {~in~} 4 \\mathbf {~time~units}\\\\\\end{array}\\right\\rbrace \\\\$ Performing a simulation of k = 5 steps with ntok = 20 max tokens with a fluidity based delay of 2, we find that the average total production of H+ in the intermembrane space (h at location is) reduces from 16 to 10.", "Lower quantity of H+ going into the intermembrane space means lower efficiency, where we define the efficiency as the total amount of $H+$ ions transferred to the intermembrane space over the simulation run.", "Example Encoding with Conditional Actions Next, we illustrate how conditional actions would be encoded in our high-level language with an example.", "Consider the pathway from question REF .", "Say, the reaction step $t4$ has developed a fault, in which it has two modes of operation, in the first mode, when $f16bp$ has less than 3 units available, the reaction proceeds normally, but when $f16bp$ is available in 3 units or higher, the reaction continues to produce $g3p$ but not $dhap$ directly.", "$dhap$ can still be produced by subsequent step from $g3p$ .", "The modified pathway is given in our pathway specification language below: $\\begin{array}{llll}&t3 \\mathbf {~may~execute~causing~} & f16bp \\mathbf {~change~value~by~} +1\\\\&t4 \\mathbf {~may~execute~causing~} & f16bp \\mathbf {~change~value~by~} -1, & dhap \\mathbf {~change~value~by~} +1,\\\\ && g3p \\mathbf {~change~value~by~} +1\\\\&~~~~~~~~~~~~~~~~\\mathbf {if~}& g3p \\mathbf {~has~value~lower~than~} 3\\\\&t4 \\mathbf {~may~execute~causing~} & f16bp \\mathbf {~change~value~by~} -1, & dhap \\mathbf {~change~value~by~} +1\\\\&~~~~~~~~~~~~~~~~\\mathbf {~if}& g3p \\mathbf {~has~value~} 3 \\mathbf {~or~higher}\\\\&t5a \\mathbf {~may~execute~causing~} & dhap \\mathbf {~change~value~by~} -1, & g3p \\mathbf {~change~value~by~} +1\\\\&t5b \\mathbf {~may~execute~causing~} & g3p \\mathbf {~change~value~by~} -1, & dhap \\mathbf {~change~value~by~} +1\\\\&t6 \\mathbf {~may~execute~causing~} & g3p \\mathbf {~change~value~by~} -1, & bpg13 \\mathbf {~change~value~by~} +2\\\\& \\mathbf {initially~} & f16bp \\mathbf {~has~value~} 0, & dhap \\mathbf {~has~value~} 0,\\\\ && g3p \\mathbf {~has~value~} 0, & bpg13 \\mathbf {~has~value~} 0\\\\&\\mathbf {firing~style~} & max\\end{array}$ We ask the same question $\\mathbf {Q}$ : $&\\mathbf {direction~of~change~in~} average \\mathbf {~rate~of~production~of~} bpg13 \\mathbf {~is~} d\\\\&~~~~~~\\mathbf {when~observed~between~time~step~} 0 \\mathbf {~and~time~step~} k; \\\\&~~~~~~\\mathbf {comparing~nominal~pathway~with~modified~pathway~obtained~}\\\\&~~~~~~~~~~~~\\mathbf {due~to~interventions:~} \\mathbf {remove~} dhap \\mathbf {~as~soon~as~produced};\\\\&~~~~~~~~~~~~\\mathbf {using~initial~setup:~} \\mathbf {continuously~supply~} f16bp \\mathbf {~in~quantity~} 1;$ Since this is a comparative quantitative query statement, we decompose it into two queries, $\\mathbf {Q_0}$ capturing the nominal case of average rate of production w.r.t.", "given initial conditions: $\\begin{array}{l}average \\mathbf {~rate~of~production~of~} bpg13 \\mathbf {~is~} n\\\\~~~~~~~~~~~~\\mathbf {when~observed~between~time~step~} 0 \\mathbf {~and~time~step~} k;\\\\~~~~~~\\mathbf {using~initial~setup:~} \\mathbf {continuously~supply~} f16bp \\mathbf {~in~quantity~} 1;\\end{array}$ and $\\mathbf {Q_1}$ the modified case w.r.t.", "initial conditions, modified to include interventions and subject to observations: $\\begin{array}{l}average \\mathbf {~rate~of~production~of~} bpg13 \\mathbf {~is~} n\\\\~~~~~~~~~~~~\\mathbf {when~observed~between~time~step~} 0 \\mathbf {~and~time~step~} k;\\\\~~~~~~\\mathbf {due~to~interventions:~} \\mathbf {remove~} dhap \\mathbf {~as~soon~as~produced};\\\\~~~~~~\\mathbf {using~initial~setup:~} \\mathbf {continuously~supply~} f16bp \\mathbf {~in~quantity~} 1;\\end{array}$ Then the task is to determine $d$ , such that $\\mathbf {D} \\models \\mathbf {Q_0}$ for some value of $n_{avg}$ , $\\mathbf {D} \\models \\mathbf {Q_1}$ for some value of $n^{\\prime }_{avg}$ , and $n^{\\prime }_{avg} \\; d \\; n_{avg}$ .", "To answer the sub-queries $\\mathbf {Q_0}$ and $\\mathbf {Q_1}$ we build modified domain descriptions $\\mathbf {D_0}$ and $\\mathbf {D_1}$ , where, $\\mathbf {D_0} \\equiv \\mathbf {D} \\diamond (\\mathbf {continuously~supply~} f16bp $ $\\mathbf {~in~quantity~} 1)$ is the nominal domain description $\\mathbf {D}$ modified according to $\\mathbf {Q_0}$ to include the initial conditions; and $\\mathbf {D_1} \\equiv \\mathbf {D_0} \\diamond (\\mathbf {remove~} dhap \\mathbf {~as~soon~as~produced})$ is the modified domain description $\\mathbf {D}$ modified according to $\\mathbf {Q_1}$ to include the initial conditions as well as the interventions.", "The $\\diamond $ operator modifies the domain description to its left by adding, removing or modifying pathway specification language statements to add the interventions and initial conditions description to its right.", "Thus, $\\mathbf {D_0} &= \\mathbf {D} + \\left\\lbrace \\begin{array}{llll}tf_{f16bp} \\mathbf {~may~execute~causing~} &f16bp \\mathbf {~change~value~by~} +1\\\\\\end{array}\\right.\\\\\\mathbf {D_1} &= \\mathbf {D_0} + \\left\\lbrace \\begin{array}{llll}tr \\mathbf {~may~execute~causing~} &dhap \\mathbf {~change~value~by~} \\\\\\end{array}\\right.$ ASP Program Next we briefly outline how the pathway specification and the query statement components are encoded in ASP (using Clingo syntax).", "In the following section, we will illustrate the process using an example.", "As evident from the previous sections, we need to simulate non-comparative queries only.", "Any comparative queries are translated into non-comparative sub-queries, each of which is simulated and their results compared to evaluate the comparative query.", "The ASP program is a concatenation of the translation of a pathway specification (domain description which includes the firing style) and internal observations.", "Any initial setup conditions and interventions are pre-applied to the pathway specification using intervention semantics in section REF before it is translated to ASP using the translation in chapter as our basis.", "The encoded pathway specification has the semantics defined in section REF .", "Internal observations in the `due to observations:' portion of query statement are translated into ASP constraints using the internal observation semantics defined in section REF and added to the encoding of the pathway specification.", "The program if simulated for a specified simulation length $k$ produces all trajectories of the pathway for the specified firing style, filtered by the internal observations.", "The query description specified in the query statement is then evaluated w.r.t.", "these trajectories.", "Although this part can be done in ASP, we have currently implemented it outside ASP in our implementation for ease of using floating point math.", "Next, we describe an implementation of our high level language and illustrate the construction of an ASP program, its simulation, and query statement evaluation.", "Implementation We have developed an implementation Implementation available at: https://sites.google.com/site/deepqa2014/ of a subset of our high level (Pathway and Query specification) language in Python.", "We use the Clingo ASP implementation for our simulation.", "In this section we describe various components of this implementation.", "An architectural overview of our implementation is shown in figure REF .", "Figure: BioPathQA Implementation System ArchitectureThe Pathway Specification Language (BioPathQA-PL) Parser component is responsible for parsing the Pathway Specification Language (BioPathQA-PL).", "It use PLY (Python Lex-Yacc)http://www.dabeaz.com/ply to parse a given pathway specification using grammar based on section REF .", "On a successful parse, a Guarded-Arc Petri Net pathway model based on section REF is constructed for the pathway specification.", "The Query Language Parser component is responsible for parsing the Query Specification Language (BioPathQA-QL).", "It uses PLY to parse a given query statement using grammar based on section REF .", "On a successful parse, an internal representation of the query statement is constructed.", "Elements of this internal representation include objects representing the query description, the list of interventions, the list of internal observations, and the list of initial setup conditions.", "Each intervention and initial setup condition object has logic in it to modify a given pathway per the intervention semantics described in section REF .", "The Query Statement Model component is also responsible for generating basic queries for aggregate queries and implementing interventions in the Petri Net Pathway Model.", "The Dictionary of fluents, locations, and actions is consulted by the ASP code generator to standardize symbol names in the ASP code produced for the pathway specification and the internal observations.", "The ASP Translator component is responsible for translating the Guarded-Arc Petri Net model into ASP facts and rules; and the driver needed to simulate the model using the firing semantics specified in the pathway model.", "The code generated is based on the ASP translation of Petri Nets and its various extensions given in chapter .", "To reduce the ASP code and its complexity, the translator limits the output model to the extensions used in the Petri Net model to be translated.", "Thus, the colored tokens extension code is not produced unless colored tokens have been used.", "Similarly, guarded-arcs code is not produced if no arc-guards are used in the model.", "The ASP Translator component is also responsible for translating internal observations from the Query Statement into ASP constraints to filter Petri Net trajectories based on the observation semantics in section REF .", "Following examples illustrate our encoding scheme.", "The observation `$a_1 \\mathbf {~switches~to~} a_2$ ' is encoded as a constraint using the following rules: obs_1_occurred(TS+1) :- time(TS;TS+1), trans(a1;a2), fires(a1,TS), not fires(a2,TS), not fires(a1,TS+1), fires(a2,TS+1).", "obs_1_occurred :- obs_1_occurred(TS), time(TS).", "obs_1_had_occurred(TSS) :- obs_1_occurred(TS), TS<=TSS, time(TSS;TS).", ":- not obs_1_occurred.", "The observation `$a_1 \\mathbf {~occurs~at~time~step~} 5$ ' is encoded as a constraint using the following rules: obs_2_occurred(TS) :- fires(a1,TS), trans(a1), time(TS), TS=5.", "obs_2_occurred :- obs_2_occurred(TS), time(TS).", "obs_2_had_occurred(TSS) :- obs_2_occurred(TS), TS<=TSS, time(TSS;TS).", ":- not obs_2_occurred.", "The observation `$s_1 \\mathbf {~is~decreasing~atloc~} l_1 \\mathbf {~when~observed~between~time~step~} 0 $ $\\mathbf {~and~time~step~} 5$ ' is encoded as a constraint using the following rules: obs_3_violated(TS) :- place(l1), col(s1), holds(l1,Q1,s1,TS), holds(l1,Q2,s1,TS+1), num(Q1;Q2), Q2 > Q1, time(TS;TS+1), TS=0, TS+1=5.", "obs_3_violated :- obs_3_violated(TS), time(TS).", "obs_3_occurred(TS+1) :- not obs_3_violated, holds(l1,Q1,s1,TS), holds(l1,Q2,s1,TS+1), time(TS;TS+1), num(Q1;Q2), Q2<Q1, TS=0, TS+1=5.", "obs_3_occurred :- obs_3_occurred(TS), time(TS).", "obs_3_had_occurred(TSS) :- obs_3_occurred(TS), TS<=TSS, time(TSS;TS).", ":- obs_3_occurred.", "In addition, the translator is also responsible for any rules needed to ease post-processing of the query description.", "For example, for qualitative queries, a generic predicate tgt_obs_occurred(TS) is generated that is true when the given qualitative description holds in an answer-set at time step $TS$ .", "The output of the translator is an ASP program, which when simulated using Clingo produces the (possibly) filtered trajectories of the pathway.", "The Post Processor component is responsible for parsing the ASP answer sets, identifying the correct atoms from it, extracting quantities from atom-bodies as necessary, organizing them into a matrix form, and aggregating them as needed.", "Figure: BioPathQA Graphical User InterfaceThe User Interface component is responsible for coordinating the processing of query statement.", "It presents the user with a graphical user interface shown in figure REF .", "The user types a Pathway Specification (in BioPathQA-PL syntax), a Query Specification (in BioPathQA-QL syntax), and simulation parameters.", "On pressing “Execute Query”, the user interface component processes the query as prints results in the bottom box.", "Query evaluation differs by the type of query.", "We describe the query evaluation methodology used below.", "For non-comparative quantitative queries: Pathway specification is parsed into a Guarded-Arc Petri Net model.", "Query statement is parsed into an internal form.", "Initial conditions from the query are applied to the pathway model.", "Interventions are applied to the pathway model.", "Modified pathway model is translated to ASP.", "Internal observations are added to the ASP code as ASP constraints.", "Answer sets of the ASP code are computed using Clingo.", "Relevant atoms are extracted: fires/2 predicate for firing rate, holds/3 (or holds/4 – colored tokens) predicate for fluent quantity or rate formulas.", "Fluent value or firing-count values are extracted and organized as matrices with rows representing answer-sets and columns representing time-steps.", "Within answer-set interval or point value sub-select is done and the values converted to rates or totals as needed.", "If aggregation, such as average, minimum, or maximum is desired, it is performed over rows of values from the last step.", "If a value was specified in the query, it is compared against the computed aggregate for boolean result.", "If a value was not specified, the computed value is returned as the value satisfying the query statement.", "For queries over all trajectories, the same value must hold over all trajectories, otherwise, only one match is required to satisfy the query.", "For non-comparative qualitative queries: Follow steps (REF )-(REF ) of non-comparative quantitative queries.", "Add rules for the query description for post-processing.", "Answer sets of the ASP code are computed using Clingo.", "Relevant atoms are extracted: tgt_obs_occurred/1 identifying the time step when the observation within the query description is satisfied.", "Truth value of the query observation is determined, including determining the truth value over all trajectories.", "For comparative quantitative queries: Query statement is decomposed into two non-comparative quantitative sub-query statements as illustrated in section REF : A nominal sub-query which has the same initial conditions as the comparative query, but none of its interventions or observations A modified sub-query which has the same initial conditions, interventions, and observations as the comparative query both sub-query statements have the same query description, which is the non-aggregate form of the comparative query description.", "Thus, a comparative average rate query is translated to non-comparative average rate sub-queries.", "Each sub-query statements is evaluated using steps (REF )-(REF ) from the non-comparative quantitative query processing.", "A direction of change is computed by comparing the computed aggregate value for the modified query statement to the nominal query statement.", "If the comparative quantitative query has a direction specified, it is the compared against the computed value for a boolean result.", "If the comparative quantitative query did not have a direction specified, the computed value is returned as the value satisfying the query statement.", "For explanation queries with query description with formula of the form (), it is expected that the number of answer-sets will be quite large.", "So, we avoid generating all answer-sets before processing them, instead we process them in-line as they are generated.", "It is a bit slower, but leads to a smaller memory foot print.", "Follow steps (REF )-(REF ) of non-comparative quantitative queries.", "Add rules for the query description for post-processing.", "Compute answer sets of the ASP code using Clingo.", "Extract relevant atoms: extract tgt_obs_occurred/1 identifying the time step when the query description is satisfied extract holds/3 (or holds/4 – for colored tokens) at the same time-step as tgt_obs_occurred/1 to construct fluent-based conditions Construct fluent-based conditions as explanation of the query observation.", "If the query is over all trajectories, fluent-based conditions for each trajectory are intersected across trajectories to determine the minimum set of conditions explaining the query observation.", "Next we illustrate query processing through an execution trace of question (REF ).", "The following shows the encoding of the base case domain, which includes the pathway specification from (REF ) with initial setup conditions from the query statement (REF ) applied: #const nts=1.", "time(0..nts).", "#const ntok=1.", "num(0..ntok).", "place(bpg13).", "place(dhap).", "place(f16bp).", "place(g3p).", "trans(src_f16bp_1).", "trans(t3).", "trans(t4).", "trans(t5a).", "trans(t5b).", "trans(t6).", "tparc(src_f16bp_1,f16bp,1,TS) :- time(TS).", "tparc(t3,f16bp,1,TS) :- time(TS).", "ptarc(f16bp,t4,1,TS) :- time(TS).", "tparc(t4,g3p,1,TS) :- time(TS).", "tparc(t4,dhap,1,TS) :- time(TS).", "ptarc(dhap,t5a,1,TS) :- time(TS).", "tparc(t5a,g3p,1,TS) :- time(TS).", "ptarc(g3p,t5b,1,TS) :- time(TS).", "tparc(t5b,dhap,1,TS) :- time(TS).", "ptarc(g3p,t6,1,TS) :- time(TS).", "tparc(t6,bpg13,2,TS) :- time(TS).", "holds(bpg13,0,0).", "holds(dhap,0,0).", "holds(f16bp,0,0).", "holds(g3p,0,0).", "could_not_have(T,TS) :- enabled(T,TS), not fires(T,TS), ptarc(P,T,Q,TS), holds(P,QQ,TS), tot_decr(P,QQQ,TS), Q > QQ - QQQ.", ":- not could_not_have(T,TS), time(TS), enabled(T,TS), not fires(T,TS), trans(T).", "min(A,B,A) :- A<=B, num(A;B).", "min(A,B,B) :- B<=A, num(A;B).", "#hide min/3.", "holdspos(P):- holds(P,N,0), place(P), num(N), N > 0. holds(P,0,0) :- place(P), not holdspos(P).", "notenabled(T,TS) :- ptarc(P,T,N,TS), holds(P,Q,TS), Q < N, place(P), trans(T), time(TS), num(N), num(Q).", "enabled(T,TS) :- trans(T), time(TS), not notenabled(T, TS).", "{ fires(T,TS) } :- enabled(T,TS), trans(T), time(TS).", "add(P,Q,T,TS) :- fires(T,TS), tparc(T,P,Q,TS), time(TS).", "del(P,Q,T,TS) :- fires(T,TS), ptarc(P,T,Q,TS), time(TS).", "tot_incr(P,QQ,TS) :- QQ = #sum[add(P,Q,T,TS) = Q : num(Q) : trans(T)], time(TS), num(QQ), place(P).", "tot_decr(P,QQ,TS) :- QQ = #sum[del(P,Q,T,TS) = Q : num(Q) : trans(T)], time(TS), num(QQ), place(P).", "holds(P,Q,TS+1) :- holds(P,Q1,TS), tot_incr(P,Q2,TS), tot_decr(P,Q3,TS), Q=Q1+Q2-Q3, place(P), num(Q;Q1;Q2;Q3), time(TS), time(TS+1).", "consumesmore(P,TS) :- holds(P,Q,TS), tot_decr(P,Q1,TS), Q1 > Q. consumesmore :- consumesmore(P,TS).", ":- consumesmore.", "The following shows encoding of the alternate case domain, which consists of the pathway specification from (REF ) with initial setup conditions and interventions applied; and any internal observations from the query statement (REF ) added: #const nts=1.", "time(0..nts).", "#const ntok=1.", "num(0..ntok).", "place(bpg13).", "place(dhap).", "place(f16bp).", "place(g3p).", "trans(reset_dhap_1).", "trans(src_f16bp_1).", "trans(t3).", "trans(t4).", "trans(t5a).", "trans(t5b).", "trans(t6).", "ptarc(dhap,reset_dhap_1,Q,TS) :- holds(dhap,Q,TS), Q>0, time(TS).", ":- enabled(reset_dhap_1,TS), not fires(reset_dhap_1,TS), time(TS).", "tparc(src_f16bp_1,f16bp,1,TS) :- time(TS).", "tparc(t3,f16bp,1,TS) :- time(TS).", "ptarc(f16bp,t4,1,TS) :- time(TS).", "tparc(t4,g3p,1,TS) :- time(TS).", "tparc(t4,dhap,1,TS) :- time(TS).", "ptarc(dhap,t5a,1,TS) :- time(TS).", "tparc(t5a,g3p,1,TS) :- time(TS).", "ptarc(g3p,t5b,1,TS) :- time(TS).", "tparc(t5b,dhap,1,TS) :- time(TS).", "ptarc(g3p,t6,1,TS) :- time(TS).", "tparc(t6,bpg13,2,TS) :- time(TS).", "holds(bpg13,0,0).", "holds(dhap,0,0).", "holds(f16bp,0,0).", "holds(g3p,0,0).", "could_not_have(T,TS) :- enabled(T,TS), not fires(T,TS), ptarc(P,T,Q,TS), holds(P,QQ,TS), tot_decr(P,QQQ,TS), Q > QQ - QQQ.", ":- not could_not_have(T,TS), time(TS), enabled(T,TS), not fires(T,TS), trans(T).", "min(A,B,A) :- A<=B, num(A;B).", "min(A,B,B) :- B<=A, num(A;B).", "#hide min/3.", "holdspos(P):- holds(P,N,0), place(P), num(N), N > 0. holds(P,0,0) :- place(P), not holdspos(P).", "notenabled(T,TS) :- ptarc(P,T,N,TS), holds(P,Q,TS), Q < N, place(P), trans(T), time(TS), num(N), num(Q).", "enabled(T,TS) :- trans(T), time(TS), not notenabled(T, TS).", "{ fires(T,TS) } :- enabled(T,TS), trans(T), time(TS).", "add(P,Q,T,TS) :- fires(T,TS), tparc(T,P,Q,TS), time(TS).", "del(P,Q,T,TS) :- fires(T,TS), ptarc(P,T,Q,TS), time(TS).", "tot_incr(P,QQ,TS) :- QQ = #sum[add(P,Q,T,TS) = Q : num(Q) : trans(T)], time(TS), num(QQ), place(P).", "tot_decr(P,QQ,TS) :- QQ = #sum[del(P,Q,T,TS) = Q : num(Q) : trans(T)], time(TS), num(QQ), place(P).", "holds(P,Q,TS+1) :- holds(P,Q1,TS), tot_incr(P,Q2,TS), tot_decr(P,Q3,TS), Q=Q1+Q2-Q3, place(P), num(Q;Q1;Q2;Q3), time(TS), time(TS+1).", "consumesmore(P,TS) :- holds(P,Q,TS), tot_decr(P,Q1,TS), Q1 > Q. consumesmore :- consumesmore(P,TS).", ":- consumesmore.", "Both programs are simulated for 5 time-steps and 20 max tokens using the following Clingo command: clingo 0 -cntok=20 -cnts=5 program.lp Answer sets of the base case are as follows: Answer: 1 holds(bpg13,0,0) holds(dhap,0,0) holds(f16bp,0,0) holds(g3p,0,0) fires(src_f16bp_1,0) fires(t3,0) holds(bpg13,0,1) holds(dhap,0,1) holds(f16bp,2,1) holds(g3p,0,1) fires(src_f16bp_1,1) fires(t3,1) fires(t4,1) holds(bpg13,0,2) holds(dhap,1,2) holds(f16bp,3,2) holds(g3p,1,2) fires(src_f16bp_1,2) fires(t3,2) fires(t4,2) fires(t5a,2) fires(t5b,2) holds(bpg13,0,3) holds(dhap,2,3) holds(f16bp,4,3) holds(g3p,2,3) fires(src_f16bp_1,3) fires(t3,3) fires(t4,3) fires(t5a,3) fires(t5b,3) fires(t6,3) holds(bpg13,2,4) holds(dhap,3,4) holds(f16bp,5,4) holds(g3p,2,4) fires(src_f16bp_1,4) fires(t3,4) fires(t4,4) fires(t5a,4) fires(t5b,4) fires(t6,4) holds(bpg13,4,5) holds(dhap,4,5) holds(f16bp,6,5) holds(g3p,2,5) fires(src_f16bp_1,5) fires(t3,5) fires(t4,5) fires(t5a,5) fires(t5b,5) fires(t6,5) Answer: 2 holds(bpg13,0,0) holds(dhap,0,0) holds(f16bp,0,0) holds(g3p,0,0) fires(src_f16bp_1,0) fires(t3,0) holds(bpg13,0,1) holds(dhap,0,1) holds(f16bp,2,1) holds(g3p,0,1) fires(src_f16bp_1,1) fires(t3,1) fires(t4,1) holds(bpg13,0,2) holds(dhap,1,2) holds(f16bp,3,2) holds(g3p,1,2) fires(src_f16bp_1,2) fires(t3,2) fires(t4,2) fires(t5a,2) fires(t6,2) holds(bpg13,2,3) holds(dhap,1,3) holds(f16bp,4,3) holds(g3p,2,3) fires(src_f16bp_1,3) fires(t3,3) fires(t4,3) fires(t5a,3) fires(t5b,3) fires(t6,3) holds(bpg13,4,4) holds(dhap,2,4) holds(f16bp,5,4) holds(g3p,2,4) fires(src_f16bp_1,4) fires(t3,4) fires(t4,4) fires(t5a,4) fires(t5b,4) fires(t6,4) holds(bpg13,6,5) holds(dhap,3,5) holds(f16bp,6,5) holds(g3p,2,5) fires(src_f16bp_1,5) fires(t3,5) fires(t4,5) fires(t5a,5) fires(t5b,5) fires(t6,5) Answer sets of the alternate case are as follows: Answer: 1 holds(bpg13,0,0) holds(dhap,0,0) holds(f16bp,0,0) holds(g3p,0,0) fires(reset_dhap_1,0) fires(src_f16bp_1,0) fires(t3,0) holds(bpg13,0,1) holds(dhap,0,1) holds(f16bp,2,1) holds(g3p,0,1) fires(reset_dhap_1,1) fires(src_f16bp_1,1) fires(t3,1) fires(t4,1) holds(bpg13,0,2) holds(dhap,1,2) holds(f16bp,3,2) holds(g3p,1,2) fires(reset_dhap_1,2) fires(src_f16bp_1,2) fires(t3,2) fires(t4,2) fires(t6,2) holds(bpg13,2,3) holds(dhap,1,3) holds(f16bp,4,3) holds(g3p,1,3) fires(reset_dhap_1,3) fires(src_f16bp_1,3) fires(t3,3) fires(t4,3) fires(t5b,3) holds(bpg13,2,4) holds(dhap,2,4) holds(f16bp,5,4) holds(g3p,1,4) fires(reset_dhap_1,4) fires(src_f16bp_1,4) fires(t3,4) fires(t4,4) fires(t5b,4) holds(bpg13,2,5) holds(dhap,2,5) holds(f16bp,6,5) holds(g3p,1,5) fires(reset_dhap_1,5) fires(src_f16bp_1,5) fires(t3,5) fires(t4,5) fires(t5b,5) Answer: 2 holds(bpg13,0,0) holds(dhap,0,0) holds(f16bp,0,0) holds(g3p,0,0) fires(reset_dhap_1,0) fires(src_f16bp_1,0) fires(t3,0) holds(bpg13,0,1) holds(dhap,0,1) holds(f16bp,2,1) holds(g3p,0,1) fires(reset_dhap_1,1) fires(src_f16bp_1,1) fires(t3,1) fires(t4,1) holds(bpg13,0,2) holds(dhap,1,2) holds(f16bp,3,2) holds(g3p,1,2) fires(reset_dhap_1,2) fires(src_f16bp_1,2) fires(t3,2) fires(t4,2) fires(t6,2) holds(bpg13,2,3) holds(dhap,1,3) holds(f16bp,4,3) holds(g3p,1,3) fires(reset_dhap_1,3) fires(src_f16bp_1,3) fires(t3,3) fires(t4,3) fires(t5b,3) holds(bpg13,2,4) holds(dhap,2,4) holds(f16bp,5,4) holds(g3p,1,4) fires(reset_dhap_1,4) fires(src_f16bp_1,4) fires(t3,4) fires(t4,4) fires(t5b,4) holds(bpg13,2,5) holds(dhap,2,5) holds(f16bp,6,5) holds(g3p,1,5) fires(reset_dhap_1,5) fires(src_f16bp_1,5) fires(t3,5) fires(t4,5) fires(t6,5) Answer: 3 holds(bpg13,0,0) holds(dhap,0,0) holds(f16bp,0,0) holds(g3p,0,0) fires(reset_dhap_1,0) fires(src_f16bp_1,0) fires(t3,0) holds(bpg13,0,1) holds(dhap,0,1) holds(f16bp,2,1) holds(g3p,0,1) fires(reset_dhap_1,1) fires(src_f16bp_1,1) fires(t3,1) fires(t4,1) holds(bpg13,0,2) holds(dhap,1,2) holds(f16bp,3,2) holds(g3p,1,2) fires(reset_dhap_1,2) fires(src_f16bp_1,2) fires(t3,2) fires(t4,2) fires(t6,2) holds(bpg13,2,3) holds(dhap,1,3) holds(f16bp,4,3) holds(g3p,1,3) fires(reset_dhap_1,3) fires(src_f16bp_1,3) fires(t3,3) fires(t4,3) fires(t5b,3) holds(bpg13,2,4) holds(dhap,2,4) holds(f16bp,5,4) holds(g3p,1,4) fires(reset_dhap_1,4) fires(src_f16bp_1,4) fires(t3,4) fires(t4,4) fires(t6,4) holds(bpg13,4,5) holds(dhap,1,5) holds(f16bp,6,5) holds(g3p,1,5) fires(reset_dhap_1,5) fires(src_f16bp_1,5) fires(t3,5) fires(t4,5) fires(t5b,5) Answer: 4 holds(bpg13,0,0) holds(dhap,0,0) holds(f16bp,0,0) holds(g3p,0,0) fires(reset_dhap_1,0) fires(src_f16bp_1,0) fires(t3,0) holds(bpg13,0,1) holds(dhap,0,1) holds(f16bp,2,1) holds(g3p,0,1) fires(reset_dhap_1,1) fires(src_f16bp_1,1) fires(t3,1) fires(t4,1) holds(bpg13,0,2) holds(dhap,1,2) holds(f16bp,3,2) holds(g3p,1,2) fires(reset_dhap_1,2) fires(src_f16bp_1,2) fires(t3,2) fires(t4,2) fires(t6,2) holds(bpg13,2,3) holds(dhap,1,3) holds(f16bp,4,3) holds(g3p,1,3) fires(reset_dhap_1,3) fires(src_f16bp_1,3) fires(t3,3) fires(t4,3) fires(t5b,3) holds(bpg13,2,4) holds(dhap,2,4) holds(f16bp,5,4) holds(g3p,1,4) fires(reset_dhap_1,4) fires(src_f16bp_1,4) fires(t3,4) fires(t4,4) fires(t6,4) holds(bpg13,4,5) holds(dhap,1,5) holds(f16bp,6,5) holds(g3p,1,5) fires(reset_dhap_1,5) fires(src_f16bp_1,5) fires(t3,5) fires(t4,5) fires(t6,5) Answer: 5 holds(bpg13,0,0) holds(dhap,0,0) holds(f16bp,0,0) holds(g3p,0,0) fires(reset_dhap_1,0) fires(src_f16bp_1,0) fires(t3,0) holds(bpg13,0,1) holds(dhap,0,1) holds(f16bp,2,1) holds(g3p,0,1) fires(reset_dhap_1,1) fires(src_f16bp_1,1) fires(t3,1) fires(t4,1) holds(bpg13,0,2) holds(dhap,1,2) holds(f16bp,3,2) holds(g3p,1,2) fires(reset_dhap_1,2) fires(src_f16bp_1,2) fires(t3,2) fires(t4,2) fires(t5b,2) holds(bpg13,0,3) holds(dhap,2,3) holds(f16bp,4,3) holds(g3p,1,3) fires(reset_dhap_1,3) fires(src_f16bp_1,3) fires(t3,3) fires(t4,3) fires(t5b,3) holds(bpg13,0,4) holds(dhap,2,4) holds(f16bp,5,4) holds(g3p,1,4) fires(reset_dhap_1,4) fires(src_f16bp_1,4) fires(t3,4) fires(t4,4) fires(t5b,4) holds(bpg13,0,5) holds(dhap,2,5) holds(f16bp,6,5) holds(g3p,1,5) fires(reset_dhap_1,5) fires(src_f16bp_1,5) fires(t3,5) fires(t4,5) fires(t5b,5) Answer: 6 holds(bpg13,0,0) holds(dhap,0,0) holds(f16bp,0,0) holds(g3p,0,0) fires(reset_dhap_1,0) fires(src_f16bp_1,0) fires(t3,0) holds(bpg13,0,1) holds(dhap,0,1) holds(f16bp,2,1) holds(g3p,0,1) fires(reset_dhap_1,1) fires(src_f16bp_1,1) fires(t3,1) fires(t4,1) holds(bpg13,0,2) holds(dhap,1,2) holds(f16bp,3,2) holds(g3p,1,2) fires(reset_dhap_1,2) fires(src_f16bp_1,2) fires(t3,2) fires(t4,2) fires(t5b,2) holds(bpg13,0,3) holds(dhap,2,3) holds(f16bp,4,3) holds(g3p,1,3) fires(reset_dhap_1,3) fires(src_f16bp_1,3) fires(t3,3) fires(t4,3) fires(t5b,3) holds(bpg13,0,4) holds(dhap,2,4) holds(f16bp,5,4) holds(g3p,1,4) fires(reset_dhap_1,4) fires(src_f16bp_1,4) fires(t3,4) fires(t4,4) fires(t5b,4) holds(bpg13,0,5) holds(dhap,2,5) holds(f16bp,6,5) holds(g3p,1,5) fires(reset_dhap_1,5) fires(src_f16bp_1,5) fires(t3,5) fires(t4,5) fires(t6,5) Answer: 7 holds(bpg13,0,0) holds(dhap,0,0) holds(f16bp,0,0) holds(g3p,0,0) fires(reset_dhap_1,0) fires(src_f16bp_1,0) fires(t3,0) holds(bpg13,0,1) holds(dhap,0,1) holds(f16bp,2,1) holds(g3p,0,1) fires(reset_dhap_1,1) fires(src_f16bp_1,1) fires(t3,1) fires(t4,1) holds(bpg13,0,2) holds(dhap,1,2) holds(f16bp,3,2) holds(g3p,1,2) fires(reset_dhap_1,2) fires(src_f16bp_1,2) fires(t3,2) fires(t4,2) fires(t5b,2) holds(bpg13,0,3) holds(dhap,2,3) holds(f16bp,4,3) holds(g3p,1,3) fires(reset_dhap_1,3) fires(src_f16bp_1,3) fires(t3,3) fires(t4,3) fires(t5b,3) holds(bpg13,0,4) holds(dhap,2,4) holds(f16bp,5,4) holds(g3p,1,4) fires(reset_dhap_1,4) fires(src_f16bp_1,4) fires(t3,4) fires(t4,4) fires(t6,4) holds(bpg13,2,5) holds(dhap,1,5) holds(f16bp,6,5) holds(g3p,1,5) fires(reset_dhap_1,5) fires(src_f16bp_1,5) fires(t3,5) fires(t4,5) fires(t5b,5) Answer: 8 holds(bpg13,0,0) holds(dhap,0,0) holds(f16bp,0,0) holds(g3p,0,0) fires(reset_dhap_1,0) fires(src_f16bp_1,0) fires(t3,0) holds(bpg13,0,1) holds(dhap,0,1) holds(f16bp,2,1) holds(g3p,0,1) fires(reset_dhap_1,1) fires(src_f16bp_1,1) fires(t3,1) fires(t4,1) holds(bpg13,0,2) holds(dhap,1,2) holds(f16bp,3,2) holds(g3p,1,2) fires(reset_dhap_1,2) fires(src_f16bp_1,2) fires(t3,2) fires(t4,2) fires(t5b,2) holds(bpg13,0,3) holds(dhap,2,3) holds(f16bp,4,3) holds(g3p,1,3) fires(reset_dhap_1,3) fires(src_f16bp_1,3) fires(t3,3) fires(t4,3) fires(t5b,3) holds(bpg13,0,4) holds(dhap,2,4) holds(f16bp,5,4) holds(g3p,1,4) fires(reset_dhap_1,4) fires(src_f16bp_1,4) fires(t3,4) fires(t4,4) fires(t6,4) holds(bpg13,2,5) holds(dhap,1,5) holds(f16bp,6,5) holds(g3p,1,5) fires(reset_dhap_1,5) fires(src_f16bp_1,5) fires(t3,5) fires(t4,5) fires(t6,5) Answer: 9 holds(bpg13,0,0) holds(dhap,0,0) holds(f16bp,0,0) holds(g3p,0,0) fires(reset_dhap_1,0) fires(src_f16bp_1,0) fires(t3,0) holds(bpg13,0,1) holds(dhap,0,1) holds(f16bp,2,1) holds(g3p,0,1) fires(reset_dhap_1,1) fires(src_f16bp_1,1) fires(t3,1) fires(t4,1) holds(bpg13,0,2) holds(dhap,1,2) holds(f16bp,3,2) holds(g3p,1,2) fires(reset_dhap_1,2) fires(src_f16bp_1,2) fires(t3,2) fires(t4,2) fires(t5b,2) holds(bpg13,0,3) holds(dhap,2,3) holds(f16bp,4,3) holds(g3p,1,3) fires(reset_dhap_1,3) fires(src_f16bp_1,3) fires(t3,3) fires(t4,3) fires(t6,3) holds(bpg13,2,4) holds(dhap,1,4) holds(f16bp,5,4) holds(g3p,1,4) fires(reset_dhap_1,4) fires(src_f16bp_1,4) fires(t3,4) fires(t4,4) fires(t5b,4) holds(bpg13,2,5) holds(dhap,2,5) holds(f16bp,6,5) holds(g3p,1,5) fires(reset_dhap_1,5) fires(src_f16bp_1,5) fires(t3,5) fires(t4,5) fires(t5b,5) Answer: 10 holds(bpg13,0,0) holds(dhap,0,0) holds(f16bp,0,0) holds(g3p,0,0) fires(reset_dhap_1,0) fires(src_f16bp_1,0) fires(t3,0) holds(bpg13,0,1) holds(dhap,0,1) holds(f16bp,2,1) holds(g3p,0,1) fires(reset_dhap_1,1) fires(src_f16bp_1,1) fires(t3,1) fires(t4,1) holds(bpg13,0,2) holds(dhap,1,2) holds(f16bp,3,2) holds(g3p,1,2) fires(reset_dhap_1,2) fires(src_f16bp_1,2) fires(t3,2) fires(t4,2) fires(t5b,2) holds(bpg13,0,3) holds(dhap,2,3) holds(f16bp,4,3) holds(g3p,1,3) fires(reset_dhap_1,3) fires(src_f16bp_1,3) fires(t3,3) fires(t4,3) fires(t6,3) holds(bpg13,2,4) holds(dhap,1,4) holds(f16bp,5,4) holds(g3p,1,4) fires(reset_dhap_1,4) fires(src_f16bp_1,4) fires(t3,4) fires(t4,4) fires(t5b,4) holds(bpg13,2,5) holds(dhap,2,5) holds(f16bp,6,5) holds(g3p,1,5) fires(reset_dhap_1,5) fires(src_f16bp_1,5) fires(t3,5) fires(t4,5) fires(t6,5) Answer: 11 holds(bpg13,0,0) holds(dhap,0,0) holds(f16bp,0,0) holds(g3p,0,0) fires(reset_dhap_1,0) fires(src_f16bp_1,0) fires(t3,0) holds(bpg13,0,1) holds(dhap,0,1) holds(f16bp,2,1) holds(g3p,0,1) fires(reset_dhap_1,1) fires(src_f16bp_1,1) fires(t3,1) fires(t4,1) holds(bpg13,0,2) holds(dhap,1,2) holds(f16bp,3,2) holds(g3p,1,2) fires(reset_dhap_1,2) fires(src_f16bp_1,2) fires(t3,2) fires(t4,2) fires(t5b,2) holds(bpg13,0,3) holds(dhap,2,3) holds(f16bp,4,3) holds(g3p,1,3) fires(reset_dhap_1,3) fires(src_f16bp_1,3) fires(t3,3) fires(t4,3) fires(t6,3) holds(bpg13,2,4) holds(dhap,1,4) holds(f16bp,5,4) holds(g3p,1,4) fires(reset_dhap_1,4) fires(src_f16bp_1,4) fires(t3,4) fires(t4,4) fires(t6,4) holds(bpg13,4,5) holds(dhap,1,5) holds(f16bp,6,5) holds(g3p,1,5) fires(reset_dhap_1,5) fires(src_f16bp_1,5) fires(t3,5) fires(t4,5) fires(t5b,5) Answer: 12 holds(bpg13,0,0) holds(dhap,0,0) holds(f16bp,0,0) holds(g3p,0,0) fires(reset_dhap_1,0) fires(src_f16bp_1,0) fires(t3,0) holds(bpg13,0,1) holds(dhap,0,1) holds(f16bp,2,1) holds(g3p,0,1) fires(reset_dhap_1,1) fires(src_f16bp_1,1) fires(t3,1) fires(t4,1) holds(bpg13,0,2) holds(dhap,1,2) holds(f16bp,3,2) holds(g3p,1,2) fires(reset_dhap_1,2) fires(src_f16bp_1,2) fires(t3,2) fires(t4,2) fires(t5b,2) holds(bpg13,0,3) holds(dhap,2,3) holds(f16bp,4,3) holds(g3p,1,3) fires(reset_dhap_1,3) fires(src_f16bp_1,3) fires(t3,3) fires(t4,3) fires(t6,3) holds(bpg13,2,4) holds(dhap,1,4) holds(f16bp,5,4) holds(g3p,1,4) fires(reset_dhap_1,4) fires(src_f16bp_1,4) fires(t3,4) fires(t4,4) fires(t6,4) holds(bpg13,4,5) holds(dhap,1,5) holds(f16bp,6,5) holds(g3p,1,5) fires(reset_dhap_1,5) fires(src_f16bp_1,5) fires(t3,5) fires(t4,5) fires(t6,5) Answer: 13 holds(bpg13,0,0) holds(dhap,0,0) holds(f16bp,0,0) holds(g3p,0,0) fires(reset_dhap_1,0) fires(src_f16bp_1,0) fires(t3,0) holds(bpg13,0,1) holds(dhap,0,1) holds(f16bp,2,1) holds(g3p,0,1) fires(reset_dhap_1,1) fires(src_f16bp_1,1) fires(t3,1) fires(t4,1) holds(bpg13,0,2) holds(dhap,1,2) holds(f16bp,3,2) holds(g3p,1,2) fires(reset_dhap_1,2) fires(src_f16bp_1,2) fires(t3,2) fires(t4,2) fires(t6,2) holds(bpg13,2,3) holds(dhap,1,3) holds(f16bp,4,3) holds(g3p,1,3) fires(reset_dhap_1,3) fires(src_f16bp_1,3) fires(t3,3) fires(t4,3) fires(t6,3) holds(bpg13,4,4) holds(dhap,1,4) holds(f16bp,5,4) holds(g3p,1,4) fires(reset_dhap_1,4) fires(src_f16bp_1,4) fires(t3,4) fires(t4,4) fires(t5b,4) holds(bpg13,4,5) holds(dhap,2,5) holds(f16bp,6,5) holds(g3p,1,5) fires(reset_dhap_1,5) fires(src_f16bp_1,5) fires(t3,5) fires(t4,5) fires(t6,5) Answer: 14 holds(bpg13,0,0) holds(dhap,0,0) holds(f16bp,0,0) holds(g3p,0,0) fires(reset_dhap_1,0) fires(src_f16bp_1,0) fires(t3,0) holds(bpg13,0,1) holds(dhap,0,1) holds(f16bp,2,1) holds(g3p,0,1) fires(reset_dhap_1,1) fires(src_f16bp_1,1) fires(t3,1) fires(t4,1) holds(bpg13,0,2) holds(dhap,1,2) holds(f16bp,3,2) holds(g3p,1,2) fires(reset_dhap_1,2) fires(src_f16bp_1,2) fires(t3,2) fires(t4,2) fires(t6,2) holds(bpg13,2,3) holds(dhap,1,3) holds(f16bp,4,3) holds(g3p,1,3) fires(reset_dhap_1,3) fires(src_f16bp_1,3) fires(t3,3) fires(t4,3) fires(t6,3) holds(bpg13,4,4) holds(dhap,1,4) holds(f16bp,5,4) holds(g3p,1,4) fires(reset_dhap_1,4) fires(src_f16bp_1,4) fires(t3,4) fires(t4,4) fires(t5b,4) holds(bpg13,4,5) holds(dhap,2,5) holds(f16bp,6,5) holds(g3p,1,5) fires(reset_dhap_1,5) fires(src_f16bp_1,5) fires(t3,5) fires(t4,5) fires(t5b,5) Answer: 15 holds(bpg13,0,0) holds(dhap,0,0) holds(f16bp,0,0) holds(g3p,0,0) fires(reset_dhap_1,0) fires(src_f16bp_1,0) fires(t3,0) holds(bpg13,0,1) holds(dhap,0,1) holds(f16bp,2,1) holds(g3p,0,1) fires(reset_dhap_1,1) fires(src_f16bp_1,1) fires(t3,1) fires(t4,1) holds(bpg13,0,2) holds(dhap,1,2) holds(f16bp,3,2) holds(g3p,1,2) fires(reset_dhap_1,2) fires(src_f16bp_1,2) fires(t3,2) fires(t4,2) fires(t6,2) holds(bpg13,2,3) holds(dhap,1,3) holds(f16bp,4,3) holds(g3p,1,3) fires(reset_dhap_1,3) fires(src_f16bp_1,3) fires(t3,3) fires(t4,3) fires(t6,3) holds(bpg13,4,4) holds(dhap,1,4) holds(f16bp,5,4) holds(g3p,1,4) fires(reset_dhap_1,4) fires(src_f16bp_1,4) fires(t3,4) fires(t4,4) fires(t6,4) holds(bpg13,6,5) holds(dhap,1,5) holds(f16bp,6,5) holds(g3p,1,5) fires(reset_dhap_1,5) fires(src_f16bp_1,5) fires(t3,5) fires(t4,5) fires(t5b,5) Answer: 16 holds(bpg13,0,0) holds(dhap,0,0) holds(f16bp,0,0) holds(g3p,0,0) fires(reset_dhap_1,0) fires(src_f16bp_1,0) fires(t3,0) holds(bpg13,0,1) holds(dhap,0,1) holds(f16bp,2,1) holds(g3p,0,1) fires(reset_dhap_1,1) fires(src_f16bp_1,1) fires(t3,1) fires(t4,1) holds(bpg13,0,2) holds(dhap,1,2) holds(f16bp,3,2) holds(g3p,1,2) fires(reset_dhap_1,2) fires(src_f16bp_1,2) fires(t3,2) fires(t4,2) fires(t6,2) holds(bpg13,2,3) holds(dhap,1,3) holds(f16bp,4,3) holds(g3p,1,3) fires(reset_dhap_1,3) fires(src_f16bp_1,3) fires(t3,3) fires(t4,3) fires(t6,3) holds(bpg13,4,4) holds(dhap,1,4) holds(f16bp,5,4) holds(g3p,1,4) fires(reset_dhap_1,4) fires(src_f16bp_1,4) fires(t3,4) fires(t4,4) fires(t6,4) holds(bpg13,6,5) holds(dhap,1,5) holds(f16bp,6,5) holds(g3p,1,5) fires(reset_dhap_1,5) fires(src_f16bp_1,5) fires(t3,5) fires(t4,5) fires(t6,5) Atoms selected for $bpg13$ quantity extraction for the nominal case: [holds(bpg13,0,0),holds(bpg13,0,1),holds(bpg13,0,2), \tholds(bpg13,0,3),holds(bpg13,2,4),holds(bpg13,4,5)], [holds(bpg13,0,0),holds(bpg13,0,1),holds(bpg13,0,2), \tholds(bpg13,2,3),holds(bpg13,4,4),holds(bpg13,6,5)] Atoms selected for $bpg13$ quantity extraction for the modified case: [holds(bpg13,0,0),holds(bpg13,0,1),holds(bpg13,0,2), \tholds(bpg13,2,3),holds(bpg13,2,4),holds(bpg13,2,5)], [holds(bpg13,0,0),holds(bpg13,0,1),holds(bpg13,0,2), \tholds(bpg13,2,3),holds(bpg13,2,4),holds(bpg13,2,5)], [holds(bpg13,0,0),holds(bpg13,0,1),holds(bpg13,0,2), \tholds(bpg13,2,3),holds(bpg13,2,4),holds(bpg13,4,5)], [holds(bpg13,0,0),holds(bpg13,0,1),holds(bpg13,0,2), \tholds(bpg13,2,3),holds(bpg13,2,4),holds(bpg13,4,5)], [holds(bpg13,0,0),holds(bpg13,0,1),holds(bpg13,0,2), \tholds(bpg13,0,3),holds(bpg13,0,4),holds(bpg13,0,5)], [holds(bpg13,0,0),holds(bpg13,0,1),holds(bpg13,0,2), \tholds(bpg13,0,3),holds(bpg13,0,4),holds(bpg13,0,5)], [holds(bpg13,0,0),holds(bpg13,0,1),holds(bpg13,0,2), \tholds(bpg13,0,3),holds(bpg13,0,4),holds(bpg13,2,5)], [holds(bpg13,0,0),holds(bpg13,0,1),holds(bpg13,0,2), \tholds(bpg13,0,3),holds(bpg13,0,4),holds(bpg13,2,5)], [holds(bpg13,0,0),holds(bpg13,0,1),holds(bpg13,0,2), \tholds(bpg13,0,3),holds(bpg13,2,4),holds(bpg13,2,5)], [holds(bpg13,0,0),holds(bpg13,0,1),holds(bpg13,0,2), \tholds(bpg13,0,3),holds(bpg13,2,4),holds(bpg13,2,5)], [holds(bpg13,0,0),holds(bpg13,0,1),holds(bpg13,0,2), \tholds(bpg13,0,3),holds(bpg13,2,4),holds(bpg13,4,5)], [holds(bpg13,0,0),holds(bpg13,0,1),holds(bpg13,0,2), \tholds(bpg13,0,3),holds(bpg13,2,4),holds(bpg13,4,5)], [holds(bpg13,0,0),holds(bpg13,0,1),holds(bpg13,0,2), \tholds(bpg13,2,3),holds(bpg13,4,4),holds(bpg13,4,5)], [holds(bpg13,0,0),holds(bpg13,0,1),holds(bpg13,0,2), \tholds(bpg13,2,3),holds(bpg13,4,4),holds(bpg13,4,5)], [holds(bpg13,0,0),holds(bpg13,0,1),holds(bpg13,0,2), \tholds(bpg13,2,3),holds(bpg13,4,4),holds(bpg13,6,5)], [holds(bpg13,0,0),holds(bpg13,0,1),holds(bpg13,0,2), \tholds(bpg13,2,3),holds(bpg13,4,4),holds(bpg13,6,5)] Progression from raw matrix of $bpg13$ quantities in various answer-sets (rows) at various simulation steps (columns) to rates and finally to the average aggregate for the nominal case: $\\begin{bmatrix}0 & 0 & 0 & 0 & 2 & 4\\\\0 & 0 & 0 & 2 & 4 & 6\\\\\\end{bmatrix}=rate=>\\begin{bmatrix}0.8\\\\1.2\\\\\\end{bmatrix}=average=>1.0$ Progression from raw matrix of $bpg13$ quantities in various answer-sets (rows) at various simulation steps (columns) to rates and finally to the average aggregate for the modified case: $\\begin{bmatrix}0 & 0 & 0 & 2 & 2 & 2\\\\0 & 0 & 0 & 2 & 2 & 2\\\\0 & 0 & 0 & 2 & 2 & 4\\\\0 & 0 & 0 & 2 & 2 & 4\\\\0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 2\\\\0 & 0 & 0 & 0 & 0 & 2\\\\0 & 0 & 0 & 0 & 2 & 2\\\\0 & 0 & 0 & 0 & 2 & 2\\\\0 & 0 & 0 & 0 & 2 & 4\\\\0 & 0 & 0 & 0 & 2 & 4\\\\0 & 0 & 0 & 2 & 4 & 4\\\\0 & 0 & 0 & 2 & 4 & 4\\\\0 & 0 & 0 & 2 & 4 & 6\\\\0 & 0 & 0 & 2 & 4 & 6\\\\\\end{bmatrix}=rate=>\\begin{bmatrix}0.4\\\\0.4\\\\0.8 \\\\0.8 \\\\0.", "\\\\0.", "\\\\0.4 \\\\0.4 \\\\0.4 \\\\0.4 \\\\0.8 \\\\0.8 \\\\0.8 \\\\0.8 \\\\1.2 \\\\1.2\\\\\\end{bmatrix}=average=>0.6$ We find that $d=^{\\prime }<^{\\prime }$ comparing the modified case rate of $0.6$ to the nominal case rate of $1.0$ .", "Since the direction $d$ was an unknown in the query statement, our system generates produces the full query specification with $d$ replaced by $^{\\prime }<^{\\prime }$ as follows: direction of change in average rate of production of 'bpg13' is '<' (0.6<1) when observed between time step 0 and time step 5 comparing nominal pathway with modified pathway obtained; due to interventions: remove 'dhap' as soon as produced; using initial setup: continuously supply 'f16bp' in quantity 1; Evaluation Methodolgy A direct comparison against other tools is not possible, since most other programs explore one state evolution, while we explore all possible state evolutions.", "In addition ASP has to ground the program completely, irrespective of whether we are computing one answer or all.", "So, to evaluate our system, we compare our results for the questions from the 2nd Deep KR Challenge against the answers they have provided.", "Our results in essence match the responses given for the questions.", "Related Work In this section, we relate our high level language with other high level action languages.", "Comparison with $\\pi $ -Calculus $\\pi $ -calculus is a formalism that is used to model biological systems and pathways by modeling biological systems as mobile communication systems.", "We use the biological model described by [65] for comparison against our system.", "In their model they represent molecules and their domains as computational processes, interacting elements of molecules as communication channels (two molecules interact if they fit together as in a lock-and-key mechanism), and reactions as communication through channel transmission.", "$\\pi $ -calculus models have the ability of changing their structure during simulation.", "Our system on the other hand only allows modification of the pathway at the start of simulation.", "Regular $\\pi $ -calculus models appear qualitative in nature.", "However, stochastic extensions allow representation of quantitative data [62].", "In contrast, the focus of our system is on the quantitative+qualitative representation using numeric fluents.", "It is unclear how one can easily implement maximal-parallelism of our system in $\\pi $ -calculus, where a maximum number of simultaneous actions occur such that they do not cause a conflict.", "Where, a set of actions is said to be in conflict if their simultaneous execution will cause a fluent to become negative.", "Comparison with Action Language $\\mathcal {A}$ Action language $\\mathcal {A}$ [26] is a formalism that has been used to model biological systems and pathways.", "First we give a brief overview of $\\mathcal {A}$ in an intuitive manner.", "Assume two sets of disjoint symbols containing fluent names and action names, then a fluent expression is either a fluent name $F$ or $\\lnot F$ .", "A domain description is composed of propositions of the following form: value proposition: $F \\mathbf {~after~} A_1;\\dots ;A_m$ , where $(m \\ge 0)$ , $F$ is a fluent and $A_1,\\dots ,A_m$ are fluents.", "effect propostion: $A \\mathbf {~causes~} F \\mathbf {~if~} P_1,\\dots ,P_n$ , where $(n \\ge 0)$ , $A$ is an action, $F,P_1,\\dots ,P_n$ are fluent expressions.", "$P_1,\\dots ,P_n$ are called preconditions of $A$ and the effect proposition describes the effect on $F$ .", "We relate it to our work: Fluents are boolean.", "We support numeric valued fluents, with binary fluents.", "Fluents are non-inertial, but inertia can be added.", "Our fluents are always intertial.", "Action description specifies the effect of an action.", "Our domain description specifies `natural'-actions, which execute automatically when their pre-conditions are satisfied, subject to certain conditions.", "As a result our domain description represents trajectories.", "No built in support for aggregates exists.", "We support a selected set of aggregates, on a single trajectory and over multiple trajectories.", "Value propositions in $\\mathcal {A}$ are representable as observations in our query language.", "Comparison with Action Language $\\mathcal {B}$ Action language $\\mathcal {B}$ extends $\\mathcal {A}$ by adding static causal laws, which allows one to specify indirect effects or ramifications of an action [28].", "We relate it to our work below: Inertia is built into the semantics of $\\mathcal {B}$ [28].", "Our language also has intertia built in.", "$\\mathcal {B}$ supports static causal laws that allow defining a fluent in terms of other fluents.", "We do not support static causal laws.", "Comparison with Action Language $\\mathcal {C}$ Action language $\\mathcal {C}$ is based on the theory of causal explanation, i.e.", "a formula is true if there is a cause for it to be true [30].", "It has been previously used to represent and reason about biological pathways [23].", "We relate it to our work below: $\\mathcal {C}$ supports boolean fluents only.", "We support numeric valued fluents, and binary fluents.", "$\\mathcal {C}$ allows both inertial and non-inertial fluents.", "While our fluents are always inertial.", "$\\mathcal {C}$ support static causal laws (or ramifications), that allow defining a fluent in terms of other fluents.", "We do not support them.", "$\\mathcal {C}$ describes causal relationships between fluents and actions.", "Our language on the other hand describes trajectories.", "Comparison with Action Language $\\mathcal {C+}$ First, we give a brief overview of $\\mathcal {C+}$ [29].", "Intuitively, atoms $\\mathcal {C+}$ are of the form $c=v$ , where $c$ is either a fluent or an action constant, $v$ belongs to the domain of $c$ , and fluents and actions form a disjoint set.", "A formula is a propositional combination of atoms.", "A fluent formula is a formula in which all constants are fluent constants; and an action formula is a formula with one action constant but no fluent constants.", "An action description in $\\mathcal {C+}$ is composed of causal laws of the following forms: static law: $\\mathbf {caused~} F \\mathbf {~if~} G$ , where $F$ and $G$ are fluent formulas action dynamic law: $\\mathbf {caused~} F \\mathbf {~if~} G$ , where $F$ is an action formula and $G$ is a formula fluent dynamic law: $\\mathbf {caused~} F \\mathbf {~if~} G \\mathbf {~after~} H$ , where $F$ and $G$ are fluent formulas and $H$ is a formula Concise forms of these laws exist, e.g.", "`$\\mathbf {intertial~} f \\equiv \\mathbf {caused~} f=v \\mathbf {~if~} f=v \\mathbf {~after~} f=v, \\forall v \\in \\text{ domain of } f$ ' that allow a more intuitive program declaration.", "We now relate it to our work.", "Some of the main differences include: Fluents are multi-valued, fluent values can be integer, boolean or other types.", "We support integer and binary valued fluents only.", "Actions are multi-valued.", "We do not support multi-valued actions.", "Both inertial and non-inertial fluents are supported.", "In comparison we allow inertial fluents only.", "Static causal laws are supported that allow changing the value of a fluent based on other fluents (ramifications).", "We do not allow static causal laws.", "Effect of parallel actions on numeric fluents is not additive.", "However, the additive fluents extension [51] adds the capability of additive fluents through new rules.", "The extended language, however, imposes certain restrictions on additive fluents and also restricts the domain of additive actions to boolean actions only.", "Our fluents are always additive.", "Supports defaults.", "We do not have the same notion as defaults, but allow initial values for fluents in our domain description.", "Action's occurrence and its effect are defined in separate statements.", "In our case, the action's occurrence and effect are generally combined in one statement.", "Although parallel actions are supported, it is unclear how one can concisely describe the condition implicit in our system that simultaneously occurring actions may not conflict.", "Two actions conflict if their simultaneous execution will cause a fluent to become negative.", "Exogenous actions seem the closest match to our may execute actions.", "However, our actions are `natural', in that they execute automatically when their pre-conditions are satisfied, they are not explicitly inhibited, and they do not conflict.", "Actions conflict when their simultaneous execution will cause one of the fluents to become negative.", "The exogenous-style character of our actions holds when the firing style is `$$ '.", "When the firing style changes, the execution semantics change as well.", "Consider the following two may execute statements in our language: $a_1 \\mathbf {~may~execute~causing~} f_1 \\mathbf {~change~value~by~} -5 \\mathbf {~if~} f_2 \\mathbf {~has~value~} 3 \\mathbf {~or~higher} \\\\a_2 \\mathbf {~may~execute~causing~} f_1 \\mathbf {~change~value~by~} -3 \\mathbf {~if~} f_2 \\mathbf {~has~value~} 2 \\mathbf {~or~higher}$ and two states: (i) $f_1=10, f_2=5$ , (ii) $f_1=6,f_2=5$ .", "In state (i) both $a_1,a_2$ can occur simultaneously (at one point) resulting in firing-choices $\\big \\lbrace \\lbrace a_1,a_2 \\rbrace ,\\lbrace a_1\\rbrace ,\\lbrace a_2\\rbrace ,\\lbrace \\rbrace ,\\big \\rbrace $ ; whereas, in state (ii) only one of $a_1$ or $a_2$ can occur at one point resulting in the firing-choices: $\\big \\lbrace \\lbrace a_1\\rbrace ,\\lbrace a_2\\rbrace ,\\lbrace \\rbrace ,\\big \\rbrace $ because of a conflict due to the limited amount of $f_1$ .", "These firing choices apply for firing style `', which allows any combination of fireable actions to occur.", "If the firing style is set to `max', the maximum set of non-conflicting actions may fire, and the firing choices for state (i) change to $\\big \\lbrace \\lbrace a_1,a_2 \\rbrace \\big \\rbrace $ and the firing choices for state (ii) change to $\\big \\lbrace \\lbrace a_1\\rbrace ,\\lbrace a_2\\rbrace \\big \\rbrace $ .", "If the firing style is set to `1', at most one action may fire at one point, and the firing choices for both state (i) and state (ii) reduce to $\\big \\lbrace \\lbrace a_1\\rbrace ,\\lbrace a_2\\rbrace ,\\lbrace \\rbrace \\big \\rbrace $ .", "So, the case with `' firing style can be represented in $\\mathcal {C+}$ with exogenous actions $a_1,a_2$ ; the case with `1' firing style can be represented in $\\mathcal {C+}$ with exogenous actions $a_1,a_2$ and a constraint requiring that both $a_1,a_2$ do not occur simultaneously; while the case with `max' firing style can be represented by exogenous actions $a_1,a_2$ with additional action dynamic laws.", "They will still be subject to the conflict checking.", "Action dynamic laws can be used to force actions similar to our must execute actions.", "Specification of initial values of fluents seem possible through the query language.", "The default statement comes close, but it does not have the same notion as setting a fluent to a value once.", "We support specifying initial values both in the domain description as well as the query.", "There does not appear built-in support for aggregation of fluent values within the same answer set, such as sum, count, rate, minimum, maximum etc.", "Although some of it could be implemented using the additive fluents extension.", "We support a set of aggregates, such as total, and rate.", "Additional aggregates can be easily added.", "We support queries over aggregates (such as minimum, maximum, average) of single-trajectory aggregates (such as total, and rate etc.)", "over a set of trajectories.", "We also support comparative queries over two sets of trajectories.", "Our queries allow modification of the domain description as part of query evaluation.", "Comparison with $\\mathcal {BC}$ Action language $\\mathcal {BC}$ combines features of $\\mathcal {B}$ and $\\mathcal {C+}$ [52].", "First we give a brief overview of $\\mathcal {BC}$ .", "Intuitively, $\\mathcal {BC}$ has actions and valued fluents.", "A valued fluent, called an atom, is of the form `$f=v$ ', where $f$ is a fluent constant and $v \\in domain(f)$ .", "A fluent can be regular or statically determined.", "An action description in $\\mathcal {BC}$ is composed of static and dynamic laws of the following form: static law: $A_0 \\mathbf {~if~} A_1,\\dots ,A_m \\mathbf {~ifcons~} A_{m+1},\\dots ,A_n$ , where $(n \\ge m \\ge 0)$ , and each $A_i$ is an atom.", "dynamic law: $A_0 \\mathbf {~after~} A_1,\\dots ,A_m \\mathbf {~ifcons~} A_{m+1},\\dots ,A_n$ , where $(n \\ge m \\ge 0)$ , $A_0$ is a regular fluent atom, each of $A_1,\\dots ,A_m$ is an atom or an action constant, and $A_{m+1},\\dots ,A_n$ are atoms.", "Concise forms of these laws exist that allow a more intuitive program declaration.", "Now we relate it to our work.", "Some of the main differences include: Fluents are multi-valued, fluent values can which can be integer, boolean, or other types.", "We only support integer and binary fluents.", "Static causal laws are allowed.", "We do not support static causal laws.", "Similar to $\\mathcal {C+}$ numeric fluent accumulation is not supported.", "It is supported in our system.", "It is unclear how aggregate queries within a trajectory can be concisely represented.", "Aggregate queries such as rate are supported in our system.", "It does not seem that queries over multiple trajectories or sets of trajectories are supported.", "Such queries are supported in our system.", "Comparison with ASPMT ASPMT combines ASP with Satisfiability Modulo Theories.", "We relate the work in [53] where $\\mathcal {C+}$ is extended using ASPMT with our work.", "It adds support for real valued fluents to $\\mathcal {C+}$ including additive fluents.", "Thus, it allows reasoning with continuous and discrete processes simultaneously.", "Our language does not support real numbers directly.", "Several systems also exist to model and reason with biological pathway.", "For example: Comparison with BioSigNet-RR BioSigNet-RR [5] is a system for representing and reasoning with signaling networks.", "We relate it to our work in an intuitive manner.", "Fluents are boolean, so qualitative queries are possible.", "We support both integer and binary fluents, so quantiative queries are also possible.", "Indirect effects (or ramifications) are supported.", "We do not support these.", "Action effects are captured separately in `$\\mathbf {~causes~}$ statement' from action triggering statements `$\\mathbf {~triggers~}$ ' and `$\\mathbf {~n\\_triggers~}$ '.", "We capture both components in a `$\\mathbf {~may~execute~causing~}$ ' or `$\\mathbf {~normally~must~execute~causing~}$ ' statement.", "Their action triggering semantics have some similarity to our actions.", "Just like their actions get triggered when the action's pre-conditions are satisfied, our actions are also triggered when their pre-conditions are satisfied.", "However, the triggering semantics are different, e.g.", "their triggers statement causes an action to occur even if it is disabled, we do not have an equivalent for it; and their n_triggers is similar in semantics to normally must execute causing statement.", "It is not clear how loops in biological systems can be modeled in their system.", "Loops are possible in our by virtue of the Petri Net semantics.", "Their queries can have time-points and their precedence relations as part of the query.", "Though our queries allow the specification of some time points for interval queries, time-points are not supported in a similar way.", "However, we do support certain types of observation relative queries.", "The intervention in their planning queries has similarities to interventions in our system.", "However, it appears that our intervention descriptions are higher level.", "Conclusion In this chapter we presented the BioPathQA system and the languages to represent and query biological pathways.", "We also presented a new type of Petri Net, the so called Guarded-Arc Petri Net that is used as a model behind our pathway specification language, which shares certain aspects with CPNs [39], but our semantics for reset arcs is different, and we allow must-fire actions that prioritize actions for firing over other actions.", "We also showed how the system can be applied to questions from college level text books that require deeper reasoning and cannot be answered by using surface level knowledge.", "Although our system is developed with respect to the biological domain, it can be applied to non-biological domain as well.", "Some of the features of our language include: natural-actions that automatically fire when their prerequisite conditions are met (subject to certain restrictions); an automatic default constraint that ensures fluents do not go negative, since they model natural systems substances; a more natural representation of locations; and control of the level of parallelism to a certain degree.", "Our query language also allows interventions similar to Pearl's surgeries [59], which are more general than actions.", "Next we want to apply BioPathQA to a real world application by doing text extraction.", "Knowledge for real world is extracted from research papers.", "In the next chapter we show how such text extraction is done for pathway construction and drug development.", "We will then show how we can apply BioPathQA to the extracted knowledge to answer questions about the extracted knowledge.", "Text Extraction for Real World Applications In the previous chapter we looked at the BioPathQA system and how it answers simulation based reasoning questions about biological pathways, including questions that require comparison of alternate scenarios through simulation.", "These so called `what-if' questions arise in biological activities such as drug development, drug interaction, and personalized medicine.", "We will now put our system and language in context of such activities.", "Cutting-edge knowledge about pathways for activities such as drug development, drug interaction, and personalized medicine comes in the form of natural language research papers, thousands of which are published each year.", "To use this knowledge with our system, we need to perform extraction.", "In this chapter we describe techniques we use for such knowledge extraction for discovering drug interactions.", "We illustrate with an example extraction how we organize the extracted knowledge into a pathway specification and give examples of relevant what-if questions that a researcher performing may ask in the drug development domain.", "Introduction Thousands of research papers are published each year about biological systems and pathways over a broad spectrum of activities, including interactions between dugs and diseases, the associated pathways, and genetic variation.", "Thus, one has to perform text extraction to extract relationships between the biochemical processes, their involvement in diseases, and their interaction with drugs.", "For personalized medicine, one is also interested in how these interrelationships change in presence of genetic variation.", "In short, we are looking for relationships between various components of the biochemical processes and their internal and external stimuli.", "Many approaches exist for extracting relationships from a document.", "Most rely on some form of co-occurrence, relative distance, or order of words in a single document.", "Some use shallow parsing as well.", "Although these techniques tend to have a higher recall, they focus on extracting explicit relationships, which are relationships that are fully captured in a sentence or a document.", "These techniques also do not capture implicit relationships that may be spread across multiple documents.", "are spread across multiple documents relating to different species.", "Additional issues arise from the level of detail from in older vs. newer texts and seemingly contradictory information due to various levels of confidence in the techniques used.", "Many do not handle negative statements.", "We primarily use a system called PTQL [75] to extract these relationships, which allows combining the syntactic structure (parse tree), semantic elements, and word order in a relationship query.", "The sentences are pre-processed by using named-entity recognition, and entity normalization to allow querying on classes of entity types, such as drugs, and diseases; and also to allow cross-linking relationships across documents when they refer to the same entity with a different name.", "Queries that use such semantic association between words/phrases are likely to produce higher precision results.", "Source knowledge for extraction primarily comes from thousands of biological abstracts published each year in PubMed http://www.ncbi.nlm.nih.gov/pubmed.", "Next we briefly describe how we extract relationships about drug interactions.", "Following that we briefly describe how we extract association of drugs, and diseases with genetic variation.", "We conclude this chapter with an illustrative example of how the drug interaction relationships are used with our system to answer questions about drug interactions and how genetic variation could be utilized in our system.", "Extracting Relationships about Drug Interactions We summarize the extraction of relationships for our work on drug-drug interactions from [73].", "Studying drug-drug interactions are a major activity in drug development.", "Drug interactions occur due to the interactions between the biological processes / pathways that are responsible metabolizing and transporting drugs.", "Metabolic processes remove a drug from the system within a certain time period.", "For a drug to remain effective, it must be maintained within its therapeutic window for the period of treatment, requiring taking the drug periodically.", "Outside the therapeutic window, a drug can become toxic if a quantity greater than the therapeutic window is retained; or it can become ineffective if a quantity less than the therapeutic window is retained.", "Since liver enzymes metabolize most drugs, it is the location where most metabolic-interaction takes place.", "Induction or inhibition of these enzymes can affect the bioavailability of a drug through transcriptional regulation, either directly or indirectly.", "For example, if drug $A$ inhibits enzyme $E$ , which metabolizes drug $B$ , then the bioavailability of drug $B$ will be higher than normal, rendering it toxic.", "On the other hand, if drug $A$ induces enzyme $E$ , which metabolizes drug $B$ , then drug $B$ 's bioavailability will be lesser than normal, rendering it ineffective.", "Inhibition of enzymes is a common form of drug-drug interactions [10].", "In direct inhibition, a drug $A$ inhibit enzyme $E$ , which is responsible for metabolism of drug $B$ .", "Drug $A$ , leads to a decrease in the level of enzyme $E$ , which in turn can increase bioavailability of drug $B$ potentially leading to toxicity.", "Alternatively, insufficient metabolism of drug $B$ can lead to smaller amount of drug $B$ 's metabolites being produced, leading to therapeutic failure.", "An example of one such direct inhibition is the interaction between CYP2D6 inhibitor quinidine and CYP2D6 substrates (i.e.", "substances metabolized by CYP2D6), such as Codeine.", "The inhibition of CYP2D6 increases the bioavailability of drugs metabolized by CYP2D6 leading to adverse side effects.", "Another form of drug interactions is through induction of enzymes [10].", "In direct induction, a drug $A$ induces enzyme $E$ , which is responsible for metabolism of drug $B$ .", "An example of such direct induction is between phenobarbital, a CYP2C9 inducer and warfarin (a CYP2C9 substrate).", "Phenobarbital leads to increased metabolism of warfarin, decreasing warfarinÕs bioavailability.", "Direct interaction due to induction though possible is not as common as indirect interaction through transcription factors, which regulate the drug metabolizing enzymes.", "In such an interaction, drug $A$ activates a transcription factor $TF$ , which regulates and induces enzyme $E$ , where enzyme $E$ metabolizes drug $B$ .", "Transcription factors are referred to as regulators of xenobiotic-metabolizing enzymes.", "Examples of such regulators include aryl hydrocarbon receptor AhR, pregnane X receptor PXR and constitutive androstane receptor CAR.", "Drug interactions can also occur due to the induction or inhibition of transporters.", "Transporters are mainly responsible for cellular uptake or efflux (elimination) of drugs.", "They play an important part in drug disposition, by transporting drugs into the liver cells, for example.", "Transporter-based drug interactions, however, are not as well studies as metabolism-based interactions [10].", "Method Extraction of drug-drug interactions from the text can either be explicit or implicit.", "Explicit extraction refers to extraction of drug-drug interaction mentioned within a single sentence, while implicit extraction requires extraction of bio-properties of drug transport, metabolism and elimination that can lead to drug-drug interaction.", "This type of indirect extraction combines background information about biological processes, identification of protein families and the interactions that are involved in drug metabolism.", "Our approach is to extract both explicit and implicit drug interactions as summarized in Fig REF and it builds upon the work done in [74].", "Figure: This figure from outlines the effects of drug A on drug B through (a) direct induction/inhibition of enzymes; (b) indirect induction/inhibition of transportation factors that regulate the drug-metabolizing enzymes.Explicit Drug Interaction Extraction Explicit extraction mainly extracts drug-drug interactions directly mentioned in PubMed (or Medline) abstracts.", "For example, the following sentences each have a metabolic interaction mentioned within the same sentence: Ciprofloxacin strongly inhibits clozapine metabolism.", "(PMID: 19067475) Enantioselective induction of cyclophosphamide metabolism by phenytoin.", "which can be extracted by using the following PTQL query using the underlined keywords from above sentences: //S{//?", "[Tag=`Drug'](d1) => //?", "[Value IN {`induce',`induces',`inhibit',`inhibits'}](v) => //?[Tag=`Drug'](d2) => //?", "[Value=`metabolism'](w)} ::: [d1 v d2 w] 5 : d1.value, v.value, d2.value.", "This PTQL query specifies that a drug (denoted by d1) must be followed by one of the keywords from $\\lbrace `induce^{\\prime },`inhibit^{\\prime }, `inhibits^{\\prime }\\rbrace $ (denoted by v), which in turn must be followed by another drug (denoted by d2) followed the keyword $`metabolism^{\\prime }$ (denoted by w); all found within a proximity of 5 words of each other.", "The query produces tripes of $\\langle d1, v, d2 \\rangle $ values as output.", "Thus the results will produce triples $\\langle d1, induces, d2 \\rangle $ and $\\langle d1, inhibits, d2 \\rangle $ which mean that the drug d1 increases the effect of d2 (i.e.", "$\\langle d1, increases, d2 \\rangle $ ) and decreases the effect of d2 (i.e.", "$\\langle d1, decreases, d2 \\rangle $ ) respectively.", "For example, the sentence S1 above matches this PTQL query and the query will produce the triplet $\\langle \\text{ciprofloxacin}, \\text{increases}, \\text{clozapine} \\rangle $ .", "Implicit Drug Interaction Extraction Implicit extraction mainly extracts drug-drug interactions not yet published, but which can be inferred from published articles and properties of drug metabolism.", "The metabolic properties themselves have their origin in various publications.", "The metabolic interactions extracted from published articles and the background knowledge of properties of drug metabolism are reasoned with in an automated fashion to infer drug interactions.", "The following table outlines the kinds of interactions extracted from the text: Table: This table from shows the triplets representing various properties relevant to the extraction of implicit drug interactions and their description.which require multiple PTQL queries for extraction.", "As an example, the following PTQL query is used to extract $\\langle protein, metabolizes, drug \\rangle $ triplets: //S{/?", "[Tag=`Drug'](d1) => //VP{//?", "[Value IN {`metabolized',`metabolised'}](rel) => //?", "[Tag=`GENE'](g1)}} ::: g1.value, rel.value, d1.value which specifies that the extracted triplets must have a drug (denoted by d1) followed by a verb phrase (denoted by VP) with the verb in $\\lbrace `metabolized^{\\prime },`metabolised^{\\prime } \\rbrace $ , followed by a gene (denoted by g1).", "Table REF shows examples of extracted triplets.", "Table: This table from shows the triplets representing various properties relevant to the extraction of implicit drug interactions and their description.", "Data Cleaning The protein-protein and protein-drug relationships extracted from the parse tree database need an extra step of refinement to ensure that they correspond to the known properties of drug metabolism.", "For instance, for a protein to metabolize a drug, the protein must be an enzyme.", "Similarly, for a protein to regulate an enzyme, the protein must be a transcription factor.", "Thus, the $\\langle protein, metabolizes, drug \\rangle $ facts get refined to $\\langle enzyme, metabolizes, drug \\rangle $ and $\\langle protein, regulates, protein \\rangle $ gets refined to $\\langle transcription factor, regulates, enzyme \\rangle $ respectively.", "Classification of proteins is done using UniProt, the Gene Ontology (GO) and Entrez Gene summary by applying rules such as: A protein p is an enzyme if it belongs to a known enzyme family, such as CYP, UGT or SULT gene families; or is annotated under UniProt with the hydrolase, ligase, lyase or transferase keywords; or is listed under the “metabolic process” GO-term; or its Entrez Gene summary mentions key phrases like “drug metabolism” or roots for “enzyme” or “catalyzes”.", "A protein p is considered as a transcription factor if it is annotated with keywords transcription, transcription-regulator or activator under UniProt; or it is listed under the “transcription factor activity” category in GO; or its Entrez Gene summary contains the phrase “transcription factor”.", "Additional rules are applied to remove conflicting information, such as, favoring negative extractions (such as `$P$ does not metabolize $D$ ') over positive extractions (such as `$P$ metabolizes $D$ ').", "For details, see [73].", "Results The correctness of extracted interactions was determined by manually compiling a gold standard for each type of interaction using co-occurrence queries.", "For example, for $\\langle protein, metabolizes, drug \\rangle $ relations, we examined sentences that contain co-occurrence of protein, drug and one of the keywords “metabolized”, “metabolize”, “metabolises”, “metabolise”, “substrate” etc.", "Table REF summarizes the performance of our extraction approach.", "Table: Performance of interactions extracted from Medline abstracts.", "TP represents true-positives, while FN represents false-negatives Extracting Knowledge About Genetic Variants We summarize the relevant portion of our work on associating genetic variants with drugs, diseases and adverse reactions as presented in [33].", "Incorrect drug dosage is the leading cause of adverse drug reactions.", "Doctors prescribe the same amount of medicine to a patient for most drugs using the average drug response, even though a particular person's drug response may be higher or lower than the average case.", "A large part of the difference in drug response can be attributed to single nucleotide polymorphisms (SNPs).", "For example, the enzyme CYP2D6 has 70 known allelic variations, 50 of which are non-functional [31].", "Patients with poor metabolizer variations may retain higher concentration of drug for typical dosage, while patients with rapid metabolizers may not even reach therapeutic level of drug or toxic level of drug metabolites [68].", "Thus, it is important to consider the individual's genetic composition for dosage determination, especially for narrow therapeutic index drugs.", "Scientists studying these variations have grouped metabolizers into categories of poor (PM), intermediate (IM), rapid (RM) and ultra-rapid metabolizers (UM) and found that for some drugs, only 20% of usual dosage was required for PM and up to 140% for UM [38].", "Information about SNPs, their frequency in various population groups, their effect on genes (enzymic activity) and related data is stored in research papers, clinical studies and surveys.", "However, it is spread-out among them.", "Various databases collect this information in different forms.", "PharmGKB collects information such information and how it related to drug response [76].", "However, it is a small percentage of the total number of articles on PharmGKB, due to time consuming manual curation of data.", "Our work focuses on automatically extracting genetic variations and their impact on drug responses from PubMed abstracts to catch up with the current state of research in the biological domain, producing a repository of knowledge to assist in personalized medicine.", "Our approach leverages on as many existing tools as possible.", "Methods Next, we describe the methods used in our extraction, including: named entity recognition, entity normalization, and relation extraction.", "Named Entity Recognition We want to identify entities including genes (also proteins and enzymes), drugs, diseases, ADRs (adverse drug reactions), SNPs, RefSNPs (rs-numbers), names of alleles, populations and frequencies.", "For genes, we use BANNER [49] trained on BioCreative II GM training set [45].", "For genotypes (genetic variations including SNPs) we used a combination of MutationFinder [15] and custom components.", "Custom components were targeted mostly on non-SNPs (“c.76_78delACT”, 11MV324KF”) and insertions/deletions (“1707 del T”, “c.76_77insG”), RefSNPs (rs-numbers) and names of alleles/haplotypes (“CYP2D64”, “T allele”, “null allele”).", "For diseases (and ADRs), we used BANNER trained on a corpus of 3000 sentences with disease annotations [50].", "An additional 200 random sentences containing no disease were added from BioCreative II GM data to offset the low percentage (10%) of sentences without disease in the 3000 sentence corpus.", "In addition to BANNER, we used a dictionary extracted from UMLS.", "This dictionary consisted of 162k terms for 54k concepts from the six categories “Disease or Syndrome”, ”Neoplastic Process”,“Congenital Abnormality”,“Mental or Behavioral Dysfunction”,“Experimental Model of Disease” and “Acquired Abnormality”.", "The list was filtered to remove unspecific as well as spurious disease names such as “symptoms”, “disorder”, .... A dictionary for adverse drug reactions originated from SIDER Side Effect Resource [47], which provides a mapping of ADR terminology to UMLS CUIs.", "It consisted of 1.6k UMLS concepts with 6.5k terms.", "For drugs, we used a dictionary based on DrugBank[80] containing about 29k different drug names including both generic as well as brand names.", "We used the cross-linking information from DrugBank to collect additional synonyms and IDs from PharmGKB.", "We cross linked to Compound and Substance IDs from PubChem to provide hyperlinks to additional information.", "For population, we collected a dictionary of terms referring to countries, regions, regions inhabitants and their ethnicities from WikiPedia, e.g.", "“Caucasian”, “Italian”, “North African”, .... We filtered out irrelevant phrases like “Chinese [hamster]”.", "For frequencies, we extract all numbers and percentages as well as ranges from sentences that contain the word “allele”, “variant”, “mutation”, or “population”.", "The output is filtered in this case as well to remove false positives referring to p-values, odd ratios, confidence intervals and common trigger words.", "Entity Normalization Genes, diseases and drugs can appear with many different names in the text.", "For example, “CYP2D6” can appear as “Cytochrome p450 2D6” or “P450 IID6” among others, but they all refer to the same enzyme (EntrezGene ID 1565).", "We use GNAT on recognized genes [32], but limit them to human, mouse and rat genes.", "The gene name recognized by BANNER is filtered by GNAT to remove non-useful modifiers and looked up against EntrezGene restricted to human, mouse and rat genes to find candidate IDs for each gene name.", "Ambiguity (multiple matches) is resolved by matching the text surrounding the gene mention with gene's annotation from a set of resources, like EntrezGene, UniProt.", "Drugs and diseases/ADRs are resolved to their official IDs from DrugBank or UMLS.", "If none is found, we choose an ID for it based on its name.", "Genetic variants Genetic variations are converted to HGVScite [21] recommended format.", "Alleles were converted to the star notation (e.g.", "“CYP2D61”) and the genotype (“TT allele”) or fixed terms such as “null allele” and “variant allele”.", "Populations mentions are mapped to controlled vocabulary to remove orthographic and lexical variations.", "Relation Extraction Twelve type of relations were extracted between the detected entities as given in Table REF .", "Different methods were applied to detect different relations depending upon relation type, sentence structure and whether another method was able to extraction a relation beforehand.", "Gene-drug, gene-disease, drug-disease were extracted using sentence based co-occurrence (possibly refined by using relation-specific keywords) due to its good precision yield of this method for these relations.", "For other relations additional extraction methods were implemented.", "These include: High-confidence co-occurrence that includes keywords These co-occurences have the relation keyword in them.", "This method is applied to gene-drug, gene-disease, drug-ADR, drug-disease and mutation-disease associations.", "It uses keywords from PolySearch [79] as well as our own.", "Co-occurrence without keywords Such co-occurrences do not require any relationship keyword.", "This method is used for allele-population and variant-population relationships.", "This method can misidentify negative relationships.", "High-confidence relationships, if not found with a keyword drop down to this method for a lower confidence result.", "1:n co-occurrence Relationships where one entity has one instance in a given sentence and the other occurs one or more times.", "Single instance entity may have more than one occurrence.", "This method is useful in identifying gene mutations, where a gene is mentioned in a sentence along with a number of its mutations.", "The gene itself may be repeated.", "Enumerations with matching counts Captures entities in sentences where a list of entities is followed by an equal number of counts.", "This method is useful in capturing alleles and their associated frequencies, e.g.", "“The frequencies of CYP1B11, 2, 3, and 4 alleles were 0.087, 0.293, 0.444, and 0.175, respectively.” Least common ancestor (LCA) sub-tree Assigns associations based on distance in parse tree.", "We used Stanford parser [42] to get grammatical structure of a sentence as a dependency tree.", "This allows relating verb to its subject and noun to its modifiers.", "This method picks the closest pair in the lowest common ancestors (dependency) sub-tree of the entities.", "Maximum distance in terms of edges connecting the entity nodes was set to 10, which was determined empirically to provide the best balance between precision and recall.", "This method associates frequencies with alleles in the sentence “The allele frequencies were 18.3% (-24T) and 21.2% (1249A)”.", "m:n co-occurrence This method builds associations between all pairs of entities.", "Low confidence co-occurrence This acts as the catch-all case if none of the above methods work.", "Table: Unique binary relations identified between detected entities from .These methods were applied in order to each sentence, stoping at the first method that extracted the desired relationship.", "Order of these methods was determined empirically based of their precision.", "The order of the method used determines our confidence in the result.", "If none of the higher confidence methods are successful, a co-occurrence based method is used for extraction with low confidence.", "Abstract-level co-occurrence are also extracted to provide hits on potential relations.", "They appear in the database only when they appear in more than a pre-set threshold number of abstracts.", "Results Performance was evaluated by evaluating the precision and recall of individual components and coverage of existing results.", "Precision and recall were tested by processing 3500 PubMed abstracts found via PharmGKB relations and manually checking the 2500 predictions.", "Coverage was tested against DrugBank and PharmGKB.", "Extracted relations went through manual evaluation for correctness.", "Each extraction was also assigned a confidence value based on the confidence in the method of extraction used.", "We got a coverage of 91% of data in DrugBank and 94% in PharmGKB.", "Taking into false positive rates for genes, drugs and gene-drug relations, SNPshot has more than 10,000 new relations.", "Applying BioPathQA to Drug-Drug Interaction Discovery Now we use our BioPathQA system from chapter to answer questions about drug-drug interaction using knowledge extracted from research publications using the approach in sections REF ,REF .", "We supplement the extracted knowledge with domain knowledge as needed.", "Let the extracted facts be as follows: The drug $gefitinib$ is metabolized by $CYP3A4$ .", "The drug $phenytoin$ induces $CYP3A4$ .", "Following additional facts have been provided about a new drug currently in development: A new drug being developed $test\\_drug$ is a CYP3A4 inhibitor We show the pathway specification based on the above facts and background knowledge, then elaborate on each component: $&\\mathbf {domain~of~} gefitinib \\mathbf {~is~} integer, cyp3a4 \\mathbf {~is~} integer, \\nonumber \\\\&~~~~~~~~~~~~~~~~~~~~ phenytoin \\mathbf {~is~} integer, test\\_drug \\mathbf {~is~} integer\\\\&t1 \\mathbf {~may~execute~causing~} \\nonumber \\\\&~~~~~~~~~~~~~~~~~~~~ gefitinib \\mathbf {~change~value~by~} -1, \\nonumber \\\\&~~~~~~~~~~~~~~~~~~~~ cyp3a4 \\mathbf {~change~value~by~} -1, \\nonumber \\\\&~~~~~~~~~~~~~~~~~~~~ cyp3a4 \\mathbf {~change~value~by~} +1\\\\&\\mathbf {normally~stimulate~} t1 \\mathbf {~by~factor~} 2 \\nonumber \\\\&~~~~~~~~~~~~~~~~~~~~ \\mathbf {~if~} phenytoin \\mathbf {~has~value~} 1 \\mathbf {~or~higher~} \\\\&\\mathbf {inhibit~} t1 \\mathbf {~if~} test\\_drug \\mathbf {~has~value~} 1 \\mathbf {~or~higher~} \\\\&\\mathbf {initially~} gefitinib \\mathbf {~has~value~} 20, cyp3a4 \\mathbf {~has~value~} 60, \\nonumber \\\\&~~~~~~~~~~~~~~~~~~~~ phenytoin \\mathbf {~has~value~} 0, test\\_drug \\mathbf {~has~value~} 0 \\\\&\\mathbf {firing~style~} max$ Line REF declares the domain of the fluents as integer numbers.", "Line represents the activity of enzyme $cyp3a4$ as the action $t1$ .", "Due to the enzymic action $t1$ , one unit of $gefitinib$ is metabolized, and thus converted to various metabolites (not shown here).", "The enzymic action uses one unit of $cyp3a4$ as catalyst, which is used in the reaction and released afterwards.", "Line represents the knowledge that $phenytoin$ induces the activity of $cyp3a4$ .", "From background knowledge we find out that the stimulation in the activity can be as high as 2-times [55].", "Line represents the knowledge that there is a new drug $test\\_drug$ being tested that is known to inhibit the activity of $cyp3a4$ .", "Line specifies the initial distribution of the drugs and enzymes in the pathway.", "Assuming the patient has been given some fixed dose, say 20 units, of the medicine $gefitinib$ .", "It also specifies there is a large 60 units quantity of $cyp3a4$ available to ensure reactions do not slow down due to unavailability of enzyme availability.", "Additionaly, the drug $phenytoin$ is absent from the system and a new drug $test\\_drug$ to be tested is not in the system either.", "This gives us our pathway specification.", "Now we consider two application scenarios for drug development.", "Drug Administration A patient is taking 20 units of $gefitinib$ , and is being prescribed additional drugs to be co-administered.", "The drug administrator wants to know if there will be an interaction with $gefitinib$ if 5 units of $phenytoin$ are co-administered.", "If there is an interaction, what will be the bioavailability of $gefitinib$ so that its dosage could be appropriately adjusted.", "The first question is asking whether giving the patient 5-units of $phenytoin$ in addition to the existing $gefitinib$ dose will cause a drug-interaction.", "It is encoded as the following query statement $\\mathbf {Q}$ for a $k$ -step simulation: $&\\mathbf {direction~of~change~in~} average \\mathbf {~value~of~} gefitinib \\mathbf {~is~} d\\\\&~~~~~~\\mathbf {when~observed~at~time~step~} k; \\\\&~~~~~~\\mathbf {comparing~nominal~pathway~with~modified~pathway~obtained~}\\\\&~~~~~~~~~~~~\\mathbf {due~to~interventions:~} \\mathbf {set~value~of~} phenytoin \\mathbf {~to~} 5;$ If the direction of change is “$=$ ” then there was no drug-interaction.", "Otherwise, an interaction was noticed.", "For a simulation of length $k=5$ , we find 15 units of $gefitinib$ remained at the end of simulation in the nominal case when no $phenytoin$ is administered.", "The amount drops to 10 units of $gefitinib$ when $phenytoin$ is co-administered.", "The change in direction is “$<$ ”.", "Thus there is an interaction.", "The second question is asking about the bioavailability of the drug $gefitinib$ after some after giving $phenytoin$ in 5 units.", "If this bioavailability falls below the efficacy level of the drug, then the drug would not treat the disease effectively.", "It is encoded as the following query statement $\\mathbf {Q}$ for a $k$ -step simulation: $&average \\mathbf {~value~of~} gefitinib \\mathbf {~is~} n\\\\&~~~~\\mathbf {when~observed~at~time~step~} k; \\\\&~~~~\\mathbf {due~to~interventions:~} \\\\&~~~~~~~~\\mathbf {set~value~of~} phenytoin \\mathbf {~to~} 5;$ For a simulation of length $k=5$ , we find 10 units of $gefitinib$ remain.", "A drug administrator (such as a pharmacist) can adjust the drug accordingly.", "Drug Development A drug manufacturer is developing a new drug $test\\_drug$ that is known to inhibit CYP3A4 that will be co-administered with drugs $gefitinib$ and $phenytoin$ .", "He wants to determine the bioavailability of $gefitinib$ over time to determine the risk of toxicity.", "The question is asking about the bioavailability of the drug $gefitinib$ after 10 time units after giving $phenytoin$ in 5 units and the new drug $test\\_drug$ in 5 units.", "If this bioavailability remains high, there is chance for toxicity due to the drug at the subsequent dosage intervals.", "It is encoded as the following query statement $\\mathbf {Q}$ for a $k$ -step simulation: $&average \\mathbf {~value~of~} gefitinib \\mathbf {~is~} n\\\\&~~~~\\mathbf {when~observed~at~time~step~} k; \\\\&~~~~\\mathbf {due~to~interventions:~} \\\\&~~~~~~~~\\mathbf {set~value~of~} phenytoin \\mathbf {~to~} 5,\\\\&~~~~~~~~\\mathbf {set~value~of~} test\\_drug \\mathbf {~to~} 5;$ For a simulation of length $k=5$ , we find all 20 units of $gefitinib$ remain.", "This could lead to toxicity by building high concentration of $gefitinib$ in the body.", "Drug Administration in Presence of Genetic Variation A drug administrator wants to establish the dosage of $morphine$ for a person based on its genetic profile using its bioavailability.", "Consider the following facts extracted about a simplified morphine pathway: $codeine$ is metabolized by $CYP2D6$ producing $morphine$ $CYP2D6$ has three allelic variations “1” – (EM) effective metabolizer (normal case) “2” – (UM) ultra rapid metabolizer “9” – (PM) poor metabolizer For simplicity, assume UM allele doubles the metabolic rate, while PM allele halves the metabolic rate of CYP2D6.", "Then, the resulting pathway is given by: $&\\mathbf {domain~of~} cyp2d6\\_allele \\mathbf {~is~} integer, cyp2d6 \\mathbf {~is~} integer\\\\&\\mathbf {domain~of~} codeine \\mathbf {~is~} integer, morphine \\mathbf {~is~} integer\\\\&cyp2d6\\_action \\mathbf {~may~execute~causing}\\\\&~~~~codeine \\mathbf {~change~value~by~} -2, morphine \\mathbf {~change~value~by~} +2\\\\&~~~~\\mathbf {if~} cyp2d6\\_allele \\mathbf {~has~value~equal~to~} 1\\\\&cyp2d6\\_action \\mathbf {~may~execute~causing}\\\\&~~~~codeine \\mathbf {~change~value~by~} -4, morphine \\mathbf {~change~value~by~} +4\\\\&~~~~\\mathbf {if~} cyp2d6\\_allele \\mathbf {~has~value~equal~to~} 2\\\\&cyp2d6\\_action \\mathbf {~may~execute~causing~}\\\\&~~~~codeine \\mathbf {~change~value~by~} -1, morphine \\mathbf {~change~value~by~} +1\\\\&~~~~\\mathbf {if~} cyp2d6\\_allele \\mathbf {~has~value~equal~to~} 9\\\\&\\mathbf {initially~}\\\\&~~~~codeine \\mathbf {~has~value~} 0, morphine \\mathbf {~has~value~} 0,\\\\&~~~~cyp2d6 \\mathbf {~has~value~} 20, cyp2d6\\_allele \\mathbf {~has~value~} 1\\\\&\\mathbf {firing~style~} max\\\\$ Then, the bioavailability of $morphine$ can be determined by the following query: $&average \\mathbf {~value~of~} morphine \\mathbf {~is~} n\\\\&~~~~\\mathbf {when~observed~at~time~step~} k; \\\\&~~~~\\mathbf {due~to~interventions:~} \\\\&~~~~~~~~\\mathbf {set~value~of~} codeine \\mathbf {~to~} 20,\\\\&~~~~~~~~\\mathbf {set~value~of~} cyp2d6 \\mathbf {~to~} 20,\\\\&~~~~~~~~\\mathbf {set~value~of~} cyp2d6\\_allele \\mathbf {~to~} 9;$ Simulation for 5 time steps reveal that the average bioavailability of $morphine$ after 5 time-steps is 5 for PM (down from 10 for EM).", "Although this is a toy example, it is easy to see the potential of capturing known genetic variations in the pathway and setting the complete genetic profile of a person in the intervention part of the query.", "Conclusion In this chapter we presented how we extract biological pathway knowledge from text, including knowledge about drug-gene interactions and their relationship to genetic variation.", "We showed how the information extracted is used to build pathway specification and illustrated how biologically relevant questions can be answered about drug-drug interaction using the BioPathQA system developed in chapter .", "Next we look at the future directions in which the research work done in this thesis can be extended.", "Conclusion and Future Work The field of knowledge representation and reasoning (KR) is currently one of the most active research areas.", "It represents the next step in the evolution of systems that know how to organize knowledge, and have the ability to intelligently respond to questions about this knowledge.", "Such questions could be about static knowledge or the dynamic processes.", "Biological systems are uniquely positioned as role models for this next evolutionary step due to their precise vocabulary and mechanical nature.", "As a result, a number of recent research challenges in the KR field are focused on it.", "The biological field itself needs systems that can intelligently answer questions about such biological processes and systems in an automated fashion, given the large number of research papers published each year.", "Curating these publications is time consuming and expensive, as a result the state of over all knowledge about biological systems lags behind the cutting edge research.", "An important class of questions asked about biological systems are the so called “what-if” questions that compare alternate scenarios of a biological pathway.", "To answer such questions, one has to perform simulation on a nominal pathway against a pathway modified due to the interventions specified for the alternate scenario.", "Often, this means creating two pathways (for nominal and alternate cases) and simulate them separately.", "This opens up the possibility that the two pathways can become out of synchronization.", "A better approach is to allow the user to specify the needed interventions in the query statement itself.", "In addition, to understand the full spread of possible outcomes, given the parallel nature of biological pathways, one must consider all possible pathway evolutions, otherwise, some outcomes may remain hidden.", "If a system is to be used by biologists, it must have a simple interface, lowering the barrier of entry.", "Since biological pathway knowledge can arrive from different sources, including books, published articles, and lab experiments, a common input format is desired.", "Such a format allows specification of pathways due to automatic extraction, as well as any changes / additions due to locally available information.", "A comprehensive end-to-end system that accomplish all the goals would take a natural language query along with any additional specific knowledge about the pathway as input, extract the relevant portion of the relevant pathway from published material (and background knowledge), simulate it based on the query, and generate the results in a visual format.", "Each of these tasks comes with its own challenges, some of which have been addressed in this thesis.", "In this thesis, we have developed a system and a high level language to specify a biological pathway and answer simulation based reasoning questions about it.", "The high level language uses controlled-English vocabulary to make it more natural for a researcher to use directly.", "The high level language has two components: a pathway specification language, and a query specification language.", "The pathway specification language allows the user to specify a pathway in a source independent form, thus locally obtained knowledge (e.g.", "from lab) can be combined with automatically extracted knowledge.", "We believe that our pathway specification language is easy for a person to understand and encode, lowering the bar to using our system.", "Our pathway specification language allows conditional actions, enabling the encoding of alternate action outcomes due to genetic variation.", "An important aspect of our pathway specification language is that it specifies trajectories, which includes specifying the initial configuration of substances, as well as state evolution style, such as maximal firing of actions, or serialized actions etc.", "Our query specification language provides a bridge between natural language questions and their formal representation.", "It is English-like but with precise semantics.", "A main feature of our query language is its support for comparative queries over alternate scenarios, which is not currently supported by any of the query languages (associated with action languages) we have reviewed.", "Our specification of alternate scenarios uses interventions (a general form of actions), that allow the user to modify the pathway as part of the query processing.", "We believe our query language is easier for a biologist to understand without requiring formal training.", "To model the pathways, we use Petri Nets, which have been used in the past to model and simulate biological pathways.", "Petri Nets have a simple visual representation, which closely matches biological pathways; and they inherently support parallelism.", "We extended the Petri Nets to add features that we needed to suit our domain, e.g., reset arcs that remove all quantity of a substance as soon as it is produced, and conditional arcs that specify the conditional outcome of an action.", "For simulation, we use ASP, which allowed us straight forward way to implement Petri Nets.", "It also gave us the ability to add extensions to the Petri Net by making local edits, implement different firing semantics, filter trajectories based on observations, and reason with the results.", "One of the major advantage of using Petri Net based simulation is the ability to generate all possible state evolutions, enabling us to process queries that determine the conditions when a certain observation becomes true.", "Our post-processing step is done in Python, which allows strong text processing capabilities using regular expressions, as well as libraries to easy process large matrices of numbers for summarization of results.", "Now we present additional challenges that need to be addressed.", "Pathway Extraction In Chapter we described how we extract facts for drug-drug interaction and gene variation.", "This work needs to be extended to include some of the newer databases that have come online recently.", "This may provide us with enzyme reaction rates, and substance quantities used in various reactions.", "The relation extraction for pathways must also be cognizant of any genetic variation mentioned in the text.", "Since the knowledge about the pathway appears in relationships at varying degree of detail, a process needs to be devised to assemble the pathway from the same level to granularity together, while also maintaining pathways at different levels of granularities.", "Since pathway extraction is a time consuming task, it would be best to create a catalog of the pathways.", "The cataloged pathways could be manually edited by the user as necessary.", "Storing pathways in this way means that would have to be updated periodically, requiring merging of new knowledge into existing pathways.", "Manual edits would have to be identified, such that the updated pathway does not overwrite them without the user's knowledge.", "Pathway Selection Questions presented in biological texts do not explicitly mention the relevant pathway to use for answering the question.", "One way to address this issue is to maintain a catalog of pre-defined pathways with keywords associated with them.", "Such keywords can include names of the substances, names of the processes, and other relevant meta-data about the pathway.", "The catalog can be searched to find the closest match to the query being asked.", "An additional aspect in proper pathway selection is to use the proper abstraction level.", "If our catalog contains a pathway at different abstraction levels, the coarsest pathway that contains the processes and substances in the query should be selected.", "Any higher fidelity will increase the processing time and generate too much irrelevant data.", "Alternatively, the catalog could contain the pathway in a hierarchical form, allowing extraction of all elements of a pathway at the same depth.", "A common way to hierarchically organize the pathway related to our system is to have hierarchical actions, which is the approach taken by hierarchical Petri nets.", "Lastly, the question may only ask about a small subsection of a much larger pathway.", "For better performance, it is beneficial to extract the smallest biological pathway network model that can answer the question.", "Pathway Modeling In Chapter , we presented our modeling of biological questions using Petri Nets and their extensions encoded in ASP.", "We came across concepts like allosteric regulation, inhibition of inhibition, and inhibition of activation that we currently do not model.", "In allosteric regulation, an enzyme is not fully enabled or disabled, the enzyme's shape changes, making it more or less active.", "The level of its activity depends upon concentrations of activators and inhibitors.", "In inhibition of inhibition, the inhibition of a reaction is itself inhibited by another inhibition; while in inhibition of activation (or stimulation), a substance inhibits the stimulation produced by a substance.", "Both of these appear to be actions on actions, something that Petri Nets do not allow.", "An alternate coding for these would have to be devised.", "As more detailed information about pathways becomes available, the reactions and processes that we have in current pathways may get replaced with more detailed sub-pathways themselves.", "However, such refinement may not come at the same time for separate legs of the pathway.", "Just replacing the coarse transition with a refined transition may not be sufficient due to relative timing constraints.", "Hence, a hierarchical Petri Net model may need to be implemented (see , ).", "Pathway Simulation In Chapter we presented our approach to encode Petri Nets and their extensions.", "We used a discrete solver called clingo for our ASP encoding.", "As the number of simulation length increases in size or larger quantities are involved, the solver slows down significantly.", "This is due to an increased grounding cost of the program.", "Incremental solving (using iclingo) does not help, since the program size still increases, and the increments merely delays the slow down but does not stop it.", "Systems such as constraint logic solvers (such as ) could be used for discrete cases.", "Alternatively, a system developed on the ASPMT [53] approach could be used, since it can represent longer runs, larger quantities, and real number quantities.", "Extend High Level Language In Chapter we described the BioPathQA system, the pathway specification and the query specification high level languages.", "As we enhance the modeling of the biological pathways, we will need to improve or extend the system as well as the high level language.", "We give a few examples of such extensions.", "Our pathway specification language currently does not support continuous quantities (real numbers).", "Extending to real numbers will improve the coverage of the pathways that can be modeled.", "In addition, derived quantities (fluents) can be added, e.g.", "pH could be defined as a formula that is read-only in nature.", "Certain observations and queries can be easily specified using a language such as LTL, especially for questions requiring conditions leading to an action or a state.", "As a result, it may be useful to add LTL formulas to the query language.", "We did not take this approach because it would have presented an additional non-English style syntax for the biologists.", "Our substance production / consumption rates and amounts are currently tied to the fluents.", "In certain situations it is desirable to analyze the quantity of a substance produced / consumed by a specific action, e.g.", "one is interested in finding the amount of H+ ions produced by a multi-protein complex IV only.", "Interventions (that are a part of the query statement) presented in this thesis are applied at the start of the simulation.", "Eliminating this restriction would allow applying surgeries to the pathway mid execution.", "Thus, instead of specifying the steady state conditions in the query statement, one could apply the intervention when a certain steady state is reached.", "Result Formatting and Visualization In Chapter we described our system that answers questions specified in our high level language.", "At the end of its process, it outputs the final result.", "This output method can be enhanced by allowing to look at the progression of results in addition to the final result.", "This provides the biologist with the whole spread of possible outcomes.", "An example of such a spread is shown in Fig.", "fig:q1:result for question REF .", "A graphical representation of the simulation progression is also beneficial in enhancing the confidence of the biologist.", "Indeed many existing tools do this.", "A similar effect can be achieved by parsing and showing the relevant portion of the answer set.", "Summary In Chapter we introduced the thesis topic and summarized specific research contributions In Chapter we introduced the foundational material of this thesis including Petri Nets and ASP.", "We showed how ASP could be used to encode basic Petri Nets.", "We also showed how ASP's elaboration tolerance and declarative syntax allows us to encode various Petri Net extensions with small localized changes.", "We also introduced a new firing semantics, the so called maximal firing set semantics to simulate a Petri Net with maximum parallel activity.", "In Chapter we showed how the Petri Net extensions and the ASP encoding can be used to answer simulation based deep reasoning questions.", "This and the work in Chapter was published in [1], [2].", "In Chapter we developed a system called BioPathQA to allow users to specify a pathway and answer queries against it.", "We also developed a pathway specification language and a query language for this system in order to avoid the vagaries of natural language.", "We introduced a new type of Guarded-arc Petri Nets to model conditional actions as a model for pathway simulation.", "We also described our implementation developed around a subset of the pathway specification language.", "In Chapter we briefly described how text extraction is done to extract real world knowledge about pathways and drug interactions.", "We then used the extracted knowledge to answer question using BioPathQA.", "The text extraction work was published in [73], [72], [33].", "Proofs of Various Propositions Assumption: The definitions in this section assume the programs $\\Pi $ do not have recursion through aggregate atoms.", "Our ASP translation ensures this due to the construction of programs $\\Pi $ .", "First we extend some definitions and properties related to ASP, such that they apply to rules with aggregate atoms.", "We will refer to the non-aggregate atoms as basic atoms.", "Recall the definitions of an ASP program given in section REF .", "Proposition 9 (Forced Atom Proposition) Let $S$ be an answer set of a ground ASP program $\\Pi $ as defined in definition REF .", "For any ground instance of a rule R in $\\Pi $ of the form $A_0 \\text{:-} A_1,\\dots ,$ $A_m,\\mathbf {not~} B_{1},\\dots ,$ $\\mathbf {not~} B_n, C_1,\\dots ,$ $C_k.$ if $\\forall A_i, 1 \\le i \\le m, S \\models A_i$ , and $\\forall B_j, 1 \\le j \\le n, S \\lnot \\models B_j$ , $\\forall C_l, 1 \\le l \\le k, S \\models C_l$ then $S \\models A_0$ .", "Proof: Let $S$ be an answer set of a ground ASP program $\\Pi $ , $R \\in \\Pi $ be a ground rule such that $\\forall B_j, 1 \\le j \\le n, S \\lnot \\models B_j$ ; and $\\forall C_l, 1 \\le l \\le k, S \\models C_l$ .", "Then, the reduct $R^S \\equiv \\lbrace p_1 \\text{:-} A_1,\\dots ,A_m.", "; \\dots ; p_h \\text{:-} A_1,\\dots ,A_m.", "\\; \| \\; \\lbrace p_1,\\dots ,p_h \\rbrace = S \\cap lit(A_0) \\rbrace $ .", "Since $S$ is an answer set of $\\Pi $ , it is a model of $\\Pi ^S$ .", "As a result, whenever, $\\forall A_i, 1 \\le i \\le m, S \\models A_i$ , $\\lbrace p_1,\\dots ,p_h \\rbrace \\subseteq S$ and $S \\models A_0$ .", "Proposition 10 (Supporting Rule Proposition) If $S$ is an answer set of a ground ASP program $\\Pi $ as defined in definition REF then $S$ is supported by $\\Pi $ .", "That is, if $S \\models A_0$ , then there exists a ground instance of a rule R in $\\Pi $ of the type $A_0 \\text{:-} A_1,\\dots ,$ $A_m,$ $\\mathbf {not~} B_{1},\\dots ,$ $\\mathbf {not~} B_n, $ $C_1,\\dots ,C_k.$ such that $\\forall A_i, 1 \\le i \\le m, S \\models A_i$ , $\\forall B_j, 1 \\le j \\le n, S \\lnot \\models B_j$ , and $\\forall C_l, 1\\le l \\le k, S \\models C_l$ .", "Proof: For $S$ to be an answer set of $\\Pi $ , it must be the deductive closure of reduct $\\Pi ^S$ .", "The deductive closure $S$ of $\\Pi ^S$ is iteratively built by starting from an empty set $S$ , and adding head atoms of rules $R_h^S \\equiv p_h \\text{:-} A_1,\\dots ,A_m., R_h^S \\in \\Pi ^S$ , whenever, $S \\models A_i, 1 \\le i \\le m$ , where, $R_h^S$ is a rule in the reduct of ground rule $R \\in \\Pi $ with $p_h \\in lit(A_0) \\cap S$ .", "Thus, there is a rule $R \\equiv A_0 \\text{:-} A_1,\\dots ,$ $A_m,$ $\\mathbf {not~} B_{1},\\dots ,$ $\\mathbf {not~}B_n, $ $C_1,\\dots ,$ $C_k.$ , $R \\in \\Pi $ , such that $\\forall C_l, 1 \\le l \\le k$ and $\\forall B_j, 1 \\le j \\le n, S \\lnot \\models B_j$ .", "Nothing else belongs in $S$ .", "Next, we extend the splitting set theorem to include aggregate atoms.", "Definition 88 (Splitting Set) A Splitting Set for a program $\\Pi $ is any set $U$ of literals such that, for every rule $R \\in \\Pi $ , if $U \\models head(R)$ then $lit(R) \\subset U$ .", "The set $U$ splits $\\Pi $ into upper and lower parts.", "The set of rules $R \\in \\Pi $ s.t.", "$lit(R) \\subset U$ is called the bottom of $\\Pi $ w.r.t.", "$U$ , denoted by $bot_U(\\Pi )$ .", "The rest of the rules, i.e.", "$\\Pi \\setminus bot_U(\\Pi )$ is called the top of $\\Pi $ w.r.t.", "$U$ , denoted by $top_U(\\Pi )$ .", "Proposition 11 Let $U$ be a splitting set of $\\Pi $ with answer set $S$ and let $X = S \\cap U$ and $Y = S \\setminus U$ .", "Then, the reduct of $\\Pi $ w.r.t.", "$S$ , i.e.", "$\\Pi ^S$ is equal to $bot_U(\\Pi )^X \\cup (\\Pi \\setminus bot_U(\\Pi ))^{X \\cup Y}$ .", "Proof: We can rewrite $\\Pi $ as $bot_U(\\Pi ) \\cup (\\Pi \\setminus bot_U(\\Pi ))$ using the definition of splitting set.", "Then the reduct of $\\Pi $ w.r.t.", "$S$ can be written in terms of $X$ and $Y$ , since $S = X \\cup Y$ .", "$\\Pi ^S =$ $\\Pi ^{X \\cup Y} =$ $(bot_U(\\Pi ) \\cup (\\Pi \\setminus bot_U(\\Pi )))^{X \\cup Y} =$ $bot_U(\\Pi )^{X \\cup Y} \\cup $ $(\\Pi \\setminus bot_U(\\Pi ))^{X \\cup Y}$ .", "Since $lit(bot_U(\\Pi )) \\subseteq U$ and $Y \\cap U = \\emptyset $ , the reduct of $bot_U(\\Pi )^{X \\cup Y} = bot_U(\\Pi )^X$ .", "Thus, $\\Pi ^{X \\cup Y} = bot_U(\\Pi )^X \\cup (\\Pi \\setminus bot_U(\\Pi ))^{X \\cup Y}$ .", "Proposition 12 Let $S$ be an answer set of a program $\\Pi $ , then $S \\subseteq lit(head(\\Pi ))$ .", "Proof: If $S$ is an answer set of a program $\\Pi $ then $S$ is produced by the deductive closure of $\\Pi ^S$ (the reduct of $\\Pi $ w.r.t $S$ ).", "By definition of the deductive closure, nothing can be in $S$ unless it is the head of some rule supported by $S$ .", "Splitting allow computing the answer set of a program $\\Pi $ in layers.", "Answer sets of the bottom layer are first used to partially evaluate the top layer, and then answer sets of the top layer are computed.", "Next, we define how a program is partially evaluated.", "Intuitively, the partial evaluation of an aggregate atom $c$ given splitting set $U$ w.r.t.", "a set of literals $X$ removes all literals that are part of the splitting set $U$ from $c$ and updates $c$ 's lower and upper bounds based on the literals in $X$ , which usually come from $bot_U$ of a program.", "The set $X$ represents our knowledge about the positive literals, while the set $U \\setminus X$ represents our knowledge about naf-literals at this stage.", "We can remove all literals in $U$ from $c$ , since the literals in $U$ will not appear in the head of any rule in $top_U$ .", "Definition 89 (Partial Evaluation of Aggregate Atom) The partial evaluation of an aggregate atom $c = l \\; [ B_0=w_0,\\dots , B_m=w_m ] \\; u$ , given splitting set $U$ w.r.t.", "a set of literals $X$ , written $eval_U(c,X)$ is a new aggregate atom $c^{\\prime }$ constructed from $c$ as follows: $pos(c^{\\prime }) = pos(c) \\setminus U$ $d=\\sum _{B_i \\in pos(c) \\cap U \\cap X}{w_i} $ $l^{\\prime } = l-d$ , $u^{\\prime } = u-d$ are the lower and upper limits of $c^{\\prime }$ Next, we define how a program is partially evaluated given a splitting set $U$ w.r.t.", "a set of literals $X$ that form the answer-set of the lower layer.", "Intuitively, a partial evaluation deletes all rules from the partial evaluation for which the body of the rule is determined to be not supported by $U$ w.r.t.", "$X$ .", "This includes rules which have an aggregate atom $c$ in their body s.t.", "$lit(c) \\subseteq U$ , but $X \\lnot \\models c$ Note that we can fully evaluate an aggregate atom $c$ w.r.t.", "answer-set $X$ if $lit(c) \\subseteq U$ ..", "In the remaining rules, the positive and negative literals that overlap with $U$ are deleted, and so are the aggregate atoms that have $lit(c) \\subseteq U$ (since such a $c$ can be fully evaluated w.r.t.", "$X$ ).", "Each remaining aggregate atom is updated by removing atoms that belong to $U$ Since the atoms in $U$ will not appear in the head of any atoms in $top_U$ and hence will not form a basis in future evaluations of $c$ ., and updating its limits based on the answer-set $X$ The limit update requires knowledge of the current answer-set to update limit values..", "The head atom is not modified, since $eval_U(...)$ is performed on $\\Pi \\setminus bot_U(\\Pi )$ , which already removes all rules with heads atoms that intersect $U$ .", "Definition 90 (Partial Evaluation) The partial evaluation of $\\Pi $ , given splitting set $U$ w.r.t.", "a set of literals $X$ is the program $eval_U(\\Pi ,X)$ composed of rules $R^{\\prime }$ for each $R \\in \\Pi $ that satisfies all the following conditions: $pos(R) \\cap U \\subseteq X,$ $((neg(R) \\cap U) \\cap X) = \\emptyset , \\text{ and }$ if there is a $c \\in agg(R)$ s.t.", "$lit(c) \\subseteq U$ , then $X \\models c$ A new rule $R^{\\prime }$ is constructed from a rule $R$ as follows: $head(R^{\\prime }) = head(R)$ , $pos(R^{\\prime }) = pos(R) \\setminus U$ , $neg(R^{\\prime }) = neg(R) \\setminus U$ , $agg(R^{\\prime }) = \\lbrace eval_U(c,X) : c \\in agg(R), lit(c) \\lnot \\subseteq U \\rbrace $ Proposition 13 Let $U$ be a splitting set for $\\Pi $ , $X$ be an answer set of $bot_U(\\Pi )$ , and $Y$ be an answer set of $eval_U(\\Pi \\setminus bot_U(\\Pi ),X)$ .", "Then, $X \\subseteq lit(\\Pi ) \\cap U$ and $Y \\subseteq lit(\\Pi ) \\setminus U$ .", "Proof: By proposition REF , $X \\subseteq lit(head(bot_U(\\Pi )))$ , and $Y \\subseteq lit(head(eval_U(\\Pi \\setminus bot_U(\\Pi ),X)))$ .", "In addition, $lit(head(bot_U(\\Pi ))) \\subseteq lit(bot_U(\\Pi ))$ and $lit(bot_U(\\Pi )) \\subseteq lit(\\Pi ) \\cap U$ by definition of $bot_U(\\Pi )$ .", "Then $X \\subseteq lit(\\Pi ) \\cap U$ , and $Y \\subseteq lit(\\Pi ) \\setminus U$ .", "Definition 91 (Solution) [4] Let $U$ be a splitting set for a program $\\Pi $ .", "A solution to $\\Pi $ w.r.t.", "$U$ is a pair $\\langle X,Y \\rangle $ of literals such that: $X$ is an answer set for $bot_U(\\Pi )$ $Y$ is an answer set for $eval_U(top_U(\\Pi ),X)$ ; and $X \\cup Y$ is consistent.", "Proposition 14 (Splitting Theorem) [4] Let $U$ be a splitting set for a program $\\Pi $ .", "A set $S$ of literals is a consistent answer set for $\\Pi $ iff $S = X \\cup Y$ for some solution $\\langle X,Y \\rangle $ of $\\Pi $ w.r.t.", "$U$ .", "Lemma 1 Let $U$ be a splitting set of $\\Pi $ , $C$ be an aggregate atom in $\\Pi $ , and $X$ and $Y$ be subsets of $lit(\\Pi )$ s.t.", "$X \\subseteq U$ , and $Y \\cap U = \\emptyset $ .", "Then, $X \\cup Y \\models C$ iff $Y \\models eval_U(C,X)$ .", "Proof: Let $C^{\\prime } = eval_U(C,X)$ , then by definition of partial evaluation of aggregate atom, $pos(C^{\\prime }) = pos(C) \\setminus U$ , with lower limit $l^{\\prime } = l-d$ , and upper limit $u^{\\prime } = u-d$ , computed from $l,u$ , the lower and upper limits of $C$ , where $d=\\displaystyle \\sum _{B_i \\in pos(C) \\cap U \\cap X}{w_i}$ $Y \\models C^{\\prime }$ iff $l^{\\prime } \\le \\left( \\displaystyle \\sum _{B^{\\prime }_i \\in pos(C^{\\prime }) \\cap Y}{w^{\\prime }_i} \\right) \\le u^{\\prime }$ – by definition of aggregate atom satisfaction.", "then $Y \\models C$ iff $l \\le \\left( \\displaystyle \\sum _{B_i \\in pos(C) \\cap U \\cap X}{w_i} +\\displaystyle \\sum _{B^{\\prime }_i \\in (pos(C) \\setminus U) \\cap Y}{w^{\\prime }_i} \\right) \\le u$ however, $(pos(C) \\cap U) \\cap X$ and $(pos(C) \\setminus U) \\cap Y$ combined represent $pos(C) \\cap (X \\cup Y)$ – since $pos(C) \\cap (X \\cup Y) &= ((pos(C) \\cap U) \\cup (pos(C) \\setminus U)) \\cap (X \\cup Y) \\\\&= [((pos(C) \\cap U) \\cup (pos(C) \\setminus U)) \\cap X] \\\\&~~~~~~\\cup [((pos(C) \\cap U) \\cup (pos(C) \\setminus U)) \\cap Y]\\\\&= [(pos(C) \\cap U) \\cap X) \\cup ((pos(C) \\setminus U) \\cap X)]\\\\&~~~~~~\\cup [(pos(C) \\cap U) \\cap Y) \\cup ((pos(C) \\setminus U) \\cap Y)]\\\\&= [((pos(C) \\cap U) \\cap X) \\cup \\emptyset ] \\cup [\\emptyset \\cup ((pos(C) \\setminus U) \\cap Y)]\\\\&= ((pos(C) \\cap U) \\cap X) \\cup ((pos(C) \\setminus U) \\cap Y)$ where $X \\subseteq U \\text{ and } Y \\cap U = \\emptyset $ thus, $Y \\models C$ iff $l \\le \\left( \\displaystyle \\sum _{B_i \\in pos(C) \\cap (X \\cup Y)}{w_i} \\right) \\le u$ which is the same as $X \\cup Y \\models C$ Lemma 2 Let $U$ be a splitting set for $\\Pi $ , and $X, Y$ be subsets of $lit(\\Pi )$ s.t.", "$X \\subseteq U$ and $Y \\cap U = \\emptyset $ .", "Then the body of a rule $R^{\\prime } \\in eval_U(\\Pi ,X)$ is satisfied by $Y$ iff the body of the rule $R \\in \\Pi $ it was constructed from is satisfied by $X \\cup Y$ .", "Proof: $Y$ satisfies $body(R^{\\prime })$ iff $pos(R^{\\prime }) \\subseteq Y$ , $neg(R^{\\prime }) \\cap Y = \\emptyset $ , $Y \\models C^{\\prime }$ for each $C^{\\prime } \\in agg(R^{\\prime })$ – by definition of rule satisfaction iff $(pos(R) \\cap U) \\subseteq X$ , $(pos(R) \\setminus U) \\subseteq Y$ , $(neg(R) \\cap U) \\cap X) = \\emptyset $ , $(neg(R) \\setminus U) \\cap Y) = \\emptyset $ , $X$ satisfies $C$ for all $C \\in agg(C)$ in which $lit(C) \\subseteq U$ , and $Y$ satisfies $eval_U(C,X)$ for all $ C \\in agg(C)$ in which $lit(C) \\lnot \\subseteq U$ – using definition of partial evaluation iff $pos(R) \\subseteq X \\cup Y$ , $neg(R) \\cap (X \\cup Y) = \\emptyset $ , $X \\cup Y \\models C$ – using $(A \\cap U) \\cup (A \\setminus U) = A$ $A \\cap (X \\cup Y) = ((A \\cap U) \\cup (A \\setminus U)) \\cap (X \\cup Y) = ((A \\cap U) \\cap (X \\cup Y)) \\cup ((A \\setminus U) \\cap (X \\cup Y)) = (A \\cap U) \\cap X) \\cup ((A \\setminus U) \\cap Y)$ – given $X \\subseteq U$ and $Y \\cap U = \\emptyset $ .", "and lemma REF Proof of Splitting Theorem: Let $U$ be a splitting set of $\\Pi $ , then a consistent set of literals $S$ is an answer set of $\\Pi $ iff it can be written as $S = X \\cup Y$ , where $X$ is an answer set of $bot_U(\\Pi )$ ; and $Y$ is an answer set of $eval_U(\\Pi \\setminus bot_U(\\Pi ),Y)$ .", "($\\Leftarrow $ ) Let $X$ is an answer set of $bot_U(\\Pi )$ ; and $Y$ is an answer set of $eval_U(\\Pi \\setminus bot_U(\\Pi ),X)$ ; we show that $X \\cup Y$ is an answer set of $\\Pi $ .", "By definition of $bot_U(\\Pi )$ , $lit(bot_U(\\Pi )) \\subseteq U$ .", "In addition, by proposition REF , $Y \\cap U = \\emptyset $ .", "Then, $\\Pi ^{X \\cup Y} = (bot_U(\\Pi ) \\cup (\\Pi \\setminus bot_U(\\Pi )))^{X \\cup Y} = bot_U(\\Pi )^{X \\cup Y} \\cup (\\Pi \\setminus bot_U(\\Pi ))^{X \\cup Y} = bot_U(\\Pi )^X \\cup (\\Pi \\setminus bot_U(\\Pi ))^{X \\cup Y}$ .", "Let $r$ be a rule in $\\Pi ^{X \\cup Y}$ , s.t.", "$X \\cup Y \\models body(r)$ then we show that $X \\cup Y \\models head(r)$ .", "The rule $r$ either belongs to $bot_U(\\Pi )^X$ or $(\\Pi \\setminus bot_U(\\Pi ))^{X \\cup Y}$ .", "Case 1: say $r \\in bot_U(\\Pi )^X$ be a rule whose body is satisfied by $X \\cup Y$ then there is a rule $R \\in bot_U(\\Pi )$ s.t.", "$r \\in R^X$ then $X \\models body(R)$ – since $X \\cup Y \\models body(r)$ ; $lit(bot_U(\\Pi )) \\subseteq U$ and $Y \\cap U = \\emptyset $ we already have $X \\models head(R)$ – since $X$ is an answer set of $bot_U(\\Pi )$ ; given then $X \\cup Y \\models head(R)$ – because $lit(R) \\subseteq U$ and $Y \\cap U = \\emptyset $ consequently, $X \\cup Y \\models head(r)$ Case 2: say $r \\in (\\Pi \\setminus bot_U(\\Pi ))^{X \\cup Y}$ be a rule whose body is satisfied by $X \\cup Y$ then there is a rule $R \\in (\\Pi \\setminus bot_U(\\Pi ))$ s.t.", "$r \\in R^{X \\cup Y}$ then $lit(head(R)) \\cap U = \\emptyset $ – otherwise, $R$ would have belonged to $bot_U(\\Pi )$ , by definition of splitting set then $head(r) \\in Y$ – since $X \\subseteq U$ in addition, $pos(R) \\subseteq X \\cup Y$ , $neg(R) \\cap (X \\cup Y) = \\emptyset $ , $X \\cup Y \\models C$ for each $C \\in agg(R)$ – using definition of reduct then $pos(R) \\cap U \\subseteq X$ or $pos(R) \\setminus U \\subseteq Y$ ; $(neg(R) \\cap U) \\cap X = \\emptyset $ and $(neg(R) \\setminus U) \\cap Y = \\emptyset $ ; and for each $C \\in agg(R)$ , either $X \\models C$ if $lit(C) \\subseteq U$ , or $Y \\models eval_U(C,X)$ if $lit(C) \\lnot \\subseteq U$ – by rearranging, lemma REF , $X \\subseteq U$ , $Y \\cap U = \\emptyset $ , and definition of partial evaluation of an aggregate atom note that $pos(R) \\cap U \\subseteq X$ , $(neg(R) \\cap U) \\cap X = \\emptyset $ , and for each $C \\in agg(R)$ , s.t.", "$lit(C) \\subseteq U$ , $X \\models C$ , represent conditions satisfied by each rule that become part of a partial evaluation – using definition of partial evaluation and $pos(R) \\setminus U$ , $neg(R) \\setminus U$ , and for each $C \\in agg(R)$ , $eval_U(C,X)$ are the modifications made to the rule during partial evaluation given splitting set $U$ w.r.t.", "$X$ – using definition of partial evaluation and $pos(R) \\setminus U \\subseteq Y$ , $(neg(R) \\setminus U) \\cap Y = \\emptyset $ , and for each $C \\in agg(R)$ , $Y \\models eval_U(C,X)$ if $lit(C) \\lnot \\subseteq U$ represent conditions satisfied by rules that become part of the reduct w.r.t $Y$ – using definition of partial evaluation and reduct then $r$ is a rule in reduct $eval_U(\\Pi \\setminus bot_U(\\Pi ),X)^Y$ – using (REF ), (REF ) above in addition, given that $Y$ satisfies $eval_U(\\Pi \\setminus bot_U(\\Pi ),X)$ , and $head(r) \\cap U = \\emptyset $ , we have $X \\cup Y \\models head(r)$ Next we show that $X \\cup Y$ satisfies all rules of $\\Pi $ .", "Say, $R$ is a rule in $\\Pi $ not satisfied by $X \\cup Y$ .", "Then, either it belongs to $bot_U(\\Pi )$ or $(\\Pi \\setminus bot_U(\\Pi ))$ .", "If it belongs to $bot_U(\\Pi )$ , it must not be satisfied by $X$ , since $lit(bot_U(\\Pi )) \\subseteq U$ and $Y \\cap U = \\emptyset $ .", "However, the contrary is given to be true.", "On the other hand if it belongs to $(\\Pi \\setminus bot_U(\\Pi ))$ , then $X \\cup Y$ satisfies $body(R)$ but not $head(R)$ .", "That would mean that its $head(R)$ is not satisfied by $Y$ , since $head(R) \\cap U = \\emptyset $ by definition of splitting set.", "However, from lemma REF we know that if $body(R)$ is satisfied by $X \\cup Y$ , $body(R^{\\prime })$ is satisfied by $Y$ for $R^{\\prime } \\in eval_U(\\Pi \\setminus bot_U(\\Pi ),X)$ .", "We also know that $Y$ satisfies all rules in $eval_U(\\Pi \\setminus bot_U(\\Pi ),X)$ .", "So, $R^{\\prime }$ must be satisfied by $Y$ contradicting our assumption.", "Thus, all rules of $\\Pi $ are satisfied by $X \\cup Y$ and $X \\cup Y$ is an answer set of $\\Pi $ .", "($\\Rightarrow $ ) Let $S$ be a consistent answer set of $\\Pi $ , we show that $S = X \\cup Y$ for sets $X$ and $Y$ s.t.", "$X$ is an answer set of $bot_U(\\Pi )$ and $Y$ is an answer set of $eval_U(\\Pi \\setminus bot_U(\\Pi ),X)$ .", "We take $X=S \\cap U$ , $Y=S \\setminus U$ , then $S=X \\cup Y$ .", "Case 1: We show that $X$ is answer set of $bot_U(\\Pi )$ $\\Pi $ can be split into $bot_U(\\Pi ) \\cup (\\Pi \\setminus bot_U(\\Pi ))$ – by definition of splitting then $X \\cup Y$ satisfies $bot_U(\\Pi )$ – $X \\cup Y$ is an answer set of $\\Pi $ ; given however $lit(bot_U(\\Pi )) \\subseteq U$ , $Y \\cap U = \\emptyset $ – by definition of splitting then $X$ satisfies $bot_U(\\Pi )$ – since elements of $Y$ do not appear in the rules of $bot_U(\\Pi )$ then $X$ is an answer set of $bot_U(\\Pi )$ Case 2: We show that $Y$ is answer set of $eval_U(\\Pi \\setminus bot_U(\\Pi ),X)$ let $r$ be a rule in $eval_U(\\Pi \\setminus bot_U(\\Pi ),X)^Y$ , s.t.", "its body is satisfied by $Y$ then $r \\in R^Y$ for an $R \\in eval_U(\\Pi \\setminus bot_U(\\Pi ),X)$ s.t.", "[(i)] (3) $pos(R) \\subseteq Y$ (4) $neg(R) \\cap Y = \\emptyset $ (5) $Y \\models C$ for all $C \\in agg(R)$ (6) $head(R) \\cap Y \\ne \\emptyset $ – using definition of reduct each $R$ is constructed from $R^{\\prime } \\in \\Pi $ that satisfies all the following conditions [(i)] (8) $pos(R^{\\prime }) \\subseteq U \\cap X$ (9) $(neg(R^{\\prime }) \\cap U) \\cap X = \\emptyset $ (10) if there is a $C^{\\prime } \\in agg(R^{\\prime })$ s.t.", "$lit(C^{\\prime }) \\subseteq U$ , then $X \\models C^{\\prime }$ ; and each $C \\in agg(R)$ is a partial evaluation of $C^{\\prime } \\in agg(R^{\\prime })$ s.t.", "$C = eval_U(C^{\\prime },X)$ – using definition of partial evaluation then the $body(R^{\\prime })$ satisfies all the following conditions: $pos(R^{\\prime }) \\subseteq X \\cup Y$ – since $X \\subseteq U$ , $X \\cap Y = \\emptyset $ $neg(R^{\\prime }) \\cap (X \\cup Y) = \\emptyset $ – since $X \\subseteq U$ , $X \\cap Y = \\emptyset $ $X \\cup Y \\models C^{\\prime }$ for each $C^{\\prime } \\in agg(R^{\\prime })$ – since [(i)] (d) each $C^{\\prime } \\in agg(R^{\\prime })$ with $lit(C^{\\prime }) \\subseteq U$ satisfied by $X$ is also satisfied by $X \\cup Y$ as $lit(Y) \\cap lit(C^{\\prime }) = \\emptyset $ ; and (e) each $C^{\\prime } \\in agg(R^{\\prime })$ with $lit(C^{\\prime }) \\lnot \\subseteq U$ is satisfied by $X \\cup Y$ – using partial evaluation, reduct construction, and $X \\cap Y = \\emptyset $ then $X \\cup Y$ satisfies $body(R^{\\prime })$ – from previous line in addition, $lit(head(R^{\\prime })) \\cap U = \\emptyset $ , otherwise, $R^{\\prime }$ would have belonged to $bot_U(\\Pi )$ by definition of splitting set then $R^{\\prime }$ is a rule in $\\Pi \\setminus bot_U(\\Pi )$ – from the last two lines we know that $X \\cup Y$ satisfies every rule in $(\\Pi \\setminus bot_U(\\Pi ))$ – given; and that elements of $U$ do not appear in the head of rules in $(\\Pi \\setminus bot_U(\\Pi ))$ – from definition of splitting; then $Y$ must satisfy the head of these rules then $Y$ satisfies $head(R^{\\prime })$ – from (REF ) Next we show that $Y$ satisfies all rules of $eval_U(\\Pi \\setminus bot_U(\\Pi ),X)$ .", "Let $R^{\\prime }$ be a rule in $eval_U(\\Pi \\setminus bot_U(\\Pi ),X)$ such that $body(R^{\\prime })$ is satisfied by $Y$ but not $head(R^{\\prime })$ .", "Since $head(R^{\\prime }) \\cap Y = \\emptyset $ , $head(R^{\\prime })$ is not satisfied by $X \\cup Y$ either.", "Then, there is an $R \\in (\\Pi \\setminus bot_U(\\Pi ))$ such that $X \\cup Y$ satisfies $body(R)$ but not $head(R)$ , which contradicts given.", "Thus, $Y$ satisfies all rules of $eval_U(\\Pi \\setminus bot_U(\\Pi ),X)$ .", "Then $Y$ is an answer set of $eval_U(\\Pi \\setminus bot_U(\\Pi ),X)$ Definition 92 (Splitting Sequence) [4] A splitting sequence for a program $\\Pi $ is a monotone, continuous sequence ${\\langle U_{\\alpha } \\rangle }_{\\alpha < \\mu }$ of splitting sets of $\\Pi $ such that $\\bigcup _{\\alpha < \\mu }{U_{\\mu }} = lit(\\Pi )$ .", "Definition 93 (Solution) [4] Let $U={\\langle U_{\\alpha } \\rangle }_{\\alpha < \\mu }$ be a splitting sequence for a program $\\Pi $ .", "A solution to $\\Pi $ w.r.t $U$ is a sequence ${\\langle X_{\\alpha } \\rangle }_{\\alpha < \\mu }$ of sets of literals such that: $X_0$ is an answer set for $bot_{U_0}(\\Pi )$ for any ordinal $\\alpha + 1 < \\mu $ , $X_{\\alpha +1}$ is an answer set of the program: $eval_{U_{\\alpha }}(bot_{U_{\\alpha +1}}(\\Pi ) \\setminus bot_{U_{\\alpha }}(\\Pi ), \\bigcup _{\\nu \\le \\alpha }{X_{\\nu }})$ for any limit ordinal $\\alpha < \\mu , X_{\\alpha } = \\emptyset $ , and $\\bigcup _{\\alpha \\le \\mu }(X_{\\alpha })$ is consistent Proposition 15 (Splitting Sequence Theorem) [4] Let $U={\\langle U_{\\alpha } \\rangle }_{\\alpha < \\mu }$ be a splitting sequence for a program $\\Pi $ .", "A set $S$ of literals is a consistent answer set for $\\Pi $ iff $S=\\bigcup _{\\alpha < \\mu }{X_{\\alpha }}$ for some solution ${\\langle X_{\\alpha } \\rangle }_{\\alpha < \\mu }$ to $\\Pi $ w.r.t $U$ .", "Proof: Let $U = \\langle U_\\alpha \\rangle _{\\alpha < \\mu }$ be a splitting sequence of $\\Pi $ , then a consistent set of literals $S=\\bigcup _{\\alpha < \\mu }{X_{\\alpha }}$ is an answer set of $\\Pi ^S$ iff $X_0$ is an answer set of $bot_{U_0}(\\Pi )$ and for any ordinal $\\alpha + 1 < \\mu $ , $X_{\\alpha +1}$ is an answer set of $eval_{U_{\\alpha }}(bot_{U_{\\alpha +1}}(\\Pi ) \\setminus bot_{U_{\\alpha }}(\\Pi ), \\bigcup _{\\nu \\le \\alpha }X_{\\nu })$ .", "Note that every literal in $bot_{U_0}(\\Pi )$ belongs to $lit(\\Pi ) \\cap U_0$ , and every literal occurring in $eval_{U_{\\alpha }}(bot_{U_{\\alpha +1}}(\\Pi ) \\setminus bot_{U_{\\alpha }}(\\Pi ), \\bigcup _{\\nu \\le \\alpha }X_{\\nu })$ , $(\\alpha + 1 < \\mu )$ belongs to $lit(\\Pi ) \\cap (U_{\\alpha +1} \\setminus U_{\\alpha })$ .", "In addition, $X_0$ , and all $X_{\\alpha +1}$ are pairwise disjoint.", "We prove the theorem by induction over the splitting sequence.", "Base case: $\\alpha = 1$ .", "The splitting sequence is $U_0 \\subseteq U_1$ .", "Then the sub-program $\\Pi _1 = bot_{U_1}(\\Pi )$ contains all literals in $U_1$ ; and $U_0$ splits $\\Pi _1$ into $bot_{U_0}(\\Pi _1)$ and $bot_{U_1}(\\Pi _1) \\setminus bot_{U_0}(\\Pi _1)$ .", "Then, $S_1 = X_0 \\cup X_1$ is a consistent answer set of $\\Pi _1$ iff $X_0 = S_1 \\cap U_0$ is an answer set of $bot_{U_0}(\\Pi _1)$ and $X_1 = S_1 \\setminus U_0$ is an answer set of $eval_{U_0}(\\Pi _1 \\setminus bot_{U_0}(\\Pi _1),X_1)$ – by the splitting theorem Since $bot_{U_0}(\\Pi _1) = bot_{U_0}(\\Pi )$ and $bot_{U_1}(\\Pi _1) \\setminus bot_{U_0}(\\Pi _1) = bot_{U_1}(\\Pi ) \\setminus bot_{U_0}(\\Pi )$ ; $S_1 = X_0 \\cup X_1$ is an answer set for $\\Pi \\setminus bot_{U_1}(\\Pi )$ .", "Induction: Assume theorem holds for $\\alpha = k$ , show theorem holds for $\\alpha = k+1$ .", "The inductive assumption holds for the splitting sequence $U_0 \\subseteq \\dots \\subseteq U_k$ .", "Then the sub-program $\\Pi _k = bot_{U_k}(\\Pi )$ contains all literals in $U_k$ and $S_k = X_0 \\cup \\dots \\cup X_k$ is a consistent answer set of $\\Pi ^{S_k}$ iff $X_0$ is an answer set for $bot_{U_0}(\\Pi _k)$ and for any $\\alpha \\le k$ , $X_{\\alpha +1}$ is answer set of $eval_{U_{\\alpha }}(bot_{U_{\\alpha +1}}(\\Pi ) \\setminus bot_{U_{\\alpha }}(\\Pi ), X_0 \\cup \\dots \\cup X_{\\alpha })$ We show that the theorem holds for $\\alpha = k+1$ .", "The splitting sequence is $U_0 \\subseteq U_{k+1}$ .", "Then the sub-program $\\Pi _{k+1} = bot_{U_{k+1}}(\\Pi )$ contains all literals $U_{k+1}$ .", "We have $U_k$ split $\\Pi _{k+1}$ into $bot_{U_k}(\\Pi _{k+1})$ and $bot_{U_{k+1}}(\\Pi _{k+1}) \\setminus bot_{U_k}(\\Pi _{k+1})$ .", "Then, $S_{k+1} = X_{0:k} \\cup X_{k+1}$ is a consistent answer set of $\\Pi _{k+1}$ iff $X_{0:k} = S_{k+1} \\cap U_k$ is an answer set of $bot_{U_k}(\\Pi _{k+1})$ and $X_{k+1} = S_{k+1} \\setminus U_k$ is an answer set of $eval_{U_k}(\\Pi _{k+1} \\setminus bot_{U_k}(\\Pi _{k+1},X_{k+1})$ – by the splitting theorem Since $bot_{U_k}(\\Pi _{k+1}) = bot_{U_k}(\\Pi )$ and $bot_{U_{k+1}}(\\Pi _{k+1}) \\setminus bot_{U_k}(\\Pi _{k+1}) = bot_{U_{k+1}}(\\Pi ) \\setminus bot_{U_k}(\\Pi )$ ; $S_{k+1} = X_{0:k} \\cup X_1$ is an answer set for $\\Pi \\setminus bot_{U_{k+1}}(\\Pi )$ .", "From the inductive assumption we know that $X_0 \\cup \\dots \\cup X_k$ is a consistent answer set of $bot_{U_k}(\\Pi )$ , $X_0$ is the answer set of $bot_{U_0}(\\Pi )$ , and for each $0 \\le \\alpha \\le k$ , $X_{\\alpha +1}$ is answer set of $eval_{U_{\\alpha }}(bot_{U_{\\alpha +1}}(\\Pi ) \\setminus bot_{U_{\\alpha }}(\\Pi ), X_0 \\cup \\dots \\cup X_{\\alpha })$ .", "Thus, $X_{0:k} = X_0 \\cup \\dots \\cup X_k$ .", "Combining above with the inductive assumption, we get $S_{k+1} = X_0 \\cup \\dots \\cup X_{k+1}$ is a consistent answer set of $\\Pi ^{S_{k+1}}$ iff $X_0$ is an answer set for $bot_{U_0}(\\Pi _{k+1})$ and for any $\\alpha \\le k+1$ , $X_{\\alpha +1}$ is answer set of $eval_{U_{\\alpha }}(bot_{U_{\\alpha +1}}(\\Pi ) \\setminus bot_{U_{\\alpha }}(\\Pi ), X_0 \\cup \\dots \\cup X_{\\alpha })$ .", "In addition, for some $\\alpha < \\mu $ , where $\\mu $ is the length of the splitting sequence $U = \\langle U_{\\alpha } \\rangle _{\\alpha < \\mu }$ of $\\Pi $ , $bot_{U_{\\alpha }}(\\Pi )$ will be the entire $\\Pi $ , i.e.", "$lit(\\Pi ) = U_{\\alpha }$ .", "Then the set $S$ of literals is a consistent answer set of $\\Pi $ iff $S=\\bigcup _{\\alpha < \\mu }(X_{\\alpha })$ for some solution $\\langle X_{\\alpha } \\rangle _{\\alpha < \\mu }$ to $\\Pi $ w.r.t $U$ .", "Proof of Proposition REF Let $PN=(P,T,E,W)$ be a Petri Net, $M_0$ be its initial marking and let $\\Pi ^0(PN,M_0,$ $k,ntok)$ be the ASP encoding of $PN$ and $M_0$ over a simulation length $k$ , with maximum $ntok$ tokens on any place node, as defined in section REF .", "Then $X=M_0,T_0,M_1,\\dots ,M_k,T_k,M_{k+1}$ is an execution sequence of a $PN$ (w.r.t.", "$M_0$ ) iff there is an answer set $A$ of $\\Pi ^0(PN,M_0,k,ntok)$ such that: $\\lbrace fires(t,ts) : t \\in T_{ts}, 0 \\le ts \\le k \\rbrace = \\lbrace fires(t,ts) : fires(t,ts) \\in A \\rbrace $ $\\begin{split}\\lbrace holds(p,q,ts) &: p \\in P, q = M_{ts}(p), 0 \\le ts \\le k+1 \\rbrace \\\\&= \\lbrace holds(p,q,ts) : holds(p,q,ts) \\in A \\rbrace \\end{split}$ We prove this by showing that: Given an execution sequence $X$ , we create a set $A$ such that it satisfies (REF ) and (REF ) and show that $A$ is an answer set of $\\Pi ^0$ Given an answer set $A$ of $\\Pi ^0$ , we create an execution sequence $X$ such that (REF ) and (REF ) are satisfied First we show (REF ): Given $PN$ and its execution sequence $X$ , we create a set $A$ as a union of the following sets: $A_1=\\lbrace num(n) : 0 \\le n \\le ntok \\rbrace $ $A_2=\\lbrace time(ts) : 0 \\le ts \\le k\\rbrace $ $A_3=\\lbrace place(p) : p \\in P \\rbrace $ $A_4=\\lbrace trans(t) : t \\in T \\rbrace $ $A_5=\\lbrace ptarc(p,t,n) : (p,t) \\in E^-, n=W(p,t) \\rbrace $ , where $E^- \\subseteq E$ $A_6=\\lbrace tparc(t,p,n) : (t,p) \\in E^+, n=W(t,p) \\rbrace $ , where $E^+ \\subseteq E$ $A_7=\\lbrace holds(p,q,0) : p \\in P, q=M_{0}(p) \\rbrace $ $A_8=\\lbrace notenabled(t,ts) : t \\in T, 0 \\le ts \\le k, \\exists p \\in \\bullet t, M_{ts}(p) < W(p,t) \\rbrace $ per definition REF (enabled transition) $A_9=\\lbrace enabled(t,ts) : t \\in T, 0 \\le ts \\le k, \\forall p \\in \\bullet t, W(p,t) \\le M_{ts}(p) \\rbrace $ per definition REF (enabled transition) $A_{10}=\\lbrace fires(t,ts) : t \\in T_{ts}, 0 \\le ts \\le k \\rbrace $ from definition REF (firing set) , only an enabled transition may fire $A_{11}=\\lbrace add(p,q,t,ts) : t \\in T_{ts}, p \\in t \\bullet , q=W(t,p), 0 \\le ts \\le k \\rbrace $ per definition REF (transition execution) $A_{12}=\\lbrace del(p,q,t,ts) : t \\in T_{ts}, p \\in \\bullet t, q=W(p,t), 0 \\le ts \\le k \\rbrace $ per definition REF (transition execution) $A_{13}=\\lbrace tot\\_incr(p,q,ts) : p \\in P, q=\\sum _{t \\in T_{ts}, p \\in t \\bullet }{W(t,p)}, 0 \\le ts \\le k \\rbrace $ per definition REF (firing set execution) $A_{14}=\\lbrace tot\\_decr(p,q,ts): p \\in P, q=\\sum _{t \\in T_{ts}, p \\in \\bullet t}{W(p,t)}, 0 \\le ts \\le k \\rbrace $ per definition REF (firing set execution) $A_{15}=\\lbrace consumesmore(p,ts) : p \\in P, q=M_{ts}(p), q1=\\sum _{t \\in T_{ts}, p \\in \\bullet t}{W(p,t)}, q1 > q, 0 \\le ts \\le k \\rbrace $ per definition REF (conflicting transitions) for enabled transition set $T_{ts}$ $A_{16}=\\lbrace consumesmore : \\exists p \\in P : q=M_{ts}(p), q1=\\sum _{t \\in T_{ts}, p \\in \\bullet t}{W(p,t)}, q1 > q, 0 \\le ts \\le k \\rbrace $ per definition REF (conflicting transitions) $A_{17}=\\lbrace holds(p,q,ts+1) : p \\in P, q=M_{ts+1}(p), 0 \\le ts < k\\rbrace $ , where $M_{ts+1}(p) = M_{ts}(p) - \\sum _{\\begin{array}{c}t \\in T_{ts}, p \\in \\bullet t\\end{array}}{W(p,t)} + \\sum _{\\begin{array}{c}t \\in T_{ts}, p \\in t \\bullet \\end{array}}{W(t,p)}$ according to definition REF (firing set execution) We show that $A$ satisfies (REF ) and (REF ), and $A$ is an answer set of $\\Pi ^0$ .", "$A$ satisfies (REF ) and (REF ) by its construction (given above).", "We show $A$ is an answer set of $\\Pi ^0$ by splitting.", "We split $lit(\\Pi ^0)$ (literals of $\\Pi ^0$ ) into a sequence of $6(k+1)+2$ sets: $U_0= head(f\\ref {f:place}) \\cup head(f\\ref {f:trans}) \\cup head(f\\ref {f:ptarc}) \\cup head(f\\ref {f:tparc}) \\cup head(f\\ref {f:time}) \\cup head(f\\ref {f:num}) \\cup head(i\\ref {i:holds}) = \\lbrace place(p) : p \\in P\\rbrace \\cup \\lbrace trans(t) : t \\in T\\rbrace \\cup \\lbrace ptarc(p,t,n) : (p,t) \\in E^-, n=W(p,t) \\rbrace \\cup \\lbrace tparc(t,p,n) : (t,p) \\in E^+, n=W(t,p) \\rbrace \\cup $ $\\lbrace time(0), \\dots , $ $time(k)\\rbrace \\cup \\lbrace num(0), \\dots , num(ntok)\\rbrace \\cup \\lbrace holds(p,q,0) : p \\in P, q=M_0(p) \\rbrace $ $U_{6k+1}=U_{6k+0} \\cup head(e\\ref {e:ne:ptarc})^{ts=k} = U_{6k+0} \\cup \\lbrace notenabled(t,k) : t \\in T \\rbrace $ $U_{6k+2}=U_{6k+1} \\cup head(e\\ref {e:enabled})^{ts=k} = U_{6k+1} \\cup \\lbrace enabled(t,k) : t \\in T \\rbrace $ $U_{6k+3}=U_{6k+2} \\cup head(a\\ref {a:fires})^{ts=k} = U_{6k+2} \\cup \\lbrace fires(t,k) : t \\in T \\rbrace $ $U_{6k+4}=U_{6k+3} \\cup head(r\\ref {r:add})^{ts=k} \\cup head(r\\ref {r:del})^{ts=k} = U_{6k+3} \\cup \\lbrace add(p,q,t,k) : p \\in P, t \\in T, q=W(t,p) \\rbrace \\cup \\lbrace del(p,q,t,k) : p \\in P, t \\in T, q=W(p,t) \\rbrace $ $U_{6k+5}=U_{6k+4} \\cup head(r\\ref {r:totincr})^{ts=k} \\cup head(r\\ref {r:totdecr})^{ts=k} = U_{6k+4} \\cup \\lbrace tot\\_incr(p,q,k) : p \\in P, 0 \\le q \\le ntok \\rbrace \\cup \\lbrace tot\\_decr(p,q,k) : p \\in P, 0 \\le q \\le ntok \\rbrace $ $U_{6k+6}=U_{6k+5} \\cup head(r\\ref {r:nextstate})^{ts=k} \\cup head(a\\ref {a:overc:place})^{ts=k} = U_{6k+5} \\cup \\lbrace holds(p,q,k+1) : p \\in P, 0 \\le q \\le ntok \\rbrace \\cup \\lbrace consumesmore(p,k) : p \\in P\\rbrace $ $U_{6k+7}=U_{6k+6} \\cup head(a\\ref {a:overc:gen}) = U_{7k+6} \\cup \\lbrace consumesmore \\rbrace $ where $head(r_i)^{ts=k}$ are head atoms of ground rule $r_i$ in which $ts=k$ .", "We write $A_i^{ts=k} = \\lbrace a(\\dots ,ts) : a(\\dots ,ts) \\in A_i, ts=k \\rbrace $ as short hand for all atoms in $A_i$ with $ts=k$ .", "$U_{\\alpha }, 0 \\le \\alpha \\le 6k+7$ form a splitting sequence, since each $U_i$ is a splitting set of $\\Pi ^0$ , and $\\langle U_{\\alpha }\\rangle _{\\alpha < \\mu }$ is a monotone continuous sequence, where $U_0 \\subseteq U_1 \\dots \\subseteq U_{6k+7}$ and $\\bigcup _{\\alpha < \\mu }{U_{\\alpha }} = lit(\\Pi ^0)$ .", "We compute the answer set of $\\Pi ^0$ using the splitting sets as follows: $bot_{U_0}(\\Pi ^0) = f\\ref {f:place} \\cup f\\ref {f:trans} \\cup f\\ref {f:ptarc} \\cup f\\ref {f:tparc} \\cup f\\ref {f:time} \\cup i\\ref {i:holds} \\cup f\\ref {f:num}$ and $X_0 = A_1 \\cup \\dots \\cup A_7$ ($= U_0$ ) is its answer set – using forced atom proposition $eval_{U_0}(bot_{U_1}(\\Pi ^0) \\setminus bot_{U_0}(\\Pi ^0), X_0) = \\lbrace notenabled(t,0) \\text{:-} .", "\| \\lbrace trans(t), $ $ ptarc(p,t,n), $ $ holds(p,q,0) \\rbrace \\subseteq X_0, \\text{~where~} q < n \\rbrace $ .", "Its answer set $X_1=A_8^{ts=0}$ – using forced atom proposition and construction of $A_8$ .", "where, $q=M_0(p)$ , $n=W(p,t)$ , and for an arc $(p,t) \\in E^-$ – by construction of $i\\ref {i:holds}$ and $f\\ref {f:ptarc}$ in $\\Pi ^0$ , and in an arc $(p,t) \\in E^-$ , $p \\in \\bullet t$ (by definition REF of preset) thus, $notenabled(t,0) \\in X_1$ represents $\\exists p \\in \\bullet t : M_0(p) < W(p,t)$ .", "$eval_{U_1}(bot_{U_2}(\\Pi ^0) \\setminus bot_{U_1}(\\Pi ^0), X_0 \\cup X_1) = \\lbrace enabled(t,0) \\text{:-}.", "\| trans(t) \\in X_0 \\cup X_1, $ $notenabled(t,0) \\notin X_0 \\cup X_1 \\rbrace $ .", "Its answer set is $X_2 = A_9^{ts=0}$ – using forced atom proposition and construction of $A_9$ .", "since an $enabled(t,0) \\in X_2$ if $\\nexists ~notenabled(t,0) \\in X_0 \\cup X_1$ , which is equivalent to $\\nexists p \\in \\bullet t : M_0(p) < W(p,t) \\equiv \\forall p \\in \\bullet t: M_0(p) \\ge W(p,t)$ .", "$eval_{U_2}(bot_{U_3}(\\Pi ^0) \\setminus bot_{U_2}(\\Pi ^0), X_0 \\cup X_1 \\cup X_2) = \\lbrace \\lbrace fires(t,0)\\rbrace \\text{:-}.", "\| enabled(t,0) \\\\ \\text{~holds in~} X_0 \\cup X_1 \\cup X_2 \\rbrace $ .", "It has multiple answer sets $X_{3.1}, \\dots , X_{3.n}$ , corresponding to elements of power set of $fires(t,0)$ atoms in $eval_{U_2}(...)$ – using supported rule proposition.", "Since we are showing that the union of answer sets of $\\Pi ^0$ determined using splitting is equal to $A$ , we only consider the set that matches the $fires(t,0)$ elements in $A$ and call it $X_3$ , ignoring the rest.", "Thus, $X_3 = A_{10}^{ts=0}$ , representing $T_0$ .", "$eval_{U_3}(bot_{U_4}(\\Pi ^0) \\setminus bot_{U_3}(\\Pi ^0), X_0 \\cup \\dots \\cup X_3) = \\lbrace add(p,n,t,0) \\text{:-}.", "\| \\lbrace fires(t,0), \\\\ tparc(t,p,n) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_3 \\rbrace \\cup \\lbrace del(p,n,t,0) \\text{:-}.", "\| \\lbrace fires(t,0), ptarc(p,t,n) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_3 \\rbrace $ .", "It's answer set is $X_4 = A_{11}^{ts=0} \\cup A_{12}^{ts=0}$ – using forced atom proposition and definitions of $A_{11}$ and $A_{12}$ .", "where each $add$ atom is equivalent to $n=W(t,p) : p \\in t \\bullet $ , and each $del$ atom is equivalent to $n=W(p,t) : p \\in \\bullet t$ , representing the effect of transitions in $T_0$ – by construction $eval_{U_4}(bot_{U_5}(\\Pi ^0) \\setminus bot_{U_4}(\\Pi ^0), X_0 \\cup \\dots \\cup X_4) = \\lbrace tot\\_incr(p,qq,0) \\text{:-}.", "\| \\\\ qq=\\sum _{add(p,q,t,0) \\in X_0 \\cup \\dots \\cup X_4}{q} \\rbrace \\cup $ $\\lbrace tot\\_decr(p,qq,0) \\text{:-}.", "\| qq=\\sum _{del(p,q,t,0) \\in X_0 \\cup \\dots \\cup X_4}{q} \\rbrace $ .", "It's answer set is $X_5 = A_{13}^{ts=0} \\cup A_{14}^{ts=0}$ – using forced atom proposition and definitions of $A_{13}$ and $A_{14}$ , ad definition REF (semantics of aggregate assignment atom).", "where each $tot\\_incr(p,qq,0)$ , $qq=\\sum _{add(p,q,t,0) \\in X_0 \\cup \\dots X_4}{q} \\\\ \\equiv $ $qq=\\sum _{t \\in X_3, p \\in t \\bullet }{W(p,t)}$ , and each $tot\\_decr(p,qq,0)$ , $qq=\\sum _{del(p,q,t,0) \\in X_0 \\cup \\dots X_4}{q} \\\\ \\equiv $ $qq=\\sum _{t \\in X_3, p \\in \\bullet t}{W(t,p)}$ , represent the net effect of transitions in $T_0$ – by construction $eval_{U_5}(bot_{U_6}(\\Pi ^0) \\setminus bot_{U_5}(\\Pi ^0), X_0 \\cup \\dots \\cup X_5) = \\lbrace consumesmore(p,0) \\text{:-}.", "\| \\\\ \\lbrace holds(p,q,0), $ $tot\\_decr(p,q1,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_5, $ $q1 > q \\rbrace \\cup $ $\\lbrace holds(p,q,1) \\text{:-}.", "\| $ $ \\lbrace holds(p,q1,0), $ $tot\\_incr(p,q2,0), $ $tot\\_decr(p,q3,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_5, $ $q=q1+q2-q3 \\rbrace $ .", "It's answer set is $X_6 = A_{15}^{ts=0} \\cup A_{17}^{ts=0}$ – using forced atom proposition.", "where, $consumesmore(p,0)$ represents $\\exists p : q=M_0(p), \\\\ q1=\\sum _{t \\in T_0, p \\in \\bullet t}{W(p,t)}, q1 > q$ – indicating place $p$ will be overconsumed if $T_0$ is fired as defined in definition REF (conflicting transitions) and $holds(p,q,1)$ represents $q=M_1(p)$ – by construction $ \\vdots $ $eval_{U_{6k+0}}(bot_{U_{6k+1}}(\\Pi ^0) \\setminus bot_{U_{6k+0}}(\\Pi ^0), X_0 \\cup \\dots \\cup X_{6k+0}) = \\\\ \\lbrace notenabled(t,k) \\text{:-} .", "\| \\lbrace trans(t), ptarc(p,t,n), holds(p,q,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{6k+0}, \\\\ \\text{~where~} q < n \\rbrace $ .", "Its answer set $X_{6k+1}=A_8^{ts=k}$ – using forced atom proposition and construction of $A_8$ .", "where, $q=M_k(p)$ , and $n=W(p,t)$ for an arc $(p,t) \\in E^-$ – by construction of $holds$ and $ptarc$ predicates in $\\Pi ^0$ , and in an arc $(p,t) \\in E^-$ , $p \\in \\bullet t$ (by definition REF of preset) thus, $notenabled(t,k) \\in X_{6k+1}$ represents $\\exists p \\in \\bullet t : M_k(p) < W(p,t)$ .", "$eval_{U_{6k+1}}(bot_{U_{6k+2}}(\\Pi ^0) \\setminus bot_{U_{6k+1}}(\\Pi ^0), X_0 \\cup \\dots \\cup X_{6k+1}) = \\lbrace enabled(t,k) \\text{:-}.", "\| \\\\ trans(t) \\in $ $X_0 \\cup \\dots \\cup X_{6k+1}, notenabled(t,k) \\notin X_0 \\cup \\dots \\cup X_{6k+1} \\rbrace $ .", "Its answer set is $X_{6k+2} = A_9^{ts=k}$ – using forced atom proposition and construction of $A_9$ .", "since an $enabled(t,k) \\in X_{6k+2}$ if $\\nexists ~notenabled(t,k) \\in X_0 \\cup \\dots \\cup X_{6k+1}$ , which is equivalent to $\\nexists p \\in \\bullet t : M_k(p) < W(p,t) \\equiv \\forall p \\in \\bullet t: M_k(p) \\ge W(p,t)$ .", "$eval_{U_{6k+2}}(bot_{U_{6k+3}}(\\Pi ^0) \\setminus bot_{U_{6k+2}}(\\Pi ^0), X_0 \\cup \\dots \\cup X_{6k+2}) = \\\\ \\lbrace \\lbrace fires(t,k)\\rbrace \\text{:-}.", "\| $ $enabled(t,k) \\text{~holds in~} $ $X_0 \\cup \\dots \\cup X_{6k+2} \\rbrace $ .", "It has multiple answer sets $X_{{6k+3}.1}, \\dots , X_{{6k+3}.n}$ , corresponding to elements of power set of $fires(t,k)$ atoms in $eval_{U_{6k+2}}(...)$ – using supported rule proposition.", "Since we are showing that the union of answer sets of $\\Pi ^0$ determined using splitting is equal to $A$ , we only consider the set that matches the $fires(t,k)$ elements in $A$ and call it $X_{6k+3}$ , ignoring the reset.", "Thus, $X_{6k+3} = A_{10}^{ts=k}$ , representing $T_k$ .", "$eval_{U_{6k+3}}(bot_{U_{6k+4}}(\\Pi ^0) \\setminus bot_{U_{6k+3}}(\\Pi ^0), X_0 \\cup \\dots \\cup X_{6k+3}) = $ $ \\lbrace add(p,n,t,k) \\text{:-}.", "\| \\\\ \\lbrace fires(t,k), tparc(t,p,n) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{6k+3} \\rbrace \\cup $ $ \\lbrace del(p,n,t,k) \\text{:-}.", "\| $ $\\lbrace fires(t,k), $ $ptarc(p,t,n) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{6k+3} \\rbrace $ .", "It's answer set is $X_{6k+4} = A_{11}^{ts=k} \\cup A_{12}^{ts=k}$ – using forced atom proposition and definitions of $A_{11}$ and $A_{12}$ .", "where, each $add$ atom is equivalent to $n=W(t,p) : p \\in t \\bullet $ , and each $del$ atom is equivalent to $n=W(p,t) : p \\in \\bullet t$ representing the effect of transitions in $T_k$ $eval_{U_{6k+4}}(bot_{U_{6k+5}}(\\Pi ^0) \\setminus bot_{U_{6k+4}}(\\Pi ^0), X_0 \\cup \\dots \\cup X_{6k+4}) = $ $\\lbrace tot\\_incr(p,qq,k) \\text{:-}.", "\| $ $ qq=\\sum _{add(p,q,t,k) \\in X_0 \\cup \\dots \\cup X_{6k+4}}{q} \\rbrace \\cup $ $\\lbrace tot\\_decr(p,qq,k) \\text{:-}.", "\| \\\\ qq=\\sum _{del(p,q,t,k) \\in X_0 \\cup \\dots \\cup X_{6k+4}}{q} \\rbrace $ .", "It's answer set is $X_5 = A_{13}^{ts=k} \\cup A_{14}^{ts=k}$ – using forced atom proposition and definitions of $A_{13}$ and $A_{14}$ .", "where, each $tot\\_incr(p,qq,k)$ , $qq=\\sum _{add(p,q,t,k) \\in X_0 \\cup \\dots X_{7k+4}}{q}$ $\\equiv qq=\\sum _{t \\in X_{6k+3}, p \\in t \\bullet }{W(p,t)}$ , and each $tot\\_decr(p,qq,k)$ , $qq=\\sum _{del(p,q,t,k) \\in X_0 \\cup \\dots X_{7k+4}}{q}$ $\\equiv qq=\\sum _{t \\in X_{6k+3}, p \\in \\bullet t}{W(t,p)}$ , represent the net effect of transitions in $T_k$ $eval_{U_{6k+5}}(bot_{U_{6k+6}}(\\Pi ^0) \\setminus bot_{U_{6k+5}}(\\Pi ^0), X_0 \\cup \\dots \\cup X_{6k+5}) = $ $ \\lbrace consumesmore(p,k) \\text{:-}.", "\| $ $ \\lbrace holds(p,q,k), $ $ tot\\_decr(p,q1,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{6k+5} , q1 > q \\rbrace \\cup \\lbrace holds(p,q,k+1) \\text{:-}., \| \\lbrace holds(p,q1,k), tot\\_incr(p,q2,k), \\\\ tot\\_decr(p,q3,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{6k+5}, q=q1+q2-q3 \\rbrace $ .", "It's answer set is $X_{6k+6} = A_{15}^{ts=k} \\cup A_{17}^{ts=k}$ – using forced atom proposition.", "where, $holds(p,q,k+1)$ represents the marking of place $p$ in the next time step due to firing $T_k$ , and, $consumesmore(p,k)$ represents $\\exists p : q=M_{k}(p), q1=\\sum _{t \\in T_{k}, p \\in \\bullet t}{W(p,t)}, $ $q1 > q$ indicating place $p$ that will be overconsumed if $T_k$ is fired as defined in definition REF (conflicting transitions) $eval_{U_{6k+6}}(bot_{U_{6k+7}}(\\Pi ^0) \\setminus bot_{U_{6k+6}}(\\Pi ^0), X_0 \\cup \\dots \\cup X_{6k+6}) = \\lbrace consumesmore \\text{:-}.", "\| $ $\\lbrace consumesmore(p,0),$ $ \\dots , $ $consumesmore(p,k)\\rbrace \\cap (X_0 \\cup \\dots \\cup X_{6k+7}) \\ne \\emptyset \\rbrace $ .", "It's answer set is $X_{6k+7} = A_{16}^{ts=k}$ – using forced atom proposition $X_{6k+7}$ will be empty since none of $consumesmore(p,0),\\dots , \\\\ consumesmore(p,k)$ hold in $X_0 \\cup \\dots \\cup X_{6k+6}$ due to the construction of $A$ and encoding of $a\\ref {a:overc:place}$ , and it is not eliminated by the constraint $a\\ref {a:overc:elim}$ .", "The set $X=X_{0} \\cup \\dots \\cup X_{6k+7}$ is the answer set of $\\Pi ^0$ by the splitting sequence theorem REF .", "Each $X_i, 0 \\le i \\le 6k+7$ matches a distinct partition of $A$ , and $X = A$ , thus $A$ is an answer set of $\\Pi ^0$ .", "Next we show (REF ): Given $\\Pi ^0$ be the encoding of a Petri Net $PN(P,T,E,W)$ with initial marking $M_0$ , and $A$ be an answer set of $\\Pi ^0$ that satisfies (REF ) and (REF ), then we can construct $X=M_0,T_0,\\dots ,M_k,T_k,M_{k+1}$ from $A$ , such that it is an execution sequence of $PN$ .", "We construct the $X$ as follows: $M_i = (M_i(p_0), \\dots , M_i(p_n))$ , where $\\lbrace holds(p_0,M_i(p_0),i), \\dots holds(p_n,M_i(p_n),i) \\rbrace \\\\ \\subseteq A$ , for $0 \\le i \\le k+1$ $T_i = \\lbrace t : fires(t,i) \\in A\\rbrace $ , for $0 \\le i \\le k$ and show that $X$ is indeed an execution sequence of $PN$ .", "We show this by induction over $k$ (i.e.", "given $M_k$ , $T_k$ is a valid firing set and its firing produces marking $M_{k+1}$ ).", "Base case: Let $k=0$ , show [(1)] $T_0$ is a valid firing set for $M_0$ , and $T_0$ 's firing in $M_0$ producing marking $M_1$ .", "We show $T_0$ is a valid firing set w.r.t.", "$M_0$ .", "Let $\\lbrace fires(t_0,0), \\dots , fires(t_x,0) \\rbrace $ be the set of all $fires(\\dots ,0)$ atoms in $A$ , Then for each $fires(t_i,0) \\in A$ $enabled(t_i,0) \\in A$ – from rule $a\\ref {a:fires}$ and supported rule proposition Then $notenabled(t_i,0) \\notin A$ – from rule $e\\ref {e:enabled}$ and supported rule proposition Then $body(e\\ref {e:ne:ptarc})$ must not hold in $A$ – from rule $e\\ref {e:ne:ptarc}$ and forced atom proposition Then $q \\lnot < n_i \\equiv q \\ge n_i$ in $e\\ref {e:ne:ptarc}$ for all $\\lbrace holds(p,q,0), ptarc(p,t_i,n_i)\\rbrace \\subseteq A$ – from $e\\ref {e:ne:ptarc}$ , forced atom proposition, and the following: $holds(p,q,0) \\in A$ represents $q=M_0(p)$ – rule $i\\ref {i:holds}$ construction $ptarc(p,t_i,n_i) \\in A$ represents $n_i=W(p,t_i)$ – rule $f\\ref {f:ptarc} $ construction Then $\\forall p \\in \\bullet t_i, M_0(p) > W(p,t_i)$ – from definition REF of preset $\\bullet t_i$ in PN Then $t_i$ is enabled and can fire in $PN$ , as a result it can belong to $T_0$ – from definition REF of enabled transition And $consumesmore \\notin A$ , since $A$ is an answer set of $\\Pi ^0$ – from rule $a\\ref {a:overc:elim}$ and supported rule proposition Then $\\nexists consumesmore(p,0) \\in A$ – from rule $a\\ref {a:overc:gen}$ and supported rule proposition Then $\\nexists \\lbrace holds(p,q,0), tot\\_decr(p,q1,0) \\rbrace \\subseteq A : q1>q$ in $body(a\\ref {a:overc:place})$ – from $a\\ref {a:overc:place}$ and forced atom proposition Then $\\nexists p : \\sum _{t_i \\in \\lbrace t_0,\\dots ,t_x\\rbrace , p \\in \\bullet t_i}{W(p,t_i)} > M_0(p)$ – from the following $holds(p,q,0)$ represents $q=M_0(p)$ – from rule $i\\ref {i:holds}$ construction, given $tot\\_decr(p,q1,0) \\in A$ if $\\lbrace del(p,q1_0,t_0,0), \\dots del(p,q1_x,t_x,0) \\rbrace \\subseteq A$ , where $q1=q1_0+\\dots +q1_x$ – from $r\\ref {r:totdecr}$ and forced atom proposition $del(p,q1_i,t_i,0) \\in A$ if $\\lbrace fires(t_i,0), ptarc(p,t_i,q1_i) \\rbrace \\subseteq A$ – from $r\\ref {r:del}$ and supported rule proposition $del(p,q1_i,t_i,0) \\in A$ represents removal of $q1_i = W(p,t_i)$ tokens from $p \\in \\bullet t_i$ – from $r\\ref {r:del}$ , supported rule proposition, and definition REF of transition execution in PN Then the set of transitions in $T_0$ do not conflict – by the definition REF of conflicting transitions Then $\\lbrace t_0, \\dots , t_x\\rbrace = T_0$ – using 1(a),1(b) above, and definition of firing set We show $M_1$ is produced by firing $T_0$ in $M_0$ .", "Let $holds(p,q,1) \\in A$ Then $\\lbrace holds(p,q1,0), tot\\_incr(p,q2,0), tot\\_decr(p,q3,0) \\rbrace \\subseteq A : q=q1+q2-q3$ – from rule $r\\ref {r:nextstate}$ and supported rule proposition Then $holds(p,q1,0) \\in A$ represents $q1=M_0(p)$ – given, rule $i\\ref {i:holds}$ construction; and $\\lbrace add(p,q_0,t_0,0), \\dots , $ $add(p,q_j,t_j,0)\\rbrace \\subseteq A : q_0 + \\dots + q_j = q2$ ; and $\\lbrace del(p,q_0,t_0,0), \\dots , $ $del(p,q_l,t_l,0)\\rbrace \\subseteq A : q_0 + \\dots + q_l = q3$ – rules $r\\ref {r:totincr},r\\ref {r:totdecr}$ and supported rule proposition, respectively Then $\\lbrace fires(t_0,0), \\dots , fires(t_j,0) \\rbrace \\subseteq A$ and $\\lbrace fires(t_0,0), \\dots , fires(t_l,0) \\rbrace $ $\\subseteq A$ – rules $r\\ref {r:add},r\\ref {r:del}$ and supported rule proposition; and the following $tparc(t_y,p,q_y) \\in A, 0 \\le y \\le j$ represents $q_y=W(t_y,p)$ – given $ptarc(p,t_z,q_z) \\in A, 0 \\le z \\le l$ represents $q_z=W(p,t_z)$ – given Then $\\lbrace fires(t_0,0), \\dots , fires(t_j,0) \\rbrace \\cup \\lbrace fires(t_0,0), \\dots , fires(t_l,0) \\rbrace \\subseteq A $ $= \\lbrace fires(t_0,0), \\dots , fires(t_x,0) \\rbrace \\subseteq A$ – set union of subsets Then for each $fires(t_x,0) \\in A$ we have $t_x \\in T_0$ – already shown in item REF above Then $q = M_0(p) + \\sum _{t_x \\in T_0 \\wedge p \\in t_x \\bullet }{W(t_x,p)} - \\sum _{t_x \\in T_0 \\wedge p \\in \\bullet t_x}{W(p,t_x)}$ – from (REF ) above and the following Each $add(p,q_j,t_j,0) \\in A$ represents $q_j=W(t_j,p)$ for $p \\in t_j \\bullet $ – rule $r\\ref {r:add}$ encoding, and definition REF of transition execution in $PN$ Each $del(p,t_y,q_y,0) \\in A$ represents $q_y=W(p,t_y)$ for $p \\in \\bullet t_y$ – from rule $r\\ref {r:del}$ encoding, and definition REF of transition execution in $PN$ Each $tot\\_incr(p,q2,0) \\in A$ represents $q2=\\sum _{t_x \\in T_0 \\wedge p \\in t_x \\bullet }{W(t_x,p)}$ – aggregate assignment atom semantics in rule $r\\ref {r:totincr}$ Each $tot\\_decr(p,q3,0) \\in A$ represents $q3=\\sum _{t_x \\in T_0 \\wedge p \\in \\bullet t_x}{W(p,t_x)}$ – aggregate assignment atom semantics in rule $r\\ref {r:totdecr}$ Then, $M_1(p) = q$ – since $holds(p,q,1) \\in A$ encodes $q=M_1(p)$ – by construction Inductive Step: Assume $M_k$ is a valid marking in $X$ for $PN$ , show [(1)] $T_k$ is a valid firing set for $M_k$ , and firing $T_k$ in $M_k$ produces marking $M_{k+1}$ .", "We show that $T_k$ is a valid firing set for $M_k$ .", "Let $\\lbrace fires(t_0,k),\\dots ,fires(t_x,k) \\rbrace $ be the set of all $fires(\\dots ,k)$ atoms in $A$ , Then for each $fires(t_i,k) \\in A$ $enabled(t_i,k) \\in A$ – from rule $a\\ref {a:fires}$ and supported rule proposition Then $notenabled(t_i,k) \\notin A$ – from rule $e\\ref {e:enabled}$ and supported rule proposition Then body of $e\\ref {e:ne:ptarc}$ must hold in $A$ – from rule $e\\ref {e:ne:ptarc}$ and forced proposition Then $q \\lnot < n_i \\equiv q \\ge n_i$ in $e\\ref {e:ne:ptarc}$ for all $\\lbrace holds(p,q,k), ptarc(p,t_i,n_i) \\rbrace \\subseteq A$ – from $e\\ref {e:ne:ptarc}$ using forced atom proposition, and the following $holds(p,q,k) \\in A$ represents $q=M_k(p)$ – construction, inductive assumption $ptarc(p,t,n_i) \\in A$ represents $n_i=W(p,t)$ – rule $f\\ref {f:tparc}$ construction Then $\\forall p \\in \\bullet t_i, M_k(p) > W(p,t_i)$ – from definition REF of preset $\\bullet t_i$ in PN Then $t_i$ is enabled and can fire in $PN$ , as a result it can belong to $T_k$ – from definition REF of enabled transition And $consumesmore \\notin A$ , since $A$ is an answer set of $\\Pi ^0$ – from rule $a\\ref {a:overc:elim}$ and supported rule proposition Then $\\nexists consumesmore(p,k) \\in A$ – from rule $a\\ref {a:overc:gen}$ and forced atom proposition Then $\\nexists \\lbrace holds(p,q,k), tot\\_decr(p,q1,k) \\rbrace \\subseteq A : q1 > q$ in $body(a\\ref {a:overc:place})$ – from $a\\ref {a:overc:place}$ and forced atom proposition Then $\\nexists p : \\sum _{t_i \\in \\lbrace t_0,\\dots ,t_x \\rbrace , p \\in \\bullet t_i}{W(p,t_i)} > M_k(p)$ – from the following $holds(p,q,k) \\in A$ represents $q=M_k(p)$ – by construction, and the inductive assumption $tot\\_decr(p,q1,k) \\in A$ if $\\lbrace del(p,q1_0,t_0,k), \\dots , del(p,q1_x,t_x,k) \\rbrace \\subseteq A$ , where $q1=q1_0+\\dots +q1_x$ – from $r\\ref {r:totdecr}$ and forced atom proposition $del(p,q1_i,t_i,k) \\in A$ if $\\lbrace fires(t_i,k), ptarc(p,t_i,q1_i) \\rbrace \\subseteq A$ – from $r\\ref {r:del}$ and supported rule proposition $del(p,q1_i,t_i,k)$ represents removal of $q1_i = W(p,t_i)$ tokens from $p \\in \\bullet t_i$ – from construction rule $r\\ref {r:del}$ , supported rule proposition, and definition REF of transition execution in $PN$ Then the set of transitions $T_k$ does conflict – by the definition REF of conflicting transitions Then $\\lbrace t_0,\\dots ,t_x\\rbrace = T_k$ – using 1(a),1(b) above We show $M_{k+1}$ is produced by firing $T_k$ in $M_k$ .", "Let $holds(p,q,k+1) \\in A$ Then $\\lbrace holds(p,q1,k), tot\\_incr(p,q2,k), tot\\_decr(p,q3,k) \\rbrace \\in A : q=q2+q2-q3$ – from rule $r\\ref {r:nextstate}$ and supported rule proposition Then $holds(p,q1,k) \\in A$ represents $q1=M_k(p)$ – inductive assumption and construction; and $\\lbrace add(p,q2_0,t_0,k), \\dots , add(p,q2_j,t_j,k) \\rbrace \\subseteq A : q2_0+\\dots +q2_j=q2$ and $\\lbrace del(p,q3_0,t_0,k), \\dots , del(p,q3_l,t_l,k) \\rbrace \\subseteq A : q3_0+\\dots +q3_l=q3$ – from rules $r\\ref {r:totincr},r\\ref {r:totdecr}$ using supported rule proposition, respectively Then $\\lbrace fires(t_0,k), \\dots , fires(t_j,k) \\rbrace \\subseteq A$ and $\\lbrace fires(t_0,k), \\dots , fires(t_l,k) \\rbrace $ $\\subseteq A$ – from rules $r\\ref {r:add} ,r\\ref {r:del} $ using supported rule proposition, respectively Then $\\lbrace fires(t_0,k), \\dots , fires(t_j,k) \\rbrace \\cup \\lbrace fires(t_0,k), \\dots , fires(t_l,k) \\rbrace $ $= \\lbrace fires(t_0,k), \\dots , fires(t_x,k) \\rbrace \\subseteq A$ – subset union property Then for each $fires(t_x,k) \\in A$ we have $t_x \\in T_k$ - already shown in item REF above Then $q = M_k(p) + \\sum _{t_x \\in T_k \\wedge p \\in t_x \\bullet }{W(t_x,p)} - \\sum _{t_x \\in T_k \\wedge p \\in \\bullet t_x}{W(p,t_x}$ – from (REF ) above and the following Each $add(p,q_j,t_j,0) \\in A$ represents $q_j = W(t_j,p)$ for $p \\in t_j \\bullet $ – encoding of $r\\ref {r:add}$ and definition REF of transition execution in PN Each $del(p,t_y,q_y,0) \\in A$ represents $q_y = W(p,t_y)$ for $p \\in \\bullet t_y$ – encoding of $r\\ref {r:del}$ and definition REF of transition execution in PN Each $tot\\_incr(p,q2,0) \\in A$ represents $q2 = \\sum _{t_x \\in T_k \\wedge p \\in t_x \\bullet }{W(t_x,p)}$ – aggregate assignment atom semantics in rule $r\\ref {r:totincr}$ Each $tot\\_decr(p,q3,0) \\in A$ represents $q3 = \\sum _{t_x \\in T_k \\wedge p \\in \\bullet t_x}{W(p,t_x)}$ – aggregate assignment atom semantics in rule $r\\ref {r:totdecr}$ Then $M_{k+1}(p) = q$ – since $holds(p,q,k+1) \\in A$ encodes $q=M_{k+1}(p)$ by construction As a result, for any $n > k$ , $T_n$ is a valid firing set w.r.t.", "$M_n$ and its firing produces marking $M_{n+1}$ .", "Conclusion: Since both REF and REF hold, $X=M_0,T_0,M_1,\\dots ,M_k,T_k,M_{k+1}$ is an execution sequence of $PN(P,T,E,W)$ (w.r.t.", "$M_0$ ) iff there is an answer set $A$ of $\\Pi ^0(PN,M_0,k,ntok)$ such that (REF ) and (REF ) hold.", "Proof of Proposition REF Let $PN=(P,T,E,W)$ be a Petri Net, $M_0$ be its initial marking and let $\\Pi ^1(PN,M_0,$ $k,ntok)$ be the ASP encoding of $PN$ and $M_0$ over a simulation length $k$ , with maximum $ntok$ tokens on any place node, as defined in section REF .", "Then $X=M_0,T_0,M_1,\\dots ,M_k,T_k,M_{k+1}$ is an execution sequence of $PN$ (w.r.t.", "$M_0$ ) iff there is an answer set $A$ of $\\Pi ^1(PN,M_0,k,ntok)$ such that: $\\lbrace fires(t,ts) : t \\in T_{ts}, 0 \\le ts \\le k\\rbrace = \\lbrace fires(t,ts) : fires(t,ts) \\in A \\rbrace $ $\\begin{split}\\lbrace holds(p,q,ts) &: p \\in P, q = M_{ts}(p), 0 \\le ts \\le k+1 \\rbrace \\\\&= \\lbrace holds(p,q,ts) : holds(p,q,ts) \\in A \\rbrace \\end{split}$ We prove this by showing that: Given an execution sequence $X$ , we create a set $A$ such that it satisfies (REF ) and (REF ) and show that $A$ is an answer set of $\\Pi ^1$ Given an answer set $A$ of $\\Pi ^1$ , we create an execution sequence $X$ such that (REF ) and (REF ) are satisfied.", "First we show (REF ): Given a $PN$ and an execution sequence $X$ of $PN$ , we create a set $A$ as a union of the following sets: $A_1=\\lbrace num(n) : 0 \\le n \\le ntok \\rbrace $ $A_2=\\lbrace time(ts) : 0 \\le ts \\le k\\rbrace $ $A_3=\\lbrace place(p) : p \\in P \\rbrace $ $A_4=\\lbrace trans(t) : t \\in T \\rbrace $ $A_5=\\lbrace ptarc(p,t,n) : (p,t) \\in E^-, n=W(p,t) \\rbrace $ , where $E^- \\subseteq E$ $A_6=\\lbrace tparc(t,p,n) : (t,p) \\in E^+, n=W(t,p) \\rbrace $ , where $E^+ \\subseteq E$ $A_7=\\lbrace holds(p,q,0) : p \\in P, q=M_{0}(p) \\rbrace $ $A_8=\\lbrace notenabled(t,ts) : t \\in T, 0 \\le ts \\le k, \\exists p \\in \\bullet t, M_{ts}(p) < W(p,t) \\rbrace $ per definition REF (enabled transition) $A_9=\\lbrace enabled(t,ts) : t \\in T, 0 \\le ts \\le k, \\forall p \\in \\bullet t, W(p,t) \\le M_{ts}(p) \\rbrace $ per definition REF (enabled transition) $A_{10}=\\lbrace fires(t,ts) : t \\in T_{ts}, 0 \\le ts \\le k \\rbrace $ per definition REF (firing set), only an enabled transition may fire $A_{11}=\\lbrace add(p,q,t,ts) : t \\in T_{ts}, p \\in t \\bullet , q=W(t,p), 0 \\le ts \\le k \\rbrace $ per definition REF (transition execution) $A_{12}=\\lbrace del(p,q,t,ts) : t \\in T_{ts}, p \\in \\bullet t, q=W(p,t), 0 \\le ts \\le k \\rbrace $ per definition REF (transition execution) $A_{13}=\\lbrace tot\\_incr(p,q,ts) : p \\in P, q=\\sum _{t \\in T_{ts}, p \\in t \\bullet }{W(t,p)}, 0 \\le ts \\le k \\rbrace $ per definition REF (firing set execution) $A_{14}=\\lbrace tot\\_decr(p,q,ts): p \\in P, q=\\sum _{t \\in T_{ts}, p \\in \\bullet t}{W(p,t)}, 0 \\le ts \\le k \\rbrace $ per definition REF (firing set execution) $A_{15}=\\lbrace consumesmore(p,ts) : p \\in P, q=M_{ts}(p), q1=\\sum _{t \\in T_{ts}, p \\in \\bullet t}{W(p,t)}, q1 > q, 0 \\le ts \\le k \\rbrace $ per definition REF (conflicting transitions) for enabled transition set $T_{ts}$ $A_{16}=\\lbrace consumesmore : \\exists p \\in P : q=M_{ts}(p), q1=\\sum _{t \\in T_{ts}, p \\in \\bullet t}{W(p,t)}, q1 > q, 0 \\le ts \\le k \\rbrace $ per definition REF (conflicting transitions) $A_{17}=\\lbrace could\\_not\\_have(t,ts) : t \\in T, (\\forall p \\in \\bullet t, W(p,t) \\le M_{ts}(p)), t \\notin T_{ts}, (\\exists p \\in \\bullet t : W(p,t) > M_{ts}(p) - \\sum _{t^{\\prime } \\in T_{ts}, p \\in \\bullet t^{\\prime }}{W(p,t^{\\prime })}), 0 \\le ts \\le k \\rbrace $ per the maximal firing set semantics $A_{18}=\\lbrace holds(p,q,ts+1) : p \\in P, q=M_{ts+1}(p), 0 \\le ts < k\\rbrace $ , where $M_{ts+1}(p) = M_{ts}(p) - \\sum _{\\begin{array}{c}t \\in T_{ts}, p \\in \\bullet t\\end{array}}{W(p,t)} + \\sum _{\\begin{array}{c}t \\in T_{ts}, p \\in t \\bullet \\end{array}}{W(t,p)}$ according to definition REF (firing set execution) We show that $A$ satisfies (REF ) and (REF ), and $A$ is an answer set of $\\Pi ^1$ .", "$A$ satisfies (REF ) and (REF ) by its construction above.", "We show $A$ is an answer set of $\\Pi ^1$ by splitting.", "We split $lit(\\Pi ^1)$ into a sequence of $6k+8$ sets: $U_0= head(f\\ref {f:place}) \\cup head(f\\ref {f:trans}) \\cup head(f\\ref {f:ptarc}) \\cup head(f\\ref {f:tparc}) \\cup head(f\\ref {f:time}) \\cup head(f\\ref {f:num}) $ $\\cup head(i\\ref {i:holds}) = \\lbrace place(p) : p \\in P\\rbrace \\cup \\lbrace trans(t) : t \\in T\\rbrace \\cup \\lbrace ptarc(p,t,n) : (p,t) \\in E^-, n=W(p,t) \\rbrace \\cup \\lbrace tparc(t,p,n) : (t,p) \\in E^+, n=W(t,p) \\rbrace \\cup $ $ \\lbrace time(0), \\dots , time(k)\\rbrace \\cup \\lbrace num(0), \\dots , num(ntok)\\rbrace \\cup \\lbrace holds(p,q,0) : p \\in P, q=M_0(p) \\rbrace $ $U_{6k+1}=U_{6k+0} \\cup head(e\\ref {e:ne:ptarc})^{ts=k} = U_{6k+0} \\cup \\lbrace notenabled(t,k) : t \\in T \\rbrace $ $U_{6k+2}=U_{6k+1} \\cup head(e\\ref {e:enabled})^{ts=k} = U_{6k+1} \\cup \\lbrace enabled(t,k) : t \\in T \\rbrace $ $U_{6k+3}=U_{6k+2} \\cup head(a\\ref {a:fires})^{ts=k} = U_{6k+2} \\cup \\lbrace fires(t,k) : t \\in T \\rbrace $ $U_{6k+4}=U_{6k+3} \\cup head(r\\ref {r:add})^{ts=k} \\cup head(r\\ref {r:del})^{ts=k} = U_{6k+3} \\cup \\lbrace add(p,q,t,k) : p \\in P, t \\in T, q=W(t,p) \\rbrace \\cup \\lbrace del(p,q,t,k) : p \\in P, t \\in T, q=W(p,t) \\rbrace $ $U_{6k+5}=U_{6k+4} \\cup head(r\\ref {r:totincr})^{ts=k} \\cup head(r\\ref {r:totdecr})^{ts=k} = U_{6k+4} \\cup \\lbrace tot\\_incr(p,q,k) : p \\in P, 0 \\le q \\le ntok \\rbrace \\cup \\lbrace tot\\_decr(p,q,k) : p \\in P, 0 \\le q \\le ntok \\rbrace $ $U_{6k+6}=U_{6k+5} \\cup head(r\\ref {r:nextstate})^{ts=k} \\cup head(a\\ref {a:overc:place})^{ts=k} \\cup head(a\\ref {a:maxfire:cnh})^{ts=k} $ $= U_{6k+5} \\\\ \\cup \\lbrace consumesmore(p,k) : p \\in P\\rbrace \\cup \\lbrace holds(p,q,k+1) : p \\in P, 0 \\le q \\le ntok \\rbrace \\cup $ $\\lbrace could\\_not\\_have(t,k) : t \\in T \\rbrace $ $U_{6k+7}=U_{6k+6} \\cup head(a\\ref {a:overc:gen}) = U_{6k+6} \\cup \\lbrace consumesmore \\rbrace $ where $head(r_i)^{ts=k}$ are head atoms of ground rule $r_i$ in which $ts=k$ .", "We write $A_i^{ts=k} = \\lbrace a(\\dots ,ts) : a(\\dots ,ts) \\in A_i, ts=k \\rbrace $ as short hand for all atoms in $A_i$ with $ts=k$ .", "$U_{\\alpha }, 0 \\le \\alpha \\le 6k+7$ form a splitting sequence, since each $U_i$ is a splitting set of $\\Pi ^1$ , and $\\langle U_{\\alpha }\\rangle _{\\alpha < \\mu }$ is a monotone continuous sequence, where $U_0 \\subseteq U_1 \\dots \\subseteq U_{6k+7}$ and $\\bigcup _{\\alpha < \\mu }{U_{\\alpha }} = lit(\\Pi ^1)$ .", "We compute the answer set of $\\Pi ^1$ using the splitting sets as follows: $bot_{U_0}(\\Pi ^1) = f\\ref {f:place} \\cup f\\ref {f:trans} \\cup f\\ref {f:ptarc} \\cup f\\ref {f:tparc} \\cup f\\ref {f:time} \\cup i\\ref {i:holds} \\cup f\\ref {f:num}$ and $X_0 = A_1 \\cup \\dots \\cup A_7$ ($= U_0$ ) is its answer set – using forced atom proposition $eval_{U_0}(bot_{U_1}(\\Pi ^1) \\setminus bot_{U_0}(\\Pi ^1), X_0) = \\lbrace notenabled(t,0) \\text{:-} .", "\| \\lbrace trans(t), \\\\ ptarc(p,t,n), holds(p,q,0) \\rbrace \\subseteq X_0, \\text{~where~} q < n \\rbrace $ .", "Its answer set $X_1=A_8^{ts=0}$ – using forced atom proposition and construction of $A_8$ .", "where, $q=M_0(p)$ , and $n=W(p,t)$ for an arc $(p,t) \\in E^-$ – by construction of $i\\ref {i:holds}$ and $f\\ref {f:ptarc}$ in $\\Pi ^1$ , and in an arc $(p,t) \\in E^-$ , $p \\in \\bullet t$ (by definition REF of preset) thus, $notenabled(t,0) \\in X_1$ represents $\\exists p \\in \\bullet t : M_0(p) < W(p,t)$ .", "$eval_{U_1}(bot_{U_2}(\\Pi ^1) \\setminus bot_{U_1}(\\Pi ^1), X_0 \\cup X_1) = \\lbrace enabled(t,0) \\text{:-}.", "\| trans(t) \\in X_0 \\cup X_1, $ $notenabled(t,0) \\notin X_0 \\cup X_1 \\rbrace $ .", "Its answer set is $X_2 = A_9^{ts=0}$ – using forced atom proposition and construction of $A_9$ .", "since an $enabled(t,0) \\in X_2$ if $\\nexists ~notenabled(t,0) \\in X_0 \\cup X_1$ , which is equivalent to $\\nexists p \\in \\bullet t : M_0(p) < W(p,t) \\equiv \\forall p \\in \\bullet t: M_0(p) \\ge W(p,t)$ .", "$eval_{U_2}(bot_{U_3}(\\Pi ^1) \\setminus bot_{U_2}(\\Pi ^1), X_0 \\cup X_1 \\cup X_2) = \\lbrace \\lbrace fires(t,0)\\rbrace \\text{:-}.", "\| enabled(t,0) \\\\ \\text{~holds in~} X_0 \\cup X_1 \\cup X_2 \\rbrace $ .", "It has multiple answer sets $X_{3.1}, \\dots , X_{3.n}$ , corresponding to elements of power set of $fires(t,0)$ atoms in $eval_{U_2}(...)$ – using supported rule proposition.", "Since we are showing that the union of answer sets of $\\Pi ^1$ determined using splitting is equal to $A$ , we only consider the set that matches the $fires(t,0)$ elements in $A$ and call it $X_3$ , ignoring the rest.", "Thus, $X_3 = A_{10}^{ts=0}$ , representing $T_0$ .", "$eval_{U_3}(bot_{U_4}(\\Pi ^1) \\setminus bot_{U_3}(\\Pi ^1), X_0 \\cup \\dots \\cup X_3) = \\lbrace add(p,n,t,0) \\text{:-}.", "\| \\lbrace fires(t,0), \\\\ tparc(t,p,n) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_3 \\rbrace \\cup \\lbrace del(p,n,t,0) \\text{:-}.", "\| \\lbrace fires(t,0), ptarc(p,t,n) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_3 \\rbrace $ .", "It's answer set is $X_4 = A_{11}^{ts=0} \\cup A_{12}^{ts=0}$ – using forced atom proposition and definitions of $A_{11}$ and $A_{12}$ .", "where, each $add$ atom encodes $n=W(t,p) : p \\in t \\bullet $ , and each $del$ atom encodes $n=W(p,t) : p \\in \\bullet t$ representing the effect of transitions in $T_0$ – by construction $eval_{U_4}(bot_{U_5}(\\Pi ^1) \\setminus bot_{U_4}(\\Pi ^1), X_0 \\cup \\dots \\cup X_4) = \\lbrace tot\\_incr(p,qq,0) \\text{:-}.", "\| $ $qq=\\sum _{add(p,q,t,0) \\in X_0 \\cup \\dots \\cup X_4}{q} \\rbrace \\cup \\lbrace tot\\_decr(p,qq,0) \\text{:-}.", "\| qq=\\sum _{del(p,q,t,0) \\in X_0 \\cup \\dots \\cup X_4}{q} \\rbrace $ .", "It's answer set is $X_5 = A_{13}^{ts=0} \\cup A_{14}^{ts=0}$ – using forced atom proposition and definitions of $A_{13}$ , $A_{14}$ , and semantics of aggregate assignment atom where, each $tot\\_incr(p,qq,0)$ , $qq=\\sum _{add(p,q,t,0) \\in X_0 \\cup \\dots X_4}{q}$ $\\equiv qq=\\sum _{t \\in X_3, p \\in t \\bullet }{W(p,t)}$ , and, each $tot\\_decr(p,qq,0)$ , $qq=\\sum _{del(p,q,t,0) \\in X_0 \\cup \\dots X_4}{q}$ $\\equiv qq=\\sum _{t \\in X_3, p \\in \\bullet t}{W(t,p)}$ , represent the net effect of actions in $T_0$ – by construction $eval_{U_5}(bot_{U_6}(\\Pi ^1) \\setminus bot_{U_5}(\\Pi ^1), X_0 \\cup \\dots \\cup X_5) = \\lbrace consumesmore(p,0) \\text{:-}.", "\| \\\\ \\lbrace holds(p,q,0), tot\\_decr(p,q1,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_5 : q1 > q \\rbrace \\cup \\lbrace holds(p,q,1) \\text{:-}., \| \\\\ \\lbrace holds(p,q1,0), tot\\_incr(p,q2,0), tot\\_decr(p,q3,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_5, q=q1+q2-q3 \\rbrace \\cup \\lbrace could\\_not\\_have(t,0) \\text{:-}.", "\| \\lbrace enabled(t,0), ptarc(s,t,q), holds(s,qq,0), \\\\ tot\\_decr(s,qqq,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_5, fires(t,0) \\notin (X_0 \\cup \\dots \\cup X_5), q > qq-qqq \\rbrace $ .", "It's answer set is $X_6 = A_{15}^{ts=0} \\cup A_{17}^{ts=0} \\cup A_{18}^{ts=0}$ – using forced atom proposition and definitions of $A_{15}, A_{17}, A_{18}$ .", "where, $consumesmore(p,0)$ represents $\\exists p : q=M_0(p), q1=\\sum _{t \\in T_0, p \\in \\bullet t}{W(p,t)}, $ $q1 > q$ indicating place $p$ will be overconsumed if $T_0$ is fired, as defined in definition REF (conflicting transitions) and, $holds(p,q,1)$ encodes $q=M_1(p)$ – by construction and $could\\_not\\_have(t,0)$ represents an enabled transition $t$ in $T_0$ that could not fire due to insufficient tokens $X_6$ does not contain $could\\_not\\_have(t,0)$ , when $enabled(t,0) \\in X_0 \\cup \\dots \\cup X_5$ and $fires(t,0) \\notin X_0 \\cup \\dots \\cup X_5$ due to construction of $A$ , encoding of $a\\ref {a:maxfire:cnh}$ and its body atoms.", "As a result it is not eliminated by the constraint $a\\ref {a:maxfire:elim}$ $ \\vdots $ $eval_{U_{6k+0}}(bot_{U_{6k+1}}(\\Pi ^1) \\setminus bot_{U_{6k+0}}(\\Pi ^1), X_0 \\cup \\dots \\cup X_{6k+0}) = \\\\ \\lbrace notenabled(t,k) \\text{:-} .", "\| \\lbrace trans(t), ptarc(p,t,n), holds(p,q,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{6k+0}, \\\\ \\text{~where~} q < n \\rbrace $ .", "Its answer set $X_{6k+1}=A_8^{ts=k}$ – using forced atom proposition and construction of $A_8$ .", "where, $q=M_k(p)$ , and $n=W(p,t)$ for an arc $(p,t) \\in E^-$ – by construction of $holds$ and $ptarc$ predicates in $\\Pi ^1$ , and in an arc $(p,t) \\in E^-$ , $p \\in \\bullet t$ (by definition REF of preset) thus, $notenabled(t,k) \\in X_{6k+1}$ represents $\\exists p \\in \\bullet t : M_k(p) < W(p,t)$ .", "$eval_{U_{6k+1}}(bot_{U_{6k+2}}(\\Pi ^1) \\setminus bot_{U_{6k+1}}(\\Pi ^1), X_0 \\cup \\dots \\cup X_{6k+1}) = \\lbrace enabled(t,k) \\text{:-}.", "\| \\\\ trans(t) \\in X_0 \\cup \\dots \\cup X_{6k+1}, notenabled(t,k) \\notin X_0 \\cup \\dots \\cup X_{6k+1} \\rbrace $ .", "Its answer set is $X_{6k+2} = A_9^{ts=k}$ – using forced atom proposition and construction of $A_9$ .", "since an $enabled(t,k) \\in X_{6k+2}$ if $\\nexists ~notenabled(t,k) \\in X_0 \\cup \\dots \\cup X_{6k+1}$ , which is equivalent to $\\nexists p \\in \\bullet t : M_k(p) < W(p,t) \\equiv \\forall p \\in \\bullet t: M_k(p) \\ge W(p,t)$ .", "$eval_{U_{6k+2}}(bot_{U_{6k+3}}(\\Pi ^1) \\setminus bot_{U_{6k+2}}(\\Pi ^1), X_0 \\cup \\dots \\cup X_{6k+2}) = \\\\ \\lbrace \\lbrace fires(t,k)\\rbrace \\text{:-}.", "\| enabled(t,k) \\text{~holds in~} X_0 \\cup \\dots \\cup X_{6k+2} \\rbrace $ .", "It has multiple answer sets $X_{{6k+3}.1}, \\dots , X_{{6k+3}.n}$ , corresponding to elements of power set of $fires(t,k)$ atoms in $eval_{U_{6k+2}}(...)$ – using supported rule proposition.", "Since we are showing that the union of answer sets of $\\Pi ^1$ determined using splitting is equal to $A$ , we only consider the set that matches the $fires(t,k)$ elements in $A$ and call it $X_{6k+3}$ , ignoring the reset.", "Thus, $X_{6k+3} = A_{10}^{ts=k}$ , representing $T_k$ .", "$eval_{U_{6k+3}}(bot_{U_{6k+4}}(\\Pi ^1) \\setminus bot_{U_{6k+3}}(\\Pi ^1), X_0 \\cup \\dots \\cup X_{6k+3}) = \\\\ \\lbrace add(p,n,t,k) \\text{:-}.", "\| \\lbrace fires(t,k), tparc(t,p,n) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{6k+3} \\rbrace \\cup \\\\ \\lbrace del(p,n,t,k) \\text{:-}.", "\| \\lbrace fires(t,k), ptarc(p,t,n) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{6k+3} \\rbrace $ .", "It's answer set is $X_{6k+4} = A_{11}^{ts=k} \\cup A_{12}^{ts=k}$ – using forced atom proposition and definitions of $A_{11}$ and $A_{12}$ .", "where, each $add$ atom is equivalent to $n=W(t,p) : p \\in t \\bullet $ , and, each $del$ atom is equivalent to $n=W(p,t) : p \\in \\bullet t$ , representing the effect of transitions in $T_k$ $eval_{U_{6k+4}}(bot_{U_{6k+5}}(\\Pi ^1) \\setminus bot_{U_{6k+4}}(\\Pi ^1), X_0 \\cup \\dots \\cup X_{6k+4}) = \\lbrace tot\\_incr(p,qq,k) \\text{:-}.", "\| \\\\ qq=\\sum _{add(p,q,t,k) \\in X_0 \\cup \\dots \\cup X_{6k+4}}{q} \\rbrace \\cup \\lbrace tot\\_decr(p,qq,k) \\text{:-}.", "\| \\\\ qq=\\sum _{del(p,q,t,k) \\in X_0 \\cup \\dots \\cup X_{6k+4}}{q} \\rbrace $ .", "It's answer set is $X_{6k+5} = A_{13}^{ts=k} \\cup A_{14}^{ts=k}$ – using forced atom proposition and definitions of $A_{13}$ and $A_{14}$ .", "where, each $tot\\_incr(p,qq,k)$ , $qq=\\sum _{add(p,q,t,k) \\in X_0 \\cup \\dots X_{6k+4}}{q}$ $\\equiv \\\\ qq=\\sum _{t \\in X_{6k+3}, p \\in t \\bullet }{W(p,t)}$ , and, each $tot\\_decr(p,qq,k)$ , $qq=\\sum _{del(p,q,t,k) \\in X_0 \\cup \\dots X_{6k+4}}{q}$ $\\equiv \\\\ qq=\\sum _{t \\in X_{6k+3}, p \\in \\bullet t}{W(t,p)}$ , represent the net effect of transitions in $T_k$ $eval_{U_{6k+5}}(bot_{U_{6k+6}}(\\Pi ^1) \\setminus bot_{U_{6k+5}}(\\Pi ^1), X_0 \\cup \\dots \\cup X_{6k+5}) = \\\\ \\lbrace consumesmore(p,k) \\text{:-}.", "\| \\lbrace holds(p,q,k), tot\\_decr(p,q1,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{6k+5} : q1 > q \\rbrace \\cup \\lbrace holds(p,q,k+1) \\text{:-}., \| \\lbrace holds(p,q1,k), tot\\_incr(p,q2,k), \\\\ tot\\_decr(p,q3,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{6k+5} : q=q1+q2-q3 \\rbrace \\cup \\lbrace could\\_not\\_have(t,k) \\text{:-} \\\\ \\lbrace enabled(t,k), ptarc(s,t,q), holds(s,qq,k), tot\\_decr(s,qqq,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{6k+5}, $ $fires(t,k) \\notin (X_0 \\cup \\dots \\cup X_{6k+5}), q > qq-qqq \\rbrace $ .", "It's answer set is $X_{6k+6} = A_{15}^{ts=k} \\cup A_{17}^{ts=k} \\cup A_{18}^{ts=k}$ – using forced atom proposition.", "where, $consumesmore(p,k)$ represents $\\exists p : q=M_{k}(p), \\\\ q1=\\sum _{t \\in T_{k}, p \\in \\bullet t}{W(p,t)}, q1 > q$ $holds(p,q,k+1)$ represents $q=M_{k+1}(p)$ indicating place $p$ that will be over consumed if $T_k$ is fired, as defined in definition REF (conflicting transitions), $holds(p,q,k+1)$ represents $q=M_{k+1}(p)$ – by construction and $could\\_not\\_have(t,k)$ represents an enabled transition $t$ in $T_k$ that could not fire due to insufficient tokens $X_{6k+6}$ does not contain $could\\_not\\_have(t,k)$ , when $enabled(t,k) \\in X_0 \\cup \\dots \\cup X_{6k+5}$ and $fires(t,k) \\notin X_0 \\cup \\dots \\cup X_{6k+5}$ due to construction of $A$ , encoding of $a\\ref {a:maxfire:cnh}$ and its body atoms.", "As a result it is note eliminated by the constraint $a\\ref {a:maxfire:elim}$ $eval_{U_{6k+6}}(bot_{U_{6k+7}}(\\Pi ^1) \\setminus bot_{U_{6k+6}}(\\Pi ^1), X_0 \\cup \\dots \\cup X_{6k+6}) = \\lbrace consumesmore \\text{:-}.", "\| $ $\\lbrace consumesmore(p,0),$ $\\dots , $ $consumesmore(p,k) \\rbrace \\cap $ $(X_0 \\cup \\dots \\cup X_{6k+6}) \\ne \\emptyset \\rbrace $ .", "It's answer set is $X_{6k+7} = A_{16}$ – using forced atom proposition $X_{6k+7}$ will be empty since none of $consumesmore(p,0),\\dots , \\\\ consumesmore(p,k)$ hold in $X_0 \\cup \\dots \\cup X_{6k+6}$ due to the construction of $A$ , encoding of $a\\ref {a:overc:place}$ and its body atoms.", "As a result, it is not eliminated by the constraint $a\\ref {a:overc:elim}$ The set $X = X_0 \\cup \\dots \\cup X_{6k+7}$ is the answer set of $\\Pi ^0$ by the splitting sequence theorem REF .", "Each $X_i, 0 \\le i \\le 6k+7$ matches a distinct portion of $A$ , and $X = A$ , thus $A$ is an answer set of $\\Pi ^1$ .", "Next we show (REF ): Given $\\Pi ^1$ be the encoding of a Petri Net $PN(P,T,E,W)$ with initial marking $M_0$ , and $A$ be an answer set of $\\Pi ^1$ that satisfies (REF ) and (REF ), then we can construct $X=M_0,T_0,\\dots ,M_k,T_k,M_{k+1}$ from $A$ , such that it is an execution sequence of $PN$ .", "We construct the $X$ as follows: $M_i = (M_i(p_0), \\dots , M_i(p_n))$ , where $\\lbrace holds(p_0,M_i(p_0),i), \\dots holds(p_n,M_i(p_n),i) \\rbrace \\\\ \\subseteq A$ , for $0 \\le i \\le k+1$ $T_i = \\lbrace t : fires(t,i) \\in A\\rbrace $ , for $0 \\le i \\le k$ and show that $X$ is indeed an execution sequence of $PN$ .", "We show this by induction over $k$ (i.e.", "given $M_k$ , $T_k$ is a valid firing set and its firing produces marking $M_{k+1}$ ).", "Base case: Let $k=0$ , and $M_0$ is a valid marking in $X$ for $PN$ , show [(1)] $T_0$ is a valid firing set for $M_0$ , and $T_0$ 's firing w.r.t.", "marking $M_0$ produces $M_1$ .", "We show $T_0$ is a valid firing set for $M_0$ .", "Let $\\lbrace fires(t_0,0), \\dots , fires(t_x,0) \\rbrace $ be the set of all $fires(\\dots ,0)$ atoms in $A$ , Then for each $fires(t_i,0) \\in A$ $enabled(t_i,0) \\in A$ – from rule $a\\ref {a:fires}$ and supported rule proposition Then $notenabled(t_i,0) \\notin A$ – from rule $e\\ref {e:enabled}$ and supported rule proposition Then $body(e\\ref {e:ne:ptarc})$ must not hold in $A$ – from rule $e\\ref {e:ne:ptarc}$ and forced atom proposition Then $q \\lnot < n_i \\equiv q \\ge n_i$ in $e\\ref {e:ne:ptarc}$ for all $\\lbrace holds(p,q,0), ptarc(p,t_i,n_i)\\rbrace \\subseteq A$ – from $e\\ref {e:ne:ptarc}$ , forced atom proposition, and the following $holds(p,q,0) \\in A$ represents $q=M_0(p)$ – rule $i\\ref {i:holds}$ construction $ptarc(p,t_i,n_i) \\in A$ represents $n_i=W(p,t_i)$ – rule $f\\ref {f:ptarc} $ construction Then $\\forall p \\in \\bullet t_i, M_0(p) > W(p,t_i)$ – from definition REF of preset $\\bullet t_i$ in PN Then $t_i$ is enabled and can fire in $PN$ , as a result it can belong to $T_0$ – from definition REF of enabled transition And $consumesmore \\notin A$ , since $A$ is an answer set of $\\Pi ^1$ – from rule $a\\ref {a:overc:elim}$ and supported rule proposition Then $\\nexists consumesmore(p,0) \\in A$ – from rule $a\\ref {a:overc:gen}$ and supported rule proposition Then $\\nexists \\lbrace holds(p,q,0), tot\\_decr(p,q1,0) \\rbrace \\subseteq A : q1>q$ in $body(a\\ref {a:overc:place})$ – from $a\\ref {a:overc:place}$ and forced atom proposition Then $\\nexists p : \\sum _{t_i \\in \\lbrace t_0,\\dots ,t_x\\rbrace , p \\in \\bullet t_i}{W(p,t_i)} > M_0(p)$ – from the following $holds(p,q,0)$ represents $q=M_0(p)$ – from rule $i\\ref {i:holds}$ construction, given $tot\\_decr(p,q1,0) \\in A$ if $\\lbrace del(p,q1_0,t_0,0), \\dots , del(p,q1_x,t_x,0) \\rbrace \\subseteq A$ , where $q1 = q1_0+\\dots +q1_x$ – from $r\\ref {r:totdecr}$ and forced atom proposition $del(p,q1_i,t_i,0) \\in A$ if $\\lbrace fires(t_i,0), ptarc(p,t_i,q1_i) \\rbrace \\subseteq A$ – from $r\\ref {r:del}$ and supported rule proposition $del(p,q1_i,t_i,0)$ represents removal of $q1_i = W(p,t_i)$ tokens from $p \\in \\bullet t_i$ – from rule $r\\ref {r:del}$ , supported rule proposition, and definition REF of transition execution in $PN$ Then the set of transitions in $T_0$ do not conflict – by the definition REF of conflicting transitions And for each $enabled(t_j,0) \\in A$ and $fires(t_j,0) \\notin A$ , $could\\_not\\_have(t_j,0) \\in A$ , since $A$ is an answer set of $\\Pi ^1$ - from rule $a\\ref {a:maxfire:elim}$ and supported rule proposition Then $\\lbrace enabled(t_j,0), holds(s,qq,0), ptarc(s,t_j,q,0), \\\\ tot\\_decr(s,qqq,0) \\rbrace \\subseteq A$ , such that $q > qq - qqq$ and $fires(t_j,0) \\notin A$ - from rule $a\\ref {a:maxfire:cnh}$ and supported rule proposition Then for an $s \\in \\bullet t_j$ , $W(s,t_j) > M_0(s) - \\sum _{t_i \\in T_0, s \\in \\bullet t_i}{W(s,t_i)}$ - from the following: $ptarc(s,t_i,q)$ represents $q=W(s,t_i)$ – from rule $f\\ref {f:r:ptarc}$ construction $holds(s,qq,0)$ represents $qq=M_0(s)$ – from $i\\ref {i:holds}$ construction $tot\\_decr(s,qqq,0) \\in A$ if $\\lbrace del(s,qqq_0,t_0,0), \\dots , del(s,qqq_x,t_x,0) \\rbrace \\subseteq A$ – from rule $r\\ref {r:totdecr}$ construction and supported rule proposition $del(s,qqq_i,t_i,0) \\in A$ if $\\lbrace fires(t_i,0), ptarc(s,t_i,qqq_i) \\rbrace \\subseteq A$ – from rule $r\\ref {r:r:del} $ and supported rule proposition $del(s,qqq_i,t_i,0)$ represents $qqq_i = W(s,t_i) : t_i \\in T_0, (s,t_i) \\in E^-$ – from rule $f\\ref {f:r:ptarc}$ construction $tot\\_decr(q,qqq,0)$ represents $\\sum _{t_i \\in T_0, s \\in \\bullet t_i}{W(s,t_i)}$ – from (C,D,E) above Then firing $T_0 \\cup \\lbrace t_j \\rbrace $ would have required more tokens than are present at its source place $s \\in \\bullet t_j$ .", "Thus, $T_0$ is a maximal set of transitions that can simultaneously fire.", "Then $\\lbrace t_0, \\dots , t_x\\rbrace = T_0$ – using 1(a),1(b) above; and using 1(c) it is a maximal firing set We show $M_1$ is produced by firing $T_0$ in $M_0$ .", "Let $holds(p,q,1) \\in A$ Then $\\lbrace holds(p,q1,0), tot\\_incr(p,q2,0), tot\\_decr(p,q3,0) \\rbrace \\subseteq A : q=q1+q2-q3$ – from rule $r\\ref {r:nextstate}$ and supported rule proposition Then, $holds(p,q1,0) \\in A$ represents $q1=M_0(p)$ – given, rule $i\\ref {i:holds}$ construction; and $\\lbrace add(p,q2_0,t_0,0), \\dots , $ $add(p,q2_j,t_j,0)\\rbrace \\subseteq A : q2_0 + \\dots + q2_j = q2$ and $\\lbrace del(p,q3_0,t_0,0), \\dots , $ $del(p,q3_l,t_l,0)\\rbrace \\subseteq A : q3_0 + \\dots + q3_l = q3$ – rules $r\\ref {r:totincr},r\\ref {r:totdecr}$ using supported rule proposition Then $\\lbrace fires(t_0,0), \\dots , fires(t_j,0) \\rbrace \\subseteq A$ and $\\lbrace fires(t_0,0), \\dots , fires(t_l,0) \\rbrace \\\\ \\subseteq A$ – rules $r\\ref {r:add},r\\ref {r:del}$ and supported rule proposition, respectively Then $\\lbrace fires(t_0,0), \\dots , $ $fires(t_j,0) \\rbrace \\cup \\lbrace fires(t_0,0), \\dots , $ $fires(t_l,0) \\rbrace \\subseteq A = \\lbrace fires(t_0,0), \\dots , $ $fires(t_x,0) \\rbrace \\subseteq A$ – set union of subsets Then for each $fires(t_x,0) \\in A$ we have $t_x \\in T_0$ – already shown in item REF above Then $q = M_0(p) + \\sum _{t_x \\in T_0 \\wedge p \\in t_x \\bullet }{W(t_x,p)} - \\sum _{t_x \\in T_0 \\wedge p \\in \\bullet t_x}{W(p,t_x)}$ – from (REF ) above and the following Each $add(p,q_j,t_j,0) \\in A$ represents $q_j=W(t_j,p)$ for $p \\in t_j \\bullet $ – rule $r\\ref {r:add}$ encoding, and definition REF of transition execution in $PN$ Each $del(p,t_y,q_y,0) \\in A$ represents $q_y=W(p,t_y)$ for $p \\in \\bullet t_y$ – from rule $r\\ref {r:del}$ encoding, and definition REF of transition execution in $PN$ Each $tot\\_incr(p,q2,0) \\in A$ represents $q2=\\sum _{t_x \\in T_0 \\wedge p \\in t_x \\bullet }{W(t_x,p)}$ – aggregate assignment atom semantics in rule $r\\ref {r:totincr}$ Each $tot\\_decr(p,q3,0) \\in A$ represents $q3=\\sum _{t_x \\in T_0 \\wedge p \\in \\bullet t_x}{W(p,t_x)}$ – aggregate assignment atom semantics in rule $r\\ref {r:totdecr}$ Then, $M_1(p) = q$ – since $holds(p,q,1) \\in A$ encodes $q=M_1(p)$ from construction Inductive Step: Let $k > 0$ , and $M_k$ is a valid marking in $X$ for $PN$ , show [(1)] $T_k$ is a valid firing set for $M_k$ , and $T_k$ 's firing in $M_k$ produces marking $M_{k+1}$ .", "We show $T_k$ is a valid firing set.", "Let $\\lbrace fires(t_0,k),\\dots ,fires(t_x,k) \\rbrace $ be the set of all $fires(\\dots ,k)$ atoms in $A$ , Then for each $fires(t_i,k) \\in A$ $enabled(t_i,k) \\in A$ – from rule $a\\ref {a:fires}$ and supported rule proposition Then $notenabled(t_i,k) \\notin A$ – from rule $e\\ref {e:enabled}$ and supported rule proposition Then $body(e\\ref {e:ne:ptarc})$ must hold in $A$ – from rule $e\\ref {e:ne:ptarc}$ and forced atom proposition Then $q \\lnot < n_i \\equiv q \\ge n_i$ in $e\\ref {e:ne:ptarc}$ for all $\\lbrace holds(p,q,k), ptarc(p,t_i,n) \\rbrace \\subseteq A$ – from $e\\ref {e:ne:ptarc}$ , forced atom proposition, and the following $holds(p,q,k) \\in A$ represents $q=M_k(p)$ – construction, inductive assumption $ptarc(p,t,n_i) \\in A$ represents $n_i=W(p,t_i)$ – rule $f\\ref {f:ptarc}$ construction Then $\\forall p \\in \\bullet t_i$ , $M_k(p) \\ge W(p,t_i)$ – from definition REF of preset $\\bullet t_i$ in $PN$ Then $t_i$ is enabled and can fire in $PN$ , as a result it can belong to $T_k$ – from definition REF of enabled transition And $consumesmore \\notin A$ , since $A$ is an answer set of $\\Pi ^1$ – from rule $a\\ref {a:overc:elim}$ and supported rule proposition Then $\\nexists consumesmore(p,k) \\in A$ – from rule $a\\ref {a:overc:gen}$ and forced atom proposition Then $\\nexists \\lbrace holds(p,q,k), tot\\_decr(p,q1,k) \\rbrace \\subseteq A : q1 > q$ in $body(a\\ref {a:overc:place})$ – from $a\\ref {a:overc:place}$ and forced atom proposition Then $\\nexists p : \\sum _{t_i \\in \\lbrace t_0,\\dots ,t_x \\rbrace , p \\in \\bullet t_i}{W(p,t_i)} > M_k(p)$ – from the following $holds(p,q,k) \\in A$ represents $q=M_k(p)$ – by construction of $\\Pi ^1$ , and the inductive assumption about $M_k(p)$ $tot\\_decr(p,q1,k) \\in A$ if $\\lbrace del(p,q1_0,t_0,k), \\dots , del(p,q1_x,t_x,k) \\rbrace \\subseteq A$ , where $q1=q1_0+\\dots +q1_x$ – from $r\\ref {r:totdecr}$ and forced atom proposition $del(p,q1_i,t_i,k) \\in A$ if $\\lbrace fires(t_i,k), ptarc(p,t_i,q1_i) \\rbrace \\subseteq A$ – from $r\\ref {r:del}$ and supported rule proposition $del(p,q1_i,t_i,k)$ represents removal of $q1_i = W(p,t_i)$ tokens from $p \\in \\bullet t_i$ – from construction rule $r\\ref {r:del}$ , supported rule proposition, and definition REF of transition execution in $PN$ Then the set of transitions $T_k$ do not conf – by the definition REF of conflicting transitions And for each $enabled(t_j,k) \\in A$ and $fires(t_j,k) \\notin A$ , $could\\_not\\_have(t_j,k) \\in A$ , since $A$ is an answer set of $\\Pi ^1$ - from rule $a\\ref {a:maxfire:elim}$ and supported rule proposition Then $\\lbrace enabled(t_j,k), holds(s,qq,k), ptarc(s,t_j,q,k), \\\\ tot\\_decr(s,qqq,k) \\rbrace \\subseteq A$ , such that $q > qq - qqq$ and $fires(t_j,0) \\notin A$ - from rule $a\\ref {a:maxfire:cnh}$ and supported rule proposition Then for an $s \\in \\bullet t_j$ , $W(s,t_j) > M_k(s) - \\sum _{t_i \\in T_k, s \\in \\bullet t_i}{W(s,t_i)}$ - from the following: $ptarc(s,t_i,q)$ represents $q=W(s,t_i)$ – from rule $f\\ref {f:r:ptarc}$ construction $holds(s,qq,k)$ represents $qq=M_k(s)$ – from $i\\ref {i:holds}$ construction $tot\\_decr(s,qqq,k) \\in A$ if $\\lbrace del(s,qqq_0,t_0,k), \\dots , del(s,qqq_x,t_x,k) \\rbrace \\subseteq A$ – from rule $r\\ref {r:totdecr}$ construction and supported rule proposition $del(s,qqq_i,t_i,k) \\in A$ if $\\lbrace fires(t_i,k), ptarc(s,t_i,qqq_i) \\rbrace \\subseteq A$ – from rule $r\\ref {r:r:del} $ and supported rule proposition $del(s,qqq_i,t_i,k)$ represents $qqq_i = W(s,t_i) : t_i \\in T_k, (s,t_i) \\in E^-$ – from rule $f\\ref {f:r:ptarc}$ construction $tot\\_decr(q,qqq,k)$ represents $\\sum _{t_i \\in T_k, s \\in \\bullet t_i}{W(s,t_i)}$ – from (C,D,E) above Then firing $T_k \\cup \\lbrace t_j \\rbrace $ would have required more tokens than are present at its source place $s \\in \\bullet t_j$ .", "Thus, $T_k$ is a maximal set of transitions that can simultaneously fire.", "Then $\\lbrace t_0,\\dots ,t_x \\rbrace = T_k$ – using 1(a),1(b) above; and using 1(c) it is a maximal firing set We show $M_{k+1}$ is produced by firing $T_k$ in $M_k$ .", "Let $holds(p,q,k+1) \\in A$ Then $\\lbrace holds(p,q1,k), tot\\_incr(p,q2,k), tot\\_decr(p,q3,k) \\rbrace \\in A : q=q2+q2-q3$ – from rule $r\\ref {r:nextstate}$ and supported rule proposition $holds(p,q1,k) \\in A$ represents $q1=M_k(p)$ – construction, inductive assumption; and $\\lbrace add(p,q2_0,t_0,k), \\dots , $ $add(p,q2_j,t_j,k) \\rbrace \\subseteq A : q2_0+\\dots +q2_j=q2$ and $\\lbrace del(p,q3_0,t_0,k), \\dots , $ $del(p,q3_l,t_l,k) \\rbrace \\subseteq A : q3_0+\\dots +q3_l=q3$ – from rules $r\\ref {r:totincr},r\\ref {r:totdecr}$ using supported rule proposition, respectively Then $\\lbrace fires(t_0,k), \\dots , fires(t_j,k) \\rbrace \\subseteq A$ and $\\lbrace fires(t_0,k), \\dots , fires(t_l,k) \\rbrace \\\\ \\subseteq A$ – from rules $r\\ref {r:add} ,r\\ref {r:del} $ using supported rule proposition, respectively Then $\\lbrace fires(t_0,k), \\dots , $ $fires(t_j,k) \\rbrace \\cup \\lbrace fires(t_0,k), \\dots , $ $fires(t_l,k) \\rbrace = \\lbrace fires(t_0,k), \\dots , $ $fires(t_x,k) \\rbrace \\subseteq A$ – subset union property Then for each $fires(t_x,k) \\in A$ we have $t_x \\in T_k$ - already shown in item (REF ) above Then $q = M_k(p) + \\sum _{t_x \\in T_k \\wedge p \\in t_x \\bullet }{W(t_x,p)} - \\sum _{t_x \\in T_k \\wedge p \\in \\bullet t_x}{W(p,t_x)}$ – from (REF ) above and the following Each $add(p,q_j,t_j,k) \\in A$ represents $q_j = W(t_j,p)$ for $p \\in t_j \\bullet $ – rule $r\\ref {r:add}$ encoding and definition REF of transition execution in PN Each $del(p,t_y,q_y,k) \\in A$ represents $q_y = W(p,t_y)$ for $p \\in \\bullet t_y$ – rule $r\\ref {r:del}$ encoding and definition REF of transition execution in PN Each $tot\\_incr(p,q2,k) \\in A$ represents $q2 = \\sum _{t_x \\in T_k \\wedge p \\in t_x \\bullet }{W(t_x,p)}$ – aggregate assignment atom semantics in rule $r\\ref {r:totincr}$ Each $tot\\_decr(p,q3,k) \\in A$ represents $q3 = \\sum _{t_x \\in T_k \\wedge p \\in \\bullet t_x}{W(p,t_x)}$ – aggregate assignment atom semantics in rule $r\\ref {r:totdecr}$ Then $M_{k+1}(p) = q$ – since $holds(p,q,k+1) \\in A$ encodes $q=M_{k+1}(p)$ – from construction As a result, for any $n > k$ , $T_n$ will be a valid firing set w.r.t.", "$M_n$ and its firing produces marking $M_{n+1}$ .", "Conclusion: Since both REF and REF hold, $X=M_0,T_0,M_1,\\dots ,M_k,T_k,M_{k+1}$ is an execution sequence of $PN(P,T,E,W)$ (w.r.t.", "$M_0$ ) iff there is an answer set $A$ of $\\Pi ^1(PN,M_0,k,ntok)$ such that (REF ) and (REF ) hold.", "Proof of Proposition REF Let $PN=(P,T,E,W,R)$ be a Petri Net, $M_0$ be its initial marking and let $\\Pi ^2(PN,M_0,k,ntok)$ by the ASP encoding of $PN$ and $M_0$ over a simulation length $k$ , with maximum $ntok$ tokens on any place node, as defined in section REF .", "Then $X=M_0,T_0,M_1,\\dots ,M_k,T_k,M_{k+1}$ is an execution sequence of $PN$ (w.r.t $M_0$ ) iff there is an answer set $A$ of $\\Pi ^2(PN,M_0,k,ntok)$ such that: $\\lbrace fires(t,ts) : t \\in T_{ts}, 0 \\le ts \\le k\\rbrace = \\lbrace fires(t,ts) : fires(t,ts) \\in A \\rbrace $ $\\begin{split}\\lbrace holds(p,q,ts) &: p \\in P, q = M_{ts}(p), 0 \\le ts \\le k+1 \\rbrace \\\\&= \\lbrace holds(p,q,ts) : holds(p,q,ts) \\in A \\rbrace \\end{split}$ We prove this by showing that: Given an execution sequence $X$ , we create a set $A$ such that it satisfies (REF ) and (REF ) and show that $A$ is an answer set of $\\Pi ^2$ Given an answer set $A$ of $\\Pi ^2$ , we create an execution sequence $X$ such that (REF ) and (REF ) are satisfied.", "First we show (REF ): Given $PN$ and an execution sequence $X$ of $PN$ , we create a set $A$ as a union of the following sets: $A_1=\\lbrace num(n) : 0 \\le n \\le ntok \\rbrace $ $A_2=\\lbrace time(ts) : 0 \\le ts \\le k\\rbrace $ $A_3=\\lbrace place(p) : p \\in P \\rbrace $ $A_4=\\lbrace trans(t) : t \\in T \\rbrace $ $A_5=\\lbrace ptarc(p,t,n,ts) : (p,t) \\in E^-, n=W(p,t), 0 \\le ts \\le k\\rbrace $ , where $E^- \\subseteq E$ $A_6=\\lbrace tparc(t,p,n,ts) : (t,p) \\in E^+, n=W(t,p), 0 \\le ts \\le k\\rbrace $ , where $E^+ \\subseteq E$ $A_7=\\lbrace holds(p,q,0) : p \\in P, q=M_{0}(p) \\rbrace $ $A_8=\\lbrace notenabled(t,ts) : t \\in T, 0 \\le ts \\le k, \\exists p \\in \\bullet t, M_{ts}(p) < W(p,t) \\rbrace $ per definition REF (enabled transition) $A_9=\\lbrace enabled(t,ts) : t \\in T, 0 \\le ts \\le k, \\forall p \\in \\bullet t, W(p,t) \\le M_{ts}(p) \\rbrace $ per definition REF (enabled transition) $A_{10}=\\lbrace fires(t,ts) : t \\in T_{ts}, 0 \\le ts \\le k \\rbrace $ per definition REF (firing set), only an enabled transition may fire $A_{11}=\\lbrace add(p,q,t,ts) : t \\in T_{ts}, p \\in t \\bullet , q=W(t,p), 0 \\le ts \\le k \\rbrace $ per definition REF (transition execution) $A_{12}=\\lbrace del(p,q,t,ts) : t \\in T_{ts}, p \\in \\bullet t, q=W(p,t), 0 \\le ts \\le k \\rbrace \\cup \\lbrace del(p,q,t,ts) : t \\in T_{ts}, p \\in R(t), q=M_{ts}(p), 0 \\le ts \\le k \\rbrace $ per definition REF (transition execution) $A_{13}=\\lbrace tot\\_incr(p,q,ts) : p \\in P, q=\\sum _{t \\in T_{ts}, p \\in t \\bullet }{W(t,p)}, 0 \\le ts \\le k \\rbrace $ per definition REF (execution) $A_{14}=\\lbrace tot\\_decr(p,q,ts): p \\in P, q=\\sum _{t \\in T_{ts}, p \\in \\bullet t}{W(p,t)}+\\sum _{t \\in T_{ts}, p \\in R(t)}{M_{ts}(p)}, \\\\ 0 \\le ts \\le k \\rbrace $ per definition REF (execution) $A_{15}=\\lbrace consumesmore(p,ts) : p \\in P, q=M_{ts}(p), q1=\\sum _{t \\in T_{ts}, p \\in \\bullet t}{W(p,t)}+\\sum _{t \\in T_{ts}, p \\in R(t)}{M_{ts}(p)}, q1 > q, 0 \\le ts \\le k \\rbrace $ per definition REF (conflicting transitions) $A_{16}=\\lbrace consumesmore : \\exists p \\in P : q=M_{ts}(p), q1=\\sum _{t \\in T_{ts}, p \\in \\bullet t}{W(p,t)}+\\sum _{t \\in T_{ts}, p \\in R(t)}{M_{ts}(p)}, q1 > q, 0 \\le ts \\le k \\rbrace $ per definition REF (conflicting transitions) $A_{17}=\\lbrace could\\_not\\_have(t,ts) : t \\in T, (\\forall p \\in \\bullet t, W(p,t) \\le M_{ts}(p)), t \\notin T_{ts}, (\\exists p \\in \\bullet t \\cup R(t) : q > M_{ts}(p) - (\\sum _{t^{\\prime } \\in T_{ts}, p \\in \\bullet t^{\\prime }}{W(p,t^{\\prime })} + \\sum _{t^{\\prime } \\in T_{ts}, p \\in R(t^{\\prime })}{M_{ts}(p)}), q=W(p,t) \\text{~if~} p \\in \\bullet t \\text{~or~} M_{ts}(p) \\text{~otherwise~}), 0 \\le ts \\le k \\rbrace $ per the maximal firing set semantics $A_{18}=\\lbrace holds(p,q,ts+1) : p \\in P, q=M_{ts+1}(p), 0 \\le ts < k\\rbrace $ , where $M_{ts+1}(p) = M_{ts}(p) - (\\sum _{\\begin{array}{c}t \\in T_{ts}, p \\in \\bullet t\\end{array}}{W(p,t)} + \\sum _{\\begin{array}{c}t \\in T_{ts}, p \\in R(t)\\end{array}}{M_{ts}(p)}) + \\sum _{\\begin{array}{c}t \\in T_{ts}, p \\in t \\bullet \\end{array}}{W(t,p)}$ according to definition REF (firing set execution) $A_{19}=\\lbrace ptarc(p,t,n,ts) : p \\in R(t), n=M_{ts}(p), n > 0, 0 \\le ts \\le k\\rbrace $ We show that $A$ satisfies (REF ) and (REF ), and $A$ is an answer set of $\\Pi ^1$ .", "$A$ satisfies (REF ) and (REF ) by its construction above.", "We show $A$ is an answer set of $\\Pi ^1$ by splitting.", "We split $lit(\\Pi ^1)$ into a sequence of $7(k+1)+2$ sets: $U_0= head(f\\ref {f:place}) \\cup head(f\\ref {f:trans}) \\cup head(f\\ref {f:time}) \\cup head(f\\ref {f:num}) \\cup head(i\\ref {i:holds}) = \\lbrace place(p) : p \\in P\\rbrace \\cup \\lbrace trans(t) : t \\in T\\rbrace \\cup \\lbrace time(0), \\dots , time(k)\\rbrace \\cup \\lbrace num(0), \\dots , num(ntok)\\rbrace \\cup \\lbrace holds(p,q,0) : p \\in P, q=M_0(p) \\rbrace $ $U_{7k+1}=U_{7k+0} \\cup head(f\\ref {f:r:ptarc})^{ts=k} \\cup head(f\\ref {f:r:tparc})^{ts=k} \\cup head(f\\ref {f:rptarc})^{ts=k} = U_{7k+0} \\cup \\\\ \\lbrace ptarc(p,t,n,k) : (p,t) \\in E^-, n=W(p,t) \\rbrace \\cup \\lbrace tparc(t,p,n,k) : (t,p) \\in E^+, n=W(t,p) \\rbrace \\cup \\lbrace ptarc(p,t,n,k) : p \\in R(t), n=M_{k}(p), n > 0 \\rbrace $ $U_{7k+2}=U_{7k+1} \\cup head(e\\ref {e:r:ne:ptarc} )^{ts=k} = U_{7k+1} \\cup \\lbrace notenabled(t,k) : t \\in T \\rbrace $ $U_{7k+3}=U_{7k+2} \\cup head(e\\ref {e:enabled})^{ts=k} = U_{7k+2} \\cup \\lbrace enabled(t,k) : t \\in T \\rbrace $ $U_{7k+4}=U_{7k+3} \\cup head(a\\ref {a:fires})^{ts=k} = U_{7k+3} \\cup \\lbrace fires(t,k) : t \\in T \\rbrace $ $U_{7k+5}=U_{7k+4} \\cup head(r\\ref {r:r:add} )^{ts=k} \\cup head(r\\ref {r:r:del} )^{ts=k} = U_{7k+4} \\cup \\lbrace add(p,q,t,k) : p \\in P, t \\in T, q=W(t,p) \\rbrace \\cup \\lbrace del(p,q,t,k) : p \\in P, t \\in T, q=W(p,t) \\rbrace \\cup \\lbrace del(p,q,t,k) : p \\in P, t \\in T, q=M_{k}(p) \\rbrace $ $U_{7k+6}=U_{7k+5} \\cup head(r\\ref {r:totincr})^{ts=k} \\cup head(r\\ref {r:totdecr})^{ts=k} = U_{7k+5} \\cup \\lbrace tot\\_incr(p,q,k) : p \\in P, 0 \\le q \\le ntok \\rbrace \\cup \\lbrace tot\\_decr(p,q,k) : p \\in P, 0 \\le q \\le ntok \\rbrace $ $U_{7k+7}=U_{7k+6} \\cup head(r\\ref {r:nextstate})^{ts=k} \\cup head(a\\ref {a:overc:place})^{ts=k} \\cup head(a\\ref {a:r:maxfire:cnh})^{ts=k} $ $= U_{7k+6} \\cup \\\\ \\lbrace consumesmore(p,k) : p \\in P\\rbrace \\cup \\lbrace holds(p,q,k+1) : p \\in P, 0 \\le q \\le ntok \\rbrace \\cup \\lbrace could\\_not\\_have(t,k) : t \\in T \\rbrace $ $U_{7k+8}=U_{7k+7} \\cup head(a\\ref {a:overc:gen}) = U_{7k+7} \\cup \\lbrace consumesmore \\rbrace $ where $head(r_i)^{ts=k}$ are head atoms of ground rule $r_i$ in which $ts=k$ .", "We write $A_i^{ts=k} = \\lbrace a(\\dots ,ts) : a(\\dots ,ts) \\in A_i, ts=k \\rbrace $ as short hand for all atoms in $A_i$ with $ts=k$ .", "$U_{\\alpha }, 0 \\le \\alpha \\le 7k+8$ form a splitting sequence, since each $U_i$ is a splitting set of $\\Pi ^1$ , and $\\langle U_{\\alpha }\\rangle _{\\alpha < \\mu }$ is a monotone continuous sequence, where $U_0 \\subseteq U_1 \\dots \\subseteq U_{7k+8}$ and $\\bigcup _{\\alpha < \\mu }{U_{\\alpha }} = lit(\\Pi ^1)$ .", "We compute the answer set of $\\Pi ^2$ using the splitting sets as follows: $bot_{U_0}(\\Pi ^2) = f\\ref {f:place} \\cup f\\ref {f:trans} \\cup f\\ref {f:time} \\cup i\\ref {i:holds} \\cup f\\ref {f:num}$ and $X_0 = A_1 \\cup \\dots \\cup A_4 \\cup A_7$ ($= U_0$ ) is its answer set – using forced atom proposition $eval_{U_0}(bot_{U_1}(\\Pi ^2) \\setminus bot_{U_0}(\\Pi ^2), X_0) = \\lbrace ptarc(p,t,q,0) \\text{:-}.", "\| q=W(p,t) \\rbrace \\cup \\\\ \\lbrace tparc(t,p,q,0) \\text{:-}.", "\| $ $q=W(t,p) \\rbrace \\cup \\lbrace ptarc(p,t,q,0) \\text{:-}.", "\| $ $q=M_0(p) \\rbrace $ .", "Its answer set $X_1=A_5^{ts=0} \\cup A_6^{ts=0} \\cup A_{19}^{ts=0}$ – using forced atom proposition and construction of $A_5, A_6, A_{19}$ .", "$eval_{U_1}(bot_{U_2}(\\Pi ^2) \\setminus bot_{U_1}(\\Pi ^2), X_0 \\cup X_1) = \\lbrace notenabled(t,0) \\text{:-} .", "\| \\lbrace trans(t), \\\\ ptarc(p,t,n,0), holds(p,q,0) \\rbrace \\subseteq X_0 \\cup X_1, \\text{~where~} q < n \\rbrace $ .", "Its answer set $X_2=A_8^{ts=0}$ – using forced atom proposition and construction of $A_8$ .", "where, $q=M_0(p)$ , and $n=W(p,t)$ for an arc $(p,t) \\in E^-$ – by construction of $i\\ref {i:holds}$ and $f\\ref {f:r:ptarc}$ in $\\Pi ^2$ , and in an arc $(p,t) \\in E^-$ , $p \\in \\bullet t$ (by definition REF of preset) thus, $notenabled(t,0) \\in X_1$ represents $\\exists p \\in \\bullet t : M_0(p) < W(p,t)$ .", "$eval_{U_2}(bot_{U_3}(\\Pi ^2) \\setminus bot_{U_2}(\\Pi ^2), X_0 \\cup \\dots \\cup X_2) = \\lbrace enabled(t,0) \\text{:-}.", "\| trans(t) \\in X_0 \\cup \\dots \\cup X_2, notenabled(t,0) \\notin X_0 \\cup \\dots \\cup X_2 \\rbrace $ .", "Its answer set is $X_3 = A_9^{ts=0}$ – using forced atom proposition and construction of $A_9$ .", "since an $enabled(t,0) \\in X_3$ if $\\nexists ~notenabled(t,0) \\in X_0 \\cup \\dots \\cup X_2$ ; which is equivalent to $\\nexists p \\in \\bullet t : M_0(p) < W(p,t) \\equiv \\forall p \\in \\bullet t: M_0(p) \\ge W(p,t)$ .", "$eval_{U_3}(bot_{U_4}(\\Pi ^2) \\setminus bot_{U_3}(\\Pi ^2), X_0 \\cup \\dots \\cup X_3) = \\lbrace \\lbrace fires(t,0)\\rbrace \\text{:-}.", "\| enabled(t,0) \\\\ \\text{~holds in~} X_0 \\cup \\dots \\cup X_3 \\rbrace $ .", "It has multiple answer sets $X_{4.1}, \\dots , X_{4.n}$ , corresponding to elements of power set of $fires(t,0)$ atoms in $eval_{U_3}(...)$ – using supported rule proposition.", "Since we are showing that the union of answer sets of $\\Pi ^2$ determined using splitting is equal to $A$ , we only consider the set that matches the $fires(t,0)$ elements in $A$ and call it $X_4$ , ignoring the rest.", "Thus, $X_4 = A_{10}^{ts=0}$ , representing $T_0$ .", "in addition, for every $t$ such that $enabled(t,0) \\in X_0 \\cup \\dots \\cup X_3, R(t) \\ne \\emptyset $ ; $fires(t,0) \\in X_4$ – per definition REF (firing set); requiring that a reset transition is fired when enabled thus, the firing set $T_0$ will not be eliminated by the constraint $f\\ref {f:c:rptarc:elim}$ $eval_{U_4}(bot_{U_5}(\\Pi ^2) \\setminus bot_{U_4}(\\Pi ^2), X_0 \\cup \\dots \\cup X_4) = \\lbrace add(p,n,t,0) \\text{:-}.", "\| \\lbrace fires(t,0), \\\\ tparc(t,p,n,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_4 \\rbrace \\cup \\lbrace del(p,n,t,0) \\text{:-}.", "\| \\lbrace fires(t,0), ptarc(p,t,n,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_4 \\rbrace $ .", "It's answer set is $X_5 = A_{11}^{ts=0} \\cup A_{12}^{ts=0}$ – using forced atom proposition and definitions of $A_{11}$ and $A_{12}$ .", "where, each $add$ atom is equivalent to $n=W(t,p) : p \\in t \\bullet $ , and each $del$ atom is equivalent to $n=W(p,t) : p \\in \\bullet t$ ; or $n=M_k(p) : p \\in R(t)$ , representing the effect of transitions in $T_0$ – by construction $eval_{U_5}(bot_{U_6}(\\Pi ^2) \\setminus bot_{U_5}(\\Pi ^2), X_0 \\cup \\dots \\cup X_5) = \\lbrace tot\\_incr(p,qq,0) \\text{:-}.", "\| $ $qq=\\sum _{add(p,q,t,0) \\in X_0 \\cup \\dots \\cup X_5}{q} \\rbrace \\cup \\lbrace tot\\_decr(p,qq,0) \\text{:-}.", "\| qq=\\sum _{del(p,q,t,0) \\in X_0 \\cup \\dots \\cup X_5}{q} \\rbrace $ .", "It's answer set is $X_6 = A_{13}^{ts=0} \\cup A_{14}^{ts=0}$ – using forced atom proposition, definitions of $A_{13}$ , $A_{14}$ , and definition REF (semantics of aggregate assignment atom).", "where, each for $tot\\_incr(p,qq,0)$ , $qq=\\sum _{add(p,q,t,0) \\in X_0 \\cup \\dots X_5}{q}$ $\\equiv qq=\\sum _{t \\in X_4, p \\in t \\bullet }{W(p,t)}$ , and each $tot\\_decr(p,qq,0)$ , $qq=\\sum _{del(p,q,t,0) \\in X_0 \\cup \\dots X_5}{q}$ $\\equiv qq=\\sum _{t \\in X_4, p \\in \\bullet t}{W(t,p)} + \\sum _{t \\in X_4, p \\in R(t)}{M_{k}(p)}$ , represent the net effect of transitions in $T_0$ – by construction $eval_{U_6}(bot_{U_7}(\\Pi ^2) \\setminus bot_{U_6}(\\Pi ^2), X_0 \\cup \\dots \\cup X_6) = \\lbrace consumesmore(p,0) \\text{:-}.", "\| \\\\ \\lbrace holds(p,q,0), tot\\_decr(p,q1,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_6, q1 > q \\rbrace \\cup \\lbrace holds(p,q,1) \\text{:-}., \| \\\\ \\lbrace holds(p,q1,0), tot\\_incr(p,q2,0), tot\\_decr(p,q3,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_6, q=q1+q2-q3 \\rbrace \\cup \\lbrace could\\_not\\_have(t,0) \\text{:-}.", "\| \\lbrace enabled(t,0), ptarc(s,t,q), holds(s,qq,0), \\\\ tot\\_decr(s,qqq,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_6, fires(t,0) \\notin (X_0 \\cup \\dots \\cup X_6), q > qq-qqq \\rbrace $ .", "It's answer set is $X_7 = A_{15}^{ts=0} \\cup A_{17}^{ts=0} \\cup A_{18}^{ts=0}$ – using forced atom proposition and definitions of $A_{15}, A_{17}, A_{18}$ .", "where, $consumesmore(p,0)$ represents $\\exists p : q=M_0(p), q1= \\\\ \\sum _{t \\in T_0, p \\in \\bullet t}{W(p,t)}+$ $\\sum _{t \\in T_0, p \\in R(t)}{M_k(p)}, $ $q1 > q$ , indicating place $p$ will be overconsumed if $T_0$ is fired – as defined in definition REF (conflicting transitions) and, $holds(p,q,1)$ represents $q=M_1(p)$ – by construction of $\\Pi ^2$ and $could\\_not\\_have(t,0)$ represents an enabled transition $t$ in $T_0$ that could not fire due to insufficient tokens $X_7$ does not contain $could\\_not\\_have(t,0)$ , when $enabled(t,0) \\in X_0 \\cup \\dots \\cup X_6$ and $fires(t,0) \\notin X_0 \\cup \\dots \\cup X_6$ due to construction of $A$ , encoding of $a\\ref {a:r:maxfire:cnh}$ and its body atoms.", "As a result it is not eliminated by the constraint $a\\ref {a:maxfire:elim}$ $ \\vdots $ $eval_{U_{7k+0}}(bot_{U_{7k+1}}(\\Pi ^2) \\setminus bot_{U_{7k+0}}(\\Pi ^2), X_0 \\cup \\dots \\cup X_{7k+0}) = \\lbrace ptarc(p,t,q,k) \\text{:-}.", "\| q=W(p,t) \\rbrace \\cup \\lbrace tparc(t,p,q,k) \\text{:-}.", "\| q=W(t,p) \\rbrace \\cup \\lbrace ptarc(p,t,q,k) \\text{:-}.", "\| q=M_0(p) \\rbrace $ .", "Its answer set $X_{7k+1}=A_5^{ts=k} \\cup A_6^{ts=k} \\cup A_{19}^{ts=k}$ – using forced atom proposition and construction of $A_5, A_6, A_{19}$ .", "$eval_{U_{7k+1}}(bot_{U_{7k+2}}(\\Pi ^2) \\setminus bot_{U_{7k+1}}(\\Pi ^2), X_0 \\cup X_{7k+1}) = \\lbrace notenabled(t,k) \\text{:-} .", "\| \\lbrace trans(t), \\\\ ptarc(p,t,n,k), holds(p,q,k) \\rbrace \\subseteq X_0 \\cup X_{7k+1}, \\text{~where~} q < n \\rbrace $ .", "Its answer set $X_{7k+2}=A_8^{ts=k}$ – using forced atom proposition and construction of $A_8$ .", "where, $q=M_0(p)$ , and $n=W(p,t)$ for an arc $(p,t) \\in E^-$ – by construction of $holds$ and $ptarc$ predicates in $\\Pi ^2$ , and in an arc $(p,t) \\in E^-$ , $p \\in \\bullet t$ (by definition REF of preset) thus, $notenabled(t,k) \\in X_{7k+1}$ represents $\\exists p \\in \\bullet t : M_0(p) < W(p,t)$ .", "$eval_{U_{7k+2}}(bot_{U_{7k+3}}(\\Pi ^2) \\setminus bot_{U_{7k+2}}(\\Pi ^2), X_0 \\cup \\dots \\cup X_{7k+2}) = \\lbrace enabled(t,k) \\text{:-}.", "\| trans(t) \\in X_0 \\cup \\dots \\cup X_{7k+2} , notenabled(t,k) \\notin X_0 \\cup \\dots \\cup X_{7k+2} \\rbrace $ .", "Its answer set is $X_{7k+3} = A_9^{ts=k}$ – using forced atom proposition and construction of $A_9$ .", "since an $enabled(t,k) \\in X_{7k+3}$ if $\\nexists ~notenabled(t,k) \\in X_0 \\cup \\dots \\cup X_{7k+2}$ ; which is equivalent to $\\nexists p \\in \\bullet t : M_0(p) < W(p,t) \\equiv \\forall p \\in \\bullet t: M_0(p) \\ge W(p,t)$ .", "$eval_{U_{7k+3}}(bot_{U_{7k+4}}(\\Pi ^2) \\setminus bot_{U_{7k+3}}(\\Pi ^2), X_0 \\cup \\dots \\cup X_{7k+3}) = \\lbrace \\lbrace fires(t,k)\\rbrace \\text{:-}.", "\| \\\\ enabled(t,k) \\\\ \\text{~holds in~} X_0 \\cup \\dots \\cup X_{7k+3} \\rbrace $ .", "It has multiple answer sets $X_{7k+4.1}, \\dots , X_{1k+4.n}$ , corresponding to elements of power set of $fires(t,k)$ atoms in $eval_{U_{7k+3}}(...)$ – using supported rule proposition.", "Since we are showing that the union of answer sets of $\\Pi ^2$ determined using splitting is equal to $A$ , we only consider the set that matches the $fires(t,k)$ elements in $A$ and call it $X_{7k+4}$ , ignoring the rest.", "Thus, $X_{7k+4} = A_{10}^{ts=k}$ , representing $T_k$ .", "in addition, for every $t$ such that $enabled(t,k) \\in X_0 \\cup \\dots \\cup X_{7k+3}, R(t) \\ne \\emptyset $ ; $fires(t,k) \\in X_{7k+4}$ – per definition REF (firing set); requiring that a reset transition is fired when enabled thus, the firing set $T_k$ will not be eliminated by the constraint $f\\ref {f:c:rptarc:elim}$ $eval_{U_{7k+4}}(bot_{U_{7k+5}}(\\Pi ^2) \\setminus bot_{U_{7k+4}}(\\Pi ^2), X_0 \\cup \\dots \\cup X_{7k+4}) = \\\\ \\lbrace add(p,n,t,k) \\text{:-}.", "\| \\lbrace fires(t,k), tparc(t,p,n,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{7k+4} \\rbrace \\cup \\\\ \\lbrace del(p,n,t,k) \\text{:-}.", "\| \\lbrace fires(t,k), ptarc(p,t,n,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{7k+4} \\rbrace $ .", "It's answer set is $X_{7k+5} = A_{11}^{ts=k} \\cup A_{12}^{ts=k}$ – using forced atom proposition and definitions of $A_{11}$ and $A_{12}$ .", "where, each $add$ atom is equivalent to $n=W(t,p) : p \\in \\bullet t$ , and each $del$ atom is equivalent to $n=W(p,t) : p \\in t \\bullet $ ; or $n=M_k(p) : p \\in R(t)$ , representing the effect of transitions in $T_k$ $eval_{U_{7k+5}}(bot_{U_{7k+6}}(\\Pi ^2) \\setminus bot_{U_{7k+5}}(\\Pi ^2), X_0 \\cup \\dots \\cup X_{7k+5}) = \\\\ \\lbrace tot\\_incr(p,qq,k) \\text{:-}.", "\| qq=\\sum _{add(p,q,t,k) \\in X_0 \\cup \\dots \\cup X_{7k+5}}{q} \\rbrace \\cup \\\\ \\lbrace tot\\_decr(p,qq,k) \\text{:-}.", "\| qq=\\sum _{del(p,q,t,k) \\in X_0 \\cup \\dots \\cup X_{7k+5}}{q} \\rbrace $ .", "It's answer set is $X_{7k+6} = A_{13}^{ts=k} \\cup A_{14}^{ts=k}$ – using forced atom proposition and definitions of $A_{13}$ and $A_{14}$ .", "where, each $tot\\_incr(p,qq,k)$ , $qq=\\sum _{add(p,q,t,k) \\in X_0 \\cup \\dots X_{7k+5}}{q}$ $\\equiv qq=\\sum _{t \\in X_{7k+4}, p \\in t \\bullet }{W(p,t)}$ , and each $tot\\_decr(p,qq,k)$ , $qq=\\sum _{del(p,q,t,k) \\in X_0 \\cup \\dots X_{7k+5}}{q}$ $\\equiv qq=\\sum _{t \\in X_{7k+4}, p \\in \\bullet t}{W(t,p)} + \\sum _{t \\in X_{7k+4}, p \\in R(t)}{M_{k}(p)}$ , represent the net effect of transitions in $T_k$ $eval_{U_{7k+6}}(bot_{U_{7k+7}}(\\Pi ^2) \\setminus bot_{U_{7k+6}}(\\Pi ^2), X_0 \\cup \\dots \\cup X_{7k+6}) = \\lbrace consumesmore(p,k) \\text{:-}.", "\| \\\\ \\lbrace holds(p,q,k), tot\\_decr(p,q1,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{7k+6}, q1 > q \\rbrace \\cup \\lbrace holds(p,q,k+1) \\text{:-}., \| \\\\ \\lbrace holds(p,q1,k), tot\\_incr(p,q2,k), tot\\_decr(p,q3,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{7k+6}, q=q1+q2-q3 \\rbrace \\cup \\lbrace could\\_not\\_have(t,k) \\text{:-}.", "\| \\lbrace enabled(t,k), ptarc(s,t,q), holds(s,qq,k), \\\\ tot\\_decr(s,qqq,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{7k+6}, fires(t,k) \\notin (X_0 \\cup \\dots \\cup X_{7k+6}), q > qq-qqq \\rbrace $ .", "It's answer set is $X_{7k+7} = A_{15}^{ts=k} \\cup A_{17}^{ts=k} \\cup A_{18}^{ts=k}$ – using forced atom proposition and definitions of $A_{15}, A_{17}, A_{18}$ .", "where, $consumesmore(p,k)$ represents $\\exists p : q=M_0(p), q1= \\\\ \\sum _{t \\in T_0, p \\in \\bullet t}{W(p,t)}+\\sum _{t \\in T_0, p \\in R(t)}{M_k(p)}, q1 > q$ , indicating place $p$ that will be over consumed if $T_k$ is fired, as defined in definition REF (conflicting transitions) $holds(p,q,k+1)$ represents $q=M_{k+1}(p)$ – by construction of $\\Pi ^2$ , and $could\\_not\\_have(t,k)$ represents enabled transition $t$ in $T_k$ that could not be fired due to insufficient tokens $X_{7k+7}$ does not contain $could\\_not\\_have(t,k)$ , when $enabled(t,k) \\in X_0 \\cup \\dots \\cup X_{7k+6}$ and $fires(t,k) \\notin X_0 \\cup \\dots \\cup X_{7k+6}$ due to the construction of $A$ , encoding of $a\\ref {a:r:maxfire:cnh}$ and its body atoms.", "As a result it is not eliminated by the constraint $a\\ref {a:maxfire:elim}$ $eval_{U_{7k+7}}(bot_{U_{7k+8}}(\\Pi ^2) \\setminus bot_{U_{7k+7}}(\\Pi ^2), X_0 \\cup \\dots \\cup X_{7k+7}) = \\lbrace consumesmore \\text{:-}.", "\| $ $ \\lbrace consumesmore(p,0),\\dots ,$ $consumesmore(p,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{7k+7} \\rbrace $ .", "It's answer set is $X_{7k+8} = A_{16}$ – using forced atom proposition $X_{7k+8}$ will be empty since none of $consumesmore(p,0),\\dots , \\\\ consumesmore(p,k)$ hold in $X_0 \\cup \\dots \\cup X_{7k+7}$ due to the construction of $A$ , encoding of $a\\ref {a:overc:place}$ and its body atoms.", "As a result, it is not eliminated by the constraint $a\\ref {a:overc:elim}$ The set $X = X_0 \\cup \\dots \\cup X_{7k+8}$ is the answer set of $\\Pi ^2$ by the splitting sequence theorem REF .", "Each $X_i, 0 \\le i \\le 7k+8$ matches a distinct portion of $A$ , and $X = A$ , thus $A$ is an answer set of $\\Pi ^2$ .", "Next we show (REF ): Given $\\Pi ^2$ be the encoding of a Petri Net $PN(P,T,E,W,R)$ with initial marking $M_0$ , and $A$ be an answer set of $\\Pi ^2$ that satisfies (REF ) and (REF ), then we can construct $X=M_0,T_0,\\dots ,M_k,T_k,M_{k+1}$ from $A$ , such that it is an execution sequence of $PN$ .", "We construct the $X$ as follows: $M_i = (M_i(p_0), \\dots , M_i(p_n))$ , where $\\lbrace holds(p_0,M_i(p_0),i), \\dots holds(p_n,M_i(p_n),i) \\rbrace \\\\ \\subseteq A$ , for $0 \\le i \\le k+1$ $T_i = \\lbrace t : fires(t,i) \\in A\\rbrace $ , for $0 \\le i \\le k$ and show that $X$ is indeed an execution sequence of $PN$ .", "We show this by induction over $k$ (i.e.", "given $M_k$ , $T_k$ is a valid firing set and its firing produces marking $M_{k+1}$ ).", "Base case: Let $k=0$ , and $M_0$ is a valid marking in $X$ for $PN$ , show [(1)] $T_0$ is a valid firing set for $M_0$ , and $T_0$ 's firing in $M_0$ produces marking $M_1$ .", "We show $T_0$ is a valid firing set for $M_0$ .", "Let $\\lbrace fires(t_0,0), \\dots , fires(t_x,0) \\rbrace $ be the set of all $fires(\\dots ,0)$ atoms in $A$ , Then for each $fires(t_i,0) \\in A$ $enabled(t_i,0) \\in A$ – from rule $a\\ref {a:fires}$ and supported rule proposition Then $notenabled(t_i,0) \\notin A$ – from rule $e\\ref {e:enabled}$ and supported rule proposition Then $body(e\\ref {e:r:ne:ptarc})$ must not hold in $A$ – from rule $e\\ref {e:r:ne:ptarc}$ and forced atom proposition Then $q \\lnot < n_i \\equiv q \\ge n_i$ in $e\\ref {e:r:ne:ptarc}$ for all $\\lbrace holds(p,q,0), ptarc(p,t_i,n_i,0)\\rbrace \\subseteq A$ – from $e\\ref {e:r:ne:ptarc}$ , forced atom proposition, and the following $holds(p,q,0) \\in A$ represents $q=M_0(p)$ – rule $i\\ref {i:holds}$ construction $ptarc(p,t_i,n_i,0) \\in A$ represents $n_i=W(p,t_i)$ – rule $f\\ref {f:r:ptarc}$ construction; or it represents $n_i = M_0(p)$ – rule $f\\ref {f:rptarc}$ construction; the construction of $f\\ref {f:rptarc}$ ensures that $notenabled(t,0)$ is never true due to the reset arc Then $\\forall p \\in \\bullet t_i, M_0(p) \\ge W(p,t_i)$ – from definition REF of preset $\\bullet t_i$ in PN Then $t_i$ is enabled and can fire in $PN$ , as a result it can belong to $T_0$ – from definition REF of enabled transition And $consumesmore \\notin A$ , since $A$ is an answer set of $\\Pi ^2$ – from rule $a\\ref {a:overc:elim}$ and supported rule proposition Then $\\nexists consumesmore(p,0) \\in A$ – from rule $a\\ref {a:overc:gen}$ and supported rule proposition Then $\\nexists \\lbrace holds(p,q,0), tot\\_decr(p,q1,0) \\rbrace \\subseteq A : q1>q$ in $body(a\\ref {a:overc:place})$ – from $a\\ref {a:overc:place}$ and forced atom proposition Then $\\nexists p : (\\sum _{t_i \\in \\lbrace t_0,\\dots ,t_x\\rbrace , p \\in \\bullet t_i}{W(p,t_i)}+\\sum _{t_i \\in \\lbrace t_0,\\dots ,t_x\\rbrace , p \\in R(t_i)}{M_0(p)}) > M_0(p)$ – from the following $holds(p,q,0)$ represents $q=M_0(p)$ – from rule $i\\ref {i:holds}$ construction, given $tot\\_decr(p,q1,0) \\in A$ if $\\lbrace del(p,q1_0,t_0,0), \\dots , del(p,q1_x,t_x,0) \\rbrace \\subseteq A$ , where $q1 = q1_0+\\dots +q1_x$ – from $r\\ref {r:totdecr}$ and forced atom proposition $del(p,q1_i,t_i,0) \\in A$ if $\\lbrace fires(t_i,0), ptarc(p,t_i,q1_i,0) \\rbrace \\subseteq A$ – from $r\\ref {r:r:del} $ and supported rule proposition $del(p,q1_i,t_i,0)$ either represents removal of $q1_i = W(p,t_i)$ tokens from $p \\in \\bullet t_i$ ; or it represents removal of $q1_i = M_0(p)$ tokens from $p \\in R(t_i)$ – from rule $r\\ref {r:r:del} $ , supported rule proposition, and definition REF of transition execution in $PN$ Then the set of transitions in $T_0$ do not conflict – by the definition REF of conflicting transitions And for each $enabled(t_j,0) \\in A$ and $fires(t_j,0) \\notin A$ , $could\\_not\\_have(t_j,0) \\in A$ , since $A$ is an answer set of $\\Pi ^2$ - from rule $a\\ref {a:maxfire:elim}$ and supported rule proposition Then $\\lbrace enabled(t_j,0), holds(s,qq,0), ptarc(s,t_j,q,0), \\\\ tot\\_decr(s,qqq,0) \\rbrace \\subseteq A$ , such that $q > qq - qqq$ and $fires(t_j,0) \\notin A$ - from rule $a\\ref {a:r:maxfire:cnh}$ and supported rule proposition Then for an $s \\in \\bullet t_j \\cup R(t_j)$ , $q > M_0(s) - (\\sum _{t_i \\in T_0, s \\in \\bullet t_i}{W(s,t_i)} + \\sum _{t_i \\in T_0, s \\in R(t_i)}{M_0(s))}$ , where $q=W(s,t_j) \\text{~if~} s \\in \\bullet t_j, \\text{~or~} M_0(s) \\text{~otherwise}$ - from the following: $ptarc(s,t_i,q,0)$ represents $q=W(s,t_i)$ if $(s,t_i) \\in E^-$ or $q=M_0(s)$ if $s \\in R(t_i)$ – from rule $f\\ref {f:r:ptarc},f\\ref {f:rptarc}$ construction $holds(s,qq,0)$ represents $qq=M_0(s)$ – from $i\\ref {i:holds}$ construction $tot\\_decr(s,qqq,0) \\in A$ if $\\lbrace del(s,qqq_0,t_0,0), \\dots , del(s,qqq_x,t_x,0) \\rbrace \\subseteq A$ – from rule $r\\ref {r:totdecr}$ construction and supported rule proposition $del(s,qqq_i,t_i,0) \\in A$ if $\\lbrace fires(t_i,0), ptarc(s,t_i,qqq_i,0) \\rbrace \\subseteq A$ – from rule $r\\ref {r:r:del} $ and supported rule proposition $del(s,qqq_i,t_i,0)$ represents $qqq_i = W(s,t_i) : t_i \\in T_0, (s,t_i) \\in E^-$ , or $qqq_i = M_0(t_i) : t_i \\in T_0, s \\in R(t_i)$ – from rule $f\\ref {f:r:ptarc},f\\ref {f:rptarc}$ construction $tot\\_decr(q,qqq,0)$ represents $\\sum _{t_i \\in T_0, s \\in \\bullet t_i}{W(s,t_i)} + \\\\ \\sum _{t_i \\in T_0, s \\in R(t_i)}{M_0(s)}$ – from (C,D,E) above Then firing $T_0 \\cup \\lbrace t_j \\rbrace $ would have required more tokens than are present at its source place $s \\in \\bullet t_j \\cup R(t_j)$ .", "Thus, $T_0$ is a maximal set of transitions that can simultaneously fire.", "And for each reset transition $t_r$ with $enabled(t_r,0) \\in A$ , $fires(t_r,0) \\in A$ , since $A$ is an answer set of $\\Pi ^2$ - from rule $f\\ref {f:c:rptarc:elim}$ and supported rule proposition Then, the firing set $T_0$ satisfies the reset-transition requirement of definition REF (firing set) Then $\\lbrace t_0, \\dots , t_x\\rbrace = T_0$ – using 1(a),1(b),1(d) above; and using 1(c) it is a maximal firing set We show $M_1$ is produced by firing $T_0$ in $M_0$ .", "Let $holds(p,q,1) \\in A$ Then $\\lbrace holds(p,q1,0), tot\\_incr(p,q2,0), tot\\_decr(p,q3,0) \\rbrace \\subseteq A : q=q1+q2-q3$ – from rule $r\\ref {r:nextstate}$ and supported rule proposition $holds(p,q1,0) \\in A$ represents $q1=M_0(p)$ – given, rule $i\\ref {i:holds}$ construction; and $\\lbrace add(p,q2_0,t_0,0), \\dots , $ $add(p,q2_j,t_j,0)\\rbrace \\subseteq A : q2_0 + \\dots + q2_j = q2$ and $\\lbrace del(p,q3_0,t_0,0), \\dots , $ $del(p,q3_l,t_l,0)\\rbrace \\subseteq A : q3_0 + \\dots + q3_l = q3$ – rules $r\\ref {r:totincr},r\\ref {r:totdecr}$ using supported rule proposition Then $\\lbrace fires(t_0,0), \\dots , fires(t_j,0) \\rbrace \\subseteq A$ and $\\lbrace fires(t_0,0), \\dots , fires(t_l,0) \\rbrace \\\\ \\subseteq A$ – rules $r\\ref {r:r:add},r\\ref {r:r:del}$ and supported rule proposition, respectively Then $\\lbrace fires(t_0,0), \\dots , fires(t_j,0) \\rbrace \\cup \\lbrace fires(t_0,0), \\dots , fires(t_l,0) \\rbrace \\subseteq A = \\lbrace fires(t_0,0), \\dots , fires(t_x,0) \\rbrace \\subseteq A$ – set union of subsets Then for each $fires(t_x,0) \\in A$ we have $t_x \\in T_0$ – already shown in item REF above Then $q = M_0(p) + \\sum _{t_x \\in T_0 \\wedge p \\in t_x \\bullet }{W(t_x,p)} - (\\sum _{t_x \\in T_0 \\wedge p \\in \\bullet t_x}{W(p,t_x)} + \\\\ \\sum _{t_x \\in T_0 \\wedge p \\in R(t_x)}{M_0(p)})$ – from (REF ),(REF ) (REF ) above and the following Each $add(p,q_j,t_j,0) \\in A$ represents $q_j=W(t_j,p)$ for $p \\in t_j \\bullet $ – rule $r\\ref {r:r:add} $ encoding, and definition REF of postset in $PN$ Each $del(p,t_y,q_y,0) \\in A$ represents either $q_y=W(p,t_y)$ for $p \\in \\bullet t_y$ , or $q_y=M_0(p)$ for $p \\in R(t_y)$ – from rule $r\\ref {r:r:del} ,f\\ref {f:r:ptarc} $ encoding and definition REF of preset in $PN$ ; or from rule $r\\ref {r:r:del} ,f\\ref {f:rptarc}$ encoding and definition of reset arc in $PN$ Each $tot\\_incr(p,q2,0) \\in A$ represents $q2=\\sum _{t_x \\in T_0 \\wedge p \\in t_x \\bullet }{W(t_x,p)}$ – aggregate assignment atom semantics in rule $r\\ref {r:totincr}$ Each $tot\\_decr(p,q3,0) \\in A$ represents $q3=\\sum _{t_x \\in T_0 \\wedge p \\in \\bullet t_x}{W(p,t_x)} + $ $\\sum _{t_x \\in T_0 \\wedge p \\in R(t_x)}{M_0(p)}$ – aggregate assignment atom semantics in rule $r\\ref {r:totdecr}$ Then $M_1(p) = q$ – since $holds(p,q,1) \\in A$ encodes $q=M_1(p)$ – from construction Inductive Step: Let $k > 0$ , and $M_k$ is a valid marking in $X$ for $PN$ , show [(1)] $T_k$ is a valid firing set in $M_k$ , and firing $T_k$ in $M_k$ produces marking $M_{k+1}$ .", "We show that $T_k$ is a valid firing set in $M_k$ .", "Let $\\lbrace fires(t_0,k), \\dots , fires(t_x,k) \\rbrace $ be the set of all $fires(\\dots ,k)$ atoms in $A$ , Then for each $fires(t_i,k) \\in A$ $enabled(t_i,k) \\in A$ – from rule $a\\ref {a:fires}$ and supported rule proposition Then $notenabled(t_i,k) \\notin A$ – from rule $e\\ref {e:enabled}$ and supported rule proposition Then $body(e\\ref {e:r:ne:ptarc})$ must not hold in $A$ – from rule $e\\ref {e:r:ne:ptarc}$ and forced atom proposition Then $q \\lnot < n_i \\equiv q \\ge n_i$ in $e\\ref {e:r:ne:ptarc}$ for all $\\lbrace holds(p,q,k), ptarc(p,t_i,n_i,k)\\rbrace \\subseteq A$ – from $e\\ref {e:r:ne:ptarc}$ , forced atom proposition, and the following $holds(p,q,k) \\in A$ represents $q=M_k(p)$ – construction, inductive assumption $ptarc(p,t_i,n_i,k) \\in A$ represents $n_i=W(p,t_i)$ – rule $f\\ref {f:r:ptarc}$ construction; or it represents $n_i=M_k(p)$ – rule $f\\ref {f:rptarc}$ construction; the construction of $f\\ref {f:rptarc}$ ensures that $notenabled(t,0)$ is never true due to the reset arc Then $\\forall p \\in \\bullet t_i, M_k(p) \\ge W(p,t_i)$ – from definition REF of preset $\\bullet t_i$ in PN Then $t_i$ is enabled and can fire in $PN$ , as a result it can belong to $T_k$ – from definition REF of enabled transition And $consumesmore \\notin A$ , since $A$ is an answer set of $\\Pi ^2$ – from rule $a\\ref {a:overc:elim}$ and supported rule proposition Then $\\nexists consumesmore(p,k) \\in A$ – from rule $a\\ref {a:overc:gen}$ and supported rule proposition Then $\\nexists \\lbrace holds(p,q,k), tot\\_decr(p,q1,k) \\rbrace \\subseteq A : q1>q$ in $body(a\\ref {a:overc:place})$ – from $a\\ref {a:overc:place}$ and forced atom proposition Then $\\nexists p : (\\sum _{t_i \\in \\lbrace t_0,\\dots ,t_x\\rbrace , p \\in \\bullet t_i}{W(p,t_i)}+\\sum _{t_i \\in \\lbrace t_0,\\dots ,t_x\\rbrace , p \\in R(t_i)}{M_k(p)}) > M_k(p)$ – from the following $holds(p,q,k)$ represents $q=M_k(p)$ – inductive assumption, given $tot\\_decr(p,q1,k) \\in A$ if $\\lbrace del(p,q1_0,t_0,k), \\dots , del(p,q1_x,t_x,k) \\rbrace \\subseteq A$ , where $q1 = q1_0+\\dots +q1_x$ – from $r\\ref {r:totdecr}$ and forced atom proposition $del(p,q1_i,t_i,k) \\in A$ if $\\lbrace fires(t_i,k), ptarc(p,t_i,q1_i,k) \\rbrace \\subseteq A$ – from $r\\ref {r:r:del} $ and supported rule proposition $del(p,q1_i,t_i,k)$ either represents removal of $q1_i = W(p,t_i)$ tokens from $p \\in \\bullet t_i$ ; or it represents removal of $q1_i = M_k(p)$ tokens from $p \\in R(t_i)$ – from rule $r\\ref {r:r:del} $ , supported rule proposition, and definition REF of transition execution in $PN$ Then the set of transitions in $T_k$ do not conflict – by the definition REF of conflicting transitions And for each $enabled(t_j,k) \\in A$ and $fires(t_j,k) \\notin A$ , $could\\_not\\_have(t_j,k) \\in A$ , since $A$ is an answer set of $\\Pi ^2$ - from rule $a\\ref {a:maxfire:elim}$ and supported rule proposition Then $\\lbrace enabled(t_j,k), holds(s,qq,k), ptarc(s,t_j,q,k), \\\\ tot\\_decr(s,qqq,k) \\rbrace \\subseteq A$ , such that $q > qq - qqq$ and $fires(t_j,k) \\notin A$ - from rule $a\\ref {a:r:maxfire:cnh}$ and supported rule proposition Then for an $s \\in \\bullet t_j \\cup R(t_j)$ , $q > M_k(s) - (\\sum _{t_i \\in T_k, s \\in \\bullet t_i}{W(s,t_i)} + \\sum _{t_i \\in T_k, s \\in R(t_i)}{M_k(s))}$ , where $q=W(s,t_j) \\text{~if~} s \\in \\bullet t_j, \\text{~or~} M_k(s) \\text{~otherwise}$ - from the following: $ptarc(s,t_i,q,k)$ represents $q=W(s,t_i)$ if $(s,t_i) \\in E^-$ or $q=M_k(s)$ if $s \\in R(t_i)$ – from rule $f\\ref {f:r:ptarc},f\\ref {f:rptarc}$ construction $holds(s,qq,k)$ represents $qq=M_k(s)$ – construction $tot\\_decr(s,qqq,k) \\in A$ if $\\lbrace del(s,qqq_0,t_0,k), \\dots , del(s,qqq_x,t_x,k) \\rbrace \\subseteq A$ – from rule $r\\ref {r:totdecr}$ construction and supported rule proposition $del(s,qqq_i,t_i,k) \\in A$ if $\\lbrace fires(t_i,k), ptarc(s,t_i,qqq_i,k) \\rbrace \\subseteq A$ – from rule $r\\ref {r:r:del} $ and supported rule proposition $del(s,qqq_i,t_i,k)$ represents $qqq_i = W(s,t_i) : t_i \\in T_k, (s,t_i) \\in E^-$ , or $qqq_i = M_k(t_i) : t_i \\in T_k, s \\in R(t_i)$ – from rule $f\\ref {f:r:ptarc},f\\ref {f:rptarc}$ construction $tot\\_decr(q,qqq,k)$ represents $\\sum _{t_i \\in T_k, s \\in \\bullet t_i}{W(s,t_i)} + \\\\ \\sum _{t_i \\in T_k, s \\in R(t_i)}{M_k(s)}$ – from (C,D,E) above Then firing $T_k \\cup \\lbrace t_j \\rbrace $ would have required more tokens than are present at its source place $s \\in \\bullet t_j \\cup R(t_j)$ .", "Thus, $T_k$ is a maximal set of transitions that can simultaneously fire.", "And for each reset transition $t_r$ with $enabled(t_r,k) \\in A$ , $fires(t_r,k) \\in A$ , since $A$ is an answer set of $\\Pi ^2$ - from rule $f\\ref {f:c:rptarc:elim}$ and supported rule proposition Then the firing set $T_k$ satisfies the reset transition requirement of definition REF (firing set) Then $\\lbrace t_0, \\dots , t_x\\rbrace = T_k$ – using 1(a),1(b), 1(d) above; and using 1(c) it is a maximal firing set We show that $M_{k+1}$ is produced by firing $T_k$ in $M_k$ .", "Let $holds(p,q,k+1) \\in A$ Then $\\lbrace holds(p,q1,k), tot\\_incr(p,q2,k), tot\\_decr(p,q3,k) \\rbrace \\subseteq A : q=q1+q2-q3$ – from rule $r\\ref {r:nextstate}$ and supported rule proposition $holds(p,q1,k) \\in A$ represents $q1=M_k(p)$ – construction, inductive assumption; and $\\lbrace add(p,q2_0,t_0,k), \\dots , $ $add(p,q2_j,t_j,k)\\rbrace \\subseteq A : q2_0 + \\dots + q2_j = q2$ and $\\lbrace del(p,q3_0,t_0,k), \\dots , $ $del(p,q3_l,t_l,k)\\rbrace \\subseteq A : q3_0 + \\dots + q3_l = q3$ – rules $r\\ref {r:totincr},r\\ref {r:totdecr}$ using supported rule proposition Then $\\lbrace fires(t_0,k), \\dots , $ $fires(t_j,k) \\rbrace \\subseteq A$ and $\\lbrace fires(t_0,k), \\dots , $ $fires(t_l,k) \\rbrace \\\\ \\subseteq A$ – rules $r\\ref {r:r:add},r\\ref {r:r:del}$ and supported rule proposition, respectively Then $\\lbrace fires(t_0,k), \\dots , fires(t_j,k) \\rbrace \\cup \\lbrace fires(t_0,k), \\dots , $ $fires(t_l,k) \\rbrace \\subseteq $ $A = \\lbrace fires(t_0,k), \\dots , $ $ fires(t_x,k) \\rbrace \\subseteq A$ – set union of subsets Then for each $fires(t_x,k) \\in A$ we have $t_x \\in T_k$ – already shown in item REF above Then $q = M_k(p) + \\sum _{t_x \\in T_0 \\wedge p \\in t_x \\bullet }{W(t_x,p)} - (\\sum _{t_x \\in T_k \\wedge p \\in \\bullet t_x}{W(p,t_x)} + \\\\ \\sum _{t_x \\in T_k \\wedge p \\in R(t_x)}{M_k(p)})$ – from (REF ) above and the following Each $add(p,q_j,t_j,k) \\in A$ represents $q_j=W(t_j,p)$ for $p \\in t_j \\bullet $ – rule $r\\ref {r:r:add} $ encoding, and definition REF of transition execution in $PN$ Each $del(p,t_y,q_y,k) \\in A$ represents either $q_y=W(p,t_y)$ for $p \\in \\bullet t_y$ , or $q_y=M_k(p)$ for $p \\in R(t_y)$ – from rule $r\\ref {r:r:del} ,f\\ref {f:r:ptarc} $ encoding and definition REF of transition execution in $PN$ ; or from rule $r\\ref {r:r:del} ,f\\ref {f:rptarc}$ encoding and definition of reset arc in $PN$ Each $tot\\_incr(p,q2,k) \\in A$ represents $q2=\\sum _{t_x \\in T_k \\wedge p \\in t_x \\bullet }{W(t_x,p)}$ – aggregate assignment atom semantics in rule $r\\ref {r:totincr}$ Each $tot\\_decr(p,q3,0) \\in A$ represents $q3=\\sum _{t_x \\in T_k \\wedge p \\in \\bullet t_x}{W(p,t_x)} + $ $\\sum _{t_x \\in T_k \\wedge p \\in R(t_x)}{M_0(p)}$ – aggregate assignment atom semantics in rule $r\\ref {r:totdecr}$ Then, $M_{k+1}(p) = q$ – since $holds(p,q,k+1) \\in A$ encodes $q=M_{k+1}(p)$ – from construction As a result, for any $n > k$ , $T_n$ is a valid firing set w.r.t.", "$M_n$ and its firing produces marking $M_{n+1}$ .", "Conclusion: Since both (REF ) and (REF ) hold, $X=M_0,T_0,M_1,\\dots ,M_k,T_{k+1}$ is an execution sequence of $PN(P,T,E,W,R)$ (w.r.t $M_0$ ) iff there is an answer set $A$ of $\\Pi ^2(PN,M_0,k,ntok)$ such that (REF ) and (REF ) hold.", "Poof of Proposition REF Let $PN=(P,T,E,W,R,I)$ be a Petri Net, $M_0$ be its initial marking and let $\\Pi ^3(PN,M_0,k,ntok)$ be the ASP encoding of $PN$ and $M_0$ over a simulation length $k$ , with maximum $ntok$ tokens on any place node, as defined in section REF .", "Then $X=M_0,T_0,M_1,\\dots ,M_k,T_k,M_{k+1}$ is an execution sequence of $PN$ (w.r.t.", "$M_0$ ) iff there is an answer set $A$ of $\\Pi ^3(PN,M_0,k,ntok)$ such that: $\\lbrace fires(t,ts) : t \\in T_{ts}, 0 \\le ts \\le k\\rbrace = \\lbrace fires(t,ts) : fires(t,ts) \\in A \\rbrace $ $\\begin{split}\\lbrace holds(p,q,ts) &: p \\in P, q = M_{ts}(p), 0 \\le ts \\le k+1 \\rbrace \\\\&= \\lbrace holds(p,q,ts) : holds(p,q,ts) \\in A \\rbrace \\end{split}$ We prove this by showing that: Given an execution sequence $X$ , we create a set $A$ such that it satisfies (REF ) and (REF ) and show that $A$ is an answer set of $\\Pi ^3$ Given an answer set $A$ of $\\Pi ^3$ , we create an execution sequence $X$ such that (REF ) and (REF ) are satisfied.", "First we show (REF ): Given $PN$ and an execution sequence $X$ of $PN$ , we create a set $A$ as a union of the following sets: $A_1=\\lbrace num(n) : 0 \\le n \\le ntok \\rbrace $ $A_2=\\lbrace time(ts) : 0 \\le ts \\le k\\rbrace $ $A_3=\\lbrace place(p) : p \\in P \\rbrace $ $A_4=\\lbrace trans(t) : t \\in T \\rbrace $ $A_5=\\lbrace ptarc(p,t,n,ts) : (p,t) \\in E^-, n=W(p,t), 0 \\le ts \\le k \\rbrace $ , where $E^- \\subseteq E$ $A_6=\\lbrace tparc(t,p,n,ts) : (t,p) \\in E^+, n=W(t,p), 0 \\le ts \\le k \\rbrace $ , where $E^+ \\subseteq E$ $A_7=\\lbrace holds(p,q,0) : p \\in P, q=M_{0}(p) \\rbrace $ $A_8=\\lbrace notenabled(t,ts) : t \\in T, 0 \\le ts \\le k, (\\exists p \\in \\bullet t, M_{ts}(p) < W(p,t)) \\vee (\\exists p \\in I(t), M_{ts}(p) \\ne 0) \\rbrace $ per definition REF (enabled transition) $A_9=\\lbrace enabled(t,ts) : t \\in T, 0 \\le ts \\le k, (\\forall p \\in \\bullet t, W(p,t) \\le M_{ts}(p)) \\wedge (\\forall p \\in I(t), M_{ts}(p) = 0) \\rbrace $ per definition REF (enabled transition) $A_{10}=\\lbrace fires(t,ts) : t \\in T_{ts}, 0 \\le ts \\le k \\rbrace $ per definition REF (firing set), only an enabled transition may fire $A_{11}=\\lbrace add(p,q,t,ts) : t \\in T_{ts}, p \\in t \\bullet , q=W(t,p), 0 \\le ts \\le k \\rbrace $ per definition REF (transition execution) $A_{12}=\\lbrace del(p,q,t,ts) : t \\in T_{ts}, p \\in \\bullet t, q=W(p,t), 0 \\le ts \\le k \\rbrace \\cup \\lbrace del(p,q,t,ts) : t \\in T_{ts}, p \\in R(t), q=M_{ts}(p), 0 \\le ts \\le k \\rbrace $ per definition REF (transition execution) $A_{13}=\\lbrace tot\\_incr(p,q,ts) : p \\in P, q=\\sum _{t \\in T_{ts}, p \\in t \\bullet }{W(t,p)}, 0 \\le ts \\le k \\rbrace $ per definition REF (firing set execution) $A_{14}=\\lbrace tot\\_decr(p,q,ts): p \\in P, q=\\sum _{t \\in T_{ts}, p \\in \\bullet t}{W(p,t)}+\\sum _{t \\in T_{ts}, p \\in R(t) }{M_{ts}(p)}, 0 \\le ts \\le k \\rbrace $ per definition REF (firing set execution) $A_{15}=\\lbrace consumesmore(p,ts) : p \\in P, q=M_{ts}(p), q1=\\sum _{t \\in T_{ts}, p \\in \\bullet t}{W(p,t)} + \\sum _{t \\in T_{ts}, p \\in R(t)}{M_{ts}(p)}, q1 > q, 0 \\le ts \\le k \\rbrace $ per definition REF (conflicting transitions) $A_{16}=\\lbrace consumesmore : \\exists p \\in P : q=M_{ts}(p), q1=\\sum _{t \\in T_{ts}, p \\in \\bullet t}{W(p,t)} + \\sum _{t \\in T_{ts}, p \\in R(t)}(M_{ts}(p)), q1 > q, 0 \\le ts \\le k \\rbrace $ per definition REF (conflicting transitions) $A_{17}=\\lbrace could\\_not\\_have(t,ts) : t \\in T, (\\forall p \\in \\bullet t, W(p,t) \\le M_{ts}(p)), t \\notin T_{ts}, (\\exists p \\in \\bullet t : W(p,t) > M_{ts}(p) - (\\sum _{t^{\\prime } \\in T_{ts}, p \\in \\bullet t^{\\prime }}{W(p,t^{\\prime })} + \\sum _{t^{\\prime } \\in T_{ts}, p \\in R(t^{\\prime })} M_{ts}(p)), 0 \\le ts \\le k \\rbrace $ per the maximal firing set semantics $A_{18}=\\lbrace holds(p,q,ts+1) : p \\in P, q=M_{ts+1}(p), 0 \\le ts < k\\rbrace $ , where $M_{ts+1}(p) = M_{ts}(p) - (\\sum _{\\begin{array}{c}t \\in T_{ts}, p \\in \\bullet t\\end{array}}{W(p,t)} + $ $\\sum _{t \\in T_{ts}, p \\in R(t)}M_{ts}(p)) + \\\\ \\sum _{\\begin{array}{c}t \\in T_{ts}, p \\in t \\bullet \\end{array}}{W(t,p)}$ according to definition REF (firing set execution) $A_{19}=\\lbrace ptarc(p,t,n,ts) : p \\in R(t), n = M_{ts}(p), n > 0, 0 \\le ts \\le k \\rbrace $ $A_{20}=\\lbrace iptarc(p,t,1,ts) : p \\in P, 0 \\le ts < k \\rbrace $ We show that $A$ satisfies (REF ) and (REF ), and $A$ is an answer set of $\\Pi ^3$ .", "$A$ satisfies (REF ) and (REF ) by its construction above.", "We show $A$ is an answer set of $\\Pi ^3$ by splitting.", "We split $lit(\\Pi ^3)$ into a sequence of $7k+9$ sets: $U_0= head(f\\ref {f:place}) \\cup head(f\\ref {f:trans}) \\cup head(f\\ref {f:time}) \\cup head(f\\ref {f:num}) \\cup head(i\\ref {i:holds}) = \\lbrace place(p) : p \\in P\\rbrace \\cup \\lbrace trans(t) : t \\in T\\rbrace \\cup \\lbrace time(0), \\dots , time(k)\\rbrace \\cup \\lbrace num(0), \\dots , num(ntok)\\rbrace \\cup \\lbrace holds(p,q,0) : p \\in P, q=M_0(p) \\rbrace $ $U_{7k+1}=U_{7k+0} \\cup head(f\\ref {f:r:ptarc})^{ts=k} \\cup head(f\\ref {f:r:tparc})^{ts=k} \\cup head(f\\ref {f:rptarc})^{ts=k} \\cup head(f\\ref {f:iptarc})^{ts=k} = U_{7k+0} \\cup \\lbrace ptarc(p,t,n,k) : (p,t) \\in E^-, n=W(p,t) \\rbrace \\cup \\lbrace tparc(t,p,n,k) : (t,p) \\in E^+, n=W(t,p) \\rbrace \\cup \\lbrace ptarc(p,t,n,k) : p \\in R(t), n=M_{k}(p), n > 0 \\rbrace \\cup \\\\ \\lbrace iptarc(p,t,1,k) : p \\in I(t) \\rbrace $ $U_{7k+2}=U_{7k+1} \\cup head(e\\ref {e:r:ne:ptarc} )^{ts=k} \\cup head(e\\ref {e:ne:iptarc})^{ts=k} = U_{7k+1} \\cup \\lbrace notenabled(t,k) : t \\in T \\rbrace $ $U_{7k+3}=U_{7k+2} \\cup head(e\\ref {e:enabled})^{ts=k} = U_{7k+2} \\cup \\lbrace enabled(t,k) : t \\in T \\rbrace $ $U_{7k+4}=U_{7k+3} \\cup head(a\\ref {a:fires})^{ts=k} = U_{7k+3} \\cup \\lbrace fires(t,k) : t \\in T \\rbrace $ $U_{7k+5}=U_{7k+4} \\cup head(r\\ref {r:r:add} )^{ts=k} \\cup head(r\\ref {r:r:del} )^{ts=k} = U_{7k+4} \\cup \\lbrace add(p,q,t,k) : p \\in P, t \\in T, q=W(t,p) \\rbrace \\cup \\lbrace del(p,q,t,k) : p \\in P, t \\in T, q=W(p,t) \\rbrace \\cup \\lbrace del(p,q,t,k) : p \\in P, t \\in T, q=M_{k}(p) \\rbrace $ $U_{7k+6}=U_{7k+5} \\cup head(r\\ref {r:totincr})^{ts=k} \\cup head(r\\ref {r:totdecr})^{ts=k} = U_{7k+5} \\cup \\lbrace tot\\_incr(p,q,k) : p \\in P, 0 \\le q \\le ntok \\rbrace \\cup \\lbrace tot\\_decr(p,q,k) : p \\in P, 0 \\le q \\le ntok \\rbrace $ $U_{7k+7}=U_{7k+6} \\cup head(r\\ref {r:nextstate})^{ts=k} \\cup head(a\\ref {a:overc:place})^{ts=k} \\cup head(a\\ref {a:r:maxfire:cnh})^{ts=k} = $ $U_{7k+6} \\cup \\\\ \\lbrace consumesmore(p,k) : p \\in P\\rbrace \\cup \\lbrace holds(p,q,k+1) : p \\in P, 0 \\le q \\le ntok \\rbrace \\cup \\lbrace could\\_not\\_have(t,k) : t \\in T \\rbrace $ $U_{7k+8}=U_{7k+7} \\cup head(a\\ref {a:overc:gen})^{ts=k} = U_{7k+7} \\cup \\lbrace consumesmore \\rbrace $ where $head(r_i)^{ts=k}$ are head atoms of ground rule $r_i$ in which $ts=k$ .", "We write $A_i^{ts=k} = \\lbrace a(\\dots ,ts) : a(\\dots ,ts) \\in A_i, ts=k \\rbrace $ as short hand for all atoms in $A_i$ with $ts=k$ .", "$U_{\\alpha }, 0 \\le \\alpha \\le 7k+8$ form a splitting sequence, since each $U_i$ is a splitting set of $\\Pi ^1$ , and $\\langle U_{\\alpha }\\rangle _{\\alpha < \\mu }$ is a monotone continuous sequence, where $U_0 \\subseteq U_1 \\dots \\subseteq U_{7k+8}$ and $\\bigcup _{\\alpha < \\mu }{U_{\\alpha }} = lit(\\Pi ^1)$ .", "We compute the answer set of $\\Pi ^3$ using the splitting sets as follows: $bot_{U_0}(\\Pi ^3) = f\\ref {f:place} \\cup f\\ref {f:trans} \\cup f\\ref {f:time} \\cup i\\ref {i:holds} \\cup f\\ref {f:num}$ and $X_0 = A_1 \\cup \\dots \\cup A_4 \\cup A_7$ ($= U_0$ ) is its answer set – using forced atom proposition $eval_{U_0}(bot_{U_1}(\\Pi ^3) \\setminus bot_{U_0}(\\Pi ^3), X_0) = \\lbrace ptarc(p,t,q,0) \\text{:-}.", "\| q=W(p,t) \\rbrace \\cup \\\\ \\lbrace tparc(t,p,q,0) \\text{:-}.", "\| q=W(t,p) \\rbrace \\cup \\lbrace ptarc(p,t,q,0) \\text{:-}.", "\| q=M_0(p) \\rbrace \\cup \\\\ \\lbrace iptarc(p,t,1,0) \\text{:-}.", "\\rbrace $ .", "Its answer set $X_1=A_5^{ts=0} \\cup A_6^{ts=0} \\cup A_{19}^{ts=0} \\cup A_{20}^{ts=0}$ – using forced atom proposition and construction of $A_5, A_6, A_{19}, A_{20}$ .", "$eval_{U_1}(bot_{U_2}(\\Pi ^3) \\setminus bot_{U_1}(\\Pi ^3), X_0 \\cup X_1) = \\lbrace notenabled(t,0) \\text{:-} .", "\| (\\lbrace trans(t), \\\\ ptarc(p,t,n,0), holds(p,q,0) \\rbrace \\subseteq X_0 \\cup X_1, \\text{~where~} q < n) \\text{~or~} \\lbrace notenabled(t,0) \\text{:-} .", "\| \\\\ (\\lbrace trans(t), iptarc(p,t,n2,0), holds(p,q,0) \\rbrace \\subseteq X_0 \\cup X_1, \\text{~where~} q \\ge n2 \\rbrace \\rbrace $ .", "Its answer set $X_2=A_8^{ts=0}$ – using forced atom proposition and construction of $A_8$ .", "where, $q=M_0(p)$ , and $n=W(p,t)$ for an arc $(p,t) \\in E^-$ – by construction $i\\ref {i:holds}$ and $f\\ref {f:r:ptarc}$ in $\\Pi ^3$ , and in an arc $(p,t) \\in E^-$ , $p \\in \\bullet t$ (by definition REF of preset) $n2=1$ – by construction of $iptarc$ predicates in $\\Pi ^3$ , meaning $q \\ge n2 \\equiv q \\ge 1 \\equiv q > 0$ , thus, $notenabled(t,0) \\in X_1$ represents $(\\exists p \\in \\bullet t : M_0(p) < W(p,t)) \\vee (\\exists p \\in I(t) : M_0(p) > 0)$ .", "$eval_{U_2}(bot_{U_3}(\\Pi ^3) \\setminus bot_{U_2}(\\Pi ^3), X_0 \\cup \\dots \\cup X_2) = \\lbrace enabled(t,0) \\text{:-}.", "\| trans(t) \\in X_0 \\cup \\dots \\cup X_2, notenabled(t,0) \\notin X_0 \\cup \\dots \\cup X_2 \\rbrace $ .", "Its answer set is $X_3 = A_9^{ts=0}$ – using forced atom proposition and construction of $A_9$ .", "since an $enabled(t,0) \\in X_3$ if $\\nexists ~notenabled(t,0) \\in X_0 \\cup \\dots \\cup X_2$ ; which is equivalent to $(\\nexists p \\in \\bullet t : M_0(p) < W(p,t)) \\wedge (\\nexists p \\in I(t) : M_0(p) > 0) \\equiv (\\forall p \\in \\bullet t: M_0(p) \\ge W(p,t)) \\wedge (\\forall p \\in I(t) : M_0(p) = 0)$ .", "$eval_{U_3}(bot_{U_4}(\\Pi ^3) \\setminus bot_{U_3}(\\Pi ^3), X_0 \\cup \\dots \\cup X_3) = \\lbrace \\lbrace fires(t,0)\\rbrace \\text{:-}.", "\| enabled(t,0) \\\\ \\text{~holds in~} X_0 \\cup \\dots \\cup X_3 \\rbrace $ .", "It has multiple answer sets $X_{4.1}, \\dots , X_{4.n}$ , corresponding to elements of power set of $fires(t,0)$ atoms in $eval_{U_3}(...)$ – using supported rule proposition.", "Since we are showing that the union of answer sets of $\\Pi ^3$ determined using splitting is equal to $A$ , we only consider the set that matches the $fires(t,0)$ elements in $A$ and call it $X_4$ , ignoring the rest.", "Thus, $X_4 = A_{10}^{ts=0}$ , representing $T_0$ .", "in addition, for every $t$ such that $enabled(t,0) \\in X_0 \\cup \\dots \\cup X_3, R(t) \\ne \\emptyset $ ; $fires(t,0) \\in X_4$ – per definition REF (firing set); requiring that a reset transition is fired when enabled thus, the firing set $T_0$ will not be eliminated by the constraint $f\\ref {f:c:rptarc:elim}$ $eval_{U_4}(bot_{U_5}(\\Pi ^3) \\setminus bot_{U_4}(\\Pi ^3), X_0 \\cup \\dots \\cup X_4) = \\lbrace add(p,n,t,0) \\text{:-}.", "\| \\lbrace fires(t,0), \\\\ tparc(t,p,n,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_4 \\rbrace \\cup \\lbrace del(p,n,t,0) \\text{:-}.", "\| \\lbrace fires(t,0), ptarc(p,t,n,0) \\rbrace \\\\ \\subseteq X_0 \\cup \\dots \\cup X_4 \\rbrace $ .", "It's answer set is $X_5 = A_{11}^{ts=0} \\cup A_{12}^{ts=0}$ – using forced atom proposition and definitions of $A_{11}$ and $A_{12}$ .", "where, each $add$ atom is equivalent to $n=W(t,p) : p \\in t \\bullet $ , and each $del$ atom is equivalent to $n=W(p,t) : p \\in \\bullet t$ ; or $n=M_k(p) : p \\in R(t)$ , representing the effect of transitions in $T_0$ .", "$eval_{U_5}(bot_{U_6}(\\Pi ^3) \\setminus bot_{U_5}(\\Pi ^3), X_0 \\cup \\dots \\cup X_5) = \\lbrace tot\\_incr(p,qq,0) \\text{:-}.", "\| $ $qq=\\sum _{add(p,q,t,0) \\in X_0 \\cup \\dots \\cup X_5}{q} \\rbrace \\cup \\lbrace tot\\_decr(p,qq,0) \\text{:-}.", "\| qq=\\sum _{del(p,q,t,0) \\in X_0 \\cup \\dots \\cup X_5}{q} \\rbrace $ .", "It's answer set is $X_6 = A_{13}^{ts=0} \\cup A_{14}^{ts=0}$ – using forced atom proposition and definitions of $A_{13}$ and $A_{14}$ .", "where, each $tot\\_incr(p,qq,0)$ , $qq=\\sum _{add(p,q,t,0) \\in X_0 \\cup \\dots X_5}{q}$ $\\equiv qq=\\sum _{t \\in X_4, p \\in t \\bullet }{W(p,t)}$ , and each $tot\\_decr(p,qq,0)$ , $qq=\\sum _{del(p,q,t,0) \\in X_0 \\cup \\dots X_5}{q}$ $\\equiv qq=\\sum _{t \\in X_4, p \\in \\bullet t}{W(t,p)} + \\sum _{t \\in X_4, p \\in R(t)}{M_{k}(p)}$ , represent the net effect of transitions in $T_0$ .", "$eval_{U_6}(bot_{U_7}(\\Pi ^3) \\setminus bot_{U_6}(\\Pi ^3), X_0 \\cup \\dots \\cup X_6) = \\lbrace consumesmore(p,0) \\text{:-}.", "\| \\\\ \\lbrace holds(p,q,0), tot\\_decr(p,q1,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_6, q1 > q \\rbrace \\cup \\lbrace holds(p,q,1) \\text{:-}., \| \\\\ \\lbrace holds(p,q1,0), tot\\_incr(p,q2,0), tot\\_decr(p,q3,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_6, q=q1+q2-q3 \\rbrace \\cup \\lbrace could\\_not\\_have(t,0) \\text{:-}.", "\| \\lbrace enabled(t,0), ptarc(s,t,q), holds(s,qq,0), \\\\ tot\\_decr(s,qqq,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_6, fires(t,0) \\notin (X_0 \\cup \\dots \\cup X_6), q > qq-qqq \\rbrace $ .", "It's answer set is $X_7 = A_{15}^{ts=0} \\cup A_{17}^{ts=0} \\cup A_{18}^{ts=0}$ – using forced atom proposition and definitions of $A_{15}, A_{17}, A_{18}, A_9$ .", "where, $consumesmore(p,0)$ represents $\\exists p : q=M_0(p), q1= \\\\ \\sum _{t \\in T_0, p \\in \\bullet t}{W(p,t)}+\\sum _{t \\in T_0, p \\in R(t)}{M_k(p)}, q1 > q$ , indicating place $p$ will be over consumed if $T_0$ is fired, as defined in definition REF (conflicting transitions) $holds(p,q,1)$ represents $q=M_1(p)$ – by construction of $\\Pi ^3$ and $could\\_not\\_have(t,0)$ represents enabled transition $t \\in T_0$ that could not fire due to insufficient tokens $X_7$ does not contain $could\\_not\\_have(t,0)$ , when $enabled(t,0) \\in X_0 \\cup \\dots \\cup X_5$ and $fires(t,0) \\notin X_0 \\cup \\dots \\cup X_6$ due to construction of $A$ , encoding of $a\\ref {a:r:maxfire:cnh}$ and its body atoms.", "As a result it is not eliminated by the constraint $a\\ref {a:maxfire:elim}$ $ \\vdots $ $eval_{U_{7k+0}}(bot_{U_{7k+1}}(\\Pi ^3) \\setminus bot_{U_{7k+0}}(\\Pi ^3), X_0 \\cup \\dots \\cup X_{7k+0}) = \\lbrace ptarc(p,t,q,k) \\text{:-}.", "\| q=W(p,t) \\rbrace \\cup \\lbrace tparc(t,p,q,k) \\text{:-}.", "\| q=W(t,p) \\rbrace \\cup \\lbrace ptarc(p,t,q,k) \\text{:-}.", "\| q=M_k(p) \\rbrace \\cup \\lbrace iptarc(p,t,1,k) \\text{:-}.", "\\rbrace $ .", "Its answer set $X_{7k+1}=A_5^{ts=k} \\cup A_6^{ts=k} \\cup A_{19}^{ts=k} \\cup A_{20}^{ts=k}$ – using forced atom proposition and construction of $A_5, A_6, A_{19}, A_{20}$ .", "$eval_{U_{7k+1}}(bot_{U_{7k+2}}(\\Pi ^3) \\setminus bot_{U_{7k+1}}(\\Pi ^3), X_0 \\cup \\dots \\cup X_{7k+1}) = \\lbrace notenabled(t,k) \\text{:-} .", "\| \\\\ (\\lbrace trans(t), ptarc(p,t,n,k), holds(p,q,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{7k+1}, \\text{~where~} q < n) \\rbrace \\text{~or~} $ $\\lbrace notenabled(t,k) \\text{:-} .", "\| (\\lbrace trans(t), iptarc(p,t,n2,k), holds(p,q,k) \\rbrace \\subseteq \\\\ X_0 \\cup \\dots \\cup X_{7k+1}, \\text{~where~} q \\ge n2 \\rbrace \\rbrace $ .", "Its answer set $X_{7k+2}=A_8^{ts=k}$ – using forced atom proposition and construction of $A_8$ .", "where, $q=M_k(p)$ , and $n=W(p,t)$ for an arc $(p,t) \\in E^-$ – by construction of $holds$ and $ptarc$ predicates in $\\Pi ^3$ , and in an arc $(p,t) \\in E^-$ , $p \\in \\bullet t$ (by definition REF of preset) $n2=1$ – by construction of $iptarc$ predicates in $\\Pi ^3$ , meaning $q \\ge n2 \\equiv q \\ge 1 \\equiv q > 0$ , thus, $notenabled(t,k) \\in X_{7k+1}$ represents $(\\exists p \\in \\bullet t : M_k(p) < W(p,t)) \\vee (\\exists p \\in I(t) : M_k(p) > k)$ .", "$eval_{U_{7k+2}}(bot_{U_{7k+3}}(\\Pi ^3) \\setminus bot_{U_{7k+2}}(\\Pi ^3), X_0 \\cup \\dots \\cup X_{7k+2}) = \\lbrace enabled(t,k) \\text{:-}.", "\| \\\\ trans(t) \\in X_0 \\cup \\dots \\cup X_{7k+2} \\wedge notenabled(t,k) \\notin X_0 \\cup \\dots \\cup X_{7k+2} \\rbrace $ .", "Its answer set is $X_{7k+3} = A_9^{ts=k}$ – using forced atom proposition and construction of $A_9$ .", "since an $enabled(t,k) \\in X_{7k+3}$ if $\\nexists ~notenabled(t,k) \\in X_0 \\cup \\dots \\cup X_{7k+2}$ ; which is equivalent to $(\\nexists p \\in \\bullet t : M_k(p) < W(p,t)) \\wedge (\\nexists p \\in I(t) : M_k(p) > k) \\equiv (\\forall p \\in \\bullet t: M_k(p) \\ge W(p,t)) \\wedge (\\forall p \\in I(t) : M_k(p) = k)$ .", "$eval_{U_{7k+3}}(bot_{U_{7k+4}}(\\Pi ^3) \\setminus bot_{U_{7k+3}}(\\Pi ^3), X_0 \\cup \\dots \\cup X_{7k+3}) = \\lbrace \\lbrace fires(t,k)\\rbrace \\text{:-}.", "\| \\\\ enabled(t,k) \\\\ \\text{~holds in~} X_0 \\cup \\dots \\cup X_{7k+3} \\rbrace $ .", "It has multiple answer sets $X_{7k+4.1}, \\dots , X_{1k+4.n}$ , corresponding to elements of power set of $fires(t,k)$ atoms in $eval_{U_{7k+3}}(...)$ – using supported rule proposition.", "Since we are showing that the union of answer sets of $\\Pi ^3$ determined using splitting is equal to $A$ , we only consider the set that matches the $fires(t,k)$ elements in $A$ and call it $X_{7k+4}$ , ignoring the rest.", "Thus, $X_{7k+4} = A_{10}^{ts=k}$ , representing $T_k$ .", "in addition, for every $t$ such that $enabled(t,k) \\in X_0 \\cup \\dots \\cup X_{7k+3}, R(t) \\ne \\emptyset $ ; $fires(t,k) \\in X_{7k+4}$ – per definition REF (firing set); requiring that a reset transition is fired when enabled thus, the firing set $T_k$ will not be eliminated by the constraint $f\\ref {f:c:rptarc:elim}$ $eval_{U_{7k+4}}(bot_{U_{7k+5}}(\\Pi ^3) \\setminus bot_{U_{7k+4}}(\\Pi ^3), X_0 \\cup \\dots \\cup X_{7k+4}) = \\lbrace add(p,n,t,k) \\text{:-}.", "\| \\\\ \\lbrace fires(t,k), tparc(t,p,n,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{7k+4} \\rbrace \\cup \\lbrace del(p,n,t,k) \\text{:-}.", "\| \\lbrace fires(t,k), \\\\ ptarc(p,t,n,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{7k+4} \\rbrace $ .", "It's answer set is $X_{7k+5} = A_{11}^{ts=k} \\cup A_{12}^{ts=k}$ – using forced atom proposition and definitions of $A_{11}$ and $A_{12}$ .", "where, each $add$ atom is equivalent to $n=W(t,p) : p \\in t \\bullet $ , and each $del$ atom is equivalent to $n=W(p,t) : p \\in \\bullet t$ ; or $n=M_k(p) : p \\in R(t)$ , representing the effect of transitions in $T_k$ $eval_{U_{7k+5}}(bot_{U_{7k+6}}(\\Pi ^3) \\setminus bot_{U_{7k+5}}(\\Pi ^3), X_0 \\cup \\dots \\cup X_{7k+5}) = \\\\ \\lbrace tot\\_incr(p,qq,k) \\text{:-}.", "\| $ $qq=\\sum _{add(p,q,t,k) \\in X_0 \\cup \\dots \\cup X_{7k+5}}{q} \\rbrace \\cup $ $\\lbrace tot\\_decr(p,qq,k) \\text{:-}.", "\| $ $qq=\\sum _{del(p,q,t,k) \\in X_0 \\cup \\dots \\cup X_{7k+5}}{q} \\rbrace $ .", "It's answer set is $X_{7k+6} = A_{13}^{ts=k} \\cup A_{14}^{ts=k}$ – using forced atom proposition and definitions of $A_{13}$ and $A_{14}$ .", "where, each $tot\\_incr(p,qq,k)$ , $qq=\\sum _{add(p,q,t,k) \\in X_0 \\cup \\dots X_{7k+5}}{q}$ $\\equiv qq=\\sum _{t \\in X_{7k+4}, p \\in t \\bullet }{W(p,t)}$ , and each $tot\\_decr(p,qq,k)$ , $qq=\\sum _{del(p,q,t,k) \\in X_0 \\cup \\dots X_{7k+5}}{q}$ $\\equiv qq=\\sum _{t \\in X_{7k+4}, p \\in \\bullet t}{W(t,p)} + \\sum _{t \\in X_{7k+4}, p \\in R(t)}{M_{k}(p)}$ , representing the net effect of transitions in $T_k$ $eval_{U_{7k+6}}(bot_{U_{7k+7}}(\\Pi ^3) \\setminus bot_{U_{7k+6}}(\\Pi ^3), X_0 \\cup \\dots \\cup X_{7k+6}) = \\lbrace consumesmore(p,k) \\text{:-}.", "\| \\\\ \\lbrace holds(p,q,k), tot\\_decr(p,q1,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{7k+6}, q1 > q \\rbrace \\cup \\lbrace holds(p,q,1) \\text{:-}., \| \\\\ \\lbrace holds(p,q1,k), tot\\_incr(p,q2,k), tot\\_decr(p,q3,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{7k+6}, q=q1+q2-q3 \\rbrace \\cup \\lbrace could\\_not\\_have(t,k) \\text{:-}.", "\| \\lbrace enabled(t,k), ptarc(s,t,q), holds(s,qq,k), \\\\ tot\\_decr(s,qqq,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{7k+6}, fires(t,k) \\notin (X_0 \\cup \\dots \\cup X_{7k+6}), q > qq-qqq \\rbrace $ .", "It's answer set is $X_{7k+7} = A_{15}^{ts=k} \\cup A_{17}^{ts=k} \\cup A_{18}^{ts=k}$ – using forced atom proposition and definitions of $A_{15}, A_{17}, A_{18}, A_9$ .", "where, $consumesmore(p,k)$ represents $\\exists p : q=M_k(p), \\\\ q1=\\sum _{t \\in T_k, p \\in \\bullet t}{W(p,t)}+\\sum _{t \\in T_k, p \\in R(t)}{M_k(p)}, q1 > q$ , indicating place $p$ that will be over consumed if $T_k$ is fired, as defined in definition REF (conflicting transitions), $holds(p,q,k+1)$ represents $q=M_{k+1}(p)$ – by construction of $\\Pi ^3$ and $could\\_not\\_have(t,k)$ represents enabled transition $t$ in $T_k$ that could not be fired due to insufficient tokens $X_{7k+7}$ does not contain $could\\_not\\_have(t,k)$ , when $enabled(t,k) \\in X_0 \\cup \\dots \\cup X_{7k+6}$ and $fires(t,k) \\notin X_0 \\cup \\dots \\cup X_{7k+6}$ due to construction of $A$ , encoding of $a\\ref {a:r:maxfire:cnh}$ and its body atoms.", "As a result it is not eliminated by the constraint $a\\ref {a:maxfire:elim}$ $eval_{U_{7k+7}}(bot_{U_{7k+8}}(\\Pi ^3) \\setminus bot_{U_{7k+7}}(\\Pi ^3), X_0 \\cup \\dots \\cup X_{7k+7}) = \\lbrace consumesmore \\text{:-}.", "\| \\\\ \\lbrace consumesmore(p,0),\\dots ,$ $consumesmore(p,k) \\rbrace \\cap (X_0 \\cup \\dots \\cup X_{7k+7}) \\ne \\emptyset \\rbrace $ .", "It's answer set is $X_{7k+8} = A_{16}$ – using forced atom proposition $X_{7k+8}$ will be empty since none of $consumesmore(p,0),\\dots , \\\\ consumesmore(p,k)$ hold in $X_0 \\cup \\dots \\cup X_{7k+7}$ due to the construction of $A$ , encoding of $a\\ref {a:overc:place}$ and its body atoms.", "As a result, it is not eliminated by the constraint $a\\ref {a:overc:elim}$ The set $X = X_0 \\cup \\dots \\cup X_{7k+8}$ is the answer set of $\\Pi ^3$ by the splitting sequence theorem REF .", "Each $X_i, 0 \\le i \\le 7k+8$ matches a distinct portion of $A$ , and $X = A$ , thus $A$ is an answer set of $\\Pi ^3$ .", "Next we show (REF ): Given $\\Pi ^3$ be the encoding of a Petri Net $PN(P,T,E,W,R,I)$ with initial marking $M_0$ , and $A$ be an answer set of $\\Pi ^3$ that satisfies (REF ) and (REF ), then we can construct $X=M_0,T_0,\\dots ,M_k,T_k,M_{k+1}$ from $A$ , such that it is an execution sequence of $PN$ .", "We construct the $X$ as follows: $M_i = (M_i(p_0), \\dots , M_i(p_n))$ , where $\\lbrace holds(p_0,M_i(p_0),i), \\dots holds(p_n,M_i(p_n),i) \\rbrace \\\\ \\subseteq A$ , for $0 \\le i \\le k+1$ $T_i = \\lbrace t : fires(t,i) \\in A\\rbrace $ , for $0 \\le i \\le k$ and show that $X$ is indeed an execution sequence of $PN$ .", "We show this by induction over $k$ (i.e.", "given $M_k$ , $T_k$ is a valid firing set and its firing produces marking $M_{k+1}$ ).", "Base case: Let $k=0$ , and $M_0$ is a valid marking in $X$ for $PN$ , show [(1)] $T_0$ is a valid firing set for $M_0$ , and $M_1$ is $T_0$ 's target marking w.r.t.", "$M_0$ .", "We show $T_0$ is a valid firing set for $M_0$ .", "Let $\\lbrace fires(t_0,0), \\dots , fires(t_x,0) \\rbrace $ be the set of all $fires(\\dots ,0)$ atoms in $A$ , Then for each $fires(t_i,0) \\in A$ $enabled(t_i,0) \\in A$ – from rule $a\\ref {a:fires}$ and supported rule proposition Then $notenabled(t_i,0) \\notin A$ – from rule $e\\ref {e:enabled}$ and supported rule proposition Then $body(e\\ref {e:r:ne:ptarc} )$ must not hold in $A$ and $body(e\\ref {e:ne:iptarc} )$ must not hold in $A$ – from rules $e\\ref {e:r:ne:ptarc} ,e\\ref {e:ne:iptarc} $ and forced atom proposition Then $q \\lnot < n_i \\equiv q \\ge n_i$ in $e\\ref {e:r:ne:ptarc}$ for all $\\lbrace holds(p,q,0), ptarc(p,t_i,n_i,0)\\rbrace \\subseteq A$ – from $e\\ref {e:r:ne:ptarc}$ , forced atom proposition, and given facts ($holds(p,q,0) \\in A, ptarc(p,t_i,n_i,0) \\in A$ ) And $q \\lnot \\ge n_i \\equiv q < n_i$ in $e\\ref {e:ne:iptarc} $ for all $\\lbrace holds(p,q,0), iptarc(p,t_i,n_i,0) \\rbrace \\subseteq A, n_i=1$ ; $q > n_i \\equiv q = 0$ – from $e\\ref {e:ne:iptarc} $ , forced atom proposition, given facts ($holds(p,q,0) \\in A, iptarc(p,t_i,1,0) \\in A$ ), and $q$ is a positive integer Then $(\\forall p \\in \\bullet t_i, M_0(p) \\ge W(p,t_i)) \\wedge (\\forall p \\in I(t_i), M_0(p) = 0)$ – from $holds(p,q,0) \\in A$ represents $q=M_0(p)$ – rule $i\\ref {i:holds}$ construction $ptarc(p,t_i,n_i,0) \\in A$ represents $n_i=W(p,t_i)$ – rule $f\\ref {f:r:ptarc}$ construction; or it represents $n_i = M_0(p)$ – rule $f\\ref {f:rptarc}$ construction; the construction of $f\\ref {f:rptarc}$ ensures that $notenabled(t,0)$ is never true due to the reset arc definition REF of preset $\\bullet t_i$ in PN Then $t_i$ is enabled and can fire in $PN$ , as a result it can belong to $T_0$ – from definition REF of enabled transition And $consumesmore \\notin A$ , since $A$ is an answer set of $\\Pi ^3$ – from rule $a\\ref {a:overc:elim}$ and supported rule proposition Then $\\nexists consumesmore(p,0) \\in A$ – from rule $a\\ref {a:overc:gen}$ and supported rule proposition Then $\\nexists \\lbrace holds(p,q,0), tot\\_decr(p,q1,0) \\rbrace \\subseteq A : q1>q$ in $body(a\\ref {a:overc:place})$ – from $a\\ref {a:overc:place}$ and forced atom proposition Then $\\nexists p : (\\sum _{t_i \\in \\lbrace t_0,\\dots ,t_x\\rbrace , p \\in \\bullet t_i}{W(p,t_i)}+\\sum _{t_i \\in \\lbrace t_0,\\dots ,t_x\\rbrace , p \\in R(t_i)}{M_0(p)}) > M_0(p)$ – from the following $holds(p,q,0)$ represents $q=M_0(p)$ – from rule $i\\ref {i:holds}$ encoding, given $tot\\_decr(p,q1,0) \\in A$ if $\\lbrace del(p,q1_0,t_0,0), \\dots , del(p,q1_x,t_x,0) \\rbrace \\subseteq A$ , where $q1 = q1_0+\\dots +q1_x$ – from $r\\ref {r:totdecr}$ and forced atom proposition $del(p,q1_i,t_i,0) \\in A$ if $\\lbrace fires(t_i,0), ptarc(p,t_i,q1_i,0) \\rbrace \\subseteq A$ – from $r\\ref {r:r:del} $ and supported rule proposition $del(p,q1_i,t_i,0)$ either represents removal of $q1_i = W(p,t_i)$ tokens from $p \\in \\bullet t_i$ ; or it represents removal of $q1_i = M_0(p)$ tokens from $p \\in R(t_i)$ – from rule $r\\ref {r:r:del} $ , supported rule proposition, and definition REF of transition execution in $PN$ Then the set of transitions in $T_0$ do not conflict – by the definition REF of conflicting transitions And for each $enabled(t_j,0) \\in A$ and $fires(t_j,0) \\notin A$ , $could\\_not\\_have(t_j,0) \\in A$ , since $A$ is an answer set of $\\Pi ^3$ - from rule $a\\ref {a:maxfire:elim}$ and supported rule proposition Then $\\lbrace enabled(t_j,0), holds(s,qq,0), ptarc(s,t_j,q,0), \\\\ tot\\_decr(s,qqq,0) \\rbrace \\subseteq A$ , such that $q > qq - qqq$ and $fires(t_j,0) \\notin A$ - from rule $a\\ref {a:r:maxfire:cnh}$ and supported rule proposition Then for an $s \\in \\bullet t_j \\cup R(t_j)$ , $q > M_0(s) - (\\sum _{t_i \\in T_0, s \\in \\bullet t_i}{W(s,t_i)} + \\sum _{t_i \\in T_0, s \\in R(t_i)}{M_0(s))}$ , where $q=W(s,t_j) \\text{~if~} s \\in \\bullet t_j, \\text{~or~} M_0(s) \\text{~otherwise}$ - from the following: $ptarc(s,t_i,q,0)$ represents $q=W(s,t_i)$ if $(s,t_i) \\in E^-$ or $q=M_0(s)$ if $s \\in R(t_i)$ – from rule $f\\ref {f:r:ptarc},f\\ref {f:rptarc}$ construction $holds(s,qq,0)$ represents $qq=M_0(s)$ – from $i\\ref {i:holds}$ construction $tot\\_decr(s,qqq,0) \\in A$ if $\\lbrace del(s,qqq_0,t_0,0), \\dots , del(s,qqq_x,t_x,0) \\rbrace \\subseteq A$ – from rule $r\\ref {r:totdecr}$ construction and supported rule proposition $del(s,qqq_i,t_i,0) \\in A$ if $\\lbrace fires(t_i,0), ptarc(s,t_i,qqq_i,0) \\rbrace \\subseteq A$ – from rule $r\\ref {r:r:del} $ and supported rule proposition $del(s,qqq_i,t_i,0)$ represents $qqq_i = W(s,t_i) : t_i \\in T_0, (s,t_i) \\in E^-$ , or $qqq_i = M_0(t_i) : t_i \\in T_0, s \\in R(t_i)$ – from rule $f\\ref {f:r:ptarc},f\\ref {f:rptarc}$ construction $tot\\_decr(q,qqq,0)$ represents $\\sum _{t_i \\in T_0, s \\in \\bullet t_i}{W(s,t_i)} + \\\\ \\sum _{t_i \\in T_0, s \\in R(t_i)}{M_0(s)}$ – from (C,D,E) above Then firing $T_0 \\cup \\lbrace t_j \\rbrace $ would have required more tokens than are present at its source place $s \\in \\bullet t_j \\cup R(t_j)$ .", "Thus, $T_0$ is a maximal set of transitions that can simultaneously fire.", "And for each reset transition $t_r$ with $enabled(t_r,0) \\in A$ , $fires(t_r,0) \\in A$ , since $A$ is an answer set of $\\Pi ^2$ - from rule $f\\ref {f:c:rptarc:elim}$ and supported rule proposition Then, the firing set $T_0$ satisfies the reset-transition requirement of definition REF (firing set) Then $\\lbrace t_0, \\dots , t_x\\rbrace = T_0$ – using 1(a),1(b),1(d) above; and using 1(c) it is a maximal firing set We show $M_1$ is produced by firing $T_0$ in $M_0$ .", "Let $holds(p,q,1) \\in A$ Then $\\lbrace holds(p,q1,0), tot\\_incr(p,q2,0), tot\\_decr(p,q3,0) \\rbrace \\subseteq A : q=q1+q2-q3$ – from rule $r\\ref {r:nextstate}$ and supported rule proposition Then $holds(p,q1,0) \\in A$ represents $q1=M_0(p)$ – given, rule $i\\ref {i:holds}$ construction; and $\\lbrace add(p,q2_0,t_0,0), \\dots , $ $add(p,q2_j,t_j,0)\\rbrace \\subseteq A : q2_0 + \\dots + q2_j = q2$ and $\\lbrace del(p,q3_0,t_0,0), \\dots , $ $del(p,q3_l,t_l,0)\\rbrace \\subseteq A : q3_0 + \\dots + q3_l = q3$ – rules $r\\ref {r:totincr},r\\ref {r:totdecr}$ and supported rule proposition, respectively Then $\\lbrace fires(t_0,0), \\dots , fires(t_j,0) \\rbrace \\subseteq A$ and $\\lbrace fires(t_0,0), \\dots , fires(t_l,0) \\rbrace \\\\ \\subseteq A$ – rules $r\\ref {r:r:add},r\\ref {r:r:del}$ and supported rule proposition, respectively Then $\\lbrace fires(t_0,0), \\dots , $ $fires(t_j,0) \\rbrace \\cup \\lbrace fires(t_0,0), \\dots , $ $fires(t_l,0) \\rbrace \\subseteq A = \\lbrace fires(t_0,0), \\dots , $ $fires(t_x,0) \\rbrace \\subseteq A$ – set union of subsets Then for each $fires(t_x,0) \\in A$ we have $t_x \\in T_0$ – already shown in item REF above Then $q = M_0(p) + \\sum _{t_x \\in T_0 \\wedge p \\in t_x \\bullet }{W(t_x,p)} - (\\sum _{t_x \\in T_0 \\wedge p \\in \\bullet t_x}{W(p,t_x)} + \\\\ \\sum _{t_x \\in T_0 \\wedge p \\in R(t_x)}{M_0(p)})$ – from (REF ) above and the following Each $add(p,q_j,t_j,0) \\in A$ represents $q_j=W(t_j,p)$ for $p \\in t_j \\bullet $ – rule $r\\ref {r:r:add} $ encoding, and definition REF of transition execution in $PN$ Each $del(p,t_y,q_y,0) \\in A$ represents either $q_y=W(p,t_y)$ for $p \\in \\bullet t_y$ , or $q_y=M_0(p)$ for $p \\in R(t_y)$ – from rule $r\\ref {r:r:del} ,f\\ref {f:r:ptarc} $ encoding and definition REF of transition execution in $PN$ ; or from rule $r\\ref {r:r:del} ,f\\ref {f:rptarc}$ encoding and definition of reset arc in $PN$ Each $tot\\_incr(p,q2,0) \\in A$ represents $q2=\\sum _{t_x \\in T_0 \\wedge p \\in t_x \\bullet }{W(t_x,p)}$ – aggregate assignment atom semantics in rule $r\\ref {r:totincr}$ Each $tot\\_decr(p,q3,0) \\in A$ represents $q3=\\sum _{t_x \\in T_0 \\wedge p \\in \\bullet t_x}{W(p,t_x)} + $ $\\sum _{t_x \\in T_0 \\wedge p \\in R(t_x)}{M_0(p)}$ – aggregate assignment atom semantics in rule $r\\ref {r:totdecr}$ Then, $M_1(p) = q$ – since $holds(p,q,1) \\in A$ encodes $q=M_1(p)$ – from construction Inductive Step: Let $k > 0$ , and $M_k$ is a valid marking in $X$ for $PN$ , show [(1)] $T_k$ is a valid firing set for $M_k$ , and firing $T_k$ in $M_k$ produces marking $M_{k+1}$ .", "We show that $T_k$ is a valid firing set in $M_k$ .", "Let $\\lbrace fires(t_0,k), \\dots , fires(t_x,k) \\rbrace $ be the set of all $fires(\\dots ,k)$ atoms in $A$ , Then for each $fires(t_i,k) \\in A$ $enabled(t_i,k) \\in A$ – from rule $a\\ref {a:fires}$ and supported rule proposition Then $notenabled(t_i,k) \\notin A$ – from rule $e\\ref {e:enabled}$ and supported rule proposition Then $body(e\\ref {e:r:ne:ptarc} )$ must not hold in $A$ and $body(e\\ref {e:ne:iptarc} )$ must not hold in $A$ – from rule $e\\ref {e:r:ne:ptarc} ,e\\ref {e:ne:iptarc} $ and forced atom proposition Then $q \\lnot < n_i \\equiv q \\ge n_i$ in $e\\ref {e:r:ne:ptarc}$ for all $\\lbrace holds(p,q,k), ptarc(p,t_i,n_i,k)\\rbrace \\subseteq A$ – from $e\\ref {e:r:ne:ptarc}$ , forced atom proposition, and given facts ($holds(p,q,k) \\in A, ptarc(p,t_i,n_i,k) \\in A$ ) And $q \\lnot \\ge n_i \\equiv q < n_i$ in $e\\ref {e:ne:iptarc} $ for all $\\lbrace holds(p,q,k), iptarc(p,t_i,n_i,k) \\rbrace \\subseteq A, n_i=1; q > n_i \\equiv q = 0$ – from $e\\ref {e:ne:iptarc} $ , forced atom proposition, given facts $(holds(p,q,k) \\in A, iptarc(p,t_i,1,k) \\in A)$ , and $q$ is a positive integer Then $(\\forall p \\in \\bullet t_i, M_k(p) \\ge W(p,t_i)) \\wedge (\\forall p \\in I(t_i), M_k(p) = 0)$ – from $holds(p,q,k) \\in A$ represents $q=M_k(p)$ – inductive assumption, given $ptarc(p,t_i,n_i,k) \\in A$ represents $n_i=W(p,t_i)$ – rule $f\\ref {f:r:ptarc}$ construction; or it represents $n_i=M_k(p)$ – rule $f\\ref {f:rptarc}$ construction; the construction of $f\\ref {f:rptarc}$ ensures that $notenabled(t,0)$ is never true due to the reset arc definition REF of preset $\\bullet t_i$ in PN Then $t_i$ is enabled and can fire in $PN$ , as a result it can belong to $T_k$ – from definition REF of enabled transition And $consumesmore \\notin A$ , since $A$ is an answer set of $\\Pi ^3$ – from rule $a\\ref {a:overc:elim}$ and supported rule proposition Then $\\nexists consumesmore(p,k) \\in A$ – from rule $a\\ref {a:overc:gen}$ and supported rule proposition Then $\\nexists \\lbrace holds(p,q,k), tot\\_decr(p,q1,k) \\rbrace \\subseteq A : q1>q$ in $body(a\\ref {a:overc:place})$ – from $a\\ref {a:overc:place}$ and forced atom proposition Then $\\nexists p : (\\sum _{t_i \\in \\lbrace t_0,\\dots ,t_x\\rbrace , p \\in \\bullet t_i}{W(p,t_i)}+\\sum _{t_i \\in \\lbrace t_0,\\dots ,t_x\\rbrace , p \\in R(t_i)}{M_k(p)}) > M_k(p)$ – from the following $holds(p,q,k)$ represents $q=M_k(p)$ – inductive assumption, construction $tot\\_decr(p,q1,k) \\in A$ if $\\lbrace del(p,q1_0,t_0,k), \\dots , del(p,q1_x,t_x,k) \\rbrace \\subseteq A$ , where $q1 = q1_0+\\dots +q1_x$ – from $r\\ref {r:totdecr}$ and forced atom proposition $del(p,q1_i,t_i,k) \\in A$ if $\\lbrace fires(t_i,k), ptarc(p,t_i,q1_i,k) \\rbrace \\subseteq A$ – from $r\\ref {r:r:del} $ and supported rule proposition $del(p,q1_i,t_i,k)$ either represents removal of $q1_i = W(p,t_i)$ tokens from $p \\in \\bullet t_i$ ; or it represents removal of $q1_i = M_k(p)$ tokens from $p \\in R(t_i)$ – from rule $r\\ref {r:r:del} $ , supported rule proposition, and definition REF of transition execution in $PN$ Then $T_k$ does not contain conflicting transitions – by the definition REF of conflicting transitions And for each $enabled(t_j,k) \\in A$ and $fires(t_j,k) \\notin A$ , $could\\_not\\_have(t_j,k) \\in A$ , since $A$ is an answer set of $\\Pi ^3$ - from rule $a\\ref {a:maxfire:elim}$ and supported rule proposition Then $\\lbrace enabled(t_j,k), holds(s,qq,k), ptarc(s,t_j,q,k), \\\\ tot\\_decr(s,qqq,k) \\rbrace \\subseteq A$ , such that $q > qq - qqq$ and $fires(t_j,k) \\notin A$ - from rule $a\\ref {a:r:maxfire:cnh}$ and supported rule proposition Then for an $s \\in \\bullet t_j \\cup R(t_j)$ , $q > M_k(s) - (\\sum _{t_i \\in T_k, s \\in \\bullet t_i}{W(s,t_i)} + \\sum _{t_i \\in T_k, s \\in R(t_i)}{M_k(s))}$ , where $q=W(s,t_j) \\text{~if~} s \\in \\bullet t_j, \\text{~or~} M_k(s) \\text{~otherwise}$ - from the following: $ptarc(s,t_i,q,k)$ represents $q=W(s,t_i)$ if $(s,t_i) \\in E^-$ or $q=M_k(s)$ if $s \\in R(t_i)$ – from rule $f\\ref {f:r:ptarc},f\\ref {f:rptarc}$ construction $holds(s,qq,k)$ represents $qq=M_k(s)$ – construction $tot\\_decr(s,qqq,k) \\in A$ if $\\lbrace del(s,qqq_0,t_0,k), \\dots , del(s,qqq_x,t_x,k) \\rbrace \\\\ \\subseteq A$ – from rule $r\\ref {r:totdecr}$ construction and supported rule proposition $del(s,qqq_i,t_i,k) \\in A$ if $\\lbrace fires(t_i,k), ptarc(s,t_i,qqq_i,k) \\rbrace \\subseteq A$ – from rule $r\\ref {r:r:del} $ and supported rule proposition $del(s,qqq_i,t_i,k)$ represents $qqq_i = W(s,t_i) : t_i \\in T_k, (s,t_i) \\in E^-$ , or $qqq_i = M_k(t_i) : t_i \\in T_k, s \\in R(t_i)$ – from rule $f\\ref {f:r:ptarc},f\\ref {f:rptarc}$ construction $tot\\_decr(q,qqq,k)$ represents $\\sum _{t_i \\in T_k, s \\in \\bullet t_i}{W(s,t_i)} + \\\\ \\sum _{t_i \\in T_k, s \\in R(t_i)}{M_k(s)}$ – from (C,D,E) above Then firing $T_k \\cup \\lbrace t_j \\rbrace $ would have required more tokens than are present at its source place $s \\in \\bullet t_j \\cup R(t_j)$ .", "Thus, $T_k$ is a maximal set of transitions that can simultaneously fire.", "And for each reset transition $t_r$ with $enabled(t_r,k) \\in A$ , $fires(t_r,k) \\in A$ , since $A$ is an answer set of $\\Pi ^2$ - from rule $f\\ref {f:c:rptarc:elim}$ and supported rule proposition Then the firing set $T_k$ satisfies the reset transition requirement of definition REF (firing set) Then $\\lbrace t_0, \\dots , t_x\\rbrace = T_k$ – using 1(a),1(b), 1(d) above; and using 1(c) it is a maximal firing set We show that $M_{k+1}$ is produced by firing $T_k$ in $M_k$ .", "Let $holds(p,q,k+1) \\in A$ Then $\\lbrace holds(p,q1,k), tot\\_incr(p,q2,k), tot\\_decr(p,q3,k) \\rbrace \\subseteq A : q=q1+q2-q3$ – from rule $r\\ref {r:nextstate}$ and supported rule proposition Then, $holds(p,q1,k) \\in A$ represents $q1=M_k(p)$ – inductive assumption, construction ; and $\\lbrace add(p,q2_0,t_0,k), \\dots , $ $add(p,q2_j,t_j,k)\\rbrace \\subseteq A : q2_0 + \\dots + q2_j = q2$ and $\\lbrace del(p,q3_0,t_0,k), \\dots , $ $del(p,q3_l,t_l,k)\\rbrace \\subseteq A : q3_0 + \\dots + q3_l = q3$ – rules $r\\ref {r:totincr},r\\ref {r:totdecr}$ and supported rule proposition, respectively Then $\\lbrace fires(t_0,k), \\dots , fires(t_j,k) \\rbrace \\subseteq A$ and $\\lbrace fires(t_0,k), \\dots , fires(t_l,k) \\rbrace \\\\ \\subseteq A$ – rules $r\\ref {r:r:add},r\\ref {r:r:del}$ and supported rule proposition, respectively Then $\\lbrace fires(t_0,k), \\dots , $ $fires(t_j,k) \\rbrace \\cup \\lbrace fires(t_0,k), \\dots , $ $fires(t_l,k) \\rbrace \\subseteq A = $ $\\lbrace fires(t_0,k), \\dots , $ $fires(t_x,k) \\rbrace \\subseteq A$ – set union of subsets Then for each $fires(t_x,k) \\in A$ we have $t_x \\in T_k$ – already shown in item REF above Then $q = M_k(p) + \\sum _{t_x \\in T_0 \\wedge p \\in t_x \\bullet }{W(t_x,p)} - (\\sum _{t_x \\in T_k \\wedge p \\in \\bullet t_x}{W(p,t_x)} + \\\\ \\sum _{t_x \\in T_k \\wedge p \\in R(t_x)}{M_k(p)})$ – from (REF ) above and the following Each $add(p,q_j,t_j,k) \\in A$ represents $q_j=W(t_j,p)$ for $p \\in t_j \\bullet $ – rule $r\\ref {r:r:add} $ encoding, and definition REF of transition execution in $PN$ Each $del(p,t_y,q_y,k) \\in A$ represents either $q_y=W(p,t_y)$ for $p \\in \\bullet t_y$ , or $q_y=M_k(p)$ for $p \\in R(t_y)$ – from rule $r\\ref {r:r:del} ,f\\ref {f:r:ptarc} $ encoding and definition REF of transition execution in $PN$ ; or from rule $r\\ref {r:r:del} ,f\\ref {f:rptarc}$ encoding and definition of reset arc in $PN$ Each $tot\\_incr(p,q2,k) \\in A$ represents $q2=\\sum _{t_x \\in T_k \\wedge p \\in t_x \\bullet }{W(t_x,p)}$ – aggregate assignment atom semantics in rule $r\\ref {r:totincr}$ Each $tot\\_decr(p,q3,0) \\in A$ represents $q3=\\sum _{t_x \\in T_k \\wedge p \\in \\bullet t_x}{W(p,t_x)} + $ $\\sum _{t_x \\in T_k \\wedge p \\in R(t_x)}{M_k(p)}$ – aggregate assignment atom semantics in rule $r\\ref {r:totdecr}$ Then, $M_{k+1}(p) = q$ – since $holds(p,q,k+1) \\in A$ encodes $q=M_{k+1}(p)$ – from construction As a result, for any $n > k$ , $T_n$ will be a valid firing set for $M_n$ and $M_{n+1}$ will be its target marking.", "Conclusion: Since both (REF ) and (REF ) hold, $X=M_0,T_0,M_1,\\dots ,M_k,T_{k+1}$ is an execution sequence of $PN(P,T,E,W,R)$ (w.r.t $M_0$ ) iff there is an answer set $A$ of $\\Pi ^3(PN,M_0,k,ntok)$ such that (REF ) and (REF ) hold.", "Proof of Proposition REF Let $PN=(P,T,E,W,R,I,Q,QW)$ be a Petri Net, $M_0$ be its initial marking and let $\\Pi ^4(PN,M_0,k,ntok)$ be the ASP encoding of $PN$ and $M_0$ over a simulation length $k$ , with maximum $ntok$ tokens on any place node, as defined in section REF .", "Then $X=M_0,T_0,M_1,\\dots ,M_k,T_k,M_{k+1}$ is an execution sequence of $PN$ (w.r.t.", "$M_0$ ) iff there is an answer set $A$ of $\\Pi ^4(PN,M_0,k,ntok)$ such that: $\\lbrace fires(t,ts) : t \\in T_{ts}, 0 \\le ts \\le k\\rbrace = \\lbrace fires(t,ts) : fires(t,ts) \\in A \\rbrace $ $\\begin{split}\\lbrace holds(p,q,ts) &: p \\in P, q = M_{ts}(p), 0 \\le ts \\le k+1 \\rbrace \\\\&= \\lbrace holds(p,q,ts) : holds(p,q,ts) \\in A \\rbrace \\end{split}$ We prove this by showing that: Given an execution sequence $X$ , we create a set $A$ such that it satisfies (REF ) and (REF ) and show that $A$ is an answer set of $\\Pi ^4$ Given an answer set $A$ of $\\Pi ^4$ , we create an execution sequence $X$ such that (REF ) and (REF ) are satisfied.", "First we show (REF ): Given $PN$ and an execution sequence $X$ of $PN$ , we create a set $A$ as a union of the following sets: $A_1=\\lbrace num(n) : 0 \\le n \\le ntok \\rbrace $ $A_2=\\lbrace time(ts) : 0 \\le ts \\le k\\rbrace $ $A_3=\\lbrace place(p) : p \\in P \\rbrace $ $A_4=\\lbrace trans(t) : t \\in T \\rbrace $ $A_5=\\lbrace ptarc(p,t,n,ts) : (p,t) \\in E^-, n=W(p,t), 0 \\le ts \\le k \\rbrace $ , where $E^- \\subseteq E$ $A_6=\\lbrace tparc(t,p,n,ts) : (t,p) \\in E^+, n=W(t,p), 0 \\le ts \\le k \\rbrace $ , where $E^+ \\subseteq E$ $A_7=\\lbrace holds(p,q,0) : p \\in P, q=M_{0}(p) \\rbrace $ $A_8=\\lbrace notenabled(t,ts) : t \\in T, 0 \\le ts \\le k, (\\exists p \\in \\bullet t, M_{ts}(p) < W(p,t)) \\vee (\\exists p \\in I(t), M_{ts}(p) \\ne 0) \\vee (\\exists (p,t) \\in Q, M_{ts}(p) < QW(p,t)) \\rbrace $ per definition REF (enabled transition) $A_9=\\lbrace enabled(t,ts) : t \\in T, 0 \\le ts \\le k, (\\forall p \\in \\bullet t, W(p,t) \\le M_{ts}(p)) \\wedge (\\forall p \\in I(t), M_{ts}(p) = 0) \\wedge (\\forall (p,t) \\in Q, M_{ts}(p) \\ge QW(p,t)) \\rbrace $ per definition REF (enabled transition) $A_{10}=\\lbrace fires(t,ts) : t \\in T_{ts}, 0 \\le ts \\le k \\rbrace $ per definition REF (enabled transitions), only an enabled transition may fire $A_{11}=\\lbrace add(p,q,t,ts) : t \\in T_{ts}, p \\in t \\bullet , q=W(t,p), 0 \\le ts \\le k \\rbrace $ per definition REF (transition execution) $A_{12}=\\lbrace del(p,q,t,ts) : t \\in T_{ts}, p \\in \\bullet t, q=W(p,t), 0 \\le ts \\le k \\rbrace \\cup \\lbrace del(p,q,t,ts) : t \\in T_{ts}, p \\in R(t), q=M_{ts}(p), 0 \\le ts \\le k \\rbrace $ per definition REF (transition execution) $A_{13}=\\lbrace tot\\_incr(p,q,ts) : p \\in P, q=\\sum _{t \\in T_{ts}, p \\in t \\bullet }{W(t,p)}, 0 \\le ts \\le k \\rbrace $ per definition REF (firing set execution) $A_{14}=\\lbrace tot\\_decr(p,q,ts): p \\in P, q=\\sum _{t \\in T_{ts}, p \\in \\bullet t}{W(p,t)}+\\sum _{t \\in T_{ts}, p \\in R(t) }{M_{ts}(p)}, 0 \\le ts \\le k \\rbrace $ per definition REF (firing set execution) $A_{15}=\\lbrace consumesmore(p,ts) : p \\in P, q=M_{ts}(p), q1=\\sum _{t \\in T_{ts}, p \\in \\bullet t}{W(p,t)} + \\sum _{t \\in T_{ts}, p \\in R(t)}{M_{ts}(p)}, q1 > q, 0 \\le ts \\le k \\rbrace $ per definition REF (conflicting transitions) $A_{16}=\\lbrace consumesmore : \\exists p \\in P : q=M_{ts}(p), q1=\\sum _{t \\in T_{ts}, p \\in \\bullet t}{W(p,t)} + \\sum _{t \\in T_{ts}, p \\in R(t)}(M_{ts}(p)), q1 > q, 0 \\le ts \\le k \\rbrace $ per definition REF (conflicting transitions) $A_{17}=\\lbrace could\\_not\\_have(t,ts) : t \\in T, (\\forall p \\in \\bullet t, W(p,t) \\le M_{ts}(p)), t \\notin T_{ts}, (\\exists p \\in \\bullet t : W(p,t) > M_{ts}(p) - (\\sum _{t^{\\prime } \\in T_{ts}, p \\in \\bullet t^{\\prime }}{W(p,t^{\\prime })} + \\sum _{t^{\\prime } \\in T_{ts}, p \\in R(t^{\\prime })} M_{ts}(p)), 0 \\le ts \\le k \\rbrace $ per the maximal firing set semantics $A_{18}=\\lbrace holds(p,q,ts+1) : p \\in P, q=M_{ts+1}(p), 0 \\le ts < k\\rbrace $ , where $M_{ts+1}(p) = M_{ts}(p) - (\\sum _{\\begin{array}{c}t \\in T_{ts}, p \\in \\bullet t\\end{array}}{W(p,t)} + \\\\ \\sum _{t \\in T_{ts}, p \\in R(t)}M_{ts}(p)) + $ $\\sum _{\\begin{array}{c}t \\in T_{ts}, p \\in t \\bullet \\end{array}}{W(t,p)}$ according to definition REF (firing set execution) $A_{19}=\\lbrace ptarc(p,t,n,ts) : p \\in R(t), n = M_{ts}(p), n > 0, 0 \\le ts \\le k \\rbrace $ $A_{20}=\\lbrace iptarc(p,t,1,ts) : p \\in P, 0 \\le ts < k \\rbrace $ $A_{21}=\\lbrace tptarc(p,t,n,ts) : (p,t) \\in Q, n=QW(p,t), 0 \\le ts \\le k \\rbrace $ We show that $A$ satisfies (REF ) and (REF ), and $A$ is an answer set of $\\Pi ^4$ .", "$A$ satisfies (REF ) and (REF ) by its construction above.", "We show $A$ is an answer set of $\\Pi ^4$ by splitting.", "We split $lit(\\Pi ^4)$ into a sequence of $7k+9$ sets: $U_0= head(f\\ref {f:place}) \\cup head(f\\ref {f:trans}) \\cup head(f\\ref {f:time}) \\cup head(f\\ref {f:num}) \\cup head(i\\ref {i:holds}) = \\lbrace place(p) : p \\in P\\rbrace \\cup \\lbrace trans(t) : t \\in T\\rbrace \\cup \\lbrace time(0), \\dots , time(k)\\rbrace \\cup \\lbrace num(0), \\dots , num(ntok)\\rbrace \\cup \\lbrace holds(p,q,0) : p \\in P, q=M_0(p) \\rbrace $ $U_{7k+1}=U_{7k+0} \\cup head(f\\ref {f:r:ptarc})^{ts=k} \\cup head(f\\ref {f:r:tparc})^{ts=k} \\cup head(f\\ref {f:rptarc})^{ts=k} \\cup head(f\\ref {f:iptarc})^{ts=k} \\cup head(f\\ref {f:tptarc})^{ts=k} = U_{7k+0} \\cup \\lbrace ptarc(p,t,n,k) : (p,t) \\in E^-, n=W(p,t) \\rbrace \\cup \\\\ \\lbrace tparc(t,p,n,k) : (t,p) \\in E^+, n=W(t,p) \\rbrace \\cup $ $\\lbrace ptarc(p,t,n,k) : p \\in R(t), n=M_{k}(p), n > 0 \\rbrace \\cup $ $\\lbrace iptarc(p,t,1,k) : p \\in I(t) \\rbrace \\cup $ $\\lbrace tptarc(p,t,n,k) : (p,t) \\in Q, n=QW(p,t) \\rbrace $ $U_{7k+2}=U_{7k+1} \\cup head(e\\ref {e:r:ne:ptarc} )^{ts=k} = U_{7k+1} \\cup \\lbrace notenabled(t,k) : t \\in T \\rbrace $ $U_{7k+3}=U_{7k+2} \\cup head(e\\ref {e:enabled})^{ts=k} = U_{7k+2} \\cup \\lbrace enabled(t,k) : t \\in T \\rbrace $ $U_{7k+4}=U_{7k+3} \\cup head(a\\ref {a:fires})^{ts=k} = U_{7k+3} \\cup \\lbrace fires(t,k) : t \\in T \\rbrace $ $U_{7k+5}=U_{7k+4} \\cup head(r\\ref {r:r:add} )^{ts=k} \\cup head(r\\ref {r:r:del} )^{ts=k} = U_{7k+4} \\cup \\lbrace add(p,q,t,k) : p \\in P, t \\in T, q=W(t,p) \\rbrace \\cup \\lbrace del(p,q,t,k) : p \\in P, t \\in T, q=W(p,t) \\rbrace \\cup \\lbrace del(p,q,t,k) : p \\in P, t \\in T, q=M_{k}(p) \\rbrace $ $U_{7k+6}=U_{7k+5} \\cup head(r\\ref {r:totincr})^{ts=k} \\cup head(r\\ref {r:totdecr})^{ts=k} = U_{7k+5} \\cup \\lbrace tot\\_incr(p,q,k) : p \\in P, 0 \\le q \\le ntok \\rbrace \\cup \\lbrace tot\\_decr(p,q,k) : p \\in P, 0 \\le q \\le ntok \\rbrace $ $U_{7k+7}=U_{7k+6} \\cup head(r\\ref {r:nextstate})^{ts=k} \\cup head(a\\ref {a:overc:place})^{ts=k} \\cup head(a\\ref {a:r:maxfire:cnh})^{ts=k} = U_{7k+6} \\cup \\\\ \\lbrace consumesmore(p,k) : p \\in P\\rbrace \\cup \\lbrace holds(p,q,k+1) : p \\in P, 0 \\le q \\le ntok \\rbrace \\cup \\lbrace could\\_not\\_have(t,k) : t \\in T \\rbrace $ $U_{7k+8}=U_{7k+7} \\cup head(a\\ref {a:overc:gen}) = U_{7k+7} \\cup \\lbrace consumesmore \\rbrace $ where $head(r_i)^{ts=k}$ are head atoms of ground rule $r_i$ in which $ts=k$ .", "We write $A_i^{ts=k} = \\lbrace a(\\dots ,ts) : a(\\dots ,ts) \\in A_i, ts=k \\rbrace $ as short hand for all atoms in $A_i$ with $ts=k$ .", "$U_{\\alpha }, 0 \\le \\alpha \\le 7k+8$ form a splitting sequence, since each $U_i$ is a splitting set of $\\Pi ^4$ , and $\\langle U_{\\alpha }\\rangle _{\\alpha < \\mu }$ is a monotone continuous sequence, where $U_0 \\subseteq U_1 \\dots \\subseteq U_{8(k+1)}$ and $\\bigcup _{\\alpha < \\mu }{U_{\\alpha }} = lit(\\Pi ^4)$ .", "We compute the answer set of $\\Pi ^4$ using the splitting sets as follows: $bot_{U_0}(\\Pi ^4) = f\\ref {f:place} \\cup f\\ref {f:trans} \\cup f\\ref {f:time} \\cup i\\ref {i:holds} \\cup f\\ref {f:num}$ and $X_0 = A_1 \\cup \\dots \\cup A_4 \\cup A_7$ ($= U_0$ ) is its answer set – using forced atom proposition $eval_{U_0}(bot_{U_1}(\\Pi ^4) \\setminus bot_{U_0}(\\Pi ^4), X_0) = \\lbrace ptarc(p,t,q,0) \\text{:-}.", "\| q=W(p,t) \\rbrace \\cup \\\\ \\lbrace tparc(t,p,q,0) \\text{:-}.", "\| q=W(t,p) \\rbrace \\cup \\lbrace ptarc(p,t,q,0) \\text{:-}.", "\| q=M_0(p) \\rbrace \\cup \\\\ \\lbrace iptarc(p,t,1,0) \\text{:-}.", "\\rbrace \\cup \\lbrace tptarc(p,t,q,0) \\text{:-}.", "\| q = QW(p,t) \\rbrace $ .", "Its answer set $X_1=A_5^{ts=0} \\cup A_6^{ts=0} \\cup A_{19}^{ts=0} \\cup A_{20}^{ts=0} \\cup A_{21}^{ts=0}$ – using forced atom proposition and construction of $A_5, A_6, A_{19}, A_{20}, A_{21}$ .", "$eval_{U_1}(bot_{U_2}(\\Pi ^4) \\setminus bot_{U_1}(\\Pi ^4), X_0 \\cup X_1) = \\lbrace notenabled(t,0) \\text{:-} .", "\| (\\lbrace trans(t), \\\\ ptarc(p,t,n,0), holds(p,q,0) \\rbrace \\subseteq X_0 \\cup X_1, \\text{~where~} q < n) \\text{~or~} (\\lbrace notenabled(t,0) \\text{:-} .", "\| \\\\ (\\lbrace trans(t), iptarc(p,t,n2,0), holds(p,q,0) \\rbrace \\subseteq X_0 \\cup X_1, \\text{~where~} q \\ge n2 \\rbrace ) \\text{~or~} \\\\ (\\lbrace trans(t), tptarc(p,t,n3,0), holds(p,q,0) \\rbrace \\subseteq X_0 \\cup X_1, \\text{~where~} q < n3) \\rbrace $ .", "Its answer set $X_2=A_8^{ts=0}$ – using forced atom proposition and construction of $A_8$ .", "where, $q=M_0(p)$ , and $n=W(p,t)$ for an arc $(p,t) \\in E^-$ – by construction of $i\\ref {i:holds}$ and $f\\ref {f:r:ptarc}$ in $\\Pi ^4$ , and in an arc $(p,t) \\in E^-$ , $p \\in \\bullet t$ (by definition REF of preset) $n2=1$ – by construction of $iptarc$ predicates in $\\Pi ^4$ , meaning $q \\ge n2 \\equiv q \\ge 1 \\equiv q > 0$ , $tptarc(p,t,n3,0)$ represents $n3=QW(p,t)$ , where $(p,t) \\in Q$ thus, $notenabled(t,0) \\in X_1$ represents $(\\exists p \\in \\bullet t : M_0(p) < W(p,t)) \\vee (\\exists p \\in I(t) : M_0(p) > 0) \\vee (\\exists (p,t) \\in Q : M_{ts}(p) < QW(p,t))$ .", "$eval_{U_2}(bot_{U_3}(\\Pi ^4) \\setminus bot_{U_2}(\\Pi ^4), X_0 \\cup \\dots \\cup X_2) = \\lbrace enabled(t,0) \\text{:-}.", "\| trans(t) \\in X_0 \\cup \\dots \\cup X_2, notenabled(t,0) \\notin X_0 \\cup \\dots \\cup X_2 \\rbrace $ .", "Its answer set is $X_3 = A_9^{ts=0}$ – using forced atom proposition and construction of $A_9$ .", "since an $enabled(t,0) \\in X_3$ if $\\nexists ~notenabled(t,0) \\in X_0 \\cup \\dots \\cup X_2$ ; which is equivalent to $(\\nexists p \\in \\bullet t : M_0(p) < W(p,t)) \\wedge (\\nexists p \\in I(t) : M_0(p) > 0) \\wedge (\\nexists (p,t) \\in Q : M_0(p) < QW(p,t) ) \\equiv (\\forall p \\in \\bullet t: M_0(p) \\ge W(p,t)) \\wedge (\\forall p \\in I(t) : M_0(p) = 0)$ .", "$eval_{U_3}(bot_{U_4}(\\Pi ^4) \\setminus bot_{U_3}(\\Pi ^4), X_0 \\cup \\dots \\cup X_3) = \\lbrace \\lbrace fires(t,0)\\rbrace \\text{:-}.", "\| enabled(t,0) \\\\ \\text{~holds in~} X_0 \\cup \\dots \\cup X_3 \\rbrace $ .", "It has multiple answer sets $X_{4.1}, \\dots , X_{4.n}$ , corresponding to elements of power set of $fires(t,0)$ atoms in $eval_{U_3}(...)$ – using supported rule proposition.", "Since we are showing that the union of answer sets of $\\Pi ^4$ determined using splitting is equal to $A$ , we only consider the set that matches the $fires(t,0)$ elements in $A$ and call it $X_4$ , ignoring the rest.", "Thus, $X_4 = A_{10}^{ts=0}$ , representing $T_0$ .", "in addition, for every $t$ such that $enabled(t,0) \\in X_0 \\cup \\dots \\cup X_3, R(t) \\ne \\emptyset $ ; $fires(t,0) \\in X_4$ – per definition REF (firing set); requiring that a reset transition is fired when enabled thus, the firing set $T_0$ will not be eliminated by the constraint $f\\ref {f:c:rptarc:elim}$ $eval_{U_4}(bot_{U_5}(\\Pi ^4) \\setminus bot_{U_4}(\\Pi ^4), X_0 \\cup \\dots \\cup X_4) = \\lbrace add(p,n,t,0) \\text{:-}.", "\| \\lbrace fires(t,0), \\\\ tparc(t,p,n,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_4 \\rbrace \\cup \\lbrace del(p,n,t,0) \\text{:-}.", "\| \\lbrace fires(t,0), ptarc(p,t,n,0) \\rbrace \\\\ \\subseteq X_0 \\cup \\dots \\cup X_4 \\rbrace $ .", "It's answer set is $X_5 = A_{11}^{ts=0} \\cup A_{12}^{ts=0}$ – using forced atom proposition and definitions of $A_{11}$ and $A_{12}$ .", "where, each $add$ atom is equivalent to $n=W(t,p),p \\in t \\bullet $ , and each $del$ atom is equivalent to $n=W(p,t), p \\in \\bullet t$ ; or $n=M_k(p), p \\in R(t)$ , representing the effect of transitions in $T_0$ – by construction $eval_{U_5}(bot_{U_6}(\\Pi ^4) \\setminus bot_{U_5}(\\Pi ^4), X_0 \\cup \\dots \\cup X_5) = \\lbrace tot\\_incr(p,qq,0) \\text{:-}.", "\| $ $qq=\\sum _{add(p,q,t,0) \\in X_0 \\cup \\dots \\cup X_5}{q} \\rbrace \\cup \\lbrace tot\\_decr(p,qq,0) \\text{:-}.", "\| qq=\\sum _{del(p,q,t,0) \\in X_0 \\cup \\dots \\cup X_5}{q} \\rbrace $ .", "It's answer set is $X_6 = A_{13}^{ts=0} \\cup A_{14}^{ts=0}$ – using forced atom proposition and definitions of $A_{13}$ and $A_{14}$ .", "where, each $tot\\_incr(p,qq,0)$ , $qq=\\sum _{add(p,q,t,0) \\in X_0 \\cup \\dots X_5}{q}$ $\\equiv qq=\\sum _{t \\in X_4, p \\in t \\bullet }{W(p,t)}$ , and each $tot\\_decr(p,qq,0)$ , $qq=\\sum _{del(p,q,t,0) \\in X_0 \\cup \\dots X_5}{q}$ $\\equiv qq=\\sum _{t \\in X_4, p \\in \\bullet t}{W(t,p)} + \\sum _{t \\in X_4, p \\in R(t)}{M_{k}(p)}$ , represent the net effect of transitions in $T_0$ – by construction $eval_{U_6}(bot_{U_7}(\\Pi ^4) \\setminus bot_{U_6}(\\Pi ^4), X_0 \\cup \\dots \\cup X_6) = \\lbrace consumesmore(p,0) \\text{:-}.", "\| \\\\ \\lbrace holds(p,q,0), tot\\_decr(p,q1,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_6, q1 > q \\rbrace \\cup \\lbrace holds(p,q,1) \\text{:-}., \| \\\\ \\lbrace holds(p,q1,0), tot\\_incr(p,q2,0), tot\\_decr(p,q3,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_6, q=q1+q2-q3 \\rbrace \\cup \\lbrace could\\_not\\_have(t,0) \\text{:-}.", "\| \\lbrace enabled(t,0), ptarc(s,t,q), holds(s,qq,0), \\\\ tot\\_decr(s,qqq,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_6, fires(t,0) \\notin (X_0 \\cup \\dots \\cup X_6), q > qq-qqq \\rbrace $ .", "It's answer set is $X_7 = A_{15}^{ts=0} \\cup A_{17}^{ts=0} \\cup A_{18}^{ts=0}$ – using forced atom proposition and definitions of $A_{15}, A_{17}, A_{18}, A_9$ .", "where, $consumesmore(p,0)$ represents $\\exists p : q=M_0(p), q1= \\\\ \\sum _{t \\in T_0, p \\in \\bullet t}{W(p,t)}+\\sum _{t \\in T_0, p \\in R(t)}{M_0(p)}, q1 > q$ , indicating place $p$ will be over consumed if $T_0$ is fired, as defined in definition REF (conflicting transitions), $holds(p,q,1)$ represents $q=M_1(p)$ – by construction of $\\Pi ^4$ , and $could\\_not\\_have(t,0)$ represents enabled transition $t \\in T_0$ that could not fire due to insufficient tokens $X_7$ does not contain $could\\_not\\_have(t,0)$ , when $enabled(t,0) \\in X_0 \\cup \\dots \\cup X_6$ and $fires(t,0) \\notin X_0 \\cup \\dots \\cup X_6$ due to construction of $A$ , encoding of $a\\ref {a:r:maxfire:cnh}$ and its body atoms.", "As a result it is not eliminated by the constraint $a\\ref {a:maxfire:elim}$ $ \\vdots $ $eval_{U_{7k+0}}(bot_{U_{7k+1}}(\\Pi ^4) \\setminus bot_{U_{7k+0}}(\\Pi ^4), X_0 \\cup \\dots \\cup X_{7k+0}) = \\lbrace ptarc(p,t,q,k) \\text{:-}.", "\| q=W(p,t) \\rbrace \\cup \\lbrace tparc(t,p,q,k) \\text{:-}.", "\| q=W(t,p) \\rbrace \\cup \\lbrace ptarc(p,t,q,k) \\text{:-}.", "\| q=M_k(p) \\rbrace \\cup \\lbrace iptarc(p,t,1,k) \\text{:-}.", "\\rbrace $ .", "Its answer set $X_{7k+1}=A_5^{ts=k} \\cup A_6^{ts=k} \\cup A_{19}^{ts=k} \\cup A_{20}^{ts=k}$ – using forced atom proposition and construction of $A_5, A_6, A_{19}, A_{20}$ .", "$eval_{U_{7k+1}}(bot_{U_{7k+2}}(\\Pi ^4) \\setminus bot_{U_{7k+1}}(\\Pi ^4), X_0 \\cup \\dots \\cup X_{7k+1}) = \\lbrace notenabled(t,k) \\text{:-} .", "\| $ $(\\lbrace trans(t), ptarc(p,t,n,k), holds(p,q,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{7k+1}, \\text{~where~} q < n) \\text{~or~} \\\\ \\lbrace notenabled(t,k) \\text{:-} .", "\| (\\lbrace trans(t), iptarc(p,t,n2,k), holds(p,q,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{7k+1}, \\\\ \\text{~where~} q \\ge n2 \\rbrace \\rbrace $ .", "Its answer set $X_{7k+2}=A_8^{ts=k}$ – using forced atom proposition and construction of $A_8$ .", "where, $q=M_k(p)$ , and $n=W(p,t)$ for an arc $(p,t) \\in E^-$ – by construction of $holds$ and $ptarc$ predicates in $\\Pi ^4$ , and in an arc $(p,t) \\in E^-$ , $p \\in \\bullet t$ (by definition REF of preset) $n2=1$ – by construction of $iptarc$ predicates in $\\Pi ^4$ , meaning $q \\ge n2 \\equiv q \\ge 1 \\equiv q > 0$ , thus, $notenabled(t,k) \\in X_{7k+1}$ represents $(\\exists p \\in \\bullet t : M_k(p) < W(p,t)) \\vee (\\exists p \\in I(t) : M_k(p) > k)$ .", "$eval_{U_{7k+2}}(bot_{U_{7k+3}}(\\Pi ^4) \\setminus bot_{U_{7k+2}}(\\Pi ^4), X_0 \\cup \\dots \\cup X_{7k+2}) = \\lbrace enabled(t,k) \\text{:-}.", "\| \\\\ trans(t) \\in X_0 \\cup \\dots \\cup X_{7k+2} \\wedge notenabled(t,k) \\notin X_0 \\cup \\dots \\cup X_{7k+2} \\rbrace $ .", "Its answer set is $X_{7k+3} = A_9^{ts=k}$ – using forced atom proposition and construction of $A_9$ .", "since an $enabled(t,k) \\in X_{7k+3}$ if $\\nexists ~notenabled(t,k) \\in X_0 \\cup \\dots \\cup X_{7k+2}$ ; which is equivalent to $(\\nexists p \\in \\bullet t : M_k(p) < W(p,t)) \\wedge (\\nexists p \\in I(t) : M_k(p) > k) \\equiv (\\forall p \\in \\bullet t: M_k(p) \\ge W(p,t)) \\wedge (\\forall p \\in I(t) : M_k(p) = k)$ .", "$eval_{U_{7k+3}}(bot_{U_{7k+4}}(\\Pi ^4) \\setminus bot_{U_{7k+3}}(\\Pi ^4), X_0 \\cup \\dots \\cup X_{7k+3}) = \\lbrace \\lbrace fires(t,k)\\rbrace \\text{:-}.", "\| \\\\ enabled(t,k) \\\\ \\text{~holds in~} X_0 \\cup \\dots \\cup X_{7k+3} \\rbrace $ .", "It has multiple answer sets $X_{7k+4.1}, \\dots , X_{7k+4.n}$ , corresponding to elements of power set of $fires(t,k)$ atoms in $eval_{U_{7k+3}}(...)$ – using supported rule proposition.", "Since we are showing that the union of answer sets of $\\Pi ^4$ determined using splitting is equal to $A$ , we only consider the set that matches the $fires(t,k)$ elements in $A$ and call it $X_{7k+4}$ , ignoring the rest.", "Thus, $X_{7k+4} = A_{10}^{ts=k}$ , representing $T_k$ .", "in addition, for every $t$ such that $enabled(t,k) \\in X_0 \\cup \\dots \\cup X_{7k+3}, R(t) \\ne \\emptyset $ ; $fires(t,k) \\in X_{7k+4}$ – per definition REF (firing set); requiring that a reset transition is fired when enabled thus, the firing set $T_k$ will not be eliminated by the constraint $f\\ref {f:c:rptarc:elim}$ $eval_{U_{7k+4}}(bot_{U_{7k+5}}(\\Pi ^4) \\setminus bot_{U_{7k+4}}(\\Pi ^4), X_0 \\cup \\dots \\cup X_{7k+4}) = \\lbrace add(p,n,t,k) \\text{:-}.", "\| \\\\ \\lbrace fires(t,k), tparc(t,p,n,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{7k+4} \\rbrace \\cup \\lbrace del(p,n,t,k) \\text{:-}.", "\| \\lbrace fires(t,k), \\\\ ptarc(p,t,n,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{7k+4} \\rbrace $ .", "It's answer set is $X_{7k+5} = A_{11}^{ts=k} \\cup A_{12}^{ts=k}$ – using forced atom proposition and definitions of $A_{11}$ and $A_{12}$ .", "where, each $add$ atom is equivalent to $n=W(t,p) : p \\in t \\bullet $ , and each $del$ atom is equivalent to $n=W(p,t) : p \\in \\bullet t$ ; or $n=M_k(p) : p \\in R(t)$ , representing the effect of transitions in $T_k$ $eval_{U_{7k+5}}(bot_{U_{7k+6}}(\\Pi ^4) \\setminus bot_{U_{7k+5}}(\\Pi ^4), X_0 \\cup \\dots \\cup X_{7k+5}) = \\\\ \\lbrace tot\\_incr(p,qq,k) \\text{:-}.", "\| qq=\\sum _{add(p,q,t,k) \\in X_0 \\cup \\dots \\cup X_{7k+5}}{q} \\rbrace \\cup \\\\ \\lbrace tot\\_decr(p,qq,k) \\text{:-}.", "\| qq=\\sum _{del(p,q,t,k) \\in X_0 \\cup \\dots \\cup X_{7k+5}}{q} \\rbrace $ .", "It's answer set is $X_{7k+6} = A_{13}^{ts=k} \\cup A_{14}^{ts=k}$ – using forced atom proposition and definitions of $A_{13}$ and $A_{14}$ .", "where, each $tot\\_incr(p,qq,k)$ , $qq=\\sum _{add(p,q,t,k) \\in X_0 \\cup \\dots X_{7k+5}}{q}$ $\\equiv qq=\\sum _{t \\in X_{7k+4}, p \\in t \\bullet }{W(p,t)}$ , and each $tot\\_decr(p,qq,k)$ , $qq=\\sum _{del(p,q,t,k) \\in X_0 \\cup \\dots X_{7k+5}}{q}$ $\\equiv qq=\\sum _{t \\in X_{7k+4}, p \\in \\bullet t}{W(t,p)} + \\sum _{t \\in X_{7k+4}, p \\in R(t)}{M_{k}(p)}$ , represent the net effect of transition in $T_k$ $eval_{U_{7k+6}}(bot_{U_{7k+7}}(\\Pi ^4) \\setminus bot_{U_{7k+6}}(\\Pi ^4), X_0 \\cup \\dots \\cup X_{7k+6}) = \\lbrace consumesmore(p,k) \\text{:-}.", "\| \\\\ \\lbrace holds(p,q,k), tot\\_decr(p,q1,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{7k+6}, q1 > q \\rbrace \\cup \\lbrace holds(p,q,1) \\text{:-}., \| \\\\ \\lbrace holds(p,q1,k), tot\\_incr(p,q2,k), tot\\_decr(p,q3,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{7k+6}, q=q1+q2-q3 \\rbrace \\cup \\lbrace could\\_not\\_have(t,k) \\text{:-}.", "\| \\lbrace enabled(t,k), ptarc(s,t,q), holds(s,qq,k), \\\\ tot\\_decr(s,qqq,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{7k+6}, fires(t,k) \\notin (X_0 \\cup \\dots \\cup X_{7k+6}), q > qq-qqq \\rbrace $ .", "It's answer set is $X_{7k+7} = A_{15}^{ts=k} \\cup A_{17}^{ts=k} \\cup A_{18}^{ts=k}$ – using forced atom proposition and definitions of $A_{15}, A_{17}, A_{18}, A_9$ .", "where, $consumesmore(p,k)$ represents $\\exists p : q=M_k(p), q1= \\\\ \\sum _{t \\in T_k, p \\in \\bullet t}{W(p,t)}+\\sum _{t \\in T_k, p \\in R(t)}{M_k(p)}, q1 > q$ , indicating place $p$ that will be over consumed if $T_k$ is fired, as defined in definition REF (conflicting transitions), $holds(p,q,k+1)$ represents $q=M_{k+1}(p)$ – by construction of $\\Pi ^4$ , and $could\\_not\\_have(t,k)$ represents enabled transition $t$ in $T_k$ that could not be fired due to insufficient tokens $X_{7k+7}$ does not contain $could\\_not\\_have(t,k)$ , when $enabled(t,k) \\in X_0 \\cup \\dots \\cup X_{7k+6}$ and $fires(t,k) \\notin X_0 \\cup \\dots \\cup X_{7k+6}$ due to construction of $A$ , encoding of $a\\ref {a:r:maxfire:cnh}$ and its body atoms.", "As a result it is not eliminated by the constraint $a\\ref {a:maxfire:elim}$ $eval_{U_{7k+7}}(bot_{U_{7k+8}}(\\Pi ^4) \\setminus bot_{U_{7k+7}}(\\Pi ^4), X_0 \\cup \\dots \\cup X_{7k+7}) = \\lbrace consumesmore \\text{:-}.", "\| \\\\ \\lbrace consumesmore(p,0),\\dots , $ $consumesmore(p,k) \\rbrace \\cap (X_0 \\cup \\dots \\cup X_{7k+7}) \\ne \\emptyset \\rbrace $ .", "It's answer set is $X_{7k+8} = A_{16}$ – using forced atom proposition $X_{7k+8}$ will be empty since none of $consumesmore(p,0),\\dots , \\\\ consumesmore(p,k)$ hold in $X_0 \\cup \\dots \\cup X_{7k+8}$ due to the construction of $A$ , encoding of $a\\ref {a:overc:place}$ and its body atoms.", "As a result, it is not eliminated by the constraint $a\\ref {a:overc:elim}$ The set $X = X_0 \\cup \\dots \\cup X_{7k+8}$ is the answer set of $\\Pi ^4$ by the splitting sequence theorem REF .", "Each $X_i, 0 \\le i \\le 7k+8$ matches a distinct portion of $A$ , and $X = A$ , thus $A$ is an answer set of $\\Pi ^4$ .", "Next we show (REF ): Given $\\Pi ^4$ be the encoding of a Petri Net $PN(P,T,E,W,R,I)$ with initial marking $M_0$ , and $A$ be an answer set of $\\Pi ^4$ that satisfies (REF ) and (REF ), then we can construct $X=M_0,T_0,\\dots ,M_k,T_k,M_{k+1}$ from $A$ , such that it is an execution sequence of $PN$ .", "We construct the $X$ as follows: $M_i = (M_i(p_0), \\dots , M_i(p_n))$ , where $\\lbrace holds(p_0,M_i(p_0),i), \\dots holds(p_n,M_i(p_n),i) \\rbrace \\\\ \\subseteq A$ , for $0 \\le i \\le k+1$ $T_i = \\lbrace t : fires(t,i) \\in A\\rbrace $ , for $0 \\le i \\le k$ and show that $X$ is indeed an execution sequence of $PN$ .", "We show this by induction over $k$ (i.e.", "given $M_k$ , $T_k$ is a valid firing set and its firing produces marking $M_{k+1}$ ).", "Base case: Let $k=0$ , and $M_0$ is a valid marking in $X$ for $PN$ , show [(1)] $T_0$ is a valid firing set for $M_0$ , and firing of $T_0$ in $M_0$ produces marking $M_1$ .", "We show $T_0$ is a valid firing set for $M_0$ .", "Let $\\lbrace fires(t_0,0), \\dots , fires(t_x,0) \\rbrace $ be the set of all $fires(\\dots ,0)$ atoms in $A$ , Then for each $fires(t_i,0) \\in A$ $enabled(t_i,0) \\in A$ – from rule $a\\ref {a:fires}$ and supported rule proposition Then $notenabled(t_i,0) \\notin A$ – from rule $e\\ref {e:enabled}$ and supported rule proposition Then either of $body(e\\ref {e:r:ne:ptarc} )$ , $body(e\\ref {e:ne:iptarc} )$ , or $body(e\\ref {e:ne:tptarc} )$ must not hold in $A$ – from rules $e\\ref {e:r:ne:ptarc} ,e\\ref {e:ne:iptarc} ,e\\ref {e:ne:tptarc} $ and forced atom proposition Then $q \\lnot < n_i \\equiv q \\ge n_i$ in $e\\ref {e:r:ne:ptarc} $ for all $\\lbrace holds(p,q,0), ptarc(p,t_i,n_i,0)\\rbrace \\subseteq A$ – from $e\\ref {e:r:ne:ptarc} $ , forced atom proposition, and given facts ($holds(p,q,0) \\in A, ptarc(p,t_i,n_i,0) \\in A$ ) And $q \\lnot \\ge n_i \\equiv q < n_i$ in $e\\ref {e:ne:iptarc} $ for all $\\lbrace holds(p,q,0), iptarc(p,t_i,n_i,0) \\rbrace \\subseteq A, n_i=1$ ; $q > n_i \\equiv q = 0$ – from $e\\ref {e:ne:iptarc} $ , forced atom proposition, given facts ($holds(p,q,0) \\in A, iptarc(p,t_i,1,0) \\in A$ ), and $q$ is a positive integer And $q \\lnot < n_i \\equiv q \\ge n_i$ in $e\\ref {e:ne:tptarc} $ for all $\\lbrace holds(p,q,0), tptarc(p,t_i,n_i,0) \\rbrace \\subseteq A$ – from $e\\ref {e:ne:tptarc} $ , forced atom proposition, and given facts Then $(\\forall p \\in \\bullet t_i, M_0(p) \\ge W(p,t_i)) \\wedge (\\forall p \\in I(t_i), M_0(p) = 0) \\wedge (\\forall (p,t_i) \\in Q, M_0(p) \\ge QW(p,t_i))$ – from the following $holds(p,q,0) \\in A$ represents $q=M_0(p)$ – rule $i\\ref {i:holds}$ construction $ptarc(p,t_i,n_i,0) \\in A$ represents $n_i=W(p,t_i)$ – rule $f\\ref {f:r:ptarc}$ construction; or it represents $n_i = M_0(p)$ – rule $f\\ref {f:rptarc}$ construction; the construction of $f\\ref {f:rptarc}$ ensures that $notenabled(t,0)$ is never true due to the reset arc definition REF of preset $\\bullet t_i$ in $PN$ definition REF of enabled transition in $PN$ Then $t_i$ is enabled and can fire in $PN$ , as a result it can belong to $T_0$ – from definition REF of enabled transition And $consumesmore \\notin A$ , since $A$ is an answer set of $\\Pi ^4$ – from rule $a\\ref {a:overc:elim}$ and supported rule proposition Then $\\nexists consumesmore(p,0) \\in A$ – from rule $a\\ref {a:overc:gen}$ and supported rule proposition Then $\\nexists \\lbrace holds(p,q,0), tot\\_decr(p,q1,0) \\rbrace \\subseteq A : q1>q$ in $body(a\\ref {a:overc:place})$ – from $a\\ref {a:overc:place}$ and forced atom proposition Then $\\nexists p : (\\sum _{t_i \\in \\lbrace t_0,\\dots ,t_x\\rbrace , p \\in \\bullet t_i}{W(p,t_i)}+\\sum _{t_i \\in \\lbrace t_0,\\dots ,t_x\\rbrace , p \\in R(t_i)}{M_0(p)}) > M_0(p)$ – from the following $holds(p,q,0)$ represents $q=M_0(p)$ – from rule $i\\ref {i:holds}$ construction, given $tot\\_decr(p,q1,0) \\in A$ if $\\lbrace del(p,q1_0,t_0,0), \\dots , del(p,q1_x,t_x,0) \\rbrace \\subseteq A$ , where $q1 = q1_0+\\dots +q1_x$ – from $r\\ref {r:totdecr}$ and forced atom proposition $del(p,q1_i,t_i,0) \\in A$ if $\\lbrace fires(t_i,0), ptarc(p,t_i,q1_i,0) \\rbrace \\subseteq A$ – from $r\\ref {r:r:del} $ and supported rule proposition $del(p,q1_i,t_i,0)$ represents removal of $q1_i = W(p,t_i)$ tokens from $p \\in \\bullet t_i$ ; or it represents removal of $q1_i = M_0(p)$ tokens from $p \\in R(t_i)$ – from rule $r\\ref {r:r:del} $ , supported rule proposition, and definition REF of transition execution in $PN$ Then the set of transitions in $T_0$ do not conflict – by the definition REF of conflicting transitions And for each $enabled(t_j,0) \\in A$ and $fires(t_j,0) \\notin A$ , $could\\_not\\_have(t_j,0) \\in A$ , since $A$ is an answer set of $\\Pi ^4$ - from rule $a\\ref {a:maxfire:elim}$ and supported rule proposition Then $\\lbrace enabled(t_j,0), holds(s,qq,0), ptarc(s,t_j,q,0), \\\\ tot\\_decr(s,qqq,0) \\rbrace \\subseteq A$ , such that $q > qq - qqq$ and $fires(t_j,0) \\notin A$ - from rule $a\\ref {a:r:maxfire:cnh}$ and supported rule proposition Then for an $s \\in \\bullet t_j \\cup R(t_j)$ , $q > M_0(s) - (\\sum _{t_i \\in T_0, s \\in \\bullet t_i}{W(s,t_i)} + \\sum _{t_i \\in T_0, s \\in R(t_i)}{M_0(s))}$ , where $q=W(s,t_j) \\text{~if~} s \\in \\bullet t_j, \\text{~or~} M_0(s) \\text{~otherwise}$ - from the following: $ptarc(s,t_i,q,0)$ represents $q=W(s,t_i)$ if $(s,t_i) \\in E^-$ or $q=M_0(s)$ if $s \\in R(t_i)$ – from rule $f\\ref {f:r:ptarc},f\\ref {f:rptarc}$ construction $holds(s,qq,0)$ represents $qq=M_0(s)$ – from $i\\ref {i:holds}$ construction $tot\\_decr(s,qqq,0) \\in A$ if $\\lbrace del(s,qqq_0,t_0,0), \\dots , del(s,qqq_x,t_x,0) \\rbrace \\subseteq A$ – from rule $r\\ref {r:totdecr}$ construction and supported rule proposition $del(s,qqq_i,t_i,0) \\in A$ if $\\lbrace fires(t_i,0), ptarc(s,t_i,qqq_i,0) \\rbrace \\subseteq A$ – from rule $r\\ref {r:r:del} $ and supported rule proposition $del(s,qqq_i,t_i,0)$ represents $qqq_i = W(s,t_i) : t_i \\in T_0, (s,t_i) \\in E^-$ , or $qqq_i = M_0(t_i) : t_i \\in T_0, s \\in R(t_i)$ – from rule $f\\ref {f:r:ptarc},f\\ref {f:rptarc}$ construction $tot\\_decr(q,qqq,0)$ represents $\\sum _{t_i \\in T_0, s \\in \\bullet t_i}{W(s,t_i)} + \\\\ \\sum _{t_i \\in T_0, s \\in R(t_i)}{M_0(s)}$ – from (C,D,E) above Then firing $T_0 \\cup \\lbrace t_j \\rbrace $ would have required more tokens than are present at its source place $s \\in \\bullet t_j \\cup R(t_j)$ .", "Thus, $T_0$ is a maximal set of transitions that can simultaneously fire.", "And for each reset transition $t_r$ with $enabled(t_r,0) \\in A$ , $fires(t_r,0) \\in A$ , since $A$ is an answer set of $\\Pi ^2$ - from rule $f\\ref {f:c:rptarc:elim}$ and supported rule proposition Then, the firing set $T_0$ satisfies the reset-transition requirement of definition REF (firing set) Then $\\lbrace t_0, \\dots , t_x\\rbrace = T_0$ – using 1(a),1(b),1(d) above; and using 1(c) it is a maximal firing set We show $M_1$ is produced by firing $T_0$ in $M_0$ .", "Let $holds(p,q,1) \\in A$ Then $\\lbrace holds(p,q1,0), tot\\_incr(p,q2,0), tot\\_decr(p,q3,0) \\rbrace \\subseteq A : q=q1+q2-q3$ – from rule $r\\ref {r:nextstate}$ and supported rule proposition Then $holds(p,q1,0) \\in A$ represents $q1=M_0(p)$ – given, rule $i\\ref {i:holds}$ construction; Then $\\lbrace add(p,q2_0,t_0,0), \\dots , add(p,q2_j,t_j,0)\\rbrace \\subseteq A : q2_0 + \\dots + q2_j = q2$ and $\\lbrace del(p,q3_0,t_0,0), \\dots , del(p,q3_l,t_l,0)\\rbrace \\subseteq A : q3_0 + \\dots + q3_l = q3$ – rules $r\\ref {r:totincr},r\\ref {r:totdecr}$ and supported rule proposition, respectively Then $\\lbrace fires(t_0,0), \\dots , fires(t_j,0) \\rbrace \\subseteq A$ and $\\lbrace fires(t_0,0), \\dots , fires(t_l,0) \\rbrace \\\\ \\subseteq A$ – rules $r\\ref {r:r:add},r\\ref {r:r:del}$ and supported rule proposition, respectively Then $\\lbrace fires(t_0,0), \\dots , fires(t_j,0) \\rbrace \\cup \\lbrace fires(t_0,0), \\dots , fires(t_l,0) \\rbrace \\subseteq A = \\lbrace fires(t_0,0), \\dots , $ $fires(t_x,0) \\rbrace \\subseteq A$ – set union of subsets Then for each $fires(t_x,0) \\in A$ we have $t_x \\in T_0$ – already shown in item REF above Then $q = M_0(p) + \\sum _{t_x \\in T_0 \\wedge p \\in t_x \\bullet }{W(t_x,p)} - (\\sum _{t_x \\in T_0 \\wedge p \\in \\bullet t_x}{W(p,t_x)} + \\\\ \\sum _{t_x \\in T_0 \\wedge p \\in R(t_x)}{M_0(p)})$ – from (REF ) above and the following Each $add(p,q_j,t_j,0) \\in A$ represents $q_j=W(t_j,p)$ for $p \\in t_j \\bullet $ – rule $r\\ref {r:r:add} $ encoding, and definition REF of transition execution in $PN$ Each $del(p,t_y,q_y,0) \\in A$ represents either $q_y=W(p,t_y)$ for $p \\in \\bullet t_y$ , or $q_y=M_0(p)$ for $p \\in R(t_y)$ – from rule $r\\ref {r:r:del} ,f\\ref {f:r:ptarc} $ encoding and definition REF of transition execution in $PN$ ; or from rule $r\\ref {r:r:del} ,f\\ref {f:rptarc}$ encoding and definition of reset arc in $PN$ Each $tot\\_incr(p,q2,0) \\in A$ represents $q2=\\sum _{t_x \\in T_0 \\wedge p \\in t_x \\bullet }{W(t_x,p)}$ – aggregate assignment atom semantics in rule $r\\ref {r:totincr}$ Each $tot\\_decr(p,q3,0) \\in A$ represents $q3=\\sum _{t_x \\in T_0 \\wedge p \\in \\bullet t_x}{W(p,t_x)} + \\sum _{t_x \\in T_0 \\wedge p \\in R(t_x)}{M_0(p)}$ – aggregate assignment atom semantics in rule $r\\ref {r:totdecr}$ Then, $M_1(p) = q$ – since $holds(p,q,1) \\in A$ encodes $q=M_1(p)$ – from construction Inductive Step: Let $k > 0$ , and $M_k$ is a valid marking in $X$ for $PN$ , show [(1)] $T_k$ is a valid firing set for $M_k$ , and firing $T_k$ in $M_k$ produces marking $M_{k+1}$ .", "We show that $T_k$ is a valid firing set in $M_k$ .", "Let $\\lbrace fires(t_0,k), \\dots , fires(t_x,k) \\rbrace $ be the set of all $fires(\\dots ,k)$ atoms in $A$ , Then for each $fires(t_i,k) \\in A$ $enabled(t_i,k) \\in A$ – from rule $a\\ref {a:fires}$ and supported rule proposition Then $notenabled(t_i,k) \\notin A$ – from rule $e\\ref {e:enabled}$ and supported rule proposition Then either of $body(e\\ref {e:r:ne:ptarc} )$ , $body(e\\ref {e:ne:iptarc} )$ , or $body(e\\ref {e:ne:tptarc} )$ must not hold in $A$ – from rule $e\\ref {e:r:ne:ptarc} ,e\\ref {e:ne:iptarc} ,e\\ref {e:ne:tptarc} $ and forced atom proposition Then $q \\lnot < n_i \\equiv q \\ge n_i$ in $e\\ref {e:r:ne:ptarc}$ for all $\\lbrace holds(p,q,k), ptarc(p,t_i,n_i,k)\\rbrace \\subseteq A$ – from $e\\ref {e:r:ne:ptarc}$ , forced atom proposition, and given facts ($holds(p,q,k) \\in A, ptarc(p,t_i,n_i,k) \\in A$ ) And $q \\lnot \\ge n_i \\equiv q < n_i$ in $e\\ref {e:ne:iptarc} $ for all $\\lbrace holds(p,q,k), iptarc(p,t_i,n_i,k) \\rbrace \\subseteq A, n_i=1; q > n_i \\equiv q = 0$ – from $e\\ref {e:ne:iptarc} $ , forced atom proposition, given facts $(holds(p,q,k) \\in A, iptarc(p,t_i,1,k) \\in A)$ , and $q$ is a positive integer And $q \\lnot < n_i \\equiv q \\ge n_i$ in $e\\ref {e:ne:tptarc} $ for all $\\lbrace holds(p,q,k), tptarc(p,t_i,n_i,k) \\rbrace \\subseteq A$ – from $e\\ref {e:ne:tptarc} $ , forced atom proposition, and given facts Then $(\\forall p \\in \\bullet t_i, M_k(p) \\ge W(p,t_i)) \\wedge (\\forall p \\in I(t_i), M_k(p) = 0) \\wedge (\\forall (p,t_i) \\in Q, M_k(p) \\ge QW(p,t_i))$ – from $holds(p,q,k) \\in A$ represents $q=M_k(p)$ – inductive assumption, given $ptarc(p,t_i,n_i,k) \\in A$ represents $n_i=W(p,t_i)$ – rule $f\\ref {f:r:ptarc}$ construction; or it represents $n_i=M_k(p)$ – rule $f\\ref {f:rptarc}$ construction; the construction of $f\\ref {f:rptarc}$ ensures that $notenabled(t,k)$ is never true due to the reset arc definition REF of preset $\\bullet t_i$ in $PN$ definition REF of enabled transition in $PN$ Then $t_i$ is enabled and can fire in $PN$ , as a result it can belong to $T_k$ – from definition REF of enabled transition And $consumesmore \\notin A$ , since $A$ is an answer set of $\\Pi ^4$ – from rule $a\\ref {a:overc:elim}$ and supported rule proposition Then $\\nexists consumesmore(p,k) \\in A$ – from rule $a\\ref {a:overc:gen}$ and supported rule proposition Then $\\nexists \\lbrace holds(p,q,k), tot\\_decr(p,q1,k) \\rbrace \\subseteq A : q1>q$ in $body(e\\ref {e:r:ne:ptarc})$ – from $a\\ref {a:overc:place}$ and forced atom proposition Then $\\nexists p : (\\sum _{t_i \\in \\lbrace t_0,\\dots ,t_x\\rbrace , p \\in \\bullet t_i}{W(p,t_i)}+\\sum _{t_i \\in \\lbrace t_0,\\dots ,t_x\\rbrace , p \\in R(t_i)}{M_k(p)}) > M_k(p)$ – from the following $holds(p,q,k)$ represents $q=M_k(p)$ – inductive assumption, given $tot\\_decr(p,q1,k) \\in A$ if $\\lbrace del(p,q1_0,t_0,k), \\dots , del(p,q1_x,t_x,k) \\rbrace \\subseteq A$ , where $q1 = q1_0+\\dots +q1_x$ – from $r\\ref {r:totdecr}$ and forced atom proposition $del(p,q1_i,t_i,k) \\in A$ if $\\lbrace fires(t_i,k), ptarc(p,t_i,q1_i,k) \\rbrace \\subseteq A$ – from $r\\ref {r:r:del} $ and supported rule proposition $del(p,q1_i,t_i,k)$ either represents removal of $q1_i = W(p,t_i)$ tokens from $p \\in \\bullet t_i$ ; or it represents removal of $q1_i = M_k(p)$ tokens from $p \\in R(t_i)$ – from rule $r\\ref {r:r:del} $ , supported rule proposition, and definition REF of transition execution in $PN$ Then $T_k$ does not contain conflicting transitions – by the definition REF of conflicting transitions And for each $enabled(t_j,k) \\in A$ and $fires(t_j,k) \\notin A$ , $could\\_not\\_have(t_j,k) \\in A$ , since $A$ is an answer set of $\\Pi ^4$ - from rule $a\\ref {a:maxfire:elim}$ and supported rule proposition Then $\\lbrace enabled(t_j,k), holds(s,qq,k), ptarc(s,t_j,q,k), \\\\ tot\\_decr(s,qqq,k) \\rbrace \\subseteq A$ , such that $q > qq - qqq$ and $fires(t_j,k) \\notin A$ - from rule $a\\ref {a:r:maxfire:cnh}$ and supported rule proposition Then for an $s \\in \\bullet t_j \\cup R(t_j)$ , $q > M_k(s) - (\\sum _{t_i \\in T_k, s \\in \\bullet t_i}{W(s,t_i)} + \\sum _{t_i \\in T_k, s \\in R(t_i)}{M_k(s))}$ , where $q=W(s,t_j) \\text{~if~} s \\in \\bullet t_j, \\text{~or~} M_k(s) \\text{~otherwise}$ - from the following: $ptarc(s,t_i,q,k)$ represents $q=W(s,t_i)$ if $(s,t_i) \\in E^-$ or $q=M_k(s)$ if $s \\in R(t_i)$ – from rule $f\\ref {f:r:ptarc},f\\ref {f:rptarc}$ construction $holds(s,qq,k)$ represents $qq=M_k(s)$ – construction $tot\\_decr(s,qqq,k) \\in A$ if $\\lbrace del(s,qqq_0,t_0,k), \\dots , del(s,qqq_x,t_x,k) \\rbrace \\\\ \\subseteq A$ – from rule $r\\ref {r:totdecr}$ construction and supported rule proposition $del(s,qqq_i,t_i,k) \\in A$ if $\\lbrace fires(t_i,k), ptarc(s,t_i,qqq_i,k) \\rbrace \\subseteq A$ – from rule $r\\ref {r:r:del} $ and supported rule proposition $del(s,qqq_i,t_i,k)$ represents $qqq_i = W(s,t_i) : t_i \\in T_k, (s,t_i) \\in E^-$ , or $qqq_i = M_k(t_i) : t_i \\in T_k, s \\in R(t_i)$ – from rule $f\\ref {f:r:ptarc},f\\ref {f:rptarc}$ construction $tot\\_decr(q,qqq,k)$ represents $\\sum _{t_i \\in T_k, s \\in \\bullet t_i}{W(s,t_i)} + \\\\ \\sum _{t_i \\in T_k, s \\in R(t_i)}{M_k(s)}$ – from (C,D,E) above Then firing $T_k \\cup \\lbrace t_j \\rbrace $ would have required more tokens than are present at its source place $s \\in \\bullet t_j \\cup R(t_j)$ .", "Thus, $T_k$ is a maximal set of transitions that can simultaneously fire.", "And for each reset transition $t_r$ with $enabled(t_r,k) \\in A$ , $fires(t_r,k) \\in A$ , since $A$ is an answer set of $\\Pi ^2$ - from rule $f\\ref {f:c:rptarc:elim}$ and supported rule proposition Then the firing set $T_k$ satisfies the reset transition requirement of definition REF (firing set) Then $\\lbrace t_0, \\dots , t_x\\rbrace = T_k$ – using 1(a),1(b), 1(d) above; and using 1(c) it is a maximal firing set We show that $M_{k+1}$ is produced by firing $T_k$ in $M_k$ .", "Let $holds(p,q,k+1) \\in A$ Then $\\lbrace holds(p,q1,k), tot\\_incr(p,q2,k), tot\\_decr(p,q3,k) \\rbrace \\subseteq A : q=q1+q2-q3$ – from rule $r\\ref {r:nextstate}$ and supported rule proposition Then $holds(p,q1,k) \\in A$ represents $q1=M_k(p)$ – construction, inductive assumption; and $\\lbrace add(p,q2_0,t_0,k), \\dots , add(p,q2_j,t_j,k)\\rbrace \\subseteq A : q2_0 + \\dots + q2_j = q2$ and $\\lbrace del(p,q3_0,t_0,k), \\dots , del(p,q3_l,t_l,k)\\rbrace \\subseteq A : q3_0 + \\dots + q3_l = q3$ – rules $r\\ref {r:totincr},r\\ref {r:totdecr}$ and supported rule proposition, respectively Then $\\lbrace fires(t_0,k), \\dots , fires(t_j,k) \\rbrace \\subseteq A$ and $\\lbrace fires(t_0,k), \\dots , fires(t_l,k) \\rbrace \\\\ \\subseteq A$ – rules $r\\ref {r:r:add},r\\ref {r:r:del}$ and supported rule proposition, respectively Then $\\lbrace fires(t_0,k), \\dots , $ $fires(t_j,k) \\rbrace \\cup \\lbrace fires(t_0,k), \\dots , $ $fires(t_l,k) \\rbrace \\subseteq A = \\lbrace fires(t_0,k), \\dots , $ $fires(t_x,k) \\rbrace \\subseteq A$ – set union of subsets Then for each $fires(t_x,k) \\in A$ we have $t_x \\in T_k$ – already shown in item REF above Then $q = M_k(p) + \\sum _{t_x \\in T_0 \\wedge p \\in t_x \\bullet }{W(t_x,p)} - (\\sum _{t_x \\in T_k \\wedge p \\in \\bullet t_x}{W(p,t_x)} + \\\\ \\sum _{t_x \\in T_k \\wedge p \\in R(t_x)}{M_k(p)})$ – from (REF ) above and the following Each $add(p,q_j,t_j,k) \\in A$ represents $q_j=W(t_j,p)$ for $p \\in t_j \\bullet $ – rule $r\\ref {r:r:add} $ encoding, and definition REF of transition execution in $PN$ Each $del(p,t_y,q_y,k) \\in A$ represents either $q_y=W(p,t_y)$ for $p \\in \\bullet t_y$ , or $q_y=M_k(p)$ for $p \\in R(t_y)$ – from rule $r\\ref {r:r:del} ,f\\ref {f:r:ptarc} $ encoding and definition REF of transition execution in $PN$ ; or from rule $r\\ref {r:r:del} ,f\\ref {f:rptarc}$ encoding and definition of reset arc in $PN$ Each $tot\\_incr(p,q2,k) \\in A$ represents $q2=\\sum _{t_x \\in T_k \\wedge p \\in t_x \\bullet }{W(t_x,p)}$ – aggregate assignment atom semantics in rule $r\\ref {r:totincr}$ Each $tot\\_decr(p,q3,0) \\in A$ represents $q3=\\sum _{t_x \\in T_k \\wedge p \\in \\bullet t_x}{W(p,t_x)} + \\sum _{t_x \\in T_k \\wedge p \\in R(t_x)}{M_0(p)}$ – aggregate assignment atom semantics in rule $r\\ref {r:totdecr}$ Then, $M_{k+1}(p) = q$ – since $holds(p,q,k+1) \\in A$ encodes $q=M_{k+1}(p)$ – from construction As a result, for any $n > k$ , $T_n$ will be a valid firing set for $M_n$ and $M_{n+1}$ will be its target marking.", "Conclusion: Since both (REF ) and (REF ) hold, $X=M_0,T_0,M_1,\\dots ,M_k,T_{k+1}$ is an execution sequence of $PN(P,T,E,W,R)$ (w.r.t $M_0$ ) iff there is an answer set $A$ of $\\Pi ^4(PN,M_0,k,ntok)$ such that (REF ) and (REF ) hold.", "Proof of Proposition REF Let $PN=(P,T,E,C,W,R,I,Q,QW)$ be a Petri Net, $M_0$ be its initial marking and let $\\Pi ^5(PN,M_0,k,ntok)$ be the ASP encoding of $PN$ and $M_0$ over a simulation length $k$ , with maximum $ntok$ tokens on any place node, as defined in section REF .", "Then $X=M_0,T_k,M_1,\\dots ,M_k,T_k,M_{k+1}$ is an execution sequence of $PN$ (w.r.t.", "$M_0$ ) iff there is an answer set $A$ of $\\Pi ^5(PN,M_0,k,ntok)$ such that: $\\lbrace fires(t,ts) : t \\in T_{ts}, 0 \\le ts \\le k\\rbrace = \\lbrace fires(t,ts) : fires(t,ts) \\in A \\rbrace $ $\\begin{split}\\lbrace holds(p,q,c,ts) &: p \\in P, c/q = M_{ts}(p), 0 \\le ts \\le k+1 \\rbrace \\\\&= \\lbrace holds(p,q,c,ts) : holds(p,q,c,ts) \\in A \\rbrace \\end{split}$ We prove this by showing that: Given an execution sequence $X$ , we create a set $A$ such that it satisfies (REF ) and (REF ) and show that $A$ is an answer set of $\\Pi ^5$ Given an answer set $A$ of $\\Pi ^5$ , we create an execution sequence $X$ such that (REF ) and (REF ) are satisfied.", "First we show (REF ): Given $PN$ and an execution sequence $X$ of $PN$ , we create a set $A$ as a union of the following sets: $A_1=\\lbrace num(n) : 0 \\le n \\le ntok \\rbrace $ $A_2=\\lbrace time(ts) : 0 \\le ts \\le k\\rbrace $ $A_3=\\lbrace place(p) : p \\in P \\rbrace $ $A_4=\\lbrace trans(t) : t \\in T \\rbrace $ $A_5=\\lbrace color(c) : c \\in C \\rbrace $ $A_6=\\lbrace ptarc(p,t,n_c,c,ts) : (p,t) \\in E^-, c \\in C, n_c=m_{W(p,t)}(c), n_c > 0, 0 \\le ts \\le k \\rbrace $ , where $E^- \\subseteq E$ $A_7=\\lbrace tparc(t,p,n_c,c,ts) : (t,p) \\in E^+, c \\in C, n_c=m_{W(t,p)}(c), n_c > 0, 0 \\le ts \\le k \\rbrace $ , where $E^+ \\subseteq E$ $A_8=\\lbrace holds(p,q_c,c,0) : p \\in P, c \\in C, q_c=m_{M_{0}(p)}(c) \\rbrace $ $A_9=\\lbrace ptarc(p,t,n_c,c,ts) : p \\in R(t), c \\in C, n_c = m_{M_{ts}(p)}, n_c > 0, 0 \\le ts \\le k \\rbrace $ $A_{10}=\\lbrace iptarc(p,t,1,c,ts) : p \\in I(t), c \\in C, 0 \\le ts < k \\rbrace $ $A_{11}=\\lbrace tptarc(p,t,n_c,c,ts) : (p,t) \\in Q, c \\in C, n_c=m_{QW(p,t)}(c), n_c > 0, 0 \\le ts \\le k \\rbrace $ $A_{12}=\\lbrace notenabled(t,ts) : t \\in T, 0 \\le ts \\le k, \\exists c \\in C, (\\exists p \\in \\bullet t, m_{M_{ts}(p)}(c) < m_{W(p,t)}(c)) \\vee (\\exists p \\in I(t), m_{M_{ts}(p)}(c) > 0) \\vee (\\exists (p,t) \\in Q, m_{M_{ts}(p)}(c) < m_{QW(p,t)}(c)) \\rbrace $ per definition REF (enabled transition) $A_{13}=\\lbrace enabled(t,ts) : t \\in T, 0 \\le ts \\le k, \\forall c \\in C, (\\forall p \\in \\bullet t, m_{W(p,t)}(c) \\le m_{M_{ts}(p)}(c)) \\wedge (\\forall p \\in I(t), m_{M_{ts}(p)}(c) = 0) \\wedge (\\forall (p,t) \\in Q, m_{M_{ts}(p)}(c) \\ge m_{QW(p,t)}(c)) \\rbrace $ per definition REF (enabled transition) $A_{14}=\\lbrace fires(t,ts) : t \\in T_{ts}, 0 \\le ts \\le k \\rbrace $ per definition REF (firing set), only an enabled transition may fire $A_{15}=\\lbrace add(p,q_c,t,c,ts) : t \\in T_{ts}, p \\in t \\bullet , c \\in C, q_c=m_{W(t,p)}(c), 0 \\le ts \\le k \\rbrace $ per definition REF (transition execution) $A_{16}=\\lbrace del(p,q_c,t,c,ts) : t \\in T_{ts}, p \\in \\bullet t, c \\in C, q_c=m_{W(p,t)}(c), 0 \\le ts \\le k \\rbrace \\cup \\lbrace del(p,q_c,t,c,ts) : t \\in T_{ts}, p \\in R(t), c \\in C, q_c=m_{M_{ts}(p)}(c), 0 \\le ts \\le k \\rbrace $ per definition REF (transition execution) $A_{17}=\\lbrace tot\\_incr(p,q_c,c,ts) : p \\in P, c \\in C, q_c=\\sum _{t \\in T_{ts}, p \\in t \\bullet }{m_{W(t,p)}(c)}, 0 \\le ts \\le k \\rbrace $ per definition REF (firing set execution) $A_{18}=\\lbrace tot\\_decr(p,q_c,c,ts): p \\in P, c \\in C, q_c=\\sum _{t \\in T_{ts}, p \\in \\bullet t}{m_{W(p,t)}(c)}+ \\\\ \\sum _{t \\in T_{ts}, p \\in R(t) }{m_{M_{ts}(p)}(c)}, 0 \\le ts \\le k \\rbrace $ per definition REF (firing set execution) $A_{19}=\\lbrace consumesmore(p,ts) : p \\in P, c \\in C, q_c=m_{M_{ts}(p)}(c), \\\\ q1_c=\\sum _{t \\in T_{ts}, p \\in \\bullet t}{m_{W(p,t)}(c)} + \\sum _{t \\in T_{ts}, p \\in R(t)}{m_{M_{ts}(p)}(c)}, q1 > q, 0 \\le ts \\le k \\rbrace $ per definition REF (conflicting transitions) $A_{20}=\\lbrace consumesmore : \\exists p \\in P, c \\in C, q_c=m_{M_{ts}(p)}(c), \\\\q1_c=\\sum _{t \\in T_{ts}, p \\in \\bullet t}{m_{W(p,t)}(c)} + \\sum _{t \\in T_{ts}, p \\in R(t)}(m_{M_{ts}(p)}(c)), q1 > q, 0 \\le ts \\le k \\rbrace $ per definition REF (conflicting transitions) per the maximal firing set semantics $A_{21}=\\lbrace holds(p,q_c,c,ts+1) : p \\in P, c \\in C, q_c=m_{M_{ts+1}(p)}(c), 0 \\le ts < k\\rbrace $ , where $M_{ts+1}(p) = M_{ts}(p) - (\\sum _{\\begin{array}{c}t \\in T_{ts}, p \\in \\bullet t\\end{array}}{W(p,t)} + \\\\ \\sum _{t \\in T_{ts}, p \\in R(t)}M_{ts}(p)) + \\sum _{\\begin{array}{c}t \\in T_{ts}, p \\in t \\bullet \\end{array}}{W(t,p)}$ according to definition REF (firing set execution) We show that $A$ satisfies (REF ) and (REF ), and $A$ is an answer set of $\\Pi ^5$ .", "$A$ satisfies (REF ) and (REF ) by its construction above.", "We show $A$ is an answer set of $\\Pi ^5$ by splitting.", "We split $lit(\\Pi ^5)$ into a sequence of $7k+9$ sets: $U_0= head(f\\ref {f:c:place}) \\cup head(f\\ref {f:c:trans}) \\cup head(f\\ref {f:c:col}) \\cup head(f\\ref {f:c:time}) \\cup head(f\\ref {f:c:num}) \\cup head(i\\ref {i:c:init}) = \\lbrace place(p) : p \\in P\\rbrace \\cup \\lbrace trans(t) : t \\in T\\rbrace \\cup \\lbrace col(c) : c \\in C \\rbrace \\cup \\lbrace time(0), \\dots , time(k)\\rbrace \\cup \\lbrace num(0), \\dots , num(ntok)\\rbrace \\cup \\lbrace holds(p,q_c,c,0) : p \\in P, c \\in C, q_c=m_{M_0(p)}(c) \\rbrace $ $U_{7k+1}=U_{7k+0} \\cup head(f\\ref {f:c:ptarc})^{ts=k} \\cup head(f\\ref {f:c:tparc})^{ts=k} \\cup head(f\\ref {f:c:rptarc})^{ts=k} \\cup head(f\\ref {f:c:iptarc})^{ts=k} \\cup head(f\\ref {f:c:tptarc})^{ts=k} = U_{7k+0} \\cup \\lbrace ptarc(p,t,n_c,c,k) : (p,t) \\in E^-, c \\in C, n_c=m_{W(p,t)}(c) \\rbrace \\cup \\lbrace tparc(t,p,n_c,c,k) : (t,p) \\in E^+, c \\in C, n_c=m_{W(t,p)}(c) \\rbrace \\cup \\lbrace ptarc(p,t,n_c,c,k) : p \\in R(t), c \\in C, n_c=m_{M_{k}(p)}(c), n > 0 \\rbrace \\cup \\lbrace iptarc(p,t,1,c,k) : p \\in I(t), c \\in C \\rbrace \\cup \\lbrace tptarc(p,t,n_c,c,k) : (p,t) \\in Q, c \\in C, n_c=m_{QW(p,t)}(c) \\rbrace $ $U_{7k+2}=U_{7k+1} \\cup head(e\\ref {e:c:ne:ptarc})^{ts=k} \\cup head(e\\ref {e:c:ne:iptarc})^{ts=k} \\cup head(e\\ref {e:c:ne:tptarc})^{ts=k} = U_{7k+1} \\cup \\\\ \\lbrace notenabled(t,k) : t \\in T \\rbrace $ $U_{7k+3}=U_{7k+2} \\cup head(e\\ref {e:c:enabled})^{ts=k} = U_{7k+2} \\cup \\lbrace enabled(t,k) : t \\in T \\rbrace $ $U_{7k+4}=U_{7k+3} \\cup head(a\\ref {a:c:fires})^{ts=k} = U_{7k+3} \\cup \\lbrace fires(t,k) : t \\in T \\rbrace $ $U_{7k+5}=U_{7k+4} \\cup head(r\\ref {r:c:add})^{ts=k} \\cup head(r\\ref {r:c:del})^{ts=k} = U_{7k+4} \\cup \\lbrace add(p,q_c,t,c,k) : p \\in P, t \\in T, c \\in C, q_c=m_{W(t,p)}(c) \\rbrace \\cup \\lbrace del(p,q_c,t,c,k) : p \\in P, t \\in T, c \\in C, q_c=m_{W(p,t)}(c) \\rbrace \\cup \\lbrace del(p,q_c,t,c,k) : p \\in P, t \\in T, c \\in C, q_c=m_{M_{k}(p)}(c) \\rbrace $ $U_{7k+6}=U_{7k+5} \\cup head(r\\ref {r:c:totincr})^{ts=k} \\cup head(r\\ref {r:c:totdecr})^{ts=k} = U_{7k+5} \\cup \\lbrace tot\\_incr(p,q_c,c,k) : p \\in P, c \\in C, 0 \\le q_c \\le ntok \\rbrace \\cup \\lbrace tot\\_decr(p,q_c,c,k) : p \\in P, c \\in C, 0 \\le q_c \\le ntok \\rbrace $ $U_{7k+7}=U_{7k+6} \\cup head(a\\ref {a:c:overc:place})^{ts=k} \\cup head(r\\ref {r:c:nextstate})^{ts=k} = U_{7k+6} \\cup \\lbrace consumesmore(p,k) : p \\in P\\rbrace \\cup \\lbrace holds(p,q,k+1) : p \\in P, 0 \\le q \\le ntok \\rbrace $ $U_{7k+8}=U_{7k+7} \\cup head(a\\ref {a:c:overc:gen}) = U_{7k+7} \\cup \\lbrace consumesmore \\rbrace $ where $head(r_i)^{ts=k}$ are head atoms of ground rule $r_i$ in which $ts=k$ .", "We write $A_i^{ts=k} = \\lbrace a(\\dots ,ts) : a(\\dots ,ts) \\in A_i, ts=k \\rbrace $ as short hand for all atoms in $A_i$ with $ts=k$ .", "$U_{\\alpha }, 0 \\le \\alpha \\le 7k+8$ form a splitting sequence, since each $U_i$ is a splitting set of $\\Pi ^5$ , and $\\langle U_{\\alpha }\\rangle _{\\alpha < \\mu }$ is a monotone continuous sequence, where $U_0 \\subseteq U_1 \\dots \\subseteq U_{7k+8}$ and $\\bigcup _{\\alpha < \\mu }{U_{\\alpha }} = lit(\\Pi ^5)$ .", "We compute the answer set of $\\Pi ^5$ using the splitting sets as follows: $bot_{U_0}(\\Pi ^5) = f\\ref {f:place} \\cup f\\ref {f:trans} \\cup f\\ref {f:c:ptarc} \\cup f9 \\cup f\\ref {f:c:tptarc} \\cup i\\ref {i:c:init}$ and $X_0 = A_1 \\cup \\dots \\cup A_5 \\cup A_8$ ($= U_0$ ) is its answer set – using forced atom proposition $eval_{U_0}(bot_{U_1}(\\Pi ^5) \\setminus bot_{U_0}(\\Pi ^5), X_0) = \\lbrace ptarc(p,t,q_c,c,0) \\text{:-}.", "\| c \\in C, q_c=m_{W(p,t)}(c), q_c > 0 \\rbrace \\cup \\lbrace tparc(t,p,q_c,c,0) \\text{:-}.", "\| c \\in C, q_c=m_{W(t,p)}(c), q_c > 0 \\rbrace \\cup \\lbrace ptarc(p,t,q_c,c,0) \\text{:-}.", "\| c \\in C, q_c=m_{M_0(p)}(c), q_c > 0 \\rbrace \\cup \\lbrace iptarc(p,t,1,c,0) \\text{:-}.", "\| c \\in C \\rbrace \\cup \\lbrace tptarc(p,t,q_c,c,0) \\text{:-}.", "\| c \\in C, q_c = m_{QW(p,t)}(c), q_c > 0 \\rbrace $ .", "Its answer set $X_1=A_6^{ts=0} \\cup A_7^{ts=0} \\cup A_9^{ts=0} \\cup A_{10}^{ts=0} \\cup A_{11}^{ts=0}$ – using forced atom proposition and construction of $A_6, A_7, A_9, A_{10}, A_{11}$ .", "$eval_{U_1}(bot_{U_2}(\\Pi ^5) \\setminus bot_{U_1}(\\Pi ^5), X_0 \\cup X_1) = \\lbrace notenabled(t,0) \\text{:-} .", "\| (\\lbrace trans(t), \\\\ ptarc(p,t,n_c,c,0), holds(p,q_c,c,0) \\rbrace \\subseteq X_0 \\cup X_1, \\text{~where~} q_c < n_c) \\text{~or~} \\\\ (\\lbrace notenabled(t,0) \\text{:-} .", "\| (\\lbrace trans(t), iptarc(p,t,n2_c,c,0), holds(p,q_c,c,0) \\rbrace \\subseteq X_0 \\cup X_1, \\\\ \\text{~where~} q_c \\ge n2_c \\rbrace ) \\text{~or~} $ $(\\lbrace trans(t), tptarc(p,t,n3_c,c,0), holds(p,q_c,c,0) \\rbrace \\subseteq X_0 \\cup X_1, \\text{~where~} q_c < n3_c) \\rbrace $ .", "Its answer set $X_2=A_{12}^{ts=0}$ – using forced atom proposition and construction of $A_{12}$ .", "where, $q_c=m_{M_0(p)}(c)$ , and $n_c=m_{W(p,t)}(c)$ for an arc $(p,t) \\in E^-$ – by construction of $i\\ref {i:c:init}$ and $f\\ref {f:c:ptarc}$ in $\\Pi ^5$ , and in an arc $(p,t) \\in E^-$ , $p \\in \\bullet t$ (by definition REF of preset) $n2_c=1$ – by construction of $iptarc$ predicates in $\\Pi ^5$ , meaning $q_c \\ge n2_c \\equiv q_c \\ge 1 \\equiv q_c > 0$ , $tptarc(p,t,n3_c,c,0)$ represents $n3_c=m_{QW(p,t)}(c)$ , where $(p,t) \\in Q$ thus, $notenabled(t,0) \\in X_1$ represents $\\exists c \\in C, (\\exists p \\in \\bullet t : m_{M_0(p)}(c) < m_{W(p,t)}(c)) \\vee (\\exists p \\in I(t) : m_{M_0(p)}(c) > 0) \\vee (\\exists (p,t) \\in Q : m_{M_0(p)}(c) < m_{QW(p,t)}(c))$ .", "$eval_{U_2}(bot_{U_3}(\\Pi ^5) \\setminus bot_{U_2}(\\Pi ^5), X_0 \\cup \\dots \\cup X_2) = \\lbrace enabled(t,0) \\text{:-}.", "\| trans(t) \\in X_0 \\cup \\dots \\cup X_2, notenabled(t,0) \\notin X_0 \\cup \\dots \\cup X_2 \\rbrace $ .", "Its answer set is $X_3 = A_{13}^{ts=0}$ – using forced atom proposition and construction of $A_{13}$ .", "where, an $enabled(t,0) \\in X_3$ if $\\nexists ~notenabled(t,0) \\in X_0 \\cup \\dots \\cup X_2$ ; which is equivalent to $\\forall c \\in C, (\\nexists p \\in \\bullet t : m_{M_0(p)}(c) < m_{W(p,t)}(c)) \\wedge (\\nexists p \\in I(t) : m_{M_0(p)}(c) > 0) \\wedge (\\nexists (p,t) \\in Q : m_{M_0(p)}(c) < m_{QW(p,t)}(c) ) \\equiv \\forall c \\in C, (\\forall p \\in \\bullet t: m_{M_0(p)}(c) \\ge m_{W(p,t)}(c)) \\wedge (\\forall p \\in I(t) : m_{M_0(p)}(c) = 0)$ .", "$eval_{U_3}(bot_{U_4}(\\Pi ^5) \\setminus bot_{U_3}(\\Pi ^5), X_0 \\cup \\dots \\cup X_3) = \\lbrace \\lbrace fires(t,0)\\rbrace \\text{:-}.", "\| enabled(t,0) \\\\ \\text{~holds in~} X_0 \\cup \\dots \\cup X_3 \\rbrace $ .", "It has multiple answer sets $X_{4.1}, \\dots , X_{4.n}$ , corresponding to elements of power set of $fires(t,0)$ atoms in $eval_{U_3}(...)$ – using supported rule proposition.", "Since we are showing that the union of answer sets of $\\Pi ^5$ determined using splitting is equal to $A$ , we only consider the set that matches the $fires(t,0)$ elements in $A$ and call it $X_4$ , ignoring the rest.", "Thus, $X_4 = A_{14}^{ts=0}$ , representing $T_0$ .", "in addition, for every $t$ such that $enabled(t,0) \\in X_0 \\cup \\dots \\cup X_3, R(t) \\ne \\emptyset $ ; $fires(t,0) \\in X_4$ – per definition REF (firing set); requiring that a reset transition is fired when enabled thus, the firing set $T_0$ will not be eliminated by the constraint $f\\ref {f:c:rptarc:elim}$ $eval_{U_4}(bot_{U_5}(\\Pi ^5) \\setminus bot_{U_4}(\\Pi ^5), X_0 \\cup \\dots \\cup X_4) = \\lbrace add(p,n_c,t,c,0) \\text{:-}.", "\| \\lbrace fires(t,0), \\\\ tparc(t,p,n_c,c,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_4 \\rbrace \\cup \\lbrace del(p,n_c,t,c,0) \\text{:-}.", "\| \\lbrace fires(t,0), \\\\ ptarc(p,t,n_c,c,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_4 \\rbrace $ .", "It's answer set is $X_5 = A_{15}^{ts=0} \\cup A_{16}^{ts=0}$ – using forced atom proposition and definitions of $A_{15}$ and $A_{16}$ .", "where, each $add$ atom is equivalent to $n_c=m_{W(t,p)}(c),c \\in C, p \\in t \\bullet $ , and each $del$ atom is equivalent to $n_c=m_{W(p,t)}(c), c \\in C, p \\in \\bullet t$ ; or $n_c=m_{M_0(p)}(c), c \\in C, p \\in R(t)$ , representing the effect of transitions in $T_0$ $eval_{U_5}(bot_{U_6}(\\Pi ^5) \\setminus bot_{U_5}(\\Pi ^5), X_0 \\cup \\dots \\cup X_5) = \\lbrace tot\\_incr(p,qq_c,c,0) \\text{:-}.", "\| qq_c=\\sum _{add(p,q_c,t,c,0) \\in X_0 \\cup \\dots \\cup X_5}{q_c} \\rbrace \\cup \\lbrace tot\\_decr(p,qq_c,c,0) \\text{:-}.", "\| qq_c= \\\\ \\sum _{del(p,q_c,t,c,0) \\in X_0 \\cup \\dots \\cup X_5}{q_c} \\rbrace $ .", "It's answer set is $X_6 = A_{17}^{ts=0} \\cup A_{18}^{ts=0}$ – using forced atom proposition and definitions of $A_{17}$ and $A_{18}$ .", "where, each $tot\\_incr(p,qq_c,c,0)$ , $qq_c=\\sum _{add(p,q_c,t,c,0) \\in X_0 \\cup \\dots X_5}{q_c}$ $\\equiv qq_c=\\sum _{t \\in X_4, p \\in t \\bullet }{m_{W(p,t)}(c)}$ , and each $tot\\_decr(p,qq_c,c,0)$ , $qq_c=\\sum _{del(p,q_c,t,c,0) \\in X_0 \\cup \\dots X_5}{q_c}$ $\\equiv qq=\\sum _{t \\in X_4, p \\in \\bullet t}{m_{W(t,p)}(c)} + \\sum _{t \\in X_4, p \\in R(t)}{m_{M_0(p)}(c)}$ , represent the net effect of transitions in $T_0$ $eval_{U_6}(bot_{U_7}(\\Pi ^5) \\setminus bot_{U_6}(\\Pi ^5), X_0 \\cup \\dots \\cup X_6) = \\lbrace consumesmore(p,0) \\text{:-}.", "\| \\\\ \\lbrace holds(p,q_c,c,0), tot\\_decr(p,q1_c,c,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_6, q1_c > q_c \\rbrace \\cup \\\\ \\lbrace holds(p,q_c,c,1) \\text{:-}., \| \\lbrace holds(p,q1_c,c,0), $ $tot\\_incr(p,q2_c,c,0), $ $tot\\_decr(p,q3_c,c,0) \\rbrace \\\\ \\subseteq X_0 \\cup \\dots \\cup X_6, q_c=q1_c+q2_c-q3_c \\rbrace $ .", "It's answer set is $X_7 = A_{19}^{ts=0} \\cup A_{21}^{ts=0}$ – using forced atom proposition and definitions of $A_{19}, A_{21}$ .", "where, $consumesmore(p,0)$ represents $\\exists p : q_c=m_{M_0(p)}(c), q1_c= \\\\ \\sum _{t \\in T_0, p \\in \\bullet t}{m_{W(p,t)}(c)}+\\sum _{t \\in T_0, p \\in R(t)}{m_{M_0(p)}(c)}, q1_c > q_c, c \\in C$ , indicating place $p$ will be over consumed if $T_0$ is fired, as defined in definition REF (conflicting transitions), $holds(p,q_c,c,1)$ represents $q_c=m_{M_1(p)}(c)$ – by construction of $\\Pi ^5$ $ \\vdots $ $eval_{U_{7k+0}}(bot_{U_{7k+1}}(\\Pi ^5) \\setminus bot_{U_{7k+0}}(\\Pi ^5), X_0 \\cup \\dots \\cup X_{7k+0}) = \\lbrace ptarc(p,t,q_c,c,k) \\text{:-}.", "\| c \\in C, q_c=m{W(p,t)}(c), q_c > 0 \\rbrace \\cup \\lbrace tparc(t,p,q_c,c,k) \\text{:-}.", "\| c \\in C, q_c=m_{W(t,p)}(c), q_c > 0 \\rbrace \\cup $ $\\lbrace ptarc(p,t,q_c,c,k) \\text{:-}.", "\| $ $c \\in C, q_c=m_{M_k(p)}(c), q_c > 0 \\rbrace \\cup \\lbrace iptarc(p,t,1,c,k) \\text{:-}.", "\| c \\in C \\rbrace \\cup \\lbrace tptarc(p,t,q_c,c,k) \\text{:-}.", "\| c \\in C, q_c = m_{QW(p,t)}(c), q_c > 0 \\rbrace $ .", "Its answer set $X_{7k+1}=A_6^{ts=k} \\cup A_7^{ts=k} \\cup A_9^{ts=k} \\cup A_{10}^{ts=k} \\cup A_{11}^{ts=k}$ – using forced atom proposition and construction of $A_6, A_7, A_9, A_{10}, A_{11}$ .", "$eval_{U_{7k+1}}(bot_{U_{7k+2}}(\\Pi ^5) \\setminus bot_{U_{7k+1}}(\\Pi ^5), X_0 \\cup \\dots \\cup X_{7k+1}) = \\lbrace notenabled(t,k) \\text{:-} .", "\| \\\\ (\\lbrace trans(t), ptarc(p,t,n_c,c,k), holds(p,q_c,c,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{7k+1}, \\text{~where~} q_c < n_c) \\text{~or~} $ $ (\\lbrace notenabled(t,k) \\text{:-} .", "\| (\\lbrace trans(t), iptarc(p,t,n2_c,c,k), holds(p,q_c,c,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{7k+1}, \\text{~where~} q_c \\ge n2_c \\rbrace ) \\text{~or~} (\\lbrace trans(t), tptarc(p,t,n3_c,c,k), \\\\ holds(p,q_c,c,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{7k+1}, \\text{~where~} q_c < n3_c) \\rbrace $ .", "Its answer set $X_{7k+2}=A_{12}^{ts=k}$ – using forced atom proposition and construction of $A_{12}$ .", "since $q_c=m_{M_k(p)}(c)$ , and $n_c=m_{W(p,t)}(c)$ for an arc $(p,t) \\in E^-$ – by construction of $holds$ and $ptarc$ predicates in $\\Pi ^5$ , and in an arc $(p,t) \\in E^-$ , $p \\in \\bullet t$ (by definition REF of preset) $n2_c=1$ – by construction of $iptarc$ predicates in $\\Pi ^5$ , meaning $q_c \\ge n2_c \\equiv q_c \\ge 1 \\equiv q_c > 0$ , $tptarc(p,t,n3_c,c,k)$ represents $n3_c=m_{QW(p,t)}(c)$ , where $(p,t) \\in Q$ thus, $notenabled(t,k) \\in X_{8k+1}$ represents $\\exists c \\in C (\\exists p \\in \\bullet t : m_{M_k(p)}(c) < m_{W(p,t)}(c)) \\vee (\\exists p \\in I(t) : m_{M_k(p)}(c) > 0) \\vee (\\exists (p,t) \\in Q : m_{M_{ts}(p)}(c) < m_{QW(p,t)}(c))$ .", "$eval_{U_{7k+2}}(bot_{U_{7k+3}}(\\Pi ^5) \\setminus bot_{U_{7k+2}}(\\Pi ^5), X_0 \\cup \\dots \\cup X_{7k+2}) = \\lbrace enabled(t,k) \\text{:-}.", "\| trans(t) \\in X_0 \\cup \\dots \\cup X_{7k+2} \\wedge notenabled(t,k) \\notin X_0 \\cup \\dots \\cup X_{7k+2} \\rbrace $ .", "Its answer set is $X_{7k+3} = A_{13}^{ts=k}$ – using forced atom proposition and construction of $A_{13}$ .", "since an $enabled(t,k) \\in X_{7k+3}$ if $\\nexists ~notenabled(t,k) \\in X_0 \\cup \\dots \\cup X_{7k+2}$ ; which is equivalent to $\\forall c \\in C, (\\nexists p \\in \\bullet t : m_{M_k(p)}(c) < m_{W(p,t)}(c)) \\wedge (\\nexists p \\in I(t) : m_{M_k(p)}(c) > 0) \\wedge (\\nexists (p,t) \\in Q : m_{M_k(p)}(c) < m_{QW(p,t)}(c) ) \\equiv \\forall c \\in C, (\\forall p \\in \\bullet t: m_{M_k(p)}(c) \\ge m_{W(p,t)}(c)) \\wedge (\\forall p \\in I(t) : m_{M_k(p)}(c) = 0)$ .", "$eval_{U_{7k+3}}(bot_{U_{7k+4}}(\\Pi ^5) \\setminus bot_{U_{7k+3}}(\\Pi ^5), X_0 \\cup \\dots \\cup X_{7k+3}) = \\lbrace \\lbrace fires(t,k)\\rbrace \\text{:-}.", "\| \\\\ enabled(t,k) \\text{~holds in~} X_0 \\cup \\dots \\cup X_{7k+3} \\rbrace $ .", "It has multiple answer sets $X_{7k+}{4.1}, \\dots , X_{7k+}{4.n}$ , corresponding to elements of power set of $fires(t,k)$ atoms in $eval_{U_3}(...)$ – using supported rule proposition.", "Since we are showing that the union of answer sets of $\\Pi ^5$ determined using splitting is equal to $A$ , we only consider the set that matches the $fires(t,k)$ elements in $A$ and call it $X_{7k+4}$ , ignoring the rest.", "Thus, $X_{7k+4} = A_{14}^{ts=k}$ , representing $T_k$ .", "in addition, for every $t$ such that $enabled(t,k) \\in X_0 \\cup \\dots \\cup X_{7k+3}, R(t) \\ne \\emptyset $ ; $fires(t,k) \\in X_{7k+4}$ – per definition REF (firing set); requiring that a reset transition is fired when enabled thus, the firing set $T_k$ will not be eliminated by the constraint $f\\ref {f:c:rptarc:elim}$ $eval_{U_{7k+4}}(bot_{U_{7k+5}}(\\Pi ^5) \\setminus bot_{U_{7k+4}}(\\Pi ^5), X_0 \\cup \\dots \\cup X_{7k+4}) = \\lbrace add(p,n_c,t,c,k) \\text{:-}.", "\| $ $ \\lbrace fires(t,k), \\\\ tparc(t,p,n_c,c,k) \\rbrace \\subseteq $ $X_0 \\cup \\dots \\cup X_{7k+4} \\rbrace \\cup $ $\\lbrace del(p,n_c,t,c,k) \\text{:-}.", "\| $ $\\lbrace fires(t,k), \\\\ ptarc(p,t,n_c,c,k) \\rbrace \\subseteq $ $X_0 \\cup \\dots \\cup X_{7k+4} \\rbrace $ .", "It's answer set is $X_{7k+5} = A_{15}^{ts=k} \\cup A_{16}^{ts=k}$ – using forced atom proposition and definitions of $A_{15}$ and $A_{16}$ .", "where each $add$ atom is equivalent to $n_c=m_{W(t,p)}(c),c \\in C, p \\in t \\bullet $ , and each $del$ atom is equivalent to $n_c=m_{W(p,t)}(c), c \\in C, p \\in \\bullet t$ ; or $n_c=m_{M_k(p)}(c), c \\in C, p \\in R(t)$ , representing the effect of transitions in $T_k$ $eval_{U_{7k+5}}(bot_{U_{7k+6}}(\\Pi ^5) \\setminus bot_{U_{7k+5}}(\\Pi ^5), X_0 \\cup \\dots \\cup X_{7k+5}) = \\lbrace tot\\_incr(p,qq_c,c,k) \\text{:-}.", "\| $ $qq=\\sum _{add(p,q_c,t,c,k) \\in X_0 \\cup \\dots \\cup X_{7k+5}}{q_c} \\rbrace \\cup $ $ \\lbrace tot\\_decr(p,qq_c,c,k) \\text{:-}.", "\| \\\\ qq_c=\\sum _{del(p,q_c,t,c,k) \\in X_0 \\cup \\dots \\cup X_{7k+5}}{q_c} \\rbrace $ .", "It's answer set is $X_{7k+6} = A_{17}^{ts=k} \\cup A_{18}^{ts=k}$ – using forced atom proposition and definitions of $A_{17}$ and $A_{18}$ .", "where, each $tot\\_incr(p,qq_c,c,k)$ , $qq_c=\\sum _{add(p,q_c,t,c,k) \\in X_0 \\cup \\dots X_{8k+5}}{q_c}$ $\\equiv qq_c=\\sum _{t \\in X_{7k+4}, p \\in t \\bullet }{m_{W(p,t)}(c)}$ , and each $tot\\_decr(p,qq_c,c,k)$ , $qq_c=\\sum _{del(p,q_c,t,c,k) \\in X_0 \\cup \\dots X_{8k+5}}{q_c}$ $\\equiv qq=\\sum _{t \\in X_{7k+4}, p \\in \\bullet t}{m_{W(t,p)}(c)} + \\sum _{t \\in X_{7k+4}, p \\in R(t)}{m_{M_{k}(p)}(c)}$ , represent the net effect of transitions in $T_k$ $eval_{U_{7k+6}}(bot_{U_{7k+7}}(\\Pi ^5) \\setminus bot_{U_{7k+6}}(\\Pi ^5), X_0 \\cup \\dots \\cup X_{7k+6}) = \\lbrace consumesmore(p,k) \\text{:-}.", "\| \\\\ \\lbrace holds(p,q_c,c,k), tot\\_decr(p,q1_c,c,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{7k+6}, q1 > q \\rbrace \\cup \\\\ \\lbrace holds(p,q_c,c,k+1) \\text{:-}.", "\| \\lbrace holds(p,q1_c,c,k), tot\\_incr(p,q2_c,c,k), \\\\ tot\\_decr(p,q3_c,c,k) \\rbrace \\subseteq $ $X_0 \\cup \\dots \\cup X_{7k+6}, q_c=q1_c+q2_c-q3_c \\rbrace $ .", "It's answer set is $X_{7k+7} = A_{19}^{ts=k} \\cup A_{21}^{ts=k}$ – using forced atom proposition and definitions of $A_{19}, A_{21}$ .", "where, $consumesmore(p,k)$ represents $\\exists p : q_c=m_{M_k(p)}(c), q1_c= \\\\ \\sum _{t \\in T_k, p \\in \\bullet t}{m_{W(p,t)}(c)}+\\sum _{t \\in T_k, p \\in R(t)}{m_{M_k(p)}(c)}, q1_c > q_c, c \\in C$ , indicating place $p$ that will be over consumed if $T_k$ is fired, as defined in definition REF (conflicting transitions), and $holds(p,q_c,c,k+1)$ represents $q_c=m_{M_{k+1}(p)}(c)$ – by construction of $\\Pi ^5$ $eval_{U_{7k+7}}(bot_{U_{7k+8}}(\\Pi ^5) \\setminus bot_{U_{7k+7}}(\\Pi ^5), X_0 \\cup \\dots \\cup X_{7k+7}) = \\lbrace consumesmore \\text{:-}.", "\| \\\\ \\lbrace consumesmore(p,0),\\dots ,$ $consumesmore(p,k) \\rbrace \\cap (X_0 \\cup \\dots \\cup X_{7k+7}) \\ne \\emptyset \\rbrace $ .", "It's answer set is $X_{7k+8} = A_{20}^{ts=k}$ – using forced atom proposition and the definition of $A_{20}$ $X_{7k+7}$ will be empty since none of $consumesmore(p,0),\\dots , \\\\ consumesmore(p,k)$ hold in $X_0 \\cup \\dots \\cup X_{7k+7}$ due to the construction of $A$ , encoding of $a\\ref {a:c:overc:place}$ and its body atoms.", "As a result, it is not eliminated by the constraint $a\\ref {a:c:overc:elim}$ The set $X = X_0 \\cup \\dots \\cup X_{7k+8}$ is the answer set of $\\Pi ^5$ by the splitting sequence theorem REF .", "Each $X_i, 0 \\le i \\le 7k+8$ matches a distinct portion of $A$ , and $X = A$ , thus $A$ is an answer set of $\\Pi ^5$ .", "Next we show (REF ): Given $\\Pi ^5$ be the encoding of a Petri Net $PN(P,T,E,C,W,R,I,$ $Q,WQ)$ with initial marking $M_0$ , and $A$ be an answer set of $\\Pi ^5$ that satisfies (REF ) and (REF ), then we can construct $X=M_0,T_k,\\dots ,M_k,T_k,M_{k+1}$ from $A$ , such that it is an execution sequence of $PN$ .", "We construct the $X$ as follows: $M_i = (M_i(p_0), \\dots , M_i(p_n))$ , where $\\lbrace holds(p_0,m_{M_i(p_0)}(c),c,i), \\dots \\\\ holds(p_n,m_{M_i(p_n)}(c),c,i) \\rbrace \\subseteq A$ , for $c \\in C, 0 \\le i \\le k+1$ $T_i = \\lbrace t : fires(t,i) \\in A\\rbrace $ , for $0 \\le i \\le k$ and show that $X$ is indeed an execution sequence of $PN$ .", "We show this by induction over $k$ (i.e.", "given $M_k$ , $T_k$ is a valid firing set and its firing produces marking $M_{k+1}$ ).", "Base case: Let $k=0$ , and $M_0$ is a valid marking in $X$ for $PN$ , show [(1)] $T_0$ is a valid firing set for $M_0$ , and firing $T_0$ in $M_0$ results in marking $M_1$ .", "We show $T_0$ is a valid firing set for $M_0$ .", "Let $\\lbrace fires(t_0,0), \\dots , fires(t_x,0) \\rbrace $ be the set of all $fires(\\dots ,0)$ atoms in $A$ , Then for each $fires(t_i,0) \\in A$ $enabled(t_i,0) \\in A$ – from rule $a\\ref {a:c:fires}$ and supported rule proposition Then $notenabled(t_i,0) \\notin A$ – from rule $e\\ref {e:c:enabled}$ and supported rule proposition Then either of $body(e\\ref {e:c:ne:ptarc})$ , $body(e\\ref {e:c:ne:iptarc})$ , or $body(e\\ref {e:c:ne:tptarc})$ must not hold in $A$ – from rules $e\\ref {e:c:ne:ptarc},e\\ref {e:c:ne:iptarc},e\\ref {e:c:ne:tptarc}$ and forced atom proposition Then $q_c \\lnot < {n_i}_c \\equiv q_c \\ge {n_i}_c$ in $e\\ref {e:c:ne:ptarc}$ for all $\\lbrace holds(p,q_c,c,0), \\\\ ptarc(p,t_i,{n_i}_c,c,0)\\rbrace \\subseteq A$ – from $e\\ref {e:c:ne:ptarc}$ , forced atom proposition, and given facts ($holds(p,q_c,c,0) \\in A, ptarc(p,t_i,{n_i}_c,0) \\in A$ ) And $q_c \\lnot \\ge {n_i}_c \\equiv q_c < {n_i}_c$ in $e\\ref {e:c:ne:iptarc}$ for all $\\lbrace holds(p,q_c,c,0), \\\\ iptarc(p,t_i,{n_i}_c,c,0) \\rbrace \\subseteq A, {n_i}_c=1$ ; $q_c > {n_i}_c \\equiv q_c = 0$ – from $e\\ref {e:c:ne:iptarc}$ , forced atom proposition, given facts ($holds(p,q_c,c,0) \\in A, iptarc(p,t_i,1,c,0) \\in A$ ), and $q_c$ is a positive integer And $q_c \\lnot < {n_i}_c \\equiv q_c \\ge {n_i}_c$ in $e\\ref {e:c:ne:tptarc}$ for all $\\lbrace holds(p,q_c,c,0), \\\\ tptarc(p,t_i,{n_i}_c,c,0) \\rbrace \\subseteq A$ – from $e\\ref {e:c:ne:tptarc}$ , forced atom proposition, and given facts Then $\\forall c \\in C, (\\forall p \\in \\bullet t_i, m_{M_0(p)}(c) \\ge m_{W(p,t_i)}(c)) \\wedge (\\forall p \\in I(t_i), m_{M_0(p)}(c) = 0) \\wedge (\\forall (p,t_i) \\in Q, m_{M_0(p)}(c) \\ge m_{QW(p,t_i)}(c))$ – from the following $holds(p,q_c,c,0) \\in A$ represents $q_c=m_{M_0(p)}(c)$ – rule $i\\ref {i:c:init}$ construction $ptarc(p,t_i,{n_i}_c,0) \\in A$ represents ${n_i}_c=m_{W(p,t_i)}(c)$ – rule $f\\ref {f:c:ptarc}$ construction; or it represents ${n_i}_c = m_{M_0(p)}(c)$ – rule $f\\ref {f:c:rptarc}$ construction; the construction of $f\\ref {f:c:rptarc}$ ensures that $notenabled(t,0)$ is never true for a reset arc definition REF of preset $\\bullet t_i$ in $PN$ definition REF of enabled transition in $PN$ Then $t_i$ is enabled and can fire in $PN$ , as a result it can belong to $T_0$ – from definition REF of enabled transition And $consumesmore \\notin A$ , since $A$ is an answer set of $\\Pi ^5$ – from rule $a\\ref {a:c:overc:gen}$ and supported rule proposition Then $\\nexists consumesmore(p,0) \\in A$ – from rule $a\\ref {a:c:overc:place}$ and supported rule proposition Then $\\nexists \\lbrace holds(p,q_c,c,0), tot\\_decr(p,q1_c,c,0) \\rbrace \\subseteq A, q1_c>q_c$ in $body(a\\ref {a:c:overc:place})$ – from $a\\ref {a:c:overc:place}$ and forced atom proposition Then $\\nexists c \\in C \\nexists p \\in P, (\\sum _{t_i \\in \\lbrace t_0,\\dots ,t_x\\rbrace , p \\in \\bullet t_i}{m_{W(p,t_i)}(c)}+ \\\\ \\sum _{t_i \\in \\lbrace t_0,\\dots ,t_x\\rbrace , p \\in R(t_i)}{m_{M_0(p)}(c)}) > m_{M_0(p)}(c)$ – from the following $holds(p,q_c,c,0)$ represents $q_c=m_{M_0(p)}(c)$ – from rule $i\\ref {i:c:init}$ construction, given $tot\\_decr(p,q1_c,c,0) \\in A$ if $\\lbrace del(p,{q1_0}_c,t_0,c,0), \\dots , \\\\ del(p,{q1_x}_c,t_x,c,0) \\rbrace \\subseteq A$ , where $q1_c = {q1_0}_c+\\dots +{q1_x}_c$ – from $r\\ref {r:c:totdecr}$ and forced atom proposition $del(p,{q1_i}_c,t_i,c,0) \\in A$ if $\\lbrace fires(t_i,0), ptarc(p,t_i,{q1_i}_c,c,0) \\rbrace \\subseteq A$ – from $r\\ref {r:c:del}$ and supported rule proposition $del(p,{q1_i}_c,t_i,c,0)$ either represents removal of ${q1_i}_c = m_{W(p,t_i)}(c)$ tokens from $p \\in \\bullet t_i$ ; or it represents removal of ${q1_i}_c = m_{M_0(p)}(c)$ tokens from $p \\in R(t_i)$ – from rules $r\\ref {r:c:del},f\\ref {f:c:ptarc},f\\ref {f:c:rptarc}$ , supported rule proposition, and definition REF of transition execution in $PN$ Then the set of transitions in $T_0$ do not conflict – by the definition REF of conflicting transitions And for each reset transition $t_r$ with $enabled(t_r,0) \\in A$ , $fires(t_r,0) \\in A$ , since $A$ is an answer set of $\\Pi ^5$ - from rule $f\\ref {f:c:rptarc:elim}$ and supported rule proposition Then, the firing set $T_0$ satisfies the reset-transition requirement of definition REF (firing set) Then $\\lbrace t_0, \\dots , t_x\\rbrace = T_0$ – using 1(a),1(b),1(d) above; and using 1(c) it is a maximal firing set We show $M_1$ is produced by firing $T_0$ in $M_0$ .", "Let $holds(p,q_c,c,1) \\in A$ Then $\\lbrace holds(p,q1_c,c,0), tot\\_incr(p,q2_c,c,0), tot\\_decr(p,q3_c,c,0) \\rbrace \\subseteq A : q_c=q1_c+q2_c-q3_c$ – from rule $r\\ref {r:c:nextstate}$ and supported rule proposition Then, $holds(p,q1_c,c,0) \\in A$ represents $q1_c=m_{M_0(p)}(c)$ – given, rule $i\\ref {i:c:init}$ construction; and $\\lbrace add(p,{q2_0}_c,t_0,c,0), \\dots , $ $add(p,{q2_j}_c,t_j,c,0)\\rbrace \\subseteq A : {q2_0}_c + \\dots + {q2_j}_c = q2_c$ and $\\lbrace del(p,{q3_0}_c,t_0,c,0), \\dots , $ $del(p,{q3_l}_c,t_l,c,0)\\rbrace \\subseteq A : {q3_0}_c + \\dots + {q3_l}_c = q3_c$ – rules $r\\ref {r:c:totincr},r\\ref {r:c:totdecr}$ and supported rule proposition, respectively Then $\\lbrace fires(t_0,0), \\dots , fires(t_j,0) \\rbrace \\subseteq A$ and $\\lbrace fires(t_0,0), \\dots , fires(t_l,0) \\rbrace $ $\\subseteq A$ – rules $r\\ref {r:c:add},r\\ref {r:c:del}$ and supported rule proposition, respectively Then $\\lbrace fires(t_0,0), \\dots , $ $fires(t_j,0) \\rbrace \\cup $ $\\lbrace fires(t_0,0), \\dots , $ $fires(t_l,0) \\rbrace \\subseteq $ $A = \\lbrace fires(t_0,0), \\dots , $ $fires(t_x,0) \\rbrace \\subseteq A$ – set union of subsets Then for each $fires(t_x,0) \\in A$ we have $t_x \\in T_0$ – already shown in item REF above Then $q_c = m_{M_0(p)}(c) + \\sum _{t_x \\in T_0 \\wedge p \\in t_x \\bullet }{m_{W(t_x,p)}(c)} - (\\sum _{t_x \\in T_0 \\wedge p \\in \\bullet t_x}{m_{W(p,t_x)}(c)} + \\sum _{t_x \\in T_0 \\wedge p \\in R(t_x)}{m_{M_0(p)}(c)})$ – from (REF ) above and the following Each $add(p,{q_j}_c,t_j,c,0) \\in A$ represents ${q_j}_c=m_{W(t_j,p)}(c)$ for $p \\in t_j \\bullet $ – rule $r\\ref {r:c:add},f\\ref {f:c:tparc}$ encoding, and definition REF of transition execution in $PN$ Each $del(p,t_y,{q_y}_c,c,0) \\in A$ represents either ${q_y}_c=m_{W(p,t_y)}(c)$ for $p \\in \\bullet t_y$ , or ${q_y}_c=m_{M_0(p)}(c)$ for $p \\in R(t_y)$ – from rule $r\\ref {r:c:del},f\\ref {f:c:ptarc}$ encoding and definition REF of transition execution in $PN$ ; or from rule $r\\ref {r:c:del},f\\ref {f:c:rptarc}$ encoding and definition of reset arc in $PN$ Each $tot\\_incr(p,q2_c,c,0) \\in A$ represents $q2_c=\\sum _{t_x \\in T_0 \\wedge p \\in t_x \\bullet }{m_{W(t_x,p)}(c)}$ – aggregate assignment atom semantics in rule $r\\ref {r:c:totincr}$ Each $tot\\_decr(p,q3_c,c,0) \\in A$ represents $q3_c=\\sum _{t_x \\in T_0 \\wedge p \\in \\bullet t_x}{m_{W(p,t_x)}(c)} + \\sum _{t_x \\in T_0 \\wedge p \\in R(t_x)}{m_{M_0(p)}(c)}$ – aggregate assignment atom semantics in rule $r\\ref {r:c:totdecr}$ Then, $m_{M_1(p)}(c) = q_c$ – since $holds(p,q_c,c,1) \\in A$ encodes $q_c=m_{M_1(p)}(c)$ – from construction Inductive Step: Let $k > 0$ , and $M_k$ is a valid marking in $X$ for $PN$ , show [(1)] $T_k$ is a valid firing set for $M_k$ , and firing $T_k$ in $M_k$ produces marking $M_{k+1}$ .", "We show that $T_k$ is a valid firing set in $M_k$ .", "Let $\\lbrace fires(t_0,k), \\dots , fires(t_x,k) \\rbrace $ be the set of all $fires(\\dots ,k)$ atoms in $A$ , Then for each $fires(t_i,k) \\in A$ $enabled(t_i,k) \\in A$ – from rule $a\\ref {a:c:fires}$ and supported rule proposition Then $notenabled(t_i,k) \\notin A$ – from rule $e\\ref {e:c:enabled}$ and supported rule proposition Then either of $body(e\\ref {e:c:ne:ptarc})$ , $body(e\\ref {e:c:ne:iptarc})$ , or $body(e\\ref {e:c:ne:tptarc})$ must not hold in $A$ – from rules $e\\ref {e:c:ne:ptarc},e\\ref {e:c:ne:iptarc},e\\ref {e:c:ne:tptarc}$ and forced atom proposition Then $q_c \\lnot < {n_i}_c \\equiv q_c \\ge {n_i}_c$ in $e\\ref {e:c:ne:ptarc}$ for all $\\lbrace holds(p,q_c,c,k), \\\\ ptarc(p,t_i,{n_i}_c,c,k)\\rbrace \\subseteq A$ – from $e\\ref {e:c:ne:ptarc}$ , forced atom proposition, given facts, and the inductive assumption And $q_c \\lnot \\ge {n_i}_c \\equiv q_c < {n_i}_c$ in $e\\ref {e:c:ne:iptarc}$ for all $\\lbrace holds(p,q_c,c,k), \\\\ iptarc(p,t_i,{n_i}_c,c,k) \\rbrace \\subseteq A, {n_i}_c=1$ ; $q_c > {n_i}_c \\equiv q_c = 0$ – from $e\\ref {e:c:ne:iptarc}$ , forced atom proposition, given facts ($holds(p,q_c,c,k) \\in A, iptarc(p,t_i,1,c,k) \\in A$ ), and $q_c$ is a positive integer And $q_c \\lnot < {n_i}_c \\equiv q_c \\ge {n_i}_c$ in $e\\ref {e:c:ne:tptarc}$ for all $\\lbrace holds(p,q_c,c,k), \\\\ tptarc(p,t_i,{n_i}_c,c,k) \\rbrace \\subseteq A$ – from $e\\ref {e:c:ne:tptarc}$ , forced atom proposition, and inductive assumption Then $(\\forall p \\in \\bullet t_i, m_{M_k(p)}(c) \\ge m_{W(p,t_i)}(c)) \\wedge (\\forall p \\in I(t_i), m_{M_k(p)}(c) = 0) \\wedge (\\forall (p,t_i) \\in Q, m_{M_k(p)}(c) \\ge m_{QW(p,t_i)}(c))$ – from $holds(p,q_c,c,k) \\in A$ represents $q_c=m_{M_k(p)}(c)$ – construction of $\\Pi ^5$ $ptarc(p,t_i,{n_i}_c,k) \\in A$ represents ${n_i}_c=m_{W(p,t_i)}(c)$ – rule $f\\ref {f:c:ptarc}$ construction; or it represents ${n_i}_c = m_{M_k(p)}(c)$ – rule $f\\ref {f:c:rptarc}$ construction; the construction of $f\\ref {f:c:rptarc}$ ensures that $notenabled(t,k)$ is never true for a reset arc definition REF of preset $\\bullet t_i$ in $PN$ definition REF of enabled transition in $PN$ Then $t_i$ is enabled and can fire in $PN$ , as a result it can belong to $T_k$ – from definition REF of enabled transition And $consumesmore \\notin A$ , since $A$ is an answer set of $\\Pi ^5$ – from rule $a\\ref {a:c:overc:gen}$ and supported rule proposition Then $\\nexists consumesmore(p,k) \\in A$ – from rule $a\\ref {a:c:overc:place}$ and supported rule proposition Then $\\nexists \\lbrace holds(p,q_c,c,k), tot\\_decr(p,q1_c,c,k) \\rbrace \\subseteq A, q1_c>q_c$ in $body(a\\ref {a:c:overc:place})$ – from $a\\ref {a:c:overc:place}$ and forced atom proposition Then $\\nexists c \\in C, \\nexists p \\in P, (\\sum _{t_i \\in \\lbrace t_0,\\dots ,t_x\\rbrace , p \\in \\bullet t_i}{m_{W(p,t_i)}(c)}+ \\\\ \\sum _{t_i \\in \\lbrace t_0,\\dots ,t_x\\rbrace , p \\in R(t_i)}{m_{M_k(p)}(c)}) > m_{M_k(p)}(c)$ – from the following $holds(p,q_c,c,k)$ represents $q_c=m_{M_k(p)}(c)$ – construction of $\\Pi ^5$ , inductive assumption $tot\\_decr(p,q1_c,c,k) \\in A$ if $\\lbrace del(p,{q1_0}_c,t_0,c,k), \\dots , \\\\ del(p,{q1_x}_c,t_x,c,k) \\rbrace \\subseteq A$ , where $q1_c = {q1_0}_c+\\dots +{q1_x}_c$ – from $r\\ref {r:c:totdecr}$ and forced atom proposition $del(p,{q1_i}_c,t_i,c,k) \\in A$ if $\\lbrace fires(t_i,k), ptarc(p,t_i,{q1_i}_c,c,k) \\rbrace \\subseteq A$ – from $r\\ref {r:c:del}$ and supported rule proposition $del(p,{q1_i}_c,t_i,c,k)$ either represents removal of ${q1_i}_c = m_{W(p,t_i)}(c)$ tokens from $p \\in \\bullet t_i$ ; or it represents removal of ${q1_i}_c = m_{M_k(p)}(c)$ tokens from $p \\in R(t_i)$ – from rules $r\\ref {r:c:del},f\\ref {f:c:ptarc},f\\ref {f:c:rptarc}$ , supported rule proposition, and definition REF of transition execution in $PN$ Then the set of transitions in $T_k$ do not conflict – by the definition REF of conflicting transitions And for each reset transition $t_r$ with $enabled(t_r,k) \\in A$ , $fires(t_r,k) \\in A$ , since $A$ is an answer set of $\\Pi ^5$ - from rule $f\\ref {f:c:rptarc:elim}$ and supported rule proposition Then the firing set $T_k$ satisfies the reset transition requirement of definition REF (firing set) Then $\\lbrace t_0, \\dots , t_x\\rbrace = T_k$ – using 1(a),1(b), 1(d) above; and using 1(c) it is a maximal firing set We show that $M_{k+1}$ is produced by firing $T_k$ in $M_k$ .", "Let $holds(p,q_c,c,k+1) \\in A$ Then $\\lbrace holds(p,q1_c,c,k), tot\\_incr(p,q2_c,c,k), tot\\_decr(p,q3_c,c,k) \\rbrace \\subseteq A : q_c=q1_c+q2_c-q3_c$ – from rule $r\\ref {r:c:nextstate}$ and supported rule proposition Then, $holds(p,q1_c,c,k) \\in A$ represents $q1_c=m_{M_k(p)}(c)$ – construction, inductive assumption; and $\\lbrace add(p,{q2_0}_c,t_0,c,k), \\dots , add(p,{q2_j}_c,t_j,c,k)\\rbrace \\subseteq A : {q2_0}_c + \\dots + {q2_j}_c = q2_c$ and $\\lbrace del(p,{q3_0}_c,t_0,c,k), \\dots , del(p,{q3_l}_c,t_l,c,k)\\rbrace \\subseteq A : {q3_0}_c + \\dots + {q3_l}_c = q3_c$ – rules $r\\ref {r:c:totincr},r\\ref {r:c:totdecr}$ and supported rule proposition, respectively Then $\\lbrace fires(t_0,k), \\dots , fires(t_j,k) \\rbrace \\subseteq A$ and $\\lbrace fires(t_0,k), \\dots , fires(t_l,k) \\rbrace \\\\ \\subseteq A$ – rules $r\\ref {r:c:add},r\\ref {r:c:del}$ and supported rule proposition, respectively Then $\\lbrace fires(t_0,k), \\dots , $ $fires(t_j,k) \\rbrace \\cup \\lbrace fires(t_0,k), \\dots , $ $fires(t_l,k) \\rbrace \\subseteq $ $A = \\lbrace fires(t_0,k), \\dots , $ $fires(t_x,k) \\rbrace \\subseteq A$ – set union of subsets Then for each $fires(t_x,k) \\in A$ we have $t_x \\in T_k$ – already shown in item REF above Then $q_c = m_{M_k(p)}(c) + \\sum _{t_x \\in T_k \\wedge p \\in t_x \\bullet }{m_{W(t_x,p)}(c)} - (\\sum _{t_x \\in T_k \\wedge p \\in \\bullet t_x}{m_{W(p,t_x)}(c)} + \\sum _{t_x \\in T_k \\wedge p \\in R(t_x)}{m_{M_k(p)}(c)})$ – from (REF ) above and the following Each $add(p,{q_j}_c,t_j,c,k) \\in A$ represents ${q_j}_c=m_{W(t_j,p)}(c)$ for $p \\in t_j \\bullet $ – rule $r\\ref {r:c:add},f\\ref {f:c:tparc}$ encoding, and definition REF of transition execution in $PN$ Each $del(p,t_y,{q_y}_c,c,k) \\in A$ represents either ${q_y}_c=m_{W(p,t_y)}(c)$ for $p \\in \\bullet t_y$ , or ${q_y}_c=m_{M_k(p)}(c)$ for $p \\in R(t_y)$ – from rule $r\\ref {r:c:del},f\\ref {f:c:ptarc}$ encoding and definition REF of transition execution in $PN$ ; or from rule $r\\ref {r:c:del},f\\ref {f:c:rptarc}$ encoding and definition of reset arc in $PN$ Each $tot\\_incr(p,q2_c,c,k) \\in A$ represents $q2_c=\\sum _{t_x \\in T_k \\wedge p \\in t_x \\bullet }{m_{W(t_x,p)}(c)}$ – aggregate assignment atom semantics in rule $r\\ref {r:c:totincr}$ Each $tot\\_decr(p,q3_c,c,k) \\in A$ represents $q3_c=\\sum _{t_x \\in T_k \\wedge p \\in \\bullet t_x}{m_{W(p,t_x)}(c)} + \\sum _{t_x \\in T_k \\wedge p \\in R(t_x)}{m_{M_k(p)}(c)}$ – aggregate assignment atom semantics in rule $r\\ref {r:c:totdecr}$ Then, $m_{M_{k+1}(p)}(c) = q_c$ – since $holds(p,q_c,c,k+1) \\in A$ encodes $q_c=m_{M_{k+1}(p)}(c)$ – from construction As a result, for any $n > k$ , $T_n$ will be a valid firing set for $M_n$ and $M_{n+1}$ will be its target marking.", "Conclusion: Since both (REF ) and (REF ) hold, $X=M_0,T_k,M_1,\\dots ,M_k,T_{k+1}$ is an execution sequence of $PN(P,T,E,C,W,R,I,Q,QW)$ (w.r.t $M_0$ ) iff there is an answer set $A$ of $\\Pi ^5(PN,M_0,k,ntok)$ such that (REF ) and (REF ) hold.", "Proof of Proposition REF Let $PN=(P,T,E,C,W,R,I,Q,QW,Z)$ be a Petri Net, $M_0$ be its initial marking and let $\\Pi ^6(PN,M_0,k,ntok)$ be the ASP encoding of $PN$ and $M_0$ over a simulation length $k$ , with maximum $ntok$ tokens on any place node, as defined in section REF .", "Then $X=M_0,T_k,M_1,\\dots ,M_k,T_k,M_{k+1}$ is an execution sequence of $PN$ (w.r.t.", "$M_0$ ) iff there is an answer set $A$ of $\\Pi ^6(PN,M_0,k,ntok)$ such that: $\\lbrace fires(t,ts) : t \\in T_{ts}, 0 \\le ts \\le k\\rbrace = \\lbrace fires(t,ts) : fires(t,ts) \\in A \\rbrace $ $\\begin{split}\\lbrace holds(p,q,c,ts) &: p \\in P, c/q = M_{ts}(p), 0 \\le ts \\le k+1 \\rbrace \\\\&= \\lbrace holds(p,q,c,ts) : holds(p,q,c,ts) \\in A \\rbrace \\end{split}$ We prove this by showing that: Given an execution sequence $X$ , we create a set $A$ such that it satisfies (REF ) and (REF ) and show that $A$ is an answer set of $\\Pi ^6$ Given an answer set $A$ of $\\Pi ^6$ , we create an execution sequence $X$ such that (REF ) and (REF ) are satisfied.", "First we show (REF ): We create a set $A$ as a union of the following sets: $A_1=\\lbrace num(n) : 0 \\le n \\le ntok \\rbrace $ $A_2=\\lbrace time(ts) : 0 \\le ts \\le k\\rbrace $ $A_3=\\lbrace place(p) : p \\in P \\rbrace $ $A_4=\\lbrace trans(t) : t \\in T \\rbrace $ $A_5=\\lbrace color(c) : c \\in C \\rbrace $ $A_6=\\lbrace ptarc(p,t,n_c,c,ts) : (p,t) \\in E^-, c \\in C, n_c=m_{W(p,t)}(c), n_c > 0, 0 \\le ts \\le k \\rbrace $ , where $E^- \\subseteq E$ $A_7=\\lbrace tparc(t,p,n_c,c,ts) : (t,p) \\in E^+, c \\in C, n_c=m_{W(t,p)}(c), n_c > 0, 0 \\le ts \\le k \\rbrace $ , where $E^+ \\subseteq E$ $A_8=\\lbrace holds(p,q_c,c,0) : p \\in P, c \\in C, q_c=m_{M_{0}(p)}(c) \\rbrace $ $A_9=\\lbrace ptarc(p,t,n_c,c,ts) : p \\in R(t), c \\in C, n_c = m_{M_{ts}(p)}, n_c > 0, 0 \\le ts \\le k \\rbrace $ $A_{10}=\\lbrace iptarc(p,t,1,c,ts) : p \\in I(t), c \\in C, 0 \\le ts < k \\rbrace $ $A_{11}=\\lbrace tptarc(p,t,n_c,c,ts) : (p,t) \\in Q, c \\in C, n_c=m_{QW(p,t)}(c), n_c > 0, 0 \\le ts \\le k \\rbrace $ $A_{12}=\\lbrace notenabled(t,ts) : t \\in T, 0 \\le ts \\le k, \\exists c \\in C, (\\exists p \\in \\bullet t, m_{M_{ts}(p)}(c) < m_{W(p,t)}(c)) \\vee (\\exists p \\in I(t), m_{M_{ts}(p)}(c) > 0) \\vee (\\exists (p,t) \\in Q, m_{M_{ts}(p)}(c) < m_{QW(p,t)}(c)) \\rbrace $ per definition REF (enabled transition) $A_{13}=\\lbrace enabled(t,ts) : t \\in T, 0 \\le ts \\le k, \\forall c \\in C, (\\forall p \\in \\bullet t, m_{W(p,t)}(c) \\le m_{M_{ts}(p)}(c)) \\wedge (\\forall p \\in I(t), m_{M_{ts}(p)}(c) = 0) \\wedge (\\forall (p,t) \\in Q, m_{M_{ts}(p)}(c) \\ge m_{QW(p,t)}(c)) \\rbrace $ per definition REF (enabled transition) $A_{14}=\\lbrace fires(t,ts) : t \\in T_{ts}, 0 \\le ts \\le k \\rbrace $ per definition REF (firing set), only an enabled transition may fire $A_{15}=\\lbrace add(p,q_c,t,c,ts) : t \\in T_{ts}, p \\in t \\bullet , c \\in C, q_c=m_{W(t,p)}(c), 0 \\le ts \\le k \\rbrace $ per definition REF (transition execution) $A_{16}=\\lbrace del(p,q_c,t,c,ts) : t \\in T_{ts}, p \\in \\bullet t, c \\in C, q_c=m_{W(p,t)}(c), 0 \\le ts \\le k \\rbrace \\cup \\lbrace del(p,q_c,t,c,ts) : t \\in T_{ts}, p \\in R(t), c \\in C, q_c=m_{M_{ts}(p)}(c), 0 \\le ts \\le k \\rbrace $ per definition REF (transition execution) $A_{17}=\\lbrace tot\\_incr(p,q_c,c,ts) : p \\in P, c \\in C, q_c=\\sum _{t \\in T_{ts}, p \\in t \\bullet }{m_{W(t,p)}(c)}, 0 \\le ts \\le k \\rbrace $ per definition REF (firing set execution) $A_{18}=\\lbrace tot\\_decr(p,q_c,c,ts): p \\in P, c \\in C, q_c=\\sum _{t \\in T_{ts}, p \\in \\bullet t}{m_{W(p,t)}(c)}+ \\\\ \\sum _{t \\in T_{ts}, p \\in R(t) }{m_{M_{ts}(p)}(c)}, 0 \\le ts \\le k \\rbrace $ per definition REF (firing set execution) $A_{19}=\\lbrace consumesmore(p,ts) : p \\in P, c \\in C, q_c=m_{M_{ts}(p)}(c), \\\\ q1_c=\\sum _{t \\in T_{ts}, p \\in \\bullet t}{m_{W(p,t)}(c)} + \\sum _{t \\in T_{ts}, p \\in R(t)}{m_{M_{ts}(p)}(c)}, q1_c > q_c, 0 \\le ts \\le k \\rbrace $ per definition REF (conflicting transitions) $A_{20}=\\lbrace consumesmore : \\exists p \\in P, c \\in C, q_c=m_{M_{ts}(p)}(c), \\\\ q1_c=\\sum _{t \\in T_{ts}, p \\in \\bullet t}{m_{W(p,t)}(c)} + \\sum _{t \\in T_{ts}, p \\in R(t)}(m_{M_{ts}(p)}(c)), q1_c > q_c, 0 \\le ts \\le k \\rbrace $ per definition REF (conflicting transitions) $A_{21}=\\lbrace holds(p,q_c,c,ts+1) : p \\in P, c \\in C, q_c=m_{M_{ts+1}(p)}(c), 0 \\le ts < k\\rbrace $ , where $M_{ts+1}(p) = M_j(p) - (\\sum _{\\begin{array}{c}t \\in T_{ts}, p \\in \\bullet t\\end{array}}{W(p,t)} + \\sum _{t \\in T_{ts}, p \\in R(t)}M_{ts}(p)) + \\\\ \\sum _{\\begin{array}{c}t \\in T_{ts}, p \\in t \\bullet \\end{array}}{W(t,p)}$ according to definition REF (firing set execution) $A_{22}=\\lbrace transpr(t,pr) : t \\in T, pr=Z(t) \\rbrace $ $A_{23}=\\lbrace notprenabled(t,ts) : t \\in T, enabled(t,ts) \\in A_{13}, (\\exists tt \\in T, enabled(tt,ts) \\in A_{13}, Z(tt) < Z(t)), 0 \\le ts \\le k \\rbrace \\\\ = \\lbrace notprenabled(t,ts) : t \\in T, (\\forall p \\in \\bullet t, W(p,t) \\le M_{ts}(p)), (\\forall p \\in I(t), M_{ts}(p) = 0), (\\forall (p,t) \\in Q, M_0(p) \\ge WQ(p,t)), \\exists tt \\in T, (\\forall pp \\in \\bullet tt, W(pp,tt) \\le M_{ts}(pp)), \\\\ (\\forall pp \\in I(tt), M_{ts}(pp) = 0), (\\forall (pp,tt) \\in Q, M_{ts}(pp) \\ge QW(pp,tt)), Z(tt) < Z(t), 0 \\le ts \\le k \\rbrace $ $A_{24}=\\lbrace prenabled(t,ts) : t \\in T, enabled(t,ts) \\in A_{13}, (\\nexists tt \\in T: enabled(tt,ts) \\in A_{13}, Z(tt) < Z(t)), 0 \\le ts \\le k \\rbrace = \\lbrace prenabled(t,ts) : t \\in T, (\\forall p \\in \\bullet t, W(p,t) \\le M_{ts}(p)), (\\forall p \\in I(t), M_{ts}(p) = 0), (\\forall (p,t) \\in Q, M_0(p) \\ge WQ(p,t)), \\nexists tt \\in T, (\\forall pp \\in \\bullet tt, W(pp,tt) \\le M_{ts}(pp)), (\\forall pp \\in I(tt), M_{ts}(pp) = 0), (\\forall (pp,tt) \\in Q, M_{ts}(pp) \\ge QW(pp,tt)), Z(tt) < Z(t), 0 \\le ts \\le k \\rbrace $ $A_{25}=\\lbrace could\\_not\\_have(t,ts): t \\in T, prenabled(t,ts) \\in A_{24}, fires(t,ts) \\notin A_{14}, (\\exists p \\in \\bullet t \\cup R(t): q > M_{ts}(p) - (\\sum _{t^{\\prime } \\in T_{ts}, p \\in \\bullet t^{\\prime }}{W(p,t^{\\prime })} + \\sum _{t^{\\prime } \\in T_{ts}, p \\in R(t^{\\prime })}{M_{ts}(p)}), q=W(p,t) \\text{~if~} (p,t) \\in E^- \\text{~or~} R(t) \\text{~otherwise}), 0 \\le ts \\le k\\rbrace =\\lbrace could\\_not\\_have(t,ts) : t \\in T, (\\forall p \\in \\bullet t, W(p,t) \\le M_{ts}(p)), (\\forall p \\in I(t), M_{ts}(p) = 0), (\\forall (p,t) \\in Q, M_0(p) \\ge QW(p,t)), (\\nexists tt \\in T, $ $(\\forall pp \\in \\bullet tt, W(pp,tt) \\le M_{ts}(pp)), (\\forall pp \\in I(tt), M_{ts}(pp) = 0), (\\forall (pp,tt) \\in Q, M_{ts}(pp) \\ge QW(pp,tt)), Z(tt) < Z(t)), t \\notin T_{ts}, (\\exists p \\in \\bullet t \\cup R(t): q > M_{ts}(p) - (\\sum _{t^{\\prime } \\in T_{ts}, p \\in \\bullet t^{\\prime }}{W(p,t^{\\prime })} + \\sum _{t^{\\prime } \\in T_{ts}, p \\in R(t^{\\prime })}{M_{ts}(p)}), q=W(p,t) \\text{~if~} \\\\ (p,t) \\in E^- \\text{~or~} R(t) \\text{~otherwise}), 0 \\le ts \\le k \\rbrace $ per the maximal firing set semantics We show that $A$ satisfies (REF ) and (REF ), and $A$ is an answer set of $\\Pi ^6$ .", "$A$ satisfies (REF ) and (REF ) by its construction above.", "We show $A$ is an answer set of $\\Pi ^6$ by splitting.", "We split $lit(\\Pi ^6)$ into a sequence of $9k+11$ sets: $U_0= head(f\\ref {f:c:place}) \\cup head(f\\ref {f:c:trans}) \\cup head(f\\ref {f:c:col}) \\cup head(f\\ref {f:c:time}) \\cup head(f\\ref {f:c:num}) \\cup head(i\\ref {i:c:init}) \\cup head(f\\ref {f:c:pr}) = \\lbrace place(p) : p \\in P\\rbrace \\cup \\lbrace trans(t) : t \\in T\\rbrace \\cup \\lbrace col(c) : c \\in C \\rbrace \\cup \\lbrace time(0), \\dots , time(k)\\rbrace \\cup \\lbrace num(0), \\dots , num(ntok)\\rbrace \\cup \\lbrace holds(p,q_c,c,0) : p \\in P, c \\in C, q_c=m_{M_0(p)}(c) \\rbrace \\cup \\lbrace transpr(t,pr) : t \\in T, pr=Z(t) \\rbrace $ $U_{9k+1}=U_{9k+0} \\cup head(f\\ref {f:c:ptarc})^{ts=k} \\cup head(f\\ref {f:c:tparc})^{ts=k} \\cup head(f\\ref {f:c:rptarc})^{ts=k} \\cup head(f\\ref {f:c:iptarc})^{ts=k} \\cup head(f\\ref {f:c:tptarc})^{ts=k} = U_{10k+0} \\cup \\lbrace ptarc(p,t,n_c,c,k) : (p,t) \\in E^-, c \\in C, n_c=m_{W(p,t)}(c) \\rbrace \\\\ \\cup \\lbrace tparc(t,p,n_c,c,k) : (t,p) \\in E^+, c \\in C, n_c=m_{W(t,p)}(c) \\rbrace \\cup \\lbrace ptarc(p,t,n_c,c,k) : p \\in R(t), c \\in C, n_c=m_{M_{k}(p)}(c), n > 0 \\rbrace \\cup \\lbrace iptarc(p,t,1,c,k) : p \\in I(t), c \\in C \\rbrace \\cup \\lbrace tptarc(p,t,n_c,c,k) : (p,t) \\in Q, c \\in C, n_c=m_{QW(p,t)}(c) \\rbrace $ $U_{9k+2}=U_{9k+1} \\cup head(e\\ref {e:c:ne:ptarc})^{ts=k} \\cup head(e\\ref {e:c:ne:iptarc})^{ts=k} \\cup head(e\\ref {e:c:ne:tptarc})^{ts=k} = U_{9k+1} \\\\ \\cup \\lbrace notenabled(t,k) : t \\in T \\rbrace $ $U_{9k+3}=U_{9k+2} \\cup head(e\\ref {e:c:enabled})^{ts=k} = U_{9k+2} \\cup \\lbrace enabled(t,k) : t \\in T \\rbrace $ $U_{9k+4}=U_{9k+3} \\cup head(a\\ref {a:c:prne})^{ts=k} = U_{9k+3} \\cup \\lbrace notprenabled(t,k) : t \\in T \\rbrace $ $U_{9k+5}=U_{9k+4} \\cup head(a\\ref {a:c:prenabled})^{ts=k} = U_{9k+4} \\cup \\lbrace prenabled(t,k) : t \\in T \\rbrace $ $U_{9k+6}=U_{9k+5} \\cup head(a\\ref {a:c:prfires})^{ts=k} = U_{9k+5} \\cup \\lbrace fires(t,k) : t \\in T \\rbrace $ $U_{9k+7}=U_{9k+6} \\cup head(r\\ref {r:c:add})^{ts=k} \\cup head(r\\ref {r:c:del})^{ts=k} = U_{9k+6} \\cup \\lbrace add(p,q_c,t,c,k) : p \\in P, t \\in T, c \\in C, q_c=m_{W(t,p)}(c) \\rbrace \\cup \\lbrace del(p,q_c,t,c,k) : p \\in P, t \\in T, c \\in C, q_c=m_{W(p,t)}(c) \\rbrace \\cup \\lbrace del(p,q_c,t,c,k) : p \\in P, t \\in T, c \\in C, q_c=m_{M_{k}(p)}(c) \\rbrace $ $U_{9k+8}=U_{9k+7} \\cup head(r\\ref {r:c:totincr})^{ts=k} \\cup head(r\\ref {r:c:totdecr})^{ts=k} = U_{9k+7} \\cup \\lbrace tot\\_incr(p,q_c,c,k) : p \\in P, c \\in C, 0 \\le q_c \\le ntok \\rbrace \\cup \\lbrace tot\\_decr(p,q_c,c,k) : p \\in P, c \\in C, 0 \\le q_c \\le ntok \\rbrace $ $U_{9k+9}=U_{9k+8} \\cup head(a\\ref {a:c:overc:place})^{ts=k} \\cup head(r\\ref {r:c:nextstate})^{ts=k} \\cup head(a\\ref {a:c:prmaxfire:cnh})^{ts=k} = U_{9k+8} \\cup \\\\ \\lbrace consumesmore(p,k) : p \\in P\\rbrace \\cup \\lbrace holds(p,q,k+1) : p \\in P, 0 \\le q \\le ntok \\rbrace \\cup \\lbrace could\\_not\\_have(t,k) : t \\in T \\rbrace $ $U_{9k+10}=U_{9k+9} \\cup head(a\\ref {a:c:overc:gen})^{ts=k} = U_{9k+9} \\cup \\lbrace consumesmore \\rbrace $ where $head(r_i)^{ts=k}$ are head atoms of ground rule $r_i$ in which $ts=k$ .", "We write $A_i^{ts=k} = \\lbrace a(\\dots ,ts) : a(\\dots ,ts) \\in A_i, ts=k \\rbrace $ as short hand for all atoms in $A_i$ with $ts=k$ .", "$U_{\\alpha }, 0 \\le \\alpha \\le 9k+10$ form a splitting sequence, since each $U_i$ is a splitting set of $\\Pi ^6$ , and $\\langle U_{\\alpha }\\rangle _{\\alpha < \\mu }$ is a monotone continuous sequence, where $U_0 \\subseteq U_1 \\dots \\subseteq U_{9k+10}$ and $\\bigcup _{\\alpha < \\mu }{U_{\\alpha }} = lit(\\Pi ^6)$ .", "We compute the answer set of $\\Pi ^6$ using the splitting sets as follows: $bot_{U_0}(\\Pi ^6) = f\\ref {f:c:place} \\cup f\\ref {f:c:trans} \\cup f\\ref {f:c:col} \\cup f\\ref {f:c:time} \\cup f\\ref {f:c:num} \\cup i\\ref {i:c:init} \\cup f\\ref {f:c:pr}$ and $X_0 = A_1 \\cup \\dots \\cup A_5 \\cup A_8$ ($= U_0$ ) is its answer set – using forced atom proposition $eval_{U_0}(bot_{U_1}(\\Pi ^6) \\setminus bot_{U_0}(\\Pi ^6), X_0) = \\lbrace ptarc(p,t,q_c,c,0) \\text{:-}.", "\| c \\in C, $ $q_c=m_{W(p,t)}(c), q_c > 0 \\rbrace \\cup \\lbrace tparc(t,p,q_c,c,0) \\text{:-}.", "\| c \\in C, q_c=m_{W(t,p)}(c), q_c > 0 \\rbrace \\cup \\lbrace ptarc(p,t,q_c,c,0) \\text{:-}.", "\| c \\in C, $ $q_c=m_{M_0(p)}(c), q_c > 0 \\rbrace \\cup \\lbrace iptarc(p,t,1,c,0) \\text{:-}.", "\| c \\in C \\rbrace \\cup \\lbrace tptarc(p,t,q_c,c,0) \\text{:-}.", "\| c \\in C, q_c = m_{QW(p,t)}(c), q_c > 0 \\rbrace $ .", "Its answer set $X_1=A_6^{ts=0} \\cup A_7^{ts=0} \\cup A_9^{ts=0} \\cup A_{10}^{ts=0} \\cup A_{11}^{ts=0}$ – using forced atom proposition and construction of $A_6, A_7, A_9, A_{10}, A_{11}$ .", "$eval_{U_1}(bot_{U_2}(\\Pi ^6) \\setminus bot_{U_1}(\\Pi ^6), X_0 \\cup X_1) = \\lbrace notenabled(t,0) \\text{:-} .", "\| (\\lbrace trans(t), \\\\ ptarc(p,t,n_c,c,0), holds(p,q_c,c,0) \\rbrace \\subseteq X_0 \\cup X_1, \\text{~where~} q_c < n_c) \\text{~or~} \\\\ (\\lbrace notenabled(t,0) \\text{:-} .", "\| (\\lbrace trans(t), iptarc(p,t,n2_c,c,0), holds(p,q_c,c,0) \\rbrace \\subseteq X_0 \\cup X_1, \\\\ \\text{~where~} q_c \\ge n2_c \\rbrace ) \\text{~or~} (\\lbrace trans(t), tptarc(p,t,n3_c,c,0), $ $holds(p,q_c,c,0) \\rbrace \\subseteq X_0 \\cup X_1, \\text{~where~} $ $q_c < n3_c) \\rbrace $ .", "Its answer set $X_2=A_{12}^{ts=0}$ – using forced atom proposition and construction of $A_{12}$ .", "where, $q_c=m_{M_0(p)}(c)$ , and $n_c=m_{W(p,t)}(c)$ for an arc $(p,t) \\in E^-$ – by construction of $i\\ref {i:c:init}$ and $f\\ref {f:c:ptarc}$ in $\\Pi ^6$ , and in an arc $(p,t) \\in E^-$ , $p \\in \\bullet t$ (by definition REF of preset) $n2_c=1$ – by construction of $iptarc$ predicates in $\\Pi ^6$ , meaning $q_c \\ge n2_c \\equiv q_c \\ge 1 \\equiv q_c > 0$ , $tptarc(p,t,n3_c,c,0)$ represents $n3_c=m_{QW(p,t)}(c)$ , where $(p,t) \\in Q$ thus, $notenabled(t,0) \\in X_1$ represents $\\exists c \\in C, (\\exists p \\in \\bullet t : m_{M_0(p)}(c) < m_{W(p,t)}(c)) \\vee (\\exists p \\in I(t) : m_{M_0(p)}(c) > 0) \\vee (\\exists (p,t) \\in Q : m_{M_0(p)}(c) < m_{QW(p,t)}(c))$ .", "$eval_{U_2}(bot_{U_3}(\\Pi ^6) \\setminus bot_{U_2}(\\Pi ^6), X_0 \\cup \\dots \\cup X_2) = \\lbrace enabled(t,0) \\text{:-}.", "\| trans(t) \\in X_0 \\cup \\dots \\cup X_2, notenabled(t,0) \\notin X_0 \\cup \\dots \\cup X_2 \\rbrace $ .", "Its answer set is $X_3 = A_{13}^{ts=0}$ – using forced atom proposition and construction of $A_{13}$ .", "since an $enabled(t,0) \\in X_3$ if $\\nexists ~notenabled(t,0) \\in X_0 \\cup \\dots \\cup X_2$ ; which is equivalent to $\\nexists t, \\forall c \\in C, (\\nexists p \\in \\bullet t, m_{M_0(p)}(c) < m_{W(p,t)}(c)), (\\nexists p \\in I(t), $ $m_{M_0(p)}(c) > 0), (\\nexists (p,t) \\in Q : m_{M_0(p)}(c) < m_{QW(p,t)}(c) ), \\forall c \\in C, (\\forall p \\in \\bullet t: m_{M_0(p)}(c) \\ge m_{W(p,t)}(c)), (\\forall p \\in I(t) : m_{M_0(p)}(c) = 0)$ .", "$eval_{U_3}(bot_{U_4}(\\Pi ^6) \\setminus bot_{U_3}(\\Pi ^6), X_0 \\cup \\dots \\cup X_3) = \\lbrace notprenabled(t,0) \\text{:-}.", "\| \\\\ \\lbrace enabled(t,0), transpr(t,p), enabled(tt,0), transpr(tt,pp) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_3, pp < p \\rbrace $ .", "Its answer set is $X_4 = A_{23}^{ts=k}$ – using forced atom proposition and construction of $A_{23}$ .", "$enabled(t,0)$ represents $\\exists t \\in T, \\forall c \\in C, (\\forall p \\in \\bullet t, m_{M_0(p)}(c) \\ge m_{W(p,t)}(c)), $ $(\\forall p \\in I(t), m_{M_0(p)}(c) = 0), (\\forall (p,t) \\in Q, m_{M_0(p)}(c) \\ge m_{QW(p,t)}(c))$ $enabled(tt,0)$ represents $\\exists tt \\in T, \\forall c \\in C, (\\forall pp \\in \\bullet tt, m_{M_0(pp)}(c) \\ge \\\\ m_{W(pp,tt)}(c)) \\wedge (\\forall pp \\in I(tt), m_{M_0(pp)}(c) = 0), (\\forall (pp,tt) \\in Q, m_{M_0(pp)}(c) \\ge m_{QW(pp,tt)}(c))$ $transpr(t,p)$ represents $p=Z(t)$ – by construction $transpr(tt,pp)$ represents $pp=Z(tt)$ – by construction thus, $notprenabled(t,0)$ represents $\\forall c \\in C, (\\forall p \\in \\bullet t, m_{M_0(p)}(c) \\ge \\\\ m_{W(p,t)}(c)) \\wedge (\\forall p \\in I(t), m_{M_0(p)}(c) = 0), \\exists tt \\in T, (\\forall pp \\in \\bullet tt, m_{M_0(pp)}(c) \\ge m_{W(pp,tt)}(c)) \\wedge (\\forall pp \\in I(tt), m_{M_0(pp)}(c) = 0), Z(tt) < Z(t)$ which is equivalent to $(\\forall p \\in \\bullet t: M_0(p) \\ge W(p,t)) \\wedge (\\forall p \\in I(t), M_0(p) = 0), \\exists tt \\in T, (\\forall pp \\in \\bullet tt, M_0(pp) \\ge W(pp,tt)), (\\forall pp \\in I(tt), M_0(pp) = 0), Z(tt) < Z(t)$ – assuming multiset domain $C$ for all operations $eval_{U_4}(bot_{U_5}(\\Pi ^6) \\setminus bot_{U_4}(\\Pi ^6), X_0 \\cup \\dots \\cup X_4) = \\lbrace prenabled(t,0) \\text{:-}.", "\| enabled(t,0) \\in X_0 \\cup \\dots \\cup X_4, notprenabled(t,0) \\notin X_0 \\cup \\dots \\cup X_4 \\rbrace $ .", "Its answer set is $X_5 = A_{24}^{ts=k}$ – using forced atom proposition and construction of $A_{24}$ $enabled(t,0)$ represents $\\forall c \\in C, (\\forall p \\in \\bullet t, m_{M_0(p)}(c) \\ge m_{W(p,t)}(c)), (\\forall p \\in I(t), m_{M_0(p)}(c) = 0), (\\forall (p,t) \\in Q, m_{M_0(p)}(c) \\ge m_{QW(p,t)}(c)) \\equiv (\\forall p \\in \\bullet t, \\\\ M_0(p) \\ge W(p,t)), (\\forall p \\in I(t), M_0(p) = 0), (\\forall (p,t) \\in Q, M_0(p) \\ge QW(p,t))$ – from REF above and assuming multiset domain $C$ for all operations $notprenabled(t,0)$ represents $(\\forall p \\in \\bullet t, M_0(p) \\ge W(p,t)), (\\forall p \\in I(t), \\\\ M_0(p) = 0), (\\forall (p,t) \\in Q, M_0(p) \\ge QW(p,t)), \\exists tt \\in T, (\\forall pp \\in \\bullet tt, M_0(pp) \\ge W(pp,tt)), (\\forall pp \\in I(tt), M_0(pp) = 0), (\\forall (pp,tt) \\in Q, M_0(pp) \\ge W(pp,tt)), \\\\ Z(tt) < Z(t)$ – from REF above and assuming multiset domain $C$ for all operations then, $prenabled(t,0)$ represents $(\\forall p \\in \\bullet t, M_0(p) \\ge W(p,t)), (\\forall p \\in I(t), \\\\ M_0(p) = 0), (\\forall (p,t) \\in Q, M_0(p) \\ge QW(p,t)), \\nexists tt \\in T, ((\\forall pp \\in \\bullet tt, M_0(pp) \\ge W(pp,tt)), (\\forall pp \\in I(tt), M_0(pp) = 0), (\\forall (pp,tt) \\in Q, M_0(pp) \\ge W(pp,tt)), \\\\ Z(tt) < Z(t))$ – from (a), (b) and $enabled(t,0) \\in X_0 \\cup \\dots \\cup X_4$ $eval_{U_5}(bot_{U_6}(\\Pi ^6) \\setminus bot_{U_5}(\\Pi ^6), X_0 \\cup \\dots \\cup X_5) = \\lbrace \\lbrace fires(t,0)\\rbrace \\text{:-}.", "\| prenabled(t,0) \\\\ \\text{~holds in~} X_0 \\cup \\dots \\cup X_5 \\rbrace $ .", "It has multiple answer sets $X_{6.1}, \\dots , X_{6.n}$ , corresponding to elements of power set of $fires(t,0)$ atoms in $eval_{U_5}(...)$ – using supported rule proposition.", "Since we are showing that the union of answer sets of $\\Pi ^6$ determined using splitting is equal to $A$ , we only consider the set that matches the $fires(t,0)$ elements in $A$ and call it $X_6$ , ignoring the rest.", "Thus, $X_6 = A_{14}^{ts=0}$ , representing $T_0$ .", "in addition, for every $t$ such that $prenabled(t,0) \\in X_0 \\cup \\dots \\cup X_5, R(t) \\ne \\emptyset $ ; $fires(t,0) \\in X_6$ – per definition REF (firing set); requiring that a reset transition is fired when enabled thus, the firing set $T_0$ will not be eliminated by the constraint $f\\ref {f:c:pr:rptarc:elim}$ $eval_{U_6}(bot_{U_7}(\\Pi ^6) \\setminus bot_{U_6}(\\Pi ^6), X_0 \\cup \\dots \\cup X_6) = \\lbrace add(p,n_c,t,c,0) \\text{:-}.", "\| \\lbrace fires(t,0), \\\\ tparc(t,p,n_c,c,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_6 \\rbrace \\cup \\lbrace del(p,n_c,t,c,0) \\text{:-}.", "\| \\lbrace fires(t,0), \\\\ ptarc(p,t,n_c,c,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_6 \\rbrace $ .", "It's answer set is $X_7 = A_{15}^{ts=0} \\cup A_{16}^{ts=0}$ – using forced atom proposition and definitions of $A_{15}$ and $A_{16}$ .", "where, each $add$ atom is equivalent to $n_c=m_{W(t,p)}(c),c \\in C, p \\in t \\bullet $ , and each $del$ atom is equivalent to $n_c=m_{W(p,t)}(c), c \\in C, p \\in \\bullet t$ ; or $n_c=m_{M_0(p)}(c), c \\in C, p \\in R(t)$ , representing the effect of transitions in $T_0$ – by construction $eval_{U_7}(bot_{U_8}(\\Pi ^6) \\setminus bot_{U_7}(\\Pi ^6), X_0 \\cup \\dots \\cup X_7) = \\lbrace tot\\_incr(p,qq_c,c,0) \\text{:-}.", "\| qq_c=\\sum _{add(p,q_c,t,c,0) \\in X_0 \\cup \\dots \\cup X_7}{q_c} \\rbrace \\cup \\lbrace tot\\_decr(p,qq_c,c,0) \\text{:-}.", "\| qq_c=\\sum _{del(p,q_c,t,c,0) \\in X_0 \\cup \\dots \\cup X_7}{q_c} \\rbrace $ .", "It's answer set is $X_8 = A_{17}^{ts=0} \\cup A_{18}^{ts=0}$ – using forced atom proposition and definitions of $A_{17}$ and $A_{18}$ .", "where, each $tot\\_incr(p,qq_c,c,0)$ , $qq_c=\\sum _{add(p,q_c,t,c,0) \\in X_0 \\cup \\dots X_7}{q_c}$ $\\equiv qq_c=\\sum _{t \\in X_6, p \\in t \\bullet }{m_{W(p,t)}(c)}$ , where, each $tot\\_decr(p,qq_c,c,0)$ , $qq_c=\\sum _{del(p,q_c,t,c,0) \\in X_0 \\cup \\dots X_7}{q_c}$ $\\equiv qq=\\sum _{t \\in X_6, p \\in \\bullet t}{m_{W(t,p)}(c)} + \\sum _{t \\in X_6, p \\in R(t)}{m_{M_0(p)}(c)}$ , represent the net effect of transitions in $T_0$ $eval_{U_8}(bot_{U_9}(\\Pi ^6) \\setminus bot_{U_8}(\\Pi ^6), X_0 \\cup \\dots \\cup X_8) = \\lbrace consumesmore(p,0) \\text{:-}.", "\| \\\\ \\lbrace holds(p,q_c,c,0), tot\\_decr(p,q1_c,c,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_8, q1_c > q_c \\rbrace \\cup \\\\ \\lbrace holds(p,q_c,c,1) \\text{:-}.", "\| \\lbrace holds(p,q1_c,c,0), tot\\_incr(p,q2_c,c,0), tot\\_decr(p,q3_c,c,0) \\rbrace \\\\ \\subseteq X_0 \\cup \\dots \\cup X_6, q_c=q1_c+q2_c-q3_c \\rbrace \\cup \\lbrace could\\_not\\_have(t,0) \\text{:-}.", "\| \\lbrace prenabled(t,0), \\\\ ptarc(s,t,q,c,0), holds(s,qq,c,0), tot\\_decr(s,qqq,c,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_8, \\\\ fires(t,0) \\notin X_0 \\cup \\dots \\cup X_8, q > qq - qqq \\rbrace $ .", "It's answer set is $X_9 = A_{19}^{ts=0} \\cup A_{21}^{ts=0} \\cup A_{25}^{ts=0}$ – using forced atom proposition and definitions of $A_{19}, A_{21}, A_{25}$ .", "where, $consumesmore(p,0)$ represents $\\exists p : q_c=m_{M_0(p)}(c), q1_c= \\\\ \\sum _{t \\in T_0, p \\in \\bullet t}{m_{W(p,t)}(c)}+\\sum _{t \\in T_0, p \\in R(t)}{m_{M_0(p)}(c)}, q1_c > q_c, c \\in C$ , indicating place $p$ will be over consumed if $T_0$ is fired, as defined in definition REF (conflicting transitions), $holds(p,q_c,c,1)$ if $q_c=m_{M_0(p)}(c)+\\sum _{t \\in T_0, p \\in t \\bullet }{m_{W(t,p)}(c)}- \\\\ (\\sum _{t \\in T_0, p \\in \\bullet t}{m_{W(p,t)}(c)}+ \\sum _{t \\in T_0, p \\in R(t)}{m_{M_0(p)}(c)})$ represented by $q_c=m_{M_1(p)}(c)$ for some $c \\in C$ – by construction of $\\Pi ^6$ $could\\_not\\_have(t,0)$ if $(\\forall p \\in \\bullet t, W(p,t) \\le M_0(p)), (\\forall p \\in I(t), M_0(p) = 0), (\\forall (p,t) \\in Q, \\\\ M_0(p) \\ge WQ(p,t)), \\nexists tt \\in T, (\\forall pp \\in \\bullet tt, W(pp,tt) \\le M_{ts}(pp)), (\\forall pp \\in I(tt), M_0(pp) = 0), (\\forall (pp,tt) \\in Q, M_0(pp) \\ge QW(pp,tt)), Z(tt) < Z(t)$ , and $q_c > m_{M_0(s)}(c) - (\\sum _{t^{\\prime } \\in T_0, s \\in \\bullet t^{\\prime }}{m_{W(s,t^{\\prime })}(c)}+ \\sum _{t^{\\prime } \\in T_0, s \\in R(t)}{m_{M_0(s)}(c)}), \\\\ q_c = m_{W(s,t)}(c) \\text{~if~} s \\in \\bullet t \\text{~or~} m_{M_0(s)}(c) \\text{~otherwise}$ for some $c \\in C$ , which becomes $q > M_0(s) - (\\sum _{t^{\\prime } \\in T_0, s \\in \\bullet t^{\\prime }}{W(s,t^{\\prime })}+ \\sum _{t^{\\prime } \\in T_0, s \\in R(t)}{M_0(s)}), q = W(s,t) \\text{~if~} s \\in \\bullet t \\text{~or~} M_0(s) \\text{~otherwise}$ for all $c \\in C$ (i), (ii) above combined match the definition of $A_{25}$ $X_9$ does not contain $could\\_not\\_have(t,0)$ , when $prenabled(t,0) \\in X_0 \\cup \\dots \\cup X_8$ and $fires(t,0) \\notin X_0 \\cup \\dots \\cup X_8$ due to construction of $A$ , encoding of $a\\ref {a:c:overc:place}$ and its body atoms.", "As a result, it is not eliminated by the constraint $a\\ref {a:c:maxfire:elim}$ $ \\vdots $ $eval_{U_{9k+0}}(bot_{U_{9k+1}}(\\Pi ^6) \\setminus bot_{U_{9k+0}}(\\Pi ^6), X_0 \\cup \\dots \\cup X_{9k+0}) = \\lbrace ptarc(p,t,q_c,c,k) \\text{:-}.", "\| c \\in C, q_c=m_{W(p,t)}(c), q_c > 0 \\rbrace \\cup \\lbrace tparc(t,p,q_c,c,k) \\text{:-}.", "\| c \\in C, q_c=m_{W(t,p)}(c), q_c > 0 \\rbrace \\cup \\lbrace ptarc(p,t,q_c,c,k) \\text{:-}.", "\| c \\in C, q_c=m_{M_0(p)}(c), q_c > 0 \\rbrace \\cup \\lbrace iptarc(p,t,1,c,k) \\text{:-}.", "\| c \\in C \\rbrace \\cup \\lbrace tptarc(p,t,q_c,c,k) \\text{:-}.", "\| c \\in C, q_c = m_{QW(p,t)}(c), q_c > 0 \\rbrace $ .", "Its answer set $X_{9k+1}=A_6^{ts=k} \\cup A_7^{ts=k} \\cup A_9^{ts=k} \\cup A_{10}^{ts=k} \\cup A_{11}^{ts=k}$ – using forced atom proposition and construction of $A_6, A_7, A_9, A_{10}, A_{11}$ .", "$eval_{U_{9k+1}}(bot_{U_{9k+2}}(\\Pi ^6) \\setminus bot_{U_{9k+1}}(\\Pi ^6), X_0 \\cup \\dots \\cup X_{9k+1}) = $ $\\lbrace notenabled(t,k) \\text{:-} .", "\| $ $(\\lbrace trans(t), $ $ptarc(p,t,n_c,c,k), $ $holds(p,q_c,c,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{10k+1}, \\text{~where~} $ $q_c < n_c) \\text{~or~} $ $(\\lbrace notenabled(t,k) \\text{:-} .", "\| $ $(\\lbrace trans(t), $ $iptarc(p,t,n2_c,c,k), $ $holds(p,q_c,c,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{10k+1}, \\text{~where~} $ $q_c \\ge n2_c \\rbrace ) \\text{~or~} $ $(\\lbrace trans(t), $ $tptarc(p,t,n3_c,c,k), \\\\ holds(p,q_c,c,k) \\rbrace \\subseteq X_0 \\cup X_{9k+1}, \\text{~where~} q_c < n3_c) \\rbrace $ .", "Its answer set $X_{9k+2}=A_{12}^{ts=k}$ – using forced atom proposition and construction of $A_{12}$ .", "where, $q_c=m_{M_0(p)}(c)$ , and $n_c=m_{W(p,t)}(c)$ for an arc $(p,t) \\in E^-$ – by construction of $i\\ref {i:c:init}$ and $f\\ref {f:c:ptarc}$ predicates in $\\Pi ^6$ , and in an arc $(p,t) \\in E^-$ , $p \\in \\bullet t$ (by definition REF of preset) $n2_c=1$ – by construction of $iptarc$ predicates in $\\Pi ^6$ , meaning $q_c \\ge n2_c \\equiv q_c \\ge 1 \\equiv q_c > 0$ , $tptarc(p,t,n3_c,c,k)$ represents $n3_c=m_{QW(p,t)}(c)$ , where $(p,t) \\in Q$ thus, $notenabled(t,k) \\in X_{9k+1}$ represents $\\exists c \\in C, (\\exists p \\in \\bullet t : m_{M_0(p)}(c) < m_{W(p,t)}(c)) \\vee (\\exists p \\in I(t) : m_{M_0(p)}(c) > k) \\vee (\\exists (p,t) \\in Q : m_{M_0(p)}(c) < m_{QW(p,t)}(c))$ .", "$eval_{U_{9k+2}}(bot_{U_{9k+3}}(\\Pi ^6) \\setminus bot_{U_{9k+2}}(\\Pi ^6), X_0 \\cup \\dots \\cup X_{9k+2}) = \\lbrace enabled(t,k) \\text{:-}.", "\| trans(t) \\in X_0 \\cup \\dots \\cup X_{9k+2}, notenabled(t,k) \\notin X_0 \\cup \\dots \\cup X_{9k+2} \\rbrace $ .", "Its answer set is $X_{9k+3} = A_{13}^{ts=k}$ – using forced atom proposition and construction of $A_{13}$ .", "since an $enabled(t,k) \\in X_{9k+3}$ if $\\nexists ~notenabled(t,k) \\in X_0 \\cup \\dots \\cup X_{9k+2}$ ; which is equivalent to $\\nexists t, \\forall c \\in C, (\\nexists p \\in \\bullet t, m_{M_0(p)}(c) < m_{W(p,t)}(c)), (\\nexists p \\in I(t), m_{M_0(p)}(c) > k), (\\nexists (p,t) \\in Q : m_{M_0(p)}(c) < m_{QW(p,t)}(c) ), \\forall c \\in C, (\\forall p \\in \\bullet t: m_{M_0(p)}(c) \\ge m_{W(p,t)}(c)), (\\forall p \\in I(t) : m_{M_0(p)}(c) = k)$ .", "$eval_{U_{9k+3}}(bot_{U_{9k+4}}(\\Pi ^6) \\setminus bot_{U_{9k+3}}(\\Pi ^6), X_0 \\cup \\dots \\cup X_{9k+3}) = \\lbrace notprenabled(t,k) \\text{:-}.", "\| \\\\ \\lbrace enabled(t,k), transpr(t,p), enabled(tt,k), transpr(tt,pp) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{9k+3}, \\\\ pp < p \\rbrace $ .", "Its answer set is $X_{9k+4} = A_{23}^{ts=k}$ – using forced atom proposition and construction of $A_{23}$ .", "$enabled(t,k)$ represents $\\exists t \\in T, \\forall c \\in C, (\\forall p \\in \\bullet t, m_{M_0(p)}(c) \\ge m_{W(p,t)}(c)), \\\\ (\\forall p \\in I(t), m_{M_0(p)}(c) = k), (\\forall (p,t) \\in Q, m_{M_0(p)}(c) \\ge m_{QW(p,t)}(c))$ $enabled(tt,k)$ represents $\\exists tt \\in T, \\forall c \\in C, (\\forall pp \\in \\bullet tt, m_{M_0(pp)}(c) \\ge \\\\ m_{W(pp,tt)}(c)) \\wedge (\\forall pp \\in I(tt), m_{M_0(pp)}(c) = k), (\\forall (pp,tt) \\in Q, m_{M_0(pp)}(c) \\ge m_{QW(pp,tt)}(c))$ $transpr(t,p)$ represents $p=Z(t)$ – by construction $transpr(tt,pp)$ represents $pp=Z(tt)$ – by construction thus, $notprenabled(t,k)$ represents $\\forall c \\in C, (\\forall p \\in \\bullet t, m_{M_0(p)}(c) \\ge m_{W(p,t)}(c)) \\\\ \\wedge (\\forall p \\in I(t), m_{M_0(p)}(c) = k), \\exists tt \\in T, (\\forall pp \\in \\bullet tt, m_{M_0(pp)}(c) \\ge m_{W(pp,tt)}(c)) \\wedge (\\forall pp \\in I(tt), m_{M_0(pp)}(c) = k), Z(tt) < Z(t)$ which is equivalent to $(\\forall p \\in \\bullet t: M_0(p) \\ge W(p,t)) \\wedge (\\forall p \\in I(t), M_0(p) = k), \\exists tt \\in T, (\\forall pp \\in \\bullet tt, M_0(pp) \\ge W(pp,tt)), (\\forall pp \\in I(tt), M_0(pp) = k), Z(tt) < Z(t)$ – assuming multiset domain $C$ for all operations $eval_{U_{9k+4}}(bot_{U_{9k+5}}(\\Pi ^6) \\setminus bot_{U_{9k+4}}(\\Pi ^6), X_0 \\cup \\dots \\cup X_{9k+4}) = \\lbrace prenabled(t,k) \\text{:-}.", "\| $ $enabled(t,k) \\in X_0 \\cup \\dots \\cup X_{9k+4}, notprenabled(t,k) \\notin X_0 \\cup \\dots \\cup X_{9k+4} \\rbrace $ .", "Its answer set is $X_{9k+5} = A_{24}^{ts=k}$ – using forced atom proposition and construction of $A_{24}$ $enabled(t,k)$ represents $\\forall c \\in C, (\\forall p \\in \\bullet t, m_{M_0(p)}(c) \\ge m_{W(p,t)}(c)), $ $(\\forall p \\in I(t), m_{M_0(p)}(c) = k), (\\forall (p,t) \\in Q, m_{M_0(p)}(c) \\ge m_{QW(p,t)}(c)) \\equiv (\\forall p \\in \\bullet t, M_0(p) \\ge W(p,t)), $ $(\\forall p \\in I(t), M_0(p) = k), (\\forall (p,t) \\in Q, M_0(p) \\ge QW(p,t))$ – from REF above and assuming multiset domain $C$ for all operations $notprenabled(t,k)$ represents $(\\forall p \\in \\bullet t, M_0(p) \\ge W(p,t)), (\\forall p \\in I(t), \\\\ M_0(p) = k), (\\forall (p,t) \\in Q, M_0(p) \\ge QW(p,t)), \\exists tt \\in T, $ $(\\forall pp \\in \\bullet tt, M_0(pp) \\ge W(pp,tt)), $ $(\\forall pp \\in I(tt), M_0(pp) = k), (\\forall (pp,tt) \\in Q, M_0(pp) \\ge W(pp,tt)), \\\\ Z(tt) < Z(t)$ – from REF above and assuming multiset domain $C$ for all operations then, $prenabled(t,k)$ represents $(\\forall p \\in \\bullet t, M_0(p) \\ge W(p,t)), (\\forall p \\in I(t), \\\\ M_0(p) = k), (\\forall (p,t) \\in Q, M_0(p) \\ge QW(p,t)), \\nexists tt \\in T, $ $((\\forall pp \\in \\bullet tt, M_0(pp) \\ge W(pp,tt)), $ $(\\forall pp \\in I(tt), M_0(pp) = k), (\\forall (pp,tt) \\in Q, M_0(pp) \\ge W(pp,tt)), \\\\ Z(tt) < Z(t))$ – from (a), (b) and $enabled(t,k) \\in X_0 \\cup \\dots \\cup X_{9k+4}$ $eval_{U_{9k+5}}(bot_{U_{9k+6}}(\\Pi ^6) \\setminus bot_{U_{9k+5}}(\\Pi ^6), X_0 \\cup \\dots \\cup X_{9k+5}) = \\lbrace \\lbrace fires(t,k)\\rbrace \\text{:-}.", "\| \\\\ prenabled(t,k) \\text{~holds in~} X_0 \\cup \\dots \\cup X_{9k+5} \\rbrace $ .", "It has multiple answer sets $X_{9k+6.1}, \\dots , \\\\ X_{9k+6.n}$ , corresponding to elements of power set of $fires(t,k)$ atoms in $eval_{U_{9k+5}}(...)$ – using supported rule proposition.", "Since we are showing that the union of answer sets of $\\Pi ^6$ determined using splitting is equal to $A$ , we only consider the set that matches the $fires(t,k)$ elements in $A$ and call it $X_{9k+6}$ , ignoring the rest.", "Thus, $X_{9k+6} = A_{14}^{ts=k}$ , representing $T_k$ .", "in addition, for every $t$ such that $prenabled(t,k) \\in X_0 \\cup \\dots \\cup X_{9k+5}, R(t) \\ne \\emptyset $ ; $fires(t,k) \\in X_{9k+6}$ – per definition REF (firing set); requiring that a reset transition is fired when enabled thus, the firing set $T_k$ will not be eliminated by the constraint $f\\ref {f:c:pr:rptarc:elim}$ $eval_{U_{9k+6}}(bot_{U_{9k+7}}(\\Pi ^6) \\setminus bot_{U_{9k+6}}(\\Pi ^6), X_0 \\cup \\dots \\cup X_{9k+6}) = \\lbrace add(p,n_c,t,c,k) \\text{:-}.", "\| \\\\ \\lbrace fires(t,k), tparc(t,p,n_c,c,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{9k+6} \\rbrace \\cup \\lbrace del(p,n_c,t,c,k) \\text{:-}.", "\| \\\\ \\lbrace fires(t,k), ptarc(p,t,n_c,c,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{9k+6} \\rbrace $ .", "It's answer set is $X_{9k+7} = A_{15}^{ts=k} \\cup A_{16}^{ts=k}$ – using forced atom proposition and definitions of $A_{15}$ and $A_{16}$ .", "where, each $add$ atom is equivalent to $n_c=m_{W(t,p)}(c),c \\in C, p \\in t \\bullet $ , and each $del$ atom is equivalent to $n_c=m_{W(p,t)}(c), c \\in C, p \\in \\bullet t$ ; or $n_c=m_{M_0(p)}(c), c \\in C, p \\in R(t)$ , representing the effect of transitions in $T_k$ .", "$eval_{U_{9k+7}}(bot_{U_{9k+8}}(\\Pi ^6) \\setminus bot_{U_{9k+7}}(\\Pi ^6), X_0 \\cup \\dots \\cup X_{9k+7}) = \\lbrace tot\\_incr(p,qq_c,c,k) \\text{:-}.", "\| $ $qq_c=\\sum _{add(p,q_c,t,c,k) \\in X_0 \\cup \\dots \\cup X_{10k+7}}{q_c} \\rbrace \\cup \\lbrace tot\\_decr(p,qq_c,c,k) \\text{:-}.", "\| qq_c= \\\\ \\sum _{del(p,q_c,t,c,k) \\in X_0 \\cup \\dots \\cup X_{9k+7}}{q_c} \\rbrace $ .", "It's answer set is $X_{10k+8} = A_{17}^{ts=k} \\cup A_{18}^{ts=k}$ – using forced atom proposition and definitions of $A_{17}$ and $A_{18}$ .", "where, each $tot\\_incr(p,qq_c,c,k)$ , $qq_c=\\sum _{add(p,q_c,t,c,k) \\in X_0 \\cup \\dots X_{10k+7}}{q_c}$ $\\equiv qq_c=\\sum _{t \\in X_{9k+6}, p \\in t \\bullet }{m_{W(p,t)}(c)}$ , and each $tot\\_decr(p,qq_c,c,k)$ , $qq_c=\\sum _{del(p,q_c,t,c,k) \\in X_0 \\cup \\dots X_{10k+7}}{q_c}$ $\\equiv qq=\\sum _{t \\in X_{10k+6}, p \\in \\bullet t}{m_{W(t,p)}(c)} + \\sum _{t \\in X_{9k+6}, p \\in R(t)}{m_{M_0(p)}(c)}$ , represent the net effect of transitions in $T_k$ $eval_{U_{9k+8}}(bot_{U_{9k+9}}(\\Pi ^6) \\setminus bot_{U_{9k+8}}(\\Pi ^6), X_0 \\cup \\dots \\cup X_{9k+8}) = \\lbrace consumesmore(p,k) \\text{:-}.", "\| \\\\ \\lbrace holds(p,q_c,c,k), tot\\_decr(p,q1_c,c,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{9k+8}, q1_c > q_c \\rbrace \\cup \\\\ \\lbrace holds(p,q_c,c,1) \\text{:-}.", "\| \\lbrace holds(p,q1_c,c,k), tot\\_incr(p,q2_c,c,k), tot\\_decr(p,q3_c,c,k) \\rbrace \\\\ \\subseteq $ $X_0 \\cup \\dots \\cup X_{9k+6}, q_c=q1_c+q2_c-q3_c \\rbrace \\cup \\lbrace could\\_not\\_have(t,k) \\text{:-}.", "\| \\lbrace prenabled(t,k), \\\\ ptarc(s,t,q,c,k), holds(s,qq,c,k), tot\\_decr(s,qqq,c,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{9k+8}, \\\\ fires(t,k) \\notin X_0 \\cup \\dots \\cup X_{9k+8}, q > qq - qqq \\rbrace $ .", "It's answer set is $X_{10k+9} = A_{19}^{ts=k} \\cup A_{21}^{ts=k} \\cup A_{25}^{ts=k}$ – using forced atom proposition and definitions of $A_{19}, A_{21}, A_{25}$ .", "where, $consumesmore(p,k)$ represents $\\exists p : q_c=m_{M_0(p)}(c), \\\\ q1_c=$ $\\sum _{t \\in T_k, p \\in \\bullet t}{m_{W(p,t)}(c)}+\\sum _{t \\in T_k, p \\in R(t)}{m_{M_0(p)}(c)}, q1_c > q_c, c \\in C$ , indicating place $p$ that will be over consumed if $T_k$ is fired, as defined in definition REF (conflicting transition) $holds(p,q_c,c,k+1)$ if $q_c=m_{M_0(p)}(c)+\\sum _{t \\in T_k, p \\in t \\bullet }{m_{W(t,p)}(c)}- \\\\ (\\sum _{t \\in T_k, p \\in \\bullet t}{m_{W(p,t)}(c)}+ \\sum _{t \\in T_k, p \\in R(t)}{m_{M_0(p)}(c)})$ represented by $q_c=m_{M_1(p)}(c)$ for some $c \\in C$ – by construction of $\\Pi ^6$ $could\\_not\\_have(t,k)$ if $(\\forall p \\in \\bullet t, W(p,t) \\le M_0(p)), (\\forall p \\in I(t), M_0(p) = k), (\\forall (p,t) \\in Q, $ $M_0(p) \\ge WQ(p,t)), \\nexists tt \\in T, (\\forall pp \\in \\bullet tt, W(pp,tt) \\le M_{ts}(pp)), (\\forall pp \\in I(tt), M_0(pp) = k), (\\forall (pp,tt) \\in Q, $ $M_0(pp) \\ge QW(pp,tt)), Z(tt) < Z(t)$ , and $q_c > m_{M_0(s)}(c) - (\\sum _{t^{\\prime } \\in T_k, s \\in \\bullet t^{\\prime }}{m_{W(s,t^{\\prime })}(c)}+ \\sum _{t^{\\prime } \\in T_k, s \\in R(t)}{m_{M_0(s)}(c)}), $ $q_c = m_{W(s,t)}(c) \\text{~if~} s \\in \\bullet t \\text{~or~} m_{M_0(s)}(c) \\text{~otherwise}$ for some $c \\in C$ , which becomes $q > M_0(s) - (\\sum _{t^{\\prime } \\in T_k, s \\in \\bullet t^{\\prime }}{W(s,t^{\\prime })}+ \\sum _{t^{\\prime } \\in T_k, s \\in R(t)}{M_0(s)}), q = W(s,t) \\text{~if~} s \\in \\bullet t \\text{~or~} M_0(s) \\text{~otherwise}$ for all $c \\in C$ (i), (ii) above combined match the definition of $A_{25}$ $X_{9k+9}$ does not contain $could\\_not\\_have(t,k)$ , when $prenabled(t,k) \\in X_0 \\cup \\dots \\cup X_{9k+6}$ and $fires(t,k) \\notin X_0 \\cup \\dots \\cup X_{9k+5}$ due to construction of $A$ , encoding of $a\\ref {a:c:maxfire:cnh}$ and its body atoms.", "As a result it is not eliminated by the constraint $a\\ref {a:c:maxfire:elim}$ $eval_{U_{9k+9}}(bot_{U_{9k+10}}(\\Pi ^6) \\setminus bot_{U_{9k+9}}(\\Pi ^6), X_0 \\cup \\dots \\cup X_{9k+9}) = \\lbrace consumesmore \\text{:-}.", "\| \\\\ \\lbrace consumesmore(p,k) \\rbrace \\subseteq A \\rbrace $ .", "It's answer set is $X_{9k+10} = A_{20}$ – using forced atom proposition and the definition of $A_{20}$ $X_{9k+10}$ will be empty since none of $consumesmore(p,0),\\dots , \\\\ consumesmore(p,k)$ hold in $X_0 \\cup \\dots \\cup X_{9k+9}$ due to the construction of $A$ , encoding of $a\\ref {a:c:overc:place}$ and its body atoms.", "As a result, it is not eliminated by the constraint $a\\ref {a:c:overc:elim}$ The set $X = X_0 \\cup \\dots \\cup X_{9k+10}$ is the answer set of $\\Pi ^6$ by the splitting sequence theorem REF .", "Each $X_i, 0 \\le i \\le 9k+10$ matches a distinct portion of $A$ , and $X = A$ , thus $A$ is an answer set of $\\Pi ^6$ .", "Next we show (REF ): Given $\\Pi ^6$ be the encoding of a Petri Net $PN(P,T,E,C,W,R,I,$ $Q,QW,Z)$ with initial marking $M_0$ , and $A$ be an answer set of $\\Pi ^6$ that satisfies (REF ) and (REF ), then we can construct $X=M_0,T_k,\\dots ,M_k,T_k,M_{k+1}$ from $A$ , such that it is an execution sequence of $PN$ .", "We construct the $X$ as follows: $M_i = (M_i(p_0), \\dots , M_i(p_n))$ , where $\\lbrace holds(p_0,m_{M_i(p_0)}(c),c,i), \\dots ,\\\\ holds(p_n,m_{M_i(p_n)}(c),c,i) \\rbrace \\subseteq A$ , for $c \\in C, 0 \\le i \\le k+1$ $T_i = \\lbrace t : fires(t,i) \\in A\\rbrace $ , for $0 \\le i \\le k$ and show that $X$ is indeed an execution sequence of $PN$ .", "We show this by induction over $k$ (i.e.", "given $M_k$ , $T_k$ is a valid firing set and its firing produces marking $M_{k+1}$ ).", "Base case: Let $k=0$ , and $M_0$ is a valid marking in $X$ for $PN$ , show [(1)] $T_0$ is a valid firing set for $M_0$ , and firing $T_0$ in $M_0$ produces marking $M_1$ .", "We show $T_0$ is a valid firing set for $M_0$ .", "Let $\\lbrace fires(t_0,0), \\dots , fires(t_x,0) \\rbrace $ be the set of all $fires(\\dots ,0)$ atoms in $A$ , Then for each $fires(t_i,0) \\in A$ $prenabled(t_i,0) \\in A$ – from rule $a\\ref {a:c:prfires}$ and supported rule proposition Then $enabled(t_i,0) \\in A$ – from rule $a\\ref {a:c:prenabled}$ and supported rule proposition And $notprenabled(t_i,0) \\notin A$ – from rule $a\\ref {a:c:prenabled}$ and supported rule proposition For $enabled(t_i,0) \\in A$ $notenabled(t_i,0) \\notin A$ – from rule $e\\ref {e:c:enabled}$ and supported rule proposition Then either of $body(e\\ref {e:c:ne:ptarc})$ , $body(e\\ref {e:c:ne:iptarc})$ , or $body(e\\ref {e:c:ne:tptarc})$ must not hold in $A$ for $t_i$ – from rules $e\\ref {e:c:ne:ptarc},e\\ref {e:c:ne:iptarc},e\\ref {e:c:ne:tptarc}$ and forced atom proposition Then $q_c \\lnot < {n_i}_c \\equiv q_c \\ge {n_i}_c$ in $e\\ref {e:c:ne:ptarc}$ for all $\\lbrace holds(p,q_c,c,0), \\\\ ptarc(p,t_i,{n_i}_c,c,0)\\rbrace \\subseteq A$ – from $e:\\ref {e:c:ne:ptarc}$ , forced atom proposition, and given facts ($holds(p,q_c,c,0) \\in A, ptarc(p,t_i,{n_i}_c,0) \\in A$ ) And $q_c \\lnot \\ge {n_i}_c \\equiv q_c < {n_i}_c$ in $e\\ref {e:c:ne:iptarc}$ for all $\\lbrace holds(p,q_c,c,0), \\\\ iptarc(p,t_i,{n_i}_c,c,0) \\rbrace \\subseteq A, {n_i}_c=1$ ; $q_c > {n_i}_c \\equiv q_c = 0$ – from $e\\ref {e:c:ne:iptarc}$ , forced atom proposition, given facts ($holds(p,q_c,c,0) \\in A, \\\\ iptarc(p,t_i,1,c,0) \\in A$ ), and $q_c$ is a positive integer And $q_c \\lnot < {n_i}_c \\equiv q_c \\ge {n_i}_c$ in $e\\ref {e:c:ne:tptarc}$ for all $\\lbrace holds(p,q_c,c,0), \\\\ tptarc(p,t_i,{n_i}_c,c,0) \\rbrace \\subseteq A$ – from $e\\ref {e:c:ne:tptarc}$ , forced atom proposition, and given facts Then $\\forall c \\in C, (\\forall p \\in \\bullet t_i, m_{M_0(p)}(c) \\ge m_{W(p,t_i)}(c)) \\wedge (\\forall p \\in I(t_i), \\\\ m_{M_0(p)}(c) = 0) \\wedge (\\forall (p,t_i) \\in Q, m_{M_0(p)}(c) \\ge m_{QW(p,t_i)}(c))$ – from $i\\ref {i:c:init},f\\ref {f:c:ptarc},f\\ref {f:c:iptarc},f\\ref {f:c:tptarc}$ construction, definition REF of preset $\\bullet t_i$ in $PN$ , definition REF of enabled transition in $PN$ , and that the construction of reset arcs by $f\\ref {f:c:rptarc}$ ensures $notenabled(t,0)$ is never true for a reset arc, where $holds(p,q_c,c,0) \\in A$ represents $q_c=m_{M_0(p)}(c)$ , $ptarc(p,t_i,{n_i}_c,0) \\in A$ represents ${n_i}_c=m_{W(p,t_i)}(c)$ , ${n_i}_c = m_{M_0(p)}(c)$ .", "Which is equivalent to $(\\forall p \\in \\bullet t_i, M_0(p) \\ge W(p,t_i)) \\wedge (\\forall p \\in I(t_i), M_0(p) = 0) \\wedge (\\forall (p,t_i) \\in Q, M_0(p) \\ge QW(p,t_i))$ – assuming multiset domain $C$ For $notprenabled(t_i,0) \\notin A$ Either $(\\nexists enabled(tt,0) \\in A : pp < p_i)$ or $(\\forall enabled(tt,0) \\in A : pp \\lnot < p_i)$ where $pp = Z(tt), p_i = Z(t_i)$ – from rule $a\\ref {a:c:prne},f\\ref {f:c:pr}$ and forced atom proposition This matches the definition of an enabled priority transition Then $t_i$ is enabled and can fire in $PN$ , as a result it can belong to $T_0$ – from definition REF of enabled transition And $consumesmore \\notin A$ , since $A$ is an answer set of $\\Pi ^6$ – from rule $a\\ref {a:c:overc:gen}$ and supported rule proposition Then $\\nexists consumesmore(p,0) \\in A$ – from rule $a\\ref {a:c:overc:place}$ and supported rule proposition Then $\\nexists \\lbrace holds(p,q_c,c,0), tot\\_decr(p,q1_c,c,0) \\rbrace \\subseteq A, q1_c>q_c$ in $body(a\\ref {a:c:overc:place})$ – from $a\\ref {a:c:overc:place}$ and forced atom proposition Then $\\nexists c \\in C \\nexists p \\in P, (\\sum _{t_i \\in \\lbrace t_0,\\dots ,t_x\\rbrace , p \\in \\bullet t_i}{m_{W(p,t_i)}(c)}+ \\\\ \\sum _{t_i \\in \\lbrace t_0,\\dots ,t_x\\rbrace , p \\in R(t_i)}{m_{M_0(p)}(c)}) > m_{M_0(p)}(c)$ – from the following $holds(p,q_c,c,0)$ represents $q_c=m_{M_0(p)}(c)$ – from rule $i\\ref {i:c:init}$ encoding, given $tot\\_decr(p,q1_c,c,0) \\in A$ if $\\lbrace del(p,{q1_0}_c,t_0,c,0), \\dots , \\\\ del(p,{q1_x}_c,t_x,c,0) \\rbrace \\subseteq A$ , where $q1_c = {q1_0}_c+\\dots +{q1_x}_c$ – from $r\\ref {r:c:totdecr}$ and forced atom proposition $del(p,{q1_i}_c,t_i,c,0) \\in A$ if $\\lbrace fires(t_i,0), ptarc(p,t_i,{q1_i}_c,c,0) \\rbrace \\subseteq A$ – from $r\\ref {r:c:del}$ and supported rule proposition $del(p,{q1_i}_c,t_i,c,0)$ either represents removal of ${q1_i}_c = m_{W(p,t_i)}(c)$ tokens from $p \\in \\bullet t_i$ ; or it represents removal of ${q1_i}_c = m_{M_0(p)}(c)$ tokens from $p \\in R(t_i)$ – from rules $r\\ref {r:c:del},f\\ref {f:c:ptarc},f\\ref {f:c:rptarc}$ , supported rule proposition, and definition REF of transition execution in $PN$ Then the set of transitions in $T_0$ do not conflict – by the definition REF of conflicting transitions And for each $prenabled(t_j,0) \\in A$ and $fires(t_j,0) \\notin A$ , $could\\_not\\_have(t_j,0) \\in A$ , since $A$ is an answer set of $\\Pi ^6$ - from rule $a\\ref {a:c:prmaxfire:elim}$ and supported rule proposition Then $\\lbrace prenabled(t_j,0), holds(s,qq_c,c,0), ptarc(s,t_j,q_c,c,0), \\\\ tot\\_decr(s,qqq_c,c,0) \\rbrace \\subseteq A$ , such that $q_c > qq_c - qqq_c$ and $fires(t_j,0) \\notin A$ - from rule $a\\ref {a:c:prmaxfire:cnh}$ and supported rule proposition Then for an $s \\in \\bullet t_j \\cup R(t_j)$ , $q_c > m_{M_0(s)}(c) - (\\sum _{t_i \\in T_0, s \\in \\bullet t_i}{m_{W(s,t_i)}(c)} + \\sum _{t_i \\in T_0, s \\in R(t_i)}{m_{M_0(s)}(c))}$ , where $q_c=m_{W(s,t_j)}(c) \\text{~if~} s \\in \\bullet t_j, \\text{~or~} m_{M_0(s)}(c)$ $ \\text{~otherwise}$ – from the following $ptarc(s,t_i,q_c,c,0)$ represents $q_c=m_{W(s,t_i)}(c)$ if $(s,t_i) \\in E^-$ or $q_c=m_{M_0(s)}(c)$ if $s \\in R(t_i)$ – from rule $f\\ref {f:c:ptarc},f\\ref {f:c:rptarc}$ construction $holds(s,qq_c,c,0)$ represents $qq_c=m_{M_0(s)}(c)$ – from $i\\ref {i:c:init}$ construction $tot\\_decr(s,qqq_c,c,0) \\in A$ if $\\lbrace del(s,{qqq_0}_c,t_0,c,0), \\dots , \\\\ del(s,{qqq_x}_c,t_x,c,0) \\rbrace \\subseteq A$ – from rule $r\\ref {r:c:totdecr}$ construction and supported rule proposition $del(s,{qqq_i}_c,t_i,c,0) \\in A$ if $\\lbrace fires(t_i,0), ptarc(s,t_i,{qqq_i}_c,c,0) \\rbrace \\subseteq A$ – from rule $r\\ref {r:c:del}$ and supported rule proposition $del(s,{qqq_i}_c,t_i,c,0)$ either represents ${qqq_i}_c = m_{W(s,t_i)}(c) : t_i \\in T_0, (s,t_i) \\in E^-$ , or ${qqq_i}_c = m_{M_0(t_i)}(c) : t_i \\in T_0, s \\in R(t_i)$ – from rule $f\\ref {f:c:ptarc},f\\ref {f:c:rptarc}$ construction $tot\\_decr(q,qqq_c,c,0)$ represents $\\sum _{t_i \\in T_0, s \\in \\bullet t_i}{m_{W(s,t_i)}(c)} + \\\\ \\sum _{t_i \\in T_0, s \\in R(t_i)}{m_{M_0(s)}(c)}$ – from (C,D,E) above Then firing $T_0 \\cup \\lbrace t_j \\rbrace $ would have required more tokens than are present at its source place $s \\in \\bullet t_j \\cup R(t_j)$ .", "Thus, $T_0$ is a maximal set of transitions that can simultaneously fire.", "And for each reset transition $t_r$ with $prenabled(t_r,0) \\in A$ , $fires(t_r,0) \\in A$ , since $A$ is an answer set of $\\Pi ^6$ - from rule $f\\ref {f:c:pr:rptarc:elim}$ and supported rule proposition Then, the firing set $T_0$ satisfies the reset-transition requirement of definition REF (firing set) Then $\\lbrace t_0, \\dots , t_x\\rbrace = T_0$ – using 1(a),1(b),1(d) above; and using 1(c) it is a maximal firing set We show $M_1$ is produced by firing $T_0$ in $M_0$ .", "Let $holds(p,q_c,c,1) \\in A$ Then $\\lbrace holds(p,q1_c,c,0), tot\\_incr(p,q2_c,c,0), tot\\_decr(p,q3_c,c,0) \\rbrace \\subseteq A : q_c=q1_c+q2_c-q3_c$ – from rule $r\\ref {r:c:nextstate}$ and supported rule proposition Then, $holds(p,q1_c,c,0) \\in A$ represents $q1_c=m_{M_0(p)}(c)$ – given, construction; and $\\lbrace add(p,{q2_0}_c,t_0,c,0), \\dots , $ $add(p,{q2_j}_c,t_j,c,0)\\rbrace \\subseteq A : {q2_0}_c + \\dots + {q2_j}_c = q2_c$ and $\\lbrace del(p,{q3_0}_c,t_0,c,0), \\dots , $ $del(p,{q3_l}_c,t_l,c,0)\\rbrace \\subseteq A : {q3_0}_c + \\dots + {q3_l}_c = q3_c$ – rules $r\\ref {r:c:totincr},r\\ref {r:c:totdecr}$ and supported rule proposition, respectively Then $\\lbrace fires(t_0,0), \\dots , fires(t_j,0) \\rbrace \\subseteq A$ and $\\lbrace fires(t_0,0), \\dots , fires(t_l,0) \\rbrace \\subseteq A$ – rules $r\\ref {r:c:add},r\\ref {r:c:del}$ and supported rule proposition, respectively Then $\\lbrace fires(t_0,0), \\dots , $ $fires(t_j,0) \\rbrace \\cup $ $\\lbrace fires(t_0,0), \\dots , $ $fires(t_l,0) \\rbrace \\subseteq $ $A = \\lbrace fires(t_0,0), \\dots , $ $fires(t_x,0) \\rbrace \\subseteq A$ – set union of subsets Then for each $fires(t_x,0) \\in A$ we have $t_x \\in T_0$ – already shown in item REF above Then $q_c = m_{M_0(p)}(c) + \\sum _{t_x \\in T_0 \\wedge p \\in t_x \\bullet }{m_{W(t_x,p)}(c)} - (\\sum _{t_x \\in T_0 \\wedge p \\in \\bullet t_x}{m_{W(p,t_x)}(c)} + \\sum _{t_x \\in T_0 \\wedge p \\in R(t_x)}{m_{M_0(p)}(c)})$ – from (REF ) above and the following Each $add(p,{q_j}_c,t_j,c,0) \\in A$ represents ${q_j}_c=m_{W(t_j,p)}(c)$ for $p \\in t_j \\bullet $ – rule $r\\ref {r:c:add},f\\ref {f:c:tparc}$ encoding, and definition REF of transition execution in $PN$ Each $del(p,t_y,{q_y}_c,c,0) \\in A$ represents either ${q_y}_c=m_{W(p,t_y)}(c)$ for $p \\in \\bullet t_y$ , or ${q_y}_c=m_{M_0(p)}(c)$ for $p \\in R(t_y)$ – from rule $r\\ref {r:c:del},f\\ref {f:c:ptarc}$ encoding and definition REF of transition execution in $PN$ ; or from rule $r\\ref {r:c:del},f\\ref {f:c:rptarc}$ encoding and definition of reset arc in $PN$ Each $tot\\_incr(p,q2_c,c,0) \\in A$ represents $q2_c=\\sum _{t_x \\in T_0 \\wedge p \\in t_x \\bullet }{m_{W(t_x,p)}(c)}$ – aggregate assignment atom semantics in rule $r\\ref {r:c:totincr}$ Each $tot\\_decr(p,q3_c,c,0) \\in A$ represents $q3_c=\\sum _{t_x \\in T_0 \\wedge p \\in \\bullet t_x}{m_{W(p,t_x)}(c)} + $ $\\sum _{t_x \\in T_0 \\wedge p \\in R(t_x)}{m_{M_0(p)}(c)}$ – aggregate assignment atom semantics in rule $r\\ref {r:c:totdecr}$ Then, $m_{M_1(p)}(c) = q_c$ – since $holds(p,q_c,c,1) \\in A$ encodes $q_c=m_{M_1(p)}(c)$ – from construction Inductive Step: Let $k > 0$ , and $M_k$ is a valid marking in $X$ for $PN$ , show [(1)] $T_k$ is a valid firing set for $M_k$ , and firing $T_k$ in $M_k$ produces marking $M_{k+1}$ .", "We show that $T_k$ is a valid firing set in $M_k$ .", "Let $\\lbrace fires(T_k,k), \\dots , fires(t_x,k) \\rbrace $ be the set of all $fires(\\dots ,k)$ atoms in $A$ , Then for each $fires(t_i,k) \\in A$ $prenabled(t_i,k) \\in A$ – from rule $a\\ref {a:c:prfires}$ and supported rule proposition Then $enabled(t_i,k) \\in A$ – from rule $a\\ref {a:c:prenabled}$ and supported rule proposition And $notprenabled(t_i,k) \\notin A$ – from rule $a\\ref {a:c:prenabled}$ and supported rule proposition For $enabled(t_i,k) \\in A$ $notenabled(t_i,k) \\notin A$ – from rule $e\\ref {e:c:enabled}$ and supported rule proposition Then either of $body(e\\ref {e:c:ne:ptarc})$ , $body(e\\ref {e:c:ne:iptarc})$ , or $body(e\\ref {e:c:ne:tptarc})$ must not hold in $A$ for $t_i$ – from rules $e\\ref {e:c:ne:ptarc},e\\ref {e:c:ne:iptarc},e\\ref {e:c:ne:tptarc}$ and forced atom proposition Then $q_c \\lnot < {n_i}_c \\equiv q_c \\ge {n_i}_c$ in $e\\ref {e:c:ne:ptarc}$ for all $\\lbrace holds(p,q_c,c,k), \\\\ ptarc(p,t_i,{n_i}_c,c,k)\\rbrace \\subseteq A$ – from $e\\ref {e:c:ne:ptarc}$ , forced atom proposition, and given facts ($holds(p,q_c,c,k) \\in A, ptarc(p,t_i,{n_i}_c,k) \\in A$ ) And $q_c \\lnot \\ge {n_i}_c \\equiv q_c < {n_i}_c$ in $e\\ref {e:c:ne:iptarc}$ for all $\\lbrace holds(p,q_c,c,k), \\\\ iptarc(p,t_i,{n_i}_c,c,k) \\rbrace \\subseteq A, {n_i}_c=1$ ; $q_c > {n_i}_c \\equiv q_c = 0$ – from $e\\ref {e:c:ne:iptarc}$ , forced atom proposition, given facts ($holds(p,q_c,c,k) \\in A, \\\\ iptarc(p,t_i,1,c,k) \\in A$ ), and $q_c$ is a positive integer And $q_c \\lnot < {n_i}_c \\equiv q_c \\ge {n_i}_c$ in $e\\ref {e:c:ne:tptarc}$ for all $\\lbrace holds(p,q_c,c,k), \\\\ tptarc(p,t_i,{n_i}_c,c,k) \\rbrace \\subseteq A$ – from $e\\ref {e:c:ne:tptarc}$ , forced atom proposition, and given facts Then $\\forall c \\in C, (\\forall p \\in \\bullet t_i, m_{M_k(p)}(c) \\ge m_{W(p,t_i)}(c)) \\wedge (\\forall p \\in I(t_i), $ $m_{M_k(p)}(c) = k) \\wedge (\\forall (p,t_i) \\in Q, m_{M_k(p)}(c) \\ge m_{QW(p,t_i)}(c))$ – from $f\\ref {f:c:ptarc},f\\ref {f:c:iptarc},f\\ref {f:c:tptarc}$ construction, inductive assumption, definition REF of preset $\\bullet t_i$ in $PN$ , definition REF of enabled transition in $PN$ , and that the construction of reset arcs by $f\\ref {f:c:rptarc}$ ensures $notenabled(t,k)$ is never true for a reset arc, where $holds(p,q_c,c,k) \\in A$ represents $q_c=m_{M_k(p)}(c)$ , $ptarc(p,t_i,{n_i}_c,k) \\in A$ represents ${n_i}_c=m_{W(p,t_i)}(c)$ , ${n_i}_c = m_{M_k(p)}(c)$ .", "Which is equivalent to $(\\forall p \\in \\bullet t_i, M_k(p) \\ge W(p,t_i)) \\wedge (\\forall p \\in I(t_i), M_k(p) = k) \\wedge (\\forall (p,t_i) \\in Q, M_k(p) \\ge QW(p,t_i))$ – assuming multiset domain $C$ For $notprenabled(t_i,k) \\notin A$ Either $(\\nexists enabled(tt,k) \\in A : pp < p_i)$ or $(\\forall enabled(tt,k) \\in A : pp \\lnot < p_i)$ where $pp = Z(tt), p_i = Z(t_i)$ – from rule $a\\ref {a:c:prne}, f\\ref {f:c:pr}$ and forced atom proposition This matches the definition of an enabled priority transition Then $t_i$ is enabled and can fire in $PN$ , as a result it can belong to $T_k$ – from definition REF of enabled transition And $consumesmore \\notin A$ , since $A$ is an answer set of $\\Pi ^6$ – from rule $a\\ref {a:c:overc:elim}$ and supported rule proposition Then $\\nexists consumesmore(p,k) \\in A$ – from rule $a\\ref {a:c:overc:elim}$ and supported rule proposition Then $\\nexists \\lbrace holds(p,q_c,c,k), tot\\_decr(p,q1_c,c,k) \\rbrace \\subseteq A, q1_c>q_c$ in $body(a\\ref {a:c:overc:place})$ – from $a\\ref {a:c:overc:place}$ and forced atom proposition Then $\\nexists c \\in C \\nexists p \\in P, (\\sum _{t_i \\in \\lbrace T_k,\\dots ,t_x\\rbrace , p \\in \\bullet t_i}{m_{W(p,t_i)}(c)}+ \\\\ \\sum _{t_i \\in \\lbrace T_k,\\dots ,t_x\\rbrace , p \\in R(t_i)}{m_{M_k(p)}(c)}) > m_{M_k(p)}(c)$ – from the following $holds(p,q_c,c,k)$ represents $q_c=m_{M_k(p)}(c)$ – from inductive assumption and construction, given $tot\\_decr(p,q1_c,c,k) \\in A$ if $\\lbrace del(p,{q1_0}_c,T_k,c,k), \\dots , \\\\ del(p,{q1_x}_c,t_x,c,k) \\rbrace \\subseteq A$ , where $q1_c = {q1_0}_c+\\dots +{q1_x}_c$ – from $r\\ref {r:c:totdecr}$ and forced atom proposition $del(p,{q1_i}_c,t_i,c,k) \\in A$ if $\\lbrace fires(t_i,k), ptarc(p,t_i,{q1_i}_c,c,k) \\rbrace \\subseteq A$ – from $r\\ref {r:c:del}$ and supported rule proposition $del(p,{q1_i}_c,t_i,c,k)$ either represents removal of ${q1_i}_c = m_{W(p,t_i)}(c)$ tokens from $p \\in \\bullet t_i$ ; or it represents removal of ${q1_i}_c = m_{M_k(p)}(c)$ tokens from $p \\in R(t_i)$ – from rules $r\\ref {r:c:del},f\\ref {f:c:ptarc},f\\ref {f:c:rptarc}$ , supported rule proposition, and definition REF of transition execution in $PN$ Then the set of transitions in $T_k$ do not conflict – by the definition REF of conflicting transitions And for each $prenabled(t_j,k) \\in A$ and $fires(t_j,k) \\notin A$ , $could\\_not\\_have(t_j,k) \\in A$ , since $A$ is an answer set of $\\Pi ^6$ - from rule $a\\ref {a:c:prmaxfire:elim}$ and supported rule proposition Then $\\lbrace prenabled(t_j,k), holds(s,qq_c,c,k), ptarc(s,t_j,q_c,c,k), \\\\ tot\\_decr(s,qqq_c,c,k) \\rbrace \\subseteq A$ , such that $q_c > qq_c - qqq_c$ and $fires(t_j,k) \\notin A$ - from rule $a\\ref {a:c:prmaxfire:cnh}$ and supported rule proposition Then for an $s \\in \\bullet t_j \\cup R(t_j)$ , $q_c > m_{M_k(s)}(c) - (\\sum _{t_i \\in T_k, s \\in \\bullet t_i}{m_{W(s,t_i)}(c)} + \\sum _{t_i \\in T_k, s \\in R(t_i)}{m_{M_k(s)}(c))}$ , where $q_c=m_{W(s,t_j)}(c) \\text{~if~} s \\in \\bullet t_j, \\text{~or~} m_{M_k(s)}(c) $ $\\text{~otherwise}$ .", "$ptarc(s,t_i,q_c,c,k)$ represents $q_c=m_{W(s,t_i)}(c)$ if $(s,t_i) \\in E^-$ or $q_c=m_{M_k(s)}(c)$ if $s \\in R(t_i)$ – from rule $f\\ref {f:c:ptarc},f\\ref {f:c:rptarc}$ construction $holds(s,qq_c,c,k)$ represents $qq_c=m_{M_k(s)}(c)$ – from inductive assumption and construction $tot\\_decr(s,qqq_c,c,k) \\in A$ if $\\lbrace del(s,{qqq_0}_c,T_k,c,k), \\dots , \\\\ del(s,{qqq_x}_c,t_x,c,k) \\rbrace \\subseteq A$ – from rule $r\\ref {r:c:totdecr}$ construction and supported rule proposition $del(s,{qqq_i}_c,t_i,c,k) \\in A$ if $\\lbrace fires(t_i,k), ptarc(s,t_i,{qqq_i}_c,c,k) \\rbrace \\subseteq A$ – from rule $r\\ref {r:c:del}$ and supported rule proposition $del(s,{qqq_i}_c,t_i,c,k)$ either represents ${qqq_i}_c = m_{W(s,t_i)}(c) : t_i \\in T_k, (s,t_i) \\in E^-$ , or ${qqq_i}_c = m_{M_k(t_i)}(c) : t_i \\in T_k, s \\in R(t_i)$ – from rule $f\\ref {f:c:ptarc},f\\ref {f:c:rptarc}$ construction $tot\\_decr(q,qqq_c,c,k)$ represents $\\sum _{t_i \\in T_k, s \\in \\bullet t_i}{m_{W(s,t_i)}(c)} + \\\\ \\sum _{t_i \\in T_k, s \\in R(t_i)}{m_{M_k(s)}(c)}$ – from (C,D,E) above Then firing $T_k \\cup \\lbrace t_j \\rbrace $ would have required more tokens than are present at its source place $s \\in \\bullet t_j \\cup R(t_j)$ .", "Thus, $T_k$ is a maximal set of transitions that can simultaneously fire.", "And for each reset transition $t_r$ with $prenabled(t_r,k) \\in A$ , $fires(t_r,k) \\in A$ , since $A$ is an answer set of $\\Pi ^6$ - from rule $f\\ref {f:c:pr:rptarc:elim}$ and supported rule proposition Then the firing set $T_k$ satisfies the reset transition requirement of definition REF (firing set) Then $\\lbrace t_0, \\dots , t_x\\rbrace = T_k$ – using 1(a),1(b), 1(d) above; and using 1(c) it is a maximal firing set We show that $M_{k+1}$ is produced by firing $T_k$ in $M_k$ .", "Let $holds(p,q_c,c,k+1) \\in A$ Then $\\lbrace holds(p,q1_c,c,k), tot\\_incr(p,q2_c,c,k), tot\\_decr(p,q3_c,c,k) \\rbrace \\subseteq A : q_c=q1_c+q2_c-q3_c$ – from rule $r\\ref {r:c:nextstate}$ and supported rule proposition Then, $holds(p,q1_c,c,k) \\in A$ represents $q1_c=m_{M_k(p)}(c)$ – inductive assumption; and $\\lbrace add(p,{q2_0}_c,T_k,c,k), \\dots , $ $add(p,{q2_j}_c,t_j,c,k)\\rbrace \\subseteq A : {q2_0}_c + \\dots + {q2_j}_c = q2_c$ and $\\lbrace del(p,{q3_0}_c,T_k,c,k), \\dots , $ $del(p,{q3_l}_c,t_l,c,k)\\rbrace \\subseteq A : {q3_0}_c + \\dots + {q3_l}_c = q3_c$ – rules $r\\ref {r:c:totincr},r\\ref {r:c:totdecr}$ and supported rule proposition, respectively Then $\\lbrace fires(T_k,k), \\dots , fires(t_j,k) \\rbrace \\subseteq A$ and $\\lbrace fires(T_k,k), \\dots , \\\\ fires(t_l,k) \\rbrace \\subseteq A$ – rules $r\\ref {r:c:add},r\\ref {r:c:del}$ and supported rule proposition, respectively Then $\\lbrace fires(T_k,k), \\dots , $ $fires(t_j,k) \\rbrace \\cup $ $\\lbrace fires(T_k,k), \\dots , $ $fires(t_l,k) \\rbrace \\subseteq $ $A = \\lbrace fires(T_k,k), \\dots , fires(t_x,k) \\rbrace \\subseteq A$ – set union of subsets Then for each $fires(t_x,k) \\in A$ we have $t_x \\in T_k$ – already shown in item REF above Then $q_c = m_{M_k(p)}(c) + \\sum _{t_x \\in T_k \\wedge p \\in t_x \\bullet }{m_{W(t_x,p)}(c)} - (\\sum _{t_x \\in T_k \\wedge p \\in \\bullet t_x}{m_{W(p,t_x)}(c)} + \\sum _{t_x \\in T_k \\wedge p \\in R(t_x)}{m_{M_k(p)}(c)})$ – from (REF ) above and the following Each $add(p,{q_j}_c,t_j,c,k) \\in A$ represents ${q_j}_c=m_{W(t_j,p)}(c)$ for $p \\in t_j \\bullet $ – rule $r\\ref {r:c:add},f\\ref {f:c:tparc}$ encoding, and definition REF of transition execution in $PN$ Each $del(p,t_y,{q_y}_c,c,k) \\in A$ represents either ${q_y}_c=m_{W(p,t_y)}(c)$ for $p \\in \\bullet t_y$ , or ${q_y}_c=m_{M_k(p)}(c)$ for $p \\in R(t_y)$ – from rule $r\\ref {r:c:del},f\\ref {f:c:ptarc}$ encoding and definition REF of transition execution in $PN$ ; or from rule $r\\ref {r:c:del},f\\ref {f:c:rptarc}$ encoding and definition of reset arc in $PN$ Each $tot\\_incr(p,q2_c,c,k) \\in A$ represents $q2_c=\\sum _{t_x \\in T_k \\wedge p \\in t_x \\bullet }{m_{W(t_x,p)}(c)}$ – aggregate assignment atom semantics in rule $r\\ref {r:c:totincr}$ Each $tot\\_decr(p,q3_c,c,k) \\in A$ represents $q3_c=\\sum _{t_x \\in T_k \\wedge p \\in \\bullet t_x}{m_{W(p,t_x)}(c)} + \\sum _{t_x \\in T_k \\wedge p \\in R(t_x)}{m_{M_k(p)}(c)}$ – aggregate assignment atom semantics in rule $r\\ref {r:c:totdecr}$ Then, $m_{M_{k+1}(p)}(c) = q_c$ – since $holds(p,q_c,c,k+1) \\in A$ encodes $q_c=m_{M_{k+1}(p)}(c)$ – from construction As a result, for any $n > k$ , $T_n$ will be a valid firing set for $M_n$ and $M_{n+1}$ will be its target marking.", "Conclusion: Since both (REF ) and (REF ) hold, $X=M_0,T_k,M_1,\\dots ,M_k,T_{k+1}$ is an execution sequence of $PN(P,T,E,C,W,R,I,Q,QW,Z)$ (w.r.t $M_0$ ) iff there is an answer set $A$ of $\\Pi ^6(PN,M_0,k,ntok)$ such that (REF ) and (REF ) hold.", "Proof of Proposition REF Let $PN=(P,T,E,C,W,R,I,Q,QW,Z,D)$ be a Petri Net, $M_0$ be its initial marking and let $\\Pi ^7(PN,M_0,k,ntok)$ be the ASP encoding of $PN$ and $M_0$ over a simulation length $k$ , with maximum $ntok$ tokens on any place node, as defined in section REF .", "Then $X=M_0,T_k,M_1,\\dots ,M_k,T_k,M_{k+1}$ is an execution sequence of $PN$ (w.r.t.", "$M_0$ ) iff there is an answer set $A$ of $\\Pi ^7(PN,M_0,k,ntok)$ such that: $\\lbrace fires(t,ts) : t \\in T_{ts}, 0 \\le ts \\le k\\rbrace = \\lbrace fires(t,ts) : fires(t,ts) \\in A \\rbrace $ $\\begin{split}\\lbrace holds(p,q,c,ts) &: p \\in P, c/q = M_{ts}(p), 0 \\le ts \\le k+1 \\rbrace \\\\&= \\lbrace holds(p,q,c,ts) : holds(p,q,c,ts) \\in A \\rbrace \\end{split}$ We prove this by showing that: Given an execution sequence $X$ , we create a set $A$ such that it satisfies (REF ) and (REF ) and show that $A$ is an answer set of $\\Pi ^7$ Given an answer set $A$ of $\\Pi ^7$ , we create an execution sequence $X$ such that (REF ) and (REF ) are satisfied.", "First we show (REF ): We create a set $A$ as a union of the following sets: $A_1=\\lbrace num(n) : 0 \\le n \\le ntok \\rbrace $ $A_2=\\lbrace time(ts) : 0 \\le ts \\le k\\rbrace $ $A_3=\\lbrace place(p) : p \\in P \\rbrace $ $A_4=\\lbrace trans(t) : t \\in T \\rbrace $ $A_5=\\lbrace color(c) : c \\in C \\rbrace $ $A_6=\\lbrace ptarc(p,t,n_c,c,ts) : (p,t) \\in E^-, c \\in C, n_c=m_{W(p,t)}(c), n_c > 0, 0 \\le ts \\le k \\rbrace $ , where $E^- \\subseteq E$ $A_7=\\lbrace tparc(t,p,n_c,c,ts,d) : (t,p) \\in E^+, c \\in C, n_c=m_{W(t,p)}(c), n_c > 0, d=D(t), 0 \\le ts \\le k \\rbrace $ , where $E^+ \\subseteq E$ $A_8=\\lbrace holds(p,q_c,c,0) : p \\in P, c \\in C, q_c=m_{M_{0}(p)}(c) \\rbrace $ $A_9=\\lbrace ptarc(p,t,n_c,c,ts) : p \\in R(t), c \\in C, n_c = m_{M_{ts}(p)}, n_c > 0, 0 \\le ts \\le k \\rbrace $ $A_{10}=\\lbrace iptarc(p,t,1,c,ts) : p \\in I(t), c \\in C, 0 \\le ts < k \\rbrace $ $A_{11}=\\lbrace tptarc(p,t,n_c,c,ts) : (p,t) \\in Q, c \\in C, n_c=m_{QW(p,t)}(c), n_c > 0, 0 \\le ts \\le k \\rbrace $ $A_{12}=\\lbrace notenabled(t,ts) : t \\in T, 0 \\le ts \\le k, \\exists c \\in C, (\\exists p \\in \\bullet t, m_{M_{ts}(p)}(c) < m_{W(p,t)}(c)) \\vee (\\exists p \\in I(t), m_{M_{ts}(p)}(c) > 0) \\vee (\\exists (p,t) \\in Q, m_{M_{ts}(p)}(c) < m_{QW(p,t)}(c)) \\rbrace $ per definition REF (enabled transition) $A_{13}=\\lbrace enabled(t,ts) : t \\in T, 0 \\le ts \\le k, \\forall c \\in C, (\\forall p \\in \\bullet t, m_{W(p,t)}(c) \\le m_{M_{ts}(p)}(c)) \\wedge (\\forall p \\in I(t), m_{M_{ts}(p)}(c) = 0) \\wedge (\\forall (p,t) \\in Q, m_{M_{ts}(p)}(c) \\ge m_{QW(p,t)}(c)) \\rbrace $ per definition REF (enabled transition) $A_{14}=\\lbrace fires(t,ts) : t \\in T_{ts}, 0 \\le ts \\le k \\rbrace $ per definition REF (enabled transitions), only an enabled transition may fire $A_{15}=\\lbrace add(p,q_c,t,c,ts+d-1) : t \\in T_{ts}, p \\in t \\bullet , c \\in C, q_c=m_{W(t,p)}(c), \\\\d = D(t), 0 \\le ts \\le k \\rbrace $ per definition REF (transition execution) $A_{16}=\\lbrace del(p,q_c,t,c,ts) : t \\in T_{ts}, p \\in \\bullet t, c \\in C, q_c=m_{W(p,t)}(c), 0 \\le ts \\le k \\rbrace \\cup \\lbrace del(p,q_c,t,c,ts) : t \\in T_{ts}, p \\in R(t), c \\in C, q_c=m_{M_{ts}(p)}(c), 0 \\le ts \\le k \\rbrace $ per definition REF (transition execution) $A_{17}=\\lbrace tot\\_incr(p,q_c,c,ts) : p \\in P, c \\in C, \\\\ q_c=\\sum _{t \\in T_{l}, p \\in t \\bullet , l \\le ts, l+D(t)=ts+1}{m_{W(t,p)}(c)}, 0 \\le ts \\le k \\rbrace $ per definition REF (firing set execution) $A_{18}=\\lbrace tot\\_decr(p,q_c,c,ts): p \\in P, c \\in C, q_c=\\sum _{t \\in T_{ts}, p \\in \\bullet t}{m_{W(p,t)}(c)}+ \\\\ \\sum _{t \\in T_{ts}, p \\in R(t) }{m_{M_{ts}(p)}(c)}, 0 \\le ts \\le k \\rbrace $ per definition REF (firing set execution) $A_{19}=\\lbrace consumesmore(p,ts) : p \\in P, c \\in C, q_c=m_{M_{ts}(p)}(c), \\\\ q1_c=\\sum _{t \\in T_{ts}, p \\in \\bullet t}{m_{W(p,t)}(c)} + \\sum _{t \\in T_{ts}, p \\in R(t)}{m_{M_{ts}(p)}(c)}, q1_c > q_c, 0 \\le ts \\le k \\rbrace $ per definition REF (conflicting transitions) $A_{20}=\\lbrace consumesmore : \\exists p \\in P, c \\in C, q_c=m_{M_{ts}(p)}(c), \\\\ q1_c=\\sum _{t \\in T_{ts}, p \\in \\bullet t}{m_{W(p,t)}(c)} + \\sum _{t \\in T_{ts}, p \\in R(t)}(m_{M_{ts}(p)}(c)), q1_c > q_c, 0 \\le ts \\le k \\rbrace $ per definition REF (conflicting transitions) $A_{21}=\\lbrace holds(p,q_c,c,ts+1) : p \\in P, c \\in C, q_c=m_{M_{ts+1}(p)}(c), 0 \\le ts < k\\rbrace $ , where $M_{ts+1}(p) = M_{ts}(p) - (\\sum _{\\begin{array}{c}t \\in T_{ts}, p \\in \\bullet t\\end{array}}{W(p,t)} + \\sum _{t \\in T_{ts}, p \\in R(t)}M_{ts}(p)) + \\\\ \\sum _{\\begin{array}{c}t \\in T_l, p \\in t \\bullet , l \\le ts, l+D(t)-1=ts\\end{array}}{W(t,p)}$ according to definition REF (firing set execution) $A_{22}=\\lbrace transpr(t,pr) : t \\in T, pr=Z(t) \\rbrace $ $A_{23}=\\lbrace notprenabled(t,ts) : t \\in T, enabled(t,ts) \\in A_{13}, (\\exists tt \\in T, enabled(tt,ts) \\in A_{13}, Z(tt) < Z(t)), 0 \\le ts \\le k \\rbrace \\\\ = \\lbrace notprenabled(t,ts) : t \\in T, (\\forall p \\in \\bullet t, W(p,t) \\le M_{ts}(p)), (\\forall p \\in I(t), M_{ts}(p) = 0), (\\forall (p,t) \\in Q, M_0(p) \\ge WQ(p,t)), \\exists tt \\in T, $ $(\\forall pp \\in \\bullet tt, W(pp,tt) \\le M_{ts}(pp)), $ $(\\forall pp \\in I(tt), M_{ts}(pp) = 0), (\\forall (pp,tt) \\in Q, M_{ts}(pp) \\ge QW(pp,tt)), Z(tt) < Z(t), 0 \\le ts \\le k \\rbrace $ $A_{24}=\\lbrace prenabled(t,ts) : t \\in T, enabled(t,ts) \\in A_{13}, (\\nexists tt \\in T: enabled(tt,ts) \\in A_{13}, Z(tt) < Z(t)), 0 \\le ts \\le k \\rbrace \\\\ = \\lbrace prenabled(t,ts) : t \\in T, (\\forall p \\in \\bullet t, W(p,t) \\le M_{ts}(p)), (\\forall p \\in I(t), M_{ts}(p) = 0), (\\forall (p,t) \\in Q, M_0(p) \\ge WQ(p,t)), \\nexists tt \\in T, $ $(\\forall pp \\in \\bullet tt, W(pp,tt) \\le M_{ts}(pp)), $ $(\\forall pp \\in I(tt), M_{ts}(pp) = 0), (\\forall (pp,tt) \\in Q, M_{ts}(pp) \\ge QW(pp,tt)), Z(tt) < Z(t), 0 \\le ts \\le k \\rbrace $ $A_{25}=\\lbrace could\\_not\\_have(t,ts): t \\in T, prenabled(t,ts) \\in A_{24}, fires(t,ts) \\notin A_{14}, (\\exists p \\in \\bullet t \\cup R(t): q > M_{ts}(p) - (\\sum _{t^{\\prime } \\in T_{ts}, p \\in \\bullet t^{\\prime }}{W(p,t^{\\prime })} + \\sum _{t^{\\prime } \\in T_{ts}, p \\in R(t^{\\prime })}{M_{ts}(p)}), q=W(p,t) \\text{~if~} (p,t) \\in E^- \\text{~or~} R(t) \\text{~otherwise}), 0 \\le ts \\le k\\rbrace \\\\ =\\lbrace could\\_not\\_have(t,ts) : t \\in T, (\\forall p \\in \\bullet t, W(p,t) \\le M_{ts}(p)), (\\forall p \\in I(t), M_{ts}(p) = 0), (\\forall (p,t) \\in Q, M_0(p) \\ge QW(p,t)), (\\nexists tt \\in T, $ $(\\forall pp \\in \\bullet tt, W(pp,tt) \\le M_{ts}(pp)), $ $(\\forall pp \\in I(tt), M_{ts}(pp) = 0), (\\forall (pp,tt) \\in Q, M_{ts}(pp) \\ge QW(pp,tt)), Z(tt) < Z(t)), t \\notin T_{ts}, (\\exists p \\in \\bullet t \\cup R(t): q > M_{ts}(p) - (\\sum _{t^{\\prime } \\in T_{ts}, p \\in \\bullet t^{\\prime }}{W(p,t^{\\prime })} + \\\\ \\sum _{t^{\\prime } \\in T_{ts}, p \\in R(t^{\\prime })}{M_{ts}(p)}), q=W(p,t) \\text{~if~} (p,t) \\in E^- \\text{~or~} R(t) \\text{~otherwise}), 0 \\le ts \\le k \\rbrace $ per the maximal firing set semantics We show that $A$ satisfies (REF ) and (REF ), and $A$ is an answer set of $\\Pi ^7$ .", "$A$ satisfies (REF ) and (REF ) by its construction above.", "We show $A$ is an answer set of $\\Pi ^7$ by splitting.", "We split $lit(\\Pi ^7)$ into a sequence of $9k+11$ sets: $U_0= head(f\\ref {f:c:place}) \\cup head(f\\ref {f:c:trans}) \\cup head(f\\ref {f:c:col}) \\cup head(f\\ref {f:c:time}) \\cup head(f\\ref {f:c:num}) \\cup head(i\\ref {i:c:init}) \\cup head(f\\ref {f:c:pr}) = \\lbrace place(p) : p \\in P\\rbrace \\cup \\lbrace trans(t) : t \\in T\\rbrace \\cup \\lbrace col(c) : c \\in C \\rbrace \\cup \\lbrace time(0), \\dots , time(k)\\rbrace \\cup \\lbrace num(0), \\dots , num(ntok)\\rbrace \\cup \\lbrace holds(p,q_c,c,0) : p \\in P, c \\in C, q_c=m_{M_0(p)}(c) \\rbrace \\cup \\lbrace transpr(t,pr) : t \\in T, pr=Z(t) \\rbrace $ $U_{9k+1}=U_{9k+0} \\cup head(f\\ref {f:c:ptarc})^{ts=k} \\cup head(f\\ref {f:c:tparc})^{ts=k} \\cup head(f\\ref {f:c:rptarc})^{ts=k} \\cup head(f\\ref {f:c:iptarc})^{ts=k} \\cup head(f\\ref {f:c:tptarc})^{ts=k} = U_{9k+0} \\cup $ $\\lbrace ptarc(p,t,n_c,c,k) : (p,t) \\in E^-, c \\in C, n_c=m_{W(p,t)}(c) \\rbrace \\\\ \\cup \\lbrace tparc(t,p,n_c,c,k,d) : (t,p) \\in E^+, c \\in C, n_c=m_{W(t,p)}(c), d=D(t) \\rbrace \\cup $ $\\lbrace ptarc(p,t,n_c,c,k) : p \\in R(t), c \\in C, n_c=m_{M_{k}(p)}(c), n > 0 \\rbrace \\cup \\lbrace iptarc(p,t,1,c,k) : \\\\ p \\in I(t), c \\in C \\rbrace \\cup $ $\\lbrace tptarc(p,t,n_c,c,k) : (p,t) \\in Q, c \\in C, n_c=m_{QW(p,t)}(c) \\rbrace $ $U_{9k+2}=U_{9k+1} \\cup head(e\\ref {e:c:ne:ptarc})^{ts=k} \\cup head(e\\ref {e:c:ne:iptarc})^{ts=k} \\cup head(e\\ref {e:c:ne:tptarc})^{ts=k} = U_{10k+1} \\cup \\\\ \\lbrace notenabled(t,k) : t \\in T \\rbrace $ $U_{9k+3}=U_{9k+2} \\cup head(e\\ref {e:c:enabled})^{ts=k} = U_{9k+2} \\cup \\lbrace enabled(t,k) : t \\in T \\rbrace $ $U_{9k+4}=U_{9k+3} \\cup head(a\\ref {a:c:prne})^{ts=k} = U_{9k+3} \\cup \\lbrace notprenabled(t,k) : t \\in T \\rbrace $ $U_{9k+5}=U_{9k+4} \\cup head(a\\ref {a:c:prenabled})^{ts=k} = U_{9k+4} \\cup \\lbrace prenabled(t,k) : t \\in T \\rbrace $ $U_{9k+6}=U_{9k+5} \\cup head(a\\ref {a:c:prfires})^{ts=k} = U_{9k+5} \\cup \\lbrace fires(t,k) : t \\in T \\rbrace $ $U_{9k+7}=U_{9k+6} \\cup head(r\\ref {r:c:dur:add})^{ts=k} \\cup head(r\\ref {r:c:del})^{ts=k} = U_{9k+6} \\cup \\lbrace add(p,q_c,t,c,k) : p \\in P, t \\in T, c \\in C, q_c=m_{W(t,p)}(c) \\rbrace \\cup \\lbrace del(p,q_c,t,c,k) : p \\in P, t \\in T, c \\in C, q_c=m_{W(p,t)}(c) \\rbrace \\cup \\lbrace del(p,q_c,t,c,k) : p \\in P, t \\in T, c \\in C, q_c=m_{M_{k}(p)}(c) \\rbrace $ $U_{9k+8}=U_{9k+7} \\cup head(r\\ref {r:c:totincr})^{ts=k} \\cup head(r\\ref {r:c:totdecr})^{ts=k} = U_{9k+7} \\cup \\lbrace tot\\_incr(p,q_c,c,k) : p \\in P, c \\in C, 0 \\le q_c \\le ntok \\rbrace \\cup \\lbrace tot\\_decr(p,q_c,c,k) : p \\in P, c \\in C, 0 \\le q_c \\le ntok \\rbrace $ $U_{9k+9}=U_{9k+8} \\cup head(a\\ref {a:c:overc:place})^{ts=k} \\cup head(r\\ref {r:c:nextstate})^{ts=k} \\cup head(a\\ref {a:c:prmaxfire:cnh})^{ts=k} = U_{9k+8} \\cup \\\\ \\lbrace consumesmore(p,k) : p \\in P\\rbrace \\cup \\lbrace holds(p,q,k+1) : p \\in P, 0 \\le q \\le ntok \\rbrace \\cup \\lbrace could\\_not\\_have(t,k) : t \\in T \\rbrace $ $U_{9k+10}=U_{9k+9} \\cup head(a\\ref {a:c:overc:gen}) = U_{9k+9} \\cup \\lbrace consumesmore \\rbrace $ where $head(r_i)^{ts=k}$ are head atoms of ground rule $r_i$ in which $ts=k$ .", "We write $A_i^{ts=k} = \\lbrace a(\\dots ,ts) : a(\\dots ,ts) \\in A_i, ts=k \\rbrace $ as short hand for all atoms in $A_i$ with $ts=k$ .", "$U_{\\alpha }, 0 \\le \\alpha \\le 9k+10$ form a splitting sequence, since each $U_i$ is a splitting set of $\\Pi ^7$ , and $\\langle U_{\\alpha }\\rangle _{\\alpha < \\mu }$ is a monotone continuous sequence, where $U_0 \\subseteq U_1 \\dots \\subseteq U_{9k+10}$ and $\\bigcup _{\\alpha < \\mu }{U_{\\alpha }} = lit(\\Pi ^7)$ .", "We compute the answer set of $\\Pi ^7$ using the splitting sets as follows: $bot_{U_0}(\\Pi ^7) = f\\ref {f:c:place} \\cup f\\ref {f:c:trans} \\cup f\\ref {f:c:col} \\cup f\\ref {f:c:time} \\cup f\\ref {f:c:num} \\cup i\\ref {i:c:init} \\cup f\\ref {f:c:pr}$ and $X_0 = A_1 \\cup \\dots \\cup A_5 \\cup A_8$ ($= U_0$ ) is its answer set – using forced atom proposition $eval_{U_0}(bot_{U_1}(\\Pi ^7) \\setminus bot_{U_0}(\\Pi ^7), X_0) = \\lbrace ptarc(p,t,q_c,c,0) \\text{:-}.", "\| c \\in C, q_c=m_{W(p,t)}(c), q_c > 0 \\rbrace \\cup $ $\\lbrace tparc(t,p,q_c,c,0,d) \\text{:-}.", "\| c \\in C, q_c=m_{W(t,p)}(c), q_c > 0, d=D(t) \\rbrace \\cup \\\\ \\lbrace ptarc(p,t,q_c,c,0) \\text{:-}.", "\| c \\in C, q_c=m_{M_0(p)}(c), q_c > 0 \\rbrace \\cup $ $\\lbrace iptarc(p,t,1,c,0) \\text{:-}.", "\| c \\in C \\rbrace \\cup $ $\\lbrace tptarc(p,t,q_c,c,0) \\text{:-}.", "\| c \\in C, q_c = m_{QW(p,t)}(c), q_c > 0 \\rbrace $ .", "Its answer set $X_1=A_6^{ts=0} \\cup A_7^{ts=0} \\cup A_9^{ts=0} \\cup A_{10}^{ts=0} \\cup A_{11}^{ts=0}$ – using forced atom proposition and construction of $A_6, A_7, A_9, A_{10}, A_{11}$ .", "$eval_{U_1}(bot_{U_2}(\\Pi ^7) \\setminus bot_{U_1}(\\Pi ^7), X_0 \\cup X_1) = \\lbrace notenabled(t,0) \\text{:-} .", "\| (\\lbrace trans(t), \\\\ ptarc(p,t,n_c,c,0), $ $holds(p,q_c,c,0) \\rbrace \\subseteq X_0 \\cup X_1, \\text{~where~} q_c < n_c) \\text{~or~} \\\\ (\\lbrace notenabled(t,0) \\text{:-} .", "\| $ $(\\lbrace trans(t), $ $iptarc(p,t,n2_c,c,0), $ $ holds(p,q_c,c,0) \\rbrace \\subseteq X_0 \\cup X_1, \\text{~where~} q_c \\ge n2_c \\rbrace ) \\text{~or~} $ $(\\lbrace trans(t), $ $tptarc(p,t,n3_c,c,0), $ $holds(p,q_c,c,0) \\rbrace \\subseteq X_0 \\cup X_1, \\text{~where~} q_c < n3_c) \\rbrace $ .", "Its answer set $X_2=A_{12}^{ts=0}$ – using forced atom proposition and construction of $A_{12}$ .", "where, $q_c=m_{M_0(p)}(c)$ , and $n_c=m_{W(p,t)}(c)$ for an arc $(p,t) \\in E^-$ – by construction of $i\\ref {i:c:init}$ and $f\\ref {f:c:ptarc}$ in $\\Pi ^7$ , and in an arc $(p,t) \\in E^-$ , $p \\in \\bullet t$ (by definition REF of preset) $n2_c=1$ – by construction of $iptarc$ predicates in $\\Pi ^7$ , meaning $q_c \\ge n2_c \\equiv q_c \\ge 1 \\equiv q_c > 0$ , $tptarc(p,t,n3_c,c,0)$ represents $n3_c=m_{QW(p,t)}(c)$ , where $(p,t) \\in Q$ thus, $notenabled(t,0) \\in X_1$ means $\\exists c \\in C, (\\exists p \\in \\bullet t : m_{M_0(p)}(c) < m_{W(p,t)}(c)) \\\\ \\vee (\\exists p \\in I(t) : m_{M_0(p)}(c) > 0) \\vee (\\exists (p,t) \\in Q : m_{M_0(p)}(c) < m_{QW(p,t)}(c))$ .", "$eval_{U_2}(bot_{U_3}(\\Pi ^7) \\setminus bot_{U_2}(\\Pi ^7), X_0 \\cup \\dots \\cup X_2) = \\lbrace enabled(t,0) \\text{:-}.", "\| trans(t) \\in X_0 \\cup \\dots \\cup X_2, notenabled(t,0) \\notin X_0 \\cup \\dots \\cup X_2 \\rbrace $ .", "Its answer set is $X_3 = A_{13}^{ts=0}$ – using forced atom proposition and construction of $A_{13}$ .", "since an $enabled(t,0) \\in X_3$ if $\\nexists ~notenabled(t,0) \\in X_0 \\cup \\dots \\cup X_2$ ; which is equivalent to $\\nexists t, \\forall c \\in C, (\\nexists p \\in \\bullet t, m_{M_0(p)}(c) < m_{W(p,t)}(c)), (\\nexists p \\in I(t), \\\\ m_{M_0(p)}(c) > 0), (\\nexists (p,t) \\in Q : m_{M_0(p)}(c) < m_{QW(p,t)}(c) ), \\forall c \\in C, (\\forall p \\in \\bullet t: m_{M_0(p)}(c) \\ge m_{W(p,t)}(c)), (\\forall p \\in I(t) : m_{M_0(p)}(c) = 0)$ .", "$eval_{U_3}(bot_{U_4}(\\Pi ^7) \\setminus bot_{U_3}(\\Pi ^7), X_0 \\cup \\dots \\cup X_3) = \\lbrace notprenabled(t,0) \\text{:-}.", "\| \\\\ \\lbrace enabled(t,0), transpr(t,p), enabled(tt,0), transpr(tt,pp) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_3, pp < p \\rbrace $ .", "Its answer set is $X_4 = A_{23}^{ts=k}$ – using forced atom proposition and construction of $A_{23}$ .", "$enabled(t,0)$ represents $\\exists t \\in T, \\forall c \\in C, (\\forall p \\in \\bullet t, m_{M_0(p)}(c) \\ge m_{W(p,t)}(c)), $ $(\\forall p \\in I(t), m_{M_0(p)}(c) = 0), (\\forall (p,t) \\in Q, m_{M_0(p)}(c) \\ge m_{QW(p,t)}(c))$ $enabled(tt,0)$ represents $\\exists tt \\in T, \\forall c \\in C, $ $(\\forall pp \\in \\bullet tt, m_{M_0(pp)}(c) \\ge \\\\ m_{W(pp,tt)}(c)) \\wedge $ $(\\forall pp \\in I(tt), m_{M_0(pp)}(c) = 0), (\\forall (pp,tt) \\in Q, m_{M_0(pp)}(c) \\ge m_{QW(pp,tt)}(c))$ $transpr(t,p)$ represents $p=Z(t)$ – by construction $transpr(tt,pp)$ represents $pp=Z(tt)$ – by construction thus, $notprenabled(t,0)$ represents $\\forall c \\in C, $ $(\\forall p \\in \\bullet t, m_{M_0(p)}(c) \\ge m_{W(p,t)}(c)), $ $(\\forall p \\in I(t), m_{M_0(p)}(c) = 0), (\\forall (p,t) \\in Q, m_{M_0(p)}(c) \\ge m_{QW(p,t)}(c)), \\exists tt \\in T, (\\forall pp \\in \\bullet tt, m_{M_0(pp)}(c) \\ge m_{W(pp,tt)}(c)), $ $(\\forall pp \\in I(tt), m_{M_0(pp)}(c) = 0), \\\\ (\\forall (pp,tt) \\in Q, m_{M_0(pp)}(c) \\ge m_{QW(pp,tt)}(c)), Z(tt) < Z(t)$ which is equivalent to $(\\forall p \\in \\bullet t: M_0(p) \\ge W(p,t)), $ $(\\forall p \\in I(t), M_0(p) = 0), (\\forall (p,t) \\in Q, M_0(p) \\ge QW(p,t)), \\exists tt \\in T, $ $(\\forall pp \\in \\bullet tt, M_0(pp) \\ge \\\\ W(pp,tt)), $ $(\\forall pp \\in I(tt), M_0(pp) = 0), (\\forall (pp,tt) \\in Q, M_0 \\ge QW(pp,tt)), \\\\ Z(tt) < Z(t)$ – assuming multiset domain $C$ for all operations $eval_{U_4}(bot_{U_5}(\\Pi ^7) \\setminus bot_{U_4}(\\Pi ^7), X_0 \\cup \\dots \\cup X_4) = \\lbrace prenabled(t,0) \\text{:-}.", "\| enabled(t,0) \\in X_0 \\cup \\dots \\cup X_4, notprenabled(t,0) \\notin X_0 \\cup \\dots \\cup X_4 \\rbrace $ .", "Its answer set is $X_5 = A_{24}^{ts=k}$ – using forced atom proposition and construction of $A_{24}$ $enabled(t,0)$ represents $\\forall c \\in C, (\\forall p \\in \\bullet t, m_{M_0(p)}(c) \\ge m_{W(p,t)}(c)), (\\forall p \\in I(t), m_{M_0(p)}(c) = 0), (\\forall (p,t) \\in Q, m_{M_0(p)}(c) \\ge m_{QW(p,t)}(c)) \\equiv $ $(\\forall p \\in \\bullet t, \\\\ M_0(p) \\ge W(p,t)), $ $(\\forall p \\in I(t), M_0(p) = 0), (\\forall (p,t) \\in Q, M_0(p) \\ge QW(p,t))$ – from REF above and assuming multiset domain $C$ for all operations $notprenabled(t,0)$ represents $(\\forall p \\in \\bullet t, M_0(p) \\ge W(p,t)), $ $(\\forall p \\in I(t), \\\\ M_0(p) = 0), (\\forall (p,t) \\in Q, M_0(p) \\ge QW(p,t)), \\exists tt \\in T, $ $(\\forall pp \\in \\bullet tt, M_0(pp) \\ge W(pp,tt)), $ $(\\forall pp \\in I(tt), M_0(pp) = 0), (\\forall (pp,tt) \\in Q, M_0(pp) \\ge QW(pp,tt)), \\\\ Z(tt) < Z(t)$ – from REF above and assuming multiset domain $C$ for all operations then, $prenabled(t,0)$ represents $(\\forall p \\in \\bullet t, M_0(p) \\ge W(p,t)), $ $(\\forall p \\in I(t), \\\\ M_0(p) = 0), (\\forall (p,t) \\in Q, M_0(p) \\ge QW(p,t)), \\nexists tt \\in T, $ $((\\forall pp \\in \\bullet tt, \\\\ M_0(pp) \\ge W(pp,tt)), $ $(\\forall pp \\in I(tt), M_0(pp) = 0), (\\forall (pp,tt) \\in Q, M_0(pp) \\ge QW(pp,tt)), Z(tt) < Z(t))$ – from (a), (b) and $enabled(t,0) \\in X_0 \\cup \\dots \\cup X_4$ $eval_{U_5}(bot_{U_6}(\\Pi ^7) \\setminus bot_{U_5}(\\Pi ^7), X_0 \\cup \\dots \\cup X_5) = \\lbrace \\lbrace fires(t,0)\\rbrace \\text{:-}.", "\| prenabled(t,0) \\\\ \\text{~holds in~} X_0 \\cup \\dots \\cup X_5 \\rbrace $ .", "It has multiple answer sets $X_{6.1}, \\dots , X_{6.n}$ , corresponding to elements of power set of $fires(t,0)$ atoms in $eval_{U_5}(...)$ – using supported rule proposition.", "Since we are showing that the union of answer sets of $\\Pi ^7$ determined using splitting is equal to $A$ , we only consider the set that matches the $fires(t,0)$ elements in $A$ and call it $X_6$ , ignoring the rest.", "Thus, $X_6 = A_{14}^{ts=0}$ , representing $T_0$ .", "in addition, for every $t$ such that $prenabled(t,0) \\in X_0 \\cup \\dots \\cup X_5, R(t) \\ne \\emptyset $ ; $fires(t,0) \\in X_6$ – per definition REF (firing set); requiring that a reset transition is fired when enabled thus, the firing set $T_0$ will not be eliminated by the constraint $f\\ref {f:c:pr:rptarc:elim}$ $eval_{U_6}(bot_{U_7}(\\Pi ^7) \\setminus bot_{U_6}(\\Pi ^7), X_0 \\cup \\dots \\cup X_6) = \\lbrace add(p,n_c,t,c,0) \\text{:-}.", "\| $ $\\lbrace fires(t,0-d+1), $ $tparc(t,p,n_c,c,0,d) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_6 \\rbrace \\cup \\lbrace del(p,n_c,t,c,0) \\text{:-}.", "\| $ $\\lbrace fires(t,0), $ $ptarc(p,t,n_c,c,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_6 \\rbrace $ .", "It's answer set is $X_7 = A_{15}^{ts=0} \\cup A_{16}^{ts=0}$ – using forced atom proposition and definitions of $A_{15}$ and $A_{16}$ .", "where, each $add$ atom is equivalent to $n_c=m_{W(t,p)}(c),c \\in C, p \\in t \\bullet $ , and each $del$ atom is equivalent to $n_c=m_{W(p,t)}(c), c \\in C, p \\in \\bullet t$ ; or $n_c=m_{M_0(p)}(c), c \\in C, p \\in R(t)$ , representing the effect of transitions in $T_0$ $eval_{U_7}(bot_{U_8}(\\Pi ^7) \\setminus bot_{U_7}(\\Pi ^7), X_0 \\cup \\dots \\cup X_7) = \\\\ \\lbrace tot\\_incr(p,qq_c,c,0) \\text{:-}.", "\| qq_c=\\sum _{add(p,q_c,t,c,0) \\in X_0 \\cup \\dots \\cup X_7}{q_c} \\rbrace \\cup \\\\ \\lbrace tot\\_decr(p,qq_c,c,0) \\text{:-}.", "\| qq_c=\\sum _{del(p,q_c,t,c,0) \\in X_0 \\cup \\dots \\cup X_7}{q_c} \\rbrace $ .", "It's answer set is $X_8 = A_{17}^{ts=0} \\cup A_{18}^{ts=0}$ – using forced atom proposition and definitions of $A_{17}$ and $A_{18}$ .", "where, each $tot\\_incr(p,qq_c,c,0)$ , $qq_c=\\sum _{add(p,q_c,t,c,0) \\in X_0 \\cup \\dots X_7}{q_c}$ $\\equiv qq_c=\\sum _{t \\in X_6, p \\in t \\bullet , 0+D(t)-1=0}{m_{W(p,t)}(c)}$ , $tot\\_decr(p,qq_c,c,0)$ , $qq_c=\\sum _{del(p,q_c,t,c,0) \\in X_0 \\cup \\dots X_7}{q_c}$ $\\equiv qq=\\sum _{t \\in X_6, p \\in \\bullet t}{m_{W(t,p)}(c)} + \\sum _{t \\in X_6, p \\in R(t)}{m_{M_0(p)}(c)}$ , represent the net effect of transitions in $T_0$ $eval_{U_8}(bot_{U_9}(\\Pi ^7) \\setminus bot_{U_8}(\\Pi ^7), X_0 \\cup \\dots \\cup X_8) = \\lbrace consumesmore(p,0) \\text{:-}.", "\| \\\\ \\lbrace holds(p,q_c,c,0), tot\\_decr(p,q1_c,c,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_8, q1_c > q_c \\rbrace \\cup $ $\\lbrace holds(p,q_c,c,1) \\text{:-}.", "\| \\lbrace holds(p,q1_c,c,0), tot\\_incr(p,q2_c,c,0), \\\\ tot\\_decr(p,q3_c,c,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_6, q_c=q1_c+q2_c-q3_c \\rbrace \\cup \\lbrace could\\_not\\_have(t,0) \\text{:-}.", "\| \\\\ \\lbrace prenabled(t,0), ptarc(s,t,q,c,0), holds(s,qq,c,0), tot\\_decr(s,qqq,c,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_8, fires(t,0) \\notin X_0 \\cup \\dots \\cup X_8, q > qq - qqq \\rbrace $ .", "It's answer set is $X_9 = A_{19}^{ts=0} \\cup A_{21}^{ts=0} \\cup A_{25}^{ts=0}$ – using forced atom proposition and definitions of $A_{19}, A_{21}, A_{25}$ .", "where $consumesmore(p,0)$ represents $\\exists p : q_c=m_{M_0(p)}(c), q1_c= \\\\ \\sum _{t \\in T_0, p \\in \\bullet t}{m_{W(p,t)}(c)}+\\sum _{t \\in T_0, p \\in R(t)}{m_{M_0(p)}(c)}, q1_c > q_c, c \\in C$ , indicating place $p$ will be over consumed if $T_0$ is fired, as defined in definition REF (conflicting transitions), $holds(p,q_c,c,1)$ if $q_c=m_{M_0(p)}(c)+\\sum _{t \\in T_0, p \\in t \\bullet , 0+D(t)-1=0}{m_{W(t,p)}(c)}- \\\\ (\\sum _{t \\in T_0, p \\in \\bullet t}{m_{W(p,t)}(c)}+ $ $\\sum _{t \\in T_0, p \\in R(t)}{m_{M_0(p)}(c)})$ represented by $q_c=m_{M_1(p)}(c)$ for some $c \\in C$ – by construction of $\\Pi ^7$ and $consumesmore(p,0)$ if $\\sum _{t \\in T_0, p \\in \\bullet t}{m_{W(p,t)}(c)}+ $ $\\sum _{t \\in T_0, p \\in R(t)}{m_{M_0(p)}(c)} > m_{M_0(p)}(c)$ for any $c \\in C$ $could\\_not\\_have(t,0)$ if $(\\forall p \\in \\bullet t, W(p,t) \\le M_0(p)), (\\forall p \\in I(t), M_0(p) = 0), (\\forall (p,t) \\in Q, \\\\ M_0(p) \\ge WQ(p,t)), \\nexists tt \\in T, (\\forall pp \\in \\bullet tt, W(pp,tt) \\le M_{ts}(pp)), (\\forall pp \\in I(tt), M_0(pp) = 0), (\\forall (pp,tt) \\in Q, M_0(pp) \\ge QW(pp,tt)), Z(tt) < Z(t)$ , and $q_c > m_{M_0(s)}(c) - (\\sum _{t^{\\prime } \\in T_0, s \\in \\bullet t^{\\prime }}{m_{W(s,t^{\\prime })}(c)}+ \\sum _{t^{\\prime } \\in T_0, s \\in R(t)}{m_{M_0(s)}(c)}), \\\\ q_c = m_{W(s,t)}(c) \\text{~if~} s \\in \\bullet t \\text{~or~} m_{M_0(s)}(c) \\text{~otherwise}$ for some $c \\in C$ , which becomes $q > M_0(s) - (\\sum _{t^{\\prime } \\in T_0, s \\in \\bullet t^{\\prime }}{W(s,t^{\\prime })}+ \\sum _{t^{\\prime } \\in T_0, s \\in R(t)}{M_0(s)}), q = W(s,t) \\text{~if~} s \\in \\bullet t \\text{~or~} M_0(s) \\text{~otherwise}$ for all $c \\in C$ (i), (ii) above combined match the definition of $A_{25}$ $X_9$ does not contain $could\\_not\\_have(t,0)$ , when $prenabled(t,0) \\in X_0 \\cup \\dots \\cup X_6$ and $fires(t,0) \\notin X_0 \\cup \\dots \\cup X_5$ due to construction of $A$ , encoding of $a\\ref {a:c:maxfire:cnh}$ and its body atoms.", "As a result, it is not eliminated by the constraint $a\\ref {a:c:prmaxfire:elim}$ $ \\vdots $ $eval_{U_{9k+0}}(bot_{U_{9k+1}}(\\Pi ^7) \\setminus bot_{U_{9k+0}}(\\Pi ^7), X_0 \\cup \\dots \\cup X_{9k+0}) = \\lbrace ptarc(p,t,q_c,c,k) \\text{:-}.", "\| c \\in C, q_c=m_{W(p,t)}(c), q_c > 0 \\rbrace \\cup \\lbrace tparc(t,p,q_c,c,k,d) \\text{:-}.", "\| c \\in C, q_c=m_{W(t,p)}(c), q_c > 0, d=D(t) \\rbrace \\cup $ $\\lbrace ptarc(p,t,q_c,c,k) \\text{:-}.", "\| c \\in C, q_c=m_{M_k(p)}(c), q_c > 0 \\rbrace \\cup \\\\ \\lbrace iptarc(p,t,1,c,k) \\text{:-}.", "\| c \\in C \\rbrace \\cup $ $\\lbrace tptarc(p,t,q_c,c,k) \\text{:-}.", "\| c \\in C, q_c = m_{QW(p,t)}(c), q_c > 0 \\rbrace $ .", "Its answer set $X_{9k+1}=A_6^{ts=k} \\cup A_7^{ts=k} \\cup A_9^{ts=k} \\cup A_{10}^{ts=k} \\cup A_{11}^{ts=k}$ – using forced atom proposition and construction of $A_6, A_7, A_9, A_{10}, A_{11}$ .", "$eval_{U_{9k+1}}(bot_{U_{10k+2}}(\\Pi ^7) \\setminus bot_{U_{9k+1}}(\\Pi ^7), X_0 \\cup \\dots \\cup X_{9k+1}) = \\lbrace notenabled(t,k) \\text{:-} .", "\| \\\\ (\\lbrace trans(t), ptarc(p,t,n_c,c,k), holds(p,q_c,c,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{9k+1}, \\text{~where~} q_c < n_c) \\text{~or~} (\\lbrace notenabled(t,k) \\text{:-} .", "\| (\\lbrace trans(t), iptarc(p,t,n2_c,c,k), holds(p,q_c,c,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{9k+1}, $ $\\text{~where~} q_c \\ge n2_c \\rbrace ) \\text{~or~} $ $(\\lbrace trans(t), tptarc(p,t,n3_c,c,k), \\\\ holds(p,q_c,c,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{9k+1}, $ $\\text{~where~} q_c < n3_c) \\rbrace $ .", "Its answer set $X_{9k+2}=A_{12}^{ts=k}$ – using forced atom proposition and construction of $A_{12}$ .", "where, $q_c=m_{M_k(p)}(c)$ , and $n_c=m_{W(p,t)}(c)$ for an arc $(p,t) \\in E^-$ – by construction of $i\\ref {i:c:init}$ and $f\\ref {f:c:ptarc}$ predicates in $\\Pi ^7$ , and in an arc $(p,t) \\in E^-$ , $p \\in \\bullet t$ (by definition REF of preset) $n2_c=1$ – by construction of $iptarc$ predicates in $\\Pi ^7$ , meaning $q_c \\ge n2_c \\equiv q_c \\ge 1 \\equiv q_c > 0$ , $tptarc(p,t,n3_c,c,k)$ represents $n3_c=m_{QW(p,t)}(c)$ , where $(p,t) \\in Q$ thus, $notenabled(t,k) \\in X_{9k+1}$ represents $\\exists c \\in C, (\\exists p \\in \\bullet t : m_{M_k(p)}(c) < m_{W(p,t)}(c)) \\vee (\\exists p \\in I(t) : m_{M_k(p)}(c) > 0) \\vee (\\exists (p,t) \\in Q : m_{M_k(p)}(c) < m_{QW(p,t)}(c))$ .", "$eval_{U_{9k+2}}(bot_{U_{9k+3}}(\\Pi ^7) \\setminus bot_{U_{9k+2}}(\\Pi ^7), X_0 \\cup \\dots \\cup X_{9k+2}) = \\lbrace enabled(t,k) \\text{:-}.", "\| trans(t) \\in X_0 \\cup \\dots \\cup X_{9k+2}, notenabled(t,k) \\notin X_0 \\cup \\dots \\cup X_{9k+2} \\rbrace $ .", "Its answer set is $X_{9k+3} = A_{13}^{ts=k}$ – using forced atom proposition and construction of $A_{13}$ .", "Since an $enabled(t,k) \\in X_{9k+3}$ if $\\nexists ~notenabled(t,k) \\in X_0 \\cup \\dots \\cup X_{9k+2}$ ; which is equivalent to $\\nexists t, \\forall c \\in C, (\\nexists p \\in \\bullet t, m_{M_k(p)}(c) < m_{W(p,t)}(c)), (\\nexists p \\in I(t), m_{M_k(p)}(c) > 0), (\\nexists (p,t) \\in Q : m_{M_k(p)}(c) < m_{QW(p,t)}(c) ), \\forall c \\in C, (\\forall p \\in \\bullet t: m_{M_k(p)}(c) \\ge m_{W(p,t)}(c)), (\\forall p \\in I(t) : m_{M_k(p)}(c) = 0)$ .", "$eval_{U_{9k+3}}(bot_{U_{9k+4}}(\\Pi ^7) \\setminus bot_{U_{9k+3}}(\\Pi ^7), X_0 \\cup \\dots \\cup X_{9k+3}) = \\lbrace notprenabled(t,k) \\text{:-}.", "\| \\\\ \\lbrace enabled(t,k), transpr(t,p), enabled(tt,k), transpr(tt,pp) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{9k+3}, \\\\ pp < p \\rbrace $ .", "Its answer set is $X_{9k+4} = A_{23}^{ts=k}$ – using forced atom proposition and construction of $A_{23}$ .", "$enabled(t,k)$ represents $\\exists t \\in T, \\forall c \\in C, (\\forall p \\in \\bullet t, m_{M_k(p)}(c) \\ge m_{W(p,t)}(c)), $ $(\\forall p \\in I(t), m_{M_k(p)}(c) = 0), (\\forall (p,t) \\in Q, m_{M_k(p)}(c) \\ge m_{QW(p,t)}(c))$ $enabled(tt,k)$ represents $\\exists tt \\in T, \\forall c \\in C, (\\forall pp \\in \\bullet tt, m_{M_k(pp)}(c) \\ge \\\\ m_{W(pp,tt)}(c)) \\wedge (\\forall pp \\in I(tt), m_{M_k(pp)}(c) = 0), (\\forall (pp,tt) \\in Q, m_{M_k(pp)}(c) \\ge m_{QW(pp,tt)}(c))$ $transpr(t,p)$ represents $p=Z(t)$ – by construction $transpr(tt,pp)$ represents $pp=Z(tt)$ – by construction thus, $notprenabled(t,k)$ represents $\\forall c \\in C, (\\forall p \\in \\bullet t, m_{M_k(p)}(c) \\ge \\\\ m_{W(p,t)}(c)), $ $(\\forall p \\in I(t), m_{M_k(p)}(c) = 0), (\\forall (p,t) \\in Q, m_{M_k(p)}(c) \\ge \\\\ m_{QW(p,t)}(c)), \\exists tt \\in T, $ $(\\forall pp \\in \\bullet tt, m_{M_k(pp)}(c) \\ge m_{W(pp,tt)}(c)), $ $(\\forall pp \\in I(tt), \\\\ m_{M_k(pp)}(c) = 0), (\\forall (pp,tt) \\in Q, m_{M_k(pp)}(c) \\ge m_{QW(pp,tt)}(c)), Z(tt) < Z(t)$ which is equivalent to $(\\forall p \\in \\bullet t: M_k(p) \\ge W(p,t)), $ $(\\forall p \\in I(t), \\\\ M_k(p) = 0), (\\forall (p,t) \\in Q, M_k(p) \\ge QW(p,t)), \\exists tt \\in T, $ $(\\forall pp \\in \\bullet tt, \\\\ M_k(pp) \\ge W(pp,tt)), $ $(\\forall pp \\in I(tt), M_k(pp) = 0), (\\forall (pp,tt) \\in Q, M_k(pp) \\ge QW(pp,tt)), Z(tt) < Z(t)$ – assuming multiset domain $C$ for all operations $eval_{U_{9k+4}}(bot_{U_{9k+5}}(\\Pi ^7) \\setminus bot_{U_{9k+4}}(\\Pi ^7), X_0 \\cup \\dots \\cup X_{9k+4}) = \\lbrace prenabled(t,k) \\text{:-}.", "\| \\\\ enabled(t,k) \\in X_0 \\cup \\dots \\cup X_{9k+4}, notprenabled(t,k) \\notin X_0 \\cup \\dots \\cup X_{9k+4} \\rbrace $ .", "Its answer set is $X_{9k+5} = A_{24}^{ts=k}$ – using forced atom proposition and construction of $A_{24}$ $enabled(t,k)$ represents $\\forall c \\in C, (\\forall p \\in \\bullet t, m_{M_k(p)}(c) \\ge m_{W(p,t)}(c)), $ $(\\forall p \\in I(t), m_{M_k(p)}(c) = 0), (\\forall (p,t) \\in Q, m_{M_k(p)}(c) \\ge m_{QW(p,t)}(c)) \\equiv (\\forall p \\in \\bullet t, M_k(p) \\ge W(p,t)), $ $(\\forall p \\in I(t), M_k(p) = 0), (\\forall (p,t) \\in Q, M_k(p) \\ge QW(p,t))$ – from REF above and assuming multiset domain $C$ for all operations $notprenabled(t,k)$ represents $(\\forall p \\in \\bullet t, M_k(p) \\ge W(p,t)), (\\forall p \\in I(t), \\\\ M_k(p) = 0), (\\forall (p,t) \\in Q, M_k(p) \\ge QW(p,t)), \\exists tt \\in T, $ $(\\forall pp \\in \\bullet tt, M_k(pp) \\ge W(pp,tt)), (\\forall pp \\in I(tt), M_k(pp) = 0), (\\forall (pp,tt) \\in Q, M_k(pp) \\ge QW(pp,tt)), \\\\ Z(tt) < Z(t)$ – from REF above and assuming multiset domain $C$ for all operations then, $prenabled(t,k)$ represents $(\\forall p \\in \\bullet t, M_k(p) \\ge W(p,t)), (\\forall p \\in I(t), \\\\ M_k(p) = 0), (\\forall (p,t) \\in Q, M_k(p) \\ge QW(p,t)), \\nexists tt \\in T, $ $((\\forall pp \\in \\bullet tt, M_k(pp) \\ge W(pp,tt)), $ $(\\forall pp \\in I(tt), M_k(pp) = 0), (\\forall (pp,tt) \\in Q, \\\\ M_k(pp) \\ge QW(pp,tt)), Z(tt) < Z(t))$ – from (a), (b) and $enabled(t,k) \\in X_0 \\cup \\dots \\cup X_{9k+4}$ $eval_{U_{9k+5}}(bot_{U_{9k+6}}(\\Pi ^7) \\setminus bot_{U_{9k+5}}(\\Pi ^7), X_0 \\cup \\dots \\cup X_{9k+5}) = \\lbrace \\lbrace fires(t,k)\\rbrace \\text{:-}.", "\| \\\\ prenabled(t,k) \\text{~holds in~} X_0 \\cup \\dots \\cup X_{9k+5} \\rbrace $ .", "It has multiple answer sets $X_{9k+6.1}, \\dots , X_{9k+6.n}$ , corresponding to elements of power set of $fires(t,k)$ atoms in $eval_{U_{9k+5}}(...)$ – using supported rule proposition.", "Since we are showing that the union of answer sets of $\\Pi ^7$ determined using splitting is equal to $A$ , we only consider the set that matches the $fires(t,k)$ elements in $A$ and call it $X_{9k+6}$ , ignoring the rest.", "Thus, $X_{9k+6} = A_{14}^{ts=k}$ , representing $T_k$ .", "in addition, for every $t$ such that $prenabled(t,k) \\in X_0 \\cup \\dots \\cup X_{9k+5}, R(t) \\ne \\emptyset $ ; $fires(t,k) \\in X_{9k+6}$ – per definition REF (firing set); requiring that a reset transition is fired when enabled thus, the firing set $T_k$ will not be eliminated by the constraint $f\\ref {f:c:pr:rptarc:elim}$ $eval_{U_{9k+6}}(bot_{U_{9k+7}}(\\Pi ^7) \\setminus bot_{U_{9k+6}}(\\Pi ^7), X_0 \\cup \\dots \\cup X_{9k+6}) = \\lbrace add(p,n_c,t,c,k) \\text{:-}.", "\| $ $\\lbrace fires(t,k-d+1), tparc(t,p,n_c,c,0,d) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{9k+6} \\rbrace \\cup \\\\ \\lbrace del(p,n_c,t,c,k) \\text{:-}.", "\| \\lbrace fires(t,k), ptarc(p,t,n_c,c,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{9k+6} \\rbrace $ .", "It's answer set is $X_{9k+7} = A_{15}^{ts=k} \\cup A_{16}^{ts=k}$ – using forced atom proposition and definitions of $A_{15}$ and $A_{16}$ .", "where, each $add$ atom is equivalent to $n_c=m_{W(t,p)}(c),c \\in C, p \\in t \\bullet $ , and each $del$ atom is equivalent to $n_c=m_{W(p,t)}(c), c \\in C, p \\in \\bullet t$ ; or $n_c=m_{M_k(p)}(c), c \\in C, p \\in R(t)$ , representing the effect of transitions in $T_k$ $eval_{U_{9k+7}}(bot_{U_{9k+8}}(\\Pi ^7) \\setminus bot_{U_{9k+7}}(\\Pi ^7), X_0 \\cup \\dots \\cup X_{9k+7}) = \\lbrace tot\\_incr(p,qq_c,c,k) \\text{:-}.", "\| $ $ qq_c=\\sum _{add(p,q_c,t,c,k) \\in X_0 \\cup \\dots \\cup X_{9k+7}}{q_c} \\rbrace \\cup \\lbrace tot\\_decr(p,qq_c,c,k) \\text{:-}.", "\| $ $ qq_c=\\sum _{del(p,q_c,t,c,k) \\in X_0 \\cup \\dots \\cup X_{9k+7}}{q_c} \\rbrace $ .", "It's answer set is $X_{9k+8} = A_{17}^{ts=k} \\cup A_{18}^{ts=k}$ – using forced atom proposition and definitions of $A_{17}$ and $A_{18}$ .", "where, each $tot\\_incr(p,qq_c,c,k)$ , $qq_c=\\sum _{add(p,q_c,t,c,k) \\in X_0 \\cup \\dots X_{9k+7}}{q_c}$ $\\equiv qq_c=\\sum _{t \\in X_{9k+6}, p \\in t \\bullet , 0 \\le l \\le k, l+D(t)-1=k }{m_{W(p,t)}(c)}$ , and each $tot\\_decr(p,qq_c,c,k)$ , $qq_c=\\sum _{del(p,q_c,t,c,k) \\in X_0 \\cup \\dots X_{9k+7}}{q_c}$ $\\equiv qq=\\sum _{t \\in X_{9k+6}, p \\in \\bullet t}{m_{W(t,p)}(c)} + \\sum _{t \\in X_{9k+6}, p \\in R(t)}{m_{M_k(p)}(c)}$ , represent the net effect of transitions in $T_k$ $eval_{U_{9k+8}}(bot_{U_{9k+9}}(\\Pi ^7) \\setminus bot_{U_{9k+8}}(\\Pi ^7), X_0 \\cup \\dots \\cup X_{9k+8}) = $ $\\lbrace consumesmore(p,k) \\text{:-}.", "\| \\\\ \\lbrace holds(p,q_c,c,k), tot\\_decr(p,q1_c,c,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{9k+8}, q1_c > q_c \\rbrace \\cup \\\\ \\lbrace holds(p,q_c,c,k+1) \\text{:-}., \| \\lbrace holds(p,q1_c,c,k), tot\\_incr(p,q2_c,c,k), \\\\ tot\\_decr(p,q3_c,c,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{9k+6}, q_c=q1_c+q2_c-q3_c \\rbrace \\cup \\\\ \\lbrace could\\_not\\_have(t,k) \\text{:-}.", "\| \\lbrace prenabled(t,k), \\\\ ptarc(s,t,q,c,k), holds(s,qq,c,k), tot\\_decr(s,qqq,c,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{9k+8}, \\\\ fires(t,k) \\notin X_0 \\cup \\dots \\cup X_{10k+8}, q > qq - qqq \\rbrace $ .", "It's answer set is $X_{9k+9} = A_{19}^{ts=k} \\cup A_{21}^{ts=k} \\cup A_{25}^{ts=k}$ – using forced atom proposition and definitions of $A_{19}, A_{21}, A_{25}$ .", "where, $consumesmore(p,k)$ represents $\\exists p : q_c=m_{M_k(p)}(c), \\\\ q1_c=\\sum _{t \\in T_k, p \\in \\bullet t}{m_{W(p,t)}(c)}+\\sum _{t \\in T_k, p \\in R(t)}{m_{M_k(p)}(c)}, q1_c > q_c, c \\in C$ , indicating place $p$ that will be over consumed if $T_k$ is fired, as defined in definition REF (conflicting transitions), $holds(p,q_c,c,k+1)$ if $q_c=m_{M_k(p)}(c)+\\sum _{t \\in T_l, p \\in t \\bullet , 0 \\le l \\le k, l+D(t)-1=k}{m_{W(t,p)}(c)}-(\\sum _{t \\in T_k, p \\in \\bullet t}{m_{W(p,t)}(c)}+ \\sum _{t \\in T_k, p \\in R(t)}{m_{M_k(p)}(c)})$ represented by $q_c=m_{M_1(p)}(c)$ for some $c \\in C$ – by construction of $\\Pi ^7$ , and $could\\_not\\_have(t,k)$ if $(\\forall p \\in \\bullet t, W(p,t) \\le M_k(p)), (\\forall p \\in I(t), M_k(p) = 0), (\\forall (p,t) \\in Q, \\\\ M_k(p) \\ge WQ(p,t)), \\nexists tt \\in T, (\\forall pp \\in \\bullet tt, W(pp,tt) \\le M_{ts}(pp)), (\\forall pp \\in I(tt), M_k(pp) = 0), (\\forall (pp,tt) \\in Q, M_k(pp) \\ge QW(pp,tt)), Z(tt) < Z(t)$ , and $q_c > m_{M_k(s)}(c) - (\\sum _{t^{\\prime } \\in T_k, s \\in \\bullet t^{\\prime }}{m_{W(s,t^{\\prime })}(c)}+ \\sum _{t^{\\prime } \\in T_k, s \\in R(t)}{m_{M_k(s)}(c)}), \\\\ q_c = m_{W(s,t)}(c) \\text{~if~} s \\in \\bullet t \\text{~or~} m_{M_k(s)}(c) \\text{~otherwise}$ for some $c \\in C$ , which becomes $q > M_k(s) - (\\sum _{t^{\\prime } \\in T_k, s \\in \\bullet t^{\\prime }}{W(s,t^{\\prime })}+ \\sum _{t^{\\prime } \\in T_k, s \\in R(t)}{M_k(s)}), q = W(s,t) \\text{~if~} s \\in \\bullet t \\text{~or~} M_k(s) \\text{~otherwise}$ for all $c \\in C$ (i), (ii) above combined match the definition of $A_{25}$ $X_{9k+9}$ does not contain $could\\_not\\_have(t,k)$ , when $prenabled(t,k) \\in X_0 \\cup \\dots \\cup X_{9k+6}$ and $fires(t,k) \\notin X_0 \\cup \\dots \\cup X_{9k+5}$ due to construction of $A$ , encoding of $a\\ref {a:c:maxfire:cnh}$ and its body atoms.", "As a result it is not eliminated by the constraint $a\\ref {a:c:prmaxfire:elim}$ $eval_{U_{9k+9}}(bot_{U_{9k+10}}(\\Pi ^7) \\setminus bot_{U_{9k+9}}(\\Pi ^7), X_0 \\cup \\dots \\cup X_{9k+9}) = \\lbrace consumesmore \\text{:-}.", "\| \\\\ \\lbrace consumesmore(p,0),\\dots ,$ $consumesmore(p,k) \\rbrace \\cap (X_0 \\cup \\dots \\cup X_{9k+9}) \\ne \\emptyset \\rbrace $ .", "It's answer set is $X_{9k+10} = A_{20}$ – using forced atom proposition and the definition of $A_{20}$ $X_{9k+10}$ will be empty since none of $consumesmore(p,0),\\dots , \\\\ consumesmore(p,k)$ hold in $X_0 \\cup \\dots \\cup X_{9k+10}$ due to the construction of $A$ , encoding of $a\\ref {a:overc:place}$ and its body atoms.", "As a result, it is not eliminated by the constraint $a\\ref {a:overc:elim}$ The set $X = X_0 \\cup \\dots \\cup X_{9k+10}$ is the answer set of $\\Pi ^7$ by the splitting sequence theorem REF .", "Each $X_i, 0 \\le i \\le 9k+10$ matches a distinct portion of $A$ , and $X = A$ , thus $A$ is an answer set of $\\Pi ^7$ .", "Next we show (REF ): Given $\\Pi ^7$ be the encoding of a Petri Net $PN(P,T,E,C,W,R,I,$ $Q,QW,Z,D)$ with initial marking $M_0$ , and $A$ be an answer set of $\\Pi ^7$ that satisfies (REF ) and (REF ), then we can construct $X=M_0,T_k,\\dots ,M_k,T_k,M_{k+1}$ from $A$ , such that it is an execution sequence of $PN$ .", "We construct the $X$ as follows: $M_i = (M_i(p_0), \\dots , M_i(p_n))$ , where $\\lbrace holds(p_0,m_{M_i(p_0)}(c),c,i), \\dots ,\\\\ holds(p_n,m_{M_i(p_n)}(c),c,i) \\rbrace \\subseteq A$ , for $c \\in C, 0 \\le i \\le k+1$ $T_i = \\lbrace t : fires(t,i) \\in A\\rbrace $ , for $0 \\le i \\le k$ and show that $X$ is indeed an execution sequence of $PN$ .", "We show this by induction over $k$ (i.e.", "given $M_k$ , $T_k$ is a valid firing set and its firing produces marking $M_{k+1}$ ).", "Base case: Let $k=0$ , and $M_0$ is a valid marking in $X$ for $PN$ , show [(1)] $T_0$ is a valid firing set for $M_0$ , and firing $T_0$ in $M_0$ produces marking $M_1$ .", "We show $T_0$ is a valid firing set for $M_0$ .", "Let $\\lbrace fires(t_0,0), \\dots , fires(t_x,0) \\rbrace $ be the set of all $fires(\\dots ,0)$ atoms in $A$ , Then for each $fires(t_i,0) \\in A$ $prenabled(t_i,0) \\in A$ – from rule $a\\ref {a:c:prfires}$ and supported rule proposition Then $enabled(t_i,0) \\in A$ – from rule $a\\ref {a:c:prenabled}$ and supported rule proposition And $notprenabled(t_i,0) \\notin A$ – from rule $a\\ref {a:c:prenabled}$ and supported rule proposition For $enabled(t_i,0) \\in A$ $notenabled(t_i,0) \\notin A$ – from rule $e\\ref {e:c:enabled}$ and supported rule proposition Then either of $body(e\\ref {e:c:ne:ptarc})$ , $body(e\\ref {e:c:ne:iptarc})$ , or $body(e\\ref {e:c:ne:tptarc})$ must not hold in $A$ for $t_i$ – from rules $body(e\\ref {e:c:ne:ptarc}), body(e\\ref {e:c:ne:iptarc}), body(e\\ref {e:c:ne:tptarc})$ and forced atom proposition Then $q_c \\lnot < {n_i}_c \\equiv q_c \\ge {n_i}_c$ in $e\\ref {e:c:ne:ptarc}$ for all $\\lbrace holds(p,q_c,c,0), \\\\ ptarc(p,t_i,{n_i}_c,c,0)\\rbrace \\subseteq A$ – from $e\\ref {e:c:ne:ptarc}$ , forced atom proposition, and given facts ($holds(p,q_c,c,0) \\in A, ptarc(p,t_i,{n_i}_c,0) \\in A$ ) And $q_c \\lnot \\ge {n_i}_c \\equiv q_c < {n_i}_c$ in $e\\ref {e:c:ne:iptarc}$ for all $\\lbrace holds(p,q_c,c,0), \\\\ iptarc(p,t_i,{n_i}_c,c,0) \\rbrace \\subseteq A, {n_i}_c=1$ ; $q_c > {n_i}_c \\equiv q_c = 0$ – from $e\\ref {e:c:ne:iptarc}$ , forced atom proposition, given facts ($holds(p,q_c,c,0) \\in A, \\\\ iptarc(p,t_i,1,c,0) \\in A$ ), and $q_c$ is a positive integer And $q_c \\lnot < {n_i}_c \\equiv q_c \\ge {n_i}_c$ in $e\\ref {e:c:ne:tptarc}$ for all $\\lbrace holds(p,q_c,c,0), \\\\ tptarc(p,t_i,{n_i}_c,c,0) \\rbrace \\subseteq A$ – from $e\\ref {e:c:ne:tptarc}$ , forced atom proposition, and given facts Then $\\forall c \\in C, (\\forall p \\in \\bullet t_i, m_{M_0(p)}(c) \\ge m_{W(p,t_i)}(c)) \\wedge (\\forall p \\in I(t_i), \\\\ m_{M_0(p)}(c) = 0) \\wedge (\\forall (p,t_i) \\in Q, m_{M_0(p)}(c) \\ge m_{QW(p,t_i)}(c))$ – from $i\\ref {i:c:init},f\\ref {f:c:ptarc},f\\ref {f:c:rptarc}$ construction, definition REF of preset $\\bullet t_i$ in $PN$ , definition REF of enabled transition in $PN$ , and that the construction of reset arcs by $f\\ref {f:c:rptarc}$ ensures $notenabled(t,0)$ is never true for a reset arc, where $holds(p,q_c,c,0) \\in A$ represents $q_c=m_{M_0(p)}(c)$ , $ptarc(p,t_i,{n_i}_c,0) \\in A$ represents ${n_i}_c=m_{W(p,t_i)}(c)$ , ${n_i}_c = m_{M_0(p)}(c)$ .", "Which is equivalent to $(\\forall p \\in \\bullet t_i, M_0(p) \\ge W(p,t_i)) \\wedge (\\forall p \\in I(t_i), M_0(p) = 0) \\wedge (\\forall (p,t_i) \\in Q, M_0(p) \\ge QW(p,t_i))$ – assuming multiset domain $C$ For $notprenabled(t_i,0) \\notin A$ Either $(\\nexists enabled(tt,0) \\in A : pp < p_i)$ or $(\\forall enabled(tt,0) \\in A : pp \\lnot < p_i)$ where $pp = Z(tt), p_i = Z(t_i)$ – from rule $a\\ref {a:c:prne}, f\\ref {f:c:pr}$ and forced atom proposition This matches the definition of an enabled priority transition Then $t_i$ is enabled and can fire in $PN$ , as a result it can belong to $T_0$ – from definition REF of enabled transition And $consumesmore \\notin A$ , since $A$ is an answer set of $\\Pi ^7$ – from rule $a\\ref {a:c:overc:elim}$ and supported rule proposition Then $\\nexists consumesmore(p,0) \\in A$ – from rule $a\\ref {a:c:overc:gen}$ and supported rule proposition Then $\\nexists \\lbrace holds(p,q_c,c,0), tot\\_decr(p,q1_c,c,0) \\rbrace \\subseteq A, q1_c>q_c$ in $body(a\\ref {a:c:overc:place})$ – from $a\\ref {a:c:overc:place}$ and forced atom proposition Then $\\nexists c \\in C \\nexists p \\in P, (\\sum _{t_i \\in \\lbrace t_0,\\dots ,t_x\\rbrace , p \\in \\bullet t_i}{m_{W(p,t_i)}(c)}+ \\\\ \\sum _{t_i \\in \\lbrace t_0,\\dots ,t_x\\rbrace , p \\in R(t_i)}{m_{M_0(p)}(c)}) > m_{M_0(p)}(c)$ – from the following $holds(p,q_c,c,0)$ represents $q_c=m_{M_0(p)}(c)$ – from rule $i\\ref {i:c:init}$ encoding, given $tot\\_decr(p,q1_c,c,0) \\in A$ if $\\lbrace del(p,{q1_0}_c,t_0,c,0), \\dots , \\\\ del(p,{q1_x}_c,t_x,c,0) \\rbrace \\subseteq A$ , where $q1_c = {q1_0}_c+\\dots +{q1_x}_c$ – from $r\\ref {r:c:totdecr}$ and forced atom proposition $del(p,{q1_i}_c,t_i,c,0) \\in A$ if $\\lbrace fires(t_i,0), ptarc(p,t_i,{q1_i}_c,c,0) \\rbrace \\subseteq A$ – from $r\\ref {r:c:del}$ and supported rule proposition $del(p,{q1_i}_c,t_i,c,0)$ represents removal of ${q1_i}_c = m_{W(p,t_i)}(c)$ tokens from $p \\in \\bullet t_i$ ; or it represents removal of ${q1_i}_c = m_{M_0(p)}(c)$ tokens from $p \\in R(t_i)$ – from rules $r\\ref {r:c:del},f\\ref {f:c:ptarc},f\\ref {f:c:rptarc}$ , supported rule proposition, and definition REF of preset in $PN$ Then the set of transitions in $T_0$ do not conflict – by the definition REF of conflicting transitions And for each $prenabled(t_j,0) \\in A$ and $fires(t_j,0) \\notin A$ , $could\\_not\\_have(t_j,0) \\in A$ , since $A$ is an answer set of $\\Pi ^7$ - from rule $a\\ref {a:c:prmaxfire:elim}$ and supported rule proposition Then $\\lbrace prenabled(t_j,0), holds(s,qq_c,c,0), ptarc(s,t_j,q_c,c,0), \\\\ tot\\_decr(s,qqq_c,c,0) \\rbrace \\subseteq A$ , such that $q_c > qq_c - qqq_c$ and $fires(t_j,0) \\notin A$ - from rule $a\\ref {a:c:prmaxfire:cnh}$ and supported rule proposition Then for an $s \\in \\bullet t_j \\cup R(t_j)$ , $q_c > m_{M_0(s)}(c) - (\\sum _{t_i \\in T_0, s \\in \\bullet t_i}{m_{W(s,t_i)}(c)} + \\sum _{t_i \\in T_0, s \\in R(t_i)}{m_{M_0(s)}(c))}$ , where $q_c=m_{W(s,t_j)}(c) \\text{~if~} s \\in \\bullet t_j, \\text{~or~} m_{M_0(s)}(c)$ $ \\text{~otherwise}$ .", "$ptarc(s,t_i,q_c,c,0)$ represents $q_c=m_{W(s,t_i)}(c)$ if $(s,t_i) \\in E^-$ or $q_c=m_{M_0(s)}(c)$ if $s \\in R(t_i)$ – from rule $f\\ref {f:c:ptarc},f\\ref {f:c:rptarc}$ construction $holds(s,qq_c,c,0)$ represents $qq_c=m_{M_0(s)}(c)$ – from $i\\ref {i:c:init}$ construction $tot\\_decr(s,qqq_c,c,0) \\in A$ if $\\lbrace del(s,{qqq_0}_c,t_0,c,0), \\dots , \\\\ del(s,{qqq_x}_c,t_x,c,0) \\rbrace \\subseteq A$ – from rule $r\\ref {r:c:totdecr}$ construction and supported rule proposition $del(s,{qqq_i}_c,t_i,c,0) \\in A$ if $\\lbrace fires(t_i,0), ptarc(s,t_i,{qqq_i}_c,c,0) \\rbrace \\subseteq A$ – from rule $r\\ref {r:c:del}$ and supported rule proposition $del(s,{qqq_i}_c,t_i,c,0)$ either represents ${qqq_i}_c = m_{W(s,t_i)}(c) : t_i \\in T_0, (s,t_i) \\in E^-$ , or ${qqq_i}_c = m_{M_0(t_i)}(c) : t_i \\in T_0, s \\in R(t_i)$ – from rule $f\\ref {f:c:ptarc},f\\ref {f:c:rptarc}$ construction $tot\\_decr(q,qqq_c,c,0)$ represents $\\sum _{t_i \\in T_0, s \\in \\bullet t_i}{m_{W(s,t_i)}(c)} + \\\\ \\sum _{t_i \\in T_0, s \\in R(t_i)}{m_{M_0(s)}(c)}$ – from (C,D,E) above Then firing $T_0 \\cup \\lbrace t_j \\rbrace $ would have required more tokens than are present at its source place $s \\in \\bullet t_j \\cup R(t_j)$ .", "Thus, $T_0$ is a maximal set of transitions that can simultaneously fire.", "And for each reset transition $t_r$ with $prenabled(t_r,0) \\in A$ , $fires(t_r,0) \\in A$ , since $A$ is an answer set of $\\Pi ^7$ - from rule $f\\ref {f:c:pr:rptarc:elim}$ and supported rule proposition Then, the firing set $T_0$ satisfies the reset-transition requirement of definition REF (firing set) Then $\\lbrace t_0, \\dots , t_x\\rbrace = T_0$ – using 1(a),1(b),1(d) above; and using 1(c) it is a maximal firing set Let $holds(p,q_c,c,1) \\in A$ Then $\\lbrace holds(p,q1_c,c,0), tot\\_incr(p,q2_c,c,0), tot\\_decr(p,q3_c,c,0) \\rbrace \\subseteq A : q_c=q1_c+q2_c-q3_c$ – from rule $r\\ref {r:c:nextstate}$ and supported rule proposition Then, $holds(p,q1_c,c,0) \\in A$ represents $q1_c=m_{M_0(p)}(c)$ – given ; and $\\lbrace add(p,{q2_0}_c,t_0,c,0), \\dots , $ $add(p,{q2_j}_c,t_j,c,0)\\rbrace \\subseteq A : {q2_0}_c + \\dots + {q2_j}_c = q2_c$ and $\\lbrace del(p,{q3_0}_c,t_0,c,0), \\dots , $ $del(p,{q3_l}_c,t_l,c,0)\\rbrace \\subseteq A : {q3_0}_c + \\dots + {q3_l}_c = q3_c$ – rules $r\\ref {r:c:totincr},r\\ref {r:c:totdecr}$ and supported rule proposition, respectively Then $\\lbrace fires(t_0,0), \\dots , fires(t_j,0) \\rbrace \\subseteq A$ and $\\lbrace fires(t_0,0), \\dots , fires(t_l,0) \\rbrace \\\\ \\subseteq A$ – rules $r\\ref {r:c:dur:add},r\\ref {r:c:del}$ and supported rule proposition, respectively Then $\\lbrace fires(t_0,0), \\dots , $ $fires(t_j,0) \\rbrace \\cup \\lbrace fires(t_0,0), \\dots , $ $fires(t_l,0) \\rbrace \\subseteq $ $A = \\lbrace fires(t_0,0), \\dots , $ $fires(t_x,0) \\rbrace \\subseteq A$ – set union of subsets Then for each $fires(t_x,0) \\in A$ we have $t_x \\in T_0$ – already shown in item REF above Then $q_c = m_{M_0(p)}(c) + \\sum _{t_x \\in T_0, p \\in t_x \\bullet , 0+D(t_x)-1=0}{m_{W(t_x,p)}(c)} - \\\\ (\\sum _{t_x \\in T_0 \\wedge p \\in \\bullet t_x}{m_{W(p,t_x)}(c)} + $ $\\sum _{t_x \\in T_0 \\wedge p \\in R(t_x)}{m_{M_0(p)}(c)})$ – from (REF ) above and the following Each $add(p,{q_j}_c,t_j,c,0) \\in A$ represents ${q_j}_c=m_{W(t_j,p)}(c)$ for $p \\in t_j \\bullet $ – rule $r\\ref {r:c:dur:add},f\\ref {f:c:dur:tparc}$ encoding, and definition REF of transition execution in $PN$ Each $del(p,t_y,{q_y}_c,c,0) \\in A$ represents either ${q_y}_c=m_{W(p,t_y)}(c)$ for $p \\in \\bullet t_y$ , or ${q_y}_c=m_{M_0(p)}(c)$ for $p \\in R(t_y)$ – from rule $r\\ref {r:c:del},f\\ref {f:c:ptarc}$ encoding and definition REF of transition execution in $PN$ ; or from rule $r\\ref {r:c:del},f\\ref {f:c:rptarc}$ encoding and definition of reset arc in $PN$ Each $tot\\_incr(p,q2_c,c,0) \\in A$ represents $q2_c=\\sum _{t_x \\in T_0 \\wedge p \\in t_x \\bullet , 0+D(t_x)-1=0}{m_{W(t_x,p)}(c)}$ – aggregate assignment atom semantics in rule $r\\ref {r:c:totincr}$ Each $tot\\_decr(p,q3_c,c,0) \\in A$ represents $q3_c=\\sum _{t_x \\in T_0 \\wedge p \\in \\bullet t_x}{m_{W(p,t_x)}(c)} \\\\ + \\sum _{t_x \\in T_0 \\wedge p \\in R(t_x)}{m_{M_0(p)}(c)}$ – aggregate assignment atom semantics in rule $r\\ref {r:c:totdecr}$ Then, $m_{M_1(p)}(c) = q_c$ – since $holds(p,q_c,c,1) \\in A$ encodes $q_c=m_{M_1(p)}(c)$ – from construction Inductive Step: Let $k > 0$ , and $M_k$ is a valid marking in $X$ for $PN$ , show [(1)] $T_k$ is a valid firing set for $M_k$ , and firing $T_k$ in $M_k$ produces marking $M_{k+1}$ .", "We show that $T_k$ is a valid firing set in $M_k$ .", "Let $\\lbrace fires(t_0,k), \\dots , fires(t_x,k) \\rbrace $ be the set of all $fires(\\dots ,k)$ atoms in $A$ , We show that $T_k$ is a valid firing set in $M_k$ .", "Then for each $fires(t_i,k) \\in A$ $prenabled(t_i,k) \\in A$ – from rule $a\\ref {a:c:prfires}$ and supported rule proposition Then $enabled(t_i,k) \\in A$ – from rule $a\\ref {a:c:prenabled}$ and supported rule proposition And $notprenabled(t_i,k) \\notin A$ – from rule $a\\ref {a:c:prenabled}$ and supported rule proposition For $enabled(t_i,k) \\in A$ $notenabled(t_i,k) \\notin A$ – from rule $e\\ref {e:c:enabled}$ and supported rule proposition Then either of $body(e\\ref {e:c:ne:ptarc})$ , $body(e\\ref {e:c:ne:iptarc})$ , or $body(e\\ref {e:c:ne:tptarc})$ must not hold in $A$ for $t_i$ – from rules $body(e\\ref {e:c:ne:ptarc}), body(e\\ref {e:c:ne:iptarc}), body(e\\ref {e:c:ne:tptarc})$ and forced atom proposition Then $q_c \\lnot < {n_i}_c \\equiv q_c \\ge {n_i}_c$ in $e\\ref {e:c:ne:ptarc}$ for all $\\lbrace holds(p,q_c,c,k), \\\\ ptarc(p,t_i,{n_i}_c,c,k)\\rbrace \\subseteq A$ – from $e\\ref {e:c:ne:ptarc}$ , forced atom proposition, and given facts ($holds(p,q_c,c,k) \\in A, ptarc(p,t_i,{n_i}_c,k) \\in A$ ) And $q_c \\lnot \\ge {n_i}_c \\equiv q_c < {n_i}_c$ in $e\\ref {e:c:ne:iptarc}$ for all $\\lbrace holds(p,q_c,c,k), \\\\ iptarc(p,t_i,{n_i}_c,c,k) \\rbrace \\subseteq A, {n_i}_c=1$ ; $q_c > {n_i}_c \\equiv q_c = 0$ – from $e\\ref {e:c:ne:iptarc}$ , forced atom proposition, given facts ($holds(p,q_c,c,k) \\in A, \\\\ iptarc(p,t_i,1,c,k) \\in A$ ), and $q_c$ is a positive integer And $q_c \\lnot < {n_i}_c \\equiv q_c \\ge {n_i}_c$ in $e\\ref {e:c:ne:tptarc}$ for all $\\lbrace holds(p,q_c,c,k), \\\\ tptarc(p,t_i,{n_i}_c,c,k) \\rbrace \\subseteq A$ – from $e\\ref {e:c:ne:tptarc}$ , forced atom proposition, and given facts Then $\\forall c \\in C, (\\forall p \\in \\bullet t_i, m_{M_k(p)}(c) \\ge m_{W(p,t_i)}(c)) \\wedge (\\forall p \\in I(t_i), \\\\ m_{M_k(p)}(c) = 0) \\wedge (\\forall (p,t_i) \\in Q, m_{M_k(p)}(c) \\ge m_{QW(p,t_i)}(c))$ – from the inductive assumption, $f\\ref {f:c:ptarc},f\\ref {f:c:rptarc}$ construction, definition REF of preset $\\bullet t_i$ in $PN$ , definition REF of enabled transition in $PN$ , and that the construction of reset arcs by $f\\ref {f:c:rptarc}$ ensures $notenabled(t,k)$ is never true for a reset arc, where $holds(p,q_c,c,k) \\in A$ represents $q_c=m_{M_k(p)}(c)$ , $ptarc(p,t_i,{n_i}_c,k) \\in A$ represents ${n_i}_c=m_{W(p,t_i)}(c)$ , ${n_i}_c = m_{M_k(p)}(c)$ .", "Which is equivalent to $(\\forall p \\in \\bullet t_i, M_k(p) \\ge W(p,t_i)) \\wedge (\\forall p \\in I(t_i), M_k(p) = 0) \\wedge (\\forall (p,t_i) \\in Q, M_k(p) \\ge QW(p,t_i))$ – assuming multiset domain $C$ For $notprenabled(t_i,k) \\notin A$ Either $(\\nexists enabled(tt,k) \\in A : pp < p_i)$ or $(\\forall enabled(tt,k) \\in A : pp \\lnot < p_i)$ where $pp = Z(tt), p_i = Z(t_i)$ – from rule $a\\ref {a:c:prne}, f\\ref {f:c:pr}$ and forced atom proposition This matches the definition of an enabled priority transition Then $t_i$ is enabled and can fire in $PN$ , as a result it can belong to $T_k$ – from definition REF of enabled transition And $consumesmore \\notin A$ , since $A$ is an answer set of $\\Pi ^7$ – from rule $a\\ref {a:c:overc:elim}$ and supported rule proposition Then $\\nexists consumesmore(p,k) \\in A$ – from rule $a\\ref {a:c:overc:gen}$ and supported rule proposition Then $\\nexists \\lbrace holds(p,q_c,c,k), tot\\_decr(p,q1_c,c,k) \\rbrace \\subseteq A, q1_c>q_c$ in $body(a\\ref {a:c:overc:place})$ – from $a\\ref {a:c:overc:place}$ and forced atom proposition Then $\\nexists c \\in C \\nexists p \\in P, (\\sum _{t_i \\in \\lbrace t_0,\\dots ,t_x\\rbrace , p \\in \\bullet t_i}{m_{W(p,t_i)}(c)}+ \\\\ \\sum _{t_i \\in \\lbrace t_0,\\dots ,t_x\\rbrace , p \\in R(t_i)}{m_{M_k(p)}(c)}) > m_{M_k(p)}(c)$ – from the following $holds(p,q_c,c,k)$ represents $q_c=m_{M_k(p)}(c)$ – from rule $PN$ encoding, given $tot\\_decr(p,q1_c,c,k) \\in A$ if $\\lbrace del(p,{q1_0}_c,t_0,c,k), \\dots , \\\\ del(p,{q1_x}_c,t_x,c,k) \\rbrace \\subseteq A$ , where $q1_c = {q1_0}_c+\\dots +{q1_x}_c$ – from $r\\ref {r:c:totdecr}$ and forced atom proposition $del(p,{q1_i}_c,t_i,c,k) \\in A$ if $\\lbrace fires(t_i,k), ptarc(p,t_i,{q1_i}_c,c,k) \\rbrace \\subseteq A$ – from $r\\ref {r:c:del}$ and supported rule proposition $del(p,{q1_i}_c,t_i,c,k)$ either represents removal of ${q1_i}_c = m_{W(p,t_i)}(c)$ tokens from $p \\in \\bullet t_i$ ; or it represents removal of ${q1_i}_c = m_{M_k(p)}(c)$ tokens from $p \\in R(t_i)$ – from rules $r\\ref {r:c:del},f\\ref {f:c:ptarc},f\\ref {f:c:rptarc}$ , supported rule proposition, and definition REF of transition execution in $PN$ Then the set of transitions in $T_k$ do not conflict – by the definition REF of conflicting transitions And for each $prenabled(t_j,k) \\in A$ and $fires(t_j,k) \\notin A$ , $could\\_not\\_have(t_j,k) \\in A$ , since $A$ is an answer set of $\\Pi ^7$ - from rule $a\\ref {a:c:prmaxfire:elim}$ and supported rule proposition Then $\\lbrace prenabled(t_j,k), holds(s,qq_c,c,k), ptarc(s,t_j,q_c,c,k), \\\\ tot\\_decr(s,qqq_c,c,k) \\rbrace \\subseteq A$ , such that $q_c > qq_c - qqq_c$ and $fires(t_j,k) \\notin A$ - from rule $a\\ref {a:c:prmaxfire:cnh}$ and supported rule proposition Then for an $s \\in \\bullet t_j \\cup R(t_j)$ , $q_c > m_{M_k(s)}(c) - (\\sum _{t_i \\in T_k, s \\in \\bullet t_i}{m_{W(s,t_i)}(c)} + \\sum _{t_i \\in T_k, s \\in R(t_i)}{m_{M_k(s)}(c))}$ , where $q_c=m_{W(s,t_j)}(c) \\text{~if~} s \\in \\bullet t_j, \\text{~or~} m_{M_k(s)}(c) $ $\\text{~otherwise}$ .", "$ptarc(s,t_i,q_c,c,k)$ represents $q_c=m_{W(s,t_i)}(c)$ if $(s,t_i) \\in E^-$ or $q_c=m_{M_k(s)}(c)$ if $s \\in R(t_i)$ – from rule $f\\ref {f:c:ptarc},f\\ref {f:c:rptarc}$ construction $holds(s,qq_c,c,k)$ represents $qq_c=m_{M_k(s)}(c)$ – from $i\\ref {i:c:init}$ construction $tot\\_decr(s,qqq_c,c,k) \\in A$ if $\\lbrace del(s,{qqq_0}_c,t_0,c,k), \\dots , \\\\ del(s,{qqq_x}_c,t_x,c,k) \\rbrace \\subseteq A$ – from rule $r\\ref {r:c:totdecr}$ construction and supported rule proposition $del(s,{qqq_i}_c,t_i,c,k) \\in A$ if $\\lbrace fires(t_i,k), ptarc(s,t_i,{qqq_i}_c,c,k) \\rbrace \\subseteq A$ – from rule $r\\ref {r:c:del}$ and supported rule proposition $del(s,{qqq_i}_c,t_i,c,k)$ represents ${qqq_i}_c = m_{W(s,t_i)}(c) : t_i \\in T_k, (s,t_i) \\in E^-$ , or ${qqq_i}_c = m_{M_k(t_i)}(c) : t_i \\in T_k, s \\in R(t_i)$ – from rule $f\\ref {f:c:ptarc},f\\ref {f:c:rptarc}$ construction $tot\\_decr(q,qqq_c,c,k)$ represents $\\sum _{t_i \\in T_k, s \\in \\bullet t_i}{m_{W(s,t_i)}(c)} + \\\\ \\sum _{t_i \\in T_k, s \\in R(t_i)}{m_{M_k(s)}(c)}$ – from (C,D,E) above Then firing $T_k \\cup \\lbrace t_j \\rbrace $ would have required more tokens than are present at its source place $s \\in \\bullet t_j \\cup R(t_j)$ .", "Thus, $T_k$ is a maximal set of transitions that can simultaneously fire.", "And for each reset transition $t_r$ with $enabled(t_r,k) \\in A$ , $fires(t_r,k) \\in A$ , since $A$ is an answer set of $\\Pi ^7$ - from rule $f\\ref {f:c:pr:rptarc:elim}$ and supported rule proposition Then the firing set $T_k$ satisfies the reset transition requirement of definition REF (firing set) Then $\\lbrace t_0, \\dots , t_x\\rbrace = T_k$ – using 1(a),1(b), 1(d) above; and using 1(c) it is a maximal firing set We show that $M_{k+1}$ is produced by firing $T_k$ in $M_k$ .", "Let $holds(p,q_c,c,k+1) \\in A$ Then $\\lbrace holds(p,q1_c,c,k), tot\\_incr(p,q2_c,c,k), tot\\_decr(p,q3_c,c,k) \\rbrace \\subseteq A : q_c=q1_c+q2_c-q3_c$ – from rule $r\\ref {r:c:nextstate}$ and supported rule proposition Then $holds(p,q1_c,c,k) \\in A$ represents $q1_c=m_{M_k(p)}(c)$ – inductive assumption; and $\\lbrace add(p,{q2_0}_c,t_0,c,k), \\dots , $ $add(p,{q2_j}_c,t_j,c,k)\\rbrace \\subseteq A : {q2_0}_c + \\dots + {q2_j}_c = q2_c$ and $\\lbrace del(p,{q3_0}_c,t_0,c,k), \\dots , $ $del(p,{q3_l}_c,t_l,c,k)\\rbrace \\subseteq A : {q3_0}_c + \\dots + {q3_l}_c = q3_c$ – rules $r\\ref {r:c:totincr},r\\ref {r:c:totdecr}$ and supported rule proposition, respectively Then $\\lbrace fires(t_0,k), \\dots , fires(t_j,k) \\rbrace \\subseteq A$ and $\\lbrace fires(t_0,k), \\dots , fires(t_l,k) \\rbrace \\\\ \\subseteq A$ – rules $r\\ref {r:c:dur:add},r\\ref {r:c:del}$ and supported rule proposition, respectively Then $\\lbrace fires(t_0,k), \\dots , fires(t_j,k) \\rbrace \\cup \\lbrace fires(t_0,k), \\dots , fires(t_l,k) \\rbrace \\subseteq A = \\lbrace fires(t_0,k), \\dots , fires(t_x,k) \\rbrace \\subseteq A$ – set union of subsets Then for each $fires(t_x,k) \\in A$ we have $t_x \\in T_k$ – already shown in item REF above Then $q_c = m_{M_k(p)}(c) + \\sum _{t_x \\in T_l, p \\in t_x \\bullet , 0 \\le l \\le k, l+D(t_x)-1=k}{m_{W(t_x,p)}(c)} - \\\\ (\\sum _{t_x \\in T_k \\wedge p \\in \\bullet t_x}{m_{W(p,t_x)}(c)} + \\sum _{t_x \\in T_k \\wedge p \\in R(t_x)}{m_{M_k(p)}(c)})$ – from (REF ) above and the following Each $add(p,{q_j}_c,t_j,c,k) \\in A$ represents ${q_j}_c=m_{W(t_j,p)}(c)$ for $p \\in t_j \\bullet $ – rule $r\\ref {r:c:dur:add},f\\ref {f:c:dur:tparc}$ encoding, and definition REF of postset in $PN$ Each $del(p,t_y,{q_y}_c,c,k) \\in A$ represents either ${q_y}_c=m_{W(p,t_y)}(c)$ for $p \\in \\bullet t_y$ , or ${q_y}_c=m_{M_k(p)}(c)$ for $p \\in R(t_y)$ – from rule $r\\ref {r:c:del},f\\ref {f:c:ptarc}$ encoding and definition REF of preset in $PN$ ; or from rule $r\\ref {r:c:del},f\\ref {f:c:rptarc}$ encoding and definition of reset arc in $PN$ Each $tot\\_incr(p,q2_c,c,k) \\in A$ represents $q2_c=\\sum _{t_x \\in T_l, p \\in t_x \\bullet , 0 \\le l \\le k, l+D(t_x)-1=k}{m_{W(t_x,p)}(c)}$ – aggregate assignment atom semantics in rule $r\\ref {r:c:totincr}$ Each $tot\\_decr(p,q3_c,c,k) \\in A$ represents $q3_c=\\sum _{t_x \\in T_k \\wedge p \\in \\bullet t_x}{m_{W(p,t_x)}(c)} \\\\ + \\sum _{t_x \\in T_k \\wedge p \\in R(t_x)}{m_{M_k(p)}(c)}$ – aggregate assignment atom semantics in rule $r\\ref {r:c:totdecr}$ Then, $m_{M_{k+1}(p)}(c) = q_c$ – since $holds(p,q_c,c,k+1) \\in A$ encodes $q_c=m_{M_{k+1}(p)}(c)$ – from construction As a result, for any $n > k$ , $T_n$ will be a valid firing set for $M_n$ and $M_{n+1}$ will be its target marking.", "Conclusion: Since both (REF ) and (REF ) hold, $X=M_0,T_k,M_1,\\dots ,M_k,T_{k+1}$ is an execution sequence of $PN(P,T,E,C,W,R,I,Q,QW,Z,D)$ (w.r.t $M_0$ ) iff there is an answer set $A$ of $\\Pi ^7(PN,M_0,k,ntok)$ such that (REF ) and (REF ) hold.", "Complete Set of Queries Used for Drug-Drug InteractionDrug-Drug Interaction Queries Drug Activates Gene 1 //S{/NP{//?[Value='activation'](kw1)=>//?[Value='of'](kw2)=>//?[Tag='GENE'](kw0)=>//?[Value='by'](kw4)=>//?", "[Tag='DRUG'](kw3)}} ::: distinct sent.cid, kw3.value, kw1.value, kw0.value, sent.value 2 //NP{/NP{/?[Tag='DRUG'](kw3)=>/?[Value='activation'](kw1)}=>/PP{/?[Value='of'](kw2)=>//?", "[Tag='GENE'](kw0)}} ::: distinct sent.cid, kw3.value, kw1.value, kw0.value, sent.value 3 //NP{/?[Value='activation'](kw1)=>/PP{/?[Value='of'](kw2)=>//?[Tag='GENE'](kw0)=>//?[Value='by'](kw4)=>//?", "[Tag='DRUG'](kw3)}} ::: distinct sent.cid, kw3.value, kw1.value, kw0.value, sent.value 4 //S{/NP{//?[Tag='GENE'](kw2)}=>/VP{//?[Value='activation'](kw1)=>//?", "[Tag='DRUG'](kw0)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 5 //S{/NP{//?[Tag='DRUG'](kw0)}=>/VP{//?[Value='activation'](kw1)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 6 //NP{/NP{/?[Tag='DRUG'](kw0)=>/?[Value='activation'](kw1)}=>/PP{//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 7 //S{/NP{//?[Value='activation'](kw1)=>//?[Tag='GENE'](kw2)}=>/VP{//?", "[Tag='DRUG'](kw0)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 8 //S{/NP{//?[Value='activation'](kw1)=>//?[Value='of'](kw2)=>//?[Tag='GENE'](kw0)=>//?[Value='by'](kw4)=>//?", "[Tag='DRUG'](kw3)}} ::: distinct sent.cid, kw3.value, kw1.value, kw0.value, sent.value 9 //NP{/NP{/?[Tag='DRUG'](kw3)=>/?[Value='activation'](kw1)}=>/PP{/?[Value='of'](kw2)=>//?", "[Tag='GENE'](kw0)}} ::: distinct sent.cid, kw3.value, kw1.value, kw0.value, sent.value 10 //NP{/?[Value='activation'](kw1)=>/PP{/?[Value='of'](kw2)=>//?[Tag='GENE'](kw0)=>//?[Value='by'](kw4)=>//?", "[Tag='DRUG'](kw3)}} ::: distinct sent.cid, kw3.value, kw1.value, kw0.value, sent.value Gene Induces Gene 1 //S{/NP{//?[Tag='GENE'](kw0)}=>/VP{//?[Value='induced'](kw1)=>//?[Value='by'](kw3)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 2 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{//?[Value='induced'](kw1)=>//?[Value='by'](kw3)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 3 //S{/NP{//?[Tag='GENE'](kw0)}=>/VP{//?[Value IN {'increase','increased'}](kw1)=>//?[Tag='GENE'](kw2)=>//?", "[Value='activity'](kw3)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 4 //S{/NP{//?[Tag='GENE'](kw0)=>//?[Value='activity'](kw3)}=>/VP{//?[Value='increase'](kw1)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 5 //S{/NP{//?[Tag='GENE'](kw0)}=>/VP{//?[Value IN {'stimulates','stimulate', 'stimulated'}](kw1)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 6 //S{/NP{//?[Tag='GENE'](kw0)}=>/VP{/?[Value IN {'stimulates','stimulate'}](kw1)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 7 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/?[Value='stimulates'](kw1)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 8 //S{/NP{//?[Tag='GENE'](kw0)}=>/VP{/?[Value='activated'](kw1)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 9 //NP{/NP{/?[Value='induction'](kw1)}=>/PP{/?[Value='of'](kw3)=>/NP{//?[Tag='GENE'](kw0)=>//?[Value='by'](kw4)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 10 //NP{/NP{/?[Value='stimulation'](kw1)}=>/PP{/?[Value='of'](kw3)=>/NP{//?[Tag='GENE'](kw0)=>//?[Value='by'](kw4)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 11 //NP{/NP{/?[Value='activation'](kw1)}=>/PP{/?[Value IN {'of'}](kw3)=>/NP{//?[Tag='GENE'](kw0)=>//?[Value='by'](kw4)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 12 //S{/NP{//?[Tag='GENE'](kw0)}=>/VP{//?[Value='inducible'](kw1)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 13 //S{/NP{//?[Tag='GENE'](kw0)}=>/VP{//?[Value='induced'](kw1)=>//?[Value='by'](kw3)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 14 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{//?[Value='induced'](kw1)=>//?[Value='by'](kw3)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 15 //S{/NP{//?[Tag='GENE'](kw0)}=>/VP{//?[Value IN {'increase','increased'}](kw1)=>//?[Tag='GENE'](kw2)=>//?", "[Value='activity'](kw3)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 16 //S{/NP{//?[Tag='GENE'](kw0)=>//?[Value='activity'](kw3)}=>/VP{//?[Value='increase'](kw1)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 17 //S{/NP{//?[Tag='GENE'](kw0)}=>/VP{//?[Value='stimulated'](kw1)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 18 //S{/NP{//?[Tag='GENE'](kw0)}=>/VP{/?[Value IN {'stimulates','stimulate'}](kw1)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 19 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/?[Value='stimulates'](kw1)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 20 //S{/NP{//?[Tag='GENE'](kw0)}=>/VP{/?[Value='activated'](kw1)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 21 //VP{/?[Value='activated'](kw1)=>/PP{//?[Tag='GENE'](kw0)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 22 //NP{/NP{/?[Value='induction'](kw1)}=>/PP{/?[Value='of'](kw3)=>/NP{//?[Tag='GENE'](kw0)=>//?[Value='by'](kw4)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 23 //NP{/NP{/?[Value='stimulation'](kw1)}=>/PP{/?[Value='of'](kw3)=>/NP{//?[Tag='GENE'](kw0)=>//?[Value='by'](kw4)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 24 //NP{/NP{/?[Value='activation'](kw1)}=>/PP{/?[Value IN {'of'}](kw3)=>/NP{//?[Tag='GENE'](kw0)=>//?[Value='by'](kw4)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 25 //S{/NP{//?[Tag='GENE'](kw0)}=>/VP{//?[Value='inducible'](kw1)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value Gene Inhibits Gene 1 //S{/NP{//?[Tag='GENE'](kw0)}=>/VP{//?[Value='induced'](kw1)=>//?[Value='by'](kw3)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 2 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{//?[Value='induced'](kw1)=>//?[Value='by'](kw3)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 3 //S{/NP{//?[Tag='GENE'](kw0)}=>/VP{//?[Value IN {'increase','increased'}](kw1)=>//?[Tag='GENE'](kw2)=>//?", "[Value='activity'](kw3)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 4 //S{/NP{//?[Tag='GENE'](kw0)=>//?[Value='activity'](kw3)}=>/VP{//?[Value='increase'](kw1)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 5 //S{/NP{//?[Tag='GENE'](kw0)}=>/VP{//?[Value IN {'stimulates','stimulate', 'stimulated'}](kw1)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 6 //S{/NP{//?[Tag='GENE'](kw0)}=>/VP{/?[Value IN {'stimulates','stimulate'}](kw1)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 7 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/?[Value='stimulates'](kw1)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 8 //S{/NP{//?[Tag='GENE'](kw0)}=>/VP{/?[Value='activated'](kw1)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 9 //NP{/NP{/?[Value='induction'](kw1)}=>/PP{/?[Value='of'](kw3)=>/NP{//?[Tag='GENE'](kw0)=>//?[Value='by'](kw4)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 10 //NP{/NP{/?[Value='stimulation'](kw1)}=>/PP{/?[Value='of'](kw3)=>/NP{//?[Tag='GENE'](kw0)=>//?[Value='by'](kw4)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 11 //NP{/NP{/?[Value='activation'](kw1)}=>/PP{/?[Value IN {'of'}](kw3)=>/NP{//?[Tag='GENE'](kw0)=>//?[Value='by'](kw4)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 12 //S{/NP{//?[Tag='GENE'](kw0)}=>/VP{//?[Value='inducible'](kw1)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value Drug Changes Gene Expression/Activity 1 //S{/?[Tag='DRUG'](kw2)=>/?[Value IN {'increased','increase','increases'}](kw1)=>/?[Value IN {'levels','level'}](kw3)=>/?", "[Tag='GENE'](kw0)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 2 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/?[Value IN {'increased','increases'}](kw1)=>/NP{//?[Tag='GENE'](kw0)=>//?", "[Value IN {'expression','level','activity','activities','levels'}](kw3)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 3 //S{/?[Tag='DRUG'](kw2)=>/VP{/NP{//?[Value IN {'increases','increase'}](kw1)=>//?[Tag='GENE'](kw0)=>//?", "[Value IN {'levels','activity'}](kw3)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 4 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value IN {'increased','increases'}](kw1)=>/NP{//?[Tag='GENE'](kw0)=>//?", "[Value IN {'activity','activities','levels'}](kw3)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 5 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/?[Value='increased'](kw1)=>/NP{//?[Value IN {'activity','levels','expression'}](kw3)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 6 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{//?[Value IN {'increased','increases','increase'}](kw1)=>//?[Tag='GENE'](kw0)=>//?", "[Value IN {'activity','levels'}](kw3)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 7 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{/?[Value='increased'](kw1)=>//?[Tag='GENE'](kw0)=>//?", "[Value IN {'expression','activity'}](kw3)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 8 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value IN {'increased','increases'}](kw1)=>/NP{//?[Value IN {'expression','activity','activities','level','levels'}](kw3)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 9 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value IN {'decreased','decreases'}](kw1)=>/NP{//?[Tag='GENE'](kw0)=>//?", "[Value IN {'activity','expression'}](kw3)}}} ::: distinct sent.cid, kw2.value, kw1,value, kw0.value, sent.value 10 //S{/?[Tag='DRUG'](kw2)=>/?[Value IN {'increased','increase','increases'}](kw1)=>/?[Value IN {'levels','level'}](kw3)=>/?", "[Tag='GENE'](kw0)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 11 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/?[Value IN {'increased','increases'}](kw1)=>/NP{//?[Tag='GENE'](kw0)=>//?", "[Value IN {'expression','level','activity','activities','levels'}](kw3)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 12 //S{/?[Tag='DRUG'](kw2)=>/VP{/NP{//?[Value IN {'increases','increase'}](kw1)=>//?[Tag='GENE'](kw0)=>//?", "[Value IN {'levels','activity'}](kw3)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 13 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value IN {'increased','increases'}](kw1)=>/NP{//?[Tag='GENE'](kw0)=>//?", "[Value IN {'activity','activities','levels'}](kw3)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 14 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/?[Value='increased'](kw1)=>/NP{//?[Value IN {'activity','levels','expression'}](kw3)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 15 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{//?[Value IN {'increased','increases','increase'}](kw1)=>//?[Tag='GENE'](kw0)=>//?", "[Value IN {'activity','levels'}](kw3)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 16 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{/?[Value='increased'](kw1)=>//?[Tag='GENE'](kw0)=>//?", "[Value IN {'expression','activity'}](kw3)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 17 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value IN {'increased','increases'}](kw1)=>/NP{//?[Value IN {'expression','activity','activities','level','levels'}](kw3)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 18 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value IN {'decreased','decreases'}](kw1)=>/NP{//?[Tag='GENE'](kw0)=>//?", "[Value IN {'activity','expression'}](kw3)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value Drug Induces/Stimulates Gene 1 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value IN {'stimulated','induced'}](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 2 //VP{/?[Value IN {'stimulated','induced'}](kw1)=>/PP{/NP{//?[Tag='DRUG'](kw2)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 3 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/?[Value IN {'stimulated','induced'}](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 4 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/?[Value IN {'stimulated','induced'}](kw1)=>/PP{//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 5 //S{/?[Tag='DRUG'](kw2)=>/VP{/NP{//?[Tag='GENE'](kw0)=>//?", "[Value IN {'stimulated','induced'}](kw1)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 6 //S{/NP{/PP{//?[Tag='DRUG'](kw2)}}=>/VP{/?[Value IN {'stimulated','induced'}](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 7 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/VP{/?[Value IN {'stimulated','induced'}](kw1)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 8 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/VP{//?[Value='induced'](kw1)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 9 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value IN {'stimulated','induced'}](kw1)=>/NP{/?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 10 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{//?[Value IN {'stimulate','induce'}](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 11 //NP{/NP{/?[Value IN {'induction','stimulation'}](kw1)}=>/PP{/NP{//?[Tag='GENE'](kw0)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 12 //S{/NP{/?[Value IN {'stimulation','induction'}](kw1)=>/PP{//?[Tag='GENE'](kw0)}}=>/VP{/VP{//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 13 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/NP{//?[Value IN {'stimulation','induction'}](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 14 //NP{/NP{/NP{/?[Value='induction'](kw1)}=>/PP{//?[Tag='GENE'](kw0)}}=>/PP{/NP{/?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 15 //NP{/NP{/?[Tag='GENE'](kw0)=>/?[Value IN {'stimulation','induction'}](kw1)}=>/PP{/NP{/?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 16 //NP{/?[Value IN {'stimulation','induction'}](kw1)=>/PP{/NP{//?[Tag='GENE'](kw0)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 17 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/?[Value IN {'stimulates','induces'}](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 18 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value IN {'stimulates','induces'}](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 19 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value IN {'stimulated','induced'}](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 20 //VP{/?[Value IN {'stimulated','induced'}](kw1)=>/PP{/NP{//?[Tag='DRUG'](kw2)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 21 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/?[Value IN {'stimulated','induced'}](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 22 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/?[Value IN {'stimulated','induced'}](kw1)=>/PP{//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 23 //S{/?[Tag='DRUG'](kw2)=>/VP{/NP{//?[Tag='GENE'](kw0)=>//?", "[Value IN {'stimulated','induced'}](kw1)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 24 //S{/NP{/PP{//?[Tag='DRUG'](kw2)}}=>/VP{/?[Value IN {'stimulated','induced'}](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 25 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/VP{/?[Value IN {'stimulated','induced'}](kw1)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 26 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/VP{//?[Value='induced'](kw1)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 27 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value IN {'stimulated','induced'}](kw1)=>/NP{/?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 28 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{//?[Value IN {'stimulate','induce'}](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 29 //NP{/NP{/?[Value IN {'induction','stimulation'}](kw1)}=>/PP{/NP{//?[Tag='GENE'](kw0)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 30 //S{/NP{/?[Value IN {'stimulation','induction'}](kw1)=>/PP{//?[Tag='GENE'](kw0)}}=>/VP{/VP{//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 31 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/NP{//?[Value IN {'stimulation','induction'}](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 32 //NP{/NP{/NP{/?[Value='induction'](kw1)}=>/PP{//?[Tag='GENE'](kw0)}}=>/PP{/NP{/?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 33 //NP{/NP{/?[Tag='GENE'](kw0)=>/?[Value IN {'stimulation','induction'}](kw1)}=>/PP{/NP{/?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 34 //NP{/?[Value IN {'stimulation','induction'}](kw1)=>/PP{/NP{//?[Tag='GENE'](kw0)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 35 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/?[Value IN {'stimulates','induces'}](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 36 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value IN {'stimulates','induces'}](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value Drug Inhibits Gene 1 //NP{/NP{/NP{/?[Value='inhibitory'](kw1)}=>/PP{//?[Tag='DRUG'](kw2)}}=>/PP{/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 2 //S{/?[Tag='DRUG'](kw2)=>/VP{/NP{/?[Tag='GENE'](kw0)=>/?", "[Value='inhibitor'](kw1)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 3 //NP{/?[Tag='DRUG'](kw2)=>/NP{/?[Tag='GENE'](kw0)=>/?", "[Value='inhibitor'](kw1)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 4 //NP{/NP{/?[Tag='GENE'](kw0)=>/?[Value='inhibitor'](kw1)}=>/?", "[Tag='DRUG'](kw2)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 5 //NP{/NP{/NP{/?[Tag='GENE'](kw0)=>/?[Tag='DRUG'](kw2)}}=>/?", "[Value='inhibitor'](kw1)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 6 //NP{/?[Tag='GENE'](kw0)=>/?[Value='inhibitor'](kw1)=>/?", "[Tag='DRUG'](kw2)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 7 //NP{/NP{/?[Value='inhibitor'](kw1)}=>/PP{/NP{//?[Tag='GENE'](kw0)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 8 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{//?[Tag='GENE'](kw0)=>//?", "[Value='inhibitor'](kw1)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 9 //S{/?[Tag='DRUG'](kw2)=>/VP{/NP{//?[Value='inhibitor'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 10 //NP{/?[Tag='DRUG'](kw2)=>/NP{/NP{/?[Tag='GENE'](kw0)}}=>/?", "[Value='inhibitor'](kw1)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 11 //S{/?[Tag='DRUG'](kw2)=>/?[Value='inhibitor'](kw1)=>/?", "[Tag='GENE'](kw0)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 12 //NP{/NP{/?[Tag='DRUG'](kw2)=>/NP{/?[Value='inhibitor'](kw1)}=>/PP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 13 //S{/?[Tag='GENE'](kw0)=>/?[Value='inhibitor'](kw1)=>/?", "[Tag='DRUG'](kw2)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 14 //NP{/?[Tag='DRUG'](kw2)=>/NP{/NP{/?[Value='inhibitor'](kw1)}=>/PP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 15 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{//?[Value='inhibitor'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 16 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/NP{//?[Value='inhibitor'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 17 //NP{/?[Tag='DRUG'](kw2)=>/NP{/NP{/?[Tag='GENE'](kw0)=>/?", "[Value='inhibitor'](kw1)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 18 //NP{/NP{/PP{//?[Tag='DRUG'](kw2)=>//?[Tag='GENE'](kw0)}=>/NP{/?", "[Value='inhibitor'](kw1)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 19 //NP{/NP{/?[Tag='GENE'](kw0)=>/?[Value='inhibitor'](kw1)=>/?", "[Tag='DRUG'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 20 //S{/?[Tag='DRUG'](kw2)=>/?[Tag='GENE'](kw0)=>/?", "[Value='inhibitor'](kw1)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 21 //NP{/NP{/?[Tag='DRUG'](kw2)=>/?[Tag='GENE'](kw0)=>/?", "[Value='inhibitor'](kw1)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 22 //NP{/NP{/?[Tag='DRUG'](kw2)=>/NP{/?[Tag='GENE'](kw0)=>/?", "[Value='inhibitor'](kw1)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 23 //NP{/NP{/PP{//?[Tag='DRUG'](kw2)}=>/?[Tag='GENE'](kw0)=>/?", "[Value='inhibitor'](kw1)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 24 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value='inhibited'](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 25 //S{/NP{/NP{/?[Tag='DRUG'](kw2)}}=>/VP{/?[Value='inhibited'](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 26 //S{/?[Tag='DRUG'](kw2)=>/?[Value='inhibited'](kw1)=>/?", "[Tag='GENE'](kw0)} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 27 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/?[Value='inhibited'](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 28 //S{/NP{/NP{//?[Tag='DRUG'](kw2)}}=>/VP{/?[Value='inhibited'](kw1)=>/NP{/?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 29 //S{/NP{/PP{//?[Tag='GENE'](kw0)}}=>/VP{/VP{/?[Value='inhibited'](kw1)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 30 //S{/NP{/NP{//?[Tag='DRUG'](kw2)}}=>/VP{/?[Value='inhibited'](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 31 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{/?[Value='inhibited'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 32 //S{/NP{/PP{//?[Tag='DRUG'](kw2)}}=>/VP{/?[Value='inhibited'](kw1)=>/NP{/?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 33 //S{/NP{/NP{/?[Tag='GENE'](kw0)}}=>/VP{/VP{/?[Value='inhibited'](kw1)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 34 //S{/NP{/NP{/?[Tag='GENE'](kw0)}}=>/VP{/?[Value='inhibited'](kw1)=>/PP{//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 35 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value='inhibited'](kw1)=>/PP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 36 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/VP{/?[Value='inhibited'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 37 //S{/NP{/NP{/?[Tag='DRUG'](kw2)}}=>/VP{/?[Value='inhibited'](kw1)=>/NP{/?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 38 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/VP{/?[Value='inhibited'](kw1)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 39 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/?[Value='inhibited'](kw1)=>/NP{/?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 40 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/VP{//?[Value='inhibited'](kw1)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 41 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value='inhibited'](kw1)=>/NP{/?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 42 //S{/NP{/NP{/?[Tag='GENE'](kw0)=>/?[Tag='DRUG'](kw2)}}=>/VP{/?", "[Value='inhibit'](kw1)}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 43 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/VP{//?[Value='inhibit'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 44 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{//?[Value='inhibit'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 45 //NP{/NP{/?[Tag='GENE'](kw0)=>/?[Value='inhibition'](kw1)}=>/PP{/NP{/?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 46 //S{/?[Tag='DRUG'](kw2)=>/VP{/PP{//?[Value='inhibition'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 47 //S{/NP{/PP{//?[Tag='DRUG'](kw2)}}=>/VP{/NP{//?[Value='inhibition'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 48 //NP{/?[Value='inhibition'](kw1)=>/PP{/NP{//?[Tag='GENE'](kw0)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 49 //S{/?[Value='inhibition'](kw1)=>/PP{/NP{//?[Tag='GENE'](kw0)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 50 //NP{/NP{/?[Tag='DRUG'](kw2)=>/?[Value='inhibition'](kw1)}=>/PP{/NP{/?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 51 //NP{/NP{/?[Value='inhibition'](kw1)}=>/PP{/NP{/?[Tag='GENE'](kw0)=>/?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 52 //S{/NP{/?[Value='inhibition'](kw1)=>/PP{//?[Tag='GENE'](kw0)}}=>/VP{/PP{//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 53 //S{/NP{/NP{/?[Value='inhibition'](kw1)}=>/PP{//?[Tag='GENE'](kw0)}}=>/VP{/NP{//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 54 //S{/?[Value IN {'inhibition','Inhibition'}](kw1)=>/?[Tag='GENE'](kw0)=>/?", "[Tag='DRUG'](kw2)} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 55 //NP{/NP{/NP{/?[Value='inhibition'](kw1)}=>/PP{//?[Tag='GENE'](kw0)}}=>/PP{/NP{//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 56 //NP{/?[Tag='DRUG'](kw2)=>/NP{/?[Value='inhibition'](kw1)}=>/PP{/NP{/?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 57 //NP{/NP{/?[Value='inhibition'](kw1)}=>/PP{/NP{//?[Tag='GENE'](kw0)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 58 //S{/?[Tag='DRUG'](kw2)=>/VP{/NP{//?[Value='inhibition'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 59 //S{/NP{/NP{/?[Value='inhibition'](kw1)}=>/PP{//?[Tag='DRUG'](kw2)}}=>/VP{/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 60 //NP{/NP{/?[Value='inhibition'](kw1)=>/PP{//?[Tag='GENE'](kw0)}}=>/PP{/NP{//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 61 //NP{/NP{/?[Tag='DRUG'](kw2)=>/?[Value='inhibition'](kw1)}=>/PP{/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 62 //NP{/NP{/?[Value='inhibition'](kw1)=>/PP{//?[Tag='GENE'](kw0)}}=>/PP{/NP{/?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 63 //NP{/NP{/NP{/?[Value='inhibition'](kw1)}=>/PP{//?[Tag='GENE'](kw0)}}=>/PP{/NP{/?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 64 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/?[Value='inhibits'](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 65 /S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value='inhibits'](kw1)=>/NP{/?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 66 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{/?[Value='inhibits'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 67 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value='inhibits'](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 68 //NP{/NP{/NP{/?[Value='inhibitory'](kw1)}=>/PP{//?[Tag='DRUG'](kw2)}}=>/PP{/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 69 //S{/?[Tag='DRUG'](kw2)=>/VP{/NP{/?[Tag='GENE'](kw0)=>/?", "[Value='inhibitor'](kw1)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 70 //NP{/?[Tag='DRUG'](kw2)=>/NP{/?[Tag='GENE'](kw0)=>/?", "[Value='inhibitor'](kw1)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 71 //NP{/NP{/?[Tag='GENE'](kw0)=>/?[Value='inhibitor'](kw1)}=>/?", "[Tag='DRUG'](kw2)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 72 //NP{/NP{/NP{/?[Tag='GENE'](kw0)=>/?[Tag='DRUG'](kw2)}}=>/?", "[Value='inhibitor'](kw1)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 73 //NP{/?[Tag='GENE'](kw0)=>/?[Value='inhibitor'](kw1)=>/?", "[Tag='DRUG'](kw2)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 74 //NP{/NP{/?[Value='inhibitor'](kw1)}=>/PP{/NP{//?[Tag='GENE'](kw0)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 75 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{//?[Tag='GENE'](kw0)=>//?", "[Value='inhibitor'](kw1)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 76 //S{/?[Tag='DRUG'](kw2)=>/VP{/NP{//?[Value='inhibitor'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 77 //NP{/?[Tag='DRUG'](kw2)=>/NP{/NP{/?[Tag='GENE'](kw0)}}=>/?", "[Value='inhibitor'](kw1)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 78 //S{/?[Tag='DRUG'](kw2)=>/?[Value='inhibitor'](kw1)=>/?", "[Tag='GENE'](kw0)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 79 //NP{/NP{/?[Tag='DRUG'](kw2)=>/NP{/?[Value='inhibitor'](kw1)}=>/PP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 80 //S{/?[Tag='GENE'](kw0)=>/?[Value='inhibitor'](kw1)=>/?", "[Tag='DRUG'](kw2)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 81 //NP{/?[Tag='DRUG'](kw2)=>/NP{/NP{/?[Value='inhibitor'](kw1)}=>/PP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 82 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{//?[Value='inhibitor'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 83 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/NP{//?[Value='inhibitor'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 84 //NP{/?[Tag='DRUG'](kw2)=>/NP{/NP{/?[Tag='GENE'](kw0)=>/?", "[Value='inhibitor'](kw1)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 85 //NP{/NP{/PP{//?[Tag='DRUG'](kw2)=>//?[Tag='GENE'](kw0)}=>/NP{/?", "[Value='inhibitor'](kw1)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 86 //NP{/NP{/?[Tag='GENE'](kw0)=>/?[Value='inhibitor'](kw1)=>/?", "[Tag='DRUG'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 87 //S{/?[Tag='DRUG'](kw2)=>/?[Tag='GENE'](kw0)=>/?", "[Value='inhibitor'](kw1)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 88 //NP{/NP{/?[Tag='DRUG'](kw2)=>/?[Tag='GENE'](kw0)=>/?", "[Value='inhibitor'](kw1)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 89 //NP{/NP{/?[Tag='DRUG'](kw2)=>/NP{/?[Tag='GENE'](kw0)=>/?", "[Value='inhibitor'](kw1)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 90 //NP{/NP{/PP{//?[Tag='DRUG'](kw2)}=>/?[Tag='GENE'](kw0)=>/?", "[Value='inhibitor'](kw1)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 91 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value='inhibited'](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 92 //S{/NP{/NP{/?[Tag='DRUG'](kw2)}}=>/VP{/?[Value='inhibited'](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 93 //S{/?[Tag='DRUG'](kw2)=>/?[Value='inhibited'](kw1)=>/?", "[Tag='GENE'](kw0)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 94 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/?[Value='inhibited'](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 95 //S{/NP{/NP{//?[Tag='DRUG'](kw2)}}=>/VP{/?[Value='inhibited'](kw1)=>/NP{/?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 96 //S{/NP{/PP{//?[Tag='GENE'](kw0)}}=>/VP{/VP{/?[Value='inhibited'](kw1)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 97 //S{/NP{/NP{//?[Tag='DRUG'](kw2)}}=>/VP{/?[Value='inhibited'](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 98 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{/?[Value='inhibited'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 99 //S{/NP{/PP{//?[Tag='DRUG'](kw2)}}=>/VP{/?[Value='inhibited'](kw1)=>/NP{/?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 100 //S{/NP{/NP{/?[Tag='GENE'](kw0)}}=>/VP{/VP{/?[Value='inhibited'](kw1)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 101 //S{/NP{/NP{/?[Tag='GENE'](kw0)}}=>/VP{/?[Value='inhibited'](kw1)=>/PP{//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 102 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value='inhibited'](kw1)=>/PP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 103 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/VP{/?[Value='inhibited'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 104 //S{/NP{/NP{/?[Tag='DRUG'](kw2)}}=>/VP{/?[Value='inhibited'](kw1)=>/NP{/?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 105 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/VP{/?[Value='inhibited'](kw1)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 106 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/?[Value='inhibited'](kw1)=>/NP{/?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 107 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/VP{//?[Value='inhibited'](kw1)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 108 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value='inhibited'](kw1)=>/NP{/?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 109 //S{/NP{/NP{/?[Tag='GENE'](kw0)=>/?[Tag='DRUG'](kw2)}}=>/VP{/?", "[Value='inhibit'](kw1)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 110 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/VP{//?[Value='inhibit'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 111 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{//?[Value='inhibit'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 112 //NP{/NP{/?[Tag='GENE'](kw0)=>/?[Value='inhibition'](kw1)}=>/PP{/NP{/?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 113 //S{/?[Tag='DRUG'](kw2)=>/VP{/PP{//?[Value='inhibition'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 114 //S{/NP{/PP{//?[Tag='DRUG'](kw2)}}=>/VP{/NP{//?[Value='inhibition'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 115 //NP{/?[Value='inhibition'](kw1)=>/PP{/NP{//?[Tag='GENE'](kw0)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 116 //S{/?[Value='inhibition'](kw1)=>/PP{/NP{//?[Tag='GENE'](kw0)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 117 //NP{/NP{/?[Tag='DRUG'](kw2)=>/?[Value='inhibition'](kw1)}=>/PP{/NP{/?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 118 //NP{/NP{/?[Value='inhibition'](kw1)}=>/PP{/NP{/?[Tag='GENE'](kw0)=>/?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 119 //S{/NP{/?[Value='inhibition'](kw1)=>/PP{//?[Tag='GENE'](kw0)}}=>/VP{/PP{//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 120 //S{/NP{/NP{/?[Value='inhibition'](kw1)}=>/PP{//?[Tag='GENE'](kw0)}}=>/VP{/NP{//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 121 //S{/?[Value IN {'inhibition','Inhibition'}](kw1)=>/?[Tag='GENE'](kw0)=>/?", "[Tag='DRUG'](kw2)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 122 //NP{/NP{/NP{/?[Value='inhibition'](kw1)}=>/PP{//?[Tag='GENE'](kw0)}}=>/PP{/NP{//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 123 //NP{/?[Tag='DRUG'](kw2)=>/NP{/?[Value='inhibition'](kw1)}=>/PP{/NP{/?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 124 //NP{/NP{/?[Value='inhibition'](kw1)}=>/PP{/NP{//?[Tag='GENE'](kw0)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 125 //S{/?[Tag='DRUG'](kw2)=>/VP{/NP{//?[Value='inhibition'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 126 //S{/NP{/NP{/?[Value='inhibition'](kw1)}=>/PP{//?[Tag='DRUG'](kw2)}}=>/VP{/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 127 //NP{/NP{/?[Value='inhibition'](kw1)=>/PP{//?[Tag='GENE'](kw0)}}=>/PP{/NP{//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 128 //NP{/NP{/?[Tag='DRUG'](kw2)=>/?[Value='inhibition'](kw1)}=>/PP{/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 129 //NP{/NP{/?[Value='inhibition'](kw1)=>/PP{//?[Tag='GENE'](kw0)}}=>/PP{/NP{/?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 130 //NP{/NP{/NP{/?[Value='inhibition'](kw1)}=>/PP{//?[Tag='GENE'](kw0)}}=>/PP{/NP{/?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 131 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/?[Value='inhibits'](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 132 /S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value='inhibits'](kw1)=>/NP{/?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 133 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{/?[Value='inhibits'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 134 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value='inhibits'](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value Gene Metabolized Drug 1 //S{/NP{/NP{/?[Tag='DRUG'](kw2)}}=>/VP{/VP{/?[Value IN {'metabolised','metabolized'}](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 2 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{//?[Value IN {'metabolised','metabolized'}](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 3 //S{/NP{/SBAR{//?[Tag='DRUG'](kw2)}}=>/VP{/?[Value IN {'metabolised','metabolized'}](kw1)=>/PP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 4 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/VP{/?[Value IN {'metabolised','metabolized'}](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 5 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/VP{//?[Value IN {'metabolized','metabolised'}](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 6 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{/?[Value IN {'metabolised','metabolized'}](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 7 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value='metabolized'](kw1)=>/PP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 8 //S{/?[Tag='DRUG'](kw2)=>/NP{/?[Value='metabolism'](kw1)}=>/VP{/VP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 9 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/NP{//?[Tag='DRUG'](kw2)=>//?", "[Value='metabolism'](kw1)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 10 //NP{/NP{/?[Tag='GENE'](kw0)=>/?[Value='metabolism'](kw1)}=>/PP{/NP{/?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 11 //NP{/?[Value='metabolism'](kw1)=>/PP{/NP{//?[Tag='DRUG'](kw2)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 12 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/VP{//?[Value='metabolism'](kw1)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 13 //S{/NP{/NP{/?[Tag='GENE'](kw0)}}=>/VP{/NP{//?[Value='metabolism'](kw1)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 14 //NP{/NP{/PP{//?[Tag='GENE'](kw0)}}=>/PP{/NP{/?[Tag='DRUG'](kw2)=>/?", "[Value='metabolism'](kw1)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 15 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/PP{//?[Value='metabolism'](kw1)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 16 //NP{/NP{/?[Tag='DRUG'](kw2)=>/?[Value='metabolism'](kw1)}=>/PP{/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 17 //NP{/NP{/?[Tag='GENE'](kw0)=>/?[Value='metabolism'](kw1)}=>/PP{/NP{//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 18 //S{/NP{/NP{/?[Value='metabolism'](kw1)}=>/PP{//?[Tag='DRUG'](kw2)}}=>/VP{/VP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 19 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/PP{//?[Tag='DRUG'](kw2)=>//?", "[Value='metabolism'](kw1)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 20 //NP{/NP{/PP{//?[Tag='GENE'](kw0)}}=>/PP{/NP{//?[Value='metabolism'](kw1)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 21 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/NP{//?[Value='substrate'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 22 //NP{/NP{/?[Tag='DRUG'](kw2)=>/NP{/?[Value='substrate'](kw1)}=>/PP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 23 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{//?[Value='substrate'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 24 //NP{/?[Tag='GENE'](kw0)=>/?[Value='substrate'](kw1)=>/?", "[Tag='DRUG'](kw2)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 25 //NP{/?[Tag='DRUG'](kw2)=>/NP{/NP{/?[Value='substrate'](kw1)}=>/PP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 26 //S{/NP{/NP{/?[Tag='DRUG'](kw2)}}=>/VP{/VP{/?[Value IN {'metabolised','metabolized'}](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 27 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{//?[Value IN {'metabolised','metabolized'}](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 28 //S{/NP{/SBAR{//?[Tag='DRUG'](kw2)}}=>/VP{/?[Value IN {'metabolised','metabolized'}](kw1)=>/PP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 29 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/VP{/?[Value IN {'metabolised','metabolized'}](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 30 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/VP{//?[Value IN {'metabolized','metabolised'}](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 31 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{/?[Value IN {'metabolised','metabolized'}](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 32 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value='metabolized'](kw1)=>/PP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 33 //S{/?[Tag='DRUG'](kw2)=>/NP{/?[Value='metabolism'](kw1)}=>/VP{/VP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 34 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/NP{//?[Tag='DRUG'](kw2)=>//?", "[Value='metabolism'](kw1)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 35 //NP{/NP{/?[Tag='GENE'](kw0)=>/?[Value='metabolism'](kw1)}=>/PP{/NP{/?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 36 //NP{/?[Value='metabolism'](kw1)=>/PP{/NP{//?[Tag='DRUG'](kw2)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 37 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/VP{//?[Value='metabolism'](kw1)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 38 //S{/NP{/NP{/?[Tag='GENE'](kw0)}}=>/VP{/NP{//?[Value='metabolism'](kw1)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 39 //NP{/NP{/PP{//?[Tag='GENE'](kw0)}}=>/PP{/NP{/?[Tag='DRUG'](kw2)=>/?", "[Value='metabolism'](kw1)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 40 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/PP{//?[Value='metabolism'](kw1)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 41 //NP{/NP{/?[Tag='DRUG'](kw2)=>/?[Value='metabolism'](kw1)}=>/PP{/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 42 //NP{/NP{/?[Tag='GENE'](kw0)=>/?[Value='metabolism'](kw1)}=>/PP{/NP{//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 43 //S{/NP{/NP{/?[Value='metabolism'](kw1)}=>/PP{//?[Tag='DRUG'](kw2)}}=>/VP{/VP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 44 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/PP{//?[Tag='DRUG'](kw2)=>//?", "[Value='metabolism'](kw1)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 45 //NP{/NP{/PP{//?[Tag='GENE'](kw0)}}=>/PP{/NP{//?[Value='metabolism'](kw1)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 46 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/NP{//?[Value='substrate'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 47 //NP{/NP{/?[Tag='DRUG'](kw2)=>/NP{/?[Value='substrate'](kw1)}=>/PP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 48 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{//?[Value='substrate'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 49 //NP{/?[Tag='GENE'](kw0)=>/?[Value='substrate'](kw1)=>/?", "[Tag='DRUG'](kw2)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 50 //NP{/?[Tag='DRUG'](kw2)=>/NP{/NP{/?[Value='substrate'](kw1)}=>/PP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value Gene Regulates Gene 1 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/VP{//?[Value IN {'regulated', 'upregulated', 'downregulated', 'up-regulated', 'down-regulated'}](kw1)=>//?[Value='by'](kw3)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 2 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/VP{/?[Value IN {'regulated', 'upregulated', 'downregulated', 'up-regulated', 'down-regulated'}](kw1)=>//?[Value='by'](kw3)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 3 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/?[Value IN {'regulated','down-regulated'}](kw1)=>/NP{//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 4 //NP{/NP{/?[Value IN {'regulation', 'upregulation', 'downregulation', 'up-regulation', 'down-regulation'}](kw1)}=>/PP{/NP{//?[Tag='GENE'](kw0)=>//?[Value='by'](kw3)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 5 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/?[Value IN {'regulates', 'upregulates', 'downregulates', 'up-regulates', 'down-regulates'}](kw1)=>/NP{//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 6 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/VP{//?[Value='in'](kw3)=>//?[Value='regulating'](kw1)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value Gene Regulate Gene (Xenobiotic Metabolism) 1 //S{//?[Tag='GENE' AND Canonical LIKE 'CYP\\%'](kw0)<=>//?", "[Tag='GENE' AND Canonical IN {'AhR', 'CASR', 'CAR', 'PXR', 'NR1I2', 'NR1I3'}](kw1)} ::: distinct sent.cid, kw0.value, kw1.value, sent.value 2 //S{//?[Tag='GENE' AND Value LIKE 'cytochrome\\%'](kw0)<=>//?", "[Tag='GENE' AND Canonical IN {'AhR', 'CASR', 'CAR', 'PXR', 'NR1I2', 'NR1I3'}](kw1)} ::: distinct sent.cid, kw0.value, kw1.value, sent.value Negative Drug Induces/Metabolizes/Inhibits Gene 1 //S{/?[Tag='DRUG'](kw0)=>/VP{/VP{/?[Value IN {'induced', 'inhibited', 'metabolized', 'metabolised'}](kw1)=>//?[Tag='GENE'](kw2)=>//?[Value='not'](kw3)=>//?", "[Tag='GENE'](kw4)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw4.value, kw3.value, sent.value 2 //S{/?[Tag='DRUG'](kw0)=>/?[Value='not'](kw3)=>/?[Tag='DRUG'](kw4)=>/?[Value='inhibited'](kw1)=>/?", "[Tag='GENE'](kw2)} ::: distinct sent.cid, kw4.value, kw1.value, kw2.value, kw3.value, sent.value 3 //S{/SBAR{/S{//?[Tag='DRUG'](kw0)}}=>/S{/S{//?[Value='metabolized'](kw1)=>//?[Tag='GENE'](kw2)=>//?[Value='not'](kw3)=>//?", "[Tag='GENE'](kw4)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw4.value, kw3.value, sent.value 4 //S{/NP{/PP{//?[Tag='GENE'](kw2)}}=>/VP{/VP{/?[Value IN {'induced', 'inhibited', 'metabolized', 'metabolised'}](kw1)=>//?[Tag='DRUG'](kw0)=>//?[Value='not'](kw3)=>//?", "[Tag='DRUG'](kw4)}}} ::: distinct sent.cid, kw4.value, kw1.value, kw2.value, kw3.value, sent.value 5 //S{/NP{/PP{//?[Tag='DRUG'](kw0)}}=>/VP{/VP{/?[Value IN {'induced', 'inhibited', 'metabolized', 'metabolised'}](kw1)=>//?[Tag='GENE'](kw2)=>//?[Value IN {'not','no'}](kw3)=>//?", "[Tag='GENE'](kw4)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw4.value, kw3.value, sent.value 6 //S{/?[Tag='DRUG'](kw0)=>/VP{/?[Value='not'](kw3)=>/VP{//?[Value IN {'induced', 'inhibited', 'metabolized', 'metabolised'}](kw1)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 7 //S{/NP{/?[Tag='GENE'](kw2)}=>/VP{/VP{/?[Value IN {'induced', 'inhibited', 'metabolized', 'metabolised'}](kw1)=>//?[Tag='DRUG'](kw0)=>//?[Value='not'](kw3)=>//?", "[Tag='DRUG'](kw4)}}} ::: distinct sent.cid, kw4.value, kw1.value, kw2.value, kw3.value, sent.value 8 //S{/NP{/NP{/?[Value='not'](kw3)=>/?[Tag='DRUG'](kw0)}}=>/VP{/?[Value IN {'inhibited','induced'}](kw1)=>/NP{//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 9 //S{/NP{/NP{/?[Value IN {'no','not'}](kw3)=>/?[Tag='DRUG'](kw0)}}=>/VP{/?[Value IN {'inhibited','induced'}](kw1)=>/NP{/?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 10 //S{/?[Tag='DRUG'](kw0)=>/S{/S{//?[Value='not'](kw3)=>//?[Value IN {'induce', 'inhibit'}](kw1)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 11 //S{/?[Tag='DRUG'](kw0)=>/?[Tag='GENE'](kw2)=>/?[Value IN {'not'}](kw3)=>/?", "[Value IN {'inhibit','induce'}](kw1)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 12 //S{/?[Tag='DRUG'](kw0)=>/VP{/?[Value='not'](kw3)=>/VP{/?[Value IN {'inhibit','induce'}](kw1)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 13 //S{/NP{/?[Tag='DRUG'](kw0)}=>/VP{/VP{/?[Value='not'](kw3)=>//?[Value='inhibit'](kw1)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 14 //S{/NP{/NP{/?[Tag='GENE'](kw2)}}=>/VP{/?[Value='not'](kw3)=>/VP{/?[Value='metabolize'](kw1)=>//?", "[Tag='DRUG'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 15 //S{/NP{/NP{//?[Tag='DRUG'](kw0)}}=>/VP{/?[Value='not'](kw3)=>/VP{/?[Value IN {'inhibit','induce'}](kw1)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 16 //S{/NP{/?[Tag='DRUG'](kw0)}=>/VP{/?[Value='not'](kw3)=>/VP{/?[Value='inhibit'](kw1)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 17 //S{/NP{/?[Tag='DRUG'](kw0)}=>/VP{/?[Value IN {'induces', 'inhibits'}](kw1)=>/NP{//?[Tag='GENE'](kw2)=>//?[Value='not'](kw3)=>//?", "[Tag='GENE'](kw4)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw4.value, kw3.value, sent.value 18 //S{/?[Tag='DRUG'](kw0)=>/VP{/?[Value='inhibits'](kw1)=>/NP{//?[Tag='GENE'](kw2)=>//?[Value='not'](kw3)=>//?", "[Tag='GENE'](kw4)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw4.value, kw3.value, sent.value 19 //S{/?[Tag='DRUG'](kw0)=>/VP{/NP{//?[Value='no'](kw3)=>//?[Value IN {'induction','metabolism','inhibition'}](kw1)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 20 //S{/NP{/NP{//?[Tag='DRUG'](kw0)=>//?[Tag='GENE'](kw2)=>//?[Value IN {'induction','metabolism','inhibition'}](kw1)}}=>/VP{/?", "[Value='not'](kw3)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 21 //S{/?[Tag='DRUG'](kw0)=>/VP{/?[Value='not'](kw3)=>/VP{//?[Value IN {'induction','metabolism','inhibition'}](kw1)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 22 //S{/NP{/PP{//?[Tag='DRUG'](kw0)}}=>/VP{/?[Value='not'](kw3)=>/VP{//?[Value IN {'induction','metabolism','inhibition'}](kw1)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value Negative Drug Induces Gene 1 //S{//?[Tag='GENE'](kw0)=>//?[Value IN {'induced','increased','stimulated'}](kw1)=>//?[Value='by'](kw3)=>//?[Tag='DRUG'](kw2)=>//?[Value='not'](kw4)=>//?", "[Tag='DRUG'](kw5)} ::: distinct sent.cid, kw5.value, kw1.value, kw0.value, kw4.value, sent.value 2 //S{//?[Tag='GENE'](kw0)=>//?[Value='not'](kw4)=>//?[Tag='GENE'](kw5)=>//?[Value IN {'induced','increased','stimulated'}](kw1)=>//?[Value='by'](kw3)=>//?", "[Tag='DRUG'](kw2)} ::: distinct sent.cid, kw2.value, kw1.value, kw5.value, kw4.value, sent.value 3 //S{//?[Tag='DRUG'](kw0)=>//?[Value='not'](kw4)=>//?[Value IN {'induce','induced','increase','increased','stimulate','stimulated'}](kw1)=>//?", "[Tag='GENE'](kw2)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw4.value, sent.value 4 //S{//?[Tag='DRUG'](kw0)=>//?[Value='not'](kw4)=>//?[Tag='DRUG'](kw5)=>//?[Value IN {'induces','increases','stimulates','induced','increased','stimulated'}](kw1)=>//?", "[Tag='GENE'](kw2)} ::: distinct sent.cid, kw5.value, kw1.value, kw2.value, kw4.value, sent.value 5 //S{//?[Value='no'](kw5)=>//?[value IN {'induction','stimulation'}](kw1)=>//?[Value IN {'of'}](kw2)=>//?[Tag='GENE'](kw0)=>//?[value IN {'by'}](kw3)=>//?", "[Tag='DRUG'](kw4)} ::: distinct sent.cid, kw4.value, kw1.value, kw0.value, kw5.value, sent.value Negative Drug Inhibits Gene 1 //S{//?[Tag='GENE'](kw0)=>//?[Value IN {'inhibited','decreased'}](kw1)=>//?[Value='by'](kw3)=>//?[Tag='DRUG'](kw2)=>//?[Value='not'](kw4)=>//?", "[Tag='DRUG'](kw5)} ::: distinct sent.cid, kw5.value, kw1.value, kw0.value, kw4.value, sent.value 2 //S{//?[Tag='GENE'](kw0)=>//?[Value='not'](kw4)=>//?[Tag='GENE'](kw5)=>//?[Value IN {'inhibited','decreased'}](kw1)=>//?[Value='by'](kw3)=>//?", "[Tag='DRUG'](kw2)} ::: distinct sent.cid, kw2.value, kw1.value, kw5.value, kw4.value, sent.value 3 //S{//?[Tag='DRUG'](kw0)=>//?[Value='not'](kw4)=>//?[Value IN {'inhibit','inhibited','decrease','decreased'}](kw1)=>//?", "[Tag='GENE'](kw2)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw4.value, sent.value 4 //S{//?[Tag='DRUG'](kw0)=>//?[Value='not'](kw4)=>//?[Tag='DRUG'](kw5)=>//?[Value IN {'inhibits','decreases','inhibited','decreased'}](kw1)=>//?", "[Tag='GENE'](kw2)} ::: distinct sent.cid, kw5.value, kw1.value, kw2.value, kw4.value, sent.value 5 //S{//?[Value='no'](kw5)=>//?[value IN {'inhibition'}](kw1)=>//?[Value IN {'of'}](kw2)=>//?[Tag='GENE'](kw0)=>//?[value IN {'by'}](kw3)=>//?", "[Tag='DRUG'](kw4)} ::: distinct sent.cid, kw4.value, kw1.value, kw0.value, kw5.value, sent.value Negative Gene Metabolizes Drug 1 //S{//?[Tag='DRUG'](kw0)=>//?[Value IN {'metabolized','metabolised'}](kw1)=>//?[Value='by'](kw3)=>//?[Tag='GENE'](kw2)=>//?[Value='not'](kw4)=>//?", "[Tag='GENE'](kw5)} ::: distinct sent.cid, kw0.value, kw1.value, kw5.value, kw4.value, sent.value 2 //S{//?[Tag='DRUG'](kw0)=>//?[Value='not'](kw4)=>//?[Tag='DRUG'](kw5)=>//?[Value IN {'metabolized','metabolised'}](kw1)=>//?[Value='by'](kw3)=>//?", "[Tag='GENE'](kw2)} ::: distinct sent.cid, kw5.value, kw1.value, kw2.value, kw4.value, sent.value 3 //S{//?[Tag='GENE'](kw0)=>//?[Value='not'](kw4)=>//?[Value IN {'metabolize','metabolise'}](kw1)=>//?", "[Tag='DRUG'](kw2)} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, kw4.value, sent.value 4 //S{//?[Tag='GENE'](kw0)=>//?[Value='not'](kw4)=>//?[Tag='GENE'](kw5)=>//?[Value IN {'metabolize','metabolise','metabolizes','metabolises'}](kw1)=>//?", "[Tag='DRUG'](kw2)} ::: distinct sent.cid, kw2.value, kw1.value, kw5.value, kw4.value, sent.value Negative Gene Downregulates Gene 1 //S{//?[Tag='GENE'](kw0)=>//?[Value IN {'suppressed','downregulated','inhibited','down-regulated','repressed','disrupted'}](kw1)=>//?[Value='by'](kw3)=>//?[Tag='GENE'](kw2)=>//?[Value='not'](kw4)=>//?", "[Tag='GENE'](kw5)} ::: distinct sent.cid, kw5.value, kw1.value, kw0.value, kw4.value, sent.value 2 //S{//?[Tag='GENE'](kw0)=>//?[Value='not'](kw4)=>//?[Tag='GENE'](kw5)=>//?[Value IN {'suppressed','downregulated','inhibited','down-regulated','repressed','disrupted'}](kw1)=>//?[Value='by'](kw3)=>//?", "[Tag='GENE'](kw2)} ::: distinct sent.cid, kw2.value, kw1.value, kw5.value, kw4.value, sent.value 3 //S{//?[Tag='GENE'](kw0)=>//?[Value='not'](kw4)=>//?[Value IN {'suppressed','suppress','downregulated','downregulate','inhibited','inhibit','down-regulated','down-regulate','repressed','repress','disrupted','disrupt'}](kw1)=>//?", "[Tag='GENE'](kw2)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw4.value, sent.value 4 //S{//?[Tag='GENE'](kw0)=>//?[Value='not'](kw4)=>//?[Tag='GENE'](kw5)=>//?[Value IN {'suppresses','downregulates','inhibits','down-regulates','represses','disrupts','suppressed','downregulated','inhibited','down-regulated','repressed','disrupted'}](kw1)=>//?", "[Tag='GENE'](kw2)} ::: distinct sent.cid, kw5.value, kw1.value, kw2.value, kw4.value, sent.value 5 //S{//?[Value='no'](kw5)=>//?[value IN {'inhibition','downregulation','down-regulation'}](kw1)=>//?[Value IN {'of'}](kw2)=>//?[Tag='GENE'](kw0)=>//?[value IN {'on'}](kw3)=>//?", "[Tag='GENE'](kw4)} ::: distinct sent.cid, kw0.value, kw1.value, kw4.value, kw5.value, sent.value Negative Gene Upregulates Gene 1 //S{//?[Tag='GENE'](kw0)=>//?[Value IN {'activated','induced','stimulated','regulated','upregulated','up-regulated'}](kw1)=>//?[Value='by'](kw3)=>//?[Tag='GENE'](kw2)=>//?[Value='not'](kw4)=>//?", "[Tag='GENE'](kw5)} ::: distinct sent.cid, kw5.value, kw1.value, kw0.value, kw4.value, sent.value 2 //S{//?[Tag='GENE'](kw0)=>//?[Value='not'](kw4)=>//?[Tag='GENE'](kw5)=>//?", "[Value IN {'activated','induced','stimulated','regulated','upregulated','up-regulated'}](kw1) ::: distinct sent.cid, kw2.value, kw1.value, kw5.value, kw4.value, sent.value 3 //S{//?[Tag='GENE'](kw0)=>//?[Value='not'](kw4)=>//?[Value IN {'activated','induced','stimulated','regulated','upregulated','up-regulated','activate','induce','stimulate','regulate','upregulate','up-regulate'}](kw1)=>//?", "[Tag='GENE'](kw2)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw4.value, sent.value 4 //S{//?[Tag='GENE'](kw0)=>//?[Value='not'](kw4)=>//?[Tag='GENE'](kw5)=>//?[Value IN {'activates','induces','stimulates','regulates','upregulates','up-regulates','activate','induce','stimulate','regulate','upregulate','up-regulate'}](kw1)=>//?", "[Tag='GENE'](kw2)} ::: distinct sent.cid, kw5.value, kw1.value, kw2.value, kw4.value, sent.value 5 //S{//?[Value='no'](kw5)=>//?[value IN {'induction','activation','stimulation','regulation','upregulation','up-regulation'}](kw1)=>//?[Value IN {'of'}](kw2)=>//?[Tag='GENE'](kw0)=>//?[value IN {'by'}](kw3)=>//?", "[Tag='GENE'](kw4)} ::: distinct sent.cid, kw4.value, kw1.value, kw0.value, kw5.value, sent.value Drug Gene Co-Occurrence 1 //S{//?[Tag='DRUG'](kw0)<=>//?", "[Tag='GENE'](kw1)} ::: distinct sent.cid, kw0.value, kw1.value, kw0.type, kw1.type, sent.value" ], [ "Introduction", "In this chapter we use various Petri Net extensions presented in Chapter and their ASP encoding to answer question from [64] that were a part of the Second Deep Knowledge Representation Challengehttps://sites.google.com/site/2nddeepkrchallenge/.", "Definition 45 (Rate) Rate of product P is defined as the quantity of P produced per unit-time.", "Rate of an action A is defined as the number of time A occurs per unit-time." ], [ "Comparing Altered Trajectories due to Reset Intervention", "Question 1 At one point in the process of glycolysis, both dihydroxyacetone phosphate (DHAP) and glyceraldehyde 3-phosphate (G3P) are produced.", "Isomerase catalyzes the reversible conversion between these two isomers.", "The conversion of DHAP to G3P never reaches equilibrium and G3P is used in the next step of glycolysis.", "What would happen to the rate of glycolysis if DHAP were removed from the process of glycolysis as quickly as it was produced?", "Provided Answer: “Glycolysis is likely to stop, or at least slow it down.", "The conversion of the two isomers is reversible, and the removal of DHAP will cause the reaction to shift in that direction so more G3P is converted to DHAP.", "If less (or no) G3P were available, the conversion of G3P into DHAP would slow down (or be unable to occur).” Solution 1 The process of glycolysis is shown in Fig 9.9 of Campbell's book.", "Glycolysis splits Glucose into Pyruvate.", "In the process it produces ATP and NADH.", "Any one of these can be used to gauge the glycolysis rate, since they will be produced in proportion to the input Glucose.", "The amount of pyruvate produced is the best choice since it is the direct end product of glycolysis.", "The ratio of the quantity of pyruvate produced over a representative span of time gives us the glycolysis rate.", "We assume a steady supply of Glucose is available and also assume that sufficient quantity of various enzymes used in glycolysis is available, since the question does not place any restriction on these substances.", "We narrow our focus to a subsection from Fructose 1,6-bisphosphate (F16BP) to 1,3-Bisphosphoglycerate (BPG13) as shown in Figure REF since that is the part the question is concerned with.", "We can ignore the linear chain up-stream of F16BP as well as the linear chain down-stream of BPG13 since the amount of F16BP available will be equal to Glucose and the amount of BPG13 will be equal to the amount of Pyruvate given our steady supply assumption.", "Figure: Petri Net graph relevant to question .", "“f16bp” is the compound Fructose 1,6-biphosphate, “bpg13” is 1,3-Bisphosphoglycerate.", "Transition trtr shown in dotted lines is added to model the elimination of dhapdhap as soon as it is produced.We fulfill the steady supply requirement of Glucose by a source transition-node $t3$ .", "We fulfill sufficient enzyme supply by a fixed quantity for each enzyme such that this quantity is in excess of what can be consumed during our simulation interval.", "Where the simulation interval is the number of time-steps over which we will measure the rate of glycolysis.", "We model the elimination of DHAP as soon as it is produced with a reset arc, shown with a dotted style in Figures REF .", "Such an arc removes all tokens from its source place when it fires.", "Since we have added it as an unconditional arc, it is always enabled for firing.", "We encode both situations in ASP with the maximal firing set policy.", "Both situations (without and with reset arc) are encoded in ASP and run for 10 steps.", "At the end of those 10 steps the amount of BPG13 is compared to determine the difference in the rate of glycolysis.", "In normal situation (without $(dhap,tr)$ reset arc), unique quantities of “bpg13” from all (2) answer-sets after 10 steps were as follows: holds(bpg13,14,10) holds(bpg13,16,10) with reset arc $tr$ , unique quantities of “bpg13” from all (512) answer-sets after 10 steps were as follows: holds(bpg13,0,10) holds(bpg13,10,10) holds(bpg13,12,10) holds(bpg13,14,10) holds(bpg13,16,10) holds(bpg13,2,10) holds(bpg13,4,10) holds(bpg13,6,10) holds(bpg13,8,10) Figure: Amount of “bpg13” produced in unique answer-sets produced by a 10 step simulation.", "The graph shows two situations, without the (dhap,tr)(dhap,tr) reset arc (normal situation) and with the reset arc (abnormal situation).", "The purpose of this graph is to depict the variation in the amounts of glycolysis produced in various answer sets.Note that the rate of glycolysis is generally lower when DHAP is immediately consumed.", "It is as low as zero essentially stopping glycolysis.", "The range of values are due to the choice between G3P being converted to DHAP or BPG13.", "If more G3P is converted to DHAP, then less BPG13 is produced and vice versa.", "Also, note that if G3P is not converted to BPG13, no NADH or ATP is produced either due to the liner chain from G3P to Pyruvate.", "The unique quantities of BPG13 are shown in a graphical format in Figure REF , while a trend of average quantity of BPG13 produced is shown in Figure REF .", "Figure: Average amount of “bpg13” produced during the 10-step simulation at various time steps.", "The average is over all answer-sets.", "The graph shows two situations, without the (dhap,tr)(dhap,tr) reset arc (normal situation) and with the reset arc (abnormal situation).", "The divergence in “bpg13” production is clearly shown.We created a minimal model of the Petri Net in Figure REF by removing enzymes and reactants that were not relevant to the question and did not contribute to the estimation of glycolysis.", "This is shown in Figure REF .", "Figure: Minimal version of the Petri Net graph in Figure .", "All reactants that do not contribute to the estimation of the rate of glycolysis have been removed.Simulating it for 10 steps with the same initial marking as the Petri Net in Figure REF produced the same results as for Figure REF ." ], [ "Determining Conditions Leading to an Observation", "Question 2 When and how does the body switch to B oxidation versus glycolysis as the major way of burning fuel?", "Provided Answer: “The relative volumes of the raw materials for B oxidation and glycolysis indicate which of these two processes will occur.", "Glycolysis uses the raw material glucose, and B oxidation uses Acyl CoA from fatty acids.", "When the blood sugar level decreases below its homeostatic level, then B oxidation will occur with available fatty acids.", "If no fatty acids are immediately available, glucagon and other hormones regulate release of stored sugar and fat, or even catabolism of proteins and nucleic acids, to be used as energy sources.” Solution 2 The answer provided requires background knowledge about the mechanism that regulates which source of energy will be used.", "This information is not presented in Chapter 9 of Campbell's book, which is the source material of this exercise.", "However, we can model it based on background information combined with Figure 9.19 of Campbell's book.", "Our model is presented in Figure REFWe can extend this model by adding expressions to inhibition arcs that compare available substances..", "Figure: Petri Net graph relevant to question .", "“fats” are fats, “prot” are proteins, “fac” are fatty acids, “sug” are sugars, “amin” are amino acids and “acoa” is ACoA.", "Transition “box” is the beta oxidation, “t5” is glycolysis, “t1” is fat digestion into fatty acids, and “t9” is protein deamination.We can test the model by simulating it for a time period and testing whether beta oxidation (box) is started when sugar (sug) is finished.", "We do not need a steady supply of sugar in this case, just enough to be consumed in a few time steps to capture the switch over.", "Fats and proteins may or may not modeled as a steady supply, since their bioavailability is dependent upon a number of external factors.", "We assume a steady supply of both and model it with large enough initial quantity that will last beyond the simulation period.", "We translate the petri net model into ASP and run it for 10 iterations.", "Following are the results: holds(acoa,0,0) holds(amin,0,0) holds(fac,0,0) holds(fats,5,0) holds(prot,3,0) holds(sug,4,0) fires(t1,0) fires(t5,0) fires(t9,0) holds(acoa,1,1) holds(amin,1,1) holds(fac,1,1) holds(fats,4,1) holds(prot,3,1) holds(sug,3,1) fires(t5,1) holds(acoa,2,2) holds(amin,1,2) holds(fac,1,2) holds(fats,4,2) holds(prot,3,2) holds(sug,2,2) fires(t5,2) holds(acoa,3,3) holds(amin,1,3) holds(fac,1,3) holds(fats,4,3) holds(prot,3,3) holds(sug,1,3) fires(t5,3) holds(acoa,4,4) holds(amin,1,4) holds(fac,1,4) holds(fats,4,4) holds(prot,3,4) holds(sug,0,4) fires(box,4) holds(acoa,5,5) holds(amin,1,5) holds(fac,0,5) holds(fats,4,5) holds(prot,3,5) holds(sug,0,5) fires(t1,5) fires(t9,5) holds(acoa,5,6) holds(amin,2,6) holds(fac,1,6) holds(fats,3,6) holds(prot,3,6) holds(sug,0,6) fires(box,6)c holds(acoa,6,7) holds(amin,2,7) holds(fac,0,7) holds(fats,3,7) holds(prot,3,7) holds(sug,0,7) fires(t1,7) fires(t9,7) holds(acoa,6,8) holds(amin,3,8) holds(fac,1,8) holds(fats,2,8) holds(prot,3,8) holds(sug,0,8) fires(box,8) holds(acoa,7,9) holds(amin,3,9) holds(fac,0,9) holds(fats,2,9) holds(prot,3,9) holds(sug,0,9) fires(t1,9) fires(t9,9) holds(acoa,7,10) holds(amin,4,10) holds(fac,1,10) holds(fats,1,10) holds(prot,3,10) holds(sug,0,10) fires(box,10) We can see that by time-step 4, the sugar supply is depleted and beta oxidation starts occurring." ], [ "Comparing Altered Trajectories due to Accumulation Intervention", "Question 3 ATP is accumulating in the cell.", "What affect would this have on the rate of glycolysis?", "Explain.", "Provided Answer: “ATP and AMP regulate the activity of phosphofructokinase.", "When there is an abundance of AMP in the cell, this indicates that the rate of ATP consumption is high.", "The cell is in need for more ATP.", "If ATP is accumulating in the cell, this indicates that the cell's demand for ATP had decreased.", "The cell can decrease its production of ATP.", "Therefore, the rate of glycolysis will decrease.” Solution 3 Control of cellular respiration is summarized in Fig 9.20 of Campbell's book.", "We can gauge the rate of glycolysis by the amount of Pyruvate produced, which is the end product of glycolysis.", "We assume a steady supply of glucose is available.", "Its availability is not impacted by any of the feedback mechanism depicted in Fig 9.20 of Campbell's book or restricted by the question.", "We can ignore the respiration steps after glycolysis, since they are directly dependent upon the end product of glycolysis, i.e.", "Pyruvate.", "These steps only reinforce the negative effect of ATP.", "The Citrate feed-back shown in Campbell's Fig 9.20 is also not relevant to the question, so we can assume a constant level of it and leave it out of the picture.", "Another simplification that we do is to treat the inhibition of Phosphofructokinase (PFK) by ATP as the inhibition of glycolysis itself.", "This is justified, since PFK is on a linear path from Glucose to Fructose 1,6-bisphosphate (F16BP), and all downstream product quantities are directly dependent upon the amount of F16BP (as shown in Campbell's Fig 9.9), given steady supply of substances involved in glycolysis.", "Our assumption also applies to ATP consumed in Fig 9.9.", "Our simplified picture is shown in Figure REF as a Petri Net.", "Figure: Petri Net graph relevant to question .", "“glu” is Glucose, “pyr” is Pyruvate.", "Transitions “gly1” represents glycolysis and “cw1” is cellular work that consumes ATP and produces AMP.", "Transition “gly1” is inhibited only when the number of atpatp tokens is greater than 4.We model cellular work that recycles ATP to AMP (see p/181 of Campbell's book) by the $cw1$ transition, shown in dotted style.", "In normal circumstances, this arc does not let ATP to collect.", "If we reduce the arc-weights incident on $cw1$ to 1, we get the situation where less work is being done and some ATP will collect, as a result glycolysis will pause and resume.", "If we remove $cw1$ (representing no cellular work), ATP will start accumulating and glycolysis will stop.", "We use an arbitrary arc-weight of 4 on the inhibition arc $(atp,gly1)$ to model an elevated level of ATP beyond normal that would cause inhibition An alternate modeling would be compare the number of tokens on the $amp$ node and the $atp$ node and set a level-threshold that inhibits $gly1$ .", "Such technique is common in colored-peri nets.. We encode all three situations in ASP with maximal firing set policy.", "We run them for 10 steps and compare the quantity of pyruvate produced to determine the difference in the rate of glycolysis.", "In normal situation when cellular work is being performed ($cw1$ arc is present), unique quantities of “pyr” after 10 step are as follows: holds(pyr,20,10) when the cellular work is reduced, i.e.", "($(atp,cw1)$ , $(cw1,amp)$ arc weights changed to 1), unique quantities of “pyr” after 10 steps are as follows: holds(pyr,14,10) with no cellular work ($cw1$ arc removed), unique quantities of “pyr” after 10 steps are as follows: holds(pyr,6,10) The results show the rate of glycolysis reducing as the cellular work decreases to the point where it stops once ATP reaches the inhibition threshold.", "Higher numbers of ATP produced in later steps of cellular respiration will reinforce this inhibition even more quickly.", "Trend of answers from various runs is shown in Figure REF .", "Figure: Amount of pyruvate produced from various lengths of runs." ], [ "Comparing Altered Trajectories due to Initial Value Intervention", "Question 4 A muscle cell had used up its supply of oxygen and ATP.", "Explain what affect would this have on the rate of cellular respiration and glycolysis?", "Provided Answer: “Oxygen is needed for cellular respiration to occur.", "Therefore, cellular respiration would stop.", "The cell would generate ATP by glycolysis only.", "Decrease in the concentration of ATP in the cell would stimulate an increased rate of glycolysis in order to produce more ATP.” Solution 4 Figure 9.18 of Campbell's book gives the general idea of what happens when oxygen is not present.", "Figure 9.20 of Campbell's book shows the control of glycolysis by ATP.", "To formulate the answer, we need pieces from both.", "ATP inhibits Phosphofructokinase (Fig 9.20 of Campbell), which is an enzyme used in glycolysis.", "No ATP means that enzyme is no longer inhibited and glycolysis can proceed at full throttle.", "Pyruvate either goes through aerobic respiration when oxygen is present or it goes through fermentation when oxygen is absent (Fig 9.18 of Campbell).", "We can monitor the rate of glycolysis and cellular respiration by observing these operations occurring (by looking at corresponding transition firing) over a simulation time period.", "Our simplified Petri Net model is shown in Figure REF .", "We ignore the details of processes following glycolysis, except that these steps produce additional ATP.", "We do not need an exact number of ATP produced as long as we keep it higher than the ATP produced by glycolysis.", "Higher numbers will just have a higher negative feed-back (or inhibition) effect on glycolysis.", "We ignore citrate's inhibition of glycolysis since that is not relevant to the question and since it gets recycled by the citric acid cycle (see Fig 9.12 of Campbell).", "We also ignore AMP, since it is not relevant to the question, by assuming sufficient supply to maintain glycolysis.", "We also assume continuous cellular work consuming ATP, without that ATP will accumulate almost immediately and stop glycolysis.", "We assume a steady supply of glucose is available to carry out glycolysis and fulfill this requirement by having a quantity in excess of the consumption during our simulation interval.", "All other substances participating in glycolysis are assumed to be available in a steady supply so that glycolysis can continue.", "Figure: Petri Net graph relevant to question .", "“glu” is Glucose, “pyr” is Pyruvate, “atp” is ATP, “eth” is ethenol or other products of fermentation, and “o2” is Oxygen.", "Transitions “gly1” represents glycolysis, “res1” is respiration in presence of oxygen, “fer1” is fermentation when no oxygen is present, and “cw1” is cellular work that consumes ATP.", "Transition “gly1” is inhibited only when the number of atpatp tokens is greater than 4.We then consider two scenarios, one where oxygen is present and where oxygen is absent and determine the change in rate of glycolysis and respiration by counting the firings of their respective transitions.", "We encode both situations in ASP with maximal firing set policy.", "Both situations are executed for 10 steps.", "At the end of those steps the firing count of “gly1” and “res1” is computed and compared to determine the difference in the rates of glycolysis and respiration respectively.", "In the normal situation (when oxygen is present), we get the following answer sets: fires(gly1,0) fires(cw1,1) fires(gly1,1) fires(res1,1) fires(cw1,2) fires(res1,2) fires(cw1,3) fires(cw1,4) fires(cw1,5) fires(gly1,5) fires(cw1,6) fires(gly1,6) fires(res1,6) fires(cw1,7) fires(res1,7) fires(cw1,8) fires(cw1,9) fires(cw1,10) while in the abnormal situation (when oxygen is absent), we get the following firings: fires(gly1,0) fires(cw1,1) fires(fer1,1) fires(gly1,1) fires(cw1,2) fires(fer1,2) fires(gly1,2) fires(cw1,3) fires(fer1,3) fires(gly1,3) fires(cw1,4) fires(fer1,4) fires(gly1,4) fires(cw1,5) fires(fer1,5) fires(gly1,5) fires(cw1,6) fires(fer1,6) fires(gly1,6) fires(cw1,7) fires(fer1,7) fires(gly1,7) fires(cw1,8) fires(fer1,8) fires(gly1,8) fires(cw1,9) fires(fer1,9) fires(gly1,9) fires(cw1,10) fires(fer1,10) fires(gly1,10) Note that the number of firings of glycolysis for normal situation is lower when oxygen is present and higher when oxygen is absent.", "While, the number of firings is zero when no oxygen is present.", "Thus, respiration stops when no oxygen is present and the need of ATP by cellular work is fulfilled by a higher amount of glycolysis.", "Trend from various runs is shown in Figure REF .", "Figure: Firing counts of glycolysis (gly1) and respiration (res1) for different simulation lengths for the petri net in Figure" ], [ "Comparing Altered Trajectories due to Inhibition Intervention", "Question 5 The final protein complex in the electron transport chain of the mitochondria is non-functional.", "Explain the effect of this on pH of the intermembrane space of the mitochondria.", "Provided Answer: “The H+ ion gradient would gradually decrease and the pH would gradually increase.", "The other proteins in the chain are still able to produce the H+ ion gradient.", "However, a non-functional, final protein in the electron transport chain would mean that oxygen is not shuttling electrons away from the electron transport chain.", "This would cause a backup in the chain, and the other proteins in the electron transport chain would no longer be able to accept electrons and pump H+ ions into the intermembrane space.", "A concentration decrease in the H+ ions means an increase in the pH.” Solution 5 The electron transport chain is shown in Fig 9.15 (1) of Campbell's book.", "In order to explain the effect on pH, we will show the change in the execution of the electron transport chain with both a functioning and non-functioning final protein.", "Since pH depends upon the concentration of H+ ions, we will quantify the difference its quantity in the intermembrane space in both scenarios as well.", "We assume that a steady input of NADH, FADH2, H+ and O2 are available in the mitochondrial matrix.", "We also assume an electron carrying capacity of 2 for both ubiquinone (Q)http://www.benbest.com/nutrceut/CoEnzymeQ.html and cytochrome c (Cyt c).", "This carrying capacity is background information not provided in Campbell's Chapter 9.", "As with previous questions, we fulfill the steady supply requirement of substances by having input quantities in excess of what would be consumed during our simulation interval.", "Figure: Petri Net graph relevant to question .", "“is” is the intermembrane space, “mm” is mitochondrial matrix, “q” is ubiquinone and “cytc” is cytochrome c. The inhibition arcs (q,t1)(q,t1), (q,t2)(q,t2) and (cytc,t3)(cytc,t3) capture the electron carrying capacities of qq and cytccytc.", "Over capacity will cause backup in electron transport chain.", "Tokens are colored, e.g.", "“h:1, nadh:1” specify one token of hh and nadhnadh each.", "Token types are “h” for H+, “nadh” for NADH, “nadp” for NADP, “fadh2” for FADH2, “fad” for FAD, “e” for electrons, “o2” for oxygen and “h2o” for water.", "We remove t4t4 to model non-functioning protein complex IVIV.We model this problem as a colored petri net shown in Figure REF .", "The normal situation is made up of the entire graph.", "The abnormal situation (with non-functional final protein complex) is modeled by removing transition $t4$ from the graphAlternatively, we can model a non-functioning transition by attaching an inhibition arc to it with one token at its source place.", "We encode both situations in ASP with maximal firing set policy.", "Both are run for 10 steps.", "The amount of $h$ (H+) is compared in the $is$ (intermembrane space) to determine change in pH and the firing sequence is compared to explain the effect.", "In normal situation (entire graph), we get the following: fires(t1,0) fires(t2,0) fires(t3,1) fires(t1,2) fires(t2,2) fires(t3,2) fires(t4,2) fires(t3,3) fires(t4,3) fires(t6,3) fires(t1,4) fires(t2,4) fires(t3,4) fires(t4,4) fires(t6,4) fires(t3,5) fires(t4,5) fires(t6,5) fires(t1,6) fires(t2,6) fires(t3,6) fires(t4,6) fires(t6,6) fires(t3,7) fires(t4,7) fires(t6,7) fires(t1,8) fires(t2,8) fires(t3,8) fires(t4,8) fires(t6,8) fires(t3,9) fires(t4,9) fires(t6,9) fires(t1,10) fires(t2,10) fires(t3,10) fires(t4,10) fires(t6,10) holds(is,15,h,10) with $t4$ removed, we get the following: fires(t1,0) fires(t2,0) fires(t3,1) fires(t1,2) fires(t2,2) fires(t3,2) fires(t6,3) fires(t6,4) holds(is,2,h,10) Note that the amount of H+ ($h$ ) produced in the intermembrane space ($is$ ) is much smaller when the final protein complex is non-functional ($t4$ removed).", "Lower H+ translates to higher pH.", "Thus, the pH of intermembrane space will increase as a result of nonfunctional final protein.", "Also, note that the firing of $t3$ , $t1$ and $t2$ responsible for shuttling electrons also stop very quickly when $t4$ no longer removes the electrons ($e$ ) from Cyt c ($cytc$ ) to produce $H_2O$ .", "This is because $cytc$ and $q$ are at their capacity on electrons that they can carry and stop the electron transport chain by inhibiting transitions $t3$ , $t2$ and $t1$ .", "Trend for various runs is shown in Figure REF .", "Figure: Simulation of Petri Net in Figure .", "In a complete model of the biological system, there will be a mechanism that keeps the quantity of H+ in check in the intermembrane space and will plateau at some point." ], [ "Comparing Altered Trajectories due to Gradient Equilization Intervention", "Question 6 Exposure to a toxin caused the membranes to become permeable to ions.", "In a mitochondrion, how would this affect the pH in the intermembrane space and also ATP production?", "Provided Answer: “The pH of the intermembrane space would decrease as H+ ions diffuse through the membrane, and the H+ ion gradient is lost.", "The H+ gradient is essential in ATP production b/c facilitated diffusion of H+ through ATP synthase drives ATP synthesis.", "Decreasing the pH would lead to a decrease in the rate of diffusion through ATP synthase and therefore a decrease in the production of ATP.” Solution 6 Oxidative phosphorylation is shown in Fig 9.15 of Campbell's book.", "In order to explain the effect on pH in the intermembrane space and the ATP production we will show the change in the amount of H+ ions in the intermembrane space as well as the amount of ATP produced when the inner mitochondrial membrane is impermeable and permeable.", "Note that the concentration of H+ determines the pH.", "we have chosen to simplify the diagram by not having FADH2 in the picture.", "Its removal does not change the response, since it provides an alternate input mechanism to electron transport chain.", "We will assume that a steady input of NADH, H+, O2, ADP and P is available in the mitochondrial matrix.", "We also assume an electron carrying capacity of 2 for both ubiquinone (Q) and cytochrome c (Cyt c).", "We fulfill the steady supply requirement of substances by having input quantities in excess of what would be consumed during our simulation interval.", "We model this problem as a colored petri net shown in Figure REF .", "Transition $t6,t7$ shown in dotted style are added to model the abnormal situationIf reverse permeability is also desired additional arcs may be added from mm to is.", "They capture the diffusion of H+ ions back from Intermembrane Space to the Mitochondrial matrix.", "One or both may be enabled to capture degrees of permeability.", "we have added a condition on the firing of $t5$ (ATP Synthase activation) to enforce gradient to pump ATP Synthase.", "Figure: Petri Net graph relevant to question .", "“is” is the intermembrane space, “mm” is mitochondrial matrix, “q” is ubiquinone and “cytc” is cytochrome c. Tokens are colored, e.g.", "“h:1, nadh:1” specify one token of hh and nadhnadh each.", "Token types are “h” for H+, “nadh” for NADH, “e” for electrons, “o2” for oxygen, “h2o” for water, “atp” for ATP and “adp” for ADP.", "We add t6,t7t6,t7 to model cross domain diffusion from intermembrane space to mitochondrial matrix.", "One or both of t6,t7t6,t7 may be enabled at a time to control the degree of permeability.", "The text above “t5” is an additional condition which must be satisfied for “t5” to be enabled.We encode both situations in ASP with maximal firing set policy.", "Both are run for 10 steps each and the amount of $h$ and $atp$ is compared to determine the effect of pH and ATP production.", "We capture the gradient requirement as the following ASP codeWe can alternatively model this by having a threshold arc from “is” to “t5” if only a minimum trigger quantity is required in the intermembrane space.", ": notenabled(T,TS) :- T==t5, C==h, trans(T), col(C), holds(is,Qis,C,TS), holds(mm,Qmm,C,TS), Qmm+3 > Qis, num(Qis;Qmm), time(TS).", "In the normal situation, we get the following $h$ token distribution after 10 steps: holds(is,11,h,10) holds(mm,1,h,10) holds(mm,6,atp,10) we change the permeability to 1 ($t6$ enabled), we get the following token distribution instead: holds(is,10,h,10) holds(mm,2,h,10) holds(mm,5,atp,10) we change the permeability to 2 ($t6,t7$ enabled), the distribution changes as follows: holds(is,8,h,10) holds(mm,4,h,10) holds(mm,2,atp,10) Note that as the permeability increases, the amount of H+ ($h$ ) in intermembrane space ($is$ ) decreases and so does the amount of ATP ($h$ ) in mitochondrial matrix.", "Thus, an increase in permeability will increase the pH.", "If the permeability increases even beyond 2, no ATP will be produced from ADP due to insufficient H+ gradient.", "Trend from various runs is shown in Figure REF .", "Figure: Quantities of H+ and ATP at various run lengths and permeabilities for the Petri Net model in Figure ." ], [ "Comparing Altered Trajectories due to Delay Intervention", "Question 7 Membranes must be fluid to function properly.", "How would decreased fluidity of the membrane affect the efficiency of the electron transport chain?", "Provided Answer: “Some of the components of the electron transport chain are mobile electron carriers, which means they must be able to move within the membrane.", "If fluidity decreases, these movable components would be encumbered and move more slowly.", "This would cause decreased efficiency of the electron transport chain.” Solution 7 The answer deals requires background knowledge about fluidity and how it relates to mobile carriers not presented in the source chapter.", "From background knowledge we find that the higher the fluidity, higher the mobility.", "The electron transport chain is presented in Fig 9.15 of Campbell's book.", "From background knowledge, we know that the efficiency of the electron transport chain is measured by the amount of ATP produced per NADH/FADH2.", "The ATP production happens due to the gradient of H+ ions across the mitochondrial membrane.", "The higher the number of H+ ions in the intermembrane space, the higher would be the gradient and the resulting efficiency.", "So we measure the efficiency of the chain by the amount of H+ transported to intermembrane space, assuming all other (fixed) molecules behave normally.", "This is a valid assumption since H+ transported from mitochondrial matrix is directly proportional to the amount of electrons shuttled through the non-mobile complexes and there is a linear chain from the electron carrier to oxygen.", "We model this chain using a Petri Net with durative transitions shown in Figure REF .", "Higher the duration of transitions, lower the fluidity of the membrane.", "We assume that a steady supply of NADH and H+ is available in the mitochondrial membrane.", "We fulfill this requirement by having quantities in excess of what will be consumed during the simulation.", "We ignore FADH2 from the diagram, since it is just an alternate path to the electron chain.", "Using it by itself will produce a lower number of H+ transporter to intermembrane space, but it will not change the result.", "We compare the amount of H+ transported into the intermembrane space to gauge the efficiency of the electron transport chain.", "More efficient the chain is, more H+ will it transport.", "We model three scenarios: normal fluidity, low fluidity with transitions $t3$ and $t4$ having an execution time of 2 and an lower fluidity with transitions $t3,t4$ having execution time of 4.", "Figure: Petri Net graph relevant to question .", "“is” is the intermembrane space, “mm” is mitochondrial matrix, “q” is ubiquinone and “cytc” is cytochrome c. Tokens are colored, e.g.", "“h:1, nadh:1” specify one token of hh and nadhnadh each.", "Token types are “h” for H+, “nadh” for NADH, “e” for electrons.", "Numbers in square brackets below the transition represent transition durations with default of one time unit, if the number is missing.We encode these cases in ASP with maximal firing set semantics and simulate them for 10 time steps.", "For the normal fluidity we get: holds(is,27,h,10) for low fluidity we get: holds(is,24,h,10) for lower fluidity we get: holds(is,18,h,10) Note that as the fluidity decreases, so does the amount of H+ transported to intermembrane space, pointing to lower efficiency of electron transport chain.", "Trend of various runs is shown in Figure REF .", "Figure: Quantities of H+ produced in the intermembrane space at various run lengths and fluidities for the Petri Net model in Figure ." ], [ "Comparing Altered Trajectories due to Priority and Read Interventions", "Question 8 Phosphofructokinase (PFK) is allosterically regulated by ATP.", "Considering the result of glycolysis, is the allosteric regulation of PFK likely to increase or decrease the rate of activity for this enzyme?", "Provided Answer: “Considering that one of the end products of glycolysis is ATP, PFK is inhibited when ATP is abundant and bound to the enzyme.", "The inhibition decreases ATP production along this pathway.” Solution 8 Regulation of Phosphofructokinase (PFK) is presented in Figure 9.20 of Campbell's book.", "We ignore substances upstream of Fructose 6-phosphate (F6P) by assuming they are available in abundance.", "We also ignore AMP by assuming normal supply of it.", "We also ignore any output of glycolysis other than ATP production since the downstream processes ultimately produce additional ATP.", "Citric acid is also ignored since it is not relevant to the question at hand.", "We monitor the rate of activity of PFK by the number of times it gets used for glycolysis.", "We model this problem as a Petri Net shown in Figure REF .", "Allosteric regulation of PFK is modeled by a compound “pfkatp” which represents PFK's binding with ATP to form a compound.", "Details of allosteric regulation are not provided in the same chapter, they are background knowledge from external sources.", "Higher than normal quantity of ATP is modeled by a threshold arc (shown with arrow-heads at both ends) with an arbitrary threshold value of 4.", "This number can be increased as necessary.", "The output of glycolysis and down stream processes “t3” has been set to 2 to run the simulation in a reasonable amount of time.", "It can be made larger as necessary.", "The allosteric regulation transition “t4” has also been given a higher priority than glycolysis transition “t3”.", "This way, ATP in excess will cause PFK to be converted to PFK+ATP compound, reducing action of PFK.", "We assume that F6P is available in sufficient quantity and so is PFK.", "This requirement is fulfilled by having more quantity than can be consumed in the simulation duration.", "We model both the normal situation including transition $t4$ shown in dotted style and the abnormal situation where $t4$ is removed.", "Figure: Petri Net graph relevant to question .", "“pfk” is phosphofructokinase, “f6p” is fructose 6-phosphate, “atp” is ATP and “pfkatp” is the pfk bound with atp for allosteric regulation.", "Transition “t3” represents enzymic action of pfk, “t4” represents the binding of pfk with atp.", "The double arrowed arc represents a threshold arc, which enables “t4” when there are at least 4 tokens available at “atp”.", "Numbers above transitions in angular brackets represent arc priorities.We encode both situations in ASP with maximal firing set policy and run them for 10 time steps.", "At the end of the run we compare the firing count of transition $t3$ for both cases.", "For the normal case (with $t4$ ), we get the following results: holds(atp,0,c,0) holds(f6p,20,c,0) holds(pfk,20,c,0) holds(pfkatp,0,c,0) fires(t3,0) holds(atp,2,c,1) holds(f6p,19,c,1) holds(pfk,19,c,1) holds(pfkatp,0,c,1) fires(t3,1) holds(atp,4,c,2) holds(f6p,18,c,2) holds(pfk,18,c,2) holds(pfkatp,0,c,2) fires(t4,2) holds(atp,3,c,3) holds(f6p,18,c,3) holds(pfk,17,c,3) holds(pfkatp,1,c,3) fires(t3,3) holds(atp,5,c,4) holds(f6p,17,c,4) holds(pfk,16,c,4) holds(pfkatp,1,c,4) fires(t4,4) holds(atp,4,c,5) holds(f6p,17,c,5) holds(pfk,15,c,5) holds(pfkatp,2,c,5) fires(t4,5) holds(atp,3,c,6) holds(f6p,17,c,6) holds(pfk,14,c,6) holds(pfkatp,3,c,6) fires(t3,6) holds(atp,5,c,7) holds(f6p,16,c,7) holds(pfk,13,c,7) holds(pfkatp,3,c,7) fires(t4,7) holds(atp,4,c,8) holds(f6p,16,c,8) holds(pfk,12,c,8) holds(pfkatp,4,c,8) fires(t4,8) holds(atp,3,c,9) holds(f6p,16,c,9) holds(pfk,11,c,9) holds(pfkatp,5,c,9) fires(t3,9) holds(atp,5,c,10) holds(f6p,15,c,10) holds(pfk,10,c,10) holds(pfkatp,5,c,10) fires(t4,10) Note that $t3$ fires only when the ATP falls below our set threshold, above it PFK is converted to PFK+ATP compound via $t4$ .", "For the abnormal case (without $t4$ ) we get the following results: holds(atp,0,c,0) holds(f6p,20,c,0) holds(pfk,20,c,0) holds(pfkatp,0,c,0) fires(t3,0) holds(atp,2,c,1) holds(f6p,19,c,1) holds(pfk,19,c,1) holds(pfkatp,0,c,1) fires(t3,1) holds(atp,4,c,2) holds(f6p,18,c,2) holds(pfk,18,c,2) holds(pfkatp,0,c,2) fires(t3,2) holds(atp,6,c,3) holds(f6p,17,c,3) holds(pfk,17,c,3) holds(pfkatp,0,c,3) fires(t3,3) holds(atp,8,c,4) holds(f6p,16,c,4) holds(pfk,16,c,4) holds(pfkatp,0,c,4) fires(t3,4) holds(atp,10,c,5) holds(f6p,15,c,5) holds(pfk,15,c,5) holds(pfkatp,0,c,5) fires(t3,5) holds(atp,12,c,6) holds(f6p,14,c,6) holds(pfk,14,c,6) holds(pfkatp,0,c,6) fires(t3,6) holds(atp,14,c,7) holds(f6p,13,c,7) holds(pfk,13,c,7) holds(pfkatp,0,c,7) fires(t3,7) holds(atp,16,c,8) holds(f6p,12,c,8) holds(pfk,12,c,8) holds(pfkatp,0,c,8) fires(t3,8) holds(atp,18,c,9) holds(f6p,11,c,9) holds(pfk,11,c,9) holds(pfkatp,0,c,9) fires(t3,9) holds(atp,20,c,10) holds(f6p,10,c,10) holds(pfk,10,c,10) holds(pfkatp,0,c,10) fires(t3,10) Note that when ATP is not abundant, transition $t3$ fires continuously, which represents the enzymic activity that converts F6P to downstream substances.", "Trend of various runs is shown in Figure REF .", "Figure: Petri Net model in Figure ." ], [ "Comparing Altered Trajectories due to Automatic Conversion Intervention", "Question 9 How does the oxidation of NADH affect the rate of glycolysis?", "Provided Answer: “NADH must be oxidized back to NAD+ in order to be used in glycolysis.", "Without this molecule, glycolysis cannot occur.” Solution 9 Cellular respiration is summarized in Fig 9.6 of Campbell's book.", "NAD+ is reduced to NADH during glycolysis (see Campbell's Fig 9.9) during the process of converting Glyceraldehyde 3-phosphate (G3P) to 1,3-Bisphosphoglycerate (BPG13).", "NADH is oxidized back to NAD+ during oxidative phosphorylation by the electron transport chain (see Campbell's Fig 9.15).", "We can gauge the rate of glycolysis by the amount of Pyruvate produced, which is the end product of glycolysis.", "We simplify our model by abstracting glycolysis as a black-box that takes Glucose and NAD+ as input and produces NADH and Pyruvate as output, since there is a linear chain from Glucose to Pyruvate that depends upon the availability of NAD+.", "We also abstract oxidative phosphorylation as a black-box which takes NADH as input and produces NAD+ as output.", "None of the other inner workings of oxidative phosphorylation play a role in answering the question assuming they are functioning normally.", "We also ignore the pyruvate oxidation and citric acid cycle stages of cellular respiration since their end products only provide additional raw material for oxidative phosphorylation and do not add value to answering the question.", "We assume a steady supply of Glucose and all other substances used in glycolysis but a limited supply of NAD+, since it can be recycled from NADH and we want to model its impact.", "We fulfill the steady supply requirement of Glucose with sufficient initial quantity in excess of what will be consumed during our simulation interval.", "We also ensure that we have sufficient initial quantity of NAD+ to maintain glycolysis as long as it can be recycled.", "Figure: Petri Net graph relevant to question .", "“glu” represents glucose, “gly1” represents glycolysis, “pyr” represents pyruvate, “ox1” represents oxidative phosphorylation, “nadh” represents NADH and “nadp” represents NAD+.", "“ox1” is removed to model stoppage of oxidation of NADH to NAD+.Figure REF is a Petri Net representation of our simplified model.", "Normal situation is modeled by the entire graph, where NADH is recycled back to NAD+, while the abnormal situation is modeled by the graph with the transition $ox1$ (shown in dotted style) removed.", "We encode both situations in ASP with the maximal firing set policy.", "Both situations are run for 5 steps and the amount of pyruvate is compared to determine the difference in the rate of glycolysis.", "In normal situation (with $ox1$ transition), unique quantities of pyruvate ($pyr$ ) are as follows: fires(gly1,0) fires(gly1,1) fires(gly1,2) fires(gly1,3) fires(gly1,4) holds(pyr,10,5) while in abnormal situation (without $ox1$ transition), unique quantities of pyruvate are as follows: fires(gly1,0) fires(gly1,1) fires(gly1,2) holds(pyr,6,5) Note that the rate of glycolysis is lower when NADH is not recycled back to NAD+, as the glycolysis stops after the initial quantity of 6 NAD+ is consumed.", "Also, the $gly1$ transition does not fire after time-step 2, indicating glycolysis has stopped.", "Trend of various runs is shown in Figure REF .", "Figure: Amount of “pyr” produced by runs of various lengths of Petri Net in Figure .", "It shows results for both normal situation where “nadh” is recycled to “nadp” as well as the abnormal situation where this recycling is stopped." ], [ "Comparing Altered Trajectories due to Initial Value Intervention", "Question 10 During intense exercise, can a muscle cell use fat as a concentrated source of chemical energy?", "Explain.", "Provided Answer: “When oxygen is present, the fatty acid chains containing most of the energy of a fat are oxidized and fed into the citric acid cycle and the electron transport chain.", "During intense exercise, however, oxygen is scarce in muscle cells, so ATP must be generated by glycolysis alone.", "A very small part of the fat molecule, the glycerol backbone, can be oxidized via glycolysis, but the amount of energy released by this portion is insignificant compared to that released by the fatty acid chains.", "(This is why moderate exercise, staying below 70% maximum heart rate, is better for burning fat because enough oxygen remains available to the muscles.", ")” Solution 10 The process of fat consumption in glycolysis and citric acid cycle is summarized in Fig 9.19 of Campbell's book.", "Fats are digested into glycerol and fatty acids.", "Glycerol gets fed into glycolysis after being converted into Gyceraldehyde 3-phosphate (G3P), while fatty acids get fed into citric acid cycle after being broken down through beta oxidation and converted into Acetyl CoA.", "Campbell's Fig 9.18 identify a junction in catabolism where aerobic respiration or fermentation take place depending upon whether oxygen is present or not.", "Energy produced at various steps is in terms of ATP produced.", "In order to explain whether fat can be used as a concentrated source of chemical energy or not, we have to show the different ways of ATP production and when they kick in.", "We combine the various pieces of information collected from Fig 9.19, second paragraph on second column of p/180, Fig 9.15, Fig 9.16 and Fig 9.18 of Campbell's book into Figure REF .", "We model two situations when oxygen is not available in the muscle cells (at the start of a intense exercise) and when oxygen is available in the muscle cells (after the exercise intensity is plateaued).", "We then compare and contrast them on the amount of ATP produced and the reasons for the firing sequences.", "Figure: Petri Net graph relevant to question .", "“fats” are fats, “dig” is digestion of fats, “gly” is glycerol, “fac” is fatty acid, “g3p” is Glyceraldehyde 3-phosphate, “pyr” is pyruvate, “o2” is oxygen, “nadh” is NADH, “acoa” is Acyl CoA, “atp” is ATP, “op1” is oxidative phosphorylation, “cac1” is citric acid cycle, “fer1” is fermentation, “ox1” is oxidation of pyruvate to Acyl CoA and “box1” is beta oxidation.Figure REF is a petri net representation of our simplified model.", "Our edge labels have lower numbers on them than the yield in Fig 9.16 of Campbell's book but they still capture the difference in volume that would be produced due to oxidative phosphorylation vs. glycolysis.", "Using exact amounts will only increase the difference of ATP production due to the two mechanisms.", "We encode both situations (when oxygen is present and when it is not) in ASP with maximal firing set policy.", "We run them for 10 steps.", "The firing sequence and the resulting yield of ATP explain what the possible use of fat as a source of chemical energy.", "At he start of intense exercise, when oxygen is in short supply: holds(acoa,0,0) holds(atp,0,0) holds(fac,0,0) holds(fats,5,0) holds(g3p,0,0) holds(gly,0,0) holds(nadh,0,0) holds(o2,0,0) holds(pyr,0,0) fires(dig,0) holds(acoa,0,1) holds(atp,0,1) holds(fac,1,1) holds(fats,4,1) holds(g3p,0,1) holds(gly,1,1) holds(nadh,0,1) holds(o2,0,1) holds(pyr,0,1) fires(box1,1) fires(dig,1) fires(t2,1) holds(acoa,1,2) holds(atp,0,2) holds(fac,1,2) holds(fats,3,2) holds(g3p,1,2) holds(gly,1,2) holds(nadh,1,2) holds(o2,0,2) holds(pyr,0,2) fires(box1,2) fires(cac1,2) fires(dig,2) fires(gly6,2) fires(t2,2) holds(acoa,1,3) holds(atp,2,3) holds(fac,1,3) holds(fats,2,3) holds(g3p,1,3) holds(gly,1,3) holds(nadh,3,3) holds(o2,0,3) holds(pyr,1,3) fires(box1,3) fires(cac1,3) fires(dig,3) fires(fer1,3) fires(gly6,3) fires(t2,3) holds(acoa,1,4) holds(atp,4,4) holds(fac,1,4) holds(fats,1,4) holds(g3p,1,4) holds(gly,1,4) holds(nadh,5,4) holds(o2,0,4) holds(pyr,1,4) fires(box1,4) fires(cac1,4) fires(dig,4) fires(fer1,4) fires(gly6,4) fires(t2,4) holds(acoa,1,5) holds(atp,6,5) holds(fac,1,5) holds(fats,0,5) holds(g3p,1,5) holds(gly,1,5) holds(nadh,7,5) holds(o2,0,5) holds(pyr,1,5) fires(box1,5) fires(cac1,5) fires(fer1,5) fires(gly6,5) fires(t2,5) holds(acoa,1,6) holds(atp,8,6) holds(fac,0,6) holds(fats,0,6) holds(g3p,1,6) holds(gly,0,6) holds(nadh,9,6) holds(o2,0,6) holds(pyr,1,6) fires(cac1,6) fires(fer1,6) fires(gly6,6) holds(acoa,0,7) holds(atp,10,7) holds(fac,0,7) holds(fats,0,7) holds(g3p,0,7) holds(gly,0,7) holds(nadh,10,7) holds(o2,0,7) holds(pyr,1,7) fires(fer1,7) holds(acoa,0,8) holds(atp,10,8) holds(fac,0,8) holds(fats,0,8) holds(g3p,0,8) holds(gly,0,8) holds(nadh,10,8) holds(o2,0,8) holds(pyr,0,8) holds(acoa,0,9) holds(atp,10,9) holds(fac,0,9) holds(fats,0,9) holds(g3p,0,9) holds(gly,0,9) holds(nadh,10,9) holds(o2,0,9) holds(pyr,0,9) holds(acoa,0,10) holds(atp,10,10) holds(fac,0,10) holds(fats,0,10) holds(g3p,0,10) holds(gly,0,10) holds(nadh,10,10) holds(o2,0,10) holds(pyr,0,10) when the exercise intensity has plateaued and oxygen is no longer in short supply: holds(acoa,0,0) holds(atp,0,0) holds(fac,0,0) holds(fats,5,0) holds(g3p,0,0) holds(gly,0,0) holds(nadh,0,0) holds(o2,10,0) holds(pyr,0,0) fires(dig,0) holds(acoa,0,1) holds(atp,0,1) holds(fac,1,1) holds(fats,4,1) holds(g3p,0,1) holds(gly,1,1) holds(nadh,0,1) holds(o2,10,1) holds(pyr,0,1) fires(box1,1) fires(dig,1) fires(t2,1) holds(acoa,1,2) holds(atp,0,2) holds(fac,1,2) holds(fats,3,2) holds(g3p,1,2) holds(gly,1,2) holds(nadh,1,2) holds(o2,10,2) holds(pyr,0,2) fires(box1,2) fires(cac1,2) fires(dig,2) fires(gly6,2) fires(op1,2) fires(t2,2) holds(acoa,1,3) holds(atp,5,3) holds(fac,1,3) holds(fats,2,3) holds(g3p,1,3) holds(gly,1,3) holds(nadh,2,3) holds(o2,9,3) holds(pyr,1,3) fires(box1,3) fires(cac1,3) fires(dig,3) fires(gly6,3) fires(op1,3) fires(ox1,3) fires(t2,3) holds(acoa,2,4) holds(atp,10,4) holds(fac,1,4) holds(fats,1,4) holds(g3p,1,4) holds(gly,1,4) holds(nadh,4,4) holds(o2,8,4) holds(pyr,1,4) fires(box1,4) fires(cac1,4) fires(dig,4) fires(gly6,4) fires(op1,4) fires(ox1,4) fires(t2,4) holds(acoa,3,5) holds(atp,15,5) holds(fac,1,5) holds(fats,0,5) holds(g3p,1,5) holds(gly,1,5) holds(nadh,6,5) holds(o2,7,5) holds(pyr,1,5) fires(box1,5) fires(cac1,5) fires(gly6,5) fires(op1,5) fires(ox1,5) fires(t2,5) holds(acoa,4,6) holds(atp,20,6) holds(fac,0,6) holds(fats,0,6) holds(g3p,1,6) holds(gly,0,6) holds(nadh,8,6) holds(o2,6,6) holds(pyr,1,6) fires(cac1,6) fires(gly6,6) fires(op1,6) fires(ox1,6) holds(acoa,4,7) holds(atp,25,7) holds(fac,0,7) holds(fats,0,7) holds(g3p,0,7) holds(gly,0,7) holds(nadh,9,7) holds(o2,5,7) holds(pyr,1,7) fires(cac1,7) fires(op1,7) fires(ox1,7) holds(acoa,4,8) holds(atp,29,8) holds(fac,0,8) holds(fats,0,8) holds(g3p,0,8) holds(gly,0,8) holds(nadh,10,8) holds(o2,4,8) holds(pyr,0,8) fires(cac1,8) fires(op1,8) holds(acoa,3,9) holds(atp,33,9) holds(fac,0,9) holds(fats,0,9) holds(g3p,0,9) holds(gly,0,9) holds(nadh,10,9) holds(o2,3,9) holds(pyr,0,9) fires(cac1,9) fires(op1,9) holds(acoa,2,10) holds(atp,37,10) holds(fac,0,10) holds(fats,0,10) holds(g3p,0,10) holds(gly,0,10) holds(nadh,10,10) holds(o2,2,10) holds(pyr,0,10) fires(cac1,10) fires(op1,10) We see that more ATP is produced when oxygen is available.", "Most ATP (energy) is produced by the oxidative phosphorylation which requires oxygen.", "When oxygen is not available, small amount of energy is produced due to glycolysis of glycerol ($gly$ ).", "With oxygen a lot more energy is produced, most of it due to fatty acids ($fac$ ).", "Trend of various runs is shown in Figure REF .", "Figure: Amount of “atp” produced by runs of various lengths of Petri Net in Figure .", "Two situations are shown: when oxygen is in short supply and when it is abundant." ], [ "Conclusion", "In this chapter we presented how to model biological systems as Petri Nets, translated them into ASP, reasoned with them and answered questions about them.", "We used diagrams from Campbell's book, background knowledge and assumptions to facilitate our modeling work.", "However, source knowledge for real world applications comes from published papers, magazines and books.", "This means that we have to do text extraction.", "In one of the following chapters we look at some of the real applications that we have worked on in the past in collaboration with other researchers to develop models using text extraction.", "But first, we look at how we use the concept of answering questions using Petri Nets to build a question answering system.", "We will extend the Petri Nets even more for this." ], [ "BioPathQA - A System for Modeling, Simulating, and Querying Biological Pathways", "The BioPathQA System" ], [ "Introduction", "In this chapter we combine the methods from Chapter , notions from action languages, and ASP to build a system BioPathQA and a language to specify pathways and query them.", "We show how various biological pathways are encoded in BioPathQA and how it computes answers of queries against them." ], [ "Description of BioPathQA", "Our system has the following components: [(i)] a pathway specification language a query language to specify the deep reasoning question, an ASP program that encodes the pathway model and its extensions for simulation.", "Knowledge about biological pathways comes in many different forms, such as cartoon diagrams, maps with well defined syntax and semantics (e.g.", "Kohn's maps [43]), and biological publications.", "Similar to other technical domains, some amount of domain knowledge is also required.", "Users want to collect information from disparate sources and encode it in a pathway specification.", "We have developed a language to allow users to describe their pathway.", "This description includes describing the substances and actions that make up the pathway, the initial state of the substances, and how the state of the pathway changes due to the actions.", "An evolution of a pathway's state from the initial state, through a set of actions is called a trajectory.", "Being a specification language targeted at biological systems, multiple actions autonomously execute in parallel as soon as their preconditions are satisfied.", "The amount of parallelism is dictated by any resource conflicts between the actions.", "When that occurs, only one sub-set of the possible actions can execute, leading to multiple outcomes from that point on.", "Questions are usually provided in natural language, which is vague.", "To avoid the vagaries of natural language, we developed a language with syntax close to natural language but with a well defined formal semantics.", "The query language allows a user to make changes to the pathway through interventions, and restrict its trajectories through observations and query on aggregate values in a trajectory, across a set of trajectories and even over two sets of trajectories.", "This allows the user to compare a base case of a pathway specification with an alternate case modified due to interventions and observations.", "This new feature is a major contribution of our research.", "Inspiration for our high level language comes from action languages and query languages such as [26].", "While action languages generally describe transition systems [27], our language describes trajectories.", "In addition, our language is geared towards modeling natural systems, in which actions occur autonomously [66] when their pre-conditions are satisfied; and we do not allow the quantities to become negative (as the quantities represent amounts of physical entities).", "Next we describe the syntax of our pathway specification language and the query language.", "Following that we will describe the syntax of our language and how we encode it in ASP.", "Syntax of Pathway Specification Language (BioPathQA-PL) The alphabet of pathway specification language $\\mathcal {P}$ consists of disjoint nonempty domain-dependent sets $A$ , $F$ , $L$ representing actions, fluents, and locations, respectively; a fixed set $S$ of firing styles; a fixed set $K$ of keywords providing syntactic sugar (shown in bold face in pathway specification language below); a fixed set of punctuations $\\lbrace `,^{\\prime } \\rbrace $ ; and a fixed set of special constants $\\lbrace `1^{\\prime },`^{\\prime },`max^{\\prime }\\rbrace $ ; and integers.", "Each fluent $f \\in F$ has a domain $dom(f)$ which is either integer or binary and specifies the values $f$ can take.", "A fluent is either simple, such as $f$ or locational, such as $f[l]$ , where $l \\in L$ .", "A state $s$ is an interpretation of $F$ that maps fluents to their values.", "We write $s(f)=v$ to represent “$f$ has the value $v$ in state $s$ ”.", "States are indexed, such that consecutive states $s_i$ and $s_{i+1}$ represent an evolution over one time step from $i$ to $i+1$ due to firing of an action set $T_i$ in $s_i$ .", "We illustrate the role of various symbols in the alphabet with examples from the biological domain.", "Consider the following example of a hypothetical pathway specification: $&\\mathbf {domain~of~} sug \\mathbf {~is~} integer, fac \\mathbf {~is~} integer, acoa \\mathbf {~is~} integer, h2o \\mathbf {~is~} integer\\\\&gly \\mathbf {~may~execute~causing~} sug \\mathbf {~change~value~by~} -1, acoa \\mathbf {~change~value~by~} +1\\\\&box \\mathbf {~may~execute~causing~} fac \\mathbf {~change~value~by~} -1, acoa \\mathbf {~change~value~by~} +1 \\\\&\\mathbf {~~~~~~~~~~~if~} h2o \\mathbf {~has~value~} 1 \\mathbf {~or~higher~}\\\\&\\mathbf {inhibit~} box \\mathbf {~if~} sug \\mathbf {~has~value~} 1 \\mathbf {~or~higher~}\\\\&\\mathbf {initially~} sug \\mathbf {~has~value~} 3, fac \\mathbf {~has~value~} 4, acoa \\mathbf {~has~value~} 0 &$ It describes two processes glycolysis and beta-oxidation represented by actions `$gly$ ' and `$box$ ' in lines () and ()-() respectively.", "Substances used by the pathway, i.e.", "sugar, fatty-acids, acetyl-CoA, and water are represented by numeric fluents `$sug$ ',`$fac$ ',`$acoa$ ', and `$h2o$ ' respectively in line (REF ).", "When glycolysis occurs, it consumes 1 unit of sugar and produces 1 unit of acetyl-CoA (line ().", "When beta-oxidation occurs, it consumes 1 unit of fatty-acids and produces 1 unit of acetyl-CoA (line ()).", "The inputs of glycolysis implicitly impose a requirement that glycolysis can only occur when at least 1 unit of sugar is available.", "Similarly, the input of beta-oxidation implicitly a requirement that beta-oxidation can only occur when at least 1 unit of fatty-acids is available.", "Beta oxidation has an additional condition imposed on it in line () that there must be at least 1 unit of water available.", "We call this a guard condition on beta-oxidation.", "Line () explictly inhibits beta-oxidation when there is any sugar available; and line () sets up the initial conditions of the pathway, i.e.", "Initially 3 units of each sugar, 4 units of fatty-acids are available and no acetyl-CoA is available.", "The words `$\\lbrace domain,$ $is,$ $may,$ $execute,$ $causing,$ $change,$ $value,$ $by,$ $has,$ $or,$ $higher,$ $inhibit,$ $if,$ $initially\\rbrace $ ' are keywords.", "When locations are involved, locational fluents take place of simple fluents and our representation changes to include locations.", "For example: $&gly \\mathbf {~may~execute~causing~} \\nonumber \\\\&~~~~sug \\mathbf {~atloc~} mm \\mathbf {~change~value~by~} -1, \\nonumber \\\\&~~~~acoa \\mathbf {~atloc~} mm \\mathbf {~change~value~by~} +1$ represents glycolysis taking 1 unit of sugar from mitochondrial matrix (represented by `$mm$ ') and produces acetyl-CoA in the mitochondrial matrix.", "Here `$atloc$ ' is an additional keyword.", "A pathway is composed of a collection of different types of statements and clauses.", "We first introduce their syntax, following that we will give intuitive definitions, and following that we will show how they are combined together to construct a pathway specification.", "Definition 46 (Fluent domain declaration statement) A fluent domain declaration statement declares the values a fluent can take.", "It has the form: $\\mathbf {domain~of~} f \\mathbf {~is~} `integer^{\\prime }\|`binary^{\\prime } \\\\\\mathbf {domain~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} `integer^{\\prime }\|`binary^{\\prime }$ for simple fluent “$f$ ”, and locational fluent “$f[l]$ ”.", "Multiple domain statements are compactly written as: $\\mathbf {domain~of~} f_1 \\mathbf {~is~} `integer^{\\prime }\|`binary^{\\prime } , \\dots , f_n \\mathbf {~is~} `integer^{\\prime }\|`binary^{\\prime } \\\\\\mathbf {domain~of~} f_1 \\mathbf {~atloc~} l_1 \\mathbf {~is~} `integer^{\\prime }\|`binary^{\\prime } , \\dots , f_n \\mathbf {~atloc~} l_n \\mathbf {~is~} `integer^{\\prime }\|`binary^{\\prime }$ Binary domain is usually used for representing substances in a signaling pathway, while a metabolic pathways take positive numeric values.", "Since the domain is for a physical entity, we do not allow negative values for fluents.", "Definition 47 (Guard condition) A guard condition takes one of the following forms: $&f \\mathbf {~has~value~} w \\mathbf {~or~higher~} \\\\&f \\mathbf {~atloc~} l \\mathbf {~has~value~} w \\mathbf {~or~higher~}\\\\&f \\mathbf {~has~value~lower~than~} w \\\\&f \\mathbf {~atloc~} l \\mathbf {~has~value~lower~than~} w \\\\&f \\mathbf {~has~value~equal~to~} w \\\\&f \\mathbf {~atloc~} l \\mathbf {~has~value~equal~to~} w \\\\&f_1 \\mathbf {~has~value~higher~than~} f_2\\\\&f_1 \\mathbf {~atloc~} l_1 \\mathbf {~has~value~higher~than~} f_2 \\mathbf {~atloc~} l_2$ where, each $f$ in (REF ), (), (), () is a simple fluent, each $f[l]$ in (), (), (), () is a locational fluent with location $l$ , and each $w \\in \\mathbb {N}^+ \\cup \\lbrace 0 \\rbrace $ .", "Definition 48 (Effect clause) An effect clause can take one of the following forms: $&f \\mathbf {~change~value~by~} e\\\\&f \\mathbf {~atloc~} l \\mathbf {~change~value~by~} e$ where $a$ is an action, $f$ in (REF ) is a simple fluent, $f[l]$ in () is a locational fluent with location $l$ , $e \\in \\mathbb {N}^+ \\cup \\lbrace \\rbrace $ for integer fluents or $e \\in \\lbrace 1,-1,\\rbrace $ for binary fluents.", "Definition 49 (May-execute statement) A may-execute statement captures the conditions for firing an action $a$ and its impact.", "It is of the form: $a \\mathbf {~may~execute~causing~} & \\mathit {effect}_1, \\dots , \\mathit {effect}_m \\nonumber \\\\\\mathbf {~if~} &guard\\_cond_1, \\dots , guard\\_cond_n$ where $\\mathit {effect}_i$ is an effect clause; and $\\mathit {guard\\_cond}_j$ is a guard condition clause, $m > 0$ , and $n \\ge 0$ .", "If $n = 0$ , the effect statement is unconditional (guarded by $\\top $ ) and the $\\mathbf {if}$ is dropped.", "A single may-execute statement must not have $\\mathit {effect}_i, \\mathit {effect}_j$ with $e_i < 0, e_j < 0$ for the same fluent; or $e_i > 0, e_j > 0$ for the same fluent.", "Definition 50 (Must-execute statement) An must-execute statement captures the impact of firing of an action $a$ that must fire when enabled (as long as it is not inhibited).", "It is an expression of the form: $a \\mathbf {~normally~must~execute~causing~} & \\mathit {effect}_1, \\dots , \\mathit {effect}_m \\nonumber \\\\\\mathbf {~if~} &guard\\_cond_{1}, \\dots , guard\\_cond_n$ where $\\mathit {effect}_i$ , and $\\mathit {guard\\_cond}_j$ are as in (REF ) above.", "Definition 51 (Inhibit statement) An inhibit statement captures the conditions that inhibit an action from occurring.", "It is an expression of the form: $\\mathbf {inhibit~} a \\mathbf {~if~} guard\\_cond_1, \\dots , guard\\_cond_n$ where $a$ is an action, $guard\\_cond_i$ is a guard condition clause, and $n \\ge 0$ .", "if $n = 0$ , the inhibition of action `$a$ ' is unconditional `$\\mathbf {if}$ ' is dropped.", "Definition 52 (Initial condition statement) An initial condition statement captures the initial state of pathway.", "It is of the form: $\\mathbf {initially~} & f \\mathbf {~has~value~} w\\\\\\mathbf {initially~} & f \\mathbf {~atloc~} l \\mathbf {~has~value~} w$ where each $f$ in (REF ) is a simple fluent, $f[l]$ in () is a locational fluent with location $l$ , and each $w$ is a non-negative integer.", "Multiple initial condition statements are written compactly as: $\\mathbf {initially~} & f_1\\mathbf {~has~value~} w_1, \\dots , f_n \\mathbf {~has~value~} w_n\\\\\\mathbf {initially~} & f_1 \\mathbf {~atloc~} l_1 \\mathbf {~has~value~} w_1, \\dots , f_n \\mathbf {~atloc~} l_n \\mathbf {~has~value~} w_n$ Definition 53 (Duration Statement) A duration statement represents the duration of an action that takes longer than a single time unit to execute.", "It is of the form: $a \\mathbf {~executes~in~} d \\mathbf {~time~units}$ where $d$ is a positive integer representing the action duration.", "Definition 54 (Stimulate Statement) A stimulate statement changes the rate of an action.", "It is an expression of the form: $\\mathbf {normally~stimulate~} a \\mathbf {~by~factor~} n \\mathbf {~if~} guard\\_cond_1,\\dots ,guard\\_cond_n$ where $guard\\_cond_i$ is a condition, $n > 0$ .", "When $n=0$ , the stimulation is unconditional and $\\mathbf {~if~}$ is dropped.", "A stimulation causes the $\\mathit {effect}$ in may-cause and must-fire multiplied by $n$ .", "Actions execute automatically when fireable, subject to the available fluent quantities.", "Definition 55 (Firing Style Statement) A firing style statement specifies how many actions execute simultaneously (or action parallelism).", "It is of the form: $\\mathbf {firing~style~} S$ where, $S$ is either “1”, “$$ ”, or “$max$ ” for serial execution, interleaved execution, and maximum parallelism.", "We will now give the intuitive meaning of these statements and put them into context w.r.t.", "the biological domain.", "Though our description below uses simple fluents only, it applies to locational fluents in a obvious manner.", "The reason for having locational fluents at all is that they allow a more natural pathway specification when substance locations are involved instead of devising one's own encoding scheme.", "For example, in a mitochondria, hydrogen ions (H+) appear in multiple locations (intermembrane space and mitochondrial matrix), with each location carrying its distinct quantity separate from other locations.", "Intuitively, a may-execute statement (REF ) represents an action $a$ that may fire if all conditions `$\\mathit {guard\\_cond}_{1},\\dots ,\\mathit {guard\\_cond}_n$ ' hold in the current state.", "When it executes, it impacts the state as specified in $\\mathit {effect}$ s. In biological context, action $a$ represents a process, such as a reaction, $\\mathit {effect}$ s represent the inputs / ingredients of the reaction, and $guard\\_cond$ represent additional preconditions necessary for the reaction to proceed.", "Condition (REF ) holds in a state $s$ if $s(f) \\ge w$ .", "It could represent an initiation concentration $w$ of a substance $f$ which is higher than the quantity consumed by the reaction $a$ .", "Condition () holds in a state $s$ if $s(f) < w$ .", "Condition () holds in a state $s$ if $s(f) = w$ .", "Condition () holds in a state $s$ if $s(f_1) > s(f_2)$ capturing a situation where a substance gradient is required for a biological process to occur.", "An example of one such process is the synthesis of ATP by ATP Synthase, which requires a H+ (Hydrogen ion) gradient across the inner mitochondrial matrix [64].", "Intuitively, the effect clause (REF ) of an action describes the impact of an action on a fluent.", "When an action $a$ fires in a state $s$ , the value of $f$ changes according to the effect clause for $f$ .", "The value of $f$ increases by $e$ if $e > 0$ , decreases by $e$ if $e < 0$ , or decreases by $s(f)$ if $e = `^{\\prime }$ (where $`^{\\prime }$ can be interpreted as $-s(f)$ ).", "For a reaction $a$ , a fluent with $e < 0$ represents an ingredient consumed in quantity $\|e\|$ by the reaction; a fluent with $e > 0$ represents a product of the reaction in quantity $e$ ; a fluent with $e = `^{\\prime }$ represents consuming all quantity of the substance due to the reaction.", "Since the fluents represent physical substances, their quantities cannot become negative.", "As a result, any action that may cause a fluent quantity to go below zero is disallowed.", "Intuitively, a must-execute statement (REF ) is similar to a may-exec, except that when enabled, it preferentially fires over other actions as long as there isn't an inhibit proposition that will cause the action to become inactive.", "It captures the effect of an action that must happen whenever enabled.", "Intuitively, an inhibit statement (REF ) specifies the conditions that inhibits an action.", "In a biological context, it defines inhibition of reactions, e.g., through biological feedback control.", "Though we could have added these conditions to may-exec, it is more intuitive to keep them separate as inhibition conditions are usually discovered separately in a biological domain.", "Including them as part of may-fire would constitute a surgery of existing knowledge bases.", "Intuitively, an initial condition statement (REF ) specifies the initial values of fluents.", "The collection of such propositions defines the initial state $s_0$ of the pathway.", "In a biological context, this defines the initial distribution of substances in the biological system.", "Intuitively, an action duration statement (REF ) represents action durations, especially when an action takes longer to execute.", "When an action $a$ with duration $d$ fires in state $s_k$ , it immediately decreases the values of fluents with $e < 0$ and $e = $ upon execution, however, it does not increase the value of fluents with $e > 0$ until time the end of its execution in state $s_{k+d}$ .", "In a biological context the action duration captures a reaction's duration.", "A reaction consumes its ingredients immediately on firing, processes them for duration $d$ and generates its products at the end of this duration.", "Intuitively, a stimulate statement (REF ) represents a change in the rate of an action $a$ .", "The stimulation causes the action to change its rate of consumption of its ingredients and production of its products by a factor $n$ .", "In biological context, this stimulation can be a result of an enzyme or a stimulant's availability, causing a reaction that normally proceeds slowly to occur faster.", "Intuitively, a firing style statement (REF ) specifies the parallelism of actions.", "When it is “1”, at most one action may fire, when it is “$max$ ”, the maximum allowable actions must fire, and when it is “$$ ”, any subset of fireable actions may fire simultaneously.", "In a biological domain the firing style allows one to model serial operations, parallel operations and maximally parallel operations.", "The maximum parallelism is also useful in quickly discovering changes that occur in a biological system.", "Definition 56 (Pathway Specification) A pathway specification is composed of one or more may-execute, must-execute, effect, inhibit, stimulate, initially, priority, duration statements, and one firing style statement.", "When a duration statement is not specified for an action, it is assumed to be 1.", "Any fluents for which an initial quantity is not specified are assumed to have a value of zero.", "A pathway specification is consistent if [(i)] there is at most one firing style, priority, duration statement for each action $a$ ; the $guard\\_cond_1,\\dots ,guard\\_cond_n$ from a may-execute or must-execute are disjoint from any other may-execute or must-execute Note that `$f1 \\text{ has value } 5 \\text{ or higher }$ ' overlaps with `$f1 \\text{ has value } 7 \\text{ or higher}$ ' and the two conditions are not considered disjoint.", "; locational and non-locational fluents may not be intermixed; domain of fluents, effects, conditions and numeric values are consistent, i.e., effects and conditions on binary fluents must be binary; and the pathway specification does not cause it to violate fluent domains by producing non-binary values for binary fluents.", "Each pathway specification $\\mathbf {D}$ represents a collection of trajectories of the form: $\\sigma = s_0, T_0, s_1, \\dots , s_{k-1}, T_{k-1}, s_k$ .", "Each trajectory encodes an evolution of the pathway starting from an initial state $s_0$ , where $s_i$ 's are states, and $T_i$ 's are sets of actions that fired in state $s_i$ .", "Intuitively, a trajectory starts from the initial state $s_0$ .", "Each $s_i,s_{i+1}$ pair represents the state evolution in one time step due to the action set $T_i$ .", "An action set $T_i$ is only executable in state $s_i$ , if the sum of changes to fluents due to $e_i < 0$ and $e_i = $ will not result in any of the fluents going negative.", "Changes to fluents due to $e_i > 0$ for the action set $T_i$ occur over subsequent time-steps depending upon the durations of actions involved.", "Thus, the state $s_i(f_i)$ is the sum of $e_i > 0$ for actions of duration $d$ that occurred $d$ time steps before (current time step) $i$ , i.e.", "$a \\in T_{i-d}$ , where the default duration $d$ of an action is 1 if none specified.", "Next we describe the semantics of the pathway specification language, which describes how these trajectories are generated.", "Semantics of Pathway Specification Language (BioPathQA-PL) The semantics of the pathway specification language are defined in terms of the trajectories of the domain description $\\mathbf {D}$ .", "Since our pathway specification language is inspired by Petri Nets, we use Petri Nets execution semantics to define its trajectories.", "However, some constructs in our pathway language specification are not directly representable in standard Petri Nets, as a result, we will have to extend them.", "Let an arc-guard be a conjunction of guard conditions of the form (REF )-(), such that it is wholly constructed of either locational or non-locational fluents, but not both.", "We introduce a new type of Guarded-arc Petri Net in which each arc has an arc-guard expression associated with it.", "Arcs with the same arc-guard are traversed when a transition connected to them fires and the arc-guard is found to hold in the current state.", "The arc-guards of arcs connected to the same transition form an exclusive set, such that only arcs corresponding to one guard expression may fire (for one transition).", "This setup can lead to different outcomes of an actionArcs for different guard expressions emanating / terminating at a place can further be combined into a single conditional arc with conditional arc-weights.", "If none of the condition applies then the arc is assumed to be missing..", "The transitions in this new type of Petri Net can have the following inscriptions on them: Propositional formula, specifying the executability conditions of the transition.", "Arc-durations, represented as “$dur(n)$ ”, where $n \\in \\mathbb {N}^+$ A must-execute inscription, “$must\\text{-}execute(guard)$ ”, requires that when the $guard$ holds in a state where this transition is enabled, it must fire, unless explicitly inhibited.", "The $guard$ has the same form as an $arc\\text{-}guard$ A stimulation inscription, “$stimulate(n,guard)$ ”, applies a multiplication factor $n \\in \\mathbb {N}^+$ to the input and output quantities consumed and produced by the transition, when $guard$ hold in the current state, where $guard$ has the same form as an $arc\\text{-}guard$ .", "Certain aspects of our nets are similar to CPNs [39].", "However, the CPNs do not allow our semantics of the reset arcs, or must-fire guards.", "Guarded-Arc Petri Net Figure: Example of a guarded-arc Petri Net.Figure REF shows an example of a guarded-arc Petri Net.", "There are two arc-guard expressions $f1<5$ and $f1>5$ .", "When $f1<5$ , $t1$ consumes one token from place $f1$ and produces one token in place $f2$ .", "When $f1>5$ , $t1$ 's consumption and production of the same tokens doubles.", "The transition $t1$ implicitly gets the guards for each arc represented the or-ed conditions $(f1<5) \\vee (f1>5))$ .", "The arc also has two conditions inhibiting it, they are represented by the and-ed conditions $\\lnot (f1>7) \\wedge \\lnot (f1<3)$ , where `$\\lnot $ ' represents logical not.", "Transition $t1$ is stimulated by factor 3 when $f1>5$ and it has a duration of 10 time units.", "A transition cannot fire even though one of its arc-guards is enabled, unless the token requirements on the arc itself are also fulfilled, e.g.", "if $f1$ has value 0 in the current state, even though $f1 < 5$ guard is satisfied, the transition cannot execute, because the input arc $(f1,t1)$ for this guard needs 1 token.", "Definition 57 (Guard) A condition is of the form: $(f < v), (f \\le v), (f > v), (f \\ge v), (f = v)$ , where $f$ is a fluent and $v$ either a fluent or a numeric constant.", "Then, a guard is a propositional formula of conditions, with each condition treated as a proposition, subject to the restriction that all fluents in all conditions in a guard are either locational or simple, but not both.", "Definition 58 (Interpretation of a Guard) An interpretation of a guard $G$ is a possible assignment of a value to each fleuent $f \\in G$ from the domain of $f$ .", "Definition 59 (Guard Satisfaction) A guard $G$ with simple fluents is satisfied w.r.t.", "a state $s$ , written $s \\models G$ iff $G$ has an interpretation in which each of its fluents $f$ has the value $s(f)$ and $G$ is true.", "A guard $G$ with locational fluents is satisfied w.r.r.", "a state $s$ , written $s \\models G$ iff $G$ has an interpretation in which each of its fluents $f[l]$ has the value $m_{s(l)}(f)$ and $G$ is true, where $m_X(f)$ is the multiplicity of $f$ in $X$ .", "Definition 60 (Guarded-Arc Petri Net) A Guarded-Arc Petri Net is a tuple $PN^G=(P,T,G,E,R,W,D,B,TG,MF,L)$ , where: $P & \\text{ is a finite set of places}\\\\T & \\text{ is a finite set of transitions}\\\\G & \\text{ is a set of guards as defined in definition~(\\ref {def:guard})}\\\\TG &: T \\rightarrow G \\text{ are the transition guards }\\\\E &\\subseteq (T \\times P \\times G) \\cup (P \\times T \\times G) \\text{ are the guarded arcs }\\\\R &\\subseteq P \\times T \\times G \\text{ are the guarded reset arcs }\\\\W &: E \\rightarrow \\mathbb {N}^+ \\text{ are arc weights }\\\\D &: T \\rightarrow \\mathbb {N}^+ \\text{ are the transition durations}\\\\B &: T \\rightarrow G \\times \\mathbb {N}^+ \\text{ transition stimulation or boost }\\\\MF &: T \\rightarrow 2^{G} \\text{ must-fire guards for a transition}\\\\L &: P \\rightarrow \\mathbb {N}^+ \\text{ specifies maximum number to tokens for each place }$ subject to constraints: $P \\cap T = \\emptyset $ $R \\cap E = \\emptyset $ Let $t_1 \\in T$ and $t_2 \\in T$ be any two distinct transitions, then $g1 \\in MF(t_1)$ and $g2 \\in MF(t_2)$ must not have an interpretation that make both $g1$ and $g2$ true.", "Let $t \\in T$ be a transition, and $gg_t = \\lbrace g: (t,p,g) \\in E \\rbrace \\cup \\lbrace g: (p,t,g) \\in E \\rbrace \\cup \\lbrace g: (p,t,g) \\in R \\rbrace $ be the set of arc-guards for normal and reset arcs connected to it, then $g_1 \\in gg_t, g_2 \\in gg_t$ must not have an interpretation that makes both $g_1$ and $g_2$ true.", "Let $t \\in T$ be a transition, and $gg_t = \\lbrace g: (g,n) \\in B(t) \\rbrace $ be its stimulation arc-guards, then $g_1 \\in gg_t, g_2 \\in gg_t$ must not have an interpretation that makes both $g_1$ and $g_2$ true.", "Let $t \\in T$ be a transition, and $(g,n) \\in B(t) : n>1$ , then there must not exist a place $p \\in P : (p,t,g) \\in E, L(p) = 1$ .", "Intuitively, stimulation of binary inputs is not supported.", "We will make a simplifying assumption that all places are readable by using their place names.", "Execution of the $PN^G$ occurs in discrete time steps.", "Definition 61 (Marking (or State)) Marking (or State) of a Guarded-Arc Petri Net $PN^G$ is the token assignment of each place $p_i \\in P$ .", "Initial marking $M_0 : P \\rightarrow \\mathbb {N}^0$ , while the token assignment at step $k$ is written as $M_k$ .", "Next we define the execution semantics of $PN^G$ .", "First we introduce some terminology that will be used below.", "Let $s_0 = M_0$ represent the the initial marking (or state), $s_k = M_k$ represent the marking (or state) at time step $k$ , $s_k(p)$ represent the marking of place $p$ at time step $k$ , such that $s_k = [s_k(p_0), \\dots , $ $s_k(p_n)]$ , where $P=\\lbrace p_0, \\dots , p_n\\rbrace $ $T_k$ be the firing-set that fired in step $k$ , $b_k(t)$ be the stimulation value applied to a transition $t$ w.r.t.", "step $k$ $en_k$ be the set of enabled transitions in state $s_k$ , $mf_k$ be the set of must-execute transitions in state $s_k$ , $consume_k(p,\\lbrace t_1,\\dots ,t_n\\rbrace )$ be the sum of tokens that will be consumed from place $p$ if transitions $t_1,\\dots ,t_n$ fired in state $s_k$ , $overc_k(\\lbrace t_1,\\dots ,t_n\\rbrace )$ be the set of places that will have over-consumption of tokens if transitions $t_1,\\dots ,t_n$ were to fire simultaneously in state $s_k$ , $sel_k(fs)$ be the set of possible firing-set choices in state $s_k$ using $\\mathit {fs}$ firing style $produce_k(p)$ be the total production of tokens in place $p$ (in state $s_k$ ) due to actions terminating in state $s_k$ , $s_{k+1}$ be the next state computed from current state $s_k$ due to firing of transition-set $T_k$ $min(a,b)$ gives the minimum of numbers $a,b$ , such that $min(a,b) = a \\text{ if } a < b \\text{ or } b \\text{ otherwise }$ .", "Then, the execution semantics of the guarded-arc Petri Net starting from state $s_0$ using firing-style $\\mathit {fs}$ is given as follows: $b_k(t) &={\\left\\lbrace \\begin{array}{ll}n &\\text{ if } (g,n) \\in B(t), s_k \\models g\\\\1 &\\text{ otherwise }\\end{array}\\right.", "}\\nonumber \\\\en_k &= \\lbrace t : t \\in T, s_k \\models TG(t), \\forall (p,t,g) \\in E, \\nonumber \\\\&~~~~~~~~(s_k \\models g, s_k(p) \\ge W(p,t,g) * b_k(t)) \\rbrace \\nonumber \\\\mf_k &= \\lbrace t : t \\in en_k, \\exists g \\in MF(t), s_k \\models g \\rbrace \\nonumber \\\\consume_k(p,\\lbrace t_1,\\dots ,t_n\\rbrace ) &= \\sum _{i=1,\\dots ,n}{W(p,t_i,g) * b_k(t) : (p,t_i,g) \\in E, s_k \\models g} \\nonumber \\\\&+ \\sum _{i=1,\\dots ,n}{s_k(p) : (p,t_i,g) \\in R, s_k \\models g} \\nonumber \\\\overc_k(\\lbrace t_1,\\dots ,t_n\\rbrace ) &= \\lbrace p : p \\in P, s_k(p) < consume_k(p,\\lbrace t_1,\\dots ,t_n\\rbrace ) \\rbrace \\nonumber \\\\sel_k(1) &={\\left\\lbrace \\begin{array}{ll}mf_k & \\text{ if } \|mf_k\| = 1\\\\\\lbrace \\lbrace t \\rbrace : t \\in en_k \\rbrace & \\text{ if } \|mf_k\| < 1\\end{array}\\right.", "}\\nonumber \\\\sel_k() &= \\lbrace ss : ss \\in 2^{en_k}, mf_k \\subseteq ss, overc_k(ss) = \\emptyset \\rbrace \\nonumber \\\\sel_k(max) &= \\lbrace ss : ss \\in 2^{en_k}, mf_k \\subseteq ss, overc_k(p,ss) = \\emptyset , \\nonumber \\\\&~~~~~~(\\nexists ss^{\\prime } \\in 2^{en_k} : ss \\subset ss^{\\prime }, mf_k \\subseteq ss^{\\prime }, overc_k(ss^{\\prime }) = \\emptyset ) \\rbrace \\nonumber \\\\T_k &= T^{\\prime }_k : T^{\\prime }_k \\in sel_k(\\mathit {fs}), (\\nexists t \\in en_k \\setminus T^{\\prime }_k, t \\text{ is a reset transition } ) \\nonumber \\\\produce_k(p) &= \\sum _{j=0,\\dots ,k}{W(t_i,p,g) b_j(t_i) : (t_i,p,g) \\in E, t_i \\in T_j, D(t_i)+j = k+1}\\nonumber \\\\s_{k+1}(p) &= min(s_k(p) - consume_k(p,T_k) + produce_k(p), L(p))$ Definition 62 (Trajectory) $\\sigma = s_0, T_0, s_1, \\dots , s_{k}, T_{k}, s_{k+1}$ is a trajectory of $PN^G$ iff given $s_0 = M_0$ , each $T_i$ is a possible firing-set in $s_i$ whose firing produces $s_{i+1}$ per $PN^G$ 's execution semantics in (REF ).", "Construction of Guarded-Arc Petri Net from a Pathway Specification Now we describe how to construct such a Petri Net from a given pathway specification $\\mathbf {D}$ with locational fluents.", "We proceed as follows: The set of transitions $T = \\lbrace a : a \\text{ is an action in } \\mathbf {D}\\rbrace $ .", "The set of places $P = \\lbrace f : f \\text{ is a fluent in } \\mathbf {D}\\rbrace $ .", "The limit relation for places $L(f) ={\\left\\lbrace \\begin{array}{ll}1 & \\text{ if } `\\mathbf {domain~of~} f \\mathbf {~is~} binary^{\\prime } \\in \\mathbf {D}\\\\\\infty & \\text{ otherwise }\\\\\\end{array}\\right.", "}$ .", "An arc-guard expression $guard\\_cond_1,\\dots ,guard\\_cond_n$ is translated into the conjunction $(guard_1,\\dots ,guard_n)$ , where $guard_i$ is obtained from $guard\\_cond_i$ as follows: A guard condition (REF ) is translated to $f \\ge w$ A guard condition () is translated to $f < w$ A guard condition () is translated to $f = w$ A guard condition () is translated to $f_1 > f_2$ A may-execute statement (REF ) is translated into guarded arcs as follows: Let guard $G$ be the translation of arc-guard conditions $guard\\_cond_1,\\dots ,$ $guard\\_cond_n$ specified in the may-execute proposition.", "The effect clause (REF ) are translated into arcs as follows: An effect clause with $e < 0$ is translated into an input arc $(f,a,G)$ , with arc-weight $W(f,a,G) = \|e\|$ .", "An effect clause with $e = `^{\\prime }$ is translated into a reset set $(f,a,G)$ with arc-weight $W(f,a,G) = $ .", "An effect clause with $e > 0$ is translated into an output arc $(a,f,G)$ , with arc-weight $W(a,f,G) = e$ .", "A must-execute statement (REF ) is translated into guarded arcs in the same way as may execute.", "In addition, it adds an arc-inscription $must\\text{-}exec(G)$ , where $G$ is the translation of the arc-guard.", "An inhibit statement (REF ) is translated into $IG = (guard_1,\\dots ,$ $guard_n)$ , where $(guard_1,\\dots ,$ $guard_n)$ is the translation of $(guard\\_cond\\_1,\\dots ,$ $guard\\_cond_n)$ An initial condition statement () sets the initial marking of a specific place $p$ to $w$ , i.e.", "$M_0(p) = w$ .", "An duration statement (REF ) adds a $dur(d)$ inscription to transition $a$ .", "A stimulate statement (REF ) adds a $stimulate(n,G)$ inscription to transition $a$ , where $G$ is the translation of the stimulate guard, a conjunction of $guard\\_cond_1,$ $\\dots ,guard\\_cond_n$ .", "A guard $(G_1 \\vee \\dots \\vee G_n) \\wedge (\\lnot IG_1 \\wedge \\dots \\wedge \\lnot IG_m)$ is added to each transition $a$ , where $G_i, 1 \\le i \\le n$ is a guard for a may-execute or a must-execute statement and $IG_i, 1 \\le i \\le m$ is a guard for an inhibit statement.", "A firing style statement (REF ) does not visibly appear on a Petri Net diagram, but it specifies the transition firing regime the Petri Net follows.", "Example: Consider the following pathway specification: $\\begin{array}{llll}\\mathbf {domain~of~} &f_1 \\mathbf {~is~} integer, &f_2 \\mathbf {~is~} integer\\\\t_1 \\mathbf {~may~execute~}\\\\\\;\\;\\;\\;\\;\\;\\;\\;\\;\\mathbf {causing~} & f_1 \\mathbf {~change~value~by~} -1, & f_2 \\mathbf {~change~value~by~} +1\\\\\\;\\;\\;\\;\\;\\;\\;\\;\\;\\mathbf {if~} & f_1 \\mathbf {~has~value~lower~than~} 5\\\\t_1 \\mathbf {~may~execute~}\\\\\\;\\;\\;\\;\\;\\;\\;\\;\\;\\mathbf {causing~} & f_1 \\mathbf {~change~value~by~} -2, & f_2 \\mathbf {~change~value~by~} +2\\\\\\;\\;\\;\\;\\;\\;\\;\\;\\;\\mathbf {if~} & f_1 \\mathbf {~has~value~higher~than~} 5\\\\\\mathbf {duration~of~} & t1 \\mathbf {~is~} 10\\\\\\mathbf {inhibit~} t1 \\mathbf {~if~} & f_1 \\mathbf {~has~value~higher~than~} 7\\\\\\mathbf {inhibit~} t1 \\mathbf {~if~} & f_1 \\mathbf {~has~value~lower~than~} 3\\\\\\mathbf {normally~stimulate~} t1 & \\mathbf {~by~factor~} 3 \\\\\\;\\;\\;\\;\\;\\;\\;\\;\\;\\mathbf {~if~} & f_2 \\mathbf {~has~value~higher~than~} 5\\\\\\mathbf {initially~}& f_1 \\mathbf {~has~value~} 0, & f_2 \\mathbf {~has~value~} 0,\\\\\\mathbf {firing~style~} & max\\end{array}$ This pathway specification is encoded as the Petri Net in figure REF .", "Guarded-Arc Petri Net with Colored Tokens Next we extend the Guarded-arc Petri Nets to add Colored tokens.", "We will use this extension to model pathways with locational fluents.", "Figure: Example of a guarded-arc Petri Net with colored tokens.Figure REF shows an example of a guarded-arc Petri Net with colored tokens.", "There are two arc-guard expressions $f3[p1]<5$ and $f3[p1]>5$ .", "When $f3[p1]<5$ , $t1$ consumes one token of color $f1$ from place $p1$ and produces one token of color $f2$ in place $p2$ .", "When $f3[p1]>5$ , $t1$ 's consumption and production of the same colored tokens doubles.", "The transition $t1$ implicitly gets the guards for each arc represented the or-ed conditions $((f3[p1]<5) \\vee (f3[p1]>5))$ .", "The arc also has two conditions inhibiting it, they are represented by the and-ed conditions $\\lnot (f3[p1]>7) \\wedge \\lnot (f3[p1]<3)$ , where `$\\lnot $ ' represents logical not.", "Transition $t1$ is stimulated by factor 3 when $f3[p1]>5$ and it has a duration of 10 time units.", "Definition 63 (Guarded-Arc Petri Net with Colored Tokens) A Guarded-Arc Petri Net with Colored Tokens is a tuple $PN^{GC}=(P,T,C,G,E,R,W,D,B,TG,$ $MF,L)$ , such that: $P &: \\text{finite set of places}\\\\T &: \\text{finite set of transitions}\\\\C &: \\text{finite set of colors}\\\\G &: \\text{set of guards as defined in definition~(\\ref {def:guard}) with locational fluents}\\\\TG &: T \\rightarrow G \\text{ are the transition guards }\\\\E &\\subseteq (T \\times P \\times G) \\cup (P \\times T \\times G) \\text{ are the guarded arcs }\\\\R &\\subseteq P \\times T \\times G \\text{ are the guarded reset arcs }\\\\W &: E \\rightarrow \\langle C,m \\rangle \\text{ are arc weights; each arc weight is a multiset over } C\\\\D &: T \\rightarrow \\mathbb {N}^+ \\text{ are the transition durations}\\\\B &: T \\rightarrow G \\times \\mathbb {N}^+ \\text{ transition stimulation or boost }\\\\MF &: T \\rightarrow 2^{G} \\text{ must-fire guards for a transition}\\\\L &: P \\times C \\rightarrow \\mathbb {N}^+ \\text{ specifies maximum number of tokens for each color in each place}\\\\$ subject to constraints: $P \\cap T = \\emptyset $ $R \\cap E = \\emptyset $ Let $t_1 \\in T$ and $t_2 \\in T$ be any two distinct transitions, then $g1 \\in MF(t_1)$ and $g2 \\in MF(t_2)$ must not have an interpretation that make both $g1$ and $g2$ true.", "Let $t \\in T$ be a transition, and $gg_t = \\lbrace g: (t,p,g) \\in E \\rbrace \\cup \\lbrace g: (p,t,g) \\in E \\rbrace \\cup \\lbrace g: (p,t,g) \\in R \\rbrace $ be the set of arc-guards for normal and reset arcs connected to it, then $g_1 \\in gg_t, g_2 \\in gg_t$ must not have an interpretation that makes both $g_1$ and $g_2$ true.", "Let $t \\in T$ be a transition, and $gg_t = \\lbrace g: (g,n) \\in B(t) \\rbrace $ be its stimulation guards, then $g_1 \\in gg_t, g_2 \\in gg_t$ must not have an interpretation that makes both $g_1$ and $g_2$ true.", "Let $t \\in T$ be a transition, and $(g,n) \\in B(t) : n>1$ , then there must not exist a place $p \\in P$ and a color $c \\in C$ such that $(p,t,g) \\in E, L(p,c) = 1$ .", "Intuitively, stimulation of binary inputs is not supported.", "Definition 64 (Marking (or State)) Marking (or State) of a Guarded-Arc Petri Net with Colored Tokens $PN^{GC}$ is the colored token assignment of each place $p_i \\in P$ .", "Initial marking is written as $M_0 : P \\rightarrow \\langle C,m \\rangle $ , while the token assignment at step $k$ is written as $M_k$ .", "We make a simplifying assumption that all places are readable by using their place name.", "Next we define the execution semantics of the guarded-arc Petri Net.", "First we introduce some terminology that will be used below.", "Let $s_0 = M_0$ represent the the initial marking (or state), $s_k = M_k$ represent the marking (or state) at time step $k$ , $s_k(p)$ represent the marking of place $p$ at time step $k$ , such that $s_k = [s_k(p_0),$ $\\dots ,s_k(p_n)]$ , where $P=\\lbrace p_0,\\dots ,p_n\\rbrace $ .", "$T_k$ be the firing-set that fired in state $s_k$ , $b_k(t)$ be the stimulation value applied to transition $t$ w.r.t.", "state $s_k$ , $en_k$ be the set of enabled transitions in state $s_k$ , $mf_k$ be the set of must-fire transitions in state $s_k$ , $consume_k(p,\\lbrace t_1,\\dots ,t_n\\rbrace )$ be the sum of colored tokens that will be consumed from place $p$ if transitions $t_1,\\dots ,t_n$ fired in state $s_k$ , $overc_k(\\lbrace t_1,\\dots ,t_n\\rbrace )$ be the set of places that will have over-consumption of tokens if transitions $t_1,\\dots ,t_n$ were to fire simultaneously in state $s_k$ , $sel_k(fs)$ be the set of possible firing-sets in state $s_k$ using $\\mathit {fs}$ firing style, $produce_k(p)$ be the total production of tokens in place $p$ in state $s_k$ due to actions terminating in state $s_k$ , $s_{k+1}$ be the next state computed from current state $s_k$ and $T_k$ , $m_X(c)$ represents the multiplicity of $c \\in C$ in multiset $X=\\langle C,m \\rangle $ , $c/n$ represents repetition of an element $c$ of a multi-set $n$ -times, multiplication of multiset $X= \\langle C,m \\rangle $ with a number $n$ be defined in terms of multiplication of element multiplicities by $n$ , i.e.", "$\\forall c \\in C, m_X(c)n$ , and $min(a,b)$ gives the minimum of numbers $a,b$ , such that $min(a,b) = a \\text{ if } a < b \\text{ or } b \\text{ otherwise }$ .", "Then, the execution semantics of the guarded-arc Petri Net starting from state $s_0$ using firing-style $\\mathit {fs}$ is given as follows: $b_k(t) &={\\left\\lbrace \\begin{array}{ll}n & \\text{ if } (g,n) \\in B(t), s_k \\models g\\\\1 &\\text{ otherwise }\\end{array}\\right.", "}\\nonumber \\\\en_k &= \\lbrace t \\in T, s_k \\models TG(t), \\forall (p,t,g) \\in E, \\nonumber \\\\&~~~~~~(s_k \\models g, s_k(p) \\ge W(p,t,g) b_k(t)) \\rbrace \\nonumber \\\\mf_k &= \\lbrace t \\in en_k, \\exists g \\in MF(t), s_k \\models g \\rbrace \\nonumber \\\\consume_k(p,\\lbrace t_1,\\dots ,t_n\\rbrace ) &= \\sum _{i=1,\\dots ,n}{W(p,t_i,g) * b_k(t) : (p,t_i,g) \\in E, s_k \\models g} \\nonumber \\\\&+ \\sum _{i=1,\\dots ,n}{s_k(p) : (p,t_i,g) \\in R, s_k \\models g} \\nonumber \\\\overc_k(\\lbrace t_1,\\dots ,t_n\\rbrace ) &= \\lbrace p \\in P : \\exists c \\in C, m_{s_k(p)}(c) < m_{consume_k(p,\\lbrace t_1,\\dots ,t_n\\rbrace )}(c) \\rbrace \\nonumber \\\\sel_k(1) &={\\left\\lbrace \\begin{array}{ll}mf_k & \\text{ if } \|mf_k\| = 1\\\\\\lbrace \\lbrace t \\rbrace : t \\in en_k \\rbrace & \\text{ if } \|mf_k\| < 1\\end{array}\\right.", "}\\nonumber \\\\sel_k() &={\\left\\lbrace \\begin{array}{ll}ss \\in 2^{en_k} : mf_k \\subseteq ss, overc_k(ss) = \\emptyset \\end{array}\\right.", "}\\nonumber \\\\sel_k(max) &={\\left\\lbrace \\begin{array}{ll}ss \\in 2^{en_k} : mf_k \\subseteq ss, overc_k(p,ss) = \\emptyset , \\\\(\\nexists ss^{\\prime } \\in 2^{en_k} : ss \\subset ss^{\\prime }, mf_k \\subseteq ss^{\\prime }, overc_k(ss^{\\prime }) = \\emptyset )\\end{array}\\right.", "}\\nonumber \\\\T_k &= T^{\\prime }_k : T^{\\prime }_k \\in sel_k(\\mathit {fs}), (\\nexists t \\in en_k \\setminus T^{\\prime }_k, t \\text{ is a reset transition } ) \\nonumber \\\\produce_k(p) &= \\sum _{j=0,\\dots ,k}{W(t_i,p,g) b_j(t_i) : (t_i,p,g) \\in E, t_i \\in T_j), D(t_i)+j = k+1}\\nonumber \\\\s_{k+1} &= [ c/n : c \\in C, \\nonumber \\\\&~~~~n=min(m_{s_k(p)}(c) \\nonumber \\\\&~~~~~~~~~~~~ - m_{consume_k(p,T_k)}(c) \\nonumber \\\\&~~~~~~~~~~~~ + m_{produce_k(p)}(c), L(p,c)) ]$ Definition 65 (Trajectory) $\\sigma = s_0, T_0, s_1, \\dots , s_{k}, T_{k}, s_{k+1}$ is a trajectory of $PN^{GC}$ iff given $s_0=M_0$ , each $T_i$ is a possible firing-set in $s_i$ whose firing produces $s_{i+1}$ per $PN^{GC}$ 's execution semantics in (REF ).", "Construction of Guarded-Arc Petri Net with Colored Tokens from a Pathway Specification with Locational Fluents Now we describe how to construct such a Petri Net from a given pathway specification $\\mathbf {D}$ with locational fluents.", "We proceed as follows: The set of transitions $T = \\lbrace a : a \\text{ is an action in } \\mathbf {D}\\rbrace $ .", "The set of colors $C = \\lbrace f : f[l] \\text{ is a fluent in } \\mathbf {D}\\rbrace $ .", "The set of places $P = \\lbrace l : f[l] \\text{ is a fluent in } \\mathbf {D}\\rbrace $ .", "The limit relation for each colored token in a place $L(f,c) ={\\left\\lbrace \\begin{array}{ll}1 & \\text{ if } `\\mathbf {domain~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} binary^{\\prime } \\in \\mathbf {D}\\\\\\infty & \\text{ otherwise }\\\\\\end{array}\\right.", "}$ .", "A guard expression $guard\\_cond_1,\\dots ,guard\\_cond_n$ is translated into the conjunction $(guard_1,\\dots ,$ $guard_n)$ , where $guard_i$ is obtained from $guard\\_cond_i$ as follows: A guard condition () is translated to $f[l] < w$ A guard condition () is translated to $f[l] = w$ A guard condition () is translated to $f[l] \\ge w$ A guard condition () is translated to $f_1[l_1] > f_2[l_2]$ A may-execute statement (REF ) is translated into guarded arcs as follows: Let guard $G$ be the translation of guard conditions $guard\\_cond_1,$ $\\dots ,guard\\_cond_n$ specified in the may-execute proposition.", "The effect clauses of the form () are grouped into input, reset and output effect sets for an action as follows: The clauses with $e < 0$ for the same place $l$ are grouped together into an input set of $a$ requiring input from place $l$ .", "The clauses with $e = `^{\\prime }$ for the same place $l$ are grouped together into a reset set of $a$ requiring input from place $l$ .", "The clauses with $e > 0$ for the same place $l$ are grouped together into an output set of $a$ to place $l$ .", "A group of input effect clauses $\\mathit {effect}_1,\\dots ,\\mathit {effect}_m, m > 0$ of the form () is translated into an input arc $(l,a,G)$ , with arc-weight $W(l,a,G) = w^+$ , where $w^+$ is the multi-set union of $f_i/\|e_i\|$ in $\\mathit {effect}_i, 1 \\le i \\le m$ .", "A group of output effect clauses $\\mathit {effect}_1,\\dots ,\\mathit {effect}_m, m > 0$ of the form () is translated into an output arc $(a,l,G)$ , with arc-weight $W(a,l,G) = w^-$ , where $w^-$ is the multi-set union of $f_i/e_i$ in $\\mathit {effect}_i, 1 \\le i \\le m$ .", "A group of reset effect clauses $\\mathit {effect}_1,\\dots ,\\mathit {effect}_m, m > 0$ of the form () is translated into a reset arc $(l,a,G)$ with arc-weight $W(l,a,G) = $ .", "A must-execute statement (REF ) is translated into guarded arcs in the same way as may execute.", "In addition, it adds an arc-inscription $must\\text{-}exec(G)$ , where $G$ is the guard, which is the translation of $guard\\_cond_1,\\dots ,guard\\_cond_n$ .", "An inhibit statement (REF ) is translated into $IG = (guard_1,\\dots ,guard_n)$ , where $(guard_1,\\dots ,guard_n)$ is the translation of $(guard\\_cond\\_1,\\dots ,guard\\_cond_n)$ An initial condition statement () sets the initial marking of a specific place $l$ for a specific color $f$ to $w$ , i.e.", "$m_{(M_0(l))}(f) = w$ .", "An duration statement (REF ) adds a $dur(d)$ inscription to transition $a$ .", "A stimulate statement (REF ) adds a $stimulate(n,G)$ inscription to transition $a$ , where guard $G$ is the translation of its guard expression $guard\\_cond_1,$ $\\dots ,guard\\_cond_n$ .", "A guard $(G_1 \\vee \\dots \\vee G_n) \\wedge (\\lnot IG_1 \\wedge \\dots \\wedge \\lnot IG_m)$ is added to each transition $a$ , where $G_i, 1 \\le i \\le n$ is the guard for a may-execute or a must-execute proposition and $IG_i, 1 \\le i \\le m$ is a guard for an inhibit proposition.", "A firing style statement (REF ) does not visibly show on a Petri Net, but it specifies the transition firing regime the Petri Net follows.", "Example Consider the following pathway specification: $&\\mathbf {domain~of~} \\nonumber \\\\&~~~~f_1 \\mathbf {~atloc~} l_1 \\mathbf {~is~} integer, f_2 \\mathbf {~atloc~} l_1 \\mathbf {~is~} integer, \\nonumber \\\\&~~~~f_3 \\mathbf {~atloc~} l_1 \\mathbf {~is~} integer, f_1 \\mathbf {~atloc~} l_2 \\mathbf {~is~} integer, \\nonumber \\\\&~~~~f_2 \\mathbf {~atloc~} l_2 \\mathbf {~is~} integer, f_3 \\mathbf {~atloc~} l_2 \\mathbf {~is~} integer \\nonumber \\\\&t_1 \\mathbf {~may~execute~causing} \\nonumber \\\\&~~~~~~~~f_1 \\mathbf {~atloc~} l_1 \\mathbf {~change~value~by~} -1, \\nonumber \\\\&~~~~~~~~f_2 \\mathbf {~atloc~} l_2 \\mathbf {~change~value~by~} +1 \\nonumber \\\\&~~~~\\mathbf {if~} f_3 \\mathbf {~atloc~} l_1 \\mathbf {~has~value~lower~than~} 5 \\nonumber \\\\&t_1 \\mathbf {~may~execute~causing} \\nonumber \\\\&~~~~~~~~f_1 \\mathbf {~atloc~} l_1 \\mathbf {~change~value~by~} -2, \\nonumber \\\\&~~~~~~~~f_2 \\mathbf {~atloc~} l_2 \\mathbf {~change~value~by~} +2 \\nonumber \\\\&~~~~\\mathbf {if~} f_3 \\mathbf {~atloc~} l_1 \\mathbf {~has~value~higher~than~} 5 \\nonumber \\\\&\\mathbf {duration~of~} t1 \\mathbf {~is~} 10 \\nonumber \\\\&\\mathbf {inhibit~} t1 \\mathbf {~if~} f_3 \\mathbf {~atloc~} l_1 \\mathbf {~has~value~higher~than~} 7 \\nonumber \\\\&\\mathbf {inhibit~} t1 \\mathbf {~if~} f_3 \\mathbf {~atloc~} l_1 \\mathbf {~has~value~lower~than~} 3 \\nonumber \\\\&\\mathbf {normally~stimulate~} t1 \\mathbf {~by~factor~} 3 \\nonumber \\\\&~~~~\\mathbf {~if~} f_2 \\mathbf {~atloc~} l_2 \\mathbf {~has~value~higher~than~} 5 \\nonumber \\\\&\\mathbf {initially~} \\nonumber \\\\&~~~~f_1 \\mathbf {~atloc~} l_1 \\mathbf {~has~value~} 0, f_1 \\mathbf {~atloc~} l_2 \\mathbf {~has~value~} 0, \\nonumber \\\\&~~~~f_2 \\mathbf {~atloc~} l_1 \\mathbf {~has~value~} 0, f_2 \\mathbf {~atloc~} l_2 \\mathbf {~has~value~} 0, \\nonumber \\\\&~~~~f_3 \\mathbf {~atloc~} l_1 \\mathbf {~has~value~} 0, f_3 \\mathbf {~atloc~} l_2 \\mathbf {~has~value~} 0 \\nonumber \\\\&\\mathbf {firing~style~} max$ Syntax of Query Language (BioPathQA-QL) The alphabet of query language $\\mathcal {Q}$ consists of the same sets $A,F,L$ from $\\mathcal {P}$ representing actions, fluents, and locations, respectively; a fixed set of reserved keywords $K$ shown in bold in syntax below; a fixed set $\\lbrace `:^{\\prime }, `;^{\\prime }, `,^{\\prime } , `^{\\prime \\prime }\\rbrace $ of punctuations; a fixed set of $\\lbrace `<^{\\prime },`>^{\\prime },`=^{\\prime }\\rbrace $ of directions; and constants.", "Our query language asks questions about biological entities and processes in a biological pathway described through the pathway specification language.", "This is our domain description.", "A query statement is composed of a query description (the quantity, observation, or condition being sought by the question), interventions (changes to the pathway), observations (about states and actions of the pathway), and initial setup conditions.", "The query statement is evaluated against the trajectories of the pathway, generated by simulating the pathway.", "These trajectories are modified by the initial setup and interventions.", "The resulting trajectories are then filtered to retain only those which satisfy the observations specified in the query statement.", "A query statement can take various forms: The simplest queries do not modify the pathway and check if a specific observation is true on a trajectory or not.", "An observation can be a point observation or an interval observation depending upon whether they can be evaluated w.r.t.", "a point or an interval on a trajectory.", "More complex queries modify the pathway in various ways and ask for comparison of an observation before and after such modification.", "Following query statements about the rate of production of `$bpg13$ ' illustrate the kind of queries that can be asked from our system about the specified glycolysis pathway as given in [64].", "Determine if `$n$ ' is a possible rate of production of substance `$bpg13$ ': $\\mathbf {rate~} & \\mathbf {of~production~of~} ^{\\prime }bpg13^{\\prime } \\mathbf {~is~} n \\nonumber \\\\&\\mathbf {when~observed~between~time~step~} 0 \\mathbf {~and~} \\mathbf {~time~step~} k ; &$ Determine if `$n$ ' is a possible rate of production of substance `$bpg13$ ' in a pathway when it is being supplied with a limited supply of an upstream substance `$f16bp$ ': $\\mathbf {rate}&\\mathbf {~of~production~of~} ^{\\prime }bpg13^{\\prime } \\mathbf {~is~} n \\nonumber \\\\&\\mathbf {when~observed~between~time~step~} 0 \\mathbf {~and~} \\mathbf {~time~step~} k ; \\nonumber \\\\&\\mathbf {using~initial~setup:~} \\mathbf {set~value~of~} `f16bp^{\\prime } \\mathbf {~to~} 5 ; &$ Determine if `$n$ ' is a possible rate of production of substance `$bpg13$ ' in a pathway when it is being supplied with a steady state supply of an upstream substance `$f16bp$ ' at the rate of 5 units per time-step: $\\mathbf {rate}&\\mathbf {~of~production~of~} ^{\\prime }bpg13^{\\prime } \\mathbf {~is~} n \\nonumber \\\\&\\mathbf {when~observed~between~time~step~} 0 \\mathbf {~and~} \\mathbf {~time~step~} k ; \\nonumber \\\\&\\mathbf {using~initial~setup:~} \\mathbf {continuously~supply~} `f16bp^{\\prime } \\mathbf {~in~quantity~} 5 ; &$ Determine if `$n$ ' is a possible rate of production of substance `$bpg13$ ' in a pathway when it is being supplied with a steady state supply of an upstream substance `$f16bp$ ' at the rate of 5 units per time-step and the pathway is modified to remove all quantity of the substance '$dhap$ ' as soon as it is produced: $\\mathbf {rate}&\\mathbf {~of~production~of~} ^{\\prime }bpg13^{\\prime } \\mathbf {~is~} n \\nonumber \\\\&\\mathbf {when~observed~between~time~step~} 0 \\mathbf {~and~} \\mathbf {~time~step~} k ; \\nonumber \\\\&\\mathbf {due~to~interventions:~} \\mathbf {remove~} `dhap^{\\prime } \\mathbf {~as~soon~as~produced} ;\\nonumber \\\\&\\mathbf {using~initial~setup:~} \\mathbf {continuously~supply~} `f16bp^{\\prime } \\mathbf {~in~quantity~} 5 ; &$ Determine if `$n$ ' is a possible rate of production of substance `$bpg13$ ' in a pathway when it is being supplied with a steady state supply of an upstream substance `$f16bp$ ' at the rate of 5 units per time-step and the pathway is modified to remove all quantity of the substance '$dhap$ ' as soon as it is produced and a non-functional pathway process / reaction named `$t5b$ ': $\\mathbf {rate}&\\mathbf {~of~production~of~} ^{\\prime }bpg13^{\\prime } \\mathbf {~is~} n \\nonumber \\\\&\\mathbf {when~observed~between~time~step~} 0 \\mathbf {~and~} \\mathbf {~time~step~} k ; \\nonumber \\\\&\\mathbf {due~to~interventions:~} \\mathbf {remove~} `dhap^{\\prime } \\mathbf {~as~soon~as~produced} ;\\nonumber \\\\&\\mathbf {due~to~observations:~} `t5b^{\\prime } \\mathbf {~does~not~occur} ; \\nonumber \\\\&\\mathbf {using~initial~setup:~} \\mathbf {continuously~supply~} `f16bp^{\\prime } \\mathbf {~in~quantity~} 5 ; &$ Determine if `$n$ ' is the average rate of production of substance `$bpg13$ ' in a pathway when it is being supplied with a steady state supply of an upstream substance `$f16bp$ ' at the rate of 5 units per time-step and the pathway is modified to remove all quantity of the substance '$dhap$ ' as soon as it is produced and a non-functional pathway process / reaction named `$t5b$ ': $\\mathbf {average}&\\mathbf {~rate~of~production~of~} ^{\\prime }bpg13^{\\prime } \\mathbf {~is~} n \\nonumber \\\\&\\mathbf {when~observed~between~time~step~} 0 \\mathbf {~and~} \\mathbf {~time~step~} k ; \\nonumber \\\\&\\mathbf {due~to~interventions:~} \\mathbf {remove~} `dhap^{\\prime } \\mathbf {~as~soon~as~produced} ;\\nonumber \\\\&\\mathbf {due~to~observations:~} `t5b^{\\prime } \\mathbf {~does~not~occur} ; \\nonumber \\\\&\\mathbf {using~initial~setup:~} \\mathbf {continuously~supply~} `f16bp^{\\prime } \\mathbf {~in~quantity~} 5 ; &$ Determine if `$d$ ' is the direction of change in the average rate of production of substance `$bpg13$ ' with a steady state supply of an upstream pathway input when compared with a pathway with the same steady state supply of an upstream pathway input, but in which the substance `$dhap$ ' is removed from the pathway as soon as it is produced and pathway process / reaction called `$t5b$ ' is non-functional: $\\mathbf {dir}&\\mathbf {ection~of~change~in~average~rate~of~production~of~} ^{\\prime }bpg13^{\\prime } \\mathbf {~is~} d \\nonumber \\\\&\\mathbf {when~observed~between~time~step~} 0 \\mathbf {~and~} \\mathbf {~time~step~} k ; \\nonumber \\\\&\\mathbf {comparing~nominal~pathway~with~modified~pathway~obtained~} \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\mathbf {due~to~interventions:~} \\mathbf {remove~} `dhap^{\\prime } \\mathbf {~as~soon~as~produced~} ;\\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\mathbf {due~to~observations:~} `t5b^{\\prime } \\mathbf {~does~not~occur} ; \\nonumber \\\\&\\mathbf {using~initial~setup:~} \\mathbf {continuously~supply~} `f16bp^{\\prime } \\mathbf {~in~quantity~} 5 ; &$ Queries can also be about actions, as illustrated in the following examples.", "Determine if action `$t5b$ ' ever occurs when there is a continuous supply of `$f16bp$ ' is available and `$t5a$ ' is disabled: $`t5b^{\\prime } &\\mathbf {~occurs~} ;\\nonumber \\\\&\\mathbf {due~to~interventions:~} \\mathbf {disable~} `t5a^{\\prime } ; \\nonumber \\\\&\\mathbf {using~initial~setup:~} \\mathbf {continuously~produce~} `f16bp^{\\prime } \\mathbf {~in~quantity~} 5 ; &$ Determine if glycolysis ($`gly^{\\prime }$ ) gets replaced with beta-oxidation ($`box^{\\prime }$ ) when sugar ($`sug^{\\prime }$ ) is exhausted but fatty acids ($`fac^{\\prime }$ ) are available, when starting with a fixed initial supply of sugar and fatty acids in quantity 5: $&`gly^{\\prime } \\mathbf {~switches~to~} `box^{\\prime } \\mathbf {~when~} \\nonumber \\\\&~~~~\\mathbf {value~of~} ^{\\prime }sug^{\\prime } \\mathbf {~is~} 0, \\nonumber \\\\&~~~~\\mathbf {value~of~} ^{\\prime }fac^{\\prime } \\mathbf {~is~higher~than~} 0 \\nonumber \\\\&~~~~\\mathbf {in~all~trajectories} ; \\nonumber \\\\&\\mathbf {due~to~observations:} \\nonumber \\\\&~~~~`gly^{\\prime } \\mathbf {~switches~to~} `box^{\\prime } \\nonumber \\\\&\\mathbf {using~initial~setup:~} \\nonumber \\\\&~~~~\\mathbf {set~value~of~} `sug^{\\prime } \\mathbf {~to~} 5, \\nonumber \\\\&~~~~\\mathbf {set~value~of~} `fac^{\\prime } \\mathbf {~to~} 5 ; &$ Next we define various syntactic elements of a query statement, give their intuitive meaning, and how these components fit together to form a query statement.", "We will define the formal semantics in a later section.", "Note that some of the single-trajectory queries can be represented as LTL formulas.", "However, we have chosen to keep the current representation as it is more intuitive for our biological domain.", "In the following description, $f_i$ 's are fluents, $l_i$ 's are locations, $n$ 's are numbers, $q$ 's are positive integer numbers, $d$ is one of the directions from $\\lbrace <,>,=\\rbrace $ .", "Definition 66 (Point) A point is a time-step on the trajectory.", "It has the form: $&\\mathbf {time~step~} ts$ Definition 67 (Interval) An interval is a sub-sequence of time-steps on a trajectory.", "It has the form: $&\\langle point \\rangle \\mathbf {~and~} \\langle point \\rangle $ Definition 68 (Aggregate Operator (aggop)) An aggregate operator computes an aggregate quantity over a sequence of values.", "It can be one of the following: $& \\mathbf {minimum}\\\\& \\mathbf {maximum}\\\\& \\mathbf {average}$ Definition 69 (Quantitative Interval Formula) A quantitative interval formula is a formula that is evaluated w.r.t.", "an interval on a trajectory for some quantity $n$ .", "$&\\mathbf {rate~of~production~of~} f \\mathbf {~is~} n\\\\&\\mathbf {rate~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n\\\\& \\mathbf {rate~of~firing~of~} a \\mathbf {~is~} n\\\\&\\mathbf {total~production~of~} f \\mathbf {~is~} n\\\\&\\mathbf {total~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n$ Intuitively, the rate of production of a fluent $f$ in interval $s_i,\\dots ,s_j$ on a trajectory $s_0,T_0,\\dots ,T_{k-1},s_k$ is $n=(s_j(f)-s_i(f))/(j-i)$ ; rate of firing of an action $a$ in interval $s_i,\\dots ,s_j$ is $n=\|\\lbrace T_l : a \\in T_l, i \\le l \\le j-1\\rbrace \|/(j-i)$ ; and total production of a fluent $f$ in interval $s_i,\\dots ,s_j$ is $n=s_j(f)-s_i(f)$ .", "If the given $n$ equals the computed $n$ , then the formula holds.", "The same intuition extends to locational fluents, except that fluent $f$ is replaced by $f[l]$ , e.g.", "rate of production of fluent $f$ at location $l$ in interval $s_i,\\dots ,s_j$ on a trajectory is $n=(s_j(f[l])-s_i(f[l]))/(j-i)$ .", "In biological context, the actions represent reactions and fluents substances used in these reactions.", "The rate and total production formulas are used in aggregate observations to determine if reactions are slowing down or speeding up during various portions of a simulation.", "Definition 70 (Quantitative Point Formula) A quantitative point formula is a formula that is evaluated w.r.t.", "a point on a trajectory for some quantity $n$ .", "$&\\mathbf {value~of~} f \\mathbf {~is~higher~than~} n\\\\&\\mathbf {value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~higher~than~} n\\\\&\\mathbf {value~of~} f \\mathbf {~is~lower~than~} n\\\\&\\mathbf {value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~lower~than~} n\\\\&\\mathbf {value~of~} f \\mathbf {~is~} n\\\\&\\mathbf {value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n$ Definition 71 (Qualitative Interval Formula) A qualitative interval formula is a formula that is evaluated w.r.t.", "an interval on a trajectory.", "$& f \\mathbf {~is~accumulating~}\\\\& f \\mathbf {~is~accumulating~} \\mathbf {~atloc~} l \\\\& f \\mathbf {~is~decreasing~}\\\\& f \\mathbf {~is~decreasing~} \\mathbf {~atloc~} l$ Intuitively, a fluent $f$ is accumulating in interval $s_i,\\dots ,s_j$ on a trajectory if $f$ 's value monotonically increases during the interval.", "A fluent $f$ is decreasing in interval $s_i,\\dots ,s_j$ on a trajectory if $f$ 's value monotonically decreases during the interval.", "The same intuition extends to locational fluents by replacing $f$ with $f[l]$ .", "Definition 72 (Qualitative Point Formula) A qualitative point formula is a formula that is evaluated w.r.t.", "a point on a trajectory.", "$& a \\mathbf {~occurs}\\\\& a \\mathbf {~does~not~occur}\\\\& a1 \\mathbf {~switches~to~} a2\\\\$ Intuitively, an action occurs at a point $i$ on the trajectory if $a \\in T_i$ ; an action does not occur at point $i$ if $a \\notin T_i$ ; an action $a1$ switches to $a2$ at point $i$ if $a1 \\in T_{i-1}$ , $a2 \\notin T_{i-1}$ , $a1 \\notin T_i$ , $a2 \\in T_i$ .", "Definition 73 (Quantitative All Interval Formula) A quantitative all interval formula is a formula that is evaluated w.r.t.", "an interval on a set of trajectories $\\sigma _1,\\dots ,\\sigma _m$ and corresponding quantities $r_1,\\dots ,r_m$ .", "$&\\mathbf {~rates~of~production~of~} f \\mathbf {~are~} [r_1,\\dots ,r_m]\\\\&\\mathbf {~rates~of~production~of~} f \\mathbf {~altoc~} l \\mathbf {~are~} [r_1,\\dots ,r_m]\\\\&\\mathbf {~rates~of~firing~of~} f \\mathbf {~are~} [r_1,\\dots ,r_m]\\\\&\\mathbf {~totals~of~production~of~} f \\mathbf {~are~} [r_1,\\dots ,r_m]\\\\&\\mathbf {~totals~of~production~of~} f \\mathbf {~altoc~} l \\mathbf {~are~} [r_1,\\dots ,r_m]$ Intuitively, a quantitative all interval formula holds on some interval $[i,j]$ over a set of trajectories $\\sigma _1,\\dots ,\\sigma _m$ for values $[r_1,\\dots ,r_m]$ if for each $r_x$ the corresponding quantitative interval formula holds in interval $[i,j]$ in trajectory $\\sigma _x$ .", "For example, $\\mathbf {rates~of~production~of~} f \\mathbf {~are~} [r_1,\\dots ,r_m]$ in interval $[i,j]$ over a set of trajectories $\\sigma _1,\\dots ,\\sigma _m$ if for each $x \\in \\lbrace 1 \\dots m\\rbrace $ , $\\textbf {rate~of~production~of~} f \\mathbf {~is~} r_x$ in interval $[i,j]$ in trajectory $\\sigma _x$ .", "Definition 74 (Quantitative All Point Formula) A quantitative all point formula is a formula that is evaluated w.r.t.", "a point on a set of trajectories $\\sigma _1,\\dots ,\\sigma _m$ and corresponding quantities $r_1,\\dots ,r_m$ .", "$&\\mathbf {values~of~} f \\mathbf {~are~} [r_1,\\dots ,r_m]\\\\&\\mathbf {values~of~} f \\mathbf {~atloc~} l \\mathbf {~are~} [r_1,\\dots ,r_m]$ Intuitively, a quantitative all point formula holds at some point $i$ over a set of trajectories $\\sigma _1,\\dots ,\\sigma _m$ for values $[r_1,\\dots ,r_m]$ if for each $r_x$ the corresponding quantitative point formula holds at point $i$ in trajectory $\\sigma _x$ .", "For example, $\\mathbf {values~of~} f \\mathbf {~are~} [r_1,\\dots ,r_m]$ at point $i$ over a set of trajectories $\\sigma _1,\\dots ,\\sigma _m$ if for each $x \\in \\lbrace 1\\dots m\\rbrace $ , $\\textbf {value~of~} f \\mathbf {~is~} r_x$ at point $i$ in trajectory $\\sigma _x$ .", "Definition 75 (Quantitative Aggregate Interval Formula) A quantitative aggregate interval formula is a formula that is evaluated w.r.t.", "an interval on a set of trajectories $\\sigma _1,\\dots ,\\sigma _m$ and an aggregate value $r$ , where $r$ is the aggregate of $[r_1,\\dots ,r_m]$ using $aggop$ .", "$&\\langle aggop \\rangle \\mathbf {~rate~of~production~of~} f \\mathbf {~is~} n\\\\&\\langle aggop \\rangle \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n\\\\&\\langle aggop \\rangle \\mathbf {~rate~of~firing~of~} a \\mathbf {~is~} n\\\\&\\langle aggop \\rangle \\mathbf {~total~production~of~} f \\mathbf {~is~} n\\\\&\\langle aggop \\rangle \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n$ Intuitively, a quantitative aggregate interval formula holds on some interval $[i,j]$ over a set of trajectories $\\sigma _1,\\dots ,\\sigma _m$ for a value $r$ if there exist $[r_1,\\dots ,r_m]$ whose aggregate value per $aggop$ is $r$ , such that for each $r_x$ the quantitative interval formula (corresponding to the quantitative aggregate interval formual) holds in interval $[i,j]$ in trajectory $\\sigma _x$ .", "For example, $average \\mathbf {~rate~of~}$ $\\mathbf {production~of~} f \\mathbf {~is~} r$ in interval $[i,j]$ over a set of trajectories $\\sigma _1,\\dots ,\\sigma _m$ if $r=(r_1+\\dots +r_m)/m$ and for each $x \\in \\lbrace 1\\dots m\\rbrace $ , $\\textbf {rate~of~production~of~} f \\mathbf {~is~} r_x$ in interval $[i,j]$ in trajectory $\\sigma _x$ .", "Definition 76 (Quantitative Aggregate Point Formula) A quantitative aggregate point formula is a formula that is evaluated w.r.t.", "a point on a set of trajectories $\\sigma _1,\\dots ,\\sigma _m$ and an aggregate value $r$ , where $r$ is the aggregate of $[r_1,\\dots ,r_m]$ using $aggop$ .", "$&\\langle aggop \\rangle \\mathbf {~value~of~} f \\mathbf {~is~} r\\\\&\\langle aggop \\rangle \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} r$ Intuitively, a quantitative aggregate point formula holds at some point $i$ over a set of trajectories $\\sigma _1,\\dots ,\\sigma _m$ for a value $r$ if there exist $[r_1,\\dots ,r_m]$ whose aggregate value per $aggop$ is $r$ , such that for each $r_x$ the quantitative point formula (corresponding to the quantitative aggregate point formual) holds at point $i$ in trajectory $\\sigma _x$ .", "For example, $average \\mathbf {~value~of~} f \\mathbf {~is~} r$ at point $i$ over a set of trajectories $\\sigma _1,\\dots ,\\sigma _m$ if $r=(r_1+\\dots +r_m)/m$ and for each $x \\in \\lbrace 1\\dots m\\rbrace $ , $\\textbf {value~of~} f \\mathbf {~is~} r_x$ at point $i$ in trajectory $\\sigma _x$ .", "Definition 77 (Quantitative Comparative Aggregate Interval Formula) A quantitative comparative aggregate interval formula is a formula that is evaluated w.r.t.", "an interval over two sets of trajectories and a direction `$d$ ' such that `$d$ ' relates the two sets of trajectories.", "$&\\mathbf {direction~of~change~in~} \\langle aggop \\rangle \\mathbf {~rate~of~production~of~} f \\mathbf {~is~} d\\\\&\\mathbf {direction~of~change~in~} \\langle aggop \\rangle \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} d\\\\&\\mathbf {direction~of~change~in~} \\langle aggop \\rangle \\mathbf {~rate~of~firing~of~} a \\mathbf {~is~} d\\\\&\\mathbf {direction~of~change~in~} \\langle aggop \\rangle \\mathbf {~total~production~of~} f \\mathbf {~is~} d\\\\&\\mathbf {direction~of~change~in~} \\langle aggop \\rangle \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} d$ Intuitively, a comparative quantitative aggregate interval formula compares two quantitative interval formulas over using the direction $d$ over a given interval.", "Definition 78 (Quantitative Comparative Aggregate Point Formula) A quantitative comparative aggregate point formula is a formula that is evaluated w.r.t.", "a point over two sets of trajectories and a direction `$d$ ' such that `$d$ ' relates the two sets of trajectories.", "$&\\mathbf {direction~of~change~in~} \\langle aggop \\rangle \\mathbf {~value~of~} f \\mathbf {~is~} d\\\\&\\mathbf {direction~of~change~in~} \\langle aggop \\rangle \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} d$ Intuitively, a comparative quantitative aggregate point formula compares two quantitative point formulas over using the direction $d$ at a given point.", "Definition 79 (Simple Interval Formula) A simple interval formula takes the following forms: $&\\langle \\text{quantitative interval formula} \\rangle \\\\&\\langle \\text{qualitative interval formula} \\rangle $ Definition 80 (Simple Point Formula) A simple interval formula takes the following forms: $&\\langle \\text{quantitative point formula} \\rangle \\\\&\\langle \\text{qualitative point formula} \\rangle $ Definition 81 (Internal Observation Description) An internal observation description takes the following form: $&\\langle \\text{simple point formula} \\rangle \\\\&\\langle \\text{simple point formula} \\rangle \\mathbf {~at~} \\langle point \\rangle \\\\&\\langle \\text{simple interval formula} \\rangle \\\\&\\langle \\text{simple interval formula} \\rangle \\mathbf {~when~observed~between~} \\langle interval \\rangle $ Definition 82 (Simple Point Formula Cascade) A simple point formula cascade takes the following form: $&\\langle \\text{simple point formula} \\rangle _0 \\nonumber \\\\&~~~~~~~~\\mathbf {~after~} \\langle \\text{simple point formula} \\rangle _{1,1}, \\dots , \\langle \\text{simple point formula} \\rangle _{1,n_1} \\nonumber \\\\&~~~~~~~~\\vdots \\nonumber \\\\&~~~~~~~~\\mathbf {~after~} \\langle \\text{simple point formula} \\rangle _{u,1}, \\dots , \\langle \\text{simple point formula} \\rangle _{u,n_u} \\\\&\\langle \\text{simple point formula} \\rangle _0 \\nonumber \\\\&~~~~~~~~\\mathbf {~when~} \\langle \\text{simple point formula} \\rangle _{1,1}, \\dots , \\langle \\text{simple point formula} \\rangle _{1,n_1}\\\\&\\langle \\text{simple point formula} \\rangle _0 \\nonumber \\\\&~~~~~~~~\\mathbf {~when~} \\langle \\text{cond} \\rangle $ where $u \\ge 1$ and `$\\text{cond}$ ' is a conjunction of $\\text{simple point formula}$ s that is true in the same point as the $\\text{simple point formula}$ .", "Intuitively, the simple point formula cascade (REF ) holds if a given sequence of point formulas hold in order in a trajectory.", "Intuitively, simple point formula cascade () holds if a given point formula occurs at the same point as a set of simple point formulas in a trajectory.", "Note that these formulas and many other of our single trajectory formulas can be replaced by an LTL [56] formula, but we have kept this syntax as it is more relevant to the question answering needs in the biological domain.", "Definition 83 (Query Description) A query description specifies a non-comparative observation that can be made either on a trajectory or a set of trajectories.", "$& \\langle \\text{quantitative aggregate interval formula} \\rangle \\mathbf {~when~observed~between~} \\langle \\text{interval} \\rangle \\\\& \\langle \\text{quantitative aggregate point formula} \\rangle \\mathbf {~when~observed~at~} \\langle \\text{point} \\rangle \\\\& \\langle \\text{quantitative all interval formula} \\rangle \\mathbf {~when~observed~between~} \\langle \\text{interval} \\rangle \\\\& \\langle \\text{quantitative all point formula} \\rangle \\mathbf {~when~observed~at~} \\langle \\text{point} \\rangle \\\\& \\langle \\text{internal observation description} \\rangle \\\\& \\langle \\text{internal observation description} \\rangle \\mathbf {~in~all~trajectories}\\\\& \\langle \\text{simple point formula cascade} \\rangle \\\\& \\langle \\text{simple point formula cascade} \\rangle \\mathbf {~in~all~trajectories}$ The single trajectory observations are can be represented using LTL formulas, but we have chosen to keep them in this form for ease of use by users from the biological domain.", "Definition 84 (Comparative Query Description) A comparative query description specifies a comparative observation that can be made w.r.t.", "two sets of trajectories.", "$& \\langle \\text{quantitative comparative aggregate interval formula} \\rangle \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\;\\mathbf {~when~observed~between~} \\langle \\text{interval} \\rangle \\\\& \\langle \\text{quantitative comparative aggregate point formula} \\rangle \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\;\\mathbf {~when~observed~at~} \\langle \\text{point} \\rangle $ Definition 85 (Intervention) Interventions define modifications to domain descriptions.", "$&\\mathbf {remove~} f_1 \\mathbf {~as~soon~as~produced} \\\\&\\mathbf {remove~} f_1 \\mathbf {~atloc~} l_1 \\mathbf {~as~soon~as~produced} \\\\&\\mathbf {disable~} a_2 \\\\&\\mathbf {continuously~transform~} f_1\\mathbf {~in~quantity~} q_1 \\mathbf {~to~} f_2 \\\\&\\mathbf {continuously~transform~} f_1\\mathbf {~atloc~} l_1 \\mathbf {~in~quantity~} q_1 \\mathbf {~to~} f_2 \\mathbf {~atloc~} l_2 \\\\&\\mathbf {make~} f_3 \\mathbf {~inhibit~} a_3 \\\\&\\mathbf {make~} f_3 \\mathbf {~atloc~} l_3 \\mathbf {~inhibit~} a_3 \\\\&\\mathbf {continuously~supply~} f_4 \\mathbf {~in~quantity~} q_4 \\\\&\\mathbf {contiunously~supply~} f_4 \\mathbf {~atloc~} l_4 \\mathbf {~in~quantity~} q_4 \\\\&\\mathbf {continuously~transfer~} f_1 \\mathbf {~in~quantity~} q_1 \\mathbf {~across~} l_1,l_2 \\mathbf {~to~lower~gradient~} \\\\&\\mathbf {add~delay~of~} q_1 \\mathbf {~time~units~in~availability~of~} f_1 \\\\&\\mathbf {add~delay~of~} q_1 \\mathbf {~time~units~in~availability~of~} f_1 \\mathbf {~atloc~} l_1 \\\\&\\mathbf {set~value~of~} f_4 \\mathbf {~to~} q_4 \\\\&\\mathbf {set~value~of~} f_4 \\mathbf {~atloc~} l_4 \\mathbf {~to~} q_4$ Intuitively, intervention (REF ) modifies the pathway such that all quantity of $f_1$ is removed as soon as it is produced; intervention () modifies the pathway such that all quantity of $f_1[l_1]$ is removed as soon as it is produced; intervention () disables the action $a_2$ ; intervention () modifies the pathway such that $f_1$ gets converted to $f_2$ at the rate of $q_1$ units per time-unit; intervention () modifies the pathway such that $f_1[l_1]$ gets converted to $f_2[l_2]$ at the rate of $q_1$ units per time-unit; intervention () modifies the pathway such that action $a_3$ is now inhibited each time there is 1 or more units of $f_3$ and sets value of $f_3$ to 1 to initially inhibit $a_3$ ; intervention () modifies the pathway such that action $a_3$ is now inhibited each time there is 1 or more units of $f_3[l_3]$ and sets value of $f_3[l_3]$ to 1 to initially inhibit $a_3$ ; intervention () modifies the pathway to continuously supply $f_4$ at the rate of $q_4$ units per time-unit; intervention () modifies the pathway to continuously supply $f_4[l_4]$ at the rate of $q_4$ units per time-unit; intervention () modifies the pathway to transfer $f_1[l_1]$ to $f_1[l_2]$ in quantity $q_1$ or back depending upon whether $f_1[l_1]$ is higher than $f_1[l_2]$ or lower; intervention () modifies the pathway to add delay of $q_1$ time units between when $f_1$ is produced to when it is made available to next action; intervention () modifies the pathway to add delay of $q_1$ time units between when $f_1[l_1]$ is produced to when it is made available to next action; intervention () modifies the pathway to set the initial value of $f_4$ to $q_4$ ; and intervention () modifies the pathway to set the initial value of $f_4[l_4]$ to $q_4$ .", "Definition 86 (Initial Condition) An initial condition is one of the intervention (), (), (), () as given in definition REF .", "Intuitively, initial conditions are interventions that setup fixed or continuous supply of substances participating in a pathway.", "Definition 87 (Query Statement) A query statement can be of the following forms: $& \\langle \\text{query description} \\rangle ; \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\mathbf {due~to~interventions:} \\; \\langle intervention \\rangle _1, \\dots , \\langle intervention \\rangle _{N1}; \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\mathbf {due~to~observations:} \\; \\langle \\text{internal observation} \\rangle _1, \\dots , \\langle \\text{internal observation} \\rangle _{N2}; \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\mathbf {using~initial~setup:} \\; \\langle \\text{initial condition} \\rangle _1, \\dots , \\langle \\text{initial condition} \\rangle _{N3};\\\\\\nonumber \\\\& \\langle \\text{comparative query description} \\rangle ; \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\;\\mathbf {comparing~nominal~pathway~with~modified~pathway~obtained~} \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\mathbf {due~to~interventions:} \\; \\langle intervention \\rangle _1, \\dots , \\langle intervention \\rangle _{N1} ; \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\mathbf {due~to~observations:} \\; \\langle \\text{internal observation} \\rangle _1, \\dots , \\langle \\text{internal observation} \\rangle _{N2}; \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\mathbf {using~initial~setup:} \\; \\langle \\text{initial condition} \\rangle _1, \\dots , \\langle \\text{initial condition} \\rangle _{N3}; &$ where interventions, observations, and initial setup are optional.", "Intuitively, a query statement asks whether a query description holds in a pathway, perhaps after modifying it with initial setup, interventions and observations.", "Intuitively, a comparative query statement asks whether a comparative query description holds with a nominal pathway is compared against a modified pathway, where both pathways have the same initial setup, but only the modified pathway has been modified with interventions and observations.", "Semantics of the Query Language (BioPathQA-QL) In this section we give the semantics of our pathway specification language and the query language.", "The semantics of the query language is in terms of the trajectories of a domain description $\\mathbf {D}$ that satisfy a query $\\mathbf {Q}$ .", "We will present the semantics using LTL-style formulas.", "First, we informally define the semantics of the query language as follows.", "Let $\\mathbf {Q}$ be a query statement of the form (REF ) with a query description $U$ , interventions $V_1,\\dots ,V_{\|V\|}$ , internal observations $O_1,\\dots ,O_{\|O\|}$ , and initial setup conditions $I_1,\\dots ,I_{\|I\|}$ .", "We construct a modified domain description $\\mathbf {D_1}$ by applying $I_1,\\dots ,I_{\|I\|}$ and$V_1,\\dots ,V_{\|V\|}$ to $\\mathbf {D}$ .", "We filter the trajectories of $\\mathbf {D_1}$ to retain only those trajectories that satisfy the observations $O_1,\\dots ,O_{\|O\|}$ .", "Then we determine if $U$ holds on any of the retained trajectories.", "If it does, then we say that $\\mathbf {D}$ satisfies $\\mathbf {Q}$ .", "Let $\\mathbf {Q}$ be a comparative query statement of the form () with quantitative comparative aggregate query description $U$ , interventions $V_1,\\dots ,V_{\|V\|}$ , internal observations $O_1,\\dots ,O_{\|O\|}$ , and initial conditions $I_1,\\dots ,I_{\|I\|}$ .", "Then we evaluate $\\mathbf {Q}$ by deriving two sub-query statements.", "$\\mathbf {Q_0}$ is constructed by removing the interventions $V_1,\\dots ,V_{\|V\|}$ and observations $O_1,\\dots ,O_{\|O\|}$ from $\\mathbf {Q}$ and replacing the quantitative comparative aggregate query description $U$ with the corresponding quantitative aggregate query description $U^{\\prime }$ , $\\mathbf {Q_1}$ is constructed by replacing the quantitative comparative aggregate query description $U$ with the corresponding quantitative aggregate query description $U^{\\prime }$ .", "Then $\\mathbf {D}$ satisfies $\\mathbf {Q}$ iff we can find $d \\in \\lbrace <,>,= \\rbrace $ s.t.", "$n \\; d \\; n^{\\prime }$ , where $\\mathbf {D}$ satisfies $\\mathbf {Q_0}$ for some value $n$ and $\\mathbf {D}$ satisfies $\\mathbf {Q_1}$ for some value $n^{\\prime }$ .", "An Illustrative Example In this section, we illustrate with an example how we intuitively evaluate a comparative query statement.", "In the later sections, we will give the formal semantics of query satisfaction.", "Consider the following simple pathway specification: $\\begin{array}{llll}&\\mathbf {domain~of~} & f_1 \\mathbf {~is~} integer, & f_2 \\mathbf {~is~} integer \\\\&t_1 \\mathbf {~may~fire~causing~} & f_1 \\mathbf {~change~value~by~} -1, & f_2 \\mathbf {~change~value~by~} +1\\\\& \\mathbf {initially~} & f_1 \\mathbf {~has~value~} 0, & f_2 \\mathbf {~has~value~} 0,\\\\&\\mathbf {firing~style~} & max\\end{array}$ Let the following specify a query statement $\\mathbf {Q}$ : $\\mathbf {dir}&\\mathbf {ection~of~change~in~} average \\mathbf {~rate~of~production~of~} f_2 \\mathbf {~is~} d\\\\&\\mathbf {when~observed~between~time~step~} 0 \\mathbf {~and~time~step~} k;\\\\&\\mathbf {comparing~nominal~pathway~with~modified~pathway~obtained~}\\\\&\\;\\;\\;\\;\\;\\;\\mathbf {due~to~interventions:~} \\mathbf {remove~} f_2 \\mathbf {~as~soon~as~produced} ;\\\\&\\mathbf {using~initial~setup:~} \\mathbf {continuously~supply~} f_1 \\mathbf {~in~quantity~} 1;$ that we want to evaluate against $\\mathbf {D}$ using a simulation length $k$ with maximum $ntok$ tokens at any place to determine `$d$ ' that satisfies it.", "We construct the baseline query $\\mathbf {Q_0}$ by removing interventions and observations, and replacing the comparative aggregate quantitative query description with the corresponding aggregate quantitative query description as follows: $average &\\mathbf {~rate~of~production~of~} f_2 \\mathbf {~is~} n\\\\&\\mathbf {when~observed~between~time~step~} 0 \\mathbf {~and~time~step~} k ;\\\\&\\mathbf {using~initial~setup:~} \\mathbf {continuously~supply~} f_1 \\mathbf {~in~quantity~} 1;\\\\$ We construct the alternate query $\\mathbf {Q_1}$ by replacing the comparative aggregate quantitative query description with the corresponding aggregate quantitative query description as follows: $average&\\mathbf {~rate~of~production~of~} f_2 \\mathbf {~is~} n^{\\prime }\\\\&\\mathbf {when~observed~between~time~step~} 0 \\mathbf {~and~time~step~} k ;\\\\&\\mathbf {due~to~interventions:~} \\mathbf {remove~} f_2 \\mathbf {~as~soon~as~produced} ;\\\\&\\mathbf {using~initial~setup:~} \\mathbf {continuously~supply~} f_1 \\mathbf {~in~quantity~} 1;\\\\$ We build a modified domain description $\\mathbf {D_0}$ as $\\mathbf {D} \\diamond (\\mathbf {continuously~supply~} f_1 $ $\\mathbf {~in~quantity~} 1)$ based on initial conditions in $\\mathbf {Q_0}$ .", "We get: $\\begin{array}{llll}&\\mathbf {domain~of~} & f_1 \\mathbf {~is~} integer, & f_2 \\mathbf {~is~} integer \\\\&t_1 \\mathbf {~may~fire~causing~} & f_1 \\mathbf {~change~value~by~} -1, & f_2 \\mathbf {~change~value~by~} +1\\\\&t_{f_1} \\mathbf {~may~fire~causing~} & f_1 \\mathbf {~change~value~by~} +1\\\\& \\mathbf {initially~} & f_1 \\mathbf {~has~value~} 0, & f_2 \\mathbf {~has~value~} 0,\\\\&\\mathbf {firing~style~} & max\\end{array}$ We evaluate $\\mathbf {Q_0}$ against $\\mathbf {D_0}$ .", "It results in $m_0$ trajectories with rate of productions $n_j=(s_k(f_2)-s_0(f_2))/k$ on trajectory $\\tau _j=s_0,\\dots ,s_k, 1 \\le j \\le m_0$ using time interval $[0,k]$ .", "The rate of productions are averaged to produce $n=(n_1+\\dots +n_{m_0})/m_0$ .", "Next, we construct the alternate domain description $\\mathbf {D_1}$ as $\\mathbf {D_0} \\diamond (\\mathbf {remove~} f_2 \\mathbf {~as~soon~} $ $\\mathbf {as~produced~})$ based on initial conditions and interventions in $\\mathbf {Q_1}$ .", "We get: $\\begin{array}{llll}&\\mathbf {domain~of~} & f_1 \\mathbf {~is~} integer, & f_2 \\mathbf {~is~} integer \\\\&t_1 \\mathbf {~may~fire~causing~} & f_1 \\mathbf {~change~value~by~} -1, & f_2 \\mathbf {~change~value~by~} +1\\\\&t_{f_1} \\mathbf {~may~fire~causing~} & f_1 \\mathbf {~change~value~by~} +1\\\\&t_{f_2} \\mathbf {~may~fire~causing~} & f_2 \\mathbf {~change~value~by~} \\\\& \\mathbf {initially~} & f_1 \\mathbf {~has~value~} 0, & f_2 \\mathbf {~has~value~} 0\\\\&\\mathbf {firing~style~} & max\\end{array}$ We evaluate $\\mathbf {Q_1}$ against $\\mathbf {D_1}$ .", "Since there are no observations, no filtering is required.", "This results in $m_1$ trajectories, each with rate of production $n^{\\prime }_j=(s_k(f_2)-s_0(f_2))/k$ on trajectory $\\tau ^{\\prime }_j = s_0,\\dots ,s_k, 1 \\le j \\le m_1$ using time interval $[0,k]$ .", "The rate of productions are averaged to produce $n^{\\prime }=(n^{\\prime }_1+\\dots +n^{\\prime }_{m_1})/m_1$ .", "Due to the simple nature of our domain description, it has only one trajectory for each of the two domains.", "As a result, for any $k > 1$ , $n^{\\prime } < n$ .", "Thus, $\\mathbf {D}$ satisfies $\\mathbf {Q}$ iff $d = ``<^{\\prime \\prime }$ .", "We will now define the semantics of how a domain description $\\mathbf {D}$ is modified according to the interventions and initial conditions, the semantics of conditions imposed by the internal observations.", "We will then formally define how $\\mathbf {Q}$ is entailed in $\\mathbf {D}$ .", "Domain Transformation due to Interventions and Initial Conditions An intervention $I$ modifies a given domain description $\\mathbf {D}$ , potentially resulting in a different set of trajectories than $\\mathbf {D}$ .", "We define a binary operator $\\diamond $ that transforms $\\mathbf {D}$ by applying an intervention $I$ as a set of edits to $\\mathbf {D}$ using the pathway specification language.", "The trajectories of the modified domain description $\\mathbf {D^{\\prime }} = \\mathbf {D} \\diamond I$ are given by the semantics of the pathway specification language.", "Below, we give the intuitive impact and edits required by each of the interventions.", "Domain modification by intervention (REF ) $\\mathbf {D^{\\prime }} = \\mathbf {D} \\diamond (\\mathbf {remove~} f_1 \\mathbf {~as~soon~}$ $\\mathbf {as~produced})$ modifies the pathway by removing all existing quantity of $f_1$ at each time step: $\\mathbf {D^{\\prime }} &= \\mathbf {D}+ \\left\\lbrace \\begin{array}{l}tr \\mathbf {~may~execute~causing~} f_1 \\mathbf {~change~value~by~} \\end{array}\\right\\rbrace $ Domain modification by intervention () $\\mathbf {D^{\\prime }} = \\mathbf {D} \\diamond (\\mathbf {remove~} f_1 \\mathbf {~atloc~} l_1)$ modifies the pathway by removing all existing quantity of $f_1$ at each time step: $\\mathbf {D^{\\prime }} &= \\mathbf {D}+ \\left\\lbrace \\begin{array}{l}tr \\mathbf {~may~execute~causing~} f_1 \\mathbf {~atloc~} l_1 \\mathbf {~change~value~by~} \\end{array}\\right\\rbrace $ Domain modification by intervention () $\\mathbf {D^{\\prime }} = \\mathbf {D} \\diamond (\\mathbf {disable~} a_2)$ modifies the pathway such that its trajectories have $a_2 \\notin T_i$ , where $i \\ge 0$ .", "$\\mathbf {D^{\\prime }} &= \\mathbf {D}+ \\left\\lbrace \\begin{array}{ll}\\mathbf {inhibit~} a_2\\end{array}\\right\\rbrace $ Domain modification by intervention () $\\mathbf {D^{\\prime }} = \\mathbf {D} \\diamond (\\mathbf {continuously~transform~} f_1 $ $\\mathbf {~in~quantity~} q_1 $ $\\mathbf {~to~} f_2)$ where $s_{i+1}(f_1)$ decreases, and $s_{i+1}(f_2)$ increases by $q_1$ at each time step $i \\ge 0$ , when $s_i(f_1) \\ge q_1$ .", "$\\mathbf {D^{\\prime }} = \\mathbf {D}&+ \\left\\lbrace \\begin{array}{ll}a_{f_{1,2}} \\mathbf {~may~execute~causing~} & f_1 \\mathbf {~change~value~by~} -q_1, \\\\ & f_2 \\mathbf {~change~value~by~} +q_1\\end{array}\\right\\rbrace $ Domain modification by intervention () $\\mathbf {D^{\\prime }} = \\mathbf {D} \\diamond (\\mathbf {continuously~transform~} $ $f_1 \\mathbf {~atloc~} l_1$ $ \\mathbf {~in~quantity~} q_1 $ $\\mathbf {~to~} f_2 \\mathbf {~atloc~} l_2)$ where $s_{i+1}(f_1[l_1])$ decreases, and $s_{i+1}(f_2[l_2])$ increases by $q_1$ at each time step $i \\ge 0$ , when $s_i(f_1[l_1]) \\ge q_1$ .", "$\\mathbf {D^{\\prime }} = \\mathbf {D}&+ \\left\\lbrace \\begin{array}{ll}a_{f_{1,2}} \\mathbf {~may~execute~causing~} & f_1 \\mathbf {~atloc~} l_1 \\mathbf {~change~value~by~} -q_1, \\\\ & f_2 \\mathbf {~atloc~} l_2 \\mathbf {~change~value~by~} +q_1\\end{array}\\right\\rbrace $ Domain modification by intervention () $\\mathbf {D^{\\prime }} = \\mathbf {D} \\diamond (\\mathbf {make~} f_3 \\mathbf {~inhibit~} a_3 )$ modifies the pathway such that it has $a_3$ inhibited due to $f_3$ .", "$\\mathbf {D^{\\prime }} = \\mathbf {D}&- \\left\\lbrace \\begin{array}{l}\\mathbf {initially~} f_3 \\mathbf {~has~value~} q \\in \\mathbf {D}\\end{array}\\right\\rbrace \\\\&+ \\left\\lbrace \\begin{array}{ll}\\mathbf {inhibit~} a_3 \\mathbf {~if~} & f_3 \\mathbf {~has~value~} 1 \\mathbf {~or~higher}\\\\\\mathbf {initially~} & f_3 \\mathbf {~has~value~} 1\\end{array}\\right\\rbrace $ Domain modification by intervention () $\\mathbf {D^{\\prime }} = \\mathbf {D} \\diamond (\\mathbf {make~} f_3 \\mathbf {~atloc~} l_3 \\mathbf {~inhibit~} a_3)$ modifies the pathway such that it has $a_3$ inhibited due to $f_3[l_3]$ .", "$\\mathbf {D^{\\prime }} = \\mathbf {D}&- \\left\\lbrace \\begin{array}{l}\\mathbf {initially~} f_3 \\mathbf {~atloc~} l_3 \\mathbf {~has~value~} q \\in \\mathbf {D}\\end{array}\\right\\rbrace \\\\&+ \\left\\lbrace \\begin{array}{ll}\\mathbf {inhibit~} a_3 \\mathbf {~if~} & f_3 \\mathbf {~atloc~} l_3 \\mathbf {~has~value~} 1 \\mathbf {~or~higher}, \\\\\\mathbf {initially~} & f_3 \\mathbf {~atloc~} l_3 \\mathbf {~has~value~} 1\\end{array}\\right\\rbrace $ Domain modification by intervention () $\\mathbf {D^{\\prime }} = \\mathbf {D} \\diamond (\\mathbf {~continuously~supply~} f_4 $ $\\mathbf {~in~quantity~} q_4)$ modifies the pathway such that a quantity $q_4$ of substance $f_4$ is supplied at each time step.", "$\\mathbf {D^{\\prime }} = \\mathbf {D}&+ \\left\\lbrace \\begin{array}{ll}t_{f_4} \\mathbf {~may~execute~causing~} & f_4 \\mathbf {~change~value~by~} +q_4\\end{array}\\right\\rbrace $ Domain modification by intervention () $\\mathbf {D^{\\prime }} = \\mathbf {D} \\diamond (\\mathbf {~continuously~supply~} f_4 $ $\\mathbf {~atloc~} l_4 $ $\\mathbf {~in~quantity~} q_4 )$ modifies the pathway such that a quantity $q_4$ of substance $f_4$ at location $l_4$ is supplied at each time step.", "$\\mathbf {D^{\\prime }} = \\mathbf {D}&+ \\left\\lbrace \\begin{array}{ll}t_{f_4} \\mathbf {~may~execute~causing~} & f_4 \\mathbf {~atloc~} l_4 \\mathbf {~change~value~by~} +q_4\\end{array}\\right\\rbrace $ Domain modification by intervention () $\\mathbf {D^{\\prime }} = \\mathbf {D} \\diamond (\\mathbf {continuously~transfer~} f_1 $ $\\mathbf {~in~quantity~} q_1 $ $\\mathbf {~across~} l_1, l_2 $ $\\mathbf {~to~lower~gradient})$ modifies the pathway such that substance represented by $f_1$ is transferred from location $l_1$ to $l_2$ or $l_2$ to $l_1$ depending upon whether it is at a higher quantity at $l_1$ or $l_2$ .", "$\\mathbf {D^{\\prime }} = \\mathbf {D}&+ \\left\\lbrace \\begin{array}{ll}t_{f_1} \\mathbf {~may~execute~causing~} & f_1 \\mathbf {~atloc~} l_1 \\mathbf {~change~value~by~} -q_1, \\\\ & f_1 \\mathbf {~atloc~} l_2 \\mathbf {~change~value~by~} +q_1 \\\\\\mathbf {~~~~~~~if}& f_1 \\mathbf {~atloc~} l_1 \\mathbf {~has~higher~value~than~} f_1 \\mathbf {~atloc~} l_2, \\\\t^{\\prime }_{f_1} \\mathbf {~may~execute~causing~} & f_1 \\mathbf {~atloc~} l_2 \\mathbf {~change~value~by~} -q_1, \\\\ & f_1 \\mathbf {~atloc~} l_1 \\mathbf {~change~value~by~} +q_1\\\\\\mathbf {~~~~~~~if}& f_1 \\mathbf {~atloc~} l_2 \\mathbf {~has~higher~value~than~} f_1 \\mathbf {~atloc~} l_1, \\\\\\end{array}\\right\\rbrace $ Domain modification by intervention () $\\mathbf {D^{\\prime }} = \\mathbf {D} \\diamond (\\mathbf {add~delay~of~} q_1 $ $\\mathbf {~time~units~} $ $\\mathbf {in~availability~of~} f_1)$ modifies the pathway such that $f_1$ 's arrival is delayed by $q_1$ time units.", "We create additional cases for all actions that produce $f_1$ , such that it goes through an additional delay action.", "$\\mathbf {D^{\\prime }} = \\mathbf {D}& - \\left\\lbrace \\begin{array}{ll}a \\mathbf {~may~execute~causing~} & f_1 \\mathbf {~change~value~by~} +w_1, \\\\ & \\mathit {effect}_1,\\dots ,\\mathit {effect}_n \\\\ & \\mathbf {~if~} cond_1,\\dots ,cond_m \\in \\mathbf {D}\\\\\\end{array}\\right\\rbrace \\\\& + \\left\\lbrace \\begin{array}{ll}a \\mathbf {~may~execute~causing~} & f^{\\prime }_1 \\mathbf {~change~value~by~} +w_1, \\\\ & \\mathit {effect}_1,\\dots ,\\mathit {effect}_n, \\\\ & \\mathbf {~if~} cond_1,\\dots ,cond_m\\\\a_{f_1} \\mathbf {~may~execute~causing~} & f^{\\prime }_1 \\mathbf {~change~value~by~} -w_1, \\\\ & f_1 \\mathbf {~change~value~by~} +w_1\\\\a_{f_1} \\mathbf {~executes~} & \\mathbf {~in~} q_1 \\mathbf {~time~units}\\\\\\end{array}\\right\\rbrace $ Domain modification by intervention () $\\mathbf {D^{\\prime }} = \\mathbf {D} \\diamond (\\mathbf {add~delay~of} q_1 $ $\\mathbf {~time~units~} $ $\\mathbf {in~availability~of~} f_1 \\mathbf {~atloc~} l_1)$ modifies trajectories such that $f_1[l_1]$ 's arrival is delayed by $q_1$ time units.", "We create additional cases for all actions that produce $f_1 \\mathbf {~atloc~} l_1$ , such that it goes through an additional delay action.", "$\\mathbf {D^{\\prime }} = \\mathbf {D}&- \\left\\lbrace \\begin{array}{ll}a \\mathbf {~may~execute~causing~} & f_1 \\mathbf {~atloc~} l_1 \\mathbf {~change~value~by~} +w_1, \\\\ & \\mathit {effect}_1,\\dots ,\\mathit {effect}_n \\\\ & \\mathbf {~if~} cond_1,\\dots ,cond_m \\in \\mathbf {D}\\end{array}\\right\\rbrace \\\\&+ \\left\\lbrace \\begin{array}{ll}a \\mathbf {~may~execute~causing~} & f_1 \\mathbf {~atloc~} l^{\\prime }_1 \\mathbf {~change~value~by~} +w_1, \\\\ & \\mathit {effect}_1,\\dots ,\\mathit {effect}_n \\\\ & \\mathbf {~if~} cond_1,\\dots ,cond_n\\\\a_{f_1} \\mathbf {~may~execute~causing~} & f_1 \\mathbf {~atloc~} l^{\\prime }_1 \\mathbf {~change~value~by~} -w_1,\\\\ & f_1 \\mathbf {~atloc~} l_1 \\mathbf {~change~value~by~} +w_1\\\\a_{f_1} \\mathbf {~executes~} & \\mathbf {~in~} q_1 \\mathbf {~time~units}\\\\\\end{array}\\right\\rbrace $ Domain modification by intervention () $\\mathbf {D^{\\prime }} = \\mathbf {D} \\diamond (\\mathbf {~set~value~of~} f_4 $ $\\mathbf {~to~} q_4)$ modifies the pathway such that its trajectories have $s_0(f_4) = q_4$ .", "$\\mathbf {D^{\\prime }} = \\mathbf {D}& - \\left\\lbrace \\begin{array}{l}\\mathbf {initially~} f_4 \\mathbf {~has~value~} n \\in \\mathbf {D}\\end{array}\\right\\rbrace \\\\& + \\left\\lbrace \\begin{array}{l}\\mathbf {initially~} f_4 \\mathbf {~has~value~} q_4\\end{array}\\right\\rbrace $ Domain modification by intervention () $\\mathbf {D^{\\prime }} = \\mathbf {D} \\diamond (\\mathbf {~set~value~of~} f_4 $ $\\mathbf {~atloc~} l_4 $ $\\mathbf {~to~} q_4 )$ modifies the pathway such that its trajectories have $s_0({f_4}_{l_4}) = q_4$ .", "$\\mathbf {D^{\\prime }} = \\mathbf {D}& - \\left\\lbrace \\begin{array}{l}\\mathbf {initially~} f_4 \\mathbf {~atloc~} l_4 \\mathbf {~has~value~} n \\in \\mathbf {D}\\end{array}\\right\\rbrace \\\\& + \\left\\lbrace \\begin{array}{l}\\mathbf {initially~} f_4 \\mathbf {~atloc~} l_4 \\mathbf {~has~value~} q_4\\end{array}\\right\\rbrace $ Formula Semantics We will now define the semantics of some common formulas that we will use in the following sections.", "First we introduce the LTL-style formulas that we will be using to define the syntax.", "A formula $\\langle s_i,\\sigma \\rangle \\models F$ represents that $F$ holds at point $i$ .", "A formula $\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace \\models F$ represents that $F$ holds at point $i$ on a set of trajectories $\\sigma _1,\\dots ,\\sigma _m$ .", "A formula $\\big \\lbrace \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\big \\rbrace \\models F$ represents that $F$ holds at point $i$ on two sets of trajectories $\\sigma _1,\\dots ,\\sigma _m$ and $\\lbrace \\bar{\\sigma }_1,\\dots ,\\bar{\\sigma }_{\\bar{m}} \\rbrace $ .", "A formula $(\\langle s_i,\\sigma \\rangle ,j) \\models F$ represents that $F$ holds in the interval $[i,j]$ on trajectory $\\sigma $ .", "A formula $(\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models F$ represents that $F$ holds in the interval $[i,j]$ on a set of trajectories $\\sigma _1,\\dots ,\\sigma _m$ .", "A formula $(\\big \\lbrace \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\big \\rbrace , j) \\models F$ represents that $F$ holds in the interval $[i,j]$ over two sets of trajectories $\\lbrace \\sigma _1,\\dots , \\sigma _m \\rbrace $ and $\\lbrace \\bar{\\sigma }_1,\\dots ,\\bar{\\sigma }_{\\bar{m}} \\rbrace $ .", "Given a domain description $\\mathbf {D}$ with simple fluents represented by a Guarded-Arc Petri Net as defined in definition REF .", "Let $\\sigma = s_0,T_0,s_1,\\dots ,T_{k-1},s_{k}$ be its trajectory as defined in (REF ), and $s_i$ be a state on that trajectory.", "Let actions $T_i$ firing in state $s_i$ be observable in $s_i$ such that $T_i \\subseteq s_i$ .", "First we define how interval formulas are satisfied on a trajectory $\\sigma $ , starting state $s_i$ and an ending point $j$ : $(\\langle s_i,\\sigma \\rangle , j) \\models & \\mathbf {~rate~of~production~of~} f \\mathbf {~is~} n \\nonumber \\\\&\\text{ if } n=(s_j(f)-s_i(f))/(j-i)\\\\(\\langle s_i,\\sigma \\rangle , j) \\models & \\mathbf {~rate~of~firing~of~} a \\mathbf {~is~} n \\nonumber \\\\&\\text{ if } n=\\sum _{i\\le k \\le j, \\langle s_k,\\sigma \\rangle \\models a \\mathbf {~occurs~}}{1}/(j-i)\\\\(\\langle s_i,\\sigma \\rangle , j) \\models & \\mathbf {~total~production~of~} f \\mathbf {~is~} n \\nonumber \\\\&\\text{ if } n=(s_j(f)-s_i(f))\\\\(\\langle s_i,\\sigma \\rangle , j) \\models & f \\mathbf {~is~accumulating~} \\nonumber \\\\&\\text{ if } (\\nexists k, i \\le k \\le j : s_{k+1}(f) < s_k(f)) \\text{ and } s_j(f) > s_i(f)\\\\(\\langle s_i,\\sigma \\rangle , j) \\models & f \\mathbf {~is~decreasing~} \\nonumber \\\\&\\text{ if } (\\nexists k, i \\le k \\le j : s_{k+1}(f) > s_k(f)) \\text{ and } s_j(f) < s_i(f)$ Next we define how a point formula is satisfied on a trajectory $\\sigma $ , in a state $s_i$ : $\\langle s_i,\\sigma \\rangle \\models & \\mathbf {~value~of~} f \\mathbf {~is~higher~than~} n \\nonumber \\\\&\\text{ if } s_i(f) > n\\\\\\langle s_i,\\sigma \\rangle \\models & \\mathbf {~value~of~} f \\mathbf {~is~lower~than~} n \\nonumber \\\\&\\text{ if } s_i(f) < n\\\\\\langle s_i,\\sigma \\rangle \\models & \\mathbf {~value~of~} f \\mathbf {~is~} n \\nonumber \\\\&\\text{ if } s_i(f) = n\\\\\\langle s_i,\\sigma \\rangle \\models & a \\mathbf {~occurs~} \\nonumber \\\\&\\text{ if } a \\in s_i\\\\\\langle s_i,\\sigma \\rangle \\models & a \\mathbf {~does~not~occur~} \\nonumber \\\\&\\text{ if } a \\notin s_i\\\\\\langle s_i,\\sigma \\rangle \\models & a_1 \\mathbf {~switches~to~} a_2 \\nonumber \\\\&\\text{ if } a_1 \\in s_{i-1} \\text{ and } a_2 \\notin s_{i-1} \\text{ and } a_1 \\notin s_i \\text{ and } a_2 \\in s_i$ Next we define how a quantitative all interval formula is satisfied on a set of trajectories $\\sigma _1,\\dots ,\\sigma _m$ with starting states $s^1_i,\\dots ,s^m_i$ and end point $j$ : $(\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models & \\mathbf {~rates~of~production~of~} f \\mathbf {~are~} [r_1,\\dots ,r_m] \\nonumber \\\\&\\text{ if } (\\langle s^1_i,\\sigma _1 \\rangle , j) \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~is~} r_1\\nonumber \\\\&\\vdots \\nonumber \\\\&\\text{ and } (\\langle s^m_i,\\sigma _m \\rangle , j) \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~is~} r_m \\\\(\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models & \\mathbf {~rates~of~firing~of~} a \\mathbf {~are~} [r_1,\\dots ,r_m] \\nonumber \\\\&\\text{ if } (\\langle s^1_i,\\sigma _1 \\rangle , j) \\models \\mathbf {~rate~of~firing~of~} a \\mathbf {~is~} r_1\\nonumber \\\\&\\vdots \\nonumber \\\\&\\text{ and } (\\langle s^m_i,\\sigma _m \\rangle , j) \\models \\mathbf {~rate~of~firing~of~} a \\mathbf {~is~} r_m \\\\(\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models & \\mathbf {~totals~of~production~of~} f \\mathbf {~are~} [r_1,\\dots ,r_m] \\nonumber \\\\&\\text{ if } (\\langle s^1_i,\\sigma _1 \\rangle , j) \\models \\mathbf {~total~production~of~} f \\mathbf {~is~} r_1\\nonumber \\\\&\\vdots \\nonumber \\\\&\\text{ and } (\\langle s^m_i,\\sigma _m \\rangle , j) \\models \\mathbf {~total~production~of~} f \\mathbf {~is~} r_m$ Next we define how a quantitative all point formula is satisfied on a set of trajectories $\\sigma _1,\\dots ,\\sigma _m$ in states $s^1_i,\\dots ,s^m_i$ : $\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace \\models & \\mathbf {~values~of~} f \\mathbf {~are~} [r_1,\\dots ,r_m] \\nonumber \\\\&\\text{ if } \\langle s^1_i,\\sigma _1 \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~is~} r_1\\nonumber \\\\&\\vdots \\nonumber \\\\&\\text{ and } \\langle s^m_i,\\sigma _m \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~is~} r_m$ Next we define how a quantitative aggregate interval formula is satisfied on a set of trajectories $\\sigma _1,\\dots ,\\sigma _m$ with starting states $s^1_i,\\dots ,s^m_i$ and end point $j$ : $(\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models & \\; average \\mathbf {~rate~of~production~of~} f \\mathbf {~is~} r \\nonumber \\\\&\\text{ if } \\exists [r_1,\\dots ,r_m] : \\nonumber \\\\&~~~~~~(\\langle s^1_i,\\sigma _1 \\rangle , j) \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~is~} r_1\\nonumber \\\\&~~~~~~\\vdots \\nonumber \\\\&~~~~~~(\\langle s^m_i,\\sigma _m \\rangle , j) \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~is~} r_m \\nonumber \\\\&\\text{ and } r=(r_1+\\dots +r_m)/m\\\\(\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models & \\; minimum \\mathbf {~rate~of~production~of~} f \\mathbf {~is~} r \\nonumber \\\\&\\text{ if } \\exists [r_1,\\dots ,r_m] : \\nonumber \\\\&~~~~~~(\\langle s^1_i,\\sigma _1 \\rangle , j) \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~is~} r_1\\nonumber \\\\&~~~~~~\\vdots \\nonumber \\\\&~~~~~~(\\langle s^m_i,\\sigma _m \\rangle , j) \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~is~} r_m \\nonumber \\\\&\\text{ and } \\exists k, 1 \\le k \\le m : r=r_k \\text{ and } \\forall x, 1 \\le x \\le m, r_k \\le r_x\\\\(\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models & \\; maximum \\mathbf {~rate~of~production~of~} f \\mathbf {~is~} r \\nonumber \\\\&\\text{ if } \\exists [r_1,\\dots ,r_m] : \\nonumber \\\\&~~~~~~(\\langle s^1_i,\\sigma _1 \\rangle , j) \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~is~} r_1\\nonumber \\\\&~~~~~~\\vdots \\nonumber \\\\&~~~~~~(\\langle s^m_i,\\sigma _m \\rangle , j) \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~is~} r_m \\nonumber \\\\&\\text{ and } \\exists k, 1 \\le k \\le m : r=r_k \\text{ and } \\forall x, 1 \\le x \\le m, r_k \\ge r_x\\\\\\vspace{20.0pt}\\nonumber \\\\(\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models & \\; average \\mathbf {~rate~of~firing~of~} f \\mathbf {~is~} r \\nonumber \\\\&\\text{ iff } \\exists [r_1,\\dots ,r_m] : \\nonumber \\\\&~~~~~~(\\langle s^1_i,\\sigma _1 \\rangle , j) \\models \\mathbf {~rate~of~firing~of~} f \\mathbf {~is~} r_1\\nonumber \\\\&~~~~~~\\vdots \\nonumber \\\\&~~~~~~(\\langle s^m_i,\\sigma _m \\rangle , j) \\models \\mathbf {~rate~of~firing~of~} f \\mathbf {~is~} r_m \\nonumber \\\\&\\text{ and } r=(r_1+\\dots +r_m)/m\\\\(\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models & \\; minimum \\mathbf {~rate~of~firing~of~} f \\mathbf {~is~} r \\nonumber \\\\&\\text{ iff } \\exists [r_1,\\dots ,r_m] : \\nonumber \\\\&~~~~~~(\\langle s^1_i,\\sigma _1 \\rangle , j) \\models \\mathbf {~rate~of~firing~of~} f \\mathbf {~is~} r_1\\nonumber \\\\&~~~~~~\\vdots \\nonumber \\\\&~~~~~~(\\langle s^m_i,\\sigma _m \\rangle , j) \\models \\mathbf {~rate~of~firing~of~} f \\mathbf {~is~} r_m \\nonumber \\\\&\\text{ and } \\exists k, 1 \\le k \\le m : r=r_k \\text{ and } \\forall x, 1 \\le x \\le m, r_k \\le r_x\\\\(\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models & \\; maximum \\mathbf {~rate~of~firing~of~} f \\mathbf {~is~} r \\nonumber \\\\&\\text{ iff } \\exists [r_1,\\dots ,r_m] : \\nonumber \\\\&~~~~~~(\\langle s^1_i,\\sigma _1 \\rangle , j) \\models \\mathbf {~rate~of~firing~of~} f \\mathbf {~is~} r_1\\nonumber \\\\&~~~~~~\\vdots \\nonumber \\\\&~~~~~~(\\langle s^m_i,\\sigma _m \\rangle , j) \\models \\mathbf {~rate~of~firing~of~} f \\mathbf {~is~} r_m \\nonumber \\\\&\\text{ and } \\exists k, 1 \\le k \\le m : r=r_k \\text{ and } \\forall x, 1 \\le x \\le m, r_k \\ge r_x\\\\\\vspace{20.0pt}\\nonumber \\\\(\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models & \\; average \\mathbf {~total~production~of~} f \\mathbf {~is~} r \\nonumber \\\\&\\text{ if } \\exists [r_1,\\dots ,r_m] : \\nonumber \\\\&~~~~~~(\\langle s^1_i,\\sigma _1 \\rangle , j) \\models \\mathbf {~total~production~of~} f \\mathbf {~is~} r_1\\nonumber \\\\&~~~~~~\\vdots \\nonumber \\\\&~~~~~~(\\langle s^m_i,\\sigma _m \\rangle , j) \\models \\mathbf {~total~production~of~} f \\mathbf {~is~} r_m \\nonumber \\\\&\\text{ and } r=(r_1+\\dots +r_m)/m\\\\(\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models & \\; minimum \\mathbf {~total~production~of~} f \\mathbf {~is~} r \\nonumber \\\\&\\text{ if } \\exists [r_1,\\dots ,r_m] : \\nonumber \\\\&~~~~~~(\\langle s^1_i,\\sigma _1 \\rangle , j) \\models \\mathbf {~total~production~of~} f \\mathbf {~is~} r_1\\nonumber \\\\&~~~~~~\\vdots \\nonumber \\\\&~~~~~~(\\langle s^m_i,\\sigma _m \\rangle , j) \\models \\mathbf {~total~production~of~} f \\mathbf {~is~} r_m \\nonumber \\\\&\\text{ and } \\exists k, 1 \\le k \\le m : r=r_k \\text{ and } \\forall x, 1 \\le x \\le m, r_k \\le r_x\\\\(\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models & \\; maximum \\mathbf {~total~production~of~} f \\mathbf {~is~} r \\nonumber \\\\&\\text{ if } \\exists [r_1,\\dots ,r_m] : \\nonumber \\\\&~~~~~~(\\langle s^1_i,\\sigma _1 \\rangle , j) \\models \\mathbf {~total~production~of~} f \\mathbf {~is~} r_1\\nonumber \\\\&~~~~~~\\vdots \\nonumber \\\\&~~~~~~(\\langle s^m_i,\\sigma _m \\rangle , j) \\models \\mathbf {~total~production~of~} f \\mathbf {~is~} r_m \\nonumber \\\\&\\text{ and } \\exists k, 1 \\le k \\le m : r=r_k \\text{ and } \\forall x, 1 \\le x \\le m, r_k \\ge r_x$ Next we define how a quantitative aggregate point formula is satisfied on a set of trajectories $\\sigma _1,\\dots ,\\sigma _m$ in states $s^1_i,\\dots ,s^m_i$ : $\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace \\models & \\; average \\mathbf {~value~of~} f \\mathbf {~is~} r \\nonumber \\\\&\\text{ if } \\exists [r_1,\\dots ,r_m] : \\nonumber \\\\&~~~~~~\\langle s^1_i,\\sigma _1 \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~is~} r_1\\nonumber \\\\&~~~~~~\\vdots \\nonumber \\\\&~~~~~~\\langle s^m_i,\\sigma _m \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~is~} r_m \\nonumber \\\\&\\text{ and } r=(r_1+\\dots +r_m)/m\\\\\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace \\models & \\; minimum \\mathbf {~value~of~} f \\mathbf {~is~} r \\nonumber \\\\&\\text{ if } \\exists [r_1,\\dots ,r_m] : \\nonumber \\\\&~~~~~~\\langle s^1_i,\\sigma _1 \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~is~} r_1\\nonumber \\\\&~~~~~~\\vdots \\nonumber \\\\&~~~~~~\\langle s^m_i,\\sigma _m \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~is~} r_m \\nonumber \\\\&\\text{ and } \\exists k, 1 \\le k \\le m : r=r_k \\text{ and } \\forall x, 1 \\le x \\le m, r_k \\le r_x\\\\\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace \\models & \\; maximum \\mathbf {~value~of~} f \\mathbf {~is~} r \\nonumber \\\\&\\text{ if } \\exists [r_1,\\dots ,r_m] : \\nonumber \\\\&~~~~~~\\langle s^1_i,\\sigma _1 \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~is~} r_1\\nonumber \\\\&~~~~~~\\vdots \\nonumber \\\\&~~~~~~\\langle s^m_i,\\sigma _m \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~is~} r_m \\nonumber \\\\&\\text{ and } \\exists k, 1 \\le k \\le m : r=r_k \\text{ and } \\forall x, 1 \\le x \\le m, r_k \\ge r_x$ Next we define how a comparative quantitative aggregate interval formula is satisfied on two sets of trajectories $\\sigma _1,\\dots ,\\sigma _m$ , and $\\bar{\\sigma }_1,\\dots ,\\bar{\\sigma }_{\\bar{m}}$ a starting point $i$ and an ending point $j$ : $&\\Big (\\Big \\lbrace \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace , j \\Big ) \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\models \\; \\mathbf {~direction~of~change~in~} average \\mathbf {~rate~of~production~of~} f \\mathbf {~is~} d \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ if } \\exists \\; n_1 : (\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models \\; average \\mathbf {~rate~of~production~of~} f \\text{~is~} n_1 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } \\exists \\; n_2 : (\\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace , j) \\models \\; average \\mathbf {~rate~of~production~of~} f \\text{~is~} n_2 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } n_2 \\; d \\; n_1\\\\&\\Big (\\Big \\lbrace \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace , j \\Big ) \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\models \\; \\mathbf {~direction~of~change~in~} minimum \\mathbf {~rate~of~production~of~} f \\mathbf {~is~} d \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ if } \\exists \\; n_1 : (\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models \\; minimum \\mathbf {~rate~of~production~of~} f \\text{~is~} n_1 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } \\exists \\; n_2 : (\\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace , j) \\models \\; minimum \\mathbf {~rate~of~production~of~} f \\text{~is~} n_2 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } n_2 \\; d \\; n_1\\\\&\\Big (\\Big \\lbrace \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace , j \\Big ) \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\models \\; \\mathbf {~direction~of~change~in~} maximum \\mathbf {~rate~of~production~of~} f \\mathbf {~is~} d \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ if } \\exists \\; n_1 : (\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models \\; maximum \\mathbf {~rate~of~production~of~} f \\text{~is~} n_1 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } \\exists \\; n_2 : (\\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace , j) \\models \\; maximum \\mathbf {~rate~of~production~of~} f \\text{~is~} n_2 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } n_2 \\; d \\; n_1\\\\\\vspace{20.0pt}\\nonumber \\\\&\\Big (\\Big \\lbrace \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace , j \\Big ) \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\models \\; \\mathbf {~direction~of~change~in~} average \\mathbf {~rate~of~firing~of~} f \\mathbf {~is~} d \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ if } \\exists \\; n_1 : (\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models \\; average \\mathbf {~rate~of~firing~of~} f \\text{~is~} n_1 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } \\exists \\; n_2 : (\\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace , j) \\models \\; average \\mathbf {~rate~of~firing~of~} f \\text{~is~} n_2 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } n_2 \\; d \\; n_1\\\\&\\Big (\\Big \\lbrace \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace , j \\Big ) \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\models \\; \\mathbf {~direction~of~change~in~} minimum \\mathbf {~rate~of~firing~of~} f \\mathbf {~is~} d \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ if } \\exists \\; n_1 : (\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models \\; minimum \\mathbf {~rate~of~firing~of~} f \\text{~is~} n_1 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } \\exists \\; n_2 : (\\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace , j) \\models \\; minimum \\mathbf {~rate~of~firing~of~} f \\text{~is~} n_2 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } n_2 \\; d \\; n_1\\\\&\\Big (\\Big \\lbrace \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace , j \\Big ) \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\models \\; \\mathbf {~direction~of~change~in~} maximum \\mathbf {~rate~of~firing~of~} f \\mathbf {~is~} d \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ if } \\exists \\; n_1 : (\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models \\; maximum \\mathbf {~rate~of~firing~of~} f \\text{~is~} n_1 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } \\exists \\; n_2 : (\\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace , j) \\models \\; maximum \\mathbf {~rate~of~firing~of~} f \\text{~is~} n_2 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } n_2 \\; d \\; n_1\\\\\\vspace{20.0pt}\\nonumber \\\\&\\Big (\\Big \\lbrace \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace , j \\Big ) \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\models \\; \\mathbf {~direction~of~change~in~} average \\mathbf {~total~production~of~} f \\mathbf {~is~} d \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ if } \\exists \\; n_1 : (\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models \\; average \\mathbf {~total~production~of~} f \\text{~is~} n_1 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } \\exists \\; n_2 : (\\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace , j) \\models \\; average \\mathbf {~total~production~of~} f \\text{~is~} n_2 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } n_2 \\; d \\; n_1\\\\&\\Big (\\Big \\lbrace \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace , j \\Big ) \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\models \\; \\mathbf {~direction~of~change~in~} minimum \\mathbf {~total~production~of~} f \\mathbf {~is~} d \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ if } \\exists \\; n_1 : (\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models \\; minimum \\mathbf {~total~production~of~} f \\text{~is~} n_1 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } \\exists \\; n_2 : (\\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace , j) \\models \\; minimum \\mathbf {~total~production~of~} f \\text{~is~} n_2 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } n_2 \\; d \\; n_1\\\\&\\Big (\\Big \\lbrace \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace , j \\Big ) \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\models \\; \\mathbf {~direction~of~change~in~} maximum \\mathbf {~total~production~of~} f \\mathbf {~is~} d \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ if } \\exists \\; n_1 : (\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models \\; maximum \\mathbf {~total~production~of~} f \\text{~is~} n_1 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } \\exists \\; n_2 : (\\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace , j) \\models \\; maximum \\mathbf {~total~production~of~} f \\text{~is~} n_2 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } n_2 \\; d \\; n_1$ Next we define how a comparative quantitative aggregate point formula is satisfied on two sets of trajectories $\\sigma _1,\\dots ,\\sigma _m$ , and $\\bar{\\sigma }_1,\\dots ,\\bar{\\sigma }_{\\bar{m}}$ at point $i$ : $&\\Big \\lbrace \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\models \\; \\mathbf {~direction~of~change~in~} average \\mathbf {~value~of~} f \\mathbf {~is~} d \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ if } \\exists \\; n_1 : \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace \\models \\; average \\mathbf {~value~of~} f \\text{~is~} n_1 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } \\exists \\; n_2 : \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\models \\; average \\mathbf {~value~of~} f \\text{~is~} n_2 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } n_2 \\; d \\; n_1\\\\&\\Big \\lbrace \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\models \\; \\mathbf {~direction~of~change~in~} minimum \\mathbf {~value~of~} f \\mathbf {~is~} d \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ if } \\exists \\; n_1 : \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace \\models \\; minimum \\mathbf {~value~of~} f \\text{~is~} n_1 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } \\exists \\; n_2 : \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\models \\; minimum \\mathbf {~value~of~} f \\text{~is~} n_2 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } n_2 \\; d \\; n_1\\\\&\\Big \\lbrace \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\models \\; \\mathbf {~direction~of~change~in~} maximum \\mathbf {~value~of~} f \\mathbf {~is~} d \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ if } \\exists \\; n_1 : \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace \\models \\; maximum \\mathbf {~value~of~} f \\text{~is~} n_1 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } \\exists \\; n_2 : \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\models \\; maximum \\mathbf {~value~of~} f \\text{~is~} n_2 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } n_2 \\; d \\; n_1$ Given a domain description $\\mathbf {D}$ with simple fluents represented by a Guarded-Arc Petri Net with Colored tokens as defined in definition REF .", "Let $\\sigma = s_0,T_0,s_1,\\dots ,T_{k-1},s_k$ be its trajectory as defined in definition REF , and $s_i$ be a state on that trajectory.", "Let actions $T_i$ firing in state $s_i$ be observable in $s_i$ such that $T_i \\subseteq s_i$ .", "We define observation semantics using LTL below.", "We will use $s_i(f[l])$ to represent $m_{s_i(l)}(f)$ (multiplicity / value of $f$ in location $l$ ) in state $s_i$ .", "First we define how interval formulas are satisfied on a trajectory $\\sigma $ , starting state $s_i$ and an ending point $j$ : $(\\langle s_i,\\sigma \\rangle , j) \\models & \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n \\nonumber \\\\&\\text{ if } n=(s_j(f[l])-s_i(f[l]))/(j-i)\\\\(\\langle s_i,\\sigma \\rangle , j) \\models & \\mathbf {~rate~of~firing~of~} a \\mathbf {~is~} n \\nonumber \\\\&\\text{ if } n=\\sum _{i\\le k \\le j, \\langle s_k,\\sigma \\rangle \\models a \\mathbf {~occurs~}}{1}/(j-i)\\\\(\\langle s_i,\\sigma \\rangle , j) \\models & \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n \\nonumber \\\\&\\text{ if } n=(s_j(f[l])-s_i(f[l]))\\\\(\\langle s_i,\\sigma \\rangle , j) \\models & f \\mathbf {~is~accumulating~} \\mathbf {~atloc~} l \\nonumber \\\\&\\text{ if } (\\nexists k, i \\le k \\le j : s_{k+1}(f[l]) < s_k(f[l])) \\text{ and } s_j(f[l]) > s_i(f[l])\\\\(\\langle s_i,\\sigma \\rangle , j) \\models & f \\mathbf {~is~decreasing~} \\mathbf {~atloc~} l \\nonumber \\\\&\\text{ if } (\\nexists k, i \\le k \\le j : s_{k+1}(f[l]) > s_k(f[l])) \\text{ and } s_j(f[l]) < s_i(f[l])$ Next we define how a point formula is satisfied on a trajectory $\\sigma $ , in a state $s_i$ : $\\langle s_i,\\sigma \\rangle \\models & \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~higher~than~} n \\nonumber \\\\&\\text{ if } s_i(f[l]) > n\\\\\\langle s_i,\\sigma \\rangle \\models & \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~lower~than~} n \\nonumber \\\\&\\text{ if } s_i(f[l]) < n\\\\\\langle s_i,\\sigma \\rangle \\models & \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n \\nonumber \\\\&\\text{ if } s_i(f[l]) = n\\\\\\langle s_i,\\sigma \\rangle \\models & a \\mathbf {~occurs~} \\nonumber \\\\&\\text{ if } a \\in s_i\\\\\\langle s_i,\\sigma \\rangle \\models & a \\mathbf {~does~not~occur~} \\nonumber \\\\&\\text{ if } a \\notin s_i\\\\\\langle s_i,\\sigma \\rangle \\models & a_1 \\mathbf {~switches~to~} a_2 \\nonumber \\\\&\\text{ if } a_1 \\in s_{i-1} \\text{ and } a_2 \\notin s_{i-1} \\text{ and } a_1 \\notin s_i \\text{ and } a_2 \\in s_i$ Next we define how a quantitative all interval formula is satisfied on a set of trajectories $\\sigma _1,\\dots ,\\sigma _m$ with starting states $s^1_i,\\dots ,s^m_i$ and end point $j$ : $(\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models & \\mathbf {~rates~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~are~} [r_1,\\dots ,r_m] \\nonumber \\\\&\\text{ if } (\\langle s^1_i,\\sigma _1 \\rangle , j) \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} r_1\\nonumber \\\\&\\vdots \\nonumber \\\\&\\text{ and } (\\langle s^m_i,\\sigma _m \\rangle , j) \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} r_m \\\\(\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models & \\mathbf {~rates~of~firing~of~} a \\mathbf {~are~} [r_1,\\dots ,r_m] \\nonumber \\\\&\\text{ if } (\\langle s^1_i,\\sigma _1 \\rangle , j) \\models \\mathbf {~rate~of~firing~of~} a \\mathbf {~is~} r_1\\nonumber \\\\&\\vdots \\nonumber \\\\&\\text{ and } (\\langle s^m_i,\\sigma _m \\rangle , j) \\models \\mathbf {~rate~of~firing~of~} a \\mathbf {~is~} r_m \\\\(\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models & \\mathbf {~totals~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~are~} [r_1,\\dots ,r_m] \\nonumber \\\\&\\text{ if } (\\langle s^1_i,\\sigma _1 \\rangle , j) \\models \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} r_1\\nonumber \\\\&\\vdots \\nonumber \\\\&\\text{ and } (\\langle s^m_i,\\sigma _m \\rangle , j) \\models \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} r_m$ Next we define how a quantitative all point formula is satisfied w.r.t.", "a set of trajectories $\\sigma _1,\\dots ,\\sigma _m$ in states $s^1_i,\\dots ,s^m_i$ : $\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace \\models & \\mathbf {~values~of~} f \\mathbf {~atloc~} l\\mathbf {~are~} [r_1,\\dots ,r_m] \\nonumber \\\\&\\text{ if } \\langle s^1_i,\\sigma _1 \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} r_1\\nonumber \\\\&\\vdots \\nonumber \\\\&\\text{ and } \\langle s^m_i,\\sigma _m \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} r_m$ Next we define how a quantitative aggregate interval formula is satisfied w.r.t.", "a set of trajectories $\\sigma _1,\\dots ,\\sigma _m$ with starting states $s^1_i,\\dots ,s^m_i$ and end point $j$ : $(\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models & \\; average \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} r \\nonumber \\\\&\\text{ if } \\exists [r_1,\\dots ,r_m] : \\nonumber \\\\&~~~~~~(\\langle s^1_i,\\sigma _1 \\rangle , j) \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} r_1\\nonumber \\\\&~~~~~~\\vdots \\nonumber \\\\&~~~~~~(\\langle s^m_i,\\sigma _m \\rangle , j) \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} r_m \\nonumber \\\\&\\text{ and } r=(r_1+\\dots +r_m)/m\\\\(\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models & \\; minimum \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} r \\nonumber \\\\&\\text{ if } \\exists [r_1,\\dots ,r_m] : \\nonumber \\\\&~~~~~~(\\langle s^1_i,\\sigma _1 \\rangle , j) \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} r_1\\nonumber \\\\&~~~~~~\\vdots \\nonumber \\\\&~~~~~~(\\langle s^m_i,\\sigma _m \\rangle , j) \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} r_m \\nonumber \\\\&\\text{ and } \\exists k, 1 \\le k \\le m : r=r_k \\text{ and } \\forall x, 1 \\le x \\le m, r_k \\le r_x\\\\(\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models & \\; maximum \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} r \\nonumber \\\\&\\text{ if } \\exists [r_1,\\dots ,r_m] : \\nonumber \\\\&~~~~~~(\\langle s^1_i,\\sigma _1 \\rangle , j) \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} r_1\\nonumber \\\\&~~~~~~\\vdots \\nonumber \\\\&~~~~~~(\\langle s^m_i,\\sigma _m \\rangle , j) \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} r_m \\nonumber \\\\&\\text{ and } \\exists k, 1 \\le k \\le m : r=r_k \\text{ and } \\forall x, 1 \\le x \\le m, r_k \\ge r_x\\\\\\vspace{20.0pt}\\nonumber \\\\(\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models & \\; average \\mathbf {~rate~of~firing~of~} f \\mathbf {~is~} r \\nonumber \\\\&\\text{ iff } \\exists [r_1,\\dots ,r_m] : \\nonumber \\\\&~~~~~~(\\langle s^1_i,\\sigma _1 \\rangle , j) \\models \\mathbf {~rate~of~firing~of~} f \\mathbf {~is~} r_1\\nonumber \\\\&~~~~~~\\vdots \\nonumber \\\\&~~~~~~(\\langle s^m_i,\\sigma _m \\rangle , j) \\models \\mathbf {~rate~of~firing~of~} f \\mathbf {~is~} r_m \\nonumber \\\\&\\text{ and } r=(r_1+\\dots +r_m)/m\\\\(\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models & \\; minimum \\mathbf {~rate~of~firing~of~} f \\mathbf {~is~} r \\nonumber \\\\&\\text{ iff } \\exists [r_1,\\dots ,r_m] : \\nonumber \\\\&~~~~~~(\\langle s^1_i,\\sigma _1 \\rangle , j) \\models \\mathbf {~rate~of~firing~of~} f \\mathbf {~is~} r_1\\nonumber \\\\&~~~~~~\\vdots \\nonumber \\\\&~~~~~~(\\langle s^m_i,\\sigma _m \\rangle , j) \\models \\mathbf {~rate~of~firing~of~} f \\mathbf {~is~} r_m \\nonumber \\\\&\\text{ and } \\exists k, 1 \\le k \\le m : r=r_k \\text{ and } \\forall x, 1 \\le x \\le m, r_k \\le r_x\\\\(\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models & \\; maximum \\mathbf {~rate~of~firing~of~} f \\mathbf {~is~} r \\nonumber \\\\&\\text{ iff } \\exists [r_1,\\dots ,r_m] : \\nonumber \\\\&~~~~~~(\\langle s^1_i,\\sigma _1 \\rangle , j) \\models \\mathbf {~rate~of~firing~of~} f \\mathbf {~is~} r_1\\nonumber \\\\&~~~~~~\\vdots \\nonumber \\\\&~~~~~~(\\langle s^m_i,\\sigma _m \\rangle , j) \\models \\mathbf {~rate~of~firing~of~} f \\mathbf {~is~} r_m \\nonumber \\\\&\\text{ and } \\exists k, 1 \\le k \\le m : r=r_k \\text{ and } \\forall x, 1 \\le x \\le m, r_k \\ge r_x\\\\\\vspace{20.0pt}\\nonumber \\\\(\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models & \\; average \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} r \\nonumber \\\\&\\text{ if } \\exists [r_1,\\dots ,r_m] : \\nonumber \\\\&~~~~~~(\\langle s^1_i,\\sigma _1 \\rangle , j) \\models \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} r_1\\nonumber \\\\&~~~~~~\\vdots \\nonumber \\\\&~~~~~~(\\langle s^m_i,\\sigma _m \\rangle , j) \\models \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} r_m \\nonumber \\\\&\\text{ and } r=(r_1+\\dots +r_m)/m\\\\(\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models & \\; minimum \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} r \\nonumber \\\\&\\text{ if } \\exists [r_1,\\dots ,r_m] : \\nonumber \\\\&~~~~~~(\\langle s^1_i,\\sigma _1 \\rangle , j) \\models \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} r_1\\nonumber \\\\&~~~~~~\\vdots \\nonumber \\\\&~~~~~~(\\langle s^m_i,\\sigma _m \\rangle , j) \\models \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} r_m \\nonumber \\\\&\\text{ and } \\exists k, 1 \\le k \\le m : r=r_k \\text{ and } \\forall x, 1 \\le x \\le m, r_k \\le r_x\\\\(\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models & \\; maximum \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} r \\nonumber \\\\&\\text{ if } \\exists [r_1,\\dots ,r_m] : \\nonumber \\\\&~~~~~~(\\langle s^1_i,\\sigma _1 \\rangle , j) \\models \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} r_1\\nonumber \\\\&~~~~~~\\vdots \\nonumber \\\\&~~~~~~(\\langle s^m_i,\\sigma _m \\rangle , j) \\models \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} r_m \\nonumber \\\\&\\text{ and } \\exists k, 1 \\le k \\le m : r=r_k \\text{ and } \\forall x, 1 \\le x \\le m, r_k \\ge r_x$ Next we define how a quantitative aggregate point formula is satisfied w.r.t.", "a set of trajectories $\\sigma _1,\\dots ,\\sigma _m$ in states $s^1_i,\\dots ,s^m_i$ : $\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace \\models & \\; average \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} r \\nonumber \\\\&\\text{ if } \\exists [r_1,\\dots ,r_m] : \\nonumber \\\\&~~~~~~\\langle s^1_i,\\sigma _1 \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} r_1\\nonumber \\\\&~~~~~~\\vdots \\nonumber \\\\&~~~~~~\\langle s^m_i,\\sigma _m \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} r_m \\nonumber \\\\&\\text{ and } r=(r_1+\\dots +r_m)/m\\\\\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace \\models & \\; minimum \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} r \\nonumber \\\\&\\text{ if } \\exists [r_1,\\dots ,r_m] : \\nonumber \\\\&~~~~~~\\langle s^1_i,\\sigma _1 \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} r_1\\nonumber \\\\&~~~~~~\\vdots \\nonumber \\\\&~~~~~~\\langle s^m_i,\\sigma _m \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} r_m \\nonumber \\\\&\\text{ and } \\exists k, 1 \\le k \\le m : r=r_k \\text{ and } \\forall x, 1 \\le x \\le m, r_k \\le r_x\\\\\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace \\models & \\; maximum \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} r \\nonumber \\\\&\\text{ if } \\exists [r_1,\\dots ,r_m] : \\nonumber \\\\&~~~~~~\\langle s^1_i,\\sigma _1 \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} r_1\\nonumber \\\\&~~~~~~\\vdots \\nonumber \\\\&~~~~~~\\langle s^m_i,\\sigma _m \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} r_m \\nonumber \\\\&\\text{ and } \\exists k, 1 \\le k \\le m : r=r_k \\text{ and } \\forall x, 1 \\le x \\le m, r_k \\ge r_x$ Next we define how a comparative quantitative aggregate interval formula is satisfied w.r.t.", "two sets of trajectories $\\sigma _1,\\dots ,\\sigma _m$ , and $\\bar{\\sigma }_1,\\dots ,\\bar{\\sigma }_{\\bar{m}}$ a starting point $i$ and an ending point $j$ : $&\\Big (\\Big \\lbrace \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace , j \\Big ) \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\models \\; \\mathbf {~direction~of~change~in~} average \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} d \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ if } \\exists \\; n_1 : (\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models \\; average \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\text{~is~} n_1 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } \\exists \\; n_2 : (\\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace , j) \\models \\; average \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\text{~is~} n_2 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } n_2 \\; d \\; n_1\\\\&\\Big (\\Big \\lbrace \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace , j \\Big ) \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\models \\; \\mathbf {~direction~of~change~in~} minimum \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} d \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ if } \\exists \\; n_1 : (\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models \\; minimum \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\text{~is~} n_1 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } \\exists \\; n_2 : (\\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace , j) \\models \\; minimum \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\text{~is~} n_2 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } n_2 \\; d \\; n_1\\\\&\\Big (\\Big \\lbrace \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace , j \\Big ) \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\models \\; \\mathbf {~direction~of~change~in~} maximum \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} d \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ if } \\exists \\; n_1 : (\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models \\; maximum \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\text{~is~} n_1 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } \\exists \\; n_2 : (\\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace , j) \\models \\; maximum \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\text{~is~} n_2 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } n_2 \\; d \\; n_1\\\\\\vspace{20.0pt}\\nonumber \\\\&\\Big (\\Big \\lbrace \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace , j \\Big ) \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\models \\; \\mathbf {~direction~of~change~in~} average \\mathbf {~rate~of~firing~of~} f \\mathbf {~is~} d \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ if } \\exists \\; n_1 : (\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models \\; average \\mathbf {~rate~of~firing~of~} f \\text{~is~} n_1 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } \\exists \\; n_2 : (\\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace , j) \\models \\; average \\mathbf {~rate~of~firing~of~} f \\text{~is~} n_2 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } n_2 \\; d \\; n_1\\\\&\\Big (\\Big \\lbrace \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace , j \\Big ) \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\models \\; \\mathbf {~direction~of~change~in~} minimum \\mathbf {~rate~of~firing~of~} f \\mathbf {~is~} d \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ if } \\exists \\; n_1 : (\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models \\; minimum \\mathbf {~rate~of~firing~of~} f \\text{~is~} n_1 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } \\exists \\; n_2 : (\\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace , j) \\models \\; minimum \\mathbf {~rate~of~firing~of~} f \\text{~is~} n_2 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } n_2 \\; d \\; n_1\\\\&\\Big (\\Big \\lbrace \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace , j \\Big ) \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\models \\; \\mathbf {~direction~of~change~in~} maximum \\mathbf {~rate~of~firing~of~} f \\mathbf {~is~} d \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ if } \\exists \\; n_1 : (\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models \\; maximum \\mathbf {~rate~of~firing~of~} f \\text{~is~} n_1 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } \\exists \\; n_2 : (\\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace , j) \\models \\; maximum \\mathbf {~rate~of~firing~of~} f \\text{~is~} n_2 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } n_2 \\; d \\; n_1\\\\\\vspace{20.0pt}\\nonumber \\\\&\\Big (\\Big \\lbrace \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace , j \\Big ) \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\models \\; \\mathbf {~direction~of~change~in~} average \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} d \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ if } \\exists \\; n_1 : (\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models \\; average \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\text{~is~} n_1 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } \\exists \\; n_2 : (\\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace , j) \\models \\; average \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\text{~is~} n_2 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } n_2 \\; d \\; n_1\\\\&\\Big (\\Big \\lbrace \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace , j \\Big ) \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\models \\; \\mathbf {~direction~of~change~in~} minimum \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} d \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ if } \\exists \\; n_1 : (\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models \\; minimum \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\text{~is~} n_1 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } \\exists \\; n_2 : (\\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace , j) \\models \\; minimum \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\text{~is~} n_2 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } n_2 \\; d \\; n_1\\\\&\\Big (\\Big \\lbrace \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace , j \\Big ) \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\models \\; \\mathbf {~direction~of~change~in~} maximum \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} d \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ if } \\exists \\; n_1 : (\\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , j) \\models \\; maximum \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\text{~is~} n_1 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } \\exists \\; n_2 : (\\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace , j) \\models \\; maximum \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\text{~is~} n_2 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } n_2 \\; d \\; n_1$ Next we define how a comparative quantitative aggregate point formula is satisfied w.r.t.", "two sets of trajectories $\\sigma _1,\\dots ,\\sigma _m$ , and $\\bar{\\sigma }_1,\\dots ,\\bar{\\sigma }_{\\bar{m}}$ at point $i$ : $&\\Big \\lbrace \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\models \\; \\mathbf {~direction~of~change~in~} average \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} d \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ if } \\exists \\; n_1 : \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace \\models \\; average \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\text{~is~} n_1 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } \\exists \\; n_2 : \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\models \\; average \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\text{~is~} n_2 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } n_2 \\; d \\; n_1\\\\&\\Big \\lbrace \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\models \\; \\mathbf {~direction~of~change~in~} minimum \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} d \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ if } \\exists \\; n_1 : \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace \\models \\; minimum \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\text{~is~} n_1 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } \\exists \\; n_2 : \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\models \\; minimum \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\text{~is~} n_2 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } n_2 \\; d \\; n_1\\\\&\\Big \\lbrace \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\models \\; \\mathbf {~direction~of~change~in~} maximum \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} d \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ if } \\exists \\; n_1 : \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace \\models \\; maximum \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\text{~is~} n_1 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } \\exists \\; n_2 : \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\models \\; maximum \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\text{~is~} n_2 \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ and } n_2 \\; d \\; n_1$ Trajectory Filtering due to Internal Observations The trajectories produced by the Guarded-Arc Petri Net execution are filtered to retain only the trajectories that satisfy all internal observations in a query statement.", "Let $\\sigma = s_0,\\dots ,s_k$ be a trajectory as given in definition REF .", "Then $\\sigma $ satisfies an observation: $&\\mathbf {~rate~of~production~of~} f \\mathbf {~is~} n \\nonumber \\\\&~~~~~~~~\\text{ if } (\\langle s_0,\\sigma \\rangle , k) \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~is~} n \\\\&\\mathbf {~rate~of~production~of~} f \\mathbf {~is~} n \\nonumber \\\\&~~~~~~~~\\text{ if } (\\langle s_0,\\sigma \\rangle , k) \\models \\mathbf {~rate~of~firing~of~} a \\mathbf {~is~} n \\\\&\\mathbf {~total~production~of~} f \\mathbf {~is~} n \\nonumber \\\\&~~~~~~~~\\text{ if } (\\langle s_0,\\sigma \\rangle , k) \\models \\mathbf {~total~production~of~} f \\mathbf {~is~} n \\\\&f \\mathbf {~is~accumulating~} \\nonumber \\\\&~~~~~~~~\\text{ if } (\\langle s_0,\\sigma \\rangle , k) \\models f \\mathbf {~is~accumulating~} \\\\&f \\mathbf {~is~decreasing~} \\nonumber \\\\&~~~~~~~~\\text{ if } (\\langle s_0,\\sigma \\rangle , k) \\models f \\mathbf {~is~decreasing~} \\\\\\vspace{20.0pt}\\nonumber \\\\&\\mathbf {~rate~of~production~of~} f \\mathbf {~is~} n \\nonumber \\\\&~~~~~~~~\\mathbf {when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&~~~~~~~~\\text{ if } (\\langle s_i,\\sigma \\rangle , j) \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~is~} n \\\\&\\mathbf {~rate~of~production~of~} f \\mathbf {~is~} n \\nonumber \\\\&~~~~~~~~\\mathbf {when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&~~~~~~~~\\text{ if } (\\langle s_i,\\sigma \\rangle , j) \\models \\mathbf {~rate~of~firing~of~} a \\mathbf {~is~} n \\\\&\\mathbf {~total~production~of~} f \\mathbf {~is~} n \\nonumber \\\\&~~~~~~~~\\mathbf {when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&~~~~~~~~\\text{ if } (\\langle s_i,\\sigma \\rangle , j) \\models \\mathbf {~total~production~of~} f \\mathbf {~is~} n \\\\&f \\mathbf {~is~accumulating~} \\nonumber \\\\&~~~~~~~~\\mathbf {when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&~~~~~~~~\\text{ if } (\\langle s_i,\\sigma \\rangle , j) \\models f \\mathbf {~is~accumulating~} \\\\&f \\mathbf {~is~decreasing~} \\nonumber \\\\&~~~~~~~~\\mathbf {when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&~~~~~~~~\\text{ if } (\\langle s_i,\\sigma \\rangle , j) \\models f \\mathbf {~is~decreasing~} \\\\\\vspace{20.0pt}\\nonumber \\\\&f \\mathbf {~is~higher~than~} n \\nonumber \\\\&~~~~~~~~\\text{ if } \\exists i, 0 \\le i \\le k: \\langle s_i,\\sigma \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~is~higher~than~} n \\\\&f \\mathbf {~is~lower~than~} n \\nonumber \\\\&~~~~~~~~\\text{ if } \\exists i, 0 \\le i \\le k: \\langle s_i,\\sigma \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~is~lower~than~} n \\\\&f \\mathbf {~is~} n \\nonumber \\\\&~~~~~~~~\\text{ if } \\exists i, 0 \\le i \\le k: \\langle s_i,\\sigma \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~is~} n \\\\&a \\mathbf {~occurs~} \\nonumber \\\\&~~~~~~~~\\text{ if } \\exists i, 0 \\le i \\le k: \\langle s_i,\\sigma \\rangle \\models a \\mathbf {~occurs~} \\\\&a \\mathbf {~does~not~occur~} \\nonumber \\\\&~~~~~~~~\\text{ if } \\exists i, 0 \\le i \\le k: \\langle s_i,\\sigma \\rangle \\models a \\mathbf {~does~not~occur~} \\\\&a_1 \\mathbf {~switches~to~} a_2 \\nonumber \\\\&~~~~~~~~\\text{ if } \\exists i, 0 \\le i \\le k: \\langle s_i,\\sigma \\rangle \\models a_1 \\mathbf {~switches~to~} a_2\\vspace{20.0pt}\\nonumber \\\\&f \\mathbf {~is~higher~than~} n \\nonumber \\\\&~~~~~~~~\\mathbf {at~time~step~} i \\nonumber \\\\&~~~~~~~~\\text{ if } \\langle s_i,\\sigma \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~is~higher~than~} n \\\\&f \\mathbf {~is~lower~than~} n \\nonumber \\\\&~~~~~~~~\\mathbf {at~time~step~} i \\nonumber \\\\&~~~~~~~~\\text{ if } \\langle s_i,\\sigma \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~is~lower~than~} n \\\\&f \\mathbf {~is~} n \\nonumber \\\\&~~~~~~~~\\mathbf {at~time~step~} i \\nonumber \\\\&~~~~~~~~\\text{ if } \\langle s_i,\\sigma \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~is~} n \\\\&a \\mathbf {~occurs~} \\nonumber \\\\&~~~~~~~~\\mathbf {at~time~step~} i \\nonumber \\\\&~~~~~~~~\\text{ if } \\langle s_i,\\sigma \\rangle \\models a \\mathbf {~occurs~} \\\\&a \\mathbf {~does~not~occur~} \\nonumber \\\\&~~~~~~~~\\mathbf {at~time~step~} i \\nonumber \\\\&~~~~~~~~\\text{ if } \\langle s_i,\\sigma \\rangle \\models a \\mathbf {~does~not~occur~} \\\\&a_1 \\mathbf {~switches~to~} a_2 \\nonumber \\\\&~~~~~~~~\\mathbf {at~time~step~} i \\nonumber \\\\&~~~~~~~~\\text{ if } \\langle s_i,\\sigma \\rangle \\models a_1 \\mathbf {~switches~to~} a_2$ Let $\\sigma = s_0,\\dots ,s_k$ be a trajectory of the form (REF ).", "Then $\\sigma $ satisfies an observation: $&\\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n \\nonumber \\\\&~~~~~~~~\\text{ if } (\\langle s_0,\\sigma \\rangle , k) \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n \\\\&\\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n \\nonumber \\\\&~~~~~~~~\\text{ if } (\\langle s_0,\\sigma \\rangle , k) \\models \\mathbf {~rate~of~firing~of~} a \\mathbf {~is~} n \\\\&\\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n \\nonumber \\\\&~~~~~~~~\\text{ if } (\\langle s_0,\\sigma \\rangle , k) \\models \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n \\\\&f \\mathbf {~is~accumulating~} \\mathbf {~atloc~} l \\nonumber \\\\&~~~~~~~~\\text{ if } (\\langle s_0,\\sigma \\rangle , k) \\models f \\mathbf {~is~accumulating~} \\mathbf {~atloc~} l \\\\&f \\mathbf {~is~decreasing~} \\mathbf {~atloc~} l \\nonumber \\\\&~~~~~~~~\\text{ if } (\\langle s_0,\\sigma \\rangle , k) \\models f \\mathbf {~is~decreasing~} \\mathbf {~atloc~} l \\\\\\vspace{20.0pt}\\nonumber \\\\&\\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n \\nonumber \\\\&~~~~~~~~\\mathbf {when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&~~~~~~~~\\text{ if } (\\langle s_i,\\sigma \\rangle , j) \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n \\\\&\\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n \\nonumber \\\\&~~~~~~~~\\mathbf {when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&~~~~~~~~\\text{ if } (\\langle s_i,\\sigma \\rangle , j) \\models \\mathbf {~rate~of~firing~of~} a \\mathbf {~is~} n \\\\&\\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n \\nonumber \\\\&~~~~~~~~\\mathbf {when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&~~~~~~~~\\text{ if } (\\langle s_i,\\sigma \\rangle , j) \\models \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n \\\\&f \\mathbf {~is~accumulating~} \\mathbf {~atloc~} l \\nonumber \\\\&~~~~~~~~\\mathbf {when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&~~~~~~~~\\text{ if } (\\langle s_i,\\sigma \\rangle , j) \\models f \\mathbf {~is~accumulating~} \\mathbf {~atloc~} l \\\\&f \\mathbf {~is~decreasing~} \\mathbf {~atloc~} l \\nonumber \\\\&~~~~~~~~\\mathbf {when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&~~~~~~~~\\text{ if } (\\langle s_i,\\sigma \\rangle , j) \\models f \\mathbf {~is~decreasing~} \\mathbf {~atloc~} l \\\\\\vspace{20.0pt}\\nonumber \\\\&f \\mathbf {~atloc~} l \\mathbf {~is~higher~than~} n \\nonumber \\\\&~~~~~~~~\\text{ if } \\exists i, 0 \\le i \\le k: \\langle s_i,\\sigma \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~higher~than~} n \\\\&f \\mathbf {~atloc~} l \\mathbf {~is~lower~than~} n \\nonumber \\\\&~~~~~~~~\\text{ if } \\exists i, 0 \\le i \\le k: \\langle s_i,\\sigma \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~lower~than~} n \\\\&f \\mathbf {~atloc~} l \\mathbf {~is~} n \\nonumber \\\\&~~~~~~~~\\text{ if } \\exists i, 0 \\le i \\le k: \\langle s_i,\\sigma \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n \\\\&a \\mathbf {~occurs~} \\nonumber \\\\&~~~~~~~~\\text{ if } \\exists i, 0 \\le i \\le k: \\langle s_i,\\sigma \\rangle \\models a \\mathbf {~occurs~} \\\\&a \\mathbf {~does~not~occur~} \\nonumber \\\\&~~~~~~~~\\text{ if } \\exists i, 0 \\le i \\le k: \\langle s_i,\\sigma \\rangle \\models a \\mathbf {~does~not~occur~} \\\\&a_1 \\mathbf {~switches~to~} a_2 \\nonumber \\\\&~~~~~~~~\\text{ if } \\exists i, 0 \\le i \\le k: \\langle s_i,\\sigma \\rangle \\models a_1 \\mathbf {~switches~to~} a_2\\vspace{20.0pt}\\nonumber \\\\&f \\mathbf {~atloc~} l \\mathbf {~is~higher~than~} n \\nonumber \\\\&~~~~~~~~\\mathbf {at~time~step~} i \\nonumber \\\\&~~~~~~~~\\text{ if } \\langle s_i,\\sigma \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~higher~than~} n \\\\&f \\mathbf {~atloc~} l \\mathbf {~is~lower~than~} n \\nonumber \\\\&~~~~~~~~\\mathbf {at~time~step~} i \\nonumber \\\\&~~~~~~~~\\text{ if } \\langle s_i,\\sigma \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~lower~than~} n \\\\&f \\mathbf {~atloc~} l \\mathbf {~is~} n \\nonumber \\\\&~~~~~~~~\\mathbf {at~time~step~} i \\nonumber \\\\&~~~~~~~~\\text{ if } \\langle s_i,\\sigma \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n \\\\&a \\mathbf {~occurs~} \\nonumber \\\\&~~~~~~~~\\mathbf {at~time~step~} i \\nonumber \\\\&~~~~~~~~\\text{ if } \\langle s_i,\\sigma \\rangle \\models a \\mathbf {~occurs~} \\\\&a \\mathbf {~does~not~occur~} \\nonumber \\\\&~~~~~~~~\\mathbf {at~time~step~} i \\nonumber \\\\&~~~~~~~~\\text{ if } \\langle s_i,\\sigma \\rangle \\models a \\mathbf {~does~not~occur~} \\\\&a_1 \\mathbf {~switches~to~} a_2 \\nonumber \\\\&~~~~~~~~\\mathbf {at~time~step~} i \\nonumber \\\\&~~~~~~~~\\text{ if } \\langle s_i,\\sigma \\rangle \\models a_1 \\mathbf {~switches~to~} a_2$ A trajectory $\\sigma $ is kept for further processing w.r.t.", "a set of internal observations $\\langle \\text{internal observation} \\rangle _1,\\dots ,\\langle \\text{internal observation} \\rangle _n$ if $\\sigma \\models \\langle \\text{internal observation} \\rangle _i$ , $1 \\le i \\le n$ .", "Query Description Satisfaction Now, we define query statement semantics using LTL syntax.", "Let $\\mathbf {D}$ be a domain description with simple fluents and $\\sigma = s_0,\\dots ,s_k$ be its trajectory of length $k$ as defined in (REF ).", "Let $\\sigma _1,\\dots ,\\sigma _m$ represent the set of trajectories of $\\mathbf {D}$ filtered by observations as necessary, with each trajectory has the form $\\sigma _i = s^i_0,\\dots ,s^i_k$ $1\\le i \\le m$ .", "Let $\\mathbf {\\bar{D}}$ be a modified domain description and $\\bar{\\sigma }_1,\\dots ,\\bar{\\sigma }_{\\bar{m}}$ be its trajectories of the form $\\bar{\\sigma }_i = \\bar{s}^i_0,\\dots ,\\bar{s}^i_k$ .", "Then we define query satisfaction using the formula satisfaction in section REF as follows.", "Two sets of trajectories $\\sigma _1,\\dots ,\\sigma _m$ and $\\bar{\\sigma }_1,\\dots ,\\bar{\\sigma }_{\\bar{m}}$ satisfy a comparative query description based on formula satisfaction of section REF as follows: $&\\Big \\lbrace \\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_0,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_0,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\models \\; \\mathbf {~direction~of~change~in~} aggop \\mathbf {~rate~of~production~of~} f \\mathbf {~is~} d \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\mathbf {~when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ if } \\Big (\\Big \\lbrace \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace , j \\Big ) \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\models \\mathbf {~direction~of~change~in~} aggop \\mathbf {~rate~of~production~of~} f \\mathbf {~is~} d \\\\&\\Big \\lbrace \\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_0,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_0,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\models \\; \\mathbf {~direction~of~change~in~} aggop \\mathbf {~rate~of~firing~of~} f \\mathbf {~is~} d \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\mathbf {~when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ if } \\Big (\\Big \\lbrace \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace , j\\Big ) \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\models \\mathbf {~direction~of~change~in~} aggop \\mathbf {~rate~of~firing~of~} f \\mathbf {~is~} d \\\\&\\Big \\lbrace \\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_0,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_0,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\models \\; \\mathbf {~direction~of~change~in~} aggop \\mathbf {~total~production~of~} f \\mathbf {~is~} d \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\mathbf {~when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ if } \\Big (\\Big \\lbrace \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace , j\\Big ) \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\models \\mathbf {~direction~of~change~in~} aggop \\mathbf {~total~production~of~} f \\mathbf {~is~} d \\\\&\\Big \\lbrace \\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_0,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_0,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\models \\; \\mathbf {~direction~of~change~in~} aggop \\mathbf {~value~of~} f \\mathbf {~is~} d \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\mathbf {~when~observed~at~time~step~} i \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ if } \\Big \\lbrace \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\models \\mathbf {~direction~of~change~in~} aggop \\mathbf {~value~of~} f \\mathbf {~is~} d$ A set of trajectories $\\sigma _1,\\dots ,\\sigma _m$ , each of length $k$ satisfies a quantitative or a qualitative interval query description based on formula satisfaction of section REF as follows: $&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~is~} n \\nonumber \\\\&~~~~\\text{ if } \\exists x, 1 \\le x \\le m : (\\langle s^x_0,\\sigma _x \\rangle , k) \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~is~} n \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~rate~of~firing~of~} a \\mathbf {~is~} n \\nonumber \\\\&~~~~\\text{ if } \\exists x, 1 \\le x \\le m : (\\langle s^x_0,\\sigma _x \\rangle , k) \\models \\mathbf {~rate~of~firing~of~} a \\mathbf {~is~} n \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~total~production~of~} f \\mathbf {~is~} n \\nonumber \\\\&~~~~\\text{ if } \\exists x, 1 \\le x \\le m : (\\langle s^x_0,\\sigma _x \\rangle , k) \\models \\mathbf {~total~production~of~} f \\mathbf {~is~} n \\\\\\vspace{20.0pt}\\nonumber \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~is~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&~~~~\\text{ if } \\exists x, 1 \\le x \\le m : (\\langle s^x_i,\\sigma _x \\rangle , j) \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~is~} n \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~rate~of~firing~of~} a \\mathbf {~is~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&~~~~\\text{ if } \\exists x, 1 \\le x \\le m : (\\langle s^x_i,\\sigma _x \\rangle , j) \\models \\mathbf {~rate~of~firing~of~} a \\mathbf {~is~} n \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~total~production~of~} f \\mathbf {~is~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&~~~~\\text{ if } \\exists x, 1 \\le x \\le m : (\\langle s^x_i,\\sigma _x \\rangle , j) \\models \\mathbf {~total~production~of~} f \\mathbf {~is~} n \\\\\\vspace{20.0pt}\\nonumber \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models f \\mathbf {~is~accumulating~} \\nonumber \\\\&~~~~\\text{ if } \\exists x, 1 \\le x \\le m : (\\langle s^x_0,\\sigma _x \\rangle , k) \\models f \\mathbf {~is~accumulating~} \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models f \\mathbf {~is~decreasing~} \\nonumber \\\\&~~~~\\text{ if } \\exists x, 1 \\le x \\le m : (\\langle s^x_0,\\sigma _x \\rangle , k) \\models f \\mathbf {~is~decreasing~} \\\\\\vspace{20.0pt}\\nonumber \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models f \\mathbf {~is~accumulating~} \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&~~~~\\text{ if } \\exists x, 1 \\le x \\le m : (\\langle s^x_i,\\sigma _x \\rangle , j) \\models f \\mathbf {~is~accumulating~} \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models f \\mathbf {~is~decreasing~} \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&~~~~\\text{ if } \\exists x, 1 \\le x \\le m : (\\langle s^x_i,\\sigma _x \\rangle , j) \\models f \\mathbf {~is~decreasing~} \\\\\\vspace{20.0pt}\\nonumber \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~is~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~in~all~trajectories~} \\nonumber \\\\&~~~~\\text{ if } \\forall x, 1 \\le x \\le m : (\\langle s^x_0,\\sigma _x \\rangle , k) \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~is~} n \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~rate~of~firing~of~} a \\mathbf {~is~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~in~all~trajectories~} \\nonumber \\\\&~~~~\\text{ if } \\forall x, 1 \\le x \\le m : (\\langle s^x_0,\\sigma _x \\rangle , k) \\models \\mathbf {~rate~of~firing~of~} a \\mathbf {~is~} n \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~total~production~of~} f \\mathbf {~is~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~in~all~trajectories~} \\nonumber \\\\&~~~~\\text{ if } \\forall x, 1 \\le x \\le m : (\\langle s^x_0,\\sigma _x \\rangle , k) \\models \\mathbf {~total~production~of~} f \\mathbf {~is~} n \\\\\\vspace{20.0pt}\\nonumber \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~is~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~in~all~trajectories~} \\nonumber \\\\&~~~~\\text{ if } \\forall x, 1 \\le x \\le m : (\\langle s^x_i,\\sigma _x \\rangle , j) \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~is~} n \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~rate~of~firing~of~} a \\mathbf {~is~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~in~all~trajectories~} \\nonumber \\\\&~~~~\\text{ if } \\forall x, 1 \\le x \\le m : (\\langle s^x_i,\\sigma _x \\rangle , j) \\models \\mathbf {~rate~of~firing~of~} a \\mathbf {~is~} n \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~total~production~of~} f \\mathbf {~is~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~in~all~trajectories~} \\nonumber \\\\&~~~~\\text{ if } \\forall x, 1 \\le x \\le m : (\\langle s^x_i,\\sigma _x \\rangle , j) \\models \\mathbf {~total~production~of~} f \\mathbf {~is~} n \\\\\\vspace{20.0pt}\\nonumber \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models f \\mathbf {~is~accumulating~} \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~in~all~trajectories~} \\nonumber \\\\&~~~~\\text{ if } \\forall x, 1 \\le x \\le m : (\\langle s^x_0,\\sigma _x \\rangle , k) \\models f \\mathbf {~is~accumulating~} \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models f \\mathbf {~is~decreasing~} \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~in~all~trajectories~} \\nonumber \\\\&~~~~\\text{ if } \\forall x, 1 \\le x \\le m : (\\langle s^x_0,\\sigma _x \\rangle , k) \\models f \\mathbf {~is~decreasing~} \\\\\\vspace{20.0pt}\\nonumber \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models f \\mathbf {~is~accumulating~} \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~in~all~trajectories~} \\nonumber \\\\&~~~~\\text{ if } \\forall x, 1 \\le x \\le m : (\\langle s^x_i,\\sigma _x \\rangle , j) \\models f \\mathbf {~is~accumulating~} \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models f \\mathbf {~is~decreasing~} \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~in~all~trajectories~} \\nonumber \\\\&~~~~\\text{ if } \\forall x, 1 \\le x \\le m : (\\langle s^x_i,\\sigma _x \\rangle , j) \\models f \\mathbf {~is~decreasing~}$ A set of trajectories $\\sigma _1,\\dots ,\\sigma _m$ , each of length $k$ satisfies a quantitative or a qualitative point query description based on formula satisfaction of section REF as follows: $&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~value~of~} f \\mathbf {~is~higher~than~} n \\nonumber \\\\&~~~~\\text{ if } \\exists x \\exists i, 1 \\le x \\le m, 0 \\le i \\le k : \\langle s^x_i,\\sigma _x \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~is~higher~than~} n \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~value~of~} f \\mathbf {~is~lower~than~} n \\nonumber \\\\&~~~~\\text{ if } \\exists x \\exists i, 1 \\le x \\le m, 0 \\le i \\le k : \\langle s^x_i,\\sigma _x \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~is~lower~than~} n \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~value~of~} f \\mathbf {~is~} n \\nonumber \\\\&~~~~\\text{ if } \\exists x \\exists i, 1 \\le x \\le m, 0 \\le i \\le k : \\langle s^x_i,\\sigma _x \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~is~} n \\\\\\vspace{20.0pt}\\nonumber \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~value~of~} f \\mathbf {~is~higher~than~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~at~time~step~} i \\nonumber \\\\&~~~~\\text{ if } \\exists x, 1 \\le x \\le m : \\langle s^x_i,\\sigma _x \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~is~higher~than~} n \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~value~of~} f \\mathbf {~is~lower~than~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~at~time~step~} i \\nonumber \\\\&~~~~\\text{ if } \\exists x, 1 \\le x \\le m : \\langle s^x_i,\\sigma _x \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~is~lower~than~} n \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~value~of~} f \\mathbf {~is~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~at~time~step~} i \\nonumber \\\\&~~~~\\text{ if } \\exists x, 1 \\le x \\le m : \\langle s^x_i,\\sigma _x \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~is~} n \\\\\\vspace{20.0pt}\\nonumber \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~value~of~} f \\mathbf {~is~higher~than~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~in~all~trajectories~} \\nonumber \\\\&~~~~\\text{ if } \\forall x \\exists i, 1 \\le x \\le m, 0 \\le i \\le k : \\langle s^x_i,\\sigma _x \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~is~higher~than~} n \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~value~of~} f \\mathbf {~is~lower~than~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~in~all~trajectories~} \\nonumber \\\\&~~~~\\text{ if } \\forall x \\exists i, 1 \\le x \\le m, 0 \\le i \\le k : \\langle s^x_i,\\sigma _x \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~is~lower~than~} n \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~value~of~} f \\mathbf {~is~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~in~all~trajectories~} \\nonumber \\\\&~~~~\\text{ if } \\forall x \\exists i, 1 \\le x \\le m, 0 \\le i \\le k : \\langle s^x_i,\\sigma _x \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~is~} n \\\\\\vspace{20.0pt}\\nonumber \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~value~of~} f \\mathbf {~is~higher~than~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~at~time~step~} i \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~in~all~trajectories~} \\nonumber \\\\&~~~~\\text{ if } \\forall x, 1 \\le x \\le m : \\langle s^x_i,\\sigma _x \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~is~higher~than~} n \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~value~of~} f \\mathbf {~is~lower~than~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~at~time~step~} i \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~in~all~trajectories~} \\nonumber \\\\&~~~~\\text{ if } \\forall x, 1 \\le x \\le m : \\langle s^x_i,\\sigma _x \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~is~lower~than~} n \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~value~of~} f \\mathbf {~is~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~at~time~step~} i \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~in~all~trajectories~} \\nonumber \\\\&~~~~\\text{ if } \\forall x, 1 \\le x \\le m : \\langle s^x_i,\\sigma _x \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~is~} n$ Now, we turn our attention to domain descriptions with locational fluents.", "Let $\\mathbf {D}$ be a domain description and $\\sigma = s_0,\\dots ,s_k$ be its trajectory of length $k$ as defined in (REF ).", "Let $\\sigma _1,\\dots ,\\sigma _m$ represent the set of trajectories of $\\mathbf {D}$ filtered by observations as necessary, with each trajectory has the form $\\sigma _i = s^i_0,\\dots ,s^i_k$ $1\\le i \\le m$ .", "Let $\\mathbf {\\bar{D}}$ be a modified domain description and $\\bar{\\sigma }_1,\\dots ,\\bar{\\sigma }_{\\bar{m}}$ be its trajectories of the form $\\bar{\\sigma }_i = \\bar{s}^i_0,\\dots ,\\bar{s}^i_k$ .", "Then we define query satisfaction using the formula satisfaction in section REF as follows.", "Two sets of trajectories $\\sigma _1,\\dots ,\\sigma _m$ and $\\bar{\\sigma }_1,\\dots ,\\bar{\\sigma }_{\\bar{m}}$ satisfy a comparative query description based on formula satisfaction of section REF as follows: $&\\Big \\lbrace \\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_0,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_0,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\models \\; \\mathbf {~direction~of~change~in~} \\langle \\text{aggop} \\rangle \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} d \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\mathbf {~when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ if } \\Big (\\Big \\lbrace \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace , j \\Big ) \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\models \\mathbf {~direction~of~change~in~} \\langle \\text{aggop} \\rangle \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} d \\\\&\\Big \\lbrace \\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_0,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_0,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\models \\; \\mathbf {~direction~of~change~in~} \\langle \\text{aggop} \\rangle \\mathbf {~rate~of~firing~of~} f \\mathbf {~is~} d \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\mathbf {~when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ if } \\Big (\\Big \\lbrace \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace , j\\Big ) \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\models \\mathbf {~direction~of~change~in~} \\langle \\text{aggop} \\rangle \\mathbf {~rate~of~firing~of~} f \\mathbf {~is~} d \\\\&\\Big \\lbrace \\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_0,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_0,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\models \\; \\mathbf {~direction~of~change~in~} \\langle \\text{aggop} \\rangle \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} d \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\mathbf {~when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ if } \\Big (\\Big \\lbrace \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace , j\\Big ) \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\models \\mathbf {~direction~of~change~in~} \\langle \\text{aggop} \\rangle \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} d \\\\&\\Big \\lbrace \\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_0,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_0,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\models \\; \\mathbf {~direction~of~change~in~} \\langle \\text{aggop} \\rangle \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} d \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\mathbf {~when~observed~at~time~step~} i \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\text{ if } \\Big \\lbrace \\lbrace \\langle s^1_i,\\sigma _1 \\rangle ,\\dots , \\langle s^m_i,\\sigma _m \\rangle \\rbrace , \\lbrace \\langle \\bar{s}^1_i,\\bar{\\sigma }_1 \\rangle ,\\dots , \\langle \\bar{s}^{\\bar{m}}_i,\\bar{\\sigma }_{\\bar{m}} \\rangle \\rbrace \\Big \\rbrace \\nonumber \\\\&\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\models \\mathbf {~direction~of~change~in~} \\langle \\text{aggop} \\rangle \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} d$ A set of trajectories $\\sigma _1,\\dots ,\\sigma _m$ , each of length $k$ satisfies a quantitative or a qualitative interval query description based on formula satisfaction of section REF as follows: $&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n \\nonumber \\\\&~~~~\\text{ if } \\exists x, 1 \\le x \\le m : (\\langle s^x_0,\\sigma _x \\rangle , k) \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~rate~of~firing~of~} a \\mathbf {~is~} n \\nonumber \\\\&~~~~\\text{ if } \\exists x, 1 \\le x \\le m : (\\langle s^x_0,\\sigma _x \\rangle , k) \\models \\mathbf {~rate~of~firing~of~} a \\mathbf {~is~} n \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n \\nonumber \\\\&~~~~\\text{ if } \\exists x, 1 \\le x \\le m : (\\langle s^x_0,\\sigma _x \\rangle , k) \\models \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n \\\\\\vspace{20.0pt}\\nonumber \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&~~~~\\text{ if } \\exists x, 1 \\le x \\le m : (\\langle s^x_i,\\sigma _x \\rangle , j) \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~rate~of~firing~of~} a \\mathbf {~is~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&~~~~\\text{ if } \\exists x, 1 \\le x \\le m : (\\langle s^x_i,\\sigma _x \\rangle , j) \\models \\mathbf {~rate~of~firing~of~} a \\mathbf {~is~} n \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&~~~~\\text{ if } \\exists x, 1 \\le x \\le m : (\\langle s^x_i,\\sigma _x \\rangle , j) \\models \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n \\\\\\vspace{20.0pt}\\nonumber \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models f \\mathbf {~is~accumulating~} \\mathbf {~atloc~} l \\nonumber \\\\&~~~~\\text{ if } \\exists x, 1 \\le x \\le m : (\\langle s^x_0,\\sigma _x \\rangle , k) \\models f \\mathbf {~is~accumulating~} \\mathbf {~atloc~} l \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models f \\mathbf {~is~decreasing~} \\mathbf {~atloc~} l \\nonumber \\\\&~~~~\\text{ if } \\exists x, 1 \\le x \\le m : (\\langle s^x_0,\\sigma _x \\rangle , k) \\models f \\mathbf {~is~decreasing~} \\mathbf {~atloc~} l \\\\\\vspace{20.0pt}\\nonumber \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models f \\mathbf {~is~accumulating~} \\mathbf {~atloc~} l \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&~~~~\\text{ if } \\exists x, 1 \\le x \\le m : (\\langle s^x_i,\\sigma _x \\rangle , j) \\models f \\mathbf {~is~accumulating~} \\mathbf {~atloc~} l \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models f \\mathbf {~is~decreasing~} \\mathbf {~atloc~} l \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&~~~~\\text{ if } \\exists x, 1 \\le x \\le m : (\\langle s^x_i,\\sigma _x \\rangle , j) \\models f \\mathbf {~is~decreasing~} \\mathbf {~atloc~} l \\\\\\vspace{30.0pt} \\nonumber \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~in~all~trajectories~} \\nonumber \\\\&~~~~\\text{ if } \\forall x, 1 \\le x \\le m : (\\langle s^x_0,\\sigma _x \\rangle , k) \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~rate~of~firing~of~} a \\mathbf {~is~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~in~all~trajectories~} \\nonumber \\\\&~~~~\\text{ if } \\forall x, 1 \\le x \\le m : (\\langle s^x_0,\\sigma _x \\rangle , k) \\models \\mathbf {~rate~of~firing~of~} a \\mathbf {~is~} n \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~in~all~trajectories~} \\nonumber \\\\&~~~~\\text{ if } \\forall x, 1 \\le x \\le m : (\\langle s^x_0,\\sigma _x \\rangle , k) \\models \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n \\\\\\vspace{20.0pt}\\nonumber \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~in~all~trajectories~} \\nonumber \\\\&~~~~\\text{ if } \\forall x, 1 \\le x \\le m : (\\langle s^x_i,\\sigma _x \\rangle , j) \\models \\mathbf {~rate~of~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~rate~of~firing~of~} a \\mathbf {~is~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~in~all~trajectories~} \\nonumber \\\\&~~~~\\text{ if } \\forall x, 1 \\le x \\le m : (\\langle s^x_i,\\sigma _x \\rangle , j) \\models \\mathbf {~rate~of~firing~of~} a \\mathbf {~is~} n \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~in~all~trajectories~} \\nonumber \\\\&~~~~\\text{ if } \\forall x, 1 \\le x \\le m : (\\langle s^x_i,\\sigma _x \\rangle , j) \\models \\mathbf {~total~production~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n \\text{ till } j\\\\\\vspace{20.0pt}\\nonumber \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models f \\mathbf {~is~accumulating~} \\mathbf {~atloc~} l \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~in~all~trajectories~} \\nonumber \\\\&~~~~\\text{ if } \\forall x, 1 \\le x \\le m : (\\langle s^x_0,\\sigma _x \\rangle , k) \\models f \\mathbf {~is~accumulating~} \\mathbf {~atloc~} l \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models f \\mathbf {~is~decreasing~} \\mathbf {~atloc~} l \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~in~all~trajectories~} \\nonumber \\\\&~~~~\\text{ if } \\forall x, 1 \\le x \\le m : (\\langle s^x_0,\\sigma _x \\rangle , k) \\models f \\mathbf {~is~decreasing~} \\mathbf {~atloc~} l \\\\\\vspace{20.0pt}\\nonumber \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models f \\mathbf {~is~accumulating~} \\mathbf {~atloc~} l \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~in~all~trajectories~} \\nonumber \\\\&~~~~\\text{ if } \\forall x, 1 \\le x \\le m : (\\langle s^x_i,\\sigma _x \\rangle , j) \\models f \\mathbf {~is~accumulating~} \\mathbf {~atloc~} l \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models f \\mathbf {~is~decreasing~} \\mathbf {~atloc~} l \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~when~observed~between~time~step~} i \\mathbf {~and~time~step~} j \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~in~all~trajectories~} \\nonumber \\\\&~~~~\\text{ if } \\forall x, 1 \\le x \\le m : (\\langle s^x_i,\\sigma _x \\rangle , j) \\models f \\mathbf {~is~decreasing~} \\mathbf {~atloc~} l$ A set of trajectories $\\sigma _1,\\dots ,\\sigma _m$ , each of length $k$ satisfies a quantitative or a qualitative point query description based on formula satisfaction of section REF as follows: $&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l\\mathbf {~is~higher~than~} n \\nonumber \\\\&~~~~\\text{ if } \\exists x \\exists i, 1 \\le x \\le m, 0 \\le i \\le k : \\langle s^x_i,\\sigma _x \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~higher~than~} n \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l\\mathbf {~is~lower~than~} n \\nonumber \\\\&~~~~\\text{ if } \\exists x \\exists i, 1 \\le x \\le m, 0 \\le i \\le k : \\langle s^x_i,\\sigma _x \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~lower~than~} n \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l\\mathbf {~is~} n \\nonumber \\\\&~~~~\\text{ if } \\exists x \\exists i, 1 \\le x \\le m, 0 \\le i \\le k : \\langle s^x_i,\\sigma _x \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n\\\\\\vspace{20.0pt}\\nonumber \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l\\mathbf {~is~higher~than~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~at~time~step~} i \\nonumber \\\\&~~~~\\text{ if } \\exists x, 1 \\le x \\le m : \\langle s^x_i,\\sigma _x \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~higher~than~} n \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l\\mathbf {~is~lower~than~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~at~time~step~} i \\nonumber \\\\&~~~~\\text{ if } \\exists x, 1 \\le x \\le m : \\langle s^x_i,\\sigma _x \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~lower~than~} n \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l\\mathbf {~is~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~at~time~step~} i \\nonumber \\\\&~~~~\\text{ if } \\exists x, 1 \\le x \\le m : \\langle s^x_i,\\sigma _x \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n\\\\\\vspace{20.0pt}\\nonumber \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l\\mathbf {~is~higher~than~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~in~all~trajectories~} \\nonumber \\\\&~~~~\\text{ if } \\forall x \\exists i, 1 \\le x \\le m, 0 \\le i \\le k : \\langle s^x_i,\\sigma _x \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~higher~than~} n \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~lower~than~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~in~all~trajectories~} \\nonumber \\\\&~~~~\\text{ if } \\forall x \\exists i, 1 \\le x \\le m, 0 \\le i \\le k : \\langle s^x_i,\\sigma _x \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~lower~than~} n \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l\\mathbf {~is~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~in~all~trajectories~} \\nonumber \\\\&~~~~\\text{ if } \\forall x \\exists i, 1 \\le x \\le m, 0 \\le i \\le k : \\langle s^x_i,\\sigma _x \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n\\\\\\vspace{20.0pt}\\nonumber \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~higher~than~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~at~time~step~} i \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~in~all~trajectories~} \\nonumber \\\\&~~~~\\text{ if } \\forall x, 1 \\le x \\le m : \\langle s^x_i,\\sigma _x \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~higher~than~} n \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l\\mathbf {~is~lower~than~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~at~time~step~} i \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~in~all~trajectories~} \\nonumber \\\\&~~~~\\text{ if } \\forall x, 1 \\le x \\le m : \\langle s^x_i,\\sigma _x \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~lower~than~} n \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l\\mathbf {~is~} n \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~at~time~step~} i \\nonumber \\\\&\\hspace{30.0pt}\\mathbf {~in~all~trajectories~} \\nonumber \\\\&~~~~\\text{ if } \\forall x, 1 \\le x \\le m : \\langle s^x_i,\\sigma _x \\rangle \\models \\mathbf {~value~of~} f \\mathbf {~atloc~} l \\mathbf {~is~} n$ Next, we generically define the satisfaction of a simple point formula cascade query w.r.t.", "a set of trajectories $\\sigma _1,\\dots ,\\sigma _m$ .", "The trajectories will either be as defined in definitions (REF ) or (REF ) for simple point formula cascade query statement made up of simple fluents or locational fluents, respectively.", "A set of trajectories $\\sigma _1,\\dots ,\\sigma _m$ , each of length $k$ satisfies a simple point formula cascade based on formula satisfaction of section REF as follows: $&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\langle \\text{simple point formula} \\rangle _0 \\nonumber \\\\&~~~~~~~~\\mathbf {~after~} \\langle \\text{simple point formula} \\rangle _{1,1}, \\dots , \\langle \\text{simple point formula} \\rangle _{1,n_1} \\nonumber \\\\&~~~~~~~~\\vdots \\nonumber \\\\&~~~~~~~~\\mathbf {~after~} \\langle \\text{simple point formula} \\rangle _{u,1}, \\dots , \\langle \\text{simple point formula} \\rangle _{u,n_u} \\nonumber \\\\&~~~~\\text{ if } \\exists x \\exists i_0 \\exists i_1 \\dots \\exists i_u, 1 \\le x \\le m, i_1 < i_0 \\le k, \\dots , i_u < i_{u-1} \\le k, 0 \\le i_u \\le k: \\nonumber \\\\&~~~~~~~~\\langle s^x_{i_0},\\sigma _x \\rangle \\models \\langle \\text{simple point formula} \\rangle _0 \\text{ and } \\nonumber \\\\&~~~~~~~~\\langle s^x_{i_1},\\sigma _x \\rangle \\models \\langle \\text{simple point formula} \\rangle _{1,1} \\text{ and } \\nonumber \\hdots \\text{ and } \\langle s^x_{i_1},\\sigma _x \\rangle \\models \\langle \\text{simple point formula} \\rangle _{1,n_1} \\text{ and } \\nonumber \\\\&~~~~~~~~~~~~\\vdots \\nonumber \\\\&~~~~~~~~\\langle s^x_{i_u},\\sigma _x \\rangle \\models \\langle \\text{simple point formula} \\rangle _{u,1} \\text{ and } \\nonumber \\hdots \\text{ and } \\langle s^x_{i_u},\\sigma _x \\rangle \\models \\langle \\text{simple point formula} \\rangle _{u,n_u} \\\\\\\\\\vspace{10.0pt}\\nonumber \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\langle \\text{simple point formula} \\rangle _0 \\nonumber \\\\&~~~~~~~~\\mathbf {~after~} \\langle \\text{simple point formula} \\rangle _{1,1}, \\dots , \\langle \\text{simple point formula} \\rangle _{1,n_1} \\nonumber \\\\&~~~~~~~~\\vdots \\nonumber \\\\&~~~~~~~~\\mathbf {~after~} \\langle \\text{simple point formula} \\rangle _{u,1}, \\dots , \\langle \\text{simple point formula} \\rangle _{u,n_u} \\nonumber \\\\&~~~~~~~~\\mathbf {~in~all~trajectories} \\nonumber \\\\&~~~~\\text{ if } \\forall x, 1 \\le x \\le m, \\exists i_0 \\exists i_1 \\dots \\exists i_u, i_1 < i_0 \\le k, \\dots , i_u < i_{u-1} \\le k, 0 \\le i_u \\le k: \\nonumber \\\\&~~~~~~~~\\langle s^x_{i_0},\\sigma _x \\rangle \\models \\langle \\text{simple point formula} \\rangle _0 \\text{ and } \\nonumber \\\\&~~~~~~~~\\langle s^x_{i_1},\\sigma _x \\rangle \\models \\langle \\text{simple point formula} \\rangle _{1,1} \\text{ and } \\nonumber \\hdots \\text{ and } \\langle s^x_{i_1},\\sigma _x \\rangle \\models \\langle \\text{simple point formula} \\rangle _{1,n_1} \\text{ and } \\nonumber \\\\&~~~~~~~~~~~~\\vdots \\nonumber \\\\&~~~~~~~~\\langle s^x_{i_u},\\sigma _x \\rangle \\models \\langle \\text{simple point formula} \\rangle _{u,1} \\text{ and } \\nonumber \\hdots \\text{ and } \\langle s^x_{i_u},\\sigma _x \\rangle \\models \\langle \\text{simple point formula} \\rangle _{u,n_u} \\\\\\\\\\vspace{10.0pt}\\nonumber \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\langle \\text{simple point formula} \\rangle _0 \\nonumber \\\\&~~~~~~~~\\mathbf {~when~} \\langle \\text{simple point formula} \\rangle _{1,1}, \\dots , \\langle \\text{simple point formula} \\rangle _{1,n_1} \\nonumber \\\\&~~~~\\text{ if } \\exists x \\exists i, 1 \\le x \\le m, 0 \\le i \\le k: \\nonumber \\\\&~~~~~~~~\\langle s^x_{i_0},\\sigma _x \\rangle \\models \\langle \\text{simple point formula} \\rangle _0 \\text{ and } \\nonumber \\\\&~~~~~~~~\\langle s^x_{i_1},\\sigma _x \\rangle \\models \\langle \\text{simple point formula} \\rangle _{1,1} \\text{ and } \\nonumber \\hdots \\text{ and } \\langle s^x_{i_1},\\sigma _x \\rangle \\models \\langle \\text{simple point formula} \\rangle _{1,n_1} \\\\\\\\\\vspace{10.0pt}\\nonumber \\\\&\\lbrace \\langle s^1_0,\\sigma _1 \\rangle ,\\dots , \\langle s^m_0,\\sigma _m \\rangle \\rbrace \\models \\langle \\text{simple point formula} \\rangle _0 \\nonumber \\\\&~~~~~~~~\\mathbf {~when~} \\langle \\text{simple point formula} \\rangle _{1,1}, \\dots , \\langle \\text{simple point formula} \\rangle _{1,n_1} \\nonumber \\\\&~~~~~~~~\\mathbf {~in~all~trajectories} \\nonumber \\\\&~~~~\\text{ if } \\forall x, 1 \\le x \\le m, \\exists i, 0 \\le i \\le k: \\nonumber \\\\&~~~~~~~~\\langle s^x_{i_0},\\sigma _x \\rangle \\models \\langle \\text{simple point formula} \\rangle _0 \\text{ and } \\nonumber \\\\&~~~~~~~~\\langle s^x_{i_1},\\sigma _x \\rangle \\models \\langle \\text{simple point formula} \\rangle _{1,1} \\text{ and } \\nonumber \\hdots \\text{ and } \\langle s^x_{i_1},\\sigma _x \\rangle \\models \\langle \\text{simple point formula} \\rangle _{1,n_1} \\\\$ Query Statement Satisfaction Let $\\mathbf {D}$ as defined in section REF be a domain description and $\\mathbf {Q}$ be a query statement (REF ) as defined in section REF with query description $U$ , interventions $V_1,\\dots ,V_{\|V\|}$ , internal observations $O_1,\\dots ,O_{\|O\|}$ , and initial conditions $I_1,\\dots ,I_{\|I\|}$ .", "Let $\\mathbf {D_1} \\equiv \\mathbf {D} \\diamond I_1 \\diamond \\dots \\diamond I_{\|I\|} \\diamond V_1 \\diamond \\dots \\diamond V_{\|V\|}$ be the modified domain description constructed by applying the initial conditions and interventions from $\\mathbf {Q}$ as defined in section REF .", "Let $\\sigma _1,\\dots ,\\sigma _m$ be the trajectories of $\\mathbf {D_1}$ that satisfy $O_1,\\dots ,O_{\|O\|}$ as given in section REF .", "Then, $\\mathbf {D}$ satisfies $\\mathbf {Q}$ if $\\lbrace \\sigma _1,\\dots ,\\sigma _m\\rbrace \\models U$ as defined in section REF .", "Let $\\mathbf {D}$ as defined in section REF be a domain description and $\\mathbf {Q}$ be a query statement () as defined in section REF with query description $U$ , interventions $V_1,\\dots ,V_{\|V\|}$ , internal observations $O_1,\\dots ,O_{\|O\|}$ , and initial conditions $I_1,\\dots ,I_{\|I\|}$ .", "Let $\\mathbf {D_)} \\equiv \\mathbf {D} \\diamond I_1 \\diamond \\dots \\diamond I_{\|I\|} $ be the nominal domain description constructed by applying the initial conditions from $\\mathbf {Q}$ as defined in section REF .", "Let $\\sigma _1,\\dots ,\\sigma _m$ be the trajectories of $\\mathbf {D_0}$ .", "Let $\\mathbf {D_1} \\equiv \\mathbf {D} \\diamond I_1 \\diamond \\dots \\diamond I_{\|I\|} \\diamond V_1 \\diamond \\dots \\diamond V_{\|V\|}$ be the modified domain description constructed by applying the initial conditions and interventions from $\\mathbf {Q}$ as defined in section REF .", "Let $\\bar{\\sigma }_1,\\dots ,\\bar{\\sigma }_{\\bar{m}}$ be the trajectories of $\\mathbf {D_1}$ that satisfy $O_1,\\dots ,O_{\|O\|}$ as given in section REF .", "Then, $\\mathbf {D}$ satisfies $\\mathbf {Q}$ if $\\Big \\lbrace \\lbrace \\sigma _1,\\dots ,\\sigma _m\\rbrace , \\lbrace \\bar{\\sigma }_1,\\dots ,\\bar{\\sigma }_{\\bar{m}}\\rbrace \\Big \\rbrace \\models U$ as defined in section REF .", "Example Encodings In this section we give some examples of how we will encode queries and pathways related to these queries.", "We will also show how the pathway is modified to answer questions Some of the same pathways appear in previous chapters, they have been updated here with additional background knowledge..", "Question 11 At one point in the process of glycolysis, both DHAP and G3P are produced.", "Isomerase catalyzes the reversible conversion between these two isomers.", "The conversion of DHAP to G3P never reaches equilibrium and G3P is used in the next step of glycolysis.", "What would happen to the rate of glycolysis if DHAP were removed from the process of glycolysis as quickly as it was produced?", "Figure: Petri Net for question The question is asking for the direction of change in the rate of glycolysis when the nominal glycolysis pathway is compared against a modified pathway in which dhap is removed as soon as it is produced.", "Since this rate can vary with the trajectory followed by the world evolution, we consider the average change in rate.", "From the domain knowledge [64] we know that the rate of $glycolysis$ can be measured by the rate of $pyruvate$ (the end product of glycolysis) and that the rate of $pyruvate$ is equal to the rate of $bpg13$ (due to linear chain from $bpg13$ to $pyruvate$ ).", "Thus, we can monitor the rate of $bpg13$ instead to determine the rate of glycolysis.", "To ensure that our pathway is not starved due to source ingredients, we add a continuous supply of $f16bp$ in quantity 1 to the pathway.", "Then, the following pathway specification encodes the domain description $\\mathbf {D}$ for question REF and produces the PN in Fig.", "REF minus the $tr,t3$ transitions: $&\\begin{array}{llll}&\\mathbf {domain~of~} & f16bp \\mathbf {~is~} integer, & dhap \\mathbf {~is~} integer, \\\\&& g3p \\mathbf {~is~} integer, & bpg13 \\mathbf {~is~} integer\\nonumber \\\\&t4 \\mathbf {~may~execute~causing~} & f16bp \\mathbf {~change~value~by~} -1, & dhap \\mathbf {~change~value~by~} +1,\\\\ && g3p \\mathbf {~change~value~by~} +1\\nonumber \\\\&t5a \\mathbf {~may~execute~causing~} & dhap \\mathbf {~change~value~by~} -1, & g3p \\mathbf {~change~value~by~} +1\\nonumber \\\\&t5b \\mathbf {~may~execute~causing~} & g3p \\mathbf {~change~value~by~} -1, & dhap \\mathbf {~change~value~by~} +1\\nonumber \\\\&t6 \\mathbf {~may~execute~causing~} & g3p \\mathbf {~change~value~by~} -1, & bpg13 \\mathbf {~change~value~by~} +2\\nonumber \\\\& \\mathbf {initially~} & f16bp \\mathbf {~has~value~} 0, & dhap \\mathbf {~has~value~} 0,\\\\ && g3p \\mathbf {~has~value~} 0, & bpg13 \\mathbf {~has~value~} 0\\nonumber \\\\&\\mathbf {firing~style~} & max\\end{array}\\\\$ And the following query $\\mathbf {Q}$ for a simulation of length $k$ encodes the question: $&\\mathbf {direction~of~change~in~} average \\mathbf {~rate~of~production~of~} bpg13 \\mathbf {~is~} d\\nonumber \\\\&~~~~~~\\mathbf {when~observed~between~time~step~} 0 \\mathbf {~and~time~step~} k;\\nonumber \\\\&~~~~~~\\mathbf {comparing~nominal~pathway~with~modified~pathway~obtained~}\\nonumber \\\\&~~~~~~~~~~~~\\mathbf {due~to~interventions:} \\mathbf {~remove~} dhap \\mathbf {~as~soon~as~produced};\\nonumber \\\\&~~~~~~\\mathbf {using~initial~setup:~} \\mathbf {continuously~supply~} f16bp \\mathbf {~in~quantity~} 1;$ Since this is a comparative quantitative query statement, it is decomposed into two sub-queries, $\\mathbf {Q_0}$ capturing the nominal case of average rate of production w.r.t.", "given initial conditions: $\\begin{array}{lll}average \\mathbf {~rate~of~production~of~} bpg13 \\mathbf {~is~} n_{avg}\\\\~~~~~~~~~~~~\\mathbf {when~observed~between~time~step~} 0 \\mathbf {~and~time~step~} k; \\\\~~~~~~\\mathbf {using~initial~setup:~} \\mathbf {continuously~supply~} f16bp \\mathbf {~in~quantity~} 1;\\end{array}$ and $\\mathbf {Q_1}$ capturing the modified case in which the pathway is subject to interventions and observations w.r.t.", "initial conditions: $\\begin{array}{lll}average \\mathbf {~rate~of~production~of~} bpg13 \\mathbf {~is~} n^{\\prime }_{avg}\\\\~~~~~~~~~~~~\\mathbf {when~observed~between~time~step~} 0 \\mathbf {~and~time~step~} k; \\\\~~~~~~\\mathbf {due~to~interventions:} \\mathbf {~remove~} dhap \\mathbf {~as~soon~as~produced} ;\\\\~~~~~~\\mathbf {using~initial~setup:~} \\mathbf {continuously~supply~} f16bp \\mathbf {~in~quantity~} 1;\\end{array}$ Then the task is to determine $d$ , such that $\\mathbf {D} \\models \\mathbf {Q_0}$ for some value of $n_{avg}$ , $\\mathbf {D} \\models \\mathbf {Q_1}$ for some value of $n^{\\prime }_{avg}$ , and $n^{\\prime }_{avg} \\; d \\; n_{avg}$ .", "To answer the sub-queries $\\mathbf {Q_0}$ and $\\mathbf {Q_1}$ we build modified domain descriptions $\\mathbf {D_0}$ and $\\mathbf {D_1}$ , where, $\\mathbf {D_0} \\equiv \\mathbf {D} \\diamond (\\mathbf {continuously~supply~} f16bp $ $\\mathbf {~in~quantity~} 1)$ is the nominal domain description $\\mathbf {D}$ modified according to $\\mathbf {Q_0}$ to include the initial conditions; and $\\mathbf {D_1} \\equiv \\mathbf {D_0} \\diamond (\\mathbf {remove~} dhap $ $\\mathbf {~as~soon~as~}$ $\\mathbf {produced})$ is the modified domain description $\\mathbf {D}$ modified according to $\\mathbf {Q_1}$ to include the initial conditions as well as the interventions.", "The $\\diamond $ operator modifies the domain description to its left by adding, removing or modifying pathway specification language statements to add the interventions and initial conditions description to its right.", "Thus, $\\mathbf {D_0} &= \\mathbf {D} + \\left\\lbrace \\begin{array}{llll}tf_{f16bp} \\mathbf {~may~execute~causing~} &f16bp \\mathbf {~change~value~by~} +1\\\\\\end{array}\\right.\\\\\\mathbf {D_1} &= \\mathbf {D_0} + \\left\\lbrace \\begin{array}{llll}tr \\mathbf {~may~execute~causing~} &dhap \\mathbf {~change~value~by~} \\\\\\end{array}\\right.$ Performing a simulation of $k=5$ steps with $ntok=20$ max tokens, we find that the average rate of $bpg13$ production decreases from $n_{avg}=0.83$ to $n^{\\prime }_{avg}=0.5$ .", "Thus, $\\mathbf {D} \\models \\mathbf {Q}$ iff $d = ^{\\prime }<^{\\prime }$ .", "Alternatively, we say that the rate of glycolysis decreases when DHAP is removed as quickly as it is produced.", "Question 12 When and how does the body switch to B-oxidation versus glycolysis as the major way of burning fuel?", "Figure: Petri Net for question The following pathway specification encodes the domain description $\\mathbf {D}$ for question REF and produces the PN in Fig.", "REF : $\\begin{array}{llll}&\\mathbf {domain~of~} & gly \\mathbf {~is~} integer, & sug \\mathbf {~is~} integer, \\\\ && fac \\mathbf {~is~} integer, & acoa \\mathbf {~is~} integer \\\\&gly \\mathbf {~may~execute~causing~} & sug \\mathbf {~change~value~by~} -1, & acoa \\mathbf {~change~value~by~} +1\\\\&box \\mathbf {~may~execute~causing~} & fac \\mathbf {~change~value~by~} -1, & acoa \\mathbf {~change~value~by~} +1\\\\&\\mathbf {inhibit~} box \\mathbf {~if~} & sug \\mathbf {~has~value~} 1 \\mathbf {~or~higher}\\\\&\\mathbf {initially~} & sug \\mathbf {~has~value~} 3, & fac \\mathbf {~has~value~} 3\\\\ && acoa \\mathbf {~has~value~} 0 \\\\&t1 \\mathbf {~may~execute~causing~} & sug \\mathbf {~change~value~by~} +1\\\\&t2 \\mathbf {~may~execute~causing~} & fac \\mathbf {~change~value~by~} +1\\\\& \\mathbf {firing~style~} & \\end{array}$ where, $fac$ represents fatty acids, $sug$ represents sugar, $acoa$ represents acetyl coenzyme-A, $gly$ represents the process of glycolysis, and $box$ represents the process of beta oxidation.", "The question is asking for the general conditions when glycolysis switches to beta-oxidation, which is some property “$p$ ” that holds after which the switch occurs.", "The query $\\mathbf {Q}$ is encoded as: $&gly \\mathbf {~switches~to~} box \\mathbf {~when~} p; \\nonumber \\\\&~~~~\\mathbf {due~to~observations:} \\nonumber \\\\&~~~~~~~~gly \\mathbf {~switches~to~} box \\nonumber \\\\&~~~~\\mathbf {~in~all~trajectories}$ where condition `$p$ ' is a conjunction of simple point formulas.", "Then the task is to determine a minimal such conjunction of formulas that is satisfied in the state where $`gly^{\\prime } \\mathbf {~switches~to~} `box^{\\prime }$ holds over all trajectories.Note that this could be an LTL formula that must hold in all trajectories, but we did not add it here to keep the language simple.", "Since there is no change in initial conditions of the pathway and there are no interventions, the modified domain description $\\mathbf {D_1} \\equiv \\mathbf {D}$ .", "Intuitively, $p$ is the property that holds over fluents of the transitional state $s_j$ in which the switch takes palce, such that $gly \\in T_{j-1}, box \\notin T_{j-1}, gly \\notin T_j, box \\in T_j$ and the minimal set of firings leading up to it.", "The only trajectories to consider are the ones in which the observation is true.", "Thus the condition $p$ is determined as the intersection of sets of fluent based conditions that were true at the time of the switch, such as: $\\lbrace &sug \\mathbf {~has~value~} s_j(sug), sug \\mathbf {~has~value~higher~than~} 0, \\dots \\\\&sug \\mathbf {~has~value~higher~than~} s_j(sug)-1,sug \\mathbf {~has~value~lower~than~} s_j(sug)+1,\\\\&fac \\mathbf {~has~value~} s_j(fac), fac \\mathbf {~has~value~higher~than~} 0, \\dots \\\\&fac \\mathbf {~has~value~higher~than~} s_j(fac)-1, fac \\mathbf {~has~value~lower~than~} s_j(fac)+1, \\\\&acoa \\mathbf {~has~value~} s_j(acoa), acoa \\mathbf {~has~value~higher~than~} 0, \\dots \\\\&acoa \\mathbf {~has~value~higher~than~} s_j(acoa)-1, acoa \\mathbf {~has~value~lower~than~} s_j(acoa)+1\\rbrace $ Simulating it for $k=5$ steps with $ntok=20$ max tokens, we find the condition $p = acoa \\text{ has value greater than } 0, $ $sug \\text{ has value } 0, $ $sug \\text{ has value lower than } 1, $ $fac \\text{ has value higher than } 0 \\rbrace $ .", "Thus, the state when this switch occurs must sugar ($sug$ ) depleted and available supply of fatty acids ($fac$ ).", "Question 13 The final protein complex in the electron transport chain of the mitochondria is non-functional.", "Explain the effect of this on pH of the intermembrane space of the mitochondria.", "Figure: Petri Net for question The question is asking for the direction of change in the pH of the intermembrane space when the nominal case is compared against a modified pathway in which the complex 4 ($t4$ ) is defective.", "Since pH is defined as $-log_{10}(H+)$ , we monitor the total production of $H+$ ions to determine the change in pH value.", "However, since different world evolutions can follow different trajectories, we consider the average production of H+.", "Furthermore, we model the defective $t4$ as being unable to carry out its reaction, by disabling/inhibiting it.", "Then the following pathway specification encodes the domain description $\\mathbf {D}$ for question REF and produces the PN in Fig.", "REF without the $ft4$ place node.", "In the pathway description below, we have included only one domain declaration for all fluents as an example and left the rest out to save space.", "Assume integer domain declaration for all fluents.", "$&\\mathbf {domain~of~} \\\\&~~~~ nadh \\mathbf {~atloc~} mm \\mathbf {~is~} integer \\\\&t1 \\mathbf {~may~execute~causing~} \\\\&~~~~ nadh \\mathbf {~atloc~} mm \\mathbf {~change~value~by~} -2, h \\mathbf {~atloc~} mm \\mathbf {~change~value~by~} -2,\\\\&~~~~ e \\mathbf {~atloc~} q \\mathbf {~change~value~by~} +2, h \\mathbf {~atloc~} is \\mathbf {~change~value~by~} +2, \\\\&~~~~ nadp \\mathbf {~atloc~} mm \\mathbf {~change~value~by~} +2\\\\&t2 \\mathbf {~may~execute~causing~} \\\\&~~~~ fadh2 \\mathbf {~atloc~} mm \\mathbf {~change~value~by~} -2, e \\mathbf {~atloc~} q \\mathbf {~change~value~by~} +2,\\\\&~~~~ fad \\mathbf {~atloc~} mm \\mathbf {~change~value~by~} +2\\\\&t3 \\mathbf {~may~execute~causing~} \\\\&~~~~ e \\mathbf {~atloc~} q \\mathbf {~change~value~by~} -2, h \\mathbf {~atloc~} mm \\mathbf {~change~value~by~} -2,\\\\&~~~~ e \\mathbf {~atloc~} cytc \\mathbf {~change~value~by~} +2, h \\mathbf {~atloc~} is \\mathbf {~change~value~by~} +2\\\\&t4 \\mathbf {~may~execute~causing~} \\\\&~~~~ o2 \\mathbf {~atloc~} mm \\mathbf {~change~value~by~} -1, e \\mathbf {~atloc~} cytc \\mathbf {~change~value~by~} -2,\\\\&~~~~ h \\mathbf {~atloc~} mm \\mathbf {~change~value~by~} -6, h2o \\mathbf {~atloc~} mm \\mathbf {~change~value~by~} +2,\\\\&~~~~ h \\mathbf {~atloc~} is \\mathbf {~change~value~by~} +2\\\\&t10 \\mathbf {~may~execute~causing~} \\\\&~~~~ nadh \\mathbf {~atloc~} mm \\mathbf {~change~value~by~} +2, \\\\&~~~~ h \\mathbf {~atloc~} \\mathbf {~change~value~by~} +4, o2 \\mathbf {~atloc~} mm \\mathbf {~change~value~by~} +1\\\\&\\mathbf {initially~} \\\\&~~~~ fadh2 \\mathbf {~atloc~} mm \\mathbf {~has~value~} 0, e \\mathbf {~atloc~} mm \\mathbf {~has~value~} 0,\\\\&~~~~ fad \\mathbf {~atloc~} mm \\mathbf {~has~value~} 0, o2 \\mathbf {~atloc~} mm \\mathbf {~has~value~} 0,\\\\&~~~~ h2o \\mathbf {~atloc~} mm \\mathbf {~has~value~} 0, atp \\mathbf {~atloc~} mm \\mathbf {~has~value~} 0\\\\&\\mathbf {initially~} \\\\&~~~~ fadh2 \\mathbf {~atloc~} is \\mathbf {~has~value~} 0, e \\mathbf {~atloc~} is \\mathbf {~has~value~} 0,\\\\&~~~~ fad \\mathbf {~atloc~} is \\mathbf {~has~value~} 0, o2 \\mathbf {~atloc~} is \\mathbf {~has~value~} 0,\\\\&~~~~ h2o \\mathbf {~atloc~} is \\mathbf {~has~value~} 0, atp \\mathbf {~atloc~} is \\mathbf {~has~value~} 0\\\\&\\mathbf {initially~} \\\\&~~~~ fadh2 \\mathbf {~atloc~} q \\mathbf {~has~value~} 0, e \\mathbf {~atloc~} q \\mathbf {~has~value~} 0,\\\\&~~~~ fad \\mathbf {~atloc~} q \\mathbf {~has~value~} 0, o2 \\mathbf {~atloc~} q \\mathbf {~has~value~} 0,\\\\&~~~~ h2o \\mathbf {~atloc~} q \\mathbf {~has~value~} 0, atp \\mathbf {~atloc~} q \\mathbf {~has~value~} 0\\\\&\\mathbf {initially~} \\\\&~~~~ fadh2 \\mathbf {~atloc~} cytc \\mathbf {~has~value~} 0, e \\mathbf {~atloc~} cytc \\mathbf {~has~value~} 0,\\\\&~~~~ fad \\mathbf {~atloc~} cytc \\mathbf {~has~value~} 0, o2 \\mathbf {~atloc~} cytc \\mathbf {~has~value~} 0,\\\\&~~~~ h2o \\mathbf {~atloc~} cytc \\mathbf {~has~value~} 0, atp \\mathbf {~atloc~} cytc \\mathbf {~has~value~} 0\\\\&~~~~ \\mathbf {firing~style~} max\\\\$ where $mm$ represents the mitochondrial matrix, $is$ represents the intermembrane space, $t1-t4$ represent the reaction of the four complexes making up the electron transport chain, $h$ is the $H+$ ion, $nadh$ is $NADH$ , $fadh2$ is $FADH_2$ , $fad$ is $FAD$ , $e$ is electrons, $o2$ is oxygen $O_2$ , $atp$ is $ATP$ , $h2o$ is water $H_2O$ , and $t10$ is a source transition that supply a continuous supply of source ingredients for the chain to function, such as $nadh$ , $h$ , $o2$ .", "As a result, the query $\\mathbf {Q}$ asked by the question is encoded as follows: $\\begin{array}{l}\\mathbf {direction~of~change~in~} average \\mathbf {~total~production~of~} h \\mathbf {~atloc~} is \\mathbf {~is~} d\\\\~~~~~~~~\\mathbf {when~observed~between~time~step~} 0 \\mathbf {~and~time~step~} k ;\\\\~~~~~~~~\\mathbf {comparing~nominal~pathway~with~modified~pathway~obtained}\\\\~~~~~~~~\\mathbf {due~to~intervention~} t4 \\mathbf {~disabled} ;\\end{array}$ Since this is a comparative quantitative query statement, it is decomposed into two queries, $\\mathbf {Q_0}$ capturing the nominal case of average production w.r.t.", "given initial conditions: $\\begin{array}{l}average \\mathbf {~total~production~of~} h \\mathbf {~atloc~} is \\mathbf {~is~} n_{avg}\\\\~~~~~~~~\\mathbf {when~observed~between~time~step~} 0 \\mathbf {~and~time~step~} k ;\\end{array}$ and $\\mathbf {Q_1}$ the modified case w.r.t.", "initial conditions, modified to include interventions and subject to observations: $\\begin{array}{l}average \\mathbf {~total~production~of~} h \\mathbf {~atloc~} is \\mathbf {~is~} n^{\\prime }_{avg}\\\\~~~~~~~~\\mathbf {when~observed~between~time~step~} 0 \\mathbf {~and~time~step~} k ;\\\\~~~~~~~~\\mathbf {due~to~intervention~} t4 \\mathbf {~disabled} ;\\end{array}$ The the task is to determine $d$ , such that $\\mathbf {D} \\models \\mathbf {Q_0}$ for some value of $n_{avg}$ , $\\mathbf {D} \\models \\mathbf {Q_1}$ for some value of $n^{\\prime }_{avg}$ , and $n^{\\prime }_{avg} \\; d \\; n_{avg}$ .", "To answer the sub-queries $\\mathbf {Q_0}$ and $\\mathbf {Q_1}$ we build nominal description $\\mathbf {D_0}$ and modified pathway $\\mathbf {D_1}$ , where $\\mathbf {D_0} \\equiv \\mathbf {D}$ since there are no initial conditions; and $\\mathbf {D_1} \\equiv \\mathbf {D_0} \\diamond (t4 \\mathbf {~disabled})$ is the domain description $\\mathbf {D}$ modified according to $\\mathbf {Q_1}$ to modified to include the initial conditions as well as interventions.", "The $\\diamond $ operator modifies the domain description to its left by adding, removing or modifying pathway specification language statements to add the intervention and initial conditions to its right.", "Thus, $\\mathbf {D_1} = \\mathbf {D_0} + \\left\\lbrace \\begin{array}{llll}\\mathbf {inhibit~} t4\\\\\\end{array}\\right.$ Performing a simulation of $k=5$ steps with $ntok=20$ max tokens, we find that the average total production of H+ in the intermembrane space ($h$ at location $is$ ) reduces from 16 to 14.", "Lower quantity of H+ translates to a higer numeric value of $-log_{10}$ , as a result the pH increases.", "Question 14 Membranes must be fluid to function properly.", "How would decreased fluidity of the membrane affect the efficiency of the electron transport chain?", "Figure: Petri Net with colored tokens alternate for question From background knowledge, we know that, “Establishing the H+ gradient is a major function of the electron transport chain” [64], we measure the efficiency in terms of H+ ions moved to the intermembrane space ($is$ ) over time.", "Thus, we interpret the question is asking for the direction of change in the production of H+ moved to the intermembrane space when the nominal case is compared against a modified pathway with decreased fluidity of membrane.", "Additional background knowledge from [64] tells us that the decreased fluidity reduces the speed of mobile carriers such as $q$ and $cytc$ .", "Fluidity can span a range of values, but we will consider one such value $v$ per query.", "The following pathway specification encodes the domain description $\\mathbf {D}$ for question REF and produces the PN in Fig.", "REF minus the places $q\\_3,cytc\\_4$ and transitions $tq,tcytc$ .", "In the pathway description below, we have included only one domain declaration for all fluents as an example and left the rest out to save space.", "Assume integer domain declaration for all fluents.", "$&\\mathbf {domain~of~} \\\\&~~~~nadh \\mathbf {~atloc~} mm \\mathbf {~is~} integer \\\\&t1 \\mathbf {~may~execute~causing~} \\\\&~~~~ nadh \\mathbf {~atloc~} mm \\mathbf {~change~value~by~} -2, h \\mathbf {~atloc~} mm \\mathbf {~change~value~by~} -2,\\\\&~~~~ e \\mathbf {~atloc~} q \\mathbf {~change~value~by~} +2, h \\mathbf {~atloc~} is \\mathbf {~change~value~by~} +2,\\\\&~~~~ nadp \\mathbf {~atloc~} mm \\mathbf {~change~value~by~} +2\\\\&t3 \\mathbf {~may~execute~causing~} \\\\&~~~~ e \\mathbf {~atloc~} q \\mathbf {~change~value~by~} -2, h \\mathbf {~atloc~} mm \\mathbf {~change~value~by~} -2,\\\\&~~~~ e \\mathbf {~atloc~} cytc \\mathbf {~change~value~by~} +2, h \\mathbf {~atloc~} is \\mathbf {~change~value~by~} +2\\\\&t4 \\mathbf {~may~execute~causing~} \\\\&~~~~ o2 \\mathbf {~atloc~} mm \\mathbf {~change~value~by~} -1, e \\mathbf {~atloc~} cytc \\mathbf {~change~value~by~} -2, \\\\&~~~~ h \\mathbf {~atloc~} mm \\mathbf {~change~value~by~} -2, h2o \\mathbf {~atloc~} mm \\mathbf {~change~value~by~} +2,\\\\&~~~~ h \\mathbf {~atloc~} is \\mathbf {~change~value~by~} +2\\\\&t10 \\mathbf {~may~execute~causing~} \\\\&~~~~ nadh \\mathbf {~atloc~} mm \\mathbf {~change~value~by~} +2, h \\mathbf {~atloc~} mm \\mathbf {~change~value~by~} +4,\\\\&~~~~ o2 \\mathbf {~atloc~} mm \\mathbf {~change~value~by~} +1\\\\& \\mathbf {initially~} \\\\&~~~~ nadh \\mathbf {~atloc~} mm \\mathbf {~has~value~} 0, h \\mathbf {~atloc~} mm \\mathbf {~has~value~} 0,\\\\&~~~~ nadp \\mathbf {~atloc~} mm \\mathbf {~has~value~} 0, o2 \\mathbf {~atloc~} mm \\mathbf {~has~value~} 0\\\\&~~~~ h2o \\mathbf {~atloc~} mm \\mathbf {~has~value~} 0 \\\\& \\mathbf {initially~} \\\\&~~~~ nadh \\mathbf {~atloc~} is \\mathbf {~has~value~} 0, h \\mathbf {~atloc~} is \\mathbf {~has~value~} 0,\\\\&~~~~ nadp \\mathbf {~atloc~} is \\mathbf {~has~value~} 0, o2 \\mathbf {~atloc~} is \\mathbf {~has~value~} 0\\\\&~~~~ h2o \\mathbf {~atloc~} is \\mathbf {~has~value~} 0 \\\\& \\mathbf {initially~} \\\\&~~~~ nadh \\mathbf {~atloc~} q \\mathbf {~has~value~} 0, h \\mathbf {~atloc~} q \\mathbf {~has~value~} 0,\\\\&~~~~ nadp \\mathbf {~atloc~} q \\mathbf {~has~value~} 0, o2 \\mathbf {~atloc~} q \\mathbf {~has~value~} 0\\\\&~~~~ h2o \\mathbf {~atloc~} q \\mathbf {~has~value~} 0 \\\\& \\mathbf {initially~} \\\\&~~~~ nadh \\mathbf {~atloc~} cytc \\mathbf {~has~value~} 0, h \\mathbf {~atloc~} cytc \\mathbf {~has~value~} 0,\\\\&~~~~ nadp \\mathbf {~atloc~} cytc \\mathbf {~has~value~} 0, o2 \\mathbf {~atloc~} cytc \\mathbf {~has~value~} 0\\\\&~~~~ h2o \\mathbf {~atloc~} cytc \\mathbf {~has~value~} 0 \\\\&\\mathbf {firing~style~} max\\\\$ The query $\\mathbf {Q}$ asked by the question is encoded as follows: $\\begin{array}{l}\\mathbf {direction~of~change~in~} average \\mathbf {~total~production~of~} h \\mathbf {~atloc~} is \\mathbf {~is~} d\\\\~~~~~~\\mathbf {when~observed~between~time~step~} 0 \\mathbf {~and~time~step~} k;\\\\~~~~~~\\mathbf {comparing~nominal~pathway~with~modified~pathway~obtained~}\\\\~~~~~~~~~~~~\\mathbf {due~to~interventions:} \\\\~~~~~~~~~~~~~~~~~~\\mathbf {add~delay~of~} v \\mathbf {~time~units~in~availability~of~} e \\mathbf {~atloc~} q,\\\\~~~~~~~~~~~~~~~~~~\\mathbf {add~delay~of~} v \\mathbf {~time~units~in~availability~of~} e \\mathbf {~atloc~} cytc;\\end{array}$ Since this is a comparative quantitative query statement, we decompose it into two queries, $\\mathbf {Q_0}$ capturing the nominal case of average rate of production w.r.t.", "given initial conditions: $\\begin{array}{l}average \\mathbf {~total~production~of~} h \\mathbf {~atloc~} is \\mathbf {~is~} d\\\\~~~~~~~~~~~~\\mathbf {when~observed~between~time~step~} 0 \\mathbf {~and~time~step~} k;\\\\\\end{array}$ and $\\mathbf {Q_1}$ the modified case w.r.t.", "initial conditions, modified to include interventions and subject to observations: $\\begin{array}{l}average \\mathbf {~total~production~of~} h \\mathbf {~atloc~} is \\mathbf {~is~} d\\\\~~~~~~~~~~~~\\mathbf {when~observed~between~time~step~} 0 \\mathbf {~and~time~step~} k;\\\\~~~~~~\\mathbf {due~to~interventions:} \\\\~~~~~~~~~~~~\\mathbf {add~delay~of~} v \\mathbf {~time~units~in~availability~of~} e \\mathbf {~atloc~} q,\\\\~~~~~~~~~~~~\\mathbf {add~delay~of~} v \\mathbf {~time~units~in~availability~of~} e \\mathbf {~atloc~} cytc;\\end{array}$ Then the task is to determine $d$ , such that $\\mathbf {D} \\models \\mathbf {Q_0}$ for some value of $n_{avg}$ , $\\mathbf {D} \\models \\mathbf {Q_1}$ for some value of $n^{\\prime }_{avg}$ , and $n^{\\prime }_{avg} \\; d \\; n_{avg}$ .", "To answer the sub-queries $\\mathbf {Q_0}$ and $\\mathbf {Q_1}$ we build modified domain descriptions $\\mathbf {D_0}$ and $\\mathbf {D_1}$ , where, $\\mathbf {D_0} \\equiv \\mathbf {D}$ is the nominal domain description $\\mathbf {D}$ modified according to $\\mathbf {Q_0}$ to include the initial conditions; and $\\mathbf {D_1} \\equiv \\mathbf {D_0} \\diamond (\\mathbf {add~delay~of~} v \\mathbf {~time~units~in~availability~of~} $ $e \\mathbf {~atloc~} q) \\diamond (\\mathbf {add~delay~of~} v \\mathbf {~time~units~in~availability~of~} $ $e \\mathbf {~atloc~} cytc)$ is the modified domain description $\\mathbf {D}$ modified according to $\\mathbf {Q_1}$ to include the initial conditions as well as the interventions.", "The $\\diamond $ operator modifies the domain description to its left by adding, removing or modifying pathway specification language statements to add the interventions and initial conditions description to its right.", "Thus, $\\mathbf {D_1} = \\mathbf {D} &-\\left\\lbrace \\begin{array}{llll}t1 \\mathbf {~may~execute~causing~} & nadh \\mathbf {~atloc~} mm \\mathbf {~change~value~by~} -2, \\\\& h \\mathbf {~atloc~} mm \\mathbf {~change~value~by~} -2,\\\\& e \\mathbf {~atloc~} q \\mathbf {~change~value~by~} +2, \\\\& h \\mathbf {~atloc~} is \\mathbf {~change~value~by~} +2,\\\\& nadp \\mathbf {~atloc~} mm \\mathbf {~change~value~by~} +2\\\\t3 \\mathbf {~may~execute~causing~} & e \\mathbf {~atloc~} q \\mathbf {~change~value~by~} -2, \\\\& h \\mathbf {~atloc~} mm \\mathbf {~change~value~by~} -2,\\\\& e \\mathbf {~atloc~} cytc \\mathbf {~change~value~by~} +2, \\\\& h \\mathbf {~atloc~} is \\mathbf {~change~value~by~} +2\\\\\\end{array}\\right\\rbrace \\\\& +\\left\\lbrace \\begin{array}{llll}t1 \\mathbf {~may~execute~causing~} & nadh \\mathbf {~atloc~} mm \\mathbf {~change~value~by~} -2, \\\\& h \\mathbf {~atloc~} mm \\mathbf {~change~value~by~} -2,\\\\& e \\mathbf {~atloc~} q\\_3 \\mathbf {~change~value~by~} +2, \\\\& h \\mathbf {~atloc~} is \\mathbf {~change~value~by~} +2,\\\\& nadp \\mathbf {~atloc~} mm \\mathbf {~change~value~by~} +2\\\\t3 \\mathbf {~may~execute~causing~} & e \\mathbf {~atloc~} q \\mathbf {~change~value~by~} -2, \\\\& h \\mathbf {~atloc~} mm \\mathbf {~change~value~by~} -2,\\\\& e \\mathbf {~atloc~} cytc\\_4 \\mathbf {~change~value~by~} +2, \\\\& h \\mathbf {~atloc~} is \\mathbf {~change~value~by~} +2\\\\tq \\mathbf {~may~fire~causing~} & e \\mathbf {~atloc~} q\\_3 \\mathbf {~change~value~by~} -2, \\\\& e \\mathbf {~atloc~} q \\mathbf {~change~value~by~} +2\\\\tcytc \\mathbf {~may~fire~causing~} & e \\mathbf {~atloc~} cytc\\_4 \\mathbf {~change~value~by~} -2, \\\\& e \\mathbf {~atloc~} cytc \\mathbf {~change~value~by~} +2\\\\tq \\mathbf {~executes~} & \\mathbf {~in~} 4 \\mathbf {~time~units}\\\\tcytc \\mathbf {~executes~} & \\mathbf {~in~} 4 \\mathbf {~time~units}\\\\\\end{array}\\right\\rbrace \\\\$ Performing a simulation of k = 5 steps with ntok = 20 max tokens with a fluidity based delay of 2, we find that the average total production of H+ in the intermembrane space (h at location is) reduces from 16 to 10.", "Lower quantity of H+ going into the intermembrane space means lower efficiency, where we define the efficiency as the total amount of $H+$ ions transferred to the intermembrane space over the simulation run.", "Example Encoding with Conditional Actions Next, we illustrate how conditional actions would be encoded in our high-level language with an example.", "Consider the pathway from question REF .", "Say, the reaction step $t4$ has developed a fault, in which it has two modes of operation, in the first mode, when $f16bp$ has less than 3 units available, the reaction proceeds normally, but when $f16bp$ is available in 3 units or higher, the reaction continues to produce $g3p$ but not $dhap$ directly.", "$dhap$ can still be produced by subsequent step from $g3p$ .", "The modified pathway is given in our pathway specification language below: $\\begin{array}{llll}&t3 \\mathbf {~may~execute~causing~} & f16bp \\mathbf {~change~value~by~} +1\\\\&t4 \\mathbf {~may~execute~causing~} & f16bp \\mathbf {~change~value~by~} -1, & dhap \\mathbf {~change~value~by~} +1,\\\\ && g3p \\mathbf {~change~value~by~} +1\\\\&~~~~~~~~~~~~~~~~\\mathbf {if~}& g3p \\mathbf {~has~value~lower~than~} 3\\\\&t4 \\mathbf {~may~execute~causing~} & f16bp \\mathbf {~change~value~by~} -1, & dhap \\mathbf {~change~value~by~} +1\\\\&~~~~~~~~~~~~~~~~\\mathbf {~if}& g3p \\mathbf {~has~value~} 3 \\mathbf {~or~higher}\\\\&t5a \\mathbf {~may~execute~causing~} & dhap \\mathbf {~change~value~by~} -1, & g3p \\mathbf {~change~value~by~} +1\\\\&t5b \\mathbf {~may~execute~causing~} & g3p \\mathbf {~change~value~by~} -1, & dhap \\mathbf {~change~value~by~} +1\\\\&t6 \\mathbf {~may~execute~causing~} & g3p \\mathbf {~change~value~by~} -1, & bpg13 \\mathbf {~change~value~by~} +2\\\\& \\mathbf {initially~} & f16bp \\mathbf {~has~value~} 0, & dhap \\mathbf {~has~value~} 0,\\\\ && g3p \\mathbf {~has~value~} 0, & bpg13 \\mathbf {~has~value~} 0\\\\&\\mathbf {firing~style~} & max\\end{array}$ We ask the same question $\\mathbf {Q}$ : $&\\mathbf {direction~of~change~in~} average \\mathbf {~rate~of~production~of~} bpg13 \\mathbf {~is~} d\\\\&~~~~~~\\mathbf {when~observed~between~time~step~} 0 \\mathbf {~and~time~step~} k; \\\\&~~~~~~\\mathbf {comparing~nominal~pathway~with~modified~pathway~obtained~}\\\\&~~~~~~~~~~~~\\mathbf {due~to~interventions:~} \\mathbf {remove~} dhap \\mathbf {~as~soon~as~produced};\\\\&~~~~~~~~~~~~\\mathbf {using~initial~setup:~} \\mathbf {continuously~supply~} f16bp \\mathbf {~in~quantity~} 1;$ Since this is a comparative quantitative query statement, we decompose it into two queries, $\\mathbf {Q_0}$ capturing the nominal case of average rate of production w.r.t.", "given initial conditions: $\\begin{array}{l}average \\mathbf {~rate~of~production~of~} bpg13 \\mathbf {~is~} n\\\\~~~~~~~~~~~~\\mathbf {when~observed~between~time~step~} 0 \\mathbf {~and~time~step~} k;\\\\~~~~~~\\mathbf {using~initial~setup:~} \\mathbf {continuously~supply~} f16bp \\mathbf {~in~quantity~} 1;\\end{array}$ and $\\mathbf {Q_1}$ the modified case w.r.t.", "initial conditions, modified to include interventions and subject to observations: $\\begin{array}{l}average \\mathbf {~rate~of~production~of~} bpg13 \\mathbf {~is~} n\\\\~~~~~~~~~~~~\\mathbf {when~observed~between~time~step~} 0 \\mathbf {~and~time~step~} k;\\\\~~~~~~\\mathbf {due~to~interventions:~} \\mathbf {remove~} dhap \\mathbf {~as~soon~as~produced};\\\\~~~~~~\\mathbf {using~initial~setup:~} \\mathbf {continuously~supply~} f16bp \\mathbf {~in~quantity~} 1;\\end{array}$ Then the task is to determine $d$ , such that $\\mathbf {D} \\models \\mathbf {Q_0}$ for some value of $n_{avg}$ , $\\mathbf {D} \\models \\mathbf {Q_1}$ for some value of $n^{\\prime }_{avg}$ , and $n^{\\prime }_{avg} \\; d \\; n_{avg}$ .", "To answer the sub-queries $\\mathbf {Q_0}$ and $\\mathbf {Q_1}$ we build modified domain descriptions $\\mathbf {D_0}$ and $\\mathbf {D_1}$ , where, $\\mathbf {D_0} \\equiv \\mathbf {D} \\diamond (\\mathbf {continuously~supply~} f16bp $ $\\mathbf {~in~quantity~} 1)$ is the nominal domain description $\\mathbf {D}$ modified according to $\\mathbf {Q_0}$ to include the initial conditions; and $\\mathbf {D_1} \\equiv \\mathbf {D_0} \\diamond (\\mathbf {remove~} dhap \\mathbf {~as~soon~as~produced})$ is the modified domain description $\\mathbf {D}$ modified according to $\\mathbf {Q_1}$ to include the initial conditions as well as the interventions.", "The $\\diamond $ operator modifies the domain description to its left by adding, removing or modifying pathway specification language statements to add the interventions and initial conditions description to its right.", "Thus, $\\mathbf {D_0} &= \\mathbf {D} + \\left\\lbrace \\begin{array}{llll}tf_{f16bp} \\mathbf {~may~execute~causing~} &f16bp \\mathbf {~change~value~by~} +1\\\\\\end{array}\\right.\\\\\\mathbf {D_1} &= \\mathbf {D_0} + \\left\\lbrace \\begin{array}{llll}tr \\mathbf {~may~execute~causing~} &dhap \\mathbf {~change~value~by~} \\\\\\end{array}\\right.$ ASP Program Next we briefly outline how the pathway specification and the query statement components are encoded in ASP (using Clingo syntax).", "In the following section, we will illustrate the process using an example.", "As evident from the previous sections, we need to simulate non-comparative queries only.", "Any comparative queries are translated into non-comparative sub-queries, each of which is simulated and their results compared to evaluate the comparative query.", "The ASP program is a concatenation of the translation of a pathway specification (domain description which includes the firing style) and internal observations.", "Any initial setup conditions and interventions are pre-applied to the pathway specification using intervention semantics in section REF before it is translated to ASP using the translation in chapter as our basis.", "The encoded pathway specification has the semantics defined in section REF .", "Internal observations in the `due to observations:' portion of query statement are translated into ASP constraints using the internal observation semantics defined in section REF and added to the encoding of the pathway specification.", "The program if simulated for a specified simulation length $k$ produces all trajectories of the pathway for the specified firing style, filtered by the internal observations.", "The query description specified in the query statement is then evaluated w.r.t.", "these trajectories.", "Although this part can be done in ASP, we have currently implemented it outside ASP in our implementation for ease of using floating point math.", "Next, we describe an implementation of our high level language and illustrate the construction of an ASP program, its simulation, and query statement evaluation.", "Implementation We have developed an implementation Implementation available at: https://sites.google.com/site/deepqa2014/ of a subset of our high level (Pathway and Query specification) language in Python.", "We use the Clingo ASP implementation for our simulation.", "In this section we describe various components of this implementation.", "An architectural overview of our implementation is shown in figure REF .", "Figure: BioPathQA Implementation System ArchitectureThe Pathway Specification Language (BioPathQA-PL) Parser component is responsible for parsing the Pathway Specification Language (BioPathQA-PL).", "It use PLY (Python Lex-Yacc)http://www.dabeaz.com/ply to parse a given pathway specification using grammar based on section REF .", "On a successful parse, a Guarded-Arc Petri Net pathway model based on section REF is constructed for the pathway specification.", "The Query Language Parser component is responsible for parsing the Query Specification Language (BioPathQA-QL).", "It uses PLY to parse a given query statement using grammar based on section REF .", "On a successful parse, an internal representation of the query statement is constructed.", "Elements of this internal representation include objects representing the query description, the list of interventions, the list of internal observations, and the list of initial setup conditions.", "Each intervention and initial setup condition object has logic in it to modify a given pathway per the intervention semantics described in section REF .", "The Query Statement Model component is also responsible for generating basic queries for aggregate queries and implementing interventions in the Petri Net Pathway Model.", "The Dictionary of fluents, locations, and actions is consulted by the ASP code generator to standardize symbol names in the ASP code produced for the pathway specification and the internal observations.", "The ASP Translator component is responsible for translating the Guarded-Arc Petri Net model into ASP facts and rules; and the driver needed to simulate the model using the firing semantics specified in the pathway model.", "The code generated is based on the ASP translation of Petri Nets and its various extensions given in chapter .", "To reduce the ASP code and its complexity, the translator limits the output model to the extensions used in the Petri Net model to be translated.", "Thus, the colored tokens extension code is not produced unless colored tokens have been used.", "Similarly, guarded-arcs code is not produced if no arc-guards are used in the model.", "The ASP Translator component is also responsible for translating internal observations from the Query Statement into ASP constraints to filter Petri Net trajectories based on the observation semantics in section REF .", "Following examples illustrate our encoding scheme.", "The observation `$a_1 \\mathbf {~switches~to~} a_2$ ' is encoded as a constraint using the following rules: obs_1_occurred(TS+1) :- time(TS;TS+1), trans(a1;a2), fires(a1,TS), not fires(a2,TS), not fires(a1,TS+1), fires(a2,TS+1).", "obs_1_occurred :- obs_1_occurred(TS), time(TS).", "obs_1_had_occurred(TSS) :- obs_1_occurred(TS), TS<=TSS, time(TSS;TS).", ":- not obs_1_occurred.", "The observation `$a_1 \\mathbf {~occurs~at~time~step~} 5$ ' is encoded as a constraint using the following rules: obs_2_occurred(TS) :- fires(a1,TS), trans(a1), time(TS), TS=5.", "obs_2_occurred :- obs_2_occurred(TS), time(TS).", "obs_2_had_occurred(TSS) :- obs_2_occurred(TS), TS<=TSS, time(TSS;TS).", ":- not obs_2_occurred.", "The observation `$s_1 \\mathbf {~is~decreasing~atloc~} l_1 \\mathbf {~when~observed~between~time~step~} 0 $ $\\mathbf {~and~time~step~} 5$ ' is encoded as a constraint using the following rules: obs_3_violated(TS) :- place(l1), col(s1), holds(l1,Q1,s1,TS), holds(l1,Q2,s1,TS+1), num(Q1;Q2), Q2 > Q1, time(TS;TS+1), TS=0, TS+1=5.", "obs_3_violated :- obs_3_violated(TS), time(TS).", "obs_3_occurred(TS+1) :- not obs_3_violated, holds(l1,Q1,s1,TS), holds(l1,Q2,s1,TS+1), time(TS;TS+1), num(Q1;Q2), Q2<Q1, TS=0, TS+1=5.", "obs_3_occurred :- obs_3_occurred(TS), time(TS).", "obs_3_had_occurred(TSS) :- obs_3_occurred(TS), TS<=TSS, time(TSS;TS).", ":- obs_3_occurred.", "In addition, the translator is also responsible for any rules needed to ease post-processing of the query description.", "For example, for qualitative queries, a generic predicate tgt_obs_occurred(TS) is generated that is true when the given qualitative description holds in an answer-set at time step $TS$ .", "The output of the translator is an ASP program, which when simulated using Clingo produces the (possibly) filtered trajectories of the pathway.", "The Post Processor component is responsible for parsing the ASP answer sets, identifying the correct atoms from it, extracting quantities from atom-bodies as necessary, organizing them into a matrix form, and aggregating them as needed.", "Figure: BioPathQA Graphical User InterfaceThe User Interface component is responsible for coordinating the processing of query statement.", "It presents the user with a graphical user interface shown in figure REF .", "The user types a Pathway Specification (in BioPathQA-PL syntax), a Query Specification (in BioPathQA-QL syntax), and simulation parameters.", "On pressing “Execute Query”, the user interface component processes the query as prints results in the bottom box.", "Query evaluation differs by the type of query.", "We describe the query evaluation methodology used below.", "For non-comparative quantitative queries: Pathway specification is parsed into a Guarded-Arc Petri Net model.", "Query statement is parsed into an internal form.", "Initial conditions from the query are applied to the pathway model.", "Interventions are applied to the pathway model.", "Modified pathway model is translated to ASP.", "Internal observations are added to the ASP code as ASP constraints.", "Answer sets of the ASP code are computed using Clingo.", "Relevant atoms are extracted: fires/2 predicate for firing rate, holds/3 (or holds/4 – colored tokens) predicate for fluent quantity or rate formulas.", "Fluent value or firing-count values are extracted and organized as matrices with rows representing answer-sets and columns representing time-steps.", "Within answer-set interval or point value sub-select is done and the values converted to rates or totals as needed.", "If aggregation, such as average, minimum, or maximum is desired, it is performed over rows of values from the last step.", "If a value was specified in the query, it is compared against the computed aggregate for boolean result.", "If a value was not specified, the computed value is returned as the value satisfying the query statement.", "For queries over all trajectories, the same value must hold over all trajectories, otherwise, only one match is required to satisfy the query.", "For non-comparative qualitative queries: Follow steps (REF )-(REF ) of non-comparative quantitative queries.", "Add rules for the query description for post-processing.", "Answer sets of the ASP code are computed using Clingo.", "Relevant atoms are extracted: tgt_obs_occurred/1 identifying the time step when the observation within the query description is satisfied.", "Truth value of the query observation is determined, including determining the truth value over all trajectories.", "For comparative quantitative queries: Query statement is decomposed into two non-comparative quantitative sub-query statements as illustrated in section REF : A nominal sub-query which has the same initial conditions as the comparative query, but none of its interventions or observations A modified sub-query which has the same initial conditions, interventions, and observations as the comparative query both sub-query statements have the same query description, which is the non-aggregate form of the comparative query description.", "Thus, a comparative average rate query is translated to non-comparative average rate sub-queries.", "Each sub-query statements is evaluated using steps (REF )-(REF ) from the non-comparative quantitative query processing.", "A direction of change is computed by comparing the computed aggregate value for the modified query statement to the nominal query statement.", "If the comparative quantitative query has a direction specified, it is the compared against the computed value for a boolean result.", "If the comparative quantitative query did not have a direction specified, the computed value is returned as the value satisfying the query statement.", "For explanation queries with query description with formula of the form (), it is expected that the number of answer-sets will be quite large.", "So, we avoid generating all answer-sets before processing them, instead we process them in-line as they are generated.", "It is a bit slower, but leads to a smaller memory foot print.", "Follow steps (REF )-(REF ) of non-comparative quantitative queries.", "Add rules for the query description for post-processing.", "Compute answer sets of the ASP code using Clingo.", "Extract relevant atoms: extract tgt_obs_occurred/1 identifying the time step when the query description is satisfied extract holds/3 (or holds/4 – for colored tokens) at the same time-step as tgt_obs_occurred/1 to construct fluent-based conditions Construct fluent-based conditions as explanation of the query observation.", "If the query is over all trajectories, fluent-based conditions for each trajectory are intersected across trajectories to determine the minimum set of conditions explaining the query observation.", "Next we illustrate query processing through an execution trace of question (REF ).", "The following shows the encoding of the base case domain, which includes the pathway specification from (REF ) with initial setup conditions from the query statement (REF ) applied: #const nts=1.", "time(0..nts).", "#const ntok=1.", "num(0..ntok).", "place(bpg13).", "place(dhap).", "place(f16bp).", "place(g3p).", "trans(src_f16bp_1).", "trans(t3).", "trans(t4).", "trans(t5a).", "trans(t5b).", "trans(t6).", "tparc(src_f16bp_1,f16bp,1,TS) :- time(TS).", "tparc(t3,f16bp,1,TS) :- time(TS).", "ptarc(f16bp,t4,1,TS) :- time(TS).", "tparc(t4,g3p,1,TS) :- time(TS).", "tparc(t4,dhap,1,TS) :- time(TS).", "ptarc(dhap,t5a,1,TS) :- time(TS).", "tparc(t5a,g3p,1,TS) :- time(TS).", "ptarc(g3p,t5b,1,TS) :- time(TS).", "tparc(t5b,dhap,1,TS) :- time(TS).", "ptarc(g3p,t6,1,TS) :- time(TS).", "tparc(t6,bpg13,2,TS) :- time(TS).", "holds(bpg13,0,0).", "holds(dhap,0,0).", "holds(f16bp,0,0).", "holds(g3p,0,0).", "could_not_have(T,TS) :- enabled(T,TS), not fires(T,TS), ptarc(P,T,Q,TS), holds(P,QQ,TS), tot_decr(P,QQQ,TS), Q > QQ - QQQ.", ":- not could_not_have(T,TS), time(TS), enabled(T,TS), not fires(T,TS), trans(T).", "min(A,B,A) :- A<=B, num(A;B).", "min(A,B,B) :- B<=A, num(A;B).", "#hide min/3.", "holdspos(P):- holds(P,N,0), place(P), num(N), N > 0. holds(P,0,0) :- place(P), not holdspos(P).", "notenabled(T,TS) :- ptarc(P,T,N,TS), holds(P,Q,TS), Q < N, place(P), trans(T), time(TS), num(N), num(Q).", "enabled(T,TS) :- trans(T), time(TS), not notenabled(T, TS).", "{ fires(T,TS) } :- enabled(T,TS), trans(T), time(TS).", "add(P,Q,T,TS) :- fires(T,TS), tparc(T,P,Q,TS), time(TS).", "del(P,Q,T,TS) :- fires(T,TS), ptarc(P,T,Q,TS), time(TS).", "tot_incr(P,QQ,TS) :- QQ = #sum[add(P,Q,T,TS) = Q : num(Q) : trans(T)], time(TS), num(QQ), place(P).", "tot_decr(P,QQ,TS) :- QQ = #sum[del(P,Q,T,TS) = Q : num(Q) : trans(T)], time(TS), num(QQ), place(P).", "holds(P,Q,TS+1) :- holds(P,Q1,TS), tot_incr(P,Q2,TS), tot_decr(P,Q3,TS), Q=Q1+Q2-Q3, place(P), num(Q;Q1;Q2;Q3), time(TS), time(TS+1).", "consumesmore(P,TS) :- holds(P,Q,TS), tot_decr(P,Q1,TS), Q1 > Q. consumesmore :- consumesmore(P,TS).", ":- consumesmore.", "The following shows encoding of the alternate case domain, which consists of the pathway specification from (REF ) with initial setup conditions and interventions applied; and any internal observations from the query statement (REF ) added: #const nts=1.", "time(0..nts).", "#const ntok=1.", "num(0..ntok).", "place(bpg13).", "place(dhap).", "place(f16bp).", "place(g3p).", "trans(reset_dhap_1).", "trans(src_f16bp_1).", "trans(t3).", "trans(t4).", "trans(t5a).", "trans(t5b).", "trans(t6).", "ptarc(dhap,reset_dhap_1,Q,TS) :- holds(dhap,Q,TS), Q>0, time(TS).", ":- enabled(reset_dhap_1,TS), not fires(reset_dhap_1,TS), time(TS).", "tparc(src_f16bp_1,f16bp,1,TS) :- time(TS).", "tparc(t3,f16bp,1,TS) :- time(TS).", "ptarc(f16bp,t4,1,TS) :- time(TS).", "tparc(t4,g3p,1,TS) :- time(TS).", "tparc(t4,dhap,1,TS) :- time(TS).", "ptarc(dhap,t5a,1,TS) :- time(TS).", "tparc(t5a,g3p,1,TS) :- time(TS).", "ptarc(g3p,t5b,1,TS) :- time(TS).", "tparc(t5b,dhap,1,TS) :- time(TS).", "ptarc(g3p,t6,1,TS) :- time(TS).", "tparc(t6,bpg13,2,TS) :- time(TS).", "holds(bpg13,0,0).", "holds(dhap,0,0).", "holds(f16bp,0,0).", "holds(g3p,0,0).", "could_not_have(T,TS) :- enabled(T,TS), not fires(T,TS), ptarc(P,T,Q,TS), holds(P,QQ,TS), tot_decr(P,QQQ,TS), Q > QQ - QQQ.", ":- not could_not_have(T,TS), time(TS), enabled(T,TS), not fires(T,TS), trans(T).", "min(A,B,A) :- A<=B, num(A;B).", "min(A,B,B) :- B<=A, num(A;B).", "#hide min/3.", "holdspos(P):- holds(P,N,0), place(P), num(N), N > 0. holds(P,0,0) :- place(P), not holdspos(P).", "notenabled(T,TS) :- ptarc(P,T,N,TS), holds(P,Q,TS), Q < N, place(P), trans(T), time(TS), num(N), num(Q).", "enabled(T,TS) :- trans(T), time(TS), not notenabled(T, TS).", "{ fires(T,TS) } :- enabled(T,TS), trans(T), time(TS).", "add(P,Q,T,TS) :- fires(T,TS), tparc(T,P,Q,TS), time(TS).", "del(P,Q,T,TS) :- fires(T,TS), ptarc(P,T,Q,TS), time(TS).", "tot_incr(P,QQ,TS) :- QQ = #sum[add(P,Q,T,TS) = Q : num(Q) : trans(T)], time(TS), num(QQ), place(P).", "tot_decr(P,QQ,TS) :- QQ = #sum[del(P,Q,T,TS) = Q : num(Q) : trans(T)], time(TS), num(QQ), place(P).", "holds(P,Q,TS+1) :- holds(P,Q1,TS), tot_incr(P,Q2,TS), tot_decr(P,Q3,TS), Q=Q1+Q2-Q3, place(P), num(Q;Q1;Q2;Q3), time(TS), time(TS+1).", "consumesmore(P,TS) :- holds(P,Q,TS), tot_decr(P,Q1,TS), Q1 > Q. consumesmore :- consumesmore(P,TS).", ":- consumesmore.", "Both programs are simulated for 5 time-steps and 20 max tokens using the following Clingo command: clingo 0 -cntok=20 -cnts=5 program.lp Answer sets of the base case are as follows: Answer: 1 holds(bpg13,0,0) holds(dhap,0,0) holds(f16bp,0,0) holds(g3p,0,0) fires(src_f16bp_1,0) fires(t3,0) holds(bpg13,0,1) holds(dhap,0,1) holds(f16bp,2,1) holds(g3p,0,1) fires(src_f16bp_1,1) fires(t3,1) fires(t4,1) holds(bpg13,0,2) holds(dhap,1,2) holds(f16bp,3,2) holds(g3p,1,2) fires(src_f16bp_1,2) fires(t3,2) fires(t4,2) fires(t5a,2) fires(t5b,2) holds(bpg13,0,3) holds(dhap,2,3) holds(f16bp,4,3) holds(g3p,2,3) fires(src_f16bp_1,3) fires(t3,3) fires(t4,3) fires(t5a,3) fires(t5b,3) fires(t6,3) holds(bpg13,2,4) holds(dhap,3,4) holds(f16bp,5,4) holds(g3p,2,4) fires(src_f16bp_1,4) fires(t3,4) fires(t4,4) fires(t5a,4) fires(t5b,4) fires(t6,4) holds(bpg13,4,5) holds(dhap,4,5) holds(f16bp,6,5) holds(g3p,2,5) fires(src_f16bp_1,5) fires(t3,5) fires(t4,5) fires(t5a,5) fires(t5b,5) fires(t6,5) Answer: 2 holds(bpg13,0,0) holds(dhap,0,0) holds(f16bp,0,0) holds(g3p,0,0) fires(src_f16bp_1,0) fires(t3,0) holds(bpg13,0,1) holds(dhap,0,1) holds(f16bp,2,1) holds(g3p,0,1) fires(src_f16bp_1,1) fires(t3,1) fires(t4,1) holds(bpg13,0,2) holds(dhap,1,2) holds(f16bp,3,2) holds(g3p,1,2) fires(src_f16bp_1,2) fires(t3,2) fires(t4,2) fires(t5a,2) fires(t6,2) holds(bpg13,2,3) holds(dhap,1,3) holds(f16bp,4,3) holds(g3p,2,3) fires(src_f16bp_1,3) fires(t3,3) fires(t4,3) fires(t5a,3) fires(t5b,3) fires(t6,3) holds(bpg13,4,4) holds(dhap,2,4) holds(f16bp,5,4) holds(g3p,2,4) fires(src_f16bp_1,4) fires(t3,4) fires(t4,4) fires(t5a,4) fires(t5b,4) fires(t6,4) holds(bpg13,6,5) holds(dhap,3,5) holds(f16bp,6,5) holds(g3p,2,5) fires(src_f16bp_1,5) fires(t3,5) fires(t4,5) fires(t5a,5) fires(t5b,5) fires(t6,5) Answer sets of the alternate case are as follows: Answer: 1 holds(bpg13,0,0) holds(dhap,0,0) holds(f16bp,0,0) holds(g3p,0,0) fires(reset_dhap_1,0) fires(src_f16bp_1,0) fires(t3,0) holds(bpg13,0,1) holds(dhap,0,1) holds(f16bp,2,1) holds(g3p,0,1) fires(reset_dhap_1,1) fires(src_f16bp_1,1) fires(t3,1) fires(t4,1) holds(bpg13,0,2) holds(dhap,1,2) holds(f16bp,3,2) holds(g3p,1,2) fires(reset_dhap_1,2) fires(src_f16bp_1,2) fires(t3,2) fires(t4,2) fires(t6,2) holds(bpg13,2,3) holds(dhap,1,3) holds(f16bp,4,3) holds(g3p,1,3) fires(reset_dhap_1,3) fires(src_f16bp_1,3) fires(t3,3) fires(t4,3) fires(t5b,3) holds(bpg13,2,4) holds(dhap,2,4) holds(f16bp,5,4) holds(g3p,1,4) fires(reset_dhap_1,4) fires(src_f16bp_1,4) fires(t3,4) fires(t4,4) fires(t5b,4) holds(bpg13,2,5) holds(dhap,2,5) holds(f16bp,6,5) holds(g3p,1,5) fires(reset_dhap_1,5) fires(src_f16bp_1,5) fires(t3,5) fires(t4,5) fires(t5b,5) Answer: 2 holds(bpg13,0,0) holds(dhap,0,0) holds(f16bp,0,0) holds(g3p,0,0) fires(reset_dhap_1,0) fires(src_f16bp_1,0) fires(t3,0) holds(bpg13,0,1) holds(dhap,0,1) holds(f16bp,2,1) holds(g3p,0,1) fires(reset_dhap_1,1) fires(src_f16bp_1,1) fires(t3,1) fires(t4,1) holds(bpg13,0,2) holds(dhap,1,2) holds(f16bp,3,2) holds(g3p,1,2) fires(reset_dhap_1,2) fires(src_f16bp_1,2) fires(t3,2) fires(t4,2) fires(t6,2) holds(bpg13,2,3) holds(dhap,1,3) holds(f16bp,4,3) holds(g3p,1,3) fires(reset_dhap_1,3) fires(src_f16bp_1,3) fires(t3,3) fires(t4,3) fires(t5b,3) holds(bpg13,2,4) holds(dhap,2,4) holds(f16bp,5,4) holds(g3p,1,4) fires(reset_dhap_1,4) fires(src_f16bp_1,4) fires(t3,4) fires(t4,4) fires(t5b,4) holds(bpg13,2,5) holds(dhap,2,5) holds(f16bp,6,5) holds(g3p,1,5) fires(reset_dhap_1,5) fires(src_f16bp_1,5) fires(t3,5) fires(t4,5) fires(t6,5) Answer: 3 holds(bpg13,0,0) holds(dhap,0,0) holds(f16bp,0,0) holds(g3p,0,0) fires(reset_dhap_1,0) fires(src_f16bp_1,0) fires(t3,0) holds(bpg13,0,1) holds(dhap,0,1) holds(f16bp,2,1) holds(g3p,0,1) fires(reset_dhap_1,1) fires(src_f16bp_1,1) fires(t3,1) fires(t4,1) holds(bpg13,0,2) holds(dhap,1,2) holds(f16bp,3,2) holds(g3p,1,2) fires(reset_dhap_1,2) fires(src_f16bp_1,2) fires(t3,2) fires(t4,2) fires(t6,2) holds(bpg13,2,3) holds(dhap,1,3) holds(f16bp,4,3) holds(g3p,1,3) fires(reset_dhap_1,3) fires(src_f16bp_1,3) fires(t3,3) fires(t4,3) fires(t5b,3) holds(bpg13,2,4) holds(dhap,2,4) holds(f16bp,5,4) holds(g3p,1,4) fires(reset_dhap_1,4) fires(src_f16bp_1,4) fires(t3,4) fires(t4,4) fires(t6,4) holds(bpg13,4,5) holds(dhap,1,5) holds(f16bp,6,5) holds(g3p,1,5) fires(reset_dhap_1,5) fires(src_f16bp_1,5) fires(t3,5) fires(t4,5) fires(t5b,5) Answer: 4 holds(bpg13,0,0) holds(dhap,0,0) holds(f16bp,0,0) holds(g3p,0,0) fires(reset_dhap_1,0) fires(src_f16bp_1,0) fires(t3,0) holds(bpg13,0,1) holds(dhap,0,1) holds(f16bp,2,1) holds(g3p,0,1) fires(reset_dhap_1,1) fires(src_f16bp_1,1) fires(t3,1) fires(t4,1) holds(bpg13,0,2) holds(dhap,1,2) holds(f16bp,3,2) holds(g3p,1,2) fires(reset_dhap_1,2) fires(src_f16bp_1,2) fires(t3,2) fires(t4,2) fires(t6,2) holds(bpg13,2,3) holds(dhap,1,3) holds(f16bp,4,3) holds(g3p,1,3) fires(reset_dhap_1,3) fires(src_f16bp_1,3) fires(t3,3) fires(t4,3) fires(t5b,3) holds(bpg13,2,4) holds(dhap,2,4) holds(f16bp,5,4) holds(g3p,1,4) fires(reset_dhap_1,4) fires(src_f16bp_1,4) fires(t3,4) fires(t4,4) fires(t6,4) holds(bpg13,4,5) holds(dhap,1,5) holds(f16bp,6,5) holds(g3p,1,5) fires(reset_dhap_1,5) fires(src_f16bp_1,5) fires(t3,5) fires(t4,5) fires(t6,5) Answer: 5 holds(bpg13,0,0) holds(dhap,0,0) holds(f16bp,0,0) holds(g3p,0,0) fires(reset_dhap_1,0) fires(src_f16bp_1,0) fires(t3,0) holds(bpg13,0,1) holds(dhap,0,1) holds(f16bp,2,1) holds(g3p,0,1) fires(reset_dhap_1,1) fires(src_f16bp_1,1) fires(t3,1) fires(t4,1) holds(bpg13,0,2) holds(dhap,1,2) holds(f16bp,3,2) holds(g3p,1,2) fires(reset_dhap_1,2) fires(src_f16bp_1,2) fires(t3,2) fires(t4,2) fires(t5b,2) holds(bpg13,0,3) holds(dhap,2,3) holds(f16bp,4,3) holds(g3p,1,3) fires(reset_dhap_1,3) fires(src_f16bp_1,3) fires(t3,3) fires(t4,3) fires(t5b,3) holds(bpg13,0,4) holds(dhap,2,4) holds(f16bp,5,4) holds(g3p,1,4) fires(reset_dhap_1,4) fires(src_f16bp_1,4) fires(t3,4) fires(t4,4) fires(t5b,4) holds(bpg13,0,5) holds(dhap,2,5) holds(f16bp,6,5) holds(g3p,1,5) fires(reset_dhap_1,5) fires(src_f16bp_1,5) fires(t3,5) fires(t4,5) fires(t5b,5) Answer: 6 holds(bpg13,0,0) holds(dhap,0,0) holds(f16bp,0,0) holds(g3p,0,0) fires(reset_dhap_1,0) fires(src_f16bp_1,0) fires(t3,0) holds(bpg13,0,1) holds(dhap,0,1) holds(f16bp,2,1) holds(g3p,0,1) fires(reset_dhap_1,1) fires(src_f16bp_1,1) fires(t3,1) fires(t4,1) holds(bpg13,0,2) holds(dhap,1,2) holds(f16bp,3,2) holds(g3p,1,2) fires(reset_dhap_1,2) fires(src_f16bp_1,2) fires(t3,2) fires(t4,2) fires(t5b,2) holds(bpg13,0,3) holds(dhap,2,3) holds(f16bp,4,3) holds(g3p,1,3) fires(reset_dhap_1,3) fires(src_f16bp_1,3) fires(t3,3) fires(t4,3) fires(t5b,3) holds(bpg13,0,4) holds(dhap,2,4) holds(f16bp,5,4) holds(g3p,1,4) fires(reset_dhap_1,4) fires(src_f16bp_1,4) fires(t3,4) fires(t4,4) fires(t5b,4) holds(bpg13,0,5) holds(dhap,2,5) holds(f16bp,6,5) holds(g3p,1,5) fires(reset_dhap_1,5) fires(src_f16bp_1,5) fires(t3,5) fires(t4,5) fires(t6,5) Answer: 7 holds(bpg13,0,0) holds(dhap,0,0) holds(f16bp,0,0) holds(g3p,0,0) fires(reset_dhap_1,0) fires(src_f16bp_1,0) fires(t3,0) holds(bpg13,0,1) holds(dhap,0,1) holds(f16bp,2,1) holds(g3p,0,1) fires(reset_dhap_1,1) fires(src_f16bp_1,1) fires(t3,1) fires(t4,1) holds(bpg13,0,2) holds(dhap,1,2) holds(f16bp,3,2) holds(g3p,1,2) fires(reset_dhap_1,2) fires(src_f16bp_1,2) fires(t3,2) fires(t4,2) fires(t5b,2) holds(bpg13,0,3) holds(dhap,2,3) holds(f16bp,4,3) holds(g3p,1,3) fires(reset_dhap_1,3) fires(src_f16bp_1,3) fires(t3,3) fires(t4,3) fires(t5b,3) holds(bpg13,0,4) holds(dhap,2,4) holds(f16bp,5,4) holds(g3p,1,4) fires(reset_dhap_1,4) fires(src_f16bp_1,4) fires(t3,4) fires(t4,4) fires(t6,4) holds(bpg13,2,5) holds(dhap,1,5) holds(f16bp,6,5) holds(g3p,1,5) fires(reset_dhap_1,5) fires(src_f16bp_1,5) fires(t3,5) fires(t4,5) fires(t5b,5) Answer: 8 holds(bpg13,0,0) holds(dhap,0,0) holds(f16bp,0,0) holds(g3p,0,0) fires(reset_dhap_1,0) fires(src_f16bp_1,0) fires(t3,0) holds(bpg13,0,1) holds(dhap,0,1) holds(f16bp,2,1) holds(g3p,0,1) fires(reset_dhap_1,1) fires(src_f16bp_1,1) fires(t3,1) fires(t4,1) holds(bpg13,0,2) holds(dhap,1,2) holds(f16bp,3,2) holds(g3p,1,2) fires(reset_dhap_1,2) fires(src_f16bp_1,2) fires(t3,2) fires(t4,2) fires(t5b,2) holds(bpg13,0,3) holds(dhap,2,3) holds(f16bp,4,3) holds(g3p,1,3) fires(reset_dhap_1,3) fires(src_f16bp_1,3) fires(t3,3) fires(t4,3) fires(t5b,3) holds(bpg13,0,4) holds(dhap,2,4) holds(f16bp,5,4) holds(g3p,1,4) fires(reset_dhap_1,4) fires(src_f16bp_1,4) fires(t3,4) fires(t4,4) fires(t6,4) holds(bpg13,2,5) holds(dhap,1,5) holds(f16bp,6,5) holds(g3p,1,5) fires(reset_dhap_1,5) fires(src_f16bp_1,5) fires(t3,5) fires(t4,5) fires(t6,5) Answer: 9 holds(bpg13,0,0) holds(dhap,0,0) holds(f16bp,0,0) holds(g3p,0,0) fires(reset_dhap_1,0) fires(src_f16bp_1,0) fires(t3,0) holds(bpg13,0,1) holds(dhap,0,1) holds(f16bp,2,1) holds(g3p,0,1) fires(reset_dhap_1,1) fires(src_f16bp_1,1) fires(t3,1) fires(t4,1) holds(bpg13,0,2) holds(dhap,1,2) holds(f16bp,3,2) holds(g3p,1,2) fires(reset_dhap_1,2) fires(src_f16bp_1,2) fires(t3,2) fires(t4,2) fires(t5b,2) holds(bpg13,0,3) holds(dhap,2,3) holds(f16bp,4,3) holds(g3p,1,3) fires(reset_dhap_1,3) fires(src_f16bp_1,3) fires(t3,3) fires(t4,3) fires(t6,3) holds(bpg13,2,4) holds(dhap,1,4) holds(f16bp,5,4) holds(g3p,1,4) fires(reset_dhap_1,4) fires(src_f16bp_1,4) fires(t3,4) fires(t4,4) fires(t5b,4) holds(bpg13,2,5) holds(dhap,2,5) holds(f16bp,6,5) holds(g3p,1,5) fires(reset_dhap_1,5) fires(src_f16bp_1,5) fires(t3,5) fires(t4,5) fires(t5b,5) Answer: 10 holds(bpg13,0,0) holds(dhap,0,0) holds(f16bp,0,0) holds(g3p,0,0) fires(reset_dhap_1,0) fires(src_f16bp_1,0) fires(t3,0) holds(bpg13,0,1) holds(dhap,0,1) holds(f16bp,2,1) holds(g3p,0,1) fires(reset_dhap_1,1) fires(src_f16bp_1,1) fires(t3,1) fires(t4,1) holds(bpg13,0,2) holds(dhap,1,2) holds(f16bp,3,2) holds(g3p,1,2) fires(reset_dhap_1,2) fires(src_f16bp_1,2) fires(t3,2) fires(t4,2) fires(t5b,2) holds(bpg13,0,3) holds(dhap,2,3) holds(f16bp,4,3) holds(g3p,1,3) fires(reset_dhap_1,3) fires(src_f16bp_1,3) fires(t3,3) fires(t4,3) fires(t6,3) holds(bpg13,2,4) holds(dhap,1,4) holds(f16bp,5,4) holds(g3p,1,4) fires(reset_dhap_1,4) fires(src_f16bp_1,4) fires(t3,4) fires(t4,4) fires(t5b,4) holds(bpg13,2,5) holds(dhap,2,5) holds(f16bp,6,5) holds(g3p,1,5) fires(reset_dhap_1,5) fires(src_f16bp_1,5) fires(t3,5) fires(t4,5) fires(t6,5) Answer: 11 holds(bpg13,0,0) holds(dhap,0,0) holds(f16bp,0,0) holds(g3p,0,0) fires(reset_dhap_1,0) fires(src_f16bp_1,0) fires(t3,0) holds(bpg13,0,1) holds(dhap,0,1) holds(f16bp,2,1) holds(g3p,0,1) fires(reset_dhap_1,1) fires(src_f16bp_1,1) fires(t3,1) fires(t4,1) holds(bpg13,0,2) holds(dhap,1,2) holds(f16bp,3,2) holds(g3p,1,2) fires(reset_dhap_1,2) fires(src_f16bp_1,2) fires(t3,2) fires(t4,2) fires(t5b,2) holds(bpg13,0,3) holds(dhap,2,3) holds(f16bp,4,3) holds(g3p,1,3) fires(reset_dhap_1,3) fires(src_f16bp_1,3) fires(t3,3) fires(t4,3) fires(t6,3) holds(bpg13,2,4) holds(dhap,1,4) holds(f16bp,5,4) holds(g3p,1,4) fires(reset_dhap_1,4) fires(src_f16bp_1,4) fires(t3,4) fires(t4,4) fires(t6,4) holds(bpg13,4,5) holds(dhap,1,5) holds(f16bp,6,5) holds(g3p,1,5) fires(reset_dhap_1,5) fires(src_f16bp_1,5) fires(t3,5) fires(t4,5) fires(t5b,5) Answer: 12 holds(bpg13,0,0) holds(dhap,0,0) holds(f16bp,0,0) holds(g3p,0,0) fires(reset_dhap_1,0) fires(src_f16bp_1,0) fires(t3,0) holds(bpg13,0,1) holds(dhap,0,1) holds(f16bp,2,1) holds(g3p,0,1) fires(reset_dhap_1,1) fires(src_f16bp_1,1) fires(t3,1) fires(t4,1) holds(bpg13,0,2) holds(dhap,1,2) holds(f16bp,3,2) holds(g3p,1,2) fires(reset_dhap_1,2) fires(src_f16bp_1,2) fires(t3,2) fires(t4,2) fires(t5b,2) holds(bpg13,0,3) holds(dhap,2,3) holds(f16bp,4,3) holds(g3p,1,3) fires(reset_dhap_1,3) fires(src_f16bp_1,3) fires(t3,3) fires(t4,3) fires(t6,3) holds(bpg13,2,4) holds(dhap,1,4) holds(f16bp,5,4) holds(g3p,1,4) fires(reset_dhap_1,4) fires(src_f16bp_1,4) fires(t3,4) fires(t4,4) fires(t6,4) holds(bpg13,4,5) holds(dhap,1,5) holds(f16bp,6,5) holds(g3p,1,5) fires(reset_dhap_1,5) fires(src_f16bp_1,5) fires(t3,5) fires(t4,5) fires(t6,5) Answer: 13 holds(bpg13,0,0) holds(dhap,0,0) holds(f16bp,0,0) holds(g3p,0,0) fires(reset_dhap_1,0) fires(src_f16bp_1,0) fires(t3,0) holds(bpg13,0,1) holds(dhap,0,1) holds(f16bp,2,1) holds(g3p,0,1) fires(reset_dhap_1,1) fires(src_f16bp_1,1) fires(t3,1) fires(t4,1) holds(bpg13,0,2) holds(dhap,1,2) holds(f16bp,3,2) holds(g3p,1,2) fires(reset_dhap_1,2) fires(src_f16bp_1,2) fires(t3,2) fires(t4,2) fires(t6,2) holds(bpg13,2,3) holds(dhap,1,3) holds(f16bp,4,3) holds(g3p,1,3) fires(reset_dhap_1,3) fires(src_f16bp_1,3) fires(t3,3) fires(t4,3) fires(t6,3) holds(bpg13,4,4) holds(dhap,1,4) holds(f16bp,5,4) holds(g3p,1,4) fires(reset_dhap_1,4) fires(src_f16bp_1,4) fires(t3,4) fires(t4,4) fires(t5b,4) holds(bpg13,4,5) holds(dhap,2,5) holds(f16bp,6,5) holds(g3p,1,5) fires(reset_dhap_1,5) fires(src_f16bp_1,5) fires(t3,5) fires(t4,5) fires(t6,5) Answer: 14 holds(bpg13,0,0) holds(dhap,0,0) holds(f16bp,0,0) holds(g3p,0,0) fires(reset_dhap_1,0) fires(src_f16bp_1,0) fires(t3,0) holds(bpg13,0,1) holds(dhap,0,1) holds(f16bp,2,1) holds(g3p,0,1) fires(reset_dhap_1,1) fires(src_f16bp_1,1) fires(t3,1) fires(t4,1) holds(bpg13,0,2) holds(dhap,1,2) holds(f16bp,3,2) holds(g3p,1,2) fires(reset_dhap_1,2) fires(src_f16bp_1,2) fires(t3,2) fires(t4,2) fires(t6,2) holds(bpg13,2,3) holds(dhap,1,3) holds(f16bp,4,3) holds(g3p,1,3) fires(reset_dhap_1,3) fires(src_f16bp_1,3) fires(t3,3) fires(t4,3) fires(t6,3) holds(bpg13,4,4) holds(dhap,1,4) holds(f16bp,5,4) holds(g3p,1,4) fires(reset_dhap_1,4) fires(src_f16bp_1,4) fires(t3,4) fires(t4,4) fires(t5b,4) holds(bpg13,4,5) holds(dhap,2,5) holds(f16bp,6,5) holds(g3p,1,5) fires(reset_dhap_1,5) fires(src_f16bp_1,5) fires(t3,5) fires(t4,5) fires(t5b,5) Answer: 15 holds(bpg13,0,0) holds(dhap,0,0) holds(f16bp,0,0) holds(g3p,0,0) fires(reset_dhap_1,0) fires(src_f16bp_1,0) fires(t3,0) holds(bpg13,0,1) holds(dhap,0,1) holds(f16bp,2,1) holds(g3p,0,1) fires(reset_dhap_1,1) fires(src_f16bp_1,1) fires(t3,1) fires(t4,1) holds(bpg13,0,2) holds(dhap,1,2) holds(f16bp,3,2) holds(g3p,1,2) fires(reset_dhap_1,2) fires(src_f16bp_1,2) fires(t3,2) fires(t4,2) fires(t6,2) holds(bpg13,2,3) holds(dhap,1,3) holds(f16bp,4,3) holds(g3p,1,3) fires(reset_dhap_1,3) fires(src_f16bp_1,3) fires(t3,3) fires(t4,3) fires(t6,3) holds(bpg13,4,4) holds(dhap,1,4) holds(f16bp,5,4) holds(g3p,1,4) fires(reset_dhap_1,4) fires(src_f16bp_1,4) fires(t3,4) fires(t4,4) fires(t6,4) holds(bpg13,6,5) holds(dhap,1,5) holds(f16bp,6,5) holds(g3p,1,5) fires(reset_dhap_1,5) fires(src_f16bp_1,5) fires(t3,5) fires(t4,5) fires(t5b,5) Answer: 16 holds(bpg13,0,0) holds(dhap,0,0) holds(f16bp,0,0) holds(g3p,0,0) fires(reset_dhap_1,0) fires(src_f16bp_1,0) fires(t3,0) holds(bpg13,0,1) holds(dhap,0,1) holds(f16bp,2,1) holds(g3p,0,1) fires(reset_dhap_1,1) fires(src_f16bp_1,1) fires(t3,1) fires(t4,1) holds(bpg13,0,2) holds(dhap,1,2) holds(f16bp,3,2) holds(g3p,1,2) fires(reset_dhap_1,2) fires(src_f16bp_1,2) fires(t3,2) fires(t4,2) fires(t6,2) holds(bpg13,2,3) holds(dhap,1,3) holds(f16bp,4,3) holds(g3p,1,3) fires(reset_dhap_1,3) fires(src_f16bp_1,3) fires(t3,3) fires(t4,3) fires(t6,3) holds(bpg13,4,4) holds(dhap,1,4) holds(f16bp,5,4) holds(g3p,1,4) fires(reset_dhap_1,4) fires(src_f16bp_1,4) fires(t3,4) fires(t4,4) fires(t6,4) holds(bpg13,6,5) holds(dhap,1,5) holds(f16bp,6,5) holds(g3p,1,5) fires(reset_dhap_1,5) fires(src_f16bp_1,5) fires(t3,5) fires(t4,5) fires(t6,5) Atoms selected for $bpg13$ quantity extraction for the nominal case: [holds(bpg13,0,0),holds(bpg13,0,1),holds(bpg13,0,2), \tholds(bpg13,0,3),holds(bpg13,2,4),holds(bpg13,4,5)], [holds(bpg13,0,0),holds(bpg13,0,1),holds(bpg13,0,2), \tholds(bpg13,2,3),holds(bpg13,4,4),holds(bpg13,6,5)] Atoms selected for $bpg13$ quantity extraction for the modified case: [holds(bpg13,0,0),holds(bpg13,0,1),holds(bpg13,0,2), \tholds(bpg13,2,3),holds(bpg13,2,4),holds(bpg13,2,5)], [holds(bpg13,0,0),holds(bpg13,0,1),holds(bpg13,0,2), \tholds(bpg13,2,3),holds(bpg13,2,4),holds(bpg13,2,5)], [holds(bpg13,0,0),holds(bpg13,0,1),holds(bpg13,0,2), \tholds(bpg13,2,3),holds(bpg13,2,4),holds(bpg13,4,5)], [holds(bpg13,0,0),holds(bpg13,0,1),holds(bpg13,0,2), \tholds(bpg13,2,3),holds(bpg13,2,4),holds(bpg13,4,5)], [holds(bpg13,0,0),holds(bpg13,0,1),holds(bpg13,0,2), \tholds(bpg13,0,3),holds(bpg13,0,4),holds(bpg13,0,5)], [holds(bpg13,0,0),holds(bpg13,0,1),holds(bpg13,0,2), \tholds(bpg13,0,3),holds(bpg13,0,4),holds(bpg13,0,5)], [holds(bpg13,0,0),holds(bpg13,0,1),holds(bpg13,0,2), \tholds(bpg13,0,3),holds(bpg13,0,4),holds(bpg13,2,5)], [holds(bpg13,0,0),holds(bpg13,0,1),holds(bpg13,0,2), \tholds(bpg13,0,3),holds(bpg13,0,4),holds(bpg13,2,5)], [holds(bpg13,0,0),holds(bpg13,0,1),holds(bpg13,0,2), \tholds(bpg13,0,3),holds(bpg13,2,4),holds(bpg13,2,5)], [holds(bpg13,0,0),holds(bpg13,0,1),holds(bpg13,0,2), \tholds(bpg13,0,3),holds(bpg13,2,4),holds(bpg13,2,5)], [holds(bpg13,0,0),holds(bpg13,0,1),holds(bpg13,0,2), \tholds(bpg13,0,3),holds(bpg13,2,4),holds(bpg13,4,5)], [holds(bpg13,0,0),holds(bpg13,0,1),holds(bpg13,0,2), \tholds(bpg13,0,3),holds(bpg13,2,4),holds(bpg13,4,5)], [holds(bpg13,0,0),holds(bpg13,0,1),holds(bpg13,0,2), \tholds(bpg13,2,3),holds(bpg13,4,4),holds(bpg13,4,5)], [holds(bpg13,0,0),holds(bpg13,0,1),holds(bpg13,0,2), \tholds(bpg13,2,3),holds(bpg13,4,4),holds(bpg13,4,5)], [holds(bpg13,0,0),holds(bpg13,0,1),holds(bpg13,0,2), \tholds(bpg13,2,3),holds(bpg13,4,4),holds(bpg13,6,5)], [holds(bpg13,0,0),holds(bpg13,0,1),holds(bpg13,0,2), \tholds(bpg13,2,3),holds(bpg13,4,4),holds(bpg13,6,5)] Progression from raw matrix of $bpg13$ quantities in various answer-sets (rows) at various simulation steps (columns) to rates and finally to the average aggregate for the nominal case: $\\begin{bmatrix}0 & 0 & 0 & 0 & 2 & 4\\\\0 & 0 & 0 & 2 & 4 & 6\\\\\\end{bmatrix}=rate=>\\begin{bmatrix}0.8\\\\1.2\\\\\\end{bmatrix}=average=>1.0$ Progression from raw matrix of $bpg13$ quantities in various answer-sets (rows) at various simulation steps (columns) to rates and finally to the average aggregate for the modified case: $\\begin{bmatrix}0 & 0 & 0 & 2 & 2 & 2\\\\0 & 0 & 0 & 2 & 2 & 2\\\\0 & 0 & 0 & 2 & 2 & 4\\\\0 & 0 & 0 & 2 & 2 & 4\\\\0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 2\\\\0 & 0 & 0 & 0 & 0 & 2\\\\0 & 0 & 0 & 0 & 2 & 2\\\\0 & 0 & 0 & 0 & 2 & 2\\\\0 & 0 & 0 & 0 & 2 & 4\\\\0 & 0 & 0 & 0 & 2 & 4\\\\0 & 0 & 0 & 2 & 4 & 4\\\\0 & 0 & 0 & 2 & 4 & 4\\\\0 & 0 & 0 & 2 & 4 & 6\\\\0 & 0 & 0 & 2 & 4 & 6\\\\\\end{bmatrix}=rate=>\\begin{bmatrix}0.4\\\\0.4\\\\0.8 \\\\0.8 \\\\0.", "\\\\0.", "\\\\0.4 \\\\0.4 \\\\0.4 \\\\0.4 \\\\0.8 \\\\0.8 \\\\0.8 \\\\0.8 \\\\1.2 \\\\1.2\\\\\\end{bmatrix}=average=>0.6$ We find that $d=^{\\prime }<^{\\prime }$ comparing the modified case rate of $0.6$ to the nominal case rate of $1.0$ .", "Since the direction $d$ was an unknown in the query statement, our system generates produces the full query specification with $d$ replaced by $^{\\prime }<^{\\prime }$ as follows: direction of change in average rate of production of 'bpg13' is '<' (0.6<1) when observed between time step 0 and time step 5 comparing nominal pathway with modified pathway obtained; due to interventions: remove 'dhap' as soon as produced; using initial setup: continuously supply 'f16bp' in quantity 1; Evaluation Methodolgy A direct comparison against other tools is not possible, since most other programs explore one state evolution, while we explore all possible state evolutions.", "In addition ASP has to ground the program completely, irrespective of whether we are computing one answer or all.", "So, to evaluate our system, we compare our results for the questions from the 2nd Deep KR Challenge against the answers they have provided.", "Our results in essence match the responses given for the questions.", "Related Work In this section, we relate our high level language with other high level action languages.", "Comparison with $\\pi $ -Calculus $\\pi $ -calculus is a formalism that is used to model biological systems and pathways by modeling biological systems as mobile communication systems.", "We use the biological model described by [65] for comparison against our system.", "In their model they represent molecules and their domains as computational processes, interacting elements of molecules as communication channels (two molecules interact if they fit together as in a lock-and-key mechanism), and reactions as communication through channel transmission.", "$\\pi $ -calculus models have the ability of changing their structure during simulation.", "Our system on the other hand only allows modification of the pathway at the start of simulation.", "Regular $\\pi $ -calculus models appear qualitative in nature.", "However, stochastic extensions allow representation of quantitative data [62].", "In contrast, the focus of our system is on the quantitative+qualitative representation using numeric fluents.", "It is unclear how one can easily implement maximal-parallelism of our system in $\\pi $ -calculus, where a maximum number of simultaneous actions occur such that they do not cause a conflict.", "Where, a set of actions is said to be in conflict if their simultaneous execution will cause a fluent to become negative.", "Comparison with Action Language $\\mathcal {A}$ Action language $\\mathcal {A}$ [26] is a formalism that has been used to model biological systems and pathways.", "First we give a brief overview of $\\mathcal {A}$ in an intuitive manner.", "Assume two sets of disjoint symbols containing fluent names and action names, then a fluent expression is either a fluent name $F$ or $\\lnot F$ .", "A domain description is composed of propositions of the following form: value proposition: $F \\mathbf {~after~} A_1;\\dots ;A_m$ , where $(m \\ge 0)$ , $F$ is a fluent and $A_1,\\dots ,A_m$ are fluents.", "effect propostion: $A \\mathbf {~causes~} F \\mathbf {~if~} P_1,\\dots ,P_n$ , where $(n \\ge 0)$ , $A$ is an action, $F,P_1,\\dots ,P_n$ are fluent expressions.", "$P_1,\\dots ,P_n$ are called preconditions of $A$ and the effect proposition describes the effect on $F$ .", "We relate it to our work: Fluents are boolean.", "We support numeric valued fluents, with binary fluents.", "Fluents are non-inertial, but inertia can be added.", "Our fluents are always intertial.", "Action description specifies the effect of an action.", "Our domain description specifies `natural'-actions, which execute automatically when their pre-conditions are satisfied, subject to certain conditions.", "As a result our domain description represents trajectories.", "No built in support for aggregates exists.", "We support a selected set of aggregates, on a single trajectory and over multiple trajectories.", "Value propositions in $\\mathcal {A}$ are representable as observations in our query language.", "Comparison with Action Language $\\mathcal {B}$ Action language $\\mathcal {B}$ extends $\\mathcal {A}$ by adding static causal laws, which allows one to specify indirect effects or ramifications of an action [28].", "We relate it to our work below: Inertia is built into the semantics of $\\mathcal {B}$ [28].", "Our language also has intertia built in.", "$\\mathcal {B}$ supports static causal laws that allow defining a fluent in terms of other fluents.", "We do not support static causal laws.", "Comparison with Action Language $\\mathcal {C}$ Action language $\\mathcal {C}$ is based on the theory of causal explanation, i.e.", "a formula is true if there is a cause for it to be true [30].", "It has been previously used to represent and reason about biological pathways [23].", "We relate it to our work below: $\\mathcal {C}$ supports boolean fluents only.", "We support numeric valued fluents, and binary fluents.", "$\\mathcal {C}$ allows both inertial and non-inertial fluents.", "While our fluents are always inertial.", "$\\mathcal {C}$ support static causal laws (or ramifications), that allow defining a fluent in terms of other fluents.", "We do not support them.", "$\\mathcal {C}$ describes causal relationships between fluents and actions.", "Our language on the other hand describes trajectories.", "Comparison with Action Language $\\mathcal {C+}$ First, we give a brief overview of $\\mathcal {C+}$ [29].", "Intuitively, atoms $\\mathcal {C+}$ are of the form $c=v$ , where $c$ is either a fluent or an action constant, $v$ belongs to the domain of $c$ , and fluents and actions form a disjoint set.", "A formula is a propositional combination of atoms.", "A fluent formula is a formula in which all constants are fluent constants; and an action formula is a formula with one action constant but no fluent constants.", "An action description in $\\mathcal {C+}$ is composed of causal laws of the following forms: static law: $\\mathbf {caused~} F \\mathbf {~if~} G$ , where $F$ and $G$ are fluent formulas action dynamic law: $\\mathbf {caused~} F \\mathbf {~if~} G$ , where $F$ is an action formula and $G$ is a formula fluent dynamic law: $\\mathbf {caused~} F \\mathbf {~if~} G \\mathbf {~after~} H$ , where $F$ and $G$ are fluent formulas and $H$ is a formula Concise forms of these laws exist, e.g.", "`$\\mathbf {intertial~} f \\equiv \\mathbf {caused~} f=v \\mathbf {~if~} f=v \\mathbf {~after~} f=v, \\forall v \\in \\text{ domain of } f$ ' that allow a more intuitive program declaration.", "We now relate it to our work.", "Some of the main differences include: Fluents are multi-valued, fluent values can be integer, boolean or other types.", "We support integer and binary valued fluents only.", "Actions are multi-valued.", "We do not support multi-valued actions.", "Both inertial and non-inertial fluents are supported.", "In comparison we allow inertial fluents only.", "Static causal laws are supported that allow changing the value of a fluent based on other fluents (ramifications).", "We do not allow static causal laws.", "Effect of parallel actions on numeric fluents is not additive.", "However, the additive fluents extension [51] adds the capability of additive fluents through new rules.", "The extended language, however, imposes certain restrictions on additive fluents and also restricts the domain of additive actions to boolean actions only.", "Our fluents are always additive.", "Supports defaults.", "We do not have the same notion as defaults, but allow initial values for fluents in our domain description.", "Action's occurrence and its effect are defined in separate statements.", "In our case, the action's occurrence and effect are generally combined in one statement.", "Although parallel actions are supported, it is unclear how one can concisely describe the condition implicit in our system that simultaneously occurring actions may not conflict.", "Two actions conflict if their simultaneous execution will cause a fluent to become negative.", "Exogenous actions seem the closest match to our may execute actions.", "However, our actions are `natural', in that they execute automatically when their pre-conditions are satisfied, they are not explicitly inhibited, and they do not conflict.", "Actions conflict when their simultaneous execution will cause one of the fluents to become negative.", "The exogenous-style character of our actions holds when the firing style is `$$ '.", "When the firing style changes, the execution semantics change as well.", "Consider the following two may execute statements in our language: $a_1 \\mathbf {~may~execute~causing~} f_1 \\mathbf {~change~value~by~} -5 \\mathbf {~if~} f_2 \\mathbf {~has~value~} 3 \\mathbf {~or~higher} \\\\a_2 \\mathbf {~may~execute~causing~} f_1 \\mathbf {~change~value~by~} -3 \\mathbf {~if~} f_2 \\mathbf {~has~value~} 2 \\mathbf {~or~higher}$ and two states: (i) $f_1=10, f_2=5$ , (ii) $f_1=6,f_2=5$ .", "In state (i) both $a_1,a_2$ can occur simultaneously (at one point) resulting in firing-choices $\\big \\lbrace \\lbrace a_1,a_2 \\rbrace ,\\lbrace a_1\\rbrace ,\\lbrace a_2\\rbrace ,\\lbrace \\rbrace ,\\big \\rbrace $ ; whereas, in state (ii) only one of $a_1$ or $a_2$ can occur at one point resulting in the firing-choices: $\\big \\lbrace \\lbrace a_1\\rbrace ,\\lbrace a_2\\rbrace ,\\lbrace \\rbrace ,\\big \\rbrace $ because of a conflict due to the limited amount of $f_1$ .", "These firing choices apply for firing style `', which allows any combination of fireable actions to occur.", "If the firing style is set to `max', the maximum set of non-conflicting actions may fire, and the firing choices for state (i) change to $\\big \\lbrace \\lbrace a_1,a_2 \\rbrace \\big \\rbrace $ and the firing choices for state (ii) change to $\\big \\lbrace \\lbrace a_1\\rbrace ,\\lbrace a_2\\rbrace \\big \\rbrace $ .", "If the firing style is set to `1', at most one action may fire at one point, and the firing choices for both state (i) and state (ii) reduce to $\\big \\lbrace \\lbrace a_1\\rbrace ,\\lbrace a_2\\rbrace ,\\lbrace \\rbrace \\big \\rbrace $ .", "So, the case with `' firing style can be represented in $\\mathcal {C+}$ with exogenous actions $a_1,a_2$ ; the case with `1' firing style can be represented in $\\mathcal {C+}$ with exogenous actions $a_1,a_2$ and a constraint requiring that both $a_1,a_2$ do not occur simultaneously; while the case with `max' firing style can be represented by exogenous actions $a_1,a_2$ with additional action dynamic laws.", "They will still be subject to the conflict checking.", "Action dynamic laws can be used to force actions similar to our must execute actions.", "Specification of initial values of fluents seem possible through the query language.", "The default statement comes close, but it does not have the same notion as setting a fluent to a value once.", "We support specifying initial values both in the domain description as well as the query.", "There does not appear built-in support for aggregation of fluent values within the same answer set, such as sum, count, rate, minimum, maximum etc.", "Although some of it could be implemented using the additive fluents extension.", "We support a set of aggregates, such as total, and rate.", "Additional aggregates can be easily added.", "We support queries over aggregates (such as minimum, maximum, average) of single-trajectory aggregates (such as total, and rate etc.)", "over a set of trajectories.", "We also support comparative queries over two sets of trajectories.", "Our queries allow modification of the domain description as part of query evaluation.", "Comparison with $\\mathcal {BC}$ Action language $\\mathcal {BC}$ combines features of $\\mathcal {B}$ and $\\mathcal {C+}$ [52].", "First we give a brief overview of $\\mathcal {BC}$ .", "Intuitively, $\\mathcal {BC}$ has actions and valued fluents.", "A valued fluent, called an atom, is of the form `$f=v$ ', where $f$ is a fluent constant and $v \\in domain(f)$ .", "A fluent can be regular or statically determined.", "An action description in $\\mathcal {BC}$ is composed of static and dynamic laws of the following form: static law: $A_0 \\mathbf {~if~} A_1,\\dots ,A_m \\mathbf {~ifcons~} A_{m+1},\\dots ,A_n$ , where $(n \\ge m \\ge 0)$ , and each $A_i$ is an atom.", "dynamic law: $A_0 \\mathbf {~after~} A_1,\\dots ,A_m \\mathbf {~ifcons~} A_{m+1},\\dots ,A_n$ , where $(n \\ge m \\ge 0)$ , $A_0$ is a regular fluent atom, each of $A_1,\\dots ,A_m$ is an atom or an action constant, and $A_{m+1},\\dots ,A_n$ are atoms.", "Concise forms of these laws exist that allow a more intuitive program declaration.", "Now we relate it to our work.", "Some of the main differences include: Fluents are multi-valued, fluent values can which can be integer, boolean, or other types.", "We only support integer and binary fluents.", "Static causal laws are allowed.", "We do not support static causal laws.", "Similar to $\\mathcal {C+}$ numeric fluent accumulation is not supported.", "It is supported in our system.", "It is unclear how aggregate queries within a trajectory can be concisely represented.", "Aggregate queries such as rate are supported in our system.", "It does not seem that queries over multiple trajectories or sets of trajectories are supported.", "Such queries are supported in our system.", "Comparison with ASPMT ASPMT combines ASP with Satisfiability Modulo Theories.", "We relate the work in [53] where $\\mathcal {C+}$ is extended using ASPMT with our work.", "It adds support for real valued fluents to $\\mathcal {C+}$ including additive fluents.", "Thus, it allows reasoning with continuous and discrete processes simultaneously.", "Our language does not support real numbers directly.", "Several systems also exist to model and reason with biological pathway.", "For example: Comparison with BioSigNet-RR BioSigNet-RR [5] is a system for representing and reasoning with signaling networks.", "We relate it to our work in an intuitive manner.", "Fluents are boolean, so qualitative queries are possible.", "We support both integer and binary fluents, so quantiative queries are also possible.", "Indirect effects (or ramifications) are supported.", "We do not support these.", "Action effects are captured separately in `$\\mathbf {~causes~}$ statement' from action triggering statements `$\\mathbf {~triggers~}$ ' and `$\\mathbf {~n\\_triggers~}$ '.", "We capture both components in a `$\\mathbf {~may~execute~causing~}$ ' or `$\\mathbf {~normally~must~execute~causing~}$ ' statement.", "Their action triggering semantics have some similarity to our actions.", "Just like their actions get triggered when the action's pre-conditions are satisfied, our actions are also triggered when their pre-conditions are satisfied.", "However, the triggering semantics are different, e.g.", "their triggers statement causes an action to occur even if it is disabled, we do not have an equivalent for it; and their n_triggers is similar in semantics to normally must execute causing statement.", "It is not clear how loops in biological systems can be modeled in their system.", "Loops are possible in our by virtue of the Petri Net semantics.", "Their queries can have time-points and their precedence relations as part of the query.", "Though our queries allow the specification of some time points for interval queries, time-points are not supported in a similar way.", "However, we do support certain types of observation relative queries.", "The intervention in their planning queries has similarities to interventions in our system.", "However, it appears that our intervention descriptions are higher level.", "Conclusion In this chapter we presented the BioPathQA system and the languages to represent and query biological pathways.", "We also presented a new type of Petri Net, the so called Guarded-Arc Petri Net that is used as a model behind our pathway specification language, which shares certain aspects with CPNs [39], but our semantics for reset arcs is different, and we allow must-fire actions that prioritize actions for firing over other actions.", "We also showed how the system can be applied to questions from college level text books that require deeper reasoning and cannot be answered by using surface level knowledge.", "Although our system is developed with respect to the biological domain, it can be applied to non-biological domain as well.", "Some of the features of our language include: natural-actions that automatically fire when their prerequisite conditions are met (subject to certain restrictions); an automatic default constraint that ensures fluents do not go negative, since they model natural systems substances; a more natural representation of locations; and control of the level of parallelism to a certain degree.", "Our query language also allows interventions similar to Pearl's surgeries [59], which are more general than actions.", "Next we want to apply BioPathQA to a real world application by doing text extraction.", "Knowledge for real world is extracted from research papers.", "In the next chapter we show how such text extraction is done for pathway construction and drug development.", "We will then show how we can apply BioPathQA to the extracted knowledge to answer questions about the extracted knowledge.", "Text Extraction for Real World Applications In the previous chapter we looked at the BioPathQA system and how it answers simulation based reasoning questions about biological pathways, including questions that require comparison of alternate scenarios through simulation.", "These so called `what-if' questions arise in biological activities such as drug development, drug interaction, and personalized medicine.", "We will now put our system and language in context of such activities.", "Cutting-edge knowledge about pathways for activities such as drug development, drug interaction, and personalized medicine comes in the form of natural language research papers, thousands of which are published each year.", "To use this knowledge with our system, we need to perform extraction.", "In this chapter we describe techniques we use for such knowledge extraction for discovering drug interactions.", "We illustrate with an example extraction how we organize the extracted knowledge into a pathway specification and give examples of relevant what-if questions that a researcher performing may ask in the drug development domain.", "Introduction Thousands of research papers are published each year about biological systems and pathways over a broad spectrum of activities, including interactions between dugs and diseases, the associated pathways, and genetic variation.", "Thus, one has to perform text extraction to extract relationships between the biochemical processes, their involvement in diseases, and their interaction with drugs.", "For personalized medicine, one is also interested in how these interrelationships change in presence of genetic variation.", "In short, we are looking for relationships between various components of the biochemical processes and their internal and external stimuli.", "Many approaches exist for extracting relationships from a document.", "Most rely on some form of co-occurrence, relative distance, or order of words in a single document.", "Some use shallow parsing as well.", "Although these techniques tend to have a higher recall, they focus on extracting explicit relationships, which are relationships that are fully captured in a sentence or a document.", "These techniques also do not capture implicit relationships that may be spread across multiple documents.", "are spread across multiple documents relating to different species.", "Additional issues arise from the level of detail from in older vs. newer texts and seemingly contradictory information due to various levels of confidence in the techniques used.", "Many do not handle negative statements.", "We primarily use a system called PTQL [75] to extract these relationships, which allows combining the syntactic structure (parse tree), semantic elements, and word order in a relationship query.", "The sentences are pre-processed by using named-entity recognition, and entity normalization to allow querying on classes of entity types, such as drugs, and diseases; and also to allow cross-linking relationships across documents when they refer to the same entity with a different name.", "Queries that use such semantic association between words/phrases are likely to produce higher precision results.", "Source knowledge for extraction primarily comes from thousands of biological abstracts published each year in PubMed http://www.ncbi.nlm.nih.gov/pubmed.", "Next we briefly describe how we extract relationships about drug interactions.", "Following that we briefly describe how we extract association of drugs, and diseases with genetic variation.", "We conclude this chapter with an illustrative example of how the drug interaction relationships are used with our system to answer questions about drug interactions and how genetic variation could be utilized in our system.", "Extracting Relationships about Drug Interactions We summarize the extraction of relationships for our work on drug-drug interactions from [73].", "Studying drug-drug interactions are a major activity in drug development.", "Drug interactions occur due to the interactions between the biological processes / pathways that are responsible metabolizing and transporting drugs.", "Metabolic processes remove a drug from the system within a certain time period.", "For a drug to remain effective, it must be maintained within its therapeutic window for the period of treatment, requiring taking the drug periodically.", "Outside the therapeutic window, a drug can become toxic if a quantity greater than the therapeutic window is retained; or it can become ineffective if a quantity less than the therapeutic window is retained.", "Since liver enzymes metabolize most drugs, it is the location where most metabolic-interaction takes place.", "Induction or inhibition of these enzymes can affect the bioavailability of a drug through transcriptional regulation, either directly or indirectly.", "For example, if drug $A$ inhibits enzyme $E$ , which metabolizes drug $B$ , then the bioavailability of drug $B$ will be higher than normal, rendering it toxic.", "On the other hand, if drug $A$ induces enzyme $E$ , which metabolizes drug $B$ , then drug $B$ 's bioavailability will be lesser than normal, rendering it ineffective.", "Inhibition of enzymes is a common form of drug-drug interactions [10].", "In direct inhibition, a drug $A$ inhibit enzyme $E$ , which is responsible for metabolism of drug $B$ .", "Drug $A$ , leads to a decrease in the level of enzyme $E$ , which in turn can increase bioavailability of drug $B$ potentially leading to toxicity.", "Alternatively, insufficient metabolism of drug $B$ can lead to smaller amount of drug $B$ 's metabolites being produced, leading to therapeutic failure.", "An example of one such direct inhibition is the interaction between CYP2D6 inhibitor quinidine and CYP2D6 substrates (i.e.", "substances metabolized by CYP2D6), such as Codeine.", "The inhibition of CYP2D6 increases the bioavailability of drugs metabolized by CYP2D6 leading to adverse side effects.", "Another form of drug interactions is through induction of enzymes [10].", "In direct induction, a drug $A$ induces enzyme $E$ , which is responsible for metabolism of drug $B$ .", "An example of such direct induction is between phenobarbital, a CYP2C9 inducer and warfarin (a CYP2C9 substrate).", "Phenobarbital leads to increased metabolism of warfarin, decreasing warfarinÕs bioavailability.", "Direct interaction due to induction though possible is not as common as indirect interaction through transcription factors, which regulate the drug metabolizing enzymes.", "In such an interaction, drug $A$ activates a transcription factor $TF$ , which regulates and induces enzyme $E$ , where enzyme $E$ metabolizes drug $B$ .", "Transcription factors are referred to as regulators of xenobiotic-metabolizing enzymes.", "Examples of such regulators include aryl hydrocarbon receptor AhR, pregnane X receptor PXR and constitutive androstane receptor CAR.", "Drug interactions can also occur due to the induction or inhibition of transporters.", "Transporters are mainly responsible for cellular uptake or efflux (elimination) of drugs.", "They play an important part in drug disposition, by transporting drugs into the liver cells, for example.", "Transporter-based drug interactions, however, are not as well studies as metabolism-based interactions [10].", "Method Extraction of drug-drug interactions from the text can either be explicit or implicit.", "Explicit extraction refers to extraction of drug-drug interaction mentioned within a single sentence, while implicit extraction requires extraction of bio-properties of drug transport, metabolism and elimination that can lead to drug-drug interaction.", "This type of indirect extraction combines background information about biological processes, identification of protein families and the interactions that are involved in drug metabolism.", "Our approach is to extract both explicit and implicit drug interactions as summarized in Fig REF and it builds upon the work done in [74].", "Figure: This figure from outlines the effects of drug A on drug B through (a) direct induction/inhibition of enzymes; (b) indirect induction/inhibition of transportation factors that regulate the drug-metabolizing enzymes.Explicit Drug Interaction Extraction Explicit extraction mainly extracts drug-drug interactions directly mentioned in PubMed (or Medline) abstracts.", "For example, the following sentences each have a metabolic interaction mentioned within the same sentence: Ciprofloxacin strongly inhibits clozapine metabolism.", "(PMID: 19067475) Enantioselective induction of cyclophosphamide metabolism by phenytoin.", "which can be extracted by using the following PTQL query using the underlined keywords from above sentences: //S{//?", "[Tag=`Drug'](d1) => //?", "[Value IN {`induce',`induces',`inhibit',`inhibits'}](v) => //?[Tag=`Drug'](d2) => //?", "[Value=`metabolism'](w)} ::: [d1 v d2 w] 5 : d1.value, v.value, d2.value.", "This PTQL query specifies that a drug (denoted by d1) must be followed by one of the keywords from $\\lbrace `induce^{\\prime },`inhibit^{\\prime }, `inhibits^{\\prime }\\rbrace $ (denoted by v), which in turn must be followed by another drug (denoted by d2) followed the keyword $`metabolism^{\\prime }$ (denoted by w); all found within a proximity of 5 words of each other.", "The query produces tripes of $\\langle d1, v, d2 \\rangle $ values as output.", "Thus the results will produce triples $\\langle d1, induces, d2 \\rangle $ and $\\langle d1, inhibits, d2 \\rangle $ which mean that the drug d1 increases the effect of d2 (i.e.", "$\\langle d1, increases, d2 \\rangle $ ) and decreases the effect of d2 (i.e.", "$\\langle d1, decreases, d2 \\rangle $ ) respectively.", "For example, the sentence S1 above matches this PTQL query and the query will produce the triplet $\\langle \\text{ciprofloxacin}, \\text{increases}, \\text{clozapine} \\rangle $ .", "Implicit Drug Interaction Extraction Implicit extraction mainly extracts drug-drug interactions not yet published, but which can be inferred from published articles and properties of drug metabolism.", "The metabolic properties themselves have their origin in various publications.", "The metabolic interactions extracted from published articles and the background knowledge of properties of drug metabolism are reasoned with in an automated fashion to infer drug interactions.", "The following table outlines the kinds of interactions extracted from the text: Table: This table from shows the triplets representing various properties relevant to the extraction of implicit drug interactions and their description.which require multiple PTQL queries for extraction.", "As an example, the following PTQL query is used to extract $\\langle protein, metabolizes, drug \\rangle $ triplets: //S{/?", "[Tag=`Drug'](d1) => //VP{//?", "[Value IN {`metabolized',`metabolised'}](rel) => //?", "[Tag=`GENE'](g1)}} ::: g1.value, rel.value, d1.value which specifies that the extracted triplets must have a drug (denoted by d1) followed by a verb phrase (denoted by VP) with the verb in $\\lbrace `metabolized^{\\prime },`metabolised^{\\prime } \\rbrace $ , followed by a gene (denoted by g1).", "Table REF shows examples of extracted triplets.", "Table: This table from shows the triplets representing various properties relevant to the extraction of implicit drug interactions and their description.", "Data Cleaning The protein-protein and protein-drug relationships extracted from the parse tree database need an extra step of refinement to ensure that they correspond to the known properties of drug metabolism.", "For instance, for a protein to metabolize a drug, the protein must be an enzyme.", "Similarly, for a protein to regulate an enzyme, the protein must be a transcription factor.", "Thus, the $\\langle protein, metabolizes, drug \\rangle $ facts get refined to $\\langle enzyme, metabolizes, drug \\rangle $ and $\\langle protein, regulates, protein \\rangle $ gets refined to $\\langle transcription factor, regulates, enzyme \\rangle $ respectively.", "Classification of proteins is done using UniProt, the Gene Ontology (GO) and Entrez Gene summary by applying rules such as: A protein p is an enzyme if it belongs to a known enzyme family, such as CYP, UGT or SULT gene families; or is annotated under UniProt with the hydrolase, ligase, lyase or transferase keywords; or is listed under the “metabolic process” GO-term; or its Entrez Gene summary mentions key phrases like “drug metabolism” or roots for “enzyme” or “catalyzes”.", "A protein p is considered as a transcription factor if it is annotated with keywords transcription, transcription-regulator or activator under UniProt; or it is listed under the “transcription factor activity” category in GO; or its Entrez Gene summary contains the phrase “transcription factor”.", "Additional rules are applied to remove conflicting information, such as, favoring negative extractions (such as `$P$ does not metabolize $D$ ') over positive extractions (such as `$P$ metabolizes $D$ ').", "For details, see [73].", "Results The correctness of extracted interactions was determined by manually compiling a gold standard for each type of interaction using co-occurrence queries.", "For example, for $\\langle protein, metabolizes, drug \\rangle $ relations, we examined sentences that contain co-occurrence of protein, drug and one of the keywords “metabolized”, “metabolize”, “metabolises”, “metabolise”, “substrate” etc.", "Table REF summarizes the performance of our extraction approach.", "Table: Performance of interactions extracted from Medline abstracts.", "TP represents true-positives, while FN represents false-negatives Extracting Knowledge About Genetic Variants We summarize the relevant portion of our work on associating genetic variants with drugs, diseases and adverse reactions as presented in [33].", "Incorrect drug dosage is the leading cause of adverse drug reactions.", "Doctors prescribe the same amount of medicine to a patient for most drugs using the average drug response, even though a particular person's drug response may be higher or lower than the average case.", "A large part of the difference in drug response can be attributed to single nucleotide polymorphisms (SNPs).", "For example, the enzyme CYP2D6 has 70 known allelic variations, 50 of which are non-functional [31].", "Patients with poor metabolizer variations may retain higher concentration of drug for typical dosage, while patients with rapid metabolizers may not even reach therapeutic level of drug or toxic level of drug metabolites [68].", "Thus, it is important to consider the individual's genetic composition for dosage determination, especially for narrow therapeutic index drugs.", "Scientists studying these variations have grouped metabolizers into categories of poor (PM), intermediate (IM), rapid (RM) and ultra-rapid metabolizers (UM) and found that for some drugs, only 20% of usual dosage was required for PM and up to 140% for UM [38].", "Information about SNPs, their frequency in various population groups, their effect on genes (enzymic activity) and related data is stored in research papers, clinical studies and surveys.", "However, it is spread-out among them.", "Various databases collect this information in different forms.", "PharmGKB collects information such information and how it related to drug response [76].", "However, it is a small percentage of the total number of articles on PharmGKB, due to time consuming manual curation of data.", "Our work focuses on automatically extracting genetic variations and their impact on drug responses from PubMed abstracts to catch up with the current state of research in the biological domain, producing a repository of knowledge to assist in personalized medicine.", "Our approach leverages on as many existing tools as possible.", "Methods Next, we describe the methods used in our extraction, including: named entity recognition, entity normalization, and relation extraction.", "Named Entity Recognition We want to identify entities including genes (also proteins and enzymes), drugs, diseases, ADRs (adverse drug reactions), SNPs, RefSNPs (rs-numbers), names of alleles, populations and frequencies.", "For genes, we use BANNER [49] trained on BioCreative II GM training set [45].", "For genotypes (genetic variations including SNPs) we used a combination of MutationFinder [15] and custom components.", "Custom components were targeted mostly on non-SNPs (“c.76_78delACT”, 11MV324KF”) and insertions/deletions (“1707 del T”, “c.76_77insG”), RefSNPs (rs-numbers) and names of alleles/haplotypes (“CYP2D64”, “T allele”, “null allele”).", "For diseases (and ADRs), we used BANNER trained on a corpus of 3000 sentences with disease annotations [50].", "An additional 200 random sentences containing no disease were added from BioCreative II GM data to offset the low percentage (10%) of sentences without disease in the 3000 sentence corpus.", "In addition to BANNER, we used a dictionary extracted from UMLS.", "This dictionary consisted of 162k terms for 54k concepts from the six categories “Disease or Syndrome”, ”Neoplastic Process”,“Congenital Abnormality”,“Mental or Behavioral Dysfunction”,“Experimental Model of Disease” and “Acquired Abnormality”.", "The list was filtered to remove unspecific as well as spurious disease names such as “symptoms”, “disorder”, .... A dictionary for adverse drug reactions originated from SIDER Side Effect Resource [47], which provides a mapping of ADR terminology to UMLS CUIs.", "It consisted of 1.6k UMLS concepts with 6.5k terms.", "For drugs, we used a dictionary based on DrugBank[80] containing about 29k different drug names including both generic as well as brand names.", "We used the cross-linking information from DrugBank to collect additional synonyms and IDs from PharmGKB.", "We cross linked to Compound and Substance IDs from PubChem to provide hyperlinks to additional information.", "For population, we collected a dictionary of terms referring to countries, regions, regions inhabitants and their ethnicities from WikiPedia, e.g.", "“Caucasian”, “Italian”, “North African”, .... We filtered out irrelevant phrases like “Chinese [hamster]”.", "For frequencies, we extract all numbers and percentages as well as ranges from sentences that contain the word “allele”, “variant”, “mutation”, or “population”.", "The output is filtered in this case as well to remove false positives referring to p-values, odd ratios, confidence intervals and common trigger words.", "Entity Normalization Genes, diseases and drugs can appear with many different names in the text.", "For example, “CYP2D6” can appear as “Cytochrome p450 2D6” or “P450 IID6” among others, but they all refer to the same enzyme (EntrezGene ID 1565).", "We use GNAT on recognized genes [32], but limit them to human, mouse and rat genes.", "The gene name recognized by BANNER is filtered by GNAT to remove non-useful modifiers and looked up against EntrezGene restricted to human, mouse and rat genes to find candidate IDs for each gene name.", "Ambiguity (multiple matches) is resolved by matching the text surrounding the gene mention with gene's annotation from a set of resources, like EntrezGene, UniProt.", "Drugs and diseases/ADRs are resolved to their official IDs from DrugBank or UMLS.", "If none is found, we choose an ID for it based on its name.", "Genetic variants Genetic variations are converted to HGVScite [21] recommended format.", "Alleles were converted to the star notation (e.g.", "“CYP2D61”) and the genotype (“TT allele”) or fixed terms such as “null allele” and “variant allele”.", "Populations mentions are mapped to controlled vocabulary to remove orthographic and lexical variations.", "Relation Extraction Twelve type of relations were extracted between the detected entities as given in Table REF .", "Different methods were applied to detect different relations depending upon relation type, sentence structure and whether another method was able to extraction a relation beforehand.", "Gene-drug, gene-disease, drug-disease were extracted using sentence based co-occurrence (possibly refined by using relation-specific keywords) due to its good precision yield of this method for these relations.", "For other relations additional extraction methods were implemented.", "These include: High-confidence co-occurrence that includes keywords These co-occurences have the relation keyword in them.", "This method is applied to gene-drug, gene-disease, drug-ADR, drug-disease and mutation-disease associations.", "It uses keywords from PolySearch [79] as well as our own.", "Co-occurrence without keywords Such co-occurrences do not require any relationship keyword.", "This method is used for allele-population and variant-population relationships.", "This method can misidentify negative relationships.", "High-confidence relationships, if not found with a keyword drop down to this method for a lower confidence result.", "1:n co-occurrence Relationships where one entity has one instance in a given sentence and the other occurs one or more times.", "Single instance entity may have more than one occurrence.", "This method is useful in identifying gene mutations, where a gene is mentioned in a sentence along with a number of its mutations.", "The gene itself may be repeated.", "Enumerations with matching counts Captures entities in sentences where a list of entities is followed by an equal number of counts.", "This method is useful in capturing alleles and their associated frequencies, e.g.", "“The frequencies of CYP1B11, 2, 3, and 4 alleles were 0.087, 0.293, 0.444, and 0.175, respectively.” Least common ancestor (LCA) sub-tree Assigns associations based on distance in parse tree.", "We used Stanford parser [42] to get grammatical structure of a sentence as a dependency tree.", "This allows relating verb to its subject and noun to its modifiers.", "This method picks the closest pair in the lowest common ancestors (dependency) sub-tree of the entities.", "Maximum distance in terms of edges connecting the entity nodes was set to 10, which was determined empirically to provide the best balance between precision and recall.", "This method associates frequencies with alleles in the sentence “The allele frequencies were 18.3% (-24T) and 21.2% (1249A)”.", "m:n co-occurrence This method builds associations between all pairs of entities.", "Low confidence co-occurrence This acts as the catch-all case if none of the above methods work.", "Table: Unique binary relations identified between detected entities from .These methods were applied in order to each sentence, stoping at the first method that extracted the desired relationship.", "Order of these methods was determined empirically based of their precision.", "The order of the method used determines our confidence in the result.", "If none of the higher confidence methods are successful, a co-occurrence based method is used for extraction with low confidence.", "Abstract-level co-occurrence are also extracted to provide hits on potential relations.", "They appear in the database only when they appear in more than a pre-set threshold number of abstracts.", "Results Performance was evaluated by evaluating the precision and recall of individual components and coverage of existing results.", "Precision and recall were tested by processing 3500 PubMed abstracts found via PharmGKB relations and manually checking the 2500 predictions.", "Coverage was tested against DrugBank and PharmGKB.", "Extracted relations went through manual evaluation for correctness.", "Each extraction was also assigned a confidence value based on the confidence in the method of extraction used.", "We got a coverage of 91% of data in DrugBank and 94% in PharmGKB.", "Taking into false positive rates for genes, drugs and gene-drug relations, SNPshot has more than 10,000 new relations.", "Applying BioPathQA to Drug-Drug Interaction Discovery Now we use our BioPathQA system from chapter to answer questions about drug-drug interaction using knowledge extracted from research publications using the approach in sections REF ,REF .", "We supplement the extracted knowledge with domain knowledge as needed.", "Let the extracted facts be as follows: The drug $gefitinib$ is metabolized by $CYP3A4$ .", "The drug $phenytoin$ induces $CYP3A4$ .", "Following additional facts have been provided about a new drug currently in development: A new drug being developed $test\\_drug$ is a CYP3A4 inhibitor We show the pathway specification based on the above facts and background knowledge, then elaborate on each component: $&\\mathbf {domain~of~} gefitinib \\mathbf {~is~} integer, cyp3a4 \\mathbf {~is~} integer, \\nonumber \\\\&~~~~~~~~~~~~~~~~~~~~ phenytoin \\mathbf {~is~} integer, test\\_drug \\mathbf {~is~} integer\\\\&t1 \\mathbf {~may~execute~causing~} \\nonumber \\\\&~~~~~~~~~~~~~~~~~~~~ gefitinib \\mathbf {~change~value~by~} -1, \\nonumber \\\\&~~~~~~~~~~~~~~~~~~~~ cyp3a4 \\mathbf {~change~value~by~} -1, \\nonumber \\\\&~~~~~~~~~~~~~~~~~~~~ cyp3a4 \\mathbf {~change~value~by~} +1\\\\&\\mathbf {normally~stimulate~} t1 \\mathbf {~by~factor~} 2 \\nonumber \\\\&~~~~~~~~~~~~~~~~~~~~ \\mathbf {~if~} phenytoin \\mathbf {~has~value~} 1 \\mathbf {~or~higher~} \\\\&\\mathbf {inhibit~} t1 \\mathbf {~if~} test\\_drug \\mathbf {~has~value~} 1 \\mathbf {~or~higher~} \\\\&\\mathbf {initially~} gefitinib \\mathbf {~has~value~} 20, cyp3a4 \\mathbf {~has~value~} 60, \\nonumber \\\\&~~~~~~~~~~~~~~~~~~~~ phenytoin \\mathbf {~has~value~} 0, test\\_drug \\mathbf {~has~value~} 0 \\\\&\\mathbf {firing~style~} max$ Line REF declares the domain of the fluents as integer numbers.", "Line represents the activity of enzyme $cyp3a4$ as the action $t1$ .", "Due to the enzymic action $t1$ , one unit of $gefitinib$ is metabolized, and thus converted to various metabolites (not shown here).", "The enzymic action uses one unit of $cyp3a4$ as catalyst, which is used in the reaction and released afterwards.", "Line represents the knowledge that $phenytoin$ induces the activity of $cyp3a4$ .", "From background knowledge we find out that the stimulation in the activity can be as high as 2-times [55].", "Line represents the knowledge that there is a new drug $test\\_drug$ being tested that is known to inhibit the activity of $cyp3a4$ .", "Line specifies the initial distribution of the drugs and enzymes in the pathway.", "Assuming the patient has been given some fixed dose, say 20 units, of the medicine $gefitinib$ .", "It also specifies there is a large 60 units quantity of $cyp3a4$ available to ensure reactions do not slow down due to unavailability of enzyme availability.", "Additionaly, the drug $phenytoin$ is absent from the system and a new drug $test\\_drug$ to be tested is not in the system either.", "This gives us our pathway specification.", "Now we consider two application scenarios for drug development.", "Drug Administration A patient is taking 20 units of $gefitinib$ , and is being prescribed additional drugs to be co-administered.", "The drug administrator wants to know if there will be an interaction with $gefitinib$ if 5 units of $phenytoin$ are co-administered.", "If there is an interaction, what will be the bioavailability of $gefitinib$ so that its dosage could be appropriately adjusted.", "The first question is asking whether giving the patient 5-units of $phenytoin$ in addition to the existing $gefitinib$ dose will cause a drug-interaction.", "It is encoded as the following query statement $\\mathbf {Q}$ for a $k$ -step simulation: $&\\mathbf {direction~of~change~in~} average \\mathbf {~value~of~} gefitinib \\mathbf {~is~} d\\\\&~~~~~~\\mathbf {when~observed~at~time~step~} k; \\\\&~~~~~~\\mathbf {comparing~nominal~pathway~with~modified~pathway~obtained~}\\\\&~~~~~~~~~~~~\\mathbf {due~to~interventions:~} \\mathbf {set~value~of~} phenytoin \\mathbf {~to~} 5;$ If the direction of change is “$=$ ” then there was no drug-interaction.", "Otherwise, an interaction was noticed.", "For a simulation of length $k=5$ , we find 15 units of $gefitinib$ remained at the end of simulation in the nominal case when no $phenytoin$ is administered.", "The amount drops to 10 units of $gefitinib$ when $phenytoin$ is co-administered.", "The change in direction is “$<$ ”.", "Thus there is an interaction.", "The second question is asking about the bioavailability of the drug $gefitinib$ after some after giving $phenytoin$ in 5 units.", "If this bioavailability falls below the efficacy level of the drug, then the drug would not treat the disease effectively.", "It is encoded as the following query statement $\\mathbf {Q}$ for a $k$ -step simulation: $&average \\mathbf {~value~of~} gefitinib \\mathbf {~is~} n\\\\&~~~~\\mathbf {when~observed~at~time~step~} k; \\\\&~~~~\\mathbf {due~to~interventions:~} \\\\&~~~~~~~~\\mathbf {set~value~of~} phenytoin \\mathbf {~to~} 5;$ For a simulation of length $k=5$ , we find 10 units of $gefitinib$ remain.", "A drug administrator (such as a pharmacist) can adjust the drug accordingly.", "Drug Development A drug manufacturer is developing a new drug $test\\_drug$ that is known to inhibit CYP3A4 that will be co-administered with drugs $gefitinib$ and $phenytoin$ .", "He wants to determine the bioavailability of $gefitinib$ over time to determine the risk of toxicity.", "The question is asking about the bioavailability of the drug $gefitinib$ after 10 time units after giving $phenytoin$ in 5 units and the new drug $test\\_drug$ in 5 units.", "If this bioavailability remains high, there is chance for toxicity due to the drug at the subsequent dosage intervals.", "It is encoded as the following query statement $\\mathbf {Q}$ for a $k$ -step simulation: $&average \\mathbf {~value~of~} gefitinib \\mathbf {~is~} n\\\\&~~~~\\mathbf {when~observed~at~time~step~} k; \\\\&~~~~\\mathbf {due~to~interventions:~} \\\\&~~~~~~~~\\mathbf {set~value~of~} phenytoin \\mathbf {~to~} 5,\\\\&~~~~~~~~\\mathbf {set~value~of~} test\\_drug \\mathbf {~to~} 5;$ For a simulation of length $k=5$ , we find all 20 units of $gefitinib$ remain.", "This could lead to toxicity by building high concentration of $gefitinib$ in the body.", "Drug Administration in Presence of Genetic Variation A drug administrator wants to establish the dosage of $morphine$ for a person based on its genetic profile using its bioavailability.", "Consider the following facts extracted about a simplified morphine pathway: $codeine$ is metabolized by $CYP2D6$ producing $morphine$ $CYP2D6$ has three allelic variations “1” – (EM) effective metabolizer (normal case) “2” – (UM) ultra rapid metabolizer “9” – (PM) poor metabolizer For simplicity, assume UM allele doubles the metabolic rate, while PM allele halves the metabolic rate of CYP2D6.", "Then, the resulting pathway is given by: $&\\mathbf {domain~of~} cyp2d6\\_allele \\mathbf {~is~} integer, cyp2d6 \\mathbf {~is~} integer\\\\&\\mathbf {domain~of~} codeine \\mathbf {~is~} integer, morphine \\mathbf {~is~} integer\\\\&cyp2d6\\_action \\mathbf {~may~execute~causing}\\\\&~~~~codeine \\mathbf {~change~value~by~} -2, morphine \\mathbf {~change~value~by~} +2\\\\&~~~~\\mathbf {if~} cyp2d6\\_allele \\mathbf {~has~value~equal~to~} 1\\\\&cyp2d6\\_action \\mathbf {~may~execute~causing}\\\\&~~~~codeine \\mathbf {~change~value~by~} -4, morphine \\mathbf {~change~value~by~} +4\\\\&~~~~\\mathbf {if~} cyp2d6\\_allele \\mathbf {~has~value~equal~to~} 2\\\\&cyp2d6\\_action \\mathbf {~may~execute~causing~}\\\\&~~~~codeine \\mathbf {~change~value~by~} -1, morphine \\mathbf {~change~value~by~} +1\\\\&~~~~\\mathbf {if~} cyp2d6\\_allele \\mathbf {~has~value~equal~to~} 9\\\\&\\mathbf {initially~}\\\\&~~~~codeine \\mathbf {~has~value~} 0, morphine \\mathbf {~has~value~} 0,\\\\&~~~~cyp2d6 \\mathbf {~has~value~} 20, cyp2d6\\_allele \\mathbf {~has~value~} 1\\\\&\\mathbf {firing~style~} max\\\\$ Then, the bioavailability of $morphine$ can be determined by the following query: $&average \\mathbf {~value~of~} morphine \\mathbf {~is~} n\\\\&~~~~\\mathbf {when~observed~at~time~step~} k; \\\\&~~~~\\mathbf {due~to~interventions:~} \\\\&~~~~~~~~\\mathbf {set~value~of~} codeine \\mathbf {~to~} 20,\\\\&~~~~~~~~\\mathbf {set~value~of~} cyp2d6 \\mathbf {~to~} 20,\\\\&~~~~~~~~\\mathbf {set~value~of~} cyp2d6\\_allele \\mathbf {~to~} 9;$ Simulation for 5 time steps reveal that the average bioavailability of $morphine$ after 5 time-steps is 5 for PM (down from 10 for EM).", "Although this is a toy example, it is easy to see the potential of capturing known genetic variations in the pathway and setting the complete genetic profile of a person in the intervention part of the query.", "Conclusion In this chapter we presented how we extract biological pathway knowledge from text, including knowledge about drug-gene interactions and their relationship to genetic variation.", "We showed how the information extracted is used to build pathway specification and illustrated how biologically relevant questions can be answered about drug-drug interaction using the BioPathQA system developed in chapter .", "Next we look at the future directions in which the research work done in this thesis can be extended.", "Conclusion and Future Work The field of knowledge representation and reasoning (KR) is currently one of the most active research areas.", "It represents the next step in the evolution of systems that know how to organize knowledge, and have the ability to intelligently respond to questions about this knowledge.", "Such questions could be about static knowledge or the dynamic processes.", "Biological systems are uniquely positioned as role models for this next evolutionary step due to their precise vocabulary and mechanical nature.", "As a result, a number of recent research challenges in the KR field are focused on it.", "The biological field itself needs systems that can intelligently answer questions about such biological processes and systems in an automated fashion, given the large number of research papers published each year.", "Curating these publications is time consuming and expensive, as a result the state of over all knowledge about biological systems lags behind the cutting edge research.", "An important class of questions asked about biological systems are the so called “what-if” questions that compare alternate scenarios of a biological pathway.", "To answer such questions, one has to perform simulation on a nominal pathway against a pathway modified due to the interventions specified for the alternate scenario.", "Often, this means creating two pathways (for nominal and alternate cases) and simulate them separately.", "This opens up the possibility that the two pathways can become out of synchronization.", "A better approach is to allow the user to specify the needed interventions in the query statement itself.", "In addition, to understand the full spread of possible outcomes, given the parallel nature of biological pathways, one must consider all possible pathway evolutions, otherwise, some outcomes may remain hidden.", "If a system is to be used by biologists, it must have a simple interface, lowering the barrier of entry.", "Since biological pathway knowledge can arrive from different sources, including books, published articles, and lab experiments, a common input format is desired.", "Such a format allows specification of pathways due to automatic extraction, as well as any changes / additions due to locally available information.", "A comprehensive end-to-end system that accomplish all the goals would take a natural language query along with any additional specific knowledge about the pathway as input, extract the relevant portion of the relevant pathway from published material (and background knowledge), simulate it based on the query, and generate the results in a visual format.", "Each of these tasks comes with its own challenges, some of which have been addressed in this thesis.", "In this thesis, we have developed a system and a high level language to specify a biological pathway and answer simulation based reasoning questions about it.", "The high level language uses controlled-English vocabulary to make it more natural for a researcher to use directly.", "The high level language has two components: a pathway specification language, and a query specification language.", "The pathway specification language allows the user to specify a pathway in a source independent form, thus locally obtained knowledge (e.g.", "from lab) can be combined with automatically extracted knowledge.", "We believe that our pathway specification language is easy for a person to understand and encode, lowering the bar to using our system.", "Our pathway specification language allows conditional actions, enabling the encoding of alternate action outcomes due to genetic variation.", "An important aspect of our pathway specification language is that it specifies trajectories, which includes specifying the initial configuration of substances, as well as state evolution style, such as maximal firing of actions, or serialized actions etc.", "Our query specification language provides a bridge between natural language questions and their formal representation.", "It is English-like but with precise semantics.", "A main feature of our query language is its support for comparative queries over alternate scenarios, which is not currently supported by any of the query languages (associated with action languages) we have reviewed.", "Our specification of alternate scenarios uses interventions (a general form of actions), that allow the user to modify the pathway as part of the query processing.", "We believe our query language is easier for a biologist to understand without requiring formal training.", "To model the pathways, we use Petri Nets, which have been used in the past to model and simulate biological pathways.", "Petri Nets have a simple visual representation, which closely matches biological pathways; and they inherently support parallelism.", "We extended the Petri Nets to add features that we needed to suit our domain, e.g., reset arcs that remove all quantity of a substance as soon as it is produced, and conditional arcs that specify the conditional outcome of an action.", "For simulation, we use ASP, which allowed us straight forward way to implement Petri Nets.", "It also gave us the ability to add extensions to the Petri Net by making local edits, implement different firing semantics, filter trajectories based on observations, and reason with the results.", "One of the major advantage of using Petri Net based simulation is the ability to generate all possible state evolutions, enabling us to process queries that determine the conditions when a certain observation becomes true.", "Our post-processing step is done in Python, which allows strong text processing capabilities using regular expressions, as well as libraries to easy process large matrices of numbers for summarization of results.", "Now we present additional challenges that need to be addressed.", "Pathway Extraction In Chapter we described how we extract facts for drug-drug interaction and gene variation.", "This work needs to be extended to include some of the newer databases that have come online recently.", "This may provide us with enzyme reaction rates, and substance quantities used in various reactions.", "The relation extraction for pathways must also be cognizant of any genetic variation mentioned in the text.", "Since the knowledge about the pathway appears in relationships at varying degree of detail, a process needs to be devised to assemble the pathway from the same level to granularity together, while also maintaining pathways at different levels of granularities.", "Since pathway extraction is a time consuming task, it would be best to create a catalog of the pathways.", "The cataloged pathways could be manually edited by the user as necessary.", "Storing pathways in this way means that would have to be updated periodically, requiring merging of new knowledge into existing pathways.", "Manual edits would have to be identified, such that the updated pathway does not overwrite them without the user's knowledge.", "Pathway Selection Questions presented in biological texts do not explicitly mention the relevant pathway to use for answering the question.", "One way to address this issue is to maintain a catalog of pre-defined pathways with keywords associated with them.", "Such keywords can include names of the substances, names of the processes, and other relevant meta-data about the pathway.", "The catalog can be searched to find the closest match to the query being asked.", "An additional aspect in proper pathway selection is to use the proper abstraction level.", "If our catalog contains a pathway at different abstraction levels, the coarsest pathway that contains the processes and substances in the query should be selected.", "Any higher fidelity will increase the processing time and generate too much irrelevant data.", "Alternatively, the catalog could contain the pathway in a hierarchical form, allowing extraction of all elements of a pathway at the same depth.", "A common way to hierarchically organize the pathway related to our system is to have hierarchical actions, which is the approach taken by hierarchical Petri nets.", "Lastly, the question may only ask about a small subsection of a much larger pathway.", "For better performance, it is beneficial to extract the smallest biological pathway network model that can answer the question.", "Pathway Modeling In Chapter , we presented our modeling of biological questions using Petri Nets and their extensions encoded in ASP.", "We came across concepts like allosteric regulation, inhibition of inhibition, and inhibition of activation that we currently do not model.", "In allosteric regulation, an enzyme is not fully enabled or disabled, the enzyme's shape changes, making it more or less active.", "The level of its activity depends upon concentrations of activators and inhibitors.", "In inhibition of inhibition, the inhibition of a reaction is itself inhibited by another inhibition; while in inhibition of activation (or stimulation), a substance inhibits the stimulation produced by a substance.", "Both of these appear to be actions on actions, something that Petri Nets do not allow.", "An alternate coding for these would have to be devised.", "As more detailed information about pathways becomes available, the reactions and processes that we have in current pathways may get replaced with more detailed sub-pathways themselves.", "However, such refinement may not come at the same time for separate legs of the pathway.", "Just replacing the coarse transition with a refined transition may not be sufficient due to relative timing constraints.", "Hence, a hierarchical Petri Net model may need to be implemented (see , ).", "Pathway Simulation In Chapter we presented our approach to encode Petri Nets and their extensions.", "We used a discrete solver called clingo for our ASP encoding.", "As the number of simulation length increases in size or larger quantities are involved, the solver slows down significantly.", "This is due to an increased grounding cost of the program.", "Incremental solving (using iclingo) does not help, since the program size still increases, and the increments merely delays the slow down but does not stop it.", "Systems such as constraint logic solvers (such as ) could be used for discrete cases.", "Alternatively, a system developed on the ASPMT [53] approach could be used, since it can represent longer runs, larger quantities, and real number quantities.", "Extend High Level Language In Chapter we described the BioPathQA system, the pathway specification and the query specification high level languages.", "As we enhance the modeling of the biological pathways, we will need to improve or extend the system as well as the high level language.", "We give a few examples of such extensions.", "Our pathway specification language currently does not support continuous quantities (real numbers).", "Extending to real numbers will improve the coverage of the pathways that can be modeled.", "In addition, derived quantities (fluents) can be added, e.g.", "pH could be defined as a formula that is read-only in nature.", "Certain observations and queries can be easily specified using a language such as LTL, especially for questions requiring conditions leading to an action or a state.", "As a result, it may be useful to add LTL formulas to the query language.", "We did not take this approach because it would have presented an additional non-English style syntax for the biologists.", "Our substance production / consumption rates and amounts are currently tied to the fluents.", "In certain situations it is desirable to analyze the quantity of a substance produced / consumed by a specific action, e.g.", "one is interested in finding the amount of H+ ions produced by a multi-protein complex IV only.", "Interventions (that are a part of the query statement) presented in this thesis are applied at the start of the simulation.", "Eliminating this restriction would allow applying surgeries to the pathway mid execution.", "Thus, instead of specifying the steady state conditions in the query statement, one could apply the intervention when a certain steady state is reached.", "Result Formatting and Visualization In Chapter we described our system that answers questions specified in our high level language.", "At the end of its process, it outputs the final result.", "This output method can be enhanced by allowing to look at the progression of results in addition to the final result.", "This provides the biologist with the whole spread of possible outcomes.", "An example of such a spread is shown in Fig.", "fig:q1:result for question REF .", "A graphical representation of the simulation progression is also beneficial in enhancing the confidence of the biologist.", "Indeed many existing tools do this.", "A similar effect can be achieved by parsing and showing the relevant portion of the answer set.", "Summary In Chapter we introduced the thesis topic and summarized specific research contributions In Chapter we introduced the foundational material of this thesis including Petri Nets and ASP.", "We showed how ASP could be used to encode basic Petri Nets.", "We also showed how ASP's elaboration tolerance and declarative syntax allows us to encode various Petri Net extensions with small localized changes.", "We also introduced a new firing semantics, the so called maximal firing set semantics to simulate a Petri Net with maximum parallel activity.", "In Chapter we showed how the Petri Net extensions and the ASP encoding can be used to answer simulation based deep reasoning questions.", "This and the work in Chapter was published in [1], [2].", "In Chapter we developed a system called BioPathQA to allow users to specify a pathway and answer queries against it.", "We also developed a pathway specification language and a query language for this system in order to avoid the vagaries of natural language.", "We introduced a new type of Guarded-arc Petri Nets to model conditional actions as a model for pathway simulation.", "We also described our implementation developed around a subset of the pathway specification language.", "In Chapter we briefly described how text extraction is done to extract real world knowledge about pathways and drug interactions.", "We then used the extracted knowledge to answer question using BioPathQA.", "The text extraction work was published in [73], [72], [33].", "Proofs of Various Propositions Assumption: The definitions in this section assume the programs $\\Pi $ do not have recursion through aggregate atoms.", "Our ASP translation ensures this due to the construction of programs $\\Pi $ .", "First we extend some definitions and properties related to ASP, such that they apply to rules with aggregate atoms.", "We will refer to the non-aggregate atoms as basic atoms.", "Recall the definitions of an ASP program given in section REF .", "Proposition 9 (Forced Atom Proposition) Let $S$ be an answer set of a ground ASP program $\\Pi $ as defined in definition REF .", "For any ground instance of a rule R in $\\Pi $ of the form $A_0 \\text{:-} A_1,\\dots ,$ $A_m,\\mathbf {not~} B_{1},\\dots ,$ $\\mathbf {not~} B_n, C_1,\\dots ,$ $C_k.$ if $\\forall A_i, 1 \\le i \\le m, S \\models A_i$ , and $\\forall B_j, 1 \\le j \\le n, S \\lnot \\models B_j$ , $\\forall C_l, 1 \\le l \\le k, S \\models C_l$ then $S \\models A_0$ .", "Proof: Let $S$ be an answer set of a ground ASP program $\\Pi $ , $R \\in \\Pi $ be a ground rule such that $\\forall B_j, 1 \\le j \\le n, S \\lnot \\models B_j$ ; and $\\forall C_l, 1 \\le l \\le k, S \\models C_l$ .", "Then, the reduct $R^S \\equiv \\lbrace p_1 \\text{:-} A_1,\\dots ,A_m.", "; \\dots ; p_h \\text{:-} A_1,\\dots ,A_m.", "\\; \| \\; \\lbrace p_1,\\dots ,p_h \\rbrace = S \\cap lit(A_0) \\rbrace $ .", "Since $S$ is an answer set of $\\Pi $ , it is a model of $\\Pi ^S$ .", "As a result, whenever, $\\forall A_i, 1 \\le i \\le m, S \\models A_i$ , $\\lbrace p_1,\\dots ,p_h \\rbrace \\subseteq S$ and $S \\models A_0$ .", "Proposition 10 (Supporting Rule Proposition) If $S$ is an answer set of a ground ASP program $\\Pi $ as defined in definition REF then $S$ is supported by $\\Pi $ .", "That is, if $S \\models A_0$ , then there exists a ground instance of a rule R in $\\Pi $ of the type $A_0 \\text{:-} A_1,\\dots ,$ $A_m,$ $\\mathbf {not~} B_{1},\\dots ,$ $\\mathbf {not~} B_n, $ $C_1,\\dots ,C_k.$ such that $\\forall A_i, 1 \\le i \\le m, S \\models A_i$ , $\\forall B_j, 1 \\le j \\le n, S \\lnot \\models B_j$ , and $\\forall C_l, 1\\le l \\le k, S \\models C_l$ .", "Proof: For $S$ to be an answer set of $\\Pi $ , it must be the deductive closure of reduct $\\Pi ^S$ .", "The deductive closure $S$ of $\\Pi ^S$ is iteratively built by starting from an empty set $S$ , and adding head atoms of rules $R_h^S \\equiv p_h \\text{:-} A_1,\\dots ,A_m., R_h^S \\in \\Pi ^S$ , whenever, $S \\models A_i, 1 \\le i \\le m$ , where, $R_h^S$ is a rule in the reduct of ground rule $R \\in \\Pi $ with $p_h \\in lit(A_0) \\cap S$ .", "Thus, there is a rule $R \\equiv A_0 \\text{:-} A_1,\\dots ,$ $A_m,$ $\\mathbf {not~} B_{1},\\dots ,$ $\\mathbf {not~}B_n, $ $C_1,\\dots ,$ $C_k.$ , $R \\in \\Pi $ , such that $\\forall C_l, 1 \\le l \\le k$ and $\\forall B_j, 1 \\le j \\le n, S \\lnot \\models B_j$ .", "Nothing else belongs in $S$ .", "Next, we extend the splitting set theorem to include aggregate atoms.", "Definition 88 (Splitting Set) A Splitting Set for a program $\\Pi $ is any set $U$ of literals such that, for every rule $R \\in \\Pi $ , if $U \\models head(R)$ then $lit(R) \\subset U$ .", "The set $U$ splits $\\Pi $ into upper and lower parts.", "The set of rules $R \\in \\Pi $ s.t.", "$lit(R) \\subset U$ is called the bottom of $\\Pi $ w.r.t.", "$U$ , denoted by $bot_U(\\Pi )$ .", "The rest of the rules, i.e.", "$\\Pi \\setminus bot_U(\\Pi )$ is called the top of $\\Pi $ w.r.t.", "$U$ , denoted by $top_U(\\Pi )$ .", "Proposition 11 Let $U$ be a splitting set of $\\Pi $ with answer set $S$ and let $X = S \\cap U$ and $Y = S \\setminus U$ .", "Then, the reduct of $\\Pi $ w.r.t.", "$S$ , i.e.", "$\\Pi ^S$ is equal to $bot_U(\\Pi )^X \\cup (\\Pi \\setminus bot_U(\\Pi ))^{X \\cup Y}$ .", "Proof: We can rewrite $\\Pi $ as $bot_U(\\Pi ) \\cup (\\Pi \\setminus bot_U(\\Pi ))$ using the definition of splitting set.", "Then the reduct of $\\Pi $ w.r.t.", "$S$ can be written in terms of $X$ and $Y$ , since $S = X \\cup Y$ .", "$\\Pi ^S =$ $\\Pi ^{X \\cup Y} =$ $(bot_U(\\Pi ) \\cup (\\Pi \\setminus bot_U(\\Pi )))^{X \\cup Y} =$ $bot_U(\\Pi )^{X \\cup Y} \\cup $ $(\\Pi \\setminus bot_U(\\Pi ))^{X \\cup Y}$ .", "Since $lit(bot_U(\\Pi )) \\subseteq U$ and $Y \\cap U = \\emptyset $ , the reduct of $bot_U(\\Pi )^{X \\cup Y} = bot_U(\\Pi )^X$ .", "Thus, $\\Pi ^{X \\cup Y} = bot_U(\\Pi )^X \\cup (\\Pi \\setminus bot_U(\\Pi ))^{X \\cup Y}$ .", "Proposition 12 Let $S$ be an answer set of a program $\\Pi $ , then $S \\subseteq lit(head(\\Pi ))$ .", "Proof: If $S$ is an answer set of a program $\\Pi $ then $S$ is produced by the deductive closure of $\\Pi ^S$ (the reduct of $\\Pi $ w.r.t $S$ ).", "By definition of the deductive closure, nothing can be in $S$ unless it is the head of some rule supported by $S$ .", "Splitting allow computing the answer set of a program $\\Pi $ in layers.", "Answer sets of the bottom layer are first used to partially evaluate the top layer, and then answer sets of the top layer are computed.", "Next, we define how a program is partially evaluated.", "Intuitively, the partial evaluation of an aggregate atom $c$ given splitting set $U$ w.r.t.", "a set of literals $X$ removes all literals that are part of the splitting set $U$ from $c$ and updates $c$ 's lower and upper bounds based on the literals in $X$ , which usually come from $bot_U$ of a program.", "The set $X$ represents our knowledge about the positive literals, while the set $U \\setminus X$ represents our knowledge about naf-literals at this stage.", "We can remove all literals in $U$ from $c$ , since the literals in $U$ will not appear in the head of any rule in $top_U$ .", "Definition 89 (Partial Evaluation of Aggregate Atom) The partial evaluation of an aggregate atom $c = l \\; [ B_0=w_0,\\dots , B_m=w_m ] \\; u$ , given splitting set $U$ w.r.t.", "a set of literals $X$ , written $eval_U(c,X)$ is a new aggregate atom $c^{\\prime }$ constructed from $c$ as follows: $pos(c^{\\prime }) = pos(c) \\setminus U$ $d=\\sum _{B_i \\in pos(c) \\cap U \\cap X}{w_i} $ $l^{\\prime } = l-d$ , $u^{\\prime } = u-d$ are the lower and upper limits of $c^{\\prime }$ Next, we define how a program is partially evaluated given a splitting set $U$ w.r.t.", "a set of literals $X$ that form the answer-set of the lower layer.", "Intuitively, a partial evaluation deletes all rules from the partial evaluation for which the body of the rule is determined to be not supported by $U$ w.r.t.", "$X$ .", "This includes rules which have an aggregate atom $c$ in their body s.t.", "$lit(c) \\subseteq U$ , but $X \\lnot \\models c$ Note that we can fully evaluate an aggregate atom $c$ w.r.t.", "answer-set $X$ if $lit(c) \\subseteq U$ ..", "In the remaining rules, the positive and negative literals that overlap with $U$ are deleted, and so are the aggregate atoms that have $lit(c) \\subseteq U$ (since such a $c$ can be fully evaluated w.r.t.", "$X$ ).", "Each remaining aggregate atom is updated by removing atoms that belong to $U$ Since the atoms in $U$ will not appear in the head of any atoms in $top_U$ and hence will not form a basis in future evaluations of $c$ ., and updating its limits based on the answer-set $X$ The limit update requires knowledge of the current answer-set to update limit values..", "The head atom is not modified, since $eval_U(...)$ is performed on $\\Pi \\setminus bot_U(\\Pi )$ , which already removes all rules with heads atoms that intersect $U$ .", "Definition 90 (Partial Evaluation) The partial evaluation of $\\Pi $ , given splitting set $U$ w.r.t.", "a set of literals $X$ is the program $eval_U(\\Pi ,X)$ composed of rules $R^{\\prime }$ for each $R \\in \\Pi $ that satisfies all the following conditions: $pos(R) \\cap U \\subseteq X,$ $((neg(R) \\cap U) \\cap X) = \\emptyset , \\text{ and }$ if there is a $c \\in agg(R)$ s.t.", "$lit(c) \\subseteq U$ , then $X \\models c$ A new rule $R^{\\prime }$ is constructed from a rule $R$ as follows: $head(R^{\\prime }) = head(R)$ , $pos(R^{\\prime }) = pos(R) \\setminus U$ , $neg(R^{\\prime }) = neg(R) \\setminus U$ , $agg(R^{\\prime }) = \\lbrace eval_U(c,X) : c \\in agg(R), lit(c) \\lnot \\subseteq U \\rbrace $ Proposition 13 Let $U$ be a splitting set for $\\Pi $ , $X$ be an answer set of $bot_U(\\Pi )$ , and $Y$ be an answer set of $eval_U(\\Pi \\setminus bot_U(\\Pi ),X)$ .", "Then, $X \\subseteq lit(\\Pi ) \\cap U$ and $Y \\subseteq lit(\\Pi ) \\setminus U$ .", "Proof: By proposition REF , $X \\subseteq lit(head(bot_U(\\Pi )))$ , and $Y \\subseteq lit(head(eval_U(\\Pi \\setminus bot_U(\\Pi ),X)))$ .", "In addition, $lit(head(bot_U(\\Pi ))) \\subseteq lit(bot_U(\\Pi ))$ and $lit(bot_U(\\Pi )) \\subseteq lit(\\Pi ) \\cap U$ by definition of $bot_U(\\Pi )$ .", "Then $X \\subseteq lit(\\Pi ) \\cap U$ , and $Y \\subseteq lit(\\Pi ) \\setminus U$ .", "Definition 91 (Solution) [4] Let $U$ be a splitting set for a program $\\Pi $ .", "A solution to $\\Pi $ w.r.t.", "$U$ is a pair $\\langle X,Y \\rangle $ of literals such that: $X$ is an answer set for $bot_U(\\Pi )$ $Y$ is an answer set for $eval_U(top_U(\\Pi ),X)$ ; and $X \\cup Y$ is consistent.", "Proposition 14 (Splitting Theorem) [4] Let $U$ be a splitting set for a program $\\Pi $ .", "A set $S$ of literals is a consistent answer set for $\\Pi $ iff $S = X \\cup Y$ for some solution $\\langle X,Y \\rangle $ of $\\Pi $ w.r.t.", "$U$ .", "Lemma 1 Let $U$ be a splitting set of $\\Pi $ , $C$ be an aggregate atom in $\\Pi $ , and $X$ and $Y$ be subsets of $lit(\\Pi )$ s.t.", "$X \\subseteq U$ , and $Y \\cap U = \\emptyset $ .", "Then, $X \\cup Y \\models C$ iff $Y \\models eval_U(C,X)$ .", "Proof: Let $C^{\\prime } = eval_U(C,X)$ , then by definition of partial evaluation of aggregate atom, $pos(C^{\\prime }) = pos(C) \\setminus U$ , with lower limit $l^{\\prime } = l-d$ , and upper limit $u^{\\prime } = u-d$ , computed from $l,u$ , the lower and upper limits of $C$ , where $d=\\displaystyle \\sum _{B_i \\in pos(C) \\cap U \\cap X}{w_i}$ $Y \\models C^{\\prime }$ iff $l^{\\prime } \\le \\left( \\displaystyle \\sum _{B^{\\prime }_i \\in pos(C^{\\prime }) \\cap Y}{w^{\\prime }_i} \\right) \\le u^{\\prime }$ – by definition of aggregate atom satisfaction.", "then $Y \\models C$ iff $l \\le \\left( \\displaystyle \\sum _{B_i \\in pos(C) \\cap U \\cap X}{w_i} +\\displaystyle \\sum _{B^{\\prime }_i \\in (pos(C) \\setminus U) \\cap Y}{w^{\\prime }_i} \\right) \\le u$ however, $(pos(C) \\cap U) \\cap X$ and $(pos(C) \\setminus U) \\cap Y$ combined represent $pos(C) \\cap (X \\cup Y)$ – since $pos(C) \\cap (X \\cup Y) &= ((pos(C) \\cap U) \\cup (pos(C) \\setminus U)) \\cap (X \\cup Y) \\\\&= [((pos(C) \\cap U) \\cup (pos(C) \\setminus U)) \\cap X] \\\\&~~~~~~\\cup [((pos(C) \\cap U) \\cup (pos(C) \\setminus U)) \\cap Y]\\\\&= [(pos(C) \\cap U) \\cap X) \\cup ((pos(C) \\setminus U) \\cap X)]\\\\&~~~~~~\\cup [(pos(C) \\cap U) \\cap Y) \\cup ((pos(C) \\setminus U) \\cap Y)]\\\\&= [((pos(C) \\cap U) \\cap X) \\cup \\emptyset ] \\cup [\\emptyset \\cup ((pos(C) \\setminus U) \\cap Y)]\\\\&= ((pos(C) \\cap U) \\cap X) \\cup ((pos(C) \\setminus U) \\cap Y)$ where $X \\subseteq U \\text{ and } Y \\cap U = \\emptyset $ thus, $Y \\models C$ iff $l \\le \\left( \\displaystyle \\sum _{B_i \\in pos(C) \\cap (X \\cup Y)}{w_i} \\right) \\le u$ which is the same as $X \\cup Y \\models C$ Lemma 2 Let $U$ be a splitting set for $\\Pi $ , and $X, Y$ be subsets of $lit(\\Pi )$ s.t.", "$X \\subseteq U$ and $Y \\cap U = \\emptyset $ .", "Then the body of a rule $R^{\\prime } \\in eval_U(\\Pi ,X)$ is satisfied by $Y$ iff the body of the rule $R \\in \\Pi $ it was constructed from is satisfied by $X \\cup Y$ .", "Proof: $Y$ satisfies $body(R^{\\prime })$ iff $pos(R^{\\prime }) \\subseteq Y$ , $neg(R^{\\prime }) \\cap Y = \\emptyset $ , $Y \\models C^{\\prime }$ for each $C^{\\prime } \\in agg(R^{\\prime })$ – by definition of rule satisfaction iff $(pos(R) \\cap U) \\subseteq X$ , $(pos(R) \\setminus U) \\subseteq Y$ , $(neg(R) \\cap U) \\cap X) = \\emptyset $ , $(neg(R) \\setminus U) \\cap Y) = \\emptyset $ , $X$ satisfies $C$ for all $C \\in agg(C)$ in which $lit(C) \\subseteq U$ , and $Y$ satisfies $eval_U(C,X)$ for all $ C \\in agg(C)$ in which $lit(C) \\lnot \\subseteq U$ – using definition of partial evaluation iff $pos(R) \\subseteq X \\cup Y$ , $neg(R) \\cap (X \\cup Y) = \\emptyset $ , $X \\cup Y \\models C$ – using $(A \\cap U) \\cup (A \\setminus U) = A$ $A \\cap (X \\cup Y) = ((A \\cap U) \\cup (A \\setminus U)) \\cap (X \\cup Y) = ((A \\cap U) \\cap (X \\cup Y)) \\cup ((A \\setminus U) \\cap (X \\cup Y)) = (A \\cap U) \\cap X) \\cup ((A \\setminus U) \\cap Y)$ – given $X \\subseteq U$ and $Y \\cap U = \\emptyset $ .", "and lemma REF Proof of Splitting Theorem: Let $U$ be a splitting set of $\\Pi $ , then a consistent set of literals $S$ is an answer set of $\\Pi $ iff it can be written as $S = X \\cup Y$ , where $X$ is an answer set of $bot_U(\\Pi )$ ; and $Y$ is an answer set of $eval_U(\\Pi \\setminus bot_U(\\Pi ),Y)$ .", "($\\Leftarrow $ ) Let $X$ is an answer set of $bot_U(\\Pi )$ ; and $Y$ is an answer set of $eval_U(\\Pi \\setminus bot_U(\\Pi ),X)$ ; we show that $X \\cup Y$ is an answer set of $\\Pi $ .", "By definition of $bot_U(\\Pi )$ , $lit(bot_U(\\Pi )) \\subseteq U$ .", "In addition, by proposition REF , $Y \\cap U = \\emptyset $ .", "Then, $\\Pi ^{X \\cup Y} = (bot_U(\\Pi ) \\cup (\\Pi \\setminus bot_U(\\Pi )))^{X \\cup Y} = bot_U(\\Pi )^{X \\cup Y} \\cup (\\Pi \\setminus bot_U(\\Pi ))^{X \\cup Y} = bot_U(\\Pi )^X \\cup (\\Pi \\setminus bot_U(\\Pi ))^{X \\cup Y}$ .", "Let $r$ be a rule in $\\Pi ^{X \\cup Y}$ , s.t.", "$X \\cup Y \\models body(r)$ then we show that $X \\cup Y \\models head(r)$ .", "The rule $r$ either belongs to $bot_U(\\Pi )^X$ or $(\\Pi \\setminus bot_U(\\Pi ))^{X \\cup Y}$ .", "Case 1: say $r \\in bot_U(\\Pi )^X$ be a rule whose body is satisfied by $X \\cup Y$ then there is a rule $R \\in bot_U(\\Pi )$ s.t.", "$r \\in R^X$ then $X \\models body(R)$ – since $X \\cup Y \\models body(r)$ ; $lit(bot_U(\\Pi )) \\subseteq U$ and $Y \\cap U = \\emptyset $ we already have $X \\models head(R)$ – since $X$ is an answer set of $bot_U(\\Pi )$ ; given then $X \\cup Y \\models head(R)$ – because $lit(R) \\subseteq U$ and $Y \\cap U = \\emptyset $ consequently, $X \\cup Y \\models head(r)$ Case 2: say $r \\in (\\Pi \\setminus bot_U(\\Pi ))^{X \\cup Y}$ be a rule whose body is satisfied by $X \\cup Y$ then there is a rule $R \\in (\\Pi \\setminus bot_U(\\Pi ))$ s.t.", "$r \\in R^{X \\cup Y}$ then $lit(head(R)) \\cap U = \\emptyset $ – otherwise, $R$ would have belonged to $bot_U(\\Pi )$ , by definition of splitting set then $head(r) \\in Y$ – since $X \\subseteq U$ in addition, $pos(R) \\subseteq X \\cup Y$ , $neg(R) \\cap (X \\cup Y) = \\emptyset $ , $X \\cup Y \\models C$ for each $C \\in agg(R)$ – using definition of reduct then $pos(R) \\cap U \\subseteq X$ or $pos(R) \\setminus U \\subseteq Y$ ; $(neg(R) \\cap U) \\cap X = \\emptyset $ and $(neg(R) \\setminus U) \\cap Y = \\emptyset $ ; and for each $C \\in agg(R)$ , either $X \\models C$ if $lit(C) \\subseteq U$ , or $Y \\models eval_U(C,X)$ if $lit(C) \\lnot \\subseteq U$ – by rearranging, lemma REF , $X \\subseteq U$ , $Y \\cap U = \\emptyset $ , and definition of partial evaluation of an aggregate atom note that $pos(R) \\cap U \\subseteq X$ , $(neg(R) \\cap U) \\cap X = \\emptyset $ , and for each $C \\in agg(R)$ , s.t.", "$lit(C) \\subseteq U$ , $X \\models C$ , represent conditions satisfied by each rule that become part of a partial evaluation – using definition of partial evaluation and $pos(R) \\setminus U$ , $neg(R) \\setminus U$ , and for each $C \\in agg(R)$ , $eval_U(C,X)$ are the modifications made to the rule during partial evaluation given splitting set $U$ w.r.t.", "$X$ – using definition of partial evaluation and $pos(R) \\setminus U \\subseteq Y$ , $(neg(R) \\setminus U) \\cap Y = \\emptyset $ , and for each $C \\in agg(R)$ , $Y \\models eval_U(C,X)$ if $lit(C) \\lnot \\subseteq U$ represent conditions satisfied by rules that become part of the reduct w.r.t $Y$ – using definition of partial evaluation and reduct then $r$ is a rule in reduct $eval_U(\\Pi \\setminus bot_U(\\Pi ),X)^Y$ – using (REF ), (REF ) above in addition, given that $Y$ satisfies $eval_U(\\Pi \\setminus bot_U(\\Pi ),X)$ , and $head(r) \\cap U = \\emptyset $ , we have $X \\cup Y \\models head(r)$ Next we show that $X \\cup Y$ satisfies all rules of $\\Pi $ .", "Say, $R$ is a rule in $\\Pi $ not satisfied by $X \\cup Y$ .", "Then, either it belongs to $bot_U(\\Pi )$ or $(\\Pi \\setminus bot_U(\\Pi ))$ .", "If it belongs to $bot_U(\\Pi )$ , it must not be satisfied by $X$ , since $lit(bot_U(\\Pi )) \\subseteq U$ and $Y \\cap U = \\emptyset $ .", "However, the contrary is given to be true.", "On the other hand if it belongs to $(\\Pi \\setminus bot_U(\\Pi ))$ , then $X \\cup Y$ satisfies $body(R)$ but not $head(R)$ .", "That would mean that its $head(R)$ is not satisfied by $Y$ , since $head(R) \\cap U = \\emptyset $ by definition of splitting set.", "However, from lemma REF we know that if $body(R)$ is satisfied by $X \\cup Y$ , $body(R^{\\prime })$ is satisfied by $Y$ for $R^{\\prime } \\in eval_U(\\Pi \\setminus bot_U(\\Pi ),X)$ .", "We also know that $Y$ satisfies all rules in $eval_U(\\Pi \\setminus bot_U(\\Pi ),X)$ .", "So, $R^{\\prime }$ must be satisfied by $Y$ contradicting our assumption.", "Thus, all rules of $\\Pi $ are satisfied by $X \\cup Y$ and $X \\cup Y$ is an answer set of $\\Pi $ .", "($\\Rightarrow $ ) Let $S$ be a consistent answer set of $\\Pi $ , we show that $S = X \\cup Y$ for sets $X$ and $Y$ s.t.", "$X$ is an answer set of $bot_U(\\Pi )$ and $Y$ is an answer set of $eval_U(\\Pi \\setminus bot_U(\\Pi ),X)$ .", "We take $X=S \\cap U$ , $Y=S \\setminus U$ , then $S=X \\cup Y$ .", "Case 1: We show that $X$ is answer set of $bot_U(\\Pi )$ $\\Pi $ can be split into $bot_U(\\Pi ) \\cup (\\Pi \\setminus bot_U(\\Pi ))$ – by definition of splitting then $X \\cup Y$ satisfies $bot_U(\\Pi )$ – $X \\cup Y$ is an answer set of $\\Pi $ ; given however $lit(bot_U(\\Pi )) \\subseteq U$ , $Y \\cap U = \\emptyset $ – by definition of splitting then $X$ satisfies $bot_U(\\Pi )$ – since elements of $Y$ do not appear in the rules of $bot_U(\\Pi )$ then $X$ is an answer set of $bot_U(\\Pi )$ Case 2: We show that $Y$ is answer set of $eval_U(\\Pi \\setminus bot_U(\\Pi ),X)$ let $r$ be a rule in $eval_U(\\Pi \\setminus bot_U(\\Pi ),X)^Y$ , s.t.", "its body is satisfied by $Y$ then $r \\in R^Y$ for an $R \\in eval_U(\\Pi \\setminus bot_U(\\Pi ),X)$ s.t.", "[(i)] (3) $pos(R) \\subseteq Y$ (4) $neg(R) \\cap Y = \\emptyset $ (5) $Y \\models C$ for all $C \\in agg(R)$ (6) $head(R) \\cap Y \\ne \\emptyset $ – using definition of reduct each $R$ is constructed from $R^{\\prime } \\in \\Pi $ that satisfies all the following conditions [(i)] (8) $pos(R^{\\prime }) \\subseteq U \\cap X$ (9) $(neg(R^{\\prime }) \\cap U) \\cap X = \\emptyset $ (10) if there is a $C^{\\prime } \\in agg(R^{\\prime })$ s.t.", "$lit(C^{\\prime }) \\subseteq U$ , then $X \\models C^{\\prime }$ ; and each $C \\in agg(R)$ is a partial evaluation of $C^{\\prime } \\in agg(R^{\\prime })$ s.t.", "$C = eval_U(C^{\\prime },X)$ – using definition of partial evaluation then the $body(R^{\\prime })$ satisfies all the following conditions: $pos(R^{\\prime }) \\subseteq X \\cup Y$ – since $X \\subseteq U$ , $X \\cap Y = \\emptyset $ $neg(R^{\\prime }) \\cap (X \\cup Y) = \\emptyset $ – since $X \\subseteq U$ , $X \\cap Y = \\emptyset $ $X \\cup Y \\models C^{\\prime }$ for each $C^{\\prime } \\in agg(R^{\\prime })$ – since [(i)] (d) each $C^{\\prime } \\in agg(R^{\\prime })$ with $lit(C^{\\prime }) \\subseteq U$ satisfied by $X$ is also satisfied by $X \\cup Y$ as $lit(Y) \\cap lit(C^{\\prime }) = \\emptyset $ ; and (e) each $C^{\\prime } \\in agg(R^{\\prime })$ with $lit(C^{\\prime }) \\lnot \\subseteq U$ is satisfied by $X \\cup Y$ – using partial evaluation, reduct construction, and $X \\cap Y = \\emptyset $ then $X \\cup Y$ satisfies $body(R^{\\prime })$ – from previous line in addition, $lit(head(R^{\\prime })) \\cap U = \\emptyset $ , otherwise, $R^{\\prime }$ would have belonged to $bot_U(\\Pi )$ by definition of splitting set then $R^{\\prime }$ is a rule in $\\Pi \\setminus bot_U(\\Pi )$ – from the last two lines we know that $X \\cup Y$ satisfies every rule in $(\\Pi \\setminus bot_U(\\Pi ))$ – given; and that elements of $U$ do not appear in the head of rules in $(\\Pi \\setminus bot_U(\\Pi ))$ – from definition of splitting; then $Y$ must satisfy the head of these rules then $Y$ satisfies $head(R^{\\prime })$ – from (REF ) Next we show that $Y$ satisfies all rules of $eval_U(\\Pi \\setminus bot_U(\\Pi ),X)$ .", "Let $R^{\\prime }$ be a rule in $eval_U(\\Pi \\setminus bot_U(\\Pi ),X)$ such that $body(R^{\\prime })$ is satisfied by $Y$ but not $head(R^{\\prime })$ .", "Since $head(R^{\\prime }) \\cap Y = \\emptyset $ , $head(R^{\\prime })$ is not satisfied by $X \\cup Y$ either.", "Then, there is an $R \\in (\\Pi \\setminus bot_U(\\Pi ))$ such that $X \\cup Y$ satisfies $body(R)$ but not $head(R)$ , which contradicts given.", "Thus, $Y$ satisfies all rules of $eval_U(\\Pi \\setminus bot_U(\\Pi ),X)$ .", "Then $Y$ is an answer set of $eval_U(\\Pi \\setminus bot_U(\\Pi ),X)$ Definition 92 (Splitting Sequence) [4] A splitting sequence for a program $\\Pi $ is a monotone, continuous sequence ${\\langle U_{\\alpha } \\rangle }_{\\alpha < \\mu }$ of splitting sets of $\\Pi $ such that $\\bigcup _{\\alpha < \\mu }{U_{\\mu }} = lit(\\Pi )$ .", "Definition 93 (Solution) [4] Let $U={\\langle U_{\\alpha } \\rangle }_{\\alpha < \\mu }$ be a splitting sequence for a program $\\Pi $ .", "A solution to $\\Pi $ w.r.t $U$ is a sequence ${\\langle X_{\\alpha } \\rangle }_{\\alpha < \\mu }$ of sets of literals such that: $X_0$ is an answer set for $bot_{U_0}(\\Pi )$ for any ordinal $\\alpha + 1 < \\mu $ , $X_{\\alpha +1}$ is an answer set of the program: $eval_{U_{\\alpha }}(bot_{U_{\\alpha +1}}(\\Pi ) \\setminus bot_{U_{\\alpha }}(\\Pi ), \\bigcup _{\\nu \\le \\alpha }{X_{\\nu }})$ for any limit ordinal $\\alpha < \\mu , X_{\\alpha } = \\emptyset $ , and $\\bigcup _{\\alpha \\le \\mu }(X_{\\alpha })$ is consistent Proposition 15 (Splitting Sequence Theorem) [4] Let $U={\\langle U_{\\alpha } \\rangle }_{\\alpha < \\mu }$ be a splitting sequence for a program $\\Pi $ .", "A set $S$ of literals is a consistent answer set for $\\Pi $ iff $S=\\bigcup _{\\alpha < \\mu }{X_{\\alpha }}$ for some solution ${\\langle X_{\\alpha } \\rangle }_{\\alpha < \\mu }$ to $\\Pi $ w.r.t $U$ .", "Proof: Let $U = \\langle U_\\alpha \\rangle _{\\alpha < \\mu }$ be a splitting sequence of $\\Pi $ , then a consistent set of literals $S=\\bigcup _{\\alpha < \\mu }{X_{\\alpha }}$ is an answer set of $\\Pi ^S$ iff $X_0$ is an answer set of $bot_{U_0}(\\Pi )$ and for any ordinal $\\alpha + 1 < \\mu $ , $X_{\\alpha +1}$ is an answer set of $eval_{U_{\\alpha }}(bot_{U_{\\alpha +1}}(\\Pi ) \\setminus bot_{U_{\\alpha }}(\\Pi ), \\bigcup _{\\nu \\le \\alpha }X_{\\nu })$ .", "Note that every literal in $bot_{U_0}(\\Pi )$ belongs to $lit(\\Pi ) \\cap U_0$ , and every literal occurring in $eval_{U_{\\alpha }}(bot_{U_{\\alpha +1}}(\\Pi ) \\setminus bot_{U_{\\alpha }}(\\Pi ), \\bigcup _{\\nu \\le \\alpha }X_{\\nu })$ , $(\\alpha + 1 < \\mu )$ belongs to $lit(\\Pi ) \\cap (U_{\\alpha +1} \\setminus U_{\\alpha })$ .", "In addition, $X_0$ , and all $X_{\\alpha +1}$ are pairwise disjoint.", "We prove the theorem by induction over the splitting sequence.", "Base case: $\\alpha = 1$ .", "The splitting sequence is $U_0 \\subseteq U_1$ .", "Then the sub-program $\\Pi _1 = bot_{U_1}(\\Pi )$ contains all literals in $U_1$ ; and $U_0$ splits $\\Pi _1$ into $bot_{U_0}(\\Pi _1)$ and $bot_{U_1}(\\Pi _1) \\setminus bot_{U_0}(\\Pi _1)$ .", "Then, $S_1 = X_0 \\cup X_1$ is a consistent answer set of $\\Pi _1$ iff $X_0 = S_1 \\cap U_0$ is an answer set of $bot_{U_0}(\\Pi _1)$ and $X_1 = S_1 \\setminus U_0$ is an answer set of $eval_{U_0}(\\Pi _1 \\setminus bot_{U_0}(\\Pi _1),X_1)$ – by the splitting theorem Since $bot_{U_0}(\\Pi _1) = bot_{U_0}(\\Pi )$ and $bot_{U_1}(\\Pi _1) \\setminus bot_{U_0}(\\Pi _1) = bot_{U_1}(\\Pi ) \\setminus bot_{U_0}(\\Pi )$ ; $S_1 = X_0 \\cup X_1$ is an answer set for $\\Pi \\setminus bot_{U_1}(\\Pi )$ .", "Induction: Assume theorem holds for $\\alpha = k$ , show theorem holds for $\\alpha = k+1$ .", "The inductive assumption holds for the splitting sequence $U_0 \\subseteq \\dots \\subseteq U_k$ .", "Then the sub-program $\\Pi _k = bot_{U_k}(\\Pi )$ contains all literals in $U_k$ and $S_k = X_0 \\cup \\dots \\cup X_k$ is a consistent answer set of $\\Pi ^{S_k}$ iff $X_0$ is an answer set for $bot_{U_0}(\\Pi _k)$ and for any $\\alpha \\le k$ , $X_{\\alpha +1}$ is answer set of $eval_{U_{\\alpha }}(bot_{U_{\\alpha +1}}(\\Pi ) \\setminus bot_{U_{\\alpha }}(\\Pi ), X_0 \\cup \\dots \\cup X_{\\alpha })$ We show that the theorem holds for $\\alpha = k+1$ .", "The splitting sequence is $U_0 \\subseteq U_{k+1}$ .", "Then the sub-program $\\Pi _{k+1} = bot_{U_{k+1}}(\\Pi )$ contains all literals $U_{k+1}$ .", "We have $U_k$ split $\\Pi _{k+1}$ into $bot_{U_k}(\\Pi _{k+1})$ and $bot_{U_{k+1}}(\\Pi _{k+1}) \\setminus bot_{U_k}(\\Pi _{k+1})$ .", "Then, $S_{k+1} = X_{0:k} \\cup X_{k+1}$ is a consistent answer set of $\\Pi _{k+1}$ iff $X_{0:k} = S_{k+1} \\cap U_k$ is an answer set of $bot_{U_k}(\\Pi _{k+1})$ and $X_{k+1} = S_{k+1} \\setminus U_k$ is an answer set of $eval_{U_k}(\\Pi _{k+1} \\setminus bot_{U_k}(\\Pi _{k+1},X_{k+1})$ – by the splitting theorem Since $bot_{U_k}(\\Pi _{k+1}) = bot_{U_k}(\\Pi )$ and $bot_{U_{k+1}}(\\Pi _{k+1}) \\setminus bot_{U_k}(\\Pi _{k+1}) = bot_{U_{k+1}}(\\Pi ) \\setminus bot_{U_k}(\\Pi )$ ; $S_{k+1} = X_{0:k} \\cup X_1$ is an answer set for $\\Pi \\setminus bot_{U_{k+1}}(\\Pi )$ .", "From the inductive assumption we know that $X_0 \\cup \\dots \\cup X_k$ is a consistent answer set of $bot_{U_k}(\\Pi )$ , $X_0$ is the answer set of $bot_{U_0}(\\Pi )$ , and for each $0 \\le \\alpha \\le k$ , $X_{\\alpha +1}$ is answer set of $eval_{U_{\\alpha }}(bot_{U_{\\alpha +1}}(\\Pi ) \\setminus bot_{U_{\\alpha }}(\\Pi ), X_0 \\cup \\dots \\cup X_{\\alpha })$ .", "Thus, $X_{0:k} = X_0 \\cup \\dots \\cup X_k$ .", "Combining above with the inductive assumption, we get $S_{k+1} = X_0 \\cup \\dots \\cup X_{k+1}$ is a consistent answer set of $\\Pi ^{S_{k+1}}$ iff $X_0$ is an answer set for $bot_{U_0}(\\Pi _{k+1})$ and for any $\\alpha \\le k+1$ , $X_{\\alpha +1}$ is answer set of $eval_{U_{\\alpha }}(bot_{U_{\\alpha +1}}(\\Pi ) \\setminus bot_{U_{\\alpha }}(\\Pi ), X_0 \\cup \\dots \\cup X_{\\alpha })$ .", "In addition, for some $\\alpha < \\mu $ , where $\\mu $ is the length of the splitting sequence $U = \\langle U_{\\alpha } \\rangle _{\\alpha < \\mu }$ of $\\Pi $ , $bot_{U_{\\alpha }}(\\Pi )$ will be the entire $\\Pi $ , i.e.", "$lit(\\Pi ) = U_{\\alpha }$ .", "Then the set $S$ of literals is a consistent answer set of $\\Pi $ iff $S=\\bigcup _{\\alpha < \\mu }(X_{\\alpha })$ for some solution $\\langle X_{\\alpha } \\rangle _{\\alpha < \\mu }$ to $\\Pi $ w.r.t $U$ .", "Proof of Proposition REF Let $PN=(P,T,E,W)$ be a Petri Net, $M_0$ be its initial marking and let $\\Pi ^0(PN,M_0,$ $k,ntok)$ be the ASP encoding of $PN$ and $M_0$ over a simulation length $k$ , with maximum $ntok$ tokens on any place node, as defined in section REF .", "Then $X=M_0,T_0,M_1,\\dots ,M_k,T_k,M_{k+1}$ is an execution sequence of a $PN$ (w.r.t.", "$M_0$ ) iff there is an answer set $A$ of $\\Pi ^0(PN,M_0,k,ntok)$ such that: $\\lbrace fires(t,ts) : t \\in T_{ts}, 0 \\le ts \\le k \\rbrace = \\lbrace fires(t,ts) : fires(t,ts) \\in A \\rbrace $ $\\begin{split}\\lbrace holds(p,q,ts) &: p \\in P, q = M_{ts}(p), 0 \\le ts \\le k+1 \\rbrace \\\\&= \\lbrace holds(p,q,ts) : holds(p,q,ts) \\in A \\rbrace \\end{split}$ We prove this by showing that: Given an execution sequence $X$ , we create a set $A$ such that it satisfies (REF ) and (REF ) and show that $A$ is an answer set of $\\Pi ^0$ Given an answer set $A$ of $\\Pi ^0$ , we create an execution sequence $X$ such that (REF ) and (REF ) are satisfied First we show (REF ): Given $PN$ and its execution sequence $X$ , we create a set $A$ as a union of the following sets: $A_1=\\lbrace num(n) : 0 \\le n \\le ntok \\rbrace $ $A_2=\\lbrace time(ts) : 0 \\le ts \\le k\\rbrace $ $A_3=\\lbrace place(p) : p \\in P \\rbrace $ $A_4=\\lbrace trans(t) : t \\in T \\rbrace $ $A_5=\\lbrace ptarc(p,t,n) : (p,t) \\in E^-, n=W(p,t) \\rbrace $ , where $E^- \\subseteq E$ $A_6=\\lbrace tparc(t,p,n) : (t,p) \\in E^+, n=W(t,p) \\rbrace $ , where $E^+ \\subseteq E$ $A_7=\\lbrace holds(p,q,0) : p \\in P, q=M_{0}(p) \\rbrace $ $A_8=\\lbrace notenabled(t,ts) : t \\in T, 0 \\le ts \\le k, \\exists p \\in \\bullet t, M_{ts}(p) < W(p,t) \\rbrace $ per definition REF (enabled transition) $A_9=\\lbrace enabled(t,ts) : t \\in T, 0 \\le ts \\le k, \\forall p \\in \\bullet t, W(p,t) \\le M_{ts}(p) \\rbrace $ per definition REF (enabled transition) $A_{10}=\\lbrace fires(t,ts) : t \\in T_{ts}, 0 \\le ts \\le k \\rbrace $ from definition REF (firing set) , only an enabled transition may fire $A_{11}=\\lbrace add(p,q,t,ts) : t \\in T_{ts}, p \\in t \\bullet , q=W(t,p), 0 \\le ts \\le k \\rbrace $ per definition REF (transition execution) $A_{12}=\\lbrace del(p,q,t,ts) : t \\in T_{ts}, p \\in \\bullet t, q=W(p,t), 0 \\le ts \\le k \\rbrace $ per definition REF (transition execution) $A_{13}=\\lbrace tot\\_incr(p,q,ts) : p \\in P, q=\\sum _{t \\in T_{ts}, p \\in t \\bullet }{W(t,p)}, 0 \\le ts \\le k \\rbrace $ per definition REF (firing set execution) $A_{14}=\\lbrace tot\\_decr(p,q,ts): p \\in P, q=\\sum _{t \\in T_{ts}, p \\in \\bullet t}{W(p,t)}, 0 \\le ts \\le k \\rbrace $ per definition REF (firing set execution) $A_{15}=\\lbrace consumesmore(p,ts) : p \\in P, q=M_{ts}(p), q1=\\sum _{t \\in T_{ts}, p \\in \\bullet t}{W(p,t)}, q1 > q, 0 \\le ts \\le k \\rbrace $ per definition REF (conflicting transitions) for enabled transition set $T_{ts}$ $A_{16}=\\lbrace consumesmore : \\exists p \\in P : q=M_{ts}(p), q1=\\sum _{t \\in T_{ts}, p \\in \\bullet t}{W(p,t)}, q1 > q, 0 \\le ts \\le k \\rbrace $ per definition REF (conflicting transitions) $A_{17}=\\lbrace holds(p,q,ts+1) : p \\in P, q=M_{ts+1}(p), 0 \\le ts < k\\rbrace $ , where $M_{ts+1}(p) = M_{ts}(p) - \\sum _{\\begin{array}{c}t \\in T_{ts}, p \\in \\bullet t\\end{array}}{W(p,t)} + \\sum _{\\begin{array}{c}t \\in T_{ts}, p \\in t \\bullet \\end{array}}{W(t,p)}$ according to definition REF (firing set execution) We show that $A$ satisfies (REF ) and (REF ), and $A$ is an answer set of $\\Pi ^0$ .", "$A$ satisfies (REF ) and (REF ) by its construction (given above).", "We show $A$ is an answer set of $\\Pi ^0$ by splitting.", "We split $lit(\\Pi ^0)$ (literals of $\\Pi ^0$ ) into a sequence of $6(k+1)+2$ sets: $U_0= head(f\\ref {f:place}) \\cup head(f\\ref {f:trans}) \\cup head(f\\ref {f:ptarc}) \\cup head(f\\ref {f:tparc}) \\cup head(f\\ref {f:time}) \\cup head(f\\ref {f:num}) \\cup head(i\\ref {i:holds}) = \\lbrace place(p) : p \\in P\\rbrace \\cup \\lbrace trans(t) : t \\in T\\rbrace \\cup \\lbrace ptarc(p,t,n) : (p,t) \\in E^-, n=W(p,t) \\rbrace \\cup \\lbrace tparc(t,p,n) : (t,p) \\in E^+, n=W(t,p) \\rbrace \\cup $ $\\lbrace time(0), \\dots , $ $time(k)\\rbrace \\cup \\lbrace num(0), \\dots , num(ntok)\\rbrace \\cup \\lbrace holds(p,q,0) : p \\in P, q=M_0(p) \\rbrace $ $U_{6k+1}=U_{6k+0} \\cup head(e\\ref {e:ne:ptarc})^{ts=k} = U_{6k+0} \\cup \\lbrace notenabled(t,k) : t \\in T \\rbrace $ $U_{6k+2}=U_{6k+1} \\cup head(e\\ref {e:enabled})^{ts=k} = U_{6k+1} \\cup \\lbrace enabled(t,k) : t \\in T \\rbrace $ $U_{6k+3}=U_{6k+2} \\cup head(a\\ref {a:fires})^{ts=k} = U_{6k+2} \\cup \\lbrace fires(t,k) : t \\in T \\rbrace $ $U_{6k+4}=U_{6k+3} \\cup head(r\\ref {r:add})^{ts=k} \\cup head(r\\ref {r:del})^{ts=k} = U_{6k+3} \\cup \\lbrace add(p,q,t,k) : p \\in P, t \\in T, q=W(t,p) \\rbrace \\cup \\lbrace del(p,q,t,k) : p \\in P, t \\in T, q=W(p,t) \\rbrace $ $U_{6k+5}=U_{6k+4} \\cup head(r\\ref {r:totincr})^{ts=k} \\cup head(r\\ref {r:totdecr})^{ts=k} = U_{6k+4} \\cup \\lbrace tot\\_incr(p,q,k) : p \\in P, 0 \\le q \\le ntok \\rbrace \\cup \\lbrace tot\\_decr(p,q,k) : p \\in P, 0 \\le q \\le ntok \\rbrace $ $U_{6k+6}=U_{6k+5} \\cup head(r\\ref {r:nextstate})^{ts=k} \\cup head(a\\ref {a:overc:place})^{ts=k} = U_{6k+5} \\cup \\lbrace holds(p,q,k+1) : p \\in P, 0 \\le q \\le ntok \\rbrace \\cup \\lbrace consumesmore(p,k) : p \\in P\\rbrace $ $U_{6k+7}=U_{6k+6} \\cup head(a\\ref {a:overc:gen}) = U_{7k+6} \\cup \\lbrace consumesmore \\rbrace $ where $head(r_i)^{ts=k}$ are head atoms of ground rule $r_i$ in which $ts=k$ .", "We write $A_i^{ts=k} = \\lbrace a(\\dots ,ts) : a(\\dots ,ts) \\in A_i, ts=k \\rbrace $ as short hand for all atoms in $A_i$ with $ts=k$ .", "$U_{\\alpha }, 0 \\le \\alpha \\le 6k+7$ form a splitting sequence, since each $U_i$ is a splitting set of $\\Pi ^0$ , and $\\langle U_{\\alpha }\\rangle _{\\alpha < \\mu }$ is a monotone continuous sequence, where $U_0 \\subseteq U_1 \\dots \\subseteq U_{6k+7}$ and $\\bigcup _{\\alpha < \\mu }{U_{\\alpha }} = lit(\\Pi ^0)$ .", "We compute the answer set of $\\Pi ^0$ using the splitting sets as follows: $bot_{U_0}(\\Pi ^0) = f\\ref {f:place} \\cup f\\ref {f:trans} \\cup f\\ref {f:ptarc} \\cup f\\ref {f:tparc} \\cup f\\ref {f:time} \\cup i\\ref {i:holds} \\cup f\\ref {f:num}$ and $X_0 = A_1 \\cup \\dots \\cup A_7$ ($= U_0$ ) is its answer set – using forced atom proposition $eval_{U_0}(bot_{U_1}(\\Pi ^0) \\setminus bot_{U_0}(\\Pi ^0), X_0) = \\lbrace notenabled(t,0) \\text{:-} .", "\| \\lbrace trans(t), $ $ ptarc(p,t,n), $ $ holds(p,q,0) \\rbrace \\subseteq X_0, \\text{~where~} q < n \\rbrace $ .", "Its answer set $X_1=A_8^{ts=0}$ – using forced atom proposition and construction of $A_8$ .", "where, $q=M_0(p)$ , $n=W(p,t)$ , and for an arc $(p,t) \\in E^-$ – by construction of $i\\ref {i:holds}$ and $f\\ref {f:ptarc}$ in $\\Pi ^0$ , and in an arc $(p,t) \\in E^-$ , $p \\in \\bullet t$ (by definition REF of preset) thus, $notenabled(t,0) \\in X_1$ represents $\\exists p \\in \\bullet t : M_0(p) < W(p,t)$ .", "$eval_{U_1}(bot_{U_2}(\\Pi ^0) \\setminus bot_{U_1}(\\Pi ^0), X_0 \\cup X_1) = \\lbrace enabled(t,0) \\text{:-}.", "\| trans(t) \\in X_0 \\cup X_1, $ $notenabled(t,0) \\notin X_0 \\cup X_1 \\rbrace $ .", "Its answer set is $X_2 = A_9^{ts=0}$ – using forced atom proposition and construction of $A_9$ .", "since an $enabled(t,0) \\in X_2$ if $\\nexists ~notenabled(t,0) \\in X_0 \\cup X_1$ , which is equivalent to $\\nexists p \\in \\bullet t : M_0(p) < W(p,t) \\equiv \\forall p \\in \\bullet t: M_0(p) \\ge W(p,t)$ .", "$eval_{U_2}(bot_{U_3}(\\Pi ^0) \\setminus bot_{U_2}(\\Pi ^0), X_0 \\cup X_1 \\cup X_2) = \\lbrace \\lbrace fires(t,0)\\rbrace \\text{:-}.", "\| enabled(t,0) \\\\ \\text{~holds in~} X_0 \\cup X_1 \\cup X_2 \\rbrace $ .", "It has multiple answer sets $X_{3.1}, \\dots , X_{3.n}$ , corresponding to elements of power set of $fires(t,0)$ atoms in $eval_{U_2}(...)$ – using supported rule proposition.", "Since we are showing that the union of answer sets of $\\Pi ^0$ determined using splitting is equal to $A$ , we only consider the set that matches the $fires(t,0)$ elements in $A$ and call it $X_3$ , ignoring the rest.", "Thus, $X_3 = A_{10}^{ts=0}$ , representing $T_0$ .", "$eval_{U_3}(bot_{U_4}(\\Pi ^0) \\setminus bot_{U_3}(\\Pi ^0), X_0 \\cup \\dots \\cup X_3) = \\lbrace add(p,n,t,0) \\text{:-}.", "\| \\lbrace fires(t,0), \\\\ tparc(t,p,n) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_3 \\rbrace \\cup \\lbrace del(p,n,t,0) \\text{:-}.", "\| \\lbrace fires(t,0), ptarc(p,t,n) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_3 \\rbrace $ .", "It's answer set is $X_4 = A_{11}^{ts=0} \\cup A_{12}^{ts=0}$ – using forced atom proposition and definitions of $A_{11}$ and $A_{12}$ .", "where each $add$ atom is equivalent to $n=W(t,p) : p \\in t \\bullet $ , and each $del$ atom is equivalent to $n=W(p,t) : p \\in \\bullet t$ , representing the effect of transitions in $T_0$ – by construction $eval_{U_4}(bot_{U_5}(\\Pi ^0) \\setminus bot_{U_4}(\\Pi ^0), X_0 \\cup \\dots \\cup X_4) = \\lbrace tot\\_incr(p,qq,0) \\text{:-}.", "\| \\\\ qq=\\sum _{add(p,q,t,0) \\in X_0 \\cup \\dots \\cup X_4}{q} \\rbrace \\cup $ $\\lbrace tot\\_decr(p,qq,0) \\text{:-}.", "\| qq=\\sum _{del(p,q,t,0) \\in X_0 \\cup \\dots \\cup X_4}{q} \\rbrace $ .", "It's answer set is $X_5 = A_{13}^{ts=0} \\cup A_{14}^{ts=0}$ – using forced atom proposition and definitions of $A_{13}$ and $A_{14}$ , ad definition REF (semantics of aggregate assignment atom).", "where each $tot\\_incr(p,qq,0)$ , $qq=\\sum _{add(p,q,t,0) \\in X_0 \\cup \\dots X_4}{q} \\\\ \\equiv $ $qq=\\sum _{t \\in X_3, p \\in t \\bullet }{W(p,t)}$ , and each $tot\\_decr(p,qq,0)$ , $qq=\\sum _{del(p,q,t,0) \\in X_0 \\cup \\dots X_4}{q} \\\\ \\equiv $ $qq=\\sum _{t \\in X_3, p \\in \\bullet t}{W(t,p)}$ , represent the net effect of transitions in $T_0$ – by construction $eval_{U_5}(bot_{U_6}(\\Pi ^0) \\setminus bot_{U_5}(\\Pi ^0), X_0 \\cup \\dots \\cup X_5) = \\lbrace consumesmore(p,0) \\text{:-}.", "\| \\\\ \\lbrace holds(p,q,0), $ $tot\\_decr(p,q1,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_5, $ $q1 > q \\rbrace \\cup $ $\\lbrace holds(p,q,1) \\text{:-}.", "\| $ $ \\lbrace holds(p,q1,0), $ $tot\\_incr(p,q2,0), $ $tot\\_decr(p,q3,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_5, $ $q=q1+q2-q3 \\rbrace $ .", "It's answer set is $X_6 = A_{15}^{ts=0} \\cup A_{17}^{ts=0}$ – using forced atom proposition.", "where, $consumesmore(p,0)$ represents $\\exists p : q=M_0(p), \\\\ q1=\\sum _{t \\in T_0, p \\in \\bullet t}{W(p,t)}, q1 > q$ – indicating place $p$ will be overconsumed if $T_0$ is fired as defined in definition REF (conflicting transitions) and $holds(p,q,1)$ represents $q=M_1(p)$ – by construction $ \\vdots $ $eval_{U_{6k+0}}(bot_{U_{6k+1}}(\\Pi ^0) \\setminus bot_{U_{6k+0}}(\\Pi ^0), X_0 \\cup \\dots \\cup X_{6k+0}) = \\\\ \\lbrace notenabled(t,k) \\text{:-} .", "\| \\lbrace trans(t), ptarc(p,t,n), holds(p,q,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{6k+0}, \\\\ \\text{~where~} q < n \\rbrace $ .", "Its answer set $X_{6k+1}=A_8^{ts=k}$ – using forced atom proposition and construction of $A_8$ .", "where, $q=M_k(p)$ , and $n=W(p,t)$ for an arc $(p,t) \\in E^-$ – by construction of $holds$ and $ptarc$ predicates in $\\Pi ^0$ , and in an arc $(p,t) \\in E^-$ , $p \\in \\bullet t$ (by definition REF of preset) thus, $notenabled(t,k) \\in X_{6k+1}$ represents $\\exists p \\in \\bullet t : M_k(p) < W(p,t)$ .", "$eval_{U_{6k+1}}(bot_{U_{6k+2}}(\\Pi ^0) \\setminus bot_{U_{6k+1}}(\\Pi ^0), X_0 \\cup \\dots \\cup X_{6k+1}) = \\lbrace enabled(t,k) \\text{:-}.", "\| \\\\ trans(t) \\in $ $X_0 \\cup \\dots \\cup X_{6k+1}, notenabled(t,k) \\notin X_0 \\cup \\dots \\cup X_{6k+1} \\rbrace $ .", "Its answer set is $X_{6k+2} = A_9^{ts=k}$ – using forced atom proposition and construction of $A_9$ .", "since an $enabled(t,k) \\in X_{6k+2}$ if $\\nexists ~notenabled(t,k) \\in X_0 \\cup \\dots \\cup X_{6k+1}$ , which is equivalent to $\\nexists p \\in \\bullet t : M_k(p) < W(p,t) \\equiv \\forall p \\in \\bullet t: M_k(p) \\ge W(p,t)$ .", "$eval_{U_{6k+2}}(bot_{U_{6k+3}}(\\Pi ^0) \\setminus bot_{U_{6k+2}}(\\Pi ^0), X_0 \\cup \\dots \\cup X_{6k+2}) = \\\\ \\lbrace \\lbrace fires(t,k)\\rbrace \\text{:-}.", "\| $ $enabled(t,k) \\text{~holds in~} $ $X_0 \\cup \\dots \\cup X_{6k+2} \\rbrace $ .", "It has multiple answer sets $X_{{6k+3}.1}, \\dots , X_{{6k+3}.n}$ , corresponding to elements of power set of $fires(t,k)$ atoms in $eval_{U_{6k+2}}(...)$ – using supported rule proposition.", "Since we are showing that the union of answer sets of $\\Pi ^0$ determined using splitting is equal to $A$ , we only consider the set that matches the $fires(t,k)$ elements in $A$ and call it $X_{6k+3}$ , ignoring the reset.", "Thus, $X_{6k+3} = A_{10}^{ts=k}$ , representing $T_k$ .", "$eval_{U_{6k+3}}(bot_{U_{6k+4}}(\\Pi ^0) \\setminus bot_{U_{6k+3}}(\\Pi ^0), X_0 \\cup \\dots \\cup X_{6k+3}) = $ $ \\lbrace add(p,n,t,k) \\text{:-}.", "\| \\\\ \\lbrace fires(t,k), tparc(t,p,n) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{6k+3} \\rbrace \\cup $ $ \\lbrace del(p,n,t,k) \\text{:-}.", "\| $ $\\lbrace fires(t,k), $ $ptarc(p,t,n) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{6k+3} \\rbrace $ .", "It's answer set is $X_{6k+4} = A_{11}^{ts=k} \\cup A_{12}^{ts=k}$ – using forced atom proposition and definitions of $A_{11}$ and $A_{12}$ .", "where, each $add$ atom is equivalent to $n=W(t,p) : p \\in t \\bullet $ , and each $del$ atom is equivalent to $n=W(p,t) : p \\in \\bullet t$ representing the effect of transitions in $T_k$ $eval_{U_{6k+4}}(bot_{U_{6k+5}}(\\Pi ^0) \\setminus bot_{U_{6k+4}}(\\Pi ^0), X_0 \\cup \\dots \\cup X_{6k+4}) = $ $\\lbrace tot\\_incr(p,qq,k) \\text{:-}.", "\| $ $ qq=\\sum _{add(p,q,t,k) \\in X_0 \\cup \\dots \\cup X_{6k+4}}{q} \\rbrace \\cup $ $\\lbrace tot\\_decr(p,qq,k) \\text{:-}.", "\| \\\\ qq=\\sum _{del(p,q,t,k) \\in X_0 \\cup \\dots \\cup X_{6k+4}}{q} \\rbrace $ .", "It's answer set is $X_5 = A_{13}^{ts=k} \\cup A_{14}^{ts=k}$ – using forced atom proposition and definitions of $A_{13}$ and $A_{14}$ .", "where, each $tot\\_incr(p,qq,k)$ , $qq=\\sum _{add(p,q,t,k) \\in X_0 \\cup \\dots X_{7k+4}}{q}$ $\\equiv qq=\\sum _{t \\in X_{6k+3}, p \\in t \\bullet }{W(p,t)}$ , and each $tot\\_decr(p,qq,k)$ , $qq=\\sum _{del(p,q,t,k) \\in X_0 \\cup \\dots X_{7k+4}}{q}$ $\\equiv qq=\\sum _{t \\in X_{6k+3}, p \\in \\bullet t}{W(t,p)}$ , represent the net effect of transitions in $T_k$ $eval_{U_{6k+5}}(bot_{U_{6k+6}}(\\Pi ^0) \\setminus bot_{U_{6k+5}}(\\Pi ^0), X_0 \\cup \\dots \\cup X_{6k+5}) = $ $ \\lbrace consumesmore(p,k) \\text{:-}.", "\| $ $ \\lbrace holds(p,q,k), $ $ tot\\_decr(p,q1,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{6k+5} , q1 > q \\rbrace \\cup \\lbrace holds(p,q,k+1) \\text{:-}., \| \\lbrace holds(p,q1,k), tot\\_incr(p,q2,k), \\\\ tot\\_decr(p,q3,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{6k+5}, q=q1+q2-q3 \\rbrace $ .", "It's answer set is $X_{6k+6} = A_{15}^{ts=k} \\cup A_{17}^{ts=k}$ – using forced atom proposition.", "where, $holds(p,q,k+1)$ represents the marking of place $p$ in the next time step due to firing $T_k$ , and, $consumesmore(p,k)$ represents $\\exists p : q=M_{k}(p), q1=\\sum _{t \\in T_{k}, p \\in \\bullet t}{W(p,t)}, $ $q1 > q$ indicating place $p$ that will be overconsumed if $T_k$ is fired as defined in definition REF (conflicting transitions) $eval_{U_{6k+6}}(bot_{U_{6k+7}}(\\Pi ^0) \\setminus bot_{U_{6k+6}}(\\Pi ^0), X_0 \\cup \\dots \\cup X_{6k+6}) = \\lbrace consumesmore \\text{:-}.", "\| $ $\\lbrace consumesmore(p,0),$ $ \\dots , $ $consumesmore(p,k)\\rbrace \\cap (X_0 \\cup \\dots \\cup X_{6k+7}) \\ne \\emptyset \\rbrace $ .", "It's answer set is $X_{6k+7} = A_{16}^{ts=k}$ – using forced atom proposition $X_{6k+7}$ will be empty since none of $consumesmore(p,0),\\dots , \\\\ consumesmore(p,k)$ hold in $X_0 \\cup \\dots \\cup X_{6k+6}$ due to the construction of $A$ and encoding of $a\\ref {a:overc:place}$ , and it is not eliminated by the constraint $a\\ref {a:overc:elim}$ .", "The set $X=X_{0} \\cup \\dots \\cup X_{6k+7}$ is the answer set of $\\Pi ^0$ by the splitting sequence theorem REF .", "Each $X_i, 0 \\le i \\le 6k+7$ matches a distinct partition of $A$ , and $X = A$ , thus $A$ is an answer set of $\\Pi ^0$ .", "Next we show (REF ): Given $\\Pi ^0$ be the encoding of a Petri Net $PN(P,T,E,W)$ with initial marking $M_0$ , and $A$ be an answer set of $\\Pi ^0$ that satisfies (REF ) and (REF ), then we can construct $X=M_0,T_0,\\dots ,M_k,T_k,M_{k+1}$ from $A$ , such that it is an execution sequence of $PN$ .", "We construct the $X$ as follows: $M_i = (M_i(p_0), \\dots , M_i(p_n))$ , where $\\lbrace holds(p_0,M_i(p_0),i), \\dots holds(p_n,M_i(p_n),i) \\rbrace \\\\ \\subseteq A$ , for $0 \\le i \\le k+1$ $T_i = \\lbrace t : fires(t,i) \\in A\\rbrace $ , for $0 \\le i \\le k$ and show that $X$ is indeed an execution sequence of $PN$ .", "We show this by induction over $k$ (i.e.", "given $M_k$ , $T_k$ is a valid firing set and its firing produces marking $M_{k+1}$ ).", "Base case: Let $k=0$ , show [(1)] $T_0$ is a valid firing set for $M_0$ , and $T_0$ 's firing in $M_0$ producing marking $M_1$ .", "We show $T_0$ is a valid firing set w.r.t.", "$M_0$ .", "Let $\\lbrace fires(t_0,0), \\dots , fires(t_x,0) \\rbrace $ be the set of all $fires(\\dots ,0)$ atoms in $A$ , Then for each $fires(t_i,0) \\in A$ $enabled(t_i,0) \\in A$ – from rule $a\\ref {a:fires}$ and supported rule proposition Then $notenabled(t_i,0) \\notin A$ – from rule $e\\ref {e:enabled}$ and supported rule proposition Then $body(e\\ref {e:ne:ptarc})$ must not hold in $A$ – from rule $e\\ref {e:ne:ptarc}$ and forced atom proposition Then $q \\lnot < n_i \\equiv q \\ge n_i$ in $e\\ref {e:ne:ptarc}$ for all $\\lbrace holds(p,q,0), ptarc(p,t_i,n_i)\\rbrace \\subseteq A$ – from $e\\ref {e:ne:ptarc}$ , forced atom proposition, and the following: $holds(p,q,0) \\in A$ represents $q=M_0(p)$ – rule $i\\ref {i:holds}$ construction $ptarc(p,t_i,n_i) \\in A$ represents $n_i=W(p,t_i)$ – rule $f\\ref {f:ptarc} $ construction Then $\\forall p \\in \\bullet t_i, M_0(p) > W(p,t_i)$ – from definition REF of preset $\\bullet t_i$ in PN Then $t_i$ is enabled and can fire in $PN$ , as a result it can belong to $T_0$ – from definition REF of enabled transition And $consumesmore \\notin A$ , since $A$ is an answer set of $\\Pi ^0$ – from rule $a\\ref {a:overc:elim}$ and supported rule proposition Then $\\nexists consumesmore(p,0) \\in A$ – from rule $a\\ref {a:overc:gen}$ and supported rule proposition Then $\\nexists \\lbrace holds(p,q,0), tot\\_decr(p,q1,0) \\rbrace \\subseteq A : q1>q$ in $body(a\\ref {a:overc:place})$ – from $a\\ref {a:overc:place}$ and forced atom proposition Then $\\nexists p : \\sum _{t_i \\in \\lbrace t_0,\\dots ,t_x\\rbrace , p \\in \\bullet t_i}{W(p,t_i)} > M_0(p)$ – from the following $holds(p,q,0)$ represents $q=M_0(p)$ – from rule $i\\ref {i:holds}$ construction, given $tot\\_decr(p,q1,0) \\in A$ if $\\lbrace del(p,q1_0,t_0,0), \\dots del(p,q1_x,t_x,0) \\rbrace \\subseteq A$ , where $q1=q1_0+\\dots +q1_x$ – from $r\\ref {r:totdecr}$ and forced atom proposition $del(p,q1_i,t_i,0) \\in A$ if $\\lbrace fires(t_i,0), ptarc(p,t_i,q1_i) \\rbrace \\subseteq A$ – from $r\\ref {r:del}$ and supported rule proposition $del(p,q1_i,t_i,0) \\in A$ represents removal of $q1_i = W(p,t_i)$ tokens from $p \\in \\bullet t_i$ – from $r\\ref {r:del}$ , supported rule proposition, and definition REF of transition execution in PN Then the set of transitions in $T_0$ do not conflict – by the definition REF of conflicting transitions Then $\\lbrace t_0, \\dots , t_x\\rbrace = T_0$ – using 1(a),1(b) above, and definition of firing set We show $M_1$ is produced by firing $T_0$ in $M_0$ .", "Let $holds(p,q,1) \\in A$ Then $\\lbrace holds(p,q1,0), tot\\_incr(p,q2,0), tot\\_decr(p,q3,0) \\rbrace \\subseteq A : q=q1+q2-q3$ – from rule $r\\ref {r:nextstate}$ and supported rule proposition Then $holds(p,q1,0) \\in A$ represents $q1=M_0(p)$ – given, rule $i\\ref {i:holds}$ construction; and $\\lbrace add(p,q_0,t_0,0), \\dots , $ $add(p,q_j,t_j,0)\\rbrace \\subseteq A : q_0 + \\dots + q_j = q2$ ; and $\\lbrace del(p,q_0,t_0,0), \\dots , $ $del(p,q_l,t_l,0)\\rbrace \\subseteq A : q_0 + \\dots + q_l = q3$ – rules $r\\ref {r:totincr},r\\ref {r:totdecr}$ and supported rule proposition, respectively Then $\\lbrace fires(t_0,0), \\dots , fires(t_j,0) \\rbrace \\subseteq A$ and $\\lbrace fires(t_0,0), \\dots , fires(t_l,0) \\rbrace $ $\\subseteq A$ – rules $r\\ref {r:add},r\\ref {r:del}$ and supported rule proposition; and the following $tparc(t_y,p,q_y) \\in A, 0 \\le y \\le j$ represents $q_y=W(t_y,p)$ – given $ptarc(p,t_z,q_z) \\in A, 0 \\le z \\le l$ represents $q_z=W(p,t_z)$ – given Then $\\lbrace fires(t_0,0), \\dots , fires(t_j,0) \\rbrace \\cup \\lbrace fires(t_0,0), \\dots , fires(t_l,0) \\rbrace \\subseteq A $ $= \\lbrace fires(t_0,0), \\dots , fires(t_x,0) \\rbrace \\subseteq A$ – set union of subsets Then for each $fires(t_x,0) \\in A$ we have $t_x \\in T_0$ – already shown in item REF above Then $q = M_0(p) + \\sum _{t_x \\in T_0 \\wedge p \\in t_x \\bullet }{W(t_x,p)} - \\sum _{t_x \\in T_0 \\wedge p \\in \\bullet t_x}{W(p,t_x)}$ – from (REF ) above and the following Each $add(p,q_j,t_j,0) \\in A$ represents $q_j=W(t_j,p)$ for $p \\in t_j \\bullet $ – rule $r\\ref {r:add}$ encoding, and definition REF of transition execution in $PN$ Each $del(p,t_y,q_y,0) \\in A$ represents $q_y=W(p,t_y)$ for $p \\in \\bullet t_y$ – from rule $r\\ref {r:del}$ encoding, and definition REF of transition execution in $PN$ Each $tot\\_incr(p,q2,0) \\in A$ represents $q2=\\sum _{t_x \\in T_0 \\wedge p \\in t_x \\bullet }{W(t_x,p)}$ – aggregate assignment atom semantics in rule $r\\ref {r:totincr}$ Each $tot\\_decr(p,q3,0) \\in A$ represents $q3=\\sum _{t_x \\in T_0 \\wedge p \\in \\bullet t_x}{W(p,t_x)}$ – aggregate assignment atom semantics in rule $r\\ref {r:totdecr}$ Then, $M_1(p) = q$ – since $holds(p,q,1) \\in A$ encodes $q=M_1(p)$ – by construction Inductive Step: Assume $M_k$ is a valid marking in $X$ for $PN$ , show [(1)] $T_k$ is a valid firing set for $M_k$ , and firing $T_k$ in $M_k$ produces marking $M_{k+1}$ .", "We show that $T_k$ is a valid firing set for $M_k$ .", "Let $\\lbrace fires(t_0,k),\\dots ,fires(t_x,k) \\rbrace $ be the set of all $fires(\\dots ,k)$ atoms in $A$ , Then for each $fires(t_i,k) \\in A$ $enabled(t_i,k) \\in A$ – from rule $a\\ref {a:fires}$ and supported rule proposition Then $notenabled(t_i,k) \\notin A$ – from rule $e\\ref {e:enabled}$ and supported rule proposition Then body of $e\\ref {e:ne:ptarc}$ must hold in $A$ – from rule $e\\ref {e:ne:ptarc}$ and forced proposition Then $q \\lnot < n_i \\equiv q \\ge n_i$ in $e\\ref {e:ne:ptarc}$ for all $\\lbrace holds(p,q,k), ptarc(p,t_i,n_i) \\rbrace \\subseteq A$ – from $e\\ref {e:ne:ptarc}$ using forced atom proposition, and the following $holds(p,q,k) \\in A$ represents $q=M_k(p)$ – construction, inductive assumption $ptarc(p,t,n_i) \\in A$ represents $n_i=W(p,t)$ – rule $f\\ref {f:tparc}$ construction Then $\\forall p \\in \\bullet t_i, M_k(p) > W(p,t_i)$ – from definition REF of preset $\\bullet t_i$ in PN Then $t_i$ is enabled and can fire in $PN$ , as a result it can belong to $T_k$ – from definition REF of enabled transition And $consumesmore \\notin A$ , since $A$ is an answer set of $\\Pi ^0$ – from rule $a\\ref {a:overc:elim}$ and supported rule proposition Then $\\nexists consumesmore(p,k) \\in A$ – from rule $a\\ref {a:overc:gen}$ and forced atom proposition Then $\\nexists \\lbrace holds(p,q,k), tot\\_decr(p,q1,k) \\rbrace \\subseteq A : q1 > q$ in $body(a\\ref {a:overc:place})$ – from $a\\ref {a:overc:place}$ and forced atom proposition Then $\\nexists p : \\sum _{t_i \\in \\lbrace t_0,\\dots ,t_x \\rbrace , p \\in \\bullet t_i}{W(p,t_i)} > M_k(p)$ – from the following $holds(p,q,k) \\in A$ represents $q=M_k(p)$ – by construction, and the inductive assumption $tot\\_decr(p,q1,k) \\in A$ if $\\lbrace del(p,q1_0,t_0,k), \\dots , del(p,q1_x,t_x,k) \\rbrace \\subseteq A$ , where $q1=q1_0+\\dots +q1_x$ – from $r\\ref {r:totdecr}$ and forced atom proposition $del(p,q1_i,t_i,k) \\in A$ if $\\lbrace fires(t_i,k), ptarc(p,t_i,q1_i) \\rbrace \\subseteq A$ – from $r\\ref {r:del}$ and supported rule proposition $del(p,q1_i,t_i,k)$ represents removal of $q1_i = W(p,t_i)$ tokens from $p \\in \\bullet t_i$ – from construction rule $r\\ref {r:del}$ , supported rule proposition, and definition REF of transition execution in $PN$ Then the set of transitions $T_k$ does conflict – by the definition REF of conflicting transitions Then $\\lbrace t_0,\\dots ,t_x\\rbrace = T_k$ – using 1(a),1(b) above We show $M_{k+1}$ is produced by firing $T_k$ in $M_k$ .", "Let $holds(p,q,k+1) \\in A$ Then $\\lbrace holds(p,q1,k), tot\\_incr(p,q2,k), tot\\_decr(p,q3,k) \\rbrace \\in A : q=q2+q2-q3$ – from rule $r\\ref {r:nextstate}$ and supported rule proposition Then $holds(p,q1,k) \\in A$ represents $q1=M_k(p)$ – inductive assumption and construction; and $\\lbrace add(p,q2_0,t_0,k), \\dots , add(p,q2_j,t_j,k) \\rbrace \\subseteq A : q2_0+\\dots +q2_j=q2$ and $\\lbrace del(p,q3_0,t_0,k), \\dots , del(p,q3_l,t_l,k) \\rbrace \\subseteq A : q3_0+\\dots +q3_l=q3$ – from rules $r\\ref {r:totincr},r\\ref {r:totdecr}$ using supported rule proposition, respectively Then $\\lbrace fires(t_0,k), \\dots , fires(t_j,k) \\rbrace \\subseteq A$ and $\\lbrace fires(t_0,k), \\dots , fires(t_l,k) \\rbrace $ $\\subseteq A$ – from rules $r\\ref {r:add} ,r\\ref {r:del} $ using supported rule proposition, respectively Then $\\lbrace fires(t_0,k), \\dots , fires(t_j,k) \\rbrace \\cup \\lbrace fires(t_0,k), \\dots , fires(t_l,k) \\rbrace $ $= \\lbrace fires(t_0,k), \\dots , fires(t_x,k) \\rbrace \\subseteq A$ – subset union property Then for each $fires(t_x,k) \\in A$ we have $t_x \\in T_k$ - already shown in item REF above Then $q = M_k(p) + \\sum _{t_x \\in T_k \\wedge p \\in t_x \\bullet }{W(t_x,p)} - \\sum _{t_x \\in T_k \\wedge p \\in \\bullet t_x}{W(p,t_x}$ – from (REF ) above and the following Each $add(p,q_j,t_j,0) \\in A$ represents $q_j = W(t_j,p)$ for $p \\in t_j \\bullet $ – encoding of $r\\ref {r:add}$ and definition REF of transition execution in PN Each $del(p,t_y,q_y,0) \\in A$ represents $q_y = W(p,t_y)$ for $p \\in \\bullet t_y$ – encoding of $r\\ref {r:del}$ and definition REF of transition execution in PN Each $tot\\_incr(p,q2,0) \\in A$ represents $q2 = \\sum _{t_x \\in T_k \\wedge p \\in t_x \\bullet }{W(t_x,p)}$ – aggregate assignment atom semantics in rule $r\\ref {r:totincr}$ Each $tot\\_decr(p,q3,0) \\in A$ represents $q3 = \\sum _{t_x \\in T_k \\wedge p \\in \\bullet t_x}{W(p,t_x)}$ – aggregate assignment atom semantics in rule $r\\ref {r:totdecr}$ Then $M_{k+1}(p) = q$ – since $holds(p,q,k+1) \\in A$ encodes $q=M_{k+1}(p)$ by construction As a result, for any $n > k$ , $T_n$ is a valid firing set w.r.t.", "$M_n$ and its firing produces marking $M_{n+1}$ .", "Conclusion: Since both REF and REF hold, $X=M_0,T_0,M_1,\\dots ,M_k,T_k,M_{k+1}$ is an execution sequence of $PN(P,T,E,W)$ (w.r.t.", "$M_0$ ) iff there is an answer set $A$ of $\\Pi ^0(PN,M_0,k,ntok)$ such that (REF ) and (REF ) hold.", "Proof of Proposition REF Let $PN=(P,T,E,W)$ be a Petri Net, $M_0$ be its initial marking and let $\\Pi ^1(PN,M_0,$ $k,ntok)$ be the ASP encoding of $PN$ and $M_0$ over a simulation length $k$ , with maximum $ntok$ tokens on any place node, as defined in section REF .", "Then $X=M_0,T_0,M_1,\\dots ,M_k,T_k,M_{k+1}$ is an execution sequence of $PN$ (w.r.t.", "$M_0$ ) iff there is an answer set $A$ of $\\Pi ^1(PN,M_0,k,ntok)$ such that: $\\lbrace fires(t,ts) : t \\in T_{ts}, 0 \\le ts \\le k\\rbrace = \\lbrace fires(t,ts) : fires(t,ts) \\in A \\rbrace $ $\\begin{split}\\lbrace holds(p,q,ts) &: p \\in P, q = M_{ts}(p), 0 \\le ts \\le k+1 \\rbrace \\\\&= \\lbrace holds(p,q,ts) : holds(p,q,ts) \\in A \\rbrace \\end{split}$ We prove this by showing that: Given an execution sequence $X$ , we create a set $A$ such that it satisfies (REF ) and (REF ) and show that $A$ is an answer set of $\\Pi ^1$ Given an answer set $A$ of $\\Pi ^1$ , we create an execution sequence $X$ such that (REF ) and (REF ) are satisfied.", "First we show (REF ): Given a $PN$ and an execution sequence $X$ of $PN$ , we create a set $A$ as a union of the following sets: $A_1=\\lbrace num(n) : 0 \\le n \\le ntok \\rbrace $ $A_2=\\lbrace time(ts) : 0 \\le ts \\le k\\rbrace $ $A_3=\\lbrace place(p) : p \\in P \\rbrace $ $A_4=\\lbrace trans(t) : t \\in T \\rbrace $ $A_5=\\lbrace ptarc(p,t,n) : (p,t) \\in E^-, n=W(p,t) \\rbrace $ , where $E^- \\subseteq E$ $A_6=\\lbrace tparc(t,p,n) : (t,p) \\in E^+, n=W(t,p) \\rbrace $ , where $E^+ \\subseteq E$ $A_7=\\lbrace holds(p,q,0) : p \\in P, q=M_{0}(p) \\rbrace $ $A_8=\\lbrace notenabled(t,ts) : t \\in T, 0 \\le ts \\le k, \\exists p \\in \\bullet t, M_{ts}(p) < W(p,t) \\rbrace $ per definition REF (enabled transition) $A_9=\\lbrace enabled(t,ts) : t \\in T, 0 \\le ts \\le k, \\forall p \\in \\bullet t, W(p,t) \\le M_{ts}(p) \\rbrace $ per definition REF (enabled transition) $A_{10}=\\lbrace fires(t,ts) : t \\in T_{ts}, 0 \\le ts \\le k \\rbrace $ per definition REF (firing set), only an enabled transition may fire $A_{11}=\\lbrace add(p,q,t,ts) : t \\in T_{ts}, p \\in t \\bullet , q=W(t,p), 0 \\le ts \\le k \\rbrace $ per definition REF (transition execution) $A_{12}=\\lbrace del(p,q,t,ts) : t \\in T_{ts}, p \\in \\bullet t, q=W(p,t), 0 \\le ts \\le k \\rbrace $ per definition REF (transition execution) $A_{13}=\\lbrace tot\\_incr(p,q,ts) : p \\in P, q=\\sum _{t \\in T_{ts}, p \\in t \\bullet }{W(t,p)}, 0 \\le ts \\le k \\rbrace $ per definition REF (firing set execution) $A_{14}=\\lbrace tot\\_decr(p,q,ts): p \\in P, q=\\sum _{t \\in T_{ts}, p \\in \\bullet t}{W(p,t)}, 0 \\le ts \\le k \\rbrace $ per definition REF (firing set execution) $A_{15}=\\lbrace consumesmore(p,ts) : p \\in P, q=M_{ts}(p), q1=\\sum _{t \\in T_{ts}, p \\in \\bullet t}{W(p,t)}, q1 > q, 0 \\le ts \\le k \\rbrace $ per definition REF (conflicting transitions) for enabled transition set $T_{ts}$ $A_{16}=\\lbrace consumesmore : \\exists p \\in P : q=M_{ts}(p), q1=\\sum _{t \\in T_{ts}, p \\in \\bullet t}{W(p,t)}, q1 > q, 0 \\le ts \\le k \\rbrace $ per definition REF (conflicting transitions) $A_{17}=\\lbrace could\\_not\\_have(t,ts) : t \\in T, (\\forall p \\in \\bullet t, W(p,t) \\le M_{ts}(p)), t \\notin T_{ts}, (\\exists p \\in \\bullet t : W(p,t) > M_{ts}(p) - \\sum _{t^{\\prime } \\in T_{ts}, p \\in \\bullet t^{\\prime }}{W(p,t^{\\prime })}), 0 \\le ts \\le k \\rbrace $ per the maximal firing set semantics $A_{18}=\\lbrace holds(p,q,ts+1) : p \\in P, q=M_{ts+1}(p), 0 \\le ts < k\\rbrace $ , where $M_{ts+1}(p) = M_{ts}(p) - \\sum _{\\begin{array}{c}t \\in T_{ts}, p \\in \\bullet t\\end{array}}{W(p,t)} + \\sum _{\\begin{array}{c}t \\in T_{ts}, p \\in t \\bullet \\end{array}}{W(t,p)}$ according to definition REF (firing set execution) We show that $A$ satisfies (REF ) and (REF ), and $A$ is an answer set of $\\Pi ^1$ .", "$A$ satisfies (REF ) and (REF ) by its construction above.", "We show $A$ is an answer set of $\\Pi ^1$ by splitting.", "We split $lit(\\Pi ^1)$ into a sequence of $6k+8$ sets: $U_0= head(f\\ref {f:place}) \\cup head(f\\ref {f:trans}) \\cup head(f\\ref {f:ptarc}) \\cup head(f\\ref {f:tparc}) \\cup head(f\\ref {f:time}) \\cup head(f\\ref {f:num}) $ $\\cup head(i\\ref {i:holds}) = \\lbrace place(p) : p \\in P\\rbrace \\cup \\lbrace trans(t) : t \\in T\\rbrace \\cup \\lbrace ptarc(p,t,n) : (p,t) \\in E^-, n=W(p,t) \\rbrace \\cup \\lbrace tparc(t,p,n) : (t,p) \\in E^+, n=W(t,p) \\rbrace \\cup $ $ \\lbrace time(0), \\dots , time(k)\\rbrace \\cup \\lbrace num(0), \\dots , num(ntok)\\rbrace \\cup \\lbrace holds(p,q,0) : p \\in P, q=M_0(p) \\rbrace $ $U_{6k+1}=U_{6k+0} \\cup head(e\\ref {e:ne:ptarc})^{ts=k} = U_{6k+0} \\cup \\lbrace notenabled(t,k) : t \\in T \\rbrace $ $U_{6k+2}=U_{6k+1} \\cup head(e\\ref {e:enabled})^{ts=k} = U_{6k+1} \\cup \\lbrace enabled(t,k) : t \\in T \\rbrace $ $U_{6k+3}=U_{6k+2} \\cup head(a\\ref {a:fires})^{ts=k} = U_{6k+2} \\cup \\lbrace fires(t,k) : t \\in T \\rbrace $ $U_{6k+4}=U_{6k+3} \\cup head(r\\ref {r:add})^{ts=k} \\cup head(r\\ref {r:del})^{ts=k} = U_{6k+3} \\cup \\lbrace add(p,q,t,k) : p \\in P, t \\in T, q=W(t,p) \\rbrace \\cup \\lbrace del(p,q,t,k) : p \\in P, t \\in T, q=W(p,t) \\rbrace $ $U_{6k+5}=U_{6k+4} \\cup head(r\\ref {r:totincr})^{ts=k} \\cup head(r\\ref {r:totdecr})^{ts=k} = U_{6k+4} \\cup \\lbrace tot\\_incr(p,q,k) : p \\in P, 0 \\le q \\le ntok \\rbrace \\cup \\lbrace tot\\_decr(p,q,k) : p \\in P, 0 \\le q \\le ntok \\rbrace $ $U_{6k+6}=U_{6k+5} \\cup head(r\\ref {r:nextstate})^{ts=k} \\cup head(a\\ref {a:overc:place})^{ts=k} \\cup head(a\\ref {a:maxfire:cnh})^{ts=k} $ $= U_{6k+5} \\\\ \\cup \\lbrace consumesmore(p,k) : p \\in P\\rbrace \\cup \\lbrace holds(p,q,k+1) : p \\in P, 0 \\le q \\le ntok \\rbrace \\cup $ $\\lbrace could\\_not\\_have(t,k) : t \\in T \\rbrace $ $U_{6k+7}=U_{6k+6} \\cup head(a\\ref {a:overc:gen}) = U_{6k+6} \\cup \\lbrace consumesmore \\rbrace $ where $head(r_i)^{ts=k}$ are head atoms of ground rule $r_i$ in which $ts=k$ .", "We write $A_i^{ts=k} = \\lbrace a(\\dots ,ts) : a(\\dots ,ts) \\in A_i, ts=k \\rbrace $ as short hand for all atoms in $A_i$ with $ts=k$ .", "$U_{\\alpha }, 0 \\le \\alpha \\le 6k+7$ form a splitting sequence, since each $U_i$ is a splitting set of $\\Pi ^1$ , and $\\langle U_{\\alpha }\\rangle _{\\alpha < \\mu }$ is a monotone continuous sequence, where $U_0 \\subseteq U_1 \\dots \\subseteq U_{6k+7}$ and $\\bigcup _{\\alpha < \\mu }{U_{\\alpha }} = lit(\\Pi ^1)$ .", "We compute the answer set of $\\Pi ^1$ using the splitting sets as follows: $bot_{U_0}(\\Pi ^1) = f\\ref {f:place} \\cup f\\ref {f:trans} \\cup f\\ref {f:ptarc} \\cup f\\ref {f:tparc} \\cup f\\ref {f:time} \\cup i\\ref {i:holds} \\cup f\\ref {f:num}$ and $X_0 = A_1 \\cup \\dots \\cup A_7$ ($= U_0$ ) is its answer set – using forced atom proposition $eval_{U_0}(bot_{U_1}(\\Pi ^1) \\setminus bot_{U_0}(\\Pi ^1), X_0) = \\lbrace notenabled(t,0) \\text{:-} .", "\| \\lbrace trans(t), \\\\ ptarc(p,t,n), holds(p,q,0) \\rbrace \\subseteq X_0, \\text{~where~} q < n \\rbrace $ .", "Its answer set $X_1=A_8^{ts=0}$ – using forced atom proposition and construction of $A_8$ .", "where, $q=M_0(p)$ , and $n=W(p,t)$ for an arc $(p,t) \\in E^-$ – by construction of $i\\ref {i:holds}$ and $f\\ref {f:ptarc}$ in $\\Pi ^1$ , and in an arc $(p,t) \\in E^-$ , $p \\in \\bullet t$ (by definition REF of preset) thus, $notenabled(t,0) \\in X_1$ represents $\\exists p \\in \\bullet t : M_0(p) < W(p,t)$ .", "$eval_{U_1}(bot_{U_2}(\\Pi ^1) \\setminus bot_{U_1}(\\Pi ^1), X_0 \\cup X_1) = \\lbrace enabled(t,0) \\text{:-}.", "\| trans(t) \\in X_0 \\cup X_1, $ $notenabled(t,0) \\notin X_0 \\cup X_1 \\rbrace $ .", "Its answer set is $X_2 = A_9^{ts=0}$ – using forced atom proposition and construction of $A_9$ .", "since an $enabled(t,0) \\in X_2$ if $\\nexists ~notenabled(t,0) \\in X_0 \\cup X_1$ , which is equivalent to $\\nexists p \\in \\bullet t : M_0(p) < W(p,t) \\equiv \\forall p \\in \\bullet t: M_0(p) \\ge W(p,t)$ .", "$eval_{U_2}(bot_{U_3}(\\Pi ^1) \\setminus bot_{U_2}(\\Pi ^1), X_0 \\cup X_1 \\cup X_2) = \\lbrace \\lbrace fires(t,0)\\rbrace \\text{:-}.", "\| enabled(t,0) \\\\ \\text{~holds in~} X_0 \\cup X_1 \\cup X_2 \\rbrace $ .", "It has multiple answer sets $X_{3.1}, \\dots , X_{3.n}$ , corresponding to elements of power set of $fires(t,0)$ atoms in $eval_{U_2}(...)$ – using supported rule proposition.", "Since we are showing that the union of answer sets of $\\Pi ^1$ determined using splitting is equal to $A$ , we only consider the set that matches the $fires(t,0)$ elements in $A$ and call it $X_3$ , ignoring the rest.", "Thus, $X_3 = A_{10}^{ts=0}$ , representing $T_0$ .", "$eval_{U_3}(bot_{U_4}(\\Pi ^1) \\setminus bot_{U_3}(\\Pi ^1), X_0 \\cup \\dots \\cup X_3) = \\lbrace add(p,n,t,0) \\text{:-}.", "\| \\lbrace fires(t,0), \\\\ tparc(t,p,n) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_3 \\rbrace \\cup \\lbrace del(p,n,t,0) \\text{:-}.", "\| \\lbrace fires(t,0), ptarc(p,t,n) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_3 \\rbrace $ .", "It's answer set is $X_4 = A_{11}^{ts=0} \\cup A_{12}^{ts=0}$ – using forced atom proposition and definitions of $A_{11}$ and $A_{12}$ .", "where, each $add$ atom encodes $n=W(t,p) : p \\in t \\bullet $ , and each $del$ atom encodes $n=W(p,t) : p \\in \\bullet t$ representing the effect of transitions in $T_0$ – by construction $eval_{U_4}(bot_{U_5}(\\Pi ^1) \\setminus bot_{U_4}(\\Pi ^1), X_0 \\cup \\dots \\cup X_4) = \\lbrace tot\\_incr(p,qq,0) \\text{:-}.", "\| $ $qq=\\sum _{add(p,q,t,0) \\in X_0 \\cup \\dots \\cup X_4}{q} \\rbrace \\cup \\lbrace tot\\_decr(p,qq,0) \\text{:-}.", "\| qq=\\sum _{del(p,q,t,0) \\in X_0 \\cup \\dots \\cup X_4}{q} \\rbrace $ .", "It's answer set is $X_5 = A_{13}^{ts=0} \\cup A_{14}^{ts=0}$ – using forced atom proposition and definitions of $A_{13}$ , $A_{14}$ , and semantics of aggregate assignment atom where, each $tot\\_incr(p,qq,0)$ , $qq=\\sum _{add(p,q,t,0) \\in X_0 \\cup \\dots X_4}{q}$ $\\equiv qq=\\sum _{t \\in X_3, p \\in t \\bullet }{W(p,t)}$ , and, each $tot\\_decr(p,qq,0)$ , $qq=\\sum _{del(p,q,t,0) \\in X_0 \\cup \\dots X_4}{q}$ $\\equiv qq=\\sum _{t \\in X_3, p \\in \\bullet t}{W(t,p)}$ , represent the net effect of actions in $T_0$ – by construction $eval_{U_5}(bot_{U_6}(\\Pi ^1) \\setminus bot_{U_5}(\\Pi ^1), X_0 \\cup \\dots \\cup X_5) = \\lbrace consumesmore(p,0) \\text{:-}.", "\| \\\\ \\lbrace holds(p,q,0), tot\\_decr(p,q1,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_5 : q1 > q \\rbrace \\cup \\lbrace holds(p,q,1) \\text{:-}., \| \\\\ \\lbrace holds(p,q1,0), tot\\_incr(p,q2,0), tot\\_decr(p,q3,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_5, q=q1+q2-q3 \\rbrace \\cup \\lbrace could\\_not\\_have(t,0) \\text{:-}.", "\| \\lbrace enabled(t,0), ptarc(s,t,q), holds(s,qq,0), \\\\ tot\\_decr(s,qqq,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_5, fires(t,0) \\notin (X_0 \\cup \\dots \\cup X_5), q > qq-qqq \\rbrace $ .", "It's answer set is $X_6 = A_{15}^{ts=0} \\cup A_{17}^{ts=0} \\cup A_{18}^{ts=0}$ – using forced atom proposition and definitions of $A_{15}, A_{17}, A_{18}$ .", "where, $consumesmore(p,0)$ represents $\\exists p : q=M_0(p), q1=\\sum _{t \\in T_0, p \\in \\bullet t}{W(p,t)}, $ $q1 > q$ indicating place $p$ will be overconsumed if $T_0$ is fired, as defined in definition REF (conflicting transitions) and, $holds(p,q,1)$ encodes $q=M_1(p)$ – by construction and $could\\_not\\_have(t,0)$ represents an enabled transition $t$ in $T_0$ that could not fire due to insufficient tokens $X_6$ does not contain $could\\_not\\_have(t,0)$ , when $enabled(t,0) \\in X_0 \\cup \\dots \\cup X_5$ and $fires(t,0) \\notin X_0 \\cup \\dots \\cup X_5$ due to construction of $A$ , encoding of $a\\ref {a:maxfire:cnh}$ and its body atoms.", "As a result it is not eliminated by the constraint $a\\ref {a:maxfire:elim}$ $ \\vdots $ $eval_{U_{6k+0}}(bot_{U_{6k+1}}(\\Pi ^1) \\setminus bot_{U_{6k+0}}(\\Pi ^1), X_0 \\cup \\dots \\cup X_{6k+0}) = \\\\ \\lbrace notenabled(t,k) \\text{:-} .", "\| \\lbrace trans(t), ptarc(p,t,n), holds(p,q,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{6k+0}, \\\\ \\text{~where~} q < n \\rbrace $ .", "Its answer set $X_{6k+1}=A_8^{ts=k}$ – using forced atom proposition and construction of $A_8$ .", "where, $q=M_k(p)$ , and $n=W(p,t)$ for an arc $(p,t) \\in E^-$ – by construction of $holds$ and $ptarc$ predicates in $\\Pi ^1$ , and in an arc $(p,t) \\in E^-$ , $p \\in \\bullet t$ (by definition REF of preset) thus, $notenabled(t,k) \\in X_{6k+1}$ represents $\\exists p \\in \\bullet t : M_k(p) < W(p,t)$ .", "$eval_{U_{6k+1}}(bot_{U_{6k+2}}(\\Pi ^1) \\setminus bot_{U_{6k+1}}(\\Pi ^1), X_0 \\cup \\dots \\cup X_{6k+1}) = \\lbrace enabled(t,k) \\text{:-}.", "\| \\\\ trans(t) \\in X_0 \\cup \\dots \\cup X_{6k+1}, notenabled(t,k) \\notin X_0 \\cup \\dots \\cup X_{6k+1} \\rbrace $ .", "Its answer set is $X_{6k+2} = A_9^{ts=k}$ – using forced atom proposition and construction of $A_9$ .", "since an $enabled(t,k) \\in X_{6k+2}$ if $\\nexists ~notenabled(t,k) \\in X_0 \\cup \\dots \\cup X_{6k+1}$ , which is equivalent to $\\nexists p \\in \\bullet t : M_k(p) < W(p,t) \\equiv \\forall p \\in \\bullet t: M_k(p) \\ge W(p,t)$ .", "$eval_{U_{6k+2}}(bot_{U_{6k+3}}(\\Pi ^1) \\setminus bot_{U_{6k+2}}(\\Pi ^1), X_0 \\cup \\dots \\cup X_{6k+2}) = \\\\ \\lbrace \\lbrace fires(t,k)\\rbrace \\text{:-}.", "\| enabled(t,k) \\text{~holds in~} X_0 \\cup \\dots \\cup X_{6k+2} \\rbrace $ .", "It has multiple answer sets $X_{{6k+3}.1}, \\dots , X_{{6k+3}.n}$ , corresponding to elements of power set of $fires(t,k)$ atoms in $eval_{U_{6k+2}}(...)$ – using supported rule proposition.", "Since we are showing that the union of answer sets of $\\Pi ^1$ determined using splitting is equal to $A$ , we only consider the set that matches the $fires(t,k)$ elements in $A$ and call it $X_{6k+3}$ , ignoring the reset.", "Thus, $X_{6k+3} = A_{10}^{ts=k}$ , representing $T_k$ .", "$eval_{U_{6k+3}}(bot_{U_{6k+4}}(\\Pi ^1) \\setminus bot_{U_{6k+3}}(\\Pi ^1), X_0 \\cup \\dots \\cup X_{6k+3}) = \\\\ \\lbrace add(p,n,t,k) \\text{:-}.", "\| \\lbrace fires(t,k), tparc(t,p,n) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{6k+3} \\rbrace \\cup \\\\ \\lbrace del(p,n,t,k) \\text{:-}.", "\| \\lbrace fires(t,k), ptarc(p,t,n) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{6k+3} \\rbrace $ .", "It's answer set is $X_{6k+4} = A_{11}^{ts=k} \\cup A_{12}^{ts=k}$ – using forced atom proposition and definitions of $A_{11}$ and $A_{12}$ .", "where, each $add$ atom is equivalent to $n=W(t,p) : p \\in t \\bullet $ , and, each $del$ atom is equivalent to $n=W(p,t) : p \\in \\bullet t$ , representing the effect of transitions in $T_k$ $eval_{U_{6k+4}}(bot_{U_{6k+5}}(\\Pi ^1) \\setminus bot_{U_{6k+4}}(\\Pi ^1), X_0 \\cup \\dots \\cup X_{6k+4}) = \\lbrace tot\\_incr(p,qq,k) \\text{:-}.", "\| \\\\ qq=\\sum _{add(p,q,t,k) \\in X_0 \\cup \\dots \\cup X_{6k+4}}{q} \\rbrace \\cup \\lbrace tot\\_decr(p,qq,k) \\text{:-}.", "\| \\\\ qq=\\sum _{del(p,q,t,k) \\in X_0 \\cup \\dots \\cup X_{6k+4}}{q} \\rbrace $ .", "It's answer set is $X_{6k+5} = A_{13}^{ts=k} \\cup A_{14}^{ts=k}$ – using forced atom proposition and definitions of $A_{13}$ and $A_{14}$ .", "where, each $tot\\_incr(p,qq,k)$ , $qq=\\sum _{add(p,q,t,k) \\in X_0 \\cup \\dots X_{6k+4}}{q}$ $\\equiv \\\\ qq=\\sum _{t \\in X_{6k+3}, p \\in t \\bullet }{W(p,t)}$ , and, each $tot\\_decr(p,qq,k)$ , $qq=\\sum _{del(p,q,t,k) \\in X_0 \\cup \\dots X_{6k+4}}{q}$ $\\equiv \\\\ qq=\\sum _{t \\in X_{6k+3}, p \\in \\bullet t}{W(t,p)}$ , represent the net effect of transitions in $T_k$ $eval_{U_{6k+5}}(bot_{U_{6k+6}}(\\Pi ^1) \\setminus bot_{U_{6k+5}}(\\Pi ^1), X_0 \\cup \\dots \\cup X_{6k+5}) = \\\\ \\lbrace consumesmore(p,k) \\text{:-}.", "\| \\lbrace holds(p,q,k), tot\\_decr(p,q1,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{6k+5} : q1 > q \\rbrace \\cup \\lbrace holds(p,q,k+1) \\text{:-}., \| \\lbrace holds(p,q1,k), tot\\_incr(p,q2,k), \\\\ tot\\_decr(p,q3,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{6k+5} : q=q1+q2-q3 \\rbrace \\cup \\lbrace could\\_not\\_have(t,k) \\text{:-} \\\\ \\lbrace enabled(t,k), ptarc(s,t,q), holds(s,qq,k), tot\\_decr(s,qqq,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{6k+5}, $ $fires(t,k) \\notin (X_0 \\cup \\dots \\cup X_{6k+5}), q > qq-qqq \\rbrace $ .", "It's answer set is $X_{6k+6} = A_{15}^{ts=k} \\cup A_{17}^{ts=k} \\cup A_{18}^{ts=k}$ – using forced atom proposition.", "where, $consumesmore(p,k)$ represents $\\exists p : q=M_{k}(p), \\\\ q1=\\sum _{t \\in T_{k}, p \\in \\bullet t}{W(p,t)}, q1 > q$ $holds(p,q,k+1)$ represents $q=M_{k+1}(p)$ indicating place $p$ that will be over consumed if $T_k$ is fired, as defined in definition REF (conflicting transitions), $holds(p,q,k+1)$ represents $q=M_{k+1}(p)$ – by construction and $could\\_not\\_have(t,k)$ represents an enabled transition $t$ in $T_k$ that could not fire due to insufficient tokens $X_{6k+6}$ does not contain $could\\_not\\_have(t,k)$ , when $enabled(t,k) \\in X_0 \\cup \\dots \\cup X_{6k+5}$ and $fires(t,k) \\notin X_0 \\cup \\dots \\cup X_{6k+5}$ due to construction of $A$ , encoding of $a\\ref {a:maxfire:cnh}$ and its body atoms.", "As a result it is note eliminated by the constraint $a\\ref {a:maxfire:elim}$ $eval_{U_{6k+6}}(bot_{U_{6k+7}}(\\Pi ^1) \\setminus bot_{U_{6k+6}}(\\Pi ^1), X_0 \\cup \\dots \\cup X_{6k+6}) = \\lbrace consumesmore \\text{:-}.", "\| $ $\\lbrace consumesmore(p,0),$ $\\dots , $ $consumesmore(p,k) \\rbrace \\cap $ $(X_0 \\cup \\dots \\cup X_{6k+6}) \\ne \\emptyset \\rbrace $ .", "It's answer set is $X_{6k+7} = A_{16}$ – using forced atom proposition $X_{6k+7}$ will be empty since none of $consumesmore(p,0),\\dots , \\\\ consumesmore(p,k)$ hold in $X_0 \\cup \\dots \\cup X_{6k+6}$ due to the construction of $A$ , encoding of $a\\ref {a:overc:place}$ and its body atoms.", "As a result, it is not eliminated by the constraint $a\\ref {a:overc:elim}$ The set $X = X_0 \\cup \\dots \\cup X_{6k+7}$ is the answer set of $\\Pi ^0$ by the splitting sequence theorem REF .", "Each $X_i, 0 \\le i \\le 6k+7$ matches a distinct portion of $A$ , and $X = A$ , thus $A$ is an answer set of $\\Pi ^1$ .", "Next we show (REF ): Given $\\Pi ^1$ be the encoding of a Petri Net $PN(P,T,E,W)$ with initial marking $M_0$ , and $A$ be an answer set of $\\Pi ^1$ that satisfies (REF ) and (REF ), then we can construct $X=M_0,T_0,\\dots ,M_k,T_k,M_{k+1}$ from $A$ , such that it is an execution sequence of $PN$ .", "We construct the $X$ as follows: $M_i = (M_i(p_0), \\dots , M_i(p_n))$ , where $\\lbrace holds(p_0,M_i(p_0),i), \\dots holds(p_n,M_i(p_n),i) \\rbrace \\\\ \\subseteq A$ , for $0 \\le i \\le k+1$ $T_i = \\lbrace t : fires(t,i) \\in A\\rbrace $ , for $0 \\le i \\le k$ and show that $X$ is indeed an execution sequence of $PN$ .", "We show this by induction over $k$ (i.e.", "given $M_k$ , $T_k$ is a valid firing set and its firing produces marking $M_{k+1}$ ).", "Base case: Let $k=0$ , and $M_0$ is a valid marking in $X$ for $PN$ , show [(1)] $T_0$ is a valid firing set for $M_0$ , and $T_0$ 's firing w.r.t.", "marking $M_0$ produces $M_1$ .", "We show $T_0$ is a valid firing set for $M_0$ .", "Let $\\lbrace fires(t_0,0), \\dots , fires(t_x,0) \\rbrace $ be the set of all $fires(\\dots ,0)$ atoms in $A$ , Then for each $fires(t_i,0) \\in A$ $enabled(t_i,0) \\in A$ – from rule $a\\ref {a:fires}$ and supported rule proposition Then $notenabled(t_i,0) \\notin A$ – from rule $e\\ref {e:enabled}$ and supported rule proposition Then $body(e\\ref {e:ne:ptarc})$ must not hold in $A$ – from rule $e\\ref {e:ne:ptarc}$ and forced atom proposition Then $q \\lnot < n_i \\equiv q \\ge n_i$ in $e\\ref {e:ne:ptarc}$ for all $\\lbrace holds(p,q,0), ptarc(p,t_i,n_i)\\rbrace \\subseteq A$ – from $e\\ref {e:ne:ptarc}$ , forced atom proposition, and the following $holds(p,q,0) \\in A$ represents $q=M_0(p)$ – rule $i\\ref {i:holds}$ construction $ptarc(p,t_i,n_i) \\in A$ represents $n_i=W(p,t_i)$ – rule $f\\ref {f:ptarc} $ construction Then $\\forall p \\in \\bullet t_i, M_0(p) > W(p,t_i)$ – from definition REF of preset $\\bullet t_i$ in PN Then $t_i$ is enabled and can fire in $PN$ , as a result it can belong to $T_0$ – from definition REF of enabled transition And $consumesmore \\notin A$ , since $A$ is an answer set of $\\Pi ^1$ – from rule $a\\ref {a:overc:elim}$ and supported rule proposition Then $\\nexists consumesmore(p,0) \\in A$ – from rule $a\\ref {a:overc:gen}$ and supported rule proposition Then $\\nexists \\lbrace holds(p,q,0), tot\\_decr(p,q1,0) \\rbrace \\subseteq A : q1>q$ in $body(a\\ref {a:overc:place})$ – from $a\\ref {a:overc:place}$ and forced atom proposition Then $\\nexists p : \\sum _{t_i \\in \\lbrace t_0,\\dots ,t_x\\rbrace , p \\in \\bullet t_i}{W(p,t_i)} > M_0(p)$ – from the following $holds(p,q,0)$ represents $q=M_0(p)$ – from rule $i\\ref {i:holds}$ construction, given $tot\\_decr(p,q1,0) \\in A$ if $\\lbrace del(p,q1_0,t_0,0), \\dots , del(p,q1_x,t_x,0) \\rbrace \\subseteq A$ , where $q1 = q1_0+\\dots +q1_x$ – from $r\\ref {r:totdecr}$ and forced atom proposition $del(p,q1_i,t_i,0) \\in A$ if $\\lbrace fires(t_i,0), ptarc(p,t_i,q1_i) \\rbrace \\subseteq A$ – from $r\\ref {r:del}$ and supported rule proposition $del(p,q1_i,t_i,0)$ represents removal of $q1_i = W(p,t_i)$ tokens from $p \\in \\bullet t_i$ – from rule $r\\ref {r:del}$ , supported rule proposition, and definition REF of transition execution in $PN$ Then the set of transitions in $T_0$ do not conflict – by the definition REF of conflicting transitions And for each $enabled(t_j,0) \\in A$ and $fires(t_j,0) \\notin A$ , $could\\_not\\_have(t_j,0) \\in A$ , since $A$ is an answer set of $\\Pi ^1$ - from rule $a\\ref {a:maxfire:elim}$ and supported rule proposition Then $\\lbrace enabled(t_j,0), holds(s,qq,0), ptarc(s,t_j,q,0), \\\\ tot\\_decr(s,qqq,0) \\rbrace \\subseteq A$ , such that $q > qq - qqq$ and $fires(t_j,0) \\notin A$ - from rule $a\\ref {a:maxfire:cnh}$ and supported rule proposition Then for an $s \\in \\bullet t_j$ , $W(s,t_j) > M_0(s) - \\sum _{t_i \\in T_0, s \\in \\bullet t_i}{W(s,t_i)}$ - from the following: $ptarc(s,t_i,q)$ represents $q=W(s,t_i)$ – from rule $f\\ref {f:r:ptarc}$ construction $holds(s,qq,0)$ represents $qq=M_0(s)$ – from $i\\ref {i:holds}$ construction $tot\\_decr(s,qqq,0) \\in A$ if $\\lbrace del(s,qqq_0,t_0,0), \\dots , del(s,qqq_x,t_x,0) \\rbrace \\subseteq A$ – from rule $r\\ref {r:totdecr}$ construction and supported rule proposition $del(s,qqq_i,t_i,0) \\in A$ if $\\lbrace fires(t_i,0), ptarc(s,t_i,qqq_i) \\rbrace \\subseteq A$ – from rule $r\\ref {r:r:del} $ and supported rule proposition $del(s,qqq_i,t_i,0)$ represents $qqq_i = W(s,t_i) : t_i \\in T_0, (s,t_i) \\in E^-$ – from rule $f\\ref {f:r:ptarc}$ construction $tot\\_decr(q,qqq,0)$ represents $\\sum _{t_i \\in T_0, s \\in \\bullet t_i}{W(s,t_i)}$ – from (C,D,E) above Then firing $T_0 \\cup \\lbrace t_j \\rbrace $ would have required more tokens than are present at its source place $s \\in \\bullet t_j$ .", "Thus, $T_0$ is a maximal set of transitions that can simultaneously fire.", "Then $\\lbrace t_0, \\dots , t_x\\rbrace = T_0$ – using 1(a),1(b) above; and using 1(c) it is a maximal firing set We show $M_1$ is produced by firing $T_0$ in $M_0$ .", "Let $holds(p,q,1) \\in A$ Then $\\lbrace holds(p,q1,0), tot\\_incr(p,q2,0), tot\\_decr(p,q3,0) \\rbrace \\subseteq A : q=q1+q2-q3$ – from rule $r\\ref {r:nextstate}$ and supported rule proposition Then, $holds(p,q1,0) \\in A$ represents $q1=M_0(p)$ – given, rule $i\\ref {i:holds}$ construction; and $\\lbrace add(p,q2_0,t_0,0), \\dots , $ $add(p,q2_j,t_j,0)\\rbrace \\subseteq A : q2_0 + \\dots + q2_j = q2$ and $\\lbrace del(p,q3_0,t_0,0), \\dots , $ $del(p,q3_l,t_l,0)\\rbrace \\subseteq A : q3_0 + \\dots + q3_l = q3$ – rules $r\\ref {r:totincr},r\\ref {r:totdecr}$ using supported rule proposition Then $\\lbrace fires(t_0,0), \\dots , fires(t_j,0) \\rbrace \\subseteq A$ and $\\lbrace fires(t_0,0), \\dots , fires(t_l,0) \\rbrace \\\\ \\subseteq A$ – rules $r\\ref {r:add},r\\ref {r:del}$ and supported rule proposition, respectively Then $\\lbrace fires(t_0,0), \\dots , $ $fires(t_j,0) \\rbrace \\cup \\lbrace fires(t_0,0), \\dots , $ $fires(t_l,0) \\rbrace \\subseteq A = \\lbrace fires(t_0,0), \\dots , $ $fires(t_x,0) \\rbrace \\subseteq A$ – set union of subsets Then for each $fires(t_x,0) \\in A$ we have $t_x \\in T_0$ – already shown in item REF above Then $q = M_0(p) + \\sum _{t_x \\in T_0 \\wedge p \\in t_x \\bullet }{W(t_x,p)} - \\sum _{t_x \\in T_0 \\wedge p \\in \\bullet t_x}{W(p,t_x)}$ – from (REF ) above and the following Each $add(p,q_j,t_j,0) \\in A$ represents $q_j=W(t_j,p)$ for $p \\in t_j \\bullet $ – rule $r\\ref {r:add}$ encoding, and definition REF of transition execution in $PN$ Each $del(p,t_y,q_y,0) \\in A$ represents $q_y=W(p,t_y)$ for $p \\in \\bullet t_y$ – from rule $r\\ref {r:del}$ encoding, and definition REF of transition execution in $PN$ Each $tot\\_incr(p,q2,0) \\in A$ represents $q2=\\sum _{t_x \\in T_0 \\wedge p \\in t_x \\bullet }{W(t_x,p)}$ – aggregate assignment atom semantics in rule $r\\ref {r:totincr}$ Each $tot\\_decr(p,q3,0) \\in A$ represents $q3=\\sum _{t_x \\in T_0 \\wedge p \\in \\bullet t_x}{W(p,t_x)}$ – aggregate assignment atom semantics in rule $r\\ref {r:totdecr}$ Then, $M_1(p) = q$ – since $holds(p,q,1) \\in A$ encodes $q=M_1(p)$ from construction Inductive Step: Let $k > 0$ , and $M_k$ is a valid marking in $X$ for $PN$ , show [(1)] $T_k$ is a valid firing set for $M_k$ , and $T_k$ 's firing in $M_k$ produces marking $M_{k+1}$ .", "We show $T_k$ is a valid firing set.", "Let $\\lbrace fires(t_0,k),\\dots ,fires(t_x,k) \\rbrace $ be the set of all $fires(\\dots ,k)$ atoms in $A$ , Then for each $fires(t_i,k) \\in A$ $enabled(t_i,k) \\in A$ – from rule $a\\ref {a:fires}$ and supported rule proposition Then $notenabled(t_i,k) \\notin A$ – from rule $e\\ref {e:enabled}$ and supported rule proposition Then $body(e\\ref {e:ne:ptarc})$ must hold in $A$ – from rule $e\\ref {e:ne:ptarc}$ and forced atom proposition Then $q \\lnot < n_i \\equiv q \\ge n_i$ in $e\\ref {e:ne:ptarc}$ for all $\\lbrace holds(p,q,k), ptarc(p,t_i,n) \\rbrace \\subseteq A$ – from $e\\ref {e:ne:ptarc}$ , forced atom proposition, and the following $holds(p,q,k) \\in A$ represents $q=M_k(p)$ – construction, inductive assumption $ptarc(p,t,n_i) \\in A$ represents $n_i=W(p,t_i)$ – rule $f\\ref {f:ptarc}$ construction Then $\\forall p \\in \\bullet t_i$ , $M_k(p) \\ge W(p,t_i)$ – from definition REF of preset $\\bullet t_i$ in $PN$ Then $t_i$ is enabled and can fire in $PN$ , as a result it can belong to $T_k$ – from definition REF of enabled transition And $consumesmore \\notin A$ , since $A$ is an answer set of $\\Pi ^1$ – from rule $a\\ref {a:overc:elim}$ and supported rule proposition Then $\\nexists consumesmore(p,k) \\in A$ – from rule $a\\ref {a:overc:gen}$ and forced atom proposition Then $\\nexists \\lbrace holds(p,q,k), tot\\_decr(p,q1,k) \\rbrace \\subseteq A : q1 > q$ in $body(a\\ref {a:overc:place})$ – from $a\\ref {a:overc:place}$ and forced atom proposition Then $\\nexists p : \\sum _{t_i \\in \\lbrace t_0,\\dots ,t_x \\rbrace , p \\in \\bullet t_i}{W(p,t_i)} > M_k(p)$ – from the following $holds(p,q,k) \\in A$ represents $q=M_k(p)$ – by construction of $\\Pi ^1$ , and the inductive assumption about $M_k(p)$ $tot\\_decr(p,q1,k) \\in A$ if $\\lbrace del(p,q1_0,t_0,k), \\dots , del(p,q1_x,t_x,k) \\rbrace \\subseteq A$ , where $q1=q1_0+\\dots +q1_x$ – from $r\\ref {r:totdecr}$ and forced atom proposition $del(p,q1_i,t_i,k) \\in A$ if $\\lbrace fires(t_i,k), ptarc(p,t_i,q1_i) \\rbrace \\subseteq A$ – from $r\\ref {r:del}$ and supported rule proposition $del(p,q1_i,t_i,k)$ represents removal of $q1_i = W(p,t_i)$ tokens from $p \\in \\bullet t_i$ – from construction rule $r\\ref {r:del}$ , supported rule proposition, and definition REF of transition execution in $PN$ Then the set of transitions $T_k$ do not conf – by the definition REF of conflicting transitions And for each $enabled(t_j,k) \\in A$ and $fires(t_j,k) \\notin A$ , $could\\_not\\_have(t_j,k) \\in A$ , since $A$ is an answer set of $\\Pi ^1$ - from rule $a\\ref {a:maxfire:elim}$ and supported rule proposition Then $\\lbrace enabled(t_j,k), holds(s,qq,k), ptarc(s,t_j,q,k), \\\\ tot\\_decr(s,qqq,k) \\rbrace \\subseteq A$ , such that $q > qq - qqq$ and $fires(t_j,0) \\notin A$ - from rule $a\\ref {a:maxfire:cnh}$ and supported rule proposition Then for an $s \\in \\bullet t_j$ , $W(s,t_j) > M_k(s) - \\sum _{t_i \\in T_k, s \\in \\bullet t_i}{W(s,t_i)}$ - from the following: $ptarc(s,t_i,q)$ represents $q=W(s,t_i)$ – from rule $f\\ref {f:r:ptarc}$ construction $holds(s,qq,k)$ represents $qq=M_k(s)$ – from $i\\ref {i:holds}$ construction $tot\\_decr(s,qqq,k) \\in A$ if $\\lbrace del(s,qqq_0,t_0,k), \\dots , del(s,qqq_x,t_x,k) \\rbrace \\subseteq A$ – from rule $r\\ref {r:totdecr}$ construction and supported rule proposition $del(s,qqq_i,t_i,k) \\in A$ if $\\lbrace fires(t_i,k), ptarc(s,t_i,qqq_i) \\rbrace \\subseteq A$ – from rule $r\\ref {r:r:del} $ and supported rule proposition $del(s,qqq_i,t_i,k)$ represents $qqq_i = W(s,t_i) : t_i \\in T_k, (s,t_i) \\in E^-$ – from rule $f\\ref {f:r:ptarc}$ construction $tot\\_decr(q,qqq,k)$ represents $\\sum _{t_i \\in T_k, s \\in \\bullet t_i}{W(s,t_i)}$ – from (C,D,E) above Then firing $T_k \\cup \\lbrace t_j \\rbrace $ would have required more tokens than are present at its source place $s \\in \\bullet t_j$ .", "Thus, $T_k$ is a maximal set of transitions that can simultaneously fire.", "Then $\\lbrace t_0,\\dots ,t_x \\rbrace = T_k$ – using 1(a),1(b) above; and using 1(c) it is a maximal firing set We show $M_{k+1}$ is produced by firing $T_k$ in $M_k$ .", "Let $holds(p,q,k+1) \\in A$ Then $\\lbrace holds(p,q1,k), tot\\_incr(p,q2,k), tot\\_decr(p,q3,k) \\rbrace \\in A : q=q2+q2-q3$ – from rule $r\\ref {r:nextstate}$ and supported rule proposition $holds(p,q1,k) \\in A$ represents $q1=M_k(p)$ – construction, inductive assumption; and $\\lbrace add(p,q2_0,t_0,k), \\dots , $ $add(p,q2_j,t_j,k) \\rbrace \\subseteq A : q2_0+\\dots +q2_j=q2$ and $\\lbrace del(p,q3_0,t_0,k), \\dots , $ $del(p,q3_l,t_l,k) \\rbrace \\subseteq A : q3_0+\\dots +q3_l=q3$ – from rules $r\\ref {r:totincr},r\\ref {r:totdecr}$ using supported rule proposition, respectively Then $\\lbrace fires(t_0,k), \\dots , fires(t_j,k) \\rbrace \\subseteq A$ and $\\lbrace fires(t_0,k), \\dots , fires(t_l,k) \\rbrace \\\\ \\subseteq A$ – from rules $r\\ref {r:add} ,r\\ref {r:del} $ using supported rule proposition, respectively Then $\\lbrace fires(t_0,k), \\dots , $ $fires(t_j,k) \\rbrace \\cup \\lbrace fires(t_0,k), \\dots , $ $fires(t_l,k) \\rbrace = \\lbrace fires(t_0,k), \\dots , $ $fires(t_x,k) \\rbrace \\subseteq A$ – subset union property Then for each $fires(t_x,k) \\in A$ we have $t_x \\in T_k$ - already shown in item (REF ) above Then $q = M_k(p) + \\sum _{t_x \\in T_k \\wedge p \\in t_x \\bullet }{W(t_x,p)} - \\sum _{t_x \\in T_k \\wedge p \\in \\bullet t_x}{W(p,t_x)}$ – from (REF ) above and the following Each $add(p,q_j,t_j,k) \\in A$ represents $q_j = W(t_j,p)$ for $p \\in t_j \\bullet $ – rule $r\\ref {r:add}$ encoding and definition REF of transition execution in PN Each $del(p,t_y,q_y,k) \\in A$ represents $q_y = W(p,t_y)$ for $p \\in \\bullet t_y$ – rule $r\\ref {r:del}$ encoding and definition REF of transition execution in PN Each $tot\\_incr(p,q2,k) \\in A$ represents $q2 = \\sum _{t_x \\in T_k \\wedge p \\in t_x \\bullet }{W(t_x,p)}$ – aggregate assignment atom semantics in rule $r\\ref {r:totincr}$ Each $tot\\_decr(p,q3,k) \\in A$ represents $q3 = \\sum _{t_x \\in T_k \\wedge p \\in \\bullet t_x}{W(p,t_x)}$ – aggregate assignment atom semantics in rule $r\\ref {r:totdecr}$ Then $M_{k+1}(p) = q$ – since $holds(p,q,k+1) \\in A$ encodes $q=M_{k+1}(p)$ – from construction As a result, for any $n > k$ , $T_n$ will be a valid firing set w.r.t.", "$M_n$ and its firing produces marking $M_{n+1}$ .", "Conclusion: Since both REF and REF hold, $X=M_0,T_0,M_1,\\dots ,M_k,T_k,M_{k+1}$ is an execution sequence of $PN(P,T,E,W)$ (w.r.t.", "$M_0$ ) iff there is an answer set $A$ of $\\Pi ^1(PN,M_0,k,ntok)$ such that (REF ) and (REF ) hold.", "Proof of Proposition REF Let $PN=(P,T,E,W,R)$ be a Petri Net, $M_0$ be its initial marking and let $\\Pi ^2(PN,M_0,k,ntok)$ by the ASP encoding of $PN$ and $M_0$ over a simulation length $k$ , with maximum $ntok$ tokens on any place node, as defined in section REF .", "Then $X=M_0,T_0,M_1,\\dots ,M_k,T_k,M_{k+1}$ is an execution sequence of $PN$ (w.r.t $M_0$ ) iff there is an answer set $A$ of $\\Pi ^2(PN,M_0,k,ntok)$ such that: $\\lbrace fires(t,ts) : t \\in T_{ts}, 0 \\le ts \\le k\\rbrace = \\lbrace fires(t,ts) : fires(t,ts) \\in A \\rbrace $ $\\begin{split}\\lbrace holds(p,q,ts) &: p \\in P, q = M_{ts}(p), 0 \\le ts \\le k+1 \\rbrace \\\\&= \\lbrace holds(p,q,ts) : holds(p,q,ts) \\in A \\rbrace \\end{split}$ We prove this by showing that: Given an execution sequence $X$ , we create a set $A$ such that it satisfies (REF ) and (REF ) and show that $A$ is an answer set of $\\Pi ^2$ Given an answer set $A$ of $\\Pi ^2$ , we create an execution sequence $X$ such that (REF ) and (REF ) are satisfied.", "First we show (REF ): Given $PN$ and an execution sequence $X$ of $PN$ , we create a set $A$ as a union of the following sets: $A_1=\\lbrace num(n) : 0 \\le n \\le ntok \\rbrace $ $A_2=\\lbrace time(ts) : 0 \\le ts \\le k\\rbrace $ $A_3=\\lbrace place(p) : p \\in P \\rbrace $ $A_4=\\lbrace trans(t) : t \\in T \\rbrace $ $A_5=\\lbrace ptarc(p,t,n,ts) : (p,t) \\in E^-, n=W(p,t), 0 \\le ts \\le k\\rbrace $ , where $E^- \\subseteq E$ $A_6=\\lbrace tparc(t,p,n,ts) : (t,p) \\in E^+, n=W(t,p), 0 \\le ts \\le k\\rbrace $ , where $E^+ \\subseteq E$ $A_7=\\lbrace holds(p,q,0) : p \\in P, q=M_{0}(p) \\rbrace $ $A_8=\\lbrace notenabled(t,ts) : t \\in T, 0 \\le ts \\le k, \\exists p \\in \\bullet t, M_{ts}(p) < W(p,t) \\rbrace $ per definition REF (enabled transition) $A_9=\\lbrace enabled(t,ts) : t \\in T, 0 \\le ts \\le k, \\forall p \\in \\bullet t, W(p,t) \\le M_{ts}(p) \\rbrace $ per definition REF (enabled transition) $A_{10}=\\lbrace fires(t,ts) : t \\in T_{ts}, 0 \\le ts \\le k \\rbrace $ per definition REF (firing set), only an enabled transition may fire $A_{11}=\\lbrace add(p,q,t,ts) : t \\in T_{ts}, p \\in t \\bullet , q=W(t,p), 0 \\le ts \\le k \\rbrace $ per definition REF (transition execution) $A_{12}=\\lbrace del(p,q,t,ts) : t \\in T_{ts}, p \\in \\bullet t, q=W(p,t), 0 \\le ts \\le k \\rbrace \\cup \\lbrace del(p,q,t,ts) : t \\in T_{ts}, p \\in R(t), q=M_{ts}(p), 0 \\le ts \\le k \\rbrace $ per definition REF (transition execution) $A_{13}=\\lbrace tot\\_incr(p,q,ts) : p \\in P, q=\\sum _{t \\in T_{ts}, p \\in t \\bullet }{W(t,p)}, 0 \\le ts \\le k \\rbrace $ per definition REF (execution) $A_{14}=\\lbrace tot\\_decr(p,q,ts): p \\in P, q=\\sum _{t \\in T_{ts}, p \\in \\bullet t}{W(p,t)}+\\sum _{t \\in T_{ts}, p \\in R(t)}{M_{ts}(p)}, \\\\ 0 \\le ts \\le k \\rbrace $ per definition REF (execution) $A_{15}=\\lbrace consumesmore(p,ts) : p \\in P, q=M_{ts}(p), q1=\\sum _{t \\in T_{ts}, p \\in \\bullet t}{W(p,t)}+\\sum _{t \\in T_{ts}, p \\in R(t)}{M_{ts}(p)}, q1 > q, 0 \\le ts \\le k \\rbrace $ per definition REF (conflicting transitions) $A_{16}=\\lbrace consumesmore : \\exists p \\in P : q=M_{ts}(p), q1=\\sum _{t \\in T_{ts}, p \\in \\bullet t}{W(p,t)}+\\sum _{t \\in T_{ts}, p \\in R(t)}{M_{ts}(p)}, q1 > q, 0 \\le ts \\le k \\rbrace $ per definition REF (conflicting transitions) $A_{17}=\\lbrace could\\_not\\_have(t,ts) : t \\in T, (\\forall p \\in \\bullet t, W(p,t) \\le M_{ts}(p)), t \\notin T_{ts}, (\\exists p \\in \\bullet t \\cup R(t) : q > M_{ts}(p) - (\\sum _{t^{\\prime } \\in T_{ts}, p \\in \\bullet t^{\\prime }}{W(p,t^{\\prime })} + \\sum _{t^{\\prime } \\in T_{ts}, p \\in R(t^{\\prime })}{M_{ts}(p)}), q=W(p,t) \\text{~if~} p \\in \\bullet t \\text{~or~} M_{ts}(p) \\text{~otherwise~}), 0 \\le ts \\le k \\rbrace $ per the maximal firing set semantics $A_{18}=\\lbrace holds(p,q,ts+1) : p \\in P, q=M_{ts+1}(p), 0 \\le ts < k\\rbrace $ , where $M_{ts+1}(p) = M_{ts}(p) - (\\sum _{\\begin{array}{c}t \\in T_{ts}, p \\in \\bullet t\\end{array}}{W(p,t)} + \\sum _{\\begin{array}{c}t \\in T_{ts}, p \\in R(t)\\end{array}}{M_{ts}(p)}) + \\sum _{\\begin{array}{c}t \\in T_{ts}, p \\in t \\bullet \\end{array}}{W(t,p)}$ according to definition REF (firing set execution) $A_{19}=\\lbrace ptarc(p,t,n,ts) : p \\in R(t), n=M_{ts}(p), n > 0, 0 \\le ts \\le k\\rbrace $ We show that $A$ satisfies (REF ) and (REF ), and $A$ is an answer set of $\\Pi ^1$ .", "$A$ satisfies (REF ) and (REF ) by its construction above.", "We show $A$ is an answer set of $\\Pi ^1$ by splitting.", "We split $lit(\\Pi ^1)$ into a sequence of $7(k+1)+2$ sets: $U_0= head(f\\ref {f:place}) \\cup head(f\\ref {f:trans}) \\cup head(f\\ref {f:time}) \\cup head(f\\ref {f:num}) \\cup head(i\\ref {i:holds}) = \\lbrace place(p) : p \\in P\\rbrace \\cup \\lbrace trans(t) : t \\in T\\rbrace \\cup \\lbrace time(0), \\dots , time(k)\\rbrace \\cup \\lbrace num(0), \\dots , num(ntok)\\rbrace \\cup \\lbrace holds(p,q,0) : p \\in P, q=M_0(p) \\rbrace $ $U_{7k+1}=U_{7k+0} \\cup head(f\\ref {f:r:ptarc})^{ts=k} \\cup head(f\\ref {f:r:tparc})^{ts=k} \\cup head(f\\ref {f:rptarc})^{ts=k} = U_{7k+0} \\cup \\\\ \\lbrace ptarc(p,t,n,k) : (p,t) \\in E^-, n=W(p,t) \\rbrace \\cup \\lbrace tparc(t,p,n,k) : (t,p) \\in E^+, n=W(t,p) \\rbrace \\cup \\lbrace ptarc(p,t,n,k) : p \\in R(t), n=M_{k}(p), n > 0 \\rbrace $ $U_{7k+2}=U_{7k+1} \\cup head(e\\ref {e:r:ne:ptarc} )^{ts=k} = U_{7k+1} \\cup \\lbrace notenabled(t,k) : t \\in T \\rbrace $ $U_{7k+3}=U_{7k+2} \\cup head(e\\ref {e:enabled})^{ts=k} = U_{7k+2} \\cup \\lbrace enabled(t,k) : t \\in T \\rbrace $ $U_{7k+4}=U_{7k+3} \\cup head(a\\ref {a:fires})^{ts=k} = U_{7k+3} \\cup \\lbrace fires(t,k) : t \\in T \\rbrace $ $U_{7k+5}=U_{7k+4} \\cup head(r\\ref {r:r:add} )^{ts=k} \\cup head(r\\ref {r:r:del} )^{ts=k} = U_{7k+4} \\cup \\lbrace add(p,q,t,k) : p \\in P, t \\in T, q=W(t,p) \\rbrace \\cup \\lbrace del(p,q,t,k) : p \\in P, t \\in T, q=W(p,t) \\rbrace \\cup \\lbrace del(p,q,t,k) : p \\in P, t \\in T, q=M_{k}(p) \\rbrace $ $U_{7k+6}=U_{7k+5} \\cup head(r\\ref {r:totincr})^{ts=k} \\cup head(r\\ref {r:totdecr})^{ts=k} = U_{7k+5} \\cup \\lbrace tot\\_incr(p,q,k) : p \\in P, 0 \\le q \\le ntok \\rbrace \\cup \\lbrace tot\\_decr(p,q,k) : p \\in P, 0 \\le q \\le ntok \\rbrace $ $U_{7k+7}=U_{7k+6} \\cup head(r\\ref {r:nextstate})^{ts=k} \\cup head(a\\ref {a:overc:place})^{ts=k} \\cup head(a\\ref {a:r:maxfire:cnh})^{ts=k} $ $= U_{7k+6} \\cup \\\\ \\lbrace consumesmore(p,k) : p \\in P\\rbrace \\cup \\lbrace holds(p,q,k+1) : p \\in P, 0 \\le q \\le ntok \\rbrace \\cup \\lbrace could\\_not\\_have(t,k) : t \\in T \\rbrace $ $U_{7k+8}=U_{7k+7} \\cup head(a\\ref {a:overc:gen}) = U_{7k+7} \\cup \\lbrace consumesmore \\rbrace $ where $head(r_i)^{ts=k}$ are head atoms of ground rule $r_i$ in which $ts=k$ .", "We write $A_i^{ts=k} = \\lbrace a(\\dots ,ts) : a(\\dots ,ts) \\in A_i, ts=k \\rbrace $ as short hand for all atoms in $A_i$ with $ts=k$ .", "$U_{\\alpha }, 0 \\le \\alpha \\le 7k+8$ form a splitting sequence, since each $U_i$ is a splitting set of $\\Pi ^1$ , and $\\langle U_{\\alpha }\\rangle _{\\alpha < \\mu }$ is a monotone continuous sequence, where $U_0 \\subseteq U_1 \\dots \\subseteq U_{7k+8}$ and $\\bigcup _{\\alpha < \\mu }{U_{\\alpha }} = lit(\\Pi ^1)$ .", "We compute the answer set of $\\Pi ^2$ using the splitting sets as follows: $bot_{U_0}(\\Pi ^2) = f\\ref {f:place} \\cup f\\ref {f:trans} \\cup f\\ref {f:time} \\cup i\\ref {i:holds} \\cup f\\ref {f:num}$ and $X_0 = A_1 \\cup \\dots \\cup A_4 \\cup A_7$ ($= U_0$ ) is its answer set – using forced atom proposition $eval_{U_0}(bot_{U_1}(\\Pi ^2) \\setminus bot_{U_0}(\\Pi ^2), X_0) = \\lbrace ptarc(p,t,q,0) \\text{:-}.", "\| q=W(p,t) \\rbrace \\cup \\\\ \\lbrace tparc(t,p,q,0) \\text{:-}.", "\| $ $q=W(t,p) \\rbrace \\cup \\lbrace ptarc(p,t,q,0) \\text{:-}.", "\| $ $q=M_0(p) \\rbrace $ .", "Its answer set $X_1=A_5^{ts=0} \\cup A_6^{ts=0} \\cup A_{19}^{ts=0}$ – using forced atom proposition and construction of $A_5, A_6, A_{19}$ .", "$eval_{U_1}(bot_{U_2}(\\Pi ^2) \\setminus bot_{U_1}(\\Pi ^2), X_0 \\cup X_1) = \\lbrace notenabled(t,0) \\text{:-} .", "\| \\lbrace trans(t), \\\\ ptarc(p,t,n,0), holds(p,q,0) \\rbrace \\subseteq X_0 \\cup X_1, \\text{~where~} q < n \\rbrace $ .", "Its answer set $X_2=A_8^{ts=0}$ – using forced atom proposition and construction of $A_8$ .", "where, $q=M_0(p)$ , and $n=W(p,t)$ for an arc $(p,t) \\in E^-$ – by construction of $i\\ref {i:holds}$ and $f\\ref {f:r:ptarc}$ in $\\Pi ^2$ , and in an arc $(p,t) \\in E^-$ , $p \\in \\bullet t$ (by definition REF of preset) thus, $notenabled(t,0) \\in X_1$ represents $\\exists p \\in \\bullet t : M_0(p) < W(p,t)$ .", "$eval_{U_2}(bot_{U_3}(\\Pi ^2) \\setminus bot_{U_2}(\\Pi ^2), X_0 \\cup \\dots \\cup X_2) = \\lbrace enabled(t,0) \\text{:-}.", "\| trans(t) \\in X_0 \\cup \\dots \\cup X_2, notenabled(t,0) \\notin X_0 \\cup \\dots \\cup X_2 \\rbrace $ .", "Its answer set is $X_3 = A_9^{ts=0}$ – using forced atom proposition and construction of $A_9$ .", "since an $enabled(t,0) \\in X_3$ if $\\nexists ~notenabled(t,0) \\in X_0 \\cup \\dots \\cup X_2$ ; which is equivalent to $\\nexists p \\in \\bullet t : M_0(p) < W(p,t) \\equiv \\forall p \\in \\bullet t: M_0(p) \\ge W(p,t)$ .", "$eval_{U_3}(bot_{U_4}(\\Pi ^2) \\setminus bot_{U_3}(\\Pi ^2), X_0 \\cup \\dots \\cup X_3) = \\lbrace \\lbrace fires(t,0)\\rbrace \\text{:-}.", "\| enabled(t,0) \\\\ \\text{~holds in~} X_0 \\cup \\dots \\cup X_3 \\rbrace $ .", "It has multiple answer sets $X_{4.1}, \\dots , X_{4.n}$ , corresponding to elements of power set of $fires(t,0)$ atoms in $eval_{U_3}(...)$ – using supported rule proposition.", "Since we are showing that the union of answer sets of $\\Pi ^2$ determined using splitting is equal to $A$ , we only consider the set that matches the $fires(t,0)$ elements in $A$ and call it $X_4$ , ignoring the rest.", "Thus, $X_4 = A_{10}^{ts=0}$ , representing $T_0$ .", "in addition, for every $t$ such that $enabled(t,0) \\in X_0 \\cup \\dots \\cup X_3, R(t) \\ne \\emptyset $ ; $fires(t,0) \\in X_4$ – per definition REF (firing set); requiring that a reset transition is fired when enabled thus, the firing set $T_0$ will not be eliminated by the constraint $f\\ref {f:c:rptarc:elim}$ $eval_{U_4}(bot_{U_5}(\\Pi ^2) \\setminus bot_{U_4}(\\Pi ^2), X_0 \\cup \\dots \\cup X_4) = \\lbrace add(p,n,t,0) \\text{:-}.", "\| \\lbrace fires(t,0), \\\\ tparc(t,p,n,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_4 \\rbrace \\cup \\lbrace del(p,n,t,0) \\text{:-}.", "\| \\lbrace fires(t,0), ptarc(p,t,n,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_4 \\rbrace $ .", "It's answer set is $X_5 = A_{11}^{ts=0} \\cup A_{12}^{ts=0}$ – using forced atom proposition and definitions of $A_{11}$ and $A_{12}$ .", "where, each $add$ atom is equivalent to $n=W(t,p) : p \\in t \\bullet $ , and each $del$ atom is equivalent to $n=W(p,t) : p \\in \\bullet t$ ; or $n=M_k(p) : p \\in R(t)$ , representing the effect of transitions in $T_0$ – by construction $eval_{U_5}(bot_{U_6}(\\Pi ^2) \\setminus bot_{U_5}(\\Pi ^2), X_0 \\cup \\dots \\cup X_5) = \\lbrace tot\\_incr(p,qq,0) \\text{:-}.", "\| $ $qq=\\sum _{add(p,q,t,0) \\in X_0 \\cup \\dots \\cup X_5}{q} \\rbrace \\cup \\lbrace tot\\_decr(p,qq,0) \\text{:-}.", "\| qq=\\sum _{del(p,q,t,0) \\in X_0 \\cup \\dots \\cup X_5}{q} \\rbrace $ .", "It's answer set is $X_6 = A_{13}^{ts=0} \\cup A_{14}^{ts=0}$ – using forced atom proposition, definitions of $A_{13}$ , $A_{14}$ , and definition REF (semantics of aggregate assignment atom).", "where, each for $tot\\_incr(p,qq,0)$ , $qq=\\sum _{add(p,q,t,0) \\in X_0 \\cup \\dots X_5}{q}$ $\\equiv qq=\\sum _{t \\in X_4, p \\in t \\bullet }{W(p,t)}$ , and each $tot\\_decr(p,qq,0)$ , $qq=\\sum _{del(p,q,t,0) \\in X_0 \\cup \\dots X_5}{q}$ $\\equiv qq=\\sum _{t \\in X_4, p \\in \\bullet t}{W(t,p)} + \\sum _{t \\in X_4, p \\in R(t)}{M_{k}(p)}$ , represent the net effect of transitions in $T_0$ – by construction $eval_{U_6}(bot_{U_7}(\\Pi ^2) \\setminus bot_{U_6}(\\Pi ^2), X_0 \\cup \\dots \\cup X_6) = \\lbrace consumesmore(p,0) \\text{:-}.", "\| \\\\ \\lbrace holds(p,q,0), tot\\_decr(p,q1,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_6, q1 > q \\rbrace \\cup \\lbrace holds(p,q,1) \\text{:-}., \| \\\\ \\lbrace holds(p,q1,0), tot\\_incr(p,q2,0), tot\\_decr(p,q3,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_6, q=q1+q2-q3 \\rbrace \\cup \\lbrace could\\_not\\_have(t,0) \\text{:-}.", "\| \\lbrace enabled(t,0), ptarc(s,t,q), holds(s,qq,0), \\\\ tot\\_decr(s,qqq,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_6, fires(t,0) \\notin (X_0 \\cup \\dots \\cup X_6), q > qq-qqq \\rbrace $ .", "It's answer set is $X_7 = A_{15}^{ts=0} \\cup A_{17}^{ts=0} \\cup A_{18}^{ts=0}$ – using forced atom proposition and definitions of $A_{15}, A_{17}, A_{18}$ .", "where, $consumesmore(p,0)$ represents $\\exists p : q=M_0(p), q1= \\\\ \\sum _{t \\in T_0, p \\in \\bullet t}{W(p,t)}+$ $\\sum _{t \\in T_0, p \\in R(t)}{M_k(p)}, $ $q1 > q$ , indicating place $p$ will be overconsumed if $T_0$ is fired – as defined in definition REF (conflicting transitions) and, $holds(p,q,1)$ represents $q=M_1(p)$ – by construction of $\\Pi ^2$ and $could\\_not\\_have(t,0)$ represents an enabled transition $t$ in $T_0$ that could not fire due to insufficient tokens $X_7$ does not contain $could\\_not\\_have(t,0)$ , when $enabled(t,0) \\in X_0 \\cup \\dots \\cup X_6$ and $fires(t,0) \\notin X_0 \\cup \\dots \\cup X_6$ due to construction of $A$ , encoding of $a\\ref {a:r:maxfire:cnh}$ and its body atoms.", "As a result it is not eliminated by the constraint $a\\ref {a:maxfire:elim}$ $ \\vdots $ $eval_{U_{7k+0}}(bot_{U_{7k+1}}(\\Pi ^2) \\setminus bot_{U_{7k+0}}(\\Pi ^2), X_0 \\cup \\dots \\cup X_{7k+0}) = \\lbrace ptarc(p,t,q,k) \\text{:-}.", "\| q=W(p,t) \\rbrace \\cup \\lbrace tparc(t,p,q,k) \\text{:-}.", "\| q=W(t,p) \\rbrace \\cup \\lbrace ptarc(p,t,q,k) \\text{:-}.", "\| q=M_0(p) \\rbrace $ .", "Its answer set $X_{7k+1}=A_5^{ts=k} \\cup A_6^{ts=k} \\cup A_{19}^{ts=k}$ – using forced atom proposition and construction of $A_5, A_6, A_{19}$ .", "$eval_{U_{7k+1}}(bot_{U_{7k+2}}(\\Pi ^2) \\setminus bot_{U_{7k+1}}(\\Pi ^2), X_0 \\cup X_{7k+1}) = \\lbrace notenabled(t,k) \\text{:-} .", "\| \\lbrace trans(t), \\\\ ptarc(p,t,n,k), holds(p,q,k) \\rbrace \\subseteq X_0 \\cup X_{7k+1}, \\text{~where~} q < n \\rbrace $ .", "Its answer set $X_{7k+2}=A_8^{ts=k}$ – using forced atom proposition and construction of $A_8$ .", "where, $q=M_0(p)$ , and $n=W(p,t)$ for an arc $(p,t) \\in E^-$ – by construction of $holds$ and $ptarc$ predicates in $\\Pi ^2$ , and in an arc $(p,t) \\in E^-$ , $p \\in \\bullet t$ (by definition REF of preset) thus, $notenabled(t,k) \\in X_{7k+1}$ represents $\\exists p \\in \\bullet t : M_0(p) < W(p,t)$ .", "$eval_{U_{7k+2}}(bot_{U_{7k+3}}(\\Pi ^2) \\setminus bot_{U_{7k+2}}(\\Pi ^2), X_0 \\cup \\dots \\cup X_{7k+2}) = \\lbrace enabled(t,k) \\text{:-}.", "\| trans(t) \\in X_0 \\cup \\dots \\cup X_{7k+2} , notenabled(t,k) \\notin X_0 \\cup \\dots \\cup X_{7k+2} \\rbrace $ .", "Its answer set is $X_{7k+3} = A_9^{ts=k}$ – using forced atom proposition and construction of $A_9$ .", "since an $enabled(t,k) \\in X_{7k+3}$ if $\\nexists ~notenabled(t,k) \\in X_0 \\cup \\dots \\cup X_{7k+2}$ ; which is equivalent to $\\nexists p \\in \\bullet t : M_0(p) < W(p,t) \\equiv \\forall p \\in \\bullet t: M_0(p) \\ge W(p,t)$ .", "$eval_{U_{7k+3}}(bot_{U_{7k+4}}(\\Pi ^2) \\setminus bot_{U_{7k+3}}(\\Pi ^2), X_0 \\cup \\dots \\cup X_{7k+3}) = \\lbrace \\lbrace fires(t,k)\\rbrace \\text{:-}.", "\| \\\\ enabled(t,k) \\\\ \\text{~holds in~} X_0 \\cup \\dots \\cup X_{7k+3} \\rbrace $ .", "It has multiple answer sets $X_{7k+4.1}, \\dots , X_{1k+4.n}$ , corresponding to elements of power set of $fires(t,k)$ atoms in $eval_{U_{7k+3}}(...)$ – using supported rule proposition.", "Since we are showing that the union of answer sets of $\\Pi ^2$ determined using splitting is equal to $A$ , we only consider the set that matches the $fires(t,k)$ elements in $A$ and call it $X_{7k+4}$ , ignoring the rest.", "Thus, $X_{7k+4} = A_{10}^{ts=k}$ , representing $T_k$ .", "in addition, for every $t$ such that $enabled(t,k) \\in X_0 \\cup \\dots \\cup X_{7k+3}, R(t) \\ne \\emptyset $ ; $fires(t,k) \\in X_{7k+4}$ – per definition REF (firing set); requiring that a reset transition is fired when enabled thus, the firing set $T_k$ will not be eliminated by the constraint $f\\ref {f:c:rptarc:elim}$ $eval_{U_{7k+4}}(bot_{U_{7k+5}}(\\Pi ^2) \\setminus bot_{U_{7k+4}}(\\Pi ^2), X_0 \\cup \\dots \\cup X_{7k+4}) = \\\\ \\lbrace add(p,n,t,k) \\text{:-}.", "\| \\lbrace fires(t,k), tparc(t,p,n,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{7k+4} \\rbrace \\cup \\\\ \\lbrace del(p,n,t,k) \\text{:-}.", "\| \\lbrace fires(t,k), ptarc(p,t,n,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{7k+4} \\rbrace $ .", "It's answer set is $X_{7k+5} = A_{11}^{ts=k} \\cup A_{12}^{ts=k}$ – using forced atom proposition and definitions of $A_{11}$ and $A_{12}$ .", "where, each $add$ atom is equivalent to $n=W(t,p) : p \\in \\bullet t$ , and each $del$ atom is equivalent to $n=W(p,t) : p \\in t \\bullet $ ; or $n=M_k(p) : p \\in R(t)$ , representing the effect of transitions in $T_k$ $eval_{U_{7k+5}}(bot_{U_{7k+6}}(\\Pi ^2) \\setminus bot_{U_{7k+5}}(\\Pi ^2), X_0 \\cup \\dots \\cup X_{7k+5}) = \\\\ \\lbrace tot\\_incr(p,qq,k) \\text{:-}.", "\| qq=\\sum _{add(p,q,t,k) \\in X_0 \\cup \\dots \\cup X_{7k+5}}{q} \\rbrace \\cup \\\\ \\lbrace tot\\_decr(p,qq,k) \\text{:-}.", "\| qq=\\sum _{del(p,q,t,k) \\in X_0 \\cup \\dots \\cup X_{7k+5}}{q} \\rbrace $ .", "It's answer set is $X_{7k+6} = A_{13}^{ts=k} \\cup A_{14}^{ts=k}$ – using forced atom proposition and definitions of $A_{13}$ and $A_{14}$ .", "where, each $tot\\_incr(p,qq,k)$ , $qq=\\sum _{add(p,q,t,k) \\in X_0 \\cup \\dots X_{7k+5}}{q}$ $\\equiv qq=\\sum _{t \\in X_{7k+4}, p \\in t \\bullet }{W(p,t)}$ , and each $tot\\_decr(p,qq,k)$ , $qq=\\sum _{del(p,q,t,k) \\in X_0 \\cup \\dots X_{7k+5}}{q}$ $\\equiv qq=\\sum _{t \\in X_{7k+4}, p \\in \\bullet t}{W(t,p)} + \\sum _{t \\in X_{7k+4}, p \\in R(t)}{M_{k}(p)}$ , represent the net effect of transitions in $T_k$ $eval_{U_{7k+6}}(bot_{U_{7k+7}}(\\Pi ^2) \\setminus bot_{U_{7k+6}}(\\Pi ^2), X_0 \\cup \\dots \\cup X_{7k+6}) = \\lbrace consumesmore(p,k) \\text{:-}.", "\| \\\\ \\lbrace holds(p,q,k), tot\\_decr(p,q1,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{7k+6}, q1 > q \\rbrace \\cup \\lbrace holds(p,q,k+1) \\text{:-}., \| \\\\ \\lbrace holds(p,q1,k), tot\\_incr(p,q2,k), tot\\_decr(p,q3,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{7k+6}, q=q1+q2-q3 \\rbrace \\cup \\lbrace could\\_not\\_have(t,k) \\text{:-}.", "\| \\lbrace enabled(t,k), ptarc(s,t,q), holds(s,qq,k), \\\\ tot\\_decr(s,qqq,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{7k+6}, fires(t,k) \\notin (X_0 \\cup \\dots \\cup X_{7k+6}), q > qq-qqq \\rbrace $ .", "It's answer set is $X_{7k+7} = A_{15}^{ts=k} \\cup A_{17}^{ts=k} \\cup A_{18}^{ts=k}$ – using forced atom proposition and definitions of $A_{15}, A_{17}, A_{18}$ .", "where, $consumesmore(p,k)$ represents $\\exists p : q=M_0(p), q1= \\\\ \\sum _{t \\in T_0, p \\in \\bullet t}{W(p,t)}+\\sum _{t \\in T_0, p \\in R(t)}{M_k(p)}, q1 > q$ , indicating place $p$ that will be over consumed if $T_k$ is fired, as defined in definition REF (conflicting transitions) $holds(p,q,k+1)$ represents $q=M_{k+1}(p)$ – by construction of $\\Pi ^2$ , and $could\\_not\\_have(t,k)$ represents enabled transition $t$ in $T_k$ that could not be fired due to insufficient tokens $X_{7k+7}$ does not contain $could\\_not\\_have(t,k)$ , when $enabled(t,k) \\in X_0 \\cup \\dots \\cup X_{7k+6}$ and $fires(t,k) \\notin X_0 \\cup \\dots \\cup X_{7k+6}$ due to the construction of $A$ , encoding of $a\\ref {a:r:maxfire:cnh}$ and its body atoms.", "As a result it is not eliminated by the constraint $a\\ref {a:maxfire:elim}$ $eval_{U_{7k+7}}(bot_{U_{7k+8}}(\\Pi ^2) \\setminus bot_{U_{7k+7}}(\\Pi ^2), X_0 \\cup \\dots \\cup X_{7k+7}) = \\lbrace consumesmore \\text{:-}.", "\| $ $ \\lbrace consumesmore(p,0),\\dots ,$ $consumesmore(p,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{7k+7} \\rbrace $ .", "It's answer set is $X_{7k+8} = A_{16}$ – using forced atom proposition $X_{7k+8}$ will be empty since none of $consumesmore(p,0),\\dots , \\\\ consumesmore(p,k)$ hold in $X_0 \\cup \\dots \\cup X_{7k+7}$ due to the construction of $A$ , encoding of $a\\ref {a:overc:place}$ and its body atoms.", "As a result, it is not eliminated by the constraint $a\\ref {a:overc:elim}$ The set $X = X_0 \\cup \\dots \\cup X_{7k+8}$ is the answer set of $\\Pi ^2$ by the splitting sequence theorem REF .", "Each $X_i, 0 \\le i \\le 7k+8$ matches a distinct portion of $A$ , and $X = A$ , thus $A$ is an answer set of $\\Pi ^2$ .", "Next we show (REF ): Given $\\Pi ^2$ be the encoding of a Petri Net $PN(P,T,E,W,R)$ with initial marking $M_0$ , and $A$ be an answer set of $\\Pi ^2$ that satisfies (REF ) and (REF ), then we can construct $X=M_0,T_0,\\dots ,M_k,T_k,M_{k+1}$ from $A$ , such that it is an execution sequence of $PN$ .", "We construct the $X$ as follows: $M_i = (M_i(p_0), \\dots , M_i(p_n))$ , where $\\lbrace holds(p_0,M_i(p_0),i), \\dots holds(p_n,M_i(p_n),i) \\rbrace \\\\ \\subseteq A$ , for $0 \\le i \\le k+1$ $T_i = \\lbrace t : fires(t,i) \\in A\\rbrace $ , for $0 \\le i \\le k$ and show that $X$ is indeed an execution sequence of $PN$ .", "We show this by induction over $k$ (i.e.", "given $M_k$ , $T_k$ is a valid firing set and its firing produces marking $M_{k+1}$ ).", "Base case: Let $k=0$ , and $M_0$ is a valid marking in $X$ for $PN$ , show [(1)] $T_0$ is a valid firing set for $M_0$ , and $T_0$ 's firing in $M_0$ produces marking $M_1$ .", "We show $T_0$ is a valid firing set for $M_0$ .", "Let $\\lbrace fires(t_0,0), \\dots , fires(t_x,0) \\rbrace $ be the set of all $fires(\\dots ,0)$ atoms in $A$ , Then for each $fires(t_i,0) \\in A$ $enabled(t_i,0) \\in A$ – from rule $a\\ref {a:fires}$ and supported rule proposition Then $notenabled(t_i,0) \\notin A$ – from rule $e\\ref {e:enabled}$ and supported rule proposition Then $body(e\\ref {e:r:ne:ptarc})$ must not hold in $A$ – from rule $e\\ref {e:r:ne:ptarc}$ and forced atom proposition Then $q \\lnot < n_i \\equiv q \\ge n_i$ in $e\\ref {e:r:ne:ptarc}$ for all $\\lbrace holds(p,q,0), ptarc(p,t_i,n_i,0)\\rbrace \\subseteq A$ – from $e\\ref {e:r:ne:ptarc}$ , forced atom proposition, and the following $holds(p,q,0) \\in A$ represents $q=M_0(p)$ – rule $i\\ref {i:holds}$ construction $ptarc(p,t_i,n_i,0) \\in A$ represents $n_i=W(p,t_i)$ – rule $f\\ref {f:r:ptarc}$ construction; or it represents $n_i = M_0(p)$ – rule $f\\ref {f:rptarc}$ construction; the construction of $f\\ref {f:rptarc}$ ensures that $notenabled(t,0)$ is never true due to the reset arc Then $\\forall p \\in \\bullet t_i, M_0(p) \\ge W(p,t_i)$ – from definition REF of preset $\\bullet t_i$ in PN Then $t_i$ is enabled and can fire in $PN$ , as a result it can belong to $T_0$ – from definition REF of enabled transition And $consumesmore \\notin A$ , since $A$ is an answer set of $\\Pi ^2$ – from rule $a\\ref {a:overc:elim}$ and supported rule proposition Then $\\nexists consumesmore(p,0) \\in A$ – from rule $a\\ref {a:overc:gen}$ and supported rule proposition Then $\\nexists \\lbrace holds(p,q,0), tot\\_decr(p,q1,0) \\rbrace \\subseteq A : q1>q$ in $body(a\\ref {a:overc:place})$ – from $a\\ref {a:overc:place}$ and forced atom proposition Then $\\nexists p : (\\sum _{t_i \\in \\lbrace t_0,\\dots ,t_x\\rbrace , p \\in \\bullet t_i}{W(p,t_i)}+\\sum _{t_i \\in \\lbrace t_0,\\dots ,t_x\\rbrace , p \\in R(t_i)}{M_0(p)}) > M_0(p)$ – from the following $holds(p,q,0)$ represents $q=M_0(p)$ – from rule $i\\ref {i:holds}$ construction, given $tot\\_decr(p,q1,0) \\in A$ if $\\lbrace del(p,q1_0,t_0,0), \\dots , del(p,q1_x,t_x,0) \\rbrace \\subseteq A$ , where $q1 = q1_0+\\dots +q1_x$ – from $r\\ref {r:totdecr}$ and forced atom proposition $del(p,q1_i,t_i,0) \\in A$ if $\\lbrace fires(t_i,0), ptarc(p,t_i,q1_i,0) \\rbrace \\subseteq A$ – from $r\\ref {r:r:del} $ and supported rule proposition $del(p,q1_i,t_i,0)$ either represents removal of $q1_i = W(p,t_i)$ tokens from $p \\in \\bullet t_i$ ; or it represents removal of $q1_i = M_0(p)$ tokens from $p \\in R(t_i)$ – from rule $r\\ref {r:r:del} $ , supported rule proposition, and definition REF of transition execution in $PN$ Then the set of transitions in $T_0$ do not conflict – by the definition REF of conflicting transitions And for each $enabled(t_j,0) \\in A$ and $fires(t_j,0) \\notin A$ , $could\\_not\\_have(t_j,0) \\in A$ , since $A$ is an answer set of $\\Pi ^2$ - from rule $a\\ref {a:maxfire:elim}$ and supported rule proposition Then $\\lbrace enabled(t_j,0), holds(s,qq,0), ptarc(s,t_j,q,0), \\\\ tot\\_decr(s,qqq,0) \\rbrace \\subseteq A$ , such that $q > qq - qqq$ and $fires(t_j,0) \\notin A$ - from rule $a\\ref {a:r:maxfire:cnh}$ and supported rule proposition Then for an $s \\in \\bullet t_j \\cup R(t_j)$ , $q > M_0(s) - (\\sum _{t_i \\in T_0, s \\in \\bullet t_i}{W(s,t_i)} + \\sum _{t_i \\in T_0, s \\in R(t_i)}{M_0(s))}$ , where $q=W(s,t_j) \\text{~if~} s \\in \\bullet t_j, \\text{~or~} M_0(s) \\text{~otherwise}$ - from the following: $ptarc(s,t_i,q,0)$ represents $q=W(s,t_i)$ if $(s,t_i) \\in E^-$ or $q=M_0(s)$ if $s \\in R(t_i)$ – from rule $f\\ref {f:r:ptarc},f\\ref {f:rptarc}$ construction $holds(s,qq,0)$ represents $qq=M_0(s)$ – from $i\\ref {i:holds}$ construction $tot\\_decr(s,qqq,0) \\in A$ if $\\lbrace del(s,qqq_0,t_0,0), \\dots , del(s,qqq_x,t_x,0) \\rbrace \\subseteq A$ – from rule $r\\ref {r:totdecr}$ construction and supported rule proposition $del(s,qqq_i,t_i,0) \\in A$ if $\\lbrace fires(t_i,0), ptarc(s,t_i,qqq_i,0) \\rbrace \\subseteq A$ – from rule $r\\ref {r:r:del} $ and supported rule proposition $del(s,qqq_i,t_i,0)$ represents $qqq_i = W(s,t_i) : t_i \\in T_0, (s,t_i) \\in E^-$ , or $qqq_i = M_0(t_i) : t_i \\in T_0, s \\in R(t_i)$ – from rule $f\\ref {f:r:ptarc},f\\ref {f:rptarc}$ construction $tot\\_decr(q,qqq,0)$ represents $\\sum _{t_i \\in T_0, s \\in \\bullet t_i}{W(s,t_i)} + \\\\ \\sum _{t_i \\in T_0, s \\in R(t_i)}{M_0(s)}$ – from (C,D,E) above Then firing $T_0 \\cup \\lbrace t_j \\rbrace $ would have required more tokens than are present at its source place $s \\in \\bullet t_j \\cup R(t_j)$ .", "Thus, $T_0$ is a maximal set of transitions that can simultaneously fire.", "And for each reset transition $t_r$ with $enabled(t_r,0) \\in A$ , $fires(t_r,0) \\in A$ , since $A$ is an answer set of $\\Pi ^2$ - from rule $f\\ref {f:c:rptarc:elim}$ and supported rule proposition Then, the firing set $T_0$ satisfies the reset-transition requirement of definition REF (firing set) Then $\\lbrace t_0, \\dots , t_x\\rbrace = T_0$ – using 1(a),1(b),1(d) above; and using 1(c) it is a maximal firing set We show $M_1$ is produced by firing $T_0$ in $M_0$ .", "Let $holds(p,q,1) \\in A$ Then $\\lbrace holds(p,q1,0), tot\\_incr(p,q2,0), tot\\_decr(p,q3,0) \\rbrace \\subseteq A : q=q1+q2-q3$ – from rule $r\\ref {r:nextstate}$ and supported rule proposition $holds(p,q1,0) \\in A$ represents $q1=M_0(p)$ – given, rule $i\\ref {i:holds}$ construction; and $\\lbrace add(p,q2_0,t_0,0), \\dots , $ $add(p,q2_j,t_j,0)\\rbrace \\subseteq A : q2_0 + \\dots + q2_j = q2$ and $\\lbrace del(p,q3_0,t_0,0), \\dots , $ $del(p,q3_l,t_l,0)\\rbrace \\subseteq A : q3_0 + \\dots + q3_l = q3$ – rules $r\\ref {r:totincr},r\\ref {r:totdecr}$ using supported rule proposition Then $\\lbrace fires(t_0,0), \\dots , fires(t_j,0) \\rbrace \\subseteq A$ and $\\lbrace fires(t_0,0), \\dots , fires(t_l,0) \\rbrace \\\\ \\subseteq A$ – rules $r\\ref {r:r:add},r\\ref {r:r:del}$ and supported rule proposition, respectively Then $\\lbrace fires(t_0,0), \\dots , fires(t_j,0) \\rbrace \\cup \\lbrace fires(t_0,0), \\dots , fires(t_l,0) \\rbrace \\subseteq A = \\lbrace fires(t_0,0), \\dots , fires(t_x,0) \\rbrace \\subseteq A$ – set union of subsets Then for each $fires(t_x,0) \\in A$ we have $t_x \\in T_0$ – already shown in item REF above Then $q = M_0(p) + \\sum _{t_x \\in T_0 \\wedge p \\in t_x \\bullet }{W(t_x,p)} - (\\sum _{t_x \\in T_0 \\wedge p \\in \\bullet t_x}{W(p,t_x)} + \\\\ \\sum _{t_x \\in T_0 \\wedge p \\in R(t_x)}{M_0(p)})$ – from (REF ),(REF ) (REF ) above and the following Each $add(p,q_j,t_j,0) \\in A$ represents $q_j=W(t_j,p)$ for $p \\in t_j \\bullet $ – rule $r\\ref {r:r:add} $ encoding, and definition REF of postset in $PN$ Each $del(p,t_y,q_y,0) \\in A$ represents either $q_y=W(p,t_y)$ for $p \\in \\bullet t_y$ , or $q_y=M_0(p)$ for $p \\in R(t_y)$ – from rule $r\\ref {r:r:del} ,f\\ref {f:r:ptarc} $ encoding and definition REF of preset in $PN$ ; or from rule $r\\ref {r:r:del} ,f\\ref {f:rptarc}$ encoding and definition of reset arc in $PN$ Each $tot\\_incr(p,q2,0) \\in A$ represents $q2=\\sum _{t_x \\in T_0 \\wedge p \\in t_x \\bullet }{W(t_x,p)}$ – aggregate assignment atom semantics in rule $r\\ref {r:totincr}$ Each $tot\\_decr(p,q3,0) \\in A$ represents $q3=\\sum _{t_x \\in T_0 \\wedge p \\in \\bullet t_x}{W(p,t_x)} + $ $\\sum _{t_x \\in T_0 \\wedge p \\in R(t_x)}{M_0(p)}$ – aggregate assignment atom semantics in rule $r\\ref {r:totdecr}$ Then $M_1(p) = q$ – since $holds(p,q,1) \\in A$ encodes $q=M_1(p)$ – from construction Inductive Step: Let $k > 0$ , and $M_k$ is a valid marking in $X$ for $PN$ , show [(1)] $T_k$ is a valid firing set in $M_k$ , and firing $T_k$ in $M_k$ produces marking $M_{k+1}$ .", "We show that $T_k$ is a valid firing set in $M_k$ .", "Let $\\lbrace fires(t_0,k), \\dots , fires(t_x,k) \\rbrace $ be the set of all $fires(\\dots ,k)$ atoms in $A$ , Then for each $fires(t_i,k) \\in A$ $enabled(t_i,k) \\in A$ – from rule $a\\ref {a:fires}$ and supported rule proposition Then $notenabled(t_i,k) \\notin A$ – from rule $e\\ref {e:enabled}$ and supported rule proposition Then $body(e\\ref {e:r:ne:ptarc})$ must not hold in $A$ – from rule $e\\ref {e:r:ne:ptarc}$ and forced atom proposition Then $q \\lnot < n_i \\equiv q \\ge n_i$ in $e\\ref {e:r:ne:ptarc}$ for all $\\lbrace holds(p,q,k), ptarc(p,t_i,n_i,k)\\rbrace \\subseteq A$ – from $e\\ref {e:r:ne:ptarc}$ , forced atom proposition, and the following $holds(p,q,k) \\in A$ represents $q=M_k(p)$ – construction, inductive assumption $ptarc(p,t_i,n_i,k) \\in A$ represents $n_i=W(p,t_i)$ – rule $f\\ref {f:r:ptarc}$ construction; or it represents $n_i=M_k(p)$ – rule $f\\ref {f:rptarc}$ construction; the construction of $f\\ref {f:rptarc}$ ensures that $notenabled(t,0)$ is never true due to the reset arc Then $\\forall p \\in \\bullet t_i, M_k(p) \\ge W(p,t_i)$ – from definition REF of preset $\\bullet t_i$ in PN Then $t_i$ is enabled and can fire in $PN$ , as a result it can belong to $T_k$ – from definition REF of enabled transition And $consumesmore \\notin A$ , since $A$ is an answer set of $\\Pi ^2$ – from rule $a\\ref {a:overc:elim}$ and supported rule proposition Then $\\nexists consumesmore(p,k) \\in A$ – from rule $a\\ref {a:overc:gen}$ and supported rule proposition Then $\\nexists \\lbrace holds(p,q,k), tot\\_decr(p,q1,k) \\rbrace \\subseteq A : q1>q$ in $body(a\\ref {a:overc:place})$ – from $a\\ref {a:overc:place}$ and forced atom proposition Then $\\nexists p : (\\sum _{t_i \\in \\lbrace t_0,\\dots ,t_x\\rbrace , p \\in \\bullet t_i}{W(p,t_i)}+\\sum _{t_i \\in \\lbrace t_0,\\dots ,t_x\\rbrace , p \\in R(t_i)}{M_k(p)}) > M_k(p)$ – from the following $holds(p,q,k)$ represents $q=M_k(p)$ – inductive assumption, given $tot\\_decr(p,q1,k) \\in A$ if $\\lbrace del(p,q1_0,t_0,k), \\dots , del(p,q1_x,t_x,k) \\rbrace \\subseteq A$ , where $q1 = q1_0+\\dots +q1_x$ – from $r\\ref {r:totdecr}$ and forced atom proposition $del(p,q1_i,t_i,k) \\in A$ if $\\lbrace fires(t_i,k), ptarc(p,t_i,q1_i,k) \\rbrace \\subseteq A$ – from $r\\ref {r:r:del} $ and supported rule proposition $del(p,q1_i,t_i,k)$ either represents removal of $q1_i = W(p,t_i)$ tokens from $p \\in \\bullet t_i$ ; or it represents removal of $q1_i = M_k(p)$ tokens from $p \\in R(t_i)$ – from rule $r\\ref {r:r:del} $ , supported rule proposition, and definition REF of transition execution in $PN$ Then the set of transitions in $T_k$ do not conflict – by the definition REF of conflicting transitions And for each $enabled(t_j,k) \\in A$ and $fires(t_j,k) \\notin A$ , $could\\_not\\_have(t_j,k) \\in A$ , since $A$ is an answer set of $\\Pi ^2$ - from rule $a\\ref {a:maxfire:elim}$ and supported rule proposition Then $\\lbrace enabled(t_j,k), holds(s,qq,k), ptarc(s,t_j,q,k), \\\\ tot\\_decr(s,qqq,k) \\rbrace \\subseteq A$ , such that $q > qq - qqq$ and $fires(t_j,k) \\notin A$ - from rule $a\\ref {a:r:maxfire:cnh}$ and supported rule proposition Then for an $s \\in \\bullet t_j \\cup R(t_j)$ , $q > M_k(s) - (\\sum _{t_i \\in T_k, s \\in \\bullet t_i}{W(s,t_i)} + \\sum _{t_i \\in T_k, s \\in R(t_i)}{M_k(s))}$ , where $q=W(s,t_j) \\text{~if~} s \\in \\bullet t_j, \\text{~or~} M_k(s) \\text{~otherwise}$ - from the following: $ptarc(s,t_i,q,k)$ represents $q=W(s,t_i)$ if $(s,t_i) \\in E^-$ or $q=M_k(s)$ if $s \\in R(t_i)$ – from rule $f\\ref {f:r:ptarc},f\\ref {f:rptarc}$ construction $holds(s,qq,k)$ represents $qq=M_k(s)$ – construction $tot\\_decr(s,qqq,k) \\in A$ if $\\lbrace del(s,qqq_0,t_0,k), \\dots , del(s,qqq_x,t_x,k) \\rbrace \\subseteq A$ – from rule $r\\ref {r:totdecr}$ construction and supported rule proposition $del(s,qqq_i,t_i,k) \\in A$ if $\\lbrace fires(t_i,k), ptarc(s,t_i,qqq_i,k) \\rbrace \\subseteq A$ – from rule $r\\ref {r:r:del} $ and supported rule proposition $del(s,qqq_i,t_i,k)$ represents $qqq_i = W(s,t_i) : t_i \\in T_k, (s,t_i) \\in E^-$ , or $qqq_i = M_k(t_i) : t_i \\in T_k, s \\in R(t_i)$ – from rule $f\\ref {f:r:ptarc},f\\ref {f:rptarc}$ construction $tot\\_decr(q,qqq,k)$ represents $\\sum _{t_i \\in T_k, s \\in \\bullet t_i}{W(s,t_i)} + \\\\ \\sum _{t_i \\in T_k, s \\in R(t_i)}{M_k(s)}$ – from (C,D,E) above Then firing $T_k \\cup \\lbrace t_j \\rbrace $ would have required more tokens than are present at its source place $s \\in \\bullet t_j \\cup R(t_j)$ .", "Thus, $T_k$ is a maximal set of transitions that can simultaneously fire.", "And for each reset transition $t_r$ with $enabled(t_r,k) \\in A$ , $fires(t_r,k) \\in A$ , since $A$ is an answer set of $\\Pi ^2$ - from rule $f\\ref {f:c:rptarc:elim}$ and supported rule proposition Then the firing set $T_k$ satisfies the reset transition requirement of definition REF (firing set) Then $\\lbrace t_0, \\dots , t_x\\rbrace = T_k$ – using 1(a),1(b), 1(d) above; and using 1(c) it is a maximal firing set We show that $M_{k+1}$ is produced by firing $T_k$ in $M_k$ .", "Let $holds(p,q,k+1) \\in A$ Then $\\lbrace holds(p,q1,k), tot\\_incr(p,q2,k), tot\\_decr(p,q3,k) \\rbrace \\subseteq A : q=q1+q2-q3$ – from rule $r\\ref {r:nextstate}$ and supported rule proposition $holds(p,q1,k) \\in A$ represents $q1=M_k(p)$ – construction, inductive assumption; and $\\lbrace add(p,q2_0,t_0,k), \\dots , $ $add(p,q2_j,t_j,k)\\rbrace \\subseteq A : q2_0 + \\dots + q2_j = q2$ and $\\lbrace del(p,q3_0,t_0,k), \\dots , $ $del(p,q3_l,t_l,k)\\rbrace \\subseteq A : q3_0 + \\dots + q3_l = q3$ – rules $r\\ref {r:totincr},r\\ref {r:totdecr}$ using supported rule proposition Then $\\lbrace fires(t_0,k), \\dots , $ $fires(t_j,k) \\rbrace \\subseteq A$ and $\\lbrace fires(t_0,k), \\dots , $ $fires(t_l,k) \\rbrace \\\\ \\subseteq A$ – rules $r\\ref {r:r:add},r\\ref {r:r:del}$ and supported rule proposition, respectively Then $\\lbrace fires(t_0,k), \\dots , fires(t_j,k) \\rbrace \\cup \\lbrace fires(t_0,k), \\dots , $ $fires(t_l,k) \\rbrace \\subseteq $ $A = \\lbrace fires(t_0,k), \\dots , $ $ fires(t_x,k) \\rbrace \\subseteq A$ – set union of subsets Then for each $fires(t_x,k) \\in A$ we have $t_x \\in T_k$ – already shown in item REF above Then $q = M_k(p) + \\sum _{t_x \\in T_0 \\wedge p \\in t_x \\bullet }{W(t_x,p)} - (\\sum _{t_x \\in T_k \\wedge p \\in \\bullet t_x}{W(p,t_x)} + \\\\ \\sum _{t_x \\in T_k \\wedge p \\in R(t_x)}{M_k(p)})$ – from (REF ) above and the following Each $add(p,q_j,t_j,k) \\in A$ represents $q_j=W(t_j,p)$ for $p \\in t_j \\bullet $ – rule $r\\ref {r:r:add} $ encoding, and definition REF of transition execution in $PN$ Each $del(p,t_y,q_y,k) \\in A$ represents either $q_y=W(p,t_y)$ for $p \\in \\bullet t_y$ , or $q_y=M_k(p)$ for $p \\in R(t_y)$ – from rule $r\\ref {r:r:del} ,f\\ref {f:r:ptarc} $ encoding and definition REF of transition execution in $PN$ ; or from rule $r\\ref {r:r:del} ,f\\ref {f:rptarc}$ encoding and definition of reset arc in $PN$ Each $tot\\_incr(p,q2,k) \\in A$ represents $q2=\\sum _{t_x \\in T_k \\wedge p \\in t_x \\bullet }{W(t_x,p)}$ – aggregate assignment atom semantics in rule $r\\ref {r:totincr}$ Each $tot\\_decr(p,q3,0) \\in A$ represents $q3=\\sum _{t_x \\in T_k \\wedge p \\in \\bullet t_x}{W(p,t_x)} + $ $\\sum _{t_x \\in T_k \\wedge p \\in R(t_x)}{M_0(p)}$ – aggregate assignment atom semantics in rule $r\\ref {r:totdecr}$ Then, $M_{k+1}(p) = q$ – since $holds(p,q,k+1) \\in A$ encodes $q=M_{k+1}(p)$ – from construction As a result, for any $n > k$ , $T_n$ is a valid firing set w.r.t.", "$M_n$ and its firing produces marking $M_{n+1}$ .", "Conclusion: Since both (REF ) and (REF ) hold, $X=M_0,T_0,M_1,\\dots ,M_k,T_{k+1}$ is an execution sequence of $PN(P,T,E,W,R)$ (w.r.t $M_0$ ) iff there is an answer set $A$ of $\\Pi ^2(PN,M_0,k,ntok)$ such that (REF ) and (REF ) hold.", "Poof of Proposition REF Let $PN=(P,T,E,W,R,I)$ be a Petri Net, $M_0$ be its initial marking and let $\\Pi ^3(PN,M_0,k,ntok)$ be the ASP encoding of $PN$ and $M_0$ over a simulation length $k$ , with maximum $ntok$ tokens on any place node, as defined in section REF .", "Then $X=M_0,T_0,M_1,\\dots ,M_k,T_k,M_{k+1}$ is an execution sequence of $PN$ (w.r.t.", "$M_0$ ) iff there is an answer set $A$ of $\\Pi ^3(PN,M_0,k,ntok)$ such that: $\\lbrace fires(t,ts) : t \\in T_{ts}, 0 \\le ts \\le k\\rbrace = \\lbrace fires(t,ts) : fires(t,ts) \\in A \\rbrace $ $\\begin{split}\\lbrace holds(p,q,ts) &: p \\in P, q = M_{ts}(p), 0 \\le ts \\le k+1 \\rbrace \\\\&= \\lbrace holds(p,q,ts) : holds(p,q,ts) \\in A \\rbrace \\end{split}$ We prove this by showing that: Given an execution sequence $X$ , we create a set $A$ such that it satisfies (REF ) and (REF ) and show that $A$ is an answer set of $\\Pi ^3$ Given an answer set $A$ of $\\Pi ^3$ , we create an execution sequence $X$ such that (REF ) and (REF ) are satisfied.", "First we show (REF ): Given $PN$ and an execution sequence $X$ of $PN$ , we create a set $A$ as a union of the following sets: $A_1=\\lbrace num(n) : 0 \\le n \\le ntok \\rbrace $ $A_2=\\lbrace time(ts) : 0 \\le ts \\le k\\rbrace $ $A_3=\\lbrace place(p) : p \\in P \\rbrace $ $A_4=\\lbrace trans(t) : t \\in T \\rbrace $ $A_5=\\lbrace ptarc(p,t,n,ts) : (p,t) \\in E^-, n=W(p,t), 0 \\le ts \\le k \\rbrace $ , where $E^- \\subseteq E$ $A_6=\\lbrace tparc(t,p,n,ts) : (t,p) \\in E^+, n=W(t,p), 0 \\le ts \\le k \\rbrace $ , where $E^+ \\subseteq E$ $A_7=\\lbrace holds(p,q,0) : p \\in P, q=M_{0}(p) \\rbrace $ $A_8=\\lbrace notenabled(t,ts) : t \\in T, 0 \\le ts \\le k, (\\exists p \\in \\bullet t, M_{ts}(p) < W(p,t)) \\vee (\\exists p \\in I(t), M_{ts}(p) \\ne 0) \\rbrace $ per definition REF (enabled transition) $A_9=\\lbrace enabled(t,ts) : t \\in T, 0 \\le ts \\le k, (\\forall p \\in \\bullet t, W(p,t) \\le M_{ts}(p)) \\wedge (\\forall p \\in I(t), M_{ts}(p) = 0) \\rbrace $ per definition REF (enabled transition) $A_{10}=\\lbrace fires(t,ts) : t \\in T_{ts}, 0 \\le ts \\le k \\rbrace $ per definition REF (firing set), only an enabled transition may fire $A_{11}=\\lbrace add(p,q,t,ts) : t \\in T_{ts}, p \\in t \\bullet , q=W(t,p), 0 \\le ts \\le k \\rbrace $ per definition REF (transition execution) $A_{12}=\\lbrace del(p,q,t,ts) : t \\in T_{ts}, p \\in \\bullet t, q=W(p,t), 0 \\le ts \\le k \\rbrace \\cup \\lbrace del(p,q,t,ts) : t \\in T_{ts}, p \\in R(t), q=M_{ts}(p), 0 \\le ts \\le k \\rbrace $ per definition REF (transition execution) $A_{13}=\\lbrace tot\\_incr(p,q,ts) : p \\in P, q=\\sum _{t \\in T_{ts}, p \\in t \\bullet }{W(t,p)}, 0 \\le ts \\le k \\rbrace $ per definition REF (firing set execution) $A_{14}=\\lbrace tot\\_decr(p,q,ts): p \\in P, q=\\sum _{t \\in T_{ts}, p \\in \\bullet t}{W(p,t)}+\\sum _{t \\in T_{ts}, p \\in R(t) }{M_{ts}(p)}, 0 \\le ts \\le k \\rbrace $ per definition REF (firing set execution) $A_{15}=\\lbrace consumesmore(p,ts) : p \\in P, q=M_{ts}(p), q1=\\sum _{t \\in T_{ts}, p \\in \\bullet t}{W(p,t)} + \\sum _{t \\in T_{ts}, p \\in R(t)}{M_{ts}(p)}, q1 > q, 0 \\le ts \\le k \\rbrace $ per definition REF (conflicting transitions) $A_{16}=\\lbrace consumesmore : \\exists p \\in P : q=M_{ts}(p), q1=\\sum _{t \\in T_{ts}, p \\in \\bullet t}{W(p,t)} + \\sum _{t \\in T_{ts}, p \\in R(t)}(M_{ts}(p)), q1 > q, 0 \\le ts \\le k \\rbrace $ per definition REF (conflicting transitions) $A_{17}=\\lbrace could\\_not\\_have(t,ts) : t \\in T, (\\forall p \\in \\bullet t, W(p,t) \\le M_{ts}(p)), t \\notin T_{ts}, (\\exists p \\in \\bullet t : W(p,t) > M_{ts}(p) - (\\sum _{t^{\\prime } \\in T_{ts}, p \\in \\bullet t^{\\prime }}{W(p,t^{\\prime })} + \\sum _{t^{\\prime } \\in T_{ts}, p \\in R(t^{\\prime })} M_{ts}(p)), 0 \\le ts \\le k \\rbrace $ per the maximal firing set semantics $A_{18}=\\lbrace holds(p,q,ts+1) : p \\in P, q=M_{ts+1}(p), 0 \\le ts < k\\rbrace $ , where $M_{ts+1}(p) = M_{ts}(p) - (\\sum _{\\begin{array}{c}t \\in T_{ts}, p \\in \\bullet t\\end{array}}{W(p,t)} + $ $\\sum _{t \\in T_{ts}, p \\in R(t)}M_{ts}(p)) + \\\\ \\sum _{\\begin{array}{c}t \\in T_{ts}, p \\in t \\bullet \\end{array}}{W(t,p)}$ according to definition REF (firing set execution) $A_{19}=\\lbrace ptarc(p,t,n,ts) : p \\in R(t), n = M_{ts}(p), n > 0, 0 \\le ts \\le k \\rbrace $ $A_{20}=\\lbrace iptarc(p,t,1,ts) : p \\in P, 0 \\le ts < k \\rbrace $ We show that $A$ satisfies (REF ) and (REF ), and $A$ is an answer set of $\\Pi ^3$ .", "$A$ satisfies (REF ) and (REF ) by its construction above.", "We show $A$ is an answer set of $\\Pi ^3$ by splitting.", "We split $lit(\\Pi ^3)$ into a sequence of $7k+9$ sets: $U_0= head(f\\ref {f:place}) \\cup head(f\\ref {f:trans}) \\cup head(f\\ref {f:time}) \\cup head(f\\ref {f:num}) \\cup head(i\\ref {i:holds}) = \\lbrace place(p) : p \\in P\\rbrace \\cup \\lbrace trans(t) : t \\in T\\rbrace \\cup \\lbrace time(0), \\dots , time(k)\\rbrace \\cup \\lbrace num(0), \\dots , num(ntok)\\rbrace \\cup \\lbrace holds(p,q,0) : p \\in P, q=M_0(p) \\rbrace $ $U_{7k+1}=U_{7k+0} \\cup head(f\\ref {f:r:ptarc})^{ts=k} \\cup head(f\\ref {f:r:tparc})^{ts=k} \\cup head(f\\ref {f:rptarc})^{ts=k} \\cup head(f\\ref {f:iptarc})^{ts=k} = U_{7k+0} \\cup \\lbrace ptarc(p,t,n,k) : (p,t) \\in E^-, n=W(p,t) \\rbrace \\cup \\lbrace tparc(t,p,n,k) : (t,p) \\in E^+, n=W(t,p) \\rbrace \\cup \\lbrace ptarc(p,t,n,k) : p \\in R(t), n=M_{k}(p), n > 0 \\rbrace \\cup \\\\ \\lbrace iptarc(p,t,1,k) : p \\in I(t) \\rbrace $ $U_{7k+2}=U_{7k+1} \\cup head(e\\ref {e:r:ne:ptarc} )^{ts=k} \\cup head(e\\ref {e:ne:iptarc})^{ts=k} = U_{7k+1} \\cup \\lbrace notenabled(t,k) : t \\in T \\rbrace $ $U_{7k+3}=U_{7k+2} \\cup head(e\\ref {e:enabled})^{ts=k} = U_{7k+2} \\cup \\lbrace enabled(t,k) : t \\in T \\rbrace $ $U_{7k+4}=U_{7k+3} \\cup head(a\\ref {a:fires})^{ts=k} = U_{7k+3} \\cup \\lbrace fires(t,k) : t \\in T \\rbrace $ $U_{7k+5}=U_{7k+4} \\cup head(r\\ref {r:r:add} )^{ts=k} \\cup head(r\\ref {r:r:del} )^{ts=k} = U_{7k+4} \\cup \\lbrace add(p,q,t,k) : p \\in P, t \\in T, q=W(t,p) \\rbrace \\cup \\lbrace del(p,q,t,k) : p \\in P, t \\in T, q=W(p,t) \\rbrace \\cup \\lbrace del(p,q,t,k) : p \\in P, t \\in T, q=M_{k}(p) \\rbrace $ $U_{7k+6}=U_{7k+5} \\cup head(r\\ref {r:totincr})^{ts=k} \\cup head(r\\ref {r:totdecr})^{ts=k} = U_{7k+5} \\cup \\lbrace tot\\_incr(p,q,k) : p \\in P, 0 \\le q \\le ntok \\rbrace \\cup \\lbrace tot\\_decr(p,q,k) : p \\in P, 0 \\le q \\le ntok \\rbrace $ $U_{7k+7}=U_{7k+6} \\cup head(r\\ref {r:nextstate})^{ts=k} \\cup head(a\\ref {a:overc:place})^{ts=k} \\cup head(a\\ref {a:r:maxfire:cnh})^{ts=k} = $ $U_{7k+6} \\cup \\\\ \\lbrace consumesmore(p,k) : p \\in P\\rbrace \\cup \\lbrace holds(p,q,k+1) : p \\in P, 0 \\le q \\le ntok \\rbrace \\cup \\lbrace could\\_not\\_have(t,k) : t \\in T \\rbrace $ $U_{7k+8}=U_{7k+7} \\cup head(a\\ref {a:overc:gen})^{ts=k} = U_{7k+7} \\cup \\lbrace consumesmore \\rbrace $ where $head(r_i)^{ts=k}$ are head atoms of ground rule $r_i$ in which $ts=k$ .", "We write $A_i^{ts=k} = \\lbrace a(\\dots ,ts) : a(\\dots ,ts) \\in A_i, ts=k \\rbrace $ as short hand for all atoms in $A_i$ with $ts=k$ .", "$U_{\\alpha }, 0 \\le \\alpha \\le 7k+8$ form a splitting sequence, since each $U_i$ is a splitting set of $\\Pi ^1$ , and $\\langle U_{\\alpha }\\rangle _{\\alpha < \\mu }$ is a monotone continuous sequence, where $U_0 \\subseteq U_1 \\dots \\subseteq U_{7k+8}$ and $\\bigcup _{\\alpha < \\mu }{U_{\\alpha }} = lit(\\Pi ^1)$ .", "We compute the answer set of $\\Pi ^3$ using the splitting sets as follows: $bot_{U_0}(\\Pi ^3) = f\\ref {f:place} \\cup f\\ref {f:trans} \\cup f\\ref {f:time} \\cup i\\ref {i:holds} \\cup f\\ref {f:num}$ and $X_0 = A_1 \\cup \\dots \\cup A_4 \\cup A_7$ ($= U_0$ ) is its answer set – using forced atom proposition $eval_{U_0}(bot_{U_1}(\\Pi ^3) \\setminus bot_{U_0}(\\Pi ^3), X_0) = \\lbrace ptarc(p,t,q,0) \\text{:-}.", "\| q=W(p,t) \\rbrace \\cup \\\\ \\lbrace tparc(t,p,q,0) \\text{:-}.", "\| q=W(t,p) \\rbrace \\cup \\lbrace ptarc(p,t,q,0) \\text{:-}.", "\| q=M_0(p) \\rbrace \\cup \\\\ \\lbrace iptarc(p,t,1,0) \\text{:-}.", "\\rbrace $ .", "Its answer set $X_1=A_5^{ts=0} \\cup A_6^{ts=0} \\cup A_{19}^{ts=0} \\cup A_{20}^{ts=0}$ – using forced atom proposition and construction of $A_5, A_6, A_{19}, A_{20}$ .", "$eval_{U_1}(bot_{U_2}(\\Pi ^3) \\setminus bot_{U_1}(\\Pi ^3), X_0 \\cup X_1) = \\lbrace notenabled(t,0) \\text{:-} .", "\| (\\lbrace trans(t), \\\\ ptarc(p,t,n,0), holds(p,q,0) \\rbrace \\subseteq X_0 \\cup X_1, \\text{~where~} q < n) \\text{~or~} \\lbrace notenabled(t,0) \\text{:-} .", "\| \\\\ (\\lbrace trans(t), iptarc(p,t,n2,0), holds(p,q,0) \\rbrace \\subseteq X_0 \\cup X_1, \\text{~where~} q \\ge n2 \\rbrace \\rbrace $ .", "Its answer set $X_2=A_8^{ts=0}$ – using forced atom proposition and construction of $A_8$ .", "where, $q=M_0(p)$ , and $n=W(p,t)$ for an arc $(p,t) \\in E^-$ – by construction $i\\ref {i:holds}$ and $f\\ref {f:r:ptarc}$ in $\\Pi ^3$ , and in an arc $(p,t) \\in E^-$ , $p \\in \\bullet t$ (by definition REF of preset) $n2=1$ – by construction of $iptarc$ predicates in $\\Pi ^3$ , meaning $q \\ge n2 \\equiv q \\ge 1 \\equiv q > 0$ , thus, $notenabled(t,0) \\in X_1$ represents $(\\exists p \\in \\bullet t : M_0(p) < W(p,t)) \\vee (\\exists p \\in I(t) : M_0(p) > 0)$ .", "$eval_{U_2}(bot_{U_3}(\\Pi ^3) \\setminus bot_{U_2}(\\Pi ^3), X_0 \\cup \\dots \\cup X_2) = \\lbrace enabled(t,0) \\text{:-}.", "\| trans(t) \\in X_0 \\cup \\dots \\cup X_2, notenabled(t,0) \\notin X_0 \\cup \\dots \\cup X_2 \\rbrace $ .", "Its answer set is $X_3 = A_9^{ts=0}$ – using forced atom proposition and construction of $A_9$ .", "since an $enabled(t,0) \\in X_3$ if $\\nexists ~notenabled(t,0) \\in X_0 \\cup \\dots \\cup X_2$ ; which is equivalent to $(\\nexists p \\in \\bullet t : M_0(p) < W(p,t)) \\wedge (\\nexists p \\in I(t) : M_0(p) > 0) \\equiv (\\forall p \\in \\bullet t: M_0(p) \\ge W(p,t)) \\wedge (\\forall p \\in I(t) : M_0(p) = 0)$ .", "$eval_{U_3}(bot_{U_4}(\\Pi ^3) \\setminus bot_{U_3}(\\Pi ^3), X_0 \\cup \\dots \\cup X_3) = \\lbrace \\lbrace fires(t,0)\\rbrace \\text{:-}.", "\| enabled(t,0) \\\\ \\text{~holds in~} X_0 \\cup \\dots \\cup X_3 \\rbrace $ .", "It has multiple answer sets $X_{4.1}, \\dots , X_{4.n}$ , corresponding to elements of power set of $fires(t,0)$ atoms in $eval_{U_3}(...)$ – using supported rule proposition.", "Since we are showing that the union of answer sets of $\\Pi ^3$ determined using splitting is equal to $A$ , we only consider the set that matches the $fires(t,0)$ elements in $A$ and call it $X_4$ , ignoring the rest.", "Thus, $X_4 = A_{10}^{ts=0}$ , representing $T_0$ .", "in addition, for every $t$ such that $enabled(t,0) \\in X_0 \\cup \\dots \\cup X_3, R(t) \\ne \\emptyset $ ; $fires(t,0) \\in X_4$ – per definition REF (firing set); requiring that a reset transition is fired when enabled thus, the firing set $T_0$ will not be eliminated by the constraint $f\\ref {f:c:rptarc:elim}$ $eval_{U_4}(bot_{U_5}(\\Pi ^3) \\setminus bot_{U_4}(\\Pi ^3), X_0 \\cup \\dots \\cup X_4) = \\lbrace add(p,n,t,0) \\text{:-}.", "\| \\lbrace fires(t,0), \\\\ tparc(t,p,n,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_4 \\rbrace \\cup \\lbrace del(p,n,t,0) \\text{:-}.", "\| \\lbrace fires(t,0), ptarc(p,t,n,0) \\rbrace \\\\ \\subseteq X_0 \\cup \\dots \\cup X_4 \\rbrace $ .", "It's answer set is $X_5 = A_{11}^{ts=0} \\cup A_{12}^{ts=0}$ – using forced atom proposition and definitions of $A_{11}$ and $A_{12}$ .", "where, each $add$ atom is equivalent to $n=W(t,p) : p \\in t \\bullet $ , and each $del$ atom is equivalent to $n=W(p,t) : p \\in \\bullet t$ ; or $n=M_k(p) : p \\in R(t)$ , representing the effect of transitions in $T_0$ .", "$eval_{U_5}(bot_{U_6}(\\Pi ^3) \\setminus bot_{U_5}(\\Pi ^3), X_0 \\cup \\dots \\cup X_5) = \\lbrace tot\\_incr(p,qq,0) \\text{:-}.", "\| $ $qq=\\sum _{add(p,q,t,0) \\in X_0 \\cup \\dots \\cup X_5}{q} \\rbrace \\cup \\lbrace tot\\_decr(p,qq,0) \\text{:-}.", "\| qq=\\sum _{del(p,q,t,0) \\in X_0 \\cup \\dots \\cup X_5}{q} \\rbrace $ .", "It's answer set is $X_6 = A_{13}^{ts=0} \\cup A_{14}^{ts=0}$ – using forced atom proposition and definitions of $A_{13}$ and $A_{14}$ .", "where, each $tot\\_incr(p,qq,0)$ , $qq=\\sum _{add(p,q,t,0) \\in X_0 \\cup \\dots X_5}{q}$ $\\equiv qq=\\sum _{t \\in X_4, p \\in t \\bullet }{W(p,t)}$ , and each $tot\\_decr(p,qq,0)$ , $qq=\\sum _{del(p,q,t,0) \\in X_0 \\cup \\dots X_5}{q}$ $\\equiv qq=\\sum _{t \\in X_4, p \\in \\bullet t}{W(t,p)} + \\sum _{t \\in X_4, p \\in R(t)}{M_{k}(p)}$ , represent the net effect of transitions in $T_0$ .", "$eval_{U_6}(bot_{U_7}(\\Pi ^3) \\setminus bot_{U_6}(\\Pi ^3), X_0 \\cup \\dots \\cup X_6) = \\lbrace consumesmore(p,0) \\text{:-}.", "\| \\\\ \\lbrace holds(p,q,0), tot\\_decr(p,q1,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_6, q1 > q \\rbrace \\cup \\lbrace holds(p,q,1) \\text{:-}., \| \\\\ \\lbrace holds(p,q1,0), tot\\_incr(p,q2,0), tot\\_decr(p,q3,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_6, q=q1+q2-q3 \\rbrace \\cup \\lbrace could\\_not\\_have(t,0) \\text{:-}.", "\| \\lbrace enabled(t,0), ptarc(s,t,q), holds(s,qq,0), \\\\ tot\\_decr(s,qqq,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_6, fires(t,0) \\notin (X_0 \\cup \\dots \\cup X_6), q > qq-qqq \\rbrace $ .", "It's answer set is $X_7 = A_{15}^{ts=0} \\cup A_{17}^{ts=0} \\cup A_{18}^{ts=0}$ – using forced atom proposition and definitions of $A_{15}, A_{17}, A_{18}, A_9$ .", "where, $consumesmore(p,0)$ represents $\\exists p : q=M_0(p), q1= \\\\ \\sum _{t \\in T_0, p \\in \\bullet t}{W(p,t)}+\\sum _{t \\in T_0, p \\in R(t)}{M_k(p)}, q1 > q$ , indicating place $p$ will be over consumed if $T_0$ is fired, as defined in definition REF (conflicting transitions) $holds(p,q,1)$ represents $q=M_1(p)$ – by construction of $\\Pi ^3$ and $could\\_not\\_have(t,0)$ represents enabled transition $t \\in T_0$ that could not fire due to insufficient tokens $X_7$ does not contain $could\\_not\\_have(t,0)$ , when $enabled(t,0) \\in X_0 \\cup \\dots \\cup X_5$ and $fires(t,0) \\notin X_0 \\cup \\dots \\cup X_6$ due to construction of $A$ , encoding of $a\\ref {a:r:maxfire:cnh}$ and its body atoms.", "As a result it is not eliminated by the constraint $a\\ref {a:maxfire:elim}$ $ \\vdots $ $eval_{U_{7k+0}}(bot_{U_{7k+1}}(\\Pi ^3) \\setminus bot_{U_{7k+0}}(\\Pi ^3), X_0 \\cup \\dots \\cup X_{7k+0}) = \\lbrace ptarc(p,t,q,k) \\text{:-}.", "\| q=W(p,t) \\rbrace \\cup \\lbrace tparc(t,p,q,k) \\text{:-}.", "\| q=W(t,p) \\rbrace \\cup \\lbrace ptarc(p,t,q,k) \\text{:-}.", "\| q=M_k(p) \\rbrace \\cup \\lbrace iptarc(p,t,1,k) \\text{:-}.", "\\rbrace $ .", "Its answer set $X_{7k+1}=A_5^{ts=k} \\cup A_6^{ts=k} \\cup A_{19}^{ts=k} \\cup A_{20}^{ts=k}$ – using forced atom proposition and construction of $A_5, A_6, A_{19}, A_{20}$ .", "$eval_{U_{7k+1}}(bot_{U_{7k+2}}(\\Pi ^3) \\setminus bot_{U_{7k+1}}(\\Pi ^3), X_0 \\cup \\dots \\cup X_{7k+1}) = \\lbrace notenabled(t,k) \\text{:-} .", "\| \\\\ (\\lbrace trans(t), ptarc(p,t,n,k), holds(p,q,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{7k+1}, \\text{~where~} q < n) \\rbrace \\text{~or~} $ $\\lbrace notenabled(t,k) \\text{:-} .", "\| (\\lbrace trans(t), iptarc(p,t,n2,k), holds(p,q,k) \\rbrace \\subseteq \\\\ X_0 \\cup \\dots \\cup X_{7k+1}, \\text{~where~} q \\ge n2 \\rbrace \\rbrace $ .", "Its answer set $X_{7k+2}=A_8^{ts=k}$ – using forced atom proposition and construction of $A_8$ .", "where, $q=M_k(p)$ , and $n=W(p,t)$ for an arc $(p,t) \\in E^-$ – by construction of $holds$ and $ptarc$ predicates in $\\Pi ^3$ , and in an arc $(p,t) \\in E^-$ , $p \\in \\bullet t$ (by definition REF of preset) $n2=1$ – by construction of $iptarc$ predicates in $\\Pi ^3$ , meaning $q \\ge n2 \\equiv q \\ge 1 \\equiv q > 0$ , thus, $notenabled(t,k) \\in X_{7k+1}$ represents $(\\exists p \\in \\bullet t : M_k(p) < W(p,t)) \\vee (\\exists p \\in I(t) : M_k(p) > k)$ .", "$eval_{U_{7k+2}}(bot_{U_{7k+3}}(\\Pi ^3) \\setminus bot_{U_{7k+2}}(\\Pi ^3), X_0 \\cup \\dots \\cup X_{7k+2}) = \\lbrace enabled(t,k) \\text{:-}.", "\| \\\\ trans(t) \\in X_0 \\cup \\dots \\cup X_{7k+2} \\wedge notenabled(t,k) \\notin X_0 \\cup \\dots \\cup X_{7k+2} \\rbrace $ .", "Its answer set is $X_{7k+3} = A_9^{ts=k}$ – using forced atom proposition and construction of $A_9$ .", "since an $enabled(t,k) \\in X_{7k+3}$ if $\\nexists ~notenabled(t,k) \\in X_0 \\cup \\dots \\cup X_{7k+2}$ ; which is equivalent to $(\\nexists p \\in \\bullet t : M_k(p) < W(p,t)) \\wedge (\\nexists p \\in I(t) : M_k(p) > k) \\equiv (\\forall p \\in \\bullet t: M_k(p) \\ge W(p,t)) \\wedge (\\forall p \\in I(t) : M_k(p) = k)$ .", "$eval_{U_{7k+3}}(bot_{U_{7k+4}}(\\Pi ^3) \\setminus bot_{U_{7k+3}}(\\Pi ^3), X_0 \\cup \\dots \\cup X_{7k+3}) = \\lbrace \\lbrace fires(t,k)\\rbrace \\text{:-}.", "\| \\\\ enabled(t,k) \\\\ \\text{~holds in~} X_0 \\cup \\dots \\cup X_{7k+3} \\rbrace $ .", "It has multiple answer sets $X_{7k+4.1}, \\dots , X_{1k+4.n}$ , corresponding to elements of power set of $fires(t,k)$ atoms in $eval_{U_{7k+3}}(...)$ – using supported rule proposition.", "Since we are showing that the union of answer sets of $\\Pi ^3$ determined using splitting is equal to $A$ , we only consider the set that matches the $fires(t,k)$ elements in $A$ and call it $X_{7k+4}$ , ignoring the rest.", "Thus, $X_{7k+4} = A_{10}^{ts=k}$ , representing $T_k$ .", "in addition, for every $t$ such that $enabled(t,k) \\in X_0 \\cup \\dots \\cup X_{7k+3}, R(t) \\ne \\emptyset $ ; $fires(t,k) \\in X_{7k+4}$ – per definition REF (firing set); requiring that a reset transition is fired when enabled thus, the firing set $T_k$ will not be eliminated by the constraint $f\\ref {f:c:rptarc:elim}$ $eval_{U_{7k+4}}(bot_{U_{7k+5}}(\\Pi ^3) \\setminus bot_{U_{7k+4}}(\\Pi ^3), X_0 \\cup \\dots \\cup X_{7k+4}) = \\lbrace add(p,n,t,k) \\text{:-}.", "\| \\\\ \\lbrace fires(t,k), tparc(t,p,n,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{7k+4} \\rbrace \\cup \\lbrace del(p,n,t,k) \\text{:-}.", "\| \\lbrace fires(t,k), \\\\ ptarc(p,t,n,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{7k+4} \\rbrace $ .", "It's answer set is $X_{7k+5} = A_{11}^{ts=k} \\cup A_{12}^{ts=k}$ – using forced atom proposition and definitions of $A_{11}$ and $A_{12}$ .", "where, each $add$ atom is equivalent to $n=W(t,p) : p \\in t \\bullet $ , and each $del$ atom is equivalent to $n=W(p,t) : p \\in \\bullet t$ ; or $n=M_k(p) : p \\in R(t)$ , representing the effect of transitions in $T_k$ $eval_{U_{7k+5}}(bot_{U_{7k+6}}(\\Pi ^3) \\setminus bot_{U_{7k+5}}(\\Pi ^3), X_0 \\cup \\dots \\cup X_{7k+5}) = \\\\ \\lbrace tot\\_incr(p,qq,k) \\text{:-}.", "\| $ $qq=\\sum _{add(p,q,t,k) \\in X_0 \\cup \\dots \\cup X_{7k+5}}{q} \\rbrace \\cup $ $\\lbrace tot\\_decr(p,qq,k) \\text{:-}.", "\| $ $qq=\\sum _{del(p,q,t,k) \\in X_0 \\cup \\dots \\cup X_{7k+5}}{q} \\rbrace $ .", "It's answer set is $X_{7k+6} = A_{13}^{ts=k} \\cup A_{14}^{ts=k}$ – using forced atom proposition and definitions of $A_{13}$ and $A_{14}$ .", "where, each $tot\\_incr(p,qq,k)$ , $qq=\\sum _{add(p,q,t,k) \\in X_0 \\cup \\dots X_{7k+5}}{q}$ $\\equiv qq=\\sum _{t \\in X_{7k+4}, p \\in t \\bullet }{W(p,t)}$ , and each $tot\\_decr(p,qq,k)$ , $qq=\\sum _{del(p,q,t,k) \\in X_0 \\cup \\dots X_{7k+5}}{q}$ $\\equiv qq=\\sum _{t \\in X_{7k+4}, p \\in \\bullet t}{W(t,p)} + \\sum _{t \\in X_{7k+4}, p \\in R(t)}{M_{k}(p)}$ , representing the net effect of transitions in $T_k$ $eval_{U_{7k+6}}(bot_{U_{7k+7}}(\\Pi ^3) \\setminus bot_{U_{7k+6}}(\\Pi ^3), X_0 \\cup \\dots \\cup X_{7k+6}) = \\lbrace consumesmore(p,k) \\text{:-}.", "\| \\\\ \\lbrace holds(p,q,k), tot\\_decr(p,q1,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{7k+6}, q1 > q \\rbrace \\cup \\lbrace holds(p,q,1) \\text{:-}., \| \\\\ \\lbrace holds(p,q1,k), tot\\_incr(p,q2,k), tot\\_decr(p,q3,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{7k+6}, q=q1+q2-q3 \\rbrace \\cup \\lbrace could\\_not\\_have(t,k) \\text{:-}.", "\| \\lbrace enabled(t,k), ptarc(s,t,q), holds(s,qq,k), \\\\ tot\\_decr(s,qqq,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{7k+6}, fires(t,k) \\notin (X_0 \\cup \\dots \\cup X_{7k+6}), q > qq-qqq \\rbrace $ .", "It's answer set is $X_{7k+7} = A_{15}^{ts=k} \\cup A_{17}^{ts=k} \\cup A_{18}^{ts=k}$ – using forced atom proposition and definitions of $A_{15}, A_{17}, A_{18}, A_9$ .", "where, $consumesmore(p,k)$ represents $\\exists p : q=M_k(p), \\\\ q1=\\sum _{t \\in T_k, p \\in \\bullet t}{W(p,t)}+\\sum _{t \\in T_k, p \\in R(t)}{M_k(p)}, q1 > q$ , indicating place $p$ that will be over consumed if $T_k$ is fired, as defined in definition REF (conflicting transitions), $holds(p,q,k+1)$ represents $q=M_{k+1}(p)$ – by construction of $\\Pi ^3$ and $could\\_not\\_have(t,k)$ represents enabled transition $t$ in $T_k$ that could not be fired due to insufficient tokens $X_{7k+7}$ does not contain $could\\_not\\_have(t,k)$ , when $enabled(t,k) \\in X_0 \\cup \\dots \\cup X_{7k+6}$ and $fires(t,k) \\notin X_0 \\cup \\dots \\cup X_{7k+6}$ due to construction of $A$ , encoding of $a\\ref {a:r:maxfire:cnh}$ and its body atoms.", "As a result it is not eliminated by the constraint $a\\ref {a:maxfire:elim}$ $eval_{U_{7k+7}}(bot_{U_{7k+8}}(\\Pi ^3) \\setminus bot_{U_{7k+7}}(\\Pi ^3), X_0 \\cup \\dots \\cup X_{7k+7}) = \\lbrace consumesmore \\text{:-}.", "\| \\\\ \\lbrace consumesmore(p,0),\\dots ,$ $consumesmore(p,k) \\rbrace \\cap (X_0 \\cup \\dots \\cup X_{7k+7}) \\ne \\emptyset \\rbrace $ .", "It's answer set is $X_{7k+8} = A_{16}$ – using forced atom proposition $X_{7k+8}$ will be empty since none of $consumesmore(p,0),\\dots , \\\\ consumesmore(p,k)$ hold in $X_0 \\cup \\dots \\cup X_{7k+7}$ due to the construction of $A$ , encoding of $a\\ref {a:overc:place}$ and its body atoms.", "As a result, it is not eliminated by the constraint $a\\ref {a:overc:elim}$ The set $X = X_0 \\cup \\dots \\cup X_{7k+8}$ is the answer set of $\\Pi ^3$ by the splitting sequence theorem REF .", "Each $X_i, 0 \\le i \\le 7k+8$ matches a distinct portion of $A$ , and $X = A$ , thus $A$ is an answer set of $\\Pi ^3$ .", "Next we show (REF ): Given $\\Pi ^3$ be the encoding of a Petri Net $PN(P,T,E,W,R,I)$ with initial marking $M_0$ , and $A$ be an answer set of $\\Pi ^3$ that satisfies (REF ) and (REF ), then we can construct $X=M_0,T_0,\\dots ,M_k,T_k,M_{k+1}$ from $A$ , such that it is an execution sequence of $PN$ .", "We construct the $X$ as follows: $M_i = (M_i(p_0), \\dots , M_i(p_n))$ , where $\\lbrace holds(p_0,M_i(p_0),i), \\dots holds(p_n,M_i(p_n),i) \\rbrace \\\\ \\subseteq A$ , for $0 \\le i \\le k+1$ $T_i = \\lbrace t : fires(t,i) \\in A\\rbrace $ , for $0 \\le i \\le k$ and show that $X$ is indeed an execution sequence of $PN$ .", "We show this by induction over $k$ (i.e.", "given $M_k$ , $T_k$ is a valid firing set and its firing produces marking $M_{k+1}$ ).", "Base case: Let $k=0$ , and $M_0$ is a valid marking in $X$ for $PN$ , show [(1)] $T_0$ is a valid firing set for $M_0$ , and $M_1$ is $T_0$ 's target marking w.r.t.", "$M_0$ .", "We show $T_0$ is a valid firing set for $M_0$ .", "Let $\\lbrace fires(t_0,0), \\dots , fires(t_x,0) \\rbrace $ be the set of all $fires(\\dots ,0)$ atoms in $A$ , Then for each $fires(t_i,0) \\in A$ $enabled(t_i,0) \\in A$ – from rule $a\\ref {a:fires}$ and supported rule proposition Then $notenabled(t_i,0) \\notin A$ – from rule $e\\ref {e:enabled}$ and supported rule proposition Then $body(e\\ref {e:r:ne:ptarc} )$ must not hold in $A$ and $body(e\\ref {e:ne:iptarc} )$ must not hold in $A$ – from rules $e\\ref {e:r:ne:ptarc} ,e\\ref {e:ne:iptarc} $ and forced atom proposition Then $q \\lnot < n_i \\equiv q \\ge n_i$ in $e\\ref {e:r:ne:ptarc}$ for all $\\lbrace holds(p,q,0), ptarc(p,t_i,n_i,0)\\rbrace \\subseteq A$ – from $e\\ref {e:r:ne:ptarc}$ , forced atom proposition, and given facts ($holds(p,q,0) \\in A, ptarc(p,t_i,n_i,0) \\in A$ ) And $q \\lnot \\ge n_i \\equiv q < n_i$ in $e\\ref {e:ne:iptarc} $ for all $\\lbrace holds(p,q,0), iptarc(p,t_i,n_i,0) \\rbrace \\subseteq A, n_i=1$ ; $q > n_i \\equiv q = 0$ – from $e\\ref {e:ne:iptarc} $ , forced atom proposition, given facts ($holds(p,q,0) \\in A, iptarc(p,t_i,1,0) \\in A$ ), and $q$ is a positive integer Then $(\\forall p \\in \\bullet t_i, M_0(p) \\ge W(p,t_i)) \\wedge (\\forall p \\in I(t_i), M_0(p) = 0)$ – from $holds(p,q,0) \\in A$ represents $q=M_0(p)$ – rule $i\\ref {i:holds}$ construction $ptarc(p,t_i,n_i,0) \\in A$ represents $n_i=W(p,t_i)$ – rule $f\\ref {f:r:ptarc}$ construction; or it represents $n_i = M_0(p)$ – rule $f\\ref {f:rptarc}$ construction; the construction of $f\\ref {f:rptarc}$ ensures that $notenabled(t,0)$ is never true due to the reset arc definition REF of preset $\\bullet t_i$ in PN Then $t_i$ is enabled and can fire in $PN$ , as a result it can belong to $T_0$ – from definition REF of enabled transition And $consumesmore \\notin A$ , since $A$ is an answer set of $\\Pi ^3$ – from rule $a\\ref {a:overc:elim}$ and supported rule proposition Then $\\nexists consumesmore(p,0) \\in A$ – from rule $a\\ref {a:overc:gen}$ and supported rule proposition Then $\\nexists \\lbrace holds(p,q,0), tot\\_decr(p,q1,0) \\rbrace \\subseteq A : q1>q$ in $body(a\\ref {a:overc:place})$ – from $a\\ref {a:overc:place}$ and forced atom proposition Then $\\nexists p : (\\sum _{t_i \\in \\lbrace t_0,\\dots ,t_x\\rbrace , p \\in \\bullet t_i}{W(p,t_i)}+\\sum _{t_i \\in \\lbrace t_0,\\dots ,t_x\\rbrace , p \\in R(t_i)}{M_0(p)}) > M_0(p)$ – from the following $holds(p,q,0)$ represents $q=M_0(p)$ – from rule $i\\ref {i:holds}$ encoding, given $tot\\_decr(p,q1,0) \\in A$ if $\\lbrace del(p,q1_0,t_0,0), \\dots , del(p,q1_x,t_x,0) \\rbrace \\subseteq A$ , where $q1 = q1_0+\\dots +q1_x$ – from $r\\ref {r:totdecr}$ and forced atom proposition $del(p,q1_i,t_i,0) \\in A$ if $\\lbrace fires(t_i,0), ptarc(p,t_i,q1_i,0) \\rbrace \\subseteq A$ – from $r\\ref {r:r:del} $ and supported rule proposition $del(p,q1_i,t_i,0)$ either represents removal of $q1_i = W(p,t_i)$ tokens from $p \\in \\bullet t_i$ ; or it represents removal of $q1_i = M_0(p)$ tokens from $p \\in R(t_i)$ – from rule $r\\ref {r:r:del} $ , supported rule proposition, and definition REF of transition execution in $PN$ Then the set of transitions in $T_0$ do not conflict – by the definition REF of conflicting transitions And for each $enabled(t_j,0) \\in A$ and $fires(t_j,0) \\notin A$ , $could\\_not\\_have(t_j,0) \\in A$ , since $A$ is an answer set of $\\Pi ^3$ - from rule $a\\ref {a:maxfire:elim}$ and supported rule proposition Then $\\lbrace enabled(t_j,0), holds(s,qq,0), ptarc(s,t_j,q,0), \\\\ tot\\_decr(s,qqq,0) \\rbrace \\subseteq A$ , such that $q > qq - qqq$ and $fires(t_j,0) \\notin A$ - from rule $a\\ref {a:r:maxfire:cnh}$ and supported rule proposition Then for an $s \\in \\bullet t_j \\cup R(t_j)$ , $q > M_0(s) - (\\sum _{t_i \\in T_0, s \\in \\bullet t_i}{W(s,t_i)} + \\sum _{t_i \\in T_0, s \\in R(t_i)}{M_0(s))}$ , where $q=W(s,t_j) \\text{~if~} s \\in \\bullet t_j, \\text{~or~} M_0(s) \\text{~otherwise}$ - from the following: $ptarc(s,t_i,q,0)$ represents $q=W(s,t_i)$ if $(s,t_i) \\in E^-$ or $q=M_0(s)$ if $s \\in R(t_i)$ – from rule $f\\ref {f:r:ptarc},f\\ref {f:rptarc}$ construction $holds(s,qq,0)$ represents $qq=M_0(s)$ – from $i\\ref {i:holds}$ construction $tot\\_decr(s,qqq,0) \\in A$ if $\\lbrace del(s,qqq_0,t_0,0), \\dots , del(s,qqq_x,t_x,0) \\rbrace \\subseteq A$ – from rule $r\\ref {r:totdecr}$ construction and supported rule proposition $del(s,qqq_i,t_i,0) \\in A$ if $\\lbrace fires(t_i,0), ptarc(s,t_i,qqq_i,0) \\rbrace \\subseteq A$ – from rule $r\\ref {r:r:del} $ and supported rule proposition $del(s,qqq_i,t_i,0)$ represents $qqq_i = W(s,t_i) : t_i \\in T_0, (s,t_i) \\in E^-$ , or $qqq_i = M_0(t_i) : t_i \\in T_0, s \\in R(t_i)$ – from rule $f\\ref {f:r:ptarc},f\\ref {f:rptarc}$ construction $tot\\_decr(q,qqq,0)$ represents $\\sum _{t_i \\in T_0, s \\in \\bullet t_i}{W(s,t_i)} + \\\\ \\sum _{t_i \\in T_0, s \\in R(t_i)}{M_0(s)}$ – from (C,D,E) above Then firing $T_0 \\cup \\lbrace t_j \\rbrace $ would have required more tokens than are present at its source place $s \\in \\bullet t_j \\cup R(t_j)$ .", "Thus, $T_0$ is a maximal set of transitions that can simultaneously fire.", "And for each reset transition $t_r$ with $enabled(t_r,0) \\in A$ , $fires(t_r,0) \\in A$ , since $A$ is an answer set of $\\Pi ^2$ - from rule $f\\ref {f:c:rptarc:elim}$ and supported rule proposition Then, the firing set $T_0$ satisfies the reset-transition requirement of definition REF (firing set) Then $\\lbrace t_0, \\dots , t_x\\rbrace = T_0$ – using 1(a),1(b),1(d) above; and using 1(c) it is a maximal firing set We show $M_1$ is produced by firing $T_0$ in $M_0$ .", "Let $holds(p,q,1) \\in A$ Then $\\lbrace holds(p,q1,0), tot\\_incr(p,q2,0), tot\\_decr(p,q3,0) \\rbrace \\subseteq A : q=q1+q2-q3$ – from rule $r\\ref {r:nextstate}$ and supported rule proposition Then $holds(p,q1,0) \\in A$ represents $q1=M_0(p)$ – given, rule $i\\ref {i:holds}$ construction; and $\\lbrace add(p,q2_0,t_0,0), \\dots , $ $add(p,q2_j,t_j,0)\\rbrace \\subseteq A : q2_0 + \\dots + q2_j = q2$ and $\\lbrace del(p,q3_0,t_0,0), \\dots , $ $del(p,q3_l,t_l,0)\\rbrace \\subseteq A : q3_0 + \\dots + q3_l = q3$ – rules $r\\ref {r:totincr},r\\ref {r:totdecr}$ and supported rule proposition, respectively Then $\\lbrace fires(t_0,0), \\dots , fires(t_j,0) \\rbrace \\subseteq A$ and $\\lbrace fires(t_0,0), \\dots , fires(t_l,0) \\rbrace \\\\ \\subseteq A$ – rules $r\\ref {r:r:add},r\\ref {r:r:del}$ and supported rule proposition, respectively Then $\\lbrace fires(t_0,0), \\dots , $ $fires(t_j,0) \\rbrace \\cup \\lbrace fires(t_0,0), \\dots , $ $fires(t_l,0) \\rbrace \\subseteq A = \\lbrace fires(t_0,0), \\dots , $ $fires(t_x,0) \\rbrace \\subseteq A$ – set union of subsets Then for each $fires(t_x,0) \\in A$ we have $t_x \\in T_0$ – already shown in item REF above Then $q = M_0(p) + \\sum _{t_x \\in T_0 \\wedge p \\in t_x \\bullet }{W(t_x,p)} - (\\sum _{t_x \\in T_0 \\wedge p \\in \\bullet t_x}{W(p,t_x)} + \\\\ \\sum _{t_x \\in T_0 \\wedge p \\in R(t_x)}{M_0(p)})$ – from (REF ) above and the following Each $add(p,q_j,t_j,0) \\in A$ represents $q_j=W(t_j,p)$ for $p \\in t_j \\bullet $ – rule $r\\ref {r:r:add} $ encoding, and definition REF of transition execution in $PN$ Each $del(p,t_y,q_y,0) \\in A$ represents either $q_y=W(p,t_y)$ for $p \\in \\bullet t_y$ , or $q_y=M_0(p)$ for $p \\in R(t_y)$ – from rule $r\\ref {r:r:del} ,f\\ref {f:r:ptarc} $ encoding and definition REF of transition execution in $PN$ ; or from rule $r\\ref {r:r:del} ,f\\ref {f:rptarc}$ encoding and definition of reset arc in $PN$ Each $tot\\_incr(p,q2,0) \\in A$ represents $q2=\\sum _{t_x \\in T_0 \\wedge p \\in t_x \\bullet }{W(t_x,p)}$ – aggregate assignment atom semantics in rule $r\\ref {r:totincr}$ Each $tot\\_decr(p,q3,0) \\in A$ represents $q3=\\sum _{t_x \\in T_0 \\wedge p \\in \\bullet t_x}{W(p,t_x)} + $ $\\sum _{t_x \\in T_0 \\wedge p \\in R(t_x)}{M_0(p)}$ – aggregate assignment atom semantics in rule $r\\ref {r:totdecr}$ Then, $M_1(p) = q$ – since $holds(p,q,1) \\in A$ encodes $q=M_1(p)$ – from construction Inductive Step: Let $k > 0$ , and $M_k$ is a valid marking in $X$ for $PN$ , show [(1)] $T_k$ is a valid firing set for $M_k$ , and firing $T_k$ in $M_k$ produces marking $M_{k+1}$ .", "We show that $T_k$ is a valid firing set in $M_k$ .", "Let $\\lbrace fires(t_0,k), \\dots , fires(t_x,k) \\rbrace $ be the set of all $fires(\\dots ,k)$ atoms in $A$ , Then for each $fires(t_i,k) \\in A$ $enabled(t_i,k) \\in A$ – from rule $a\\ref {a:fires}$ and supported rule proposition Then $notenabled(t_i,k) \\notin A$ – from rule $e\\ref {e:enabled}$ and supported rule proposition Then $body(e\\ref {e:r:ne:ptarc} )$ must not hold in $A$ and $body(e\\ref {e:ne:iptarc} )$ must not hold in $A$ – from rule $e\\ref {e:r:ne:ptarc} ,e\\ref {e:ne:iptarc} $ and forced atom proposition Then $q \\lnot < n_i \\equiv q \\ge n_i$ in $e\\ref {e:r:ne:ptarc}$ for all $\\lbrace holds(p,q,k), ptarc(p,t_i,n_i,k)\\rbrace \\subseteq A$ – from $e\\ref {e:r:ne:ptarc}$ , forced atom proposition, and given facts ($holds(p,q,k) \\in A, ptarc(p,t_i,n_i,k) \\in A$ ) And $q \\lnot \\ge n_i \\equiv q < n_i$ in $e\\ref {e:ne:iptarc} $ for all $\\lbrace holds(p,q,k), iptarc(p,t_i,n_i,k) \\rbrace \\subseteq A, n_i=1; q > n_i \\equiv q = 0$ – from $e\\ref {e:ne:iptarc} $ , forced atom proposition, given facts $(holds(p,q,k) \\in A, iptarc(p,t_i,1,k) \\in A)$ , and $q$ is a positive integer Then $(\\forall p \\in \\bullet t_i, M_k(p) \\ge W(p,t_i)) \\wedge (\\forall p \\in I(t_i), M_k(p) = 0)$ – from $holds(p,q,k) \\in A$ represents $q=M_k(p)$ – inductive assumption, given $ptarc(p,t_i,n_i,k) \\in A$ represents $n_i=W(p,t_i)$ – rule $f\\ref {f:r:ptarc}$ construction; or it represents $n_i=M_k(p)$ – rule $f\\ref {f:rptarc}$ construction; the construction of $f\\ref {f:rptarc}$ ensures that $notenabled(t,0)$ is never true due to the reset arc definition REF of preset $\\bullet t_i$ in PN Then $t_i$ is enabled and can fire in $PN$ , as a result it can belong to $T_k$ – from definition REF of enabled transition And $consumesmore \\notin A$ , since $A$ is an answer set of $\\Pi ^3$ – from rule $a\\ref {a:overc:elim}$ and supported rule proposition Then $\\nexists consumesmore(p,k) \\in A$ – from rule $a\\ref {a:overc:gen}$ and supported rule proposition Then $\\nexists \\lbrace holds(p,q,k), tot\\_decr(p,q1,k) \\rbrace \\subseteq A : q1>q$ in $body(a\\ref {a:overc:place})$ – from $a\\ref {a:overc:place}$ and forced atom proposition Then $\\nexists p : (\\sum _{t_i \\in \\lbrace t_0,\\dots ,t_x\\rbrace , p \\in \\bullet t_i}{W(p,t_i)}+\\sum _{t_i \\in \\lbrace t_0,\\dots ,t_x\\rbrace , p \\in R(t_i)}{M_k(p)}) > M_k(p)$ – from the following $holds(p,q,k)$ represents $q=M_k(p)$ – inductive assumption, construction $tot\\_decr(p,q1,k) \\in A$ if $\\lbrace del(p,q1_0,t_0,k), \\dots , del(p,q1_x,t_x,k) \\rbrace \\subseteq A$ , where $q1 = q1_0+\\dots +q1_x$ – from $r\\ref {r:totdecr}$ and forced atom proposition $del(p,q1_i,t_i,k) \\in A$ if $\\lbrace fires(t_i,k), ptarc(p,t_i,q1_i,k) \\rbrace \\subseteq A$ – from $r\\ref {r:r:del} $ and supported rule proposition $del(p,q1_i,t_i,k)$ either represents removal of $q1_i = W(p,t_i)$ tokens from $p \\in \\bullet t_i$ ; or it represents removal of $q1_i = M_k(p)$ tokens from $p \\in R(t_i)$ – from rule $r\\ref {r:r:del} $ , supported rule proposition, and definition REF of transition execution in $PN$ Then $T_k$ does not contain conflicting transitions – by the definition REF of conflicting transitions And for each $enabled(t_j,k) \\in A$ and $fires(t_j,k) \\notin A$ , $could\\_not\\_have(t_j,k) \\in A$ , since $A$ is an answer set of $\\Pi ^3$ - from rule $a\\ref {a:maxfire:elim}$ and supported rule proposition Then $\\lbrace enabled(t_j,k), holds(s,qq,k), ptarc(s,t_j,q,k), \\\\ tot\\_decr(s,qqq,k) \\rbrace \\subseteq A$ , such that $q > qq - qqq$ and $fires(t_j,k) \\notin A$ - from rule $a\\ref {a:r:maxfire:cnh}$ and supported rule proposition Then for an $s \\in \\bullet t_j \\cup R(t_j)$ , $q > M_k(s) - (\\sum _{t_i \\in T_k, s \\in \\bullet t_i}{W(s,t_i)} + \\sum _{t_i \\in T_k, s \\in R(t_i)}{M_k(s))}$ , where $q=W(s,t_j) \\text{~if~} s \\in \\bullet t_j, \\text{~or~} M_k(s) \\text{~otherwise}$ - from the following: $ptarc(s,t_i,q,k)$ represents $q=W(s,t_i)$ if $(s,t_i) \\in E^-$ or $q=M_k(s)$ if $s \\in R(t_i)$ – from rule $f\\ref {f:r:ptarc},f\\ref {f:rptarc}$ construction $holds(s,qq,k)$ represents $qq=M_k(s)$ – construction $tot\\_decr(s,qqq,k) \\in A$ if $\\lbrace del(s,qqq_0,t_0,k), \\dots , del(s,qqq_x,t_x,k) \\rbrace \\\\ \\subseteq A$ – from rule $r\\ref {r:totdecr}$ construction and supported rule proposition $del(s,qqq_i,t_i,k) \\in A$ if $\\lbrace fires(t_i,k), ptarc(s,t_i,qqq_i,k) \\rbrace \\subseteq A$ – from rule $r\\ref {r:r:del} $ and supported rule proposition $del(s,qqq_i,t_i,k)$ represents $qqq_i = W(s,t_i) : t_i \\in T_k, (s,t_i) \\in E^-$ , or $qqq_i = M_k(t_i) : t_i \\in T_k, s \\in R(t_i)$ – from rule $f\\ref {f:r:ptarc},f\\ref {f:rptarc}$ construction $tot\\_decr(q,qqq,k)$ represents $\\sum _{t_i \\in T_k, s \\in \\bullet t_i}{W(s,t_i)} + \\\\ \\sum _{t_i \\in T_k, s \\in R(t_i)}{M_k(s)}$ – from (C,D,E) above Then firing $T_k \\cup \\lbrace t_j \\rbrace $ would have required more tokens than are present at its source place $s \\in \\bullet t_j \\cup R(t_j)$ .", "Thus, $T_k$ is a maximal set of transitions that can simultaneously fire.", "And for each reset transition $t_r$ with $enabled(t_r,k) \\in A$ , $fires(t_r,k) \\in A$ , since $A$ is an answer set of $\\Pi ^2$ - from rule $f\\ref {f:c:rptarc:elim}$ and supported rule proposition Then the firing set $T_k$ satisfies the reset transition requirement of definition REF (firing set) Then $\\lbrace t_0, \\dots , t_x\\rbrace = T_k$ – using 1(a),1(b), 1(d) above; and using 1(c) it is a maximal firing set We show that $M_{k+1}$ is produced by firing $T_k$ in $M_k$ .", "Let $holds(p,q,k+1) \\in A$ Then $\\lbrace holds(p,q1,k), tot\\_incr(p,q2,k), tot\\_decr(p,q3,k) \\rbrace \\subseteq A : q=q1+q2-q3$ – from rule $r\\ref {r:nextstate}$ and supported rule proposition Then, $holds(p,q1,k) \\in A$ represents $q1=M_k(p)$ – inductive assumption, construction ; and $\\lbrace add(p,q2_0,t_0,k), \\dots , $ $add(p,q2_j,t_j,k)\\rbrace \\subseteq A : q2_0 + \\dots + q2_j = q2$ and $\\lbrace del(p,q3_0,t_0,k), \\dots , $ $del(p,q3_l,t_l,k)\\rbrace \\subseteq A : q3_0 + \\dots + q3_l = q3$ – rules $r\\ref {r:totincr},r\\ref {r:totdecr}$ and supported rule proposition, respectively Then $\\lbrace fires(t_0,k), \\dots , fires(t_j,k) \\rbrace \\subseteq A$ and $\\lbrace fires(t_0,k), \\dots , fires(t_l,k) \\rbrace \\\\ \\subseteq A$ – rules $r\\ref {r:r:add},r\\ref {r:r:del}$ and supported rule proposition, respectively Then $\\lbrace fires(t_0,k), \\dots , $ $fires(t_j,k) \\rbrace \\cup \\lbrace fires(t_0,k), \\dots , $ $fires(t_l,k) \\rbrace \\subseteq A = $ $\\lbrace fires(t_0,k), \\dots , $ $fires(t_x,k) \\rbrace \\subseteq A$ – set union of subsets Then for each $fires(t_x,k) \\in A$ we have $t_x \\in T_k$ – already shown in item REF above Then $q = M_k(p) + \\sum _{t_x \\in T_0 \\wedge p \\in t_x \\bullet }{W(t_x,p)} - (\\sum _{t_x \\in T_k \\wedge p \\in \\bullet t_x}{W(p,t_x)} + \\\\ \\sum _{t_x \\in T_k \\wedge p \\in R(t_x)}{M_k(p)})$ – from (REF ) above and the following Each $add(p,q_j,t_j,k) \\in A$ represents $q_j=W(t_j,p)$ for $p \\in t_j \\bullet $ – rule $r\\ref {r:r:add} $ encoding, and definition REF of transition execution in $PN$ Each $del(p,t_y,q_y,k) \\in A$ represents either $q_y=W(p,t_y)$ for $p \\in \\bullet t_y$ , or $q_y=M_k(p)$ for $p \\in R(t_y)$ – from rule $r\\ref {r:r:del} ,f\\ref {f:r:ptarc} $ encoding and definition REF of transition execution in $PN$ ; or from rule $r\\ref {r:r:del} ,f\\ref {f:rptarc}$ encoding and definition of reset arc in $PN$ Each $tot\\_incr(p,q2,k) \\in A$ represents $q2=\\sum _{t_x \\in T_k \\wedge p \\in t_x \\bullet }{W(t_x,p)}$ – aggregate assignment atom semantics in rule $r\\ref {r:totincr}$ Each $tot\\_decr(p,q3,0) \\in A$ represents $q3=\\sum _{t_x \\in T_k \\wedge p \\in \\bullet t_x}{W(p,t_x)} + $ $\\sum _{t_x \\in T_k \\wedge p \\in R(t_x)}{M_k(p)}$ – aggregate assignment atom semantics in rule $r\\ref {r:totdecr}$ Then, $M_{k+1}(p) = q$ – since $holds(p,q,k+1) \\in A$ encodes $q=M_{k+1}(p)$ – from construction As a result, for any $n > k$ , $T_n$ will be a valid firing set for $M_n$ and $M_{n+1}$ will be its target marking.", "Conclusion: Since both (REF ) and (REF ) hold, $X=M_0,T_0,M_1,\\dots ,M_k,T_{k+1}$ is an execution sequence of $PN(P,T,E,W,R)$ (w.r.t $M_0$ ) iff there is an answer set $A$ of $\\Pi ^3(PN,M_0,k,ntok)$ such that (REF ) and (REF ) hold.", "Proof of Proposition REF Let $PN=(P,T,E,W,R,I,Q,QW)$ be a Petri Net, $M_0$ be its initial marking and let $\\Pi ^4(PN,M_0,k,ntok)$ be the ASP encoding of $PN$ and $M_0$ over a simulation length $k$ , with maximum $ntok$ tokens on any place node, as defined in section REF .", "Then $X=M_0,T_0,M_1,\\dots ,M_k,T_k,M_{k+1}$ is an execution sequence of $PN$ (w.r.t.", "$M_0$ ) iff there is an answer set $A$ of $\\Pi ^4(PN,M_0,k,ntok)$ such that: $\\lbrace fires(t,ts) : t \\in T_{ts}, 0 \\le ts \\le k\\rbrace = \\lbrace fires(t,ts) : fires(t,ts) \\in A \\rbrace $ $\\begin{split}\\lbrace holds(p,q,ts) &: p \\in P, q = M_{ts}(p), 0 \\le ts \\le k+1 \\rbrace \\\\&= \\lbrace holds(p,q,ts) : holds(p,q,ts) \\in A \\rbrace \\end{split}$ We prove this by showing that: Given an execution sequence $X$ , we create a set $A$ such that it satisfies (REF ) and (REF ) and show that $A$ is an answer set of $\\Pi ^4$ Given an answer set $A$ of $\\Pi ^4$ , we create an execution sequence $X$ such that (REF ) and (REF ) are satisfied.", "First we show (REF ): Given $PN$ and an execution sequence $X$ of $PN$ , we create a set $A$ as a union of the following sets: $A_1=\\lbrace num(n) : 0 \\le n \\le ntok \\rbrace $ $A_2=\\lbrace time(ts) : 0 \\le ts \\le k\\rbrace $ $A_3=\\lbrace place(p) : p \\in P \\rbrace $ $A_4=\\lbrace trans(t) : t \\in T \\rbrace $ $A_5=\\lbrace ptarc(p,t,n,ts) : (p,t) \\in E^-, n=W(p,t), 0 \\le ts \\le k \\rbrace $ , where $E^- \\subseteq E$ $A_6=\\lbrace tparc(t,p,n,ts) : (t,p) \\in E^+, n=W(t,p), 0 \\le ts \\le k \\rbrace $ , where $E^+ \\subseteq E$ $A_7=\\lbrace holds(p,q,0) : p \\in P, q=M_{0}(p) \\rbrace $ $A_8=\\lbrace notenabled(t,ts) : t \\in T, 0 \\le ts \\le k, (\\exists p \\in \\bullet t, M_{ts}(p) < W(p,t)) \\vee (\\exists p \\in I(t), M_{ts}(p) \\ne 0) \\vee (\\exists (p,t) \\in Q, M_{ts}(p) < QW(p,t)) \\rbrace $ per definition REF (enabled transition) $A_9=\\lbrace enabled(t,ts) : t \\in T, 0 \\le ts \\le k, (\\forall p \\in \\bullet t, W(p,t) \\le M_{ts}(p)) \\wedge (\\forall p \\in I(t), M_{ts}(p) = 0) \\wedge (\\forall (p,t) \\in Q, M_{ts}(p) \\ge QW(p,t)) \\rbrace $ per definition REF (enabled transition) $A_{10}=\\lbrace fires(t,ts) : t \\in T_{ts}, 0 \\le ts \\le k \\rbrace $ per definition REF (enabled transitions), only an enabled transition may fire $A_{11}=\\lbrace add(p,q,t,ts) : t \\in T_{ts}, p \\in t \\bullet , q=W(t,p), 0 \\le ts \\le k \\rbrace $ per definition REF (transition execution) $A_{12}=\\lbrace del(p,q,t,ts) : t \\in T_{ts}, p \\in \\bullet t, q=W(p,t), 0 \\le ts \\le k \\rbrace \\cup \\lbrace del(p,q,t,ts) : t \\in T_{ts}, p \\in R(t), q=M_{ts}(p), 0 \\le ts \\le k \\rbrace $ per definition REF (transition execution) $A_{13}=\\lbrace tot\\_incr(p,q,ts) : p \\in P, q=\\sum _{t \\in T_{ts}, p \\in t \\bullet }{W(t,p)}, 0 \\le ts \\le k \\rbrace $ per definition REF (firing set execution) $A_{14}=\\lbrace tot\\_decr(p,q,ts): p \\in P, q=\\sum _{t \\in T_{ts}, p \\in \\bullet t}{W(p,t)}+\\sum _{t \\in T_{ts}, p \\in R(t) }{M_{ts}(p)}, 0 \\le ts \\le k \\rbrace $ per definition REF (firing set execution) $A_{15}=\\lbrace consumesmore(p,ts) : p \\in P, q=M_{ts}(p), q1=\\sum _{t \\in T_{ts}, p \\in \\bullet t}{W(p,t)} + \\sum _{t \\in T_{ts}, p \\in R(t)}{M_{ts}(p)}, q1 > q, 0 \\le ts \\le k \\rbrace $ per definition REF (conflicting transitions) $A_{16}=\\lbrace consumesmore : \\exists p \\in P : q=M_{ts}(p), q1=\\sum _{t \\in T_{ts}, p \\in \\bullet t}{W(p,t)} + \\sum _{t \\in T_{ts}, p \\in R(t)}(M_{ts}(p)), q1 > q, 0 \\le ts \\le k \\rbrace $ per definition REF (conflicting transitions) $A_{17}=\\lbrace could\\_not\\_have(t,ts) : t \\in T, (\\forall p \\in \\bullet t, W(p,t) \\le M_{ts}(p)), t \\notin T_{ts}, (\\exists p \\in \\bullet t : W(p,t) > M_{ts}(p) - (\\sum _{t^{\\prime } \\in T_{ts}, p \\in \\bullet t^{\\prime }}{W(p,t^{\\prime })} + \\sum _{t^{\\prime } \\in T_{ts}, p \\in R(t^{\\prime })} M_{ts}(p)), 0 \\le ts \\le k \\rbrace $ per the maximal firing set semantics $A_{18}=\\lbrace holds(p,q,ts+1) : p \\in P, q=M_{ts+1}(p), 0 \\le ts < k\\rbrace $ , where $M_{ts+1}(p) = M_{ts}(p) - (\\sum _{\\begin{array}{c}t \\in T_{ts}, p \\in \\bullet t\\end{array}}{W(p,t)} + \\\\ \\sum _{t \\in T_{ts}, p \\in R(t)}M_{ts}(p)) + $ $\\sum _{\\begin{array}{c}t \\in T_{ts}, p \\in t \\bullet \\end{array}}{W(t,p)}$ according to definition REF (firing set execution) $A_{19}=\\lbrace ptarc(p,t,n,ts) : p \\in R(t), n = M_{ts}(p), n > 0, 0 \\le ts \\le k \\rbrace $ $A_{20}=\\lbrace iptarc(p,t,1,ts) : p \\in P, 0 \\le ts < k \\rbrace $ $A_{21}=\\lbrace tptarc(p,t,n,ts) : (p,t) \\in Q, n=QW(p,t), 0 \\le ts \\le k \\rbrace $ We show that $A$ satisfies (REF ) and (REF ), and $A$ is an answer set of $\\Pi ^4$ .", "$A$ satisfies (REF ) and (REF ) by its construction above.", "We show $A$ is an answer set of $\\Pi ^4$ by splitting.", "We split $lit(\\Pi ^4)$ into a sequence of $7k+9$ sets: $U_0= head(f\\ref {f:place}) \\cup head(f\\ref {f:trans}) \\cup head(f\\ref {f:time}) \\cup head(f\\ref {f:num}) \\cup head(i\\ref {i:holds}) = \\lbrace place(p) : p \\in P\\rbrace \\cup \\lbrace trans(t) : t \\in T\\rbrace \\cup \\lbrace time(0), \\dots , time(k)\\rbrace \\cup \\lbrace num(0), \\dots , num(ntok)\\rbrace \\cup \\lbrace holds(p,q,0) : p \\in P, q=M_0(p) \\rbrace $ $U_{7k+1}=U_{7k+0} \\cup head(f\\ref {f:r:ptarc})^{ts=k} \\cup head(f\\ref {f:r:tparc})^{ts=k} \\cup head(f\\ref {f:rptarc})^{ts=k} \\cup head(f\\ref {f:iptarc})^{ts=k} \\cup head(f\\ref {f:tptarc})^{ts=k} = U_{7k+0} \\cup \\lbrace ptarc(p,t,n,k) : (p,t) \\in E^-, n=W(p,t) \\rbrace \\cup \\\\ \\lbrace tparc(t,p,n,k) : (t,p) \\in E^+, n=W(t,p) \\rbrace \\cup $ $\\lbrace ptarc(p,t,n,k) : p \\in R(t), n=M_{k}(p), n > 0 \\rbrace \\cup $ $\\lbrace iptarc(p,t,1,k) : p \\in I(t) \\rbrace \\cup $ $\\lbrace tptarc(p,t,n,k) : (p,t) \\in Q, n=QW(p,t) \\rbrace $ $U_{7k+2}=U_{7k+1} \\cup head(e\\ref {e:r:ne:ptarc} )^{ts=k} = U_{7k+1} \\cup \\lbrace notenabled(t,k) : t \\in T \\rbrace $ $U_{7k+3}=U_{7k+2} \\cup head(e\\ref {e:enabled})^{ts=k} = U_{7k+2} \\cup \\lbrace enabled(t,k) : t \\in T \\rbrace $ $U_{7k+4}=U_{7k+3} \\cup head(a\\ref {a:fires})^{ts=k} = U_{7k+3} \\cup \\lbrace fires(t,k) : t \\in T \\rbrace $ $U_{7k+5}=U_{7k+4} \\cup head(r\\ref {r:r:add} )^{ts=k} \\cup head(r\\ref {r:r:del} )^{ts=k} = U_{7k+4} \\cup \\lbrace add(p,q,t,k) : p \\in P, t \\in T, q=W(t,p) \\rbrace \\cup \\lbrace del(p,q,t,k) : p \\in P, t \\in T, q=W(p,t) \\rbrace \\cup \\lbrace del(p,q,t,k) : p \\in P, t \\in T, q=M_{k}(p) \\rbrace $ $U_{7k+6}=U_{7k+5} \\cup head(r\\ref {r:totincr})^{ts=k} \\cup head(r\\ref {r:totdecr})^{ts=k} = U_{7k+5} \\cup \\lbrace tot\\_incr(p,q,k) : p \\in P, 0 \\le q \\le ntok \\rbrace \\cup \\lbrace tot\\_decr(p,q,k) : p \\in P, 0 \\le q \\le ntok \\rbrace $ $U_{7k+7}=U_{7k+6} \\cup head(r\\ref {r:nextstate})^{ts=k} \\cup head(a\\ref {a:overc:place})^{ts=k} \\cup head(a\\ref {a:r:maxfire:cnh})^{ts=k} = U_{7k+6} \\cup \\\\ \\lbrace consumesmore(p,k) : p \\in P\\rbrace \\cup \\lbrace holds(p,q,k+1) : p \\in P, 0 \\le q \\le ntok \\rbrace \\cup \\lbrace could\\_not\\_have(t,k) : t \\in T \\rbrace $ $U_{7k+8}=U_{7k+7} \\cup head(a\\ref {a:overc:gen}) = U_{7k+7} \\cup \\lbrace consumesmore \\rbrace $ where $head(r_i)^{ts=k}$ are head atoms of ground rule $r_i$ in which $ts=k$ .", "We write $A_i^{ts=k} = \\lbrace a(\\dots ,ts) : a(\\dots ,ts) \\in A_i, ts=k \\rbrace $ as short hand for all atoms in $A_i$ with $ts=k$ .", "$U_{\\alpha }, 0 \\le \\alpha \\le 7k+8$ form a splitting sequence, since each $U_i$ is a splitting set of $\\Pi ^4$ , and $\\langle U_{\\alpha }\\rangle _{\\alpha < \\mu }$ is a monotone continuous sequence, where $U_0 \\subseteq U_1 \\dots \\subseteq U_{8(k+1)}$ and $\\bigcup _{\\alpha < \\mu }{U_{\\alpha }} = lit(\\Pi ^4)$ .", "We compute the answer set of $\\Pi ^4$ using the splitting sets as follows: $bot_{U_0}(\\Pi ^4) = f\\ref {f:place} \\cup f\\ref {f:trans} \\cup f\\ref {f:time} \\cup i\\ref {i:holds} \\cup f\\ref {f:num}$ and $X_0 = A_1 \\cup \\dots \\cup A_4 \\cup A_7$ ($= U_0$ ) is its answer set – using forced atom proposition $eval_{U_0}(bot_{U_1}(\\Pi ^4) \\setminus bot_{U_0}(\\Pi ^4), X_0) = \\lbrace ptarc(p,t,q,0) \\text{:-}.", "\| q=W(p,t) \\rbrace \\cup \\\\ \\lbrace tparc(t,p,q,0) \\text{:-}.", "\| q=W(t,p) \\rbrace \\cup \\lbrace ptarc(p,t,q,0) \\text{:-}.", "\| q=M_0(p) \\rbrace \\cup \\\\ \\lbrace iptarc(p,t,1,0) \\text{:-}.", "\\rbrace \\cup \\lbrace tptarc(p,t,q,0) \\text{:-}.", "\| q = QW(p,t) \\rbrace $ .", "Its answer set $X_1=A_5^{ts=0} \\cup A_6^{ts=0} \\cup A_{19}^{ts=0} \\cup A_{20}^{ts=0} \\cup A_{21}^{ts=0}$ – using forced atom proposition and construction of $A_5, A_6, A_{19}, A_{20}, A_{21}$ .", "$eval_{U_1}(bot_{U_2}(\\Pi ^4) \\setminus bot_{U_1}(\\Pi ^4), X_0 \\cup X_1) = \\lbrace notenabled(t,0) \\text{:-} .", "\| (\\lbrace trans(t), \\\\ ptarc(p,t,n,0), holds(p,q,0) \\rbrace \\subseteq X_0 \\cup X_1, \\text{~where~} q < n) \\text{~or~} (\\lbrace notenabled(t,0) \\text{:-} .", "\| \\\\ (\\lbrace trans(t), iptarc(p,t,n2,0), holds(p,q,0) \\rbrace \\subseteq X_0 \\cup X_1, \\text{~where~} q \\ge n2 \\rbrace ) \\text{~or~} \\\\ (\\lbrace trans(t), tptarc(p,t,n3,0), holds(p,q,0) \\rbrace \\subseteq X_0 \\cup X_1, \\text{~where~} q < n3) \\rbrace $ .", "Its answer set $X_2=A_8^{ts=0}$ – using forced atom proposition and construction of $A_8$ .", "where, $q=M_0(p)$ , and $n=W(p,t)$ for an arc $(p,t) \\in E^-$ – by construction of $i\\ref {i:holds}$ and $f\\ref {f:r:ptarc}$ in $\\Pi ^4$ , and in an arc $(p,t) \\in E^-$ , $p \\in \\bullet t$ (by definition REF of preset) $n2=1$ – by construction of $iptarc$ predicates in $\\Pi ^4$ , meaning $q \\ge n2 \\equiv q \\ge 1 \\equiv q > 0$ , $tptarc(p,t,n3,0)$ represents $n3=QW(p,t)$ , where $(p,t) \\in Q$ thus, $notenabled(t,0) \\in X_1$ represents $(\\exists p \\in \\bullet t : M_0(p) < W(p,t)) \\vee (\\exists p \\in I(t) : M_0(p) > 0) \\vee (\\exists (p,t) \\in Q : M_{ts}(p) < QW(p,t))$ .", "$eval_{U_2}(bot_{U_3}(\\Pi ^4) \\setminus bot_{U_2}(\\Pi ^4), X_0 \\cup \\dots \\cup X_2) = \\lbrace enabled(t,0) \\text{:-}.", "\| trans(t) \\in X_0 \\cup \\dots \\cup X_2, notenabled(t,0) \\notin X_0 \\cup \\dots \\cup X_2 \\rbrace $ .", "Its answer set is $X_3 = A_9^{ts=0}$ – using forced atom proposition and construction of $A_9$ .", "since an $enabled(t,0) \\in X_3$ if $\\nexists ~notenabled(t,0) \\in X_0 \\cup \\dots \\cup X_2$ ; which is equivalent to $(\\nexists p \\in \\bullet t : M_0(p) < W(p,t)) \\wedge (\\nexists p \\in I(t) : M_0(p) > 0) \\wedge (\\nexists (p,t) \\in Q : M_0(p) < QW(p,t) ) \\equiv (\\forall p \\in \\bullet t: M_0(p) \\ge W(p,t)) \\wedge (\\forall p \\in I(t) : M_0(p) = 0)$ .", "$eval_{U_3}(bot_{U_4}(\\Pi ^4) \\setminus bot_{U_3}(\\Pi ^4), X_0 \\cup \\dots \\cup X_3) = \\lbrace \\lbrace fires(t,0)\\rbrace \\text{:-}.", "\| enabled(t,0) \\\\ \\text{~holds in~} X_0 \\cup \\dots \\cup X_3 \\rbrace $ .", "It has multiple answer sets $X_{4.1}, \\dots , X_{4.n}$ , corresponding to elements of power set of $fires(t,0)$ atoms in $eval_{U_3}(...)$ – using supported rule proposition.", "Since we are showing that the union of answer sets of $\\Pi ^4$ determined using splitting is equal to $A$ , we only consider the set that matches the $fires(t,0)$ elements in $A$ and call it $X_4$ , ignoring the rest.", "Thus, $X_4 = A_{10}^{ts=0}$ , representing $T_0$ .", "in addition, for every $t$ such that $enabled(t,0) \\in X_0 \\cup \\dots \\cup X_3, R(t) \\ne \\emptyset $ ; $fires(t,0) \\in X_4$ – per definition REF (firing set); requiring that a reset transition is fired when enabled thus, the firing set $T_0$ will not be eliminated by the constraint $f\\ref {f:c:rptarc:elim}$ $eval_{U_4}(bot_{U_5}(\\Pi ^4) \\setminus bot_{U_4}(\\Pi ^4), X_0 \\cup \\dots \\cup X_4) = \\lbrace add(p,n,t,0) \\text{:-}.", "\| \\lbrace fires(t,0), \\\\ tparc(t,p,n,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_4 \\rbrace \\cup \\lbrace del(p,n,t,0) \\text{:-}.", "\| \\lbrace fires(t,0), ptarc(p,t,n,0) \\rbrace \\\\ \\subseteq X_0 \\cup \\dots \\cup X_4 \\rbrace $ .", "It's answer set is $X_5 = A_{11}^{ts=0} \\cup A_{12}^{ts=0}$ – using forced atom proposition and definitions of $A_{11}$ and $A_{12}$ .", "where, each $add$ atom is equivalent to $n=W(t,p),p \\in t \\bullet $ , and each $del$ atom is equivalent to $n=W(p,t), p \\in \\bullet t$ ; or $n=M_k(p), p \\in R(t)$ , representing the effect of transitions in $T_0$ – by construction $eval_{U_5}(bot_{U_6}(\\Pi ^4) \\setminus bot_{U_5}(\\Pi ^4), X_0 \\cup \\dots \\cup X_5) = \\lbrace tot\\_incr(p,qq,0) \\text{:-}.", "\| $ $qq=\\sum _{add(p,q,t,0) \\in X_0 \\cup \\dots \\cup X_5}{q} \\rbrace \\cup \\lbrace tot\\_decr(p,qq,0) \\text{:-}.", "\| qq=\\sum _{del(p,q,t,0) \\in X_0 \\cup \\dots \\cup X_5}{q} \\rbrace $ .", "It's answer set is $X_6 = A_{13}^{ts=0} \\cup A_{14}^{ts=0}$ – using forced atom proposition and definitions of $A_{13}$ and $A_{14}$ .", "where, each $tot\\_incr(p,qq,0)$ , $qq=\\sum _{add(p,q,t,0) \\in X_0 \\cup \\dots X_5}{q}$ $\\equiv qq=\\sum _{t \\in X_4, p \\in t \\bullet }{W(p,t)}$ , and each $tot\\_decr(p,qq,0)$ , $qq=\\sum _{del(p,q,t,0) \\in X_0 \\cup \\dots X_5}{q}$ $\\equiv qq=\\sum _{t \\in X_4, p \\in \\bullet t}{W(t,p)} + \\sum _{t \\in X_4, p \\in R(t)}{M_{k}(p)}$ , represent the net effect of transitions in $T_0$ – by construction $eval_{U_6}(bot_{U_7}(\\Pi ^4) \\setminus bot_{U_6}(\\Pi ^4), X_0 \\cup \\dots \\cup X_6) = \\lbrace consumesmore(p,0) \\text{:-}.", "\| \\\\ \\lbrace holds(p,q,0), tot\\_decr(p,q1,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_6, q1 > q \\rbrace \\cup \\lbrace holds(p,q,1) \\text{:-}., \| \\\\ \\lbrace holds(p,q1,0), tot\\_incr(p,q2,0), tot\\_decr(p,q3,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_6, q=q1+q2-q3 \\rbrace \\cup \\lbrace could\\_not\\_have(t,0) \\text{:-}.", "\| \\lbrace enabled(t,0), ptarc(s,t,q), holds(s,qq,0), \\\\ tot\\_decr(s,qqq,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_6, fires(t,0) \\notin (X_0 \\cup \\dots \\cup X_6), q > qq-qqq \\rbrace $ .", "It's answer set is $X_7 = A_{15}^{ts=0} \\cup A_{17}^{ts=0} \\cup A_{18}^{ts=0}$ – using forced atom proposition and definitions of $A_{15}, A_{17}, A_{18}, A_9$ .", "where, $consumesmore(p,0)$ represents $\\exists p : q=M_0(p), q1= \\\\ \\sum _{t \\in T_0, p \\in \\bullet t}{W(p,t)}+\\sum _{t \\in T_0, p \\in R(t)}{M_0(p)}, q1 > q$ , indicating place $p$ will be over consumed if $T_0$ is fired, as defined in definition REF (conflicting transitions), $holds(p,q,1)$ represents $q=M_1(p)$ – by construction of $\\Pi ^4$ , and $could\\_not\\_have(t,0)$ represents enabled transition $t \\in T_0$ that could not fire due to insufficient tokens $X_7$ does not contain $could\\_not\\_have(t,0)$ , when $enabled(t,0) \\in X_0 \\cup \\dots \\cup X_6$ and $fires(t,0) \\notin X_0 \\cup \\dots \\cup X_6$ due to construction of $A$ , encoding of $a\\ref {a:r:maxfire:cnh}$ and its body atoms.", "As a result it is not eliminated by the constraint $a\\ref {a:maxfire:elim}$ $ \\vdots $ $eval_{U_{7k+0}}(bot_{U_{7k+1}}(\\Pi ^4) \\setminus bot_{U_{7k+0}}(\\Pi ^4), X_0 \\cup \\dots \\cup X_{7k+0}) = \\lbrace ptarc(p,t,q,k) \\text{:-}.", "\| q=W(p,t) \\rbrace \\cup \\lbrace tparc(t,p,q,k) \\text{:-}.", "\| q=W(t,p) \\rbrace \\cup \\lbrace ptarc(p,t,q,k) \\text{:-}.", "\| q=M_k(p) \\rbrace \\cup \\lbrace iptarc(p,t,1,k) \\text{:-}.", "\\rbrace $ .", "Its answer set $X_{7k+1}=A_5^{ts=k} \\cup A_6^{ts=k} \\cup A_{19}^{ts=k} \\cup A_{20}^{ts=k}$ – using forced atom proposition and construction of $A_5, A_6, A_{19}, A_{20}$ .", "$eval_{U_{7k+1}}(bot_{U_{7k+2}}(\\Pi ^4) \\setminus bot_{U_{7k+1}}(\\Pi ^4), X_0 \\cup \\dots \\cup X_{7k+1}) = \\lbrace notenabled(t,k) \\text{:-} .", "\| $ $(\\lbrace trans(t), ptarc(p,t,n,k), holds(p,q,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{7k+1}, \\text{~where~} q < n) \\text{~or~} \\\\ \\lbrace notenabled(t,k) \\text{:-} .", "\| (\\lbrace trans(t), iptarc(p,t,n2,k), holds(p,q,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{7k+1}, \\\\ \\text{~where~} q \\ge n2 \\rbrace \\rbrace $ .", "Its answer set $X_{7k+2}=A_8^{ts=k}$ – using forced atom proposition and construction of $A_8$ .", "where, $q=M_k(p)$ , and $n=W(p,t)$ for an arc $(p,t) \\in E^-$ – by construction of $holds$ and $ptarc$ predicates in $\\Pi ^4$ , and in an arc $(p,t) \\in E^-$ , $p \\in \\bullet t$ (by definition REF of preset) $n2=1$ – by construction of $iptarc$ predicates in $\\Pi ^4$ , meaning $q \\ge n2 \\equiv q \\ge 1 \\equiv q > 0$ , thus, $notenabled(t,k) \\in X_{7k+1}$ represents $(\\exists p \\in \\bullet t : M_k(p) < W(p,t)) \\vee (\\exists p \\in I(t) : M_k(p) > k)$ .", "$eval_{U_{7k+2}}(bot_{U_{7k+3}}(\\Pi ^4) \\setminus bot_{U_{7k+2}}(\\Pi ^4), X_0 \\cup \\dots \\cup X_{7k+2}) = \\lbrace enabled(t,k) \\text{:-}.", "\| \\\\ trans(t) \\in X_0 \\cup \\dots \\cup X_{7k+2} \\wedge notenabled(t,k) \\notin X_0 \\cup \\dots \\cup X_{7k+2} \\rbrace $ .", "Its answer set is $X_{7k+3} = A_9^{ts=k}$ – using forced atom proposition and construction of $A_9$ .", "since an $enabled(t,k) \\in X_{7k+3}$ if $\\nexists ~notenabled(t,k) \\in X_0 \\cup \\dots \\cup X_{7k+2}$ ; which is equivalent to $(\\nexists p \\in \\bullet t : M_k(p) < W(p,t)) \\wedge (\\nexists p \\in I(t) : M_k(p) > k) \\equiv (\\forall p \\in \\bullet t: M_k(p) \\ge W(p,t)) \\wedge (\\forall p \\in I(t) : M_k(p) = k)$ .", "$eval_{U_{7k+3}}(bot_{U_{7k+4}}(\\Pi ^4) \\setminus bot_{U_{7k+3}}(\\Pi ^4), X_0 \\cup \\dots \\cup X_{7k+3}) = \\lbrace \\lbrace fires(t,k)\\rbrace \\text{:-}.", "\| \\\\ enabled(t,k) \\\\ \\text{~holds in~} X_0 \\cup \\dots \\cup X_{7k+3} \\rbrace $ .", "It has multiple answer sets $X_{7k+4.1}, \\dots , X_{7k+4.n}$ , corresponding to elements of power set of $fires(t,k)$ atoms in $eval_{U_{7k+3}}(...)$ – using supported rule proposition.", "Since we are showing that the union of answer sets of $\\Pi ^4$ determined using splitting is equal to $A$ , we only consider the set that matches the $fires(t,k)$ elements in $A$ and call it $X_{7k+4}$ , ignoring the rest.", "Thus, $X_{7k+4} = A_{10}^{ts=k}$ , representing $T_k$ .", "in addition, for every $t$ such that $enabled(t,k) \\in X_0 \\cup \\dots \\cup X_{7k+3}, R(t) \\ne \\emptyset $ ; $fires(t,k) \\in X_{7k+4}$ – per definition REF (firing set); requiring that a reset transition is fired when enabled thus, the firing set $T_k$ will not be eliminated by the constraint $f\\ref {f:c:rptarc:elim}$ $eval_{U_{7k+4}}(bot_{U_{7k+5}}(\\Pi ^4) \\setminus bot_{U_{7k+4}}(\\Pi ^4), X_0 \\cup \\dots \\cup X_{7k+4}) = \\lbrace add(p,n,t,k) \\text{:-}.", "\| \\\\ \\lbrace fires(t,k), tparc(t,p,n,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{7k+4} \\rbrace \\cup \\lbrace del(p,n,t,k) \\text{:-}.", "\| \\lbrace fires(t,k), \\\\ ptarc(p,t,n,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{7k+4} \\rbrace $ .", "It's answer set is $X_{7k+5} = A_{11}^{ts=k} \\cup A_{12}^{ts=k}$ – using forced atom proposition and definitions of $A_{11}$ and $A_{12}$ .", "where, each $add$ atom is equivalent to $n=W(t,p) : p \\in t \\bullet $ , and each $del$ atom is equivalent to $n=W(p,t) : p \\in \\bullet t$ ; or $n=M_k(p) : p \\in R(t)$ , representing the effect of transitions in $T_k$ $eval_{U_{7k+5}}(bot_{U_{7k+6}}(\\Pi ^4) \\setminus bot_{U_{7k+5}}(\\Pi ^4), X_0 \\cup \\dots \\cup X_{7k+5}) = \\\\ \\lbrace tot\\_incr(p,qq,k) \\text{:-}.", "\| qq=\\sum _{add(p,q,t,k) \\in X_0 \\cup \\dots \\cup X_{7k+5}}{q} \\rbrace \\cup \\\\ \\lbrace tot\\_decr(p,qq,k) \\text{:-}.", "\| qq=\\sum _{del(p,q,t,k) \\in X_0 \\cup \\dots \\cup X_{7k+5}}{q} \\rbrace $ .", "It's answer set is $X_{7k+6} = A_{13}^{ts=k} \\cup A_{14}^{ts=k}$ – using forced atom proposition and definitions of $A_{13}$ and $A_{14}$ .", "where, each $tot\\_incr(p,qq,k)$ , $qq=\\sum _{add(p,q,t,k) \\in X_0 \\cup \\dots X_{7k+5}}{q}$ $\\equiv qq=\\sum _{t \\in X_{7k+4}, p \\in t \\bullet }{W(p,t)}$ , and each $tot\\_decr(p,qq,k)$ , $qq=\\sum _{del(p,q,t,k) \\in X_0 \\cup \\dots X_{7k+5}}{q}$ $\\equiv qq=\\sum _{t \\in X_{7k+4}, p \\in \\bullet t}{W(t,p)} + \\sum _{t \\in X_{7k+4}, p \\in R(t)}{M_{k}(p)}$ , represent the net effect of transition in $T_k$ $eval_{U_{7k+6}}(bot_{U_{7k+7}}(\\Pi ^4) \\setminus bot_{U_{7k+6}}(\\Pi ^4), X_0 \\cup \\dots \\cup X_{7k+6}) = \\lbrace consumesmore(p,k) \\text{:-}.", "\| \\\\ \\lbrace holds(p,q,k), tot\\_decr(p,q1,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{7k+6}, q1 > q \\rbrace \\cup \\lbrace holds(p,q,1) \\text{:-}., \| \\\\ \\lbrace holds(p,q1,k), tot\\_incr(p,q2,k), tot\\_decr(p,q3,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{7k+6}, q=q1+q2-q3 \\rbrace \\cup \\lbrace could\\_not\\_have(t,k) \\text{:-}.", "\| \\lbrace enabled(t,k), ptarc(s,t,q), holds(s,qq,k), \\\\ tot\\_decr(s,qqq,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{7k+6}, fires(t,k) \\notin (X_0 \\cup \\dots \\cup X_{7k+6}), q > qq-qqq \\rbrace $ .", "It's answer set is $X_{7k+7} = A_{15}^{ts=k} \\cup A_{17}^{ts=k} \\cup A_{18}^{ts=k}$ – using forced atom proposition and definitions of $A_{15}, A_{17}, A_{18}, A_9$ .", "where, $consumesmore(p,k)$ represents $\\exists p : q=M_k(p), q1= \\\\ \\sum _{t \\in T_k, p \\in \\bullet t}{W(p,t)}+\\sum _{t \\in T_k, p \\in R(t)}{M_k(p)}, q1 > q$ , indicating place $p$ that will be over consumed if $T_k$ is fired, as defined in definition REF (conflicting transitions), $holds(p,q,k+1)$ represents $q=M_{k+1}(p)$ – by construction of $\\Pi ^4$ , and $could\\_not\\_have(t,k)$ represents enabled transition $t$ in $T_k$ that could not be fired due to insufficient tokens $X_{7k+7}$ does not contain $could\\_not\\_have(t,k)$ , when $enabled(t,k) \\in X_0 \\cup \\dots \\cup X_{7k+6}$ and $fires(t,k) \\notin X_0 \\cup \\dots \\cup X_{7k+6}$ due to construction of $A$ , encoding of $a\\ref {a:r:maxfire:cnh}$ and its body atoms.", "As a result it is not eliminated by the constraint $a\\ref {a:maxfire:elim}$ $eval_{U_{7k+7}}(bot_{U_{7k+8}}(\\Pi ^4) \\setminus bot_{U_{7k+7}}(\\Pi ^4), X_0 \\cup \\dots \\cup X_{7k+7}) = \\lbrace consumesmore \\text{:-}.", "\| \\\\ \\lbrace consumesmore(p,0),\\dots , $ $consumesmore(p,k) \\rbrace \\cap (X_0 \\cup \\dots \\cup X_{7k+7}) \\ne \\emptyset \\rbrace $ .", "It's answer set is $X_{7k+8} = A_{16}$ – using forced atom proposition $X_{7k+8}$ will be empty since none of $consumesmore(p,0),\\dots , \\\\ consumesmore(p,k)$ hold in $X_0 \\cup \\dots \\cup X_{7k+8}$ due to the construction of $A$ , encoding of $a\\ref {a:overc:place}$ and its body atoms.", "As a result, it is not eliminated by the constraint $a\\ref {a:overc:elim}$ The set $X = X_0 \\cup \\dots \\cup X_{7k+8}$ is the answer set of $\\Pi ^4$ by the splitting sequence theorem REF .", "Each $X_i, 0 \\le i \\le 7k+8$ matches a distinct portion of $A$ , and $X = A$ , thus $A$ is an answer set of $\\Pi ^4$ .", "Next we show (REF ): Given $\\Pi ^4$ be the encoding of a Petri Net $PN(P,T,E,W,R,I)$ with initial marking $M_0$ , and $A$ be an answer set of $\\Pi ^4$ that satisfies (REF ) and (REF ), then we can construct $X=M_0,T_0,\\dots ,M_k,T_k,M_{k+1}$ from $A$ , such that it is an execution sequence of $PN$ .", "We construct the $X$ as follows: $M_i = (M_i(p_0), \\dots , M_i(p_n))$ , where $\\lbrace holds(p_0,M_i(p_0),i), \\dots holds(p_n,M_i(p_n),i) \\rbrace \\\\ \\subseteq A$ , for $0 \\le i \\le k+1$ $T_i = \\lbrace t : fires(t,i) \\in A\\rbrace $ , for $0 \\le i \\le k$ and show that $X$ is indeed an execution sequence of $PN$ .", "We show this by induction over $k$ (i.e.", "given $M_k$ , $T_k$ is a valid firing set and its firing produces marking $M_{k+1}$ ).", "Base case: Let $k=0$ , and $M_0$ is a valid marking in $X$ for $PN$ , show [(1)] $T_0$ is a valid firing set for $M_0$ , and firing of $T_0$ in $M_0$ produces marking $M_1$ .", "We show $T_0$ is a valid firing set for $M_0$ .", "Let $\\lbrace fires(t_0,0), \\dots , fires(t_x,0) \\rbrace $ be the set of all $fires(\\dots ,0)$ atoms in $A$ , Then for each $fires(t_i,0) \\in A$ $enabled(t_i,0) \\in A$ – from rule $a\\ref {a:fires}$ and supported rule proposition Then $notenabled(t_i,0) \\notin A$ – from rule $e\\ref {e:enabled}$ and supported rule proposition Then either of $body(e\\ref {e:r:ne:ptarc} )$ , $body(e\\ref {e:ne:iptarc} )$ , or $body(e\\ref {e:ne:tptarc} )$ must not hold in $A$ – from rules $e\\ref {e:r:ne:ptarc} ,e\\ref {e:ne:iptarc} ,e\\ref {e:ne:tptarc} $ and forced atom proposition Then $q \\lnot < n_i \\equiv q \\ge n_i$ in $e\\ref {e:r:ne:ptarc} $ for all $\\lbrace holds(p,q,0), ptarc(p,t_i,n_i,0)\\rbrace \\subseteq A$ – from $e\\ref {e:r:ne:ptarc} $ , forced atom proposition, and given facts ($holds(p,q,0) \\in A, ptarc(p,t_i,n_i,0) \\in A$ ) And $q \\lnot \\ge n_i \\equiv q < n_i$ in $e\\ref {e:ne:iptarc} $ for all $\\lbrace holds(p,q,0), iptarc(p,t_i,n_i,0) \\rbrace \\subseteq A, n_i=1$ ; $q > n_i \\equiv q = 0$ – from $e\\ref {e:ne:iptarc} $ , forced atom proposition, given facts ($holds(p,q,0) \\in A, iptarc(p,t_i,1,0) \\in A$ ), and $q$ is a positive integer And $q \\lnot < n_i \\equiv q \\ge n_i$ in $e\\ref {e:ne:tptarc} $ for all $\\lbrace holds(p,q,0), tptarc(p,t_i,n_i,0) \\rbrace \\subseteq A$ – from $e\\ref {e:ne:tptarc} $ , forced atom proposition, and given facts Then $(\\forall p \\in \\bullet t_i, M_0(p) \\ge W(p,t_i)) \\wedge (\\forall p \\in I(t_i), M_0(p) = 0) \\wedge (\\forall (p,t_i) \\in Q, M_0(p) \\ge QW(p,t_i))$ – from the following $holds(p,q,0) \\in A$ represents $q=M_0(p)$ – rule $i\\ref {i:holds}$ construction $ptarc(p,t_i,n_i,0) \\in A$ represents $n_i=W(p,t_i)$ – rule $f\\ref {f:r:ptarc}$ construction; or it represents $n_i = M_0(p)$ – rule $f\\ref {f:rptarc}$ construction; the construction of $f\\ref {f:rptarc}$ ensures that $notenabled(t,0)$ is never true due to the reset arc definition REF of preset $\\bullet t_i$ in $PN$ definition REF of enabled transition in $PN$ Then $t_i$ is enabled and can fire in $PN$ , as a result it can belong to $T_0$ – from definition REF of enabled transition And $consumesmore \\notin A$ , since $A$ is an answer set of $\\Pi ^4$ – from rule $a\\ref {a:overc:elim}$ and supported rule proposition Then $\\nexists consumesmore(p,0) \\in A$ – from rule $a\\ref {a:overc:gen}$ and supported rule proposition Then $\\nexists \\lbrace holds(p,q,0), tot\\_decr(p,q1,0) \\rbrace \\subseteq A : q1>q$ in $body(a\\ref {a:overc:place})$ – from $a\\ref {a:overc:place}$ and forced atom proposition Then $\\nexists p : (\\sum _{t_i \\in \\lbrace t_0,\\dots ,t_x\\rbrace , p \\in \\bullet t_i}{W(p,t_i)}+\\sum _{t_i \\in \\lbrace t_0,\\dots ,t_x\\rbrace , p \\in R(t_i)}{M_0(p)}) > M_0(p)$ – from the following $holds(p,q,0)$ represents $q=M_0(p)$ – from rule $i\\ref {i:holds}$ construction, given $tot\\_decr(p,q1,0) \\in A$ if $\\lbrace del(p,q1_0,t_0,0), \\dots , del(p,q1_x,t_x,0) \\rbrace \\subseteq A$ , where $q1 = q1_0+\\dots +q1_x$ – from $r\\ref {r:totdecr}$ and forced atom proposition $del(p,q1_i,t_i,0) \\in A$ if $\\lbrace fires(t_i,0), ptarc(p,t_i,q1_i,0) \\rbrace \\subseteq A$ – from $r\\ref {r:r:del} $ and supported rule proposition $del(p,q1_i,t_i,0)$ represents removal of $q1_i = W(p,t_i)$ tokens from $p \\in \\bullet t_i$ ; or it represents removal of $q1_i = M_0(p)$ tokens from $p \\in R(t_i)$ – from rule $r\\ref {r:r:del} $ , supported rule proposition, and definition REF of transition execution in $PN$ Then the set of transitions in $T_0$ do not conflict – by the definition REF of conflicting transitions And for each $enabled(t_j,0) \\in A$ and $fires(t_j,0) \\notin A$ , $could\\_not\\_have(t_j,0) \\in A$ , since $A$ is an answer set of $\\Pi ^4$ - from rule $a\\ref {a:maxfire:elim}$ and supported rule proposition Then $\\lbrace enabled(t_j,0), holds(s,qq,0), ptarc(s,t_j,q,0), \\\\ tot\\_decr(s,qqq,0) \\rbrace \\subseteq A$ , such that $q > qq - qqq$ and $fires(t_j,0) \\notin A$ - from rule $a\\ref {a:r:maxfire:cnh}$ and supported rule proposition Then for an $s \\in \\bullet t_j \\cup R(t_j)$ , $q > M_0(s) - (\\sum _{t_i \\in T_0, s \\in \\bullet t_i}{W(s,t_i)} + \\sum _{t_i \\in T_0, s \\in R(t_i)}{M_0(s))}$ , where $q=W(s,t_j) \\text{~if~} s \\in \\bullet t_j, \\text{~or~} M_0(s) \\text{~otherwise}$ - from the following: $ptarc(s,t_i,q,0)$ represents $q=W(s,t_i)$ if $(s,t_i) \\in E^-$ or $q=M_0(s)$ if $s \\in R(t_i)$ – from rule $f\\ref {f:r:ptarc},f\\ref {f:rptarc}$ construction $holds(s,qq,0)$ represents $qq=M_0(s)$ – from $i\\ref {i:holds}$ construction $tot\\_decr(s,qqq,0) \\in A$ if $\\lbrace del(s,qqq_0,t_0,0), \\dots , del(s,qqq_x,t_x,0) \\rbrace \\subseteq A$ – from rule $r\\ref {r:totdecr}$ construction and supported rule proposition $del(s,qqq_i,t_i,0) \\in A$ if $\\lbrace fires(t_i,0), ptarc(s,t_i,qqq_i,0) \\rbrace \\subseteq A$ – from rule $r\\ref {r:r:del} $ and supported rule proposition $del(s,qqq_i,t_i,0)$ represents $qqq_i = W(s,t_i) : t_i \\in T_0, (s,t_i) \\in E^-$ , or $qqq_i = M_0(t_i) : t_i \\in T_0, s \\in R(t_i)$ – from rule $f\\ref {f:r:ptarc},f\\ref {f:rptarc}$ construction $tot\\_decr(q,qqq,0)$ represents $\\sum _{t_i \\in T_0, s \\in \\bullet t_i}{W(s,t_i)} + \\\\ \\sum _{t_i \\in T_0, s \\in R(t_i)}{M_0(s)}$ – from (C,D,E) above Then firing $T_0 \\cup \\lbrace t_j \\rbrace $ would have required more tokens than are present at its source place $s \\in \\bullet t_j \\cup R(t_j)$ .", "Thus, $T_0$ is a maximal set of transitions that can simultaneously fire.", "And for each reset transition $t_r$ with $enabled(t_r,0) \\in A$ , $fires(t_r,0) \\in A$ , since $A$ is an answer set of $\\Pi ^2$ - from rule $f\\ref {f:c:rptarc:elim}$ and supported rule proposition Then, the firing set $T_0$ satisfies the reset-transition requirement of definition REF (firing set) Then $\\lbrace t_0, \\dots , t_x\\rbrace = T_0$ – using 1(a),1(b),1(d) above; and using 1(c) it is a maximal firing set We show $M_1$ is produced by firing $T_0$ in $M_0$ .", "Let $holds(p,q,1) \\in A$ Then $\\lbrace holds(p,q1,0), tot\\_incr(p,q2,0), tot\\_decr(p,q3,0) \\rbrace \\subseteq A : q=q1+q2-q3$ – from rule $r\\ref {r:nextstate}$ and supported rule proposition Then $holds(p,q1,0) \\in A$ represents $q1=M_0(p)$ – given, rule $i\\ref {i:holds}$ construction; Then $\\lbrace add(p,q2_0,t_0,0), \\dots , add(p,q2_j,t_j,0)\\rbrace \\subseteq A : q2_0 + \\dots + q2_j = q2$ and $\\lbrace del(p,q3_0,t_0,0), \\dots , del(p,q3_l,t_l,0)\\rbrace \\subseteq A : q3_0 + \\dots + q3_l = q3$ – rules $r\\ref {r:totincr},r\\ref {r:totdecr}$ and supported rule proposition, respectively Then $\\lbrace fires(t_0,0), \\dots , fires(t_j,0) \\rbrace \\subseteq A$ and $\\lbrace fires(t_0,0), \\dots , fires(t_l,0) \\rbrace \\\\ \\subseteq A$ – rules $r\\ref {r:r:add},r\\ref {r:r:del}$ and supported rule proposition, respectively Then $\\lbrace fires(t_0,0), \\dots , fires(t_j,0) \\rbrace \\cup \\lbrace fires(t_0,0), \\dots , fires(t_l,0) \\rbrace \\subseteq A = \\lbrace fires(t_0,0), \\dots , $ $fires(t_x,0) \\rbrace \\subseteq A$ – set union of subsets Then for each $fires(t_x,0) \\in A$ we have $t_x \\in T_0$ – already shown in item REF above Then $q = M_0(p) + \\sum _{t_x \\in T_0 \\wedge p \\in t_x \\bullet }{W(t_x,p)} - (\\sum _{t_x \\in T_0 \\wedge p \\in \\bullet t_x}{W(p,t_x)} + \\\\ \\sum _{t_x \\in T_0 \\wedge p \\in R(t_x)}{M_0(p)})$ – from (REF ) above and the following Each $add(p,q_j,t_j,0) \\in A$ represents $q_j=W(t_j,p)$ for $p \\in t_j \\bullet $ – rule $r\\ref {r:r:add} $ encoding, and definition REF of transition execution in $PN$ Each $del(p,t_y,q_y,0) \\in A$ represents either $q_y=W(p,t_y)$ for $p \\in \\bullet t_y$ , or $q_y=M_0(p)$ for $p \\in R(t_y)$ – from rule $r\\ref {r:r:del} ,f\\ref {f:r:ptarc} $ encoding and definition REF of transition execution in $PN$ ; or from rule $r\\ref {r:r:del} ,f\\ref {f:rptarc}$ encoding and definition of reset arc in $PN$ Each $tot\\_incr(p,q2,0) \\in A$ represents $q2=\\sum _{t_x \\in T_0 \\wedge p \\in t_x \\bullet }{W(t_x,p)}$ – aggregate assignment atom semantics in rule $r\\ref {r:totincr}$ Each $tot\\_decr(p,q3,0) \\in A$ represents $q3=\\sum _{t_x \\in T_0 \\wedge p \\in \\bullet t_x}{W(p,t_x)} + \\sum _{t_x \\in T_0 \\wedge p \\in R(t_x)}{M_0(p)}$ – aggregate assignment atom semantics in rule $r\\ref {r:totdecr}$ Then, $M_1(p) = q$ – since $holds(p,q,1) \\in A$ encodes $q=M_1(p)$ – from construction Inductive Step: Let $k > 0$ , and $M_k$ is a valid marking in $X$ for $PN$ , show [(1)] $T_k$ is a valid firing set for $M_k$ , and firing $T_k$ in $M_k$ produces marking $M_{k+1}$ .", "We show that $T_k$ is a valid firing set in $M_k$ .", "Let $\\lbrace fires(t_0,k), \\dots , fires(t_x,k) \\rbrace $ be the set of all $fires(\\dots ,k)$ atoms in $A$ , Then for each $fires(t_i,k) \\in A$ $enabled(t_i,k) \\in A$ – from rule $a\\ref {a:fires}$ and supported rule proposition Then $notenabled(t_i,k) \\notin A$ – from rule $e\\ref {e:enabled}$ and supported rule proposition Then either of $body(e\\ref {e:r:ne:ptarc} )$ , $body(e\\ref {e:ne:iptarc} )$ , or $body(e\\ref {e:ne:tptarc} )$ must not hold in $A$ – from rule $e\\ref {e:r:ne:ptarc} ,e\\ref {e:ne:iptarc} ,e\\ref {e:ne:tptarc} $ and forced atom proposition Then $q \\lnot < n_i \\equiv q \\ge n_i$ in $e\\ref {e:r:ne:ptarc}$ for all $\\lbrace holds(p,q,k), ptarc(p,t_i,n_i,k)\\rbrace \\subseteq A$ – from $e\\ref {e:r:ne:ptarc}$ , forced atom proposition, and given facts ($holds(p,q,k) \\in A, ptarc(p,t_i,n_i,k) \\in A$ ) And $q \\lnot \\ge n_i \\equiv q < n_i$ in $e\\ref {e:ne:iptarc} $ for all $\\lbrace holds(p,q,k), iptarc(p,t_i,n_i,k) \\rbrace \\subseteq A, n_i=1; q > n_i \\equiv q = 0$ – from $e\\ref {e:ne:iptarc} $ , forced atom proposition, given facts $(holds(p,q,k) \\in A, iptarc(p,t_i,1,k) \\in A)$ , and $q$ is a positive integer And $q \\lnot < n_i \\equiv q \\ge n_i$ in $e\\ref {e:ne:tptarc} $ for all $\\lbrace holds(p,q,k), tptarc(p,t_i,n_i,k) \\rbrace \\subseteq A$ – from $e\\ref {e:ne:tptarc} $ , forced atom proposition, and given facts Then $(\\forall p \\in \\bullet t_i, M_k(p) \\ge W(p,t_i)) \\wedge (\\forall p \\in I(t_i), M_k(p) = 0) \\wedge (\\forall (p,t_i) \\in Q, M_k(p) \\ge QW(p,t_i))$ – from $holds(p,q,k) \\in A$ represents $q=M_k(p)$ – inductive assumption, given $ptarc(p,t_i,n_i,k) \\in A$ represents $n_i=W(p,t_i)$ – rule $f\\ref {f:r:ptarc}$ construction; or it represents $n_i=M_k(p)$ – rule $f\\ref {f:rptarc}$ construction; the construction of $f\\ref {f:rptarc}$ ensures that $notenabled(t,k)$ is never true due to the reset arc definition REF of preset $\\bullet t_i$ in $PN$ definition REF of enabled transition in $PN$ Then $t_i$ is enabled and can fire in $PN$ , as a result it can belong to $T_k$ – from definition REF of enabled transition And $consumesmore \\notin A$ , since $A$ is an answer set of $\\Pi ^4$ – from rule $a\\ref {a:overc:elim}$ and supported rule proposition Then $\\nexists consumesmore(p,k) \\in A$ – from rule $a\\ref {a:overc:gen}$ and supported rule proposition Then $\\nexists \\lbrace holds(p,q,k), tot\\_decr(p,q1,k) \\rbrace \\subseteq A : q1>q$ in $body(e\\ref {e:r:ne:ptarc})$ – from $a\\ref {a:overc:place}$ and forced atom proposition Then $\\nexists p : (\\sum _{t_i \\in \\lbrace t_0,\\dots ,t_x\\rbrace , p \\in \\bullet t_i}{W(p,t_i)}+\\sum _{t_i \\in \\lbrace t_0,\\dots ,t_x\\rbrace , p \\in R(t_i)}{M_k(p)}) > M_k(p)$ – from the following $holds(p,q,k)$ represents $q=M_k(p)$ – inductive assumption, given $tot\\_decr(p,q1,k) \\in A$ if $\\lbrace del(p,q1_0,t_0,k), \\dots , del(p,q1_x,t_x,k) \\rbrace \\subseteq A$ , where $q1 = q1_0+\\dots +q1_x$ – from $r\\ref {r:totdecr}$ and forced atom proposition $del(p,q1_i,t_i,k) \\in A$ if $\\lbrace fires(t_i,k), ptarc(p,t_i,q1_i,k) \\rbrace \\subseteq A$ – from $r\\ref {r:r:del} $ and supported rule proposition $del(p,q1_i,t_i,k)$ either represents removal of $q1_i = W(p,t_i)$ tokens from $p \\in \\bullet t_i$ ; or it represents removal of $q1_i = M_k(p)$ tokens from $p \\in R(t_i)$ – from rule $r\\ref {r:r:del} $ , supported rule proposition, and definition REF of transition execution in $PN$ Then $T_k$ does not contain conflicting transitions – by the definition REF of conflicting transitions And for each $enabled(t_j,k) \\in A$ and $fires(t_j,k) \\notin A$ , $could\\_not\\_have(t_j,k) \\in A$ , since $A$ is an answer set of $\\Pi ^4$ - from rule $a\\ref {a:maxfire:elim}$ and supported rule proposition Then $\\lbrace enabled(t_j,k), holds(s,qq,k), ptarc(s,t_j,q,k), \\\\ tot\\_decr(s,qqq,k) \\rbrace \\subseteq A$ , such that $q > qq - qqq$ and $fires(t_j,k) \\notin A$ - from rule $a\\ref {a:r:maxfire:cnh}$ and supported rule proposition Then for an $s \\in \\bullet t_j \\cup R(t_j)$ , $q > M_k(s) - (\\sum _{t_i \\in T_k, s \\in \\bullet t_i}{W(s,t_i)} + \\sum _{t_i \\in T_k, s \\in R(t_i)}{M_k(s))}$ , where $q=W(s,t_j) \\text{~if~} s \\in \\bullet t_j, \\text{~or~} M_k(s) \\text{~otherwise}$ - from the following: $ptarc(s,t_i,q,k)$ represents $q=W(s,t_i)$ if $(s,t_i) \\in E^-$ or $q=M_k(s)$ if $s \\in R(t_i)$ – from rule $f\\ref {f:r:ptarc},f\\ref {f:rptarc}$ construction $holds(s,qq,k)$ represents $qq=M_k(s)$ – construction $tot\\_decr(s,qqq,k) \\in A$ if $\\lbrace del(s,qqq_0,t_0,k), \\dots , del(s,qqq_x,t_x,k) \\rbrace \\\\ \\subseteq A$ – from rule $r\\ref {r:totdecr}$ construction and supported rule proposition $del(s,qqq_i,t_i,k) \\in A$ if $\\lbrace fires(t_i,k), ptarc(s,t_i,qqq_i,k) \\rbrace \\subseteq A$ – from rule $r\\ref {r:r:del} $ and supported rule proposition $del(s,qqq_i,t_i,k)$ represents $qqq_i = W(s,t_i) : t_i \\in T_k, (s,t_i) \\in E^-$ , or $qqq_i = M_k(t_i) : t_i \\in T_k, s \\in R(t_i)$ – from rule $f\\ref {f:r:ptarc},f\\ref {f:rptarc}$ construction $tot\\_decr(q,qqq,k)$ represents $\\sum _{t_i \\in T_k, s \\in \\bullet t_i}{W(s,t_i)} + \\\\ \\sum _{t_i \\in T_k, s \\in R(t_i)}{M_k(s)}$ – from (C,D,E) above Then firing $T_k \\cup \\lbrace t_j \\rbrace $ would have required more tokens than are present at its source place $s \\in \\bullet t_j \\cup R(t_j)$ .", "Thus, $T_k$ is a maximal set of transitions that can simultaneously fire.", "And for each reset transition $t_r$ with $enabled(t_r,k) \\in A$ , $fires(t_r,k) \\in A$ , since $A$ is an answer set of $\\Pi ^2$ - from rule $f\\ref {f:c:rptarc:elim}$ and supported rule proposition Then the firing set $T_k$ satisfies the reset transition requirement of definition REF (firing set) Then $\\lbrace t_0, \\dots , t_x\\rbrace = T_k$ – using 1(a),1(b), 1(d) above; and using 1(c) it is a maximal firing set We show that $M_{k+1}$ is produced by firing $T_k$ in $M_k$ .", "Let $holds(p,q,k+1) \\in A$ Then $\\lbrace holds(p,q1,k), tot\\_incr(p,q2,k), tot\\_decr(p,q3,k) \\rbrace \\subseteq A : q=q1+q2-q3$ – from rule $r\\ref {r:nextstate}$ and supported rule proposition Then $holds(p,q1,k) \\in A$ represents $q1=M_k(p)$ – construction, inductive assumption; and $\\lbrace add(p,q2_0,t_0,k), \\dots , add(p,q2_j,t_j,k)\\rbrace \\subseteq A : q2_0 + \\dots + q2_j = q2$ and $\\lbrace del(p,q3_0,t_0,k), \\dots , del(p,q3_l,t_l,k)\\rbrace \\subseteq A : q3_0 + \\dots + q3_l = q3$ – rules $r\\ref {r:totincr},r\\ref {r:totdecr}$ and supported rule proposition, respectively Then $\\lbrace fires(t_0,k), \\dots , fires(t_j,k) \\rbrace \\subseteq A$ and $\\lbrace fires(t_0,k), \\dots , fires(t_l,k) \\rbrace \\\\ \\subseteq A$ – rules $r\\ref {r:r:add},r\\ref {r:r:del}$ and supported rule proposition, respectively Then $\\lbrace fires(t_0,k), \\dots , $ $fires(t_j,k) \\rbrace \\cup \\lbrace fires(t_0,k), \\dots , $ $fires(t_l,k) \\rbrace \\subseteq A = \\lbrace fires(t_0,k), \\dots , $ $fires(t_x,k) \\rbrace \\subseteq A$ – set union of subsets Then for each $fires(t_x,k) \\in A$ we have $t_x \\in T_k$ – already shown in item REF above Then $q = M_k(p) + \\sum _{t_x \\in T_0 \\wedge p \\in t_x \\bullet }{W(t_x,p)} - (\\sum _{t_x \\in T_k \\wedge p \\in \\bullet t_x}{W(p,t_x)} + \\\\ \\sum _{t_x \\in T_k \\wedge p \\in R(t_x)}{M_k(p)})$ – from (REF ) above and the following Each $add(p,q_j,t_j,k) \\in A$ represents $q_j=W(t_j,p)$ for $p \\in t_j \\bullet $ – rule $r\\ref {r:r:add} $ encoding, and definition REF of transition execution in $PN$ Each $del(p,t_y,q_y,k) \\in A$ represents either $q_y=W(p,t_y)$ for $p \\in \\bullet t_y$ , or $q_y=M_k(p)$ for $p \\in R(t_y)$ – from rule $r\\ref {r:r:del} ,f\\ref {f:r:ptarc} $ encoding and definition REF of transition execution in $PN$ ; or from rule $r\\ref {r:r:del} ,f\\ref {f:rptarc}$ encoding and definition of reset arc in $PN$ Each $tot\\_incr(p,q2,k) \\in A$ represents $q2=\\sum _{t_x \\in T_k \\wedge p \\in t_x \\bullet }{W(t_x,p)}$ – aggregate assignment atom semantics in rule $r\\ref {r:totincr}$ Each $tot\\_decr(p,q3,0) \\in A$ represents $q3=\\sum _{t_x \\in T_k \\wedge p \\in \\bullet t_x}{W(p,t_x)} + \\sum _{t_x \\in T_k \\wedge p \\in R(t_x)}{M_0(p)}$ – aggregate assignment atom semantics in rule $r\\ref {r:totdecr}$ Then, $M_{k+1}(p) = q$ – since $holds(p,q,k+1) \\in A$ encodes $q=M_{k+1}(p)$ – from construction As a result, for any $n > k$ , $T_n$ will be a valid firing set for $M_n$ and $M_{n+1}$ will be its target marking.", "Conclusion: Since both (REF ) and (REF ) hold, $X=M_0,T_0,M_1,\\dots ,M_k,T_{k+1}$ is an execution sequence of $PN(P,T,E,W,R)$ (w.r.t $M_0$ ) iff there is an answer set $A$ of $\\Pi ^4(PN,M_0,k,ntok)$ such that (REF ) and (REF ) hold.", "Proof of Proposition REF Let $PN=(P,T,E,C,W,R,I,Q,QW)$ be a Petri Net, $M_0$ be its initial marking and let $\\Pi ^5(PN,M_0,k,ntok)$ be the ASP encoding of $PN$ and $M_0$ over a simulation length $k$ , with maximum $ntok$ tokens on any place node, as defined in section REF .", "Then $X=M_0,T_k,M_1,\\dots ,M_k,T_k,M_{k+1}$ is an execution sequence of $PN$ (w.r.t.", "$M_0$ ) iff there is an answer set $A$ of $\\Pi ^5(PN,M_0,k,ntok)$ such that: $\\lbrace fires(t,ts) : t \\in T_{ts}, 0 \\le ts \\le k\\rbrace = \\lbrace fires(t,ts) : fires(t,ts) \\in A \\rbrace $ $\\begin{split}\\lbrace holds(p,q,c,ts) &: p \\in P, c/q = M_{ts}(p), 0 \\le ts \\le k+1 \\rbrace \\\\&= \\lbrace holds(p,q,c,ts) : holds(p,q,c,ts) \\in A \\rbrace \\end{split}$ We prove this by showing that: Given an execution sequence $X$ , we create a set $A$ such that it satisfies (REF ) and (REF ) and show that $A$ is an answer set of $\\Pi ^5$ Given an answer set $A$ of $\\Pi ^5$ , we create an execution sequence $X$ such that (REF ) and (REF ) are satisfied.", "First we show (REF ): Given $PN$ and an execution sequence $X$ of $PN$ , we create a set $A$ as a union of the following sets: $A_1=\\lbrace num(n) : 0 \\le n \\le ntok \\rbrace $ $A_2=\\lbrace time(ts) : 0 \\le ts \\le k\\rbrace $ $A_3=\\lbrace place(p) : p \\in P \\rbrace $ $A_4=\\lbrace trans(t) : t \\in T \\rbrace $ $A_5=\\lbrace color(c) : c \\in C \\rbrace $ $A_6=\\lbrace ptarc(p,t,n_c,c,ts) : (p,t) \\in E^-, c \\in C, n_c=m_{W(p,t)}(c), n_c > 0, 0 \\le ts \\le k \\rbrace $ , where $E^- \\subseteq E$ $A_7=\\lbrace tparc(t,p,n_c,c,ts) : (t,p) \\in E^+, c \\in C, n_c=m_{W(t,p)}(c), n_c > 0, 0 \\le ts \\le k \\rbrace $ , where $E^+ \\subseteq E$ $A_8=\\lbrace holds(p,q_c,c,0) : p \\in P, c \\in C, q_c=m_{M_{0}(p)}(c) \\rbrace $ $A_9=\\lbrace ptarc(p,t,n_c,c,ts) : p \\in R(t), c \\in C, n_c = m_{M_{ts}(p)}, n_c > 0, 0 \\le ts \\le k \\rbrace $ $A_{10}=\\lbrace iptarc(p,t,1,c,ts) : p \\in I(t), c \\in C, 0 \\le ts < k \\rbrace $ $A_{11}=\\lbrace tptarc(p,t,n_c,c,ts) : (p,t) \\in Q, c \\in C, n_c=m_{QW(p,t)}(c), n_c > 0, 0 \\le ts \\le k \\rbrace $ $A_{12}=\\lbrace notenabled(t,ts) : t \\in T, 0 \\le ts \\le k, \\exists c \\in C, (\\exists p \\in \\bullet t, m_{M_{ts}(p)}(c) < m_{W(p,t)}(c)) \\vee (\\exists p \\in I(t), m_{M_{ts}(p)}(c) > 0) \\vee (\\exists (p,t) \\in Q, m_{M_{ts}(p)}(c) < m_{QW(p,t)}(c)) \\rbrace $ per definition REF (enabled transition) $A_{13}=\\lbrace enabled(t,ts) : t \\in T, 0 \\le ts \\le k, \\forall c \\in C, (\\forall p \\in \\bullet t, m_{W(p,t)}(c) \\le m_{M_{ts}(p)}(c)) \\wedge (\\forall p \\in I(t), m_{M_{ts}(p)}(c) = 0) \\wedge (\\forall (p,t) \\in Q, m_{M_{ts}(p)}(c) \\ge m_{QW(p,t)}(c)) \\rbrace $ per definition REF (enabled transition) $A_{14}=\\lbrace fires(t,ts) : t \\in T_{ts}, 0 \\le ts \\le k \\rbrace $ per definition REF (firing set), only an enabled transition may fire $A_{15}=\\lbrace add(p,q_c,t,c,ts) : t \\in T_{ts}, p \\in t \\bullet , c \\in C, q_c=m_{W(t,p)}(c), 0 \\le ts \\le k \\rbrace $ per definition REF (transition execution) $A_{16}=\\lbrace del(p,q_c,t,c,ts) : t \\in T_{ts}, p \\in \\bullet t, c \\in C, q_c=m_{W(p,t)}(c), 0 \\le ts \\le k \\rbrace \\cup \\lbrace del(p,q_c,t,c,ts) : t \\in T_{ts}, p \\in R(t), c \\in C, q_c=m_{M_{ts}(p)}(c), 0 \\le ts \\le k \\rbrace $ per definition REF (transition execution) $A_{17}=\\lbrace tot\\_incr(p,q_c,c,ts) : p \\in P, c \\in C, q_c=\\sum _{t \\in T_{ts}, p \\in t \\bullet }{m_{W(t,p)}(c)}, 0 \\le ts \\le k \\rbrace $ per definition REF (firing set execution) $A_{18}=\\lbrace tot\\_decr(p,q_c,c,ts): p \\in P, c \\in C, q_c=\\sum _{t \\in T_{ts}, p \\in \\bullet t}{m_{W(p,t)}(c)}+ \\\\ \\sum _{t \\in T_{ts}, p \\in R(t) }{m_{M_{ts}(p)}(c)}, 0 \\le ts \\le k \\rbrace $ per definition REF (firing set execution) $A_{19}=\\lbrace consumesmore(p,ts) : p \\in P, c \\in C, q_c=m_{M_{ts}(p)}(c), \\\\ q1_c=\\sum _{t \\in T_{ts}, p \\in \\bullet t}{m_{W(p,t)}(c)} + \\sum _{t \\in T_{ts}, p \\in R(t)}{m_{M_{ts}(p)}(c)}, q1 > q, 0 \\le ts \\le k \\rbrace $ per definition REF (conflicting transitions) $A_{20}=\\lbrace consumesmore : \\exists p \\in P, c \\in C, q_c=m_{M_{ts}(p)}(c), \\\\q1_c=\\sum _{t \\in T_{ts}, p \\in \\bullet t}{m_{W(p,t)}(c)} + \\sum _{t \\in T_{ts}, p \\in R(t)}(m_{M_{ts}(p)}(c)), q1 > q, 0 \\le ts \\le k \\rbrace $ per definition REF (conflicting transitions) per the maximal firing set semantics $A_{21}=\\lbrace holds(p,q_c,c,ts+1) : p \\in P, c \\in C, q_c=m_{M_{ts+1}(p)}(c), 0 \\le ts < k\\rbrace $ , where $M_{ts+1}(p) = M_{ts}(p) - (\\sum _{\\begin{array}{c}t \\in T_{ts}, p \\in \\bullet t\\end{array}}{W(p,t)} + \\\\ \\sum _{t \\in T_{ts}, p \\in R(t)}M_{ts}(p)) + \\sum _{\\begin{array}{c}t \\in T_{ts}, p \\in t \\bullet \\end{array}}{W(t,p)}$ according to definition REF (firing set execution) We show that $A$ satisfies (REF ) and (REF ), and $A$ is an answer set of $\\Pi ^5$ .", "$A$ satisfies (REF ) and (REF ) by its construction above.", "We show $A$ is an answer set of $\\Pi ^5$ by splitting.", "We split $lit(\\Pi ^5)$ into a sequence of $7k+9$ sets: $U_0= head(f\\ref {f:c:place}) \\cup head(f\\ref {f:c:trans}) \\cup head(f\\ref {f:c:col}) \\cup head(f\\ref {f:c:time}) \\cup head(f\\ref {f:c:num}) \\cup head(i\\ref {i:c:init}) = \\lbrace place(p) : p \\in P\\rbrace \\cup \\lbrace trans(t) : t \\in T\\rbrace \\cup \\lbrace col(c) : c \\in C \\rbrace \\cup \\lbrace time(0), \\dots , time(k)\\rbrace \\cup \\lbrace num(0), \\dots , num(ntok)\\rbrace \\cup \\lbrace holds(p,q_c,c,0) : p \\in P, c \\in C, q_c=m_{M_0(p)}(c) \\rbrace $ $U_{7k+1}=U_{7k+0} \\cup head(f\\ref {f:c:ptarc})^{ts=k} \\cup head(f\\ref {f:c:tparc})^{ts=k} \\cup head(f\\ref {f:c:rptarc})^{ts=k} \\cup head(f\\ref {f:c:iptarc})^{ts=k} \\cup head(f\\ref {f:c:tptarc})^{ts=k} = U_{7k+0} \\cup \\lbrace ptarc(p,t,n_c,c,k) : (p,t) \\in E^-, c \\in C, n_c=m_{W(p,t)}(c) \\rbrace \\cup \\lbrace tparc(t,p,n_c,c,k) : (t,p) \\in E^+, c \\in C, n_c=m_{W(t,p)}(c) \\rbrace \\cup \\lbrace ptarc(p,t,n_c,c,k) : p \\in R(t), c \\in C, n_c=m_{M_{k}(p)}(c), n > 0 \\rbrace \\cup \\lbrace iptarc(p,t,1,c,k) : p \\in I(t), c \\in C \\rbrace \\cup \\lbrace tptarc(p,t,n_c,c,k) : (p,t) \\in Q, c \\in C, n_c=m_{QW(p,t)}(c) \\rbrace $ $U_{7k+2}=U_{7k+1} \\cup head(e\\ref {e:c:ne:ptarc})^{ts=k} \\cup head(e\\ref {e:c:ne:iptarc})^{ts=k} \\cup head(e\\ref {e:c:ne:tptarc})^{ts=k} = U_{7k+1} \\cup \\\\ \\lbrace notenabled(t,k) : t \\in T \\rbrace $ $U_{7k+3}=U_{7k+2} \\cup head(e\\ref {e:c:enabled})^{ts=k} = U_{7k+2} \\cup \\lbrace enabled(t,k) : t \\in T \\rbrace $ $U_{7k+4}=U_{7k+3} \\cup head(a\\ref {a:c:fires})^{ts=k} = U_{7k+3} \\cup \\lbrace fires(t,k) : t \\in T \\rbrace $ $U_{7k+5}=U_{7k+4} \\cup head(r\\ref {r:c:add})^{ts=k} \\cup head(r\\ref {r:c:del})^{ts=k} = U_{7k+4} \\cup \\lbrace add(p,q_c,t,c,k) : p \\in P, t \\in T, c \\in C, q_c=m_{W(t,p)}(c) \\rbrace \\cup \\lbrace del(p,q_c,t,c,k) : p \\in P, t \\in T, c \\in C, q_c=m_{W(p,t)}(c) \\rbrace \\cup \\lbrace del(p,q_c,t,c,k) : p \\in P, t \\in T, c \\in C, q_c=m_{M_{k}(p)}(c) \\rbrace $ $U_{7k+6}=U_{7k+5} \\cup head(r\\ref {r:c:totincr})^{ts=k} \\cup head(r\\ref {r:c:totdecr})^{ts=k} = U_{7k+5} \\cup \\lbrace tot\\_incr(p,q_c,c,k) : p \\in P, c \\in C, 0 \\le q_c \\le ntok \\rbrace \\cup \\lbrace tot\\_decr(p,q_c,c,k) : p \\in P, c \\in C, 0 \\le q_c \\le ntok \\rbrace $ $U_{7k+7}=U_{7k+6} \\cup head(a\\ref {a:c:overc:place})^{ts=k} \\cup head(r\\ref {r:c:nextstate})^{ts=k} = U_{7k+6} \\cup \\lbrace consumesmore(p,k) : p \\in P\\rbrace \\cup \\lbrace holds(p,q,k+1) : p \\in P, 0 \\le q \\le ntok \\rbrace $ $U_{7k+8}=U_{7k+7} \\cup head(a\\ref {a:c:overc:gen}) = U_{7k+7} \\cup \\lbrace consumesmore \\rbrace $ where $head(r_i)^{ts=k}$ are head atoms of ground rule $r_i$ in which $ts=k$ .", "We write $A_i^{ts=k} = \\lbrace a(\\dots ,ts) : a(\\dots ,ts) \\in A_i, ts=k \\rbrace $ as short hand for all atoms in $A_i$ with $ts=k$ .", "$U_{\\alpha }, 0 \\le \\alpha \\le 7k+8$ form a splitting sequence, since each $U_i$ is a splitting set of $\\Pi ^5$ , and $\\langle U_{\\alpha }\\rangle _{\\alpha < \\mu }$ is a monotone continuous sequence, where $U_0 \\subseteq U_1 \\dots \\subseteq U_{7k+8}$ and $\\bigcup _{\\alpha < \\mu }{U_{\\alpha }} = lit(\\Pi ^5)$ .", "We compute the answer set of $\\Pi ^5$ using the splitting sets as follows: $bot_{U_0}(\\Pi ^5) = f\\ref {f:place} \\cup f\\ref {f:trans} \\cup f\\ref {f:c:ptarc} \\cup f9 \\cup f\\ref {f:c:tptarc} \\cup i\\ref {i:c:init}$ and $X_0 = A_1 \\cup \\dots \\cup A_5 \\cup A_8$ ($= U_0$ ) is its answer set – using forced atom proposition $eval_{U_0}(bot_{U_1}(\\Pi ^5) \\setminus bot_{U_0}(\\Pi ^5), X_0) = \\lbrace ptarc(p,t,q_c,c,0) \\text{:-}.", "\| c \\in C, q_c=m_{W(p,t)}(c), q_c > 0 \\rbrace \\cup \\lbrace tparc(t,p,q_c,c,0) \\text{:-}.", "\| c \\in C, q_c=m_{W(t,p)}(c), q_c > 0 \\rbrace \\cup \\lbrace ptarc(p,t,q_c,c,0) \\text{:-}.", "\| c \\in C, q_c=m_{M_0(p)}(c), q_c > 0 \\rbrace \\cup \\lbrace iptarc(p,t,1,c,0) \\text{:-}.", "\| c \\in C \\rbrace \\cup \\lbrace tptarc(p,t,q_c,c,0) \\text{:-}.", "\| c \\in C, q_c = m_{QW(p,t)}(c), q_c > 0 \\rbrace $ .", "Its answer set $X_1=A_6^{ts=0} \\cup A_7^{ts=0} \\cup A_9^{ts=0} \\cup A_{10}^{ts=0} \\cup A_{11}^{ts=0}$ – using forced atom proposition and construction of $A_6, A_7, A_9, A_{10}, A_{11}$ .", "$eval_{U_1}(bot_{U_2}(\\Pi ^5) \\setminus bot_{U_1}(\\Pi ^5), X_0 \\cup X_1) = \\lbrace notenabled(t,0) \\text{:-} .", "\| (\\lbrace trans(t), \\\\ ptarc(p,t,n_c,c,0), holds(p,q_c,c,0) \\rbrace \\subseteq X_0 \\cup X_1, \\text{~where~} q_c < n_c) \\text{~or~} \\\\ (\\lbrace notenabled(t,0) \\text{:-} .", "\| (\\lbrace trans(t), iptarc(p,t,n2_c,c,0), holds(p,q_c,c,0) \\rbrace \\subseteq X_0 \\cup X_1, \\\\ \\text{~where~} q_c \\ge n2_c \\rbrace ) \\text{~or~} $ $(\\lbrace trans(t), tptarc(p,t,n3_c,c,0), holds(p,q_c,c,0) \\rbrace \\subseteq X_0 \\cup X_1, \\text{~where~} q_c < n3_c) \\rbrace $ .", "Its answer set $X_2=A_{12}^{ts=0}$ – using forced atom proposition and construction of $A_{12}$ .", "where, $q_c=m_{M_0(p)}(c)$ , and $n_c=m_{W(p,t)}(c)$ for an arc $(p,t) \\in E^-$ – by construction of $i\\ref {i:c:init}$ and $f\\ref {f:c:ptarc}$ in $\\Pi ^5$ , and in an arc $(p,t) \\in E^-$ , $p \\in \\bullet t$ (by definition REF of preset) $n2_c=1$ – by construction of $iptarc$ predicates in $\\Pi ^5$ , meaning $q_c \\ge n2_c \\equiv q_c \\ge 1 \\equiv q_c > 0$ , $tptarc(p,t,n3_c,c,0)$ represents $n3_c=m_{QW(p,t)}(c)$ , where $(p,t) \\in Q$ thus, $notenabled(t,0) \\in X_1$ represents $\\exists c \\in C, (\\exists p \\in \\bullet t : m_{M_0(p)}(c) < m_{W(p,t)}(c)) \\vee (\\exists p \\in I(t) : m_{M_0(p)}(c) > 0) \\vee (\\exists (p,t) \\in Q : m_{M_0(p)}(c) < m_{QW(p,t)}(c))$ .", "$eval_{U_2}(bot_{U_3}(\\Pi ^5) \\setminus bot_{U_2}(\\Pi ^5), X_0 \\cup \\dots \\cup X_2) = \\lbrace enabled(t,0) \\text{:-}.", "\| trans(t) \\in X_0 \\cup \\dots \\cup X_2, notenabled(t,0) \\notin X_0 \\cup \\dots \\cup X_2 \\rbrace $ .", "Its answer set is $X_3 = A_{13}^{ts=0}$ – using forced atom proposition and construction of $A_{13}$ .", "where, an $enabled(t,0) \\in X_3$ if $\\nexists ~notenabled(t,0) \\in X_0 \\cup \\dots \\cup X_2$ ; which is equivalent to $\\forall c \\in C, (\\nexists p \\in \\bullet t : m_{M_0(p)}(c) < m_{W(p,t)}(c)) \\wedge (\\nexists p \\in I(t) : m_{M_0(p)}(c) > 0) \\wedge (\\nexists (p,t) \\in Q : m_{M_0(p)}(c) < m_{QW(p,t)}(c) ) \\equiv \\forall c \\in C, (\\forall p \\in \\bullet t: m_{M_0(p)}(c) \\ge m_{W(p,t)}(c)) \\wedge (\\forall p \\in I(t) : m_{M_0(p)}(c) = 0)$ .", "$eval_{U_3}(bot_{U_4}(\\Pi ^5) \\setminus bot_{U_3}(\\Pi ^5), X_0 \\cup \\dots \\cup X_3) = \\lbrace \\lbrace fires(t,0)\\rbrace \\text{:-}.", "\| enabled(t,0) \\\\ \\text{~holds in~} X_0 \\cup \\dots \\cup X_3 \\rbrace $ .", "It has multiple answer sets $X_{4.1}, \\dots , X_{4.n}$ , corresponding to elements of power set of $fires(t,0)$ atoms in $eval_{U_3}(...)$ – using supported rule proposition.", "Since we are showing that the union of answer sets of $\\Pi ^5$ determined using splitting is equal to $A$ , we only consider the set that matches the $fires(t,0)$ elements in $A$ and call it $X_4$ , ignoring the rest.", "Thus, $X_4 = A_{14}^{ts=0}$ , representing $T_0$ .", "in addition, for every $t$ such that $enabled(t,0) \\in X_0 \\cup \\dots \\cup X_3, R(t) \\ne \\emptyset $ ; $fires(t,0) \\in X_4$ – per definition REF (firing set); requiring that a reset transition is fired when enabled thus, the firing set $T_0$ will not be eliminated by the constraint $f\\ref {f:c:rptarc:elim}$ $eval_{U_4}(bot_{U_5}(\\Pi ^5) \\setminus bot_{U_4}(\\Pi ^5), X_0 \\cup \\dots \\cup X_4) = \\lbrace add(p,n_c,t,c,0) \\text{:-}.", "\| \\lbrace fires(t,0), \\\\ tparc(t,p,n_c,c,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_4 \\rbrace \\cup \\lbrace del(p,n_c,t,c,0) \\text{:-}.", "\| \\lbrace fires(t,0), \\\\ ptarc(p,t,n_c,c,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_4 \\rbrace $ .", "It's answer set is $X_5 = A_{15}^{ts=0} \\cup A_{16}^{ts=0}$ – using forced atom proposition and definitions of $A_{15}$ and $A_{16}$ .", "where, each $add$ atom is equivalent to $n_c=m_{W(t,p)}(c),c \\in C, p \\in t \\bullet $ , and each $del$ atom is equivalent to $n_c=m_{W(p,t)}(c), c \\in C, p \\in \\bullet t$ ; or $n_c=m_{M_0(p)}(c), c \\in C, p \\in R(t)$ , representing the effect of transitions in $T_0$ $eval_{U_5}(bot_{U_6}(\\Pi ^5) \\setminus bot_{U_5}(\\Pi ^5), X_0 \\cup \\dots \\cup X_5) = \\lbrace tot\\_incr(p,qq_c,c,0) \\text{:-}.", "\| qq_c=\\sum _{add(p,q_c,t,c,0) \\in X_0 \\cup \\dots \\cup X_5}{q_c} \\rbrace \\cup \\lbrace tot\\_decr(p,qq_c,c,0) \\text{:-}.", "\| qq_c= \\\\ \\sum _{del(p,q_c,t,c,0) \\in X_0 \\cup \\dots \\cup X_5}{q_c} \\rbrace $ .", "It's answer set is $X_6 = A_{17}^{ts=0} \\cup A_{18}^{ts=0}$ – using forced atom proposition and definitions of $A_{17}$ and $A_{18}$ .", "where, each $tot\\_incr(p,qq_c,c,0)$ , $qq_c=\\sum _{add(p,q_c,t,c,0) \\in X_0 \\cup \\dots X_5}{q_c}$ $\\equiv qq_c=\\sum _{t \\in X_4, p \\in t \\bullet }{m_{W(p,t)}(c)}$ , and each $tot\\_decr(p,qq_c,c,0)$ , $qq_c=\\sum _{del(p,q_c,t,c,0) \\in X_0 \\cup \\dots X_5}{q_c}$ $\\equiv qq=\\sum _{t \\in X_4, p \\in \\bullet t}{m_{W(t,p)}(c)} + \\sum _{t \\in X_4, p \\in R(t)}{m_{M_0(p)}(c)}$ , represent the net effect of transitions in $T_0$ $eval_{U_6}(bot_{U_7}(\\Pi ^5) \\setminus bot_{U_6}(\\Pi ^5), X_0 \\cup \\dots \\cup X_6) = \\lbrace consumesmore(p,0) \\text{:-}.", "\| \\\\ \\lbrace holds(p,q_c,c,0), tot\\_decr(p,q1_c,c,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_6, q1_c > q_c \\rbrace \\cup \\\\ \\lbrace holds(p,q_c,c,1) \\text{:-}., \| \\lbrace holds(p,q1_c,c,0), $ $tot\\_incr(p,q2_c,c,0), $ $tot\\_decr(p,q3_c,c,0) \\rbrace \\\\ \\subseteq X_0 \\cup \\dots \\cup X_6, q_c=q1_c+q2_c-q3_c \\rbrace $ .", "It's answer set is $X_7 = A_{19}^{ts=0} \\cup A_{21}^{ts=0}$ – using forced atom proposition and definitions of $A_{19}, A_{21}$ .", "where, $consumesmore(p,0)$ represents $\\exists p : q_c=m_{M_0(p)}(c), q1_c= \\\\ \\sum _{t \\in T_0, p \\in \\bullet t}{m_{W(p,t)}(c)}+\\sum _{t \\in T_0, p \\in R(t)}{m_{M_0(p)}(c)}, q1_c > q_c, c \\in C$ , indicating place $p$ will be over consumed if $T_0$ is fired, as defined in definition REF (conflicting transitions), $holds(p,q_c,c,1)$ represents $q_c=m_{M_1(p)}(c)$ – by construction of $\\Pi ^5$ $ \\vdots $ $eval_{U_{7k+0}}(bot_{U_{7k+1}}(\\Pi ^5) \\setminus bot_{U_{7k+0}}(\\Pi ^5), X_0 \\cup \\dots \\cup X_{7k+0}) = \\lbrace ptarc(p,t,q_c,c,k) \\text{:-}.", "\| c \\in C, q_c=m{W(p,t)}(c), q_c > 0 \\rbrace \\cup \\lbrace tparc(t,p,q_c,c,k) \\text{:-}.", "\| c \\in C, q_c=m_{W(t,p)}(c), q_c > 0 \\rbrace \\cup $ $\\lbrace ptarc(p,t,q_c,c,k) \\text{:-}.", "\| $ $c \\in C, q_c=m_{M_k(p)}(c), q_c > 0 \\rbrace \\cup \\lbrace iptarc(p,t,1,c,k) \\text{:-}.", "\| c \\in C \\rbrace \\cup \\lbrace tptarc(p,t,q_c,c,k) \\text{:-}.", "\| c \\in C, q_c = m_{QW(p,t)}(c), q_c > 0 \\rbrace $ .", "Its answer set $X_{7k+1}=A_6^{ts=k} \\cup A_7^{ts=k} \\cup A_9^{ts=k} \\cup A_{10}^{ts=k} \\cup A_{11}^{ts=k}$ – using forced atom proposition and construction of $A_6, A_7, A_9, A_{10}, A_{11}$ .", "$eval_{U_{7k+1}}(bot_{U_{7k+2}}(\\Pi ^5) \\setminus bot_{U_{7k+1}}(\\Pi ^5), X_0 \\cup \\dots \\cup X_{7k+1}) = \\lbrace notenabled(t,k) \\text{:-} .", "\| \\\\ (\\lbrace trans(t), ptarc(p,t,n_c,c,k), holds(p,q_c,c,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{7k+1}, \\text{~where~} q_c < n_c) \\text{~or~} $ $ (\\lbrace notenabled(t,k) \\text{:-} .", "\| (\\lbrace trans(t), iptarc(p,t,n2_c,c,k), holds(p,q_c,c,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{7k+1}, \\text{~where~} q_c \\ge n2_c \\rbrace ) \\text{~or~} (\\lbrace trans(t), tptarc(p,t,n3_c,c,k), \\\\ holds(p,q_c,c,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{7k+1}, \\text{~where~} q_c < n3_c) \\rbrace $ .", "Its answer set $X_{7k+2}=A_{12}^{ts=k}$ – using forced atom proposition and construction of $A_{12}$ .", "since $q_c=m_{M_k(p)}(c)$ , and $n_c=m_{W(p,t)}(c)$ for an arc $(p,t) \\in E^-$ – by construction of $holds$ and $ptarc$ predicates in $\\Pi ^5$ , and in an arc $(p,t) \\in E^-$ , $p \\in \\bullet t$ (by definition REF of preset) $n2_c=1$ – by construction of $iptarc$ predicates in $\\Pi ^5$ , meaning $q_c \\ge n2_c \\equiv q_c \\ge 1 \\equiv q_c > 0$ , $tptarc(p,t,n3_c,c,k)$ represents $n3_c=m_{QW(p,t)}(c)$ , where $(p,t) \\in Q$ thus, $notenabled(t,k) \\in X_{8k+1}$ represents $\\exists c \\in C (\\exists p \\in \\bullet t : m_{M_k(p)}(c) < m_{W(p,t)}(c)) \\vee (\\exists p \\in I(t) : m_{M_k(p)}(c) > 0) \\vee (\\exists (p,t) \\in Q : m_{M_{ts}(p)}(c) < m_{QW(p,t)}(c))$ .", "$eval_{U_{7k+2}}(bot_{U_{7k+3}}(\\Pi ^5) \\setminus bot_{U_{7k+2}}(\\Pi ^5), X_0 \\cup \\dots \\cup X_{7k+2}) = \\lbrace enabled(t,k) \\text{:-}.", "\| trans(t) \\in X_0 \\cup \\dots \\cup X_{7k+2} \\wedge notenabled(t,k) \\notin X_0 \\cup \\dots \\cup X_{7k+2} \\rbrace $ .", "Its answer set is $X_{7k+3} = A_{13}^{ts=k}$ – using forced atom proposition and construction of $A_{13}$ .", "since an $enabled(t,k) \\in X_{7k+3}$ if $\\nexists ~notenabled(t,k) \\in X_0 \\cup \\dots \\cup X_{7k+2}$ ; which is equivalent to $\\forall c \\in C, (\\nexists p \\in \\bullet t : m_{M_k(p)}(c) < m_{W(p,t)}(c)) \\wedge (\\nexists p \\in I(t) : m_{M_k(p)}(c) > 0) \\wedge (\\nexists (p,t) \\in Q : m_{M_k(p)}(c) < m_{QW(p,t)}(c) ) \\equiv \\forall c \\in C, (\\forall p \\in \\bullet t: m_{M_k(p)}(c) \\ge m_{W(p,t)}(c)) \\wedge (\\forall p \\in I(t) : m_{M_k(p)}(c) = 0)$ .", "$eval_{U_{7k+3}}(bot_{U_{7k+4}}(\\Pi ^5) \\setminus bot_{U_{7k+3}}(\\Pi ^5), X_0 \\cup \\dots \\cup X_{7k+3}) = \\lbrace \\lbrace fires(t,k)\\rbrace \\text{:-}.", "\| \\\\ enabled(t,k) \\text{~holds in~} X_0 \\cup \\dots \\cup X_{7k+3} \\rbrace $ .", "It has multiple answer sets $X_{7k+}{4.1}, \\dots , X_{7k+}{4.n}$ , corresponding to elements of power set of $fires(t,k)$ atoms in $eval_{U_3}(...)$ – using supported rule proposition.", "Since we are showing that the union of answer sets of $\\Pi ^5$ determined using splitting is equal to $A$ , we only consider the set that matches the $fires(t,k)$ elements in $A$ and call it $X_{7k+4}$ , ignoring the rest.", "Thus, $X_{7k+4} = A_{14}^{ts=k}$ , representing $T_k$ .", "in addition, for every $t$ such that $enabled(t,k) \\in X_0 \\cup \\dots \\cup X_{7k+3}, R(t) \\ne \\emptyset $ ; $fires(t,k) \\in X_{7k+4}$ – per definition REF (firing set); requiring that a reset transition is fired when enabled thus, the firing set $T_k$ will not be eliminated by the constraint $f\\ref {f:c:rptarc:elim}$ $eval_{U_{7k+4}}(bot_{U_{7k+5}}(\\Pi ^5) \\setminus bot_{U_{7k+4}}(\\Pi ^5), X_0 \\cup \\dots \\cup X_{7k+4}) = \\lbrace add(p,n_c,t,c,k) \\text{:-}.", "\| $ $ \\lbrace fires(t,k), \\\\ tparc(t,p,n_c,c,k) \\rbrace \\subseteq $ $X_0 \\cup \\dots \\cup X_{7k+4} \\rbrace \\cup $ $\\lbrace del(p,n_c,t,c,k) \\text{:-}.", "\| $ $\\lbrace fires(t,k), \\\\ ptarc(p,t,n_c,c,k) \\rbrace \\subseteq $ $X_0 \\cup \\dots \\cup X_{7k+4} \\rbrace $ .", "It's answer set is $X_{7k+5} = A_{15}^{ts=k} \\cup A_{16}^{ts=k}$ – using forced atom proposition and definitions of $A_{15}$ and $A_{16}$ .", "where each $add$ atom is equivalent to $n_c=m_{W(t,p)}(c),c \\in C, p \\in t \\bullet $ , and each $del$ atom is equivalent to $n_c=m_{W(p,t)}(c), c \\in C, p \\in \\bullet t$ ; or $n_c=m_{M_k(p)}(c), c \\in C, p \\in R(t)$ , representing the effect of transitions in $T_k$ $eval_{U_{7k+5}}(bot_{U_{7k+6}}(\\Pi ^5) \\setminus bot_{U_{7k+5}}(\\Pi ^5), X_0 \\cup \\dots \\cup X_{7k+5}) = \\lbrace tot\\_incr(p,qq_c,c,k) \\text{:-}.", "\| $ $qq=\\sum _{add(p,q_c,t,c,k) \\in X_0 \\cup \\dots \\cup X_{7k+5}}{q_c} \\rbrace \\cup $ $ \\lbrace tot\\_decr(p,qq_c,c,k) \\text{:-}.", "\| \\\\ qq_c=\\sum _{del(p,q_c,t,c,k) \\in X_0 \\cup \\dots \\cup X_{7k+5}}{q_c} \\rbrace $ .", "It's answer set is $X_{7k+6} = A_{17}^{ts=k} \\cup A_{18}^{ts=k}$ – using forced atom proposition and definitions of $A_{17}$ and $A_{18}$ .", "where, each $tot\\_incr(p,qq_c,c,k)$ , $qq_c=\\sum _{add(p,q_c,t,c,k) \\in X_0 \\cup \\dots X_{8k+5}}{q_c}$ $\\equiv qq_c=\\sum _{t \\in X_{7k+4}, p \\in t \\bullet }{m_{W(p,t)}(c)}$ , and each $tot\\_decr(p,qq_c,c,k)$ , $qq_c=\\sum _{del(p,q_c,t,c,k) \\in X_0 \\cup \\dots X_{8k+5}}{q_c}$ $\\equiv qq=\\sum _{t \\in X_{7k+4}, p \\in \\bullet t}{m_{W(t,p)}(c)} + \\sum _{t \\in X_{7k+4}, p \\in R(t)}{m_{M_{k}(p)}(c)}$ , represent the net effect of transitions in $T_k$ $eval_{U_{7k+6}}(bot_{U_{7k+7}}(\\Pi ^5) \\setminus bot_{U_{7k+6}}(\\Pi ^5), X_0 \\cup \\dots \\cup X_{7k+6}) = \\lbrace consumesmore(p,k) \\text{:-}.", "\| \\\\ \\lbrace holds(p,q_c,c,k), tot\\_decr(p,q1_c,c,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{7k+6}, q1 > q \\rbrace \\cup \\\\ \\lbrace holds(p,q_c,c,k+1) \\text{:-}.", "\| \\lbrace holds(p,q1_c,c,k), tot\\_incr(p,q2_c,c,k), \\\\ tot\\_decr(p,q3_c,c,k) \\rbrace \\subseteq $ $X_0 \\cup \\dots \\cup X_{7k+6}, q_c=q1_c+q2_c-q3_c \\rbrace $ .", "It's answer set is $X_{7k+7} = A_{19}^{ts=k} \\cup A_{21}^{ts=k}$ – using forced atom proposition and definitions of $A_{19}, A_{21}$ .", "where, $consumesmore(p,k)$ represents $\\exists p : q_c=m_{M_k(p)}(c), q1_c= \\\\ \\sum _{t \\in T_k, p \\in \\bullet t}{m_{W(p,t)}(c)}+\\sum _{t \\in T_k, p \\in R(t)}{m_{M_k(p)}(c)}, q1_c > q_c, c \\in C$ , indicating place $p$ that will be over consumed if $T_k$ is fired, as defined in definition REF (conflicting transitions), and $holds(p,q_c,c,k+1)$ represents $q_c=m_{M_{k+1}(p)}(c)$ – by construction of $\\Pi ^5$ $eval_{U_{7k+7}}(bot_{U_{7k+8}}(\\Pi ^5) \\setminus bot_{U_{7k+7}}(\\Pi ^5), X_0 \\cup \\dots \\cup X_{7k+7}) = \\lbrace consumesmore \\text{:-}.", "\| \\\\ \\lbrace consumesmore(p,0),\\dots ,$ $consumesmore(p,k) \\rbrace \\cap (X_0 \\cup \\dots \\cup X_{7k+7}) \\ne \\emptyset \\rbrace $ .", "It's answer set is $X_{7k+8} = A_{20}^{ts=k}$ – using forced atom proposition and the definition of $A_{20}$ $X_{7k+7}$ will be empty since none of $consumesmore(p,0),\\dots , \\\\ consumesmore(p,k)$ hold in $X_0 \\cup \\dots \\cup X_{7k+7}$ due to the construction of $A$ , encoding of $a\\ref {a:c:overc:place}$ and its body atoms.", "As a result, it is not eliminated by the constraint $a\\ref {a:c:overc:elim}$ The set $X = X_0 \\cup \\dots \\cup X_{7k+8}$ is the answer set of $\\Pi ^5$ by the splitting sequence theorem REF .", "Each $X_i, 0 \\le i \\le 7k+8$ matches a distinct portion of $A$ , and $X = A$ , thus $A$ is an answer set of $\\Pi ^5$ .", "Next we show (REF ): Given $\\Pi ^5$ be the encoding of a Petri Net $PN(P,T,E,C,W,R,I,$ $Q,WQ)$ with initial marking $M_0$ , and $A$ be an answer set of $\\Pi ^5$ that satisfies (REF ) and (REF ), then we can construct $X=M_0,T_k,\\dots ,M_k,T_k,M_{k+1}$ from $A$ , such that it is an execution sequence of $PN$ .", "We construct the $X$ as follows: $M_i = (M_i(p_0), \\dots , M_i(p_n))$ , where $\\lbrace holds(p_0,m_{M_i(p_0)}(c),c,i), \\dots \\\\ holds(p_n,m_{M_i(p_n)}(c),c,i) \\rbrace \\subseteq A$ , for $c \\in C, 0 \\le i \\le k+1$ $T_i = \\lbrace t : fires(t,i) \\in A\\rbrace $ , for $0 \\le i \\le k$ and show that $X$ is indeed an execution sequence of $PN$ .", "We show this by induction over $k$ (i.e.", "given $M_k$ , $T_k$ is a valid firing set and its firing produces marking $M_{k+1}$ ).", "Base case: Let $k=0$ , and $M_0$ is a valid marking in $X$ for $PN$ , show [(1)] $T_0$ is a valid firing set for $M_0$ , and firing $T_0$ in $M_0$ results in marking $M_1$ .", "We show $T_0$ is a valid firing set for $M_0$ .", "Let $\\lbrace fires(t_0,0), \\dots , fires(t_x,0) \\rbrace $ be the set of all $fires(\\dots ,0)$ atoms in $A$ , Then for each $fires(t_i,0) \\in A$ $enabled(t_i,0) \\in A$ – from rule $a\\ref {a:c:fires}$ and supported rule proposition Then $notenabled(t_i,0) \\notin A$ – from rule $e\\ref {e:c:enabled}$ and supported rule proposition Then either of $body(e\\ref {e:c:ne:ptarc})$ , $body(e\\ref {e:c:ne:iptarc})$ , or $body(e\\ref {e:c:ne:tptarc})$ must not hold in $A$ – from rules $e\\ref {e:c:ne:ptarc},e\\ref {e:c:ne:iptarc},e\\ref {e:c:ne:tptarc}$ and forced atom proposition Then $q_c \\lnot < {n_i}_c \\equiv q_c \\ge {n_i}_c$ in $e\\ref {e:c:ne:ptarc}$ for all $\\lbrace holds(p,q_c,c,0), \\\\ ptarc(p,t_i,{n_i}_c,c,0)\\rbrace \\subseteq A$ – from $e\\ref {e:c:ne:ptarc}$ , forced atom proposition, and given facts ($holds(p,q_c,c,0) \\in A, ptarc(p,t_i,{n_i}_c,0) \\in A$ ) And $q_c \\lnot \\ge {n_i}_c \\equiv q_c < {n_i}_c$ in $e\\ref {e:c:ne:iptarc}$ for all $\\lbrace holds(p,q_c,c,0), \\\\ iptarc(p,t_i,{n_i}_c,c,0) \\rbrace \\subseteq A, {n_i}_c=1$ ; $q_c > {n_i}_c \\equiv q_c = 0$ – from $e\\ref {e:c:ne:iptarc}$ , forced atom proposition, given facts ($holds(p,q_c,c,0) \\in A, iptarc(p,t_i,1,c,0) \\in A$ ), and $q_c$ is a positive integer And $q_c \\lnot < {n_i}_c \\equiv q_c \\ge {n_i}_c$ in $e\\ref {e:c:ne:tptarc}$ for all $\\lbrace holds(p,q_c,c,0), \\\\ tptarc(p,t_i,{n_i}_c,c,0) \\rbrace \\subseteq A$ – from $e\\ref {e:c:ne:tptarc}$ , forced atom proposition, and given facts Then $\\forall c \\in C, (\\forall p \\in \\bullet t_i, m_{M_0(p)}(c) \\ge m_{W(p,t_i)}(c)) \\wedge (\\forall p \\in I(t_i), m_{M_0(p)}(c) = 0) \\wedge (\\forall (p,t_i) \\in Q, m_{M_0(p)}(c) \\ge m_{QW(p,t_i)}(c))$ – from the following $holds(p,q_c,c,0) \\in A$ represents $q_c=m_{M_0(p)}(c)$ – rule $i\\ref {i:c:init}$ construction $ptarc(p,t_i,{n_i}_c,0) \\in A$ represents ${n_i}_c=m_{W(p,t_i)}(c)$ – rule $f\\ref {f:c:ptarc}$ construction; or it represents ${n_i}_c = m_{M_0(p)}(c)$ – rule $f\\ref {f:c:rptarc}$ construction; the construction of $f\\ref {f:c:rptarc}$ ensures that $notenabled(t,0)$ is never true for a reset arc definition REF of preset $\\bullet t_i$ in $PN$ definition REF of enabled transition in $PN$ Then $t_i$ is enabled and can fire in $PN$ , as a result it can belong to $T_0$ – from definition REF of enabled transition And $consumesmore \\notin A$ , since $A$ is an answer set of $\\Pi ^5$ – from rule $a\\ref {a:c:overc:gen}$ and supported rule proposition Then $\\nexists consumesmore(p,0) \\in A$ – from rule $a\\ref {a:c:overc:place}$ and supported rule proposition Then $\\nexists \\lbrace holds(p,q_c,c,0), tot\\_decr(p,q1_c,c,0) \\rbrace \\subseteq A, q1_c>q_c$ in $body(a\\ref {a:c:overc:place})$ – from $a\\ref {a:c:overc:place}$ and forced atom proposition Then $\\nexists c \\in C \\nexists p \\in P, (\\sum _{t_i \\in \\lbrace t_0,\\dots ,t_x\\rbrace , p \\in \\bullet t_i}{m_{W(p,t_i)}(c)}+ \\\\ \\sum _{t_i \\in \\lbrace t_0,\\dots ,t_x\\rbrace , p \\in R(t_i)}{m_{M_0(p)}(c)}) > m_{M_0(p)}(c)$ – from the following $holds(p,q_c,c,0)$ represents $q_c=m_{M_0(p)}(c)$ – from rule $i\\ref {i:c:init}$ construction, given $tot\\_decr(p,q1_c,c,0) \\in A$ if $\\lbrace del(p,{q1_0}_c,t_0,c,0), \\dots , \\\\ del(p,{q1_x}_c,t_x,c,0) \\rbrace \\subseteq A$ , where $q1_c = {q1_0}_c+\\dots +{q1_x}_c$ – from $r\\ref {r:c:totdecr}$ and forced atom proposition $del(p,{q1_i}_c,t_i,c,0) \\in A$ if $\\lbrace fires(t_i,0), ptarc(p,t_i,{q1_i}_c,c,0) \\rbrace \\subseteq A$ – from $r\\ref {r:c:del}$ and supported rule proposition $del(p,{q1_i}_c,t_i,c,0)$ either represents removal of ${q1_i}_c = m_{W(p,t_i)}(c)$ tokens from $p \\in \\bullet t_i$ ; or it represents removal of ${q1_i}_c = m_{M_0(p)}(c)$ tokens from $p \\in R(t_i)$ – from rules $r\\ref {r:c:del},f\\ref {f:c:ptarc},f\\ref {f:c:rptarc}$ , supported rule proposition, and definition REF of transition execution in $PN$ Then the set of transitions in $T_0$ do not conflict – by the definition REF of conflicting transitions And for each reset transition $t_r$ with $enabled(t_r,0) \\in A$ , $fires(t_r,0) \\in A$ , since $A$ is an answer set of $\\Pi ^5$ - from rule $f\\ref {f:c:rptarc:elim}$ and supported rule proposition Then, the firing set $T_0$ satisfies the reset-transition requirement of definition REF (firing set) Then $\\lbrace t_0, \\dots , t_x\\rbrace = T_0$ – using 1(a),1(b),1(d) above; and using 1(c) it is a maximal firing set We show $M_1$ is produced by firing $T_0$ in $M_0$ .", "Let $holds(p,q_c,c,1) \\in A$ Then $\\lbrace holds(p,q1_c,c,0), tot\\_incr(p,q2_c,c,0), tot\\_decr(p,q3_c,c,0) \\rbrace \\subseteq A : q_c=q1_c+q2_c-q3_c$ – from rule $r\\ref {r:c:nextstate}$ and supported rule proposition Then, $holds(p,q1_c,c,0) \\in A$ represents $q1_c=m_{M_0(p)}(c)$ – given, rule $i\\ref {i:c:init}$ construction; and $\\lbrace add(p,{q2_0}_c,t_0,c,0), \\dots , $ $add(p,{q2_j}_c,t_j,c,0)\\rbrace \\subseteq A : {q2_0}_c + \\dots + {q2_j}_c = q2_c$ and $\\lbrace del(p,{q3_0}_c,t_0,c,0), \\dots , $ $del(p,{q3_l}_c,t_l,c,0)\\rbrace \\subseteq A : {q3_0}_c + \\dots + {q3_l}_c = q3_c$ – rules $r\\ref {r:c:totincr},r\\ref {r:c:totdecr}$ and supported rule proposition, respectively Then $\\lbrace fires(t_0,0), \\dots , fires(t_j,0) \\rbrace \\subseteq A$ and $\\lbrace fires(t_0,0), \\dots , fires(t_l,0) \\rbrace $ $\\subseteq A$ – rules $r\\ref {r:c:add},r\\ref {r:c:del}$ and supported rule proposition, respectively Then $\\lbrace fires(t_0,0), \\dots , $ $fires(t_j,0) \\rbrace \\cup $ $\\lbrace fires(t_0,0), \\dots , $ $fires(t_l,0) \\rbrace \\subseteq $ $A = \\lbrace fires(t_0,0), \\dots , $ $fires(t_x,0) \\rbrace \\subseteq A$ – set union of subsets Then for each $fires(t_x,0) \\in A$ we have $t_x \\in T_0$ – already shown in item REF above Then $q_c = m_{M_0(p)}(c) + \\sum _{t_x \\in T_0 \\wedge p \\in t_x \\bullet }{m_{W(t_x,p)}(c)} - (\\sum _{t_x \\in T_0 \\wedge p \\in \\bullet t_x}{m_{W(p,t_x)}(c)} + \\sum _{t_x \\in T_0 \\wedge p \\in R(t_x)}{m_{M_0(p)}(c)})$ – from (REF ) above and the following Each $add(p,{q_j}_c,t_j,c,0) \\in A$ represents ${q_j}_c=m_{W(t_j,p)}(c)$ for $p \\in t_j \\bullet $ – rule $r\\ref {r:c:add},f\\ref {f:c:tparc}$ encoding, and definition REF of transition execution in $PN$ Each $del(p,t_y,{q_y}_c,c,0) \\in A$ represents either ${q_y}_c=m_{W(p,t_y)}(c)$ for $p \\in \\bullet t_y$ , or ${q_y}_c=m_{M_0(p)}(c)$ for $p \\in R(t_y)$ – from rule $r\\ref {r:c:del},f\\ref {f:c:ptarc}$ encoding and definition REF of transition execution in $PN$ ; or from rule $r\\ref {r:c:del},f\\ref {f:c:rptarc}$ encoding and definition of reset arc in $PN$ Each $tot\\_incr(p,q2_c,c,0) \\in A$ represents $q2_c=\\sum _{t_x \\in T_0 \\wedge p \\in t_x \\bullet }{m_{W(t_x,p)}(c)}$ – aggregate assignment atom semantics in rule $r\\ref {r:c:totincr}$ Each $tot\\_decr(p,q3_c,c,0) \\in A$ represents $q3_c=\\sum _{t_x \\in T_0 \\wedge p \\in \\bullet t_x}{m_{W(p,t_x)}(c)} + \\sum _{t_x \\in T_0 \\wedge p \\in R(t_x)}{m_{M_0(p)}(c)}$ – aggregate assignment atom semantics in rule $r\\ref {r:c:totdecr}$ Then, $m_{M_1(p)}(c) = q_c$ – since $holds(p,q_c,c,1) \\in A$ encodes $q_c=m_{M_1(p)}(c)$ – from construction Inductive Step: Let $k > 0$ , and $M_k$ is a valid marking in $X$ for $PN$ , show [(1)] $T_k$ is a valid firing set for $M_k$ , and firing $T_k$ in $M_k$ produces marking $M_{k+1}$ .", "We show that $T_k$ is a valid firing set in $M_k$ .", "Let $\\lbrace fires(t_0,k), \\dots , fires(t_x,k) \\rbrace $ be the set of all $fires(\\dots ,k)$ atoms in $A$ , Then for each $fires(t_i,k) \\in A$ $enabled(t_i,k) \\in A$ – from rule $a\\ref {a:c:fires}$ and supported rule proposition Then $notenabled(t_i,k) \\notin A$ – from rule $e\\ref {e:c:enabled}$ and supported rule proposition Then either of $body(e\\ref {e:c:ne:ptarc})$ , $body(e\\ref {e:c:ne:iptarc})$ , or $body(e\\ref {e:c:ne:tptarc})$ must not hold in $A$ – from rules $e\\ref {e:c:ne:ptarc},e\\ref {e:c:ne:iptarc},e\\ref {e:c:ne:tptarc}$ and forced atom proposition Then $q_c \\lnot < {n_i}_c \\equiv q_c \\ge {n_i}_c$ in $e\\ref {e:c:ne:ptarc}$ for all $\\lbrace holds(p,q_c,c,k), \\\\ ptarc(p,t_i,{n_i}_c,c,k)\\rbrace \\subseteq A$ – from $e\\ref {e:c:ne:ptarc}$ , forced atom proposition, given facts, and the inductive assumption And $q_c \\lnot \\ge {n_i}_c \\equiv q_c < {n_i}_c$ in $e\\ref {e:c:ne:iptarc}$ for all $\\lbrace holds(p,q_c,c,k), \\\\ iptarc(p,t_i,{n_i}_c,c,k) \\rbrace \\subseteq A, {n_i}_c=1$ ; $q_c > {n_i}_c \\equiv q_c = 0$ – from $e\\ref {e:c:ne:iptarc}$ , forced atom proposition, given facts ($holds(p,q_c,c,k) \\in A, iptarc(p,t_i,1,c,k) \\in A$ ), and $q_c$ is a positive integer And $q_c \\lnot < {n_i}_c \\equiv q_c \\ge {n_i}_c$ in $e\\ref {e:c:ne:tptarc}$ for all $\\lbrace holds(p,q_c,c,k), \\\\ tptarc(p,t_i,{n_i}_c,c,k) \\rbrace \\subseteq A$ – from $e\\ref {e:c:ne:tptarc}$ , forced atom proposition, and inductive assumption Then $(\\forall p \\in \\bullet t_i, m_{M_k(p)}(c) \\ge m_{W(p,t_i)}(c)) \\wedge (\\forall p \\in I(t_i), m_{M_k(p)}(c) = 0) \\wedge (\\forall (p,t_i) \\in Q, m_{M_k(p)}(c) \\ge m_{QW(p,t_i)}(c))$ – from $holds(p,q_c,c,k) \\in A$ represents $q_c=m_{M_k(p)}(c)$ – construction of $\\Pi ^5$ $ptarc(p,t_i,{n_i}_c,k) \\in A$ represents ${n_i}_c=m_{W(p,t_i)}(c)$ – rule $f\\ref {f:c:ptarc}$ construction; or it represents ${n_i}_c = m_{M_k(p)}(c)$ – rule $f\\ref {f:c:rptarc}$ construction; the construction of $f\\ref {f:c:rptarc}$ ensures that $notenabled(t,k)$ is never true for a reset arc definition REF of preset $\\bullet t_i$ in $PN$ definition REF of enabled transition in $PN$ Then $t_i$ is enabled and can fire in $PN$ , as a result it can belong to $T_k$ – from definition REF of enabled transition And $consumesmore \\notin A$ , since $A$ is an answer set of $\\Pi ^5$ – from rule $a\\ref {a:c:overc:gen}$ and supported rule proposition Then $\\nexists consumesmore(p,k) \\in A$ – from rule $a\\ref {a:c:overc:place}$ and supported rule proposition Then $\\nexists \\lbrace holds(p,q_c,c,k), tot\\_decr(p,q1_c,c,k) \\rbrace \\subseteq A, q1_c>q_c$ in $body(a\\ref {a:c:overc:place})$ – from $a\\ref {a:c:overc:place}$ and forced atom proposition Then $\\nexists c \\in C, \\nexists p \\in P, (\\sum _{t_i \\in \\lbrace t_0,\\dots ,t_x\\rbrace , p \\in \\bullet t_i}{m_{W(p,t_i)}(c)}+ \\\\ \\sum _{t_i \\in \\lbrace t_0,\\dots ,t_x\\rbrace , p \\in R(t_i)}{m_{M_k(p)}(c)}) > m_{M_k(p)}(c)$ – from the following $holds(p,q_c,c,k)$ represents $q_c=m_{M_k(p)}(c)$ – construction of $\\Pi ^5$ , inductive assumption $tot\\_decr(p,q1_c,c,k) \\in A$ if $\\lbrace del(p,{q1_0}_c,t_0,c,k), \\dots , \\\\ del(p,{q1_x}_c,t_x,c,k) \\rbrace \\subseteq A$ , where $q1_c = {q1_0}_c+\\dots +{q1_x}_c$ – from $r\\ref {r:c:totdecr}$ and forced atom proposition $del(p,{q1_i}_c,t_i,c,k) \\in A$ if $\\lbrace fires(t_i,k), ptarc(p,t_i,{q1_i}_c,c,k) \\rbrace \\subseteq A$ – from $r\\ref {r:c:del}$ and supported rule proposition $del(p,{q1_i}_c,t_i,c,k)$ either represents removal of ${q1_i}_c = m_{W(p,t_i)}(c)$ tokens from $p \\in \\bullet t_i$ ; or it represents removal of ${q1_i}_c = m_{M_k(p)}(c)$ tokens from $p \\in R(t_i)$ – from rules $r\\ref {r:c:del},f\\ref {f:c:ptarc},f\\ref {f:c:rptarc}$ , supported rule proposition, and definition REF of transition execution in $PN$ Then the set of transitions in $T_k$ do not conflict – by the definition REF of conflicting transitions And for each reset transition $t_r$ with $enabled(t_r,k) \\in A$ , $fires(t_r,k) \\in A$ , since $A$ is an answer set of $\\Pi ^5$ - from rule $f\\ref {f:c:rptarc:elim}$ and supported rule proposition Then the firing set $T_k$ satisfies the reset transition requirement of definition REF (firing set) Then $\\lbrace t_0, \\dots , t_x\\rbrace = T_k$ – using 1(a),1(b), 1(d) above; and using 1(c) it is a maximal firing set We show that $M_{k+1}$ is produced by firing $T_k$ in $M_k$ .", "Let $holds(p,q_c,c,k+1) \\in A$ Then $\\lbrace holds(p,q1_c,c,k), tot\\_incr(p,q2_c,c,k), tot\\_decr(p,q3_c,c,k) \\rbrace \\subseteq A : q_c=q1_c+q2_c-q3_c$ – from rule $r\\ref {r:c:nextstate}$ and supported rule proposition Then, $holds(p,q1_c,c,k) \\in A$ represents $q1_c=m_{M_k(p)}(c)$ – construction, inductive assumption; and $\\lbrace add(p,{q2_0}_c,t_0,c,k), \\dots , add(p,{q2_j}_c,t_j,c,k)\\rbrace \\subseteq A : {q2_0}_c + \\dots + {q2_j}_c = q2_c$ and $\\lbrace del(p,{q3_0}_c,t_0,c,k), \\dots , del(p,{q3_l}_c,t_l,c,k)\\rbrace \\subseteq A : {q3_0}_c + \\dots + {q3_l}_c = q3_c$ – rules $r\\ref {r:c:totincr},r\\ref {r:c:totdecr}$ and supported rule proposition, respectively Then $\\lbrace fires(t_0,k), \\dots , fires(t_j,k) \\rbrace \\subseteq A$ and $\\lbrace fires(t_0,k), \\dots , fires(t_l,k) \\rbrace \\\\ \\subseteq A$ – rules $r\\ref {r:c:add},r\\ref {r:c:del}$ and supported rule proposition, respectively Then $\\lbrace fires(t_0,k), \\dots , $ $fires(t_j,k) \\rbrace \\cup \\lbrace fires(t_0,k), \\dots , $ $fires(t_l,k) \\rbrace \\subseteq $ $A = \\lbrace fires(t_0,k), \\dots , $ $fires(t_x,k) \\rbrace \\subseteq A$ – set union of subsets Then for each $fires(t_x,k) \\in A$ we have $t_x \\in T_k$ – already shown in item REF above Then $q_c = m_{M_k(p)}(c) + \\sum _{t_x \\in T_k \\wedge p \\in t_x \\bullet }{m_{W(t_x,p)}(c)} - (\\sum _{t_x \\in T_k \\wedge p \\in \\bullet t_x}{m_{W(p,t_x)}(c)} + \\sum _{t_x \\in T_k \\wedge p \\in R(t_x)}{m_{M_k(p)}(c)})$ – from (REF ) above and the following Each $add(p,{q_j}_c,t_j,c,k) \\in A$ represents ${q_j}_c=m_{W(t_j,p)}(c)$ for $p \\in t_j \\bullet $ – rule $r\\ref {r:c:add},f\\ref {f:c:tparc}$ encoding, and definition REF of transition execution in $PN$ Each $del(p,t_y,{q_y}_c,c,k) \\in A$ represents either ${q_y}_c=m_{W(p,t_y)}(c)$ for $p \\in \\bullet t_y$ , or ${q_y}_c=m_{M_k(p)}(c)$ for $p \\in R(t_y)$ – from rule $r\\ref {r:c:del},f\\ref {f:c:ptarc}$ encoding and definition REF of transition execution in $PN$ ; or from rule $r\\ref {r:c:del},f\\ref {f:c:rptarc}$ encoding and definition of reset arc in $PN$ Each $tot\\_incr(p,q2_c,c,k) \\in A$ represents $q2_c=\\sum _{t_x \\in T_k \\wedge p \\in t_x \\bullet }{m_{W(t_x,p)}(c)}$ – aggregate assignment atom semantics in rule $r\\ref {r:c:totincr}$ Each $tot\\_decr(p,q3_c,c,k) \\in A$ represents $q3_c=\\sum _{t_x \\in T_k \\wedge p \\in \\bullet t_x}{m_{W(p,t_x)}(c)} + \\sum _{t_x \\in T_k \\wedge p \\in R(t_x)}{m_{M_k(p)}(c)}$ – aggregate assignment atom semantics in rule $r\\ref {r:c:totdecr}$ Then, $m_{M_{k+1}(p)}(c) = q_c$ – since $holds(p,q_c,c,k+1) \\in A$ encodes $q_c=m_{M_{k+1}(p)}(c)$ – from construction As a result, for any $n > k$ , $T_n$ will be a valid firing set for $M_n$ and $M_{n+1}$ will be its target marking.", "Conclusion: Since both (REF ) and (REF ) hold, $X=M_0,T_k,M_1,\\dots ,M_k,T_{k+1}$ is an execution sequence of $PN(P,T,E,C,W,R,I,Q,QW)$ (w.r.t $M_0$ ) iff there is an answer set $A$ of $\\Pi ^5(PN,M_0,k,ntok)$ such that (REF ) and (REF ) hold.", "Proof of Proposition REF Let $PN=(P,T,E,C,W,R,I,Q,QW,Z)$ be a Petri Net, $M_0$ be its initial marking and let $\\Pi ^6(PN,M_0,k,ntok)$ be the ASP encoding of $PN$ and $M_0$ over a simulation length $k$ , with maximum $ntok$ tokens on any place node, as defined in section REF .", "Then $X=M_0,T_k,M_1,\\dots ,M_k,T_k,M_{k+1}$ is an execution sequence of $PN$ (w.r.t.", "$M_0$ ) iff there is an answer set $A$ of $\\Pi ^6(PN,M_0,k,ntok)$ such that: $\\lbrace fires(t,ts) : t \\in T_{ts}, 0 \\le ts \\le k\\rbrace = \\lbrace fires(t,ts) : fires(t,ts) \\in A \\rbrace $ $\\begin{split}\\lbrace holds(p,q,c,ts) &: p \\in P, c/q = M_{ts}(p), 0 \\le ts \\le k+1 \\rbrace \\\\&= \\lbrace holds(p,q,c,ts) : holds(p,q,c,ts) \\in A \\rbrace \\end{split}$ We prove this by showing that: Given an execution sequence $X$ , we create a set $A$ such that it satisfies (REF ) and (REF ) and show that $A$ is an answer set of $\\Pi ^6$ Given an answer set $A$ of $\\Pi ^6$ , we create an execution sequence $X$ such that (REF ) and (REF ) are satisfied.", "First we show (REF ): We create a set $A$ as a union of the following sets: $A_1=\\lbrace num(n) : 0 \\le n \\le ntok \\rbrace $ $A_2=\\lbrace time(ts) : 0 \\le ts \\le k\\rbrace $ $A_3=\\lbrace place(p) : p \\in P \\rbrace $ $A_4=\\lbrace trans(t) : t \\in T \\rbrace $ $A_5=\\lbrace color(c) : c \\in C \\rbrace $ $A_6=\\lbrace ptarc(p,t,n_c,c,ts) : (p,t) \\in E^-, c \\in C, n_c=m_{W(p,t)}(c), n_c > 0, 0 \\le ts \\le k \\rbrace $ , where $E^- \\subseteq E$ $A_7=\\lbrace tparc(t,p,n_c,c,ts) : (t,p) \\in E^+, c \\in C, n_c=m_{W(t,p)}(c), n_c > 0, 0 \\le ts \\le k \\rbrace $ , where $E^+ \\subseteq E$ $A_8=\\lbrace holds(p,q_c,c,0) : p \\in P, c \\in C, q_c=m_{M_{0}(p)}(c) \\rbrace $ $A_9=\\lbrace ptarc(p,t,n_c,c,ts) : p \\in R(t), c \\in C, n_c = m_{M_{ts}(p)}, n_c > 0, 0 \\le ts \\le k \\rbrace $ $A_{10}=\\lbrace iptarc(p,t,1,c,ts) : p \\in I(t), c \\in C, 0 \\le ts < k \\rbrace $ $A_{11}=\\lbrace tptarc(p,t,n_c,c,ts) : (p,t) \\in Q, c \\in C, n_c=m_{QW(p,t)}(c), n_c > 0, 0 \\le ts \\le k \\rbrace $ $A_{12}=\\lbrace notenabled(t,ts) : t \\in T, 0 \\le ts \\le k, \\exists c \\in C, (\\exists p \\in \\bullet t, m_{M_{ts}(p)}(c) < m_{W(p,t)}(c)) \\vee (\\exists p \\in I(t), m_{M_{ts}(p)}(c) > 0) \\vee (\\exists (p,t) \\in Q, m_{M_{ts}(p)}(c) < m_{QW(p,t)}(c)) \\rbrace $ per definition REF (enabled transition) $A_{13}=\\lbrace enabled(t,ts) : t \\in T, 0 \\le ts \\le k, \\forall c \\in C, (\\forall p \\in \\bullet t, m_{W(p,t)}(c) \\le m_{M_{ts}(p)}(c)) \\wedge (\\forall p \\in I(t), m_{M_{ts}(p)}(c) = 0) \\wedge (\\forall (p,t) \\in Q, m_{M_{ts}(p)}(c) \\ge m_{QW(p,t)}(c)) \\rbrace $ per definition REF (enabled transition) $A_{14}=\\lbrace fires(t,ts) : t \\in T_{ts}, 0 \\le ts \\le k \\rbrace $ per definition REF (firing set), only an enabled transition may fire $A_{15}=\\lbrace add(p,q_c,t,c,ts) : t \\in T_{ts}, p \\in t \\bullet , c \\in C, q_c=m_{W(t,p)}(c), 0 \\le ts \\le k \\rbrace $ per definition REF (transition execution) $A_{16}=\\lbrace del(p,q_c,t,c,ts) : t \\in T_{ts}, p \\in \\bullet t, c \\in C, q_c=m_{W(p,t)}(c), 0 \\le ts \\le k \\rbrace \\cup \\lbrace del(p,q_c,t,c,ts) : t \\in T_{ts}, p \\in R(t), c \\in C, q_c=m_{M_{ts}(p)}(c), 0 \\le ts \\le k \\rbrace $ per definition REF (transition execution) $A_{17}=\\lbrace tot\\_incr(p,q_c,c,ts) : p \\in P, c \\in C, q_c=\\sum _{t \\in T_{ts}, p \\in t \\bullet }{m_{W(t,p)}(c)}, 0 \\le ts \\le k \\rbrace $ per definition REF (firing set execution) $A_{18}=\\lbrace tot\\_decr(p,q_c,c,ts): p \\in P, c \\in C, q_c=\\sum _{t \\in T_{ts}, p \\in \\bullet t}{m_{W(p,t)}(c)}+ \\\\ \\sum _{t \\in T_{ts}, p \\in R(t) }{m_{M_{ts}(p)}(c)}, 0 \\le ts \\le k \\rbrace $ per definition REF (firing set execution) $A_{19}=\\lbrace consumesmore(p,ts) : p \\in P, c \\in C, q_c=m_{M_{ts}(p)}(c), \\\\ q1_c=\\sum _{t \\in T_{ts}, p \\in \\bullet t}{m_{W(p,t)}(c)} + \\sum _{t \\in T_{ts}, p \\in R(t)}{m_{M_{ts}(p)}(c)}, q1_c > q_c, 0 \\le ts \\le k \\rbrace $ per definition REF (conflicting transitions) $A_{20}=\\lbrace consumesmore : \\exists p \\in P, c \\in C, q_c=m_{M_{ts}(p)}(c), \\\\ q1_c=\\sum _{t \\in T_{ts}, p \\in \\bullet t}{m_{W(p,t)}(c)} + \\sum _{t \\in T_{ts}, p \\in R(t)}(m_{M_{ts}(p)}(c)), q1_c > q_c, 0 \\le ts \\le k \\rbrace $ per definition REF (conflicting transitions) $A_{21}=\\lbrace holds(p,q_c,c,ts+1) : p \\in P, c \\in C, q_c=m_{M_{ts+1}(p)}(c), 0 \\le ts < k\\rbrace $ , where $M_{ts+1}(p) = M_j(p) - (\\sum _{\\begin{array}{c}t \\in T_{ts}, p \\in \\bullet t\\end{array}}{W(p,t)} + \\sum _{t \\in T_{ts}, p \\in R(t)}M_{ts}(p)) + \\\\ \\sum _{\\begin{array}{c}t \\in T_{ts}, p \\in t \\bullet \\end{array}}{W(t,p)}$ according to definition REF (firing set execution) $A_{22}=\\lbrace transpr(t,pr) : t \\in T, pr=Z(t) \\rbrace $ $A_{23}=\\lbrace notprenabled(t,ts) : t \\in T, enabled(t,ts) \\in A_{13}, (\\exists tt \\in T, enabled(tt,ts) \\in A_{13}, Z(tt) < Z(t)), 0 \\le ts \\le k \\rbrace \\\\ = \\lbrace notprenabled(t,ts) : t \\in T, (\\forall p \\in \\bullet t, W(p,t) \\le M_{ts}(p)), (\\forall p \\in I(t), M_{ts}(p) = 0), (\\forall (p,t) \\in Q, M_0(p) \\ge WQ(p,t)), \\exists tt \\in T, (\\forall pp \\in \\bullet tt, W(pp,tt) \\le M_{ts}(pp)), \\\\ (\\forall pp \\in I(tt), M_{ts}(pp) = 0), (\\forall (pp,tt) \\in Q, M_{ts}(pp) \\ge QW(pp,tt)), Z(tt) < Z(t), 0 \\le ts \\le k \\rbrace $ $A_{24}=\\lbrace prenabled(t,ts) : t \\in T, enabled(t,ts) \\in A_{13}, (\\nexists tt \\in T: enabled(tt,ts) \\in A_{13}, Z(tt) < Z(t)), 0 \\le ts \\le k \\rbrace = \\lbrace prenabled(t,ts) : t \\in T, (\\forall p \\in \\bullet t, W(p,t) \\le M_{ts}(p)), (\\forall p \\in I(t), M_{ts}(p) = 0), (\\forall (p,t) \\in Q, M_0(p) \\ge WQ(p,t)), \\nexists tt \\in T, (\\forall pp \\in \\bullet tt, W(pp,tt) \\le M_{ts}(pp)), (\\forall pp \\in I(tt), M_{ts}(pp) = 0), (\\forall (pp,tt) \\in Q, M_{ts}(pp) \\ge QW(pp,tt)), Z(tt) < Z(t), 0 \\le ts \\le k \\rbrace $ $A_{25}=\\lbrace could\\_not\\_have(t,ts): t \\in T, prenabled(t,ts) \\in A_{24}, fires(t,ts) \\notin A_{14}, (\\exists p \\in \\bullet t \\cup R(t): q > M_{ts}(p) - (\\sum _{t^{\\prime } \\in T_{ts}, p \\in \\bullet t^{\\prime }}{W(p,t^{\\prime })} + \\sum _{t^{\\prime } \\in T_{ts}, p \\in R(t^{\\prime })}{M_{ts}(p)}), q=W(p,t) \\text{~if~} (p,t) \\in E^- \\text{~or~} R(t) \\text{~otherwise}), 0 \\le ts \\le k\\rbrace =\\lbrace could\\_not\\_have(t,ts) : t \\in T, (\\forall p \\in \\bullet t, W(p,t) \\le M_{ts}(p)), (\\forall p \\in I(t), M_{ts}(p) = 0), (\\forall (p,t) \\in Q, M_0(p) \\ge QW(p,t)), (\\nexists tt \\in T, $ $(\\forall pp \\in \\bullet tt, W(pp,tt) \\le M_{ts}(pp)), (\\forall pp \\in I(tt), M_{ts}(pp) = 0), (\\forall (pp,tt) \\in Q, M_{ts}(pp) \\ge QW(pp,tt)), Z(tt) < Z(t)), t \\notin T_{ts}, (\\exists p \\in \\bullet t \\cup R(t): q > M_{ts}(p) - (\\sum _{t^{\\prime } \\in T_{ts}, p \\in \\bullet t^{\\prime }}{W(p,t^{\\prime })} + \\sum _{t^{\\prime } \\in T_{ts}, p \\in R(t^{\\prime })}{M_{ts}(p)}), q=W(p,t) \\text{~if~} \\\\ (p,t) \\in E^- \\text{~or~} R(t) \\text{~otherwise}), 0 \\le ts \\le k \\rbrace $ per the maximal firing set semantics We show that $A$ satisfies (REF ) and (REF ), and $A$ is an answer set of $\\Pi ^6$ .", "$A$ satisfies (REF ) and (REF ) by its construction above.", "We show $A$ is an answer set of $\\Pi ^6$ by splitting.", "We split $lit(\\Pi ^6)$ into a sequence of $9k+11$ sets: $U_0= head(f\\ref {f:c:place}) \\cup head(f\\ref {f:c:trans}) \\cup head(f\\ref {f:c:col}) \\cup head(f\\ref {f:c:time}) \\cup head(f\\ref {f:c:num}) \\cup head(i\\ref {i:c:init}) \\cup head(f\\ref {f:c:pr}) = \\lbrace place(p) : p \\in P\\rbrace \\cup \\lbrace trans(t) : t \\in T\\rbrace \\cup \\lbrace col(c) : c \\in C \\rbrace \\cup \\lbrace time(0), \\dots , time(k)\\rbrace \\cup \\lbrace num(0), \\dots , num(ntok)\\rbrace \\cup \\lbrace holds(p,q_c,c,0) : p \\in P, c \\in C, q_c=m_{M_0(p)}(c) \\rbrace \\cup \\lbrace transpr(t,pr) : t \\in T, pr=Z(t) \\rbrace $ $U_{9k+1}=U_{9k+0} \\cup head(f\\ref {f:c:ptarc})^{ts=k} \\cup head(f\\ref {f:c:tparc})^{ts=k} \\cup head(f\\ref {f:c:rptarc})^{ts=k} \\cup head(f\\ref {f:c:iptarc})^{ts=k} \\cup head(f\\ref {f:c:tptarc})^{ts=k} = U_{10k+0} \\cup \\lbrace ptarc(p,t,n_c,c,k) : (p,t) \\in E^-, c \\in C, n_c=m_{W(p,t)}(c) \\rbrace \\\\ \\cup \\lbrace tparc(t,p,n_c,c,k) : (t,p) \\in E^+, c \\in C, n_c=m_{W(t,p)}(c) \\rbrace \\cup \\lbrace ptarc(p,t,n_c,c,k) : p \\in R(t), c \\in C, n_c=m_{M_{k}(p)}(c), n > 0 \\rbrace \\cup \\lbrace iptarc(p,t,1,c,k) : p \\in I(t), c \\in C \\rbrace \\cup \\lbrace tptarc(p,t,n_c,c,k) : (p,t) \\in Q, c \\in C, n_c=m_{QW(p,t)}(c) \\rbrace $ $U_{9k+2}=U_{9k+1} \\cup head(e\\ref {e:c:ne:ptarc})^{ts=k} \\cup head(e\\ref {e:c:ne:iptarc})^{ts=k} \\cup head(e\\ref {e:c:ne:tptarc})^{ts=k} = U_{9k+1} \\\\ \\cup \\lbrace notenabled(t,k) : t \\in T \\rbrace $ $U_{9k+3}=U_{9k+2} \\cup head(e\\ref {e:c:enabled})^{ts=k} = U_{9k+2} \\cup \\lbrace enabled(t,k) : t \\in T \\rbrace $ $U_{9k+4}=U_{9k+3} \\cup head(a\\ref {a:c:prne})^{ts=k} = U_{9k+3} \\cup \\lbrace notprenabled(t,k) : t \\in T \\rbrace $ $U_{9k+5}=U_{9k+4} \\cup head(a\\ref {a:c:prenabled})^{ts=k} = U_{9k+4} \\cup \\lbrace prenabled(t,k) : t \\in T \\rbrace $ $U_{9k+6}=U_{9k+5} \\cup head(a\\ref {a:c:prfires})^{ts=k} = U_{9k+5} \\cup \\lbrace fires(t,k) : t \\in T \\rbrace $ $U_{9k+7}=U_{9k+6} \\cup head(r\\ref {r:c:add})^{ts=k} \\cup head(r\\ref {r:c:del})^{ts=k} = U_{9k+6} \\cup \\lbrace add(p,q_c,t,c,k) : p \\in P, t \\in T, c \\in C, q_c=m_{W(t,p)}(c) \\rbrace \\cup \\lbrace del(p,q_c,t,c,k) : p \\in P, t \\in T, c \\in C, q_c=m_{W(p,t)}(c) \\rbrace \\cup \\lbrace del(p,q_c,t,c,k) : p \\in P, t \\in T, c \\in C, q_c=m_{M_{k}(p)}(c) \\rbrace $ $U_{9k+8}=U_{9k+7} \\cup head(r\\ref {r:c:totincr})^{ts=k} \\cup head(r\\ref {r:c:totdecr})^{ts=k} = U_{9k+7} \\cup \\lbrace tot\\_incr(p,q_c,c,k) : p \\in P, c \\in C, 0 \\le q_c \\le ntok \\rbrace \\cup \\lbrace tot\\_decr(p,q_c,c,k) : p \\in P, c \\in C, 0 \\le q_c \\le ntok \\rbrace $ $U_{9k+9}=U_{9k+8} \\cup head(a\\ref {a:c:overc:place})^{ts=k} \\cup head(r\\ref {r:c:nextstate})^{ts=k} \\cup head(a\\ref {a:c:prmaxfire:cnh})^{ts=k} = U_{9k+8} \\cup \\\\ \\lbrace consumesmore(p,k) : p \\in P\\rbrace \\cup \\lbrace holds(p,q,k+1) : p \\in P, 0 \\le q \\le ntok \\rbrace \\cup \\lbrace could\\_not\\_have(t,k) : t \\in T \\rbrace $ $U_{9k+10}=U_{9k+9} \\cup head(a\\ref {a:c:overc:gen})^{ts=k} = U_{9k+9} \\cup \\lbrace consumesmore \\rbrace $ where $head(r_i)^{ts=k}$ are head atoms of ground rule $r_i$ in which $ts=k$ .", "We write $A_i^{ts=k} = \\lbrace a(\\dots ,ts) : a(\\dots ,ts) \\in A_i, ts=k \\rbrace $ as short hand for all atoms in $A_i$ with $ts=k$ .", "$U_{\\alpha }, 0 \\le \\alpha \\le 9k+10$ form a splitting sequence, since each $U_i$ is a splitting set of $\\Pi ^6$ , and $\\langle U_{\\alpha }\\rangle _{\\alpha < \\mu }$ is a monotone continuous sequence, where $U_0 \\subseteq U_1 \\dots \\subseteq U_{9k+10}$ and $\\bigcup _{\\alpha < \\mu }{U_{\\alpha }} = lit(\\Pi ^6)$ .", "We compute the answer set of $\\Pi ^6$ using the splitting sets as follows: $bot_{U_0}(\\Pi ^6) = f\\ref {f:c:place} \\cup f\\ref {f:c:trans} \\cup f\\ref {f:c:col} \\cup f\\ref {f:c:time} \\cup f\\ref {f:c:num} \\cup i\\ref {i:c:init} \\cup f\\ref {f:c:pr}$ and $X_0 = A_1 \\cup \\dots \\cup A_5 \\cup A_8$ ($= U_0$ ) is its answer set – using forced atom proposition $eval_{U_0}(bot_{U_1}(\\Pi ^6) \\setminus bot_{U_0}(\\Pi ^6), X_0) = \\lbrace ptarc(p,t,q_c,c,0) \\text{:-}.", "\| c \\in C, $ $q_c=m_{W(p,t)}(c), q_c > 0 \\rbrace \\cup \\lbrace tparc(t,p,q_c,c,0) \\text{:-}.", "\| c \\in C, q_c=m_{W(t,p)}(c), q_c > 0 \\rbrace \\cup \\lbrace ptarc(p,t,q_c,c,0) \\text{:-}.", "\| c \\in C, $ $q_c=m_{M_0(p)}(c), q_c > 0 \\rbrace \\cup \\lbrace iptarc(p,t,1,c,0) \\text{:-}.", "\| c \\in C \\rbrace \\cup \\lbrace tptarc(p,t,q_c,c,0) \\text{:-}.", "\| c \\in C, q_c = m_{QW(p,t)}(c), q_c > 0 \\rbrace $ .", "Its answer set $X_1=A_6^{ts=0} \\cup A_7^{ts=0} \\cup A_9^{ts=0} \\cup A_{10}^{ts=0} \\cup A_{11}^{ts=0}$ – using forced atom proposition and construction of $A_6, A_7, A_9, A_{10}, A_{11}$ .", "$eval_{U_1}(bot_{U_2}(\\Pi ^6) \\setminus bot_{U_1}(\\Pi ^6), X_0 \\cup X_1) = \\lbrace notenabled(t,0) \\text{:-} .", "\| (\\lbrace trans(t), \\\\ ptarc(p,t,n_c,c,0), holds(p,q_c,c,0) \\rbrace \\subseteq X_0 \\cup X_1, \\text{~where~} q_c < n_c) \\text{~or~} \\\\ (\\lbrace notenabled(t,0) \\text{:-} .", "\| (\\lbrace trans(t), iptarc(p,t,n2_c,c,0), holds(p,q_c,c,0) \\rbrace \\subseteq X_0 \\cup X_1, \\\\ \\text{~where~} q_c \\ge n2_c \\rbrace ) \\text{~or~} (\\lbrace trans(t), tptarc(p,t,n3_c,c,0), $ $holds(p,q_c,c,0) \\rbrace \\subseteq X_0 \\cup X_1, \\text{~where~} $ $q_c < n3_c) \\rbrace $ .", "Its answer set $X_2=A_{12}^{ts=0}$ – using forced atom proposition and construction of $A_{12}$ .", "where, $q_c=m_{M_0(p)}(c)$ , and $n_c=m_{W(p,t)}(c)$ for an arc $(p,t) \\in E^-$ – by construction of $i\\ref {i:c:init}$ and $f\\ref {f:c:ptarc}$ in $\\Pi ^6$ , and in an arc $(p,t) \\in E^-$ , $p \\in \\bullet t$ (by definition REF of preset) $n2_c=1$ – by construction of $iptarc$ predicates in $\\Pi ^6$ , meaning $q_c \\ge n2_c \\equiv q_c \\ge 1 \\equiv q_c > 0$ , $tptarc(p,t,n3_c,c,0)$ represents $n3_c=m_{QW(p,t)}(c)$ , where $(p,t) \\in Q$ thus, $notenabled(t,0) \\in X_1$ represents $\\exists c \\in C, (\\exists p \\in \\bullet t : m_{M_0(p)}(c) < m_{W(p,t)}(c)) \\vee (\\exists p \\in I(t) : m_{M_0(p)}(c) > 0) \\vee (\\exists (p,t) \\in Q : m_{M_0(p)}(c) < m_{QW(p,t)}(c))$ .", "$eval_{U_2}(bot_{U_3}(\\Pi ^6) \\setminus bot_{U_2}(\\Pi ^6), X_0 \\cup \\dots \\cup X_2) = \\lbrace enabled(t,0) \\text{:-}.", "\| trans(t) \\in X_0 \\cup \\dots \\cup X_2, notenabled(t,0) \\notin X_0 \\cup \\dots \\cup X_2 \\rbrace $ .", "Its answer set is $X_3 = A_{13}^{ts=0}$ – using forced atom proposition and construction of $A_{13}$ .", "since an $enabled(t,0) \\in X_3$ if $\\nexists ~notenabled(t,0) \\in X_0 \\cup \\dots \\cup X_2$ ; which is equivalent to $\\nexists t, \\forall c \\in C, (\\nexists p \\in \\bullet t, m_{M_0(p)}(c) < m_{W(p,t)}(c)), (\\nexists p \\in I(t), $ $m_{M_0(p)}(c) > 0), (\\nexists (p,t) \\in Q : m_{M_0(p)}(c) < m_{QW(p,t)}(c) ), \\forall c \\in C, (\\forall p \\in \\bullet t: m_{M_0(p)}(c) \\ge m_{W(p,t)}(c)), (\\forall p \\in I(t) : m_{M_0(p)}(c) = 0)$ .", "$eval_{U_3}(bot_{U_4}(\\Pi ^6) \\setminus bot_{U_3}(\\Pi ^6), X_0 \\cup \\dots \\cup X_3) = \\lbrace notprenabled(t,0) \\text{:-}.", "\| \\\\ \\lbrace enabled(t,0), transpr(t,p), enabled(tt,0), transpr(tt,pp) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_3, pp < p \\rbrace $ .", "Its answer set is $X_4 = A_{23}^{ts=k}$ – using forced atom proposition and construction of $A_{23}$ .", "$enabled(t,0)$ represents $\\exists t \\in T, \\forall c \\in C, (\\forall p \\in \\bullet t, m_{M_0(p)}(c) \\ge m_{W(p,t)}(c)), $ $(\\forall p \\in I(t), m_{M_0(p)}(c) = 0), (\\forall (p,t) \\in Q, m_{M_0(p)}(c) \\ge m_{QW(p,t)}(c))$ $enabled(tt,0)$ represents $\\exists tt \\in T, \\forall c \\in C, (\\forall pp \\in \\bullet tt, m_{M_0(pp)}(c) \\ge \\\\ m_{W(pp,tt)}(c)) \\wedge (\\forall pp \\in I(tt), m_{M_0(pp)}(c) = 0), (\\forall (pp,tt) \\in Q, m_{M_0(pp)}(c) \\ge m_{QW(pp,tt)}(c))$ $transpr(t,p)$ represents $p=Z(t)$ – by construction $transpr(tt,pp)$ represents $pp=Z(tt)$ – by construction thus, $notprenabled(t,0)$ represents $\\forall c \\in C, (\\forall p \\in \\bullet t, m_{M_0(p)}(c) \\ge \\\\ m_{W(p,t)}(c)) \\wedge (\\forall p \\in I(t), m_{M_0(p)}(c) = 0), \\exists tt \\in T, (\\forall pp \\in \\bullet tt, m_{M_0(pp)}(c) \\ge m_{W(pp,tt)}(c)) \\wedge (\\forall pp \\in I(tt), m_{M_0(pp)}(c) = 0), Z(tt) < Z(t)$ which is equivalent to $(\\forall p \\in \\bullet t: M_0(p) \\ge W(p,t)) \\wedge (\\forall p \\in I(t), M_0(p) = 0), \\exists tt \\in T, (\\forall pp \\in \\bullet tt, M_0(pp) \\ge W(pp,tt)), (\\forall pp \\in I(tt), M_0(pp) = 0), Z(tt) < Z(t)$ – assuming multiset domain $C$ for all operations $eval_{U_4}(bot_{U_5}(\\Pi ^6) \\setminus bot_{U_4}(\\Pi ^6), X_0 \\cup \\dots \\cup X_4) = \\lbrace prenabled(t,0) \\text{:-}.", "\| enabled(t,0) \\in X_0 \\cup \\dots \\cup X_4, notprenabled(t,0) \\notin X_0 \\cup \\dots \\cup X_4 \\rbrace $ .", "Its answer set is $X_5 = A_{24}^{ts=k}$ – using forced atom proposition and construction of $A_{24}$ $enabled(t,0)$ represents $\\forall c \\in C, (\\forall p \\in \\bullet t, m_{M_0(p)}(c) \\ge m_{W(p,t)}(c)), (\\forall p \\in I(t), m_{M_0(p)}(c) = 0), (\\forall (p,t) \\in Q, m_{M_0(p)}(c) \\ge m_{QW(p,t)}(c)) \\equiv (\\forall p \\in \\bullet t, \\\\ M_0(p) \\ge W(p,t)), (\\forall p \\in I(t), M_0(p) = 0), (\\forall (p,t) \\in Q, M_0(p) \\ge QW(p,t))$ – from REF above and assuming multiset domain $C$ for all operations $notprenabled(t,0)$ represents $(\\forall p \\in \\bullet t, M_0(p) \\ge W(p,t)), (\\forall p \\in I(t), \\\\ M_0(p) = 0), (\\forall (p,t) \\in Q, M_0(p) \\ge QW(p,t)), \\exists tt \\in T, (\\forall pp \\in \\bullet tt, M_0(pp) \\ge W(pp,tt)), (\\forall pp \\in I(tt), M_0(pp) = 0), (\\forall (pp,tt) \\in Q, M_0(pp) \\ge W(pp,tt)), \\\\ Z(tt) < Z(t)$ – from REF above and assuming multiset domain $C$ for all operations then, $prenabled(t,0)$ represents $(\\forall p \\in \\bullet t, M_0(p) \\ge W(p,t)), (\\forall p \\in I(t), \\\\ M_0(p) = 0), (\\forall (p,t) \\in Q, M_0(p) \\ge QW(p,t)), \\nexists tt \\in T, ((\\forall pp \\in \\bullet tt, M_0(pp) \\ge W(pp,tt)), (\\forall pp \\in I(tt), M_0(pp) = 0), (\\forall (pp,tt) \\in Q, M_0(pp) \\ge W(pp,tt)), \\\\ Z(tt) < Z(t))$ – from (a), (b) and $enabled(t,0) \\in X_0 \\cup \\dots \\cup X_4$ $eval_{U_5}(bot_{U_6}(\\Pi ^6) \\setminus bot_{U_5}(\\Pi ^6), X_0 \\cup \\dots \\cup X_5) = \\lbrace \\lbrace fires(t,0)\\rbrace \\text{:-}.", "\| prenabled(t,0) \\\\ \\text{~holds in~} X_0 \\cup \\dots \\cup X_5 \\rbrace $ .", "It has multiple answer sets $X_{6.1}, \\dots , X_{6.n}$ , corresponding to elements of power set of $fires(t,0)$ atoms in $eval_{U_5}(...)$ – using supported rule proposition.", "Since we are showing that the union of answer sets of $\\Pi ^6$ determined using splitting is equal to $A$ , we only consider the set that matches the $fires(t,0)$ elements in $A$ and call it $X_6$ , ignoring the rest.", "Thus, $X_6 = A_{14}^{ts=0}$ , representing $T_0$ .", "in addition, for every $t$ such that $prenabled(t,0) \\in X_0 \\cup \\dots \\cup X_5, R(t) \\ne \\emptyset $ ; $fires(t,0) \\in X_6$ – per definition REF (firing set); requiring that a reset transition is fired when enabled thus, the firing set $T_0$ will not be eliminated by the constraint $f\\ref {f:c:pr:rptarc:elim}$ $eval_{U_6}(bot_{U_7}(\\Pi ^6) \\setminus bot_{U_6}(\\Pi ^6), X_0 \\cup \\dots \\cup X_6) = \\lbrace add(p,n_c,t,c,0) \\text{:-}.", "\| \\lbrace fires(t,0), \\\\ tparc(t,p,n_c,c,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_6 \\rbrace \\cup \\lbrace del(p,n_c,t,c,0) \\text{:-}.", "\| \\lbrace fires(t,0), \\\\ ptarc(p,t,n_c,c,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_6 \\rbrace $ .", "It's answer set is $X_7 = A_{15}^{ts=0} \\cup A_{16}^{ts=0}$ – using forced atom proposition and definitions of $A_{15}$ and $A_{16}$ .", "where, each $add$ atom is equivalent to $n_c=m_{W(t,p)}(c),c \\in C, p \\in t \\bullet $ , and each $del$ atom is equivalent to $n_c=m_{W(p,t)}(c), c \\in C, p \\in \\bullet t$ ; or $n_c=m_{M_0(p)}(c), c \\in C, p \\in R(t)$ , representing the effect of transitions in $T_0$ – by construction $eval_{U_7}(bot_{U_8}(\\Pi ^6) \\setminus bot_{U_7}(\\Pi ^6), X_0 \\cup \\dots \\cup X_7) = \\lbrace tot\\_incr(p,qq_c,c,0) \\text{:-}.", "\| qq_c=\\sum _{add(p,q_c,t,c,0) \\in X_0 \\cup \\dots \\cup X_7}{q_c} \\rbrace \\cup \\lbrace tot\\_decr(p,qq_c,c,0) \\text{:-}.", "\| qq_c=\\sum _{del(p,q_c,t,c,0) \\in X_0 \\cup \\dots \\cup X_7}{q_c} \\rbrace $ .", "It's answer set is $X_8 = A_{17}^{ts=0} \\cup A_{18}^{ts=0}$ – using forced atom proposition and definitions of $A_{17}$ and $A_{18}$ .", "where, each $tot\\_incr(p,qq_c,c,0)$ , $qq_c=\\sum _{add(p,q_c,t,c,0) \\in X_0 \\cup \\dots X_7}{q_c}$ $\\equiv qq_c=\\sum _{t \\in X_6, p \\in t \\bullet }{m_{W(p,t)}(c)}$ , where, each $tot\\_decr(p,qq_c,c,0)$ , $qq_c=\\sum _{del(p,q_c,t,c,0) \\in X_0 \\cup \\dots X_7}{q_c}$ $\\equiv qq=\\sum _{t \\in X_6, p \\in \\bullet t}{m_{W(t,p)}(c)} + \\sum _{t \\in X_6, p \\in R(t)}{m_{M_0(p)}(c)}$ , represent the net effect of transitions in $T_0$ $eval_{U_8}(bot_{U_9}(\\Pi ^6) \\setminus bot_{U_8}(\\Pi ^6), X_0 \\cup \\dots \\cup X_8) = \\lbrace consumesmore(p,0) \\text{:-}.", "\| \\\\ \\lbrace holds(p,q_c,c,0), tot\\_decr(p,q1_c,c,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_8, q1_c > q_c \\rbrace \\cup \\\\ \\lbrace holds(p,q_c,c,1) \\text{:-}.", "\| \\lbrace holds(p,q1_c,c,0), tot\\_incr(p,q2_c,c,0), tot\\_decr(p,q3_c,c,0) \\rbrace \\\\ \\subseteq X_0 \\cup \\dots \\cup X_6, q_c=q1_c+q2_c-q3_c \\rbrace \\cup \\lbrace could\\_not\\_have(t,0) \\text{:-}.", "\| \\lbrace prenabled(t,0), \\\\ ptarc(s,t,q,c,0), holds(s,qq,c,0), tot\\_decr(s,qqq,c,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_8, \\\\ fires(t,0) \\notin X_0 \\cup \\dots \\cup X_8, q > qq - qqq \\rbrace $ .", "It's answer set is $X_9 = A_{19}^{ts=0} \\cup A_{21}^{ts=0} \\cup A_{25}^{ts=0}$ – using forced atom proposition and definitions of $A_{19}, A_{21}, A_{25}$ .", "where, $consumesmore(p,0)$ represents $\\exists p : q_c=m_{M_0(p)}(c), q1_c= \\\\ \\sum _{t \\in T_0, p \\in \\bullet t}{m_{W(p,t)}(c)}+\\sum _{t \\in T_0, p \\in R(t)}{m_{M_0(p)}(c)}, q1_c > q_c, c \\in C$ , indicating place $p$ will be over consumed if $T_0$ is fired, as defined in definition REF (conflicting transitions), $holds(p,q_c,c,1)$ if $q_c=m_{M_0(p)}(c)+\\sum _{t \\in T_0, p \\in t \\bullet }{m_{W(t,p)}(c)}- \\\\ (\\sum _{t \\in T_0, p \\in \\bullet t}{m_{W(p,t)}(c)}+ \\sum _{t \\in T_0, p \\in R(t)}{m_{M_0(p)}(c)})$ represented by $q_c=m_{M_1(p)}(c)$ for some $c \\in C$ – by construction of $\\Pi ^6$ $could\\_not\\_have(t,0)$ if $(\\forall p \\in \\bullet t, W(p,t) \\le M_0(p)), (\\forall p \\in I(t), M_0(p) = 0), (\\forall (p,t) \\in Q, \\\\ M_0(p) \\ge WQ(p,t)), \\nexists tt \\in T, (\\forall pp \\in \\bullet tt, W(pp,tt) \\le M_{ts}(pp)), (\\forall pp \\in I(tt), M_0(pp) = 0), (\\forall (pp,tt) \\in Q, M_0(pp) \\ge QW(pp,tt)), Z(tt) < Z(t)$ , and $q_c > m_{M_0(s)}(c) - (\\sum _{t^{\\prime } \\in T_0, s \\in \\bullet t^{\\prime }}{m_{W(s,t^{\\prime })}(c)}+ \\sum _{t^{\\prime } \\in T_0, s \\in R(t)}{m_{M_0(s)}(c)}), \\\\ q_c = m_{W(s,t)}(c) \\text{~if~} s \\in \\bullet t \\text{~or~} m_{M_0(s)}(c) \\text{~otherwise}$ for some $c \\in C$ , which becomes $q > M_0(s) - (\\sum _{t^{\\prime } \\in T_0, s \\in \\bullet t^{\\prime }}{W(s,t^{\\prime })}+ \\sum _{t^{\\prime } \\in T_0, s \\in R(t)}{M_0(s)}), q = W(s,t) \\text{~if~} s \\in \\bullet t \\text{~or~} M_0(s) \\text{~otherwise}$ for all $c \\in C$ (i), (ii) above combined match the definition of $A_{25}$ $X_9$ does not contain $could\\_not\\_have(t,0)$ , when $prenabled(t,0) \\in X_0 \\cup \\dots \\cup X_8$ and $fires(t,0) \\notin X_0 \\cup \\dots \\cup X_8$ due to construction of $A$ , encoding of $a\\ref {a:c:overc:place}$ and its body atoms.", "As a result, it is not eliminated by the constraint $a\\ref {a:c:maxfire:elim}$ $ \\vdots $ $eval_{U_{9k+0}}(bot_{U_{9k+1}}(\\Pi ^6) \\setminus bot_{U_{9k+0}}(\\Pi ^6), X_0 \\cup \\dots \\cup X_{9k+0}) = \\lbrace ptarc(p,t,q_c,c,k) \\text{:-}.", "\| c \\in C, q_c=m_{W(p,t)}(c), q_c > 0 \\rbrace \\cup \\lbrace tparc(t,p,q_c,c,k) \\text{:-}.", "\| c \\in C, q_c=m_{W(t,p)}(c), q_c > 0 \\rbrace \\cup \\lbrace ptarc(p,t,q_c,c,k) \\text{:-}.", "\| c \\in C, q_c=m_{M_0(p)}(c), q_c > 0 \\rbrace \\cup \\lbrace iptarc(p,t,1,c,k) \\text{:-}.", "\| c \\in C \\rbrace \\cup \\lbrace tptarc(p,t,q_c,c,k) \\text{:-}.", "\| c \\in C, q_c = m_{QW(p,t)}(c), q_c > 0 \\rbrace $ .", "Its answer set $X_{9k+1}=A_6^{ts=k} \\cup A_7^{ts=k} \\cup A_9^{ts=k} \\cup A_{10}^{ts=k} \\cup A_{11}^{ts=k}$ – using forced atom proposition and construction of $A_6, A_7, A_9, A_{10}, A_{11}$ .", "$eval_{U_{9k+1}}(bot_{U_{9k+2}}(\\Pi ^6) \\setminus bot_{U_{9k+1}}(\\Pi ^6), X_0 \\cup \\dots \\cup X_{9k+1}) = $ $\\lbrace notenabled(t,k) \\text{:-} .", "\| $ $(\\lbrace trans(t), $ $ptarc(p,t,n_c,c,k), $ $holds(p,q_c,c,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{10k+1}, \\text{~where~} $ $q_c < n_c) \\text{~or~} $ $(\\lbrace notenabled(t,k) \\text{:-} .", "\| $ $(\\lbrace trans(t), $ $iptarc(p,t,n2_c,c,k), $ $holds(p,q_c,c,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{10k+1}, \\text{~where~} $ $q_c \\ge n2_c \\rbrace ) \\text{~or~} $ $(\\lbrace trans(t), $ $tptarc(p,t,n3_c,c,k), \\\\ holds(p,q_c,c,k) \\rbrace \\subseteq X_0 \\cup X_{9k+1}, \\text{~where~} q_c < n3_c) \\rbrace $ .", "Its answer set $X_{9k+2}=A_{12}^{ts=k}$ – using forced atom proposition and construction of $A_{12}$ .", "where, $q_c=m_{M_0(p)}(c)$ , and $n_c=m_{W(p,t)}(c)$ for an arc $(p,t) \\in E^-$ – by construction of $i\\ref {i:c:init}$ and $f\\ref {f:c:ptarc}$ predicates in $\\Pi ^6$ , and in an arc $(p,t) \\in E^-$ , $p \\in \\bullet t$ (by definition REF of preset) $n2_c=1$ – by construction of $iptarc$ predicates in $\\Pi ^6$ , meaning $q_c \\ge n2_c \\equiv q_c \\ge 1 \\equiv q_c > 0$ , $tptarc(p,t,n3_c,c,k)$ represents $n3_c=m_{QW(p,t)}(c)$ , where $(p,t) \\in Q$ thus, $notenabled(t,k) \\in X_{9k+1}$ represents $\\exists c \\in C, (\\exists p \\in \\bullet t : m_{M_0(p)}(c) < m_{W(p,t)}(c)) \\vee (\\exists p \\in I(t) : m_{M_0(p)}(c) > k) \\vee (\\exists (p,t) \\in Q : m_{M_0(p)}(c) < m_{QW(p,t)}(c))$ .", "$eval_{U_{9k+2}}(bot_{U_{9k+3}}(\\Pi ^6) \\setminus bot_{U_{9k+2}}(\\Pi ^6), X_0 \\cup \\dots \\cup X_{9k+2}) = \\lbrace enabled(t,k) \\text{:-}.", "\| trans(t) \\in X_0 \\cup \\dots \\cup X_{9k+2}, notenabled(t,k) \\notin X_0 \\cup \\dots \\cup X_{9k+2} \\rbrace $ .", "Its answer set is $X_{9k+3} = A_{13}^{ts=k}$ – using forced atom proposition and construction of $A_{13}$ .", "since an $enabled(t,k) \\in X_{9k+3}$ if $\\nexists ~notenabled(t,k) \\in X_0 \\cup \\dots \\cup X_{9k+2}$ ; which is equivalent to $\\nexists t, \\forall c \\in C, (\\nexists p \\in \\bullet t, m_{M_0(p)}(c) < m_{W(p,t)}(c)), (\\nexists p \\in I(t), m_{M_0(p)}(c) > k), (\\nexists (p,t) \\in Q : m_{M_0(p)}(c) < m_{QW(p,t)}(c) ), \\forall c \\in C, (\\forall p \\in \\bullet t: m_{M_0(p)}(c) \\ge m_{W(p,t)}(c)), (\\forall p \\in I(t) : m_{M_0(p)}(c) = k)$ .", "$eval_{U_{9k+3}}(bot_{U_{9k+4}}(\\Pi ^6) \\setminus bot_{U_{9k+3}}(\\Pi ^6), X_0 \\cup \\dots \\cup X_{9k+3}) = \\lbrace notprenabled(t,k) \\text{:-}.", "\| \\\\ \\lbrace enabled(t,k), transpr(t,p), enabled(tt,k), transpr(tt,pp) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{9k+3}, \\\\ pp < p \\rbrace $ .", "Its answer set is $X_{9k+4} = A_{23}^{ts=k}$ – using forced atom proposition and construction of $A_{23}$ .", "$enabled(t,k)$ represents $\\exists t \\in T, \\forall c \\in C, (\\forall p \\in \\bullet t, m_{M_0(p)}(c) \\ge m_{W(p,t)}(c)), \\\\ (\\forall p \\in I(t), m_{M_0(p)}(c) = k), (\\forall (p,t) \\in Q, m_{M_0(p)}(c) \\ge m_{QW(p,t)}(c))$ $enabled(tt,k)$ represents $\\exists tt \\in T, \\forall c \\in C, (\\forall pp \\in \\bullet tt, m_{M_0(pp)}(c) \\ge \\\\ m_{W(pp,tt)}(c)) \\wedge (\\forall pp \\in I(tt), m_{M_0(pp)}(c) = k), (\\forall (pp,tt) \\in Q, m_{M_0(pp)}(c) \\ge m_{QW(pp,tt)}(c))$ $transpr(t,p)$ represents $p=Z(t)$ – by construction $transpr(tt,pp)$ represents $pp=Z(tt)$ – by construction thus, $notprenabled(t,k)$ represents $\\forall c \\in C, (\\forall p \\in \\bullet t, m_{M_0(p)}(c) \\ge m_{W(p,t)}(c)) \\\\ \\wedge (\\forall p \\in I(t), m_{M_0(p)}(c) = k), \\exists tt \\in T, (\\forall pp \\in \\bullet tt, m_{M_0(pp)}(c) \\ge m_{W(pp,tt)}(c)) \\wedge (\\forall pp \\in I(tt), m_{M_0(pp)}(c) = k), Z(tt) < Z(t)$ which is equivalent to $(\\forall p \\in \\bullet t: M_0(p) \\ge W(p,t)) \\wedge (\\forall p \\in I(t), M_0(p) = k), \\exists tt \\in T, (\\forall pp \\in \\bullet tt, M_0(pp) \\ge W(pp,tt)), (\\forall pp \\in I(tt), M_0(pp) = k), Z(tt) < Z(t)$ – assuming multiset domain $C$ for all operations $eval_{U_{9k+4}}(bot_{U_{9k+5}}(\\Pi ^6) \\setminus bot_{U_{9k+4}}(\\Pi ^6), X_0 \\cup \\dots \\cup X_{9k+4}) = \\lbrace prenabled(t,k) \\text{:-}.", "\| $ $enabled(t,k) \\in X_0 \\cup \\dots \\cup X_{9k+4}, notprenabled(t,k) \\notin X_0 \\cup \\dots \\cup X_{9k+4} \\rbrace $ .", "Its answer set is $X_{9k+5} = A_{24}^{ts=k}$ – using forced atom proposition and construction of $A_{24}$ $enabled(t,k)$ represents $\\forall c \\in C, (\\forall p \\in \\bullet t, m_{M_0(p)}(c) \\ge m_{W(p,t)}(c)), $ $(\\forall p \\in I(t), m_{M_0(p)}(c) = k), (\\forall (p,t) \\in Q, m_{M_0(p)}(c) \\ge m_{QW(p,t)}(c)) \\equiv (\\forall p \\in \\bullet t, M_0(p) \\ge W(p,t)), $ $(\\forall p \\in I(t), M_0(p) = k), (\\forall (p,t) \\in Q, M_0(p) \\ge QW(p,t))$ – from REF above and assuming multiset domain $C$ for all operations $notprenabled(t,k)$ represents $(\\forall p \\in \\bullet t, M_0(p) \\ge W(p,t)), (\\forall p \\in I(t), \\\\ M_0(p) = k), (\\forall (p,t) \\in Q, M_0(p) \\ge QW(p,t)), \\exists tt \\in T, $ $(\\forall pp \\in \\bullet tt, M_0(pp) \\ge W(pp,tt)), $ $(\\forall pp \\in I(tt), M_0(pp) = k), (\\forall (pp,tt) \\in Q, M_0(pp) \\ge W(pp,tt)), \\\\ Z(tt) < Z(t)$ – from REF above and assuming multiset domain $C$ for all operations then, $prenabled(t,k)$ represents $(\\forall p \\in \\bullet t, M_0(p) \\ge W(p,t)), (\\forall p \\in I(t), \\\\ M_0(p) = k), (\\forall (p,t) \\in Q, M_0(p) \\ge QW(p,t)), \\nexists tt \\in T, $ $((\\forall pp \\in \\bullet tt, M_0(pp) \\ge W(pp,tt)), $ $(\\forall pp \\in I(tt), M_0(pp) = k), (\\forall (pp,tt) \\in Q, M_0(pp) \\ge W(pp,tt)), \\\\ Z(tt) < Z(t))$ – from (a), (b) and $enabled(t,k) \\in X_0 \\cup \\dots \\cup X_{9k+4}$ $eval_{U_{9k+5}}(bot_{U_{9k+6}}(\\Pi ^6) \\setminus bot_{U_{9k+5}}(\\Pi ^6), X_0 \\cup \\dots \\cup X_{9k+5}) = \\lbrace \\lbrace fires(t,k)\\rbrace \\text{:-}.", "\| \\\\ prenabled(t,k) \\text{~holds in~} X_0 \\cup \\dots \\cup X_{9k+5} \\rbrace $ .", "It has multiple answer sets $X_{9k+6.1}, \\dots , \\\\ X_{9k+6.n}$ , corresponding to elements of power set of $fires(t,k)$ atoms in $eval_{U_{9k+5}}(...)$ – using supported rule proposition.", "Since we are showing that the union of answer sets of $\\Pi ^6$ determined using splitting is equal to $A$ , we only consider the set that matches the $fires(t,k)$ elements in $A$ and call it $X_{9k+6}$ , ignoring the rest.", "Thus, $X_{9k+6} = A_{14}^{ts=k}$ , representing $T_k$ .", "in addition, for every $t$ such that $prenabled(t,k) \\in X_0 \\cup \\dots \\cup X_{9k+5}, R(t) \\ne \\emptyset $ ; $fires(t,k) \\in X_{9k+6}$ – per definition REF (firing set); requiring that a reset transition is fired when enabled thus, the firing set $T_k$ will not be eliminated by the constraint $f\\ref {f:c:pr:rptarc:elim}$ $eval_{U_{9k+6}}(bot_{U_{9k+7}}(\\Pi ^6) \\setminus bot_{U_{9k+6}}(\\Pi ^6), X_0 \\cup \\dots \\cup X_{9k+6}) = \\lbrace add(p,n_c,t,c,k) \\text{:-}.", "\| \\\\ \\lbrace fires(t,k), tparc(t,p,n_c,c,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{9k+6} \\rbrace \\cup \\lbrace del(p,n_c,t,c,k) \\text{:-}.", "\| \\\\ \\lbrace fires(t,k), ptarc(p,t,n_c,c,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{9k+6} \\rbrace $ .", "It's answer set is $X_{9k+7} = A_{15}^{ts=k} \\cup A_{16}^{ts=k}$ – using forced atom proposition and definitions of $A_{15}$ and $A_{16}$ .", "where, each $add$ atom is equivalent to $n_c=m_{W(t,p)}(c),c \\in C, p \\in t \\bullet $ , and each $del$ atom is equivalent to $n_c=m_{W(p,t)}(c), c \\in C, p \\in \\bullet t$ ; or $n_c=m_{M_0(p)}(c), c \\in C, p \\in R(t)$ , representing the effect of transitions in $T_k$ .", "$eval_{U_{9k+7}}(bot_{U_{9k+8}}(\\Pi ^6) \\setminus bot_{U_{9k+7}}(\\Pi ^6), X_0 \\cup \\dots \\cup X_{9k+7}) = \\lbrace tot\\_incr(p,qq_c,c,k) \\text{:-}.", "\| $ $qq_c=\\sum _{add(p,q_c,t,c,k) \\in X_0 \\cup \\dots \\cup X_{10k+7}}{q_c} \\rbrace \\cup \\lbrace tot\\_decr(p,qq_c,c,k) \\text{:-}.", "\| qq_c= \\\\ \\sum _{del(p,q_c,t,c,k) \\in X_0 \\cup \\dots \\cup X_{9k+7}}{q_c} \\rbrace $ .", "It's answer set is $X_{10k+8} = A_{17}^{ts=k} \\cup A_{18}^{ts=k}$ – using forced atom proposition and definitions of $A_{17}$ and $A_{18}$ .", "where, each $tot\\_incr(p,qq_c,c,k)$ , $qq_c=\\sum _{add(p,q_c,t,c,k) \\in X_0 \\cup \\dots X_{10k+7}}{q_c}$ $\\equiv qq_c=\\sum _{t \\in X_{9k+6}, p \\in t \\bullet }{m_{W(p,t)}(c)}$ , and each $tot\\_decr(p,qq_c,c,k)$ , $qq_c=\\sum _{del(p,q_c,t,c,k) \\in X_0 \\cup \\dots X_{10k+7}}{q_c}$ $\\equiv qq=\\sum _{t \\in X_{10k+6}, p \\in \\bullet t}{m_{W(t,p)}(c)} + \\sum _{t \\in X_{9k+6}, p \\in R(t)}{m_{M_0(p)}(c)}$ , represent the net effect of transitions in $T_k$ $eval_{U_{9k+8}}(bot_{U_{9k+9}}(\\Pi ^6) \\setminus bot_{U_{9k+8}}(\\Pi ^6), X_0 \\cup \\dots \\cup X_{9k+8}) = \\lbrace consumesmore(p,k) \\text{:-}.", "\| \\\\ \\lbrace holds(p,q_c,c,k), tot\\_decr(p,q1_c,c,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{9k+8}, q1_c > q_c \\rbrace \\cup \\\\ \\lbrace holds(p,q_c,c,1) \\text{:-}.", "\| \\lbrace holds(p,q1_c,c,k), tot\\_incr(p,q2_c,c,k), tot\\_decr(p,q3_c,c,k) \\rbrace \\\\ \\subseteq $ $X_0 \\cup \\dots \\cup X_{9k+6}, q_c=q1_c+q2_c-q3_c \\rbrace \\cup \\lbrace could\\_not\\_have(t,k) \\text{:-}.", "\| \\lbrace prenabled(t,k), \\\\ ptarc(s,t,q,c,k), holds(s,qq,c,k), tot\\_decr(s,qqq,c,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{9k+8}, \\\\ fires(t,k) \\notin X_0 \\cup \\dots \\cup X_{9k+8}, q > qq - qqq \\rbrace $ .", "It's answer set is $X_{10k+9} = A_{19}^{ts=k} \\cup A_{21}^{ts=k} \\cup A_{25}^{ts=k}$ – using forced atom proposition and definitions of $A_{19}, A_{21}, A_{25}$ .", "where, $consumesmore(p,k)$ represents $\\exists p : q_c=m_{M_0(p)}(c), \\\\ q1_c=$ $\\sum _{t \\in T_k, p \\in \\bullet t}{m_{W(p,t)}(c)}+\\sum _{t \\in T_k, p \\in R(t)}{m_{M_0(p)}(c)}, q1_c > q_c, c \\in C$ , indicating place $p$ that will be over consumed if $T_k$ is fired, as defined in definition REF (conflicting transition) $holds(p,q_c,c,k+1)$ if $q_c=m_{M_0(p)}(c)+\\sum _{t \\in T_k, p \\in t \\bullet }{m_{W(t,p)}(c)}- \\\\ (\\sum _{t \\in T_k, p \\in \\bullet t}{m_{W(p,t)}(c)}+ \\sum _{t \\in T_k, p \\in R(t)}{m_{M_0(p)}(c)})$ represented by $q_c=m_{M_1(p)}(c)$ for some $c \\in C$ – by construction of $\\Pi ^6$ $could\\_not\\_have(t,k)$ if $(\\forall p \\in \\bullet t, W(p,t) \\le M_0(p)), (\\forall p \\in I(t), M_0(p) = k), (\\forall (p,t) \\in Q, $ $M_0(p) \\ge WQ(p,t)), \\nexists tt \\in T, (\\forall pp \\in \\bullet tt, W(pp,tt) \\le M_{ts}(pp)), (\\forall pp \\in I(tt), M_0(pp) = k), (\\forall (pp,tt) \\in Q, $ $M_0(pp) \\ge QW(pp,tt)), Z(tt) < Z(t)$ , and $q_c > m_{M_0(s)}(c) - (\\sum _{t^{\\prime } \\in T_k, s \\in \\bullet t^{\\prime }}{m_{W(s,t^{\\prime })}(c)}+ \\sum _{t^{\\prime } \\in T_k, s \\in R(t)}{m_{M_0(s)}(c)}), $ $q_c = m_{W(s,t)}(c) \\text{~if~} s \\in \\bullet t \\text{~or~} m_{M_0(s)}(c) \\text{~otherwise}$ for some $c \\in C$ , which becomes $q > M_0(s) - (\\sum _{t^{\\prime } \\in T_k, s \\in \\bullet t^{\\prime }}{W(s,t^{\\prime })}+ \\sum _{t^{\\prime } \\in T_k, s \\in R(t)}{M_0(s)}), q = W(s,t) \\text{~if~} s \\in \\bullet t \\text{~or~} M_0(s) \\text{~otherwise}$ for all $c \\in C$ (i), (ii) above combined match the definition of $A_{25}$ $X_{9k+9}$ does not contain $could\\_not\\_have(t,k)$ , when $prenabled(t,k) \\in X_0 \\cup \\dots \\cup X_{9k+6}$ and $fires(t,k) \\notin X_0 \\cup \\dots \\cup X_{9k+5}$ due to construction of $A$ , encoding of $a\\ref {a:c:maxfire:cnh}$ and its body atoms.", "As a result it is not eliminated by the constraint $a\\ref {a:c:maxfire:elim}$ $eval_{U_{9k+9}}(bot_{U_{9k+10}}(\\Pi ^6) \\setminus bot_{U_{9k+9}}(\\Pi ^6), X_0 \\cup \\dots \\cup X_{9k+9}) = \\lbrace consumesmore \\text{:-}.", "\| \\\\ \\lbrace consumesmore(p,k) \\rbrace \\subseteq A \\rbrace $ .", "It's answer set is $X_{9k+10} = A_{20}$ – using forced atom proposition and the definition of $A_{20}$ $X_{9k+10}$ will be empty since none of $consumesmore(p,0),\\dots , \\\\ consumesmore(p,k)$ hold in $X_0 \\cup \\dots \\cup X_{9k+9}$ due to the construction of $A$ , encoding of $a\\ref {a:c:overc:place}$ and its body atoms.", "As a result, it is not eliminated by the constraint $a\\ref {a:c:overc:elim}$ The set $X = X_0 \\cup \\dots \\cup X_{9k+10}$ is the answer set of $\\Pi ^6$ by the splitting sequence theorem REF .", "Each $X_i, 0 \\le i \\le 9k+10$ matches a distinct portion of $A$ , and $X = A$ , thus $A$ is an answer set of $\\Pi ^6$ .", "Next we show (REF ): Given $\\Pi ^6$ be the encoding of a Petri Net $PN(P,T,E,C,W,R,I,$ $Q,QW,Z)$ with initial marking $M_0$ , and $A$ be an answer set of $\\Pi ^6$ that satisfies (REF ) and (REF ), then we can construct $X=M_0,T_k,\\dots ,M_k,T_k,M_{k+1}$ from $A$ , such that it is an execution sequence of $PN$ .", "We construct the $X$ as follows: $M_i = (M_i(p_0), \\dots , M_i(p_n))$ , where $\\lbrace holds(p_0,m_{M_i(p_0)}(c),c,i), \\dots ,\\\\ holds(p_n,m_{M_i(p_n)}(c),c,i) \\rbrace \\subseteq A$ , for $c \\in C, 0 \\le i \\le k+1$ $T_i = \\lbrace t : fires(t,i) \\in A\\rbrace $ , for $0 \\le i \\le k$ and show that $X$ is indeed an execution sequence of $PN$ .", "We show this by induction over $k$ (i.e.", "given $M_k$ , $T_k$ is a valid firing set and its firing produces marking $M_{k+1}$ ).", "Base case: Let $k=0$ , and $M_0$ is a valid marking in $X$ for $PN$ , show [(1)] $T_0$ is a valid firing set for $M_0$ , and firing $T_0$ in $M_0$ produces marking $M_1$ .", "We show $T_0$ is a valid firing set for $M_0$ .", "Let $\\lbrace fires(t_0,0), \\dots , fires(t_x,0) \\rbrace $ be the set of all $fires(\\dots ,0)$ atoms in $A$ , Then for each $fires(t_i,0) \\in A$ $prenabled(t_i,0) \\in A$ – from rule $a\\ref {a:c:prfires}$ and supported rule proposition Then $enabled(t_i,0) \\in A$ – from rule $a\\ref {a:c:prenabled}$ and supported rule proposition And $notprenabled(t_i,0) \\notin A$ – from rule $a\\ref {a:c:prenabled}$ and supported rule proposition For $enabled(t_i,0) \\in A$ $notenabled(t_i,0) \\notin A$ – from rule $e\\ref {e:c:enabled}$ and supported rule proposition Then either of $body(e\\ref {e:c:ne:ptarc})$ , $body(e\\ref {e:c:ne:iptarc})$ , or $body(e\\ref {e:c:ne:tptarc})$ must not hold in $A$ for $t_i$ – from rules $e\\ref {e:c:ne:ptarc},e\\ref {e:c:ne:iptarc},e\\ref {e:c:ne:tptarc}$ and forced atom proposition Then $q_c \\lnot < {n_i}_c \\equiv q_c \\ge {n_i}_c$ in $e\\ref {e:c:ne:ptarc}$ for all $\\lbrace holds(p,q_c,c,0), \\\\ ptarc(p,t_i,{n_i}_c,c,0)\\rbrace \\subseteq A$ – from $e:\\ref {e:c:ne:ptarc}$ , forced atom proposition, and given facts ($holds(p,q_c,c,0) \\in A, ptarc(p,t_i,{n_i}_c,0) \\in A$ ) And $q_c \\lnot \\ge {n_i}_c \\equiv q_c < {n_i}_c$ in $e\\ref {e:c:ne:iptarc}$ for all $\\lbrace holds(p,q_c,c,0), \\\\ iptarc(p,t_i,{n_i}_c,c,0) \\rbrace \\subseteq A, {n_i}_c=1$ ; $q_c > {n_i}_c \\equiv q_c = 0$ – from $e\\ref {e:c:ne:iptarc}$ , forced atom proposition, given facts ($holds(p,q_c,c,0) \\in A, \\\\ iptarc(p,t_i,1,c,0) \\in A$ ), and $q_c$ is a positive integer And $q_c \\lnot < {n_i}_c \\equiv q_c \\ge {n_i}_c$ in $e\\ref {e:c:ne:tptarc}$ for all $\\lbrace holds(p,q_c,c,0), \\\\ tptarc(p,t_i,{n_i}_c,c,0) \\rbrace \\subseteq A$ – from $e\\ref {e:c:ne:tptarc}$ , forced atom proposition, and given facts Then $\\forall c \\in C, (\\forall p \\in \\bullet t_i, m_{M_0(p)}(c) \\ge m_{W(p,t_i)}(c)) \\wedge (\\forall p \\in I(t_i), \\\\ m_{M_0(p)}(c) = 0) \\wedge (\\forall (p,t_i) \\in Q, m_{M_0(p)}(c) \\ge m_{QW(p,t_i)}(c))$ – from $i\\ref {i:c:init},f\\ref {f:c:ptarc},f\\ref {f:c:iptarc},f\\ref {f:c:tptarc}$ construction, definition REF of preset $\\bullet t_i$ in $PN$ , definition REF of enabled transition in $PN$ , and that the construction of reset arcs by $f\\ref {f:c:rptarc}$ ensures $notenabled(t,0)$ is never true for a reset arc, where $holds(p,q_c,c,0) \\in A$ represents $q_c=m_{M_0(p)}(c)$ , $ptarc(p,t_i,{n_i}_c,0) \\in A$ represents ${n_i}_c=m_{W(p,t_i)}(c)$ , ${n_i}_c = m_{M_0(p)}(c)$ .", "Which is equivalent to $(\\forall p \\in \\bullet t_i, M_0(p) \\ge W(p,t_i)) \\wedge (\\forall p \\in I(t_i), M_0(p) = 0) \\wedge (\\forall (p,t_i) \\in Q, M_0(p) \\ge QW(p,t_i))$ – assuming multiset domain $C$ For $notprenabled(t_i,0) \\notin A$ Either $(\\nexists enabled(tt,0) \\in A : pp < p_i)$ or $(\\forall enabled(tt,0) \\in A : pp \\lnot < p_i)$ where $pp = Z(tt), p_i = Z(t_i)$ – from rule $a\\ref {a:c:prne},f\\ref {f:c:pr}$ and forced atom proposition This matches the definition of an enabled priority transition Then $t_i$ is enabled and can fire in $PN$ , as a result it can belong to $T_0$ – from definition REF of enabled transition And $consumesmore \\notin A$ , since $A$ is an answer set of $\\Pi ^6$ – from rule $a\\ref {a:c:overc:gen}$ and supported rule proposition Then $\\nexists consumesmore(p,0) \\in A$ – from rule $a\\ref {a:c:overc:place}$ and supported rule proposition Then $\\nexists \\lbrace holds(p,q_c,c,0), tot\\_decr(p,q1_c,c,0) \\rbrace \\subseteq A, q1_c>q_c$ in $body(a\\ref {a:c:overc:place})$ – from $a\\ref {a:c:overc:place}$ and forced atom proposition Then $\\nexists c \\in C \\nexists p \\in P, (\\sum _{t_i \\in \\lbrace t_0,\\dots ,t_x\\rbrace , p \\in \\bullet t_i}{m_{W(p,t_i)}(c)}+ \\\\ \\sum _{t_i \\in \\lbrace t_0,\\dots ,t_x\\rbrace , p \\in R(t_i)}{m_{M_0(p)}(c)}) > m_{M_0(p)}(c)$ – from the following $holds(p,q_c,c,0)$ represents $q_c=m_{M_0(p)}(c)$ – from rule $i\\ref {i:c:init}$ encoding, given $tot\\_decr(p,q1_c,c,0) \\in A$ if $\\lbrace del(p,{q1_0}_c,t_0,c,0), \\dots , \\\\ del(p,{q1_x}_c,t_x,c,0) \\rbrace \\subseteq A$ , where $q1_c = {q1_0}_c+\\dots +{q1_x}_c$ – from $r\\ref {r:c:totdecr}$ and forced atom proposition $del(p,{q1_i}_c,t_i,c,0) \\in A$ if $\\lbrace fires(t_i,0), ptarc(p,t_i,{q1_i}_c,c,0) \\rbrace \\subseteq A$ – from $r\\ref {r:c:del}$ and supported rule proposition $del(p,{q1_i}_c,t_i,c,0)$ either represents removal of ${q1_i}_c = m_{W(p,t_i)}(c)$ tokens from $p \\in \\bullet t_i$ ; or it represents removal of ${q1_i}_c = m_{M_0(p)}(c)$ tokens from $p \\in R(t_i)$ – from rules $r\\ref {r:c:del},f\\ref {f:c:ptarc},f\\ref {f:c:rptarc}$ , supported rule proposition, and definition REF of transition execution in $PN$ Then the set of transitions in $T_0$ do not conflict – by the definition REF of conflicting transitions And for each $prenabled(t_j,0) \\in A$ and $fires(t_j,0) \\notin A$ , $could\\_not\\_have(t_j,0) \\in A$ , since $A$ is an answer set of $\\Pi ^6$ - from rule $a\\ref {a:c:prmaxfire:elim}$ and supported rule proposition Then $\\lbrace prenabled(t_j,0), holds(s,qq_c,c,0), ptarc(s,t_j,q_c,c,0), \\\\ tot\\_decr(s,qqq_c,c,0) \\rbrace \\subseteq A$ , such that $q_c > qq_c - qqq_c$ and $fires(t_j,0) \\notin A$ - from rule $a\\ref {a:c:prmaxfire:cnh}$ and supported rule proposition Then for an $s \\in \\bullet t_j \\cup R(t_j)$ , $q_c > m_{M_0(s)}(c) - (\\sum _{t_i \\in T_0, s \\in \\bullet t_i}{m_{W(s,t_i)}(c)} + \\sum _{t_i \\in T_0, s \\in R(t_i)}{m_{M_0(s)}(c))}$ , where $q_c=m_{W(s,t_j)}(c) \\text{~if~} s \\in \\bullet t_j, \\text{~or~} m_{M_0(s)}(c)$ $ \\text{~otherwise}$ – from the following $ptarc(s,t_i,q_c,c,0)$ represents $q_c=m_{W(s,t_i)}(c)$ if $(s,t_i) \\in E^-$ or $q_c=m_{M_0(s)}(c)$ if $s \\in R(t_i)$ – from rule $f\\ref {f:c:ptarc},f\\ref {f:c:rptarc}$ construction $holds(s,qq_c,c,0)$ represents $qq_c=m_{M_0(s)}(c)$ – from $i\\ref {i:c:init}$ construction $tot\\_decr(s,qqq_c,c,0) \\in A$ if $\\lbrace del(s,{qqq_0}_c,t_0,c,0), \\dots , \\\\ del(s,{qqq_x}_c,t_x,c,0) \\rbrace \\subseteq A$ – from rule $r\\ref {r:c:totdecr}$ construction and supported rule proposition $del(s,{qqq_i}_c,t_i,c,0) \\in A$ if $\\lbrace fires(t_i,0), ptarc(s,t_i,{qqq_i}_c,c,0) \\rbrace \\subseteq A$ – from rule $r\\ref {r:c:del}$ and supported rule proposition $del(s,{qqq_i}_c,t_i,c,0)$ either represents ${qqq_i}_c = m_{W(s,t_i)}(c) : t_i \\in T_0, (s,t_i) \\in E^-$ , or ${qqq_i}_c = m_{M_0(t_i)}(c) : t_i \\in T_0, s \\in R(t_i)$ – from rule $f\\ref {f:c:ptarc},f\\ref {f:c:rptarc}$ construction $tot\\_decr(q,qqq_c,c,0)$ represents $\\sum _{t_i \\in T_0, s \\in \\bullet t_i}{m_{W(s,t_i)}(c)} + \\\\ \\sum _{t_i \\in T_0, s \\in R(t_i)}{m_{M_0(s)}(c)}$ – from (C,D,E) above Then firing $T_0 \\cup \\lbrace t_j \\rbrace $ would have required more tokens than are present at its source place $s \\in \\bullet t_j \\cup R(t_j)$ .", "Thus, $T_0$ is a maximal set of transitions that can simultaneously fire.", "And for each reset transition $t_r$ with $prenabled(t_r,0) \\in A$ , $fires(t_r,0) \\in A$ , since $A$ is an answer set of $\\Pi ^6$ - from rule $f\\ref {f:c:pr:rptarc:elim}$ and supported rule proposition Then, the firing set $T_0$ satisfies the reset-transition requirement of definition REF (firing set) Then $\\lbrace t_0, \\dots , t_x\\rbrace = T_0$ – using 1(a),1(b),1(d) above; and using 1(c) it is a maximal firing set We show $M_1$ is produced by firing $T_0$ in $M_0$ .", "Let $holds(p,q_c,c,1) \\in A$ Then $\\lbrace holds(p,q1_c,c,0), tot\\_incr(p,q2_c,c,0), tot\\_decr(p,q3_c,c,0) \\rbrace \\subseteq A : q_c=q1_c+q2_c-q3_c$ – from rule $r\\ref {r:c:nextstate}$ and supported rule proposition Then, $holds(p,q1_c,c,0) \\in A$ represents $q1_c=m_{M_0(p)}(c)$ – given, construction; and $\\lbrace add(p,{q2_0}_c,t_0,c,0), \\dots , $ $add(p,{q2_j}_c,t_j,c,0)\\rbrace \\subseteq A : {q2_0}_c + \\dots + {q2_j}_c = q2_c$ and $\\lbrace del(p,{q3_0}_c,t_0,c,0), \\dots , $ $del(p,{q3_l}_c,t_l,c,0)\\rbrace \\subseteq A : {q3_0}_c + \\dots + {q3_l}_c = q3_c$ – rules $r\\ref {r:c:totincr},r\\ref {r:c:totdecr}$ and supported rule proposition, respectively Then $\\lbrace fires(t_0,0), \\dots , fires(t_j,0) \\rbrace \\subseteq A$ and $\\lbrace fires(t_0,0), \\dots , fires(t_l,0) \\rbrace \\subseteq A$ – rules $r\\ref {r:c:add},r\\ref {r:c:del}$ and supported rule proposition, respectively Then $\\lbrace fires(t_0,0), \\dots , $ $fires(t_j,0) \\rbrace \\cup $ $\\lbrace fires(t_0,0), \\dots , $ $fires(t_l,0) \\rbrace \\subseteq $ $A = \\lbrace fires(t_0,0), \\dots , $ $fires(t_x,0) \\rbrace \\subseteq A$ – set union of subsets Then for each $fires(t_x,0) \\in A$ we have $t_x \\in T_0$ – already shown in item REF above Then $q_c = m_{M_0(p)}(c) + \\sum _{t_x \\in T_0 \\wedge p \\in t_x \\bullet }{m_{W(t_x,p)}(c)} - (\\sum _{t_x \\in T_0 \\wedge p \\in \\bullet t_x}{m_{W(p,t_x)}(c)} + \\sum _{t_x \\in T_0 \\wedge p \\in R(t_x)}{m_{M_0(p)}(c)})$ – from (REF ) above and the following Each $add(p,{q_j}_c,t_j,c,0) \\in A$ represents ${q_j}_c=m_{W(t_j,p)}(c)$ for $p \\in t_j \\bullet $ – rule $r\\ref {r:c:add},f\\ref {f:c:tparc}$ encoding, and definition REF of transition execution in $PN$ Each $del(p,t_y,{q_y}_c,c,0) \\in A$ represents either ${q_y}_c=m_{W(p,t_y)}(c)$ for $p \\in \\bullet t_y$ , or ${q_y}_c=m_{M_0(p)}(c)$ for $p \\in R(t_y)$ – from rule $r\\ref {r:c:del},f\\ref {f:c:ptarc}$ encoding and definition REF of transition execution in $PN$ ; or from rule $r\\ref {r:c:del},f\\ref {f:c:rptarc}$ encoding and definition of reset arc in $PN$ Each $tot\\_incr(p,q2_c,c,0) \\in A$ represents $q2_c=\\sum _{t_x \\in T_0 \\wedge p \\in t_x \\bullet }{m_{W(t_x,p)}(c)}$ – aggregate assignment atom semantics in rule $r\\ref {r:c:totincr}$ Each $tot\\_decr(p,q3_c,c,0) \\in A$ represents $q3_c=\\sum _{t_x \\in T_0 \\wedge p \\in \\bullet t_x}{m_{W(p,t_x)}(c)} + $ $\\sum _{t_x \\in T_0 \\wedge p \\in R(t_x)}{m_{M_0(p)}(c)}$ – aggregate assignment atom semantics in rule $r\\ref {r:c:totdecr}$ Then, $m_{M_1(p)}(c) = q_c$ – since $holds(p,q_c,c,1) \\in A$ encodes $q_c=m_{M_1(p)}(c)$ – from construction Inductive Step: Let $k > 0$ , and $M_k$ is a valid marking in $X$ for $PN$ , show [(1)] $T_k$ is a valid firing set for $M_k$ , and firing $T_k$ in $M_k$ produces marking $M_{k+1}$ .", "We show that $T_k$ is a valid firing set in $M_k$ .", "Let $\\lbrace fires(T_k,k), \\dots , fires(t_x,k) \\rbrace $ be the set of all $fires(\\dots ,k)$ atoms in $A$ , Then for each $fires(t_i,k) \\in A$ $prenabled(t_i,k) \\in A$ – from rule $a\\ref {a:c:prfires}$ and supported rule proposition Then $enabled(t_i,k) \\in A$ – from rule $a\\ref {a:c:prenabled}$ and supported rule proposition And $notprenabled(t_i,k) \\notin A$ – from rule $a\\ref {a:c:prenabled}$ and supported rule proposition For $enabled(t_i,k) \\in A$ $notenabled(t_i,k) \\notin A$ – from rule $e\\ref {e:c:enabled}$ and supported rule proposition Then either of $body(e\\ref {e:c:ne:ptarc})$ , $body(e\\ref {e:c:ne:iptarc})$ , or $body(e\\ref {e:c:ne:tptarc})$ must not hold in $A$ for $t_i$ – from rules $e\\ref {e:c:ne:ptarc},e\\ref {e:c:ne:iptarc},e\\ref {e:c:ne:tptarc}$ and forced atom proposition Then $q_c \\lnot < {n_i}_c \\equiv q_c \\ge {n_i}_c$ in $e\\ref {e:c:ne:ptarc}$ for all $\\lbrace holds(p,q_c,c,k), \\\\ ptarc(p,t_i,{n_i}_c,c,k)\\rbrace \\subseteq A$ – from $e\\ref {e:c:ne:ptarc}$ , forced atom proposition, and given facts ($holds(p,q_c,c,k) \\in A, ptarc(p,t_i,{n_i}_c,k) \\in A$ ) And $q_c \\lnot \\ge {n_i}_c \\equiv q_c < {n_i}_c$ in $e\\ref {e:c:ne:iptarc}$ for all $\\lbrace holds(p,q_c,c,k), \\\\ iptarc(p,t_i,{n_i}_c,c,k) \\rbrace \\subseteq A, {n_i}_c=1$ ; $q_c > {n_i}_c \\equiv q_c = 0$ – from $e\\ref {e:c:ne:iptarc}$ , forced atom proposition, given facts ($holds(p,q_c,c,k) \\in A, \\\\ iptarc(p,t_i,1,c,k) \\in A$ ), and $q_c$ is a positive integer And $q_c \\lnot < {n_i}_c \\equiv q_c \\ge {n_i}_c$ in $e\\ref {e:c:ne:tptarc}$ for all $\\lbrace holds(p,q_c,c,k), \\\\ tptarc(p,t_i,{n_i}_c,c,k) \\rbrace \\subseteq A$ – from $e\\ref {e:c:ne:tptarc}$ , forced atom proposition, and given facts Then $\\forall c \\in C, (\\forall p \\in \\bullet t_i, m_{M_k(p)}(c) \\ge m_{W(p,t_i)}(c)) \\wedge (\\forall p \\in I(t_i), $ $m_{M_k(p)}(c) = k) \\wedge (\\forall (p,t_i) \\in Q, m_{M_k(p)}(c) \\ge m_{QW(p,t_i)}(c))$ – from $f\\ref {f:c:ptarc},f\\ref {f:c:iptarc},f\\ref {f:c:tptarc}$ construction, inductive assumption, definition REF of preset $\\bullet t_i$ in $PN$ , definition REF of enabled transition in $PN$ , and that the construction of reset arcs by $f\\ref {f:c:rptarc}$ ensures $notenabled(t,k)$ is never true for a reset arc, where $holds(p,q_c,c,k) \\in A$ represents $q_c=m_{M_k(p)}(c)$ , $ptarc(p,t_i,{n_i}_c,k) \\in A$ represents ${n_i}_c=m_{W(p,t_i)}(c)$ , ${n_i}_c = m_{M_k(p)}(c)$ .", "Which is equivalent to $(\\forall p \\in \\bullet t_i, M_k(p) \\ge W(p,t_i)) \\wedge (\\forall p \\in I(t_i), M_k(p) = k) \\wedge (\\forall (p,t_i) \\in Q, M_k(p) \\ge QW(p,t_i))$ – assuming multiset domain $C$ For $notprenabled(t_i,k) \\notin A$ Either $(\\nexists enabled(tt,k) \\in A : pp < p_i)$ or $(\\forall enabled(tt,k) \\in A : pp \\lnot < p_i)$ where $pp = Z(tt), p_i = Z(t_i)$ – from rule $a\\ref {a:c:prne}, f\\ref {f:c:pr}$ and forced atom proposition This matches the definition of an enabled priority transition Then $t_i$ is enabled and can fire in $PN$ , as a result it can belong to $T_k$ – from definition REF of enabled transition And $consumesmore \\notin A$ , since $A$ is an answer set of $\\Pi ^6$ – from rule $a\\ref {a:c:overc:elim}$ and supported rule proposition Then $\\nexists consumesmore(p,k) \\in A$ – from rule $a\\ref {a:c:overc:elim}$ and supported rule proposition Then $\\nexists \\lbrace holds(p,q_c,c,k), tot\\_decr(p,q1_c,c,k) \\rbrace \\subseteq A, q1_c>q_c$ in $body(a\\ref {a:c:overc:place})$ – from $a\\ref {a:c:overc:place}$ and forced atom proposition Then $\\nexists c \\in C \\nexists p \\in P, (\\sum _{t_i \\in \\lbrace T_k,\\dots ,t_x\\rbrace , p \\in \\bullet t_i}{m_{W(p,t_i)}(c)}+ \\\\ \\sum _{t_i \\in \\lbrace T_k,\\dots ,t_x\\rbrace , p \\in R(t_i)}{m_{M_k(p)}(c)}) > m_{M_k(p)}(c)$ – from the following $holds(p,q_c,c,k)$ represents $q_c=m_{M_k(p)}(c)$ – from inductive assumption and construction, given $tot\\_decr(p,q1_c,c,k) \\in A$ if $\\lbrace del(p,{q1_0}_c,T_k,c,k), \\dots , \\\\ del(p,{q1_x}_c,t_x,c,k) \\rbrace \\subseteq A$ , where $q1_c = {q1_0}_c+\\dots +{q1_x}_c$ – from $r\\ref {r:c:totdecr}$ and forced atom proposition $del(p,{q1_i}_c,t_i,c,k) \\in A$ if $\\lbrace fires(t_i,k), ptarc(p,t_i,{q1_i}_c,c,k) \\rbrace \\subseteq A$ – from $r\\ref {r:c:del}$ and supported rule proposition $del(p,{q1_i}_c,t_i,c,k)$ either represents removal of ${q1_i}_c = m_{W(p,t_i)}(c)$ tokens from $p \\in \\bullet t_i$ ; or it represents removal of ${q1_i}_c = m_{M_k(p)}(c)$ tokens from $p \\in R(t_i)$ – from rules $r\\ref {r:c:del},f\\ref {f:c:ptarc},f\\ref {f:c:rptarc}$ , supported rule proposition, and definition REF of transition execution in $PN$ Then the set of transitions in $T_k$ do not conflict – by the definition REF of conflicting transitions And for each $prenabled(t_j,k) \\in A$ and $fires(t_j,k) \\notin A$ , $could\\_not\\_have(t_j,k) \\in A$ , since $A$ is an answer set of $\\Pi ^6$ - from rule $a\\ref {a:c:prmaxfire:elim}$ and supported rule proposition Then $\\lbrace prenabled(t_j,k), holds(s,qq_c,c,k), ptarc(s,t_j,q_c,c,k), \\\\ tot\\_decr(s,qqq_c,c,k) \\rbrace \\subseteq A$ , such that $q_c > qq_c - qqq_c$ and $fires(t_j,k) \\notin A$ - from rule $a\\ref {a:c:prmaxfire:cnh}$ and supported rule proposition Then for an $s \\in \\bullet t_j \\cup R(t_j)$ , $q_c > m_{M_k(s)}(c) - (\\sum _{t_i \\in T_k, s \\in \\bullet t_i}{m_{W(s,t_i)}(c)} + \\sum _{t_i \\in T_k, s \\in R(t_i)}{m_{M_k(s)}(c))}$ , where $q_c=m_{W(s,t_j)}(c) \\text{~if~} s \\in \\bullet t_j, \\text{~or~} m_{M_k(s)}(c) $ $\\text{~otherwise}$ .", "$ptarc(s,t_i,q_c,c,k)$ represents $q_c=m_{W(s,t_i)}(c)$ if $(s,t_i) \\in E^-$ or $q_c=m_{M_k(s)}(c)$ if $s \\in R(t_i)$ – from rule $f\\ref {f:c:ptarc},f\\ref {f:c:rptarc}$ construction $holds(s,qq_c,c,k)$ represents $qq_c=m_{M_k(s)}(c)$ – from inductive assumption and construction $tot\\_decr(s,qqq_c,c,k) \\in A$ if $\\lbrace del(s,{qqq_0}_c,T_k,c,k), \\dots , \\\\ del(s,{qqq_x}_c,t_x,c,k) \\rbrace \\subseteq A$ – from rule $r\\ref {r:c:totdecr}$ construction and supported rule proposition $del(s,{qqq_i}_c,t_i,c,k) \\in A$ if $\\lbrace fires(t_i,k), ptarc(s,t_i,{qqq_i}_c,c,k) \\rbrace \\subseteq A$ – from rule $r\\ref {r:c:del}$ and supported rule proposition $del(s,{qqq_i}_c,t_i,c,k)$ either represents ${qqq_i}_c = m_{W(s,t_i)}(c) : t_i \\in T_k, (s,t_i) \\in E^-$ , or ${qqq_i}_c = m_{M_k(t_i)}(c) : t_i \\in T_k, s \\in R(t_i)$ – from rule $f\\ref {f:c:ptarc},f\\ref {f:c:rptarc}$ construction $tot\\_decr(q,qqq_c,c,k)$ represents $\\sum _{t_i \\in T_k, s \\in \\bullet t_i}{m_{W(s,t_i)}(c)} + \\\\ \\sum _{t_i \\in T_k, s \\in R(t_i)}{m_{M_k(s)}(c)}$ – from (C,D,E) above Then firing $T_k \\cup \\lbrace t_j \\rbrace $ would have required more tokens than are present at its source place $s \\in \\bullet t_j \\cup R(t_j)$ .", "Thus, $T_k$ is a maximal set of transitions that can simultaneously fire.", "And for each reset transition $t_r$ with $prenabled(t_r,k) \\in A$ , $fires(t_r,k) \\in A$ , since $A$ is an answer set of $\\Pi ^6$ - from rule $f\\ref {f:c:pr:rptarc:elim}$ and supported rule proposition Then the firing set $T_k$ satisfies the reset transition requirement of definition REF (firing set) Then $\\lbrace t_0, \\dots , t_x\\rbrace = T_k$ – using 1(a),1(b), 1(d) above; and using 1(c) it is a maximal firing set We show that $M_{k+1}$ is produced by firing $T_k$ in $M_k$ .", "Let $holds(p,q_c,c,k+1) \\in A$ Then $\\lbrace holds(p,q1_c,c,k), tot\\_incr(p,q2_c,c,k), tot\\_decr(p,q3_c,c,k) \\rbrace \\subseteq A : q_c=q1_c+q2_c-q3_c$ – from rule $r\\ref {r:c:nextstate}$ and supported rule proposition Then, $holds(p,q1_c,c,k) \\in A$ represents $q1_c=m_{M_k(p)}(c)$ – inductive assumption; and $\\lbrace add(p,{q2_0}_c,T_k,c,k), \\dots , $ $add(p,{q2_j}_c,t_j,c,k)\\rbrace \\subseteq A : {q2_0}_c + \\dots + {q2_j}_c = q2_c$ and $\\lbrace del(p,{q3_0}_c,T_k,c,k), \\dots , $ $del(p,{q3_l}_c,t_l,c,k)\\rbrace \\subseteq A : {q3_0}_c + \\dots + {q3_l}_c = q3_c$ – rules $r\\ref {r:c:totincr},r\\ref {r:c:totdecr}$ and supported rule proposition, respectively Then $\\lbrace fires(T_k,k), \\dots , fires(t_j,k) \\rbrace \\subseteq A$ and $\\lbrace fires(T_k,k), \\dots , \\\\ fires(t_l,k) \\rbrace \\subseteq A$ – rules $r\\ref {r:c:add},r\\ref {r:c:del}$ and supported rule proposition, respectively Then $\\lbrace fires(T_k,k), \\dots , $ $fires(t_j,k) \\rbrace \\cup $ $\\lbrace fires(T_k,k), \\dots , $ $fires(t_l,k) \\rbrace \\subseteq $ $A = \\lbrace fires(T_k,k), \\dots , fires(t_x,k) \\rbrace \\subseteq A$ – set union of subsets Then for each $fires(t_x,k) \\in A$ we have $t_x \\in T_k$ – already shown in item REF above Then $q_c = m_{M_k(p)}(c) + \\sum _{t_x \\in T_k \\wedge p \\in t_x \\bullet }{m_{W(t_x,p)}(c)} - (\\sum _{t_x \\in T_k \\wedge p \\in \\bullet t_x}{m_{W(p,t_x)}(c)} + \\sum _{t_x \\in T_k \\wedge p \\in R(t_x)}{m_{M_k(p)}(c)})$ – from (REF ) above and the following Each $add(p,{q_j}_c,t_j,c,k) \\in A$ represents ${q_j}_c=m_{W(t_j,p)}(c)$ for $p \\in t_j \\bullet $ – rule $r\\ref {r:c:add},f\\ref {f:c:tparc}$ encoding, and definition REF of transition execution in $PN$ Each $del(p,t_y,{q_y}_c,c,k) \\in A$ represents either ${q_y}_c=m_{W(p,t_y)}(c)$ for $p \\in \\bullet t_y$ , or ${q_y}_c=m_{M_k(p)}(c)$ for $p \\in R(t_y)$ – from rule $r\\ref {r:c:del},f\\ref {f:c:ptarc}$ encoding and definition REF of transition execution in $PN$ ; or from rule $r\\ref {r:c:del},f\\ref {f:c:rptarc}$ encoding and definition of reset arc in $PN$ Each $tot\\_incr(p,q2_c,c,k) \\in A$ represents $q2_c=\\sum _{t_x \\in T_k \\wedge p \\in t_x \\bullet }{m_{W(t_x,p)}(c)}$ – aggregate assignment atom semantics in rule $r\\ref {r:c:totincr}$ Each $tot\\_decr(p,q3_c,c,k) \\in A$ represents $q3_c=\\sum _{t_x \\in T_k \\wedge p \\in \\bullet t_x}{m_{W(p,t_x)}(c)} + \\sum _{t_x \\in T_k \\wedge p \\in R(t_x)}{m_{M_k(p)}(c)}$ – aggregate assignment atom semantics in rule $r\\ref {r:c:totdecr}$ Then, $m_{M_{k+1}(p)}(c) = q_c$ – since $holds(p,q_c,c,k+1) \\in A$ encodes $q_c=m_{M_{k+1}(p)}(c)$ – from construction As a result, for any $n > k$ , $T_n$ will be a valid firing set for $M_n$ and $M_{n+1}$ will be its target marking.", "Conclusion: Since both (REF ) and (REF ) hold, $X=M_0,T_k,M_1,\\dots ,M_k,T_{k+1}$ is an execution sequence of $PN(P,T,E,C,W,R,I,Q,QW,Z)$ (w.r.t $M_0$ ) iff there is an answer set $A$ of $\\Pi ^6(PN,M_0,k,ntok)$ such that (REF ) and (REF ) hold.", "Proof of Proposition REF Let $PN=(P,T,E,C,W,R,I,Q,QW,Z,D)$ be a Petri Net, $M_0$ be its initial marking and let $\\Pi ^7(PN,M_0,k,ntok)$ be the ASP encoding of $PN$ and $M_0$ over a simulation length $k$ , with maximum $ntok$ tokens on any place node, as defined in section REF .", "Then $X=M_0,T_k,M_1,\\dots ,M_k,T_k,M_{k+1}$ is an execution sequence of $PN$ (w.r.t.", "$M_0$ ) iff there is an answer set $A$ of $\\Pi ^7(PN,M_0,k,ntok)$ such that: $\\lbrace fires(t,ts) : t \\in T_{ts}, 0 \\le ts \\le k\\rbrace = \\lbrace fires(t,ts) : fires(t,ts) \\in A \\rbrace $ $\\begin{split}\\lbrace holds(p,q,c,ts) &: p \\in P, c/q = M_{ts}(p), 0 \\le ts \\le k+1 \\rbrace \\\\&= \\lbrace holds(p,q,c,ts) : holds(p,q,c,ts) \\in A \\rbrace \\end{split}$ We prove this by showing that: Given an execution sequence $X$ , we create a set $A$ such that it satisfies (REF ) and (REF ) and show that $A$ is an answer set of $\\Pi ^7$ Given an answer set $A$ of $\\Pi ^7$ , we create an execution sequence $X$ such that (REF ) and (REF ) are satisfied.", "First we show (REF ): We create a set $A$ as a union of the following sets: $A_1=\\lbrace num(n) : 0 \\le n \\le ntok \\rbrace $ $A_2=\\lbrace time(ts) : 0 \\le ts \\le k\\rbrace $ $A_3=\\lbrace place(p) : p \\in P \\rbrace $ $A_4=\\lbrace trans(t) : t \\in T \\rbrace $ $A_5=\\lbrace color(c) : c \\in C \\rbrace $ $A_6=\\lbrace ptarc(p,t,n_c,c,ts) : (p,t) \\in E^-, c \\in C, n_c=m_{W(p,t)}(c), n_c > 0, 0 \\le ts \\le k \\rbrace $ , where $E^- \\subseteq E$ $A_7=\\lbrace tparc(t,p,n_c,c,ts,d) : (t,p) \\in E^+, c \\in C, n_c=m_{W(t,p)}(c), n_c > 0, d=D(t), 0 \\le ts \\le k \\rbrace $ , where $E^+ \\subseteq E$ $A_8=\\lbrace holds(p,q_c,c,0) : p \\in P, c \\in C, q_c=m_{M_{0}(p)}(c) \\rbrace $ $A_9=\\lbrace ptarc(p,t,n_c,c,ts) : p \\in R(t), c \\in C, n_c = m_{M_{ts}(p)}, n_c > 0, 0 \\le ts \\le k \\rbrace $ $A_{10}=\\lbrace iptarc(p,t,1,c,ts) : p \\in I(t), c \\in C, 0 \\le ts < k \\rbrace $ $A_{11}=\\lbrace tptarc(p,t,n_c,c,ts) : (p,t) \\in Q, c \\in C, n_c=m_{QW(p,t)}(c), n_c > 0, 0 \\le ts \\le k \\rbrace $ $A_{12}=\\lbrace notenabled(t,ts) : t \\in T, 0 \\le ts \\le k, \\exists c \\in C, (\\exists p \\in \\bullet t, m_{M_{ts}(p)}(c) < m_{W(p,t)}(c)) \\vee (\\exists p \\in I(t), m_{M_{ts}(p)}(c) > 0) \\vee (\\exists (p,t) \\in Q, m_{M_{ts}(p)}(c) < m_{QW(p,t)}(c)) \\rbrace $ per definition REF (enabled transition) $A_{13}=\\lbrace enabled(t,ts) : t \\in T, 0 \\le ts \\le k, \\forall c \\in C, (\\forall p \\in \\bullet t, m_{W(p,t)}(c) \\le m_{M_{ts}(p)}(c)) \\wedge (\\forall p \\in I(t), m_{M_{ts}(p)}(c) = 0) \\wedge (\\forall (p,t) \\in Q, m_{M_{ts}(p)}(c) \\ge m_{QW(p,t)}(c)) \\rbrace $ per definition REF (enabled transition) $A_{14}=\\lbrace fires(t,ts) : t \\in T_{ts}, 0 \\le ts \\le k \\rbrace $ per definition REF (enabled transitions), only an enabled transition may fire $A_{15}=\\lbrace add(p,q_c,t,c,ts+d-1) : t \\in T_{ts}, p \\in t \\bullet , c \\in C, q_c=m_{W(t,p)}(c), \\\\d = D(t), 0 \\le ts \\le k \\rbrace $ per definition REF (transition execution) $A_{16}=\\lbrace del(p,q_c,t,c,ts) : t \\in T_{ts}, p \\in \\bullet t, c \\in C, q_c=m_{W(p,t)}(c), 0 \\le ts \\le k \\rbrace \\cup \\lbrace del(p,q_c,t,c,ts) : t \\in T_{ts}, p \\in R(t), c \\in C, q_c=m_{M_{ts}(p)}(c), 0 \\le ts \\le k \\rbrace $ per definition REF (transition execution) $A_{17}=\\lbrace tot\\_incr(p,q_c,c,ts) : p \\in P, c \\in C, \\\\ q_c=\\sum _{t \\in T_{l}, p \\in t \\bullet , l \\le ts, l+D(t)=ts+1}{m_{W(t,p)}(c)}, 0 \\le ts \\le k \\rbrace $ per definition REF (firing set execution) $A_{18}=\\lbrace tot\\_decr(p,q_c,c,ts): p \\in P, c \\in C, q_c=\\sum _{t \\in T_{ts}, p \\in \\bullet t}{m_{W(p,t)}(c)}+ \\\\ \\sum _{t \\in T_{ts}, p \\in R(t) }{m_{M_{ts}(p)}(c)}, 0 \\le ts \\le k \\rbrace $ per definition REF (firing set execution) $A_{19}=\\lbrace consumesmore(p,ts) : p \\in P, c \\in C, q_c=m_{M_{ts}(p)}(c), \\\\ q1_c=\\sum _{t \\in T_{ts}, p \\in \\bullet t}{m_{W(p,t)}(c)} + \\sum _{t \\in T_{ts}, p \\in R(t)}{m_{M_{ts}(p)}(c)}, q1_c > q_c, 0 \\le ts \\le k \\rbrace $ per definition REF (conflicting transitions) $A_{20}=\\lbrace consumesmore : \\exists p \\in P, c \\in C, q_c=m_{M_{ts}(p)}(c), \\\\ q1_c=\\sum _{t \\in T_{ts}, p \\in \\bullet t}{m_{W(p,t)}(c)} + \\sum _{t \\in T_{ts}, p \\in R(t)}(m_{M_{ts}(p)}(c)), q1_c > q_c, 0 \\le ts \\le k \\rbrace $ per definition REF (conflicting transitions) $A_{21}=\\lbrace holds(p,q_c,c,ts+1) : p \\in P, c \\in C, q_c=m_{M_{ts+1}(p)}(c), 0 \\le ts < k\\rbrace $ , where $M_{ts+1}(p) = M_{ts}(p) - (\\sum _{\\begin{array}{c}t \\in T_{ts}, p \\in \\bullet t\\end{array}}{W(p,t)} + \\sum _{t \\in T_{ts}, p \\in R(t)}M_{ts}(p)) + \\\\ \\sum _{\\begin{array}{c}t \\in T_l, p \\in t \\bullet , l \\le ts, l+D(t)-1=ts\\end{array}}{W(t,p)}$ according to definition REF (firing set execution) $A_{22}=\\lbrace transpr(t,pr) : t \\in T, pr=Z(t) \\rbrace $ $A_{23}=\\lbrace notprenabled(t,ts) : t \\in T, enabled(t,ts) \\in A_{13}, (\\exists tt \\in T, enabled(tt,ts) \\in A_{13}, Z(tt) < Z(t)), 0 \\le ts \\le k \\rbrace \\\\ = \\lbrace notprenabled(t,ts) : t \\in T, (\\forall p \\in \\bullet t, W(p,t) \\le M_{ts}(p)), (\\forall p \\in I(t), M_{ts}(p) = 0), (\\forall (p,t) \\in Q, M_0(p) \\ge WQ(p,t)), \\exists tt \\in T, $ $(\\forall pp \\in \\bullet tt, W(pp,tt) \\le M_{ts}(pp)), $ $(\\forall pp \\in I(tt), M_{ts}(pp) = 0), (\\forall (pp,tt) \\in Q, M_{ts}(pp) \\ge QW(pp,tt)), Z(tt) < Z(t), 0 \\le ts \\le k \\rbrace $ $A_{24}=\\lbrace prenabled(t,ts) : t \\in T, enabled(t,ts) \\in A_{13}, (\\nexists tt \\in T: enabled(tt,ts) \\in A_{13}, Z(tt) < Z(t)), 0 \\le ts \\le k \\rbrace \\\\ = \\lbrace prenabled(t,ts) : t \\in T, (\\forall p \\in \\bullet t, W(p,t) \\le M_{ts}(p)), (\\forall p \\in I(t), M_{ts}(p) = 0), (\\forall (p,t) \\in Q, M_0(p) \\ge WQ(p,t)), \\nexists tt \\in T, $ $(\\forall pp \\in \\bullet tt, W(pp,tt) \\le M_{ts}(pp)), $ $(\\forall pp \\in I(tt), M_{ts}(pp) = 0), (\\forall (pp,tt) \\in Q, M_{ts}(pp) \\ge QW(pp,tt)), Z(tt) < Z(t), 0 \\le ts \\le k \\rbrace $ $A_{25}=\\lbrace could\\_not\\_have(t,ts): t \\in T, prenabled(t,ts) \\in A_{24}, fires(t,ts) \\notin A_{14}, (\\exists p \\in \\bullet t \\cup R(t): q > M_{ts}(p) - (\\sum _{t^{\\prime } \\in T_{ts}, p \\in \\bullet t^{\\prime }}{W(p,t^{\\prime })} + \\sum _{t^{\\prime } \\in T_{ts}, p \\in R(t^{\\prime })}{M_{ts}(p)}), q=W(p,t) \\text{~if~} (p,t) \\in E^- \\text{~or~} R(t) \\text{~otherwise}), 0 \\le ts \\le k\\rbrace \\\\ =\\lbrace could\\_not\\_have(t,ts) : t \\in T, (\\forall p \\in \\bullet t, W(p,t) \\le M_{ts}(p)), (\\forall p \\in I(t), M_{ts}(p) = 0), (\\forall (p,t) \\in Q, M_0(p) \\ge QW(p,t)), (\\nexists tt \\in T, $ $(\\forall pp \\in \\bullet tt, W(pp,tt) \\le M_{ts}(pp)), $ $(\\forall pp \\in I(tt), M_{ts}(pp) = 0), (\\forall (pp,tt) \\in Q, M_{ts}(pp) \\ge QW(pp,tt)), Z(tt) < Z(t)), t \\notin T_{ts}, (\\exists p \\in \\bullet t \\cup R(t): q > M_{ts}(p) - (\\sum _{t^{\\prime } \\in T_{ts}, p \\in \\bullet t^{\\prime }}{W(p,t^{\\prime })} + \\\\ \\sum _{t^{\\prime } \\in T_{ts}, p \\in R(t^{\\prime })}{M_{ts}(p)}), q=W(p,t) \\text{~if~} (p,t) \\in E^- \\text{~or~} R(t) \\text{~otherwise}), 0 \\le ts \\le k \\rbrace $ per the maximal firing set semantics We show that $A$ satisfies (REF ) and (REF ), and $A$ is an answer set of $\\Pi ^7$ .", "$A$ satisfies (REF ) and (REF ) by its construction above.", "We show $A$ is an answer set of $\\Pi ^7$ by splitting.", "We split $lit(\\Pi ^7)$ into a sequence of $9k+11$ sets: $U_0= head(f\\ref {f:c:place}) \\cup head(f\\ref {f:c:trans}) \\cup head(f\\ref {f:c:col}) \\cup head(f\\ref {f:c:time}) \\cup head(f\\ref {f:c:num}) \\cup head(i\\ref {i:c:init}) \\cup head(f\\ref {f:c:pr}) = \\lbrace place(p) : p \\in P\\rbrace \\cup \\lbrace trans(t) : t \\in T\\rbrace \\cup \\lbrace col(c) : c \\in C \\rbrace \\cup \\lbrace time(0), \\dots , time(k)\\rbrace \\cup \\lbrace num(0), \\dots , num(ntok)\\rbrace \\cup \\lbrace holds(p,q_c,c,0) : p \\in P, c \\in C, q_c=m_{M_0(p)}(c) \\rbrace \\cup \\lbrace transpr(t,pr) : t \\in T, pr=Z(t) \\rbrace $ $U_{9k+1}=U_{9k+0} \\cup head(f\\ref {f:c:ptarc})^{ts=k} \\cup head(f\\ref {f:c:tparc})^{ts=k} \\cup head(f\\ref {f:c:rptarc})^{ts=k} \\cup head(f\\ref {f:c:iptarc})^{ts=k} \\cup head(f\\ref {f:c:tptarc})^{ts=k} = U_{9k+0} \\cup $ $\\lbrace ptarc(p,t,n_c,c,k) : (p,t) \\in E^-, c \\in C, n_c=m_{W(p,t)}(c) \\rbrace \\\\ \\cup \\lbrace tparc(t,p,n_c,c,k,d) : (t,p) \\in E^+, c \\in C, n_c=m_{W(t,p)}(c), d=D(t) \\rbrace \\cup $ $\\lbrace ptarc(p,t,n_c,c,k) : p \\in R(t), c \\in C, n_c=m_{M_{k}(p)}(c), n > 0 \\rbrace \\cup \\lbrace iptarc(p,t,1,c,k) : \\\\ p \\in I(t), c \\in C \\rbrace \\cup $ $\\lbrace tptarc(p,t,n_c,c,k) : (p,t) \\in Q, c \\in C, n_c=m_{QW(p,t)}(c) \\rbrace $ $U_{9k+2}=U_{9k+1} \\cup head(e\\ref {e:c:ne:ptarc})^{ts=k} \\cup head(e\\ref {e:c:ne:iptarc})^{ts=k} \\cup head(e\\ref {e:c:ne:tptarc})^{ts=k} = U_{10k+1} \\cup \\\\ \\lbrace notenabled(t,k) : t \\in T \\rbrace $ $U_{9k+3}=U_{9k+2} \\cup head(e\\ref {e:c:enabled})^{ts=k} = U_{9k+2} \\cup \\lbrace enabled(t,k) : t \\in T \\rbrace $ $U_{9k+4}=U_{9k+3} \\cup head(a\\ref {a:c:prne})^{ts=k} = U_{9k+3} \\cup \\lbrace notprenabled(t,k) : t \\in T \\rbrace $ $U_{9k+5}=U_{9k+4} \\cup head(a\\ref {a:c:prenabled})^{ts=k} = U_{9k+4} \\cup \\lbrace prenabled(t,k) : t \\in T \\rbrace $ $U_{9k+6}=U_{9k+5} \\cup head(a\\ref {a:c:prfires})^{ts=k} = U_{9k+5} \\cup \\lbrace fires(t,k) : t \\in T \\rbrace $ $U_{9k+7}=U_{9k+6} \\cup head(r\\ref {r:c:dur:add})^{ts=k} \\cup head(r\\ref {r:c:del})^{ts=k} = U_{9k+6} \\cup \\lbrace add(p,q_c,t,c,k) : p \\in P, t \\in T, c \\in C, q_c=m_{W(t,p)}(c) \\rbrace \\cup \\lbrace del(p,q_c,t,c,k) : p \\in P, t \\in T, c \\in C, q_c=m_{W(p,t)}(c) \\rbrace \\cup \\lbrace del(p,q_c,t,c,k) : p \\in P, t \\in T, c \\in C, q_c=m_{M_{k}(p)}(c) \\rbrace $ $U_{9k+8}=U_{9k+7} \\cup head(r\\ref {r:c:totincr})^{ts=k} \\cup head(r\\ref {r:c:totdecr})^{ts=k} = U_{9k+7} \\cup \\lbrace tot\\_incr(p,q_c,c,k) : p \\in P, c \\in C, 0 \\le q_c \\le ntok \\rbrace \\cup \\lbrace tot\\_decr(p,q_c,c,k) : p \\in P, c \\in C, 0 \\le q_c \\le ntok \\rbrace $ $U_{9k+9}=U_{9k+8} \\cup head(a\\ref {a:c:overc:place})^{ts=k} \\cup head(r\\ref {r:c:nextstate})^{ts=k} \\cup head(a\\ref {a:c:prmaxfire:cnh})^{ts=k} = U_{9k+8} \\cup \\\\ \\lbrace consumesmore(p,k) : p \\in P\\rbrace \\cup \\lbrace holds(p,q,k+1) : p \\in P, 0 \\le q \\le ntok \\rbrace \\cup \\lbrace could\\_not\\_have(t,k) : t \\in T \\rbrace $ $U_{9k+10}=U_{9k+9} \\cup head(a\\ref {a:c:overc:gen}) = U_{9k+9} \\cup \\lbrace consumesmore \\rbrace $ where $head(r_i)^{ts=k}$ are head atoms of ground rule $r_i$ in which $ts=k$ .", "We write $A_i^{ts=k} = \\lbrace a(\\dots ,ts) : a(\\dots ,ts) \\in A_i, ts=k \\rbrace $ as short hand for all atoms in $A_i$ with $ts=k$ .", "$U_{\\alpha }, 0 \\le \\alpha \\le 9k+10$ form a splitting sequence, since each $U_i$ is a splitting set of $\\Pi ^7$ , and $\\langle U_{\\alpha }\\rangle _{\\alpha < \\mu }$ is a monotone continuous sequence, where $U_0 \\subseteq U_1 \\dots \\subseteq U_{9k+10}$ and $\\bigcup _{\\alpha < \\mu }{U_{\\alpha }} = lit(\\Pi ^7)$ .", "We compute the answer set of $\\Pi ^7$ using the splitting sets as follows: $bot_{U_0}(\\Pi ^7) = f\\ref {f:c:place} \\cup f\\ref {f:c:trans} \\cup f\\ref {f:c:col} \\cup f\\ref {f:c:time} \\cup f\\ref {f:c:num} \\cup i\\ref {i:c:init} \\cup f\\ref {f:c:pr}$ and $X_0 = A_1 \\cup \\dots \\cup A_5 \\cup A_8$ ($= U_0$ ) is its answer set – using forced atom proposition $eval_{U_0}(bot_{U_1}(\\Pi ^7) \\setminus bot_{U_0}(\\Pi ^7), X_0) = \\lbrace ptarc(p,t,q_c,c,0) \\text{:-}.", "\| c \\in C, q_c=m_{W(p,t)}(c), q_c > 0 \\rbrace \\cup $ $\\lbrace tparc(t,p,q_c,c,0,d) \\text{:-}.", "\| c \\in C, q_c=m_{W(t,p)}(c), q_c > 0, d=D(t) \\rbrace \\cup \\\\ \\lbrace ptarc(p,t,q_c,c,0) \\text{:-}.", "\| c \\in C, q_c=m_{M_0(p)}(c), q_c > 0 \\rbrace \\cup $ $\\lbrace iptarc(p,t,1,c,0) \\text{:-}.", "\| c \\in C \\rbrace \\cup $ $\\lbrace tptarc(p,t,q_c,c,0) \\text{:-}.", "\| c \\in C, q_c = m_{QW(p,t)}(c), q_c > 0 \\rbrace $ .", "Its answer set $X_1=A_6^{ts=0} \\cup A_7^{ts=0} \\cup A_9^{ts=0} \\cup A_{10}^{ts=0} \\cup A_{11}^{ts=0}$ – using forced atom proposition and construction of $A_6, A_7, A_9, A_{10}, A_{11}$ .", "$eval_{U_1}(bot_{U_2}(\\Pi ^7) \\setminus bot_{U_1}(\\Pi ^7), X_0 \\cup X_1) = \\lbrace notenabled(t,0) \\text{:-} .", "\| (\\lbrace trans(t), \\\\ ptarc(p,t,n_c,c,0), $ $holds(p,q_c,c,0) \\rbrace \\subseteq X_0 \\cup X_1, \\text{~where~} q_c < n_c) \\text{~or~} \\\\ (\\lbrace notenabled(t,0) \\text{:-} .", "\| $ $(\\lbrace trans(t), $ $iptarc(p,t,n2_c,c,0), $ $ holds(p,q_c,c,0) \\rbrace \\subseteq X_0 \\cup X_1, \\text{~where~} q_c \\ge n2_c \\rbrace ) \\text{~or~} $ $(\\lbrace trans(t), $ $tptarc(p,t,n3_c,c,0), $ $holds(p,q_c,c,0) \\rbrace \\subseteq X_0 \\cup X_1, \\text{~where~} q_c < n3_c) \\rbrace $ .", "Its answer set $X_2=A_{12}^{ts=0}$ – using forced atom proposition and construction of $A_{12}$ .", "where, $q_c=m_{M_0(p)}(c)$ , and $n_c=m_{W(p,t)}(c)$ for an arc $(p,t) \\in E^-$ – by construction of $i\\ref {i:c:init}$ and $f\\ref {f:c:ptarc}$ in $\\Pi ^7$ , and in an arc $(p,t) \\in E^-$ , $p \\in \\bullet t$ (by definition REF of preset) $n2_c=1$ – by construction of $iptarc$ predicates in $\\Pi ^7$ , meaning $q_c \\ge n2_c \\equiv q_c \\ge 1 \\equiv q_c > 0$ , $tptarc(p,t,n3_c,c,0)$ represents $n3_c=m_{QW(p,t)}(c)$ , where $(p,t) \\in Q$ thus, $notenabled(t,0) \\in X_1$ means $\\exists c \\in C, (\\exists p \\in \\bullet t : m_{M_0(p)}(c) < m_{W(p,t)}(c)) \\\\ \\vee (\\exists p \\in I(t) : m_{M_0(p)}(c) > 0) \\vee (\\exists (p,t) \\in Q : m_{M_0(p)}(c) < m_{QW(p,t)}(c))$ .", "$eval_{U_2}(bot_{U_3}(\\Pi ^7) \\setminus bot_{U_2}(\\Pi ^7), X_0 \\cup \\dots \\cup X_2) = \\lbrace enabled(t,0) \\text{:-}.", "\| trans(t) \\in X_0 \\cup \\dots \\cup X_2, notenabled(t,0) \\notin X_0 \\cup \\dots \\cup X_2 \\rbrace $ .", "Its answer set is $X_3 = A_{13}^{ts=0}$ – using forced atom proposition and construction of $A_{13}$ .", "since an $enabled(t,0) \\in X_3$ if $\\nexists ~notenabled(t,0) \\in X_0 \\cup \\dots \\cup X_2$ ; which is equivalent to $\\nexists t, \\forall c \\in C, (\\nexists p \\in \\bullet t, m_{M_0(p)}(c) < m_{W(p,t)}(c)), (\\nexists p \\in I(t), \\\\ m_{M_0(p)}(c) > 0), (\\nexists (p,t) \\in Q : m_{M_0(p)}(c) < m_{QW(p,t)}(c) ), \\forall c \\in C, (\\forall p \\in \\bullet t: m_{M_0(p)}(c) \\ge m_{W(p,t)}(c)), (\\forall p \\in I(t) : m_{M_0(p)}(c) = 0)$ .", "$eval_{U_3}(bot_{U_4}(\\Pi ^7) \\setminus bot_{U_3}(\\Pi ^7), X_0 \\cup \\dots \\cup X_3) = \\lbrace notprenabled(t,0) \\text{:-}.", "\| \\\\ \\lbrace enabled(t,0), transpr(t,p), enabled(tt,0), transpr(tt,pp) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_3, pp < p \\rbrace $ .", "Its answer set is $X_4 = A_{23}^{ts=k}$ – using forced atom proposition and construction of $A_{23}$ .", "$enabled(t,0)$ represents $\\exists t \\in T, \\forall c \\in C, (\\forall p \\in \\bullet t, m_{M_0(p)}(c) \\ge m_{W(p,t)}(c)), $ $(\\forall p \\in I(t), m_{M_0(p)}(c) = 0), (\\forall (p,t) \\in Q, m_{M_0(p)}(c) \\ge m_{QW(p,t)}(c))$ $enabled(tt,0)$ represents $\\exists tt \\in T, \\forall c \\in C, $ $(\\forall pp \\in \\bullet tt, m_{M_0(pp)}(c) \\ge \\\\ m_{W(pp,tt)}(c)) \\wedge $ $(\\forall pp \\in I(tt), m_{M_0(pp)}(c) = 0), (\\forall (pp,tt) \\in Q, m_{M_0(pp)}(c) \\ge m_{QW(pp,tt)}(c))$ $transpr(t,p)$ represents $p=Z(t)$ – by construction $transpr(tt,pp)$ represents $pp=Z(tt)$ – by construction thus, $notprenabled(t,0)$ represents $\\forall c \\in C, $ $(\\forall p \\in \\bullet t, m_{M_0(p)}(c) \\ge m_{W(p,t)}(c)), $ $(\\forall p \\in I(t), m_{M_0(p)}(c) = 0), (\\forall (p,t) \\in Q, m_{M_0(p)}(c) \\ge m_{QW(p,t)}(c)), \\exists tt \\in T, (\\forall pp \\in \\bullet tt, m_{M_0(pp)}(c) \\ge m_{W(pp,tt)}(c)), $ $(\\forall pp \\in I(tt), m_{M_0(pp)}(c) = 0), \\\\ (\\forall (pp,tt) \\in Q, m_{M_0(pp)}(c) \\ge m_{QW(pp,tt)}(c)), Z(tt) < Z(t)$ which is equivalent to $(\\forall p \\in \\bullet t: M_0(p) \\ge W(p,t)), $ $(\\forall p \\in I(t), M_0(p) = 0), (\\forall (p,t) \\in Q, M_0(p) \\ge QW(p,t)), \\exists tt \\in T, $ $(\\forall pp \\in \\bullet tt, M_0(pp) \\ge \\\\ W(pp,tt)), $ $(\\forall pp \\in I(tt), M_0(pp) = 0), (\\forall (pp,tt) \\in Q, M_0 \\ge QW(pp,tt)), \\\\ Z(tt) < Z(t)$ – assuming multiset domain $C$ for all operations $eval_{U_4}(bot_{U_5}(\\Pi ^7) \\setminus bot_{U_4}(\\Pi ^7), X_0 \\cup \\dots \\cup X_4) = \\lbrace prenabled(t,0) \\text{:-}.", "\| enabled(t,0) \\in X_0 \\cup \\dots \\cup X_4, notprenabled(t,0) \\notin X_0 \\cup \\dots \\cup X_4 \\rbrace $ .", "Its answer set is $X_5 = A_{24}^{ts=k}$ – using forced atom proposition and construction of $A_{24}$ $enabled(t,0)$ represents $\\forall c \\in C, (\\forall p \\in \\bullet t, m_{M_0(p)}(c) \\ge m_{W(p,t)}(c)), (\\forall p \\in I(t), m_{M_0(p)}(c) = 0), (\\forall (p,t) \\in Q, m_{M_0(p)}(c) \\ge m_{QW(p,t)}(c)) \\equiv $ $(\\forall p \\in \\bullet t, \\\\ M_0(p) \\ge W(p,t)), $ $(\\forall p \\in I(t), M_0(p) = 0), (\\forall (p,t) \\in Q, M_0(p) \\ge QW(p,t))$ – from REF above and assuming multiset domain $C$ for all operations $notprenabled(t,0)$ represents $(\\forall p \\in \\bullet t, M_0(p) \\ge W(p,t)), $ $(\\forall p \\in I(t), \\\\ M_0(p) = 0), (\\forall (p,t) \\in Q, M_0(p) \\ge QW(p,t)), \\exists tt \\in T, $ $(\\forall pp \\in \\bullet tt, M_0(pp) \\ge W(pp,tt)), $ $(\\forall pp \\in I(tt), M_0(pp) = 0), (\\forall (pp,tt) \\in Q, M_0(pp) \\ge QW(pp,tt)), \\\\ Z(tt) < Z(t)$ – from REF above and assuming multiset domain $C$ for all operations then, $prenabled(t,0)$ represents $(\\forall p \\in \\bullet t, M_0(p) \\ge W(p,t)), $ $(\\forall p \\in I(t), \\\\ M_0(p) = 0), (\\forall (p,t) \\in Q, M_0(p) \\ge QW(p,t)), \\nexists tt \\in T, $ $((\\forall pp \\in \\bullet tt, \\\\ M_0(pp) \\ge W(pp,tt)), $ $(\\forall pp \\in I(tt), M_0(pp) = 0), (\\forall (pp,tt) \\in Q, M_0(pp) \\ge QW(pp,tt)), Z(tt) < Z(t))$ – from (a), (b) and $enabled(t,0) \\in X_0 \\cup \\dots \\cup X_4$ $eval_{U_5}(bot_{U_6}(\\Pi ^7) \\setminus bot_{U_5}(\\Pi ^7), X_0 \\cup \\dots \\cup X_5) = \\lbrace \\lbrace fires(t,0)\\rbrace \\text{:-}.", "\| prenabled(t,0) \\\\ \\text{~holds in~} X_0 \\cup \\dots \\cup X_5 \\rbrace $ .", "It has multiple answer sets $X_{6.1}, \\dots , X_{6.n}$ , corresponding to elements of power set of $fires(t,0)$ atoms in $eval_{U_5}(...)$ – using supported rule proposition.", "Since we are showing that the union of answer sets of $\\Pi ^7$ determined using splitting is equal to $A$ , we only consider the set that matches the $fires(t,0)$ elements in $A$ and call it $X_6$ , ignoring the rest.", "Thus, $X_6 = A_{14}^{ts=0}$ , representing $T_0$ .", "in addition, for every $t$ such that $prenabled(t,0) \\in X_0 \\cup \\dots \\cup X_5, R(t) \\ne \\emptyset $ ; $fires(t,0) \\in X_6$ – per definition REF (firing set); requiring that a reset transition is fired when enabled thus, the firing set $T_0$ will not be eliminated by the constraint $f\\ref {f:c:pr:rptarc:elim}$ $eval_{U_6}(bot_{U_7}(\\Pi ^7) \\setminus bot_{U_6}(\\Pi ^7), X_0 \\cup \\dots \\cup X_6) = \\lbrace add(p,n_c,t,c,0) \\text{:-}.", "\| $ $\\lbrace fires(t,0-d+1), $ $tparc(t,p,n_c,c,0,d) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_6 \\rbrace \\cup \\lbrace del(p,n_c,t,c,0) \\text{:-}.", "\| $ $\\lbrace fires(t,0), $ $ptarc(p,t,n_c,c,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_6 \\rbrace $ .", "It's answer set is $X_7 = A_{15}^{ts=0} \\cup A_{16}^{ts=0}$ – using forced atom proposition and definitions of $A_{15}$ and $A_{16}$ .", "where, each $add$ atom is equivalent to $n_c=m_{W(t,p)}(c),c \\in C, p \\in t \\bullet $ , and each $del$ atom is equivalent to $n_c=m_{W(p,t)}(c), c \\in C, p \\in \\bullet t$ ; or $n_c=m_{M_0(p)}(c), c \\in C, p \\in R(t)$ , representing the effect of transitions in $T_0$ $eval_{U_7}(bot_{U_8}(\\Pi ^7) \\setminus bot_{U_7}(\\Pi ^7), X_0 \\cup \\dots \\cup X_7) = \\\\ \\lbrace tot\\_incr(p,qq_c,c,0) \\text{:-}.", "\| qq_c=\\sum _{add(p,q_c,t,c,0) \\in X_0 \\cup \\dots \\cup X_7}{q_c} \\rbrace \\cup \\\\ \\lbrace tot\\_decr(p,qq_c,c,0) \\text{:-}.", "\| qq_c=\\sum _{del(p,q_c,t,c,0) \\in X_0 \\cup \\dots \\cup X_7}{q_c} \\rbrace $ .", "It's answer set is $X_8 = A_{17}^{ts=0} \\cup A_{18}^{ts=0}$ – using forced atom proposition and definitions of $A_{17}$ and $A_{18}$ .", "where, each $tot\\_incr(p,qq_c,c,0)$ , $qq_c=\\sum _{add(p,q_c,t,c,0) \\in X_0 \\cup \\dots X_7}{q_c}$ $\\equiv qq_c=\\sum _{t \\in X_6, p \\in t \\bullet , 0+D(t)-1=0}{m_{W(p,t)}(c)}$ , $tot\\_decr(p,qq_c,c,0)$ , $qq_c=\\sum _{del(p,q_c,t,c,0) \\in X_0 \\cup \\dots X_7}{q_c}$ $\\equiv qq=\\sum _{t \\in X_6, p \\in \\bullet t}{m_{W(t,p)}(c)} + \\sum _{t \\in X_6, p \\in R(t)}{m_{M_0(p)}(c)}$ , represent the net effect of transitions in $T_0$ $eval_{U_8}(bot_{U_9}(\\Pi ^7) \\setminus bot_{U_8}(\\Pi ^7), X_0 \\cup \\dots \\cup X_8) = \\lbrace consumesmore(p,0) \\text{:-}.", "\| \\\\ \\lbrace holds(p,q_c,c,0), tot\\_decr(p,q1_c,c,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_8, q1_c > q_c \\rbrace \\cup $ $\\lbrace holds(p,q_c,c,1) \\text{:-}.", "\| \\lbrace holds(p,q1_c,c,0), tot\\_incr(p,q2_c,c,0), \\\\ tot\\_decr(p,q3_c,c,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_6, q_c=q1_c+q2_c-q3_c \\rbrace \\cup \\lbrace could\\_not\\_have(t,0) \\text{:-}.", "\| \\\\ \\lbrace prenabled(t,0), ptarc(s,t,q,c,0), holds(s,qq,c,0), tot\\_decr(s,qqq,c,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_8, fires(t,0) \\notin X_0 \\cup \\dots \\cup X_8, q > qq - qqq \\rbrace $ .", "It's answer set is $X_9 = A_{19}^{ts=0} \\cup A_{21}^{ts=0} \\cup A_{25}^{ts=0}$ – using forced atom proposition and definitions of $A_{19}, A_{21}, A_{25}$ .", "where $consumesmore(p,0)$ represents $\\exists p : q_c=m_{M_0(p)}(c), q1_c= \\\\ \\sum _{t \\in T_0, p \\in \\bullet t}{m_{W(p,t)}(c)}+\\sum _{t \\in T_0, p \\in R(t)}{m_{M_0(p)}(c)}, q1_c > q_c, c \\in C$ , indicating place $p$ will be over consumed if $T_0$ is fired, as defined in definition REF (conflicting transitions), $holds(p,q_c,c,1)$ if $q_c=m_{M_0(p)}(c)+\\sum _{t \\in T_0, p \\in t \\bullet , 0+D(t)-1=0}{m_{W(t,p)}(c)}- \\\\ (\\sum _{t \\in T_0, p \\in \\bullet t}{m_{W(p,t)}(c)}+ $ $\\sum _{t \\in T_0, p \\in R(t)}{m_{M_0(p)}(c)})$ represented by $q_c=m_{M_1(p)}(c)$ for some $c \\in C$ – by construction of $\\Pi ^7$ and $consumesmore(p,0)$ if $\\sum _{t \\in T_0, p \\in \\bullet t}{m_{W(p,t)}(c)}+ $ $\\sum _{t \\in T_0, p \\in R(t)}{m_{M_0(p)}(c)} > m_{M_0(p)}(c)$ for any $c \\in C$ $could\\_not\\_have(t,0)$ if $(\\forall p \\in \\bullet t, W(p,t) \\le M_0(p)), (\\forall p \\in I(t), M_0(p) = 0), (\\forall (p,t) \\in Q, \\\\ M_0(p) \\ge WQ(p,t)), \\nexists tt \\in T, (\\forall pp \\in \\bullet tt, W(pp,tt) \\le M_{ts}(pp)), (\\forall pp \\in I(tt), M_0(pp) = 0), (\\forall (pp,tt) \\in Q, M_0(pp) \\ge QW(pp,tt)), Z(tt) < Z(t)$ , and $q_c > m_{M_0(s)}(c) - (\\sum _{t^{\\prime } \\in T_0, s \\in \\bullet t^{\\prime }}{m_{W(s,t^{\\prime })}(c)}+ \\sum _{t^{\\prime } \\in T_0, s \\in R(t)}{m_{M_0(s)}(c)}), \\\\ q_c = m_{W(s,t)}(c) \\text{~if~} s \\in \\bullet t \\text{~or~} m_{M_0(s)}(c) \\text{~otherwise}$ for some $c \\in C$ , which becomes $q > M_0(s) - (\\sum _{t^{\\prime } \\in T_0, s \\in \\bullet t^{\\prime }}{W(s,t^{\\prime })}+ \\sum _{t^{\\prime } \\in T_0, s \\in R(t)}{M_0(s)}), q = W(s,t) \\text{~if~} s \\in \\bullet t \\text{~or~} M_0(s) \\text{~otherwise}$ for all $c \\in C$ (i), (ii) above combined match the definition of $A_{25}$ $X_9$ does not contain $could\\_not\\_have(t,0)$ , when $prenabled(t,0) \\in X_0 \\cup \\dots \\cup X_6$ and $fires(t,0) \\notin X_0 \\cup \\dots \\cup X_5$ due to construction of $A$ , encoding of $a\\ref {a:c:maxfire:cnh}$ and its body atoms.", "As a result, it is not eliminated by the constraint $a\\ref {a:c:prmaxfire:elim}$ $ \\vdots $ $eval_{U_{9k+0}}(bot_{U_{9k+1}}(\\Pi ^7) \\setminus bot_{U_{9k+0}}(\\Pi ^7), X_0 \\cup \\dots \\cup X_{9k+0}) = \\lbrace ptarc(p,t,q_c,c,k) \\text{:-}.", "\| c \\in C, q_c=m_{W(p,t)}(c), q_c > 0 \\rbrace \\cup \\lbrace tparc(t,p,q_c,c,k,d) \\text{:-}.", "\| c \\in C, q_c=m_{W(t,p)}(c), q_c > 0, d=D(t) \\rbrace \\cup $ $\\lbrace ptarc(p,t,q_c,c,k) \\text{:-}.", "\| c \\in C, q_c=m_{M_k(p)}(c), q_c > 0 \\rbrace \\cup \\\\ \\lbrace iptarc(p,t,1,c,k) \\text{:-}.", "\| c \\in C \\rbrace \\cup $ $\\lbrace tptarc(p,t,q_c,c,k) \\text{:-}.", "\| c \\in C, q_c = m_{QW(p,t)}(c), q_c > 0 \\rbrace $ .", "Its answer set $X_{9k+1}=A_6^{ts=k} \\cup A_7^{ts=k} \\cup A_9^{ts=k} \\cup A_{10}^{ts=k} \\cup A_{11}^{ts=k}$ – using forced atom proposition and construction of $A_6, A_7, A_9, A_{10}, A_{11}$ .", "$eval_{U_{9k+1}}(bot_{U_{10k+2}}(\\Pi ^7) \\setminus bot_{U_{9k+1}}(\\Pi ^7), X_0 \\cup \\dots \\cup X_{9k+1}) = \\lbrace notenabled(t,k) \\text{:-} .", "\| \\\\ (\\lbrace trans(t), ptarc(p,t,n_c,c,k), holds(p,q_c,c,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{9k+1}, \\text{~where~} q_c < n_c) \\text{~or~} (\\lbrace notenabled(t,k) \\text{:-} .", "\| (\\lbrace trans(t), iptarc(p,t,n2_c,c,k), holds(p,q_c,c,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{9k+1}, $ $\\text{~where~} q_c \\ge n2_c \\rbrace ) \\text{~or~} $ $(\\lbrace trans(t), tptarc(p,t,n3_c,c,k), \\\\ holds(p,q_c,c,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{9k+1}, $ $\\text{~where~} q_c < n3_c) \\rbrace $ .", "Its answer set $X_{9k+2}=A_{12}^{ts=k}$ – using forced atom proposition and construction of $A_{12}$ .", "where, $q_c=m_{M_k(p)}(c)$ , and $n_c=m_{W(p,t)}(c)$ for an arc $(p,t) \\in E^-$ – by construction of $i\\ref {i:c:init}$ and $f\\ref {f:c:ptarc}$ predicates in $\\Pi ^7$ , and in an arc $(p,t) \\in E^-$ , $p \\in \\bullet t$ (by definition REF of preset) $n2_c=1$ – by construction of $iptarc$ predicates in $\\Pi ^7$ , meaning $q_c \\ge n2_c \\equiv q_c \\ge 1 \\equiv q_c > 0$ , $tptarc(p,t,n3_c,c,k)$ represents $n3_c=m_{QW(p,t)}(c)$ , where $(p,t) \\in Q$ thus, $notenabled(t,k) \\in X_{9k+1}$ represents $\\exists c \\in C, (\\exists p \\in \\bullet t : m_{M_k(p)}(c) < m_{W(p,t)}(c)) \\vee (\\exists p \\in I(t) : m_{M_k(p)}(c) > 0) \\vee (\\exists (p,t) \\in Q : m_{M_k(p)}(c) < m_{QW(p,t)}(c))$ .", "$eval_{U_{9k+2}}(bot_{U_{9k+3}}(\\Pi ^7) \\setminus bot_{U_{9k+2}}(\\Pi ^7), X_0 \\cup \\dots \\cup X_{9k+2}) = \\lbrace enabled(t,k) \\text{:-}.", "\| trans(t) \\in X_0 \\cup \\dots \\cup X_{9k+2}, notenabled(t,k) \\notin X_0 \\cup \\dots \\cup X_{9k+2} \\rbrace $ .", "Its answer set is $X_{9k+3} = A_{13}^{ts=k}$ – using forced atom proposition and construction of $A_{13}$ .", "Since an $enabled(t,k) \\in X_{9k+3}$ if $\\nexists ~notenabled(t,k) \\in X_0 \\cup \\dots \\cup X_{9k+2}$ ; which is equivalent to $\\nexists t, \\forall c \\in C, (\\nexists p \\in \\bullet t, m_{M_k(p)}(c) < m_{W(p,t)}(c)), (\\nexists p \\in I(t), m_{M_k(p)}(c) > 0), (\\nexists (p,t) \\in Q : m_{M_k(p)}(c) < m_{QW(p,t)}(c) ), \\forall c \\in C, (\\forall p \\in \\bullet t: m_{M_k(p)}(c) \\ge m_{W(p,t)}(c)), (\\forall p \\in I(t) : m_{M_k(p)}(c) = 0)$ .", "$eval_{U_{9k+3}}(bot_{U_{9k+4}}(\\Pi ^7) \\setminus bot_{U_{9k+3}}(\\Pi ^7), X_0 \\cup \\dots \\cup X_{9k+3}) = \\lbrace notprenabled(t,k) \\text{:-}.", "\| \\\\ \\lbrace enabled(t,k), transpr(t,p), enabled(tt,k), transpr(tt,pp) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{9k+3}, \\\\ pp < p \\rbrace $ .", "Its answer set is $X_{9k+4} = A_{23}^{ts=k}$ – using forced atom proposition and construction of $A_{23}$ .", "$enabled(t,k)$ represents $\\exists t \\in T, \\forall c \\in C, (\\forall p \\in \\bullet t, m_{M_k(p)}(c) \\ge m_{W(p,t)}(c)), $ $(\\forall p \\in I(t), m_{M_k(p)}(c) = 0), (\\forall (p,t) \\in Q, m_{M_k(p)}(c) \\ge m_{QW(p,t)}(c))$ $enabled(tt,k)$ represents $\\exists tt \\in T, \\forall c \\in C, (\\forall pp \\in \\bullet tt, m_{M_k(pp)}(c) \\ge \\\\ m_{W(pp,tt)}(c)) \\wedge (\\forall pp \\in I(tt), m_{M_k(pp)}(c) = 0), (\\forall (pp,tt) \\in Q, m_{M_k(pp)}(c) \\ge m_{QW(pp,tt)}(c))$ $transpr(t,p)$ represents $p=Z(t)$ – by construction $transpr(tt,pp)$ represents $pp=Z(tt)$ – by construction thus, $notprenabled(t,k)$ represents $\\forall c \\in C, (\\forall p \\in \\bullet t, m_{M_k(p)}(c) \\ge \\\\ m_{W(p,t)}(c)), $ $(\\forall p \\in I(t), m_{M_k(p)}(c) = 0), (\\forall (p,t) \\in Q, m_{M_k(p)}(c) \\ge \\\\ m_{QW(p,t)}(c)), \\exists tt \\in T, $ $(\\forall pp \\in \\bullet tt, m_{M_k(pp)}(c) \\ge m_{W(pp,tt)}(c)), $ $(\\forall pp \\in I(tt), \\\\ m_{M_k(pp)}(c) = 0), (\\forall (pp,tt) \\in Q, m_{M_k(pp)}(c) \\ge m_{QW(pp,tt)}(c)), Z(tt) < Z(t)$ which is equivalent to $(\\forall p \\in \\bullet t: M_k(p) \\ge W(p,t)), $ $(\\forall p \\in I(t), \\\\ M_k(p) = 0), (\\forall (p,t) \\in Q, M_k(p) \\ge QW(p,t)), \\exists tt \\in T, $ $(\\forall pp \\in \\bullet tt, \\\\ M_k(pp) \\ge W(pp,tt)), $ $(\\forall pp \\in I(tt), M_k(pp) = 0), (\\forall (pp,tt) \\in Q, M_k(pp) \\ge QW(pp,tt)), Z(tt) < Z(t)$ – assuming multiset domain $C$ for all operations $eval_{U_{9k+4}}(bot_{U_{9k+5}}(\\Pi ^7) \\setminus bot_{U_{9k+4}}(\\Pi ^7), X_0 \\cup \\dots \\cup X_{9k+4}) = \\lbrace prenabled(t,k) \\text{:-}.", "\| \\\\ enabled(t,k) \\in X_0 \\cup \\dots \\cup X_{9k+4}, notprenabled(t,k) \\notin X_0 \\cup \\dots \\cup X_{9k+4} \\rbrace $ .", "Its answer set is $X_{9k+5} = A_{24}^{ts=k}$ – using forced atom proposition and construction of $A_{24}$ $enabled(t,k)$ represents $\\forall c \\in C, (\\forall p \\in \\bullet t, m_{M_k(p)}(c) \\ge m_{W(p,t)}(c)), $ $(\\forall p \\in I(t), m_{M_k(p)}(c) = 0), (\\forall (p,t) \\in Q, m_{M_k(p)}(c) \\ge m_{QW(p,t)}(c)) \\equiv (\\forall p \\in \\bullet t, M_k(p) \\ge W(p,t)), $ $(\\forall p \\in I(t), M_k(p) = 0), (\\forall (p,t) \\in Q, M_k(p) \\ge QW(p,t))$ – from REF above and assuming multiset domain $C$ for all operations $notprenabled(t,k)$ represents $(\\forall p \\in \\bullet t, M_k(p) \\ge W(p,t)), (\\forall p \\in I(t), \\\\ M_k(p) = 0), (\\forall (p,t) \\in Q, M_k(p) \\ge QW(p,t)), \\exists tt \\in T, $ $(\\forall pp \\in \\bullet tt, M_k(pp) \\ge W(pp,tt)), (\\forall pp \\in I(tt), M_k(pp) = 0), (\\forall (pp,tt) \\in Q, M_k(pp) \\ge QW(pp,tt)), \\\\ Z(tt) < Z(t)$ – from REF above and assuming multiset domain $C$ for all operations then, $prenabled(t,k)$ represents $(\\forall p \\in \\bullet t, M_k(p) \\ge W(p,t)), (\\forall p \\in I(t), \\\\ M_k(p) = 0), (\\forall (p,t) \\in Q, M_k(p) \\ge QW(p,t)), \\nexists tt \\in T, $ $((\\forall pp \\in \\bullet tt, M_k(pp) \\ge W(pp,tt)), $ $(\\forall pp \\in I(tt), M_k(pp) = 0), (\\forall (pp,tt) \\in Q, \\\\ M_k(pp) \\ge QW(pp,tt)), Z(tt) < Z(t))$ – from (a), (b) and $enabled(t,k) \\in X_0 \\cup \\dots \\cup X_{9k+4}$ $eval_{U_{9k+5}}(bot_{U_{9k+6}}(\\Pi ^7) \\setminus bot_{U_{9k+5}}(\\Pi ^7), X_0 \\cup \\dots \\cup X_{9k+5}) = \\lbrace \\lbrace fires(t,k)\\rbrace \\text{:-}.", "\| \\\\ prenabled(t,k) \\text{~holds in~} X_0 \\cup \\dots \\cup X_{9k+5} \\rbrace $ .", "It has multiple answer sets $X_{9k+6.1}, \\dots , X_{9k+6.n}$ , corresponding to elements of power set of $fires(t,k)$ atoms in $eval_{U_{9k+5}}(...)$ – using supported rule proposition.", "Since we are showing that the union of answer sets of $\\Pi ^7$ determined using splitting is equal to $A$ , we only consider the set that matches the $fires(t,k)$ elements in $A$ and call it $X_{9k+6}$ , ignoring the rest.", "Thus, $X_{9k+6} = A_{14}^{ts=k}$ , representing $T_k$ .", "in addition, for every $t$ such that $prenabled(t,k) \\in X_0 \\cup \\dots \\cup X_{9k+5}, R(t) \\ne \\emptyset $ ; $fires(t,k) \\in X_{9k+6}$ – per definition REF (firing set); requiring that a reset transition is fired when enabled thus, the firing set $T_k$ will not be eliminated by the constraint $f\\ref {f:c:pr:rptarc:elim}$ $eval_{U_{9k+6}}(bot_{U_{9k+7}}(\\Pi ^7) \\setminus bot_{U_{9k+6}}(\\Pi ^7), X_0 \\cup \\dots \\cup X_{9k+6}) = \\lbrace add(p,n_c,t,c,k) \\text{:-}.", "\| $ $\\lbrace fires(t,k-d+1), tparc(t,p,n_c,c,0,d) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{9k+6} \\rbrace \\cup \\\\ \\lbrace del(p,n_c,t,c,k) \\text{:-}.", "\| \\lbrace fires(t,k), ptarc(p,t,n_c,c,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{9k+6} \\rbrace $ .", "It's answer set is $X_{9k+7} = A_{15}^{ts=k} \\cup A_{16}^{ts=k}$ – using forced atom proposition and definitions of $A_{15}$ and $A_{16}$ .", "where, each $add$ atom is equivalent to $n_c=m_{W(t,p)}(c),c \\in C, p \\in t \\bullet $ , and each $del$ atom is equivalent to $n_c=m_{W(p,t)}(c), c \\in C, p \\in \\bullet t$ ; or $n_c=m_{M_k(p)}(c), c \\in C, p \\in R(t)$ , representing the effect of transitions in $T_k$ $eval_{U_{9k+7}}(bot_{U_{9k+8}}(\\Pi ^7) \\setminus bot_{U_{9k+7}}(\\Pi ^7), X_0 \\cup \\dots \\cup X_{9k+7}) = \\lbrace tot\\_incr(p,qq_c,c,k) \\text{:-}.", "\| $ $ qq_c=\\sum _{add(p,q_c,t,c,k) \\in X_0 \\cup \\dots \\cup X_{9k+7}}{q_c} \\rbrace \\cup \\lbrace tot\\_decr(p,qq_c,c,k) \\text{:-}.", "\| $ $ qq_c=\\sum _{del(p,q_c,t,c,k) \\in X_0 \\cup \\dots \\cup X_{9k+7}}{q_c} \\rbrace $ .", "It's answer set is $X_{9k+8} = A_{17}^{ts=k} \\cup A_{18}^{ts=k}$ – using forced atom proposition and definitions of $A_{17}$ and $A_{18}$ .", "where, each $tot\\_incr(p,qq_c,c,k)$ , $qq_c=\\sum _{add(p,q_c,t,c,k) \\in X_0 \\cup \\dots X_{9k+7}}{q_c}$ $\\equiv qq_c=\\sum _{t \\in X_{9k+6}, p \\in t \\bullet , 0 \\le l \\le k, l+D(t)-1=k }{m_{W(p,t)}(c)}$ , and each $tot\\_decr(p,qq_c,c,k)$ , $qq_c=\\sum _{del(p,q_c,t,c,k) \\in X_0 \\cup \\dots X_{9k+7}}{q_c}$ $\\equiv qq=\\sum _{t \\in X_{9k+6}, p \\in \\bullet t}{m_{W(t,p)}(c)} + \\sum _{t \\in X_{9k+6}, p \\in R(t)}{m_{M_k(p)}(c)}$ , represent the net effect of transitions in $T_k$ $eval_{U_{9k+8}}(bot_{U_{9k+9}}(\\Pi ^7) \\setminus bot_{U_{9k+8}}(\\Pi ^7), X_0 \\cup \\dots \\cup X_{9k+8}) = $ $\\lbrace consumesmore(p,k) \\text{:-}.", "\| \\\\ \\lbrace holds(p,q_c,c,k), tot\\_decr(p,q1_c,c,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{9k+8}, q1_c > q_c \\rbrace \\cup \\\\ \\lbrace holds(p,q_c,c,k+1) \\text{:-}., \| \\lbrace holds(p,q1_c,c,k), tot\\_incr(p,q2_c,c,k), \\\\ tot\\_decr(p,q3_c,c,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{9k+6}, q_c=q1_c+q2_c-q3_c \\rbrace \\cup \\\\ \\lbrace could\\_not\\_have(t,k) \\text{:-}.", "\| \\lbrace prenabled(t,k), \\\\ ptarc(s,t,q,c,k), holds(s,qq,c,k), tot\\_decr(s,qqq,c,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{9k+8}, \\\\ fires(t,k) \\notin X_0 \\cup \\dots \\cup X_{10k+8}, q > qq - qqq \\rbrace $ .", "It's answer set is $X_{9k+9} = A_{19}^{ts=k} \\cup A_{21}^{ts=k} \\cup A_{25}^{ts=k}$ – using forced atom proposition and definitions of $A_{19}, A_{21}, A_{25}$ .", "where, $consumesmore(p,k)$ represents $\\exists p : q_c=m_{M_k(p)}(c), \\\\ q1_c=\\sum _{t \\in T_k, p \\in \\bullet t}{m_{W(p,t)}(c)}+\\sum _{t \\in T_k, p \\in R(t)}{m_{M_k(p)}(c)}, q1_c > q_c, c \\in C$ , indicating place $p$ that will be over consumed if $T_k$ is fired, as defined in definition REF (conflicting transitions), $holds(p,q_c,c,k+1)$ if $q_c=m_{M_k(p)}(c)+\\sum _{t \\in T_l, p \\in t \\bullet , 0 \\le l \\le k, l+D(t)-1=k}{m_{W(t,p)}(c)}-(\\sum _{t \\in T_k, p \\in \\bullet t}{m_{W(p,t)}(c)}+ \\sum _{t \\in T_k, p \\in R(t)}{m_{M_k(p)}(c)})$ represented by $q_c=m_{M_1(p)}(c)$ for some $c \\in C$ – by construction of $\\Pi ^7$ , and $could\\_not\\_have(t,k)$ if $(\\forall p \\in \\bullet t, W(p,t) \\le M_k(p)), (\\forall p \\in I(t), M_k(p) = 0), (\\forall (p,t) \\in Q, \\\\ M_k(p) \\ge WQ(p,t)), \\nexists tt \\in T, (\\forall pp \\in \\bullet tt, W(pp,tt) \\le M_{ts}(pp)), (\\forall pp \\in I(tt), M_k(pp) = 0), (\\forall (pp,tt) \\in Q, M_k(pp) \\ge QW(pp,tt)), Z(tt) < Z(t)$ , and $q_c > m_{M_k(s)}(c) - (\\sum _{t^{\\prime } \\in T_k, s \\in \\bullet t^{\\prime }}{m_{W(s,t^{\\prime })}(c)}+ \\sum _{t^{\\prime } \\in T_k, s \\in R(t)}{m_{M_k(s)}(c)}), \\\\ q_c = m_{W(s,t)}(c) \\text{~if~} s \\in \\bullet t \\text{~or~} m_{M_k(s)}(c) \\text{~otherwise}$ for some $c \\in C$ , which becomes $q > M_k(s) - (\\sum _{t^{\\prime } \\in T_k, s \\in \\bullet t^{\\prime }}{W(s,t^{\\prime })}+ \\sum _{t^{\\prime } \\in T_k, s \\in R(t)}{M_k(s)}), q = W(s,t) \\text{~if~} s \\in \\bullet t \\text{~or~} M_k(s) \\text{~otherwise}$ for all $c \\in C$ (i), (ii) above combined match the definition of $A_{25}$ $X_{9k+9}$ does not contain $could\\_not\\_have(t,k)$ , when $prenabled(t,k) \\in X_0 \\cup \\dots \\cup X_{9k+6}$ and $fires(t,k) \\notin X_0 \\cup \\dots \\cup X_{9k+5}$ due to construction of $A$ , encoding of $a\\ref {a:c:maxfire:cnh}$ and its body atoms.", "As a result it is not eliminated by the constraint $a\\ref {a:c:prmaxfire:elim}$ $eval_{U_{9k+9}}(bot_{U_{9k+10}}(\\Pi ^7) \\setminus bot_{U_{9k+9}}(\\Pi ^7), X_0 \\cup \\dots \\cup X_{9k+9}) = \\lbrace consumesmore \\text{:-}.", "\| \\\\ \\lbrace consumesmore(p,0),\\dots ,$ $consumesmore(p,k) \\rbrace \\cap (X_0 \\cup \\dots \\cup X_{9k+9}) \\ne \\emptyset \\rbrace $ .", "It's answer set is $X_{9k+10} = A_{20}$ – using forced atom proposition and the definition of $A_{20}$ $X_{9k+10}$ will be empty since none of $consumesmore(p,0),\\dots , \\\\ consumesmore(p,k)$ hold in $X_0 \\cup \\dots \\cup X_{9k+10}$ due to the construction of $A$ , encoding of $a\\ref {a:overc:place}$ and its body atoms.", "As a result, it is not eliminated by the constraint $a\\ref {a:overc:elim}$ The set $X = X_0 \\cup \\dots \\cup X_{9k+10}$ is the answer set of $\\Pi ^7$ by the splitting sequence theorem REF .", "Each $X_i, 0 \\le i \\le 9k+10$ matches a distinct portion of $A$ , and $X = A$ , thus $A$ is an answer set of $\\Pi ^7$ .", "Next we show (REF ): Given $\\Pi ^7$ be the encoding of a Petri Net $PN(P,T,E,C,W,R,I,$ $Q,QW,Z,D)$ with initial marking $M_0$ , and $A$ be an answer set of $\\Pi ^7$ that satisfies (REF ) and (REF ), then we can construct $X=M_0,T_k,\\dots ,M_k,T_k,M_{k+1}$ from $A$ , such that it is an execution sequence of $PN$ .", "We construct the $X$ as follows: $M_i = (M_i(p_0), \\dots , M_i(p_n))$ , where $\\lbrace holds(p_0,m_{M_i(p_0)}(c),c,i), \\dots ,\\\\ holds(p_n,m_{M_i(p_n)}(c),c,i) \\rbrace \\subseteq A$ , for $c \\in C, 0 \\le i \\le k+1$ $T_i = \\lbrace t : fires(t,i) \\in A\\rbrace $ , for $0 \\le i \\le k$ and show that $X$ is indeed an execution sequence of $PN$ .", "We show this by induction over $k$ (i.e.", "given $M_k$ , $T_k$ is a valid firing set and its firing produces marking $M_{k+1}$ ).", "Base case: Let $k=0$ , and $M_0$ is a valid marking in $X$ for $PN$ , show [(1)] $T_0$ is a valid firing set for $M_0$ , and firing $T_0$ in $M_0$ produces marking $M_1$ .", "We show $T_0$ is a valid firing set for $M_0$ .", "Let $\\lbrace fires(t_0,0), \\dots , fires(t_x,0) \\rbrace $ be the set of all $fires(\\dots ,0)$ atoms in $A$ , Then for each $fires(t_i,0) \\in A$ $prenabled(t_i,0) \\in A$ – from rule $a\\ref {a:c:prfires}$ and supported rule proposition Then $enabled(t_i,0) \\in A$ – from rule $a\\ref {a:c:prenabled}$ and supported rule proposition And $notprenabled(t_i,0) \\notin A$ – from rule $a\\ref {a:c:prenabled}$ and supported rule proposition For $enabled(t_i,0) \\in A$ $notenabled(t_i,0) \\notin A$ – from rule $e\\ref {e:c:enabled}$ and supported rule proposition Then either of $body(e\\ref {e:c:ne:ptarc})$ , $body(e\\ref {e:c:ne:iptarc})$ , or $body(e\\ref {e:c:ne:tptarc})$ must not hold in $A$ for $t_i$ – from rules $body(e\\ref {e:c:ne:ptarc}), body(e\\ref {e:c:ne:iptarc}), body(e\\ref {e:c:ne:tptarc})$ and forced atom proposition Then $q_c \\lnot < {n_i}_c \\equiv q_c \\ge {n_i}_c$ in $e\\ref {e:c:ne:ptarc}$ for all $\\lbrace holds(p,q_c,c,0), \\\\ ptarc(p,t_i,{n_i}_c,c,0)\\rbrace \\subseteq A$ – from $e\\ref {e:c:ne:ptarc}$ , forced atom proposition, and given facts ($holds(p,q_c,c,0) \\in A, ptarc(p,t_i,{n_i}_c,0) \\in A$ ) And $q_c \\lnot \\ge {n_i}_c \\equiv q_c < {n_i}_c$ in $e\\ref {e:c:ne:iptarc}$ for all $\\lbrace holds(p,q_c,c,0), \\\\ iptarc(p,t_i,{n_i}_c,c,0) \\rbrace \\subseteq A, {n_i}_c=1$ ; $q_c > {n_i}_c \\equiv q_c = 0$ – from $e\\ref {e:c:ne:iptarc}$ , forced atom proposition, given facts ($holds(p,q_c,c,0) \\in A, \\\\ iptarc(p,t_i,1,c,0) \\in A$ ), and $q_c$ is a positive integer And $q_c \\lnot < {n_i}_c \\equiv q_c \\ge {n_i}_c$ in $e\\ref {e:c:ne:tptarc}$ for all $\\lbrace holds(p,q_c,c,0), \\\\ tptarc(p,t_i,{n_i}_c,c,0) \\rbrace \\subseteq A$ – from $e\\ref {e:c:ne:tptarc}$ , forced atom proposition, and given facts Then $\\forall c \\in C, (\\forall p \\in \\bullet t_i, m_{M_0(p)}(c) \\ge m_{W(p,t_i)}(c)) \\wedge (\\forall p \\in I(t_i), \\\\ m_{M_0(p)}(c) = 0) \\wedge (\\forall (p,t_i) \\in Q, m_{M_0(p)}(c) \\ge m_{QW(p,t_i)}(c))$ – from $i\\ref {i:c:init},f\\ref {f:c:ptarc},f\\ref {f:c:rptarc}$ construction, definition REF of preset $\\bullet t_i$ in $PN$ , definition REF of enabled transition in $PN$ , and that the construction of reset arcs by $f\\ref {f:c:rptarc}$ ensures $notenabled(t,0)$ is never true for a reset arc, where $holds(p,q_c,c,0) \\in A$ represents $q_c=m_{M_0(p)}(c)$ , $ptarc(p,t_i,{n_i}_c,0) \\in A$ represents ${n_i}_c=m_{W(p,t_i)}(c)$ , ${n_i}_c = m_{M_0(p)}(c)$ .", "Which is equivalent to $(\\forall p \\in \\bullet t_i, M_0(p) \\ge W(p,t_i)) \\wedge (\\forall p \\in I(t_i), M_0(p) = 0) \\wedge (\\forall (p,t_i) \\in Q, M_0(p) \\ge QW(p,t_i))$ – assuming multiset domain $C$ For $notprenabled(t_i,0) \\notin A$ Either $(\\nexists enabled(tt,0) \\in A : pp < p_i)$ or $(\\forall enabled(tt,0) \\in A : pp \\lnot < p_i)$ where $pp = Z(tt), p_i = Z(t_i)$ – from rule $a\\ref {a:c:prne}, f\\ref {f:c:pr}$ and forced atom proposition This matches the definition of an enabled priority transition Then $t_i$ is enabled and can fire in $PN$ , as a result it can belong to $T_0$ – from definition REF of enabled transition And $consumesmore \\notin A$ , since $A$ is an answer set of $\\Pi ^7$ – from rule $a\\ref {a:c:overc:elim}$ and supported rule proposition Then $\\nexists consumesmore(p,0) \\in A$ – from rule $a\\ref {a:c:overc:gen}$ and supported rule proposition Then $\\nexists \\lbrace holds(p,q_c,c,0), tot\\_decr(p,q1_c,c,0) \\rbrace \\subseteq A, q1_c>q_c$ in $body(a\\ref {a:c:overc:place})$ – from $a\\ref {a:c:overc:place}$ and forced atom proposition Then $\\nexists c \\in C \\nexists p \\in P, (\\sum _{t_i \\in \\lbrace t_0,\\dots ,t_x\\rbrace , p \\in \\bullet t_i}{m_{W(p,t_i)}(c)}+ \\\\ \\sum _{t_i \\in \\lbrace t_0,\\dots ,t_x\\rbrace , p \\in R(t_i)}{m_{M_0(p)}(c)}) > m_{M_0(p)}(c)$ – from the following $holds(p,q_c,c,0)$ represents $q_c=m_{M_0(p)}(c)$ – from rule $i\\ref {i:c:init}$ encoding, given $tot\\_decr(p,q1_c,c,0) \\in A$ if $\\lbrace del(p,{q1_0}_c,t_0,c,0), \\dots , \\\\ del(p,{q1_x}_c,t_x,c,0) \\rbrace \\subseteq A$ , where $q1_c = {q1_0}_c+\\dots +{q1_x}_c$ – from $r\\ref {r:c:totdecr}$ and forced atom proposition $del(p,{q1_i}_c,t_i,c,0) \\in A$ if $\\lbrace fires(t_i,0), ptarc(p,t_i,{q1_i}_c,c,0) \\rbrace \\subseteq A$ – from $r\\ref {r:c:del}$ and supported rule proposition $del(p,{q1_i}_c,t_i,c,0)$ represents removal of ${q1_i}_c = m_{W(p,t_i)}(c)$ tokens from $p \\in \\bullet t_i$ ; or it represents removal of ${q1_i}_c = m_{M_0(p)}(c)$ tokens from $p \\in R(t_i)$ – from rules $r\\ref {r:c:del},f\\ref {f:c:ptarc},f\\ref {f:c:rptarc}$ , supported rule proposition, and definition REF of preset in $PN$ Then the set of transitions in $T_0$ do not conflict – by the definition REF of conflicting transitions And for each $prenabled(t_j,0) \\in A$ and $fires(t_j,0) \\notin A$ , $could\\_not\\_have(t_j,0) \\in A$ , since $A$ is an answer set of $\\Pi ^7$ - from rule $a\\ref {a:c:prmaxfire:elim}$ and supported rule proposition Then $\\lbrace prenabled(t_j,0), holds(s,qq_c,c,0), ptarc(s,t_j,q_c,c,0), \\\\ tot\\_decr(s,qqq_c,c,0) \\rbrace \\subseteq A$ , such that $q_c > qq_c - qqq_c$ and $fires(t_j,0) \\notin A$ - from rule $a\\ref {a:c:prmaxfire:cnh}$ and supported rule proposition Then for an $s \\in \\bullet t_j \\cup R(t_j)$ , $q_c > m_{M_0(s)}(c) - (\\sum _{t_i \\in T_0, s \\in \\bullet t_i}{m_{W(s,t_i)}(c)} + \\sum _{t_i \\in T_0, s \\in R(t_i)}{m_{M_0(s)}(c))}$ , where $q_c=m_{W(s,t_j)}(c) \\text{~if~} s \\in \\bullet t_j, \\text{~or~} m_{M_0(s)}(c)$ $ \\text{~otherwise}$ .", "$ptarc(s,t_i,q_c,c,0)$ represents $q_c=m_{W(s,t_i)}(c)$ if $(s,t_i) \\in E^-$ or $q_c=m_{M_0(s)}(c)$ if $s \\in R(t_i)$ – from rule $f\\ref {f:c:ptarc},f\\ref {f:c:rptarc}$ construction $holds(s,qq_c,c,0)$ represents $qq_c=m_{M_0(s)}(c)$ – from $i\\ref {i:c:init}$ construction $tot\\_decr(s,qqq_c,c,0) \\in A$ if $\\lbrace del(s,{qqq_0}_c,t_0,c,0), \\dots , \\\\ del(s,{qqq_x}_c,t_x,c,0) \\rbrace \\subseteq A$ – from rule $r\\ref {r:c:totdecr}$ construction and supported rule proposition $del(s,{qqq_i}_c,t_i,c,0) \\in A$ if $\\lbrace fires(t_i,0), ptarc(s,t_i,{qqq_i}_c,c,0) \\rbrace \\subseteq A$ – from rule $r\\ref {r:c:del}$ and supported rule proposition $del(s,{qqq_i}_c,t_i,c,0)$ either represents ${qqq_i}_c = m_{W(s,t_i)}(c) : t_i \\in T_0, (s,t_i) \\in E^-$ , or ${qqq_i}_c = m_{M_0(t_i)}(c) : t_i \\in T_0, s \\in R(t_i)$ – from rule $f\\ref {f:c:ptarc},f\\ref {f:c:rptarc}$ construction $tot\\_decr(q,qqq_c,c,0)$ represents $\\sum _{t_i \\in T_0, s \\in \\bullet t_i}{m_{W(s,t_i)}(c)} + \\\\ \\sum _{t_i \\in T_0, s \\in R(t_i)}{m_{M_0(s)}(c)}$ – from (C,D,E) above Then firing $T_0 \\cup \\lbrace t_j \\rbrace $ would have required more tokens than are present at its source place $s \\in \\bullet t_j \\cup R(t_j)$ .", "Thus, $T_0$ is a maximal set of transitions that can simultaneously fire.", "And for each reset transition $t_r$ with $prenabled(t_r,0) \\in A$ , $fires(t_r,0) \\in A$ , since $A$ is an answer set of $\\Pi ^7$ - from rule $f\\ref {f:c:pr:rptarc:elim}$ and supported rule proposition Then, the firing set $T_0$ satisfies the reset-transition requirement of definition REF (firing set) Then $\\lbrace t_0, \\dots , t_x\\rbrace = T_0$ – using 1(a),1(b),1(d) above; and using 1(c) it is a maximal firing set Let $holds(p,q_c,c,1) \\in A$ Then $\\lbrace holds(p,q1_c,c,0), tot\\_incr(p,q2_c,c,0), tot\\_decr(p,q3_c,c,0) \\rbrace \\subseteq A : q_c=q1_c+q2_c-q3_c$ – from rule $r\\ref {r:c:nextstate}$ and supported rule proposition Then, $holds(p,q1_c,c,0) \\in A$ represents $q1_c=m_{M_0(p)}(c)$ – given ; and $\\lbrace add(p,{q2_0}_c,t_0,c,0), \\dots , $ $add(p,{q2_j}_c,t_j,c,0)\\rbrace \\subseteq A : {q2_0}_c + \\dots + {q2_j}_c = q2_c$ and $\\lbrace del(p,{q3_0}_c,t_0,c,0), \\dots , $ $del(p,{q3_l}_c,t_l,c,0)\\rbrace \\subseteq A : {q3_0}_c + \\dots + {q3_l}_c = q3_c$ – rules $r\\ref {r:c:totincr},r\\ref {r:c:totdecr}$ and supported rule proposition, respectively Then $\\lbrace fires(t_0,0), \\dots , fires(t_j,0) \\rbrace \\subseteq A$ and $\\lbrace fires(t_0,0), \\dots , fires(t_l,0) \\rbrace \\\\ \\subseteq A$ – rules $r\\ref {r:c:dur:add},r\\ref {r:c:del}$ and supported rule proposition, respectively Then $\\lbrace fires(t_0,0), \\dots , $ $fires(t_j,0) \\rbrace \\cup \\lbrace fires(t_0,0), \\dots , $ $fires(t_l,0) \\rbrace \\subseteq $ $A = \\lbrace fires(t_0,0), \\dots , $ $fires(t_x,0) \\rbrace \\subseteq A$ – set union of subsets Then for each $fires(t_x,0) \\in A$ we have $t_x \\in T_0$ – already shown in item REF above Then $q_c = m_{M_0(p)}(c) + \\sum _{t_x \\in T_0, p \\in t_x \\bullet , 0+D(t_x)-1=0}{m_{W(t_x,p)}(c)} - \\\\ (\\sum _{t_x \\in T_0 \\wedge p \\in \\bullet t_x}{m_{W(p,t_x)}(c)} + $ $\\sum _{t_x \\in T_0 \\wedge p \\in R(t_x)}{m_{M_0(p)}(c)})$ – from (REF ) above and the following Each $add(p,{q_j}_c,t_j,c,0) \\in A$ represents ${q_j}_c=m_{W(t_j,p)}(c)$ for $p \\in t_j \\bullet $ – rule $r\\ref {r:c:dur:add},f\\ref {f:c:dur:tparc}$ encoding, and definition REF of transition execution in $PN$ Each $del(p,t_y,{q_y}_c,c,0) \\in A$ represents either ${q_y}_c=m_{W(p,t_y)}(c)$ for $p \\in \\bullet t_y$ , or ${q_y}_c=m_{M_0(p)}(c)$ for $p \\in R(t_y)$ – from rule $r\\ref {r:c:del},f\\ref {f:c:ptarc}$ encoding and definition REF of transition execution in $PN$ ; or from rule $r\\ref {r:c:del},f\\ref {f:c:rptarc}$ encoding and definition of reset arc in $PN$ Each $tot\\_incr(p,q2_c,c,0) \\in A$ represents $q2_c=\\sum _{t_x \\in T_0 \\wedge p \\in t_x \\bullet , 0+D(t_x)-1=0}{m_{W(t_x,p)}(c)}$ – aggregate assignment atom semantics in rule $r\\ref {r:c:totincr}$ Each $tot\\_decr(p,q3_c,c,0) \\in A$ represents $q3_c=\\sum _{t_x \\in T_0 \\wedge p \\in \\bullet t_x}{m_{W(p,t_x)}(c)} \\\\ + \\sum _{t_x \\in T_0 \\wedge p \\in R(t_x)}{m_{M_0(p)}(c)}$ – aggregate assignment atom semantics in rule $r\\ref {r:c:totdecr}$ Then, $m_{M_1(p)}(c) = q_c$ – since $holds(p,q_c,c,1) \\in A$ encodes $q_c=m_{M_1(p)}(c)$ – from construction Inductive Step: Let $k > 0$ , and $M_k$ is a valid marking in $X$ for $PN$ , show [(1)] $T_k$ is a valid firing set for $M_k$ , and firing $T_k$ in $M_k$ produces marking $M_{k+1}$ .", "We show that $T_k$ is a valid firing set in $M_k$ .", "Let $\\lbrace fires(t_0,k), \\dots , fires(t_x,k) \\rbrace $ be the set of all $fires(\\dots ,k)$ atoms in $A$ , We show that $T_k$ is a valid firing set in $M_k$ .", "Then for each $fires(t_i,k) \\in A$ $prenabled(t_i,k) \\in A$ – from rule $a\\ref {a:c:prfires}$ and supported rule proposition Then $enabled(t_i,k) \\in A$ – from rule $a\\ref {a:c:prenabled}$ and supported rule proposition And $notprenabled(t_i,k) \\notin A$ – from rule $a\\ref {a:c:prenabled}$ and supported rule proposition For $enabled(t_i,k) \\in A$ $notenabled(t_i,k) \\notin A$ – from rule $e\\ref {e:c:enabled}$ and supported rule proposition Then either of $body(e\\ref {e:c:ne:ptarc})$ , $body(e\\ref {e:c:ne:iptarc})$ , or $body(e\\ref {e:c:ne:tptarc})$ must not hold in $A$ for $t_i$ – from rules $body(e\\ref {e:c:ne:ptarc}), body(e\\ref {e:c:ne:iptarc}), body(e\\ref {e:c:ne:tptarc})$ and forced atom proposition Then $q_c \\lnot < {n_i}_c \\equiv q_c \\ge {n_i}_c$ in $e\\ref {e:c:ne:ptarc}$ for all $\\lbrace holds(p,q_c,c,k), \\\\ ptarc(p,t_i,{n_i}_c,c,k)\\rbrace \\subseteq A$ – from $e\\ref {e:c:ne:ptarc}$ , forced atom proposition, and given facts ($holds(p,q_c,c,k) \\in A, ptarc(p,t_i,{n_i}_c,k) \\in A$ ) And $q_c \\lnot \\ge {n_i}_c \\equiv q_c < {n_i}_c$ in $e\\ref {e:c:ne:iptarc}$ for all $\\lbrace holds(p,q_c,c,k), \\\\ iptarc(p,t_i,{n_i}_c,c,k) \\rbrace \\subseteq A, {n_i}_c=1$ ; $q_c > {n_i}_c \\equiv q_c = 0$ – from $e\\ref {e:c:ne:iptarc}$ , forced atom proposition, given facts ($holds(p,q_c,c,k) \\in A, \\\\ iptarc(p,t_i,1,c,k) \\in A$ ), and $q_c$ is a positive integer And $q_c \\lnot < {n_i}_c \\equiv q_c \\ge {n_i}_c$ in $e\\ref {e:c:ne:tptarc}$ for all $\\lbrace holds(p,q_c,c,k), \\\\ tptarc(p,t_i,{n_i}_c,c,k) \\rbrace \\subseteq A$ – from $e\\ref {e:c:ne:tptarc}$ , forced atom proposition, and given facts Then $\\forall c \\in C, (\\forall p \\in \\bullet t_i, m_{M_k(p)}(c) \\ge m_{W(p,t_i)}(c)) \\wedge (\\forall p \\in I(t_i), \\\\ m_{M_k(p)}(c) = 0) \\wedge (\\forall (p,t_i) \\in Q, m_{M_k(p)}(c) \\ge m_{QW(p,t_i)}(c))$ – from the inductive assumption, $f\\ref {f:c:ptarc},f\\ref {f:c:rptarc}$ construction, definition REF of preset $\\bullet t_i$ in $PN$ , definition REF of enabled transition in $PN$ , and that the construction of reset arcs by $f\\ref {f:c:rptarc}$ ensures $notenabled(t,k)$ is never true for a reset arc, where $holds(p,q_c,c,k) \\in A$ represents $q_c=m_{M_k(p)}(c)$ , $ptarc(p,t_i,{n_i}_c,k) \\in A$ represents ${n_i}_c=m_{W(p,t_i)}(c)$ , ${n_i}_c = m_{M_k(p)}(c)$ .", "Which is equivalent to $(\\forall p \\in \\bullet t_i, M_k(p) \\ge W(p,t_i)) \\wedge (\\forall p \\in I(t_i), M_k(p) = 0) \\wedge (\\forall (p,t_i) \\in Q, M_k(p) \\ge QW(p,t_i))$ – assuming multiset domain $C$ For $notprenabled(t_i,k) \\notin A$ Either $(\\nexists enabled(tt,k) \\in A : pp < p_i)$ or $(\\forall enabled(tt,k) \\in A : pp \\lnot < p_i)$ where $pp = Z(tt), p_i = Z(t_i)$ – from rule $a\\ref {a:c:prne}, f\\ref {f:c:pr}$ and forced atom proposition This matches the definition of an enabled priority transition Then $t_i$ is enabled and can fire in $PN$ , as a result it can belong to $T_k$ – from definition REF of enabled transition And $consumesmore \\notin A$ , since $A$ is an answer set of $\\Pi ^7$ – from rule $a\\ref {a:c:overc:elim}$ and supported rule proposition Then $\\nexists consumesmore(p,k) \\in A$ – from rule $a\\ref {a:c:overc:gen}$ and supported rule proposition Then $\\nexists \\lbrace holds(p,q_c,c,k), tot\\_decr(p,q1_c,c,k) \\rbrace \\subseteq A, q1_c>q_c$ in $body(a\\ref {a:c:overc:place})$ – from $a\\ref {a:c:overc:place}$ and forced atom proposition Then $\\nexists c \\in C \\nexists p \\in P, (\\sum _{t_i \\in \\lbrace t_0,\\dots ,t_x\\rbrace , p \\in \\bullet t_i}{m_{W(p,t_i)}(c)}+ \\\\ \\sum _{t_i \\in \\lbrace t_0,\\dots ,t_x\\rbrace , p \\in R(t_i)}{m_{M_k(p)}(c)}) > m_{M_k(p)}(c)$ – from the following $holds(p,q_c,c,k)$ represents $q_c=m_{M_k(p)}(c)$ – from rule $PN$ encoding, given $tot\\_decr(p,q1_c,c,k) \\in A$ if $\\lbrace del(p,{q1_0}_c,t_0,c,k), \\dots , \\\\ del(p,{q1_x}_c,t_x,c,k) \\rbrace \\subseteq A$ , where $q1_c = {q1_0}_c+\\dots +{q1_x}_c$ – from $r\\ref {r:c:totdecr}$ and forced atom proposition $del(p,{q1_i}_c,t_i,c,k) \\in A$ if $\\lbrace fires(t_i,k), ptarc(p,t_i,{q1_i}_c,c,k) \\rbrace \\subseteq A$ – from $r\\ref {r:c:del}$ and supported rule proposition $del(p,{q1_i}_c,t_i,c,k)$ either represents removal of ${q1_i}_c = m_{W(p,t_i)}(c)$ tokens from $p \\in \\bullet t_i$ ; or it represents removal of ${q1_i}_c = m_{M_k(p)}(c)$ tokens from $p \\in R(t_i)$ – from rules $r\\ref {r:c:del},f\\ref {f:c:ptarc},f\\ref {f:c:rptarc}$ , supported rule proposition, and definition REF of transition execution in $PN$ Then the set of transitions in $T_k$ do not conflict – by the definition REF of conflicting transitions And for each $prenabled(t_j,k) \\in A$ and $fires(t_j,k) \\notin A$ , $could\\_not\\_have(t_j,k) \\in A$ , since $A$ is an answer set of $\\Pi ^7$ - from rule $a\\ref {a:c:prmaxfire:elim}$ and supported rule proposition Then $\\lbrace prenabled(t_j,k), holds(s,qq_c,c,k), ptarc(s,t_j,q_c,c,k), \\\\ tot\\_decr(s,qqq_c,c,k) \\rbrace \\subseteq A$ , such that $q_c > qq_c - qqq_c$ and $fires(t_j,k) \\notin A$ - from rule $a\\ref {a:c:prmaxfire:cnh}$ and supported rule proposition Then for an $s \\in \\bullet t_j \\cup R(t_j)$ , $q_c > m_{M_k(s)}(c) - (\\sum _{t_i \\in T_k, s \\in \\bullet t_i}{m_{W(s,t_i)}(c)} + \\sum _{t_i \\in T_k, s \\in R(t_i)}{m_{M_k(s)}(c))}$ , where $q_c=m_{W(s,t_j)}(c) \\text{~if~} s \\in \\bullet t_j, \\text{~or~} m_{M_k(s)}(c) $ $\\text{~otherwise}$ .", "$ptarc(s,t_i,q_c,c,k)$ represents $q_c=m_{W(s,t_i)}(c)$ if $(s,t_i) \\in E^-$ or $q_c=m_{M_k(s)}(c)$ if $s \\in R(t_i)$ – from rule $f\\ref {f:c:ptarc},f\\ref {f:c:rptarc}$ construction $holds(s,qq_c,c,k)$ represents $qq_c=m_{M_k(s)}(c)$ – from $i\\ref {i:c:init}$ construction $tot\\_decr(s,qqq_c,c,k) \\in A$ if $\\lbrace del(s,{qqq_0}_c,t_0,c,k), \\dots , \\\\ del(s,{qqq_x}_c,t_x,c,k) \\rbrace \\subseteq A$ – from rule $r\\ref {r:c:totdecr}$ construction and supported rule proposition $del(s,{qqq_i}_c,t_i,c,k) \\in A$ if $\\lbrace fires(t_i,k), ptarc(s,t_i,{qqq_i}_c,c,k) \\rbrace \\subseteq A$ – from rule $r\\ref {r:c:del}$ and supported rule proposition $del(s,{qqq_i}_c,t_i,c,k)$ represents ${qqq_i}_c = m_{W(s,t_i)}(c) : t_i \\in T_k, (s,t_i) \\in E^-$ , or ${qqq_i}_c = m_{M_k(t_i)}(c) : t_i \\in T_k, s \\in R(t_i)$ – from rule $f\\ref {f:c:ptarc},f\\ref {f:c:rptarc}$ construction $tot\\_decr(q,qqq_c,c,k)$ represents $\\sum _{t_i \\in T_k, s \\in \\bullet t_i}{m_{W(s,t_i)}(c)} + \\\\ \\sum _{t_i \\in T_k, s \\in R(t_i)}{m_{M_k(s)}(c)}$ – from (C,D,E) above Then firing $T_k \\cup \\lbrace t_j \\rbrace $ would have required more tokens than are present at its source place $s \\in \\bullet t_j \\cup R(t_j)$ .", "Thus, $T_k$ is a maximal set of transitions that can simultaneously fire.", "And for each reset transition $t_r$ with $enabled(t_r,k) \\in A$ , $fires(t_r,k) \\in A$ , since $A$ is an answer set of $\\Pi ^7$ - from rule $f\\ref {f:c:pr:rptarc:elim}$ and supported rule proposition Then the firing set $T_k$ satisfies the reset transition requirement of definition REF (firing set) Then $\\lbrace t_0, \\dots , t_x\\rbrace = T_k$ – using 1(a),1(b), 1(d) above; and using 1(c) it is a maximal firing set We show that $M_{k+1}$ is produced by firing $T_k$ in $M_k$ .", "Let $holds(p,q_c,c,k+1) \\in A$ Then $\\lbrace holds(p,q1_c,c,k), tot\\_incr(p,q2_c,c,k), tot\\_decr(p,q3_c,c,k) \\rbrace \\subseteq A : q_c=q1_c+q2_c-q3_c$ – from rule $r\\ref {r:c:nextstate}$ and supported rule proposition Then $holds(p,q1_c,c,k) \\in A$ represents $q1_c=m_{M_k(p)}(c)$ – inductive assumption; and $\\lbrace add(p,{q2_0}_c,t_0,c,k), \\dots , $ $add(p,{q2_j}_c,t_j,c,k)\\rbrace \\subseteq A : {q2_0}_c + \\dots + {q2_j}_c = q2_c$ and $\\lbrace del(p,{q3_0}_c,t_0,c,k), \\dots , $ $del(p,{q3_l}_c,t_l,c,k)\\rbrace \\subseteq A : {q3_0}_c + \\dots + {q3_l}_c = q3_c$ – rules $r\\ref {r:c:totincr},r\\ref {r:c:totdecr}$ and supported rule proposition, respectively Then $\\lbrace fires(t_0,k), \\dots , fires(t_j,k) \\rbrace \\subseteq A$ and $\\lbrace fires(t_0,k), \\dots , fires(t_l,k) \\rbrace \\\\ \\subseteq A$ – rules $r\\ref {r:c:dur:add},r\\ref {r:c:del}$ and supported rule proposition, respectively Then $\\lbrace fires(t_0,k), \\dots , fires(t_j,k) \\rbrace \\cup \\lbrace fires(t_0,k), \\dots , fires(t_l,k) \\rbrace \\subseteq A = \\lbrace fires(t_0,k), \\dots , fires(t_x,k) \\rbrace \\subseteq A$ – set union of subsets Then for each $fires(t_x,k) \\in A$ we have $t_x \\in T_k$ – already shown in item REF above Then $q_c = m_{M_k(p)}(c) + \\sum _{t_x \\in T_l, p \\in t_x \\bullet , 0 \\le l \\le k, l+D(t_x)-1=k}{m_{W(t_x,p)}(c)} - \\\\ (\\sum _{t_x \\in T_k \\wedge p \\in \\bullet t_x}{m_{W(p,t_x)}(c)} + \\sum _{t_x \\in T_k \\wedge p \\in R(t_x)}{m_{M_k(p)}(c)})$ – from (REF ) above and the following Each $add(p,{q_j}_c,t_j,c,k) \\in A$ represents ${q_j}_c=m_{W(t_j,p)}(c)$ for $p \\in t_j \\bullet $ – rule $r\\ref {r:c:dur:add},f\\ref {f:c:dur:tparc}$ encoding, and definition REF of postset in $PN$ Each $del(p,t_y,{q_y}_c,c,k) \\in A$ represents either ${q_y}_c=m_{W(p,t_y)}(c)$ for $p \\in \\bullet t_y$ , or ${q_y}_c=m_{M_k(p)}(c)$ for $p \\in R(t_y)$ – from rule $r\\ref {r:c:del},f\\ref {f:c:ptarc}$ encoding and definition REF of preset in $PN$ ; or from rule $r\\ref {r:c:del},f\\ref {f:c:rptarc}$ encoding and definition of reset arc in $PN$ Each $tot\\_incr(p,q2_c,c,k) \\in A$ represents $q2_c=\\sum _{t_x \\in T_l, p \\in t_x \\bullet , 0 \\le l \\le k, l+D(t_x)-1=k}{m_{W(t_x,p)}(c)}$ – aggregate assignment atom semantics in rule $r\\ref {r:c:totincr}$ Each $tot\\_decr(p,q3_c,c,k) \\in A$ represents $q3_c=\\sum _{t_x \\in T_k \\wedge p \\in \\bullet t_x}{m_{W(p,t_x)}(c)} \\\\ + \\sum _{t_x \\in T_k \\wedge p \\in R(t_x)}{m_{M_k(p)}(c)}$ – aggregate assignment atom semantics in rule $r\\ref {r:c:totdecr}$ Then, $m_{M_{k+1}(p)}(c) = q_c$ – since $holds(p,q_c,c,k+1) \\in A$ encodes $q_c=m_{M_{k+1}(p)}(c)$ – from construction As a result, for any $n > k$ , $T_n$ will be a valid firing set for $M_n$ and $M_{n+1}$ will be its target marking.", "Conclusion: Since both (REF ) and (REF ) hold, $X=M_0,T_k,M_1,\\dots ,M_k,T_{k+1}$ is an execution sequence of $PN(P,T,E,C,W,R,I,Q,QW,Z,D)$ (w.r.t $M_0$ ) iff there is an answer set $A$ of $\\Pi ^7(PN,M_0,k,ntok)$ such that (REF ) and (REF ) hold.", "Complete Set of Queries Used for Drug-Drug InteractionDrug-Drug Interaction Queries Drug Activates Gene 1 //S{/NP{//?[Value='activation'](kw1)=>//?[Value='of'](kw2)=>//?[Tag='GENE'](kw0)=>//?[Value='by'](kw4)=>//?", "[Tag='DRUG'](kw3)}} ::: distinct sent.cid, kw3.value, kw1.value, kw0.value, sent.value 2 //NP{/NP{/?[Tag='DRUG'](kw3)=>/?[Value='activation'](kw1)}=>/PP{/?[Value='of'](kw2)=>//?", "[Tag='GENE'](kw0)}} ::: distinct sent.cid, kw3.value, kw1.value, kw0.value, sent.value 3 //NP{/?[Value='activation'](kw1)=>/PP{/?[Value='of'](kw2)=>//?[Tag='GENE'](kw0)=>//?[Value='by'](kw4)=>//?", "[Tag='DRUG'](kw3)}} ::: distinct sent.cid, kw3.value, kw1.value, kw0.value, sent.value 4 //S{/NP{//?[Tag='GENE'](kw2)}=>/VP{//?[Value='activation'](kw1)=>//?", "[Tag='DRUG'](kw0)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 5 //S{/NP{//?[Tag='DRUG'](kw0)}=>/VP{//?[Value='activation'](kw1)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 6 //NP{/NP{/?[Tag='DRUG'](kw0)=>/?[Value='activation'](kw1)}=>/PP{//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 7 //S{/NP{//?[Value='activation'](kw1)=>//?[Tag='GENE'](kw2)}=>/VP{//?", "[Tag='DRUG'](kw0)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 8 //S{/NP{//?[Value='activation'](kw1)=>//?[Value='of'](kw2)=>//?[Tag='GENE'](kw0)=>//?[Value='by'](kw4)=>//?", "[Tag='DRUG'](kw3)}} ::: distinct sent.cid, kw3.value, kw1.value, kw0.value, sent.value 9 //NP{/NP{/?[Tag='DRUG'](kw3)=>/?[Value='activation'](kw1)}=>/PP{/?[Value='of'](kw2)=>//?", "[Tag='GENE'](kw0)}} ::: distinct sent.cid, kw3.value, kw1.value, kw0.value, sent.value 10 //NP{/?[Value='activation'](kw1)=>/PP{/?[Value='of'](kw2)=>//?[Tag='GENE'](kw0)=>//?[Value='by'](kw4)=>//?", "[Tag='DRUG'](kw3)}} ::: distinct sent.cid, kw3.value, kw1.value, kw0.value, sent.value Gene Induces Gene 1 //S{/NP{//?[Tag='GENE'](kw0)}=>/VP{//?[Value='induced'](kw1)=>//?[Value='by'](kw3)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 2 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{//?[Value='induced'](kw1)=>//?[Value='by'](kw3)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 3 //S{/NP{//?[Tag='GENE'](kw0)}=>/VP{//?[Value IN {'increase','increased'}](kw1)=>//?[Tag='GENE'](kw2)=>//?", "[Value='activity'](kw3)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 4 //S{/NP{//?[Tag='GENE'](kw0)=>//?[Value='activity'](kw3)}=>/VP{//?[Value='increase'](kw1)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 5 //S{/NP{//?[Tag='GENE'](kw0)}=>/VP{//?[Value IN {'stimulates','stimulate', 'stimulated'}](kw1)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 6 //S{/NP{//?[Tag='GENE'](kw0)}=>/VP{/?[Value IN {'stimulates','stimulate'}](kw1)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 7 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/?[Value='stimulates'](kw1)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 8 //S{/NP{//?[Tag='GENE'](kw0)}=>/VP{/?[Value='activated'](kw1)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 9 //NP{/NP{/?[Value='induction'](kw1)}=>/PP{/?[Value='of'](kw3)=>/NP{//?[Tag='GENE'](kw0)=>//?[Value='by'](kw4)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 10 //NP{/NP{/?[Value='stimulation'](kw1)}=>/PP{/?[Value='of'](kw3)=>/NP{//?[Tag='GENE'](kw0)=>//?[Value='by'](kw4)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 11 //NP{/NP{/?[Value='activation'](kw1)}=>/PP{/?[Value IN {'of'}](kw3)=>/NP{//?[Tag='GENE'](kw0)=>//?[Value='by'](kw4)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 12 //S{/NP{//?[Tag='GENE'](kw0)}=>/VP{//?[Value='inducible'](kw1)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 13 //S{/NP{//?[Tag='GENE'](kw0)}=>/VP{//?[Value='induced'](kw1)=>//?[Value='by'](kw3)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 14 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{//?[Value='induced'](kw1)=>//?[Value='by'](kw3)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 15 //S{/NP{//?[Tag='GENE'](kw0)}=>/VP{//?[Value IN {'increase','increased'}](kw1)=>//?[Tag='GENE'](kw2)=>//?", "[Value='activity'](kw3)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 16 //S{/NP{//?[Tag='GENE'](kw0)=>//?[Value='activity'](kw3)}=>/VP{//?[Value='increase'](kw1)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 17 //S{/NP{//?[Tag='GENE'](kw0)}=>/VP{//?[Value='stimulated'](kw1)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 18 //S{/NP{//?[Tag='GENE'](kw0)}=>/VP{/?[Value IN {'stimulates','stimulate'}](kw1)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 19 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/?[Value='stimulates'](kw1)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 20 //S{/NP{//?[Tag='GENE'](kw0)}=>/VP{/?[Value='activated'](kw1)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 21 //VP{/?[Value='activated'](kw1)=>/PP{//?[Tag='GENE'](kw0)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 22 //NP{/NP{/?[Value='induction'](kw1)}=>/PP{/?[Value='of'](kw3)=>/NP{//?[Tag='GENE'](kw0)=>//?[Value='by'](kw4)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 23 //NP{/NP{/?[Value='stimulation'](kw1)}=>/PP{/?[Value='of'](kw3)=>/NP{//?[Tag='GENE'](kw0)=>//?[Value='by'](kw4)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 24 //NP{/NP{/?[Value='activation'](kw1)}=>/PP{/?[Value IN {'of'}](kw3)=>/NP{//?[Tag='GENE'](kw0)=>//?[Value='by'](kw4)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 25 //S{/NP{//?[Tag='GENE'](kw0)}=>/VP{//?[Value='inducible'](kw1)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value Gene Inhibits Gene 1 //S{/NP{//?[Tag='GENE'](kw0)}=>/VP{//?[Value='induced'](kw1)=>//?[Value='by'](kw3)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 2 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{//?[Value='induced'](kw1)=>//?[Value='by'](kw3)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 3 //S{/NP{//?[Tag='GENE'](kw0)}=>/VP{//?[Value IN {'increase','increased'}](kw1)=>//?[Tag='GENE'](kw2)=>//?", "[Value='activity'](kw3)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 4 //S{/NP{//?[Tag='GENE'](kw0)=>//?[Value='activity'](kw3)}=>/VP{//?[Value='increase'](kw1)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 5 //S{/NP{//?[Tag='GENE'](kw0)}=>/VP{//?[Value IN {'stimulates','stimulate', 'stimulated'}](kw1)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 6 //S{/NP{//?[Tag='GENE'](kw0)}=>/VP{/?[Value IN {'stimulates','stimulate'}](kw1)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 7 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/?[Value='stimulates'](kw1)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 8 //S{/NP{//?[Tag='GENE'](kw0)}=>/VP{/?[Value='activated'](kw1)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 9 //NP{/NP{/?[Value='induction'](kw1)}=>/PP{/?[Value='of'](kw3)=>/NP{//?[Tag='GENE'](kw0)=>//?[Value='by'](kw4)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 10 //NP{/NP{/?[Value='stimulation'](kw1)}=>/PP{/?[Value='of'](kw3)=>/NP{//?[Tag='GENE'](kw0)=>//?[Value='by'](kw4)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 11 //NP{/NP{/?[Value='activation'](kw1)}=>/PP{/?[Value IN {'of'}](kw3)=>/NP{//?[Tag='GENE'](kw0)=>//?[Value='by'](kw4)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 12 //S{/NP{//?[Tag='GENE'](kw0)}=>/VP{//?[Value='inducible'](kw1)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value Drug Changes Gene Expression/Activity 1 //S{/?[Tag='DRUG'](kw2)=>/?[Value IN {'increased','increase','increases'}](kw1)=>/?[Value IN {'levels','level'}](kw3)=>/?", "[Tag='GENE'](kw0)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 2 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/?[Value IN {'increased','increases'}](kw1)=>/NP{//?[Tag='GENE'](kw0)=>//?", "[Value IN {'expression','level','activity','activities','levels'}](kw3)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 3 //S{/?[Tag='DRUG'](kw2)=>/VP{/NP{//?[Value IN {'increases','increase'}](kw1)=>//?[Tag='GENE'](kw0)=>//?", "[Value IN {'levels','activity'}](kw3)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 4 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value IN {'increased','increases'}](kw1)=>/NP{//?[Tag='GENE'](kw0)=>//?", "[Value IN {'activity','activities','levels'}](kw3)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 5 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/?[Value='increased'](kw1)=>/NP{//?[Value IN {'activity','levels','expression'}](kw3)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 6 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{//?[Value IN {'increased','increases','increase'}](kw1)=>//?[Tag='GENE'](kw0)=>//?", "[Value IN {'activity','levels'}](kw3)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 7 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{/?[Value='increased'](kw1)=>//?[Tag='GENE'](kw0)=>//?", "[Value IN {'expression','activity'}](kw3)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 8 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value IN {'increased','increases'}](kw1)=>/NP{//?[Value IN {'expression','activity','activities','level','levels'}](kw3)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 9 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value IN {'decreased','decreases'}](kw1)=>/NP{//?[Tag='GENE'](kw0)=>//?", "[Value IN {'activity','expression'}](kw3)}}} ::: distinct sent.cid, kw2.value, kw1,value, kw0.value, sent.value 10 //S{/?[Tag='DRUG'](kw2)=>/?[Value IN {'increased','increase','increases'}](kw1)=>/?[Value IN {'levels','level'}](kw3)=>/?", "[Tag='GENE'](kw0)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 11 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/?[Value IN {'increased','increases'}](kw1)=>/NP{//?[Tag='GENE'](kw0)=>//?", "[Value IN {'expression','level','activity','activities','levels'}](kw3)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 12 //S{/?[Tag='DRUG'](kw2)=>/VP{/NP{//?[Value IN {'increases','increase'}](kw1)=>//?[Tag='GENE'](kw0)=>//?", "[Value IN {'levels','activity'}](kw3)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 13 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value IN {'increased','increases'}](kw1)=>/NP{//?[Tag='GENE'](kw0)=>//?", "[Value IN {'activity','activities','levels'}](kw3)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 14 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/?[Value='increased'](kw1)=>/NP{//?[Value IN {'activity','levels','expression'}](kw3)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 15 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{//?[Value IN {'increased','increases','increase'}](kw1)=>//?[Tag='GENE'](kw0)=>//?", "[Value IN {'activity','levels'}](kw3)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 16 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{/?[Value='increased'](kw1)=>//?[Tag='GENE'](kw0)=>//?", "[Value IN {'expression','activity'}](kw3)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 17 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value IN {'increased','increases'}](kw1)=>/NP{//?[Value IN {'expression','activity','activities','level','levels'}](kw3)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 18 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value IN {'decreased','decreases'}](kw1)=>/NP{//?[Tag='GENE'](kw0)=>//?", "[Value IN {'activity','expression'}](kw3)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value Drug Induces/Stimulates Gene 1 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value IN {'stimulated','induced'}](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 2 //VP{/?[Value IN {'stimulated','induced'}](kw1)=>/PP{/NP{//?[Tag='DRUG'](kw2)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 3 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/?[Value IN {'stimulated','induced'}](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 4 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/?[Value IN {'stimulated','induced'}](kw1)=>/PP{//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 5 //S{/?[Tag='DRUG'](kw2)=>/VP{/NP{//?[Tag='GENE'](kw0)=>//?", "[Value IN {'stimulated','induced'}](kw1)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 6 //S{/NP{/PP{//?[Tag='DRUG'](kw2)}}=>/VP{/?[Value IN {'stimulated','induced'}](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 7 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/VP{/?[Value IN {'stimulated','induced'}](kw1)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 8 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/VP{//?[Value='induced'](kw1)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 9 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value IN {'stimulated','induced'}](kw1)=>/NP{/?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 10 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{//?[Value IN {'stimulate','induce'}](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 11 //NP{/NP{/?[Value IN {'induction','stimulation'}](kw1)}=>/PP{/NP{//?[Tag='GENE'](kw0)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 12 //S{/NP{/?[Value IN {'stimulation','induction'}](kw1)=>/PP{//?[Tag='GENE'](kw0)}}=>/VP{/VP{//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 13 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/NP{//?[Value IN {'stimulation','induction'}](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 14 //NP{/NP{/NP{/?[Value='induction'](kw1)}=>/PP{//?[Tag='GENE'](kw0)}}=>/PP{/NP{/?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 15 //NP{/NP{/?[Tag='GENE'](kw0)=>/?[Value IN {'stimulation','induction'}](kw1)}=>/PP{/NP{/?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 16 //NP{/?[Value IN {'stimulation','induction'}](kw1)=>/PP{/NP{//?[Tag='GENE'](kw0)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 17 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/?[Value IN {'stimulates','induces'}](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 18 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value IN {'stimulates','induces'}](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 19 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value IN {'stimulated','induced'}](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 20 //VP{/?[Value IN {'stimulated','induced'}](kw1)=>/PP{/NP{//?[Tag='DRUG'](kw2)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 21 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/?[Value IN {'stimulated','induced'}](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 22 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/?[Value IN {'stimulated','induced'}](kw1)=>/PP{//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 23 //S{/?[Tag='DRUG'](kw2)=>/VP{/NP{//?[Tag='GENE'](kw0)=>//?", "[Value IN {'stimulated','induced'}](kw1)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 24 //S{/NP{/PP{//?[Tag='DRUG'](kw2)}}=>/VP{/?[Value IN {'stimulated','induced'}](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 25 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/VP{/?[Value IN {'stimulated','induced'}](kw1)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 26 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/VP{//?[Value='induced'](kw1)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 27 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value IN {'stimulated','induced'}](kw1)=>/NP{/?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 28 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{//?[Value IN {'stimulate','induce'}](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 29 //NP{/NP{/?[Value IN {'induction','stimulation'}](kw1)}=>/PP{/NP{//?[Tag='GENE'](kw0)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 30 //S{/NP{/?[Value IN {'stimulation','induction'}](kw1)=>/PP{//?[Tag='GENE'](kw0)}}=>/VP{/VP{//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 31 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/NP{//?[Value IN {'stimulation','induction'}](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 32 //NP{/NP{/NP{/?[Value='induction'](kw1)}=>/PP{//?[Tag='GENE'](kw0)}}=>/PP{/NP{/?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 33 //NP{/NP{/?[Tag='GENE'](kw0)=>/?[Value IN {'stimulation','induction'}](kw1)}=>/PP{/NP{/?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 34 //NP{/?[Value IN {'stimulation','induction'}](kw1)=>/PP{/NP{//?[Tag='GENE'](kw0)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 35 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/?[Value IN {'stimulates','induces'}](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 36 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value IN {'stimulates','induces'}](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value Drug Inhibits Gene 1 //NP{/NP{/NP{/?[Value='inhibitory'](kw1)}=>/PP{//?[Tag='DRUG'](kw2)}}=>/PP{/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 2 //S{/?[Tag='DRUG'](kw2)=>/VP{/NP{/?[Tag='GENE'](kw0)=>/?", "[Value='inhibitor'](kw1)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 3 //NP{/?[Tag='DRUG'](kw2)=>/NP{/?[Tag='GENE'](kw0)=>/?", "[Value='inhibitor'](kw1)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 4 //NP{/NP{/?[Tag='GENE'](kw0)=>/?[Value='inhibitor'](kw1)}=>/?", "[Tag='DRUG'](kw2)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 5 //NP{/NP{/NP{/?[Tag='GENE'](kw0)=>/?[Tag='DRUG'](kw2)}}=>/?", "[Value='inhibitor'](kw1)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 6 //NP{/?[Tag='GENE'](kw0)=>/?[Value='inhibitor'](kw1)=>/?", "[Tag='DRUG'](kw2)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 7 //NP{/NP{/?[Value='inhibitor'](kw1)}=>/PP{/NP{//?[Tag='GENE'](kw0)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 8 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{//?[Tag='GENE'](kw0)=>//?", "[Value='inhibitor'](kw1)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 9 //S{/?[Tag='DRUG'](kw2)=>/VP{/NP{//?[Value='inhibitor'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 10 //NP{/?[Tag='DRUG'](kw2)=>/NP{/NP{/?[Tag='GENE'](kw0)}}=>/?", "[Value='inhibitor'](kw1)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 11 //S{/?[Tag='DRUG'](kw2)=>/?[Value='inhibitor'](kw1)=>/?", "[Tag='GENE'](kw0)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 12 //NP{/NP{/?[Tag='DRUG'](kw2)=>/NP{/?[Value='inhibitor'](kw1)}=>/PP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 13 //S{/?[Tag='GENE'](kw0)=>/?[Value='inhibitor'](kw1)=>/?", "[Tag='DRUG'](kw2)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 14 //NP{/?[Tag='DRUG'](kw2)=>/NP{/NP{/?[Value='inhibitor'](kw1)}=>/PP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 15 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{//?[Value='inhibitor'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 16 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/NP{//?[Value='inhibitor'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 17 //NP{/?[Tag='DRUG'](kw2)=>/NP{/NP{/?[Tag='GENE'](kw0)=>/?", "[Value='inhibitor'](kw1)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 18 //NP{/NP{/PP{//?[Tag='DRUG'](kw2)=>//?[Tag='GENE'](kw0)}=>/NP{/?", "[Value='inhibitor'](kw1)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 19 //NP{/NP{/?[Tag='GENE'](kw0)=>/?[Value='inhibitor'](kw1)=>/?", "[Tag='DRUG'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 20 //S{/?[Tag='DRUG'](kw2)=>/?[Tag='GENE'](kw0)=>/?", "[Value='inhibitor'](kw1)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 21 //NP{/NP{/?[Tag='DRUG'](kw2)=>/?[Tag='GENE'](kw0)=>/?", "[Value='inhibitor'](kw1)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 22 //NP{/NP{/?[Tag='DRUG'](kw2)=>/NP{/?[Tag='GENE'](kw0)=>/?", "[Value='inhibitor'](kw1)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 23 //NP{/NP{/PP{//?[Tag='DRUG'](kw2)}=>/?[Tag='GENE'](kw0)=>/?", "[Value='inhibitor'](kw1)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 24 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value='inhibited'](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 25 //S{/NP{/NP{/?[Tag='DRUG'](kw2)}}=>/VP{/?[Value='inhibited'](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 26 //S{/?[Tag='DRUG'](kw2)=>/?[Value='inhibited'](kw1)=>/?", "[Tag='GENE'](kw0)} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 27 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/?[Value='inhibited'](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 28 //S{/NP{/NP{//?[Tag='DRUG'](kw2)}}=>/VP{/?[Value='inhibited'](kw1)=>/NP{/?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 29 //S{/NP{/PP{//?[Tag='GENE'](kw0)}}=>/VP{/VP{/?[Value='inhibited'](kw1)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 30 //S{/NP{/NP{//?[Tag='DRUG'](kw2)}}=>/VP{/?[Value='inhibited'](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 31 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{/?[Value='inhibited'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 32 //S{/NP{/PP{//?[Tag='DRUG'](kw2)}}=>/VP{/?[Value='inhibited'](kw1)=>/NP{/?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 33 //S{/NP{/NP{/?[Tag='GENE'](kw0)}}=>/VP{/VP{/?[Value='inhibited'](kw1)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 34 //S{/NP{/NP{/?[Tag='GENE'](kw0)}}=>/VP{/?[Value='inhibited'](kw1)=>/PP{//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 35 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value='inhibited'](kw1)=>/PP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 36 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/VP{/?[Value='inhibited'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 37 //S{/NP{/NP{/?[Tag='DRUG'](kw2)}}=>/VP{/?[Value='inhibited'](kw1)=>/NP{/?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 38 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/VP{/?[Value='inhibited'](kw1)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 39 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/?[Value='inhibited'](kw1)=>/NP{/?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 40 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/VP{//?[Value='inhibited'](kw1)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 41 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value='inhibited'](kw1)=>/NP{/?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 42 //S{/NP{/NP{/?[Tag='GENE'](kw0)=>/?[Tag='DRUG'](kw2)}}=>/VP{/?", "[Value='inhibit'](kw1)}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 43 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/VP{//?[Value='inhibit'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 44 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{//?[Value='inhibit'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 45 //NP{/NP{/?[Tag='GENE'](kw0)=>/?[Value='inhibition'](kw1)}=>/PP{/NP{/?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 46 //S{/?[Tag='DRUG'](kw2)=>/VP{/PP{//?[Value='inhibition'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 47 //S{/NP{/PP{//?[Tag='DRUG'](kw2)}}=>/VP{/NP{//?[Value='inhibition'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 48 //NP{/?[Value='inhibition'](kw1)=>/PP{/NP{//?[Tag='GENE'](kw0)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 49 //S{/?[Value='inhibition'](kw1)=>/PP{/NP{//?[Tag='GENE'](kw0)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 50 //NP{/NP{/?[Tag='DRUG'](kw2)=>/?[Value='inhibition'](kw1)}=>/PP{/NP{/?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 51 //NP{/NP{/?[Value='inhibition'](kw1)}=>/PP{/NP{/?[Tag='GENE'](kw0)=>/?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 52 //S{/NP{/?[Value='inhibition'](kw1)=>/PP{//?[Tag='GENE'](kw0)}}=>/VP{/PP{//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 53 //S{/NP{/NP{/?[Value='inhibition'](kw1)}=>/PP{//?[Tag='GENE'](kw0)}}=>/VP{/NP{//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 54 //S{/?[Value IN {'inhibition','Inhibition'}](kw1)=>/?[Tag='GENE'](kw0)=>/?", "[Tag='DRUG'](kw2)} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 55 //NP{/NP{/NP{/?[Value='inhibition'](kw1)}=>/PP{//?[Tag='GENE'](kw0)}}=>/PP{/NP{//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 56 //NP{/?[Tag='DRUG'](kw2)=>/NP{/?[Value='inhibition'](kw1)}=>/PP{/NP{/?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 57 //NP{/NP{/?[Value='inhibition'](kw1)}=>/PP{/NP{//?[Tag='GENE'](kw0)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 58 //S{/?[Tag='DRUG'](kw2)=>/VP{/NP{//?[Value='inhibition'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 59 //S{/NP{/NP{/?[Value='inhibition'](kw1)}=>/PP{//?[Tag='DRUG'](kw2)}}=>/VP{/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 60 //NP{/NP{/?[Value='inhibition'](kw1)=>/PP{//?[Tag='GENE'](kw0)}}=>/PP{/NP{//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 61 //NP{/NP{/?[Tag='DRUG'](kw2)=>/?[Value='inhibition'](kw1)}=>/PP{/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 62 //NP{/NP{/?[Value='inhibition'](kw1)=>/PP{//?[Tag='GENE'](kw0)}}=>/PP{/NP{/?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 63 //NP{/NP{/NP{/?[Value='inhibition'](kw1)}=>/PP{//?[Tag='GENE'](kw0)}}=>/PP{/NP{/?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 64 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/?[Value='inhibits'](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 65 /S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value='inhibits'](kw1)=>/NP{/?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 66 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{/?[Value='inhibits'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 67 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value='inhibits'](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 68 //NP{/NP{/NP{/?[Value='inhibitory'](kw1)}=>/PP{//?[Tag='DRUG'](kw2)}}=>/PP{/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 69 //S{/?[Tag='DRUG'](kw2)=>/VP{/NP{/?[Tag='GENE'](kw0)=>/?", "[Value='inhibitor'](kw1)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 70 //NP{/?[Tag='DRUG'](kw2)=>/NP{/?[Tag='GENE'](kw0)=>/?", "[Value='inhibitor'](kw1)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 71 //NP{/NP{/?[Tag='GENE'](kw0)=>/?[Value='inhibitor'](kw1)}=>/?", "[Tag='DRUG'](kw2)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 72 //NP{/NP{/NP{/?[Tag='GENE'](kw0)=>/?[Tag='DRUG'](kw2)}}=>/?", "[Value='inhibitor'](kw1)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 73 //NP{/?[Tag='GENE'](kw0)=>/?[Value='inhibitor'](kw1)=>/?", "[Tag='DRUG'](kw2)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 74 //NP{/NP{/?[Value='inhibitor'](kw1)}=>/PP{/NP{//?[Tag='GENE'](kw0)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 75 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{//?[Tag='GENE'](kw0)=>//?", "[Value='inhibitor'](kw1)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 76 //S{/?[Tag='DRUG'](kw2)=>/VP{/NP{//?[Value='inhibitor'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 77 //NP{/?[Tag='DRUG'](kw2)=>/NP{/NP{/?[Tag='GENE'](kw0)}}=>/?", "[Value='inhibitor'](kw1)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 78 //S{/?[Tag='DRUG'](kw2)=>/?[Value='inhibitor'](kw1)=>/?", "[Tag='GENE'](kw0)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 79 //NP{/NP{/?[Tag='DRUG'](kw2)=>/NP{/?[Value='inhibitor'](kw1)}=>/PP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 80 //S{/?[Tag='GENE'](kw0)=>/?[Value='inhibitor'](kw1)=>/?", "[Tag='DRUG'](kw2)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 81 //NP{/?[Tag='DRUG'](kw2)=>/NP{/NP{/?[Value='inhibitor'](kw1)}=>/PP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 82 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{//?[Value='inhibitor'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 83 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/NP{//?[Value='inhibitor'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 84 //NP{/?[Tag='DRUG'](kw2)=>/NP{/NP{/?[Tag='GENE'](kw0)=>/?", "[Value='inhibitor'](kw1)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 85 //NP{/NP{/PP{//?[Tag='DRUG'](kw2)=>//?[Tag='GENE'](kw0)}=>/NP{/?", "[Value='inhibitor'](kw1)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 86 //NP{/NP{/?[Tag='GENE'](kw0)=>/?[Value='inhibitor'](kw1)=>/?", "[Tag='DRUG'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 87 //S{/?[Tag='DRUG'](kw2)=>/?[Tag='GENE'](kw0)=>/?", "[Value='inhibitor'](kw1)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 88 //NP{/NP{/?[Tag='DRUG'](kw2)=>/?[Tag='GENE'](kw0)=>/?", "[Value='inhibitor'](kw1)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 89 //NP{/NP{/?[Tag='DRUG'](kw2)=>/NP{/?[Tag='GENE'](kw0)=>/?", "[Value='inhibitor'](kw1)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 90 //NP{/NP{/PP{//?[Tag='DRUG'](kw2)}=>/?[Tag='GENE'](kw0)=>/?", "[Value='inhibitor'](kw1)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 91 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value='inhibited'](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 92 //S{/NP{/NP{/?[Tag='DRUG'](kw2)}}=>/VP{/?[Value='inhibited'](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 93 //S{/?[Tag='DRUG'](kw2)=>/?[Value='inhibited'](kw1)=>/?", "[Tag='GENE'](kw0)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 94 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/?[Value='inhibited'](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 95 //S{/NP{/NP{//?[Tag='DRUG'](kw2)}}=>/VP{/?[Value='inhibited'](kw1)=>/NP{/?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 96 //S{/NP{/PP{//?[Tag='GENE'](kw0)}}=>/VP{/VP{/?[Value='inhibited'](kw1)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 97 //S{/NP{/NP{//?[Tag='DRUG'](kw2)}}=>/VP{/?[Value='inhibited'](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 98 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{/?[Value='inhibited'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 99 //S{/NP{/PP{//?[Tag='DRUG'](kw2)}}=>/VP{/?[Value='inhibited'](kw1)=>/NP{/?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 100 //S{/NP{/NP{/?[Tag='GENE'](kw0)}}=>/VP{/VP{/?[Value='inhibited'](kw1)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 101 //S{/NP{/NP{/?[Tag='GENE'](kw0)}}=>/VP{/?[Value='inhibited'](kw1)=>/PP{//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 102 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value='inhibited'](kw1)=>/PP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 103 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/VP{/?[Value='inhibited'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 104 //S{/NP{/NP{/?[Tag='DRUG'](kw2)}}=>/VP{/?[Value='inhibited'](kw1)=>/NP{/?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 105 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/VP{/?[Value='inhibited'](kw1)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 106 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/?[Value='inhibited'](kw1)=>/NP{/?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 107 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/VP{//?[Value='inhibited'](kw1)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 108 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value='inhibited'](kw1)=>/NP{/?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 109 //S{/NP{/NP{/?[Tag='GENE'](kw0)=>/?[Tag='DRUG'](kw2)}}=>/VP{/?", "[Value='inhibit'](kw1)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 110 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/VP{//?[Value='inhibit'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 111 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{//?[Value='inhibit'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 112 //NP{/NP{/?[Tag='GENE'](kw0)=>/?[Value='inhibition'](kw1)}=>/PP{/NP{/?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 113 //S{/?[Tag='DRUG'](kw2)=>/VP{/PP{//?[Value='inhibition'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 114 //S{/NP{/PP{//?[Tag='DRUG'](kw2)}}=>/VP{/NP{//?[Value='inhibition'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 115 //NP{/?[Value='inhibition'](kw1)=>/PP{/NP{//?[Tag='GENE'](kw0)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 116 //S{/?[Value='inhibition'](kw1)=>/PP{/NP{//?[Tag='GENE'](kw0)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 117 //NP{/NP{/?[Tag='DRUG'](kw2)=>/?[Value='inhibition'](kw1)}=>/PP{/NP{/?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 118 //NP{/NP{/?[Value='inhibition'](kw1)}=>/PP{/NP{/?[Tag='GENE'](kw0)=>/?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 119 //S{/NP{/?[Value='inhibition'](kw1)=>/PP{//?[Tag='GENE'](kw0)}}=>/VP{/PP{//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 120 //S{/NP{/NP{/?[Value='inhibition'](kw1)}=>/PP{//?[Tag='GENE'](kw0)}}=>/VP{/NP{//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 121 //S{/?[Value IN {'inhibition','Inhibition'}](kw1)=>/?[Tag='GENE'](kw0)=>/?", "[Tag='DRUG'](kw2)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 122 //NP{/NP{/NP{/?[Value='inhibition'](kw1)}=>/PP{//?[Tag='GENE'](kw0)}}=>/PP{/NP{//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 123 //NP{/?[Tag='DRUG'](kw2)=>/NP{/?[Value='inhibition'](kw1)}=>/PP{/NP{/?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 124 //NP{/NP{/?[Value='inhibition'](kw1)}=>/PP{/NP{//?[Tag='GENE'](kw0)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 125 //S{/?[Tag='DRUG'](kw2)=>/VP{/NP{//?[Value='inhibition'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 126 //S{/NP{/NP{/?[Value='inhibition'](kw1)}=>/PP{//?[Tag='DRUG'](kw2)}}=>/VP{/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 127 //NP{/NP{/?[Value='inhibition'](kw1)=>/PP{//?[Tag='GENE'](kw0)}}=>/PP{/NP{//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 128 //NP{/NP{/?[Tag='DRUG'](kw2)=>/?[Value='inhibition'](kw1)}=>/PP{/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 129 //NP{/NP{/?[Value='inhibition'](kw1)=>/PP{//?[Tag='GENE'](kw0)}}=>/PP{/NP{/?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 130 //NP{/NP{/NP{/?[Value='inhibition'](kw1)}=>/PP{//?[Tag='GENE'](kw0)}}=>/PP{/NP{/?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 131 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/?[Value='inhibits'](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 132 /S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value='inhibits'](kw1)=>/NP{/?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 133 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{/?[Value='inhibits'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 134 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value='inhibits'](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value Gene Metabolized Drug 1 //S{/NP{/NP{/?[Tag='DRUG'](kw2)}}=>/VP{/VP{/?[Value IN {'metabolised','metabolized'}](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 2 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{//?[Value IN {'metabolised','metabolized'}](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 3 //S{/NP{/SBAR{//?[Tag='DRUG'](kw2)}}=>/VP{/?[Value IN {'metabolised','metabolized'}](kw1)=>/PP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 4 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/VP{/?[Value IN {'metabolised','metabolized'}](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 5 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/VP{//?[Value IN {'metabolized','metabolised'}](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 6 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{/?[Value IN {'metabolised','metabolized'}](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 7 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value='metabolized'](kw1)=>/PP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 8 //S{/?[Tag='DRUG'](kw2)=>/NP{/?[Value='metabolism'](kw1)}=>/VP{/VP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 9 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/NP{//?[Tag='DRUG'](kw2)=>//?", "[Value='metabolism'](kw1)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 10 //NP{/NP{/?[Tag='GENE'](kw0)=>/?[Value='metabolism'](kw1)}=>/PP{/NP{/?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 11 //NP{/?[Value='metabolism'](kw1)=>/PP{/NP{//?[Tag='DRUG'](kw2)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 12 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/VP{//?[Value='metabolism'](kw1)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 13 //S{/NP{/NP{/?[Tag='GENE'](kw0)}}=>/VP{/NP{//?[Value='metabolism'](kw1)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 14 //NP{/NP{/PP{//?[Tag='GENE'](kw0)}}=>/PP{/NP{/?[Tag='DRUG'](kw2)=>/?", "[Value='metabolism'](kw1)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 15 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/PP{//?[Value='metabolism'](kw1)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 16 //NP{/NP{/?[Tag='DRUG'](kw2)=>/?[Value='metabolism'](kw1)}=>/PP{/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 17 //NP{/NP{/?[Tag='GENE'](kw0)=>/?[Value='metabolism'](kw1)}=>/PP{/NP{//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 18 //S{/NP{/NP{/?[Value='metabolism'](kw1)}=>/PP{//?[Tag='DRUG'](kw2)}}=>/VP{/VP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 19 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/PP{//?[Tag='DRUG'](kw2)=>//?", "[Value='metabolism'](kw1)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 20 //NP{/NP{/PP{//?[Tag='GENE'](kw0)}}=>/PP{/NP{//?[Value='metabolism'](kw1)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 21 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/NP{//?[Value='substrate'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 22 //NP{/NP{/?[Tag='DRUG'](kw2)=>/NP{/?[Value='substrate'](kw1)}=>/PP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 23 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{//?[Value='substrate'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 24 //NP{/?[Tag='GENE'](kw0)=>/?[Value='substrate'](kw1)=>/?", "[Tag='DRUG'](kw2)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 25 //NP{/?[Tag='DRUG'](kw2)=>/NP{/NP{/?[Value='substrate'](kw1)}=>/PP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 26 //S{/NP{/NP{/?[Tag='DRUG'](kw2)}}=>/VP{/VP{/?[Value IN {'metabolised','metabolized'}](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 27 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{//?[Value IN {'metabolised','metabolized'}](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 28 //S{/NP{/SBAR{//?[Tag='DRUG'](kw2)}}=>/VP{/?[Value IN {'metabolised','metabolized'}](kw1)=>/PP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 29 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/VP{/?[Value IN {'metabolised','metabolized'}](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 30 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/VP{//?[Value IN {'metabolized','metabolised'}](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 31 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{/?[Value IN {'metabolised','metabolized'}](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 32 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value='metabolized'](kw1)=>/PP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 33 //S{/?[Tag='DRUG'](kw2)=>/NP{/?[Value='metabolism'](kw1)}=>/VP{/VP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 34 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/NP{//?[Tag='DRUG'](kw2)=>//?", "[Value='metabolism'](kw1)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 35 //NP{/NP{/?[Tag='GENE'](kw0)=>/?[Value='metabolism'](kw1)}=>/PP{/NP{/?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 36 //NP{/?[Value='metabolism'](kw1)=>/PP{/NP{//?[Tag='DRUG'](kw2)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 37 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/VP{//?[Value='metabolism'](kw1)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 38 //S{/NP{/NP{/?[Tag='GENE'](kw0)}}=>/VP{/NP{//?[Value='metabolism'](kw1)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 39 //NP{/NP{/PP{//?[Tag='GENE'](kw0)}}=>/PP{/NP{/?[Tag='DRUG'](kw2)=>/?", "[Value='metabolism'](kw1)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 40 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/PP{//?[Value='metabolism'](kw1)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 41 //NP{/NP{/?[Tag='DRUG'](kw2)=>/?[Value='metabolism'](kw1)}=>/PP{/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 42 //NP{/NP{/?[Tag='GENE'](kw0)=>/?[Value='metabolism'](kw1)}=>/PP{/NP{//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 43 //S{/NP{/NP{/?[Value='metabolism'](kw1)}=>/PP{//?[Tag='DRUG'](kw2)}}=>/VP{/VP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 44 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/PP{//?[Tag='DRUG'](kw2)=>//?", "[Value='metabolism'](kw1)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 45 //NP{/NP{/PP{//?[Tag='GENE'](kw0)}}=>/PP{/NP{//?[Value='metabolism'](kw1)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 46 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/NP{//?[Value='substrate'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 47 //NP{/NP{/?[Tag='DRUG'](kw2)=>/NP{/?[Value='substrate'](kw1)}=>/PP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 48 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{//?[Value='substrate'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 49 //NP{/?[Tag='GENE'](kw0)=>/?[Value='substrate'](kw1)=>/?", "[Tag='DRUG'](kw2)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 50 //NP{/?[Tag='DRUG'](kw2)=>/NP{/NP{/?[Value='substrate'](kw1)}=>/PP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value Gene Regulates Gene 1 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/VP{//?[Value IN {'regulated', 'upregulated', 'downregulated', 'up-regulated', 'down-regulated'}](kw1)=>//?[Value='by'](kw3)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 2 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/VP{/?[Value IN {'regulated', 'upregulated', 'downregulated', 'up-regulated', 'down-regulated'}](kw1)=>//?[Value='by'](kw3)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 3 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/?[Value IN {'regulated','down-regulated'}](kw1)=>/NP{//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 4 //NP{/NP{/?[Value IN {'regulation', 'upregulation', 'downregulation', 'up-regulation', 'down-regulation'}](kw1)}=>/PP{/NP{//?[Tag='GENE'](kw0)=>//?[Value='by'](kw3)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 5 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/?[Value IN {'regulates', 'upregulates', 'downregulates', 'up-regulates', 'down-regulates'}](kw1)=>/NP{//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 6 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/VP{//?[Value='in'](kw3)=>//?[Value='regulating'](kw1)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value Gene Regulate Gene (Xenobiotic Metabolism) 1 //S{//?[Tag='GENE' AND Canonical LIKE 'CYP\\%'](kw0)<=>//?", "[Tag='GENE' AND Canonical IN {'AhR', 'CASR', 'CAR', 'PXR', 'NR1I2', 'NR1I3'}](kw1)} ::: distinct sent.cid, kw0.value, kw1.value, sent.value 2 //S{//?[Tag='GENE' AND Value LIKE 'cytochrome\\%'](kw0)<=>//?", "[Tag='GENE' AND Canonical IN {'AhR', 'CASR', 'CAR', 'PXR', 'NR1I2', 'NR1I3'}](kw1)} ::: distinct sent.cid, kw0.value, kw1.value, sent.value Negative Drug Induces/Metabolizes/Inhibits Gene 1 //S{/?[Tag='DRUG'](kw0)=>/VP{/VP{/?[Value IN {'induced', 'inhibited', 'metabolized', 'metabolised'}](kw1)=>//?[Tag='GENE'](kw2)=>//?[Value='not'](kw3)=>//?", "[Tag='GENE'](kw4)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw4.value, kw3.value, sent.value 2 //S{/?[Tag='DRUG'](kw0)=>/?[Value='not'](kw3)=>/?[Tag='DRUG'](kw4)=>/?[Value='inhibited'](kw1)=>/?", "[Tag='GENE'](kw2)} ::: distinct sent.cid, kw4.value, kw1.value, kw2.value, kw3.value, sent.value 3 //S{/SBAR{/S{//?[Tag='DRUG'](kw0)}}=>/S{/S{//?[Value='metabolized'](kw1)=>//?[Tag='GENE'](kw2)=>//?[Value='not'](kw3)=>//?", "[Tag='GENE'](kw4)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw4.value, kw3.value, sent.value 4 //S{/NP{/PP{//?[Tag='GENE'](kw2)}}=>/VP{/VP{/?[Value IN {'induced', 'inhibited', 'metabolized', 'metabolised'}](kw1)=>//?[Tag='DRUG'](kw0)=>//?[Value='not'](kw3)=>//?", "[Tag='DRUG'](kw4)}}} ::: distinct sent.cid, kw4.value, kw1.value, kw2.value, kw3.value, sent.value 5 //S{/NP{/PP{//?[Tag='DRUG'](kw0)}}=>/VP{/VP{/?[Value IN {'induced', 'inhibited', 'metabolized', 'metabolised'}](kw1)=>//?[Tag='GENE'](kw2)=>//?[Value IN {'not','no'}](kw3)=>//?", "[Tag='GENE'](kw4)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw4.value, kw3.value, sent.value 6 //S{/?[Tag='DRUG'](kw0)=>/VP{/?[Value='not'](kw3)=>/VP{//?[Value IN {'induced', 'inhibited', 'metabolized', 'metabolised'}](kw1)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 7 //S{/NP{/?[Tag='GENE'](kw2)}=>/VP{/VP{/?[Value IN {'induced', 'inhibited', 'metabolized', 'metabolised'}](kw1)=>//?[Tag='DRUG'](kw0)=>//?[Value='not'](kw3)=>//?", "[Tag='DRUG'](kw4)}}} ::: distinct sent.cid, kw4.value, kw1.value, kw2.value, kw3.value, sent.value 8 //S{/NP{/NP{/?[Value='not'](kw3)=>/?[Tag='DRUG'](kw0)}}=>/VP{/?[Value IN {'inhibited','induced'}](kw1)=>/NP{//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 9 //S{/NP{/NP{/?[Value IN {'no','not'}](kw3)=>/?[Tag='DRUG'](kw0)}}=>/VP{/?[Value IN {'inhibited','induced'}](kw1)=>/NP{/?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 10 //S{/?[Tag='DRUG'](kw0)=>/S{/S{//?[Value='not'](kw3)=>//?[Value IN {'induce', 'inhibit'}](kw1)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 11 //S{/?[Tag='DRUG'](kw0)=>/?[Tag='GENE'](kw2)=>/?[Value IN {'not'}](kw3)=>/?", "[Value IN {'inhibit','induce'}](kw1)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 12 //S{/?[Tag='DRUG'](kw0)=>/VP{/?[Value='not'](kw3)=>/VP{/?[Value IN {'inhibit','induce'}](kw1)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 13 //S{/NP{/?[Tag='DRUG'](kw0)}=>/VP{/VP{/?[Value='not'](kw3)=>//?[Value='inhibit'](kw1)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 14 //S{/NP{/NP{/?[Tag='GENE'](kw2)}}=>/VP{/?[Value='not'](kw3)=>/VP{/?[Value='metabolize'](kw1)=>//?", "[Tag='DRUG'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 15 //S{/NP{/NP{//?[Tag='DRUG'](kw0)}}=>/VP{/?[Value='not'](kw3)=>/VP{/?[Value IN {'inhibit','induce'}](kw1)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 16 //S{/NP{/?[Tag='DRUG'](kw0)}=>/VP{/?[Value='not'](kw3)=>/VP{/?[Value='inhibit'](kw1)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 17 //S{/NP{/?[Tag='DRUG'](kw0)}=>/VP{/?[Value IN {'induces', 'inhibits'}](kw1)=>/NP{//?[Tag='GENE'](kw2)=>//?[Value='not'](kw3)=>//?", "[Tag='GENE'](kw4)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw4.value, kw3.value, sent.value 18 //S{/?[Tag='DRUG'](kw0)=>/VP{/?[Value='inhibits'](kw1)=>/NP{//?[Tag='GENE'](kw2)=>//?[Value='not'](kw3)=>//?", "[Tag='GENE'](kw4)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw4.value, kw3.value, sent.value 19 //S{/?[Tag='DRUG'](kw0)=>/VP{/NP{//?[Value='no'](kw3)=>//?[Value IN {'induction','metabolism','inhibition'}](kw1)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 20 //S{/NP{/NP{//?[Tag='DRUG'](kw0)=>//?[Tag='GENE'](kw2)=>//?[Value IN {'induction','metabolism','inhibition'}](kw1)}}=>/VP{/?", "[Value='not'](kw3)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 21 //S{/?[Tag='DRUG'](kw0)=>/VP{/?[Value='not'](kw3)=>/VP{//?[Value IN {'induction','metabolism','inhibition'}](kw1)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 22 //S{/NP{/PP{//?[Tag='DRUG'](kw0)}}=>/VP{/?[Value='not'](kw3)=>/VP{//?[Value IN {'induction','metabolism','inhibition'}](kw1)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value Negative Drug Induces Gene 1 //S{//?[Tag='GENE'](kw0)=>//?[Value IN {'induced','increased','stimulated'}](kw1)=>//?[Value='by'](kw3)=>//?[Tag='DRUG'](kw2)=>//?[Value='not'](kw4)=>//?", "[Tag='DRUG'](kw5)} ::: distinct sent.cid, kw5.value, kw1.value, kw0.value, kw4.value, sent.value 2 //S{//?[Tag='GENE'](kw0)=>//?[Value='not'](kw4)=>//?[Tag='GENE'](kw5)=>//?[Value IN {'induced','increased','stimulated'}](kw1)=>//?[Value='by'](kw3)=>//?", "[Tag='DRUG'](kw2)} ::: distinct sent.cid, kw2.value, kw1.value, kw5.value, kw4.value, sent.value 3 //S{//?[Tag='DRUG'](kw0)=>//?[Value='not'](kw4)=>//?[Value IN {'induce','induced','increase','increased','stimulate','stimulated'}](kw1)=>//?", "[Tag='GENE'](kw2)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw4.value, sent.value 4 //S{//?[Tag='DRUG'](kw0)=>//?[Value='not'](kw4)=>//?[Tag='DRUG'](kw5)=>//?[Value IN {'induces','increases','stimulates','induced','increased','stimulated'}](kw1)=>//?", "[Tag='GENE'](kw2)} ::: distinct sent.cid, kw5.value, kw1.value, kw2.value, kw4.value, sent.value 5 //S{//?[Value='no'](kw5)=>//?[value IN {'induction','stimulation'}](kw1)=>//?[Value IN {'of'}](kw2)=>//?[Tag='GENE'](kw0)=>//?[value IN {'by'}](kw3)=>//?", "[Tag='DRUG'](kw4)} ::: distinct sent.cid, kw4.value, kw1.value, kw0.value, kw5.value, sent.value Negative Drug Inhibits Gene 1 //S{//?[Tag='GENE'](kw0)=>//?[Value IN {'inhibited','decreased'}](kw1)=>//?[Value='by'](kw3)=>//?[Tag='DRUG'](kw2)=>//?[Value='not'](kw4)=>//?", "[Tag='DRUG'](kw5)} ::: distinct sent.cid, kw5.value, kw1.value, kw0.value, kw4.value, sent.value 2 //S{//?[Tag='GENE'](kw0)=>//?[Value='not'](kw4)=>//?[Tag='GENE'](kw5)=>//?[Value IN {'inhibited','decreased'}](kw1)=>//?[Value='by'](kw3)=>//?", "[Tag='DRUG'](kw2)} ::: distinct sent.cid, kw2.value, kw1.value, kw5.value, kw4.value, sent.value 3 //S{//?[Tag='DRUG'](kw0)=>//?[Value='not'](kw4)=>//?[Value IN {'inhibit','inhibited','decrease','decreased'}](kw1)=>//?", "[Tag='GENE'](kw2)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw4.value, sent.value 4 //S{//?[Tag='DRUG'](kw0)=>//?[Value='not'](kw4)=>//?[Tag='DRUG'](kw5)=>//?[Value IN {'inhibits','decreases','inhibited','decreased'}](kw1)=>//?", "[Tag='GENE'](kw2)} ::: distinct sent.cid, kw5.value, kw1.value, kw2.value, kw4.value, sent.value 5 //S{//?[Value='no'](kw5)=>//?[value IN {'inhibition'}](kw1)=>//?[Value IN {'of'}](kw2)=>//?[Tag='GENE'](kw0)=>//?[value IN {'by'}](kw3)=>//?", "[Tag='DRUG'](kw4)} ::: distinct sent.cid, kw4.value, kw1.value, kw0.value, kw5.value, sent.value Negative Gene Metabolizes Drug 1 //S{//?[Tag='DRUG'](kw0)=>//?[Value IN {'metabolized','metabolised'}](kw1)=>//?[Value='by'](kw3)=>//?[Tag='GENE'](kw2)=>//?[Value='not'](kw4)=>//?", "[Tag='GENE'](kw5)} ::: distinct sent.cid, kw0.value, kw1.value, kw5.value, kw4.value, sent.value 2 //S{//?[Tag='DRUG'](kw0)=>//?[Value='not'](kw4)=>//?[Tag='DRUG'](kw5)=>//?[Value IN {'metabolized','metabolised'}](kw1)=>//?[Value='by'](kw3)=>//?", "[Tag='GENE'](kw2)} ::: distinct sent.cid, kw5.value, kw1.value, kw2.value, kw4.value, sent.value 3 //S{//?[Tag='GENE'](kw0)=>//?[Value='not'](kw4)=>//?[Value IN {'metabolize','metabolise'}](kw1)=>//?", "[Tag='DRUG'](kw2)} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, kw4.value, sent.value 4 //S{//?[Tag='GENE'](kw0)=>//?[Value='not'](kw4)=>//?[Tag='GENE'](kw5)=>//?[Value IN {'metabolize','metabolise','metabolizes','metabolises'}](kw1)=>//?", "[Tag='DRUG'](kw2)} ::: distinct sent.cid, kw2.value, kw1.value, kw5.value, kw4.value, sent.value Negative Gene Downregulates Gene 1 //S{//?[Tag='GENE'](kw0)=>//?[Value IN {'suppressed','downregulated','inhibited','down-regulated','repressed','disrupted'}](kw1)=>//?[Value='by'](kw3)=>//?[Tag='GENE'](kw2)=>//?[Value='not'](kw4)=>//?", "[Tag='GENE'](kw5)} ::: distinct sent.cid, kw5.value, kw1.value, kw0.value, kw4.value, sent.value 2 //S{//?[Tag='GENE'](kw0)=>//?[Value='not'](kw4)=>//?[Tag='GENE'](kw5)=>//?[Value IN {'suppressed','downregulated','inhibited','down-regulated','repressed','disrupted'}](kw1)=>//?[Value='by'](kw3)=>//?", "[Tag='GENE'](kw2)} ::: distinct sent.cid, kw2.value, kw1.value, kw5.value, kw4.value, sent.value 3 //S{//?[Tag='GENE'](kw0)=>//?[Value='not'](kw4)=>//?[Value IN {'suppressed','suppress','downregulated','downregulate','inhibited','inhibit','down-regulated','down-regulate','repressed','repress','disrupted','disrupt'}](kw1)=>//?", "[Tag='GENE'](kw2)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw4.value, sent.value 4 //S{//?[Tag='GENE'](kw0)=>//?[Value='not'](kw4)=>//?[Tag='GENE'](kw5)=>//?[Value IN {'suppresses','downregulates','inhibits','down-regulates','represses','disrupts','suppressed','downregulated','inhibited','down-regulated','repressed','disrupted'}](kw1)=>//?", "[Tag='GENE'](kw2)} ::: distinct sent.cid, kw5.value, kw1.value, kw2.value, kw4.value, sent.value 5 //S{//?[Value='no'](kw5)=>//?[value IN {'inhibition','downregulation','down-regulation'}](kw1)=>//?[Value IN {'of'}](kw2)=>//?[Tag='GENE'](kw0)=>//?[value IN {'on'}](kw3)=>//?", "[Tag='GENE'](kw4)} ::: distinct sent.cid, kw0.value, kw1.value, kw4.value, kw5.value, sent.value Negative Gene Upregulates Gene 1 //S{//?[Tag='GENE'](kw0)=>//?[Value IN {'activated','induced','stimulated','regulated','upregulated','up-regulated'}](kw1)=>//?[Value='by'](kw3)=>//?[Tag='GENE'](kw2)=>//?[Value='not'](kw4)=>//?", "[Tag='GENE'](kw5)} ::: distinct sent.cid, kw5.value, kw1.value, kw0.value, kw4.value, sent.value 2 //S{//?[Tag='GENE'](kw0)=>//?[Value='not'](kw4)=>//?[Tag='GENE'](kw5)=>//?", "[Value IN {'activated','induced','stimulated','regulated','upregulated','up-regulated'}](kw1) ::: distinct sent.cid, kw2.value, kw1.value, kw5.value, kw4.value, sent.value 3 //S{//?[Tag='GENE'](kw0)=>//?[Value='not'](kw4)=>//?[Value IN {'activated','induced','stimulated','regulated','upregulated','up-regulated','activate','induce','stimulate','regulate','upregulate','up-regulate'}](kw1)=>//?", "[Tag='GENE'](kw2)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw4.value, sent.value 4 //S{//?[Tag='GENE'](kw0)=>//?[Value='not'](kw4)=>//?[Tag='GENE'](kw5)=>//?[Value IN {'activates','induces','stimulates','regulates','upregulates','up-regulates','activate','induce','stimulate','regulate','upregulate','up-regulate'}](kw1)=>//?", "[Tag='GENE'](kw2)} ::: distinct sent.cid, kw5.value, kw1.value, kw2.value, kw4.value, sent.value 5 //S{//?[Value='no'](kw5)=>//?[value IN {'induction','activation','stimulation','regulation','upregulation','up-regulation'}](kw1)=>//?[Value IN {'of'}](kw2)=>//?[Tag='GENE'](kw0)=>//?[value IN {'by'}](kw3)=>//?", "[Tag='GENE'](kw4)} ::: distinct sent.cid, kw4.value, kw1.value, kw0.value, kw5.value, sent.value Drug Gene Co-Occurrence 1 //S{//?[Tag='DRUG'](kw0)<=>//?", "[Tag='GENE'](kw1)} ::: distinct sent.cid, kw0.value, kw1.value, kw0.type, kw1.type, sent.value" ], [ "Text Extraction for Real World Applications", "In the previous chapter we looked at the BioPathQA system and how it answers simulation based reasoning questions about biological pathways, including questions that require comparison of alternate scenarios through simulation.", "These so called `what-if' questions arise in biological activities such as drug development, drug interaction, and personalized medicine.", "We will now put our system and language in context of such activities.", "Cutting-edge knowledge about pathways for activities such as drug development, drug interaction, and personalized medicine comes in the form of natural language research papers, thousands of which are published each year.", "To use this knowledge with our system, we need to perform extraction.", "In this chapter we describe techniques we use for such knowledge extraction for discovering drug interactions.", "We illustrate with an example extraction how we organize the extracted knowledge into a pathway specification and give examples of relevant what-if questions that a researcher performing may ask in the drug development domain." ], [ "Introduction", "Thousands of research papers are published each year about biological systems and pathways over a broad spectrum of activities, including interactions between dugs and diseases, the associated pathways, and genetic variation.", "Thus, one has to perform text extraction to extract relationships between the biochemical processes, their involvement in diseases, and their interaction with drugs.", "For personalized medicine, one is also interested in how these interrelationships change in presence of genetic variation.", "In short, we are looking for relationships between various components of the biochemical processes and their internal and external stimuli.", "Many approaches exist for extracting relationships from a document.", "Most rely on some form of co-occurrence, relative distance, or order of words in a single document.", "Some use shallow parsing as well.", "Although these techniques tend to have a higher recall, they focus on extracting explicit relationships, which are relationships that are fully captured in a sentence or a document.", "These techniques also do not capture implicit relationships that may be spread across multiple documents.", "are spread across multiple documents relating to different species.", "Additional issues arise from the level of detail from in older vs. newer texts and seemingly contradictory information due to various levels of confidence in the techniques used.", "Many do not handle negative statements.", "We primarily use a system called PTQL [75] to extract these relationships, which allows combining the syntactic structure (parse tree), semantic elements, and word order in a relationship query.", "The sentences are pre-processed by using named-entity recognition, and entity normalization to allow querying on classes of entity types, such as drugs, and diseases; and also to allow cross-linking relationships across documents when they refer to the same entity with a different name.", "Queries that use such semantic association between words/phrases are likely to produce higher precision results.", "Source knowledge for extraction primarily comes from thousands of biological abstracts published each year in PubMed http://www.ncbi.nlm.nih.gov/pubmed.", "Next we briefly describe how we extract relationships about drug interactions.", "Following that we briefly describe how we extract association of drugs, and diseases with genetic variation.", "We conclude this chapter with an illustrative example of how the drug interaction relationships are used with our system to answer questions about drug interactions and how genetic variation could be utilized in our system." ], [ "Extracting Relationships about Drug Interactions", "We summarize the extraction of relationships for our work on drug-drug interactions from [73].", "Studying drug-drug interactions are a major activity in drug development.", "Drug interactions occur due to the interactions between the biological processes / pathways that are responsible metabolizing and transporting drugs.", "Metabolic processes remove a drug from the system within a certain time period.", "For a drug to remain effective, it must be maintained within its therapeutic window for the period of treatment, requiring taking the drug periodically.", "Outside the therapeutic window, a drug can become toxic if a quantity greater than the therapeutic window is retained; or it can become ineffective if a quantity less than the therapeutic window is retained.", "Since liver enzymes metabolize most drugs, it is the location where most metabolic-interaction takes place.", "Induction or inhibition of these enzymes can affect the bioavailability of a drug through transcriptional regulation, either directly or indirectly.", "For example, if drug $A$ inhibits enzyme $E$ , which metabolizes drug $B$ , then the bioavailability of drug $B$ will be higher than normal, rendering it toxic.", "On the other hand, if drug $A$ induces enzyme $E$ , which metabolizes drug $B$ , then drug $B$ 's bioavailability will be lesser than normal, rendering it ineffective.", "Inhibition of enzymes is a common form of drug-drug interactions [10].", "In direct inhibition, a drug $A$ inhibit enzyme $E$ , which is responsible for metabolism of drug $B$ .", "Drug $A$ , leads to a decrease in the level of enzyme $E$ , which in turn can increase bioavailability of drug $B$ potentially leading to toxicity.", "Alternatively, insufficient metabolism of drug $B$ can lead to smaller amount of drug $B$ 's metabolites being produced, leading to therapeutic failure.", "An example of one such direct inhibition is the interaction between CYP2D6 inhibitor quinidine and CYP2D6 substrates (i.e.", "substances metabolized by CYP2D6), such as Codeine.", "The inhibition of CYP2D6 increases the bioavailability of drugs metabolized by CYP2D6 leading to adverse side effects.", "Another form of drug interactions is through induction of enzymes [10].", "In direct induction, a drug $A$ induces enzyme $E$ , which is responsible for metabolism of drug $B$ .", "An example of such direct induction is between phenobarbital, a CYP2C9 inducer and warfarin (a CYP2C9 substrate).", "Phenobarbital leads to increased metabolism of warfarin, decreasing warfarinÕs bioavailability.", "Direct interaction due to induction though possible is not as common as indirect interaction through transcription factors, which regulate the drug metabolizing enzymes.", "In such an interaction, drug $A$ activates a transcription factor $TF$ , which regulates and induces enzyme $E$ , where enzyme $E$ metabolizes drug $B$ .", "Transcription factors are referred to as regulators of xenobiotic-metabolizing enzymes.", "Examples of such regulators include aryl hydrocarbon receptor AhR, pregnane X receptor PXR and constitutive androstane receptor CAR.", "Drug interactions can also occur due to the induction or inhibition of transporters.", "Transporters are mainly responsible for cellular uptake or efflux (elimination) of drugs.", "They play an important part in drug disposition, by transporting drugs into the liver cells, for example.", "Transporter-based drug interactions, however, are not as well studies as metabolism-based interactions [10]." ], [ "Method", "Extraction of drug-drug interactions from the text can either be explicit or implicit.", "Explicit extraction refers to extraction of drug-drug interaction mentioned within a single sentence, while implicit extraction requires extraction of bio-properties of drug transport, metabolism and elimination that can lead to drug-drug interaction.", "This type of indirect extraction combines background information about biological processes, identification of protein families and the interactions that are involved in drug metabolism.", "Our approach is to extract both explicit and implicit drug interactions as summarized in Fig REF and it builds upon the work done in [74].", "Figure: This figure from outlines the effects of drug A on drug B through (a) direct induction/inhibition of enzymes; (b) indirect induction/inhibition of transportation factors that regulate the drug-metabolizing enzymes." ], [ "Explicit Drug Interaction Extraction", "Explicit extraction mainly extracts drug-drug interactions directly mentioned in PubMed (or Medline) abstracts.", "For example, the following sentences each have a metabolic interaction mentioned within the same sentence: Ciprofloxacin strongly inhibits clozapine metabolism.", "(PMID: 19067475) Enantioselective induction of cyclophosphamide metabolism by phenytoin.", "which can be extracted by using the following PTQL query using the underlined keywords from above sentences: //S{//?", "[Tag=`Drug'](d1) => //?", "[Value IN {`induce',`induces',`inhibit',`inhibits'}](v) => //?[Tag=`Drug'](d2) => //?", "[Value=`metabolism'](w)} ::: [d1 v d2 w] 5 : d1.value, v.value, d2.value.", "This PTQL query specifies that a drug (denoted by d1) must be followed by one of the keywords from $\\lbrace `induce^{\\prime },`inhibit^{\\prime }, `inhibits^{\\prime }\\rbrace $ (denoted by v), which in turn must be followed by another drug (denoted by d2) followed the keyword $`metabolism^{\\prime }$ (denoted by w); all found within a proximity of 5 words of each other.", "The query produces tripes of $\\langle d1, v, d2 \\rangle $ values as output.", "Thus the results will produce triples $\\langle d1, induces, d2 \\rangle $ and $\\langle d1, inhibits, d2 \\rangle $ which mean that the drug d1 increases the effect of d2 (i.e.", "$\\langle d1, increases, d2 \\rangle $ ) and decreases the effect of d2 (i.e.", "$\\langle d1, decreases, d2 \\rangle $ ) respectively.", "For example, the sentence S1 above matches this PTQL query and the query will produce the triplet $\\langle \\text{ciprofloxacin}, \\text{increases}, \\text{clozapine} \\rangle $ ." ], [ "Implicit Drug Interaction Extraction", "Implicit extraction mainly extracts drug-drug interactions not yet published, but which can be inferred from published articles and properties of drug metabolism.", "The metabolic properties themselves have their origin in various publications.", "The metabolic interactions extracted from published articles and the background knowledge of properties of drug metabolism are reasoned with in an automated fashion to infer drug interactions.", "The following table outlines the kinds of interactions extracted from the text: Table: This table from shows the triplets representing various properties relevant to the extraction of implicit drug interactions and their description.which require multiple PTQL queries for extraction.", "As an example, the following PTQL query is used to extract $\\langle protein, metabolizes, drug \\rangle $ triplets: //S{/?", "[Tag=`Drug'](d1) => //VP{//?", "[Value IN {`metabolized',`metabolised'}](rel) => //?", "[Tag=`GENE'](g1)}} ::: g1.value, rel.value, d1.value which specifies that the extracted triplets must have a drug (denoted by d1) followed by a verb phrase (denoted by VP) with the verb in $\\lbrace `metabolized^{\\prime },`metabolised^{\\prime } \\rbrace $ , followed by a gene (denoted by g1).", "Table REF shows examples of extracted triplets.", "Table: This table from shows the triplets representing various properties relevant to the extraction of implicit drug interactions and their description." ], [ "Data Cleaning", "The protein-protein and protein-drug relationships extracted from the parse tree database need an extra step of refinement to ensure that they correspond to the known properties of drug metabolism.", "For instance, for a protein to metabolize a drug, the protein must be an enzyme.", "Similarly, for a protein to regulate an enzyme, the protein must be a transcription factor.", "Thus, the $\\langle protein, metabolizes, drug \\rangle $ facts get refined to $\\langle enzyme, metabolizes, drug \\rangle $ and $\\langle protein, regulates, protein \\rangle $ gets refined to $\\langle transcription factor, regulates, enzyme \\rangle $ respectively.", "Classification of proteins is done using UniProt, the Gene Ontology (GO) and Entrez Gene summary by applying rules such as: A protein p is an enzyme if it belongs to a known enzyme family, such as CYP, UGT or SULT gene families; or is annotated under UniProt with the hydrolase, ligase, lyase or transferase keywords; or is listed under the “metabolic process” GO-term; or its Entrez Gene summary mentions key phrases like “drug metabolism” or roots for “enzyme” or “catalyzes”.", "A protein p is considered as a transcription factor if it is annotated with keywords transcription, transcription-regulator or activator under UniProt; or it is listed under the “transcription factor activity” category in GO; or its Entrez Gene summary contains the phrase “transcription factor”.", "Additional rules are applied to remove conflicting information, such as, favoring negative extractions (such as `$P$ does not metabolize $D$ ') over positive extractions (such as `$P$ metabolizes $D$ ').", "For details, see [73]." ], [ "Results", "The correctness of extracted interactions was determined by manually compiling a gold standard for each type of interaction using co-occurrence queries.", "For example, for $\\langle protein, metabolizes, drug \\rangle $ relations, we examined sentences that contain co-occurrence of protein, drug and one of the keywords “metabolized”, “metabolize”, “metabolises”, “metabolise”, “substrate” etc.", "Table REF summarizes the performance of our extraction approach.", "Table: Performance of interactions extracted from Medline abstracts.", "TP represents true-positives, while FN represents false-negatives" ], [ "Extracting Knowledge About Genetic Variants", "We summarize the relevant portion of our work on associating genetic variants with drugs, diseases and adverse reactions as presented in [33].", "Incorrect drug dosage is the leading cause of adverse drug reactions.", "Doctors prescribe the same amount of medicine to a patient for most drugs using the average drug response, even though a particular person's drug response may be higher or lower than the average case.", "A large part of the difference in drug response can be attributed to single nucleotide polymorphisms (SNPs).", "For example, the enzyme CYP2D6 has 70 known allelic variations, 50 of which are non-functional [31].", "Patients with poor metabolizer variations may retain higher concentration of drug for typical dosage, while patients with rapid metabolizers may not even reach therapeutic level of drug or toxic level of drug metabolites [68].", "Thus, it is important to consider the individual's genetic composition for dosage determination, especially for narrow therapeutic index drugs.", "Scientists studying these variations have grouped metabolizers into categories of poor (PM), intermediate (IM), rapid (RM) and ultra-rapid metabolizers (UM) and found that for some drugs, only 20% of usual dosage was required for PM and up to 140% for UM [38].", "Information about SNPs, their frequency in various population groups, their effect on genes (enzymic activity) and related data is stored in research papers, clinical studies and surveys.", "However, it is spread-out among them.", "Various databases collect this information in different forms.", "PharmGKB collects information such information and how it related to drug response [76].", "However, it is a small percentage of the total number of articles on PharmGKB, due to time consuming manual curation of data.", "Our work focuses on automatically extracting genetic variations and their impact on drug responses from PubMed abstracts to catch up with the current state of research in the biological domain, producing a repository of knowledge to assist in personalized medicine.", "Our approach leverages on as many existing tools as possible." ], [ "Methods", "Next, we describe the methods used in our extraction, including: named entity recognition, entity normalization, and relation extraction." ], [ "Named Entity Recognition", "We want to identify entities including genes (also proteins and enzymes), drugs, diseases, ADRs (adverse drug reactions), SNPs, RefSNPs (rs-numbers), names of alleles, populations and frequencies.", "For genes, we use BANNER [49] trained on BioCreative II GM training set [45].", "For genotypes (genetic variations including SNPs) we used a combination of MutationFinder [15] and custom components.", "Custom components were targeted mostly on non-SNPs (“c.76_78delACT”, 11MV324KF”) and insertions/deletions (“1707 del T”, “c.76_77insG”), RefSNPs (rs-numbers) and names of alleles/haplotypes (“CYP2D64”, “T allele”, “null allele”).", "For diseases (and ADRs), we used BANNER trained on a corpus of 3000 sentences with disease annotations [50].", "An additional 200 random sentences containing no disease were added from BioCreative II GM data to offset the low percentage (10%) of sentences without disease in the 3000 sentence corpus.", "In addition to BANNER, we used a dictionary extracted from UMLS.", "This dictionary consisted of 162k terms for 54k concepts from the six categories “Disease or Syndrome”, ”Neoplastic Process”,“Congenital Abnormality”,“Mental or Behavioral Dysfunction”,“Experimental Model of Disease” and “Acquired Abnormality”.", "The list was filtered to remove unspecific as well as spurious disease names such as “symptoms”, “disorder”, .... A dictionary for adverse drug reactions originated from SIDER Side Effect Resource [47], which provides a mapping of ADR terminology to UMLS CUIs.", "It consisted of 1.6k UMLS concepts with 6.5k terms.", "For drugs, we used a dictionary based on DrugBank[80] containing about 29k different drug names including both generic as well as brand names.", "We used the cross-linking information from DrugBank to collect additional synonyms and IDs from PharmGKB.", "We cross linked to Compound and Substance IDs from PubChem to provide hyperlinks to additional information.", "For population, we collected a dictionary of terms referring to countries, regions, regions inhabitants and their ethnicities from WikiPedia, e.g.", "“Caucasian”, “Italian”, “North African”, .... We filtered out irrelevant phrases like “Chinese [hamster]”.", "For frequencies, we extract all numbers and percentages as well as ranges from sentences that contain the word “allele”, “variant”, “mutation”, or “population”.", "The output is filtered in this case as well to remove false positives referring to p-values, odd ratios, confidence intervals and common trigger words." ], [ "Entity Normalization", "Genes, diseases and drugs can appear with many different names in the text.", "For example, “CYP2D6” can appear as “Cytochrome p450 2D6” or “P450 IID6” among others, but they all refer to the same enzyme (EntrezGene ID 1565).", "We use GNAT on recognized genes [32], but limit them to human, mouse and rat genes.", "The gene name recognized by BANNER is filtered by GNAT to remove non-useful modifiers and looked up against EntrezGene restricted to human, mouse and rat genes to find candidate IDs for each gene name.", "Ambiguity (multiple matches) is resolved by matching the text surrounding the gene mention with gene's annotation from a set of resources, like EntrezGene, UniProt.", "Drugs and diseases/ADRs are resolved to their official IDs from DrugBank or UMLS.", "If none is found, we choose an ID for it based on its name.", "Genetic variants Genetic variations are converted to HGVScite [21] recommended format.", "Alleles were converted to the star notation (e.g.", "“CYP2D61”) and the genotype (“TT allele”) or fixed terms such as “null allele” and “variant allele”.", "Populations mentions are mapped to controlled vocabulary to remove orthographic and lexical variations." ], [ "Relation Extraction", "Twelve type of relations were extracted between the detected entities as given in Table REF .", "Different methods were applied to detect different relations depending upon relation type, sentence structure and whether another method was able to extraction a relation beforehand.", "Gene-drug, gene-disease, drug-disease were extracted using sentence based co-occurrence (possibly refined by using relation-specific keywords) due to its good precision yield of this method for these relations.", "For other relations additional extraction methods were implemented.", "These include: High-confidence co-occurrence that includes keywords These co-occurences have the relation keyword in them.", "This method is applied to gene-drug, gene-disease, drug-ADR, drug-disease and mutation-disease associations.", "It uses keywords from PolySearch [79] as well as our own.", "Co-occurrence without keywords Such co-occurrences do not require any relationship keyword.", "This method is used for allele-population and variant-population relationships.", "This method can misidentify negative relationships.", "High-confidence relationships, if not found with a keyword drop down to this method for a lower confidence result.", "1:n co-occurrence Relationships where one entity has one instance in a given sentence and the other occurs one or more times.", "Single instance entity may have more than one occurrence.", "This method is useful in identifying gene mutations, where a gene is mentioned in a sentence along with a number of its mutations.", "The gene itself may be repeated.", "Enumerations with matching counts Captures entities in sentences where a list of entities is followed by an equal number of counts.", "This method is useful in capturing alleles and their associated frequencies, e.g.", "“The frequencies of CYP1B11, 2, 3, and 4 alleles were 0.087, 0.293, 0.444, and 0.175, respectively.” Least common ancestor (LCA) sub-tree Assigns associations based on distance in parse tree.", "We used Stanford parser [42] to get grammatical structure of a sentence as a dependency tree.", "This allows relating verb to its subject and noun to its modifiers.", "This method picks the closest pair in the lowest common ancestors (dependency) sub-tree of the entities.", "Maximum distance in terms of edges connecting the entity nodes was set to 10, which was determined empirically to provide the best balance between precision and recall.", "This method associates frequencies with alleles in the sentence “The allele frequencies were 18.3% (-24T) and 21.2% (1249A)”.", "m:n co-occurrence This method builds associations between all pairs of entities.", "Low confidence co-occurrence This acts as the catch-all case if none of the above methods work.", "Table: Unique binary relations identified between detected entities from .These methods were applied in order to each sentence, stoping at the first method that extracted the desired relationship.", "Order of these methods was determined empirically based of their precision.", "The order of the method used determines our confidence in the result.", "If none of the higher confidence methods are successful, a co-occurrence based method is used for extraction with low confidence.", "Abstract-level co-occurrence are also extracted to provide hits on potential relations.", "They appear in the database only when they appear in more than a pre-set threshold number of abstracts." ], [ "Results", "Performance was evaluated by evaluating the precision and recall of individual components and coverage of existing results.", "Precision and recall were tested by processing 3500 PubMed abstracts found via PharmGKB relations and manually checking the 2500 predictions.", "Coverage was tested against DrugBank and PharmGKB.", "Extracted relations went through manual evaluation for correctness.", "Each extraction was also assigned a confidence value based on the confidence in the method of extraction used.", "We got a coverage of 91% of data in DrugBank and 94% in PharmGKB.", "Taking into false positive rates for genes, drugs and gene-drug relations, SNPshot has more than 10,000 new relations." ], [ "Applying BioPathQA to Drug-Drug Interaction Discovery", "Now we use our BioPathQA system from chapter to answer questions about drug-drug interaction using knowledge extracted from research publications using the approach in sections REF ,REF .", "We supplement the extracted knowledge with domain knowledge as needed.", "Let the extracted facts be as follows: The drug $gefitinib$ is metabolized by $CYP3A4$ .", "The drug $phenytoin$ induces $CYP3A4$ .", "Following additional facts have been provided about a new drug currently in development: A new drug being developed $test\\_drug$ is a CYP3A4 inhibitor We show the pathway specification based on the above facts and background knowledge, then elaborate on each component: $&\\mathbf {domain~of~} gefitinib \\mathbf {~is~} integer, cyp3a4 \\mathbf {~is~} integer, \\nonumber \\\\&~~~~~~~~~~~~~~~~~~~~ phenytoin \\mathbf {~is~} integer, test\\_drug \\mathbf {~is~} integer\\\\&t1 \\mathbf {~may~execute~causing~} \\nonumber \\\\&~~~~~~~~~~~~~~~~~~~~ gefitinib \\mathbf {~change~value~by~} -1, \\nonumber \\\\&~~~~~~~~~~~~~~~~~~~~ cyp3a4 \\mathbf {~change~value~by~} -1, \\nonumber \\\\&~~~~~~~~~~~~~~~~~~~~ cyp3a4 \\mathbf {~change~value~by~} +1\\\\&\\mathbf {normally~stimulate~} t1 \\mathbf {~by~factor~} 2 \\nonumber \\\\&~~~~~~~~~~~~~~~~~~~~ \\mathbf {~if~} phenytoin \\mathbf {~has~value~} 1 \\mathbf {~or~higher~} \\\\&\\mathbf {inhibit~} t1 \\mathbf {~if~} test\\_drug \\mathbf {~has~value~} 1 \\mathbf {~or~higher~} \\\\&\\mathbf {initially~} gefitinib \\mathbf {~has~value~} 20, cyp3a4 \\mathbf {~has~value~} 60, \\nonumber \\\\&~~~~~~~~~~~~~~~~~~~~ phenytoin \\mathbf {~has~value~} 0, test\\_drug \\mathbf {~has~value~} 0 \\\\&\\mathbf {firing~style~} max$ Line REF declares the domain of the fluents as integer numbers.", "Line represents the activity of enzyme $cyp3a4$ as the action $t1$ .", "Due to the enzymic action $t1$ , one unit of $gefitinib$ is metabolized, and thus converted to various metabolites (not shown here).", "The enzymic action uses one unit of $cyp3a4$ as catalyst, which is used in the reaction and released afterwards.", "Line represents the knowledge that $phenytoin$ induces the activity of $cyp3a4$ .", "From background knowledge we find out that the stimulation in the activity can be as high as 2-times [55].", "Line represents the knowledge that there is a new drug $test\\_drug$ being tested that is known to inhibit the activity of $cyp3a4$ .", "Line specifies the initial distribution of the drugs and enzymes in the pathway.", "Assuming the patient has been given some fixed dose, say 20 units, of the medicine $gefitinib$ .", "It also specifies there is a large 60 units quantity of $cyp3a4$ available to ensure reactions do not slow down due to unavailability of enzyme availability.", "Additionaly, the drug $phenytoin$ is absent from the system and a new drug $test\\_drug$ to be tested is not in the system either.", "This gives us our pathway specification.", "Now we consider two application scenarios for drug development." ], [ "Drug Administration", "A patient is taking 20 units of $gefitinib$ , and is being prescribed additional drugs to be co-administered.", "The drug administrator wants to know if there will be an interaction with $gefitinib$ if 5 units of $phenytoin$ are co-administered.", "If there is an interaction, what will be the bioavailability of $gefitinib$ so that its dosage could be appropriately adjusted.", "The first question is asking whether giving the patient 5-units of $phenytoin$ in addition to the existing $gefitinib$ dose will cause a drug-interaction.", "It is encoded as the following query statement $\\mathbf {Q}$ for a $k$ -step simulation: $&\\mathbf {direction~of~change~in~} average \\mathbf {~value~of~} gefitinib \\mathbf {~is~} d\\\\&~~~~~~\\mathbf {when~observed~at~time~step~} k; \\\\&~~~~~~\\mathbf {comparing~nominal~pathway~with~modified~pathway~obtained~}\\\\&~~~~~~~~~~~~\\mathbf {due~to~interventions:~} \\mathbf {set~value~of~} phenytoin \\mathbf {~to~} 5;$ If the direction of change is “$=$ ” then there was no drug-interaction.", "Otherwise, an interaction was noticed.", "For a simulation of length $k=5$ , we find 15 units of $gefitinib$ remained at the end of simulation in the nominal case when no $phenytoin$ is administered.", "The amount drops to 10 units of $gefitinib$ when $phenytoin$ is co-administered.", "The change in direction is “$<$ ”.", "Thus there is an interaction.", "The second question is asking about the bioavailability of the drug $gefitinib$ after some after giving $phenytoin$ in 5 units.", "If this bioavailability falls below the efficacy level of the drug, then the drug would not treat the disease effectively.", "It is encoded as the following query statement $\\mathbf {Q}$ for a $k$ -step simulation: $&average \\mathbf {~value~of~} gefitinib \\mathbf {~is~} n\\\\&~~~~\\mathbf {when~observed~at~time~step~} k; \\\\&~~~~\\mathbf {due~to~interventions:~} \\\\&~~~~~~~~\\mathbf {set~value~of~} phenytoin \\mathbf {~to~} 5;$ For a simulation of length $k=5$ , we find 10 units of $gefitinib$ remain.", "A drug administrator (such as a pharmacist) can adjust the drug accordingly." ], [ "Drug Development", "A drug manufacturer is developing a new drug $test\\_drug$ that is known to inhibit CYP3A4 that will be co-administered with drugs $gefitinib$ and $phenytoin$ .", "He wants to determine the bioavailability of $gefitinib$ over time to determine the risk of toxicity.", "The question is asking about the bioavailability of the drug $gefitinib$ after 10 time units after giving $phenytoin$ in 5 units and the new drug $test\\_drug$ in 5 units.", "If this bioavailability remains high, there is chance for toxicity due to the drug at the subsequent dosage intervals.", "It is encoded as the following query statement $\\mathbf {Q}$ for a $k$ -step simulation: $&average \\mathbf {~value~of~} gefitinib \\mathbf {~is~} n\\\\&~~~~\\mathbf {when~observed~at~time~step~} k; \\\\&~~~~\\mathbf {due~to~interventions:~} \\\\&~~~~~~~~\\mathbf {set~value~of~} phenytoin \\mathbf {~to~} 5,\\\\&~~~~~~~~\\mathbf {set~value~of~} test\\_drug \\mathbf {~to~} 5;$ For a simulation of length $k=5$ , we find all 20 units of $gefitinib$ remain.", "This could lead to toxicity by building high concentration of $gefitinib$ in the body." ], [ "Drug Administration in Presence of Genetic Variation", "A drug administrator wants to establish the dosage of $morphine$ for a person based on its genetic profile using its bioavailability.", "Consider the following facts extracted about a simplified morphine pathway: $codeine$ is metabolized by $CYP2D6$ producing $morphine$ $CYP2D6$ has three allelic variations “1” – (EM) effective metabolizer (normal case) “2” – (UM) ultra rapid metabolizer “*9” – (PM) poor metabolizer For simplicity, assume UM allele doubles the metabolic rate, while PM allele halves the metabolic rate of CYP2D6.", "Then, the resulting pathway is given by: $&\\mathbf {domain~of~} cyp2d6\\_allele \\mathbf {~is~} integer, cyp2d6 \\mathbf {~is~} integer\\\\&\\mathbf {domain~of~} codeine \\mathbf {~is~} integer, morphine \\mathbf {~is~} integer\\\\&cyp2d6\\_action \\mathbf {~may~execute~causing}\\\\&~~~~codeine \\mathbf {~change~value~by~} -2, morphine \\mathbf {~change~value~by~} +2\\\\&~~~~\\mathbf {if~} cyp2d6\\_allele \\mathbf {~has~value~equal~to~} 1\\\\&cyp2d6\\_action \\mathbf {~may~execute~causing}\\\\&~~~~codeine \\mathbf {~change~value~by~} -4, morphine \\mathbf {~change~value~by~} +4\\\\&~~~~\\mathbf {if~} cyp2d6\\_allele \\mathbf {~has~value~equal~to~} 2\\\\&cyp2d6\\_action \\mathbf {~may~execute~causing~}\\\\&~~~~codeine \\mathbf {~change~value~by~} -1, morphine \\mathbf {~change~value~by~} +1\\\\&~~~~\\mathbf {if~} cyp2d6\\_allele \\mathbf {~has~value~equal~to~} 9\\\\&\\mathbf {initially~}\\\\&~~~~codeine \\mathbf {~has~value~} 0, morphine \\mathbf {~has~value~} 0,\\\\&~~~~cyp2d6 \\mathbf {~has~value~} 20, cyp2d6\\_allele \\mathbf {~has~value~} 1\\\\&\\mathbf {firing~style~} max\\\\$ Then, the bioavailability of $morphine$ can be determined by the following query: $&average \\mathbf {~value~of~} morphine \\mathbf {~is~} n\\\\&~~~~\\mathbf {when~observed~at~time~step~} k; \\\\&~~~~\\mathbf {due~to~interventions:~} \\\\&~~~~~~~~\\mathbf {set~value~of~} codeine \\mathbf {~to~} 20,\\\\&~~~~~~~~\\mathbf {set~value~of~} cyp2d6 \\mathbf {~to~} 20,\\\\&~~~~~~~~\\mathbf {set~value~of~} cyp2d6\\_allele \\mathbf {~to~} 9;$ Simulation for 5 time steps reveal that the average bioavailability of $morphine$ after 5 time-steps is 5 for PM (down from 10 for EM).", "Although this is a toy example, it is easy to see the potential of capturing known genetic variations in the pathway and setting the complete genetic profile of a person in the intervention part of the query." ], [ "Conclusion", "In this chapter we presented how we extract biological pathway knowledge from text, including knowledge about drug-gene interactions and their relationship to genetic variation.", "We showed how the information extracted is used to build pathway specification and illustrated how biologically relevant questions can be answered about drug-drug interaction using the BioPathQA system developed in chapter .", "Next we look at the future directions in which the research work done in this thesis can be extended." ], [ "Conclusion and Future Work", "The field of knowledge representation and reasoning (KR) is currently one of the most active research areas.", "It represents the next step in the evolution of systems that know how to organize knowledge, and have the ability to intelligently respond to questions about this knowledge.", "Such questions could be about static knowledge or the dynamic processes.", "Biological systems are uniquely positioned as role models for this next evolutionary step due to their precise vocabulary and mechanical nature.", "As a result, a number of recent research challenges in the KR field are focused on it.", "The biological field itself needs systems that can intelligently answer questions about such biological processes and systems in an automated fashion, given the large number of research papers published each year.", "Curating these publications is time consuming and expensive, as a result the state of over all knowledge about biological systems lags behind the cutting edge research.", "An important class of questions asked about biological systems are the so called “what-if” questions that compare alternate scenarios of a biological pathway.", "To answer such questions, one has to perform simulation on a nominal pathway against a pathway modified due to the interventions specified for the alternate scenario.", "Often, this means creating two pathways (for nominal and alternate cases) and simulate them separately.", "This opens up the possibility that the two pathways can become out of synchronization.", "A better approach is to allow the user to specify the needed interventions in the query statement itself.", "In addition, to understand the full spread of possible outcomes, given the parallel nature of biological pathways, one must consider all possible pathway evolutions, otherwise, some outcomes may remain hidden.", "If a system is to be used by biologists, it must have a simple interface, lowering the barrier of entry.", "Since biological pathway knowledge can arrive from different sources, including books, published articles, and lab experiments, a common input format is desired.", "Such a format allows specification of pathways due to automatic extraction, as well as any changes / additions due to locally available information.", "A comprehensive end-to-end system that accomplish all the goals would take a natural language query along with any additional specific knowledge about the pathway as input, extract the relevant portion of the relevant pathway from published material (and background knowledge), simulate it based on the query, and generate the results in a visual format.", "Each of these tasks comes with its own challenges, some of which have been addressed in this thesis.", "In this thesis, we have developed a system and a high level language to specify a biological pathway and answer simulation based reasoning questions about it.", "The high level language uses controlled-English vocabulary to make it more natural for a researcher to use directly.", "The high level language has two components: a pathway specification language, and a query specification language.", "The pathway specification language allows the user to specify a pathway in a source independent form, thus locally obtained knowledge (e.g.", "from lab) can be combined with automatically extracted knowledge.", "We believe that our pathway specification language is easy for a person to understand and encode, lowering the bar to using our system.", "Our pathway specification language allows conditional actions, enabling the encoding of alternate action outcomes due to genetic variation.", "An important aspect of our pathway specification language is that it specifies trajectories, which includes specifying the initial configuration of substances, as well as state evolution style, such as maximal firing of actions, or serialized actions etc.", "Our query specification language provides a bridge between natural language questions and their formal representation.", "It is English-like but with precise semantics.", "A main feature of our query language is its support for comparative queries over alternate scenarios, which is not currently supported by any of the query languages (associated with action languages) we have reviewed.", "Our specification of alternate scenarios uses interventions (a general form of actions), that allow the user to modify the pathway as part of the query processing.", "We believe our query language is easier for a biologist to understand without requiring formal training.", "To model the pathways, we use Petri Nets, which have been used in the past to model and simulate biological pathways.", "Petri Nets have a simple visual representation, which closely matches biological pathways; and they inherently support parallelism.", "We extended the Petri Nets to add features that we needed to suit our domain, e.g., reset arcs that remove all quantity of a substance as soon as it is produced, and conditional arcs that specify the conditional outcome of an action.", "For simulation, we use ASP, which allowed us straight forward way to implement Petri Nets.", "It also gave us the ability to add extensions to the Petri Net by making local edits, implement different firing semantics, filter trajectories based on observations, and reason with the results.", "One of the major advantage of using Petri Net based simulation is the ability to generate all possible state evolutions, enabling us to process queries that determine the conditions when a certain observation becomes true.", "Our post-processing step is done in Python, which allows strong text processing capabilities using regular expressions, as well as libraries to easy process large matrices of numbers for summarization of results.", "Now we present additional challenges that need to be addressed." ], [ "Pathway Extraction", "In Chapter we described how we extract facts for drug-drug interaction and gene variation.", "This work needs to be extended to include some of the newer databases that have come online recently.", "This may provide us with enzyme reaction rates, and substance quantities used in various reactions.", "The relation extraction for pathways must also be cognizant of any genetic variation mentioned in the text.", "Since the knowledge about the pathway appears in relationships at varying degree of detail, a process needs to be devised to assemble the pathway from the same level to granularity together, while also maintaining pathways at different levels of granularities.", "Since pathway extraction is a time consuming task, it would be best to create a catalog of the pathways.", "The cataloged pathways could be manually edited by the user as necessary.", "Storing pathways in this way means that would have to be updated periodically, requiring merging of new knowledge into existing pathways.", "Manual edits would have to be identified, such that the updated pathway does not overwrite them without the user's knowledge." ], [ "Pathway Selection", "Questions presented in biological texts do not explicitly mention the relevant pathway to use for answering the question.", "One way to address this issue is to maintain a catalog of pre-defined pathways with keywords associated with them.", "Such keywords can include names of the substances, names of the processes, and other relevant meta-data about the pathway.", "The catalog can be searched to find the closest match to the query being asked.", "An additional aspect in proper pathway selection is to use the proper abstraction level.", "If our catalog contains a pathway at different abstraction levels, the coarsest pathway that contains the processes and substances in the query should be selected.", "Any higher fidelity will increase the processing time and generate too much irrelevant data.", "Alternatively, the catalog could contain the pathway in a hierarchical form, allowing extraction of all elements of a pathway at the same depth.", "A common way to hierarchically organize the pathway related to our system is to have hierarchical actions, which is the approach taken by hierarchical Petri nets.", "Lastly, the question may only ask about a small subsection of a much larger pathway.", "For better performance, it is beneficial to extract the smallest biological pathway network model that can answer the question." ], [ "Pathway Modeling", "In Chapter , we presented our modeling of biological questions using Petri Nets and their extensions encoded in ASP.", "We came across concepts like allosteric regulation, inhibition of inhibition, and inhibition of activation that we currently do not model.", "In allosteric regulation, an enzyme is not fully enabled or disabled, the enzyme's shape changes, making it more or less active.", "The level of its activity depends upon concentrations of activators and inhibitors.", "In inhibition of inhibition, the inhibition of a reaction is itself inhibited by another inhibition; while in inhibition of activation (or stimulation), a substance inhibits the stimulation produced by a substance.", "Both of these appear to be actions on actions, something that Petri Nets do not allow.", "An alternate coding for these would have to be devised.", "As more detailed information about pathways becomes available, the reactions and processes that we have in current pathways may get replaced with more detailed sub-pathways themselves.", "However, such refinement may not come at the same time for separate legs of the pathway.", "Just replacing the coarse transition with a refined transition may not be sufficient due to relative timing constraints.", "Hence, a hierarchical Petri Net model may need to be implemented (see , )." ], [ "Pathway Simulation", "In Chapter we presented our approach to encode Petri Nets and their extensions.", "We used a discrete solver called clingo for our ASP encoding.", "As the number of simulation length increases in size or larger quantities are involved, the solver slows down significantly.", "This is due to an increased grounding cost of the program.", "Incremental solving (using iclingo) does not help, since the program size still increases, and the increments merely delays the slow down but does not stop it.", "Systems such as constraint logic solvers (such as ) could be used for discrete cases.", "Alternatively, a system developed on the ASPMT [53] approach could be used, since it can represent longer runs, larger quantities, and real number quantities." ], [ "Extend High Level Language", "In Chapter we described the BioPathQA system, the pathway specification and the query specification high level languages.", "As we enhance the modeling of the biological pathways, we will need to improve or extend the system as well as the high level language.", "We give a few examples of such extensions.", "Our pathway specification language currently does not support continuous quantities (real numbers).", "Extending to real numbers will improve the coverage of the pathways that can be modeled.", "In addition, derived quantities (fluents) can be added, e.g.", "pH could be defined as a formula that is read-only in nature.", "Certain observations and queries can be easily specified using a language such as LTL, especially for questions requiring conditions leading to an action or a state.", "As a result, it may be useful to add LTL formulas to the query language.", "We did not take this approach because it would have presented an additional non-English style syntax for the biologists.", "Our substance production / consumption rates and amounts are currently tied to the fluents.", "In certain situations it is desirable to analyze the quantity of a substance produced / consumed by a specific action, e.g.", "one is interested in finding the amount of H+ ions produced by a multi-protein complex IV only.", "Interventions (that are a part of the query statement) presented in this thesis are applied at the start of the simulation.", "Eliminating this restriction would allow applying surgeries to the pathway mid execution.", "Thus, instead of specifying the steady state conditions in the query statement, one could apply the intervention when a certain steady state is reached." ], [ "Result Formatting and Visualization", "In Chapter we described our system that answers questions specified in our high level language.", "At the end of its process, it outputs the final result.", "This output method can be enhanced by allowing to look at the progression of results in addition to the final result.", "This provides the biologist with the whole spread of possible outcomes.", "An example of such a spread is shown in Fig.", "fig:q1:result for question REF .", "A graphical representation of the simulation progression is also beneficial in enhancing the confidence of the biologist.", "Indeed many existing tools do this.", "A similar effect can be achieved by parsing and showing the relevant portion of the answer set." ], [ "Summary", "In Chapter we introduced the thesis topic and summarized specific research contributions In Chapter we introduced the foundational material of this thesis including Petri Nets and ASP.", "We showed how ASP could be used to encode basic Petri Nets.", "We also showed how ASP's elaboration tolerance and declarative syntax allows us to encode various Petri Net extensions with small localized changes.", "We also introduced a new firing semantics, the so called maximal firing set semantics to simulate a Petri Net with maximum parallel activity.", "In Chapter we showed how the Petri Net extensions and the ASP encoding can be used to answer simulation based deep reasoning questions.", "This and the work in Chapter was published in [1], [2].", "In Chapter we developed a system called BioPathQA to allow users to specify a pathway and answer queries against it.", "We also developed a pathway specification language and a query language for this system in order to avoid the vagaries of natural language.", "We introduced a new type of Guarded-arc Petri Nets to model conditional actions as a model for pathway simulation.", "We also described our implementation developed around a subset of the pathway specification language.", "In Chapter we briefly described how text extraction is done to extract real world knowledge about pathways and drug interactions.", "We then used the extracted knowledge to answer question using BioPathQA.", "The text extraction work was published in [73], [72], [33]." ], [ "Proofs of Various Propositions", "Assumption: The definitions in this section assume the programs $\\Pi $ do not have recursion through aggregate atoms.", "Our ASP translation ensures this due to the construction of programs $\\Pi $ .", "First we extend some definitions and properties related to ASP, such that they apply to rules with aggregate atoms.", "We will refer to the non-aggregate atoms as basic atoms.", "Recall the definitions of an ASP program given in section REF .", "Proposition 9 (Forced Atom Proposition) Let $S$ be an answer set of a ground ASP program $\\Pi $ as defined in definition REF .", "For any ground instance of a rule R in $\\Pi $ of the form $A_0 \\text{:-} A_1,\\dots ,$ $A_m,\\mathbf {not~} B_{1},\\dots ,$ $\\mathbf {not~} B_n, C_1,\\dots ,$ $C_k.$ if $\\forall A_i, 1 \\le i \\le m, S \\models A_i$ , and $\\forall B_j, 1 \\le j \\le n, S \\lnot \\models B_j$ , $\\forall C_l, 1 \\le l \\le k, S \\models C_l$ then $S \\models A_0$ .", "Proof: Let $S$ be an answer set of a ground ASP program $\\Pi $ , $R \\in \\Pi $ be a ground rule such that $\\forall B_j, 1 \\le j \\le n, S \\lnot \\models B_j$ ; and $\\forall C_l, 1 \\le l \\le k, S \\models C_l$ .", "Then, the reduct $R^S \\equiv \\lbrace p_1 \\text{:-} A_1,\\dots ,A_m.", "; \\dots ; p_h \\text{:-} A_1,\\dots ,A_m.", "\\; \| \\; \\lbrace p_1,\\dots ,p_h \\rbrace = S \\cap lit(A_0) \\rbrace $ .", "Since $S$ is an answer set of $\\Pi $ , it is a model of $\\Pi ^S$ .", "As a result, whenever, $\\forall A_i, 1 \\le i \\le m, S \\models A_i$ , $\\lbrace p_1,\\dots ,p_h \\rbrace \\subseteq S$ and $S \\models A_0$ .", "Proposition 10 (Supporting Rule Proposition) If $S$ is an answer set of a ground ASP program $\\Pi $ as defined in definition REF then $S$ is supported by $\\Pi $ .", "That is, if $S \\models A_0$ , then there exists a ground instance of a rule R in $\\Pi $ of the type $A_0 \\text{:-} A_1,\\dots ,$ $A_m,$ $\\mathbf {not~} B_{1},\\dots ,$ $\\mathbf {not~} B_n, $ $C_1,\\dots ,C_k.$ such that $\\forall A_i, 1 \\le i \\le m, S \\models A_i$ , $\\forall B_j, 1 \\le j \\le n, S \\lnot \\models B_j$ , and $\\forall C_l, 1\\le l \\le k, S \\models C_l$ .", "Proof: For $S$ to be an answer set of $\\Pi $ , it must be the deductive closure of reduct $\\Pi ^S$ .", "The deductive closure $S$ of $\\Pi ^S$ is iteratively built by starting from an empty set $S$ , and adding head atoms of rules $R_h^S \\equiv p_h \\text{:-} A_1,\\dots ,A_m., R_h^S \\in \\Pi ^S$ , whenever, $S \\models A_i, 1 \\le i \\le m$ , where, $R_h^S$ is a rule in the reduct of ground rule $R \\in \\Pi $ with $p_h \\in lit(A_0) \\cap S$ .", "Thus, there is a rule $R \\equiv A_0 \\text{:-} A_1,\\dots ,$ $A_m,$ $\\mathbf {not~} B_{1},\\dots ,$ $\\mathbf {not~}B_n, $ $C_1,\\dots ,$ $C_k.$ , $R \\in \\Pi $ , such that $\\forall C_l, 1 \\le l \\le k$ and $\\forall B_j, 1 \\le j \\le n, S \\lnot \\models B_j$ .", "Nothing else belongs in $S$ .", "Next, we extend the splitting set theorem to include aggregate atoms.", "Definition 88 (Splitting Set) A Splitting Set for a program $\\Pi $ is any set $U$ of literals such that, for every rule $R \\in \\Pi $ , if $U \\models head(R)$ then $lit(R) \\subset U$ .", "The set $U$ splits $\\Pi $ into upper and lower parts.", "The set of rules $R \\in \\Pi $ s.t.", "$lit(R) \\subset U$ is called the bottom of $\\Pi $ w.r.t.", "$U$ , denoted by $bot_U(\\Pi )$ .", "The rest of the rules, i.e.", "$\\Pi \\setminus bot_U(\\Pi )$ is called the top of $\\Pi $ w.r.t.", "$U$ , denoted by $top_U(\\Pi )$ .", "Proposition 11 Let $U$ be a splitting set of $\\Pi $ with answer set $S$ and let $X = S \\cap U$ and $Y = S \\setminus U$ .", "Then, the reduct of $\\Pi $ w.r.t.", "$S$ , i.e.", "$\\Pi ^S$ is equal to $bot_U(\\Pi )^X \\cup (\\Pi \\setminus bot_U(\\Pi ))^{X \\cup Y}$ .", "Proof: We can rewrite $\\Pi $ as $bot_U(\\Pi ) \\cup (\\Pi \\setminus bot_U(\\Pi ))$ using the definition of splitting set.", "Then the reduct of $\\Pi $ w.r.t.", "$S$ can be written in terms of $X$ and $Y$ , since $S = X \\cup Y$ .", "$\\Pi ^S =$ $\\Pi ^{X \\cup Y} =$ $(bot_U(\\Pi ) \\cup (\\Pi \\setminus bot_U(\\Pi )))^{X \\cup Y} =$ $bot_U(\\Pi )^{X \\cup Y} \\cup $ $(\\Pi \\setminus bot_U(\\Pi ))^{X \\cup Y}$ .", "Since $lit(bot_U(\\Pi )) \\subseteq U$ and $Y \\cap U = \\emptyset $ , the reduct of $bot_U(\\Pi )^{X \\cup Y} = bot_U(\\Pi )^X$ .", "Thus, $\\Pi ^{X \\cup Y} = bot_U(\\Pi )^X \\cup (\\Pi \\setminus bot_U(\\Pi ))^{X \\cup Y}$ .", "Proposition 12 Let $S$ be an answer set of a program $\\Pi $ , then $S \\subseteq lit(head(\\Pi ))$ .", "Proof: If $S$ is an answer set of a program $\\Pi $ then $S$ is produced by the deductive closure of $\\Pi ^S$ (the reduct of $\\Pi $ w.r.t $S$ ).", "By definition of the deductive closure, nothing can be in $S$ unless it is the head of some rule supported by $S$ .", "Splitting allow computing the answer set of a program $\\Pi $ in layers.", "Answer sets of the bottom layer are first used to partially evaluate the top layer, and then answer sets of the top layer are computed.", "Next, we define how a program is partially evaluated.", "Intuitively, the partial evaluation of an aggregate atom $c$ given splitting set $U$ w.r.t.", "a set of literals $X$ removes all literals that are part of the splitting set $U$ from $c$ and updates $c$ 's lower and upper bounds based on the literals in $X$ , which usually come from $bot_U$ of a program.", "The set $X$ represents our knowledge about the positive literals, while the set $U \\setminus X$ represents our knowledge about naf-literals at this stage.", "We can remove all literals in $U$ from $c$ , since the literals in $U$ will not appear in the head of any rule in $top_U$ .", "Definition 89 (Partial Evaluation of Aggregate Atom) The partial evaluation of an aggregate atom $c = l \\; [ B_0=w_0,\\dots , B_m=w_m ] \\; u$ , given splitting set $U$ w.r.t.", "a set of literals $X$ , written $eval_U(c,X)$ is a new aggregate atom $c^{\\prime }$ constructed from $c$ as follows: $pos(c^{\\prime }) = pos(c) \\setminus U$ $d=\\sum _{B_i \\in pos(c) \\cap U \\cap X}{w_i} $ $l^{\\prime } = l-d$ , $u^{\\prime } = u-d$ are the lower and upper limits of $c^{\\prime }$ Next, we define how a program is partially evaluated given a splitting set $U$ w.r.t.", "a set of literals $X$ that form the answer-set of the lower layer.", "Intuitively, a partial evaluation deletes all rules from the partial evaluation for which the body of the rule is determined to be not supported by $U$ w.r.t.", "$X$ .", "This includes rules which have an aggregate atom $c$ in their body s.t.", "$lit(c) \\subseteq U$ , but $X \\lnot \\models c$ Note that we can fully evaluate an aggregate atom $c$ w.r.t.", "answer-set $X$ if $lit(c) \\subseteq U$ ..", "In the remaining rules, the positive and negative literals that overlap with $U$ are deleted, and so are the aggregate atoms that have $lit(c) \\subseteq U$ (since such a $c$ can be fully evaluated w.r.t.", "$X$ ).", "Each remaining aggregate atom is updated by removing atoms that belong to $U$ Since the atoms in $U$ will not appear in the head of any atoms in $top_U$ and hence will not form a basis in future evaluations of $c$ ., and updating its limits based on the answer-set $X$ The limit update requires knowledge of the current answer-set to update limit values..", "The head atom is not modified, since $eval_U(...)$ is performed on $\\Pi \\setminus bot_U(\\Pi )$ , which already removes all rules with heads atoms that intersect $U$ .", "Definition 90 (Partial Evaluation) The partial evaluation of $\\Pi $ , given splitting set $U$ w.r.t.", "a set of literals $X$ is the program $eval_U(\\Pi ,X)$ composed of rules $R^{\\prime }$ for each $R \\in \\Pi $ that satisfies all the following conditions: $pos(R) \\cap U \\subseteq X,$ $((neg(R) \\cap U) \\cap X) = \\emptyset , \\text{ and }$ if there is a $c \\in agg(R)$ s.t.", "$lit(c) \\subseteq U$ , then $X \\models c$ A new rule $R^{\\prime }$ is constructed from a rule $R$ as follows: $head(R^{\\prime }) = head(R)$ , $pos(R^{\\prime }) = pos(R) \\setminus U$ , $neg(R^{\\prime }) = neg(R) \\setminus U$ , $agg(R^{\\prime }) = \\lbrace eval_U(c,X) : c \\in agg(R), lit(c) \\lnot \\subseteq U \\rbrace $ Proposition 13 Let $U$ be a splitting set for $\\Pi $ , $X$ be an answer set of $bot_U(\\Pi )$ , and $Y$ be an answer set of $eval_U(\\Pi \\setminus bot_U(\\Pi ),X)$ .", "Then, $X \\subseteq lit(\\Pi ) \\cap U$ and $Y \\subseteq lit(\\Pi ) \\setminus U$ .", "Proof: By proposition REF , $X \\subseteq lit(head(bot_U(\\Pi )))$ , and $Y \\subseteq lit(head(eval_U(\\Pi \\setminus bot_U(\\Pi ),X)))$ .", "In addition, $lit(head(bot_U(\\Pi ))) \\subseteq lit(bot_U(\\Pi ))$ and $lit(bot_U(\\Pi )) \\subseteq lit(\\Pi ) \\cap U$ by definition of $bot_U(\\Pi )$ .", "Then $X \\subseteq lit(\\Pi ) \\cap U$ , and $Y \\subseteq lit(\\Pi ) \\setminus U$ .", "Definition 91 (Solution) [4] Let $U$ be a splitting set for a program $\\Pi $ .", "A solution to $\\Pi $ w.r.t.", "$U$ is a pair $\\langle X,Y \\rangle $ of literals such that: $X$ is an answer set for $bot_U(\\Pi )$ $Y$ is an answer set for $eval_U(top_U(\\Pi ),X)$ ; and $X \\cup Y$ is consistent.", "Proposition 14 (Splitting Theorem) [4] Let $U$ be a splitting set for a program $\\Pi $ .", "A set $S$ of literals is a consistent answer set for $\\Pi $ iff $S = X \\cup Y$ for some solution $\\langle X,Y \\rangle $ of $\\Pi $ w.r.t.", "$U$ .", "Lemma 1 Let $U$ be a splitting set of $\\Pi $ , $C$ be an aggregate atom in $\\Pi $ , and $X$ and $Y$ be subsets of $lit(\\Pi )$ s.t.", "$X \\subseteq U$ , and $Y \\cap U = \\emptyset $ .", "Then, $X \\cup Y \\models C$ iff $Y \\models eval_U(C,X)$ .", "Proof: Let $C^{\\prime } = eval_U(C,X)$ , then by definition of partial evaluation of aggregate atom, $pos(C^{\\prime }) = pos(C) \\setminus U$ , with lower limit $l^{\\prime } = l-d$ , and upper limit $u^{\\prime } = u-d$ , computed from $l,u$ , the lower and upper limits of $C$ , where $d=\\displaystyle \\sum _{B_i \\in pos(C) \\cap U \\cap X}{w_i}$ $Y \\models C^{\\prime }$ iff $l^{\\prime } \\le \\left( \\displaystyle \\sum _{B^{\\prime }_i \\in pos(C^{\\prime }) \\cap Y}{w^{\\prime }_i} \\right) \\le u^{\\prime }$ – by definition of aggregate atom satisfaction.", "then $Y \\models C$ iff $l \\le \\left( \\displaystyle \\sum _{B_i \\in pos(C) \\cap U \\cap X}{w_i} +\\displaystyle \\sum _{B^{\\prime }_i \\in (pos(C) \\setminus U) \\cap Y}{w^{\\prime }_i} \\right) \\le u$ however, $(pos(C) \\cap U) \\cap X$ and $(pos(C) \\setminus U) \\cap Y$ combined represent $pos(C) \\cap (X \\cup Y)$ – since $pos(C) \\cap (X \\cup Y) &= ((pos(C) \\cap U) \\cup (pos(C) \\setminus U)) \\cap (X \\cup Y) \\\\&= [((pos(C) \\cap U) \\cup (pos(C) \\setminus U)) \\cap X] \\\\&~~~~~~\\cup [((pos(C) \\cap U) \\cup (pos(C) \\setminus U)) \\cap Y]\\\\&= [(pos(C) \\cap U) \\cap X) \\cup ((pos(C) \\setminus U) \\cap X)]\\\\&~~~~~~\\cup [(pos(C) \\cap U) \\cap Y) \\cup ((pos(C) \\setminus U) \\cap Y)]\\\\&= [((pos(C) \\cap U) \\cap X) \\cup \\emptyset ] \\cup [\\emptyset \\cup ((pos(C) \\setminus U) \\cap Y)]\\\\&= ((pos(C) \\cap U) \\cap X) \\cup ((pos(C) \\setminus U) \\cap Y)$ where $X \\subseteq U \\text{ and } Y \\cap U = \\emptyset $ thus, $Y \\models C$ iff $l \\le \\left( \\displaystyle \\sum _{B_i \\in pos(C) \\cap (X \\cup Y)}{w_i} \\right) \\le u$ which is the same as $X \\cup Y \\models C$ Lemma 2 Let $U$ be a splitting set for $\\Pi $ , and $X, Y$ be subsets of $lit(\\Pi )$ s.t.", "$X \\subseteq U$ and $Y \\cap U = \\emptyset $ .", "Then the body of a rule $R^{\\prime } \\in eval_U(\\Pi ,X)$ is satisfied by $Y$ iff the body of the rule $R \\in \\Pi $ it was constructed from is satisfied by $X \\cup Y$ .", "Proof: $Y$ satisfies $body(R^{\\prime })$ iff $pos(R^{\\prime }) \\subseteq Y$ , $neg(R^{\\prime }) \\cap Y = \\emptyset $ , $Y \\models C^{\\prime }$ for each $C^{\\prime } \\in agg(R^{\\prime })$ – by definition of rule satisfaction iff $(pos(R) \\cap U) \\subseteq X$ , $(pos(R) \\setminus U) \\subseteq Y$ , $(neg(R) \\cap U) \\cap X) = \\emptyset $ , $(neg(R) \\setminus U) \\cap Y) = \\emptyset $ , $X$ satisfies $C$ for all $C \\in agg(C)$ in which $lit(C) \\subseteq U$ , and $Y$ satisfies $eval_U(C,X)$ for all $ C \\in agg(C)$ in which $lit(C) \\lnot \\subseteq U$ – using definition of partial evaluation iff $pos(R) \\subseteq X \\cup Y$ , $neg(R) \\cap (X \\cup Y) = \\emptyset $ , $X \\cup Y \\models C$ – using $(A \\cap U) \\cup (A \\setminus U) = A$ $A \\cap (X \\cup Y) = ((A \\cap U) \\cup (A \\setminus U)) \\cap (X \\cup Y) = ((A \\cap U) \\cap (X \\cup Y)) \\cup ((A \\setminus U) \\cap (X \\cup Y)) = (A \\cap U) \\cap X) \\cup ((A \\setminus U) \\cap Y)$ – given $X \\subseteq U$ and $Y \\cap U = \\emptyset $ .", "and lemma REF Proof of Splitting Theorem: Let $U$ be a splitting set of $\\Pi $ , then a consistent set of literals $S$ is an answer set of $\\Pi $ iff it can be written as $S = X \\cup Y$ , where $X$ is an answer set of $bot_U(\\Pi )$ ; and $Y$ is an answer set of $eval_U(\\Pi \\setminus bot_U(\\Pi ),Y)$ .", "($\\Leftarrow $ ) Let $X$ is an answer set of $bot_U(\\Pi )$ ; and $Y$ is an answer set of $eval_U(\\Pi \\setminus bot_U(\\Pi ),X)$ ; we show that $X \\cup Y$ is an answer set of $\\Pi $ .", "By definition of $bot_U(\\Pi )$ , $lit(bot_U(\\Pi )) \\subseteq U$ .", "In addition, by proposition REF , $Y \\cap U = \\emptyset $ .", "Then, $\\Pi ^{X \\cup Y} = (bot_U(\\Pi ) \\cup (\\Pi \\setminus bot_U(\\Pi )))^{X \\cup Y} = bot_U(\\Pi )^{X \\cup Y} \\cup (\\Pi \\setminus bot_U(\\Pi ))^{X \\cup Y} = bot_U(\\Pi )^X \\cup (\\Pi \\setminus bot_U(\\Pi ))^{X \\cup Y}$ .", "Let $r$ be a rule in $\\Pi ^{X \\cup Y}$ , s.t.", "$X \\cup Y \\models body(r)$ then we show that $X \\cup Y \\models head(r)$ .", "The rule $r$ either belongs to $bot_U(\\Pi )^X$ or $(\\Pi \\setminus bot_U(\\Pi ))^{X \\cup Y}$ .", "Case 1: say $r \\in bot_U(\\Pi )^X$ be a rule whose body is satisfied by $X \\cup Y$ then there is a rule $R \\in bot_U(\\Pi )$ s.t.", "$r \\in R^X$ then $X \\models body(R)$ – since $X \\cup Y \\models body(r)$ ; $lit(bot_U(\\Pi )) \\subseteq U$ and $Y \\cap U = \\emptyset $ we already have $X \\models head(R)$ – since $X$ is an answer set of $bot_U(\\Pi )$ ; given then $X \\cup Y \\models head(R)$ – because $lit(R) \\subseteq U$ and $Y \\cap U = \\emptyset $ consequently, $X \\cup Y \\models head(r)$ Case 2: say $r \\in (\\Pi \\setminus bot_U(\\Pi ))^{X \\cup Y}$ be a rule whose body is satisfied by $X \\cup Y$ then there is a rule $R \\in (\\Pi \\setminus bot_U(\\Pi ))$ s.t.", "$r \\in R^{X \\cup Y}$ then $lit(head(R)) \\cap U = \\emptyset $ – otherwise, $R$ would have belonged to $bot_U(\\Pi )$ , by definition of splitting set then $head(r) \\in Y$ – since $X \\subseteq U$ in addition, $pos(R) \\subseteq X \\cup Y$ , $neg(R) \\cap (X \\cup Y) = \\emptyset $ , $X \\cup Y \\models C$ for each $C \\in agg(R)$ – using definition of reduct then $pos(R) \\cap U \\subseteq X$ or $pos(R) \\setminus U \\subseteq Y$ ; $(neg(R) \\cap U) \\cap X = \\emptyset $ and $(neg(R) \\setminus U) \\cap Y = \\emptyset $ ; and for each $C \\in agg(R)$ , either $X \\models C$ if $lit(C) \\subseteq U$ , or $Y \\models eval_U(C,X)$ if $lit(C) \\lnot \\subseteq U$ – by rearranging, lemma REF , $X \\subseteq U$ , $Y \\cap U = \\emptyset $ , and definition of partial evaluation of an aggregate atom note that $pos(R) \\cap U \\subseteq X$ , $(neg(R) \\cap U) \\cap X = \\emptyset $ , and for each $C \\in agg(R)$ , s.t.", "$lit(C) \\subseteq U$ , $X \\models C$ , represent conditions satisfied by each rule that become part of a partial evaluation – using definition of partial evaluation and $pos(R) \\setminus U$ , $neg(R) \\setminus U$ , and for each $C \\in agg(R)$ , $eval_U(C,X)$ are the modifications made to the rule during partial evaluation given splitting set $U$ w.r.t.", "$X$ – using definition of partial evaluation and $pos(R) \\setminus U \\subseteq Y$ , $(neg(R) \\setminus U) \\cap Y = \\emptyset $ , and for each $C \\in agg(R)$ , $Y \\models eval_U(C,X)$ if $lit(C) \\lnot \\subseteq U$ represent conditions satisfied by rules that become part of the reduct w.r.t $Y$ – using definition of partial evaluation and reduct then $r$ is a rule in reduct $eval_U(\\Pi \\setminus bot_U(\\Pi ),X)^Y$ – using (REF ), (REF ) above in addition, given that $Y$ satisfies $eval_U(\\Pi \\setminus bot_U(\\Pi ),X)$ , and $head(r) \\cap U = \\emptyset $ , we have $X \\cup Y \\models head(r)$ Next we show that $X \\cup Y$ satisfies all rules of $\\Pi $ .", "Say, $R$ is a rule in $\\Pi $ not satisfied by $X \\cup Y$ .", "Then, either it belongs to $bot_U(\\Pi )$ or $(\\Pi \\setminus bot_U(\\Pi ))$ .", "If it belongs to $bot_U(\\Pi )$ , it must not be satisfied by $X$ , since $lit(bot_U(\\Pi )) \\subseteq U$ and $Y \\cap U = \\emptyset $ .", "However, the contrary is given to be true.", "On the other hand if it belongs to $(\\Pi \\setminus bot_U(\\Pi ))$ , then $X \\cup Y$ satisfies $body(R)$ but not $head(R)$ .", "That would mean that its $head(R)$ is not satisfied by $Y$ , since $head(R) \\cap U = \\emptyset $ by definition of splitting set.", "However, from lemma REF we know that if $body(R)$ is satisfied by $X \\cup Y$ , $body(R^{\\prime })$ is satisfied by $Y$ for $R^{\\prime } \\in eval_U(\\Pi \\setminus bot_U(\\Pi ),X)$ .", "We also know that $Y$ satisfies all rules in $eval_U(\\Pi \\setminus bot_U(\\Pi ),X)$ .", "So, $R^{\\prime }$ must be satisfied by $Y$ contradicting our assumption.", "Thus, all rules of $\\Pi $ are satisfied by $X \\cup Y$ and $X \\cup Y$ is an answer set of $\\Pi $ .", "($\\Rightarrow $ ) Let $S$ be a consistent answer set of $\\Pi $ , we show that $S = X \\cup Y$ for sets $X$ and $Y$ s.t.", "$X$ is an answer set of $bot_U(\\Pi )$ and $Y$ is an answer set of $eval_U(\\Pi \\setminus bot_U(\\Pi ),X)$ .", "We take $X=S \\cap U$ , $Y=S \\setminus U$ , then $S=X \\cup Y$ .", "Case 1: We show that $X$ is answer set of $bot_U(\\Pi )$ $\\Pi $ can be split into $bot_U(\\Pi ) \\cup (\\Pi \\setminus bot_U(\\Pi ))$ – by definition of splitting then $X \\cup Y$ satisfies $bot_U(\\Pi )$ – $X \\cup Y$ is an answer set of $\\Pi $ ; given however $lit(bot_U(\\Pi )) \\subseteq U$ , $Y \\cap U = \\emptyset $ – by definition of splitting then $X$ satisfies $bot_U(\\Pi )$ – since elements of $Y$ do not appear in the rules of $bot_U(\\Pi )$ then $X$ is an answer set of $bot_U(\\Pi )$ Case 2: We show that $Y$ is answer set of $eval_U(\\Pi \\setminus bot_U(\\Pi ),X)$ let $r$ be a rule in $eval_U(\\Pi \\setminus bot_U(\\Pi ),X)^Y$ , s.t.", "its body is satisfied by $Y$ then $r \\in R^Y$ for an $R \\in eval_U(\\Pi \\setminus bot_U(\\Pi ),X)$ s.t.", "[(i)] (3) $pos(R) \\subseteq Y$ (4) $neg(R) \\cap Y = \\emptyset $ (5) $Y \\models C$ for all $C \\in agg(R)$ (6) $head(R) \\cap Y \\ne \\emptyset $ – using definition of reduct each $R$ is constructed from $R^{\\prime } \\in \\Pi $ that satisfies all the following conditions [(i)] (8) $pos(R^{\\prime }) \\subseteq U \\cap X$ (9) $(neg(R^{\\prime }) \\cap U) \\cap X = \\emptyset $ (10) if there is a $C^{\\prime } \\in agg(R^{\\prime })$ s.t.", "$lit(C^{\\prime }) \\subseteq U$ , then $X \\models C^{\\prime }$ ; and each $C \\in agg(R)$ is a partial evaluation of $C^{\\prime } \\in agg(R^{\\prime })$ s.t.", "$C = eval_U(C^{\\prime },X)$ – using definition of partial evaluation then the $body(R^{\\prime })$ satisfies all the following conditions: $pos(R^{\\prime }) \\subseteq X \\cup Y$ – since $X \\subseteq U$ , $X \\cap Y = \\emptyset $ $neg(R^{\\prime }) \\cap (X \\cup Y) = \\emptyset $ – since $X \\subseteq U$ , $X \\cap Y = \\emptyset $ $X \\cup Y \\models C^{\\prime }$ for each $C^{\\prime } \\in agg(R^{\\prime })$ – since [(i)] (d) each $C^{\\prime } \\in agg(R^{\\prime })$ with $lit(C^{\\prime }) \\subseteq U$ satisfied by $X$ is also satisfied by $X \\cup Y$ as $lit(Y) \\cap lit(C^{\\prime }) = \\emptyset $ ; and (e) each $C^{\\prime } \\in agg(R^{\\prime })$ with $lit(C^{\\prime }) \\lnot \\subseteq U$ is satisfied by $X \\cup Y$ – using partial evaluation, reduct construction, and $X \\cap Y = \\emptyset $ then $X \\cup Y$ satisfies $body(R^{\\prime })$ – from previous line in addition, $lit(head(R^{\\prime })) \\cap U = \\emptyset $ , otherwise, $R^{\\prime }$ would have belonged to $bot_U(\\Pi )$ by definition of splitting set then $R^{\\prime }$ is a rule in $\\Pi \\setminus bot_U(\\Pi )$ – from the last two lines we know that $X \\cup Y$ satisfies every rule in $(\\Pi \\setminus bot_U(\\Pi ))$ – given; and that elements of $U$ do not appear in the head of rules in $(\\Pi \\setminus bot_U(\\Pi ))$ – from definition of splitting; then $Y$ must satisfy the head of these rules then $Y$ satisfies $head(R^{\\prime })$ – from (REF ) Next we show that $Y$ satisfies all rules of $eval_U(\\Pi \\setminus bot_U(\\Pi ),X)$ .", "Let $R^{\\prime }$ be a rule in $eval_U(\\Pi \\setminus bot_U(\\Pi ),X)$ such that $body(R^{\\prime })$ is satisfied by $Y$ but not $head(R^{\\prime })$ .", "Since $head(R^{\\prime }) \\cap Y = \\emptyset $ , $head(R^{\\prime })$ is not satisfied by $X \\cup Y$ either.", "Then, there is an $R \\in (\\Pi \\setminus bot_U(\\Pi ))$ such that $X \\cup Y$ satisfies $body(R)$ but not $head(R)$ , which contradicts given.", "Thus, $Y$ satisfies all rules of $eval_U(\\Pi \\setminus bot_U(\\Pi ),X)$ .", "Then $Y$ is an answer set of $eval_U(\\Pi \\setminus bot_U(\\Pi ),X)$ Definition 92 (Splitting Sequence) [4] A splitting sequence for a program $\\Pi $ is a monotone, continuous sequence ${\\langle U_{\\alpha } \\rangle }_{\\alpha < \\mu }$ of splitting sets of $\\Pi $ such that $\\bigcup _{\\alpha < \\mu }{U_{\\mu }} = lit(\\Pi )$ .", "Definition 93 (Solution) [4] Let $U={\\langle U_{\\alpha } \\rangle }_{\\alpha < \\mu }$ be a splitting sequence for a program $\\Pi $ .", "A solution to $\\Pi $ w.r.t $U$ is a sequence ${\\langle X_{\\alpha } \\rangle }_{\\alpha < \\mu }$ of sets of literals such that: $X_0$ is an answer set for $bot_{U_0}(\\Pi )$ for any ordinal $\\alpha + 1 < \\mu $ , $X_{\\alpha +1}$ is an answer set of the program: $eval_{U_{\\alpha }}(bot_{U_{\\alpha +1}}(\\Pi ) \\setminus bot_{U_{\\alpha }}(\\Pi ), \\bigcup _{\\nu \\le \\alpha }{X_{\\nu }})$ for any limit ordinal $\\alpha < \\mu , X_{\\alpha } = \\emptyset $ , and $\\bigcup _{\\alpha \\le \\mu }(X_{\\alpha })$ is consistent Proposition 15 (Splitting Sequence Theorem) [4] Let $U={\\langle U_{\\alpha } \\rangle }_{\\alpha < \\mu }$ be a splitting sequence for a program $\\Pi $ .", "A set $S$ of literals is a consistent answer set for $\\Pi $ iff $S=\\bigcup _{\\alpha < \\mu }{X_{\\alpha }}$ for some solution ${\\langle X_{\\alpha } \\rangle }_{\\alpha < \\mu }$ to $\\Pi $ w.r.t $U$ .", "Proof: Let $U = \\langle U_\\alpha \\rangle _{\\alpha < \\mu }$ be a splitting sequence of $\\Pi $ , then a consistent set of literals $S=\\bigcup _{\\alpha < \\mu }{X_{\\alpha }}$ is an answer set of $\\Pi ^S$ iff $X_0$ is an answer set of $bot_{U_0}(\\Pi )$ and for any ordinal $\\alpha + 1 < \\mu $ , $X_{\\alpha +1}$ is an answer set of $eval_{U_{\\alpha }}(bot_{U_{\\alpha +1}}(\\Pi ) \\setminus bot_{U_{\\alpha }}(\\Pi ), \\bigcup _{\\nu \\le \\alpha }X_{\\nu })$ .", "Note that every literal in $bot_{U_0}(\\Pi )$ belongs to $lit(\\Pi ) \\cap U_0$ , and every literal occurring in $eval_{U_{\\alpha }}(bot_{U_{\\alpha +1}}(\\Pi ) \\setminus bot_{U_{\\alpha }}(\\Pi ), \\bigcup _{\\nu \\le \\alpha }X_{\\nu })$ , $(\\alpha + 1 < \\mu )$ belongs to $lit(\\Pi ) \\cap (U_{\\alpha +1} \\setminus U_{\\alpha })$ .", "In addition, $X_0$ , and all $X_{\\alpha +1}$ are pairwise disjoint.", "We prove the theorem by induction over the splitting sequence.", "Base case: $\\alpha = 1$ .", "The splitting sequence is $U_0 \\subseteq U_1$ .", "Then the sub-program $\\Pi _1 = bot_{U_1}(\\Pi )$ contains all literals in $U_1$ ; and $U_0$ splits $\\Pi _1$ into $bot_{U_0}(\\Pi _1)$ and $bot_{U_1}(\\Pi _1) \\setminus bot_{U_0}(\\Pi _1)$ .", "Then, $S_1 = X_0 \\cup X_1$ is a consistent answer set of $\\Pi _1$ iff $X_0 = S_1 \\cap U_0$ is an answer set of $bot_{U_0}(\\Pi _1)$ and $X_1 = S_1 \\setminus U_0$ is an answer set of $eval_{U_0}(\\Pi _1 \\setminus bot_{U_0}(\\Pi _1),X_1)$ – by the splitting theorem Since $bot_{U_0}(\\Pi _1) = bot_{U_0}(\\Pi )$ and $bot_{U_1}(\\Pi _1) \\setminus bot_{U_0}(\\Pi _1) = bot_{U_1}(\\Pi ) \\setminus bot_{U_0}(\\Pi )$ ; $S_1 = X_0 \\cup X_1$ is an answer set for $\\Pi \\setminus bot_{U_1}(\\Pi )$ .", "Induction: Assume theorem holds for $\\alpha = k$ , show theorem holds for $\\alpha = k+1$ .", "The inductive assumption holds for the splitting sequence $U_0 \\subseteq \\dots \\subseteq U_k$ .", "Then the sub-program $\\Pi _k = bot_{U_k}(\\Pi )$ contains all literals in $U_k$ and $S_k = X_0 \\cup \\dots \\cup X_k$ is a consistent answer set of $\\Pi ^{S_k}$ iff $X_0$ is an answer set for $bot_{U_0}(\\Pi _k)$ and for any $\\alpha \\le k$ , $X_{\\alpha +1}$ is answer set of $eval_{U_{\\alpha }}(bot_{U_{\\alpha +1}}(\\Pi ) \\setminus bot_{U_{\\alpha }}(\\Pi ), X_0 \\cup \\dots \\cup X_{\\alpha })$ We show that the theorem holds for $\\alpha = k+1$ .", "The splitting sequence is $U_0 \\subseteq U_{k+1}$ .", "Then the sub-program $\\Pi _{k+1} = bot_{U_{k+1}}(\\Pi )$ contains all literals $U_{k+1}$ .", "We have $U_k$ split $\\Pi _{k+1}$ into $bot_{U_k}(\\Pi _{k+1})$ and $bot_{U_{k+1}}(\\Pi _{k+1}) \\setminus bot_{U_k}(\\Pi _{k+1})$ .", "Then, $S_{k+1} = X_{0:k} \\cup X_{k+1}$ is a consistent answer set of $\\Pi _{k+1}$ iff $X_{0:k} = S_{k+1} \\cap U_k$ is an answer set of $bot_{U_k}(\\Pi _{k+1})$ and $X_{k+1} = S_{k+1} \\setminus U_k$ is an answer set of $eval_{U_k}(\\Pi _{k+1} \\setminus bot_{U_k}(\\Pi _{k+1},X_{k+1})$ – by the splitting theorem Since $bot_{U_k}(\\Pi _{k+1}) = bot_{U_k}(\\Pi )$ and $bot_{U_{k+1}}(\\Pi _{k+1}) \\setminus bot_{U_k}(\\Pi _{k+1}) = bot_{U_{k+1}}(\\Pi ) \\setminus bot_{U_k}(\\Pi )$ ; $S_{k+1} = X_{0:k} \\cup X_1$ is an answer set for $\\Pi \\setminus bot_{U_{k+1}}(\\Pi )$ .", "From the inductive assumption we know that $X_0 \\cup \\dots \\cup X_k$ is a consistent answer set of $bot_{U_k}(\\Pi )$ , $X_0$ is the answer set of $bot_{U_0}(\\Pi )$ , and for each $0 \\le \\alpha \\le k$ , $X_{\\alpha +1}$ is answer set of $eval_{U_{\\alpha }}(bot_{U_{\\alpha +1}}(\\Pi ) \\setminus bot_{U_{\\alpha }}(\\Pi ), X_0 \\cup \\dots \\cup X_{\\alpha })$ .", "Thus, $X_{0:k} = X_0 \\cup \\dots \\cup X_k$ .", "Combining above with the inductive assumption, we get $S_{k+1} = X_0 \\cup \\dots \\cup X_{k+1}$ is a consistent answer set of $\\Pi ^{S_{k+1}}$ iff $X_0$ is an answer set for $bot_{U_0}(\\Pi _{k+1})$ and for any $\\alpha \\le k+1$ , $X_{\\alpha +1}$ is answer set of $eval_{U_{\\alpha }}(bot_{U_{\\alpha +1}}(\\Pi ) \\setminus bot_{U_{\\alpha }}(\\Pi ), X_0 \\cup \\dots \\cup X_{\\alpha })$ .", "In addition, for some $\\alpha < \\mu $ , where $\\mu $ is the length of the splitting sequence $U = \\langle U_{\\alpha } \\rangle _{\\alpha < \\mu }$ of $\\Pi $ , $bot_{U_{\\alpha }}(\\Pi )$ will be the entire $\\Pi $ , i.e.", "$lit(\\Pi ) = U_{\\alpha }$ .", "Then the set $S$ of literals is a consistent answer set of $\\Pi $ iff $S=\\bigcup _{\\alpha < \\mu }(X_{\\alpha })$ for some solution $\\langle X_{\\alpha } \\rangle _{\\alpha < \\mu }$ to $\\Pi $ w.r.t $U$ ." ], [ "Proof of Proposition ", "Let $PN=(P,T,E,W)$ be a Petri Net, $M_0$ be its initial marking and let $\\Pi ^0(PN,M_0,$ $k,ntok)$ be the ASP encoding of $PN$ and $M_0$ over a simulation length $k$ , with maximum $ntok$ tokens on any place node, as defined in section REF .", "Then $X=M_0,T_0,M_1,\\dots ,M_k,T_k,M_{k+1}$ is an execution sequence of a $PN$ (w.r.t.", "$M_0$ ) iff there is an answer set $A$ of $\\Pi ^0(PN,M_0,k,ntok)$ such that: $\\lbrace fires(t,ts) : t \\in T_{ts}, 0 \\le ts \\le k \\rbrace = \\lbrace fires(t,ts) : fires(t,ts) \\in A \\rbrace $ $\\begin{split}\\lbrace holds(p,q,ts) &: p \\in P, q = M_{ts}(p), 0 \\le ts \\le k+1 \\rbrace \\\\&= \\lbrace holds(p,q,ts) : holds(p,q,ts) \\in A \\rbrace \\end{split}$ We prove this by showing that: Given an execution sequence $X$ , we create a set $A$ such that it satisfies (REF ) and (REF ) and show that $A$ is an answer set of $\\Pi ^0$ Given an answer set $A$ of $\\Pi ^0$ , we create an execution sequence $X$ such that (REF ) and (REF ) are satisfied First we show (REF ): Given $PN$ and its execution sequence $X$ , we create a set $A$ as a union of the following sets: $A_1=\\lbrace num(n) : 0 \\le n \\le ntok \\rbrace $ $A_2=\\lbrace time(ts) : 0 \\le ts \\le k\\rbrace $ $A_3=\\lbrace place(p) : p \\in P \\rbrace $ $A_4=\\lbrace trans(t) : t \\in T \\rbrace $ $A_5=\\lbrace ptarc(p,t,n) : (p,t) \\in E^-, n=W(p,t) \\rbrace $ , where $E^- \\subseteq E$ $A_6=\\lbrace tparc(t,p,n) : (t,p) \\in E^+, n=W(t,p) \\rbrace $ , where $E^+ \\subseteq E$ $A_7=\\lbrace holds(p,q,0) : p \\in P, q=M_{0}(p) \\rbrace $ $A_8=\\lbrace notenabled(t,ts) : t \\in T, 0 \\le ts \\le k, \\exists p \\in \\bullet t, M_{ts}(p) < W(p,t) \\rbrace $ per definition REF (enabled transition) $A_9=\\lbrace enabled(t,ts) : t \\in T, 0 \\le ts \\le k, \\forall p \\in \\bullet t, W(p,t) \\le M_{ts}(p) \\rbrace $ per definition REF (enabled transition) $A_{10}=\\lbrace fires(t,ts) : t \\in T_{ts}, 0 \\le ts \\le k \\rbrace $ from definition REF (firing set) , only an enabled transition may fire $A_{11}=\\lbrace add(p,q,t,ts) : t \\in T_{ts}, p \\in t \\bullet , q=W(t,p), 0 \\le ts \\le k \\rbrace $ per definition REF (transition execution) $A_{12}=\\lbrace del(p,q,t,ts) : t \\in T_{ts}, p \\in \\bullet t, q=W(p,t), 0 \\le ts \\le k \\rbrace $ per definition REF (transition execution) $A_{13}=\\lbrace tot\\_incr(p,q,ts) : p \\in P, q=\\sum _{t \\in T_{ts}, p \\in t \\bullet }{W(t,p)}, 0 \\le ts \\le k \\rbrace $ per definition REF (firing set execution) $A_{14}=\\lbrace tot\\_decr(p,q,ts): p \\in P, q=\\sum _{t \\in T_{ts}, p \\in \\bullet t}{W(p,t)}, 0 \\le ts \\le k \\rbrace $ per definition REF (firing set execution) $A_{15}=\\lbrace consumesmore(p,ts) : p \\in P, q=M_{ts}(p), q1=\\sum _{t \\in T_{ts}, p \\in \\bullet t}{W(p,t)}, q1 > q, 0 \\le ts \\le k \\rbrace $ per definition REF (conflicting transitions) for enabled transition set $T_{ts}$ $A_{16}=\\lbrace consumesmore : \\exists p \\in P : q=M_{ts}(p), q1=\\sum _{t \\in T_{ts}, p \\in \\bullet t}{W(p,t)}, q1 > q, 0 \\le ts \\le k \\rbrace $ per definition REF (conflicting transitions) $A_{17}=\\lbrace holds(p,q,ts+1) : p \\in P, q=M_{ts+1}(p), 0 \\le ts < k\\rbrace $ , where $M_{ts+1}(p) = M_{ts}(p) - \\sum _{\\begin{array}{c}t \\in T_{ts}, p \\in \\bullet t\\end{array}}{W(p,t)} + \\sum _{\\begin{array}{c}t \\in T_{ts}, p \\in t \\bullet \\end{array}}{W(t,p)}$ according to definition REF (firing set execution) We show that $A$ satisfies (REF ) and (REF ), and $A$ is an answer set of $\\Pi ^0$ .", "$A$ satisfies (REF ) and (REF ) by its construction (given above).", "We show $A$ is an answer set of $\\Pi ^0$ by splitting.", "We split $lit(\\Pi ^0)$ (literals of $\\Pi ^0$ ) into a sequence of $6(k+1)+2$ sets: $U_0= head(f\\ref {f:place}) \\cup head(f\\ref {f:trans}) \\cup head(f\\ref {f:ptarc}) \\cup head(f\\ref {f:tparc}) \\cup head(f\\ref {f:time}) \\cup head(f\\ref {f:num}) \\cup head(i\\ref {i:holds}) = \\lbrace place(p) : p \\in P\\rbrace \\cup \\lbrace trans(t) : t \\in T\\rbrace \\cup \\lbrace ptarc(p,t,n) : (p,t) \\in E^-, n=W(p,t) \\rbrace \\cup \\lbrace tparc(t,p,n) : (t,p) \\in E^+, n=W(t,p) \\rbrace \\cup $ $\\lbrace time(0), \\dots , $ $time(k)\\rbrace \\cup \\lbrace num(0), \\dots , num(ntok)\\rbrace \\cup \\lbrace holds(p,q,0) : p \\in P, q=M_0(p) \\rbrace $ $U_{6k+1}=U_{6k+0} \\cup head(e\\ref {e:ne:ptarc})^{ts=k} = U_{6k+0} \\cup \\lbrace notenabled(t,k) : t \\in T \\rbrace $ $U_{6k+2}=U_{6k+1} \\cup head(e\\ref {e:enabled})^{ts=k} = U_{6k+1} \\cup \\lbrace enabled(t,k) : t \\in T \\rbrace $ $U_{6k+3}=U_{6k+2} \\cup head(a\\ref {a:fires})^{ts=k} = U_{6k+2} \\cup \\lbrace fires(t,k) : t \\in T \\rbrace $ $U_{6k+4}=U_{6k+3} \\cup head(r\\ref {r:add})^{ts=k} \\cup head(r\\ref {r:del})^{ts=k} = U_{6k+3} \\cup \\lbrace add(p,q,t,k) : p \\in P, t \\in T, q=W(t,p) \\rbrace \\cup \\lbrace del(p,q,t,k) : p \\in P, t \\in T, q=W(p,t) \\rbrace $ $U_{6k+5}=U_{6k+4} \\cup head(r\\ref {r:totincr})^{ts=k} \\cup head(r\\ref {r:totdecr})^{ts=k} = U_{6k+4} \\cup \\lbrace tot\\_incr(p,q,k) : p \\in P, 0 \\le q \\le ntok \\rbrace \\cup \\lbrace tot\\_decr(p,q,k) : p \\in P, 0 \\le q \\le ntok \\rbrace $ $U_{6k+6}=U_{6k+5} \\cup head(r\\ref {r:nextstate})^{ts=k} \\cup head(a\\ref {a:overc:place})^{ts=k} = U_{6k+5} \\cup \\lbrace holds(p,q,k+1) : p \\in P, 0 \\le q \\le ntok \\rbrace \\cup \\lbrace consumesmore(p,k) : p \\in P\\rbrace $ $U_{6k+7}=U_{6k+6} \\cup head(a\\ref {a:overc:gen}) = U_{7k+6} \\cup \\lbrace consumesmore \\rbrace $ where $head(r_i)^{ts=k}$ are head atoms of ground rule $r_i$ in which $ts=k$ .", "We write $A_i^{ts=k} = \\lbrace a(\\dots ,ts) : a(\\dots ,ts) \\in A_i, ts=k \\rbrace $ as short hand for all atoms in $A_i$ with $ts=k$ .", "$U_{\\alpha }, 0 \\le \\alpha \\le 6k+7$ form a splitting sequence, since each $U_i$ is a splitting set of $\\Pi ^0$ , and $\\langle U_{\\alpha }\\rangle _{\\alpha < \\mu }$ is a monotone continuous sequence, where $U_0 \\subseteq U_1 \\dots \\subseteq U_{6k+7}$ and $\\bigcup _{\\alpha < \\mu }{U_{\\alpha }} = lit(\\Pi ^0)$ .", "We compute the answer set of $\\Pi ^0$ using the splitting sets as follows: $bot_{U_0}(\\Pi ^0) = f\\ref {f:place} \\cup f\\ref {f:trans} \\cup f\\ref {f:ptarc} \\cup f\\ref {f:tparc} \\cup f\\ref {f:time} \\cup i\\ref {i:holds} \\cup f\\ref {f:num}$ and $X_0 = A_1 \\cup \\dots \\cup A_7$ ($= U_0$ ) is its answer set – using forced atom proposition $eval_{U_0}(bot_{U_1}(\\Pi ^0) \\setminus bot_{U_0}(\\Pi ^0), X_0) = \\lbrace notenabled(t,0) \\text{:-} .", "\| \\lbrace trans(t), $ $ ptarc(p,t,n), $ $ holds(p,q,0) \\rbrace \\subseteq X_0, \\text{~where~} q < n \\rbrace $ .", "Its answer set $X_1=A_8^{ts=0}$ – using forced atom proposition and construction of $A_8$ .", "where, $q=M_0(p)$ , $n=W(p,t)$ , and for an arc $(p,t) \\in E^-$ – by construction of $i\\ref {i:holds}$ and $f\\ref {f:ptarc}$ in $\\Pi ^0$ , and in an arc $(p,t) \\in E^-$ , $p \\in \\bullet t$ (by definition REF of preset) thus, $notenabled(t,0) \\in X_1$ represents $\\exists p \\in \\bullet t : M_0(p) < W(p,t)$ .", "$eval_{U_1}(bot_{U_2}(\\Pi ^0) \\setminus bot_{U_1}(\\Pi ^0), X_0 \\cup X_1) = \\lbrace enabled(t,0) \\text{:-}.", "\| trans(t) \\in X_0 \\cup X_1, $ $notenabled(t,0) \\notin X_0 \\cup X_1 \\rbrace $ .", "Its answer set is $X_2 = A_9^{ts=0}$ – using forced atom proposition and construction of $A_9$ .", "since an $enabled(t,0) \\in X_2$ if $\\nexists ~notenabled(t,0) \\in X_0 \\cup X_1$ , which is equivalent to $\\nexists p \\in \\bullet t : M_0(p) < W(p,t) \\equiv \\forall p \\in \\bullet t: M_0(p) \\ge W(p,t)$ .", "$eval_{U_2}(bot_{U_3}(\\Pi ^0) \\setminus bot_{U_2}(\\Pi ^0), X_0 \\cup X_1 \\cup X_2) = \\lbrace \\lbrace fires(t,0)\\rbrace \\text{:-}.", "\| enabled(t,0) \\\\ \\text{~holds in~} X_0 \\cup X_1 \\cup X_2 \\rbrace $ .", "It has multiple answer sets $X_{3.1}, \\dots , X_{3.n}$ , corresponding to elements of power set of $fires(t,0)$ atoms in $eval_{U_2}(...)$ – using supported rule proposition.", "Since we are showing that the union of answer sets of $\\Pi ^0$ determined using splitting is equal to $A$ , we only consider the set that matches the $fires(t,0)$ elements in $A$ and call it $X_3$ , ignoring the rest.", "Thus, $X_3 = A_{10}^{ts=0}$ , representing $T_0$ .", "$eval_{U_3}(bot_{U_4}(\\Pi ^0) \\setminus bot_{U_3}(\\Pi ^0), X_0 \\cup \\dots \\cup X_3) = \\lbrace add(p,n,t,0) \\text{:-}.", "\| \\lbrace fires(t,0), \\\\ tparc(t,p,n) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_3 \\rbrace \\cup \\lbrace del(p,n,t,0) \\text{:-}.", "\| \\lbrace fires(t,0), ptarc(p,t,n) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_3 \\rbrace $ .", "It's answer set is $X_4 = A_{11}^{ts=0} \\cup A_{12}^{ts=0}$ – using forced atom proposition and definitions of $A_{11}$ and $A_{12}$ .", "where each $add$ atom is equivalent to $n=W(t,p) : p \\in t \\bullet $ , and each $del$ atom is equivalent to $n=W(p,t) : p \\in \\bullet t$ , representing the effect of transitions in $T_0$ – by construction $eval_{U_4}(bot_{U_5}(\\Pi ^0) \\setminus bot_{U_4}(\\Pi ^0), X_0 \\cup \\dots \\cup X_4) = \\lbrace tot\\_incr(p,qq,0) \\text{:-}.", "\| \\\\ qq=\\sum _{add(p,q,t,0) \\in X_0 \\cup \\dots \\cup X_4}{q} \\rbrace \\cup $ $\\lbrace tot\\_decr(p,qq,0) \\text{:-}.", "\| qq=\\sum _{del(p,q,t,0) \\in X_0 \\cup \\dots \\cup X_4}{q} \\rbrace $ .", "It's answer set is $X_5 = A_{13}^{ts=0} \\cup A_{14}^{ts=0}$ – using forced atom proposition and definitions of $A_{13}$ and $A_{14}$ , ad definition REF (semantics of aggregate assignment atom).", "where each $tot\\_incr(p,qq,0)$ , $qq=\\sum _{add(p,q,t,0) \\in X_0 \\cup \\dots X_4}{q} \\\\ \\equiv $ $qq=\\sum _{t \\in X_3, p \\in t \\bullet }{W(p,t)}$ , and each $tot\\_decr(p,qq,0)$ , $qq=\\sum _{del(p,q,t,0) \\in X_0 \\cup \\dots X_4}{q} \\\\ \\equiv $ $qq=\\sum _{t \\in X_3, p \\in \\bullet t}{W(t,p)}$ , represent the net effect of transitions in $T_0$ – by construction $eval_{U_5}(bot_{U_6}(\\Pi ^0) \\setminus bot_{U_5}(\\Pi ^0), X_0 \\cup \\dots \\cup X_5) = \\lbrace consumesmore(p,0) \\text{:-}.", "\| \\\\ \\lbrace holds(p,q,0), $ $tot\\_decr(p,q1,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_5, $ $q1 > q \\rbrace \\cup $ $\\lbrace holds(p,q,1) \\text{:-}.", "\| $ $ \\lbrace holds(p,q1,0), $ $tot\\_incr(p,q2,0), $ $tot\\_decr(p,q3,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_5, $ $q=q1+q2-q3 \\rbrace $ .", "It's answer set is $X_6 = A_{15}^{ts=0} \\cup A_{17}^{ts=0}$ – using forced atom proposition.", "where, $consumesmore(p,0)$ represents $\\exists p : q=M_0(p), \\\\ q1=\\sum _{t \\in T_0, p \\in \\bullet t}{W(p,t)}, q1 > q$ – indicating place $p$ will be overconsumed if $T_0$ is fired as defined in definition REF (conflicting transitions) and $holds(p,q,1)$ represents $q=M_1(p)$ – by construction $ \\vdots $ $eval_{U_{6k+0}}(bot_{U_{6k+1}}(\\Pi ^0) \\setminus bot_{U_{6k+0}}(\\Pi ^0), X_0 \\cup \\dots \\cup X_{6k+0}) = \\\\ \\lbrace notenabled(t,k) \\text{:-} .", "\| \\lbrace trans(t), ptarc(p,t,n), holds(p,q,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{6k+0}, \\\\ \\text{~where~} q < n \\rbrace $ .", "Its answer set $X_{6k+1}=A_8^{ts=k}$ – using forced atom proposition and construction of $A_8$ .", "where, $q=M_k(p)$ , and $n=W(p,t)$ for an arc $(p,t) \\in E^-$ – by construction of $holds$ and $ptarc$ predicates in $\\Pi ^0$ , and in an arc $(p,t) \\in E^-$ , $p \\in \\bullet t$ (by definition REF of preset) thus, $notenabled(t,k) \\in X_{6k+1}$ represents $\\exists p \\in \\bullet t : M_k(p) < W(p,t)$ .", "$eval_{U_{6k+1}}(bot_{U_{6k+2}}(\\Pi ^0) \\setminus bot_{U_{6k+1}}(\\Pi ^0), X_0 \\cup \\dots \\cup X_{6k+1}) = \\lbrace enabled(t,k) \\text{:-}.", "\| \\\\ trans(t) \\in $ $X_0 \\cup \\dots \\cup X_{6k+1}, notenabled(t,k) \\notin X_0 \\cup \\dots \\cup X_{6k+1} \\rbrace $ .", "Its answer set is $X_{6k+2} = A_9^{ts=k}$ – using forced atom proposition and construction of $A_9$ .", "since an $enabled(t,k) \\in X_{6k+2}$ if $\\nexists ~notenabled(t,k) \\in X_0 \\cup \\dots \\cup X_{6k+1}$ , which is equivalent to $\\nexists p \\in \\bullet t : M_k(p) < W(p,t) \\equiv \\forall p \\in \\bullet t: M_k(p) \\ge W(p,t)$ .", "$eval_{U_{6k+2}}(bot_{U_{6k+3}}(\\Pi ^0) \\setminus bot_{U_{6k+2}}(\\Pi ^0), X_0 \\cup \\dots \\cup X_{6k+2}) = \\\\ \\lbrace \\lbrace fires(t,k)\\rbrace \\text{:-}.", "\| $ $enabled(t,k) \\text{~holds in~} $ $X_0 \\cup \\dots \\cup X_{6k+2} \\rbrace $ .", "It has multiple answer sets $X_{{6k+3}.1}, \\dots , X_{{6k+3}.n}$ , corresponding to elements of power set of $fires(t,k)$ atoms in $eval_{U_{6k+2}}(...)$ – using supported rule proposition.", "Since we are showing that the union of answer sets of $\\Pi ^0$ determined using splitting is equal to $A$ , we only consider the set that matches the $fires(t,k)$ elements in $A$ and call it $X_{6k+3}$ , ignoring the reset.", "Thus, $X_{6k+3} = A_{10}^{ts=k}$ , representing $T_k$ .", "$eval_{U_{6k+3}}(bot_{U_{6k+4}}(\\Pi ^0) \\setminus bot_{U_{6k+3}}(\\Pi ^0), X_0 \\cup \\dots \\cup X_{6k+3}) = $ $ \\lbrace add(p,n,t,k) \\text{:-}.", "\| \\\\ \\lbrace fires(t,k), tparc(t,p,n) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{6k+3} \\rbrace \\cup $ $ \\lbrace del(p,n,t,k) \\text{:-}.", "\| $ $\\lbrace fires(t,k), $ $ptarc(p,t,n) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{6k+3} \\rbrace $ .", "It's answer set is $X_{6k+4} = A_{11}^{ts=k} \\cup A_{12}^{ts=k}$ – using forced atom proposition and definitions of $A_{11}$ and $A_{12}$ .", "where, each $add$ atom is equivalent to $n=W(t,p) : p \\in t \\bullet $ , and each $del$ atom is equivalent to $n=W(p,t) : p \\in \\bullet t$ representing the effect of transitions in $T_k$ $eval_{U_{6k+4}}(bot_{U_{6k+5}}(\\Pi ^0) \\setminus bot_{U_{6k+4}}(\\Pi ^0), X_0 \\cup \\dots \\cup X_{6k+4}) = $ $\\lbrace tot\\_incr(p,qq,k) \\text{:-}.", "\| $ $ qq=\\sum _{add(p,q,t,k) \\in X_0 \\cup \\dots \\cup X_{6k+4}}{q} \\rbrace \\cup $ $\\lbrace tot\\_decr(p,qq,k) \\text{:-}.", "\| \\\\ qq=\\sum _{del(p,q,t,k) \\in X_0 \\cup \\dots \\cup X_{6k+4}}{q} \\rbrace $ .", "It's answer set is $X_5 = A_{13}^{ts=k} \\cup A_{14}^{ts=k}$ – using forced atom proposition and definitions of $A_{13}$ and $A_{14}$ .", "where, each $tot\\_incr(p,qq,k)$ , $qq=\\sum _{add(p,q,t,k) \\in X_0 \\cup \\dots X_{7k+4}}{q}$ $\\equiv qq=\\sum _{t \\in X_{6k+3}, p \\in t \\bullet }{W(p,t)}$ , and each $tot\\_decr(p,qq,k)$ , $qq=\\sum _{del(p,q,t,k) \\in X_0 \\cup \\dots X_{7k+4}}{q}$ $\\equiv qq=\\sum _{t \\in X_{6k+3}, p \\in \\bullet t}{W(t,p)}$ , represent the net effect of transitions in $T_k$ $eval_{U_{6k+5}}(bot_{U_{6k+6}}(\\Pi ^0) \\setminus bot_{U_{6k+5}}(\\Pi ^0), X_0 \\cup \\dots \\cup X_{6k+5}) = $ $ \\lbrace consumesmore(p,k) \\text{:-}.", "\| $ $ \\lbrace holds(p,q,k), $ $ tot\\_decr(p,q1,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{6k+5} , q1 > q \\rbrace \\cup \\lbrace holds(p,q,k+1) \\text{:-}., \| \\lbrace holds(p,q1,k), tot\\_incr(p,q2,k), \\\\ tot\\_decr(p,q3,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{6k+5}, q=q1+q2-q3 \\rbrace $ .", "It's answer set is $X_{6k+6} = A_{15}^{ts=k} \\cup A_{17}^{ts=k}$ – using forced atom proposition.", "where, $holds(p,q,k+1)$ represents the marking of place $p$ in the next time step due to firing $T_k$ , and, $consumesmore(p,k)$ represents $\\exists p : q=M_{k}(p), q1=\\sum _{t \\in T_{k}, p \\in \\bullet t}{W(p,t)}, $ $q1 > q$ indicating place $p$ that will be overconsumed if $T_k$ is fired as defined in definition REF (conflicting transitions) $eval_{U_{6k+6}}(bot_{U_{6k+7}}(\\Pi ^0) \\setminus bot_{U_{6k+6}}(\\Pi ^0), X_0 \\cup \\dots \\cup X_{6k+6}) = \\lbrace consumesmore \\text{:-}.", "\| $ $\\lbrace consumesmore(p,0),$ $ \\dots , $ $consumesmore(p,k)\\rbrace \\cap (X_0 \\cup \\dots \\cup X_{6k+7}) \\ne \\emptyset \\rbrace $ .", "It's answer set is $X_{6k+7} = A_{16}^{ts=k}$ – using forced atom proposition $X_{6k+7}$ will be empty since none of $consumesmore(p,0),\\dots , \\\\ consumesmore(p,k)$ hold in $X_0 \\cup \\dots \\cup X_{6k+6}$ due to the construction of $A$ and encoding of $a\\ref {a:overc:place}$ , and it is not eliminated by the constraint $a\\ref {a:overc:elim}$ .", "The set $X=X_{0} \\cup \\dots \\cup X_{6k+7}$ is the answer set of $\\Pi ^0$ by the splitting sequence theorem REF .", "Each $X_i, 0 \\le i \\le 6k+7$ matches a distinct partition of $A$ , and $X = A$ , thus $A$ is an answer set of $\\Pi ^0$ .", "Next we show (REF ): Given $\\Pi ^0$ be the encoding of a Petri Net $PN(P,T,E,W)$ with initial marking $M_0$ , and $A$ be an answer set of $\\Pi ^0$ that satisfies (REF ) and (REF ), then we can construct $X=M_0,T_0,\\dots ,M_k,T_k,M_{k+1}$ from $A$ , such that it is an execution sequence of $PN$ .", "We construct the $X$ as follows: $M_i = (M_i(p_0), \\dots , M_i(p_n))$ , where $\\lbrace holds(p_0,M_i(p_0),i), \\dots holds(p_n,M_i(p_n),i) \\rbrace \\\\ \\subseteq A$ , for $0 \\le i \\le k+1$ $T_i = \\lbrace t : fires(t,i) \\in A\\rbrace $ , for $0 \\le i \\le k$ and show that $X$ is indeed an execution sequence of $PN$ .", "We show this by induction over $k$ (i.e.", "given $M_k$ , $T_k$ is a valid firing set and its firing produces marking $M_{k+1}$ ).", "Base case: Let $k=0$ , show [(1)] $T_0$ is a valid firing set for $M_0$ , and $T_0$ 's firing in $M_0$ producing marking $M_1$ .", "We show $T_0$ is a valid firing set w.r.t.", "$M_0$ .", "Let $\\lbrace fires(t_0,0), \\dots , fires(t_x,0) \\rbrace $ be the set of all $fires(\\dots ,0)$ atoms in $A$ , Then for each $fires(t_i,0) \\in A$ $enabled(t_i,0) \\in A$ – from rule $a\\ref {a:fires}$ and supported rule proposition Then $notenabled(t_i,0) \\notin A$ – from rule $e\\ref {e:enabled}$ and supported rule proposition Then $body(e\\ref {e:ne:ptarc})$ must not hold in $A$ – from rule $e\\ref {e:ne:ptarc}$ and forced atom proposition Then $q \\lnot < n_i \\equiv q \\ge n_i$ in $e\\ref {e:ne:ptarc}$ for all $\\lbrace holds(p,q,0), ptarc(p,t_i,n_i)\\rbrace \\subseteq A$ – from $e\\ref {e:ne:ptarc}$ , forced atom proposition, and the following: $holds(p,q,0) \\in A$ represents $q=M_0(p)$ – rule $i\\ref {i:holds}$ construction $ptarc(p,t_i,n_i) \\in A$ represents $n_i=W(p,t_i)$ – rule $f\\ref {f:ptarc} $ construction Then $\\forall p \\in \\bullet t_i, M_0(p) > W(p,t_i)$ – from definition REF of preset $\\bullet t_i$ in PN Then $t_i$ is enabled and can fire in $PN$ , as a result it can belong to $T_0$ – from definition REF of enabled transition And $consumesmore \\notin A$ , since $A$ is an answer set of $\\Pi ^0$ – from rule $a\\ref {a:overc:elim}$ and supported rule proposition Then $\\nexists consumesmore(p,0) \\in A$ – from rule $a\\ref {a:overc:gen}$ and supported rule proposition Then $\\nexists \\lbrace holds(p,q,0), tot\\_decr(p,q1,0) \\rbrace \\subseteq A : q1>q$ in $body(a\\ref {a:overc:place})$ – from $a\\ref {a:overc:place}$ and forced atom proposition Then $\\nexists p : \\sum _{t_i \\in \\lbrace t_0,\\dots ,t_x\\rbrace , p \\in \\bullet t_i}{W(p,t_i)} > M_0(p)$ – from the following $holds(p,q,0)$ represents $q=M_0(p)$ – from rule $i\\ref {i:holds}$ construction, given $tot\\_decr(p,q1,0) \\in A$ if $\\lbrace del(p,q1_0,t_0,0), \\dots del(p,q1_x,t_x,0) \\rbrace \\subseteq A$ , where $q1=q1_0+\\dots +q1_x$ – from $r\\ref {r:totdecr}$ and forced atom proposition $del(p,q1_i,t_i,0) \\in A$ if $\\lbrace fires(t_i,0), ptarc(p,t_i,q1_i) \\rbrace \\subseteq A$ – from $r\\ref {r:del}$ and supported rule proposition $del(p,q1_i,t_i,0) \\in A$ represents removal of $q1_i = W(p,t_i)$ tokens from $p \\in \\bullet t_i$ – from $r\\ref {r:del}$ , supported rule proposition, and definition REF of transition execution in PN Then the set of transitions in $T_0$ do not conflict – by the definition REF of conflicting transitions Then $\\lbrace t_0, \\dots , t_x\\rbrace = T_0$ – using 1(a),1(b) above, and definition of firing set We show $M_1$ is produced by firing $T_0$ in $M_0$ .", "Let $holds(p,q,1) \\in A$ Then $\\lbrace holds(p,q1,0), tot\\_incr(p,q2,0), tot\\_decr(p,q3,0) \\rbrace \\subseteq A : q=q1+q2-q3$ – from rule $r\\ref {r:nextstate}$ and supported rule proposition Then $holds(p,q1,0) \\in A$ represents $q1=M_0(p)$ – given, rule $i\\ref {i:holds}$ construction; and $\\lbrace add(p,q_0,t_0,0), \\dots , $ $add(p,q_j,t_j,0)\\rbrace \\subseteq A : q_0 + \\dots + q_j = q2$ ; and $\\lbrace del(p,q_0,t_0,0), \\dots , $ $del(p,q_l,t_l,0)\\rbrace \\subseteq A : q_0 + \\dots + q_l = q3$ – rules $r\\ref {r:totincr},r\\ref {r:totdecr}$ and supported rule proposition, respectively Then $\\lbrace fires(t_0,0), \\dots , fires(t_j,0) \\rbrace \\subseteq A$ and $\\lbrace fires(t_0,0), \\dots , fires(t_l,0) \\rbrace $ $\\subseteq A$ – rules $r\\ref {r:add},r\\ref {r:del}$ and supported rule proposition; and the following $tparc(t_y,p,q_y) \\in A, 0 \\le y \\le j$ represents $q_y=W(t_y,p)$ – given $ptarc(p,t_z,q_z) \\in A, 0 \\le z \\le l$ represents $q_z=W(p,t_z)$ – given Then $\\lbrace fires(t_0,0), \\dots , fires(t_j,0) \\rbrace \\cup \\lbrace fires(t_0,0), \\dots , fires(t_l,0) \\rbrace \\subseteq A $ $= \\lbrace fires(t_0,0), \\dots , fires(t_x,0) \\rbrace \\subseteq A$ – set union of subsets Then for each $fires(t_x,0) \\in A$ we have $t_x \\in T_0$ – already shown in item REF above Then $q = M_0(p) + \\sum _{t_x \\in T_0 \\wedge p \\in t_x \\bullet }{W(t_x,p)} - \\sum _{t_x \\in T_0 \\wedge p \\in \\bullet t_x}{W(p,t_x)}$ – from (REF ) above and the following Each $add(p,q_j,t_j,0) \\in A$ represents $q_j=W(t_j,p)$ for $p \\in t_j \\bullet $ – rule $r\\ref {r:add}$ encoding, and definition REF of transition execution in $PN$ Each $del(p,t_y,q_y,0) \\in A$ represents $q_y=W(p,t_y)$ for $p \\in \\bullet t_y$ – from rule $r\\ref {r:del}$ encoding, and definition REF of transition execution in $PN$ Each $tot\\_incr(p,q2,0) \\in A$ represents $q2=\\sum _{t_x \\in T_0 \\wedge p \\in t_x \\bullet }{W(t_x,p)}$ – aggregate assignment atom semantics in rule $r\\ref {r:totincr}$ Each $tot\\_decr(p,q3,0) \\in A$ represents $q3=\\sum _{t_x \\in T_0 \\wedge p \\in \\bullet t_x}{W(p,t_x)}$ – aggregate assignment atom semantics in rule $r\\ref {r:totdecr}$ Then, $M_1(p) = q$ – since $holds(p,q,1) \\in A$ encodes $q=M_1(p)$ – by construction Inductive Step: Assume $M_k$ is a valid marking in $X$ for $PN$ , show [(1)] $T_k$ is a valid firing set for $M_k$ , and firing $T_k$ in $M_k$ produces marking $M_{k+1}$ .", "We show that $T_k$ is a valid firing set for $M_k$ .", "Let $\\lbrace fires(t_0,k),\\dots ,fires(t_x,k) \\rbrace $ be the set of all $fires(\\dots ,k)$ atoms in $A$ , Then for each $fires(t_i,k) \\in A$ $enabled(t_i,k) \\in A$ – from rule $a\\ref {a:fires}$ and supported rule proposition Then $notenabled(t_i,k) \\notin A$ – from rule $e\\ref {e:enabled}$ and supported rule proposition Then body of $e\\ref {e:ne:ptarc}$ must hold in $A$ – from rule $e\\ref {e:ne:ptarc}$ and forced proposition Then $q \\lnot < n_i \\equiv q \\ge n_i$ in $e\\ref {e:ne:ptarc}$ for all $\\lbrace holds(p,q,k), ptarc(p,t_i,n_i) \\rbrace \\subseteq A$ – from $e\\ref {e:ne:ptarc}$ using forced atom proposition, and the following $holds(p,q,k) \\in A$ represents $q=M_k(p)$ – construction, inductive assumption $ptarc(p,t,n_i) \\in A$ represents $n_i=W(p,t)$ – rule $f\\ref {f:tparc}$ construction Then $\\forall p \\in \\bullet t_i, M_k(p) > W(p,t_i)$ – from definition REF of preset $\\bullet t_i$ in PN Then $t_i$ is enabled and can fire in $PN$ , as a result it can belong to $T_k$ – from definition REF of enabled transition And $consumesmore \\notin A$ , since $A$ is an answer set of $\\Pi ^0$ – from rule $a\\ref {a:overc:elim}$ and supported rule proposition Then $\\nexists consumesmore(p,k) \\in A$ – from rule $a\\ref {a:overc:gen}$ and forced atom proposition Then $\\nexists \\lbrace holds(p,q,k), tot\\_decr(p,q1,k) \\rbrace \\subseteq A : q1 > q$ in $body(a\\ref {a:overc:place})$ – from $a\\ref {a:overc:place}$ and forced atom proposition Then $\\nexists p : \\sum _{t_i \\in \\lbrace t_0,\\dots ,t_x \\rbrace , p \\in \\bullet t_i}{W(p,t_i)} > M_k(p)$ – from the following $holds(p,q,k) \\in A$ represents $q=M_k(p)$ – by construction, and the inductive assumption $tot\\_decr(p,q1,k) \\in A$ if $\\lbrace del(p,q1_0,t_0,k), \\dots , del(p,q1_x,t_x,k) \\rbrace \\subseteq A$ , where $q1=q1_0+\\dots +q1_x$ – from $r\\ref {r:totdecr}$ and forced atom proposition $del(p,q1_i,t_i,k) \\in A$ if $\\lbrace fires(t_i,k), ptarc(p,t_i,q1_i) \\rbrace \\subseteq A$ – from $r\\ref {r:del}$ and supported rule proposition $del(p,q1_i,t_i,k)$ represents removal of $q1_i = W(p,t_i)$ tokens from $p \\in \\bullet t_i$ – from construction rule $r\\ref {r:del}$ , supported rule proposition, and definition REF of transition execution in $PN$ Then the set of transitions $T_k$ does conflict – by the definition REF of conflicting transitions Then $\\lbrace t_0,\\dots ,t_x\\rbrace = T_k$ – using 1(a),1(b) above We show $M_{k+1}$ is produced by firing $T_k$ in $M_k$ .", "Let $holds(p,q,k+1) \\in A$ Then $\\lbrace holds(p,q1,k), tot\\_incr(p,q2,k), tot\\_decr(p,q3,k) \\rbrace \\in A : q=q2+q2-q3$ – from rule $r\\ref {r:nextstate}$ and supported rule proposition Then $holds(p,q1,k) \\in A$ represents $q1=M_k(p)$ – inductive assumption and construction; and $\\lbrace add(p,q2_0,t_0,k), \\dots , add(p,q2_j,t_j,k) \\rbrace \\subseteq A : q2_0+\\dots +q2_j=q2$ and $\\lbrace del(p,q3_0,t_0,k), \\dots , del(p,q3_l,t_l,k) \\rbrace \\subseteq A : q3_0+\\dots +q3_l=q3$ – from rules $r\\ref {r:totincr},r\\ref {r:totdecr}$ using supported rule proposition, respectively Then $\\lbrace fires(t_0,k), \\dots , fires(t_j,k) \\rbrace \\subseteq A$ and $\\lbrace fires(t_0,k), \\dots , fires(t_l,k) \\rbrace $ $\\subseteq A$ – from rules $r\\ref {r:add} ,r\\ref {r:del} $ using supported rule proposition, respectively Then $\\lbrace fires(t_0,k), \\dots , fires(t_j,k) \\rbrace \\cup \\lbrace fires(t_0,k), \\dots , fires(t_l,k) \\rbrace $ $= \\lbrace fires(t_0,k), \\dots , fires(t_x,k) \\rbrace \\subseteq A$ – subset union property Then for each $fires(t_x,k) \\in A$ we have $t_x \\in T_k$ - already shown in item REF above Then $q = M_k(p) + \\sum _{t_x \\in T_k \\wedge p \\in t_x \\bullet }{W(t_x,p)} - \\sum _{t_x \\in T_k \\wedge p \\in \\bullet t_x}{W(p,t_x}$ – from (REF ) above and the following Each $add(p,q_j,t_j,0) \\in A$ represents $q_j = W(t_j,p)$ for $p \\in t_j \\bullet $ – encoding of $r\\ref {r:add}$ and definition REF of transition execution in PN Each $del(p,t_y,q_y,0) \\in A$ represents $q_y = W(p,t_y)$ for $p \\in \\bullet t_y$ – encoding of $r\\ref {r:del}$ and definition REF of transition execution in PN Each $tot\\_incr(p,q2,0) \\in A$ represents $q2 = \\sum _{t_x \\in T_k \\wedge p \\in t_x \\bullet }{W(t_x,p)}$ – aggregate assignment atom semantics in rule $r\\ref {r:totincr}$ Each $tot\\_decr(p,q3,0) \\in A$ represents $q3 = \\sum _{t_x \\in T_k \\wedge p \\in \\bullet t_x}{W(p,t_x)}$ – aggregate assignment atom semantics in rule $r\\ref {r:totdecr}$ Then $M_{k+1}(p) = q$ – since $holds(p,q,k+1) \\in A$ encodes $q=M_{k+1}(p)$ by construction As a result, for any $n > k$ , $T_n$ is a valid firing set w.r.t.", "$M_n$ and its firing produces marking $M_{n+1}$ .", "Conclusion: Since both REF and REF hold, $X=M_0,T_0,M_1,\\dots ,M_k,T_k,M_{k+1}$ is an execution sequence of $PN(P,T,E,W)$ (w.r.t.", "$M_0$ ) iff there is an answer set $A$ of $\\Pi ^0(PN,M_0,k,ntok)$ such that (REF ) and (REF ) hold.", "Proof of Proposition REF Let $PN=(P,T,E,W)$ be a Petri Net, $M_0$ be its initial marking and let $\\Pi ^1(PN,M_0,$ $k,ntok)$ be the ASP encoding of $PN$ and $M_0$ over a simulation length $k$ , with maximum $ntok$ tokens on any place node, as defined in section REF .", "Then $X=M_0,T_0,M_1,\\dots ,M_k,T_k,M_{k+1}$ is an execution sequence of $PN$ (w.r.t.", "$M_0$ ) iff there is an answer set $A$ of $\\Pi ^1(PN,M_0,k,ntok)$ such that: $\\lbrace fires(t,ts) : t \\in T_{ts}, 0 \\le ts \\le k\\rbrace = \\lbrace fires(t,ts) : fires(t,ts) \\in A \\rbrace $ $\\begin{split}\\lbrace holds(p,q,ts) &: p \\in P, q = M_{ts}(p), 0 \\le ts \\le k+1 \\rbrace \\\\&= \\lbrace holds(p,q,ts) : holds(p,q,ts) \\in A \\rbrace \\end{split}$ We prove this by showing that: Given an execution sequence $X$ , we create a set $A$ such that it satisfies (REF ) and (REF ) and show that $A$ is an answer set of $\\Pi ^1$ Given an answer set $A$ of $\\Pi ^1$ , we create an execution sequence $X$ such that (REF ) and (REF ) are satisfied.", "First we show (REF ): Given a $PN$ and an execution sequence $X$ of $PN$ , we create a set $A$ as a union of the following sets: $A_1=\\lbrace num(n) : 0 \\le n \\le ntok \\rbrace $ $A_2=\\lbrace time(ts) : 0 \\le ts \\le k\\rbrace $ $A_3=\\lbrace place(p) : p \\in P \\rbrace $ $A_4=\\lbrace trans(t) : t \\in T \\rbrace $ $A_5=\\lbrace ptarc(p,t,n) : (p,t) \\in E^-, n=W(p,t) \\rbrace $ , where $E^- \\subseteq E$ $A_6=\\lbrace tparc(t,p,n) : (t,p) \\in E^+, n=W(t,p) \\rbrace $ , where $E^+ \\subseteq E$ $A_7=\\lbrace holds(p,q,0) : p \\in P, q=M_{0}(p) \\rbrace $ $A_8=\\lbrace notenabled(t,ts) : t \\in T, 0 \\le ts \\le k, \\exists p \\in \\bullet t, M_{ts}(p) < W(p,t) \\rbrace $ per definition REF (enabled transition) $A_9=\\lbrace enabled(t,ts) : t \\in T, 0 \\le ts \\le k, \\forall p \\in \\bullet t, W(p,t) \\le M_{ts}(p) \\rbrace $ per definition REF (enabled transition) $A_{10}=\\lbrace fires(t,ts) : t \\in T_{ts}, 0 \\le ts \\le k \\rbrace $ per definition REF (firing set), only an enabled transition may fire $A_{11}=\\lbrace add(p,q,t,ts) : t \\in T_{ts}, p \\in t \\bullet , q=W(t,p), 0 \\le ts \\le k \\rbrace $ per definition REF (transition execution) $A_{12}=\\lbrace del(p,q,t,ts) : t \\in T_{ts}, p \\in \\bullet t, q=W(p,t), 0 \\le ts \\le k \\rbrace $ per definition REF (transition execution) $A_{13}=\\lbrace tot\\_incr(p,q,ts) : p \\in P, q=\\sum _{t \\in T_{ts}, p \\in t \\bullet }{W(t,p)}, 0 \\le ts \\le k \\rbrace $ per definition REF (firing set execution) $A_{14}=\\lbrace tot\\_decr(p,q,ts): p \\in P, q=\\sum _{t \\in T_{ts}, p \\in \\bullet t}{W(p,t)}, 0 \\le ts \\le k \\rbrace $ per definition REF (firing set execution) $A_{15}=\\lbrace consumesmore(p,ts) : p \\in P, q=M_{ts}(p), q1=\\sum _{t \\in T_{ts}, p \\in \\bullet t}{W(p,t)}, q1 > q, 0 \\le ts \\le k \\rbrace $ per definition REF (conflicting transitions) for enabled transition set $T_{ts}$ $A_{16}=\\lbrace consumesmore : \\exists p \\in P : q=M_{ts}(p), q1=\\sum _{t \\in T_{ts}, p \\in \\bullet t}{W(p,t)}, q1 > q, 0 \\le ts \\le k \\rbrace $ per definition REF (conflicting transitions) $A_{17}=\\lbrace could\\_not\\_have(t,ts) : t \\in T, (\\forall p \\in \\bullet t, W(p,t) \\le M_{ts}(p)), t \\notin T_{ts}, (\\exists p \\in \\bullet t : W(p,t) > M_{ts}(p) - \\sum _{t^{\\prime } \\in T_{ts}, p \\in \\bullet t^{\\prime }}{W(p,t^{\\prime })}), 0 \\le ts \\le k \\rbrace $ per the maximal firing set semantics $A_{18}=\\lbrace holds(p,q,ts+1) : p \\in P, q=M_{ts+1}(p), 0 \\le ts < k\\rbrace $ , where $M_{ts+1}(p) = M_{ts}(p) - \\sum _{\\begin{array}{c}t \\in T_{ts}, p \\in \\bullet t\\end{array}}{W(p,t)} + \\sum _{\\begin{array}{c}t \\in T_{ts}, p \\in t \\bullet \\end{array}}{W(t,p)}$ according to definition REF (firing set execution) We show that $A$ satisfies (REF ) and (REF ), and $A$ is an answer set of $\\Pi ^1$ .", "$A$ satisfies (REF ) and (REF ) by its construction above.", "We show $A$ is an answer set of $\\Pi ^1$ by splitting.", "We split $lit(\\Pi ^1)$ into a sequence of $6k+8$ sets: $U_0= head(f\\ref {f:place}) \\cup head(f\\ref {f:trans}) \\cup head(f\\ref {f:ptarc}) \\cup head(f\\ref {f:tparc}) \\cup head(f\\ref {f:time}) \\cup head(f\\ref {f:num}) $ $\\cup head(i\\ref {i:holds}) = \\lbrace place(p) : p \\in P\\rbrace \\cup \\lbrace trans(t) : t \\in T\\rbrace \\cup \\lbrace ptarc(p,t,n) : (p,t) \\in E^-, n=W(p,t) \\rbrace \\cup \\lbrace tparc(t,p,n) : (t,p) \\in E^+, n=W(t,p) \\rbrace \\cup $ $ \\lbrace time(0), \\dots , time(k)\\rbrace \\cup \\lbrace num(0), \\dots , num(ntok)\\rbrace \\cup \\lbrace holds(p,q,0) : p \\in P, q=M_0(p) \\rbrace $ $U_{6k+1}=U_{6k+0} \\cup head(e\\ref {e:ne:ptarc})^{ts=k} = U_{6k+0} \\cup \\lbrace notenabled(t,k) : t \\in T \\rbrace $ $U_{6k+2}=U_{6k+1} \\cup head(e\\ref {e:enabled})^{ts=k} = U_{6k+1} \\cup \\lbrace enabled(t,k) : t \\in T \\rbrace $ $U_{6k+3}=U_{6k+2} \\cup head(a\\ref {a:fires})^{ts=k} = U_{6k+2} \\cup \\lbrace fires(t,k) : t \\in T \\rbrace $ $U_{6k+4}=U_{6k+3} \\cup head(r\\ref {r:add})^{ts=k} \\cup head(r\\ref {r:del})^{ts=k} = U_{6k+3} \\cup \\lbrace add(p,q,t,k) : p \\in P, t \\in T, q=W(t,p) \\rbrace \\cup \\lbrace del(p,q,t,k) : p \\in P, t \\in T, q=W(p,t) \\rbrace $ $U_{6k+5}=U_{6k+4} \\cup head(r\\ref {r:totincr})^{ts=k} \\cup head(r\\ref {r:totdecr})^{ts=k} = U_{6k+4} \\cup \\lbrace tot\\_incr(p,q,k) : p \\in P, 0 \\le q \\le ntok \\rbrace \\cup \\lbrace tot\\_decr(p,q,k) : p \\in P, 0 \\le q \\le ntok \\rbrace $ $U_{6k+6}=U_{6k+5} \\cup head(r\\ref {r:nextstate})^{ts=k} \\cup head(a\\ref {a:overc:place})^{ts=k} \\cup head(a\\ref {a:maxfire:cnh})^{ts=k} $ $= U_{6k+5} \\\\ \\cup \\lbrace consumesmore(p,k) : p \\in P\\rbrace \\cup \\lbrace holds(p,q,k+1) : p \\in P, 0 \\le q \\le ntok \\rbrace \\cup $ $\\lbrace could\\_not\\_have(t,k) : t \\in T \\rbrace $ $U_{6k+7}=U_{6k+6} \\cup head(a\\ref {a:overc:gen}) = U_{6k+6} \\cup \\lbrace consumesmore \\rbrace $ where $head(r_i)^{ts=k}$ are head atoms of ground rule $r_i$ in which $ts=k$ .", "We write $A_i^{ts=k} = \\lbrace a(\\dots ,ts) : a(\\dots ,ts) \\in A_i, ts=k \\rbrace $ as short hand for all atoms in $A_i$ with $ts=k$ .", "$U_{\\alpha }, 0 \\le \\alpha \\le 6k+7$ form a splitting sequence, since each $U_i$ is a splitting set of $\\Pi ^1$ , and $\\langle U_{\\alpha }\\rangle _{\\alpha < \\mu }$ is a monotone continuous sequence, where $U_0 \\subseteq U_1 \\dots \\subseteq U_{6k+7}$ and $\\bigcup _{\\alpha < \\mu }{U_{\\alpha }} = lit(\\Pi ^1)$ .", "We compute the answer set of $\\Pi ^1$ using the splitting sets as follows: $bot_{U_0}(\\Pi ^1) = f\\ref {f:place} \\cup f\\ref {f:trans} \\cup f\\ref {f:ptarc} \\cup f\\ref {f:tparc} \\cup f\\ref {f:time} \\cup i\\ref {i:holds} \\cup f\\ref {f:num}$ and $X_0 = A_1 \\cup \\dots \\cup A_7$ ($= U_0$ ) is its answer set – using forced atom proposition $eval_{U_0}(bot_{U_1}(\\Pi ^1) \\setminus bot_{U_0}(\\Pi ^1), X_0) = \\lbrace notenabled(t,0) \\text{:-} .", "\| \\lbrace trans(t), \\\\ ptarc(p,t,n), holds(p,q,0) \\rbrace \\subseteq X_0, \\text{~where~} q < n \\rbrace $ .", "Its answer set $X_1=A_8^{ts=0}$ – using forced atom proposition and construction of $A_8$ .", "where, $q=M_0(p)$ , and $n=W(p,t)$ for an arc $(p,t) \\in E^-$ – by construction of $i\\ref {i:holds}$ and $f\\ref {f:ptarc}$ in $\\Pi ^1$ , and in an arc $(p,t) \\in E^-$ , $p \\in \\bullet t$ (by definition REF of preset) thus, $notenabled(t,0) \\in X_1$ represents $\\exists p \\in \\bullet t : M_0(p) < W(p,t)$ .", "$eval_{U_1}(bot_{U_2}(\\Pi ^1) \\setminus bot_{U_1}(\\Pi ^1), X_0 \\cup X_1) = \\lbrace enabled(t,0) \\text{:-}.", "\| trans(t) \\in X_0 \\cup X_1, $ $notenabled(t,0) \\notin X_0 \\cup X_1 \\rbrace $ .", "Its answer set is $X_2 = A_9^{ts=0}$ – using forced atom proposition and construction of $A_9$ .", "since an $enabled(t,0) \\in X_2$ if $\\nexists ~notenabled(t,0) \\in X_0 \\cup X_1$ , which is equivalent to $\\nexists p \\in \\bullet t : M_0(p) < W(p,t) \\equiv \\forall p \\in \\bullet t: M_0(p) \\ge W(p,t)$ .", "$eval_{U_2}(bot_{U_3}(\\Pi ^1) \\setminus bot_{U_2}(\\Pi ^1), X_0 \\cup X_1 \\cup X_2) = \\lbrace \\lbrace fires(t,0)\\rbrace \\text{:-}.", "\| enabled(t,0) \\\\ \\text{~holds in~} X_0 \\cup X_1 \\cup X_2 \\rbrace $ .", "It has multiple answer sets $X_{3.1}, \\dots , X_{3.n}$ , corresponding to elements of power set of $fires(t,0)$ atoms in $eval_{U_2}(...)$ – using supported rule proposition.", "Since we are showing that the union of answer sets of $\\Pi ^1$ determined using splitting is equal to $A$ , we only consider the set that matches the $fires(t,0)$ elements in $A$ and call it $X_3$ , ignoring the rest.", "Thus, $X_3 = A_{10}^{ts=0}$ , representing $T_0$ .", "$eval_{U_3}(bot_{U_4}(\\Pi ^1) \\setminus bot_{U_3}(\\Pi ^1), X_0 \\cup \\dots \\cup X_3) = \\lbrace add(p,n,t,0) \\text{:-}.", "\| \\lbrace fires(t,0), \\\\ tparc(t,p,n) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_3 \\rbrace \\cup \\lbrace del(p,n,t,0) \\text{:-}.", "\| \\lbrace fires(t,0), ptarc(p,t,n) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_3 \\rbrace $ .", "It's answer set is $X_4 = A_{11}^{ts=0} \\cup A_{12}^{ts=0}$ – using forced atom proposition and definitions of $A_{11}$ and $A_{12}$ .", "where, each $add$ atom encodes $n=W(t,p) : p \\in t \\bullet $ , and each $del$ atom encodes $n=W(p,t) : p \\in \\bullet t$ representing the effect of transitions in $T_0$ – by construction $eval_{U_4}(bot_{U_5}(\\Pi ^1) \\setminus bot_{U_4}(\\Pi ^1), X_0 \\cup \\dots \\cup X_4) = \\lbrace tot\\_incr(p,qq,0) \\text{:-}.", "\| $ $qq=\\sum _{add(p,q,t,0) \\in X_0 \\cup \\dots \\cup X_4}{q} \\rbrace \\cup \\lbrace tot\\_decr(p,qq,0) \\text{:-}.", "\| qq=\\sum _{del(p,q,t,0) \\in X_0 \\cup \\dots \\cup X_4}{q} \\rbrace $ .", "It's answer set is $X_5 = A_{13}^{ts=0} \\cup A_{14}^{ts=0}$ – using forced atom proposition and definitions of $A_{13}$ , $A_{14}$ , and semantics of aggregate assignment atom where, each $tot\\_incr(p,qq,0)$ , $qq=\\sum _{add(p,q,t,0) \\in X_0 \\cup \\dots X_4}{q}$ $\\equiv qq=\\sum _{t \\in X_3, p \\in t \\bullet }{W(p,t)}$ , and, each $tot\\_decr(p,qq,0)$ , $qq=\\sum _{del(p,q,t,0) \\in X_0 \\cup \\dots X_4}{q}$ $\\equiv qq=\\sum _{t \\in X_3, p \\in \\bullet t}{W(t,p)}$ , represent the net effect of actions in $T_0$ – by construction $eval_{U_5}(bot_{U_6}(\\Pi ^1) \\setminus bot_{U_5}(\\Pi ^1), X_0 \\cup \\dots \\cup X_5) = \\lbrace consumesmore(p,0) \\text{:-}.", "\| \\\\ \\lbrace holds(p,q,0), tot\\_decr(p,q1,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_5 : q1 > q \\rbrace \\cup \\lbrace holds(p,q,1) \\text{:-}., \| \\\\ \\lbrace holds(p,q1,0), tot\\_incr(p,q2,0), tot\\_decr(p,q3,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_5, q=q1+q2-q3 \\rbrace \\cup \\lbrace could\\_not\\_have(t,0) \\text{:-}.", "\| \\lbrace enabled(t,0), ptarc(s,t,q), holds(s,qq,0), \\\\ tot\\_decr(s,qqq,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_5, fires(t,0) \\notin (X_0 \\cup \\dots \\cup X_5), q > qq-qqq \\rbrace $ .", "It's answer set is $X_6 = A_{15}^{ts=0} \\cup A_{17}^{ts=0} \\cup A_{18}^{ts=0}$ – using forced atom proposition and definitions of $A_{15}, A_{17}, A_{18}$ .", "where, $consumesmore(p,0)$ represents $\\exists p : q=M_0(p), q1=\\sum _{t \\in T_0, p \\in \\bullet t}{W(p,t)}, $ $q1 > q$ indicating place $p$ will be overconsumed if $T_0$ is fired, as defined in definition REF (conflicting transitions) and, $holds(p,q,1)$ encodes $q=M_1(p)$ – by construction and $could\\_not\\_have(t,0)$ represents an enabled transition $t$ in $T_0$ that could not fire due to insufficient tokens $X_6$ does not contain $could\\_not\\_have(t,0)$ , when $enabled(t,0) \\in X_0 \\cup \\dots \\cup X_5$ and $fires(t,0) \\notin X_0 \\cup \\dots \\cup X_5$ due to construction of $A$ , encoding of $a\\ref {a:maxfire:cnh}$ and its body atoms.", "As a result it is not eliminated by the constraint $a\\ref {a:maxfire:elim}$ $ \\vdots $ $eval_{U_{6k+0}}(bot_{U_{6k+1}}(\\Pi ^1) \\setminus bot_{U_{6k+0}}(\\Pi ^1), X_0 \\cup \\dots \\cup X_{6k+0}) = \\\\ \\lbrace notenabled(t,k) \\text{:-} .", "\| \\lbrace trans(t), ptarc(p,t,n), holds(p,q,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{6k+0}, \\\\ \\text{~where~} q < n \\rbrace $ .", "Its answer set $X_{6k+1}=A_8^{ts=k}$ – using forced atom proposition and construction of $A_8$ .", "where, $q=M_k(p)$ , and $n=W(p,t)$ for an arc $(p,t) \\in E^-$ – by construction of $holds$ and $ptarc$ predicates in $\\Pi ^1$ , and in an arc $(p,t) \\in E^-$ , $p \\in \\bullet t$ (by definition REF of preset) thus, $notenabled(t,k) \\in X_{6k+1}$ represents $\\exists p \\in \\bullet t : M_k(p) < W(p,t)$ .", "$eval_{U_{6k+1}}(bot_{U_{6k+2}}(\\Pi ^1) \\setminus bot_{U_{6k+1}}(\\Pi ^1), X_0 \\cup \\dots \\cup X_{6k+1}) = \\lbrace enabled(t,k) \\text{:-}.", "\| \\\\ trans(t) \\in X_0 \\cup \\dots \\cup X_{6k+1}, notenabled(t,k) \\notin X_0 \\cup \\dots \\cup X_{6k+1} \\rbrace $ .", "Its answer set is $X_{6k+2} = A_9^{ts=k}$ – using forced atom proposition and construction of $A_9$ .", "since an $enabled(t,k) \\in X_{6k+2}$ if $\\nexists ~notenabled(t,k) \\in X_0 \\cup \\dots \\cup X_{6k+1}$ , which is equivalent to $\\nexists p \\in \\bullet t : M_k(p) < W(p,t) \\equiv \\forall p \\in \\bullet t: M_k(p) \\ge W(p,t)$ .", "$eval_{U_{6k+2}}(bot_{U_{6k+3}}(\\Pi ^1) \\setminus bot_{U_{6k+2}}(\\Pi ^1), X_0 \\cup \\dots \\cup X_{6k+2}) = \\\\ \\lbrace \\lbrace fires(t,k)\\rbrace \\text{:-}.", "\| enabled(t,k) \\text{~holds in~} X_0 \\cup \\dots \\cup X_{6k+2} \\rbrace $ .", "It has multiple answer sets $X_{{6k+3}.1}, \\dots , X_{{6k+3}.n}$ , corresponding to elements of power set of $fires(t,k)$ atoms in $eval_{U_{6k+2}}(...)$ – using supported rule proposition.", "Since we are showing that the union of answer sets of $\\Pi ^1$ determined using splitting is equal to $A$ , we only consider the set that matches the $fires(t,k)$ elements in $A$ and call it $X_{6k+3}$ , ignoring the reset.", "Thus, $X_{6k+3} = A_{10}^{ts=k}$ , representing $T_k$ .", "$eval_{U_{6k+3}}(bot_{U_{6k+4}}(\\Pi ^1) \\setminus bot_{U_{6k+3}}(\\Pi ^1), X_0 \\cup \\dots \\cup X_{6k+3}) = \\\\ \\lbrace add(p,n,t,k) \\text{:-}.", "\| \\lbrace fires(t,k), tparc(t,p,n) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{6k+3} \\rbrace \\cup \\\\ \\lbrace del(p,n,t,k) \\text{:-}.", "\| \\lbrace fires(t,k), ptarc(p,t,n) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{6k+3} \\rbrace $ .", "It's answer set is $X_{6k+4} = A_{11}^{ts=k} \\cup A_{12}^{ts=k}$ – using forced atom proposition and definitions of $A_{11}$ and $A_{12}$ .", "where, each $add$ atom is equivalent to $n=W(t,p) : p \\in t \\bullet $ , and, each $del$ atom is equivalent to $n=W(p,t) : p \\in \\bullet t$ , representing the effect of transitions in $T_k$ $eval_{U_{6k+4}}(bot_{U_{6k+5}}(\\Pi ^1) \\setminus bot_{U_{6k+4}}(\\Pi ^1), X_0 \\cup \\dots \\cup X_{6k+4}) = \\lbrace tot\\_incr(p,qq,k) \\text{:-}.", "\| \\\\ qq=\\sum _{add(p,q,t,k) \\in X_0 \\cup \\dots \\cup X_{6k+4}}{q} \\rbrace \\cup \\lbrace tot\\_decr(p,qq,k) \\text{:-}.", "\| \\\\ qq=\\sum _{del(p,q,t,k) \\in X_0 \\cup \\dots \\cup X_{6k+4}}{q} \\rbrace $ .", "It's answer set is $X_{6k+5} = A_{13}^{ts=k} \\cup A_{14}^{ts=k}$ – using forced atom proposition and definitions of $A_{13}$ and $A_{14}$ .", "where, each $tot\\_incr(p,qq,k)$ , $qq=\\sum _{add(p,q,t,k) \\in X_0 \\cup \\dots X_{6k+4}}{q}$ $\\equiv \\\\ qq=\\sum _{t \\in X_{6k+3}, p \\in t \\bullet }{W(p,t)}$ , and, each $tot\\_decr(p,qq,k)$ , $qq=\\sum _{del(p,q,t,k) \\in X_0 \\cup \\dots X_{6k+4}}{q}$ $\\equiv \\\\ qq=\\sum _{t \\in X_{6k+3}, p \\in \\bullet t}{W(t,p)}$ , represent the net effect of transitions in $T_k$ $eval_{U_{6k+5}}(bot_{U_{6k+6}}(\\Pi ^1) \\setminus bot_{U_{6k+5}}(\\Pi ^1), X_0 \\cup \\dots \\cup X_{6k+5}) = \\\\ \\lbrace consumesmore(p,k) \\text{:-}.", "\| \\lbrace holds(p,q,k), tot\\_decr(p,q1,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{6k+5} : q1 > q \\rbrace \\cup \\lbrace holds(p,q,k+1) \\text{:-}., \| \\lbrace holds(p,q1,k), tot\\_incr(p,q2,k), \\\\ tot\\_decr(p,q3,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{6k+5} : q=q1+q2-q3 \\rbrace \\cup \\lbrace could\\_not\\_have(t,k) \\text{:-} \\\\ \\lbrace enabled(t,k), ptarc(s,t,q), holds(s,qq,k), tot\\_decr(s,qqq,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{6k+5}, $ $fires(t,k) \\notin (X_0 \\cup \\dots \\cup X_{6k+5}), q > qq-qqq \\rbrace $ .", "It's answer set is $X_{6k+6} = A_{15}^{ts=k} \\cup A_{17}^{ts=k} \\cup A_{18}^{ts=k}$ – using forced atom proposition.", "where, $consumesmore(p,k)$ represents $\\exists p : q=M_{k}(p), \\\\ q1=\\sum _{t \\in T_{k}, p \\in \\bullet t}{W(p,t)}, q1 > q$ $holds(p,q,k+1)$ represents $q=M_{k+1}(p)$ indicating place $p$ that will be over consumed if $T_k$ is fired, as defined in definition REF (conflicting transitions), $holds(p,q,k+1)$ represents $q=M_{k+1}(p)$ – by construction and $could\\_not\\_have(t,k)$ represents an enabled transition $t$ in $T_k$ that could not fire due to insufficient tokens $X_{6k+6}$ does not contain $could\\_not\\_have(t,k)$ , when $enabled(t,k) \\in X_0 \\cup \\dots \\cup X_{6k+5}$ and $fires(t,k) \\notin X_0 \\cup \\dots \\cup X_{6k+5}$ due to construction of $A$ , encoding of $a\\ref {a:maxfire:cnh}$ and its body atoms.", "As a result it is note eliminated by the constraint $a\\ref {a:maxfire:elim}$ $eval_{U_{6k+6}}(bot_{U_{6k+7}}(\\Pi ^1) \\setminus bot_{U_{6k+6}}(\\Pi ^1), X_0 \\cup \\dots \\cup X_{6k+6}) = \\lbrace consumesmore \\text{:-}.", "\| $ $\\lbrace consumesmore(p,0),$ $\\dots , $ $consumesmore(p,k) \\rbrace \\cap $ $(X_0 \\cup \\dots \\cup X_{6k+6}) \\ne \\emptyset \\rbrace $ .", "It's answer set is $X_{6k+7} = A_{16}$ – using forced atom proposition $X_{6k+7}$ will be empty since none of $consumesmore(p,0),\\dots , \\\\ consumesmore(p,k)$ hold in $X_0 \\cup \\dots \\cup X_{6k+6}$ due to the construction of $A$ , encoding of $a\\ref {a:overc:place}$ and its body atoms.", "As a result, it is not eliminated by the constraint $a\\ref {a:overc:elim}$ The set $X = X_0 \\cup \\dots \\cup X_{6k+7}$ is the answer set of $\\Pi ^0$ by the splitting sequence theorem REF .", "Each $X_i, 0 \\le i \\le 6k+7$ matches a distinct portion of $A$ , and $X = A$ , thus $A$ is an answer set of $\\Pi ^1$ .", "Next we show (REF ): Given $\\Pi ^1$ be the encoding of a Petri Net $PN(P,T,E,W)$ with initial marking $M_0$ , and $A$ be an answer set of $\\Pi ^1$ that satisfies (REF ) and (REF ), then we can construct $X=M_0,T_0,\\dots ,M_k,T_k,M_{k+1}$ from $A$ , such that it is an execution sequence of $PN$ .", "We construct the $X$ as follows: $M_i = (M_i(p_0), \\dots , M_i(p_n))$ , where $\\lbrace holds(p_0,M_i(p_0),i), \\dots holds(p_n,M_i(p_n),i) \\rbrace \\\\ \\subseteq A$ , for $0 \\le i \\le k+1$ $T_i = \\lbrace t : fires(t,i) \\in A\\rbrace $ , for $0 \\le i \\le k$ and show that $X$ is indeed an execution sequence of $PN$ .", "We show this by induction over $k$ (i.e.", "given $M_k$ , $T_k$ is a valid firing set and its firing produces marking $M_{k+1}$ ).", "Base case: Let $k=0$ , and $M_0$ is a valid marking in $X$ for $PN$ , show [(1)] $T_0$ is a valid firing set for $M_0$ , and $T_0$ 's firing w.r.t.", "marking $M_0$ produces $M_1$ .", "We show $T_0$ is a valid firing set for $M_0$ .", "Let $\\lbrace fires(t_0,0), \\dots , fires(t_x,0) \\rbrace $ be the set of all $fires(\\dots ,0)$ atoms in $A$ , Then for each $fires(t_i,0) \\in A$ $enabled(t_i,0) \\in A$ – from rule $a\\ref {a:fires}$ and supported rule proposition Then $notenabled(t_i,0) \\notin A$ – from rule $e\\ref {e:enabled}$ and supported rule proposition Then $body(e\\ref {e:ne:ptarc})$ must not hold in $A$ – from rule $e\\ref {e:ne:ptarc}$ and forced atom proposition Then $q \\lnot < n_i \\equiv q \\ge n_i$ in $e\\ref {e:ne:ptarc}$ for all $\\lbrace holds(p,q,0), ptarc(p,t_i,n_i)\\rbrace \\subseteq A$ – from $e\\ref {e:ne:ptarc}$ , forced atom proposition, and the following $holds(p,q,0) \\in A$ represents $q=M_0(p)$ – rule $i\\ref {i:holds}$ construction $ptarc(p,t_i,n_i) \\in A$ represents $n_i=W(p,t_i)$ – rule $f\\ref {f:ptarc} $ construction Then $\\forall p \\in \\bullet t_i, M_0(p) > W(p,t_i)$ – from definition REF of preset $\\bullet t_i$ in PN Then $t_i$ is enabled and can fire in $PN$ , as a result it can belong to $T_0$ – from definition REF of enabled transition And $consumesmore \\notin A$ , since $A$ is an answer set of $\\Pi ^1$ – from rule $a\\ref {a:overc:elim}$ and supported rule proposition Then $\\nexists consumesmore(p,0) \\in A$ – from rule $a\\ref {a:overc:gen}$ and supported rule proposition Then $\\nexists \\lbrace holds(p,q,0), tot\\_decr(p,q1,0) \\rbrace \\subseteq A : q1>q$ in $body(a\\ref {a:overc:place})$ – from $a\\ref {a:overc:place}$ and forced atom proposition Then $\\nexists p : \\sum _{t_i \\in \\lbrace t_0,\\dots ,t_x\\rbrace , p \\in \\bullet t_i}{W(p,t_i)} > M_0(p)$ – from the following $holds(p,q,0)$ represents $q=M_0(p)$ – from rule $i\\ref {i:holds}$ construction, given $tot\\_decr(p,q1,0) \\in A$ if $\\lbrace del(p,q1_0,t_0,0), \\dots , del(p,q1_x,t_x,0) \\rbrace \\subseteq A$ , where $q1 = q1_0+\\dots +q1_x$ – from $r\\ref {r:totdecr}$ and forced atom proposition $del(p,q1_i,t_i,0) \\in A$ if $\\lbrace fires(t_i,0), ptarc(p,t_i,q1_i) \\rbrace \\subseteq A$ – from $r\\ref {r:del}$ and supported rule proposition $del(p,q1_i,t_i,0)$ represents removal of $q1_i = W(p,t_i)$ tokens from $p \\in \\bullet t_i$ – from rule $r\\ref {r:del}$ , supported rule proposition, and definition REF of transition execution in $PN$ Then the set of transitions in $T_0$ do not conflict – by the definition REF of conflicting transitions And for each $enabled(t_j,0) \\in A$ and $fires(t_j,0) \\notin A$ , $could\\_not\\_have(t_j,0) \\in A$ , since $A$ is an answer set of $\\Pi ^1$ - from rule $a\\ref {a:maxfire:elim}$ and supported rule proposition Then $\\lbrace enabled(t_j,0), holds(s,qq,0), ptarc(s,t_j,q,0), \\\\ tot\\_decr(s,qqq,0) \\rbrace \\subseteq A$ , such that $q > qq - qqq$ and $fires(t_j,0) \\notin A$ - from rule $a\\ref {a:maxfire:cnh}$ and supported rule proposition Then for an $s \\in \\bullet t_j$ , $W(s,t_j) > M_0(s) - \\sum _{t_i \\in T_0, s \\in \\bullet t_i}{W(s,t_i)}$ - from the following: $ptarc(s,t_i,q)$ represents $q=W(s,t_i)$ – from rule $f\\ref {f:r:ptarc}$ construction $holds(s,qq,0)$ represents $qq=M_0(s)$ – from $i\\ref {i:holds}$ construction $tot\\_decr(s,qqq,0) \\in A$ if $\\lbrace del(s,qqq_0,t_0,0), \\dots , del(s,qqq_x,t_x,0) \\rbrace \\subseteq A$ – from rule $r\\ref {r:totdecr}$ construction and supported rule proposition $del(s,qqq_i,t_i,0) \\in A$ if $\\lbrace fires(t_i,0), ptarc(s,t_i,qqq_i) \\rbrace \\subseteq A$ – from rule $r\\ref {r:r:del} $ and supported rule proposition $del(s,qqq_i,t_i,0)$ represents $qqq_i = W(s,t_i) : t_i \\in T_0, (s,t_i) \\in E^-$ – from rule $f\\ref {f:r:ptarc}$ construction $tot\\_decr(q,qqq,0)$ represents $\\sum _{t_i \\in T_0, s \\in \\bullet t_i}{W(s,t_i)}$ – from (C,D,E) above Then firing $T_0 \\cup \\lbrace t_j \\rbrace $ would have required more tokens than are present at its source place $s \\in \\bullet t_j$ .", "Thus, $T_0$ is a maximal set of transitions that can simultaneously fire.", "Then $\\lbrace t_0, \\dots , t_x\\rbrace = T_0$ – using 1(a),1(b) above; and using 1(c) it is a maximal firing set We show $M_1$ is produced by firing $T_0$ in $M_0$ .", "Let $holds(p,q,1) \\in A$ Then $\\lbrace holds(p,q1,0), tot\\_incr(p,q2,0), tot\\_decr(p,q3,0) \\rbrace \\subseteq A : q=q1+q2-q3$ – from rule $r\\ref {r:nextstate}$ and supported rule proposition Then, $holds(p,q1,0) \\in A$ represents $q1=M_0(p)$ – given, rule $i\\ref {i:holds}$ construction; and $\\lbrace add(p,q2_0,t_0,0), \\dots , $ $add(p,q2_j,t_j,0)\\rbrace \\subseteq A : q2_0 + \\dots + q2_j = q2$ and $\\lbrace del(p,q3_0,t_0,0), \\dots , $ $del(p,q3_l,t_l,0)\\rbrace \\subseteq A : q3_0 + \\dots + q3_l = q3$ – rules $r\\ref {r:totincr},r\\ref {r:totdecr}$ using supported rule proposition Then $\\lbrace fires(t_0,0), \\dots , fires(t_j,0) \\rbrace \\subseteq A$ and $\\lbrace fires(t_0,0), \\dots , fires(t_l,0) \\rbrace \\\\ \\subseteq A$ – rules $r\\ref {r:add},r\\ref {r:del}$ and supported rule proposition, respectively Then $\\lbrace fires(t_0,0), \\dots , $ $fires(t_j,0) \\rbrace \\cup \\lbrace fires(t_0,0), \\dots , $ $fires(t_l,0) \\rbrace \\subseteq A = \\lbrace fires(t_0,0), \\dots , $ $fires(t_x,0) \\rbrace \\subseteq A$ – set union of subsets Then for each $fires(t_x,0) \\in A$ we have $t_x \\in T_0$ – already shown in item REF above Then $q = M_0(p) + \\sum _{t_x \\in T_0 \\wedge p \\in t_x \\bullet }{W(t_x,p)} - \\sum _{t_x \\in T_0 \\wedge p \\in \\bullet t_x}{W(p,t_x)}$ – from (REF ) above and the following Each $add(p,q_j,t_j,0) \\in A$ represents $q_j=W(t_j,p)$ for $p \\in t_j \\bullet $ – rule $r\\ref {r:add}$ encoding, and definition REF of transition execution in $PN$ Each $del(p,t_y,q_y,0) \\in A$ represents $q_y=W(p,t_y)$ for $p \\in \\bullet t_y$ – from rule $r\\ref {r:del}$ encoding, and definition REF of transition execution in $PN$ Each $tot\\_incr(p,q2,0) \\in A$ represents $q2=\\sum _{t_x \\in T_0 \\wedge p \\in t_x \\bullet }{W(t_x,p)}$ – aggregate assignment atom semantics in rule $r\\ref {r:totincr}$ Each $tot\\_decr(p,q3,0) \\in A$ represents $q3=\\sum _{t_x \\in T_0 \\wedge p \\in \\bullet t_x}{W(p,t_x)}$ – aggregate assignment atom semantics in rule $r\\ref {r:totdecr}$ Then, $M_1(p) = q$ – since $holds(p,q,1) \\in A$ encodes $q=M_1(p)$ from construction Inductive Step: Let $k > 0$ , and $M_k$ is a valid marking in $X$ for $PN$ , show [(1)] $T_k$ is a valid firing set for $M_k$ , and $T_k$ 's firing in $M_k$ produces marking $M_{k+1}$ .", "We show $T_k$ is a valid firing set.", "Let $\\lbrace fires(t_0,k),\\dots ,fires(t_x,k) \\rbrace $ be the set of all $fires(\\dots ,k)$ atoms in $A$ , Then for each $fires(t_i,k) \\in A$ $enabled(t_i,k) \\in A$ – from rule $a\\ref {a:fires}$ and supported rule proposition Then $notenabled(t_i,k) \\notin A$ – from rule $e\\ref {e:enabled}$ and supported rule proposition Then $body(e\\ref {e:ne:ptarc})$ must hold in $A$ – from rule $e\\ref {e:ne:ptarc}$ and forced atom proposition Then $q \\lnot < n_i \\equiv q \\ge n_i$ in $e\\ref {e:ne:ptarc}$ for all $\\lbrace holds(p,q,k), ptarc(p,t_i,n) \\rbrace \\subseteq A$ – from $e\\ref {e:ne:ptarc}$ , forced atom proposition, and the following $holds(p,q,k) \\in A$ represents $q=M_k(p)$ – construction, inductive assumption $ptarc(p,t,n_i) \\in A$ represents $n_i=W(p,t_i)$ – rule $f\\ref {f:ptarc}$ construction Then $\\forall p \\in \\bullet t_i$ , $M_k(p) \\ge W(p,t_i)$ – from definition REF of preset $\\bullet t_i$ in $PN$ Then $t_i$ is enabled and can fire in $PN$ , as a result it can belong to $T_k$ – from definition REF of enabled transition And $consumesmore \\notin A$ , since $A$ is an answer set of $\\Pi ^1$ – from rule $a\\ref {a:overc:elim}$ and supported rule proposition Then $\\nexists consumesmore(p,k) \\in A$ – from rule $a\\ref {a:overc:gen}$ and forced atom proposition Then $\\nexists \\lbrace holds(p,q,k), tot\\_decr(p,q1,k) \\rbrace \\subseteq A : q1 > q$ in $body(a\\ref {a:overc:place})$ – from $a\\ref {a:overc:place}$ and forced atom proposition Then $\\nexists p : \\sum _{t_i \\in \\lbrace t_0,\\dots ,t_x \\rbrace , p \\in \\bullet t_i}{W(p,t_i)} > M_k(p)$ – from the following $holds(p,q,k) \\in A$ represents $q=M_k(p)$ – by construction of $\\Pi ^1$ , and the inductive assumption about $M_k(p)$ $tot\\_decr(p,q1,k) \\in A$ if $\\lbrace del(p,q1_0,t_0,k), \\dots , del(p,q1_x,t_x,k) \\rbrace \\subseteq A$ , where $q1=q1_0+\\dots +q1_x$ – from $r\\ref {r:totdecr}$ and forced atom proposition $del(p,q1_i,t_i,k) \\in A$ if $\\lbrace fires(t_i,k), ptarc(p,t_i,q1_i) \\rbrace \\subseteq A$ – from $r\\ref {r:del}$ and supported rule proposition $del(p,q1_i,t_i,k)$ represents removal of $q1_i = W(p,t_i)$ tokens from $p \\in \\bullet t_i$ – from construction rule $r\\ref {r:del}$ , supported rule proposition, and definition REF of transition execution in $PN$ Then the set of transitions $T_k$ do not conf – by the definition REF of conflicting transitions And for each $enabled(t_j,k) \\in A$ and $fires(t_j,k) \\notin A$ , $could\\_not\\_have(t_j,k) \\in A$ , since $A$ is an answer set of $\\Pi ^1$ - from rule $a\\ref {a:maxfire:elim}$ and supported rule proposition Then $\\lbrace enabled(t_j,k), holds(s,qq,k), ptarc(s,t_j,q,k), \\\\ tot\\_decr(s,qqq,k) \\rbrace \\subseteq A$ , such that $q > qq - qqq$ and $fires(t_j,0) \\notin A$ - from rule $a\\ref {a:maxfire:cnh}$ and supported rule proposition Then for an $s \\in \\bullet t_j$ , $W(s,t_j) > M_k(s) - \\sum _{t_i \\in T_k, s \\in \\bullet t_i}{W(s,t_i)}$ - from the following: $ptarc(s,t_i,q)$ represents $q=W(s,t_i)$ – from rule $f\\ref {f:r:ptarc}$ construction $holds(s,qq,k)$ represents $qq=M_k(s)$ – from $i\\ref {i:holds}$ construction $tot\\_decr(s,qqq,k) \\in A$ if $\\lbrace del(s,qqq_0,t_0,k), \\dots , del(s,qqq_x,t_x,k) \\rbrace \\subseteq A$ – from rule $r\\ref {r:totdecr}$ construction and supported rule proposition $del(s,qqq_i,t_i,k) \\in A$ if $\\lbrace fires(t_i,k), ptarc(s,t_i,qqq_i) \\rbrace \\subseteq A$ – from rule $r\\ref {r:r:del} $ and supported rule proposition $del(s,qqq_i,t_i,k)$ represents $qqq_i = W(s,t_i) : t_i \\in T_k, (s,t_i) \\in E^-$ – from rule $f\\ref {f:r:ptarc}$ construction $tot\\_decr(q,qqq,k)$ represents $\\sum _{t_i \\in T_k, s \\in \\bullet t_i}{W(s,t_i)}$ – from (C,D,E) above Then firing $T_k \\cup \\lbrace t_j \\rbrace $ would have required more tokens than are present at its source place $s \\in \\bullet t_j$ .", "Thus, $T_k$ is a maximal set of transitions that can simultaneously fire.", "Then $\\lbrace t_0,\\dots ,t_x \\rbrace = T_k$ – using 1(a),1(b) above; and using 1(c) it is a maximal firing set We show $M_{k+1}$ is produced by firing $T_k$ in $M_k$ .", "Let $holds(p,q,k+1) \\in A$ Then $\\lbrace holds(p,q1,k), tot\\_incr(p,q2,k), tot\\_decr(p,q3,k) \\rbrace \\in A : q=q2+q2-q3$ – from rule $r\\ref {r:nextstate}$ and supported rule proposition $holds(p,q1,k) \\in A$ represents $q1=M_k(p)$ – construction, inductive assumption; and $\\lbrace add(p,q2_0,t_0,k), \\dots , $ $add(p,q2_j,t_j,k) \\rbrace \\subseteq A : q2_0+\\dots +q2_j=q2$ and $\\lbrace del(p,q3_0,t_0,k), \\dots , $ $del(p,q3_l,t_l,k) \\rbrace \\subseteq A : q3_0+\\dots +q3_l=q3$ – from rules $r\\ref {r:totincr},r\\ref {r:totdecr}$ using supported rule proposition, respectively Then $\\lbrace fires(t_0,k), \\dots , fires(t_j,k) \\rbrace \\subseteq A$ and $\\lbrace fires(t_0,k), \\dots , fires(t_l,k) \\rbrace \\\\ \\subseteq A$ – from rules $r\\ref {r:add} ,r\\ref {r:del} $ using supported rule proposition, respectively Then $\\lbrace fires(t_0,k), \\dots , $ $fires(t_j,k) \\rbrace \\cup \\lbrace fires(t_0,k), \\dots , $ $fires(t_l,k) \\rbrace = \\lbrace fires(t_0,k), \\dots , $ $fires(t_x,k) \\rbrace \\subseteq A$ – subset union property Then for each $fires(t_x,k) \\in A$ we have $t_x \\in T_k$ - already shown in item (REF ) above Then $q = M_k(p) + \\sum _{t_x \\in T_k \\wedge p \\in t_x \\bullet }{W(t_x,p)} - \\sum _{t_x \\in T_k \\wedge p \\in \\bullet t_x}{W(p,t_x)}$ – from (REF ) above and the following Each $add(p,q_j,t_j,k) \\in A$ represents $q_j = W(t_j,p)$ for $p \\in t_j \\bullet $ – rule $r\\ref {r:add}$ encoding and definition REF of transition execution in PN Each $del(p,t_y,q_y,k) \\in A$ represents $q_y = W(p,t_y)$ for $p \\in \\bullet t_y$ – rule $r\\ref {r:del}$ encoding and definition REF of transition execution in PN Each $tot\\_incr(p,q2,k) \\in A$ represents $q2 = \\sum _{t_x \\in T_k \\wedge p \\in t_x \\bullet }{W(t_x,p)}$ – aggregate assignment atom semantics in rule $r\\ref {r:totincr}$ Each $tot\\_decr(p,q3,k) \\in A$ represents $q3 = \\sum _{t_x \\in T_k \\wedge p \\in \\bullet t_x}{W(p,t_x)}$ – aggregate assignment atom semantics in rule $r\\ref {r:totdecr}$ Then $M_{k+1}(p) = q$ – since $holds(p,q,k+1) \\in A$ encodes $q=M_{k+1}(p)$ – from construction As a result, for any $n > k$ , $T_n$ will be a valid firing set w.r.t.", "$M_n$ and its firing produces marking $M_{n+1}$ .", "Conclusion: Since both REF and REF hold, $X=M_0,T_0,M_1,\\dots ,M_k,T_k,M_{k+1}$ is an execution sequence of $PN(P,T,E,W)$ (w.r.t.", "$M_0$ ) iff there is an answer set $A$ of $\\Pi ^1(PN,M_0,k,ntok)$ such that (REF ) and (REF ) hold.", "Proof of Proposition REF Let $PN=(P,T,E,W,R)$ be a Petri Net, $M_0$ be its initial marking and let $\\Pi ^2(PN,M_0,k,ntok)$ by the ASP encoding of $PN$ and $M_0$ over a simulation length $k$ , with maximum $ntok$ tokens on any place node, as defined in section REF .", "Then $X=M_0,T_0,M_1,\\dots ,M_k,T_k,M_{k+1}$ is an execution sequence of $PN$ (w.r.t $M_0$ ) iff there is an answer set $A$ of $\\Pi ^2(PN,M_0,k,ntok)$ such that: $\\lbrace fires(t,ts) : t \\in T_{ts}, 0 \\le ts \\le k\\rbrace = \\lbrace fires(t,ts) : fires(t,ts) \\in A \\rbrace $ $\\begin{split}\\lbrace holds(p,q,ts) &: p \\in P, q = M_{ts}(p), 0 \\le ts \\le k+1 \\rbrace \\\\&= \\lbrace holds(p,q,ts) : holds(p,q,ts) \\in A \\rbrace \\end{split}$ We prove this by showing that: Given an execution sequence $X$ , we create a set $A$ such that it satisfies (REF ) and (REF ) and show that $A$ is an answer set of $\\Pi ^2$ Given an answer set $A$ of $\\Pi ^2$ , we create an execution sequence $X$ such that (REF ) and (REF ) are satisfied.", "First we show (REF ): Given $PN$ and an execution sequence $X$ of $PN$ , we create a set $A$ as a union of the following sets: $A_1=\\lbrace num(n) : 0 \\le n \\le ntok \\rbrace $ $A_2=\\lbrace time(ts) : 0 \\le ts \\le k\\rbrace $ $A_3=\\lbrace place(p) : p \\in P \\rbrace $ $A_4=\\lbrace trans(t) : t \\in T \\rbrace $ $A_5=\\lbrace ptarc(p,t,n,ts) : (p,t) \\in E^-, n=W(p,t), 0 \\le ts \\le k\\rbrace $ , where $E^- \\subseteq E$ $A_6=\\lbrace tparc(t,p,n,ts) : (t,p) \\in E^+, n=W(t,p), 0 \\le ts \\le k\\rbrace $ , where $E^+ \\subseteq E$ $A_7=\\lbrace holds(p,q,0) : p \\in P, q=M_{0}(p) \\rbrace $ $A_8=\\lbrace notenabled(t,ts) : t \\in T, 0 \\le ts \\le k, \\exists p \\in \\bullet t, M_{ts}(p) < W(p,t) \\rbrace $ per definition REF (enabled transition) $A_9=\\lbrace enabled(t,ts) : t \\in T, 0 \\le ts \\le k, \\forall p \\in \\bullet t, W(p,t) \\le M_{ts}(p) \\rbrace $ per definition REF (enabled transition) $A_{10}=\\lbrace fires(t,ts) : t \\in T_{ts}, 0 \\le ts \\le k \\rbrace $ per definition REF (firing set), only an enabled transition may fire $A_{11}=\\lbrace add(p,q,t,ts) : t \\in T_{ts}, p \\in t \\bullet , q=W(t,p), 0 \\le ts \\le k \\rbrace $ per definition REF (transition execution) $A_{12}=\\lbrace del(p,q,t,ts) : t \\in T_{ts}, p \\in \\bullet t, q=W(p,t), 0 \\le ts \\le k \\rbrace \\cup \\lbrace del(p,q,t,ts) : t \\in T_{ts}, p \\in R(t), q=M_{ts}(p), 0 \\le ts \\le k \\rbrace $ per definition REF (transition execution) $A_{13}=\\lbrace tot\\_incr(p,q,ts) : p \\in P, q=\\sum _{t \\in T_{ts}, p \\in t \\bullet }{W(t,p)}, 0 \\le ts \\le k \\rbrace $ per definition REF (execution) $A_{14}=\\lbrace tot\\_decr(p,q,ts): p \\in P, q=\\sum _{t \\in T_{ts}, p \\in \\bullet t}{W(p,t)}+\\sum _{t \\in T_{ts}, p \\in R(t)}{M_{ts}(p)}, \\\\ 0 \\le ts \\le k \\rbrace $ per definition REF (execution) $A_{15}=\\lbrace consumesmore(p,ts) : p \\in P, q=M_{ts}(p), q1=\\sum _{t \\in T_{ts}, p \\in \\bullet t}{W(p,t)}+\\sum _{t \\in T_{ts}, p \\in R(t)}{M_{ts}(p)}, q1 > q, 0 \\le ts \\le k \\rbrace $ per definition REF (conflicting transitions) $A_{16}=\\lbrace consumesmore : \\exists p \\in P : q=M_{ts}(p), q1=\\sum _{t \\in T_{ts}, p \\in \\bullet t}{W(p,t)}+\\sum _{t \\in T_{ts}, p \\in R(t)}{M_{ts}(p)}, q1 > q, 0 \\le ts \\le k \\rbrace $ per definition REF (conflicting transitions) $A_{17}=\\lbrace could\\_not\\_have(t,ts) : t \\in T, (\\forall p \\in \\bullet t, W(p,t) \\le M_{ts}(p)), t \\notin T_{ts}, (\\exists p \\in \\bullet t \\cup R(t) : q > M_{ts}(p) - (\\sum _{t^{\\prime } \\in T_{ts}, p \\in \\bullet t^{\\prime }}{W(p,t^{\\prime })} + \\sum _{t^{\\prime } \\in T_{ts}, p \\in R(t^{\\prime })}{M_{ts}(p)}), q=W(p,t) \\text{~if~} p \\in \\bullet t \\text{~or~} M_{ts}(p) \\text{~otherwise~}), 0 \\le ts \\le k \\rbrace $ per the maximal firing set semantics $A_{18}=\\lbrace holds(p,q,ts+1) : p \\in P, q=M_{ts+1}(p), 0 \\le ts < k\\rbrace $ , where $M_{ts+1}(p) = M_{ts}(p) - (\\sum _{\\begin{array}{c}t \\in T_{ts}, p \\in \\bullet t\\end{array}}{W(p,t)} + \\sum _{\\begin{array}{c}t \\in T_{ts}, p \\in R(t)\\end{array}}{M_{ts}(p)}) + \\sum _{\\begin{array}{c}t \\in T_{ts}, p \\in t \\bullet \\end{array}}{W(t,p)}$ according to definition REF (firing set execution) $A_{19}=\\lbrace ptarc(p,t,n,ts) : p \\in R(t), n=M_{ts}(p), n > 0, 0 \\le ts \\le k\\rbrace $ We show that $A$ satisfies (REF ) and (REF ), and $A$ is an answer set of $\\Pi ^1$ .", "$A$ satisfies (REF ) and (REF ) by its construction above.", "We show $A$ is an answer set of $\\Pi ^1$ by splitting.", "We split $lit(\\Pi ^1)$ into a sequence of $7(k+1)+2$ sets: $U_0= head(f\\ref {f:place}) \\cup head(f\\ref {f:trans}) \\cup head(f\\ref {f:time}) \\cup head(f\\ref {f:num}) \\cup head(i\\ref {i:holds}) = \\lbrace place(p) : p \\in P\\rbrace \\cup \\lbrace trans(t) : t \\in T\\rbrace \\cup \\lbrace time(0), \\dots , time(k)\\rbrace \\cup \\lbrace num(0), \\dots , num(ntok)\\rbrace \\cup \\lbrace holds(p,q,0) : p \\in P, q=M_0(p) \\rbrace $ $U_{7k+1}=U_{7k+0} \\cup head(f\\ref {f:r:ptarc})^{ts=k} \\cup head(f\\ref {f:r:tparc})^{ts=k} \\cup head(f\\ref {f:rptarc})^{ts=k} = U_{7k+0} \\cup \\\\ \\lbrace ptarc(p,t,n,k) : (p,t) \\in E^-, n=W(p,t) \\rbrace \\cup \\lbrace tparc(t,p,n,k) : (t,p) \\in E^+, n=W(t,p) \\rbrace \\cup \\lbrace ptarc(p,t,n,k) : p \\in R(t), n=M_{k}(p), n > 0 \\rbrace $ $U_{7k+2}=U_{7k+1} \\cup head(e\\ref {e:r:ne:ptarc} )^{ts=k} = U_{7k+1} \\cup \\lbrace notenabled(t,k) : t \\in T \\rbrace $ $U_{7k+3}=U_{7k+2} \\cup head(e\\ref {e:enabled})^{ts=k} = U_{7k+2} \\cup \\lbrace enabled(t,k) : t \\in T \\rbrace $ $U_{7k+4}=U_{7k+3} \\cup head(a\\ref {a:fires})^{ts=k} = U_{7k+3} \\cup \\lbrace fires(t,k) : t \\in T \\rbrace $ $U_{7k+5}=U_{7k+4} \\cup head(r\\ref {r:r:add} )^{ts=k} \\cup head(r\\ref {r:r:del} )^{ts=k} = U_{7k+4} \\cup \\lbrace add(p,q,t,k) : p \\in P, t \\in T, q=W(t,p) \\rbrace \\cup \\lbrace del(p,q,t,k) : p \\in P, t \\in T, q=W(p,t) \\rbrace \\cup \\lbrace del(p,q,t,k) : p \\in P, t \\in T, q=M_{k}(p) \\rbrace $ $U_{7k+6}=U_{7k+5} \\cup head(r\\ref {r:totincr})^{ts=k} \\cup head(r\\ref {r:totdecr})^{ts=k} = U_{7k+5} \\cup \\lbrace tot\\_incr(p,q,k) : p \\in P, 0 \\le q \\le ntok \\rbrace \\cup \\lbrace tot\\_decr(p,q,k) : p \\in P, 0 \\le q \\le ntok \\rbrace $ $U_{7k+7}=U_{7k+6} \\cup head(r\\ref {r:nextstate})^{ts=k} \\cup head(a\\ref {a:overc:place})^{ts=k} \\cup head(a\\ref {a:r:maxfire:cnh})^{ts=k} $ $= U_{7k+6} \\cup \\\\ \\lbrace consumesmore(p,k) : p \\in P\\rbrace \\cup \\lbrace holds(p,q,k+1) : p \\in P, 0 \\le q \\le ntok \\rbrace \\cup \\lbrace could\\_not\\_have(t,k) : t \\in T \\rbrace $ $U_{7k+8}=U_{7k+7} \\cup head(a\\ref {a:overc:gen}) = U_{7k+7} \\cup \\lbrace consumesmore \\rbrace $ where $head(r_i)^{ts=k}$ are head atoms of ground rule $r_i$ in which $ts=k$ .", "We write $A_i^{ts=k} = \\lbrace a(\\dots ,ts) : a(\\dots ,ts) \\in A_i, ts=k \\rbrace $ as short hand for all atoms in $A_i$ with $ts=k$ .", "$U_{\\alpha }, 0 \\le \\alpha \\le 7k+8$ form a splitting sequence, since each $U_i$ is a splitting set of $\\Pi ^1$ , and $\\langle U_{\\alpha }\\rangle _{\\alpha < \\mu }$ is a monotone continuous sequence, where $U_0 \\subseteq U_1 \\dots \\subseteq U_{7k+8}$ and $\\bigcup _{\\alpha < \\mu }{U_{\\alpha }} = lit(\\Pi ^1)$ .", "We compute the answer set of $\\Pi ^2$ using the splitting sets as follows: $bot_{U_0}(\\Pi ^2) = f\\ref {f:place} \\cup f\\ref {f:trans} \\cup f\\ref {f:time} \\cup i\\ref {i:holds} \\cup f\\ref {f:num}$ and $X_0 = A_1 \\cup \\dots \\cup A_4 \\cup A_7$ ($= U_0$ ) is its answer set – using forced atom proposition $eval_{U_0}(bot_{U_1}(\\Pi ^2) \\setminus bot_{U_0}(\\Pi ^2), X_0) = \\lbrace ptarc(p,t,q,0) \\text{:-}.", "\| q=W(p,t) \\rbrace \\cup \\\\ \\lbrace tparc(t,p,q,0) \\text{:-}.", "\| $ $q=W(t,p) \\rbrace \\cup \\lbrace ptarc(p,t,q,0) \\text{:-}.", "\| $ $q=M_0(p) \\rbrace $ .", "Its answer set $X_1=A_5^{ts=0} \\cup A_6^{ts=0} \\cup A_{19}^{ts=0}$ – using forced atom proposition and construction of $A_5, A_6, A_{19}$ .", "$eval_{U_1}(bot_{U_2}(\\Pi ^2) \\setminus bot_{U_1}(\\Pi ^2), X_0 \\cup X_1) = \\lbrace notenabled(t,0) \\text{:-} .", "\| \\lbrace trans(t), \\\\ ptarc(p,t,n,0), holds(p,q,0) \\rbrace \\subseteq X_0 \\cup X_1, \\text{~where~} q < n \\rbrace $ .", "Its answer set $X_2=A_8^{ts=0}$ – using forced atom proposition and construction of $A_8$ .", "where, $q=M_0(p)$ , and $n=W(p,t)$ for an arc $(p,t) \\in E^-$ – by construction of $i\\ref {i:holds}$ and $f\\ref {f:r:ptarc}$ in $\\Pi ^2$ , and in an arc $(p,t) \\in E^-$ , $p \\in \\bullet t$ (by definition REF of preset) thus, $notenabled(t,0) \\in X_1$ represents $\\exists p \\in \\bullet t : M_0(p) < W(p,t)$ .", "$eval_{U_2}(bot_{U_3}(\\Pi ^2) \\setminus bot_{U_2}(\\Pi ^2), X_0 \\cup \\dots \\cup X_2) = \\lbrace enabled(t,0) \\text{:-}.", "\| trans(t) \\in X_0 \\cup \\dots \\cup X_2, notenabled(t,0) \\notin X_0 \\cup \\dots \\cup X_2 \\rbrace $ .", "Its answer set is $X_3 = A_9^{ts=0}$ – using forced atom proposition and construction of $A_9$ .", "since an $enabled(t,0) \\in X_3$ if $\\nexists ~notenabled(t,0) \\in X_0 \\cup \\dots \\cup X_2$ ; which is equivalent to $\\nexists p \\in \\bullet t : M_0(p) < W(p,t) \\equiv \\forall p \\in \\bullet t: M_0(p) \\ge W(p,t)$ .", "$eval_{U_3}(bot_{U_4}(\\Pi ^2) \\setminus bot_{U_3}(\\Pi ^2), X_0 \\cup \\dots \\cup X_3) = \\lbrace \\lbrace fires(t,0)\\rbrace \\text{:-}.", "\| enabled(t,0) \\\\ \\text{~holds in~} X_0 \\cup \\dots \\cup X_3 \\rbrace $ .", "It has multiple answer sets $X_{4.1}, \\dots , X_{4.n}$ , corresponding to elements of power set of $fires(t,0)$ atoms in $eval_{U_3}(...)$ – using supported rule proposition.", "Since we are showing that the union of answer sets of $\\Pi ^2$ determined using splitting is equal to $A$ , we only consider the set that matches the $fires(t,0)$ elements in $A$ and call it $X_4$ , ignoring the rest.", "Thus, $X_4 = A_{10}^{ts=0}$ , representing $T_0$ .", "in addition, for every $t$ such that $enabled(t,0) \\in X_0 \\cup \\dots \\cup X_3, R(t) \\ne \\emptyset $ ; $fires(t,0) \\in X_4$ – per definition REF (firing set); requiring that a reset transition is fired when enabled thus, the firing set $T_0$ will not be eliminated by the constraint $f\\ref {f:c:rptarc:elim}$ $eval_{U_4}(bot_{U_5}(\\Pi ^2) \\setminus bot_{U_4}(\\Pi ^2), X_0 \\cup \\dots \\cup X_4) = \\lbrace add(p,n,t,0) \\text{:-}.", "\| \\lbrace fires(t,0), \\\\ tparc(t,p,n,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_4 \\rbrace \\cup \\lbrace del(p,n,t,0) \\text{:-}.", "\| \\lbrace fires(t,0), ptarc(p,t,n,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_4 \\rbrace $ .", "It's answer set is $X_5 = A_{11}^{ts=0} \\cup A_{12}^{ts=0}$ – using forced atom proposition and definitions of $A_{11}$ and $A_{12}$ .", "where, each $add$ atom is equivalent to $n=W(t,p) : p \\in t \\bullet $ , and each $del$ atom is equivalent to $n=W(p,t) : p \\in \\bullet t$ ; or $n=M_k(p) : p \\in R(t)$ , representing the effect of transitions in $T_0$ – by construction $eval_{U_5}(bot_{U_6}(\\Pi ^2) \\setminus bot_{U_5}(\\Pi ^2), X_0 \\cup \\dots \\cup X_5) = \\lbrace tot\\_incr(p,qq,0) \\text{:-}.", "\| $ $qq=\\sum _{add(p,q,t,0) \\in X_0 \\cup \\dots \\cup X_5}{q} \\rbrace \\cup \\lbrace tot\\_decr(p,qq,0) \\text{:-}.", "\| qq=\\sum _{del(p,q,t,0) \\in X_0 \\cup \\dots \\cup X_5}{q} \\rbrace $ .", "It's answer set is $X_6 = A_{13}^{ts=0} \\cup A_{14}^{ts=0}$ – using forced atom proposition, definitions of $A_{13}$ , $A_{14}$ , and definition REF (semantics of aggregate assignment atom).", "where, each for $tot\\_incr(p,qq,0)$ , $qq=\\sum _{add(p,q,t,0) \\in X_0 \\cup \\dots X_5}{q}$ $\\equiv qq=\\sum _{t \\in X_4, p \\in t \\bullet }{W(p,t)}$ , and each $tot\\_decr(p,qq,0)$ , $qq=\\sum _{del(p,q,t,0) \\in X_0 \\cup \\dots X_5}{q}$ $\\equiv qq=\\sum _{t \\in X_4, p \\in \\bullet t}{W(t,p)} + \\sum _{t \\in X_4, p \\in R(t)}{M_{k}(p)}$ , represent the net effect of transitions in $T_0$ – by construction $eval_{U_6}(bot_{U_7}(\\Pi ^2) \\setminus bot_{U_6}(\\Pi ^2), X_0 \\cup \\dots \\cup X_6) = \\lbrace consumesmore(p,0) \\text{:-}.", "\| \\\\ \\lbrace holds(p,q,0), tot\\_decr(p,q1,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_6, q1 > q \\rbrace \\cup \\lbrace holds(p,q,1) \\text{:-}., \| \\\\ \\lbrace holds(p,q1,0), tot\\_incr(p,q2,0), tot\\_decr(p,q3,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_6, q=q1+q2-q3 \\rbrace \\cup \\lbrace could\\_not\\_have(t,0) \\text{:-}.", "\| \\lbrace enabled(t,0), ptarc(s,t,q), holds(s,qq,0), \\\\ tot\\_decr(s,qqq,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_6, fires(t,0) \\notin (X_0 \\cup \\dots \\cup X_6), q > qq-qqq \\rbrace $ .", "It's answer set is $X_7 = A_{15}^{ts=0} \\cup A_{17}^{ts=0} \\cup A_{18}^{ts=0}$ – using forced atom proposition and definitions of $A_{15}, A_{17}, A_{18}$ .", "where, $consumesmore(p,0)$ represents $\\exists p : q=M_0(p), q1= \\\\ \\sum _{t \\in T_0, p \\in \\bullet t}{W(p,t)}+$ $\\sum _{t \\in T_0, p \\in R(t)}{M_k(p)}, $ $q1 > q$ , indicating place $p$ will be overconsumed if $T_0$ is fired – as defined in definition REF (conflicting transitions) and, $holds(p,q,1)$ represents $q=M_1(p)$ – by construction of $\\Pi ^2$ and $could\\_not\\_have(t,0)$ represents an enabled transition $t$ in $T_0$ that could not fire due to insufficient tokens $X_7$ does not contain $could\\_not\\_have(t,0)$ , when $enabled(t,0) \\in X_0 \\cup \\dots \\cup X_6$ and $fires(t,0) \\notin X_0 \\cup \\dots \\cup X_6$ due to construction of $A$ , encoding of $a\\ref {a:r:maxfire:cnh}$ and its body atoms.", "As a result it is not eliminated by the constraint $a\\ref {a:maxfire:elim}$ $ \\vdots $ $eval_{U_{7k+0}}(bot_{U_{7k+1}}(\\Pi ^2) \\setminus bot_{U_{7k+0}}(\\Pi ^2), X_0 \\cup \\dots \\cup X_{7k+0}) = \\lbrace ptarc(p,t,q,k) \\text{:-}.", "\| q=W(p,t) \\rbrace \\cup \\lbrace tparc(t,p,q,k) \\text{:-}.", "\| q=W(t,p) \\rbrace \\cup \\lbrace ptarc(p,t,q,k) \\text{:-}.", "\| q=M_0(p) \\rbrace $ .", "Its answer set $X_{7k+1}=A_5^{ts=k} \\cup A_6^{ts=k} \\cup A_{19}^{ts=k}$ – using forced atom proposition and construction of $A_5, A_6, A_{19}$ .", "$eval_{U_{7k+1}}(bot_{U_{7k+2}}(\\Pi ^2) \\setminus bot_{U_{7k+1}}(\\Pi ^2), X_0 \\cup X_{7k+1}) = \\lbrace notenabled(t,k) \\text{:-} .", "\| \\lbrace trans(t), \\\\ ptarc(p,t,n,k), holds(p,q,k) \\rbrace \\subseteq X_0 \\cup X_{7k+1}, \\text{~where~} q < n \\rbrace $ .", "Its answer set $X_{7k+2}=A_8^{ts=k}$ – using forced atom proposition and construction of $A_8$ .", "where, $q=M_0(p)$ , and $n=W(p,t)$ for an arc $(p,t) \\in E^-$ – by construction of $holds$ and $ptarc$ predicates in $\\Pi ^2$ , and in an arc $(p,t) \\in E^-$ , $p \\in \\bullet t$ (by definition REF of preset) thus, $notenabled(t,k) \\in X_{7k+1}$ represents $\\exists p \\in \\bullet t : M_0(p) < W(p,t)$ .", "$eval_{U_{7k+2}}(bot_{U_{7k+3}}(\\Pi ^2) \\setminus bot_{U_{7k+2}}(\\Pi ^2), X_0 \\cup \\dots \\cup X_{7k+2}) = \\lbrace enabled(t,k) \\text{:-}.", "\| trans(t) \\in X_0 \\cup \\dots \\cup X_{7k+2} , notenabled(t,k) \\notin X_0 \\cup \\dots \\cup X_{7k+2} \\rbrace $ .", "Its answer set is $X_{7k+3} = A_9^{ts=k}$ – using forced atom proposition and construction of $A_9$ .", "since an $enabled(t,k) \\in X_{7k+3}$ if $\\nexists ~notenabled(t,k) \\in X_0 \\cup \\dots \\cup X_{7k+2}$ ; which is equivalent to $\\nexists p \\in \\bullet t : M_0(p) < W(p,t) \\equiv \\forall p \\in \\bullet t: M_0(p) \\ge W(p,t)$ .", "$eval_{U_{7k+3}}(bot_{U_{7k+4}}(\\Pi ^2) \\setminus bot_{U_{7k+3}}(\\Pi ^2), X_0 \\cup \\dots \\cup X_{7k+3}) = \\lbrace \\lbrace fires(t,k)\\rbrace \\text{:-}.", "\| \\\\ enabled(t,k) \\\\ \\text{~holds in~} X_0 \\cup \\dots \\cup X_{7k+3} \\rbrace $ .", "It has multiple answer sets $X_{7k+4.1}, \\dots , X_{1k+4.n}$ , corresponding to elements of power set of $fires(t,k)$ atoms in $eval_{U_{7k+3}}(...)$ – using supported rule proposition.", "Since we are showing that the union of answer sets of $\\Pi ^2$ determined using splitting is equal to $A$ , we only consider the set that matches the $fires(t,k)$ elements in $A$ and call it $X_{7k+4}$ , ignoring the rest.", "Thus, $X_{7k+4} = A_{10}^{ts=k}$ , representing $T_k$ .", "in addition, for every $t$ such that $enabled(t,k) \\in X_0 \\cup \\dots \\cup X_{7k+3}, R(t) \\ne \\emptyset $ ; $fires(t,k) \\in X_{7k+4}$ – per definition REF (firing set); requiring that a reset transition is fired when enabled thus, the firing set $T_k$ will not be eliminated by the constraint $f\\ref {f:c:rptarc:elim}$ $eval_{U_{7k+4}}(bot_{U_{7k+5}}(\\Pi ^2) \\setminus bot_{U_{7k+4}}(\\Pi ^2), X_0 \\cup \\dots \\cup X_{7k+4}) = \\\\ \\lbrace add(p,n,t,k) \\text{:-}.", "\| \\lbrace fires(t,k), tparc(t,p,n,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{7k+4} \\rbrace \\cup \\\\ \\lbrace del(p,n,t,k) \\text{:-}.", "\| \\lbrace fires(t,k), ptarc(p,t,n,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{7k+4} \\rbrace $ .", "It's answer set is $X_{7k+5} = A_{11}^{ts=k} \\cup A_{12}^{ts=k}$ – using forced atom proposition and definitions of $A_{11}$ and $A_{12}$ .", "where, each $add$ atom is equivalent to $n=W(t,p) : p \\in \\bullet t$ , and each $del$ atom is equivalent to $n=W(p,t) : p \\in t \\bullet $ ; or $n=M_k(p) : p \\in R(t)$ , representing the effect of transitions in $T_k$ $eval_{U_{7k+5}}(bot_{U_{7k+6}}(\\Pi ^2) \\setminus bot_{U_{7k+5}}(\\Pi ^2), X_0 \\cup \\dots \\cup X_{7k+5}) = \\\\ \\lbrace tot\\_incr(p,qq,k) \\text{:-}.", "\| qq=\\sum _{add(p,q,t,k) \\in X_0 \\cup \\dots \\cup X_{7k+5}}{q} \\rbrace \\cup \\\\ \\lbrace tot\\_decr(p,qq,k) \\text{:-}.", "\| qq=\\sum _{del(p,q,t,k) \\in X_0 \\cup \\dots \\cup X_{7k+5}}{q} \\rbrace $ .", "It's answer set is $X_{7k+6} = A_{13}^{ts=k} \\cup A_{14}^{ts=k}$ – using forced atom proposition and definitions of $A_{13}$ and $A_{14}$ .", "where, each $tot\\_incr(p,qq,k)$ , $qq=\\sum _{add(p,q,t,k) \\in X_0 \\cup \\dots X_{7k+5}}{q}$ $\\equiv qq=\\sum _{t \\in X_{7k+4}, p \\in t \\bullet }{W(p,t)}$ , and each $tot\\_decr(p,qq,k)$ , $qq=\\sum _{del(p,q,t,k) \\in X_0 \\cup \\dots X_{7k+5}}{q}$ $\\equiv qq=\\sum _{t \\in X_{7k+4}, p \\in \\bullet t}{W(t,p)} + \\sum _{t \\in X_{7k+4}, p \\in R(t)}{M_{k}(p)}$ , represent the net effect of transitions in $T_k$ $eval_{U_{7k+6}}(bot_{U_{7k+7}}(\\Pi ^2) \\setminus bot_{U_{7k+6}}(\\Pi ^2), X_0 \\cup \\dots \\cup X_{7k+6}) = \\lbrace consumesmore(p,k) \\text{:-}.", "\| \\\\ \\lbrace holds(p,q,k), tot\\_decr(p,q1,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{7k+6}, q1 > q \\rbrace \\cup \\lbrace holds(p,q,k+1) \\text{:-}., \| \\\\ \\lbrace holds(p,q1,k), tot\\_incr(p,q2,k), tot\\_decr(p,q3,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{7k+6}, q=q1+q2-q3 \\rbrace \\cup \\lbrace could\\_not\\_have(t,k) \\text{:-}.", "\| \\lbrace enabled(t,k), ptarc(s,t,q), holds(s,qq,k), \\\\ tot\\_decr(s,qqq,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{7k+6}, fires(t,k) \\notin (X_0 \\cup \\dots \\cup X_{7k+6}), q > qq-qqq \\rbrace $ .", "It's answer set is $X_{7k+7} = A_{15}^{ts=k} \\cup A_{17}^{ts=k} \\cup A_{18}^{ts=k}$ – using forced atom proposition and definitions of $A_{15}, A_{17}, A_{18}$ .", "where, $consumesmore(p,k)$ represents $\\exists p : q=M_0(p), q1= \\\\ \\sum _{t \\in T_0, p \\in \\bullet t}{W(p,t)}+\\sum _{t \\in T_0, p \\in R(t)}{M_k(p)}, q1 > q$ , indicating place $p$ that will be over consumed if $T_k$ is fired, as defined in definition REF (conflicting transitions) $holds(p,q,k+1)$ represents $q=M_{k+1}(p)$ – by construction of $\\Pi ^2$ , and $could\\_not\\_have(t,k)$ represents enabled transition $t$ in $T_k$ that could not be fired due to insufficient tokens $X_{7k+7}$ does not contain $could\\_not\\_have(t,k)$ , when $enabled(t,k) \\in X_0 \\cup \\dots \\cup X_{7k+6}$ and $fires(t,k) \\notin X_0 \\cup \\dots \\cup X_{7k+6}$ due to the construction of $A$ , encoding of $a\\ref {a:r:maxfire:cnh}$ and its body atoms.", "As a result it is not eliminated by the constraint $a\\ref {a:maxfire:elim}$ $eval_{U_{7k+7}}(bot_{U_{7k+8}}(\\Pi ^2) \\setminus bot_{U_{7k+7}}(\\Pi ^2), X_0 \\cup \\dots \\cup X_{7k+7}) = \\lbrace consumesmore \\text{:-}.", "\| $ $ \\lbrace consumesmore(p,0),\\dots ,$ $consumesmore(p,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{7k+7} \\rbrace $ .", "It's answer set is $X_{7k+8} = A_{16}$ – using forced atom proposition $X_{7k+8}$ will be empty since none of $consumesmore(p,0),\\dots , \\\\ consumesmore(p,k)$ hold in $X_0 \\cup \\dots \\cup X_{7k+7}$ due to the construction of $A$ , encoding of $a\\ref {a:overc:place}$ and its body atoms.", "As a result, it is not eliminated by the constraint $a\\ref {a:overc:elim}$ The set $X = X_0 \\cup \\dots \\cup X_{7k+8}$ is the answer set of $\\Pi ^2$ by the splitting sequence theorem REF .", "Each $X_i, 0 \\le i \\le 7k+8$ matches a distinct portion of $A$ , and $X = A$ , thus $A$ is an answer set of $\\Pi ^2$ .", "Next we show (REF ): Given $\\Pi ^2$ be the encoding of a Petri Net $PN(P,T,E,W,R)$ with initial marking $M_0$ , and $A$ be an answer set of $\\Pi ^2$ that satisfies (REF ) and (REF ), then we can construct $X=M_0,T_0,\\dots ,M_k,T_k,M_{k+1}$ from $A$ , such that it is an execution sequence of $PN$ .", "We construct the $X$ as follows: $M_i = (M_i(p_0), \\dots , M_i(p_n))$ , where $\\lbrace holds(p_0,M_i(p_0),i), \\dots holds(p_n,M_i(p_n),i) \\rbrace \\\\ \\subseteq A$ , for $0 \\le i \\le k+1$ $T_i = \\lbrace t : fires(t,i) \\in A\\rbrace $ , for $0 \\le i \\le k$ and show that $X$ is indeed an execution sequence of $PN$ .", "We show this by induction over $k$ (i.e.", "given $M_k$ , $T_k$ is a valid firing set and its firing produces marking $M_{k+1}$ ).", "Base case: Let $k=0$ , and $M_0$ is a valid marking in $X$ for $PN$ , show [(1)] $T_0$ is a valid firing set for $M_0$ , and $T_0$ 's firing in $M_0$ produces marking $M_1$ .", "We show $T_0$ is a valid firing set for $M_0$ .", "Let $\\lbrace fires(t_0,0), \\dots , fires(t_x,0) \\rbrace $ be the set of all $fires(\\dots ,0)$ atoms in $A$ , Then for each $fires(t_i,0) \\in A$ $enabled(t_i,0) \\in A$ – from rule $a\\ref {a:fires}$ and supported rule proposition Then $notenabled(t_i,0) \\notin A$ – from rule $e\\ref {e:enabled}$ and supported rule proposition Then $body(e\\ref {e:r:ne:ptarc})$ must not hold in $A$ – from rule $e\\ref {e:r:ne:ptarc}$ and forced atom proposition Then $q \\lnot < n_i \\equiv q \\ge n_i$ in $e\\ref {e:r:ne:ptarc}$ for all $\\lbrace holds(p,q,0), ptarc(p,t_i,n_i,0)\\rbrace \\subseteq A$ – from $e\\ref {e:r:ne:ptarc}$ , forced atom proposition, and the following $holds(p,q,0) \\in A$ represents $q=M_0(p)$ – rule $i\\ref {i:holds}$ construction $ptarc(p,t_i,n_i,0) \\in A$ represents $n_i=W(p,t_i)$ – rule $f\\ref {f:r:ptarc}$ construction; or it represents $n_i = M_0(p)$ – rule $f\\ref {f:rptarc}$ construction; the construction of $f\\ref {f:rptarc}$ ensures that $notenabled(t,0)$ is never true due to the reset arc Then $\\forall p \\in \\bullet t_i, M_0(p) \\ge W(p,t_i)$ – from definition REF of preset $\\bullet t_i$ in PN Then $t_i$ is enabled and can fire in $PN$ , as a result it can belong to $T_0$ – from definition REF of enabled transition And $consumesmore \\notin A$ , since $A$ is an answer set of $\\Pi ^2$ – from rule $a\\ref {a:overc:elim}$ and supported rule proposition Then $\\nexists consumesmore(p,0) \\in A$ – from rule $a\\ref {a:overc:gen}$ and supported rule proposition Then $\\nexists \\lbrace holds(p,q,0), tot\\_decr(p,q1,0) \\rbrace \\subseteq A : q1>q$ in $body(a\\ref {a:overc:place})$ – from $a\\ref {a:overc:place}$ and forced atom proposition Then $\\nexists p : (\\sum _{t_i \\in \\lbrace t_0,\\dots ,t_x\\rbrace , p \\in \\bullet t_i}{W(p,t_i)}+\\sum _{t_i \\in \\lbrace t_0,\\dots ,t_x\\rbrace , p \\in R(t_i)}{M_0(p)}) > M_0(p)$ – from the following $holds(p,q,0)$ represents $q=M_0(p)$ – from rule $i\\ref {i:holds}$ construction, given $tot\\_decr(p,q1,0) \\in A$ if $\\lbrace del(p,q1_0,t_0,0), \\dots , del(p,q1_x,t_x,0) \\rbrace \\subseteq A$ , where $q1 = q1_0+\\dots +q1_x$ – from $r\\ref {r:totdecr}$ and forced atom proposition $del(p,q1_i,t_i,0) \\in A$ if $\\lbrace fires(t_i,0), ptarc(p,t_i,q1_i,0) \\rbrace \\subseteq A$ – from $r\\ref {r:r:del} $ and supported rule proposition $del(p,q1_i,t_i,0)$ either represents removal of $q1_i = W(p,t_i)$ tokens from $p \\in \\bullet t_i$ ; or it represents removal of $q1_i = M_0(p)$ tokens from $p \\in R(t_i)$ – from rule $r\\ref {r:r:del} $ , supported rule proposition, and definition REF of transition execution in $PN$ Then the set of transitions in $T_0$ do not conflict – by the definition REF of conflicting transitions And for each $enabled(t_j,0) \\in A$ and $fires(t_j,0) \\notin A$ , $could\\_not\\_have(t_j,0) \\in A$ , since $A$ is an answer set of $\\Pi ^2$ - from rule $a\\ref {a:maxfire:elim}$ and supported rule proposition Then $\\lbrace enabled(t_j,0), holds(s,qq,0), ptarc(s,t_j,q,0), \\\\ tot\\_decr(s,qqq,0) \\rbrace \\subseteq A$ , such that $q > qq - qqq$ and $fires(t_j,0) \\notin A$ - from rule $a\\ref {a:r:maxfire:cnh}$ and supported rule proposition Then for an $s \\in \\bullet t_j \\cup R(t_j)$ , $q > M_0(s) - (\\sum _{t_i \\in T_0, s \\in \\bullet t_i}{W(s,t_i)} + \\sum _{t_i \\in T_0, s \\in R(t_i)}{M_0(s))}$ , where $q=W(s,t_j) \\text{~if~} s \\in \\bullet t_j, \\text{~or~} M_0(s) \\text{~otherwise}$ - from the following: $ptarc(s,t_i,q,0)$ represents $q=W(s,t_i)$ if $(s,t_i) \\in E^-$ or $q=M_0(s)$ if $s \\in R(t_i)$ – from rule $f\\ref {f:r:ptarc},f\\ref {f:rptarc}$ construction $holds(s,qq,0)$ represents $qq=M_0(s)$ – from $i\\ref {i:holds}$ construction $tot\\_decr(s,qqq,0) \\in A$ if $\\lbrace del(s,qqq_0,t_0,0), \\dots , del(s,qqq_x,t_x,0) \\rbrace \\subseteq A$ – from rule $r\\ref {r:totdecr}$ construction and supported rule proposition $del(s,qqq_i,t_i,0) \\in A$ if $\\lbrace fires(t_i,0), ptarc(s,t_i,qqq_i,0) \\rbrace \\subseteq A$ – from rule $r\\ref {r:r:del} $ and supported rule proposition $del(s,qqq_i,t_i,0)$ represents $qqq_i = W(s,t_i) : t_i \\in T_0, (s,t_i) \\in E^-$ , or $qqq_i = M_0(t_i) : t_i \\in T_0, s \\in R(t_i)$ – from rule $f\\ref {f:r:ptarc},f\\ref {f:rptarc}$ construction $tot\\_decr(q,qqq,0)$ represents $\\sum _{t_i \\in T_0, s \\in \\bullet t_i}{W(s,t_i)} + \\\\ \\sum _{t_i \\in T_0, s \\in R(t_i)}{M_0(s)}$ – from (C,D,E) above Then firing $T_0 \\cup \\lbrace t_j \\rbrace $ would have required more tokens than are present at its source place $s \\in \\bullet t_j \\cup R(t_j)$ .", "Thus, $T_0$ is a maximal set of transitions that can simultaneously fire.", "And for each reset transition $t_r$ with $enabled(t_r,0) \\in A$ , $fires(t_r,0) \\in A$ , since $A$ is an answer set of $\\Pi ^2$ - from rule $f\\ref {f:c:rptarc:elim}$ and supported rule proposition Then, the firing set $T_0$ satisfies the reset-transition requirement of definition REF (firing set) Then $\\lbrace t_0, \\dots , t_x\\rbrace = T_0$ – using 1(a),1(b),1(d) above; and using 1(c) it is a maximal firing set We show $M_1$ is produced by firing $T_0$ in $M_0$ .", "Let $holds(p,q,1) \\in A$ Then $\\lbrace holds(p,q1,0), tot\\_incr(p,q2,0), tot\\_decr(p,q3,0) \\rbrace \\subseteq A : q=q1+q2-q3$ – from rule $r\\ref {r:nextstate}$ and supported rule proposition $holds(p,q1,0) \\in A$ represents $q1=M_0(p)$ – given, rule $i\\ref {i:holds}$ construction; and $\\lbrace add(p,q2_0,t_0,0), \\dots , $ $add(p,q2_j,t_j,0)\\rbrace \\subseteq A : q2_0 + \\dots + q2_j = q2$ and $\\lbrace del(p,q3_0,t_0,0), \\dots , $ $del(p,q3_l,t_l,0)\\rbrace \\subseteq A : q3_0 + \\dots + q3_l = q3$ – rules $r\\ref {r:totincr},r\\ref {r:totdecr}$ using supported rule proposition Then $\\lbrace fires(t_0,0), \\dots , fires(t_j,0) \\rbrace \\subseteq A$ and $\\lbrace fires(t_0,0), \\dots , fires(t_l,0) \\rbrace \\\\ \\subseteq A$ – rules $r\\ref {r:r:add},r\\ref {r:r:del}$ and supported rule proposition, respectively Then $\\lbrace fires(t_0,0), \\dots , fires(t_j,0) \\rbrace \\cup \\lbrace fires(t_0,0), \\dots , fires(t_l,0) \\rbrace \\subseteq A = \\lbrace fires(t_0,0), \\dots , fires(t_x,0) \\rbrace \\subseteq A$ – set union of subsets Then for each $fires(t_x,0) \\in A$ we have $t_x \\in T_0$ – already shown in item REF above Then $q = M_0(p) + \\sum _{t_x \\in T_0 \\wedge p \\in t_x \\bullet }{W(t_x,p)} - (\\sum _{t_x \\in T_0 \\wedge p \\in \\bullet t_x}{W(p,t_x)} + \\\\ \\sum _{t_x \\in T_0 \\wedge p \\in R(t_x)}{M_0(p)})$ – from (REF ),(REF ) (REF ) above and the following Each $add(p,q_j,t_j,0) \\in A$ represents $q_j=W(t_j,p)$ for $p \\in t_j \\bullet $ – rule $r\\ref {r:r:add} $ encoding, and definition REF of postset in $PN$ Each $del(p,t_y,q_y,0) \\in A$ represents either $q_y=W(p,t_y)$ for $p \\in \\bullet t_y$ , or $q_y=M_0(p)$ for $p \\in R(t_y)$ – from rule $r\\ref {r:r:del} ,f\\ref {f:r:ptarc} $ encoding and definition REF of preset in $PN$ ; or from rule $r\\ref {r:r:del} ,f\\ref {f:rptarc}$ encoding and definition of reset arc in $PN$ Each $tot\\_incr(p,q2,0) \\in A$ represents $q2=\\sum _{t_x \\in T_0 \\wedge p \\in t_x \\bullet }{W(t_x,p)}$ – aggregate assignment atom semantics in rule $r\\ref {r:totincr}$ Each $tot\\_decr(p,q3,0) \\in A$ represents $q3=\\sum _{t_x \\in T_0 \\wedge p \\in \\bullet t_x}{W(p,t_x)} + $ $\\sum _{t_x \\in T_0 \\wedge p \\in R(t_x)}{M_0(p)}$ – aggregate assignment atom semantics in rule $r\\ref {r:totdecr}$ Then $M_1(p) = q$ – since $holds(p,q,1) \\in A$ encodes $q=M_1(p)$ – from construction Inductive Step: Let $k > 0$ , and $M_k$ is a valid marking in $X$ for $PN$ , show [(1)] $T_k$ is a valid firing set in $M_k$ , and firing $T_k$ in $M_k$ produces marking $M_{k+1}$ .", "We show that $T_k$ is a valid firing set in $M_k$ .", "Let $\\lbrace fires(t_0,k), \\dots , fires(t_x,k) \\rbrace $ be the set of all $fires(\\dots ,k)$ atoms in $A$ , Then for each $fires(t_i,k) \\in A$ $enabled(t_i,k) \\in A$ – from rule $a\\ref {a:fires}$ and supported rule proposition Then $notenabled(t_i,k) \\notin A$ – from rule $e\\ref {e:enabled}$ and supported rule proposition Then $body(e\\ref {e:r:ne:ptarc})$ must not hold in $A$ – from rule $e\\ref {e:r:ne:ptarc}$ and forced atom proposition Then $q \\lnot < n_i \\equiv q \\ge n_i$ in $e\\ref {e:r:ne:ptarc}$ for all $\\lbrace holds(p,q,k), ptarc(p,t_i,n_i,k)\\rbrace \\subseteq A$ – from $e\\ref {e:r:ne:ptarc}$ , forced atom proposition, and the following $holds(p,q,k) \\in A$ represents $q=M_k(p)$ – construction, inductive assumption $ptarc(p,t_i,n_i,k) \\in A$ represents $n_i=W(p,t_i)$ – rule $f\\ref {f:r:ptarc}$ construction; or it represents $n_i=M_k(p)$ – rule $f\\ref {f:rptarc}$ construction; the construction of $f\\ref {f:rptarc}$ ensures that $notenabled(t,0)$ is never true due to the reset arc Then $\\forall p \\in \\bullet t_i, M_k(p) \\ge W(p,t_i)$ – from definition REF of preset $\\bullet t_i$ in PN Then $t_i$ is enabled and can fire in $PN$ , as a result it can belong to $T_k$ – from definition REF of enabled transition And $consumesmore \\notin A$ , since $A$ is an answer set of $\\Pi ^2$ – from rule $a\\ref {a:overc:elim}$ and supported rule proposition Then $\\nexists consumesmore(p,k) \\in A$ – from rule $a\\ref {a:overc:gen}$ and supported rule proposition Then $\\nexists \\lbrace holds(p,q,k), tot\\_decr(p,q1,k) \\rbrace \\subseteq A : q1>q$ in $body(a\\ref {a:overc:place})$ – from $a\\ref {a:overc:place}$ and forced atom proposition Then $\\nexists p : (\\sum _{t_i \\in \\lbrace t_0,\\dots ,t_x\\rbrace , p \\in \\bullet t_i}{W(p,t_i)}+\\sum _{t_i \\in \\lbrace t_0,\\dots ,t_x\\rbrace , p \\in R(t_i)}{M_k(p)}) > M_k(p)$ – from the following $holds(p,q,k)$ represents $q=M_k(p)$ – inductive assumption, given $tot\\_decr(p,q1,k) \\in A$ if $\\lbrace del(p,q1_0,t_0,k), \\dots , del(p,q1_x,t_x,k) \\rbrace \\subseteq A$ , where $q1 = q1_0+\\dots +q1_x$ – from $r\\ref {r:totdecr}$ and forced atom proposition $del(p,q1_i,t_i,k) \\in A$ if $\\lbrace fires(t_i,k), ptarc(p,t_i,q1_i,k) \\rbrace \\subseteq A$ – from $r\\ref {r:r:del} $ and supported rule proposition $del(p,q1_i,t_i,k)$ either represents removal of $q1_i = W(p,t_i)$ tokens from $p \\in \\bullet t_i$ ; or it represents removal of $q1_i = M_k(p)$ tokens from $p \\in R(t_i)$ – from rule $r\\ref {r:r:del} $ , supported rule proposition, and definition REF of transition execution in $PN$ Then the set of transitions in $T_k$ do not conflict – by the definition REF of conflicting transitions And for each $enabled(t_j,k) \\in A$ and $fires(t_j,k) \\notin A$ , $could\\_not\\_have(t_j,k) \\in A$ , since $A$ is an answer set of $\\Pi ^2$ - from rule $a\\ref {a:maxfire:elim}$ and supported rule proposition Then $\\lbrace enabled(t_j,k), holds(s,qq,k), ptarc(s,t_j,q,k), \\\\ tot\\_decr(s,qqq,k) \\rbrace \\subseteq A$ , such that $q > qq - qqq$ and $fires(t_j,k) \\notin A$ - from rule $a\\ref {a:r:maxfire:cnh}$ and supported rule proposition Then for an $s \\in \\bullet t_j \\cup R(t_j)$ , $q > M_k(s) - (\\sum _{t_i \\in T_k, s \\in \\bullet t_i}{W(s,t_i)} + \\sum _{t_i \\in T_k, s \\in R(t_i)}{M_k(s))}$ , where $q=W(s,t_j) \\text{~if~} s \\in \\bullet t_j, \\text{~or~} M_k(s) \\text{~otherwise}$ - from the following: $ptarc(s,t_i,q,k)$ represents $q=W(s,t_i)$ if $(s,t_i) \\in E^-$ or $q=M_k(s)$ if $s \\in R(t_i)$ – from rule $f\\ref {f:r:ptarc},f\\ref {f:rptarc}$ construction $holds(s,qq,k)$ represents $qq=M_k(s)$ – construction $tot\\_decr(s,qqq,k) \\in A$ if $\\lbrace del(s,qqq_0,t_0,k), \\dots , del(s,qqq_x,t_x,k) \\rbrace \\subseteq A$ – from rule $r\\ref {r:totdecr}$ construction and supported rule proposition $del(s,qqq_i,t_i,k) \\in A$ if $\\lbrace fires(t_i,k), ptarc(s,t_i,qqq_i,k) \\rbrace \\subseteq A$ – from rule $r\\ref {r:r:del} $ and supported rule proposition $del(s,qqq_i,t_i,k)$ represents $qqq_i = W(s,t_i) : t_i \\in T_k, (s,t_i) \\in E^-$ , or $qqq_i = M_k(t_i) : t_i \\in T_k, s \\in R(t_i)$ – from rule $f\\ref {f:r:ptarc},f\\ref {f:rptarc}$ construction $tot\\_decr(q,qqq,k)$ represents $\\sum _{t_i \\in T_k, s \\in \\bullet t_i}{W(s,t_i)} + \\\\ \\sum _{t_i \\in T_k, s \\in R(t_i)}{M_k(s)}$ – from (C,D,E) above Then firing $T_k \\cup \\lbrace t_j \\rbrace $ would have required more tokens than are present at its source place $s \\in \\bullet t_j \\cup R(t_j)$ .", "Thus, $T_k$ is a maximal set of transitions that can simultaneously fire.", "And for each reset transition $t_r$ with $enabled(t_r,k) \\in A$ , $fires(t_r,k) \\in A$ , since $A$ is an answer set of $\\Pi ^2$ - from rule $f\\ref {f:c:rptarc:elim}$ and supported rule proposition Then the firing set $T_k$ satisfies the reset transition requirement of definition REF (firing set) Then $\\lbrace t_0, \\dots , t_x\\rbrace = T_k$ – using 1(a),1(b), 1(d) above; and using 1(c) it is a maximal firing set We show that $M_{k+1}$ is produced by firing $T_k$ in $M_k$ .", "Let $holds(p,q,k+1) \\in A$ Then $\\lbrace holds(p,q1,k), tot\\_incr(p,q2,k), tot\\_decr(p,q3,k) \\rbrace \\subseteq A : q=q1+q2-q3$ – from rule $r\\ref {r:nextstate}$ and supported rule proposition $holds(p,q1,k) \\in A$ represents $q1=M_k(p)$ – construction, inductive assumption; and $\\lbrace add(p,q2_0,t_0,k), \\dots , $ $add(p,q2_j,t_j,k)\\rbrace \\subseteq A : q2_0 + \\dots + q2_j = q2$ and $\\lbrace del(p,q3_0,t_0,k), \\dots , $ $del(p,q3_l,t_l,k)\\rbrace \\subseteq A : q3_0 + \\dots + q3_l = q3$ – rules $r\\ref {r:totincr},r\\ref {r:totdecr}$ using supported rule proposition Then $\\lbrace fires(t_0,k), \\dots , $ $fires(t_j,k) \\rbrace \\subseteq A$ and $\\lbrace fires(t_0,k), \\dots , $ $fires(t_l,k) \\rbrace \\\\ \\subseteq A$ – rules $r\\ref {r:r:add},r\\ref {r:r:del}$ and supported rule proposition, respectively Then $\\lbrace fires(t_0,k), \\dots , fires(t_j,k) \\rbrace \\cup \\lbrace fires(t_0,k), \\dots , $ $fires(t_l,k) \\rbrace \\subseteq $ $A = \\lbrace fires(t_0,k), \\dots , $ $ fires(t_x,k) \\rbrace \\subseteq A$ – set union of subsets Then for each $fires(t_x,k) \\in A$ we have $t_x \\in T_k$ – already shown in item REF above Then $q = M_k(p) + \\sum _{t_x \\in T_0 \\wedge p \\in t_x \\bullet }{W(t_x,p)} - (\\sum _{t_x \\in T_k \\wedge p \\in \\bullet t_x}{W(p,t_x)} + \\\\ \\sum _{t_x \\in T_k \\wedge p \\in R(t_x)}{M_k(p)})$ – from (REF ) above and the following Each $add(p,q_j,t_j,k) \\in A$ represents $q_j=W(t_j,p)$ for $p \\in t_j \\bullet $ – rule $r\\ref {r:r:add} $ encoding, and definition REF of transition execution in $PN$ Each $del(p,t_y,q_y,k) \\in A$ represents either $q_y=W(p,t_y)$ for $p \\in \\bullet t_y$ , or $q_y=M_k(p)$ for $p \\in R(t_y)$ – from rule $r\\ref {r:r:del} ,f\\ref {f:r:ptarc} $ encoding and definition REF of transition execution in $PN$ ; or from rule $r\\ref {r:r:del} ,f\\ref {f:rptarc}$ encoding and definition of reset arc in $PN$ Each $tot\\_incr(p,q2,k) \\in A$ represents $q2=\\sum _{t_x \\in T_k \\wedge p \\in t_x \\bullet }{W(t_x,p)}$ – aggregate assignment atom semantics in rule $r\\ref {r:totincr}$ Each $tot\\_decr(p,q3,0) \\in A$ represents $q3=\\sum _{t_x \\in T_k \\wedge p \\in \\bullet t_x}{W(p,t_x)} + $ $\\sum _{t_x \\in T_k \\wedge p \\in R(t_x)}{M_0(p)}$ – aggregate assignment atom semantics in rule $r\\ref {r:totdecr}$ Then, $M_{k+1}(p) = q$ – since $holds(p,q,k+1) \\in A$ encodes $q=M_{k+1}(p)$ – from construction As a result, for any $n > k$ , $T_n$ is a valid firing set w.r.t.", "$M_n$ and its firing produces marking $M_{n+1}$ .", "Conclusion: Since both (REF ) and (REF ) hold, $X=M_0,T_0,M_1,\\dots ,M_k,T_{k+1}$ is an execution sequence of $PN(P,T,E,W,R)$ (w.r.t $M_0$ ) iff there is an answer set $A$ of $\\Pi ^2(PN,M_0,k,ntok)$ such that (REF ) and (REF ) hold.", "Poof of Proposition REF Let $PN=(P,T,E,W,R,I)$ be a Petri Net, $M_0$ be its initial marking and let $\\Pi ^3(PN,M_0,k,ntok)$ be the ASP encoding of $PN$ and $M_0$ over a simulation length $k$ , with maximum $ntok$ tokens on any place node, as defined in section REF .", "Then $X=M_0,T_0,M_1,\\dots ,M_k,T_k,M_{k+1}$ is an execution sequence of $PN$ (w.r.t.", "$M_0$ ) iff there is an answer set $A$ of $\\Pi ^3(PN,M_0,k,ntok)$ such that: $\\lbrace fires(t,ts) : t \\in T_{ts}, 0 \\le ts \\le k\\rbrace = \\lbrace fires(t,ts) : fires(t,ts) \\in A \\rbrace $ $\\begin{split}\\lbrace holds(p,q,ts) &: p \\in P, q = M_{ts}(p), 0 \\le ts \\le k+1 \\rbrace \\\\&= \\lbrace holds(p,q,ts) : holds(p,q,ts) \\in A \\rbrace \\end{split}$ We prove this by showing that: Given an execution sequence $X$ , we create a set $A$ such that it satisfies (REF ) and (REF ) and show that $A$ is an answer set of $\\Pi ^3$ Given an answer set $A$ of $\\Pi ^3$ , we create an execution sequence $X$ such that (REF ) and (REF ) are satisfied.", "First we show (REF ): Given $PN$ and an execution sequence $X$ of $PN$ , we create a set $A$ as a union of the following sets: $A_1=\\lbrace num(n) : 0 \\le n \\le ntok \\rbrace $ $A_2=\\lbrace time(ts) : 0 \\le ts \\le k\\rbrace $ $A_3=\\lbrace place(p) : p \\in P \\rbrace $ $A_4=\\lbrace trans(t) : t \\in T \\rbrace $ $A_5=\\lbrace ptarc(p,t,n,ts) : (p,t) \\in E^-, n=W(p,t), 0 \\le ts \\le k \\rbrace $ , where $E^- \\subseteq E$ $A_6=\\lbrace tparc(t,p,n,ts) : (t,p) \\in E^+, n=W(t,p), 0 \\le ts \\le k \\rbrace $ , where $E^+ \\subseteq E$ $A_7=\\lbrace holds(p,q,0) : p \\in P, q=M_{0}(p) \\rbrace $ $A_8=\\lbrace notenabled(t,ts) : t \\in T, 0 \\le ts \\le k, (\\exists p \\in \\bullet t, M_{ts}(p) < W(p,t)) \\vee (\\exists p \\in I(t), M_{ts}(p) \\ne 0) \\rbrace $ per definition REF (enabled transition) $A_9=\\lbrace enabled(t,ts) : t \\in T, 0 \\le ts \\le k, (\\forall p \\in \\bullet t, W(p,t) \\le M_{ts}(p)) \\wedge (\\forall p \\in I(t), M_{ts}(p) = 0) \\rbrace $ per definition REF (enabled transition) $A_{10}=\\lbrace fires(t,ts) : t \\in T_{ts}, 0 \\le ts \\le k \\rbrace $ per definition REF (firing set), only an enabled transition may fire $A_{11}=\\lbrace add(p,q,t,ts) : t \\in T_{ts}, p \\in t \\bullet , q=W(t,p), 0 \\le ts \\le k \\rbrace $ per definition REF (transition execution) $A_{12}=\\lbrace del(p,q,t,ts) : t \\in T_{ts}, p \\in \\bullet t, q=W(p,t), 0 \\le ts \\le k \\rbrace \\cup \\lbrace del(p,q,t,ts) : t \\in T_{ts}, p \\in R(t), q=M_{ts}(p), 0 \\le ts \\le k \\rbrace $ per definition REF (transition execution) $A_{13}=\\lbrace tot\\_incr(p,q,ts) : p \\in P, q=\\sum _{t \\in T_{ts}, p \\in t \\bullet }{W(t,p)}, 0 \\le ts \\le k \\rbrace $ per definition REF (firing set execution) $A_{14}=\\lbrace tot\\_decr(p,q,ts): p \\in P, q=\\sum _{t \\in T_{ts}, p \\in \\bullet t}{W(p,t)}+\\sum _{t \\in T_{ts}, p \\in R(t) }{M_{ts}(p)}, 0 \\le ts \\le k \\rbrace $ per definition REF (firing set execution) $A_{15}=\\lbrace consumesmore(p,ts) : p \\in P, q=M_{ts}(p), q1=\\sum _{t \\in T_{ts}, p \\in \\bullet t}{W(p,t)} + \\sum _{t \\in T_{ts}, p \\in R(t)}{M_{ts}(p)}, q1 > q, 0 \\le ts \\le k \\rbrace $ per definition REF (conflicting transitions) $A_{16}=\\lbrace consumesmore : \\exists p \\in P : q=M_{ts}(p), q1=\\sum _{t \\in T_{ts}, p \\in \\bullet t}{W(p,t)} + \\sum _{t \\in T_{ts}, p \\in R(t)}(M_{ts}(p)), q1 > q, 0 \\le ts \\le k \\rbrace $ per definition REF (conflicting transitions) $A_{17}=\\lbrace could\\_not\\_have(t,ts) : t \\in T, (\\forall p \\in \\bullet t, W(p,t) \\le M_{ts}(p)), t \\notin T_{ts}, (\\exists p \\in \\bullet t : W(p,t) > M_{ts}(p) - (\\sum _{t^{\\prime } \\in T_{ts}, p \\in \\bullet t^{\\prime }}{W(p,t^{\\prime })} + \\sum _{t^{\\prime } \\in T_{ts}, p \\in R(t^{\\prime })} M_{ts}(p)), 0 \\le ts \\le k \\rbrace $ per the maximal firing set semantics $A_{18}=\\lbrace holds(p,q,ts+1) : p \\in P, q=M_{ts+1}(p), 0 \\le ts < k\\rbrace $ , where $M_{ts+1}(p) = M_{ts}(p) - (\\sum _{\\begin{array}{c}t \\in T_{ts}, p \\in \\bullet t\\end{array}}{W(p,t)} + $ $\\sum _{t \\in T_{ts}, p \\in R(t)}M_{ts}(p)) + \\\\ \\sum _{\\begin{array}{c}t \\in T_{ts}, p \\in t \\bullet \\end{array}}{W(t,p)}$ according to definition REF (firing set execution) $A_{19}=\\lbrace ptarc(p,t,n,ts) : p \\in R(t), n = M_{ts}(p), n > 0, 0 \\le ts \\le k \\rbrace $ $A_{20}=\\lbrace iptarc(p,t,1,ts) : p \\in P, 0 \\le ts < k \\rbrace $ We show that $A$ satisfies (REF ) and (REF ), and $A$ is an answer set of $\\Pi ^3$ .", "$A$ satisfies (REF ) and (REF ) by its construction above.", "We show $A$ is an answer set of $\\Pi ^3$ by splitting.", "We split $lit(\\Pi ^3)$ into a sequence of $7k+9$ sets: $U_0= head(f\\ref {f:place}) \\cup head(f\\ref {f:trans}) \\cup head(f\\ref {f:time}) \\cup head(f\\ref {f:num}) \\cup head(i\\ref {i:holds}) = \\lbrace place(p) : p \\in P\\rbrace \\cup \\lbrace trans(t) : t \\in T\\rbrace \\cup \\lbrace time(0), \\dots , time(k)\\rbrace \\cup \\lbrace num(0), \\dots , num(ntok)\\rbrace \\cup \\lbrace holds(p,q,0) : p \\in P, q=M_0(p) \\rbrace $ $U_{7k+1}=U_{7k+0} \\cup head(f\\ref {f:r:ptarc})^{ts=k} \\cup head(f\\ref {f:r:tparc})^{ts=k} \\cup head(f\\ref {f:rptarc})^{ts=k} \\cup head(f\\ref {f:iptarc})^{ts=k} = U_{7k+0} \\cup \\lbrace ptarc(p,t,n,k) : (p,t) \\in E^-, n=W(p,t) \\rbrace \\cup \\lbrace tparc(t,p,n,k) : (t,p) \\in E^+, n=W(t,p) \\rbrace \\cup \\lbrace ptarc(p,t,n,k) : p \\in R(t), n=M_{k}(p), n > 0 \\rbrace \\cup \\\\ \\lbrace iptarc(p,t,1,k) : p \\in I(t) \\rbrace $ $U_{7k+2}=U_{7k+1} \\cup head(e\\ref {e:r:ne:ptarc} )^{ts=k} \\cup head(e\\ref {e:ne:iptarc})^{ts=k} = U_{7k+1} \\cup \\lbrace notenabled(t,k) : t \\in T \\rbrace $ $U_{7k+3}=U_{7k+2} \\cup head(e\\ref {e:enabled})^{ts=k} = U_{7k+2} \\cup \\lbrace enabled(t,k) : t \\in T \\rbrace $ $U_{7k+4}=U_{7k+3} \\cup head(a\\ref {a:fires})^{ts=k} = U_{7k+3} \\cup \\lbrace fires(t,k) : t \\in T \\rbrace $ $U_{7k+5}=U_{7k+4} \\cup head(r\\ref {r:r:add} )^{ts=k} \\cup head(r\\ref {r:r:del} )^{ts=k} = U_{7k+4} \\cup \\lbrace add(p,q,t,k) : p \\in P, t \\in T, q=W(t,p) \\rbrace \\cup \\lbrace del(p,q,t,k) : p \\in P, t \\in T, q=W(p,t) \\rbrace \\cup \\lbrace del(p,q,t,k) : p \\in P, t \\in T, q=M_{k}(p) \\rbrace $ $U_{7k+6}=U_{7k+5} \\cup head(r\\ref {r:totincr})^{ts=k} \\cup head(r\\ref {r:totdecr})^{ts=k} = U_{7k+5} \\cup \\lbrace tot\\_incr(p,q,k) : p \\in P, 0 \\le q \\le ntok \\rbrace \\cup \\lbrace tot\\_decr(p,q,k) : p \\in P, 0 \\le q \\le ntok \\rbrace $ $U_{7k+7}=U_{7k+6} \\cup head(r\\ref {r:nextstate})^{ts=k} \\cup head(a\\ref {a:overc:place})^{ts=k} \\cup head(a\\ref {a:r:maxfire:cnh})^{ts=k} = $ $U_{7k+6} \\cup \\\\ \\lbrace consumesmore(p,k) : p \\in P\\rbrace \\cup \\lbrace holds(p,q,k+1) : p \\in P, 0 \\le q \\le ntok \\rbrace \\cup \\lbrace could\\_not\\_have(t,k) : t \\in T \\rbrace $ $U_{7k+8}=U_{7k+7} \\cup head(a\\ref {a:overc:gen})^{ts=k} = U_{7k+7} \\cup \\lbrace consumesmore \\rbrace $ where $head(r_i)^{ts=k}$ are head atoms of ground rule $r_i$ in which $ts=k$ .", "We write $A_i^{ts=k} = \\lbrace a(\\dots ,ts) : a(\\dots ,ts) \\in A_i, ts=k \\rbrace $ as short hand for all atoms in $A_i$ with $ts=k$ .", "$U_{\\alpha }, 0 \\le \\alpha \\le 7k+8$ form a splitting sequence, since each $U_i$ is a splitting set of $\\Pi ^1$ , and $\\langle U_{\\alpha }\\rangle _{\\alpha < \\mu }$ is a monotone continuous sequence, where $U_0 \\subseteq U_1 \\dots \\subseteq U_{7k+8}$ and $\\bigcup _{\\alpha < \\mu }{U_{\\alpha }} = lit(\\Pi ^1)$ .", "We compute the answer set of $\\Pi ^3$ using the splitting sets as follows: $bot_{U_0}(\\Pi ^3) = f\\ref {f:place} \\cup f\\ref {f:trans} \\cup f\\ref {f:time} \\cup i\\ref {i:holds} \\cup f\\ref {f:num}$ and $X_0 = A_1 \\cup \\dots \\cup A_4 \\cup A_7$ ($= U_0$ ) is its answer set – using forced atom proposition $eval_{U_0}(bot_{U_1}(\\Pi ^3) \\setminus bot_{U_0}(\\Pi ^3), X_0) = \\lbrace ptarc(p,t,q,0) \\text{:-}.", "\| q=W(p,t) \\rbrace \\cup \\\\ \\lbrace tparc(t,p,q,0) \\text{:-}.", "\| q=W(t,p) \\rbrace \\cup \\lbrace ptarc(p,t,q,0) \\text{:-}.", "\| q=M_0(p) \\rbrace \\cup \\\\ \\lbrace iptarc(p,t,1,0) \\text{:-}.", "\\rbrace $ .", "Its answer set $X_1=A_5^{ts=0} \\cup A_6^{ts=0} \\cup A_{19}^{ts=0} \\cup A_{20}^{ts=0}$ – using forced atom proposition and construction of $A_5, A_6, A_{19}, A_{20}$ .", "$eval_{U_1}(bot_{U_2}(\\Pi ^3) \\setminus bot_{U_1}(\\Pi ^3), X_0 \\cup X_1) = \\lbrace notenabled(t,0) \\text{:-} .", "\| (\\lbrace trans(t), \\\\ ptarc(p,t,n,0), holds(p,q,0) \\rbrace \\subseteq X_0 \\cup X_1, \\text{~where~} q < n) \\text{~or~} \\lbrace notenabled(t,0) \\text{:-} .", "\| \\\\ (\\lbrace trans(t), iptarc(p,t,n2,0), holds(p,q,0) \\rbrace \\subseteq X_0 \\cup X_1, \\text{~where~} q \\ge n2 \\rbrace \\rbrace $ .", "Its answer set $X_2=A_8^{ts=0}$ – using forced atom proposition and construction of $A_8$ .", "where, $q=M_0(p)$ , and $n=W(p,t)$ for an arc $(p,t) \\in E^-$ – by construction $i\\ref {i:holds}$ and $f\\ref {f:r:ptarc}$ in $\\Pi ^3$ , and in an arc $(p,t) \\in E^-$ , $p \\in \\bullet t$ (by definition REF of preset) $n2=1$ – by construction of $iptarc$ predicates in $\\Pi ^3$ , meaning $q \\ge n2 \\equiv q \\ge 1 \\equiv q > 0$ , thus, $notenabled(t,0) \\in X_1$ represents $(\\exists p \\in \\bullet t : M_0(p) < W(p,t)) \\vee (\\exists p \\in I(t) : M_0(p) > 0)$ .", "$eval_{U_2}(bot_{U_3}(\\Pi ^3) \\setminus bot_{U_2}(\\Pi ^3), X_0 \\cup \\dots \\cup X_2) = \\lbrace enabled(t,0) \\text{:-}.", "\| trans(t) \\in X_0 \\cup \\dots \\cup X_2, notenabled(t,0) \\notin X_0 \\cup \\dots \\cup X_2 \\rbrace $ .", "Its answer set is $X_3 = A_9^{ts=0}$ – using forced atom proposition and construction of $A_9$ .", "since an $enabled(t,0) \\in X_3$ if $\\nexists ~notenabled(t,0) \\in X_0 \\cup \\dots \\cup X_2$ ; which is equivalent to $(\\nexists p \\in \\bullet t : M_0(p) < W(p,t)) \\wedge (\\nexists p \\in I(t) : M_0(p) > 0) \\equiv (\\forall p \\in \\bullet t: M_0(p) \\ge W(p,t)) \\wedge (\\forall p \\in I(t) : M_0(p) = 0)$ .", "$eval_{U_3}(bot_{U_4}(\\Pi ^3) \\setminus bot_{U_3}(\\Pi ^3), X_0 \\cup \\dots \\cup X_3) = \\lbrace \\lbrace fires(t,0)\\rbrace \\text{:-}.", "\| enabled(t,0) \\\\ \\text{~holds in~} X_0 \\cup \\dots \\cup X_3 \\rbrace $ .", "It has multiple answer sets $X_{4.1}, \\dots , X_{4.n}$ , corresponding to elements of power set of $fires(t,0)$ atoms in $eval_{U_3}(...)$ – using supported rule proposition.", "Since we are showing that the union of answer sets of $\\Pi ^3$ determined using splitting is equal to $A$ , we only consider the set that matches the $fires(t,0)$ elements in $A$ and call it $X_4$ , ignoring the rest.", "Thus, $X_4 = A_{10}^{ts=0}$ , representing $T_0$ .", "in addition, for every $t$ such that $enabled(t,0) \\in X_0 \\cup \\dots \\cup X_3, R(t) \\ne \\emptyset $ ; $fires(t,0) \\in X_4$ – per definition REF (firing set); requiring that a reset transition is fired when enabled thus, the firing set $T_0$ will not be eliminated by the constraint $f\\ref {f:c:rptarc:elim}$ $eval_{U_4}(bot_{U_5}(\\Pi ^3) \\setminus bot_{U_4}(\\Pi ^3), X_0 \\cup \\dots \\cup X_4) = \\lbrace add(p,n,t,0) \\text{:-}.", "\| \\lbrace fires(t,0), \\\\ tparc(t,p,n,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_4 \\rbrace \\cup \\lbrace del(p,n,t,0) \\text{:-}.", "\| \\lbrace fires(t,0), ptarc(p,t,n,0) \\rbrace \\\\ \\subseteq X_0 \\cup \\dots \\cup X_4 \\rbrace $ .", "It's answer set is $X_5 = A_{11}^{ts=0} \\cup A_{12}^{ts=0}$ – using forced atom proposition and definitions of $A_{11}$ and $A_{12}$ .", "where, each $add$ atom is equivalent to $n=W(t,p) : p \\in t \\bullet $ , and each $del$ atom is equivalent to $n=W(p,t) : p \\in \\bullet t$ ; or $n=M_k(p) : p \\in R(t)$ , representing the effect of transitions in $T_0$ .", "$eval_{U_5}(bot_{U_6}(\\Pi ^3) \\setminus bot_{U_5}(\\Pi ^3), X_0 \\cup \\dots \\cup X_5) = \\lbrace tot\\_incr(p,qq,0) \\text{:-}.", "\| $ $qq=\\sum _{add(p,q,t,0) \\in X_0 \\cup \\dots \\cup X_5}{q} \\rbrace \\cup \\lbrace tot\\_decr(p,qq,0) \\text{:-}.", "\| qq=\\sum _{del(p,q,t,0) \\in X_0 \\cup \\dots \\cup X_5}{q} \\rbrace $ .", "It's answer set is $X_6 = A_{13}^{ts=0} \\cup A_{14}^{ts=0}$ – using forced atom proposition and definitions of $A_{13}$ and $A_{14}$ .", "where, each $tot\\_incr(p,qq,0)$ , $qq=\\sum _{add(p,q,t,0) \\in X_0 \\cup \\dots X_5}{q}$ $\\equiv qq=\\sum _{t \\in X_4, p \\in t \\bullet }{W(p,t)}$ , and each $tot\\_decr(p,qq,0)$ , $qq=\\sum _{del(p,q,t,0) \\in X_0 \\cup \\dots X_5}{q}$ $\\equiv qq=\\sum _{t \\in X_4, p \\in \\bullet t}{W(t,p)} + \\sum _{t \\in X_4, p \\in R(t)}{M_{k}(p)}$ , represent the net effect of transitions in $T_0$ .", "$eval_{U_6}(bot_{U_7}(\\Pi ^3) \\setminus bot_{U_6}(\\Pi ^3), X_0 \\cup \\dots \\cup X_6) = \\lbrace consumesmore(p,0) \\text{:-}.", "\| \\\\ \\lbrace holds(p,q,0), tot\\_decr(p,q1,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_6, q1 > q \\rbrace \\cup \\lbrace holds(p,q,1) \\text{:-}., \| \\\\ \\lbrace holds(p,q1,0), tot\\_incr(p,q2,0), tot\\_decr(p,q3,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_6, q=q1+q2-q3 \\rbrace \\cup \\lbrace could\\_not\\_have(t,0) \\text{:-}.", "\| \\lbrace enabled(t,0), ptarc(s,t,q), holds(s,qq,0), \\\\ tot\\_decr(s,qqq,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_6, fires(t,0) \\notin (X_0 \\cup \\dots \\cup X_6), q > qq-qqq \\rbrace $ .", "It's answer set is $X_7 = A_{15}^{ts=0} \\cup A_{17}^{ts=0} \\cup A_{18}^{ts=0}$ – using forced atom proposition and definitions of $A_{15}, A_{17}, A_{18}, A_9$ .", "where, $consumesmore(p,0)$ represents $\\exists p : q=M_0(p), q1= \\\\ \\sum _{t \\in T_0, p \\in \\bullet t}{W(p,t)}+\\sum _{t \\in T_0, p \\in R(t)}{M_k(p)}, q1 > q$ , indicating place $p$ will be over consumed if $T_0$ is fired, as defined in definition REF (conflicting transitions) $holds(p,q,1)$ represents $q=M_1(p)$ – by construction of $\\Pi ^3$ and $could\\_not\\_have(t,0)$ represents enabled transition $t \\in T_0$ that could not fire due to insufficient tokens $X_7$ does not contain $could\\_not\\_have(t,0)$ , when $enabled(t,0) \\in X_0 \\cup \\dots \\cup X_5$ and $fires(t,0) \\notin X_0 \\cup \\dots \\cup X_6$ due to construction of $A$ , encoding of $a\\ref {a:r:maxfire:cnh}$ and its body atoms.", "As a result it is not eliminated by the constraint $a\\ref {a:maxfire:elim}$ $ \\vdots $ $eval_{U_{7k+0}}(bot_{U_{7k+1}}(\\Pi ^3) \\setminus bot_{U_{7k+0}}(\\Pi ^3), X_0 \\cup \\dots \\cup X_{7k+0}) = \\lbrace ptarc(p,t,q,k) \\text{:-}.", "\| q=W(p,t) \\rbrace \\cup \\lbrace tparc(t,p,q,k) \\text{:-}.", "\| q=W(t,p) \\rbrace \\cup \\lbrace ptarc(p,t,q,k) \\text{:-}.", "\| q=M_k(p) \\rbrace \\cup \\lbrace iptarc(p,t,1,k) \\text{:-}.", "\\rbrace $ .", "Its answer set $X_{7k+1}=A_5^{ts=k} \\cup A_6^{ts=k} \\cup A_{19}^{ts=k} \\cup A_{20}^{ts=k}$ – using forced atom proposition and construction of $A_5, A_6, A_{19}, A_{20}$ .", "$eval_{U_{7k+1}}(bot_{U_{7k+2}}(\\Pi ^3) \\setminus bot_{U_{7k+1}}(\\Pi ^3), X_0 \\cup \\dots \\cup X_{7k+1}) = \\lbrace notenabled(t,k) \\text{:-} .", "\| \\\\ (\\lbrace trans(t), ptarc(p,t,n,k), holds(p,q,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{7k+1}, \\text{~where~} q < n) \\rbrace \\text{~or~} $ $\\lbrace notenabled(t,k) \\text{:-} .", "\| (\\lbrace trans(t), iptarc(p,t,n2,k), holds(p,q,k) \\rbrace \\subseteq \\\\ X_0 \\cup \\dots \\cup X_{7k+1}, \\text{~where~} q \\ge n2 \\rbrace \\rbrace $ .", "Its answer set $X_{7k+2}=A_8^{ts=k}$ – using forced atom proposition and construction of $A_8$ .", "where, $q=M_k(p)$ , and $n=W(p,t)$ for an arc $(p,t) \\in E^-$ – by construction of $holds$ and $ptarc$ predicates in $\\Pi ^3$ , and in an arc $(p,t) \\in E^-$ , $p \\in \\bullet t$ (by definition REF of preset) $n2=1$ – by construction of $iptarc$ predicates in $\\Pi ^3$ , meaning $q \\ge n2 \\equiv q \\ge 1 \\equiv q > 0$ , thus, $notenabled(t,k) \\in X_{7k+1}$ represents $(\\exists p \\in \\bullet t : M_k(p) < W(p,t)) \\vee (\\exists p \\in I(t) : M_k(p) > k)$ .", "$eval_{U_{7k+2}}(bot_{U_{7k+3}}(\\Pi ^3) \\setminus bot_{U_{7k+2}}(\\Pi ^3), X_0 \\cup \\dots \\cup X_{7k+2}) = \\lbrace enabled(t,k) \\text{:-}.", "\| \\\\ trans(t) \\in X_0 \\cup \\dots \\cup X_{7k+2} \\wedge notenabled(t,k) \\notin X_0 \\cup \\dots \\cup X_{7k+2} \\rbrace $ .", "Its answer set is $X_{7k+3} = A_9^{ts=k}$ – using forced atom proposition and construction of $A_9$ .", "since an $enabled(t,k) \\in X_{7k+3}$ if $\\nexists ~notenabled(t,k) \\in X_0 \\cup \\dots \\cup X_{7k+2}$ ; which is equivalent to $(\\nexists p \\in \\bullet t : M_k(p) < W(p,t)) \\wedge (\\nexists p \\in I(t) : M_k(p) > k) \\equiv (\\forall p \\in \\bullet t: M_k(p) \\ge W(p,t)) \\wedge (\\forall p \\in I(t) : M_k(p) = k)$ .", "$eval_{U_{7k+3}}(bot_{U_{7k+4}}(\\Pi ^3) \\setminus bot_{U_{7k+3}}(\\Pi ^3), X_0 \\cup \\dots \\cup X_{7k+3}) = \\lbrace \\lbrace fires(t,k)\\rbrace \\text{:-}.", "\| \\\\ enabled(t,k) \\\\ \\text{~holds in~} X_0 \\cup \\dots \\cup X_{7k+3} \\rbrace $ .", "It has multiple answer sets $X_{7k+4.1}, \\dots , X_{1k+4.n}$ , corresponding to elements of power set of $fires(t,k)$ atoms in $eval_{U_{7k+3}}(...)$ – using supported rule proposition.", "Since we are showing that the union of answer sets of $\\Pi ^3$ determined using splitting is equal to $A$ , we only consider the set that matches the $fires(t,k)$ elements in $A$ and call it $X_{7k+4}$ , ignoring the rest.", "Thus, $X_{7k+4} = A_{10}^{ts=k}$ , representing $T_k$ .", "in addition, for every $t$ such that $enabled(t,k) \\in X_0 \\cup \\dots \\cup X_{7k+3}, R(t) \\ne \\emptyset $ ; $fires(t,k) \\in X_{7k+4}$ – per definition REF (firing set); requiring that a reset transition is fired when enabled thus, the firing set $T_k$ will not be eliminated by the constraint $f\\ref {f:c:rptarc:elim}$ $eval_{U_{7k+4}}(bot_{U_{7k+5}}(\\Pi ^3) \\setminus bot_{U_{7k+4}}(\\Pi ^3), X_0 \\cup \\dots \\cup X_{7k+4}) = \\lbrace add(p,n,t,k) \\text{:-}.", "\| \\\\ \\lbrace fires(t,k), tparc(t,p,n,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{7k+4} \\rbrace \\cup \\lbrace del(p,n,t,k) \\text{:-}.", "\| \\lbrace fires(t,k), \\\\ ptarc(p,t,n,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{7k+4} \\rbrace $ .", "It's answer set is $X_{7k+5} = A_{11}^{ts=k} \\cup A_{12}^{ts=k}$ – using forced atom proposition and definitions of $A_{11}$ and $A_{12}$ .", "where, each $add$ atom is equivalent to $n=W(t,p) : p \\in t \\bullet $ , and each $del$ atom is equivalent to $n=W(p,t) : p \\in \\bullet t$ ; or $n=M_k(p) : p \\in R(t)$ , representing the effect of transitions in $T_k$ $eval_{U_{7k+5}}(bot_{U_{7k+6}}(\\Pi ^3) \\setminus bot_{U_{7k+5}}(\\Pi ^3), X_0 \\cup \\dots \\cup X_{7k+5}) = \\\\ \\lbrace tot\\_incr(p,qq,k) \\text{:-}.", "\| $ $qq=\\sum _{add(p,q,t,k) \\in X_0 \\cup \\dots \\cup X_{7k+5}}{q} \\rbrace \\cup $ $\\lbrace tot\\_decr(p,qq,k) \\text{:-}.", "\| $ $qq=\\sum _{del(p,q,t,k) \\in X_0 \\cup \\dots \\cup X_{7k+5}}{q} \\rbrace $ .", "It's answer set is $X_{7k+6} = A_{13}^{ts=k} \\cup A_{14}^{ts=k}$ – using forced atom proposition and definitions of $A_{13}$ and $A_{14}$ .", "where, each $tot\\_incr(p,qq,k)$ , $qq=\\sum _{add(p,q,t,k) \\in X_0 \\cup \\dots X_{7k+5}}{q}$ $\\equiv qq=\\sum _{t \\in X_{7k+4}, p \\in t \\bullet }{W(p,t)}$ , and each $tot\\_decr(p,qq,k)$ , $qq=\\sum _{del(p,q,t,k) \\in X_0 \\cup \\dots X_{7k+5}}{q}$ $\\equiv qq=\\sum _{t \\in X_{7k+4}, p \\in \\bullet t}{W(t,p)} + \\sum _{t \\in X_{7k+4}, p \\in R(t)}{M_{k}(p)}$ , representing the net effect of transitions in $T_k$ $eval_{U_{7k+6}}(bot_{U_{7k+7}}(\\Pi ^3) \\setminus bot_{U_{7k+6}}(\\Pi ^3), X_0 \\cup \\dots \\cup X_{7k+6}) = \\lbrace consumesmore(p,k) \\text{:-}.", "\| \\\\ \\lbrace holds(p,q,k), tot\\_decr(p,q1,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{7k+6}, q1 > q \\rbrace \\cup \\lbrace holds(p,q,1) \\text{:-}., \| \\\\ \\lbrace holds(p,q1,k), tot\\_incr(p,q2,k), tot\\_decr(p,q3,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{7k+6}, q=q1+q2-q3 \\rbrace \\cup \\lbrace could\\_not\\_have(t,k) \\text{:-}.", "\| \\lbrace enabled(t,k), ptarc(s,t,q), holds(s,qq,k), \\\\ tot\\_decr(s,qqq,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{7k+6}, fires(t,k) \\notin (X_0 \\cup \\dots \\cup X_{7k+6}), q > qq-qqq \\rbrace $ .", "It's answer set is $X_{7k+7} = A_{15}^{ts=k} \\cup A_{17}^{ts=k} \\cup A_{18}^{ts=k}$ – using forced atom proposition and definitions of $A_{15}, A_{17}, A_{18}, A_9$ .", "where, $consumesmore(p,k)$ represents $\\exists p : q=M_k(p), \\\\ q1=\\sum _{t \\in T_k, p \\in \\bullet t}{W(p,t)}+\\sum _{t \\in T_k, p \\in R(t)}{M_k(p)}, q1 > q$ , indicating place $p$ that will be over consumed if $T_k$ is fired, as defined in definition REF (conflicting transitions), $holds(p,q,k+1)$ represents $q=M_{k+1}(p)$ – by construction of $\\Pi ^3$ and $could\\_not\\_have(t,k)$ represents enabled transition $t$ in $T_k$ that could not be fired due to insufficient tokens $X_{7k+7}$ does not contain $could\\_not\\_have(t,k)$ , when $enabled(t,k) \\in X_0 \\cup \\dots \\cup X_{7k+6}$ and $fires(t,k) \\notin X_0 \\cup \\dots \\cup X_{7k+6}$ due to construction of $A$ , encoding of $a\\ref {a:r:maxfire:cnh}$ and its body atoms.", "As a result it is not eliminated by the constraint $a\\ref {a:maxfire:elim}$ $eval_{U_{7k+7}}(bot_{U_{7k+8}}(\\Pi ^3) \\setminus bot_{U_{7k+7}}(\\Pi ^3), X_0 \\cup \\dots \\cup X_{7k+7}) = \\lbrace consumesmore \\text{:-}.", "\| \\\\ \\lbrace consumesmore(p,0),\\dots ,$ $consumesmore(p,k) \\rbrace \\cap (X_0 \\cup \\dots \\cup X_{7k+7}) \\ne \\emptyset \\rbrace $ .", "It's answer set is $X_{7k+8} = A_{16}$ – using forced atom proposition $X_{7k+8}$ will be empty since none of $consumesmore(p,0),\\dots , \\\\ consumesmore(p,k)$ hold in $X_0 \\cup \\dots \\cup X_{7k+7}$ due to the construction of $A$ , encoding of $a\\ref {a:overc:place}$ and its body atoms.", "As a result, it is not eliminated by the constraint $a\\ref {a:overc:elim}$ The set $X = X_0 \\cup \\dots \\cup X_{7k+8}$ is the answer set of $\\Pi ^3$ by the splitting sequence theorem REF .", "Each $X_i, 0 \\le i \\le 7k+8$ matches a distinct portion of $A$ , and $X = A$ , thus $A$ is an answer set of $\\Pi ^3$ .", "Next we show (REF ): Given $\\Pi ^3$ be the encoding of a Petri Net $PN(P,T,E,W,R,I)$ with initial marking $M_0$ , and $A$ be an answer set of $\\Pi ^3$ that satisfies (REF ) and (REF ), then we can construct $X=M_0,T_0,\\dots ,M_k,T_k,M_{k+1}$ from $A$ , such that it is an execution sequence of $PN$ .", "We construct the $X$ as follows: $M_i = (M_i(p_0), \\dots , M_i(p_n))$ , where $\\lbrace holds(p_0,M_i(p_0),i), \\dots holds(p_n,M_i(p_n),i) \\rbrace \\\\ \\subseteq A$ , for $0 \\le i \\le k+1$ $T_i = \\lbrace t : fires(t,i) \\in A\\rbrace $ , for $0 \\le i \\le k$ and show that $X$ is indeed an execution sequence of $PN$ .", "We show this by induction over $k$ (i.e.", "given $M_k$ , $T_k$ is a valid firing set and its firing produces marking $M_{k+1}$ ).", "Base case: Let $k=0$ , and $M_0$ is a valid marking in $X$ for $PN$ , show [(1)] $T_0$ is a valid firing set for $M_0$ , and $M_1$ is $T_0$ 's target marking w.r.t.", "$M_0$ .", "We show $T_0$ is a valid firing set for $M_0$ .", "Let $\\lbrace fires(t_0,0), \\dots , fires(t_x,0) \\rbrace $ be the set of all $fires(\\dots ,0)$ atoms in $A$ , Then for each $fires(t_i,0) \\in A$ $enabled(t_i,0) \\in A$ – from rule $a\\ref {a:fires}$ and supported rule proposition Then $notenabled(t_i,0) \\notin A$ – from rule $e\\ref {e:enabled}$ and supported rule proposition Then $body(e\\ref {e:r:ne:ptarc} )$ must not hold in $A$ and $body(e\\ref {e:ne:iptarc} )$ must not hold in $A$ – from rules $e\\ref {e:r:ne:ptarc} ,e\\ref {e:ne:iptarc} $ and forced atom proposition Then $q \\lnot < n_i \\equiv q \\ge n_i$ in $e\\ref {e:r:ne:ptarc}$ for all $\\lbrace holds(p,q,0), ptarc(p,t_i,n_i,0)\\rbrace \\subseteq A$ – from $e\\ref {e:r:ne:ptarc}$ , forced atom proposition, and given facts ($holds(p,q,0) \\in A, ptarc(p,t_i,n_i,0) \\in A$ ) And $q \\lnot \\ge n_i \\equiv q < n_i$ in $e\\ref {e:ne:iptarc} $ for all $\\lbrace holds(p,q,0), iptarc(p,t_i,n_i,0) \\rbrace \\subseteq A, n_i=1$ ; $q > n_i \\equiv q = 0$ – from $e\\ref {e:ne:iptarc} $ , forced atom proposition, given facts ($holds(p,q,0) \\in A, iptarc(p,t_i,1,0) \\in A$ ), and $q$ is a positive integer Then $(\\forall p \\in \\bullet t_i, M_0(p) \\ge W(p,t_i)) \\wedge (\\forall p \\in I(t_i), M_0(p) = 0)$ – from $holds(p,q,0) \\in A$ represents $q=M_0(p)$ – rule $i\\ref {i:holds}$ construction $ptarc(p,t_i,n_i,0) \\in A$ represents $n_i=W(p,t_i)$ – rule $f\\ref {f:r:ptarc}$ construction; or it represents $n_i = M_0(p)$ – rule $f\\ref {f:rptarc}$ construction; the construction of $f\\ref {f:rptarc}$ ensures that $notenabled(t,0)$ is never true due to the reset arc definition REF of preset $\\bullet t_i$ in PN Then $t_i$ is enabled and can fire in $PN$ , as a result it can belong to $T_0$ – from definition REF of enabled transition And $consumesmore \\notin A$ , since $A$ is an answer set of $\\Pi ^3$ – from rule $a\\ref {a:overc:elim}$ and supported rule proposition Then $\\nexists consumesmore(p,0) \\in A$ – from rule $a\\ref {a:overc:gen}$ and supported rule proposition Then $\\nexists \\lbrace holds(p,q,0), tot\\_decr(p,q1,0) \\rbrace \\subseteq A : q1>q$ in $body(a\\ref {a:overc:place})$ – from $a\\ref {a:overc:place}$ and forced atom proposition Then $\\nexists p : (\\sum _{t_i \\in \\lbrace t_0,\\dots ,t_x\\rbrace , p \\in \\bullet t_i}{W(p,t_i)}+\\sum _{t_i \\in \\lbrace t_0,\\dots ,t_x\\rbrace , p \\in R(t_i)}{M_0(p)}) > M_0(p)$ – from the following $holds(p,q,0)$ represents $q=M_0(p)$ – from rule $i\\ref {i:holds}$ encoding, given $tot\\_decr(p,q1,0) \\in A$ if $\\lbrace del(p,q1_0,t_0,0), \\dots , del(p,q1_x,t_x,0) \\rbrace \\subseteq A$ , where $q1 = q1_0+\\dots +q1_x$ – from $r\\ref {r:totdecr}$ and forced atom proposition $del(p,q1_i,t_i,0) \\in A$ if $\\lbrace fires(t_i,0), ptarc(p,t_i,q1_i,0) \\rbrace \\subseteq A$ – from $r\\ref {r:r:del} $ and supported rule proposition $del(p,q1_i,t_i,0)$ either represents removal of $q1_i = W(p,t_i)$ tokens from $p \\in \\bullet t_i$ ; or it represents removal of $q1_i = M_0(p)$ tokens from $p \\in R(t_i)$ – from rule $r\\ref {r:r:del} $ , supported rule proposition, and definition REF of transition execution in $PN$ Then the set of transitions in $T_0$ do not conflict – by the definition REF of conflicting transitions And for each $enabled(t_j,0) \\in A$ and $fires(t_j,0) \\notin A$ , $could\\_not\\_have(t_j,0) \\in A$ , since $A$ is an answer set of $\\Pi ^3$ - from rule $a\\ref {a:maxfire:elim}$ and supported rule proposition Then $\\lbrace enabled(t_j,0), holds(s,qq,0), ptarc(s,t_j,q,0), \\\\ tot\\_decr(s,qqq,0) \\rbrace \\subseteq A$ , such that $q > qq - qqq$ and $fires(t_j,0) \\notin A$ - from rule $a\\ref {a:r:maxfire:cnh}$ and supported rule proposition Then for an $s \\in \\bullet t_j \\cup R(t_j)$ , $q > M_0(s) - (\\sum _{t_i \\in T_0, s \\in \\bullet t_i}{W(s,t_i)} + \\sum _{t_i \\in T_0, s \\in R(t_i)}{M_0(s))}$ , where $q=W(s,t_j) \\text{~if~} s \\in \\bullet t_j, \\text{~or~} M_0(s) \\text{~otherwise}$ - from the following: $ptarc(s,t_i,q,0)$ represents $q=W(s,t_i)$ if $(s,t_i) \\in E^-$ or $q=M_0(s)$ if $s \\in R(t_i)$ – from rule $f\\ref {f:r:ptarc},f\\ref {f:rptarc}$ construction $holds(s,qq,0)$ represents $qq=M_0(s)$ – from $i\\ref {i:holds}$ construction $tot\\_decr(s,qqq,0) \\in A$ if $\\lbrace del(s,qqq_0,t_0,0), \\dots , del(s,qqq_x,t_x,0) \\rbrace \\subseteq A$ – from rule $r\\ref {r:totdecr}$ construction and supported rule proposition $del(s,qqq_i,t_i,0) \\in A$ if $\\lbrace fires(t_i,0), ptarc(s,t_i,qqq_i,0) \\rbrace \\subseteq A$ – from rule $r\\ref {r:r:del} $ and supported rule proposition $del(s,qqq_i,t_i,0)$ represents $qqq_i = W(s,t_i) : t_i \\in T_0, (s,t_i) \\in E^-$ , or $qqq_i = M_0(t_i) : t_i \\in T_0, s \\in R(t_i)$ – from rule $f\\ref {f:r:ptarc},f\\ref {f:rptarc}$ construction $tot\\_decr(q,qqq,0)$ represents $\\sum _{t_i \\in T_0, s \\in \\bullet t_i}{W(s,t_i)} + \\\\ \\sum _{t_i \\in T_0, s \\in R(t_i)}{M_0(s)}$ – from (C,D,E) above Then firing $T_0 \\cup \\lbrace t_j \\rbrace $ would have required more tokens than are present at its source place $s \\in \\bullet t_j \\cup R(t_j)$ .", "Thus, $T_0$ is a maximal set of transitions that can simultaneously fire.", "And for each reset transition $t_r$ with $enabled(t_r,0) \\in A$ , $fires(t_r,0) \\in A$ , since $A$ is an answer set of $\\Pi ^2$ - from rule $f\\ref {f:c:rptarc:elim}$ and supported rule proposition Then, the firing set $T_0$ satisfies the reset-transition requirement of definition REF (firing set) Then $\\lbrace t_0, \\dots , t_x\\rbrace = T_0$ – using 1(a),1(b),1(d) above; and using 1(c) it is a maximal firing set We show $M_1$ is produced by firing $T_0$ in $M_0$ .", "Let $holds(p,q,1) \\in A$ Then $\\lbrace holds(p,q1,0), tot\\_incr(p,q2,0), tot\\_decr(p,q3,0) \\rbrace \\subseteq A : q=q1+q2-q3$ – from rule $r\\ref {r:nextstate}$ and supported rule proposition Then $holds(p,q1,0) \\in A$ represents $q1=M_0(p)$ – given, rule $i\\ref {i:holds}$ construction; and $\\lbrace add(p,q2_0,t_0,0), \\dots , $ $add(p,q2_j,t_j,0)\\rbrace \\subseteq A : q2_0 + \\dots + q2_j = q2$ and $\\lbrace del(p,q3_0,t_0,0), \\dots , $ $del(p,q3_l,t_l,0)\\rbrace \\subseteq A : q3_0 + \\dots + q3_l = q3$ – rules $r\\ref {r:totincr},r\\ref {r:totdecr}$ and supported rule proposition, respectively Then $\\lbrace fires(t_0,0), \\dots , fires(t_j,0) \\rbrace \\subseteq A$ and $\\lbrace fires(t_0,0), \\dots , fires(t_l,0) \\rbrace \\\\ \\subseteq A$ – rules $r\\ref {r:r:add},r\\ref {r:r:del}$ and supported rule proposition, respectively Then $\\lbrace fires(t_0,0), \\dots , $ $fires(t_j,0) \\rbrace \\cup \\lbrace fires(t_0,0), \\dots , $ $fires(t_l,0) \\rbrace \\subseteq A = \\lbrace fires(t_0,0), \\dots , $ $fires(t_x,0) \\rbrace \\subseteq A$ – set union of subsets Then for each $fires(t_x,0) \\in A$ we have $t_x \\in T_0$ – already shown in item REF above Then $q = M_0(p) + \\sum _{t_x \\in T_0 \\wedge p \\in t_x \\bullet }{W(t_x,p)} - (\\sum _{t_x \\in T_0 \\wedge p \\in \\bullet t_x}{W(p,t_x)} + \\\\ \\sum _{t_x \\in T_0 \\wedge p \\in R(t_x)}{M_0(p)})$ – from (REF ) above and the following Each $add(p,q_j,t_j,0) \\in A$ represents $q_j=W(t_j,p)$ for $p \\in t_j \\bullet $ – rule $r\\ref {r:r:add} $ encoding, and definition REF of transition execution in $PN$ Each $del(p,t_y,q_y,0) \\in A$ represents either $q_y=W(p,t_y)$ for $p \\in \\bullet t_y$ , or $q_y=M_0(p)$ for $p \\in R(t_y)$ – from rule $r\\ref {r:r:del} ,f\\ref {f:r:ptarc} $ encoding and definition REF of transition execution in $PN$ ; or from rule $r\\ref {r:r:del} ,f\\ref {f:rptarc}$ encoding and definition of reset arc in $PN$ Each $tot\\_incr(p,q2,0) \\in A$ represents $q2=\\sum _{t_x \\in T_0 \\wedge p \\in t_x \\bullet }{W(t_x,p)}$ – aggregate assignment atom semantics in rule $r\\ref {r:totincr}$ Each $tot\\_decr(p,q3,0) \\in A$ represents $q3=\\sum _{t_x \\in T_0 \\wedge p \\in \\bullet t_x}{W(p,t_x)} + $ $\\sum _{t_x \\in T_0 \\wedge p \\in R(t_x)}{M_0(p)}$ – aggregate assignment atom semantics in rule $r\\ref {r:totdecr}$ Then, $M_1(p) = q$ – since $holds(p,q,1) \\in A$ encodes $q=M_1(p)$ – from construction Inductive Step: Let $k > 0$ , and $M_k$ is a valid marking in $X$ for $PN$ , show [(1)] $T_k$ is a valid firing set for $M_k$ , and firing $T_k$ in $M_k$ produces marking $M_{k+1}$ .", "We show that $T_k$ is a valid firing set in $M_k$ .", "Let $\\lbrace fires(t_0,k), \\dots , fires(t_x,k) \\rbrace $ be the set of all $fires(\\dots ,k)$ atoms in $A$ , Then for each $fires(t_i,k) \\in A$ $enabled(t_i,k) \\in A$ – from rule $a\\ref {a:fires}$ and supported rule proposition Then $notenabled(t_i,k) \\notin A$ – from rule $e\\ref {e:enabled}$ and supported rule proposition Then $body(e\\ref {e:r:ne:ptarc} )$ must not hold in $A$ and $body(e\\ref {e:ne:iptarc} )$ must not hold in $A$ – from rule $e\\ref {e:r:ne:ptarc} ,e\\ref {e:ne:iptarc} $ and forced atom proposition Then $q \\lnot < n_i \\equiv q \\ge n_i$ in $e\\ref {e:r:ne:ptarc}$ for all $\\lbrace holds(p,q,k), ptarc(p,t_i,n_i,k)\\rbrace \\subseteq A$ – from $e\\ref {e:r:ne:ptarc}$ , forced atom proposition, and given facts ($holds(p,q,k) \\in A, ptarc(p,t_i,n_i,k) \\in A$ ) And $q \\lnot \\ge n_i \\equiv q < n_i$ in $e\\ref {e:ne:iptarc} $ for all $\\lbrace holds(p,q,k), iptarc(p,t_i,n_i,k) \\rbrace \\subseteq A, n_i=1; q > n_i \\equiv q = 0$ – from $e\\ref {e:ne:iptarc} $ , forced atom proposition, given facts $(holds(p,q,k) \\in A, iptarc(p,t_i,1,k) \\in A)$ , and $q$ is a positive integer Then $(\\forall p \\in \\bullet t_i, M_k(p) \\ge W(p,t_i)) \\wedge (\\forall p \\in I(t_i), M_k(p) = 0)$ – from $holds(p,q,k) \\in A$ represents $q=M_k(p)$ – inductive assumption, given $ptarc(p,t_i,n_i,k) \\in A$ represents $n_i=W(p,t_i)$ – rule $f\\ref {f:r:ptarc}$ construction; or it represents $n_i=M_k(p)$ – rule $f\\ref {f:rptarc}$ construction; the construction of $f\\ref {f:rptarc}$ ensures that $notenabled(t,0)$ is never true due to the reset arc definition REF of preset $\\bullet t_i$ in PN Then $t_i$ is enabled and can fire in $PN$ , as a result it can belong to $T_k$ – from definition REF of enabled transition And $consumesmore \\notin A$ , since $A$ is an answer set of $\\Pi ^3$ – from rule $a\\ref {a:overc:elim}$ and supported rule proposition Then $\\nexists consumesmore(p,k) \\in A$ – from rule $a\\ref {a:overc:gen}$ and supported rule proposition Then $\\nexists \\lbrace holds(p,q,k), tot\\_decr(p,q1,k) \\rbrace \\subseteq A : q1>q$ in $body(a\\ref {a:overc:place})$ – from $a\\ref {a:overc:place}$ and forced atom proposition Then $\\nexists p : (\\sum _{t_i \\in \\lbrace t_0,\\dots ,t_x\\rbrace , p \\in \\bullet t_i}{W(p,t_i)}+\\sum _{t_i \\in \\lbrace t_0,\\dots ,t_x\\rbrace , p \\in R(t_i)}{M_k(p)}) > M_k(p)$ – from the following $holds(p,q,k)$ represents $q=M_k(p)$ – inductive assumption, construction $tot\\_decr(p,q1,k) \\in A$ if $\\lbrace del(p,q1_0,t_0,k), \\dots , del(p,q1_x,t_x,k) \\rbrace \\subseteq A$ , where $q1 = q1_0+\\dots +q1_x$ – from $r\\ref {r:totdecr}$ and forced atom proposition $del(p,q1_i,t_i,k) \\in A$ if $\\lbrace fires(t_i,k), ptarc(p,t_i,q1_i,k) \\rbrace \\subseteq A$ – from $r\\ref {r:r:del} $ and supported rule proposition $del(p,q1_i,t_i,k)$ either represents removal of $q1_i = W(p,t_i)$ tokens from $p \\in \\bullet t_i$ ; or it represents removal of $q1_i = M_k(p)$ tokens from $p \\in R(t_i)$ – from rule $r\\ref {r:r:del} $ , supported rule proposition, and definition REF of transition execution in $PN$ Then $T_k$ does not contain conflicting transitions – by the definition REF of conflicting transitions And for each $enabled(t_j,k) \\in A$ and $fires(t_j,k) \\notin A$ , $could\\_not\\_have(t_j,k) \\in A$ , since $A$ is an answer set of $\\Pi ^3$ - from rule $a\\ref {a:maxfire:elim}$ and supported rule proposition Then $\\lbrace enabled(t_j,k), holds(s,qq,k), ptarc(s,t_j,q,k), \\\\ tot\\_decr(s,qqq,k) \\rbrace \\subseteq A$ , such that $q > qq - qqq$ and $fires(t_j,k) \\notin A$ - from rule $a\\ref {a:r:maxfire:cnh}$ and supported rule proposition Then for an $s \\in \\bullet t_j \\cup R(t_j)$ , $q > M_k(s) - (\\sum _{t_i \\in T_k, s \\in \\bullet t_i}{W(s,t_i)} + \\sum _{t_i \\in T_k, s \\in R(t_i)}{M_k(s))}$ , where $q=W(s,t_j) \\text{~if~} s \\in \\bullet t_j, \\text{~or~} M_k(s) \\text{~otherwise}$ - from the following: $ptarc(s,t_i,q,k)$ represents $q=W(s,t_i)$ if $(s,t_i) \\in E^-$ or $q=M_k(s)$ if $s \\in R(t_i)$ – from rule $f\\ref {f:r:ptarc},f\\ref {f:rptarc}$ construction $holds(s,qq,k)$ represents $qq=M_k(s)$ – construction $tot\\_decr(s,qqq,k) \\in A$ if $\\lbrace del(s,qqq_0,t_0,k), \\dots , del(s,qqq_x,t_x,k) \\rbrace \\\\ \\subseteq A$ – from rule $r\\ref {r:totdecr}$ construction and supported rule proposition $del(s,qqq_i,t_i,k) \\in A$ if $\\lbrace fires(t_i,k), ptarc(s,t_i,qqq_i,k) \\rbrace \\subseteq A$ – from rule $r\\ref {r:r:del} $ and supported rule proposition $del(s,qqq_i,t_i,k)$ represents $qqq_i = W(s,t_i) : t_i \\in T_k, (s,t_i) \\in E^-$ , or $qqq_i = M_k(t_i) : t_i \\in T_k, s \\in R(t_i)$ – from rule $f\\ref {f:r:ptarc},f\\ref {f:rptarc}$ construction $tot\\_decr(q,qqq,k)$ represents $\\sum _{t_i \\in T_k, s \\in \\bullet t_i}{W(s,t_i)} + \\\\ \\sum _{t_i \\in T_k, s \\in R(t_i)}{M_k(s)}$ – from (C,D,E) above Then firing $T_k \\cup \\lbrace t_j \\rbrace $ would have required more tokens than are present at its source place $s \\in \\bullet t_j \\cup R(t_j)$ .", "Thus, $T_k$ is a maximal set of transitions that can simultaneously fire.", "And for each reset transition $t_r$ with $enabled(t_r,k) \\in A$ , $fires(t_r,k) \\in A$ , since $A$ is an answer set of $\\Pi ^2$ - from rule $f\\ref {f:c:rptarc:elim}$ and supported rule proposition Then the firing set $T_k$ satisfies the reset transition requirement of definition REF (firing set) Then $\\lbrace t_0, \\dots , t_x\\rbrace = T_k$ – using 1(a),1(b), 1(d) above; and using 1(c) it is a maximal firing set We show that $M_{k+1}$ is produced by firing $T_k$ in $M_k$ .", "Let $holds(p,q,k+1) \\in A$ Then $\\lbrace holds(p,q1,k), tot\\_incr(p,q2,k), tot\\_decr(p,q3,k) \\rbrace \\subseteq A : q=q1+q2-q3$ – from rule $r\\ref {r:nextstate}$ and supported rule proposition Then, $holds(p,q1,k) \\in A$ represents $q1=M_k(p)$ – inductive assumption, construction ; and $\\lbrace add(p,q2_0,t_0,k), \\dots , $ $add(p,q2_j,t_j,k)\\rbrace \\subseteq A : q2_0 + \\dots + q2_j = q2$ and $\\lbrace del(p,q3_0,t_0,k), \\dots , $ $del(p,q3_l,t_l,k)\\rbrace \\subseteq A : q3_0 + \\dots + q3_l = q3$ – rules $r\\ref {r:totincr},r\\ref {r:totdecr}$ and supported rule proposition, respectively Then $\\lbrace fires(t_0,k), \\dots , fires(t_j,k) \\rbrace \\subseteq A$ and $\\lbrace fires(t_0,k), \\dots , fires(t_l,k) \\rbrace \\\\ \\subseteq A$ – rules $r\\ref {r:r:add},r\\ref {r:r:del}$ and supported rule proposition, respectively Then $\\lbrace fires(t_0,k), \\dots , $ $fires(t_j,k) \\rbrace \\cup \\lbrace fires(t_0,k), \\dots , $ $fires(t_l,k) \\rbrace \\subseteq A = $ $\\lbrace fires(t_0,k), \\dots , $ $fires(t_x,k) \\rbrace \\subseteq A$ – set union of subsets Then for each $fires(t_x,k) \\in A$ we have $t_x \\in T_k$ – already shown in item REF above Then $q = M_k(p) + \\sum _{t_x \\in T_0 \\wedge p \\in t_x \\bullet }{W(t_x,p)} - (\\sum _{t_x \\in T_k \\wedge p \\in \\bullet t_x}{W(p,t_x)} + \\\\ \\sum _{t_x \\in T_k \\wedge p \\in R(t_x)}{M_k(p)})$ – from (REF ) above and the following Each $add(p,q_j,t_j,k) \\in A$ represents $q_j=W(t_j,p)$ for $p \\in t_j \\bullet $ – rule $r\\ref {r:r:add} $ encoding, and definition REF of transition execution in $PN$ Each $del(p,t_y,q_y,k) \\in A$ represents either $q_y=W(p,t_y)$ for $p \\in \\bullet t_y$ , or $q_y=M_k(p)$ for $p \\in R(t_y)$ – from rule $r\\ref {r:r:del} ,f\\ref {f:r:ptarc} $ encoding and definition REF of transition execution in $PN$ ; or from rule $r\\ref {r:r:del} ,f\\ref {f:rptarc}$ encoding and definition of reset arc in $PN$ Each $tot\\_incr(p,q2,k) \\in A$ represents $q2=\\sum _{t_x \\in T_k \\wedge p \\in t_x \\bullet }{W(t_x,p)}$ – aggregate assignment atom semantics in rule $r\\ref {r:totincr}$ Each $tot\\_decr(p,q3,0) \\in A$ represents $q3=\\sum _{t_x \\in T_k \\wedge p \\in \\bullet t_x}{W(p,t_x)} + $ $\\sum _{t_x \\in T_k \\wedge p \\in R(t_x)}{M_k(p)}$ – aggregate assignment atom semantics in rule $r\\ref {r:totdecr}$ Then, $M_{k+1}(p) = q$ – since $holds(p,q,k+1) \\in A$ encodes $q=M_{k+1}(p)$ – from construction As a result, for any $n > k$ , $T_n$ will be a valid firing set for $M_n$ and $M_{n+1}$ will be its target marking.", "Conclusion: Since both (REF ) and (REF ) hold, $X=M_0,T_0,M_1,\\dots ,M_k,T_{k+1}$ is an execution sequence of $PN(P,T,E,W,R)$ (w.r.t $M_0$ ) iff there is an answer set $A$ of $\\Pi ^3(PN,M_0,k,ntok)$ such that (REF ) and (REF ) hold.", "Proof of Proposition REF Let $PN=(P,T,E,W,R,I,Q,QW)$ be a Petri Net, $M_0$ be its initial marking and let $\\Pi ^4(PN,M_0,k,ntok)$ be the ASP encoding of $PN$ and $M_0$ over a simulation length $k$ , with maximum $ntok$ tokens on any place node, as defined in section REF .", "Then $X=M_0,T_0,M_1,\\dots ,M_k,T_k,M_{k+1}$ is an execution sequence of $PN$ (w.r.t.", "$M_0$ ) iff there is an answer set $A$ of $\\Pi ^4(PN,M_0,k,ntok)$ such that: $\\lbrace fires(t,ts) : t \\in T_{ts}, 0 \\le ts \\le k\\rbrace = \\lbrace fires(t,ts) : fires(t,ts) \\in A \\rbrace $ $\\begin{split}\\lbrace holds(p,q,ts) &: p \\in P, q = M_{ts}(p), 0 \\le ts \\le k+1 \\rbrace \\\\&= \\lbrace holds(p,q,ts) : holds(p,q,ts) \\in A \\rbrace \\end{split}$ We prove this by showing that: Given an execution sequence $X$ , we create a set $A$ such that it satisfies (REF ) and (REF ) and show that $A$ is an answer set of $\\Pi ^4$ Given an answer set $A$ of $\\Pi ^4$ , we create an execution sequence $X$ such that (REF ) and (REF ) are satisfied.", "First we show (REF ): Given $PN$ and an execution sequence $X$ of $PN$ , we create a set $A$ as a union of the following sets: $A_1=\\lbrace num(n) : 0 \\le n \\le ntok \\rbrace $ $A_2=\\lbrace time(ts) : 0 \\le ts \\le k\\rbrace $ $A_3=\\lbrace place(p) : p \\in P \\rbrace $ $A_4=\\lbrace trans(t) : t \\in T \\rbrace $ $A_5=\\lbrace ptarc(p,t,n,ts) : (p,t) \\in E^-, n=W(p,t), 0 \\le ts \\le k \\rbrace $ , where $E^- \\subseteq E$ $A_6=\\lbrace tparc(t,p,n,ts) : (t,p) \\in E^+, n=W(t,p), 0 \\le ts \\le k \\rbrace $ , where $E^+ \\subseteq E$ $A_7=\\lbrace holds(p,q,0) : p \\in P, q=M_{0}(p) \\rbrace $ $A_8=\\lbrace notenabled(t,ts) : t \\in T, 0 \\le ts \\le k, (\\exists p \\in \\bullet t, M_{ts}(p) < W(p,t)) \\vee (\\exists p \\in I(t), M_{ts}(p) \\ne 0) \\vee (\\exists (p,t) \\in Q, M_{ts}(p) < QW(p,t)) \\rbrace $ per definition REF (enabled transition) $A_9=\\lbrace enabled(t,ts) : t \\in T, 0 \\le ts \\le k, (\\forall p \\in \\bullet t, W(p,t) \\le M_{ts}(p)) \\wedge (\\forall p \\in I(t), M_{ts}(p) = 0) \\wedge (\\forall (p,t) \\in Q, M_{ts}(p) \\ge QW(p,t)) \\rbrace $ per definition REF (enabled transition) $A_{10}=\\lbrace fires(t,ts) : t \\in T_{ts}, 0 \\le ts \\le k \\rbrace $ per definition REF (enabled transitions), only an enabled transition may fire $A_{11}=\\lbrace add(p,q,t,ts) : t \\in T_{ts}, p \\in t \\bullet , q=W(t,p), 0 \\le ts \\le k \\rbrace $ per definition REF (transition execution) $A_{12}=\\lbrace del(p,q,t,ts) : t \\in T_{ts}, p \\in \\bullet t, q=W(p,t), 0 \\le ts \\le k \\rbrace \\cup \\lbrace del(p,q,t,ts) : t \\in T_{ts}, p \\in R(t), q=M_{ts}(p), 0 \\le ts \\le k \\rbrace $ per definition REF (transition execution) $A_{13}=\\lbrace tot\\_incr(p,q,ts) : p \\in P, q=\\sum _{t \\in T_{ts}, p \\in t \\bullet }{W(t,p)}, 0 \\le ts \\le k \\rbrace $ per definition REF (firing set execution) $A_{14}=\\lbrace tot\\_decr(p,q,ts): p \\in P, q=\\sum _{t \\in T_{ts}, p \\in \\bullet t}{W(p,t)}+\\sum _{t \\in T_{ts}, p \\in R(t) }{M_{ts}(p)}, 0 \\le ts \\le k \\rbrace $ per definition REF (firing set execution) $A_{15}=\\lbrace consumesmore(p,ts) : p \\in P, q=M_{ts}(p), q1=\\sum _{t \\in T_{ts}, p \\in \\bullet t}{W(p,t)} + \\sum _{t \\in T_{ts}, p \\in R(t)}{M_{ts}(p)}, q1 > q, 0 \\le ts \\le k \\rbrace $ per definition REF (conflicting transitions) $A_{16}=\\lbrace consumesmore : \\exists p \\in P : q=M_{ts}(p), q1=\\sum _{t \\in T_{ts}, p \\in \\bullet t}{W(p,t)} + \\sum _{t \\in T_{ts}, p \\in R(t)}(M_{ts}(p)), q1 > q, 0 \\le ts \\le k \\rbrace $ per definition REF (conflicting transitions) $A_{17}=\\lbrace could\\_not\\_have(t,ts) : t \\in T, (\\forall p \\in \\bullet t, W(p,t) \\le M_{ts}(p)), t \\notin T_{ts}, (\\exists p \\in \\bullet t : W(p,t) > M_{ts}(p) - (\\sum _{t^{\\prime } \\in T_{ts}, p \\in \\bullet t^{\\prime }}{W(p,t^{\\prime })} + \\sum _{t^{\\prime } \\in T_{ts}, p \\in R(t^{\\prime })} M_{ts}(p)), 0 \\le ts \\le k \\rbrace $ per the maximal firing set semantics $A_{18}=\\lbrace holds(p,q,ts+1) : p \\in P, q=M_{ts+1}(p), 0 \\le ts < k\\rbrace $ , where $M_{ts+1}(p) = M_{ts}(p) - (\\sum _{\\begin{array}{c}t \\in T_{ts}, p \\in \\bullet t\\end{array}}{W(p,t)} + \\\\ \\sum _{t \\in T_{ts}, p \\in R(t)}M_{ts}(p)) + $ $\\sum _{\\begin{array}{c}t \\in T_{ts}, p \\in t \\bullet \\end{array}}{W(t,p)}$ according to definition REF (firing set execution) $A_{19}=\\lbrace ptarc(p,t,n,ts) : p \\in R(t), n = M_{ts}(p), n > 0, 0 \\le ts \\le k \\rbrace $ $A_{20}=\\lbrace iptarc(p,t,1,ts) : p \\in P, 0 \\le ts < k \\rbrace $ $A_{21}=\\lbrace tptarc(p,t,n,ts) : (p,t) \\in Q, n=QW(p,t), 0 \\le ts \\le k \\rbrace $ We show that $A$ satisfies (REF ) and (REF ), and $A$ is an answer set of $\\Pi ^4$ .", "$A$ satisfies (REF ) and (REF ) by its construction above.", "We show $A$ is an answer set of $\\Pi ^4$ by splitting.", "We split $lit(\\Pi ^4)$ into a sequence of $7k+9$ sets: $U_0= head(f\\ref {f:place}) \\cup head(f\\ref {f:trans}) \\cup head(f\\ref {f:time}) \\cup head(f\\ref {f:num}) \\cup head(i\\ref {i:holds}) = \\lbrace place(p) : p \\in P\\rbrace \\cup \\lbrace trans(t) : t \\in T\\rbrace \\cup \\lbrace time(0), \\dots , time(k)\\rbrace \\cup \\lbrace num(0), \\dots , num(ntok)\\rbrace \\cup \\lbrace holds(p,q,0) : p \\in P, q=M_0(p) \\rbrace $ $U_{7k+1}=U_{7k+0} \\cup head(f\\ref {f:r:ptarc})^{ts=k} \\cup head(f\\ref {f:r:tparc})^{ts=k} \\cup head(f\\ref {f:rptarc})^{ts=k} \\cup head(f\\ref {f:iptarc})^{ts=k} \\cup head(f\\ref {f:tptarc})^{ts=k} = U_{7k+0} \\cup \\lbrace ptarc(p,t,n,k) : (p,t) \\in E^-, n=W(p,t) \\rbrace \\cup \\\\ \\lbrace tparc(t,p,n,k) : (t,p) \\in E^+, n=W(t,p) \\rbrace \\cup $ $\\lbrace ptarc(p,t,n,k) : p \\in R(t), n=M_{k}(p), n > 0 \\rbrace \\cup $ $\\lbrace iptarc(p,t,1,k) : p \\in I(t) \\rbrace \\cup $ $\\lbrace tptarc(p,t,n,k) : (p,t) \\in Q, n=QW(p,t) \\rbrace $ $U_{7k+2}=U_{7k+1} \\cup head(e\\ref {e:r:ne:ptarc} )^{ts=k} = U_{7k+1} \\cup \\lbrace notenabled(t,k) : t \\in T \\rbrace $ $U_{7k+3}=U_{7k+2} \\cup head(e\\ref {e:enabled})^{ts=k} = U_{7k+2} \\cup \\lbrace enabled(t,k) : t \\in T \\rbrace $ $U_{7k+4}=U_{7k+3} \\cup head(a\\ref {a:fires})^{ts=k} = U_{7k+3} \\cup \\lbrace fires(t,k) : t \\in T \\rbrace $ $U_{7k+5}=U_{7k+4} \\cup head(r\\ref {r:r:add} )^{ts=k} \\cup head(r\\ref {r:r:del} )^{ts=k} = U_{7k+4} \\cup \\lbrace add(p,q,t,k) : p \\in P, t \\in T, q=W(t,p) \\rbrace \\cup \\lbrace del(p,q,t,k) : p \\in P, t \\in T, q=W(p,t) \\rbrace \\cup \\lbrace del(p,q,t,k) : p \\in P, t \\in T, q=M_{k}(p) \\rbrace $ $U_{7k+6}=U_{7k+5} \\cup head(r\\ref {r:totincr})^{ts=k} \\cup head(r\\ref {r:totdecr})^{ts=k} = U_{7k+5} \\cup \\lbrace tot\\_incr(p,q,k) : p \\in P, 0 \\le q \\le ntok \\rbrace \\cup \\lbrace tot\\_decr(p,q,k) : p \\in P, 0 \\le q \\le ntok \\rbrace $ $U_{7k+7}=U_{7k+6} \\cup head(r\\ref {r:nextstate})^{ts=k} \\cup head(a\\ref {a:overc:place})^{ts=k} \\cup head(a\\ref {a:r:maxfire:cnh})^{ts=k} = U_{7k+6} \\cup \\\\ \\lbrace consumesmore(p,k) : p \\in P\\rbrace \\cup \\lbrace holds(p,q,k+1) : p \\in P, 0 \\le q \\le ntok \\rbrace \\cup \\lbrace could\\_not\\_have(t,k) : t \\in T \\rbrace $ $U_{7k+8}=U_{7k+7} \\cup head(a\\ref {a:overc:gen}) = U_{7k+7} \\cup \\lbrace consumesmore \\rbrace $ where $head(r_i)^{ts=k}$ are head atoms of ground rule $r_i$ in which $ts=k$ .", "We write $A_i^{ts=k} = \\lbrace a(\\dots ,ts) : a(\\dots ,ts) \\in A_i, ts=k \\rbrace $ as short hand for all atoms in $A_i$ with $ts=k$ .", "$U_{\\alpha }, 0 \\le \\alpha \\le 7k+8$ form a splitting sequence, since each $U_i$ is a splitting set of $\\Pi ^4$ , and $\\langle U_{\\alpha }\\rangle _{\\alpha < \\mu }$ is a monotone continuous sequence, where $U_0 \\subseteq U_1 \\dots \\subseteq U_{8(k+1)}$ and $\\bigcup _{\\alpha < \\mu }{U_{\\alpha }} = lit(\\Pi ^4)$ .", "We compute the answer set of $\\Pi ^4$ using the splitting sets as follows: $bot_{U_0}(\\Pi ^4) = f\\ref {f:place} \\cup f\\ref {f:trans} \\cup f\\ref {f:time} \\cup i\\ref {i:holds} \\cup f\\ref {f:num}$ and $X_0 = A_1 \\cup \\dots \\cup A_4 \\cup A_7$ ($= U_0$ ) is its answer set – using forced atom proposition $eval_{U_0}(bot_{U_1}(\\Pi ^4) \\setminus bot_{U_0}(\\Pi ^4), X_0) = \\lbrace ptarc(p,t,q,0) \\text{:-}.", "\| q=W(p,t) \\rbrace \\cup \\\\ \\lbrace tparc(t,p,q,0) \\text{:-}.", "\| q=W(t,p) \\rbrace \\cup \\lbrace ptarc(p,t,q,0) \\text{:-}.", "\| q=M_0(p) \\rbrace \\cup \\\\ \\lbrace iptarc(p,t,1,0) \\text{:-}.", "\\rbrace \\cup \\lbrace tptarc(p,t,q,0) \\text{:-}.", "\| q = QW(p,t) \\rbrace $ .", "Its answer set $X_1=A_5^{ts=0} \\cup A_6^{ts=0} \\cup A_{19}^{ts=0} \\cup A_{20}^{ts=0} \\cup A_{21}^{ts=0}$ – using forced atom proposition and construction of $A_5, A_6, A_{19}, A_{20}, A_{21}$ .", "$eval_{U_1}(bot_{U_2}(\\Pi ^4) \\setminus bot_{U_1}(\\Pi ^4), X_0 \\cup X_1) = \\lbrace notenabled(t,0) \\text{:-} .", "\| (\\lbrace trans(t), \\\\ ptarc(p,t,n,0), holds(p,q,0) \\rbrace \\subseteq X_0 \\cup X_1, \\text{~where~} q < n) \\text{~or~} (\\lbrace notenabled(t,0) \\text{:-} .", "\| \\\\ (\\lbrace trans(t), iptarc(p,t,n2,0), holds(p,q,0) \\rbrace \\subseteq X_0 \\cup X_1, \\text{~where~} q \\ge n2 \\rbrace ) \\text{~or~} \\\\ (\\lbrace trans(t), tptarc(p,t,n3,0), holds(p,q,0) \\rbrace \\subseteq X_0 \\cup X_1, \\text{~where~} q < n3) \\rbrace $ .", "Its answer set $X_2=A_8^{ts=0}$ – using forced atom proposition and construction of $A_8$ .", "where, $q=M_0(p)$ , and $n=W(p,t)$ for an arc $(p,t) \\in E^-$ – by construction of $i\\ref {i:holds}$ and $f\\ref {f:r:ptarc}$ in $\\Pi ^4$ , and in an arc $(p,t) \\in E^-$ , $p \\in \\bullet t$ (by definition REF of preset) $n2=1$ – by construction of $iptarc$ predicates in $\\Pi ^4$ , meaning $q \\ge n2 \\equiv q \\ge 1 \\equiv q > 0$ , $tptarc(p,t,n3,0)$ represents $n3=QW(p,t)$ , where $(p,t) \\in Q$ thus, $notenabled(t,0) \\in X_1$ represents $(\\exists p \\in \\bullet t : M_0(p) < W(p,t)) \\vee (\\exists p \\in I(t) : M_0(p) > 0) \\vee (\\exists (p,t) \\in Q : M_{ts}(p) < QW(p,t))$ .", "$eval_{U_2}(bot_{U_3}(\\Pi ^4) \\setminus bot_{U_2}(\\Pi ^4), X_0 \\cup \\dots \\cup X_2) = \\lbrace enabled(t,0) \\text{:-}.", "\| trans(t) \\in X_0 \\cup \\dots \\cup X_2, notenabled(t,0) \\notin X_0 \\cup \\dots \\cup X_2 \\rbrace $ .", "Its answer set is $X_3 = A_9^{ts=0}$ – using forced atom proposition and construction of $A_9$ .", "since an $enabled(t,0) \\in X_3$ if $\\nexists ~notenabled(t,0) \\in X_0 \\cup \\dots \\cup X_2$ ; which is equivalent to $(\\nexists p \\in \\bullet t : M_0(p) < W(p,t)) \\wedge (\\nexists p \\in I(t) : M_0(p) > 0) \\wedge (\\nexists (p,t) \\in Q : M_0(p) < QW(p,t) ) \\equiv (\\forall p \\in \\bullet t: M_0(p) \\ge W(p,t)) \\wedge (\\forall p \\in I(t) : M_0(p) = 0)$ .", "$eval_{U_3}(bot_{U_4}(\\Pi ^4) \\setminus bot_{U_3}(\\Pi ^4), X_0 \\cup \\dots \\cup X_3) = \\lbrace \\lbrace fires(t,0)\\rbrace \\text{:-}.", "\| enabled(t,0) \\\\ \\text{~holds in~} X_0 \\cup \\dots \\cup X_3 \\rbrace $ .", "It has multiple answer sets $X_{4.1}, \\dots , X_{4.n}$ , corresponding to elements of power set of $fires(t,0)$ atoms in $eval_{U_3}(...)$ – using supported rule proposition.", "Since we are showing that the union of answer sets of $\\Pi ^4$ determined using splitting is equal to $A$ , we only consider the set that matches the $fires(t,0)$ elements in $A$ and call it $X_4$ , ignoring the rest.", "Thus, $X_4 = A_{10}^{ts=0}$ , representing $T_0$ .", "in addition, for every $t$ such that $enabled(t,0) \\in X_0 \\cup \\dots \\cup X_3, R(t) \\ne \\emptyset $ ; $fires(t,0) \\in X_4$ – per definition REF (firing set); requiring that a reset transition is fired when enabled thus, the firing set $T_0$ will not be eliminated by the constraint $f\\ref {f:c:rptarc:elim}$ $eval_{U_4}(bot_{U_5}(\\Pi ^4) \\setminus bot_{U_4}(\\Pi ^4), X_0 \\cup \\dots \\cup X_4) = \\lbrace add(p,n,t,0) \\text{:-}.", "\| \\lbrace fires(t,0), \\\\ tparc(t,p,n,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_4 \\rbrace \\cup \\lbrace del(p,n,t,0) \\text{:-}.", "\| \\lbrace fires(t,0), ptarc(p,t,n,0) \\rbrace \\\\ \\subseteq X_0 \\cup \\dots \\cup X_4 \\rbrace $ .", "It's answer set is $X_5 = A_{11}^{ts=0} \\cup A_{12}^{ts=0}$ – using forced atom proposition and definitions of $A_{11}$ and $A_{12}$ .", "where, each $add$ atom is equivalent to $n=W(t,p),p \\in t \\bullet $ , and each $del$ atom is equivalent to $n=W(p,t), p \\in \\bullet t$ ; or $n=M_k(p), p \\in R(t)$ , representing the effect of transitions in $T_0$ – by construction $eval_{U_5}(bot_{U_6}(\\Pi ^4) \\setminus bot_{U_5}(\\Pi ^4), X_0 \\cup \\dots \\cup X_5) = \\lbrace tot\\_incr(p,qq,0) \\text{:-}.", "\| $ $qq=\\sum _{add(p,q,t,0) \\in X_0 \\cup \\dots \\cup X_5}{q} \\rbrace \\cup \\lbrace tot\\_decr(p,qq,0) \\text{:-}.", "\| qq=\\sum _{del(p,q,t,0) \\in X_0 \\cup \\dots \\cup X_5}{q} \\rbrace $ .", "It's answer set is $X_6 = A_{13}^{ts=0} \\cup A_{14}^{ts=0}$ – using forced atom proposition and definitions of $A_{13}$ and $A_{14}$ .", "where, each $tot\\_incr(p,qq,0)$ , $qq=\\sum _{add(p,q,t,0) \\in X_0 \\cup \\dots X_5}{q}$ $\\equiv qq=\\sum _{t \\in X_4, p \\in t \\bullet }{W(p,t)}$ , and each $tot\\_decr(p,qq,0)$ , $qq=\\sum _{del(p,q,t,0) \\in X_0 \\cup \\dots X_5}{q}$ $\\equiv qq=\\sum _{t \\in X_4, p \\in \\bullet t}{W(t,p)} + \\sum _{t \\in X_4, p \\in R(t)}{M_{k}(p)}$ , represent the net effect of transitions in $T_0$ – by construction $eval_{U_6}(bot_{U_7}(\\Pi ^4) \\setminus bot_{U_6}(\\Pi ^4), X_0 \\cup \\dots \\cup X_6) = \\lbrace consumesmore(p,0) \\text{:-}.", "\| \\\\ \\lbrace holds(p,q,0), tot\\_decr(p,q1,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_6, q1 > q \\rbrace \\cup \\lbrace holds(p,q,1) \\text{:-}., \| \\\\ \\lbrace holds(p,q1,0), tot\\_incr(p,q2,0), tot\\_decr(p,q3,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_6, q=q1+q2-q3 \\rbrace \\cup \\lbrace could\\_not\\_have(t,0) \\text{:-}.", "\| \\lbrace enabled(t,0), ptarc(s,t,q), holds(s,qq,0), \\\\ tot\\_decr(s,qqq,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_6, fires(t,0) \\notin (X_0 \\cup \\dots \\cup X_6), q > qq-qqq \\rbrace $ .", "It's answer set is $X_7 = A_{15}^{ts=0} \\cup A_{17}^{ts=0} \\cup A_{18}^{ts=0}$ – using forced atom proposition and definitions of $A_{15}, A_{17}, A_{18}, A_9$ .", "where, $consumesmore(p,0)$ represents $\\exists p : q=M_0(p), q1= \\\\ \\sum _{t \\in T_0, p \\in \\bullet t}{W(p,t)}+\\sum _{t \\in T_0, p \\in R(t)}{M_0(p)}, q1 > q$ , indicating place $p$ will be over consumed if $T_0$ is fired, as defined in definition REF (conflicting transitions), $holds(p,q,1)$ represents $q=M_1(p)$ – by construction of $\\Pi ^4$ , and $could\\_not\\_have(t,0)$ represents enabled transition $t \\in T_0$ that could not fire due to insufficient tokens $X_7$ does not contain $could\\_not\\_have(t,0)$ , when $enabled(t,0) \\in X_0 \\cup \\dots \\cup X_6$ and $fires(t,0) \\notin X_0 \\cup \\dots \\cup X_6$ due to construction of $A$ , encoding of $a\\ref {a:r:maxfire:cnh}$ and its body atoms.", "As a result it is not eliminated by the constraint $a\\ref {a:maxfire:elim}$ $ \\vdots $ $eval_{U_{7k+0}}(bot_{U_{7k+1}}(\\Pi ^4) \\setminus bot_{U_{7k+0}}(\\Pi ^4), X_0 \\cup \\dots \\cup X_{7k+0}) = \\lbrace ptarc(p,t,q,k) \\text{:-}.", "\| q=W(p,t) \\rbrace \\cup \\lbrace tparc(t,p,q,k) \\text{:-}.", "\| q=W(t,p) \\rbrace \\cup \\lbrace ptarc(p,t,q,k) \\text{:-}.", "\| q=M_k(p) \\rbrace \\cup \\lbrace iptarc(p,t,1,k) \\text{:-}.", "\\rbrace $ .", "Its answer set $X_{7k+1}=A_5^{ts=k} \\cup A_6^{ts=k} \\cup A_{19}^{ts=k} \\cup A_{20}^{ts=k}$ – using forced atom proposition and construction of $A_5, A_6, A_{19}, A_{20}$ .", "$eval_{U_{7k+1}}(bot_{U_{7k+2}}(\\Pi ^4) \\setminus bot_{U_{7k+1}}(\\Pi ^4), X_0 \\cup \\dots \\cup X_{7k+1}) = \\lbrace notenabled(t,k) \\text{:-} .", "\| $ $(\\lbrace trans(t), ptarc(p,t,n,k), holds(p,q,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{7k+1}, \\text{~where~} q < n) \\text{~or~} \\\\ \\lbrace notenabled(t,k) \\text{:-} .", "\| (\\lbrace trans(t), iptarc(p,t,n2,k), holds(p,q,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{7k+1}, \\\\ \\text{~where~} q \\ge n2 \\rbrace \\rbrace $ .", "Its answer set $X_{7k+2}=A_8^{ts=k}$ – using forced atom proposition and construction of $A_8$ .", "where, $q=M_k(p)$ , and $n=W(p,t)$ for an arc $(p,t) \\in E^-$ – by construction of $holds$ and $ptarc$ predicates in $\\Pi ^4$ , and in an arc $(p,t) \\in E^-$ , $p \\in \\bullet t$ (by definition REF of preset) $n2=1$ – by construction of $iptarc$ predicates in $\\Pi ^4$ , meaning $q \\ge n2 \\equiv q \\ge 1 \\equiv q > 0$ , thus, $notenabled(t,k) \\in X_{7k+1}$ represents $(\\exists p \\in \\bullet t : M_k(p) < W(p,t)) \\vee (\\exists p \\in I(t) : M_k(p) > k)$ .", "$eval_{U_{7k+2}}(bot_{U_{7k+3}}(\\Pi ^4) \\setminus bot_{U_{7k+2}}(\\Pi ^4), X_0 \\cup \\dots \\cup X_{7k+2}) = \\lbrace enabled(t,k) \\text{:-}.", "\| \\\\ trans(t) \\in X_0 \\cup \\dots \\cup X_{7k+2} \\wedge notenabled(t,k) \\notin X_0 \\cup \\dots \\cup X_{7k+2} \\rbrace $ .", "Its answer set is $X_{7k+3} = A_9^{ts=k}$ – using forced atom proposition and construction of $A_9$ .", "since an $enabled(t,k) \\in X_{7k+3}$ if $\\nexists ~notenabled(t,k) \\in X_0 \\cup \\dots \\cup X_{7k+2}$ ; which is equivalent to $(\\nexists p \\in \\bullet t : M_k(p) < W(p,t)) \\wedge (\\nexists p \\in I(t) : M_k(p) > k) \\equiv (\\forall p \\in \\bullet t: M_k(p) \\ge W(p,t)) \\wedge (\\forall p \\in I(t) : M_k(p) = k)$ .", "$eval_{U_{7k+3}}(bot_{U_{7k+4}}(\\Pi ^4) \\setminus bot_{U_{7k+3}}(\\Pi ^4), X_0 \\cup \\dots \\cup X_{7k+3}) = \\lbrace \\lbrace fires(t,k)\\rbrace \\text{:-}.", "\| \\\\ enabled(t,k) \\\\ \\text{~holds in~} X_0 \\cup \\dots \\cup X_{7k+3} \\rbrace $ .", "It has multiple answer sets $X_{7k+4.1}, \\dots , X_{7k+4.n}$ , corresponding to elements of power set of $fires(t,k)$ atoms in $eval_{U_{7k+3}}(...)$ – using supported rule proposition.", "Since we are showing that the union of answer sets of $\\Pi ^4$ determined using splitting is equal to $A$ , we only consider the set that matches the $fires(t,k)$ elements in $A$ and call it $X_{7k+4}$ , ignoring the rest.", "Thus, $X_{7k+4} = A_{10}^{ts=k}$ , representing $T_k$ .", "in addition, for every $t$ such that $enabled(t,k) \\in X_0 \\cup \\dots \\cup X_{7k+3}, R(t) \\ne \\emptyset $ ; $fires(t,k) \\in X_{7k+4}$ – per definition REF (firing set); requiring that a reset transition is fired when enabled thus, the firing set $T_k$ will not be eliminated by the constraint $f\\ref {f:c:rptarc:elim}$ $eval_{U_{7k+4}}(bot_{U_{7k+5}}(\\Pi ^4) \\setminus bot_{U_{7k+4}}(\\Pi ^4), X_0 \\cup \\dots \\cup X_{7k+4}) = \\lbrace add(p,n,t,k) \\text{:-}.", "\| \\\\ \\lbrace fires(t,k), tparc(t,p,n,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{7k+4} \\rbrace \\cup \\lbrace del(p,n,t,k) \\text{:-}.", "\| \\lbrace fires(t,k), \\\\ ptarc(p,t,n,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{7k+4} \\rbrace $ .", "It's answer set is $X_{7k+5} = A_{11}^{ts=k} \\cup A_{12}^{ts=k}$ – using forced atom proposition and definitions of $A_{11}$ and $A_{12}$ .", "where, each $add$ atom is equivalent to $n=W(t,p) : p \\in t \\bullet $ , and each $del$ atom is equivalent to $n=W(p,t) : p \\in \\bullet t$ ; or $n=M_k(p) : p \\in R(t)$ , representing the effect of transitions in $T_k$ $eval_{U_{7k+5}}(bot_{U_{7k+6}}(\\Pi ^4) \\setminus bot_{U_{7k+5}}(\\Pi ^4), X_0 \\cup \\dots \\cup X_{7k+5}) = \\\\ \\lbrace tot\\_incr(p,qq,k) \\text{:-}.", "\| qq=\\sum _{add(p,q,t,k) \\in X_0 \\cup \\dots \\cup X_{7k+5}}{q} \\rbrace \\cup \\\\ \\lbrace tot\\_decr(p,qq,k) \\text{:-}.", "\| qq=\\sum _{del(p,q,t,k) \\in X_0 \\cup \\dots \\cup X_{7k+5}}{q} \\rbrace $ .", "It's answer set is $X_{7k+6} = A_{13}^{ts=k} \\cup A_{14}^{ts=k}$ – using forced atom proposition and definitions of $A_{13}$ and $A_{14}$ .", "where, each $tot\\_incr(p,qq,k)$ , $qq=\\sum _{add(p,q,t,k) \\in X_0 \\cup \\dots X_{7k+5}}{q}$ $\\equiv qq=\\sum _{t \\in X_{7k+4}, p \\in t \\bullet }{W(p,t)}$ , and each $tot\\_decr(p,qq,k)$ , $qq=\\sum _{del(p,q,t,k) \\in X_0 \\cup \\dots X_{7k+5}}{q}$ $\\equiv qq=\\sum _{t \\in X_{7k+4}, p \\in \\bullet t}{W(t,p)} + \\sum _{t \\in X_{7k+4}, p \\in R(t)}{M_{k}(p)}$ , represent the net effect of transition in $T_k$ $eval_{U_{7k+6}}(bot_{U_{7k+7}}(\\Pi ^4) \\setminus bot_{U_{7k+6}}(\\Pi ^4), X_0 \\cup \\dots \\cup X_{7k+6}) = \\lbrace consumesmore(p,k) \\text{:-}.", "\| \\\\ \\lbrace holds(p,q,k), tot\\_decr(p,q1,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{7k+6}, q1 > q \\rbrace \\cup \\lbrace holds(p,q,1) \\text{:-}., \| \\\\ \\lbrace holds(p,q1,k), tot\\_incr(p,q2,k), tot\\_decr(p,q3,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{7k+6}, q=q1+q2-q3 \\rbrace \\cup \\lbrace could\\_not\\_have(t,k) \\text{:-}.", "\| \\lbrace enabled(t,k), ptarc(s,t,q), holds(s,qq,k), \\\\ tot\\_decr(s,qqq,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{7k+6}, fires(t,k) \\notin (X_0 \\cup \\dots \\cup X_{7k+6}), q > qq-qqq \\rbrace $ .", "It's answer set is $X_{7k+7} = A_{15}^{ts=k} \\cup A_{17}^{ts=k} \\cup A_{18}^{ts=k}$ – using forced atom proposition and definitions of $A_{15}, A_{17}, A_{18}, A_9$ .", "where, $consumesmore(p,k)$ represents $\\exists p : q=M_k(p), q1= \\\\ \\sum _{t \\in T_k, p \\in \\bullet t}{W(p,t)}+\\sum _{t \\in T_k, p \\in R(t)}{M_k(p)}, q1 > q$ , indicating place $p$ that will be over consumed if $T_k$ is fired, as defined in definition REF (conflicting transitions), $holds(p,q,k+1)$ represents $q=M_{k+1}(p)$ – by construction of $\\Pi ^4$ , and $could\\_not\\_have(t,k)$ represents enabled transition $t$ in $T_k$ that could not be fired due to insufficient tokens $X_{7k+7}$ does not contain $could\\_not\\_have(t,k)$ , when $enabled(t,k) \\in X_0 \\cup \\dots \\cup X_{7k+6}$ and $fires(t,k) \\notin X_0 \\cup \\dots \\cup X_{7k+6}$ due to construction of $A$ , encoding of $a\\ref {a:r:maxfire:cnh}$ and its body atoms.", "As a result it is not eliminated by the constraint $a\\ref {a:maxfire:elim}$ $eval_{U_{7k+7}}(bot_{U_{7k+8}}(\\Pi ^4) \\setminus bot_{U_{7k+7}}(\\Pi ^4), X_0 \\cup \\dots \\cup X_{7k+7}) = \\lbrace consumesmore \\text{:-}.", "\| \\\\ \\lbrace consumesmore(p,0),\\dots , $ $consumesmore(p,k) \\rbrace \\cap (X_0 \\cup \\dots \\cup X_{7k+7}) \\ne \\emptyset \\rbrace $ .", "It's answer set is $X_{7k+8} = A_{16}$ – using forced atom proposition $X_{7k+8}$ will be empty since none of $consumesmore(p,0),\\dots , \\\\ consumesmore(p,k)$ hold in $X_0 \\cup \\dots \\cup X_{7k+8}$ due to the construction of $A$ , encoding of $a\\ref {a:overc:place}$ and its body atoms.", "As a result, it is not eliminated by the constraint $a\\ref {a:overc:elim}$ The set $X = X_0 \\cup \\dots \\cup X_{7k+8}$ is the answer set of $\\Pi ^4$ by the splitting sequence theorem REF .", "Each $X_i, 0 \\le i \\le 7k+8$ matches a distinct portion of $A$ , and $X = A$ , thus $A$ is an answer set of $\\Pi ^4$ .", "Next we show (REF ): Given $\\Pi ^4$ be the encoding of a Petri Net $PN(P,T,E,W,R,I)$ with initial marking $M_0$ , and $A$ be an answer set of $\\Pi ^4$ that satisfies (REF ) and (REF ), then we can construct $X=M_0,T_0,\\dots ,M_k,T_k,M_{k+1}$ from $A$ , such that it is an execution sequence of $PN$ .", "We construct the $X$ as follows: $M_i = (M_i(p_0), \\dots , M_i(p_n))$ , where $\\lbrace holds(p_0,M_i(p_0),i), \\dots holds(p_n,M_i(p_n),i) \\rbrace \\\\ \\subseteq A$ , for $0 \\le i \\le k+1$ $T_i = \\lbrace t : fires(t,i) \\in A\\rbrace $ , for $0 \\le i \\le k$ and show that $X$ is indeed an execution sequence of $PN$ .", "We show this by induction over $k$ (i.e.", "given $M_k$ , $T_k$ is a valid firing set and its firing produces marking $M_{k+1}$ ).", "Base case: Let $k=0$ , and $M_0$ is a valid marking in $X$ for $PN$ , show [(1)] $T_0$ is a valid firing set for $M_0$ , and firing of $T_0$ in $M_0$ produces marking $M_1$ .", "We show $T_0$ is a valid firing set for $M_0$ .", "Let $\\lbrace fires(t_0,0), \\dots , fires(t_x,0) \\rbrace $ be the set of all $fires(\\dots ,0)$ atoms in $A$ , Then for each $fires(t_i,0) \\in A$ $enabled(t_i,0) \\in A$ – from rule $a\\ref {a:fires}$ and supported rule proposition Then $notenabled(t_i,0) \\notin A$ – from rule $e\\ref {e:enabled}$ and supported rule proposition Then either of $body(e\\ref {e:r:ne:ptarc} )$ , $body(e\\ref {e:ne:iptarc} )$ , or $body(e\\ref {e:ne:tptarc} )$ must not hold in $A$ – from rules $e\\ref {e:r:ne:ptarc} ,e\\ref {e:ne:iptarc} ,e\\ref {e:ne:tptarc} $ and forced atom proposition Then $q \\lnot < n_i \\equiv q \\ge n_i$ in $e\\ref {e:r:ne:ptarc} $ for all $\\lbrace holds(p,q,0), ptarc(p,t_i,n_i,0)\\rbrace \\subseteq A$ – from $e\\ref {e:r:ne:ptarc} $ , forced atom proposition, and given facts ($holds(p,q,0) \\in A, ptarc(p,t_i,n_i,0) \\in A$ ) And $q \\lnot \\ge n_i \\equiv q < n_i$ in $e\\ref {e:ne:iptarc} $ for all $\\lbrace holds(p,q,0), iptarc(p,t_i,n_i,0) \\rbrace \\subseteq A, n_i=1$ ; $q > n_i \\equiv q = 0$ – from $e\\ref {e:ne:iptarc} $ , forced atom proposition, given facts ($holds(p,q,0) \\in A, iptarc(p,t_i,1,0) \\in A$ ), and $q$ is a positive integer And $q \\lnot < n_i \\equiv q \\ge n_i$ in $e\\ref {e:ne:tptarc} $ for all $\\lbrace holds(p,q,0), tptarc(p,t_i,n_i,0) \\rbrace \\subseteq A$ – from $e\\ref {e:ne:tptarc} $ , forced atom proposition, and given facts Then $(\\forall p \\in \\bullet t_i, M_0(p) \\ge W(p,t_i)) \\wedge (\\forall p \\in I(t_i), M_0(p) = 0) \\wedge (\\forall (p,t_i) \\in Q, M_0(p) \\ge QW(p,t_i))$ – from the following $holds(p,q,0) \\in A$ represents $q=M_0(p)$ – rule $i\\ref {i:holds}$ construction $ptarc(p,t_i,n_i,0) \\in A$ represents $n_i=W(p,t_i)$ – rule $f\\ref {f:r:ptarc}$ construction; or it represents $n_i = M_0(p)$ – rule $f\\ref {f:rptarc}$ construction; the construction of $f\\ref {f:rptarc}$ ensures that $notenabled(t,0)$ is never true due to the reset arc definition REF of preset $\\bullet t_i$ in $PN$ definition REF of enabled transition in $PN$ Then $t_i$ is enabled and can fire in $PN$ , as a result it can belong to $T_0$ – from definition REF of enabled transition And $consumesmore \\notin A$ , since $A$ is an answer set of $\\Pi ^4$ – from rule $a\\ref {a:overc:elim}$ and supported rule proposition Then $\\nexists consumesmore(p,0) \\in A$ – from rule $a\\ref {a:overc:gen}$ and supported rule proposition Then $\\nexists \\lbrace holds(p,q,0), tot\\_decr(p,q1,0) \\rbrace \\subseteq A : q1>q$ in $body(a\\ref {a:overc:place})$ – from $a\\ref {a:overc:place}$ and forced atom proposition Then $\\nexists p : (\\sum _{t_i \\in \\lbrace t_0,\\dots ,t_x\\rbrace , p \\in \\bullet t_i}{W(p,t_i)}+\\sum _{t_i \\in \\lbrace t_0,\\dots ,t_x\\rbrace , p \\in R(t_i)}{M_0(p)}) > M_0(p)$ – from the following $holds(p,q,0)$ represents $q=M_0(p)$ – from rule $i\\ref {i:holds}$ construction, given $tot\\_decr(p,q1,0) \\in A$ if $\\lbrace del(p,q1_0,t_0,0), \\dots , del(p,q1_x,t_x,0) \\rbrace \\subseteq A$ , where $q1 = q1_0+\\dots +q1_x$ – from $r\\ref {r:totdecr}$ and forced atom proposition $del(p,q1_i,t_i,0) \\in A$ if $\\lbrace fires(t_i,0), ptarc(p,t_i,q1_i,0) \\rbrace \\subseteq A$ – from $r\\ref {r:r:del} $ and supported rule proposition $del(p,q1_i,t_i,0)$ represents removal of $q1_i = W(p,t_i)$ tokens from $p \\in \\bullet t_i$ ; or it represents removal of $q1_i = M_0(p)$ tokens from $p \\in R(t_i)$ – from rule $r\\ref {r:r:del} $ , supported rule proposition, and definition REF of transition execution in $PN$ Then the set of transitions in $T_0$ do not conflict – by the definition REF of conflicting transitions And for each $enabled(t_j,0) \\in A$ and $fires(t_j,0) \\notin A$ , $could\\_not\\_have(t_j,0) \\in A$ , since $A$ is an answer set of $\\Pi ^4$ - from rule $a\\ref {a:maxfire:elim}$ and supported rule proposition Then $\\lbrace enabled(t_j,0), holds(s,qq,0), ptarc(s,t_j,q,0), \\\\ tot\\_decr(s,qqq,0) \\rbrace \\subseteq A$ , such that $q > qq - qqq$ and $fires(t_j,0) \\notin A$ - from rule $a\\ref {a:r:maxfire:cnh}$ and supported rule proposition Then for an $s \\in \\bullet t_j \\cup R(t_j)$ , $q > M_0(s) - (\\sum _{t_i \\in T_0, s \\in \\bullet t_i}{W(s,t_i)} + \\sum _{t_i \\in T_0, s \\in R(t_i)}{M_0(s))}$ , where $q=W(s,t_j) \\text{~if~} s \\in \\bullet t_j, \\text{~or~} M_0(s) \\text{~otherwise}$ - from the following: $ptarc(s,t_i,q,0)$ represents $q=W(s,t_i)$ if $(s,t_i) \\in E^-$ or $q=M_0(s)$ if $s \\in R(t_i)$ – from rule $f\\ref {f:r:ptarc},f\\ref {f:rptarc}$ construction $holds(s,qq,0)$ represents $qq=M_0(s)$ – from $i\\ref {i:holds}$ construction $tot\\_decr(s,qqq,0) \\in A$ if $\\lbrace del(s,qqq_0,t_0,0), \\dots , del(s,qqq_x,t_x,0) \\rbrace \\subseteq A$ – from rule $r\\ref {r:totdecr}$ construction and supported rule proposition $del(s,qqq_i,t_i,0) \\in A$ if $\\lbrace fires(t_i,0), ptarc(s,t_i,qqq_i,0) \\rbrace \\subseteq A$ – from rule $r\\ref {r:r:del} $ and supported rule proposition $del(s,qqq_i,t_i,0)$ represents $qqq_i = W(s,t_i) : t_i \\in T_0, (s,t_i) \\in E^-$ , or $qqq_i = M_0(t_i) : t_i \\in T_0, s \\in R(t_i)$ – from rule $f\\ref {f:r:ptarc},f\\ref {f:rptarc}$ construction $tot\\_decr(q,qqq,0)$ represents $\\sum _{t_i \\in T_0, s \\in \\bullet t_i}{W(s,t_i)} + \\\\ \\sum _{t_i \\in T_0, s \\in R(t_i)}{M_0(s)}$ – from (C,D,E) above Then firing $T_0 \\cup \\lbrace t_j \\rbrace $ would have required more tokens than are present at its source place $s \\in \\bullet t_j \\cup R(t_j)$ .", "Thus, $T_0$ is a maximal set of transitions that can simultaneously fire.", "And for each reset transition $t_r$ with $enabled(t_r,0) \\in A$ , $fires(t_r,0) \\in A$ , since $A$ is an answer set of $\\Pi ^2$ - from rule $f\\ref {f:c:rptarc:elim}$ and supported rule proposition Then, the firing set $T_0$ satisfies the reset-transition requirement of definition REF (firing set) Then $\\lbrace t_0, \\dots , t_x\\rbrace = T_0$ – using 1(a),1(b),1(d) above; and using 1(c) it is a maximal firing set We show $M_1$ is produced by firing $T_0$ in $M_0$ .", "Let $holds(p,q,1) \\in A$ Then $\\lbrace holds(p,q1,0), tot\\_incr(p,q2,0), tot\\_decr(p,q3,0) \\rbrace \\subseteq A : q=q1+q2-q3$ – from rule $r\\ref {r:nextstate}$ and supported rule proposition Then $holds(p,q1,0) \\in A$ represents $q1=M_0(p)$ – given, rule $i\\ref {i:holds}$ construction; Then $\\lbrace add(p,q2_0,t_0,0), \\dots , add(p,q2_j,t_j,0)\\rbrace \\subseteq A : q2_0 + \\dots + q2_j = q2$ and $\\lbrace del(p,q3_0,t_0,0), \\dots , del(p,q3_l,t_l,0)\\rbrace \\subseteq A : q3_0 + \\dots + q3_l = q3$ – rules $r\\ref {r:totincr},r\\ref {r:totdecr}$ and supported rule proposition, respectively Then $\\lbrace fires(t_0,0), \\dots , fires(t_j,0) \\rbrace \\subseteq A$ and $\\lbrace fires(t_0,0), \\dots , fires(t_l,0) \\rbrace \\\\ \\subseteq A$ – rules $r\\ref {r:r:add},r\\ref {r:r:del}$ and supported rule proposition, respectively Then $\\lbrace fires(t_0,0), \\dots , fires(t_j,0) \\rbrace \\cup \\lbrace fires(t_0,0), \\dots , fires(t_l,0) \\rbrace \\subseteq A = \\lbrace fires(t_0,0), \\dots , $ $fires(t_x,0) \\rbrace \\subseteq A$ – set union of subsets Then for each $fires(t_x,0) \\in A$ we have $t_x \\in T_0$ – already shown in item REF above Then $q = M_0(p) + \\sum _{t_x \\in T_0 \\wedge p \\in t_x \\bullet }{W(t_x,p)} - (\\sum _{t_x \\in T_0 \\wedge p \\in \\bullet t_x}{W(p,t_x)} + \\\\ \\sum _{t_x \\in T_0 \\wedge p \\in R(t_x)}{M_0(p)})$ – from (REF ) above and the following Each $add(p,q_j,t_j,0) \\in A$ represents $q_j=W(t_j,p)$ for $p \\in t_j \\bullet $ – rule $r\\ref {r:r:add} $ encoding, and definition REF of transition execution in $PN$ Each $del(p,t_y,q_y,0) \\in A$ represents either $q_y=W(p,t_y)$ for $p \\in \\bullet t_y$ , or $q_y=M_0(p)$ for $p \\in R(t_y)$ – from rule $r\\ref {r:r:del} ,f\\ref {f:r:ptarc} $ encoding and definition REF of transition execution in $PN$ ; or from rule $r\\ref {r:r:del} ,f\\ref {f:rptarc}$ encoding and definition of reset arc in $PN$ Each $tot\\_incr(p,q2,0) \\in A$ represents $q2=\\sum _{t_x \\in T_0 \\wedge p \\in t_x \\bullet }{W(t_x,p)}$ – aggregate assignment atom semantics in rule $r\\ref {r:totincr}$ Each $tot\\_decr(p,q3,0) \\in A$ represents $q3=\\sum _{t_x \\in T_0 \\wedge p \\in \\bullet t_x}{W(p,t_x)} + \\sum _{t_x \\in T_0 \\wedge p \\in R(t_x)}{M_0(p)}$ – aggregate assignment atom semantics in rule $r\\ref {r:totdecr}$ Then, $M_1(p) = q$ – since $holds(p,q,1) \\in A$ encodes $q=M_1(p)$ – from construction Inductive Step: Let $k > 0$ , and $M_k$ is a valid marking in $X$ for $PN$ , show [(1)] $T_k$ is a valid firing set for $M_k$ , and firing $T_k$ in $M_k$ produces marking $M_{k+1}$ .", "We show that $T_k$ is a valid firing set in $M_k$ .", "Let $\\lbrace fires(t_0,k), \\dots , fires(t_x,k) \\rbrace $ be the set of all $fires(\\dots ,k)$ atoms in $A$ , Then for each $fires(t_i,k) \\in A$ $enabled(t_i,k) \\in A$ – from rule $a\\ref {a:fires}$ and supported rule proposition Then $notenabled(t_i,k) \\notin A$ – from rule $e\\ref {e:enabled}$ and supported rule proposition Then either of $body(e\\ref {e:r:ne:ptarc} )$ , $body(e\\ref {e:ne:iptarc} )$ , or $body(e\\ref {e:ne:tptarc} )$ must not hold in $A$ – from rule $e\\ref {e:r:ne:ptarc} ,e\\ref {e:ne:iptarc} ,e\\ref {e:ne:tptarc} $ and forced atom proposition Then $q \\lnot < n_i \\equiv q \\ge n_i$ in $e\\ref {e:r:ne:ptarc}$ for all $\\lbrace holds(p,q,k), ptarc(p,t_i,n_i,k)\\rbrace \\subseteq A$ – from $e\\ref {e:r:ne:ptarc}$ , forced atom proposition, and given facts ($holds(p,q,k) \\in A, ptarc(p,t_i,n_i,k) \\in A$ ) And $q \\lnot \\ge n_i \\equiv q < n_i$ in $e\\ref {e:ne:iptarc} $ for all $\\lbrace holds(p,q,k), iptarc(p,t_i,n_i,k) \\rbrace \\subseteq A, n_i=1; q > n_i \\equiv q = 0$ – from $e\\ref {e:ne:iptarc} $ , forced atom proposition, given facts $(holds(p,q,k) \\in A, iptarc(p,t_i,1,k) \\in A)$ , and $q$ is a positive integer And $q \\lnot < n_i \\equiv q \\ge n_i$ in $e\\ref {e:ne:tptarc} $ for all $\\lbrace holds(p,q,k), tptarc(p,t_i,n_i,k) \\rbrace \\subseteq A$ – from $e\\ref {e:ne:tptarc} $ , forced atom proposition, and given facts Then $(\\forall p \\in \\bullet t_i, M_k(p) \\ge W(p,t_i)) \\wedge (\\forall p \\in I(t_i), M_k(p) = 0) \\wedge (\\forall (p,t_i) \\in Q, M_k(p) \\ge QW(p,t_i))$ – from $holds(p,q,k) \\in A$ represents $q=M_k(p)$ – inductive assumption, given $ptarc(p,t_i,n_i,k) \\in A$ represents $n_i=W(p,t_i)$ – rule $f\\ref {f:r:ptarc}$ construction; or it represents $n_i=M_k(p)$ – rule $f\\ref {f:rptarc}$ construction; the construction of $f\\ref {f:rptarc}$ ensures that $notenabled(t,k)$ is never true due to the reset arc definition REF of preset $\\bullet t_i$ in $PN$ definition REF of enabled transition in $PN$ Then $t_i$ is enabled and can fire in $PN$ , as a result it can belong to $T_k$ – from definition REF of enabled transition And $consumesmore \\notin A$ , since $A$ is an answer set of $\\Pi ^4$ – from rule $a\\ref {a:overc:elim}$ and supported rule proposition Then $\\nexists consumesmore(p,k) \\in A$ – from rule $a\\ref {a:overc:gen}$ and supported rule proposition Then $\\nexists \\lbrace holds(p,q,k), tot\\_decr(p,q1,k) \\rbrace \\subseteq A : q1>q$ in $body(e\\ref {e:r:ne:ptarc})$ – from $a\\ref {a:overc:place}$ and forced atom proposition Then $\\nexists p : (\\sum _{t_i \\in \\lbrace t_0,\\dots ,t_x\\rbrace , p \\in \\bullet t_i}{W(p,t_i)}+\\sum _{t_i \\in \\lbrace t_0,\\dots ,t_x\\rbrace , p \\in R(t_i)}{M_k(p)}) > M_k(p)$ – from the following $holds(p,q,k)$ represents $q=M_k(p)$ – inductive assumption, given $tot\\_decr(p,q1,k) \\in A$ if $\\lbrace del(p,q1_0,t_0,k), \\dots , del(p,q1_x,t_x,k) \\rbrace \\subseteq A$ , where $q1 = q1_0+\\dots +q1_x$ – from $r\\ref {r:totdecr}$ and forced atom proposition $del(p,q1_i,t_i,k) \\in A$ if $\\lbrace fires(t_i,k), ptarc(p,t_i,q1_i,k) \\rbrace \\subseteq A$ – from $r\\ref {r:r:del} $ and supported rule proposition $del(p,q1_i,t_i,k)$ either represents removal of $q1_i = W(p,t_i)$ tokens from $p \\in \\bullet t_i$ ; or it represents removal of $q1_i = M_k(p)$ tokens from $p \\in R(t_i)$ – from rule $r\\ref {r:r:del} $ , supported rule proposition, and definition REF of transition execution in $PN$ Then $T_k$ does not contain conflicting transitions – by the definition REF of conflicting transitions And for each $enabled(t_j,k) \\in A$ and $fires(t_j,k) \\notin A$ , $could\\_not\\_have(t_j,k) \\in A$ , since $A$ is an answer set of $\\Pi ^4$ - from rule $a\\ref {a:maxfire:elim}$ and supported rule proposition Then $\\lbrace enabled(t_j,k), holds(s,qq,k), ptarc(s,t_j,q,k), \\\\ tot\\_decr(s,qqq,k) \\rbrace \\subseteq A$ , such that $q > qq - qqq$ and $fires(t_j,k) \\notin A$ - from rule $a\\ref {a:r:maxfire:cnh}$ and supported rule proposition Then for an $s \\in \\bullet t_j \\cup R(t_j)$ , $q > M_k(s) - (\\sum _{t_i \\in T_k, s \\in \\bullet t_i}{W(s,t_i)} + \\sum _{t_i \\in T_k, s \\in R(t_i)}{M_k(s))}$ , where $q=W(s,t_j) \\text{~if~} s \\in \\bullet t_j, \\text{~or~} M_k(s) \\text{~otherwise}$ - from the following: $ptarc(s,t_i,q,k)$ represents $q=W(s,t_i)$ if $(s,t_i) \\in E^-$ or $q=M_k(s)$ if $s \\in R(t_i)$ – from rule $f\\ref {f:r:ptarc},f\\ref {f:rptarc}$ construction $holds(s,qq,k)$ represents $qq=M_k(s)$ – construction $tot\\_decr(s,qqq,k) \\in A$ if $\\lbrace del(s,qqq_0,t_0,k), \\dots , del(s,qqq_x,t_x,k) \\rbrace \\\\ \\subseteq A$ – from rule $r\\ref {r:totdecr}$ construction and supported rule proposition $del(s,qqq_i,t_i,k) \\in A$ if $\\lbrace fires(t_i,k), ptarc(s,t_i,qqq_i,k) \\rbrace \\subseteq A$ – from rule $r\\ref {r:r:del} $ and supported rule proposition $del(s,qqq_i,t_i,k)$ represents $qqq_i = W(s,t_i) : t_i \\in T_k, (s,t_i) \\in E^-$ , or $qqq_i = M_k(t_i) : t_i \\in T_k, s \\in R(t_i)$ – from rule $f\\ref {f:r:ptarc},f\\ref {f:rptarc}$ construction $tot\\_decr(q,qqq,k)$ represents $\\sum _{t_i \\in T_k, s \\in \\bullet t_i}{W(s,t_i)} + \\\\ \\sum _{t_i \\in T_k, s \\in R(t_i)}{M_k(s)}$ – from (C,D,E) above Then firing $T_k \\cup \\lbrace t_j \\rbrace $ would have required more tokens than are present at its source place $s \\in \\bullet t_j \\cup R(t_j)$ .", "Thus, $T_k$ is a maximal set of transitions that can simultaneously fire.", "And for each reset transition $t_r$ with $enabled(t_r,k) \\in A$ , $fires(t_r,k) \\in A$ , since $A$ is an answer set of $\\Pi ^2$ - from rule $f\\ref {f:c:rptarc:elim}$ and supported rule proposition Then the firing set $T_k$ satisfies the reset transition requirement of definition REF (firing set) Then $\\lbrace t_0, \\dots , t_x\\rbrace = T_k$ – using 1(a),1(b), 1(d) above; and using 1(c) it is a maximal firing set We show that $M_{k+1}$ is produced by firing $T_k$ in $M_k$ .", "Let $holds(p,q,k+1) \\in A$ Then $\\lbrace holds(p,q1,k), tot\\_incr(p,q2,k), tot\\_decr(p,q3,k) \\rbrace \\subseteq A : q=q1+q2-q3$ – from rule $r\\ref {r:nextstate}$ and supported rule proposition Then $holds(p,q1,k) \\in A$ represents $q1=M_k(p)$ – construction, inductive assumption; and $\\lbrace add(p,q2_0,t_0,k), \\dots , add(p,q2_j,t_j,k)\\rbrace \\subseteq A : q2_0 + \\dots + q2_j = q2$ and $\\lbrace del(p,q3_0,t_0,k), \\dots , del(p,q3_l,t_l,k)\\rbrace \\subseteq A : q3_0 + \\dots + q3_l = q3$ – rules $r\\ref {r:totincr},r\\ref {r:totdecr}$ and supported rule proposition, respectively Then $\\lbrace fires(t_0,k), \\dots , fires(t_j,k) \\rbrace \\subseteq A$ and $\\lbrace fires(t_0,k), \\dots , fires(t_l,k) \\rbrace \\\\ \\subseteq A$ – rules $r\\ref {r:r:add},r\\ref {r:r:del}$ and supported rule proposition, respectively Then $\\lbrace fires(t_0,k), \\dots , $ $fires(t_j,k) \\rbrace \\cup \\lbrace fires(t_0,k), \\dots , $ $fires(t_l,k) \\rbrace \\subseteq A = \\lbrace fires(t_0,k), \\dots , $ $fires(t_x,k) \\rbrace \\subseteq A$ – set union of subsets Then for each $fires(t_x,k) \\in A$ we have $t_x \\in T_k$ – already shown in item REF above Then $q = M_k(p) + \\sum _{t_x \\in T_0 \\wedge p \\in t_x \\bullet }{W(t_x,p)} - (\\sum _{t_x \\in T_k \\wedge p \\in \\bullet t_x}{W(p,t_x)} + \\\\ \\sum _{t_x \\in T_k \\wedge p \\in R(t_x)}{M_k(p)})$ – from (REF ) above and the following Each $add(p,q_j,t_j,k) \\in A$ represents $q_j=W(t_j,p)$ for $p \\in t_j \\bullet $ – rule $r\\ref {r:r:add} $ encoding, and definition REF of transition execution in $PN$ Each $del(p,t_y,q_y,k) \\in A$ represents either $q_y=W(p,t_y)$ for $p \\in \\bullet t_y$ , or $q_y=M_k(p)$ for $p \\in R(t_y)$ – from rule $r\\ref {r:r:del} ,f\\ref {f:r:ptarc} $ encoding and definition REF of transition execution in $PN$ ; or from rule $r\\ref {r:r:del} ,f\\ref {f:rptarc}$ encoding and definition of reset arc in $PN$ Each $tot\\_incr(p,q2,k) \\in A$ represents $q2=\\sum _{t_x \\in T_k \\wedge p \\in t_x \\bullet }{W(t_x,p)}$ – aggregate assignment atom semantics in rule $r\\ref {r:totincr}$ Each $tot\\_decr(p,q3,0) \\in A$ represents $q3=\\sum _{t_x \\in T_k \\wedge p \\in \\bullet t_x}{W(p,t_x)} + \\sum _{t_x \\in T_k \\wedge p \\in R(t_x)}{M_0(p)}$ – aggregate assignment atom semantics in rule $r\\ref {r:totdecr}$ Then, $M_{k+1}(p) = q$ – since $holds(p,q,k+1) \\in A$ encodes $q=M_{k+1}(p)$ – from construction As a result, for any $n > k$ , $T_n$ will be a valid firing set for $M_n$ and $M_{n+1}$ will be its target marking.", "Conclusion: Since both (REF ) and (REF ) hold, $X=M_0,T_0,M_1,\\dots ,M_k,T_{k+1}$ is an execution sequence of $PN(P,T,E,W,R)$ (w.r.t $M_0$ ) iff there is an answer set $A$ of $\\Pi ^4(PN,M_0,k,ntok)$ such that (REF ) and (REF ) hold.", "Proof of Proposition REF Let $PN=(P,T,E,C,W,R,I,Q,QW)$ be a Petri Net, $M_0$ be its initial marking and let $\\Pi ^5(PN,M_0,k,ntok)$ be the ASP encoding of $PN$ and $M_0$ over a simulation length $k$ , with maximum $ntok$ tokens on any place node, as defined in section REF .", "Then $X=M_0,T_k,M_1,\\dots ,M_k,T_k,M_{k+1}$ is an execution sequence of $PN$ (w.r.t.", "$M_0$ ) iff there is an answer set $A$ of $\\Pi ^5(PN,M_0,k,ntok)$ such that: $\\lbrace fires(t,ts) : t \\in T_{ts}, 0 \\le ts \\le k\\rbrace = \\lbrace fires(t,ts) : fires(t,ts) \\in A \\rbrace $ $\\begin{split}\\lbrace holds(p,q,c,ts) &: p \\in P, c/q = M_{ts}(p), 0 \\le ts \\le k+1 \\rbrace \\\\&= \\lbrace holds(p,q,c,ts) : holds(p,q,c,ts) \\in A \\rbrace \\end{split}$ We prove this by showing that: Given an execution sequence $X$ , we create a set $A$ such that it satisfies (REF ) and (REF ) and show that $A$ is an answer set of $\\Pi ^5$ Given an answer set $A$ of $\\Pi ^5$ , we create an execution sequence $X$ such that (REF ) and (REF ) are satisfied.", "First we show (REF ): Given $PN$ and an execution sequence $X$ of $PN$ , we create a set $A$ as a union of the following sets: $A_1=\\lbrace num(n) : 0 \\le n \\le ntok \\rbrace $ $A_2=\\lbrace time(ts) : 0 \\le ts \\le k\\rbrace $ $A_3=\\lbrace place(p) : p \\in P \\rbrace $ $A_4=\\lbrace trans(t) : t \\in T \\rbrace $ $A_5=\\lbrace color(c) : c \\in C \\rbrace $ $A_6=\\lbrace ptarc(p,t,n_c,c,ts) : (p,t) \\in E^-, c \\in C, n_c=m_{W(p,t)}(c), n_c > 0, 0 \\le ts \\le k \\rbrace $ , where $E^- \\subseteq E$ $A_7=\\lbrace tparc(t,p,n_c,c,ts) : (t,p) \\in E^+, c \\in C, n_c=m_{W(t,p)}(c), n_c > 0, 0 \\le ts \\le k \\rbrace $ , where $E^+ \\subseteq E$ $A_8=\\lbrace holds(p,q_c,c,0) : p \\in P, c \\in C, q_c=m_{M_{0}(p)}(c) \\rbrace $ $A_9=\\lbrace ptarc(p,t,n_c,c,ts) : p \\in R(t), c \\in C, n_c = m_{M_{ts}(p)}, n_c > 0, 0 \\le ts \\le k \\rbrace $ $A_{10}=\\lbrace iptarc(p,t,1,c,ts) : p \\in I(t), c \\in C, 0 \\le ts < k \\rbrace $ $A_{11}=\\lbrace tptarc(p,t,n_c,c,ts) : (p,t) \\in Q, c \\in C, n_c=m_{QW(p,t)}(c), n_c > 0, 0 \\le ts \\le k \\rbrace $ $A_{12}=\\lbrace notenabled(t,ts) : t \\in T, 0 \\le ts \\le k, \\exists c \\in C, (\\exists p \\in \\bullet t, m_{M_{ts}(p)}(c) < m_{W(p,t)}(c)) \\vee (\\exists p \\in I(t), m_{M_{ts}(p)}(c) > 0) \\vee (\\exists (p,t) \\in Q, m_{M_{ts}(p)}(c) < m_{QW(p,t)}(c)) \\rbrace $ per definition REF (enabled transition) $A_{13}=\\lbrace enabled(t,ts) : t \\in T, 0 \\le ts \\le k, \\forall c \\in C, (\\forall p \\in \\bullet t, m_{W(p,t)}(c) \\le m_{M_{ts}(p)}(c)) \\wedge (\\forall p \\in I(t), m_{M_{ts}(p)}(c) = 0) \\wedge (\\forall (p,t) \\in Q, m_{M_{ts}(p)}(c) \\ge m_{QW(p,t)}(c)) \\rbrace $ per definition REF (enabled transition) $A_{14}=\\lbrace fires(t,ts) : t \\in T_{ts}, 0 \\le ts \\le k \\rbrace $ per definition REF (firing set), only an enabled transition may fire $A_{15}=\\lbrace add(p,q_c,t,c,ts) : t \\in T_{ts}, p \\in t \\bullet , c \\in C, q_c=m_{W(t,p)}(c), 0 \\le ts \\le k \\rbrace $ per definition REF (transition execution) $A_{16}=\\lbrace del(p,q_c,t,c,ts) : t \\in T_{ts}, p \\in \\bullet t, c \\in C, q_c=m_{W(p,t)}(c), 0 \\le ts \\le k \\rbrace \\cup \\lbrace del(p,q_c,t,c,ts) : t \\in T_{ts}, p \\in R(t), c \\in C, q_c=m_{M_{ts}(p)}(c), 0 \\le ts \\le k \\rbrace $ per definition REF (transition execution) $A_{17}=\\lbrace tot\\_incr(p,q_c,c,ts) : p \\in P, c \\in C, q_c=\\sum _{t \\in T_{ts}, p \\in t \\bullet }{m_{W(t,p)}(c)}, 0 \\le ts \\le k \\rbrace $ per definition REF (firing set execution) $A_{18}=\\lbrace tot\\_decr(p,q_c,c,ts): p \\in P, c \\in C, q_c=\\sum _{t \\in T_{ts}, p \\in \\bullet t}{m_{W(p,t)}(c)}+ \\\\ \\sum _{t \\in T_{ts}, p \\in R(t) }{m_{M_{ts}(p)}(c)}, 0 \\le ts \\le k \\rbrace $ per definition REF (firing set execution) $A_{19}=\\lbrace consumesmore(p,ts) : p \\in P, c \\in C, q_c=m_{M_{ts}(p)}(c), \\\\ q1_c=\\sum _{t \\in T_{ts}, p \\in \\bullet t}{m_{W(p,t)}(c)} + \\sum _{t \\in T_{ts}, p \\in R(t)}{m_{M_{ts}(p)}(c)}, q1 > q, 0 \\le ts \\le k \\rbrace $ per definition REF (conflicting transitions) $A_{20}=\\lbrace consumesmore : \\exists p \\in P, c \\in C, q_c=m_{M_{ts}(p)}(c), \\\\q1_c=\\sum _{t \\in T_{ts}, p \\in \\bullet t}{m_{W(p,t)}(c)} + \\sum _{t \\in T_{ts}, p \\in R(t)}(m_{M_{ts}(p)}(c)), q1 > q, 0 \\le ts \\le k \\rbrace $ per definition REF (conflicting transitions) per the maximal firing set semantics $A_{21}=\\lbrace holds(p,q_c,c,ts+1) : p \\in P, c \\in C, q_c=m_{M_{ts+1}(p)}(c), 0 \\le ts < k\\rbrace $ , where $M_{ts+1}(p) = M_{ts}(p) - (\\sum _{\\begin{array}{c}t \\in T_{ts}, p \\in \\bullet t\\end{array}}{W(p,t)} + \\\\ \\sum _{t \\in T_{ts}, p \\in R(t)}M_{ts}(p)) + \\sum _{\\begin{array}{c}t \\in T_{ts}, p \\in t \\bullet \\end{array}}{W(t,p)}$ according to definition REF (firing set execution) We show that $A$ satisfies (REF ) and (REF ), and $A$ is an answer set of $\\Pi ^5$ .", "$A$ satisfies (REF ) and (REF ) by its construction above.", "We show $A$ is an answer set of $\\Pi ^5$ by splitting.", "We split $lit(\\Pi ^5)$ into a sequence of $7k+9$ sets: $U_0= head(f\\ref {f:c:place}) \\cup head(f\\ref {f:c:trans}) \\cup head(f\\ref {f:c:col}) \\cup head(f\\ref {f:c:time}) \\cup head(f\\ref {f:c:num}) \\cup head(i\\ref {i:c:init}) = \\lbrace place(p) : p \\in P\\rbrace \\cup \\lbrace trans(t) : t \\in T\\rbrace \\cup \\lbrace col(c) : c \\in C \\rbrace \\cup \\lbrace time(0), \\dots , time(k)\\rbrace \\cup \\lbrace num(0), \\dots , num(ntok)\\rbrace \\cup \\lbrace holds(p,q_c,c,0) : p \\in P, c \\in C, q_c=m_{M_0(p)}(c) \\rbrace $ $U_{7k+1}=U_{7k+0} \\cup head(f\\ref {f:c:ptarc})^{ts=k} \\cup head(f\\ref {f:c:tparc})^{ts=k} \\cup head(f\\ref {f:c:rptarc})^{ts=k} \\cup head(f\\ref {f:c:iptarc})^{ts=k} \\cup head(f\\ref {f:c:tptarc})^{ts=k} = U_{7k+0} \\cup \\lbrace ptarc(p,t,n_c,c,k) : (p,t) \\in E^-, c \\in C, n_c=m_{W(p,t)}(c) \\rbrace \\cup \\lbrace tparc(t,p,n_c,c,k) : (t,p) \\in E^+, c \\in C, n_c=m_{W(t,p)}(c) \\rbrace \\cup \\lbrace ptarc(p,t,n_c,c,k) : p \\in R(t), c \\in C, n_c=m_{M_{k}(p)}(c), n > 0 \\rbrace \\cup \\lbrace iptarc(p,t,1,c,k) : p \\in I(t), c \\in C \\rbrace \\cup \\lbrace tptarc(p,t,n_c,c,k) : (p,t) \\in Q, c \\in C, n_c=m_{QW(p,t)}(c) \\rbrace $ $U_{7k+2}=U_{7k+1} \\cup head(e\\ref {e:c:ne:ptarc})^{ts=k} \\cup head(e\\ref {e:c:ne:iptarc})^{ts=k} \\cup head(e\\ref {e:c:ne:tptarc})^{ts=k} = U_{7k+1} \\cup \\\\ \\lbrace notenabled(t,k) : t \\in T \\rbrace $ $U_{7k+3}=U_{7k+2} \\cup head(e\\ref {e:c:enabled})^{ts=k} = U_{7k+2} \\cup \\lbrace enabled(t,k) : t \\in T \\rbrace $ $U_{7k+4}=U_{7k+3} \\cup head(a\\ref {a:c:fires})^{ts=k} = U_{7k+3} \\cup \\lbrace fires(t,k) : t \\in T \\rbrace $ $U_{7k+5}=U_{7k+4} \\cup head(r\\ref {r:c:add})^{ts=k} \\cup head(r\\ref {r:c:del})^{ts=k} = U_{7k+4} \\cup \\lbrace add(p,q_c,t,c,k) : p \\in P, t \\in T, c \\in C, q_c=m_{W(t,p)}(c) \\rbrace \\cup \\lbrace del(p,q_c,t,c,k) : p \\in P, t \\in T, c \\in C, q_c=m_{W(p,t)}(c) \\rbrace \\cup \\lbrace del(p,q_c,t,c,k) : p \\in P, t \\in T, c \\in C, q_c=m_{M_{k}(p)}(c) \\rbrace $ $U_{7k+6}=U_{7k+5} \\cup head(r\\ref {r:c:totincr})^{ts=k} \\cup head(r\\ref {r:c:totdecr})^{ts=k} = U_{7k+5} \\cup \\lbrace tot\\_incr(p,q_c,c,k) : p \\in P, c \\in C, 0 \\le q_c \\le ntok \\rbrace \\cup \\lbrace tot\\_decr(p,q_c,c,k) : p \\in P, c \\in C, 0 \\le q_c \\le ntok \\rbrace $ $U_{7k+7}=U_{7k+6} \\cup head(a\\ref {a:c:overc:place})^{ts=k} \\cup head(r\\ref {r:c:nextstate})^{ts=k} = U_{7k+6} \\cup \\lbrace consumesmore(p,k) : p \\in P\\rbrace \\cup \\lbrace holds(p,q,k+1) : p \\in P, 0 \\le q \\le ntok \\rbrace $ $U_{7k+8}=U_{7k+7} \\cup head(a\\ref {a:c:overc:gen}) = U_{7k+7} \\cup \\lbrace consumesmore \\rbrace $ where $head(r_i)^{ts=k}$ are head atoms of ground rule $r_i$ in which $ts=k$ .", "We write $A_i^{ts=k} = \\lbrace a(\\dots ,ts) : a(\\dots ,ts) \\in A_i, ts=k \\rbrace $ as short hand for all atoms in $A_i$ with $ts=k$ .", "$U_{\\alpha }, 0 \\le \\alpha \\le 7k+8$ form a splitting sequence, since each $U_i$ is a splitting set of $\\Pi ^5$ , and $\\langle U_{\\alpha }\\rangle _{\\alpha < \\mu }$ is a monotone continuous sequence, where $U_0 \\subseteq U_1 \\dots \\subseteq U_{7k+8}$ and $\\bigcup _{\\alpha < \\mu }{U_{\\alpha }} = lit(\\Pi ^5)$ .", "We compute the answer set of $\\Pi ^5$ using the splitting sets as follows: $bot_{U_0}(\\Pi ^5) = f\\ref {f:place} \\cup f\\ref {f:trans} \\cup f\\ref {f:c:ptarc} \\cup f9 \\cup f\\ref {f:c:tptarc} \\cup i\\ref {i:c:init}$ and $X_0 = A_1 \\cup \\dots \\cup A_5 \\cup A_8$ ($= U_0$ ) is its answer set – using forced atom proposition $eval_{U_0}(bot_{U_1}(\\Pi ^5) \\setminus bot_{U_0}(\\Pi ^5), X_0) = \\lbrace ptarc(p,t,q_c,c,0) \\text{:-}.", "\| c \\in C, q_c=m_{W(p,t)}(c), q_c > 0 \\rbrace \\cup \\lbrace tparc(t,p,q_c,c,0) \\text{:-}.", "\| c \\in C, q_c=m_{W(t,p)}(c), q_c > 0 \\rbrace \\cup \\lbrace ptarc(p,t,q_c,c,0) \\text{:-}.", "\| c \\in C, q_c=m_{M_0(p)}(c), q_c > 0 \\rbrace \\cup \\lbrace iptarc(p,t,1,c,0) \\text{:-}.", "\| c \\in C \\rbrace \\cup \\lbrace tptarc(p,t,q_c,c,0) \\text{:-}.", "\| c \\in C, q_c = m_{QW(p,t)}(c), q_c > 0 \\rbrace $ .", "Its answer set $X_1=A_6^{ts=0} \\cup A_7^{ts=0} \\cup A_9^{ts=0} \\cup A_{10}^{ts=0} \\cup A_{11}^{ts=0}$ – using forced atom proposition and construction of $A_6, A_7, A_9, A_{10}, A_{11}$ .", "$eval_{U_1}(bot_{U_2}(\\Pi ^5) \\setminus bot_{U_1}(\\Pi ^5), X_0 \\cup X_1) = \\lbrace notenabled(t,0) \\text{:-} .", "\| (\\lbrace trans(t), \\\\ ptarc(p,t,n_c,c,0), holds(p,q_c,c,0) \\rbrace \\subseteq X_0 \\cup X_1, \\text{~where~} q_c < n_c) \\text{~or~} \\\\ (\\lbrace notenabled(t,0) \\text{:-} .", "\| (\\lbrace trans(t), iptarc(p,t,n2_c,c,0), holds(p,q_c,c,0) \\rbrace \\subseteq X_0 \\cup X_1, \\\\ \\text{~where~} q_c \\ge n2_c \\rbrace ) \\text{~or~} $ $(\\lbrace trans(t), tptarc(p,t,n3_c,c,0), holds(p,q_c,c,0) \\rbrace \\subseteq X_0 \\cup X_1, \\text{~where~} q_c < n3_c) \\rbrace $ .", "Its answer set $X_2=A_{12}^{ts=0}$ – using forced atom proposition and construction of $A_{12}$ .", "where, $q_c=m_{M_0(p)}(c)$ , and $n_c=m_{W(p,t)}(c)$ for an arc $(p,t) \\in E^-$ – by construction of $i\\ref {i:c:init}$ and $f\\ref {f:c:ptarc}$ in $\\Pi ^5$ , and in an arc $(p,t) \\in E^-$ , $p \\in \\bullet t$ (by definition REF of preset) $n2_c=1$ – by construction of $iptarc$ predicates in $\\Pi ^5$ , meaning $q_c \\ge n2_c \\equiv q_c \\ge 1 \\equiv q_c > 0$ , $tptarc(p,t,n3_c,c,0)$ represents $n3_c=m_{QW(p,t)}(c)$ , where $(p,t) \\in Q$ thus, $notenabled(t,0) \\in X_1$ represents $\\exists c \\in C, (\\exists p \\in \\bullet t : m_{M_0(p)}(c) < m_{W(p,t)}(c)) \\vee (\\exists p \\in I(t) : m_{M_0(p)}(c) > 0) \\vee (\\exists (p,t) \\in Q : m_{M_0(p)}(c) < m_{QW(p,t)}(c))$ .", "$eval_{U_2}(bot_{U_3}(\\Pi ^5) \\setminus bot_{U_2}(\\Pi ^5), X_0 \\cup \\dots \\cup X_2) = \\lbrace enabled(t,0) \\text{:-}.", "\| trans(t) \\in X_0 \\cup \\dots \\cup X_2, notenabled(t,0) \\notin X_0 \\cup \\dots \\cup X_2 \\rbrace $ .", "Its answer set is $X_3 = A_{13}^{ts=0}$ – using forced atom proposition and construction of $A_{13}$ .", "where, an $enabled(t,0) \\in X_3$ if $\\nexists ~notenabled(t,0) \\in X_0 \\cup \\dots \\cup X_2$ ; which is equivalent to $\\forall c \\in C, (\\nexists p \\in \\bullet t : m_{M_0(p)}(c) < m_{W(p,t)}(c)) \\wedge (\\nexists p \\in I(t) : m_{M_0(p)}(c) > 0) \\wedge (\\nexists (p,t) \\in Q : m_{M_0(p)}(c) < m_{QW(p,t)}(c) ) \\equiv \\forall c \\in C, (\\forall p \\in \\bullet t: m_{M_0(p)}(c) \\ge m_{W(p,t)}(c)) \\wedge (\\forall p \\in I(t) : m_{M_0(p)}(c) = 0)$ .", "$eval_{U_3}(bot_{U_4}(\\Pi ^5) \\setminus bot_{U_3}(\\Pi ^5), X_0 \\cup \\dots \\cup X_3) = \\lbrace \\lbrace fires(t,0)\\rbrace \\text{:-}.", "\| enabled(t,0) \\\\ \\text{~holds in~} X_0 \\cup \\dots \\cup X_3 \\rbrace $ .", "It has multiple answer sets $X_{4.1}, \\dots , X_{4.n}$ , corresponding to elements of power set of $fires(t,0)$ atoms in $eval_{U_3}(...)$ – using supported rule proposition.", "Since we are showing that the union of answer sets of $\\Pi ^5$ determined using splitting is equal to $A$ , we only consider the set that matches the $fires(t,0)$ elements in $A$ and call it $X_4$ , ignoring the rest.", "Thus, $X_4 = A_{14}^{ts=0}$ , representing $T_0$ .", "in addition, for every $t$ such that $enabled(t,0) \\in X_0 \\cup \\dots \\cup X_3, R(t) \\ne \\emptyset $ ; $fires(t,0) \\in X_4$ – per definition REF (firing set); requiring that a reset transition is fired when enabled thus, the firing set $T_0$ will not be eliminated by the constraint $f\\ref {f:c:rptarc:elim}$ $eval_{U_4}(bot_{U_5}(\\Pi ^5) \\setminus bot_{U_4}(\\Pi ^5), X_0 \\cup \\dots \\cup X_4) = \\lbrace add(p,n_c,t,c,0) \\text{:-}.", "\| \\lbrace fires(t,0), \\\\ tparc(t,p,n_c,c,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_4 \\rbrace \\cup \\lbrace del(p,n_c,t,c,0) \\text{:-}.", "\| \\lbrace fires(t,0), \\\\ ptarc(p,t,n_c,c,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_4 \\rbrace $ .", "It's answer set is $X_5 = A_{15}^{ts=0} \\cup A_{16}^{ts=0}$ – using forced atom proposition and definitions of $A_{15}$ and $A_{16}$ .", "where, each $add$ atom is equivalent to $n_c=m_{W(t,p)}(c),c \\in C, p \\in t \\bullet $ , and each $del$ atom is equivalent to $n_c=m_{W(p,t)}(c), c \\in C, p \\in \\bullet t$ ; or $n_c=m_{M_0(p)}(c), c \\in C, p \\in R(t)$ , representing the effect of transitions in $T_0$ $eval_{U_5}(bot_{U_6}(\\Pi ^5) \\setminus bot_{U_5}(\\Pi ^5), X_0 \\cup \\dots \\cup X_5) = \\lbrace tot\\_incr(p,qq_c,c,0) \\text{:-}.", "\| qq_c=\\sum _{add(p,q_c,t,c,0) \\in X_0 \\cup \\dots \\cup X_5}{q_c} \\rbrace \\cup \\lbrace tot\\_decr(p,qq_c,c,0) \\text{:-}.", "\| qq_c= \\\\ \\sum _{del(p,q_c,t,c,0) \\in X_0 \\cup \\dots \\cup X_5}{q_c} \\rbrace $ .", "It's answer set is $X_6 = A_{17}^{ts=0} \\cup A_{18}^{ts=0}$ – using forced atom proposition and definitions of $A_{17}$ and $A_{18}$ .", "where, each $tot\\_incr(p,qq_c,c,0)$ , $qq_c=\\sum _{add(p,q_c,t,c,0) \\in X_0 \\cup \\dots X_5}{q_c}$ $\\equiv qq_c=\\sum _{t \\in X_4, p \\in t \\bullet }{m_{W(p,t)}(c)}$ , and each $tot\\_decr(p,qq_c,c,0)$ , $qq_c=\\sum _{del(p,q_c,t,c,0) \\in X_0 \\cup \\dots X_5}{q_c}$ $\\equiv qq=\\sum _{t \\in X_4, p \\in \\bullet t}{m_{W(t,p)}(c)} + \\sum _{t \\in X_4, p \\in R(t)}{m_{M_0(p)}(c)}$ , represent the net effect of transitions in $T_0$ $eval_{U_6}(bot_{U_7}(\\Pi ^5) \\setminus bot_{U_6}(\\Pi ^5), X_0 \\cup \\dots \\cup X_6) = \\lbrace consumesmore(p,0) \\text{:-}.", "\| \\\\ \\lbrace holds(p,q_c,c,0), tot\\_decr(p,q1_c,c,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_6, q1_c > q_c \\rbrace \\cup \\\\ \\lbrace holds(p,q_c,c,1) \\text{:-}., \| \\lbrace holds(p,q1_c,c,0), $ $tot\\_incr(p,q2_c,c,0), $ $tot\\_decr(p,q3_c,c,0) \\rbrace \\\\ \\subseteq X_0 \\cup \\dots \\cup X_6, q_c=q1_c+q2_c-q3_c \\rbrace $ .", "It's answer set is $X_7 = A_{19}^{ts=0} \\cup A_{21}^{ts=0}$ – using forced atom proposition and definitions of $A_{19}, A_{21}$ .", "where, $consumesmore(p,0)$ represents $\\exists p : q_c=m_{M_0(p)}(c), q1_c= \\\\ \\sum _{t \\in T_0, p \\in \\bullet t}{m_{W(p,t)}(c)}+\\sum _{t \\in T_0, p \\in R(t)}{m_{M_0(p)}(c)}, q1_c > q_c, c \\in C$ , indicating place $p$ will be over consumed if $T_0$ is fired, as defined in definition REF (conflicting transitions), $holds(p,q_c,c,1)$ represents $q_c=m_{M_1(p)}(c)$ – by construction of $\\Pi ^5$ $ \\vdots $ $eval_{U_{7k+0}}(bot_{U_{7k+1}}(\\Pi ^5) \\setminus bot_{U_{7k+0}}(\\Pi ^5), X_0 \\cup \\dots \\cup X_{7k+0}) = \\lbrace ptarc(p,t,q_c,c,k) \\text{:-}.", "\| c \\in C, q_c=m{W(p,t)}(c), q_c > 0 \\rbrace \\cup \\lbrace tparc(t,p,q_c,c,k) \\text{:-}.", "\| c \\in C, q_c=m_{W(t,p)}(c), q_c > 0 \\rbrace \\cup $ $\\lbrace ptarc(p,t,q_c,c,k) \\text{:-}.", "\| $ $c \\in C, q_c=m_{M_k(p)}(c), q_c > 0 \\rbrace \\cup \\lbrace iptarc(p,t,1,c,k) \\text{:-}.", "\| c \\in C \\rbrace \\cup \\lbrace tptarc(p,t,q_c,c,k) \\text{:-}.", "\| c \\in C, q_c = m_{QW(p,t)}(c), q_c > 0 \\rbrace $ .", "Its answer set $X_{7k+1}=A_6^{ts=k} \\cup A_7^{ts=k} \\cup A_9^{ts=k} \\cup A_{10}^{ts=k} \\cup A_{11}^{ts=k}$ – using forced atom proposition and construction of $A_6, A_7, A_9, A_{10}, A_{11}$ .", "$eval_{U_{7k+1}}(bot_{U_{7k+2}}(\\Pi ^5) \\setminus bot_{U_{7k+1}}(\\Pi ^5), X_0 \\cup \\dots \\cup X_{7k+1}) = \\lbrace notenabled(t,k) \\text{:-} .", "\| \\\\ (\\lbrace trans(t), ptarc(p,t,n_c,c,k), holds(p,q_c,c,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{7k+1}, \\text{~where~} q_c < n_c) \\text{~or~} $ $ (\\lbrace notenabled(t,k) \\text{:-} .", "\| (\\lbrace trans(t), iptarc(p,t,n2_c,c,k), holds(p,q_c,c,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{7k+1}, \\text{~where~} q_c \\ge n2_c \\rbrace ) \\text{~or~} (\\lbrace trans(t), tptarc(p,t,n3_c,c,k), \\\\ holds(p,q_c,c,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{7k+1}, \\text{~where~} q_c < n3_c) \\rbrace $ .", "Its answer set $X_{7k+2}=A_{12}^{ts=k}$ – using forced atom proposition and construction of $A_{12}$ .", "since $q_c=m_{M_k(p)}(c)$ , and $n_c=m_{W(p,t)}(c)$ for an arc $(p,t) \\in E^-$ – by construction of $holds$ and $ptarc$ predicates in $\\Pi ^5$ , and in an arc $(p,t) \\in E^-$ , $p \\in \\bullet t$ (by definition REF of preset) $n2_c=1$ – by construction of $iptarc$ predicates in $\\Pi ^5$ , meaning $q_c \\ge n2_c \\equiv q_c \\ge 1 \\equiv q_c > 0$ , $tptarc(p,t,n3_c,c,k)$ represents $n3_c=m_{QW(p,t)}(c)$ , where $(p,t) \\in Q$ thus, $notenabled(t,k) \\in X_{8k+1}$ represents $\\exists c \\in C (\\exists p \\in \\bullet t : m_{M_k(p)}(c) < m_{W(p,t)}(c)) \\vee (\\exists p \\in I(t) : m_{M_k(p)}(c) > 0) \\vee (\\exists (p,t) \\in Q : m_{M_{ts}(p)}(c) < m_{QW(p,t)}(c))$ .", "$eval_{U_{7k+2}}(bot_{U_{7k+3}}(\\Pi ^5) \\setminus bot_{U_{7k+2}}(\\Pi ^5), X_0 \\cup \\dots \\cup X_{7k+2}) = \\lbrace enabled(t,k) \\text{:-}.", "\| trans(t) \\in X_0 \\cup \\dots \\cup X_{7k+2} \\wedge notenabled(t,k) \\notin X_0 \\cup \\dots \\cup X_{7k+2} \\rbrace $ .", "Its answer set is $X_{7k+3} = A_{13}^{ts=k}$ – using forced atom proposition and construction of $A_{13}$ .", "since an $enabled(t,k) \\in X_{7k+3}$ if $\\nexists ~notenabled(t,k) \\in X_0 \\cup \\dots \\cup X_{7k+2}$ ; which is equivalent to $\\forall c \\in C, (\\nexists p \\in \\bullet t : m_{M_k(p)}(c) < m_{W(p,t)}(c)) \\wedge (\\nexists p \\in I(t) : m_{M_k(p)}(c) > 0) \\wedge (\\nexists (p,t) \\in Q : m_{M_k(p)}(c) < m_{QW(p,t)}(c) ) \\equiv \\forall c \\in C, (\\forall p \\in \\bullet t: m_{M_k(p)}(c) \\ge m_{W(p,t)}(c)) \\wedge (\\forall p \\in I(t) : m_{M_k(p)}(c) = 0)$ .", "$eval_{U_{7k+3}}(bot_{U_{7k+4}}(\\Pi ^5) \\setminus bot_{U_{7k+3}}(\\Pi ^5), X_0 \\cup \\dots \\cup X_{7k+3}) = \\lbrace \\lbrace fires(t,k)\\rbrace \\text{:-}.", "\| \\\\ enabled(t,k) \\text{~holds in~} X_0 \\cup \\dots \\cup X_{7k+3} \\rbrace $ .", "It has multiple answer sets $X_{7k+}{4.1}, \\dots , X_{7k+}{4.n}$ , corresponding to elements of power set of $fires(t,k)$ atoms in $eval_{U_3}(...)$ – using supported rule proposition.", "Since we are showing that the union of answer sets of $\\Pi ^5$ determined using splitting is equal to $A$ , we only consider the set that matches the $fires(t,k)$ elements in $A$ and call it $X_{7k+4}$ , ignoring the rest.", "Thus, $X_{7k+4} = A_{14}^{ts=k}$ , representing $T_k$ .", "in addition, for every $t$ such that $enabled(t,k) \\in X_0 \\cup \\dots \\cup X_{7k+3}, R(t) \\ne \\emptyset $ ; $fires(t,k) \\in X_{7k+4}$ – per definition REF (firing set); requiring that a reset transition is fired when enabled thus, the firing set $T_k$ will not be eliminated by the constraint $f\\ref {f:c:rptarc:elim}$ $eval_{U_{7k+4}}(bot_{U_{7k+5}}(\\Pi ^5) \\setminus bot_{U_{7k+4}}(\\Pi ^5), X_0 \\cup \\dots \\cup X_{7k+4}) = \\lbrace add(p,n_c,t,c,k) \\text{:-}.", "\| $ $ \\lbrace fires(t,k), \\\\ tparc(t,p,n_c,c,k) \\rbrace \\subseteq $ $X_0 \\cup \\dots \\cup X_{7k+4} \\rbrace \\cup $ $\\lbrace del(p,n_c,t,c,k) \\text{:-}.", "\| $ $\\lbrace fires(t,k), \\\\ ptarc(p,t,n_c,c,k) \\rbrace \\subseteq $ $X_0 \\cup \\dots \\cup X_{7k+4} \\rbrace $ .", "It's answer set is $X_{7k+5} = A_{15}^{ts=k} \\cup A_{16}^{ts=k}$ – using forced atom proposition and definitions of $A_{15}$ and $A_{16}$ .", "where each $add$ atom is equivalent to $n_c=m_{W(t,p)}(c),c \\in C, p \\in t \\bullet $ , and each $del$ atom is equivalent to $n_c=m_{W(p,t)}(c), c \\in C, p \\in \\bullet t$ ; or $n_c=m_{M_k(p)}(c), c \\in C, p \\in R(t)$ , representing the effect of transitions in $T_k$ $eval_{U_{7k+5}}(bot_{U_{7k+6}}(\\Pi ^5) \\setminus bot_{U_{7k+5}}(\\Pi ^5), X_0 \\cup \\dots \\cup X_{7k+5}) = \\lbrace tot\\_incr(p,qq_c,c,k) \\text{:-}.", "\| $ $qq=\\sum _{add(p,q_c,t,c,k) \\in X_0 \\cup \\dots \\cup X_{7k+5}}{q_c} \\rbrace \\cup $ $ \\lbrace tot\\_decr(p,qq_c,c,k) \\text{:-}.", "\| \\\\ qq_c=\\sum _{del(p,q_c,t,c,k) \\in X_0 \\cup \\dots \\cup X_{7k+5}}{q_c} \\rbrace $ .", "It's answer set is $X_{7k+6} = A_{17}^{ts=k} \\cup A_{18}^{ts=k}$ – using forced atom proposition and definitions of $A_{17}$ and $A_{18}$ .", "where, each $tot\\_incr(p,qq_c,c,k)$ , $qq_c=\\sum _{add(p,q_c,t,c,k) \\in X_0 \\cup \\dots X_{8k+5}}{q_c}$ $\\equiv qq_c=\\sum _{t \\in X_{7k+4}, p \\in t \\bullet }{m_{W(p,t)}(c)}$ , and each $tot\\_decr(p,qq_c,c,k)$ , $qq_c=\\sum _{del(p,q_c,t,c,k) \\in X_0 \\cup \\dots X_{8k+5}}{q_c}$ $\\equiv qq=\\sum _{t \\in X_{7k+4}, p \\in \\bullet t}{m_{W(t,p)}(c)} + \\sum _{t \\in X_{7k+4}, p \\in R(t)}{m_{M_{k}(p)}(c)}$ , represent the net effect of transitions in $T_k$ $eval_{U_{7k+6}}(bot_{U_{7k+7}}(\\Pi ^5) \\setminus bot_{U_{7k+6}}(\\Pi ^5), X_0 \\cup \\dots \\cup X_{7k+6}) = \\lbrace consumesmore(p,k) \\text{:-}.", "\| \\\\ \\lbrace holds(p,q_c,c,k), tot\\_decr(p,q1_c,c,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{7k+6}, q1 > q \\rbrace \\cup \\\\ \\lbrace holds(p,q_c,c,k+1) \\text{:-}.", "\| \\lbrace holds(p,q1_c,c,k), tot\\_incr(p,q2_c,c,k), \\\\ tot\\_decr(p,q3_c,c,k) \\rbrace \\subseteq $ $X_0 \\cup \\dots \\cup X_{7k+6}, q_c=q1_c+q2_c-q3_c \\rbrace $ .", "It's answer set is $X_{7k+7} = A_{19}^{ts=k} \\cup A_{21}^{ts=k}$ – using forced atom proposition and definitions of $A_{19}, A_{21}$ .", "where, $consumesmore(p,k)$ represents $\\exists p : q_c=m_{M_k(p)}(c), q1_c= \\\\ \\sum _{t \\in T_k, p \\in \\bullet t}{m_{W(p,t)}(c)}+\\sum _{t \\in T_k, p \\in R(t)}{m_{M_k(p)}(c)}, q1_c > q_c, c \\in C$ , indicating place $p$ that will be over consumed if $T_k$ is fired, as defined in definition REF (conflicting transitions), and $holds(p,q_c,c,k+1)$ represents $q_c=m_{M_{k+1}(p)}(c)$ – by construction of $\\Pi ^5$ $eval_{U_{7k+7}}(bot_{U_{7k+8}}(\\Pi ^5) \\setminus bot_{U_{7k+7}}(\\Pi ^5), X_0 \\cup \\dots \\cup X_{7k+7}) = \\lbrace consumesmore \\text{:-}.", "\| \\\\ \\lbrace consumesmore(p,0),\\dots ,$ $consumesmore(p,k) \\rbrace \\cap (X_0 \\cup \\dots \\cup X_{7k+7}) \\ne \\emptyset \\rbrace $ .", "It's answer set is $X_{7k+8} = A_{20}^{ts=k}$ – using forced atom proposition and the definition of $A_{20}$ $X_{7k+7}$ will be empty since none of $consumesmore(p,0),\\dots , \\\\ consumesmore(p,k)$ hold in $X_0 \\cup \\dots \\cup X_{7k+7}$ due to the construction of $A$ , encoding of $a\\ref {a:c:overc:place}$ and its body atoms.", "As a result, it is not eliminated by the constraint $a\\ref {a:c:overc:elim}$ The set $X = X_0 \\cup \\dots \\cup X_{7k+8}$ is the answer set of $\\Pi ^5$ by the splitting sequence theorem REF .", "Each $X_i, 0 \\le i \\le 7k+8$ matches a distinct portion of $A$ , and $X = A$ , thus $A$ is an answer set of $\\Pi ^5$ .", "Next we show (REF ): Given $\\Pi ^5$ be the encoding of a Petri Net $PN(P,T,E,C,W,R,I,$ $Q,WQ)$ with initial marking $M_0$ , and $A$ be an answer set of $\\Pi ^5$ that satisfies (REF ) and (REF ), then we can construct $X=M_0,T_k,\\dots ,M_k,T_k,M_{k+1}$ from $A$ , such that it is an execution sequence of $PN$ .", "We construct the $X$ as follows: $M_i = (M_i(p_0), \\dots , M_i(p_n))$ , where $\\lbrace holds(p_0,m_{M_i(p_0)}(c),c,i), \\dots \\\\ holds(p_n,m_{M_i(p_n)}(c),c,i) \\rbrace \\subseteq A$ , for $c \\in C, 0 \\le i \\le k+1$ $T_i = \\lbrace t : fires(t,i) \\in A\\rbrace $ , for $0 \\le i \\le k$ and show that $X$ is indeed an execution sequence of $PN$ .", "We show this by induction over $k$ (i.e.", "given $M_k$ , $T_k$ is a valid firing set and its firing produces marking $M_{k+1}$ ).", "Base case: Let $k=0$ , and $M_0$ is a valid marking in $X$ for $PN$ , show [(1)] $T_0$ is a valid firing set for $M_0$ , and firing $T_0$ in $M_0$ results in marking $M_1$ .", "We show $T_0$ is a valid firing set for $M_0$ .", "Let $\\lbrace fires(t_0,0), \\dots , fires(t_x,0) \\rbrace $ be the set of all $fires(\\dots ,0)$ atoms in $A$ , Then for each $fires(t_i,0) \\in A$ $enabled(t_i,0) \\in A$ – from rule $a\\ref {a:c:fires}$ and supported rule proposition Then $notenabled(t_i,0) \\notin A$ – from rule $e\\ref {e:c:enabled}$ and supported rule proposition Then either of $body(e\\ref {e:c:ne:ptarc})$ , $body(e\\ref {e:c:ne:iptarc})$ , or $body(e\\ref {e:c:ne:tptarc})$ must not hold in $A$ – from rules $e\\ref {e:c:ne:ptarc},e\\ref {e:c:ne:iptarc},e\\ref {e:c:ne:tptarc}$ and forced atom proposition Then $q_c \\lnot < {n_i}_c \\equiv q_c \\ge {n_i}_c$ in $e\\ref {e:c:ne:ptarc}$ for all $\\lbrace holds(p,q_c,c,0), \\\\ ptarc(p,t_i,{n_i}_c,c,0)\\rbrace \\subseteq A$ – from $e\\ref {e:c:ne:ptarc}$ , forced atom proposition, and given facts ($holds(p,q_c,c,0) \\in A, ptarc(p,t_i,{n_i}_c,0) \\in A$ ) And $q_c \\lnot \\ge {n_i}_c \\equiv q_c < {n_i}_c$ in $e\\ref {e:c:ne:iptarc}$ for all $\\lbrace holds(p,q_c,c,0), \\\\ iptarc(p,t_i,{n_i}_c,c,0) \\rbrace \\subseteq A, {n_i}_c=1$ ; $q_c > {n_i}_c \\equiv q_c = 0$ – from $e\\ref {e:c:ne:iptarc}$ , forced atom proposition, given facts ($holds(p,q_c,c,0) \\in A, iptarc(p,t_i,1,c,0) \\in A$ ), and $q_c$ is a positive integer And $q_c \\lnot < {n_i}_c \\equiv q_c \\ge {n_i}_c$ in $e\\ref {e:c:ne:tptarc}$ for all $\\lbrace holds(p,q_c,c,0), \\\\ tptarc(p,t_i,{n_i}_c,c,0) \\rbrace \\subseteq A$ – from $e\\ref {e:c:ne:tptarc}$ , forced atom proposition, and given facts Then $\\forall c \\in C, (\\forall p \\in \\bullet t_i, m_{M_0(p)}(c) \\ge m_{W(p,t_i)}(c)) \\wedge (\\forall p \\in I(t_i), m_{M_0(p)}(c) = 0) \\wedge (\\forall (p,t_i) \\in Q, m_{M_0(p)}(c) \\ge m_{QW(p,t_i)}(c))$ – from the following $holds(p,q_c,c,0) \\in A$ represents $q_c=m_{M_0(p)}(c)$ – rule $i\\ref {i:c:init}$ construction $ptarc(p,t_i,{n_i}_c,0) \\in A$ represents ${n_i}_c=m_{W(p,t_i)}(c)$ – rule $f\\ref {f:c:ptarc}$ construction; or it represents ${n_i}_c = m_{M_0(p)}(c)$ – rule $f\\ref {f:c:rptarc}$ construction; the construction of $f\\ref {f:c:rptarc}$ ensures that $notenabled(t,0)$ is never true for a reset arc definition REF of preset $\\bullet t_i$ in $PN$ definition REF of enabled transition in $PN$ Then $t_i$ is enabled and can fire in $PN$ , as a result it can belong to $T_0$ – from definition REF of enabled transition And $consumesmore \\notin A$ , since $A$ is an answer set of $\\Pi ^5$ – from rule $a\\ref {a:c:overc:gen}$ and supported rule proposition Then $\\nexists consumesmore(p,0) \\in A$ – from rule $a\\ref {a:c:overc:place}$ and supported rule proposition Then $\\nexists \\lbrace holds(p,q_c,c,0), tot\\_decr(p,q1_c,c,0) \\rbrace \\subseteq A, q1_c>q_c$ in $body(a\\ref {a:c:overc:place})$ – from $a\\ref {a:c:overc:place}$ and forced atom proposition Then $\\nexists c \\in C \\nexists p \\in P, (\\sum _{t_i \\in \\lbrace t_0,\\dots ,t_x\\rbrace , p \\in \\bullet t_i}{m_{W(p,t_i)}(c)}+ \\\\ \\sum _{t_i \\in \\lbrace t_0,\\dots ,t_x\\rbrace , p \\in R(t_i)}{m_{M_0(p)}(c)}) > m_{M_0(p)}(c)$ – from the following $holds(p,q_c,c,0)$ represents $q_c=m_{M_0(p)}(c)$ – from rule $i\\ref {i:c:init}$ construction, given $tot\\_decr(p,q1_c,c,0) \\in A$ if $\\lbrace del(p,{q1_0}_c,t_0,c,0), \\dots , \\\\ del(p,{q1_x}_c,t_x,c,0) \\rbrace \\subseteq A$ , where $q1_c = {q1_0}_c+\\dots +{q1_x}_c$ – from $r\\ref {r:c:totdecr}$ and forced atom proposition $del(p,{q1_i}_c,t_i,c,0) \\in A$ if $\\lbrace fires(t_i,0), ptarc(p,t_i,{q1_i}_c,c,0) \\rbrace \\subseteq A$ – from $r\\ref {r:c:del}$ and supported rule proposition $del(p,{q1_i}_c,t_i,c,0)$ either represents removal of ${q1_i}_c = m_{W(p,t_i)}(c)$ tokens from $p \\in \\bullet t_i$ ; or it represents removal of ${q1_i}_c = m_{M_0(p)}(c)$ tokens from $p \\in R(t_i)$ – from rules $r\\ref {r:c:del},f\\ref {f:c:ptarc},f\\ref {f:c:rptarc}$ , supported rule proposition, and definition REF of transition execution in $PN$ Then the set of transitions in $T_0$ do not conflict – by the definition REF of conflicting transitions And for each reset transition $t_r$ with $enabled(t_r,0) \\in A$ , $fires(t_r,0) \\in A$ , since $A$ is an answer set of $\\Pi ^5$ - from rule $f\\ref {f:c:rptarc:elim}$ and supported rule proposition Then, the firing set $T_0$ satisfies the reset-transition requirement of definition REF (firing set) Then $\\lbrace t_0, \\dots , t_x\\rbrace = T_0$ – using 1(a),1(b),1(d) above; and using 1(c) it is a maximal firing set We show $M_1$ is produced by firing $T_0$ in $M_0$ .", "Let $holds(p,q_c,c,1) \\in A$ Then $\\lbrace holds(p,q1_c,c,0), tot\\_incr(p,q2_c,c,0), tot\\_decr(p,q3_c,c,0) \\rbrace \\subseteq A : q_c=q1_c+q2_c-q3_c$ – from rule $r\\ref {r:c:nextstate}$ and supported rule proposition Then, $holds(p,q1_c,c,0) \\in A$ represents $q1_c=m_{M_0(p)}(c)$ – given, rule $i\\ref {i:c:init}$ construction; and $\\lbrace add(p,{q2_0}_c,t_0,c,0), \\dots , $ $add(p,{q2_j}_c,t_j,c,0)\\rbrace \\subseteq A : {q2_0}_c + \\dots + {q2_j}_c = q2_c$ and $\\lbrace del(p,{q3_0}_c,t_0,c,0), \\dots , $ $del(p,{q3_l}_c,t_l,c,0)\\rbrace \\subseteq A : {q3_0}_c + \\dots + {q3_l}_c = q3_c$ – rules $r\\ref {r:c:totincr},r\\ref {r:c:totdecr}$ and supported rule proposition, respectively Then $\\lbrace fires(t_0,0), \\dots , fires(t_j,0) \\rbrace \\subseteq A$ and $\\lbrace fires(t_0,0), \\dots , fires(t_l,0) \\rbrace $ $\\subseteq A$ – rules $r\\ref {r:c:add},r\\ref {r:c:del}$ and supported rule proposition, respectively Then $\\lbrace fires(t_0,0), \\dots , $ $fires(t_j,0) \\rbrace \\cup $ $\\lbrace fires(t_0,0), \\dots , $ $fires(t_l,0) \\rbrace \\subseteq $ $A = \\lbrace fires(t_0,0), \\dots , $ $fires(t_x,0) \\rbrace \\subseteq A$ – set union of subsets Then for each $fires(t_x,0) \\in A$ we have $t_x \\in T_0$ – already shown in item REF above Then $q_c = m_{M_0(p)}(c) + \\sum _{t_x \\in T_0 \\wedge p \\in t_x \\bullet }{m_{W(t_x,p)}(c)} - (\\sum _{t_x \\in T_0 \\wedge p \\in \\bullet t_x}{m_{W(p,t_x)}(c)} + \\sum _{t_x \\in T_0 \\wedge p \\in R(t_x)}{m_{M_0(p)}(c)})$ – from (REF ) above and the following Each $add(p,{q_j}_c,t_j,c,0) \\in A$ represents ${q_j}_c=m_{W(t_j,p)}(c)$ for $p \\in t_j \\bullet $ – rule $r\\ref {r:c:add},f\\ref {f:c:tparc}$ encoding, and definition REF of transition execution in $PN$ Each $del(p,t_y,{q_y}_c,c,0) \\in A$ represents either ${q_y}_c=m_{W(p,t_y)}(c)$ for $p \\in \\bullet t_y$ , or ${q_y}_c=m_{M_0(p)}(c)$ for $p \\in R(t_y)$ – from rule $r\\ref {r:c:del},f\\ref {f:c:ptarc}$ encoding and definition REF of transition execution in $PN$ ; or from rule $r\\ref {r:c:del},f\\ref {f:c:rptarc}$ encoding and definition of reset arc in $PN$ Each $tot\\_incr(p,q2_c,c,0) \\in A$ represents $q2_c=\\sum _{t_x \\in T_0 \\wedge p \\in t_x \\bullet }{m_{W(t_x,p)}(c)}$ – aggregate assignment atom semantics in rule $r\\ref {r:c:totincr}$ Each $tot\\_decr(p,q3_c,c,0) \\in A$ represents $q3_c=\\sum _{t_x \\in T_0 \\wedge p \\in \\bullet t_x}{m_{W(p,t_x)}(c)} + \\sum _{t_x \\in T_0 \\wedge p \\in R(t_x)}{m_{M_0(p)}(c)}$ – aggregate assignment atom semantics in rule $r\\ref {r:c:totdecr}$ Then, $m_{M_1(p)}(c) = q_c$ – since $holds(p,q_c,c,1) \\in A$ encodes $q_c=m_{M_1(p)}(c)$ – from construction Inductive Step: Let $k > 0$ , and $M_k$ is a valid marking in $X$ for $PN$ , show [(1)] $T_k$ is a valid firing set for $M_k$ , and firing $T_k$ in $M_k$ produces marking $M_{k+1}$ .", "We show that $T_k$ is a valid firing set in $M_k$ .", "Let $\\lbrace fires(t_0,k), \\dots , fires(t_x,k) \\rbrace $ be the set of all $fires(\\dots ,k)$ atoms in $A$ , Then for each $fires(t_i,k) \\in A$ $enabled(t_i,k) \\in A$ – from rule $a\\ref {a:c:fires}$ and supported rule proposition Then $notenabled(t_i,k) \\notin A$ – from rule $e\\ref {e:c:enabled}$ and supported rule proposition Then either of $body(e\\ref {e:c:ne:ptarc})$ , $body(e\\ref {e:c:ne:iptarc})$ , or $body(e\\ref {e:c:ne:tptarc})$ must not hold in $A$ – from rules $e\\ref {e:c:ne:ptarc},e\\ref {e:c:ne:iptarc},e\\ref {e:c:ne:tptarc}$ and forced atom proposition Then $q_c \\lnot < {n_i}_c \\equiv q_c \\ge {n_i}_c$ in $e\\ref {e:c:ne:ptarc}$ for all $\\lbrace holds(p,q_c,c,k), \\\\ ptarc(p,t_i,{n_i}_c,c,k)\\rbrace \\subseteq A$ – from $e\\ref {e:c:ne:ptarc}$ , forced atom proposition, given facts, and the inductive assumption And $q_c \\lnot \\ge {n_i}_c \\equiv q_c < {n_i}_c$ in $e\\ref {e:c:ne:iptarc}$ for all $\\lbrace holds(p,q_c,c,k), \\\\ iptarc(p,t_i,{n_i}_c,c,k) \\rbrace \\subseteq A, {n_i}_c=1$ ; $q_c > {n_i}_c \\equiv q_c = 0$ – from $e\\ref {e:c:ne:iptarc}$ , forced atom proposition, given facts ($holds(p,q_c,c,k) \\in A, iptarc(p,t_i,1,c,k) \\in A$ ), and $q_c$ is a positive integer And $q_c \\lnot < {n_i}_c \\equiv q_c \\ge {n_i}_c$ in $e\\ref {e:c:ne:tptarc}$ for all $\\lbrace holds(p,q_c,c,k), \\\\ tptarc(p,t_i,{n_i}_c,c,k) \\rbrace \\subseteq A$ – from $e\\ref {e:c:ne:tptarc}$ , forced atom proposition, and inductive assumption Then $(\\forall p \\in \\bullet t_i, m_{M_k(p)}(c) \\ge m_{W(p,t_i)}(c)) \\wedge (\\forall p \\in I(t_i), m_{M_k(p)}(c) = 0) \\wedge (\\forall (p,t_i) \\in Q, m_{M_k(p)}(c) \\ge m_{QW(p,t_i)}(c))$ – from $holds(p,q_c,c,k) \\in A$ represents $q_c=m_{M_k(p)}(c)$ – construction of $\\Pi ^5$ $ptarc(p,t_i,{n_i}_c,k) \\in A$ represents ${n_i}_c=m_{W(p,t_i)}(c)$ – rule $f\\ref {f:c:ptarc}$ construction; or it represents ${n_i}_c = m_{M_k(p)}(c)$ – rule $f\\ref {f:c:rptarc}$ construction; the construction of $f\\ref {f:c:rptarc}$ ensures that $notenabled(t,k)$ is never true for a reset arc definition REF of preset $\\bullet t_i$ in $PN$ definition REF of enabled transition in $PN$ Then $t_i$ is enabled and can fire in $PN$ , as a result it can belong to $T_k$ – from definition REF of enabled transition And $consumesmore \\notin A$ , since $A$ is an answer set of $\\Pi ^5$ – from rule $a\\ref {a:c:overc:gen}$ and supported rule proposition Then $\\nexists consumesmore(p,k) \\in A$ – from rule $a\\ref {a:c:overc:place}$ and supported rule proposition Then $\\nexists \\lbrace holds(p,q_c,c,k), tot\\_decr(p,q1_c,c,k) \\rbrace \\subseteq A, q1_c>q_c$ in $body(a\\ref {a:c:overc:place})$ – from $a\\ref {a:c:overc:place}$ and forced atom proposition Then $\\nexists c \\in C, \\nexists p \\in P, (\\sum _{t_i \\in \\lbrace t_0,\\dots ,t_x\\rbrace , p \\in \\bullet t_i}{m_{W(p,t_i)}(c)}+ \\\\ \\sum _{t_i \\in \\lbrace t_0,\\dots ,t_x\\rbrace , p \\in R(t_i)}{m_{M_k(p)}(c)}) > m_{M_k(p)}(c)$ – from the following $holds(p,q_c,c,k)$ represents $q_c=m_{M_k(p)}(c)$ – construction of $\\Pi ^5$ , inductive assumption $tot\\_decr(p,q1_c,c,k) \\in A$ if $\\lbrace del(p,{q1_0}_c,t_0,c,k), \\dots , \\\\ del(p,{q1_x}_c,t_x,c,k) \\rbrace \\subseteq A$ , where $q1_c = {q1_0}_c+\\dots +{q1_x}_c$ – from $r\\ref {r:c:totdecr}$ and forced atom proposition $del(p,{q1_i}_c,t_i,c,k) \\in A$ if $\\lbrace fires(t_i,k), ptarc(p,t_i,{q1_i}_c,c,k) \\rbrace \\subseteq A$ – from $r\\ref {r:c:del}$ and supported rule proposition $del(p,{q1_i}_c,t_i,c,k)$ either represents removal of ${q1_i}_c = m_{W(p,t_i)}(c)$ tokens from $p \\in \\bullet t_i$ ; or it represents removal of ${q1_i}_c = m_{M_k(p)}(c)$ tokens from $p \\in R(t_i)$ – from rules $r\\ref {r:c:del},f\\ref {f:c:ptarc},f\\ref {f:c:rptarc}$ , supported rule proposition, and definition REF of transition execution in $PN$ Then the set of transitions in $T_k$ do not conflict – by the definition REF of conflicting transitions And for each reset transition $t_r$ with $enabled(t_r,k) \\in A$ , $fires(t_r,k) \\in A$ , since $A$ is an answer set of $\\Pi ^5$ - from rule $f\\ref {f:c:rptarc:elim}$ and supported rule proposition Then the firing set $T_k$ satisfies the reset transition requirement of definition REF (firing set) Then $\\lbrace t_0, \\dots , t_x\\rbrace = T_k$ – using 1(a),1(b), 1(d) above; and using 1(c) it is a maximal firing set We show that $M_{k+1}$ is produced by firing $T_k$ in $M_k$ .", "Let $holds(p,q_c,c,k+1) \\in A$ Then $\\lbrace holds(p,q1_c,c,k), tot\\_incr(p,q2_c,c,k), tot\\_decr(p,q3_c,c,k) \\rbrace \\subseteq A : q_c=q1_c+q2_c-q3_c$ – from rule $r\\ref {r:c:nextstate}$ and supported rule proposition Then, $holds(p,q1_c,c,k) \\in A$ represents $q1_c=m_{M_k(p)}(c)$ – construction, inductive assumption; and $\\lbrace add(p,{q2_0}_c,t_0,c,k), \\dots , add(p,{q2_j}_c,t_j,c,k)\\rbrace \\subseteq A : {q2_0}_c + \\dots + {q2_j}_c = q2_c$ and $\\lbrace del(p,{q3_0}_c,t_0,c,k), \\dots , del(p,{q3_l}_c,t_l,c,k)\\rbrace \\subseteq A : {q3_0}_c + \\dots + {q3_l}_c = q3_c$ – rules $r\\ref {r:c:totincr},r\\ref {r:c:totdecr}$ and supported rule proposition, respectively Then $\\lbrace fires(t_0,k), \\dots , fires(t_j,k) \\rbrace \\subseteq A$ and $\\lbrace fires(t_0,k), \\dots , fires(t_l,k) \\rbrace \\\\ \\subseteq A$ – rules $r\\ref {r:c:add},r\\ref {r:c:del}$ and supported rule proposition, respectively Then $\\lbrace fires(t_0,k), \\dots , $ $fires(t_j,k) \\rbrace \\cup \\lbrace fires(t_0,k), \\dots , $ $fires(t_l,k) \\rbrace \\subseteq $ $A = \\lbrace fires(t_0,k), \\dots , $ $fires(t_x,k) \\rbrace \\subseteq A$ – set union of subsets Then for each $fires(t_x,k) \\in A$ we have $t_x \\in T_k$ – already shown in item REF above Then $q_c = m_{M_k(p)}(c) + \\sum _{t_x \\in T_k \\wedge p \\in t_x \\bullet }{m_{W(t_x,p)}(c)} - (\\sum _{t_x \\in T_k \\wedge p \\in \\bullet t_x}{m_{W(p,t_x)}(c)} + \\sum _{t_x \\in T_k \\wedge p \\in R(t_x)}{m_{M_k(p)}(c)})$ – from (REF ) above and the following Each $add(p,{q_j}_c,t_j,c,k) \\in A$ represents ${q_j}_c=m_{W(t_j,p)}(c)$ for $p \\in t_j \\bullet $ – rule $r\\ref {r:c:add},f\\ref {f:c:tparc}$ encoding, and definition REF of transition execution in $PN$ Each $del(p,t_y,{q_y}_c,c,k) \\in A$ represents either ${q_y}_c=m_{W(p,t_y)}(c)$ for $p \\in \\bullet t_y$ , or ${q_y}_c=m_{M_k(p)}(c)$ for $p \\in R(t_y)$ – from rule $r\\ref {r:c:del},f\\ref {f:c:ptarc}$ encoding and definition REF of transition execution in $PN$ ; or from rule $r\\ref {r:c:del},f\\ref {f:c:rptarc}$ encoding and definition of reset arc in $PN$ Each $tot\\_incr(p,q2_c,c,k) \\in A$ represents $q2_c=\\sum _{t_x \\in T_k \\wedge p \\in t_x \\bullet }{m_{W(t_x,p)}(c)}$ – aggregate assignment atom semantics in rule $r\\ref {r:c:totincr}$ Each $tot\\_decr(p,q3_c,c,k) \\in A$ represents $q3_c=\\sum _{t_x \\in T_k \\wedge p \\in \\bullet t_x}{m_{W(p,t_x)}(c)} + \\sum _{t_x \\in T_k \\wedge p \\in R(t_x)}{m_{M_k(p)}(c)}$ – aggregate assignment atom semantics in rule $r\\ref {r:c:totdecr}$ Then, $m_{M_{k+1}(p)}(c) = q_c$ – since $holds(p,q_c,c,k+1) \\in A$ encodes $q_c=m_{M_{k+1}(p)}(c)$ – from construction As a result, for any $n > k$ , $T_n$ will be a valid firing set for $M_n$ and $M_{n+1}$ will be its target marking.", "Conclusion: Since both (REF ) and (REF ) hold, $X=M_0,T_k,M_1,\\dots ,M_k,T_{k+1}$ is an execution sequence of $PN(P,T,E,C,W,R,I,Q,QW)$ (w.r.t $M_0$ ) iff there is an answer set $A$ of $\\Pi ^5(PN,M_0,k,ntok)$ such that (REF ) and (REF ) hold.", "Proof of Proposition REF Let $PN=(P,T,E,C,W,R,I,Q,QW,Z)$ be a Petri Net, $M_0$ be its initial marking and let $\\Pi ^6(PN,M_0,k,ntok)$ be the ASP encoding of $PN$ and $M_0$ over a simulation length $k$ , with maximum $ntok$ tokens on any place node, as defined in section REF .", "Then $X=M_0,T_k,M_1,\\dots ,M_k,T_k,M_{k+1}$ is an execution sequence of $PN$ (w.r.t.", "$M_0$ ) iff there is an answer set $A$ of $\\Pi ^6(PN,M_0,k,ntok)$ such that: $\\lbrace fires(t,ts) : t \\in T_{ts}, 0 \\le ts \\le k\\rbrace = \\lbrace fires(t,ts) : fires(t,ts) \\in A \\rbrace $ $\\begin{split}\\lbrace holds(p,q,c,ts) &: p \\in P, c/q = M_{ts}(p), 0 \\le ts \\le k+1 \\rbrace \\\\&= \\lbrace holds(p,q,c,ts) : holds(p,q,c,ts) \\in A \\rbrace \\end{split}$ We prove this by showing that: Given an execution sequence $X$ , we create a set $A$ such that it satisfies (REF ) and (REF ) and show that $A$ is an answer set of $\\Pi ^6$ Given an answer set $A$ of $\\Pi ^6$ , we create an execution sequence $X$ such that (REF ) and (REF ) are satisfied.", "First we show (REF ): We create a set $A$ as a union of the following sets: $A_1=\\lbrace num(n) : 0 \\le n \\le ntok \\rbrace $ $A_2=\\lbrace time(ts) : 0 \\le ts \\le k\\rbrace $ $A_3=\\lbrace place(p) : p \\in P \\rbrace $ $A_4=\\lbrace trans(t) : t \\in T \\rbrace $ $A_5=\\lbrace color(c) : c \\in C \\rbrace $ $A_6=\\lbrace ptarc(p,t,n_c,c,ts) : (p,t) \\in E^-, c \\in C, n_c=m_{W(p,t)}(c), n_c > 0, 0 \\le ts \\le k \\rbrace $ , where $E^- \\subseteq E$ $A_7=\\lbrace tparc(t,p,n_c,c,ts) : (t,p) \\in E^+, c \\in C, n_c=m_{W(t,p)}(c), n_c > 0, 0 \\le ts \\le k \\rbrace $ , where $E^+ \\subseteq E$ $A_8=\\lbrace holds(p,q_c,c,0) : p \\in P, c \\in C, q_c=m_{M_{0}(p)}(c) \\rbrace $ $A_9=\\lbrace ptarc(p,t,n_c,c,ts) : p \\in R(t), c \\in C, n_c = m_{M_{ts}(p)}, n_c > 0, 0 \\le ts \\le k \\rbrace $ $A_{10}=\\lbrace iptarc(p,t,1,c,ts) : p \\in I(t), c \\in C, 0 \\le ts < k \\rbrace $ $A_{11}=\\lbrace tptarc(p,t,n_c,c,ts) : (p,t) \\in Q, c \\in C, n_c=m_{QW(p,t)}(c), n_c > 0, 0 \\le ts \\le k \\rbrace $ $A_{12}=\\lbrace notenabled(t,ts) : t \\in T, 0 \\le ts \\le k, \\exists c \\in C, (\\exists p \\in \\bullet t, m_{M_{ts}(p)}(c) < m_{W(p,t)}(c)) \\vee (\\exists p \\in I(t), m_{M_{ts}(p)}(c) > 0) \\vee (\\exists (p,t) \\in Q, m_{M_{ts}(p)}(c) < m_{QW(p,t)}(c)) \\rbrace $ per definition REF (enabled transition) $A_{13}=\\lbrace enabled(t,ts) : t \\in T, 0 \\le ts \\le k, \\forall c \\in C, (\\forall p \\in \\bullet t, m_{W(p,t)}(c) \\le m_{M_{ts}(p)}(c)) \\wedge (\\forall p \\in I(t), m_{M_{ts}(p)}(c) = 0) \\wedge (\\forall (p,t) \\in Q, m_{M_{ts}(p)}(c) \\ge m_{QW(p,t)}(c)) \\rbrace $ per definition REF (enabled transition) $A_{14}=\\lbrace fires(t,ts) : t \\in T_{ts}, 0 \\le ts \\le k \\rbrace $ per definition REF (firing set), only an enabled transition may fire $A_{15}=\\lbrace add(p,q_c,t,c,ts) : t \\in T_{ts}, p \\in t \\bullet , c \\in C, q_c=m_{W(t,p)}(c), 0 \\le ts \\le k \\rbrace $ per definition REF (transition execution) $A_{16}=\\lbrace del(p,q_c,t,c,ts) : t \\in T_{ts}, p \\in \\bullet t, c \\in C, q_c=m_{W(p,t)}(c), 0 \\le ts \\le k \\rbrace \\cup \\lbrace del(p,q_c,t,c,ts) : t \\in T_{ts}, p \\in R(t), c \\in C, q_c=m_{M_{ts}(p)}(c), 0 \\le ts \\le k \\rbrace $ per definition REF (transition execution) $A_{17}=\\lbrace tot\\_incr(p,q_c,c,ts) : p \\in P, c \\in C, q_c=\\sum _{t \\in T_{ts}, p \\in t \\bullet }{m_{W(t,p)}(c)}, 0 \\le ts \\le k \\rbrace $ per definition REF (firing set execution) $A_{18}=\\lbrace tot\\_decr(p,q_c,c,ts): p \\in P, c \\in C, q_c=\\sum _{t \\in T_{ts}, p \\in \\bullet t}{m_{W(p,t)}(c)}+ \\\\ \\sum _{t \\in T_{ts}, p \\in R(t) }{m_{M_{ts}(p)}(c)}, 0 \\le ts \\le k \\rbrace $ per definition REF (firing set execution) $A_{19}=\\lbrace consumesmore(p,ts) : p \\in P, c \\in C, q_c=m_{M_{ts}(p)}(c), \\\\ q1_c=\\sum _{t \\in T_{ts}, p \\in \\bullet t}{m_{W(p,t)}(c)} + \\sum _{t \\in T_{ts}, p \\in R(t)}{m_{M_{ts}(p)}(c)}, q1_c > q_c, 0 \\le ts \\le k \\rbrace $ per definition REF (conflicting transitions) $A_{20}=\\lbrace consumesmore : \\exists p \\in P, c \\in C, q_c=m_{M_{ts}(p)}(c), \\\\ q1_c=\\sum _{t \\in T_{ts}, p \\in \\bullet t}{m_{W(p,t)}(c)} + \\sum _{t \\in T_{ts}, p \\in R(t)}(m_{M_{ts}(p)}(c)), q1_c > q_c, 0 \\le ts \\le k \\rbrace $ per definition REF (conflicting transitions) $A_{21}=\\lbrace holds(p,q_c,c,ts+1) : p \\in P, c \\in C, q_c=m_{M_{ts+1}(p)}(c), 0 \\le ts < k\\rbrace $ , where $M_{ts+1}(p) = M_j(p) - (\\sum _{\\begin{array}{c}t \\in T_{ts}, p \\in \\bullet t\\end{array}}{W(p,t)} + \\sum _{t \\in T_{ts}, p \\in R(t)}M_{ts}(p)) + \\\\ \\sum _{\\begin{array}{c}t \\in T_{ts}, p \\in t \\bullet \\end{array}}{W(t,p)}$ according to definition REF (firing set execution) $A_{22}=\\lbrace transpr(t,pr) : t \\in T, pr=Z(t) \\rbrace $ $A_{23}=\\lbrace notprenabled(t,ts) : t \\in T, enabled(t,ts) \\in A_{13}, (\\exists tt \\in T, enabled(tt,ts) \\in A_{13}, Z(tt) < Z(t)), 0 \\le ts \\le k \\rbrace \\\\ = \\lbrace notprenabled(t,ts) : t \\in T, (\\forall p \\in \\bullet t, W(p,t) \\le M_{ts}(p)), (\\forall p \\in I(t), M_{ts}(p) = 0), (\\forall (p,t) \\in Q, M_0(p) \\ge WQ(p,t)), \\exists tt \\in T, (\\forall pp \\in \\bullet tt, W(pp,tt) \\le M_{ts}(pp)), \\\\ (\\forall pp \\in I(tt), M_{ts}(pp) = 0), (\\forall (pp,tt) \\in Q, M_{ts}(pp) \\ge QW(pp,tt)), Z(tt) < Z(t), 0 \\le ts \\le k \\rbrace $ $A_{24}=\\lbrace prenabled(t,ts) : t \\in T, enabled(t,ts) \\in A_{13}, (\\nexists tt \\in T: enabled(tt,ts) \\in A_{13}, Z(tt) < Z(t)), 0 \\le ts \\le k \\rbrace = \\lbrace prenabled(t,ts) : t \\in T, (\\forall p \\in \\bullet t, W(p,t) \\le M_{ts}(p)), (\\forall p \\in I(t), M_{ts}(p) = 0), (\\forall (p,t) \\in Q, M_0(p) \\ge WQ(p,t)), \\nexists tt \\in T, (\\forall pp \\in \\bullet tt, W(pp,tt) \\le M_{ts}(pp)), (\\forall pp \\in I(tt), M_{ts}(pp) = 0), (\\forall (pp,tt) \\in Q, M_{ts}(pp) \\ge QW(pp,tt)), Z(tt) < Z(t), 0 \\le ts \\le k \\rbrace $ $A_{25}=\\lbrace could\\_not\\_have(t,ts): t \\in T, prenabled(t,ts) \\in A_{24}, fires(t,ts) \\notin A_{14}, (\\exists p \\in \\bullet t \\cup R(t): q > M_{ts}(p) - (\\sum _{t^{\\prime } \\in T_{ts}, p \\in \\bullet t^{\\prime }}{W(p,t^{\\prime })} + \\sum _{t^{\\prime } \\in T_{ts}, p \\in R(t^{\\prime })}{M_{ts}(p)}), q=W(p,t) \\text{~if~} (p,t) \\in E^- \\text{~or~} R(t) \\text{~otherwise}), 0 \\le ts \\le k\\rbrace =\\lbrace could\\_not\\_have(t,ts) : t \\in T, (\\forall p \\in \\bullet t, W(p,t) \\le M_{ts}(p)), (\\forall p \\in I(t), M_{ts}(p) = 0), (\\forall (p,t) \\in Q, M_0(p) \\ge QW(p,t)), (\\nexists tt \\in T, $ $(\\forall pp \\in \\bullet tt, W(pp,tt) \\le M_{ts}(pp)), (\\forall pp \\in I(tt), M_{ts}(pp) = 0), (\\forall (pp,tt) \\in Q, M_{ts}(pp) \\ge QW(pp,tt)), Z(tt) < Z(t)), t \\notin T_{ts}, (\\exists p \\in \\bullet t \\cup R(t): q > M_{ts}(p) - (\\sum _{t^{\\prime } \\in T_{ts}, p \\in \\bullet t^{\\prime }}{W(p,t^{\\prime })} + \\sum _{t^{\\prime } \\in T_{ts}, p \\in R(t^{\\prime })}{M_{ts}(p)}), q=W(p,t) \\text{~if~} \\\\ (p,t) \\in E^- \\text{~or~} R(t) \\text{~otherwise}), 0 \\le ts \\le k \\rbrace $ per the maximal firing set semantics We show that $A$ satisfies (REF ) and (REF ), and $A$ is an answer set of $\\Pi ^6$ .", "$A$ satisfies (REF ) and (REF ) by its construction above.", "We show $A$ is an answer set of $\\Pi ^6$ by splitting.", "We split $lit(\\Pi ^6)$ into a sequence of $9k+11$ sets: $U_0= head(f\\ref {f:c:place}) \\cup head(f\\ref {f:c:trans}) \\cup head(f\\ref {f:c:col}) \\cup head(f\\ref {f:c:time}) \\cup head(f\\ref {f:c:num}) \\cup head(i\\ref {i:c:init}) \\cup head(f\\ref {f:c:pr}) = \\lbrace place(p) : p \\in P\\rbrace \\cup \\lbrace trans(t) : t \\in T\\rbrace \\cup \\lbrace col(c) : c \\in C \\rbrace \\cup \\lbrace time(0), \\dots , time(k)\\rbrace \\cup \\lbrace num(0), \\dots , num(ntok)\\rbrace \\cup \\lbrace holds(p,q_c,c,0) : p \\in P, c \\in C, q_c=m_{M_0(p)}(c) \\rbrace \\cup \\lbrace transpr(t,pr) : t \\in T, pr=Z(t) \\rbrace $ $U_{9k+1}=U_{9k+0} \\cup head(f\\ref {f:c:ptarc})^{ts=k} \\cup head(f\\ref {f:c:tparc})^{ts=k} \\cup head(f\\ref {f:c:rptarc})^{ts=k} \\cup head(f\\ref {f:c:iptarc})^{ts=k} \\cup head(f\\ref {f:c:tptarc})^{ts=k} = U_{10k+0} \\cup \\lbrace ptarc(p,t,n_c,c,k) : (p,t) \\in E^-, c \\in C, n_c=m_{W(p,t)}(c) \\rbrace \\\\ \\cup \\lbrace tparc(t,p,n_c,c,k) : (t,p) \\in E^+, c \\in C, n_c=m_{W(t,p)}(c) \\rbrace \\cup \\lbrace ptarc(p,t,n_c,c,k) : p \\in R(t), c \\in C, n_c=m_{M_{k}(p)}(c), n > 0 \\rbrace \\cup \\lbrace iptarc(p,t,1,c,k) : p \\in I(t), c \\in C \\rbrace \\cup \\lbrace tptarc(p,t,n_c,c,k) : (p,t) \\in Q, c \\in C, n_c=m_{QW(p,t)}(c) \\rbrace $ $U_{9k+2}=U_{9k+1} \\cup head(e\\ref {e:c:ne:ptarc})^{ts=k} \\cup head(e\\ref {e:c:ne:iptarc})^{ts=k} \\cup head(e\\ref {e:c:ne:tptarc})^{ts=k} = U_{9k+1} \\\\ \\cup \\lbrace notenabled(t,k) : t \\in T \\rbrace $ $U_{9k+3}=U_{9k+2} \\cup head(e\\ref {e:c:enabled})^{ts=k} = U_{9k+2} \\cup \\lbrace enabled(t,k) : t \\in T \\rbrace $ $U_{9k+4}=U_{9k+3} \\cup head(a\\ref {a:c:prne})^{ts=k} = U_{9k+3} \\cup \\lbrace notprenabled(t,k) : t \\in T \\rbrace $ $U_{9k+5}=U_{9k+4} \\cup head(a\\ref {a:c:prenabled})^{ts=k} = U_{9k+4} \\cup \\lbrace prenabled(t,k) : t \\in T \\rbrace $ $U_{9k+6}=U_{9k+5} \\cup head(a\\ref {a:c:prfires})^{ts=k} = U_{9k+5} \\cup \\lbrace fires(t,k) : t \\in T \\rbrace $ $U_{9k+7}=U_{9k+6} \\cup head(r\\ref {r:c:add})^{ts=k} \\cup head(r\\ref {r:c:del})^{ts=k} = U_{9k+6} \\cup \\lbrace add(p,q_c,t,c,k) : p \\in P, t \\in T, c \\in C, q_c=m_{W(t,p)}(c) \\rbrace \\cup \\lbrace del(p,q_c,t,c,k) : p \\in P, t \\in T, c \\in C, q_c=m_{W(p,t)}(c) \\rbrace \\cup \\lbrace del(p,q_c,t,c,k) : p \\in P, t \\in T, c \\in C, q_c=m_{M_{k}(p)}(c) \\rbrace $ $U_{9k+8}=U_{9k+7} \\cup head(r\\ref {r:c:totincr})^{ts=k} \\cup head(r\\ref {r:c:totdecr})^{ts=k} = U_{9k+7} \\cup \\lbrace tot\\_incr(p,q_c,c,k) : p \\in P, c \\in C, 0 \\le q_c \\le ntok \\rbrace \\cup \\lbrace tot\\_decr(p,q_c,c,k) : p \\in P, c \\in C, 0 \\le q_c \\le ntok \\rbrace $ $U_{9k+9}=U_{9k+8} \\cup head(a\\ref {a:c:overc:place})^{ts=k} \\cup head(r\\ref {r:c:nextstate})^{ts=k} \\cup head(a\\ref {a:c:prmaxfire:cnh})^{ts=k} = U_{9k+8} \\cup \\\\ \\lbrace consumesmore(p,k) : p \\in P\\rbrace \\cup \\lbrace holds(p,q,k+1) : p \\in P, 0 \\le q \\le ntok \\rbrace \\cup \\lbrace could\\_not\\_have(t,k) : t \\in T \\rbrace $ $U_{9k+10}=U_{9k+9} \\cup head(a\\ref {a:c:overc:gen})^{ts=k} = U_{9k+9} \\cup \\lbrace consumesmore \\rbrace $ where $head(r_i)^{ts=k}$ are head atoms of ground rule $r_i$ in which $ts=k$ .", "We write $A_i^{ts=k} = \\lbrace a(\\dots ,ts) : a(\\dots ,ts) \\in A_i, ts=k \\rbrace $ as short hand for all atoms in $A_i$ with $ts=k$ .", "$U_{\\alpha }, 0 \\le \\alpha \\le 9k+10$ form a splitting sequence, since each $U_i$ is a splitting set of $\\Pi ^6$ , and $\\langle U_{\\alpha }\\rangle _{\\alpha < \\mu }$ is a monotone continuous sequence, where $U_0 \\subseteq U_1 \\dots \\subseteq U_{9k+10}$ and $\\bigcup _{\\alpha < \\mu }{U_{\\alpha }} = lit(\\Pi ^6)$ .", "We compute the answer set of $\\Pi ^6$ using the splitting sets as follows: $bot_{U_0}(\\Pi ^6) = f\\ref {f:c:place} \\cup f\\ref {f:c:trans} \\cup f\\ref {f:c:col} \\cup f\\ref {f:c:time} \\cup f\\ref {f:c:num} \\cup i\\ref {i:c:init} \\cup f\\ref {f:c:pr}$ and $X_0 = A_1 \\cup \\dots \\cup A_5 \\cup A_8$ ($= U_0$ ) is its answer set – using forced atom proposition $eval_{U_0}(bot_{U_1}(\\Pi ^6) \\setminus bot_{U_0}(\\Pi ^6), X_0) = \\lbrace ptarc(p,t,q_c,c,0) \\text{:-}.", "\| c \\in C, $ $q_c=m_{W(p,t)}(c), q_c > 0 \\rbrace \\cup \\lbrace tparc(t,p,q_c,c,0) \\text{:-}.", "\| c \\in C, q_c=m_{W(t,p)}(c), q_c > 0 \\rbrace \\cup \\lbrace ptarc(p,t,q_c,c,0) \\text{:-}.", "\| c \\in C, $ $q_c=m_{M_0(p)}(c), q_c > 0 \\rbrace \\cup \\lbrace iptarc(p,t,1,c,0) \\text{:-}.", "\| c \\in C \\rbrace \\cup \\lbrace tptarc(p,t,q_c,c,0) \\text{:-}.", "\| c \\in C, q_c = m_{QW(p,t)}(c), q_c > 0 \\rbrace $ .", "Its answer set $X_1=A_6^{ts=0} \\cup A_7^{ts=0} \\cup A_9^{ts=0} \\cup A_{10}^{ts=0} \\cup A_{11}^{ts=0}$ – using forced atom proposition and construction of $A_6, A_7, A_9, A_{10}, A_{11}$ .", "$eval_{U_1}(bot_{U_2}(\\Pi ^6) \\setminus bot_{U_1}(\\Pi ^6), X_0 \\cup X_1) = \\lbrace notenabled(t,0) \\text{:-} .", "\| (\\lbrace trans(t), \\\\ ptarc(p,t,n_c,c,0), holds(p,q_c,c,0) \\rbrace \\subseteq X_0 \\cup X_1, \\text{~where~} q_c < n_c) \\text{~or~} \\\\ (\\lbrace notenabled(t,0) \\text{:-} .", "\| (\\lbrace trans(t), iptarc(p,t,n2_c,c,0), holds(p,q_c,c,0) \\rbrace \\subseteq X_0 \\cup X_1, \\\\ \\text{~where~} q_c \\ge n2_c \\rbrace ) \\text{~or~} (\\lbrace trans(t), tptarc(p,t,n3_c,c,0), $ $holds(p,q_c,c,0) \\rbrace \\subseteq X_0 \\cup X_1, \\text{~where~} $ $q_c < n3_c) \\rbrace $ .", "Its answer set $X_2=A_{12}^{ts=0}$ – using forced atom proposition and construction of $A_{12}$ .", "where, $q_c=m_{M_0(p)}(c)$ , and $n_c=m_{W(p,t)}(c)$ for an arc $(p,t) \\in E^-$ – by construction of $i\\ref {i:c:init}$ and $f\\ref {f:c:ptarc}$ in $\\Pi ^6$ , and in an arc $(p,t) \\in E^-$ , $p \\in \\bullet t$ (by definition REF of preset) $n2_c=1$ – by construction of $iptarc$ predicates in $\\Pi ^6$ , meaning $q_c \\ge n2_c \\equiv q_c \\ge 1 \\equiv q_c > 0$ , $tptarc(p,t,n3_c,c,0)$ represents $n3_c=m_{QW(p,t)}(c)$ , where $(p,t) \\in Q$ thus, $notenabled(t,0) \\in X_1$ represents $\\exists c \\in C, (\\exists p \\in \\bullet t : m_{M_0(p)}(c) < m_{W(p,t)}(c)) \\vee (\\exists p \\in I(t) : m_{M_0(p)}(c) > 0) \\vee (\\exists (p,t) \\in Q : m_{M_0(p)}(c) < m_{QW(p,t)}(c))$ .", "$eval_{U_2}(bot_{U_3}(\\Pi ^6) \\setminus bot_{U_2}(\\Pi ^6), X_0 \\cup \\dots \\cup X_2) = \\lbrace enabled(t,0) \\text{:-}.", "\| trans(t) \\in X_0 \\cup \\dots \\cup X_2, notenabled(t,0) \\notin X_0 \\cup \\dots \\cup X_2 \\rbrace $ .", "Its answer set is $X_3 = A_{13}^{ts=0}$ – using forced atom proposition and construction of $A_{13}$ .", "since an $enabled(t,0) \\in X_3$ if $\\nexists ~notenabled(t,0) \\in X_0 \\cup \\dots \\cup X_2$ ; which is equivalent to $\\nexists t, \\forall c \\in C, (\\nexists p \\in \\bullet t, m_{M_0(p)}(c) < m_{W(p,t)}(c)), (\\nexists p \\in I(t), $ $m_{M_0(p)}(c) > 0), (\\nexists (p,t) \\in Q : m_{M_0(p)}(c) < m_{QW(p,t)}(c) ), \\forall c \\in C, (\\forall p \\in \\bullet t: m_{M_0(p)}(c) \\ge m_{W(p,t)}(c)), (\\forall p \\in I(t) : m_{M_0(p)}(c) = 0)$ .", "$eval_{U_3}(bot_{U_4}(\\Pi ^6) \\setminus bot_{U_3}(\\Pi ^6), X_0 \\cup \\dots \\cup X_3) = \\lbrace notprenabled(t,0) \\text{:-}.", "\| \\\\ \\lbrace enabled(t,0), transpr(t,p), enabled(tt,0), transpr(tt,pp) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_3, pp < p \\rbrace $ .", "Its answer set is $X_4 = A_{23}^{ts=k}$ – using forced atom proposition and construction of $A_{23}$ .", "$enabled(t,0)$ represents $\\exists t \\in T, \\forall c \\in C, (\\forall p \\in \\bullet t, m_{M_0(p)}(c) \\ge m_{W(p,t)}(c)), $ $(\\forall p \\in I(t), m_{M_0(p)}(c) = 0), (\\forall (p,t) \\in Q, m_{M_0(p)}(c) \\ge m_{QW(p,t)}(c))$ $enabled(tt,0)$ represents $\\exists tt \\in T, \\forall c \\in C, (\\forall pp \\in \\bullet tt, m_{M_0(pp)}(c) \\ge \\\\ m_{W(pp,tt)}(c)) \\wedge (\\forall pp \\in I(tt), m_{M_0(pp)}(c) = 0), (\\forall (pp,tt) \\in Q, m_{M_0(pp)}(c) \\ge m_{QW(pp,tt)}(c))$ $transpr(t,p)$ represents $p=Z(t)$ – by construction $transpr(tt,pp)$ represents $pp=Z(tt)$ – by construction thus, $notprenabled(t,0)$ represents $\\forall c \\in C, (\\forall p \\in \\bullet t, m_{M_0(p)}(c) \\ge \\\\ m_{W(p,t)}(c)) \\wedge (\\forall p \\in I(t), m_{M_0(p)}(c) = 0), \\exists tt \\in T, (\\forall pp \\in \\bullet tt, m_{M_0(pp)}(c) \\ge m_{W(pp,tt)}(c)) \\wedge (\\forall pp \\in I(tt), m_{M_0(pp)}(c) = 0), Z(tt) < Z(t)$ which is equivalent to $(\\forall p \\in \\bullet t: M_0(p) \\ge W(p,t)) \\wedge (\\forall p \\in I(t), M_0(p) = 0), \\exists tt \\in T, (\\forall pp \\in \\bullet tt, M_0(pp) \\ge W(pp,tt)), (\\forall pp \\in I(tt), M_0(pp) = 0), Z(tt) < Z(t)$ – assuming multiset domain $C$ for all operations $eval_{U_4}(bot_{U_5}(\\Pi ^6) \\setminus bot_{U_4}(\\Pi ^6), X_0 \\cup \\dots \\cup X_4) = \\lbrace prenabled(t,0) \\text{:-}.", "\| enabled(t,0) \\in X_0 \\cup \\dots \\cup X_4, notprenabled(t,0) \\notin X_0 \\cup \\dots \\cup X_4 \\rbrace $ .", "Its answer set is $X_5 = A_{24}^{ts=k}$ – using forced atom proposition and construction of $A_{24}$ $enabled(t,0)$ represents $\\forall c \\in C, (\\forall p \\in \\bullet t, m_{M_0(p)}(c) \\ge m_{W(p,t)}(c)), (\\forall p \\in I(t), m_{M_0(p)}(c) = 0), (\\forall (p,t) \\in Q, m_{M_0(p)}(c) \\ge m_{QW(p,t)}(c)) \\equiv (\\forall p \\in \\bullet t, \\\\ M_0(p) \\ge W(p,t)), (\\forall p \\in I(t), M_0(p) = 0), (\\forall (p,t) \\in Q, M_0(p) \\ge QW(p,t))$ – from REF above and assuming multiset domain $C$ for all operations $notprenabled(t,0)$ represents $(\\forall p \\in \\bullet t, M_0(p) \\ge W(p,t)), (\\forall p \\in I(t), \\\\ M_0(p) = 0), (\\forall (p,t) \\in Q, M_0(p) \\ge QW(p,t)), \\exists tt \\in T, (\\forall pp \\in \\bullet tt, M_0(pp) \\ge W(pp,tt)), (\\forall pp \\in I(tt), M_0(pp) = 0), (\\forall (pp,tt) \\in Q, M_0(pp) \\ge W(pp,tt)), \\\\ Z(tt) < Z(t)$ – from REF above and assuming multiset domain $C$ for all operations then, $prenabled(t,0)$ represents $(\\forall p \\in \\bullet t, M_0(p) \\ge W(p,t)), (\\forall p \\in I(t), \\\\ M_0(p) = 0), (\\forall (p,t) \\in Q, M_0(p) \\ge QW(p,t)), \\nexists tt \\in T, ((\\forall pp \\in \\bullet tt, M_0(pp) \\ge W(pp,tt)), (\\forall pp \\in I(tt), M_0(pp) = 0), (\\forall (pp,tt) \\in Q, M_0(pp) \\ge W(pp,tt)), \\\\ Z(tt) < Z(t))$ – from (a), (b) and $enabled(t,0) \\in X_0 \\cup \\dots \\cup X_4$ $eval_{U_5}(bot_{U_6}(\\Pi ^6) \\setminus bot_{U_5}(\\Pi ^6), X_0 \\cup \\dots \\cup X_5) = \\lbrace \\lbrace fires(t,0)\\rbrace \\text{:-}.", "\| prenabled(t,0) \\\\ \\text{~holds in~} X_0 \\cup \\dots \\cup X_5 \\rbrace $ .", "It has multiple answer sets $X_{6.1}, \\dots , X_{6.n}$ , corresponding to elements of power set of $fires(t,0)$ atoms in $eval_{U_5}(...)$ – using supported rule proposition.", "Since we are showing that the union of answer sets of $\\Pi ^6$ determined using splitting is equal to $A$ , we only consider the set that matches the $fires(t,0)$ elements in $A$ and call it $X_6$ , ignoring the rest.", "Thus, $X_6 = A_{14}^{ts=0}$ , representing $T_0$ .", "in addition, for every $t$ such that $prenabled(t,0) \\in X_0 \\cup \\dots \\cup X_5, R(t) \\ne \\emptyset $ ; $fires(t,0) \\in X_6$ – per definition REF (firing set); requiring that a reset transition is fired when enabled thus, the firing set $T_0$ will not be eliminated by the constraint $f\\ref {f:c:pr:rptarc:elim}$ $eval_{U_6}(bot_{U_7}(\\Pi ^6) \\setminus bot_{U_6}(\\Pi ^6), X_0 \\cup \\dots \\cup X_6) = \\lbrace add(p,n_c,t,c,0) \\text{:-}.", "\| \\lbrace fires(t,0), \\\\ tparc(t,p,n_c,c,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_6 \\rbrace \\cup \\lbrace del(p,n_c,t,c,0) \\text{:-}.", "\| \\lbrace fires(t,0), \\\\ ptarc(p,t,n_c,c,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_6 \\rbrace $ .", "It's answer set is $X_7 = A_{15}^{ts=0} \\cup A_{16}^{ts=0}$ – using forced atom proposition and definitions of $A_{15}$ and $A_{16}$ .", "where, each $add$ atom is equivalent to $n_c=m_{W(t,p)}(c),c \\in C, p \\in t \\bullet $ , and each $del$ atom is equivalent to $n_c=m_{W(p,t)}(c), c \\in C, p \\in \\bullet t$ ; or $n_c=m_{M_0(p)}(c), c \\in C, p \\in R(t)$ , representing the effect of transitions in $T_0$ – by construction $eval_{U_7}(bot_{U_8}(\\Pi ^6) \\setminus bot_{U_7}(\\Pi ^6), X_0 \\cup \\dots \\cup X_7) = \\lbrace tot\\_incr(p,qq_c,c,0) \\text{:-}.", "\| qq_c=\\sum _{add(p,q_c,t,c,0) \\in X_0 \\cup \\dots \\cup X_7}{q_c} \\rbrace \\cup \\lbrace tot\\_decr(p,qq_c,c,0) \\text{:-}.", "\| qq_c=\\sum _{del(p,q_c,t,c,0) \\in X_0 \\cup \\dots \\cup X_7}{q_c} \\rbrace $ .", "It's answer set is $X_8 = A_{17}^{ts=0} \\cup A_{18}^{ts=0}$ – using forced atom proposition and definitions of $A_{17}$ and $A_{18}$ .", "where, each $tot\\_incr(p,qq_c,c,0)$ , $qq_c=\\sum _{add(p,q_c,t,c,0) \\in X_0 \\cup \\dots X_7}{q_c}$ $\\equiv qq_c=\\sum _{t \\in X_6, p \\in t \\bullet }{m_{W(p,t)}(c)}$ , where, each $tot\\_decr(p,qq_c,c,0)$ , $qq_c=\\sum _{del(p,q_c,t,c,0) \\in X_0 \\cup \\dots X_7}{q_c}$ $\\equiv qq=\\sum _{t \\in X_6, p \\in \\bullet t}{m_{W(t,p)}(c)} + \\sum _{t \\in X_6, p \\in R(t)}{m_{M_0(p)}(c)}$ , represent the net effect of transitions in $T_0$ $eval_{U_8}(bot_{U_9}(\\Pi ^6) \\setminus bot_{U_8}(\\Pi ^6), X_0 \\cup \\dots \\cup X_8) = \\lbrace consumesmore(p,0) \\text{:-}.", "\| \\\\ \\lbrace holds(p,q_c,c,0), tot\\_decr(p,q1_c,c,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_8, q1_c > q_c \\rbrace \\cup \\\\ \\lbrace holds(p,q_c,c,1) \\text{:-}.", "\| \\lbrace holds(p,q1_c,c,0), tot\\_incr(p,q2_c,c,0), tot\\_decr(p,q3_c,c,0) \\rbrace \\\\ \\subseteq X_0 \\cup \\dots \\cup X_6, q_c=q1_c+q2_c-q3_c \\rbrace \\cup \\lbrace could\\_not\\_have(t,0) \\text{:-}.", "\| \\lbrace prenabled(t,0), \\\\ ptarc(s,t,q,c,0), holds(s,qq,c,0), tot\\_decr(s,qqq,c,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_8, \\\\ fires(t,0) \\notin X_0 \\cup \\dots \\cup X_8, q > qq - qqq \\rbrace $ .", "It's answer set is $X_9 = A_{19}^{ts=0} \\cup A_{21}^{ts=0} \\cup A_{25}^{ts=0}$ – using forced atom proposition and definitions of $A_{19}, A_{21}, A_{25}$ .", "where, $consumesmore(p,0)$ represents $\\exists p : q_c=m_{M_0(p)}(c), q1_c= \\\\ \\sum _{t \\in T_0, p \\in \\bullet t}{m_{W(p,t)}(c)}+\\sum _{t \\in T_0, p \\in R(t)}{m_{M_0(p)}(c)}, q1_c > q_c, c \\in C$ , indicating place $p$ will be over consumed if $T_0$ is fired, as defined in definition REF (conflicting transitions), $holds(p,q_c,c,1)$ if $q_c=m_{M_0(p)}(c)+\\sum _{t \\in T_0, p \\in t \\bullet }{m_{W(t,p)}(c)}- \\\\ (\\sum _{t \\in T_0, p \\in \\bullet t}{m_{W(p,t)}(c)}+ \\sum _{t \\in T_0, p \\in R(t)}{m_{M_0(p)}(c)})$ represented by $q_c=m_{M_1(p)}(c)$ for some $c \\in C$ – by construction of $\\Pi ^6$ $could\\_not\\_have(t,0)$ if $(\\forall p \\in \\bullet t, W(p,t) \\le M_0(p)), (\\forall p \\in I(t), M_0(p) = 0), (\\forall (p,t) \\in Q, \\\\ M_0(p) \\ge WQ(p,t)), \\nexists tt \\in T, (\\forall pp \\in \\bullet tt, W(pp,tt) \\le M_{ts}(pp)), (\\forall pp \\in I(tt), M_0(pp) = 0), (\\forall (pp,tt) \\in Q, M_0(pp) \\ge QW(pp,tt)), Z(tt) < Z(t)$ , and $q_c > m_{M_0(s)}(c) - (\\sum _{t^{\\prime } \\in T_0, s \\in \\bullet t^{\\prime }}{m_{W(s,t^{\\prime })}(c)}+ \\sum _{t^{\\prime } \\in T_0, s \\in R(t)}{m_{M_0(s)}(c)}), \\\\ q_c = m_{W(s,t)}(c) \\text{~if~} s \\in \\bullet t \\text{~or~} m_{M_0(s)}(c) \\text{~otherwise}$ for some $c \\in C$ , which becomes $q > M_0(s) - (\\sum _{t^{\\prime } \\in T_0, s \\in \\bullet t^{\\prime }}{W(s,t^{\\prime })}+ \\sum _{t^{\\prime } \\in T_0, s \\in R(t)}{M_0(s)}), q = W(s,t) \\text{~if~} s \\in \\bullet t \\text{~or~} M_0(s) \\text{~otherwise}$ for all $c \\in C$ (i), (ii) above combined match the definition of $A_{25}$ $X_9$ does not contain $could\\_not\\_have(t,0)$ , when $prenabled(t,0) \\in X_0 \\cup \\dots \\cup X_8$ and $fires(t,0) \\notin X_0 \\cup \\dots \\cup X_8$ due to construction of $A$ , encoding of $a\\ref {a:c:overc:place}$ and its body atoms.", "As a result, it is not eliminated by the constraint $a\\ref {a:c:maxfire:elim}$ $ \\vdots $ $eval_{U_{9k+0}}(bot_{U_{9k+1}}(\\Pi ^6) \\setminus bot_{U_{9k+0}}(\\Pi ^6), X_0 \\cup \\dots \\cup X_{9k+0}) = \\lbrace ptarc(p,t,q_c,c,k) \\text{:-}.", "\| c \\in C, q_c=m_{W(p,t)}(c), q_c > 0 \\rbrace \\cup \\lbrace tparc(t,p,q_c,c,k) \\text{:-}.", "\| c \\in C, q_c=m_{W(t,p)}(c), q_c > 0 \\rbrace \\cup \\lbrace ptarc(p,t,q_c,c,k) \\text{:-}.", "\| c \\in C, q_c=m_{M_0(p)}(c), q_c > 0 \\rbrace \\cup \\lbrace iptarc(p,t,1,c,k) \\text{:-}.", "\| c \\in C \\rbrace \\cup \\lbrace tptarc(p,t,q_c,c,k) \\text{:-}.", "\| c \\in C, q_c = m_{QW(p,t)}(c), q_c > 0 \\rbrace $ .", "Its answer set $X_{9k+1}=A_6^{ts=k} \\cup A_7^{ts=k} \\cup A_9^{ts=k} \\cup A_{10}^{ts=k} \\cup A_{11}^{ts=k}$ – using forced atom proposition and construction of $A_6, A_7, A_9, A_{10}, A_{11}$ .", "$eval_{U_{9k+1}}(bot_{U_{9k+2}}(\\Pi ^6) \\setminus bot_{U_{9k+1}}(\\Pi ^6), X_0 \\cup \\dots \\cup X_{9k+1}) = $ $\\lbrace notenabled(t,k) \\text{:-} .", "\| $ $(\\lbrace trans(t), $ $ptarc(p,t,n_c,c,k), $ $holds(p,q_c,c,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{10k+1}, \\text{~where~} $ $q_c < n_c) \\text{~or~} $ $(\\lbrace notenabled(t,k) \\text{:-} .", "\| $ $(\\lbrace trans(t), $ $iptarc(p,t,n2_c,c,k), $ $holds(p,q_c,c,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{10k+1}, \\text{~where~} $ $q_c \\ge n2_c \\rbrace ) \\text{~or~} $ $(\\lbrace trans(t), $ $tptarc(p,t,n3_c,c,k), \\\\ holds(p,q_c,c,k) \\rbrace \\subseteq X_0 \\cup X_{9k+1}, \\text{~where~} q_c < n3_c) \\rbrace $ .", "Its answer set $X_{9k+2}=A_{12}^{ts=k}$ – using forced atom proposition and construction of $A_{12}$ .", "where, $q_c=m_{M_0(p)}(c)$ , and $n_c=m_{W(p,t)}(c)$ for an arc $(p,t) \\in E^-$ – by construction of $i\\ref {i:c:init}$ and $f\\ref {f:c:ptarc}$ predicates in $\\Pi ^6$ , and in an arc $(p,t) \\in E^-$ , $p \\in \\bullet t$ (by definition REF of preset) $n2_c=1$ – by construction of $iptarc$ predicates in $\\Pi ^6$ , meaning $q_c \\ge n2_c \\equiv q_c \\ge 1 \\equiv q_c > 0$ , $tptarc(p,t,n3_c,c,k)$ represents $n3_c=m_{QW(p,t)}(c)$ , where $(p,t) \\in Q$ thus, $notenabled(t,k) \\in X_{9k+1}$ represents $\\exists c \\in C, (\\exists p \\in \\bullet t : m_{M_0(p)}(c) < m_{W(p,t)}(c)) \\vee (\\exists p \\in I(t) : m_{M_0(p)}(c) > k) \\vee (\\exists (p,t) \\in Q : m_{M_0(p)}(c) < m_{QW(p,t)}(c))$ .", "$eval_{U_{9k+2}}(bot_{U_{9k+3}}(\\Pi ^6) \\setminus bot_{U_{9k+2}}(\\Pi ^6), X_0 \\cup \\dots \\cup X_{9k+2}) = \\lbrace enabled(t,k) \\text{:-}.", "\| trans(t) \\in X_0 \\cup \\dots \\cup X_{9k+2}, notenabled(t,k) \\notin X_0 \\cup \\dots \\cup X_{9k+2} \\rbrace $ .", "Its answer set is $X_{9k+3} = A_{13}^{ts=k}$ – using forced atom proposition and construction of $A_{13}$ .", "since an $enabled(t,k) \\in X_{9k+3}$ if $\\nexists ~notenabled(t,k) \\in X_0 \\cup \\dots \\cup X_{9k+2}$ ; which is equivalent to $\\nexists t, \\forall c \\in C, (\\nexists p \\in \\bullet t, m_{M_0(p)}(c) < m_{W(p,t)}(c)), (\\nexists p \\in I(t), m_{M_0(p)}(c) > k), (\\nexists (p,t) \\in Q : m_{M_0(p)}(c) < m_{QW(p,t)}(c) ), \\forall c \\in C, (\\forall p \\in \\bullet t: m_{M_0(p)}(c) \\ge m_{W(p,t)}(c)), (\\forall p \\in I(t) : m_{M_0(p)}(c) = k)$ .", "$eval_{U_{9k+3}}(bot_{U_{9k+4}}(\\Pi ^6) \\setminus bot_{U_{9k+3}}(\\Pi ^6), X_0 \\cup \\dots \\cup X_{9k+3}) = \\lbrace notprenabled(t,k) \\text{:-}.", "\| \\\\ \\lbrace enabled(t,k), transpr(t,p), enabled(tt,k), transpr(tt,pp) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{9k+3}, \\\\ pp < p \\rbrace $ .", "Its answer set is $X_{9k+4} = A_{23}^{ts=k}$ – using forced atom proposition and construction of $A_{23}$ .", "$enabled(t,k)$ represents $\\exists t \\in T, \\forall c \\in C, (\\forall p \\in \\bullet t, m_{M_0(p)}(c) \\ge m_{W(p,t)}(c)), \\\\ (\\forall p \\in I(t), m_{M_0(p)}(c) = k), (\\forall (p,t) \\in Q, m_{M_0(p)}(c) \\ge m_{QW(p,t)}(c))$ $enabled(tt,k)$ represents $\\exists tt \\in T, \\forall c \\in C, (\\forall pp \\in \\bullet tt, m_{M_0(pp)}(c) \\ge \\\\ m_{W(pp,tt)}(c)) \\wedge (\\forall pp \\in I(tt), m_{M_0(pp)}(c) = k), (\\forall (pp,tt) \\in Q, m_{M_0(pp)}(c) \\ge m_{QW(pp,tt)}(c))$ $transpr(t,p)$ represents $p=Z(t)$ – by construction $transpr(tt,pp)$ represents $pp=Z(tt)$ – by construction thus, $notprenabled(t,k)$ represents $\\forall c \\in C, (\\forall p \\in \\bullet t, m_{M_0(p)}(c) \\ge m_{W(p,t)}(c)) \\\\ \\wedge (\\forall p \\in I(t), m_{M_0(p)}(c) = k), \\exists tt \\in T, (\\forall pp \\in \\bullet tt, m_{M_0(pp)}(c) \\ge m_{W(pp,tt)}(c)) \\wedge (\\forall pp \\in I(tt), m_{M_0(pp)}(c) = k), Z(tt) < Z(t)$ which is equivalent to $(\\forall p \\in \\bullet t: M_0(p) \\ge W(p,t)) \\wedge (\\forall p \\in I(t), M_0(p) = k), \\exists tt \\in T, (\\forall pp \\in \\bullet tt, M_0(pp) \\ge W(pp,tt)), (\\forall pp \\in I(tt), M_0(pp) = k), Z(tt) < Z(t)$ – assuming multiset domain $C$ for all operations $eval_{U_{9k+4}}(bot_{U_{9k+5}}(\\Pi ^6) \\setminus bot_{U_{9k+4}}(\\Pi ^6), X_0 \\cup \\dots \\cup X_{9k+4}) = \\lbrace prenabled(t,k) \\text{:-}.", "\| $ $enabled(t,k) \\in X_0 \\cup \\dots \\cup X_{9k+4}, notprenabled(t,k) \\notin X_0 \\cup \\dots \\cup X_{9k+4} \\rbrace $ .", "Its answer set is $X_{9k+5} = A_{24}^{ts=k}$ – using forced atom proposition and construction of $A_{24}$ $enabled(t,k)$ represents $\\forall c \\in C, (\\forall p \\in \\bullet t, m_{M_0(p)}(c) \\ge m_{W(p,t)}(c)), $ $(\\forall p \\in I(t), m_{M_0(p)}(c) = k), (\\forall (p,t) \\in Q, m_{M_0(p)}(c) \\ge m_{QW(p,t)}(c)) \\equiv (\\forall p \\in \\bullet t, M_0(p) \\ge W(p,t)), $ $(\\forall p \\in I(t), M_0(p) = k), (\\forall (p,t) \\in Q, M_0(p) \\ge QW(p,t))$ – from REF above and assuming multiset domain $C$ for all operations $notprenabled(t,k)$ represents $(\\forall p \\in \\bullet t, M_0(p) \\ge W(p,t)), (\\forall p \\in I(t), \\\\ M_0(p) = k), (\\forall (p,t) \\in Q, M_0(p) \\ge QW(p,t)), \\exists tt \\in T, $ $(\\forall pp \\in \\bullet tt, M_0(pp) \\ge W(pp,tt)), $ $(\\forall pp \\in I(tt), M_0(pp) = k), (\\forall (pp,tt) \\in Q, M_0(pp) \\ge W(pp,tt)), \\\\ Z(tt) < Z(t)$ – from REF above and assuming multiset domain $C$ for all operations then, $prenabled(t,k)$ represents $(\\forall p \\in \\bullet t, M_0(p) \\ge W(p,t)), (\\forall p \\in I(t), \\\\ M_0(p) = k), (\\forall (p,t) \\in Q, M_0(p) \\ge QW(p,t)), \\nexists tt \\in T, $ $((\\forall pp \\in \\bullet tt, M_0(pp) \\ge W(pp,tt)), $ $(\\forall pp \\in I(tt), M_0(pp) = k), (\\forall (pp,tt) \\in Q, M_0(pp) \\ge W(pp,tt)), \\\\ Z(tt) < Z(t))$ – from (a), (b) and $enabled(t,k) \\in X_0 \\cup \\dots \\cup X_{9k+4}$ $eval_{U_{9k+5}}(bot_{U_{9k+6}}(\\Pi ^6) \\setminus bot_{U_{9k+5}}(\\Pi ^6), X_0 \\cup \\dots \\cup X_{9k+5}) = \\lbrace \\lbrace fires(t,k)\\rbrace \\text{:-}.", "\| \\\\ prenabled(t,k) \\text{~holds in~} X_0 \\cup \\dots \\cup X_{9k+5} \\rbrace $ .", "It has multiple answer sets $X_{9k+6.1}, \\dots , \\\\ X_{9k+6.n}$ , corresponding to elements of power set of $fires(t,k)$ atoms in $eval_{U_{9k+5}}(...)$ – using supported rule proposition.", "Since we are showing that the union of answer sets of $\\Pi ^6$ determined using splitting is equal to $A$ , we only consider the set that matches the $fires(t,k)$ elements in $A$ and call it $X_{9k+6}$ , ignoring the rest.", "Thus, $X_{9k+6} = A_{14}^{ts=k}$ , representing $T_k$ .", "in addition, for every $t$ such that $prenabled(t,k) \\in X_0 \\cup \\dots \\cup X_{9k+5}, R(t) \\ne \\emptyset $ ; $fires(t,k) \\in X_{9k+6}$ – per definition REF (firing set); requiring that a reset transition is fired when enabled thus, the firing set $T_k$ will not be eliminated by the constraint $f\\ref {f:c:pr:rptarc:elim}$ $eval_{U_{9k+6}}(bot_{U_{9k+7}}(\\Pi ^6) \\setminus bot_{U_{9k+6}}(\\Pi ^6), X_0 \\cup \\dots \\cup X_{9k+6}) = \\lbrace add(p,n_c,t,c,k) \\text{:-}.", "\| \\\\ \\lbrace fires(t,k), tparc(t,p,n_c,c,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{9k+6} \\rbrace \\cup \\lbrace del(p,n_c,t,c,k) \\text{:-}.", "\| \\\\ \\lbrace fires(t,k), ptarc(p,t,n_c,c,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{9k+6} \\rbrace $ .", "It's answer set is $X_{9k+7} = A_{15}^{ts=k} \\cup A_{16}^{ts=k}$ – using forced atom proposition and definitions of $A_{15}$ and $A_{16}$ .", "where, each $add$ atom is equivalent to $n_c=m_{W(t,p)}(c),c \\in C, p \\in t \\bullet $ , and each $del$ atom is equivalent to $n_c=m_{W(p,t)}(c), c \\in C, p \\in \\bullet t$ ; or $n_c=m_{M_0(p)}(c), c \\in C, p \\in R(t)$ , representing the effect of transitions in $T_k$ .", "$eval_{U_{9k+7}}(bot_{U_{9k+8}}(\\Pi ^6) \\setminus bot_{U_{9k+7}}(\\Pi ^6), X_0 \\cup \\dots \\cup X_{9k+7}) = \\lbrace tot\\_incr(p,qq_c,c,k) \\text{:-}.", "\| $ $qq_c=\\sum _{add(p,q_c,t,c,k) \\in X_0 \\cup \\dots \\cup X_{10k+7}}{q_c} \\rbrace \\cup \\lbrace tot\\_decr(p,qq_c,c,k) \\text{:-}.", "\| qq_c= \\\\ \\sum _{del(p,q_c,t,c,k) \\in X_0 \\cup \\dots \\cup X_{9k+7}}{q_c} \\rbrace $ .", "It's answer set is $X_{10k+8} = A_{17}^{ts=k} \\cup A_{18}^{ts=k}$ – using forced atom proposition and definitions of $A_{17}$ and $A_{18}$ .", "where, each $tot\\_incr(p,qq_c,c,k)$ , $qq_c=\\sum _{add(p,q_c,t,c,k) \\in X_0 \\cup \\dots X_{10k+7}}{q_c}$ $\\equiv qq_c=\\sum _{t \\in X_{9k+6}, p \\in t \\bullet }{m_{W(p,t)}(c)}$ , and each $tot\\_decr(p,qq_c,c,k)$ , $qq_c=\\sum _{del(p,q_c,t,c,k) \\in X_0 \\cup \\dots X_{10k+7}}{q_c}$ $\\equiv qq=\\sum _{t \\in X_{10k+6}, p \\in \\bullet t}{m_{W(t,p)}(c)} + \\sum _{t \\in X_{9k+6}, p \\in R(t)}{m_{M_0(p)}(c)}$ , represent the net effect of transitions in $T_k$ $eval_{U_{9k+8}}(bot_{U_{9k+9}}(\\Pi ^6) \\setminus bot_{U_{9k+8}}(\\Pi ^6), X_0 \\cup \\dots \\cup X_{9k+8}) = \\lbrace consumesmore(p,k) \\text{:-}.", "\| \\\\ \\lbrace holds(p,q_c,c,k), tot\\_decr(p,q1_c,c,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{9k+8}, q1_c > q_c \\rbrace \\cup \\\\ \\lbrace holds(p,q_c,c,1) \\text{:-}.", "\| \\lbrace holds(p,q1_c,c,k), tot\\_incr(p,q2_c,c,k), tot\\_decr(p,q3_c,c,k) \\rbrace \\\\ \\subseteq $ $X_0 \\cup \\dots \\cup X_{9k+6}, q_c=q1_c+q2_c-q3_c \\rbrace \\cup \\lbrace could\\_not\\_have(t,k) \\text{:-}.", "\| \\lbrace prenabled(t,k), \\\\ ptarc(s,t,q,c,k), holds(s,qq,c,k), tot\\_decr(s,qqq,c,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{9k+8}, \\\\ fires(t,k) \\notin X_0 \\cup \\dots \\cup X_{9k+8}, q > qq - qqq \\rbrace $ .", "It's answer set is $X_{10k+9} = A_{19}^{ts=k} \\cup A_{21}^{ts=k} \\cup A_{25}^{ts=k}$ – using forced atom proposition and definitions of $A_{19}, A_{21}, A_{25}$ .", "where, $consumesmore(p,k)$ represents $\\exists p : q_c=m_{M_0(p)}(c), \\\\ q1_c=$ $\\sum _{t \\in T_k, p \\in \\bullet t}{m_{W(p,t)}(c)}+\\sum _{t \\in T_k, p \\in R(t)}{m_{M_0(p)}(c)}, q1_c > q_c, c \\in C$ , indicating place $p$ that will be over consumed if $T_k$ is fired, as defined in definition REF (conflicting transition) $holds(p,q_c,c,k+1)$ if $q_c=m_{M_0(p)}(c)+\\sum _{t \\in T_k, p \\in t \\bullet }{m_{W(t,p)}(c)}- \\\\ (\\sum _{t \\in T_k, p \\in \\bullet t}{m_{W(p,t)}(c)}+ \\sum _{t \\in T_k, p \\in R(t)}{m_{M_0(p)}(c)})$ represented by $q_c=m_{M_1(p)}(c)$ for some $c \\in C$ – by construction of $\\Pi ^6$ $could\\_not\\_have(t,k)$ if $(\\forall p \\in \\bullet t, W(p,t) \\le M_0(p)), (\\forall p \\in I(t), M_0(p) = k), (\\forall (p,t) \\in Q, $ $M_0(p) \\ge WQ(p,t)), \\nexists tt \\in T, (\\forall pp \\in \\bullet tt, W(pp,tt) \\le M_{ts}(pp)), (\\forall pp \\in I(tt), M_0(pp) = k), (\\forall (pp,tt) \\in Q, $ $M_0(pp) \\ge QW(pp,tt)), Z(tt) < Z(t)$ , and $q_c > m_{M_0(s)}(c) - (\\sum _{t^{\\prime } \\in T_k, s \\in \\bullet t^{\\prime }}{m_{W(s,t^{\\prime })}(c)}+ \\sum _{t^{\\prime } \\in T_k, s \\in R(t)}{m_{M_0(s)}(c)}), $ $q_c = m_{W(s,t)}(c) \\text{~if~} s \\in \\bullet t \\text{~or~} m_{M_0(s)}(c) \\text{~otherwise}$ for some $c \\in C$ , which becomes $q > M_0(s) - (\\sum _{t^{\\prime } \\in T_k, s \\in \\bullet t^{\\prime }}{W(s,t^{\\prime })}+ \\sum _{t^{\\prime } \\in T_k, s \\in R(t)}{M_0(s)}), q = W(s,t) \\text{~if~} s \\in \\bullet t \\text{~or~} M_0(s) \\text{~otherwise}$ for all $c \\in C$ (i), (ii) above combined match the definition of $A_{25}$ $X_{9k+9}$ does not contain $could\\_not\\_have(t,k)$ , when $prenabled(t,k) \\in X_0 \\cup \\dots \\cup X_{9k+6}$ and $fires(t,k) \\notin X_0 \\cup \\dots \\cup X_{9k+5}$ due to construction of $A$ , encoding of $a\\ref {a:c:maxfire:cnh}$ and its body atoms.", "As a result it is not eliminated by the constraint $a\\ref {a:c:maxfire:elim}$ $eval_{U_{9k+9}}(bot_{U_{9k+10}}(\\Pi ^6) \\setminus bot_{U_{9k+9}}(\\Pi ^6), X_0 \\cup \\dots \\cup X_{9k+9}) = \\lbrace consumesmore \\text{:-}.", "\| \\\\ \\lbrace consumesmore(p,k) \\rbrace \\subseteq A \\rbrace $ .", "It's answer set is $X_{9k+10} = A_{20}$ – using forced atom proposition and the definition of $A_{20}$ $X_{9k+10}$ will be empty since none of $consumesmore(p,0),\\dots , \\\\ consumesmore(p,k)$ hold in $X_0 \\cup \\dots \\cup X_{9k+9}$ due to the construction of $A$ , encoding of $a\\ref {a:c:overc:place}$ and its body atoms.", "As a result, it is not eliminated by the constraint $a\\ref {a:c:overc:elim}$ The set $X = X_0 \\cup \\dots \\cup X_{9k+10}$ is the answer set of $\\Pi ^6$ by the splitting sequence theorem REF .", "Each $X_i, 0 \\le i \\le 9k+10$ matches a distinct portion of $A$ , and $X = A$ , thus $A$ is an answer set of $\\Pi ^6$ .", "Next we show (REF ): Given $\\Pi ^6$ be the encoding of a Petri Net $PN(P,T,E,C,W,R,I,$ $Q,QW,Z)$ with initial marking $M_0$ , and $A$ be an answer set of $\\Pi ^6$ that satisfies (REF ) and (REF ), then we can construct $X=M_0,T_k,\\dots ,M_k,T_k,M_{k+1}$ from $A$ , such that it is an execution sequence of $PN$ .", "We construct the $X$ as follows: $M_i = (M_i(p_0), \\dots , M_i(p_n))$ , where $\\lbrace holds(p_0,m_{M_i(p_0)}(c),c,i), \\dots ,\\\\ holds(p_n,m_{M_i(p_n)}(c),c,i) \\rbrace \\subseteq A$ , for $c \\in C, 0 \\le i \\le k+1$ $T_i = \\lbrace t : fires(t,i) \\in A\\rbrace $ , for $0 \\le i \\le k$ and show that $X$ is indeed an execution sequence of $PN$ .", "We show this by induction over $k$ (i.e.", "given $M_k$ , $T_k$ is a valid firing set and its firing produces marking $M_{k+1}$ ).", "Base case: Let $k=0$ , and $M_0$ is a valid marking in $X$ for $PN$ , show [(1)] $T_0$ is a valid firing set for $M_0$ , and firing $T_0$ in $M_0$ produces marking $M_1$ .", "We show $T_0$ is a valid firing set for $M_0$ .", "Let $\\lbrace fires(t_0,0), \\dots , fires(t_x,0) \\rbrace $ be the set of all $fires(\\dots ,0)$ atoms in $A$ , Then for each $fires(t_i,0) \\in A$ $prenabled(t_i,0) \\in A$ – from rule $a\\ref {a:c:prfires}$ and supported rule proposition Then $enabled(t_i,0) \\in A$ – from rule $a\\ref {a:c:prenabled}$ and supported rule proposition And $notprenabled(t_i,0) \\notin A$ – from rule $a\\ref {a:c:prenabled}$ and supported rule proposition For $enabled(t_i,0) \\in A$ $notenabled(t_i,0) \\notin A$ – from rule $e\\ref {e:c:enabled}$ and supported rule proposition Then either of $body(e\\ref {e:c:ne:ptarc})$ , $body(e\\ref {e:c:ne:iptarc})$ , or $body(e\\ref {e:c:ne:tptarc})$ must not hold in $A$ for $t_i$ – from rules $e\\ref {e:c:ne:ptarc},e\\ref {e:c:ne:iptarc},e\\ref {e:c:ne:tptarc}$ and forced atom proposition Then $q_c \\lnot < {n_i}_c \\equiv q_c \\ge {n_i}_c$ in $e\\ref {e:c:ne:ptarc}$ for all $\\lbrace holds(p,q_c,c,0), \\\\ ptarc(p,t_i,{n_i}_c,c,0)\\rbrace \\subseteq A$ – from $e:\\ref {e:c:ne:ptarc}$ , forced atom proposition, and given facts ($holds(p,q_c,c,0) \\in A, ptarc(p,t_i,{n_i}_c,0) \\in A$ ) And $q_c \\lnot \\ge {n_i}_c \\equiv q_c < {n_i}_c$ in $e\\ref {e:c:ne:iptarc}$ for all $\\lbrace holds(p,q_c,c,0), \\\\ iptarc(p,t_i,{n_i}_c,c,0) \\rbrace \\subseteq A, {n_i}_c=1$ ; $q_c > {n_i}_c \\equiv q_c = 0$ – from $e\\ref {e:c:ne:iptarc}$ , forced atom proposition, given facts ($holds(p,q_c,c,0) \\in A, \\\\ iptarc(p,t_i,1,c,0) \\in A$ ), and $q_c$ is a positive integer And $q_c \\lnot < {n_i}_c \\equiv q_c \\ge {n_i}_c$ in $e\\ref {e:c:ne:tptarc}$ for all $\\lbrace holds(p,q_c,c,0), \\\\ tptarc(p,t_i,{n_i}_c,c,0) \\rbrace \\subseteq A$ – from $e\\ref {e:c:ne:tptarc}$ , forced atom proposition, and given facts Then $\\forall c \\in C, (\\forall p \\in \\bullet t_i, m_{M_0(p)}(c) \\ge m_{W(p,t_i)}(c)) \\wedge (\\forall p \\in I(t_i), \\\\ m_{M_0(p)}(c) = 0) \\wedge (\\forall (p,t_i) \\in Q, m_{M_0(p)}(c) \\ge m_{QW(p,t_i)}(c))$ – from $i\\ref {i:c:init},f\\ref {f:c:ptarc},f\\ref {f:c:iptarc},f\\ref {f:c:tptarc}$ construction, definition REF of preset $\\bullet t_i$ in $PN$ , definition REF of enabled transition in $PN$ , and that the construction of reset arcs by $f\\ref {f:c:rptarc}$ ensures $notenabled(t,0)$ is never true for a reset arc, where $holds(p,q_c,c,0) \\in A$ represents $q_c=m_{M_0(p)}(c)$ , $ptarc(p,t_i,{n_i}_c,0) \\in A$ represents ${n_i}_c=m_{W(p,t_i)}(c)$ , ${n_i}_c = m_{M_0(p)}(c)$ .", "Which is equivalent to $(\\forall p \\in \\bullet t_i, M_0(p) \\ge W(p,t_i)) \\wedge (\\forall p \\in I(t_i), M_0(p) = 0) \\wedge (\\forall (p,t_i) \\in Q, M_0(p) \\ge QW(p,t_i))$ – assuming multiset domain $C$ For $notprenabled(t_i,0) \\notin A$ Either $(\\nexists enabled(tt,0) \\in A : pp < p_i)$ or $(\\forall enabled(tt,0) \\in A : pp \\lnot < p_i)$ where $pp = Z(tt), p_i = Z(t_i)$ – from rule $a\\ref {a:c:prne},f\\ref {f:c:pr}$ and forced atom proposition This matches the definition of an enabled priority transition Then $t_i$ is enabled and can fire in $PN$ , as a result it can belong to $T_0$ – from definition REF of enabled transition And $consumesmore \\notin A$ , since $A$ is an answer set of $\\Pi ^6$ – from rule $a\\ref {a:c:overc:gen}$ and supported rule proposition Then $\\nexists consumesmore(p,0) \\in A$ – from rule $a\\ref {a:c:overc:place}$ and supported rule proposition Then $\\nexists \\lbrace holds(p,q_c,c,0), tot\\_decr(p,q1_c,c,0) \\rbrace \\subseteq A, q1_c>q_c$ in $body(a\\ref {a:c:overc:place})$ – from $a\\ref {a:c:overc:place}$ and forced atom proposition Then $\\nexists c \\in C \\nexists p \\in P, (\\sum _{t_i \\in \\lbrace t_0,\\dots ,t_x\\rbrace , p \\in \\bullet t_i}{m_{W(p,t_i)}(c)}+ \\\\ \\sum _{t_i \\in \\lbrace t_0,\\dots ,t_x\\rbrace , p \\in R(t_i)}{m_{M_0(p)}(c)}) > m_{M_0(p)}(c)$ – from the following $holds(p,q_c,c,0)$ represents $q_c=m_{M_0(p)}(c)$ – from rule $i\\ref {i:c:init}$ encoding, given $tot\\_decr(p,q1_c,c,0) \\in A$ if $\\lbrace del(p,{q1_0}_c,t_0,c,0), \\dots , \\\\ del(p,{q1_x}_c,t_x,c,0) \\rbrace \\subseteq A$ , where $q1_c = {q1_0}_c+\\dots +{q1_x}_c$ – from $r\\ref {r:c:totdecr}$ and forced atom proposition $del(p,{q1_i}_c,t_i,c,0) \\in A$ if $\\lbrace fires(t_i,0), ptarc(p,t_i,{q1_i}_c,c,0) \\rbrace \\subseteq A$ – from $r\\ref {r:c:del}$ and supported rule proposition $del(p,{q1_i}_c,t_i,c,0)$ either represents removal of ${q1_i}_c = m_{W(p,t_i)}(c)$ tokens from $p \\in \\bullet t_i$ ; or it represents removal of ${q1_i}_c = m_{M_0(p)}(c)$ tokens from $p \\in R(t_i)$ – from rules $r\\ref {r:c:del},f\\ref {f:c:ptarc},f\\ref {f:c:rptarc}$ , supported rule proposition, and definition REF of transition execution in $PN$ Then the set of transitions in $T_0$ do not conflict – by the definition REF of conflicting transitions And for each $prenabled(t_j,0) \\in A$ and $fires(t_j,0) \\notin A$ , $could\\_not\\_have(t_j,0) \\in A$ , since $A$ is an answer set of $\\Pi ^6$ - from rule $a\\ref {a:c:prmaxfire:elim}$ and supported rule proposition Then $\\lbrace prenabled(t_j,0), holds(s,qq_c,c,0), ptarc(s,t_j,q_c,c,0), \\\\ tot\\_decr(s,qqq_c,c,0) \\rbrace \\subseteq A$ , such that $q_c > qq_c - qqq_c$ and $fires(t_j,0) \\notin A$ - from rule $a\\ref {a:c:prmaxfire:cnh}$ and supported rule proposition Then for an $s \\in \\bullet t_j \\cup R(t_j)$ , $q_c > m_{M_0(s)}(c) - (\\sum _{t_i \\in T_0, s \\in \\bullet t_i}{m_{W(s,t_i)}(c)} + \\sum _{t_i \\in T_0, s \\in R(t_i)}{m_{M_0(s)}(c))}$ , where $q_c=m_{W(s,t_j)}(c) \\text{~if~} s \\in \\bullet t_j, \\text{~or~} m_{M_0(s)}(c)$ $ \\text{~otherwise}$ – from the following $ptarc(s,t_i,q_c,c,0)$ represents $q_c=m_{W(s,t_i)}(c)$ if $(s,t_i) \\in E^-$ or $q_c=m_{M_0(s)}(c)$ if $s \\in R(t_i)$ – from rule $f\\ref {f:c:ptarc},f\\ref {f:c:rptarc}$ construction $holds(s,qq_c,c,0)$ represents $qq_c=m_{M_0(s)}(c)$ – from $i\\ref {i:c:init}$ construction $tot\\_decr(s,qqq_c,c,0) \\in A$ if $\\lbrace del(s,{qqq_0}_c,t_0,c,0), \\dots , \\\\ del(s,{qqq_x}_c,t_x,c,0) \\rbrace \\subseteq A$ – from rule $r\\ref {r:c:totdecr}$ construction and supported rule proposition $del(s,{qqq_i}_c,t_i,c,0) \\in A$ if $\\lbrace fires(t_i,0), ptarc(s,t_i,{qqq_i}_c,c,0) \\rbrace \\subseteq A$ – from rule $r\\ref {r:c:del}$ and supported rule proposition $del(s,{qqq_i}_c,t_i,c,0)$ either represents ${qqq_i}_c = m_{W(s,t_i)}(c) : t_i \\in T_0, (s,t_i) \\in E^-$ , or ${qqq_i}_c = m_{M_0(t_i)}(c) : t_i \\in T_0, s \\in R(t_i)$ – from rule $f\\ref {f:c:ptarc},f\\ref {f:c:rptarc}$ construction $tot\\_decr(q,qqq_c,c,0)$ represents $\\sum _{t_i \\in T_0, s \\in \\bullet t_i}{m_{W(s,t_i)}(c)} + \\\\ \\sum _{t_i \\in T_0, s \\in R(t_i)}{m_{M_0(s)}(c)}$ – from (C,D,E) above Then firing $T_0 \\cup \\lbrace t_j \\rbrace $ would have required more tokens than are present at its source place $s \\in \\bullet t_j \\cup R(t_j)$ .", "Thus, $T_0$ is a maximal set of transitions that can simultaneously fire.", "And for each reset transition $t_r$ with $prenabled(t_r,0) \\in A$ , $fires(t_r,0) \\in A$ , since $A$ is an answer set of $\\Pi ^6$ - from rule $f\\ref {f:c:pr:rptarc:elim}$ and supported rule proposition Then, the firing set $T_0$ satisfies the reset-transition requirement of definition REF (firing set) Then $\\lbrace t_0, \\dots , t_x\\rbrace = T_0$ – using 1(a),1(b),1(d) above; and using 1(c) it is a maximal firing set We show $M_1$ is produced by firing $T_0$ in $M_0$ .", "Let $holds(p,q_c,c,1) \\in A$ Then $\\lbrace holds(p,q1_c,c,0), tot\\_incr(p,q2_c,c,0), tot\\_decr(p,q3_c,c,0) \\rbrace \\subseteq A : q_c=q1_c+q2_c-q3_c$ – from rule $r\\ref {r:c:nextstate}$ and supported rule proposition Then, $holds(p,q1_c,c,0) \\in A$ represents $q1_c=m_{M_0(p)}(c)$ – given, construction; and $\\lbrace add(p,{q2_0}_c,t_0,c,0), \\dots , $ $add(p,{q2_j}_c,t_j,c,0)\\rbrace \\subseteq A : {q2_0}_c + \\dots + {q2_j}_c = q2_c$ and $\\lbrace del(p,{q3_0}_c,t_0,c,0), \\dots , $ $del(p,{q3_l}_c,t_l,c,0)\\rbrace \\subseteq A : {q3_0}_c + \\dots + {q3_l}_c = q3_c$ – rules $r\\ref {r:c:totincr},r\\ref {r:c:totdecr}$ and supported rule proposition, respectively Then $\\lbrace fires(t_0,0), \\dots , fires(t_j,0) \\rbrace \\subseteq A$ and $\\lbrace fires(t_0,0), \\dots , fires(t_l,0) \\rbrace \\subseteq A$ – rules $r\\ref {r:c:add},r\\ref {r:c:del}$ and supported rule proposition, respectively Then $\\lbrace fires(t_0,0), \\dots , $ $fires(t_j,0) \\rbrace \\cup $ $\\lbrace fires(t_0,0), \\dots , $ $fires(t_l,0) \\rbrace \\subseteq $ $A = \\lbrace fires(t_0,0), \\dots , $ $fires(t_x,0) \\rbrace \\subseteq A$ – set union of subsets Then for each $fires(t_x,0) \\in A$ we have $t_x \\in T_0$ – already shown in item REF above Then $q_c = m_{M_0(p)}(c) + \\sum _{t_x \\in T_0 \\wedge p \\in t_x \\bullet }{m_{W(t_x,p)}(c)} - (\\sum _{t_x \\in T_0 \\wedge p \\in \\bullet t_x}{m_{W(p,t_x)}(c)} + \\sum _{t_x \\in T_0 \\wedge p \\in R(t_x)}{m_{M_0(p)}(c)})$ – from (REF ) above and the following Each $add(p,{q_j}_c,t_j,c,0) \\in A$ represents ${q_j}_c=m_{W(t_j,p)}(c)$ for $p \\in t_j \\bullet $ – rule $r\\ref {r:c:add},f\\ref {f:c:tparc}$ encoding, and definition REF of transition execution in $PN$ Each $del(p,t_y,{q_y}_c,c,0) \\in A$ represents either ${q_y}_c=m_{W(p,t_y)}(c)$ for $p \\in \\bullet t_y$ , or ${q_y}_c=m_{M_0(p)}(c)$ for $p \\in R(t_y)$ – from rule $r\\ref {r:c:del},f\\ref {f:c:ptarc}$ encoding and definition REF of transition execution in $PN$ ; or from rule $r\\ref {r:c:del},f\\ref {f:c:rptarc}$ encoding and definition of reset arc in $PN$ Each $tot\\_incr(p,q2_c,c,0) \\in A$ represents $q2_c=\\sum _{t_x \\in T_0 \\wedge p \\in t_x \\bullet }{m_{W(t_x,p)}(c)}$ – aggregate assignment atom semantics in rule $r\\ref {r:c:totincr}$ Each $tot\\_decr(p,q3_c,c,0) \\in A$ represents $q3_c=\\sum _{t_x \\in T_0 \\wedge p \\in \\bullet t_x}{m_{W(p,t_x)}(c)} + $ $\\sum _{t_x \\in T_0 \\wedge p \\in R(t_x)}{m_{M_0(p)}(c)}$ – aggregate assignment atom semantics in rule $r\\ref {r:c:totdecr}$ Then, $m_{M_1(p)}(c) = q_c$ – since $holds(p,q_c,c,1) \\in A$ encodes $q_c=m_{M_1(p)}(c)$ – from construction Inductive Step: Let $k > 0$ , and $M_k$ is a valid marking in $X$ for $PN$ , show [(1)] $T_k$ is a valid firing set for $M_k$ , and firing $T_k$ in $M_k$ produces marking $M_{k+1}$ .", "We show that $T_k$ is a valid firing set in $M_k$ .", "Let $\\lbrace fires(T_k,k), \\dots , fires(t_x,k) \\rbrace $ be the set of all $fires(\\dots ,k)$ atoms in $A$ , Then for each $fires(t_i,k) \\in A$ $prenabled(t_i,k) \\in A$ – from rule $a\\ref {a:c:prfires}$ and supported rule proposition Then $enabled(t_i,k) \\in A$ – from rule $a\\ref {a:c:prenabled}$ and supported rule proposition And $notprenabled(t_i,k) \\notin A$ – from rule $a\\ref {a:c:prenabled}$ and supported rule proposition For $enabled(t_i,k) \\in A$ $notenabled(t_i,k) \\notin A$ – from rule $e\\ref {e:c:enabled}$ and supported rule proposition Then either of $body(e\\ref {e:c:ne:ptarc})$ , $body(e\\ref {e:c:ne:iptarc})$ , or $body(e\\ref {e:c:ne:tptarc})$ must not hold in $A$ for $t_i$ – from rules $e\\ref {e:c:ne:ptarc},e\\ref {e:c:ne:iptarc},e\\ref {e:c:ne:tptarc}$ and forced atom proposition Then $q_c \\lnot < {n_i}_c \\equiv q_c \\ge {n_i}_c$ in $e\\ref {e:c:ne:ptarc}$ for all $\\lbrace holds(p,q_c,c,k), \\\\ ptarc(p,t_i,{n_i}_c,c,k)\\rbrace \\subseteq A$ – from $e\\ref {e:c:ne:ptarc}$ , forced atom proposition, and given facts ($holds(p,q_c,c,k) \\in A, ptarc(p,t_i,{n_i}_c,k) \\in A$ ) And $q_c \\lnot \\ge {n_i}_c \\equiv q_c < {n_i}_c$ in $e\\ref {e:c:ne:iptarc}$ for all $\\lbrace holds(p,q_c,c,k), \\\\ iptarc(p,t_i,{n_i}_c,c,k) \\rbrace \\subseteq A, {n_i}_c=1$ ; $q_c > {n_i}_c \\equiv q_c = 0$ – from $e\\ref {e:c:ne:iptarc}$ , forced atom proposition, given facts ($holds(p,q_c,c,k) \\in A, \\\\ iptarc(p,t_i,1,c,k) \\in A$ ), and $q_c$ is a positive integer And $q_c \\lnot < {n_i}_c \\equiv q_c \\ge {n_i}_c$ in $e\\ref {e:c:ne:tptarc}$ for all $\\lbrace holds(p,q_c,c,k), \\\\ tptarc(p,t_i,{n_i}_c,c,k) \\rbrace \\subseteq A$ – from $e\\ref {e:c:ne:tptarc}$ , forced atom proposition, and given facts Then $\\forall c \\in C, (\\forall p \\in \\bullet t_i, m_{M_k(p)}(c) \\ge m_{W(p,t_i)}(c)) \\wedge (\\forall p \\in I(t_i), $ $m_{M_k(p)}(c) = k) \\wedge (\\forall (p,t_i) \\in Q, m_{M_k(p)}(c) \\ge m_{QW(p,t_i)}(c))$ – from $f\\ref {f:c:ptarc},f\\ref {f:c:iptarc},f\\ref {f:c:tptarc}$ construction, inductive assumption, definition REF of preset $\\bullet t_i$ in $PN$ , definition REF of enabled transition in $PN$ , and that the construction of reset arcs by $f\\ref {f:c:rptarc}$ ensures $notenabled(t,k)$ is never true for a reset arc, where $holds(p,q_c,c,k) \\in A$ represents $q_c=m_{M_k(p)}(c)$ , $ptarc(p,t_i,{n_i}_c,k) \\in A$ represents ${n_i}_c=m_{W(p,t_i)}(c)$ , ${n_i}_c = m_{M_k(p)}(c)$ .", "Which is equivalent to $(\\forall p \\in \\bullet t_i, M_k(p) \\ge W(p,t_i)) \\wedge (\\forall p \\in I(t_i), M_k(p) = k) \\wedge (\\forall (p,t_i) \\in Q, M_k(p) \\ge QW(p,t_i))$ – assuming multiset domain $C$ For $notprenabled(t_i,k) \\notin A$ Either $(\\nexists enabled(tt,k) \\in A : pp < p_i)$ or $(\\forall enabled(tt,k) \\in A : pp \\lnot < p_i)$ where $pp = Z(tt), p_i = Z(t_i)$ – from rule $a\\ref {a:c:prne}, f\\ref {f:c:pr}$ and forced atom proposition This matches the definition of an enabled priority transition Then $t_i$ is enabled and can fire in $PN$ , as a result it can belong to $T_k$ – from definition REF of enabled transition And $consumesmore \\notin A$ , since $A$ is an answer set of $\\Pi ^6$ – from rule $a\\ref {a:c:overc:elim}$ and supported rule proposition Then $\\nexists consumesmore(p,k) \\in A$ – from rule $a\\ref {a:c:overc:elim}$ and supported rule proposition Then $\\nexists \\lbrace holds(p,q_c,c,k), tot\\_decr(p,q1_c,c,k) \\rbrace \\subseteq A, q1_c>q_c$ in $body(a\\ref {a:c:overc:place})$ – from $a\\ref {a:c:overc:place}$ and forced atom proposition Then $\\nexists c \\in C \\nexists p \\in P, (\\sum _{t_i \\in \\lbrace T_k,\\dots ,t_x\\rbrace , p \\in \\bullet t_i}{m_{W(p,t_i)}(c)}+ \\\\ \\sum _{t_i \\in \\lbrace T_k,\\dots ,t_x\\rbrace , p \\in R(t_i)}{m_{M_k(p)}(c)}) > m_{M_k(p)}(c)$ – from the following $holds(p,q_c,c,k)$ represents $q_c=m_{M_k(p)}(c)$ – from inductive assumption and construction, given $tot\\_decr(p,q1_c,c,k) \\in A$ if $\\lbrace del(p,{q1_0}_c,T_k,c,k), \\dots , \\\\ del(p,{q1_x}_c,t_x,c,k) \\rbrace \\subseteq A$ , where $q1_c = {q1_0}_c+\\dots +{q1_x}_c$ – from $r\\ref {r:c:totdecr}$ and forced atom proposition $del(p,{q1_i}_c,t_i,c,k) \\in A$ if $\\lbrace fires(t_i,k), ptarc(p,t_i,{q1_i}_c,c,k) \\rbrace \\subseteq A$ – from $r\\ref {r:c:del}$ and supported rule proposition $del(p,{q1_i}_c,t_i,c,k)$ either represents removal of ${q1_i}_c = m_{W(p,t_i)}(c)$ tokens from $p \\in \\bullet t_i$ ; or it represents removal of ${q1_i}_c = m_{M_k(p)}(c)$ tokens from $p \\in R(t_i)$ – from rules $r\\ref {r:c:del},f\\ref {f:c:ptarc},f\\ref {f:c:rptarc}$ , supported rule proposition, and definition REF of transition execution in $PN$ Then the set of transitions in $T_k$ do not conflict – by the definition REF of conflicting transitions And for each $prenabled(t_j,k) \\in A$ and $fires(t_j,k) \\notin A$ , $could\\_not\\_have(t_j,k) \\in A$ , since $A$ is an answer set of $\\Pi ^6$ - from rule $a\\ref {a:c:prmaxfire:elim}$ and supported rule proposition Then $\\lbrace prenabled(t_j,k), holds(s,qq_c,c,k), ptarc(s,t_j,q_c,c,k), \\\\ tot\\_decr(s,qqq_c,c,k) \\rbrace \\subseteq A$ , such that $q_c > qq_c - qqq_c$ and $fires(t_j,k) \\notin A$ - from rule $a\\ref {a:c:prmaxfire:cnh}$ and supported rule proposition Then for an $s \\in \\bullet t_j \\cup R(t_j)$ , $q_c > m_{M_k(s)}(c) - (\\sum _{t_i \\in T_k, s \\in \\bullet t_i}{m_{W(s,t_i)}(c)} + \\sum _{t_i \\in T_k, s \\in R(t_i)}{m_{M_k(s)}(c))}$ , where $q_c=m_{W(s,t_j)}(c) \\text{~if~} s \\in \\bullet t_j, \\text{~or~} m_{M_k(s)}(c) $ $\\text{~otherwise}$ .", "$ptarc(s,t_i,q_c,c,k)$ represents $q_c=m_{W(s,t_i)}(c)$ if $(s,t_i) \\in E^-$ or $q_c=m_{M_k(s)}(c)$ if $s \\in R(t_i)$ – from rule $f\\ref {f:c:ptarc},f\\ref {f:c:rptarc}$ construction $holds(s,qq_c,c,k)$ represents $qq_c=m_{M_k(s)}(c)$ – from inductive assumption and construction $tot\\_decr(s,qqq_c,c,k) \\in A$ if $\\lbrace del(s,{qqq_0}_c,T_k,c,k), \\dots , \\\\ del(s,{qqq_x}_c,t_x,c,k) \\rbrace \\subseteq A$ – from rule $r\\ref {r:c:totdecr}$ construction and supported rule proposition $del(s,{qqq_i}_c,t_i,c,k) \\in A$ if $\\lbrace fires(t_i,k), ptarc(s,t_i,{qqq_i}_c,c,k) \\rbrace \\subseteq A$ – from rule $r\\ref {r:c:del}$ and supported rule proposition $del(s,{qqq_i}_c,t_i,c,k)$ either represents ${qqq_i}_c = m_{W(s,t_i)}(c) : t_i \\in T_k, (s,t_i) \\in E^-$ , or ${qqq_i}_c = m_{M_k(t_i)}(c) : t_i \\in T_k, s \\in R(t_i)$ – from rule $f\\ref {f:c:ptarc},f\\ref {f:c:rptarc}$ construction $tot\\_decr(q,qqq_c,c,k)$ represents $\\sum _{t_i \\in T_k, s \\in \\bullet t_i}{m_{W(s,t_i)}(c)} + \\\\ \\sum _{t_i \\in T_k, s \\in R(t_i)}{m_{M_k(s)}(c)}$ – from (C,D,E) above Then firing $T_k \\cup \\lbrace t_j \\rbrace $ would have required more tokens than are present at its source place $s \\in \\bullet t_j \\cup R(t_j)$ .", "Thus, $T_k$ is a maximal set of transitions that can simultaneously fire.", "And for each reset transition $t_r$ with $prenabled(t_r,k) \\in A$ , $fires(t_r,k) \\in A$ , since $A$ is an answer set of $\\Pi ^6$ - from rule $f\\ref {f:c:pr:rptarc:elim}$ and supported rule proposition Then the firing set $T_k$ satisfies the reset transition requirement of definition REF (firing set) Then $\\lbrace t_0, \\dots , t_x\\rbrace = T_k$ – using 1(a),1(b), 1(d) above; and using 1(c) it is a maximal firing set We show that $M_{k+1}$ is produced by firing $T_k$ in $M_k$ .", "Let $holds(p,q_c,c,k+1) \\in A$ Then $\\lbrace holds(p,q1_c,c,k), tot\\_incr(p,q2_c,c,k), tot\\_decr(p,q3_c,c,k) \\rbrace \\subseteq A : q_c=q1_c+q2_c-q3_c$ – from rule $r\\ref {r:c:nextstate}$ and supported rule proposition Then, $holds(p,q1_c,c,k) \\in A$ represents $q1_c=m_{M_k(p)}(c)$ – inductive assumption; and $\\lbrace add(p,{q2_0}_c,T_k,c,k), \\dots , $ $add(p,{q2_j}_c,t_j,c,k)\\rbrace \\subseteq A : {q2_0}_c + \\dots + {q2_j}_c = q2_c$ and $\\lbrace del(p,{q3_0}_c,T_k,c,k), \\dots , $ $del(p,{q3_l}_c,t_l,c,k)\\rbrace \\subseteq A : {q3_0}_c + \\dots + {q3_l}_c = q3_c$ – rules $r\\ref {r:c:totincr},r\\ref {r:c:totdecr}$ and supported rule proposition, respectively Then $\\lbrace fires(T_k,k), \\dots , fires(t_j,k) \\rbrace \\subseteq A$ and $\\lbrace fires(T_k,k), \\dots , \\\\ fires(t_l,k) \\rbrace \\subseteq A$ – rules $r\\ref {r:c:add},r\\ref {r:c:del}$ and supported rule proposition, respectively Then $\\lbrace fires(T_k,k), \\dots , $ $fires(t_j,k) \\rbrace \\cup $ $\\lbrace fires(T_k,k), \\dots , $ $fires(t_l,k) \\rbrace \\subseteq $ $A = \\lbrace fires(T_k,k), \\dots , fires(t_x,k) \\rbrace \\subseteq A$ – set union of subsets Then for each $fires(t_x,k) \\in A$ we have $t_x \\in T_k$ – already shown in item REF above Then $q_c = m_{M_k(p)}(c) + \\sum _{t_x \\in T_k \\wedge p \\in t_x \\bullet }{m_{W(t_x,p)}(c)} - (\\sum _{t_x \\in T_k \\wedge p \\in \\bullet t_x}{m_{W(p,t_x)}(c)} + \\sum _{t_x \\in T_k \\wedge p \\in R(t_x)}{m_{M_k(p)}(c)})$ – from (REF ) above and the following Each $add(p,{q_j}_c,t_j,c,k) \\in A$ represents ${q_j}_c=m_{W(t_j,p)}(c)$ for $p \\in t_j \\bullet $ – rule $r\\ref {r:c:add},f\\ref {f:c:tparc}$ encoding, and definition REF of transition execution in $PN$ Each $del(p,t_y,{q_y}_c,c,k) \\in A$ represents either ${q_y}_c=m_{W(p,t_y)}(c)$ for $p \\in \\bullet t_y$ , or ${q_y}_c=m_{M_k(p)}(c)$ for $p \\in R(t_y)$ – from rule $r\\ref {r:c:del},f\\ref {f:c:ptarc}$ encoding and definition REF of transition execution in $PN$ ; or from rule $r\\ref {r:c:del},f\\ref {f:c:rptarc}$ encoding and definition of reset arc in $PN$ Each $tot\\_incr(p,q2_c,c,k) \\in A$ represents $q2_c=\\sum _{t_x \\in T_k \\wedge p \\in t_x \\bullet }{m_{W(t_x,p)}(c)}$ – aggregate assignment atom semantics in rule $r\\ref {r:c:totincr}$ Each $tot\\_decr(p,q3_c,c,k) \\in A$ represents $q3_c=\\sum _{t_x \\in T_k \\wedge p \\in \\bullet t_x}{m_{W(p,t_x)}(c)} + \\sum _{t_x \\in T_k \\wedge p \\in R(t_x)}{m_{M_k(p)}(c)}$ – aggregate assignment atom semantics in rule $r\\ref {r:c:totdecr}$ Then, $m_{M_{k+1}(p)}(c) = q_c$ – since $holds(p,q_c,c,k+1) \\in A$ encodes $q_c=m_{M_{k+1}(p)}(c)$ – from construction As a result, for any $n > k$ , $T_n$ will be a valid firing set for $M_n$ and $M_{n+1}$ will be its target marking.", "Conclusion: Since both (REF ) and (REF ) hold, $X=M_0,T_k,M_1,\\dots ,M_k,T_{k+1}$ is an execution sequence of $PN(P,T,E,C,W,R,I,Q,QW,Z)$ (w.r.t $M_0$ ) iff there is an answer set $A$ of $\\Pi ^6(PN,M_0,k,ntok)$ such that (REF ) and (REF ) hold.", "Proof of Proposition REF Let $PN=(P,T,E,C,W,R,I,Q,QW,Z,D)$ be a Petri Net, $M_0$ be its initial marking and let $\\Pi ^7(PN,M_0,k,ntok)$ be the ASP encoding of $PN$ and $M_0$ over a simulation length $k$ , with maximum $ntok$ tokens on any place node, as defined in section REF .", "Then $X=M_0,T_k,M_1,\\dots ,M_k,T_k,M_{k+1}$ is an execution sequence of $PN$ (w.r.t.", "$M_0$ ) iff there is an answer set $A$ of $\\Pi ^7(PN,M_0,k,ntok)$ such that: $\\lbrace fires(t,ts) : t \\in T_{ts}, 0 \\le ts \\le k\\rbrace = \\lbrace fires(t,ts) : fires(t,ts) \\in A \\rbrace $ $\\begin{split}\\lbrace holds(p,q,c,ts) &: p \\in P, c/q = M_{ts}(p), 0 \\le ts \\le k+1 \\rbrace \\\\&= \\lbrace holds(p,q,c,ts) : holds(p,q,c,ts) \\in A \\rbrace \\end{split}$ We prove this by showing that: Given an execution sequence $X$ , we create a set $A$ such that it satisfies (REF ) and (REF ) and show that $A$ is an answer set of $\\Pi ^7$ Given an answer set $A$ of $\\Pi ^7$ , we create an execution sequence $X$ such that (REF ) and (REF ) are satisfied.", "First we show (REF ): We create a set $A$ as a union of the following sets: $A_1=\\lbrace num(n) : 0 \\le n \\le ntok \\rbrace $ $A_2=\\lbrace time(ts) : 0 \\le ts \\le k\\rbrace $ $A_3=\\lbrace place(p) : p \\in P \\rbrace $ $A_4=\\lbrace trans(t) : t \\in T \\rbrace $ $A_5=\\lbrace color(c) : c \\in C \\rbrace $ $A_6=\\lbrace ptarc(p,t,n_c,c,ts) : (p,t) \\in E^-, c \\in C, n_c=m_{W(p,t)}(c), n_c > 0, 0 \\le ts \\le k \\rbrace $ , where $E^- \\subseteq E$ $A_7=\\lbrace tparc(t,p,n_c,c,ts,d) : (t,p) \\in E^+, c \\in C, n_c=m_{W(t,p)}(c), n_c > 0, d=D(t), 0 \\le ts \\le k \\rbrace $ , where $E^+ \\subseteq E$ $A_8=\\lbrace holds(p,q_c,c,0) : p \\in P, c \\in C, q_c=m_{M_{0}(p)}(c) \\rbrace $ $A_9=\\lbrace ptarc(p,t,n_c,c,ts) : p \\in R(t), c \\in C, n_c = m_{M_{ts}(p)}, n_c > 0, 0 \\le ts \\le k \\rbrace $ $A_{10}=\\lbrace iptarc(p,t,1,c,ts) : p \\in I(t), c \\in C, 0 \\le ts < k \\rbrace $ $A_{11}=\\lbrace tptarc(p,t,n_c,c,ts) : (p,t) \\in Q, c \\in C, n_c=m_{QW(p,t)}(c), n_c > 0, 0 \\le ts \\le k \\rbrace $ $A_{12}=\\lbrace notenabled(t,ts) : t \\in T, 0 \\le ts \\le k, \\exists c \\in C, (\\exists p \\in \\bullet t, m_{M_{ts}(p)}(c) < m_{W(p,t)}(c)) \\vee (\\exists p \\in I(t), m_{M_{ts}(p)}(c) > 0) \\vee (\\exists (p,t) \\in Q, m_{M_{ts}(p)}(c) < m_{QW(p,t)}(c)) \\rbrace $ per definition REF (enabled transition) $A_{13}=\\lbrace enabled(t,ts) : t \\in T, 0 \\le ts \\le k, \\forall c \\in C, (\\forall p \\in \\bullet t, m_{W(p,t)}(c) \\le m_{M_{ts}(p)}(c)) \\wedge (\\forall p \\in I(t), m_{M_{ts}(p)}(c) = 0) \\wedge (\\forall (p,t) \\in Q, m_{M_{ts}(p)}(c) \\ge m_{QW(p,t)}(c)) \\rbrace $ per definition REF (enabled transition) $A_{14}=\\lbrace fires(t,ts) : t \\in T_{ts}, 0 \\le ts \\le k \\rbrace $ per definition REF (enabled transitions), only an enabled transition may fire $A_{15}=\\lbrace add(p,q_c,t,c,ts+d-1) : t \\in T_{ts}, p \\in t \\bullet , c \\in C, q_c=m_{W(t,p)}(c), \\\\d = D(t), 0 \\le ts \\le k \\rbrace $ per definition REF (transition execution) $A_{16}=\\lbrace del(p,q_c,t,c,ts) : t \\in T_{ts}, p \\in \\bullet t, c \\in C, q_c=m_{W(p,t)}(c), 0 \\le ts \\le k \\rbrace \\cup \\lbrace del(p,q_c,t,c,ts) : t \\in T_{ts}, p \\in R(t), c \\in C, q_c=m_{M_{ts}(p)}(c), 0 \\le ts \\le k \\rbrace $ per definition REF (transition execution) $A_{17}=\\lbrace tot\\_incr(p,q_c,c,ts) : p \\in P, c \\in C, \\\\ q_c=\\sum _{t \\in T_{l}, p \\in t \\bullet , l \\le ts, l+D(t)=ts+1}{m_{W(t,p)}(c)}, 0 \\le ts \\le k \\rbrace $ per definition REF (firing set execution) $A_{18}=\\lbrace tot\\_decr(p,q_c,c,ts): p \\in P, c \\in C, q_c=\\sum _{t \\in T_{ts}, p \\in \\bullet t}{m_{W(p,t)}(c)}+ \\\\ \\sum _{t \\in T_{ts}, p \\in R(t) }{m_{M_{ts}(p)}(c)}, 0 \\le ts \\le k \\rbrace $ per definition REF (firing set execution) $A_{19}=\\lbrace consumesmore(p,ts) : p \\in P, c \\in C, q_c=m_{M_{ts}(p)}(c), \\\\ q1_c=\\sum _{t \\in T_{ts}, p \\in \\bullet t}{m_{W(p,t)}(c)} + \\sum _{t \\in T_{ts}, p \\in R(t)}{m_{M_{ts}(p)}(c)}, q1_c > q_c, 0 \\le ts \\le k \\rbrace $ per definition REF (conflicting transitions) $A_{20}=\\lbrace consumesmore : \\exists p \\in P, c \\in C, q_c=m_{M_{ts}(p)}(c), \\\\ q1_c=\\sum _{t \\in T_{ts}, p \\in \\bullet t}{m_{W(p,t)}(c)} + \\sum _{t \\in T_{ts}, p \\in R(t)}(m_{M_{ts}(p)}(c)), q1_c > q_c, 0 \\le ts \\le k \\rbrace $ per definition REF (conflicting transitions) $A_{21}=\\lbrace holds(p,q_c,c,ts+1) : p \\in P, c \\in C, q_c=m_{M_{ts+1}(p)}(c), 0 \\le ts < k\\rbrace $ , where $M_{ts+1}(p) = M_{ts}(p) - (\\sum _{\\begin{array}{c}t \\in T_{ts}, p \\in \\bullet t\\end{array}}{W(p,t)} + \\sum _{t \\in T_{ts}, p \\in R(t)}M_{ts}(p)) + \\\\ \\sum _{\\begin{array}{c}t \\in T_l, p \\in t \\bullet , l \\le ts, l+D(t)-1=ts\\end{array}}{W(t,p)}$ according to definition REF (firing set execution) $A_{22}=\\lbrace transpr(t,pr) : t \\in T, pr=Z(t) \\rbrace $ $A_{23}=\\lbrace notprenabled(t,ts) : t \\in T, enabled(t,ts) \\in A_{13}, (\\exists tt \\in T, enabled(tt,ts) \\in A_{13}, Z(tt) < Z(t)), 0 \\le ts \\le k \\rbrace \\\\ = \\lbrace notprenabled(t,ts) : t \\in T, (\\forall p \\in \\bullet t, W(p,t) \\le M_{ts}(p)), (\\forall p \\in I(t), M_{ts}(p) = 0), (\\forall (p,t) \\in Q, M_0(p) \\ge WQ(p,t)), \\exists tt \\in T, $ $(\\forall pp \\in \\bullet tt, W(pp,tt) \\le M_{ts}(pp)), $ $(\\forall pp \\in I(tt), M_{ts}(pp) = 0), (\\forall (pp,tt) \\in Q, M_{ts}(pp) \\ge QW(pp,tt)), Z(tt) < Z(t), 0 \\le ts \\le k \\rbrace $ $A_{24}=\\lbrace prenabled(t,ts) : t \\in T, enabled(t,ts) \\in A_{13}, (\\nexists tt \\in T: enabled(tt,ts) \\in A_{13}, Z(tt) < Z(t)), 0 \\le ts \\le k \\rbrace \\\\ = \\lbrace prenabled(t,ts) : t \\in T, (\\forall p \\in \\bullet t, W(p,t) \\le M_{ts}(p)), (\\forall p \\in I(t), M_{ts}(p) = 0), (\\forall (p,t) \\in Q, M_0(p) \\ge WQ(p,t)), \\nexists tt \\in T, $ $(\\forall pp \\in \\bullet tt, W(pp,tt) \\le M_{ts}(pp)), $ $(\\forall pp \\in I(tt), M_{ts}(pp) = 0), (\\forall (pp,tt) \\in Q, M_{ts}(pp) \\ge QW(pp,tt)), Z(tt) < Z(t), 0 \\le ts \\le k \\rbrace $ $A_{25}=\\lbrace could\\_not\\_have(t,ts): t \\in T, prenabled(t,ts) \\in A_{24}, fires(t,ts) \\notin A_{14}, (\\exists p \\in \\bullet t \\cup R(t): q > M_{ts}(p) - (\\sum _{t^{\\prime } \\in T_{ts}, p \\in \\bullet t^{\\prime }}{W(p,t^{\\prime })} + \\sum _{t^{\\prime } \\in T_{ts}, p \\in R(t^{\\prime })}{M_{ts}(p)}), q=W(p,t) \\text{~if~} (p,t) \\in E^- \\text{~or~} R(t) \\text{~otherwise}), 0 \\le ts \\le k\\rbrace \\\\ =\\lbrace could\\_not\\_have(t,ts) : t \\in T, (\\forall p \\in \\bullet t, W(p,t) \\le M_{ts}(p)), (\\forall p \\in I(t), M_{ts}(p) = 0), (\\forall (p,t) \\in Q, M_0(p) \\ge QW(p,t)), (\\nexists tt \\in T, $ $(\\forall pp \\in \\bullet tt, W(pp,tt) \\le M_{ts}(pp)), $ $(\\forall pp \\in I(tt), M_{ts}(pp) = 0), (\\forall (pp,tt) \\in Q, M_{ts}(pp) \\ge QW(pp,tt)), Z(tt) < Z(t)), t \\notin T_{ts}, (\\exists p \\in \\bullet t \\cup R(t): q > M_{ts}(p) - (\\sum _{t^{\\prime } \\in T_{ts}, p \\in \\bullet t^{\\prime }}{W(p,t^{\\prime })} + \\\\ \\sum _{t^{\\prime } \\in T_{ts}, p \\in R(t^{\\prime })}{M_{ts}(p)}), q=W(p,t) \\text{~if~} (p,t) \\in E^- \\text{~or~} R(t) \\text{~otherwise}), 0 \\le ts \\le k \\rbrace $ per the maximal firing set semantics We show that $A$ satisfies (REF ) and (REF ), and $A$ is an answer set of $\\Pi ^7$ .", "$A$ satisfies (REF ) and (REF ) by its construction above.", "We show $A$ is an answer set of $\\Pi ^7$ by splitting.", "We split $lit(\\Pi ^7)$ into a sequence of $9k+11$ sets: $U_0= head(f\\ref {f:c:place}) \\cup head(f\\ref {f:c:trans}) \\cup head(f\\ref {f:c:col}) \\cup head(f\\ref {f:c:time}) \\cup head(f\\ref {f:c:num}) \\cup head(i\\ref {i:c:init}) \\cup head(f\\ref {f:c:pr}) = \\lbrace place(p) : p \\in P\\rbrace \\cup \\lbrace trans(t) : t \\in T\\rbrace \\cup \\lbrace col(c) : c \\in C \\rbrace \\cup \\lbrace time(0), \\dots , time(k)\\rbrace \\cup \\lbrace num(0), \\dots , num(ntok)\\rbrace \\cup \\lbrace holds(p,q_c,c,0) : p \\in P, c \\in C, q_c=m_{M_0(p)}(c) \\rbrace \\cup \\lbrace transpr(t,pr) : t \\in T, pr=Z(t) \\rbrace $ $U_{9k+1}=U_{9k+0} \\cup head(f\\ref {f:c:ptarc})^{ts=k} \\cup head(f\\ref {f:c:tparc})^{ts=k} \\cup head(f\\ref {f:c:rptarc})^{ts=k} \\cup head(f\\ref {f:c:iptarc})^{ts=k} \\cup head(f\\ref {f:c:tptarc})^{ts=k} = U_{9k+0} \\cup $ $\\lbrace ptarc(p,t,n_c,c,k) : (p,t) \\in E^-, c \\in C, n_c=m_{W(p,t)}(c) \\rbrace \\\\ \\cup \\lbrace tparc(t,p,n_c,c,k,d) : (t,p) \\in E^+, c \\in C, n_c=m_{W(t,p)}(c), d=D(t) \\rbrace \\cup $ $\\lbrace ptarc(p,t,n_c,c,k) : p \\in R(t), c \\in C, n_c=m_{M_{k}(p)}(c), n > 0 \\rbrace \\cup \\lbrace iptarc(p,t,1,c,k) : \\\\ p \\in I(t), c \\in C \\rbrace \\cup $ $\\lbrace tptarc(p,t,n_c,c,k) : (p,t) \\in Q, c \\in C, n_c=m_{QW(p,t)}(c) \\rbrace $ $U_{9k+2}=U_{9k+1} \\cup head(e\\ref {e:c:ne:ptarc})^{ts=k} \\cup head(e\\ref {e:c:ne:iptarc})^{ts=k} \\cup head(e\\ref {e:c:ne:tptarc})^{ts=k} = U_{10k+1} \\cup \\\\ \\lbrace notenabled(t,k) : t \\in T \\rbrace $ $U_{9k+3}=U_{9k+2} \\cup head(e\\ref {e:c:enabled})^{ts=k} = U_{9k+2} \\cup \\lbrace enabled(t,k) : t \\in T \\rbrace $ $U_{9k+4}=U_{9k+3} \\cup head(a\\ref {a:c:prne})^{ts=k} = U_{9k+3} \\cup \\lbrace notprenabled(t,k) : t \\in T \\rbrace $ $U_{9k+5}=U_{9k+4} \\cup head(a\\ref {a:c:prenabled})^{ts=k} = U_{9k+4} \\cup \\lbrace prenabled(t,k) : t \\in T \\rbrace $ $U_{9k+6}=U_{9k+5} \\cup head(a\\ref {a:c:prfires})^{ts=k} = U_{9k+5} \\cup \\lbrace fires(t,k) : t \\in T \\rbrace $ $U_{9k+7}=U_{9k+6} \\cup head(r\\ref {r:c:dur:add})^{ts=k} \\cup head(r\\ref {r:c:del})^{ts=k} = U_{9k+6} \\cup \\lbrace add(p,q_c,t,c,k) : p \\in P, t \\in T, c \\in C, q_c=m_{W(t,p)}(c) \\rbrace \\cup \\lbrace del(p,q_c,t,c,k) : p \\in P, t \\in T, c \\in C, q_c=m_{W(p,t)}(c) \\rbrace \\cup \\lbrace del(p,q_c,t,c,k) : p \\in P, t \\in T, c \\in C, q_c=m_{M_{k}(p)}(c) \\rbrace $ $U_{9k+8}=U_{9k+7} \\cup head(r\\ref {r:c:totincr})^{ts=k} \\cup head(r\\ref {r:c:totdecr})^{ts=k} = U_{9k+7} \\cup \\lbrace tot\\_incr(p,q_c,c,k) : p \\in P, c \\in C, 0 \\le q_c \\le ntok \\rbrace \\cup \\lbrace tot\\_decr(p,q_c,c,k) : p \\in P, c \\in C, 0 \\le q_c \\le ntok \\rbrace $ $U_{9k+9}=U_{9k+8} \\cup head(a\\ref {a:c:overc:place})^{ts=k} \\cup head(r\\ref {r:c:nextstate})^{ts=k} \\cup head(a\\ref {a:c:prmaxfire:cnh})^{ts=k} = U_{9k+8} \\cup \\\\ \\lbrace consumesmore(p,k) : p \\in P\\rbrace \\cup \\lbrace holds(p,q,k+1) : p \\in P, 0 \\le q \\le ntok \\rbrace \\cup \\lbrace could\\_not\\_have(t,k) : t \\in T \\rbrace $ $U_{9k+10}=U_{9k+9} \\cup head(a\\ref {a:c:overc:gen}) = U_{9k+9} \\cup \\lbrace consumesmore \\rbrace $ where $head(r_i)^{ts=k}$ are head atoms of ground rule $r_i$ in which $ts=k$ .", "We write $A_i^{ts=k} = \\lbrace a(\\dots ,ts) : a(\\dots ,ts) \\in A_i, ts=k \\rbrace $ as short hand for all atoms in $A_i$ with $ts=k$ .", "$U_{\\alpha }, 0 \\le \\alpha \\le 9k+10$ form a splitting sequence, since each $U_i$ is a splitting set of $\\Pi ^7$ , and $\\langle U_{\\alpha }\\rangle _{\\alpha < \\mu }$ is a monotone continuous sequence, where $U_0 \\subseteq U_1 \\dots \\subseteq U_{9k+10}$ and $\\bigcup _{\\alpha < \\mu }{U_{\\alpha }} = lit(\\Pi ^7)$ .", "We compute the answer set of $\\Pi ^7$ using the splitting sets as follows: $bot_{U_0}(\\Pi ^7) = f\\ref {f:c:place} \\cup f\\ref {f:c:trans} \\cup f\\ref {f:c:col} \\cup f\\ref {f:c:time} \\cup f\\ref {f:c:num} \\cup i\\ref {i:c:init} \\cup f\\ref {f:c:pr}$ and $X_0 = A_1 \\cup \\dots \\cup A_5 \\cup A_8$ ($= U_0$ ) is its answer set – using forced atom proposition $eval_{U_0}(bot_{U_1}(\\Pi ^7) \\setminus bot_{U_0}(\\Pi ^7), X_0) = \\lbrace ptarc(p,t,q_c,c,0) \\text{:-}.", "\| c \\in C, q_c=m_{W(p,t)}(c), q_c > 0 \\rbrace \\cup $ $\\lbrace tparc(t,p,q_c,c,0,d) \\text{:-}.", "\| c \\in C, q_c=m_{W(t,p)}(c), q_c > 0, d=D(t) \\rbrace \\cup \\\\ \\lbrace ptarc(p,t,q_c,c,0) \\text{:-}.", "\| c \\in C, q_c=m_{M_0(p)}(c), q_c > 0 \\rbrace \\cup $ $\\lbrace iptarc(p,t,1,c,0) \\text{:-}.", "\| c \\in C \\rbrace \\cup $ $\\lbrace tptarc(p,t,q_c,c,0) \\text{:-}.", "\| c \\in C, q_c = m_{QW(p,t)}(c), q_c > 0 \\rbrace $ .", "Its answer set $X_1=A_6^{ts=0} \\cup A_7^{ts=0} \\cup A_9^{ts=0} \\cup A_{10}^{ts=0} \\cup A_{11}^{ts=0}$ – using forced atom proposition and construction of $A_6, A_7, A_9, A_{10}, A_{11}$ .", "$eval_{U_1}(bot_{U_2}(\\Pi ^7) \\setminus bot_{U_1}(\\Pi ^7), X_0 \\cup X_1) = \\lbrace notenabled(t,0) \\text{:-} .", "\| (\\lbrace trans(t), \\\\ ptarc(p,t,n_c,c,0), $ $holds(p,q_c,c,0) \\rbrace \\subseteq X_0 \\cup X_1, \\text{~where~} q_c < n_c) \\text{~or~} \\\\ (\\lbrace notenabled(t,0) \\text{:-} .", "\| $ $(\\lbrace trans(t), $ $iptarc(p,t,n2_c,c,0), $ $ holds(p,q_c,c,0) \\rbrace \\subseteq X_0 \\cup X_1, \\text{~where~} q_c \\ge n2_c \\rbrace ) \\text{~or~} $ $(\\lbrace trans(t), $ $tptarc(p,t,n3_c,c,0), $ $holds(p,q_c,c,0) \\rbrace \\subseteq X_0 \\cup X_1, \\text{~where~} q_c < n3_c) \\rbrace $ .", "Its answer set $X_2=A_{12}^{ts=0}$ – using forced atom proposition and construction of $A_{12}$ .", "where, $q_c=m_{M_0(p)}(c)$ , and $n_c=m_{W(p,t)}(c)$ for an arc $(p,t) \\in E^-$ – by construction of $i\\ref {i:c:init}$ and $f\\ref {f:c:ptarc}$ in $\\Pi ^7$ , and in an arc $(p,t) \\in E^-$ , $p \\in \\bullet t$ (by definition REF of preset) $n2_c=1$ – by construction of $iptarc$ predicates in $\\Pi ^7$ , meaning $q_c \\ge n2_c \\equiv q_c \\ge 1 \\equiv q_c > 0$ , $tptarc(p,t,n3_c,c,0)$ represents $n3_c=m_{QW(p,t)}(c)$ , where $(p,t) \\in Q$ thus, $notenabled(t,0) \\in X_1$ means $\\exists c \\in C, (\\exists p \\in \\bullet t : m_{M_0(p)}(c) < m_{W(p,t)}(c)) \\\\ \\vee (\\exists p \\in I(t) : m_{M_0(p)}(c) > 0) \\vee (\\exists (p,t) \\in Q : m_{M_0(p)}(c) < m_{QW(p,t)}(c))$ .", "$eval_{U_2}(bot_{U_3}(\\Pi ^7) \\setminus bot_{U_2}(\\Pi ^7), X_0 \\cup \\dots \\cup X_2) = \\lbrace enabled(t,0) \\text{:-}.", "\| trans(t) \\in X_0 \\cup \\dots \\cup X_2, notenabled(t,0) \\notin X_0 \\cup \\dots \\cup X_2 \\rbrace $ .", "Its answer set is $X_3 = A_{13}^{ts=0}$ – using forced atom proposition and construction of $A_{13}$ .", "since an $enabled(t,0) \\in X_3$ if $\\nexists ~notenabled(t,0) \\in X_0 \\cup \\dots \\cup X_2$ ; which is equivalent to $\\nexists t, \\forall c \\in C, (\\nexists p \\in \\bullet t, m_{M_0(p)}(c) < m_{W(p,t)}(c)), (\\nexists p \\in I(t), \\\\ m_{M_0(p)}(c) > 0), (\\nexists (p,t) \\in Q : m_{M_0(p)}(c) < m_{QW(p,t)}(c) ), \\forall c \\in C, (\\forall p \\in \\bullet t: m_{M_0(p)}(c) \\ge m_{W(p,t)}(c)), (\\forall p \\in I(t) : m_{M_0(p)}(c) = 0)$ .", "$eval_{U_3}(bot_{U_4}(\\Pi ^7) \\setminus bot_{U_3}(\\Pi ^7), X_0 \\cup \\dots \\cup X_3) = \\lbrace notprenabled(t,0) \\text{:-}.", "\| \\\\ \\lbrace enabled(t,0), transpr(t,p), enabled(tt,0), transpr(tt,pp) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_3, pp < p \\rbrace $ .", "Its answer set is $X_4 = A_{23}^{ts=k}$ – using forced atom proposition and construction of $A_{23}$ .", "$enabled(t,0)$ represents $\\exists t \\in T, \\forall c \\in C, (\\forall p \\in \\bullet t, m_{M_0(p)}(c) \\ge m_{W(p,t)}(c)), $ $(\\forall p \\in I(t), m_{M_0(p)}(c) = 0), (\\forall (p,t) \\in Q, m_{M_0(p)}(c) \\ge m_{QW(p,t)}(c))$ $enabled(tt,0)$ represents $\\exists tt \\in T, \\forall c \\in C, $ $(\\forall pp \\in \\bullet tt, m_{M_0(pp)}(c) \\ge \\\\ m_{W(pp,tt)}(c)) \\wedge $ $(\\forall pp \\in I(tt), m_{M_0(pp)}(c) = 0), (\\forall (pp,tt) \\in Q, m_{M_0(pp)}(c) \\ge m_{QW(pp,tt)}(c))$ $transpr(t,p)$ represents $p=Z(t)$ – by construction $transpr(tt,pp)$ represents $pp=Z(tt)$ – by construction thus, $notprenabled(t,0)$ represents $\\forall c \\in C, $ $(\\forall p \\in \\bullet t, m_{M_0(p)}(c) \\ge m_{W(p,t)}(c)), $ $(\\forall p \\in I(t), m_{M_0(p)}(c) = 0), (\\forall (p,t) \\in Q, m_{M_0(p)}(c) \\ge m_{QW(p,t)}(c)), \\exists tt \\in T, (\\forall pp \\in \\bullet tt, m_{M_0(pp)}(c) \\ge m_{W(pp,tt)}(c)), $ $(\\forall pp \\in I(tt), m_{M_0(pp)}(c) = 0), \\\\ (\\forall (pp,tt) \\in Q, m_{M_0(pp)}(c) \\ge m_{QW(pp,tt)}(c)), Z(tt) < Z(t)$ which is equivalent to $(\\forall p \\in \\bullet t: M_0(p) \\ge W(p,t)), $ $(\\forall p \\in I(t), M_0(p) = 0), (\\forall (p,t) \\in Q, M_0(p) \\ge QW(p,t)), \\exists tt \\in T, $ $(\\forall pp \\in \\bullet tt, M_0(pp) \\ge \\\\ W(pp,tt)), $ $(\\forall pp \\in I(tt), M_0(pp) = 0), (\\forall (pp,tt) \\in Q, M_0 \\ge QW(pp,tt)), \\\\ Z(tt) < Z(t)$ – assuming multiset domain $C$ for all operations $eval_{U_4}(bot_{U_5}(\\Pi ^7) \\setminus bot_{U_4}(\\Pi ^7), X_0 \\cup \\dots \\cup X_4) = \\lbrace prenabled(t,0) \\text{:-}.", "\| enabled(t,0) \\in X_0 \\cup \\dots \\cup X_4, notprenabled(t,0) \\notin X_0 \\cup \\dots \\cup X_4 \\rbrace $ .", "Its answer set is $X_5 = A_{24}^{ts=k}$ – using forced atom proposition and construction of $A_{24}$ $enabled(t,0)$ represents $\\forall c \\in C, (\\forall p \\in \\bullet t, m_{M_0(p)}(c) \\ge m_{W(p,t)}(c)), (\\forall p \\in I(t), m_{M_0(p)}(c) = 0), (\\forall (p,t) \\in Q, m_{M_0(p)}(c) \\ge m_{QW(p,t)}(c)) \\equiv $ $(\\forall p \\in \\bullet t, \\\\ M_0(p) \\ge W(p,t)), $ $(\\forall p \\in I(t), M_0(p) = 0), (\\forall (p,t) \\in Q, M_0(p) \\ge QW(p,t))$ – from REF above and assuming multiset domain $C$ for all operations $notprenabled(t,0)$ represents $(\\forall p \\in \\bullet t, M_0(p) \\ge W(p,t)), $ $(\\forall p \\in I(t), \\\\ M_0(p) = 0), (\\forall (p,t) \\in Q, M_0(p) \\ge QW(p,t)), \\exists tt \\in T, $ $(\\forall pp \\in \\bullet tt, M_0(pp) \\ge W(pp,tt)), $ $(\\forall pp \\in I(tt), M_0(pp) = 0), (\\forall (pp,tt) \\in Q, M_0(pp) \\ge QW(pp,tt)), \\\\ Z(tt) < Z(t)$ – from REF above and assuming multiset domain $C$ for all operations then, $prenabled(t,0)$ represents $(\\forall p \\in \\bullet t, M_0(p) \\ge W(p,t)), $ $(\\forall p \\in I(t), \\\\ M_0(p) = 0), (\\forall (p,t) \\in Q, M_0(p) \\ge QW(p,t)), \\nexists tt \\in T, $ $((\\forall pp \\in \\bullet tt, \\\\ M_0(pp) \\ge W(pp,tt)), $ $(\\forall pp \\in I(tt), M_0(pp) = 0), (\\forall (pp,tt) \\in Q, M_0(pp) \\ge QW(pp,tt)), Z(tt) < Z(t))$ – from (a), (b) and $enabled(t,0) \\in X_0 \\cup \\dots \\cup X_4$ $eval_{U_5}(bot_{U_6}(\\Pi ^7) \\setminus bot_{U_5}(\\Pi ^7), X_0 \\cup \\dots \\cup X_5) = \\lbrace \\lbrace fires(t,0)\\rbrace \\text{:-}.", "\| prenabled(t,0) \\\\ \\text{~holds in~} X_0 \\cup \\dots \\cup X_5 \\rbrace $ .", "It has multiple answer sets $X_{6.1}, \\dots , X_{6.n}$ , corresponding to elements of power set of $fires(t,0)$ atoms in $eval_{U_5}(...)$ – using supported rule proposition.", "Since we are showing that the union of answer sets of $\\Pi ^7$ determined using splitting is equal to $A$ , we only consider the set that matches the $fires(t,0)$ elements in $A$ and call it $X_6$ , ignoring the rest.", "Thus, $X_6 = A_{14}^{ts=0}$ , representing $T_0$ .", "in addition, for every $t$ such that $prenabled(t,0) \\in X_0 \\cup \\dots \\cup X_5, R(t) \\ne \\emptyset $ ; $fires(t,0) \\in X_6$ – per definition REF (firing set); requiring that a reset transition is fired when enabled thus, the firing set $T_0$ will not be eliminated by the constraint $f\\ref {f:c:pr:rptarc:elim}$ $eval_{U_6}(bot_{U_7}(\\Pi ^7) \\setminus bot_{U_6}(\\Pi ^7), X_0 \\cup \\dots \\cup X_6) = \\lbrace add(p,n_c,t,c,0) \\text{:-}.", "\| $ $\\lbrace fires(t,0-d+1), $ $tparc(t,p,n_c,c,0,d) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_6 \\rbrace \\cup \\lbrace del(p,n_c,t,c,0) \\text{:-}.", "\| $ $\\lbrace fires(t,0), $ $ptarc(p,t,n_c,c,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_6 \\rbrace $ .", "It's answer set is $X_7 = A_{15}^{ts=0} \\cup A_{16}^{ts=0}$ – using forced atom proposition and definitions of $A_{15}$ and $A_{16}$ .", "where, each $add$ atom is equivalent to $n_c=m_{W(t,p)}(c),c \\in C, p \\in t \\bullet $ , and each $del$ atom is equivalent to $n_c=m_{W(p,t)}(c), c \\in C, p \\in \\bullet t$ ; or $n_c=m_{M_0(p)}(c), c \\in C, p \\in R(t)$ , representing the effect of transitions in $T_0$ $eval_{U_7}(bot_{U_8}(\\Pi ^7) \\setminus bot_{U_7}(\\Pi ^7), X_0 \\cup \\dots \\cup X_7) = \\\\ \\lbrace tot\\_incr(p,qq_c,c,0) \\text{:-}.", "\| qq_c=\\sum _{add(p,q_c,t,c,0) \\in X_0 \\cup \\dots \\cup X_7}{q_c} \\rbrace \\cup \\\\ \\lbrace tot\\_decr(p,qq_c,c,0) \\text{:-}.", "\| qq_c=\\sum _{del(p,q_c,t,c,0) \\in X_0 \\cup \\dots \\cup X_7}{q_c} \\rbrace $ .", "It's answer set is $X_8 = A_{17}^{ts=0} \\cup A_{18}^{ts=0}$ – using forced atom proposition and definitions of $A_{17}$ and $A_{18}$ .", "where, each $tot\\_incr(p,qq_c,c,0)$ , $qq_c=\\sum _{add(p,q_c,t,c,0) \\in X_0 \\cup \\dots X_7}{q_c}$ $\\equiv qq_c=\\sum _{t \\in X_6, p \\in t \\bullet , 0+D(t)-1=0}{m_{W(p,t)}(c)}$ , $tot\\_decr(p,qq_c,c,0)$ , $qq_c=\\sum _{del(p,q_c,t,c,0) \\in X_0 \\cup \\dots X_7}{q_c}$ $\\equiv qq=\\sum _{t \\in X_6, p \\in \\bullet t}{m_{W(t,p)}(c)} + \\sum _{t \\in X_6, p \\in R(t)}{m_{M_0(p)}(c)}$ , represent the net effect of transitions in $T_0$ $eval_{U_8}(bot_{U_9}(\\Pi ^7) \\setminus bot_{U_8}(\\Pi ^7), X_0 \\cup \\dots \\cup X_8) = \\lbrace consumesmore(p,0) \\text{:-}.", "\| \\\\ \\lbrace holds(p,q_c,c,0), tot\\_decr(p,q1_c,c,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_8, q1_c > q_c \\rbrace \\cup $ $\\lbrace holds(p,q_c,c,1) \\text{:-}.", "\| \\lbrace holds(p,q1_c,c,0), tot\\_incr(p,q2_c,c,0), \\\\ tot\\_decr(p,q3_c,c,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_6, q_c=q1_c+q2_c-q3_c \\rbrace \\cup \\lbrace could\\_not\\_have(t,0) \\text{:-}.", "\| \\\\ \\lbrace prenabled(t,0), ptarc(s,t,q,c,0), holds(s,qq,c,0), tot\\_decr(s,qqq,c,0) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_8, fires(t,0) \\notin X_0 \\cup \\dots \\cup X_8, q > qq - qqq \\rbrace $ .", "It's answer set is $X_9 = A_{19}^{ts=0} \\cup A_{21}^{ts=0} \\cup A_{25}^{ts=0}$ – using forced atom proposition and definitions of $A_{19}, A_{21}, A_{25}$ .", "where $consumesmore(p,0)$ represents $\\exists p : q_c=m_{M_0(p)}(c), q1_c= \\\\ \\sum _{t \\in T_0, p \\in \\bullet t}{m_{W(p,t)}(c)}+\\sum _{t \\in T_0, p \\in R(t)}{m_{M_0(p)}(c)}, q1_c > q_c, c \\in C$ , indicating place $p$ will be over consumed if $T_0$ is fired, as defined in definition REF (conflicting transitions), $holds(p,q_c,c,1)$ if $q_c=m_{M_0(p)}(c)+\\sum _{t \\in T_0, p \\in t \\bullet , 0+D(t)-1=0}{m_{W(t,p)}(c)}- \\\\ (\\sum _{t \\in T_0, p \\in \\bullet t}{m_{W(p,t)}(c)}+ $ $\\sum _{t \\in T_0, p \\in R(t)}{m_{M_0(p)}(c)})$ represented by $q_c=m_{M_1(p)}(c)$ for some $c \\in C$ – by construction of $\\Pi ^7$ and $consumesmore(p,0)$ if $\\sum _{t \\in T_0, p \\in \\bullet t}{m_{W(p,t)}(c)}+ $ $\\sum _{t \\in T_0, p \\in R(t)}{m_{M_0(p)}(c)} > m_{M_0(p)}(c)$ for any $c \\in C$ $could\\_not\\_have(t,0)$ if $(\\forall p \\in \\bullet t, W(p,t) \\le M_0(p)), (\\forall p \\in I(t), M_0(p) = 0), (\\forall (p,t) \\in Q, \\\\ M_0(p) \\ge WQ(p,t)), \\nexists tt \\in T, (\\forall pp \\in \\bullet tt, W(pp,tt) \\le M_{ts}(pp)), (\\forall pp \\in I(tt), M_0(pp) = 0), (\\forall (pp,tt) \\in Q, M_0(pp) \\ge QW(pp,tt)), Z(tt) < Z(t)$ , and $q_c > m_{M_0(s)}(c) - (\\sum _{t^{\\prime } \\in T_0, s \\in \\bullet t^{\\prime }}{m_{W(s,t^{\\prime })}(c)}+ \\sum _{t^{\\prime } \\in T_0, s \\in R(t)}{m_{M_0(s)}(c)}), \\\\ q_c = m_{W(s,t)}(c) \\text{~if~} s \\in \\bullet t \\text{~or~} m_{M_0(s)}(c) \\text{~otherwise}$ for some $c \\in C$ , which becomes $q > M_0(s) - (\\sum _{t^{\\prime } \\in T_0, s \\in \\bullet t^{\\prime }}{W(s,t^{\\prime })}+ \\sum _{t^{\\prime } \\in T_0, s \\in R(t)}{M_0(s)}), q = W(s,t) \\text{~if~} s \\in \\bullet t \\text{~or~} M_0(s) \\text{~otherwise}$ for all $c \\in C$ (i), (ii) above combined match the definition of $A_{25}$ $X_9$ does not contain $could\\_not\\_have(t,0)$ , when $prenabled(t,0) \\in X_0 \\cup \\dots \\cup X_6$ and $fires(t,0) \\notin X_0 \\cup \\dots \\cup X_5$ due to construction of $A$ , encoding of $a\\ref {a:c:maxfire:cnh}$ and its body atoms.", "As a result, it is not eliminated by the constraint $a\\ref {a:c:prmaxfire:elim}$ $ \\vdots $ $eval_{U_{9k+0}}(bot_{U_{9k+1}}(\\Pi ^7) \\setminus bot_{U_{9k+0}}(\\Pi ^7), X_0 \\cup \\dots \\cup X_{9k+0}) = \\lbrace ptarc(p,t,q_c,c,k) \\text{:-}.", "\| c \\in C, q_c=m_{W(p,t)}(c), q_c > 0 \\rbrace \\cup \\lbrace tparc(t,p,q_c,c,k,d) \\text{:-}.", "\| c \\in C, q_c=m_{W(t,p)}(c), q_c > 0, d=D(t) \\rbrace \\cup $ $\\lbrace ptarc(p,t,q_c,c,k) \\text{:-}.", "\| c \\in C, q_c=m_{M_k(p)}(c), q_c > 0 \\rbrace \\cup \\\\ \\lbrace iptarc(p,t,1,c,k) \\text{:-}.", "\| c \\in C \\rbrace \\cup $ $\\lbrace tptarc(p,t,q_c,c,k) \\text{:-}.", "\| c \\in C, q_c = m_{QW(p,t)}(c), q_c > 0 \\rbrace $ .", "Its answer set $X_{9k+1}=A_6^{ts=k} \\cup A_7^{ts=k} \\cup A_9^{ts=k} \\cup A_{10}^{ts=k} \\cup A_{11}^{ts=k}$ – using forced atom proposition and construction of $A_6, A_7, A_9, A_{10}, A_{11}$ .", "$eval_{U_{9k+1}}(bot_{U_{10k+2}}(\\Pi ^7) \\setminus bot_{U_{9k+1}}(\\Pi ^7), X_0 \\cup \\dots \\cup X_{9k+1}) = \\lbrace notenabled(t,k) \\text{:-} .", "\| \\\\ (\\lbrace trans(t), ptarc(p,t,n_c,c,k), holds(p,q_c,c,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{9k+1}, \\text{~where~} q_c < n_c) \\text{~or~} (\\lbrace notenabled(t,k) \\text{:-} .", "\| (\\lbrace trans(t), iptarc(p,t,n2_c,c,k), holds(p,q_c,c,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{9k+1}, $ $\\text{~where~} q_c \\ge n2_c \\rbrace ) \\text{~or~} $ $(\\lbrace trans(t), tptarc(p,t,n3_c,c,k), \\\\ holds(p,q_c,c,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{9k+1}, $ $\\text{~where~} q_c < n3_c) \\rbrace $ .", "Its answer set $X_{9k+2}=A_{12}^{ts=k}$ – using forced atom proposition and construction of $A_{12}$ .", "where, $q_c=m_{M_k(p)}(c)$ , and $n_c=m_{W(p,t)}(c)$ for an arc $(p,t) \\in E^-$ – by construction of $i\\ref {i:c:init}$ and $f\\ref {f:c:ptarc}$ predicates in $\\Pi ^7$ , and in an arc $(p,t) \\in E^-$ , $p \\in \\bullet t$ (by definition REF of preset) $n2_c=1$ – by construction of $iptarc$ predicates in $\\Pi ^7$ , meaning $q_c \\ge n2_c \\equiv q_c \\ge 1 \\equiv q_c > 0$ , $tptarc(p,t,n3_c,c,k)$ represents $n3_c=m_{QW(p,t)}(c)$ , where $(p,t) \\in Q$ thus, $notenabled(t,k) \\in X_{9k+1}$ represents $\\exists c \\in C, (\\exists p \\in \\bullet t : m_{M_k(p)}(c) < m_{W(p,t)}(c)) \\vee (\\exists p \\in I(t) : m_{M_k(p)}(c) > 0) \\vee (\\exists (p,t) \\in Q : m_{M_k(p)}(c) < m_{QW(p,t)}(c))$ .", "$eval_{U_{9k+2}}(bot_{U_{9k+3}}(\\Pi ^7) \\setminus bot_{U_{9k+2}}(\\Pi ^7), X_0 \\cup \\dots \\cup X_{9k+2}) = \\lbrace enabled(t,k) \\text{:-}.", "\| trans(t) \\in X_0 \\cup \\dots \\cup X_{9k+2}, notenabled(t,k) \\notin X_0 \\cup \\dots \\cup X_{9k+2} \\rbrace $ .", "Its answer set is $X_{9k+3} = A_{13}^{ts=k}$ – using forced atom proposition and construction of $A_{13}$ .", "Since an $enabled(t,k) \\in X_{9k+3}$ if $\\nexists ~notenabled(t,k) \\in X_0 \\cup \\dots \\cup X_{9k+2}$ ; which is equivalent to $\\nexists t, \\forall c \\in C, (\\nexists p \\in \\bullet t, m_{M_k(p)}(c) < m_{W(p,t)}(c)), (\\nexists p \\in I(t), m_{M_k(p)}(c) > 0), (\\nexists (p,t) \\in Q : m_{M_k(p)}(c) < m_{QW(p,t)}(c) ), \\forall c \\in C, (\\forall p \\in \\bullet t: m_{M_k(p)}(c) \\ge m_{W(p,t)}(c)), (\\forall p \\in I(t) : m_{M_k(p)}(c) = 0)$ .", "$eval_{U_{9k+3}}(bot_{U_{9k+4}}(\\Pi ^7) \\setminus bot_{U_{9k+3}}(\\Pi ^7), X_0 \\cup \\dots \\cup X_{9k+3}) = \\lbrace notprenabled(t,k) \\text{:-}.", "\| \\\\ \\lbrace enabled(t,k), transpr(t,p), enabled(tt,k), transpr(tt,pp) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{9k+3}, \\\\ pp < p \\rbrace $ .", "Its answer set is $X_{9k+4} = A_{23}^{ts=k}$ – using forced atom proposition and construction of $A_{23}$ .", "$enabled(t,k)$ represents $\\exists t \\in T, \\forall c \\in C, (\\forall p \\in \\bullet t, m_{M_k(p)}(c) \\ge m_{W(p,t)}(c)), $ $(\\forall p \\in I(t), m_{M_k(p)}(c) = 0), (\\forall (p,t) \\in Q, m_{M_k(p)}(c) \\ge m_{QW(p,t)}(c))$ $enabled(tt,k)$ represents $\\exists tt \\in T, \\forall c \\in C, (\\forall pp \\in \\bullet tt, m_{M_k(pp)}(c) \\ge \\\\ m_{W(pp,tt)}(c)) \\wedge (\\forall pp \\in I(tt), m_{M_k(pp)}(c) = 0), (\\forall (pp,tt) \\in Q, m_{M_k(pp)}(c) \\ge m_{QW(pp,tt)}(c))$ $transpr(t,p)$ represents $p=Z(t)$ – by construction $transpr(tt,pp)$ represents $pp=Z(tt)$ – by construction thus, $notprenabled(t,k)$ represents $\\forall c \\in C, (\\forall p \\in \\bullet t, m_{M_k(p)}(c) \\ge \\\\ m_{W(p,t)}(c)), $ $(\\forall p \\in I(t), m_{M_k(p)}(c) = 0), (\\forall (p,t) \\in Q, m_{M_k(p)}(c) \\ge \\\\ m_{QW(p,t)}(c)), \\exists tt \\in T, $ $(\\forall pp \\in \\bullet tt, m_{M_k(pp)}(c) \\ge m_{W(pp,tt)}(c)), $ $(\\forall pp \\in I(tt), \\\\ m_{M_k(pp)}(c) = 0), (\\forall (pp,tt) \\in Q, m_{M_k(pp)}(c) \\ge m_{QW(pp,tt)}(c)), Z(tt) < Z(t)$ which is equivalent to $(\\forall p \\in \\bullet t: M_k(p) \\ge W(p,t)), $ $(\\forall p \\in I(t), \\\\ M_k(p) = 0), (\\forall (p,t) \\in Q, M_k(p) \\ge QW(p,t)), \\exists tt \\in T, $ $(\\forall pp \\in \\bullet tt, \\\\ M_k(pp) \\ge W(pp,tt)), $ $(\\forall pp \\in I(tt), M_k(pp) = 0), (\\forall (pp,tt) \\in Q, M_k(pp) \\ge QW(pp,tt)), Z(tt) < Z(t)$ – assuming multiset domain $C$ for all operations $eval_{U_{9k+4}}(bot_{U_{9k+5}}(\\Pi ^7) \\setminus bot_{U_{9k+4}}(\\Pi ^7), X_0 \\cup \\dots \\cup X_{9k+4}) = \\lbrace prenabled(t,k) \\text{:-}.", "\| \\\\ enabled(t,k) \\in X_0 \\cup \\dots \\cup X_{9k+4}, notprenabled(t,k) \\notin X_0 \\cup \\dots \\cup X_{9k+4} \\rbrace $ .", "Its answer set is $X_{9k+5} = A_{24}^{ts=k}$ – using forced atom proposition and construction of $A_{24}$ $enabled(t,k)$ represents $\\forall c \\in C, (\\forall p \\in \\bullet t, m_{M_k(p)}(c) \\ge m_{W(p,t)}(c)), $ $(\\forall p \\in I(t), m_{M_k(p)}(c) = 0), (\\forall (p,t) \\in Q, m_{M_k(p)}(c) \\ge m_{QW(p,t)}(c)) \\equiv (\\forall p \\in \\bullet t, M_k(p) \\ge W(p,t)), $ $(\\forall p \\in I(t), M_k(p) = 0), (\\forall (p,t) \\in Q, M_k(p) \\ge QW(p,t))$ – from REF above and assuming multiset domain $C$ for all operations $notprenabled(t,k)$ represents $(\\forall p \\in \\bullet t, M_k(p) \\ge W(p,t)), (\\forall p \\in I(t), \\\\ M_k(p) = 0), (\\forall (p,t) \\in Q, M_k(p) \\ge QW(p,t)), \\exists tt \\in T, $ $(\\forall pp \\in \\bullet tt, M_k(pp) \\ge W(pp,tt)), (\\forall pp \\in I(tt), M_k(pp) = 0), (\\forall (pp,tt) \\in Q, M_k(pp) \\ge QW(pp,tt)), \\\\ Z(tt) < Z(t)$ – from REF above and assuming multiset domain $C$ for all operations then, $prenabled(t,k)$ represents $(\\forall p \\in \\bullet t, M_k(p) \\ge W(p,t)), (\\forall p \\in I(t), \\\\ M_k(p) = 0), (\\forall (p,t) \\in Q, M_k(p) \\ge QW(p,t)), \\nexists tt \\in T, $ $((\\forall pp \\in \\bullet tt, M_k(pp) \\ge W(pp,tt)), $ $(\\forall pp \\in I(tt), M_k(pp) = 0), (\\forall (pp,tt) \\in Q, \\\\ M_k(pp) \\ge QW(pp,tt)), Z(tt) < Z(t))$ – from (a), (b) and $enabled(t,k) \\in X_0 \\cup \\dots \\cup X_{9k+4}$ $eval_{U_{9k+5}}(bot_{U_{9k+6}}(\\Pi ^7) \\setminus bot_{U_{9k+5}}(\\Pi ^7), X_0 \\cup \\dots \\cup X_{9k+5}) = \\lbrace \\lbrace fires(t,k)\\rbrace \\text{:-}.", "\| \\\\ prenabled(t,k) \\text{~holds in~} X_0 \\cup \\dots \\cup X_{9k+5} \\rbrace $ .", "It has multiple answer sets $X_{9k+6.1}, \\dots , X_{9k+6.n}$ , corresponding to elements of power set of $fires(t,k)$ atoms in $eval_{U_{9k+5}}(...)$ – using supported rule proposition.", "Since we are showing that the union of answer sets of $\\Pi ^7$ determined using splitting is equal to $A$ , we only consider the set that matches the $fires(t,k)$ elements in $A$ and call it $X_{9k+6}$ , ignoring the rest.", "Thus, $X_{9k+6} = A_{14}^{ts=k}$ , representing $T_k$ .", "in addition, for every $t$ such that $prenabled(t,k) \\in X_0 \\cup \\dots \\cup X_{9k+5}, R(t) \\ne \\emptyset $ ; $fires(t,k) \\in X_{9k+6}$ – per definition REF (firing set); requiring that a reset transition is fired when enabled thus, the firing set $T_k$ will not be eliminated by the constraint $f\\ref {f:c:pr:rptarc:elim}$ $eval_{U_{9k+6}}(bot_{U_{9k+7}}(\\Pi ^7) \\setminus bot_{U_{9k+6}}(\\Pi ^7), X_0 \\cup \\dots \\cup X_{9k+6}) = \\lbrace add(p,n_c,t,c,k) \\text{:-}.", "\| $ $\\lbrace fires(t,k-d+1), tparc(t,p,n_c,c,0,d) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{9k+6} \\rbrace \\cup \\\\ \\lbrace del(p,n_c,t,c,k) \\text{:-}.", "\| \\lbrace fires(t,k), ptarc(p,t,n_c,c,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{9k+6} \\rbrace $ .", "It's answer set is $X_{9k+7} = A_{15}^{ts=k} \\cup A_{16}^{ts=k}$ – using forced atom proposition and definitions of $A_{15}$ and $A_{16}$ .", "where, each $add$ atom is equivalent to $n_c=m_{W(t,p)}(c),c \\in C, p \\in t \\bullet $ , and each $del$ atom is equivalent to $n_c=m_{W(p,t)}(c), c \\in C, p \\in \\bullet t$ ; or $n_c=m_{M_k(p)}(c), c \\in C, p \\in R(t)$ , representing the effect of transitions in $T_k$ $eval_{U_{9k+7}}(bot_{U_{9k+8}}(\\Pi ^7) \\setminus bot_{U_{9k+7}}(\\Pi ^7), X_0 \\cup \\dots \\cup X_{9k+7}) = \\lbrace tot\\_incr(p,qq_c,c,k) \\text{:-}.", "\| $ $ qq_c=\\sum _{add(p,q_c,t,c,k) \\in X_0 \\cup \\dots \\cup X_{9k+7}}{q_c} \\rbrace \\cup \\lbrace tot\\_decr(p,qq_c,c,k) \\text{:-}.", "\| $ $ qq_c=\\sum _{del(p,q_c,t,c,k) \\in X_0 \\cup \\dots \\cup X_{9k+7}}{q_c} \\rbrace $ .", "It's answer set is $X_{9k+8} = A_{17}^{ts=k} \\cup A_{18}^{ts=k}$ – using forced atom proposition and definitions of $A_{17}$ and $A_{18}$ .", "where, each $tot\\_incr(p,qq_c,c,k)$ , $qq_c=\\sum _{add(p,q_c,t,c,k) \\in X_0 \\cup \\dots X_{9k+7}}{q_c}$ $\\equiv qq_c=\\sum _{t \\in X_{9k+6}, p \\in t \\bullet , 0 \\le l \\le k, l+D(t)-1=k }{m_{W(p,t)}(c)}$ , and each $tot\\_decr(p,qq_c,c,k)$ , $qq_c=\\sum _{del(p,q_c,t,c,k) \\in X_0 \\cup \\dots X_{9k+7}}{q_c}$ $\\equiv qq=\\sum _{t \\in X_{9k+6}, p \\in \\bullet t}{m_{W(t,p)}(c)} + \\sum _{t \\in X_{9k+6}, p \\in R(t)}{m_{M_k(p)}(c)}$ , represent the net effect of transitions in $T_k$ $eval_{U_{9k+8}}(bot_{U_{9k+9}}(\\Pi ^7) \\setminus bot_{U_{9k+8}}(\\Pi ^7), X_0 \\cup \\dots \\cup X_{9k+8}) = $ $\\lbrace consumesmore(p,k) \\text{:-}.", "\| \\\\ \\lbrace holds(p,q_c,c,k), tot\\_decr(p,q1_c,c,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{9k+8}, q1_c > q_c \\rbrace \\cup \\\\ \\lbrace holds(p,q_c,c,k+1) \\text{:-}., \| \\lbrace holds(p,q1_c,c,k), tot\\_incr(p,q2_c,c,k), \\\\ tot\\_decr(p,q3_c,c,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{9k+6}, q_c=q1_c+q2_c-q3_c \\rbrace \\cup \\\\ \\lbrace could\\_not\\_have(t,k) \\text{:-}.", "\| \\lbrace prenabled(t,k), \\\\ ptarc(s,t,q,c,k), holds(s,qq,c,k), tot\\_decr(s,qqq,c,k) \\rbrace \\subseteq X_0 \\cup \\dots \\cup X_{9k+8}, \\\\ fires(t,k) \\notin X_0 \\cup \\dots \\cup X_{10k+8}, q > qq - qqq \\rbrace $ .", "It's answer set is $X_{9k+9} = A_{19}^{ts=k} \\cup A_{21}^{ts=k} \\cup A_{25}^{ts=k}$ – using forced atom proposition and definitions of $A_{19}, A_{21}, A_{25}$ .", "where, $consumesmore(p,k)$ represents $\\exists p : q_c=m_{M_k(p)}(c), \\\\ q1_c=\\sum _{t \\in T_k, p \\in \\bullet t}{m_{W(p,t)}(c)}+\\sum _{t \\in T_k, p \\in R(t)}{m_{M_k(p)}(c)}, q1_c > q_c, c \\in C$ , indicating place $p$ that will be over consumed if $T_k$ is fired, as defined in definition REF (conflicting transitions), $holds(p,q_c,c,k+1)$ if $q_c=m_{M_k(p)}(c)+\\sum _{t \\in T_l, p \\in t \\bullet , 0 \\le l \\le k, l+D(t)-1=k}{m_{W(t,p)}(c)}-(\\sum _{t \\in T_k, p \\in \\bullet t}{m_{W(p,t)}(c)}+ \\sum _{t \\in T_k, p \\in R(t)}{m_{M_k(p)}(c)})$ represented by $q_c=m_{M_1(p)}(c)$ for some $c \\in C$ – by construction of $\\Pi ^7$ , and $could\\_not\\_have(t,k)$ if $(\\forall p \\in \\bullet t, W(p,t) \\le M_k(p)), (\\forall p \\in I(t), M_k(p) = 0), (\\forall (p,t) \\in Q, \\\\ M_k(p) \\ge WQ(p,t)), \\nexists tt \\in T, (\\forall pp \\in \\bullet tt, W(pp,tt) \\le M_{ts}(pp)), (\\forall pp \\in I(tt), M_k(pp) = 0), (\\forall (pp,tt) \\in Q, M_k(pp) \\ge QW(pp,tt)), Z(tt) < Z(t)$ , and $q_c > m_{M_k(s)}(c) - (\\sum _{t^{\\prime } \\in T_k, s \\in \\bullet t^{\\prime }}{m_{W(s,t^{\\prime })}(c)}+ \\sum _{t^{\\prime } \\in T_k, s \\in R(t)}{m_{M_k(s)}(c)}), \\\\ q_c = m_{W(s,t)}(c) \\text{~if~} s \\in \\bullet t \\text{~or~} m_{M_k(s)}(c) \\text{~otherwise}$ for some $c \\in C$ , which becomes $q > M_k(s) - (\\sum _{t^{\\prime } \\in T_k, s \\in \\bullet t^{\\prime }}{W(s,t^{\\prime })}+ \\sum _{t^{\\prime } \\in T_k, s \\in R(t)}{M_k(s)}), q = W(s,t) \\text{~if~} s \\in \\bullet t \\text{~or~} M_k(s) \\text{~otherwise}$ for all $c \\in C$ (i), (ii) above combined match the definition of $A_{25}$ $X_{9k+9}$ does not contain $could\\_not\\_have(t,k)$ , when $prenabled(t,k) \\in X_0 \\cup \\dots \\cup X_{9k+6}$ and $fires(t,k) \\notin X_0 \\cup \\dots \\cup X_{9k+5}$ due to construction of $A$ , encoding of $a\\ref {a:c:maxfire:cnh}$ and its body atoms.", "As a result it is not eliminated by the constraint $a\\ref {a:c:prmaxfire:elim}$ $eval_{U_{9k+9}}(bot_{U_{9k+10}}(\\Pi ^7) \\setminus bot_{U_{9k+9}}(\\Pi ^7), X_0 \\cup \\dots \\cup X_{9k+9}) = \\lbrace consumesmore \\text{:-}.", "\| \\\\ \\lbrace consumesmore(p,0),\\dots ,$ $consumesmore(p,k) \\rbrace \\cap (X_0 \\cup \\dots \\cup X_{9k+9}) \\ne \\emptyset \\rbrace $ .", "It's answer set is $X_{9k+10} = A_{20}$ – using forced atom proposition and the definition of $A_{20}$ $X_{9k+10}$ will be empty since none of $consumesmore(p,0),\\dots , \\\\ consumesmore(p,k)$ hold in $X_0 \\cup \\dots \\cup X_{9k+10}$ due to the construction of $A$ , encoding of $a\\ref {a:overc:place}$ and its body atoms.", "As a result, it is not eliminated by the constraint $a\\ref {a:overc:elim}$ The set $X = X_0 \\cup \\dots \\cup X_{9k+10}$ is the answer set of $\\Pi ^7$ by the splitting sequence theorem REF .", "Each $X_i, 0 \\le i \\le 9k+10$ matches a distinct portion of $A$ , and $X = A$ , thus $A$ is an answer set of $\\Pi ^7$ .", "Next we show (REF ): Given $\\Pi ^7$ be the encoding of a Petri Net $PN(P,T,E,C,W,R,I,$ $Q,QW,Z,D)$ with initial marking $M_0$ , and $A$ be an answer set of $\\Pi ^7$ that satisfies (REF ) and (REF ), then we can construct $X=M_0,T_k,\\dots ,M_k,T_k,M_{k+1}$ from $A$ , such that it is an execution sequence of $PN$ .", "We construct the $X$ as follows: $M_i = (M_i(p_0), \\dots , M_i(p_n))$ , where $\\lbrace holds(p_0,m_{M_i(p_0)}(c),c,i), \\dots ,\\\\ holds(p_n,m_{M_i(p_n)}(c),c,i) \\rbrace \\subseteq A$ , for $c \\in C, 0 \\le i \\le k+1$ $T_i = \\lbrace t : fires(t,i) \\in A\\rbrace $ , for $0 \\le i \\le k$ and show that $X$ is indeed an execution sequence of $PN$ .", "We show this by induction over $k$ (i.e.", "given $M_k$ , $T_k$ is a valid firing set and its firing produces marking $M_{k+1}$ ).", "Base case: Let $k=0$ , and $M_0$ is a valid marking in $X$ for $PN$ , show [(1)] $T_0$ is a valid firing set for $M_0$ , and firing $T_0$ in $M_0$ produces marking $M_1$ .", "We show $T_0$ is a valid firing set for $M_0$ .", "Let $\\lbrace fires(t_0,0), \\dots , fires(t_x,0) \\rbrace $ be the set of all $fires(\\dots ,0)$ atoms in $A$ , Then for each $fires(t_i,0) \\in A$ $prenabled(t_i,0) \\in A$ – from rule $a\\ref {a:c:prfires}$ and supported rule proposition Then $enabled(t_i,0) \\in A$ – from rule $a\\ref {a:c:prenabled}$ and supported rule proposition And $notprenabled(t_i,0) \\notin A$ – from rule $a\\ref {a:c:prenabled}$ and supported rule proposition For $enabled(t_i,0) \\in A$ $notenabled(t_i,0) \\notin A$ – from rule $e\\ref {e:c:enabled}$ and supported rule proposition Then either of $body(e\\ref {e:c:ne:ptarc})$ , $body(e\\ref {e:c:ne:iptarc})$ , or $body(e\\ref {e:c:ne:tptarc})$ must not hold in $A$ for $t_i$ – from rules $body(e\\ref {e:c:ne:ptarc}), body(e\\ref {e:c:ne:iptarc}), body(e\\ref {e:c:ne:tptarc})$ and forced atom proposition Then $q_c \\lnot < {n_i}_c \\equiv q_c \\ge {n_i}_c$ in $e\\ref {e:c:ne:ptarc}$ for all $\\lbrace holds(p,q_c,c,0), \\\\ ptarc(p,t_i,{n_i}_c,c,0)\\rbrace \\subseteq A$ – from $e\\ref {e:c:ne:ptarc}$ , forced atom proposition, and given facts ($holds(p,q_c,c,0) \\in A, ptarc(p,t_i,{n_i}_c,0) \\in A$ ) And $q_c \\lnot \\ge {n_i}_c \\equiv q_c < {n_i}_c$ in $e\\ref {e:c:ne:iptarc}$ for all $\\lbrace holds(p,q_c,c,0), \\\\ iptarc(p,t_i,{n_i}_c,c,0) \\rbrace \\subseteq A, {n_i}_c=1$ ; $q_c > {n_i}_c \\equiv q_c = 0$ – from $e\\ref {e:c:ne:iptarc}$ , forced atom proposition, given facts ($holds(p,q_c,c,0) \\in A, \\\\ iptarc(p,t_i,1,c,0) \\in A$ ), and $q_c$ is a positive integer And $q_c \\lnot < {n_i}_c \\equiv q_c \\ge {n_i}_c$ in $e\\ref {e:c:ne:tptarc}$ for all $\\lbrace holds(p,q_c,c,0), \\\\ tptarc(p,t_i,{n_i}_c,c,0) \\rbrace \\subseteq A$ – from $e\\ref {e:c:ne:tptarc}$ , forced atom proposition, and given facts Then $\\forall c \\in C, (\\forall p \\in \\bullet t_i, m_{M_0(p)}(c) \\ge m_{W(p,t_i)}(c)) \\wedge (\\forall p \\in I(t_i), \\\\ m_{M_0(p)}(c) = 0) \\wedge (\\forall (p,t_i) \\in Q, m_{M_0(p)}(c) \\ge m_{QW(p,t_i)}(c))$ – from $i\\ref {i:c:init},f\\ref {f:c:ptarc},f\\ref {f:c:rptarc}$ construction, definition REF of preset $\\bullet t_i$ in $PN$ , definition REF of enabled transition in $PN$ , and that the construction of reset arcs by $f\\ref {f:c:rptarc}$ ensures $notenabled(t,0)$ is never true for a reset arc, where $holds(p,q_c,c,0) \\in A$ represents $q_c=m_{M_0(p)}(c)$ , $ptarc(p,t_i,{n_i}_c,0) \\in A$ represents ${n_i}_c=m_{W(p,t_i)}(c)$ , ${n_i}_c = m_{M_0(p)}(c)$ .", "Which is equivalent to $(\\forall p \\in \\bullet t_i, M_0(p) \\ge W(p,t_i)) \\wedge (\\forall p \\in I(t_i), M_0(p) = 0) \\wedge (\\forall (p,t_i) \\in Q, M_0(p) \\ge QW(p,t_i))$ – assuming multiset domain $C$ For $notprenabled(t_i,0) \\notin A$ Either $(\\nexists enabled(tt,0) \\in A : pp < p_i)$ or $(\\forall enabled(tt,0) \\in A : pp \\lnot < p_i)$ where $pp = Z(tt), p_i = Z(t_i)$ – from rule $a\\ref {a:c:prne}, f\\ref {f:c:pr}$ and forced atom proposition This matches the definition of an enabled priority transition Then $t_i$ is enabled and can fire in $PN$ , as a result it can belong to $T_0$ – from definition REF of enabled transition And $consumesmore \\notin A$ , since $A$ is an answer set of $\\Pi ^7$ – from rule $a\\ref {a:c:overc:elim}$ and supported rule proposition Then $\\nexists consumesmore(p,0) \\in A$ – from rule $a\\ref {a:c:overc:gen}$ and supported rule proposition Then $\\nexists \\lbrace holds(p,q_c,c,0), tot\\_decr(p,q1_c,c,0) \\rbrace \\subseteq A, q1_c>q_c$ in $body(a\\ref {a:c:overc:place})$ – from $a\\ref {a:c:overc:place}$ and forced atom proposition Then $\\nexists c \\in C \\nexists p \\in P, (\\sum _{t_i \\in \\lbrace t_0,\\dots ,t_x\\rbrace , p \\in \\bullet t_i}{m_{W(p,t_i)}(c)}+ \\\\ \\sum _{t_i \\in \\lbrace t_0,\\dots ,t_x\\rbrace , p \\in R(t_i)}{m_{M_0(p)}(c)}) > m_{M_0(p)}(c)$ – from the following $holds(p,q_c,c,0)$ represents $q_c=m_{M_0(p)}(c)$ – from rule $i\\ref {i:c:init}$ encoding, given $tot\\_decr(p,q1_c,c,0) \\in A$ if $\\lbrace del(p,{q1_0}_c,t_0,c,0), \\dots , \\\\ del(p,{q1_x}_c,t_x,c,0) \\rbrace \\subseteq A$ , where $q1_c = {q1_0}_c+\\dots +{q1_x}_c$ – from $r\\ref {r:c:totdecr}$ and forced atom proposition $del(p,{q1_i}_c,t_i,c,0) \\in A$ if $\\lbrace fires(t_i,0), ptarc(p,t_i,{q1_i}_c,c,0) \\rbrace \\subseteq A$ – from $r\\ref {r:c:del}$ and supported rule proposition $del(p,{q1_i}_c,t_i,c,0)$ represents removal of ${q1_i}_c = m_{W(p,t_i)}(c)$ tokens from $p \\in \\bullet t_i$ ; or it represents removal of ${q1_i}_c = m_{M_0(p)}(c)$ tokens from $p \\in R(t_i)$ – from rules $r\\ref {r:c:del},f\\ref {f:c:ptarc},f\\ref {f:c:rptarc}$ , supported rule proposition, and definition REF of preset in $PN$ Then the set of transitions in $T_0$ do not conflict – by the definition REF of conflicting transitions And for each $prenabled(t_j,0) \\in A$ and $fires(t_j,0) \\notin A$ , $could\\_not\\_have(t_j,0) \\in A$ , since $A$ is an answer set of $\\Pi ^7$ - from rule $a\\ref {a:c:prmaxfire:elim}$ and supported rule proposition Then $\\lbrace prenabled(t_j,0), holds(s,qq_c,c,0), ptarc(s,t_j,q_c,c,0), \\\\ tot\\_decr(s,qqq_c,c,0) \\rbrace \\subseteq A$ , such that $q_c > qq_c - qqq_c$ and $fires(t_j,0) \\notin A$ - from rule $a\\ref {a:c:prmaxfire:cnh}$ and supported rule proposition Then for an $s \\in \\bullet t_j \\cup R(t_j)$ , $q_c > m_{M_0(s)}(c) - (\\sum _{t_i \\in T_0, s \\in \\bullet t_i}{m_{W(s,t_i)}(c)} + \\sum _{t_i \\in T_0, s \\in R(t_i)}{m_{M_0(s)}(c))}$ , where $q_c=m_{W(s,t_j)}(c) \\text{~if~} s \\in \\bullet t_j, \\text{~or~} m_{M_0(s)}(c)$ $ \\text{~otherwise}$ .", "$ptarc(s,t_i,q_c,c,0)$ represents $q_c=m_{W(s,t_i)}(c)$ if $(s,t_i) \\in E^-$ or $q_c=m_{M_0(s)}(c)$ if $s \\in R(t_i)$ – from rule $f\\ref {f:c:ptarc},f\\ref {f:c:rptarc}$ construction $holds(s,qq_c,c,0)$ represents $qq_c=m_{M_0(s)}(c)$ – from $i\\ref {i:c:init}$ construction $tot\\_decr(s,qqq_c,c,0) \\in A$ if $\\lbrace del(s,{qqq_0}_c,t_0,c,0), \\dots , \\\\ del(s,{qqq_x}_c,t_x,c,0) \\rbrace \\subseteq A$ – from rule $r\\ref {r:c:totdecr}$ construction and supported rule proposition $del(s,{qqq_i}_c,t_i,c,0) \\in A$ if $\\lbrace fires(t_i,0), ptarc(s,t_i,{qqq_i}_c,c,0) \\rbrace \\subseteq A$ – from rule $r\\ref {r:c:del}$ and supported rule proposition $del(s,{qqq_i}_c,t_i,c,0)$ either represents ${qqq_i}_c = m_{W(s,t_i)}(c) : t_i \\in T_0, (s,t_i) \\in E^-$ , or ${qqq_i}_c = m_{M_0(t_i)}(c) : t_i \\in T_0, s \\in R(t_i)$ – from rule $f\\ref {f:c:ptarc},f\\ref {f:c:rptarc}$ construction $tot\\_decr(q,qqq_c,c,0)$ represents $\\sum _{t_i \\in T_0, s \\in \\bullet t_i}{m_{W(s,t_i)}(c)} + \\\\ \\sum _{t_i \\in T_0, s \\in R(t_i)}{m_{M_0(s)}(c)}$ – from (C,D,E) above Then firing $T_0 \\cup \\lbrace t_j \\rbrace $ would have required more tokens than are present at its source place $s \\in \\bullet t_j \\cup R(t_j)$ .", "Thus, $T_0$ is a maximal set of transitions that can simultaneously fire.", "And for each reset transition $t_r$ with $prenabled(t_r,0) \\in A$ , $fires(t_r,0) \\in A$ , since $A$ is an answer set of $\\Pi ^7$ - from rule $f\\ref {f:c:pr:rptarc:elim}$ and supported rule proposition Then, the firing set $T_0$ satisfies the reset-transition requirement of definition REF (firing set) Then $\\lbrace t_0, \\dots , t_x\\rbrace = T_0$ – using 1(a),1(b),1(d) above; and using 1(c) it is a maximal firing set Let $holds(p,q_c,c,1) \\in A$ Then $\\lbrace holds(p,q1_c,c,0), tot\\_incr(p,q2_c,c,0), tot\\_decr(p,q3_c,c,0) \\rbrace \\subseteq A : q_c=q1_c+q2_c-q3_c$ – from rule $r\\ref {r:c:nextstate}$ and supported rule proposition Then, $holds(p,q1_c,c,0) \\in A$ represents $q1_c=m_{M_0(p)}(c)$ – given ; and $\\lbrace add(p,{q2_0}_c,t_0,c,0), \\dots , $ $add(p,{q2_j}_c,t_j,c,0)\\rbrace \\subseteq A : {q2_0}_c + \\dots + {q2_j}_c = q2_c$ and $\\lbrace del(p,{q3_0}_c,t_0,c,0), \\dots , $ $del(p,{q3_l}_c,t_l,c,0)\\rbrace \\subseteq A : {q3_0}_c + \\dots + {q3_l}_c = q3_c$ – rules $r\\ref {r:c:totincr},r\\ref {r:c:totdecr}$ and supported rule proposition, respectively Then $\\lbrace fires(t_0,0), \\dots , fires(t_j,0) \\rbrace \\subseteq A$ and $\\lbrace fires(t_0,0), \\dots , fires(t_l,0) \\rbrace \\\\ \\subseteq A$ – rules $r\\ref {r:c:dur:add},r\\ref {r:c:del}$ and supported rule proposition, respectively Then $\\lbrace fires(t_0,0), \\dots , $ $fires(t_j,0) \\rbrace \\cup \\lbrace fires(t_0,0), \\dots , $ $fires(t_l,0) \\rbrace \\subseteq $ $A = \\lbrace fires(t_0,0), \\dots , $ $fires(t_x,0) \\rbrace \\subseteq A$ – set union of subsets Then for each $fires(t_x,0) \\in A$ we have $t_x \\in T_0$ – already shown in item REF above Then $q_c = m_{M_0(p)}(c) + \\sum _{t_x \\in T_0, p \\in t_x \\bullet , 0+D(t_x)-1=0}{m_{W(t_x,p)}(c)} - \\\\ (\\sum _{t_x \\in T_0 \\wedge p \\in \\bullet t_x}{m_{W(p,t_x)}(c)} + $ $\\sum _{t_x \\in T_0 \\wedge p \\in R(t_x)}{m_{M_0(p)}(c)})$ – from (REF ) above and the following Each $add(p,{q_j}_c,t_j,c,0) \\in A$ represents ${q_j}_c=m_{W(t_j,p)}(c)$ for $p \\in t_j \\bullet $ – rule $r\\ref {r:c:dur:add},f\\ref {f:c:dur:tparc}$ encoding, and definition REF of transition execution in $PN$ Each $del(p,t_y,{q_y}_c,c,0) \\in A$ represents either ${q_y}_c=m_{W(p,t_y)}(c)$ for $p \\in \\bullet t_y$ , or ${q_y}_c=m_{M_0(p)}(c)$ for $p \\in R(t_y)$ – from rule $r\\ref {r:c:del},f\\ref {f:c:ptarc}$ encoding and definition REF of transition execution in $PN$ ; or from rule $r\\ref {r:c:del},f\\ref {f:c:rptarc}$ encoding and definition of reset arc in $PN$ Each $tot\\_incr(p,q2_c,c,0) \\in A$ represents $q2_c=\\sum _{t_x \\in T_0 \\wedge p \\in t_x \\bullet , 0+D(t_x)-1=0}{m_{W(t_x,p)}(c)}$ – aggregate assignment atom semantics in rule $r\\ref {r:c:totincr}$ Each $tot\\_decr(p,q3_c,c,0) \\in A$ represents $q3_c=\\sum _{t_x \\in T_0 \\wedge p \\in \\bullet t_x}{m_{W(p,t_x)}(c)} \\\\ + \\sum _{t_x \\in T_0 \\wedge p \\in R(t_x)}{m_{M_0(p)}(c)}$ – aggregate assignment atom semantics in rule $r\\ref {r:c:totdecr}$ Then, $m_{M_1(p)}(c) = q_c$ – since $holds(p,q_c,c,1) \\in A$ encodes $q_c=m_{M_1(p)}(c)$ – from construction Inductive Step: Let $k > 0$ , and $M_k$ is a valid marking in $X$ for $PN$ , show [(1)] $T_k$ is a valid firing set for $M_k$ , and firing $T_k$ in $M_k$ produces marking $M_{k+1}$ .", "We show that $T_k$ is a valid firing set in $M_k$ .", "Let $\\lbrace fires(t_0,k), \\dots , fires(t_x,k) \\rbrace $ be the set of all $fires(\\dots ,k)$ atoms in $A$ , We show that $T_k$ is a valid firing set in $M_k$ .", "Then for each $fires(t_i,k) \\in A$ $prenabled(t_i,k) \\in A$ – from rule $a\\ref {a:c:prfires}$ and supported rule proposition Then $enabled(t_i,k) \\in A$ – from rule $a\\ref {a:c:prenabled}$ and supported rule proposition And $notprenabled(t_i,k) \\notin A$ – from rule $a\\ref {a:c:prenabled}$ and supported rule proposition For $enabled(t_i,k) \\in A$ $notenabled(t_i,k) \\notin A$ – from rule $e\\ref {e:c:enabled}$ and supported rule proposition Then either of $body(e\\ref {e:c:ne:ptarc})$ , $body(e\\ref {e:c:ne:iptarc})$ , or $body(e\\ref {e:c:ne:tptarc})$ must not hold in $A$ for $t_i$ – from rules $body(e\\ref {e:c:ne:ptarc}), body(e\\ref {e:c:ne:iptarc}), body(e\\ref {e:c:ne:tptarc})$ and forced atom proposition Then $q_c \\lnot < {n_i}_c \\equiv q_c \\ge {n_i}_c$ in $e\\ref {e:c:ne:ptarc}$ for all $\\lbrace holds(p,q_c,c,k), \\\\ ptarc(p,t_i,{n_i}_c,c,k)\\rbrace \\subseteq A$ – from $e\\ref {e:c:ne:ptarc}$ , forced atom proposition, and given facts ($holds(p,q_c,c,k) \\in A, ptarc(p,t_i,{n_i}_c,k) \\in A$ ) And $q_c \\lnot \\ge {n_i}_c \\equiv q_c < {n_i}_c$ in $e\\ref {e:c:ne:iptarc}$ for all $\\lbrace holds(p,q_c,c,k), \\\\ iptarc(p,t_i,{n_i}_c,c,k) \\rbrace \\subseteq A, {n_i}_c=1$ ; $q_c > {n_i}_c \\equiv q_c = 0$ – from $e\\ref {e:c:ne:iptarc}$ , forced atom proposition, given facts ($holds(p,q_c,c,k) \\in A, \\\\ iptarc(p,t_i,1,c,k) \\in A$ ), and $q_c$ is a positive integer And $q_c \\lnot < {n_i}_c \\equiv q_c \\ge {n_i}_c$ in $e\\ref {e:c:ne:tptarc}$ for all $\\lbrace holds(p,q_c,c,k), \\\\ tptarc(p,t_i,{n_i}_c,c,k) \\rbrace \\subseteq A$ – from $e\\ref {e:c:ne:tptarc}$ , forced atom proposition, and given facts Then $\\forall c \\in C, (\\forall p \\in \\bullet t_i, m_{M_k(p)}(c) \\ge m_{W(p,t_i)}(c)) \\wedge (\\forall p \\in I(t_i), \\\\ m_{M_k(p)}(c) = 0) \\wedge (\\forall (p,t_i) \\in Q, m_{M_k(p)}(c) \\ge m_{QW(p,t_i)}(c))$ – from the inductive assumption, $f\\ref {f:c:ptarc},f\\ref {f:c:rptarc}$ construction, definition REF of preset $\\bullet t_i$ in $PN$ , definition REF of enabled transition in $PN$ , and that the construction of reset arcs by $f\\ref {f:c:rptarc}$ ensures $notenabled(t,k)$ is never true for a reset arc, where $holds(p,q_c,c,k) \\in A$ represents $q_c=m_{M_k(p)}(c)$ , $ptarc(p,t_i,{n_i}_c,k) \\in A$ represents ${n_i}_c=m_{W(p,t_i)}(c)$ , ${n_i}_c = m_{M_k(p)}(c)$ .", "Which is equivalent to $(\\forall p \\in \\bullet t_i, M_k(p) \\ge W(p,t_i)) \\wedge (\\forall p \\in I(t_i), M_k(p) = 0) \\wedge (\\forall (p,t_i) \\in Q, M_k(p) \\ge QW(p,t_i))$ – assuming multiset domain $C$ For $notprenabled(t_i,k) \\notin A$ Either $(\\nexists enabled(tt,k) \\in A : pp < p_i)$ or $(\\forall enabled(tt,k) \\in A : pp \\lnot < p_i)$ where $pp = Z(tt), p_i = Z(t_i)$ – from rule $a\\ref {a:c:prne}, f\\ref {f:c:pr}$ and forced atom proposition This matches the definition of an enabled priority transition Then $t_i$ is enabled and can fire in $PN$ , as a result it can belong to $T_k$ – from definition REF of enabled transition And $consumesmore \\notin A$ , since $A$ is an answer set of $\\Pi ^7$ – from rule $a\\ref {a:c:overc:elim}$ and supported rule proposition Then $\\nexists consumesmore(p,k) \\in A$ – from rule $a\\ref {a:c:overc:gen}$ and supported rule proposition Then $\\nexists \\lbrace holds(p,q_c,c,k), tot\\_decr(p,q1_c,c,k) \\rbrace \\subseteq A, q1_c>q_c$ in $body(a\\ref {a:c:overc:place})$ – from $a\\ref {a:c:overc:place}$ and forced atom proposition Then $\\nexists c \\in C \\nexists p \\in P, (\\sum _{t_i \\in \\lbrace t_0,\\dots ,t_x\\rbrace , p \\in \\bullet t_i}{m_{W(p,t_i)}(c)}+ \\\\ \\sum _{t_i \\in \\lbrace t_0,\\dots ,t_x\\rbrace , p \\in R(t_i)}{m_{M_k(p)}(c)}) > m_{M_k(p)}(c)$ – from the following $holds(p,q_c,c,k)$ represents $q_c=m_{M_k(p)}(c)$ – from rule $PN$ encoding, given $tot\\_decr(p,q1_c,c,k) \\in A$ if $\\lbrace del(p,{q1_0}_c,t_0,c,k), \\dots , \\\\ del(p,{q1_x}_c,t_x,c,k) \\rbrace \\subseteq A$ , where $q1_c = {q1_0}_c+\\dots +{q1_x}_c$ – from $r\\ref {r:c:totdecr}$ and forced atom proposition $del(p,{q1_i}_c,t_i,c,k) \\in A$ if $\\lbrace fires(t_i,k), ptarc(p,t_i,{q1_i}_c,c,k) \\rbrace \\subseteq A$ – from $r\\ref {r:c:del}$ and supported rule proposition $del(p,{q1_i}_c,t_i,c,k)$ either represents removal of ${q1_i}_c = m_{W(p,t_i)}(c)$ tokens from $p \\in \\bullet t_i$ ; or it represents removal of ${q1_i}_c = m_{M_k(p)}(c)$ tokens from $p \\in R(t_i)$ – from rules $r\\ref {r:c:del},f\\ref {f:c:ptarc},f\\ref {f:c:rptarc}$ , supported rule proposition, and definition REF of transition execution in $PN$ Then the set of transitions in $T_k$ do not conflict – by the definition REF of conflicting transitions And for each $prenabled(t_j,k) \\in A$ and $fires(t_j,k) \\notin A$ , $could\\_not\\_have(t_j,k) \\in A$ , since $A$ is an answer set of $\\Pi ^7$ - from rule $a\\ref {a:c:prmaxfire:elim}$ and supported rule proposition Then $\\lbrace prenabled(t_j,k), holds(s,qq_c,c,k), ptarc(s,t_j,q_c,c,k), \\\\ tot\\_decr(s,qqq_c,c,k) \\rbrace \\subseteq A$ , such that $q_c > qq_c - qqq_c$ and $fires(t_j,k) \\notin A$ - from rule $a\\ref {a:c:prmaxfire:cnh}$ and supported rule proposition Then for an $s \\in \\bullet t_j \\cup R(t_j)$ , $q_c > m_{M_k(s)}(c) - (\\sum _{t_i \\in T_k, s \\in \\bullet t_i}{m_{W(s,t_i)}(c)} + \\sum _{t_i \\in T_k, s \\in R(t_i)}{m_{M_k(s)}(c))}$ , where $q_c=m_{W(s,t_j)}(c) \\text{~if~} s \\in \\bullet t_j, \\text{~or~} m_{M_k(s)}(c) $ $\\text{~otherwise}$ .", "$ptarc(s,t_i,q_c,c,k)$ represents $q_c=m_{W(s,t_i)}(c)$ if $(s,t_i) \\in E^-$ or $q_c=m_{M_k(s)}(c)$ if $s \\in R(t_i)$ – from rule $f\\ref {f:c:ptarc},f\\ref {f:c:rptarc}$ construction $holds(s,qq_c,c,k)$ represents $qq_c=m_{M_k(s)}(c)$ – from $i\\ref {i:c:init}$ construction $tot\\_decr(s,qqq_c,c,k) \\in A$ if $\\lbrace del(s,{qqq_0}_c,t_0,c,k), \\dots , \\\\ del(s,{qqq_x}_c,t_x,c,k) \\rbrace \\subseteq A$ – from rule $r\\ref {r:c:totdecr}$ construction and supported rule proposition $del(s,{qqq_i}_c,t_i,c,k) \\in A$ if $\\lbrace fires(t_i,k), ptarc(s,t_i,{qqq_i}_c,c,k) \\rbrace \\subseteq A$ – from rule $r\\ref {r:c:del}$ and supported rule proposition $del(s,{qqq_i}_c,t_i,c,k)$ represents ${qqq_i}_c = m_{W(s,t_i)}(c) : t_i \\in T_k, (s,t_i) \\in E^-$ , or ${qqq_i}_c = m_{M_k(t_i)}(c) : t_i \\in T_k, s \\in R(t_i)$ – from rule $f\\ref {f:c:ptarc},f\\ref {f:c:rptarc}$ construction $tot\\_decr(q,qqq_c,c,k)$ represents $\\sum _{t_i \\in T_k, s \\in \\bullet t_i}{m_{W(s,t_i)}(c)} + \\\\ \\sum _{t_i \\in T_k, s \\in R(t_i)}{m_{M_k(s)}(c)}$ – from (C,D,E) above Then firing $T_k \\cup \\lbrace t_j \\rbrace $ would have required more tokens than are present at its source place $s \\in \\bullet t_j \\cup R(t_j)$ .", "Thus, $T_k$ is a maximal set of transitions that can simultaneously fire.", "And for each reset transition $t_r$ with $enabled(t_r,k) \\in A$ , $fires(t_r,k) \\in A$ , since $A$ is an answer set of $\\Pi ^7$ - from rule $f\\ref {f:c:pr:rptarc:elim}$ and supported rule proposition Then the firing set $T_k$ satisfies the reset transition requirement of definition REF (firing set) Then $\\lbrace t_0, \\dots , t_x\\rbrace = T_k$ – using 1(a),1(b), 1(d) above; and using 1(c) it is a maximal firing set We show that $M_{k+1}$ is produced by firing $T_k$ in $M_k$ .", "Let $holds(p,q_c,c,k+1) \\in A$ Then $\\lbrace holds(p,q1_c,c,k), tot\\_incr(p,q2_c,c,k), tot\\_decr(p,q3_c,c,k) \\rbrace \\subseteq A : q_c=q1_c+q2_c-q3_c$ – from rule $r\\ref {r:c:nextstate}$ and supported rule proposition Then $holds(p,q1_c,c,k) \\in A$ represents $q1_c=m_{M_k(p)}(c)$ – inductive assumption; and $\\lbrace add(p,{q2_0}_c,t_0,c,k), \\dots , $ $add(p,{q2_j}_c,t_j,c,k)\\rbrace \\subseteq A : {q2_0}_c + \\dots + {q2_j}_c = q2_c$ and $\\lbrace del(p,{q3_0}_c,t_0,c,k), \\dots , $ $del(p,{q3_l}_c,t_l,c,k)\\rbrace \\subseteq A : {q3_0}_c + \\dots + {q3_l}_c = q3_c$ – rules $r\\ref {r:c:totincr},r\\ref {r:c:totdecr}$ and supported rule proposition, respectively Then $\\lbrace fires(t_0,k), \\dots , fires(t_j,k) \\rbrace \\subseteq A$ and $\\lbrace fires(t_0,k), \\dots , fires(t_l,k) \\rbrace \\\\ \\subseteq A$ – rules $r\\ref {r:c:dur:add},r\\ref {r:c:del}$ and supported rule proposition, respectively Then $\\lbrace fires(t_0,k), \\dots , fires(t_j,k) \\rbrace \\cup \\lbrace fires(t_0,k), \\dots , fires(t_l,k) \\rbrace \\subseteq A = \\lbrace fires(t_0,k), \\dots , fires(t_x,k) \\rbrace \\subseteq A$ – set union of subsets Then for each $fires(t_x,k) \\in A$ we have $t_x \\in T_k$ – already shown in item REF above Then $q_c = m_{M_k(p)}(c) + \\sum _{t_x \\in T_l, p \\in t_x \\bullet , 0 \\le l \\le k, l+D(t_x)-1=k}{m_{W(t_x,p)}(c)} - \\\\ (\\sum _{t_x \\in T_k \\wedge p \\in \\bullet t_x}{m_{W(p,t_x)}(c)} + \\sum _{t_x \\in T_k \\wedge p \\in R(t_x)}{m_{M_k(p)}(c)})$ – from (REF ) above and the following Each $add(p,{q_j}_c,t_j,c,k) \\in A$ represents ${q_j}_c=m_{W(t_j,p)}(c)$ for $p \\in t_j \\bullet $ – rule $r\\ref {r:c:dur:add},f\\ref {f:c:dur:tparc}$ encoding, and definition REF of postset in $PN$ Each $del(p,t_y,{q_y}_c,c,k) \\in A$ represents either ${q_y}_c=m_{W(p,t_y)}(c)$ for $p \\in \\bullet t_y$ , or ${q_y}_c=m_{M_k(p)}(c)$ for $p \\in R(t_y)$ – from rule $r\\ref {r:c:del},f\\ref {f:c:ptarc}$ encoding and definition REF of preset in $PN$ ; or from rule $r\\ref {r:c:del},f\\ref {f:c:rptarc}$ encoding and definition of reset arc in $PN$ Each $tot\\_incr(p,q2_c,c,k) \\in A$ represents $q2_c=\\sum _{t_x \\in T_l, p \\in t_x \\bullet , 0 \\le l \\le k, l+D(t_x)-1=k}{m_{W(t_x,p)}(c)}$ – aggregate assignment atom semantics in rule $r\\ref {r:c:totincr}$ Each $tot\\_decr(p,q3_c,c,k) \\in A$ represents $q3_c=\\sum _{t_x \\in T_k \\wedge p \\in \\bullet t_x}{m_{W(p,t_x)}(c)} \\\\ + \\sum _{t_x \\in T_k \\wedge p \\in R(t_x)}{m_{M_k(p)}(c)}$ – aggregate assignment atom semantics in rule $r\\ref {r:c:totdecr}$ Then, $m_{M_{k+1}(p)}(c) = q_c$ – since $holds(p,q_c,c,k+1) \\in A$ encodes $q_c=m_{M_{k+1}(p)}(c)$ – from construction As a result, for any $n > k$ , $T_n$ will be a valid firing set for $M_n$ and $M_{n+1}$ will be its target marking.", "Conclusion: Since both (REF ) and (REF ) hold, $X=M_0,T_k,M_1,\\dots ,M_k,T_{k+1}$ is an execution sequence of $PN(P,T,E,C,W,R,I,Q,QW,Z,D)$ (w.r.t $M_0$ ) iff there is an answer set $A$ of $\\Pi ^7(PN,M_0,k,ntok)$ such that (REF ) and (REF ) hold.", "Complete Set of Queries Used for Drug-Drug InteractionDrug-Drug Interaction Queries Drug Activates Gene 1 //S{/NP{//?[Value='activation'](kw1)=>//?[Value='of'](kw2)=>//?[Tag='GENE'](kw0)=>//?[Value='by'](kw4)=>//?", "[Tag='DRUG'](kw3)}} ::: distinct sent.cid, kw3.value, kw1.value, kw0.value, sent.value 2 //NP{/NP{/?[Tag='DRUG'](kw3)=>/?[Value='activation'](kw1)}=>/PP{/?[Value='of'](kw2)=>//?", "[Tag='GENE'](kw0)}} ::: distinct sent.cid, kw3.value, kw1.value, kw0.value, sent.value 3 //NP{/?[Value='activation'](kw1)=>/PP{/?[Value='of'](kw2)=>//?[Tag='GENE'](kw0)=>//?[Value='by'](kw4)=>//?", "[Tag='DRUG'](kw3)}} ::: distinct sent.cid, kw3.value, kw1.value, kw0.value, sent.value 4 //S{/NP{//?[Tag='GENE'](kw2)}=>/VP{//?[Value='activation'](kw1)=>//?", "[Tag='DRUG'](kw0)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 5 //S{/NP{//?[Tag='DRUG'](kw0)}=>/VP{//?[Value='activation'](kw1)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 6 //NP{/NP{/?[Tag='DRUG'](kw0)=>/?[Value='activation'](kw1)}=>/PP{//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 7 //S{/NP{//?[Value='activation'](kw1)=>//?[Tag='GENE'](kw2)}=>/VP{//?", "[Tag='DRUG'](kw0)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 8 //S{/NP{//?[Value='activation'](kw1)=>//?[Value='of'](kw2)=>//?[Tag='GENE'](kw0)=>//?[Value='by'](kw4)=>//?", "[Tag='DRUG'](kw3)}} ::: distinct sent.cid, kw3.value, kw1.value, kw0.value, sent.value 9 //NP{/NP{/?[Tag='DRUG'](kw3)=>/?[Value='activation'](kw1)}=>/PP{/?[Value='of'](kw2)=>//?", "[Tag='GENE'](kw0)}} ::: distinct sent.cid, kw3.value, kw1.value, kw0.value, sent.value 10 //NP{/?[Value='activation'](kw1)=>/PP{/?[Value='of'](kw2)=>//?[Tag='GENE'](kw0)=>//?[Value='by'](kw4)=>//?", "[Tag='DRUG'](kw3)}} ::: distinct sent.cid, kw3.value, kw1.value, kw0.value, sent.value Gene Induces Gene 1 //S{/NP{//?[Tag='GENE'](kw0)}=>/VP{//?[Value='induced'](kw1)=>//?[Value='by'](kw3)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 2 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{//?[Value='induced'](kw1)=>//?[Value='by'](kw3)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 3 //S{/NP{//?[Tag='GENE'](kw0)}=>/VP{//?[Value IN {'increase','increased'}](kw1)=>//?[Tag='GENE'](kw2)=>//?", "[Value='activity'](kw3)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 4 //S{/NP{//?[Tag='GENE'](kw0)=>//?[Value='activity'](kw3)}=>/VP{//?[Value='increase'](kw1)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 5 //S{/NP{//?[Tag='GENE'](kw0)}=>/VP{//?[Value IN {'stimulates','stimulate', 'stimulated'}](kw1)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 6 //S{/NP{//?[Tag='GENE'](kw0)}=>/VP{/?[Value IN {'stimulates','stimulate'}](kw1)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 7 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/?[Value='stimulates'](kw1)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 8 //S{/NP{//?[Tag='GENE'](kw0)}=>/VP{/?[Value='activated'](kw1)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 9 //NP{/NP{/?[Value='induction'](kw1)}=>/PP{/?[Value='of'](kw3)=>/NP{//?[Tag='GENE'](kw0)=>//?[Value='by'](kw4)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 10 //NP{/NP{/?[Value='stimulation'](kw1)}=>/PP{/?[Value='of'](kw3)=>/NP{//?[Tag='GENE'](kw0)=>//?[Value='by'](kw4)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 11 //NP{/NP{/?[Value='activation'](kw1)}=>/PP{/?[Value IN {'of'}](kw3)=>/NP{//?[Tag='GENE'](kw0)=>//?[Value='by'](kw4)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 12 //S{/NP{//?[Tag='GENE'](kw0)}=>/VP{//?[Value='inducible'](kw1)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 13 //S{/NP{//?[Tag='GENE'](kw0)}=>/VP{//?[Value='induced'](kw1)=>//?[Value='by'](kw3)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 14 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{//?[Value='induced'](kw1)=>//?[Value='by'](kw3)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 15 //S{/NP{//?[Tag='GENE'](kw0)}=>/VP{//?[Value IN {'increase','increased'}](kw1)=>//?[Tag='GENE'](kw2)=>//?", "[Value='activity'](kw3)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 16 //S{/NP{//?[Tag='GENE'](kw0)=>//?[Value='activity'](kw3)}=>/VP{//?[Value='increase'](kw1)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 17 //S{/NP{//?[Tag='GENE'](kw0)}=>/VP{//?[Value='stimulated'](kw1)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 18 //S{/NP{//?[Tag='GENE'](kw0)}=>/VP{/?[Value IN {'stimulates','stimulate'}](kw1)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 19 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/?[Value='stimulates'](kw1)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 20 //S{/NP{//?[Tag='GENE'](kw0)}=>/VP{/?[Value='activated'](kw1)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 21 //VP{/?[Value='activated'](kw1)=>/PP{//?[Tag='GENE'](kw0)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 22 //NP{/NP{/?[Value='induction'](kw1)}=>/PP{/?[Value='of'](kw3)=>/NP{//?[Tag='GENE'](kw0)=>//?[Value='by'](kw4)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 23 //NP{/NP{/?[Value='stimulation'](kw1)}=>/PP{/?[Value='of'](kw3)=>/NP{//?[Tag='GENE'](kw0)=>//?[Value='by'](kw4)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 24 //NP{/NP{/?[Value='activation'](kw1)}=>/PP{/?[Value IN {'of'}](kw3)=>/NP{//?[Tag='GENE'](kw0)=>//?[Value='by'](kw4)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 25 //S{/NP{//?[Tag='GENE'](kw0)}=>/VP{//?[Value='inducible'](kw1)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value Gene Inhibits Gene 1 //S{/NP{//?[Tag='GENE'](kw0)}=>/VP{//?[Value='induced'](kw1)=>//?[Value='by'](kw3)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 2 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{//?[Value='induced'](kw1)=>//?[Value='by'](kw3)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 3 //S{/NP{//?[Tag='GENE'](kw0)}=>/VP{//?[Value IN {'increase','increased'}](kw1)=>//?[Tag='GENE'](kw2)=>//?", "[Value='activity'](kw3)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 4 //S{/NP{//?[Tag='GENE'](kw0)=>//?[Value='activity'](kw3)}=>/VP{//?[Value='increase'](kw1)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 5 //S{/NP{//?[Tag='GENE'](kw0)}=>/VP{//?[Value IN {'stimulates','stimulate', 'stimulated'}](kw1)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 6 //S{/NP{//?[Tag='GENE'](kw0)}=>/VP{/?[Value IN {'stimulates','stimulate'}](kw1)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 7 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/?[Value='stimulates'](kw1)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 8 //S{/NP{//?[Tag='GENE'](kw0)}=>/VP{/?[Value='activated'](kw1)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 9 //NP{/NP{/?[Value='induction'](kw1)}=>/PP{/?[Value='of'](kw3)=>/NP{//?[Tag='GENE'](kw0)=>//?[Value='by'](kw4)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 10 //NP{/NP{/?[Value='stimulation'](kw1)}=>/PP{/?[Value='of'](kw3)=>/NP{//?[Tag='GENE'](kw0)=>//?[Value='by'](kw4)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 11 //NP{/NP{/?[Value='activation'](kw1)}=>/PP{/?[Value IN {'of'}](kw3)=>/NP{//?[Tag='GENE'](kw0)=>//?[Value='by'](kw4)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 12 //S{/NP{//?[Tag='GENE'](kw0)}=>/VP{//?[Value='inducible'](kw1)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value Drug Changes Gene Expression/Activity 1 //S{/?[Tag='DRUG'](kw2)=>/?[Value IN {'increased','increase','increases'}](kw1)=>/?[Value IN {'levels','level'}](kw3)=>/?", "[Tag='GENE'](kw0)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 2 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/?[Value IN {'increased','increases'}](kw1)=>/NP{//?[Tag='GENE'](kw0)=>//?", "[Value IN {'expression','level','activity','activities','levels'}](kw3)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 3 //S{/?[Tag='DRUG'](kw2)=>/VP{/NP{//?[Value IN {'increases','increase'}](kw1)=>//?[Tag='GENE'](kw0)=>//?", "[Value IN {'levels','activity'}](kw3)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 4 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value IN {'increased','increases'}](kw1)=>/NP{//?[Tag='GENE'](kw0)=>//?", "[Value IN {'activity','activities','levels'}](kw3)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 5 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/?[Value='increased'](kw1)=>/NP{//?[Value IN {'activity','levels','expression'}](kw3)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 6 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{//?[Value IN {'increased','increases','increase'}](kw1)=>//?[Tag='GENE'](kw0)=>//?", "[Value IN {'activity','levels'}](kw3)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 7 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{/?[Value='increased'](kw1)=>//?[Tag='GENE'](kw0)=>//?", "[Value IN {'expression','activity'}](kw3)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 8 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value IN {'increased','increases'}](kw1)=>/NP{//?[Value IN {'expression','activity','activities','level','levels'}](kw3)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 9 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value IN {'decreased','decreases'}](kw1)=>/NP{//?[Tag='GENE'](kw0)=>//?", "[Value IN {'activity','expression'}](kw3)}}} ::: distinct sent.cid, kw2.value, kw1,value, kw0.value, sent.value 10 //S{/?[Tag='DRUG'](kw2)=>/?[Value IN {'increased','increase','increases'}](kw1)=>/?[Value IN {'levels','level'}](kw3)=>/?", "[Tag='GENE'](kw0)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 11 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/?[Value IN {'increased','increases'}](kw1)=>/NP{//?[Tag='GENE'](kw0)=>//?", "[Value IN {'expression','level','activity','activities','levels'}](kw3)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 12 //S{/?[Tag='DRUG'](kw2)=>/VP{/NP{//?[Value IN {'increases','increase'}](kw1)=>//?[Tag='GENE'](kw0)=>//?", "[Value IN {'levels','activity'}](kw3)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 13 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value IN {'increased','increases'}](kw1)=>/NP{//?[Tag='GENE'](kw0)=>//?", "[Value IN {'activity','activities','levels'}](kw3)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 14 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/?[Value='increased'](kw1)=>/NP{//?[Value IN {'activity','levels','expression'}](kw3)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 15 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{//?[Value IN {'increased','increases','increase'}](kw1)=>//?[Tag='GENE'](kw0)=>//?", "[Value IN {'activity','levels'}](kw3)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 16 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{/?[Value='increased'](kw1)=>//?[Tag='GENE'](kw0)=>//?", "[Value IN {'expression','activity'}](kw3)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 17 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value IN {'increased','increases'}](kw1)=>/NP{//?[Value IN {'expression','activity','activities','level','levels'}](kw3)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 18 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value IN {'decreased','decreases'}](kw1)=>/NP{//?[Tag='GENE'](kw0)=>//?", "[Value IN {'activity','expression'}](kw3)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value Drug Induces/Stimulates Gene 1 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value IN {'stimulated','induced'}](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 2 //VP{/?[Value IN {'stimulated','induced'}](kw1)=>/PP{/NP{//?[Tag='DRUG'](kw2)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 3 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/?[Value IN {'stimulated','induced'}](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 4 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/?[Value IN {'stimulated','induced'}](kw1)=>/PP{//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 5 //S{/?[Tag='DRUG'](kw2)=>/VP{/NP{//?[Tag='GENE'](kw0)=>//?", "[Value IN {'stimulated','induced'}](kw1)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 6 //S{/NP{/PP{//?[Tag='DRUG'](kw2)}}=>/VP{/?[Value IN {'stimulated','induced'}](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 7 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/VP{/?[Value IN {'stimulated','induced'}](kw1)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 8 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/VP{//?[Value='induced'](kw1)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 9 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value IN {'stimulated','induced'}](kw1)=>/NP{/?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 10 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{//?[Value IN {'stimulate','induce'}](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 11 //NP{/NP{/?[Value IN {'induction','stimulation'}](kw1)}=>/PP{/NP{//?[Tag='GENE'](kw0)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 12 //S{/NP{/?[Value IN {'stimulation','induction'}](kw1)=>/PP{//?[Tag='GENE'](kw0)}}=>/VP{/VP{//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 13 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/NP{//?[Value IN {'stimulation','induction'}](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 14 //NP{/NP{/NP{/?[Value='induction'](kw1)}=>/PP{//?[Tag='GENE'](kw0)}}=>/PP{/NP{/?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 15 //NP{/NP{/?[Tag='GENE'](kw0)=>/?[Value IN {'stimulation','induction'}](kw1)}=>/PP{/NP{/?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 16 //NP{/?[Value IN {'stimulation','induction'}](kw1)=>/PP{/NP{//?[Tag='GENE'](kw0)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 17 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/?[Value IN {'stimulates','induces'}](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 18 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value IN {'stimulates','induces'}](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 19 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value IN {'stimulated','induced'}](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 20 //VP{/?[Value IN {'stimulated','induced'}](kw1)=>/PP{/NP{//?[Tag='DRUG'](kw2)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 21 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/?[Value IN {'stimulated','induced'}](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 22 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/?[Value IN {'stimulated','induced'}](kw1)=>/PP{//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 23 //S{/?[Tag='DRUG'](kw2)=>/VP{/NP{//?[Tag='GENE'](kw0)=>//?", "[Value IN {'stimulated','induced'}](kw1)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 24 //S{/NP{/PP{//?[Tag='DRUG'](kw2)}}=>/VP{/?[Value IN {'stimulated','induced'}](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 25 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/VP{/?[Value IN {'stimulated','induced'}](kw1)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 26 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/VP{//?[Value='induced'](kw1)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 27 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value IN {'stimulated','induced'}](kw1)=>/NP{/?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 28 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{//?[Value IN {'stimulate','induce'}](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 29 //NP{/NP{/?[Value IN {'induction','stimulation'}](kw1)}=>/PP{/NP{//?[Tag='GENE'](kw0)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 30 //S{/NP{/?[Value IN {'stimulation','induction'}](kw1)=>/PP{//?[Tag='GENE'](kw0)}}=>/VP{/VP{//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 31 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/NP{//?[Value IN {'stimulation','induction'}](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 32 //NP{/NP{/NP{/?[Value='induction'](kw1)}=>/PP{//?[Tag='GENE'](kw0)}}=>/PP{/NP{/?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 33 //NP{/NP{/?[Tag='GENE'](kw0)=>/?[Value IN {'stimulation','induction'}](kw1)}=>/PP{/NP{/?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 34 //NP{/?[Value IN {'stimulation','induction'}](kw1)=>/PP{/NP{//?[Tag='GENE'](kw0)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 35 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/?[Value IN {'stimulates','induces'}](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 36 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value IN {'stimulates','induces'}](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value Drug Inhibits Gene 1 //NP{/NP{/NP{/?[Value='inhibitory'](kw1)}=>/PP{//?[Tag='DRUG'](kw2)}}=>/PP{/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 2 //S{/?[Tag='DRUG'](kw2)=>/VP{/NP{/?[Tag='GENE'](kw0)=>/?", "[Value='inhibitor'](kw1)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 3 //NP{/?[Tag='DRUG'](kw2)=>/NP{/?[Tag='GENE'](kw0)=>/?", "[Value='inhibitor'](kw1)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 4 //NP{/NP{/?[Tag='GENE'](kw0)=>/?[Value='inhibitor'](kw1)}=>/?", "[Tag='DRUG'](kw2)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 5 //NP{/NP{/NP{/?[Tag='GENE'](kw0)=>/?[Tag='DRUG'](kw2)}}=>/?", "[Value='inhibitor'](kw1)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 6 //NP{/?[Tag='GENE'](kw0)=>/?[Value='inhibitor'](kw1)=>/?", "[Tag='DRUG'](kw2)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 7 //NP{/NP{/?[Value='inhibitor'](kw1)}=>/PP{/NP{//?[Tag='GENE'](kw0)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 8 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{//?[Tag='GENE'](kw0)=>//?", "[Value='inhibitor'](kw1)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 9 //S{/?[Tag='DRUG'](kw2)=>/VP{/NP{//?[Value='inhibitor'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 10 //NP{/?[Tag='DRUG'](kw2)=>/NP{/NP{/?[Tag='GENE'](kw0)}}=>/?", "[Value='inhibitor'](kw1)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 11 //S{/?[Tag='DRUG'](kw2)=>/?[Value='inhibitor'](kw1)=>/?", "[Tag='GENE'](kw0)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 12 //NP{/NP{/?[Tag='DRUG'](kw2)=>/NP{/?[Value='inhibitor'](kw1)}=>/PP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 13 //S{/?[Tag='GENE'](kw0)=>/?[Value='inhibitor'](kw1)=>/?", "[Tag='DRUG'](kw2)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 14 //NP{/?[Tag='DRUG'](kw2)=>/NP{/NP{/?[Value='inhibitor'](kw1)}=>/PP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 15 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{//?[Value='inhibitor'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 16 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/NP{//?[Value='inhibitor'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 17 //NP{/?[Tag='DRUG'](kw2)=>/NP{/NP{/?[Tag='GENE'](kw0)=>/?", "[Value='inhibitor'](kw1)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 18 //NP{/NP{/PP{//?[Tag='DRUG'](kw2)=>//?[Tag='GENE'](kw0)}=>/NP{/?", "[Value='inhibitor'](kw1)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 19 //NP{/NP{/?[Tag='GENE'](kw0)=>/?[Value='inhibitor'](kw1)=>/?", "[Tag='DRUG'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 20 //S{/?[Tag='DRUG'](kw2)=>/?[Tag='GENE'](kw0)=>/?", "[Value='inhibitor'](kw1)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 21 //NP{/NP{/?[Tag='DRUG'](kw2)=>/?[Tag='GENE'](kw0)=>/?", "[Value='inhibitor'](kw1)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 22 //NP{/NP{/?[Tag='DRUG'](kw2)=>/NP{/?[Tag='GENE'](kw0)=>/?", "[Value='inhibitor'](kw1)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 23 //NP{/NP{/PP{//?[Tag='DRUG'](kw2)}=>/?[Tag='GENE'](kw0)=>/?", "[Value='inhibitor'](kw1)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 24 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value='inhibited'](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 25 //S{/NP{/NP{/?[Tag='DRUG'](kw2)}}=>/VP{/?[Value='inhibited'](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 26 //S{/?[Tag='DRUG'](kw2)=>/?[Value='inhibited'](kw1)=>/?", "[Tag='GENE'](kw0)} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 27 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/?[Value='inhibited'](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 28 //S{/NP{/NP{//?[Tag='DRUG'](kw2)}}=>/VP{/?[Value='inhibited'](kw1)=>/NP{/?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 29 //S{/NP{/PP{//?[Tag='GENE'](kw0)}}=>/VP{/VP{/?[Value='inhibited'](kw1)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 30 //S{/NP{/NP{//?[Tag='DRUG'](kw2)}}=>/VP{/?[Value='inhibited'](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 31 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{/?[Value='inhibited'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 32 //S{/NP{/PP{//?[Tag='DRUG'](kw2)}}=>/VP{/?[Value='inhibited'](kw1)=>/NP{/?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 33 //S{/NP{/NP{/?[Tag='GENE'](kw0)}}=>/VP{/VP{/?[Value='inhibited'](kw1)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 34 //S{/NP{/NP{/?[Tag='GENE'](kw0)}}=>/VP{/?[Value='inhibited'](kw1)=>/PP{//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 35 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value='inhibited'](kw1)=>/PP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 36 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/VP{/?[Value='inhibited'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 37 //S{/NP{/NP{/?[Tag='DRUG'](kw2)}}=>/VP{/?[Value='inhibited'](kw1)=>/NP{/?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 38 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/VP{/?[Value='inhibited'](kw1)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 39 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/?[Value='inhibited'](kw1)=>/NP{/?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 40 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/VP{//?[Value='inhibited'](kw1)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 41 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value='inhibited'](kw1)=>/NP{/?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 42 //S{/NP{/NP{/?[Tag='GENE'](kw0)=>/?[Tag='DRUG'](kw2)}}=>/VP{/?", "[Value='inhibit'](kw1)}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 43 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/VP{//?[Value='inhibit'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 44 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{//?[Value='inhibit'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 45 //NP{/NP{/?[Tag='GENE'](kw0)=>/?[Value='inhibition'](kw1)}=>/PP{/NP{/?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 46 //S{/?[Tag='DRUG'](kw2)=>/VP{/PP{//?[Value='inhibition'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 47 //S{/NP{/PP{//?[Tag='DRUG'](kw2)}}=>/VP{/NP{//?[Value='inhibition'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 48 //NP{/?[Value='inhibition'](kw1)=>/PP{/NP{//?[Tag='GENE'](kw0)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 49 //S{/?[Value='inhibition'](kw1)=>/PP{/NP{//?[Tag='GENE'](kw0)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 50 //NP{/NP{/?[Tag='DRUG'](kw2)=>/?[Value='inhibition'](kw1)}=>/PP{/NP{/?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 51 //NP{/NP{/?[Value='inhibition'](kw1)}=>/PP{/NP{/?[Tag='GENE'](kw0)=>/?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 52 //S{/NP{/?[Value='inhibition'](kw1)=>/PP{//?[Tag='GENE'](kw0)}}=>/VP{/PP{//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 53 //S{/NP{/NP{/?[Value='inhibition'](kw1)}=>/PP{//?[Tag='GENE'](kw0)}}=>/VP{/NP{//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 54 //S{/?[Value IN {'inhibition','Inhibition'}](kw1)=>/?[Tag='GENE'](kw0)=>/?", "[Tag='DRUG'](kw2)} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 55 //NP{/NP{/NP{/?[Value='inhibition'](kw1)}=>/PP{//?[Tag='GENE'](kw0)}}=>/PP{/NP{//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 56 //NP{/?[Tag='DRUG'](kw2)=>/NP{/?[Value='inhibition'](kw1)}=>/PP{/NP{/?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 57 //NP{/NP{/?[Value='inhibition'](kw1)}=>/PP{/NP{//?[Tag='GENE'](kw0)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 58 //S{/?[Tag='DRUG'](kw2)=>/VP{/NP{//?[Value='inhibition'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 59 //S{/NP{/NP{/?[Value='inhibition'](kw1)}=>/PP{//?[Tag='DRUG'](kw2)}}=>/VP{/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 60 //NP{/NP{/?[Value='inhibition'](kw1)=>/PP{//?[Tag='GENE'](kw0)}}=>/PP{/NP{//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 61 //NP{/NP{/?[Tag='DRUG'](kw2)=>/?[Value='inhibition'](kw1)}=>/PP{/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 62 //NP{/NP{/?[Value='inhibition'](kw1)=>/PP{//?[Tag='GENE'](kw0)}}=>/PP{/NP{/?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 63 //NP{/NP{/NP{/?[Value='inhibition'](kw1)}=>/PP{//?[Tag='GENE'](kw0)}}=>/PP{/NP{/?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 64 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/?[Value='inhibits'](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 65 /S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value='inhibits'](kw1)=>/NP{/?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 66 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{/?[Value='inhibits'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 67 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value='inhibits'](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 68 //NP{/NP{/NP{/?[Value='inhibitory'](kw1)}=>/PP{//?[Tag='DRUG'](kw2)}}=>/PP{/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 69 //S{/?[Tag='DRUG'](kw2)=>/VP{/NP{/?[Tag='GENE'](kw0)=>/?", "[Value='inhibitor'](kw1)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 70 //NP{/?[Tag='DRUG'](kw2)=>/NP{/?[Tag='GENE'](kw0)=>/?", "[Value='inhibitor'](kw1)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 71 //NP{/NP{/?[Tag='GENE'](kw0)=>/?[Value='inhibitor'](kw1)}=>/?", "[Tag='DRUG'](kw2)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 72 //NP{/NP{/NP{/?[Tag='GENE'](kw0)=>/?[Tag='DRUG'](kw2)}}=>/?", "[Value='inhibitor'](kw1)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 73 //NP{/?[Tag='GENE'](kw0)=>/?[Value='inhibitor'](kw1)=>/?", "[Tag='DRUG'](kw2)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 74 //NP{/NP{/?[Value='inhibitor'](kw1)}=>/PP{/NP{//?[Tag='GENE'](kw0)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 75 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{//?[Tag='GENE'](kw0)=>//?", "[Value='inhibitor'](kw1)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 76 //S{/?[Tag='DRUG'](kw2)=>/VP{/NP{//?[Value='inhibitor'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 77 //NP{/?[Tag='DRUG'](kw2)=>/NP{/NP{/?[Tag='GENE'](kw0)}}=>/?", "[Value='inhibitor'](kw1)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 78 //S{/?[Tag='DRUG'](kw2)=>/?[Value='inhibitor'](kw1)=>/?", "[Tag='GENE'](kw0)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 79 //NP{/NP{/?[Tag='DRUG'](kw2)=>/NP{/?[Value='inhibitor'](kw1)}=>/PP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 80 //S{/?[Tag='GENE'](kw0)=>/?[Value='inhibitor'](kw1)=>/?", "[Tag='DRUG'](kw2)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 81 //NP{/?[Tag='DRUG'](kw2)=>/NP{/NP{/?[Value='inhibitor'](kw1)}=>/PP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 82 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{//?[Value='inhibitor'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 83 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/NP{//?[Value='inhibitor'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 84 //NP{/?[Tag='DRUG'](kw2)=>/NP{/NP{/?[Tag='GENE'](kw0)=>/?", "[Value='inhibitor'](kw1)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 85 //NP{/NP{/PP{//?[Tag='DRUG'](kw2)=>//?[Tag='GENE'](kw0)}=>/NP{/?", "[Value='inhibitor'](kw1)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 86 //NP{/NP{/?[Tag='GENE'](kw0)=>/?[Value='inhibitor'](kw1)=>/?", "[Tag='DRUG'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 87 //S{/?[Tag='DRUG'](kw2)=>/?[Tag='GENE'](kw0)=>/?", "[Value='inhibitor'](kw1)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 88 //NP{/NP{/?[Tag='DRUG'](kw2)=>/?[Tag='GENE'](kw0)=>/?", "[Value='inhibitor'](kw1)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 89 //NP{/NP{/?[Tag='DRUG'](kw2)=>/NP{/?[Tag='GENE'](kw0)=>/?", "[Value='inhibitor'](kw1)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 90 //NP{/NP{/PP{//?[Tag='DRUG'](kw2)}=>/?[Tag='GENE'](kw0)=>/?", "[Value='inhibitor'](kw1)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 91 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value='inhibited'](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 92 //S{/NP{/NP{/?[Tag='DRUG'](kw2)}}=>/VP{/?[Value='inhibited'](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 93 //S{/?[Tag='DRUG'](kw2)=>/?[Value='inhibited'](kw1)=>/?", "[Tag='GENE'](kw0)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 94 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/?[Value='inhibited'](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 95 //S{/NP{/NP{//?[Tag='DRUG'](kw2)}}=>/VP{/?[Value='inhibited'](kw1)=>/NP{/?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 96 //S{/NP{/PP{//?[Tag='GENE'](kw0)}}=>/VP{/VP{/?[Value='inhibited'](kw1)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 97 //S{/NP{/NP{//?[Tag='DRUG'](kw2)}}=>/VP{/?[Value='inhibited'](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 98 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{/?[Value='inhibited'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 99 //S{/NP{/PP{//?[Tag='DRUG'](kw2)}}=>/VP{/?[Value='inhibited'](kw1)=>/NP{/?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 100 //S{/NP{/NP{/?[Tag='GENE'](kw0)}}=>/VP{/VP{/?[Value='inhibited'](kw1)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 101 //S{/NP{/NP{/?[Tag='GENE'](kw0)}}=>/VP{/?[Value='inhibited'](kw1)=>/PP{//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 102 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value='inhibited'](kw1)=>/PP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 103 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/VP{/?[Value='inhibited'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 104 //S{/NP{/NP{/?[Tag='DRUG'](kw2)}}=>/VP{/?[Value='inhibited'](kw1)=>/NP{/?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 105 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/VP{/?[Value='inhibited'](kw1)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 106 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/?[Value='inhibited'](kw1)=>/NP{/?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 107 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/VP{//?[Value='inhibited'](kw1)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 108 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value='inhibited'](kw1)=>/NP{/?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 109 //S{/NP{/NP{/?[Tag='GENE'](kw0)=>/?[Tag='DRUG'](kw2)}}=>/VP{/?", "[Value='inhibit'](kw1)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 110 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/VP{//?[Value='inhibit'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 111 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{//?[Value='inhibit'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 112 //NP{/NP{/?[Tag='GENE'](kw0)=>/?[Value='inhibition'](kw1)}=>/PP{/NP{/?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 113 //S{/?[Tag='DRUG'](kw2)=>/VP{/PP{//?[Value='inhibition'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 114 //S{/NP{/PP{//?[Tag='DRUG'](kw2)}}=>/VP{/NP{//?[Value='inhibition'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 115 //NP{/?[Value='inhibition'](kw1)=>/PP{/NP{//?[Tag='GENE'](kw0)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 116 //S{/?[Value='inhibition'](kw1)=>/PP{/NP{//?[Tag='GENE'](kw0)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 117 //NP{/NP{/?[Tag='DRUG'](kw2)=>/?[Value='inhibition'](kw1)}=>/PP{/NP{/?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 118 //NP{/NP{/?[Value='inhibition'](kw1)}=>/PP{/NP{/?[Tag='GENE'](kw0)=>/?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 119 //S{/NP{/?[Value='inhibition'](kw1)=>/PP{//?[Tag='GENE'](kw0)}}=>/VP{/PP{//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 120 //S{/NP{/NP{/?[Value='inhibition'](kw1)}=>/PP{//?[Tag='GENE'](kw0)}}=>/VP{/NP{//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 121 //S{/?[Value IN {'inhibition','Inhibition'}](kw1)=>/?[Tag='GENE'](kw0)=>/?", "[Tag='DRUG'](kw2)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 122 //NP{/NP{/NP{/?[Value='inhibition'](kw1)}=>/PP{//?[Tag='GENE'](kw0)}}=>/PP{/NP{//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 123 //NP{/?[Tag='DRUG'](kw2)=>/NP{/?[Value='inhibition'](kw1)}=>/PP{/NP{/?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 124 //NP{/NP{/?[Value='inhibition'](kw1)}=>/PP{/NP{//?[Tag='GENE'](kw0)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 125 //S{/?[Tag='DRUG'](kw2)=>/VP{/NP{//?[Value='inhibition'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 126 //S{/NP{/NP{/?[Value='inhibition'](kw1)}=>/PP{//?[Tag='DRUG'](kw2)}}=>/VP{/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 127 //NP{/NP{/?[Value='inhibition'](kw1)=>/PP{//?[Tag='GENE'](kw0)}}=>/PP{/NP{//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 128 //NP{/NP{/?[Tag='DRUG'](kw2)=>/?[Value='inhibition'](kw1)}=>/PP{/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 129 //NP{/NP{/?[Value='inhibition'](kw1)=>/PP{//?[Tag='GENE'](kw0)}}=>/PP{/NP{/?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 130 //NP{/NP{/NP{/?[Value='inhibition'](kw1)}=>/PP{//?[Tag='GENE'](kw0)}}=>/PP{/NP{/?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 131 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/?[Value='inhibits'](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 132 /S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value='inhibits'](kw1)=>/NP{/?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 133 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{/?[Value='inhibits'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 134 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value='inhibits'](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value Gene Metabolized Drug 1 //S{/NP{/NP{/?[Tag='DRUG'](kw2)}}=>/VP{/VP{/?[Value IN {'metabolised','metabolized'}](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 2 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{//?[Value IN {'metabolised','metabolized'}](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 3 //S{/NP{/SBAR{//?[Tag='DRUG'](kw2)}}=>/VP{/?[Value IN {'metabolised','metabolized'}](kw1)=>/PP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 4 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/VP{/?[Value IN {'metabolised','metabolized'}](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 5 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/VP{//?[Value IN {'metabolized','metabolised'}](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 6 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{/?[Value IN {'metabolised','metabolized'}](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 7 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value='metabolized'](kw1)=>/PP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 8 //S{/?[Tag='DRUG'](kw2)=>/NP{/?[Value='metabolism'](kw1)}=>/VP{/VP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 9 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/NP{//?[Tag='DRUG'](kw2)=>//?", "[Value='metabolism'](kw1)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 10 //NP{/NP{/?[Tag='GENE'](kw0)=>/?[Value='metabolism'](kw1)}=>/PP{/NP{/?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 11 //NP{/?[Value='metabolism'](kw1)=>/PP{/NP{//?[Tag='DRUG'](kw2)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 12 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/VP{//?[Value='metabolism'](kw1)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 13 //S{/NP{/NP{/?[Tag='GENE'](kw0)}}=>/VP{/NP{//?[Value='metabolism'](kw1)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 14 //NP{/NP{/PP{//?[Tag='GENE'](kw0)}}=>/PP{/NP{/?[Tag='DRUG'](kw2)=>/?", "[Value='metabolism'](kw1)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 15 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/PP{//?[Value='metabolism'](kw1)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 16 //NP{/NP{/?[Tag='DRUG'](kw2)=>/?[Value='metabolism'](kw1)}=>/PP{/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 17 //NP{/NP{/?[Tag='GENE'](kw0)=>/?[Value='metabolism'](kw1)}=>/PP{/NP{//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 18 //S{/NP{/NP{/?[Value='metabolism'](kw1)}=>/PP{//?[Tag='DRUG'](kw2)}}=>/VP{/VP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 19 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/PP{//?[Tag='DRUG'](kw2)=>//?", "[Value='metabolism'](kw1)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 20 //NP{/NP{/PP{//?[Tag='GENE'](kw0)}}=>/PP{/NP{//?[Value='metabolism'](kw1)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 21 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/NP{//?[Value='substrate'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 22 //NP{/NP{/?[Tag='DRUG'](kw2)=>/NP{/?[Value='substrate'](kw1)}=>/PP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 23 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{//?[Value='substrate'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 24 //NP{/?[Tag='GENE'](kw0)=>/?[Value='substrate'](kw1)=>/?", "[Tag='DRUG'](kw2)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 25 //NP{/?[Tag='DRUG'](kw2)=>/NP{/NP{/?[Value='substrate'](kw1)}=>/PP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 26 //S{/NP{/NP{/?[Tag='DRUG'](kw2)}}=>/VP{/VP{/?[Value IN {'metabolised','metabolized'}](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 27 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{//?[Value IN {'metabolised','metabolized'}](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 28 //S{/NP{/SBAR{//?[Tag='DRUG'](kw2)}}=>/VP{/?[Value IN {'metabolised','metabolized'}](kw1)=>/PP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 29 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/VP{/?[Value IN {'metabolised','metabolized'}](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 30 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/VP{//?[Value IN {'metabolized','metabolised'}](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 31 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{/?[Value IN {'metabolised','metabolized'}](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 32 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value='metabolized'](kw1)=>/PP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 33 //S{/?[Tag='DRUG'](kw2)=>/NP{/?[Value='metabolism'](kw1)}=>/VP{/VP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 34 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/NP{//?[Tag='DRUG'](kw2)=>//?", "[Value='metabolism'](kw1)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 35 //NP{/NP{/?[Tag='GENE'](kw0)=>/?[Value='metabolism'](kw1)}=>/PP{/NP{/?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 36 //NP{/?[Value='metabolism'](kw1)=>/PP{/NP{//?[Tag='DRUG'](kw2)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 37 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/VP{//?[Value='metabolism'](kw1)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 38 //S{/NP{/NP{/?[Tag='GENE'](kw0)}}=>/VP{/NP{//?[Value='metabolism'](kw1)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 39 //NP{/NP{/PP{//?[Tag='GENE'](kw0)}}=>/PP{/NP{/?[Tag='DRUG'](kw2)=>/?", "[Value='metabolism'](kw1)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 40 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/PP{//?[Value='metabolism'](kw1)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 41 //NP{/NP{/?[Tag='DRUG'](kw2)=>/?[Value='metabolism'](kw1)}=>/PP{/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 42 //NP{/NP{/?[Tag='GENE'](kw0)=>/?[Value='metabolism'](kw1)}=>/PP{/NP{//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 43 //S{/NP{/NP{/?[Value='metabolism'](kw1)}=>/PP{//?[Tag='DRUG'](kw2)}}=>/VP{/VP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 44 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/PP{//?[Tag='DRUG'](kw2)=>//?", "[Value='metabolism'](kw1)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 45 //NP{/NP{/PP{//?[Tag='GENE'](kw0)}}=>/PP{/NP{//?[Value='metabolism'](kw1)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 46 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/NP{//?[Value='substrate'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 47 //NP{/NP{/?[Tag='DRUG'](kw2)=>/NP{/?[Value='substrate'](kw1)}=>/PP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 48 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{//?[Value='substrate'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 49 //NP{/?[Tag='GENE'](kw0)=>/?[Value='substrate'](kw1)=>/?", "[Tag='DRUG'](kw2)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 50 //NP{/?[Tag='DRUG'](kw2)=>/NP{/NP{/?[Value='substrate'](kw1)}=>/PP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value Gene Regulates Gene 1 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/VP{//?[Value IN {'regulated', 'upregulated', 'downregulated', 'up-regulated', 'down-regulated'}](kw1)=>//?[Value='by'](kw3)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 2 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/VP{/?[Value IN {'regulated', 'upregulated', 'downregulated', 'up-regulated', 'down-regulated'}](kw1)=>//?[Value='by'](kw3)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 3 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/?[Value IN {'regulated','down-regulated'}](kw1)=>/NP{//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 4 //NP{/NP{/?[Value IN {'regulation', 'upregulation', 'downregulation', 'up-regulation', 'down-regulation'}](kw1)}=>/PP{/NP{//?[Tag='GENE'](kw0)=>//?[Value='by'](kw3)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 5 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/?[Value IN {'regulates', 'upregulates', 'downregulates', 'up-regulates', 'down-regulates'}](kw1)=>/NP{//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 6 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/VP{//?[Value='in'](kw3)=>//?[Value='regulating'](kw1)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value Gene Regulate Gene (Xenobiotic Metabolism) 1 //S{//?[Tag='GENE' AND Canonical LIKE 'CYP\\%'](kw0)<=>//?", "[Tag='GENE' AND Canonical IN {'AhR', 'CASR', 'CAR', 'PXR', 'NR1I2', 'NR1I3'}](kw1)} ::: distinct sent.cid, kw0.value, kw1.value, sent.value 2 //S{//?[Tag='GENE' AND Value LIKE 'cytochrome\\%'](kw0)<=>//?", "[Tag='GENE' AND Canonical IN {'AhR', 'CASR', 'CAR', 'PXR', 'NR1I2', 'NR1I3'}](kw1)} ::: distinct sent.cid, kw0.value, kw1.value, sent.value Negative Drug Induces/Metabolizes/Inhibits Gene 1 //S{/?[Tag='DRUG'](kw0)=>/VP{/VP{/?[Value IN {'induced', 'inhibited', 'metabolized', 'metabolised'}](kw1)=>//?[Tag='GENE'](kw2)=>//?[Value='not'](kw3)=>//?", "[Tag='GENE'](kw4)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw4.value, kw3.value, sent.value 2 //S{/?[Tag='DRUG'](kw0)=>/?[Value='not'](kw3)=>/?[Tag='DRUG'](kw4)=>/?[Value='inhibited'](kw1)=>/?", "[Tag='GENE'](kw2)} ::: distinct sent.cid, kw4.value, kw1.value, kw2.value, kw3.value, sent.value 3 //S{/SBAR{/S{//?[Tag='DRUG'](kw0)}}=>/S{/S{//?[Value='metabolized'](kw1)=>//?[Tag='GENE'](kw2)=>//?[Value='not'](kw3)=>//?", "[Tag='GENE'](kw4)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw4.value, kw3.value, sent.value 4 //S{/NP{/PP{//?[Tag='GENE'](kw2)}}=>/VP{/VP{/?[Value IN {'induced', 'inhibited', 'metabolized', 'metabolised'}](kw1)=>//?[Tag='DRUG'](kw0)=>//?[Value='not'](kw3)=>//?", "[Tag='DRUG'](kw4)}}} ::: distinct sent.cid, kw4.value, kw1.value, kw2.value, kw3.value, sent.value 5 //S{/NP{/PP{//?[Tag='DRUG'](kw0)}}=>/VP{/VP{/?[Value IN {'induced', 'inhibited', 'metabolized', 'metabolised'}](kw1)=>//?[Tag='GENE'](kw2)=>//?[Value IN {'not','no'}](kw3)=>//?", "[Tag='GENE'](kw4)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw4.value, kw3.value, sent.value 6 //S{/?[Tag='DRUG'](kw0)=>/VP{/?[Value='not'](kw3)=>/VP{//?[Value IN {'induced', 'inhibited', 'metabolized', 'metabolised'}](kw1)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 7 //S{/NP{/?[Tag='GENE'](kw2)}=>/VP{/VP{/?[Value IN {'induced', 'inhibited', 'metabolized', 'metabolised'}](kw1)=>//?[Tag='DRUG'](kw0)=>//?[Value='not'](kw3)=>//?", "[Tag='DRUG'](kw4)}}} ::: distinct sent.cid, kw4.value, kw1.value, kw2.value, kw3.value, sent.value 8 //S{/NP{/NP{/?[Value='not'](kw3)=>/?[Tag='DRUG'](kw0)}}=>/VP{/?[Value IN {'inhibited','induced'}](kw1)=>/NP{//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 9 //S{/NP{/NP{/?[Value IN {'no','not'}](kw3)=>/?[Tag='DRUG'](kw0)}}=>/VP{/?[Value IN {'inhibited','induced'}](kw1)=>/NP{/?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 10 //S{/?[Tag='DRUG'](kw0)=>/S{/S{//?[Value='not'](kw3)=>//?[Value IN {'induce', 'inhibit'}](kw1)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 11 //S{/?[Tag='DRUG'](kw0)=>/?[Tag='GENE'](kw2)=>/?[Value IN {'not'}](kw3)=>/?", "[Value IN {'inhibit','induce'}](kw1)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 12 //S{/?[Tag='DRUG'](kw0)=>/VP{/?[Value='not'](kw3)=>/VP{/?[Value IN {'inhibit','induce'}](kw1)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 13 //S{/NP{/?[Tag='DRUG'](kw0)}=>/VP{/VP{/?[Value='not'](kw3)=>//?[Value='inhibit'](kw1)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 14 //S{/NP{/NP{/?[Tag='GENE'](kw2)}}=>/VP{/?[Value='not'](kw3)=>/VP{/?[Value='metabolize'](kw1)=>//?", "[Tag='DRUG'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 15 //S{/NP{/NP{//?[Tag='DRUG'](kw0)}}=>/VP{/?[Value='not'](kw3)=>/VP{/?[Value IN {'inhibit','induce'}](kw1)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 16 //S{/NP{/?[Tag='DRUG'](kw0)}=>/VP{/?[Value='not'](kw3)=>/VP{/?[Value='inhibit'](kw1)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 17 //S{/NP{/?[Tag='DRUG'](kw0)}=>/VP{/?[Value IN {'induces', 'inhibits'}](kw1)=>/NP{//?[Tag='GENE'](kw2)=>//?[Value='not'](kw3)=>//?", "[Tag='GENE'](kw4)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw4.value, kw3.value, sent.value 18 //S{/?[Tag='DRUG'](kw0)=>/VP{/?[Value='inhibits'](kw1)=>/NP{//?[Tag='GENE'](kw2)=>//?[Value='not'](kw3)=>//?", "[Tag='GENE'](kw4)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw4.value, kw3.value, sent.value 19 //S{/?[Tag='DRUG'](kw0)=>/VP{/NP{//?[Value='no'](kw3)=>//?[Value IN {'induction','metabolism','inhibition'}](kw1)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 20 //S{/NP{/NP{//?[Tag='DRUG'](kw0)=>//?[Tag='GENE'](kw2)=>//?[Value IN {'induction','metabolism','inhibition'}](kw1)}}=>/VP{/?", "[Value='not'](kw3)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 21 //S{/?[Tag='DRUG'](kw0)=>/VP{/?[Value='not'](kw3)=>/VP{//?[Value IN {'induction','metabolism','inhibition'}](kw1)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 22 //S{/NP{/PP{//?[Tag='DRUG'](kw0)}}=>/VP{/?[Value='not'](kw3)=>/VP{//?[Value IN {'induction','metabolism','inhibition'}](kw1)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value Negative Drug Induces Gene 1 //S{//?[Tag='GENE'](kw0)=>//?[Value IN {'induced','increased','stimulated'}](kw1)=>//?[Value='by'](kw3)=>//?[Tag='DRUG'](kw2)=>//?[Value='not'](kw4)=>//?", "[Tag='DRUG'](kw5)} ::: distinct sent.cid, kw5.value, kw1.value, kw0.value, kw4.value, sent.value 2 //S{//?[Tag='GENE'](kw0)=>//?[Value='not'](kw4)=>//?[Tag='GENE'](kw5)=>//?[Value IN {'induced','increased','stimulated'}](kw1)=>//?[Value='by'](kw3)=>//?", "[Tag='DRUG'](kw2)} ::: distinct sent.cid, kw2.value, kw1.value, kw5.value, kw4.value, sent.value 3 //S{//?[Tag='DRUG'](kw0)=>//?[Value='not'](kw4)=>//?[Value IN {'induce','induced','increase','increased','stimulate','stimulated'}](kw1)=>//?", "[Tag='GENE'](kw2)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw4.value, sent.value 4 //S{//?[Tag='DRUG'](kw0)=>//?[Value='not'](kw4)=>//?[Tag='DRUG'](kw5)=>//?[Value IN {'induces','increases','stimulates','induced','increased','stimulated'}](kw1)=>//?", "[Tag='GENE'](kw2)} ::: distinct sent.cid, kw5.value, kw1.value, kw2.value, kw4.value, sent.value 5 //S{//?[Value='no'](kw5)=>//?[value IN {'induction','stimulation'}](kw1)=>//?[Value IN {'of'}](kw2)=>//?[Tag='GENE'](kw0)=>//?[value IN {'by'}](kw3)=>//?", "[Tag='DRUG'](kw4)} ::: distinct sent.cid, kw4.value, kw1.value, kw0.value, kw5.value, sent.value Negative Drug Inhibits Gene 1 //S{//?[Tag='GENE'](kw0)=>//?[Value IN {'inhibited','decreased'}](kw1)=>//?[Value='by'](kw3)=>//?[Tag='DRUG'](kw2)=>//?[Value='not'](kw4)=>//?", "[Tag='DRUG'](kw5)} ::: distinct sent.cid, kw5.value, kw1.value, kw0.value, kw4.value, sent.value 2 //S{//?[Tag='GENE'](kw0)=>//?[Value='not'](kw4)=>//?[Tag='GENE'](kw5)=>//?[Value IN {'inhibited','decreased'}](kw1)=>//?[Value='by'](kw3)=>//?", "[Tag='DRUG'](kw2)} ::: distinct sent.cid, kw2.value, kw1.value, kw5.value, kw4.value, sent.value 3 //S{//?[Tag='DRUG'](kw0)=>//?[Value='not'](kw4)=>//?[Value IN {'inhibit','inhibited','decrease','decreased'}](kw1)=>//?", "[Tag='GENE'](kw2)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw4.value, sent.value 4 //S{//?[Tag='DRUG'](kw0)=>//?[Value='not'](kw4)=>//?[Tag='DRUG'](kw5)=>//?[Value IN {'inhibits','decreases','inhibited','decreased'}](kw1)=>//?", "[Tag='GENE'](kw2)} ::: distinct sent.cid, kw5.value, kw1.value, kw2.value, kw4.value, sent.value 5 //S{//?[Value='no'](kw5)=>//?[value IN {'inhibition'}](kw1)=>//?[Value IN {'of'}](kw2)=>//?[Tag='GENE'](kw0)=>//?[value IN {'by'}](kw3)=>//?", "[Tag='DRUG'](kw4)} ::: distinct sent.cid, kw4.value, kw1.value, kw0.value, kw5.value, sent.value Negative Gene Metabolizes Drug 1 //S{//?[Tag='DRUG'](kw0)=>//?[Value IN {'metabolized','metabolised'}](kw1)=>//?[Value='by'](kw3)=>//?[Tag='GENE'](kw2)=>//?[Value='not'](kw4)=>//?", "[Tag='GENE'](kw5)} ::: distinct sent.cid, kw0.value, kw1.value, kw5.value, kw4.value, sent.value 2 //S{//?[Tag='DRUG'](kw0)=>//?[Value='not'](kw4)=>//?[Tag='DRUG'](kw5)=>//?[Value IN {'metabolized','metabolised'}](kw1)=>//?[Value='by'](kw3)=>//?", "[Tag='GENE'](kw2)} ::: distinct sent.cid, kw5.value, kw1.value, kw2.value, kw4.value, sent.value 3 //S{//?[Tag='GENE'](kw0)=>//?[Value='not'](kw4)=>//?[Value IN {'metabolize','metabolise'}](kw1)=>//?", "[Tag='DRUG'](kw2)} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, kw4.value, sent.value 4 //S{//?[Tag='GENE'](kw0)=>//?[Value='not'](kw4)=>//?[Tag='GENE'](kw5)=>//?[Value IN {'metabolize','metabolise','metabolizes','metabolises'}](kw1)=>//?", "[Tag='DRUG'](kw2)} ::: distinct sent.cid, kw2.value, kw1.value, kw5.value, kw4.value, sent.value Negative Gene Downregulates Gene 1 //S{//?[Tag='GENE'](kw0)=>//?[Value IN {'suppressed','downregulated','inhibited','down-regulated','repressed','disrupted'}](kw1)=>//?[Value='by'](kw3)=>//?[Tag='GENE'](kw2)=>//?[Value='not'](kw4)=>//?", "[Tag='GENE'](kw5)} ::: distinct sent.cid, kw5.value, kw1.value, kw0.value, kw4.value, sent.value 2 //S{//?[Tag='GENE'](kw0)=>//?[Value='not'](kw4)=>//?[Tag='GENE'](kw5)=>//?[Value IN {'suppressed','downregulated','inhibited','down-regulated','repressed','disrupted'}](kw1)=>//?[Value='by'](kw3)=>//?", "[Tag='GENE'](kw2)} ::: distinct sent.cid, kw2.value, kw1.value, kw5.value, kw4.value, sent.value 3 //S{//?[Tag='GENE'](kw0)=>//?[Value='not'](kw4)=>//?[Value IN {'suppressed','suppress','downregulated','downregulate','inhibited','inhibit','down-regulated','down-regulate','repressed','repress','disrupted','disrupt'}](kw1)=>//?", "[Tag='GENE'](kw2)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw4.value, sent.value 4 //S{//?[Tag='GENE'](kw0)=>//?[Value='not'](kw4)=>//?[Tag='GENE'](kw5)=>//?[Value IN {'suppresses','downregulates','inhibits','down-regulates','represses','disrupts','suppressed','downregulated','inhibited','down-regulated','repressed','disrupted'}](kw1)=>//?", "[Tag='GENE'](kw2)} ::: distinct sent.cid, kw5.value, kw1.value, kw2.value, kw4.value, sent.value 5 //S{//?[Value='no'](kw5)=>//?[value IN {'inhibition','downregulation','down-regulation'}](kw1)=>//?[Value IN {'of'}](kw2)=>//?[Tag='GENE'](kw0)=>//?[value IN {'on'}](kw3)=>//?", "[Tag='GENE'](kw4)} ::: distinct sent.cid, kw0.value, kw1.value, kw4.value, kw5.value, sent.value Negative Gene Upregulates Gene 1 //S{//?[Tag='GENE'](kw0)=>//?[Value IN {'activated','induced','stimulated','regulated','upregulated','up-regulated'}](kw1)=>//?[Value='by'](kw3)=>//?[Tag='GENE'](kw2)=>//?[Value='not'](kw4)=>//?", "[Tag='GENE'](kw5)} ::: distinct sent.cid, kw5.value, kw1.value, kw0.value, kw4.value, sent.value 2 //S{//?[Tag='GENE'](kw0)=>//?[Value='not'](kw4)=>//?[Tag='GENE'](kw5)=>//?", "[Value IN {'activated','induced','stimulated','regulated','upregulated','up-regulated'}](kw1) ::: distinct sent.cid, kw2.value, kw1.value, kw5.value, kw4.value, sent.value 3 //S{//?[Tag='GENE'](kw0)=>//?[Value='not'](kw4)=>//?[Value IN {'activated','induced','stimulated','regulated','upregulated','up-regulated','activate','induce','stimulate','regulate','upregulate','up-regulate'}](kw1)=>//?", "[Tag='GENE'](kw2)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw4.value, sent.value 4 //S{//?[Tag='GENE'](kw0)=>//?[Value='not'](kw4)=>//?[Tag='GENE'](kw5)=>//?[Value IN {'activates','induces','stimulates','regulates','upregulates','up-regulates','activate','induce','stimulate','regulate','upregulate','up-regulate'}](kw1)=>//?", "[Tag='GENE'](kw2)} ::: distinct sent.cid, kw5.value, kw1.value, kw2.value, kw4.value, sent.value 5 //S{//?[Value='no'](kw5)=>//?[value IN {'induction','activation','stimulation','regulation','upregulation','up-regulation'}](kw1)=>//?[Value IN {'of'}](kw2)=>//?[Tag='GENE'](kw0)=>//?[value IN {'by'}](kw3)=>//?", "[Tag='GENE'](kw4)} ::: distinct sent.cid, kw4.value, kw1.value, kw0.value, kw5.value, sent.value Drug Gene Co-Occurrence 1 //S{//?[Tag='DRUG'](kw0)<=>//?", "[Tag='GENE'](kw1)} ::: distinct sent.cid, kw0.value, kw1.value, kw0.type, kw1.type, sent.value" ], [ "Complete Set of Queries Used for Drug-Drug Interaction", "Drug-Drug Interaction Queries" ], [ "Drug Activates Gene", "1 //S{/NP{//?[Value='activation'](kw1)=>//?[Value='of'](kw2)=>//?[Tag='GENE'](kw0)=>//?[Value='by'](kw4)=>//?", "[Tag='DRUG'](kw3)}} ::: distinct sent.cid, kw3.value, kw1.value, kw0.value, sent.value 2 //NP{/NP{/?[Tag='DRUG'](kw3)=>/?[Value='activation'](kw1)}=>/PP{/?[Value='of'](kw2)=>//?", "[Tag='GENE'](kw0)}} ::: distinct sent.cid, kw3.value, kw1.value, kw0.value, sent.value 3 //NP{/?[Value='activation'](kw1)=>/PP{/?[Value='of'](kw2)=>//?[Tag='GENE'](kw0)=>//?[Value='by'](kw4)=>//?", "[Tag='DRUG'](kw3)}} ::: distinct sent.cid, kw3.value, kw1.value, kw0.value, sent.value 4 //S{/NP{//?[Tag='GENE'](kw2)}=>/VP{//?[Value='activation'](kw1)=>//?", "[Tag='DRUG'](kw0)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 5 //S{/NP{//?[Tag='DRUG'](kw0)}=>/VP{//?[Value='activation'](kw1)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 6 //NP{/NP{/?[Tag='DRUG'](kw0)=>/?[Value='activation'](kw1)}=>/PP{//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 7 //S{/NP{//?[Value='activation'](kw1)=>//?[Tag='GENE'](kw2)}=>/VP{//?", "[Tag='DRUG'](kw0)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 8 //S{/NP{//?[Value='activation'](kw1)=>//?[Value='of'](kw2)=>//?[Tag='GENE'](kw0)=>//?[Value='by'](kw4)=>//?", "[Tag='DRUG'](kw3)}} ::: distinct sent.cid, kw3.value, kw1.value, kw0.value, sent.value 9 //NP{/NP{/?[Tag='DRUG'](kw3)=>/?[Value='activation'](kw1)}=>/PP{/?[Value='of'](kw2)=>//?", "[Tag='GENE'](kw0)}} ::: distinct sent.cid, kw3.value, kw1.value, kw0.value, sent.value 10 //NP{/?[Value='activation'](kw1)=>/PP{/?[Value='of'](kw2)=>//?[Tag='GENE'](kw0)=>//?[Value='by'](kw4)=>//?", "[Tag='DRUG'](kw3)}} ::: distinct sent.cid, kw3.value, kw1.value, kw0.value, sent.value" ], [ "Gene Induces Gene", "1 //S{/NP{//?[Tag='GENE'](kw0)}=>/VP{//?[Value='induced'](kw1)=>//?[Value='by'](kw3)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 2 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{//?[Value='induced'](kw1)=>//?[Value='by'](kw3)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 3 //S{/NP{//?[Tag='GENE'](kw0)}=>/VP{//?[Value IN {'increase','increased'}](kw1)=>//?[Tag='GENE'](kw2)=>//?", "[Value='activity'](kw3)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 4 //S{/NP{//?[Tag='GENE'](kw0)=>//?[Value='activity'](kw3)}=>/VP{//?[Value='increase'](kw1)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 5 //S{/NP{//?[Tag='GENE'](kw0)}=>/VP{//?[Value IN {'stimulates','stimulate', 'stimulated'}](kw1)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 6 //S{/NP{//?[Tag='GENE'](kw0)}=>/VP{/?[Value IN {'stimulates','stimulate'}](kw1)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 7 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/?[Value='stimulates'](kw1)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 8 //S{/NP{//?[Tag='GENE'](kw0)}=>/VP{/?[Value='activated'](kw1)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 9 //NP{/NP{/?[Value='induction'](kw1)}=>/PP{/?[Value='of'](kw3)=>/NP{//?[Tag='GENE'](kw0)=>//?[Value='by'](kw4)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 10 //NP{/NP{/?[Value='stimulation'](kw1)}=>/PP{/?[Value='of'](kw3)=>/NP{//?[Tag='GENE'](kw0)=>//?[Value='by'](kw4)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 11 //NP{/NP{/?[Value='activation'](kw1)}=>/PP{/?[Value IN {'of'}](kw3)=>/NP{//?[Tag='GENE'](kw0)=>//?[Value='by'](kw4)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 12 //S{/NP{//?[Tag='GENE'](kw0)}=>/VP{//?[Value='inducible'](kw1)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 13 //S{/NP{//?[Tag='GENE'](kw0)}=>/VP{//?[Value='induced'](kw1)=>//?[Value='by'](kw3)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 14 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{//?[Value='induced'](kw1)=>//?[Value='by'](kw3)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 15 //S{/NP{//?[Tag='GENE'](kw0)}=>/VP{//?[Value IN {'increase','increased'}](kw1)=>//?[Tag='GENE'](kw2)=>//?", "[Value='activity'](kw3)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 16 //S{/NP{//?[Tag='GENE'](kw0)=>//?[Value='activity'](kw3)}=>/VP{//?[Value='increase'](kw1)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 17 //S{/NP{//?[Tag='GENE'](kw0)}=>/VP{//?[Value='stimulated'](kw1)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 18 //S{/NP{//?[Tag='GENE'](kw0)}=>/VP{/?[Value IN {'stimulates','stimulate'}](kw1)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 19 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/?[Value='stimulates'](kw1)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 20 //S{/NP{//?[Tag='GENE'](kw0)}=>/VP{/?[Value='activated'](kw1)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 21 //VP{/?[Value='activated'](kw1)=>/PP{//?[Tag='GENE'](kw0)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 22 //NP{/NP{/?[Value='induction'](kw1)}=>/PP{/?[Value='of'](kw3)=>/NP{//?[Tag='GENE'](kw0)=>//?[Value='by'](kw4)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 23 //NP{/NP{/?[Value='stimulation'](kw1)}=>/PP{/?[Value='of'](kw3)=>/NP{//?[Tag='GENE'](kw0)=>//?[Value='by'](kw4)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 24 //NP{/NP{/?[Value='activation'](kw1)}=>/PP{/?[Value IN {'of'}](kw3)=>/NP{//?[Tag='GENE'](kw0)=>//?[Value='by'](kw4)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 25 //S{/NP{//?[Tag='GENE'](kw0)}=>/VP{//?[Value='inducible'](kw1)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value" ], [ "Gene Inhibits Gene", "1 //S{/NP{//?[Tag='GENE'](kw0)}=>/VP{//?[Value='induced'](kw1)=>//?[Value='by'](kw3)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 2 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{//?[Value='induced'](kw1)=>//?[Value='by'](kw3)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 3 //S{/NP{//?[Tag='GENE'](kw0)}=>/VP{//?[Value IN {'increase','increased'}](kw1)=>//?[Tag='GENE'](kw2)=>//?", "[Value='activity'](kw3)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 4 //S{/NP{//?[Tag='GENE'](kw0)=>//?[Value='activity'](kw3)}=>/VP{//?[Value='increase'](kw1)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 5 //S{/NP{//?[Tag='GENE'](kw0)}=>/VP{//?[Value IN {'stimulates','stimulate', 'stimulated'}](kw1)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 6 //S{/NP{//?[Tag='GENE'](kw0)}=>/VP{/?[Value IN {'stimulates','stimulate'}](kw1)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 7 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/?[Value='stimulates'](kw1)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 8 //S{/NP{//?[Tag='GENE'](kw0)}=>/VP{/?[Value='activated'](kw1)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 9 //NP{/NP{/?[Value='induction'](kw1)}=>/PP{/?[Value='of'](kw3)=>/NP{//?[Tag='GENE'](kw0)=>//?[Value='by'](kw4)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 10 //NP{/NP{/?[Value='stimulation'](kw1)}=>/PP{/?[Value='of'](kw3)=>/NP{//?[Tag='GENE'](kw0)=>//?[Value='by'](kw4)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 11 //NP{/NP{/?[Value='activation'](kw1)}=>/PP{/?[Value IN {'of'}](kw3)=>/NP{//?[Tag='GENE'](kw0)=>//?[Value='by'](kw4)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 12 //S{/NP{//?[Tag='GENE'](kw0)}=>/VP{//?[Value='inducible'](kw1)=>//?", "[Tag='GENE'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value" ], [ "Drug Changes Gene Expression/Activity", "1 //S{/?[Tag='DRUG'](kw2)=>/?[Value IN {'increased','increase','increases'}](kw1)=>/?[Value IN {'levels','level'}](kw3)=>/?", "[Tag='GENE'](kw0)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 2 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/?[Value IN {'increased','increases'}](kw1)=>/NP{//?[Tag='GENE'](kw0)=>//?", "[Value IN {'expression','level','activity','activities','levels'}](kw3)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 3 //S{/?[Tag='DRUG'](kw2)=>/VP{/NP{//?[Value IN {'increases','increase'}](kw1)=>//?[Tag='GENE'](kw0)=>//?", "[Value IN {'levels','activity'}](kw3)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 4 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value IN {'increased','increases'}](kw1)=>/NP{//?[Tag='GENE'](kw0)=>//?", "[Value IN {'activity','activities','levels'}](kw3)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 5 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/?[Value='increased'](kw1)=>/NP{//?[Value IN {'activity','levels','expression'}](kw3)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 6 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{//?[Value IN {'increased','increases','increase'}](kw1)=>//?[Tag='GENE'](kw0)=>//?", "[Value IN {'activity','levels'}](kw3)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 7 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{/?[Value='increased'](kw1)=>//?[Tag='GENE'](kw0)=>//?", "[Value IN {'expression','activity'}](kw3)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 8 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value IN {'increased','increases'}](kw1)=>/NP{//?[Value IN {'expression','activity','activities','level','levels'}](kw3)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 9 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value IN {'decreased','decreases'}](kw1)=>/NP{//?[Tag='GENE'](kw0)=>//?", "[Value IN {'activity','expression'}](kw3)}}} ::: distinct sent.cid, kw2.value, kw1,value, kw0.value, sent.value 10 //S{/?[Tag='DRUG'](kw2)=>/?[Value IN {'increased','increase','increases'}](kw1)=>/?[Value IN {'levels','level'}](kw3)=>/?", "[Tag='GENE'](kw0)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 11 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/?[Value IN {'increased','increases'}](kw1)=>/NP{//?[Tag='GENE'](kw0)=>//?", "[Value IN {'expression','level','activity','activities','levels'}](kw3)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 12 //S{/?[Tag='DRUG'](kw2)=>/VP{/NP{//?[Value IN {'increases','increase'}](kw1)=>//?[Tag='GENE'](kw0)=>//?", "[Value IN {'levels','activity'}](kw3)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 13 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value IN {'increased','increases'}](kw1)=>/NP{//?[Tag='GENE'](kw0)=>//?", "[Value IN {'activity','activities','levels'}](kw3)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 14 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/?[Value='increased'](kw1)=>/NP{//?[Value IN {'activity','levels','expression'}](kw3)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 15 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{//?[Value IN {'increased','increases','increase'}](kw1)=>//?[Tag='GENE'](kw0)=>//?", "[Value IN {'activity','levels'}](kw3)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 16 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{/?[Value='increased'](kw1)=>//?[Tag='GENE'](kw0)=>//?", "[Value IN {'expression','activity'}](kw3)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 17 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value IN {'increased','increases'}](kw1)=>/NP{//?[Value IN {'expression','activity','activities','level','levels'}](kw3)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 18 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value IN {'decreased','decreases'}](kw1)=>/NP{//?[Tag='GENE'](kw0)=>//?", "[Value IN {'activity','expression'}](kw3)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value" ], [ "Drug Induces/Stimulates Gene", "1 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value IN {'stimulated','induced'}](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 2 //VP{/?[Value IN {'stimulated','induced'}](kw1)=>/PP{/NP{//?[Tag='DRUG'](kw2)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 3 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/?[Value IN {'stimulated','induced'}](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 4 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/?[Value IN {'stimulated','induced'}](kw1)=>/PP{//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 5 //S{/?[Tag='DRUG'](kw2)=>/VP{/NP{//?[Tag='GENE'](kw0)=>//?", "[Value IN {'stimulated','induced'}](kw1)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 6 //S{/NP{/PP{//?[Tag='DRUG'](kw2)}}=>/VP{/?[Value IN {'stimulated','induced'}](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 7 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/VP{/?[Value IN {'stimulated','induced'}](kw1)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 8 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/VP{//?[Value='induced'](kw1)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 9 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value IN {'stimulated','induced'}](kw1)=>/NP{/?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 10 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{//?[Value IN {'stimulate','induce'}](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 11 //NP{/NP{/?[Value IN {'induction','stimulation'}](kw1)}=>/PP{/NP{//?[Tag='GENE'](kw0)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 12 //S{/NP{/?[Value IN {'stimulation','induction'}](kw1)=>/PP{//?[Tag='GENE'](kw0)}}=>/VP{/VP{//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 13 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/NP{//?[Value IN {'stimulation','induction'}](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 14 //NP{/NP{/NP{/?[Value='induction'](kw1)}=>/PP{//?[Tag='GENE'](kw0)}}=>/PP{/NP{/?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 15 //NP{/NP{/?[Tag='GENE'](kw0)=>/?[Value IN {'stimulation','induction'}](kw1)}=>/PP{/NP{/?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 16 //NP{/?[Value IN {'stimulation','induction'}](kw1)=>/PP{/NP{//?[Tag='GENE'](kw0)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 17 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/?[Value IN {'stimulates','induces'}](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 18 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value IN {'stimulates','induces'}](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 19 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value IN {'stimulated','induced'}](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 20 //VP{/?[Value IN {'stimulated','induced'}](kw1)=>/PP{/NP{//?[Tag='DRUG'](kw2)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 21 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/?[Value IN {'stimulated','induced'}](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 22 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/?[Value IN {'stimulated','induced'}](kw1)=>/PP{//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 23 //S{/?[Tag='DRUG'](kw2)=>/VP{/NP{//?[Tag='GENE'](kw0)=>//?", "[Value IN {'stimulated','induced'}](kw1)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 24 //S{/NP{/PP{//?[Tag='DRUG'](kw2)}}=>/VP{/?[Value IN {'stimulated','induced'}](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 25 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/VP{/?[Value IN {'stimulated','induced'}](kw1)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 26 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/VP{//?[Value='induced'](kw1)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 27 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value IN {'stimulated','induced'}](kw1)=>/NP{/?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 28 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{//?[Value IN {'stimulate','induce'}](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 29 //NP{/NP{/?[Value IN {'induction','stimulation'}](kw1)}=>/PP{/NP{//?[Tag='GENE'](kw0)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 30 //S{/NP{/?[Value IN {'stimulation','induction'}](kw1)=>/PP{//?[Tag='GENE'](kw0)}}=>/VP{/VP{//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 31 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/NP{//?[Value IN {'stimulation','induction'}](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 32 //NP{/NP{/NP{/?[Value='induction'](kw1)}=>/PP{//?[Tag='GENE'](kw0)}}=>/PP{/NP{/?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 33 //NP{/NP{/?[Tag='GENE'](kw0)=>/?[Value IN {'stimulation','induction'}](kw1)}=>/PP{/NP{/?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 34 //NP{/?[Value IN {'stimulation','induction'}](kw1)=>/PP{/NP{//?[Tag='GENE'](kw0)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 35 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/?[Value IN {'stimulates','induces'}](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 36 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value IN {'stimulates','induces'}](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value" ], [ "Drug Inhibits Gene", "1 //NP{/NP{/NP{/?[Value='inhibitory'](kw1)}=>/PP{//?[Tag='DRUG'](kw2)}}=>/PP{/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 2 //S{/?[Tag='DRUG'](kw2)=>/VP{/NP{/?[Tag='GENE'](kw0)=>/?", "[Value='inhibitor'](kw1)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 3 //NP{/?[Tag='DRUG'](kw2)=>/NP{/?[Tag='GENE'](kw0)=>/?", "[Value='inhibitor'](kw1)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 4 //NP{/NP{/?[Tag='GENE'](kw0)=>/?[Value='inhibitor'](kw1)}=>/?", "[Tag='DRUG'](kw2)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 5 //NP{/NP{/NP{/?[Tag='GENE'](kw0)=>/?[Tag='DRUG'](kw2)}}=>/?", "[Value='inhibitor'](kw1)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 6 //NP{/?[Tag='GENE'](kw0)=>/?[Value='inhibitor'](kw1)=>/?", "[Tag='DRUG'](kw2)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 7 //NP{/NP{/?[Value='inhibitor'](kw1)}=>/PP{/NP{//?[Tag='GENE'](kw0)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 8 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{//?[Tag='GENE'](kw0)=>//?", "[Value='inhibitor'](kw1)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 9 //S{/?[Tag='DRUG'](kw2)=>/VP{/NP{//?[Value='inhibitor'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 10 //NP{/?[Tag='DRUG'](kw2)=>/NP{/NP{/?[Tag='GENE'](kw0)}}=>/?", "[Value='inhibitor'](kw1)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 11 //S{/?[Tag='DRUG'](kw2)=>/?[Value='inhibitor'](kw1)=>/?", "[Tag='GENE'](kw0)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 12 //NP{/NP{/?[Tag='DRUG'](kw2)=>/NP{/?[Value='inhibitor'](kw1)}=>/PP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 13 //S{/?[Tag='GENE'](kw0)=>/?[Value='inhibitor'](kw1)=>/?", "[Tag='DRUG'](kw2)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 14 //NP{/?[Tag='DRUG'](kw2)=>/NP{/NP{/?[Value='inhibitor'](kw1)}=>/PP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 15 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{//?[Value='inhibitor'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 16 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/NP{//?[Value='inhibitor'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 17 //NP{/?[Tag='DRUG'](kw2)=>/NP{/NP{/?[Tag='GENE'](kw0)=>/?", "[Value='inhibitor'](kw1)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 18 //NP{/NP{/PP{//?[Tag='DRUG'](kw2)=>//?[Tag='GENE'](kw0)}=>/NP{/?", "[Value='inhibitor'](kw1)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 19 //NP{/NP{/?[Tag='GENE'](kw0)=>/?[Value='inhibitor'](kw1)=>/?", "[Tag='DRUG'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 20 //S{/?[Tag='DRUG'](kw2)=>/?[Tag='GENE'](kw0)=>/?", "[Value='inhibitor'](kw1)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 21 //NP{/NP{/?[Tag='DRUG'](kw2)=>/?[Tag='GENE'](kw0)=>/?", "[Value='inhibitor'](kw1)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 22 //NP{/NP{/?[Tag='DRUG'](kw2)=>/NP{/?[Tag='GENE'](kw0)=>/?", "[Value='inhibitor'](kw1)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 23 //NP{/NP{/PP{//?[Tag='DRUG'](kw2)}=>/?[Tag='GENE'](kw0)=>/?", "[Value='inhibitor'](kw1)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 24 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value='inhibited'](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 25 //S{/NP{/NP{/?[Tag='DRUG'](kw2)}}=>/VP{/?[Value='inhibited'](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 26 //S{/?[Tag='DRUG'](kw2)=>/?[Value='inhibited'](kw1)=>/?", "[Tag='GENE'](kw0)} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 27 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/?[Value='inhibited'](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 28 //S{/NP{/NP{//?[Tag='DRUG'](kw2)}}=>/VP{/?[Value='inhibited'](kw1)=>/NP{/?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 29 //S{/NP{/PP{//?[Tag='GENE'](kw0)}}=>/VP{/VP{/?[Value='inhibited'](kw1)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 30 //S{/NP{/NP{//?[Tag='DRUG'](kw2)}}=>/VP{/?[Value='inhibited'](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 31 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{/?[Value='inhibited'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 32 //S{/NP{/PP{//?[Tag='DRUG'](kw2)}}=>/VP{/?[Value='inhibited'](kw1)=>/NP{/?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 33 //S{/NP{/NP{/?[Tag='GENE'](kw0)}}=>/VP{/VP{/?[Value='inhibited'](kw1)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 34 //S{/NP{/NP{/?[Tag='GENE'](kw0)}}=>/VP{/?[Value='inhibited'](kw1)=>/PP{//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 35 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value='inhibited'](kw1)=>/PP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 36 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/VP{/?[Value='inhibited'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 37 //S{/NP{/NP{/?[Tag='DRUG'](kw2)}}=>/VP{/?[Value='inhibited'](kw1)=>/NP{/?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 38 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/VP{/?[Value='inhibited'](kw1)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 39 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/?[Value='inhibited'](kw1)=>/NP{/?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 40 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/VP{//?[Value='inhibited'](kw1)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 41 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value='inhibited'](kw1)=>/NP{/?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 42 //S{/NP{/NP{/?[Tag='GENE'](kw0)=>/?[Tag='DRUG'](kw2)}}=>/VP{/?", "[Value='inhibit'](kw1)}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 43 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/VP{//?[Value='inhibit'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 44 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{//?[Value='inhibit'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 45 //NP{/NP{/?[Tag='GENE'](kw0)=>/?[Value='inhibition'](kw1)}=>/PP{/NP{/?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 46 //S{/?[Tag='DRUG'](kw2)=>/VP{/PP{//?[Value='inhibition'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 47 //S{/NP{/PP{//?[Tag='DRUG'](kw2)}}=>/VP{/NP{//?[Value='inhibition'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 48 //NP{/?[Value='inhibition'](kw1)=>/PP{/NP{//?[Tag='GENE'](kw0)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 49 //S{/?[Value='inhibition'](kw1)=>/PP{/NP{//?[Tag='GENE'](kw0)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 50 //NP{/NP{/?[Tag='DRUG'](kw2)=>/?[Value='inhibition'](kw1)}=>/PP{/NP{/?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 51 //NP{/NP{/?[Value='inhibition'](kw1)}=>/PP{/NP{/?[Tag='GENE'](kw0)=>/?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 52 //S{/NP{/?[Value='inhibition'](kw1)=>/PP{//?[Tag='GENE'](kw0)}}=>/VP{/PP{//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 53 //S{/NP{/NP{/?[Value='inhibition'](kw1)}=>/PP{//?[Tag='GENE'](kw0)}}=>/VP{/NP{//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 54 //S{/?[Value IN {'inhibition','Inhibition'}](kw1)=>/?[Tag='GENE'](kw0)=>/?", "[Tag='DRUG'](kw2)} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 55 //NP{/NP{/NP{/?[Value='inhibition'](kw1)}=>/PP{//?[Tag='GENE'](kw0)}}=>/PP{/NP{//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 56 //NP{/?[Tag='DRUG'](kw2)=>/NP{/?[Value='inhibition'](kw1)}=>/PP{/NP{/?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 57 //NP{/NP{/?[Value='inhibition'](kw1)}=>/PP{/NP{//?[Tag='GENE'](kw0)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 58 //S{/?[Tag='DRUG'](kw2)=>/VP{/NP{//?[Value='inhibition'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 59 //S{/NP{/NP{/?[Value='inhibition'](kw1)}=>/PP{//?[Tag='DRUG'](kw2)}}=>/VP{/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 60 //NP{/NP{/?[Value='inhibition'](kw1)=>/PP{//?[Tag='GENE'](kw0)}}=>/PP{/NP{//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 61 //NP{/NP{/?[Tag='DRUG'](kw2)=>/?[Value='inhibition'](kw1)}=>/PP{/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 62 //NP{/NP{/?[Value='inhibition'](kw1)=>/PP{//?[Tag='GENE'](kw0)}}=>/PP{/NP{/?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 63 //NP{/NP{/NP{/?[Value='inhibition'](kw1)}=>/PP{//?[Tag='GENE'](kw0)}}=>/PP{/NP{/?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 64 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/?[Value='inhibits'](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 65 /S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value='inhibits'](kw1)=>/NP{/?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 66 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{/?[Value='inhibits'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 67 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value='inhibits'](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, sent.value 68 //NP{/NP{/NP{/?[Value='inhibitory'](kw1)}=>/PP{//?[Tag='DRUG'](kw2)}}=>/PP{/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 69 //S{/?[Tag='DRUG'](kw2)=>/VP{/NP{/?[Tag='GENE'](kw0)=>/?", "[Value='inhibitor'](kw1)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 70 //NP{/?[Tag='DRUG'](kw2)=>/NP{/?[Tag='GENE'](kw0)=>/?", "[Value='inhibitor'](kw1)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 71 //NP{/NP{/?[Tag='GENE'](kw0)=>/?[Value='inhibitor'](kw1)}=>/?", "[Tag='DRUG'](kw2)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 72 //NP{/NP{/NP{/?[Tag='GENE'](kw0)=>/?[Tag='DRUG'](kw2)}}=>/?", "[Value='inhibitor'](kw1)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 73 //NP{/?[Tag='GENE'](kw0)=>/?[Value='inhibitor'](kw1)=>/?", "[Tag='DRUG'](kw2)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 74 //NP{/NP{/?[Value='inhibitor'](kw1)}=>/PP{/NP{//?[Tag='GENE'](kw0)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 75 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{//?[Tag='GENE'](kw0)=>//?", "[Value='inhibitor'](kw1)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 76 //S{/?[Tag='DRUG'](kw2)=>/VP{/NP{//?[Value='inhibitor'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 77 //NP{/?[Tag='DRUG'](kw2)=>/NP{/NP{/?[Tag='GENE'](kw0)}}=>/?", "[Value='inhibitor'](kw1)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 78 //S{/?[Tag='DRUG'](kw2)=>/?[Value='inhibitor'](kw1)=>/?", "[Tag='GENE'](kw0)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 79 //NP{/NP{/?[Tag='DRUG'](kw2)=>/NP{/?[Value='inhibitor'](kw1)}=>/PP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 80 //S{/?[Tag='GENE'](kw0)=>/?[Value='inhibitor'](kw1)=>/?", "[Tag='DRUG'](kw2)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 81 //NP{/?[Tag='DRUG'](kw2)=>/NP{/NP{/?[Value='inhibitor'](kw1)}=>/PP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 82 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{//?[Value='inhibitor'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 83 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/NP{//?[Value='inhibitor'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 84 //NP{/?[Tag='DRUG'](kw2)=>/NP{/NP{/?[Tag='GENE'](kw0)=>/?", "[Value='inhibitor'](kw1)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 85 //NP{/NP{/PP{//?[Tag='DRUG'](kw2)=>//?[Tag='GENE'](kw0)}=>/NP{/?", "[Value='inhibitor'](kw1)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 86 //NP{/NP{/?[Tag='GENE'](kw0)=>/?[Value='inhibitor'](kw1)=>/?", "[Tag='DRUG'](kw2)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 87 //S{/?[Tag='DRUG'](kw2)=>/?[Tag='GENE'](kw0)=>/?", "[Value='inhibitor'](kw1)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 88 //NP{/NP{/?[Tag='DRUG'](kw2)=>/?[Tag='GENE'](kw0)=>/?", "[Value='inhibitor'](kw1)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 89 //NP{/NP{/?[Tag='DRUG'](kw2)=>/NP{/?[Tag='GENE'](kw0)=>/?", "[Value='inhibitor'](kw1)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 90 //NP{/NP{/PP{//?[Tag='DRUG'](kw2)}=>/?[Tag='GENE'](kw0)=>/?", "[Value='inhibitor'](kw1)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 91 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value='inhibited'](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 92 //S{/NP{/NP{/?[Tag='DRUG'](kw2)}}=>/VP{/?[Value='inhibited'](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 93 //S{/?[Tag='DRUG'](kw2)=>/?[Value='inhibited'](kw1)=>/?", "[Tag='GENE'](kw0)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 94 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/?[Value='inhibited'](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 95 //S{/NP{/NP{//?[Tag='DRUG'](kw2)}}=>/VP{/?[Value='inhibited'](kw1)=>/NP{/?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 96 //S{/NP{/PP{//?[Tag='GENE'](kw0)}}=>/VP{/VP{/?[Value='inhibited'](kw1)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 97 //S{/NP{/NP{//?[Tag='DRUG'](kw2)}}=>/VP{/?[Value='inhibited'](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 98 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{/?[Value='inhibited'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 99 //S{/NP{/PP{//?[Tag='DRUG'](kw2)}}=>/VP{/?[Value='inhibited'](kw1)=>/NP{/?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 100 //S{/NP{/NP{/?[Tag='GENE'](kw0)}}=>/VP{/VP{/?[Value='inhibited'](kw1)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 101 //S{/NP{/NP{/?[Tag='GENE'](kw0)}}=>/VP{/?[Value='inhibited'](kw1)=>/PP{//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 102 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value='inhibited'](kw1)=>/PP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 103 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/VP{/?[Value='inhibited'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 104 //S{/NP{/NP{/?[Tag='DRUG'](kw2)}}=>/VP{/?[Value='inhibited'](kw1)=>/NP{/?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 105 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/VP{/?[Value='inhibited'](kw1)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 106 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/?[Value='inhibited'](kw1)=>/NP{/?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 107 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/VP{//?[Value='inhibited'](kw1)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 108 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value='inhibited'](kw1)=>/NP{/?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 109 //S{/NP{/NP{/?[Tag='GENE'](kw0)=>/?[Tag='DRUG'](kw2)}}=>/VP{/?", "[Value='inhibit'](kw1)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 110 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/VP{//?[Value='inhibit'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 111 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{//?[Value='inhibit'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 112 //NP{/NP{/?[Tag='GENE'](kw0)=>/?[Value='inhibition'](kw1)}=>/PP{/NP{/?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 113 //S{/?[Tag='DRUG'](kw2)=>/VP{/PP{//?[Value='inhibition'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 114 //S{/NP{/PP{//?[Tag='DRUG'](kw2)}}=>/VP{/NP{//?[Value='inhibition'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 115 //NP{/?[Value='inhibition'](kw1)=>/PP{/NP{//?[Tag='GENE'](kw0)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 116 //S{/?[Value='inhibition'](kw1)=>/PP{/NP{//?[Tag='GENE'](kw0)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 117 //NP{/NP{/?[Tag='DRUG'](kw2)=>/?[Value='inhibition'](kw1)}=>/PP{/NP{/?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 118 //NP{/NP{/?[Value='inhibition'](kw1)}=>/PP{/NP{/?[Tag='GENE'](kw0)=>/?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 119 //S{/NP{/?[Value='inhibition'](kw1)=>/PP{//?[Tag='GENE'](kw0)}}=>/VP{/PP{//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 120 //S{/NP{/NP{/?[Value='inhibition'](kw1)}=>/PP{//?[Tag='GENE'](kw0)}}=>/VP{/NP{//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 121 //S{/?[Value IN {'inhibition','Inhibition'}](kw1)=>/?[Tag='GENE'](kw0)=>/?", "[Tag='DRUG'](kw2)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 122 //NP{/NP{/NP{/?[Value='inhibition'](kw1)}=>/PP{//?[Tag='GENE'](kw0)}}=>/PP{/NP{//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 123 //NP{/?[Tag='DRUG'](kw2)=>/NP{/?[Value='inhibition'](kw1)}=>/PP{/NP{/?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 124 //NP{/NP{/?[Value='inhibition'](kw1)}=>/PP{/NP{//?[Tag='GENE'](kw0)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 125 //S{/?[Tag='DRUG'](kw2)=>/VP{/NP{//?[Value='inhibition'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 126 //S{/NP{/NP{/?[Value='inhibition'](kw1)}=>/PP{//?[Tag='DRUG'](kw2)}}=>/VP{/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 127 //NP{/NP{/?[Value='inhibition'](kw1)=>/PP{//?[Tag='GENE'](kw0)}}=>/PP{/NP{//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 128 //NP{/NP{/?[Tag='DRUG'](kw2)=>/?[Value='inhibition'](kw1)}=>/PP{/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 129 //NP{/NP{/?[Value='inhibition'](kw1)=>/PP{//?[Tag='GENE'](kw0)}}=>/PP{/NP{/?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 130 //NP{/NP{/NP{/?[Value='inhibition'](kw1)}=>/PP{//?[Tag='GENE'](kw0)}}=>/PP{/NP{/?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 131 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/?[Value='inhibits'](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 132 /S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value='inhibits'](kw1)=>/NP{/?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 133 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{/?[Value='inhibits'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 134 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value='inhibits'](kw1)=>/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value" ], [ "Gene Metabolized Drug", "1 //S{/NP{/NP{/?[Tag='DRUG'](kw2)}}=>/VP{/VP{/?[Value IN {'metabolised','metabolized'}](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 2 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{//?[Value IN {'metabolised','metabolized'}](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 3 //S{/NP{/SBAR{//?[Tag='DRUG'](kw2)}}=>/VP{/?[Value IN {'metabolised','metabolized'}](kw1)=>/PP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 4 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/VP{/?[Value IN {'metabolised','metabolized'}](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 5 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/VP{//?[Value IN {'metabolized','metabolised'}](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 6 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{/?[Value IN {'metabolised','metabolized'}](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 7 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value='metabolized'](kw1)=>/PP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 8 //S{/?[Tag='DRUG'](kw2)=>/NP{/?[Value='metabolism'](kw1)}=>/VP{/VP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 9 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/NP{//?[Tag='DRUG'](kw2)=>//?", "[Value='metabolism'](kw1)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 10 //NP{/NP{/?[Tag='GENE'](kw0)=>/?[Value='metabolism'](kw1)}=>/PP{/NP{/?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 11 //NP{/?[Value='metabolism'](kw1)=>/PP{/NP{//?[Tag='DRUG'](kw2)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 12 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/VP{//?[Value='metabolism'](kw1)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 13 //S{/NP{/NP{/?[Tag='GENE'](kw0)}}=>/VP{/NP{//?[Value='metabolism'](kw1)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 14 //NP{/NP{/PP{//?[Tag='GENE'](kw0)}}=>/PP{/NP{/?[Tag='DRUG'](kw2)=>/?", "[Value='metabolism'](kw1)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 15 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/PP{//?[Value='metabolism'](kw1)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 16 //NP{/NP{/?[Tag='DRUG'](kw2)=>/?[Value='metabolism'](kw1)}=>/PP{/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 17 //NP{/NP{/?[Tag='GENE'](kw0)=>/?[Value='metabolism'](kw1)}=>/PP{/NP{//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 18 //S{/NP{/NP{/?[Value='metabolism'](kw1)}=>/PP{//?[Tag='DRUG'](kw2)}}=>/VP{/VP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 19 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/PP{//?[Tag='DRUG'](kw2)=>//?", "[Value='metabolism'](kw1)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 20 //NP{/NP{/PP{//?[Tag='GENE'](kw0)}}=>/PP{/NP{//?[Value='metabolism'](kw1)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 21 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/NP{//?[Value='substrate'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 22 //NP{/NP{/?[Tag='DRUG'](kw2)=>/NP{/?[Value='substrate'](kw1)}=>/PP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 23 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{//?[Value='substrate'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 24 //NP{/?[Tag='GENE'](kw0)=>/?[Value='substrate'](kw1)=>/?", "[Tag='DRUG'](kw2)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 25 //NP{/?[Tag='DRUG'](kw2)=>/NP{/NP{/?[Value='substrate'](kw1)}=>/PP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 26 //S{/NP{/NP{/?[Tag='DRUG'](kw2)}}=>/VP{/VP{/?[Value IN {'metabolised','metabolized'}](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 27 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{//?[Value IN {'metabolised','metabolized'}](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 28 //S{/NP{/SBAR{//?[Tag='DRUG'](kw2)}}=>/VP{/?[Value IN {'metabolised','metabolized'}](kw1)=>/PP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 29 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/VP{/?[Value IN {'metabolised','metabolized'}](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 30 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/VP{//?[Value IN {'metabolized','metabolised'}](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 31 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{/?[Value IN {'metabolised','metabolized'}](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 32 //S{/?[Tag='DRUG'](kw2)=>/VP{/?[Value='metabolized'](kw1)=>/PP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 33 //S{/?[Tag='DRUG'](kw2)=>/NP{/?[Value='metabolism'](kw1)}=>/VP{/VP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 34 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/NP{//?[Tag='DRUG'](kw2)=>//?", "[Value='metabolism'](kw1)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 35 //NP{/NP{/?[Tag='GENE'](kw0)=>/?[Value='metabolism'](kw1)}=>/PP{/NP{/?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 36 //NP{/?[Value='metabolism'](kw1)=>/PP{/NP{//?[Tag='DRUG'](kw2)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 37 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/VP{//?[Value='metabolism'](kw1)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 38 //S{/NP{/NP{/?[Tag='GENE'](kw0)}}=>/VP{/NP{//?[Value='metabolism'](kw1)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 39 //NP{/NP{/PP{//?[Tag='GENE'](kw0)}}=>/PP{/NP{/?[Tag='DRUG'](kw2)=>/?", "[Value='metabolism'](kw1)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 40 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/PP{//?[Value='metabolism'](kw1)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 41 //NP{/NP{/?[Tag='DRUG'](kw2)=>/?[Value='metabolism'](kw1)}=>/PP{/NP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 42 //NP{/NP{/?[Tag='GENE'](kw0)=>/?[Value='metabolism'](kw1)}=>/PP{/NP{//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 43 //S{/NP{/NP{/?[Value='metabolism'](kw1)}=>/PP{//?[Tag='DRUG'](kw2)}}=>/VP{/VP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 44 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/PP{//?[Tag='DRUG'](kw2)=>//?", "[Value='metabolism'](kw1)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 45 //NP{/NP{/PP{//?[Tag='GENE'](kw0)}}=>/PP{/NP{//?[Value='metabolism'](kw1)=>//?", "[Tag='DRUG'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 46 //S{/NP{/?[Tag='DRUG'](kw2)}=>/VP{/NP{//?[Value='substrate'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 47 //NP{/NP{/?[Tag='DRUG'](kw2)=>/NP{/?[Value='substrate'](kw1)}=>/PP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 48 //S{/?[Tag='DRUG'](kw2)=>/VP{/VP{//?[Value='substrate'](kw1)=>//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 49 //NP{/?[Tag='GENE'](kw0)=>/?[Value='substrate'](kw1)=>/?", "[Tag='DRUG'](kw2)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 50 //NP{/?[Tag='DRUG'](kw2)=>/NP{/NP{/?[Value='substrate'](kw1)}=>/PP{//?", "[Tag='GENE'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value" ], [ "Gene Regulates Gene", "1 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/VP{//?[Value IN {'regulated', 'upregulated', 'downregulated', 'up-regulated', 'down-regulated'}](kw1)=>//?[Value='by'](kw3)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 2 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/VP{/?[Value IN {'regulated', 'upregulated', 'downregulated', 'up-regulated', 'down-regulated'}](kw1)=>//?[Value='by'](kw3)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 3 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/?[Value IN {'regulated','down-regulated'}](kw1)=>/NP{//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 4 //NP{/NP{/?[Value IN {'regulation', 'upregulation', 'downregulation', 'up-regulation', 'down-regulation'}](kw1)}=>/PP{/NP{//?[Tag='GENE'](kw0)=>//?[Value='by'](kw3)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 5 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/?[Value IN {'regulates', 'upregulates', 'downregulates', 'up-regulates', 'down-regulates'}](kw1)=>/NP{//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, sent.value 6 //S{/NP{/?[Tag='GENE'](kw0)}=>/VP{/VP{//?[Value='in'](kw3)=>//?[Value='regulating'](kw1)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value" ], [ "Gene Regulate Gene (Xenobiotic Metabolism)", "1 //S{//?[Tag='GENE' AND Canonical LIKE 'CYP\\%'](kw0)<=>//?", "[Tag='GENE' AND Canonical IN {'AhR', 'CASR', 'CAR', 'PXR', 'NR1I2', 'NR1I3'}](kw1)} ::: distinct sent.cid, kw0.value, kw1.value, sent.value 2 //S{//?[Tag='GENE' AND Value LIKE 'cytochrome\\%'](kw0)<=>//?", "[Tag='GENE' AND Canonical IN {'AhR', 'CASR', 'CAR', 'PXR', 'NR1I2', 'NR1I3'}](kw1)} ::: distinct sent.cid, kw0.value, kw1.value, sent.value" ], [ "Negative Drug Induces/Metabolizes/Inhibits Gene", "1 //S{/?[Tag='DRUG'](kw0)=>/VP{/VP{/?[Value IN {'induced', 'inhibited', 'metabolized', 'metabolised'}](kw1)=>//?[Tag='GENE'](kw2)=>//?[Value='not'](kw3)=>//?", "[Tag='GENE'](kw4)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw4.value, kw3.value, sent.value 2 //S{/?[Tag='DRUG'](kw0)=>/?[Value='not'](kw3)=>/?[Tag='DRUG'](kw4)=>/?[Value='inhibited'](kw1)=>/?", "[Tag='GENE'](kw2)} ::: distinct sent.cid, kw4.value, kw1.value, kw2.value, kw3.value, sent.value 3 //S{/SBAR{/S{//?[Tag='DRUG'](kw0)}}=>/S{/S{//?[Value='metabolized'](kw1)=>//?[Tag='GENE'](kw2)=>//?[Value='not'](kw3)=>//?", "[Tag='GENE'](kw4)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw4.value, kw3.value, sent.value 4 //S{/NP{/PP{//?[Tag='GENE'](kw2)}}=>/VP{/VP{/?[Value IN {'induced', 'inhibited', 'metabolized', 'metabolised'}](kw1)=>//?[Tag='DRUG'](kw0)=>//?[Value='not'](kw3)=>//?", "[Tag='DRUG'](kw4)}}} ::: distinct sent.cid, kw4.value, kw1.value, kw2.value, kw3.value, sent.value 5 //S{/NP{/PP{//?[Tag='DRUG'](kw0)}}=>/VP{/VP{/?[Value IN {'induced', 'inhibited', 'metabolized', 'metabolised'}](kw1)=>//?[Tag='GENE'](kw2)=>//?[Value IN {'not','no'}](kw3)=>//?", "[Tag='GENE'](kw4)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw4.value, kw3.value, sent.value 6 //S{/?[Tag='DRUG'](kw0)=>/VP{/?[Value='not'](kw3)=>/VP{//?[Value IN {'induced', 'inhibited', 'metabolized', 'metabolised'}](kw1)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 7 //S{/NP{/?[Tag='GENE'](kw2)}=>/VP{/VP{/?[Value IN {'induced', 'inhibited', 'metabolized', 'metabolised'}](kw1)=>//?[Tag='DRUG'](kw0)=>//?[Value='not'](kw3)=>//?", "[Tag='DRUG'](kw4)}}} ::: distinct sent.cid, kw4.value, kw1.value, kw2.value, kw3.value, sent.value 8 //S{/NP{/NP{/?[Value='not'](kw3)=>/?[Tag='DRUG'](kw0)}}=>/VP{/?[Value IN {'inhibited','induced'}](kw1)=>/NP{//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 9 //S{/NP{/NP{/?[Value IN {'no','not'}](kw3)=>/?[Tag='DRUG'](kw0)}}=>/VP{/?[Value IN {'inhibited','induced'}](kw1)=>/NP{/?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 10 //S{/?[Tag='DRUG'](kw0)=>/S{/S{//?[Value='not'](kw3)=>//?[Value IN {'induce', 'inhibit'}](kw1)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 11 //S{/?[Tag='DRUG'](kw0)=>/?[Tag='GENE'](kw2)=>/?[Value IN {'not'}](kw3)=>/?", "[Value IN {'inhibit','induce'}](kw1)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 12 //S{/?[Tag='DRUG'](kw0)=>/VP{/?[Value='not'](kw3)=>/VP{/?[Value IN {'inhibit','induce'}](kw1)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 13 //S{/NP{/?[Tag='DRUG'](kw0)}=>/VP{/VP{/?[Value='not'](kw3)=>//?[Value='inhibit'](kw1)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 14 //S{/NP{/NP{/?[Tag='GENE'](kw2)}}=>/VP{/?[Value='not'](kw3)=>/VP{/?[Value='metabolize'](kw1)=>//?", "[Tag='DRUG'](kw0)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 15 //S{/NP{/NP{//?[Tag='DRUG'](kw0)}}=>/VP{/?[Value='not'](kw3)=>/VP{/?[Value IN {'inhibit','induce'}](kw1)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 16 //S{/NP{/?[Tag='DRUG'](kw0)}=>/VP{/?[Value='not'](kw3)=>/VP{/?[Value='inhibit'](kw1)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 17 //S{/NP{/?[Tag='DRUG'](kw0)}=>/VP{/?[Value IN {'induces', 'inhibits'}](kw1)=>/NP{//?[Tag='GENE'](kw2)=>//?[Value='not'](kw3)=>//?", "[Tag='GENE'](kw4)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw4.value, kw3.value, sent.value 18 //S{/?[Tag='DRUG'](kw0)=>/VP{/?[Value='inhibits'](kw1)=>/NP{//?[Tag='GENE'](kw2)=>//?[Value='not'](kw3)=>//?", "[Tag='GENE'](kw4)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw4.value, kw3.value, sent.value 19 //S{/?[Tag='DRUG'](kw0)=>/VP{/NP{//?[Value='no'](kw3)=>//?[Value IN {'induction','metabolism','inhibition'}](kw1)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 20 //S{/NP{/NP{//?[Tag='DRUG'](kw0)=>//?[Tag='GENE'](kw2)=>//?[Value IN {'induction','metabolism','inhibition'}](kw1)}}=>/VP{/?", "[Value='not'](kw3)}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 21 //S{/?[Tag='DRUG'](kw0)=>/VP{/?[Value='not'](kw3)=>/VP{//?[Value IN {'induction','metabolism','inhibition'}](kw1)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value 22 //S{/NP{/PP{//?[Tag='DRUG'](kw0)}}=>/VP{/?[Value='not'](kw3)=>/VP{//?[Value IN {'induction','metabolism','inhibition'}](kw1)=>//?", "[Tag='GENE'](kw2)}}} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw3.value, sent.value" ], [ "Negative Drug Induces Gene", "1 //S{//?[Tag='GENE'](kw0)=>//?[Value IN {'induced','increased','stimulated'}](kw1)=>//?[Value='by'](kw3)=>//?[Tag='DRUG'](kw2)=>//?[Value='not'](kw4)=>//?", "[Tag='DRUG'](kw5)} ::: distinct sent.cid, kw5.value, kw1.value, kw0.value, kw4.value, sent.value 2 //S{//?[Tag='GENE'](kw0)=>//?[Value='not'](kw4)=>//?[Tag='GENE'](kw5)=>//?[Value IN {'induced','increased','stimulated'}](kw1)=>//?[Value='by'](kw3)=>//?", "[Tag='DRUG'](kw2)} ::: distinct sent.cid, kw2.value, kw1.value, kw5.value, kw4.value, sent.value 3 //S{//?[Tag='DRUG'](kw0)=>//?[Value='not'](kw4)=>//?[Value IN {'induce','induced','increase','increased','stimulate','stimulated'}](kw1)=>//?", "[Tag='GENE'](kw2)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw4.value, sent.value 4 //S{//?[Tag='DRUG'](kw0)=>//?[Value='not'](kw4)=>//?[Tag='DRUG'](kw5)=>//?[Value IN {'induces','increases','stimulates','induced','increased','stimulated'}](kw1)=>//?", "[Tag='GENE'](kw2)} ::: distinct sent.cid, kw5.value, kw1.value, kw2.value, kw4.value, sent.value 5 //S{//?[Value='no'](kw5)=>//?[value IN {'induction','stimulation'}](kw1)=>//?[Value IN {'of'}](kw2)=>//?[Tag='GENE'](kw0)=>//?[value IN {'by'}](kw3)=>//?", "[Tag='DRUG'](kw4)} ::: distinct sent.cid, kw4.value, kw1.value, kw0.value, kw5.value, sent.value" ], [ "Negative Drug Inhibits Gene", "1 //S{//?[Tag='GENE'](kw0)=>//?[Value IN {'inhibited','decreased'}](kw1)=>//?[Value='by'](kw3)=>//?[Tag='DRUG'](kw2)=>//?[Value='not'](kw4)=>//?", "[Tag='DRUG'](kw5)} ::: distinct sent.cid, kw5.value, kw1.value, kw0.value, kw4.value, sent.value 2 //S{//?[Tag='GENE'](kw0)=>//?[Value='not'](kw4)=>//?[Tag='GENE'](kw5)=>//?[Value IN {'inhibited','decreased'}](kw1)=>//?[Value='by'](kw3)=>//?", "[Tag='DRUG'](kw2)} ::: distinct sent.cid, kw2.value, kw1.value, kw5.value, kw4.value, sent.value 3 //S{//?[Tag='DRUG'](kw0)=>//?[Value='not'](kw4)=>//?[Value IN {'inhibit','inhibited','decrease','decreased'}](kw1)=>//?", "[Tag='GENE'](kw2)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw4.value, sent.value 4 //S{//?[Tag='DRUG'](kw0)=>//?[Value='not'](kw4)=>//?[Tag='DRUG'](kw5)=>//?[Value IN {'inhibits','decreases','inhibited','decreased'}](kw1)=>//?", "[Tag='GENE'](kw2)} ::: distinct sent.cid, kw5.value, kw1.value, kw2.value, kw4.value, sent.value 5 //S{//?[Value='no'](kw5)=>//?[value IN {'inhibition'}](kw1)=>//?[Value IN {'of'}](kw2)=>//?[Tag='GENE'](kw0)=>//?[value IN {'by'}](kw3)=>//?", "[Tag='DRUG'](kw4)} ::: distinct sent.cid, kw4.value, kw1.value, kw0.value, kw5.value, sent.value" ], [ "Negative Gene Metabolizes Drug", "1 //S{//?[Tag='DRUG'](kw0)=>//?[Value IN {'metabolized','metabolised'}](kw1)=>//?[Value='by'](kw3)=>//?[Tag='GENE'](kw2)=>//?[Value='not'](kw4)=>//?", "[Tag='GENE'](kw5)} ::: distinct sent.cid, kw0.value, kw1.value, kw5.value, kw4.value, sent.value 2 //S{//?[Tag='DRUG'](kw0)=>//?[Value='not'](kw4)=>//?[Tag='DRUG'](kw5)=>//?[Value IN {'metabolized','metabolised'}](kw1)=>//?[Value='by'](kw3)=>//?", "[Tag='GENE'](kw2)} ::: distinct sent.cid, kw5.value, kw1.value, kw2.value, kw4.value, sent.value 3 //S{//?[Tag='GENE'](kw0)=>//?[Value='not'](kw4)=>//?[Value IN {'metabolize','metabolise'}](kw1)=>//?", "[Tag='DRUG'](kw2)} ::: distinct sent.cid, kw2.value, kw1.value, kw0.value, kw4.value, sent.value 4 //S{//?[Tag='GENE'](kw0)=>//?[Value='not'](kw4)=>//?[Tag='GENE'](kw5)=>//?[Value IN {'metabolize','metabolise','metabolizes','metabolises'}](kw1)=>//?", "[Tag='DRUG'](kw2)} ::: distinct sent.cid, kw2.value, kw1.value, kw5.value, kw4.value, sent.value" ], [ "Negative Gene Downregulates Gene", "1 //S{//?[Tag='GENE'](kw0)=>//?[Value IN {'suppressed','downregulated','inhibited','down-regulated','repressed','disrupted'}](kw1)=>//?[Value='by'](kw3)=>//?[Tag='GENE'](kw2)=>//?[Value='not'](kw4)=>//?", "[Tag='GENE'](kw5)} ::: distinct sent.cid, kw5.value, kw1.value, kw0.value, kw4.value, sent.value 2 //S{//?[Tag='GENE'](kw0)=>//?[Value='not'](kw4)=>//?[Tag='GENE'](kw5)=>//?[Value IN {'suppressed','downregulated','inhibited','down-regulated','repressed','disrupted'}](kw1)=>//?[Value='by'](kw3)=>//?", "[Tag='GENE'](kw2)} ::: distinct sent.cid, kw2.value, kw1.value, kw5.value, kw4.value, sent.value 3 //S{//?[Tag='GENE'](kw0)=>//?[Value='not'](kw4)=>//?[Value IN {'suppressed','suppress','downregulated','downregulate','inhibited','inhibit','down-regulated','down-regulate','repressed','repress','disrupted','disrupt'}](kw1)=>//?", "[Tag='GENE'](kw2)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw4.value, sent.value 4 //S{//?[Tag='GENE'](kw0)=>//?[Value='not'](kw4)=>//?[Tag='GENE'](kw5)=>//?[Value IN {'suppresses','downregulates','inhibits','down-regulates','represses','disrupts','suppressed','downregulated','inhibited','down-regulated','repressed','disrupted'}](kw1)=>//?", "[Tag='GENE'](kw2)} ::: distinct sent.cid, kw5.value, kw1.value, kw2.value, kw4.value, sent.value 5 //S{//?[Value='no'](kw5)=>//?[value IN {'inhibition','downregulation','down-regulation'}](kw1)=>//?[Value IN {'of'}](kw2)=>//?[Tag='GENE'](kw0)=>//?[value IN {'on'}](kw3)=>//?", "[Tag='GENE'](kw4)} ::: distinct sent.cid, kw0.value, kw1.value, kw4.value, kw5.value, sent.value" ], [ "Negative Gene Upregulates Gene", "1 //S{//?[Tag='GENE'](kw0)=>//?[Value IN {'activated','induced','stimulated','regulated','upregulated','up-regulated'}](kw1)=>//?[Value='by'](kw3)=>//?[Tag='GENE'](kw2)=>//?[Value='not'](kw4)=>//?", "[Tag='GENE'](kw5)} ::: distinct sent.cid, kw5.value, kw1.value, kw0.value, kw4.value, sent.value 2 //S{//?[Tag='GENE'](kw0)=>//?[Value='not'](kw4)=>//?[Tag='GENE'](kw5)=>//?", "[Value IN {'activated','induced','stimulated','regulated','upregulated','up-regulated'}](kw1) ::: distinct sent.cid, kw2.value, kw1.value, kw5.value, kw4.value, sent.value 3 //S{//?[Tag='GENE'](kw0)=>//?[Value='not'](kw4)=>//?[Value IN {'activated','induced','stimulated','regulated','upregulated','up-regulated','activate','induce','stimulate','regulate','upregulate','up-regulate'}](kw1)=>//?", "[Tag='GENE'](kw2)} ::: distinct sent.cid, kw0.value, kw1.value, kw2.value, kw4.value, sent.value 4 //S{//?[Tag='GENE'](kw0)=>//?[Value='not'](kw4)=>//?[Tag='GENE'](kw5)=>//?[Value IN {'activates','induces','stimulates','regulates','upregulates','up-regulates','activate','induce','stimulate','regulate','upregulate','up-regulate'}](kw1)=>//?", "[Tag='GENE'](kw2)} ::: distinct sent.cid, kw5.value, kw1.value, kw2.value, kw4.value, sent.value 5 //S{//?[Value='no'](kw5)=>//?[value IN {'induction','activation','stimulation','regulation','upregulation','up-regulation'}](kw1)=>//?[Value IN {'of'}](kw2)=>//?[Tag='GENE'](kw0)=>//?[value IN {'by'}](kw3)=>//?", "[Tag='GENE'](kw4)} ::: distinct sent.cid, kw4.value, kw1.value, kw0.value, kw5.value, sent.value" ], [ "Drug Gene Co-Occurrence", "1 //S{//?[Tag='DRUG'](kw0)<=>//?", "[Tag='GENE'](kw1)} ::: distinct sent.cid, kw0.value, kw1.value, kw0.type, kw1.type, sent.value" ] ]	1403.0541
[ [ "Volumes of convex lattice polytopes and a question of V. I. Arnold" ], [ "Abstract We show by a direct construction that there are at least $\\exp\\{cV^{(d-1)/(d+1)}\\}$ convex lattice polytopes in $\\mathbb{R}^d$ of volume $V$ that are different in the sense that none of them can be carried to an other one by a lattice preserving affine transformation.", "This is achieved by considering the family $\\mathcal{P}^d(r)$ (to be defined in the text) of convex lattice polytopes whose volumes are between $0$ and $r^d/d!$.", "Namely we prove that for $P \\in \\mathcal{P}^d(r)$, $d!\\mathrm{vol\\;} P$ takes all possible integer values between $cr^{d-1}$ and $r^d$ where $c>0$ is a constant depending only on $d$." ], [ "Introduction and main result", "Let $e_1,\\ldots ,e_d$ be the standard basis of $\\mathbb {R}^d$ , $d \\ge 2$ and for $r \\in \\mathbb {N}$ define $A(r)=\\mathrm {conv}\\lbrace re_1,\\ldots ,re_d\\rbrace \\mbox{ and } S(r)=\\mathrm {conv}\\lbrace A(r)\\cup \\lbrace 0\\rbrace \\rbrace .$ We will write $A^d(r)$ and $S^d(r)$ in case of need.", "We further define a family $\\mathcal {P}(r)=\\mathcal {P}^d(r)$ as the collection of all convex lattice polytopes $P$ with $A(r)\\subset P\\subset S(r)$ .", "As is well known, $v(P)=d!", "\\mathrm {vol\\;}P$ is an integer for $P \\in \\mathcal {P}(r)$ , and $0\\le v(P) \\le r^d$ since $A(r)$ and $S(r)$ belong to the family $\\mathcal {P}(r)$ .", "It is also clear that if $v(P)\\ne 0$ , then it is at least $r^{d-1}$ since $P\\in \\mathcal {P}(r)$ with $v(P)>0$ contains $d$ -dimensional simplex whose base is $A(r)$ .", "The question addressed in this paper concerns the set of values of $\\lbrace v(P):P \\in \\mathcal {P}^d(r)\\rbrace $ .", "It has emerged in connection with a problem of V. I. Arnold [1] (see also [2]) as we will explain soon.", "Our main result is Theorem 1.1 Given $d\\ge 2$ there is a number $c=c_d>0$ such that for all integers $v \\in [cr^{d-1},r^d]$ there is a polytope $P \\in \\mathcal {P}^d(r)$ with $v(P)=v$ .", "We will prove this result in a more precise form in Section .", "The case $d=2$ is simple: it is easy to see that $\\lbrace v(P):P \\in \\mathcal {P}^2(r)\\rbrace =\\lbrace 0,r,r+1,\\ldots ,r^2\\rbrace $ .", "We will show in Section that there are other gaps besides $(0,r^{d-1})$ in the set $\\lbrace v(P): P\\in \\mathcal {P}^d(r)\\rbrace $ when $d\\ge 3$ ." ], [ "Arnold's question", "Two convex lattice polytopes are equivalent if one can be carried to the other by a lattice preserving affine transformation.", "This is an equivalence relation and equivalent polytopes have the same volume.", "Let $N_d(V)$ denote the number of equivalence classes of convex lattice polytopes in $\\mathbb {R}^d$ of volume $V$ .", "(Of course, $d!V$ is a positive integer.)", "Arnold [1] showed that $V^{1/3} \\ll \\log N_2(V) \\ll V^{1/3} \\log V.$ After earlier results by Konyagin and Sevastyanov [6], the upper bound was improved and extended to higher dimensions to $\\log N_d(V) \\ll V^{(d-1)/(d+1)}$ by Bárány and Pach [4] (for $d=2$ ) and by Bárány and Vershik [5] (for $d\\ge 2$ ).", "The lower bound $\\log N_d(V) \\gg V^{(d-1)/(d+1)}$ for all $d\\ge 2$ has recently been proved in [2].", "More information about Arnold's question can be found in Arnold [1], Bárány [2], and Zong [8] and Liu, Zong [7].", "We obtain the same lower bound as a direct and fairly simple application of Theorem REF : Corollary 2.1 $V^{(d-1)/(d+1)} \\ll \\log N_d(V)$ .", "Some remarks are in place here about notation and terminology.", "A convex polytope $P\\subset \\mathbb {R}^d$ is a lattice polytope if its vertex set, $\\mathrm {vert\\;}P$ is a subset of ${\\mathbb {Z}^d}$ , the integer lattice.", "The number of vertices of a polytope $P$ is denoted by $f_0(P)$ .", "Throughout the paper we use, together with the usual “little oh” and “big Oh” notation, the convenient $\\ll $ symbol, which means, for functions $f,g :\\mathbb {R}_+ \\rightarrow \\mathbb {R}$ , that $f(V) \\ll g(V)$ if there are constants $V_0>0$ and $c>0$ such that $f(V) \\le cg(V)$ for all $V>V_0$ .", "These constants, to be denoted by $c,c_1,\\dots ,b,b_1\\dots $ may only depend on dimension.", "The Euclidean norm of the vector $x=(x_1,\\dots ,x_d)\\in \\mathbb {R}^d$ is $\\Vert x\\Vert =\\sqrt{x_1^2+\\dots +x_d^2}$ .", "$B^d$ denotes the Euclidean unit ball of $\\mathbb {R}^d$ , and $\\mathrm {vol\\;}B_d=\\omega _d$ .", "We write $[n]=\\lbrace 1,2,\\dots ,n\\rbrace $ .", "The paper is organized as follows.", "The main result is proved in the next section.", "The integer convex hull and some of its properties are given in Section .", "The proof of Corollary REF is the content of Section .", "Then we describe further gaps in $\\lbrace v(P):P\\in \\mathcal {P}^r(r)\\rbrace $ when $d\\ge 3$ .", "We finish with concluding remarks." ], [ "Proof of Theorem ", "As we have mentioned, $\\lbrace v(P):P \\in \\mathcal {P}^2(r)\\rbrace =\\lbrace 0,r,r+1,\\ldots ,r^2\\rbrace $ .", "We assume from now on that $d\\ge 3$ .", "Here comes the more precise version of Theorem REF .", "Theorem 3.1 Assume $d\\ge 3$ and $r> r_0=2^dd!$ .", "Then for every non-negative integer $m\\le (r-2^dd!", ")^d$ there is a $P \\in \\mathcal {P}^d(r)$ with $v(P)=r^d-m$ .", "This result implies Theorem REF for $r> r_0$ with $c=2^dd!$ for instance.", "For $r\\le r_0$ the theorem holds by choosing the constant $c$ large enough.", "It will be more convenient to work with $m(P)=d!\\mathrm {vol\\;}(S^d(r)\\setminus P)$ which we call the missed volume of $P \\in \\mathcal {P}^d(r)$ .", "(It would be more appropriate to call it missed volume times $d!$ though.)", "With this notation Theorem REF says that the set $\\lbrace m(P):P \\in \\mathcal {P}^d(r)\\rbrace $ , that is, the set of missed volumes, contains all integers between 0 and $(r-2^dd!", ")^d$ provided $r>r_0$ .", "The proof is based on the following Lemma 3.1 Assume $d\\ge 2$ and let $g(x)=(2x)^d$ and $m \\in \\mathbb {N}$ .", "Then there are integers $x_0\\ge x_1 \\ge \\ldots \\ge x_{d-1}\\ge 0$ and an integer $m_d \\in \\lbrace 0,1,\\ldots ,2^dd!\\rbrace $ such that $m=g(x_0)+g^{\\prime }(x_1)+g^{\\prime \\prime }(x_2)+\\ldots +g^{(d-1)}(x_{d-1})+m_d.$ Proof.", "We give an algorithm that outputs the numbers $x_0,\\ldots ,x_{d-1}$ and $m_0=m,m_1,\\ldots ,m_d$ .", "Start with $m_0=m$ and let $x_0$ be the unique non-negative integer with $g(x_0)\\le m_0<g(x_0+1)$ .", "If $m_{i-1}$ and $x_{i-1}$ have been defined then set $m_i=m_{i-1}-g^{(i-1)}(x_{i-1})$ and let $x_i$ be the unique non-negative integer with $g(x_i)\\le m_i<g(x_i+1)$ .", "We stop with $m_{d-1}$ and $x_{d-1}$ and define $m_d=m_{d-1}-g^{(d-1)}(x_{d-1})$ .", "We claim that $x_i\\le x_{i-1}$ .", "Note first that by construction and by the intermediate value theorem $m_i&=&m_{i-1}-g^{(i-1)}(x_{i-1})<g^{(i-1)}(x_{i-1}+1)-g^{(i-1)}(x_{i-1})\\\\&=&g^{(i)}(\\xi )\\le g^{(i)}(x_{i-1}+1),$ where $\\xi \\in [x_{i-1},x_{i-1}+1]$ , and we also used that $g^{(i)}(x)$ is increasing for $x\\ge 0$ .", "So if, contrary to the claim, we had $x_i > x_{i-1}$ , then $x_{i-1}+1\\le x_i$ .", "As $g^{(i)}(x)$ is increasing we have $m_i < g^{(i)}(x_{i-1}+1) \\le g^{(i)}(x_i)\\le m_i,$ a contradiction.", "The same method gives that $m_d\\le 2^dd!$ : $m_d&=&m_{d-1}-g^{(d-1)}(x_{d-1})<g^{(d-1)}(x_{d-1}+1)-g^{(d-1)}(x_{d-1})\\\\&=&g^{(d)}(\\xi )=2^dd!,$ for all $x$ .", "The proof is finished by adding the defining equalities $m_i=m_{i-1}-g^{(i-1)}(x_{i-1})$ for $i=1,2,\\ldots ,d$ .$\\Box $ Remark.", "The same method works for every polynomial $g$ of degree $d$ such that $g^{(i)}(x)>0$ for all $i=0,1,\\ldots ,d$ and $x>0$ .", "We return now to the proof of Theorem REF .", "So given $r >r_0$ and $m \\in \\lbrace 0,1,\\ldots ,(r-2^dd!", ")^d\\rbrace $ we are going to construct $P \\in \\mathcal {P}(r)$ with $m(P)=m$ .", "This is easy if $m \\le r$ : the simplex $\\Delta $ with vertices $me_1,e_2,\\ldots ,e_d$ has $v(\\Delta )=m$ so the missed volume of the closure of $S(r)\\setminus \\Delta $ , which is a lattice polytope, is $m$ .", "So assume $m > r$ .", "Apply Lemma REF with $m$ which is at most $(r-2^dd!", ")^d$ , to get numbers $x_0,\\ldots ,x_{d-1}$ and $m_d$ .", "Note that $2x_0 \\le r-2^dd!$ and also $x_0\\ge 2$ as $m>r\\ge r_0=2^dd!$ .", "Next we are going to define simplices $\\Delta _0,\\ldots ,\\Delta _d$ that are lattice polytopes contained in $S^d(r)$ , are pairwise internally disjoint, and $v(\\Delta _i)=g^{(i)}(x_i)$ for $i=0,1,\\ldots ,d-1$ and $v(\\Delta _d)=m_d$ .", "We set $e_i^=2x_0e_i$ and define $\\Delta _0=\\mathrm {conv}\\lbrace e_1^,\\ldots ,e_d^\\rbrace $ , a non-degenerate simplex because $x_0\\ge 2$ .", "Clearly, $v(\\Delta _0)=2^dx_0^d=g(x_0)$ and $\\Delta _0 \\subset S(r)$ .", "Next, for $i \\in [d]$ , we let $\\Delta _i$ be the convex hull of vectors $&\\;&e_i^,e_i^+2^dd(d-1)\\ldots (d-i+1)e_i \\mbox{ and }\\\\&\\;&e_i^+(e_j-e_i)\\; (j<i) \\mbox{ and } e_i^+x_i(e_j-e_i)\\; (j>i),$ this is a simplex whose vertices are lattice points in $S(r)$ , as one can check easily.", "Its edges starting from vertex $e_i^$ are the vectors $e_j-e_i$ for $j<i$ , $2^dd(d-1)\\ldots (d-i+1)e_i$ , and $x_i(e_j-e_i)$ for $j>i$ .", "As the vectors $e_1-e_i,\\ldots ,e_{i-1}-e_i,e_i,e_{i+1}-e_i,\\ldots ,e_d-e_i$ form a basis of the lattice ${\\mathbb {Z}^d}$ , $v(\\Delta _i)=2^dd(d-1)\\ldots (d-i+1)x_i^{(d-i)}=g^{(i)}(x_i).$ Note that $\\Delta _i$ may be degenerate (exactly when $x_i=0$ ) but only for $i>0$ .", "These simplices are internally disjoint.", "Indeed, $\\Delta _0$ lies on one side of the hyperplane $\\textrm {aff\\;}\\lbrace e_1^,\\ldots ,e_d^\\rbrace $ and all other simplices are on the other side.", "Further, as $2x_0 \\le r-2^dd!$ and $x_i \\le x_0$ , every $\\Delta _i$ with $i>0$ is contained in the simplex whose vertices are $e_i^,e_i^+2^dd!e_i$ and $\\frac{1}{2} (e_i^+e_j^)$ for $j\\ne i$ , and these larger simplices are internally disjoint.", "We check next that the closure of $S(r) \\setminus \\bigcup _0^d \\Delta _i$ is a lattice polytope $P$ in $\\mathcal {P}(r)$ .", "Write $X_0$ for the set of vertices of $\\Delta _0$ except 0, and $X_i$ for the set of vertices of $\\Delta _i$ except $e_i^$ .", "Let $Y$ be the set of vertices of $A(r)$ .", "$P$ is obtained from $S(r)$ by deleting $\\Delta _0,\\Delta _1,\\ldots ,\\Delta _d$ in this order.", "Deleting $\\Delta _0$ results in $P_0=\\mathrm {conv}(Y \\cup X_0)\\in \\mathcal {P}(r)$ .", "Deleting $\\Delta _1$ from $P_0$ gives $P_1=\\mathrm {conv}(Y \\cup X_0\\cup X_1) \\in \\mathcal {P}(r)$ .", "Similarly, deleting $\\Delta _i$ from $P_{i-1}$ results in a convex lattice polytope $P_i \\in \\mathcal {P}(r)$ whose vertices are $(Y \\cup \\bigcup _0^iX_i)$ .", "This works even if $\\Delta _i$ is degenerate; then $P_i=P_{i-1}$ .", "Finally we have $P=P_d \\in \\mathcal {P}(r)$ and the missed volume of $P$ is $m(P)=\\sum _0^d v(\\Delta _i)=\\sum _0^{d-1}g^{(i)}(x_i-1)+m_d=m.$ This finishes the construction and gives a polytope $P \\in \\mathcal {P}(r)$ with $m(P)=m$ if $r\\ge r_0$ .", "$\\Box $ Remark.", "The same construction with minor modification works for all $m\\le (r-2^dd)^d$ and $r\\ge r_0=2^dd$ ." ], [ "The integer convex hull", "Suppose $K \\subset \\mathbb {R}^d$ is a bounded convex set.", "Its integer convex hull, $I(K)$ , is defined as $I(K)=\\mathrm {conv}(K \\cap {\\mathbb {Z}^d}),$ which is a convex lattice polytope if nonempty.", "To avoid some trivial complications we assume that $r$ is large enough, $r\\ge r_d$ say.", "One important ingredient of our construction is $Q_r=I(rB^d)=\\mathrm {conv}({\\mathbb {Z}^d}\\cap rB^d).$ Trivially $\\mathrm {vol\\;}Q_r \\le \\omega _dr^d$ .", "It is proved in Bárány and Larman in [3] that $\\mathrm {vol\\;}(rB_d \\setminus Q_r) \\ll r^{d\\frac{d-1}{d+1}}$ .", "The last exponent will appear so often that we write $D=d\\frac{d-1}{d+1}$ .", "The number of vertices of $Q_r$ is estimated in [3] as $r^D \\ll f_0(Q_r) \\ll r^D.$ The vertices of $Q_r$ are very close to the boundary of $rB^d$ .", "More precisely, we have the following estimate which is also used in [3].", "Lemma 4.1 If $x$ is a vertex of $Q_r$ , then $r-\\Vert x\\Vert \\ll r^{-(d-1)/(d+1)}$ .", "Proof.", "The set $rB^d \\cap (2x-rB^d)$ is convex and centrally symmetric with center $x \\in {\\mathbb {Z}^d}$ .", "It does not contain any lattice point $z\\ne x$ : if it does, then both $z$ and $2x-z$ are lattice points in $rB^d$ and $x=\\frac{1}{2} \\left(z+(2x-z)\\right)$ is not a vertex of $Q_r$ .", "By Minkowksi's classical theorem, $\\mathrm {vol\\;}(rB^d \\cap (2x-rB^d))< 2^d$ .", "The estimate in the lemma follows from this by a simple computation.$\\Box $ We let $\\mathbb {R}^d_+$ denote the set of $x=(x_1,\\ldots ,x_d)\\in \\mathbb {R}^d$ with $x_i\\ge 0$ for every $i\\in [d]$ .", "In the proof of Corollary REF we will consider $Q^r=Q_r \\cap \\mathbb {R}^d_+$ .", "It is clear that $Q^r=I(rB^d \\cap \\mathbb {R}^d_+)$ and further, that $\\mathrm {vol\\;}\\left((rB^d \\cap \\mathbb {R}^d_+)\\setminus Q^r\\right) \\ll r^D.$ Let $X$ be the set of those vertices $x=(x_1,\\ldots ,x_d)$ of $Q^r$ for which $x_i>0$ for all $i \\in [d]$ .", "We claim that $r^D \\ll \|X\| \\ll r^D$ Only the lower bound needs some explanation.", "The number of vertices of $Q_r$ with $x_i=0$ for some $i \\in [d]$ is less than $d$ times the number of vertices of $Q_r^{d-1}$ which is of order $r^{(d-1)(d-2)/d}=o(r^D)$ .", "Then $\|X\|\\ge 2^{-d}\\left((f_0(Q_r)-o(r^D)\\right)$ , so indeed $r^D \\ll \|X\|$ .", "Lemma REF says that $\\Vert x\\Vert \\ge r-b_1r^{-(d-1)/(d+1)}$ where $b_1>0$ depends only on $d$ .", "Define $r_0=r-b_1r^{-(d-1)/(d+1)}$ , so $X$ lies in the annulus $rB^d \\setminus r_0B^d$ .", "Consequently all lattice points in $r_0B^d\\cap \\mathbb {R}^d_+$ are contained in $Q^r\\setminus X$ and so $I(r_0B^d\\cap \\mathbb {R}^d_+)\\subset I(Q^r\\setminus X)$ .", "The result from [3] cited above applies to $I(r_0B^d \\cap \\mathbb {R}^d_+)$ and gives that $\\mathrm {vol\\;}\\left((r_0B^d \\cap \\mathbb {R}^d_+) \\setminus I(r_0B^d \\cap \\mathbb {R}^d_+)\\right) \\ll r_0^D\\ll r^D$ as $r_0<r$ .", "This implies that with a suitable constant $b>0$ $2^{-d}\\omega _d r^d -br^D \\le \\mathrm {vol\\;}I(Q^r\\setminus X) \\le 2^{-d}\\omega _d r^d.$ After these preparation we are ready for the proof of Corollary REF ." ], [ "Proof of Corollary ", "Given a (large enough) number $V$ with $d!V \\in \\mathbb {N}$ we are going to construct many non-equivalent convex lattice polytopes whose volume equals $V$ .", "As a first step, we define $r$ via the equation $V=2^{-d}\\omega _d r-br^D.$ Consider $Q^r$ from the previous section.", "For $Z \\subset X$ we define $Q(Z)=I(Q^r \\setminus Z)$ , $Q(Z)$ is a convex lattice polytope, it contains $Q(X)$ .", "So $V \\le \\mathrm {vol\\;}Q(X) \\le \\mathrm {vol\\;}Q(Z) \\le 2^{-d}\\omega _d r^d$ .", "Set $m(Z)=\\mathrm {vol\\;}Q(Z) -V$ .", "Then $m(Z)=\\mathrm {vol\\;}Q(Z)-V \\le 2^{-d}\\omega _dr^d -V=br^D.$ Since $r^D\\ll \|X\|$ , the number of polytopes $Q(Z)$ is $2^{\|X\|} \\ge \\exp \\lbrace b_2r^D\\rbrace =\\exp \\lbrace b_3 V^{(d-1)/(d+1)}\\rbrace $ with suitable positive constants $b_2,b_3$ .", "This is what we need in Corollary REF .", "But the volumes of the $Q(Z)$ are larger than $V$ .", "So we are going to cut off volume $\\mathrm {vol\\;}Q(Z)-V=m(Z)/d!$ from $Q(Z)$ so that what is left is a lattice polytope of volume exactly $V$ .", "We will do so using Theorem REF or rather Theorem REF the following way.", "Set $\\rho = \\lfloor r/10 \\rfloor \\in \\mathbb {N}$ and assume $r_d$ is so large that $\\rho >r_0=2^dd!$ .", "Consider $A(\\rho )$ , $S(\\rho )$ and $\\mathcal {P}(\\rho )$ from Section .", "Theorem REF says that given if $m=d!m(Z)\\le (\\rho -2^dd!", ")^d$ , there is a polytope $P=P(Z) \\in \\mathcal {P}$ with $m(P)=m$ .", "Then $P^(Z)=\\left[Q(Z)\\setminus S(r)\\right] \\cup P(Z)=Q(Z)\\setminus \\left[S(r) \\setminus P(Z)\\right]$ is a convex lattice polytope, and its volume is exactly $V$ .", "Now we have $\\exp \\lbrace b V^{(d-1)/(d+1)}\\rbrace $ convex lattice polytopes $P^(Z)$ , each of volume $V$ .", "We claim that any one of them is equivalent to at most $d!$ other $P^(W)$ .", "This will clearly finish that proof of $\\log N_d(V) \\gg V^{(d-1)/(d+1)}$ .", "To prove this note first that $P^(Z)$ has a long edge on the segment $[0,re_i]$ for all $i\\in [d]$ .", "This edge is of the form $E_i=E_i(Z)=[\\alpha _i e_i,\\lfloor r \\rfloor e_i]$ and $\\alpha _i\\le \\rho \\le r/10$ .", "It is quite easy to check (we omit the details) that $P^(Z)$ has exactly these $d$ edges whose length is larger than $0.9r$ .", "So if $P^(Z)$ and $P^(W)$ are equivalent, then the lattice preserving affine transformation $T$ that carries $P^(Z)$ to $P^(W)$ , has to map each $E_i(Z)$ to a uniquely determined $E_j(W)$ .", "As the lines containing $E_i(Z)$ all pass through the origin, $T(0)=0$ and so $T$ is a linear map.", "Thus $T$ permutes the axes and is lattice preserving.", "Then it has to permute the vectors $e_1,\\ldots ,e_d$ .", "There are $d!$ such transformations.", "Consequently, $P^(Z)$ is equivalent with at most $d!$ other polytopes of the form $P^(W)$ .$\\Box $ Remark.", "The above construction can be modified so that all $P^(Z)$ are non-equivalent.", "Namely, set $t=\\lfloor r \\rfloor $ and replace $Z\\subset X$ by $Z^0=Z\\cup \\bigcup _1^d \\lbrace te_i,(t-1)e_i,\\ldots , (t-i+1)e_i\\rbrace .$ Set $Q(Z^0)=I(Q^r \\setminus Z^0)$ and $P^(Z^0)=\\left[Q(Z^0)\\setminus S(r)\\right]\\cup P(Z^0)$ where $P(Z^0) \\in \\mathcal {P}(\\rho )$ is again chosen so that $\\mathrm {vol\\;}P^(Z^0)=V$ .", "The long edges of $P^(Z^0)$ are almost the same $E_i(Z)$ except that this time each carries a `marker', namely the last $i$ lattice points are missing from $E_i(Z)$ .", "So if $P^(Z^0)$ and $P^(W^0)$ are equivalent, then the corresponding lattice preserving affine transformation has to be the identity." ], [ "Gaps in $v(\\mathcal {P})$", "Let $A_k=\\lbrace (x_1, x_2, \\cdots , x_d): x_1+x_2+\\cdots x_d=k,x_i\\in \\lbrace 0, 1, 2, \\cdots , \\\\ k \\rbrace , i\\in [d] \\rbrace $ , where $k=0, 1,\\cdots , r$ .", "For any $P \\in \\mathcal {P}(r)$ , if $P \\cap A_{r-2} \\ne \\emptyset $ , then clearly $v(P)\\ge 2 r^{d-1}$ .", "Now suppose that $P\\cap A_{r-2} = \\emptyset $ but $P\\cap A_{r-1} \\ne \\emptyset $ .", "Assume that $P=\\mathrm {conv}\\lbrace A(r)\\cup B\\rbrace $ , where $B \\subset A_{r-1}$ , $B=\\lbrace b_1, b_2, \\cdots , b_t\\rbrace $ , $b_i=(b^i_1, b^i_2, \\cdots , b^i_d)$ , $b^i_j\\in \\lbrace 0, 1, 2, \\cdots , r-1 \\rbrace $ for all $i\\in [t]$ and $j\\in [d]$ .", "For any $b_i, b_k \\in B$ , let $q_{ik}=\\max \\lbrace \|b^i_j - b^k_j\| : j\\in [d]\\rbrace $ and $q= \\max \\lbrace q_{ik} : b_i, b_k\\in B\\rbrace $ .", "Theorem 6.1 For any $d\\ge 3$ and $P \\in \\mathcal {P}(r)$ , $v(P)\\notin [r^{d-1}+1,r^{d-1}+r^{d-2}-1]$ .", "If $\|B\|=1$ , then $P$ is actually a $d$ -dimensional simplex with base $A(r)$ , which means that $v(P)=r^{d-1}$ .", "If $\|B\|=2$ , then suppose that $B=\\lbrace b_1, b_2\\rbrace $ .", "In this case $q=q_{12}=\\max \\lbrace \|b^1_j-b^2_j\|: j\\in [d] \\rbrace $ and $q \\ge 1$ as $b_1\\ne b_2$ .", "Thus we may assume without loss of generality, that $b^1_d - b^2_d= q\\ge 1$ .", "Let $P_1=\\mathrm {conv}\\lbrace A(r)\\cup \\lbrace b_1\\rbrace \\rbrace $ , $P_2=\\mathrm {conv}$ $\\lbrace re_1, re_2,\\cdots ,$ $re_{d-1}, b_1, b_2\\rbrace $ .", "Since $b^1_d \\ne b^2_d$ , $P_2$ is a $d$ -dimensional simplex.", "It is easy to check that $P_1$ and $P_2$ are internally disjoint which implies $v(P)=v(P_1)+v(P_2)$ .", "Clearly, $v(P_1)=r^{d-1}$ .", "Furthermore, we have $v(P_2)= \\left\|\\det \\left(\\begin{array}{ccccc}1 & \\ldots & 1 & 1 & 1\\\\re_1 & \\ldots & re_{d-1} & b_1 & b_2\\end{array}\\right)\\right\|=\|b^1_d-b^2_d\|r^{d-2},$ which means that $v(P)=r^{d-1}+q r^{d-2}\\ge r^{d-1}+r^{d-2}$ .", "Clearly $v(P)\\ge r^{d-1}+r^{d-2}$ still holds for any $P \\in \\mathcal {P}(r)$ satisfying $\|B\|\\ge 3$ .", "The proof is complete.", "Now we consider the special case when $d=3$ .", "Define a graph $G=(V(G), E(G))$ such that $V(G)=A_{r-1}$ , $E(G)=\\lbrace b_ib_k:\|b^i_1 - b^k_1\|+ \|b^i_2 - b^k_2\|+ \|b^i_3 - b^k_3\|=2 \\rbrace $ .", "Clearly $G$ is a triangular grid graph with boundary $\\mathrm {conv}\\lbrace (r-1)e_1,(r-1)e_2\\rbrace \\cup \\mathrm {conv}\\lbrace (r-1)e_1, (r-1)e_3\\rbrace \\cup \\mathrm {conv}\\lbrace (r-1)e_2,(r-1)e_3\\rbrace $ .", "Furthermore, it is not difficult to see that for $b_i,b_k\\in A_{r-1}$ , $b_ib_k\\in E(G)$ if and only if $q_{ik}=1$ .", "For any $b_i, b_k \\in V(G)$ , let $l_{b_ib_k}$ denote the line determined by $b_i, b_k$ .", "In the hyperplane determined by $A_{r-1}$ , let $H^+_{ik}$ denote the open halfplane bounded by $l_{b_ib_k}$ such that $\|H^+_{ik}\\cap \\lbrace (r-1)e_1, (r-1)e_2, (r-1)e_3\\rbrace \|=2$ and $H^-_{ik}$ the open halfplane bounded by $l_{b_ib_k}$ satisfying $\|H^-_{ik}\\cap \\lbrace (r-1)e_1, (r-1)e_2, (r-1)e_3\\rbrace \|=1$ .", "Furthermore, let $d_G(b_i,b_k)$ denote the graphic distance between $b_i$ and $b_k$ , i.e., the length of the shortest paths between $b_i$ and $b_k$ .", "Clearly, $d_G(b_i, b_k)=q_{ik}$ .", "For any $b_i\\in A_{r-1}$ , let $D_{b_i}(s)=\\lbrace b_k: b_k\\in A_{r-1}, d_G(b_i, b_k) \\le s\\rbrace $ .", "Theorem 6.2 If $d=3$ , then for any $P \\in \\mathcal {P}(r)$ , $v(P)\\notin [r^{2}+r+2,r^{2}+2r-1]$ .", "If $\|B\|=1$ , clearly $v(P)=r^2$ .", "If $\|B\|=2$ , then by the proof of Theorem REF we know that $v(P)\\in \\lbrace r^2+kr: k=1, 2, \\cdots , r-1\\rbrace $ .", "Now suppose that $\|B\|=t$ $(t\\ge 3)$ and $B=\\lbrace b_1, b_2, \\cdots , b_t\\rbrace $ , where $b_i=(b^i_1, b^i_2, b^i_3)$ , $b^i_j\\in \\lbrace 0, 1, 2, \\cdots , r-1 \\rbrace $ for all $i\\in [t]$ and $j\\in [3]$ .", "If $q\\ge 2$ , assume without loss of generality that $q_{12}=q\\ge 2$ , then we have $v(P)\\ge v(\\mathrm {conv}\\lbrace A(r)\\cup \\lbrace b_1, b_2\\rbrace \\rbrace )$ $= r^{2}+qr$ $\\ge r^{2}+2r$ .", "Now suppose that $q=1$ .", "Then any two points in $B$ are adjacent in $G$ , which forces $t=3$ and $\\mathrm {conv}B$ is a 2-dimensional simplex homothetic to $A(r)$ .", "Furthermore, $\\mathrm {conv}B=\\theta _B+ \\frac{\\lambda _B}{r}A(r)$ , where $\\lambda _B=\\pm 1$ .", "Suppose without loss of generality that $l_{b_1b_2}\\parallel l_{re_1,re_2}$ and the vectors $\\overrightarrow{b_1b_2}$ and $\\overrightarrow{re_1,re_2}$ have the same direction.", "If $b_3 \\in H^-_{12}$ , then $\\lambda _B=1$ and $v(P)=v(\\mathrm {conv}\\lbrace A(r)\\cup \\lbrace b_1, b_2\\rbrace \\rbrace )$ $+v(\\mathrm {conv}\\lbrace re_3, b_1, b_2,b_3\\rbrace )$ $=r^2+r+1$ .", "If $b_3 \\in H^+_{12}$ , then $\\lambda _B=-1$ and $v(P)=v(\\mathrm {conv}\\lbrace A(r)\\cup \\lbrace b_1, b_2\\rbrace \\rbrace )$ $+v(\\mathrm {conv}\\lbrace re_1,re_2, b_1,b_3\\rbrace )$ $+v(\\mathrm {conv}\\lbrace re_2, b_1, b_2,b_3\\rbrace )$ $=(r^2+r)+r+1=r^2+2r+1$ .", "Combining the above discussions we see that there is no $v(P)$ of $P\\in \\mathcal {P}(r)$ lying in the interval $[r^{2}+r+2, r^{2}+2r-1]$ , and the proof is complete.", "Theorem 6.3 If $d=3$ and $r\\ge 6$ , then for any $P \\in \\mathcal {P}(r)$ , $v(P)\\notin [r^{2}+2r+5, r^{2}+3r-1]$ .", "Now still suppose that $\|B\|=t$ and $B=\\lbrace b_1, b_2, \\cdots , b_t\\rbrace $ , where $b_i=(b^i_1, b^i_2, b^i_3)$ , $b^i_j\\in \\lbrace 0, 1, 2, \\cdots , r-1 \\rbrace $ for all $i\\in [t]$ and $j\\in [3]$ .", "If $q\\ge 3$ , say, $q_{12}=q\\ge 3$ , then $v(P)\\ge v(\\mathrm {conv}\\lbrace A(r)\\cup \\lbrace b_1, b_2\\rbrace )$ $=r^2+qr \\ge r^2+3r$ .", "If $q\\le 2$ , then combining the proof of Theorem REF , we only need to consider the cases when $B$ satisfies $t\\ge 3$ and $q=2$ .", "Figure: t≥3t\\ge 3,q=2q=2.Case 1.", "$\\exists \\ b_i, b_k \\in B$ such that $q_{ik}=2$ , $l_{b_ib_k}$ is parallel to one side of $A(r-1)$ .", "Assume without loss of generality that $b_1, b_2 \\in B$ such that $q_{12}=2$ , $l_{b_1b_2}\\parallel l_{(r-1)e_1,(r-1)e_2}\\parallel l_{re_1,re_2}$ and the vectors $\\overrightarrow{b_1b_2}$ and $\\overrightarrow{re_1,re_2}$ have the same direction.", "Since $q=2$ , $B\\subset D_{b_1}(2)\\cap D_{b_2}(2)$ $=\\lbrace b_1, b_2, c_0, c_1, c_2,c_3,$ $c_4, c_5, c_6\\rbrace $ , as shown in Figure REF $(a)$ , where $c_0\\in l_{b_1b_2}$ , $\\lbrace c_1, c_2, c_3\\rbrace \\subset H^+_{12}$ and $\\lbrace c_4, c_5, c_6\\rbrace \\subset H^-_{12}$ .", "Clearly, $v(\\mathrm {conv}\\lbrace A(r)\\cup \\lbrace b_1, b_2, c_1\\rbrace \\rbrace )$ $=v(\\mathrm {conv}\\lbrace A(r)\\cup \\lbrace b_1,b_2\\rbrace \\rbrace )$ $+v(\\mathrm {conv}\\lbrace b_1, c_1, re_1, re_2\\rbrace )$ $+v(\\mathrm {conv}\\lbrace b_1, c_1, b_2,re_2\\rbrace )$ $=(r^2+2r)+r+2=r^2+3r+2.$ Similarly, $v(\\mathrm {conv}\\lbrace A(r)\\cup \\lbrace b_1, b_2,c_2\\rbrace \\rbrace )$ $=r^2+3r+2.$ As a result, if $B\\cap H^+_{12} \\ne \\emptyset $ , then $v(P)\\ge r^2+3r+2$ .", "If $B\\cap H^+_{12} =\\emptyset $ , then all the possible values of $v(P)$ are: $\\begin{array}{l}v(\\mathrm {conv}\\lbrace A(r)\\cup \\lbrace b_1, b_2,c_0\\rbrace \\rbrace )=r^2+2r;\\\\v(\\mathrm {conv}\\lbrace A(r)\\cup \\lbrace b_1, b_2,c_4\\rbrace \\rbrace )=r^2+2r+2;\\\\v(\\mathrm {conv}\\lbrace A(r)\\cup \\lbrace b_1, b_2,c_5\\rbrace \\rbrace )=r^2+2r+2;\\\\v(\\mathrm {conv}\\lbrace A(r)\\cup \\lbrace b_1, b_2,c_4, c_5\\rbrace \\rbrace )=r^2+2r+3;\\\\v(\\mathrm {conv}\\lbrace A(r)\\cup \\lbrace b_1, b_2,c_6\\rbrace \\rbrace )=r^2+2r+4.\\end{array}$ Case 2.", "Otherwise, suppose $b_1, b_2\\in B$ such that $l_{b_1b_2}\\perp l_{(r-1)e_1, (r-1)e_2}$ .", "Then we have $B\\subset \\lbrace b_1, b_2,c_1, c_2, c_3, c_4\\rbrace $ and $B$ contains at most two adjacent points among $\\lbrace c_1, c_2, c_3, c_4\\rbrace $ , as shown in Figure REF $(b)$ .", "Clearly, $ v(\\mathrm {conv}\\lbrace A(r)\\cup \\lbrace b_1, b_2,c_2\\rbrace \\rbrace )=v(\\mathrm {conv}\\lbrace A(r)\\cup \\lbrace b_1, b_2\\rbrace )$ $+v(\\mathrm {conv}\\lbrace b_1, b_2,$ $ c_2, re_1\\rbrace )$ $=r^2+2r+1$ .", "Similarly, $v(\\mathrm {conv}\\lbrace A(r)\\cup \\lbrace b_1, b_2,c_3\\rbrace \\rbrace ) =r^2+2r+1$ .", "Furthermore, $v(\\mathrm {conv}\\lbrace A(r)\\cup \\lbrace b_1, b_2,c_1\\rbrace \\rbrace )$ $=v(\\mathrm {conv}\\lbrace A(r)\\cup \\lbrace b_1, b_2,c_1, c_2\\rbrace \\rbrace )$ $=v(\\mathrm {conv}\\lbrace A(r)\\cup \\lbrace b_2, c_1\\rbrace )$ $+v(\\mathrm {conv}\\lbrace b_1, b_2, re_2, re_3\\rbrace )+$ $v(\\mathrm {conv}\\lbrace b_1, b_2, c_1, re_3\\rbrace )$ $=(r^2+2r)+r+3=r^2+3r+3$ .", "Similarly, $v(\\mathrm {conv}\\lbrace A(r)\\cup \\lbrace b_1, b_2,c_4\\rbrace \\rbrace )$ $=v(\\mathrm {conv}\\lbrace A(r)\\cup \\lbrace b_1, b_2,c_3, c_4\\rbrace \\rbrace )$ $=r^2+3r+3$ .", "Finally, $v(\\mathrm {conv}\\lbrace A(r)\\cup \\lbrace b_1, b_2,c_2, c_3\\rbrace \\rbrace )$ $=v(\\mathrm {conv}\\lbrace A(r)\\cup \\lbrace b_1, b_2\\rbrace )$ $+v(\\mathrm {conv}\\lbrace b_1, b_2, c_2, re_1\\rbrace )$ $+v(\\mathrm {conv}\\lbrace b_1, b_2, c_3, re_2\\rbrace )$ $=r^2+2r+2$ .", "Combining all the discussions above, we have the following facts: if $\|B\|=1$ , then $v(P)=r^2$ ; if $\|B\|=2$ , then $v(P)\\in \\lbrace r^2+kr: k=1, 2, \\cdots , r-1\\rbrace $ ; if $\|B\|\\ge 3$ and $q=1$ , then $v(P)\\in \\lbrace r^2+r+1, r^2+2r+1\\rbrace $ ; if $\|B\|\\ge 3$ and $q=2$ , then either $v(P)\\in [r^2+2r, r^2+2r+4]$ or $v(P)\\ge r^2+3r$ ; if $\|B\|\\ge 3$ and $q\\ge 3$ , then $v(P)\\ge r^2+3r$ .", "If $r\\ge 6$ , then $(r^2+3r-1)-(r^2+2r+5)=r-6\\ge 0$ , which means that $[r^2+2r+5, r^2+3r-1]$ is nonempty.", "Clearly there is no $P \\in \\mathcal {P}(r)$ satisfying $v(P)\\in [r^{2}+2r+5, r^{2}+3r-1]$ .", "The proof is complete.", "Corollary 6.1 For every $v\\in [r^{2}+2r, r^{2}+2r+4]$ , there exists a $P \\in \\mathcal {P}(r)$ such that $v(P)=v$ ." ], [ "Concluding remarks", "We mention further that, as Arnold [1] suggests, the paraboloid $D_r=\\lbrace x \\in \\mathbb {R}^d_+:x_1^2+\\dots +x_{d-1}^2 \\le x_d \\le r^2\\rbrace $ can be used to give the lower bound in Corollary REF as the following proof, or rather sketch of a proof, shows.", "Given $V>0$ with $d!v\\in \\mathbb {N}$ we are going to construct many non-equivalent convex lattice polytopes of volume $V$ .", "The integer convex hull $I(D_r)$ of $D_r$ has a vertex corresponding to each lattice point $z=(z_1,\\ldots ,z_{d-1},0) \\in rB^d\\cap \\mathbb {R}^d_+$ , namely the point $z+\\Vert z\\Vert ^2e_d$ .", "Denoting this set of vertices by $X$ we have $r^{d-1} \\ll \|X\|\\ll r^{d-1}$ .", "Also, $\\mathrm {vol\\;}I(D_r)$ is of order $r^{d+1}$ , and $\\mathrm {vol\\;}I(D_r) - \\mathrm {vol\\;}I(D_r \\setminus X)\\ll r^{d-1}$ as one can check easily.", "Define $r$ by $V=\\mathrm {vol\\;}I(D_r \\setminus X)$ and note that $V\\gg r^{d+1} \\gg \|X\|^{(d+1)/(d-1)}$ .", "For $Z \\subset X$ consider the polytopes $D(Z)=I(D_r \\setminus Z)$ .", "This is $2^{\|X\|}\\ge \\exp \\lbrace c_1 V^{(d-1)/(d+1)}\\rbrace $ convex lattice polytopes, each having volume between $V$ and $V+c_2r^{d-1}$ (where $c_1,c_2>0$ are constants depending only on $d$ ).", "At the vertex $(0,\\ldots ,0,\\lfloor r^2 \\rfloor )$ of $D(Z)$ one can place a congruent copy of $S(\\rho )$ ; here one chooses $\\rho \\in \\mathbb {N}$ to be of order $r^{(d-1)/d}$ .", "We denote this copy by $S(\\rho )$ as well.", "Given $Z \\subset X$ set $m=m(Z)=d!", "(\\mathrm {vol\\;}D(Z)-V)$ .", "Theorem REF implies the existence of $P(Z) \\in S(\\rho )$ with $m(P(Z))=m(Z)$ .", "We define $D^(Z)=\\left[D(Z)\\setminus S(\\rho )\\right] \\cup P(Z),$ which is a convex lattice polytope and $\\mathrm {vol\\;}D^(Z)=V$ .", "One shows again that the number of $D^(W)$ s equivalent to a fixed $D^(Z)$ is at most $d!$ .", "We leave the details to the interested reader.", "Acknowledgements.", "Support from ERC Advanced Research Grant no 267165 (DISCONV) and from Hungarian National Research Grant K 83767 is acknowledged with thanks.", "The second author gratefully acknowledges financial supports by NNSF of China (11071055); NSF of Hebei Province (A2012205080, A2013205189); Program for New Century Excellent Talents in University, Ministry of Education of China (NCET-10-0129); the project of Outstanding Experts' Overseas Training of Hebei Province.", "Imre Bárány Rényi Institute of Mathematics, Hungarian Academy of Sciences H-1364 Budapest PoB.", "127 Hungary [email protected] and Department of Mathematics University College London Gower Street, London, WC1E 6BT, UK Liping Yuan College of Mathematics and Information Science, Hebei Normal University, 050024 Shijiazhuang, P. R. China.", "[email protected], [email protected] and Hebei Key Laboratory of Computational Mathematics and Applications, 050024 Shijiazhuang, P. R. China." ] ]	1403.0437
[ [ "A search for quantum coin-flipping protocols using optimization\n techniques" ], [ "Abstract Coin-flipping is a cryptographic task in which two physically separated, mistrustful parties wish to generate a fair coin-flip by communicating with each other.", "Chailloux and Kerenidis (2009) designed quantum protocols that guarantee coin-flips with near optimal bias.", "The probability of any outcome in these protocols is provably at most $1/\\sqrt{2} + \\delta$ for any given $\\delta > 0$.", "However, no explicit description of these protocols is known, and the number of rounds in the protocols tends to infinity as $\\delta$ goes to 0.", "In fact, the smallest bias achieved by known explicit protocols is $1/4$ (Ambainis, 2001).", "We take a computational optimization approach, based mostly on convex optimization, to the search for simple and explicit quantum strong coin-flipping protocols.", "We present a search algorithm to identify protocols with low bias within a natural class, protocols based on bit-commitment (Nayak and Shor, 2003) restricting to commitment states used by Mochon (2005).", "An analysis of the resulting protocols via semidefinite programs (SDPs) unveils a simple structure.", "For example, we show that the SDPs reduce to second-order cone programs.", "We devise novel cheating strategies in the protocol by restricting the semidefinite programs and use the strategies to prune the search.", "The techniques we develop enable a computational search for protocols given by a mesh over the parameter space.", "The protocols have up to six rounds of communication, with messages of varying dimension and include the best known explicit protocol (with bias 1/4).", "We conduct two kinds of search: one for protocols with bias below 0.2499, and one for protocols in the neighbourhood of protocols with bias 1/4.", "Neither of these searches yields better bias.", "Based on the mathematical ideas behind the search algorithm, we prove a lower bound on the bias of a class of four-round protocols." ], [ "Introduction", "Some fundamental problems in the area of Quantum Cryptography allow formulations in the language of convex optimization in the space of hermitian matrices over the complex numbers, in particular, in the language of semidefinite optimization.", "These formulations enable us to take a computational optimization approach towards solutions of some of these problems.", "In the rest of this section, we describe quantum coin-flipping and introduce our approach." ], [ "Quantum coin-flipping", "Coin-flipping is a classic cryptographic task introduced by Blum .", "In this task, two remotely situated parties, Alice and Bob, would like to agree on a uniformly random bit by communicating with each other.", "The complication is that neither party trusts the other.", "If Alice were to toss a coin and send the outcome to Bob, Bob would have no means to verify whether this was a uniformly random outcome.", "In particular, if Alice wishes to cheat, she could send the outcome of her choice without any possibility of being caught cheating.", "We are interested in a communication protocol that is designed to protect an honest party from being cheated.", "More precisely, a “strong coin-flipping protocol” with bias $\\epsilon $ is a two-party communication protocol in the style of Yao , .", "In the protocol, the two players, Alice and Bob, start with no inputs and compute a value $c_\\mathrm {A}, c_\\mathrm {B}\\in \\left\\lbrace 0,1 \\right\\rbrace $ , respectively, or declare that the other player is cheating.", "If both players are honest, i.e., follow the protocol, then they agree on the outcome of the protocol ($c_\\mathrm {A}= c_\\mathrm {B}$ ), and the coin toss is fair ($\\Pr (c_\\mathrm {A}= c_\\mathrm {B}= b) = 1/2$ , for any $b\\in \\left\\lbrace 0,1 \\right\\rbrace $ ).", "Moreover, if one of the players deviates arbitrarily from the protocol in his or her local computation, i.e., is “dishonest” (and the other party is honest), then the probability of either outcome 0 or 1 is at most $1/2 + \\epsilon $ .", "Other variants of coin-flipping have also been studied in the literature.", "However, in the rest of the article, by “coin-flipping” (without any modifiers) we mean strong coin flipping.", "A straightforward game-theoretic argument proves that if the two parties in a coin-flipping protocol communicate classically and are computationally unbounded, at least one party can cheat perfectly (with bias $1/2$ ).", "In other words, there is at least one party, say Bob, and at least one outcome $b \\in \\left\\lbrace 0,1 \\right\\rbrace $ such that Bob can ensure outcome $b$ with probability 1 by choosing his messages in the protocol appropriately.", "Consequently, classical coin-flipping protocols with bias $\\epsilon <1/2$ are only possible under complexity-theoretic assumptions, and when Alice and Bob have limited computational resources.", "Quantum communication offers the possibility of “unconditionally secure” cryptography, wherein the security of a protocol rests solely on the validity of quantum mechanics as a faithful description of nature.", "The first few proposals for quantum information processing, namely the Wiesner quantum money scheme and the Bennett-Brassard quantum key expansion protocol were motivated by precisely this idea.", "These schemes were indeed eventually shown to be unconditionally secure in principle , , , .", "In light of these results, several researchers have studied the possibility of quantum coin-flipping protocols, as a step towards studying more general secure multi-party computations.", "Lo and Chau and Mayers were the first to consider quantum protocols for coin-flipping without any computational assumptions.", "They proved that no protocol with a finite number of rounds could achieve 0 bias.", "Nonetheless, Aharonov, Ta-Shma, Vazirani, and Yao designed a simple, three-round quantum protocol that achieved bias $\\approx 0.4143 < 1/2$ .", "This is impossible classically, even with an unbounded number of rounds.", "Ambainis designed a protocol with bias $1/4$ à la Aharonov et al., and proved that it is optimal within a class (see also Refs.", ", for a simpler version of the protocol and a complete proof of security).", "Shortly thereafter, Kitaev proved that any strong coin-flipping protocol with a finite number of rounds of communication has bias at least $(\\sqrt{2}-1)/2 \\approx 0.207$ (see Ref.", "for an alternative proof).", "Kitaev's seminal work uses semidefinite optimization in a central way.", "This argument extends to protocols with an unbounded number of rounds.", "This remained the state of the art for several years, with inconclusive evidence in either direction as to whether $1/4 = 0.25$ or $(\\sqrt{2}-1)/2$ is optimal.", "In 2009, Chailloux and Kerenidis settled this question through an elegant protocol scheme that has bias at most $(\\sqrt{2}-1)/2 + \\delta $ for any $\\delta > 0$ of our choice (building on , see below).", "We refer to this as the CK protocol.", "The CK protocol uses breakthrough work by Mochon , which itself builds upon the “point game” framework proposed by Kitaev.", "Mochon shows there are weak coin-flipping protocols with arbitrarily small bias.", "(This work has appeared only in the form of an unpublished manuscript, but has been verified by experts on the topic; see e.g. .)", "A weak coin-flipping protocol is a variant of coin-flipping in which each party favours a distinct outcome, say Alice favours 0 and Bob favours 1.", "The requirement when they are honest is the same as before.", "We say it has bias $\\epsilon $ if the following condition holds.", "When Alice is dishonest and Bob honest, we only require that Bob's outcome is 0 (Alice's favoured outcome) with probability at most $1/2+\\epsilon $ .", "A similar condition to protect Alice holds, when she is honest and Bob is dishonest.", "The weaker requirement of security against a dishonest player allows us to circumvent the Kitaev lower bound.", "While Mochon's work pins down the optimal bias for weak coin-flipping, it does this in a non-constructive fashion: we only know of the existence of protocols with arbitrarily small bias, not of its explicit description.", "Moreover, the number of rounds tends to infinity as the bias decreases to 0.", "As a consequence, the CK protocol for strong coin-flipping is also existential, and the number of rounds tends to infinity as the bias decreases to $(\\sqrt{2}-1)/2$ .", "It is perhaps very surprising that no progress on finding better explicit protocols has been made in over a decade." ], [ "Search for explicit protocols", "This work is driven by the quest to find explicit and simple strong coin-flipping protocols with bias smaller than $1/4$ .", "There are two main challenges in this quest.", "First, there seems to be little insight into the structure (if any) that protocols with small bias have; knowledge of such structure might help narrow our search for an optimal protocol.", "Second, the analysis of protocols, even those of a restricted form, with more than three rounds of communication is technically quite difficult.", "As the first step in deriving the $(\\sqrt{2}-1)/2$ lower bound, Kitaev proved that the optimal cheating probability of any dishonest party in a protocol with an explicit description is characterized by a semidefinite program (SDP).", "While this does not entirely address the second challenge, it reduces the analysis of a protocol to that of a well-studied optimization problem.", "In fact this formulation as an SDP enabled Mochon to analyze an important class of weak coin-flipping protocols , and later discover the optimal weak coin flipping protocol .", "SDPs resulting from strong coin-flipping protocols, however, do not appear to be amenable to similar analysis.", "We take a computational optimization approach to the search for explicit strong coin-flipping protocols.", "We focus on a class of protocols studied by Nayak and Shor that are based on “bit commitment”.", "This is a natural class of protocols that generalizes those due to Aharonov et al.", "and Ambainis, and provides a rich test bed for our search.", "(See Section for a description of such protocols.)", "Early proposals of multi-round protocols in this class were all shown to have bias at least $1/4$ , without eliminating the possibility of smaller bias (see, e.g., Ref. ).", "A characterization of the smallest bias achievable in this class would be significant progress on the problem: it would either lead to simple, explicit protocols with bias smaller than $1/4$ , or we would learn that protocols with smaller bias take some other, yet to be discovered form.", "Chailloux and Kerenidis have studied a version of quantum bit-commitment that may have implications for coin-flipping.", "They proved that in any quantum bit-commitment protocol with computationally unbounded players, at least one party can cheat with bias at least $\\approx 0.239$ .", "Since the protocols we study involve two interleaved commitments to independently chosen bits, this lower bound does not apply to the class.", "Chailloux and Kerenidis also give a protocol scheme for bit-commitment that guarantees bias arbitrarily close to $0.239$ .", "The protocol scheme is non-constructive as it uses the Mochon weak coin-flipping protocol.", "It is possible that any explicit protocols we discover for coin-flipping could also lead to explicit bit-commitment with bias smaller than $1/4$ .", "We present an algorithm for finding protocols with low bias.", "Each bit-commitment based coin-flipping protocol is specified by a 4-tuple of quantum states.", "At a high level, the algorithm iterates through a suitably fine mesh of such 4-tuples, and computes the bias of the resulting protocols.", "The size of the mesh scales faster than $1/\\nu ^{\\kappa D}$ , where $\\nu $ is a precision parameter, $\\kappa $ is a universal constant, and $D$ is the dimension of the states.", "The dimension itself scales as $2^n$ , where $n$ is the number of quantum bits involved.", "In order to minimize the doubly exponential size of the set of 4-tuples we examine, we further restrict our attention to states of the form introduced by Mochon for weak coin-flipping .", "The additional advantage of this kind of state is that the SDPs in the analysis of the protocols simplify drastically.", "In fact, all but a few constraints reduce to linear equalities so that the SDPs may be solved more efficiently.", "Next, we employ two techniques to prune the search space of 4-tuples.", "First, we use a sequence of strategies for dishonest players whose bias is given by a closed form expression determined by the four states.", "The idea is that if the bias for any of these strategies is higher than $1/4$ for any 4-tuple of states, we may safely rule it out as a candidate optimal protocol.", "This also has the advantage of avoiding a call to the SDP solver, the computationally most intensive step in the search algorithm.", "The second technique is to invoke symmetries in the search space as well as in the problem to identify protocols with the same bias.", "The idea here is to compute the bias for as few members of an equivalence class of protocols as possible.", "These techniques enable a computational search for protocols with up to six rounds of communication, with messages of varying dimension.", "The Ambainis protocol with bias $1/4$ has three rounds, and it is entirely possible that a strong coin-flipping protocol with a small number of rounds be optimal.", "Thus, the search non-trivially extends our understanding of this cryptographic primitive.", "We elaborate on this next." ], [ "The results", "We performed two types of search.", "The first was an optimistic search that sought protocols within the mesh with bias at most $1/4$ minus a small constant.", "We chose the constant to be $0.001$ .", "The rationale here was that if the mesh contains protocols with bias close to the lower bound of $\\approx 0.207$ , we would find protocols that have bias closer to $0.25$ (but smaller than it) relatively quickly.", "We searched for four-round protocols in which each message is of dimension ranging from 2 to 9, each with varying fineness for the mesh.", "We found that our heuristics, i.e., the filtering by fixed cheating strategies, performed so well that they eliminated every protocol: all of the protocols given by the mesh were found to have bias larger than $0.2499$ without the need to solve any SDP.", "Inspired by the search algorithm, we give an analytical proof that four-round qubit protocols have bias at least $0.2487$ .", "The initial search for four-round protocols helped us fine-tune the filter by a careful selection of the order in which the cheating strategies were tried.", "The idea was to eliminate most protocols with the least amount of computation.", "This made it feasible for us to search for protocols in finer meshes, with messages of higher dimension, and with a larger number of rounds.", "In particular, we were able to check six-round protocols with messages of dimension 2 and 3.", "Our heuristics again performed very well, eliminating almost every protocol before any SDP needed to be solved.", "Even during this search, not a single protocol with bias less than $0.2499$ was found.", "We also performed a search over meshes shifted by a randomly chosen parameter.", "This was to avoid potential anomalies caused by any special properties of the mesh we used.", "No protocols with bias less than $0.2499$ were found in this search either.", "The second kind of search focused on protocols with bias close to $0.25$ .", "We first identified protocols in the mesh with the least bias.", "Not surprisingly, these protocols all had computationally verified bias $1/4$ .", "We zoned in on the neighbourhood of these protocols.", "The idea here was to see if there are perturbations to the 4-tuple that lead to a decrease in bias.", "This search revealed 2 different equivalence classes of protocols for the four-round version and 6 for the six-round version.", "Four of these eight protocols are equivalent to optimal three-round protocols (within this class).", "However, the four remaining six-round protocols bear no resemblance to any known protocol with bias $1/4$ .", "A search in the neighbourhoods of all these protocols revealed no protocols with bias less than $1/4$ (details in Section ).", "It may not immediately be evident that the above searches involved a computational examination of extremely large sets of protocols and that the techniques described above in Section REF , were crucial in enabling this search.", "The symmetry arguments pruned the searches drastically, and in some cases only 1 in every $1,000,000$ protocols needed to be checked.", "In most cases, the cheating strategies (developed in Section ) filtered out the rest of the protocols entirely.", "To give an example of the efficiency of our search, we were able to check $2.74 \\times 10^{16}$ protocols in a matter of days.", "Without the symmetry arguments and the use of cheating strategies as a filter, this same search would have taken well over 69 million years, even using the very simplified forms of the SDPs.", "Further refinement of these ideas may make a more thorough search of protocols with four or more rounds feasible.", "The search algorithm, if implemented with exact feasibility guarantees, has the potential to give us computer aided proofs that certain classes of protocols in the family do not achieve optimal bias.", "Suppose we use a mesh such that given any 4-tuple $S$ of states, there is a 4-tuple $S^{\\prime }$ in the mesh such that the pairwise fidelity between corresponding distributions is at least $1 - \\delta $ .", "Further suppose the numerical approximation to the bias for $S^{\\prime }$ has additive error $\\tau $ due to the filter or SDP solver, and finite precision arithmeticNote that in our experiments feasibility is guaranteed only up to a tolerance, so as a result we do not have an independently verifiable upper bound on the additive error in terms of the objective value.", "Indeed, efficiently obtaining an exact feasible solution to SDPs, in general, is still an open problem at the time of this writing..", "If the algorithm reports that there are no tuples in the mesh with bias at most $\\epsilon ^$ , then it holds that there are no 4-tuples, even outside the mesh, with bias at most $\\epsilon ^ - \\sqrt{8 \\, \\delta } - \\tau $ .", "The fineness of the mesh we are able to support currently is not sufficient for such proofs.", "A refinement of the search algorithm along the lines described above, however, would yield lower bounds for new classes of bit-commitment based protocols." ] ]	1403.0505
[ [ "Electromagnetic signals from bare strange stars" ], [ "Abstract The crystalline color superconducting phase is believed to be the ground state of deconfined quark matter for sufficiently large values of the strange quark mass.", "This phase has the remarkable property of being more rigid than any known material.", "It can therefore sustain large shear stresses, supporting torsional oscillations of large amplitude.", "The torsional oscillations could lead to observable electromagnetic signals if strange stars have a crystalline color superconducting crust.", "Indeed, considering a simple model of strange star with a bare quark matter surface, it turns out that a positive charge is localized in a narrow shell about ten Fermi thick beneath the star surface.", "The electrons needed to neutralize the positive charge of quarks spill in the star exterior forming an electromagnetically bounded atmosphere hundreds of Fermi thick.", "When a torsional oscillation is excited, for example by a stellar glitch, the positive charge oscillates with typical kHz frequencies, for a crust thickness of about one-tenth of the stellar radius, to hundreds of Hz, for a crust thickness of about nine-tenths of the stellar radius.", "Higher frequencies, of the order of few GHz, can be reached if the star crust is of the order of few centimeters thick.", "We estimate the emitted power considering emission by an oscillating magnetic dipole, finding that it can be quite large, of the order of $10^{45}$ erg/s for a thin crust.", "The associated relaxation times are very uncertain, with values ranging between microseconds and minutes, depending on the crust thickness.", "The radiated photons will be in part absorbed by the electronic atmosphere, but a sizable fraction of them should be emitted by the star." ], [ "Introduction", "One of the routes for studying the properties of matter at very high densities is by the inspection of the properties of compact stellar objects (CSOs).", "These are stars having a mass of $1-2 M_\\odot $ and a radius of about 10 km, typically observed as pulsars.", "Baryonic matter inside a CSO is squeezed at densities about a factor 3-5 larger than in heavy nuclei.", "From a simple geometrical reasoning one can argue that in these conditions baryons are likely to lose their identity [1] and a new form of matter should be realized.", "One possibility is that the extremely high densities and low temperatures may favor the transition from nuclear matter to deconfined quark matter in the core of the CSO [2], [3], [4], [5].", "In this case compact (hybrid) stars featuring quark cores and a crust of standard nuclear matter would exist.", "A second possibility is that strange matter is the ground state of the hadrons [6].", "In this case at high densities there should exist the possibility of converting nuclear matter to deconfined matter.", "The resulting CSO would be a strange star [7], [8], i.e a CSO completely constituted of deconfined matter, see [9] for a review.", "Unfortunately these two possibilities cannot be checked by first principle calculations.", "Indeed at the densities relevant for CSOs, quantum chromodynamics (QCD) is nonperturbative, because the typical energy scale is about $\\Lambda _\\text{QCD}$ .", "Moreover, lattice QCD simulations at large baryonic densities are unfeasible because of the so-called sign problem [10], see [11] for a recent review and [12] for a study of an inhomogeneous phase.", "Although not firmly established by first principles, it is reasonable to expect that if deconfined quark matter is present, it should be in a color superconducting (CSC) phase [13], [14], [15].", "The reason is that the critical temperature of color superconductors is large, $T_{c} \\simeq 0.57 \\Delta $ , where $\\Delta \\sim 5 -100$ MeV is the gap parameter.", "For the greatest part of the CSO lifetime, the temperature is much lower than this critical temperature and the CSC phase is thermodynamically favored.", "It is widely accepted that at asymptotic densities, when the up, down and strange quarks can be treated as massless, the color-flavor locked (CFL) phase [16] is the ground state of matter.", "This phase is energetically favored because quarks of all flavors and of all colors form standard Cooper pairs, thus maximizing the free energy gain.", "However, considering realistic conditions realizable within CSOs a different CSC phase could be realized.", "The reason is that the nonzero and possibly large value of the strange quark mass, $M_s$ , combined with the requirement of beta equilibrium, electromagnetic and color neutrality, tends to pull apart the Fermi spheres of quarks with different flavors [17].", "The mismatch between the Fermi spheres is proportional to $M_s^2/\\mu $ , where $ \\mu = \\frac{\\mu _u+\\mu _d+\\mu _s}{3}\\,, $ is the average quark chemical potential.", "The free energy price of having simultaneous pairing of three-flavor quark matter increases with increasing values of $M_s^2/\\mu $ .", "Since the free energy gain is proportional to the CFL gap parameter, $\\Delta _{\\text{CFL}}$ , if $M_s^2/\\mu > c \\Delta _{\\text{CFL}}$ , with $c$ a number of order 1 [18], a different and less symmetric CSC phase should be favored.", "One possibility is that the crystalline color superconducting (CCSC) phase is realized [19], [20], [15], [21].", "In this phase quarks form Cooper pairs with nonzero total momentum, and there is no free energy cost proportional to $M_s^2/\\mu $ .", "The only free energy cost is due to the formation of counterpropagating currents; see for example the qualitative discussion in [21].", "Actually, with increasing values of $M_s^2/\\mu $ various inhomogeneous CSC phases can be realized, because the system has many degrees of freedom [15].", "The CCSC phase should be favored for certain values of the chemical potential mismatch.", "In reality, the CCSC phase is not one single phase but a collection of phases, characterized by their crystalline arrangements, which are favored for different values of $M_s^2/\\mu $ .", "The Ginzburg-Landau (GL) analysis of [20] has shown that in three-flavor quark matter there are two good candidate structures that are energetically favored for $2.9 \\Delta _{\\text{CFL}} \\lesssim \\frac{M_s^2}{\\mu } \\lesssim 10.4 \\Delta _{\\text{CFL}}\\,.$ This range of values is certainly model dependent, moreover the GL expansion is under poor quantitative control [15].", "For this reason we shall consider strange star models in which both the CFL phase and the CCSC phase are realized.", "Since the CFL phase is expected to be favored at high densities, we shall assume that it is realized in the core of the CSO.", "The CCSC phase is favored at smaller densities and constitutes the crust of the CSO.", "The radius, $R_{\\it c}$ , at which the CFL core turns into the CCSC crust will be used as a free parameter.", "We shall restrict our analysis to bare strange stars [7], meaning that we shall assume that on the top of the strange star surface there is no other layer of baryonic matter.", "Our model of strange star resembles the typical onion structure of a standard neutron star with a solid crust and a superfluid core.", "It is similar to the model discussed in [22] for studying $r-$ mode oscillations.", "In that work the core radius was determined using a microscopic approach; instead we treat $R_{\\it c}$ as a free parameter.", "One quantitative difference between our model and standard neutron star models, is that the CCSC phase is extremely rigid, much more rigid than the ironlike crust.", "The shear modulus of the energetically favored phase can be obtained studying the low energy oscillations of the condensate modulation [23], [24], [25], [26].", "In particular, the low energy expansion of the GL Lagrangian of [26] leads to a shear modulus $\\nu \\simeq \\nu _0\\left(\\frac{\\Delta }{10 \\text{ MeV}}\\right)^2\\left(\\frac{\\mu }{400 \\text{ MeV}}\\right)^2\\,,$ where $\\nu _0= 2.47 \\frac{\\text{MeV}}{\\text{fm}^3}\\,,$ will be our reference value.", "The reader is warned that the actual value of the shear modulus might differ from $\\nu _0$ by a large amount because of the various approximations used in [26].", "The value of $\\Delta $ is also uncertain, with reasonable values ranging between 5 MeV and 25 MeV, see the discussions in [26], [15].", "Regarding the quark chemical potential, we shall consider the values obtained in the construction of hydrodynamically stable strange stars.", "The shear modulus of different crystalline structures is proportional to $\\nu $ , with corrections of the order unity.", "Since in our treatment we shall only exploit the rigidness of the CCSC phase giving the order of magnitude estimates for the various computed quantities, the actual crystalline pattern is irrelevant for our purposes.", "Taking into account the uncertainty in the gap parameter and in the quark chemical potential, it can be estimated that the value of $\\nu $ is larger than in conventional neutron star crust (see for example [27]), by at least a factor of $20-1000$ [26].", "This large value of the shear modulus is due to the fact that the typical energy density associated with the oscillations of the condensate modulation is $\\mu ^2 \\Delta ^2$ , where $\\Delta $ is determined by the strong interaction in the antitriplet channel.", "Instead, in conventional neutron star the associated energy is at the electromagnetic scale.", "Given the large shear modulus, one immediate consequence is that CCSC matter can sustain large deformations.", "In Refs.", "[28], [29], [30], [31], [22] it has been studied the emission of gravitational waves by various mechanisms that induce a quadrupole deformation of the CCSC structure.", "See also [32] for a discussion of a different mechanism of gravitational wave emission from strange stars.", "In the present paper we shall instead consider the electromagnetic (EM) emission by strange stars with a CCSC crust.", "Since the strange star surface confines baryonic matter but allows the leakage of electrons, it follows that at the star surface there is a charge separation at the hundreds of Fermi scale [7].", "Because of the large shear modulus, our model of bare strange star can sustain large and fast torsional oscillations, leading to a periodic displacement of the surface charge.", "We shall see that the frequencies of torsional oscillations are of the order of MHz if the crust is hundreds of meters thick.", "Lower frequencies are reached if the crust is a few kilometers thick; GHz frequencies are reached if the crust is few centimeters thick.", "The amplitude of the oscillations at the star surface is in any case of the order of centimeters, leading to an enormous emitted radiation, of the order of $10^{41}$ erg/s, steeply increasing for thin crusts.", "Thus the oscillation energy should be radiated away very efficiently, on time scales of milliseconds or even microseconds for a thin crust and of the order of hundreds of seconds for a thick crust.", "More in detail, we shall determine the frequency, the amplitude, the damping times and the emitted power as a function of the various parameters that characterize the strange star.", "This paper is organized as follows.", "In Sec.", "we discuss spherically symmetric strange stars in hydrodynamical equilibrium.", "In Sec.", "we determine the charge distribution close to the surface of the strange star.", "In Sec.", "we study the torsional oscillations of the strange star, estimating the frequencies, the emitted power and the decay time as a function of the various parameters of the model.", "We draw our conclusions and a possible connection with astronomical observations in Sec.", "." ], [ "Equilibrium configurations of spherically symmetric strange stars", "For a spherically symmetric nonrotating star, the unperturbed background can be described by the static metric $ds^2 = g_{\\mu \\nu } d x^\\mu d x^\\nu = - e^{2 \\Phi (r)} dt^2 +e^{2 \\Lambda (r)} dr^2 + r^2 d\\Omega ^2\\,.$ The relation between the function $\\Lambda (r)$ and the mass distribution $m(r)$ is given by the solution of Einstein's equations inside the star, namely $e^{2 \\Lambda (r)} = \\left[1- \\frac{2 m(r) G}{r} \\right]^{-1},\\,\\,m(r) = \\int ^r_0 dr^{\\prime } \\, r^{\\prime 2} \\, \\rho (r^{\\prime }) \\,,$ where $\\rho (r)$ is the energy density of the fluid.", "The equilibrium structure is obtained by solving the Tolman-Oppenheimer-Volkov (TOV) equation $\\frac{\\partial p}{\\partial r}=-\\frac{G (p+ \\rho ) \\left(m+4 \\pi p \\, r^3\\right)}{r (r-2 G \\,m)} \\,,$ once the equation of state (EoS) $p(\\rho )$ is specified.", "The star radius, $R$ , is determined by the boundary condition on the pressure $ p(R) =0$ , simply meaning that the pressure at the surface of the star should vanish.", "The gravitational potential $\\Phi $ can be found from $\\frac{\\partial \\Phi }{\\partial r} = \\frac{G \\left(m+4 \\pi \\, p \\, r^3\\right)}{r (r-2 G m)}\\,,$ once $ p$ is derived from the solution of Eq.", "(REF ).", "Outside the star, for $ r > R$ , defining $m(R)= M$ , we have that $e^{2 \\Lambda (r)} =\\left(1- \\frac{2 M G}{r} \\right)^{-1} \\, , \\qquad e^{2 \\Phi (r)} = 1- \\frac{2 M G}{r} \\, .$ In the present work we consider a simple strange star model, entirely composed of deconfined three-flavor quark matter in the CSC phase.", "The detailed form of the CSC phase is not important here, because quark pairing should account for a small variation of the quark matter EoS.", "Since for the range of densities attainable in compact stars QCD perturbative calculations are not trustable, we use the general parameterization of the EoS given in [33] $\\Omega _{\\text{QM}} = -\\frac{3}{4 \\pi ^2} a_4 \\mu ^4 + \\frac{3}{4 \\pi ^2} a_2 \\mu ^2 + B_{\\text{eff}}\\,,$ where $a_4$ , $a_2$ and $B_{\\text{eff}}$ are independent of the average quark chemical potential $\\mu $ .", "This parameterization can be seen as a Taylor expansion of the grand potential, with phenomenological coefficients (see [33] for a discussion of the relevant range of values of each parameter).", "In order to take into account the impact of the uncertainty of these coefficients on our results, we consider two extreme situations, namely A ($a_4=0.7$ , $a_2=(200$ MeV)$^2$ and $B_{\\text{eff}}=(165$ MeV)$^4$ ) and B ($a_4=0.7$ , $a_2=0$ and $B_{\\text{eff}}=(145$ MeV)$^4$ ).", "In Fig.", "REF we report the mass-radius sequences obtained solving the TOV equations using the above EoS for the two different sets of parameters.", "The largest attainable mass with each set will be our reference model, represented as a black dot in Fig.", "REF .", "In detail: - Model A, with a total mass of $M=1.27 M_\\odot $ , $R\\simeq 7.1$ km and $\\rho _c \\simeq 510^{15} \\text{g}/\\text{cm}^3$ ; - Model B, with $M\\simeq 2.0 M_\\odot $ , $R\\simeq 10.9$ km and $\\rho _c\\simeq 2 10^{15} \\text{g}/\\text{cm}^3$ .", "The presence of electric charge is expected to produce corrections on masses and radii of strange quark stars at the $15\\%$ and $5\\%$ levels, respectively [34].", "Figure: Mass radius relation for strange stars with the EoS given inEq.", "() for the two sets of parameters values discussed in the text.Changing these parameters it is possible to span a large range ofvalues of mass and radius.", "The black dots represent the equilibriumstructure assumed as reference models in this paper.In both models we assume that at a certain radial distance, $R_{\\it c}=a R$ with $0 \\le a \\le 1$ , there is a phase transition between the CFL phase and the CCSC phase.", "In Fig.", "REF we show a pictorial description of the star structure.", "Since the values of $M_s$ and of the gap parameters are unknown, it is not possible to determine from first principles the radial distance at which the CFL phase turns into the CCSC phase.", "For this reason we treat $a$ as a parameter.", "More in detail, in our model we are assuming that the CFL-CCSC phase transition does not change in an appreciable way the EoS.", "Thus, our assumption is that at a given $R_{\\it c}$ the pressure of the CCSC phase and of the CFL phase are equal, but the difference between the pressure of these two phases is always small in the sense that computing the star mass and radius using only a CFL EoS or only a CCSC EoS does not change the results in an appreciable way.", "This is a fair approximation as far as the gap parameter in both phases are similar and much less than the average chemical potential.", "Our model is basically the model discussed in [22], but we treat $a$ as a free parameter, whereas they compute it by a microscopic theory.", "Figure: Sketch of the structure of the considered bare strange star model.", "The star core extends up to a radius R 𝑐 R_{\\it c}, and is made by color-flavor locked matter.The crust is made by the extremely rigid crystalline color superconducting matter.", "The star radius, RR, is determined by the solution of the TOV equation () with the EoS in Eq. ().", "We treat the core radius, R 𝑐 =aRR_{\\it c}= a R, as a free parameter.", "The strange star is surrounded by a cloud of electrons, the electrosphere, having a width (not in scale in the figure) of hundreds of Fermi, see Sec.", "." ], [ "Charge distribution", "It is very interesting to study the charge distribution close to the surface of the strange star.", "The reason is that quarks are confined inside the strange star by the strong interaction, while electrons can leak by a certain distance outside the star and are bound only by the electromagnetic interaction [7].", "As we shall see in detail below, electrons form an electrosphere hundreds of Fermi thick on the top of the star surface.", "This negative charge is compensated by a positive charge of quarks in a narrow layer, about 10 fm thick, beneath the star surface.", "Let us see how this charge separation happens.", "At equilibrium, the chemical potentials associated to any free particle species must be constant, i.e.", "space independent, otherwise particles would move to compensate the chemical potential difference.", "However, in the presence of an electrostatic potential, $\\phi $ , the density of particles can be space dependent.", "In the local density approximation this fact can be taken into account defining the space dependent effective chemical potential $\\mu _i(\\mathbf {x}) = \\mu _i + e Q_i \\phi (\\mathbf {x})\\,,$ where $Q_i$ is the charge of the species $i$ , in units of the electric charge, $e$ .", "Note that because of the weak process $u + d \\leftrightarrow u + s$ one has $\\mu _s(\\mathbf {x})=\\mu _d(\\mathbf {x})$ .", "Note also that the average chemical potential, $\\mu $ , see Eq.", "(REF ), is space independent; indeed $\\mu _u(\\mathbf {x}) + \\mu _d(\\mathbf {x}) + \\mu _s(\\mathbf {x})= \\mu _u + \\mu _d +\\mu _s=3 \\mu $ .", "To simplify the charge distribution treatment, we shall assume that the leading effect of color interactions is to provide confinement of quarks in the interior of the star [7].", "This is a good approximation if the subleading effect of color interactions is the quark condensation.", "Indeed quark condensation is expected to produce corrections to our results of the order $\\Delta /M_s$ .", "Since the surface of the star is in the CCSC phase it amounts to less than $10 \\%$ corrections.", "We also neglect the fact that in the CCSC phase the $U(1)_\\text{em}$ is rotated to a $\\tilde{U}(1)$ , because the mixing angle between the photon and the pertinent color field is small [16], [14], [15].", "Therefore, we approximate the number density of the fermionic species, $n_i$ , as a free Fermi gas, meaning that $n_i(\\mathbf {x}) = C_i\\frac{k_{F,i}(\\mathbf {x})^3}{3 \\pi ^2}\\,,$ where $C_i$ is a factor taking into account the color degrees of freedom and $k_{F,i}(\\mathbf {x})=\\sqrt{\\mu _i( \\mathbf {x})^2 - m_i^2} $ is the Fermi momentum, with $m_i$ the mass.", "The number density of quarks ends abruptly at the surface of the star, but the number density of electrons extends over distances $r>R$ , determining the thickness of the electrosphere.", "If the charge distribution varies in a region much smaller than the star radius, it is possible to approximate the geometry of the interface as planar.", "We shall assume that this is the case and then check that it is a good approximation.", "For a planar interface Poisson's equation reads $\\frac{d^2 \\phi }{d z^2} = e \\sum _i Q_i\\, n_i(z) \\,,$ where $z$ measures the distance from the quark matter discontinuity, located at $z=0$ ; the star interior corresponding to $z <0$ .", "Two boundary conditions are obtained requiring that the charge density vanishes far from the interface.", "For the sake of notation we define $V(z)=\\mu _e(z)=\\mu _e- e \\phi (z)\\,,$ and we can rewrite the Poisson's equation as $\\frac{d^2 V}{d z^2} = - \\frac{4 \\alpha _\\text{em}}{3 \\pi } \\sum _i Q_i\\, C_i\\, k_{F,i}^3 \\,.$ Considering the weak equilibrium processes, the effective quark chemical potentials can be written as $\\mu _u(z) = \\mu -\\frac{2}{3} V(z)\\,, \\,\\,\\, \\mu _d(z) = \\mu _s(z) = \\mu + \\frac{1}{3} V(z)\\,,$ and the corresponding number densities can be obtained substituting these expressions in Eq.", "(REF ).", "In principle the average quark chemical potential depends on the radial coordinate as well.", "Indeed from the EoS we can determine for any value of $r$ the function $\\mu (r)$ .", "However, the average quark chemical potential varies on the length scale of hundreds of meters at least, much larger than the length scale of the quark charge distribution, which as we shall see below is of few tens of Fermi at most.", "Since the $z=0$ region corresponds to the star surface, we can take $\\mu \\simeq \\mu _R \\equiv \\mu (R)$ .", "We report in Table REF the values of these quantities for the two considered models.", "No net charge is present far from the boundary, therefore we require that $n_{e}(z)_{z \\rightarrow +\\infty } =0\\,,$ and that $\\left[\\frac{2}{3} n_u(z) -\\frac{1}{3} n_d(z) -\\frac{1}{3}n_s(z) - n_e (z)\\right]_{z \\rightarrow -\\infty }=0\\,.$ Neglecting the electron mass the first condition leads to $V(+\\infty )= 0$ ; neglecting also the light quark masses the second condition leads to $V_q^3 &=& 2 \\left(\\mu _R - \\frac{2}{3} V_q\\right)^3 -\\left(\\mu _R +\\frac{1}{3} V_q\\right)^3\\nonumber \\\\ &-& \\left[\\left(\\mu _R +\\frac{1}{3} V_q\\right)^2- M_s^2\\right]^{\\frac{3}{2}}$ that fixes a relation among $M_s$ , $\\mu _R$ and $V_q=V(-\\infty )$ .", "Assuming that at the crust-electrosphere interface there is no surface charge, we obtain a third boundary condition requiring that the electric field is a continuous function at $z=0$ .", "This boundary condition can be used to obtain an expression of the effective electron chemical potential at the surface that depends on $V_q$ and $\\mu $ , approximately given by $V_0 = V_q -\\frac{V_q^2}{2 \\sqrt{3} \\mu } + {\\cal O}(V_q^3/\\mu ^2)\\,.$ Note that in most of the studies it is assumed that the quark distribution is constant, leading to $V_0 = 3/4 V_q$ , see [7].", "Here we have instead used the free Fermi gas distribution for quarks, but the quantitative result remains basically the same: the surface potential $V_0$ is smaller than $V_q$ by a not great amount.", "The Poisson's equation (REF ) can be analytically solved for positive values of $z$ and we find $V(z) = \\frac{V_0}{1+\\sqrt{\\frac{2 \\alpha _\\text{em}}{3 \\pi }} V_0 z} \\qquad \\text{for $z>0$}\\,.$ Upon substituting the expression above in Eq.", "(REF ) (taking $C_e=1$ ), one readily obtains the electron distribution for positive $z$ .", "In the interior of the star the Poisson's equation must be solved numerically.", "For negative values of $z$ we obtain by a fit of the total charge distribution, $\\ \\sum _i Q_i\\, n_i(z) = b\\, e^{z/d} \\qquad z \\le 0 \\,,$ where the $b$ and $d$ are two parameters describing the maximum charge density and the Debye screening length of the total charge distribution, respectively.", "The screening length has been computed in a different way in [35], finding results analogous to ours.", "The values of $V_0$ and of the fitting parameters $b$ and $d$ for the considered models and for two values of $M_s$ are reported in Table REF .", "In the interior of the star, the positive charge distribution corresponds to a shell of thickness less than 10 fm peaked at $r=R$ .", "This result is basically independent of the considered star model and of the strange quark mass.", "It relies on the fact that we are considering that the dominant charge carriers in the interior of the star are gapless quarks.", "Indeed, in the relevant CCSC phases quarks have a linear direction dependent dispersion law, see [20], [15].", "Moreover, unpaired quarks are as present as well.", "For this reason the Debye screening length is approximately given by the free Fermi gas expression for three massless flavors, $d \\simeq d_\\text{Fermi gas}= \\sqrt{\\frac{\\pi }{8 \\alpha _\\text{em} \\mu ^2} }\\, \\sim 5\\, \\text{fm} $ for $\\mu = 300$ MeV.", "The number density at the surface is approximately independent of the considered model, but depends on the chosen value of the strange quark mass.", "The reason is that with decreasing strange quark mass the electronic density decreases.", "This effect can also be seen from the values of the positive surface charge density beneath the star surface $Q_+ = e \\sum _i Q_i \\int _0^R dr \\, n_i(r) \\,,$ reported in the last column of Table REF .", "This positive charge is balanced by the electron negative charge outside the star.", "The electron distribution extends outside the star for a distance approximately given by $(\\sqrt{\\frac{4 \\alpha _\\text{em}}{3 \\pi } } M_s^2/\\mu )^{-1}$ , see Eqs.", "(REF ), (REF ) and (REF ), of the order of hundreds of Fermi.", "Therefore, the length scales of both charge distributions are much less than the star radius.", "Table: Values of the parameters characterizing the surface of the two considered star models.", "The second and third columns represent the average quark chemical potential and the matter density at the surface of the star, respectively.", "Considering two different values of the strange quark mass we report for each stellar model the parameters d,bd, b and V 0 V_0 characterizing the charge distributions, see Eqs.", "() and (), and the positive surface charge density, Q + Q_+, see Eq.", "().Note that, in principle, a charge separation should occur as well at $R_{\\it c}$ , at the CFL-CCSC interface; the reason being that the bulk CFL matter has vanishing electron chemical potential [16], [14], [15].", "However, whatever phenomenon takes place at the CFL-CCSC interface should be screened by the overlying CCSC layer." ], [ "Nonradial oscillations", "We now consider the possible oscillations of the above determined charge distribution.", "In particular, we are interested in nonradial oscillations, which can generate an EM current at the star surface.", "Stars have a large number of nonradial oscillations, which can be classified as spheroidal and toroidal oscillations, see for example [36].", "For definiteness, we focus on torsional oscillations [37], [38], [39], [40], a particular class of toroidal oscillations.", "The torsional oscillations are the only toroidal oscillations in nonrotating stars with a negligible magnetic field [36].", "These oscillations can be produced by acting with a torque on a rigid structure, as shown in Fig.", "REF for a simple rigid slab.", "When the applied forces are parallel to the sides of the slab they produce a deformation of the structure.", "As the external torque vanishes, the slab starts to oscillate around the equilibrium configuration.", "The restoring force is proportional to the shear modulus and the frequency of the small amplitude oscillations is given by $ \\omega \\propto \\frac{1}{D}\\sqrt{\\frac{\\nu }{\\rho }}\\,,$ where $D$ is the thickness of the slab.", "The slab can be thought as a local approximation of the CCSC crust, and therefore we expect that the frequency of the crust torsional oscillations has the same qualitative dependence on $\\nu $ , $\\rho $ and the crust thickness, $D=R-R_{\\it c}$ as in Eq.", "(REF ).", "Our interest in the torsional oscillations is clearly due to the fact that in the CCSC phase the shear modulus is extremely large and can therefore sustain large amplitude oscillations.", "We shall show that for a sufficiently thin CCSC crust, the frequency of the oscillations lies in the MHz radio-frequency range, whereas for a thick CCSC crust the frequency of the oscillations lies in the hundreds of Hz range.", "Figure: Pictorial description of the torsional oscillations of a homogeneous two-dimensional slab with D≪LD \\ll L. The equilibrium configuration corresponds to the one in which all the horizontal lines are parallel (left panel).", "A torque applied at the surfaces of the slab slightly deforms it by an angle θ\\theta (central panel).", "As the applied forces vanish the slab starts to oscillate around the equilibrium configuration, reaching the configuration with deforming angle -θ-\\theta (right panel).", "The restoring force governing the oscillation is proportional to the shear modulus, ν\\nu .", "The applied force determines the amplitude of the oscillation; the frequency of the oscillation is proportional to ν/(ρD 2 )\\sqrt{\\nu /(\\rho D^2)}, where ρ\\rho is the matter density of the slab.Given the spherical symmetry of the nonrotating stars, it is useful to use spherical coordinates for the displacement vector $\\xi _{nl}^r =0 \\qquad \\xi _{nl}^\\theta = 0 \\qquad \\xi _{nl}^{\\phi } = \\frac{W_{nl}(r)}{r \\sin \\theta } \\frac{\\partial P_l(\\cos \\theta )}{\\partial \\theta } e^{i \\omega _{nl} t}\\,,$ where $l$ is the angular momentum and $n$ is the principal quantum number indicating the number of nodes.", "In the following we shall study these oscillations in the Newtonian approximation.", "We expect that a treatment with full general relativity (GR) should give correction factors of order unity.", "Given the large uncertainties of the various parameters of the model, it seems appropriate to neglect GR corrections.", "We shall investigate the GR corrections in a future work.", "In the Newtonian limit the velocity perturbationEulerian and Lagrangian perturbations are identical for toroidal oscillations [36].", "is given by $\\delta \\mathbf {u}_{nl} = i \\omega _{nl} \\mathbf {\\xi }_{nl} \\,,$ and the amplitude of the horizontal oscillation satisfies the following differential equation $\\omega _{nl}^2 W_{nl} &=& \\frac{\\nu }{\\rho } \\left[-\\frac{1}{\\nu } \\frac{d \\nu }{d r} \\left(\\frac{d W_{nl}}{ d r}-\\frac{W_{nl}}{ r}\\right) -\\frac{1}{r^2}\\frac{d}{dr}\\left( r^2\\frac{d W_{nl}}{ d r} \\right) \\right.", "\\nonumber \\\\ &+& \\left.", "\\frac{l(l+1)}{r^2}W_{nl} \\right]\\,.$ The torsional oscillation extends in the CCSC crust and disappears suddenly in both the CFL phase and the electrosphere.", "The displacement is discontinuous at these interfaces because both the CFL and the electrosphere have a vanishing shear modulus.", "We shall simplify the discussion assuming that the quark matter in the CCSC crust has constant density, $\\rho \\simeq \\rho _R=\\rho (R)$ .", "This is a good approximation because in all the considered cases correspond to a CCSC crust a few kilometers thick at most.", "In any case, the density of quark matter in strange stars does not sharply change, because strange stars are self-bound CSOs.", "We shall as well neglect the radial dependence of the shear modulus and take it as a constant.", "In this way the displacement satisfies the following differential equation: $\\frac{d^2 W_{nl}}{ d r^2} + \\frac{2}{r} \\frac{d W_{nl}}{ d r}+ \\left(\\frac{\\omega _{nl}^2}{v_s^2} - \\frac{l(l+1)}{r^2}\\right)W_{nl}=0\\,,$ where $v_s= \\sqrt{\\nu /\\rho _R}$ is the shear wave velocity.", "It is useful to switch to the adimensional variable $y= \\omega _{nl} r/v_s$ , and to define $W_{nl}= U_{nl}/y$ .", "In this way the above differential equation can be written as $U_{nl}^{\\prime \\prime }(y) + \\left(1 - \\frac{l(l+1)}{y^2}\\right) U_{nl}(y) =0 \\,.$ The solution of this equation can be expressed as a sum of spherical Bessel and Neumann functions $U_{nl}(y) = c_1 j_{l}(y) + c_2 n_l(y)\\,.$ After an initial stage in which the crust has been excited by some external agency, we assume that there is no torque acting on both interfaces of the crust.", "This corresponds to assuming the no-traction condition [36], leading to $U_{nl}^{\\prime }(y_1) = 2\\frac{U_{nl}(y_1)}{y_1} \\qquad U_{nl}^{\\prime }(y_2) = 2\\frac{U_{nl}(y_2)}{y_2} \\,,$ where $y_2=\\omega _{nl} R/v_s$ corresponds to the CCSC-electrosphere interface and $y_1 =\\omega _{nl} aR/v_s = a y_2$ corresponds to the CFL-CCSC interface.", "One of the two conditions can be used to eliminate one of the two coefficients in Eq.", "(REF ).", "The other condition determines the quantized frequencies.", "For definiteness, we shall hereafter assume that the only excited mode is the one with $l=1$ and with one nodeThe mode with $n=1$ is the first nontrivial mode for $l=1$ .", "Indeed the mode with no nodes $(n=0) $ corresponds to a global rotation of the star., $n=1$ .", "In this case we find that for $a \\gtrsim 0.3$ one can use the approximate functions $y_2 \\simeq \\frac{\\pi }{1-a} \\qquad y_1 \\simeq \\frac{a \\pi }{1-a}\\,,$ and the corresponding oscillation frequency is given by $\\omega _{11} \\simeq 0.06 \\left( \\frac{\\nu }{\\nu _0} \\right)^{1/2}\\!\\!\\!\\left( \\frac{\\delta R}{1 \\text{km}} \\right)^{-1} \\left(\\frac{\\rho _R}{\\rho _0} \\right)^{-1/2} \\text{MHz}\\,,$ where $\\delta R= (1-a)R$ and $\\rho _0=10^{15} \\text{g/cm}^{3}$ .", "Note that for each set of parameters the equation above gives the smaller attainable frequency.", "Radial overtones, having higher frequencies, could as well be excited by the external agency triggering the oscillation of the crust.", "As in the simple example of the slab (see the caption of Fig.", "REF ) the amplitude of the oscillations is determined by the external agency, which fixes the amount of energy of each mode.", "The frequency of the oscillations is proportional to the shear velocity divided by the crust width.", "For definiteness, we shall assume that a fraction $\\alpha $ of the energy of a glitch excites the $l=1, n=1$ mode; thus $\\alpha E_{\\text{glitch}} = \\frac{\\rho _R}{2} \\int \|\\delta \\mathbf {u}_{11}\|^2 dV \\,,$ where we shall consider as a reference value $E_{\\text{glitch}}^{\\text{Vela}} = 3\\times 10^{-12} M_\\odot $ as estimated for the giant Vela glitches.", "Less energetic glitches, as for the Crab, simply correspond to smaller values of $\\alpha $ .", "Of particular relevance for us is the amplitude of the oscillation at the star surface, because it determines the displacement of the quark electric charge.", "The displacement does in general depend on the thickness of the crust, on the shear modulus and can be expressed as $W_{11}(R) = A(a) \\left( \\frac{\\nu }{\\nu _0}\\right)^{-1/2} \\left(\\frac{R}{10 \\text{km}}\\right)^{-1/2}\\left(\\frac{\\alpha E_{\\text{glitch}}}{E_{\\text{glitch}}^{\\text{Vela}} }\\right)^{1/2}\\,,\\,$ where $A(a)$ is reported in Fig.REF .", "Figure: Function determining the horizontal displacement at the surface of the star associated with the considered torsional oscillation, see Eq.", "().The amplitude of the oscillations is in general quite large, of the order of centimeters, except for $a \\simeq 1$ .", "Indeed, the amplitude vanishes for $a=1$ , because this case corresponds to a star completely made of CFL matter.", "In the following we shall always consider the case in which the crust has a macroscopic extension, much larger than the extension of the positive charge distribution.", "Notice that considering a perturbation that excites more modes does not qualitatively change the above results.", "The only effect is a distribution of the total energy among modes with higher frequency.", "The fluctuation of the current density induced by periodic horizontal displacement associated with the torsional oscillation is given by $\\delta {\\mathbf {J}}= e \\sum _i Q_i\\, n_i(z) {\\delta \\mathbf {u}_{11}}$ .", "The EM vector field in a point outside the source is given by $\\delta {\\bf A}({\\bf r}, t)=\\int \\frac{\\delta {\\bf J}({\\bf {r^{\\prime }}},t_R)}{\|{\\bf r}-{\\bf r^{\\prime }}\|}d V^{\\prime }\\,,$ where $t_R=t-\|\\mathbf {r} - \\mathbf {r}^{\\prime }\|$ .", "We estimate the emitted power by considering the moving electric charges as an oscillating magnetic dipole.", "In the far field approximation ($\|\\mathbf {r}\| \\gg \|\\mathbf {r}^{\\prime }\|$ ), and for a coherent emission ($\\omega _{11} \\ll 1/ \|\\mathbf {r}^{\\prime }\|$ ), we obtain that $P &=& \\frac{e^2 \\pi ^4 \\omega _{11}^6}{6} \\left( \\sum _i Q_i \\int _0^R dr \\, n_i(r) r^3 W_{11}(r)\\right)^2 \\sin ^2(\\omega _{11} t) \\nonumber \\\\ &\\simeq & \\frac{\\pi ^4 (\\omega _{11} R)^6}{6} W_{11}(R)^2 Q_+^2 \\sin ^2(\\omega _{11} t)\\,,$ where $Q_+$ is given in Eq.", "(REF ).", "An approximate value of the radiated power is given by $P(a) \\simeq 6.4 \\times 10^{41} \\left(\\frac{y_2(a)^6 A^2(a)}{y_2(0)^6 A^2(0)}\\right) \\left(\\frac{\\nu }{\\nu _0} \\right)^2 \\left(\\frac{\\rho _R}{\\rho _0} \\right)^{-3} \\left(\\frac{R}{10 \\text{km}}\\right)^{-1} \\left(\\frac{\\alpha E_{\\text{glitch}}}{E_{\\text{glitch}}^{\\text{Vela}} }\\right) \\left( \\frac{Q_+}{Q}\\right)^2 \\text{erg/s}\\,,$ where we have averaged over time and considered as a reference value for the surface charge density $Q= 10^5$ MeV$^3$ fm.", "The values of $Q_+$ for the two considered models and for two different values of $M_s$ are reported in Table REF .", "The radiated power increases with increasing $a$ , meaning that the thinner the crust, the larger the radiated power (as far as the crust remains larger than the region in which there is a positive electric charge).", "For example considering $a=0.9$ we obtain a radiated power of about $10^{45}$ erg/s.", "The radiated power increases because the oscillation frequency increases with increasing $a$ .", "It certainly happens that for $a \\rightarrow 1$ the amplitude of the oscillation decreases, see Fig.", "REF , but it does not decrease fast enough to compensate for the increase of the frequency, and thus the product $y_2(a)^6 A^2(a)$ in Eq.", "(REF ) increases with increasing $a$ .", "Figure: Radiated power, see Eq.", "(), as a function of the ratio between the core radius and the star radius, aa.", "The radiated power is normalized to the value for a=0a=0, corresponding to a strange star with a rigid crust extending from the surface down to the center of the star.Table: Approximate values of the damping times for the two considered models for two different values of the strange quark mass and of the gap parameter.", "In the third column we have assumed that a=0.9a=0.9, meaning that the CCSC crust is about 0.70.7 km (1.11.1 km) thick for Model A (Model B).", "In the fourth column we have taken a=0.1a=0.1, meaning that the CCSC crust is about 6.46.4 km (9.89.8 km) thick for Model A (Model B).Approximate values of the damping time, $\\tau $ , for the two considered models and for different values of $M_s$ and $\\Delta $ are reported in Table REF , for two values of $a$ .", "The damping time is computed simply dividing the energy of the oscillation for the corresponding emitted power.", "This certainly gives a rough, order of magnitude, estimate of the time needed for emitting all the energy.", "We find that $\\tau $ decreases with increasing values of $M_s$ and/or $\\Delta $ .", "The reason is that with increasing values of $M_s$ the positive charge close to the star surface increases, see Eq.", "(REF ) and Table REF .", "With increasing values of $\\Delta $ the shear modulus increases, see Eq.", "(REF ), and therefore the frequency of the oscillations increases.", "Since $P \\propto Q_+^2 \\omega _{11}^6$ , the dependence on $\\Delta $ is particularly strong, see Eq.", "(REF ) and Eq.", "(REF ).", "So far we have assumed that the electrosphere does not screen the radiated photons.", "However, it is conceivable that a sizable fraction of the emitted energy will be scattered by the electrosphere.", "The detailed modelization of the electrosphere and of the interaction with the photons emitted by the positive charged quark layer is a nontrivial problem, see for example [41], [42], [43], [22].", "We shall estimate the absorbed fraction considering a completely degenerate electron gas at small temperature [41].", "The absorption is due to the Thomson scattering of photons off degenerate electrons and leads to an exponential reduction of the emitted power.", "Assuming that electrons are degenerate and considering that temperature corrections, of order $T/m_e$ , can be neglected, one finds that the intensity of the emitted radiation is suppressed by [41] $\\eta = \\exp \\left(-\\frac{1.129 }{6 \\pi ^{2}} \\sqrt{\\frac{3 \\pi }{2\\alpha _\\text{em}}} \\sigma _{\\it T} V_0^2\\right)\\,,$ where $\\sigma _{\\it T}=8\\pi /3 (\\alpha _\\text{em}^2/m_e^2)$ is the total Thomson cross section.", "For the considered values of the surface potential, see Table REF , we obtain suppression factors of order $0.1$ ." ], [ "Conclusions", "We have discussed two very simple strange star models entirely composed of deconfined color superconducting matter.", "Our star models are very similar to those proposed in [22] for the discussion of $r-$ mode oscillations.", "We assume that the star core is composed by CFL matter and there is a crust of rigid CCSC quark matter.", "The size of the star crust is unknown, because it depends on the detailed values of the strange quark mass and of the gap parameters, which are very uncertain.", "For this reason we have treated the ratio between the core radius and the star radius as a free parameter.", "In our treatment of the crust we have determined the equilibrium charge configuration, in Sec.", ", using the free Fermi gas distributions but at the same time we have considered CCSC matter, in Sec.", ", as a rigid crystalline structure.", "These two facts may seem to be in contradiction.", "However, in the relevant crystalline phases, quarks close to the Fermi sphere have a linear dispersion law [44], [15], which indeed mimics the behavior of free quarks.", "The effect of the condensate is to induce a direction dependent Fermi velocity.", "Moreover, not all quarks on the top of the Fermi sphere are paired.", "For these reasons we have assumed that the EM properties of the CCSC phase are similar to those of unpaired quark matter.", "What is rigid is the modulation of the underlying quark condensate that can be seen as the structure on the top of which quarks propagate.", "This treatment of the EM properties of the CCSC phase is in our opinion an educated assumption.", "A detailed study of the EM properties of the CCSC phase is necessary to substantiate this approach.", "The discussion of the quark matter surface can certainly be improved including condensation effects, following for example the discussion in [45], or viscous damping below the star crust as in [46].", "Moreover it would be interesting to study whether strangelet crystals could form on the star surface [47].", "One should investigate whether these strangelet nuggets might coexist with the CCSC phase, presumably assuming that the surface tension of quark matter is not too large, eventually leading to a drastic reduction of the surface charge density.", "In our simple treatment, we estimate the emitted energy of the torsional oscillations using an oscillating magnetic dipole.", "The emitted power is extremely large, for stars with a small CFL core it is of the order of $10^{41} \\eta $ erg/s, where $\\eta $ is a screening factor due to the presence of the electrosphere, estimated in Eq.", "(REF ).", "The emitted power steeply increases with increasing values of $a$ , see Fig.", "REF , meaning that stars with a large CFL core and a thin CCSC crust would probably emit all the oscillation energy in milliseconds.", "For a sufficiently thin crust, say hundreds of meters thick, we expect that the emission is at the MHz frequency.", "Given the large emitted power, it is tempting to compare our results with the most powerful observed EM emissions.", "The radio bursts observed from Rotating Radio Transients have a duration of few milliseconds and the associated flux energy is extremely large [48].", "However, the observed frequencies are of the order of GHz, see [49], [50], [51], whereas we found that the frequencies associated with the $n=1, l=1$ mode of a star with a 1 km CCSC crust are of the order of tens of kHz.", "If the crust is thinner, say a few centimeters thick, then GHz frequencies can be attained, but in this case the emitted power, estimated by Eq.", "(REF ), is extremely large and the damping time should be much less than the observed milliseconds.", "A loophole might be that by Eq.", "(REF ) we are overestimating the emitted power by orders of magnitude.", "The reason is that in Eq.", "(REF ) we are using the coherent emission approximation, which conceivably breaks down at such large frequencies.", "Moreover, we are assuming the presence of a net positive charge, with electrons only providing a screen for the emitted power.", "A more refined treatment should include the effect of the star magnetic field which might strongly couple the oscillation of the star and of the electrosphere.", "Therefore, future work in this direction is needed to clarify how the presented discussion of the emitted EM power changes in the presence of strong magnetic fields, see for example [52], [53].", "A different possibility is that there exists a mechanism for exciting predominantly modes with higher angular momentum and/or higher principal quantum numbers.", "In this case, frequencies larger than tens of kHz could be reached for stars with a thick crust, as well.", "Different powerful phenomena of great interest are giant magnetar x-rays flares [54].", "The observation of these flares has posed a challenge to strange stars with no crust [55].", "The standard explanation of these flares is indeed related with the seismic vibrations of the crust triggered by a starquake.", "Typical frequencies are of the order of hundreds of Hz at most and the emitted luminosities is of the order of $10^{44} - 10^{46}$ erg/s.", "The measured decaying time is of order of minutes.", "In our model, oscillations of hundreds of Hz can be reached only if the shear modulus is sufficiently small, of the order of $\\nu _0 10^{-4}$ , making it comparable with standard nuclear crusts, and if the CCSC crust is sufficiently thick, say of the order of a few kilometers, meaning that $a \\sim 0.1$ .", "For these small values of $a$ the damping time can be of the order of hundreds of seconds, see the last column in Table REF .", "Basically, our bare strange star model has frequency and decay times compatible with magnetar flares only if it has the same structure of a standard neutron star.", "The caveat is that these flares are observed in magnetars, which are CSOs expected to have a large magnetic field.", "In our simple treatment we have neglected the effect of the background magnetic field, which however in the case of magnetars could be sizable.", "We have neglected the effect of the temperature, as well.", "Although temperature effects are negligible for strange stars older than $\\sim 10$ s [56], when the temperature has dropped below the MeV scale, it would be interesting to see what is the effect of a large, say $\\sim 10$ MeV temperature, see for example the discussion in [41], [57].", "Unfortunately, the shear modulus has only been computed at vanishing temperature [26].", "A detailed study of the temperature dependence of the shear modulus and of the response of the CCSC structure to the temperature is needed to ascertain the correct temperature dependence of the torsional oscillations.", "However, let us assume that the CCSC structure has already formed when the temperature is of the order of few MeV, and that it responds to the temperature as a standard material, meaning that with increasing temperature the shear modulus decreases.", "From Eq.", "(REF ) it follows that the frequency of the torsional oscillation decreases, and, from Eq.", "(REF ), that the amplitude increases.", "Moreover, an increasing temperature should lead to an increase of the number densities of the light quarks and electrons, leading to a larger $Q_+$ .", "The overall effect on the radiated power is not obvious, because in Eq.", "(REF ) we have that $\\omega _{11}$ decreases, but $W_{11}$ and $Q_+$ increase; therefore a careful study of the various contributions is necessary.", "Regarding the emission mechanisms of the electrosphere, one should consider the various processes that become relevant at nonvanishing temperature.", "In particular, $e^+e^-$ production is believed to be the dominant process at high temperature [58], [59], [60], [61].", "Finally, note that in the evaluation of the horizontal displacement one should include the radial dependence of the shear modulus and of the matter densities as well as GR corrections, which are expected to be small, but should nevertheless be considered for a more refined study.", "We thank M. Alford and I. Bombaci for insightful discussions." ] ]	1403.0128
[ [ "Isoperimetry and stability properties of balls with respect to nonlocal\n energies" ], [ "Abstract We obtain a sharp quantitative isoperimetric inequality for nonlocal $s$-perimeters, uniform with respect to $s$ bounded away from $0$.", "This allows us to address local and global minimality properties of balls with respect to the volume-constrained minimization of a free energy consisting of a nonlocal $s$-perimeter plus a non-local repulsive interaction term.", "In the particular case $s =1$ the $s$-perimeter coincides with the classical perimeter, and our results improve the ones of Kn\\\"upfer and Muratov concerning minimality of balls of small volume in isoperimetric problems with a competition between perimeter and a nonlocal potential term.", "More precisely, their result is extended to its maximal range of validity concerning the type of nonlocal potentials considered, and is also generalized to the case where local perimeters are replaced by their nonlocal counterparts." ], [ "Introduction", "In the recent paper [7], Caffarelli, Roquejoffre, and Savin have initiated the study of Plateau-type problems with respect to a family of nonlocal perimeter functionals.", "A regularity theory for such nonlocal minimal surfaces has been developed by several authors [10], [4], [18], [35], [12], while the relation of nonlocal perimeters with their local counterpart has been investigated in [8], [3].", "The isoperimetry of balls in nonlocal isoperimetric problems has been addressed in [19].", "Precisely, given $s\\in (0,1)$ and $n\\ge 2$ , one defines the $s$ -perimeter of a set $E\\subset \\mathbb {R}^n$ as $P_s(E):=\\int _E\\int _{E^c}\\frac{dx\\,dy}{\|x-y\|^{n+s}}\\in [0,\\infty ]\\,.$ As proved in [19], if $0<\|E\|<\\infty $ then we have the nonlocal isoperimetric inequality $P_s(E)\\ge \\frac{P_s(B)}{\|B\|^{(n-s)/n}}\\,\|E\|^{(n-s)/n}\\,,$ where $B_r:=\\lbrace x\\in \\mathbb {R}^n:\|x\|<r\\rbrace $ , $B:=B_1$ , and $\|E\|$ is the Lebesgue measure of $E$ .", "Notice that the right-hand side of (REF ) is equal to $P_s(B_{r_E})$ , the $s$ -perimeter of a ball of radius $r_E=(\|E\|/\|B\|)^{1/n}$ – so that $\|E\|=\|B_{r_E}\|$ .", "Moreover, again in [19] it is shown that equality holds in (REF ) if and only if $E=x+B_{r_E}$ for some $x\\in \\mathbb {R}^n$ .", "In [24] the following stronger form of (REF ) was proved: $P_s(E)\\ge \\frac{P_s(B)}{\|B\|^{(n-s)/n}}\\,\|E\|^{(n-s)/n}\\Big \\lbrace 1+\\frac{A(E)^{4/s}}{C(n,s)}\\Big \\rbrace \\,,$ where $C(n,s)$ is a non-explicit positive constant depending on $n$ and $s$ only, while $A(E):=\\inf \\Big \\lbrace \\frac{\|E\\Delta (x+B_{r_E})\|}{\|E\|}:x\\in \\mathbb {R}^n\\Big \\rbrace $ measures the $L^1$ -distance of $E$ from the set of balls of volume $\|E\|$ and is commonly known as the Fraenkel asymmetry of $E$ (recall that, given two sets $E$ and $F$ , $\|E\\Delta F\|:=\|E\\setminus F\|+\|F\\setminus E\|$ ).", "Our first main result improves (REF ) by providing the sharp decay rate for $A(E)$ in (REF ).", "Moreover, we control the constant $C(n,s)$ appearing in (REF ) and make sure it does not degenerate as long as $s$ stays away from 0.", "Theorem 1.1 For every $n\\ge 2$ and $s_0\\in (0,1)$ there exists a positive constant $C(n,s_0)$ such that $P_s(E)\\ge \\frac{P_s(B)}{\|B\|^{(n-s)/n}}\\,\|E\|^{(n-s)/n}\\,\\Big \\lbrace 1+\\frac{A(E)^2}{C(n,s_0)}\\Big \\rbrace \\,,$ whenever $s\\in [s_0,1]$ and $0<\|E\|<\\infty $ .", "Remark 1.2 The constant $C(n,s_0)$ we obtain in (REF ) is not explicit.", "It is natural to conjecture that $C(n,s_0)\\approx 1/s_0$ as $s_0\\rightarrow 0^+$ , see (REF ) below.", "Letting $s\\rightarrow 1$ we recover the sharp stability result for the classical perimeter, that was first proved in [22] by symmetrization methods and later extended to anisotropic perimeters in [17] by mass transportation.", "The latter approach yields an explicit constant $C(n)$ in (REF ) when $s=1$ , that grows polynomially in $n$ .", "It remains an open problem to prove (REF ) with an explicit constant $C(n,s)$ .", "We next turn to consider nonlocal isoperimetric problems in presence of nonlocal repulsive interaction terms.", "The starting point is provided by Gamow model for the nucleus, which consists in the volume constraint minimization of the energy $P(E)+V_\\alpha (E)$ , where $P(E)$ is the (distributional) perimeter of $E\\subset \\mathbb {R}^n$ defines as $P(E):=\\sup \\Bigl \\lbrace \\int _E {\\rm div} X(x)\\,dx : X \\in C^1_c(\\mathbb {R}^n;\\mathbb {R}^n),\\,\|X\|\\le 1 \\Bigr \\rbrace \\,,$ while, given $\\alpha \\in (0,n)$ , $V_\\alpha (E)$ is the Riesz potential $V_\\alpha (E):=\\int _E\\,\\int _E\\,\\frac{dx\\,dy}{\|x-y\|^{n-\\alpha }}\\,.$ By minimizing $P(E)+V_\\alpha (E)$ with $\|E\|=m$ fixed, we observe a competition between the perimeter term, that tries to round up candidate minimizers into balls, and the Riesz potential, that tries to smear them around.", "(Notice also that, by Riesz inequality, balls are actually the volume constrained maximizers of $V_\\alpha $ .)", "It was recently proved by Knüpfer and Muratov that: (a) If $n=2$ and $\\alpha \\in (0,2)$ , then there exists $m_0=m_0(n,\\alpha )$ such that Euclidean balls of volume $m \\le m_0$ are the only minimizers of $P(E)+V_\\alpha (E)$ under the volume constraint $\|E\|=m$ [25] .", "(b) If $n=2$ and $\\alpha $ is sufficiently close to 2, then balls are the unique minimizers for $m \\le m_0$ while for $m>m_0$ there are no minimizers [25].", "(c) If $3\\le n\\le 7$ and $\\alpha \\in (1,n)$ , then the result in (a) holds [26].", "In [6], Bonacini and Cristoferi have recently extended both (b) and (c) above to the case $n\\ge 3$ , and have also shown that balls of volume $m$ are volume-constrained $L^1$ -local minimizers of $P(E)+V_\\alpha (E)$ if $m<m_\\star (n,\\alpha )$ , while they are never volume-constrained $L^1$ -local minimizers if $m>m_\\star (n,\\alpha )$ .", "The constant $m_\\star (n,\\alpha )$ is characterized in terms of a minimization problem, that is explicitly solved in the case $n=3$ (in particular, in the physically relevant case $n=3$ , $s=1$ , and $\\alpha =2$ (Coulomb kernel), one finds $m_\\star (3,1,2)=5$ , a result that was actually already known in the physics literature since the 30's [5], [14], [21]).", "Let us also mention that, in addition to (b), further nonexistence results are contained in [26], [30].", "We stress that, apart from the special case $n=2$ , all these results are limited to the case $\\alpha \\in (1,n)$ , named the far-field dominated regime by Knüpfer and Muratov to mark its contrast to the near-field dominated regime $\\alpha \\in (0,1]$ .", "Our second and third main results extend (a) and (c) above in two directions: first, by covering the full range $\\alpha \\in (0,n)$ for all $n \\ge 3$ , and second, by including the possibility for the dominant perimeter term to be a nonlocal $s$ -perimeter.", "The global minimality threshold $m_0$ is shown to be uniformly positive with respect to $s$ and $\\alpha $ provided they both stay away from zero.", "The local minimality threshold $m_\\star (n,s,\\alpha )$ is characterized in terms of a minimization problem.", "In order to include the classical perimeter as a limiting case when $s \\rightarrow 1$ , we recall that, by combining [8] with [3], one finds that $\\lim _{s\\rightarrow 1^-}(1-s)\\,P_s(E)=\\omega _{n-1}\\,P(E)$ whenever $E$ is an open set with $C^{1,\\gamma }$ -boundary for some $\\gamma >0$ (from now on, $\\omega _n$ denotes the volume of the $n$ -dimensional ball of radius 1).", "Hence, to recover the classical perimeter we need to suitably renormalize the $s$ -perimeter.", "Theorem 1.3 For every $n\\ge 2$ , $s_0\\in (0,1)$ , and $\\alpha _0\\in (0,n)$ , there exists $m_0=m_0(n,s_0,\\alpha _0)>0$ such that, if $m\\in (0,m_0)$ , $s\\in (s_0,1)$ , and $\\alpha \\in (\\alpha _0,n)$ , then the variational problems $&&\\inf \\bigg \\lbrace \\frac{1-s}{\\omega _{n-1}}\\,P_s(E)+V_\\alpha (E):\|E\|=m\\bigg \\rbrace \\,,\\\\&&\\inf \\Big \\lbrace P(E)+V_\\alpha (E):\|E\|=m\\Big \\rbrace \\,,$ admit balls of volume $m$ as their (unique up to translations) minimizers.", "Remark 1.4 An important open problem is, of course, to provide explicit lower bounds on $m_0$ .", "Let us now define a positive constant $m_\\star $ by setting $m_\\star (n,s,\\alpha ):={\\left\\lbrace \\begin{array}{ll}\\displaystyle \\omega _n\\,\\bigg (\\frac{n+s}{n-\\alpha }\\,\\frac{s\\,(1-s)\\,P_s(B)}{\\omega _{n-1}\\,\\alpha \\,V_\\alpha (B)}\\bigg )^{n/(\\alpha +s)}\\,, &\\hspace{28.45274pt} \\text{if $s\\in (0,1)$}\\,,\\\\[10pt]\\displaystyle \\omega _n\\,\\bigg (\\frac{n+1}{n-\\alpha }\\,\\frac{P(B)}{\\alpha \\,V_\\alpha (B)}\\bigg )^{n/(\\alpha +1)}\\,, &\\hspace{28.45274pt} \\text{if $s=1$}\\,.\\end{array}\\right.", "}$ The constant $m_\\star (n,s,\\alpha )$ is the threshold for volume-constrained $L^1$ -local minimality of balls with respect to the functional $\\frac{1-s}{\\omega _{n-1}}\\,P_s+V_\\alpha $ , as shown in the next theorem: Theorem 1.5 For every $n\\ge 2$ , $s\\in (0,1)$ , and $\\alpha \\in (0,n)$ , let $m_\\star =m_\\star (n,s,\\alpha )$ be as in (REF ).", "For every $m\\in (0,m_\\star )$ there exists $\\varepsilon _\\star =\\varepsilon _\\star (n,s,\\alpha ,m)>0$ such that, if $B[m]$ denotes a ball of volume $m$ , then $\\frac{1-s}{\\omega _{n-1}}\\,P_s(B[m])+V_\\alpha (B[m])\\le \\frac{1-s}{\\omega _{n-1}}\\,P_s(E)+V_\\alpha (E)\\,,$ whenever $\|E\|=m$ and $\|E\\Delta B[m]\|\\le \\varepsilon _\\star \\,m$ .", "Moreover, if $m>m_\\star $ , then there exists a sequence of sets $\\lbrace E_h\\rbrace _{h\\in \\mathbb {N}}$ with $\|E_h\|=m$ and $\|E_h\\Delta B[m]\|\\rightarrow 0$ as $h\\rightarrow \\infty $ such that (REF ) fails with $E=E_h$ for every $h\\in \\mathbb {N}$ .", "Both Theorem REF and Theorem REF are obtained by combining a Taylor's expansion of nonlocal perimeters near balls, discussed in section , with a uniform version of the regularity theory developed in [7], [10], presented in section .", "In the case of Theorem REF , these two tools are combined in section through a suitable version of Ekeland's variational principle.", "We implement this approach, that was introduced in the case $s=1$ by Cicalese and Leonardi [11], through a penalization argument closer to the one adopted in [1].", "Due to the nonlocality of $s$ -perimeters, the implementation itself will not be straightforward, and will require to develop some lemmas of independent interest, like the nucleation lemma (Lemma REF ) and the truncation lemma (Lemma REF ).", "Concerning Theorem REF , our proof is inspired by the strategy used in [16] (see also [15] for a related argument) to show the isoperimetry of balls in isoperimetric problems with log-convex densities.", "Starting from the results in sections and , the proof of Theorem REF is given in section .", "Finally, the proof of Theorem REF is based on some second variation formulae for nonlocal functionals (discussed in section ), which are then exploited to characterize the threshold for volume-constrained stability (in the sense of second variation) of balls in section .", "The passage from stability to $L^1$ -local minimality is finally addressed in section .", "The proof of this last result is pretty delicate since we do not know that the ball is a global minimizer, a fact that usually plays a crucial role in this kind of arguments." ], [ "Acknowledgment", "The work of NF and MM was supported by ERC under FP7 Advanced Grant n. 226234.", "The work of NF was partially carried on at the University of Jyväskylä under the FiDiPro Program.", "The work of AF was supported by NSF Grant DMS-1262411.", "The work of FM was supported by NSF Grant DMS-1265910.", "The work of VM was supported by ANR Grant 10-JCJC 0106." ], [ "A Fuglede-type result for the fractional perimeter", "In this section we are going to prove Theorem REF on nearly spherical sets.", "Precisely, we shall consider bounded open sets $E$ with $\|E\|=\|B\|$ , $\\int _Ex\\,dx=0$ , and whose boundary satisfies $\\partial E=\\lbrace (1+u(x))x:\\,x\\in \\partial B\\rbrace \\,,\\qquad \\mbox{where $u\\in C^1(\\partial B)$}\\,,$ for some $u$ with $\\Vert u\\Vert _{C^1(\\partial B)}$ small.", "We correspondingly seek for a control on some fractional Sobolev norm of $u$ in terms of $P_s(E)-P_s(B)$ .", "More precisely, we shall control $[u]_{\\frac{1+s}{2}}^2:= [u]_{H^{\\frac{1+s}{2}}(\\partial B)}^2=\\iint _{\\partial B\\times \\partial B}\\frac{\|u(x)-u(y)\|^2}{\|x-y\|^{n+s}}\\,d{\\mathcal {H}}^{n-1}_x\\,d{\\mathcal {H}}^{n-1}_y\\,,$ as well as the $L^2$ -norm of $u$ .", "This kind of result is well-known in the local case (see Fuglede [23]), and takes the following form in the nonlocal case.", "Theorem 2.1 There exist constants $\\varepsilon _0\\in (0,1/2)$ and $c_0>0$ , depending only on $n$ , with the following property: If $E$ is a nearly spherical set as in (REF ), with $\|E\|=\|B\|$ , $\\int _Ex\\,dx=0$ , and $\\Vert u\\Vert _{C^1(\\partial B)}<\\varepsilon _0$ , then $P_s(E)-P_s(B)\\ge c_0\\,\\Big ([u]_{\\frac{1+s}{2}}^2+s\\,P_s(B)\\,\\Vert u\\Vert _{L^2(\\partial B)}^2\\Big )\\,,\\qquad \\forall s\\in (0,1)\\,.$ Remark 2.2 If we multiply by $1-s$ in (REF ) and then take the limit $s\\rightarrow 1^-$ , then by (REF ) and (REF ) we get $P(E)-P(B)\\ge c(n)\\,\\Vert u\\Vert ^2_{H^1}$ whenever $u\\in C^{1,\\gamma }(\\partial B)$ for some $\\gamma \\in (0,1)$ (thus, on every Lipschitz function $u:\\partial B\\rightarrow \\mathbb {R}$ by density).", "Thus Theorem REF implies [23].", "In order to prove Theorem REF , we need to premise some facts concerning hypersingular Riesz operators on the sphere.", "Following [34], one defines the hypersingular Riesz operator on the sphere of order $\\gamma \\in (0,1) \\cup (1,2)$ as $\\mathcal {D}^\\gamma u(x):=\\frac{\\gamma \\,2^{\\gamma -1}}{\\pi ^{\\frac{n-1}{2}}}\\,\\frac{\\Gamma (\\frac{n-1+\\gamma }{2})}{\\Gamma (1-\\frac{\\gamma }{2})}\\,\\,{\\rm p.v.", "}\\left(\\int _{\\partial B}\\frac{u(x)-u(y)}{\|x-y\|^{n-1+\\gamma }}\\,d{\\mathcal {H}}^{n-1}_y\\right)\\,,\\qquad x\\in \\partial B\\,,$ cf.", "[34].", "(Here, $\\Gamma $ denotes the usual Euler's Gamma function, and the symbol p.v.", "means that the integral is taken in the Cauchy principal value sense.)", "By [34], the $k$ -th eigenvalue of $\\mathcal {D}^\\gamma $ is given by $\\lambda _{k}^(\\gamma ):=\\frac{\\Gamma (k+\\frac{n-1+\\gamma }{2})}{\\Gamma (k+\\frac{n-1-\\gamma }{2})}-\\frac{\\Gamma (\\frac{n-1+\\gamma }{2})}{\\Gamma (\\frac{n-1-\\gamma }{2})}\\,,\\qquad k\\in \\mathbb {N}\\cup \\lbrace 0\\rbrace \\,,$ (so that $\\lambda _{k}^(\\gamma )\\ge 0$ , $\\lambda _{k}^(\\gamma )$ is strictly increasing in $k$ , and $\\lambda _{k}^(\\gamma )\\uparrow \\infty $ as $k\\rightarrow \\infty $ ).", "Moreover, if we denote by ${\\mathcal {S}}_k$ the finite dimensional subspace of spherical harmonics of degree $k$ , and by $\\lbrace Y_k^i\\rbrace _{i=1}^{d(k)}$ an orthonormal basis for ${\\mathcal {S}}_k$ in $L^2(\\partial B)$ , then $\\mathcal {D}^\\gamma Y_k=\\lambda _{k}^(\\gamma )\\,Y_k\\,,\\qquad \\forall k\\in \\mathbb {N}\\cup \\lbrace 0\\rbrace \\,.$ When no confusion arises, we shall often denote by $Y_k$ a generic element in ${\\mathcal {S}}_k$ .", "Given $s\\in (0,1)$ , let us now introduce the operator ${I}_su(x):=2\\,{\\rm p.v.", "}\\left(\\int _{\\partial B}\\frac{u(x)-u(y)}{\|x-y\|^{n+s}}\\,d{\\mathcal {H}}^{n-1}_y\\right)\\,,\\qquad u\\in C^2(\\partial B)\\,,$ so that, for every $u\\in C^2(\\partial B)$ , ${I}_su=\\frac{2^{1-s}\\,\\pi ^{\\frac{n-1}{2}}}{1+s}\\,\\frac{\\Gamma (\\frac{1-s}{2})}{\\Gamma (\\frac{n+s}{2})}\\,\\mathcal {D}^{1+s}u\\,,$ and $[u]_{\\frac{1+s}{2}}^2=\\int _{\\partial B}u\\,{I}_su\\,d{\\mathcal {H}}^{n-1}\\,.$ Let us denote by $\\lambda _k^s$ the $k$ -th eigenvalue of ${I}_s$ .", "By (REF ), (REF ), and (REF ) we find that $\\lambda _k^s$ satisfies $\\lambda _0^s=0\\,,\\qquad \\lambda _{k+1}^s>\\lambda _k^s\\,,\\qquad {I}_sY_k=\\lambda _k^s\\,Y_k\\,,\\qquad \\forall k\\in \\mathbb {N}\\cup \\lbrace 0\\rbrace \\,,$ and $\\lambda _k^s\\uparrow \\infty $ as $k\\rightarrow \\infty $ .", "If we denote by $a_k^i(u):=\\int _{\\partial B}u\\,Y_k^i\\,d{\\mathcal {H}}^{n-1}$ the Fourier coefficient of $u$ corresponding to $Y_k^i$ , then we obtain $[u]_{\\frac{1+s}{2}}^2=\\sum _{k=0}^\\infty \\sum _{i=1}^{d(k)}\\,\\lambda _k^s\\,a_k^i(u)^2\\,.$ Concerning the value of $\\lambda _1^s$ and $\\lambda _2^s$ , we shall need the following proposition.", "Proposition 2.3 One has $\\lambda _1^s&=&s(n-s)\\frac{P_s(B)}{P(B)}\\,.\\\\\\lambda _2^s&=&\\frac{2n}{n-s}\\lambda _1^s\\,.$ Since each coordinate function $x_i$ , $i=1,\\dots ,n$ , belongs to ${\\mathcal {S}}_1$ , we have ${I}_sx_i=\\lambda _1^sx_i$ .", "Hence, inserting $x_i$ in (REF ) and adding up over $i$ , yields $\\lambda _1^s=\\frac{1}{P(B)}\\iint _{\\partial B\\times \\partial B}\\frac{d{\\mathcal {H}}^{n-1}_x\\,d{\\mathcal {H}}^{n-1}_y}{\|x-y\|^{n+s-2}}\\,.$ For $z\\in \\mathbb {R}^n\\setminus \\lbrace 0\\rbrace $ , we now set $\\mathcal {K}(z):=-\\frac{1}{n+s-2}\\frac{1}{\|z\|^{n+s-2}}\\,.$ Splitting $\\nabla \\mathcal {K}$ into its tangential and normal components to $\\partial B$ , we compute for $y\\notin {\\overline{B}}$ the integral $L(y):=&\\int _{\\partial B}\\frac{(x-y)\\cdot (x-y)}{\|x-y\|^{n+s}}\\,d{\\mathcal {H}}^{n-1}_x\\\\=&\\int _{\\partial B}\\nabla _x \\mathcal {K}(x-y)\\cdot x\\,d{\\mathcal {H}}^{n-1}_x-\\int _{\\partial B}\\nabla _x\\mathcal {K}(x-y)\\cdot y\\,d{\\mathcal {H}}^{n-1}_x\\nonumber \\\\=&\\int _{\\partial B}(1-x\\cdot y)\\frac{\\partial \\mathcal {K}}{\\partial \\nu (x)}(x-y)\\,d{\\mathcal {H}}^{n-1}_x-\\int _{\\partial B}\\nabla _{\\tau }\\mathcal {K}(x-y)\\nabla _{\\tau }(x\\cdot y)\\,d{\\mathcal {H}}^{n-1}_x\\nonumber \\\\=&\\!", ":\\mathcal {A}(y)-\\mathcal {B}(y)\\,.\\nonumber $ We now evaluate separately $\\mathcal {A}(y)$ and $\\mathcal {B}(y)$ .", "Noticing that $\\Delta {\\mathcal {K}}(z)=-s/\|z\|^{n+s}$ , we first integrate $\\mathcal {A}(y)$ by parts to obtain $\\mathcal {A}(y)&=\\int _B\\Delta _x \\mathcal {K}(x-y)(1-x\\cdot y)\\,dx+\\int _B\\nabla _x \\mathcal {K}(x-y)\\nabla _x(1-x\\cdot y)\\,dx\\\\&=-s\\int _B\\frac{1-x\\cdot y}{\|x-y\|^{n+s}}\\,dx+\\int _B\\frac{\|y\|^2-x\\cdot y}{\|x-y\|^{n+s}}\\,dx\\\\&=(1-s)\\int _B\\frac{1-x\\cdot y}{\|x-y\|^{n+s}}\\,dx+\\int _B\\frac{\|y\|^2-1}{\|x-y\|^{n+s}}\\,dx\\,.$ We now denote by $\\Delta _{{\\mathbb {S}}^{n-1}}$ the standard Laplace–Beltrami operator on the sphere and recall that $-\\Delta _{{\\mathbb {S}}^{n-1}}x_i=(n-1)x_i$ for $i=1,\\dots ,n$ .", "Integrating $\\mathcal {B}(y)$ by parts leads to $\\mathcal {B}(y)&=-\\int _{\\partial B}\\mathcal {K}(x-y)\\Delta _{{\\mathbb {S}}^{n-1}}(x\\cdot y)\\,d{\\mathcal {H}}^{n-1}_x=(n-1)\\int _{\\partial B}\\mathcal {K}(x-y)x\\cdot y\\,d{\\mathcal {H}}^{n-1}_x\\\\&=-\\frac{n-1}{n+s-2}\\int _{\\partial B}\\frac{x\\cdot y}{\|x-y\|^{n+s-2}}\\,d{\\mathcal {H}}^{n-1}_x\\,.$ From the above expressions of $\\mathcal {A}$ and $\\mathcal {B}$ , we can let $y$ converge to a point on $\\partial B$ to find $L(y)=(1-s)\\int _B\\frac{1-x\\cdot y}{\|x-y\|^{n+s}}\\,dx+\\frac{n-1}{n+s-2}\\int _{\\partial B}\\frac{x\\cdot y}{\|x-y\|^{n+s-2}}\\,d{\\mathcal {H}}^{n-1}_x\\,,\\quad y\\in \\partial B\\,.$ Integrating over $\\partial B$ the first integral on the right hand side of the previous equality, and using the divergence theorem again, we get $\\int _Bdx\\int _{\\partial B}\\frac{1-x\\cdot y}{\|x-y\|^{n+s}}\\,d{\\mathcal {H}}^{n-1}_y&= \\int _Bdx\\int _{\\partial B}\\frac{(y-x)\\cdot y}{\|x-y\|^{n+s}}\\,d{\\mathcal {H}}^{n-1}_y\\,dx\\\\&=\\int _Bdx\\int _{\\partial B}\\frac{\\partial \\mathcal {K}}{\\partial \\nu }(y-x)\\,d{\\mathcal {H}}^{n-1}_y=-\\int _Bdx\\int _{B^c}\\Delta _y\\mathcal {K}(y-x)\\,dy\\\\&=s\\int _B\\int _{B^c}\\frac{1}{\|x-y\|^{n+s}}\\,dx\\,dy=sP_s(B)\\,.$ From this formula, integrating both sides of (REF ) and recalling (REF ) and (REF ), we obtain $\\lambda _1^s=s(1-s)\\frac{P_s(B)}{P(B)}+\\frac{n-1}{(n+s-2)P(B)}\\iint _{\\partial B\\times \\partial B}\\frac{x\\cdot y}{\|x-y\|^{n+s-2}}\\,d{\\mathcal {H}}^{n-1}_x\\,d{\\mathcal {H}}^{n-1}_y\\,.$ To deal with the last integral of the previous equality we need to rewrite $P_s(B)$ as follows $P_s(B)&=\\int _{B^c}dy\\int _B\\frac{(x-y)\\cdot (x-y)}{\|x-y\|^{n+s+2}}\\,dx=-\\frac{1}{n+s}\\int _{B^c}dy\\int _B\\nabla _x\\Bigl (\\frac{1}{\|x-y\|^{n+s}}\\Bigr )\\cdot (x-y)\\,dx\\\\&=-\\frac{1}{n+s}\\int _{B^c}\\biggl (-n\\int _B\\frac{dx}{\|x-y\|^{n+s}}+\\int _{\\partial B}\\frac{(x-y)\\cdot x}{\|x-y\|^{n+s}}\\,d{\\mathcal {H}}^{n-1}_x\\biggr )\\,dy\\\\&=\\frac{n}{n+s}P_s(B)-\\frac{1}{n+s}\\int _{B^c}dy\\int _{\\partial B}\\frac{(x-y)\\cdot x}{\|x-y\|^{n+s}}\\,d{\\mathcal {H}}^{n-1}_x\\,.$ Therefore $P_s(B)&=\\frac{1}{s}\\int _{\\partial B}d{\\mathcal {H}}^{n-1}_x\\int _{B^c}\\frac{(y-x)\\cdot x}{\|x-y\|^{n+s}}\\,dy \\\\&=-\\frac{1}{s(n+s-2)}\\int _{\\partial B}d{\\mathcal {H}}^{n-1}_x\\int _{B^c}\\nabla _y\\Bigl (\\frac{1}{\|x-y\|^{n+s-2}}\\Bigr )\\cdot x\\,dy\\\\&=\\frac{1}{s(n+s-2)}\\iint _{\\partial B\\times \\partial B}\\frac{x\\cdot y}{\|x-y\|^{n+s-2}}\\,d{\\mathcal {H}}^{n-1}_x\\,d{\\mathcal {H}}^{n-1}_y\\,.$ Combining this last equality with (REF ) leads to the proof of (REF ).", "Finally, using (REF ) and exploiting the factorial property of the Gamma function $\\Gamma (z+1)=\\Gamma (z)\\,z$ for every $z\\in \\mathbb {C}\\setminus \\lbrace -k:k\\in \\mathbb {N}\\cup \\lbrace 0\\rbrace \\rbrace $ , we see that $\\lambda _1^(\\alpha )=\\frac{\\alpha }{\\kappa }\\, \\frac{\\Gamma (\\alpha +\\kappa )}{\\Gamma (\\kappa )} \\,,\\quad \\lambda _2^(\\alpha )=\\frac{1+\\alpha +2\\kappa }{1+\\kappa }\\,\\lambda _1^(\\alpha )\\,,\\quad \\kappa :=\\frac{n-1-\\alpha }{2}\\,.$ Since $\\alpha =1+s$ , we infer from (REF ) and (REF ) that $\\lambda _2^s/\\lambda _1^s=\\lambda _2^(\\alpha )/\\lambda _1^(\\alpha )=\\frac{2n}{n-s}$ which is precisely identity ().", "Step 1.", "We start by slightly rephrasing the assumption.", "Precisely, we consider a function $u\\in C^1(\\partial B)$ with $\\Vert u\\Vert _{C^1(\\partial B)}\\le 1/2$ such that there exists $t\\in (0,2\\varepsilon _0)$ with the property that the bounded open set $F_t$ whose boundary is given by $\\partial F_t=\\lbrace (1+tu(x))x:\\,x\\in \\partial B\\rbrace \\,,$ satisfies $\|F_t\|=\|B\|\\,,\\qquad \\int _{F_t}x\\,dx=0\\,.$ We thus aim to prove that, if $\\varepsilon _0$ and $c_0$ are small enough, then $P_s(F_t)-P_s(B)\\ge c_0\\,t^2\\,\\Big ([u]_{\\frac{1+s}{2}}^2+s\\,P_s(B)\\,\\Vert u\\Vert _{L^2}^2\\Big )\\,,\\qquad \\forall s\\in (0,1)\\,.$ Changing to polar coordinates, we first rewrite $ P_s(F_t)=\\iint _{\\partial B\\times \\partial B}\\biggl (\\int _{0}^{1+tu(x)}\\int _{1+tu(y)}^{+\\infty }\\frac{r^{n-1}\\varrho ^{n-1}}{(\|r-\\varrho \|^2+r\\varrho \|x-y\|^2)^{\\frac{n+s}{2}}}\\,dr\\,d\\varrho \\biggr )\\,d{\\mathcal {H}}^{n-1}_x\\,d{\\mathcal {H}}^{n-1}_y\\,.$ Then, symmetrizing this formula leads to $P_s(F_t)=\\frac{1}{2}\\iint _{\\partial B\\times \\partial B}\\biggl (\\int _{0}^{1+tu(x)}\\int _{1+tu(y)}^{+\\infty } f_{\|x-y\|}(r,\\varrho )\\,dr\\,d\\varrho \\\\+\\int _{0}^{1+tu(y)}\\int _{1+tu(x)}^{+\\infty } f_{\|x-y\|}(r,\\varrho )\\,dr\\,d\\varrho \\biggr )\\,d{\\mathcal {H}}^{n-1}_x\\,d{\\mathcal {H}}^{n-1}_y\\,,$ where, for $r,\\varrho ,\\theta >0$ , we have set $f_{\\theta }(r,\\varrho ):=\\frac{r^{n-1}\\varrho ^{n-1}}{(\|r-\\varrho \|^2+r\\varrho \\,\\theta ^2)^{\\frac{n+s}{2}}}\\,.$ Using the convention $\\int _a^b=-\\int _b^a$ , we formally have $\\int _0^b\\int _a^{+\\infty }+ \\int _0^a\\int _b^{+\\infty } = \\int _a^b\\int _a^{b} + \\int _0^a\\int _a^{+\\infty } + \\int _0^b\\int _b^{+\\infty }\\,,$ so that $P_s(F_t)=\\frac{1}{2}\\iint _{\\partial B\\times \\partial B}\\biggl (\\int _{1+tu(y)}^{1+tu(x)}\\int _{1+tu(y)}^{1+tu(x)} f_{\|x-y\|}(r,\\varrho )\\,dr\\,d\\varrho \\biggr )\\,d{\\mathcal {H}}^{n-1}_x\\,d{\\mathcal {H}}^{n-1}_y\\\\+\\iint _{\\partial B\\times \\partial B}\\biggl (\\int _{0}^{1+tu(x)}\\int _{1+tu(x)}^{+\\infty } f_{\|x-y\|}(r,\\varrho )\\,dr\\,d\\varrho \\biggr )\\,d{\\mathcal {H}}^{n-1}_x\\,d{\\mathcal {H}}^{n-1}_y\\,.$ Rescaling variables, we find that $\\int _{\\partial B}\\biggl (\\int _0^{1+tu(x)}\\int _{1+tu(x)}^{+\\infty }f_{\|x-y\|}(r,\\varrho )\\,dr\\,d\\varrho \\biggr )\\,d{\\mathcal {H}}^{n-1}_y\\\\=(1+tu(x))^{n-s}\\int _{\\partial B}\\int _0^1\\int _1^{+\\infty }f_{\|x-y\|}(r,\\varrho )\\,dr\\,d\\varrho \\,d{\\mathcal {H}}^{n-1}_y\\,,\\qquad \\forall x\\in \\partial B\\,.$ By symmetry, the triple integral on the right hand side of this identity does not depend on $x\\in \\partial B$ .", "Its constant value is easily deduced by evaluating (REF ) at $t=0$ and yields $ P_s(B)=P(B)\\, \\int _{\\partial B}\\int _0^1\\int _1^{+\\infty }f_{\|x-y\|}(r,\\varrho )\\,dr\\,d\\varrho \\,d{\\mathcal {H}}^{n-1}_y\\,,\\qquad \\forall x\\in \\partial B\\,.$ Combining the last two identities with (REF ), we conclude that $P_s(F_t)=\\frac{1}{2}\\iint _{\\partial B\\times \\partial B}\\biggl (\\int _{1+tu(y)}^{1+tu(x)}\\int _{1+tu(y)}^{1+tu(x)} f_{\|x-y\|}(r,\\varrho )\\,dr\\,d\\varrho \\biggr )\\,d{\\mathcal {H}}^{n-1}_x\\,d{\\mathcal {H}}^{n-1}_y\\\\+\\frac{P_s(B)}{P(B)}\\int _{\\partial B}(1+tu(x))^{n-s}\\,d{\\mathcal {H}}^{n-1}_x\\,.$ With a last change of variable in the first term on the right hand side of this identity, we reach the following formula for $P_s(F_t)$ : $P_s(F_t)=\\frac{t^2}{2}\\,g(t)+\\frac{P_s(B)}{P(B)}\\,h(t)\\,,$ where we have set $g(t):=\\iint _{\\partial B\\times \\partial B}\\biggl (\\int _{u(y)}^{u(x)}\\int _{u(y)}^{u(x)} f_{\|x-y\|}(1+tr,1+t\\varrho )\\,dr\\,d\\varrho \\biggr )\\,d{\\mathcal {H}}^{n-1}_x\\,d{\\mathcal {H}}^{n-1}_y\\,,$ and $h(t):=\\int _{\\partial B}(1+tu(x))^{n-s}\\,d{\\mathcal {H}}^{n-1}_x\\,.$ Since $g$ depends smoothly on $t$ , we can find $\\tau \\in (0,t)$ such that $g(t)=g(0)+t\\,g^{\\prime }(\\tau )$ .", "In addition, observing that $\\Big \|r\\,\\frac{\\partial f_\\theta }{\\partial r}(1+\\tau \\,r,1+\\tau \\,\\varrho )+\\varrho \\,\\frac{\\partial f_\\theta }{\\partial \\varrho }(1+\\tau \\,r,1+\\tau \\,\\varrho )\\Big \|\\le \\frac{C(n)}{\\theta ^{n+s}}\\,,\\qquad \\forall r,\\varrho \\in \\Big (-\\frac{1}{2},\\frac{1}{2}\\Big )\\,,$ for a suitable dimensional constant $C(n)$ (whose value is allowed to change from line to line), one can estimate $\|g^{\\prime }(\\tau )\|\\le C(n)\\iint _{\\partial B\\times \\partial B}\\frac{\|u(x)-u(y)\|^2}{\|x-y\|^{n+s}}\\,d{\\mathcal {H}}^{n-1}_x\\,d{\\mathcal {H}}^{n-1}_y=C(n)\\,[u]_{\\frac{1+s}{2}}^2\\,.$ Taking into account that $g(0)=[u]_{\\frac{1+s}{2}}^2$ and $h(0)=P(B)$ , we then infer from (REF ) that $P_s(F_t)-P_s(B)\\ge \\frac{t^2}{2}[u]_{\\frac{1+s}{2}}^2+\\frac{P_s(B)}{P(B)}\\,\\big (h(t)-h(0)\\big )-C(n)\\,t^3\\,[u]_{\\frac{1+s}{2}}^2\\,.$ We now exploit the volume constraint $\|F_t\|=\|B\|$ to deduce that $\\int _{\\partial B}(1+t\\,u)^n\\,d{\\mathcal {H}}^{n-1}=n\\,\|F_{t}\|=n\\,\|B\|=P(B)=h(0)\\,,$ so that $h(t)-h(0)=\\int _{\\partial B}(1+t\\,u)^n\\big ((1+t\\,u)^{-s}-1\\big )\\,d{\\mathcal {H}}^{n-1}_x\\,.$ By a Taylor expansion, we find that for every $\|z\|\\le 1/2$ , $\\big ((1+z)^{-s}-1\\big )(1+z)^n=\\Bigl (-sz+\\frac{s(s+1)}{2}z^2+sR_1(z)\\Bigr )\\Bigl (1+nz+\\frac{n(n-1)}{2}z^2+R_2(z)\\Bigr )\\,,$ with $\|R_1(z)\|+\|R_2(z)\|\\le C(n)\|z\|^3$ .", "Thus $h(t)-h(0)\\ge -s\\int _{\\partial B}\\Bigl [t\\,u+\\Bigl (n-\\frac{s+1}{2}\\Bigr )t^2\\,u^2\\Bigr ]\\,d{\\mathcal {H}}^{n-1}-C(n)s\\,t^3\\Vert u\\Vert ^2_{L^2}\\,.$ Exploiting the volume constraint again, i.e., $\\int _{\\partial B}\\big ((1+t\\,u)^n-1\\big )=0$ , and expanding the term $(1+t\\,u)^n$ , we get $-\\int _{\\partial B}t\\,u\\,d{\\mathcal {H}}^{n-1}\\ge \\frac{(n-1)}{2}\\int _{\\partial B}t^2\\,u^2\\,d{\\mathcal {H}}^{n-1}-C(n)\\,t^3\\,\\Vert u\\Vert ^2_{L^2}\\,.$ We may now combine (REF ) with (REF ) and (REF ) to obtain $\\frac{P_s(B)}{P(B)}\\big (h(t)-h(0)\\big )&\\ge -\\frac{t^2}{2}\\,\\frac{s(n-s)P_s(B)}{P(B)}\\,\\int _{\\partial B}u^2\\,d{\\mathcal {H}}^{n-1}-C(n)\\,\\frac{s\\,P_s(B)}{P(B)}\\,t^3\\Vert u\\Vert ^2_{L^2}\\\\&=-\\frac{t^2}{2}\\,\\lambda _1^s\\,\\int _{\\partial B}u^2\\,d{\\mathcal {H}}^{n-1}-\\frac{C(n)}{n-s}\\,\\lambda _1^s\\,t^3\\Vert u\\Vert ^2_{L^2}\\,.$ We plug this last inequality into (REF ) to find that $P_s(F_t)-P_s(B)&\\ge &\\frac{t^2}{2}\\Big ([u]_{\\frac{1+s}{2}}^2-\\lambda _1^s\\,\\Vert u\\Vert _{L^2}^2\\Big )-C(n)\\,t^3\\,\\Big ([u]_{\\frac{1+s}{2}}^2+\\lambda _1^s\\,\\Vert u\\Vert _{L^2}^2\\Big )\\,.$ Setting for brevity $a^i_k:=a^i_k(u)$ , we now apply (REF ) to deduce that, for every $\\eta \\in (0,1)$ , $\\nonumber [u]_{\\frac{1+s}{2}}^2-\\lambda _1^s\\,\\Vert u\\Vert _{L^2}^2&\\ge \\sum _{k=1}^{\\infty }\\sum _{i=1}^{d(k)}\\lambda _k^s\|a^i_k\|^2-\\lambda _1^s\\sum _{k=0}^{\\infty }\\sum _{i=1}^{d(k)}\|a^i_k\|^2\\\\&=\\frac{1}{4}\\,\\sum _{k=2}^{\\infty }\\sum _{i=1}^{d(k)}\\lambda _k^s\|a^i_k\|^2+\\sum _{k=2}^{\\infty }\\sum _{i=1}^{d(k)}\\bigg (\\frac{3}{4}\\lambda _k^s-\\lambda _1^s\\bigg )\|a^i_k\|^2-\\lambda _1^s\|a_0\|^2\\nonumber \\\\&\\ge \\frac{1}{4}\\,[u]_{\\frac{1+s}{2}}^2+\\sum _{k=2}^{\\infty }\\sum _{i=1}^{d(k)}\\bigg (\\frac{3}{4}\\lambda _k^s-\\lambda _1^s\\bigg )\|a^i_k\|^2-\\lambda _1^s\\sum _{i=1}^n\|a^i_1\|^2-\\lambda _1^s\|a_0\|^2\\nonumber \\,.$ Thanks to (REF ) and (), $\\frac{3}{4}\\lambda _k^s-\\lambda _1^s\\ge \\lambda _1^s/2$ for every $k\\ge 2$ .", "Hence, $[u]_{\\frac{1+s}{2}}^2-\\lambda _1^s\\,\\Vert u\\Vert _{L^2}^2\\ge \\frac{1}{4}\\,[u]_{\\frac{1+s}{2}}^2+\\lambda _1^s\\bigg (\\frac{1}{2}\\,\\sum _{k=2}^{\\infty }\\sum _{i=1}^{d(k)}\|a^i_k\|^2-\\sum _{i=1}^n\|a^i_1\|^2-\|a_0\|^2\\bigg )\\,.$ Using the volume constraint again and taking into account that $a_0=P(B)^{-1/2}\\int _{\\partial B}u\\,$ , one easily estimates for a suitably small value of $\\varepsilon _0$ , $\|a_0\|\\le C(n)\\,t\\,\\Vert u\\Vert ^2_{L^2}\\,.$ Similarly, the barycenter constraint $0=\\int _{\\partial B}x_i\\,(1+t\\,u)^{n+1}\\,d{\\mathcal {H}}^{n-1}$ yields $\\Big \|\\int _{\\partial B}x_i\\,u\\,d{\\mathcal {H}}^{n-1}\\Big \|\\le C(n)\\,t\\,\\Vert u\\Vert _{L^2}^2\\,,$ so that, taking into account that $Y_1^i=c(n)\\,x_i$ for some constant $c(n)$ depending on $n$ only, $\|a_1^i\|\\le C(n)\\,t\\Vert u\\Vert _{2}^2\\,,\\qquad i=1,...,n\\,.$ We can now combine (REF ) and (REF ) with $\\Vert u\\Vert _{L^2}^2=\\sum _{k=0}^\\infty \\sum _{i=1}^{d(k)}\|a_k^i\|^2$ , to conclude that $\|a_0\|^2+\\sum _{i=1}^n\|a^i_1\|^2\\le C(n)\\,t\\,\\sum _{k=2}^\\infty \\sum _{i=1}^{d(k)}\|a_k^i\|^2\\,.$ This last inequality implies of course that, for $\\varepsilon _0$ small, $\\frac{1}{2}\\,\\sum _{k=2}^{\\infty }\\sum _{i=1}^{d(k)}\|a^i_k\|^2-\\sum _{i=1}^n\|a^i_1\|^2-\|a_0\|^2\\ge \\frac{\\Vert u\\Vert _{L^2}^2}{4}\\,.$ By (REF ), (REF ), and (REF ) we thus find $P_s(F_t)-P_s(B)&\\ge & \\frac{t^2}{8}\\Big ([u]_{\\frac{1+s}{2}}^2+\\lambda _1^s\\,\\Vert u\\Vert _{L^2}^2\\Big )-C(n)\\,t^3\\,\\Big ([u]_{\\frac{1+s}{2}}^2+\\lambda _1^s\\,\\Vert u\\Vert _{L^2}^2\\Big )\\\\&\\ge &\\frac{t^2}{16}\\,\\Big ([u]_{\\frac{1+s}{2}}^2+\\lambda _1^s\\,\\Vert u\\Vert _{L^2}^2\\Big )\\,,$ provided $\\varepsilon _0$ , hence $t$ , is small enough with respect to $n$ .", "Since $\\lambda _1^s\\ge s\\,P_s(B)$ , we have completed the proof of (REF ), thus of Theorem REF ." ], [ "Uniform estimates for almost-minimizers of nonlocal perimeters", "A crucial step in our proof of Theorem REF and Theorem REF is the application of the regularity theory for nonlocal perimeter minimizers: indeed, this is the step where we reduce to consider small normal deformations of balls, and thus become able to apply Theorem REF .", "The parts of the regularity theory for nonlocal perimeter minimizers that are relevant to us have been developed in [7], [10] with the parameter $s$ fixed.", "In other words, there is no explicit discussion on how the regularity estimates should behave as $s$ approaches the limit values 0 or 1, although it is pretty clear [8], [3], [13] that they should degenerate when $s\\rightarrow 0^+$ , and that they should be stable, after scaling $s$ -perimeter by the factor $(1-s)$ , in the limit $s\\rightarrow 1^-$ .", "Since we shall need to exploit these natural uniformity properties, in this section we explain how to deduce these results from the results contained in [7], [10], with the aim of proving Corollary REF below.", "In order to minimize the amount of technicalities, we shall discuss these issues working with a rather special notion of almost-minimality, that we now introduce.", "It goes without saying, the results we present should hold true in the more general class of almost-minimizers considered in [10].", "We thus introduce the special class of almost-minimizers we shall consider.", "Given $\\Lambda \\ge 0$ , $s\\in (0,1)$ , and a bounded Borel set $E\\subset \\mathbb {R}^n$ , we say that $E$ is a (global) $\\Lambda $ -minimizer of the $s$ -perimeter if $P_s(E)\\le P_s(F)+\\frac{\\Lambda }{1-s}\\,\|E\\Delta F\|\\,,$ for every bounded set $F\\subset \\mathbb {R}^n$ .", "Since the validity of (REF ) is not affected if we replace $E$ with some $E^{\\prime }$ with $\|E\\Delta E^{\\prime }\|=0$ , we shall always assume that a $\\Lambda $ -minimizer of the $s$ -perimeter has been normalized so that $\\mbox{$E$ is Borel, with}\\,\\, \\partial E=\\Big \\lbrace x\\in \\mathbb {R}^n:\\mbox{$0<\|E\\cap B(x,r)\|<\\omega _n\\,r^n$ for every $r>0$}\\Big \\rbrace $ (as show for instance in [31], this can always be done).", "As explained, we shall need some regularity estimates for $\\Lambda $ -minimizers of the $s$ -perimeter to be uniform with respect to $s\\in [s_0,1]$ , for $s_0\\in (0,1)$ fixed.", "We start with the following uniform density estimates.", "(The proof is classical, compare with [31] for the local case, and with [7] for the nonlocal case, but we give the details here in order to keep track of the constants.)", "Lemma 3.1 If $s\\in (0,1)$ , $\\Lambda \\ge 0$ , and $E$ satisfies the minimality property (REF ) and the normalization condition (REF ), then we have $\|B\|\\,(1-c_0)\\,r^n\\ge \|E\\cap B(x_0,r)\|\\ge \|B\|\\,c_0\\,r^n\\,,$ whenever $x_0\\in \\partial E$ and $r\\le r_0$ , where $c_0=\\Big (\\frac{s}{8\\,\|B\|\\,2^{n/s}}\\frac{(1-s)P_s(B)}{P(B)}\\Big )^{n/s}\\,,\\qquad r_0=\\Big (\\frac{(1-s)\\,P_s(B)}{2\\,\\Lambda \\,\|B\|}\\Big )^{1/s}\\,.$ The following elementary lemma (De Giorgi iteration) is needed in the proof.", "Lemma 3.2 Let $\\alpha \\in (0,1)$ , $N>1$ , $M>0$ , and $\\lbrace u_k\\rbrace _{k\\in \\mathbb {N}}$ be a decreasing sequence of positive numbers such that $u_{k+1}^{1-\\alpha }\\le N^k\\,M\\,u_k\\,,\\qquad \\forall k\\in \\mathbb {N}\\,.$ If $u_0\\le \\frac{1}{N^{(1-\\alpha )/\\alpha ^2}\\,M^{1/\\alpha }}\\,,$ then $u_k\\rightarrow 0$ as $k\\rightarrow \\infty $ .", "By (REF ) and (REF ), induction proves that $u_k\\le N^{-k/\\alpha }\\,u_0$ for every $k\\in \\mathbb {N}$ .", "Being the two proofs analogous, we only prove the lower bound in (REF ).", "Up to a translation we may also assume that $x_0=0$ .", "We fix $r>0$ , set $u(r):=\|E\\cap B_r\|$ , and apply (REF ) with $F=E\\setminus B_r$ to find $(1-s)\\int _E\\int _{E^c}\\frac{dx\\,dy}{\|x-y\|^{n+s}}\\le (1-s)\\int _{E\\setminus B_r}\\int _{E^c\\cup (E\\cap B_r)}\\frac{dx\\,dy}{\|x-y\|^{n+s}}+\\Lambda u(r)\\,,$ As a consequence $(1-s)\\int _{E\\cap B_r}\\int _{E^c}\\frac{dx\\,dy}{\|x-y\|^{n+s}}\\le (1-s)\\int _{E\\setminus B_r}\\int _{E\\cap B_r}\\frac{dx\\,dy}{\|x-y\|^{n+s}}+\\Lambda u(r)\\,,$ hence, by adding up $(1-s)\\int _{E\\setminus B_r}\\int _{E\\cap B_r}\\frac{dx\\,dy}{\|x-y\|^{n+s}}$ to both sides we immediately get, for every $r>0$ , $P_s(E\\cap B_r)\\le 2\\,\\int _{E\\setminus B_r}\\int _{E\\cap B_r}\\frac{dx\\,dy}{\|x-y\|^{n+s}}+\\frac{\\Lambda }{1-s}\\,u(r)\\,.$ On the one hand, $P_s(E\\cap B_r)\\ge P_s(B)\\,(u(r)/\|B\|)^{(n-s)/n}$ by the isoperimetric inequality (REF ); on the other hand, by the coarea formula $\\nonumber \\int _{E\\setminus B_r}\\int _{E\\cap B_r}\\frac{dx\\,dy}{\|x-y\|^{n+s}}&\\le & \\int _{E\\cap B_r}dx\\int _{B(x,r-\|x\|)^c}\\frac{dy}{\|x-y\|^{n+s}}\\\\&=&\\frac{P(B)}{s}\\int _{E\\cap B_r}\\frac{dx}{(r-\|x\|)^s}=\\frac{P(B)}{s}\\,\\int _0^r\\frac{u^{\\prime }(t)}{(r-t)^s}\\,dt\\,,$ where we have also taken into account that $u^{\\prime }(t)={\\mathcal {H}}^{n-1}(E\\cap \\partial B_t)$ for a.e.", "$t>0$ .", "By combining these two facts with (REF ) we find $\\frac{P_s(B)}{\|B\|^{(n-s)/n}}\\,u(r)^{(n-s)/n}\\le \\frac{2\\,P(B)}{s}\\int _0^r\\,\\frac{u^{\\prime }(t)}{(r-t)^s}\\,dt+\\frac{\\Lambda }{1-s}\\,u(r)\\,,\\qquad \\forall r>0\\,.$ Since $u(r)\\le \|B\|\\,r^n$ for every $r>0$ , our choice of $r_0$ implies that $\\frac{\\Lambda }{1-s}\\,u(r)\\le \\frac{P_s(B)}{2\\,\|B\|^{(n-s)/n}}\\,u(r)^{(n-s)/n}\\,,\\qquad \\forall r\\le r_0\\,,$ and enables us to deduce from (REF ) that $u(r)^{(n-s)/n}\\le \\frac{4\\,P(B)\\,\|B\|^{(n-s)/n}}{s\\,P_s(B)}\\int _0^r\\,\\frac{u^{\\prime }(t)}{(r-t)^s}\\,dt\\,,\\qquad \\forall r\\le r_0\\,.$ By integrating (REF ) on $(0,\\ell )\\subset (0,r_0)$ and by Fubini's theorem, we thus obtain $\\int _0^\\ell \\,u(r)^{(n-s)/n}\\,dr\\le \\frac{4\\,P(B)\\,\|B\|^{(n-s)/n}}{s\\,(1-s)\\,P_s(B)}\\,\\ell ^{1-s}\\,u(\\ell ),\\qquad \\forall \\ell \\le r_0\\,.$ We now argue by contradiction, and assume the existence of $\\ell _0\\le r_0$ such that $u(\\ell _0)\\le c_0\\,\|B\|\\,\\ell _0^n$ .", "Correspondingly we set $\\ell _k:=\\frac{\\ell _0}{2}+\\frac{\\ell _0}{2^{k+1}}\\,,\\qquad u_k:=u(\\ell _k)\\,,\\qquad C_1:=\\frac{4\\,P(B)\\,\|B\|^{(n-s)/n}}{s\\,(1-s)\\,P_s(B)}\\,,$ and notice that (REF ) implies $\\frac{\\ell _0}{2^{k+2}}\\,u_{k+1}^{(n-s)/n}= (\\ell _k-\\ell _{k+1})u_{k+1}^{(n-s)/n}\\le \\int _{\\ell _{k+1}}^{\\ell _k}\\,u^{(n-s)/n}\\le C_1\\,\\ell _k^{1-s}\\,u_k\\le C_1\\ell _0^{1-s}\\,u_k\\,,$ that is, $u_{k+1}^{1-\\alpha }\\le 2^k\\,M\\,u_k$ for $M:=4\\,C_1\\,\\ell _0^{-s}$ and $\\alpha =s/n$ .", "Since $u_k\\rightarrow u(\\ell _0/2)=\|E\\cap B_{\\ell _0/2}\|>0$ (indeed, $0\\in \\partial E$ and (REF ) is in force), by Lemma REF we deduce that $u(\\ell _0)=u_0>\\frac{1}{2^{(1-\\alpha )/\\alpha ^2}\\,M^{1/\\alpha }}=\\frac{2^{n/s}\\,\\ell _0^n}{2^{(n/s)^2}\\,(4\\,C_1)^{n/s}}=c_0\\,\|B\|\\,\\ell _0^n\\,.$ However, this is a contradiction to $u(\\ell _0)\\le c_0\\,\|B\|\\,\\ell _0^n$ , and the lemma is proved.", "Introducing a further bit of special terminology, we say that a bounded Borel set $E\\subset \\mathbb {R}^n$ is a $\\Lambda $ -minimizer of the 1-perimeter if $P(E)\\le P(F)+\\frac{\\Lambda }{\\omega _{n-1}}\\,\|E\\Delta F\|\\,,$ for every bounded $F\\subset \\mathbb {R}^n$ , and if (REF ) holds true.", "We have the following compactness theorem.", "Theorem 3.3 If $R>0$ , $s_0\\in (0,1)$ , and $E_h$ ($h\\in \\mathbb {N}$ ) is a $\\Lambda $ -minimizer of the $s_h$ -perimeter with $s_h\\in [s_0,1)$ and $E_h\\subset B_R$ for every $h\\in \\mathbb {N}$ , then there exist $s_\\in [s_0,1]$ and a $\\Lambda $ -minimizer of the $s_$ -perimeter $E$ such that, up to extracting subsequences, $s_h\\rightarrow s_$ , $\|E_h\\Delta E\|\\rightarrow 0$ and $\\partial E_h$ converges to $\\partial E$ in Hausdorff distance as $h\\rightarrow \\infty $ .", "Up to extracting subsequences we may obviously assume that $s_h\\rightarrow s_$ as $h\\rightarrow \\infty $ , where $s_\\in [s_0,1]$ .", "By exploiting (REF ) with $F=B_R$ we see that $\\sup _{h\\in \\mathbb {N}}(1-s_h)\\,P_{s_h}(E_h)\\le 2\\Lambda \\,\|B_R\|+\\sup _{h\\in \\mathbb {N}}(1-s_h)\\,P_{s_h}(B_R)<\\infty \\,,$ where we have used the fact that $(1-s)\\,P_s(B)\\rightarrow \\omega _{n-1}P(B)$ as $s\\rightarrow 1^+$ (recall (REF )).", "Step one: We prove the theorem in the case $s_=1$ .", "By (REF ) and by [3], we find that, up to extracting subsequences, $\|E_h\\Delta E\|\\rightarrow 0$ as $h\\rightarrow \\infty $ for some set $E\\subset B_R$ with finite perimeter.", "By [3], $\\omega _{n-1}\\,P(E)\\le \\liminf _{h\\rightarrow \\infty }(1-s_h)\\,P_{s_h}(E_h)\\,,$ and, if $F\\subset \\mathbb {R}^n$ is bounded, then we can find bounded set $F_h$ ($h\\in \\mathbb {N}$ ) such that $\|F_h\\Delta F\|\\rightarrow 0$ as $h\\rightarrow \\infty $ and $\\omega _{n-1}\\,P(F)=\\liminf _{h\\rightarrow \\infty }(1-s_h)\\,P_{s_h}(F_h)\\,.$ By (REF ), $(1-s_h)\\,P_{s_h}(E_h)\\le (1-s_h)\\,P_{s_h}(F_h)+\\Lambda \\,\|E_h\\Delta F_h\|$ ; by letting $h\\rightarrow \\infty $ , we find that $E$ is a $\\Lambda $ -minimizer of the 1-perimeter.", "The fact that $\\partial E_h$ converges to $\\partial E$ in Hausdorff distance as $h\\rightarrow \\infty $ is now a standard consequences of the uniform density estimates proved in Lemma REF .", "Step two: We address the case $s_<1$ .", "In this case we may notice that (REF ) together with the assumption that $E_h\\subset B_R$ allows us to say that $\\lbrace P_s(E_h)\\rbrace _{h\\in \\mathbb {N}}$ is bounded in $\\mathbb {R}$ for some $s\\in (0,1)$ .", "By compactness of the embedding of $H^{s/2}$ in $L^1_{loc}$ and by the assumption $E_h\\subset B_R$ we find a set $E\\subset B_R$ such that, up to extracting subsequences, $\|E_h\\Delta E\|\\rightarrow 0$ as $h\\rightarrow \\infty $ .", "If we pick any bounded set $F\\subset \\mathbb {R}^n$ , then by Appendix there exists a sequence of bounded sets $\\lbrace F_h\\rbrace _{h\\in \\mathbb {N}}$ such that $\\lim _{h\\rightarrow \\infty }\|F_h\\Delta F\|=0\\,,\\qquad \\limsup _{h\\rightarrow \\infty }P_{s_h}(F_h)\\le P_{s_}(F)\\,.$ By applying (REF ) to $E_h$ and $F_h$ , and then by letting $h\\rightarrow \\infty $ , we find that $P_{s_}(E)\\le \\liminf _{h\\rightarrow \\infty }P_{s_h}(E_h)\\le \\limsup _{h\\rightarrow \\infty }P_{s_h}(F)+\\frac{\\Lambda }{1-s_h}\\,\|E_h\\Delta F_h\|\\le P_{s_}(F)+\\frac{\\Lambda }{1-s_}\\,\|E\\Delta F\|\\,,$ where the first inequality follows by Fatou's lemma, and the last one by (REF ).", "Since the Hausdorff convergence of $\\partial E_h$ to $\\partial E$ is again consequence of Lemma REF , the proof is complete.", "The next result is a uniform (with respect to $s$ ) version of the classical “improvement of flatness” statement.", "Theorem 3.4 Given $n\\ge 2$ , $\\Lambda \\ge 0$ , and $s_0\\in (0,1)$ , there exist $\\tau ,\\eta ,q\\in (0,1)$ , depending on $n$ , $\\Lambda $ and $s_0$ only, with the following property: If $E$ is a $\\Lambda $ -minimizer of the $s$ -perimeter for some $s\\in [s_0,1]$ with $0\\in \\partial E$ and $B\\cap \\partial E\\subset \\Big \\lbrace y\\in \\mathbb {R}^n:\|(y-x)\\cdot e\|<\\tau \\Big \\rbrace $ for some $e\\in S^{n-1}$ , then there exists $e_0\\in S^{n-1}$ such that $B_{\\eta }\\cap \\partial E\\subset \\Big \\lbrace y\\in \\mathbb {R}^n:\|(y-x)\\cdot e_0\|<q\\,\\tau \\,\\eta \\Big \\rbrace \\,.$ Step one: We prove that if $\\bar{s}\\in (0,1]$ , then there exist $\\delta >0$ and $\\bar{\\tau },\\bar{\\eta },\\bar{q}\\in (0,1)$ (depending on $n$ , $\\bar{s}$ and $\\Lambda $ only), such that if $s\\in (\\bar{s}-\\delta ,\\bar{s}+\\delta )\\cap (0,1]$ and $E$ is a $\\Lambda $ -minimizer of the $s$ -perimeter with $0\\in \\partial E$ and $B\\cap \\partial E\\subset \\Big \\lbrace y\\in \\mathbb {R}^n:\|(y-x)\\cdot e\|<\\bar{\\tau }\\Big \\rbrace $ for some $e\\in S^{n-1}$ , then there exists $e_0\\in S^{n-1}$ such that $B_{\\bar{\\eta }}\\cap \\partial E\\subset \\Big \\lbrace y\\in \\mathbb {R}^n:\|(y-x)\\cdot e_0\|<\\bar{q}\\,\\bar{\\tau }\\,\\bar{\\eta }\\Big \\rbrace \\,.$ Indeed, it follows from [31] in the case $\\bar{s}=1$ , and from [10] if $\\bar{s}<1$ , that there exist $\\bar{\\tau },\\bar{\\eta },\\bar{q}\\in (0,1/2)$ (depending on $n$ , $\\bar{s}$ and $\\Lambda $ only) such that if $F$ is a $\\Lambda $ -minimizer of the $\\bar{s}$ -perimeter with $0\\in \\partial F\\,,\\qquad B\\cap \\partial F\\subset \\Big \\lbrace y\\in \\mathbb {R}^n:\|(y-x)\\cdot e\|<2\\,\\bar{\\tau }\\Big \\rbrace $ for some $e\\in S^{n-1}$ , then there exists $e_0\\in S^{n-1}$ such that $B_{\\bar{\\eta }}\\cap \\partial F\\subset \\Big \\lbrace y\\in \\mathbb {R}^n:\|(y-x)\\cdot e_0\|<\\frac{\\bar{q}}{4}\\,(2\\,\\bar{\\tau })\\,\\bar{\\eta }\\,\\Big \\rbrace \\,.$ Let us now assume by contradiction that our claim is false.", "Then we can find a sequence $s_h\\rightarrow \\bar{s}$ as $h\\rightarrow \\infty $ , and, for every $h\\in \\mathbb {N}$ , $E_h$ $\\Lambda $ -minimizer of the $s_h$ -perimeter such that, for some $e_h\\in S^{n-1}$ , $0\\in \\partial E_h\\,,\\qquad B\\cap \\partial E_h\\subset \\Big \\lbrace y\\in \\mathbb {R}^n:\|(y-x)\\cdot e_h\|<\\bar{\\tau }\\Big \\rbrace \\,,\\qquad \\forall h\\in \\mathbb {N}\\,,$ but $B_{\\bar{\\eta }}\\cap \\partial E_h\\lnot \\subset \\Big \\lbrace y\\in \\mathbb {R}^n:\|(y-x)\\cdot e_0\|<\\bar{q}\\,\\bar{\\tau }\\,\\bar{\\eta }\\Big \\rbrace \\,,\\qquad \\forall h\\in \\mathbb {N}\\,,\\forall e_0\\in S^{n-1}\\,.$ By the compactness theorem, there exists a $\\Lambda $ -minimizer of the $\\bar{s}$ -perimeter $F$ such that $\\partial E_h$ converges to $\\partial F$ with respect to the Hausdorff distance on compact sets.", "By the latter information we have $0\\in \\partial F$ , and we find from (REF ) that $F$ is a $\\Lambda $ -minimizer of the $\\bar{s}$ -perimeter such that (REF ) holds true.", "In particular, there exists $e_0\\in S^{n-1}$ such that (REF ) holds true.", "By exploiting the local Hausdorff convergence of $\\partial E_h$ to $\\partial F$ one more time, we thus find that, if $h$ is large enough, then $B_{\\bar{\\eta }}\\cap \\partial E_h\\subset \\Big \\lbrace y\\in \\mathbb {R}^n:\|(y-x)\\cdot e_0\|<\\bar{q}\\,\\bar{\\tau }\\,\\bar{\\eta }\\Big \\rbrace \\,,$ a contradiction to (REF ).", "We have completed the proof of step one.", "Step two: We complete the proof of the theorem by covering $[s_0,1]$ with a finite number of intervals $(\\bar{s}_i-\\delta _i,\\bar{s}_i+\\delta _i)$ of the form constructed in step one.", "Improvement of flatness implies $C^{1,\\alpha }$ -regularity by a standard argument.", "By exploiting the uniformity of the constants obtained in Theorem REF one thus gets the following uniform regularity criterion.", "Corollary 3.5 If $n\\ge 2$ , $\\Lambda \\ge 0$ and $s_0\\in (0,1)$ , then there exist positive constants $\\varepsilon _0<1$ , $C_0>0$ , and $\\alpha <1$ , depending on $n$ , $\\Lambda $ and $s_0$ only, with the following property: If $E$ is a $\\Lambda $ -minimizer of the $s$ -perimeter for some $s\\in [s_0,1)$ and $0\\in \\partial E\\,,\\qquad B\\cap \\partial E\\subset \\Big \\lbrace y\\in \\mathbb {R}^n:\|(y-x)\\cdot e\|<\\varepsilon _0\\Big \\rbrace $ for some $e\\in S^{n-1}$ , then $B_{1/2}\\cap \\partial E$ is the graph of a function with $C^{1,\\alpha }$ -norm bounded by $C_0$ .", "Finally, by Hausdorff convergence of sequences of minimizers, we can exploit the regularity criterion (REF ) and the smoothness of the limit set $B$ via a standard argument (see, e.g., [31]) in order to obtain the following result, that plays a crucial role in the proof of our main results.", "Corollary 3.6 If $n\\ge 2$ , $\\Lambda \\ge 0$ , $s_0\\in (0,1)$ , $E_h$ ($h\\in \\mathbb {N}$ ) is a $\\Lambda $ -minimizer of the $s_h$ -perimeter for some $s_h\\in [s_0,1)$ , and $E_h$ converges in measure to $B$ , then there exists a bounded sequence $\\lbrace u_h\\rbrace _{h\\in \\mathbb {N}}\\subset C^{1,\\alpha }(\\partial B)$ (for some $\\alpha \\in (0,1)$ independent of $h$ ) such that $\\partial E_h=\\Big \\lbrace (1+u_h(x))x:x\\in \\partial B\\Big \\rbrace \\,,\\qquad \\lim _{h\\rightarrow \\infty }\\Vert u_h\\Vert _{C^1(\\partial B)}=0\\,.$" ], [ "Proof of Theorem ", "Given $s\\in (0,1]$ , we introduce the fractional isoperimetric gap of $E\\subset \\mathbb {R}^n$ (with $0<\|E\|<\\infty $ ) $D_s(E):=\\frac{P_s(E)}{P_s(B_{r_E})}-1\\,,$ where $r_E=(\|E\|/\|B\|)^{1/n}$ and $P_1(E)=P(E)$ denotes the distributional perimeter of $E$ .", "We shall also set $\\delta _{s_0}(E):=\\inf _{s_0\\le s< 1}D_s(E)\\,.$ With this notation at hand, the quantitative isoperimetric inequality (REF ) takes the form $A(E)^2\\le C(n,s_0)\\,\\delta _{s_0}(E)\\,.$ We begin by noticing that we can easily obtain (REF ) in the case of nearly spherical sets as a consequence of Theorem REF .", "Remark 4.1 Starting from Corollary REF , we shall coherently enumerate the constants appearing in the various statements of this section.", "For example, thorough this section, the symbol $C_0$ will always denote the constant appearing in (REF ).", "No confusion will arise as we shall not need to refer to constants defined in other sections of the paper.", "Symbols like $C(n,s)$ shall be used to denote generic constants (depending on $n$ and $s$ only) whose precise value shall be inessential to us.", "Corollary 4.2 For every $n\\ge 2$ there exist positive constants $C_0(n)$ and $\\varepsilon _0(n)$ such that $\\frac{C_0(n)}{s}\\,D_s(E)\\ge A(E)^2$ whenever $s\\in (0,1)$ and $E$ is a nearly spherical set as in (REF ), with $\|E\|=\|B\|$ , $\\int _Exdx=0$ , and $\\Vert u\\Vert _{C^1(\\partial B)}\\le \\varepsilon _0(n)$ .", "In particular, under these assumptions on $E$ , we have that $\\frac{C_0(n)}{s_0}\\,\\delta _{s_0}(E)\\ge A(E)^2\\,,\\qquad \\forall s_0\\in (0,1)\\,.$ This follows immediately by (REF ) since $A(E)\\le C(n)\\int _{\\partial B}\|u\|\\,d{\\mathcal {H}}^{n-1}\\le C(n)\\sqrt{\\int _{\\partial B}\|u\|^2\\,d{\\mathcal {H}}^{n-1}}\\,.$ The proof of Theorem REF is thus based on a reduction argument to the case considered in Corollary REF , much as in the spirit of what done [11] in the case $s=1$ .", "To this end, we argue by contradiction and assume (REF ) to fail.", "This gives us a sequence $\\lbrace E_h\\rbrace _{h\\in \\mathbb {N}}$ of almost-isoperimetric sets (that is, $D_{s_h}(E_h)\\rightarrow 0$ as $h\\rightarrow \\infty $ for some $s_h\\in [s_0,1)$ ) with $\|E_h\|=\|B\|$ such that $D_{s_h}(E_h)< M\\,A(E_h)^2$ , for a constant $M$ as large as we want.", "By Lemma REF below, the first information allows us to deduce that, up to translations, $\|E_h\\Delta B\|\\rightarrow 0$ as $h\\rightarrow \\infty $ .", "We next “round-up” our sets $E_h$ by solving a penalized isoperimetric problem, see Lemma REF , to obtain a new sequence $\\lbrace F_h\\rbrace _{h\\in \\mathbb {N}}$ – with the same properties of $\\lbrace E_h\\rbrace _{h\\in \\mathbb {N}}$ concerning isoperimetric gaps and asymmetry – but with the additional feature of being nearly spherical sets associated to functions $\\lbrace u_h\\rbrace _{h\\in \\mathbb {N}}\\subset C^1(\\partial B)$ with $\\Vert u_h\\Vert _{C^1(\\partial B)}\\rightarrow 0$ as $h\\rightarrow \\infty $ .", "By (REF ) this means that $C_0(n)/s_0\\ge M$ , which gives a contradiction if we started the argument with $M$ large enough.", "In order to make this argument rigorous we need to premise a series of remarks that seem interesting in their own.", "The first one is a nucleation lemma for nonlocal perimeters in the spirit of [2], see also [31].", "Here, $E^{(1)}$ stands for the set of points of density 1 of a measurable set $E$ .", "Lemma 4.3 If $n\\ge 2$ , $s\\in (0,1)$ , $P_s(E)<\\infty $ , and $0<\|E\|<\\infty $ , then there exists $x\\in E^{(1)}$ such that $\|E\\cap B(x,1)\|\\ge \\min \\Big \\lbrace \\frac{\\chi _1\\,\|E\|}{(1-s)\\,P_s(E)},\\frac{1}{\\chi _2}\\Big \\rbrace ^{n/s}\\,,$ where $\\chi _1(n,s):=\\frac{(1-s)\\,P_s(B)}{4\\,\|B\|^{(n-s)/n}\\,\\xi (n)}\\,,$ $\\chi _2(n,s):=\\frac{2^{3+(n/s)}\\,\|B\|^{(n-s)/n}\\,P(B)}{s(1-s)\\,P_s(B)}\\,,$ and where $\\xi (n)$ is Besicovitch's covering constant (see for instance [31]).", "In particular, $0<\\inf \\lbrace \\chi _1(n,s),\\chi _2(n,s)^{-1}:s\\in [s_0,1)\\rbrace <\\infty $ for every $s_0\\in (0,1)$ .", "Step one: We show that if $x\\in E^{(1)}$ with $\|E\\cap B(x,1)\|\\le \\Big (\\frac{(1-s)\\,P_s(B)}{2\\,\|B\|^{(n-s)/n}\\,\\alpha } \\Big )^{n/s}$ for some $\\alpha $ satisfying $\\alpha \\ge \\frac{2^{2+(n/s)}\\,P(B)}{s}\\,,$ then there exists $r_x\\in (0,1]$ such that $\|E\\cap B(x,r_x)\|\\le \\frac{(1-s)}{\\alpha }\\,\\int _{E\\cap B(x,r_x)}\\int _{E^c}\\frac{dz\\,dy}{\|z-y\|^{n+s}}\\,.$ Indeed, if not, setting for brevity $u(r):=\|E\\cap B(x,r)\|$ we have $(1-s)\\int _{E\\cap B(x,r)}\\int _{E^c}\\frac{dz\\,dy}{\|z-y\|^{n+s}}\\le \\alpha \\,u(r)$ for every $r\\le 1$ .", "By adding up $(1-s)\\int _{E\\setminus B(x,r)}\\int _{E\\cap B(x,r)}\\frac{dz\\,dy}{\|z-y\|^{n+s}}$ to both sides, we get $P_s(E\\cap B(x,r))\\le \\int _{E\\setminus B(x,r)}\\int _{E\\cap B(x,r)}\\frac{dz\\,dy}{\|z-y\|^{n+s}}+\\frac{\\alpha }{1-s}\\,u(r)$ for every $r\\le 1$ so that, arguing as in the proof of Lemma REF , we get $\\frac{P_s(B)}{\|B\|^{(n-s)/n}}\\,u(r)^{(n-s)/n}\\le \\frac{P(B)}{s}\\int _0^r\\,\\frac{u^{\\prime }(t)}{(r-t)^s}\\,dt+\\frac{\\alpha }{1-s}\\,u(r)\\,,\\qquad \\forall r\\le 1\\,,$ cf.", "with (REF ).", "By (REF ) we have $\\frac{\\alpha }{1-s}\\,u(r)\\le \\frac{\\alpha }{1-s}\\,u(1)^{s/n}\\,u(r)^{(n-s)/n}\\le \\frac{P_s(B)}{2\\,\|B\|^{(n-s)/n}}\\,u(r)^{(n-s)/n}\\,,$ so that (REF ) gives $u(r)^{(n-s)/n}\\le \\frac{2\\,P(B)\\,\|B\|^{(n-s)/n}}{s\\,P_s(B)}\\int _0^r\\,\\frac{u^{\\prime }(t)}{(r-t)^s}\\,dt\\,,\\qquad \\forall r\\le 1\\,.$ Notice that (REF ) implies (REF ) with 1 in place of $r_0$ .", "Moreover, (REF ) implies that $u(1)\\le c_0\|B\|$ , where $c_0=\\Big (\\frac{s}{8\\,\|B\|\\,2^{n/s}}\\frac{(1-s)P_s(B)}{P(B)}\\Big )^{n/s}\\,,$ is the constant defined in Lemma REF .", "Therefore, by repeating the very same iteration argument seen in the proof of Lemma REF (notice that $u(r)>0$ for every $r>0$ since $x\\in E^{(1)}$ ), we see that $u(1)> c_0\|B\|$ , and thus find a contradiction.", "This completes the proof of step one.", "Step two: We complete the proof of the lemma.", "We argue by contradiction, and assume that for every $x\\in E^{(1)}$ we have $\|E\\cap B(x,1)\|\\le \\min \\Big \\lbrace \\frac{\\chi _1\\,\|E\|}{(1-s)\\,P_s(E)},\\frac{1}{\\chi _2}\\Big \\rbrace ^{n/s}\\,.$ If we set $\\alpha :=\\frac{(1-s)\\,P_s(B)}{2\\,\|B\|^{(n-s)/n}}\\,\\min \\Big \\lbrace \\frac{\\chi _1\\,\|E\|}{(1-s)\\,P_s(E)},\\frac{1}{\\chi _2}\\Big \\rbrace ^{-1}\\,,$ then (REF ) takes the form of (REF ) for a value of $\\alpha $ that (by definition of $\\chi _2$ ) satisfies (REF ).", "Hence, by step one, for every $x\\in E^{(1)}$ there exists $r_x\\in (0,1]$ such that (REF ) holds true with $\\alpha $ as in (REF ).", "By applying Besicovitch covering theorem, see [31], we find a countable disjoint family of balls $\\lbrace B(x_h,r_h)\\rbrace _{h\\in \\mathbb {N}}$ such that $x_h\\in E^{(1)}$ , $r_h=r_{x_h}$ is such that (REF ) holds true with $x=x_h$ , and thus $\|E\|&\\le &\\xi (n)\\sum _{h\\in \\mathbb {N}}\|E\\cap B(x_h,r_h)\|\\le \\frac{\\xi (n)(1-s)}{\\alpha }\\sum _{h\\in \\mathbb {N}}\\int _{E\\cap B(x_h,r_h)}\\int _{E^c}\\frac{dz\\,dy}{\|z-y\|^{n+s}}\\\\&\\le &\\frac{\\xi (n)(1-s)P_s(E)}{\\alpha }\\le \\chi _1\\,\\frac{\\xi (n)\\,2\\,\|B\|^{(n-s)/n}}{(1-s)\\,P_s(B)}\\,\|E\|=\\frac{\|E\|}{2}\\,,$ by definition of $\\chi _1$ .", "This is a contradiction, and the lemma is proved.", "Next, we prove the following “soft” stability lemma.", "An analogous statement was proved in [24] in the case one works with $D_{s_0}(E)$ in place of $\\delta _{s_0}(E)$ , and under the additional assumption that $A(E)\\le 3/2$ .", "This last assumption was not a real restriction in [24], as the soft stability lemma was applied to sets enjoying certain symmetry properties that, in turn, were granting that $A(E)\\le 3/2$ .", "We avoid here the use of symmetrization arguments by exploiting the more general tool provided us by the nucleation lemma, Lemma REF .", "Lemma 4.4 If $n\\ge 2$ and $s_0\\in (0,1)$ , then for every $\\varepsilon >0$ there exists $\\delta >0$ (depending on $n$ , $s_0$ , and $\\varepsilon $ ) such that if $\\delta _{s_0}(E)<\\delta $ then $A(E)<\\varepsilon $ .", "By contradiction, we assume the existence of a sequence of sets $E_h\\subset \\mathbb {R}^n$ , $h\\in \\mathbb {N}$ , such that $\|E_h\|=\|B\|\\,,\\qquad A(E_h)\\ge \\varepsilon \\,,\\qquad \\lim _{h\\rightarrow \\infty }\\delta _{s_0}(E_h)=0\\,,$ where $\\varepsilon $ is a positive constant.", "In particular there exist $s_h\\in [s_0,1)$ , $h\\in \\mathbb {N}$ , such that $\\lim _{h\\rightarrow \\infty }\\frac{P_{s_h}(E_h)}{P_{s_h}(B)}=1\\,.$ Without loss of generality, we assume that $s_h\\rightarrow s_\\in [s_0,1]$ as $h\\rightarrow \\infty $ .", "Since $(1-s)\\,P_s(B)\\rightarrow \\omega _{n-1}\\,P(B)$ as $s\\rightarrow 1^-$ , we find that $\\sup _{h\\in \\mathbb {N}}(1-s_h)\\,P_{s_h}(E_h)<\\infty \\,.$ By Lemma REF , see (REF ), we find that, up to translations, $\|E_h\\cap B\|\\ge \\min \\Big \\lbrace \\frac{\\chi _1(n,s_h)\\,\|B\|}{(1-s_h)\\,P_{s_h}(E_h)},\\frac{1}{\\chi _2(n,s_h)}\\Big \\rbrace ^{n/s_h}\\ge \\kappa _\\,,$ for some positive constant $\\kappa _$ independent of $h$ .", "By compactness of the embedding of $H^{s/2}(\\mathbb {R}^n)$ into $L^1_{loc}(\\mathbb {R}^n)$ when $s_<1$ , or by [3] in case $s_=1$ , we exploit (REF ) to deduce that, up to extracting subsequences, there exists a measurable set $E$ such that for every $K\\subset \\subset \\mathbb {R}^n$ we have $\|(E_h\\Delta E)\\cap K\|\\rightarrow 0$ as $h\\rightarrow \\infty $ .", "By local convergence of $E_h$ to $E$ and by (REF ), we have $\|E\|\\le \|B\|$ .", "If $s_=1$ , then by [3] and by (REF ) we find $\\omega _{n-1}\\,P(E)\\le \\liminf _{h\\rightarrow \\infty }(1-s_h)\\,P_{s_h}(E_h)=\\liminf _{h\\rightarrow \\infty }(1-s_h)\\,P_{s_h}(B)=\\omega _{n-1}\\,P(B)\\,,$ that is, $P(E)\\le P(B)$ .", "If, instead, $s_<1$ , then (REF ) gives $P_{s_}(B)=\\lim _{h\\rightarrow \\infty }P_{s_h}(E_h)=\\lim _{h\\rightarrow \\infty }\\int _{\\mathbb {R}^n}\\int _{\\mathbb {R}^n}\\frac{1_{E_h}(x)1_{E_h^c}(y)}{\|x-y\|^{n+s_h}}\\,dxdy\\ge P_{s_}(E)\\,,$ where the last inequality follows by Fatou's lemma.", "In both cases, $P_{s_}(E)\\le P_{s_}(B)$ .", "Should it be $\|E\|=\|B\|$ , then, by the (nonlocal, if $s_<1$ ) isoperimetric theorem, we would be able to conclude that $A(E)=0$ , against $A(E_h)\\ge \\varepsilon $ for every $h\\in \\mathbb {N}$ .", "Should it be $\|E\|=0$ , then we would get a contradiction with (REF ).", "Therefore, it must be $0<\|E\|<\|B\|$ .", "By a standard application of the concentration-compactness lemma (see, e.g., [24]), $0<\|E\|<\|B\|$ can happen only if there exists $\\lambda \\in (0,1)$ such that for every $\\sigma >0$ and $h$ large enough there exist $F_h^\\sigma , G_h^\\sigma \\subset E_h$ with the property that $\|E_h\\setminus (F_h^\\sigma \\cup G_h^\\sigma )\|<\\sigma \\,,\\quad \|\|F_h^\\sigma \|-\\lambda \\,\|B\|\|<\\sigma \\,,\\quad \|\|G_h^\\sigma \|-(1-\\lambda )\\,\|B\|\|<\\sigma \\,,$ and ${\\rm dist}(F_h^\\sigma ,G_h^\\sigma )\\rightarrow +\\infty $ as $h\\rightarrow \\infty $ .", "Let us now set $K_{s,\\eta }(z):=\\frac{1_{\\lbrace \\eta <\|z\|<\\eta ^{-1}\\rbrace }}{\|z\|^{n+s}}+\\frac{1_{\\lbrace \|z\|<\\eta \\rbrace }}{\\eta ^{n+s}}\\,,\\qquad z\\in \\mathbb {R}^n\\,,$ so that $K_{s,\\eta }(x-y)\\le \|x-y\|^{-(n+s)}$ , and thus $P_{s_h}(E_h)&\\ge & \\int _{F_h^\\sigma }\\int _{E_h^c}K_{s_h,\\eta }(x-y)\\,dxdy+\\int _{G_h^\\sigma }\\int _{E_h^c}K_{s_h,\\eta }(x-y)\\,dxdy\\\\&\\ge & \\int _{F_h^\\sigma }\\int _{(F_h^\\sigma )^c}K_{s_h,\\eta }(x-y)\\,dxdy+\\int _{G_h^\\sigma }\\int _{(G_h^\\sigma )^c}K_{s_h,\\eta }(x-y)\\,dxdy-\\frac{C(n)\\sigma }{\\eta ^{n+s_h}}\\\\&\\ge & \\int _{B_{a_h^\\sigma }}\\int _{(B_{a_h^\\sigma })^c}K_{s_h,\\eta }(x-y)\\,dxdy+\\int _{B_{b_h^\\sigma }}\\int _{(B_{b_h^\\sigma })^c}K_{s_h,\\eta }(x-y)\\,dxdy-\\frac{C(n)\\sigma }{\\eta ^{n+s_h}}\\,,$ where in the last inequality we have used [20] and we have chosen $a_h^\\sigma ,b_h^\\sigma >0$ in such a way that $\|B_{a_h^\\sigma }\|=\|F_h^\\sigma \|$ and $\|B_{b_h^\\sigma }\|=\|G_h^\\sigma \|$ .", "We now first let $\\sigma \\rightarrow 0^+$ , to obtain $P_{s_h}(E_h)\\ge \\int _{B_{a}}\\int _{(B_{a})^c}K_{s_h,\\eta }(x-y)\\,dxdy+\\int _{B_{b}}\\int _{(B_{b})^c}K_{s_h,\\eta }(x-y)\\,dxdy\\,,$ where $a$ and $b$ are such that $\|B_a\|=\\lambda \\,\|B\|$ and $\|B_b\|=(1-\\lambda )\|B\|$ .", "Next we let $\\eta \\rightarrow 0^+$ , divide by $P_{s_h}(B)$ , and then let $h\\rightarrow \\infty $ to reach the contradiction $1\\ge \\frac{P_{s_}(B_a)}{P_{s_}(B)}+\\frac{P_{s_}(B_b)}{P_{s_}(B)}=\\lambda ^{(n-s_)/n}+(1-\\lambda )^{(n-s_)/n}>1\\,.$ This completes the proof of the lemma.", "Next, we introduce the variational problems with penalization needed to round-up the nearly-isoperimetric sets $E_h$ into nearly-spherical sets $F_h$ .", "Precisely, we shall consider the problems $\\inf \\Big \\lbrace (1-s)\\,P_s(E)+\\Lambda \\,\\bigl \| \|E\|-\|B\|\\bigr \|+\|{\\alpha }(E)-{\\alpha }\|:E\\subset \\mathbb {R}^n\\Big \\rbrace \\,,$ where $s\\in (0,1)$ , $\\Lambda \\ge 0$ , ${\\alpha }>0$ , and ${\\alpha }(E):=\\inf \\Big \\lbrace \|E\\Delta (x+B)\|:x\\in \\mathbb {R}^n\\Big \\rbrace \\,,\\qquad E\\subset \\mathbb {R}^n\\,.$ Notice that the existence of minimizers in (REF ) is a non-trivial issue.", "Indeed, minimizing sequences, in general, are compact only with respect to local convergence in measure, with respect to which $\\Lambda \\,\\bigl \| \|E\|-\|B\|\\bigr \|$ is just upper semicontinuous if $\|E\| \\le \|B\|$ .", "In addition, we cannot obtain global convergence through the isoperimetric argument used in the proof of Lemma REF , since (as we shall see in the proof of Lemma REF ) a minimizing sequence in (REF ) will not be in general a sequence with vanishing isoperimetric gap (because $ {\\alpha }(E)$ has to stay close to $ {\\alpha }$ ).", "Therefore we have to resort to a finer argument, and show how to modify an arbitrary minimizing sequence into a uniformly bounded minimizing sequence.", "We base our argument on the following truncation lemma: the proof by contradiction is inspired by [2], see also [31].", "Lemma 4.5 Let $n\\ge 2$ , $s\\in (0,1)$ , and $E\\subset \\mathbb {R}^n$ .", "If $\|E\\setminus B\|\\le \\eta <1$ , then there exists $1\\le r_E\\le 1+ C_1(n,s)\\,\\eta ^{1/n}$ such that $(1-s)\\,P_s(E\\cap B_{r_E})\\le (1-s)\\,P_s(E)-\\frac{\|E\\setminus B_{r_E}\|}{C_2(n,s)\\,\\eta ^{s/n}}\\,,$ where $C_1(n,s):=2^{1+(n-s)/s}\\,\\Big (\\frac{4\\,\|B\|^{(n-s)/n}\\,P(B)}{s\\,(1-s)\\,P_s(B)}\\Big )^{1/s} \\,,\\qquad C_2(n,s):=\\frac{2\|B\|^{(n-s)/n}}{(1-s)\\,P_s(B)}\\,.$ In particular, $\\sup \\lbrace C_1(n,s)+C_2(n,s):s_0\\le s<1\\rbrace <\\infty $ .", "Without loss of generality we consider a set $E$ with $\|E\\setminus B\|\\le \\eta <1$ and $\|E\\setminus B_{1+C_1\\,\\eta ^{1/n}}\|>0$ .", "Correspondingly, if we set $u(r):=\|E\\setminus B_r\|$ , $r>0$ , then $u$ is a decreasing function with $[0,1+c_1\\,\\eta ^{1/n}]\\subset {\\rm spt}\\,u\\,\\qquad u(1)\\le \\eta \\,,\\qquad u^{\\prime }(r)=-{\\mathcal {H}}^{n-1}(E\\cap \\partial B_r)\\quad \\mbox{for a.e.", "$r>0$}\\,.$ Arguing by contradiction, we now assume that $(1-s)\\,P_s(E)\\le (1-s)\\,P_s(E\\cap B_r)+\\frac{u(r)}{C_2\\,\\eta ^{s/n}}\\,,\\qquad \\forall r\\in (1,1+C_1\\,\\eta ^{1/n})\\,.$ First, we notice that we have the identity $P_s(E\\cap B_r)-P_s(E)=2\\,\\int _{E\\cap B_r}\\int _{E\\cap B_r^c}\\frac{dx\\,dy}{\|x-y\|^{n+s}}-P_s(E\\setminus B_r)\\,,\\qquad \\forall r>0\\,;$ second, by arguing as in the proof of (REF ), and by (REF ), we see that $\\int _{E\\cap B_r}\\int _{E\\cap B_r^c}\\frac{dx\\,dy}{\|x-y\|^{n+s}}\\le \\frac{P(B)}{s}\\,\\int _r^\\infty \\,\\frac{-u^{\\prime }(t)}{(t-r)^s}\\,dt\\,,\\qquad \\forall r>0\\,;$ finally, by (REF ), $P_s(E\\setminus B_r)\\ge P_s(B)\|B\|^{(s-n)/n}\\,u(r)^{(n-s)/n}$ .", "We may thus combine these three remarks with (REF ) to conclude that, if $r\\in (1,1+C_1\\,\\eta ^{1/n})$ , then $\\nonumber 0&\\le & \\frac{2\\,P(B)}{s}\\,\\int _r^\\infty \\frac{-u^{\\prime }(t)}{(t-r)^s}\\,dt-\\frac{P_s(B)}{\|B\|^{(n-s)/n}}\\,u(r)^{(n-s)/n}+\\frac{u(r)}{(1-s)\\,C_2\\,\\eta ^{s/n}}\\\\&\\le &\\frac{2\\,P(B)}{s}\\,\\int _r^\\infty \\frac{-u^{\\prime }(t)}{(t-r)^s}\\,dt-\\frac{P_s(B)}{2\|B\|^{(n-s)/n}}\\,u(r)^{(n-s)/n}\\,,$ where in the last inequality we have used our choice of $C_2$ and the fact that $u(r)\\le \\eta $ for every $r>1$ .", "We rewrite (REF ) in the more convenient form $u(r)^{(n-s)/n}\\le C_3\\,\\int _r^\\infty \\frac{-u^{\\prime }(t)}{(t-r)^s}\\,dt\\,,\\qquad \\forall r\\in (1,1+c\\,\\eta ^{1/n})\\,,$ where we have set $C_3(n,s):=\\frac{4\\,\|B\|^{(n-s)/n}\\,P(B)}{s\\,P_s(B)}\\,.$ Let us set $r_k:=1+(1-2^{-k})\\,C_1\\,\\eta ^{1/n}$ , so that $r_0=1$ , $r_k<r_{k+1}$ , and $r_\\infty =1+C_1\\,\\eta ^{1/n}$ .", "Correspondingly, if we set $u_k=u(r_k)$ , then by (REF ) we find that $u_0\\le \\eta $ , $u_k\\ge u_{k+1}$ , and $u_\\infty =\\lim _{k\\rightarrow \\infty }u_k>0$ .", "We are now going to show that (REF ) implies $u_\\infty =0$ , thus obtaining a contradiction and proving the lemma.", "Indeed, if we integrate (REF ) on $(r_k,r_{k+1})$ we get $(r_{k+1}-r_k)\\,u_{k+1}^{(n-s)/n}&\\le &C_3 \\int _{r_k}^{r_{k+1}}\\,dr\\int _r^\\infty \\frac{-u^{\\prime }(t)}{(t-r)^s}\\,dt\\\\\\nonumber &=&C_3\\int _{r_k}^{r_{k+1}}(-u^{\\prime }(t))\\,dt\\int _{r_k}^t\\,\\frac{dr}{(t-r)^s}+C_3\\int _{r_{k+1}}^{\\infty }(-u^{\\prime }(t))\\,dt\\int _{r_k}^{r_{k+1}}\\,\\frac{dr}{(t-r)^s}\\,.$ On the one hand we easily find that $\\int _{r_k}^{r_{k+1}}(-u^{\\prime }(t))\\,dt\\int _{r_k}^t\\,\\frac{dr}{(t-r)^s}\\le \\frac{(r_{k+1}-r_k)^{1-s}}{1-s}\\,(u_k-u_{k+1})\\,;$ on the other hand we notice that, for every $t>r_{k+1}$ , since $\|b^{1-s}-a^{1-s}\|\\le \|b-a\|^{1-s}$ for $a,b\\ge 0$ , $\\int _{r_k}^{r_{k+1}}\\,\\frac{dr}{(t-r)^s}=\\frac{(t-r_k)^{1-s}-(t-r_{k+1})^{1-s}}{1-s}\\le \\frac{(r_{k+1}-r_k)^{1-s}}{1-s}\\,.$ Hence, since $\|E\|<\\infty $ implies $\\lim _{r\\rightarrow \\infty }u(r)=0$ , $\\int _{r_{k+1}}^{\\infty }(-u^{\\prime }(t))\\,dt\\int _{r_k}^{r_{k+1}}\\,\\frac{dr}{(t-r)^s}\\le \\frac{(r_{k+1}-r_k)^{1-s}}{1-s}\\,u_{k+1}\\,.$ We combine (REF ), (REF ), and (REF ) to find $(r_{k+1}-r_k)\\,u_{k+1}^{(n-s)/n}\\le \\frac{C_3}{1-s}\\,(r_{k+1}-r_k)^{1-s}\\,u_k\\,.$ Since $r_{k+1}-r_k=C_1\\,\\eta ^{1/n}\\,2^{-k-1}$ , we conclude that $u_{k+1}^{1-\\alpha }\\le N^k\\,M\\,u_k$ , where $\\alpha =\\frac{s}{n}\\,,\\qquad N=2^s\\,,\\qquad M=\\Big (\\frac{2}{C_1\\,\\eta ^{1/n}}\\Big )^s\\,\\frac{C_3}{1-s}\\,.$ We notice that, since $u_0\\le \\eta <1$ , we have $u_0\\le (N^{(1-\\alpha )/a^2}M^{1/\\alpha })^{-1}$ thanks to our choice of $C_1$ .", "We are thus in the position to apply Lemma REF to get $u_\\infty =0$ and obtain the required contradiction.", "Given $n\\ge 2$ , $s\\in (0,1)$ , $\\alpha >0$ , and $E\\subset \\mathbb {R}^n$ , let us set for the sake of brevity ${\\mathcal {F}}_{s,\\Lambda ,{\\alpha }}(E):=(1-s)\\,P_s(E)+\\Lambda \\,\\Big \|\|E\|-\|B\|\\Big \|+\|{\\alpha }(E)-{\\alpha }\|\\,.$ We now prove the existence of global minimizers of ${\\mathcal {F}}_{s,\\Lambda ,{\\alpha }}$ .", "Lemma 4.6 If $n\\ge 2$ , $s\\in (0,1)$ , $\\Lambda >\\Lambda _0(n,s)$ and ${\\alpha }<\\varepsilon _1(n,s)$ , then there exists a minimizer $E$ in the variational problem (REF ), that is, ${\\mathcal {F}}_{s,\\Lambda ,{\\alpha }}(E)\\le {\\mathcal {F}}_{s,\\Lambda ,{\\alpha }}(F)$ for every $F\\subset \\mathbb {R}^n$ .", "Moreover, up to a translation, this minimizer satisfies $E\\subset B_{C_4(n,s)}\\,.$ Here we have set $\\Lambda _0(n,s)&:=&\\frac{(1-s)\\, P_s(B)}{\|B\|}\\,,\\\\\\varepsilon _1(n,s)&:=&\\frac{1}{2}\\min \\Big \\lbrace 1,\\Big (\\frac{1}{(\\Lambda +1)C_2(n,s)}\\Big )^{n/s},4\|B\|\\Big \\rbrace \\,,\\\\C_4(n,s)&:=&1+C_1(n,s)\\,(2\\varepsilon _1(n,s))^{1/n}\\,.$ In particular, $\\inf \\lbrace \\varepsilon _1(n,s):s_0\\le s<1\\rbrace >0$ and $\\sup \\lbrace \\Lambda _0(n,s)+C_4(n,s):s_0\\le s<1\\rbrace <\\infty $ .", "Step one: We first show that, since $s\\in (0,1)$ and $\\Lambda >(1-s)\\, P_s(B)/\|B\|$ , then the unit ball $B$ is the unique solution, up to a translation, of the minimization problem $\\min \\bigl \\lbrace (1-s)\\,P_s(E)+\\Lambda \\bigl \|\|E\|-\|B\|\\bigr \|:\\,E\\subset \\mathbb {R}^n\\bigr \\rbrace \\,.$ Indeed, by comparing any set $E$ with a ball having its same volume and thanks to (REF ), we immediately reduce the competition class in (REF ) to the family of balls in $\\mathbb {R}^n$ .", "Note that, if $r>1$ , then $P_s(B)<P_s(B_r)$ , so that only balls with radius $r\\le 1$ have to be considered.", "At the same time, if $\\Lambda > (1-s)\\,P_s(B)/\\omega _n$ , then one immediately gets that $(1-s)\\,P_s(B_r)+\\Lambda \\bigl \|\|B_r\|-\|B\|\\bigr \|=r^{n-s}(1-s)\\,P_s(B)+\\Lambda \\omega _n(1-r^n)$ as a function of $r\\in [0,1]$ attains its minimum at $r=1$ .", "Step two: Let us denote by $\\gamma $ the infimum value in (REF ), and let us consider sets $E_h$ ($h\\in \\mathbb {N}$ ) with ${\\mathcal {F}}_{s,\\Lambda ,{\\alpha }}(E_h)\\le \\gamma +h^{-1}\\,{\\alpha }$ .", "Since ${\\alpha }<\\varepsilon _1\\le 2\|B\|$ , we immediately get that $\\gamma \\le (1-s)P_s(B)$ .", "Therefore, since by step one $(1-s)\\,P_s(B)\\le (1-s)\\,P_s(E_h)+\\Lambda \\,\|\|E_h\|-\|B\|\|$ , we conclude that $\|{\\alpha }(E_h)-{\\alpha }\|\\le \\,h^{-1}{\\alpha }$ .", "Hence, up to translations, we obtain that $\|E_h\\setminus B\|\\le \|E_h\\Delta B\|\\le 2\\,{\\alpha }<2\\varepsilon _1<1\\,,\\qquad \\forall h\\in \\mathbb {N}\\,.$ If we set $\\eta :=2\\,{\\alpha }$ , then by Lemma REF we can find $1\\le r_h\\le 1+ C_1(n,s)\\,\\eta ^{1/n}$ such that $(1-s)\\,P_s(E_h\\cap B_{r_h})\\le (1-s)\\,P_s(E_h)-\\frac{\|E_h\\setminus B_{r_h}\|}{C_2(n,s)\\,\\eta ^{s/n}}\\,.$ Since $\|{\\alpha }(I)-{\\alpha }(J)\|\\le \|I\\Delta J\|$ for every $I,J\\subset \\mathbb {R}^n$ , if we set $F_h:=E_h\\cap B_{r_h}$ then $\\Lambda \\,\|\|F_h\|-\|B\|\|+\|{\\alpha }(F_h)-{\\alpha }\|\\le \\Lambda \\,\|\|E_h\|-\|B\|\|+\|{\\alpha }(E_h)-{\\alpha }\|+(\\Lambda +1)\\,\|E_h\\setminus B_{r_h}\|\\,,$ so that (REF ) implies (by our choice of $\\varepsilon _1>\\eta /2$ ) ${\\mathcal {F}}_{s,\\Lambda ,{\\alpha }}(F_h)\\le {\\mathcal {F}}_{s,\\Lambda ,{\\alpha }}(E_h)+\\bigg ((\\Lambda +1)-\\frac{1}{C_2(n,s)\\,\\eta ^{s/n}}\\bigg )\|E_h\\setminus B_{r_h}\|\\le {\\mathcal {F}}_{s,\\Lambda ,{\\alpha }}(E_h)\\,.$ From this we conclude that ${\\mathcal {F}}_{s,\\Lambda ,{\\alpha }}(F_h)\\rightarrow \\gamma $ as $h\\rightarrow \\infty $ , that is, $\\lbrace F_h\\rbrace _{h\\in \\mathbb {N}}$ is still a minimizing sequence for (REF ) with the additional feature that, by construction, $F_h\\subset B_{1+C_1\\,(2\\varepsilon _1)^{1/n}}\\,,\\qquad \\forall h\\in \\mathbb {N}\\,.$ It is now easy to prove the existence of a minimizer in (REF ).", "Since both sides of (REF ) are scaling invariant, we may assume that $\|E\|=\|B\|$ .", "We want to show the existence of $\\delta _0=\\delta _0(n,s_0)>0$ such that, if $M>0$ is large enough, then $A(E)^2\\le M\\,\\delta _{s_0}(E)\\,,\\qquad \\mbox{whenever $\\delta _{s_0}(E)\\le \\delta _0$}\\,.$ (Notice that, since we always have $A(E)\\le 2$ , then $A(E)^2\\le (4/\\delta _0)\\delta _{s_0}(E)$ whenever $\\delta _{s_0}(E)\\ge \\delta _0$ : in other words, (REF ) immediately implies (REF ).)", "To prove (REF ) we argue by contradiction, assuming that there exists a sequence $E_h$ of sets with $\|E_h\|=\|B\|$ , $\\delta _{s_0}(E_h)\\rightarrow 0$ as $h\\rightarrow \\infty $ , but $\\delta _{s_0}(E_h)< \\frac{A(E_h)^2}{M}\\,.$ By Lemma REF (and since $\|E_h\|=\|B\|$ ) we can thus find $s_h\\in [s_0,1)$ and $h\\in \\mathbb {N}$ such that $\\lim _{h\\rightarrow \\infty }\\frac{P_{s_h}(E_h)}{P_{s_h}(B)}=1\\,,\\qquad D_{s_h}(E_h)\\le \\frac{\|E_h\\Delta B\|^2}{M\|B\|^2}\\,,\\qquad \\lim _{h\\rightarrow \\infty }{\\alpha }(E_h)=0\\,.$ We set ${\\alpha }_h:={\\alpha }(E_h)$ (so that, up to translations, ${\\alpha }_h=\|E_h\\Delta B\|$ ) and consider the minimization problems $\\inf \\Big \\lbrace (1-s_h)\\,P_{s_h}(E)+\\Lambda \\bigl \|\|E\|-\|B\|\\bigr \|+\|{\\alpha }(E)-{\\alpha }_h\|:\\,E\\subset \\mathbb {R}^n\\Big \\rbrace \\,,$ where $\\Lambda $ is chosen so that $\\Lambda >\\sup _{s\\in [s_0,1)}\\frac{(1-s)\\,P_{s}(B)}{\|B\|}\\,;$ notice that the right-hand side of (REF ) is finite since $(1-s)\\,P_s(B)\\rightarrow \\omega _{n-1}\\,P(B)$ as $s\\rightarrow 1^-$ .", "For the same reason, $\\inf _{s\\in [s_0,1)}\\varepsilon _1(n,s)>0$ , and thus for every $h$ large enough we may entail that ${\\alpha }_h<\\inf _{s\\in [s_0,1)}\\varepsilon _1(n,s)\\,.$ We can thus apply Lemma REF to prove the existence of minimizers $F_h$ in (REF ) with $F_h\\subset B_{C_4(n,s_h)}\\subset B_{C_5(n,s_0)}\\,,\\qquad \\mbox{with}\\quad C_5(n,s_0):=\\sup _{s\\in [s_0,1)}C_4(n,s)<\\infty \\,.$ We shall assume (as we can do up to translations) that $\\int _{F_h}x\\,dx=0\\,,\\qquad \\forall h\\in \\mathbb {N}\\,.$ By the minimality of each $F_h$ , recalling (REF ) and (REF ) we have that ${\\mathcal {F}}_{s_h,\\Lambda ,{\\alpha }_h}(F_h)&\\le F_{s_h,\\Lambda ,{\\alpha }_h}(E_h)=(1-s_h)\\, P_{s_h}(E_h)\\le (1-s_h)\\,P_{s_h}(B)+\\frac{(1-s_h){\\alpha }_h^2P_{s_h}(B)}{M\|B\|^2}\\\\&\\le (1-s_h)\\,P_{s_h}(F_h)+\\Lambda \\bigl \|\|F_h\|-\|B\|\\bigr \|+\\frac{(1-s_h){\\alpha }_h^2P_{s_h}(B)}{M\|B\|^2}\\,, \\nonumber $ where in the last inequality we used step one in the proof of Lemma REF .", "Since ${\\alpha }_h\\rightarrow 0$ , we infer that ${\\alpha }(F_h)/{\\alpha }_h\\rightarrow 1$ as $h\\rightarrow \\infty $ .", "By taking into account (REF ), this implies in particular that $\\lim _{h\\rightarrow \\infty }\|F_h\\Delta B\|=0\\,.$ If we now exploit the minimality property of each $F_h$ together with the Lipschitz properties of $t\\mapsto \|t-\|B\|\|$ , $t\\mapsto \|t-{\\alpha }_h\|$ , and the inequality $\|{\\alpha }(I)-{\\alpha }(J)\|\\le \|I\\Delta J\|$ for every $I,J\\subset \\mathbb {R}^n$ , then we find that each $F_h$ enjoy a uniform global almost-minimality property of the form $(1-s_h)P_{s_h}(F_h)\\le (1-s_h)P_{s_h}(G)+(\\Lambda +1)\|F_h\\triangle G\|\\,,\\qquad \\forall G\\subset \\mathbb {R}^n\\,.$ By (REF ), (REF ), (REF ), and Corollary REF , we find that $F_h$ is nearly spherical, in the sense that $\\partial F_h=\\lbrace x\\,(1+u_h(x)):\\,x\\in \\partial B\\rbrace $ , where $\\Vert u_h\\Vert _{C^1(\\partial B)}\\rightarrow 0$ as $h\\rightarrow \\infty $ .", "Let now $\\lambda _h>0$ be such that $\|\\lambda _h\\,F_h\|=\|B\|$ , and set $G_h=\\lambda _h\\,F_h$ .", "We notice that, by (REF ), $(1-s_h)\\,\\bigl (P_{s_h}(G_h)-P_{s_h}(B)\\bigr )&=(1-s_h)\\,P_{s_h}(F_h)\\,(\\lambda _h^{n-s}-1)+(1-s_h)\\big (P_{s_h}(F_h)-P_{s_h}(B)\\big )\\\\&\\le (1-s_h)\\,P_{s_h}(F_h)\\,(\\lambda _h^{n-s}-1)-\\Lambda \\,\|\|F_h\|-\|B\|\|+\\frac{(1-s_h){\\alpha }_h^2P_{s_h}(B)}{M\|B\|^2}\\,.$ Again by (REF ), we have $(1-s_h)\\,P_{s_h}(F_h)\\le (1-s_h)\\,P_{s_h}(B)+(1-s_h){\\alpha }_h^2P_{s_h}(B)/(M\|B\|^2)\\le C_6$ , provided we set $C_6(n,s_0):=\\sup _{s\\in [s_0,1)}(1-s)\\,P_s(B)\\bigl (1+\|B\|^{-2}\\inf _{s\\in [s_0,1)}\\varepsilon _1(n,s)^2\\bigr )\\,,$ and assume $M\\ge 1$ .", "Thus, by taking into account that $\\lambda ^{n-s}-1\\le \|\\lambda ^n-1\|$ for every $\\lambda >0$ and that $\\lambda _h\\rightarrow 1$ , we get $(1-s_h)\\,\\big (P_{s_h}(G_h)-P_{s_h}(B)\\big )&\\le &C_6\\,(\\lambda _h^{n-s}-1)-\\frac{\\Lambda }{2}\\,\|B\|\\,\|\\lambda _h^n-1\|+\\frac{(1-s_h){\\alpha }_h^2P_{s_h}(B)}{M\|B\|^2}\\\\&\\le &\\Big (C_6\\,-\\frac{\\Lambda }{2}\\,\|B\|\\Big )\|\\lambda _h^n-1\|+\\frac{(1-s_h){\\alpha }_h^2P_{s_h}(B)}{M\|B\|^2}\\,.$ We thus strengthen (REF ) into $\\Lambda >C_6/\|B\|$ to find that $P_{s_h}(G_h)-P_{s_h}(B)\\le {\\alpha }_h^2P_{s_h}(B)/(M\|B\|^2)$ , that is $D_{s_h}(G_h)\\le \\frac{{\\alpha }_h^2}{M\|B\|^2}\\,,$ that we combine with Corollary REF to get $A(G_h)^2\\le \\frac{C_0(n)}{s_0}\\,D_{s_h}(G_h)\\le \\frac{C_0}{s_0\\,M\|B\|^2}\\,{\\alpha }_h^2\\,.$ Now, by scale invariance $A(G_h)=A(F_h)$ ; moreover, by (REF ), $\|F_h\|\\rightarrow \|B\|$ as $h\\rightarrow \\infty $ , and thus $A(F_h)^2\\ge {\\alpha }(F_h)^2/(2\|B\|^2)$ for $h$ large enough; finally, as noticed in proving (REF ), ${\\alpha }(F_h)/{\\alpha }_h\\rightarrow 1$ as $h\\rightarrow \\infty $ , so that $A(F_h)^2\\ge {\\alpha }_h/(4\|B\|^2)$ for every $h$ large enough, and we conclude that $\\frac{{\\alpha }_h^2}{4}\\le \\frac{C_0}{s_0\\,M}\\,{\\alpha }_h^2\\,.$ We may thus choose $M>\\max \\Big \\lbrace 1,\\frac{4\\,C_0(n)}{s_0}\\Big \\rbrace \\,,$ in order to find a contradiction.", "This completes the proof of Theorem REF ." ], [ "Proof of Theorem ", "This section is devoted to the proof of Theorem REF .", "We shall continue the enumeration of constants that we started in section , working with the same convention set in Remark REF .", "We begin with an existence result.", "In the following, given a set $E\\subset \\mathbb {R}^n$ we shall set ${\\rm Per}_s(E):=\\left\\lbrace \\begin{array}{l l}\\frac{1-s}{\\omega _{n-1}}\\,P_s(E)\\,,&\\mbox{if $s\\in (0,1)$}\\,,\\\\P(E)\\,,&\\mbox{if $s=1$}\\,.\\end{array}\\right.$ Notice that, by (REF ), at least on smooth sets ${\\rm Per}_s$ is continuous as a function of $s\\in (0,1]$ .", "Recall that $V_\\alpha $ denotes the Riesz potential defined in (REF ).", "Lemma 5.1 If $n\\ge 2$ , $s\\in (0,1]$ , and $\\alpha \\in (0,n)$ , then there exist positive constants $m_1(n,\\alpha ,s)$ and $R_0(n,s)$ with the following property: For every $m<m_1$ , the variational problem $\\inf \\Big \\lbrace {\\rm Per}_s(E)+V_\\alpha (E):\|E\|=m\\Big \\rbrace $ admits minimizers, and every minimizer $E$ in (REF ) satisfies (up to a translation) the uniform bound $E\\subset B_{(m/\|B\|)^{1/n}\\,R_0}\\,.$ Moreover, $\\sup \\Big \\lbrace \\frac{1}{m_1(n,\\alpha ,s)}+R_0(n,s):\\alpha \\in [\\alpha _0,n)\\,, s\\in [s_0,1]\\Big \\rbrace <\\infty \\,,\\qquad \\forall s_0\\in (0,1),\\,\\alpha _0\\in (0,n)\\,.$ We first notice that, as expected, the truncation lemma for nonlocal perimeters, namely Lemma REF , holds true as well for classical perimeters.", "This can be seen either by adapting the argument of Lemma REF to the local case, or can be inferred as a particular case of [31].", "Either ways, one ends up showing that if $n\\ge 2$ and $E\\subset \\mathbb {R}^n$ is such that $\|E\\setminus B\|\\le \\eta <1$ , then there exists $1\\le r_E\\le 1+ C_1^\\,\\eta ^{1/n}$ such that $P(E\\cap B_{r_E})\\le P(E)-\\frac{\|E\\setminus B_{r_E}\|}{C_2^\\,\\eta ^{1/n}}\\,,$ where $C_1^$ and $C_2^$ are positive constants that depend on the dimension $n$ only.", "We then extend the definition of $C_1(n,s)$ and $C_2(n,s)$ given in (REF ) to the case $s=1$ by setting $C_1(n,1)=C_1^$ and $C_2(n,1)=C_2^$ .", "In conclusion, this shows that for every $n\\ge 2$ , $s\\in (0,1]$ and $E\\subset \\mathbb {R}^n$ is such that $\|E\\setminus B\|\\le \\eta <1$ , there exists $1\\le r_E\\le 1+ C_1(n,s)\\,\\eta ^{1/n}$ such that ${\\rm Per}_s(E\\cap B_{r_E})\\le {\\rm Per}_s(E)-\\frac{\|E\\setminus B_{r_E}\|}{C_2(n,s)\\,\\eta ^{1/n}}\\,,$ where $C_1(n,s)$ a $C_2(n,s)$ are such that $\\sup \\Big \\lbrace C_1(n,s)+C_2(n,s):s\\in [s_0,1]\\Big \\rbrace <\\infty \\,,\\qquad \\forall s_0\\in (0,1)\\,.$ With this tool at hand, we now pick $n\\ge 2$ , $\\alpha \\in (0,n)$ , $s\\in (0,1]$ , and denote by $\\gamma $ the infimum in (REF ).", "We claim that for every $m<m_1$ , $\\gamma =\\inf \\Big \\lbrace {\\rm Per}_s(E)+V_\\alpha (E):\|E\|=m\\,,E\\subset B_{(m/\|B\|)^{1/n}\\,R_0}\\Big \\rbrace \\,,$ where $m_1=m_1(n,s,\\alpha ):=\|B\|\\,\\min \\bigg \\lbrace 1,\\frac{{\\rm Per}_s(B)}{8\|B\|^2C(n,s)\\,V_\\alpha (B)},\\frac{{\\rm Per}_s(B)}{2\|B\|^2C(n,s)\\,V_\\alpha (B)}\\,\\Big (\\frac{\|B\|}{8\\,C_2\\,C_7}\\Big )^{2n/s}\\bigg \\rbrace ^{n/(\\alpha +s)} \\,,$ $R_0(n,s):=3(1+C_1)\\,,$ $C(n,s)$ is a constant such that (REF ) holds, and $C_7$ is defined as $C_7(n,s,\\alpha ):=2\\,\\Big ({\\rm Per}_s(B)+V_\\alpha (B)\\Big )\\,.$ (Note that (REF ) follows immediately from $(1-s)\\,P_s(B)\\rightarrow \\omega _{n-1}\\,P(B)$ as $s\\rightarrow 1^+$ and from the fact that $C(n,s)\\le C(n,s_0)$ if $s\\ge s_0$ .)", "We start noting that if $B[m]$ denotes the ball of volume $m$ then, since $m\\le \|B\|$ , $\\nonumber \\gamma &\\le & {\\rm Per}_s(B[m])+V_\\alpha (B[m])\\\\&=&\\Big (\\frac{m}{\|B\|}\\Big )^{(n-s)/n}\\,{\\rm Per}_s(B)+\\Big (\\frac{m}{\|B\|}\\Big )^{(n+\\alpha )/n}\\,V_\\alpha (B)\\hspace{11.38092pt}\\\\\\nonumber &\\le & C_7\\,\\Big (\\frac{m}{\|B\|}\\Big )^{(n-s)/n}\\,,$ where in the last inequality we have used the definition of $C_7$ .", "If $E$ is a generic set with $\|E\|=m\\,,\\qquad {\\rm Per}_s(E)+V_\\alpha (E)\\le \\gamma +V_\\alpha (B)\\,\\Big (\\frac{m}{\|B\|}\\Big )^{(n+\\alpha )/n}\\,,$ then by (REF ) we find $D_s(E)\\le \\frac{2\\,(m/\|B\|)^{(n+\\alpha )/n}\\,V_\\alpha (B)}{(m/\|B\|)^{(n-s)/n}\\,{\\rm Per}_s(B)}=\\frac{2\\,V_\\alpha (B)}{{\\rm Per}_s(B)}\\,\\Big (\\frac{m}{\|B\|}\\Big )^{(\\alpha +s)/n}\\,.$ Let us set $E_:=\\lambda \\,E$ where $\\lambda :=(\|B\|/m)^{1/n}$ , so that $\|E_\|=\|B\|$ .", "Since $D_s(E)=D_s(E_)$ , up to a translation we have, recalling (REF ), $\|E_\\Delta B\|\\le \|B\|\\bigg (C(n,s)\\Big (\\frac{m}{\|B\|}\\Big )^{(\\alpha +s)/n}\\frac{\\,2\\,V_\\alpha (B)}{{\\rm Per}_s(B)}\\bigg )^{1/2}=:\\eta \\,.$ By Lemma REF we can find $r_\\le 1+C_1\\,\\eta ^{1/n}$ such that ${\\rm Per}_s(E_\\cap B_{r_})\\le {\\rm Per}_s(E_)-\\frac{\|E_\\setminus B_{r_}\|}{C_2\\,\\eta ^{s/n}}\\,.$ In particular, scaling back to $E$ and setting $r_m=r_/\\lambda $ , we find ${\\rm Per}_s(E\\cap B_{r_m})\\le {\\rm Per}_s(E)-\\Big (\\frac{m}{\|B\|}\\Big )^{(n-s)/n}\\frac{\|B\|}{C_2\\,\\eta ^{s/n}}\\,\\frac{\|E\\setminus B_{r_m}\|}{m}\\,.$ Since trivially $V_\\alpha (E\\cap B_{r_m})\\le V_\\alpha (E)$ , we conclude that ${\\rm Per}_s(E\\cap B_{r_m})+V_\\alpha (E\\cap B_{r_m})\\le {\\rm Per}_s(E)+V_\\alpha (E)-\\Big (\\frac{m}{\|B\|}\\Big )^{(n-s)/n}\\frac{u\|B\|}{C_2\\,\\eta ^{s/n}}\\,,$ where we have set $u:=\|E\\setminus B_{r_m}\|/m$ .", "Let us now consider $F:=\\mu (E\\cap B_{r_m})$ for $\\mu >0$ such that $\|F\|=m$ .", "Since $\\mu =(1-u)^{-1/n}$ with $u<\\eta $ , if we assume that $\\eta \\le 1/2$ , and take into account that $\\frac{1}{(1-u)^p}\\le 1+2^{p+1}\\,u\\qquad \\forall u\\in [0,1/2]\\,,$ then, by $\\max \\lbrace \\mu ^{n-s},\\mu ^{n+\\alpha }\\rbrace =\\mu ^{n+\\alpha }\\le 1+8\\,u$ and by (REF ), we conclude that ${\\rm Per}_s(F)+V_\\alpha (F)&=&\\mu ^{n-s}{\\rm Per}_s(E\\cap B_{r_m})+\\mu ^{n+\\alpha }\\,V_\\alpha (E\\cap B_{r_m})\\\\&\\le &(1+8\\,u)\\Big ({\\rm Per}_s(E\\cap B_{r_m})+V_\\alpha (E\\cap B_{r_m})\\Big )\\\\&\\le &{\\rm Per}_s(E)+V_\\alpha (E)+\\Big (8\\,C_7-\\,\\frac{\|B\|}{C_2\\,\\eta ^{s/n}}\\Big )\\,\\Big (\\frac{m}{\|B\|}\\Big )^{(n-s)/n}\\,u\\,,$ where we have also taken into account that, by (REF ), (REF ), (REF ), and $m\\le \|B\|$ , ${\\rm Per}_s(E\\cap B_{r_m})+V_\\alpha (E\\cap B_{r_m})&\\le &\\Big (\\frac{m}{\|B\|}\\Big )^{(n-s)/n}\\,{\\rm Per}_s(B)+2\\Big (\\frac{m}{\|B\|}\\Big )^{(n+\\alpha )/n}\\,V_\\alpha (B)\\\\&\\le &C_7\\,\\Big (\\frac{m}{\|B\|}\\Big )^{(n-s)/n}\\,.$ Since the definition of $m_1$ implies that $\\eta ^{s/n}\\le \|B\|/(8\\,C_2\\,C_7)$ , we have proved that for every set $E$ as in (REF ) we can find a set $F$ with $\|F\|=m$ and $F\\subset B_{\\mu r_m}$ such that ${\\rm Per}_s(F)+V_\\alpha (F)\\le {\\rm Per}_s(E)+V_\\alpha (E)$ .", "This implies (REF ) and completes the proof of the lemma by observing that $\\mu \\le 1+2^{1+1/n}u<3$ and $r_m=r_/\\lambda \\le (1+C_1)(m/\|B\|)^\\frac{1}{n}$ .", "Next, we want to show that minimizers in (REF ), once rescaled to have the volume of the unit ball, are $\\Lambda $ -minimizers of the $s$ -perimeter for some uniform value of $\\Lambda $ .", "Lemma 5.2 If $n\\ge 2$ , $s\\in (0,1]$ , $\\alpha \\in (0,n)$ , $E$ is a minimizer in (REF ) for $m<m_1$ , and $E_=\\lambda \\,E$ for $\\lambda >0$ such that $\|E_\|=\|B\|$ , then $E_\\subset B_{R_0}$ and ${\\rm Per}_s(E_)\\le {\\rm Per}_s(F)+ \\Lambda _1\\,\|E_\\Delta F\|\\,,$ for every $F\\subset \\mathbb {R}^n$ .", "Here, $\\Lambda _1(n,\\alpha ,s)&:=&\\frac{4\\,C_7}{\|B\|}+\\frac{6\\,\|B\|\\,(1+C_8)\\,C_8^{\\alpha /n}}{\\alpha }\\,,\\\\C_8(n,\\alpha ,s)&:=&\\Big (1+\\frac{V_\\alpha (B)}{{\\rm Per}_s(B)}\\Big )^{n/(n-s)}\\,.$ In particular, $\\sup _{s\\in [s_0,1],\\alpha \\in [\\alpha _0,n)}\\Lambda _1(n,s,\\alpha )<\\infty \\,,\\qquad \\forall s_0\\in (0,1)\\,,\\alpha _0\\in (0,n)\\,.$ We first notice that, if $F,G\\subset \\mathbb {R}^n$ with $\|F\|<\\infty $ , then $V_\\alpha (F)-V_\\alpha (G)\\le \\frac{2\\,P(B)}{\\alpha }\\,\\Big (\\frac{\|F\|}{\|B\|}\\Big )^{\\alpha /n}\\,\|F\\setminus G\|\\,.$ (This is a more precise version of [33].)", "Indeed, if $r_F=(\|F\|/\|B\|)^{1/n}$ is the radius of the ball of volume $\|F\|$ , then $V_\\alpha (F)-V_\\alpha (G)\\le 2\\int _F\\int _{F\\setminus G}\\frac{dx\\,dy}{\|x-y\|^{n-\\alpha }}=2\\int _{F\\setminus G}dx\\,\\int _F\\,\\frac{dy}{\|x-y\|^{n-\\alpha }}\\le 2\|F\\setminus G\|\\int _{B_{r_F}}\\,\\frac{dz}{\|z\|^{n-\\alpha }}\\,,$ that is (REF ).", "We now prove that $E_$ satisfies (REF ).", "Of course, we may directly assume that ${\\rm Per}_s(F)\\le {\\rm Per}_s(E_)$ .", "We also claim that we can reduce to prove (REF ) in the case that $\\frac{1}{2}\\le \\frac{\|F\|}{\|B\|}\\le C_8\\,.$ Indeed, if we compare $E$ with a ball of volume $m$ (see (REF )) and then multiply the resulting inequality by $\\lambda ^{n-s}$ , we find $ {\\rm Per}_s(E_)+\\frac{V_\\alpha (E_)}{\\lambda ^{\\alpha +s}}\\le {\\rm Per}_s(B)+\\frac{V_\\alpha (B)}{\\lambda ^{\\alpha +s}}\\le {\\rm Per}_s(B)+V_\\alpha (B)\\,,$ where in the last inequality we have taken into account that $\\lambda \\ge 1$ (because $m\\le m_1\\le \|B\|$ ).", "If now $F$ is such that $\|F\|\\le \|B\|/2$ , then $\|E_\\Delta F\|\\ge \|B\|/2$ , and thus (REF ) trivially holds true by (REF ) and our definition of $\\Lambda _1$ .", "If instead the upper bound in (REF ) does not hold, then we obtain a contradiction by combining ${\\rm Per}_s(F)\\le {\\rm Per}_s(E_)$ , (REF ) (or the classical isoperimetric inequality if $s=1$ ), and (REF ).", "We have thus reduced to prove (REF ) in the case that (REF ) holds true.", "If we now set $\\mu =(m/\|F\|)^{1/n}$ , then $\|\\mu \\,F\|=m$ , and by minimality of $E$ in (REF ) and by (REF ) we find that ${\\rm Per}_s(E)&\\le &{\\rm Per}_s(\\mu \\,F)+V_\\alpha (\\mu \\,F)-V_\\alpha (E)\\\\&=&{\\rm Per}_s(\\mu \\,F)+\\mu ^{n+\\alpha }\\Big (V_\\alpha (F)-V_\\alpha (E_)\\Big )+\\Big ((\\lambda \\,\\mu )^{n+\\alpha }-1\\Big )\\,V_\\alpha (E)\\,,$ where in the last identity we have added and subtracted $V_\\alpha (\\lambda \\,\\mu \\,E)$ .", "We multiply this inequality by $\\lambda ^{n-s}$ , apply (REF ) and (REF ) to the second term on the right-hand side, and take into account that $\\lambda ^{n-s}\\,V_\\alpha (E)=\\lambda ^{-s-\\alpha }\\,V_\\alpha (E_)$ , to find that ${\\rm Per}_s(E_)&\\le &(\\lambda \\,\\mu )^{n-s}\\,{\\rm Per}_s(F)+\\lambda ^{n-s}\\mu ^{n+\\alpha }\\,\\frac{2\\,P(B)\\,C_8^{\\alpha /n}}{\\alpha }\\,\|F\\setminus E_\|\\\\\\nonumber &&+\\Big ((\\lambda \\,\\mu )^{n+\\alpha }-1\\Big )\\,\\frac{V_\\alpha (E_)}{\\lambda ^{\\alpha +s}}\\,.$ We now estimate the various terms on the right-hand side of (REF ).", "Since $\|F\|\\ge \|B\|/2$ and $\|B\|-\|F\|=\|E_\|-\|F\|\\le \|E_\\Delta F\|$ give $(\\lambda \\,\\mu )^{n-s}=\\Big (1+\\frac{\|B\|-\|F\|}{\|F\|}\\Big )^{(n-s)/n}\\le 1+\\frac{n-s}{n}\\,\\frac{\|E_\\Delta F\|}{\|B\|/2}\\le 1+\\frac{2}{\|B\|}\|E_\\Delta F\|\\,,$ by ${\\rm Per}_s(F)\\le {\\rm Per}_s(E_)$ and (REF ) we find $(\\lambda \\,\\mu )^{n-s}\\,{\\rm Per}_s(F)\\le {\\rm Per}_s(F)+\\frac{C_7}{\|B\|}\\,\|E_\\Delta F\|\\,.$ Since (REF ) also gives $\|E_\\Delta F\|\\le (1+C_8)\\,\|B\|$ , by (REF ) and $m\\le \|B\|$ we have $\\nonumber \\lambda ^{n-s}\\,\\mu ^{n+\\alpha }&=&\\mu ^{\\alpha +s}\\,(\\lambda \\mu )^{n-s}\\le \\Big (\\frac{m}{\|B\|}\\Big )^{(\\alpha +s)/n}\\Big (1+\\frac{2(n-s)}{P(B)}\\,\|E_\\Delta F\|\\Big )\\\\&\\le & 1+\\frac{2(n-s)}{n}\\,(1+C_8)\\le 3\\,(1+C_8)\\,.$ Finally, by $\|F\|\\ge \|B\|/2$ we find that $(\\lambda \\,\\mu )^{n+\\alpha }=\\Big (1+\\frac{\|B\|-\|F\|}{\|F\|}\\Big )^{1+(\\alpha /n)}\\le 1+(2^{1+(\\alpha /n)}-1)\\,\\Big \|\\frac{\|B\|-\|F\|}{\|F\|}\\Big \|\\le 1+\\frac{6}{\|B\|}\\,\|E_\\Delta F\|\\,,$ that combined with (REF ) gives $\\Big ((\\lambda \\,\\mu )^{n+\\alpha }-1\\Big )\\,\\frac{V_\\alpha (E_)}{\\lambda ^{\\alpha +s}}\\le \\frac{3\\,C_7}{\|B\|}\\,\|E_\\Delta F\|\\,.$ We now plug (REF ), (REF ), (REF ), and (REF ) into (REF ) to complete the proof of (REF ).", "Let us fix $s_0\\in (0,1)$ and $\\alpha _0\\in (0,n)$ , and let $\\bar{m}_1:=\\inf \\Big \\lbrace m_1(n,\\alpha ,s):\\alpha \\in [\\alpha _0,n)\\,,s\\in [s_0,1)\\Big \\rbrace \\,,$ so that, by Lemma REF and Lemma REF , $\\bar{m}_1>0$ and for every $m<\\bar{m}_1$ , $\\alpha \\in [\\alpha _0,n)$ , and $s\\in [s_0,1)$ , there exists a minimizer $E_{m,\\alpha ,s}$ of $\\inf \\Big \\lbrace {\\rm Per}_s(E)+V_\\alpha (E):\|E\|=m\\Big \\rbrace $ such that ${\\rm Per}_s(E_{m,\\alpha ,s})\\le {\\rm Per}_s(F)+\\bar{\\Lambda }_1\\,\|E_{m,\\alpha ,s}\\Delta F\|\\,,\\qquad \\forall F\\subset \\mathbb {R}^n\\,,$ where $\\bar{\\Lambda }_1:=\\sup \\Big \\lbrace \\Lambda _1(n,\\alpha ,s):\\alpha \\in [\\alpha _0,n)\\,,s\\in [s_0,1)\\Big \\rbrace <\\infty \\,.$ We now want to show the existence of $m_0\\le \\bar{m}_1$ such that $A(E_{m,\\alpha ,s})=0$ for $m<m_0$ , which implies that $E_{m,\\alpha ,s}$ is a ball (recall (REF )).", "We argue by contradiction and construct sequences $\\lbrace s_h\\rbrace _{h\\in \\mathbb {N}}\\subset [s_0,1]$ , $\\lbrace \\alpha _h\\rbrace _{h\\in \\mathbb {N}}\\subset [\\alpha _0,n)$ , and $\\lbrace E_h\\rbrace _{h\\in \\mathbb {N}}$ minimizers of ${\\rm Per}_{s_h}+V_{\\alpha _h}$ at volume $m_h$ , such that $m_h\\rightarrow 0^+$ as $h\\rightarrow \\infty $ and, if we set $\\lambda _h=(\|B\|/m_h)^{1/n}$ , then $E_{h,}=\\lambda _h\\,E_h$ is a $\\bar{\\Lambda }_1$ -minimizers of the $s_h$ -perimeter with $\|E_{h,}\|=\|B\|\\,,\\qquad A(E_{h,})=A(E_h)>0\\,,\\qquad \\forall h\\in \\mathbb {N}\\,.$ By (REF ) and either by Theorem REF if $s_h<1$ , or by [22] in the case $s_h=1$ , we have that, for a suitable positive constant $C(n,s_0)$ , $\\frac{A(E_h)^2}{C_0(n,s_0)}\\le D_{s_h}(E_h)\\le \\frac{2\\,V_{\\alpha _h}(B)}{{\\rm Per}_{s_h}(B)}\\,\\Big (\\frac{m_h}{\|B\|}\\Big )^{(\\alpha _h+s_h)/n}\\,,$ so that $A(E_{h,})\\le C(n,s_0,\\alpha _0)\\,m_h^{(\\alpha _0+s_0)/2n}\\,,\\qquad \\forall h\\in \\mathbb {N}\\,.$ Up to translations, we may thus assume $\\lim _{h\\rightarrow \\infty }\|E_{h,}\\Delta B\|=0\\,.$ By Corollary REF , we thus have $\\partial E_{h,}=\\Big \\lbrace (1+u_h(x))\\,x:x\\in \\partial B\\Big \\rbrace \\,,\\qquad u_h\\in C^1(\\partial B)\\,,\\qquad \\forall h\\in \\mathbb {N}\\,,$ where $\\Vert u_h\\Vert _{C^1(\\partial B)}\\rightarrow 0$ as $h\\rightarrow \\infty $ .", "Since $\|E_{h,}\|=\|B\|$ , by Lemma REF below we find that $V_{\\alpha }(B)-V_{\\alpha }(E_{h,})\\le C(n)\\,\\Big ([u_h]^2_{\\frac{1-\\alpha }{2}}+\\Vert u_h\\Vert _{L^2(\\partial B)}^2\\Big )\\,,\\qquad \\forall \\alpha \\in (0,n)\\,,$ where $[u]^2_{\\frac{1-\\alpha }{2}}:=\\iint _{\\partial B\\times \\partial B}\\frac{\|u(x)-u(y)\|^2}{\|x-y\|^{n-\\alpha }}d{\\mathcal {H}}^{n-1}_x\\,d{\\mathcal {H}}^{n-1}_y\\,.$ Notice, in particular, that $[u]^2_{\\frac{1-\\alpha }{2}}\\le 2^{\\alpha +s}\\,[u]^2_{\\frac{1+s}{2}}\\,,\\qquad \\forall \\alpha \\in (0,n)\\,,s\\in (0,1)\\,.$ At the same time, by ${\\rm Per}_{s_h}(E_h)+V_{\\alpha _h}(E_h)\\le {\\rm Per}_{s_h}(B_{r_h})+V_{\\alpha _h}(B_{r_h})$ , where $\|B_{r_h}\|=m_h$ , we have $\\delta _{s_0}(E_h)&\\le & D_{s_h}(E_h)\\le \\frac{V_{\\alpha _h}(B_{r_h})-V_{\\alpha _h}(E_h)}{{\\rm Per}_{s_h}(B_{r_h})}\\\\&\\le &m_h^{(\\alpha _h+s_h)/n}\\,\\frac{C(n)\\,\\Big ([u_h]^2_{\\frac{1-\\alpha }{2}}+\\Vert u_h\\Vert _{L^2(\\partial B)}^2\\Big )}{\\inf _{s\\in [s_0,1)}{\\rm Per}_{s}(B)}\\\\&\\le &C(n,s_0)\\,m_h^{(\\alpha _h+s_h)/n}\\,\\Big ([u_h]^2_{\\frac{1+s_0}{2}}+\\Vert u_h\\Vert _{L^2(\\partial B)}^2\\Big )\\,,$ where we used (REF ).", "On the other hand, by Theorem REF (notice that we can assume without loss of generality that $\\int _{E_h}x\\,dx=0$ for every $h\\in \\mathbb {N}$ ) $\\delta _{s_0}(E_h)\\ge \\frac{s_0}{C(n)}\\,\\Big ([u_h]^2_{\\frac{1+s_0}{2}}+\\Vert u_h\\Vert _{L^2(\\partial B)}^2\\Big )\\,.$ We have thus proved $\\frac{s_0}{C(n)}\\le C(n,s_0)\\,m_h^{(\\alpha _h+s_h)/n}\\,,$ and since $\\alpha _h\\ge \\alpha _0$ , $s_h\\ge s_0$ , and $m_h \\rightarrow 0$ , this inequality leads to a contradiction for $h$ sufficiently large.", "Let us recall that, by Riesz's rearrangement inequality, for every $\\alpha \\in (0,n)$ $V_\\alpha (B)\\ge V_\\alpha (E)\\qquad \\mbox{ whenever $\|E\|=\|B\|$}\\,,$ with equality if and only if $E=x+B$ for some $x\\in \\mathbb {R}^n$ .", "(Indeed, the radial convolution kernel $\|z\|^{\\alpha -n}$ is strictly decreasing.)", "Due to the maximality property of balls expressed in (REF ), one expect the quantity $V_\\alpha $ to satisfy an estimate of the form $V_\\alpha (E)\\ge V_\\alpha (B)-C(n,\\alpha )\\,\\Vert u\\Vert ^2$ on nearly spherical sets of volume $\|B\|$ , for some suitable norm $\\Vert \\cdot \\Vert $ .", "This is exactly the content of the following lemma.", "Lemma 5.3 There exist positive constants $\\varepsilon _0$ and $C_0$ , depending on $n$ only, with the following property: If $E\\subset \\mathbb {R}^n$ is an open set such that $\|E\|=\|B\|$ and $\\partial E=\\Big \\lbrace (1+u(x))\\,x:x\\in \\partial B\\Big \\rbrace \\,,$ for some function $u\\in C^1(\\partial B)$ with $\\Vert u\\Vert _{C^1(\\partial B)}\\le \\varepsilon _0$ , then $V_\\alpha (B)-V_\\alpha (E)\\le C_0\\,\\Big ([u]^2_{\\frac{1-\\alpha }{2}}+\\alpha V_\\alpha (B)\\Vert u\\Vert _{L^2(\\partial B)}^2\\Big )\\,,\\qquad \\forall \\alpha \\in (0,n)\\,.$ The proof of this result is very similar to the one of Theorem REF .", "As in that proof, we slightly change notation and assume that $E_t$ is an open set with $\|E_t\|=\|B\|$ and $E_t=\\Big \\lbrace (1+t\\,u(x))\\,x:x\\in \\partial B\\Big \\rbrace \\,,\\qquad \\Vert u\\Vert _{C^1(\\partial B)}\\le \\frac{1}{2}\\,,\\qquad t\\in (0,2\\varepsilon _0)\\,.$ Given $r,\\rho ,\\theta \\ge 0$ we now set $f_{\\theta }(r,\\rho ):=\\frac{r^{n-1}\\,\\rho ^{n-1}}{(\|r-\\rho \|^2+r\\,\\rho \\,\\theta ^2)^{(n-\\alpha )/2}}\\,,$ so that $V_\\alpha (E_t)=\\int _{\\partial B}d{\\mathcal {H}}^{n-1}_x\\int _{\\partial B}d{\\mathcal {H}}^{n-1}_y\\int _0^{1+t\\,u(x)}dr\\int _0^{1+t\\,u(y)}f_{\|x-y\|}(r,\\rho )\\,d\\rho \\,.$ By exploiting the identity $2\\,\\int _0^a\\int _0^b=\\int _0^a\\int _0^a+\\int _0^b\\int _0^b-\\int _a^b\\int _a^b\\,,\\qquad a,b\\in \\mathbb {R}\\,,$ we find that $V_\\alpha (E_t)&=&\\int _{\\partial B}d{\\mathcal {H}}^{n-1}_x\\int _{\\partial B}d{\\mathcal {H}}^{n-1}_y\\int _0^{1+t\\,u(x)}dr\\int _0^{1+t\\,u(x)}f_{\|x-y\|}(r,\\rho )\\,d\\rho \\\\\\nonumber &&-\\frac{1}{2}\\int _{\\partial B}d{\\mathcal {H}}^{n-1}_x\\int _{\\partial B}d{\\mathcal {H}}^{n-1}_y\\int _{1+t\\,u(y)}^{1+t\\,u(x)}dr\\int _{1+t\\,u(y)}^{1+t\\,u(x)}f_{\|x-y\|}(r,\\rho )\\,d\\rho \\,.$ By a change of variable, for every $x\\in \\partial B$ we find $&&\\int _{\\partial B}d{\\mathcal {H}}^{n-1}_y\\int _0^{1+t\\,u(x)}dr\\int _0^{1+t\\,u(x)}f_{\|x-y\|}(r,\\rho )\\,d\\rho \\\\&=&(1+t\\,u(x))^{n+\\alpha }\\,\\int _{\\partial B}d{\\mathcal {H}}^{n-1}_y\\int _0^{1}dr\\int _0^{1}f_{\|x-y\|}(r,\\rho )\\,d\\rho =(1+t\\,u(x))^{n+\\alpha }\\,\\frac{V_\\alpha (B)}{P(B)}\\,,$ where in the last identity we have used (REF ) with $u=0$ .", "Hence, $\\nonumber V_\\alpha (E_t)&=&-\\frac{1}{2}\\int _{\\partial B}d{\\mathcal {H}}^{n-1}_x\\int _{\\partial B}d{\\mathcal {H}}^{n-1}_y\\int _{1+t\\,u(y)}^{1+t\\,u(x)}dr\\int _{1+t\\,u(y)}^{1+t\\,u(x)}f_{\|x-y\|}(r,\\rho )\\,d\\rho \\\\\\nonumber &&+\\frac{V_\\alpha (B)}{P(B)}\\,\\int _{\\partial B}(1+t\\,u)^{n+\\alpha }\\,d{\\mathcal {H}}^{n-1}\\,,$ from which we conclude that $V_\\alpha (B)-V_\\alpha (E_t)=\\frac{t^2}{2}\\,g(t)+\\frac{V_\\alpha (B)}{P(B)}\\,(h(0)-h(t))\\,,$ provided we set $h(t):=\\int _{\\partial B}(1+t\\,u)^{n+\\alpha }\\,d{\\mathcal {H}}^{n-1}$ and $g(t):=\\int _{\\partial B}d{\\mathcal {H}}^{n-1}_x\\int _{\\partial B}d{\\mathcal {H}}^{n-1}_y\\int _{u(y)}^{u(x)}dr\\int _{u(y)}^{u(x)}f_{\|x-y\|}(1+t\\,r,1+t\\,\\rho )\\,d\\rho \\,.$ Since $\|B\|=\|E_t\|$ implies $\\int _{\\partial B}(1+t\\,u)^n=n\\,\|E_t\|=n\\,\|B\|=P(B)=h(0)$ , we get $h(0)-h(t)&=\\int _{\\partial B}(1+tu)^n\\big (1-(1+tu)^{\\alpha }\\big )\\,d{\\mathcal {H}}^{n-1}\\\\&\\le -\\alpha \\,t\\,\\int _{\\partial B}u\\,d{\\mathcal {H}}^{n-1}-\\alpha \\,(2n+\\alpha -1)\\,\\frac{t^2}{2}\\int _{\\partial B}u^2\\,d{\\mathcal {H}}^{n-1}+C(n)\\,\\alpha \\,t^3\\,\\Vert u\\Vert _{L^2}^2\\,.$ In addition, because $\|B\|=\|E_t\|$ also gives $0=\\int _{\\partial B}\\big ((1+t\\,u)^n-1\\big )$ , we can likewise deduce that $-t\\int _{\\partial B}u\\,d{\\mathcal {H}}^{n-1}\\le (n-1)\\frac{t^2}{2}\\int _{\\partial B}u^2\\,d{\\mathcal {H}}^{n-1}+ C(n)\\,t^3\\,\\Vert u\\Vert _{L^2}^2\\,,$ therefore $h(0)-h(t)\\le -\\alpha \\,(n+\\alpha )\\,\\frac{t^2}{2}\\,\\int _{\\partial B}u^2\\,d{\\mathcal {H}}^{n-1}+\\alpha \\,C(n)\\,t^3\\,\\Vert u\\Vert _{L^2}^2\\,.$ Furthermore, we notice that $g(0)=\\iint _{\\partial B\\times \\partial B}\\frac{\|u(x)-u(y)\|^2}{\|x-y\|^{n-\\alpha }}d{\\mathcal {H}}^{n-1}_x\\,d{\\mathcal {H}}^{n-1}_y=[u]^2_{\\frac{1-\\alpha }{2}}\\,.$ Arguing as in the proof of Theorem REF , we infer that $g(t)=g(0)+t\\,g^{\\prime }(\\tau )$ for some $\\tau \\in (0,t)$ and with $\|g^{\\prime }(\\tau )\|\\le C(n)\\,g(0)$ .", "Hence, $V_\\alpha (B)-V_\\alpha (E_t)\\le \\frac{t^2}{2} \\left([u]^2_{\\frac{1-\\alpha }{2}}-\\alpha (n+\\alpha )\\frac{V_\\alpha (B)}{P(B)}\\,\\Vert u\\Vert ^2_{L^2} \\right)\\\\+C(n)\\,t^3\\Big ([u]^2_{\\frac{1-\\alpha }{2}}+\\alpha V_\\alpha (B) \\Vert u\\Vert _{L^2}^2\\Big )\\,.$ This last estimate obviously implies the announced result." ], [ "First and second variation formulae and local minimizers", "In this section we provide first and second variation formulae for the functionals $P_s$ (compare with [12]) and $V_\\alpha $ , and actually for generic nonlocal functionals behaving like $P_s$ and $V_\\alpha $ .", "Before introducing our precise setting, let us recall what is the situation in the case of the classical perimeter functional (see, e.g., [36], [27] or [31]), and set some useful terminology.", "Given an open set $\\Omega $ and a vector field $X\\in C^\\infty _c(\\Omega ;\\mathbb {R}^n)$ , we denote by $\\lbrace \\Phi _t\\rbrace _{t\\in \\mathbb {R}}$ the flow induced by $X$, that is the smooth map $(t,x)\\in \\mathbb {R}\\times \\mathbb {R}^n\\mapsto \\Phi _t(x)\\in \\mathbb {R}^n$ defined by solving the family of ODEs (parameterized by $x\\in \\mathbb {R}^n$ ) ${\\left\\lbrace \\begin{array}{ll}\\partial _t\\Phi _t(x)=X(\\Phi _t(x))\\,,& t\\in \\mathbb {R}\\,, \\\\\\Phi _0(x)=x\\,.\\end{array}\\right.", "}$ By the implicit function theorem, there always exists $\\varepsilon >0$ such that $\\lbrace \\Phi _t\\rbrace _{\|t\|<\\varepsilon }$ is a smooth family of diffeomorphisms.", "Given $E\\subset \\mathbb {R}^n$ with $\|E\|<\\infty $ , one says that $X$ induces a volume-preserving flow on $E$ if $\|\\Phi _t(E)\|=\|E\|$ for every $\|t\|<\\varepsilon $ .", "If $E$ is a set of finite perimeter in $\\Omega $ and $E_t:=\\Phi _t(E)$ , then $\\lbrace E_t\\rbrace _{\|t\|<\\varepsilon }$ is a family of sets of finite perimeter in $\\Omega $ , $t\\mapsto P(E_t;\\Omega )$ is a smooth function on $\|t\|<\\varepsilon $ (thanks to the area formula for rectifiable sets), and it makes sense to define the first and second variations of the perimeter at $E$ along $X$ (or, more precisely, along the flow induced by $X$ via (REF )) as $\\delta P(E;\\Omega )[X]:=\\frac{d}{dt}P(E_t;\\Omega )_{\\bigl \|t=0}\\,,\\qquad \\delta ^2P(E;\\Omega )[X]:=\\frac{d^2}{dt^2}P(E_t;\\Omega )_{\\bigl \|t=0}\\,.$ One says that $E$ is a volume-constrained stationary set for the perimeter in $\\Omega $ if $\\delta P(E;\\Omega )[X]=0$ whenever $X$ induces a volume-preserving flow on $E$ ; if in addition $\\delta ^2 P(E;\\Omega )[X]\\ge 0$ for every $X$ inducing a volume-preserving flow on $E$ , then $E$ is said to be a volume-constrained stable set for the perimeter in $\\Omega $.", "The interest into these properties stems from the immediate fact that if $E$ is a local volume-constrained perimeter minimizer in $\\Omega $, that is, if $P(E;\\Omega )<\\infty $ and, for some $\\delta >0$ , $P(E;\\Omega )\\le P(F;\\Omega )\\,,\\qquad \\forall F\\subset \\Omega \\,,\\quad \|E\|=\|F\|\\,,\\quad \|E\\Delta F\|<\\delta \\,,$ then $E$ is automatically a volume-constrained stable set for the perimeter in $\\Omega $ .", "In order to effectively exploit stability one needs explicit formulas for $\\delta P(E;\\Omega )[X]$ and $\\delta ^2 P(E;\\Omega )[X]$ in terms of $X$ .", "When $\\partial E\\cap \\Omega $ is a $C^2$ -hypersurface one can obtain such formulas by using the area formula, Taylor's expansions, and the divergence theorem on $\\partial E\\cap \\Omega $ .", "Denoting by ${\\mathrm {H}}_{\\partial E}$ the scalar mean curvature of $\\partial E\\cap \\Omega $ (with respect to the orientation induced by the outer unit normal $\\nu _E$ to $E$ ), by ${\\mathrm {c}}^2_{\\partial E}$ the sum of the squares of the principal curvatures of $\\partial E\\cap \\Omega $ , and setting $\\zeta =X\\cdot \\nu _E$ for the normal component of $X$ with respect to $\\nu _E$ , one gets the classical formulae $\\delta P(E;\\Omega )[X]&=&\\int _{\\partial E\\cap \\Omega }{\\rm H}_{\\partial E}\\,\\zeta \\,d{\\mathcal {H}}^{n-1}\\,,\\\\\\delta ^2 P(E;\\Omega )[X]&=&\\int _{\\partial E} \|\\nabla _\\tau \\zeta \|^2-{\\rm c}^2_{\\partial E}\\,\\zeta ^2\\,d{\\mathcal {H}}^{n-1}\\\\\\nonumber &&+\\int _{\\partial E} {\\rm H}_{\\partial E} \\big ( ({\\rm div}X)\\,\\zeta -{\\rm div}_{\\tau }\\bigl (\\zeta \\,X_\\tau \\bigl )\\big )\\,d{\\mathcal {H}}^{n-1}\\,.$ (Here, $X_\\tau =X-\\zeta \\,\\nu _E$ is the tangential projection of $X$ along $\\partial E$ , while $\\nabla _\\tau $ and ${\\rm div}_\\tau $ denote the tangential gradient and the tangential divergence operators to $\\partial E$ .)", "If $E$ is a volume-constrained stationary set for the perimeter in $\\Omega $ , then ${\\rm H}_{\\partial E}$ is constant on $\\partial E\\cap \\Omega $ and $\\delta ^2 P(E;\\Omega )[X]=\\int _{\\partial E} \|\\nabla _\\tau \\zeta \|^2-{\\rm c}^2_{\\partial E}\\,\\zeta ^2\\,d{\\mathcal {H}}^{n-1}$ whenever $X$ induces a volume-preserving flow on $E$ .", "Indeed, $\|E_t\|=\|E\|$ for every $\|t\|<\\varepsilon $ implies $0=\\frac{d}{dt}\|E_t\|_{\\bigl \|t=0}=\\int _{\\partial E} \\zeta \\,d{\\mathcal {H}}^{n-1}\\,,\\qquad 0=\\frac{d^2}{dt^2}\|E_t\|_{\\bigl \|t=0}=\\int _{\\partial E}\\,({\\rm div}X)\\,\\zeta \\,d{\\mathcal {H}}^{n-1}\\,.$ By combining the first condition in (REF ) with $\\delta P(E;\\Omega )[X]=0$ and (REF ), one finds that ${\\rm H}_{\\partial E}$ is constant on $\\partial E\\cap \\Omega $ .", "By combining (), the second condition in (REF ), the fact that ${\\rm H}_{\\partial E}$ is constant on $\\partial E\\cap \\Omega $ , and the identity $\\int _{\\partial E}{\\rm div}_{\\tau }\\bigl (\\zeta \\,X_\\tau \\bigl )\\,d{\\mathcal {H}}^{n-1}=0$ (which follows by the tangential divergence theorem), one deduces (REF ).", "We now want to obtain these kind of variation formulas for the nonlocal functionals considered in this paper.", "We shall actually work in a broader framework.", "Precisely, given $s\\in (0,1)$ and $\\alpha \\in (0,n)$ , we fix thorough this section two convolution kernels $K,G\\in C^1(\\mathbb {R}^n\\setminus \\lbrace 0\\rbrace ;[0,\\infty ))$ which are symmetric by the origin (i.e., $K(-z)=K(z)$ and $G(-z)=G(z)$ for every $z\\in \\mathbb {R}^n\\setminus \\lbrace 0\\rbrace $ ) and satisfy the pointwise bounds $K(z)\\le \\frac{C_K}{\|z\|^{n+s}}\\,, \\qquad G(z)\\le \\frac{C_G}{\|z\|^{n-\\alpha }}\\,,\\qquad \\forall z\\in \\mathbb {R}^n\\setminus \\lbrace 0\\rbrace \\,,$ for some constants $C_K$ and $C_G$ .", "Correspondingly, given $E\\subset \\mathbb {R}^n$ , we consider the nonlocal functionals (defined in $[0,\\infty ]$ ) $P_K(E)=\\iint _{E\\times E^c}K(x-y)\\,dx\\,dy\\,,\\qquad V_G(E)=\\iint _{E\\times E}G(x-y)\\,dx\\,dy\\,.$ Notice that the two functionals are substantially different only in presence of the singularities allowed in (REF ).", "Indeed, by virtue of (REF ), $K$ is possibly singular only close to the origin, while $G$ is possibly singular only at infinity (in the sense that the integral of $G$ may diverge at infinity).", "When no singularity is present, then the two functionals are essentially equivalent in the sense that one has $P_K(E)=\|E\|\\,\\Vert K\\Vert _{L^1(\\mathbb {R}^n)}-V_K(E)\\,,\\qquad \\mbox{if $K\\in L^1(\\mathbb {R}^n)$ and $\|E\|<\\infty $.", "}$ We next introduce the restrictions of $P_K$ and $V_G$ to a given open set $\\Omega $ .", "Following [7], we set $P_K(E,\\Omega )&:=&\\int _{E\\cap \\Omega }\\int _{E^c\\cap \\Omega }K(x-y)\\,dx\\,dy+\\int _{E\\cap \\Omega }\\int _{E^c\\setminus \\Omega }K(x-y)\\,dx\\,dy\\\\&&+\\int _{E\\setminus \\Omega }\\int _{E^c\\cap \\Omega }K(x-y)\\,dx\\,dy\\,,\\\\V_G(E,\\Omega )&:=&\\int _{E\\cap \\Omega }\\int _{E\\cap \\Omega }G(x-y)\\,dx\\,dy+2\\int _{E\\cap \\Omega }\\int _{E\\setminus \\Omega }G(x-y)\\,dx\\,dy\\,.$ If $P_K(E;\\Omega )<\\infty $ , $X\\in C^\\infty _c(\\Omega ;\\mathbb {R}^n)$ , and $E_t=\\Phi _t(E)$ as before, then one finds from the area formula that $t\\mapsto P_K(E_t;\\Omega )$ is a smooth function for $\|t\|<\\varepsilon $ , and correspondingly is able to define the first and second variations of $P_K(\\cdot ,\\Omega )$ at $E$ along $X$ as $\\delta P_K(E;\\Omega )[X]= \\frac{d }{dt}P_K(E_t;\\Omega )_{\\bigl \|t=0}\\,,\\qquad \\delta ^2 P_K(E;\\Omega )[X]= \\frac{d^2 }{dt^2}P_K(E_t;\\Omega )_{\\bigl \|t=0}\\,.$ Identical definitions are adopted when $V_G$ is considered in place of $P_K$ and $E$ is such that $V_G(E;\\Omega )<\\infty $ (as it is the case, for example, whenever $E$ is bounded).", "Having set our terminology, we now turn to the problem of expressing first and second variations along $X$ in terms of boundary integrals involving $X$ and its derivatives, in the spirit of (REF ) and ().", "These formulas involve some “nonlocal” variants of the quantities ${\\rm H}_{\\partial E}$ and ${\\rm c}^2_{\\partial E}$ , that are introduced as follows.", "Given $E\\subset \\mathbb {R}^n$ , $x\\in \\mathbb {R}^n$ , and a non-negative Borel function $J$ on $\\mathbb {R}^n$ , we define (as elements of $[-\\infty ,\\infty ]$ ) ${\\rm H}_{J,\\partial E}(x)&:=&{\\rm p.v.", "}\\left(\\int _{\\mathbb {R}^n}\\bigl (\\chi _{E^c}(y)-\\chi _E(y)\\bigr )\\,J(x-y)\\,dy\\right)\\\\\\nonumber &=&\\limsup _{\\varepsilon \\rightarrow 0^+}\\int _{\\mathbb {R}^n\\setminus B(x,\\varepsilon )}\\bigl (\\chi _{E^c}(y)-\\chi _E(y)\\bigr )\\,J(x-y)\\,dy\\,,\\\\{\\rm H}^_{J,\\partial E}(x)&:=&2\\int _{E} J(x-y)\\,dy\\,.$ Moreover, given an orientable hypersurface $M$ of class $C^1$ in $\\mathbb {R}^n$ , and denoting by $\\nu _M$ an orientation of $M$ , we define ${\\rm c}^2_{J,M}:M\\rightarrow [0,\\infty ]$ by setting ${\\rm c}^2_{J,M}(x)&:=&\\int _{M}\\,J(x-y)\|\\nu _M(x)-\\nu _M(y)\|^2\\,d{\\mathcal {H}}^{n-1}_y\\,,\\qquad \\forall x\\in M\\,.$ The functions ${\\rm H}_{J,\\partial E}$ and ${\\rm H}^_{J,\\partial E}$ will play the role of nonlocal mean curvatures for $P_K$ when $J=K$ and for $V_G$ when $J=G$ , respectively.", "As it turns out, if $J\\in L^1(\\mathbb {R}^n)$ then the two quantities are equivalent up to a constant and a change of sign, that is, ${\\rm H}_{J,\\partial E}(x)=\\Vert J\\Vert _{L^1(\\mathbb {R}^n)}-{\\rm H}^_{J,\\partial E}(x)\\,,\\qquad \\forall x\\in \\mathbb {R}^n\\,,$ a result that, of course, is in accord with (REF ).", "We are now in the position to the state the main theorem of this section.", "Theorem 6.1 Let $K,G\\in C^1(\\mathbb {R}^n\\setminus \\lbrace 0\\rbrace ;[0,\\infty ))$ be even functions satisfying (REF ) for some $s\\in (0,1)$ and $\\alpha \\in (0,n)$ , let $\\Omega $ be an open set in $\\mathbb {R}^n$ , let $E\\subset \\mathbb {R}^n$ be an open set with $C^{1,1}$ -boundary such that $\\partial E\\cap \\Omega $ is a $C^2$ -hypersurface, and, given $X\\in C^\\infty _c(\\Omega ;\\mathbb {R}^n)$ , set $\\zeta =X\\cdot \\nu _E$ .", "If $P_K(E;\\Omega )<\\infty $ and $\\int _{\\partial E}(1+\|z\|)^{-n-s}\\,d{\\mathcal {H}}^{n-1}_z<\\infty $ , then $\\delta P_K(E;\\Omega )[X]&=&\\int _{\\partial E}{\\rm H}_{K,\\partial E}\\,\\zeta \\,d{\\mathcal {H}}^{n-1}\\,,\\\\\\nonumber \\delta ^2 P_K(E;\\Omega )[X]&=&\\iint _{\\partial E\\times \\partial E}K(x-y)\|\\zeta (x)-\\zeta (y)\|^2\\,d{\\mathcal {H}}^{n-1}_x\\,d{\\mathcal {H}}^{n-1}_y-\\int _{\\partial E}{\\rm c}^2_{K,\\partial E}\\,\\zeta ^2\\,d{\\mathcal {H}}^{n-1}\\\\&&+\\int _{\\partial E}{\\rm H}_{K,\\partial E}\\,\\Big (({\\rm div}X)\\,\\zeta -{\\rm div}_{\\tau }\\bigl (\\zeta \\,X_\\tau \\bigr )\\Big )\\,d{\\mathcal {H}}^{n-1}\\,.$ If $V_G(E;\\Omega )<\\infty $ and $\\int _E \|z\|^{-n+\\alpha }\\,dz<\\infty $ , then $\\nonumber \\delta V_G(E;\\Omega )[X]&=&\\int _{\\partial E}{\\rm H}^_{G,\\partial E}\\,\\zeta \\,d{\\mathcal {H}}^{n-1}\\,.\\\\\\nonumber \\delta ^2 V_G(E;\\Omega )[X]&=&-\\iint _{\\partial E\\times \\partial E}G(x-y)\|\\zeta (x)-\\zeta (y)\|^2\\,d{\\mathcal {H}}^{n-1}_x\\,d{\\mathcal {H}}^{n-1}_y+\\int _{\\partial E}{\\rm c}^2_{G,\\partial E}\\,\\zeta ^2\\,d{\\mathcal {H}}^{n-1}\\\\&&+\\int _{\\partial E}{\\rm H}^_{G,\\partial E}\\,\\Big (({\\rm div}X)\\,\\zeta -{\\rm div}_{\\tau }\\bigl (\\zeta \\,X_\\tau \\bigr )\\Big )\\, d{\\mathcal {H}}^{n-1}\\,.$ Remark 6.2 Let $E$ be as in Theorem REF .", "By arguing as in the deduction of (REF ) from (REF ) and (), we see that if $E$ is a volume-constrained stationary set for $P_K$ , then $\\delta ^2 P_K(E;\\Omega )[X]=\\iint _{\\partial E\\times \\partial E}K(x-y)\|\\zeta (x)-\\zeta (y)\|^2\\,d{\\mathcal {H}}^{n-1}_x\\,d{\\mathcal {H}}^{n-1}_y-\\int _{\\partial E}{\\rm c}^2_{K,\\partial E}\\,\\zeta ^2\\,d{\\mathcal {H}}^{n-1}\\,.$ whenever $X$ is volume-preserving on $E$ .", "Similarly, if $E$ is a volume-constrained stationary set for $V_G$ , then $\\delta ^2 V_G(E;\\Omega )[X]=-\\iint _{\\partial E\\times \\partial E}G(x-y)\|\\zeta (x)-\\zeta (y)\|^2\\,d{\\mathcal {H}}^{n-1}_x\\,d{\\mathcal {H}}^{n-1}_y+\\int _{\\partial E}{\\rm c}^2_{G,\\partial E}\\,\\zeta ^2\\,d{\\mathcal {H}}^{n-1}\\,,$ whenever $X$ is volume-preserving on $E$ .", "The fact that $\\partial E$ is of class $C^{1,1}$ guarantees that ${\\rm c}^2_{K,\\partial E}(x)\\in \\mathbb {R}$ for every $x\\in \\partial E$ .", "It also implies that $\\zeta =X\\cdot \\nu _E$ is a Lipschitz function, which in turn guarantees that the first-integral on the right-hand side of () converge.", "The convergence of ${\\rm c}^2_{G,\\partial E}$ and of the first integral on the right-hand side of () is trivial.", "In the next two propositions we address the continuity properties of ${\\rm H}_{K,\\partial E}$ and ${\\rm H}^_{G,\\partial E}$ .", "Proposition 6.3 If $s\\in (0,1)$ , $K\\in C^1(\\mathbb {R}^n\\setminus \\lbrace 0\\rbrace ;[0,\\infty ))$ is even and satisfies $K(z)\\le C_K/\|z\|^{n+s}$ for every $z\\in \\mathbb {R}^n\\setminus \\lbrace 0\\rbrace $ , $\\Omega $ and $E$ are open sets, and $\\partial E\\cap \\Omega $ is an hypersurface of class $C^{1,\\sigma }$ for some $\\sigma \\in (s,1)$ , then (REF ) defines a continuous real-valued function ${\\rm H}_{K,\\partial E}$ on $\\partial E\\cap \\Omega $ .", "Given $\\delta \\in [0,1/2)$ , let $\\eta _\\delta \\in C^\\infty ([0,\\infty );[0,1])$ be such that $\\eta _\\delta =1$ on $[0,\\delta )\\cup (1/\\delta ,\\infty )$ , $\\eta _\\delta =0$ on $[2\\delta ,1/2\\delta )$ , and $\|\\eta _\\delta ^{\\prime }\|\\le 2/\\delta $ on $[0,\\infty )$ , and $\\eta _\\delta (s)\\downarrow 0$ for every $s>0$ as $\\delta \\rightarrow 0^+$ .", "If we set $K_\\delta (z)=(1-\\eta _\\delta (\|z\|))\\,K(z)$ , $z\\in \\mathbb {R}^n$ , then $K_\\delta \\in C^1_c(\\mathbb {R}^n)\\subset L^1(\\mathbb {R}^n)$ , so that ${\\rm H}_{K_\\delta ,\\partial E}(x)=\\int _{E^c}K_\\delta (x-y)\\,dy-\\int _{E}K_\\delta (x-y)\\,dy\\,,\\qquad \\forall x\\in \\mathbb {R}^n\\,,$ and thus ${\\rm H}_{K_\\delta ,\\partial E}$ is a continuous function on $\\mathbb {R}^n$ for every $\\delta >0$ .", "In fact, we notice for future use that ${\\rm H}_{K_\\delta ,\\partial E}\\in C^1(\\mathbb {R}^n)$ , with $\\nabla {\\rm H}_{{K_{\\delta }},\\partial E}(x)=\\int _{E^c}\\nabla K_\\delta (x-y)\\,dy-\\int _{E}\\nabla K_\\delta (x-y)\\,dy\\,,\\qquad \\forall x\\in \\mathbb {R}^n\\,.$ Let us now decompose $x\\in \\mathbb {R}^n$ as $(x^{\\prime },x_n)\\in \\mathbb {R}^{n-1}\\times \\mathbb {R}$ , and set $C_r=\\Big \\lbrace x\\in \\mathbb {R}^n:\|x^{\\prime }\|<r\\,,\|x_n\|<r\\Big \\rbrace \\,,\\qquad P_{r,\\gamma }=\\Big \\lbrace x\\in C_r:\\gamma \\,\|x^{\\prime }\|^{1+\\sigma }<x_n\\Big \\rbrace \\,,$ for $r>0$ and $\\gamma >0$ .", "If $\\Omega ^{\\prime }\\subset \\subset \\Omega $ , then we can find $r>0$ and $\\gamma >0$ such that for every $x\\in \\partial E\\cap \\Omega ^{\\prime }$ there exists a rotation around the origin followed by a translation, denoted by $Q_x$ , such that $\\Big (C_r\\setminus P_{r,\\gamma }\\Big )\\cap \\lbrace x_n>0\\rbrace \\subset Q_x(E^c)\\,,\\qquad \\Big (C_r\\setminus P_{r,\\gamma }\\Big )\\cap \\lbrace x_n<0\\rbrace \\subset Q_x(E)\\,,$ see Figure REF .", "Figure: The sets defined in ().", "The region P r,γ P_{r,\\gamma } is that part of C r C_r encolosed by the graphs x n =±γ\|x ' \| 1+σ x_n=\\pm \\gamma \|x^{\\prime }\|^{1+\\sigma }.Provided $\\varepsilon <\\delta <2\\delta <r$ , we thus find that $&&\\Big \|\\int _{\\mathbb {R}^n\\setminus B(x,\\varepsilon )}(\\chi _{E^c}(y)-\\chi _{E}(y))\\,K(x-y)\\,dy-\\int _{\\mathbb {R}^n\\setminus B(x,\\varepsilon )}(\\chi _{E^c}(y)-\\chi _{E}(y))\\,K_\\delta (x-y)\\,dy\\Big \|\\\\&=&\\Big \|\\int _{E^c\\setminus B(x,\\varepsilon )}\\eta _\\delta (\|x-y\|)\\,K(x-y)\\,dy-\\int _{E\\setminus B(x,\\varepsilon )}\\eta _\\delta (\|x-y\|)\\,K(x-y)\\,dy\\Big \|\\\\&\\le &\\Big \|\\int _{(C_r\\cap E^c)\\setminus B(x,\\varepsilon )}\\eta _\\delta (\|x-y\|)\\,K(x-y)\\,dy-\\int _{(C_r\\cap E)\\setminus B(x,\\varepsilon )}\\eta _\\delta (\|x-y\|)\\,K(x-y)\\,dy\\Big \|\\\\&&+2\\,\\int _{\\mathbb {R}^n\\setminus B_{1/2\\delta }}K(z)\\,dz\\\\&\\le &\\int _{Q_x^{-1}(P_{r,\\gamma })\\setminus B(x,\\varepsilon )}\\eta _\\delta (\|x-y\|)\\,K(x-y)\\,dy+2\\,\\int _{\\mathbb {R}^n\\setminus B_{1/2\\delta }}K(z)\\,dz\\,,$ where in the last inequality we have used (REF ) and the symmetry of $K$ to cancel out opposite contributions from the points in $E^c$ and in $E$ lying in $Q_x^{-1}(C_r\\setminus P_{r,\\gamma })$ .", "We now notice that $\\omega (\\delta )&:=&\\int _{Q_x^{-1}(P_{r,\\gamma })\\setminus B(x,\\varepsilon )}\\eta _\\delta (\|x-y\|)\\,K(x-y)\\,dy\\\\&\\le &\\int _{Q_x^{-1}(P_{r,\\gamma })}\\eta _\\delta (\|x-y\|)\\,K(x-y)\\,dy\\\\&=&\\int _{\|z^{\\prime }\|<r}dz^{\\prime }\\int _{-\\gamma \\,\|z^{\\prime }\|^{1+\\sigma }}^{\\gamma \\,\|z^{\\prime }\|^{1+\\sigma }}\\eta _\\delta (\|z\|)\\,K(z)\\,dz_n\\\\&\\le &C_K\\,\\int _{\|z^{\\prime }\|<r}dz^{\\prime }\\int _{-\\gamma \\,\|z^{\\prime }\|^{1+\\sigma }}^{\\gamma \\,\|z^{\\prime }\|^{1+\\sigma }}\\frac{dz_n}{(\|z^{\\prime }\|^2+\|z_n\|^2)^{(n+s)/2}}\\,.$ Since $\\eta _\\delta (z)\\rightarrow 0$ for every $z\\in \\mathbb {R}^n\\setminus \\lbrace 0\\rbrace $ as $\\delta \\rightarrow 0^+$ , and since $\\int _{\|z^{\\prime }\|<r}dz^{\\prime }\\int _{-\\gamma \\,\|z^{\\prime }\|^{1+\\sigma }}^{\\gamma \\,\|z^{\\prime }\|^{1+\\sigma }}\\frac{dz_n}{(\|z^{\\prime }\|^2+\|z_n\|^2)^{(n+s)/2}}<\\infty \\,,$ we conclude that $\\omega (\\delta )\\rightarrow 0$ as $\\delta \\rightarrow 0$ (with a velocity that depends on $C_K$ , $s$ , $r$ , $\\gamma $ and $\\sigma $ only).", "Since $\\int _{\\mathbb {R}^n\\setminus B_{1/2\\delta }}K(z)\\,dz\\rightarrow 0$ as $\\delta \\rightarrow 0^+$ (with a velocity that depends on $C_K$ and $s$ only), we conclude that, if $\\omega _0(\\delta )=\\omega (\\delta )+2\\int _{\\mathbb {R}^n\\setminus B_{1/2\\delta }}K(z)\\,dz$ , then $\\Big \|\\int _{\\mathbb {R}^n\\setminus B(x,\\varepsilon )}(\\chi _{E^c}(y)-\\chi _{E}(y))\\,K(x-y)\\,dy-\\int _{\\mathbb {R}^n\\setminus B(x,\\varepsilon )}(\\chi _{E^c}(y)-\\chi _{E}(y))\\,K_\\delta (x-y)\\,dy\\Big \|\\le \\omega _0(\\delta )\\,,$ for every $x\\in \\partial E\\cap \\Omega ^{\\prime }$ and every $\\varepsilon <\\delta <2\\delta <r$ .", "We thus conclude that ${\\rm H}_{K,\\partial E}(x)\\in \\mathbb {R}$ for every $x\\in \\partial E\\cap \\Omega ^{\\prime }$ , and that ${\\rm H}_{K_\\delta ,\\partial E}\\rightarrow {\\rm H}_{K,\\partial E}$ uniformly on $\\partial E\\cap \\Omega ^{\\prime }$ .", "In particular, ${\\rm H}_{K,\\partial E}$ is real-valued and continuous on $\\partial E\\cap \\Omega $ .", "Since the function $z \\mapsto \|z\|^{-n+\\alpha }$ belongs to $L^1_{\\rm loc}(\\mathbb {R}^n)$ , we also have the following result: Proposition 6.4 If $G\\in C^1(\\mathbb {R}^n\\setminus \\lbrace 0\\rbrace ;[0,\\infty ))$ is even and satisfies (REF ) for some $\\alpha \\in (0,n)$ and $\\int _E \|z\|^{-n+\\alpha }\\,dz<\\infty $ (this is the case for instance if $E$ is bounded), then () defines a continuous real-valued function ${\\rm H}^_{G,\\partial E}$ on $\\mathbb {R}^n$ .", "We shall detail the proof of the theorem only in the case of $P_K$ , being the discussion for $V_G$ similar.", "We denote by $\\varepsilon $ the positive number such that $\\lbrace \\Phi _t\\rbrace _{\|t\|<\\varepsilon }$ is a smooth family of diffeomorphisms of $\\mathbb {R}^n$ .", "Step one: Given $\\delta \\ge 0$ , we define ${K_{\\delta }}$ as in the proof of Proposition REF .", "Our goal here is proving (REF ) and () with $K_\\delta $ in place of $K$ .", "We first claim that ${\\rm H}_{{K_{\\delta }},\\partial E}\\in C^1(\\mathbb {R}^n)$ , and that $\\nabla {\\rm H}_{{K_{\\delta }},\\partial E}$ can be expressed both as in (REF ) and as in (REF ) below.", "Since $E$ is an open set with Lipschitz boundary and $K_\\delta \\in C^1_c(\\mathbb {R}^n)$ , by the Gauss–Green theorem, the symmetry of $K_\\delta $ , and (REF ), we find that $\\nabla {\\rm H}_{{K_{\\delta }},\\partial E}(x)=2\\,\\int _{\\partial E}K_{\\delta }(y-x)\\,\\nu _E(y)\\,d{\\mathcal {H}}^{n-1}_y\\,,\\qquad \\forall x\\in \\mathbb {R}^n\\,.$ We now notice that, since $E_{t+h}=\\Phi _h(E_t)$ , by the area formula we get, whenever $\|t\|<\\varepsilon $ and $\|t+h\|<\\varepsilon $ , $P_{K_{\\delta }}(E_{t+h},\\Omega )=\\int _{E_t\\cap \\Omega }\\int _{E^c_t\\cap \\Omega }{K_{\\delta }}(\\Phi _h(x)-\\Phi _h(y))J_{\\Phi _h}(x)J_{\\Phi _h}(y)\\,dx\\,dy\\\\+\\int _{E_t\\cap \\Omega }\\int _{E^c_t\\setminus \\Omega }{K_{\\delta }}(\\Phi _h(x)-y)J_{\\Phi _h}(x)\\,dx\\,dy+\\int _{E_t\\setminus \\Omega }\\int _{E^c_t\\cap \\Omega }{K_{\\delta }}(x-\\Phi _h(y))J_{\\Phi _h}(y)\\,dx\\,dy\\,,$ where $J_{\\Phi _h}$ stands for the Jacobian of the map $\\Phi _h$ .", "Since $\\Phi _h={\\rm Id}\\,+h\\,X+O(h^2)$ and $J_{\\Phi _h}=1+h\\,{\\rm div}X+O(h^2)$ uniformly on $\\mathbb {R}^n$ as $h\\rightarrow 0$ , we deduce from $\\frac{d }{dt}P_{K_{\\delta }}(E_t,\\Omega )= \\frac{d }{dh}P_{K_{\\delta }}(E_{t+h},\\Omega )_{\\bigl \|h=0}\\,,$ and by the smoothness of ${K_{\\delta }}$ that $\\frac{d }{dt}P_{K_{\\delta }}(E_t,\\Omega )&=\\int _{E_t\\cap \\Omega }\\int _{E^c_t\\cap \\Omega }\\nabla {K_{\\delta }}(x-y)\\cdot (X(x)-X(y))\\,dx\\,dy \\nonumber \\\\&\\qquad +\\int _{E_t\\cap \\Omega }\\int _{E^c_t\\cap \\Omega } {K_{\\delta }}(x-y)({\\rm div}X(x)+{\\rm div}X(y))\\,dx\\,dy \\nonumber \\\\&\\qquad +\\int _{E_t\\cap \\Omega }\\int _{E^c_t\\setminus \\Omega }\\Big (\\nabla {K_{\\delta }}(x-y)\\cdot X(x)+{K_{\\delta }}(x-y){\\rm div}X(x)\\Big ) \\,dx\\,dy\\nonumber \\\\&\\qquad +\\int _{E_t\\setminus \\Omega }\\int _{E^c_t\\cap \\Omega }\\Big (\\nabla {K_{\\delta }}(x-y)\\cdot X(y)+{K_{\\delta }}(x-y){\\rm div}X(y)\\Big ) \\,dx\\,dy\\nonumber \\\\ &=I_1+I_2+I_3+I_4\\,.\\nonumber $ By symmetry of ${K_{\\delta }}$ and by the divergence theorem, we find $I_1 &=\\displaystyle \\int _{E_t^c\\cap \\Omega }\\biggl (\\int _{E_t}\\nabla {K_{\\delta }}(x-y)\\cdot X(x)\\,dx\\biggr )\\,dy+\\displaystyle \\int _{E_t\\cap \\Omega }\\biggl (\\int _{E_t^c}\\nabla {K_{\\delta }}(y-x)\\cdot X(y)\\,dy\\biggr )\\,dx \\\\&=-\\displaystyle \\int _{E_t^c\\cap \\Omega }\\biggl (\\int _{E_t} {K_{\\delta }}(x-y){\\rm div}X(x)\\,dx\\biggr )\\,dy+ \\displaystyle \\int _{E_t^c\\cap \\Omega }\\biggl (\\int _{\\partial E_t} {K_{\\delta }}(x-y)X(x)\\cdot \\nu _{E_t}(x)\\,d{\\mathcal {H}}^{n-1}_x\\biggr )\\,dy\\\\& \\quad -\\displaystyle \\int _{E_t\\cap \\Omega }\\biggl (\\int _{E_t^c} {K_{\\delta }}(x-y){\\rm div}X(y)\\,dy\\biggr )\\,dx-\\displaystyle \\int _{E_t\\cap \\Omega }\\biggl (\\int _{\\partial E_t} {K_{\\delta }}(x-y)X(y)\\cdot \\nu _{E_t}(y)\\,d{\\mathcal {H}}^{n-1}_y\\biggr )\\,dx\\,,$ which leads to $I_1+I_2=\\!\\displaystyle \\int _{E_t^c\\cap \\Omega }\\biggl (\\int _{\\partial E_t}\\!", "{K_{\\delta }}(x-y)X(x)\\cdot \\nu _{E_t}(x)d{\\mathcal {H}}^{n-1}_x\\!\\biggr )dy-\\displaystyle \\int _{E_t\\cap \\Omega }\\biggl (\\int _{\\partial E_t}\\!", "{K_{\\delta }}(x-y)X(y)\\cdot \\nu _{E_t}(y)d{\\mathcal {H}}^{n-1}_y\\!\\biggr )dx\\,.$ Similarly, we get that $I_3&=\\displaystyle \\int _{E_t^c\\setminus \\Omega }\\biggl (\\int _{\\partial E_t} {K_{\\delta }}(x-y)X(x)\\cdot \\nu _{E_t}(x)\\,d{\\mathcal {H}}^{n-1}_x\\biggr )\\,dy,\\\\I_4&=-\\displaystyle \\int _{E_t\\setminus \\Omega }\\biggl (\\int _{\\partial E_t} {K_{\\delta }}(x-y)X(y)\\cdot \\nu _{E_t}(y)\\,d{\\mathcal {H}}^{n-1}_y\\biggr )\\,dx\\,.$ By exploiting once more the symmetry of ${K_{\\delta }}$ we thus conclude that (for every $t$ small enough) $\\frac{d }{dt}P_{K_{\\delta }}(E_t,\\Omega )=\\int _{\\partial E_t}\\,{\\rm H}_{{K_{\\delta }},\\partial E_t}\\,(X\\cdot \\nu _{E_t})\\,d{\\mathcal {H}}^{n-1}\\,,$ which of course implies (REF ) with $K_\\delta $ in place of $K$ by setting $t=0$ .", "Having in mind to differentiate (REF ), we now notice that, by the area formula, $\\int _{\\partial E_t}\\,{\\rm H}_{{K_{\\delta }},\\partial E_t}\\,(X\\cdot \\nu _{E_t})\\,d{\\mathcal {H}}^{n-1}=\\int _{\\partial E}\\,{\\rm H}_{{K_{\\delta }},\\partial E_t}(\\Phi _t)\\,(X(\\Phi _t)\\cdot \\nu _{E_t}(\\Phi _t))\\,J_{\\Phi _t}^{\\partial E}\\,d{\\mathcal {H}}^{n-1}\\,,$ where $J_{\\Phi _t}^{\\partial E}$ denotes the tangential Jacobian of $\\Phi _t$ with respect to $\\partial E$ .", "Therefore, $\\frac{d^2 }{dt^2}P_{K_{\\delta }}(E_t, \\Omega )_{\\bigl \|t=0}&=& \\frac{d}{dt}\\biggl (\\int _{\\partial E_t}{\\rm H}_{{K_{\\delta }},\\partial E_t}\\,(X\\cdot \\nu _{E_t})\\,d{\\mathcal {H}}^{n-1}\\biggr )_{\\bigl \|t=0}\\\\&=&\\int _{\\partial E}\\frac{d}{dt}\\bigl ({\\rm H}_{{K_{\\delta }},\\partial E_t}(\\Phi _t)\\bigr )_{\\bigl \|t=0}\\,(X\\cdot \\nu _E)\\,d{\\mathcal {H}}^{n-1}\\nonumber \\\\&&+\\int _{\\partial E}{\\rm H}_{{K_{\\delta }},\\partial E}\\,\\frac{d}{dt}\\bigl (X(\\Phi _t)\\cdot \\nu _{E_t}(\\Phi _t))\\,J^{\\partial E_t}_{\\Phi _t}\\bigr )_{\\bigl \|t=0}\\,d{\\mathcal {H}}^{n-1}\\nonumber \\\\&=&J_1+J_2\\,.", "\\nonumber $ In order to compute $J_1$ we begin noticing that, by the area formula and since ${K_{\\delta }}\\in L^1(\\mathbb {R}^n)$ , ${\\rm H}_{{K_{\\delta }},\\partial E_t}(\\Phi _t(x))=\\int _{\\mathbb {R}^n}\\bigl (\\chi _{E^c}(y)-\\chi _E(y)\\bigr ){K_{\\delta }}(\\Phi _t(x)-\\Phi _t(y))\\,J_{\\Phi _t}(y)\\,dy\\,.$ By symmetry and smoothness of ${K_{\\delta }}$ , by the Taylor's expansions in $t$ of $\\Phi _t$ and $J_{\\Phi _t}$ mentioned above, by recalling that ${\\rm H}_{{K_{\\delta }},\\partial E}\\in C^1(\\mathbb {R}^n)$ and (REF ), and by the divergence theorem, we get $&&\\frac{d}{dt}\\bigl ({\\rm H}_{{K_{\\delta }},\\partial E_t}(\\Phi _t(x))\\bigr )_{\\bigl \|t=0}=\\int _{\\mathbb {R}^n}(\\chi _{E^c}(y)-\\chi _E(y))\\nabla {K_{\\delta }}(x-y)\\cdot (X(x)-X(y))\\,dy \\\\&& \\qquad \\qquad \\qquad \\qquad \\qquad \\quad +\\int _{\\mathbb {R}^n}(\\chi _{E^c}(y)-\\chi _E(y)){K_{\\delta }}(x-y){\\rm div}X(y)\\,dy\\nonumber \\\\&& \\qquad \\qquad =\\nabla {\\rm H}_{{K_{\\delta }},\\partial E}(x)\\cdot X(x)+\\int _{E^c}\\nabla {K_{\\delta }}(y-x)\\cdot X(y)\\,dy-\\int _{E}\\nabla {K_{\\delta }}(y-x)\\cdot X(y)\\,dy\\nonumber \\\\&& \\qquad \\qquad \\qquad \\qquad \\qquad \\quad +\\int _{\\mathbb {R}^n}(\\chi _{E^c}(y)-\\chi _E(y)){K_{\\delta }}(x-y){\\rm div}X(y)\\,dy \\nonumber \\\\&& \\qquad \\qquad =\\nabla {\\rm H}_{{K_{\\delta }},\\partial E}(x)\\cdot X(x)-2\\int _{\\partial E}{K_{\\delta }}(x-y)X(y)\\cdot \\nu _E(y)\\,dy\\,.", "\\nonumber $ By this last identity and by the symmetry of ${K_{\\delta }}$ , setting $\\zeta =X\\cdot \\nu _E$ we find that $\\nonumber J_1&=&-2\\iint _{\\partial E\\times \\partial E}{K_{\\delta }}(x-y)\\,\\zeta (x)\\,\\zeta (y)\\,d{\\mathcal {H}}^{n-1}_x\\,d{\\mathcal {H}}^{n-1}_y+\\int _{\\partial E}\\big (\\nabla {\\rm H}_{{K_{\\delta }},\\partial E}\\cdot X\\big )\\,\\zeta \\,d{\\mathcal {H}}^{n-1}\\\\&=& \\iint _{\\partial E\\times \\partial E}{K_{\\delta }}(x-y)\|\\zeta (x)-\\zeta (y)\|^2\\,d{\\mathcal {H}}^{n-1}_x\\,d{\\mathcal {H}}^{n-1}_y-2\\iint _{\\partial E\\times \\partial E} {K_{\\delta }}(x-y)\\,\\zeta (x)^2\\,d{\\mathcal {H}}^{n-1}_x\\,d{\\mathcal {H}}^{n-1}_y \\nonumber \\\\&& \\qquad +\\int _{\\partial E}\\big (\\nabla {\\rm H}_{{K_{\\delta }},\\partial E}\\cdot \\nu _E\\big )\\,\\zeta ^2\\,d{\\mathcal {H}}^{n-1}+\\int _{\\partial E}\\big (\\nabla _\\tau {\\rm H}_{{K_{\\delta }},\\partial E}\\cdot X_\\tau \\big )\\,\\zeta \\,d{\\mathcal {H}}^{n-1}\\,, $ where in the last identities we have simply completed a square and used the identity $X=\\zeta \\,\\nu +X_\\tau $ .", "By (REF ) we also get $\\nabla {\\rm H}_{{K_{\\delta }},\\partial E}(x)\\cdot \\nu _E(x)&=&-\\int _{\\partial E} {K_{\\delta }}(x-y)\|\\nu _E(x)-\\nu _E(y)\|^2\\,d{\\mathcal {H}}^{n-1}_y+2\\int _{\\partial E} {K_{\\delta }}(x-y)\\,d{\\mathcal {H}}^{n-1}_y\\\\&=&-{\\rm c}^2_{{K_{\\delta }},\\partial E}(x)+2\\int _{\\partial E} {K_{\\delta }}(x-y)\\,d{\\mathcal {H}}^{n-1}_y\\,,$ and thus we conclude from (REF ) that $\\nonumber J_1&=&\\iint _{\\partial E\\times \\partial E}{K_{\\delta }}(x-y)\\,\|\\zeta (x)-\\zeta (y)\|^2\\,d{\\mathcal {H}}^{n-1}_x\\,d{\\mathcal {H}}^{n-1}_y-\\int _{\\partial E}\\,{\\rm c}^2_{{K_{\\delta }},\\partial E}\\,\\zeta ^2\\,d{\\mathcal {H}}^{n-1}\\\\&&+\\int _{\\partial E}\\big (\\nabla _\\tau {\\rm H}_{{K_{\\delta }},\\partial E}\\cdot X_\\tau \\big )\\,\\zeta \\,d{\\mathcal {H}}^{n-1}\\,.$ In order to compute $J_2$ , we notice that, by arguing as in [9] (see also [36]), one finds $\\frac{d}{dt}\\Big (X(\\Phi _t)\\cdot \\nu _{E_t}(\\Phi _t)J^{\\partial E}_{\\Phi _t}\\Big )_{\\bigl \|t=0}=Z\\cdot \\nu _E-2X_\\tau \\cdot \\nabla _\\tau \\zeta +{\\rm B}_{\\partial E}[X_\\tau ,X_\\tau ]+{\\rm div}_\\tau \\bigl (\\zeta \\,X\\bigr )\\,,$ where $Z$ is the vector field defined by $Z(x)=\\partial _{tt}^2\\Phi _t(x)_{\\bigl \|t=0}\\,,\\qquad x\\in \\mathbb {R}^n\\,,$ and where ${\\rm B}_{\\partial E}$ denotes the second fundamental form of $\\partial E$ .", "Hence, $J_2&=&\\int _{\\partial E}{\\rm H}_{{K_{\\delta }},\\partial E}\\Big (Z\\cdot \\nu _E-2X_\\tau \\cdot \\nabla _\\tau \\zeta +{\\rm B}_{\\partial E}[X_\\tau ,X_\\tau ]\\Big )\\,d{\\mathcal {H}}^{n-1}\\\\\\nonumber &&+\\int _{\\partial E}{\\rm H}_{{K_{\\delta }},\\partial E}\\,{\\rm div}_\\tau \\bigl (\\zeta \\,X\\bigr )\\,d{\\mathcal {H}}^{n-1}\\,.$ By the tangential divergence theorem $\\int _{\\partial E}{\\rm div}_\\tau Y\\,d{\\mathcal {H}}^{n-1}=\\int _{\\partial E}Y\\cdot \\nu _E\\,{\\rm H}_E\\,d{\\mathcal {H}}^{n-1}\\qquad \\forall Y\\in C^1_c(\\Omega ;\\mathbb {R}^n)$ (recall that ${\\rm H}_{\\partial E}$ denotes the scalar mean curvature of $\\partial E$ taken with respect to $\\nu _E$ ), so that the sum of the second lines of (REF ) and (REF ) is equal to $\\int _{\\partial E}{\\rm H}_{{K_{\\delta }},\\partial E}\\,{\\rm div}_\\tau \\bigl (\\zeta \\,X\\bigr )\\,d{\\mathcal {H}}^{n-1}+\\int _{\\partial E}\\big (\\nabla _\\tau {\\rm H}_{{K_{\\delta }},\\partial E}\\cdot X_\\tau \\big )\\,\\zeta \\,d{\\mathcal {H}}^{n-1}\\\\=\\int _{\\partial E}{\\rm div}_\\tau \\bigl ({\\rm H}_{{K_{\\delta }},\\partial E}\\zeta \\,X\\bigr )\\,d{\\mathcal {H}}^{n-1}=\\int _{\\partial E}\\,{\\rm H}_{{K_{\\delta }},\\partial E}\\,{\\rm H}_{\\partial E}\\,\\zeta ^2\\,d{\\mathcal {H}}^{n-1}\\,.$ We thus deduce from (REF ), (REF ), and (REF ), that $\\frac{d^2 }{dt^2}P_{K_{\\delta }}(E_t, \\Omega )_{\\bigl \|t=0}=\\iint _{\\partial E\\times \\partial E}{K_{\\delta }}(x-y)\\,\|\\zeta (x)-\\zeta (y)\|^2\\,d{\\mathcal {H}}^{n-1}_x\\,d{\\mathcal {H}}^{n-1}_y-\\int _{\\partial E}\\,{\\rm c}^2_{{K_{\\delta }},\\partial E}\\,\\zeta ^2\\,d{\\mathcal {H}}^{n-1}\\\\+\\int _{\\partial E}{\\rm H}_{{K_{\\delta }},\\partial E}\\Big (Z\\cdot \\nu _E-2X_\\tau \\cdot \\nabla _\\tau \\zeta +{\\rm B}_{\\partial E}[X_\\tau ,X_\\tau ]+{\\rm H}_{\\partial E}\\,\\zeta ^2\\Big )\\,d{\\mathcal {H}}^{n-1}\\,.$ By exploiting the identity $Z\\cdot \\nu _E-2X_\\tau \\cdot \\nabla _\\tau \\zeta +{\\rm B}_{\\partial E}[X_\\tau ,X_\\tau ]+{\\rm H}_{\\partial E}\\zeta ^2=-{\\rm div}_{\\tau }\\bigl (\\zeta \\,X_\\tau )+({\\rm div}X)\\,\\zeta $ (see, for example, [1]), we thus come to prove () with $K_\\delta $ in place of $K$ .", "Step two: We now prove (REF ) and () by taking the limit as $\\delta \\rightarrow 0^+$ in (REF ) and () with $K_\\delta $ in place of $K$ .", "Let us set $\\varphi _\\delta (t):=P_{K_\\delta }(E_t;\\Omega )$ and $\\varphi (t):=P_K(E_t;\\Omega )$ , so that $\\varphi _\\delta $ and $\\varphi $ are smooth functions on $(-\\varepsilon ,\\varepsilon )$ with $\\lim _{\\delta \\rightarrow 0^+}\\varphi _\\delta (t)=\\varphi (t)\\,,\\qquad \\forall \|t\|<\\varepsilon \\,.$ (This follows by monotone convergence, as $\\eta _\\delta \\downarrow 0^+$ as $\\delta \\rightarrow 0^+$ on $(0,\\infty )$ .)", "Let $\\Omega ^{\\prime }\\subset \\subset \\Omega $ be an open set such that ${\\rm spt}X\\subset \\subset \\Omega ^{\\prime }$ .", "Thanks to the smoothness of $\\lbrace \\Phi _t\\rbrace _{\|t\|<\\varepsilon }$ , the argument in the proof of Proposition REF can be repeated for every set $E_t=\\Phi _t(E)$ corresponding to $\|t\|<\\varepsilon $ with the same constants $r$ and $\\gamma $ , thus showing that $\\lim _{\\delta \\rightarrow 0^+}\\sup _{\|t\|<\\varepsilon }\\sup _{\\partial E_t\\cap \\Omega ^{\\prime }}\|{\\rm H}_{K_\\delta ,\\partial E_t}-{\\rm H}_{K,\\partial E_t}\|=0\\,.$ At the same time, by step one, $\\varphi _\\delta ^{\\prime }(t)=\\int _{\\partial E_t}{\\rm H}_{K_\\delta ,\\partial E_t}\\,\\zeta \\,d{\\mathcal {H}}^{n-1}\\,,\\qquad \\forall \|t\|<\\varepsilon \\,,$ so that (REF ) and (REF ) imply that $\\lim _{\\delta \\rightarrow 0^+}\\sup _{\|t\|<\\varepsilon }\\Big \|\\varphi _\\delta ^{\\prime }(t)-\\int _{\\partial E_t}\\,{\\rm H}_{K,\\partial E_t}\\,\\zeta \\,d{\\mathcal {H}}^{n-1}\\Big \|=0\\,.$ By the mean value theorem, (REF ) and (REF ) give $\\varphi ^{\\prime }(t)=\\int _{\\partial E_t}{\\rm H}_{K,\\partial E_t}\\,\\zeta \\,d{\\mathcal {H}}^{n-1}\\,,\\qquad \\forall \|t\|<\\varepsilon \\,,$ which implies (REF ) for $t=0$ .", "In order to prove (), we first notice that, by step one, $\\nonumber \\varphi _\\delta ^{\\prime \\prime }(t)&=&\\iint _{\\partial E_t\\times \\partial E_t}K_\\delta (x-y)\|\\zeta (x)-\\zeta (y)\|^2\\,d{\\mathcal {H}}^{n-1}_x\\,d{\\mathcal {H}}^{n-1}_y-\\int _{\\partial E_t}{\\rm c}^2_{K_\\delta ,\\partial E_t}\\,\\zeta ^2\\,d{\\mathcal {H}}^{n-1}\\\\&&+\\int _{\\partial E_t}{\\rm H}_{K_\\delta ,\\partial E_t}\\,\\Big (({\\rm div}X)\\,\\zeta -{\\rm div}_{\\tau }\\bigl (\\zeta \\,X_\\tau \\bigr )\\Big )\\,d{\\mathcal {H}}^{n-1}\\,,\\qquad \\forall \|t\|<\\varepsilon \\,.$ Let $A_1(t,\\delta )$ , $A_2(t,\\delta )$ and $A_3(t,\\delta )$ denote the three integrals on the right-hand side of (REF ), and let $A_1(t)$ , $A_2(t)$ and $A_3(t)$ stand for the corresponding integrals obtained by replacing $K_\\delta $ with $K$ .", "By arguing as above, we just need to prove that for $i=1,2,3$ we have $A_i(t,\\delta )\\rightarrow A_i(t)$ uniformly on $\|t\|<\\varepsilon $ as $\\delta \\rightarrow 0^+$ .", "The fact that $A_3(t,\\delta )\\rightarrow A_3(t)$ uniformly on $\|t\|<\\varepsilon $ as $\\delta \\rightarrow 0^+$ follows from (REF ) and of the smoothness of $X$ .", "Finally, when $i=1,2$ , the uniform convergence of $A_i(t,\\delta )\\rightarrow A_i(t)$ for $\|t\|<\\varepsilon $ as $\\delta \\rightarrow 0^+$ is a simple consequence of the fact that $\\zeta $ is Lipschitz and compactly supported in $\\Omega ^{\\prime }$ , and that $\\lbrace \\Omega ^{\\prime }\\cap \\partial E_t\\rbrace _{\|t\|<\\varepsilon }$ is a uniform family of $C^2$ -hypersurfaces.", "This completes the proof of the theorem." ], [ "The stability threshold", "In this section we consider the family of functionals ${\\rm Per}_s+\\beta \\,V_\\alpha $ ($\\beta >0$ ) and discuss in terms of the value of $\\beta $ the volume-constrained stability of ${\\rm Per}_s+\\beta \\,V_\\alpha $ around the unit ball $B$ .", "Our interest in this problem lies in the fact that, as we shall prove in section , stability is actually a necessary and sufficient condition for volume-constrained local minimality.", "Therefore the analysis carried on in this section will provide the basis for the proof of Theorem REF .", "We set $\\beta _\\star (n,s,\\alpha ):={\\left\\lbrace \\begin{array}{ll}\\displaystyle \\frac{1-s}{\\omega _{n-1}}\\inf _{k\\ge 2}\\, \\frac{\\lambda _k^s-\\lambda _1^s}{\\mu _k^\\alpha -\\mu _1^\\alpha }\\,, &\\hspace{28.45274pt} \\text{if $s\\in (0,1)$}\\,,\\\\[10pt]\\displaystyle \\inf _{k\\ge 2} \\,\\frac{\\lambda _k^1-\\lambda _1^1}{\\mu _k^\\alpha -\\mu _1^\\alpha }\\,, &\\hspace{28.45274pt} \\text{if $s=1$}\\,,\\end{array}\\right.", "}$ where, for every $k\\in \\mathbb {N}\\cup \\lbrace 0\\rbrace $ , $\\lambda _k^1&=&k(k+n-2)\\,,\\\\\\lambda _k^s&=&\\frac{2^{1-s}\\,\\pi ^{\\frac{n-1}{2}}}{1+s}\\,\\frac{\\Gamma (\\frac{1-s}{2})}{\\Gamma (\\frac{n+s}{2})}\\,\\bigg (\\frac{\\Gamma (k+\\frac{n+s}{2})}{\\Gamma (k+\\frac{n-2-s}{2})}-\\frac{\\Gamma (\\frac{n+s}{2})}{\\Gamma (\\frac{n-2-s}{2})}\\bigg )\\,,\\qquad \\hspace{5.69046pt}s\\in (0,1)\\,,\\\\\\mu _k^\\alpha &=&\\frac{2^{1+\\alpha }\\,\\pi ^{\\frac{n-1}{2}}}{1-\\alpha }\\,\\frac{\\Gamma (\\frac{1+\\alpha }{2})}{\\Gamma (\\frac{n-\\alpha }{2})}\\,\\bigg (\\frac{\\Gamma (k+\\frac{n-\\alpha }{2})}{\\Gamma (k+\\frac{n-2+\\alpha }{2})}-\\frac{\\Gamma (\\frac{n-\\alpha }{2})}{\\Gamma (\\frac{n-2+\\alpha }{2})}\\bigg )\\,,\\qquad \\alpha \\in (0,1)\\,,\\\\\\mu _k^\\alpha &=&2^\\alpha \\,\\pi ^{\\frac{n-1}{2}}\\,\\frac{\\Gamma (\\frac{\\alpha -1}{2})}{\\Gamma (\\frac{n-\\alpha }{2})}\\,\\bigg (\\frac{\\Gamma (\\frac{n-\\alpha }{2})}{\\Gamma (\\frac{n-2+\\alpha }{2})}-\\frac{\\Gamma (k+\\frac{n-\\alpha }{2})}{\\Gamma (k+\\frac{n-2+\\alpha }{2})}\\bigg )\\,,\\qquad \\hspace{11.38092pt}\\alpha \\in (1,n)\\,,\\\\\\mu _k^1&=&\\frac{4\\,\\pi ^{\\frac{n-1}{2}}}{\\Gamma (\\frac{n-1}{2})}\\,\\left(\\frac{\\Gamma ^\\prime (k+\\frac{n-1}{2})}{\\Gamma (k+\\frac{n-1}{2})} - \\frac{\\Gamma ^\\prime (\\frac{n-1}{2})}{\\Gamma (\\frac{n-1}{2})} \\right)\\,.$ Here $\\Gamma $ denotes the Euler's Gamma function, while $\\Gamma ^{\\prime }$ is the derivative of $\\Gamma $ , so that $\\Gamma ^{\\prime }/\\Gamma $ is the digamma function.", "By exploiting basic properties of the Gamma function, it is straightforward to check that $\\lambda _k^s/\\mu _k^\\alpha \\rightarrow \\infty $ as $k\\rightarrow \\infty $ , so that the infimum in (REF ) is achieved, and $\\beta _\\star >0$ .", "We shall actually prove that the infimum is always achieved at $k=2$ and the formula for $\\beta _\\star $ considerably simplifies (see Proposition REF ).", "Theorem 7.1 The unit ball $B$ is a volume-constrained stable set for ${\\rm Per}_s+\\beta \\,V_\\alpha $ if and only if $\\beta \\in (0,\\beta _\\star ]$ .", "Let us first of all explain the origin of the formula (REF ) for $\\beta _\\star $ .", "Since $B$ is a volume-constrained stationary set for $P$ , $P_s$ , and $V_\\alpha $ (indeed, $B$ is a global volume-constrained minimizer of $P$ and $P_s$ , and a global volume-constrained maximizer of $V_\\alpha $ ), by Remark REF we find that (setting $K_s(z)=\|z\|^{-(n+s)}$ and $G_\\alpha (z)=\|z\|^{-(n-\\alpha )}$ for every $z\\in \\mathbb {R}^n\\setminus \\lbrace 0\\rbrace $ ) $\\delta ^2P(B)[X]&=&\\iint _{\\partial B}\|\\nabla _\\tau \\zeta \|^2 d{\\mathcal {H}}^{n-1}-\\int _{\\partial B}{\\rm c}^2_{\\partial B}\\,\\zeta ^2\\,d{\\mathcal {H}}^{n-1}\\,,\\\\\\delta ^2P_s(B)[X]&=&\\iint _{\\partial B\\times \\partial B}\\frac{\|\\zeta (x)-\\zeta (y)\|^2}{\|x-y\|^{n+s}}\\,d{\\mathcal {H}}^{n-1}_x\\,d{\\mathcal {H}}^{n-1}_y-\\int _{\\partial B}{\\rm c}^2_{K_s,\\partial B}\\,\\zeta ^2\\,d{\\mathcal {H}}^{n-1}\\,,\\\\\\delta ^2 V_\\alpha (B)[X]&=&-\\iint _{\\partial B\\times \\partial B}\\frac{\|\\zeta (x)-\\zeta (y)\|^2}{\|x-y\|^{n-\\alpha }}\\,d{\\mathcal {H}}^{n-1}_x\\,d{\\mathcal {H}}^{n-1}_y+\\int _{\\partial B}{\\rm c}^2_{G_\\alpha ,\\partial B}\\,\\zeta ^2\\,d{\\mathcal {H}}^{n-1}\\,,$ for every $X$ inducing a volume-preserving flow on $B$ (here, $\\zeta =X\\cdot \\nu _B$ ).", "The reason why we are able to discuss the volume-constrained stability of ${\\rm Per}_s+\\beta \\,V_\\alpha $ at $B$ is that the Sobolev semi-norms $[u]_{H^1(\\partial B)}$ , $[u]_{H^{(1+s)/2}(\\partial B)}$ , and $[u]_{H^{(1-\\alpha )/2}(\\partial B)}$ , can all be decomposed in terms of the Fourier coefficients of $u$ with respect to a orthonormal basis of spherical harmonics.", "Indeed, recalling our notation $\\lbrace Y_k^i\\rbrace _{i=1}^{d(k)}$ for an orthonormal basis in $L^2(\\partial B)$ of the space $\\mathcal {S}_k$ of spherical harmonics of degree $k$ , we have proved in (REF ) that $\\iint _{\\partial B\\times \\partial B}\\frac{\|u(x)-u(y)\|^2}{\|x-y\|^{n+s}}\\,d{\\mathcal {H}}^{n-1}_x\\,d{\\mathcal {H}}^{n-1}_y=\\sum _{k=0}^\\infty \\sum _{i=1}^{d(k)}\\,\\lambda _k^s\\,a_k^i(u)^2\\,,$ where $a_k^i(u)=\\int _{\\partial B}u\\,Y_k^i\\,d{\\mathcal {H}}^{n-1}$ .", "Similarly, it is well-known that $\\int _{\\partial B} \|\\nabla _\\tau u\|^2\\,d\\mathcal {H}^{n-1}= \\sum _{k=0}^\\infty \\sum _{i=1}^{d(k)}\\,\\lambda _k^1\\,a_k^i(u)^2\\,,$ with $\\lambda _k^1$ defined as in (REF ); see, for example, [32].", "We finally claim that for every $\\alpha \\in (0,n)$ we have $\\iint _{\\partial B\\times \\partial B}\\frac{\|u(x)-u(y)\|^2}{\|x-y\|^{n-\\alpha }}\\,d\\mathcal {H}^{n-1}_x\\mathcal {H}^{n-1}_y=\\sum _{k=0}^\\infty \\sum _{i=1}^{d(k)}\\,\\mu _k^\\alpha \\,a_k^i(u)^2\\,,$ for $\\mu _k^\\alpha $ defined as in (), (), and ().", "Indeed, following [34], one defines the Riesz operator on the sphere of order $\\gamma \\in (0,n-1)$ as $\\mathcal {R}^\\gamma u(x):=\\frac{1}{2^\\gamma \\,\\pi ^{\\frac{n-1}{2}}}\\,\\frac{\\Gamma (\\frac{n-1-\\gamma }{2})}{\\Gamma (\\frac{\\gamma }{2})}\\,\\int _{\\partial B}\\frac{u(y)}{\|x-y\|^{n-1-\\gamma }}\\,d{\\mathcal {H}}^{n-1}_y\\,,\\qquad x\\in \\partial B\\,.$ By [34], the $k$ -th eigenvalue of ${\\mathcal {R}}^\\gamma $ is given by $\\mu _k^(\\gamma )=\\frac{\\Gamma (k+\\frac{n-1-\\gamma }{2})}{\\Gamma (k+\\frac{n-1+\\gamma }{2})} \\,,\\qquad k\\in \\mathbb {N}\\cup \\lbrace 0\\rbrace \\,,$ so that $\\mu _k^(\\gamma )> 0$ , $\\mu _k^(\\gamma )$ is strictly decreasing in $k$ , and $\\mu ^_k(\\gamma )\\downarrow 0$ as $k\\rightarrow \\infty $ .", "Moreover ${\\mathcal {R}}^\\gamma Y_k=\\mu _k^(\\gamma )\\,Y_k\\,,\\qquad \\forall k\\in \\mathbb {N}\\cup \\lbrace 0\\rbrace \\,,$ where $Y_k$ denotes a generic spherical harmonic of degree $k$ .", "In particular $\\frac{1}{2^\\gamma \\,\\pi ^{\\frac{n-1}{2}}}\\,\\frac{\\Gamma (\\frac{n-1-\\gamma }{2})}{\\Gamma (\\frac{\\gamma }{2})}\\,\\int _{\\partial B}\\frac{d{\\mathcal {H}}^{n-1}_y}{\|x-y\|^{n-1-\\gamma }}=\\mu ^_0(\\gamma )\\,\\qquad \\text{for every $x\\in \\partial B$}\\,.$ Next, similarly to what we have done in section , we introduce for every $\\alpha \\in (0,n)$ the operator ${R}_\\alpha u(x):=2\\int _{\\partial B}\\frac{u(x)-u(y)}{\|x-y\|^{n-\\alpha }}\\,d{\\mathcal {H}}^{n-1}_y\\,,\\qquad u\\in C^1(\\partial B)\\,,$ so that, for every $u\\in C^1(\\partial B)$ , $[u]_{\\frac{1-\\alpha }{2}}^2= \\iint _{\\partial B\\times \\partial B}\\frac{\|u(x)-u(y)\|^2}{\|x-y\|^{n-\\alpha }}\\,d\\mathcal {H}^{n-1}_x\\mathcal {H}^{n-1}_y=\\int _{\\partial B}u\\,{R}_\\alpha u\\,d{\\mathcal {H}}^{n-1}\\,.$ If $\\alpha \\in (1,n)$ then $\\gamma =\\alpha -1\\in (0,n-1)$ , and thus we can deduce from (REF ) and (REF ) that ${R}_\\alpha =2^{\\alpha }\\,\\pi ^{\\frac{n-1}{2}}\\,\\frac{\\Gamma (\\frac{\\alpha -1}{2})}{\\Gamma (\\frac{n-\\alpha }{2})}\\,\\Big (\\mu _0^(\\alpha -1){\\rm Id}-{\\mathcal {R}}^{\\alpha -1}\\Big )\\,,\\qquad \\alpha \\in (1,n)\\,.$ In particular, we deduce from (REF ) and (REF ) that (REF ) holds true with $\\mu _k^\\alpha $ defined as in () whenever $\\alpha \\in (1,n)$ .", "If $\\alpha \\in (0,1)$ , then ${R}_\\alpha $ becomes singular and by applying (REF ) with $\\gamma =1-\\alpha \\in (0,1)$ we have ${R}_\\alpha =\\frac{2^{1+\\alpha }\\pi ^{\\frac{n-1}{2}}}{1-\\alpha }\\,\\frac{\\Gamma (\\frac{1+\\alpha }{2})}{\\Gamma (\\frac{n-\\alpha }{2})}\\,\\mathcal {D}^{1-\\alpha }\\,,\\qquad \\alpha \\in (0,1)\\,.$ In particular, it follows from (REF ) and (REF ) that (REF ) holds true with $\\mu _k^\\alpha $ defined as in ().", "Finally, to prove (REF ) in the case $\\alpha =1$ , it just suffice to notice that ${R}_\\alpha Y\\rightarrow {R}_1 Y$ as $\\alpha \\rightarrow 1$ for every spherical harmonic $Y$ : therefore the eigenvalue $\\mu _k^1$ of ${R}_1$ can be simply computed by taking the limit of $\\mu _k^\\alpha $ as $\\alpha \\rightarrow 1^+$ in () or as $\\alpha \\rightarrow 1^-$ in ().", "In both ways one verifies the validity of (REF ) with $\\alpha =1$ and with $\\mu _k^1$ defined as in ().", "As a last preparatory remark to the proof of Theorem REF , let us notice that by (), (), and () (and by exploiting some classical properties of the Gamma and digamma functions), one has $\\mu _0^\\alpha =0\\,,\\qquad \\mu _{k+1}^\\alpha >\\mu _k^\\alpha \\,,\\qquad {R}_\\alpha Y_k=\\mu _k^\\alpha \\, Y_k\\,,\\qquad \\forall k\\in \\mathbb {N}\\cup \\lbrace 0\\rbrace \\,,\\quad \\forall \\alpha \\in (0,n)\\,.$ In addition, $\\lbrace \\mu _k^\\alpha \\rbrace $ is bounded for $\\alpha \\in (1,n)$ , and $\\mu _k^\\alpha \\uparrow \\infty $ as $k\\rightarrow \\infty $ for $\\alpha \\in (0,1]$ .", "Finally, we notice that since the coordinate functions $x_i$ , $i=1,\\dots ,n$ , belong to ${\\mathcal {S}}_1$ , we have ${R}_\\alpha x_i=\\mu _1^\\alpha x_i$ by (REF ).", "Inserting $x_i$ in (REF ) and adding up over $i$ , yields $\\mu _1^\\alpha =\\frac{1}{P(B)}\\iint _{\\partial B\\times \\partial B}\\frac{d{\\mathcal {H}}^{n-1}_x\\,d{\\mathcal {H}}^{n-1}_y}{\|x-y\|^{n-2-\\alpha }}=\\int _{\\partial B}\\frac{\\,d{\\mathcal {H}}^{n-1}_y}{\|z-y\|^{n-2-\\alpha }}\\,,\\qquad \\forall z\\in \\partial B\\,.$ We can thus conclude that ${\\rm c}_{\\partial B}^2=n-1\\,,\\qquad {\\rm c}^2_{K_s,\\partial B}=\\lambda _1^s\\,,\\qquad {\\rm c}^2_{V_\\alpha ,\\partial B}=\\mu _1^\\alpha \\,,$ for every $s\\in (0,1)$ and $\\alpha \\in (0,n)$ : indeed, the first identity is trivial, while the second and the third one follow from (REF ), (REF ), and (REF ).", "Starting from the above considerations, given $s\\in (0,1]$ and $\\alpha \\in (0,n)$ we are led to consider the following quadratic functionals $\\mathcal {QP}_1(u)&:=&\\int _{\\partial B}\|\\nabla _\\tau u\|^2\\,d\\mathcal {H}^{n-1}-(n-1)\\int _{\\partial B}u^2\\,d\\mathcal {H}^{n-1}\\,,\\\\\\mathcal {QP}_s(u)&:=&\\frac{1-s}{\\omega _{n-1}}\\bigg (\\iint _{\\partial B\\times \\partial B}\\frac{\|u(x)-u(y)\|^2}{\|x-y\|^{n+s}}\\,d\\mathcal {H}^{n-1}_x\\mathcal {H}^{n-1}_y- \\lambda _1^s\\int _{\\partial B}u^2\\,d{\\mathcal {H}}^{n-1}\\bigg )\\,,\\\\\\mathcal {QV}_\\alpha (u)&:=&\\iint _{\\partial B\\times \\partial B}\\frac{\|u(x)-u(y)\|^2}{\|x-y\|^{n-\\alpha }}\\,d\\mathcal {H}^{n-1}_x\\mathcal {H}^{n-1}_y-\\mu _1^\\alpha \\,\\int _{\\partial B}u^2\\,d\\mathcal {H}^{n-1}\\,.$ We set $\\widetilde{H}^{\\frac{1+s}{2}}(\\partial B):=\\Big \\lbrace u\\in H^{\\frac{1+s}{2}}(\\partial B):\\int _{\\partial B}u\\,d{\\mathcal {H}}^{n-1}=0\\Big \\rbrace \\,,$ and notice the validity of the following proposition.", "Proposition 7.2 If $s\\in (0,1]$ , $\\alpha \\in (0,n)$ , and $\\beta >0$ , then $\\mathcal {QP}_s(u)-\\beta \\,\\mathcal {QV}_\\alpha (u)={\\left\\lbrace \\begin{array}{ll}\\displaystyle \\sum _{k=2}^\\infty \\sum _{i=1}^{d(k)}\\bigg (\\frac{1-s}{\\omega _{n-1}}(\\lambda _k^s-\\lambda _1^s)-\\beta (\\mu _k^\\alpha -\\mu _1^\\alpha )\\bigg )\\,a_k^i(u)^2\\,, & \\text{if $s\\in (0,1)$}\\,,\\\\[12pt]\\displaystyle \\sum _{k=2}^\\infty \\sum _{i=1}^{d(k)}\\bigg ((\\lambda _k^1-\\lambda _1^1)-\\beta (\\mu _k^\\alpha -\\mu _1^\\alpha )\\bigg )\\,a_k^i(u)^2\\,, & \\text{if $s=1$}\\,.\\end{array}\\right.", "}$ for every $u\\in \\widetilde{H}^{\\frac{1+s}{2}}(\\partial B)$ .", "In particular, $\\mathcal {QP}_s-\\beta \\,\\mathcal {QV}_\\alpha \\ge 0$ on $\\widetilde{H}^{\\frac{1+s}{2}}(\\partial B)$ if and only if $\\beta \\in (0,\\beta _\\star ]$ .", "This is immediate from the definition of $\\beta _\\star $ and from (REF ), (REF ), and (REF ), once one takes into account that $a_0(u)=0$ for every $u\\in L^2(\\partial B)$ with $\\int _{\\partial B}u\\,d{\\mathcal {H}}^{n-1}=0$ .", "(Indeed, $\\mathcal {S}_0$ is the space of constant functions on $\\partial B$ .)", "We premise a final lemma to the proof of Theorem REF .", "Lemma 7.3 Given $n\\ge 2$ , there exist positive constants $C_0$ and $\\delta _0$ , depending on $n$ only, with the following property: If $v\\in C^\\infty (\\partial B)$ and $\\Vert v\\Vert _{C^1(\\partial B)}\\le \\delta _0$ , then there exists $X\\in C^\\infty _c(\\mathbb {R}^n;\\mathbb {R}^n)$ such that ${\\rm div}X=0$ on $B_2\\setminus B_{1/2}$ ; the flow $\\Phi _t$ induced by $X$ satisfies $\\Phi _1(x)=(1+v(x)) x$ for every $x\\in \\partial B$ ; $\\Vert X\\cdot \\nu _B-v\\Vert _{C^1(\\partial B)}\\le C_0\\,\\Vert v\\Vert ^2_{C^1(\\partial B)}$ .", "If in addition $\|\\Phi _1(B)\|=\|B\|$ , then $\|\\Phi _t(B)\|=\|B\|$ for every $t\\in (-1,1)$ .", "Let $\\chi :[0,\\infty )\\rightarrow [0,1]$ be a smooth cut-off function such that $\\chi (r)=1$ for $r\\in [1/2,2]$ and $\\chi (r)=0$ for $r\\in [0,1/4]\\cup [3,\\infty )$ , and define $X\\in C^\\infty _c(\\mathbb {R}^n;\\mathbb {R}^n)$ by setting $X(x)=\\frac{\\chi (\|x\|)}{n}\\bigg (\\Big (1+ v\\Big (\\frac{x}{\|x\|}\\Big )\\Big )^n-1\\bigg ) \\frac{x}{\|x\|^n}\\,,\\qquad x\\in \\mathbb {R}^n\\,.$ Direct computations show the validity of (i) and (iii) (the latter with a constant $C_0$ that depends on $\\delta _0$ ).", "Up to further decrease the value of $\\delta _0$ we can ensure that $\\Phi _t$ is a diffeomorphism for every $\|t\|\\le 1$ .", "By a direct computation we see that $\\Phi _t(x)=\\bigg (1+ t\\big (\\big (1+ v(x)\\big )^n-1\\big )\\bigg )^{\\frac{1}{n}}x\\,,$ for every $x\\in \\partial B$ and $\|t\|\\le 1$ .", "In particular, (ii) holds true.", "By (REF ) and by (i) we infer that $\\frac{d^2}{dt^2}\|E_t\|=\\int _{\\partial E_t}({\\rm div}X)(X\\cdot \\nu _{ E_t})\\,d\\mathcal {H}^{n-1}=0\\,\\qquad \\forall \|t\|\\le 1\\,,$ that is, $t\\mapsto \|E_t\|$ is affine on $[-1,1]$ .", "In particular, if $\|E_1\|=\|B\|=\|E_0\|$ , then $\|E_t\|=\|B\|$ for every $t\\in [-1,1]$ .", "We fix $\\beta >0$ and claim that $B$ is a volume-constrained stable set for ${\\rm Per}_s+\\beta \\,V_\\alpha $ if and only if $\\mathcal {QP}_s(u)-\\beta \\,\\mathcal {QV}_\\alpha (u)\\ge 0\\,,\\qquad \\mbox{$\\forall u\\in C^\\infty (\\partial B)$ with $\\int _{\\partial B}u\\,d{\\mathcal {H}}^{n-1}=0$}\\,;$ the theorem will then follow by a standard density argument and by Proposition REF .", "By (REF ), (), and (), we see that $B$ is a volume-constrained stable set for ${\\rm Per}_s+\\beta \\,V_\\alpha $ if and only if $&&\\mathcal {QP}_s(X\\cdot \\nu _B)-\\beta \\,\\mathcal {QV}_\\alpha (X\\cdot \\nu _B)\\ge 0\\,,\\qquad \\mbox{$\\forall X\\in C^\\infty _c(\\mathbb {R}^n;\\mathbb {R}^n)$ inducing }\\\\\\nonumber &&\\hspace{170.71652pt}\\mbox{ a volume-preserving flow on $B$}\\,.$ Now, the fact that (REF ) implies (REF ) is obvious: indeed, recall (REF ), $u=X\\cdot \\nu _B$ satisfies $\\int _{\\partial B}u\\,d{\\mathcal {H}}^{n-1}=0$ whenever $X$ induces a volume-preserving flow on $B$ .", "To prove the reverse implication, let us fix $u\\in C^\\infty (\\partial B)$ with $\\int _{\\partial B}u\\,d{\\mathcal {H}}^{n-1}=0$ , and consider the open sets $E_\\delta =\\Big \\lbrace (1+\\delta \\,u(x))\\,x:x\\in \\partial B\\Big \\rbrace \\,,\\qquad \\delta \\in (0,1)\\,.$ Since $\\int _{\\partial B}u\\,d{\\mathcal {H}}^{n-1}=0$ , we have that $\\bigl \| \|E_\\delta \|-\|B\|\\bigr \| \\le C\\delta ^2$ for some constant $C$ depending on $u$ only.", "Therefore, if $F_\\delta =(\|B\|/\|E_\\delta \|)^{1/n}\\,E_\\delta $ , then we have $F_\\delta =\\Big \\lbrace (1+v_\\delta (x))\\,x:x\\in \\partial B\\Big \\rbrace \\,,\\qquad \\delta \\in (0,1)\\,,$ for some $v_\\delta \\in C^\\infty (\\partial B)$ with $\\Vert v_\\delta \\Vert _{C^1(\\partial B)}\\le C\\,\\delta $ and $\\Vert v_\\delta -\\delta \\,u\\Vert _{C^1(\\partial B)}\\le C\\,\\delta ^2$ (again, the constant $C$ does not depend on $\\delta $ ).", "Provided $\\delta $ is small enough we can apply Lemma REF to find a vector field $X_\\delta \\in C^\\infty _c(\\mathbb {R}^n;\\mathbb {R}^n)$ inducing a volume-preserving flow on $B$ , and with the property that $\\Vert X_\\delta \\cdot \\nu _B-v_\\delta \\Vert _{C^1(\\partial B)}\\le C\\,\\Vert v_\\delta \\Vert _{C^1(\\partial B)}^2\\le C\\,\\delta ^2\\,.$ In particular, $\\Vert X_\\delta \\cdot \\nu _B-\\delta \\,u\\Vert _{C^1(\\partial B)}\\le C\\,\\delta ^2$ , and thus by (REF ) we have (recall that $\\mathcal {QP}$ and $\\mathcal {QV}$ are quadratic forms) $0\\le \\mathcal {QP}_s(X_\\delta \\cdot \\nu _B)-\\beta \\,\\mathcal {QV}_\\alpha (X_\\delta \\cdot \\nu _B)\\le \\mathcal {QP}_s(\\delta \\,u)-\\beta \\,\\mathcal {QV}_\\alpha (\\delta \\,u)+C\\,\\delta ^3\\,.$ We divide by $\\delta ^2$ and let $\\delta \\rightarrow 0^+$ to find that $\\mathcal {QP}_s(u)-\\beta \\,\\mathcal {QV}_\\alpha (u)\\ge 0$ .", "This shows that (REF ) implies (REF ), and thus completes the proof of the theorem.", "We close this section with the following result.", "Proposition 7.4 For every $n\\ge 2$ , $s\\in (0,1]$ and $\\alpha \\in (0,n)$ one has $\\beta _\\star (n,s,\\alpha )={\\left\\lbrace \\begin{array}{ll}\\displaystyle \\frac{n+s}{n-\\alpha }\\,\\frac{s\\,(1-s)\\,P_s(B)}{\\alpha \\,\\omega _{n-1}V_\\alpha (B)}\\,, &\\hspace{28.45274pt} \\text{if $s\\in (0,1)$}\\,,\\\\[10pt]\\displaystyle \\frac{n+1}{n-\\alpha }\\,\\frac{P(B)}{\\alpha \\,V_\\alpha (B)}\\,, &\\hspace{28.45274pt} \\text{if $s=1$}\\,.\\end{array}\\right.", "}$ By appendix $\\nonumber \\beta _\\star (n,s,\\alpha )={\\left\\lbrace \\begin{array}{ll}\\displaystyle (1-s)\\,\\frac{\\lambda _2^s-\\lambda _1^s}{\\mu _2^\\alpha -\\mu _1^\\alpha }\\,, &\\hspace{28.45274pt} \\text{if $s\\in (0,1)$}\\,,\\\\[10pt]\\displaystyle \\frac{\\lambda _2^1-\\lambda _1^1}{\\mu _2^\\alpha -\\mu _1^\\alpha }\\,, &\\hspace{28.45274pt} \\text{if $s=1$}\\,,\\end{array}\\right.", "}$ We then find (REF ) by Proposition REF and by Proposition REF below." ], [ "Proof of Theorem ", "We are now in the position of proving Theorem REF .", "We begin with the following result, which extends Theorem REF to the family of functionals ${\\rm Per}_s+\\beta \\,V_\\alpha $ with $\\beta \\in (0,\\beta _\\star )$ .", "Theorem 8.1 For every $s\\in (0,1)$ , $\\alpha \\in (0,n)$ , and $\\beta \\in (0,\\beta _\\star (n,s,\\alpha )),$ there exist positive constants $c_0=c_0(n)$ and $\\varepsilon _\\beta =\\varepsilon _\\beta (n,s,\\alpha )$ with the following property: If $E$ is a nearly spherical set as in (REF ) with $\|E\|=\|B\|$ , $\\int _Ex\\,dx=0$ , and $\\Vert u\\Vert _{C^1(\\partial B)}<\\varepsilon _\\beta $ , then $\\big ({\\rm Per}_s+\\beta V_\\alpha \\big )(E)-\\big ({\\rm Per}_s+\\beta V_\\alpha \\big )(B)\\ge c_0\\Big (1-\\frac{\\beta }{\\beta _\\star }\\Big )\\,\\Big ((1-s)[u]_{\\frac{1+s}{2}}^2+\\Vert u\\Vert _{L^2(\\partial B)}^2\\Big )\\,.$ Moreover, we can take $\\varepsilon _\\beta $ of the form $\\varepsilon _\\beta =\\Big (1-\\frac{\\beta }{\\beta _\\star }\\Big )\\,\\varepsilon _0(n)\\,,$ for a suitable positive constant $\\varepsilon _0(n)$ .", "Remark 8.2 If $\\beta \\in (0,\\beta _\\star (n,1,\\alpha ))$ and $u$ satisfies the assumptions of Theorem REF , then $\\big (P+\\beta V_\\alpha \\big )(E)-\\big (P +\\beta V_\\alpha \\big )(B)\\ge c_0\\Big (1-\\frac{\\beta }{\\beta _\\star (n,1,\\alpha )}\\Big )\\, \\Vert u\\Vert _{H^1(\\partial B)}^2\\,.$ To prove this observe that, by a standard approximation argument, it suffices to consider the case when $u\\in C^{1,\\gamma }(\\partial B)$ for some $\\gamma \\in (0,1)$ , and thus ${\\rm Per}_s(E)\\rightarrow P(E)$ as $s\\rightarrow 1^-$ by (REF ).", "By (REF ) and again by (REF ), $\\beta _\\star (n,s,\\alpha )\\rightarrow \\beta _\\star (n,1,\\alpha )$ as $s\\rightarrow 1^-$ .", "In particular, we can find $\\tau >0$ such that $\\,\\beta <\\beta _\\star (n,s,\\alpha )$ and $\\varepsilon _\\beta (n,1,\\alpha )<\\varepsilon _{\\beta }(n,s,\\alpha )$ for every $s\\in (1-\\tau ,1)$ .", "We may thus apply (REF ) with $s\\in (1-\\tau ,1)$ and then let $\\tau \\rightarrow 0^+$ , to find that $\\big (P+\\beta V_\\alpha \\big )(E)-\\big (P +\\beta V_\\alpha \\big )(B)\\ge c_0\\Big (1-\\frac{\\beta }{\\beta _\\star (n,1,\\alpha )}\\Big )\\,\\limsup _{s\\rightarrow 1^{-1}}\\,(1-s)[u]_{\\frac{1+s}{2}}^2\\,.$ Finally, by (REF ) and () we find that $\\lambda _k^s\\rightarrow \\omega _{n-1}\\lambda _k^1$ as $s\\rightarrow 1^-$ , hence recalling (REF ) and (REF ) we get $\\lim _{s\\rightarrow 1^-}\\,(1-s)[u]^2_{\\frac{1+s}{2}}=\\omega _{n-1}\\int _{\\partial B}\|\\nabla _\\tau u\|^2\\,$ and (REF ) is proved.", "Remark 8.3 Theorem REF follows from Theorem REF by letting $\\alpha \\rightarrow n^-$ in (REF ).", "Indeed, denoting by $C$ a generic constant depending on $n$ only, we notice that (), (), and () give $\\mu _k^\\alpha -\\mu _1^\\alpha \\le C\\,(n-\\alpha )$ for all $k\\ge 2$ .", "At the same time, by exploiting (REF ), (REF ), and (REF ) we find that $(1-s)\\,\\lambda _1^s\\ge \\frac{1}{C}\\,,\\qquad \\forall s\\in (0,1)\\,,$ so that by Proposition REF , again for every $k\\ge 2$ , $(1-s)(\\lambda _k^s-\\lambda _1^s)\\ge (1-s)(\\lambda _2^s-\\lambda _1^s)=\\frac{n+s}{n-s} (1-s)\\lambda _1^s\\ge \\frac{1}{C}\\,.$ We thus conclude from (REF ) that $\\beta _\\star (n,s,\\alpha )\\ge \\frac{c(n)}{n-\\alpha }\\,,$ for a suitable positive constant $c(n)$ .", "In particular, $\\beta _\\star (n,s,\\alpha )\\rightarrow \\infty $ as $\\alpha \\rightarrow n^-$ uniformly with respect to $s\\in (0,1)$ , and (REF ) follows by letting $\\alpha \\rightarrow n^-$ in (REF ).", "Before discussing the proof of Theorem REF we need the following observation, which parallels Proposition REF .", "Proposition 8.4 For every $\\alpha \\in (0,n)$ , one has $\\mu _1^\\alpha &=&\\alpha (n+\\alpha )\\frac{V_\\alpha (B)}{P(B)}\\,,\\\\\\mu _2^\\alpha &=&\\frac{2n}{n+\\alpha }\\,\\mu _1^\\alpha \\,.$ By scaling, $V_\\alpha (B_r)=r^{n+\\alpha }V_\\alpha (B)$ .", "Hence, $(n+\\alpha )V_\\alpha (B)=\\frac{d}{dr}\\Big \|_{r=1}V_\\alpha (B_r)=2\\int _{B}\\,dx\\int _{\\partial B}\\frac{d{\\mathcal {H}}^{n-1}_y}{\|x-y\|^{n-\\alpha }}\\,.$ Since $\\frac{1}{\|x-y\|^{n-\\alpha }}=\\frac{1}{\\alpha } \\,{\\rm div}_x\\biggl (\\frac{x-y}{\|x-y\|^{n-\\alpha }}\\biggr )$ by the divergence theorem we get $\\alpha (n+\\alpha )V_\\alpha (B)=2\\iint _{\\partial B\\times \\partial B}\\frac{(x-y)\\cdot x}{\|x-y\|^{n-\\alpha }}\\,d{\\mathcal {H}}^{n-1}_x\\,d{\\mathcal {H}}^{n-1}_y\\,.$ By symmetry, the right-hand side of the last identity is equal to $&&\\iint _{\\partial B\\times \\partial B}\\frac{(x-y)\\cdot x}{\|x-y\|^{n+s}}\\,d{\\mathcal {H}}^{n-1}_x\\,d{\\mathcal {H}}^{n-1}_y+ \\iint _{\\partial B\\times \\partial B}\\frac{(y-x)\\cdot y}{\|x-y\|^{n+s}} \\,d{\\mathcal {H}}^{n-1}_y\\,d{\\mathcal {H}}^{n-1}_x\\\\&&=\\iint _{\\partial B\\times \\partial B}\\frac{d{\\mathcal {H}}^{n-1}_x\\,d{\\mathcal {H}}^{n-1}_y}{\|x-y\|^{n+s-2}}\\,,$ so that (REF ) follows from (REF ).", "One can deduce () from (), (), and () (depending on whether $\\alpha \\in (1,n)$ , $\\alpha =1$ or $\\alpha \\in (0,1)$ ) by exploiting the factorial property of the Gamma function.", "Since a similar argument was presented in Proposition REF , we omit the details.", "We consider $u\\in C^1(\\partial B)$ with $\\Vert u\\Vert _{C^1(\\partial B)}\\le 1/2$ and assume the existence of $t\\in (0,2\\,\\varepsilon _\\beta )$ such that the open set $E_t$ whose boundary is given by $\\partial E_t=\\Big \\lbrace (1+t\\,u(x))\\,x:x\\in \\partial B\\Big \\rbrace $ satisfies $\|E_t\|=\|B\|$ and $\\int _{E_t}x\\,dx=0$ .", "If $\\varepsilon _\\beta $ is small enough then (REF ), (REF ), and (REF ) imply that $\\big ({\\rm Per}_s+\\beta V_\\alpha \\big )(E_t)-\\big ({\\rm Per}_s+\\beta V_\\alpha \\big )(B)\\ge \\frac{t^2}{2}\\Big (\\mathcal {QP}_s(u)-\\beta \\mathcal {QV}_\\alpha (u)\\Big )\\\\-C(n)t^3\\bigg (\\frac{1-s}{\\omega _{n-1}}\\big ([u]^2_{\\frac{1+s}{2}}+\\lambda _1^s\\Vert u\\Vert ^2_{L^2}\\big )+\\beta \\big ([u]^2_{\\frac{1-\\alpha }{2}}+\\mu _1^\\alpha \\Vert u\\Vert ^2_{L^2}\\big )\\bigg )\\,.$ By Proposition REF and by definition of $\\beta _\\star $ we have $\\mathcal {QP}_s(u)-\\beta \\mathcal {QV}_\\alpha (u) & = \\sum _{k=2}^\\infty \\sum _{i=1}^{d(k)}\\bigg (\\frac{1-s}{\\omega _{n-1}}(\\lambda _k^s-\\lambda _1^s)-\\beta (\\mu _k^\\alpha -\\mu _1^\\alpha )\\bigg )\\,\|a_k^i\|^2 \\\\& \\ge \\frac{1-s}{\\omega _{n-1}}\\Big (1-\\frac{\\beta }{\\beta _\\star } \\Big )\\sum _{k=2}^\\infty \\sum _{i=1}^{d(k)}(\\lambda _k^s-\\lambda _1^s)\|a_k^i\|^2\\\\& = \\frac{1-s}{\\omega _{n-1}}\\Big (1-\\frac{\\beta }{\\beta _\\star } \\Big ) \\Big ([u]^2_{\\frac{1+s}{2}} - \\lambda _1^s\\Vert u\\Vert ^2_{L^2}\\Big )\\,,$ thus using (REF ) and (REF ) we find $\\mathcal {QP}_s(u)-\\beta \\mathcal {QV}_\\alpha (u) \\ge \\frac{1-s}{4}\\Big (1-\\frac{\\beta }{\\beta _\\star } \\Big )\\Big ([u]^2_{\\frac{1+s}{2}} + \\lambda _1^s\\Vert u\\Vert ^2_{L^2}\\Big )\\, .$ Choosing $\\varepsilon _\\beta $ small enough, we can apply (REF ) and () to estimate $\\mu _1^\\alpha \\Vert u\\Vert ^2_{L^2}\\le 2\\mu _1^\\alpha \\sum _{k=2}^\\infty \\sum _{i=1}^{d(k)}\|a_k^i\|^2\\le \\frac{2(n+\\alpha )}{n-\\alpha } \\sum _{k=2}^\\infty \\sum _{i=1}^{d(k)}(\\mu _k^\\alpha -\\mu ^\\alpha _1)\|a_k^i\|^2\\le C(n)\\mathcal {QV}_\\alpha (u)\\,,$ where in the last inequality we have used the temporary assumption that $\\alpha \\le n-\\frac{1}{2}\\,.$ By (REF ) and by (REF ) (which gives, in particular, $\\mathcal {QP}_s(u)\\ge \\beta \\mathcal {QV}_\\alpha (u)$ ), we find $\\beta \\Big ([u]^2_{\\frac{1-\\alpha }{2}}+\\mu _1^\\alpha \\Vert u\\Vert ^2_{L^2}\\Big ) = \\beta \\mathcal {QV}_\\alpha (u) +2\\beta \\mu _1^\\alpha \\Vert u\\Vert ^2_{L^2} \\le C(n)\\beta \\, \\mathcal {QV}_\\alpha (u) \\le C(n) \\mathcal {QP}_s(u)\\,.$ By gathering (REF ), (REF ), and (REF ) we end up with $\\big ({\\rm Per}_s+\\beta V_\\alpha \\big )(E_t)-\\big ({\\rm Per}_s+\\beta V_\\alpha \\big )(B)\\ge \\frac{1-s}{\\omega _{n-1}}\\Big (\\frac{t^2}{8}\\Big (1-\\frac{\\beta }{\\beta _\\star } \\Big )-C(n)t^3\\Big )\\,\\Big ([u]^2_{\\frac{1+s}{2}} +\\lambda _1^s \\Vert u\\Vert ^2_{L^2}\\Big ).$ By choosing $\\varepsilon _0(n)$ suitably small in (REF ), and by exploiting (REF ), (REF ), and (REF ) to deduce that $(1-s)\\,\\lambda _1^s\\ge c(n)>0$ for a suitable positive constant $c(n)$ , we deduce that $\\big ({\\rm Per}_s+\\beta V_\\alpha \\big )(E_t)-\\big ({\\rm Per}_s+\\beta V_\\alpha \\big )(B)\\ge c_0 t^2\\Big (1-\\frac{\\beta }{\\beta _\\star } \\Big )\\Big ((1-s)[u]^2_{\\frac{1+s}{2}} + \\Vert u\\Vert ^2_{L^2}\\Big )\\,,$ for a constant $c_0$ which only depends on $n$ .", "This completes the proof of the theorem in the case (REF ) holds true.", "Let us now assume that $\\alpha \\in (n-1/2,n)$ , and prove a stronger version of (REF ).", "Since $\|E_t\|=\|B\|$ , we can write $V_\\alpha (B)-V_\\alpha (E_t)=\\big (V_\\alpha (B)-\|B\|^2\\big )-\\big (V_\\alpha (E_t)-\|E_t\|^2\\big )\\,.$ If we set $f_{\\theta }(r,\\rho ):=\\frac{r^{n-1}\\varrho ^{n-1}}{(\|r-\\varrho \|^2+r\\,\\varrho \\,\\theta ^2)^{\\frac{n-\\alpha }{2}}}-r^{n-1}\\rho ^{n-1}\\,,\\qquad r,\\rho ,\\theta \\ge 0\\,,$ then we find $V_\\alpha (E_t)-\|E_t\|^2=\\iint _{\\partial B\\times \\partial B}\\biggl (\\int _{0}^{1+tu(x)}\\int _0^{1+tu(y)} f_{\|x-y\|}(r,\\rho ) \\,dr\\,d\\varrho \\biggr )\\,d{\\mathcal {H}}^{n-1}_x\\,d{\\mathcal {H}}^{n-1}_y\\,,$ Arguing as in the proof of Lemma REF , we derive that $V_\\alpha (E_t)-\|E_t\|^2=&-\\frac{t^2}{2}\\tilde{g}(t) +\\frac{V_\\alpha (B)}{P(B)}\\int _{\\partial B}(1+t u)^{n+\\alpha }\\,d{\\mathcal {H}}^{n-1}-\\frac{\|B\|^2}{P(B)}\\int _{\\partial B}(1+t u)^{2n}\\,d{\\mathcal {H}}^{n-1}\\\\=& -\\frac{t^2}{2}\\tilde{g}(t) +\\frac{V_\\alpha (B)-\|B\|^2}{P(B)}\\int _{\\partial B}(1+t u)^{n+\\alpha }\\,d{\\mathcal {H}}^{n-1}\\\\&\\hspace{100.0pt}-\\frac{\|B\|^2}{P(B)}\\int _{\\partial B}(1+t u)^{2n}\\Big (1-(1+t u)^{\\alpha -n}\\Big )\\,d{\\mathcal {H}}^{n-1}\\,,$ with $\\tilde{g}(t):=\\iint _{\\partial B\\times \\partial B} \\left( \\int _{u(y)}^{u(x)} \\int _{u(y)}^{u(x)} f_{\|x-y\|}(1+tr,1+t\\rho )\\,drd\\rho \\right)\\,d{\\mathcal {H}}^{n-1}_x\\,d{\\mathcal {H}}^{n-1}_y\\,.", "$ Setting $h(t):=\\int _{\\partial B}(1+tu)^{n+\\alpha }$ and $\\ell (t):= \\int _{\\partial B}(1+t u)^{2n}\\Big (1-(1+t u)^{\\alpha -n}\\Big )\\,d{\\mathcal {H}}^{n-1}\\,,$ we conclude that $ V_\\alpha (B)-V_\\alpha (E_t)=\\frac{t^2}{2}\\tilde{g}(t) + \\frac{V_\\alpha (B)-\|B\|^2}{P(B)}\\big (h(0)-h(t) \\big ) +\\frac{\|B\|^2}{P(B)}\\,\\ell (t)\\,.$ In the proof of Lemma REF we showed that $h(0)-h(t)\\le -\\alpha \\,(n+\\alpha )\\,\\frac{t^2}{2}\\,\\int _{\\partial B}u^2\\,d{\\mathcal {H}}^{n-1}+C(n)\\,t^3\\,\\Vert u\\Vert _{L^2}^2\\,.$ In the same way (using Taylor expansion and $\|E_t\|=\|B\|$ ) we obtain that $\\ell (t)\\le (n-\\alpha )(2n+\\alpha )\\frac{t^2}{2}\\Vert u\\Vert ^2_{L^2}+(n-\\alpha )C(n)t^3\\Vert u\\Vert ^2_{L^2}\\,.$ Then, noticing that $\\alpha (n+\\alpha )=2n^2-(n-\\alpha )(2n+\\alpha ) $ and using (REF ), we compute $\\frac{V_\\alpha (B)-\|B\|^2}{P(B)}=\\frac{1}{\\alpha (n+\\alpha )}\\Big (\\mu _1^\\alpha -\\alpha (n+\\alpha )\\frac{\|B\|^2}{P(B)}\\Big )\\\\=\\frac{1}{\\alpha (n+\\alpha )}\\Big (\\mu _1^\\alpha -2n^2\\frac{\|B\|^2}{P(B)}\\Big ) +(n-\\alpha )\\frac{(2n+\\alpha )\|B\|^2}{\\alpha (n+\\alpha )P(B)}\\,.$ On the other hand, (REF ) implies $\\mu _1^\\alpha \\,\\mathop {\\longrightarrow }\\limits _{\\alpha \\rightarrow n}\\, 2n^2\\frac{\|B\|^2}{P(B)}=:\\mu _1^n \\,.$ From the explicit value of $\\mu _1^\\alpha $ given by (), we easily infer that $\|\\mu _1^\\alpha -\\mu _1^n\|\\le (n-\\alpha )C(n)$ .", "Hence, $\\Big \| \\frac{V_\\alpha (B)-\|B\|^2}{P(B)}\\Big \|\\le (n-\\alpha )C(n)\\,.$ Gathering (REF ), (REF ), (REF ), and (REF ), we are led to $\\frac{V_\\alpha (B)-\|B\|^2}{P(B)}\\big (h(0)-h(t) \\big ) +\\frac{\|B\|^2}{P(B)}\\,\\ell (t)\\le -(\\mu _1^\\alpha -\\mu _1^n)\\frac{t^2}{2}\\Vert u\\Vert ^2_{L^2} +(n-\\alpha )C(n)t^3\\Vert u\\Vert ^2_{L^2}\\,.", "$ Next, from the smooth dependence $\\tilde{g}$ on $t$ , we can find $\\tau \\in (0,t)$ such that $\\tilde{g}(t)=\\tilde{g}(0)+t\\,\\tilde{g}^{\\prime }(\\tau )$ .", "Since $\\alpha \\in (n-1/2,n)$ , we have the estimate $\\Big \|r\\,\\frac{\\partial f_\\theta }{\\partial r}(1+\\tau \\,r,1+\\tau \\,\\varrho )+\\varrho \\,\\frac{\\partial f_\\theta }{\\partial \\varrho }(1+\\tau \\,r,1+\\tau \\,\\varrho )\\Big \|\\le (n-\\alpha )\\frac{C(n)}{\\theta ^{n-\\alpha }}\\big (1+\|\\log (\\theta )\|\\big )\\le (n-\\alpha )\\frac{C(n)}{\\theta ^{3/4}}\\,,$ for all $r,\\varrho \\in (-\\frac{1}{2},\\frac{1}{2})$ , all $\\theta \\in (0,2]$ , and a suitable constant $C(n)$ .", "In turn, the sequence $\\lbrace \\mu _k^{n-3/4}\\rbrace $ is bounded and one can estimate $\|g^{\\prime }(\\tau )\|\\le (n-\\alpha )C(n)\\iint _{\\partial B\\times \\partial B}\\frac{\|u(x)-u(y)\|^2}{\|x-y\|^{3/4}}\\,d{\\mathcal {H}}^{n-1}_x\\,d{\\mathcal {H}}^{n-1}_y\\le (n-\\alpha )C(n)\\,\\Vert u\\Vert _{L^2}^2\\,,$ therefore $V_\\alpha (B)-V_\\alpha (E_t)\\le \\frac{t^2}{2}\\tilde{g}(0) -(\\mu _1^\\alpha -\\mu _1^n)\\frac{t^2}{2}\\Vert u\\Vert ^2_{L^2}+(n-\\alpha )C(n)t^3\\Vert u\\Vert ^2_{L^2}\\,.", "$ Then, we notice that $\\tilde{g}(0)=[u]^2_{\\frac{1-\\alpha }{2}} -\\iint _{\\partial B\\times \\partial B}\|u(x)-u(y)\|^2\\,d{\\mathcal {H}}^{n-1}_x\\,d{\\mathcal {H}}^{n-1}_y\\,.", "$ Also, from () we infer that $\\lim _{\\alpha \\rightarrow n}\\mu _k^\\alpha =\\mu _1^n\\,,\\qquad \\forall k\\ge 1\\,.", "$ Hence, by dominated convergence we have $[u]^2_{\\frac{1-\\alpha }{2}}=\\sum _{k=1}^\\infty \\sum _{i=1}^{d(k)}\\mu _k^\\alpha \|a_k^i\|^2\\mathop {\\longrightarrow }\\limits _{\\alpha \\rightarrow n} \\mu _1^n\\sum _{k=1}^\\infty \\sum _{i=1}^{d(k)}\|a_k^i\|^2\\,.$ Since we obviously have $[u]^2_{\\frac{1-\\alpha }{2}} \\mathop {\\longrightarrow }\\limits _{\\alpha \\rightarrow n} \\iint _{\\partial B\\times \\partial B}\|u(x)-u(y)\|^2\\,d{\\mathcal {H}}^{n-1}_x\\,d{\\mathcal {H}}^{n-1}_y\\,,$ we have thus proved that $ \\iint _{\\partial B\\times \\partial B}\|u(x)-u(y)\|^2\\,d{\\mathcal {H}}^{n-1}_x\\,d{\\mathcal {H}}^{n-1}_y = \\mu _1^n\\sum _{k=1}^\\infty \\sum _{i=1}^{d(k)}\|a_k^i\|^2\\,.$ As a consequence, $V_\\alpha (B)-V_\\alpha (E_t)\\le \\frac{t^2}{2} \\sum _{k=2}^\\infty \\sum _{i=1}^{d(k)}(\\mu _k^\\alpha -\\mu _1^\\alpha ) \|a_k^i\|^2 - (\\mu _1^\\alpha -\\mu _1^n)\\frac{t^2}{2}\|a_0\|^2+(n-\\alpha )C(n)t^3\\Vert u\\Vert ^2_{L^2}\\,.", "$ Recalling (REF ) and the fact that $\|\\mu _1^\\alpha -\\mu _1^n\|\\le (n-\\alpha )C(n)$ , we conclude that $V_\\alpha (B)-V_\\alpha (E_t)\\le \\frac{t^2}{2}\\mathcal {QV}_\\alpha (u)+ (n-\\alpha )C(n)t^3\\Vert u\\Vert ^2_{L^2}\\,,$ that is the required strengthening of (REF ).", "Now we can apply (REF ) together with (REF ) to find that $\\big ({\\rm Per}_s+\\beta V_\\alpha \\big )(E_t)-\\big ({\\rm Per}_s+\\beta V_\\alpha \\big )(B)\\ge \\frac{t^2}{2}\\Big (\\mathcal {QP}_s(u)-\\beta \\mathcal {QV}_\\alpha (u)\\Big )\\\\-C(n)t^3\\bigg (\\frac{1-s}{\\omega _{n-1}}\\big ([u]^2_{\\frac{1+s}{2}}+\\lambda _1^s\\Vert u\\Vert ^2_{L^2}\\big )+(n-\\alpha )\\beta \\Vert u\\Vert ^2_{L^2}\\bigg )\\,.$ Arguing as in the previous case, it yields $\\big ({\\rm Per}_s+\\beta V_\\alpha \\big )(E_t)-\\big ({\\rm Per}_s+\\beta V_\\alpha \\big )(B)\\ge \\frac{t^2}{8}\\Big (1-\\frac{\\beta }{\\beta _\\star } \\Big )\\bigg (\\frac{1-s}{\\omega _{n-1}}[u]^2_{\\frac{1+s}{2}} + (1-s)\\lambda _1^s\\Vert u\\Vert ^2_{L^2}\\bigg ) \\\\-C(n)t^3\\bigg (\\frac{1-s}{\\omega _{n-1}}[u]^2_{\\frac{1+s}{2}}+(1-s)\\lambda _1^s\\Vert u\\Vert ^2_{L^2}+(n-\\alpha )\\beta _\\star \\Vert u\\Vert ^2_{L^2}\\bigg )\\,.$ Since $(n-\\alpha )\\beta _\\star \\le C(n)$ by (REF ), we conclude as in the previous case.", "As a last tool in the proof of Theorem REF we prove the following lemma.", "Lemma 8.5 Let $s\\in (0,1]$ and $\\alpha \\in (0,n)$ .", "If $\\beta <\\beta _\\star $ , then $B$ is a local volume-constrained minimizer of ${\\rm Per}_s+\\beta \\,V_\\alpha $ .", "If $\\beta >\\beta _\\star $ , then $B$ is not a local volume-constrained minimizer of ${\\rm Per}_s+\\beta \\,V_\\alpha $ .", "If $B$ is a local volume-constrained minimizer of ${\\rm Per}_s+\\beta \\,V_\\alpha $ , then $B$ is automatically a volume-constrained stable set for ${\\rm Per}_s+\\beta \\,V_\\alpha $ , and thus $\\beta \\le \\beta _\\star $ by Theorem REF .", "We are thus left to prove that if $\\beta <\\beta _\\star $ , then $B$ is a local volume-constrained minimizer of ${\\rm Per}_s+\\beta \\,V_\\alpha $ .", "To this end, we argue by contradiction and assume the existence of some $\\beta <\\beta _$ such that there exists a sequence $\\lbrace E_h\\rbrace _{h\\in \\mathbb {N}}$ with $\|E_h\|=\|B\|\\,,\\qquad \\lim _{h\\rightarrow \\infty }\|E_h\\Delta B\|=0\\,,\\qquad {\\rm Per}_s(E_h)+\\beta \\,V_\\alpha (E_h)<{\\rm Per}_s(B)+\\beta \\,V_\\alpha (B)\\,,\\quad \\forall h\\in \\mathbb {N}\\,.$ We divide the proof in two steps.", "Step one: We show the existence of a radius $R>0$ (depending on $n$ , $s$ and $\\alpha $ only) such that the sequence $E_h$ in (REF ) can actually be assumed to satisfy the additional constraint $E_h\\subset B_R\\,,\\qquad \\forall h\\in \\mathbb {N}\\,.$ To show this, let us introduce a parameter $\\eta <1$ (whose precise value will be chosen shortly depending on $n$ , $s$ and $\\alpha $ ) and let us assume without loss of generality and thanks to (REF ) that $\|E_h\\Delta B\|<\\eta $ for every $h\\in \\mathbb {N}$ .", "By Lemma REF , see in particular (REF ), there exists a sequence $\\lbrace r_h\\rbrace _{h\\in \\mathbb {N}}$ with $1\\le r_h\\le 1+ C_1\\,\\eta ^{1/n}$ such that ${\\rm Per}_s(E_h\\cap B_{r_h})\\le {\\rm Per}_s(E_h)-\\frac{\|E_h\\setminus B_{r_h}\|}{C_2\\,\\eta ^{1/n}}\\,,$ where $C_1$ and $C_2$ depend on $n$ and $s$ only.", "Next, we consider $\\mu _h>0$ such that $F_h:=\\mu _h\\, (E_h\\cap B_{r_h})$ satisfies $\|F_h\|=\|B\|$ .", "Since $\|E_h\\Delta B\|\\rightarrow 0$ as $h\\rightarrow \\infty $ , it must be that $\\mu _h\\rightarrow 1$ and $\|F_h\\Delta B\|\\rightarrow 0$ as $h\\rightarrow \\infty $ .", "In particular, we can assume without loss of generality that $F_h\\subset B_R$ for every $h\\in \\mathbb {N}$ , provided we set $R:=2+ C_1\\,\\eta ^{1/n}$ .", "We finally show that ${\\rm Per}_s(F_h)+\\beta \\,V_\\alpha (F_h)\\le {\\rm Per}_s(E_h)+\\beta \\,V_\\alpha (E_h)\\,.$ Indeed, by setting $u_h:=\|E_h\\setminus B_{r_h}\|$ we find that ${\\rm Per}_s(F_h)+\\beta \\,V_\\alpha (F_h)&=&\\mu _h^{n-s}{\\rm Per}_s(E_h\\cap B_{r_h})+\\mu _h^{n+\\alpha }\\beta \\,V_\\alpha (E_h\\cap B_{r_h})\\\\&\\le &(1+C\\,u_h)\\,\\Big ({\\rm Per}_s(E_h\\cap B_{r_h})+\\beta \\,V_\\alpha (E_h\\cap B_{r_h})\\Big )\\,,$ where $C=C(n,s,\\alpha )$ .", "By $V_\\alpha (E_h\\cap B_{r_h})\\le V_\\alpha (E_h)$ , (REF ), and (REF ), we conclude that ${\\rm Per}_s(F_h)+\\beta \\,V_\\alpha (F_h)&\\le &{\\rm Per}_s(E_h)+\\beta \\,V_\\alpha (E_h)\\\\&&+\\bigg (C\\,\\Big ({\\rm Per}_s(B)+\\beta _\\star \\,V_\\alpha (B)\\Big )-\\frac{1}{C_2\\,\\eta ^{1/n}}\\bigg )\\,u_h\\,,$ so that (REF ) follows provided $\\eta $ was suitably chosen in terms of $n$ , $s$ and $\\alpha $ only.", "Step two: Given $M>0$ and a sequence $E_h$ satisfying (REF ) and (REF ) we now consider the variational problems $\\gamma _h:=\\inf \\bigg \\lbrace {\\rm Per}_s(E)+\\beta \\,V_\\alpha (E)+M\\,\|E\\Delta E_h\|:E\\subset \\mathbb {R}^n\\bigg \\rbrace \\,,\\qquad h\\in \\mathbb {N}\\,,$ and prove the existence of minimizers.", "Indeed, if $R$ is as in (REF ), then $V_\\alpha (E\\cap B_R)\\le V_\\alpha (E)$ by set inclusion, ${\\rm Per}_s(E\\cap B_R)\\le {\\rm Per}_s(E)$ by Lemma REF , while, if we set $F=E\\cap B_{R}$ , then by (REF ), $\\nonumber \|F\\Delta E_h\|&=&\|F\\setminus E_h\|+\|E_h\\setminus F\|\\le \|E\\setminus E_h\|+\|(E_h\\cap B_{R})\\setminus F\|+\|E_h\\setminus B_{R}\|\\\\\\nonumber &=&\|E\\setminus E_h\|+\|(E_h\\cap B_{R})\\setminus E\|\\\\&\\le &\|E\\Delta E_h\|\\,.\\nonumber $ Thus the value of $\\gamma _h$ is not changed if we restrict the minimization class by imposing $E\\subset B_R$ .", "By the Direct Method, there exists a minimizer $F_h$ in (REF ) for every $h\\in \\mathbb {N}$ , with $F_h\\subset B_R$ .", "We now claim that there exists $\\Lambda >0$ such that ${\\rm Per}_s(F_h)\\le {\\rm Per}_s(E)+\\Lambda \\,\|E\\Delta F_h\|\\,,\\qquad \\forall E\\subset \\mathbb {R}^n\\,,$ for every $h\\in \\mathbb {N}$ .", "Indeed, by minimality of $F_h$ in (REF ) and by (REF ), we find that for every bounded set $E\\subset \\mathbb {R}^n$ one has ${\\rm Per}_s(F_h)-{\\rm Per}_s(E)\\le \\frac{2\\,P(B)\\,\\beta }{\\alpha }\\,\\Big (\\frac{\|E\|}{\|B\|}\\Big )^{\\alpha /n}\\,\|E\\setminus F_h\|+M\\,\\Big (\|E\\Delta E_h\|-\|F_h\\Delta E_h\|\\Big )\\,.$ In particular, (REF ) follows provided $\\Lambda \\ge \\frac{2^{1+\\alpha }\\,P(B)\\,\\beta \\,R^\\alpha }{\\alpha }+M\\,,$ whenever $\|E\|\\le \|B_{2R}\|$ .", "To address the complementary case, we just notice that, setting for the sake of brevity ${\\mathcal {F}}:={\\rm Per}_s+\\beta \\,V_\\alpha $ , by (REF ) and by minimality of $F_h$ one has ${\\mathcal {F}}(B)>{\\mathcal {F}}(E_h)\\ge {\\mathcal {F}}(F_h)+M\\,\|F_h\\Delta E_h\|\\,.$ In particular ${\\rm Per}_s(F_h)\\le {\\mathcal {F}}(B)$ for every $h\\in \\mathbb {N}$ .", "Hence, if $\|E\|\\ge \|B_{2R}\|$ then by $F_h\\subset B_R$ we have ${\\rm Per}_s(E)+\\Lambda \\,\|E\\Delta F_h\|\\ge \\Lambda (\|E\|-\|F_h\|)\\ge \\Lambda \\,\|B\|(2^n-1)\\,R^n\\ge {\\rm Per}_s(F_h)\\,,$ provided $\\Lambda \\ge \\frac{{\\mathcal {F}}(B)}{\|B\|(2^n-1)\\,R^n}\\,.$ We choose $\\Lambda $ to be the maximum between the right-hand sides of (REF ) and (REF ), and in this way (REF ) is proved.", "We now notice that by (REF ), (REF ), and up to discard finitely many $h$ 's, we can assume that $\|F_h\\Delta B\|\\le \\frac{2\\,{\\mathcal {F}}(B)}{M}\\,,\\qquad \\forall h\\in \\mathbb {N}\\,.$ Let now $\\varepsilon _\\beta $ be defined as in Theorem REF .", "By Corollary REF there exist $\\alpha \\in (0,1)$ and $\\delta >0$ (depending on $n$ , $s$ and $\\alpha $ only) such that the following holds: If $F$ is a $\\Lambda $ -minimizer of the $s$ -perimeter with $\|F\\Delta B\|<\\delta $ ($\\Lambda $ as in (REF )), then there is $u\\in C^{1,\\alpha }(\\partial B)$ such that $\\partial F=\\Big \\lbrace (1+u(x))\\,x:x\\in \\partial B\\Big \\rbrace \\,,\\qquad \\Vert u\\Vert _{C^1(\\partial B)}<\\varepsilon _\\beta \\,.$ Hence, by (REF ) and (REF ), we can choose $M$ large enough (depending on $n$ , $s$ and $\\alpha $ ) in such a way that, for every $h\\in \\mathbb {N}$ , there exists $u_h\\in C^{1,\\alpha }(\\partial B)$ with $\\partial F_h=\\Big \\lbrace (1+u_h(x))\\,x:x\\in \\partial B\\Big \\rbrace \\,,\\qquad \\Vert u_h\\Vert _{C^1(\\partial B)}<\\varepsilon _\\beta \\,.$ Let us set $t_h:=(\|F_h\|/\|B\|)^{1/n}$ and $G_h:=x_h+t_h\\,F_h$ for $x_h$ such that $\\int _{G_h}x\\,dx=0$ .", "By (REF ), we can make $\|t_h-1\|$ small enough in terms of $\\varepsilon _\\beta $ to entail that for every $h\\in \\mathbb {N}$ there exists $v_h\\in C^{1,\\alpha }(\\partial B)$ with $\\partial G_h=\\Big \\lbrace (1+v_h(x))\\,x:x\\in \\partial B\\Big \\rbrace \\,,\\qquad \\Vert v_h\\Vert _{C^1(\\partial B)}<\\varepsilon _\\beta \\,.$ By Theorem REF we conclude that ${\\mathcal {F}}(B)\\le {\\mathcal {F}}(G_h)=t_h^{n-s}\\,{\\rm Per}_s(F_h)+t_h^{n+\\alpha }\\,\\beta \\,V_\\alpha (F_h)\\le \\max \\lbrace t_h^{n-s},t_h^{n+\\alpha }\\rbrace \\,{\\mathcal {F}}(F_h)\\,,$ which in turn gives, in combination with (REF ), $\\frac{{\\mathcal {F}}(B)}{\\max \\lbrace t_h^{n-s},t_h^{n+\\alpha }\\rbrace }+M\\,\|F_h\\Delta E_h\|\\le {\\mathcal {F}}(B)\\,.$ If $t_h=1$ for a value of $h$ , then by (REF ) we find $F_h=E_h$ and thus ${\\mathcal {F}}(F_h)={\\mathcal {F}}(E_h)<{\\mathcal {F}}(B)$ , a contradiction to (REF ).", "At the same time, since ${\\mathcal {F}}(B)>0$ , (REF ) implies that $t_h\\ge 1$ for every $h\\in \\mathbb {N}$ .", "We may thus assume that $t_h>1$ for every $h\\in \\mathbb {N}$ .", "Since $\|F_h\\Delta E_h\|\\ge \|\|F_h\|-\|B\|\|=\|B\|\\,(t_h^n-1)$ , by (REF ) we find $M\\,\|B\|\\,(t_h^n-1)\\le {\\mathcal {F}}(B)\\,\\Big (1-\\frac{1}{t_h^{n+\\alpha }}\\Big )\\,,$ where, say, $t_h\\in (1,3/2)$ for every $h\\in \\mathbb {N}$ .", "However, if $M$ is large enough depending on $n$ , $s$ , and $\\alpha $ only, we actually have that $M\\,\|B\|\\,(t^n-1)>{\\mathcal {F}}(B)\\,\\Big (1-\\frac{1}{t^{n+\\alpha }}\\Big )\\,,\\qquad \\forall t\\in (1,3/2)\\,.$ We thus find a contradiction also in the case that $t_h>1$ for every $h\\in \\mathbb {N}$ .", "This completes the proof of the lemma.", "Given $m>0$ let us define $\\beta >0$ by setting $\\beta =\\Big (\\frac{m}{\|B\|}\\Big )^{(n+\\alpha )/n}\\,\\Big (\\frac{\|B\|}{m}\\Big )^{(n-s)/n}=\\Big (\\frac{m}{\|B\|}\\Big )^{(s+\\alpha )/n}\\,.$ (Notice that $\\beta <\\beta _\\star $ if and only if $m<m_\\star $ , since by (REF ) and (REF ) we have $m_\\star =\|B\|\\,\\beta _\\star ^{n/(s+\\alpha )}$ .)", "By exploiting this identity and the scaling properties of ${\\rm Per}_s$ and $V_\\alpha $ , and denoting by $B[m]$ a ball of volume $m$ , given $\\delta >0$ we notice that ${\\rm Per}_s(B)+\\beta \\,V_\\alpha (B)\\le {\\rm Per}_s(F)+\\beta \\,V_\\alpha (F)\\,,\\qquad \\mbox{whenever $\|F\|=\|B\|$ and $\|F\\Delta B\|<\\delta $}$ if and only if ${\\rm Per}_s(B[m])+V_\\alpha (B[m])\\le {\\rm Per}_s(E)+V_\\alpha (E)\\,,\\qquad \\mbox{whenever $\|E\|=m$ and $\|E\\Delta B[m]\|<\\frac{m}{\|B\|}\\,\\delta $}\\,.$ As a consequence, Theorem REF is equivalent to Lemma REF ." ], [ "A simple $\\Gamma $ -convergence result", "Here we prove the $\\Gamma $ -convergence of $P_s$ to $P_{s_}$ in the limit $s\\rightarrow s_$ , with $s_\\in (0,1)$ fixed.", "Of course, if $\|(E_h\\Delta E)\\cap K\|\\rightarrow 0$ for every $K\\subset \\subset \\mathbb {R}^n$ and $s_h\\rightarrow s_\\in (0,1)$ as $h\\rightarrow \\infty $ , then by Fatou's lemma one easily obtains $P_{s_}(E)\\le \\liminf _{h\\rightarrow \\infty }P_{s_h}(E_h)\\,,$ that is the $\\Gamma $ -liminf inequality.", "The proof of the $\\Gamma $ -limsup inequality is only slightly longer.", "For the sake of simplicity, we shall limit ourselves to work with bounded sets (this is the case we need in the paper).", "Precisely, given a bounded set $F\\subset \\mathbb {R}^n$ , we want to construct a sequence $\\lbrace F_h\\rbrace _{h\\in \\mathbb {N}}$ of bounded sets such that $\|F_h\\Delta F\|\\rightarrow \\infty $ as $h\\rightarrow \\infty $ and $\\limsup _{h\\rightarrow \\infty }P_{s_h}(F_h)\\le P_{s_}(F)\\,.$ We now prove (REF ).", "We start by recalling the following nonlocal coarea formula due to Visintin [37], $\\int _{\\mathbb {R}^n}dx\\int _{\\mathbb {R}^n}\\frac{\|u(x)-u(y)\|}{\|x-y\|^{n+s}}\\,dy=2\\,\\int _0^1\\,P_s(\\lbrace u>t\\rbrace )\\,dt\\,,\\qquad s\\in (0,1)\\,,$ that holds true (as an identity in $[0,\\infty ]$ ) whenever $u:\\mathbb {R}^n\\rightarrow [0,1]$ is Borel measurable; see [3].", "Next we use [29] to infer that if $P_{s_}(F)<\\infty $ and we set $u_\\varepsilon =1_F\\star \\rho _\\varepsilon $ , $\\rho _\\varepsilon $ a standard $\\varepsilon $ -mollifier, then $\\lim _{\\varepsilon \\rightarrow 0^+}\\int _{\\mathbb {R}^n}dx\\int _{\\mathbb {R}^n}\\frac{\|u_\\varepsilon (x)-u_\\varepsilon (y)\|}{\|x-y\|^{n+s_}}\\,dy=2\\,P_{s_}(F)\\,.$ Combining (REF ) and (REF ) with a classical argument by De Giorgi, see, e.g.", "[31], we reduce the proof of (REF ) to the case that $F$ is a bounded, smooth set.", "This implies that $P_s(F)<\\infty $ for every $s\\in (0,1)$ .", "In particular, if we let $s_{}\\in (0,1)$ be such that $s_h<s_{}$ for every $h\\in \\mathbb {N}$ , then we trivially find that, for every $(x,y)\\in \\mathbb {R}^n\\times \\mathbb {R}^n$ , $\\frac{1_{F\\times F^c}(x,y)}{\|x-y\|^{n+s_h}}\\le 1_{(F\\times F^c)\\cap \\lbrace \|x-y\|>1\\rbrace }(x,y)+\\frac{1_{F\\times F^c\\cap \\lbrace \|x-y\|\\le 1\\rbrace }(x,y)}{\|x-y\|^{n+s_{}}}=:g(x,y)\\,,$ where $g\\in L^1(\\mathbb {R}^n\\times \\mathbb {R}^n)$ thanks to the fact that $P_{s_{}}(F)<\\infty $ .", "In particular, $\\lim _{h\\rightarrow \\infty }P_{s_h}(F)=P_{s_}(F)\\,,$ whenever $s_h\\rightarrow s_\\in (0,1)$ as $h\\rightarrow \\infty $ and $F$ is a smooth bounded set.", "This proves (REF )." ], [ "A geometric lemma", "The following natural fact, which is well-known in the case of the classical perimeter, was used in the proof of Lemma REF .", "We give a proof since it may be useful elsewhere.", "Lemma 2.1 If $s\\in (0,1]$ and $E\\subset \\mathbb {R}^n$ is such that $P_s(E)<\\infty $ , then $P_s(E\\cap K)\\le P_s(E)$ for every convex set $K\\subset \\mathbb {R}^n$ .", "The case $s=1$ being classical, we can assume $s<1$ .", "Since any convex set can be written as a countable intersections of half-space, it is enough to prove that $P_s(E\\cap H)\\le P_s(E)$ whenever $H$ is an half-space.", "By approximation, it suffices to prove this estimate when $E$ is bounded.", "We now observe that, if we set $F:=E\\cup H$ , using that $E\\subset F$ , $E\\setminus H=F\\setminus H$ , and $F\\cap H=H$ , we get $P_s(E)-P_s(E\\cap H)&=&\\int _E \\int _{E^c} \\frac{dx\\,dy}{\|x-y\|^{n+s}}-\\int _{E\\cap H}\\int _{(E\\cap H)^c}\\frac{dx\\,dy}{\|x-y\|^{n+s}}\\\\&=&\\biggl ( \\int _{E\\cap H} + \\int _{E\\setminus H}\\biggr )\\int _{E^c}\\frac{dx\\,dy}{\|x-y\|^{n+s}}- \\biggl ( \\int _{E^c} + \\int _{E\\setminus H}\\biggr )\\int _{E\\cap H}\\frac{dx\\,dy}{\|x-y\|^{n+s}}\\\\&=&\\biggl (\\int _{E^c} - \\int _{E\\cap H} \\biggr )\\int _{E\\setminus H} \\frac{dx\\,dy}{\|x-y\|^{n+s}}\\\\&\\ge &\\biggl (\\int _{F^c} - \\int _{F\\cap H} \\biggr )\\int _{F\\setminus H} \\frac{dx\\,dy}{\|x-y\|^{n+s}}.$ We now observe that (just by doing the above steps backward) the last term is formally equal to $P_s(F)-P_s(H)$ .", "However, this does not really make sense as both $P_s(F)$ and $P_s(H)$ are actually infinite.", "For this reason, we have to consider a local version of $P_s$ : given a set $G$ and a domain $A$ , we define the $s$ -perimeter of $G$ inside $A$ as $P_s(G;A):=\\biggl (\\int _{G\\cap A}\\int _{G^c\\cap A}+\\int _{G\\cap A}\\int _{G^c\\cap A^c}+\\int _{G\\cap A^c}\\int _{G^c\\cap A}\\biggr )\\frac{dx\\,dy}{\|x-y\|^{n+s}}.$ With this notation, if $B_R$ is a large ball which contains $E$ (recall that $E$ is bounded), since $F$ is equal to $H$ outside $B_R$ it is easy to check that $\\biggl (\\int _{F^c} - \\int _{F\\cap H} \\biggr )\\int _{F\\setminus H} \\frac{dx\\,dy}{\|x-y\|^{n+s}}= P_s(F;B_R)-P_s(H;B_R).$ Applying [3] we deduce that $P_s(F;B_R)-P_s(H;B_R)\\ge 0$ , concluding the proof." ], [ "About the constant $\\beta _\\star $", "We have already noticed that, in order to show the equivalence between the two formulas (REF ) and (REF ) for $\\beta _\\star $ , it suffices to show that, for every $s\\in (0,1]$ and $\\alpha \\in (0,n)$ , one has $\\frac{\\lambda _k^s-\\lambda _1^s}{\\mu _k^\\alpha -\\mu _1^\\alpha }\\ge \\frac{\\lambda _2^s-\\lambda _1^s}{\\mu _2^\\alpha -\\mu _1^\\alpha }\\qquad \\forall k \\ge 2\\,.$ Proof of (REF ) in the case that $s\\in (0,1)$ and $\\alpha \\in (0,1)$ .", "In this case, the repeated application of the factorial property of the gamma function shows that (REF ) is equivalent in proving that the quantity $X_k:=\\frac{\\frac{\\prod _{j=1}^k\\left(j+\\frac{n+s}{2}\\right)}{\\prod _{j=1}^k\\left(j+\\frac{n-2-s}{2}\\right)} -1}{\\frac{\\prod _{j=1}^k\\left(j+\\frac{n-\\alpha }{2}\\right)}{\\prod _{j=1}^k\\left(j+\\frac{n-2+\\alpha }{2}\\right)} -1}$ attains its minimal value on $k\\ge 1$ at $k=1$ .", "To this end it is convenient to rewrite $X_k$ as follows: first, we notice that $X_k=\\frac{\\frac{\\prod _{j=1}^k\\left(j+\\frac{n-1}{2}+t\\right)}{\\prod _{j=1}^k\\left(j+\\frac{n-1}{2}-t\\right)} -1}{\\frac{\\prod _{j=1}^k\\left(j+\\frac{n-1}{2}+\\tau \\right)}{\\prod _{j=1}^k\\left(j+\\frac{n-1}{2}-\\tau \\right)} -1},\\qquad \\text{where}\\quad t:=\\frac{1+s}{2},\\quad \\tau :=\\frac{1-\\alpha }{2}\\,,$ (and thus, $0<\\tau <t$ ); second, we set $a_k:=\\prod _{j=2}^k\\alpha _j,\\qquad b_k:=\\prod _{j=2}^k\\beta _j,\\qquad c_k:=\\prod _{j=2}^k\\gamma _j,\\qquad d_k:=\\prod _{j=2}^k\\delta _j\\,,$ where $\\alpha _k:=k+\\frac{n-1}{2}+t,\\qquad \\beta _k:=k+\\frac{n-1}{2}-t,$ $\\gamma _k:=k+\\frac{n-1}{2}+\\tau ,\\qquad \\delta _k:=k+\\frac{n-1}{2}-\\tau .$ In this way, $X_k\\ge X_1$ for every $k\\ge 2$ can be rephrased into $\\frac{\\frac{a_k\\alpha _1 }{b_k\\beta _1} - 1}{\\frac{c_k\\gamma _1}{d_k\\delta _1}-1} \\ge \\frac{\\frac{\\alpha _1 }{\\beta _1} - 1}{\\frac{\\gamma _1}{\\delta _1}-1}\\,,\\qquad \\forall k\\ge 2\\,.$ It is useful to rearrange the terms in (REF ) and rewrite it as $a_k d_k \\alpha _1(\\gamma _1-\\delta _1)+b_kd_k(\\alpha _1\\delta _1 -\\beta _1\\gamma _1)+b_kc_k\\gamma _1(\\beta _1 - \\alpha _1) \\ge 0\\,,\\qquad \\forall k\\ge 2\\,.$ We now observe that, setting $\\ell :=(n+1)/2$ , we have $\\alpha _1=\\ell +t,\\quad \\beta _1=\\ell -t,\\quad \\gamma _1=\\ell +\\tau ,\\quad \\delta _1=\\ell -\\tau ,\\quad \\alpha _1\\delta _1 -\\beta _1\\gamma _1=2\\ell (t-\\tau ).$ Hence, substituting these formulas into the above expression we find that $\\mbox{left-hand side of (\\ref {due ridotta x})}&=&2a_kd_k(\\ell +t)\\tau + 2 b_kd_k\\ell (t-\\tau ) -2b_kc_k(\\ell +\\tau )t\\\\&=&2(a_kd_k - b_kc_k)t\\tau +2(a_k-b_k)d_k\\ell \\tau - 2 (c_k-d_k)b_k\\ell t\\,.$ Therefore (REF ) follows by showing that $&a_kd_k \\ge b_kc_k\\,,&\\qquad \\forall \\,k \\ge 2\\,,\\\\&(a_k-b_k)d_k \\tau \\ge (c_k-d_k)b_k t\\,,&\\qquad \\forall \\,k \\ge 2\\,.$ To prove (REF ) it suffices to observe that $\\alpha _j \\delta _j -\\beta _j \\gamma _j=2\\Bigl (j+\\frac{n-1}{2}\\Bigr )(t-\\tau ) \\ge 0\\qquad \\forall \\,j \\ge 1,$ so that $a_kd_k =\\prod _{j=2}^k\\alpha _j\\delta _j \\ge \\prod _{j=2}^k\\beta _j\\gamma _j=b_kc_k\\,,\\qquad \\forall k\\ge 2\\,,$ as desired.", "We now prove () by induction.", "A simple manipulation shows that () in the case $k=2$ is equivalent to $d_2\\ge b_2$ , which is true, so that we directly focus on the inductive hypothesis.", "By noticing that $a_{k+1}=a_k\\alpha _{k+1}$ , and that analogous identities hold for $\\beta _k$ , $\\gamma _k$ and $\\delta _k$ , we can equivalently reformulate the $(k+1)$ -case of () as $(a_k\\alpha _{k+1}-b_k\\beta _{k+1})d_k\\delta _{k+1}\\tau \\ge (c_k\\gamma _{k+1}-d_k\\delta _{k+1})b_k\\beta _{k+1} t\\,.$ This last inequality can be conveniently rewritten as $&&a_k(\\alpha _{k+1}-\\beta _{k+1}) d_k\\delta _{k+1}\\tau +\\beta _{k+1}\\delta _{k+1}(a_k-b_k)d_k\\tau \\\\&&\\hspace{85.35826pt}\\ge c_k(\\gamma _{k+1}-\\delta _{k+1})b_k\\beta _{k+1}t + \\beta _{k+1}\\delta _{k+1}(c_k-d_k)b_k t\\,.$ Indeed, by the inductive hypothesis $(a_k-b_k)d_k \\tau \\ge (c_k-d_k)b_k t$ , it is clear that a sufficient condition for this last inequality (and thus, for ()) to hold true, is that $a_k(\\alpha _{k+1}-\\beta _{k+1}) d_k\\delta _{k+1}\\tau \\ge c_k(\\gamma _{k+1}-\\delta _{k+1})b_k\\beta _{k+1}t\\,.$ By $\\alpha _{k+1}-\\beta _{k+1}=2t$ and $\\gamma _{k+1}-\\delta _{k+1}=2\\tau $ , (REF ) is equivalent to $2 (a_kd_k \\delta _{k+1} - b_k c_k \\beta _{k+1})t \\tau \\ge 0\\,.$ Finally, this inequality holds true because of (REF ) and the fact that $\\delta _{k+1}\\ge \\beta _{k+1}$ .", "This complete the proof of (), and thus of (REF ) in the case that $\\sigma \\in (0,1)$ and $\\alpha \\in (0,1)$ .", "Proof of (REF ) in the case that $s\\in (0,1)$ and $\\alpha \\in (1,n)$ .", "By the factorial property of the gamma function (REF ) is now equivalent in proving that $X_k\\ge X_1$ for every $k\\ge 2$ , where we have now set $X_k:=\\frac{\\frac{\\prod _{j=1}^k\\left(j+\\frac{n+s}{2}\\right)}{\\prod _{j=1}^k\\left(j+\\frac{n-2-s}{2}\\right)} -1}{1-\\frac{\\prod _{j=1}^k\\left(j+\\frac{n-\\alpha }{2}\\right)}{\\prod _{j=1}^k\\left(j+\\frac{n-2+\\alpha }{2}\\right)}}\\,.$ We notice that $X_k=\\frac{\\frac{\\prod _{j=1}^k\\left(j+\\frac{n-1}{2}+t\\right)}{\\prod _{j=1}^k\\left(j+\\frac{n-1}{2}-t\\right)} -1}{1-\\frac{\\prod _{j=1}^k\\left(j+\\frac{n-1}{2}-\\tau \\right)}{\\prod _{j=1}^k\\left(j+\\frac{n-1}{2}+\\tau \\right)}}\\,,\\qquad \\text{where}\\quad t:=\\frac{1+s}{2},\\quad \\tau :=\\frac{\\alpha -1}{2}\\,.$ We next define $a_k$ , $b_k$ , $c_k$ and $d_k$ as in (REF ), with $\\alpha _k$ , $\\beta _k$ , $\\gamma _k$ and $\\delta _k$ given by $\\alpha _k:=k+\\frac{n-1}{2}+t,\\qquad \\beta _k:=k+\\frac{n-1}{2}-t,$ $\\gamma _k:=k+\\frac{n-1}{2}-\\tau ,\\qquad \\delta _k:=k+\\frac{n-1}{2}+\\tau .$ We have thus reformulated (REF ) as $\\frac{\\frac{a_k\\alpha _1 }{b_k\\beta _1} - 1}{1-\\frac{c_k\\gamma _1}{d_k\\delta _1}} \\ge \\frac{\\frac{\\alpha _1 }{\\beta _1} - 1}{1-\\frac{\\gamma _1}{\\delta _1}}\\,,\\qquad \\forall k\\ge 2\\,,$ which is in turn equivalent to $a_k d_k \\alpha _1(\\delta _1-\\gamma _1)+b_kd_k(\\beta _1\\gamma _1-\\alpha _1\\delta _1)+b_kc_k\\gamma _1(\\alpha _1-\\beta _1) \\ge 0\\,,\\qquad \\forall k\\ge 2\\,.$ If we set $\\ell =(n+1)/2$ , then we find $\\alpha _1=\\ell +t,\\quad \\beta _1=\\ell -t,\\quad \\gamma _1=\\ell -\\tau ,\\quad \\delta _1=\\ell +\\tau ,\\quad \\alpha _1\\delta _1 -\\beta _1\\gamma _1=2\\ell (t+\\tau )\\,,$ so that $\\mbox{left-hand side of (\\ref {due ridotta xxx})}&=&2a_kd_k(\\ell +t)\\tau - 2 b_kd_k\\ell (t+\\tau ) +2b_kc_kt(\\ell -\\tau )\\\\&=&2(a_kd_k - b_kc_k)t\\tau +2(a_k-b_k)d_k\\ell \\tau + 2 (c_k-d_k)b_k\\ell t\\,.$ We are thus left to prove that $&a_kd_k \\ge b_kc_k\\,,&\\qquad \\forall \\,k \\ge 2\\,,\\\\&(a_k-b_k)d_k \\tau \\ge (d_k-c_k)b_k t\\,,&\\qquad \\forall \\,k \\ge 2\\,.$ To prove (REF ) it suffices to observe that $\\alpha _j \\delta _j -\\beta _j \\gamma _j=2\\Bigl (j+\\frac{n-1}{2}\\Bigr )(t+\\tau ) \\ge 0\\qquad \\forall \\,j \\ge 1,$ where $t>0$ and $\\tau >0$ .", "To prove () we argue once again by induction.", "One easily sees that () in the case $k=2$ is equivalent to say that $d_2\\ge b_2$ , which is true also in the present case.", "We now check the inductive hypothesis.", "The $(k+1)$ -case of () is now equivalent to $(a_k\\alpha _{k+1}-b_k\\beta _{k+1})d_k\\delta _{k+1}\\tau \\ge (d_k\\delta _{k+1}-c_k\\gamma _{k+1})b_k\\beta _{k+1} t\\,.$ We reformulate this as $&&a_k(\\alpha _{k+1}-\\beta _{k+1}) d_k\\delta _{k+1}\\tau +\\beta _{k+1}\\delta _{k+1}(a_k-b_k)d_k\\tau \\\\&&\\hspace{85.35826pt}\\ge c_k(\\delta _{k+1}-\\gamma _{k+1})b_k\\beta _{k+1}t + \\beta _{k+1}\\delta _{k+1}(d_k-c_k)b_k t\\,.$ By the inductive hypothesis $(a_k-b_k)d_k \\tau \\ge (d_k-c_k)b_k t$ , thus we are left to check that $a_k(\\alpha _{k+1}-\\beta _{k+1}) d_k\\delta _{k+1}\\tau \\ge c_k(\\delta _{k+1}-\\gamma _{k+1})b_k\\beta _{k+1}t\\,.$ By $\\alpha _{k+1}-\\beta _{k+1}=2t$ and $\\delta _{k+1}-\\gamma _{k+1}=2\\tau $ , (REF ) is equivalent to $2 (a_kd_k \\delta _{k+1} - b_k c_k \\beta _{k+1})t \\tau \\ge 0$ , which is true thanks to (REF ) and $\\delta _{k+1}\\ge \\beta _{k+1}$ .", "The proof of (), thus of (REF ) in the case that $\\sigma \\in (0,1)$ and $\\alpha \\in (1,n)$ , is now complete.", "Proof of (REF ) in the remaining cases.", "The case that $s\\in (0,1)$ and $\\alpha =1$ is covered by taking the limit as $\\alpha \\rightarrow 1^-$ with $s$ fixed in (REF ) for $\\alpha \\in (0,1)$ .", "This proves (REF ) for every $s\\in (0,1)$ and $\\alpha \\in (0,n)$ .", "The case $s=1$ is recovered by multiplying (REF ) by $1-s$ when $s\\in (0,1)$ and then taking the limit as $s\\rightarrow 1^-$ with $\\alpha $ fixed.", "The proof of (REF ) is now complete." ] ]	1403.0516
[ [ "Self-forced evolutions of an implicit rotating source: A natural\n framework to model comparable and intermediate mass-ratio systems from\n inspiral through ringdown" ], [ "Abstract We develop a waveform model to describe the inspiral, merger and ringdown of binary systems with comparable and intermediate mass-ratios.", "This model incorporates first-order conservative self-force corrections to the energy and angular momentum, which are valid in the strong-field regime [1].", "We model the radiative part of the self-force by deriving second-order radiative corrections to the energy flux.", "These corrections are obtained by minimizing the phase discrepancy between our self-force model and the effective one body model [2, 3] for a variety of mass-ratios.", "We show that our model performs substantially better than post-Newtonian approximants currently used to model neutron star-black hole mergers from early inspiral to the innermost stable circular orbit.", "In order to match the late inspiral evolution onto the plunge regime, we extend the 'transition phase' developed by Ori and Thorne [4] by including finite mass-ratio corrections and modelling the orbital phase evolution using an implicit rotating source [5].", "We explicitly show that the implicit rotating source approach provides a natural transition from late-time radiation to ringdown that is equivalent to ringdown waveform modelling based on a sum of quasinormal modes." ], [ "Introduction", "The black hole (BH) mass function in the local Universe is a strongly bi-modal distribution that identifies two main families: stellar-mass BHs with typical masses $\\sim 10M_{\\odot }$ observed in Galactic X-ray binaries [6] and, more recently, in globular clusters [7], and supermassive BHs with masses $\\mathrel {\\hbox{\\unknown.", "{\\hbox{$\\sim $}}}\\hbox{$>$}}$ 105 M$ observed to be present in most galactic nuclei~\\cite {Merloni:2008, Fukugita:2004}.", "However, a population of X-ray sources with luminosities in excess of $ 1039 erg s-1 $ has recently been observed, and {\\it {Chandra}} and {\\it {XMM-Newton}} spectral observations of these ultra-luminous X-ray sources (ULXs) revealed cool disc signatures that were consistent with the presence of intermediate mass BHs (IMBHs) with masses $ 102-4 M$~\\cite {Miller:2004,Miller:2004b,Miller:2006_BOOK}.", "Subsequent observations have shown that these ULXs have spectral and temporal signatures that are not consistent with the sub-Eddington accretion regime that is expected for IMBHs at typical ULX luminosities.", "Rather, these later studies suggest that many ULXs are powered by super-Eddington accretion onto $$\\sim $$<$ 100 M$ BH remnants.", "Nevertheless, recent work by Swartz et al.~\\cite {Swartz:2011} has demonstrated that as well as the high mass X-ray binaries that characterise most ULXs, there is a subpopulation of ULXs that seem to be powered by a separate physical mechanism.", "These objects have typical luminosities $ L$\\sim $$>$ 1041 erg s-1 $, which cannot be explained by close to maximal radiation from super-Eddington accretion onto massive BHs formed in low metallicity regions~\\cite {Zampieri:2009, Belczynski:2010, Ohsuga:2011}.", "Several hyper-luminous X-ray sources, including M82 X-1, ESO 243-49 HLX-1, Cartwheel N10 and CXO J122518.6+144545, present the best indirect evidence for the existence of IMBHs~\\cite {Matsumoto:2001,Farrel:2009,Wolter:2010,Jonker:2010}.", "In particular, the colocation of M82 X-1 with a massive, young stellar cluster, the features if its power spectrum, and some reported transitions between a hard state and a thermal dominant state, make this object a strong IMBH candidate~\\cite {Portegies:2004,Strohmayer:2003,Kaaret:2007,Feng:2010}.", "Recent searches of archival {\\it {Chandra}} and {\\it {XMM-Newton}} data sets have also uncovered two new hyper-luminous X-ray sources with luminosities in excess of $ 1039 erg s-1 $.", "These sources are the most promising IMBH candidates currently known, although the highest possible super-Eddington accretion rate onto the largest permitted BH remnant cannot yet be ruled out~\\cite {Sutton:2012}.", "This increasing body of observational evidence~\\cite {Trenti:2006,Coleman:2004}, and the fact that the existence of IMBHs provides a compelling explanation for the initial seeding of supermassive BHs present in most galactic nuclei~\\cite {Volonteri:2010,Schneider:2002,Yu:2002} has revived the quest for these elusive objects.$ Since hyper-luminous X-ray sources are rare, and our knowledge about their astrophysical properties is still limited, we may have to use a different means to search for IMBHs in order to improve our knowledge about the channels that lead to the formation of these objects, and to shed light on their astrophysical properties, such as mass and spin distributions [31].", "In this paper we explore the use of observations in the gravitational wave (GW) spectrum to gain insight into the properties of IMBHs.", "The current upgrade of the Laser Interferometer Gravitational Wave Observatory (LIGO) and its international partners Virgo and Kagra [32], [33], [34], will enable the detection of GWs from coalescences involving IMBHs with masses $50 M_{\\odot } \\mathrel {\\hbox{\\unknown.", "{\\hbox{$\\sim $}}}\\hbox{$<$}}$ M $\\sim $$<$ 500 M$, if these instruments achieve their target sensitivity down to the low-frequency cutoff at 10Hz (See Figure~\\ref {ZDHP_promise})~\\cite {ZDHP:2010}.", "Advanced LIGO (aLIGO) and Advanced Virgo are expected to have greatest sensitivity in the 60Hz - 500Hz range, with a peak at $ 60$ Hz (see Figure~\\ref {ZDHP_promise}).", "Proposed third generation detectors, such as the Einstein Telescope~\\cite {Freise:2009}, which aim to extend the frequency range of ground-based detectors down to 1Hz, while also maintaining high frequency sensitivity up to 10kHz, will enhance our ability to search for GWs emitted by sources that involve BHs with masses between $ 102-4M$~\\cite {etgair,Huerta:2011a,Huerta:2011b}.$ A promising channel for detection of IMBHs is through the emission of gravitational radiation during the coalescence of stellar-mass compact remnants — neutron stars (NSs) or BHs — with IMBHs in core-collapsed globular clusters.", "This expectation is backed up by numerical simulations of globular clusters [40], [41], [42], [43], [44], [45], [46], [47] which suggest that IMBHs could undergo several collisions with stellar-mass compact remnants during the lifetime of the cluster through a variety of mechanisms, including gravitational radiation, Kozai resonances and binary exchange processes.", "As discussed in [48], the most likely mechanism for the formation of binaries involving a stellar-mass compact remnant and an IMBH is hardening via three body interactions, with an expected detection rate of $\\sim 1-10\\, {\\rm {yr}}^{-1}$ with ground-based observatories [48], [49].", "Figure: The panel shows the expected sensitivity for two configurations of the Einstein Telescope (ET), namely, ETD (black), ETB (blue) and LIGO's Zero Detuned High Power (ZDHP) configuration (red).", "The vertical axis measures F normalized =f/f max -7/6 S n (f max )/S n (f)F_{\\rm {normalized}} = \\left(f/f_{\\rm {max}}\\right)^{-7/6}\\sqrt{S_n(f_{\\rm {max}})/S_n(f)}, where f max f_{\\rm {max}} is the maximum of the corresponding power spectral density, S n (f)S_n(f).The frequency of the dominant quadrupolar harmonic in the GWs emitted at the innermost stable circular orbit (ISCO) for a binary of non-spinning objects is $f_{\\rm {ISCO}}= 4.4 {\\rm {kHz}} \\left(\\frac{M_{\\odot }}{M}\\right),$ so for a typical intermediate mass–ratio coalescence (IMRC) with total mass $M\\mathrel {\\hbox{\\unknown.", "{\\hbox{$\\sim $}}}\\hbox{$>$}}$ 100 M$, advanced detectors will observe the late inspiral, merger and ringdown.", "However, the heaviest IMRCs, with masses $$\\sim $$>$ 500M$, will coalesce out of band or at the low frequency limit of the bandwidth.", "Hence, merger and ringdown --- which intrinsically generate a signal-to-noise ratio (SNR) suppressed by a factor of symmetric mass-ratio --- see Table~\\ref {length} below --- relative to the SNR generated during the inspiral phase--- will significantly contribute to the SNR of IMRCs over a considerable portion of the detectable mass-range~\\cite {Smith:2013}.", "In order to get the most information from GW observations of IMRCs, it is therefore necessary to develop waveform models that incorporate the inspiral, merger and ringdown in a physically consistent way.", "In order to make progress in this direction, we previously developed a waveform model that combined results from Black Hole Perturbation Theory (BHPT) and post-Newtonian (PN) theory to explore the information that could be obtained from observations of IMRCs with the EinsteinTelescope~\\cite {Huerta:2011a,Huerta:2011b}.", "Although this model provided an important step in exploring the science that could be done with IMRC observations, the model was limited.$ In [54], [55], [56], [51] we explored using the self-force formalism [52], [53] to develop a waveform model with a robust description of the dynamical evolution of IMRCs during the inspiral phase.", "These were found to be effective when used to carry out matched-filter based searches for inspiral-only IMRCs [50], but searches for IMRCs in the advanced detector era will require waveform models that include not only the inspiral but also the merger and ringdown [50].", "The model described in [38] included merger and ring down but without the self-force driven inspiral.", "It would be ideal to develop a consistent inspiral, merger, ringdown waveform model by comparison to numerical relativity simulations, as described in [2], [3].", "However, numerical relativity simulations for systems with mass-ratios $q=m_1/m_2 \\mathrel {\\hbox{\\unknown.", "{\\hbox{$\\sim $}}}\\hbox{$<$}}$ 1/10$ are very computationally expensive at present~\\cite {Mroue:2013}.", "We can circumvent this problem by making use of recent breakthroughs in the self-force program that have shown that the conservative part of the self-force can reproduce, with good accuracy, results from numerical relativity simulations of comparable-mass binary systems~\\cite {LeTiec:2012}.", "Furthermore, the recent computation of the self-force inside the ISCO equips us to now develop models that better reproduce the true dynamics of black hole binaries in the strong field regime~\\cite {Akcay:2012}.$ In addition to IMRCs, which may be detected by second generation detectors, and with more likelihood by third generation detectors, we have explored the suitability of using self-forced evolutions to model the mergers of systems involving NSs and stellar mass BHs [59].", "Since NSBH mergers are promising GW sources for second generation detectors, we need to develop accurate and computationally inexpensive templates appropriate for these events.", "Current efforts to model NSBH systems have generally used PN approximants, evaluated to the highest PN order available, but these approximants are not sufficiently reliable to model these events [60], [61].", "In Figure REF , we show the phase difference between the PN approximant TaylorT4 [62] and the EOB model introduced in [2], [3].", "This exhibits a substantial discrepancy near the ISCO.", "These considerations have impelled us to develop a different approach to model events with the typical mass-ratios expected for NSBH binaries.", "Figure: The phase discrepancy in radians between the PN approximant TaylorT4, and the Effective One Body model, shown as a function of time from r=30Mr=30M to the point when the TaylorT4 model reaches the ISCO.", "The systems have mass-ratio, qq, total mass, MM, and final phase discrepancy, ΔΦ\\Delta \\Phi : (q,M,ΔΦ)=(1/6,7M ⊙ ,21.5 rads )(q, M, \\Delta \\Phi ) = (1/6, 7M_{\\odot },21.5\\,{\\rm rads}) (top-left), (1/8,9M ⊙ ,30.2 rads )(1/8, 9M_{\\odot },30.2\\,{\\rm rads}) (top-right), (1/10,11M ⊙ ,70.1 rads )(1/10, 11M_{\\odot },70.1\\,{\\rm rads}) (bottom-left) and (1/15,16M ⊙ ,83.2 rads )(1/15, 16M_{\\odot },83.2\\,{\\rm rads}) (bottom-right) respectively.In this paper we combine recent developments in the self-force program, in PN theory and in numerical relativity to develop a model that describes the inspiral, merger and ringdown of IMRCs and comparable mass-ratio systems that could be detected by second and third generation ground-based GW detectors.", "In Section we discuss the modelling of the conservative part of the self-force.", "We show that using the linear in mass-ratio self force results for binaries with mass-ratios $q\\mathrel {\\hbox{\\unknown.", "{\\hbox{$\\sim $}}}\\hbox{$>$}}$ 1/6$ gives a system without an ISCO.", "We then discuss the implications of this result for the modelling of comparable and intermediate mass--ratio binaries.", "Thereafter, we describe the approach we use to model the radiative part of the self-force for the inspiral evolution.", "Having constructed the inspiral part of the self-forced waveform model, we extend the transition scheme of Ori and Thorne~\\cite {ori} by including finite mass-ratio corrections, and modelling the orbital phase evolution using the implicit rotating source (IRS) model.", "We adopt this description for the late-time radiation in order to provide a smooth progression from late inspiral to rindgown.", "We show that this approach provides the correct orbital frequency evolution in the vicinity of the light-ring.", "Finally, we construct the ringdown waveform using both a sum of quasinormal modes and the late-time radiation waveform evolution predicted by the IRS model, and show their equivalence.", "Section~\\ref {conclu} presents a summary of our findings and future directions of work.$ Throughout this paper we will use units with $G=c=1$ , unless otherwise stated.", "We will consider BH binaries on circular orbits with component masses $m_1, m_2$ , such that $m_1 < m_2$ .", "We assume that the binary components are non spinning.", "We will use several combinations of the masses $m_{1\\, ,2}$ in the following Sections, which are summarized in Table REF .", "Table: The table summarizes the nomenclature we will use throughout our analysis.Having defined the variables to be used in the subsequent sections, we shall now describe the construction of the self-forced waveform model.", "The model consists of four building blocks — the inspiral, the transition, the plunge and the ringdown phases.", "The next section describes the inspiral evolution." ], [ "Inspiral evolution", "We model the inspiral phase evolution in the context of the Effective One Body (EOB) formalism [64], i.e., we consider the scenario in which the dynamics of a binary system is mapped onto the motion of a test particle in a time-independent and spherically symmetric Schwarzschild space-time with total mass $M$ : $\\mathrm {d}s^{2}_{\\rm {EOB}} = -A(r)\\mathrm {d}t^2 + B(r)\\mathrm {d}t^2 + r^2\\mathrm {d}\\Omega ^2\\,,$ where the potentials $A, \\, B$ are known to 3PN order [65], [66].", "In the test-mass particle limit $\\eta \\rightarrow 0$ , these potentials recover the Schwarzschild results, namely: $A(u, \\eta \\rightarrow 0) = B^{-1}(u, \\eta \\rightarrow 0)= 1-2\\,u,\\quad {\\rm {with}} \\quad u=\\frac{M}{r}.$ In the EOB formalism, the orbital frequency evolution can be derived from a Hamiltonian, $H_{\\rm {EOB}}$ [64], given by: $H_{\\rm {EOB}} = M\\sqrt{1+2\\,\\eta \\left(H_{\\rm {eff}} -1\\right)},$ using the general Hamiltonian equation: $\\frac{d \\phi }{d \\mathrm {t} } = M\\Omega = \\frac{\\partial H_{\\rm {EOB}}}{\\partial L} = \\frac{u^2\\,L(x)\\,A(u)}{H(u)\\,H_{\\rm {eff}}(u)},$ where $H_{\\rm {eff}}(u) &=& \\frac{A(u)}{\\sqrt{\\tilde{A}(u)}}\\,, \\qquad \\tilde{A}(u)= A(u) + \\frac{1}{2}\\,u\\,A^{\\prime }(u), \\\\\\nonumber &&{\\rm {and}} \\,\\, \\quad H(u)=\\sqrt{1+2\\,\\eta \\left(H_{\\rm {eff}} -1\\right)}.\\\\\\nonumber $ Recent work in the self-force formalism has enabled the derivation of gravitational self-force corrections to the EOB potential $A(u)\\rightarrow 1-2u+\\eta \\, a(u) + {\\cal {O}}(\\eta ^2)$ [67].", "Deriving this gravitational self-force contribution, $a(u)$ , is equivalent to including all PN corrections to the EOB potential $A(u)$ at linear order in $\\eta $ .", "We shall now briefly describe the construction of the gravitational self-force contribution $a(u)$ , emphasizing the fact that this contribution encodes information about the strong-field regime of the gravitational field.", "As shown by Detweiler and Whiting [68], the gravitational self-force corrected worldline can be interpreted as a geodesic in a smooth perturbed spacetime with metric $g_{\\alpha \\beta } = g^{0}_{\\alpha \\beta }(m_2) + h^{R}_{\\alpha \\beta },$ where the regularized $R$ field is a smooth perturbation associated with $m_1$ .", "Detweiler proposed a gauge invariant relation to handle the conservative effect of the gravitational self-force in circular motion [69], [70]: $z_1(\\Omega )= \\sqrt{1-3x}\\left(1- \\frac{1}{2} h^{R,\\,F}_{uu} + q \\frac{x}{1-3x}\\right),$ where $x$ is the gauge-invariant dimensionless frequency parameter given by $x=\\left(M \\Omega \\right)^{2/3}$ , $h^{R,\\,G}_{uu}$ is a double contraction of the regularised metric perturbation with the four-velocity, $u^{\\mu }$ , $ h^{R,\\, G}_{uu} = h^{R,\\,G}_{\\mu \\nu }u^{\\mu } u^{\\nu } $ , the label $G$ indicates the gauge used to evaluate the metric perturbation and the label $F$ indicates that this expression is valid within the class of asymptotically flat gauges.", "In [1], $z_1(\\Omega )$ was calculated in Lorenz gauge and the following gauge transformation can be used to link the asymptotically flat $h^{R,\\,F}_{uu} $ metric perturbation to its Lorenz-gauge counterpart $h^{R,\\,L}_{uu} $ : $h^{R,\\,F}_{uu} = h^{R,\\,L}_{uu} + 2q\\frac{x(1-2x)}{\\left(1-3x)\\right)^{3/2}}.$ Hence, inserting Eq.", "(REF ) into Eq.", "(REF ) leads to $z_1(\\Omega )= \\sqrt{1-3x}\\left(1- \\frac{1}{2} h^{R,\\,L}_{uu} - 2q\\frac{x(1-2x)}{\\left(1-3x)\\right)^{3/2}} + q \\frac{x}{1-3x}\\right).$ The numerical data obtained in [1] for $h^{R,\\,L}_{uu} $ from $x>1/5$ was new and the numerical accuracy for $x<1/5$ was also much improved compared to previous results [69], [71].", "The EoB potential $a(x)$ can be constructed from $h_{uu}^{R,L}$ via $a(x) = -\\frac{1}{2}\\left(1-3x\\right)\\tilde{h}^{R,\\,L}_{uu} - 2x \\sqrt{1-3x},$ with $\\tilde{h}^{R,\\,L}_{uu}= q^{-1}h^{R,\\,L}_{uu}$ .", "In [1], a global fit formula for $a(x)$ was given a dense sample of numerical values over the entire range $0 < x < \\frac{1}{3}$ .", "The numerical fit for $a(x)$ that we use in this study is taken from Eq.", "(54) of [1].", "This global analytic fit for $a(x)$ can be recast using the relation $a(x)= 2x^3\\, \\frac{(1-2x)}{\\sqrt{1 - 3 x}}\\,a_{E}(x).$ Using the above dictionary, the model for $a(x)$ in this paper reproduces the numerical data points for the function $a_{E}(x)$ to within a maximal absolute difference of $1.2\\times 10^{-5}$ over the domain $0<x<\\frac{1}{3}$ .", "The corresponding self-force corrected energy and angular momentum are given by [1], [67] $E(u(x))&=&E_0(x) + \\eta \\left(-\\frac{1}{3}\\frac{x}{\\sqrt{1-3x}}a^{\\prime }(x) + \\frac{1}{2}\\frac{1-4x}{\\left(1-3x\\right)^{3/2}} a(x) -E_0(x)\\left( \\frac{1}{2}E_0(x) + \\frac{x}{3}\\frac{1-6x}{\\left(1-3x\\right)^{3/2}} \\right)\\right),\\\\L_z (u(x))&=& L_0(x) + \\eta \\left( -\\frac{1}{3}\\frac{x}{\\sqrt{x(1-3x)}}a^{\\prime }(x) -\\frac{1}{2}\\frac{1}{\\sqrt{x}\\left(1-3x\\right)^{3/2}} a(x) -\\frac{1}{3 }\\frac{1-6x}{ \\sqrt{x} \\left(1-3x\\right)^{3/2}}\\left(E_0(x)-1\\right) \\right)\\,,\\\\&&{\\rm {with}}\\qquad \\qquad u(x)= x\\left( 1+ \\eta \\Bigg [\\frac{1}{6}a^{\\prime }(x) + \\frac{2}{3}\\left(\\frac{1-2x}{\\sqrt{1-3x}} -1 \\right) \\Bigg ] \\right)\\,,$ where $E_0(x)$ and $L_0(x)$ are given by $E_0(x) &=& \\frac{1-2x}{\\sqrt{1 - 3 x}} -1,\\\\L_0(x)&=& \\frac{1}{\\sqrt{x (1 - 3 x)}}.$ Figure: The panels show the energy and angular momentum given by Eqs.", "()-(), respectively.", "We show the functional form of these parameters for binary systems with mass-ratio values, from top to bottom, q∈[0,1/100,1/20,1/10,1/6,1/5,1]q \\in [0,\\, 1/100, \\,1/20, \\,1/10, \\,1/6, \\,1/5, \\,1 ].In Figure REF , we show the effect of these conservative corrections on the orbital parameters.", "As discussed in [67], if one makes use of the self-force expression for the energy given by Eq.", "(REF ), and minimize it with respect to the orbital frequency, then one finds that binary systems with mass-ratios $q\\in [1, \\,1/2, \\,1/3]$ do not have an ISCO in this model.", "It was argued in [67] that deriving self-force results in the strong field regime may alleviate this problem.", "We have explored this issue, and have found that using linear self-force corrections that are valid all the way to the light ring (last unstable circular orbit for massless particles) does not fix this problem for comparable mass-ratio systems.", "In Figure REF , we show that the existence of an ISCO is guaranteed for BH binaries with symmetric mass-ratio $\\eta \\mathrel {\\hbox{\\unknown.", "{\\hbox{$\\sim $}}}\\hbox{$<$}}$ 6/49 (or q $\\sim $$<$ 1/6)$, and its location may be approximated by\\begin{equation}x_{\\mathrm {ISCO}}=\\frac{1}{6}\\left(1+ 0.83401\\eta +4.59483\\eta ^2\\right).\\end{equation}It remains to be seen whether the inclusion of second-order conservative corrections gives an ISCO for binaries with mass-ratios $ q$\\sim $$>$ 1/6$.$ Figure: The location of the innermost stable circular orbit is determined by the condition dE/dx=0{\\rm {d}}E/\\rm {d}x = 0.", "The panel shows dE/dx{\\rm {d}}E/\\rm {d}x as a function of the gauge invariant quantity x=MΩ 2/3 x=\\left(M\\,\\Omega \\right)^{2/3}.", "The various curves represent binary systems with mass-ratios, from top to bottom, q∈[0,1/100,1/20,1/10,1/6,1/5,1]q \\in [0,\\, 1/100, \\,1/20, \\,1/10, \\,1/6, \\,1/5, \\,1 ].", "Note that binaries with mass-ratios qbad hbox>q\\mathrel {\\hbox{\\unknown.", "{\\hbox{$\\sim $}}}\\hbox{$>$}}1/6donothaveanISCOinthismodel.", "do not have an ISCO in this model.", "In summary, the building blocks we use to construct the conservative dynamics are The orbital frequency evolution is computed using Eq.", "(REF ) with the gravitational self-force contribution included in the potential $A(u)= 1-2u + \\eta \\, a(u)$ .", "Eq.", "(REF ) is evaluated using the self-force-corrected expression for the angular momentum, $L(x)$ , given by Eq. ().", "The self-force-corrected expression for the energy, given in Eq.", "(REF ) is only used to determine the point at which the inspiral ends and the transition region begins.", "Eq.", "(REF ) provides a very accurate modeling of the orbital frequency from early inspiral through the ISCO.", "However, the post-ISCO time evolution of this prescription does not render an accurate representation of the orbital frequency as compared to numerical relativity simulations.", "This is a problem that has been addressed in the EOB formalism by introducing a phenomenological approach —the so-called non-quasi-circular coefficients— that enabled them to reproduce the orbital evolution extracted from numerical simulations [2].", "The approach we will follow to circumvent this problem is described in detail in Section REF , and consists of embedding the self-force formalism in the context of the implicit rotating source model [5] after the ISCO.", "This completes the description of the conservative part of the self-force.", "We now describe how to couple this with the radiative part of the self-force to model the inspiral evolution." ], [ "Dissipative dynamics", "A consistent self-force evolution model that incorporates first-order in mass-ratio conservative corrections should also include second-order radiative corrections.", "The model we described in the previous section was constructed including first order conservative corrections.", "However, second-order self-force radiative corrections are not known at present.", "Several studies have demonstrated the importance of including the missing second order corrections to the radiative part of the self-force, both for source detection and for parameter estimation [72], [73], [51], [55], [54].", "We use a new prescription for the energy flux that uses the first-order in mass ratio terms derived in [74], including PN corrections up to $22^\\mathrm {nd}$ PN order $\\left(\\dot{E}\\right)_{\\rm PN} &=& -\\frac{32}{5}\\frac{\\mu ^2}{M}x^{7/2}\\Bigg [1-\\frac{1247}{336}x + 4\\pi x^{3/2} - \\frac{44711}{9072}x^2-\\frac{8191}{672}\\pi x^{5/2} \\\\\\nonumber &+&x^3 \\bigg \\lbrace \\frac{6\\,643\\,739\\,519}{69\\,854\\,400} +\\frac{16}{3}\\pi ^2 -\\frac{1712}{105}\\gamma _{\\rm E} -\\frac{856}{105}\\ln (16x) \\Big \\rbrace -\\frac{16285}{504} \\pi x^{7/2} \\\\\\nonumber &+& x^4 \\Big \\lbrace -\\frac{323105549467}{3178375200} + \\frac{232597}{4410}\\gamma _{\\rm E} -\\frac{1369}{126}\\pi + \\frac{39931}{294}\\ln (2) -\\frac{47385}{1568}\\ln (3) +\\frac{232597}{4410}\\ln (x) \\Big \\rbrace \\\\\\nonumber &+& x^{9/2}\\Big \\lbrace \\frac{265978667519}{745113600}\\pi -\\frac{6848}{105}\\gamma _{\\rm E}\\pi -\\frac{13696}{105}\\pi \\ln (2) -\\frac{6848}{105}\\pi \\ln (x)\\Big \\rbrace \\\\\\nonumber &+& {\\rm { higher \\, order \\, corrections \\, up\\, to \\, 22PN \\, order}} \\Bigg ].$ We include higher-order in mass-ratio terms using the exponential resummation approach described in [72].", "In this approach, the energy flux is $\\left(\\frac{{\\mathrm {d}}E}{{\\mathrm {d}}t}\\right)_{\\rm hybrid} &=& {\\cal {L}}_{\\rm 0} \\exp \\left( {\\cal {L}}_{\\eta } \\right)\\, ,$ where $ {\\cal {L}}_{\\rm 0} $ denotes the leading-order in mass-ratio PN energy flux given in Eq.", "(REF ), and $ {\\cal {L}}_{\\eta } $ incorporates mass-ratio corrections to the highest PN order available [75], [76], [72], and additional corrections characterised by a set of unknown coefficients, $b_i$ ${\\cal {L}}_{\\eta } &=& \\Bigg [x\\bigg [-\\frac{35}{12}\\eta + b_1\\,\\eta ^2\\bigg ] + 4\\pi x^{3/2}\\bigg [ b_2\\, \\eta + b_3 \\eta ^2 \\bigg ] + x^2 \\bigg [\\frac{9271}{504}\\eta + \\frac{65}{18}\\eta ^2\\bigg ] + \\pi x^{5/2}\\bigg [ -\\frac{583}{24} \\eta + b_4\\, \\eta ^2\\bigg ] \\\\\\nonumber &+& x^3 \\bigg [\\eta \\left(-\\frac{134\\,543}{7\\,776} + \\frac{41}{48}\\pi ^2\\right) -\\frac{94403}{3024}\\eta ^2 - \\frac{775}{324}\\eta ^3\\bigg ] + \\pi x^{7/2} \\bigg [\\frac{214745}{1728} \\eta + \\frac{193385}{3024}\\eta ^2\\bigg ] \\Bigg ].$ The coefficients $b_i$ were taken to be constant in [72], but we found that a better match to the EOB phase evolution could be obtained by allowing an additional dependence on mass-ratio in these terms (see Eqs.", "(REF )-() below).", "We constrain the $b_i$ coefficients by ensuring that the phase evolution of this model reproduces the phase evolution predicted by the EOB model introduced in [2], [3], which was calibrated to the phase evolution of compact binaries observed in numerical relativity simulations.", "To do so, we implemented the EOB model [2] and performed a Monte Carlo simulation to optimize the values of the $b_i$ coefficients (see Figure REF ).", "The optimization was done in two stages.", "We started by considering the three coefficients $b_1,\\, b_2$ and $b_4$ , sampling a wide range of parameter space, namely $b_i\\in [-200,200]$ .", "We constrained the duration of the waveform from early inspiral to the light-ring to be similar to its EOB counterpart.", "Waveforms that differed from their EOB counterparts by more than $10^{-4}$ seconds were discarded.", "Once the region under consideration had been sparsely sampled, we focused on regions of parameter space where the orbital phase evolution was closest to the EOB evolution, and finely sampled these to obtain the optimal values for the coefficients.", "We found that this approach enabled us to reproduce the EOB phase evolution with a phase discrepancy of the order $\\sim 1$ rad.", "After constraining $b_1,\\, b_2$ and $b_4$ , we explored whether including additional corrections could further improve the phase evolution, by adding $\\eta $ corrections beyond 3PN order.", "Such corrections were found to have a negligible impact on the actual phase evolution.", "This is not difficult to understand, since such corrections are of order $({\\cal {O}}(\\eta ^4),\\, {\\cal {O}}(\\eta ^3))$ , at (3PN, 3.5PN) respectively.", "We found a similar behavior when we added leading order mass-ratios corrections beyond 4PN order.", "Thus, we took a different approach: having derived the optimal value for $b_1,\\, b_2$ and $b_4$ , we took these results as initial seeds for an additional MC simulation in which $b_3$ was also included in Eq.", "(REF ), and repeated the optimization procedure.", "The results of these simulations are shown in Figure REF .", "We carried out several different Monte Carlo runs to find the `optimal' optimization interval, meaning the range of radial separations over which we tried to best match the phase evolution relative to the EOB model.", "We found that starting the optimization at $r=30M$ gave results that performed moderately well at early inspiral, but that underperformed at late inspiral, leading to phase discrepancies of order $\\sim 3$ rads.", "Starting the optimization at $r=20M$ instead decreased the phase discrepancy with respect to the former case by a factor of 10 during early inspiral, and enabled us to reproduce the phase evolution in the EOB model for all the mass-ratios considered to within the accuracy of the numerical waveforms used to calibrate the EOB model in [2], [3].", "Implementing these numerically optimized higher-order $\\eta $ corrections in Eq.", "(REF ) leads to: ${\\cal {L}}_{\\eta } &=& \\Bigg [x\\bigg [-\\frac{35}{12}\\eta + B_1\\bigg ] + 4\\pi x^{3/2}B_2 + x^2 \\bigg [\\frac{9271}{504}\\eta + \\frac{65}{18}\\eta ^2\\bigg ] + \\pi x^{5/2}\\bigg [ -\\frac{583}{24} \\eta + B_3\\bigg ] \\\\\\nonumber &+& x^3 \\bigg [\\eta \\left(-\\frac{134\\,543}{7\\,776} + \\frac{41}{48}\\pi ^2\\right) -\\frac{94403}{3024}\\eta ^2 - \\frac{775}{324}\\eta ^3\\bigg ] + \\pi x^{7/2} \\bigg [\\frac{214745}{1728} \\eta + \\frac{193385}{3024}\\eta ^2\\bigg ] \\Bigg ].$ where: $B_1&=& \\frac{1583.650 - 11760.507\\, \\eta }{1 + 142.389\\, \\eta - 981.723\\, \\eta ^2}\\,\\eta ^2\\,,\\\\B_2 &=& \\frac{-12.081 + 35.482\\, \\eta }{1 - 4.678 \\eta + 13.280\\, \\eta ^2}\\,\\eta + \\frac{19.045 - 240.031\\, \\eta }{1 - 18.461\\, \\eta + 74.142\\, \\eta ^2}\\,\\eta ^2\\,,\\\\B_3 &=& \\frac{51.814 - 980.100\\, \\eta }{1 - 13.912\\, \\eta + 88.797\\, \\eta ^2}\\,\\eta ^2\\,.$ This improved prescription for the energy flux, which incorporates second-order mass-ratio corrections to the PN expansion up to 3.5PN order, is sufficient to generate a model whose phase evolution reproduces with excellent accuracy the phase evolution predicted by EOB throughout inspiral and merger (see Figure REF ).", "Given the energy flux defined by Eqs (REF )–(REF ), we generate the inspiral trajectory using the simple prescription $\\frac{{\\mathrm {d}}x}{{\\mathrm {d}}t}= \\frac{{\\mathrm {d}} E}{{\\mathrm {d}} t}\\frac{{\\mathrm {d}} x}{{\\mathrm {d}}E}\\,,$ where we have used the mass-ratio corrected energy —Eq.", "(REF )— to compute ${\\mathrm {d}}E/{\\mathrm {d}}x$ .", "Figure REF shows that for binaries with mass-ratio $q=1/6$ , the phase discrepancy between our self-force model and EOB is $\\mathrel {\\hbox{\\unknown.", "{\\hbox{$\\sim $}}}\\hbox{$<$}}$ 0.5$ rads at the light-ring, which is within the numerical accuracy of the simulations used to calibrate EOB.", "It has been shown recently that EOB remains accurate for mass-ratios up to $ q=1/8$~\\cite {Pan:2013}.", "In that regime the phase discrepancy between this model and EOB at the light-ring is less than 1 rad, as shown in Figure~\\ref {PNoptimized}.", "For binaries with $ q=1/10$, the phase discrepancy at the light-ring is $$\\sim $$<$ 1.2$ rads, which is still within the numerical accuracy of available simulations~\\cite {carlosI, carlosII}.$ Figure: The (top, bottom) panels show the results of the optimization runs that were used to constrain the values of the b i b_i coefficients given in Eq. ().", "The panels show the results for binaries of mass–ratio q∈[1/6,1/8]q\\in [1/6,\\, 1/8], and total mass M=[7M ⊙ ,9M ⊙ ]M= [7M_{\\odot },\\, 9M_{\\odot }].", "The `optimal' value for the coefficients has been chosen by ensuring that the flux prescription minimizes the phase discrepancy between the EOB model and our self-force model.", "The color bar shows the phase difference squared between both models, which is integrated from r=20Mr=20M all the way to the light-ring.It must be emphasized that even if we only use the inspiral evolution to model binaries with mass-ratios that typically describe NSBH binaries, our self-force evolution model performs better than TaylorT4, since we can reduce the phase discrepancy between TaylorT4 and EOB at the last stable circular orbit by a factor of $(\\sim 40, \\, \\sim 70)$ for binaries with $q=(1/6,\\,1/10)$ and total mass $M\\in (7M_{\\odot } ,\\, 11M_{\\odot } )$ (see Figure REF ).", "Figure: The panels show the orbital phase evolution of a self-force model making use of optimized PN energy flux given by Eq.", "() and the phase evolution as predicted by the EOB model.", "The [top/bottom] panels exhibit this evolution for a compact binary with mass–ratio q=[(1/6,1/8),(1/10,1/15)]q=[(1/6,\\,1/8), \\,( 1/10,\\,1/15)], and total mass M=[(7M ⊙ ,9M ⊙ ),(11M ⊙ ,16M ⊙ )]M=[ (7M_{\\odot } ,\\, 9M_{\\odot } ), \\, ( 11M_{\\odot },\\, 16M_{\\odot }) ], respectively.Figure REF conveys an important message — deriving the second order corrections to the radiative part of the self-force may well provide a robust framework to describe in a single model not only events that are naturally described by BHPT, such as the mergers of stellar mass compact objects with supermassive BHs in galactic nuclei [51], [55], [80], [81], [82], [52], [83], but also events that are more naturally described by PN or numerical methods, in particular the coalescences of compact object binaries with comparable or intermediate mass-ratios [51], [84], [38], [39], [85].", "To finish this Section, we describe the approach followed to construct the gravitational waveform from the inspiral trajectory.", "At leading PN order, a general inspiral waveform can be written as $h(t) = -(h_{+} - i h_{\\times }) = \\sum _{\\ell =2}^{\\infty } \\sum _{m=-\\ell }^{l} h^{\\ell m} {}_{-\\!2}Y_{\\ell m}(\\iota ,\\Phi ).$ If only the leading-order modes $(\\ell ,m)=(2, \\pm 2)$ and included, the inspiral waveform components are given by $h_{+}(t)&=& \\frac{4\\, \\mu \\, r^2\\, \\dot{\\phi }^2 }{D}\\left(\\frac{1+\\cos ^2 \\iota }{2}\\right)\\cos \\left[2(\\phi (t) + \\Phi )\\right],\\\\h_{\\times }(t)&=& \\frac{4\\, \\mu \\, r^2\\, \\dot{\\phi }^2}{D} \\cos \\iota \\sin \\left[2(\\phi (t) + \\Phi )\\right],$ where $D$ is the distance to the source.", "Since the orbital evolution will deviate from a circular trajectory during late inspiral ($\\dot{r}\\ne 0$ ), we must consider more general orbits in which both $\\dot{r}$ and $\\dot{r}\\dot{\\phi }$ are non-negligible.", "For such orbits, the Newtonian GW polarizations are given by [86]: $h_{+}(t)&=& \\frac{2 \\mu }{D}\\Bigg \\lbrace \\left(1+\\cos ^2 \\iota \\right) \\Bigg [ \\cos \\left[2(\\phi (t) + \\Phi )\\right]\\left(-\\dot{r}^2 + r^2 \\dot{\\phi }^2 + \\frac{1}{r}\\right) \\nonumber \\\\ &+& 2r\\,\\dot{r}\\,\\dot{\\phi }\\,\\sin \\left[2(\\phi (t) + \\Phi )\\right]\\Bigg ] + \\left(-\\dot{r}^2 - r^2\\dot{\\phi }^2 + \\frac{1}{r}\\right)\\sin ^2 \\iota \\Bigg \\rbrace \\,,\\\\h_{\\times }(t)&=&\\frac{4 \\mu }{D}\\cos \\iota \\Bigg \\lbrace \\sin \\left[2(\\phi (t) + \\Phi )\\right]\\left(-\\dot{r}^2 + r^2 \\dot{\\phi }^2 + \\frac{1}{r}\\right) \\nonumber \\\\ &-& 2r\\,\\dot{r}\\,\\dot{\\phi }\\,\\cos \\left[2(\\phi (t) + \\Phi )\\right]\\Bigg \\rbrace ,$ where $\\dot{r}$ can be computed using $\\frac{{\\mathrm {d}}r}{{\\mathrm {d}}t} = -\\frac{1}{u^2}\\frac{{\\mathrm {d}u}}{{\\mathrm {d}}x}\\frac{{\\mathrm {d}x}}{{\\mathrm {d}}t}\\, .$ Having described the construction of the inspiral evolution, we shall now describe the approach followed to smoothly connect the late inspiral evolution onto the plunge phase.", "The adiabatic prescription given by Eq.", "(REF ) breaks down when $dE/dx\\rightarrow 0$ .", "Hence, we need a scheme that enables us to match the late inspiral phase onto the plunge phase.", "We will do this by modifying the “transition” phase developed by Ori and Thorne [4] by including finite mass–ratio corrections." ], [ "Transition and plunge phases", "In this Section we describe an extension of the transition phase model introduced by Ori and Thorne [4].", "The basic idea behind this approach can be understood by studying the motion of an inspiralling object in terms of the effective potential, $V(r, L)$ , which takes the following simple form for a Schwarzschild BH [87]: $V(r, L) = \\left(1-\\frac{2}{r} \\right)\\left(1+ \\frac{L^2}{r^2}\\right).$ Throughout the inspiral, the body moves along a nearly circular orbit, and hence the radio of the energy flux to the angular momentum flux is given by: $\\frac{{\\mathrm {d}}E}{{\\mathrm {d}}\\tau } = \\Omega \\frac{{\\mathrm {d}}L}{{\\mathrm {d}}\\tau }.$ Hence, near the ISCO, the energy and angular momentum of the body satisfy the following relations: $E & \\rightarrow & E_{\\rm {ISCO}} + \\Omega _{\\rm {ISCO}}\\, \\xi ,\\\\L & \\rightarrow & L_{\\rm {ISCO}}+ \\xi .$ Re-writing the effective potential, Eq.", "(REF ), in terms of $\\xi = L-L_{\\rm {ISCO}}$ , one notices that during early inspiral, $\\xi \\gg 0$ , the motion of the object is adiabatic, and the object sits at the minimum of the potential —as shown in the left panel of Figure REF .", "However, as the object nears the ISCO, the minimum of the potential moves inward due to radiation reaction.", "At some point, the body's inertia prevents the body from staying at the minimum of the potential, and adiabatic inspiral breaks down [4] — illustrated in the right-hand panel of Figure REF .", "Figure: Left panel: The object sits at the minimum of the effective potential, Eq.", "(), which corresponds to the case ξ=L-L ISCO ≫0\\xi = L-L_{\\rm {ISCO}} \\gg 0.", "Right panel.", "Blue (top) curve: radial geodesic motion, which corresponds to ξ=L-L ISCO ≫0\\xi = L-L_{\\rm {ISCO}} \\gg 0; Red (middle) curve: the object nears the ISCO and the orbit shrinks due to radiation reaction.", "Note that the minimum of the potential has moved inwards (ξ=0.35\\xi = 0.35).", "Yellow (bottom) curve: body's inertia prevents it from staying at the minimum of the potential, and adiabatic inspiral breaks down (ξ=0\\xi = 0).", "At this point the transition regime takes over the late inspiral evolution .", "Note: this plot is based on Figure 1 of .The equation that governs the radial motion during the transition regime is found by linearising the equation $\\left(\\frac{{\\mathrm {d}}r}{{\\mathrm {d}}\\tau }\\right)^2 = E(r)^2 - V(r,\\,L),$ using Eqs.", "(REF ), (), and $\\frac{{\\mathrm {d}}\\xi }{{\\mathrm {d}}\\tau }= \\kappa \\, \\eta , \\quad {\\rm {with}} \\quad \\kappa =\\bigg [\\frac{32}{5}\\Omega ^{7/3} \\frac{\\dot{{\\cal {E}}}}{\\sqrt{1-3u}}\\bigg ]_{\\rm {ISCO}} \\,,$ where $\\dot{{\\cal {E}}}$ is the general relativistic correction to the Newtonian, quadrupole-moment formula [63].", "We now extend these Eqs.", "by including finite mass-ratio corrections.", "Eq.", "(REF ) can be replaced by $\\frac{{\\mathrm {d}}x}{{\\mathrm {d}}t}= \\frac{u^2(1-2u)}{E(x) }\\left(\\frac{{\\mathrm {d}}u}{{\\mathrm {d}}x}\\right)^{-1} \\bigg [E(x)^2 - V\\left(u(x),L(x)\\right)\\bigg ]^{1/2},$ where we have used $\\frac{{\\mathrm {d}}\\tau }{{\\mathrm {d}}t}= \\frac{1-2\\,u(x)}{E(x)}\\,,$ and the expressions for the energy and angular momentum are given by Eqs.", "(REF ), ().", "In order to linearize Eq.", "(REF ) we replace $E(x)$ and $L(x)$ by Eqs (REF ) and () respectively.", "Figure: The left panel shows the effective potential for a Schwarzschild BH without including finite mass–ratio corrections.", "Note that the minimum of the potential takes place at the ISCO, which can be determined using Eq. ().", "The right panel exhibits the influence of finite mass-ratio corrections on the effective potential used to modify Ori and Thorne transition regime .", "The curves represent binaries, from top to bottom, with mass-ratios q∈[0,1/100,1/20,1/10,1/6,1/5,1]q \\in [0,\\, 1/100, \\,1/20, \\,1/10, \\,1/6, \\,1/5, \\,1 ].As discussed in [4], since these equations use the $\\eta $ -corrected values for $E(x_{\\rm {ISCO}} )$ , $L(x_{\\rm {ISCO}}) $ and $ \\Omega _{\\rm {ISCO}}$ , then they remain valid even for finite mass-ratio $\\eta $ [4].", "In Figure REF we show the effect that these finite mass-ratio $\\eta $ corrections have on the effective potential $V(x, L(x))$ .", "We determine the point at which the transition regime starts by carrying out a stability analysis near the ISCO using ${\\mathrm {d}} E/ {\\mathrm {d}} x$ .", "As shown in Figure REF , the ISCO is determined by the relation ${\\mathrm {d}} E/{\\mathrm {d}} x =0$ .", "We have found that the relation $\\left(\\frac{{\\mathrm {d}}E}{{\\mathrm {d}}x}\\right)\\Bigg \|_{\\rm {transition}} = -0.054 + \\frac{1.757\\times 10^{-4}}{\\eta }\\,,$ provides a robust criterion to mark the start of the transition regime for binaries with mass-ratios $1/100<q<1/6$ .", "In [63], the authors only kept terms linear in $\\xi $ , but we have explored which higher order terms had a noticeable impact on the evolution by examining their impact on the length and phasing of the waveform.", "We found that terms $\\propto \\xi $ and $ \\propto (u- u_{\\rm {ISCO}})\\xi $ were important, but corrections at order ${\\cal {O}}(\\xi ^2)$ could be ignored even for comparable mass-ratio systems.", "We model the evolution of the orbital frequency during the transition regime and thereafter in a different manner to that proposed by Ori and Thorne [4].", "In order to ensure that the late-time evolution of the orbital frequency of our self-force model is as close as possible to the orbital evolution extracted from numerical relativity simulations, we incorporate the late-time frequency evolution that was derived by Baker et al [5] in their implicit rotating source (IRS) model, namely: $\\frac{d \\phi }{d \\mathrm {t} } = \\Omega _{\\rm {i}}+ \\left(\\Omega _{\\rm {f}}\\ -\\Omega _{\\rm {i}}\\right)\\left(\\frac{1 + \\tanh ( \\ln \\sqrt{\\varkappa }+ (t-t_0)/b)}{2}\\right)^{\\varkappa }\\, ,$ where $ \\Omega _{\\rm {i}}$ is the value of the orbital frequency when the transition regime begins, and $ \\Omega _{\\rm {f}}$ is the value of the frequency at the light ring, which corresponds to $\\omega _{\\rm {\\ell m n}}/m$ , where $\\omega _{\\rm {\\ell m n}}$ is the fundamental quasi-normal ringing frequency $(n=0)$ for the fundamental mode $(\\ell , m) = (2,2)$ of the post-merger black hole (see Eq.", "(REF ) below).", "The constant mass-dependent coefficient $t_0$ is computed by ensuring that $d\\Omega /dt$ peaks at a time $t=t_0$ .", "The parameter $\\varkappa $ is computed by enforcing continuity between the first order time derivative of the orbital frequency as predicted by the self-force evolution —Eq.", "(REF )— and that given by the first order time derivative of Eq.", "(REF ).", "At the end of the plunge phase, we match the plunge waveforms, which are generated using Eqs.", "(REF ), (), onto the $l=m=2$ , $n=0$ ringdown mode since this dominates the ringdown radiation.", "In the following Section we will describe in detail the procedure followed to attach the ringdowm waveform.", "After the transition regime, the equations of motion we use to model the plunge phase are: the second order time derivative of Eq.", "(REF ) which gives the radial evolution, and Eq.", "(REF ) which describes the orbital frequency evolution.", "We determine the point at which to attach the plunge phase by integrating Eq.", "(REF ) backwards in time, and finding the point at which the transition and plunge equations of motion smoothly match.", "Figure: (Top, bottom) panels: the left panel shows the inspiral, transition and plunge radial evolution for a BH binary of mass-ratio q=(1/6,1/8)q=(1/6,\\,1/8) — and total mass M∈(7M ⊙ ,9M ⊙ )M\\in (7M_{\\odot } ,\\, 9M_{\\odot } ) — using the coordinate transformation given by Eq. ().", "The right panel shows the orbital frequency MΩM\\Omega from late inspiral all the way to the light ring.", "The evolution starts from an initial radial value r=30Mr=30M.Figure: As in Figure , but with the (top, bottom) panels showing the radial and orbital frequency evolution for binaries with mass–ratios q=(1/10,1/15)q=(1/10,\\,1/15), and total mass M∈(11M ⊙ ,16M ⊙ )M\\in ( 11M_{\\odot },\\, 16M_{\\odot }) , respectively.", "As before, the evolution starts from an initial radial value r=30Mr=30M.In Figures REF and REF we show the evolution obtained by combining Ori and Thorne's [4] transition approach for the radial motion with the frequency evolution proposed by Baker et al [5].", "In all the cases shown in Figures REF and REF , the orbital frequency peaks and saturates at a value given by $\\omega _{\\ell m n}/m$ .", "This can be understood if we analyze the asymptotic behavior of Eq.", "(REF ) near the light-ring, i.e., $\\frac{d \\phi }{d \\mathrm {t} } \\approx \\Omega _{\\rm {f}} - \\left(\\Omega _{\\rm {f}}\\ -\\Omega _{\\rm {i}}\\right)e^{ -2(t - t_0)/b}.$ Recasting Eq.", "(REF ) in this form, enables us to identify the constant coefficient $b$ with the e-folding rate for the decay of the fundamental quasinormal mode (QNM).", "As discussed previously, Eq.", "(REF ) predicts the expected orbital evolution during late inspiral and onward.", "To provide a unified description from late inspiral through to ringdown, we have decided to adopt the IRS approach, since this framework allows us to smoothly transition from late inspiral onto the plunge phase, and finally describe the ringdown waveform as a natural consequence of the IRS strain-rate amplitude decay relation $A^2(t) \\propto \\Omega \\dot{\\Omega }$ [5].", "In other words, since Eq.", "(REF ) has the correct behavior near the light-ring as predicted by BHPT, the IRS model provides a natural framework to attach the ringdowm waveform at the end of the plunge phase.", "We will discuss this feature in further detail in the following Section." ], [ "Ringdown Waveform", "Numerical relativity simulations have shown that coalescing binary BHs in general relativity lead to the formation of a distorted rotating remnant, which radiates GWs while it settles down into a stationary Kerr BH [88], [89].", "The GWs emitted during this intermediate phase resemble a ringing bell.", "Hence, this type of radiation is commonly known in the literature as ringdown radiation, and consists of a superposition of QNMs — first discovered in numerical studies of the scattering of GWs in the Schwarzschild spacetime by Vishveshwara [90].", "QNMs are damped oscillations whose frequencies and damping times are uniquely determined by the mass and spin of the post-merger Kerr BH.", "The frequency $\\hat{\\omega }$ of each QNM has two components: the real part represents the oscillation frequency, and the imaginary part corresponds to the inverse of the damping time: $\\hat{\\omega }= \\omega _{\\ell m n} - i/\\tau _{\\ell m n}.$ As discussed above, the observables $\\omega _{\\ell m n}, \\, \\tau _{\\ell m n}$ are uniquely determined by the final mass, $M_{\\rm {f}}$ , and final spin, $q_{\\rm {f}}$ , of the post-merger Kerr BH.", "To determine $M_{\\rm {f}}$ , we use the analytic phenomenological expression for the final mass of the BH remnant that results from the merger of generic binary BHs on circular quasi-orbits introduced in [91], namely, $\\frac{M_{\\rm {f}}}{M} = 1- \\left(1-\\frac{2\\sqrt{2}}{3}\\right)\\eta - 0.543763\\, \\eta ^2.$ This expression reproduces the expected result in the test mass particle limit, and also reproduces results from currently available numerical relativity simulations [91], [92].", "We determine the final spin of the BH remnant $q_{\\rm {f}}$ using the fit proposed in [93]: $q_{\\rm {f}} = \\sqrt{12}\\, \\eta + s_1\\,\\eta ^2 + s_2\\,\\eta ^3,$ with: $s_1=-3.454\\pm 0.132, \\qquad s_1=2.353\\pm 0.548.$ This prescription is consistent with the numerical relativity simulations described in [93], [92], and reproduces test mass limit predictions.", "This compact formula is also consistent with the prescriptions introduced in [94], [95].", "The largest discrepancy between Eq.", "(REF ) and those derived in [94], [95] is $\\mathrel {\\hbox{\\unknown.", "{\\hbox{$\\sim $}}}\\hbox{$<$}}$ 2.5%$ for binaries with $ q$\\sim $$<$ 1/6$.", "The ringdown waveform is given by~\\cite {Berti:2006b, Baker:2008}\\begin{eqnarray}h(t)&=& -\\left(h_{+} - i h_{\\times }\\right) = \\frac{ M_{\\rm {f}} }{D} \\sum _{\\ell m n} {\\cal {A}}_{\\ell m n}\\,e^{-i \\left( \\omega _{\\ell m n}\\, t + \\phi _{\\ell m n} \\right)} \\, e^{-t/\\tau _{\\ell m n} }\\,,\\end{eqnarray}where $ Am n $ and $ m n$ are constants to be determined by smoothly matching the plunge waveform onto the subsequent ringdown.", "The ringdown portion of the self-force waveform model constructed in this paper includes the mode $ =m=2$ and the tones $ n=0, 1, 2$.", "The approach we follow to attach the leading mode and overtones is the following:$ In order to ensure continuity between the plunge and ringdown waveforms, we use the end of the plunge waveform — Eqs.", "(REF ), () — to construct an interpolation function $F(t)$ .", "The interpolation method used to construct $F(t)$ is a cubic spline.", "We match the plunge waveform onto the leading mode $\\ell =m=2$ , $n=0$ of the ringdown waveform, Eq.", "(), at the point where the amplitude of the plunge waveform peaks, $t_{\\rm {max}}$ .", "Attaching the mode requires $F(t=t_{\\rm {max}})$ and $F^{\\prime }(t=t_{\\rm {max}})$ which are computed from the interpolation function.", "These conditions fix two constants per polarisation.", "To attach the first overtone, $\\ell =m=2,\\, n=1$ , we insert into Eq.", "() the constants determined by attaching the leading mode as seeds to compute the amplitude and phase coefficients for the first overtone by enforcing continuity at $t_{\\rm {max}} + dt$ .", "Finally, we insert into Eq.", "() the value of the amplitude and phase coefficients previously determined for the leading mode and first overtone, and determine the four remaining constants by enforcing continuity at $t_{\\rm {max}} + 2\\,dt$ .", "Having described the methodology followed to construct complete waveforms for comparable and intermediate mass-ratio systems, we finish this Section by putting together all these various pieces to construct sample waveforms for a few systems with mass-ratio $q\\in [1/6,\\,1/8,\\, 1/10,\\, 1/15]$ , and total mass $M\\in ( 7M_{\\odot },\\, 9M_{\\odot },\\, 11M_{\\odot },\\, 16M_{\\odot }) $ in Figure REF .", "Figure: The panels show sample waveforms from inspiral to ringdown for systems with mass-ratios q∈[1/6,1/8]q\\in [1/6,\\,1/8] —and total mass M∈(7M ⊙ ,9M ⊙ )M\\in ( 7M_{\\odot },\\, 9M_{\\odot }) — (top panels—from left to right) and q∈[1/10,1/15]q\\in [1/10,\\, 1/15] —and total mass M∈(11M ⊙ ,16M ⊙ )M\\in ( 11M_{\\odot },\\, 16M_{\\odot }) — (bottom panels —from left to right).", "The inspiral evolution for the [top, bottom] panels starts from r=[30M,25M]r=[30M,\\, 25M]." ], [ " Ringdown waveform construction in the context of an Implicit Rotating Source", "Having described the ringdown waveform construction as a sum of quasinormal modes, we finish this Section by exhibiting the power of the IRS model to describe the ringdown evolution.", "In the IRS model, the strain-rate amplitude decay is given by [5]: $A^2(t) = 16\\,\\pi \\, \\dot{E} \\approx 16\\,\\pi \\,\\xi \\,\\Omega \\,\\dot{\\Omega }\\,.$ Using Eq.", "(REF ), in the limit $\\Omega \\rightarrow \\Omega _{\\rm {f}}$ , the amplitude decay is given by $A_0^2\\,e^{-2t/\\tau } \\approx \\frac{32\\, \\pi \\,\\xi \\, \\Omega _{\\rm {f}}}{b}\\left(\\Omega _{\\rm {f}} - \\Omega _{\\rm {i}}\\right) e^{-2(t-t_0)/\\tau }\\,.$ Hence, the late-time amplitude in the IRS model is given by $A^2_{\\ell m} \\approx 16\\,\\pi \\,\\xi _{\\ell m}\\,\\omega _{\\ell m}\\,\\dot{\\omega }_{\\ell m}\\,,$ where the value of $\\xi _{\\ell m}$ is set by ensuring amplitude continuity at the light-ring.", "In Figure REF we explicitly show the equivalence of the ringdown waveform construction both in the IRS approach and using the sum of QNMs utilized in the previous Section.", "This detailed analysis shows that the IRS is a powerful tool to model the late time portions of the waveforms in a unified way.", "Figure: The panels show sample the late-time evolution of waveforms whose ringdown phase is modeled using the implicit rotating source (IRS) model and a sum of quasinormal modes (QNMs).", "The systems shown correspond to binaries with mass-ratios q∈[1/6,1/8]q\\in [1/6,\\,1/8] —and total mass M∈(7M ⊙ ,9M ⊙ )M\\in ( 7M_{\\odot },\\, 9M_{\\odot }) — (top panels—from left to right) and q∈[1/10,1/15]q\\in [1/10,\\, 1/15] —and total mass M∈(11M ⊙ ,16M ⊙ )M\\in ( 11M_{\\odot },\\, 16M_{\\odot }) — (bottom panels —from left to right).", "The inspiral evolution for the [top, bottom] panels starts from r=[30M,25M]r=[30M,\\, 25M]." ], [ "Summary", "In this Section we briefly summarize the key ingredients that were used to develop the waveform model described in this paper: Inspiral evolution The building blocks of the inspiral evolution are the expressions for the energy, $E$ , and angular momentum, $L$ , — Eqs.", "(REF ) and () — that include gravitational self-force corrections and are valid over the domain $0<x<\\frac{1}{3}$ [1].", "The orbital frequency is modeled using Eq.", "(REF ).", "This prescription encapsulates gravitational self-force corrections that render a better phase evolution when calibrated against EOB.", "The inspiral trajectory is modeled using the simple prescription (REF ).", "This scheme is no longer valid near ISCO, where $dE/dx =0$ for binaries with $q\\le 6$ .", "We construct the inspiral waveform using Eqs.", "(REF ), ().", "When the inspiralling object nears the ISCO, we need to invoke the `transition scheme' introduced by Ori and Thorne [4], which enables us to smoothly attach the late inspiral evolution onto the plunge phase.", "Transition phase The transition regime starts at a point when $ \\mathrm {d} E/ \\mathrm {d} x$ satisfies Eq.", "(REF ).", "The equations of motion that govern the transition phase are (REF ), ().", "These relations are valid, since the motion near the ISCO is nearly-circular.", "Using Eqs.", "(REF ), (), we linearize the second order time derivative of Eq.", "(REF ).", "In order to reproduce the orbital phase evolution predicted by numerical simulations from the ISCO to the light-ring, we modify the original transition phase by smoothly matching the inspiral orbital phase evolution, Eq.", "(REF ), onto the IRS model, Eq.", "(REF ) at the start of the transition phase.", "Plunge phase The equations of motion that govern the plunge phase are given by the second order time derivative of Eq.", "(REF ), and Eq.", "(REF ).", "We integrate these relations backwards in time to find the point at which both the transition and plunge equations of motion smoothly match.", "The transition phase ends at this point.", "Near the light-ring Eq.", "(REF ) has the behavior predicted by BHPT, which enables us to smoothly match the plunge phase onto the ringdown.", "Both the transition and plunge waveforms are constructed using Eqs.", "(REF ) and ().", "Having constructed the waveform from inspiral to ringdown, we derived second-order radiative corrections to improve the radiative evolution of the waveform model.", "This was necessary in order to construct a waveform model that is internally consistent, i..e, since the orbital elements include first-order conservative corrections, then radiative corrections should enter the fluxes at second order.", "We have derived these corrections by enforcing a close agreement between our self-force model and EOB, and have shown that our model can reproduce the orbital phase evolution predicted by EOB within the numerical error of the simulations used to calibrate this model for a variety of mass-ratios.", "Ringdown phase The ringdown waveform is constructed using Eq. ().", "We use the plunge waveform to construct an interpolation function $F(t)$ , and then use this function to attach the leading mode $\\ell =m=2,\\, n=0$ at the point where the amplitude of the plunge waveform peaks, $t_{\\rm {max}}$ .", "We enforce continuity by ensuring that $F( t_{\\rm {max}}) = h^{n=0}_{\\rm {RD}}$ and $F^{\\prime }( t_{\\rm {max}}) = h^{\\prime n=0}_{\\rm {RD}}$ .", "We include the first and second overtone $n=1,\\, n=2$ in the ringdown waveform.", "Using the IRS model, we have shown that having knowledge of the time evolution of the orbital frequency provides sufficient information to construct the amplitude decay during ringdowm.", "Hence, we can construct an alternative ringdowm waveform using only Eq.", "(REF ), and ensuring smooth continuity with the plunge waveform.", "Finally, we have explicitly shown that the implicit rotating source approach provides a natural transition from late-time radiation to ringdown that is equivalent to ringdown waveform modeling based on a sum of QNMs.", "Throughout the paper we have emphasized the fact that our model provides a more reliable framework to model binaries whose components are non-spinning, and with mass-ratios $q\\le 1/6$ , as compared to available PN approximants.", "It is worth emphasizing that our model is also computationally inexpensive.", "All the waveforms we generated to constrain the higher-order $\\eta $ corrections in the energy flux —Eq.", "(REF )— can be generated in fractions of a second.", "A direct comparison between our code and EOB shows that, averaged over 500 realizations, our code is $\\sim 20\\%$ faster than the optimized version of the EOB code currently available in the LIGO Scientific Collaboration LAL library.", "It should be emphasized, though, that our code at present has not been optimized, and hence, compared to EOB our model is expected to significantly reduce the cost of waveform generation, making it relatively more viable for parameter estimation efforts.", "This is a key feature in our model that enabled us to sample a wide region of parameter space to constrain the $B_i$ coefficients in Eq.", "(REF ).", "Furthermore, these self-forced waveforms do not need to be highly sampled near the light-ring, hence decreasing the speed with which they can be generated, because the prescription we have used to model the late-time orbital frequency evolution provides the correct evolution near the light-ring.", "This particular feature also enables us to match the plunge waveform onto its ringdown counterpart without having to interpolate the orbital frequency evolution using a phenomenological approach.", "The model is internally consistent and the only phenomenology invoked during its construction is related to currently unknown physics, i.e., higher order radiative corrections to the energy flux.", "Once these corrections are formally derived in the near future, the flexibility of our model will enable us to replace the radiative corrections that we have currently computed by numerical optimization.", "At that stage, we will be able to describe in a single unified model the dynamical evolution of binaries whose mass-ratios range from the extreme to the comparable regime." ], [ "Conclusions", "In this paper we have developed a self-force waveform model that is capable of reproducing with good accuracy the inspiral, merger and ringdown of binaries with mass-ratios $q\\le 1/6$ .", "This work suggests that a model that incorporates first order conservative self-force corrections in the orbital elements, and second-order radiative corrections in the dissipative piece of the self-force may suffice to describe in a unified manner the coalescence of binaries with mass-ratios that range from the comparable to the extreme.", "Our model includes conservative self-force corrections that have been derived in the strong–field regime [1].", "Using these results, we find that binaries with mass-ratios $q\\mathrel {\\hbox{\\unknown.", "{\\hbox{$\\sim $}}}\\hbox{$>$}}$ 1/6$ do not have an ISCO.", "For systems with $ q1/6$, we have derived a simple relation that provides the location of the ISCO in terms of the symmetric mass-ratio (see Eq.~(\\ref {xisco_eq})).", "To describe the inspiral evolution, we have derived second-order corrections to the energy flux by minimizing the phase discrepancy between our self-force model and the EOB model~\\cite {buho, Damour:2013} for a variety of mass-ratios.", "We have shown that our model reproduces the phase evolution of the EOB model within the accuracy of available numerical simulations for a variety of mass-ratios.$ This paper also presents an extension of the “transition regime” developed by Ori and Thorne [4] to smoothly match the late inspiral evolution onto the plunge phase.", "We have found that using the inspiral phase expression for the orbital frequency during the plunge phase does not render an accurate description of the actual orbital evolution as predicted by numerical simulations.", "Hence, we have embedded the self-force framework in the IRS model proposed by Baker et al [5] to ensure that our model is as close as possible to the orbital evolution predicted by numerical relativity simulations.", "The implementation of this prescription ensures that the orbital frequency saturates near the light-ring, which facilitates matching onto the ringdown phase.", "We have shown that the IRS model provides a natural transition onto the ringdown phase that is equivalent to a ringdown waveform construction based on a sum of QNMs.", "The motivation for constructing this model is two-fold: to exhibit the versatility of the self-force formalism to accurately describe the evolution of binaries beyond the extreme mass-ratio limit; and to provide a tool that can be used to explore the information that could be extracted by GW detectors that target binaries with comparable and intermediate mass-ratios.", "Current studies have only explored the use of PN approximants to model the merger of NSBH binaries, despite their inadequacy to capture the evolution of these systems [96], [60], [61] (see Figure REF ).", "Comparing Figures REF and REF , we conclude that even if second generation GW detectors were only capable of capturing the inspiral evolution of NSBH mergers, our self-force model would be better equipped to describe these events.", "The construction of this IRS self-force model constitutes an important step towards the construction of more reliable waveforms to describe IMRCs.", "In [50], it was shown that Huerta–Gair (HG) waveforms — which are closely related to the model developed in this paper, and which were also based on a consistent combination of BHPT and self-force corrections — work in the relevant mass range for advanced detectors.", "The new model introduced in this article, improves upon these waveforms and should therefore not only work in the regime of interest to advanced detectors but over a wider parameter space that could be applicable to third generation detectors or later observations.", "Having developed a strong foundation to model binaries on circular orbits whose components are not rotating, it is necessary to incorporate more ingredients to the waveform model to capture GW signals from binaries whose components have significant spin [97], [2], [98], [99], [85], [100], [101], [62], or systems that form in core-collapsed globular clusters, and hence are expected to have non-negligible eccentricity at merger [102], [103].", "In order to do so, we require input both from the self-force program — which is making substantial progress towards the computation of the self-force in a Kerr background [104], [105], [106]— and from numerical simulations [57], in particular from binaries with typical mass-ratios $1/20<q<1/10$ to explore the performance of our self-force model in this currently unconstrained regime.", "Undoubtedly, the development and implementation of improved numerical algorithms [107] to carry out these simulations will facilitate the realization of these studies in the near future.", "Recent observational discoveries [7] suggest that NSBH mergers may also occur in globular clusters.", "Hence, in these dense stellar environments we may expect that multiple $n$ -body interactions and binary exchange processes may lead to the formation of binaries on eccentric orbits.", "Detectors such as the Einstein Telescope, operating at low frequencies $\\sim 1$ Hz, may be capable of detecting the signature of eccentricity during the early inspiral.", "In order to assess these effects, we aim to extend the model introduced in this paper by including eccentricity, making use of self-force corrections for generic orbits in a Schwarzschild background [80]." ], [ "Acknowledgments", "EH thanks the hospitality of the TAPIR group at Caltech where part of this work was carried out, and the generosity of Prof Peter Saulson, who made this visit possible trough his National Science Foundation (NSF) grant number PHY-120583.", "We are pleased to thank Chad Galley, Alex Huerta Gago, Anıl Zenginoğlu and Fan Zhang for useful interactions.", "JG's work is supported by the Royal Society.", "PK acknowledges support from the NSF grant number PHY-0847611.", "Some calculations were performed on the Syracuse University Gravitation and Relativity cluster, which is supported by NSF grants PHY-1040231, PHY-1104371 and Syracuse University ITS." ] ]	1403.0561
[ [ "Orbits of hybrid systems as qualitative indicators of quantum dynamics" ], [ "Abstract Hamiltonian theory of hybrid quantum-classical systems is used to study dynamics of the classical subsystem coupled to different types of quantum systems.", "It is shown that the qualitative properties of orbits of the classical subsystem clearly indicate if the quantum subsystem does or does not have additional conserved observables." ], [ "Introduction", "Linear Schrödinger equation of any quantum mechanical system is equivalent to an integrable Hamiltonian dynamical system [1], [2], [3], [4], [5], [6].", "As such, the linear Schrödinger equation of a bounded system has only periodic or quasi-periodic orbits.", "However, integrable systems are exceptional [7].", "Typical Hamiltonian system has also plenty of irregular, i.e.", "chaotic orbits [7], but these do not appear in standard quantum mechanics.", "Integrability, or the lack of it, of Hamiltonian dynamical systems is related to the symmetries of the model and to the existence of a sufficient number of integrals of motion.", "The difference between integrable and non-integrable systems is clearly manifested in the qualitative properties of orbits.", "The former have only regular, periodic or quasi-periodic orbits, and in the latter the chaotic orbits dominate.", "Classification of quantum system into regular or irregular such as ergodic or chaotic, is possible using different plausible and variously motivated criteria without reference to the orbital properties.", "Usually, the criteria are formulated in terms of the properties of the energy spectrum, and the connection with the classical, well developed, notions of regular or chaotic dynamics, formulated in terms of orbital properties, is obscured.", "The purpose of our work was to investigate qualitative properties of orbits of a hybrid quantum-classical system, where the classical part is integrable when isolated and the quantum part is characterized as symmetric or non-symmetric by the existence of constant observables.", "In particular, we want to see if the symmetry, or the lack of it, might be displayed in the qualitative properties of orbits of the classical part.", "To this end we utilized recently developed Hamiltonian hybrid theory of quantum-classical (QC) systems [8], [9], [10], [11], [12].", "Our main result is that indeed quantum systems, characterized as nonsymmetric imply chaotic orbits of the classical degrees of freedom (CDF) coupled to the quantum system.", "On the other hand, CDF show regular dynamics if coupled to a symmetric quantum system, i.e.", "a quantum system with sufficient number of constant observables.", "One of the first to introduce some sort of dynamical distinction between quantum systems was von Neumann [13] with his definition of quantum ergodicity based on the properties of the Hamiltonian eigenspectrum.", "Further developments and different approaches to the problems of quantum irregular dynamics can be divided into three groups.", "The literature on the topic is enormous, and we shall give only a few examples or a relevant review for each of the approaches.", "The most popular was the type of studies analyzing the spectral properties of quantum systems obtained by quantization of chaotic classical systems (see the reviews collected in [14]).", "Still in the framework of systems whose classical analog is chaotic, there were studies of semi-classical dynamics [14] and phase space distributions [14].", "The second group of studies consists of those works where an intrinsic definition of quantum chaoticity is attempted [15].", "Neither the works in the first nor those in the second group rely on the topological properties of pure state orbits of quantum systems.", "The third group originates from the studies of open quantum systems, and here the properties of orbits of an open quantum system are important.", "Classical property of chaoticity defined in terms of orbital properties was analyzed in quantum systems interacting with different types of environments [16], [17], [18].", "It was observed that orbits of such open quantum systems in the macro-limit might be chaotic.", "In the next section we shall briefly recapitulate the Hamiltonian theory of hybrid systems.", "In section 3 we present the hybrid models consisting of qualitatively different pairs of qubits as the quantum part and the linear oscillator as the classical part.", "Section 4 will describe numerical computations of hybrid dynamics and our main results.", "Brief summary will be given in section 5." ], [ "Hamiltonian hybrid theory", "There is no unique generally accepted theory of interaction between micro and macro degrees of freedom, where the former are described by quantum and the latter by classical theory (see [8] for an informative review).", "Some of the suggested hybrid theories are mathematically inconsistent, and “no go\" type theorems have been formulated [19], suggesting that no consistent hybrid theory can be formulated.", "Nevertheless, mathematically consistent but inequivalent hybrid theories exist [8], [20], [21], [22], [23].", "The Hamiltonian hybrid theory, as formulated and discussed for example in [8], [11], [12], has many of the properties commonly expected of a good hybrid theory, but has also some controversial features.", "It's physical content is equivalent to the standard mean field approximation, but it is formulated entirely in terms of the Hamiltonian framework, which provides useful insights such as the one presented in this communication.", "The theory is based on the equivalence of the Schrödinger equation on ${\\cal H}^N$ and the corresponding Hamiltonian system on ${\\mathbb {R}}^{2N}$ .", "The Riemannian $g$ and the symplectic $\\omega $ structures on the phase space ${\\cal M}_q={\\mathbb {R}}^{2N}$ are given by the real and imaginary parts of the Hermitian scalar product on ${\\cal H}^N$ : $\\langle \\psi \|\\phi \\rangle =g(\\psi ,\\phi )+i\\omega (\\psi ,\\phi )$ .", "Schrödinger equation in an abstract basis $\\lbrace \|n\\rangle \\rbrace $ of ${\\cal H}^N$ $i\\hbar \\frac{\\partial c_n}{\\partial t}=\\sum _ m H_{nm}c_m$ where $\|\\psi \\rangle =\\sum _n c_n \|n\\rangle $ and $H_{nm}=\\langle n\|\\hat{H}\|m\\rangle $ is equivalent to Hamiltonian equations $\\dot{x}_n=\\frac{\\partial H(x,y)}{\\partial y_n},\\quad \\dot{y}_n=-\\frac{\\partial H(x,y)}{\\partial x_n}$ where $c_n=(x_n+iy_n)/\\sqrt{2\\hbar }$ and $H(x,y)=\\langle \\psi _{xy}\|\\hat{H}\|\\psi _{xy}\\rangle ,$ where $(x,y)$ stands for $(x_1,x_2\\dots x_N,y_1,y_2\\dots y_N)$ .", "Only quadratic functions $A(x,y)$ of the form $A(x,y)=\\langle \\psi _{xy}\|\\hat{A}\|\\psi _{xy}\\rangle $ are related to the physical observables $\\hat{A}$ .", "In particular, the canonical coordinates $(x,y)$ of quantum degrees of freedom (QDF) do not have such interpretation.", "Hamiltonian hybrid theory uses the Hamiltonian formulations of quantum and classical dynamics, and couples the classical and quantum systems as they would be coupled in the theory of Hamiltonian systems.", "The phase space of QC system is given by the Cartesian product ${\\cal M}_{qc}={\\cal M}_q\\times {\\cal M}_c,$ and the total Hamiltonian is of the form $H_{qc}(x,y,q,p)=H_q(x,y)+H_{cl}(q,p)+H_{int}(x,y,q,p).$ The dynamical equations of the hybrid theory are just the Hamiltonian equations with the Hamiltonian (REF ).", "Observe two fundamental properties of the Hamiltonian hybrid theory: a) There is no entanglement between QDF and CDF and b) the canonical coordinates of CDF have the interpretation of conjugate physical variables and have sharp values in any pure state $(x,y,q,p)$ of the hybrid.", "Hamiltonian theory of hybrid systems can be developed starting from the Hamiltonian formulation of a composite quantum system and imposing a constraint that one of the components is behaving as a classical system [11]." ], [ "Qualitatively different quantum systems coupled to the classical harmonic oscillator", "We shall consider the following three examples of quantum system with different symmetry properties.", "All three examples involve a pair of interacting qubits, where $\\sigma _{x,y,z}^{1,2}$ denote $x,y$ or $z$ Pauli matrix of the qubit 1 or the qubit 2, and $\\omega ,\\:\\mu $ and $\\beta $ are parameters.", "The simplest is given by $\\hat{H}_s=\\hbar \\omega \\sigma _z^1+\\hbar \\omega \\sigma _z^2+\\hbar \\mu \\sigma _z^1 \\sigma _z^2.$ The system has two additional independent constant observables $\\sigma _z^1$ and $\\sigma _z^2$ corresponding to the $SO(2)\\times SO(2)$ symmetry of the model.", "Next two models are examples of non-symmetric systems.", "The system $\\hat{H}_{ns1}=\\hat{H}_s+\\hbar \\beta \\sigma _y^1$ has only $\\sigma _z^2$ as the additional constant observable, and in the system $\\hat{H}_{ns2}=\\hbar \\omega \\sigma _z^1+\\hbar \\omega \\sigma _z^2+\\hbar \\mu \\sigma _x^1 \\sigma _x^2,$ there are no additional dynamical constant observables.", "Let us stress that the Hamiltonian systems with the Hamiltonian functions given by $\\langle \\psi \|\\hat{H}\|\\psi \\rangle $ are integrable with only the regular (non-chaotic) orbits irrespective of their symmetry properties.", "Figure: Figures illustrate the time series q(τ)q(\\tau ) (a,c) and the corresponding amplitudes of the Fourier spectra (b,d), of the classical oscillator subpart of the hybrid system with the quantum subpart given by symmetric () (a,b) and non-symmetric () (c,d) systems.", "The values of the parameters areω=1,μ=5,m=k=1,c 1 =15,c 2 =1\\omega =1,\\: \\mu =5, m=k=1,\\: c_1=15,c_2=1.Figure: Figures illustrate the time series (a,c) and the corresponding amplitudes of the Fourier spectra (b,d), of the x 1 x_1 canonical coordinate of the quantum subpart of the hybrid system given by symmetric H s H_{s}() (a,b) and non-symmetric H ns2 H_{ns2}() (c,d) systems.", "The values of the parametersare the same as in fig.1.Figure: Figures illustrate the time series q(τ)q(\\tau ) (a) and x 1 (τ)x_1(\\tau ) and the corresponding amplitudes of the Fourier spectra (b,d).", "Thehamiltonian is non-symmetric H ns1 H_{ns1}().", "The values of the parameters are the same as in fig.1.The Hamilton functions corresponding to the three quantum systems (REF ),(REF ) and (REF ) are given by the general rule (REF ).", "In the computational basis $\|1\\rangle =\|1,1\\rangle ,\\: \|2\\rangle =\|1,-1\\rangle , \\: \|3\\rangle =\|-1,1\\rangle , \\: \|4\\rangle =\|-1,-1\\rangle $ , where for example $\|1,1\\rangle =\|1\\rangle \\otimes \|1\\rangle $ and $\|\\pm 1\\rangle $ are the eigenvectors of $\\sigma _z$ , the Hamilton functions are $&& H_{s}(x,y)=\\omega (x_1^2+y_1^2-x_4^2-y_4^2)\\nonumber \\\\&+&\\frac{\\mu }{2} (x_1^2-x_2^2-x_3^2+x_4^2+y_1^2-y_2^2-y_3^2+y_4^2),$ $&&H_{ns1}(x,y)=\\omega (x_1^2+y_1^2-x_4^2-y_4^2)\\nonumber \\\\&+&\\frac{\\mu }{2} (x_1^2-x_2^2-x_3^2+x_4^2+y_1^2-y_2^2-y_3^2+y_4^2)\\nonumber \\\\&+&\\beta (y_3 x_1 + y_4 x_2 - y_1 x_3 - y_2 x_4)$ and $&&H_{ns2}(x,y)=\\omega (x_1^2+y_1^2-x_4^2-y_4^2)\\nonumber \\\\&+&\\mu (x_2x_3+x_1x_4+y_2y_3+y_1y_4).$ Observe that, due to the $1/\\sqrt{2\\hbar }$ scaling of the canonical coordinates $(x,y)$ , $\\hbar $ does not appear in the Hamilton's functions (REF ), (REF ) and (REF ) nor in the corresponding Hmilton's equations and their solutions $x(t)\\dots $ .", "Of course, $\\hbar $ reappears in the functions $\\langle \\sigma _{x}^1\\rangle \\dots $ .", "The classical system that we want to couple with quantum systems (REF ), (REF ) or (REF ) is one-dimensional linear oscillator with the Hamiltonian $H_{cl}(q,p)=\\frac{p^2}{2m}+kq^2,$ which of course has only regular periodic orbits.", "The $QC$ interaction term is taken to be such that it does not interfere with the existence of operators commuting with the Hamiltonian of the quantum part.", "In other words, the operator $\\hat{H}_q+\\hat{H}_{int}$ has the same additional constant observables as the quantum part $\\hat{H}_q$ .", "Furthermore, $\\hat{H}_{int}$ must depend on observables of the qubit 1 and of the qubit 2.", "For example $\\hat{H}_{int}= q(c_1\\hbar \\sigma _z^1+c_2\\hbar \\sigma _z^2)$ implying $H_{int}(x,y,q,p)=q(c_1\\hbar \\langle \\sigma _z^1\\rangle +c_2\\hbar \\langle \\sigma _z^2\\rangle )$ or explicitly $&&H_{int}=\\frac{c_1 q}{2} (x_1^2+x_2^2-x_3^2-x_4^2+y_1^2+y_2^2-y_3^2-y_4^2)\\nonumber \\\\&+&\\frac{c_2q}{2} (x_1^2-x_2^2+x_3^2-x_4^2+y_1^2-y_2^2+y_3^2-y_4^2).$ The total Hamiltonian is given by the sum of (REF ), (REF ) and one of (REF ), (REF ) or (REF ).", "Observe that the functions $\\langle \\sigma _z^1\\rangle $ and $\\langle \\sigma _z^2\\rangle $ are constants of motion for the hybrid $H_s+H_{int}+H_{cl}$ , as is the function $\\langle \\sigma _z^2\\rangle $ constant for the hybrid $H_{ns1}+H_{int}+H_{cl}$ .", "Thus, $H_{int}$ given by (REF ) satisfies the general condition that we impose on the QC interaction." ], [ "Numerical computations and the results", "Hamiltonian equations are solved numerically and the dynamics of CDF, illustrated in fig.", "1 and fig.", "3a,b and of QDF illustrated in fig.", "2 and fig.", "3c,d, is observed in the cases corresponding to the symmetric or non-symmetric quantum parts for different values of the parameters $\\mu $ and $c$ .", "Let us first stress again that if there is no classical system then all orbits are regular for either of the quantum systems.", "On the other hand the hybrid system displays different behavior.", "Consider first the time series generated by the CDF.", "Figures 1a,b,c,d and figures 3a,b show the time series $q(\\tau )$ (fig.", "1a,c and fig.", "3a), where $\\tau =\\omega t$ is the dimensionless time, and the corresponding Fourier amplitude spectra (fig.", "1b,d and fig.3b).", "Fig.", "1a,b are obtained with the quantum symmetric system (REF ), fig.", "1c,d with quantum non-symmetric system (REF ) and fig.", "3 with quantum non-symmetric system (REF ).", "Obviously, the orbits of the CDF are periodic, with single frequency, in the symmetric case, and chaotic with a broad-band spectrum in the non-symmetric cases.", "We can conclude that the qualitative properties of orbits of a classical system coupled with a quantum system are excellent indicators of the symmetries of the quantum system.", "Consider now the dynamics of QDF illustrated in fig.", "2a,b,c,d.", "and fig.", "3c,d by plotting the time series generated by $x_1(t)$ and the corresponding Fourier amplitudes spectra.", "Qualitatively the same properties are displayed by dynamics of other canonical coordinates $x_2,x_3,x_4,y_1,y_2,y_3,y_4$ or, for example, by the dynamics of expectation values $\\langle \\sigma _x^1(t)\\rangle ,\\dots $ .", "Again, the time series are regular if the quantum systems are symmetric and are chaotic in the quantum non-symmetric case.", "The same conclusion is obtained with $H_{ns2}$ replaced by $H_{ns1}$ .", "We can conclude that the orbits of the hybrid system, are regular or chaotic, in the sense of Hamiltonian dynamics, depending on the quantum subpart being symmetric or non-symmetric.", "Thus, the relation between symmetry and existence of independent constants of motion on one hand and the qualitative properties of orbits on the other, which is the characteristic feature of classical mechanics and is not a feature of isolated quantum systems, is restored by appropriate coupling of the quantum and a classical integrable system.", "Observe that such behavior can not be obtained by coupling two quantum systems (instead of quantum-classical coupling).", "In this case, and even for the simplest quantum system in place of the classical one, the phase space of the quantum composite system is much larger than ${\\cal M}_{qc}$ because of the degrees of freedom corresponding to the possibility of entanglement, and the total system is always linear.", "All degrees of freedom of a quantum-quantum system in the Hamiltonian formulation display only regular dynamics, independently of the symmetries of the quantum Hamiltonian.", "On the other hand, the hybrid systems are nonlinear, due to the QC coupling and the phase space of the form (REF ), and the relation between the symmetries and the qualitative properties of orbits is like in the general Hamiltonian theory.", "Explanation of the observed properties relies on the fact that the five degrees of freedom hybrid Hamiltonian system with quantum symmetric subpart has enough independent constants of motion in involution.", "These are given by $H(x,y,q,p),H_{s}(x,y),\\langle \\sigma _z^1\\rangle ,\\langle \\sigma _z^2\\rangle $ and the norm of the state of the quantum subpart.", "On the other hand $H_{ns1}+H_{int}+H_{cl}$ , or $H_{ns2}+H_{int}+H_{cl}$ do not have enough such constants of motion since the quantum part $\\hat{H}_{ns2}$ does not commute with $\\sigma _z^1$ and $\\sigma _z^2$ and $\\hat{H}_{ns1}$ with $\\sigma _z^1$ .", "Only $H_s+H_{int}+H_{cl}$ is integrable while those obtained with non-symmetric quantum subparts are not and thus have some chaotic orbits." ], [ "Summary", "In summary, we have shown that the orbits of an integrable classical system when coupled to a quantum system in an appropriate way remain regular or become chaotic depending on the presence or lack of symmetries in the quantum part.", "To this end we used the Hamiltonian theory of quantum-classical systems and examples of qubit systems.", "The first fact is an important restriction on our work.", "On the second point, the nature of our results is qualitative and is therefore expected to be valid generically, and not only for the considered examples.", "Considering the choice of Hamiltonian theory to describe QC interaction, we were motivated by the mathematical consistency of the theory and the fact that the theory describes orbits of pure states of a deterministic Hamiltonian system.", "There are other consistent hybrid theories, but they are either formulated in terms of probability densities [21], [22] or in terms of stochastic pure state evolution [20], [23].", "Of course, the significance of our result could be properly judged only after the status of Hamiltonian hybrid theory is sufficiently understood.", "We acknowledge support of the Ministry of Education and Science of the Republic of Serbia, contracts No.", "171006, 171017, 171020, 171038 and 45016 and COST (Action MP1006)." ] ]	1403.0455
[ [ "Existence of Chaos in Plane $\\mathbb{R}^2$ and its Application in\n Macroeconomics" ], [ "Abstract The Devaney, Li-Yorke and distributional chaos in plane $\\mathbb{R}^2$ can occur in the continuous dynamical system generated by Euler equation branching.", "Euler equation branching is a type of differential inclusion $\\dot x \\in \\{f(x),g(x) \\} $, where $f,g:X \\subset \\mathbb{R}^n \\rightarrow \\mathbb{R}^n$ are continuous and $f(x) \\neq g(x)$ in every point $x \\in X$.", "Stockman and Raines (Stockman, D. R.; Raines, B. R.: Chaotic sets and Euler equation branching, Journal of Mathematical Economics, 2010, Volume 46, pp.", "1173-1193) defined so-called chaotic set in plane $\\mathbb{R}^2$ which existence leads to an existence of Devaney, Li-Yorke and distributional chaos.", "In this paper, we follow up on Stockman, Raines and we show that chaos in plane $\\mathbb{R}^2$ with two \"classical\" (with non-zero determinant of Jacobi's matrix) hyperbolic singular points of both branches not lying in the same point in $\\mathbb{R}^2$ is always admitted.", "But the chaos existence is caused also by set of solutions of Euler equation branching which have to fulfil conditions following from the definition of so-called chaotic set.", "So, we research this set of solutions.", "In the second part we create new overall macroeconomic equilibrium model called IS-LM/QY-ML.", "The construction of this model follows from the fundamental macroeconomic equilibrium model called IS-LM but we include every important economic phenomena like inflation effect, endogenous money supply, economic cycle etc.", "in contrast with the original IS-LM model.", "We research the dynamical behaviour of this new IS-LM/QY-ML model and show when a chaos exists with relevant economic interpretation." ], [ "1.3" ] ]	1403.0111
[ [ "Stable maps and branched shadows of 3-manifolds" ], [ "Abstract Turaev's shadow can be seen locally as the Stein factorization of a stable map.", "In this paper, we define the notion of stable map complexity for a compact orientable 3-manifold bounded by (possibly empty) tori counting, with some weights, the minimal number of singular fibers of codimension 2 of stable maps into the real plane, and prove that this number equals the minimal number of vertices of its branched shadows.", "In consequence, we give a complete characterization of hyperbolic links in the 3-sphere whose exteriors have stable map complexity 1 in terms of Dehn surgeries, and also give an observation concerning the coincidence of the stable map complexity and shadow complexity using estimations of hyperbolic volumes." ], [ "citing..5em3" ] ]	1403.0596
[ [ "On the Interaction between Turbulence and a Planar Rarefaction" ], [ "Abstract The modeling of turbulence, whether it be numerical or analytical, is a difficult challenge.", "Turbulence is amenable to analysis with linear theory if it is subject to rapid distortions, i.e., motions occurring on a time scale that is short compared to the time scale for non-linear interactions.", "Such an approach could prove useful for understanding aspects of astrophysical turbulence, which is often subject to rapid distortions, such as supernova explosions or the free-fall associated with gravitational instability.", "As a proof of principle, a particularly simple problem is considered here: the evolution of vorticity due to a planar rarefaction in an ideal gas.", "Vorticity can either grow or decay in the wake of a rarefaction front, and there are two competing effects that determine which outcome occurs: entropy fluctuations couple to the mean pressure gradient to produce vorticity via baroclinic effects, whereas vorticity is damped due to the conservation of angular momentum as the fluid expands.", "In the limit of purely entropic fluctuations in the ambient fluid, a strong rarefaction generates vorticity with a turbulent Mach number on the order of the root-mean square of the ambient entropy fluctuations.", "The analytical results are shown to compare well with results from two- and three-dimensional numerical simulations.", "Analytical solutions are also derived in the linear regime of Reynolds-averaged turbulence models.", "This highlights an inconsistency in standard turbulence models that prevents them from accurately capturing the physics of rarefaction-turbulence interaction.", "Finally, dimensional analysis of the equations indicates that rapid distortion of turbulence can give rise to two distinct regimes in the turbulent spectrum: a distortion range at large scales where linear distortion effects dominate, and an inertial range at small scales where non-linear effects dominate." ], [ "Introduction", "Subsonic turbulence is important in both the intracluster and intergalactic medium [39], [41].", "Supersonic turbulence is present in the interstellar medium and plays an indispensable role in star formation [27].", "Numerical modeling of turbulence is difficult and fraught with uncertainty [4], and any analytical results that can be obtained provide both a check for numerical codes and a wider view of parameter space (restricted by the assumptions underlying the analytical results).", "Although analytical modeling comes with its own set of difficulties, significant progress can be made for turbulence subjected to rapid distortions.", "Rapid distortion theory (RDT) is an analytical approach to the study of turbulence for conditions under which non-linear effects can be neglected [37].", "Such conditions pertain, for example, to a supernova explosion propagating through a turbulent medium or to turbulent eddies in gravitational free-fall.", "The purpose of this work is to investigate a particularly simple problem using RDT as a proof-of-principle for its application to more realistic astrophysical flows.", "The problem to be studied is the evolution of subsonic turbulence in an ideal gas subject to a centered rarefaction.", "Such a flow occurs, for example, when a shock propagates from heavy to light material in an interaction with a contact discontinuity [28].", "To make the problem analytically tractable, the analysis is restricted to modes that are oriented perpendicular to the distortion; these are the incompressive modes that couple most strongly to the mean flow.", "A complete RDT analysis of rarefaction-turbulence interaction would need to take into account the full spectrum of linear modes.", "Despite this restriction, the one-dimensional analytical solution derived here captures the essential physics.", "Favorable comparisons are made with both two- and three-dimensional numerical simulations.", "Another approach to modeling turbulence when sufficient resolution is not available is to employ a Reynolds-averaged turbulence model [12], [10], [29], [30].", "These models have been used, for example, to capture mixing in interactions between active galactic nuclei and bubbles [38], between high-redshift galactic outflows and clouds [16], between shocks and clouds [31], [32], and between galactic haloes and the intergalactic medium [8].", "Turbulence modeling comes with its own set of uncertainties, due to multiple closures and poorly-constrained model coefficients.", "Analytical solutions are derived here for Reynolds-averaged models in the linear regime which can serve as a verification test for these models.", "In addition, comparison to the analytical linear theory highlights an inconsistency in standard models that prevents them from correctly capturing the physics of rarefaction-turbulence interaction.", "A simple proposal for correcting this inconsistency will be provided.", "§ outlines the basic equations and provides the well-known expressions for a centered rarefaction.", "An overview of RDT along with its application to the problem at hand is given in §.", "Comparisons between RDT and numerical simulations are provided in §, Reynolds-averaged models are discussed in §, and § summarizes the analysis and gives suggestions for future work." ], [ "Basic Equations and Mean Flow", "The Euler equations for an ideal fluid are $\\frac{d\\rho }{dt} + \\rho \\mbox{$\\nabla $}\\cdot \\mbox{$v$} = 0.$ $\\rho \\frac{d\\mbox{$v$}}{dt} + \\mbox{$\\nabla $}p = 0,$ $\\frac{ds}{dt} = 0,$ where $d/dt = \\partial /\\partial t + \\mbox{$v$}\\cdot \\mbox{$\\nabla $}$ is a Lagrangian derivative, $\\mbox{$v$}$ is the fluid velocity, $\\rho $ is the mass density, $p$ is the pressure, and $s$ is the specific entropy.", "Viscosity is negligible due to the large Reynolds numbers of astrophysical flows; application of the results obtained below to terrestrial flows will be valid for a more restrictive range of length scales.", "For an ideal gas equation of state, $s = \\ln \\left(\\frac{p}{\\rho ^\\gamma }\\right),$ where $\\gamma $ is the adiabatic index.", "A centered rarefaction is a self-similar flow, the analysis of which can be found in standard references (e.g., [24]).", "The density and pressure obey the following isentropic relations: $\\frac{\\rho }{\\rho _0} = \\left(\\frac{c_a}{c_0}\\right)^{\\frac{2}{\\gamma - 1}},\\;\\;\\frac{p}{p_0} = \\left(\\frac{c_a}{c_0}\\right)^{\\frac{2\\gamma }{\\gamma - 1}},$ where ${c}_a= \\sqrt{\\gamma p/\\rho }$ is the adiabatic sound speed and a zero subscript denotes an ambient fluid quantity.", "The sound speed and velocity vary with the self-similar variable $\\xi \\equiv x/t$ as $c_a = \\frac{\\gamma - 1}{\\gamma + 1}\\xi + \\frac{2}{\\gamma + 1}c_0,\\;\\;v_x = \\frac{2}{\\gamma + 1}\\left(\\xi - c_0\\right),$ where $x$ is the direction in which the rarefaction propagates.", "The velocity is taken to be in the frame of the ambient fluid, so that $v_x = 0$ when $x = {c}_0t$ ; this defines the front of the rarefaction." ], [ "Rapid Distortion Theory", "RDT is the application of linear theory to distorted turbulent flows, valid when the time scale for nonlinear interactions is longer than the time scale over which the distortion operates.", "Turbulence in that limit can be approximated as a superposition of linear modes driven by the mean distortion.", "Such an approach has a long history in application to both incompressible [42], [43], [2], [3], [21], [15], [37] and compressible [33], [34], [44], [13], [14], [36], [25], [26], [17], [46], [19], [20], [23] fluids.", "While it may seem counter-intuitive to model turbulence using linear theory, any given snapshot of a turbulent flow field can be completely characterized as a superposition of linear modes.", "It is only on time scales over which non-linear interactions between modes become important that this simple picture breaks down.", "Over the time scales considered, and absent the rapid distortion by the mean flow, the turbulence is essentially frozen.", "While the notion of turbulence generally implies the dominance of non-linear interactions, in RDT the mean flow is distorting a snapshot of developed turbulence." ], [ "General Considerations", "RDT is valid when turbulence is distorted on a time scale that is much shorter than the eddy turnover time (the time scale for non-linear interactions between scales due to the velocity advection term), i.e., $t_d \\equiv \\frac{\\ell _d}{v_d} \\ll \\frac{\\lambda }{v_\\lambda } \\equiv t_{nl},$ where $\\ell _d$ and $v_d$ are the length and velocity scales of the distortion, and $\\lambda $ and $v_\\lambda $ are the length and velocity scales of an eddy.", "Assuming a Kolmogorov velocity spectrum, $v_\\lambda \\sim v_\\ell \\left(\\frac{\\lambda }{\\ell }\\right)^{1/3},$ where $\\ell $ is the integral scale and $v_\\ell $ is the eddy speed at that scale, it can be seen from (REF ) that RDT is valid for ${\\cal A} \\ll \\left(\\frac{\\lambda }{\\ell }\\right)^{2/3}.$ where ${\\cal A} \\equiv \\frac{t_d}{t_\\ell } = \\frac{v_\\ell \\ell _d}{v_d \\ell }$ is the ratio of the distortion time scale to the integral time scale $t_\\ell \\equiv \\ell /v_\\ell $ .", "Evaluating expression (REF ) at the integral scale and assuming that the fluid is distorted at the integral scale ($\\ell _d \\sim \\ell $ ) gives ${\\cal M}_t \\ll {\\cal M},$ where ${\\cal M}_t \\equiv v_\\ell /c_a$ is the turbulent Mach number at the integral scale and ${\\cal M} \\equiv v_d/c_a$ is the Mach number of the distortion.", "Expression (REF ) says that subsonic turbulence distorted by a sonic mean flow can be analyzed with RDT.", "Larger distortion scales ($\\ell _d > \\ell $ ) would make expression (REF ) more restrictive (the turbulent Mach number at the integral scale would need to be smaller for RDT to apply).", "Smaller distortion scales would make expression (REF ) less restrictive and would allow for supersonic turbulence; the latter, however, would require a reanalysis in terms of a Burger's spectrum.", "Notice that RDT is not in general valid all the way down to the dissipation scale, since $t_d > t_{nl}$ for $\\lambda < \\lambda _{nl}$ , where $\\lambda _{nl} \\equiv {\\cal A}^{3/2} \\ell = \\frac{v_{\\lambda _{nl}}}{v_d} \\ell _d$ is the length scale at which non-linear interactions become important.", "RDT is valid for $\\lambda _{nl} < \\lambda < \\ell $ , a range of length scales that can be referred to as the distortion range.", "The inertial range is reduced to $\\lambda _0 < \\lambda < \\lambda _{nl}$ , where $\\lambda _0 \\sim R^{-3/4} \\ell $ is the dissipation scale (here $R \\equiv v_\\ell \\ell /\\nu $ is the integral scale Reynolds number and $\\nu $ is the kinematic viscosity).", "Whether or not a distinct inertial range is present thus depends upon the Reynolds number of the flow as well as the size and speed of the integral-scale eddies.", "For $\\lambda _{nl} > \\lambda _0$ , i.e., for ${\\cal A} > R^{-1/2}$ , an inertial range persists at small scales under distortion, whereas for ${\\cal A} < R^{-1/2}$ the distortion range extends down to the dissipation scale.", "Figure: Frequencies in RDT as a function of scale for R=10 5 R = 10^5.", "Shown are the non-linear frequency ω ˜ nl \\widetilde{\\omega }_{nl} (solid line), the dissipation frequency ω ˜ diss \\widetilde{\\omega }_{diss} (dashed line), and the distortion frequency ω ˜ d \\widetilde{\\omega }_d for two cases: ℓ d =ℓ\\ell _d = \\ell and ℳ t =0.1ℳ{\\cal M}_t = 0.1 {\\cal M} (dotted line), and ℓ d =ℓ/R\\ell _d = \\ell /R and ℳ t =ℳ{\\cal M}_t = {\\cal M} (dot-dashed line).", "See text for discussion.Figure REF compares the relevant frequencies in RDT: the non-linear frequency $\\widetilde{\\omega }_{nl} \\equiv t_\\ell /t_{nl} = \\widetilde{k}^{2/3}$ , the dissipation frequency $\\widetilde{\\omega }_{diss} \\equiv t_\\ell /t_{diss} = R^{-1} \\widetilde{k}^{2}$ , and the distortion frequency $\\widetilde{\\omega }_d \\equiv t_\\ell /t_d = {\\cal A}^{-1}$ , where the frequencies have all been normalized to the integral frequency.", "Here $t_{diss} = \\lambda ^2/\\nu $ is the dissipation time scale and $\\widetilde{k} \\equiv \\ell /\\lambda $ .", "A larger value for one of these frequencies implies the dominance of that physical effect.", "The dotted line in Figure REF represents the case where the distortion only operates on a subset of scales (the distortion range), with the remainder of the scales being dominated by non-linear effects (the inertial range).Despite the fact that subsonic turbulence at the integral scale remains subsonic at smaller scales, RDT eventually breaks down because the size of an eddy decreases with scale more strongly than its speed.", "The size of an eddy decreases as $\\lambda $ whereas its speed decreases as $\\lambda ^{1/3}$ , so that even though smaller eddies move more slowly, they turn over more rapidly.", "The dashed line represents the case where the entire range of scales is dominated by distortion and non-linear effects are nowhere important; this is relevant to shock-turbulence interaction (see below).", "Due to the large Reynolds numbers of astrophysical flows, the physical inertial range will generally be distinct from the distortion range.", "Due to the much smaller effective Reynolds numbers of numerical calculations, however, numerically capturing both the distortion range and a significant portion of the inertial range can be a severe challenge.", "For a fixed-grid numerical calculation, well-resolving the scale at which non-linear interactions become important requires $\\lambda _{nl} \\gg \\Delta $ , where $\\Delta \\equiv \\ell _d/N$ is the spatial resolution of the calculation and $N$ is the number of grid cells.", "Using expression (REF ), this implies $N \\gg \\left(\\frac{{\\cal M}_t}{{\\cal M}}\\right)^{-3/2} \\left(\\frac{\\ell _d}{\\ell }\\right)^{-1/2}.$ For ${\\cal M}_t \\sim 0.01$ , ${\\cal M} \\sim 1$ and $\\ell _d \\sim \\ell $ , a numerical calculation with $N^3 \\gg 10^9$ cells would be required to capture both the distortion range and a non-negligible portion of the inertial range.", "Studies of shock-turbulence interaction [33] are a form of inhomogeneous RDT, where $\\ell _d$ and $v_d$ are the shock width and speed, respectively.", "The width of a steady shock is given by $\\ell _d \\sim \\frac{\\nu }{v_d} = \\frac{{\\cal M}_t}{{\\cal M}R} \\ell ,$ which implies that the distortion scale in this case is much smaller than the integral scale; $\\ell _d$ is also the scale at which dissipation takes place.", "Using (REF ) in (REF ) gives $\\frac{{\\cal M}_t}{{\\cal M}} \\ll R^{1/2}\\left(\\frac{\\lambda }{\\ell }\\right)^{1/3},$ or ${\\cal M}_t \\ll {\\cal M} R^{1/2}$ at the integral scale.", "Since ${\\cal M} R^{1/2}$ is large, expression (REF ) implies that RDT is valid for a shock interacting with any level of turbulence, although again these results would have to be reanalyzed for supersonic turbulence.", "The scale at which non-linear effects are important in a shock-turbulence interaction is $\\lambda _{nl} = R^{-3/2}\\left(\\frac{{\\cal M}_t}{{\\cal M}}\\right)^3 \\ell = R^{-1/2}\\left(\\frac{{\\cal M}_t}{{\\cal M}}\\right)^2 \\ell _d,$ which implies $\\lambda _{nl} \\ll \\ell _d$ , i.e., the non-linear scale is smaller than the dissipation scale.", "There is therefore no range of length scales in a shock-turbulence interaction for which non-linear effects dominate over linear distortion effects (see the dot-dashed line in Figure REF .)" ], [ "Linear equations", "To proceed quantitatively with RDT, fluid quantities are decomposed into a mean and a fluctuation, e.g., $\\rho = \\overline{\\rho } + \\rho ^\\prime $ , where a bar denotes a mean and a prime denotes a fluctuation (defined to have zero mean).", "The mean flow is given by expressions (REF ) and (REF ), where a mean is taken here to be a spatial average over $y$ and $z$ , i.e., over the directions perpendicular to the mean flow.", "Fluctuations are governed by the linearized versions of equations (REF )–(REF ): $\\frac{\\partial \\rho ^\\prime }{\\partial t} + \\mbox{$v$}\\cdot \\mbox{$\\nabla $}\\rho ^\\prime + \\rho ^\\prime \\mbox{$\\nabla $}\\cdot \\mbox{$v$}+ \\mbox{$v$}^\\prime \\cdot \\mbox{$\\nabla $}{\\rho }+ {\\rho }\\mbox{$\\nabla $}\\cdot \\mbox{$v$}^\\prime = 0,$ $\\frac{\\partial \\mbox{$v$}^\\prime }{\\partial t} + \\mbox{$v$}\\cdot \\mbox{$\\nabla $}\\mbox{$v$}^\\prime + \\mbox{$v$}^\\prime \\cdot \\mbox{$\\nabla $}\\mbox{$v$}- \\frac{\\rho ^\\prime }{{\\rho }^2} \\mbox{$\\nabla $}{p}+ \\frac{1}{{\\rho }}\\mbox{$\\nabla $}p^\\prime = 0,$ $\\frac{\\partial s^\\prime }{\\partial t} + \\mbox{$v$}\\cdot \\mbox{$\\nabla $}s^\\prime + \\mbox{$v$}^\\prime \\cdot \\mbox{$\\nabla $}{s}= 0.$ For simplicity of notation, basic fluid variables rather than mean flow quantities are used in these equations and in what follows; this notation is precise to linear order.", "Incompressive modes in a compressible fluid are captured by invoking the Boussinesq approximation, valid for short-wave length, low-frequency fluctuations.", "These modes have pressure fluctuations that are small compared to density fluctuations, so that $s^\\prime \\approx -\\gamma \\rho ^\\prime /{\\rho }$ .", "The evolution of incompressive density fluctuations is thus governed by the perturbed entropy equation (REF ) rather than the perturbed continuity equation (REF ); the latter reduces simply to the incompressive condition.", "Under the Boussinesq approximation, then, the governing linear equations become $\\mbox{$\\nabla $}\\cdot \\mbox{$v$}^\\prime = 0,$ $\\frac{d\\mbox{$v$}^\\prime }{dt} = -\\mbox{$v$}^\\prime \\cdot \\mbox{$\\nabla $}\\mbox{$v$}+ \\frac{\\rho ^\\prime }{{\\rho }^2} \\mbox{$\\nabla $}{p}- \\frac{1}{{\\rho }}\\mbox{$\\nabla $}p^\\prime ,$ $\\gamma \\frac{d}{dt}\\left(\\frac{\\rho ^\\prime }{{\\rho }}\\right) = \\mbox{$v$}^\\prime \\cdot \\mbox{$\\nabla $}{s}.$" ], [ "Vorticity Evolution Under a Planar Rarefaction", "Before proceeding with RDT for the problem at hand, some physical intuition can be built by taking a qualitative look at the evolution of vorticity.", "Consider a subsonic turbulent flow undergoing rapid distortion by a planar rarefaction.", "The vorticity in that case can be considered to be a perturbation on the mean flow, and the vorticity equation in component form reduces to $\\frac{d\\omega _x}{dt} = 0 , \\;\\; \\frac{d}{dt}\\left(\\frac{\\mbox{$\\omega $}_\\perp }{\\rho }\\right) = \\frac{\\left(\\mbox{$\\nabla $}p \\times \\mbox{$\\nabla $}\\rho ^\\prime \\right)_\\perp }{\\rho ^3}.$ A detailed derivation of these expressions is given in Appendix .", "The vorticity component parallel to the mean flow is conserved for a fluid element (the stretching and dilatation terms cancel).", "If the baroclinic term can be neglected (this requires turbulent density fluctuations to be much smaller than turbulent velocity fluctuations), the vorticity component perpendicular to the mean flow scales with the mean density.", "This physical behavior is illustrated in Figure REF , which shows a rotating cylindrical vortex expanded parallel and perpendicular to its rotation axis.", "When the vortex is expanded along its rotation axis (the upper portion of Figure REF ), its circular cross-sectional area is unchanged, and its rotation rate therefore remains the same.", "This accounts for the conservation of parallel vorticity.", "When the vortex is expanded perpendicular to its axis (the lower portion of Figure REF ), its cross-sectional area increases, along with the path length that each fluid element must traverse in a rotation; this along with local angular momentum conservation ensures that the rotation rate of the vortex decreases in proportion to the amount of expansion.", "This accounts for the scaling of the perpendicular vorticity with density.", "Figure: A rotating cylindrical vortex before (left) and after (right) expansion parallel (top) and perpendicular (bottom) to its rotation axis.", "See text for discussion.Additional considerations demonstrate that planar expansion will interact primarily with turbulent structures oriented along the expansion direction (i.e., with wave vectors perpendicular to the expansion direction).", "Figure REF shows a rotating elliptical vortex expanded perpendicular to its rotation axis.", "Fluid elements spend most of their time traversing the major axis of the ellipse, so that expansion along this direction (the upper portion of Figure REF ) results in a greater speed-up than expansion along the minor axis (the lower portion of Figure REF ).", "Figure: A rotating elliptical vortex before (left) and after (right) expansion perpendicular to its rotation axis, with the major axis oriented either parallel (top) or perpendicular (bottom) to the expansion direction.", "See text for discussion.If the ambient fluid is dominated by entropy (pressure-less density) fluctuations rather than vorticity fluctuations, the second expression in (REF ) indicates that a planar rarefaction will generate perpendicular vorticity at the level of the ambient entropy fluctuations.", "The growth of vorticity will continue until the vortical fluctuations approach the level of the entropic fluctuations, at which point the decrease of vorticity due to expansion described above will begin to take over.", "This physical behavior is illustrated in Figure REF , which shows an entropy fluctuation consisting of two fluid parcels, one heavier than the other.", "If the density gradient is oriented perpendicular to the expansion direction (the upper portion of Figure REF ), the pressure force will accelerate the light fluid parcel more than the heavy one, and the resulting baroclinic torque will rotate the fluid parcels about their center of mass.", "For a density gradient oriented parallel to the expansion direction (the lower portion of Figure REF ), no baroclinic torque is applied and therefore no vorticity is generated.", "Figure: A stratified entropy fluctuation before (left) and after (right) expansion, with the density gradient oriented either perpendicular (top) or parallel (bottom) to the expansion direction.", "The vertical block arrows represent the pressure force associated with the rarefaction.", "See text for discussion.A word of clarification is in order here on the generation of vorticity due to baroclinic effects.", "It might be tempting to dismiss this effect because equations (REF ) indicate that $p = p(\\rho )$ and the baroclinic term therefore vanishes.", "In addition, the adiabatic condition (REF ) implies that $p/\\rho ^\\gamma $ is conserved for a fluid element.", "Equations (REF ) and (REF ), however, are statements about the flow, not properties of the fluid.", "For the non-isothermal ideal-gas equation of state considered here, $p = p(\\rho ,T)$ in general, even while $p/\\rho ^\\gamma $ is conserved for a fluid element.", "In addition, the equilibrium flow is barotropic (the baroclinic term vanishes at leading order), but the fluid is not (vorticity can be generated at higher orders).", "A barotropic equilibrium and adiabatic flow are consistent with the generation of vorticity; indeed, most basic fluid instabilities are analyzed under the same conditions.The density blob that appears in canonical descriptions of buoyancy instability, for example, is a superposition of entropy fluctuations, and a baroclinic torque is applied to it by gravity.", "Finally, the behavior of entropy fluctuations under expansion can be estimated by the following considerations.", "In the absence of pressure fluctuations, a density fluctuation will expand in the same manner as the mean fluid, and should therefore scale with the local density.", "In addition, the same considerations as those surrounding Figure REF apply here: entropy fluctuations will be elongated in the direction parallel to the mean flow.", "A planar rarefaction will thus generate anisotropy in both vortical and entropic fluctuations.", "The considerations of this section can be used to justify the use of a two-dimensional numerical model to capture these effects.", "The parallel vorticity is unaffected by expansion, and both perpendicular components behave in the same manner.", "All that is required to capture the essential physics is a wave vector perpendicular to the expansion direction.", "The stretching term, which only exists in three-dimensions for a planar geometry, is negligible for the perpendicular vorticity components (see Figure REF and Appendix ).", "In addition, the dominance of perpendicular wave vectors implies rotational velocites that are primarily in the same direction as the expansion (fluid velocities in the upper portion of Figure REF are primarily along the major axis of the ellipse).", "This suggests that the fluctuations can be captured with a one-dimensional model; subsequent sections will demonstrate this to be the case." ], [ "One-dimensional Linear Theory", "The standard approach in RDT is to decompose the fluctuations into Fourier modes and study their evolution using either the full set of linear equations (REF )–(REF ) or the incompressive set (REF )–(REF ).", "The mean flow represented by (REF ) and (REF ), however, precludes such an approach and necessitates the much more difficult task of performing an RDT analysis that is inhomogeneous in the $x$ -direction.", "The problem can be made analytically tractable, however, by further restricting the analysis to incompressive modes that have a wave vector perpendicular to the gradient direction and a dominant velocity component in the gradient direction (the former implies the latter for $\\mbox{$\\nabla $}\\cdot \\mbox{$v$}^\\prime = 0$ ).", "As discussed in §REF , these are the incompressive modes that are most affected by the expansion.", "In this limit, the pressure fluctuation in the equation for the dominant velocity component can be ignored, and equations (REF )–(REF ) further reduce to the one-dimensional form $\\frac{dv_x^\\prime }{dt} = -v_x^\\prime \\frac{\\partial v_x}{\\partial x} + \\frac{\\rho ^\\prime }{{\\rho }^2} \\frac{\\partial {p}}{\\partial x},$ $\\frac{d}{dt}\\left(\\frac{\\rho ^\\prime }{{\\rho }}\\right) = \\frac{v_x^\\prime }{\\gamma } \\frac{\\partial {s}}{\\partial x}.$ Transforming to the self-similar variable $\\xi $ , these equations become $\\left({v}_x- \\xi \\right)\\frac{dv_x^\\prime }{d\\xi } = -v_x^\\prime \\frac{d{v}_x}{d\\xi } + \\frac{\\rho ^\\prime }{{\\rho }^2} \\frac{d{p}}{d\\xi },$ $\\frac{d}{d\\xi }\\left(\\frac{\\rho ^\\prime }{{\\rho }}\\right) = \\frac{v_x^\\prime }{\\gamma } \\frac{d{s}}{d\\xi }.$ Applying the mean flow conditions appropriate for a rarefaction ($ds = 0$ , $dp = \\rho c_a dv_x$ , $dv_x/d\\xi = 2/[\\gamma + 1]$ , $v_x + {c}_a= \\xi $ ), the equations are finally given by $\\frac{\\gamma + 1}{2}\\frac{dv_x^\\prime }{d\\xi } = \\frac{v_x^\\prime }{{c}_a} - \\frac{\\rho ^\\prime }{{\\rho }}.$ $\\frac{d}{d\\xi }\\left(\\frac{\\rho ^\\prime }{{\\rho }}\\right) = 0.$ Equation (REF ) can be trivially solved to give $\\rho ^\\prime = \\rho ^\\prime _0\\frac{{\\rho }}{{\\rho }_0}.$ To solve equation (REF ), transform to the dependent variable $v_x^\\prime /{c}_a$ and the independent variable $\\ln {c}_a$ : $\\left(\\gamma - 1\\right)\\frac{d}{d\\ln {c}_a}\\left(\\frac{v_x^\\prime }{{c}_a}\\right) = \\left(3-\\gamma \\right)\\frac{v_x^\\prime }{{c}_a} - 2\\frac{\\rho ^\\prime }{{\\rho }}.$ This can be readily integrated, using (REF ), to give $\\frac{v_x^\\prime }{{c}_0} = \\frac{v_{x0}^\\prime }{{c}_0} \\frac{{\\rho }}{{\\rho }_0} + \\frac{2}{3-\\gamma }\\frac{\\rho ^\\prime _0}{{\\rho }_0}\\left(\\frac{{c}_a}{{c}_0} - \\frac{{\\rho }}{{\\rho }_0}\\right).$ Expressions (REF ) and (REF ) confirm the qualitative analysis of §REF .", "They demonstrate that incompressive density fluctuations scale with the mean density, and that incompressive velocity fluctuations are subject to two competing effects.", "The first term in expression (REF ) represents the reduction of vorticity due to fluid expansion: as a vortex expands, its rotational velocity decreases due to the conservation of angular momentum.", "In the limit of negligible ambient entropic fluctuations, the vortical fluctuations scale with the mean density.", "The second term in expression (REF ) represents the baroclinic production of vorticity due to the interaction between the entropic fluctuations and the mean pressure gradient.", "The relative importance of these two terms is determined by the ratio of entropic and vortical fluctuations in the ambient fluid.", "An expression for the turbulent kinetic energy can be constructed by averaging the square of expression (REF ), $K_x \\equiv \\overline{v_x^{\\prime 2}}$ : $\\frac{K_x}{K_{x0}} &=& \\left(\\frac{{\\rho }}{{\\rho }_0}\\right)^2 + \\frac{4\\Phi _{x0} {\\cal A}_{x0}}{3-\\gamma }\\frac{{\\rho }}{{\\rho }_0}\\left(\\frac{{c}_a}{{c}_0} - \\frac{{\\rho }}{{\\rho }_0}\\right) \\nonumber \\\\ && + \\frac{4{\\cal A}_{x0}^2}{\\left(3 - \\gamma \\right)^2}\\left(\\frac{{c}_a}{{c}_0} - \\frac{{\\rho }}{{\\rho }_0}\\right)^2,$ where $\\Phi _{x0} \\equiv \\frac{\\overline{\\rho ^\\prime _0 v_{x0}^\\prime }}{\\sqrt{\\overline{\\rho _0^{\\prime 2}}\\,\\overline{v_{x0}^{\\prime 2}}}}\\;\\;,\\;\\; {\\cal A}_{x0} \\equiv \\sqrt{\\frac{\\overline{\\rho _0^{\\prime 2}}/{\\rho }_0^2}{ \\overline{v_{x0}^{\\prime 2}}/{c}_0^2}}$ are the Pearson correlation coefficient and amplitude ratio for the entropic and vortical fluctuations ahead of the rarefaction front.", "Figure REF delineates the regions of $({\\cal A}_{x0},\\Phi _{x0})$ parameter space where $K_x$ grows and decays in the wake of a rarefaction, and Figure REF shows sample spatial profiles for each of these regions.", "In region I, $K_x$ decays throughout the rarefaction.", "In region II, $K_x$ grows directly behind the rarefaction front and reaches a maximum $K_{+}$ before finally decaying.", "In regions III and IV, $K_x$ experiences a period of decay to $K_{-}$ followed by growth to $K_{+}$ followed by decay, so that $K_x$ has both a local minimum and a local maximum.", "In region III, $K_{+}$ is a local but not a global maximum ($K_{+} < K_{x0}$ ), so that $K_x$ never grows larger than its initial amplitude.", "In region IV, $K_{+}$ is a global maximum ($K_{+} > K_{x0}$ ), so that $K_x$ grows but the growth is delayed due to the initial decay phase.", "Figure: Phase diagram of the growth/decay of subsonic turbulence in the wake of a rarefaction for γ=5/3\\gamma = 5/3.", "See text for discussion.Figure: Sample profiles of K x K_x for the regions defined in Figure : decay (I, dotted line, 𝒜 x0 =0.5{\\cal A}_{x0} = 0.5, Φ x0 =0.5\\Phi _{x0} = 0.5), growth (II, solid line, 𝒜 x0 =2.5{\\cal A}_{x0} = 2.5, Φ x0 =0.8\\Phi _{x0} = 0.8), delayed decay (III, dot-dashed line, 𝒜 x0 =1{\\cal A}_{x0} = 1, Φ x0 =-0.8\\Phi _{x0} = -0.8) and delayed growth (IV, dashed line, 𝒜 x0 =2.5{\\cal A}_{x0} = 2.5, Φ x0 =-0.8\\Phi _{x0} = -0.8).The borders of the regions defined in Figure REF can be determined from an analysis of expression (REF ), the details of which are given in Appendix .", "The lower boundary of region II is given by $\\Phi _{x0} {\\cal A}_{x0} = 1$ , the boundary between regions I and III is a portion of the ellipse defined by ${\\cal A}_{x0}^2 + \\left(\\frac{\\gamma +1}{2}\\right)^2\\Phi _{x0}^2 - \\left(3-\\gamma \\right){\\cal A}_{x0} \\Phi _{x0} - 2\\left(\\gamma - 1\\right) = 0,$ and the boundary between regions III and IV is given by $K_{+} = K_{x0}$ , where $K_{+}$ is defined in expression (REF ).", "This boundary can be determined analytically for specific values of $\\gamma $ ; for $\\gamma = 5/3$ it is given by $\\Phi _{x0} = \\frac{16 + 72{\\cal A}_{x0}^2 - 27{\\cal A}_{x0}^4}{64 {\\cal A}_{x0}},$ which intersects $\\Phi _{x0} = -1$ at ${\\cal A}_{x0} = 2$ .", "The critical point where the three boundaries intersect is $\\left({\\cal A}_{x0},\\Phi _{x0}\\right) = \\left(\\sqrt{\\frac{\\gamma + 1}{2}}, \\sqrt{\\frac{2}{\\gamma + 1}}\\right).$ Figure REF in Appendix shows the phase diagrams for $\\gamma = 7/5$ and $\\gamma = 1$ .", "As $\\gamma $ decreases from $5/3$ to 1, the boundary between regions I and III approaches the line $\\Phi _{x0} = {\\cal A}_{x0}$ , the boundary between regions III and IV approaches the vertical line ${\\cal A}_{x0} = 1$ , and the critical point approaches $(\\Phi _{x0},{\\cal A}_{x0}) = (1,1)$ .", "It can be seen from Figures REF and REF that ${\\cal A}_{x0}>1$ is a necessary condition for vorticity amplification by a rarefaction.", "Expressed physically, this is the requirement that ambient entropic fluctuations (in units of the ambient density) be larger than ambient vortical fluctuations (in units of the ambient sound speed).", "Figures REF and REF also show that for $\\gamma \\le 5/3$ , ${\\cal A}_{x0} >2$ is a sufficient condition for vorticity amplification by a rarefaction.", "In the quiescent limit (${\\cal A}_{x0} \\rightarrow \\infty $ ), the ambient fluid is dominated by entropic fluctuations and the ambient vortical fluctuations are negligible.", "Vorticity can be generated by a rarefaction in that case due to baroclinic production, as the incompressive density fluctuations interact with the mean pressure gradient.", "The vortical energy generated by this mechanism peaks at $K_{+} = \\frac{1}{2} \\left(\\frac{\\gamma - 1}{2}\\right)^\\frac{2(\\gamma - 1)}{3-\\gamma } \\frac{\\overline{\\rho _0^{\\prime 2}}}{{\\rho }_0^2} {c}_0^2,$ obtained by taking the ${\\cal A}_{x0} \\rightarrow \\infty $ limit of expressions (REF ) and (REF ).", "This corresponds to an upper limit on the turbulent kinetic energy that can be generated by a planar rarefaction.", "This upper limit is comparable to the turbulence generated by a shock interacting with ambient density fluctuations [23].", "For $\\gamma = 5/3$ , it is $K_{+}\\left(\\gamma = 5/3\\right) = \\frac{1}{6} \\frac{\\overline{\\rho _0^{\\prime 2}}}{{\\rho }_0^2} {c}_0^2.$ The vortical energy generated by a rarefaction depends upon the strength of the rarefaction: reaching the upper limit given by expression (REF ) requires a rarefaction that reduces the mean density to $\\rho _+ = \\rho _0 \\left(\\frac{\\gamma - 1}{2}\\right)^\\frac{2}{3-\\gamma }$ ($\\rho _+ \\approx 0.2\\rho _0$ for $\\gamma = 5/3$ ).", "This is equivalent to a piston velocity (or, equivalently, a velocity jump across the rarefaction) of $\\left\|v_{p}\\right\| = \\frac{2{c}_0}{\\gamma - 1}\\left(1 - \\left[\\frac{\\gamma - 1}{2}\\right]^\\frac{\\gamma - 1}{3-\\gamma }\\right)$ ($\\left\|v_p\\right\| \\approx 1.3 {c}_0$ for $\\gamma = 5/3$ ).", "For stronger rarefactions, the vortical energy will peak at $K_+$ when $\\rho = \\rho _+$ and then decay.", "The piston velocity required to generate the maximum level of vorticity increases as $\\gamma \\rightarrow 1$ , indicating that for more compressible fluids, more energy is required in order to generate turbulence by this mechanism.", "This section compares the results of § to numerical simulations of equations (REF )–(REF ) using the Zeus algorithm [40].", "Details of the numerical algorithm are given in Appendix .", "Subsonic turbulence was generated in these calculations by initializing a random vorticity field and allowing it to evolve for many sound crossing times.", "An outgoing piston boundary condition was then applied to one face of the computational domain, generating a planar rarefaction.", "Three-dimensional results are shown in §REF , followed by two-dimensional results in §REF ." ], [ "Three-dimensional Results", "To qualitatively confirm the results of §REF , Figures REF and REF show snapshots of vorticity components parallel and perpendicular to the expansion direction in a three-dimensional calculation after the rarefaction front has propagated partway across the computational domain.", "The piston is applied to the upper face in these figures, and the rarefaction front propagates from top to bottom.", "Results for the other perpendicular vorticity component are similar to Figure REF .", "To aid in interpretation, the cylindrical vortices from Figure REF are superimposed on the three-dimensional results in Figures REF and REF .", "As expected from equation (REF ), the parallel vorticity component is advected by the mean flow but remains unchanged in amplitude, whereas the perpendicular component is damped by the expansion.", "Notice also that after expansion the turbulent structures have a wave vector component that is predominately in the perpendicular direction, consistent with the discussion surrounding Figure REF .", "Figure REF also justifies the use of RDT for this problem.", "These calculations were performed in the frame of the rarefaction rear, located at the upper face of the computational domain (see Appendix ).", "Fluid below the upper face in Figures REF and REF is under expansion, whereas the upper face itself is at rest and not undergoing expansion.", "Comparison of this face at the initial and final times in Figure REF clearly demonstrates that the turbulent structures remain essentially unchanged over the time scales considered.", "As discussed in §, absent the rapid distortion the turbulence is frozen.", "Figure: Component of vorticity parallel to a rarefaction propagating from top to bottom, at the initial (top) and final (bottom) times.", "See text for discussion.", "(See the online version for a colored copy of this figure.", "A higher resolution copy is available upon request.", ")Figure: Component of vorticity perpendicular to a rarefaction propagating from top to bottom, at the initial (top) and final (bottom) times.", "See text for discussion.", "(See the online version for a colored copy of this figure.", "A higher resolution copy is available upon request.", ")As a quantitative comparison, the one-dimensional theory of §REF should capture average turbulent quantities from both two- and three-dimensional simulations.", "To verify this, Figures REF and REF show results from a couple of representative three-dimensional calculations at moderate resolution ($128^3$ with $L_x = L_y = L_z$ ).", "Figure REF shows the profile of vortical kinetic energy $K_x$ for a rarefaction applied to developed turbulence, and Figure REF shows $K_x$ for a rarefaction applied to random entropy fluctuations.", "The comparison between the analytical solution and the three-dimensional numerical results is remarkably good, considering the low resolution used and the fact that the Boussinesq approximation is only marginally satisfied.", "One reason for this is that the considerations of §REF should apply to all wavelengths if the effects of pressure fluctuations can be ignored or averaged over.", "The discrepancies in Figures REF and REF are likely due to the low resolution employed and the fact that the Boussinesq approximation is not well-satisfied.", "Figure: Numerical (solid line) and analytical (dashed line) profiles of K x K_x for a three-dimensional simulation with 𝒜 x0 =0.5{\\cal A}_{x0} = 0.5 and Φ x0 =-0.1\\Phi _{x0} = -0.1 (region I).Figure: Numerical (solid line) and analytical (dashed line) profiles of K x K_x for a three-dimensional simulation with 𝒜 x0 =∞{\\cal A}_{x0} = \\infty (region II).The validity of the Boussinesq approximation has been checked for a single mode in three dimensions; a good match can be obtained for $L_y = L_z = 0.01 L_x$ , with one perpendicular wavelength across the computational domain.", "Figure REF demonstrates an improved match to theory for smaller length scales: shown in this figure are results from a three-dimensional calculation with a box size $1\\%$ of that used to generate the results in Figure REF .", "Figure: Numerical (solid line) and analytical (dashed line) profiles of K x K_x for a three-dimensional simulation with shorter wave lengths than Figure .It was argued in § and Appendix that certain terms in the vorticity equation should dominate in RDT.", "In particular, for a planar mean flow in the $x$ -direction, the $x$ -component of the stretching and dilation terms should cancel, and the $y$ - and $z$ -components of the stretching term should be negligible.", "Figure REF demonstrates the former and Figure REF demonstrates the latter for the calculation shown in Figure REF .", "Both figures also include the baroclinic term for comparison.", "The quantities plotted in these figures are the volume-integrated terms in the vorticity equation: $D_i \\equiv -\\int dV\\,\\omega _i\\mbox{$\\nabla $}\\cdot \\mbox{$v$},\\;\\;S_i \\equiv \\int dV\\, \\mbox{$\\omega $} \\cdot \\mbox{$\\nabla $}v_i,$ $B_i \\equiv \\int dV\\,\\frac{\\left(\\mbox{$\\nabla $}p \\times \\mbox{$\\nabla $}\\rho \\right)_i}{\\rho ^2}.$ Figure: Time evolution of the xx-component of terms in the vorticity equation for the calculation in Figure .", "Shown are the dilatation term D x D_x (solid line), stretching term S x S_x (dashed line), sum of the stretching and dilatation terms D x +S x D_x + S_x (dotted line), and baroclinic term B x B_x (dot-dashed line).Figure: Time evolution of the yy- and zz-components of terms in the vorticity equation for the calculation in Figure .", "Shown are the dilatation terms D y D_y and D z D_z (solid lines), stretching terms S y S_y and S z S_z (dashed lines), and baroclinic terms B y B_y and B z B_z (dot-dashed lines)." ], [ "Two-dimensional Results", "This section compares the results of §REF with numerical calculations using a two-dimensional version of the Zeus algorithm [40].", "Simulating only two dimensions enables better resolution of the short-wavelength incompressive modes at lower computational cost.", "The simulations were initialized with random vortical and entropic fluctuations in a computational box of size $L$ , and an outgoing piston boundary condition was applied to one side of the computational domain (the left side of the figures shown below).", "All results shown here were obtained at a numerical resolution of $2048^2$ .", "Figures REF and REF are snapshots of vorticity from simulations that demonstrate the two competing effects described in §REF : Figure REF shows the damping of vorticity due to fluid expansion, and Figure REF shows the production of vorticity due to baroclinicity.", "The former simulation was initialized with random vortical fluctuations (${\\cal A}_{x0} = 0$ ), and the latter with random entropic fluctuations (${\\cal A}_{x0} = \\infty $ ).", "Notice that the wave vector of the vorticity in the rarefaction region in Figures REF and REF is aligned primarily in the direction perpendicular to the mean flow, consistent with the estimates made in §REF .This is not an artifact of the initial conditions as the initial random fluctuations were isotropic.", "Figure: Two-dimensional example of vorticity decay due to a planar rarefaction.", "Shown is the vorticity when the rarefaction front is at x=0.7Lx = 0.7L.", "(See the online version for a colored copy of this figure.", "A higher resolution copy is available upon request.", ")Figure: Two-dimensional example of baroclinic vorticity production due to a planar rarefaction.", "Shown is the vorticity when the rarefaction front is at x=0.7Lx = 0.7L.", "(See the online version for a colored copy of this figure.", "A higher resolution copy is available upon request.", ")For a more quantitative comparison with theory, Figures REF –REF show profiles of the vortical kinetic energy $K_x$ from a series of simulations with varying ${\\cal A}_{x0}$ and $\\Phi _{x0}$ .", "These simulations were initialized with both vortical and entropic fluctuations, where the relative amplitudes of the initial random fields were controlled but no attempt was made to control the correlation between them.", "The values for ${\\cal A}_{x0}$ and $\\Phi _{x0}$ quoted in the figure captions were obtained by numerical measurement, i.e., by a spatial average over the $y$ -direction in the ambient fluid at the current time.", "The profiles in all of these figures are shown when the rarefaction front is at $x = 0.7L$ .", "It is clear from these results that the analytical theory of §REF captures the essential physics.", "The noise on these plots is a manifestation of compressive motions; restricting the comparison to a single incompressive mode yields a better match with theory, although even in that case oscillations are generated at the front and back of the rarefaction.Some of these oscillations are numerical due to the weak discontinuities at these locations and can be removed with either a physical or linear artificial viscosity.", "It is remarkable that all four regions (Figures REF and REF ) from the one-dimensional theory of §REF can be accessed in a numerical simulation simply by varying the ratio of ambient vortical and entropic fluctuations.", "Figure: Numerical (solid line) and analytical (dashed line) profiles of K x K_x for a two-dimensional simulation with 𝒜 x0 =0.5{\\cal A}_{x0} = 0.5 and Φ x0 =0.235\\Phi _{x0} = 0.235 (region I).Figure: Numerical (solid line) and analytical (dashed line) profiles of K x K_x for a two-dimensional simulation with 𝒜 x0 =∞{\\cal A}_{x0} = \\infty (region II).Figure: Numerical (solid line) and analytical (dashed line) profiles of K x K_x for a two-dimensional simulation with 𝒜 x0 =1.3{\\cal A}_{x0} = 1.3 and Φ x0 =0.1\\Phi _{x0} = 0.1 (region III).Figure: Numerical (solid line) and analytical (dashed line) profiles of K x K_x for a two-dimensional simulation with 𝒜 x0 =2.75{\\cal A}_{x0} = 2.75 and Φ x0 =0.1\\Phi _{x0} = 0.1 (region IV).Figure REF demonstrates the validity of the Boussinesq approximation in two-dimensions by plotting the density, pressure and entropy fluctuations for the results shown in Figure REF (obtained by taking a slice through the computational domain).", "Departures from the Boussinesq approximation are significant only near the piston.", "Producing a plot similar to Figure REF in three dimensions would require a high-resolution calculation, since the Boussinesq approximation is only valid for short wavelengths.", "Figure: Slice plot of ρ ' /ρ\\rho ^\\prime /\\rho (solid line), p ' /pp^\\prime /p (dashed line) and -s ' /(γs)-s^\\prime /(\\gamma s) (dotted line) for the results shown in Figure .Two-dimensional calculations were also run in which the initial state was allowed to develop into turbulence, but the inverse cascade that is present in two dimensions due to the conservation of potential vorticity results in a turbulent state with the bulk of the power on large scales; this increases the relative importance of compressibility as well as compromises the ability to obtain clean averages over the inhomogeneities.", "Both of these issues complicate comparison with the linear theory of §REF , which neglects compressive modes and assumes that ambient quantities can be characterized by a single value.", "These issues could be avoided by either performing ensemble averages over a series of calculations or increasing the numerical resolution so that the initial small scales could develop into turbulence at an intermediate scale before the piston was applied.", "The computational cost of both of these approaches would be fairly severe, however, and since turbulence under rapid distortion evolves in the same manner as a random vorticity field under rapid distortion, the approach taken here is entirely appropriate." ], [ "Reynolds-averaged Models", "Reynolds-averaged models are a class of turbulence models obtained by averaging the Euler or Navier-Stokes equations and postulating closures for high-order correlations among fluctuations.", "They consist of evolution equations for the turbulent kinetic energy and, typically, a turbulent dissipation rate or a turbulent length scale.", "Common instantiations are the $K$ –$\\epsilon $ model of [12], where $K$ is the turbulent kinetic energy and $\\epsilon $ is the dissipation rate, and the $K$ –$\\ell $ model of [10], where here $\\ell $ is a turbulent length scale.", "These are two-equation models; three-or-more equation models have also been developed, an example of which is the model described in [5].", "None of these models correctly capture the physical theory described above.", "This section will discuss the reasons for this, outline the regions of parameter space that current models do capture, and provide guidance towards a better model.", "The primary reason for the failure of Reynolds-averaged models to capture rarefaction-turbulence interaction is that they assume incompressive density fluctuations are driven by density gradients rather than entropy gradients.", "As discussed in §REF , incompressive density fluctuations are governed by entropy conservation rather than mass conservation; in other words, they obey equation (REF ) rather than equation (REF ) with $\\mbox{$\\nabla $}\\cdot \\mbox{$v$}^\\prime = 0$ .", "In developing their transport equations for variable-density turbulence, [5] derive the evolution equation for density fluctuations from mass conservation.", "This is inconsistent with the Boussinesq approximation as well as with the notion of $K$ as a source of turbulent diffusivity.", "To use equation (REF ) with $\\mbox{$\\nabla $}\\cdot \\mbox{$v$}^\\prime = 0$ rather than equation (REF ) is to ignore the low-frequency character of subsonic turbulence as compared to compressive motions.", "The Boussinesq approximation is essentially $\\partial /\\partial t \\ll {c}_a\\partial /\\partial x$ , which leads directly to $\\mbox{$\\nabla $}\\cdot \\mbox{$v$}^\\prime \\approx 0$ .", "The remainder of the continuity equation, if considered at all, is taken up by compressive motions with $\\partial /\\partial t \\sim {c}_a\\partial /\\partial x$ and $\\mbox{$\\nabla $}\\cdot \\mbox{$v$}^\\prime \\ne 0$ .", "Using equation (REF ) with $\\mbox{$\\nabla $}\\cdot \\mbox{$v$}^\\prime = 0$ introduces a compressive component into the turbulent diffusivity, and that inconsistently.", "Appendix outlines linear theory under this erroneous assumption; $K_x$ obtained in this manner is given by equation (REF ) and is clearly inconsistent with expression (REF ).", "One can demonstrate that (REF ) is the solution to the [5] model in the linear regime, and that this model will therefore not correctly capture rarefaction-turbulence interaction as described above.", "The general linear solution for a two-equation model in the presence of a rarefied mean flow is derived in Appendix , where it is shown that a two-equation model with a density-gradient closure and a particular set of model coefficients matches the [5] model for $\\Phi _{x0} = -1$ and ${\\cal A}_{x0} = 2/(\\gamma +1)$ .", "Two-equation models therefore also fail to capture the physical results of §REF .", "Figures REF –REF compare linear theory results from each of the regions in Figures REF and REF with the inconsistent linear theory expression (REF ) from [5] as well as expressions (REF ) and (REF ) from a two-equation model; the latter represent two separate initial conditions and bracket the results that can be obtained with a two-equation model using standard coefficient values ($C_{K2} = 1$ , $C_{\\epsilon 1} = 1.44$ , $C_{\\epsilon 2} = 1.92$ ).", "Whereas the [5] model generally underestimates the growth of turbulence in a rarefaction, the two-equation model (with standard settings) can either underestimate or overestimate the growth.", "Notice also that the discrepancies between the consistent and inconsistent linear theory results only become significant for strong rarefactions.", "Figure: Profiles of K x K_x in region I (𝒜 x0 =0.5{\\cal A}_{x0} = 0.5, Φ x0 =0.5\\Phi _{x0} = 0.5) using the consistent linear theory expression () (solid line), the inconsistent linear theory expression () (dotted line), the two-equation model expression () (dashed line), and the two-equation model expression () (dot-dashed line).Figure: Profiles of K x K_x in region II (𝒜 x0 =2.5{\\cal A}_{x0} = 2.5, Φ x0 =0.8\\Phi _{x0} = 0.8) using the consistent linear theory expression () (solid line), the inconsistent linear theory expression () (dotted line), the two-equation model expression () (dashed line), and the two-equation model expression () (dot-dashed line).Figure: Profiles of K x K_x in region III (𝒜 x0 =1{\\cal A}_{x0} = 1, Φ x0 =-0.8\\Phi _{x0} = -0.8) using the consistent linear theory expression () (solid line), the inconsistent linear theory expression () (dotted line), the two-equation model expression () (dashed line), and the two-equation model expression () (dot-dashed line).Figure: Profiles of K x K_x in region IV (𝒜 x0 =2.5{\\cal A}_{x0} = 2.5, Φ x0 =-0.8\\Phi _{x0} = -0.8) using the consistent linear theory expression () (solid line), the inconsistent linear theory expression () (dotted line), the two-equation model expression () (dashed line), and the two-equation model expression () (dot-dashed line).The [5] model can be made consistent with the results of §REF by replacing mean-density gradients with $-\\mbox{$\\nabla $}s/\\gamma $ , i.e., $\\mbox{$\\nabla $}\\ln {\\rho }\\rightarrow \\mbox{$\\nabla $}\\ln {\\rho }- \\frac{1}{\\gamma } \\mbox{$\\nabla $}\\ln {p}.$ This prescription is valid only for an ideal-gas equation of state; for a general equation of state one would need to re-derive the equation for density fluctuations from the internal energy equation under the assumption of zero pressure fluctuations.", "In a similar manner, replacing the mean-density gradient in the standard closure for a two-equation model with $-\\mbox{$\\nabla $}s/\\gamma $ , $\\overline{\\mbox{$v$}^{\\prime \\prime }} \\equiv \\frac{C_{\\mu }\\sqrt{K} \\ell }{\\sigma _{\\rho }}\\left(\\mbox{$\\nabla $}\\ln {\\rho }- \\frac{1}{\\gamma } \\mbox{$\\nabla $}\\ln {p}\\right),$ results in a model that is consistent with the Boussinesq approximation [1], [9].", "The two primes here denote a departure from a Favre-averaged (density-weighted Reynolds-averaged) quantity.", "A similar closure was recently used in Reynolds-averaged Navier-Stokes modeling of re-shocked Richtmyer-Meshkov instability experiments [29], [30].", "The baroclinic production of vorticity in this context is not directly captured by the buoyancy production term (rarefactions are Rayleigh-Taylor stable), but is captured rather by the coupling of the density-velocity correlation to $K$ and depends upon the evolution of that correlation.", "This physical effect is thus missing from a two-equation model, so that only turbulent decay can be modeled.", "Even with an entropy-gradient closure, a two-equation turbulence model can thus capture the results of §REF only when vortical fluctuations dominate over entropic fluctuations ahead of the rarefaction (region I).", "It is standard practice in two-equation models to turn off buoyancy production in Rayleigh-Taylor-stable flows (see Appendix ), so that standard models mimic an entropy-gradient closure for isentropic flows (buoyancy production is zero in both cases).", "A two-equation model with an entropy-gradient closure would thus give the same result as the dashed lines in Figures REF –REF .", "As discussed in Appendix , an exact match between a two-equation model and linear theory can only occur for $\\Phi _{x0} = -1$ and ${\\cal A}_0 \\ll 1$ .", "A final subtlety associated with Reynolds-averaged modeling of rarefaction-turbulence interaction should be mentioned.", "For $\\Phi _{x0} = -1$ (which corresponds to a particular set of ambient conditions in the [5] model), $v_x^\\prime $ crosses zero at ${c}_a^{\\ast } = {c}_0\\left(1 + \\frac{3-\\gamma }{2{\\cal A}_{x0}}\\right)^{-\\frac{3-\\gamma }{\\gamma - 1}}.$ In the quiescent limit (${\\cal A}_{x0} \\gg 1$ ), this implies that $K_{x0}$ touches zero close to the rarefaction front before growing.", "Such behavior is difficult to capture with a numerical model, particularly if a floor is implemented to keep $K$ from becoming too small." ], [ "Summary and Discussion", "A one-dimensional analytical solution for vortical and entropic fluctuations subject to a planar rarefaction has been derived and compared to two- and three-dimensional numerical simulations.", "Despite some restrictive assumptions, the consistency between the analytical and numerical results indicates that the analysis has captured the essential physics.", "The primary results are given by expressions (REF ), (REF ) and (REF ), and they demonstrate that 1) entropic fluctuations (i.e., incompressive density fluctuations) scale with the mean density in a planar rarefaction and 2) vortical fluctuations can grow or decay depending upon the correlation between and relative amplitude of the ambient entropic and vortical fluctuations.", "Growth occurs when ambient entropic fluctuations dominate over ambient vortical fluctuations, and decay occurs in the opposite limit.", "The peak turbulent Mach number that can be produced by a rarefaction scales with the ambient entropic fluctuations, and purely-decaying vortical fluctuations scale with the mean density.", "Detailed phase spaces outlining regions of growth and decay are given in Figures REF and REF .", "It should be emphasized that the growth and decay described in this work occurs only for the velocity component parallel to the rarefaction.", "The other two components are unchanged due to the conservation of parallel vorticity (§REF ).", "Isotropic turbulence with equal power in all three velocity components will therefore see only one-third of its total energy impacted by a planar rarefaction.", "Analytical solutions have also been derived for Reynolds-averaged turbulence models in the same context, and it has been demonstrated that in their standard incarnations, these models fail to capture rarefaction-turbulence interaction correctly.", "Reynolds-averaged models typically employ density gradients in their buoyancy source terms or derive the evolution equation for density fluctuations from mass conservation.", "Incompressive density fluctuations, however, are governed by entropy conservation and are therefore driven by entropy gradients, not density gradients.", "While Reynolds-averaged models are often used to model flows where the difference between these gradients is negligible (such as the classical Rayleigh-Taylor instability between two fluids of different densities), the difference can be pronounced in an isentropic flow such as a rarefaction.", "Reynolds-averaged models for astrophysics applications should incorporate one of the more general expressions (REF ) and (REF ) [1], [9].", "The RDT analysis presented here captures the behavior of subsonic turbulence under rarefaction, and may be a step towards understanding certain aspects of astrophysical turbulence without resorting to numerical simulation or turbulence modeling.", "In addition, the derived solutions can be used to verify algorithms used to model such turbulence.", "The general approach can be applied to any scenario in which turbulence is subject to rapid distortion.", "The scenario analyzed in this paper was a supersonic bulk flow propagating through subsonic turbulence, but linear theory could just as rigorously be applied to a subsonic bulk flow or to supersonic turbulence, provided the bulk flow is more rapid than the turbulence; an example of the latter would be a supernova explosion propagating through the supersonic turbulence of the interstellar medium [45].", "The recent study by [35] is similar in spirit to the one performed here, although the focus there was on the behavior of turbulence under compression rather than expansion.", "The spin down of turbulent eddies due to angular momentum conservation described in §REF is essentially the adiabatic cooling mechanism discussed by [35].", "The compression and expansion in [35] was implemented by applying a scale transformation to the basic equations, whereas here it has been implemented hydrodynamically in a self-consistent manner.", "It is straightforward to extend the theory derived here to an isentropic planar compression; this will be explored in a future publication.", "Finally, a time-scale analysis (§REF and Figure REF ) demonstrates the existence of distinct distortion and inertial ranges in a turbulent flow undergoing rapid distortion [6], [7].", "While capturing both of these ranges in a numerical calculation is a challenge (see expression REF ), it would be interesting to explore the observational implications of the presence of an additional length scale $\\lambda _{nl}$ in distorted turbulent flows.", "This too is an avenue for future work.", "I thank Oleg Schilling and Matthew Kunz for their comments, and the referee for several helpful suggestions that greatly improved the manuscript.", "This work was performed under the auspices of Lawrence Livermore National Security, LLC, (LLNS) under Contract No.$\\;$ DE-AC52-07NA27344." ], [ "A. Vorticity equation under RDT", "The vorticity equation for an ideal fluid is $\\frac{d\\mbox{$\\omega $}}{dt} = \\left(\\mbox{$\\omega $} \\cdot \\mbox{$\\nabla $}\\right) \\mbox{$v$} - \\mbox{$\\omega $} \\left(\\mbox{$\\nabla $}\\cdot \\mbox{$v$}\\right) + \\frac{\\mbox{$\\nabla $}p \\times \\mbox{$\\nabla $}\\rho }{\\rho ^2},$ where the terms on the right hand side are, in order, the stretching/tilting term, the dilatation term and the baroclinic term.", "Decomposing the fluid quantities into a mean plus a fluctuation, and assuming an irrotational ($\\overline{\\mbox{$\\omega $}} = 0$ ), barotropic ($\\mbox{$\\nabla $}\\overline{p} \\times \\mbox{$\\nabla $}\\overline{\\rho } = 0$ ) equilibrium, this can be expressed as $\\frac{d\\mbox{$\\omega $}}{dt} = \\left(\\mbox{$\\omega $} \\cdot \\mbox{$\\nabla $}\\right) \\overline{\\mbox{$v$}} + \\left(\\mbox{$\\omega $} \\cdot \\mbox{$\\nabla $}\\right) \\mbox{$v$}^\\prime - \\mbox{$\\omega $} \\left(\\mbox{$\\nabla $}\\cdot \\overline{\\mbox{$v$}}\\right) - \\mbox{$\\omega $} \\left(\\mbox{$\\nabla $}\\cdot \\mbox{$v$}^\\prime \\right) + \\frac{\\mbox{$\\nabla $}p^\\prime \\times \\mbox{$\\nabla $}\\overline{\\rho }}{\\overline{\\rho }^2} + \\frac{\\mbox{$\\nabla $}\\overline{p} \\times \\mbox{$\\nabla $}\\rho ^\\prime }{\\overline{\\rho }^2} + \\frac{\\mbox{$\\nabla $}p^\\prime \\times \\mbox{$\\nabla $}\\rho ^\\prime }{\\overline{\\rho }^2}.$ There are three non-linear terms on the right-hand side of equation (REF ), one each from the stretching, dilatation and baroclinic terms (the vorticity is of the same order as $\\mbox{$v$}^\\prime $ ).", "There is also a non-linear part of the advection term: $\\frac{d\\mbox{$\\omega $}}{dt} = \\frac{\\partial \\mbox{$\\omega $}}{\\partial t} + \\overline{\\mbox{$v$}}\\cdot \\mbox{$\\nabla $}\\mbox{$\\omega $} + \\mbox{$v$}^\\prime \\cdot \\mbox{$\\nabla $}\\mbox{$\\omega $}.$ Neglecting the non-linear terms (assuming fluctuations are much less than means) yields $\\frac{d\\mbox{$\\omega $}}{dt} = \\left(\\mbox{$\\omega $} \\cdot \\mbox{$\\nabla $}\\right) \\overline{\\mbox{$v$}} - \\mbox{$\\omega $} \\left(\\mbox{$\\nabla $}\\cdot \\overline{\\mbox{$v$}}\\right) + \\frac{\\mbox{$\\nabla $}p^\\prime \\times \\mbox{$\\nabla $}\\overline{\\rho }}{\\overline{\\rho }^2} + \\frac{\\mbox{$\\nabla $}\\overline{p} \\times \\mbox{$\\nabla $}\\rho ^\\prime }{\\overline{\\rho }^2}.$ It can be seen from equation (REF ) that both pressure and density fluctuations can generate vorticity at linear order.", "Under the Boussinesq approximation, however, the pressure fluctuations can be neglected relative to the density fluctuations, and equation (REF ) becomes $\\frac{d\\mbox{$\\omega $}}{dt} = \\left(\\mbox{$\\omega $} \\cdot \\mbox{$\\nabla $}\\right) \\overline{\\mbox{$v$}} - \\mbox{$\\omega $} \\left(\\mbox{$\\nabla $}\\cdot \\overline{\\mbox{$v$}}\\right) + \\frac{\\mbox{$\\nabla $}\\overline{p} \\times \\mbox{$\\nabla $}\\rho ^\\prime }{\\overline{\\rho }^2}.$ This is the vorticity equation for small-amplitude, incompressive perturbations about an irrotational, barotropic equilibrium.", "Aligning the $x$ -coordinate with the direction of the mean velocity ($\\overline{\\mbox{$v$}} = \\overline{v}_x \\mbox{$\\hat{x}$}$ ) and mean pressure gradient, equation (REF ) in component form is $\\frac{d\\omega _x}{dt} = \\omega _x \\frac{\\partial \\overline{v}_x}{\\partial x} - \\omega _x \\frac{\\partial \\overline{v}_x}{\\partial x} = 0$ and $\\frac{d\\mbox{$\\omega $}_\\perp }{dt} = -\\mbox{$\\omega $}_\\perp \\left(\\mbox{$\\nabla $}\\cdot \\overline{\\mbox{$v$}}\\right) + \\frac{\\left(\\mbox{$\\nabla $}\\overline{p} \\times \\mbox{$\\nabla $}\\rho ^\\prime \\right)_\\perp }{\\overline{\\rho }^2}.$ Using the mean continuity equation (REF ), equation (REF ) is equivalent to $\\frac{d}{dt}\\left(\\frac{\\mbox{$\\omega $}_\\perp }{\\overline{\\rho }}\\right) = \\frac{\\left(\\mbox{$\\nabla $}\\overline{p} \\times \\mbox{$\\nabla $}\\rho ^\\prime \\right)_\\perp }{\\overline{\\rho }^3}.$ Removing the bars, equations (REF ) and (REF ) are equivalent to expressions (REF ) in the text.", "The $z$ -component of equation (REF ) is also equivalent to equation (REF ) for $\\mbox{$v$}^\\prime = v_x^\\prime \\mbox{$\\hat{x}$}$ ." ], [ "B. Analysis of $K$ growth and decay", "Extrema in $K_x$ occur for $\\frac{\\gamma -1}{\\gamma +1}\\alpha ^2f^2 + \\left(\\Phi _{x0} - \\alpha \\right)\\alpha f + \\frac{2}{\\gamma +1}\\left(\\alpha ^2-2\\Phi _{x0} \\alpha + 1\\right) = 0,$ where $f \\equiv \\frac{{c}_a\\rho _0}{{c}_0{\\rho }}, \\;\\; \\alpha \\equiv \\frac{2{\\cal A}_{x0}}{3-\\gamma }.$ The solutions to equation (REF ) are $f_\\pm = \\frac{\\left(\\gamma + 1\\right)}{2\\left(\\gamma - 1\\right)\\alpha }\\left[\\alpha - \\Phi _{x0} \\pm \\sqrt{\\Phi _{x0}^2 - 1 + \\left(\\frac{3-\\gamma }{\\gamma +1}\\right)^2\\left(\\alpha ^2-2\\Phi _{x0} \\alpha + 1\\right)}\\,\\right],$ where the positive root is associated with a local maximum in $K_x$ and the negative root is associated with a local minimum.", "Using (REF ) in (REF ), the extrema in $K_x$ can be expressed as $K_{\\pm } = K_{x0}\\frac{3-\\gamma }{\\gamma + 1}f_\\pm ^{\\frac{4}{\\gamma -3}}\\left(\\alpha ^2f_\\pm ^2 - \\alpha ^2+2\\Phi _{x0} \\alpha - 1\\right).$ Expression (REF ) for the portion of the ellipse between regions I and III of Figure REF is obtained by setting the discriminant of expression (REF ) to zero.", "Inside the ellipse, the discriminant is negative and there are no local extrema in $K_x$ .", "Outside the ellipse, there are one or two extrema depending on the value of $f$ .", "When $f < 1$ , the extrema in $K_x$ are unphysical as they occur outside of the rarefaction: ${c}_a> {c}_0$ for $0 < f < 1$ and ${c}_a< 0$ for $f < 0$ .", "Region I thus includes the inside of the ellipse, as well as two regions outside the ellipse where the extrema are unphysical: for ${\\cal A}_{x0} < \\left(3 - \\gamma \\right)\\Phi _{x0}/2$ (the upper left corner of Figure REF ) $f < 0$ , and for $\\left(3 - \\gamma \\right)\\Phi _{x0}/2 < {\\cal A}_{x0} < \\Phi _{x0}^{-1}$ (a small region outside the ellipse but below $\\Phi _{x0} {\\cal A}_{x0} = 1$ ) $f < 1$ .", "The former follows from $f < 0$ for $\\alpha < \\Phi _{x0}$ , and the latter follows from $f < 1$ for $\\Phi _{x0} {\\cal A}_{x0} < 1$ .", "By definition, the local maximum is contained within the rarefaction in region II ($f_+ > 1$ ), and both extrema are contained within the rarefaction in regions III and IV ($f_+ > 1$ and $f_- > 1$ ).", "One can show that the second derivative of $K_x$ is proportional to $2{\\cal A}_{x0} - \\left(\\gamma + 1\\right)$ when $f = 1$ , so that the extremum is a maximum when $f = 1$ and ${\\cal A}_{x0} < \\sqrt{(\\gamma +1)/2}$ , i.e., to the left of the critical point in Figure REF .", "This implies that the boundary between regions I and II is $f_+ = 1$ and the boundary between regions II and IV is $f_- = 1$ .", "In either case, setting $f = 1$ in expression (REF ) yields $\\left(3-\\gamma \\right)\\Phi _{x0} \\alpha = 2$ , i.e., $\\Phi _{x0} {\\cal A}_{x0} = 1$ .", "Determining the border between regions III and IV is somewhat more involved.", "The presence of a local maximum does not guarantee overall growth of $K_x$ ; $K_+ > K_{x0}$ is an additional constraint for $K_+$ to be a global maximum.", "Setting $K_+ = K_{x0}$ in expression (REF ) thus gives the boundary between regions III and IV.", "Combining equations (REF ) and (REF ) with $K_\\pm = K_{x0}$ gives $\\left(p - 1\\right) f_\\pm ^p - p f_\\pm ^{p-1} - \\alpha ^2 f_+^2 + 2\\alpha ^2 f_\\pm + 1- \\alpha ^2 = 0,$ and $2\\alpha \\Phi _{x0} = pf_\\pm ^{p-1} + 2\\alpha ^2\\left(1-f_\\pm \\right),$ where $p \\equiv 4/(3-\\gamma )$ .", "Analytical solutions to equations (REF ) and (REF ) can be found for $p = 2$ , $5/2$ and 3 (corresponding to $\\gamma = 1$ , $7/5$ and $5/3$ ).", "For $\\gamma = 5/3$ ($p = 3$ ), equation (REF ) is $\\left(f_\\pm - 1\\right)^2\\left(f_\\pm - \\frac{\\alpha ^2 - 1}{2}\\right) = 0.$ Two of these roots ($f_\\pm = 1$ ) correspond to the peak occurring at the rarefaction front and have already been discussed.", "The other root defines the border between regions III and IV for $\\gamma = 5/3$ and corresponds to (using REF ) $\\Phi _{x0} = \\frac{3 + 6\\alpha ^2 - \\alpha ^4}{8\\alpha },$ which is equivalent to expression (REF ) for $\\alpha = 3{\\cal A}_{x0}/2$ .", "For $\\gamma = 1$ ($p = 2$ ), equation (REF ) is $(f_\\pm - 1)^2(1 - \\alpha ^2) = 0$ , so that the border between regions III and IV for $\\gamma = 1$ is given by ${\\cal A}_{x0} = 1$ .", "For $\\gamma = 7/5$ ($p = 5/2$ ), equation (REF ) is $\\left(\\sqrt{f_\\pm } - 1\\right)^2\\left(\\frac{3}{2}f_\\pm ^{3/2} + \\left[3 - \\alpha ^2\\right]f_\\pm + 2\\left[1 - \\alpha ^2\\right] f_\\pm ^{1/2} + 1 - \\alpha ^2\\right) = 0,$ so that the border between regions III and IV for $\\gamma = 7/5$ is given by $\\Phi _{x0} = \\frac{1}{{\\cal A}_{x0}}f_+^{3/2} + \\frac{5}{4}{\\cal A}_{x0}\\left(1-f_+\\right).$ Here $f_+$ is given by $f_+^{1/2} = -2\\sqrt{Q}\\cos \\left(\\frac{\\theta + 2\\pi }{3}\\right) - \\frac{2}{3}\\left(1 - \\frac{25{\\cal A}_{x0}^2}{48}\\right),$ where $\\theta \\equiv \\cos ^{-1}\\left(\\frac{R}{\\sqrt{Q^3}}\\right), \\;\\; Q \\equiv \\left(\\frac{5{\\cal A}_{x0}}{72}\\right)^2\\left(48 + 25{\\cal A}_{x0}^2\\right), \\;\\; R \\equiv - \\frac{5}{373248}\\left(-13824 + 4320{\\cal A}_{x0}^2 + 9000 {\\cal A}_{x0}^4 + 3125 {\\cal A}_{x0}^6\\right).$ Figure: Phase diagram of the growth/decay of subsonic turbulence in the wake of a planar rarefaction for γ=7/5\\gamma = 7/5 (left) and γ=1\\gamma = 1 (right).", "See text for discussion." ], [ "C. Details of numerical algorithm", "The numerical results shown in § were obtained with Zeus [40], a second-order finite-difference Eulerian algorithm.", "The specific code used was a hydrodynamic version of the code used in [18].", "All runs were performed with $\\gamma = 5/3$ .", "A Courant number of $0.1$ [11] was sometimes required at high resolution to avoid numerical instability at the front and rear of the rarefaction, both of which locations are weak discontinuities in the flow.", "A strong rarefaction was generated by applying a piston velocity $v_{p,lab} = -1.5 {c}_0$ (in the lab frame) to the lower $x$ boundary (for $\\gamma = 5/3$ , this is half the value required to evacuate the fluid at the rear of the rarefaction, and $20\\%$ greater in magnitude than the value required to reach peak growth, expression REF ).", "In order to get the rarefaction to propagate, it was necessary to apply the piston boundary condition to the first set of $x$ zones inside the computational domain.", "Zero-slope boundary conditions in $x$ were applied for the other fluid variables, and periodic boundary conditions were used in $y$ and $z$ .", "If turbulence was allowed to develop first, periodic boundary conditions were used everywhere until the piston was applied.", "For the strong rarefactions that were generated, the rear of the rarefaction is located off the grid if the calculations are performed in the frame of the ambient fluid (the lab frame).", "To capture the entire rarefaction, it was necessary to transition to a frame with a speed greater than or equal to the speed of the rarefaction rear (in the lab frame).", "In the interest of minimizing numerical diffusion, a frame following the rarefaction rear was chosen.", "This was accomplished by adding a constant speed $\\left\|v_{r,lab}\\right\|$ to the entire computational domain as well as to the piston velocity ($v_{p,rear} = \\left\|v_{r,lab}\\right\| + v_{p,lab}$ ).", "With the piston velocity chosen, the velocity of the rarefaction rear in the lab frame is $v_{r,lab} = {c}_0- \\frac{\\gamma +1}{2} \\left\|v_{p,lab}\\right\| = -{c}_0\\;\\; \\mathrm {for} \\;\\; \\gamma = 5/3,$ and the new piston velocity is $v_{p,rear} = -0.5{c}_0$ .", "In the plots of numerical results, the self-similar variable on the horizontal axis is given by $\\xi = \\frac{x}{t} - \\left\|v_{r,lab}\\right\| = \\frac{x}{t} - {c}_0,$ where $t$ is relative to the time the piston was applied.", "This translates the results back to the lab frame.", "The ambient quantities in expression (REF ) were calculated by performing an average over $y$ (for two-dimensional calculations) or over $y$ and $z$ (for three-dimensional calculations) in the ambient region (i.e., for $\\xi \\ge {c}_0$ ) at the current time." ], [ "D. Entropy fluctuations from the continuity equation", "Using equation (REF ) with $\\mbox{$\\nabla $}\\cdot \\mbox{$v$}^\\prime = 0$ rather than equation (REF ), $\\frac{d}{dt}\\left(\\frac{\\rho ^\\prime }{{\\rho }}\\right) = -\\mbox{$v$}^\\prime \\cdot \\mbox{$\\nabla $}\\ln {\\rho },$ equation (REF ) is replaced with $\\left(\\gamma - 1\\right)\\frac{d}{d\\ln {c}_a}\\left(\\frac{\\rho ^\\prime }{{\\rho }}\\right) = 2\\frac{v_x^\\prime }{{c}_a}.$ Equations (REF ) and (REF ) comprise a system of equations with constant coefficients, leading to a characteristic equation with eigenvalues $\\lambda = \\frac{3 - \\gamma }{2\\left(\\gamma - 1\\right)} \\pm i \\beta , \\;\\; \\beta \\equiv \\frac{\\sqrt{\\left(7 - \\gamma \\right)\\left(\\gamma + 1\\right)}}{2\\left(\\gamma - 1\\right)}.$ The general (inconsistent) linear theory solution is then $v_x^\\prime = {c}_0\\left(\\frac{{c}_a}{{c}_0}\\right)^{\\frac{\\gamma +1}{2\\left(\\gamma - 1\\right)}}\\left(c_1\\cos \\beta \\eta + c_2 \\sin \\beta \\eta \\right),$ $\\rho ^\\prime = {\\rho }_0 \\left(\\frac{{c}_a}{{c}_0}\\right)^{\\frac{7-\\gamma }{2\\left(\\gamma - 1\\right)}}\\left(c_3\\cos \\beta \\eta + c_4 \\sin \\beta \\eta \\right),$ where $c_1 = \\frac{v_{x0}^\\prime }{{c}_0} , \\;\\; c_2 = \\frac{\\left(3 - \\gamma \\right)v_{x0}^\\prime /{c}_0- 4\\rho ^\\prime _0/{\\rho }_0}{2\\left(\\gamma - 1\\right)\\beta },\\;\\;c_3 = \\frac{\\rho ^\\prime _0}{{\\rho }_0} , \\;\\; c_4 = \\frac{4v_{x0}^\\prime /{c}_0- \\left(3 - \\gamma \\right)\\rho ^\\prime _0/{\\rho }_0}{2\\left(\\gamma - 1\\right)\\beta }, \\;\\; \\eta \\equiv \\ln \\left(\\frac{{c}_a}{{c}_0}\\right).$ Taking the square of expression (REF ) and averaging yields an expression for the vortical energy: $\\frac{K_x}{K_{x0}} = \\left(\\frac{{c}_a}{{c}_0}\\right)^{\\frac{\\gamma +1}{\\gamma - 1}}\\left(\\cos ^2 \\beta \\eta + \\frac{3 - \\gamma - 4\\Phi _{x0} {\\cal A}_{x0}}{\\left[\\gamma - 1\\right]\\beta } \\cos \\beta \\eta \\sin \\beta \\eta + \\frac{\\left[3 - \\gamma \\right]^2 - 8\\left[3 - \\gamma \\right]\\Phi _{x0} {\\cal A}_{x0} + 16{\\cal A}_{x0}^2}{4\\left[\\gamma - 1\\right]^2\\beta ^2} \\sin ^2 \\beta \\eta \\right).$ As mentioned in the text, this is the solution to the [5] model in the linear regime (with $K = K_x $ ).", "In the notation of [5], $\\Phi _{x0} = a_{x0}/\\sqrt{2 b_0 K_{x0}}$ and ${\\cal A}_{x0} = \\sqrt{b_0{c}_0^2/(2K_{x0})}$ ." ], [ "E. Derivation of two-equation Reynolds-averaged model solution", "Subsonic turbulence evolving under a sonic mean flow implies negligible turbulent diffusion; a two-equation model under these conditions (and assuming zero mean shear) takes the form [22] $\\frac{dK}{dt} =C_\\mu P \\left(\\frac{2f_t-2}{f_t}\\left[\\mbox{$\\nabla $}\\cdot \\mbox{$v$}\\right]^{2}-\\frac{\\mbox{$\\nabla $}p \\cdot \\mbox{$\\nabla $}\\rho }{\\sigma _{\\rho }\\rho ^2}\\right) K \\tau -\\frac{2}{f_t}\\left(\\mbox{$\\nabla $}\\cdot {\\mbox{$v$}}\\right)K-C_{K2}\\epsilon ,$ $\\frac{d\\epsilon }{dt} =C_\\mu P \\left( \\frac{2f_t-2}{f_t}C_{\\epsilon 1}\\left[\\mbox{$\\nabla $}\\cdot \\mbox{$v$}\\right]^{2} -C_{\\epsilon 0}\\frac{\\mbox{$\\nabla $}p \\cdot \\mbox{$\\nabla $}\\rho }{\\sigma _{\\rho }\\rho ^2}\\right)K -\\frac{2}{f_t}C_{\\epsilon 1}\\left(\\mbox{$\\nabla $}\\cdot {\\mbox{$v$}}\\right)\\epsilon -C_{\\epsilon 2}\\frac{\\epsilon ^{2}}{K},$ where $K$ is the turbulent kinetic energy, $\\epsilon $ is the turbulent dissipation rate, $\\tau \\equiv K/\\epsilon $ is a turbulent time scale, $\\sigma _{\\rho }$ and the $C$ 's are model coefficients, and the fluid quantities are assumed to take on their mean values.", "The parameter $P$ in these equations is a switch that can take on the value 0 or 1; due to numerical issues associated with the production terms, Reynolds-averaged models often operate with $P = 0$ , setting both the anisotropic portion of the Reynolds stress and the buoyancy production in stable regions to zero [10].", "$K \\equiv \\overline{v^{\\prime 2}}$ here includes all of the velocity components (Reynolds- and Favre-averages are equivalent in the linear regime).", "The parameter $f_t$ is part of the Reynolds-stress closure and is a measure of the turbulent degrees-of-freedom.", "It is generally set equal to 3, a value appropriate for isotropic turbulence, but is kept general here in order to make contact with linear theory.", "A value closer to 1 is more appropriate for the anisotropic turbulence associated with gradient-driven instabilities and other flows, such as the one considered here, that have a preferred direction.", "In deriving a solution to these equations, it is useful to combine them into an equation for $\\tau $ : $\\frac{d\\tau }{dt} =C_{\\mu }P\\left(\\frac{2f_t-2}{f_t}C_{\\tau 1}\\left[\\mbox{$\\nabla $}\\cdot \\mbox{$v$}\\right]^{2} -C_{\\tau 0}\\frac{\\mbox{$\\nabla $}p \\cdot \\mbox{$\\nabla $}\\rho }{\\sigma _{\\rho }\\rho ^2}\\right) \\tau ^{2}-\\frac{2}{f_t}C_{\\tau 1}\\left(\\mbox{$\\nabla $}\\cdot {\\mbox{$v$}}\\right)\\tau -C_{\\tau 2},$ where $C_{\\tau 0} = 1 - C_{\\epsilon 0}$ , $C_{\\tau 1} = 1 - C_{\\epsilon 1}$ and $C_{\\tau 2} = C_{K2} - C_{\\epsilon 2}$ .", "A self-similar solution to these equations must be able to satisfy the boundary conditions in the ambient fluid, which are given by the solution to equations (REF ) and (REF ) with all of the gradients set to zero: $K_0(t) =K_i\\left(1 - \\frac{C_{\\tau 2}}{\\tau _i}t\\right)^\\frac{C_{K2}}{C_{\\tau 2}},\\;\\;\\tau _0(t) = \\tau _i -C_{\\tau 2}t,$ where $\\tau _i$ and $K_i$ are the values of the model variables in the ambient fluid when the piston is applied.", "This suggests the self-similar form $\\tau \\equiv \\tau _0(t) T(\\xi )$ , $K \\equiv K_0(t) \\kappa (\\xi )$ , with $T_0 = \\kappa _0 = 1$ .", "Under this assumption, and for the mean flow associated with a rarefaction, equations (REF ) and (REF ) become $-\\frac{\\gamma - 1}{\\gamma + 1} \\frac{d\\ln \\kappa }{d\\eta } = C_\\mu P \\left(\\frac{2f_t-2}{f_t}-\\frac{1}{\\sigma _{\\rho }}\\right) \\left(\\frac{2}{\\gamma +1}\\right)^{2}\\left(\\frac{\\tau _0}{t}\\right) T-\\frac{4}{f_t\\left(\\gamma + 1\\right)} + C_{K2}\\left(\\frac{t}{\\tau _0}\\right)\\left(1 - \\frac{1}{T}\\right),$ $-\\left(\\frac{\\tau _0}{t}\\right)\\frac{\\gamma -1}{\\gamma +1}\\frac{dT}{d\\eta } = C_{\\mu } P \\left(\\frac{2f_t-2}{f_t}C_{\\tau 1} - \\frac{C_{\\tau 0}}{\\sigma _{\\rho }}\\right) \\left(\\frac{2}{\\gamma +1}\\right)^{2}\\left(\\frac{\\tau _0}{t}\\right)^2T^{2}- \\frac{4C_{\\tau 1}}{f_t\\left(\\gamma +1\\right)}\\left(\\frac{\\tau _0}{t}\\right) T + C_{\\tau 2}\\left(T - 1\\right),$ where again $\\eta \\equiv \\ln \\left({c}_a/{c}_0\\right)$ .", "The presence of $\\tau _0(t)$ and $t$ in these equations indicates that a general self-similar solution is not available; the turbulent time scale $\\tau _i$ sets a characteristic scale.", "There are two limiting cases, however, in which a self-similar solution can be obtained: $\\tau _i \\gg t$ with $P = 0$ (so that $\\tau _0 \\approx \\tau _i$ ), and $\\tau _i \\ll t$ (so that $\\tau _0 \\approx -C_{\\tau 2} t$ ).", "Case 1: $\\tau _i \\gg t$ and $P = 0$ .", "Under these conditions, equations (REF ) and (REF ) become $\\frac{d\\ln \\kappa }{d\\eta } = \\frac{4}{f_t\\left(\\gamma - 1\\right)}, \\;\\; \\frac{d\\ln T}{d\\eta } = \\frac{4C_{\\tau 1}}{f_t\\left(\\gamma -1\\right)},$ so that $K = K_i \\left(\\frac{\\rho }{\\rho _0}\\right)^{\\frac{2}{f_t}}, \\;\\; \\tau = \\tau _i \\left(\\frac{\\rho }{\\rho _0}\\right)^{\\frac{2C_{\\tau 1}}{f_t}}.$ For $f_t = 1$ , this is equivalent to the ${\\cal A}_{x0} \\ll 1$ limit of expression (REF ).", "Case 2: $\\tau _i \\ll t $ .", "Under this condition, equations (REF ) and (REF ) become $\\frac{\\gamma - 1}{\\gamma + 1} \\frac{d\\ln \\kappa }{d\\eta } = C_\\mu C_{\\tau 2} P \\left(\\frac{2f_t-2}{f_t}-\\frac{1}{\\sigma _{\\rho }}\\right) \\left(\\frac{2}{\\gamma +1}\\right)^{2} T+ \\frac{4}{f_t\\left(\\gamma + 1\\right)} + \\left(\\frac{C_{K2}}{C_{\\tau 2}}\\right)\\left(1 - \\frac{1}{T}\\right),$ $\\frac{\\gamma -1}{\\gamma +1}\\frac{dT}{d\\eta } = C_{\\mu } P \\left(\\frac{2f_t-2}{f_t}C_{\\tau 1} - \\frac{C_{\\tau 0}}{\\sigma _{\\rho }}\\right) \\left(\\frac{2}{\\gamma +1}\\right)^{2}C_{\\tau 2}T^{2}+ \\frac{4C_{\\tau 1}}{f_t\\left(\\gamma +1\\right)} T + T - 1.$ The latter equation can be written as $\\frac{\\gamma -1}{\\gamma +1}\\frac{dT}{d\\eta } = P\\Gamma ^{2}T^{2} + 2F T - 1,$ where $\\Gamma ^2 \\equiv \\left(\\frac{2}{\\gamma + 1}\\right)^2 C_\\mu C_{\\tau 2}\\left(\\frac{2f_t-2}{f_t}C_{\\tau 1}-\\frac{C_{\\tau 0}}{\\sigma _\\rho }\\right),\\;\\; F \\equiv \\frac{1}{2} + \\frac{2C_{\\tau 1}}{f_t\\left(\\gamma + 1\\right)}.$ This can in turn be rewritten as $dU/dS = U^2 - 1$ , where $U \\equiv \\left(P\\Gamma ^2 T + F\\right)/G$ , $G \\equiv \\sqrt{P\\Gamma ^2 + F^2}$ , $S \\equiv G (\\gamma + 1)\\eta /(\\gamma - 1)$ , which yields the solution $T = f_1/f_2$ , where $f_1 = \\cosh S + \\frac{F - 1}{G}\\sinh S,\\;\\; f_2 = \\cosh S - \\frac{P\\Gamma ^2 + F}{G} \\sinh S.$ Notice that $G$ can be real or imaginary depending upon the values of the model coefficients.", "Under the same set of transformations, the equation for $\\kappa $ becomes $G \\frac{d\\ln \\kappa }{dS} = \\frac{P}{C_{\\tau \\ast }}\\Gamma ^{2}T+ \\frac{4}{f_t\\left(\\gamma + 1\\right)} + \\frac{C_{K2}}{C_{\\tau 2}} - \\frac{C_{K2}}{C_{\\tau 2}}\\frac{1}{T},$ where $C_{\\tau \\ast } \\equiv \\frac{\\frac{2f_t-2}{f_t}C_{\\tau 1}-\\frac{1}{\\sigma _\\rho }C_{\\tau 0}}{\\frac{2f_t-2}{f_t}-\\frac{1}{\\sigma _{\\rho }}}.$ Using $\\int \\Gamma ^2 T dS = - F S - G \\ln f_2$ and $\\int dS/T = F S - G \\ln f_1$ , the general solution in this limit is ${\\rm \\,K}= K_i\\left(-\\frac{C_{\\tau 2}}{\\tau _i}t\\right)^\\frac{C_{K2}}{C_{\\tau 2}} \\left(\\frac{\\rho }{\\rho _0}\\right)^{\\frac{2}{f_t} -\\frac{P}{C_{\\tau \\ast }}\\left(\\frac{\\gamma + 1}{4} + \\frac{C_{\\tau 1}}{f_t}\\right) + \\frac{C_{K2}}{C_{\\tau 2}}\\left(\\frac{\\gamma + 1}{4} - \\frac{C_{\\tau 1}}{f_t}\\right)}\\frac{f_1^{C_{K2}/C_{\\tau 2}}}{f_2^{P/C_{\\tau \\ast }}},\\;\\;\\tau = -C_{\\tau 2} t \\frac{f_1}{f_2},$ where $f_1$ and $f_2$ are defined in (REF ).", "For $P = 0$ , $C_{K2} = 0$ and $f_t = 1$ , this is equivalent to the Case 1 solution as well as the ${\\cal A}_{x0} \\ll 1$ limit of expression (REF ).", "As demonstrated by [22], two-equation models [12], [10] capture the results of linear theory under a specific choice of model coefficients and an interpretation of the model length scale as a Lagrangian fluid displacement, plus negligible turbulent diffusion and dissipation.", "For a $K$ –$\\epsilon $ model, the coefficient choice is $C_{\\epsilon 0} = 3/2$ , $C_{\\epsilon 1} = 2$ , $C_{\\epsilon 2} = \\sqrt{2}$ , $C_\\mu = \\sqrt{2} \\sigma _\\rho $ , $C_{K2} = 0$ , and $P = 1$ ; in addition, a Reynolds-stress closure appropriate for anisotropic turbulence is required to capture the anisotropy associated with the modes analyzed in §REF ($f_t = 1$ ).", "Under these conditions, $\\Gamma = i2/(\\gamma + 1)$ , $F = (\\gamma - 3)/(2[\\gamma + 1])$ , $G = i \\beta (\\gamma - 1)/(\\gamma +1)$ and $S = i\\beta \\eta $ , and equation (REF ) reduces to $K = K_i\\left(\\frac{{c}_a}{{c}_0}\\right)^{\\frac{\\gamma + 1}{\\gamma - 1}}\\left(\\cos \\beta \\eta + \\frac{3 - \\gamma + \\frac{8}{\\gamma +1}}{2[\\gamma - 1]\\beta } \\sin \\beta \\eta \\right)^2.$ Expression (REF ) with $\\Phi _{x0} = -1$ is $K_x = K_{x0} \\left(\\frac{{c}_a}{{c}_0}\\right)^{\\frac{\\gamma +1}{\\gamma - 1}}\\left(\\cos \\beta \\eta + \\frac{3 - \\gamma + 4{\\cal A}_{x0}}{2\\left[\\gamma - 1\\right]\\beta } \\sin \\beta \\eta \\right)^2.$ Comparing these two expressions, it can be seen that a two-equation model with the settings described above gives the same result as the [5] model with $\\Phi _{x0} = -1$ and ${\\cal A}_{x0} = 2/(\\gamma +1)$ .", "In the notation of [5], the latter two conditions correspond to $a_{x0} = -(\\gamma +1)b_0{c}_0/2$ and $K_{x0} = (\\gamma +1)^2 b_0{c}_0^2/8$ .", "As mentioned above, however, Reynolds-averaged models typically set $P = 0$ in a rarefaction, along with $f_t = 3$ and $C_{K2} = 1$ .", "Under these conditions, solutions (REF ) and (REF ) become $K = K_i \\left(\\frac{\\rho }{\\rho _0}\\right)^{\\frac{2}{3}}, \\;\\; \\tau = \\tau _i \\left(\\frac{\\rho }{\\rho _0}\\right)^{\\frac{2C_{\\tau 1}}{3}}$ and ${\\rm \\,K}= K_i\\left(-\\frac{C_{\\tau 2}}{\\tau _i}t\\right)^\\frac{1}{C_{\\tau 2}} \\left(\\frac{{c}_a}{{c}_0}\\right)^{\\frac{1}{\\gamma - 1}\\left(\\frac{4}{3} + \\frac{1}{C_{\\tau 2}}\\left[\\frac{\\gamma + 1}{2} - \\frac{2C_{\\tau 1}}{3}\\right]\\right)}\\left(\\cosh S + \\frac{F - 1}{F}\\sinh S\\right)^{1/C_{\\tau 2}},$ with $S \\rightarrow F\\eta (\\gamma +1)/(\\gamma -1)$ and $F \\rightarrow 1/2 + 2C_{\\tau 1}/(3[\\gamma +1])$ .", "These expressions are useful for verifying a standard Reynolds-averaged model implementation." ] ]	1403.0564
[ [ "On repellers in quasi-periodically forced logistic map system" ], [ "Abstract We propose a method to identify and to locate \"repellers'' in quasi-periodically forced logistic map (QPLM), using a kind of Morse decomposition of nested attracting invariant sets.", "In order to obtain the invariant sets, we use an auxiliary 1+2-dimensional skew-product map system describing the evolution of a line segment in the phase space of QPLM.", "With this method, detailed structure of repellers can be visualized, and the emergence of a repeller in QPLM can be detected as an easily observable bifurcation in the auxiliary system.", "In addition to the method to detect the repellers, we propose a new numerical method for distinguishing a strange non-chaotic attractor (SNA) from a smooth torus attractor, using a correspondence between SNAs in QPLM and attractors with riddled basin in the auxiliary system." ], [ "introduction", "Quasi-periodically forced systems could be considered as one of the simplest class of “essentially non-autonomous” dynamical systems.", "In this class of systems, surprisingly complex dynamical phenomena, including rather common existence of strange non-chaotic attractors(SNA), were reported in early 1980s [1], [2], [3].", "Since then the feature and the mechanism of such phenomena have been extensively studied by many researchers with various background, and related phenomena have been uncovered in relatively wide variety of systems [4], [5], [6], [7], [8], [9], [10].", "Major part of the theoretical researches on the phenomena have been done mainly for the cases with systems with “forced” invariant set induced by something like symmetry [14], [15], [16], [17], [18], [19], [20], [21].", "On the other hand, also for the systems without such forced invariant sets, rigorous results are obtained however mainly in reversible 1+1-dimensional systems: the existence of SNA is indicated for the case where the existence of continuous attractor is excluded by topological constraint[11], [12] and the emergence of SNA at non-uniform saddle-node bifurcation is proven [13].", "As for higher dimensional and/or irreversible systems, also where large part of the numerical studies have been done, there still many phenomena which have not been understood clearly.", "When a quasi-periodic external force is applied to such systems that possibly exhibit chaotic behavior even without external forcing, it seems natural to expect that unstable invariant sets with complicated structure would play essential role in the dynamics[22], [23].", "In quasi periodically forced circle map system with relatively strong modulation (thus it is no longer reversible), existence of such complicated structure is indicated using densely distributed winding number of the orbits[24].", "Also in numerical researches, repelling invariant sets with complicated structure are observed, for example, as a basin boundaries between two distinct attractors, and the boundary crisis has been identified as one of the routes to SNA.", "Thus such repelling invariant sets are considered to be directly relevant with the emergence of SNA at least for such cases.", "In some other cases, the existence of repelling sets is also strongly suggested from observation of the parameter dependence of the size or the smoothness of the attractor (i.e., crisis-like drastic enlargement of an attractor or “fractalization” of a torus), while there seem to be no visible basin boundary in its neighborhood.", "In such cases it is not trivial task to identify and to locate such presumed repellers.", "One possible approach to locate such invariant sets is to consider a perturbation to the invariant sets in the system without external forcing.", "However, this method is applicable only for limited class of invariant sets (i.e., a torus or composition of multiple tori).", "Another useful approach is to approximate the quasi-periodic forcing with periodic one[25], [26], and thus approximate 1+1-dimensional skew-product system with 1-parameter family of 1-dimensional autonomous systems.", "This method, known as rational approximation (RA) method, have been used as a major tool to observe such invariant sets in numerical researches.", "It has been used for the classification of the bifurcation to create SNA in QPLM [27], [28], [29], [30], [31], [32], and corresponding types of bifurcation are observed also in QP-forced higher dimensional systems [33], [34], [35].", "Though the observed features of the “repeller” obtained in RA are consistent with the qualitative changes of the behavior of the attractors on the whole, RA cannot reproduce the topological structure of the genuine “repeller” verbatim, and the correspondence of the bifurcations in QP-forced systems and those obtained by RA is somewhat obscure.", "In this report, we propose a method to locate and to obtain the images of repellers in QP-forced logistic map system(QPLM) including those with non-trivial structures, and exhibit some of the numerical results obtained with this method.", "Our results are consistent with the bifurcation scenario that have been inferred from RA method or from direct observation of the attractors, and provide some novel perspective to bifurcation phenomena where various repellers are involved.", "Our method is presented in section 2.", "We introduce an auxiliary dynamical system describing the evolution of a segment, that is given as a 1+2 dimensional skew-product map system, and describe outline of the method to identify “repeller” of the QPLM.", "Some details of the algorithm is given in the the appendix.", "Some of the numerical results obtained with this method are demonstrated in section 3, including detailed phase diagram exhibiting a complicated bifurcation structure near TDT(torus doubling termination) critical point, as well as images of repellers with non-trivial structure.", "Summary and short discussion will be given in the last section." ], [ "Method", "In this section, we will explain our method to obtain the images of “repellers” in QPLM.", "In the first place, we introduce concrete form of the system and auxiliary “segment map”, a 1+2- dimensional skew-product map that describes the evolution of a “segment” in the phase space of QPLM.", "Then we describe the method to obtain repellers in QPLM.", "Some technical details of the algorithm to obtain the images of repellers, together with a method to distinguish SNA from “smooth” torus will be given in appendix." ], [ "QPLM and “segment map”", "Here we consider the quasi-periodically forced logistic (quadratic) map in the following form, $M\|\\Omega \\rightarrow \\Omega $ ,$(\\Omega :=T^1([0,1))\\times {\\mathbf {R}})$ , $M(\\theta _n,x_n)=(\\theta _{n+1},x_{n+1})$ $\\theta _{n+1} &=& (\\theta _n + \\omega ) \\ \\mbox{mod}\\ 1,\\cr x_{n+1} &=& a - x_n^2 + \\epsilon \\cos (2\\pi \\theta _n).$ $a$ and $\\epsilon $ are treated as control parameters, and $\\omega $ is fixed as $(\\sqrt{5}-1)/2$ in this paper.", "In order to examine the stability of a point in $\\Omega $ , it would be a natural approach to consider the temporal evolution of its neighborhood.", "Regarding the trivial neutral stability in the $\\theta $ direction, it seems reasonable to think about the evolution of a vertical segment that contains the target point.", "Fortunately the evolution of a vertical segment can be written concretely as a (1+2)-dimensional piece-wise polynomial skew-product map system.", "Thus here we introduce an auxiliary “segment map” system, given as $\\tilde{M}\\ \|\\ \\tilde{\\Omega }\\rightarrow \\tilde{\\Omega },\\ \\tilde{\\Omega }=T^1([0,1))\\times ({\\mathbf {R}}\\times {\\mathbf {R}}_+), {\\mathbf {R}}_+=\\lbrace x\\in {\\mathbf {R}}\|x\\ge 0\\rbrace $ , $\\tilde{M}(\\theta _n,z_n,w_{n})=(\\theta _{n+1},z_{n+1},w_{n+1})$ $\\theta _{n+1} &=& (\\theta _n + \\omega ) \\ \\mbox{mod}\\ 1,\\cr z_{n+1} &=& \\left\\lbrace \\begin{array}{ll}a - z_n^2 -w_n^2 + \\epsilon \\cos (2\\pi \\theta _n),& (\|z_n\|\\ge w_n)\\cr a - (\|z_n\|+w_n)^2/2 + \\epsilon \\cos (2\\pi \\theta _n),& (\|z_n\|<w_n)\\end{array}\\right.\\cr w_{n+1} &=& \\left\\lbrace \\begin{array}{ll}2\|z_n\|w_n,& \\ (\|z_n\|\\ge w_n)\\cr (\|z_n\|+w_n)^2/2,& \\ (\|z_n\|<w_n)\\end{array}\\right.$ Here $\\theta $ is the “phase” of the external force as in the QPLM(REF ).", "$z$ and $w$ correspond to the center position and the half-length of the segment respectively.", "Although the derivative of this map has discontinuity on $z=0$ , we have observed no symptom of pathological phenomena due to this discontinuity.", "Note that, the subspace of zero-length segments ($\\tilde{\\Omega }_0=\\lbrace w=0\\rbrace \\subset \\tilde{\\Omega }$ ) is kept invariant by the map $\\tilde{M}$ , and the restriction of $\\tilde{M}$ on $\\tilde{\\Omega }_0$ could be naturally identified with QPLM ($M$ on $\\Omega $ ).", "We sometimes call a subset of the phase space with common $\\theta $ value as a “fiber” (both for QPLM and for the segment map system).", "For a point $u=(\\theta ,z,w)\\in \\tilde{\\Omega }$ , the length of the corresponding segment in $\\Omega $ ($= 2w$ ) is written as ${\\rm length}\\ (u)$ .", "For a point $u=(\\theta ,z,w)$ in $\\tilde{\\Omega }$ , we let $\\underline{u}$ to represent a set (segment) $\\lbrace (\\theta ,x)\\in \\Omega \\ \|\\ z-w\\le x\\le z+w\\rbrace $ .", "Similarly, for $U$ which is a subset of $\\tilde{\\Omega }$ , let $\\underline{U}$ stands for $\\bigcup _{u\\in U}\\underline{u} \\subset \\Omega $ .", "We call $\\underline{U}$ as the “shadow” of $U$ .", "For an invariant set (that may or may not be an attractor) $X\\in W$ where $W$ may be either $\\Omega $ or $\\tilde{\\Omega }$ , we let $B(X)$ denote the basin of attraction $\\lbrace u\\in W\\ \|\\ \\omega (u)\\subset X\\rbrace $ , where $\\omega (u)$ denotes the $\\omega $ -limit set of the orbit starting from $u$ ." ], [ "A method to locate repellers", "Now let us describe the outline of the method to detect and to locate “repellers”.", "Our basic strategy is to find a set of points that are transversely unstable and non-wandering, using the map describing temporal evolution of a line segment.", "As for the transverse stability of a point, we observe asymptotic behavior of an orbit (of the segment map) starting from a segment (with positive length) containing the point, and check if the length of the segment would converge to 0 or not.", "If it converges to 0, that indicates no transversely unstable point is contained in the initial segment.", "On the other hand, if the length of the orbit starting from the segment has a positive limsup value, transversely unstable point(s) presumably exist in the initial segment.", "The asymptotic behavior of the orbit of the segment map gives some information about the recurrence property of the points in the initial segment as well.", "As the $\\omega $ -limit set of an orbit of the segment map is an invariant set, its shadow determines a corresponding invariant set of the original QPLM.", "If the initial segment (i.e., a segment in QPLM corresponding to the shadow of the initial point of the segment map) is a subset of this invariant set, there should exist non-wandering points in the initial segment.", "Thus we are interested in such point ($p$ ) that satisfies $\\omega $ -limit set of any orbits (of $\\tilde{M}$ ) starting from a point corresponding to non-zero length segment containing the point $p$ is detached from $w=0$ plane, the shadow of the $\\omega $ -limit set contains the point $p$ Above mentioned $\\omega $ -limit set for starting point corresponding to sufficiently short segments is expected to be one of the attractors of the segment map.", "Let $A$ denote the attractor.", "The ensemble of orbits starting from a neighborhood of the original point $p$ should be dense in the shadow of the attractor $A$ .", "Thus the shadow of the attractor $A$ could be regarded as a “basin of repulsion” of the repelling non-wandering point $p$ .", "There can be multiple repelling non-wandering points with the common “basin of repulsion”.", "The set of those repelling non-wandering points consist a transitive set.", "Thus the set as a whole could be identified as a “repeller”.", "We let $R(A)$ denotes a repeller with “basin of repulsion” $A$ .", "Basically what we try to obtain is a image of the section of repellers on a target fiber.", "In order to do this, we try to carry out the following steps.", "Obtain the list of pull back attractors of the segment map on the target fiber, Find pull back attractors that correspond to a segment whose proper subset corresponds to another pull back attractor, Look for such points which are in the shadow of the outer attractor and whose corresponding $\\omega $ -limit set does not belong to any of inner attractor's shadow.", "The obtained set of points would be the section of the repeller whose basin of repulsion is given by the shadow of the outer attractor.", "In practice, we try to obtain the image of multiple sections of the repellers using the information of pull back attractors on one fiber.", "The detail of the procedure is described in the appendix.", "It should be noted that a pull back attractor on a fiber does not simply correspond to a section of the attractor in autonomous 2-dimensional system.", "[36] If $A\\subset \\Omega $ is an asymptotically stable attractor of $M$ in the view the 2-dim autonomous system, boundary of $A$ would consist of graph(s) of continuous and piece-wise smooth function of $\\theta $ .", "In such cases, the intersection of $A$ with a certain fiber $\\lbrace \\theta =c\\rbrace $ should coincide with a pull back attracting set on the fiber (which may consists of multiple pull back attractors).", "On the other hand, for the cases with SNA, although pull-back attractor(s) for almost all fibers can be obtained just like the above case, the shape of their union over the fibers is not continuous.", "Thus SNA does not simply correspond to an asymptotically stable attractor of the 2-dimensional autonomous systems.", "It is not easy to distinguish these two types of pull back attractors from simple observation.", "For the case of SNA, however, there should exist a repelling orbit (that should be a part of a repeller) in arbitrarily small neighborhood of the pull back attractor[37].", "Thus we could expect that there exist trajectories of $\\tilde{M}$ starting from a point in an arbitrarily small neighborhood of an “SNA” but finally attracted to another attractor (of $\\tilde{M}$ ) that corresponds to the basin of repulsion of the repeller which is located in touch with the attractor(SNA).", "Thus, it is expected that SNAs exhibit a kind of sensitivity of final state against arbitrarily small perturbation in the neighborhood of the attractor in $\\tilde{M}$ , like those attractors with “riddled” basin[38], [16].", "We try to detect this presumed type of sensitivity of final state in $\\tilde{M}$ expected for SNAs, by checking whether a small perturbation for $w$ component of $\\tilde{M}$ would make the destination of the trajectory to another attractor (in $w>0$ region) or not.", "Thus we introduce a “perturbed” version of segment map $\\tilde{M}^+$ , given as $\\tilde{M}^+ = \\tilde{W}^+ \\circ \\tilde{M}$ where $\\tilde{W}^+ \\ \| \\ \\tilde{\\Omega }\\rightarrow \\tilde{\\Omega }$ , $\\tilde{W}^+(\\theta ,z,w)=(\\theta ,z,\\min \\lbrace w,\\delta ^+\\rbrace )$ , where $\\delta ^+$ denotes the amplitude of the perturbation.", "( $\\delta $ is set as $5.0\\times 10^{-7}$ in the examples shown in this report.)" ], [ "Numerical results", "In this section, we present some of the numerical results, namely, phase diagrams in $(a,\\epsilon )$ -plane and visualized images of attractors and repellers for some selected parameter values.", "Figure: Phase diagram in aa-ϵ\\epsilon planeThe obtained phase diagram (Fig.REF ) seems similar to those obtained in the preceding studies.", "However, since our criterion to identify an attractor as SNA is different from those used in the preceding studies (based on the divergence of the estimated phase sensitivity), the position of the obtained borderline between smooth attracting torus and SNA has some discrepancy.", "It would be notable that we observed no SNA near the border of the boundary crisis which is associated with the disappearance of bounded attractor in the bump-like region near $(a,e)=(1.2, 0.42)$ (as is shown in box (c)), which is consistent with the natural expectation that SNA appears only AT the boundary crisis for such cases.", "An example of the repellers with non-trivial structure (observed as “ring shaped” repellers in RA method) is exhibited in Fig.", "REF .", "Besides the apparently fractal layered structure, thin apertures can be observed in the rim of the rings (in boxes b and c).", "These thin apertures are mapped to the large gap near $\\theta =0.35$ (indicated by a two-way arrow in box a), after 15 (for box b) and 28 (for box c) iterations.", "The above result suggests that the repeller has a family of infinitely many apertures (with progressively thin widths), and thus when adequately coarse grained it could appear to consist of finite number of chunks.", "Thus the approximation for the repeller obtained by RA could be a good approximation for the repeller in quasi-periodically forced system, in spite of the fact that the set of finite number of rings cannot be invariant in the quasi-periodically forced system.", "Figure: An example of repeller with non-trivial structure observed at(a,ϵ)=(1.0,0.35)(a,\\epsilon )=(1.0,0.35).", "Thin apertures in the rim of ring likestructures are visible in enlarged view in boxes (b) and (c).", "Dataof pull-back attractors and repellers on 10000 target fibers inθ∈[0,1]\\theta \\in [0,1] are used.In the next figure (Fig.", "REF ), we chose the parameter near the emergence of the ring-like shaped repeller, and visualized the shape of the repeller and the attracting invariant set corresponding to the basin of repulsion of the repeller.", "Sequence of inward spikes are observed in the boundary curve of the basin.", "We think that these spikes are a symptom of a non-uniform saddle-node bifurcation, where the attractor of $\\tilde{M}$ (which corresponds to the basin of the ring-like shaped repeller) collide with a saddle.", "Thus it is surmised that, at the bifurcation of the disappearance of the repeller, the boundary would touch the last component of the repeller on infinitely many fibers simultaneously, while on typical fibers the repeller of QPLM with fractal-like structure is still retained.", "Thus it is conjectured that the repeller would have strictly positive lyapunov exponent even at the onset.", "Figure: Repeller and its basin of repulsion, near the emergence ofthe repeller (a,ϵ)=(1.0,0.336057731)(a,\\epsilon )=(1.0,0.336057731).", "Data of 10000 targetfibers are plotted together.An interesting repetitive bifurcation structure has been reported near the torus doubling termination (TDT) critical point[39], [40].", "Near this critical point, multiply nested structure of invariant sets is observed as shown in Figs.REF and REF .", "The obtained result is consistent with coexistence of multiple noisy attractors with different scales[40] and also with the features of the parameter dependence of the attractor[4], i.e., the birth of SNA via internal crisis like behavior with decreasing $a$ and via fractalization like behavior with increasing $a$ .", "Figure: Phase diagram near TDT critical pointFigure: Nested structure of absorbing areas (basin of repulsion) andcorresponding repellers observed at (a,ϵ)=(1.1581,0.361)(a,\\epsilon )=(1.1581,0.361)(indicated by the black + in fig. )", "Plots in box(a), (b) are obtained with 2000, 1500 target fibers of θ\\theta values within [0,1][0,1], [0.31,0.46][0.31,0.46]respectively.In the next figure (Fig.", "REF ), the parameter is set as $(a,\\epsilon )=(0.75,0.46672)$ , a little after the onset of SNA by “fractalization” on the path corresponding to the one investigated by Kaneko and Nishikawa[3], [28].", "We can see a kind of fractal like complicated structure with some apparent smoothness, however, the images of the repeller is “dirty” due to apparently superimposed irregular vertical stripe like structure.", "We think that the repeller would be dense in its basin of repulsion due to these stripes and correspondingly the basin of SNA in $\\tilde{M}$ should be riddled, while structures with some smoothness would be apparent as far as we observe finite number of fibers.", "Figure: Repeller and SNA emerged via fractalization route,(a,ϵ)=(0.75,0.46673)(a,\\epsilon )=(0.75, 0.46673).", "The pull-back attractor (SNA) andthe repeller, as well as boundary of absorbing areas, are plottedfor 4000 (a), 2000 (b and c) target fibers, with θ\\theta values in[0,1][0,1] for (a), [0.8,0.9][0.8,0.9] for (b), and [0.84,0.85][0.84,0.85] for (c)respectively." ], [ "summary", "We proposed a method to observe a kind of “repeller” in quasi-periodically forced logistic map system and a method to distinguish “smooth” attractors and SNAs, using auxiliary dynamical system describing evolution of a “line segment” on a fiber.", "This method enables to obtain images of repellers and bifurcation diagrams efficiently.", "Some numerical results obtained with these methods are exhibited, namely, phase diagram with some revision in the borderline between smooth torus attractor and SNA, detailed images of a repellers with non-trivial ring-like structure, absorbing area surrounded by curves with many peaks observed nearly at the emergence of a non-trivial repeller, which exhibits a symptom of non-uniform saddle-node bifurcation in the segment map system, nested structure of multiple absorbing areas and associated multiple repellers observed near TDT critical point, image of repeller with apparently noisy structure which is the counterpart of SNA.", "By the use of segment map a similarity between attractors with riddled basin and SNAs become apparent, and the relevance between various phenomena including “fractalization of torus” and “emergence of ring-like shaped repeller” and (easily observable) bifurcations in the segment map (i.e., bifurcations accompanied with qualitative changes of attractors) are also indicated.", "At parameter values where the “fractalization” occur, it seems that the transformation from a smooth torus to SNA and the emergence of a new repeller occur simultaneously.", "We will report detailed analysis on this phenomenon in the forthcoming paper.", "Our method make the most of the fact that the dynamics on the fiber is one dimensional map.", "Thus it would not be straight forward to apply our method to systems with higher dimensional fiber dynamics.", "On the other hand, the dynamics of the forcing could be easily generalized, for example, to chaotic ones.", "As the attraction to SNA in quasi-periodically forced systems could be regarded as a special class of the weak generalized synchronization phenomena [41], [42], [43], application of our method to systems with chaotic driving would help to obtain some new perspective for generalized synchronization phenomena between non-identical chaotic elements.", "We appreciate N. Takahashi, T. Mitsui, and Y. Sato for helpful suggestion and discussion.", "This research is supported by JSPS KAKENHI Grant Number 60115938." ], [ "numerical algorithm", "Here we describe some details of the algorithm used in our numerical calculation.", "Our calculation consists of 3 parts.", "(1) Search for the invariant sets with nested structure, which gives information of the existence/absence of repellers.", "(2) Decomposition of the invariant sets into the repeller and the basin of smaller attracting invariant sets, to obtain the approximate images of the repellers.", "(3) Check for the behavior of perturbed orbits starting from a point on the attractors obtained in (1), in order to distinguish “smooth” attracting torus and SNA.", "Both (2) and (3) need the result of (1), but (2) and (3) could be carried out mutually independently.", "In part (1), we fix a target fiber and will make a list of pull back attractors of $\\tilde{M}$ on the fiber.", "Here we intend to list up all the attractors that may attract some of the sufficiently short vertical segments.", "The obtained list would contain those corresponding to the basin of repulsion of repellers as well as those directly corresponding to the attractors of $M$ .", "Each of the obtained pull-back attractors is represented by a segment on the fiber (whose length might be zero).", "In this step, we use evolution towards the target fiber from sufficiently long ago.", "In part (2), we choose multiple target fibers that can be reached by forward iteration from the fiber chosen in (1).", "For each target fiber, we locate the images of the segments obtained in (1) and then try to decompose them to obtain the images of the repellers on the fiber.", "In the decomposition, we use forward iteration from the target fiber.", "In part (3), we choose the zero-length segments obtained in (1) that represent pull-back attractors with negative lyapunov exponent, and calculate forward trajectory from each of the segments using segment map with small perturbation to $w$ component.", "We also calculate the forward iteration stating from segments that correspond to the basin of repellers that possibly has contact with the attractor in question.", "If these trajectories coincide after sufficiently long transient steps, that indicates the existence of a fiber on which the distance between the attractor and the repeller is smaller than the amplitude of the perturbation, which could be regarded as a signature of the absence of the asymptotic stability of the attractor, implying that the attractor would presumably be an “SNA”.", "In the following, we try to illustrate outline of the algorithm in the form of a pseudo program.", "global variables (for parameters): Iinit: a segment on the initial fiber chosen so that any invariant set has non empty intersection with this segment) TList: set of integer used to specify theta values of fibers to be observed global variables (for results): InvList: set of segments list of pull-back attractor of the segment map on the target fiber of Part1 Unity[J] (J: a member of InvList): boolean true if the invariant set represented by J do not have proper subset which is an attracting invariant set RepellerImage[J,t] (J: a member of InvList, t: a member of TList): set of segments image of a section of repeller whose basin of repulsion corresponds to J SNA[J] (J : a member of InvList): boolean true if J is classified as SNA working variables: I,J,K: segments on a fiber I: on initial fiber J: on target fiber K: on future fiber" ], [ "Part 1: Search for the attracting sets of $\\tilde{M}$", "Part1Main clear InvList call P1CheckImage(Iinit) Procedure P1CheckImage(I) J=IntialTransient(I) if (J is not in InvList) then append J in InvList if length(J)>0 then Unity[J]=P1UnityCheck(I,J) else Unity[J]=true if ( (Unity[J] is false) and (length(I) > param_cutoff) ) then divide I into (I_1, I_2) call P1CheckImage(I_1) call P1CheckImage(I_2) boolean P1UnityCheck(I,J) \"Take an arbitrary point v in segment I\" if (\"LyapunovExponent(v)\"<0) then return false \"Take a segment V, as theta(V)=theta(J), center(V)=InitialTransient(v), length(V)=very short\" if (IterateMany(V) and IterateMany(J) coincide) then return true else return false In this part, we try to obtain a list of pull back attractors of $\\tilde{M}$ on a target fiber.", "We are interested in such attractors which are also attractors of $M$ (i.e., stable torus/tori, SNA, chaotic attractor) or which attract orbits with initial conditions that correspond to short segments representing sufficiently small neighborhoods of a point on repellers of $M$ .", "Let the target fiber be specified as $\\lbrace \\theta =\\theta _0\\rbrace $ ($\\theta _0\\in [0,1)$ ), $C_-$ be a constant satisfying $a-C_-^2+\\epsilon < C_-$ , and $\\tau ^-$ be a sufficiently large integer parameter that gives the duration of initial transient.", "Then we take “initial segment” (a point in $\\tilde{\\Omega }$ ) $I_{init}=(\\theta _-,0,\|C_-\|)$ on the initial fiber specified with $\\theta _-=(\\theta _0 - \\tau ^- \\omega )\\ \\rm {mod}\\ 1$ .", "With this choice of $I_{init}$ , every invariant set of $M$ should have non-empty intersection with $\\underline{I_{init}}$ .", "We calculate the trajectory starting from a segment $I$ (which is set as $I_{init}$ for the first trial) on the initial fiber to obtain its image ($J$ ) on the target fiber, then record $J$ as a member of the list of pull back attractor on the target fiber (“InvList”).", "If it has non-zero length and does not correspond to “transitive” chaotic attractor, it is presumed that there exist pull back attractor(s) which corresponds to a proper subset of $J$ .", "In such case, we divide the initial segment $I$ into two, and recursively repeat this calculation with using each fragment of $I$ as initial segment, until the initial segment becomes sufficiently short." ], [ "Part 2: Visualization of repellers", "Part2Main foreach J in InvList if (Unity[J] is false) then foreach t in TList clear tmpRepellerImage JT=\"Iterate_t_times(J)\" call P2Decomposition(JT) RepellerImage[J,t]=tmpRepellerImage Procedure P2Decomposition(J) K=IterateManytimes(J) call P2DecompSub(J,K) Procedure P2DecompSub(JX,K) L=IterateManytimes(JX) if (L matches K) then if (length(JX)<param_cutoff) then \"append JX to tmpRepellerImage\" else \"divide JX into (JX1,JX2)\" call P2DecompSub(JX1,K) call P2DecompSub(JX2,K) In this part, we choose multiple target fibers, and try to obtain the image of each repeller on each target fiber.", "The target fibers are chosen to be located on the future of the target fiber of part 1.", "We firstly obtain a segments ($J$ ) on the target fiber that corresponds to the basin of a repeller, and also its image in a sufficient future($K$ ).", "Then divide the segment on the target fiber and check if the image of the fragment in the future is expanded up to coincide with the image of the original segment.", "If their images in the future matches, it implies that the fragment possibly contains point(s) of the repeller, and in such case we recursively divide the fragments again and check their future image.", "After a sufficient number of recursive dividing of the segment, we would obtain an approximate image of the repeller on the fiber, i.e., a set of sufficiently short fragments that covers the intersection of the repeller and the target fiber." ], [ "Part 3: Sensitivity check for discriminating SNA and torus", "Part3Main foreach J in InvList if (length(J) > 0) then SNA[J]=false K[J]=IterateManytimes(J) foreach J in InvList if (length(J) is 0) then L[J]=PerturbedIterateManytimes(J) if (L[J] matches one of K[*]) then SNA[J]=true else SNA[J]=false In this part, we try to discriminate SNA from smooth torus by checking the sensitivity of final state against a perturbation in a small neighborhood of the attractor.", "We calculate a sufficiently long trajectory of perturbed map $\\tilde{M}^+$ starting from a segment (which is the member of InvList in question).", "If the segment corresponds to an asymptotically stable attractor (i.e., smooth torus/tori attractor), the perturbation would not kick the trajectory out of the basin of the original attractor, but if it corresponds to an SNA, the trajectory would be attracted to another attractor corresponding to the basin of a repeller after a sufficiently long iteration.", "Note that the perturbation $\\tilde{W}^+$ affects the map only in a neighborhood of $\\lbrace w=0\\rbrace $ plane, thus attractors detached from $\\lbrace w=0\\rbrace $ are not affected by the perturbation." ] ]	1403.0321
[ [ "Ground-based detection of the near-infrared emission from the dayside of\n WASP-5b" ], [ "Abstract (Abridged) WASP-5b is a highly irradiated dense hot Jupiter orbiting a G4V star every 1.6 days.", "We observed two secondary eclipses of WASP-5b in the J, H and K bands simultaneously.", "Thermal emission of WASP-5b is detected in the J and K bands.", "The retrieved planet-to-star flux ratios in the J and K bands are 0.168 +0.050/-0.052% and 0.269+/-0.062%, corresponding to brightness temperatures of 2996 +212/-261K and 2890 +246/-269K, respectively.", "No thermal emission is detected in the H band, with a 3-sigma upper limit of 0.166%, corresponding to a maximum temperature of 2779K.", "On the whole, our J, H, K results can be explained by a roughly isothermal temperature profile of ~2700K in the deep layers of the planetary dayside atmosphere that are probed at these wavelengths.", "Together with Spitzer observations, which probe higher layers that are found to be at ~1900K, a temperature inversion is ruled out in the range of pressures probed by the combined data set.", "While an oxygen-rich model is unable to explain all the data, a carbon-rich model provides a reasonable fit but violates energy balance." ], [ "Introduction", "Currently, the most fruitful results on the characterization of exoplanetary atmospheres come from transiting planets.", "Since the first transiting planet HD 209458b was discovered in 1999 [6], more than 400 are confirmedhttp://exoplanet.eu/.", "The orbital parameters of these planets are well constrained when transit observations were combined with radial velocity measurements.", "Precise planetary parameters such as mass and radius can be determined as well, which leads to a preliminary view of the internal structure of a planet, and thus to constrain the formation and evolutionary history of the planet [25], [17].", "Furthermore, transiting planets provide unprecedented opportunities to probe their atmospheres, not only from wavelength-dependent effective radius variations determined from the transit [7], but also from differential planetary photon measurements from occultation [12].", "In the latter case, the planet passes behind the star, which leaves us only stellar emission during a total eclipse.", "As a subset of transiting planets that are exposed to high irradiation in close orbits around their host stars, hot Jupiters are the most favorable targets for thermal emission detection through secondary-eclipse observation.", "Their close orbits translate into a high occultation probability and frequency, while their high temperatures and large sizes make the planet-to-star flux ratio favorable.", "The first thermal emission detections of hot Jupiters have been achieved with the Spitzer Space Telescope [12], [8], which operates in the mid-infrared (MIR) wavelength range.", "Since then, a flood of such detections have been made with Spitzer observations, resulting in better knowledge of the chemical composition and thermal structure of the planetary atmosphere.", "Compared with the MIR, the near-infrared (NIR) wavelength range covers the peak of the spectral energy distribution (SED) of a planet and probes deeper into the atmosphere, therefore it can be used to constrain the atmosphere's temperature structure and energy budget.", "While the Hubble Space Telescope has contributed much to the NIR observation on planetary secondary eclipses [48], [49], now more observations with high precision are starting to come from ground-based mid-to-large aperture telescopes thanks to the atmospheric window in the NIR [9], [10], [11], [5], [23].", "As shown for example by [39], these ground-based NIR measurements play a crucial role in determining the C/O ratio when combined with measurements from Spitzer observations.", "WASP-5b was first detected by [1] as a hot Jupiter orbiting a 12.3 mag G4V type star every 1.628 days.", "Its mass and radius are derived to be 1.58 and 1.09 times of the Jovian values, respectively, which places it among the relatively dense hot Jupiters.", "Several follow-up transit observations have refined its density to be nearly the same as our Jupiter [47], [19].", "Its host star has a slightly supersolar metallicity ([Fe/H]=+0.09$\\pm $ 0.09), according to the high-resolution VLT/UVES spectroscopy of [21].", "The planetary orbit might have a marginally nonezero eccentricity based on the joint analysis of RV and photometric measurements [21], [50], [29].", "[50] studied the Rossiter-McLaughlin effect in the WASP-5 system and found a sky-projected spin-orbit angle compatible with zero ($\\lambda $ =12.1$^{+10.0\\circ }_{-8.0}$ ), indicating an orbit aligned with the stellar rotation axis.", "Furthermore, several studies focused on the potential transit-timing variations (TTVs) of this system.", "[21] first noticed that a linear fit cannot explain the transit ephemeris very well, which was later suspected to be caused by the poor quality of one timing [47].", "[19] studied its TTVs in detail with an additional seven new transit observations and calculated a TTV rms of 68 s, only marginally larger than their mean timing uncertainty of 41 s. [28] revisited this TTV signal by combining their nine new epochs and suggested that this TTV might be introduced by data uncertainties and systematics and not by gravitational perturbations.", "From these intensive previous studies, WASP-5b has become an intriguing target for atmospheric characterization.", "It is not bloated, although it receives a relatively high irradiation of $\\sim $ 2.1$\\times $ 10$^9$ erg s$^{-1}$ cm$^{-2}$ from its 5700 K [21] host star [19], which would place it in the pM class in the scheme proposed by [18].", "Its proximity to the host star results in an equilibrium temperature of 1739 K assuming zero albedo and isotropic redistribution of heat across the whole planet, which could be as high as 2223 K in the extreme case of zero heat-redistribution.", "Its CaII H and K line strength [50] suggests that the activity of the host star might prevent it from having an inverted atmosphere, given the correlation proposed by [31].", "Recently, [3] reported thermal detections from the Warm Spitzer mission, suggesting a weak thermal inversion or no inversion at all, with poor day-to-night energy redistribution.", "In this paper, we present the first ground-based detections of thermal emission from the atmosphere of WASP-5b in the $J$ and $K$ bands through observations of secondary eclipse.", "Section describes our observations of two secondary-eclipse events and the process of data reduction.", "Section summarizes the approaches that we employed to remove the systematics and to optimally retrieve the flux ratios.", "In Sect.", ", we discuss remaining systematic uncertainties and orbital eccentricity, and we also offer explanations for the thermal emission of WASP-5b with planetary atmosphere models.", "Finally, we conclude in Sect.", "." ], [ "Observations and data reduction", "We observed two secondary-eclipse events of WASP-5b with the GROND instrument mounted on the MPG/ESO 2.2 meter telescope at La Silla, Chile.", "This imaging instrument was designed primarily for the simultaneous observation of gamma-ray burst afterglows and other transients in seven filters: the Sloan $g^{\\prime }$ , $r^{\\prime }$ , $i^{\\prime }$ , $z^{\\prime }$ and the NIR $J$ , $H$ , $K$ [24].", "Dichroics are used to split the incident light into seven optical and NIR channels.", "Photons of the four optical channels are recorded by backside-illuminated $2048\\times 2048$ E2V CCDs and stored in FITS files with four extensions.", "Photons of the three NIR channels are recorded by Rockwell HAWAII-1 arrays ($1024\\times 1024$ ) and stored in a single FITS file with a size of $3072\\times 1024$ .", "The optical arm has a field of view (FOV) of $5.4\\times 5.4$ arcmin$^2$ with a pixel scale of 0158, while the NIR arm has an FOV of $10\\times 10$ arcmin$^2$ with a pixel scale of 060.", "The guiding system employs a camera placed outside the main GROND vessel, $23^{\\prime }$ south of the scientific FOV, which has a crucial impact on the choice of science pointing especially in the case of defocused observations.To avoid poor guiding, we did not employ the defocusing technique.", "The capability of simultaneous optical-to-NIR multiband observation makes GROND a potentially good instrument for transit and occultation observations.", "For secondary-eclipse observations, the optical arm provides the opportunity to detect scattered light in favorable cases, while the NIR arm allows one to construct an SED for the thermal emission of a planetary atmosphere.", "In both of our observations, we only used the NIR arm (i.e.", "WASP-5 was not in the optical FOV) to include as many potential reference stars as possible in the NIR FOV.", "The first secondary-eclipse event was observed continuously for four hours on UT July 26 2011, from 04:13 to 08:16.", "The observation was performed in an ABAB nodding pattern.", "Four exposures were taken on each nodding position during the first one third of the observing time, and 12 exposures each were taken in the remaining time.", "Each exposure was composed of two integrations of 3 seconds each (DIT=3 s), which were averaged together.", "However, the actual nodding pattern was far more complicated.", "The telescope operation GUI software crashed several times, and a new ABAB pattern was re-started on each crashed position.", "The resulting time series of each band was full of red noise, which is difficult to correct since systematic effects affect the recorded signals differently depending on location on the detector.", "Time series of each location did not cover the whole occultation duration, which makes the systematic correction problem even worse.", "Therefore we decided to discard this dataset in our further analysis.", "Only the result in the $K$ band is shown for comparison in Sect.", "and Fig.", "REF .", "Figure: KK-band occultation light-curve of WASP-5observed on UT July 26 2011 (in nodding mode).", "As described in the text,instrumental crashes led to four groups of nods, none of which coveredthe whole eclipse.", "The top four panels show the raw light-curve foreach nod, overplotted with the best-fit model.", "Bottom panel showsthe phase-folded KK-band light curve, which has been corrected for baselinetrend and binned every 10 minutes for display purposes.The second event of the secondary eclipse was observed continuously for 4.6 hours on UT September 8 2011, from 02:41 to 07:17 in staring mode.", "Before and after the science time series, the sky around the scientific FOV was measured using a 20-position dither pattern, which was used to construct the sky emission model in the subsequent reduction.", "During the science observation, four integrations of 3 seconds each were averaged into one exposure, resulting in 707 frames recorded and a duty cycle of $\\sim $ 53%.", "The peak count level of the target star is well below the saturation level.", "The airmass started at 1.22, decreased to 1.02, and rose to 1.11 in the end; the seeing was unstable during the eclipse, ranging from $1^{\\prime \\prime }$ to $3^{\\prime \\prime }$ as measured from the point spread functions (PSFs) of the stars.", "The moon was illuminated around 82%, and had a minimum distance of 55$^{\\circ }$ to WASP-5 at the end of the observation.", "In the following text, we always refer to this second dataset unless specified otherwise.", "We reduced the acquired data with our IDLIDL is an acronym for Interactive Data Language, for details we refer to http://www.exelisvis.com/idl/ pipeline in a standard way, which mainly makes use of NASA IDL Astronomy User's LibrarySee http://idlastro.gsfc.nasa.gov/.", "General image calibration stepsIn principle, NIR data need nonlinearity correction.", "However, we did not include the correction in our final calibration steps to avoid introducing additional noise [2].", "According to our experiment, the derived eclipse depths did not change significantly ($\\ll 1\\sigma $ ) when the data were reduced with or without nonlinearity correction.", "include dark subtraction, read-out pattern removal, flat division, and sky subtraction.", "We made DARK master files by median-combining 20 individual dark-current measurements and subtracted them from all the raw images.", "To correct the electronic odd-even readout pattern along the X-axis, each dark-subtracted image was smoothed with a boxcar median filter and compared with the unsmoothed one.", "The amplitudes of readout patterns were obtained from the resulting difference image and were corrected in the unsmoothed dark-subtracted image.", "Finally, SKYFLAT master files were generated by median-combining 48 individual twilight sky flat measurements, which first had the star masked out and were then normalized and combined.", "The dark- and pattern-corrected images were divided by these SKYFLAT files for flat-field correction.", "To eliminate the sky contribution in our staring-mode data, we constructed a sky emission model for each science image using the 20-position dithering sky measurements.", "These sky images were star-masked and normalized and then dark- and flat-calibrated in the same way as the science images.", "We median-combined on the sky stack images to generate basic sky emission models.", "The pre- and post-science sky models were scaled to the background level of each science image.", "A final sky model was created by combining the pre- and post-science sky models while taking the inverse square of the fitted $\\chi ^2$ as the weight.", "The final sky model was later subtracted from the corresponding science image.", "Due to the long time-scale of our observation, the sky is expected to be variable.", "Thus this sky correction is only a first-order correction.", "Nevertheless, it results in light curves of slightly better precision than the approach without sky subtraction, according to our experiment.", "We performed aperture photometry on the calibrated images with the IDL DAOPHOT package.", "We first determined the locations of WASP-5 as well as several nearby comparison stars of similar brightness using IDL/FIND, which calculates the centroids by fitting Gaussians to the marginal $x$ and $y$ distributions.", "The FWHMs for each star, which were used to indicate the seeing during our observation, were calculated in a similar way.", "We carefully chose the best comparison-star ensemble to normalize the WASP-5 time series as follows: various combinations of comparison stars were tried.", "For each ensemble, time series of chosen comparison stars (as well as WASP-5) were individually normalized by the median of their out-of-eclipse flux levels, and then weighted-combined according to the inverse square of uncertainties.", "The ensemble that made the normalized WASP-5 light curve show the smallest scatter was considered as the optimal reference.", "We also experimented to find the best photometric results by placing 30 apertures on each star in a step of 0.5 pixel, each aperture again with 10 annuli of different sizes in a step of 1 pixel.", "The aperture and annulus that made reference-corrected WASP-5 light curve behave with the smallest scatter was chosen as the optimal aperture setting.", "As a result, we used six comparison stars for the $J$ band, three for the $H$ band, and four for the $K$ band.", "The aperture settings for the $J$ , $H$ , $K$ bands are (6.5, 13.5-22.0) pixels, (6.0, 6.0-19.0) pixels, (5.0, 7.0-19.0) pixels in the format of (aperture size, sky annulus inner/outer sizes), respectively.", "Finally, we extracted the time stamp stored in the header of the FITS file.", "The default time stamp was the starting UTC time of each frame.", "We took into account the readout time and the arm-waiting timeThe optical and NIR arms of GROND are not operated independently.", "to make the final time stamp centered on the central point of each total integration.", "We converted this UTC time stamp into Barycentric Julian Date in the Barycentric Dynamical Time standard (BJD$_{\\rm {TDB}}$ ) using the IDL procedure written by [14].", "Figure: Near-infrared occultation light-curves of WASP-5observed on UT Sep 08 2011 (in staring mode).", "Each panel from top to bottomshows for the JJ, HH and KK bands.", "Top left: raw light-curvesoverplotted with the best-fit models, which consist of a theoretical light-curvemultiplied with a decorrelation function.", "Top right: light curvesafter baseline correction.", "Bottom left: light curves binned everyseven minutes for display purposes.", "Bottom right: residuals oflight-curve fitting.Table: Results of the MCMC analysis on the secondary eclipse (Sep-08-2011) of WASP-5b" ], [ "Light-curve analysis", "As shown in the top left panel of Fig.", "REF , the WASP-5 light curves exhibited obvious red noise even after normalization by the composite reference light-curve as described above.", "Part of this red noise arises from instrumental systematics, such as different star locations on the detector, seeing variation (thus different number of pixels within the volume of the star's FWHM), which can be inferred from the correlation between each parameter and the normalized flux (see Fig.", "REF –REF in the appendix).", "In the literature, some authors chose to construct a systematics model using out-of-eclipse data and applied these relationships to the whole light curve for correction [9].", "This requires that the range of instrumental parameters during in-eclipse is repeatable in the out-of-eclipse data, otherwise it would lead to extrapolation.", "Since most of our instrumental parameters were not necessarily repeatable between in-eclipse and out-of-eclipse (e.g.", "slow drift of star location on the detector, variation of seeing), we decided to fit the whole light curve with an analytic occultation model multiplied by a baseline correction model.", "We adopted the [40] formulae without limb-darkening as our occultation model.", "System parameters such as period $P$ , planet-to-star radius ratio $R_p/R_$ , inclination $i$ , and scaled semi-major axis $a/R_$ were obtained from [19] and were fixed in the formulae, while mid-occultation time $T_{\\rm {mid}}$ and flux ratio $F_p/F_$ were set as free parameters.", "The baseline detrending model was a sum of polynomials of star positions ($x$ , $y$ ), seeings ($s$ ), airmass ($z$ ), and time ($t$ ).", "We varied the combination of these instrumental and atmospheric terms to generate different baseline models.", "We searched for the best-fit solutions by minimizing the chi-square: $\\chi ^2=\\sum \\limits _{i=1}^{N}\\frac{[f_i(\\mathrm {obs}) - f_i(\\mathrm {mod})]^2}{\\sigma _{f,i}^2},$ in which $f_i(\\rm {obs})$ and $\\sigma _{f,i}$ are the light-curve data and its uncertainty, while $f_i(\\rm {mod})$ is the light-curve model, in the form of $f(\\mathrm {mod})=E(T_{\\mathrm {mid}},F_p/F_)B(x,y,s,z,t),$ We experimented with a set of baseline models to find the model that can best remove the instrumental systematics.", "We calculated the Bayesian information criterion [45] for the results from different baseline models: $BIC=\\chi ^2+k\\log (N),$ where $k$ is the number of free parameters and $N$ is the number of data points.", "The baseline model that generated the smallest BIC value was considered as our final choice.", "With this approach, we used as few free parameters as possible to prevent overinterpreting the baseline function.", "In our experiments, linear baseline functions in most cases failed to fit the eclipse depth and produced very large BIC values.", "Among the baseline functions that have BICs similar to that of the chosen one, the measured eclipse depths agree well with each other (see e.g.", "Fig.", "REF and REF ).", "The final adopted baseline functions are $B_J=c_0+c_1x+c_2y+c_3xy+c_4x^2+c_5y^2+c_6s_x+c_7s_y+c_8t,$ $B_H=c_0+c_1x+c_2y+c_3xy+c_4y^2+c_5s_x,$ and $B_K=c_0+c_1xy+c_2x^2+c_3y^2+c_4s+c_5t,$ where $s_x$ and $s_y$ refer to the FWHMs of marginal $x$ and $y$ distributions, respectively, while $s$ is their quadratic mean.", "We employed the Markov chain Monte Carlo (MCMC) technique with the Metropolis-Hastings algorithm with Gibbs sampling to determine the posterior probability distribution function (PDF) for each parameter [15], [16].", "Following the approach of [22], only parameters in the analytic occultation model $E(T_{\\rm {mid}},F_p/F_)$ are perturbed, while the coefficients of baseline function are solved using the singular value decomposition [43] algorithm.", "At each MCMC step, a jump parameter was randomly selected, and the light curve was divided by the resulting analytic occultation model.", "The coefficients in the baseline function were then solved by linear least-squares minimization using the SVD.", "This jump was accepted if the resulting $\\chi ^2$ is lower than the previous $\\chi ^2$ , or accepted according to the probability $\\exp {(-\\Delta \\chi ^2/2)}$ if the resulting $\\chi ^2$ is higher.", "We optimized the step scale so that the acceptance rate was $\\sim $ 0.44 before a chain starts [16].", "After running a chain of MCMC, the first 10% links were discarded and the remaining were used to determine the best-fit values and uncertainties of jump parameters (as well as the baseline coefficients).", "Several chains were run to check that they were well mixed and converged using the [20] statistics.", "We adopted the median values of the marginalized distributions as the final parameter values and the 15.865%/84.135% values of the distributions as the 1-$\\sigma $ lower/upper uncertainties, respectively.", "Figure: Joint probability distributions between mid-eclipse time and flux ratiofrom our MCMC analysis for the JJ (left), HH (middle), and KK bands(right).", "The mid-eclipse time has been converted to phase in these plots.", "Thecontour lines mark the 68.3% (1σ1\\sigma ) and 95.5% (2σ2\\sigma ) confidence regionsof the joint posterior distributions, respectively.", "The gray scale indicates thedensity of distributions.We performed the MCMC-based light-curve modeling in three scenarios.", "In the first scenario, we tried to find the best-fit mid-occultation times and flux ratios.", "$T_{\\rm {mid}}$ and $F_p/F_$ were allowed to vary freely for the $J$ and $K$ bands.", "Since there was no detection in the $H$ band, we chose to adopt the PDF of $T_{\\rm {mid}}$ coming from the $K$ band as a Gaussian prior.", "We first ran a chain of 1 000 000 links to find the scaling factors, since the photometric uncertainties might not represent the real uncertainties in the light curves well.", "We calculated the reduced chi-square ($\\chi ^2_{\\nu }$ ) for the best-fit model and recorded the scaling factor $\\beta _1$ =$\\sqrt{\\chi ^2_v}$ .", "We then calculated the standard deviations for the best-fit residuals without binning, also for those with time bins ranging from 10 minutes to the ingress/egress duration of WASP-5b.", "The median value of the factor $\\beta _2=\\frac{\\sigma _N}{\\sigma _1}\\sqrt{\\frac{N(M-1)}{M}}$ was recorded as the second scaling factor, where $N$ is the number of individual binned points, $M$ is the number of bins, $\\sigma _N$ is the standard deviation of $N$ -point binned residuals, and $\\sigma _1$ is the un-binned version.", "These two scaling factors were multiplied with the original uncertainties to account for the under-/overestimated noise.", "Normally, the first scaling factor would make the fitting to have a reduced chi-square close to 1, while the second would take into account the time-correlated red noise.", "This approach has been widely applied in transit light-curve modeling [42], [51].", "The derived ($\\beta _1$ , $\\beta _2$ ) for the $J$ , $H$ , $K$ bands were (1.79, 1.28), (1.25, 1.46), (1.05,1.33), respectively.", "After rescaling the uncertainties, we ran another five chains of 1 000 000 links to finalize the modeling.", "The $J$ -band flux ratio changed from 0.175$\\pm $ 0.021% to 0.168$^{+0.050}_{-0.052}$ %, while the $K$ -band flux ratio changed from 0.272$\\pm $ 0.044% to 0.269$\\pm $ 0.062%.", "The achieved light-curve quality for the $J$ , $H$ and $K$ bands are 1847, 1813 and 1777 ppm per two-minute interval in terms of $rms$ of O–C (observed minus calculated) residuals.", "The estimated photon noise limits in the $J$ , $H$ , $K$ bands are 2.3$\\times $ 10$^{-4}$ , 2.4$\\times $ 10$^{-4}$ , and 3.7$\\times $ 10$^{-4}$ per two-minute interval, respectively.", "This uncertainties rescaling has barely changed the best-fit values, but enlarged their uncertainties, thus decreased the detection significance.", "The derived jump parameters and coefficients for each band are listed in Table REF , while the posterior joint probability distributions between $T_{\\rm {mid}}$ and $F_p/F_$ are shown in Fig.", "REF .", "In the second scenario, we changed the form of occultation model to $E(T_{58},T_{\\rm {mid}},F_p/F_)$ so that we could fit the occultation duration $T_{58}$ .", "Since our light curves are of poor quality, we decided to adopt the PDFs of $T_{\\rm {mid}}$ and $F_p/F_$ from the light-curve modeling in the first scenario as Gaussian priors input to the light-curve modeling in the second scenario.", "Another five chains of 1 000 000 links were run in search for the best-fit occultation duration of the $J$ and $K$ light curves.", "The $H$ band was not fitted because there was no detection.", "We obtained an occultation duration of $0.1001^{+0.0063}_{-0.0070}$ days for the $J$ band and $0.1026^{+0.0054}_{-0.0058}$ days for the $K$ band.", "They are both consistent with the primary transit duration $T_{14}$ =0.1004 days [19] within their large uncertainties.", "In the third scenario, we tried to model the light curves from the nodding observation (the Jul-26-2011 dataset) to directly compare the staring and nodding mode observations.", "The $J$ and $H$ bands in the nodding mode could fail to be fitted because of their extremely poor data quality.", "Thus we only modeled the $K$ band.", "The light curve was divided into four groups of nods according to their nodding positions.", "In the modeling, all four sub-light-curves share the same $F_p/F_$ and have the mid-eclipse time fixed on the expected mid-point assuming zero eccentricity, while they are allowed to have different coefficients in the baseline from nod to nod.", "The adopted baseline function is $B_K=c_0+c_1x+c_2y+c_3xy+c_4x^2+c_5y^2.$ We performed the MCMC-based modeling in the same manner as in the first scenario to find the scaling factors ($\\beta _1$ =1.47 and $\\beta _2$ =1.49) and to determine the flux ratio.", "This resulted in a flux ratio of 0.268$\\pm $ 0.076% and $0.27^{+0.16}_{-0.15}$ % for the unscaled and rescaled versions, respectively.", "The $rms$ of O–C residuals for this nodding light-curve is 4214 ppm per two-minute interval, twice as high as the staring mode.", "Considering that the nodding observations featured crashes, a better comparison would be using the un-crashed nodding pair.", "The $rms$ for the first 2-hour parts of nod A and B is 4738 ppm, while for the last 1-hour parts of nod C and D it is 2638 ppm.", "The main difference between these two un-crashed nodding pairs is their nodding pattern, that is, the locations of the same star on the detector are different, which results in instrumental systematics of very different levels.", "A location change also exists within one nodding pair.", "In contrast, the star's location on the detector is relatively stable in the staring observation.", "Without the risk of introducing unexpected systematics from a different location, it is easier to model the light curve, which results in higher precision.", "This reconfirms that the staring mode is a better suited strategy than the nodding mode in exoplanet observations, which has been noted in several previous observations [13], [10].", "In addition to this analysis, we also examined our light curves to determine the correlations between measured eclipse depth and the choices of aperture size and reference ensemble (see Fig.", "REF and REF in the appendix).", "As the aperture radius increases, the $rms$ of light-curve O–C residuals first decreases to a minimum and then rises, which is expected because smaller apertures might lose partial stellar flux while larger apertures would include more sky noise.", "Correspondingly, the measured eclipse depth first changes greatly with the aperture size and then stabilizes when the aperture size approaches our chosen value.", "For aperture sizes that result in $rms$ similar to that of the chosen aperture, the measured eclipse depths agree well with our reported result within 1-$\\sigma $ error bars.", "Furthermore, the measured eclipse depths derived from different combinations of reference stars are consistent with each other when they produce light curves with relatively low red noise.", "Therefore, we confirm that our choices of photometry and reference ensemble are ideal, in contrast, the measured eclipse depths are relatively insensitive to the choice of aperture size and reference ensemble.", "Figure: RMSRMS of the binned residuals v.s.", "bin size for the JJ,HH and KK occultation light-curves.", "The red lines showsthe prediction for Gaussian white noise (1/N1/\\sqrt{N}).The vertical dashed line show the corresponding ingress/egressduration time.", "We can still see the strong effect of correlatednoise in all three bands even after the baseline correction.Table: Adopted parameters in the modeling process" ], [ "Results and discussions", "We list the derived jump parameters of our MCMC analysis in Table REF , along with the coefficients of the baseline functions from the modeling in the first scenario.", "Figure REF shows the $K$ -band light curve from the nodding observation (Jul-26-2011), while Fig.", "REF shows all three light curves from the staring observation (Sep-08-2011).", "The adopted parameters that were used in our MCMC analysis are given in Table REF ." ], [ "Correlated noise", "In our MCMC analysis, we have propagated the uncertainties of the baseline detrending function into the PDFs of jump parameters by solving its coefficients at each MCMC step.", "We also tried to account for the red noise by rescaling the photometric uncertainties with the $\\beta $ factors.", "The time averaging processes are shown in Fig.REF .", "The $rms$ of the binned residuals clearly deviates from the predicted Gaussian white noise, as shown by the red lines, indicating the presence of correlated noise in our light curves.", "Here we employed another commonly used method [46], the \"prayer-bead\" residual permutation method, which preserves the shape of time-correlated noise, to investigate whether there is still excess red noise that is not included in our MCMC analysis.", "Firstly, the best-fit model was removed from the light curve.", "The residuals were cyclically shifted from the $i$ th to the $i$ +1th positions (of the $N$ data points), while off-position data points in the end were wrapped at the beginning.", "Then the best-fit model was added back to the permuted residuals to form a new synthetic light curve.", "This synthetic light curve was then fitted in the same way as the real light curve, as described in Sect. .", "We inverted the light-curve sequence to perform another series of cycling, thus achieving 2$N$$-$ 1 synthetic light curves in total.", "We also calculated the median and 68.3% confidence level of the resulting distribution as the best-fit value and $1\\sigma $ uncertainties.", "This residual-permutation (RP)-based analysis leads to a flux ratio of $0.268^{+0.062}_{-0.054}$ % for the $K$ band and $0.167^{+0.033}_{-0.038}$ % for the $J$ band.", "The RP-based flux ratios have smaller uncertainties than those of the $\\beta $ -based MCMC analysis (0.269$\\pm $ 0.062% and $0.168^{+0.050}_{-0.052}$ %, correspondingly).", "The differences in best-fit values are very small.", "This indicates that our $\\beta $ -based MCMC analysis has included the potential impact from the time-correlated noise.", "We adopted the $\\beta $ -based MCMC results as our final results." ], [ "Orbital eccentricity", "We obtain an average mid-occultation offset time of 10.5$\\pm $ 3.1 minutes and an average occultation duration time of $0.1016^{+0.0041}_{-0.0045}$ days by combining values of the $J$ and $K$ bands with weights according to the inverse square of their uncertainties.", "The secondary eclipse of WASP-5b is expected to occur at phase $\\phi $ =0.5002 if it is in a circular orbit.", "This value has taken into account the delayed light travel time of $\\sim $ 27 s [34].", "However, our average mid-occultation time occurs at a delayed offset of 10.1$\\pm $ 3.1 minutes to this expected phase, which might indicate a nonzero eccentricity.", "We used equations from [44] to derive the values of $e\\cos \\omega $ and $e\\sin \\omega $ : $e\\cos \\omega \\simeq \\pi \\Delta \\phi /2$ $e\\sin \\omega =\\frac{D_{\\mathrm {II}}-D_{\\mathrm {I}}}{D_{\\mathrm {II}}+D_{\\mathrm {I}}}\\frac{\\alpha ^2-\\cos ^2i}{\\alpha ^2-2\\cos ^2i},$ where $\\alpha $ =$(R_*/a+R_{\\rm {p}}/a)/\\sqrt{1-e^2}$ , while $D_{\\rm {II}}$ and $D_{\\rm {I}}$ refer to the durations of secondary eclipse and primary transit.", "Thus $e\\cos \\omega $ and $e\\sin \\omega $ can be constrained if we can measure the mid-eclipse time and duration of a secondary eclipse with sufficient precision.", "We calculated an $e\\cos \\omega $ =0.0067$\\pm $ 0.0021 and an $e\\sin \\omega $ =0.007$\\pm $ 0.026.", "While the former parameter barely deviates from zero by $3\\sigma $ , the latter is consistent with zero within its large uncertainty.", "The 68.3% confidence level for eccentricity is $e$ =0.020$^{+0.019}_{-0.011}$ , with corresponding argument of periastron $\|\\omega \|$ =71$^{+11\\circ }_{-31}$ .", "Our derived eccentricity is only slightly larger than zero at a significance lower than $2\\sigma $ .", "From the previous radial velocity studies, [21] found a tentatively nonzero value of $e$ =0.038$^{+0.026}_{-0.018}$ , while [29] claimed that its eccentricity is compatible with zero ($e$ =0.012$\\pm $ 0.007) based on more RV measurements.", "Our result, derived from a different approach, lies between them and is consistent with both results within their errorbars.", "Recent Warm Spitzer measurements resulted in a mean value of $e\\cos \\omega $ =0.0025$\\pm $ 0.0012 [3], which is 1.74$\\sigma $ lower than our average value (c.f.", "0.86$\\sigma $ lower than our $K$ -band result, see Table REF ).", "However, we are cautious to draw any conclusion on nonzero eccentricity here.", "The shapes of our light curves are complicated due to the existence of instrumental and atmospheric systematics, as can be seen in Fig.REF .", "It is very likely that these systematic effects bias the mid-eclipse time.", "Furthermore, the occultation duration is poorly constrained by our measurements." ], [ "Eclipse depths and brightness temperatures", "To preliminarily probe the atmosphere, we first calculated the brightness temperatures corresponding to the measured flux ratios.", "We assumed blackbody emission for the planet and interpolated the stellar spectrum in the Kurucz stellar models [33] for the host star (using $T_{\\rm {eff}}$ =5700 K, $\\log g$ =4.395 and [Fe/H]=0.0).", "The blackbody spectrum and the stellar spectrum were both integrated over the bandpass of our three NIR bands individually.", "The blackbody temperature that yields the resulting flux ratio best-fit was adopted as the corresponding brightness temperature in each band.", "For the Sep-08-2011 secondary eclipse, the measured flux ratios are $0.168^{+0.050}_{-0.052}$ % and 0.269$\\pm $ 0.062% in the $J$ and $K$ band, and a $3\\sigma $ upper limit of 0.166% in the $H$ band, which translates to brightness temperatures of $2996^{+212}_{-261}$ K, $2890^{+246}_{-269}$ K and $<2779$ K ($3\\sigma $ ), respectively.", "We used the same approach to calculate the brightness temperature of the Warm Spitzer data, where [3] reported 0.197$\\pm $ 0.028% at 3.6 $\\mu $ m and 0.237$\\pm $ 0.024% at 4.5 $\\mu $ m. In this way, the temperatures in the NIR and MIR were derived with the same stellar atmosphere models and parameters [3].", "As a result, the Spitzer eclipse depths translate into brightness temperatures of 1982$^{+117}_{-122}$ K and 1900$^{+92}_{-94}$ K, respectively.", "The temperature derived from our NIR data ($\\sim $ 2700 K) completely disagrees with that derived from the Warm Spitzer data ($\\sim $ 1900 K)." ], [ "Atmospheric models", "To investigate possible scenarios of the atmospheric properties, we modeled the emerging spectrum of the dayside atmosphere of WASP-5b using the exoplanetary atmospheric modeling and retrieval method of [35], [36].", "Our model performs line-by-line radiative transfer in a plane-parallel atmosphere, with constraints on local thermodynamic equilibrium, hydrostatic equilibrium, and global energy balance.", "The pressure-temperature ($P$ -$T$ ) profile and the molecular composition are free parameters of the model, allowing exploration of models with and without thermal inversions, and with oxygen-rich as well as carbon-rich compositions [39].", "The model includes all the primary sources of opacity expected in hydrogen-dominated atmospheres in the temperature regimes of hot Jupiters, such as molecular line absorption due to various molecules (H$_2$ O, CO, CH$_4$ , CO$_2$ , HCN, C$_2$ H$_2$ , TiO, VO) and collision-induced absorption (CIA) due to H$_2$ [39].", "The volume-mixing ratios of all the molecules are free parameters in the model.", "Given that the number of model parameters ($N$ =10-14, depending on the C/O ratio) is much higher than the number of available data points, our goal is to nominally constrain the regions of model space favored by the data rather than determine a unique fit." ], [ "Constraints from our NIR data", "Our observations place a stringent constraint on the temperature structure of the lower atmosphere of the planetary dayside.", "The $J$ , $H$ , and $K$ bands contain only weak molecular features due to spectroscopically dominant molecules in hot-Jupiter atmospheres.", "As such, photometric observations in these bands probe deep into the lower regions of the planetary atmosphere until the high pressures make the atmosphere optically thick (around $P \\sim 0.1-1$ bar) due to H$_2$ -H$_2$ CIA continuum absorption [39].", "Our observed brightness temperatures in the $J$ , $H$ , and $K$ bands can be explained by a roughly isothermal temperature profile of $\\sim $ 2700 K in the lower atmosphere of WASP-5b, consistent with the fact that for highly irradiated hot Jupiters the dayside temperature structure at $\\tau $$\\sim $ 1 tends to be isothermal [27], [35], [26].", "In principle, our $J$ and $K$ band data allow for significantly higher temperatures, up to $\\sim $ 3200 K, but our $H$ -band observation rules out temperatures above $\\sim $ 2700 K. As shown in Fig.", "REF , a blackbody spectrum of 2700 K representing the continuum blackbody of the lower atmosphere provides a reasonable fit to the $J$ , $H$ , $K$ data.", "However, an isothermal temperature profile at $\\sim $ 2700 K over the entire vertical extent of the atmosphere is unlikely.", "This would violate global energy balance since the planet would radiate substantially more energy than it receives.", "We assume that the internal source of energy is negligible compared to the incident irradiation [4].", "Figure: Dayside thermal emission spectrum from the hot Jupiter WASP-5b in terms ofplanet-to-star flux ratios (left) and planetary dayside flux (right).", "Theblue circles with error bars at 1.26 μ\\mu m and 2.15 μ\\mu m and the upper-limit at1.65 μ\\mu m, show our measured planet-star flux ratios in the photometric JJ, HH,and KK bands.", "The Spitzer photometric observations reported by are shown at 3.6 μ\\mu m and 4.5 μ\\mu m (black squares).", "The dotted gray curves showtwo planetary blackbody spectra with temperatures of 1900 K and 2700 K. While ourJJ, HH, KK band data can be fit with a ∼\\sim 2700 K blackbody spectrum, theSpitzer data is consistent with a ∼\\sim 1900 K blackbody.", "The red and greencurves show two model spectra of the dayside atmosphere of WASP-5b withoxygen-rich and carbon-rich compositions, respectively.", "The inset shows the correspondingpressure-temperature profiles without thermal inversions; a thermal inversion in thedayside atmosphere of WASP-5b is ruled out by the data.", "An oxygen-rich modelis unable to explain all the data.", "On the other hand, while a carbon-rich model providesa good fit to the data, it radiates 70% more energy than the incident irradiation,which may be unphysical unless there are additional absorbing species in theatmosphere that are not accounted for in the current C-rich model.", "SeeSect.", "for discussion." ], [ "Constraints from NIR and ", "Additional constraints on the temperature profile and on the chemical composition of the dayside atmosphere of WASP-5b are obtained by combining our data with new photometric observations obtained with the Spitzer Space Telescope at 3.6 $\\mu $ m and 4.5 $\\mu $ m [3].", "Our data together with the Spitzer data rule out a thermal inversion in WASP-5b irrespective of its chemical composition, consistent with the finding of [3] based on the Spitzer data alone.", "The Spitzer data probe higher atmospheric layers than the ground-based data due to strong molecular absorption in the two Spitzer bands, and are consistent with brightness temperatures of $\\sim $ 1900 K, which is much lower than the $\\sim $ 2700 K temperatures in the ground-based channels.", "Consequently, the two data sets suggest temperatures decreasing outward in the atmosphere.", "Previous work has shown that the Spitzer data also provide good diagnostics of the C/O ratio of the atmosphere as the bandpasses overlap with broad spectroscopic features of several dominant C- and O-bearing molecules [39].", "Chemical compositions of hot-Jupiter atmospheres can be extremely different depending on whether they are oxygen-rich (C/O $<$ 1) or carbon-rich (C/O$\\ge $ 1).", "Whereas in O-rich atmospheres (e.g.", "of solar composition, with C/O = 0.5), H$_2$ O and CO, and possibly TiO and VO, are the dominant sources of opacity, C-rich atmospheres are depleted in H$_2$ O and abundant in CO, CH$_4$ , HCN, and C$_2$ H$_2$ [38], [32], [39], [41].", "We investigated both O-rich and C-rich scenarios in the present work and found that neither composition simultaneously provides a good fit to the data and satisfies energy balance, as shown in Fig.", "REF .", "However, the chemical composition is poorly constrained by the current data.", "First, we found that an O-rich solar composition atmosphere can neither fit all the data to within the 1-$\\sigma $ errors nor satisfy energy balance; it radiates more energy than it receives.", "A composition with enhanced metallicity ($5\\times $ solar), but still O-rich, can satisfy energy balance, but still does not fit all the data, predicting the planet-star flux contrast at 3.6 $\\mu $ m to be $\\gtrsim $ 3-$\\sigma $ higher than the observed value, as shown in Fig.", "REF .", "On the other hand, a C-rich model can fit all the data reasonably well, but radiates 70% more energy than it receives from incident radiation, thereby violating global energy balance.", "Both the O-rich and C-rich models were consistent with the lack of a thermal inversion in the planet.", "While the high chromospheric activity of the host star could destroy inversion-causing species in the atmosphere irrespective of its C/O ratio [31], [3], a C-rich atmosphere would be naturally depleted in oxygen-rich inversion-causing compounds such as TiO and VO, which designates WASP-5b as a C2-class hot Jupiter in the classification of [39]." ], [ "Possible scenarios and future prospects", "Our data agree with two possible scenarios for WASP-5b: a carbon-rich and an oxygen-rich atmosphere.", "However, we caution that new observations are required to conclusively constrain its chemical composition.", "The C-rich scenario, while providing a good fit to all available data, requires an explanation for the apparent energy excess in the emergent spectrum.", "This could be mitigated by an additional absorber in the atmosphere, which has high opacity blueward of the $J$ band ($\\lesssim $ 1.1 $\\mu $ m) and/or a strong feature in the $H$ band ($\\sim $ 1.5–1.8 $\\mu $ m).", "The presence of such a component is currently merely speculative, but could be seen or ruled out using follow-up spectroscopic observations, for instance with HST/WFC3.", "Such observations would additionally constrain the energy budget of the dayside atmosphere of WASP-5b.", "The O-rich model satisfies global energy balance but does less well at simultaneously fitting the $J$ , $H$ , $K$ data and the Spitzer 3.6 $\\mu $ m point.", "One explanation is that the planet shows substantial temporal variability in its emerging spectrum, but the magnitude of the inferred variability seems implausibly high.", "Another possibility is that different systematic effects between the ground-based and Spitzer data bias the derived thermal emission measurements.", "Spectroscopy with HST/WFC3 in the 1.1–1.7 $\\mu $ m bandpass would allow us to conclusively constrain the chemical composition of the atmosphere, since our two model spectra in Fig.", "REF predict very different spectral shapes in that bandpass.", "In addition, observations of thermal phase curves with warm Spitzer [30] will also allow us to place stringent constraints on the day-to-night energy redistribution, since all models fitting our current data predict extremely low redistributions implying strong day-to-night thermal contrasts." ], [ "Conclusions", "We observed two secondary eclipses of WASP-5b simultaneously in the $J$ , $H$ and $K$ bands with GROND on the MPG/ESO 2.2 meter telescope, one in nodding mode and the other in staring mode.", "Although we failed to extract useful results from the nodding-mode observation due to the associated complicated systematics, we did measure the occultation dips from the staring-mode observation with reasonable precision, reconfirming that the staring mode is more suited than the nodding mode for exoplanet observations.", "We have successfully detected the thermal emission from the dayside of WASP-5b in the $J$ and $K$ bands, with flux ratios of $0.168^{+0.050}_{-0.052}$ % and 0.269$\\pm $ 0.062%, respectively.", "In the $H$ band we derived a 3-$\\sigma $ upper limit of 0.166%.", "The brightness temperatures inferred from the $J$ and $K$ bands are consistent with each other ($2996^{+212}_{-261}$ K and $2890^{+246}_{-269}$ K, respectively), but the upper limit in the $H$ band rules out temperatures above 2779 K at $3\\sigma $ level.", "While a slight difference might exist, together they indicate a roughly isothermal lower atmosphere of $\\sim $ 2700 K. We modeled the GROND data together with the Warm Spitzer data using the spectral retrieval technique, ruling out a thermal inversion.", "We fit our data with two different models: an oxygen-rich atmosphere and a carbon-rich atmosphere.", "The O-rich model requires a very low day-to-night-side heat redistribution but satisfies energy balance.", "The C-rich model fits our data better, but violates energy balance in that it radiates 70% more energy than it receives.", "To constrain the chemical composition of WASP-5b and to distinguish atmospheric models, more observations in the NIR, in particular spectroscopy, are required.", "We thank the referee Bryce Croll for his careful reading and helpful comments that improved the manuscript.", "We acknowledge Timo Anguita for technical support of the observations.", "G.C.", "acknowledges the Chinese Academy of Sciences and the Max Planck Society for the support of doctoral training in the program.", "N.M. acknowledges support from the Yale Center for Astronomy and Astrophysics (YCAA) at Yale University through the YCAA prize postdoctoral fellowship.", "H.W.", "acknowledges the support by NSFC grants 11173060, 11127903, and 11233007.", "This work is supported by the Strategic Priority Research Program \"The Emergence of Cosmological Structures\" of the Chinese Academy of Sciences, Grant No.", "XDB09000000.", "Part of the funding for GROND (both hardware and personnel) was generously granted from the Leibniz-Prize to G. Hasinger (DFG grant HA 1850/28-1)." ], [ "Additional figures", "In this appendix, we present figures that show the dependence of measured eclipse depth on the choice of aperture radii and on the choice of different reference star combinations.", "We also display the correlations between raw light-curve flux and detrending parameters.", "Figure: Dependence of rmsrms, measured eclipse depth, and red noise factor on aperture size and reference ensemble for the KK band.", "The left panel shows the dependence on aperture size.", "The top subpanel displays the rmsrms of light-curve O–C residuals.", "The middle subpanel displays corresponding eclipse depths.", "The bottom subpanel displays the time-averaging red noise factor (i.e.", "β 2 \\beta _2 as denoted in Sect. ).", "The results derived from three cases of baseline functions (BF) are shown for comparison: the chosen best BF, a candidate BF that results in similar BIC to the best BF, and a linear BF that produces a poor fit.", "The vertical dotted line refers to the chosen aperture size, while the dash-shaded area refers to the 1-σ\\sigma confidence level of our reported result.", "The right panel shows the dependence of rmsrms, measured eclipse depth, and red noise factor on different reference star ensembles.", "These ensembles have been sorted according to rmsrms for display purposes, with the best one being #1.", "The subpanels on the right are organized in the same manner as for the left.Figure: Dependence of rmsrms, measured eclipse depth, and red noise factor on aperture size and reference ensemble for the JJ band.", "The subpanels in this figure are organized in the same manner as Fig.", ".Figure: Correlation between instrumental parameters (i.e.", "variables in the baseline function) and the normalized flux for the KK band.", "The first row shows the flux before baseline correction, while the second row shows the flux after baseline correction as a comparison.", "In these plots, xx and yy refer to the relative positions, and FWHM x _x and FHWM y _y refer to the full-width at half maximum of the marginalized PSF, both in pixels.Figure: Correlation between instrumental parameters and the normalized flux for the JJ band.", "The figure is displayed in the same manner as Fig.", "." ] ]	1403.0586
[ [ "Topological and measure properties of some self-similar sets" ], [ "Abstract Given a finite subset $\\Sigma\\subset\\mathbb{R}$ and a positive real number $q<1$ we study topological and measure-theoretic properties of the self-similar set $K(\\Sigma;q)=\\big\\{\\sum_{n=0}^\\infty a_nq^n:(a_n)_{n\\in\\omega}\\in\\Sigma^\\omega\\big\\}$, which is the unique compact solution of the equation $K=\\Sigma+qK$.", "The obtained results are applied to studying partial sumsets $E(x)=\\big\\{\\sum_{n=0}^\\infty x_n\\varepsilon_n:(\\varepsilon_n)_{n\\in\\omega}\\in\\{0,1\\}^\\omega\\big\\}$ of some (multigeometric) sequences $x=(x_n)_{n\\in\\omega}$." ], [ "Introduction", "Suppose that $x=\\left( x_{n}\\right) _{n=1}^{\\infty }$ is an absolutely summable sequence with infinitely many nonzero terms and let $E( x) =\\Big \\lbrace \\sum _{n=1}^\\infty \\varepsilon _{n}x_{n}:(\\varepsilon _{n})_{n=1}^{\\infty }\\in \\lbrace 0,1\\rbrace ^{\\mathbb {N}}\\Big \\rbrace $ denote the set of all subsums of the series $\\sum _{n=1}^{\\infty }x_{n},$ called the achievement set (or a partial sumset) of $x$ .", "The investigation of topological properties of achievement sets was initiated almost one hundred years ago.", "In 1914 Soichi Kakeya [10] presented the following result: Theorem 1.1 (Kakeya) For any sequence $x\\in l_{1}\\setminus c_{00}$ $E(x)$ is a perfect compact set.", "If $\|x_{n}\|>\\sum _{i>n}\|x_{i}\|$ for almost all $n$ , then $E(x)$ is homeomorphic to the ternary Cantor set.", "If $\|x_{n}\|\\le \\sum _{i>n}\|x_{i}\|$ for almost all $n$ , then $E(x)$ is a finite union of closed intervals.", "In the case of non-increasing sequence $x $ , the last inequality is also necessary for $E(x)$ to be a finite union of intervals.", "Moreover, Kakeya conjectured that $E(x)$ is either nowhere dense or a finite union of intervals.", "Probably, the first counterexample to this conjecture was given by Weinstein and Shapiro ([17]) and, independently, by Ferens ([6]).", "The simplest example was presented by Guthrie and Nymann [7]: for the sequence $c=\\big (\\frac{5+(-1)^n}{4^n}\\big )_{n=1}^\\infty $ , the set $T=E(c)$ contains an interval but is not a finite union of intervals.", "In the same paper they formulated the following theorem, finally proved in [13]: Theorem 1.2 For any sequence $x\\in l_{1}\\setminus c_{00}$ , $E(x)$ is one of the following sets: a finite union of closed intervals; homeomorphic to the Cantor set; homeomorphic to the set $T$ .", "Note, that the set $T=E(c)$ is homeomorphic to $C\\cup \\bigcup _{n=1}^{\\infty }S_{2n-1}$ , where $S_{n}$ denotes the union of the $2^{n-1}$ open middle thirds which are removed from $[0,1]$ at the $n$ -th step in the construction of the Cantor ternary set $C$ .", "Such sets are called Cantorvals (to emphasize their similarity to unions of intervals and to the Cantor set simultaneously).", "Formally, a Cantorval (more precisely, an $\\mathcal {M}$ -Cantorval, see [11]) is a non-empty compact subset $S$ of the real line such that $S$ is the closure of its interior, and both endpoints of any non-degenerated component are accumulation points of one-point components of $S$ .", "A non-empty subset $C$ of the real line $\\mathbb {R}$ will be called a Cantor set if it is compact, zero-dimensional, and has no isolated points.", "Let us observe that Theorem REF says, that $l_{1}$ can be devided into 4 sets: $c_{00}$ and the sets connected with cases (1), (2) and (3).", "Some algebraic and topological properties of these sets have been recently considered in [1].", "We will describe sequences constructed by Weinstein and Shapiro, Ferens and Guthrie and Nymann using the notion of multigeometric sequence.", "We call a sequence multigeometric if it is of the form $(k_{0},k_{1},\\dots ,k_{m},k_{0}q,k_{1}q,\\dots ,k_{m}q,k_{0}q^{2},k_{1}q^{2},\\dots ,k_{m}q^{2},k_{0}q^{3}\\dots )$ for some positive numbers $k_{0},\\dots ,k_{m}$ and $q\\in \\left( 0,1\\right) $ .", "We will denote such a sequence by $(k_{0},k_{1},\\dots ,k_{m};q)$ .", "Keeping in mind that the type of $E\\left( x\\right) $ is the same as $E\\left( \\alpha x\\right) $ , for any $\\alpha >0$ , we can describe the Weinstein-Shapiro sequence as $a=(8,7,6,5,4;\\tfrac{1}{10}),$ the Ferens sequence as $b=(7,6,5,4,3;\\tfrac{2}{27})$ and the Guthrie-Nymann sequence as $c=(3,2;\\tfrac{1}{4}).$ Another interesting example of a sequence $d$ with $E(d)$ being Cantorval was presented by R. Jones in ([9]).", "The sequence is of the form $d=(3,2,2,2;\\tfrac{19}{109}).$ In fact, Jones constructed continuum many sequences generating Cantorvals, indexed by a parameter $q$ , by proving that, for any positive number $q$ with $\\frac{1}{5}\\leqslant \\sum _{n=1}^{\\infty }q^{n}<\\frac{2}{9}$ (i.e.", "$\\frac{1}{6}\\leqslant q<\\frac{2}{11}$ ) the achievement set of the sequence $(3,2,2,2;q)$ is a Cantorval.", "The structure of the achievement sets $E(x)$ for multigeometric sequences $x$ was studied in the paper [3], which contains a necessary condition for the achivement set $E(x)$ to be an interval and sufficient conditions for $E(x)$ to contain an interval or have Lebesgue measure zero.", "In the case of a Guthrie-Nymann-Jones sequence $x_{q}=(3,2,\\dots ,2;q),$ of rank $m$ (i.e., with $m$ repeated 2's), the set $E(x_{q})\\ $ is an interval if and only if $q\\geqslant \\frac{2}{2m+5}$ , $E(x_{q})$ is a Cantor set of measure zero if $q<\\frac{1}{2m+2}$ , and $E(x_{q})$ is a Cantorval if $q\\in \\lbrace \\frac{1}{2m+2}\\rbrace \\cup \\big [\\frac{1}{2m},\\frac{1}{2m+5}\\big )$ .", "In this paper we reveal some structural properties of the sets $E(x_{q})$ for $q$ belonging to the “misterious” interval $(\\frac{1}{2m+2},\\frac{1}{2m})$ .", "In particular, we shall show that for almost all $q$ in this interval the set $E(x_{q})$ has positive Lebesgue measure and there is a decreasing sequence $(q_{n})$ convergent to $\\frac{1}{2m+2}$ for which $E(x_{q_{n}})$ is a Cantor set of zero Lebesgue measure.", "The above description of the structure of $E(x_{q})$ can be presented as follows: 0$\\mathcal {C}_0$$\\frac{1}{2m+2}$$\\mathcal {MC}$$\\lambda ^+$$\\frac{1}{2m}$$\\mathcal {MC}$$\\frac{2}{2m+5}$$\\mathcal {I}$1where $\\mathcal {C}_{0}$ (resp.", "$\\mathcal {MC}$ , $\\mathcal {I}$ ) indicates sets of numbers $q$ for which the set $E(x_{q})$ is a Cantor set of zero Lebesgue measure (resp.", "a Cantorval, an interval).", "The symbol $\\lambda ^{+}$ indicates that for almost all $q$ in a given interval the sets $E(x_{q})$ have positive Lebesgue measure, which means that the set $Z=\\lbrace q\\in \\big (\\frac{1}{2m+2},\\frac{1}{2m}\\big ):\\lambda (E(x_{q}))=0\\rbrace $ has Lebesgue measure $\\lambda (Z)=0$ .", "Similar diagrams we use later in this paper.", "The achievement sets of multigeometric sequences are partial cases of self-similar sets of the form $K(\\Sigma ;q)=\\Big \\lbrace \\sum _{n=0}^\\infty a_nq^n:(a_n)_{n=0}^\\infty \\in \\Sigma ^\\omega \\Big \\rbrace $ where $\\Sigma \\subset \\mathbb {R}$ is a set of real numbers and $q\\in (0,1)$ .", "The set $K(\\Sigma ;q)$ is self-similar in the sense that $K(\\Sigma ;q)=\\Sigma +q\\cdot K(\\Sigma ;q)$ .", "Moreover, the set $K(\\Sigma ;q)$ can be found as a unique compact solution $K\\subset \\mathbb {R}$ of the equation $K=\\Sigma +qK$ .", "It follows that for a multigeometric sequence $x_{q}=(k_{0},\\dots ,k_{m};q)$ the achievement set $E(x)$ coincides with the self-similar set $K(\\Sigma ;q)$ for the set $\\Sigma =\\Big \\lbrace \\sum _{n=0}^{m}k_{n}\\varepsilon _{n}:(\\varepsilon _{n})_{n=0}^{m}\\in \\lbrace 0,1\\rbrace ^{m+1}\\Big \\rbrace $ of all possible sums of the numbers $k_{0},\\dots ,k_{m}$ .", "This makes possible to apply for studying the achievement sets $E(x_{q})$ the theory of self-similar sets developed in [8], [14] and, first of all, in [5].", "In this paper we shall describe some topological and measure properties of the self-similar sets $K(\\Sigma ;q)$ depending on the value of the similarity ratio $q\\in (0,1)$ , and shall apply the obtained result to establishing topological and measure properties of achievement sets of multigeometric progressions.", "To formulate the principal results we need to introduce some number characteristics of compact subsets $A\\subset \\mathbb {R}$ .", "Given a compact subset $A\\subset \\mathbb {R}$ containing more than one point let $\\mathrm {diam}\\,A=\\sup \\lbrace \|a-b\|:a,b\\in A\\rbrace $ be the diameter of $A$ and $\\delta (A)=\\inf \\lbrace \|a-b\|:a,b\\in A,\\;a\\ne b\\rbrace \\mbox{ and }\\Delta (A)=\\sup \\lbrace \|a-b\|:a,b\\in A,\\;(a,b)\\cap A=\\emptyset \\rbrace $ be the smallest and largest gaps in $A$ , respectively.", "Observe that $A$ is an interval (equal to $[\\min A,\\max A]$ ) if and only if $\\Delta (A)=0$ .", "Also put $I(A)=\\frac{\\Delta (A)}{\\Delta (A)+\\mathrm {diam}\\,A}\\mbox{ \\ \\ and \\ \\ }i(A)=\\inf \\lbrace I(B):B\\subset A,\\;\\;2\\le \|B\|<\\omega \\rbrace .$ In particular, given a finite subset $\\Sigma \\subset \\mathbb {R}$ of cardinality $\|\\Sigma \|\\ge 2$ , we will write it as $\\Sigma =\\lbrace \\sigma _{1},\\dots ,\\sigma _{s}\\rbrace $ for real numbers $\\sigma _{1}<\\dots <\\sigma _{s}$ .", "Then we have $\\mathrm {diam}(\\Sigma )=\\sigma _{s}-\\sigma _{1},\\;\\;\\delta (\\Sigma )=\\min _{i<s}(\\sigma _{i+1}-\\sigma _{i}), \\mbox{ \\ and \\ }\\Delta (\\Sigma )=\\max _{i<s}(\\sigma _{i+1}-\\sigma _{i}).$ Theorem 1.3 Let $\\Sigma =\\lbrace \\sigma _1,\\dots ,\\sigma _s\\rbrace $ for some real numbers $\\sigma _1<\\dots <\\sigma _s$ .", "The self-similar sets $K(\\Sigma ;q)$ where $q\\in (0,1)$ have the following properties: $K(\\Sigma ;q)$ is an interval if and only if $q\\ge I(\\Sigma )$ ; $K(\\Sigma ;q)$ is not a finite union of intervals if $q<I(\\Sigma )$ and $\\Delta (\\Sigma )\\in \\lbrace \\sigma _2-\\sigma _1,\\sigma _s-\\sigma _{s-1}\\rbrace $ ; $K(\\Sigma ;q)$ contains an interval if $q\\ge i(\\Sigma )$ ; If $d=\\frac{\\delta (\\Sigma )}{\\mathrm {diam}(\\Sigma )}<\\frac{1}{3+2\\sqrt{2}}$ and $\\frac{1}{\|\\Sigma \|}<\\frac{\\sqrt{d}}{1+\\sqrt{d}}$ , then for almost all $q\\in \\big (\\frac{1}{\|\\Sigma \|},\\frac{\\sqrt{d}}{1+\\sqrt{d}}\\big )$ the set $K(\\Sigma ;q)$ has positive Lebesgue measure and the set $K(\\Sigma ;\\sqrt{q})$ contains an interval; $K(\\Sigma ;q)$ is a Cantor set of zero Lebesgue measure if $q<\\frac{1}{\|\\Sigma \|}$ or, more generally, if $q^n<\\frac{1}{\|\\Sigma _n\|}$ for some $n\\in \\mathbb {N}$ where $\\Sigma _n=\\big \\lbrace \\sum _{k=0}^{n-1}a_kq^k:(a_k)_{k=0}^{n-1}\\in \\Sigma ^n\\big \\rbrace $ .", "If $\\Sigma \\supset \\lbrace a,a+1,b+1,c+1,b+\|\\Sigma \|,c+\|\\Sigma \|\\rbrace $ for some real numbers $a,b,c\\in \\mathbb {R}$ with $b\\ne c$ , then there is a strictly decreasing sequence $(q_n)_{n\\in \\omega }$ with $\\lim _{n\\rightarrow \\infty }q_n=\\frac{1}{\|\\Sigma \|}$ such that the sets $K(\\Sigma ;q_n)$ has Lebesgue mesure zero.", "The statements (1)–(3) from this theorem will be proved in Section , the statement (4) in Section and (5),(6) in Section .", "Writing that for almost all $q$ in an interval $(a,b)$ some property $\\mathcal {P}(q)$ holds we have in mind that the set $Z=\\lbrace q\\in (a,b):\\mathcal {P}(q)$ does not hold$\\rbrace $ has Lebesgue measure $\\lambda (Z)=0$ ." ], [ "Intervals and Cantorvals", "In this section we generalize results of [3] detecting the self-similar sets $K(\\Sigma ;q)$ which are intervals or Cantorvals.", "In the following theorem we prove the statements (1)–(3) of Theorem REF .", "Theorem 2.1 Let $q\\in (0,1)$ and $\\Sigma =\\lbrace \\sigma _1,\\dots ,\\sigma _s\\rbrace \\subset \\mathbb {R}$ be a finite set with $\\sigma _1<\\dots <\\sigma _s$ .", "The self-similar set $K(\\Sigma ;q)=\\big \\lbrace \\sum _{i=0}^{\\infty }a_{i}q^{i}:(a_{i})_{i\\in \\omega }\\in \\Sigma ^{\\omega }\\big \\rbrace $ is an interval if and only if $q\\ge I(\\Sigma )$ ; contains an interval if $q\\ge i(\\Sigma )$ ; is not a finite union of intervals if $q<I(\\Sigma )$ and $\\Delta (\\Sigma )\\in \\lbrace \\sigma _2-\\sigma _1,\\sigma _s-\\sigma _{s-1}\\rbrace $ .", "1.", "Observe that $\\mathrm {diam} K(\\Sigma ;q)=\\mathrm {diam}(\\Sigma )/(1-q)$ .", "Assuming that $q\\ge I(\\Sigma )=\\Delta (\\Sigma )/(\\Delta (\\Sigma )+\\mathrm {diam}\\Sigma )$ , we conclude that $\\Delta (\\Sigma )\\le q\\cdot \\mathrm {diam}(\\Sigma )/(1-q)=q\\cdot \\mathrm {diam} K(\\Sigma ;q)$ , which implies that $\\Delta (K(\\Sigma ;q))=\\Delta (\\Sigma +q\\cdot K(\\Sigma ;q))\\le \\Delta (q\\cdot K(\\Sigma ;q))=q\\cdot \\Delta (K,\\Sigma ;q).$ Since $q<1$ this inequality is possible only in case $\\Delta (K(\\Sigma ;q))=0$ , which means that $K(\\Sigma ;q)$ is an interval.", "If $q<\\Delta (\\Sigma )/(\\Delta (\\Sigma )+\\mathrm {diam}\\Sigma )$ , then $\\Delta (\\Sigma )>q\\cdot \\mathrm {diam}(\\Sigma )/(1-q)=q\\cdot \\mathrm {diam}(K(\\Sigma ;q))$ and we can find two consequtive points $a<b$ in $\\Sigma $ with $b=a+\\Delta (\\Sigma )>a+\\mathrm {diam}(q K(\\Sigma ;q))$ and conclude that $[a,b]\\cap K(\\Sigma ;q)=[a,b]\\cap (\\Sigma +qK(\\Sigma ;q))\\subset [a,a+\\mathrm {diam}(q\\,K(\\Sigma ;q))]\\ne [a,b]$ , so $K( \\Sigma ;q) $ is not an interval.", "2.", "Now assume that $q\\ge i(\\Sigma )$ and find a subset $B\\subset \\Sigma $ such that $I(B)=i(\\Sigma )<q$ .", "By the preceding item, the self-similar set $K(B;q)=B+q K(B;q)$ is an interval.", "Consequently, $K(\\Sigma ;q)$ contains the interval $K(B;q)$ .", "3.", "Finally assume that $\\Delta (\\Sigma )=\\sigma _2-\\sigma _1$ and $q<I(\\Sigma )$ .", "Since for every $a\\in \\Sigma $ we get $K(\\Sigma -a;q)=K(\\Sigma ;q)-\\frac{a}{1-q}$ , we can replace $\\Sigma $ by its shift and assume that $\\sigma _1=0$ and hence $\\Delta (\\Sigma )=\\sigma _2-\\sigma _1=\\sigma _2$ .", "It follows from $q<I(\\Sigma )=\\sigma _2/(\\sigma _2+\\mathrm {diam} \\Sigma )$ that for any $j\\in \\mathbb {N}$ , the interval $\\big ( \\sum _{n=j+1}^{\\infty }q^{n}\\sigma _{s},q^{j}\\sigma _{2}\\big ) $ is nonempty and disjoint from $K\\left( \\Sigma ;q\\right) $ .", "Hence, no interval of the form $\\left[ 0,\\varepsilon \\right] $ is included in $K\\left( \\Sigma ;q\\right) $ .", "But $0\\in K\\left( \\Sigma ;q\\right) $ , so $K\\left( \\Sigma ;q\\right) $ is not a finite union of closed intervals.", "By analogy we can consider the case $\\Delta (\\Sigma )=\\sigma _s-\\sigma _{s-1}$ .", "In particular, Theorem REF implies: Corollary 2.2 For $\\Sigma =\\lbrace 0,1,2,\\dots ,s-1\\rbrace $ the set $K\\left( \\Sigma ;q\\right) $ is an interval if and only if $q\\ge I(\\Sigma )=\\frac{1}{\|\\Sigma \|}$ .", "Corollary 2.3 If $\\lbrace k,k+1,\\dots ,k+n-1\\rbrace \\subset \\Sigma $ , then $i(\\Sigma )\\le \\frac{1}{n}$ and for every $q\\ge \\frac{1}{n}$ the set $K(\\Sigma ;q)$ contains an interval.", "In particular, for the Guthrie-Nymann-Jones multigeometric sequence $x_{q}=(3,2,\\dots ,2;q)$ of rank $m$ the sumset $\\Sigma =\\lbrace 0,2,\\dots ,2m+1,2m+3\\rbrace $ has cardinality $\|\\Sigma \|=2m+2$ , $I(\\Sigma )=\\frac{\\Delta (\\Sigma )}{\\Delta (\\Sigma )+\\mathrm {diam}\\Sigma }=\\frac{2}{2m+5}$ , $i(\\Sigma )=\\min \\big \\lbrace \\frac{1}{2m},\\frac{2}{2m+5}\\big \\rbrace $ , and $d=\\frac{\\delta (\\Sigma )}{\\mathrm {diam}(\\Sigma )}=\\frac{1}{2m+3}$ .", "So, for $q\\in \\big [\\frac{2}{2m+5},1\\big )$ the set $E(x_{q})=K(\\Sigma ;q)$ is an interval and for $q\\in \\big [\\frac{1}{2m},\\frac{2}{2m+5}\\big )$ a Cantorval." ], [ "Sets of positive measure", "In this section we shall prove the statement (4) of Theorem REF detecting numbers $q$ for which the self-similar set $K(\\Sigma ;q)$ has positive Lebesgue measure $\\lambda (K(\\Sigma ;q))$ .", "For this we shall apply the deep results of Boris Solomyak [15] related to the distribution of the random series $\\sum _{n=0}^{\\infty }a_n\\lambda ^{n},$ where the coefficients $a_n\\in \\Sigma $ are chosen independently with probability $\\frac{1}{\|\\Sigma \|}$ each.", "Given a finite subset $\\Sigma \\subset \\mathbb {R}$ consider the number $\\alpha (\\Sigma )=\\inf \\big \\lbrace x\\in (0,1):\\exists (a_n)_{n\\in \\omega }\\in (\\Sigma -\\Sigma )^\\omega \\setminus \\lbrace 0\\rbrace ^\\omega \\mbox{ suchthat }\\sum _{n=0}^\\infty a_nx^n=0\\mbox{ and }\\sum _{n=1}^\\infty na_nx^{n-1}=0\\big \\rbrace .$ The first part of the following theorem was proved by Solomyak in [15]: Theorem 3.1 Let $\\Sigma \\subset \\mathbb {R}$ be a finite subset.", "If $\\frac{1}{\|\\Sigma \|}<\\alpha (\\Sigma )$ , then for almost all $q$ in the interval $\\big (\\frac{1}{\|\\Sigma \|},\\alpha (\\Sigma )\\big )$ the self-similar set $K(\\Sigma ;q)$ has positive Lebesgue measure and the set $K(\\Sigma ;\\sqrt{q})$ contains an interval.", "By Theorem 1.2 of [15], for almost all $q\\in \\big (\\frac{1}{\|\\Sigma \|},\\alpha (\\Sigma )\\big )$ the self-similar set $K(\\Sigma ;q)$ has positive Lebesgue measure.", "Since $K(\\Sigma ;\\sqrt{q})=K(\\Sigma ;q)+\\sqrt{q}\\cdot K(\\Sigma ;q)$ , the set $K(\\Sigma ;q)$ contains an interval, being the sum of two sets of positive Lebesque measure (according to the famous Steinhaus Theorem [16]).", "The definition of Solomyak's constant $\\alpha (\\Sigma )$ does not suggest any efficient way of its calculation.", "In [15] Solomyak found an efficient lower bound on $\\alpha (\\Sigma )$ based on the notion of a $()$ -function, i.e., a function of the form $g(x)=-\\sum _{k=1}^{n-1}x^k+\\gamma x^n+\\sum _{k=n+1}^\\infty x^k$ for some $n\\in \\mathbb {N}$ and $\\gamma \\in [-1,1]$ .", "In Lemma 3.1 [15] Solomyak proved that every $()$ -function $g(x)$ has a unique critical point on $[0,1)$ at which $g$ takes its minimal value.", "Moreover, for every $d>0$ there is a unique $()$ -function $g_d(x)$ such that $\\min _{[0,1)}g_d=-d$ .", "The unique critical point $x_d\\in g_d^{-1}(-d)\\in [0,1)$ of $g_d$ will be denoted by $\\underline{\\alpha }(d)$ .", "The following lower bound on the number $\\alpha (\\Sigma )$ follows from Proposition 3.2 and inequality (15) in [15].", "Lemma 3.2 For every finite set $\\Sigma \\subset \\mathbb {R}$ of cardinality $\|\\Sigma \|\\ge 2$ we get $\\alpha (\\Sigma )\\ge \\underline{\\alpha }(d)\\mbox{ \\ where \\ }d=\\frac{\\delta (\\Sigma )}{\\mathrm {diam}(\\Sigma )}.$ The function $\\underline{\\alpha }(d)$ can be calculated effectively (at least for $d\\le \\frac{1}{2}$ ).", "Lemma 3.3 If $0<d\\le \\frac{1}{3+2\\sqrt{2}}$ , then $\\underline{\\alpha }(d)=\\frac{\\sqrt{d}}{1+\\sqrt{d}}.$ Observe that the minimal value of the $()$ -function $g(x)=-x+\\sum _{k=2}^\\infty x^k=-x+\\frac{x^2}{1-x}$ is equal to $-\\frac{1}{3+2\\sqrt{2}}$ , which implies that for $d\\in \\big (0,\\frac{1}{3+2\\sqrt{2}}\\big ]$ the number $\\underline{\\alpha }(d)$ is equal to the critical point of the unique $()$ -function $g(x)=\\gamma x+\\sum _{k=2}^\\infty x^k=-1+(\\gamma -1)x+\\frac{1}{1-x}$ with $\\min _{[0,1)}g=-d$ .", "This $()$ -function has derivative $g^{\\prime }(x)=(\\gamma -1)+\\frac{1}{(1-x)^2}$ .", "If $x$ is the critical point of $g $ , then $1-\\gamma =\\frac{1}{(1-x)^2}$ and the equality $d=-1+(\\gamma -1)x+\\frac{1}{1-x}=-1-\\frac{x}{(1-x)^2}+\\frac{1}{1-x}$ has the solution $x=1-\\frac{1}{1+\\sqrt{d}}=\\frac{\\sqrt{d}}{1+\\sqrt{d}}$ which is equal to $\\underline{\\alpha }(d)$ .", "For $d>\\frac{1}{3+2\\sqrt{2}}$ the formula for $\\underline{\\alpha }(d)$ is more complex.", "Lemma 3.4 If $\\frac{1}{3+2\\sqrt{2}}\\le d\\le \\frac{1}{2}$ , then the value $\\underline{\\alpha }(d)=\\frac{1+d}{3}+\\frac{\\@root 3 \\of {2}\\cdot R}{6}+\\frac{2d^2-8d-1}{3\\@root 3 \\of {2}\\cdot R}$ where $R=\\@root 3 \\of {4d^3-24d^2+21d-5+3\\sqrt{3}\\sqrt{1-8d^3+39d^2-6d}}$ can be found as the unique real solution of the qubic equation $2(x-1)^3+(4-2d)(x-1)^2+3(x-1)+1=0.$ Since the minimal values of the $()$ -functions $g_1(x)=-x+\\sum _{k=2}^\\infty x^k$ and $g(x)=-x-x^2+\\sum _{k=3}^\\infty x^k$ are equal to $-\\frac{1}{3+2\\sqrt{2}}$ and $-\\frac{1}{2}$ , respectively, for $d\\in \\big [\\frac{1}{3+2\\sqrt{2}},\\frac{1}{2}\\big ]$ the number $\\underline{\\alpha }(d)$ is equal to the critical point of a unique $()$ -function $g(x)=-x+\\gamma x^2+\\sum _{k=3}^\\infty x^k=-1-2x+(\\gamma -1)x^2+\\frac{1}{1-x}$ with $\\min _{[0,1)}g=-d$ .", "At the critical point $x$ the derivative of $g$ equals zero: $0=g^{\\prime }(x)=-2+2(\\gamma -1)x+\\frac{1}{(1-x)^2}$ which implies that $\\gamma -1=\\frac{1}{2x}\\Big (2-\\frac{1}{(1-x)^2}\\Big )=\\frac{2x^2-4x+1}{2x(1-x)^2}.$ After substitution of $\\gamma -1$ to the formula of the function $g(x)$ , we get $-d=-1-2x-\\frac{2x^3-4x^2+x}{2(1-x)^2}+\\frac{1}{1-x}.$ This equation is equivalent to the qubic equation $2(x-1)^3+(4-2d)(x-1)^2+3(x-1)+1=0.$ Solving this equation with the Cardano formulas we can get the solution $\\underline{\\alpha }(d)$ written in the lemma.", "Remark 3.5 Calculating the value $\\underline{\\alpha }(d)$ for some concrete numbers $d$ , we get $\\underline{\\alpha }(\\tfrac{1}{5})\\approx 0.32482,\\;\\;\\underline{\\alpha }(\\tfrac{1}{4})\\approx 0.37097, \\; \\underline{\\alpha }(\\tfrac{1}{3})\\approx 0.42773,\\;\\underline{\\alpha }(\\tfrac{1}{2})=0.5 .$ Theorem REF and Lemma REF imply: Corollary 3.6 Let $\\Sigma \\subset \\mathbb {R}$ be a finite subset containing more than three points and $d=\\delta (\\Sigma )/\\mathrm {diam}(\\Sigma )$ .", "If $d\\le \\frac{1}{3+2\\sqrt{2}}$ and $\\frac{\\sqrt{d}}{1+\\sqrt{d}}>\\frac{1}{\|\\Sigma \|}$ , then for almost all $q$ in the interval $\\big (\\frac{1}{\|\\Sigma \|},\\frac{\\sqrt{d}}{1+\\sqrt{d}}\\big )$ the self-similar set $K(\\Sigma ;q)$ has positive Lebesgue measure and the set $K(\\Sigma ;\\sqrt{q})$ contains an interval.", "Remark 3.7 Theorem REF says that for $q\\in [i(\\Sigma ),1)$ the set $K(\\Sigma ;q)$ contains an interval.", "By Theorem REF under certain conditions the same is true for almost all $q\\in \\big [\\frac{1}{\\sqrt{\|\\Sigma \|}},\\sqrt{\\alpha (\\Sigma )}\\big )$ .", "Let us remark that the numbers $i(\\Sigma ) $ and $\\frac{1}{\\sqrt{\|\\Sigma \|}}$ are incomparable in general.", "Indeed, for the multigeometric sequence $(1,\\dots ,1;q)$ containing $k>1$ units the set $\\Sigma =\\lbrace 0,\\dots ,k\\rbrace $ has $i(\\Sigma )=I(\\Sigma )=\\frac{1}{k+1}=\\frac{1}{\|\\Sigma \|}<\\frac{1}{\\sqrt{\|\\Sigma \|}}.$ On the other hand, for the multigeometric sequence $(3^{k-1},3^{k-2},\\dots ,3,1;q)$ the set $\\Sigma =\\lbrace \\sum _{n=0}^{k-1}3^{n}\\varepsilon _{n}:(\\varepsilon _{n})_{n<k}\\in \\lbrace 0,1\\rbrace ^{k}\\rbrace $ has cardinality $\|\\Sigma \|=2^{k}$ , diameter $\\mathrm {diam}(\\Sigma )=(3^{k}-1)/2$ , $d=\\frac{\\delta (\\Sigma )}{\\mathrm {diam}(\\Sigma )}=\\frac{2}{3^{k}-1}$ and $i(\\Sigma )=I(\\Sigma )=\\frac{1}{4}+\\frac{1}{4\\cdot 3^{k-1}}>\\frac{1}{\\sqrt{2}^{k}}=\\frac{1}{\\sqrt{\|\\Sigma \|}}$ .", "Corollary REF guarantees that for almost all $q\\in \\big (\\frac{1}{\\sqrt{2}^{k}},\\frac{\\@root 4 \\of {d}}{\\sqrt{1+\\sqrt{d}}}\\big )$ the set $K(\\Sigma ;q)$ contains an interval.", "Multigeometric sequences of the form $\\left( k+m,\\dots ,k+1,k;q\\right)$ with $m\\ge k$ we will call, after [2], Ferens-like sequences.", "The achievement set $E\\left( x\\right) $ for a Ferens-like sequence coincides with the self-similar set $K(\\Sigma ;q)$ for the set $\\Sigma =\\lbrace 0,k,k+1,\\dots ,n-k,n\\rbrace \\text{.", "}$ where $n=(m+1)(2k+m)/2$ .", "Sets $K\\left( \\Sigma ;q\\right) $ with $\\Sigma $ of this form will be called Ferens-like fractals.", "Note that Guthrie-Nymann-Jones sequence of rank $m$ generates a Ferens-like fractal (with $\\Sigma =\\lbrace 0,2,3,\\dots ,2m+1,2m+3\\rbrace $ .", "There are also Ferens-like fractals which are not originated by any multigeometric sequence (for example $K(\\Sigma ;q)$ with $\\Sigma =\\left\\lbrace 0,4,5,6,7,11\\right\\rbrace $ ).", "However, as an easy consequence of the main theorem of [12], we obtain for Ferens-like fractals “trichotomy\" analogous to that formulated in Theorem REF .", "Moreover, some theorems formulated for multigeometric sequences are in fact proved for $K(\\Sigma ;q)$ (see for example Theorem 2 in [3]).", "Example 3.8 For the Ferens-like sequence $x_{q}=(4,3,2;q)$ we get $\\Sigma =\\lbrace 0,2,3,4,5,6,7,9\\rbrace $ , $d=\\frac{\\delta (\\Sigma )}{\\mathrm {diam}(\\Sigma )}=\\frac{1}{9}<\\frac{1}{3+2\\sqrt{2}}\\mbox{ \\ and \\ }\\frac{\\sqrt{d}}{1+\\sqrt{d}}=\\frac{1}{4}>\\frac{1}{6}=i(\\Sigma ).$ By Corollary REF (and Theorem REF ), for almost all numbers $q\\in \\big (\\frac{1}{8},1\\big )$ the achievement set $E(x_{q})=K(\\Sigma ;q)$ has positive Lebesgue measure (for $q<\\frac{2}{11}=I(\\Sigma )$ it is not a finite union of intervals).", "By Theorem REF , for any $q\\in [i(\\Sigma ),I(\\Sigma ))=[\\frac{1}{6};\\frac{2}{11})$ the set $K(\\Sigma ;q)$ is a Cantorval.", "The structure of the sets $E(x_{q})=K(\\Sigma ;q)$ is described in the diagram: $\\mathcal {C}_0$$\\frac{1}{8}$$\\lambda ^+$$\\frac{1}{6}$$\\mathcal {MC}$$\\frac{2}{11}$$\\mathcal {I}$More generally, for any Ferens-like fractal, $\|\\Sigma \|=n-2k+3$ , $\\Delta (\\Sigma )=k$ , $\\delta \\left( \\Sigma \\right) =1$ , $I(\\Sigma )=\\frac{k}{n+k}$ , $i(\\Sigma )=\\min \\big (\\frac{1}{\\left\|\\Sigma \\right\|-2},I(\\Sigma )\\big )$ and $d=\\frac{1}{n}$ .", "Moreover, if $n\\ge 7$ then $\\underline{\\alpha }(d)=\\frac{1}{\\sqrt{n}+1}$ .", "Therefore, one can check that for any Ferens-like sequence we have $\\underline{\\alpha }(d)>i(\\Sigma )$ , and we can draw an analogous diagram.", "The same result we can obtain for any Ferens-like fractal with $k=2$ (even if it is not originated by any Ferens-like sequence).", "However, there are Ferens-like fractals with $\\underline{\\alpha }(d)<i(\\Sigma )$ (for example $K(\\Sigma ;q)$ with $\\Sigma =\\left\\lbrace 0,3,4,7\\right\\rbrace $ or $\\Sigma =\\left\\lbrace 0,4,5,6,7,11\\right\\rbrace $ ).", "Example 3.9 For the Guthrie-Nymann-Jones sequence $x_{q}=(3,2,\\dots ,2;q)$ of rank $m\\ge 2$ we get $\\Sigma =\\lbrace 0,2,3,\\dots ,2m+1,2m+3\\rbrace $ , $\|\\Sigma \|=2m+2$ , $I(\\Sigma )=\\frac{2}{2m+5}$ , $i(\\Sigma )=\\min \\big \\lbrace \\frac{1}{2m},\\frac{2}{2m+5}\\big \\rbrace $ , $d=\\frac{1}{2m+3}$ and $\\underline{\\alpha }(d)=1/(1+\\sqrt{2m+3})$ .", "Moreover, we have $d<\\frac{1}{3+2\\sqrt{2}}$ and $\\underline{\\alpha }(d)\\ge i(\\Sigma )>\\frac{1}{2m+2}=\\frac{1}{\|\\Sigma \|}$ .", "So, we can apply Corollary REF and conclude that for almost all numbers $q\\in \\big (\\frac{1}{2m+2},\\frac{1}{2m}\\big )$ the self-similar set $K(\\Sigma ;q)$ has positive Lebesgue measure.", "By Theorem REF , for any $q\\in [i(\\Sigma ),\\frac{2}{2m+5})$ the set $K(\\Sigma ;q)$ is a Cantorval and for all $q\\in [\\frac{2}{2m+5},1)$ it is an interval.", "For $m=1$ we obtain $\\underline{\\alpha }(d)=\\underline{\\alpha }(\\frac{1}{5})>\\frac{2}{7}$ .", "Therefore, for almost all numbers $q\\in \\big (\\frac{1}{4},\\frac{2}{7}\\big )$ the set $K(\\Sigma ;q)$ has positive Lebesgue measure." ], [ "Self-similar sets of zero Lebesgue measure", "The results of the preceding section yields conditions under which for almost all $q$ in an interval $\\big [\\frac{1}{\|\\Sigma \|},\\alpha (\\Sigma )\\big )$ the set $K( \\Sigma ;q) $ has positive Lebesgue measure.", "In this section we shall show that this interval can contain infinitely many numbers $q$ with $\\lambda (K(\\Sigma ;q)) =0$ thus proving the statements (5) and (6) of Theorem REF .", "Theorem 4.1 If there exists $n\\in \\mathbb {N}$ such that $\\Big \\vert \\sum _{i=0}^{n-1}q^{i}\\Sigma \\Big \\vert \\cdot q^{n}<1$ then the set $K(\\Sigma ,q)$ has measure zero.", "Denote $K:=K(\\Sigma ,q)$ .", "From the equality $K=\\Sigma +qK$ we obtain, by induction, that $K=\\sum _{i=0}^{n-1}q^{i}\\Sigma +q^{n}K.$ Let $\\Sigma _{n}=\\sum _{i=0}^{n-1}q^{i}\\Sigma $ .", "If $\|\\Sigma _{n}\|\\cdot q^{n}<1$ , then $\\lambda (K)\\le \|\\Sigma _{n}\|\\cdot q^{n}\\cdot \\lambda (K)<1\\cdot \\lambda (K)$ which is possible only if $\\lambda (K)=0$ .", "To use the latter theorem we need a technical lemma: Lemma 4.2 For any integer numbers $s>1$ and $n>1$ the unique positive solution $q$ of the equation $x+x^{2}+\\dots +x^{n-1}=\\frac{1}{s-1} $ is greater than $\\frac{1}{s}$ .", "Moreover, there is $n_{0}\\in \\mathbb {N}$ such that for any $n>n_{0}$ $\\left( s^{n}-2^{n-1}\\right) \\cdot q^{n}<1\\text{.}", "$ Clearly $\\sum _{i=1}^{n-1}\\left( \\frac{1}{s}\\right) ^{i}=\\frac{1}{s-1}\\cdot \\left( 1-\\frac{1}{s^{n-1}}\\right) <\\frac{1}{s-1}\\text{,}$ so $q>\\frac{1}{s}$ .", "From the equality $\\frac{1}{s-1}=\\sum _{i=1}^{n-2}\\left( \\frac{1}{s}\\right) ^{i}+\\frac{1}{\\left(s-1\\right) s^{n-2}}$ we obtain $q^{n-1}=\\frac{1}{s-1}-\\sum _{i=1}^{n-2}q^{i}<\\frac{1}{s-1}-\\sum _{i=1}^{n-2}\\left( \\frac{1}{s}\\right) ^{i}=\\frac{1}{\\left( s-1\\right) s^{n-2}}\\text{.", "}$ Using the latter inequality and the equality $\\frac{1}{s-1}=\\frac{q-q^{n}}{1-q}$ we have $\\frac{1-q}{s-1}=q\\left( 1-q^{n-1}\\right) >q\\left( 1-\\frac{1}{\\left(s-1\\right) s^{n-2}}\\right) \\text{.", "}$ Therefore, $1-q>\\left( s-1\\right) q-\\frac{q}{s^{n-2}}$ (which means that $sq-\\frac{q}{s^{n-2}}<1$ ) and finally $q<\\frac{1}{s\\left( 1-\\frac{1}{s^{n-1}}\\right) }\\text{.}", "$ From Bernoulli's inequality it follows that $\\left( 1-\\frac{1}{s^{n-1}}\\right) ^{n}\\ge 1-\\frac{n}{s^{n-1}}$ and, by (REF ), we have $q^{n}<\\frac{1}{s^{n}\\cdot \\left( 1-\\frac{n}{s^{n-1}}\\right) }\\text{.", "}$ Consequently, $\\left( s^{n}-2^{n-1}\\right) \\cdot q^{n}<\\frac{s^{n}\\cdot \\left( 1-\\frac{2^{n-1}}{s^{n}}\\right) }{s^{n}\\cdot \\left( 1-\\frac{n}{s^{n-1}}\\right) }$ Obviously, for $n$ greater then some $n_{0}$ $\\frac{2^{n-1}}{s}>n$ and hence $\\frac{2^{n-1}}{s^{n}}>\\frac{n}{s^{n-1}}$ which proves (REF ).", "Theorem 4.3 If a finite subset $\\Sigma \\subset \\mathbb {R}$ contains the set $\\lbrace a,a+1,b+1,c+1,b+\|\\Sigma \|,c+\|\\Sigma \|\\rbrace $ for some real numbers $a,b,c$ with $b\\ne c$ , then there is a decreasing sequence $(q_{n})_{n=1}^{\\infty }$ tending to $\\frac{1}{\|\\Sigma \|}$ such that, for any $n\\in \\mathbb {N}$ , the self-similar set $K(\\Sigma ,q_{n})$ has Lebesgue measure zero.", "Let $s=\|\\Sigma \|$ and for every $n$ denote by $q_{n}$ the unique positive solution of the equation (REF ) from Lemma REF .", "Let $n_{0}$ be a natural number such that $\\left( s^{n}-2^{n-1}\\right) \\cdot \\left( q_{n}\\right) ^{n}<1$ for any $n>n_{0}$ .", "Clearly $(q_{n}) _{n=n_{0}}^{\\infty }$ is a decreasing sequence and $\\lim _{n\\rightarrow \\infty }q_{n}=\\frac{1}{s}$ .", "It suffices to show that $K(\\Sigma ,q)$ has measure zero for $n>n_{0}$ .", "Taking into account that each $q_{n}$ is a solution of (REF ), we conclude that $a+\\sum _{i=1}^{n-1}(s-1+\\varepsilon _{i})( q_{n})^{i}=(a+1)+\\sum _{i=1}^{n-1}\\varepsilon _{i}( q_{n}) ^{i}$ for any $\\varepsilon _{i}\\in \\lbrace b+1,c+1\\rbrace \\subset \\Sigma $ .", "Therefore $\\left\|\\sum _{i=1}^{n-1}\\left( q_{n}\\right) ^{i}\\Sigma \\right\|\\le s^{n}-2^{n-1}\\text{.", "}$ Hence, by Lemma REF , $\\left\|\\sum _{i=1}^{n-1}\\left( q_{n}\\right) ^{i}\\Sigma \\right\|\\cdot \\left( q_{n}\\right) ^{n}<1.$ and we can apply Theorem REF to conclude that $K(\\Sigma ,q)$ has Lebesgue measure zero.", "The condition $\\big \\lbrace a,a+1,b+1,c+1,b+\|\\Sigma \|,c+\|\\Sigma \|\\big \\rbrace \\subset \\Sigma \\qquad \\mathrm {(*)}$ looks a bit artificial but it can be easily verified for many sumsets $\\Sigma $ of multigeometric sequences.", "In particular, for the Guthrie-Nymann-Jones sequence of rank $m\\ge 1$ $x_{q}=(3,2,\\dots ,2;q),$ the sumset $\\Sigma =\\lbrace 0,2,3,\\dots ,2m+1,2m+3\\rbrace $ has cardinality $\|\\Sigma \|=2m+2$ .", "Observe that for the set $\\Sigma $ the condition $(\\ast )$ holds for $a=2$ , $b=1$ and $c=-1$ .", "Because of that Theorem REF yields a sequence $(q_{n})_{n=1}^{\\infty }\\searrow \\frac{1}{2m+2}$ such that for every $n\\in \\mathbb {N}$ the self-similar set $E(x_{q_{n}})$ is a Cantor sets of zero Lebesgue measure.", "By [3], for $q=\\frac{1}{2m+2}$ the achievement set $E(x_{q})$ is a Cantorval.", "Therefore, if $m>2$ , there are three ratios $p<q<r$ such that $E(x_{p})$ and $E(x_{r})$ are Cantor sets while $E(x_{q})$ is a Cantorval.", "By our best knowledge it is the first result of this type for multigeometric sequences.", "Now we will focus on Ferens-like sequences $x_{q}=(m+k,\\dots ,k;q)$ where $m\\ge k$ .", "For $k=1$ the Ferens-like sequence $x_{q}=(m+1,\\dots ,2,1;q)$ has $\\Sigma =\\big \\lbrace 0,1,2,\\dots ,(m+2)\\left( m+1\\right) /2\\big \\rbrace \\text{.", "}$ The set $E(x_{q})$ is a Cantor set (for $q<\\frac{1}{\|\\Sigma \|}$ ) or an interval (for $q\\ge \\frac{1}{\|\\Sigma \|}$ ); see Theorem 7 in [3]), Theorem REF or Theorem REF .", "For $k=2$ , the “shortest\" Ferens-like sequence is $x_{q}=(4,3,2;q)$ .", "For this sequence $\\Sigma =\\left\\lbrace 0,2,3,4,5,6,7,9\\right\\rbrace \\text{.", "}$ Note that the same $\\Sigma $ has Guthrie-Nymann-Jones sequence $(3,2,2,2;q)$ (see Example REF ).", "It follows that $E(x_{q})$ is a Cantor set for $q\\in \\big (0,\\frac{1}{8}\\big )$ and $E(x_{q})$ is a Cantorval for $q=\\frac{1}{8}$ .", "By Theorem REF , $K(\\Sigma ;q)$ is an interval for $q\\ge I(\\Sigma )=\\frac{2}{11}$ and a Cantorval for $q\\in \\big (\\frac{1}{6},\\frac{2}{11}\\big )$ .", "As shown in Example REF , for almost all $q\\in \\big (\\frac{1}{8},\\frac{1}{6}\\big )$ the set $K(\\Sigma ;q)$ has positive Lebesgue measure.", "Using Theorem REF , we can find a decreasing sequence $(q_{n})$ tending to $\\frac{1}{8}$ for which the sets $K(\\Sigma ;q_{n})$ have zero Lebesgue measure.", "For $k=3$ the “shortest\" Ferens-like sequence is $x_{q}=(6,5,4,3;q)$ .", "For this sequence $\\Sigma =\\left\\lbrace 0,3,\\dots ,15,18\\right\\rbrace $ and $\|\\Sigma \|=15$ .", "Since $1\\in \\frac{1}{15}\\Sigma $ the set $\\Sigma _{2}=\\Sigma +\\frac{1}{15}\\Sigma $ has less than $\\left\|15\\right\|^{2}$ elements (for example 4 can be presented as $4+0$ or as $3+1$ ).", "Therefore $\\frac{1}{15^{2}}\|\\Sigma _{2}\|<1$ and for $q=\\frac{1}{15}$ the set $E(x_{q})$ is a Cantor set according to Theorem REF .", "Moreover, calculating for $q=\\frac{1}{14}>\\frac{1}{15}$ the cardinality $\|\\Sigma _3\|=\|\\Sigma +q\\Sigma +q^{2}\\Sigma \|=2655<14^{3}$ and applying Theorem REF , we conclude that the achievement set $E(x_{q})$ is a Cantor set of zero Lebesgue measure for $q=\\frac{1}{14}$ .", "On the other hand, Corollary REF implies that for almost all $q\\in \\big (\\frac{1}{15},\\frac{1}{1+\\sqrt{18}})$ the achievement set $E(x_{q})$ has positive Lebesque measure.", "The set $\\Sigma $ has $i(\\Sigma )=\\frac{1}{13}$ and $I(\\Sigma )=\\frac{3}{21}=\\frac{1}{7}$ .", "So, in this case we have the diagram: $\\mathcal {C}_0$$\\frac{1}{15}$$\\lambda ^+$$\\lambda ^+$$\\mathcal {C}_0$$\\frac{1}{14}$$\\frac{1}{13}$$\\mathcal {MC}$$\\frac{1}{7}$$\\mathcal {I}$01As in the previous case, we can use Theorem REF (taking $a=b=3$ and $c=-1$ ) and find a decreasing sequence $(q_{n})$ tending to $\\frac{1}{15}$ such that all $E(x_{q_{n}})$ have zero Lebesgue measure.", "Suppose now that $k>3$ .", "For the Ferens-like sequence $x_{q}=(k+m,\\dots ,k+1,k;q)$ its sumset $\\Sigma $ contains the number $\|\\Sigma \|$ , which implies that $\|\\Sigma +q\\Sigma \|<\|\\Sigma \|^{2}$ for $q=\\frac{1}{\|\\Sigma \|}$ and therefore $E(x_{q})$ is a Cantor set of zero measure according to Theorem REF ." ], [ "Rational ratios", "For a contraction ratio $q\\in \\lbrace \\frac{1}{n+1}:n\\in \\mathbb {N}\\rbrace $ self-similar sets of positive Lebesgue measure can be characterized as follows: Theorem 5.1 Let $\\Sigma \\subset \\mathbb {Z}$ be a finite set, $q\\in \\lbrace \\frac{1}{n+1}:n\\in \\mathbb {N}\\rbrace $ and $\\Sigma _{n}=\\sum _{i=0}^{n-1}q^{i}\\Sigma $ for $n\\in \\mathbb {N}$ .", "For the compact set $K=K(\\Sigma ;q)$ the following conditions are equivalent: (i) $\|\\Sigma _n\| \\cdot q^n \\ge 1$ for all $n\\in \\mathbb {N}$ ; (ii) $\\inf _{n\\in \\mathbb {N}}\|\\Sigma _{n}\|\\cdot q^n>0$ , (iii) $\\lambda (K)>0.$ The implication (iii)$\\Rightarrow $ (i) follows from Theorem REF while (i)$\\Rightarrow $ (ii) is trivial.", "It remains to prove (ii)$\\Rightarrow $ (iii).", "Suppose that $\\lambda (K)=0$ .", "Given any $r>0$ consider the $r$ -neighborhood $H(K,r) =\\lbrace h\\in \\mathbb {R}:\\mathrm {dist}(h,K) <r\\rbrace $ of the set $K=K(\\Sigma ;q)$ .", "Take any point $z\\in \\big \\lbrace \\sum _{i=n}^\\infty x_iq^i:\\forall i\\ge n\\;x_i\\in \\Sigma \\big \\rbrace $ and observe that $\\Sigma _n+z\\subset K=\\big \\lbrace \\sum _{i=0}^\\infty x_iq^i:(x_i)_{i\\in \\omega }\\in \\Sigma ^\\omega \\big \\rbrace $ , which implies that $H(\\Sigma _{n}+z,r) \\subset H(K,r)$ for all $r>0$ .", "The continuity of the Lebesgue measure implies that $\\lambda (H(K,r))\\rightarrow 0$ when $r$ tends to zero.", "It follows from $\\Sigma \\subset \\mathbb {Z}$ and $\\frac{1}{q}\\in \\mathbb {N}$ that $\\Sigma _{n}\\subset q^{n-1}\\cdot \\mathbb {Z}\\text{.", "}$ Hence, for any two different points $x$ and $y$ from $\\Sigma _{n}$ , the distance between $x$ and $y$ is no less then $q^{n-1}>q^n$ .", "Therefore, for any $n\\in \\mathbb {N}$ , $\|\\Sigma _n\|\\cdot q^n=\\lambda \\big ( H\\big ( \\Sigma _{n},\\tfrac{1}{2}q^{n}\\big )\\big )=\\lambda \\big (H\\big (\\Sigma _n+z,\\tfrac{1}{2}q^n\\big )\\big )\\le \\lambda (K,\\tfrac{1}{2}q^n)$ which means that $\\lim _{n\\rightarrow \\infty }\|\\Sigma _{n}\|\\cdot q^n=0$ .", "Theorems REF combined with Corollary 2.3 of [14] imply the following corollary.", "Corollary 5.2 For a finite subset $\\Sigma \\subset \\mathbb {Z}$ and the number $q=\\frac{1}{\|\\Sigma \|}<1$ the following conditions are equivalent: (1) $K(\\Sigma ;q)$ has positive Lebesgue measure; (2) $K(\\Sigma ;q)$ contains an interval; (3) for every $n\\in \\mathbb {N}$ the set $\\Sigma _{n}=\\sum _{k=0}^{n-1}q^k\\Sigma $ has cardinality $\|\\Sigma _{n}\|=\|\\Sigma \|^{n}$ .", "Problem 5.3 Is it true that for a finite set $\\Sigma \\subset \\mathbb {Z}$ and any (rational) $q\\in (0,1)$ the self-similar set $K(\\Sigma ;q)$ has positive Lebesgue measure if and only if it contains an interval?", "Remark 5.4 According to [4], there exists a 10-element set $\\Sigma $ on the complex plane $\\mathbb {C}$ such that for $q=\\frac{1}{3}$ the self-similar compact set $K(\\Sigma ;q)=\\Sigma +qK(\\Sigma ;q)\\subset \\mathbb {C}$ has positive Lebesgue measure and empty interior in $\\mathbb {C}$ ." ] ]	1403.0098
[ [ "Pairing, Pseudogap and Fermi Arcs in Cuprates" ], [ "Abstract We use Angle Resolved Photoemission Spectroscopy (ARPES) to study the relationship between the pseudogap, pairing and Fermi arcs in cuprates.", "High quality data measured over a wide range of dopings reveals a consistent picture of Fermiology and pairing in these materials.", "The pseudogap is due to an ordered state that competes with superconductivity rather then preformed pairs.", "Pairing does occur below Tpair~150K and significantly above Tc, but well below T* and the doping dependence of this temperature scale is distinct from that of the pseudogap.", "The d-wave gap is present below Tpair, and its interplay with strong scattering creates \"artificial\" Fermi arcs for Tc<T<Tpair.", "However, above Tpair, the pseudogap exists only at the antipodal region.", "This leads to presence of real, gapless Fermi arcs close to the node.", "The length of these arcs remains constant up to T, where the full Fermi surface is recovered.", "We demonstrate that these findings resolve a number of seemingly contradictory scenarios." ], [ "Introduction", "The pseudogap is one of the most interesting aspects of the physics of the cuprates, second only to their unusually high transition temperatures.", "Since its discovery [1], [2], [3], [4], the origin of the pseudogap and its relationship to high temperature superconductivity has been a subject to intense debate over the last two decades[5].", "One of the most popular models is based on the existence of pre-formed pairs [6], but other models invoking the presence of an ordered state have been proposed too[7], [8], [9], [10].", "A circular dichroism in the pseudogap state was discovered by early ARPES experiments [11] and presence of staggered magnetic field was detected using high precision magneto-optical Kerr effect [12] as well as neutron scattering measurements[13].", "Early STM measurements revealed presence of checker board pattern consistent with charge ordering in the cores of the vortexes [14] then in heavily under doped samples[15] and moderately doped samples[16].", "More recently series of high precision X-ray scattering experiments detected fluctuating charge density order within pseudo gap state[17], [18], [19].", "All these findings point to the fact that pseudo gap is a manifestation of an ordered states or perhaps interplay of two ordered states involving charge and magnetic moment.", "Angle Resolved Photoemission Spectroscopy (ARPES) has played a very important role in elucidating the electronic properties of the cuprates [20], [21].", "Examples of ARPES data measured on Bi2201 samples along with a diagram of the Fermi surface are shown in Fig.", "1.", "Indeed, due to it's unique momentum resolution, ARPES was one of the techniques of choice in studies of the pseudogap [22], [23] and the one that led to discovery of Fermi arcs [24].", "The classical definition of the Fermi surface is a set of closed contours in momentum space that separates the occupied and unoccupied states.", "Fermi arcs, being disconnected and possessing by definition “end points” are a highly unusual concept in \"classical\" condensed matter physics.", "Many of the early studies of the pseudo gap and Fermi arcs relied on line shape analysis, primarily Energy Distribution Curves (EDCs).", "The signature of the pseudogap in symmetrized EDCs is shown in Fig.", "2.", "The spectra is considered as “gapped”, when a dip or flat top are present at E$_f$ .", "A single peak at E$_f$ is considered evidence for no gap.", "This conventional approach led to several important advances such as measurements of the superconducting gap anisotropy.", "Yet, at the same time, it has limitations, particularly when broad spectral peaks are present in the pseudogap state at the antinode.", "Here we revisit these issues using very high quality data and a novel quantitative analysis.", "This approach is more objective, has higher sensitivity to spectral changes and is rooted in the basic definition of the energy gap, based on the density of states.", "We demonstrate that at low temperature in optimally and under doped cuprates, two distinct gaps are present: the superconducting gap and a pseudogap.", "They represent different energy scales that evolve in distinct ways with momentum, temperature and doping.", "Furthermore, we show that the pseudogap competes with superconductivity by depleting the low energy electrons in the antinodal region, that otherwise would form pairs below T$_c$ .", "In the under doped materials therefore, only a small portion of the Fermi surface close to the nodal region contributes to the superfluid density.", "Our quantitative approach provides evidence of pairing above T$_c$ and allows us to detect the onset temperature of pairing T$_{pair}$ that is significantly lower than the pseudo gap temperature T.", "This naturally explains earlier conclusions about the extension of the d-wave gap above T$_c$ .", "Our data strongly supports theoretical proposals that Fermi arcs below T$_{pair}$ are an artifact due to the interplay between a d-wave gap and strong scattering effects [52].", "Using this quantitive approach we are also able to demonstrate that above T$_{pair}$ , real, gapless Fermi arcs exist in the nodal region and are likely due to an ordered state[38], [39], [16], [40], [15] responsible for the pseudogap." ], [ "Experiments", "Optimally doped Bi$_2$ Sr$_2$ CaCu$_2$ O$_{8+\\delta }$ (Bi2212) single crystals with $T_{\\rm c}$ =93K (OP93K) and (Bi,Pb)$_2$ (Sr,La)$_2$ CuO$_{6+\\delta }$ (Bi2201) single crystals with $T_{\\rm c}$ =32K (OP32K) were grown by the conventional floating-zone (FZ) technique.", "We partially substituted Pb for Bi in Bi2201 to suppress the modulation in the BiO plane, and avoid contamination of the ARPES signal with diffraction images due to the superlattice: the outgoing photoelectrons are diffracted at the modulated BiO layer, creating multiple images of the bands and Fermi surface that are shifted in momentum.", "The modulation-free samples enable us to precisely analyze the ARPES spectra.", "ARPES data was acquired using a laboratory-based system consisting of a Scienta SES2002 electron analyzer and GammaData helium UV lamp.", "All data were acquired using the HeI line with a photon energy of 21.2 eV.", "The angular resolution was $0.13^\\circ $ and $\\sim 0.5^\\circ $ along and perpendicular to the direction of the analyzer slits, respectively.", "The energy corresponding to the chemical potential was determined from the Fermi edge of a polycrystalline Au reference in electrical contact with the sample.", "The energy resolution was set at $\\sim 10$ meV - confirmed by measuring the energy width between 90$\\%$ and 10 $\\%$ of the Fermi edge from the same Au reference.", "Figure: a) Plot of the Brillouin zone and Fermi surface contour for Bi2201 showing the definition of the Fermi surface angle.", "b) ARPES intensity plot along the antinodal cut (marked with a red line for φ=0\\phi =0.", "c) ARPES data along the nodal direction (φ=45\\phi =45).", "The data was measured at T=11K.Custom designed refocusing optics enabled us to accumulate high statistics spectra in a short period of time with no sample surface aging from the absorption or loss of oxygen.", "Special care was taken to purify the helium gas supply for the UV source to remove even the smallest trace of contaminants that could contribute to surface contamination.", "Typically no changes in the spectral lineshape of samples were observed in consecutive measurements performed over several days.", "We constructed a sample manipulator with the tilt and azimuth motions mounted on a two stage closed cycle He refrigerator.", "To measure the partial density of states along a direction perpendicular to the FS we controlled the sample orientation in-situ.", "Specially designed temperature management system allowed us to rapidly change sample temperature.", "This was critical for obtaining high density of data points in temperature scan and further limit sample aging.", "Measurements were performed on several samples and we confirmed that all yielded consistent results." ], [ "Two energy gaps", "The issue of one vs two energy gaps is directly tied to the origin of the pseudogap.", "If the pseudogap is a state of pre-formed pairs, then naturally the energy gap arises from pairing and has the same form for all temperatures below T* as reported in number of studies [25], [26], [27], [28].", "However several other studies including Raman Spectroscopy, ARPES and STM [29], [30], [31], [32], [33] reveal certain differences in the behavior of the energy gap close to the node compared to the antipodal region.", "More precisely, the spectral gap follows the behavior expected of d-wave pairing gap only in regions of momentum space close to the node.", "In the antinodal regions there are significant deviations from d-wave behavior.", "For example, the antipodal gap magnitude continuously increases with decreased doping, despite a decrease of T$_c$ and it is almost temperature independent up to T.", "We illustrate this in Fig.", "3, where the magnitude of the “spectral gap” (we use that term to refer to the gap extracted from the data to avoid bias) is plotted for three doping levels, below and above T$_c$ .", "The dotted lines mark the expected behavior of a d-wave order parameter.", "In the overdoped sample (panel c), at low temperatures, the spectral gap follows exactly the predictions based on a d-wave order parameter.", "Above T$_c$ , the gap has a similar magnitude in the antipodal region, but vanishes near the node, unlike a d-wave gap.", "At optimal doping, the low temperature data close to the node, fits d-wave very well, but deviates strongly in the antinodal region.", "This becomes more extreme in underdoped samples (panel a).", "It is also worth noting that the maximum value of the spectral gap at the antinode for the UD23K sample is larger by a factor of 2 than that of the OP35K sample, despite an actual decrease of the superconducting critical temperature.", "These data suggest that the spectral gap in under doped cuprates has two components, namely a d-wave superconducting gap and a pseudogap.", "The two gaps have different momentum, doping and temperature dependences.", "The superconducting gap follows the doping dependence of T$_c$ , vanishes well below T and has pure d-wave momentum dependence.", "The pseudogap, in contrast, continuously increases with decreased doping, persists up to T* and only exists close to the antinode.", "Figure: Momentum dependence of the spectral gap for three doping levels measured below and above T c _c.", "The dotted line is expectation of d-wave order parameter.Perhaps the most convincing evidence for the existence of two distinct gaps in the cuprates is found by examining very carefully the temperature dependence of very high quality EDC data (Fig.", "4).", "Blue lines mark the peak positions and therefore the magnitude of the gap.", "As the temperature is increased, the gap actually increases in magnitude, only to vanish close to the T.", "This behavior is illustrated in panel b, where we plot the magnitude of the spectral gap as a function of temperature for several doping levels.", "Such a violation of monotonicity is completely inconsistent with a single gap picture and can be only explained by the two gap scenario.", "More precisely, below T$_c$ , sharp quasiparticle peaks mark the location of the d-wave pairing gap.", "As T$_c$ is approached, those peaks vanish and reveal a second gap of larger magnitude that persists up to T and it is this therefore that is the proper pseudogap.", "In heavily underdoped samples (e. g. UD25K) the pseudogap is so large, that no quasiparticle peaks are present at the antinode even at the lowest temperature.", "In overdoped samples, the pseudogap is much smaller than the pairing gap, which dominates the spectra and preserves the monotonicity.", "In the range of doping where the magnitude of the pseudogap is larger than pairing gap, but not large enough to wipe out the quasiparticle peaks, non-monotonic behavior is observed.", "Figure: Detailed temperature dependence of the spectral gap.", "a) symmetrized EDCs as a function of temperature at the antinodal crossing in optimally doped sample.", "The value of the gap extracted from the data is marked by blue lines.", "Value of the spectral gap as a function of temperature for several doping levels.", "Note that in data close to optimal doping, the gap first increases upon warming, then decreases to zero at temperatures much higher than T c _c." ], [ "Competition between the pseudogap and superconductivity", "Now that we have demonstrated the existence of two energy scales in the cuprates, a natural question arises about their mutual relationship.", "The most direct way to answer this is to look how these two energy scales affect the spectral properties.", "As we mentioned in the introduction, conventional line shape analysis is difficult to utilize here.", "Instead, we use the quantitative analysis shown in Fig.", "5ab [36].", "In the pseudogap state, the spectral weight near E$_f$ decreases upon cooling.", "The change in this spectral weight (marked in red in the insert of panel 5c) is a good measure of the “strength” of the psudogap.", "In the superconducting state, however, the formation of quasiparticle peaks occurs at the pairing gap edge (marked in blue areas in panel 5d).", "These peaks signify the contribution to a superfluid density from this region of momentum space[35].", "The temperature dependence of both quantities is shown in panels 5 c and d. The pseudogap weight decreases linearly to zero at T, while the quasiparticle weight vanishes at T$_c$ .", "To study the relationship between the two quantities we plot in Fig.", "5e the weight of the quasiparticle peaks versus pseudogap weight for momentum points, where they are non-zero.", "For all three dopings, there is a strong anti-correlation between the two quantities.", "The superconducting quasiparticle peak weight decreases as the pseudogap weight increases at all momentum points and all three dopings.", "Such a strong anti-correlation is definitive proof that the two phenomena compete for available states near E$_f$ .", "In Fig.", "5f we look at the momentum dependence of the normalized weight of the quasiparticle peaks.", "If the pseudogap really was a state of pre-formed pairs, the weight transferred from near E$_f$ , should re-condense into coherent pairs below T$_c$ , thus the quantity shown in Fig.", "5f should remain at 100% for all momentum points.", "Instead we observe a strong depletion of the quasiparticle weight in the antipodal region (where the pseudogap is strongest), that increases with decreased doping.", "In heavily underoped sample (green curve), the pseudogap completely dominates the antipodal region and there are no quasiparticle peaks, thus only areas close to the node participate in superconductivity.", "This again signifies that the pseudogap competes with the formation of coherent pairs.", "These conclusions are in good agreement with thermodynamics studies[34] and were recently confirmed by another ARPES study [37]." ], [ "Pairing above T$_c$", "Competition of the pseudogap with superconductivity does not preclude the formation of Cooper pairs above T$_c$ .", "Indeed, several thermodynamical and transport studies report signatures of pairing above T$_c$ , which do not however persist up to T [41], [42], [44], [45], [46], [48].", "We find spectroscopic evidence of pair formation and were able to establish the value of the pairing temperature using our quantitative approach[49].", "In Fig.", "6 we plot a detailed temperature dependence of the spectral weight in Bi2201 at E$_f$ (integrated within 10 meV) and antinodal crossing for several doping levels.", "In the heavily underdoped sample (panel a), where the spectrum is completely dominated by the pseudo gap (as demonstrated in the section above), this weight is linear with temperature.", "Such linear behavior is a signature of the pseudogap.", "As we increase the doping, there is a deviation from such linear form, and the weight at E$_f$ decreases faster below certain temperature (marked by the red arrow and referred to as T$_{pair}$ ).", "We note that there are no other striking features in this dependence as we cross T$_c$ .", "This is a strong evidence that pairing occurs already at T$_{pair}$ and partially coexists with the pseudogap.", "With increased doping, the weight lost to the pseudogap (blue area) decreases and the area related to pairing increases - signifying an increase in superfluid density.", "This is consistent with direct measurements of the superfluid density by $\\mu $ SR [34].", "We also note that the pairing temperature reaches 150K in Bi2212, which likely sets the maximum achievable critical temperature in cuprates.", "This also resolves the controversy between the existence of two distinct energy scales and reports of a continuous evolution of the gap across T$_c$ [25], [26], [27], [28].", "Below T$_{pair}$ , a d-wave pairing gap is present in the spectra, therefore no dramatic changes in the gap properties are observed.", "Only at higher temperatures, do distinct features of the pseudogap manifest themselves." ], [ "Real Fermi arcs", "The discovery of spectroscopic signatures of pairing is very closely related to the issue of Fermi arcs.", "Earlier studies reported a linear dependence of the length of these arcs with temperature [50].", "This was interpreted in a quite natural way as the interplay of a d-wave pairing gap and strong scattering that exists above T$_c$ [52].", "When the scattering rate exceeds the magnitude of the gap, symmetrized spectra lack the dip at E$_f$ and appear gapless.", "Since the d-wave gap is smallest in the nodal region, this part of the Fermi surface appears gapless, while spectra away from the node still bear the signature of the gap.", "As the temperature increases, so does the scattering rate and a larger portion of the Fermi surface appears \"normal\".", "This gives the appearance of arcs expanding with increasing temperature.", "[52].", "This scenario was recently confirmed by follow-up ARPES experiments[53], which demonstrated that a d-wave gap is still present at all Fermi momentum points up to 150K (which is of the order of our T$_{pair}$ ).", "An interesting question arises.", "What is the Fermi surface at higher temperatures in the absence of pairing, where the pseudo gap is still present?", "Figure: Measurements of the gap opening temperature using a DOS approach.", "panels (a-d) show the temperature dependence of the MDC weight.", "When this quantity begins to decrease it implies the opening of the gap.", "In the antinode region (panels a, c) this occurs at T, while in the nodal region, the gap opens at a much lower temperature T pair _{pair}, however still above T c _c.", "Panel e) shows the length of the real Fermi arc as a function of temperature determined using this method.", "Open red circles are plotted based on conclusions from Ref.", "To examine this issue in detail we again utilize our quantitative approach [51].", "The most sensitive probe of the opening of an energy gap is the density of states (DOS) at E$_f$ .", "To obtain momentum dependent information we note that the area of the MDC at E$_f$ along a cut in momentum space represents a contribution to the DOS(E$_f$ ).", "When this quantity is integrated over the Brillouin zone, it is directly proportional to DOS(E$_f$ ) modulo matrix elements.", "To detect the opening of an energy gap we monitor the area of the MDC (A$_{MDC}$ ) along a cut as a function of temperature.", "In the absence of gap, this quantity should be constant, since it is independent of the scattering rate (only the width of the MDC increases with the scattering rate).", "As the gap opens, this quantity should decrease, and any departure from a constant value marks the gap opening temperature.", "We use this method to detect the gap opening temperature along the Fermi surface of Bi2201 and Bi2212.", "Examples of such plots are shown in Fig.", "7 a-d. Close to the antinode, the pseudo gap opens at T.", "A$_{MDC}$ is constant above T* ($\\sim $ 250K signifying the absence of a gap and it decreases below this temperature as the pseudogap opens.", "This quantity decreases faster once T$_{pair}$ is reached and is consistent with the conclusions drawn in the previous section.", "Close to the node, A$_{MDC}$ remains constant well below T* and it only starts to decrease once T$_{pair}$ is reached.", "This occurs at all momentum points for $\\phi \\ge 20 ^\\circ $ .", "In contrast, the pseudogap opens over the remainder of the Fermi surface exactly at T.", "This proves that real, gapless Fermi arcs exist above T$_{pair}$ .", "Furthermore, since the pairing gap opens at all points in this part of the Fermi surface below T$_{pair}$ , the length of this arc remains constant between T and T$_{pair}$ .", "Indeed in Fig.", "7e we plot the length of the Fermi arc, extracted by examining the gap opening temperature at 8 points along the Fermi surface.", "As the temperature is lowered below T, the full metallic Fermi surface collapses to gapless Fermi arcs.", "The length of these arcs remains constant down to T$_{pair}$ , below which the arcs collapse to the node of a d-wave gap.", "The density of our data points close to the node allow us to put an upper limit on the size of the arc below T$_{pair}$ , but we know from Ref.", "[53], that the Fermi surface below 150K has a full d-wave gap.", "To demonstrate the universality of our findings, we plot in Fig.", "7f, the weight lost to the pseudogap as a function of momentum and temperature (in a similar fashion to Fig.", "6, but extracted from MDC areas).", "All data points extrapolate to $\\phi \\sim 20 ^\\circ $ , demonstrating that this endpoint of the arc is independent of temperature.", "Similarly in Fig.", "7g, we plot the area representing paired states.", "Again the points extrapolate to the nodal point for all temperatures below T$_{pair}$ , demonstrating that a d-wave gap is present with a single gapless node per quadrant.", "Figure: Phase diagram of Bi2201 with superconducting, pairing and pseudo gap temperatures.", "In the blue area only the pseudo gap is present in the samples and real, gapless Fermi are exist.", "In the red area, a pseudogap coexists with pairing and a d-wave order parameter gaps Fermi surface.", "Panels (b-d) schematically show gapless Fermi surface in three key areas.In summary, we arrived at a consistent Fermiology of pseudogap state in cuprates by utilizing a quantitative approach to ARPES data.", "This picture reconciles most major controversies and is summarized in Fig.", "8.", "In panel A we plot the partial phase diagram with measured superconducting, pairing and pseudogap temperatures.", "The Fermi surface in key regions are plotted schematically in panels (b-d).", "We demonstrated that the pseudogap and pairing gap are two distinct energy scales that compete for low energy states.", "Pairing in the cuprates occurs well above T$_c$ , but also significantly below T.", "The pairing temperature T$_{pair}\\sim 150K$ likely sets maximum achievable superconducting transition temperature in the cuprates.", "Below T$_{pair}$ a d-wave gap is present at all points of the Fermi surface and gapless Fermi arcs are absent.", "Above T$_{pair}$ , the pseudogap dominates the spectral properties and leads to the formation of real, gapless Fermi arcs in the nodal region.", "The length of these arcs do not change with temperature and their tips define a set of vectors that are likely the key to understanding the order responsible for the origin of the pseudogap." ] ]	1403.0492
[ [ "Summarisation of Short-Term and Long-Term Videos using Texture and\n Colour" ], [ "Abstract We present a novel approach to video summarisation that makes use of a Bag-of-visual-Textures (BoT) approach.", "Two systems are proposed, one based solely on the BoT approach and another which exploits both colour information and BoT features.", "On 50 short-term videos from the Open Video Project we show that our BoT and fusion systems both achieve state-of-the-art performance, obtaining an average F-measure of 0.83 and 0.86 respectively, a relative improvement of 9% and 13% when compared to the previous state-of-the-art.", "When applied to a new underwater surveillance dataset containing 33 long-term videos, the proposed system reduces the amount of footage by a factor of 27, with only minor degradation in the information content.", "This order of magnitude reduction in video data represents significant savings in terms of time and potential labour cost when manually reviewing such footage." ], [ "Introduction", "Video abstraction aims at providing concise representations of long videos.", "It has applications in browsing and retrieval of large volumes of videos [1] and also in improving the effectiveness and efficiency of video storage [21].", "Video abstraction can be categorised into two general groups: video summarisation and video skimming [10], [21].", "Video summarisation, also known as still image abstraction, static storyboard or static video abstract, is a compilation of representative frames selected from the original video [6].", "Video skimming, also known as moving image abstraction or moving/dynamic storyboard, is a collection of short video clips [2], [10].", "Both approaches should preserve the most important content from the video in order to present a comprehensible and understandable description for the end user.", "In general, video skimming provides a more coherent and visually attractive result.", "It often retains a high-level of linguistic meaning due to its capacity to combine audio and moving elements [14], [21].", "However, video summarisation is easier to generate and is not constrained in terms of timing and synchronisation [2], [21].", "Video summarisation is an active area of research within the computer vision community and it has been applied in various video categories such as Wildlife Videos [23], sports videos [15], TV documentaries [2], among others.", "In [1] the various approaches to video summarisation are divided into six techniques consisting of: feature selection, clustering algorithms, event detection methods, shot selection, trajectory analysis and the use of mosaics.", "Often a combination of techniques is used, for example one of the most common approaches is to combine feature selection with a form of clustering [2], [6], [13].", "In [24] a video summary is obtained by extracting a feature vector from each frame and then clustering the resulting set of feature vectors.", "The smallest clusters are then removed.", "A keyframe – a frame that forms part of the video summary – is selected for each cluster centroid by taking the frame whose feature vector is closest to the centroid.", "Similar approaches are adopted in [2], [6], [7], [10] where the major difference is in the choice of feature vector used to represent each frame.", "Colour histograms are used in [2], [6], motion-based features are used in [7], and saliency maps are used in [10].", "Each of the previously proposed feature vectors has its drawbacks.", "For instance, the colour histogram approach used in [2], [6] retains only coarse information about the frame.", "Motion-based features of [7] fail when the motion in the videos is too large.", "Finally, the saliency maps used in [10] perform poorly for cluttered and textured backgrounds.", "To date, limited work has been done on incorporating texture information to perform video summarisation.", "Contributions.", "In this paper we first propose the use of texture information to improve video summarisation.", "We propose the use of the computationally efficient and effective bag-of-textures approach; we conjecture that this will improve video summarisation as it has been successfully applied to a range of image processing tasks, such as matching and classification of natural scenes and faces [12], [19], [22].", "The bag-of-textures model divides an image into small patches, extracts appearance descriptors from each patch, quantises each descriptor into a discrete “visual word”, and then computes a compact histogram representation [8], providing considerably different information than colour histograms.", "In addition, we propose a fusion based system for video summarisation, where both colour and texture information is exploited.", "This will allow us to overcome the shortcomings of either approach.", "Similar approaches have been shown to be advantageous in object classification tasks [11].", "We show that our system may be applied not only to short-term videos but also to long-term videos, helping in the detection of the existence of a rare species of fish.", "The layout of this paper is as follows.", "In Section we describe in detail our proposed video summarisation method that exploits the benefits of using texture histograms based on the bag-of-textures model.", "In Section we present our improved video summarisation method that fuses the visual information provided by both the colour and texture histograms.", "In Section we describe how we evaluate the video summaries of short-term and long-term videos.", "In Section , we present experiments which show that the proposed methods obtain higher performance than existing methods based on colour histograms.", "Section summarises the main findings." ], [ "Bag-of-Textures for Video Summarisation", "This section describes our proposed bag-of-textures (BoT) approach.", "There are four main stages: Pre-processing: The input video is sub-sampled after which each frame is filtered and rescaled.", "BoT representation: Local Texture Features.", "Each frame is divided into small patches (blocks) and from each block we extract 2D-DCT features, which is an effective and compact representation [16].", "Dictionary Training.", "A generic visual dictionary is trained to describe the most commonly occurring textures in an independent training set.", "Generation of BoT Histogram.", "Each frame is represented by a histogram which is obtained by matching the feature vectors from each block to the dictionary.", "Keyframe selection: Similar frames are grouped into an automatically determined number of clusters.", "One keyframe is selected per cluster.", "Post-processing: In this final stage, we eliminate possible repetitive frames and create the static video summary.", "Each of these stages is elucidated in the following sections." ], [ "Sampling and Rescaling", "The original input video is re-sampled to one frame per second in order to reduce the number of video frames to be examined.", "Each frame is then converted into gray-scale and re-scaled to be a quarter of its original size, in order to reduce the computational cost of the following stages." ], [ "Noise Filtering", "There are often uninformative frames that appear at the beginning and/or the end of a segment that may affect the appearance of a video summary [6].", "These frames are usually colour-homogeneous due to fade-in and fade-out effects, and have a small standard deviation of their pixel values.", "Frames with a standard deviation below a threshold are eliminated." ], [ "Local Texture Features", "Each frame is divided into $N$ overlapping blocks.", "To each block we apply the 2D discrete cosine transform (2D-DCT) to obtain a $D$ -dimensional feature vector that represents the local texture information [16].", "Thus, the local texture feature for the $n$ -th block of the $i$ -th frame is ${\\bf {x}}_{i,n}$ ." ], [ "Dictionary Training", "The dictionary is trained using the k-means algorithm [3] by pooling the local texture features from a set of training frames.", "The resulting $G$ cluster centers $\\lbrace {\\mu }_1, \\cdots , {\\mu }_{G}\\rbrace $ represent the local textures (codewords) of the dictionary." ], [ "Generation of BoT Histogram", "In the BoT approach the $i$ -th frame is represented by a histogram, ${\\bf {h}}^\\text{BoT}_{i}$ .", "This $G$ -dimensional histogram represents the relative frequency of the local texture features within the frame.", "The $g$ -th dimension of ${\\bf {h}}^\\text{BoT}_{i}$ is the relative frequency of the $g$ -th local texture feature from the dictionary, similar to [5].", "The histogram is normalised to sum to one.", "Thus, each local texture feature can be converted to a local histogram, ${\\bf {h}}^{\\text{BoT}}_{i,n}$ , of dimension $G$ where each dimension $g$ is given by, ${h}^{\\text{BoT}}_{g,i,n} = {\\left\\lbrace \\begin{array}{ll} 1 & \\text{if }g = \\underset{{k \\in 1, \\cdots , G}}{\\arg \\ \\min } \\Vert {\\bf {x}}_{i,n} - {\\mu }_k \\Vert _{2} \\\\ 0 & \\text{otherwise} \\end{array}\\right.", "}.$ These $N$ local histograms can then be summed and normalised to produce the final BoT histogram, ${\\bf {h}}^{\\text{BoT}}_{i} = \\frac{1}{N} \\sum \\nolimits _{n=1}^{N}{\\bf {h}}^{\\text{BoT}}_{i,n}.$" ], [ "Keyframe Selection", "To obtain a set of keyframes we adopt an approach similar to that of [6].", "A keyframe is a frame that forms part of the video summarisation.", "The k-means algorithm is used to cluster similar frames into $K$ segments, and the resultant centroids are then used to select the keyframes.", "Initially, the frames are grouped consecutively, assuming that sequential frames share similar content.", "To automatically determine the number of clusters, $K$ , we calculate the Euclidean distance between two consecutive frames.", "If the distance is greater than a threshold $\\tau $ then $K$ is incremented.", "For each cluster centroid the frame whose BoT histogram is closest is selected as a keyframe.", "A total of $K$ keyframes is then reached." ], [ "Post-processing", "Having obtained the initial set of $K$ keyframes we then attempt to discard those keyframes which are too similar.", "This is achieved by comparing all keyframes against each other.", "If the Euclidean distance between the BoT histograms of the keyframes is smaller than a threshold $\\tau $ then one of the two keyframes under consideration is discarded.", "This gives the final static video summary that consists of $N_{as}$ keyframes, where $N_{as}\\le K$ , with $as$ standing for automatic summary.", "Lastly, the static video summary is obtained after organising the resulting keyframes in temporal order." ], [ "Fusion of Colour and BoT", "In this section, we present a hybrid system that fuses colour histograms [6] and BoT texture information, termed as CaT (for Colour and Texture).", "The proposed CaT approach to video summarisation has the same 4 stages as our proposed BoT video summarisation approach, but with additions in order to obtain colour histograms.", "We describe these additions below.", "Pre-processing: The input video is processed in two independent ways.", "First, we obtain the BoT histograms as described in Section REF .", "Second, to obtain the colour histograms we extract the Hue component, from the HSV colour space, of the unscaled input frame similar to [6].", "In both cases we remove uninformative frames by employing the noise filtering process described in Section REF .", "Texture and Colour Histogram: The BoT histogram is the same as explained in Section REF .", "The colour histogram, ${\\bf {h}}^\\text{hue}_{i}$ , of the $i$ -th frame is computed using only the Hue component as in [6].", "Keyframe Selection: The BoT and colour histograms are clustered using $k$ -means.", "This stage is similar to Section REF .", "The difference lies in the distance measure used to compare all frames against each other.", "To select the number of keyframes $K$ we combine the information from the BoT and colour histograms.", "When calculating the distance between frame $a$ and $b$ we use the weighted summation of Euclidean distances: $\\alpha \\Vert {\\bf {h}}^{\\text{BoT}}_{a} - {\\bf {h}}^{\\text{BoT}}_{b} \\Vert _{2} + \\beta \\Vert {\\bf {h}}^{\\text{hue}}_{a} - {\\bf {h}}^{\\text{hue}}_{b} \\Vert _{2}$ under the constraints $\\alpha +\\beta =1$ , $\\alpha \\ge 0$ , $\\beta \\ge 0$ .", "Each keyframe is selected by finding the frame which is closest to each cluster centroid.", "For the CaT approach the distance between a frame and a centroid is calculated as a weighted summation of the Euclidean distances, as per (REF ).", "Post-processing: To eliminate similar frames we use the procedure described in Section REF but replace the Euclidean distance with the weighted summation of the Euclidean distances, as per (REF )." ], [ "Datasets and Evaluation Metrics", "To evaluate the performance of video summarisation we use two datasets consisting of short- and long-term video data.", "The short-term data is obtained from the Open Video ProjectOpen Video Project: http://www.open-video.org.", "The long-term data is a new dataset that consists of 14 hours of underwater video surveillance which monitors the behaviour of marine wildlife." ], [ "Short-Term Videos", "We use the 50 videos from the Open Video Project which contain ground truth [6].", "Each ground truth consists of the summary provided by $P=5$ users.", "The users provided the summaries under no restrictions upon length nor appearance of the summaries.", "To evaluate the performance on the short-term video data we use the “Comparison of User Summaries” (CUS) method [6].", "This method compares the automatic video summarisation and ground truth by exhaustively calculating the distance between the frames from the automatic summarisation and the ground truth.", "Two frames are similar if the distance between their respective feature vectors (histograms) is less than an evaluation threshold $\\delta $ .", "If the frames match they are removed from the next iteration of the comparison process.", "For performance evaluation, the distance measure used for the BoT approach is the Euclidean distance, however, to be consistent with prior work [6], the distance measure for the colour histograms is the $L_{1}$ -norm.", "Therefore, the distance measure used for CaT is the weighted summation of the Euclidean distance for the BoT histograms and the $L_{1}$ -norm for the colour histograms: $\\alpha \\Vert {\\bf {h}}^{\\text{bof}}_{a} - {\\bf {h}}^{\\text{bof}}_{b} \\Vert _{2} + \\beta \\Vert {\\bf {h}}^{\\text{hue}}_{a} - {\\bf {h}}^{\\text{hue}}_{b} \\Vert _{1}.$ Various evaluation metrics exist to measure the quality of an automatic video summary.", "We use three evaluation metrics so that we can compare our proposed approaches with two state-of-the-art methods [6], [2].", "To compare with [6] we use accuracy ($acc$ ) and error ($err$ ), and to compare with [2] we use the $F$ -measure.", "To calculate $acc$ and $err$ , each frame in the automatic video summary is compared with all frames in the user summary and then the number of matching frames ($N_{m}$ ) and non-matching frames ($N_{nm}$ ) are calculated: $\\begin{array}{cc}acc = \\frac{N_{m}}{N_{u}}, & err = \\frac{N_{nm}}{N_{u}}\\end{array}$ where $N_{as}$ and $N_{u}$ are the total number of frames from the automatic and user summary, respectively.", "The $F$ -measure, defined as $F= \\frac{2\\times \\text{precision}\\times \\text{recall}}{\\text{precision} ~+~ \\text{recall}}$ is used to to provide a single number that balances $\\text{precision} = N_m/N_{as}$ and $\\text{recall} = N_m/N_{u}$ .", "The evaluation metrics are presented as an average.", "First, we take the average from the $P$ users to obtain $acc_P$ , $err_P$ , and $F_P$ ; for each video there are $P=5$ users.", "Then we take the average across all of the videos to obtain $\\overline{acc}$ , $\\overline{err}$ , and $\\overline{F}$ .", "In terms of $\\overline{acc}$ it is desirable to have a high value as it measures the number of matching frames.", "In terms of $\\overline{err}$ it is desirable to have a small value as it measures the number of non-matching frames.", "With regards to $\\overline{F}$ it is desirable to obtain a high value, which occurs when the $\\text{precision}$ and $\\text{recall}$ are large." ], [ "Long-Term Videos", "The long-term videos consist of 14 hours of underwater footage from 33 videos which are on average 25 minutes in duration.", "This data was obtained from the NSW-DPINew South Wales Department of Primary Industries, Australia., courtesy of David Harasti.", "Example images are shown in Figure REF .", "In each video there is always at least one segment where a rare species of fish, the black cod, is within view.", "Normally these videos would be inspected by a human expert to determine if there is an instance of the rare fish within.", "We propose that video summarisation can be used to reduce the amount of footage to be viewed in order to detect the existence of this rare species of fish.", "Using ground truth which provides time-stamps when this rare species is within view, we examine the effectiveness of video summarisation to provide at least one keyframe in each static video summary with the rare species of interest within view.", "This is useful as it presents a way to reduce the time and cost of manually viewing a large amount of video data.", "Figure: Example images from the long-term underwater surveillance videos; the added red ellipsoids highlight the rare species of interest.To calculate the performance of long-term videos we present results in terms of detection accuracy and the average compression ratio ($R_{\\text{c}}$ ).", "Detection accuracy refers to whether an instance of the rare species is among any of the chosen keyframes for a static video summary; $75\\%$ would mean that there is at least 1 keyframe of the rare species in $75\\%$ of the static video summaries.", "To calculate the average compression ratio we first note that because we have long-term videos then for each video there might be many hundreds of keyframes.", "To present all of these keyframes effectively to the user we re-encode them into a static video summary by presenting each keyframe for $0.25$ seconds.", "This gives the user time to effectively view the keyframe.", "Thus the $t$ -th long-term video ${\\bf {V}}_{t}$ is converted to a static video summary ${\\bf {S}}_{t}$ with a compression ratio given by: $R_{c,t} = 4 \\times \\frac{\\text{Duration} ({\\bf {V}}_{t}) }{\\text{Duration} ({\\bf {S}}_{t}) }$ where $\\text{Duration}$ is the duration of a video and the factor of 4 is introduced as there are 4 keyframes per second of the shortened video." ], [ "Experiments", "An important part of both the BoT and CaT approaches is the training of the dictionary to obtain the texture histograms.", "To train this dictionary we use 10 frames randomly selected from videos taken from the Open Video Project that have no user summaries, ensuring they are independent of the evaluation dataset.", "In addition, the frames selected to train the dictionary look significantly different to the ground truth provided by the users.", "To obtain the proposed local texture features we divide each frame into a set of overlapping blocks.", "Similar to [19] we use a block size of $8 \\times 8$ with an overlap margin of 6 pixels, and represent each block as a $D=15$ dimensional feature vector containing 2D-DCT coefficients.", "We extract the first 16 2D-DCT coefficients, which represent low-frequency information [16], and omit the first coefficient as it is the most sensitive to illumination changes.", "With regards to the colour histogram, we quantise the Hue component into 16 bins as per [6].", "These parameters are the same for all experiments.", "The values for the threshold $\\tau $ , fusion weight $\\alpha $ and evaluation threshold $\\delta $ were determined experimentally.", "For all of the experiments we search for the optimal fusion parameter $\\alpha =\\lbrace 0.0, 0.1, \\cdots , 1.0 \\rbrace $ .", "Our proposed methods were implemented using the OpenCV [4] and Armadillo [18] C++ libraries." ], [ "Short-Term Videos", "We compare the performance against two baseline systems from literature, VSUMM [6] and VISON [2].", "The two baseline systems both use colour information as their primary feature.", "VSUMM uses colour information by retaining only the Hue component of HSV and generating a histogram of 16 bins.", "VISON is a state-of-the-art approach and consists of a histogram of the HSV representation of each frame.", "It combines the HSV information in a compressed form such that the Hue component is treated with greater importance and results in a histogram of 256 bins.", "An initial set of experiments were performed to find the optimal number of components for the dictionary of our proposed texture features.", "Using a fixed number of components $G=\\lbrace 8,16,32 \\rbrace $ and a fixed number of thresholds $\\tau =\\lbrace 0.05,0.10,\\cdots ,0.5 \\rbrace $ , we found that using just $G=8$ components provided optimal performance.", "We kept the number of components constant for the remainder of our experiments.", "In Figure REF we present a summary of the average performance for 50 short-videos of our proposed systems, BoT and CaT, and the two baselines.", "Two interesting results can be seen from this figure.", "First, it can be seen that the texture-only BoT system performs better than either the VSUMM or VISON approaches which primarily use colour information.", "The BoT system obtains an average $F$ -measure of $\\overline{F}=0.83$ , which is a relative improvement of $9\\%$ when compared to VISON, $\\overline{F}=0.76$ .", "Furthermore, the $\\overline{acc}$ and $\\overline{err}$ of the BoT system shows that it produces a more accurate summarisation than VSUMM and also has the lowest $\\overline{err}$ of any systemNo results in terms of $\\overline{acc}$ and $\\overline{err}$ were supplied for VISON in [2]..", "This suggests that texture information is either equally or more important than colour information for the task of video summarisation.", "Second, the proposed CaT system (fusing colour histograms and the proposed texture histograms) performs better than the two baseline systems and the proposed texture-only BoT system.", "The CaT system has an average $F$ -measure of $\\overline{F}=0.86$ , which is a relative improvement of $13\\%$ when compared to VISON $\\overline{F}=0.76$ , the previous state-of-the-art approach.", "Figure: Comparative evaluation of our proposed methods with VSUMM and VISON .Lower values of err ¯\\overline{err} as well as higher values of acc ¯\\overline{acc} and F ¯\\overline{F} are desired.Figure: Static video summary for “the future of energy gases - segment 09”,using(a) VSUMM,(b) VISON,(c) proposed BoT,and(d) proposed CaT.Figure REF shows the qualitative results for the automatic summarisation provided by VSUMM and VISON as well as our proposed BoT and CaT systems.", "It can be seen that VSUMM (Figure REF a) with ${F_P}=0.83$ , VISON (Figure REF b) with ${F_P}=0.78$ , and our proposed BoT (Figure REF c) with ${F_P}=0.74$ contain some keyframes that may not be of interest and/or are repetitive.", "In contrast, the proposed CaT system (Figure REF d) provides the most consistent video summary with ${F_P}=0.86$ ." ], [ "Long-Term Videos", "In this section we present results on 33 long-term videos which last on average for 25 minutes.", "We examine the applicability of video summarisation to long-term videos to efficiently detect a rare species of fish and measure performance in terms of detection accuracy and compression rate (see Section REF ).", "The accuracy and average compression ratio of the algorithm for various thresholds, $\\tau =\\lbrace 0.025, 0.05, \\dots , 0.1 \\rbrace $ , is presented in Figure REF .", "It can be seen in Figure REF a that the CaT algorithm consistently outperforms the BoT and VSUMM algorithms.", "We attribute this to the fact that the background in these videos is relatively stable and so the colour histograms used in VSUMM do not change as often compared to the short-term videos used in [6].", "In Figure REF b it can be seen that while using the VSUMM algorithm provides better average compression ratio than either the BoT or CaT approaches, it comes at the cost of accuracy.", "In general the proposed fusion approach provides the most consistent trade-off between accuracy and average compression ratio.", "We take the optimal system at the threshold $\\tau =0.05$ as this provides a high degree of detection accuracy, $85\\%$ , and a good average compression ratio of 27.", "This system will allow a user to see the fish of interest in $85\\%$ of the summarised videos while reducing the amount of video data to view by 27 times, more than an order of magnitude.", "Such an approach would reduce the 14 hours of video data to just 31 minutes, thus enabling significantly more efficient reviewing of the data.", "Figure: Demonstration of the trade-off between (a) the detection accuracy and (b) the average compression ratio R c R_{c} for the 33 long-term videos using the CaT, BoT and VSUMM approaches." ], [ "Summary and Future Work", "In this paper, we have proposed the novel use of textures to perform video summarisation.", "We proposed to use a visual-bag-of-textures (BoT) in two ways.", "First, a BoT system which uses only texture features is proposed and it is shown to outperform two state-of-the-art systems which use colour only, VSUMM and VISON.", "Second, a fused system that combines Colour and Texture (CaT) is proposed and it is shown to provide further improvements.", "Both of our proposed systems outperform two state-of-the-art approaches, VSUMM and VISON, which use colour features.", "Experiments on 50 short-term videos, obtained from the Open Video Project, show that our proposed texture-only system (BoT) obtains an $F$ -measure of $0.83$ , which is better than either VSUMM or VISON which obtain an average $F$ -measure of $0.73$ and $0.76$ , respectively.", "Furthermore, our fused system (CaT) demonstrates that combining colour and texture features yields state-of-the-art performance with an average $F$ -measure of $0.86$ .", "We have also shown that video summarisation can be applied effectively to long-term videos.", "Using 33 long-term surveillance videos, in our case underwater surveillance footage, we have shown that video summarisation can be used to significantly reduce the amount of footage to view, by up to a factor of 27, with only a minor degradation in the information content.", "Future work should examine alternative features and application settings with a particular emphasis for long-term videos.", "For instance, emphasising the importance of foreground objects [17] should be explored, as well as explicit modelling of movement (or actions) of such objects [9], [20].", "Moreover, the applicability of video summarisation to CCTV surveillance footage should also be considered." ] ]	1403.0315
[ [ "On the production properties of the Doubly-Charmed Baryons" ], [ "Abstract This paper focuses on disagreement between theoretical predictions and experimental results of the production properties of Doubly Charmed Baryons.", "The kinematic dependencies were used to clarify the discrepancy between the SELEX data and the theory.", "The production ratio of the $\\Xi_{cc}^+$ baryon in the SELEX kinematic region is presented.", "The recent experimental results are reviewed." ], [ "Introduction", "In 2002 the SELEX collaboration published the first observation of $\\Xi _{cc}^+$ baryon in the charged decay mode $\\Xi _{cc}^+\\rightarrow \\Lambda _c^+K^- \\pi ^+$ from $\\Lambda _c^+\\rightarrow p K^- \\pi ^+$ (1630 events) sample [1].", "In 2005 the SELEX collaboration reported an observation of $\\Xi _{cc}^+\\rightarrow p D^+ K^-$ decay mode from 1450 $D^+ \\rightarrow K^- \\pi ^+ \\pi ^+$ decays to complement the previously reported decay [2].", "The mass and lifetime also have been measured by SELEX (see Table REF ).", "Table: This table summarizes the SELEX results on measurements of the mass and lifetime of Ξ cc + \\Xi _{cc}^+ baryon.The production properties of $\\Xi _{cc}^+$ baryon can be obtained from the measurements and compared to that of $\\Lambda _c^+$ baryon and $D^+$ meson: $R_{\\Lambda _c^+} = \\frac{\\sigma (\\Xi _{cc}^+) \\cdot Br(\\Xi _{cc}^+\\rightarrow \\Lambda _c^+K^- \\pi ^+)}{\\sigma (\\Lambda _c^+)} \\approx 0.015$ and $R_{D^+} = \\frac{\\sigma (\\Xi _{cc}^+) \\cdot Br(\\Xi _{cc}^+\\rightarrow p D^+ K^-)}{\\sigma (D^+)} \\approx 0.004$ in kinematic region $x_F > 0.3$ [3].", "Using known fragmentation ratio $f(c \\rightarrow \\Lambda _c^+) = 0.071 \\pm 0.003~\\text{(exp.", ")}\\pm 0.018~\\text{(br.", ")}$ [4] and assuming $Br(\\Xi _{cc}^+\\rightarrow \\Lambda _c^+K^- \\pi ^+) \\approx Br(\\Lambda _c^+\\rightarrow p K^- \\pi ^+) = (5.0 \\pm 1.3)~\\%$ [5], one can obtain the ratio of the production cross-section: $\\frac{\\sigma (\\Xi _{cc}^+)}{\\sigma (c\\bar{c})} = R_{\\Lambda _c^+} \\cdot \\frac{f(c \\rightarrow \\Lambda _c^+)}{Br(\\Xi _{cc}^+\\rightarrow \\Lambda _c^+K^- \\pi ^+)} \\\\ \\simeq 2.1 \\cdot 10^{-2}$ Comparing this result with theoretically predicted $\\sigma (\\Xi _{cc}^+) / \\sigma (c\\bar{c}) \\sim 10^{-6} - 10^{-5}$ [6], [7] production ratio for fixed-target experiments, we see that measured $\\Xi _{cc}^+$ production cross-section is at least $10^3$ times larger than theoretical prediction.", "This is a significant discrepancy between theory and experiment.", "Recent searches in different production environments at BaBar [8], Belle [9] and LHCb [10] experiments also have not shown evidence (see Table REF ) for the production properties reported by SELEX.", "Table: This table shows the production properties on σ(Ξ cc + )·Br(Ξ cc + →Λ c + K - π + )/σ(Λ c + )\\sigma (\\Xi _{cc}^+) \\cdot Br(\\Xi _{cc}^+\\rightarrow \\Lambda _c^+K^- \\pi ^+)/\\sigma (\\Lambda _c^+) in different production environment.", "The LHCb results based on the assumption that Ξ cc + \\Xi _{cc}^+ lifetime is 100 fs and 400 fs respectively.To clarify this issue we research kinematic dependencies of hadronic production properties of the doubly-charmed baryon.", "The SELEX experiment is a fixed-target experiment used the Fermilab charged hyperon beam at 600 GeV/c to produce charm particles in a set of thin foil of Cu or in a diamond and operated in the $x_F > 0.3$ kinematic region.", "The negative beam composition was about 50% $\\Sigma ^-$ , 50% $\\pi ^-$ .", "The positive beam was 90% protons.", "The Born approximation of the hadronic production of $\\Xi _{cc}^+$ baryon and the limitation of the model have been discussed in Ref. [7].", "Below we will use the results without any additional discussions.", "Following [7] the parton level production cross-sections of $\\Xi _{cc}^+$ baryons: $\\hat{\\sigma }_{gg} = 213 \\cdot \\left( 1 - \\frac{4 \\cdot m_c}{\\sqrt{\\hat{s}}} \\right)^{1.9} \\left( \\frac{4 \\cdot m_c}{\\sqrt{\\hat{s}}} \\right)^{1.35}~~\\text{pb},\\\\\\hat{\\sigma }_{q\\bar{q}} = 206 \\cdot \\left( 1 - \\frac{4 \\cdot m_c}{\\sqrt{\\hat{s}}} \\right)^{1.8} \\left( \\frac{4 \\cdot m_c}{\\sqrt{\\hat{s}}} \\right)^{2.9}~~\\text{pb}.$ One has to mention that numerical coefficients depend on the model parameters, so that $\\hat{\\sigma } \\sim \\alpha _s \|R(0)\|^2 / m_c^2$ , where $\\alpha _s = 0.2$ , $R(0) = 0.601 \\text{GeV}^{3/2}$ and $m_c = 1.7$ GeV.", "Let us remind the reader that the hadron level cross-section can be presented as $\\sigma = \\sum \\int dx_1^{(k)} dx_2^{(m)} f_k (x_1, \\mu ) f_m (x_2, \\mu ) \\hat{\\sigma }(x_1, x_2),$ where $f_i (x, \\mu )$ is a parton distribution function, $x$ is the ratio of the parton momentum to the momentum of the hadron and $\\mu $ is the energy scale of the interaction.", "Combining Eqs.", "REF , , REF and using CTEQ6L [11] parametrization for parton distribution functions, we may expect $\\Xi _{cc}^+$ production cross-section in the kinematic region $x_F >0.4$ [3] to be $\\sigma (\\Xi _{cc}^+) \\gtrsim 25~\\text{pb}.$ Upon contracting this result with predictions of the Born approximation of two-quark hadronic production in the same kinematic region, $\\Xi _{cc}^+$ production ratio at the SELEX is $\\frac{\\sigma (\\Xi _{cc}^+)}{\\sigma (c\\bar{c})} \\sim (10^{-3} - 10^{-2})$ This result can be easily interpreted taking into account to the four-quark production properties.", "Assuming that $\\hat{\\sigma }(\\Xi _{cc}^+)$ is proportional to the parton level of the four-quark production cross-section $\\hat{\\sigma }(4Q)$ and following [12] we can see that $\\hat{\\sigma }(4Q)$ is growing up with the energy and turns out to be a leading process instead of two-quark production cross-section which dominates at the small energies.", "The uncertainty of the result came from the two-quark production cross-section.", "The SELEX analysis strategy [3] requires $x_F > 0.4$ for a final state ($\\Lambda _c^+$ plus the positively charged track and the negatively charged track) which leads to uncertainty of the final charm $x_F$ distribution and cannot be considered without full modeling of the production environment." ], [ "Belle and LHCb", "The Belle experiment presented the upper limit on the $\\sigma (e^+e^- \\rightarrow \\Xi _{cc}^+X)$ is 82-500 fb for the decay mode with the $\\Lambda _c^+$ at $\\sqrt{s} = 10.58$ GeV using 980 $\\text{fb}^{-1}$ .", "The most realistic calculations [6], [13] predict $\\sigma (\\Xi _{cc}^+) \\simeq (35 \\pm 10) \\times 10^{-3}$ pb what turns out to be at least twice as less as the given limit.", "Another recent result from the LHCb experiment provides the upper limits at 95% C.L.", "on the ratio $\\sigma (\\Xi _{cc}^+) \\cdot Br(\\Xi _{cc}^+\\rightarrow \\Lambda _c^+K^- \\pi ^+)/\\sigma (\\Lambda _c^+)$ to be $1.5 \\times 10^{-2}$ and $3.9 \\times 10^{-4}$ for lifetimes 100 fs and 400 fs respectively, for an integrated luminosity of 0.65 $\\text{fb}^{-1}$ .", "It is compared with result from Ref.", "[6] $\\sim 10^{-4} - 10^{-3}$ .", "However, the LHCb did not reach the lifetime measured by the SELEX experiment yet." ], [ "Conclusion", "In our paper we recalculated the hadronic production cross-section of the $\\Xi _{cc}^+$ baryon in the SELEX kinematic region and made the comparative analysis of experimental data and theoretical predictions.", "We found no significant discrepancy between the theory and the experimental results of the production properties of the doubly charmed baryons.", "The authors would like to thank Dr. Alexander Rakitin for his friendly support and proofreading the manuscript." ] ]	1403.0264
[ [ "Lyndon words for Artin-Schelter regular algebras" ], [ "Abstract We show certain invariants of graded algebras of which all obstructions are Lyndon words and provide some methods to construct Artin-Schelter regular algebras from a closed set of Lyndon words." ], [ "Introduction", "A Lyndon word is a non-empty word greater in lexicographical order than all of its rotations.", "Lyndon words share remarkable combinatoric and algebraic properties.", "For instance, they are wildly used in the context of Lie algebras and their universal enveloping algebras.", "Recently, monomial algebras defined by Lyndon words are studied in [7].", "It will be interesting to explore Lyndon words in a more general context of associative algebras.", "This work is a practice of the idea for Artin-Schelter regular algebras.", "Artin-Schelter regular algebras are a class of graded algebras which may be thought of as homogeneous coordinate rings of noncommutative spaces.", "They were introduced by Artin and Schelter [3] in late 1980's, and have been extensively studied since then.", "The class of such algebras of which all the obstructions are Lyndon words is of particular interest.", "This class includes all low dimensional Artin-Schelter regular bigraded algebras in two generators [18] and the universal enveloping algebras of positively graded finite dimensional Lie algebras.", "Invariants of algebras in this class are relatively easy to deal with, one of our goals is to find accessible approaches for constructing of new such algebras.", "It is well-known that the standard bracketing of Lyndon words on an alphabet $X$ form a basis of the free Lie algebra ${\\rm Lie}(X)$ , and their monotonic products form a basis of the free associative algebra $k\\langle X\\rangle $ .", "Given a Gröbner set $G$ of Lie polynomials, then the algebra $k\\langle X\\rangle /(G)$ , which is the universal enveloping algebra of the Lie algebra $\\mathfrak {g}=\\text{Lie}(X)/(G)_L$ , is determined by irreducible Lyndon words modulo $G$ (see [14], [5]).", "The main tool of our approaches is a deformation of the standard bracketing of Lyndon words through a matrix $q$ .", "We associate a closed set $U$ of Lyndon words with a graded algebra $A(U,q)$ (Definition REF ) and give a practical criterion for determining the regularity of $A(U,q)$ .", "As an example, we are able to obtain the algebras $D(v,p)$ in [13] as a deformation of the universal enveloping algebra $D(-2,-1)$ in this sense, which is a motivation of this paper.", "A natural question for Artin-Schelter regular algebras is the determination of their Gorenstein parameters.", "In [6], the authors give an optimal estimation for the universal enveloping algebras of finite dimensional graded Lie algebras which are generated in degree one.", "In this paper, we obtain a counterpart of such estimation for a larger class of Artin-Schelter regular algebras which are generated in degree one and of which all the obstructions are Lyndon words (Corollary REF ).", "The organization of this paper is as follows.", "In Section 1, we introduce notations, recall the definitions of Lyndon words and Artin-Schelter regular algebras, and review some basic facts of Lyndon words.", "In Section 2, we give a picture of the graph of chains on antichains of Lyndon words, calculate algebraic and homological invariants of graded algebras of which the obstructions are Lyndon words.", "We devote Section 3 to the construction of Artin-Schelter regular algebras in terms of the generalized bracketing of Lyndon words.", "The results, in particular, give a connection between Artin-Schelter regular algebras and Hopf algebras.", "We work over a fixed field $k$ of characteristic 0.", "All vector spaces, algebras and unadorned tensor $\\otimes $ are over $k$ .", "The notation $\\mathbb {N}$ denotes the set of non-negative integers." ], [ "Noncommutative Gröbner Bases", "We begin by introducing some notations and terminologies that will be used in the sequel.", "For noncommutative Gröbner bases theory, we refer the details to [11], [15].", "Throughout $X=\\lbrace x_1,\\cdots ,x_n\\rbrace $ stands for a finite enumerated alphabet of letters.", "Denote by $X^$ the set of all words on $X$ including the empty word 1.", "The length of a word $u$ is denoted by $l(u)$ , and the constitute of $u$ is the tuple $ (r_1,\\cdots ,r_n)\\in \\mathbb {N}^n$ , where $r_i$ is the number of occurrences of $x_i$ in $u$ for $1\\le i \\le n$ , in particular, the constitute of 1 is $(0,\\cdots ,0)$ .", "With each $x_i$ we associate a positive integer $d_i$ .", "Define the degree of $u$ by $\\deg (u)=r_1d_1 +\\cdots + r_nd_n$ .", "The free algebra on $X$ is denoted by $k\\langle X\\rangle $ , and it is considered as a graded algebra $k\\langle X\\rangle =\\bigoplus _{m\\ge 0}k\\langle X\\rangle _m$ , where $k\\langle X\\rangle _m$ is the linear span of all words of degree $m$ .", "We say that a word $u$ is a factor of another word $v$ if $w_1uw_2=v$ for some words $w_1,w_2\\in X^$ .", "If $w_1=1$ then $u$ is called a prefix of $v$ , and if $w_2=1$ then $u$ is called a suffix of $v$ .", "A factor $u$ of $v$ is called proper if $u\\ne v$ .", "We fix an ordering $x_1<x_{2}<\\cdots < x_n$ on the letters.", "The lexicographical order (lex order, for short) $<_{\\rm lex}$ on $X^$ is then given as follows: for $u, v\\in X^$ , $u <_{\\rm lex} v$ iff either there are factorizations $u=rx_is,v=rx_jt$ with $x_i<x_j$ , or $v$ is a proper prefix of $u$ .", "Clearly $<_{\\rm lex}$ is a total order, which is preserved by left multiplication, but not always by right multiplication.", "For example, $x_2^2 <_{\\rm lex} x_2$ , but $x_2^2x_1 >_{\\rm lex} x_2x_1$ .", "However, if $u<_{\\rm lex}v$ and $v$ is not a prefix of $u$ , then the order is preserved by right multiplication, even right multiplication by different words.", "In particular, this holds when $\\deg (u)=\\deg (v)$ .", "In the sequel we use the deglex order $<_{\\rm deglex}$ on $X^$ given by, for $u, v\\in X^$ , $u <_{\\rm deglex} v\\quad \\text{ iff }\\quad \\left\\lbrace \\begin{array}{llll}\\deg (u) < \\deg (v),\\quad \\text{or}&&\\\\\\deg (u) = \\deg (v) and u<_{\\rm lex}v.\\end{array}\\right.$ It is an admissible order on $X^$ , which means a well order on $X^$ preserved by left and right multiplications.", "For any nonzero polynomial $f\\in k\\langle X\\rangle $ , the leading word $\\operatorname{\\rm LW}(f)$ of $f$ is the largest word occurs in $f$ , and the leading coefficient $\\operatorname{\\rm LC}(f)$ of $f$ is the coefficient of $\\operatorname{\\rm LW}(f)$ in $f$ .", "A nonzero polynomial with leading coefficient 1 is said to be monic.", "Let $G\\subseteq k\\langle X\\rangle $ be a set of polynomials.", "The set of all leading words of nonzero polynomials in $G$ is denoted by $\\operatorname{\\rm LW}(G)$ .", "A word $u$ is reducible modulo $G$ (resp.", "irreducible modulo $G$) provided that $u$ has a factor in $\\operatorname{\\rm LW}(G)$ (resp.", "has no factor in $\\operatorname{\\rm LW}(G)$ ).", "The set of all irreducible words modulo $G$ is denoted by ${\\rm Irr}(G)$ .", "If $0\\notin G$ , every $f\\in G$ is monic and all words occurred in $f$ are irreducible modulo $G\\backslash \\lbrace f\\rbrace $ , then we say $G$ is a reduced set.", "A 4-tuple $(l_1,r_1,l_2,r_2)$ of words is called an ambiguity of a pair of words $(u_1,u_2)$ in case $l_1u_1r_1=l_2u_2r_2$ and one of the following conditions holds: (1) $l_1=r_1=1$ ; (2) $l_1=r_2=1$ , $r_1$ is a nontrivial proper suffix of $u_2$ and $l_2$ is a nontrivial proper prefix of $u_1$ .", "We define a composition of nonzero polynomials $f_1,f_2$ to be a polynomial of the form $S(f_1,f_2)[l_1,r_1,l_2,r_2]=\\frac{l_1f_1r_1}{\\operatorname{\\rm LC}(f_1)}-\\frac{l_2f_2r_2}{\\operatorname{\\rm LC}(f_2)},$ where $(l_1,r_1,l_2,r_2)$ is an ambiguity of $\\big (\\operatorname{\\rm LW}(f_1),\\operatorname{\\rm LW}(f_2)\\big )$ .", "Definition 1.1 Let $G$ be a subset of $k\\langle X\\rangle $ .", "A polynomial $f$ is trivial modulo $G$ if it has a presentation $f=\\sum _{i\\in I}a_iu_ig_iv_i,\\quad a_i\\in k,\\; u_i, v_i\\in X^,\\; g_i\\in G\\backslash \\lbrace 0\\rbrace $ with $u_i\\operatorname{\\rm LW}(g_i)v_i\\le _{\\rm deglex}\\operatorname{\\rm LW}(f)$ .", "The set $G$ is called a Gröbner set in $k\\langle X\\rangle $ if any composition of nonzero polynomials in $G$ is trivial modulo $G$ .", "Let $\\mathfrak {a}$ be an ideal of $k\\langle X\\rangle $ .", "A word $u$ is called an obstruction of the algebra $A=k\\langle X\\rangle /\\mathfrak {a}$ if $\\operatorname{\\rm LW}(\\mathfrak {a})$ contains $u$ but no proper factor of $u$ .", "If $G$ is a Gröbner set that generates $\\mathfrak {a}$ , then the set of obstructions of $A$ coincides with the set $\\lbrace u\\in \\operatorname{\\rm LW}(G)\\,\|\\, u \\text{ has no proper factor in }\\operatorname{\\rm LW}(G)\\rbrace $ .", "Note that there exists a unique reduced Gröbner set $G_0$ that generates $\\mathfrak {a}$ .", "If $\\mathfrak {a}$ is homogeneous, then $G_0$ consists of homogeneous polynomials and it contains a minimal generating set of $\\mathfrak {a}$ ([18]).", "The following version of Bergman's Diamond Lemma assures the usefulness of Gröbner bases theory.", "Proposition 1.2 [4] Let $\\mathfrak {a}$ be an ideal of $k\\langle X\\rangle $ and $G\\subseteq \\mathfrak {a}$ .", "Then the following statements are equivalent: $G$ is a Gröbner set that generates $\\mathfrak {a}$ .", "The leading word of every nonzero polynomial in $\\mathfrak {a}$ is reducible modulo $G$ .", "Every polynomial in $\\mathfrak {a}$ is trivial modulo $G$ .", "${\\rm Irr}(G)$ forms a basis of the quotient algebra $k\\langle X\\rangle /\\mathfrak {a}$ ." ], [ "Lyndon Words", "In this subsection we give a short summary of facts about Lyndon words which we shall use.", "All of the facts in this subsection are well-known.", "Our main references are [5], [7], [10], [12].", "Definition 1.3 A word $u\\in X^$ is called a Lyndon word if $u\\ne 1$ and $u>_{\\rm lex} wv$ for every factorization $u=vw$ with $v,w\\ne 1$ .", "The set of Lyndon words on $X$ is denoted $\\mathbb {L}=\\mathbb {L}(X)$ .", "Immediately, by definition, there is no non-empty word that is both a proper prefix and a proper suffix of a Lyndon word.", "In particular, every Lyndon word is not a power of another word.", "There are alternative characterizations for checking whether a word is Lyndon.", "Proposition 1.4 The following statements are equivalent for any non-empty word $u$ : $u$ is a Lyndon word.", "$u >_{\\rm lex} w$ for every factorization $u=vw$ with $v,w \\ne 1$ .", "$v >_{\\rm lex} w$ for every factorization $u = vw$ with $v,w\\ne 1$ .", "By [10] one has $(1) \\Leftrightarrow (2)$ and it is easy to see $(2) \\Rightarrow (3)$ .", "Assume that $(3)$ holds and let $u = vw$ with $v,w\\ne 1$ be an arbitrary factorization.", "Note that $v>_{\\rm lex}w$ .", "If $v$ is not a prefix of $w$ , then $u = vw >_{\\rm lex} w$ .", "Otherwise $v$ is a prefix of $w$ , and then $w=v^kw^{\\prime }$ for some $k\\ge 1$ and $w^{\\prime }\\in X^$ such that $v$ is not a prefix of $w^{\\prime }$ .", "Therefore $u=v^{k+1}w^{\\prime }$ and $v\\ge _{\\rm lex} v^{k+1} >_{\\rm lex} w^{\\prime }$ .", "Observe that $v^{k+1} >_{\\rm lex} v^kw^{\\prime } = w$ and $v^{k+1}$ is not a prefix of $w$ , one gets $u>_{\\rm lex}w$ .", "Now the implication $(3) \\Rightarrow (2)$ is completed.", "Example 1.5 The Lyndon words of length five or less in two letters $x_1,x_2$ are as follows: $x_1,\\,x_2,\\, x_2x_1,\\, x_2x_1^2,\\, x_2^2x_1,\\, x_2x_1^3,\\, x_2^2x_1^2, \\,x_2^3x_1,\\, x_2x_1^4,\\, x_2x_1x_2x_1^2,\\, x_2^2x_1^3,\\, x_2^2x_1x_2x_1, \\,x_2^3x_1^2,\\,x_2^4x_1.$ Lyndon words have many good combinatorial features.", "We list below the corresponding version of [7], [12] and [7] respectively in our context.", "Proposition 1.6 Let $u=w_1w_2,\\, v=w_2w_3$ be Lyndon words.", "If $u >_{\\rm lex} v$ (which holds in priori when $w_2\\ne 1$ ), then $w=w_1w_2w_3$ is a Lyndon word.", "If $u=u^{\\prime }u^{\\prime \\prime }$ with $u$ a Lyndon word and $u^{\\prime \\prime }$ its longest proper suffix that is Lyndon, then $u^{\\prime }$ is a Lyndon word.", "We call the pair $\\operatorname{\\rm Sh }(u)=(u^{\\prime },u^{\\prime \\prime })$ the Shirshov factorization of $u$ .", "Suppose that $u >_{\\rm lex} v$ are Lyndon words.", "Then the Shirshov factorization of $uv$ is $(u,v)$ iff either $u$ is a letter or $u^{\\prime \\prime }\\le _{\\rm lex} v$ when $\\operatorname{\\rm Sh }(u)=(u^{\\prime },u^{\\prime \\prime })$ .", "Every word $u\\ne 1$ can be written uniquely as a product $u=w_1\\cdots w_r$ of Lyndon words with $w_1\\le _{\\rm lex} \\cdots \\le _{\\rm lex}w_r$ .", "We call such a decomposition the Lyndon decomposition of $u$ .", "If $v$ is a Lyndon word and it is a factor of $u$ with Lyndon decomposition $u=w_1\\cdots w_r$ , then $v$ is a factor of one of the words $w_1,\\cdots ,w_r$ .", "Remark 1.7 Parts $(1),(2)$ of Proposition REF allow us to get every Lyndon word by starting with $X$ and concatenating inductively each pair of Lyndon words $v,w$ with $v>_{\\rm lex}w$ .", "Note that in the Lyndon decomposition $u=w_1\\cdots w_r$ , $w_r$ is the longest Lyndon suffix of $u$ .", "Lemma 1.8 Assume that $u$ is a Lyndon word with $\\operatorname{\\rm Sh }(u)=(u^{\\prime },u^{\\prime \\prime })$ .", "If $v$ is a Lyndon word that is a factor of $u$ , then either $v$ is a factor of $u^{\\prime }$ , or $v$ is a factor of $u^{\\prime \\prime }$ , or $v$ is a prefix of $u$ such that $l(v)> l(u^{\\prime })$ .", "It is suffice to exclude the possibility that there are factorizations $u^{\\prime }=pl,\\, v=lr,\\, u^{\\prime \\prime }=rq$ with $p,l,r\\ne 1$ .", "Otherwise one would have a proper Lyndon suffix $lu^{\\prime \\prime }$ of $u$ by the part (1) of Proposition REF , which is impossible by the part $(2)$ of Proposition REF ." ], [ "Artin-Schelter regular algebras", "An Artin-Schelter regular algebra (AS-regular, for short) is a positively $\\mathbb {Z}$ -graded algebra $A=\\bigoplus _{m\\ge 0}A_m$ which is connected ($A_0=k$ ) and satisfies the following three conditions: $A$ has finite global dimension $d$ ; $A$ has finite Gelfand-Kirillov dimension ($\\operatorname{\\rm GKdim }$ , for short); $A$ is Gorenstein; that is, for some $l\\in \\mathbb {Z}$ , $\\operatorname{\\underline{Ext}}^i_A(k_A,A)=\\left\\lbrace \\begin{array}{ll}0,&i\\ne d,\\\\k(l),& i=d,\\end{array}\\right.$ where $k_A$ is the trivial right $A$ -module $A/A_{>0}$ , and the notation $(l)$ is the degree $l$ -shifting on graded modules.", "The index $l$ will be called the Gorenstein parameter of $A$ .", "All known examples of AS-regular algebras are strongly Noetherian, Auslander-regular and Cohen-Macaulay.", "We refer to [20] for a review of the definitions." ], [ "Invariants of graded algebras", "In this section we focus on the calculation of invariants of graded algebras of which all obstructions are Lyndon words.", "We provide the details in the way for a consistency.", "The readers will note some of familiar results, for example in [7], [8], with different forms and different certificates.", "Definition 2.1 We call a set $U$ of Lyndon words closed if $U \\supseteq X$ and $U$ contains each Lyndon word that is a factor of some word in $U$ .", "Also we call a set $V$ of words an antichain if any word in $V$ has no proper factor in $V$ .", "Let $\\mathcal {Y}$ be the set of all closed subsets of $\\mathbb {L}$ and let $\\mathcal {Z}$ be the set of all subsets of $\\mathbb {L}$ that contain no letters.", "Define three set maps as follows: $\\Phi : \\mathcal {Y} \\rightarrow \\mathcal {Z}, & \\quad &\\Phi (U) = \\lbrace \\ v\\in \\mathbb {L}\\backslash U\\;\|\\; v \\text{ has no proper factor in }\\mathbb {L}\\backslash U\\ \\rbrace .", "\\\\\\bar{\\Phi }: \\mathcal {Y} \\rightarrow \\mathcal {Z}, & \\quad &\\bar{\\Phi }(U) = \\lbrace \\ v\\in \\mathbb {L}\\backslash U\\;\|\\; v^{\\prime },v^{\\prime \\prime }\\in U \\text{ whenever }\\operatorname{\\rm Sh }(v)=(v^{\\prime },v^{\\prime \\prime })\\ \\rbrace .", "\\\\\\Psi : \\mathcal {Z} \\rightarrow \\mathcal {Y}, &\\quad &\\Psi (U) = \\lbrace \\ v\\in \\mathbb {L}\\;\\;\|\\;v \\text{ has no factor in }U\\ \\rbrace .$ Clearly $\\Phi (U)$ is always an antichain for any $U\\in \\mathcal {Y}$ , and $\\Phi (\\Psi (V))=V$ for any antichain $V\\in \\mathcal {Z}$ .", "Also one has $\\Phi (U)\\subseteq \\bar{\\Phi }(U)$ for any $U\\in \\mathcal {Y}$ and $\\Psi \\circ \\Phi = \\Psi \\circ \\bar{\\Phi } = \\operatorname{\\rm id }_{\\mathcal {Y}}$ .", "Example 2.2 We list some typical examples of closed subsets of $\\mathbb {L}(\\lbrace x_1,x_2\\rbrace )$ , which occur in the presentation of low dimensional $\\mathbb {Z}^2$ -graded Artin-Schelter regular algebras [18].", "Table: NO_CAPTIONCase $(4)$ shows that $\\Phi \\ne \\bar{\\Phi }$ in general.", "Example 2.3 (Fibonacci words) The sequence of Fibonacci words $\\lbrace f_m\\rbrace _{m\\ge 0}$ on two words $x_1,x_2$ is given by the initial conditions $f_0=x_1$ , $f_1=x_2$ , and then for $r\\ge 1$ , $f_{2r} = f_{2r-1}f_{2r-2}$ and $f_{2r+1} = f_{2r-1}f_{2r}$ .", "Obviously the length of the $p$ -th Fibonacci word is the $p$ -th Fibonacci number.", "A simple induction on $r$ gives that every Fibonacci word is a Lyndon word and they are sorted as $f_0 <_{\\rm lex} f_2 <_{\\rm lex} \\cdots <_{\\rm lex} f_{2r} <_{\\rm lex} \\cdots <_{\\rm lex} f_{2r+1} <_{\\rm lex} \\cdots <_{\\rm lex} f_3 <_{\\rm lex} f_1.$ Also one has the following facts: $ \\operatorname{\\rm Sh }(f_{2r})=(f_{2r-1},f_{2r-2})$ and $\\operatorname{\\rm Sh }(f_{2r+1}) = (f_{2r-1}, f_{2r})$ for any $r\\ge 1$ .", "The set $U$ of all Fibonacci words is a closed set with $\\Phi (U)=\\lbrace f_{2r-1}f_{2r+1}\\,\|\\, r\\ge 1\\rbrace \\cup \\lbrace f_{2r}f_{2r-2}\\,\|\\, r\\ge 1\\rbrace .$ For any $p\\ge 2$ , the set $U_p=\\lbrace f_0,f_1,\\cdots , f_{p-1}\\rbrace $ is a closed set with $\\Phi (U_p)=\\lbrace f_{2r-1}f_{2r+1}\\,\|\\, r\\ge 1,\\, 2r+1\\le p\\rbrace \\cup \\lbrace f_{2r}f_{2r-2}\\,\|\\, r\\ge 1,\\,2r\\le p\\rbrace \\cup \\lbrace f_p\\rbrace ,$ in particular, $\\#(\\Phi (U_p))=p-1$ .", "Clearly, the sets $U_2,U_3,U_4,U_5$ coincide with that given in Example REF .", "(1) A simple induction on $r$ together with the part (3) of Proposition REF gives the result.", "(2) We firstly show that any proper Lyndon factor $v$ of $f_{2r}$ is a factor of $f_{2r-1}$ or of $f_{2r-2}$ .", "By Lemma REF and the part (1), it suffices to exclude the possibility that $v$ is a prefix of $f_{2r}$ such that $l(v)>l(f_{2r-1})$ .", "Otherwise there would exist $1\\le k\\le r-1$ such that $f_{2r}=f_{2r-2k-1}wf_{2r-2k-1}f_{2r-2k-2}$ and $l(f_{2r-1})\\le l(f_{2r-2k-1}w) < l(v) \\le l(f_{2r-2k-1}wf_{2r-2k-1})$ .", "Let $v=f_{2r-2k-1}wt$ , where $t\\ne 1$ .", "Then $t$ is both a prefix and a suffix of $v$ , which is impossible.", "A similar discussion gives that any proper Lyndon factor $v$ of $f_{2r+1}$ is a factor of $f_{2r-1}$ or of $f_{2r}$ .", "Thus Lyndon factors of Fibonacci words are Fibonacci words and so $U$ is a closed set.", "Now set $W=\\lbrace f_{2r-1}f_{2r+1}\\,\|\\, r\\ge 1\\rbrace \\cup \\lbrace f_{2r}f_{2r-2}\\,\|\\, r\\ge 1\\rbrace $ .", "We have $W\\subseteq \\Phi (U)$ by Proposition REF .", "Note that every $f_p$ has a proper prefix $f_{2r-1}$ (resp.", "a suffix $f_{2r}$ ) whenever $2r-1<p$ (resp.", "$2r<p$ ).", "Therefore if $i>j+1$ , then $f_{2i}f_{2j}$ has a proper suffix $f_{2j+2}f_{2j}$ and $f_{2j-1}f_{2i-1}$ has a proper prefix $f_{2j-1}f_{2j+1}$ .", "Also if $i>j$ then $f_{2i-1}f_{2j-2}$ has a proper suffix $f_{2j}f_{2j-2}$ , if $i=j$ then $f_{2j-1}f_{2j-2}=f_{2j}$ , if $i=j-1$ then $f_{2j-3}f_{2j-2}=f_{2j-1}$ , and if $i<j-1$ then $f_{2i-1}f_{2j-2}$ has a proper prefix $f_{2i-1}f_{2i+1}$ .", "Since every element $u\\in \\Phi (U)$ is of the form $f_rf_s$ with $f_r>_{\\rm lex}f_s$ , the above discussion implies that $u\\in W$ .", "Thus $\\Phi (U)=W$ .", "(3) Clearly $U_p$ is a closed set by the part (2) above.", "The equality is obtained by a similar discussion of that for $U$ given in the previous paragraph.", "The next result gives a bound of $\\Phi (U)$ and $\\bar{\\Phi }(U)$ for any $U\\in \\mathcal {Y}$ .", "It generalizes [7].", "For any closed set $U$ , let $\\Upsilon (U)=\\lbrace (u,v)\\,\|\\, u,v\\in U,\\, u>_{\\rm lex}v\\, \\text{and there is no}\\, w \\in U \\,\\text{such that}\\, u>_{\\rm lex}w>_{\\rm lex}v \\rbrace .$ Proposition 2.4 Let $U$ be a closed set of Lyndon words.", "Then $\\operatorname{\\rm Sh }(uv)=(u,v)$ for any $(u,v)\\in \\Upsilon (U)$ and the map $\\Upsilon (U) \\rightarrow \\mathbb {L}$ , given by assigning each $(u,v)\\in \\Upsilon (U)$ to the concatenating $uv$ , is an injection with image contained in $\\Phi (U)$ .", "In particular, $\\lbrace uv\\,\|\\,(u,v)\\in \\Upsilon (U)\\rbrace \\subseteq \\Phi (U)\\subseteq \\bar{\\Phi }(U)\\subseteq \\lbrace uv\\,\|\\,u>_{\\rm lex}v\\in U\\rbrace \\backslash U.$ Firstly we show that $\\operatorname{\\rm Sh }(uv)=(u,v)$ for any $(u,v)\\in \\Upsilon (U)$ .", "If $u$ is a letter, then it is true.", "If $u$ is not a letter, write $\\operatorname{\\rm Sh }(u)=(u^{\\prime },u^{\\prime \\prime })$ .", "Then $u>_{\\rm lex}v\\ge _{\\rm lex} u^{\\prime \\prime }$ and thus $\\operatorname{\\rm Sh }(u)=(u,v)$ by the part (3) of Proposition REF .", "Secondly we show that the map is injective.", "Otherwise, assume that $uv=u^{\\prime }v^{\\prime }$ for some distinct pairs $(u,v),\\,(u^{\\prime }v^{\\prime })\\in \\Upsilon (U)$ , then without lost of generality, let $u$ be a proper prefix of $u^{\\prime }$ .", "Observe that there is a factorization $u^{\\prime }=rw,\\, v=ws$ with $r,w,s\\ne 1$ , one gets a contradiction $u>_{\\rm lex} u^{\\prime } >_{\\rm lex} w >_{\\rm lex} v$ by Proposition REF .", "Thirdly we show $uv\\in \\Phi (U)$ for any $(u,v)\\in \\Upsilon (U)$ .", "Note that $uv\\notin U$ for otherwise one would have $u>_{\\rm lex}uv>_{\\rm lex}v$ .", "Assume that $uv\\notin \\Phi (U)$ , then $uv$ has a proper factor $w\\in \\Phi (U)$ .", "Let $\\operatorname{\\rm Sh }(w)=(w^{\\prime },w^{\\prime \\prime })$ , then $w^{\\prime },w^{\\prime \\prime }\\in U$ and $w$ is a prefix of $uv$ with $l(w^{\\prime })\\ge l(u)$ by Lemma REF .", "If $l(w^{\\prime })=l(u)$ , then $w^{\\prime \\prime }$ is a proper prefix of $v$ and hence $u=w^{\\prime }>_{\\rm lex} w^{\\prime \\prime } >_{\\rm lex} v$ which is impossible.", "If $l(w^{\\prime })>l(u)$ , then there are factorizations $w^{\\prime }=us,\\, v=sv^{\\prime \\prime }$ with $s,v^{\\prime \\prime }\\ne 1$ , which gives rise to a contradiction $u>_{\\rm lex} w^{\\prime } >_{\\rm lex} s >_{\\rm lex} v$ by Proposition REF .", "The inclusion relations now follows immediately.", "Corollary 2.5 Let $U$ be a finite closed set of Lyndon words with $\\#(U)=d$ .", "Then $d-1 \\le \\#(\\Phi (U)) \\le \\#(\\bar{\\Phi }(U)) \\le \\frac{d(d-1)}{2}.$ Definition 2.6 Let $V$ be an antichain of words.", "The graph of chains on $V$ is the directed graph $\\Gamma (V)$ , whose set of vertices is $\\lbrace 1\\rbrace \\cup (X\\backslash V) \\cup S$ , where $S$ is the set of all proper suffix of words in $V$ that is of length $\\ge 2$ , and whose set of arrows is defined as follows: for any $x_i\\in X\\backslash V$ there is an arrow $1\\rightarrow x_i$ ; for any two vertices $u,v\\in (X\\backslash V) \\cup S$ , there is an arrow $u\\rightarrow v$ if and only $uv$ has a unique factor in $V$ which is a suffix of $uv$ .", "Clearly if $1\\rightarrow v_1\\rightarrow v_2\\rightarrow \\cdots \\rightarrow v_p$ and $1\\rightarrow v_1^{\\prime }\\rightarrow v_2^{\\prime }\\rightarrow \\cdots \\rightarrow v_q^{\\prime }$ are two paths in $\\Gamma (V)$ such that $v_1v_2\\cdots v_p= v_1^{\\prime }v_2^{\\prime }\\cdots v_q^{\\prime }$ , then $p=q$ and $v_i=v_i^{\\prime }$ for $i=1,\\cdots , p$ .", "For $p\\ge 1$ , a word $w$ is called a $p$ -chain on $V$ if there is path $1\\rightarrow v_1\\rightarrow v_2\\rightarrow \\cdots \\rightarrow v_p$ in $\\Gamma (V)$ such that $w=v_1v_2\\cdots v_p$ .", "The empty word 1 is called the 0-chain on $V$ .", "Denote by $C_p(V)$ the set of all $p$ -chains on $V$ for all $p\\ge 0$ .", "Readily one has $C_{0}(V)=\\lbrace 1\\rbrace $ , $C_{1}(V)=X\\backslash V$ and $C_{2}(V)=V\\backslash X$ .", "Lemma 2.7 Let $U$ be a closed set of Lyndon words and let $V=\\Phi (U)$ .", "Then for any path $v_1\\rightarrow v_2\\rightarrow v_3$ in $\\Gamma (V)$ , if there exist $u_1 >_{\\rm lex} \\bar{u}_2\\in U$ and a non-empty suffix $u_1^{\\prime }$ of $u_1$ such that $u_1^{\\prime }\\bar{u}_2$ is a suffix of $v_1v_2$ and $l(\\bar{u}_2)\\le l(v_2) \\le l(u_1^{\\prime })+l(\\bar{u}_2)$ , then there exist $u_2,\\bar{u}_3\\in U$ and a word $t_2$ such that $v_3=t_2\\bar{u}_3$ , $u_2$ is a suffix of $v_2t_2$ , $u_2^{\\prime }:=\\bar{u}_2t_2$ is a suffix of $u_2$ and $u_1>_{\\rm lex}u_2>_{\\rm lex}\\bar{u}_3$ .", "for any $p\\ge 2$ and any path $v_1\\rightarrow v_2\\rightarrow \\cdots \\rightarrow v_p$ in $\\Gamma (V)$ , if there exist $u_1,\\bar{u}_2\\in U$ such that $u_1\\bar{u}_2 = v_1v_2$ and $l(\\bar{u}_2)\\le l(v_2)$ , then there exist $u_2,\\cdots ,u_{p-1},\\bar{u}_p$ in $U$ and a non-empty suffix $u_i^{\\prime }$ of $u_i$ for $i=2,\\cdots , p-1$ such that $u_1>_{\\rm lex}u_2>_{\\rm lex}\\cdots >_{\\rm lex}\\bar{u}_p$ and $v_1v_2\\cdots v_p=u_1u_2^{\\prime }u_3^{\\prime }\\cdots u_{p-1}^{\\prime }\\bar{u}_p$ .", "for any $p\\ge 2$ and any path $1\\rightarrow v_1\\rightarrow v_2\\rightarrow \\cdots \\rightarrow v_p$ in $\\Gamma (V)$ , there exists a sequence $u_1>_{\\rm lex}u_2 >_{\\rm lex}\\cdots >_{\\rm lex}u_p$ in $U$ and a non-empty suffix $u_i^{\\prime }$ of $u_i$ for $i=2,\\cdots , p-1$ such that $v_1v_2\\cdots v_p=u_1u_2^{\\prime }u_3^{\\prime }\\cdots u_{p-1}^{\\prime }u_p$ .", "(1) Let $w_2$ be the unique factor of $v_2v_3$ in $V$ with $\\operatorname{\\rm Sh }(w_2)=(w_2^{\\prime },w_2^{\\prime \\prime })$ .", "There are four cases: (i) $l(w_2^{\\prime \\prime })\\le l(v_3)$ and $l(w_2) \\le l(\\bar{u}_2)+ l(v_3)$ .", "Let $v_3=tw_2^{\\prime \\prime }$ .", "Set $u_2=\\bar{u}_2t_2$ and $\\bar{u}_3=w_2^{\\prime \\prime }$ .", "Then $u_2$ is a Lyndon word by the part (1) of Proposition REF , and $u_1>_{\\rm lex} \\bar{u}_2\\ge _{\\rm lex} u_2\\ge _{\\rm lex} w_2^{\\prime }>_{\\rm lex} w_2^{\\prime \\prime }=\\bar{u}_3$ .", "(ii) $l(w_2^{\\prime \\prime })\\le l(v_3)$ and $l(w_2) > l(\\bar{u}_2)+ l(v_3)$ .", "Then $w_2^{\\prime }=a\\bar{u}_2t_2$ and $v_3=t_2w_2^{\\prime \\prime }$ for some words $a \\ (\\ne 1)$ and $t_2$ .", "Set $u_2=w_2^{\\prime }$ , $\\bar{u}_3=w_2^{\\prime \\prime }$ .", "Then $u_1 >_{\\rm lex} u_2>_{\\rm lex} \\bar{u}_3$ .", "(iii) $l(v_3) < l(w_2^{\\prime \\prime })\\le l(\\bar{u}_2)+ l(v_3)$ .", "Clearly one can find a Lyndon suffix $\\bar{w}$ of $w_2^{\\prime \\prime }$ such that $l(\\bar{w})>l(v_3)$ and $l(\\bar{w}^{\\prime \\prime })\\le l(v_3)$ , where $\\operatorname{\\rm Sh }(\\bar{w}) = (\\bar{w}^{\\prime },\\bar{w}^{\\prime \\prime })$ .", "Let $v_3=t_2\\bar{w}^{\\prime \\prime }$ .", "Set $u_2= \\bar{u}_2t_2$ and $\\bar{u}_3=\\bar{w}^{\\prime \\prime }$ .", "Then again $u_2$ is a Lyndon word by the part (1) of Proposition REF , and $u_1 >_{\\rm lex} \\bar{u}_2 \\ge _{\\rm lex} u_2 \\ge _{\\rm lex} \\bar{w}^{\\prime } >_{\\rm lex} \\bar{w}^{\\prime \\prime }=\\bar{u}_3$ .", "(iv) $l(w_2^{\\prime \\prime })> l(\\bar{u}_2)+ l(v_3)$ .", "Then one can find a Lyndon suffix $\\bar{w}$ of $w_2^{\\prime \\prime }$ such that $l(\\bar{w})>l(\\bar{u}_2)+ l(v_3)$ and $l(\\bar{w}^{\\prime \\prime })\\le l(\\bar{u}_2)+ l(v_3)$ , where $\\operatorname{\\rm Sh }(\\bar{w}) = (\\bar{w}^{\\prime },\\bar{w}^{\\prime \\prime })$ .", "In this situation, there are two subcases: one is $l(\\bar{w}^{\\prime \\prime })\\le l(v_3)$ , which reduces to the case (ii); and the other one is $l(\\bar{w}^{\\prime \\prime })>l(v_3)$ , which reduces to the case (iii).", "We have exhausted all possibilities and the result (1) follows.", "(2) Let $u_1^{\\prime }=u_1$ .", "Apply the part (1) iteratively $p-2$ times, one gets the result.", "(3) Let $u_1, \\bar{u}_2$ be given by $\\operatorname{\\rm Sh }(v_1v_2)=(u_1,\\bar{u}_2)$ .", "Readily $l(\\bar{u}_2)\\le l(v_2)$ because $v_1$ is a letter.", "The result now follows from the part (2) by setting $u_p=\\bar{u}_p$ .", "Lemma 2.8 Let $U$ be a closed set of Lyndon words and let $V=\\Phi (U)$ .", "Let $p\\ge 2$ and $u_1,u_2,\\cdots ,u_p\\in U$ with $(u_1,u_2),\\cdots ,(u_{p-1},u_p)\\in \\Upsilon (U)$ .", "Let $u_1=x_iu_1^{\\prime }$ .", "Then $1\\rightarrow x_i\\rightarrow u_1^{\\prime }u_2\\rightarrow u_3\\rightarrow \\cdots \\rightarrow u_p$ is a path in $\\Gamma (V)$ .", "In particular the concatenating $u_1u_2\\cdots u_p$ is a $p$ -chain on $V$ .", "Note that $x_i,u_1^{\\prime }u_2, u_3,\\cdots ,u_p$ are vertices of $\\Gamma (V)$ and there are arrows $x_i\\rightarrow u_1^{\\prime }u_2$ , $u_3\\rightarrow u_4$ ,$\\cdots $ , $u_{p-1}\\rightarrow u_p$ in $\\Gamma (V)$ by Proposition REF .", "Thus it remains to show that there is an arrow $u_1^{\\prime }u_2\\rightarrow u_3$ in $\\Gamma (V)$ .", "Otherwise, there is a non-empty suffix $l$ of $u^{\\prime }$ and a non-empty proper prefix $r$ of $u_3$ such that $w=lu_2r\\in V$ .", "Then by Lemma REF , there are four cases: (i) $\\operatorname{\\rm Sh }(w)=(l,u_2r)$ .", "One has $u_2>_{\\rm lex} u_2r>_{\\rm lex}u_3$ .", "(ii) $\\operatorname{\\rm Sh }(w)=(l^{\\prime },l^{\\prime \\prime }u_2r)$ for some non-empty proper prefix $l^{\\prime }$ of $l$ .", "One has $u_1>_{\\rm lex} l^{\\prime \\prime }u_2r >_{\\rm lex} u_3$ .", "(iii) $\\operatorname{\\rm Sh }(w)=(lu_2, r)$ .", "One has $u_1 >_{\\rm lex} lu_2 >_{\\rm lex} u_2$ .", "(iv) $\\operatorname{\\rm Sh }(w)=(lu_2r^{\\prime }, r^{\\prime \\prime })$ for some non-empty proper suffix $r^{\\prime \\prime }$ of $r$ .", "One has $u_1 >_{\\rm lex} lu_2r^{\\prime } >_{\\rm lex} u_3$ .", "Since there is no member in $U$ other than $u_2$ between $u_1$ and $u_3$ , all cases listed above is impossible and hence there is an arrow $u_1^{\\prime }u_2\\rightarrow u_3$ in $\\Gamma (V)$ .", "Remark 2.9 In [8], the author characterized the graph of chains on an antichain $V$ , for which the monomial algebra $k\\langle X\\rangle /(V)$ is of finite global dimension and of finite GK-dimension.", "It is worth to mention that Lemma REF is equivalent to [8], while the part (3) of Lemma REF is stronger than [8].", "Proposition 2.10 Let $U=\\lbrace z_1 >_{\\rm lex} z_2 >_{\\rm lex} \\cdots >_{\\rm lex} z_d\\rbrace $ be a finite closed set of Lyndon words and let $V=\\Phi (U)$ .", "Then $1\\rightarrow z_1\\rightarrow z_2 \\rightarrow \\cdots \\rightarrow z_d$ is the unique path of length $d$ in $\\Gamma (V)$ started at 1 and there is no path of length $p > d$ in $\\Gamma (V)$ started at 1.", "In particular, $C_d(V)=\\lbrace z_1z_2\\cdots z_d\\rbrace $ and $C_p(V)=\\emptyset $ for any $p>d$ .", "Note that $z_1=x_1$ and so $1\\rightarrow z_1\\rightarrow z_2\\rightarrow \\cdots \\rightarrow z_d$ is a path in $\\Gamma (V)$ by Lemma REF .", "Also there are no paths of length $p>d$ in $\\Gamma (V)$ started at 1 by the part (3) of Lemma REF .", "Assume that $1\\rightarrow v_1\\rightarrow v_2\\rightarrow \\cdots \\rightarrow v_d$ is a path in $\\Gamma (V)$ .", "Write $\\operatorname{\\rm Sh }(v_1v_2)=(u_1,\\bar{u}_2)$ .", "Then $u_1=z_1$ by the part (2) of Lemma REF and so it is a letter.", "Hence $v_1=z_1$ .", "Apply the part (1) of Lemma REF on the path $z_1\\rightarrow v_2\\rightarrow v_3$ with $u_1^{\\prime }=u_1=z_1$ and $\\bar{u}_2=v_2$ , one gets $u_2=v_2t_2$ and $u_2\\bar{u}_3=v_2v_3$ .", "Thus $u_2=z_2$ by applying the part (2) of Lemma REF on the path $v_2\\rightarrow v_3\\rightarrow \\cdots \\rightarrow v_p$ .", "Since $z_2\\ge _{\\rm lex} v_2$ , one has $v_2=z_2$ .", "Now by induction suppose $v_j=z_j$ for $j\\le i$ , where $i\\ge 2$ .", "Then apply the part (1) of Lemma REF on the path $z_{i-1}\\rightarrow z_i \\rightarrow v_{i+1}$ with $u_{i-1}^{\\prime }=u_{i-1}=z_{i-1}$ and $\\bar{u}_i=z_i$ , then $u_i >_{\\rm lex} \\bar{u}_{i+1}$ , $v_{i+1} = t_i\\bar{u}_{i+1}$ and $u_i^{\\prime }=u_i=z_it_i$ .", "If $t_i\\ne 1$ , the part (2) of Lemma REF for the path $z_i\\rightarrow v_{i+1}\\rightarrow \\cdots \\rightarrow v_p$ gives that $z_i>_{\\rm lex}u_i >_{\\rm lex} u_{i+1}>_{\\rm lex}\\cdots >_{\\rm lex} u_d$ and thus there are $d-i+1$ elements in $U$ that is less than $z_i$ , which is impossible.", "So $t_i=1$ and hence $z_i=u_i, v_{i+1}=\\bar{u}_{i+1}$ .", "Note that $\\bar{u}_{i+1}$ is a prefix of $u_{i+1}$ in this case and hence $z_i>_{\\rm lex} \\bar{u}_{i+1} \\ge _{\\rm lex} u_{i+1} >_{\\rm lex} u_{i+2}>_{\\rm lex}\\cdots >_{\\rm lex} u_d$ .", "Therefore, one has $v_{i+1} = \\bar{u}_{i+1} = u_{i+1} = z_{i+1}$ .", "The last statement is obtained by definition.", "Now we turn to the main result of this section, which provides an efficient way to calculate invariants of certain graded algebras.", "Theorem 2.11 Let $U$ be a closed set of Lyndon words and let $A=k\\langle X\\rangle /(G)$ , where $G$ is a set of homogeneous polynomials such that $\\Phi (U)\\subseteq \\operatorname{\\rm LW}(G) \\subseteq \\mathbb {L}\\backslash U$ .", "If $G$ is a Gröbner set then one has: The set of obstructions of $A$ is $\\Phi (U)$ .", "${\\rm Irr}(G) = \\lbrace u_1u_2\\cdots u_s\\,\|\\,u_1\\le _{\\rm lex} u_2\\le _{\\rm lex} \\cdots \\le _{\\rm lex} u_s \\in U,\\ s\\ge 1\\rbrace \\cup \\lbrace 1\\rbrace $ .", "The Hilbert series of $A$ is $H_A(t)=\\prod _{u\\in U}\\big (1-t^{\\deg (u)}\\big )^{-1}$ and $\\operatorname{\\rm GKdim }A=\\#(U)$ .", "The global dimension of $A$ satisfies $\\operatorname{\\rm gldim}(A)\\le \\#(\\Phi (U))+1$ .", "If $U$ is a finite set then $\\operatorname{\\rm gldim}A = \\#(U)$ and the last nonzero term of the minimal free resolution of $k_A$ is $A(-l)$ , where $l=\\sum _{u\\in U}\\deg (u)$ .", "(1) By the definition, the set of obstructions of $A$ coincides with the set $\\lbrace u\\in \\operatorname{\\rm LW}(G)\\,\|\\,u \\text{ has no proper factor in }\\operatorname{\\rm LW}(G)\\rbrace ,$ which equals to $\\Phi (U)$ .", "(2) Note that $U=\\Psi (\\operatorname{\\rm LW}(G))$ .", "Apply the parts $(4),(5)$ of Proposition REF , one gets the result.", "(3) It is an easy combinatoric exercise to see $H_A(t)=\\prod _{u\\in U}\\big (1-t^{\\deg (u)}\\big )^{-1}$ .", "Then by [17] one obtains $\\operatorname{\\rm GKdim }A=\\#(U)$ .", "(4) Assume that $\\Phi (U)$ is a finite set.", "Suppose $1\\rightarrow v_1\\rightarrow v_2\\rightarrow \\cdots \\rightarrow v_p$ is a path in the graph $\\Gamma (\\Phi (U))$ .", "Let $w_i$ be the unique factor of $v_iv_{i+1}$ in $\\Phi (U)$ for $i=1,\\cdots , p-1$ .", "Immediately one has $w_1>_{\\rm lex}w_2>_{\\rm lex}\\cdots >_{\\rm lex}w_{p-1}$ .", "Thus $p-1\\le \\#(\\Phi (U))$ .", "Therefore there is no $p$ -chain on $\\Phi (U)$ if $p > \\#(\\Phi (U))+1$ .", "Thus [1] gives that $\\operatorname{\\rm gldim}(A) =\\text{pd}(k_A) \\le \\#(\\Phi (U))+1$ .", "(5) Let $\\#(U)=d$ and let $0\\rightarrow F_d\\rightarrow F_{d-1} \\rightarrow \\cdots \\rightarrow F_1\\rightarrow A\\rightarrow k\\rightarrow 0$ be the minimal free resolution of $k_A$ .", "Consider the Anick's resolution $\\mathcal {E}$ of $k_A$ : $&&\\cdots \\rightarrow kC_p\\otimes A\\rightarrow \\cdots \\rightarrow kC_d\\otimes A\\xrightarrow{} kC_{d-1}\\otimes A \\rightarrow \\cdots \\rightarrow kC_1\\otimes A \\rightarrow A\\rightarrow k_A\\rightarrow 0,$ where $C_i= C_i(\\Phi (U))$ .", "By Proposition REF , one has $kC_p=0$ for any $p>d$ and $kC_d \\cong k(-l)$ , where $l=\\sum _{u\\in U}\\deg (u)$ .", "Also by the part (3) of Lemma REF , one has $kC_{d-1} \\cong \\bigoplus _{c\\in C_{d-1}}k(-r_c)$ with $r_c < l$ for each $c\\in C_{d-1}$ .", "Thus in the complex $\\mathcal {E}\\otimes _A k$ , the differential $\\delta _d\\otimes _A k=0$ and hence $\\operatorname{\\underline{Tor}}^A_d(k,k)=k(-l)$ and $\\operatorname{\\underline{Tor}}^A_p(k,k)=0$ for any $p>d$ .", "The result now follows from the facts $\\operatorname{\\rm gldim}A = \\sup \\lbrace n\\,\|\\, \\operatorname{\\underline{Tor}}^A_n(k_A,{}_Ak)\\ne 0\\rbrace $ and $F_p\\cong \\operatorname{\\underline{Tor}}^A_{p}(k_A,{}_Ak)\\otimes A$ .", "Lemma 2.12 Let $U$ be a finite closed set of Lyndon words on $X=\\lbrace x_1,\\cdots ,x_n\\rbrace $ with $\\#(U)=d$ .", "Assume that $\\deg (x_1)=\\cdots =\\deg (x_n)=1$ , then $\\sum _{u\\in U} \\deg (u) \\le \\frac{\\phi ^{d-n+4}-\\psi ^{d-n+4}}{\\phi -\\psi }+n-3, \\quad \\text{ where }\\, \\phi =\\frac{1+\\sqrt{5}}{2},\\, \\psi =\\frac{1-\\sqrt{5}}{2}.$ Denote $a_i$ the $i$ -th Fibonacci number and $s_i=\\sum _{j=0}^ia_j$ .", "Then, by induction, $s_i=a_{i+2}-1$ .", "Enumerating elements in $U$ in deglex order as $x_1 \\le _{\\rm deglex} \\cdots \\le _{\\rm deglex} x_{n-2} \\le _{\\rm deglex} x_{n-1}=g_0\\le _{\\rm deglex} x_n=g_1\\le _{\\rm deglex} g_2 \\le _{\\rm deglex} \\cdots \\le _{\\rm deglex} g_{d-n+1},$ then $\\deg (g_i)\\le a_i$ for $i=0,\\cdots ,d-n+1$ as showed in [7].", "Therefore $\\sum _{u\\in U} \\deg (u) \\le s_{d-n+1} + n-2 = a_{d-n+3}+n-3 = \\frac{\\phi ^{d-n+4}-\\psi ^{d-n+4}}{\\phi -\\psi }+n-3,$ where the last equality follows from the general formula $a_i=\\frac{\\phi ^{i+1}-\\psi ^{i+1}}{\\phi -\\psi }$ for $i\\ge 0$ .", "Combine the parts (3), (5) of Theorem REF and Lemma REF , we get: Corollary 2.13 Let $A$ be an AS-regular algebra of global dimension $d$ and of Gorenstein parameter $l$ which is generated in degree one with $\\dim A_1=n$ .", "If all obstructions of $A$ with respect to some choice of a basis $x_1,\\cdots ,x_n$ of $A_1$ are Lyndon words, then $\\operatorname{\\rm GKdim }A=d$ and $l\\le \\frac{\\phi ^{d-n+4}-\\psi ^{d-n+4}}{\\phi -\\psi }+n-3.$ Remark 2.14 Note that AS-regular algebras of which the obstructions are Lyndon words is a context more general than the universal enveloping algebras of positively graded Lie algebras.", "Thus Corollary REF can be thought of as a counterpart of [6].", "However, we are unable to decide whether or not the given estimation is optimal.", "Indeed, suppose that $\\mathfrak {g}$ is a graded Lie algebra of dimension $d$ which is generated in degree one with $\\dim \\mathfrak {g}_1=n$ .", "Then alike the discussion in [6], the highest possible Gorenstein parameter of the universal enveloping algebra $U(\\mathfrak {g})$ is $\\binom{d-n+2}{2}+n-1 &= (1+1+2+\\cdots +d-n+1) + n-2\\\\&\\le (a_0+a_1+a_2+\\cdots +a_{d-n+1})+n-2\\\\&=\\frac{\\phi ^{d-n+4}-\\psi ^{d-n+4}}{\\phi -\\psi }+n-3.$ It is interesting to find AS-regular algebras which is generated in degree one and of which the Gorenstein parameter is larger than $\\binom{d-n+2}{2}+n-1$ .", "Note that the only examples known to us are [18]." ], [ "Construction of Artin-Schelter regular algebras from Lyndon words", "In this section we turn to the construction of AS-regular algebras from Lyndon words.", "The major tool is the generalized bracketing on Lyndon words defined inductively by Shirshov factorization.", "Throughout this section, $X=\\lbrace x_1,\\cdots , x_n\\rbrace $ , $q=[q_{i,j}]_{n\\times n}$ is a matrix with nonzero entries in $k$ .", "Also $u,v,w$ denote words on $X$ .", "Let $q_{u,1}=q_{1,u}=1$ , then define $q_{x_i,x_ju}=q_{i,j}q_{x_i,u}$ and $q_{ux_i,v}=q_{u,v}q_{x_i,v}$ inductively.", "Obviously, the following equalities hold: $q_{u,vw}=q_{u,v}q_{u,w},\\quad q_{uv,w}=q_{u,v}q_{v,w}.$ Let $[-,-]: k\\langle X\\rangle \\otimes k\\langle X\\rangle \\rightarrow k\\langle X\\rangle $ be the bilinear operation given by $[u,v]=uv-q_{u,v}vu$ .", "Then $[[u,v],w] &= [u,[v,w]]-q_{u,v} v[u,w]+q_{v,w}[u,w]v, \\\\[uv,w] &= u[v,w] + q_{v,w}[u,w]v, \\\\[u,vw] &= [u,v]w + q_{u,v} v[u,w].$ Now set $[x_i]=x_i$ for letters $x_i\\in X$ .", "If $u$ is a Lyndon word with $l(u)>1$ and $\\operatorname{\\rm Sh }(u)=(u^{\\prime },u^{\\prime \\prime })$ , define $[u] := [[u^{\\prime }],[u^{\\prime \\prime }]].$ Note that, inductively, $[u]$ is a monic homogeneous polynomial with $\\operatorname{\\rm LW}([u])=u$ and all words occur in $[u]$ have the same constitute.", "In particular $[u]=[u^{\\prime }][u^{\\prime \\prime }]-q_{u^{\\prime },u^{\\prime \\prime }}[u^{\\prime \\prime }][u^{\\prime }]$ if $\\operatorname{\\rm Sh }(u)=(u^{\\prime },u^{\\prime \\prime })$ .", "Definition 3.1 A polynomial of the form $[u]$ , where $u$ is a Lyndon word, is called a super-letter.", "A finite product of super-letters is called a super-word.", "Obviously, if $D=[u_1]\\cdots [u_r]$ is a super-word, then $D$ is a monic homogeneous polynomial with $\\operatorname{\\rm LW}(D)=u_1\\cdots u_r$ and every word occurs in $D$ has the same constitute with $u_1u_2\\cdots u_r$ .", "Moreover, super-letters are in one-to-one correspondence with Lyndon words, and every super-word has a unique factorization in super-letters [9].", "Thus one can define a total order on the set of all super-letters by $[u]> [v]\\;\\; \\text{ iff }\\;\\; u>_{\\rm lex}v,$ and it then extends to a lexicographical order on the set of super-words in a natural way.", "Definition 3.2 A super-word $D=[u_1][u_2]\\cdots [u_r]$ is called monotonic if $[u_1]\\le [u_2]\\le \\cdots \\le [u_r]$ .", "It is easy to see that all monotonic super-words form a basis of $k\\langle X\\rangle $ by the part (4) of Proposition REF .", "Moreover, if $D=[u_1]\\cdots [r_r]$ and $D^{\\prime }=[u^{\\prime }_1]\\cdots [u^{\\prime }_s]$ are monotonic super-words, then $D>_{\\rm lex}D^{\\prime }$ iff $u_1\\cdots u_r >_{\\rm lex} u^{\\prime }_1\\cdots u^{\\prime }_s$ [10].", "Thus with the part (4) of Proposition REF at hand, the lexicographical order on super-words is compatible with that on ordinary words.", "In the sequel, we will associate to each tuple $I=(i_1,\\cdots ,i_r)$ a new variable $x_I=x_{(i_1,\\cdots ,i_r)}$ .", "We will frequently write $x_I=x_{i_1i_2\\cdots i_r}$ if there is no risk of confusions.", "In particular, we identify $x_{(i)}=x_i$ for $i=1, \\cdots , n$ .", "For any Lyndon word $u=x_{i_1}\\cdots x_{i_r}$ let $\\rho (u)=(i_1,\\cdots , i_r)$ , and for any closed set $U$ of Lyndon words let $X_U=\\lbrace x_{\\rho (u)}\\,\|\\,u\\in U\\rbrace .$ The degree function on $X_U^$ is given by assigning $x_{\\rho (u)}$ to the number $\\deg (u)$ .", "Clearly $X\\subseteq X_U$ and $k\\langle X\\rangle $ is a graded subalgebra of $k\\langle X_U\\rangle $ .", "The total order on $X$ extends to a total order on $X_U$ by saying $x_{\\rho (u)}> x_{\\rho (v)}$ iff $u>_{\\rm lex}v$ for any $u,v\\in U$ .", "Thus the lex order and the deglex order on $X_U^$ are extensions of that on $X^$ respectively.", "Define $[-,-]:k\\langle X_U\\rangle \\otimes k\\langle X_U\\rangle \\rightarrow k\\langle X_U\\rangle $ to be the linear map by $[D,D^{\\prime }]=DD^{\\prime }-q_{u_1\\cdots u_r, v_1\\cdots v_s}D^{\\prime }D,$ where $D=x_{\\rho (u_1)}\\cdots x_{\\rho (u_r)},\\, D^{\\prime }=x_{\\rho (v_1)}\\cdots x_{\\rho (v_s)}$ .", "It is an extending of the linear map $[-,-]:k\\langle X\\rangle \\otimes k\\langle X\\rangle \\rightarrow k\\langle X\\rangle $ .", "Let $\\beta =\\beta _U:k\\langle X_U\\rangle \\rightarrow k\\langle X\\rangle \\ $ be the homomorphism of graded algebras given by $\\beta (x_{\\rho (u)})=[u]$ for each $u\\in U$ .", "Note that $\\beta \|_{k\\langle X\\rangle }=\\operatorname{\\rm id }_{k\\langle X\\rangle }.$ For any $u\\in U\\backslash X$ with $\\operatorname{\\rm Sh }(u)=(v,w)$ , we denote $f_u=[x_{\\rho (v)},x_{\\rho (w)}] - x_{\\rho (u)}.$ Then a simple induction on the length of $u\\in U$ implies that $[u]-x_{\\rho (u)}\\in (F_U), \\text{ where } F_U=\\lbrace f_u\\,\|\\, u\\in U\\backslash X\\rbrace .$ Therefore, for any set $G$ of homogeneous polynomials, $\\beta $ induces an isomorphism of graded algebras $k\\langle X_U\\rangle /(F_U\\cup G) \\xrightarrow{} k\\langle X\\rangle /(G).$ The inverse is the homomorphism induced by the injection map $k\\langle X\\rangle \\subseteq k\\langle X_U\\rangle $ .", "In particular, when $G=\\emptyset $ , one gets $k\\langle X_U\\rangle /(F_U)\\cong k\\langle X\\rangle $ .", "Lemma 3.3 Let $U$ be a closed set of Lyndon words.", "For each $v\\in \\bar{\\Phi }(U)$ we associate with a homogeneous polynomial $g_v\\in k\\langle X\\rangle $ of degree $\\deg (v)$ that is a linear combination of super-words $[u_1]\\cdots [u_r]<_{\\rm lex}[v]$ with $u_1,\\cdots , u_r\\in U$ .", "Let $G=\\lbrace \\overline{g}_v=[v]-g_v\\,\|\\,v\\in \\bar{\\Phi }(U)\\rbrace $ .", "Then for each pair $u>_{\\rm lex}u^{\\prime }\\in U$ , there is a homogeneous polynomial $h_{u,u^{\\prime }}\\in k\\langle X_U\\rangle $ of degree $\\deg (uu^{\\prime })$ such that $h_{u,u^{\\prime }}$ is a linear combination of lexicographical lesser words than $x_{\\rho (u)}x_{\\rho (u^{\\prime })}$ , $(H)=(F_U\\cup G)$ , where $H=\\lbrace \\bar{h}_{u,u^{\\prime }}=[x_{\\rho (u)},x_{\\rho (u^{\\prime })}]-h_{u,u^{\\prime }}\\,\|\\,u>_{\\rm lex}u^{\\prime }\\in U\\rbrace $ .", "Moreover, $k\\langle X\\rangle /(G)\\cong k\\langle X_U\\rangle /(H)$ as graded algebras and the following statements are equivalent: $G$ is a Gröbner set with respect to the deglex order on $X^$ .", "$H$ is a Gröbner set with respect to the deglex order on $X_U^$ .", "$J(u,v,w) = [h_{u,v},x_{\\rho (w)}] - [x_{\\rho (u)}, h_{v,w}] + q_{u,v} x_{\\rho (v)}h_{u,w} - q_{v,w} h_{u,w}x_{\\rho (v)}$ is trivial modulo $H$ with respect to the deglex order on $X_U^*$ for each triple $u>_{\\rm lex}v>_{\\rm lex}w \\in U$ .", "By a simpler discussion as that given in the proof of [9], there is a homogeneous polynomials $c_{(u\|u^{\\prime })}\\in k\\langle X\\rangle $ of degree $\\deg (uu^{\\prime })$ for each pair $u>_{\\rm lex}u^{\\prime }\\in U$ such that $\\bar{c}_{(u\|u^{\\prime })}=[[u],[u^{\\prime }]]-c_{(u\|u^{\\prime })} \\in (G)$ , $c_{(u\|u^{\\prime })}$ is a linear combination of super-words $[u_1]\\cdots [u_r]\\le _{\\rm lex}[uu^{\\prime }]$ with $u_1,\\cdots ,u_r\\in U$ .", "In particular, if $\\operatorname{\\rm Sh }(uu^{\\prime })=(u,u^{\\prime })$ and $uu^{\\prime }\\in U$ then $c_{(u\|u^{\\prime })}=[uu^{\\prime }]$ , and if $\\operatorname{\\rm Sh }(uu^{\\prime })=(u,u^{\\prime })$ but $uu^{\\prime }\\notin U$ then $c_{(u\|u^{\\prime })}=g_{uu^{\\prime }}$ .", "Let $h_{u,u^{\\prime }}$ be the polynomial in $k\\langle X_U\\rangle $ obtained from $c_{(u\|u^{\\prime })}$ by replacing each super-word $[u_1]\\cdots [u_r]$ occurs in $c_{(u\|u^{\\prime })}$ with $x_{\\rho (u_1)}\\cdots x_{\\rho (u_r)}$ .", "Clearly $h_{u,u^{\\prime }}$ is a linear combination of lexicographical lesser words than $x_{\\rho (u)}x_{\\rho (u^{\\prime })}$ .", "By construction, we have the following observations: $f_u=\\bar{h}_{u^{\\prime },u^{\\prime \\prime }}$ for any word $u\\in U\\backslash X$ with $\\operatorname{\\rm Sh }(u)=(u^{\\prime },u^{\\prime \\prime })$ and so $F_U\\subseteq H$ , $\\beta (\\bar{h}_{u,u^{\\prime }}) = \\bar{c}_{(u\|u^{\\prime })}$ and so $\\bar{h}_{u,u^{\\prime }}-\\bar{c}_{(u\|u^{\\prime })} \\in (F_U)$ .", "Hence one has $(F_U\\cup G) = (F_U \\cup \\lbrace \\bar{c}_{(u\|u^{\\prime })}\\,\|\\,u>_{\\rm lex}u^{\\prime }\\in U\\rbrace ) = (H)$ .", "Note that $\\beta $ induces an isomorphism of graded algebras $k\\langle X_U\\rangle /(H) \\xrightarrow{} k\\langle X\\rangle /(G)$ .", "Thus all monotonic super-words in super-letters $[u], u\\in U$ form a basis of $k\\langle X\\rangle /(G)$ iff all monotonic words on $X_U$ form a basis of $k\\langle X_U\\rangle /(H)$ .", "Then Proposition REF implies the final equivalence relations.", "Remark 3.4 Lemma REF is used in the sequel to simplify the computation in determining whether $G$ is a Gröbner set (see Example REF ).", "However, it is interesting in its own right.", "Given a Lie algebra $\\mathfrak {g}=\\text{Lie}(X)/(G)_L$ , where $G$ is a set of Lie polynomials and $(G)_L$ is the Lie ideal of $\\text{Lie}(X)$ generated by $G$ .", "Assume that $\\lbrace e_i\\rbrace _{i\\in I}$ is an ordered basis of $\\mathfrak {g}$ with structure constants $\\lbrace [e_i,e_j]-\\sum a_{i,j}^re_r\\rbrace _{i>j}$ , then Lemma REF is a generalization of the following formula: $k\\langle X\\rangle /(G) \\cong U(\\mathfrak {g}) \\cong k\\langle x_i\\;\|\\;i\\in I\\rangle /(\\lbrace [x_i,x_j]- \\textstyle \\sum a_{i,j}^r x_r\\rbrace _{i>j}),$ where $U(g)$ denotes the universal enveloping algebra of $\\mathfrak {g}$ .", "Definition 3.5 Let $G \\subseteq k\\langle X\\rangle $ be a set of homogeneous polynomials.", "A super-letter $[u]$ is said to be hard modulo $G$ if, in the algebra $k\\langle X \\rangle /(G)$ , $[u]$ is not a linear combination of super-words of the same degree in lexicographical lesser super-letters than $[u]$ .", "Clearly if $u$ is a Lyndon word that is irreducible modulo $G$ , then $[u]$ is hard modulo $G$ .", "Lemma 3.6 Let $U$ be a finite closed set of Lyndon words and let $A=k\\langle X\\rangle /(G)$ , where $G$ is a set of homogeneous polynomials such that $\\Phi (U)\\subseteq \\operatorname{\\rm LW}(G) \\subseteq \\mathbb {L}\\backslash U$ .", "If $G$ is a Gröbner set modulo which there are no hard super-letters other than $[u], u\\in U$ , then one has: There exists an algebra filtration $\\mathcal {F} = \\lbrace F_i\\rbrace _{i\\ge 0}$ one $A$ such that, as graded algebras, $\\text{\\rm gr}_{\\mathcal {F}}(A)\\, \\cong \\, k\\langle X_U\\rangle /([x_{\\rho (u)},x_{\\rho (u^{\\prime })}]: u>_{\\rm lex}u^{\\prime }\\in U).$ $A$ is AS-regular of global dimension $d=\\#(U)$ and of Gorenstein parameter $l=\\sum _{u\\in U} \\deg (u)$ .", "Moreover, $A$ is strongly Noetherian, Auslander-regular and Cohen-Macaulay.", "(1) By Theorem REF , the set of all monotonic super-words in super-letters $[u],u\\in U$ is a basis of $A$ .", "Since there are no hard super-letters other than $[u],u\\in U$ , if $v$ is a reducible word, then $[v]$ is not hard and hence it is a linear combination of super-words of the same degree in lexicographical lesser super-letters than $[v]$ in $A$ .", "A similar, but simpler, discussion of that given in [10] implies the result.", "(2) By [19], [20] and the part (1), we obtain that $\\text{gr}_{\\mathcal {F}}(A)$ is AS-regular, strongly Noetherian, Auslander-regular and Cohen-Macaulay, and so is $A$ by [19] and [16].", "Apply Theorem REF , we have $d=\\#(U)$ and $l=\\sum _{u\\in U} \\deg (u)$ .", "Note that the sufficient condition given in Lemma REF for a graded algebra $k\\langle X\\rangle /(G)$ to be AS-regular is not easy to check in general.", "In the remaining of the paper, we focus on providing practical approaches for constructing AS-regular algebras.", "Definition 3.7 To each closed set $U$ of Lyndon words we associate with a graded algebra $A(U,q)=k\\langle X\\rangle /(G(U,q)),\\quad \\text{where}\\quad G(U,q)=\\lbrace [v]\\,\|\\,v\\in \\bar{\\Phi }(U)\\rbrace .$ In general, $(G(U,q)) \\ne (\\lbrace [v]\\,\|\\,v\\in \\Phi (U)\\rbrace )$ , however, if $G(U,q)$ is a Gröbner set, then the obstruction of $A(U,q)$ is $\\Phi (U)$ and $(G(U,q))= (\\lbrace [v]\\,\|\\,v\\in \\Phi (U)\\rbrace )$ .", "Theorem 3.8 Let $U$ be a finite closed set of Lyndon words.", "If $G(U,q)$ is a Gröbner set, then $A(U,q)$ is AS-regular of global dimension $d=\\#(U)$ and of Gorenstein parameter $l=\\sum _{u\\in U} \\deg (u)$ .", "Moreover, $A(U,q)$ is strongly Noetherian, Auslander-regular and Cohen-Macaulay.", "Let $A=A(U,q)$ .", "Note that there are no reducible Lyndon words of length 1, and if $v$ is a reducible Lyndon word of length 2 then $[v]=0$ in $A$ because $v\\in \\bar{\\Phi }(U)$ .", "Assume that $[v]=0$ in $A$ for any reducible Lyndon word $v$ of length less than $p$ .", "Now let $v_0$ be an arbitrary reducible Lyndon word of length $p$ with $\\operatorname{\\rm Sh }(v_0)=(v^{\\prime }_0,v^{\\prime \\prime }_0)$ .", "If $v_0^{\\prime },v_0^{\\prime \\prime }\\in U$ , then $v_0\\in \\bar{\\Phi }(U)$ and hence $[v_0]=0$ in $A$ .", "If $v_0^{\\prime }\\notin U$ (resp.", "$v_0^{\\prime \\prime }\\notin U$ ), then by induction one has $[v_0^{\\prime }]=0$ (resp, $[v_0^{\\prime \\prime }]=0$ ) in $A$ , and both implies that $[v_0]=0$ in $A$ .", "Therefore $[v]=0$ in $A$ for any reducible Lyndon word $v$ and so there are no hard super-letters other than $[u],u\\in U$ .", "Now the result follows from Lemma REF .", "Remark 3.9 There is no counterpart of Theorem REF if we set $G(U,q)=\\lbrace [v]\\,\|\\,v\\in \\Phi (U)\\rbrace $ in Definition REF , since in this situation there may exists some $v\\in \\bar{\\Phi }(U)\\backslash \\Phi (U)$ such that $[v]\\notin (G(U,q))$ .", "However, for any Lyndon words $u,v$ with $u=avb$ there exists a factorization $b=cd$ such that $[u]=[a[vc]d]$ .", "This fact follows immediately from Lemma REF through a simple induction on the length of $u$ .", "It is crucial in the Gröbner bases theory for Lie algebras [5].", "Example 3.10 Let $U=X$ .", "Then $G(U,q)=\\lbrace \\ [x_j,x_i]\\;\|\\;1\\le i<j\\le n\\ \\rbrace $ .", "Clearly $G(U,q)$ is a Gröbner set.", "The algebra $A(U,q)=\\mathcal {O}_q(k^n)$ is called the multiparameter quantum affine space.", "It is AS-regular of global dimension $n$ and of Gorenstein parameter $n$ .", "Example 3.11 We continue to use the symbols given in Example REF and give some AS-regular algebras below by Theorem REF .", "The computation is based on Lemma REF .", "In each case, the equation system obtained from all $J(u,v,w)$ with $u>_{\\rm lex}v>_{\\rm lex}w$ are not very complicated and so it is solved by hand.", "(1) $A(U_2,q)=k\\langle x_1,x_2\\rangle /([x_2,x_1])$ .", "$G(U_2,q)$ is always a Gröbner set.", "$A$ is the quantum plane and it is AS-regular of global dimension 2 and of Gorenstein parameter 2.", "(2) $A(U_3,q)\\cong k\\langle x_1,x_{21}, x_2\\rangle /(f_1,f_2,f_3)$ , where $f_1=[x_2,x_{21}],\\quad f_2=[x_2, x_1] - x_{21},\\quad f_3=[x_{21},x_1].$ $G(U_3,q)$ is a Gröbner set iff $q_{22}=q_{11}$ .", "In this case, $A(U_3,q)$ is the algebra of type $S_1$ in [3].", "It is AS-regular of global dimension 3 and of Gorenstein parameter 4.", "(3) $A(U_4,q) \\cong k\\langle x_1,x_{21},x_{221},x_2\\rangle /(f_1,f_2,\\cdots ,f_6)$ , where $&f_1=[x_2,x_{221}],& &f_2=[x_2,x_{21}]-x_{221},& & f_3=[x_2,x_1]-x_{21},\\\\&f_4=[x_{221},x_{21}],& &f_5=[x_{221},x_1]- q_{21}(q_{11}-q_{22})x_{21}^2,& &f_6=[x_{21},x_1].$ $G(U_4,q)$ is a Gröner set iff one of the following conditions holds: $q_{22}=1,\\, q_{11}=1,\\, q_{21}q_{12}=1$ .", "$q_{22}=\\zeta ,\\, q_{11}=\\zeta ,\\, q_{21}q_{12}=\\zeta ^2$ , where $\\zeta ^2+\\zeta +1=0$ .", "$q_{11}=q_{22}^2,\\, \\, q_{21}q_{12}q_{22}^2=1$ .", "Taking Case (a), Case (b) and Case (c), $A(U_4,q)$ are corresponding to $D(-2q_{21},-q_{21})$ , $D(q_{21}\\zeta ^2, -q_{21}\\zeta ^2)$ and $D(-q_{21}-q_{21}q_{22}^2,-q_{21}q_{22})$ in [13] respectively.", "They are AS-regular of global dimension 4 and of Gorenstein parameter 7.", "Remark 3.12 In the case (c), let $q_{22}$ be a root of the polynomial $x^2-\\frac{v}{p}x+1$ and let $q_{21}=pq_{22}-v$ , then we obtain the algebra $D(v,p)$ ; that is, we can reconstruct $D(v,p)$ as a deformation of the universal enveloping algebra $D(-2,-1)$ .", "(4) $A(U_5,q) \\cong k\\langle x_1,x_{21},x_{22121},x_{221},x_2\\rangle /(f_1,f_2,\\cdots ,f_{10})$ , where $&&f_1=[x_2,x_{221}],\\qquad f_2=[x_2,x_{22121}],\\qquad f_3=[x_2,x_{21}]-x_{221},\\qquad f_4=[x_2,x_1]-x_{21},\\\\&&f_5=[x_{221},x_{22121}],\\;\\; f_6=[x_{221},x_{21}]-x_{22121},\\qquad f_7=[x_{221},x_1]- q_{21}(q_{11}-q_{22})x_{21}^2,\\\\&&f_8=[x_{22121},x_{21}], \\quad f_9=[x_{22121},x_1]- q_{21}^2q_{11}(q_{11}-q_{22})(1-q_{22}^2q_{21}q_{12})x_{21}^3,\\quad f_{10}=[x_{21},x_1].$ There does not exist $q$ such that $G(U_5,q)$ is a Gröbner set.", "Indeed, consider $J(x_2,x_{221},x_{21})$ , $J(x_2,x_{221},x_1)$ and $J(x_2,x_{22121},x_{21})$ , one gets a system of equations with no solutions: $q_{21}q_{12}q_{11}=1,\\quad q_{22}^2+q_{22}+1=0,\\quad q_{11}=-q_{22}^3=-1,\\quad q_{22}^6(q_{21}q_{12})^5q_{11}^4=1.$ (5) $A(U^{\\prime }_5,q) \\cong k\\langle x_1,x_{21},x_{221},x_{2221},x_2\\rangle /(f_1,f_2,\\cdots ,f_{10})$ , where $&&f_1=[x_2,x_{2221}],\\;\\qquad f_2=[x_2,x_{221}]-x_{2221},\\qquad f_3=[x_2,x_{21}]-x_{221},\\\\&&f_4=[x_2,x_1]-x_{21},\\quad f_5=[x_{2221},x_{221}],\\quad f_6=[x_{2221},x_{21}]-q_{22}^2q_{21}(q_{21}q_{12}q_{11}-1)x_{221}^2,\\\\&&f_7=[x_{2221},x_1]- \\big (q_{22}q_{21}^2(q_{22}q_{21}q_{12}q_{11}+1)(q_{11}-q_{22}) + q_{21}^2q_{11}(1-q_{22}^4q_{21}q_{12})\\big ) x_{21}x_{221},\\\\&&f_8=[x_{221},x_{21}], \\;\\qquad f_9=[x_{221},x_1]- q_{21}(q_{11}-q_{22})x_{21}^2,\\qquad f_{10}=[x_{21},x_1].$ $G(U_5^{\\prime },q)$ is a Gröbner set iff $q_{22}=q_{11}=1$ and $q_{21}q_{12}=1$ .", "In this case, $A(U_5^{\\prime },q)$ corresponds to $\\mathcal {D}(q_{21})$ in [18].", "It is AS-regular of global dimension 5 and of Gorenstein parameter 11.", "(6) $A(U^{\\prime \\prime }_5,q) \\cong k\\langle x_1,x_{211},x_{21},x_{221},x_2\\rangle /(f_1,f_2,\\cdots ,f_{10})$ , where $&&f_1=[x_2,x_{221}],\\;\\quad f_2=[x_2,x_{21}]-x_{221},\\quad f_3=[x_2,x_{211}],\\quad f_4=[x_2,x_1]-x_{21},\\\\&&f_5=[x_{221},x_{21}],\\quad f_6=[x_{221},x_{211}],\\;\\;\\qquad f_7=[x_{221},x_1]- q_{21}(q_{11}-q_{22})x_{21}^2,\\\\&&f_8=[x_{21},x_{211}],\\quad f_9=[x_{21},x_1]-x_{211},\\quad f_{10}=[x_{211},x_1].$ $G(U_5^{\\prime \\prime },q)$ is a Gröbner set iff one of the following conditions holds: $q_{22}=1,\\, q_{11}=1,\\, \\, q_{21}q_{12}=1$ .", "$q_{22}=\\zeta ,\\, q_{11}=\\zeta ,\\, q_{21}q_{12}=\\zeta ^2$ , where $\\zeta ^2+\\zeta +1=0$ .", "In Case (a) and Case (b), $A(U_5^{\\prime \\prime },q)$ correspond to $\\mathcal {L}(-2q_{21},\\frac{1}{2},\\frac{1}{2})$ and $\\mathcal {L}(q_{21}\\zeta ^2,-1,q)$ where $q^2-q+1=0$ , in [18] respectively.", "They are AS-regular of global dimension 5 and of Gorenstein parameter 10.", "Proposition 3.13 Assume the notations given in Example REF .", "Then, for any $r\\ge 5$ , there is no matrix $q$ such that $G(U_r,q)$ is a Gröbner set.", "By Part (4) of Example REF , it suffices to show that there is no matrix $q$ such that $G(U_r,q)$ is a Gröbner set for $r\\ge 6$ .", "Assume the notations given in Lemma REF .", "Then, for $r\\ge 6$ , one has $A(U_r,q)\\cong k\\langle x_u \\, \|\\, u\\in U_r\\rangle /(H).$ Clearly, $[x_2,x_{22121}],\\; [x_{22121},x_{21}],\\; [x_{221},x_{22121}]-x_{22122121},\\; [x_2,x_{21}]-x_{221}\\in H$ .", "Thus, modulo $H$ , $J(x_2,x_{22121},x_{21})&=q_{22}^3q_{21}^2 x_{22121}x_{221}-q_{22}^3q_{21}^3q_{12}^2q_{11}^2x_{221}x_{22121}\\\\&= q_{22}^3q_{21}^2\\big (1-q_{22}^6(q_{21}q_{12})^5 q_{11}^4\\big )x_{22121}x_{221} - q_{22}^6q_{21}^4q_{12}^3q_{11}^2x_{22122121}\\\\&\\ne 0.$ Therefore, $G(U_r,q)$ is not a Gröbner set for any $r\\ge 6$ .", "Remark 3.14 The existence of AS-regular algebras of which the set of obstructions is $\\Phi (U_r)$ is of particular interesting, for it relates to that whether or not the estimation in Corollary REF is optimal.", "The existence of such algebras is realized in Parts (1), (2), (3) of Example REF for $r=2,3,4$ , and in [18] for $r=5$ .", "However, [7] tells that there is no such bigraded algebras for $r=6$ .", "Naturally we have two questions: Does there exist a bigraded AS-regular algebra with two generators of which the set of obstructions is $U_r$ for $r\\ge 7$ ?", "Does there exist an AS-regular algebra with two generators of which the set of obstructions is $U_r$ for $r\\ge 6$ ?", "With Proposition REF at hand, we may get a negative answer for the first question.", "Now we turn to another approach which can be thought of as a generalization of the universal enveloping algebras of Lie algebras.", "Let $c:k\\langle X\\rangle \\otimes k\\langle X\\rangle \\rightarrow k\\langle X\\rangle \\otimes k\\langle X\\rangle $ be the linear map given by $c(u\\otimes v) = q_{u,v}v\\otimes u.$ Then $k\\langle X\\rangle \\otimes k\\langle X\\rangle $ is an associative algebra with multiplication $(\\mu \\otimes \\mu ) \\circ (\\operatorname{\\rm id }\\otimes c\\otimes \\operatorname{\\rm id })$ and unit $1\\otimes 1$ , where $\\operatorname{\\rm id }=\\operatorname{\\rm id }_{k\\langle X\\rangle }$ and $\\mu $ is the multiplication of $k\\langle X\\rangle $ .", "This algebra will be denoted $k\\langle X\\rangle \\otimes ^c k\\langle X\\rangle $ .", "Set $\\Delta : k\\langle X\\rangle \\rightarrow k\\langle X\\rangle \\otimes ^c k\\langle X\\rangle $ and $\\varepsilon :k\\langle X\\rangle \\rightarrow k$ to be the homomorphisms of algebras given by $\\Delta (x_i)=x_i\\otimes 1 + 1\\otimes x_i$ and $\\varepsilon (x_i)=0$ for every $x_i\\in X$ .", "The structure $k\\langle X\\rangle ^c = (k\\langle X\\rangle , \\Delta , \\varepsilon , c)$ is a braided bialgebra [2].", "By definition, a polynomial $f\\in k\\langle X\\rangle ^c$ is called primitive if $\\Delta (f)=f\\otimes 1+1\\otimes f$ .", "Theorem 3.15 Let $U$ be a finite closed set of Lyndon words and let $A=k\\langle X\\rangle /(G)$ , where $G$ is a set of homogeneous polynomials such that $\\Phi (U)\\subseteq \\operatorname{\\rm LW}(G) \\subseteq \\mathbb {L}\\backslash U$ .", "If $G$ is a Gröbner set and it consists of primitive elements in $k\\langle X\\rangle ^c$ such that $c(\\,kG\\otimes k\\langle X\\rangle + k\\langle X\\rangle \\otimes kG\\,) \\subseteq kG\\otimes k\\langle X\\rangle + k\\langle X\\rangle \\otimes kG$ .", "Then $A$ is AS-regular of global dimension $d=\\#(U)$ and of Gorenstein parameter $l=\\sum _{u\\in U} \\deg (u)$ .", "Moreover, $A$ is strongly Noetherian, Auslander-regular and Cohen-Macaulay.", "By [2], $(G)$ is a braided biideal of $k\\langle X\\rangle ^c$ and so $A$ is a braided bialgebra.", "Therefore, by [10], a super-letter $[u]$ is hard modulo $G$ iff $u$ is irreducible modulo $G$ .", "Thus there are no hard super-letters other than $[u],u\\in U$ .", "Now Lemma REF gives the result.", "Remark 3.16 Assume the notations given in Theorem REF .", "We consider the special case when $q_{i,j}=1$ for all $1\\le i,j\\le n$ .", "Then $c$ is the usual flip map, the bracket $[-,-]$ is the Lie operation and the set of primitive elements in $k\\langle X\\rangle ^c$ is the free Lie algebra $\\text{Lie}(X)$ .", "Note that $A=k\\langle X\\rangle /(G)$ is the universal enveloping algebra of the positively graded Lie algebra $\\mathfrak {g}=\\text{Lie}(X)/(G)_L$ .", "By [5], the set $\\lbrace \\ [u]\\;\|\\;u\\in U\\ \\rbrace $ is a basis of $\\mathfrak {g}$ .", "Thereby, in this case, Theorem REF is the well-known result that the universal enveloping algebra of a finite-dimensional positively graded Lie algebra is AS-regular.", "Lemma 3.17 Assume that $q_{i,j}q_{j,i}=1$ for any $1\\le i,j\\le n$ .", "Then a polynomial $f\\in k\\langle X\\rangle ^c$ is primitive iff it is a linear combination of super-letters.", "Note that $q_{u,v}q_{v,u}=1$ for any two words $u,v$ .", "The result follows from an essentially the same discussion of that for [12].", "Combine Lemma REF and Theorem REF , we conclude: Corollary 3.18 Assume that $q_{i,j}q_{j,i}=1$ for any $1\\le i,j\\le n$ .", "Let $U$ be a finite closed set of Lyndon words and $A=k\\langle X\\rangle /(G)$ , where $G$ is a set of homogeneous polynomials such that $\\Phi (U)\\subseteq \\operatorname{\\rm LW}(G) \\subseteq \\mathbb {L}\\backslash U$ .", "If $G$ is a Gröbner set in which every polynomial is a linear combination of super-letters of the same constitute.", "Then $A$ is AS-regular of global dimension $d=\\#(U)$ and of Gorenstein parameter $l=\\sum _{u\\in U}^d \\deg (u)$ .", "Moreover, $A$ is strongly Noetherian, Auslander-regular and Cohen-Macaulay." ] ]	1403.0385
[ [ "Bayes Merging of Multiple Vocabularies for Scalable Image Retrieval" ], [ "Abstract The Bag-of-Words (BoW) representation is well applied to recent state-of-the-art image retrieval works.", "Typically, multiple vocabularies are generated to correct quantization artifacts and improve recall.", "However, this routine is corrupted by vocabulary correlation, i.e., overlapping among different vocabularies.", "Vocabulary correlation leads to an over-counting of the indexed features in the overlapped area, or the intersection set, thus compromising the retrieval accuracy.", "In order to address the correlation problem while preserve the benefit of high recall, this paper proposes a Bayes merging approach to down-weight the indexed features in the intersection set.", "Through explicitly modeling the correlation problem in a probabilistic view, a joint similarity on both image- and feature-level is estimated for the indexed features in the intersection set.", "We evaluate our method through extensive experiments on three benchmark datasets.", "Albeit simple, Bayes merging can be well applied in various merging tasks, and consistently improves the baselines on multi-vocabulary merging.", "Moreover, Bayes merging is efficient in terms of both time and memory cost, and yields competitive performance compared with the state-of-the-art methods." ], [ "Introduction", "This paper considers the task of Bag-of-Words (BoW) based image retrieval, especially on multi-vocabulary merging.", "We aim at improving the retrieval accuracy while maintaining affordable memory and time cost.", "Figure: An Illustration of vocabulary correlation.", "Given a query feature, it is quantized to two visual words in two vocabularies.", "Then, two sets of indexed features, 𝒜\\mathcal {A} and ℬ\\mathcal {B}, are identified from the two inverted files, respectively.", "The area of the intersection set 𝒜∩ℬ\\mathcal {A}\\cap \\mathcal {B} (denoted as Card(𝒜∩ℬ)Card(\\mathcal {A}\\cap \\mathcal {B})) encodes the extent of correlation between the two sets.", "In this paper, we focus on the indexed features in 𝒜∩ℬ\\mathcal {A}\\cap \\mathcal {B}.The vocabulary (also called the codebook or quantizer) lies at the core of the BoW based image retrieval system.", "It functions by quantizing SIFT descriptors [9] to discrete visual words.", "The quantized visual words are the nearest centers to the feature vectors in the feature space.", "In order to reduce quantization error and improve recall, multiple vocabularies are often generated, and each feature is quantized to different visual words from multiple vocabularies.", "The primary benefit of using multiple vocabularies is that more candidate features are recalled, which corrects quantization artifacts to some extent.", "However, the routine of multi-vocabulary merging is affected by a crucial problem, , vocabulary correlation [3] (see Fig.", "REF ).", "Given a query feature, based on the inverted files with two individual vocabularies, two sets of indexed features $\\mathcal {A}$ and $\\mathcal {B}$ are identified, sharing an intersection set $\\mathcal {A}\\cap \\mathcal {B}$ .", "In this paper, the area of $\\mathcal {A}\\cap \\mathcal {B}$ is approximated by $Card (\\mathcal {A}\\cap \\mathcal {B})$ .", "The larger $Card(\\mathcal {A}\\cap \\mathcal {B})$ is, the larger the correlation will be.", "In an extreme case, total correlation occurs if $Card(\\mathcal {A}\\cap \\mathcal {B}) = Card(\\mathcal {A}\\cup \\mathcal {B})$ , and merging $\\mathcal {A}$ and $\\mathcal {B}$ brings no benefit.", "A straightforward method for multi-vocabulary merging consists in concatenating the BoW histograms of different vocabularies [11].", "In a microscopic view of this method, the indexed features in $\\mathcal {A}\\cap \\mathcal {B}$ are counted twice in Fig.", "REF .", "Nevertheless, since images in this area are mostly irrelevant ones (the number of relevant images is always very small), the over-counting may actually compromise the retrieval accuracy [12].", "In this paper, we consider the situation in which the given vocabularies are correlated, and we aim to reduce the impact of correlation.", "To address this problem, this paper proposes to model the vocabulary correlation problem from a probabilistic view.", "In a nutshell, we jointly estimate an image- and feature-level similarity for the indexed features in the intersection set (or overlapping area).", "Given a query feature, lists of indexed features are extracted from multiple inverted files.", "Then, we identify the intersection and union sets of the lists, from which the cardinality ratio is calculated.", "This ratio thus encodes the extent of correlation (see Fig.", "REF ).", "For the indexed images in the intersection set, its similarity with the query is estimated as a function of the cardinality ratio, and subsequently added to the matching score.", "Experiments on several benchmark datasets demonstrate that Bayes merging is effective, and yields competitive results with the state-of-the-art methods." ], [ "Related Work", "Vocabulary Generation The vocabulary provides a discrete partitioning of the feature space by visual words.", "Typically, either flat kmeans [12], [4] or hierarchical kmeans [11] is employed to train a vocabulary in an unsupervised manner.", "Improved methods include incorporating contextual information into the vocabulary [19], building super-sized vocabulary [16], [20], [10], making use of the active points [15], etc.", "Matching Refinement Feature-to-feature matching is a key issue in the BoW model.", "The baseline approach employs a coarse word-to-word matching, resulting in undesirable low precision.", "To improve precision, some works analyze the spatial contexts [16], [21], [24] of SIFT features, and use the spatial constraints as solution to refining matching.", "Another line of works extracts binary signatures from SIFT descriptors [4] or its contexts [23], [8].", "The feature matching is thus refined by a further check of the Hamming distance between binary signatures.", "In this paper, however, we argue that even if two features are adjacent in the feature space, the corresponding images are probably very different.", "Therefore, we are supposed to look one step further by estimating a joint similarity on both image- and feature-level from clues in multiple vocabularies.", "Multiple Vocabularies It is well known that multi-vocabulary merging is effective in improving recall [3], [18].", "Typically, multi-vocabulary merging can be performed either at score level, , by concatenating the BoW histograms [11], or at rank level, , by rank aggregation [7].", "On the other hand, some works also provide clues that multiple vocabularies also improve precision [1], [17].", "To address the problem of vocabulary correlation, Xia [18] propose to create the vocabularies jointly and reduce correlation from the view of vocabulary generation.", "A more relevant work includes [3], which uses PCA to implicitly remove correlation of given vocabularies, resulting in a low dimensional image representation.", "Our work departs from previous works in two aspects.", "First, we explicitly model the vocabulary correlation problem from a probabilistic view.", "Second, our work is proposed for the BoW based image retrieval task, which differs from NN search problems." ], [ "Notations", "Assume that the $K$ vocabularies are denoted as $\\mathcal {V}^{(k)} = \\lbrace v_1^{(k)}, v_2^{(k)}, ..., v_{s_k}^{(k)}\\rbrace , k = 1,...,K$ , where $v_i^{(k)}$ represents a visual word and $s_k$ is the vocabulary size.", "Correspondingly, built on $\\mathcal {V}^{(k)}$ , $K$ inverted files are organized as $\\mathcal {W}^{(k)} = \\lbrace W_1^{(k)}, W_2^{(k)}, ..., W_{s_k}^{(k)}\\rbrace , k = 1,...,K$ , where each entry $W_i^{(k)}$ contains a list of indexed features.", "Given a query SIFT feature $x$ , it is quantized to a visual word tuple $\\left(v^{(1)}, v^{(2)},...,v^{(K)}\\right)$ , where $v^{(k)}, k=1,...,K$ is the nearest centroid in $\\mathcal {V}^{(k)}$ to $x$ .", "With the $K$ visual words we can identify $K$ sets of indexed features in entries $\\lbrace W_{i_k}^{(k)}\\rbrace _{k=1}^K$ .", "From the $K$ sets, we can define three types of sets to be used in this paper.", "Definition 1 ($\\mathbf {k}^{th}$ -order intersection set) The intersection set of $k$ , and only $k$ sets, denoted as $\\cap ^{(k)}$ , $k\\ge 2$ .", "Definition 2 ($\\mathbf {k}^{th}$ -order union set) The union set of $k$ , and only $k$ sets, denoted as $\\cup ^{(k)}$ , $k\\ge 2$ .", "Definition 3 (difference set) The set in which no overlapping exists, i.e., $\\cup ^{(K)} - \\sum _{k = 2}^{K} \\cap ^{(k)}$ , $K\\ge 2$ ." ], [ "Baselines", "Single vocabulary baseline (B$_0$ ) For a single vocabulary, we adopt the baseline introduced in [12], [4].", "Specifically, vocabularies are trained by AKM on the independent Flickr60K data [4], and average IDF [22] weighting scheme is used.", "We replace the original SIFT descriptor with rootSIFT [13].", "In this scenario, we denote the matching function between two features $x$ and $y$ as, $f_0(x, y) = \\delta _{v_x, v_y}$ where $v_x$ and $v_y$ are visual words of $x$ and $y$ in the vocabulary, respectively, and $\\delta (\\cdot )$ is the Kronecker delta response.", "Conventional vocabulary merging (B$_1$ ) Given $K$ vocabularies, B$_1$ simply concatenates multiple BoW histograms [11].", "It is equivalent to a simple score-level addition of the outputs of multiple vocabularies.", "The matching function between features $x$ and $y$ can be defined as $f_1(x, y) = \\sum _{k = 1}^{K}\\delta _{v_x^{(k)}, v_y^{(k)}}$ where $v_x^{(k)}$ and $v_y^{(k)}$ are visual words in vocabulary $\\mathcal {V}^{(k)}$ for $x$ and $y$ , respectively.", "Eq.", "REF shows that in baseline B$_1$ , an indexed feature is counted $k$ times if it is in the $k^{th}$ -order intersection set $\\cap ^{(k)}$ , and only once if in the difference set (since there is no overlapping).", "Multi-index based vocabulary merging (B$_2$ ) In [1], a multi-index is organized as a multi-dimensional structure.", "In its nature, given $K$ vocabularies, two features are considered as a match iff they are in the $K^{th}$ -order intersection set $\\cap ^{(K)}$ of the indexed feature lists.", "Therefore, in baseline B$_2$ , the matching function is defined as $f_2(x, y) = \\prod _{k=1}^{K}\\delta _{v_x^{(k)}, v_y^{(k)}}$ Eq.", "REF only counts the indexed features in $\\cap ^{(K)}$ , discarding the rest.", "Therefore, the recall is low for B$_2$ ." ], [ "Proposed Method", "For multi-vocabulary merging, the major problem is the over-counting of the intersection sets $\\cap ^{(k)}, k = 1,...,K$ .", "On the other hand, the major benefit is a high recall, which is encoded in the difference set.", "Taking both issues into consideration, we propose to exert a likelihood on the intersection sets and preserve the difference set (scored as B$_1$ ).", "Without loss of generality, we start from the case of two vocabularies and then generalize it to multiple vocabularies.", "Figure: The distribution of the cardinality ratio on 10K, 100K, 1000K images, respectively.", "We use two vocabularies of size 20K." ], [ "Model Formulation", "Given a query feature $x$ in image $Q$ , two sets of indexed features $\\mathcal {A}$ and $\\mathcal {B}$ are identified in two inverted files, respectively.", "Here, we want to evaluate the likelihood that a SIFT feature $y$ is a true neighbor of $x$ given that $y$ belongs to the intersection set of $\\mathcal {A}$ and $\\mathcal {B}$ .", "This likelihood can be modeled as the following conditional probability, $w(x, y) = p(y \\in T_x \\left.\\right\| y \\in \\mathcal {A} \\cap \\mathcal {B}).$ In Eq.", "REF , we define $T_x$ as the set of features which are visually similar to $x$ (locally) and belong to the ground truth images of $Q$ (globally).", "On the other hand, $F_x$ is defined as the features which violate any of the two criteria.", "Therefore, $T_x$ and $F_x$ satisfy the follows $p(y\\in T_x) + p(y \\in F_x) = 1.$ For simplicity, we denote $y \\in \\mathcal {A} \\cap \\mathcal {B}$ as $\\mathcal {A} \\cap \\mathcal {B}$ , $y \\in T_x$ as $T_x$ , and $y \\in T_y$ as $T_y$ , Then, using the formula of Bayes' theorem as well as Eq.", "REF , we get $\\begin{aligned}p(T_x &\\left.\\right\| \\mathcal {A} \\cap \\mathcal {B}) = \\frac{p( \\mathcal {A} \\cap \\mathcal {B}\\left.\\right\| T_x)\\cdot p( T_x)}{p( \\mathcal {A} \\cap \\mathcal {B})}\\\\&= \\frac{p( \\mathcal {A} \\cap \\mathcal {B}\\left.\\right\| T_x)\\cdot p(T_x)}{p( \\mathcal {A} \\cap \\mathcal {B} \\left.\\right\| T_x)\\cdot p(T_x) + p(\\mathcal {A} \\cap \\mathcal {B} \\left.\\right\| F_x)\\cdot p(F_x)}.\\end{aligned}$ Then, re-formulating Eq.", "REF , we have $p(T_x \\left.\\right\| \\mathcal {A} \\cap \\mathcal {B}) = \\left(1 + \\frac{p( \\mathcal {A} \\cap \\mathcal {B}\\left.\\right\| F_x)}{p( \\mathcal {A} \\cap \\mathcal {B}\\left.\\right\| T_x)}\\cdot \\frac{p(F_x)}{p(T_x)}\\right)^{-1}.$ In Eq.", "REF , there are actually three random variables to estimate, , $p( \\mathcal {A} \\cap \\mathcal {B}\\left.\\right\| F_x)$ (term 1), $p( \\mathcal {A} \\cap \\mathcal {B}\\left.\\right\| T_x)$ (term 2), and $p(F_x) \\slash p(T_x)$ (term 3).", "In Section REF , we will exploit the estimation of these probabilities." ], [ "Probability Estimation", "Estimation of term 1 In Eq.", "REF , the term $p( \\mathcal {A} \\cap \\mathcal {B}\\left.\\right\| F_x)$ encodes the probability that feature $y$ lies in the set $\\mathcal {A} \\cap \\mathcal {B}$ given that $y$ is a false match of query feature $x$ .", "In this case, we should consider the distribution of the $x$ ' false matches in sets $\\mathcal {A}$ and $\\mathcal {B}$ .", "In large databases, the number of true matches (both locally and globally) is limited.", "In other words, false matches dominate the space covered by $\\mathcal {A}$ and $\\mathcal {B}$ .", "Therefore, we assume that false matches are uniformly distributed in $\\mathcal {A}$ and $\\mathcal {B}$ , and term 1 can be estimated as $p( \\mathcal {A} \\cap \\mathcal {B}\\left.\\right\| F_x) = \\frac{Card(\\mathcal {A} \\cap \\mathcal {B})}{Card(\\mathcal {A} \\cup \\mathcal {B})},$ where $Card(\\cdot )$ represents the cardinality of a set.", "Eq.", "REF implies that, the probability that a false match falls into $\\mathcal {A}\\cap \\mathcal {B}$ is proportional to the cardinality ratio $\\frac{Card(\\mathcal {A} \\cap \\mathcal {B})}{Card(\\mathcal {A}\\cup \\mathcal {B})}$ .", "Intuitively, the larger the intersection set is, the more probable that a false match will fall into it.", "Fig.", "REF depicts the distribution of this cardinality ratio on different database scales.", "Estimation of term 2 In contrast to term 1, the probability encoded in term 2 reflects the likelihood that $y$ , a true neighbor of query $x$ , falls into the intersection set $\\mathcal {A} \\cap \\mathcal {B}$ .", "Still, we estimate this probability as a function of the cardinality ratio $\\frac{Card(\\mathcal {A} \\cap \\mathcal {B})}{Card(\\mathcal {A}\\cup \\mathcal {B})}$ .", "However, since the number of true matches is very small compared to false ones, we do not adopt the method in estimating term 1.", "Instead, image data with ground truth is used to analyze the distribution.", "Specifically, empirical analysis is performed on Oxford and Holidays datasets.", "Given a feature $x$ in the query image $Q$ , true matches are defined as the features which have a Hamming distance [4] smaller than 20 to $x$ and which appear in the ground truth images of $Q$ .", "Then we calculate the ratio of the number of true matches in $\\mathcal {A}\\cap \\mathcal {B}$ to the number of true matches in $\\mathcal {A}\\cup \\mathcal {B}$ .", "Finally, the relationship between the ratio and $\\frac{Card(\\mathcal {A} \\cap \\mathcal {B})}{Card(\\mathcal {A} \\cup \\mathcal {B})}$ is depicted in Fig.", "REF .", "A surprising fact from Fig.", "REF is that $p( \\mathcal {A} \\cap \\mathcal {B}\\left.\\right\| F_x)$ increases linearly with $\\frac{Card(\\mathcal {A} \\cap \\mathcal {B})}{Card(\\mathcal {A} \\cup \\mathcal {B})}$ .", "Contrary to our expectation, true matches do not aggregate around the query point.", "Instead, they tend to scatter in the high-dimensional feature space.", "Otherwise, the curves in Fig.", "REF would take on a $\\log (\\cdot )$ -like profile.", "On the other hand, Fig.", "REF also implies that the indexed features in $\\mathcal {A} \\cap \\mathcal {B}$ are mostly false matches.", "This explains why the over-counting compromises the retrieval accuracy.", "Moreover, we also find that the trend in Fig.", "REF seems to be database-independent.", "Figure: Distribution of term 2 as a function of Card(𝒜∩ℬ) Card(𝒜∪ℬ)\\frac{Card(\\mathcal {A} \\cap \\mathcal {B})}{Card(\\mathcal {A} \\cup \\mathcal {B})} on Oxford (left) and Holidays (right) datasets.", "Least Square Fitting of degree 1 is performed on Oxford, plotted as the red line.", "We find that the same line also fits the trends of Holidays dataset.Estimation of term 3 Term 3, , $p(F_x) \\slash p(T_x)$ , can be interpreted as the ratio of the probability of $y$ being a false match to $y$ being a true match.", "Typically, as the database grows, the number of false images will become larger, and the value of term 3 will increase.", "To model this property, and thus making our system adjustable to large scale settings, we set term 3 as $\\frac{p(F_x)}{p(T_x)} = \\log \\left(N\\cdot c\\right),$ where $N$ is the number of images in the database, and $c$ is a weighting parameter.", "Note that we add a $\\log (\\cdot )$ operator due to numerical considerations." ], [ "Similarity Interpretation", "Using the estimation methods introduced in Section.", "REF , we are able to provide an explicit implementation of the probability model (Eq.", "REF ).", "Specifically, we assume four database sizes are involved, , 5K, 10K, 100K, 1M, and we set the parameter $c$ to 1 for better illustration.", "The derived probability function is plotted against $\\frac{Card(\\mathcal {A} \\cap \\mathcal {B})}{Card(\\mathcal {A} \\cup \\mathcal {B})}$ in Fig.", "REF .", "From the curves in Fig.", "REF , we can get several implications in terms of physical interpretation.", "First, when the intersection area is very small (the cardinality ratio is close to zero), it is very likely that $y$ is a true match if it falls into this area.", "In this scenario, the discriminative power of the intersection set is high, and can be trusted when merging vocabularies.", "Figure: The estimation of Eq.", "as a function of Card(𝒜∩ℬ) Card(𝒜∪ℬ)\\frac{Card(\\mathcal {A} \\cap \\mathcal {B})}{Card(\\mathcal {A} \\cup \\mathcal {B})}.", "Four curves are presented, corresponding to N=N = 5K, 10K, 100K, and 1M, respectively.", "The vocabulary sizes are both 20K.Second, when the cardinality ratio approaches 1, , sets $\\mathcal {A}$ and $\\mathcal {B}$ share a large overlap, the probability of $y$ being a true match is small.", "This makes more sense if we take into consideration the fact that false images dominate the entire feature space.", "Moreover, a larger intersection means a larger dependency (or correlation) between two vocabularies, in which situation our method exerts a punishment (low weight) and overcomes this problem to some extent.", "Third, as the database becomes larger, the curves lean towards the origin.", "In fact, for large databases, the chances that $y$ is a true match will be more remote under each cardinality ratio.", "Nevertheless, the cardinality ratio tends to get smaller (see Fig.", "REF ) as the database grows, so the estimated probability will be compensated to some extent.", "As a summary, Fig.", "REF reveals some interesting properties of $\\mathcal {A} \\cap \\mathcal {B}$ .", "The formula Eq.", "REF will be adopted into the BoW-based image retrieval framework in Section.", "REF ." ], [ "Generalization to Multiple Vocabularies", "In this section, we generalize our method to the case of multiple vocabularies ($K\\ge 2$ ).", "Given $K$ vocabularies, a query feature $x$ is quantized to $K$ visual words, and subsequently $K$ sets of indexed features are identified, , $\\lbrace \\mathcal {A}_i\\rbrace _{i = 1}^K$ .", "If a database feature $y$ falls into the $k^{th}$ -order intersection set of $\\lbrace \\mathcal {A}_i\\rbrace _{i = 1}^K$ , the probability of it being a true match to $x$ is defined as $w(x, y) = p(y \\in T_x \\left.\\right\| y \\in \\mathcal {A}_1 \\cap \\mathcal {A}_2 \\cap ... \\cap \\mathcal {A}_k).$ Using the similarity function derived in Section REF , we can estimate Eq.", "REF as a function of the cardinality ratio of the $k^{th}$ -order intersection and union sets $ \\frac{Card\\left(\\mathcal {A}_1 \\cap \\mathcal {A}_2 \\cap ... \\cap \\mathcal {A}_k \\right)}{ Card\\left(\\mathcal {A}_1 \\cup \\mathcal {A}_2 \\cup ... \\cup \\mathcal {A}_k\\right)}$ ." ], [ "Proposed Image Retrieval Pipeline", "[t] Bayes merging for image retrieval [1] The query image $Q$ with $L$ descriptors $x_1, x_2,..., x_L$ ; The $K$ vocabularies $\\mathcal {V}^{(1)}, \\mathcal {V}^{(2)},..., \\mathcal {V}^{(K)}$ ; The $K$ inverted files $\\mathcal {W}^{(1)}, \\mathcal {W}^{(2)},..., \\mathcal {W}^{(K)}$ ; $n=1: L$ Quantize $x_n$ into $K$ visual words $v^{(1)},...,v^{(K)}$ ; Identify $K$ lists of indexed features $\\mathcal {A}^{(1)},...,\\mathcal {A}^{(K)}$ ; Find all $k^{th}$ -order intersection sets, $k = 2,...,K$ ; Find all $k^{th}$ -order union sets, $k = 2,...,K$ ; each indexed feature in $\\cup ^K$ Find the $k^{th}$ -order intersection set it falls in; Find the $k^{th}$ -order union set it falls in; Calculate $\\frac{Card\\left(\\cap ^{(k)}\\right)}{Card\\left(\\cup ^{(k)}\\right)}$ ; Calculate matching strength using Eq.", "REF ; Vote for the candidate image using Eq.", "REF ; In this section, the matching function of the Bayes merging method is defined as follows, $f(x, y) ={\\left\\lbrace \\begin{array}{ll}kw(x, y), &\\mbox{if } y \\in \\cap ^k, k\\ge 2\\\\\\sum _{i=1}^{K}\\delta _{v_x^{(k)}, v_y^{(k)}}, &\\mbox{otherwise}\\end{array}\\right.", "}$ where $w(x, y)$ is the similarity function defined in Eq.", "REF .", "If $w(x, y) = 1$ , Bayes merging reduces to the baseline B$_1$ .", "The pipeline of Bayes merging is summarized in Algorithm REF .", "In the offline steps, $K$ vocabularies are trained and the corresponding $K$ inverted files are organized.", "During online retrieval, given a query image $Q$ with $L$ descriptors, for each feature $x_n$ , we quantize it to $K$ visual words (step REF ).", "Then, $K$ lists of indexed features are identified (step REF ), from which all $k^{th}$ -order intersection and union sets are identified (step REF , REF ).", "For each indexed feature in $\\cup ^K$ , we find the $k^{th}$ -order intersection and union sets it falls in (step REF , REF ), and calculate the cardinality ratio (step REF ).", "Finally, matching strength is calculated according to Eq.", "REF and used in the matching function as Eq.", "REF (steps REF and REF ).", "For one query feature, we have to traverse $\\cup ^K$ twice in Algorithm REF , which doubles the query time.", "However, in the supplementary material, we demonstrate that we can accomplish this process by traversing $\\cup ^K$ only once, thus solving the efficiency problem of Bayes merging." ], [ "Experiments", "In this section, the proposed Bayes merging is evaluated on three benchmark datasets, , Holidays [4], Oxford [12], and Ukbench [11].", "The details of the datasets are summarized in Table REF .", "We also add the Flickr 1M dataset [4] of one million images to test the scalability of our method.", "All the vocabularies are trained independently on the Flickr60K dataset [4] using AKM [12] with different initial seeds.", "Table: Details of the datasets in the experiments." ], [ "Parameter Analysis", "One parameter, , the weighting parameter $c$ in Eq.", "REF is involved in the probabilistic model.", "We evaluate $c$ on the Holidays and Oxford datasets, and record in Table REF the mAP results against different values of $c$ .", "We can see that the mAP results remain stable when $c$ ranges from 10 to 50, probably due to the effect of the log operator in Eq.", "REF .", "We therefore set $c$ to 30 in the following experiments.", "Table: The impact of parameter cc on image retrieval accuracy.", "Results (mAP in percent) on Oxford 5K and Holidays datasets are presented.", "We set c=30c = 30 from these results." ], [ "Evaluation", "Comparison with the baselines We first compare Bayes merging with the baselines, , $\\mbox{B}_0$ , $\\mbox{B}_1$ , $\\mbox{B}_2$ defined in Section REF .", "The results are demonstrated in Fig.", "REF and Fig.", "REF .", "From these results we find that baseline $\\mbox{B}_2$ does not benefit from introducing multiple vocabularies, and that its performance drops when merging more vocabularies, because the recall further decreases.", "We speculate that Multiple Assignment will bring benefit [1], [17] to B$_2$ .", "Moreover, baseline B$_1$ brings limited improvements over B$_0$ .", "In fact, B$_1$ has a higher recall than B$_0$ , but this benefit is impaired by vocabulary correlation in which many irrelevant images are over-counted.", "In comparison, it is clear that Bayes merging yields great improvements.", "Take Holidays for example, when merging two vocabularies of size 20K, the gains in mAP over the three baselines are $9.28\\%$ , $8.64\\%$ , and $21.21\\%$ , respectively.", "The improvement is even higher for three vocabularies.", "Nevertheless, we favor two vocabularies due to the fact that the marginal improvement is prominent, while introducing little computational complexity.", "Table: Results on three benchmark datasets for different methods: baselines B 0 \\mbox{B}_0 and B 1 \\mbox{B}_1, the proposed method (Bayes), Hamming Embedding (HE) , and burstiness weighting (Burst) .", "We consider the merging of 2×\\times 20K, 3×\\times 20K, and 2×\\times 50K vocabularies, respectively.Impact of vocabulary sizes The vocabulary size may have an impact on the effectiveness of Bayes merging.", "To this end, we generate vocabularies of size 10K, 20K, 50K, 100K, and 200K on the independent Flickr60K data.", "In Fig.", "REF , we demonstrate the results obtained from various vocabulary sizes on the three datasets.", "Except for the three baselines, we also report results obtained by Bayes merging of two or three vocabularies.", "From Fig.", "REF , we can see that B$_1$ still yields limited improvement over B$_0$ .", "Moreover, B$_1$ and B$_2$ perform better under those larger vocabularies.", "This is due to the fact that larger vocabularies reduce correlation.", "But for large databases, vocabularies are never large enough, so the correlation problem would be more severe in the large-scale case.", "Moreover, it is clear that the Bayes merging method exceeds the baselines consistently under different vocabulary sizes.", "Meanwhile, Bayes merging of three vocabularies has a slightly higher performance than two vocabularies.", "Merging vocabularies of different sizes Bayes merging can also be generalized to merging vocabularies of different sizes, and the procedure is essentially the same with Algorithm REF .", "As with the contribution of each vocabulary, we adopt the same unit weight for all vocabularies, as it is shown to yield satisfying performance in [3].", "In this paper, we report the merging results on Oxford dataset in Table REF .", "Table REF demonstrates that merging vocabularies of different sizes marginally improves mAP on Oxford.", "For example, Bayes merging of two vocabularies of size 10K and 20K improves over the 2$\\times $ 10K and 2$\\times $ 20K Bayes methods by $1.07\\%$ and $0.34\\%$ , respectively.", "We speculate that vocabularies of different sizes provide extra complementary information, which can be captured by our method.", "However, since the smaller vocabulary introduces more noise, the benefit is limited.", "Table: The mAP of Bayes merging of vocabularies of different sizes on Oxford dataset.", "In comparison, Bayes merging of two vocabularies of the same size yields an mAP of 46.04%46.04\\%, 46.77%46.77\\%, and 47.79%47.79\\% for the 10K, 20K, 50K vocabularies, respectively.Combination with Hamming Embedding To test whether Bayes merging is complementary to some prior arts, we combine it with Hamming Embedding (HE) [4] and burstiness weighting [5] using the default parameters.", "HE effectively improves the precision of feature matching.", "In our experiment, HE with a single vocabulary achieves an mAP of $76.24\\%$ and $56.65\\%$ on Holidays and Oxford, and an N-S score of $3.49$ on Ukbench, respectively.", "The results in Fig.", "REF and Table REF indicate that Bayes merging yields consistent improvements of the B$_0$ + HE method.", "Specifically, when merging two vocabularies of 20K, the mAP is improved from $76.24\\%$ to $81.20\\%$ and from $56.65\\%$ to $63.32\\%$ on Holidays and Oxford, respectively.", "Similar trend can be observed on Ukbench: N-S score rises from 3.49 to 3.61.", "In its nature, HE results in refined matching in the feature space (locally).", "Complementarily, the Bayes merging jointly considers the image- and feature-level similarity.", "Therefore, while good matching in the feature space can be guaranteed by HE, our method punishes those of a false match in the image space.", "In this scenario, we actually raise an interesting question: can we simply trust feature-to-feature similarity in image retrieval?", "In addition, combining burstiness weights brings about extra, though limited improvement (see Table REF ).", "Our implementation differs from [5] in that we do not apply the weights on images in the intersection set, but instead on the difference set ($\\mathcal {A}\\cup \\mathcal {B} - \\mathcal {A}\\cap \\mathcal {B}$ ) only.", "A performance summary of various methods is presented in Table REF .", "Large-scale experiments To test the scalability of our method, we add the Flickr1M distractor images [4] to the Holidays and Oxford datasets.", "For comparison, we report the results of baselines $\\mbox{B}_0$ and $\\mbox{B}_1$ .", "From Fig.", "REF , it is clear that Bayes merging outperforms the two baselines significantly.", "On Holidays dataset mixed with one million images, Bayes merging achieves mAP of 39.60$\\%$ , compared with 28.19$\\%$ and 29.26$\\%$ of baseline B$_0$ and B$_1$ , respectively.", "In terms of efficiency, the baseline method $\\mbox{B}_0$ consumes 4 bytes per feature, and 1.9 GB for indexing one million images.", "The Bayes merging of two vocabularies doubles the memory cost to about 3.8 GB on Flickr1M.", "On the other hand, it takes 2.52s and 4.87s for $\\mbox{B}_0$ and $\\mbox{B}_1$ to perform one query on 1 million image size, respectively, using a server with 3.46 GHz and 64GB memory.", "Bayes merging involves identifying the intersection set and calculate the cardinality ratio.", "In fact, the cardinality ratio can be computed and stored offline.", "Moreover, as shown in the supplementary material, we are able to perform both the identification and the voting tasks by traversing the two lists of indexed features only once.", "Therefore, our method only marginally increases the query time to 5.12s.", "Figure: The mAP results as a function of the database size on Holidays and Oxford datasets.", "Three methods are compared, , baselines B 0 _0, B 1 _1, and Bayes merging of two vocabularies.", "The vocabulary size is 20K for all methods.Comparison with state-of-the-arts We first compare our method with [3] which employs PCA to addresses the correlation problem implicitly.", "In [3], merging four 16K vocabularies and eight 8K vocabularies yield an mAP of $55.8\\%$ and $56.7\\%$ , respectively.", "Moreover, merging vocabularies of multiple sizes obtains a best mAP of $58.8\\%$ on Holidays.", "In comparison, the result obtained by Bayes merging is $58.5\\%$ and $60.4\\%$ for two and three vocabularies of size 20K, respectively.", "Second, we compare the Bayes merging with the Rank Aggregation (RA) method [2], [7] in Table REF .", "Following [7], we take the median of multiple ranks as the final rank.", "Since RA works on the rank level, it does not address the correlation problem, so its performance is limited.", "The results demonstrate the superiority of Bayes merging.", "Finally, we compare the results of Bayes merging with state-of-the-arts in Table REF .", "On the three datasets, we achive mAP = $\\bf {81.9\\%}$ on Holidays, mAP = $\\bf {65.0\\%}$ on Oxford, and N-S = $\\bf 3.62$ on Ukbench.", "We have also tested on the data provided by [14], where the codebook size is 65K.", "On Oxford datastet, the mAP is 77.3%.", "Note that some sophisticated techniques are absent in our system, such as spatial constraints [6], [16], semantic consistency [20], etc.", "Still, the results demonstrate that the performance of Bayes merging is very competitive.", "We also provide some sample retrieval results in the supplementary material.", "Table: Comparisons with rank aggregation (RA).", "Different numbers of vocabularies are trained to test RA.", "Vocabulary size is 20K.Table: Comparisons with the state-of-the-art methods." ], [ "Conclusion", "Multi-vocabulary merging is an effective method to improve the recall of visual matching.", "However, this process is impaired by vocabulary correlation.", "To address the problem, this paper proposes a Bayes merging approach to explicitly estimate the matching strength of the indexed features in the intersection sets, while preserving those in the difference set.", "In a probabilistic view, Bayes merging is capable of jointly modeling an image- and feature-level similarity from multiple sets of indexed features.", "Specifically, we exploit the probability that an indexed feature is a true match (both locally and globally) if it is located in the intersection sets of multiple inverted files.", "Extensive experiments demonstrate that Bayes merging effectively reduces the impact of vocabulary correlation, thus improving the retrieval accuracy significantly.", "Further, our method is efficient, and yields competitive results with state-of-the-arts.", "Acknowledgement This work was supported by the National High Technology Research and Development Program of China (863 program) under Grant No.", "2012AA011004 and the National Science and Technology Support Program under Grant No.", "2013BAK02B04.", "This work also was supported in part to Dr. Qi Tian by ARO grant W911NF-12-1-0057, Faculty Research Awards by NEC Laboratories of America, and 2012 UTSA START-R Research Award respectively.", "This work was supported in part by National Science Foundation of China (NSFC) 61128007." ] ]	1403.0284
[ [ "Equation of state of a relativistic theory from a moving frame" ], [ "Abstract We propose a new strategy for determining the equation of state of a relativistic thermal quantum field theory by considering it in a moving reference system.", "In this frame an observer can measure the entropy density of the system directly from its average total momentum.", "In the Euclidean path integral formalism, this amounts to compute the expectation value of the off-diagonal components T_{0k} of the energy-momentum tensor in presence of shifted boundary conditions.", "The entropy is thus easily measured from the expectation value of a local observable computed at the target temperature T only.", "At large T, the temperature itself is the only scale which drives the systematic errors, and the lattice spacing can be tuned to perform a reliable continuum limit extrapolation while keeping finite-size effects under control.", "We test this strategy for the four-dimensional SU(3) Yang-Mills theory.", "We present precise results for the entropy density and its step-scaling function in the temperature range 0.9 T_c - 20 T_c.", "At each temperature, we consider four lattice spacings in order to extrapolate the results to the continuum limit.", "As a byproduct we also determine the ultraviolet finite renormalization constant of T_{0k} by imposing suitable Ward identities.", "These findings establish this strategy as a solid, simple and efficient method for an accurate determination of the equation of state of a relativistic thermal field theory over several orders of magnitude in T." ], [ "Equation of state of a relativistic theory from a moving frame Leonardo Giusti$^{a,b}$ , Michele Pepe$^b$ $^a$ Dipartimento di Fisica, Università di Milano-Bicocca, Piazza della Scienza 3, I-20126 Milano, Italy $^b$ INFN, Sezione di Milano-Bicocca, Piazza della Scienza 3, I-20126 Milano, Italy We propose a new strategy for determining the equation of state of a relativistic thermal quantum field theory by considering it in a moving reference system.", "In this frame an observer can measure the entropy density of the system directly from its average total momentum.", "In the Euclidean path integral formalism, this amounts to compute the expectation value of the off-diagonal components $T_{0k}$ of the energy-momentum tensor in presence of shifted boundary conditions.", "The entropy is thus easily measured from the expectation value of a local observable computed at the target temperature $T$ only.", "At large $T$ , the temperature itself is the only scale which drives the systematic errors, and the lattice spacing can be tuned to perform a reliable continuum limit extrapolation while keeping finite-size effects under control.", "We test this strategy for the four-dimensional $SU(3)$ Yang-Mills theory.", "We present precise results for the entropy density and its step-scaling function in the temperature range $0.9 \\,T_c - 20\\, T_c$ .", "At each temperature, we consider four lattice spacings in order to extrapolate the results to the continuum limit.", "As a byproduct we also determine the ultraviolet finite renormalization constant of $T_{0k}$ by imposing suitable Ward identities.", "These findings establish this strategy as a solid, simple and efficient method for an accurate determination of the equation of state of a relativistic thermal field theory over several orders of magnitude in $T$ .", "Introduction.— Relativistic thermal quantum field theories are of central importance in many areas of research in physics.", "The equation of state (EOS) of Quantum Chromo Dynamics (QCD) is a very basic property of strongly-interacting matter that is of absolute interest in particle and nuclear physics, and in cosmology.", "It is also a crucial input in the analysis of data collected at the heavy-ion colliders.", "Lattice QCD is the only known theoretical framework where the EOS can be determined from first principles in the interesting range of temperature values.", "Since the perturbative expansion converges very slowly, the full computation of the EOS has to be done numerically over several orders of magnitude in $T$ .", "Severe unphysical contributions hinder the standard way of computing the pressure and the energy density.", "The expansion of the free energy in the bare parameters, and the subtraction of ultraviolet power divergences make the computation of the EOS technically difficult and numerically very demanding [1], [2], [3], [4] (see Ref.", "[5] for a recent review).", "Temperatures higher than a few hundreds MeV are still unreachable with staggered fermions.", "The computation remains prohibitive with Wilson fermions.", "The obstacles, however, are not rooted in the physics content of the EOS, but in the strategy adopted for its computation.", "This calls for a conceptual progress able to trigger new computational strategies, which in turn are capable to reach the goal of a precise computation of the EOS in a generic discretization of the theory.", "The underlying Lorentz symmetry of relativistic thermal theories offers an elegant and simple solution to this problem.", "In these theories the entropy is proportional to the total momentum of the system as measured by an observer in a moving frame.", "Remarkably, the corresponding Euclidean path integral formulation is rather simple.", "It corresponds to inserting a shift ${\\xi }$ in the spatial directions when closing the boundary conditions of a field $\\phi $ in the compact direction of length $L_0$ [6], [7], [8], [9] $\\phi (L_0,{x}) = \\phi (0,{x} - L_0\\, {\\xi })\\; .$ In the thermodynamic limit, the invariance of the dynamics under the $SO(4)$ group implies that the free energy density $f(L_0 ,{\\xi })$ satisfies [6], [7], [8] $f(L_0 ,{\\xi }) = f(L_0 \\sqrt{1+{\\xi }^2},{0})\\; .$ Hence the free energy does not depend on $L_0$ and ${\\xi }$ separately but on the combination $L_0 \\sqrt{1+{\\xi }^2}= T^{-1}$ which fixes the inverse temperature of the system.", "This redundancy implies that the thermal distributions of the total energy and momentum are related, and interesting Ward identities (WIs) follow.", "In particular, the entropy density can be written as [6] $\\frac{s(T)}{T^3} = -\\frac{(1+ {\\xi }^2)}{\\xi _k}\\frac{\\langle T_{0k} \\rangle _{\\xi }}{T^4}\\; ,$ where $\\langle \\cdot \\rangle _{\\xi }$ stands for the expectation value computed with a non-zero shift ${\\xi }$ .", "No ultraviolet power-divergent contributions need to be subtracted from $\\langle T_{0k} \\rangle _{\\xi }$ .", "In this Letter we explore a new computational strategy for determining the EOS of a relativistic thermal quantum field theory based on Eq.", "(REF ).", "We illustrate the power of the method in the $SU(3)$ Yang-Mills theory, where we determine the entropy density of the system in the range $0.9\\, T_c - 20\\,T_c$ .", "This is a particularly interesting theory since it is the limit of QCD in absence of fermions (or with infinitely heavy fermions), and it can be used to test new ideas and numerical methods without facing the problems of simulating dynamical fermions.", "Since it relies on Lorentz invariance only, the strategy is directly applicable to any relativistic thermal theory and, in particular, to QCD.", "Entropy density from the lattice.— We regularize the four-dimensional $SU(3)$ Yang–Mills theory on a square lattice of size $ L_0 \\times L^3$ and of spacing $a$ .", "The link variables $U_\\mu (x) \\in SU(3)$ represent the gauge field and the Wilson action $S$ is, up to a constant, given by $S[U] = -\\frac{\\beta }{6}\\!", "\\sum _{x,\\mu \\nu }\\!\\mbox{Re} \\,\\hbox{\\rm Tr}[U_\\mu (x)U_\\nu (x+\\hat{\\mu })U^\\dagger _\\mu (x+\\hat{\\nu })U^\\dagger _\\nu (x)]\\nonumber $ where $\\beta =6/g_0^2$ , and $g_0$ is the bare coupling.", "We impose periodic boundary conditions in the spatial directions and shifted boundary conditions along the compact direction, $U_\\mu (L_0,{x}) = U_\\mu (0,{x}- L_0\\, {\\xi })$ , where $(L_0/a)\\, {\\xi }$ is a vector with integer components.", "We consider the clover definition of the energy-momentum tensor on the lattice [10] $T_{\\mu \\nu } = \\frac{\\beta }{6}\\Big \\lbrace F^a_{\\mu \\alpha }F^a_{\\nu \\alpha }- \\frac{1}{4} \\delta _{\\mu \\nu } F^a_{\\alpha \\beta }F^a_{\\alpha \\beta } \\Big \\rbrace \\; .$ The field strength tensor is defined as $F^a_{\\mu \\nu }(x) = - \\frac{i}{4 a^2}\\,\\hbox{\\rm Tr}\\Big \\lbrace \\Big [Q_{\\mu \\nu }(x) - Q_{\\nu \\mu }(x)\\Big ]T^a\\Big \\rbrace \\; ,$ where $T^a=\\lambda ^a/2$ with $\\lambda ^a$ being the Gell-Mann matrices, and (see Ref.", "[10] for more details) $Q_{\\mu \\nu }(x) = P_{\\mu \\nu }(x) + P_{\\nu -\\mu }(x) + P_{-\\mu -\\nu }(x) +P_{-\\nu \\mu }(x)\\; .$ The matrix $P_{\\mu \\nu }(x)$ is the parallel transport along an elementary plaquette at the lattice site $x$ along the directions $\\mu $ and $\\nu $ , and the minus sign stands for the negative orientation.", "The lattice regularization breaks explicitly translation invariance down to a discrete sub-group.", "As a consequence the off-diagonal components of the energy-momentum tensor renormalize multiplicatively [10], and Eq.", "(REF ) becomes $\\frac{s(T)}{T^3} = -\\frac{(1+{\\xi }^2)}{\\xi _k} \\,\\frac{Z_T \\langle T_{0k} \\rangle _{\\xi }}{T^4}\\; .$ The renormalization constant $Z_T$ of $T_{0k}$ can be fixed by imposing suitable WIs [7], [6].", "$Z_T$ depends only on the bare coupling constant and, up to discretization effects, it is independent of the kinematic parameters e.g., $L$ , $T$ , ${\\xi }$ .", "These parameters can be chosen at will, with the condition that they remain constant in physical units when approaching the continuum limit, or that they generate in $Z_T$ negligible discretization effects compared to the statistical errors.", "Ultimately, which WI and/or kinematics are the most effective has to be investigated numerically.", "We have found that for the $SU(3)$ Yang–Mills theory discretized with the Wilson action, $Z_T$ can be determined with small discretization effects and with a limited numerical effort as $Z_T = \\frac{1}{2 a L^{3}}\\frac{1}{\\langle T_{0k} \\rangle _{{\\xi }}}\\ln {\\frac{Z(L_0,{\\xi } + a/L_0 \\hat{k})}{Z(L_0,{\\xi } - a/L_0 \\hat{k})}}\\; ,$ where $Z(L_0,{\\xi })$ is the partition function of the theory.", "Once $Z_T$ is known, the lattice size and spacing can be adjusted so to carry out a reliable continuum limit extrapolation of the entropy density at any given value of $T$ with moderate computational resources.", "This is possible thanks to the fact that at large $T$ the temperature itself is the only relevant scale that drives discretization and finite volume effects.", "The mass gap of the theory is proportional to $T$ , and small pre-factors in its expression do not invalidate the strategy.", "Indeed increasing the spatial size of the lattice does not increase the computational effort at fixed statistical accuracy since $T_{0k}$ is a local observable.", "A slightly different approach is to define a step-scaling function $\\Sigma (T,r)$ for the entropy density as $\\Sigma (T,r) = \\frac{ T^3 s(T^{\\prime })}{T^{\\prime 3} s(T)} =\\frac{(1+{\\xi } ^{^{\\prime } 2})^3\\,\\, \\xi _k}{(1+\\xi ^2)^3\\,\\, \\xi ^{\\prime }_k}\\frac{\\langle T_{0k} \\rangle _{\\xi ^{\\prime }} }{\\langle T_{0k} \\rangle _{\\xi } }\\; ,$ where ${\\xi }$ and ${\\xi }^{\\prime }$ are two different shifts.", "The factor $Z_T$ drops out and the step-scaling function has a universal continuum limit as it stands.", "When $L_0$ and $\\beta $ are kept fixed, the step $r$ in the temperature is given by the ratio $r=T^{\\prime }/T=\\sqrt{1+{\\xi }^2}/\\sqrt{1+{\\xi } ^{^{\\prime } 2}}$ .", "Once $\\Sigma (T,r)$ is known, the entropy density at a given temperature can be obtained from its value at a single reference temperature $T_0$ by solving the straightforward recursion relation.", "Thus, $Z_T$ has to be determined only at the values of $\\beta $ where $s(T_0)/T_0^3$ is being measured.", "Figure: Left: continuum limit extrapolation of the entropy step-scaling function.", "Right:continuum limit extrapolation of entropy density at the reference temperature T 0 T_0,normalized to the Stefan-Boltzmann (SB) value s SB /T 3 =32π 2 /45s_{\\rm SB}/T^3= 32\\pi ^2/45.Numerical computation.— We have measured the entropy density (preliminary results were presented in [11]) in the range $0.9\\, T_c - 20\\, T_c$ , where $T_c$ is the critical temperature.", "We opted for computing the step-scaling function at 9 temperatures in the range $T_0/2\\,$ –$\\,8\\, T_0$ , with values separated by a step-factor of about $\\sqrt{2}$ .", "The reference temperature has been fixed to $T_0=L^{-1}_{\\rm max}$ , where $L_{\\rm max}$ in units of the standard reference scale $r_0$ corresponds to $L_{\\rm max}/r_0=0.738(16)$ [12], [13].", "The critical temperature is $r_0 T_c=0.750(4)$ [1], [14], and therefore $T_0\\simeq 1.807\\, T_c$ .", "At this temperature we have computed also the renormalization constant $Z_T$ .", "At each value of the lattice spacing and of $L_0/a$ , we have measured $\\langle T_{0k} \\rangle _{\\xi }$ for two shifts, ${\\xi }=(1,0,0)$ and $(1,1,1)$ with standard numerical techniques.", "The step-scaling function is then computed by using Eq.", "(REF ) as $\\Sigma \\big (1/(2 L_0),\\sqrt{2}\\big )=\\langle T_{0k} \\rangle _{(1,0,0)}/(8 \\langle T_{0k} \\rangle _{(1,1,1)})$ At each $T$ we have collected data at four different values of the lattice spacing, corresponding to $L_0/a=3$ , 4, 5 and 6.", "At the first four temperatures, $\\beta $ has been fixed from $r_0/a$ by requiring that $L_{\\rm max}=0.738\\, r_0$ [13].", "For the other data sets, we have determined $\\beta $ by interpolating quadratically in $\\ln {(L/a)}$ the data listed in Tables A.1 and A.4 of Ref.", "[12] corresponding to fixed values of $\\bar{g}^2(L)$ .", "In order to keep finite volume effects below the statistical errors, we have considered $T L \\ge 12$ .", "Taking into account the present estimate of the lightest screening mass, finite size effects are expected to be negligible compared to our statistical errors [6].", "On the coarsest lattice of each data set, we have performed numerical simulations at a smaller volume.", "No finite size corrections were observed within errors.", "All the details of the simulations will be reported elsewhere [15].", "We just note that the $\\beta $ values range from $5.85$ to $8.6$ , and the number of lattice points in the spatial directions goes from $64^3$ to $128^3$ .", "Table: Continuum limit extrapolated values of the step-scalingfunction and of the entropy density.In Fig.", "REF we show the results for $\\bar{\\Sigma }=\\Sigma -\\Sigma _0+1$ as a function of $(a/L_0)^2$ for the 8 highest temperatures, where $(\\Sigma _0-1)$ are the tree-level discretization effects that are subtracted analytically [6].", "The statistical errors range from 1 per-mille up to 3.5 per-mille.", "For these data sets the residual lattice artifacts turn out to be very small, and at most of $2\\%$ already at $L_0/a=3$ .", "A continuum linear extrapolation in $(a/L_0)^2$ of the three points with finer lattice spacings works very well for all data sets as shown in Fig.", "REF .", "The intercepts of these fits are our best estimate of the step-scaling function in the continuum limit.", "A quadratic fit of all four points give always compatible results within the statistical errors.", "The same applies for a combined fit of all data with discretization effects parametrized as expected in the weak coupling expansion.", "For the last 5 temperatures we interpolate the results for $\\Sigma _s(T,\\sqrt{2})$ in the renormalized coupling, and use the fit function to correct for the slight mismatch in the scales from Ref. [12].", "The best values for the step-scaling function are given in Tab.", "REF , and shown in the left plot of Fig.", "REF .", "The renormalization constant $Z_T$ has been determined from Eq.", "(REF ).", "In this case it is not necessary to consider large spatial volumes, and the numerical simulations have been performed with $L/a=12$ and 16.", "The finite-volume $\\langle T_{0k} \\rangle _{\\xi }$ in the denominator has been computed as described above.", "The derivative in the numerator requires the calculation of a ratio of two partition functions which cannot be computed in a single Monte Carlo simulation due to the very poor overlap of the relevant phase space of the two integrals.", "In this case we have used the Monte Carlo procedure of Refs.", "[8], [9].", "We consider a set of $(n+1)$ systems with action $\\overline{S}(U,r_i)= r_i S(U^{({\\xi } - a/L_0 \\hat{k})}) + (1-r_i) S(U^{({\\xi } + a/L_0 \\hat{k})})$ ($r_i=i/n$ , $i=0,1,\\dots ,n$ ), where the superscript indicates the shift in the boundary conditions.", "The relevant phase space of two successive systems with $r_i$ and $r_{i+1}$ is very similar and the ratio of their partition functions, ${\\cal Z}(\\beta ,r_i)/{\\cal Z}(\\beta ,r_{i+1})$ , can be efficiently measured as the expectation value of the observable $O(U,r_{i+1}) = \\exp {({\\overline{S}}(U,r_{i+1})-{\\overline{S}}(U,r_i))}$ on the ensemble of gauge configurations generated with the action ${\\overline{S}}(U,r_{i+1})$ [16].", "The discrete derivative is then written as $\\frac{1}{2 a}\\ln {\\frac{Z(L_0,{\\xi } + a/L_0 \\hat{k})}{Z(L_0,{\\xi } - a/L_0 \\hat{k})}}= \\frac{1}{2 a} \\sum _{i=0}^{n-1}\\ln {\\frac{{\\cal Z}(\\beta ,r_i)}{{\\cal Z}(\\beta ,r_{i+1})}}\\; .$ All the details and the results of the computation of $Z_T$ will be presented elsewhere [15].", "In Tab.", "REF we report the values of $Z_T$ at the 8 values of $\\beta $ needed to renormalize the entropy density at the temperature $T_0$ computed with shift ${\\xi }=(1,0,0)$ and $(1,1,1)$ .", "Albeit with smaller statistical errors, our values are in agreement with those found in Ref. [17].", "Also in this case we have subtracted the discretiazion effects of the free theory.", "In each of the two sets of data we keep $L_0$ fixed in physical units, so that residual (small) discretization effects in $Z_T$ will be removed in the continuum limit extrapolation of the renormalized entropy density.", "Discretization effects due to finite volume are negligible within our errors.", "For completeness, in the same Table we also report the corresponding expectation values of $\\langle T_{0k} \\rangle _{\\xi }$ in the large volume which enters Eq.", "(REF ).", "The results for $s(T_0)/T^3_0$ as defined in Eq.", "(REF ) are shown in the right plot of Fig.", "REF .", "Table: The bare vacuum expectation values of〈T 0k 〉 ξ \\langle T_{0k} \\rangle _{\\xi } at the reference temperatureT 0 T_0 for ξ=(1,0,0){\\xi }=(1,0,0) and (1,1,1)(1,1,1).", "The renormalizationconstant Z T Z_T at the corresponding eight β\\beta values is alsoreported.Figure: The step-scaling function (left) and the entropy density normalized to theSB value (right) versusthe temperature.", "The dashed lines (red) are the SB values, while the dotted-dashed lines (blue)are the perturbation theory ones from Ref.", ".The typical statistical error is just below half a percent, while the largest discretization error is roughly $3\\%$ .", "The continuum limit extrapolation of the data with ${\\xi }=(1,0,0)$ and $(1,1,1)$ at the three finer lattices are in excellent agreement among themselves.", "A combined extrapolation gives $s(T_0)/s_{\\rm SB}(T_0)=0.788(4)$ with a $\\chi ^2/{\\rm dof}=0.74$ , see Tab.", "REF .", "Results and conclusions.— Once the entropy density has been measured at $T_0$ , $s(T)$ at the other temperatures is computed by solving the straightforward recursive relation for the step-scaling function.", "The values obtained for the entropy density are reported in Tab.", "REF and shown in Fig.", "REF .", "The precision reached for $s(T)$ is half a percent at $T_0$ , and becomes at most $1.5\\%$ at $T/T_0=8\\sqrt{2}$ .", "We expect to reduce the latter error to the same level of the former once the renormalization constant is determined in the full range $0\\le g_0^2\\le 1$ [15].", "Taking into account that the entire computation required a few million of core hours on BG/Q, the precision reached shows the potentiality of the strategy.", "The results for the entropy density are in agreement with those in Refs.", "[1], [18], and for $T > 2\\, T_c$ with the more precise ones in Ref. [2].", "Our data differ by several standard deviations from those in Ref.", "[2] in the interval $T_c < T < 2\\, T_c$ .", "A more detailed comparison will be presented in Ref.", "[15], where more points will be added in this low-temperature region.", "The step-scaling function at $T\\sim \\!", "15 T_c$ is already compatible with the high-temperature limit within the half a percent uncertainty quoted.", "The entropy density, however, still differs from the Stefan-Boltzmann value by rougly $5\\%$ at $T\\simeq 20\\, T_c$ .", "To compare with the known perturbative formula [19], we use $\\Lambda _{{\\overline{\\mbox{\\scriptsize MS}}}}\\, r_0=0.586(48)$ [12], [13] and we fix the $O(g^6)$ undetermined coefficient by matching the perturbative value of the entropy density with our data at the largest temperature $T\\simeq 20\\, T_c$ .", "The results are shown in Fig.", "REF .", "Despite the good agreement, it must be said that the contribution from the various orders in the perturbative series is oscillating.", "At our largest temperature the contribution of $O(g^6)$ is roughly $40\\%$ of the total correction to the entropy density given by the other terms, see Ref.", "[15] for more details.", "On a more theoretical side, the results presented in this Letter are a direct non-perturbative verification of the consequences of Lorentz invariance at finite T. Acknowledgments.— We thank H. B. Meyer and D. Robaina for discussions.", "The simulations were performed on the BG/Q at CINECA (INFN and LISA agreement), and on PC clusters at the Physics Department of the University of Milano-Bicocca.", "We thankfully acknowledge the computer resources and technical support provided by these institutions.", "This work was partially supported by the INFN SUMA project." ] ]	1403.0360
[ [ "Several variants of the Dumont differential system and permutation\n statistics" ], [ "Abstract The Dumont differential system on the Jacobi elliptic functions was introduced by Dumont (Math Comp, 1979, 33: 1293--1297) and was extensively studied by Dumont, Viennot, Flajolet and so on.", "In this paper, we first present a labeling scheme for the cycle structure of permutations.", "We then introduce two types of Jacobi-pairs of differential equations.", "We present a general method to derive the solutions of these differential equations.", "As applications, we present some characterizations for several permutation statistics." ], [ "Introduction", "Let ${\\rm sn\\,}(u,k),{\\rm cn\\,}(u,k)$ and ${\\rm dn\\,}(u,k)$ be three basic Jacobi elliptic functions.", "These functions occur naturally in geometry, analysis, number theory, algebra and combinatorics (see [6], [9], [22], [23]).", "Finding the connection between Jacobi elliptic functions and combinatorics has been a hot research topic in mathematics for more than forties years (see, e.g., [8], [9], [12], [23], [32] and references therein).", "As pointed out by Flajolet and Françon [12], the question of possible combinatorial significance of the coefficients appearing in the Maclaurin expansions of the Jacobi elliptic functions has first been raised by Schützenberger.", "This paper presents several answers related to Dumont's [9] work.", "In particular, in Section , we show that the statistics of permutations, such as the longest alternating subsequences and alternating runs are both closely related to these functions.", "It should be noted that the study of the distribution of the length of the longest alternating subsequences of permutations was recently initiated by Stanley [28], [29], [30].", "The function ${\\rm sn\\,}(u,k)$ is defined as the inverse of the elliptic integral $u=F(x,k)=\\int _{0}^{x}\\frac{d t}{\\sqrt{(1-t^2)(1-k^2t^2)}},$ where $k$ is known as the modulus and satisfies $0<k<1$ (see [15]).", "Hence $x={\\rm sn\\,}(u,k)$ .", "The functions ${\\rm cn\\,}(u,k)$ and ${\\rm dn\\,}(u,k)$ are given as follows: ${\\rm cn\\,}(u,k)=\\sqrt{1-{\\rm sn\\,}^2(u,k)},\\qquad {\\rm dn\\,}(u,k)=\\sqrt{1-k^2{\\rm sn\\,}^2(u,k)}.$ Moreover, Jacobi elliptic functions can also be defined in terms of Jacobi theta functions (see [14], [17]).", "Using formal methods, Abel [1] discovered that these functions satisfy the following system of differential equations: $\\begin{split}\\frac{d}{du}{\\rm sn\\,}(u,k)&={\\rm cn\\,}(u,k){\\rm dn\\,}(u,k),\\\\\\frac{d}{du}{\\rm cn\\,}(u,k)&=-{\\rm sn\\,}(u,k){\\rm dn\\,}(u,k),\\\\\\frac{d}{du}{\\rm dn\\,}(u,k)&=-k^2{\\rm sn\\,}(u,k){\\rm cn\\,}(u,k).\\end{split}$ Let $\\mathfrak {S}_n$ denote the symmetric group of all permutations of $[n]$ , where $[n]=\\lbrace 1,2,\\ldots ,n\\rbrace $ .", "An interior peak in $\\pi $ is an index $i\\in \\lbrace 2,3,\\ldots ,n-1\\rbrace $ such that $\\pi (i-1)<\\pi (i)>\\pi (i+1)$ .", "Given a permutation $\\pi \\in \\mathfrak {S}_n$ , a value $i\\in [n]$ is called a cycle peak if $\\pi ^{-1}(i)<i>\\pi (i)$ .", "Let $X(\\pi )$ (resp., $Y(\\pi )$ ) be the number of odd (resp., even) cycle peaks of $\\pi $ .", "Let $D$ be the derivative operator on the polynomials in three variables.", "The Dumont differential system on the Jacobi elliptic functions is defined by $D(x)=yz,\\quad D(y)=xz,\\quad D(z)=xy.$ Define the numbers $s_{n,i,j}$ by $\\begin{split}D^{2n}(x)&=\\sum _{i,j\\ge 0}s_{2n,i,j}x^{2i+1}y^{2j}z^{2n-2i-2j},\\\\D^{2n+1}(x)&=\\sum _{i,j\\ge 0}s_{2n+1,i,j}x^{2i}y^{2j+1}z^{2n-2i-2j+1}.\\end{split}$ The study of (REF ) was initiated by Schett [26] in a slightly different form.", "Schett found that $\\sum _{j\\ge 0}s_{n,i,j}=P_{n,\\lfloor (n-1)/{2}\\rfloor -i}$ , where $P_{n,k}$ is the number of permutations in $\\mathfrak {S}_n$ with $k$ interior peaks.", "Dumont [8] deduced the recurrence relation $\\begin{split}s_{2n,i,j}&=(2j+1)s_{2n-1,i,j}+(2i+2)s_{2n-1,i+1,j-1}+(2n-2i-2j+1)s_{2n-1,i,j-1},\\\\s_{2n+1,i,j}&=(2i+1))s_{2n,i,j}+(2j+2)s_{2n,i-1,j+1}+(2n-2i-2j+2)s_{2n,i-1,j},\\end{split}$ and established that $s_{n,i,j}=\|\\lbrace \\pi \\in \\mathfrak {S}_n: X(\\pi )=i, Y(\\pi )=j\\rbrace \|.$ Subsequently, Dumont [9] studied the symmetric variants of (REF ): $\\begin{split}\\frac{d}{du}{\\rm sn\\,}(u;a,b)&={\\rm cn\\,}(u;a,b){\\rm dn\\,}(u;a,b),\\\\\\frac{d}{du}{\\rm cn\\,}(u;a,b)&=a^2{\\rm sn\\,}(u;a,b){\\rm dn\\,}(u;a,b),\\\\\\frac{d}{du}{\\rm dn\\,}(u;a,b)&=b^2{\\rm sn\\,}(u;a,b){\\rm cn\\,}(u;a,b),\\end{split}$ with the initial conditions ${\\rm sn\\,}(0;a,b)=0$ , ${\\rm cn\\,}(0;a,b)=1$ and ${\\rm dn\\,}(0;a,b)=1$ .", "In particular, for (REF ), Dumont [9] showed the following generating function $\\sum _{n\\ge 0}D^n(x)\\frac{u^n}{n!", "}=\\frac{yz{\\rm sn\\,}(u;y^{\\prime },z^{\\prime })+x{\\rm cn\\,}(u;y^{\\prime },z^{\\prime }){\\rm dn\\,}(u;y^{\\prime },z^{\\prime })}{1-x^2{\\rm sn\\,}^2(u;y^{\\prime },z^{\\prime })},$ where $y^{\\prime }=\\sqrt{y^2-x^2}$ and $z^{\\prime }=\\sqrt{z^2-x^2}$ .", "This paper is organized as follows.", "In Section , we collect some notation, definitions and results that will be needed in the sequel.", "In Section , we give the solutions of several systems of partial differential equations (PDEs for short) associated with (REF ), although it has always been a challenging problem to find explicit formulas for solutions of most PDEs, as pointed out by Evans [11].", "In Section , we present new characterizations for several combinatorial structures as applications." ], [ "Preliminaries", "Recall that $F(x,k)$ is defined in (REF ).", "We define $\\begin{split}h_{p,q}&=F\\left(\\sqrt{\\frac{q(1-p)}{q-p}},\\sqrt{\\frac{q-p}{1-p}}\\right),\\\\\\ell _{p,q}&=F\\left(q\\sqrt{\\frac{1-p}{q^2-p}},\\sqrt{\\frac{q^2-p}{1-p}}\\right),\\\\k_{p,q}&=\\sqrt{\\frac{p-1}{q-p}}\\arctan \\left(\\sqrt{\\frac{q(p-1)}{q-p}}\\right),\\\\x_\\pm &=(p-1)x\\pm k_{p,q}.\\end{split}$ For any sequence $a_{n,i,j}$ , we define the following generating functions $A=A(x,p,q)&=\\sum _{n,i,j\\ge 0}a_{n,i,j}\\frac{x^{n}}{n!", "}p^iq^j,\\\\AE=AE(x,p,q)&=\\sum _{n,i,j\\ge 0}a_{2n,i,j}\\frac{x^{2n}}{(2n)!", "}p^iq^j=\\frac{1}{2}\\bigl (A(x,p,q)+A(-x,p,q)\\bigr ),\\\\AO=AO(x,p,q)&=\\sum _{n,i,j\\ge 0}a_{2n+1,i,j}\\frac{x^{2n+1}}{(2n+1)!", "}p^iq^j=\\frac{1}{2}\\bigl (A(x,p,q)-A(-x,p,q)\\bigr ),$ where we use the small letters $a,b,c,\\ldots $ for sequences, capital letters $A,B,C,\\ldots $ for generating functions, and $AE,BE,CE,\\ldots ,AO,BO,CO,\\ldots $ for the even and odd parts of the generating functions, respectively.", "We denote by $H_y$ the partial derivative of the function $H$ with respect to $y$ .", "From (REF ), we get the following comparable result of (REF ).", "Theorem 1 We have $\\left\\lbrace \\begin{array}{ll}SO(x,p,q)&=\\frac{\\sqrt{p-1}}{2\\sqrt{q}}\\left(K\\left(\\frac{1-q}{1-p},\\sqrt{p-1}x-h_{p,q}\\right)-K\\left(\\frac{1-q}{1-p},\\sqrt{p-1}x+h_{p,q}\\right)\\right),\\\\[5pt]SE(x,p,q)&=\\frac{\\sqrt{p-1}}{2\\sqrt{p}}\\left(K\\left(\\frac{1-q}{1-p},\\sqrt{p-1}x-h_{p,q}\\right)+K\\left(\\frac{1-q}{1-p},\\sqrt{p-1}x+h_{p,q}\\right)\\right).\\end{array}\\right.$ where $K(p,x)=\\sqrt{1-p}{\\rm cn\\,}(\\sqrt{p}x,\\sqrt{1-1/p}), -1<p<1$ and $0<q<1$ .", "By (REF ), we have $\\left\\lbrace \\begin{array}{ll}SO_x&=SE+2p(1-p)SE_p+2p(1-q)SE_q+pxSE_x,\\\\SE_x&=SO+2q(1-p)SO_p+2q(1-q)SO_q+qxSO_x.\\end{array}\\right.$ Equivalently, $(SO^{\\prime },SE^{\\prime })=\\frac{1}{\\sqrt{p-1}}(\\sqrt{q}SO,\\sqrt{p}SE)$ satisfies $\\left\\lbrace \\begin{array}{ll}SO^{\\prime }_x&=2\\sqrt{pq}(1-p)SE^{\\prime }_p+2\\sqrt{pq}(1-q)SE^{\\prime }_q+\\sqrt{pq}xSE^{\\prime }_x,\\\\SE^{\\prime }_x&=2\\sqrt{pq}(1-p)SO^{\\prime }_p+2\\sqrt{pq}(1-q)SO^{\\prime }_q+\\sqrt{pq}xSO^{\\prime }_x.\\end{array}\\right.$ Solving (REF ) for $SO^{\\prime }_x-SE^{\\prime }_x$ and $SO^{\\prime }_x+SE^{\\prime }_x$ (with the help of maple), we obtain that there exist two (analytical) functions $K_1$ and $K_2$ such that $\\left\\lbrace \\begin{array}{ll}SO^{\\prime }-SE^{\\prime }&=K_1\\left(\\frac{1-q}{1-p},\\sqrt{p-1}x+h_{p,q}\\right),\\\\[5pt]SO^{\\prime }+SE^{\\prime }&=K_2\\left(\\frac{1-q}{1-p},\\sqrt{p-1}x-h_{p,q}\\right).\\end{array}\\right.$ In order to provide explicit formulas for the generating functions $SO^{\\prime }_x$ and $SE^{\\prime }_x$ , we first solve (REF ) for $q=0$ .", "In this case, we obtain $\\left\\lbrace \\begin{array}{ll}SO_x(x,p,0)&=SE(x,p,0)+2p(1-p)SE_p(x,p,0)+2pSE_q(x,p,0)+pxSE_x(x,p,0),\\\\SE_x(x,p,0)&=SO(x,p,0).\\end{array}\\right.$ Note that the initial conditions are $SO(0,p,q)=0,\\quad SE(0,p,q)=1,\\quad SO(x,0,0)=\\frac{e^x-e^{-x}}{2},\\quad SE(x,0,0)=\\frac{e^x+e^{-x}}{2}.$ Thus it is easy to see that the solution of this system of PDEs is $SO(x,p,0)=-I{\\rm dn\\,}(Ix, \\sqrt{p}){\\rm sn\\,}(Ix, \\sqrt{p})\\quad \\mbox{ and }\\quad SE(x,p,0)={\\rm cn\\,}(Ix,\\sqrt{p}),$ with $I$ is the imaginary unit.", "Therefore, when $q=0$ , (REF ) gives $-\\frac{\\sqrt{p}}{\\sqrt{p-1}}{\\rm cn\\,}(Ix,\\sqrt{p})&=K_1\\left(\\frac{1}{1-p},\\sqrt{p-1}x\\right)\\\\\\frac{\\sqrt{p}}{\\sqrt{p-1}}{\\rm cn\\,}(Ix,\\sqrt{p})&=K_2\\left(\\frac{1}{1-p},\\sqrt{p-1}x\\right),$ which leads to $K_2(p,x)=-K_1(p,x)=K(p,x)$ .", "Hence, by (REF ) we get (REF ), as desired.", "In order to provide a unified approach to the sequences discussed in this paper, we introduce the following definitions.", "Definition 2 A pair $(F,G)=(F(x,p,q),G(x,p,q))$ of functions is called a $J$ -pair of the first type if they satisfy the following system of PDEs: $\\left\\lbrace \\begin{array}{ll}F_x=2p\\sqrt{q}(1-p)G_p+2p\\sqrt{q}(1-q)G_q+2p\\sqrt{q}xG_x,\\\\G_x=2p\\sqrt{q}(1-p)F_p+2p\\sqrt{q}(1-q)F_q+2p\\sqrt{q}xF_x.\\end{array}\\right.$ Remark 3 By defining $P(x,p,q)=F(x,p,q)-G(x,p,q), ~Q(x,p,q)=F(x,p,q)+G(x,p,q),$ we have $\\left\\lbrace \\begin{array}{ll}P_x(x,p,q)+2p\\sqrt{q}((1-p)P_p(x,p,q)+(1-q)P_q(x,p,q)+xP_x(x,p,q))=0,\\\\Q_x(x,p,q)-2p\\sqrt{q}((1-p)Q_p(x,p,q)+(1-q)Q_q(x,p,q)+xQ_x(x,p,q))=0.\\end{array}\\right.$ It is not hard to check that the solution (with $p,q\\ne 1$ and $q\\ne 0$ ) of these PDEs is given by $P(x,p,q)&=V\\left(\\frac{1-q}{1-p},\\,x_+\\right),\\\\Q(x,p,q)&=V\\left(\\frac{1-q}{1-p},\\,x_-\\right),$ for any two functions $V$ and $\\widetilde{V}$ .", "Definition 4 A pair $(\\widetilde{F},\\widetilde{G})=(\\widetilde{F}(x,p,q),\\widetilde{G}(x,p,q))$ of functions is called a $J$ -pair of the second type if they satisfy the following system of PDEs: $\\left\\lbrace \\begin{array}{ll}\\widetilde{F}_x=2q\\sqrt{p}(1-p)\\widetilde{G}_p+\\sqrt{p}(1-q^2)\\widetilde{G}_q+xq\\sqrt{p}\\widetilde{G}_x,\\\\\\widetilde{G}_x=2q\\sqrt{p}(1-p)\\widetilde{F}_p+\\sqrt{p}(1-q^2)\\widetilde{F}_q+xq\\sqrt{p}\\widetilde{F}_x.\\end{array}\\right.$ Remark 5 We shall give a remark on the solution of (REF ).", "By defining $\\widetilde{P}(x,p,q)=\\widetilde{F}(x,p,q)-\\widetilde{G}(x,p,q), \\qquad \\widetilde{Q}(x,p,q)=\\widetilde{F}(x,p,q)+\\widetilde{G}(x,p,q),$ we have $\\left\\lbrace \\begin{array}{ll}\\widetilde{P}_x(x,p,q)+\\sqrt{q}(2q(1-p)\\widetilde{P}_p(x,p,q)+(1-q^2)\\widetilde{P}_q(x,p,q)+xq\\widetilde{P}_x(x,p,q))=0,\\\\\\widetilde{Q}_x(x,p,q)-\\sqrt{q}(2q(1-p)\\widetilde{Q}_p(x,p,q)+(1-q^2)\\widetilde{Q}_q(x,p,q)+xq\\widetilde{Q}_x(x,p,q))=0.\\end{array}\\right.$ It is not hard to check that the solution (with $p,q\\ne 1$ and $q\\ne 0$ ) of these PDEs is given by $\\left\\lbrace \\begin{array}{ll}\\widetilde{P}(x,p,q)&=W\\left(\\frac{1-q^2}{1-p},\\sqrt{p-1}x-\\ell _{p,q}\\right),\\\\\\widetilde{Q}(x,p,q)&=\\widetilde{W}\\left(\\frac{1-q^2}{1-p},\\sqrt{p-1}x+\\ell _{p,q}\\right),\\\\\\end{array}\\right.$ for any two functions $W$ and $\\widetilde{W}$ ." ], [ "$J$ -pair of the first type", "There are countless combinatorial structures related to the differential operators $xD$ and $Dx$ ; see, e.g., [13], [16], [20].", "It is natural to further study (REF ) via these differential operators.", "Assume that $(xD)^{n+1}(x)=(xD)(xD)^n(x)=xD((xD)^n(x)),$ $(Dx)^{n+1}(x)=(Dx)(Dx)^n(x)=D(x(Dx)^n(x)),$ $(Dx)^{n+1}(y)=(Dx)(Dx)^n(y)=D(x(Dx)^n(y)).$ In particular, from (REF ), we have $\\begin{split}(xD)(x)&=xyz,\\quad (xD)^2(x)=xy^2z^2+x^3y^2+x^3z^2,\\\\(Dx)(x)&=2xyz,\\quad (Dx)^2(x)=4xy^2z^2+2x^3y^2+2x^3z^2,\\\\(Dx)(y)&=y^2z+x^2z,\\quad (Dx)^2(y)=y^3z^2+5x^2yz^2+x^2y^3+x^4y.\\end{split}$ For $n\\ge 0$ , we define $(xD)^{2n}(x)&=\\sum _{i,j\\ge 0}a_{2n,i,j}x^{2i+1}y^{2j}z^{4n-2i-2j},\\\\(xD)^{2n+1}(x)&=\\sum _{i,j\\ge 0}a_{2n+1,i,j}x^{2i+1}y^{2j+1}z^{4n-2i-2j+1},\\\\(Dx)^{2n}(x)&=\\sum _{i,j\\ge 0}c_{2n,i,j}x^{2i+1}y^{2j}z^{4n-2i-2j},\\\\(Dx)^{2n+1}(x)&=\\sum _{i,j\\ge 0}c_{2n+1,i,j}x^{2i+1}y^{2j+1}z^{4n-2i-2j+1},\\\\(Dx)^{2n}(y)&=\\sum _{i,j\\ge 0}d_{2n,i,j}x^{2i}y^{2j+1}z^{4n-2i-2j},\\\\(Dx)^{2n+1}(y)&=\\sum _{i,j\\ge 0}d_{2n+1,i,j}x^{2i}y^{2j}z^{4n-2i-2j+3}.$ For convenience, we list the first terms of the corresponding generating functions: $A(x,p,q)&=1+x+(p(1+q)+q)\\frac{x^2}{2!}+(4p^2+5p(1+q)+q)\\frac{x^3}{3!}\\\\&+(p^3(4+4q)+p^2(5+50q+5q^2)+p(18q^2+18q)+q^2)\\frac{x^4}{4!}\\\\&+(16p^4+p^3(148+148q)+p^2(61+394q+61q^2)+p(58q+58q^2)+q^2)\\frac{x^5}{5!", "}+\\cdots ,$ $C(x,p,q)&=1+2x+2(p(1+q)+2q)\\frac{x^2}{2!}+8(p^2+2p(1+q)+q)\\frac{x^3}{3!}\\\\&+8(p^3(1+q)+2p^2(1+9q+q^2)+11pq(1+q)+2q^2)\\frac{x^4}{4!}\\\\&+16(2p^4+26p^3(1+q)+p^2(17+98q+17q^2)+26pq(1+q)+2q^2)\\frac{x^5}{5!", "}+\\cdots ,$ $D(x,p,q)&=1+(p+q)x+(p^2+p(5+q)+q)\\frac{x^2}{2!}+(p^3+p^2(5+18q)+pq(18+5q)+q^2)\\frac{x^3}{3!}\\\\&+(p^4+p^3(58+18q)+p^2(61+164q+5q^2)+pq(58+18q)+q^2)\\frac{x^4}{4!", "}+\\cdots ,$ Since $(xD)^{2n+1}(x)&=(xD)(xD)^{2n}(x)\\\\&=xD\\left(\\sum _{i,j\\ge 0}a_{2n,i,j}x^{2i+1}y^{2j}z^{4n-2i-2j}\\right)\\\\&=\\sum _{i,j\\ge 0}(2i+1)a_{2n,i,j}x^{2i+1}y^{2j+1}z^{4n-2i-2j+1}+\\sum _{i,j\\ge 0}2ja_{2n,i,j}x^{2i+3}y^{2j-1}z^{4n-2i-2j+1}+\\\\&\\quad \\sum _{i,j\\ge 0}(4n-2i-2j)a_{2n,i,j}x^{2i+3}y^{2j+1}z^{4n-2i-2j-1},$ we have $a_{2n+1,i,j}=(2i+1)a_{2n,i,j}+(2j+2)a_{2n,i-1,j+1}+(4n-2i-2j+2)a_{2n,i-1,j}.$ Similarly, $a_{2n,i,j}=(2i+1)a_{2n-1,i,j-1}+(2j+1)a_{2n-1,i-1,j}+(4n-2i-2j+1)a_{2n-1,i-1,j-1}.$ The recurrences (REF ) and (REF ) can be written as the following lemma.", "Lemma 6 We have $\\left\\lbrace \\begin{array}{ll}AO_x&=AE+2p(1-p)AE_p+2p(1-q)AE_q+2xpAE_x,\\\\AE_x&=(q+p-pq)AO+2pq(1-p)AO_p+2pq(1-q)AO_q+2xpqAO_x.\\end{array}\\right.$ Equivalently, $(AO^{\\prime },AE^{\\prime })$ is a $J$ -pair of the first type, where $AO^{\\prime }=\\sqrt{\\frac{pq}{p-1}}AO$ and $AE^{\\prime }=\\sqrt{\\frac{p}{p-1}}AE$ .", "Theorem 7 Let $y=\\frac{1-q}{1-p}$ .", "Define $G(x,p)=\\sqrt{\\frac{1-p}{\\cos ^2(x\\sqrt{p(1-p)})-p}}\\mbox{ and }H(x,p)=\\frac{(1-p)\\sin (2x\\sqrt{p(1-p)})}{2\\sqrt{p}(\\cos ^2(x\\sqrt{p(1-p)})-p)^{3/2}}.$ Then $AO(x,p,q)=\\frac{1}{2}\\sqrt{\\frac{p-q}{pq}}(H(yx_-,1-1/y)-G(yx_+,1-1/y)),\\\\AE(x,p,q)=\\frac{1}{2}\\sqrt{\\frac{p-q}{p}}(H(yx_-,1-1/y)+G(yx_+,1-1/y)).$ By Remark REF and Lemma REF , we obtain that $\\sqrt{\\frac{pq}{p-1}}AO(x,p,q)-\\sqrt{\\frac{p}{p-1}}AE(x,p,q)=V(y,x_+),$ and $\\sqrt{\\frac{pq}{p-1}}AO(x,p,q)+\\sqrt{\\frac{p}{p-1}}AE(x,p,q)=\\tilde{V}(y,x_-),$ for some functions $V$ and $\\tilde{V}$ .", "Moreover, at $q=0$ , then above equations reduce to $-V(p,x)=\\tilde{V}(p,x)=\\sqrt{1-p}AE(-px,1-1/p,0).$ Hence, if we guess that $AE(x,p,0)=G(x,p)$ and $AO(x,p,0)=H(x,p)$ , then we get $\\sqrt{\\frac{pq}{p-1}}AO(x,p,q)-\\sqrt{\\frac{p}{p-1}}AE(x,p,q)=-\\sqrt{1-y}G(yx_+,1-1/y),\\\\\\sqrt{\\frac{pq}{p-1}}AO(x,p,q)+\\sqrt{\\frac{p}{p-1}}AE(x,p,q)=\\sqrt{1-y}H(yx_-,1-1/y),$ which imply $AO(x,p,q)=\\frac{1}{2}\\sqrt{\\frac{(1-y)(p-1)}{pq}}(H(yx_-,1-1/y)-G(yx_+,1-1/y)),\\\\AE(x,p,q)=\\frac{1}{2}\\sqrt{\\frac{(p-1)(1-y)}{p}}(H(yx_-,1-1/y)+G(yx_+,1-1/y)).$ It is routine to check that the functions $AO$ and $AE$ satisfy Lemma REF .", "This completes the proof.", "Along the same lines, we get $\\begin{split}c_{2n,i,j}&=(2i+2)c_{2n-1,i,j-1}+(2j+1)c_{2n-1,i-1,j}+(4n-2i-2j+1)c_{2n-1,i-1,j-1},\\\\c_{2n+1,i,j}&=(2i+2)c_{2n,i,j}+(2j+2)c_{2n,i-1,j+1}+(4n-2i-2j+2)c_{2n,i-1,j}.\\end{split}$ This leads us to the following result.", "Lemma 8 We have $\\left\\lbrace \\begin{array}{ll}CO_x=2CE+2p(1-p)CE_p+2p(1-q)CE_q+2xpCE_x,\\\\CE_x=(p+2q-pq)CO+2pq(1-p)CO_p+2pq(1-q)CO_q+2xpqCO_x.\\end{array}\\right.$ Equivalently, $(CO^{\\prime },CE^{\\prime })$ is a $J$ -pair of the first type, where $CO^{\\prime }=\\frac{p\\sqrt{q}}{p-1}CO$ and $CE^{\\prime }=\\frac{p}{p-1}CE$ .", "Theorem 9 Define $y=\\frac{1-q}{1-p}$ and $G(x,p)=\\frac{1-p}{p\\cos ^2(x\\sqrt{p-1})+1-p}$ .", "Then $\\begin{array}{ll}CO(x,p,q)=\\frac{p-1}{2p\\sqrt{q}}\\bigl (G(x_-,y)-G(x_+,y)\\bigr ),\\\\[5pt]CE(x,p,q)=\\frac{p-1}{2p}\\bigl (G(x_-,y)+G(x_+,y)\\bigr ),\\\\[5pt]C(x,p,q)=\\frac{p-1}{2p\\sqrt{q}}\\bigl (G(x_-,y)-G(x_+,y)\\bigr )+\\frac{p-1}{2p}\\bigl (G(x_-,y)+G(x_+,y)\\bigr ).\\end{array}$ By Remark REF and Lemma REF , we obtain that $\\frac{p\\sqrt{q}}{p-1}CO(x,p,q)-\\frac{p}{p-1}CE(x,p,q)=\\tilde{V}(y,x_+),$ and $\\frac{p\\sqrt{q}}{p-1}CO(x,p,q)+\\frac{p}{p-1}CE(x,p,q)=V(y,x_-)$ for some functions $V$ and $\\tilde{V}$ .", "Moreover, at $q=0$ , then above equations reduce to $V(1/(1-p),(p-1)x)=-\\tilde{V}(1/(1-p),(p-1)x).$ Hence, if we take $(1-p)CE(-px,1-1/p,0)&=G(x,p),\\\\[5pt]CO(x,p,q)&=\\frac{p-1}{2p\\sqrt{q}}(G(x_-,y)-G(x_+,y)),\\\\[5pt]CE(x,p,q)&=\\frac{p-1}{2p}(G(x_-,y)+G(x_+,y)),$ then (REF ) is a solution for (REF ), where $V\\bigl (1/(1-p),(p-1)x\\bigr )=-\\tilde{V}\\bigl (1/(1-p),(p-1)x\\bigr )=(1-p)CE(-px,1-1/p,0)=G(x,p).$ To complete the proof, we need to check that the functions $CO$ and $CE$ satisfy Lemma REF , which is routine.", "Corollary 10 We have $C(x,0,q)&=\\cosh (2\\sqrt{q}x)+\\frac{1}{\\sqrt{q}}\\sinh (2\\sqrt{q}x),\\\\C(x,1,q)&=\\frac{(x^2(q-1)+2x+1)}{(x^2(1-q)-2x+1)(x^2(1-q)+2x+1)},\\\\C(x,p,0)&=\\frac{(1-p)\\sqrt{1-p}\\sin (2x\\sqrt{p(1-p)})}{\\sqrt{p}(\\cos ^2(x\\sqrt{p(1-p)})-p)^2}+\\frac{1-p}{\\cos ^2(x\\sqrt{p(1-p)})-p},\\\\C(x,p,1)&=\\frac{p-1}{p-e^{2x(p-1)}}.$ By applying Theorem REF for $q=0$ or $p=1$ , we obtain the formulas of $C(x,p,0)$ and $C(x,1,q)$ .", "Solving (REF ) for $p=0$ we obtain $CE(x,0,q)&=\\alpha _q e^{2\\sqrt{q}x}+\\beta _q e^{-2\\sqrt{q}x},\\\\CO(x,0,q)&=\\frac{1}{\\sqrt{q}}(\\alpha _q e^{2\\sqrt{q}x}-\\beta _q e^{-2\\sqrt{q}x}).$ By using the initial conditions $CE(0,p,q)=1$ and $CO(0,p,q)=0$ , we obtain $CE(x,0,p)=\\cosh (2\\sqrt{q}x)$ and $CO(x,0,q)=\\frac{1}{\\sqrt{q}}\\sinh (2\\sqrt{q}x)$ , which completes the first part of the proof.", "Again, solving (REF ) with $q=1$ for $CO(x,p,1)-CE(x,p,1)$ and $CO(x,p,1)+CE(x,p,1)$ , we obtain $CO(x,p,1)-CE(x,p,1)&=\\frac{p-1}{p}V(x(p-1)+\\frac{1}{2}\\ln p),\\\\CO(x,p,1)+CE(x,p,1)&=\\frac{p-1}{p}\\tilde{V}(x(p-1)-\\frac{1}{2}\\ln p),$ where $V,\\tilde{V}$ are two fixed functions.", "By the initial values $CE(0,p,q)=1$ and $CO(0,p,q)=0$ , we get $V(y)=\\frac{e^{2y}}{1-e^{2y}}\\mbox{ and }\\tilde{V}(y)=\\frac{1}{1-e^{2y}}.$ Hence, $CO(x,p,1)-CE(x,p,1)&=\\frac{(p-1)e^{2x(p-1)}}{1-pe^{2x(p-1)}},\\\\CO(x,p,1)+CE(x,p,1)&=\\frac{p-1}{p-e^{2x(p-1)})}.$ This completes the proof.", "Along the same lines, we get $\\begin{split}d_{2n,i,j}&=(2i+1)d_{2n-1,i,j}+(2j+2)d_{2n-1,i-1,j+1}+(4n-2i-2j+1)d_{2n-1,i-1,j},\\\\d_{2n+1,i,j}&=(2i+1)d_{2n,i,j-1}+(2j+1)d_{2n,i-1,j}+(4n-2i-2j+4)d_{2n,i-1,j-1},\\end{split}$ which leads to the following result.", "Lemma 11 We have $\\left\\lbrace \\begin{array}{ll}DO_x=(p+q)DE+2pq(1-p)DE_p+2pq(1-q)DE_q+2pqxDE_x,\\\\DE_x=(1+p)DO+2p(1-p)DO_p+2p(1-q)DO_q+2pxDO_x.\\end{array}\\right.$ Equivalently, $(DO^{\\prime },DE^{\\prime })$ is a $J$ -pair of the first type, where $DO^{\\prime }=\\sqrt{\\frac{p}{p-1}}DO$ and $DE^{\\prime }=\\sqrt{\\frac{pq}{p-1}}DE$ .", "By similar arguments as in the proof of Theorem REF with help of Remark REF and Lemma REF , we obtain the following result.", "Theorem 12 Define $y=\\frac{1-q}{1-p}$ and $G(x,p)=\\frac{\\sinh (x\\sqrt{p-1})}{1-\\frac{p}{p-1}\\cosh ^2(x\\sqrt{p-1})}$ .", "Then $DO(x,p,q)&=\\frac{\\sqrt{p-1}}{2\\sqrt{p}}(G(x_-,y)+G(x_+,y)),\\\\DE(x,p,q)&=\\frac{\\sqrt{p-1}}{2\\sqrt{pq}}(G(x_-,y)-G(x_+,y)),\\\\D(x,p,q)&=\\frac{\\sqrt{p-1}}{2\\sqrt{pq}}(G(x_-,y)-G(x_+,y))+\\frac{\\sqrt{p-1}}{2\\sqrt{p}}(G(x_-,y)+G(x_+,y)).$ Corollary 13 Let $p^{\\prime }=\\sqrt{p(p-1)}$ .", "Then we have $D(x,p,0)&=\\frac{(p-1)\\cosh (xp^{\\prime })(\\cosh ^2(xp^{\\prime })-2+p)}{((p-1)\\cosh ^2(xp^{\\prime })-p\\sinh ^2(xp^{\\prime }))^2}+\\frac{p^{\\prime }\\sinh (xp^{\\prime })}{p-\\cosh ^2(xp^{\\prime })},\\\\D(x,1,q)&=\\frac{(x^2(q-1)+2x-1)(x^2(1-q)+2x+1)(x^3(q-1)^2+x^2(q-1)-x(q+1)-1)}{(x^2(q-1)-2x\\sqrt{q}+1)^2(x^2(q-1)+2x\\sqrt{q}+1)^2},\\\\D(x,p,1)&=\\frac{(1 - p)e^{(1 - p)x}}{1-pe^{2(1 - p)x}}.$ From Corollary REF and Corollary REF , it is easy to verify that $C(x,1,q)&=\\sum _{n\\ge 0}\\sum _{k\\ge 0}\\binom{2n+1}{2k}q^kx^{2n}+\\sum _{n\\ge 1}\\sum _{k\\ge 0}\\binom{2n}{2k+1}q^kx^{2n-1},\\\\D(x,1,q)&=\\sum _{n\\ge 0}\\sum _{k\\ge 0}\\binom{2n+1}{2k+1}q^kx^{2n}+\\sum _{n\\ge 1}\\sum _{k\\ge 0}\\binom{2n}{2k}q^kx^{2n-1}.$" ], [ "$J$ -pair of the second type", "In his study [5] of exponential structures in combinatorics, Chen introduced the grammatical method systematically.", "Let $A$ be an alphabet whose letters are regarded as independent commutative indeterminates.", "Following Chen, a context-free grammar $G$ over $A$ is defined as a set of substitution rules that replace a letter in $A$ by a formal function over $A$ .", "The formal derivative $D$ is a linear operator defined with respect to a context-free grammar $G$ .", "Therefore, an equivalent form of (REF ) is given by the context-free grammar $G=\\lbrace x\\rightarrow yz,\\, y\\rightarrow xz,\\,z\\rightarrow xy\\rbrace .$ Dumont [10] considered chains of general substitution rules on words.", "In particular, he discovered the following result.", "Proposition 14 ([10]) If $G=\\lbrace w\\rightarrow wx,\\, x\\rightarrow wx\\rbrace ,$ then $D^n(w)=\\sum _{k=0}^{n-1}\\genfrac<>{0.0pt}{}{n}{k}w^{k+1}x^{n-k},$ where $\\genfrac<>{0.0pt}{}{n}{k}$ is the Eulerian number, i.e., the number of permutations in $\\mathfrak {S}_n$ with $k$ descents.", "As a conjunction of (REF ) and (REF ), it is natural to consider the context-free grammar $G=\\lbrace w\\rightarrow wx,\\, x\\rightarrow yz,\\, y\\rightarrow xz,\\,z\\rightarrow xy\\rbrace .$ From (REF ), we have $\\begin{split}D(w)&=wx,\\quad D^2(w)=w(x^2+yz),\\quad D^3(x)=w(x^3+xz^2+3xyz+xy^2),\\\\D^4(w)&=w(x^4+10x^2yz+4x^2z^2+4x^2y^2+3y^2z^2+y^3z+yz^3),\\\\D(w^2)&=2w^2x,\\quad D^2(w^2)=w^2(4x^2+2yz),\\quad D^3(w^2)=w^2(8x^3+12xyz+2xz^2+2xy^2).\\end{split}$ For $n\\ge 0$ , we define $D^{2n}(w)&=w\\sum _{i,j\\ge 0}t_{2n,i,j}x^{2i}y^{j}z^{2n-2i-j},\\\\D^{2n+1}(w)&=w\\sum _{i,j\\ge 0}t_{2n+1,i,j}x^{2i+1}y^{j}z^{2n-2i-j},\\\\D^{2n}(w^2)&=w^2\\sum _{i,j\\ge 0}r_{2n,i,j}x^{2i}y^{j}z^{2n-2i-j},\\\\D^{2n+1}(w^2)&=w^2\\sum _{i,j\\ge 0}r_{2n+1,i,j}x^{2i+1}y^{j}z^{2n-2i-j}.$ The first terms of the corresponding generating functions are given as follows: $T(x,p,q)&=1+x+(p+q)\\frac{x^2}{2!}+(1+p+3q+q^2)\\frac{x^3}{3!", "}\\\\&\\quad +(p^2+4p+(10p+1)q+(4p+3)q^2+q^3)\\frac{x^4}{4!", "}\\\\&\\quad +(p^2+14p+1+(30p+15)q+(14p+29)q^2+15q^3+q^4)\\frac{x^5}{5!", "}+\\cdots .\\\\R(x,p,q)&=1+2x+(4p+2q)\\frac{x^2}{2!}+(2+8p+12q+2q^2)\\frac{x^3}{3!", "}\\\\&\\quad +(16p+16p^2+(2+56p)q+(12+16p)q^2+2q^3)\\frac{x^4}{4!", "}\\\\&\\quad +(2+88p+32p^2+(60+240p)q+(148+88p)q^2+60q^3+2q^4)\\frac{x^5}{5!", "}+\\cdots .$ Since $D^{2n+1}(w)&=D(D^{2n}(w))\\\\&=D\\left(w\\sum _{i,j\\ge 0}t_{2n,i,j}x^{2i}y^{j}z^{2n-2i-j}\\right)\\\\&=w\\sum _{i,j\\ge 0}t_{2n,i,j}x^{2i+1}y^{j}z^{2n-2i-j}+w\\sum _{i,j\\ge 0}2it_{2n,i,j}x^{2i-1}y^{j+1}z^{2n-2i-j+1}\\\\&\\quad +w\\sum _{i,j\\ge 0}jt_{2n,i,j}x^{2i+1}y^{j-1}z^{2n-2i-j+1}+w\\sum _{i,j\\ge 0}(2n-2i-j)t_{2n,i,j}x^{2i+1}y^{j+1}z^{2n-2i-j-1},$ we have $t_{2n+1,i,j}=t_{2n,i,j}+(2i+2)t_{2n,i+1,j-1}+(j+1)t_{2n,i,j+1}+(2n-2i-j+1)t_{2n,i,j-1}.$ Similarly, $t_{2n,i,j}=t_{2n-1,i-1,j}+(2i+1)t_{2n-1,i,j-1}+(j+1)t_{2n-1,i-1,j+1}+(2n-2i-j+1)t_{2n-1,i-1,j-1}.$ By rewriting these recurrence relations in terms of the generating functions $TE$ and $TO$ , we obtain the following result.", "Lemma 15 We have $\\left\\lbrace \\begin{array}{ll}TO_x&=TE+2q(1-p)TE_p+(1-q^2)TE_q+xqTE_x,\\\\TE_x&=(p+q-qp)TO+2pq(1-p)TO_p+p(1-q^2)TO_q+xqpTO_x.\\end{array}\\right.$ Equivalently, $(TO^{\\prime },TE^{\\prime })$ is a $J$ -pair of the second type, where $TO^{\\prime }=\\sqrt{\\frac{p(1+q)}{1-q}}TO$ and $TE^{\\prime }=\\sqrt{\\frac{1+q}{1-q}}TE$ .", "Theorem 16 Let $\\ell ^{\\prime }_{p,q}=\\sqrt{\\frac{1-q^2}{1-p}}\\ell _{p,q}$ .", "Then we have $\\left\\lbrace \\begin{array}{ll}TO(x,p,q)=\\frac{q-1}{\\sqrt{p(p-1)}}{\\rm sn\\,}(-\\sqrt{q^2-1}x+\\ell ^{\\prime }_{p,q},\\sqrt{\\frac{p-q^2}{1-q^2}}),\\\\[5pt]TE(x,p,q)=\\sqrt{\\frac{1-q}{1+q}}{\\rm dn\\,}(-\\sqrt{q^2-1}x-\\ell ^{\\prime }_{p,q},\\sqrt{\\frac{p-q^2}{1-q^2}}).\\end{array}\\right.$ By Remark REF , we see that Lemma REF leads to $\\left\\lbrace \\begin{array}{ll}\\sqrt{\\frac{p(1+q)}{1-q}}TO(x,p,q)-\\sqrt{\\frac{1+q}{1-q}}TE(x,p,q)=W\\left(\\frac{1-q^2}{1-p},\\sqrt{p-1}x-\\ell _{p,q}\\right),\\\\[5pt]\\sqrt{\\frac{p(1+q)}{1-q}}TO(x,p,q)+\\sqrt{\\frac{1+q}{1-q}}TE(x,p,q)=\\widetilde{W}\\left(\\frac{1-q^2}{1-p},\\sqrt{p-1}x+\\ell _{p,q}\\right),\\end{array}\\right.$ for some functions $W$ and $\\tilde{W}$ .", "Thus, at $q=0$ we have that $\\left\\lbrace \\begin{array}{ll}\\sqrt{p}TO(I\\sqrt{p}x,1-1/p,0)-TE(I\\sqrt{p}x,1-1/p,0)&=W\\left(p,x\\right),\\\\\\sqrt{p}TO(I\\sqrt{p}x,1-1/p,0)+TE(I\\sqrt{p}x,1-1/p,0)&=\\widetilde{W}\\left(p,x\\right),\\\\\\end{array}\\right.$ Therefore, if we set $TE(x,p,0)={\\rm dn\\,}(Ix,\\sqrt{p})\\quad \\mbox{ and }\\quad TO(x,p,0)=-I{\\rm sn\\,}(Ix,\\sqrt{p}),$ then $\\left\\lbrace \\begin{array}{ll}-I\\sqrt{p}{\\rm sn\\,}(-\\sqrt{p}x,\\sqrt{1-1/p})-{\\rm dn\\,}(-\\sqrt{p}x,\\sqrt{1-1/p})&=W\\left(p,x\\right),\\\\-I\\sqrt{p}{\\rm sn\\,}(-\\sqrt{p}x,\\sqrt{1-1/p})+{\\rm dn\\,}(-\\sqrt{p}x,\\sqrt{1-1/p})&=\\widetilde{W}\\left(p,x\\right).\\end{array}\\right.$ By (REF ), we have $\\left\\lbrace \\begin{array}{ll}&\\sqrt{\\frac{p(1+q)}{1-q}}TO(x,p,q)-\\sqrt{\\frac{1+q}{1-q}}TE(x,p,q)\\\\&\\qquad =-\\sqrt{\\frac{q^2-1}{1-p}}{\\rm sn\\,}(-\\sqrt{q^2-1}x+\\ell ^{\\prime }_{p,q},\\sqrt{\\frac{p-q^2}{1-q^2}})-{\\rm dn\\,}(-\\sqrt{q^2-1}x+\\ell ^{\\prime }_{p,q},\\sqrt{\\frac{p-q^2}{1-q^2}}),\\\\ &\\sqrt{\\frac{p(1+q)}{1-q}}TO(x,p,q)+\\sqrt{\\frac{1+q}{1-q}}TE(x,p,q)\\\\&\\qquad =-\\sqrt{\\frac{q^2-1}{1-p}}{\\rm sn\\,}(-\\sqrt{q^2-1}x-\\ell ^{\\prime }_{p,q},\\sqrt{\\frac{p-q^2}{1-q^2}})+{\\rm dn\\,}(-\\sqrt{q^2-1}x-\\ell ^{\\prime }_{p,q},\\sqrt{\\frac{p-q^2}{1-q^2}}),\\end{array}\\right.$ which implies $\\left\\lbrace \\begin{array}{ll}TO(x,p,q)=\\frac{q-1}{\\sqrt{p(p-1)}}{\\rm sn\\,}(-\\sqrt{q^2-1}x+\\ell ^{\\prime }_{p,q},\\sqrt{\\frac{p-q^2}{1-q^2}}),\\\\TE(x,p,q)=\\sqrt{\\frac{1-q}{1+q}}{\\rm dn\\,}(-\\sqrt{q^2-1}x-\\ell ^{\\prime }_{p,q},\\sqrt{\\frac{p-q^2}{1-q^2}}),\\end{array}\\right.$ which agrees with the case $q=0$ .", "It suffices and is routine to check that the functions $TO$ and $TE$ satisfy Lemma REF .", "This completes the proof.", "By the above theorem (or by a direct check using Lemma REF ), we realize the following result.", "Corollary 17 Let $h(x,p)=\\frac{\\sqrt{p-1}}{\\sqrt{p-1}\\cosh (x\\sqrt{p-1})-\\sqrt{p}\\sinh (x\\sqrt{p-1})}$ .", "Then, we have $T(x,p,1)&=\\frac{1}{2}(h(x,p)+h(-x,p))+\\frac{1}{2\\sqrt{p}}(h(x,p)-h(-x,p)),\\\\T(x,1,q)&=\\frac{q^2-1+\\sqrt{q^2-1}\\sinh (x\\sqrt{q^2-1})}{(1+q)(q-\\cosh (x\\sqrt{q^2-1}))}.$ Along the same lines, we have $\\begin{split}r_{2n+1,i,j}&=2r_{2n,i,j}+(2i+2)r_{2n,i+1,j-1}+(j+1)r_{2n,i,j+1}+(2n-2i-j+1)r_{2n,i,j-1},\\\\r_{2n,i,j}&=2r_{2n-1,i-1,j}+(2i+1)r_{2n-1,i,j-1}+(j+1)r_{2n-1,i-1,j+1}\\\\&\\quad +(2n-2i-j+1)r_{2n-1,i-1,j-1},\\end{split}$ which implies the following result.", "Lemma 18 We have $\\left\\lbrace \\begin{array}{ll}RO_x=2RE+2q(1-p)RE_p+(1-q^2)RE_q+xqRE_x,\\\\RE_x=(2p+q-pq)RO+2pq(1-p)RO_p+p(1-q^2)RO_q+xpqRO_x.\\end{array}\\right.$ Equivalently, $(RO^{\\prime },RE^{\\prime })$ is a $J$ -pair of the second type, where $RO^{\\prime }=\\frac{\\sqrt{p}(1+q)}{1-q}RO\\quad \\mbox{ and }\\quad RE^{\\prime }=\\frac{1+q}{1-q}RE.$ Along the line of the proof of Theorem REF we state the following result.", "Theorem 19 Let $\\left\\lbrace \\begin{array}{ll}U\\left(p,x\\right)=-2I\\sqrt{p}{\\rm dn\\,}(-\\sqrt{p}x,p^{\\prime }){\\rm sn\\,}(-\\sqrt{p}x,p^{\\prime })-2p{\\rm cn\\,}^2(-\\sqrt{p}x,p^{\\prime })+1-2/p,\\\\\\widetilde{U}\\left(p,x\\right)=-2I\\sqrt{p}{\\rm dn\\,}(-\\sqrt{p}x,p^{\\prime }){\\rm sn\\,}(-\\sqrt{p}x,p^{\\prime })+2p{\\rm cn\\,}^2(-\\sqrt{p}x,p^{\\prime })-1+2/p,\\end{array}\\right.$ where $p^{\\prime }=\\sqrt{1-1/p}$ .", "Then $\\left\\lbrace \\begin{array}{ll}RO(x,p,q)&=\\frac{\\sqrt{p}(1-q)}{2(1+q)}\\left(U\\left(\\frac{1-q^2}{1-p},\\sqrt{p-1}x-\\ell _{p,q}\\right)+\\widetilde{U}\\left(\\frac{1-q^2}{1-p},\\sqrt{p-1}x+\\ell _{p,q}\\right)\\right),\\\\[5pt]RE(x,p,q)&=\\frac{1-q}{2(1+q)}\\left(\\widetilde{U}\\left(\\frac{1-q^2}{1-p},\\sqrt{p-1}x+\\ell _{p,q}\\right)-U\\left(\\frac{1-q^2}{1-p},\\sqrt{p-1}x-\\ell _{p,q}\\right)\\right).\\end{array}\\right.$ By Remark REF , we obtain $\\left\\lbrace \\begin{array}{ll}\\frac{\\sqrt{p}(1+q)}{1-q}RO(x,p,q)-\\frac{1+q}{1-q}RE(x,p,q)&=W\\left(\\frac{1-q^2}{1-p},\\sqrt{p-1}x-\\ell _{p,q}\\right),\\\\[5pt]\\frac{\\sqrt{p}(1+q)}{1-q}RO(x,p,q)+\\frac{1+q}{1-q}RE(x,p,q)&=\\widetilde{W}\\left(\\frac{1-q^2}{1-p},\\sqrt{p-1}x+\\ell _{p,q}\\right),\\\\\\end{array}\\right.$ for some functions $W$ and $\\tilde{W}$ .", "Thus, at $q=0$ we have that $\\left\\lbrace \\begin{array}{ll}\\sqrt{p}RO(I\\sqrt{p}x,1-1/p,0)-RE(I\\sqrt{p}x,1-1/p,0)&=W\\left(p,x\\right),\\\\\\sqrt{p}RO(I\\sqrt{p}x,1-1/p,0)+RE(I\\sqrt{p}x,1-1/p,0)&=\\widetilde{W}\\left(p,x\\right),\\\\\\end{array}\\right.$ Therefore, if we set $RE(x,p,0)=2p{\\rm cn\\,}^2(Ix,\\sqrt{p})-2p+1\\quad \\mbox{ and }\\quad RO(x,p,0)=-2I{\\rm dn\\,}(Ix,\\sqrt{p}){\\rm sn\\,}(Ix,\\sqrt{p}),$ then $\\left\\lbrace \\begin{array}{ll}-2I\\sqrt{p}{\\rm dn\\,}(-\\sqrt{p}x,p^{\\prime }){\\rm sn\\,}(-\\sqrt{p}x,p^{\\prime })-2p{\\rm cn\\,}^2(-\\sqrt{p}x,p^{\\prime })+1-2/p&=W\\left(p,x\\right),\\\\[5pt]-2I\\sqrt{p}{\\rm dn\\,}(-\\sqrt{p}x,p^{\\prime }){\\rm sn\\,}(-\\sqrt{p}x,p^{\\prime })+2p{\\rm cn\\,}^2(-\\sqrt{p}x,p^{\\prime })-1+2/p&=\\widetilde{W}\\left(p,x\\right),\\end{array}\\right.$ where $p^{\\prime }=\\sqrt{1-1/p}$ .", "By (REF ), we deduce that $\\left\\lbrace \\begin{array}{ll}RO(x,p,q)&=\\frac{\\sqrt{p}(1-q)}{2(1+q)}\\biggl (W\\left(\\frac{1-q^2}{1-p},\\sqrt{p-1}x-\\ell _{p,q}\\right)+\\widetilde{W}\\left(\\frac{1-q^2}{1-p},\\sqrt{p-1}x+\\ell _{p,q}\\right)\\biggr ),\\\\[5pt]RE(x,p,q)&=\\frac{1-q}{2(1+q)}\\biggl (\\widetilde{W}\\left(\\frac{1-q^2}{1-p},\\sqrt{p-1}x+\\ell _{p,q}\\right)-W\\left(\\frac{1-q^2}{1-p},\\sqrt{p-1}x-\\ell _{p,q}\\right)\\biggr ),\\end{array}\\right.$ which agrees with the case $q=0$ .", "To complete the proof, we still need to check that the functions $RO$ and $RE$ satisfy Lemma REF , which is routine." ], [ "Applications", "In this section we apply the results obtained in the previous section to present new characterizations for several combinatorial structures." ], [ "Peaks, descents and perfect matchings", "One of the most interesting permutation statistics is the peaks statistic (see, e.g., [2], [7], [18], [19], [21], [25], [31] and the references contained therein).", "A left peak in $\\pi $ is an index $i\\in [n-1]$ such that $\\pi (i-1)<\\pi (i)>\\pi (i+1)$ , where we take $\\pi (0)=0$ .", "Denote by $\\widetilde{P}_{n,k}$ the number of permutations in $\\mathfrak {S}_n$ with $k$ left peaks.", "Recall that $P_{n,k}$ is the number of permutations in $\\mathfrak {S}_n$ with $k$ interior peaks.", "Define $P_n(x)=\\sum _{k=0}^{\\lfloor \\frac{n-1}{2}\\rfloor }P_{n,k}x^k\\quad \\mbox{ and }\\quad \\widetilde{P}_n(x)=\\sum _{k=0}^{\\lfloor \\frac{n}{2}\\rfloor }\\widetilde{P}_{n,k}x^k.$ The polynomial $P_n(x)$ satisfies recurrence relation $P_{n+1}(x)=(nx-x+2)P_n(x)+2x(1-x)\\frac{d}{dx}P_n(x),$ with the initial values $P_1(x)=1$ , $P_2(x)=2$ , $P_3(x)=4+2x$ , and the polynomial $\\widetilde{P}_n(x)$ satisfies recurrence relation $\\widetilde{P}_{n+1}(x)=(nx+1)\\widetilde{P}_{n}(x)+2x(1-x)\\frac{d}{dx}\\widetilde{P}_{n}(x),$ with the initial values $\\widetilde{P}_1(x)=1,\\widetilde{P}_2(x)=1+x,\\widetilde{P}_3(x)=1+5x$ (see [27]).", "A descent of a permutation $\\pi \\in \\mathfrak {S}_n$ is a position $i$ such that $\\pi (i)>\\pi (i+1)$ .", "Denote by ${\\rm des\\,}(\\pi )$ the number of descents of $\\pi $ .", "Let $A_n(x)=\\sum _{\\pi \\in \\mathfrak {S}_n}x^{{\\rm des\\,}(\\pi )}=\\sum _{k=0}^{n-1}\\genfrac<>{0.0pt}{}{n}{k}x^{k}.$ The polynomial $A_n(x)$ is called an Eulerian polynomial.", "Let $B_n$ denote the set of signed permutations of $\\pm [n]$ such that $\\pi (-i)=-\\pi (i)$ for all $i$ , where $\\pm [n]=\\lbrace \\pm 1,\\pm 2,\\ldots ,\\pm n\\rbrace $ .", "Let ${B}_n(x)=\\sum _{k=0}^nB(n,k)x^{k}=\\sum _{\\pi \\in B_n}x^{{\\rm des\\,}_B(\\pi )},$ where ${\\rm des\\,}_B=\|\\lbrace i\\in [n]:\\omega (i-1)>\\omega ({i})\\rbrace \|$ with $\\pi (0)=0$ .", "The polynomial $B_n(x)$ is said to be the Eulerian polynomial of type $B$, while $B(n,k)$ is called an Eulerian number of type $B$.", "It is well known that Eulerian polynomials are closely retated to peak statistics.", "In particular, Stembridge [31] showed that $P_n\\left(\\frac{4x}{(1+x)^2}\\right)=\\frac{2^{n-1}}{(1+x)^{n-1}}A_n(x).$ In [24], Petersen observed that $\\widetilde{P}_n\\left(\\frac{4x}{(1+x)^2}\\right)=\\frac{1}{(1+x)^n}\\left((1-x)^n+\\sum _{i=1}^n\\binom{n}{i}(1-x)^{n-i}2^ixA_i(x)\\right).$ It should be noted that $\\widetilde{P}_n\\left(\\frac{4x}{(1+x)^2}\\right)=\\frac{B_n(x)}{(1+x)^{n}}.$ Recall that a perfect matching of $[2n]$ is a partition of $[2n]$ into $n$ blocks of size 2.", "Denote by $N({n,k})$ the number of perfect matchings of $[2n]$ with the restriction that only $k$ matching pairs have odd smaller entries (see [27]).", "It is easy to verify that $N({n+1,k})=2kN({n,k})+(2n-2k+3)N({n,k-1}).$ Let $N_n(x)=\\sum _{k=1}^nN({n,k})x^k$ .", "It follows from (REF ) that $N_{n+1}(x)=(2n+1)xN_n(x)+2x(1-x)\\frac{d}{dx}N_n(x),$ with initial values $N_0(x)=1$ , $N_1(x)=x$ , $N_2(x)=2x+x^2$ and $N_3(x)=4x+10x^2+x^3$ .", "There is an expansion of the Eulerian polynomial $A_n(x)$ in terms of $N_n(x)$ (see [20]): $2^nA_n(x)=\\sum _{k=0}^n\\binom{n}{k}N_k(x)N_{n-k}(x).$ We conclude the following theorem from the discussion above.", "Theorem 20 For $n\\ge 1$ , we have $\\sum _{i,j\\ge 0}a_{n,i,j}=(2n-1)!", "!$ .", "$\\sum _{j\\ge 0}a_{n,i,j}=N(n,n-i)$ .", "$\\sum _{j\\ge 0}a_{n,i,\\lfloor \\frac{n}{2}\\rfloor }x^i=\\sum _{j\\ge 0}a_{n,i,\\lfloor \\frac{n}{2}\\rfloor -i}x^i=\\widetilde{P}_n(x)$ .", "$\\sum _{j\\ge 0}c_{n,i,j}=2^n\\genfrac<>{0.0pt}{}{n}{i}$ .", "$\\sum _{j\\ge 0}d_{n,i,j}=B(n,i)$ .", "$\\sum _{i\\ge 0}c_{n,i,\\lfloor \\frac{n}{2}\\rfloor }x^i=\\sum _{i\\ge 0}c_{n,i,\\lfloor \\frac{n}{2}\\rfloor -i}x^i=P_{n+1}(x)$ .", "$\\sum _{i\\ge 0}c_{2n-1,i,0}x^{2n-2-i}=\\sum _{i\\ge 0}c_{2n,i,0}x^{2n-1-i}=P_{2n}(x)$ .", "$\\sum _{i\\ge 0}d_{n,i,\\lceil \\frac{n}{2}\\rceil }x^i=\\widetilde{P}_{n}(x)$ and $\\sum _{i\\ge 0}d_{n,i,\\lceil \\frac{n}{2}\\rceil -i}x^i=\\widetilde{P}_{n+1}(x)$ .", "$\\sum _{i\\ge 0}d_{2n,i,0}x^{2n-i}=\\sum _{i\\ge 0}d_{2n+1,i,0}x^{2n+1-i}=\\widetilde{P}_{2n+1}(x)$ .", "We only prove the assertion for the sequence $a_{n,i,j}$ and the corresponding assertion for the other sequences follow from similar consideration.", "(A) Setting $p,q=1$ in Lemma REF gives $\\left\\lbrace \\begin{array}{ll}AO_x(x,1,1)=AE(x,1,1)+2xAE_x(x,1,1),\\\\AE_x(x,1,1)=AO(x,1,1)+2xAO_x(x,1,1),\\end{array}\\right.$ which implies $A_x(x,1,1)=A(x,1,1)+2xA_x(x,1,1)$ .", "Therefore, $A(x,1,1)=\\frac{A(0,1,1)}{\\sqrt{1-2x}}=\\frac{1}{\\sqrt{1-2x}}=\\sum _{n\\ge 0}\\frac{n!}{2^n}\\binom{2n}{n}\\frac{x^n}{n!", "}.$ Hence, $\\sum _{i,j\\ge 0}a_{n,i,j}=\\frac{n!}{2^n}\\binom{2n}{n}=(2n-1)!", "!,$ as required.", "(B) Setting $q=1$ in Lemma REF gives $A_x(x,p,1)=A(x,p,1)+2p(1-p)A_p(x,p,1)+2xpA_x(x,1,1).$ By $A(0,1,p)=1$ , it is a simple routine to check that $A(x,p,1)=\\frac{\\sqrt{1-p}e^{x(1-p)}}{\\sqrt{1-pe^{2x(1-p)}}}$ .", "Therefore, by [20] we have that $A(px,1/p,1)=\\frac{\\sqrt{1-p}}{\\sqrt{1-pe^{2x(1-p)}}}=\\sum _{n,k\\ge 0}N(n,k)x^np^k,$ which implies that $A(x,p,1)=\\sum _{n,k\\ge 0}N(n,n-k)x^np^k$ .", "Hence $\\sum _{j\\ge 0}a_{n,k,j}=N(n,n-k)$ , as claimed.", "(C) Let $f_{n,i}=a_{n,i,\\lfloor n/2\\rfloor }$ .", "By (REF ) and (REF ), we have $f_{n,i}=(2i+1)f_{n-1,i}+(n-2i+1)f_{n-1,i-1},\\quad 0\\le i\\le \\lfloor n/2\\rfloor ,$ with $f_{0,0}=1$ .", "Define $f_n(x)=\\sum _{i\\ge 0}f_{n,i}x^i$ .", "Then $f_{n+1}(x)=(nx+1)f_{n}(x)+2x(1-x)\\frac{d}{dx}f_{n}(x),$ with the initial condition $f_0(x)=1$ .", "By comparing (REF ) with (REF ), we see that the polynomials $f_n(x)$ satisfy the same recurrence relation and initial conditions as $\\widetilde{P}_n(x)$ , so they coincide to each other.", "Similarly, it is easy to verify that $\\sum _{j\\ge 0}a_{n,i,\\lfloor \\frac{n}{2}\\rfloor -i}x^i=\\widetilde{P}_n(x),$ which completes the proof.", "We end this subsection by giving another characterization of the numbers $N(n,k)$ and the polynomials $P_n(x)$ .", "Assume that $(xD)^{n+1}(y)=xD(xD)^n(y)=xD\\bigl ((xD)^n(y)\\bigr ).$ As a “dual\" of $(xD)^n(x)$ , we define $\\begin{split}(xD)^{2n}(y)&=\\sum _{i,j\\ge 0}b_{2n,i,j}x^{2i+2}y^{4n-1-2i-2j}z^{2j},\\\\(xD)^{2n+1}(y)&=\\sum _{i,j\\ge 0}b_{2n+1,i,j}x^{2i+2}y^{4n-2i-2j}z^{2j+1}.\\end{split}$ From (REF ), we have $(xD)(y)=x^2z,\\quad (xD)^2(y)=2x^2yz^2+x^4y.$ The first terms of the generating function $B(x,p,q)$ are given as follows: $B(x,p,q)&=1+x+(p+2q)\\frac{x^2}{2!", "}+\\bigl (p^2+p(2q+8)+4q\\bigr )\\frac{x^3}{3!", "}\\\\&\\quad +\\bigl (p^3+p^2(8+28q)+p(16q^2+44q)+8q^2\\bigr )\\frac{x^4}{4!", "}\\\\&\\quad +\\bigl (p^4+p^3(88+28q)+p^2(136+364q+16q^2)+p(208q+88q^2)+16q^2\\bigr )\\frac{x^5}{5!", "}+\\cdots .$ It is easy to verify that $\\begin{split}b_{2n,i,j}&=(4n-2i-2j)b_{2n-1,i-1,j-1}+(2i+2)b_{2n-1,i,j-1}+(2j+1)b_{2n-1,i-1,j},\\\\b_{2n+1,i,j}&=(4n-2i-2j+1)b_{2n,i-1,j}+(2i+2)b_{2n,i,j}+(2j+2)b_{2n,i-1,j+1},\\end{split}$ which leads to the following result.", "Lemma 21 We have $\\left\\lbrace \\begin{array}{ll}BO_x=p-1+(2-p)BE+2p(1-p)BE_p+2p(1-q)BE_q+2xpBE_x,\\\\BE_x=(p+2q-2pq)BO+2pq(1-p)BO_p+2pq(1-q)BO_q+2xpqBO_x.\\end{array}\\right.$ Equivalently, $(BO^{\\prime },BE^{\\prime })$ is a $J$ -pair of the first type, where $BO^{\\prime }=p\\sqrt{\\frac{q}{p-1}}BO\\quad \\mbox{ and }\\quad BE^{\\prime }=\\frac{p-1}{p}+\\frac{p}{\\sqrt{p-1}}BE.$ In the same way as the proof of Theorem REF , one can easily show the following result.", "Theorem 22 For $n\\ge 1$ , we have $\\sum _{i,j\\ge 0}b_{n,i,j}=(2n-1)!", "!$ .", "$\\sum _{j\\ge 0}b_{n,i,j}=N(n,i+1)$ .", "$\\sum _{i\\ge 0}b_{n,i,\\lfloor \\frac{n}{2}\\rfloor }x^i=P_n(x)$ and $\\sum _{i\\ge 0}b_{n,i,\\lfloor \\frac{n}{2}\\rfloor -i}x^i=\\frac{1}{2}P_{n+1}(x)$ ." ], [ "Alternating runs and the longest alternating subsequences", "Let $\\pi =\\pi (1)\\pi (2)\\cdots \\pi (n)\\in \\mathfrak {S}_n$ .", "We say that $\\pi $ changes direction at position $i$ if either $\\pi ({i-1})<\\pi (i)>\\pi (i+1)$ , or $\\pi (i-1)>\\pi (i)<\\pi (i+1)$ , where $i\\in \\lbrace 2,3,\\ldots ,n-1\\rbrace $ .", "We say that $\\pi $ has $k$ alternating runs if there are $k-1$ indices $i$ such that $\\pi $ changes direction at these positions [27].", "Let $R(n,k)$ denote the number of permutations in $\\mathfrak {S}_n$ with $k$ alternating runs.", "In recent years, Deutsch and Gessel [27], Stanley [29] also studied the generating function for these numbers.", "As pointed out by Canfield and Wilf [4], the generating function for the numbers $R(n,k)$ can be elusive.", "Let $R_n(x)=\\sum _{k=1}^{n-1}R(n,k)x^k$ .", "The polynomial $R_n(x)$ is closely related to $P_n(x)$ : $R_n(x)=\\frac{x(1+x)^{n-2}}{2^{n-2}}P_n\\left(\\frac{2x}{1+x}\\right).$ for $n\\ge 2$ , which was established in [21].", "An alternating subsequence of $\\pi $ is a subsequence $\\pi ({i_1})\\cdots \\pi ({i_k})$ satisfying $\\pi ({i_1})>\\pi ({i_2})<\\pi ({i_3})>\\cdots \\pi ({i_k}).$ Denote by ${\\rm as\\,}(\\pi )$ the length of the longest alternating subsequence of $\\pi $ .", "There is a large literature devoted to ${\\rm as\\,}(\\pi )$ (see [3], [30], [33]).", "Define $a_k(n)=\\#\\lbrace \\pi \\in \\mathfrak {S}_n:{\\rm as\\,}(\\pi )=k\\rbrace .$ It should be noted that $a_k(n)$ is also the number of permutations in $\\mathfrak {S}_n$ with $k$ up-down runs.", "The up-down runs of a permutation $\\pi $ are the alternating runs of $\\pi $ endowed with a 0 in the front (see [27]).", "For example, the permutation $\\pi =514632$ has 3 alternating runs and 4 up-down runs.", "In the same way as the proof of Theorem REF , it is a routine exercise to show the following.", "Theorem 23 For $n\\ge 1$ , we have $\\sum _{i,j\\ge 0}t_{n,i,j}=\\sum _{i,j\\ge 0}r_{n-1,i,j}=n!$ .", "$\\sum _{i\\ge 0}t_{n,i,j}=a_{n-j}(n)$ .", "$\\sum _{j\\ge 0}t_{n,i,j}=\\widetilde{P}_{n,\\lfloor \\frac{n}{2}\\rfloor -i}$ .", "$\\sum _{i\\ge 0}r_{n,i,j}=R(n+1,n-j)$ .", "$\\sum _{j\\ge 0}r_{n,i,j}=P_{n+1,\\lfloor \\frac{n}{2}\\rfloor -i}$ .", "Recall that a permutation $\\pi $ is alternating if $\\pi (1)>\\pi (2)<\\pi (3)>\\cdots \\pi (n).$ In other words, $\\pi (i)<\\pi ({i+1})$ if $i$ is even and $\\pi (i)>\\pi ({i+1})$ if $i$ is odd.", "Let $E_n$ denote the number of alternating permutations in $\\mathfrak {S}_n$ .", "For instance, $E_4=5$ , corresponding to the permutations 2143, 3142, 3241, 4132 and 4231.", "Similarly, define $\\pi $ to be reverse alternating if $\\pi (1)<\\pi (2)>\\pi (3)<\\cdots \\pi (n)$ .", "The bijection $\\pi \\mapsto \\pi ^c$ on $\\mathfrak {S}_n$ defined by $\\pi ^c(i)=n+1-\\pi (i)$ shows that $E_n$ is also the number of reverse alternating permutations in $\\mathfrak {S}_n$ .", "The number $E_n$ is called an Euler number (see [30]).", "By definition, we have $P_{2n+1,n}=E_{2n+1},~\\widetilde{P}_{2n,n}=E_{2n}.$ From (REF ), (REF ) and (REF ), it is easy to verify that $t_{2n,0,0}=r_{2n,0,0}=0,t_{2n-1,0,0}=1,r_{2n-1,0,0}=2$ and for $1\\le j\\le 2n-1$ , we have $\\begin{split}t_{2n,0,j}&=t_{2n,0,2n-j},\\\\t_{2n,0,j}&=t_{2n-1,0,j-1},\\\\r_{2n,0,j}&=r_{2n,0,2n-j},\\\\r_{2n,0,j}&=r_{2n-1,0,j-1}.\\end{split}$ Therefore, from Theorem REF , we see that the numbers $t_{2n,0,j}$ and $r_{2n,0,j}$ construct a symmetric array with row sums equaling the Euler numbers, as demonstrated in the following array: $\\begin{array}{ccccc}& & 1& & \\\\& & 2& & \\\\& 1 & 3 & 1 & \\\\& 2 & 12 &2 & \\\\1 & 15 & 29 & 15 & 1 \\\\2 & 60 & 148 & 60 & 2\\\\& & \\cdots & & \\\\\\end{array}$ For convenience, we list the tables of the values of $t_{n,i,j}$ and $r_{n,i,j}$ for $1\\le n\\le 4$ .", "Table: NO_CAPTIONTable: NO_CAPTIONTable: NO_CAPTIONTable: NO_CAPTIONTable: NO_CAPTIONTable: NO_CAPTIONTable: NO_CAPTIONTable: NO_CAPTIONDefine $\\begin{split}{\\rm sn\\,}(x,k)&=\\sum _{n\\ge 0}(-1)^nJ_{2n+1}(k^2)\\frac{x^{2n+1}}{(2n+1)!", "},\\\\{\\rm cn\\,}(x,k)&=1+\\sum _{n\\ge 0}(-1)^nJ_{2n}(k^2)\\frac{x^{2n}}{(2n)!", "}.\\end{split}$ Note that $J_n(k^2)=\\sum _{0\\le 2i\\le n-1}J_{n,2i}k^{2i}.$ Dumont [8] found that $s_{2n,i,0}=J_{2n,2i}$ and $s_{2n+1,i,0}=J_{2n+2,2i}$ .", "By comparing (REF ) with (REF ) and (REF ), we get the following result immediately.", "Theorem 24 For $n\\ge 1$ , we have $J_{n,2i}=t_{n,\\lfloor \\frac{n}{2}\\rfloor -i,0}$ .", "It follows from Leibniz's formula that $\\begin{split}D^{2n+1}(w)&=D^{2n}(wx)\\\\&=\\sum _{k\\ge 0}\\binom{2n}{2k}D^{2k}(w)D^{2n-2k}(x)+\\sum _{k\\ge 0}\\binom{2n}{2k+1}D^{2k+1}(w)D^{2n-2k-1}(x),\\end{split}$ and similarly, $\\begin{split}D^{2n+2}(w)&=D^{2n+1}(wx)\\\\&=\\sum _{k\\ge 0}\\binom{2n+1}{2k}D^{2k}(w)D^{2n+1-2k}(x)+\\sum _{k\\ge 0}\\binom{2n+1}{2k+1}D^{2k+1}(w)D^{2n-2k}(x).\\end{split}$ Therefore, combining (REF ), we get $\\begin{split}t_{2n+1,i,0}&=\\sum _{k\\ge 0}\\binom{2n}{2k}\\sum _{j=0}^it_{2k,j,0}s_{2n-2k,i-j,0},\\\\t_{2n+2,i+1,0}&=\\sum _{k\\ge 0}\\binom{2n+1}{2k+1}\\sum _{j=0}^it_{2k+1,j,0}s_{2n-2k,i-j,0}.\\end{split}$ Therefore, as a corollary of Theorem REF , we get the following.", "Corollary 25 ([32]) For $n\\ge 0$ , we have $\\begin{split}J_{2n+1,2n-2i}&=\\sum _{k\\ge 0}\\binom{2n}{2k}\\sum _{j=0}^iJ_{2k,2k-2j}J_{2n-2k,2i-2j},\\\\J_{2n+2,2n-2i}&=\\sum _{k\\ge 0}\\binom{2n+1}{2k+1}\\sum _{j=0}^iJ_{2k+1,2k-2j}J_{2n-2k,2i-2j}.\\end{split}$ We end our paper by giving the following.", "Conjecture 26 Let $s_{n,i,j}$ be the numbers defined by (REF ).", "Set $\\widetilde{s}_{n,i,j}=s_{n,j,i}$ , i.e., $\\widetilde{s}_{n,i,j}=\|\\lbrace \\pi \\in \\mathfrak {S}_n: X(\\pi )=j, Y(\\pi )=i\\rbrace \|,$ where $X(\\pi )$ (resp., $Y(\\pi )$ ) is the number of odd (resp., even) cycle peaks of $\\pi $ .", "Then $\\begin{split}\\widetilde{s}_{2n+1,i,0}&=t_{2n+1,i,0},\\\\\\widetilde{s}_{2n+1,i,j}&=t_{2n+1,i,2j-1}+t_{2n+1,i,2j}\\quad \\textrm {for j\\ge 1},\\\\\\widetilde{s}_{2n,i,j}&=t_{2n,i,2j}+t_{2n,i,2j+1}\\quad \\textrm {for j\\ge 0}.\\end{split}$" ] ]	1403.0233
[ [ "Pseudo 2D chemical model of hot Jupiter atmospheres: application to HD\n 209458b and HD 189733b" ], [ "Abstract We have developed a pseudo two-dimensional model of a planetary atmosphere, which takes into account thermochemical kinetics, photochemistry, vertical mixing, and horizontal transport, the latter being modeled as a uniform zonal wind.", "We have applied the model to the atmospheres of the hot Jupiters HD 209458b and HD 189733b.", "The adopted eddy diffusion coefficients are calculated by following the behaviour of passive tracers in three-dimensional general circulation models, which results in eddy values significantly below previous estimates.", "We find that the distribution of molecules with altitude and longitude in the atmospheres of these two hot Jupiters is complex because of the interplay of the various physical and chemical processes at work.", "Much of the distribution of molecules is driven by the strong zonal wind and the limited extent of vertical transport, resulting in an important homogenisation of the chemical composition with longitude.", "In general, molecular abundances are quenched horizontally to values typical of the hottest dayside regions, and thus the composition in the cooler nightside regions is highly contaminated by that of warmer dayside regions.", "As a consequence, the abundance of methane remains low, even below the predictions of previous one-dimensional models, which is likely to be in conflict with the high CH4 content inferred from observations of the dayside of HD 209458b.", "Another consequence of the important longitudinal homogenisation of the abundances is that the variability of the chemical composition has little effect on the way the emission spectrum is modified with phase and on the changes in the transmission spectrum from the transit ingress to the egress, these variations in the spectra being mainly due to changes in the temperature rather than in the composition between the different sides of the planet." ], [ "Introduction", "Strongly irradiated by their close host star, hot Jupiters reside in extreme environments and represent a class of planets without analogue in our solar system.", "This type of exoplanets, the first to be discovered around main-sequence stars, remains the best available to study through observations and challenge a variety of models in the area of planetary science (see reviews on the subject by [4] 2010; [90] 2010; [15] 2011; [94] 2011).", "In recent years, multiwavelength observations of transiting hot Jupiters have allowed scientists to put constraints on the physical and chemical state of their atmospheres.", "Among these hot Jupiters, the best characterized are probably HD 209458b and HD 189733b, which belong to some of the brightest and closest transiting systems, and for which primary transit, secondary eclipse, and phase curve measurements have been used to probe, though often with controversial interpretations, various characteristics of their atmospheres, such as the thermal structure ([26] 2005, 2006; [51] 2008; [16] 2008), winds and day-night heat redistribution ([50] 2007, 2012; [20] 2007; [98] 2010), and mixing ratios of some of the main molecular constituents ([107] 2007; [100] 2008, 2009a, 2009b, 2010; [41] 2008; [97] 2009; [31] 2009; [6] 2010; [38] 2011; [110] 2012; [59] 2012; [87] 2013; [24] 2013).", "The ability of observations at infrared and visible wavelengths to characterize the physical and chemical state of exoplanet atmospheres has motivated the development of various types of theoretical models.", "On the one hand, there are those aiming at investigating the physical structure of hot-Jupiter atmospheres, either one-dimensional radiative models ([47] 2005; [36] 2008; [74] 2014) or three-dimensional general circulation models ([17] 2008; [93] 2009; [43] 2011a,b; [33] 2012; [83] 2013; [73] 2013).", "These models have shown how fascinating the climates of hot Jupiters are, with atmospheric temperatures usually in excess of 1000 K, and helped to understand some global observed trends.", "Some hot Jupiters are found to display a strong thermal inversion in the dayside while others do not (e.g.", "[36] 2008).", "Strong winds with velocities of a few km s$^{-1}$ develop and transport the heat from the dayside to the nightside, reducing the temperature contrast between the two hemispheres.", "The circulation pattern in these planets is characterized by an equatorial superrotating eastward jet.", "On the other hand, the chemical composition of hot Jupiters has been investigated by one-dimensional models, which currently account for thermochemical kinetics, vertical mixing, and photochemistry ([112] 2009; [61] 2010, 2011; [71] 2011, 2013; [56] 2012; [108] 2012, 2014; [2] 2014).", "These models have revealed the existence of three different chemical regimes in the vertical direction.", "A first one at the bottom of the atmosphere, where the high temperatures and pressures ensure a chemical equilibrium composition.", "A second one located above this, where the transport of material between deep regions and higher layers occurs faster than chemical kinetics so that abundances are quenched at the chemical equilibrium values of the quench level.", "And a third one located in the upper atmosphere, where the exposure to ultraviolet (UV) stellar radiation drives photochemistry.", "The exact boundaries between these three zones depend on the physical conditions of the atmosphere and on each species.", "In addition to the retrieval of average atmospheric quantities from observations, there is a growing interest in the physical and chemical differences that may exist between different longitudes and latitudes in hot-Jupiter atmospheres, and in the possibility of probing these gradients through observations.", "Indeed, important temperature contrasts between different planetary sides of hot Jupiters, noticeably between day and night sides, have been predicted ([91] 2002), observed for a dozen hot Jupiters (see [50] 2007 for the first one), qualitatively understood ([22] 2011; [75] 2013), and confirmed by three-dimensional general circulation models (e.g.", "[77] 2012).", "These temperature gradients, together with the fact that photochemistry switches on and off in the day and night sides, are at the origin of a potential chemical differentiation in the atmosphere along the horizontal dimension, especially along longitude.", "On the other hand, strong eastward jets with speeds of a few km s$^{-1}$ are believed to dominate the atmospheric circulation in the equatorial regions, as predicted by [91] (2002), theorized in [95] (2011), potentially observed by [98] (2010), and confirmed by almost all general circulation models of hot Jupiters.", "These strong horizontal winds are an important potential source of homogenization of the chemical composition between locations with different temperatures and UV illumination.", "The existence of winds and horizontal gradients in the temperature and chemical composition of hot-Jupiter atmospheres has mainly been considered from a theoretical point of view, although some of these effects can be studied through phase curve observations ([35] 2006; [21] 2008; [69] 2012; [25] 2012), monitoring of the transit ingress and egress ([37] 2010), and Doppler shifts of spectral lines during the primary transit ([98] 2010; [70] 2012; [96] 2013).", "The existence of horizontal chemical gradients has been addressed in the frame of a series of one-dimensional models in the vertical direction at different longitudes (e.g.", "[71] 2011).", "An attempt to understand the interplay between circulation dynamics and chemistry was undertaken by [19] (2006), who coupled a three-dimensional general circulation model of HD 209458b to a simple chemical kinetics scheme dealing with the interconversion between CO and CH$_4$ .", "These authors found that, even in the presence of strong temperature gradients, the mixing ratios of CO and CH$_4$ are homogenized throughout the planet's atmosphere in the 1 bar to 1 mbar pressure regime.", "In our team, we have recently adopted a different approach in which we coupled a robust chemical kinetics scheme to a simplified dynamical model of HD 209458b's atmosphere ([1] 2012).", "In this approach the atmosphere was assumed to rotate as a solid body, mimicking a uniform zonal wind, while vertical mixing and photochemistry were neglected.", "We found that the zonal wind acts as a powerful disequilibrium process that tends to homogenize the chemical composition, bringing molecular abundances at the limb and nightside regions close to chemical equilibrium values characteristic of the dayside.", "Here we present an improved model that simultaneously takes into account thermochemical kinetics, photochemistry, vertical mixing, and horizontal transport in the form of a uniform zonal wind.", "We apply our model to study the interplay between atmospheric dynamics and chemical processes, and the distribution of the main atmospheric constituents in the atmosphere of the hot Jupiters HD 209458b and HD 189733b." ], [ "Model", "We modeled the atmospheres of HD 209458b and HD 189733b, for which we adopted the parameters derived by [99] (2010).", "For the system of HD 209458 we took a stellar radius of 1.162 $R_{\\odot }$ , a planetary radius and mass of 1.38 $R_J$ and 0.714 $M_J$ (where $R_J$ and $M_J$ stand for Jupiter radius and mass), and an orbital distance of 0.04747 au.", "For the system of HD 189733 the adopted parameters are a stellar radius of 0.752 $R_{\\odot }$ , a planetary radius and mass of 1.151 $R_J$ and 1.150 $M_J$ , and a planet-to-star distance of 0.03142 au.", "The atmosphere model is based on some of the outcomes of three-dimensional general circulation models (GCMs) developed for HD 209458b and HD 189733b ([93] 2009; [73] 2013, in preparation), which indicate that circulation dynamics is dominated by a broad eastward equatorial jet.", "On the assumption that the eastward jet dominates the circulation pattern, it seems well justified to model the atmosphere as a vertical column that rotates along the equator, which mimicks a uniform zonal wind.", "The main shortcoming of this approach is that it reduces the whole circulation dynamics to a uniform zonal wind, although it has the clear advantage over more traditional one-dimensional models in the vertical direction of simultaneously taking into account the mixing and transport of material in the vertical and horizontal directions." ], [ "Pseudo two-dimensional chemical model", "In one-dimensional models of planetary atmospheres, the distribution of each species in the vertical direction is governed by the coupled continuity-transport equation $\\frac{\\partial f_i}{\\partial t} = \\frac{P_i}{n} - f_i L_i - \\frac{1}{n r^2} \\frac{\\partial (r^2 \\phi _i)}{\\partial r}, $ where $f_i$ is the mixing ratio of species $i$ , $t$ the time, $n$ the total number density of particles, $r$ the radial distance to the center of the planet, $P_i$ and $L_i$ the rates of production and loss, respectively, of species $i$ , and $\\phi _i$ the vertical transport flux of particles of species $i$ (positive upward and negative downward).", "The first two terms on the right side of Eq.", "(REF ) account for the formation and destruction of species $i$ by chemical and photochemical processes, while the third term accounts for the vertical transport in a spherical atmosphere.", "In this way, thermochemical kinetics, photochemistry, and vertical mixing can be taken into account through Eq.", "(REF ).", "The transport flux can be described by eddy and molecular diffusion as $\\phi _i = - K_{zz} n \\frac{\\partial f_i}{\\partial z} - D_i n \\Big ( \\frac{\\partial f_i}{\\partial z} + \\frac{f_i}{H_i} - \\frac{f_i}{H_0} + \\frac{\\alpha _i}{T} \\frac{d T}{ d z} f_i \\Big ), $ where $z$ is the altitude in the atmosphere with respect to some reference level (typically set at a pressure of 1 bar), $T$ is the gas kinetic temperature, $K_{zz}$ is the eddy diffusion coefficient, $D_i$ is the coefficient of molecular diffusion of species $i$ , $H_i$ is the scale height of species $i$ , $H_0$ is the mean scale height of the atmosphere, and $\\alpha _i$ is the thermal diffusion factor of species $i$ .", "More details on Eqs.", "(REF ) and (REF ) can be found, for instance, in [5] (1973) and [111] (1999).", "The coefficient of molecular diffusion $D_i$ is estimated from the kinetic theory of gases (see [84] 1988), while the factor of thermal diffusion $\\alpha _i$ is set to $-0.25$ for the light species H, H$_2$ , and He ([5] 1973), and to 0 for the rest of species.", "The eddy diffusion coefficient $K_{zz}$ is a rather empirical formalism to take into account advective and turbulent mixing processes in the vertical direction, and is discussed in more detail in section REF .", "To compute the abundances of the different species as a function of altitude, the atmosphere is divided into a certain number of layers and the continuous variables in Eqs.", "(REF ) and (REF ) are discretized as a function of altitude.", "After the discretization, Eq.", "(REF ) reads $\\frac{\\partial f_i^j}{\\partial t} = \\frac{P_i^j}{n^j} - f_i^j L_i^j - \\frac{\\big (r^{j+1/2}\\big )^2 \\phi _i^{j+1/2} - \\big (r^{j-1/2}\\big )^2 \\phi _i^{j-1/2}}{n^j \\big (r^j\\big )^2 \\big (z^{j+1/2} - z^{j-1/2}\\big )}, $ where the superscript $j$ refers to the $j^{\\rm th}$ layer, while $j+1/2$ and $j-1/2$ refer to its upper and lower boundaries, respectively, so that layers are ordered from bottom to top.", "The transport fluxes of species $i$ at the upper and lower boundaries of layer $j$ , $\\phi _i^{j+1/2}$ and $\\phi _i^{j-1/2}$ , are then given by $\\phi _i^{j \\pm 1/2} = - K_{zz}^{j \\pm 1/2} n^{j \\pm 1/2} \\frac{\\partial f_i}{\\partial z} \\Big \|_{j \\pm 1/2} - D_i^{j \\pm 1/2} n^{j \\pm 1/2} \\Bigg [ \\frac{\\partial f_i}{\\partial z} \\Big \|_{j \\pm 1/2} \\nonumber \\\\+ \\bigg ( \\frac{f_i^{j \\pm 1/2}}{H_i^{j \\pm 1/2}} - \\frac{f_i^{j \\pm 1/2}}{H_0^{j \\pm 1/2}} + \\frac{\\alpha _i}{T^{j \\pm 1/2}} \\frac{d T}{d z} \\Big \|_{j \\pm 1/2} f_i^{j \\pm 1/2} \\bigg ) \\Bigg ], $ where the variables evaluated at $j+1/2$ and $j-1/2$ boundaries are approximated as the arithmetic mean of the values at layers $j$ and $j+1$ and at layers $j-1$ and $j$ , respectively.", "We assume that there is neither gain nor loss of material in the atmosphere, and thus the transport fluxes at the bottom and top boundaries of the atmosphere are set to zero.", "The rates of production and loss of each species in Eqs.", "(REF ) and (REF ) are given by chemical and photochemical processes.", "Thermochemical kinetics is taken into account with a chemical network, which consists of 104 neutral species composed of C, H, N, and O linked by 1918 chemical reactions, that has been validated in the area of combustion chemistry by numerous experiments over the 300-2500 K temperature range and the 0.01-100 bar pressure regime, and has been found suitable to model the atmospheres of hot Jupiters.", "Most reactions are reversed with their rate constants fulfilling detailed balance to ensure that, in the absence of disequilibrium processes such as photochemistry or mixing, thermochemical equilibrium is achieved at sufficiently long times.", "The reaction scheme is described in [108] (2012), with some minor modifications given in [1] (2012).", "As photochemical processes we consider photodissociations, whose rates depend on the incident UV flux and the relevant cross sections.", "The incident UV flux is calculated by solving the radiative transfer in the vertical direction for a given zenith angle, where the spherical geometry of layers is taken into account when computing the path length along each of them.", "Absorption and Rayleigh scattering, the latter being treated through a two-ray iterative algorithm ([48] 1977), both contribute to the attenuation of UV light throughout the atmosphere.", "Absorption and photodissociation cross-sections are described in detail in [108] (2012).", "Rayleigh-scattering cross-sections are calculated for the most abundant species from their polarizability (see e.g.", "[104] 1969).", "As UV spectrum for the host star HD 209458, we adopt the spectrum of the Sun (mean between minimum and maximum activity from [106] 2004) below 168 nm and a Kurucz synthetic spectrumSee http://kurucz.harvard.edu/stars/hd209458 at longer wavelengths.", "For HD 189733b, below 335 nm we adopt a UV spectrum of $\\epsilon $ Eridani based on the CAB X-exoplanets archive ([89] 2011) and observations with FUSE and HST (see details in [108] 2012), and a Kurucz synthetic spectrumSee http://kurucz.harvard.edu/stars/hd189733 above 335 nm.", "In one-dimensional vertical models of planetary atmospheres, the system of differential equations given by Eq.", "(REF ), with as many equations as the number of layers times the number of species, is integrated as a function of time, starting from some initial composition, usually given by thermochemical equilibrium, until a steady state is reached.", "During the evolution, the physical conditions of the vertical atmosphere column remain static.", "In the pseudo two-dimensional approach adopted here, we consider that the vertical atmosphere column rotates around the planet's equator, and thus the system of differential equations is integrated as a function of time with physical conditions varying with time, according to the periodic changes experienced during this travel.", "A vertical atmosphere column rotating around the equator mimics a uniform zonal wind, which is an idealization of the equatorial superrotating jet structure found by three-dimensional GCMs for hot-Jupiter atmospheres.", "This approach may be seen as a pseudo two-dimensional model in which the second dimension, which corresponds to the longitude (the first one being the altitude), is in fact treated as a time dependence in the frame of an atmosphere column rotating around the equator.", "To build the pseudo two-dimensional chemical models of HD 209458b and HD 189733b the vertical atmosphere column is divided into 100-200 layers spanning the pressure range 500-10$^{-8}$ bar.", "The evolution of the vertical atmosphere column starts at the substellar point with an initial composition given by either thermochemical equilibrium or a one-dimensional vertical model, the latter usually resulting in shorter integration times before a periodic state is reached.", "The convenience of starting with the composition of the hottest substellar regions is discussed in [1] (2012).", "Thermochemical equilibrium calculations were carried out using a code that minimizes the Gibbs energy based on the algorithm of [40] (1994) and the thermochemical data described in [108] (2012) for the 102 species included.", "A solar elemental composition ([3] 2009) was adopted for the atmospheres of both HD 209458b and HD 189733b.", "The planetary sphere was then discretized into a certain number of longitudes (typically 100) and the system of differential equations given by Eq.", "(REF ) was integrated as the atmosphere column moves from one longitude to the next, at a constant angular velocity.", "To speed up the numerical calculations, the physical variables that vary with longitude (in our case these are the vertical structures of temperature and incident UV flux) were discretized as a function of longitude, that is, they were assumed to remain constant within each discretized longitude interval.", "As long as there are important longitudinal temperature gradients, the atmospheric scale height also varies with longitude, so that the atmosphere expands or shrinks depending on whether it gets warmer or cooler.", "To incorporate this effect, which may have important consequences for transit spectra, the vertical atmosphere column was enlarged or compressed (the radius at the base of the atmosphere remaining fixed) to fulfill hydrostatic equilibrium at any longitude.", "The variation of the incident UV flux with longitude was taken into account through the zenith angle.", "At the limbs we considered a zenith angle slightly different from a right angle because of the finite apparent size of the star and because of atmospheric refraction, for which we adopted a refraction angle of half a degree as in the case of visible light at Earth.", "The nonlinear system of first-order ordinary differential equations given by Eq.", "(REF ) was integrated as a function of time using a backward differentiation formula implicit method for stiff problems implemented in the Fortran solver DLSODES within the ODEPACK packageSee http://computation.llnl.gov/casc/odepack ([45] 1983; [78] 1993).", "The evolution of the vertical atmosphere column was followed during several rotation cycles until the abundances of the main atmospheric constituents achieved a periodic behavior, which for HD 209458b and HD 189733b, occurs after some tens or hundreds of rotation periods." ], [ "Atmospheric dynamics and temperature (GCMs)", "The pseudo two-dimensional chemical model needs some key input data related to the zonal wind speed, thermal structure, and strength of vertical mixing.", "These data are calculated with the three-dimensional general circulation model SPARC/MITgcm developed by [93] (2009), in which dynamics and radiative transfer are coupled.", "The data used here for HD 209458b are based on the simulations by [73] (2013), while those for HD 189733b are based on calculations by Parmentier et al.", "(in preparation), both of which cover a pressure range from about 200 bar to 2 $\\mu $ bar.", "These GCM simulations provide a wealth of detailed information regarding the physical structure of the atmosphere, although they remain limited with respect to the chemical structure as long as the composition is assumed to be given by local chemical equilibrium.", "Figure: Zonal-mean zonal wind speed as a function of latitude and pressure calculated with a GCM simulation of HD 189733b (Parmentier et al.", "in preparation).", "A strong superrotating equatorial jet is clearly present in the 10-10 -3 ^{-3} bar pressure range." ], [ "Wind structure", "Circulation dynamics in the atmospheres of hot Jupiters is dominated by a fast eastward (or superrotating) jet stream at the equator.", "This superrotating jet was first predicted by [91] (2002), has been found to emerge from almost all GCM simulations of hot Jupiters ([18] 2005; [92] 2008, 2009, 2013; [32] 2008; [80] 2010, 2012a,b; [76] 2010, 2012; [43] 2011a,b; [60] 2010; [49] 2013; [73] 2013), and is also understood theoretically ([95] 2011).", "A shift of the hottest point of the planet eastward from the substellar point has been directly observed in several exoplanets ([50] 2007, 2009a, 2012; [23] 2010) and interpreted as a direct consequence of this jet.", "The superrotating jet in HD 209458b and HD 189733b spans over all longitudes and has a well-defined location in latitude (around $\\pm $ 20$^\\circ $ ) and pressure (between 1-10 bar and 10$^{-6}$ -10$^{-3}$ bar), as illustrated in Figs.", "REF and REF , where the zonally averaged zonal wind speed is depicted as a function of latitude and pressure for each planet.", "To derive a mean speed of the jet for our pseudo two-dimensional chemical model, we averaged the zonal wind speed longitudinally over the whole planet, latitudinally over $\\pm $ 20$^\\circ $ , and vertically between 1 and 10$^{-6}$ bar, the latter corresponding to the top of the atmosphere in the GCM simulations.", "We find mean zonal wind speeds of 3.85 km s$^{-1}$ for HD 209458b and 2.43 km s$^{-1}$ for HD 189733b, in both cases in the eastward direction.", "These values were adopted in the pseudo two-dimensional chemical model as the speed of the zonal wind at the equator and 1 bar pressure level, thus setting the angular velocity of the rotating vertical atmosphere column (i.e., its rotation period).", "At high latitudes, above 50$^\\circ $ , the circulation is no longer dominated by the superrotating jet; the zonal-mean zonal wind is westward and the flow exhibits a complex structure, with westward and eastward winds, and a substantial day-to-night flow over the poles ([93] 2009).", "At low latitudes, the zonal-mean zonal wind is eastward over most of the vertical structure (above the 1-10 bar pressure level), although the shape of the superrotating jet changes gradually with altitude, from a well-defined banded flow with little longitudinal variability of the jet speed in the deep atmosphere to a less banded flow with important longitudinal variations of the wind speed in the upper levels ([93] 2009).", "It is also worth noting that according to [96] (2013), the circulation regime in the atmospheres of hot Jupiters changes from a superrotating one to a high-altitude day-to-night flow when the radiative or the frictional time scales become short, as occurs at low pressures under intense insolation or strong drag forces.", "In this regard, we note that the GCM simulations by [73] (2013, in preparation) used here are based on a drag-free case, and are thus the most favorable for the presence of a strong equatorial jet.", "That is, we would have found a somewhat slower equatorial jet if drag forces were included in the GCM simulations ([81] 2012a, 2013; [96] 2013).", "Figure: Temperature structure averaged latitudinally over ±\\pm 20 ∘ ^\\circ around the equator of HD 209458b, as calculated with a GCM simulation ( 2013).", "Note the extremely hot dayside stratosphere above the 1 mbar pressure level.In view of the discussion above, our pseudo two-dimensional chemical model based on a uniform zonal wind probably is a good approximation for the equatorial region ($\\pm $ 20$^\\circ $ ) in the 1 bar to 1 mbar pressure regime, and may still provide a reasonable description of upper equatorial layers, where an eastward jet is still present although with a less uniform structure.", "In the polar regions our formalism may not be adequate since the circulation regime is more complex, and thus the interplay between dynamics and chemistry may lead to a very different distribution of the chemical composition from that predicted by our model.", "It is interesting to note that low latitudes contribute more to the projected area of the planet's disk than polar regions, and thus planetary emission is to a large extent dominated by the equatorial regions modeled here.", "The same is not true for transmission spectra however, where low and high latitudes are both important." ], [ "Temperature structure", "Among the dozen hot Jupiters for which we have good observational constraints on their atmospheric properties ([90] 2010), half of them are believed to have a strong thermal inversion at low pressure in the dayside, while the other half are thought to lack such an inversion.", "The presence of a stratosphere in hot Jupiters is commonly attributed to the survival in the gas phase of the strong absorbers at visible wavelengths TiO and VO ([36] 2008; [93] 2009; [73] 2013).", "In this theoretical framework, planets that are warm enough to have an appreciable opacity due to TiO and VO (pM class planets) host a stratosphere, while those that are cooler (pL class planets) do not develop a temperature inversion in their atmospheres ([36] 2008).", "This, not yet firmly established however because no unambigous detection of TiO has been obtained ([30] 2008), the nature of the absorbers that cause temperature inversions in hot Jupiters is still debated.", "For example, photochemical products of some undetermined nature or arising from the photochemical destruction of H$_2$ S have also been postulated as possible absorbers responsible for these stratospheres ([13] 2008; [112] 2009).", "Moreover, not all planets fit into this pM/pL scheme, and other parameters such as the atmospheric elemental C/O abundance ratio ([68] 2012) or the stellar activity ([54] 2010) might control whether there are stratospheres in hot Jupiters.", "Figure: Temperature structure averaged latitudinally over ±\\pm 20 ∘ ^\\circ around the equator of HD 189733b, as calculated with a GCM simulation (Parmentier et al.", "in preparation).HD 209458b and HD 189733b are good examples of these two types of hot Jupiters, the former hosting a strong thermal inversion in the dayside, while the latter does not.", "The temperature resulting from the GCM simulations and averaged latitudinally over an equatorial band $\\pm $ 20$^\\circ $ in latitude is shown as a function of longitude and pressure in Fig.", "REF for HD 209458b and in Fig.", "REF for HD 189733b.", "This equatorial band of $\\pm $ 20$^\\circ $ in latitude corresponds to the region where the equatorial jet is present in the GCM simulations (see Figs.", "REF and REF ).", "For the pseudo two-dimensional chemical model, which focuses on the equatorial region where the eastward jet develops, we adopted the temperature distribution shown in Figs.", "REF and REF , assuming an isothermal atmosphere at pressures lower than 2 $\\mu $ bar.", "In our previous study ([1] 2012), the temperature structure of HD 209458b's atmosphere was calculated with a one-dimensional time-dependent radiative model and resulted in an atmosphere without a strong temperature inversion.", "Here, the temperature structure calculated for the atmospheres of HD 209458b and HD 189733b comes from GCM simulations, which result in a strong temperature inversion for the former planet and an atmosphere without stratosphere for the latter one.", "This permits us to explore the chemistry of hot Jupiters with and without a stratosphere.", "It is also worth noting that in the case of HD 209458b, there is evidence of a dayside temperature inversion from observations of the planetary emission spectrum at infrared wavelengths ([51] 2008)." ], [ "Vertical eddy diffusion coefficient", "Another important outcome of GCM simulations is the quantification of the strength with which material is transported in the vertical direction in the atmosphere.", "Although this mixing is not diffusive in a rigorous sense, once averaged over the whole planet, it can be well represented by an effective eddy diffusion coefficient that varies with pressure ([73] 2013).", "This variable enters directly as input into one-dimensional and pseudo two-dimensional chemical models of planetary atmospheres such as ours.", "The eddy diffusion coefficient is commonly estimated in the literature as the root mean square of the vertical velocity times the vertical scale height ([61] 2010; [71] 2011).", "Recently, [73] (2013) have used a more rigorous approach to estimate an effective eddy diffusion coefficient in HD 209458b by following the behavior of passive tracers in a GCM.", "These authors have shown that vertical mixing in hot-Jupiter atmospheres is driven by large-scale circulation patterns.", "There are large regions with ascending motions and large regions with descending motions, some of them contributing more to the global mixing than others.", "It has been also shown that a diffusion coefficient is a good representation of the vertical mixing that takes place in the three-dimensional model of the atmosphere.", "The resulting values for HD 209458b are 10-100 times lower than those obtained with the previous method (see Fig.", "REF ), and are used here.", "The vertical profile of the eddy diffusion coefficient for HD 209458b can be approximated by the expression $K_{zz}$ (cm$^2$ s$^{-1}$ ) = 5 $\\times $ 10$^8$ $p^{-0.5}$ , where the pressure $p$ is expressed in bar (see [73] 2013).", "In the case of HD 189733b we used the expression $K_{zz}$ (cm$^2$ s$^{-1}$ ) = 10$^7$ $p^{-0.65}$ , where the pressure $p$ is again expressed in bar.", "This expression is based on preliminary results by Parmentier et al.", "(in preparation) using the method involving passive tracers, and results in values up to 1000 times lower than those obtained with the previous more crude method (see Fig.", "REF ).", "In both HD 209458b and HD 189733b we considered a constant $K_{zz}$ value at pressures lower than 10$^{-5}$ bar." ], [ "Dynamical time scales", "To assess the relative strengths of horizontal transport and vertical mixing in the atmospheres of HD 209458b and HD 189733b it is useful to argue in terms of dynamical time scales.", "The dynamical time scale of horizontal transport may be roughly estimated as $\\tau _{dyn}^h = \\pi R_p / u$ , where $R_p$ is the planetary radius and $u$ the zonal wind speed, while that related to vertical mixing can be approximated as $\\tau _{dyn}^v = H^2/K_{zz}$ , where $H$ is the atmospheric scale height and $K_{zz}$ the eddy diffusion coefficient.", "If we take a zonal wind speed uniform with altitude and equal to the mean value given in section REF , we find that horizontal transport occurs faster than vertical mixing over most of the vertical structure of the atmospheres of HD 209458b and HD 189733b (see dotted and dashed lines in Fig.", "REF ).", "Only in the upper layers, at pressures below 10$^{-3}$ -10$^{-4}$ bar, the high eddy diffusion coefficient makes vertical mixing faster than horizontal transport.", "In the deep atmosphere, however, the equatorial superrotating jet vanishes and zonal winds become slower (see Figs.", "REF and REF ), although horizontal transport still remains faster than or at least similar to vertical mixing (see solid and dashed lines in Fig.", "REF ).", "In these deep layers, below the 1-10 bar pressure level, our assumption of a zonal wind speed uniform with altitude and with values as high as a few km s$^{-1}$ is not valid.", "This is clearly a limitation of the pseudo two-dimensional model, although the implications for the resulting two-dimensional distribution of atmospheric constituents are not strong because in these deep layers the temperature remains rather uniform with longitude, and molecular abundances are largely controlled by thermochemical equilibrium, which makes them quite insensitive to the strength of horizontal transport.", "This has been verified by running models for HD 209458b and HD 189733b with zonal wind speeds down to 1000 times slower than the nominal mean values given in section REF .", "In the way the pseudo two-dimensional chemical model is conceived, it clearly deals with the equatorial region of hot Jupiter atmospheres.", "First, the formalism adopted, in which a vertical atmosphere column rotates around the equator at a constant angular velocity, is adequate for the equatorial region ($\\pm $ 20$^\\circ $ in latitude), where a strong eastward jet is found to dominate the circulation according to GCM simulations (see Figs.", "REF and REF ).", "Second, the temperature structure adopted (see Figs.", "REF and REF ) corresponds to the average over an equatorial band of width $\\pm $ 20$^\\circ $ in latitude.", "Third, the rotation period of the atmosphere column is calculated from the wind speed retrieved from the GCM (which is also an average over an equatorial band $\\pm $ 20$^\\circ $ in latitude) and the equatorial circumference.", "And fourth, the longitude-dependent zenith angle adopted to compute the penetration of stellar UV photons corresponds to the equatorial latitude.", "The adopted formalism is therefore adequate for the equatorial region as long as circulation is dominated by an eastward jet.", "With these limitations in mind, we now present and discuss the chemical composition distribution resulting from the pseudo two-dimensional chemical model for the atmospheres of HD 209458b and HD 189733b." ], [ "Overview", "A first glance at the calculated distribution of the chemical composition with altitude and longitude in the atmospheres of HD 209458b and HD 189733b can be obtained by examining the ranges over which the vertical abundance profiles vary with longitude.", "This information is shown in Figs.", "REF and REF for some of the most abundant species, after H$_2$ and He.", "We can see that some molecules such as CO, H$_2$ O, and N$_2$ show little abundance variation with longitude, while some others such as CH$_4$ , CO$_2$ , NH$_3$ , and HCN experience important changes in their abundances as longitude varies.", "Abundance variations are usually restricted to the upper regions of the atmosphere (above the 10$^{-1}$ -10$^{-3}$ bar pressure level, depending on the molecule) but not to the lower atmosphere, where molecules maintain rather uniform abundances with longitude.", "On the one hand, longitudinal gradients in the temperature and incident stellar UV flux drive the abundance variations with longitude, while on the other, the zonal wind tends to homogenize the chemical composition in the longitudinal direction, resulting in the complex abundance distributions shown in Figs.", "REF and REF .", "These results agree with the predictions of [19] (2006) concerning the CO distribution in HD 209458b's atmosphere.", "These authors coupled a GCM to a simple chemical kinetics scheme dealing with the interconversion between CO and CH$_4$ and found that CO shows a rather homogeneous distribution with longitude and latitude in spite of the strong variations predicted by chemical equilibrium.", "We also find a rather homogeneous distribution of CO with longitude, although the same is not true for other molecules that display important longitudinal abundance gradients.", "Figure: Vertical cuts of the abundance distributions of some of the most abundant molecules at longitudes spanning the 0-360 ∘ ^\\circ range, as calculated with the pseudo two-dimensional chemical model for HD 189733b's atmosphere.Figure: Distribution of H 2 _2O, CO 2 _2, CH 4 _4, and HCN as a function of longitude and pressure in the equatorial band of HD 209458b's atmosphere, as calculated with the pseudo two-dimensional chemical model.Figure: Same as Fig.", ", but for HD 189733b.In Figs.", "REF and REF we show the atmospheric distribution of selected molecules that may influence planetary spectra as a function of longitude and pressure.", "Water vapor illustrates the case of a molecule with a rather uniform distribution throughout the atmosphere of both planets, except for a slight enhancement at high pressures ($>$ 10 bar) and a small depletion, which in HD 209458b occurs at about 10$^{-5}$ bar eastward of the substellar point and is induced by the warm stratosphere, and in HD 189733b takes place in the upper dayside layers (above the 10$^{-7}$ bar pressure level) through photochemical destruction.", "Carbon monoxide also has a quite uniform distribution and is not shown in Figs.", "REF and REF .", "Carbon dioxide is perhaps the most abundant molecule showing important longitudinal abundance variations, with a marked day-to-night contrast.", "In HD 209458b this molecule is enhanced in the cooler nightside, where it is thermodynamically favored.", "In HD 189733b the nightside enhancement is only barely apparent in the 10$^{-5}$ -10$^{-1}$ bar pressure range, while in upper layers the situation is reversed and CO$_2$ becomes depleted in the nightside regions because of a complex interplay between chemistry and dynamics.", "In the atmospheres of both planets, CO$_2$ maintains a mixing ratio between a few 10$^{-8}$ and a few 10$^{-5}$ .", "The hydrides CH$_4$ and NH$_3$ show important abundance variations in the vertical direction, their abundance decrease when moving toward upper low-pressure layers, and also some longitudinal variability, which is only important at low abundance levels, however.", "In HD 209458b, methane is largely suppressed above the 1 mbar pressure level because of the stratosphere.", "In HD 189733b it is present at a more important level, except in the very upper layers where its depletion in the warmer dayside regions is propagated by the jet to the east, contaminating the nightside regions to a large extent.", "Hydrogen cyanide also shows important abundance variations with both longitude and altitude.", "This molecule is greatly enhanced by the action of photochemistry, and thus becomes quite abundant in the upper dayside regions of HD 189733b and to a lower extent in the upper dayside layers of HD 209458b, where photochemistry is largely supressed by the presence of the stratosphere ([71] 2011; [108] 2012).", "The distribution of HCN in the upper atmosphere shows that the eastward jet results in a contamination of nightside regions with HCN formed in the dayside.", "As long as there is an important departure from chemical equilibrium in the atmospheric composition of both planets, the assumption of local chemical equilibrium in the GCM simulations may be an issue and one potential source of inconsistency between the GCM and the chemical model.", "Much of the thermal budget of these atmospheres, however, is controlled by water vapor, whose abundance is rather uniform and close to chemical equilibrium.", "This fact may justify to some extent the assumption of local chemical equilibrium in GCMs.", "We note however that other atmospheric constituents such as CO and CO$_2$ can also play an important role in the thermal balance of hot-Jupiter atmospheres, especially for elemental compositions far from solar, in which case the hypothesis of chemical equilibrium usually adopted in GCMs may not be adequate.", "Obviously, a more accurate and self-consistent approach would be to couple a robust chemical kinetics network to a GCM, although this is a very challenging computational task." ], [ "Comparison with limiting cases", "To obtain insight into the predicted distribution of molecules in the atmospheres of HD 209458b and HD 189733b, a useful and pedagogical exercise is to compare the abundance distributions calculated by the pseudo two-dimensional chemical model with those predicted in various limiting cases.", "A first one in which vertical mixing is neglected and therefore the only disequilibrium processes are horizontal advection and photochemistry (horizontal transport case), a second one consisting of a one-dimensional vertical model including vertical mixing and photochemistry, which neglects horizontal transport (vertical mixing case), and a third one which is given by local thermochemical equilibrium.", "Figs.", "REF and REF show the vertical distributions of some of the most abundant species, after H$_2$ and He, in the atmospheres of HD 209458b and HD 189733b, respectively, at four longitudes (substellar and antistellar points, and evening and morning limbsWe use the terms morning and evening limb to refer to the situation encountered by the traveling wind when crossing each of the two meridians of the planet's terminator.", "Morning, also called west or leading, and evening, also called east or trailing, limbs are probed by transmission spectra at the ingress and egress, respectively, during primary transit.", "), as calculated by the pseudo two-dimensional model and the three aforementioned limiting cases.", "We may summarize the effects of horizontal transport (modeled as a uniform zonal wind) and vertical mixing (modeled as an eddy diffusion process) by saying that horizontal transport tends to homogenize abundances in the horizontal direction, bringing them close to chemical equilibrium values of the hottest dayside regions, while vertical mixing tends to homogenize abundances in the vertical direction, bringing them close to chemical equilibrium values of hot bottom regions.", "Figure: Vertical distributions of the most abundant atmospheric constituents, after H 2 _2 and He, at four longitudes: substellar point (0 ∘ 0^\\circ ), evening limb (+90 ∘ +90^\\circ ), antistellar point (±180 ∘ \\pm 180^\\circ ), and morning limb (-90 ∘ -90^\\circ ) in the atmosphere of HD 209458b.", "We show the mole fractions calculated by the pseudo two-dimensional chemical model (solid lines), by a model that neglects vertical mixing (horizontal transport case; dashed-dotted lines), by a one-dimensional vertical model that neglects horizontal transport (vertical mixing case; dashed lines), and by local thermochemical equilibrium (dotted lines).", "Photochemistry is taken into account in all cases but the last.Figure: Same as Fig.", ", but for HD 189733b.The effect of horizontal transport is perfectly illustrated in the case of methane.", "In both HD 209458b and HD 189733b, the abundance profile of CH$_4$ given by the horizontal transport case (blue dashed-dotted lines in Figs.", "REF and REF ) almost perfectly resembles the chemical equilibrium profile at the substellar point (blue dotted lines in upper left panel of Figs.", "REF and REF ), and remains almost invariant with longitude in spite of the important abundance enhancement predicted by chemical equilibrium in the cooler nightside and morning limb regions.", "The existence of a strong stratosphere in HD 209458b introduces important differences with respect to HD 189733b.", "The hot temperatures in the upper dayside layers of HD 209458b result in short chemical time scales and therefore allows chemical kinetics to mitigate to some extent the horizontal quenching induced by the zonal wind.", "This is clearly seen for CO$_2$ (magenta dashed-dotted lines in Figs.", "REF and REF ), whose abundance varies longitudinally within 2-3 orders of magnitude in HD 209458b, while in HD 189733b it shows an almost uniform distribution with longitude.", "The abundance distributions obtained in the horizontal transport case are qualitatively similar to those presented by [1] (2012), although there are some quantitative differences due to the lack of photochemistry and a temperature inversion for HD 209458b in that previous study.", "Photochemistry plays in fact an important role in the horizontal transport case (dashed-dotted lines in Figs.", "REF and REF ), as it causes molecular abundances to vary with longitude in the upper layers due to the switch on/off of photochemistry in the day and night sides, as the wind surrounds the planet.", "Note also that the lack of vertical mixing in this case causes the photochemically active region to shift down to the level where, in the absence of vertical transport, chemical kinetics is able to counterbalance photodissociations, that is, to synthesize during the night the molecules that have been photodissociated during the day.", "Another interesting consequence of photochemistry in the horizontal transport case is that molecules such as HCN (gray dashed-dotted lines in Figs.", "REF and REF ), which are formed by photochemistry in the upper dayside regions, remain present in the upper nightside regions as a consequence of the continuous horizontal transport, and can in fact increase their abundances through the molecular synthesis ocurring during the night.", "In the extreme case where vertical transport completely dominates over any kind of horizontal transport, the homogenization is produced in the vertical, and not longitudinal, direction.", "The value at which a given molecular abundance is quenched vertically corresponds to the chemical equilibrium abundance at the altitude where the rates of chemical reactions and vertical transport become similar, the so-called quench region.", "This quench region may be located at a different altitude for each species, although in hot Jupiters such as HD 209458b and HD 189733b it is usually located in the 10-10$^{-2}$ bar pressure range ([71] 2011; [108] 2012; also this study).", "Assuming the strength of vertical mixing does not vary with longitude (as done in this study), the vertical mixing case would yield uniform abundances with longitude if temperatures do not vary much with longitude in the range of altitudes where abundances are usually quenched vertically.", "In this case, the quench region for a given species would be the same at all longitudes, and so would the vertically quenched abundance.", "According to the GCM simulations of HD 209458b and HD 189733b, the temperature varies significantly with longitude above the 1 bar pressure level, and thus the exact values at which the abundances of the different species are quenched vertically vary with longitude.", "The temperature constrast between day and nightside regions is therefore one of the main causes of abundance variations with longitude, as illustrated by CH$_4$ in both planets (blue dashed lines in Figs.", "REF and REF ).", "Another factor that drives longitudinal abundance gradients in the vertical mixing case is photochemistry, which switches on and off in the day and nightsides, respectively.", "Without horizontal transport that connects the day and nightsides, abundances become rather flat in the vertical direction in the nightside, where photochemistry is suppressed, and display more complicated vertical abundance profiles in the dayside, where photochemistry causes molecules such as NH$_3$ to be depleted while some others such as HCN are enhanced.", "Note that because we used eddy diffusion coefficients significantly below those adopted in previous studies (e.g.", "[71] 2011; [108] 2012), the vertical quench of abundances in the dayside is not as apparent because it is strongly counterbalanced by photochemistry.", "In the pseudo two-dimensional model, in which both horizontal transport and vertical mixing are simultaneously taken into account, the distribution of atmospheric constituents (solid lines in Figs.", "REF and REF ) results from the combined effect of various processes that tend to drive the chemical composition to a variety of distributions.", "On the one hand, chemical kinetics proceeds to drive the composition close to local chemical equilibrium.", "On the other hand, horizontal transport tends to homogenize abundances longitudinally, while vertical mixing does the same in the vertical direction.", "Finally, stellar UV photons tend to photodissociate molecules in the upper dayside layers, and new molecules are formed through chemical reactions involving the radicals produced in the photodissociations.", "Among these processes, horizontal transport and vertical mixing compete in homogenizing the chemical composition in the longitudinal and vertical directions, respectively.", "In the atmospheres of HD 209458b and HD 189733b horizontal transport occurs faster than vertical mixing below the $\\sim $ 1 mbar pressure level (see section REF ), and therefore molecular abundances are strongly homogenized in the longitudinal direction in this region.", "In upper layers the competition of mixing and photochemical processes results in a more complex distribution of atmospheric constituents.", "Molecular abundances show a wide variety of behaviors when both horizontal transport and vertical mixing are considered simultaneously.", "The abundances of molecules such as CH$_4$ , NH$_3$ , and HCN tend to follow those given by the vertical mixing case at the substellar region, but at other longitudes the situation is quite different depending on the molecule (blue, green, and gray solid lines in Figs.", "REF and REF ).", "At the antistellar point, for example, the abundance profiles of CH$_4$ and NH$_3$ are closer to those predicted by the pure horizontal transport case than by the vertical mixing one, but HCN does follow a behavior completely different from each of these two limiting cases.", "The abundances of CO, H$_2$ O, and N$_2$ show little variation with longitude or altitude and are therefore not affected by whether horizontal transport or vertical mixing dominates.", "Nevertheless, the coupling of horizontal transport and vertical mixing results in some curious behaviors, such as that of water vapor at the substellar point of HD 209458b (red lines in Fig.", "REF ).", "The two limiting cases of pure horizontal transport and pure vertical mixing predict a decline in its abundance in the upper layers because of photodissociation and because of a low chemical equilibrium abundance at these low pressures.", "However, horizontal and vertical dynamics working simultaneously bring water from more humid regions so that there is no decline in its abundance up to the top of the atmosphere (at 10$^{-8}$ bar in our model).", "In summary, taking into account both horizontal transport and vertical mixing produces complex abundance distributions that in many cases cannot be predicted a priori.", "Figure: Vertical cuts of the abundance distributions of some of the most abundant molecules in HD 209458b's atmosphere at longitudes spanning the 0-360 ∘ ^\\circ range, as calculated (from top to bottom) with the pseudo two-dimensional model, in the horizontal transport and vertical mixing cases, and under local chemical equilibrium.Figure: Same as Fig.", ", but for HD 189733b.We may have a different view of the situation by looking at the ranges over which the vertical abundance profiles vary with longitude in the various limiting cases (see Figs.", "REF and REF ).", "Our attention first focuses on the fact that local chemical equilibrium predicts strong variations of the chemical composition with longitude in the atmospheres of both HD 209458b and HD 189733b.", "This is especially true for CH$_4$ and CO in the latter planet, where methane becomes more abundant than carbon monoxide in the cooler nightside regions.", "Disequilibrium processes, however, in particular horizontal transport and vertical mixing, reduce to a large extent the longitudinal variability of molecular abundances.", "As already stated, although perhaps more clearly seen in Figs.", "REF and REF , horizontal transport tends to homogenize abundances with longitude.", "The effect of a purely horizontal transport is perfectly illustrated in HD 189733b's atmosphere, where, except for the photochemically active region in the upper layers, the distribution of molecules is remarkably homogeneous with longitude (see horizontal transport panel in Fig.", "REF ).", "In the atmosphere of HD 209458b, on the other hand, a pure horizontal transport allows for some longitudinal variability in the abundances of CO$_2$ and NH$_3$ above the 10$^{-3}$ bar pressure level (see horizontal transport panel in Fig.", "REF ), mainly because of the activation of chemical kinetics in the dayside stratosphere and its ability to counterbalance the homogenization driven by horizontal transport.", "In the vertical mixing case (i.e., no horizontal transport), abundances are more uniform in the vertical direction but show important longitudinal variations, with a marked day/night asymmetry characterized by rather flat vertical abundance profiles in the nightside and abundances varying with altitude in the dayside because of the influence of photochemistry (see e.g.", "CH$_4$ , NH$_3$ , and HCN in vertical mixing panels of Figs.", "REF and REF ).", "When horizontal transport and vertical mixing are considered simultaneously (top panels of Figs.", "REF and REF ), the distribution of molecules in the lower atmosphere of both HD 209458b and HD 189733b, below the 10$^{-3}$ bar pressure level, remains remarkably homogeneous with longitude and close to that given by the pure vertical mixing case at the substellar regions.", "That is, the chemical composition of the hottest dayside regions propagates to the remaining longitudes, which indicates that the zonal wind transports material faster than vertical mixing processes do.", "In the upper atmosphere the abundance profiles become more complicated because of the combined effect of the photochemistry that takes place in the dayside and the mixing of material ocurring in both the vertical and horizontal directions." ], [ "Comparison with previous one-dimensional models", "It is interesting to compare the results obtained with the pseudo two-dimensional model with previous results from one-dimensional vertical models ([71] 2011; [108] 2012).", "There are two main differences between our model and these previous ones.", "The first is related to the eddy diffusion coefficients adopted, which are noticeably lower in this study because they are calculated by following the behavior of passive tracers in GCM simulations, while those used previously were also estimated from GCM simulations but as the root mean square of the vertical velocity times the vertical scale height.", "The second is related to the very nature of the model, which in our case is a pseudo two-dimensional model that simultaneously takes into account horizontal transport and vertical mixing, while in these previous studies horizontal transport is neglected.", "To isolate the differences caused by each of these factors we compare in Figs.", "REF and REF the vertical abundance distributions of some of the most abundant molecules at the substellar point and at the two limbs, as calculated with our pseudo two-dimensional model using the nominal vertical profile of the eddy diffusion coefficient (see section REF ), as given by the same pseudo two-dimensional model but using the high eddy diffusion coefficient profiles derived by [71] (2011), which are about 10-100 times higher than ours for HD 209458b and about 10-1000 times higher than ours for HD 189733b, and as computed with a one-dimensional vertical model using the high eddy diffusivity values of [71] (2011).", "The main effect of increasing the strength of vertical mixing in the frame of a pseudo two-dimensional model is that the quench region shifts down to lower altitudes.", "For molecules such as CH$_4$ , NH$_3$ , and HCN, this implies that their vertically quenched abundances increase (compare solid and dashed lines in Figs.", "REF and REF ).", "If horizontal transport is completely supressed, that is, moving from a pseudo two-dimensional model to a one-dimensional vertical model (from dashed to dotted lines in Figs.", "REF and REF ), the horizontal homogenization of abundances is completely lost and thus the abundances of species such as CH$_4$ experience more important variations with longitude.", "Another interesting consequence of suppressing horizontal transport concerns water vapor, carbon monoxide, and molecular nitrogen, whose abundances decrease in the upper layers of the dayside regions (see upper panel in Figs.", "REF and REF ).", "In HD 209458b the depletion of these molecules is caused by the hot stratosphere, where neutral O and C atoms are favored over molecules, while in HD 189733b it is caused by photodissociation by UV photons.", "This loss of H$_2$ O, CO, and N$_2$ molecules in the upper dayside layers is shifted to higher altitudes when horizontal transport, which brings molecules from other longitudes, is taken into account.", "The vertical abundance profiles calculated with the one-dimensional vertical model at the substellar point and at the two limbs (dotted lines in Figs.", "REF and REF ) can be compared with the one-dimensional results of [71] (2011) using averaged thermal profiles for the dayside and terminator regions (see also [108] 2012).", "There is a good overall agreement between our substellar point results and their dayside results, on the one hand and on the other, between our results at the two limbs and their results at the terminator region, except for CH$_4$ , NH$_3$ , and HCN, for which we find vertically quenched abundances lower by about one order of magnitude.", "Although there are some differences in the adopted elemental abundances, stellar UV spectra, and zenith angles, the main source of the discrepancies is attributed to the different temperature profiles adopted.", "On the one hand there are slight differences between the GCM results of [93] (2009), adopted by [71] (2011) and [108] (2012), and those of [73] (2013, in preparation), which are adopted here.", "On the other, and more importantly, the temperature profiles have a different nature.", "They are averages over the dayside and terminator regions in their case, while in ours they correspond to specific longitudes.", "The dayside average temperature profile of [71] (2011) is cooler than our substellar temperature profile by about 100 K around the 1 bar pressure level in both planets, which results in vertically quenched abundances of CH$_4$ and NH$_3$ higher than ours by about one order of magnitude (part of the abundance differences are also due to the different chemical network adopted; see Fig.", "7 of [108] 2012).", "This serves to illustrate how relatively small changes of temperature in the 0.1-10 bar pressure regime –the quench region for most molecules– may induce important variations in the vertically quenched abundance of certain molecules.", "This also raises the question of whether it is convenient to use a temperature profile averaged over the dayside in one-dimensional chemical models that aim at obtaining a vertical distribution of molecules representative of the dayside.", "Although it may be a reasonable choice if one is limited by the one-dimensional character of the model, averaging the temperature over the whole dayside masks the temperatures of the hottest regions, near the substellar point, which are in fact the most important as they control much of the chemical composition at other longitudes if horizontal transport becomes important.", "In summary, the main implications of using a pseudo two-dimensional approach and of the downward revision of the eddy values in the atmospheres of HD 209458b and HD 189733b are that, on the the one hand, the longitudinal variability of the chemical composition is greatly reduced compared with the expectations of pure chemical equilibrium or one-dimensional vertical models and, on the other hand, the mixing ratios of CH$_4$ , NH$_3$ , and HCN are significantly reduced compared with results of previous one-dimensional models (by one order of magnitude or more with respect to the results of [71] 2011), down to levels at which their influence on the planetary spectra are probably minor." ], [ "Calculated vs. observed molecular abundances", "We now proceed to a discussion in which we compare the molecular abundances calculated with the pseudo two-dimensional chemical model and those derived from observations.", "Our main aim here is to evaluate whether or not the calculated composition, which is based on plausible physical and chemical grounds, is compatible with the mixing ratios derived by retrieval methods used to interpret the observations.", "The molecules H$_2$ O, CO, CO$_2$ , and CH$_4$ have all been claimed to be detected in the atmospheres of HD 209458b and HD 189733b either in the terminator region of the planet from primary transit observations, in the dayside from secondary eclipse observations, or in both regions using the two methods.", "Although we are not in a position to cast doubt on any of these detections, given the controversial results often found by different authors in the interpretation of spectra of exoplanets it is advisable to be cautious when using the derived mixing ratios to argue in any direction.", "Having this in mind, hereafter we use the term detection instead of claim of detection.", "Water vapor and carbon monoxide are calculated with nearly their maximum possible abundances in both planets and show a rather homogeneous distribution as a function of both altitude and longitude (see Figs.", "REF and REF ).", "Adopting a solar elemental composition, as done here, the calculated mixing ratios of both H$_2$ O and CO are around 5 $\\times $ 10$^{-4}$ .", "Water vapor being the species that provides most of the atmospheric opacity at infrared wavelengths, it was the first molecule to be detected in the atmosphere of an extrasolar planet, concretely in the transmission spectrum of HD 189733b ([107] 2007), and H$_2$ O mixing ratios derived from observations for both HD 189733b and HD 209458b are usually in the range of the calculated value of 5 $\\times $ 10$^{-4}$ ([107] 2007; [41] 2008; [100] 2008, 2009a,b; [66] 2009; [6] 2010; [59] 2012; [63] 2013; [28] 2013).", "Carbon monoxide, although less evident than water vapor, has also been detected in both planets and the mixing ratios derived are in the range of the values inferred for H$_2$ O and expected from the chemical model ([101] 2009a; [31] 2009; [66] 2009; [59] 2012; [63] 2013).", "The calculated mixing ratio of carbon dioxide in the two hot Jupiters is in the range 10$^{-7}$ - 10$^{-6}$ depending on the pressure level, with a more important longitudinal variation in the atmosphere of HD 209458b than in that of HD 189733b (see Figs.", "REF and REF ).", "This molecule has been also detected through secondary-eclipse observations in the dayside of HD 189733b, with mixing ratios spanning a wide range from 10$^{-7}$ up to more than 10$^{-3}$ ([101] 2009a; [66] 2009; [59] 2012; [63] 2013), and in the dayside of HD 209458b, with a mixing ratio in the range 10$^{-6}$ - 10$^{-5}$ ([102] 2009b).", "Taking into account the uncertainties associated with the values retrieved from observations, the agreement with the calculated abundance is reasonably good for CO$_2$ .", "The most important discrepancies between calculated and observed abundances are probably found for methane.", "This molecule is predicted to be very abundant in the cooler nightside regions of both planets, especially in HD 189733b, according to chemical equilibrium (see lower panels in Figs.", "REF and REF ), but reaches quite low abundances everywhere in the atmosphere according to the pseudo two-dimensional non-equilibrium model (see upper panels in Figs.", "REF and REF ).", "In both hot Jupiters, the calculated mixing ratio of CH$_4$ is in fact significantly lower than the predictions of previous one-dimensional models ([71] 2011; [108] 2012), a finding that strengthens the conflict with observations.", "We find that the mixing ratio of CH$_4$ above the 1 bar pressure level is below 10$^{-7}$ in HD 209458b and below 10$^{-6}$ in HD 189733b, whatever the side of the planet.", "In HD 209458b, secondary-eclipse observations have been interpreted as evidence of methane being present in the dayside with a mixing ratio between 2 $\\times $ 10$^{-5}$ and 2 $\\times $ 10$^{-4}$ ([102] 2009b), or within the less constraining range 4 $\\times $ 10$^{-8}$ - 3 $\\times $ 10$^{-2}$ ([66] 2009).", "In fact, the abundance of CH$_4$ retrieved in these studies is similar or even higher than that retrieved for H$_2$ O, which is clearly not the case according to our predictions.", "It seems difficult to reconcile the low abundance of CH$_4$ calculated by the pseudo two-dimensional chemical model with the high methane content inferred from observations, which points to some fundamental problem in either of the two sides.", "As concerns the chemical model, an enhancement of the vertical transport to the levels adopted by [71] (2011) or the supression of horizontal transport would increase the abundance of CH$_4$ only slightly (see Fig.", "REF ).", "Photochemistry, which might potentially enhance the abundance of CH$_4$ , is largely supressed by the stratosphere in the dayside atmosphere of HD 209458b.", "An elemental composition of the planetary atmosphere far from the solar one with, for example, an elemental C/O abundance ratio higher than 1, or some unidentified disequilibrium process, which might be related to, for instance, clouds or hazes, might lead to a high methane content in the warm atmospheric layers of HD 209458b's dayside.", "Some problems on the observational side cannot be ruled out, taking into account the difficulties associated to the acquisition of photometric fluxes of exoplanets and the possibility of incomplete spectroscopic line lists of some molecules relevant to the interpretation of spectra of exoplanets (see e.g.", "the recently published line list for hot methane by [42] 2012).", "In HD 189733b, contradictory results exist on the detection of methane in both the terminator and dayside regions.", "[100] (2008) reported the detection of CH$_4$ through primary-transit observations, with a derived mixing ratio of about 5 $\\times $ 10$^{-5}$ , although [97] (2009) did not find evidence of its presence in the transmission spectrum.", "These contradictory results obtained using NICMOS data could point to non-negligible systematics in the data (e.g., [39] 2012).", "Controversial results also exist on the detection of CH$_4$ in the dayside emission spectrum of HD 189733b ([101] 2009a, 2010; [66] 2009; [110] 2012; [59] 2012; [63] 2013; [7] 2013).", "Until observations can draw more reliable conclusions it is difficult to decide whether or not observations and models are in conflict regarding the abundance of CH$_4$ in HD 189733b." ], [ "Variations in the planetary spectra", "The calculated distribution of molecules in the atmospheres of HD 209458b and HD 189733b may be probed by observations.", "Instead of comparing synthetic spectra and available observations of these two planets, we are here mainly interested in evaluating whether the longitudinal variability of the chemical composition may be probed by observations.", "For example, the monitoring of the emission spectrum of the planet at different phases during an orbital period would probe the composition in the different sides of the planet.", "In addition, the observation of the transmission spectrum at the ingress and egress during primary transit conditions would allow one to probe possible chemical differentiation between the morning and evening limbs of the planet's terminator.", "Planetary emission and transmission spectra were computed using the line-by-line radiative transfer code described in Appendix .", "Since the code is currently limited because it is one-dimensional in the vertical direction, we adopted mean vertical profiles by averaging the temperature structure in longitude and latitude given by the GCM simulations of [73] (2013, in preparation) and the longitudinal distribution of abundances obtained with the pseudo two-dimensional chemical model.", "In the case of emission spectra, we adopted a weighted average profile of temperature and of mixing ratios over the hemisphere facing the observer (weighted by the projected area on the planetary disk to better represent the situation encountered by an observer), where mixing ratios were assumed to be uniform with latitude.", "In transmission spectra, vertical profiles are simply obtained by averaging over the whole terminator, or over the morning or evening limb.", "After adopting an average pressure-temperature profile, the planetary radius (see values in section ) is assigned to the 1 bar pressure level and the altitude of each layer in the atmosphere is computed according to hydrostatic equilibrium.", "We note that similarly to the case of one-dimensional chemical models, the use of average vertical profiles in calculating planetary spectra is an approximation that masks the longitudinal and latitudinal structure of temperature and chemical composition and may result in non-negligible inaccuracies in the appearance of the spectra, which we plan to investigate in the future.", "Is is useful to begin our discussion on planetary spectra with a pedagogical plot that shows the pressure level probed by transmission and emission spectra for HD 209458b and HD 189733b (see Fig.", "REF ).", "We first note that transmission and emission spectra probe different pressure levels, with transmission spectra being sensitive to upper atmospheric layers than emission spectra.", "At infrared wavelengths (1-30 $\\mu $ m), and for the thermal and chemical composition profiles adopted by us for these two hot Jupiters, emission spectra probe pressures between 10 and 10$^{-2}$ bar, while transmission spectra probes the 1-10$^{-3}$ bar pressure regime.", "A second aspect worth noting is that there are strong variations with wavelength in both types of spectra, which implies that observations at different wavelengths are sensitive to the physical and chemical conditions of different pressure levels.", "It is always useful to keep these ideas in mind when analyzing the vertical distribution of molecules calculated with a chemical model, because only a very specific region of the atmosphere becomes relevant to planetary spectra." ], [ "Variation of emission spectra with phase", "A modulation of the planetary emission with the orbital phase has been observed for HD 189733b by monitoring the photometric flux in the 8 $\\mu $ m band of Spitzer IRAC during a good part of the orbit of the planet ([50] 2007).", "This has served to evidence the important temperature contrast between the different sides of the planet, noticeably between day and night, and indirectly the presence of strong winds that can redistribute the energy from the day to the night side, due to an observed shift between the hot spot and the substellar point.", "Various theoretical studies have also been interested in predicting the variation of the planetary flux with the orbital phase in HD 209458b and HD 189733b using the temperature structure calculated with GCM simulations ([35] 2006; [92] 2008, 2009; [79] 2008; [14] 2010; [83] 2013).", "Most previous studies have focused on the link between light curves and variations of temperature between the different planetary sides, and on the comparison between predicted and observed photometric fluxes.", "Here we are instead interested in discussing the influence of the temperature but also that of the chemical composition (assumed to be given by chemical equilibrium in previous studies) on the variation of the planetary emission with phase.", "Figure: Calculated transmission spectra for the evening and morning limbs of HD 209458b (left panel) and HD 189733b (right panel), where the vertical structure is obtained by averaging the temperature over each limb and adopting the abundance profiles at each limb from the nominal pseudo two-dimensional chemical model.", "Spectra have been smoothed to a resolving power RR = 300.", "The transit depth is simply calculated as (R p (λ)/R * ) 2 (R_p (\\lambda ) / R_)^2, where R p (λ)R_p (\\lambda ) is the calculated radius of the planet as a function of wavelength and R R_* is the stellar radius (see values in section ).", "The absolute scale of the transmission spectrum is set by our choice of assigning the value of the planetary radius given in section to the 1 bar pressure level.", "Since no attempt has been made to reproduce the absolute scale indicated by primary transit observations, calculated transit depths are somewhat higher than given by observations.", "The insets in both panels compare the transmission spectrum around 4.3 μ\\mu m as calculated for the evening limb and as computed using the mean temperature profile of the evening limb and the chemical composition corresponding to the morning limb.", "The most important differences between both spectra occur around 4.3 and 15 μ\\mu m, due to CO 2 _2.We show in Fig.", "REF how the calculated emission spectra of HD 209458b and HD 189733b vary with the phase of the planet.", "Important variations with phase are apparent in HD 209458b, whose strong dayside stratosphere causes the dayside emission spectrum to be significantly brighter and to have a noticeably different spectral shape than at other phases.", "In HD 189733b, the modulation of the flux and the variation of the spectral shape with phase are also important although less pronounced.", "Emission spectra are controlled on the one hand, by the vertical temperature structure, and on the other, by the abundances of the main atmospheric constituents providing opacity.", "The sensitivity of emission spectra to the thermal structure is illustrated in HD 209458b, whose dayside (facing a temperature inversion to the observer) shows some spectral features that appear in emission and not in absorption, as occurs for the other planetary sides of HD 209458b and for HD 189733b.", "These differences in the spectra can be used to infer whether there is a stratosphere in the atmosphere of a hot Jupiter from observations of its dayside emission spectrum ([51] 2008, 2009b; [13] 2008; [64] 2008, 2010; [105] 2010, [67] 2010).", "It is also interesting to note how similar the emission spectra of night and morning sides are in the two planets, as are the day and evening sides in the case of HD 189733b.", "This is a consequence of the eastward transport of energy by the superrotating jet, which shifts the hottest and coldest regions to the east of the substellar and antistellar points, respectively.", "In the calculated emission spectra of both HD 209458b and HD 189733b, most of the atmospheric opacity along the 1-30 $\\mu $ m wavelength range is provided by water vapor, with carbon monoxide contributing at 2.3 and 4.6 $\\mu $ m, CO$_2$ at 4.3 and 15 $\\mu $ m, and collision-induced absorption by H$_2$ -H$_2$ in certain wavelength ranges below 4 $\\mu $ m. No other species leaves appreciable signatures in the calculated emission spectra of HD 209458b, although in that of HD 189733b CH$_4$ contributes around 3.3 and 7.7 $\\mu $ m, NH$_3$ around 10.6 $\\mu $ m, and HCN at 14 $\\mu $ m. An interesting question that arises from the change in the emission spectrum with phase is whether it is entirely caused by the variation of temperature in the different sides of the planet or whether the longitudinal variation of the chemical composition contributes to an important extent.", "To illustrate this point we compare in the inset of HD 209458b's panel in Fig.", "REF the dayside emission spectrum with a synthetic spectrum calculated using the mean vertical temperature structure of the dayside and the mean chemical composition of the nightside.", "The two spectra are nearly identical except for a slight difference around 4.3 $\\mu $ m, a spectral region where atmospheric opacity is to a large extent dominated by CO$_2$ .", "Similar models in which the temperature structure and the chemical composition are adopted from different planetary sides indicate that variations of HD 209458b's emission spectrum with phase are almost entirely caused by changes of temperature, with the only effect that can be purely adscribed to variations in the chemical composition being restricted to the tiny variation (less than 0.02 % in the planet-to-star flux ratio) at 4.3 $\\mu $ m, which is caused by the longitudinal variation of about one order of magnitude in the abundance of CO$_2$ (see Fig.", "REF ).", "The reasons of the small impact of the chemical composition on the variation of emission spectra with phase are related to the important longitudinal homogenization of the abundances driven by the zonal wind in HD 209458b (see Fig.", "REF ).", "In fact, most of the atmospheric opacity affecting the emission spectrum comes from H$_2$ O, CO, and CO$_2$ , in order of decreasing importance, and the two former molecules show remarkably uniform abundances with longitude, while only the abundance of the latter molecule experiences some longitudinal variation, leading to a slight variation of the planetary flux with phase around 4.3 $\\mu $ m. In HD 189733b, the homogenization of the chemical composition with longitude is even more marked than in HD 209458b because of the lack of a stratosphere and the rather low eddy coefficient values (compare Figs.", "REF and REF ).", "Because the abundances of H$_2$ O, CO, CO$_2$ , CH$_4$ , NH$_3$ , and HCN (the main molecules providing opacity, in order of decreasing importance) vary little between the different sides of HD 189733b at the pressures probed by emission spectra ($>$ 10$^{-2}$ bar), the impact of the chemical composition on the change of the emission spectrum with phase becomes almost negligible, even around 4.3 $\\mu $ m because of the reduced longitudinal variation of the abundance of CO$_2$ ." ], [ "Transmission spectra of evening and morning limbs", "Variations in the composition of the atmosphere between the different sides of the planet may also be probed by transmission spectroscopy.", "Indeed, it is a priori possible to probe differences in the thermal and chemical structure of the two limbs if observations are able to obtain the transmission spectrum during the first half of the primary transit ingress, which would probe the leading or morning limb, and during the second half of the transit egress, which would probe the trailing or evening limb.", "Although a non-zero impact parameter during the transit would complicate the situation somewhat and such observations are very challenging today, they may be feasible in the near future.", "The subject has been addressed theoretically for hot Jupiters such as HD 189733b and HD 209458b in some studies in which the differences between the transmission spectra of leading and trailing limbs are evaluated under different assumptions for the chemical composition of each of the two limbs, either chemical equilibrium or some disequilibrium estimation ([37] 2010; [14] 2010).", "Here we revisit the subject in the light of the molecular abundances calculated in this study with the pseudo two-dimensional chemical model.", "To illustrate the possibility that transmission spectroscopy might be able to distinguish between the two limbs of HD 209458b and HD 189733b we show in Fig.", "REF the transmission spectrum calculated by adopting the chemical composition and mean temperature of the evening and morning limbs of these two exoplanets.", "Since we are mainly interested in comparing the spectra at the two limbs and not in comparing with observations, we set the absolute scale of transmission spectra by simply assigning the value of the planetary radius given in section to the 1 bar pressure level and made no attempt to reproduce the absolute scale of the photometric transit depths derived from observations.", "In HD 209458b and HD 189733b, the transmission spectrum of the evening limb shows a higher degree of absorption and also the variations of the transit depth with wavelength have a larger amplitude than at the morning limb, whose transmission spectrum is flatter.", "These differences are mainly due to the different temperature profile of the two limbs.", "Because the atmosphere at the morning limb is cooler and thus has a smaller scale height than at the evening limb, it becomes more compact, resulting in smaller apparent radii at all wavelengths and a flatter transmission spectrum.", "In addition to this dependence of the transmission spectrum on temperature, which causes it to shift up or down and to have a more elongated or flattened overall shape, the spectral structure is controlled by the relative abundances of the main species that provide opacity in the atmosphere.", "Fig.", "REF shows the relative contributions of the different sources of opacity taken into account in calculating the transmission spectra.", "Similarly to the emission spectra, in the calculated transmission spectra of HD 209458b and HD 189733b most of the atmospheric opacity at infrared wavelengths is provided by H$_2$ O, with CO being important at 2.3 and 4.6 $\\mu $ m, CO$_2$ at 4.3 and 15 $\\mu $ m, the H$_2$ -H$_2$ continuum at certain wavelengths below 3 $\\mu $ m, and, in the case of HD 189733b, CH$_4$ having some contribution around 3.3 and 7.7 $\\mu $ m, NH$_3$ around 10.6 $\\mu $ m, and HCN at 14 $\\mu $ m. Similarly to emission spectra, we evaluated to which extent differences in the chemical composition of evening and morning limbs contribute to the change of the transmission spectrum from one limb to the other.", "To this purpose, we computed transmission spectra in which we switched the temperature and chemical profiles between the two different limbs.", "As an example we compare in the insets of left and right panels in Fig.", "REF the transmission spectrum of the evening limb with a synthetic spectrum calculated using the temperature structure of the evening limb and the chemical composition of the morning limb.", "In HD 209458b the two spectra are very similar, except for a different degree of absorption around 4.3 and 15 $\\mu $ m, which is due to the difference of nearly one order of magnitude in the abundance of CO$_2$ between the two limbs (see Fig.", "REF ).", "Similarly to emission spectra, the longitudinal homogenization driven by horizontal transport is at the origin of the weak impact of other molecules on the variation of transmission spectra between both limbs.", "Because the abundances of CO and H$_2$ O are very similar in both limbs, and other molecules such as CH$_4$ , NH$_3$ , and HCN contribute little to the atmospheric opacity at infrared wavelengths because of their rather low abundances, the only chemical effect contributing to the change of the transmission spectrum from one limb to the other of HD 209458b is restricted to carbon dioxide.", "In HD 189733b, the even stronger longitudinal homogenization of abundances (compare Figs.", "REF and REF ) diminishes the extent of chemical effects, which are now restricted to a very weak change of the absorption around 4.3 and 15 $\\mu $ m (see inset in HD 189733b's panel of Fig.", "REF ), again due to a slight increase in the abundance of CO$_2$ when moving from the evening limb to the morning one (see Fig.", "REF )." ], [ "Summary", "We have developed a pseudo two-dimensional model of a planetary atmosphere that takes into account thermochemical kinetics, photochemistry, vertical mixing, and horizontal transport, and allows one to calculate the distribution with altitude and longitude of the main atmospheric constituents.", "Horizontal transport was modeled through a uniform zonal wind and thus the model is best suited for studying the atmosphere of planets whose circulation dynamics is dominated by an equatorial superrotating jet, as is expected to be the case of hot Jupiters.", "We therefore applied the model to study the atmospheres of the well-known exoplanets HD 209458b and HD 189733b.", "We used the temperature structure from GCM simulations and parameterized the turbulent mixing in the vertical direction using an eddy coefficient profile, which was calculated by following the behavior of passive tracers in GCM simulations, a method that results in substantially lower eddy values, by a factor of 10-100 in HD 209458b and of 10-1000 in HD 189733b, than previous estimates based on cruder methods.", "Molecular abundances homogenized with longitude to values typical of the hottest dayside regions.", "– We found that the distribution of molecules in the atmospheres of HD 209458b and HD 189733b is quite complex because of the interplay of the various (photo)chemical and dynamical processes at work, which form, destroy, and transport molecules throughout the atmosphere.", "Much of the distribution of the atmospheric constituents is driven by the strong zonal wind, which reaches speeds of a few km s$^{-1}$ , and the limited extent of vertical transport, with relatively low eddy diffusion coefficients below 10$^9$ cm$^2$ s$^{-1}$ around the 1 bar pressure level, resulting in an important homogenization of molecular abundances with longitude, in particular in the atmosphere of HD 189733b, which lacks a stratosphere and has quite low eddy diffusion coefficients.", "Moreover, molecular abundances are quenched horizontally to values typical of the hottest dayside regions, and therefore the composition of the cooler nightside regions is highly contaminated by that of warmer dayside regions.", "In hot Jupiters with a temperature inversion, such as HD 209458b, the longitudinal homogenization of molecular abundances is not as marked as in planets lacking a stratosphere, such as HD 189733b.", "In general, the cooler the planet, the stronger the homogenization of the chemical composition with longitude.", "Furthermore, in cooler planets such as hot Neptunes orbiting M dwarfs (e.g., GJ 436b) the temperature contrast between day and nightsides decreases because the cooling rate scales with the cube of temperature (e.g., [60] 2010), and therefore the composition is expected to be even more homogeneous with longitude than in warmer planets such as HD 209458b and HD 189733b.", "However, unlike hot Jupiters, hot Neptunes may have an atmospheric metallicity much higher than solar ([62] 2011; [72] 2013; [2] 2014; [109] 2014), which makes it interesting to investigate the extent of the spatial variation of molecular abundances in their atmospheres.", "Low methane content.", "– A major consequence of our pseudo two-dimensional chemical model is that methane reaches quite low abundances in the atmospheres of HD 209458b and HD 189733b, lower than the values calculated by previous one-dimensional models.", "The main reason for the low CH$_4$ abundance is that most of the atmosphere is contaminated by the hottest dayside regions, where the chemical equilibrium abundance of CH$_4$ is the lowest.", "The calculated mixing ratio of CH$_4$ in the dayside of HD 209458b is significantly below the values inferred from observations, which points to some fundamental problem in either the chemical model or the observational side.", "If the strength of vertical transport is substantially higher than in our nominal model, the calculated abundance of some molecules such as CH$_4$ and NH$_3$ would experience significant enhancement, especially in HD 189733b, although a conflict with observations would still exist regarding CH$_4$ in the dayside of HD 209458b.", "Variability of planetary spectra driven by thermal, rather than chemical, gradients.", "– An important consequence of the strong longitudinal homogenization of molecular abundances in the atmospheres of HD 209458b and HD 189733b is that the variability of the chemical composition has little effect on the way the emission spectrum is modified with phase and on the changes of the transmission spectrum from the transit ingress to the egress.", "Temperature variations and not chemical gradients are therefore at the origin of these types of variations in the planetary spectra.", "Only the longitudinal variation of the abundance of CO$_2$ , of nearly one order of magnitude, in the atmosphere of HD 209458b, is predicted to induce variations in the planetary spectra around 4.3 and 15 $\\mu $ m. We note, however, that an inhomogenous distribution of clouds and/or hazes (none of them included in our model) may induce important variations in the emission spectra with phase and in the transmission spectra from one limb to the other.", "These variations are best characterized at short wavelengths.", "Indeed, there is evidence of the presence of hazes in the atmosphere of HD 189733b ([58] 2008; [97] 2009), and an inhomogeneous distribution of clouds has recently been inferred for the hot Jupiter Kepler 7b ([29] 2013).", "The main drawback of our pseudo two-dimensional chemical model is the oversimplification of atmospheric dynamics, which is probably adequate for equatorial regions, but not at high latitudes.", "Ideally, GCM simulations coupled to a robust chemical network would provide an even more realistic view of the distribution of molecules in the atmospheres of HD 209458b and HD 189733b, but such calculations are very challenging from a computational point of view.", "Telescope facilities planned for the near or more distant future, such as the James Webb Space Telescope, Spica, and EChO, will be able to test some of the predictions of our pseudo two-dimensional model, in particular the low abundance of methane in the two planets and the important longitudinal homogenization of the chemical composition.", "We thank our anonymous referee for insightful comments which helped to improve this article.", "We acknowledge Adam P. Showman and Jonathan J. Fortney for the use of the SPARC/MITgcm code, Vincent Hue for useful discussions on photochemical models, Sergio Blanco-Cuaresma and Christophe Cossou for their help with Python and Fortran, and Vincent Eymet and Philip von Paris for kindly helping to validate the line–by–line radiative transfer code.", "M. A. and F. S. acknowledge support from the European Research Council (ERC Grant 209622: E$_3$ ARTHs).", "O.V.", "acknowledges support from the KU Leuven IDO project IDO/10/2013 and from the FWO Postdoctoral Fellowship Program.", "Computer time for this study was provided by the computing facilities MCIA (Mésocentre de Calcul Intensif Aquitain) of the Université de Bordeaux and of the Université de Pau et des Pays de l'Adour." ], [ "Calculation of planetary spectra", "To investigate the influence of the physical and chemical structure of the atmospheres of HD 209458b and HD 189733b on their transmission and emission spectra, we developed a line-by-line radiative transfer code that is independent of the pseudo two-dimensional chemical code.", "Currently, the code is limited because it is one-dimensional in the sense that the atmosphere is divided into various layers in the vertical direction (typically 60 spanning the 10-10$^{-6}$ bar pressure range) and each layer is assumed to be homogeneous with longitude and latitude.", "Therefore, each layer is characterized by a given pressure, temperature, and chemical composition, and longitudinal and latitudinal gradients are neglected.", "For transmission spectra, the physical and chemical profile in the vertical direction at either the east or west limb, or a mean of the profiles at both limbs, can be used.", "For emission spectra, the limitations caused by the one-dimensional character of the code can be partially alleviated by adopting thermal and chemical vertical profiles averaged in some manner over the hemisphere facing the observer.", "It is common in infrared spectroscopy to use the wavenumber with units of cm$^{-1}$ , instead of frequency or wavelength, and we therefore adopt this choice hereafter as well.", "At this stage, there are various sources of opacity included in the code.", "On the one hand, we consider collision induced absorption (CIA) by H$_2$ -H$_2$ , for which available absorption coefficients cover the wavelength range 10-25,000 cm$^{-1}$ and temperatures between 60 and 7000 K ([11] 2001; [12] 2002), and by H$_2$ -He, in which case absorption coefficients in the wavelength range 10-25,000 cm$^{-1}$ and for temperatures in the range 100-7000 K are available ([8] 1989, 1997; [9] 1989).", "CIA absorption coefficients scale with the square of pressure and thus become the dominant source of opacity at high pressures, usually above 1 bar.", "On the other hand, we consider spectroscopic transitions (mostly ro-vibrational transitions lying at infrared wavelengths) of H$_2$ O, CO, and CO$_2$ , whose data are taken from HITEMP ([86] 2010), and of CH$_4$ , NH$_3$ , and HCN, for which data from HITRAN ([85] 2009) are adopted.", "The spectral region of interest is divided into a certain number of spectral bins, whose widths are determined by the spectral resolution imposed.", "In each layer of the atmosphere, the contribution of a spectroscopic transition $j$ (centered on a wavenumber $\\tilde{\\nu _j}$ and which belongs to a species $i$ ) to the absorption coefficient $k(\\tilde{\\nu }_l)$ in a spectral bin $l$ (centered on a wavenumber $\\tilde{\\nu }_l$ and having a width $\\Delta \\tilde{\\nu }_l$ ), which we may label as $k_{ij}^S(\\tilde{\\nu }_l)$ , can be expressed as $k_{ij}^S(\\tilde{\\nu }_l) = S_j(T) n_i \\int _{\\tilde{\\nu }_l-\\Delta \\tilde{\\nu }_l / 2}^{\\tilde{\\nu }_l+\\Delta \\tilde{\\nu }_l / 2} \\phi _j (\\tilde{\\nu }^{\\prime } - \\tilde{\\nu _j}) d \\tilde{\\nu }^{\\prime } \\frac{1}{\\Delta \\tilde{\\nu }_l}, $ where $S_j(T)$ is the line intensity of the spectroscopic transition $j$ , which depends on the temperature $T$ and is usually given with units of cm$^{-1}$ /(molecule cm$^{-2}$ ) in the HITRAN and HITEMP databases, $n_i$ is the number density of species $i$ in the atmospheric layer where the absorption coefficient is to be evaluated, and $\\phi _j$ is the line profile function of transition $j$ , which has units of inverse of wavenumber (i.e., cm) and must be normalized such that the integral of $\\phi _j(\\tilde{\\nu }^{\\prime } - \\tilde{\\nu _j})$ from $\\tilde{\\nu }^{\\prime } - \\tilde{\\nu _j}=-\\infty $ to $\\tilde{\\nu }^{\\prime } - \\tilde{\\nu _j}=+\\infty $ yields unity.", "The absorption coefficient $k(\\tilde{\\nu }_l)$ has units of cm$^{-1}$ .", "The integral in Eq.", "(REF ) extends between the lower and upper wavenumber edges of the spectral bin $l$ .", "The line profile function $\\phi _j(\\tilde{\\nu }^{\\prime } - \\tilde{\\nu _j})$ is taken as a Voigt profile, which results from the convolution of a Gaussian and a Lorentzian profile, and thus accounts for the Doppler and pressure broadening of spectral lines in each layer of the atmosphere due to thermal motions and collisions, respectively.", "The Voigt profile function is calculated numerically with a routine based on an implementation of [46]'s algorithm by [57] (1997).", "Ideally, extremely high spectral resolution would be desirable to properly resolve the narrowest line profiles, although in practice this is too expensive in terms of computing time.", "For the calculations carried out here we adopted a spectral resolution of 0.03 cm$^{-1}$ , which is on the order of the line widths in the layers that provide most atmospheric opacity and has been found to be high enough to yield relative errors below 1 % in the computed spectra.", "Spectroscopic transitions lying farther away than 50 cm$^{-1}$ of a given spectral bin $l$ have not been taken into account when computing the absorption coefficient $k(\\tilde{\\nu }_l)$ .", "In addition, a cutoff in the line intensity $S_j$ (296 K) of 10$^{-40}$ cm$^{-1}$ /(molecule cm$^{-2}$ ) was adopted to neglect weak lines and speed up the calculations.", "This is perhaps the most delicate aspect because weak lines are numerous, especially in the HITEMP line lists, and at high temperatures the sum of all them results in non-negligible opacity enhancements in certain spectral regions.", "Calculations carried out with different line intensity cutoffs in selected spectral regions indicate that the relative error in the calculated spectra is at most 10 % in the hottest case studied here (the dayside emission spectrum of HD 209458b), and lower than 5 % in the remaining computed spectra.", "Currently, the code does not take into account light scattering, which becomes important at wavelengths shorter than $\\sim $ 1 $\\mu $ m, and therefore calculated spectra are reliable at infrared wavelengths, but not in the visible region of the electromagnetic spectrum.", "The code does not take into account the Doppler shift of spectral lines due to atmospheric winds either, an effect that may be observable in high-resolution spectra of hot Jupiters, where winds are strong ([98] 2010; [70] 2012; [96] 2013).", "At each atmospheric layer, the absorption coefficient $k(\\tilde{\\nu }_l)$ in each wavenumber bin $l$ is calculated as the sum of the contributions from all the spectroscopic transitions of the various absorbing species included, together with the contributions of the CIA couples, that is, $k(\\tilde{\\nu }_l) = \\sum _i \\sum _j k_{ij}^S (\\tilde{\\nu }_l) + \\sum _m k_m^{CIA} (\\tilde{\\nu }_l), $ where the sum in $m$ extends to the H$_2$ -H$_2$ and H$_2$ -He couples.", "After calculating the absorption coefficient in each wavenumber bin and at each atmospheric layer, computing the transmission and emission planetary spectra becomes straightforward, provided scattering is not considered.", "To calculate the transmission spectrum, the optical depth $\\tau (\\tilde{\\nu }_l,b)$ along a tangential line of sight intersecting the planet's atmosphere is computed as a function of the impact parameter $b$ in each spectral interval $l$ as $\\tau (\\tilde{\\nu }_l,b) = \\sum _{h=1}^{N(b)} k_h(\\tilde{\\nu }_l) \\Delta \\ell _h(b), $ where $\\Delta \\ell _h(b)$ is the path length (in cm) intersected by layer $h$ along the tangential line of sight at impact parameter $b$ , and the sum in $h$ extends to all atmospheric layers $N(b)$ intersected by the tangential line of sight.", "The apparent radius of the planet in each wavenumber interval is then retrieved as the impact parameter for which the optical depth along the tangential line of sight becomes 2/3.", "This latter value is rather arbitrary and is mainly chosen for similarity with the definition of a stellar photosphere, although it is very close to the value of 0.56 inferred by [58] (2008) and is not critical to derive the apparent radius of the planet.", "To obtain the emission spectrum, the emergent specific intensity along the observer's line of sight is computed as a function of impact parameter in each wavenumber bin.", "We thus need to solve the equation of radiative transfer along the various paths pointing toward the observer that pass through the planetary atmosphere at different impact parameters.", "As long as the different atmospheric layers are homogeneous, the equation of radiative transfer can be solved sequentially from the back to the front for each of the atmospheric layers intersected by the path.", "For each intersected layer, the equation of radiative transfer reads (see e.g.", "[88] 2004) $I_{\\tilde{\\nu }} = e^{-\\tau _{\\tilde{\\nu }}} \\big [ I_{\\tilde{\\nu }}^0 + B_{\\tilde{\\nu }}(T) \\big (e^{\\tau _{\\tilde{\\nu }}} - 1\\big ) \\big ], $ where the subscript $\\tilde{\\nu }$ indicates wavenumber dependence, $I_{\\tilde{\\nu }}^0$ and $I_{\\tilde{\\nu }}$ are the incoming and outgoing specific intensities that enter and emerge, respectively, from the current layer along a given path, $\\tau _{\\tilde{\\nu }}$ is the optical depth along the path within the current layer, and $B_{\\tilde{\\nu }}(T)$ is Planck's function.", "The final emission spectrum is then computed by averaging the wavenumber-dependent specific intensity, calculated as a function of the impact parameter, over the projected area of the emitting hemisphere.", "The line-by-line radiative transfer code was checked against the suite of radiative transfer tools 'kspectrum'See http://code.google.com/p/kspectrum/, which has been widely used to model the atmosphere of solar system planets such as Venus ([34] 2009)." ] ]	1403.0121
[ [ "A Generalisation of Isomorphisms with Applications" ], [ "Abstract In this paper, we study the behaviour of TF-isomorphisms, a natural generalisation of isomorphisms.", "TF-isomorphisms allow us to simplify the approach to seemingly unrelated problems.", "In particular, we mention the Neighbourhood Reconstruction problem, the Matrix Symmetrization problem and Stability of Graphs.", "We start with a study of invariance under TF-isomorphisms.", "In particular, we show that alternating trails and incidence double covers are conserved by TF-isomorphisms, irrespective of whether they are TF-isomorphisms between graphs or digraphs.", "We then define an equivalence relation and subsequently relate its equivalence classes to the incidence double cover of a graph.", "By directing the edges of an incidence double cover from one colour class to the other and discarding isolated vertices we obtain an invariant under TF-isomorphisms which gathers a number of invariants.", "This can be used to study TF-orbitals, an analogous generalisation of the orbitals of a permutation group." ], [ "Introduction", "Consider the two graphs shown in Figure REF .", "One is the well-known Petersen graph which we denote by $\\Pi $ and the other is a graph which is not so well-known which we sometimes refer to as Petersen's cousin, for reasons which will soon become apparent, and which we denote by $\\Lambda $ .", "What relationship could there be between them?", "Consider the set of neighbourhoods of the vertices in the two graphs.", "These are: Figure: The Petersen graph Π\\Pi and its less well-known cousin Λ\\Lambda .Neighbourhoods of $\\Pi $ : {2,5,6}, {1,3,7}, {2,4,8}, {3,5,9}, {1,4,10}, {1,8,9}, {2,9,10}, {3,6,10}, {4,6,7}, {5,7,8}.", "Neighbourhoods of $\\Lambda $ : {4,6,7}, {3,5,9}, {2,4,8}, {1,3,7}, {2,9,10}, {1,8,9}, {1,4,10}, {3,6,10}, {2,5,6}, {5,7,8}.", "Up to a re-ordering, both graphs have the same family of neighbourhoods.", "It is therefore clear that if one were given just the family of neighbourhoods of the Petersen graph one would not be able to determine that the graph they came from was Petersen—it could have been the second graph which also has the same neighbours.", "In the literature the following problem (the Neighbourhood Reconstruction Problem) has been proposed (for example, in [1] and [2]): given the neighbourhoods of the vertices of of $G$ , can $G$ be determined uniquely up to isomorphism?", "The two graphs above clearly show that the answer to this question is “no” in general.", "The Petersen graph is not reconstructible this way because the second graph shown in the figure is a reconstruction of the Petersen which is not isomorphic to it.", "Why does this happen?", "We shall explain this below.", "How many other reconstructions of the Petersen graph can be obtained this way?", "We shall see later on that this second graph is the only such reconstruction of the Petersen graph.", "There are a few other problems which have been considered in the graph theory literature which, as we shall see, are closely related to the neighbourhood reconstruction problem.", "The Realisability Problem.", "When is a given family of vertices the neighbourhood family of a graph or a digraph?", "What is the computational complexity of determining whether such a given family is the neighbourhood family of a graph or a digraph?", "The Matrix Symmetrization Problem.", "Given a $(0,1)$ -matrix $A$ , is it possible to change it into a symmetric matrix using (independent) row and column permutations?", "Although it is not immediately obvious, we shall see that this problem is related to the Realisability Problem.", "This problem was first studied in the paper [13] starting with a matrix $A$ which is already symmetric.", "Stability.", "This problem was first raised and studied in [13].", "Given the categorical product $G \\times K_2$ of a graph or digraph $G$ with the complete graph $K_2$ , the graph $G$ is said to be unstable when the automorphism group of the product is not isomorphic to Aut$(G) \\times \\mathbb {Z}$ .", "When is a graph unstable?", "This question was heavily studied in [14], [18], [19], [20], and again, although not immediately clear why, it is strongly related to the previous questions.", "The excellent survey paper [4] gives a good historical picture of work done on these problems.", "In this paper we shall present a new type of isomorphism between graphs and digraphs which, we believe, has independent interest but also unifies the above problems, as we shall demonstrate along the way while presenting our results." ], [ "Notation", "A mixed graph is a pair $G=(\\mbox{$V$}(G),\\mbox{$A$}(G))$ where $V$$(G)$ is a set and $A(G)$ is a set of ordered pairs of elements of $V$$(G)$ .", "The elements of $V$$(G)$ are called vertices and the elements of $A(G)$ are called arcs.", "When referring to an arc $(u,v)$ , we say that $u$ is adjacent to $v$ and $v$ is adjacent from $u$ .", "The vertex $u$ is the tail and $v$ is the head of a given arc $(u,v)$ .", "An arc of the form $(u,u)$ is called a loop.", "A mixed graph cannot contain multiple arcs, that is, it cannot contain the arc $(u,v)$ more than once.", "A set $S$ of arcs is self-paired if, whenever $(u,v) \\in $ $S$ , $(v,u)$ is also in $S$ .", "If $S$ $\\ =\\lbrace (u,v), (v,u)\\rbrace $ , then we identify $S$ with the unordered pair $\\lbrace u,v\\rbrace $ ; this unordered pair is called an edge.", "It is useful to consider two special cases of mixed graphs.", "A graph is a mixed graph without loops whose arc-set is self-paired.", "The edge set of a graph is denoted by $E$$(G)$ .", "A digraph is a mixed graph with no loops in which no set of arcs is self-paired.", "The inverse $G^{\\prime }$ of a mixed graph $G$ is obtained from $G$ by reversing all its arcs, that is $V$$(G^{\\prime }) =$$V$$(G)$ and $(v,u)$ is an arc of $G^{\\prime }$ if and only if $(u,v)$ is an arc of $G$ .", "A digraph $G$ may therefore be characterised as a mixed graph for which $A(G)$ and $A(G^{\\prime })$ are disjoint and a graph as one for which $A(G)=A(G^{\\prime })$ .", "The underlying graph $\\widehat{G}$ of a mixed graph $G$ is a graph with the vertex set $V$$(\\widehat{G})$ $=$ $V$$(G)$ and the edge set $E$$(\\widehat{G})$ defined by $\\lbrace x,y\\rbrace \\in $ $E$$(\\widehat{G})$ if and only if either $(x,y)$ or $(y,x)$ is an element of $A(G)$ .", "Two arcs are incident in $G$ if the corresponding edges in the underlying graph $\\widehat{G}$ have a common vertex.", "When we say that a mixed graph is connected, we mean that the underlying graph is connected.", "Given a mixed graph $G$ and a vertex $v \\in $ $V$$(G)$ , we define the in-neighbourhood $N_{in}(v)$ by $N_{in}(v) = \\lbrace x \\in \\mbox{$V$}(G)- (x,v) \\in \\mbox{A}(G)\\rbrace $ .", "Similarly we define the out-neighbourhood $N_{out}(v)$ by $N_{out}(v) = \\lbrace x \\in \\mbox{$V$}(G)-(v,x) \\in \\mbox{A}(G)\\rbrace $ .", "The in-degree $ \\rho _{in}(v)$ of a vertex $v$ is defined by $ \\rho _{in}(v) = \|N_{in}(v)\|$ and the out-degree $ \\rho _{out}(v)$ of a vertex $v$ is defined by $ \\rho _{out}(v) = \|N_{out}(v)\|$ .", "When $G$ is a graph, these notions reduce to the usual neighbourhood $N(v)=N_{in}(v)=N_{out}(v)$ and degree $\\rho (v)=\\rho _{in}(v)=\\rho _{out}(v)$ .", "A vertex $v$ is called a source if $ \\rho _{in}(v)= 0$ and a sink if $ \\rho _{out}(v)=0$ .", "A vertex is said to be isolated when it is both a source and a sink, that is, it is not adjacent to or from any vertex.", "A mixed graph $G$ is called bipartite if there is a partition of $V$$(G)$ into two sets $X$ and $Y$ , which we call colour classes, such that for each arc $(u,v)$ of $G$ the set $\\lbrace u,v\\rbrace $ intersects both $X$ and $Y$ .", "We call a bipartite digraph having one colour class consisting of sources and the other colour class consisting of sinks as a strongly bipartite digraph.", "Let $G$ be a digraph and let $(u,v)$ be an arc of $G$ .", "If in $G-(u,v)$ , the vertices $u$ , $v$ are either both sources or both sinks, then we call $(u,v)$ an S-arc of $G$ .", "A set $P$ of arcs of $G$ is called a trail if its elements can be ordered in a sequence $a_{1},\\ a_{2}, \\dots ,\\ a_{k}$ such that each $a_{i}$ is incident with $a_{i+1}$ for all $i = 1,\\ \\dots ,\\ k-1$ .", "If $u$ is the vertex of $a_{1}$ , that is not in $a_{2}$ and $v$ is the vertex of $a_{k}$ which is not in $a_{k-1}$ , then we say that $P$ joins $u$ and $v$ ; $u$ is called the first vertex of $P$ and $v$ is called the last vertex with respect to the sequence $a_{1},\\ a_{2},\\ \\dots ,\\ a_{k}$ .", "If, whenever $a_{i}=(x,y)$ , either $a_{i+1}=(x,z)$ or $a_{i+1}=(z,y)$ for some new vertex $z$ , $P$ is called an alternating trail or A-trail.", "If the first vertex $u$ and the start-vertex $v$ of an A-trail $P$ are different, then $P$ is said to be open.", "If they are equal then we have to distinguish between two cases.", "When the number of arcs is even then $P$ is called closed while when the number of arcs is odd then $P$ is called semi-closed.", "Note that if $P$ is semi-closed then either (i) $a_{1}=(u,x)$ for some vertex $x$ and $a_{k} = (y,u)$ for some vertex $y$ or (ii) $a_{1}=(x,u)$ and $a_{k} = (u,y)$ .", "If $P$ is closed then either $a_{1} =(u,x)$ or $a_{k}=(u,y)$ or $a_{1}=(x,u)$ and $a_{k} = (y,u)$ .", "Observe also that the choice of the first (equal to the last) vertex for a closed A-trail is not unique but depends on the ordering of the arcs.", "However, this choice is unique for semi-closed A-trails as this simple argument shows.", "Suppose $P$ is semi-closed and the arcs of $P$ are ordered such that $u$ is the unique (in that ordering) first and last vertex, that is, it is the unique vertex such as the first and the last arcs in the ordering in $P$ do not alternate in direction at the meeting point $u$ .", "Therefore, it is easy to see that both $\\rho _{in}(u)$ and $\\rho _{out}(u)$ (degrees taken in $P$ as a subgraph induced by its arcs) are odd whereas any other vertex $v$ in the trail has both $\\rho _{in}(v)$ and $\\rho _{out}(v)$ even.", "This is because, in the given ordering, arcs have to alternate in direction at $v$ and therefore in-arcs of the form $(x,v)$ are paired with out-arcs of the form $(v,y)$ .", "Therefore, in no ordering of the arcs of $P$ can $u$ be anything but the only vertex at which the first and last arcs do not alternate.", "The same argument holds for open A-trails.", "Therefore, open and semi-closed A-trails are similar at least in the sense that the first and last vertices are uniquely determined regardless of the sequence of the arcs.", "This similarity will be strengthened by the results which we shall shortly present.", "Let $G$ be a mixed graph.", "If, for every $u$ , $v \\in $ $V$$(G)$ , there exists an A-trail which joins them, then we say that $G$ is A-connected.", "Clearly, a connected graph $G$ with at least two edges is always A-connected since we can always choose any orientation for a given edge.", "In the case of a mixed graph, A-connectedness is not guaranteed.", "In fact, there are easy counterexamples also among digraphs.", "In a connected graph, the length (that is, number of edges) of a shortest path between two given vertices $u$ , $v$ is denoted by $d(u,v)$ .", "Any other graph theoretical terms which we use are standard and can be found in textbooks such as [3] and [5].", "Information on automorphism groups of a graph can be found in [12].", "Let $G$ and $H$ be two mixed graphs and $\\alpha $ , $\\beta $ be bijections from $V$$(G)$ to $V$$(H)$ .", "The pair $(\\alpha ,\\beta )$ is said to be a two-fold isomorphism (or TF-isomorphism) if the following holds: $(u,v)$ is an arc of $G$ if and only if $(\\alpha (u),\\beta (v))$ is an arc of $H$ .", "We then say that $G$ and $H$ are TF-isomorphic and write $G\\cong ^{\\mbox{{\\tiny {\\textbf {TF}}}}} H$ .", "Note that when $\\alpha =\\beta $ the pair $(\\alpha ,\\beta )$ is a TF-isomorphism if and only if $\\alpha $ itself is an isomorphism.", "If $\\alpha \\ne \\beta $ , then the given TF-isomophism $(\\alpha ,\\beta )$ is essentially different from a usual isomorphism and hence we call $(\\alpha ,\\beta )$ a non-trivial TF-isomorphism.", "In this case, we also say that $G$ and $H$ are non-trivially TF-isomorphic.", "If $(\\alpha ,\\beta )$ is a non-trivial TF-isomorphism from a mixed graph $G$ to a mixed graph $H$ , the bijections $\\alpha $ and $\\beta $ need not necessarily be isomorphisms from $G$ to $H$ .", "This is illustrated by examples found in [10], and also others found below.", "When $G=H$ , $(\\alpha ,\\beta )$ is said to be a TF-automorphism and it is again called non-trivial if $\\alpha \\ne \\beta $ .", "The set of all TF-automorphisms of $G$ with multiplication defined by $(\\alpha ,\\beta )(\\gamma ,\\delta ) = (\\alpha \\gamma , \\beta \\delta )$ is a subgroup of $S_{V(G)} \\times S_{V(G) }$ and it is called the two-fold automorphism group of $G$ and is denoted by $\\mbox{Aut}^{\\mbox{{\\tiny {\\textbf {TF}}}}}(G)$ .", "Note that if we identify an automorphism $\\alpha $ with the TF-automorphism $(\\alpha ,\\alpha )$ , then Aut$(G) \\subseteq $ $\\mbox{Aut}^{\\mbox{{\\tiny {\\textbf {TF}}}}}(G)$ .", "When a graph has no non-trivial TF-automorphisms, Aut$(G)= $$\\mbox{Aut}^{\\mbox{{\\tiny {\\textbf {TF}}}}}(G)$ .", "It is possible for an asymmetric graph $G$ , that is a graph with $\|$ Aut$(G)\| = 1$ , to have non-trivial TF-automorphisms [10]." ], [ "Some double covers and invariants under TF-isomorphisms", "Let $G$ be a mixed graph.", "The incidence double cover of $G$ , denoted by IDC$(G)$ is a bipartite graph with vertex set $V$ (IDC$(G)$ )$\\subseteq $ $V$$(G) \\times \\lbrace 0,\\ 1\\rbrace $ and edge set $E$ (IDC$(G)$ ) $=$ $\\lbrace (u,0),(v,1)\\rbrace \\ \| \\ (u,v) \\in $ $A(G)\\rbrace $ .", "The reader may refer to [8] for more information regarding the incidence double cover of graphs and its relevance to the study of association schemes.", "The A-cover of $G$ , denoted by $\\mbox{\\textbf {ADC}}(G)$ is a strongly bipartite digraph with vertex set $V$$(\\mbox{\\textbf {ADC}}(G))$ $\\subseteq $ $V$$(G) \\times \\lbrace 0,\\ 1\\rbrace $ and arc set $A$ ($\\mbox{\\textbf {ADC}}(G)$ ) $=$ $\\lbrace (u,0),(v,1)\\rbrace \\ $ $\| \\ (u,v)$ $ \\in $ $A(G)\\rbrace $ .", "For a more concise notation, very often we use $u_{0}$ or $u_{1}$ to label the elements of $V$$(G) \\times \\lbrace 0,\\ 1\\rbrace $ instead of $(u,0)$ or $(u,1)$ .", "It is clear that ADC$(G)$ is obtained from IDC$(G)$ by removing isolated vertices and changing every edge $\\lbrace u_{0},v_{1}\\rbrace $ into an arc $(u_{0},v_{1})$ .", "Theorem 3.1 Let $G$ , $H$ be mixed graphs.", "Then $G$ and $H$ are two-fold isomorphic if and only if IDC$(G)$ and IDC$(H)$ are isomorphic.", "Let $(\\alpha ,\\beta )$ be any two-fold isomorphism from $G$ to $H$ .", "This implies that, for any $(u,v) \\in $ $A(G)$ , $(\\alpha (u),\\beta (v)) \\in $ $A(H)$ .", "Consequently, given the corresponding $\\lbrace (u,0),(v,1)\\rbrace \\in $ $E$ (IDC$(G))$ , $\\lbrace (\\alpha (u),0),(\\beta (v),1)\\rbrace $ $\\in $ $E$ (IDC$(H))$ .", "Define $\\phi :$ $V$ (IDC$(G)$ ) $\\rightarrow $ $V$ (IDC$(H)$ ) such that $\\phi (x,0) = (\\alpha (x),0)$ and $\\phi (x,1) = (\\beta (x),1)$ for any $x \\in $ $A(G)$ such that $\\phi $ is an isomorphism from IDC$(G)$ to IDC$(H)$ .", "Conversely, let $\\alpha $ be any isomorphism from IDC$(G)$ to IDC$(H)$ such that $\\alpha \\lbrace (u,0),$ $(v,1)\\rbrace $ $=$ $\\lbrace (u^{\\prime },0),$ $(v^{\\prime },1)\\rbrace $ .", "Clearly $(u,v)\\in $ $A(G)$ and $(u^{\\prime },v^{\\prime }) \\in $ $A(H)$ by the definition of IDC.", "Define $\\alpha :$ $V$ ($G$ ) $\\rightarrow $ $V$ ($H$ ) such that $\\alpha (u) = u^{\\prime }$ if and only if $\\phi (u,0) = (u^{\\prime },0)$ .", "Similarly define $\\beta :$ $V$ ($G$ ) $\\rightarrow $ $V$ ($H$ ) such that $\\beta (v) = v^{\\prime }$ if and only if $\\phi (v,1) = (v^{\\prime },1)$ .", "Given any $(x,y) \\in $ $A(G)$ , $(\\alpha (x),\\beta (y)) \\in $ $A(H)$ since $\\lbrace (x,0),(y,1)\\rbrace \\in $ $E$ (IDC$(G)$ ) if and only if $\\lbrace \\phi (x,0),\\phi (y,1)\\rbrace = \\lbrace (x^{\\prime },0),(y^{\\prime },1)\\rbrace $ in $E$ (IDC$(H)$ ) if and only if $(x^{\\prime },y^{\\prime })$ $\\in $ $A(H)$ .", "$\\Box $ We now present what was Theorem 3.7 in [9], one of our main results in [9], as a corollary to Theorem REF .", "The canonical double cover (CDC) of a graph or digraph $G$ (also called its duplex especially in computational chemistry literature, for example, [17]) is the graph or digraph whose vertex set is $V(G) \\times \\lbrace 0,1\\rbrace $ and in which there is an arc from $(u,i)$ to $(v,j)$ if and only if $i \\ne j$ and there is an arc from $u$ to $v$ in $G$ .", "The canonical double cover of $G$ is often described as the direct or categorical product $G \\times K_{2}$ [7], [6], and is sometimes also called the bipartite double cover of $G$ .", "For graphs, the canonical double cover is identical to the incidence double cover.", "Corollary 3.2 Two graphs $G$ , $H$ are TF-isomorphic if and only if CDC$(G)$ and CDC$(H)$ are isomorphic.", "In fact, since $G$ and $H$ are graphs, IDC$(G)$ $\\cong $ CDC$(G)$ and IDC$(H)$ $\\cong $ CDC$(H)$ .", "$\\Box $ Therefore, in general the IDC of a mixed graph $G$ is a structure which is invariant under the action of a TF-isomorphism acting on $G$ .", "In the case of mixed graphs which are not graphs, Theorem REF is a significant improvement over Theorem3.7 in [9] which only considered TF-isomorphic graphs (not mixed graphs) and the canonical double cover." ], [ "Digression ", "Now we can see how the neighbourhood reconstruction problem and the other problems we discussed in the first section can be described in terms of TF-isomorphisms.", "First, consider this alternative way of looking at TF-isomorphisms.", "An incidence structure or, alternatively, a hypergraph, is a finite set of vertices with a system of subsets (blocks) some of which can be repeated.", "Number the $n$ vertices of a hypergraph in some arbitrary but fixed way, and do similarly for the $b$ blocks of the hypergraph.", "The incidence matrix of the hypergraph is the $n\\times b$ matrix whose $ij$ entry is 1 if the $i$ th vertex is in the $j$ block, and is zero otherwise.", "Let $H_1, H_2$ be two such hypergraphs with incidence matrices $B_1, B_2$ , respectively.", "Then usually $H_1$ and $H_2$ are said to be isomorphic if there is a bijection $\\alpha $ from $V(H_1)$ to $V(H_2)$ (effectively, a relabelling of the vertices of $H_1$ ) such that, under the resulting relabelling, the blocks of $\\alpha (H_1)$ are the same as the blocks of $H_2$ , possibly in a different order.", "Similarly, an automorphism of a hypergraph $H$ is a permutation of $V(H)$ (a relabelling of the vertices) such that the new blocks are a re-odering of the old blocks.", "In other words, we have a permutation $\\alpha $ of the rows of the incidence matrix $B_1$ such that the columns become a permutation of the columns of $B_2$ .", "We can remove this last detail and make even the columns the same as those of $B_2$ by saying that an isomorphism from $H_1$ to $H_2$ is an independent re-ordering $\\alpha $ of the rows and $\\beta $ of the columns of $B_1$ such that it becomes $B_2$ .", "Similarly, an automorphism of $H$ is an independent re-ordering of the rows and columns of $B$ which leaves $B$ unchanged.", "Therefore if we consider the adjacency matrix $A$ of a graph $G$ as an incidence matrix of a hypergraph with $n$ vertices (corresponding to the rows) and $b=n$ blocks (corresponding to the columns), a TF-isomorphism (TF-automorphism) is an isomorphism (automorphism) of the hypergraph represented by $A$ .", "Looking back at the example of the Petersen graph $\\Pi $ and what we have called its cousin $\\Lambda $ we see that their neighbourhoods considered as the blocks of two hypergraphs give isomorphic hypergraphs which means, according to the previous discussions, that $\\Pi $ and $\\Lambda $ are non-trivially TF-isomorphic, and that is why one is a neighbourhood reconstruction of the other!", "What non-trivial TF-isomorphism can we write from one to the other?", "Looking at how the list of neighbourhoods of the vertices $\\lbrace 1,2,\\ldots ,10\\rbrace $ of the second graph appear as a permutation of the same list of neighbourhoods of the first graph easily indicates that if $\\alpha =\\mbox{id}$ and $\\beta =(1\\ 9)(2\\ 4)(5\\ 7)$ then $(\\alpha ,\\beta )$ is a TF-isomorphism from the $\\Pi $ to $\\Lambda $ as labelled in Figure REF .", "But how do we know that $\\Lambda $ is the only graph which is a neighbourhood reconstruction of (that is, TF-isomorphic to) the Petersen graph?", "We shall soon see this when below, we present one more result on canonical double covers.", "The Matrix Symmetrization Problem can also be described in terms of TF-isomorphisms: given a digraph $D$ , is there a graph $G$ to which $D$ is non-trivially TF-isomorphic?", "In the case when the matrix $A$ is already symmetric, as the problem was originally posed in [13], this question becomes: given a graph $G$ is it non-trivially isomorphic to some other graph (possibly $G$ itself)?", "Figure: The Desargues graph.Let us now return to the Petersen graph $\\Pi $ and its cousin $\\Lambda $ from Figure REF .", "Since these two graphs are TF-isomorphic then they have the same CDC, and in fact, their common CDC is the well known Desargues graph shown in Figure REF .", "(We have labelled Petersen's cousin by $\\Lambda $ in honour of Livio Porcu who seems to have been the first one to observe in [16] that $\\Pi $ and $\\Lambda $ have the same CDC.)", "But now we can explain why these two graphs are the only ones with the same neighbourhood family.", "First we recall this result proved by Pacco and Scapellato in [15].", "As an easy reference, Theorem 5.3 of [15] may be restated as follows using our current terminology.", "Theorem 3.3 Given a connected bipartite mixed graph $H$ , the number of non-isomorphic mixed graphs $G$ such that ${\\emph {\\mbox{\\textbf {CDC}}(G)}} \\cong H$ is equal to the number of conjugacy classes of involutions in Aut$(H)$ that interchange the two colour classes of $H$ .", "The number of non-isomorphic loopless mixed graphs $H$ such that ${\\emph {\\mbox{\\textbf {CDC}}(G)}} \\cong H$ is equal to the number of conjugacy classes of involutions in Aut$(H)$ that interchange the two colour classes of $H$ and do not take any vertex $u$ to a vertex $v$ such that $(u,v)$ is an arc.", "$\\Box $ Now, the automorphism group of the Desergues graph $D$ is isomorphic to $S_5 \\times Z_2$ , and has order 240.", "Letting $\\beta $ be the automorphism of $D$ taking $(v,0)$ into $(v,1)$ and vice versa, note that $\\beta $ belongs to the centre of the group.", "Hence, each involution of Aut$(D)$ takes the form $(\\alpha ,{\\rm id})$ or $(\\alpha ,\\beta )$ , where $\\alpha $ is an involution of $S_5$ .", "Only the latter swaps the two colour classes; its conjugacy classes are as many as those of involutions of $S_5$ .", "The number of conjugacy classes of involutions of $S_5$ is exactly 2, corresponding to transpositions and double transpositions.", "Therefore, by Theorem REF , there are exactly two non-isomorphic mixed graphs whose CDC is $D$ .", "One of them must be Petersen itself, while the other one is $\\Lambda $ , for which we know already that the CDC is $D$ .", "So in this case, only these proper graphs occur, not more general mixed graphs.", "Observe that since the Petersen graph's automorphism group is isomorphic to $S_5$ , this graph is stable.", "However, Aut($\\Lambda $ ) is isomorphic to $S_3\\times Z_2$ .", "Thus the index of the automorphism group of $\\Lambda $ in Aut$(D)$ is 20 and so it is unstable." ], [ "TF-isomorphism and alternating trails", "We shall consider isomorphisms and TF-isomorphisms between pairs of mixed graphs, that is, we shall allow loops, directed arcs and edges.", "Configurations conserved by TF-isomorphisms must also be conserved by isomorphisms since the latter are just a special case of the former.", "However, the converse does not necessarily hold.", "It is well known that loops, paths and cycles are all conserved by isomorphisms, but it is easy to see that they are not necessarily conserved by TF-isomorphisms.", "For example, an arc $(u,v)$ can be mapped into a loop by a TF-isomorphism $(\\alpha ,\\beta )$ if $\\alpha (u)=\\beta (u)$ .", "An isomorphism conserves degrees, in-degrees and out-degrees.", "In the case of TF-isomorphisms, the situation is slightly more elaborate.", "First note that $\\alpha $ must conserve the out-degree of each vertex but not the in-degree.", "Likewise, $\\beta $ must conserve the in-degree of each vertex but not the out-degree.", "Therefore, if some vertex $u \\in $ $V(G)$ is a source, $\\alpha (u)$ might not be a source but it is certainly not a sink.", "If $u \\in $ $V(G)$ is a sink, then $\\alpha (u)$ must also be a sink.", "An analogous argument holds for $\\beta $ .", "Hence in the case of a digraph whose vertex set consists only of sources and sinks, $\\alpha $ and $\\beta $ must take sources to sources and sinks to sinks.", "Also, if $G$ and $H$ are graphs and $(\\alpha ,\\beta )$ is a TF-isomorphism from $G$ to some graph $H$ , then $\\alpha $ , $\\beta $ must preserve the degree $\\rho (v)$ of any vertex $v \\in $ $V(G)$ since for every vertex $v$ in a graph, $\\rho (v) = \\rho _{in}(v) = \\rho _{out}(v)$ .", "The definitions of the term path found in the literature tacitly imply a specific direction from one vertex to the subsequent vertex in a sequence.", "For example, if $u ,v, w$ is a path in graph $G$ and $\\alpha $ is an isomorphism from $G$ to $H$ , then the arcs $(u,v)$ and $(v,w)$ are mapped into the arcs $(\\alpha (u),\\alpha (v))$ and $(\\alpha (v),\\alpha (w))$ in $H$ , with the common vertex $\\alpha (v)$ .", "But, if $(\\alpha ,\\beta )$ is a TF-isomorphism from $G$ to $H$ then the arc $(u,v)$ is mapped into $(\\alpha (u),\\beta (v))$ and it is the arc $(v,w)$ which is mapped into $(\\alpha (w),\\beta (v))$ containing the common vertex $\\beta (v)$ with the previous arc.", "That is, to obtain a common vertex between images of successive arcs, we need to alternate the directions in the original path as $(u,v)$ , $(w,v)$ .", "This motivates our definition of A-trails and indicates the trend of our next results which show what type of A-trails are conserved by TF-isomorphisms.", "Proposition 4.1 Let $G$ and $G^{\\prime }$ be mixed graphs and $P$ be an A-trail in $G$ .", "Let $(\\alpha ,\\beta )$ be any TF-isomorphism from $G$ to $G^{\\prime }$ .", "Then there exists an A-trail $P^{\\prime }$ in $G^{\\prime }$ such that $(\\alpha ,\\beta )$ restricted to $P$ maps $P$ to $P^{\\prime }$ .", "For an A-trail consisting of just one arc, the result is trivial.", "Let us therefore consider an A-trail consisting of $k$ arcs with $k\\ge 2$ .", "Let the start vertex of a given A-trail $P$ be $x_{0}$ and label the successive vertices by $x_{1}, \\dots ,\\ x_{k}$ .", "Assume without loss of generality that $x_{0}$ is the tail of $a_{1}$ .", "The TF-isomorphism maps the arc $a_{1}= (x_{0},x_{1})$ into the arc $a_{1}^{\\prime }=(\\alpha (x_{0}),\\beta (x_{1}))$ .", "The next arc in $P$ is $a_{2}=(x_{2},x_{1})$ which is mapped by the TF-isomorphism to the arc $a_{2}^{\\prime }=(\\alpha (x_{2}),\\beta (x_{1}))$ with $\\beta (x_{1})$ as a common vertex with $a_{1}^{\\prime }$ .", "By repeating the process until all arcs of $P$ have been included, we obtain an A-trail $P^{\\prime }$ of $G^{\\prime }$ .", "Then, by restricting the action of the pair $(\\alpha ,\\beta )$ to $P$ we obtain $P^{\\prime }$ as its image.", "$\\Box $ This proposition immediately gives the following corollary.", "Corollary 4.2 If $G$ is a A-connected mixed graph which is TF-isomorphic to $H$ , then $H$ is also A-connected.", "$\\Box $ Proposition REF implies that $Z$ -trails are invariant under the action of a TF-isomorphism.", "The following remarks are aimed to present a clearer picture to the reader.", "Recall that in an A-trail vertices may be repeated so that different alternating trails such as the A-trails $P$ and $P^{\\prime }$ described in Proposition REF , when taken as digraphs in their own right, may not necessarily be TF-isomorphic.", "This is illustrated in Figure REF .", "Figure: GG and G ' G^{\\prime } are TF-isomorphic digraphs but PP and P ' P^{\\prime } are not.For $G$ and $G^{\\prime }$ as in Figure REF , let $\\alpha $ map 5, 7, 3 into $5^{\\prime }$ , $3^{\\prime }$ , $7^{\\prime }$ respectively and let it map arbitrarily the rest of the vertices of $G$ to the rest of the vertices of $G^{\\prime }$ .", "Let $\\beta $ map 6, 5, 1, 4, 2 into $4^{\\prime }$ , $2^{\\prime }$ , $1^{\\prime }$ , $6^{\\prime }$ , $5^{\\prime }$ respectively and let it map the rest of the vertices of $G$ to the rest of the vertices of $G^{\\prime }$ .", "The maps $\\alpha $ and $\\beta $ may be represented as shown below where the entries labelled by $$ may be replaced arbitrarily but without repetitions by any of the vertices to which there is no defined mapping.", "$ {\\alpha = \\left( \\begin{tabular}{ccccccc}1&2&3&4&5&6&7\\\\ \\ && 7^{\\prime }&&5^{\\prime }&&3^{\\prime } \\end{tabular} \\right)}\\qquad {\\beta = \\left( \\begin{tabular}{ccccccc}1&2&3&4&5&6&7\\\\1^{\\prime }&5^{\\prime }&&6^{\\prime }&2^{\\prime }&4^{\\prime }&* \\end{tabular} \\right)} $ The pair $(\\alpha ,\\beta )$ is then a TF-isomorphism from $G$ to $G^{\\prime }$ .", "However, the alternating trails $P$ and $P^{\\prime }$ in Figure REF are not TF-isomorphic digraphs.", "On the other hand, as stated in Proposition REF , any A-trail of a given graph, mixed graph or digraph $G$ is mapped by a TF-isomorphism to some A-trail of graph $G^{\\prime }$ whenever $G$ and $G^{\\prime }$ are TF-isomorphic.", "This is also the case of the trails $P$ and $P^{\\prime }$ in Figure REF .", "In fact it is easy to check that the open trail $P$ is mapped to the semi-closed trail $P^{\\prime }$ by the pair $(\\alpha ,\\beta )$ as defined above.", "However $P$ and $P^{\\prime }$ are not TF-isomorphic.", "Proposition 4.3 Let $G$ and $H$ be mixed graphs.", "Then a TF-isomorphism $(\\alpha ,\\beta )$ from $G$ to $H$ takes closed A-trails of $G$ to closed A-trails of $H$ .", "A closed trail $P$ has an even number of arcs and so it cannot be mapped to a semi-closed trail.", "Besides, if $P$ were mapped to an open trail, then $\\alpha $ or $\\beta $ must map some vertex of $P$ to both the first vertex and last vertex of the open A-trail, which is a contradiction since $\\alpha $ and $\\beta $ are bijections.", "As regards the latter case, note that a semi-closed A-trail has an odd number of arcs and a closed A-trail has an even number of arcs.", "$\\Box $ Therefore, closed A-trails are preserved by TF-isomorphisms just as they are by isomorphisms, but the situation is different for open and semi-closed A-trails.", "Proposition 4.4 Let $G$ and $H$ be A-connected mixed graphs.", "Then any non-trivial TF-isomorphism $(\\alpha ,\\beta )$ from $G$ to $H$ takes at least one open A-trail into a semi-closed A-trail and vice-versa.", "As $(\\alpha ,\\beta )$ is non-trivial, there is at least one vertex $u \\in $ $V(G)$ such that $\\alpha (u)\\ne \\beta (u)$ .", "Since both $\\alpha $ and $\\beta $ are bijections, we get $\\alpha (u)=\\beta (v)$ for some $v\\ne u$ .", "Since $G$ is Z-connected, there exists an A-trail joining $u$ and $v$ .", "Clearly $P$ is open.", "Its image $P^{\\prime }$ under $(\\alpha ,\\beta )$ is an A-trail of $H$ , that starts by $\\alpha (u)$ and ends by $\\beta (v)$ , but since they are equal, $P^{\\prime }$ is semi-closed.", "Since $(\\alpha ,\\beta )$ is a non-trivial TF-isomorphism from $G$ to $H$ , $(\\alpha ^{-1},\\beta ^{-1})$ is a non-trivial TF-isomorphism from $H$ to $G$ .", "Therefore, we may use the same arguments to show that $(\\alpha ^{-1},\\beta ^{-1})$ must take an open A-trail of $H$ to a semi-closed A-trail of $G$ .", "This implies that $(\\alpha ,\\beta )$ must take some semi-closed A-trail in $G$ to an open A-trail in $H$ .", "$\\Box $ Consider the following example.", "Let $G$ be a closed A-trail with 6 vertices and let $H$ consist of a $K_{3}$ and 3 isolated vertices.", "Note that $G$ is A-connected whereas $H$ is not.", "It is straightforward to check that $G$ and $H$ are TF-isomorphic.", "However, any TF-isomorphism from $G$ to $H$ is clearly non-trivial and maps an open A-trail of length 3 in $G$ to a semi-closed A-trail of $H$ .", "Therefore, the result of Proposition REF is false if the hypothesis, namely that both $G$ and $H$ are A-connected, is dropped.", "As an application of Proposition REF we get the following result.", "Corollary 4.5 A bipartite graph and a non-bipartite graph cannot be TF-isomorphic.", "Indeed if $G$ is bipartite and $(\\alpha ,\\beta )$ is a TF-isomorphism from $G$ to some other graph $H$ , then $\\alpha =\\beta $ .", "Let $G$ be a graph and let $(\\alpha ,\\beta )$ be a non-trivial TF-isomorphism from $G$ to some graph $H$ .", "Then, in view of Proposition REF , there is an open A-trail of $G$ that is taken to a semi-closed A-trail of $H$ .", "Therefore $H$ has an odd cycle and is non-bipartite.", "Conversely, $(\\alpha ^{-1},\\beta ^{-1})$ is a non-trivial TF-isomorphism from $H$ to $G$ and therefore, by the same argument $G$ cannot be bipartite.", "$\\Box $ The next section contains a more detailed study of how, using A-trails, a mixed graph $G$ can be made to correspond to a strongly bipartite digraphs, extending the results of Zelinka, particularly those exposed in [21].", "It will turn out that this digraph is a double cover of $G$ which we have already encountered." ], [ "Alternating double covers and an equivalence relation on arcs", "Let $G$ be any mixed graph.", "Consider the relation $R$ on the set $A(G)$ defined by: $xRy$ if and only if $x$ and $y$ are the first and last arcs of an A-trail of $G$ .", "Clearly $xRx$ since any given arc is the first and also the last arc of an A-trail containing only one arc.", "If $xRy$ then $yRx$ since if $x$ is the first arc of an A-trail, then $y$ is the last arc and vice-versa.", "Now suppose that $xRy$ and $yRz$ .", "If $x$ is the first arc of an A-trail $P$ , then $y$ is the last arc of the $P$ .", "Then if $y$ is the first arc of an A-trail $Q$ and $z$ is the last arc of $Q$ , then the set-theoretical union of $P$ and $Q$ is an A-trail which has $x$ as first arc and $z$ as last arc.", "Lemma 5.1 Let $G$ be a connected graph.", "Then every two edges of G are joined by trails of both odd and even length if and only if $G$ is not bipartite.", "Let $G$ be non-bipartite.", "Take any two edges $e_1$ , $ e_2$ and fix an odd cycle $C$ of $G$ and two vertices $v_1$ , $v_2$ of $C$ .", "Choose two trails $P_1$ , $P_2$ joining $e_1$ , $e_2$ with $v_1$ , $v_2$ respectively.", "Let $P^{\\prime }$ and $P^{\\prime \\prime }$ the two trails that join $v_1$ and $v_2$ using the edges of $C$ .", "Then there are two trails joining $e_1$ , $e_2$ , namely $P_1,\\ P^{\\prime },\\ P_2$ and $P_1,\\ P^{\\prime \\prime },\\ P_2$ .", "One of them has odd length and the other even because if $P^{\\prime \\prime }$ is odd, $P^{\\prime }$ is even and vice versa and therefore, the inclusion of one instead of the other switches parity.", "Conversely, suppose that there are trails of odd and even length between two fixed edges.", "In the subgraph induced by these paths, the vertices cannot be partitioned in two distinct colour classes and hence this subgraph must be non-bipartite and hence must contain an odd circuit.", "Hence $G$ contains an odd circuit and is also non-bipartite.", "$\\Box $ Corollary 5.2 Let $G$ be a connected graph.", "Then $R$ has one equivalence class if $G$ is not bipartite and two if it is bipartite.", "Suppose that $G$ is non-bipartite.", "Consider two arcs $x_1$ and $x_2$ and take the corresponding edges as $e_1$ and $e_2$ .", "Start from $x_1$ .", "On each edge of trails joining $e_1$ and $e_2$ , choose the arc to obtain an A-trail.", "Note that before this process may continue for all edges except at most $e_2$ .", "When $e_2$ is reached, the arc corresponding to edge incident with $e_2$ may form an A-trail of order 2 or a directed path.", "But this depends on whether the concerned trail has odd or even length, so one of the two will give a whole A-trail containing both $x$ and $y$ .", "On the other hand, if $G$ is bipartite, given that $x$ and $y$ form a directed path, which always happens, then every trail joining $x$ and $y$ will be of even length; but an A-trail of even length is open and can't allow directed paths.", "$\\Box $ Each equivalence class of $R$ is a set, to which one can naturally associate an A-connected sub-digraph, whose arcs are the elements of the class and whose vertices are those incident to at least one of such arcs.", "In general, the relation $R$ may yield any number of classes, not just one or two as in the case of graphs, as we shall see in Theorem REF below.", "If $v$ is any vertex, two different arcs that have $v$ as a first vertex form an A-trail; the same can be said for two different arcs having $v$ as a head.", "Therefore, the arcs incident with $v$ belong to only one class or two.", "In the latter case, we say that $v$ is a frontier vertex.", "Let $F(G)$ be the set of all frontier vertices of $G$ .", "In view of Corollary REF , if $G$ is a graph then $F(G)$ is either empty (if $G$ is not bipartite) or $F(G)$ $=$ $V$$(G)$ (if $G$ is bipartite).", "The proof of the next result is straightforward.", "Proposition 5.3 Let $G$ be a connected mixed graph.", "The following are equivalent: (i) All classes of $R$ are singletons.", "(ii) All A-trails of $G$ are singletons.", "(iii) Each vertex of $G$ has both in-degree and out-degree less than or equal to 1.", "(iv) $G$ is a directed path or a directed cycle.", "$\\Box $ Proposition 5.4 Let $G$ be a connected mixed graph.", "Then $R$ has only one class if and only if the set $F(G)$ is empty.", "The condition is clearly necessary, for if $v$ were an element of $F(G)$ then by definition we would have at least two different classes.", "On the other hand, if there is more than one class, let $x$ and $y$ be arcs that belong to different classes.", "Since $G$ is connected, there is a trail $P$ that joins $x$ and $y$ .", "Somewhere in $P$ there must be $x^{\\prime }$ and $y^{\\prime }$ that belong to different classes and are incident with a vertex $v$ .", "Thus $v \\in F(G)$ .", "$\\Box $ Proposition 5.5 Let $G$ be a connected mixed graph.", "Then $F(G)$ is empty or $F(G)$ $=$ $V$$(G)$ or $F(G)$ is a disconnected set of the underlying graph.", "We can assume that $F(G)$ is a proper subset of $V$$(G)$ .", "By Proposition REF , there are at least two classes for $R$ .", "Letting $x$ , $y$ be elements of different classes, by the same argument as in Proposition REF we infer that each trial joining $x$ and $y$ must pass through a vertex $v\\in F(G)$ .", "Therefore, removing $F(G)$ the arcs $x$ and $y$ end up in different connected components.", "$\\Box $ Theorem 5.6 For every pair $(m,k)$ of positive integers, there exists a mixed graph on which the equivalence relation $R$ induces $m$ classes and having $k$ frontier vertices if and only if $m-1\\le k$ .", "Let us first construct a mixed graph with $m$ classes and $k$ frontier vertices whenever $m-1 \\le k$ .", "Note that if $m-1=k$ a directed path satisfies the statement (each class consists of a single arc).", "The same holds for $m-2=k$ and a directed cycle.", "Assume then that $m-3\\le k$ .", "Consider the 4-set $\\lbrace a,b,c,d\\rbrace $ and consider $m-2$ mixed graphs $H_i$ for $i=1,\\ \\dots ,m-2$ where $V$$(H_i)$ $=$ $\\lbrace (a,i),(b,i),(c,i),(d,i),(a,i+1)\\rbrace $ and $A(H_i)$ contains all the arcs of the triangle $(b,i)$ , $(c,i)$ , $(d,i)$ , plus the additional arcs $((a,i),(b,i))$ and $((d,i),(a,i+1))$ .", "Take any connected bipartite graph $K$ with $k-m+2$ vertices and fix a vertex $u$ of $K$ .", "Let $L$ be the digraph consisting of the single arc $(u,(a,1))$ .", "Let $G$ be the (standard graph-theoretical) union of $K$ , $L$ , $H_1,\\ \\dots , H_{m-2}$ .", "Then $G$ is a connected mixed graph.", "The classes for $R$ in $G$ are: (i) the class of $K$ containing the arcs incident to $u$ ; (ii) the class of $K$ containing the arcs incident from $u$ , together with the extra arc $(u(a,1))$ ; (iii) each of the $H_i$ 's for $i=1,\\dots ,\\ m-2$ .", "Hence, their number is $m$ .", "Moreover, $F(G)=V(K) \\cup \\lbrace (a,1),(a,2),...,(a,m-2)\\rbrace $ , then $\|F(G)\|=(k-m+2)+(m-2)=k$ .", "Therefore, for all cases where the stated inequality holds, there is a mixed graph $G$ as claimed.", "Conversely, consider now any mixed graph $G$ and define a graph $X$ such that $V$$(X)$ $=$ $V$$(G)/R$ (that is, the set of classes of $R$ in $G$ ) and two vertices are adjacent when the associated mixed graphs share a frontier vertex.", "Then $m=\|X\|$ , while the number $k^{\\prime }$ of edges of $X$ is less or equal to $k=\|F(G)\|$ (because two classes might share more than a frontier vertex).", "The known inequality $m-1\\le k^{\\prime }$ implies $m-1 \\le k$ as claimed.", "$\\Box $ As remarked earlier, a strongly bipartite digraph can be associated with each equivalence class of $R$ .", "Now let these strongly bipartite digraphs $D_{1}, D_{2}, \\dots , D_{k}$ corresponding to the different classes of $R$ obtained from the mixed graph $G$ .", "Let any vertex $u$ of $V(G)$ which appears as a source in $D_{i}$ be labelled $u_{0}$ and let any vertex $v$ of $V(G)$ which appears as a sink in $D_{i}$ be labelled $v_{1}$ .", "Therefore, an arc $(u,v)$ in $D_{i}$ now becomes $(u_{0},v_{1})$ .", "It turns out that the strongly bipartite digraph consisting of the components $D_{i}$ labelled this way is ADC$(G)$ which we have already defined earlier.", "Figure REF shows an example which may be used to illustrate the following remarks which highlight certain properties of $\\mbox{\\textbf {ADC}}(G)$ in relation to the mixed graph $G$ : 1.", "We know that, for any vertex $u$ of $G$ , all incoming arcs $(x,u)$ of $G$ are in the same component of ADC$(G)$ and similarly all outgoing arcs $(u,x)$ of $G$ are in the same component of ADC$(G)$ .", "Therefore $u_0$ if present in ADC$(G)$ , cannot appear in two different components.", "Similarly for $u_1$ .", "However, as we see in examples below, $u_0,$ and $u_1$ can, in some cases, appear in the same component and they can, in other cases, appear in different components.", "In particular, if $G$ is a bipartite graph they appear in different components as shown in Figure REF $(ii)$ and if $G$ is a non-bipartite graph, they are in the same component as shown in Figure REF $(ii)$ .", "2.", "ADC$(G)$ is a strongly bipartite digraph.", "3.", "By definition, there is no $u_0$ in ADC($G$ ) if $u$ is a sink in $G$ , and there is no $u_1$ if $u$ is a source.", "Figure: ADC(G)\\mbox{\\textbf {ADC}}(G) obtained from a digraph GG." ], [ "TF-isomorphisms and mixed graph covers", "The following result can be seen as a corollary to Theorem REF and the proof is easy since the IDC of a mixed graph $G$ can be obtained from $\\mbox{\\textbf {ADC}}(G)$ simply by removing the directions of the arcs and isolated vertices are irrelevant.", "Here we give an direct proof because it will help us in later constructions.", "Theorem 6.1 Let $G$ , $H$ be mixed graphs.", "The $G$ and $H$ are TF-isomorphic if and only if $\\mbox{\\textbf {\\emph {ADC}}}(G)$ and $\\mbox{\\textbf {\\emph {ADC}}}(H)$ are isomorphic.", "Let $(\\alpha ,\\beta )$ be a TF-isomorphism from $G$ to $H$ .", "Let $(u,v)$ be an arc of $G$ .", "First note that if $(\\alpha (u),\\beta (v))$ is an arc of $H$ , then $(u_{0},v_{1})$ is an arc of $\\mbox{\\textbf {ADC}}(G)$ and $(\\alpha (u)_{0},\\beta (u)_{1})$ is an arc of $\\mbox{\\textbf {ADC}}(H)$ .", "Let $f$ be a map from $V$$(\\mbox{\\textbf {ADC}}(G))$ to $V$$(\\mbox{\\textbf {ADC}}(H))$ such that $f: u_{0} \\mapsto x_{0}$ if $x = \\alpha (u)$ and $f: v_{1} \\mapsto y_{1}$ if $y =\\beta (v)$ .", "Consider any arc $(u,v)$ of $G$ and consider the corresponding arc $(u_{0},v_{1})$ in $A(\\mbox{\\textbf {ADC}}(G))$ .", "Let $(\\alpha ,\\beta )(u,v) = (x,y)$ .", "Then by definition $f$ takes $(u_{0},v_{1})$ to $(x_{0},y_{1})$ in $A(\\mbox{\\textbf {ADC}}(H))$ .", "The function $f$ maps arcs of $\\mbox{\\textbf {ADC}}(G)$ to arcs of $\\mbox{\\textbf {ADC}}(H)$ and it is clearly bijective.", "Hence, $f$ is an isomorphism from $\\mbox{\\textbf {ADC}}(G)$ to $\\mbox{\\textbf {ADC}}(H)$ .", "Now suppose that $\\mbox{\\textbf {ADC}}(G)$ and $\\mbox{\\textbf {ADC}}(H)$ are isomorphic.", "This implies that there exists a map $f$ such that $f(u_{0},v_{1})$ $=$ $(x_{0},y_{1})$ .", "Note that the arcs must always start from a vertex whose label has 0 as subscript and incident to a vertex whose label has 1 as subscript, by virtue of the construction presented above.", "Define $\\alpha $ , $\\beta $ from $V$$(G)$ to $V$$(H)$ as follows.", "Let $\\alpha (u) = x$ if $f(u_{0}) = x_{0}$ where $u \\in $ $V$$(G)$ and $x \\in $ $V$$(H)$ and let $\\beta (v) = y$ if $f(v_{1}) = y_{1}$ where $v \\in $ $V$$(G)$ and $y \\in $ $V$$(H)$ .", "Then $(\\alpha ,\\beta )$ takes any arc $(u,v) \\in $ $A(G)$ to some $(x,y)$ in $A(H)$ .", "This two-fold mapping is bijective and hence $(\\alpha ,\\beta )$ is a TF-isomorphism from $G$ to $H$ .", "$\\Box $ Corollary 6.2 Let $(\\alpha ,\\beta )$ be a TF-isomorphism from a mixed graph $G$ to a mixed graph $H$ .", "Then there exists an isomorphism $f_{\\alpha ,\\beta }$ from $\\mbox{\\textbf {\\emph {ADC}}}(G)$ to $\\mbox{\\textbf {\\emph {ADC}}}(H)$ such that $f_{\\alpha ,\\beta }(u_{0},v_{1}) = (x_{0},y_{1})$ if and only if $x = \\alpha (u)$ and $y =\\beta (v)$ for some TF-isomorphism $(\\alpha ,\\beta )$ from $G$ to $H$ .", "The result follows from the proof of Theorem REF .", "$\\Box $ Refer to Figure REF .", "An isomorphism $f$ from $\\mbox{\\textbf {ADC}}(G)$ to $\\mbox{\\textbf {ADC}}(H)$ and the corresponding maps $\\alpha $ and $\\beta $ from V$(G)$ onto V$(H)$ , which are derived from $f$ as described in the proof of Theorem REF , are given below.", "$f: 1_{0} \\mapsto 1_{0}^{\\prime } & \\ & f: 1_{1} \\mapsto 1_{1}^{\\prime }\\\\f: 2_{1} \\mapsto 2_{0}^{\\prime } & \\ & f: 2_{1} \\mapsto 3_{1}^{\\prime }\\\\f: 3_{0} \\mapsto 3_{0}^{\\prime } & \\ & f: 3_{1} \\mapsto 2_{1}^{\\prime }\\\\f: 4_{0} \\mapsto 6_{0}^{\\prime } &\\ & f: 4_{1} \\mapsto 5_{1}^{\\prime }\\\\f: 5_{0} \\mapsto 7_{0}^{\\prime } &\\ & f: 5_{1} \\mapsto 4_{1}^{\\prime }\\\\f: 6_{0} \\mapsto 5_{0}^{\\prime } & \\ & f: 6_{1} \\mapsto 6_{1}^{\\prime }\\\\f: 7_{0} \\mapsto 4_{0}^{\\prime } & \\ & f: 7_{1} \\mapsto 7_{1}^{\\prime }$ $\\alpha : 1 \\mapsto 1^{\\prime } & \\ & \\beta : 1 \\mapsto 1^{\\prime }\\\\\\alpha : 2 \\mapsto 2^{\\prime } & \\ & \\beta : 2 \\mapsto 3^{\\prime }\\\\\\alpha : 3 \\mapsto 3^{\\prime } & \\ & \\beta : 3 \\mapsto 2^{\\prime }\\\\\\alpha : 4 \\mapsto 6^{\\prime } &\\ & \\beta : 4 \\mapsto 5^{\\prime }\\\\\\alpha : 5 \\mapsto 7^{\\prime } &\\ & \\beta : 5 \\mapsto 4^{\\prime }\\\\\\alpha : 6 \\mapsto 5^{\\prime } & \\ & \\beta : 6 \\mapsto 6^{\\prime }\\\\\\alpha : 7 \\mapsto 4^{\\prime } & \\ & \\beta : 7 \\mapsto 7^{\\prime }$ Figure REF shows a digraph $G$ and its alternating double cover $\\mbox{\\textbf {ADC}}(G)$ which in this case has three components, namely $D_{1}$ , $D_{2}$ and $D_{3}$ .", "Figure REF also shows how the components of $\\mbox{\\textbf {ADC}}(G)$ can be combined by associating vertices of the form $u_{0}$ with vertices of the form $v_{1}$ , irrespective of whether $u=v$ or $u \\ne v$ , to form $G$ or other digraphs such as $G_{1}$ , $G_{2}$ and $G_{3}$ having the same number of vertices as $G$ .", "It is easy to check that $G$ , $G_{1}$ , $G_{2}$ and $G_{3}$ are pairwise two-fold isomorphic as expected from the result of Theorem REF since each of these digraphs have the same number of vertices and have isomorphic ADCs.", "Proposition 6.3 (i) A digraph $H$ is isomorphic to $\\mbox{\\emph {\\textbf {ADC}}}(G)$ for some $G$ if and only if $H$ is strongly bipartite.", "(ii) For every digraph $G$ , ${\\mbox{\\emph {\\textbf {ADC}}}}(\\mbox{\\emph {\\textbf {ADC}}}(G))$ is isomorphic to $\\mbox{\\emph {\\textbf {ADC}}}(G)$ .", "We already know that the condition stated in $(i)$ is necessary in order to have $H$ isomorphic to some $\\mbox{\\textbf {ADC}}(G)$ .", "Conversely, if $H$ has this property, define map $f:$ $V(H)$ $\\rightarrow V (\\mbox{\\textbf {ADC}}(H))$ as follows: $f(u) = u_{0}$ if $u$ is a source, $f(u) = u_{1}$ if $u$ is a sink.", "Clearly $f$ is a bijection If $(u,v)$ is an arc of $H$ then by our assumption $u$ is a source and $v$ is a sink of $H$ .", "Then $(u_{0},v_{1})$ $=$ $(f(u),f(v))$ is an arc of $\\mbox{\\textbf {ADC}}(H)$ .", "Likewise, each arc of $\\mbox{\\textbf {ADC}}(H)$ takes the form $(u_{0},v_{1})$ , with $u$ source and $v$ sink of $H$ and hence $(u_{0},v_{1})$ is the image of $(u,v)$ under $f$ .", "This proves that $f$ is an isomorphism from $H$ to $\\mbox{\\textbf {ADC}}(H)$ , so $(i)$ is satisfied with $G=H$ .", "Now $(ii)$ is a straightforward consequence of $(i)$ , taking $H = \\mbox{\\textbf {ADC}}(G)$ .", "$\\Box $ Figure: GG and HH are TF-isomorphic graphs and have isomorphic ADCs." ], [ "Two-fold orbitals", "Let $\\mathbf {\\Gamma } \\le \\mathcal {S} = S_{\\mbox{\\scriptsize {$\|V\|$}}} \\times S_{\\mbox{\\scriptsize {$\|V\|$}}}$ .", "For a fixed element $(u,v)$ of $V\\times V$ let $ \\mathbf {\\Gamma }(u,v) = \\lbrace (\\alpha (u),\\beta (v)\\ \|\\ (\\alpha ,\\beta ) \\in \\mathbf {\\Gamma }\\rbrace .", "$ The set $\\mathbf {\\Gamma }(u,v)$ is called a two-fold orbital or TF-orbital.", "A two-fold orbital is the set of arcs of a digraph $G$ having vertex set $V$ which we call two-fold orbital digraph or TF-orbital digraph.", "If for every arc $(x,y)$ in $\\mathbf {\\Gamma }(u,v)$ , the oppositely directed arc $(y,x)$ is also contained in $\\mathbf {\\Gamma }(u,v)$ , then $G$ is a two-fold orbital graph or TF-orbital graph.", "This generalisation of the well-known concept of orbital (di)graph has been discussed in [9].", "Proposition 7.1 Let $G$ be a strongly bipartite digraph.", "Then (i) There is a homomorphism $\\psi $ of Aut$^{\\mbox{\\tiny \\textbf {{TF}}}}(G)$ onto Aut$(G)$ .", "(ii) If $G$ is a TF-orbital digraph, then it is also an orbital digraph.", "If $(\\alpha ,\\beta )$ is a TF-automorphism of $G$ , define $\\psi (\\alpha ,\\beta )$ $=$ $f:$ $V(G) \\rightarrow $ $V(G)$ as follows: $f(u) = \\alpha (u)$ if $u$ is a source and $f(u)=\\beta (u)$ if $u$ is a sink.", "Since $\\alpha $ preserves sources then $f$ takes sources to sources.", "Similarly, since $\\beta $ preserves sinks, then $f$ takes sinks to sinks.", "Since both $\\alpha $ and $\\beta $ are permutations, the restrictions of $f$ to the set of sources and to the set of sinks are also permutations.", "Hence $f$ is a permutation of V$(G)$ .", "Given any arc $(u,v)$ of $G$ , note that $(\\alpha ,\\beta )$ takes $(u,v)$ to $(\\alpha (u),\\beta (v))$ , which is equal to $(f(u),f(v))$ because $u$ is a source and $v$ is a sink.", "Hence $f$ is an automorphism of $G$ .", "so $\\psi $ maps $\\mbox{Aut}^{\\mbox{{\\tiny {\\textbf {TF}}}}}(G)$$\\ $ to Aut$(G)$ .", "A direct computation proves that $\\psi $ is a group homomorphism, hence $(i)$ holds.", "Assume now that $G = \\mathbf {\\Gamma }(u,v)$ for some $\\mathbf {\\Gamma }$ .", "Then $\\mathbf {\\Gamma }$ is a subgroup of $\\mbox{Aut}^{\\mbox{{\\tiny {\\textbf {TF}}}}}(G)$$\\ $ and $\\psi (\\mathbf {\\Gamma })$ is a subgroup of Aut $G$ .", "Each arc of $G$ takes the form $(\\alpha (u),\\beta (v))$ , where $(\\alpha ,\\beta )$ $\\in $ $\\mathbf {\\Gamma }$ and $u$ , $v$ are a source and a sink respectively.", "Letting $f = \\psi (\\alpha ,\\beta )$ this arc is $(f(u),f(v))$ , so it belongs to the orbital digraph $\\psi (\\mathbf {\\Gamma })(u,v)$ .", "This proves that $G$ is contained in this orbital digraph.", "The opposite inclusion can be shown the same way, so that $G = \\psi (\\Gamma )(u,v)$ and $(ii)$ follows.", "$\\Box $ Corollary 7.2 Let $G$ be a strongly bipartite digraph.", "Then $G$ is a two-fold orbital digraph if and only if ${\\emph {\\mbox{\\textbf {CDC}}(G)}}$ is an orbital digraph.", "By Proposition REF , $G$ and $\\mbox{\\textbf {ADC}}(G)$ are isomorphic.", "If either of them is a TF-orbital, then of course the same holds for the other one, but by Proposition REF in this case these TF-orbitals are both orbitals.", "$\\Box $" ], [ "Conclusion", "We believe that TF-isomorphisms is a relatively new concept.", "The only other author who considered them was Zelinka in a short paper motivated by the concept of isotopy in semigroups [22], [23].", "Our papers ([9] and [10]) are the first attempts at a systematic study of TF-isomorphisms.", "In this paper we have shown close links between TF-isomorphisms and double covers, and how the decomposition of a particular double cover can be used to obtain TF-isomorphic graphs.", "We have also seen that TF-isomorphisms give a new angle for looking at some older problems in graph theory.", "But does the notion of TF-isomorphism add anything new to these older questions?", "We believe that it does.", "For example, in [11] we prove this result which explains instability of graphs in terms of TF-automorphisms.", "Theorem 8.1 Let Aut$^{\\mbox{\\tiny \\textbf {{TF}}}}(G)$ be the group of TF-automorphisms of a mixed graph $G$ .", "Then Aut$({\\emph {\\mbox{\\textbf {CDC}}(G)}})$ is isomorphic to the semi-direct product Aut$^{\\mbox{\\tiny \\textbf {{TF}}}}(G)\\rtimes \\mathbb {Z}_2$ .", "Therefore $G$ is unstable if and only if it has a non-trivial TF-automorphism.", "$\\Box $ Also, it is not very likely that looking at these questions without the notion of TF-isomorphisms would lead one to the notion of A-trails, a technique which we feel is very useful, or the construction of asymmetric graphs with a non-trivial TF-isomorphism, an interesting notion which would be not so natural to formulate using only matrix methods, say.", "Some results and proofs are clearer in the TF-isomorphism setting.", "For example, in some of the papers cited we find this result about graph reconstruction from neighbourhoods.", "Theorem 8.2 ([1]) If $G$ is a connected bipartite graph, then any nonisomorphic graph $H$ with the same neighbourhood family as $G$ must be a disconnected graph with two components which themselves have identical neighbourhood hypergraphs.", "$\\Box $ From the TF-isomorphism point of view, this result follows from three very basic facts: (i) two graphs have the same neighbourhood family (equivalent to being TF-isomorphic) if and only if they have the same canonical double cover; (ii) the canonical double cover of a graph $G$ is disconnected if and only if $G$ is bipartite; and (iii) when $G$ is bipartite, the canonical double cover of $G$ is simply two disjoint copies of $G$ .", "Therefore, for $H$ to have the same canonical double cover as $G$ , it must consist of two components isomorphic to $K$ , where $G$ is the canonical double cover of $K$ .", "This gives Theorem REF .", "And moreover, from these remarks we also see that the only bipartite graphs for which there are non-isomorphic graphs with the same neighbourhood hypergraph are those which are canonical double covers.", "The Realisability Problem restricted to bipartite graphs therefore becomes: given a bipartite graph $G$ , is there a graph $K$ such that $G$ is the canonical double cover of $K$ ?", "A result in this direction was proved in [14], where graphs whose canonical double covers are Cayley graphs are characterised So it seems that the TF-isomorphism point of view can give a new handle on some of these problems.", "We intend to pursue this line of research in a forthcoming work." ], [ "Acknowledgement", "We are grateful to M. Muzychuk for first pointing out to us the usefulness of considering TF-isomorphisms as isomorphisms between incidence structures." ], [ "TF-isomorphism and alternating trails", "We shall consider isomorphisms and TF-isomorphisms between pairs of mixed graphs, that is, we shall allow loops, directed arcs and edges.", "Configurations conserved by TF-isomorphisms must also be conserved by isomorphisms since the latter are just a special case of the former.", "However, the converse does not necessarily hold.", "It is well known that loops, paths and cycles are all conserved by isomorphisms, but it is easy to see that they are not necessarily conserved by TF-isomorphisms.", "For example, an arc $(u,v)$ can be mapped into a loop by a TF-isomorphism $(\\alpha ,\\beta )$ if $\\alpha (u)=\\beta (u)$ .", "An isomorphism conserves degrees, in-degrees and out-degrees.", "In the case of TF-isomorphisms, the situation is slightly more elaborate.", "First note that $\\alpha $ must conserve the out-degree of each vertex but not the in-degree.", "Likewise, $\\beta $ must conserve the in-degree of each vertex but not the out-degree.", "Therefore, if some vertex $u \\in $ $V(G)$ is a source, $\\alpha (u)$ might not be a source but it is certainly not a sink.", "If $u \\in $ $V(G)$ is a sink, then $\\alpha (u)$ must also be a sink.", "An analogous argument holds for $\\beta $ .", "Hence in the case of a digraph whose vertex set consists only of sources and sinks, $\\alpha $ and $\\beta $ must take sources to sources and sinks to sinks.", "Also, if $G$ and $H$ are graphs and $(\\alpha ,\\beta )$ is a TF-isomorphism from $G$ to some graph $H$ , then $\\alpha $ , $\\beta $ must preserve the degree $\\rho (v)$ of any vertex $v \\in $ $V(G)$ since for every vertex $v$ in a graph, $\\rho (v) = \\rho _{in}(v) = \\rho _{out}(v)$ .", "The definitions of the term path found in the literature tacitly imply a specific direction from one vertex to the subsequent vertex in a sequence.", "For example, if $u ,v, w$ is a path in graph $G$ and $\\alpha $ is an isomorphism from $G$ to $H$ , then the arcs $(u,v)$ and $(v,w)$ are mapped into the arcs $(\\alpha (u),\\alpha (v))$ and $(\\alpha (v),\\alpha (w))$ in $H$ , with the common vertex $\\alpha (v)$ .", "But, if $(\\alpha ,\\beta )$ is a TF-isomorphism from $G$ to $H$ then the arc $(u,v)$ is mapped into $(\\alpha (u),\\beta (v))$ and it is the arc $(v,w)$ which is mapped into $(\\alpha (w),\\beta (v))$ containing the common vertex $\\beta (v)$ with the previous arc.", "That is, to obtain a common vertex between images of successive arcs, we need to alternate the directions in the original path as $(u,v)$ , $(w,v)$ .", "This motivates our definition of A-trails and indicates the trend of our next results which show what type of A-trails are conserved by TF-isomorphisms.", "Proposition 4.1 Let $G$ and $G^{\\prime }$ be mixed graphs and $P$ be an A-trail in $G$ .", "Let $(\\alpha ,\\beta )$ be any TF-isomorphism from $G$ to $G^{\\prime }$ .", "Then there exists an A-trail $P^{\\prime }$ in $G^{\\prime }$ such that $(\\alpha ,\\beta )$ restricted to $P$ maps $P$ to $P^{\\prime }$ .", "For an A-trail consisting of just one arc, the result is trivial.", "Let us therefore consider an A-trail consisting of $k$ arcs with $k\\ge 2$ .", "Let the start vertex of a given A-trail $P$ be $x_{0}$ and label the successive vertices by $x_{1}, \\dots ,\\ x_{k}$ .", "Assume without loss of generality that $x_{0}$ is the tail of $a_{1}$ .", "The TF-isomorphism maps the arc $a_{1}= (x_{0},x_{1})$ into the arc $a_{1}^{\\prime }=(\\alpha (x_{0}),\\beta (x_{1}))$ .", "The next arc in $P$ is $a_{2}=(x_{2},x_{1})$ which is mapped by the TF-isomorphism to the arc $a_{2}^{\\prime }=(\\alpha (x_{2}),\\beta (x_{1}))$ with $\\beta (x_{1})$ as a common vertex with $a_{1}^{\\prime }$ .", "By repeating the process until all arcs of $P$ have been included, we obtain an A-trail $P^{\\prime }$ of $G^{\\prime }$ .", "Then, by restricting the action of the pair $(\\alpha ,\\beta )$ to $P$ we obtain $P^{\\prime }$ as its image.", "$\\Box $ This proposition immediately gives the following corollary.", "Corollary 4.2 If $G$ is a A-connected mixed graph which is TF-isomorphic to $H$ , then $H$ is also A-connected.", "$\\Box $ Proposition REF implies that $Z$ -trails are invariant under the action of a TF-isomorphism.", "The following remarks are aimed to present a clearer picture to the reader.", "Recall that in an A-trail vertices may be repeated so that different alternating trails such as the A-trails $P$ and $P^{\\prime }$ described in Proposition REF , when taken as digraphs in their own right, may not necessarily be TF-isomorphic.", "This is illustrated in Figure REF .", "Figure: GG and G ' G^{\\prime } are TF-isomorphic digraphs but PP and P ' P^{\\prime } are not.For $G$ and $G^{\\prime }$ as in Figure REF , let $\\alpha $ map 5, 7, 3 into $5^{\\prime }$ , $3^{\\prime }$ , $7^{\\prime }$ respectively and let it map arbitrarily the rest of the vertices of $G$ to the rest of the vertices of $G^{\\prime }$ .", "Let $\\beta $ map 6, 5, 1, 4, 2 into $4^{\\prime }$ , $2^{\\prime }$ , $1^{\\prime }$ , $6^{\\prime }$ , $5^{\\prime }$ respectively and let it map the rest of the vertices of $G$ to the rest of the vertices of $G^{\\prime }$ .", "The maps $\\alpha $ and $\\beta $ may be represented as shown below where the entries labelled by $$ may be replaced arbitrarily but without repetitions by any of the vertices to which there is no defined mapping.", "$ {\\alpha = \\left( \\begin{tabular}{ccccccc}1&2&3&4&5&6&7\\\\ \\ && 7^{\\prime }&&5^{\\prime }&&3^{\\prime } \\end{tabular} \\right)}\\qquad {\\beta = \\left( \\begin{tabular}{ccccccc}1&2&3&4&5&6&7\\\\1^{\\prime }&5^{\\prime }&&6^{\\prime }&2^{\\prime }&4^{\\prime }&* \\end{tabular} \\right)} $ The pair $(\\alpha ,\\beta )$ is then a TF-isomorphism from $G$ to $G^{\\prime }$ .", "However, the alternating trails $P$ and $P^{\\prime }$ in Figure REF are not TF-isomorphic digraphs.", "On the other hand, as stated in Proposition REF , any A-trail of a given graph, mixed graph or digraph $G$ is mapped by a TF-isomorphism to some A-trail of graph $G^{\\prime }$ whenever $G$ and $G^{\\prime }$ are TF-isomorphic.", "This is also the case of the trails $P$ and $P^{\\prime }$ in Figure REF .", "In fact it is easy to check that the open trail $P$ is mapped to the semi-closed trail $P^{\\prime }$ by the pair $(\\alpha ,\\beta )$ as defined above.", "However $P$ and $P^{\\prime }$ are not TF-isomorphic.", "Proposition 4.3 Let $G$ and $H$ be mixed graphs.", "Then a TF-isomorphism $(\\alpha ,\\beta )$ from $G$ to $H$ takes closed A-trails of $G$ to closed A-trails of $H$ .", "A closed trail $P$ has an even number of arcs and so it cannot be mapped to a semi-closed trail.", "Besides, if $P$ were mapped to an open trail, then $\\alpha $ or $\\beta $ must map some vertex of $P$ to both the first vertex and last vertex of the open A-trail, which is a contradiction since $\\alpha $ and $\\beta $ are bijections.", "As regards the latter case, note that a semi-closed A-trail has an odd number of arcs and a closed A-trail has an even number of arcs.", "$\\Box $ Therefore, closed A-trails are preserved by TF-isomorphisms just as they are by isomorphisms, but the situation is different for open and semi-closed A-trails.", "Proposition 4.4 Let $G$ and $H$ be A-connected mixed graphs.", "Then any non-trivial TF-isomorphism $(\\alpha ,\\beta )$ from $G$ to $H$ takes at least one open A-trail into a semi-closed A-trail and vice-versa.", "As $(\\alpha ,\\beta )$ is non-trivial, there is at least one vertex $u \\in $ $V(G)$ such that $\\alpha (u)\\ne \\beta (u)$ .", "Since both $\\alpha $ and $\\beta $ are bijections, we get $\\alpha (u)=\\beta (v)$ for some $v\\ne u$ .", "Since $G$ is Z-connected, there exists an A-trail joining $u$ and $v$ .", "Clearly $P$ is open.", "Its image $P^{\\prime }$ under $(\\alpha ,\\beta )$ is an A-trail of $H$ , that starts by $\\alpha (u)$ and ends by $\\beta (v)$ , but since they are equal, $P^{\\prime }$ is semi-closed.", "Since $(\\alpha ,\\beta )$ is a non-trivial TF-isomorphism from $G$ to $H$ , $(\\alpha ^{-1},\\beta ^{-1})$ is a non-trivial TF-isomorphism from $H$ to $G$ .", "Therefore, we may use the same arguments to show that $(\\alpha ^{-1},\\beta ^{-1})$ must take an open A-trail of $H$ to a semi-closed A-trail of $G$ .", "This implies that $(\\alpha ,\\beta )$ must take some semi-closed A-trail in $G$ to an open A-trail in $H$ .", "$\\Box $ Consider the following example.", "Let $G$ be a closed A-trail with 6 vertices and let $H$ consist of a $K_{3}$ and 3 isolated vertices.", "Note that $G$ is A-connected whereas $H$ is not.", "It is straightforward to check that $G$ and $H$ are TF-isomorphic.", "However, any TF-isomorphism from $G$ to $H$ is clearly non-trivial and maps an open A-trail of length 3 in $G$ to a semi-closed A-trail of $H$ .", "Therefore, the result of Proposition REF is false if the hypothesis, namely that both $G$ and $H$ are A-connected, is dropped.", "As an application of Proposition REF we get the following result.", "Corollary 4.5 A bipartite graph and a non-bipartite graph cannot be TF-isomorphic.", "Indeed if $G$ is bipartite and $(\\alpha ,\\beta )$ is a TF-isomorphism from $G$ to some other graph $H$ , then $\\alpha =\\beta $ .", "Let $G$ be a graph and let $(\\alpha ,\\beta )$ be a non-trivial TF-isomorphism from $G$ to some graph $H$ .", "Then, in view of Proposition REF , there is an open A-trail of $G$ that is taken to a semi-closed A-trail of $H$ .", "Therefore $H$ has an odd cycle and is non-bipartite.", "Conversely, $(\\alpha ^{-1},\\beta ^{-1})$ is a non-trivial TF-isomorphism from $H$ to $G$ and therefore, by the same argument $G$ cannot be bipartite.", "$\\Box $ The next section contains a more detailed study of how, using A-trails, a mixed graph $G$ can be made to correspond to a strongly bipartite digraphs, extending the results of Zelinka, particularly those exposed in [21].", "It will turn out that this digraph is a double cover of $G$ which we have already encountered." ], [ "Alternating double covers and an equivalence relation on arcs", "Let $G$ be any mixed graph.", "Consider the relation $R$ on the set $A(G)$ defined by: $xRy$ if and only if $x$ and $y$ are the first and last arcs of an A-trail of $G$ .", "Clearly $xRx$ since any given arc is the first and also the last arc of an A-trail containing only one arc.", "If $xRy$ then $yRx$ since if $x$ is the first arc of an A-trail, then $y$ is the last arc and vice-versa.", "Now suppose that $xRy$ and $yRz$ .", "If $x$ is the first arc of an A-trail $P$ , then $y$ is the last arc of the $P$ .", "Then if $y$ is the first arc of an A-trail $Q$ and $z$ is the last arc of $Q$ , then the set-theoretical union of $P$ and $Q$ is an A-trail which has $x$ as first arc and $z$ as last arc.", "Lemma 5.1 Let $G$ be a connected graph.", "Then every two edges of G are joined by trails of both odd and even length if and only if $G$ is not bipartite.", "Let $G$ be non-bipartite.", "Take any two edges $e_1$ , $ e_2$ and fix an odd cycle $C$ of $G$ and two vertices $v_1$ , $v_2$ of $C$ .", "Choose two trails $P_1$ , $P_2$ joining $e_1$ , $e_2$ with $v_1$ , $v_2$ respectively.", "Let $P^{\\prime }$ and $P^{\\prime \\prime }$ the two trails that join $v_1$ and $v_2$ using the edges of $C$ .", "Then there are two trails joining $e_1$ , $e_2$ , namely $P_1,\\ P^{\\prime },\\ P_2$ and $P_1,\\ P^{\\prime \\prime },\\ P_2$ .", "One of them has odd length and the other even because if $P^{\\prime \\prime }$ is odd, $P^{\\prime }$ is even and vice versa and therefore, the inclusion of one instead of the other switches parity.", "Conversely, suppose that there are trails of odd and even length between two fixed edges.", "In the subgraph induced by these paths, the vertices cannot be partitioned in two distinct colour classes and hence this subgraph must be non-bipartite and hence must contain an odd circuit.", "Hence $G$ contains an odd circuit and is also non-bipartite.", "$\\Box $ Corollary 5.2 Let $G$ be a connected graph.", "Then $R$ has one equivalence class if $G$ is not bipartite and two if it is bipartite.", "Suppose that $G$ is non-bipartite.", "Consider two arcs $x_1$ and $x_2$ and take the corresponding edges as $e_1$ and $e_2$ .", "Start from $x_1$ .", "On each edge of trails joining $e_1$ and $e_2$ , choose the arc to obtain an A-trail.", "Note that before this process may continue for all edges except at most $e_2$ .", "When $e_2$ is reached, the arc corresponding to edge incident with $e_2$ may form an A-trail of order 2 or a directed path.", "But this depends on whether the concerned trail has odd or even length, so one of the two will give a whole A-trail containing both $x$ and $y$ .", "On the other hand, if $G$ is bipartite, given that $x$ and $y$ form a directed path, which always happens, then every trail joining $x$ and $y$ will be of even length; but an A-trail of even length is open and can't allow directed paths.", "$\\Box $ Each equivalence class of $R$ is a set, to which one can naturally associate an A-connected sub-digraph, whose arcs are the elements of the class and whose vertices are those incident to at least one of such arcs.", "In general, the relation $R$ may yield any number of classes, not just one or two as in the case of graphs, as we shall see in Theorem REF below.", "If $v$ is any vertex, two different arcs that have $v$ as a first vertex form an A-trail; the same can be said for two different arcs having $v$ as a head.", "Therefore, the arcs incident with $v$ belong to only one class or two.", "In the latter case, we say that $v$ is a frontier vertex.", "Let $F(G)$ be the set of all frontier vertices of $G$ .", "In view of Corollary REF , if $G$ is a graph then $F(G)$ is either empty (if $G$ is not bipartite) or $F(G)$ $=$ $V$$(G)$ (if $G$ is bipartite).", "The proof of the next result is straightforward.", "Proposition 5.3 Let $G$ be a connected mixed graph.", "The following are equivalent: (i) All classes of $R$ are singletons.", "(ii) All A-trails of $G$ are singletons.", "(iii) Each vertex of $G$ has both in-degree and out-degree less than or equal to 1.", "(iv) $G$ is a directed path or a directed cycle.", "$\\Box $ Proposition 5.4 Let $G$ be a connected mixed graph.", "Then $R$ has only one class if and only if the set $F(G)$ is empty.", "The condition is clearly necessary, for if $v$ were an element of $F(G)$ then by definition we would have at least two different classes.", "On the other hand, if there is more than one class, let $x$ and $y$ be arcs that belong to different classes.", "Since $G$ is connected, there is a trail $P$ that joins $x$ and $y$ .", "Somewhere in $P$ there must be $x^{\\prime }$ and $y^{\\prime }$ that belong to different classes and are incident with a vertex $v$ .", "Thus $v \\in F(G)$ .", "$\\Box $ Proposition 5.5 Let $G$ be a connected mixed graph.", "Then $F(G)$ is empty or $F(G)$ $=$ $V$$(G)$ or $F(G)$ is a disconnected set of the underlying graph.", "We can assume that $F(G)$ is a proper subset of $V$$(G)$ .", "By Proposition REF , there are at least two classes for $R$ .", "Letting $x$ , $y$ be elements of different classes, by the same argument as in Proposition REF we infer that each trial joining $x$ and $y$ must pass through a vertex $v\\in F(G)$ .", "Therefore, removing $F(G)$ the arcs $x$ and $y$ end up in different connected components.", "$\\Box $ Theorem 5.6 For every pair $(m,k)$ of positive integers, there exists a mixed graph on which the equivalence relation $R$ induces $m$ classes and having $k$ frontier vertices if and only if $m-1\\le k$ .", "Let us first construct a mixed graph with $m$ classes and $k$ frontier vertices whenever $m-1 \\le k$ .", "Note that if $m-1=k$ a directed path satisfies the statement (each class consists of a single arc).", "The same holds for $m-2=k$ and a directed cycle.", "Assume then that $m-3\\le k$ .", "Consider the 4-set $\\lbrace a,b,c,d\\rbrace $ and consider $m-2$ mixed graphs $H_i$ for $i=1,\\ \\dots ,m-2$ where $V$$(H_i)$ $=$ $\\lbrace (a,i),(b,i),(c,i),(d,i),(a,i+1)\\rbrace $ and $A(H_i)$ contains all the arcs of the triangle $(b,i)$ , $(c,i)$ , $(d,i)$ , plus the additional arcs $((a,i),(b,i))$ and $((d,i),(a,i+1))$ .", "Take any connected bipartite graph $K$ with $k-m+2$ vertices and fix a vertex $u$ of $K$ .", "Let $L$ be the digraph consisting of the single arc $(u,(a,1))$ .", "Let $G$ be the (standard graph-theoretical) union of $K$ , $L$ , $H_1,\\ \\dots , H_{m-2}$ .", "Then $G$ is a connected mixed graph.", "The classes for $R$ in $G$ are: (i) the class of $K$ containing the arcs incident to $u$ ; (ii) the class of $K$ containing the arcs incident from $u$ , together with the extra arc $(u(a,1))$ ; (iii) each of the $H_i$ 's for $i=1,\\dots ,\\ m-2$ .", "Hence, their number is $m$ .", "Moreover, $F(G)=V(K) \\cup \\lbrace (a,1),(a,2),...,(a,m-2)\\rbrace $ , then $\|F(G)\|=(k-m+2)+(m-2)=k$ .", "Therefore, for all cases where the stated inequality holds, there is a mixed graph $G$ as claimed.", "Conversely, consider now any mixed graph $G$ and define a graph $X$ such that $V$$(X)$ $=$ $V$$(G)/R$ (that is, the set of classes of $R$ in $G$ ) and two vertices are adjacent when the associated mixed graphs share a frontier vertex.", "Then $m=\|X\|$ , while the number $k^{\\prime }$ of edges of $X$ is less or equal to $k=\|F(G)\|$ (because two classes might share more than a frontier vertex).", "The known inequality $m-1\\le k^{\\prime }$ implies $m-1 \\le k$ as claimed.", "$\\Box $ As remarked earlier, a strongly bipartite digraph can be associated with each equivalence class of $R$ .", "Now let these strongly bipartite digraphs $D_{1}, D_{2}, \\dots , D_{k}$ corresponding to the different classes of $R$ obtained from the mixed graph $G$ .", "Let any vertex $u$ of $V(G)$ which appears as a source in $D_{i}$ be labelled $u_{0}$ and let any vertex $v$ of $V(G)$ which appears as a sink in $D_{i}$ be labelled $v_{1}$ .", "Therefore, an arc $(u,v)$ in $D_{i}$ now becomes $(u_{0},v_{1})$ .", "It turns out that the strongly bipartite digraph consisting of the components $D_{i}$ labelled this way is ADC$(G)$ which we have already defined earlier.", "Figure REF shows an example which may be used to illustrate the following remarks which highlight certain properties of $\\mbox{\\textbf {ADC}}(G)$ in relation to the mixed graph $G$ : 1.", "We know that, for any vertex $u$ of $G$ , all incoming arcs $(x,u)$ of $G$ are in the same component of ADC$(G)$ and similarly all outgoing arcs $(u,x)$ of $G$ are in the same component of ADC$(G)$ .", "Therefore $u_0$ if present in ADC$(G)$ , cannot appear in two different components.", "Similarly for $u_1$ .", "However, as we see in examples below, $u_0,$ and $u_1$ can, in some cases, appear in the same component and they can, in other cases, appear in different components.", "In particular, if $G$ is a bipartite graph they appear in different components as shown in Figure REF $(ii)$ and if $G$ is a non-bipartite graph, they are in the same component as shown in Figure REF $(ii)$ .", "2.", "ADC$(G)$ is a strongly bipartite digraph.", "3.", "By definition, there is no $u_0$ in ADC($G$ ) if $u$ is a sink in $G$ , and there is no $u_1$ if $u$ is a source.", "Figure: ADC(G)\\mbox{\\textbf {ADC}}(G) obtained from a digraph GG." ], [ "TF-isomorphisms and mixed graph covers", "The following result can be seen as a corollary to Theorem REF and the proof is easy since the IDC of a mixed graph $G$ can be obtained from $\\mbox{\\textbf {ADC}}(G)$ simply by removing the directions of the arcs and isolated vertices are irrelevant.", "Here we give an direct proof because it will help us in later constructions.", "Theorem 6.1 Let $G$ , $H$ be mixed graphs.", "The $G$ and $H$ are TF-isomorphic if and only if $\\mbox{\\textbf {\\emph {ADC}}}(G)$ and $\\mbox{\\textbf {\\emph {ADC}}}(H)$ are isomorphic.", "Let $(\\alpha ,\\beta )$ be a TF-isomorphism from $G$ to $H$ .", "Let $(u,v)$ be an arc of $G$ .", "First note that if $(\\alpha (u),\\beta (v))$ is an arc of $H$ , then $(u_{0},v_{1})$ is an arc of $\\mbox{\\textbf {ADC}}(G)$ and $(\\alpha (u)_{0},\\beta (u)_{1})$ is an arc of $\\mbox{\\textbf {ADC}}(H)$ .", "Let $f$ be a map from $V$$(\\mbox{\\textbf {ADC}}(G))$ to $V$$(\\mbox{\\textbf {ADC}}(H))$ such that $f: u_{0} \\mapsto x_{0}$ if $x = \\alpha (u)$ and $f: v_{1} \\mapsto y_{1}$ if $y =\\beta (v)$ .", "Consider any arc $(u,v)$ of $G$ and consider the corresponding arc $(u_{0},v_{1})$ in $A(\\mbox{\\textbf {ADC}}(G))$ .", "Let $(\\alpha ,\\beta )(u,v) = (x,y)$ .", "Then by definition $f$ takes $(u_{0},v_{1})$ to $(x_{0},y_{1})$ in $A(\\mbox{\\textbf {ADC}}(H))$ .", "The function $f$ maps arcs of $\\mbox{\\textbf {ADC}}(G)$ to arcs of $\\mbox{\\textbf {ADC}}(H)$ and it is clearly bijective.", "Hence, $f$ is an isomorphism from $\\mbox{\\textbf {ADC}}(G)$ to $\\mbox{\\textbf {ADC}}(H)$ .", "Now suppose that $\\mbox{\\textbf {ADC}}(G)$ and $\\mbox{\\textbf {ADC}}(H)$ are isomorphic.", "This implies that there exists a map $f$ such that $f(u_{0},v_{1})$ $=$ $(x_{0},y_{1})$ .", "Note that the arcs must always start from a vertex whose label has 0 as subscript and incident to a vertex whose label has 1 as subscript, by virtue of the construction presented above.", "Define $\\alpha $ , $\\beta $ from $V$$(G)$ to $V$$(H)$ as follows.", "Let $\\alpha (u) = x$ if $f(u_{0}) = x_{0}$ where $u \\in $ $V$$(G)$ and $x \\in $ $V$$(H)$ and let $\\beta (v) = y$ if $f(v_{1}) = y_{1}$ where $v \\in $ $V$$(G)$ and $y \\in $ $V$$(H)$ .", "Then $(\\alpha ,\\beta )$ takes any arc $(u,v) \\in $ $A(G)$ to some $(x,y)$ in $A(H)$ .", "This two-fold mapping is bijective and hence $(\\alpha ,\\beta )$ is a TF-isomorphism from $G$ to $H$ .", "$\\Box $ Corollary 6.2 Let $(\\alpha ,\\beta )$ be a TF-isomorphism from a mixed graph $G$ to a mixed graph $H$ .", "Then there exists an isomorphism $f_{\\alpha ,\\beta }$ from $\\mbox{\\textbf {\\emph {ADC}}}(G)$ to $\\mbox{\\textbf {\\emph {ADC}}}(H)$ such that $f_{\\alpha ,\\beta }(u_{0},v_{1}) = (x_{0},y_{1})$ if and only if $x = \\alpha (u)$ and $y =\\beta (v)$ for some TF-isomorphism $(\\alpha ,\\beta )$ from $G$ to $H$ .", "The result follows from the proof of Theorem REF .", "$\\Box $ Refer to Figure REF .", "An isomorphism $f$ from $\\mbox{\\textbf {ADC}}(G)$ to $\\mbox{\\textbf {ADC}}(H)$ and the corresponding maps $\\alpha $ and $\\beta $ from V$(G)$ onto V$(H)$ , which are derived from $f$ as described in the proof of Theorem REF , are given below.", "$f: 1_{0} \\mapsto 1_{0}^{\\prime } & \\ & f: 1_{1} \\mapsto 1_{1}^{\\prime }\\\\f: 2_{1} \\mapsto 2_{0}^{\\prime } & \\ & f: 2_{1} \\mapsto 3_{1}^{\\prime }\\\\f: 3_{0} \\mapsto 3_{0}^{\\prime } & \\ & f: 3_{1} \\mapsto 2_{1}^{\\prime }\\\\f: 4_{0} \\mapsto 6_{0}^{\\prime } &\\ & f: 4_{1} \\mapsto 5_{1}^{\\prime }\\\\f: 5_{0} \\mapsto 7_{0}^{\\prime } &\\ & f: 5_{1} \\mapsto 4_{1}^{\\prime }\\\\f: 6_{0} \\mapsto 5_{0}^{\\prime } & \\ & f: 6_{1} \\mapsto 6_{1}^{\\prime }\\\\f: 7_{0} \\mapsto 4_{0}^{\\prime } & \\ & f: 7_{1} \\mapsto 7_{1}^{\\prime }$ $\\alpha : 1 \\mapsto 1^{\\prime } & \\ & \\beta : 1 \\mapsto 1^{\\prime }\\\\\\alpha : 2 \\mapsto 2^{\\prime } & \\ & \\beta : 2 \\mapsto 3^{\\prime }\\\\\\alpha : 3 \\mapsto 3^{\\prime } & \\ & \\beta : 3 \\mapsto 2^{\\prime }\\\\\\alpha : 4 \\mapsto 6^{\\prime } &\\ & \\beta : 4 \\mapsto 5^{\\prime }\\\\\\alpha : 5 \\mapsto 7^{\\prime } &\\ & \\beta : 5 \\mapsto 4^{\\prime }\\\\\\alpha : 6 \\mapsto 5^{\\prime } & \\ & \\beta : 6 \\mapsto 6^{\\prime }\\\\\\alpha : 7 \\mapsto 4^{\\prime } & \\ & \\beta : 7 \\mapsto 7^{\\prime }$ Figure REF shows a digraph $G$ and its alternating double cover $\\mbox{\\textbf {ADC}}(G)$ which in this case has three components, namely $D_{1}$ , $D_{2}$ and $D_{3}$ .", "Figure REF also shows how the components of $\\mbox{\\textbf {ADC}}(G)$ can be combined by associating vertices of the form $u_{0}$ with vertices of the form $v_{1}$ , irrespective of whether $u=v$ or $u \\ne v$ , to form $G$ or other digraphs such as $G_{1}$ , $G_{2}$ and $G_{3}$ having the same number of vertices as $G$ .", "It is easy to check that $G$ , $G_{1}$ , $G_{2}$ and $G_{3}$ are pairwise two-fold isomorphic as expected from the result of Theorem REF since each of these digraphs have the same number of vertices and have isomorphic ADCs.", "Proposition 6.3 (i) A digraph $H$ is isomorphic to $\\mbox{\\emph {\\textbf {ADC}}}(G)$ for some $G$ if and only if $H$ is strongly bipartite.", "(ii) For every digraph $G$ , ${\\mbox{\\emph {\\textbf {ADC}}}}(\\mbox{\\emph {\\textbf {ADC}}}(G))$ is isomorphic to $\\mbox{\\emph {\\textbf {ADC}}}(G)$ .", "We already know that the condition stated in $(i)$ is necessary in order to have $H$ isomorphic to some $\\mbox{\\textbf {ADC}}(G)$ .", "Conversely, if $H$ has this property, define map $f:$ $V(H)$ $\\rightarrow V (\\mbox{\\textbf {ADC}}(H))$ as follows: $f(u) = u_{0}$ if $u$ is a source, $f(u) = u_{1}$ if $u$ is a sink.", "Clearly $f$ is a bijection If $(u,v)$ is an arc of $H$ then by our assumption $u$ is a source and $v$ is a sink of $H$ .", "Then $(u_{0},v_{1})$ $=$ $(f(u),f(v))$ is an arc of $\\mbox{\\textbf {ADC}}(H)$ .", "Likewise, each arc of $\\mbox{\\textbf {ADC}}(H)$ takes the form $(u_{0},v_{1})$ , with $u$ source and $v$ sink of $H$ and hence $(u_{0},v_{1})$ is the image of $(u,v)$ under $f$ .", "This proves that $f$ is an isomorphism from $H$ to $\\mbox{\\textbf {ADC}}(H)$ , so $(i)$ is satisfied with $G=H$ .", "Now $(ii)$ is a straightforward consequence of $(i)$ , taking $H = \\mbox{\\textbf {ADC}}(G)$ .", "$\\Box $ Figure: GG and HH are TF-isomorphic graphs and have isomorphic ADCs." ], [ "Two-fold orbitals", "Let $\\mathbf {\\Gamma } \\le \\mathcal {S} = S_{\\mbox{\\scriptsize {$\|V\|$}}} \\times S_{\\mbox{\\scriptsize {$\|V\|$}}}$ .", "For a fixed element $(u,v)$ of $V\\times V$ let $ \\mathbf {\\Gamma }(u,v) = \\lbrace (\\alpha (u),\\beta (v)\\ \|\\ (\\alpha ,\\beta ) \\in \\mathbf {\\Gamma }\\rbrace .", "$ The set $\\mathbf {\\Gamma }(u,v)$ is called a two-fold orbital or TF-orbital.", "A two-fold orbital is the set of arcs of a digraph $G$ having vertex set $V$ which we call two-fold orbital digraph or TF-orbital digraph.", "If for every arc $(x,y)$ in $\\mathbf {\\Gamma }(u,v)$ , the oppositely directed arc $(y,x)$ is also contained in $\\mathbf {\\Gamma }(u,v)$ , then $G$ is a two-fold orbital graph or TF-orbital graph.", "This generalisation of the well-known concept of orbital (di)graph has been discussed in [9].", "Proposition 7.1 Let $G$ be a strongly bipartite digraph.", "Then (i) There is a homomorphism $\\psi $ of Aut$^{\\mbox{\\tiny \\textbf {{TF}}}}(G)$ onto Aut$(G)$ .", "(ii) If $G$ is a TF-orbital digraph, then it is also an orbital digraph.", "If $(\\alpha ,\\beta )$ is a TF-automorphism of $G$ , define $\\psi (\\alpha ,\\beta )$ $=$ $f:$ $V(G) \\rightarrow $ $V(G)$ as follows: $f(u) = \\alpha (u)$ if $u$ is a source and $f(u)=\\beta (u)$ if $u$ is a sink.", "Since $\\alpha $ preserves sources then $f$ takes sources to sources.", "Similarly, since $\\beta $ preserves sinks, then $f$ takes sinks to sinks.", "Since both $\\alpha $ and $\\beta $ are permutations, the restrictions of $f$ to the set of sources and to the set of sinks are also permutations.", "Hence $f$ is a permutation of V$(G)$ .", "Given any arc $(u,v)$ of $G$ , note that $(\\alpha ,\\beta )$ takes $(u,v)$ to $(\\alpha (u),\\beta (v))$ , which is equal to $(f(u),f(v))$ because $u$ is a source and $v$ is a sink.", "Hence $f$ is an automorphism of $G$ .", "so $\\psi $ maps $\\mbox{Aut}^{\\mbox{{\\tiny {\\textbf {TF}}}}}(G)$$\\ $ to Aut$(G)$ .", "A direct computation proves that $\\psi $ is a group homomorphism, hence $(i)$ holds.", "Assume now that $G = \\mathbf {\\Gamma }(u,v)$ for some $\\mathbf {\\Gamma }$ .", "Then $\\mathbf {\\Gamma }$ is a subgroup of $\\mbox{Aut}^{\\mbox{{\\tiny {\\textbf {TF}}}}}(G)$$\\ $ and $\\psi (\\mathbf {\\Gamma })$ is a subgroup of Aut $G$ .", "Each arc of $G$ takes the form $(\\alpha (u),\\beta (v))$ , where $(\\alpha ,\\beta )$ $\\in $ $\\mathbf {\\Gamma }$ and $u$ , $v$ are a source and a sink respectively.", "Letting $f = \\psi (\\alpha ,\\beta )$ this arc is $(f(u),f(v))$ , so it belongs to the orbital digraph $\\psi (\\mathbf {\\Gamma })(u,v)$ .", "This proves that $G$ is contained in this orbital digraph.", "The opposite inclusion can be shown the same way, so that $G = \\psi (\\Gamma )(u,v)$ and $(ii)$ follows.", "$\\Box $ Corollary 7.2 Let $G$ be a strongly bipartite digraph.", "Then $G$ is a two-fold orbital digraph if and only if ${\\emph {\\mbox{\\textbf {CDC}}(G)}}$ is an orbital digraph.", "By Proposition REF , $G$ and $\\mbox{\\textbf {ADC}}(G)$ are isomorphic.", "If either of them is a TF-orbital, then of course the same holds for the other one, but by Proposition REF in this case these TF-orbitals are both orbitals.", "$\\Box $" ], [ "Conclusion", "We believe that TF-isomorphisms is a relatively new concept.", "The only other author who considered them was Zelinka in a short paper motivated by the concept of isotopy in semigroups [22], [23].", "Our papers ([9] and [10]) are the first attempts at a systematic study of TF-isomorphisms.", "In this paper we have shown close links between TF-isomorphisms and double covers, and how the decomposition of a particular double cover can be used to obtain TF-isomorphic graphs.", "We have also seen that TF-isomorphisms give a new angle for looking at some older problems in graph theory.", "But does the notion of TF-isomorphism add anything new to these older questions?", "We believe that it does.", "For example, in [11] we prove this result which explains instability of graphs in terms of TF-automorphisms.", "Theorem 8.1 Let Aut$^{\\mbox{\\tiny \\textbf {{TF}}}}(G)$ be the group of TF-automorphisms of a mixed graph $G$ .", "Then Aut$({\\emph {\\mbox{\\textbf {CDC}}(G)}})$ is isomorphic to the semi-direct product Aut$^{\\mbox{\\tiny \\textbf {{TF}}}}(G)\\rtimes \\mathbb {Z}_2$ .", "Therefore $G$ is unstable if and only if it has a non-trivial TF-automorphism.", "$\\Box $ Also, it is not very likely that looking at these questions without the notion of TF-isomorphisms would lead one to the notion of A-trails, a technique which we feel is very useful, or the construction of asymmetric graphs with a non-trivial TF-isomorphism, an interesting notion which would be not so natural to formulate using only matrix methods, say.", "Some results and proofs are clearer in the TF-isomorphism setting.", "For example, in some of the papers cited we find this result about graph reconstruction from neighbourhoods.", "Theorem 8.2 ([1]) If $G$ is a connected bipartite graph, then any nonisomorphic graph $H$ with the same neighbourhood family as $G$ must be a disconnected graph with two components which themselves have identical neighbourhood hypergraphs.", "$\\Box $ From the TF-isomorphism point of view, this result follows from three very basic facts: (i) two graphs have the same neighbourhood family (equivalent to being TF-isomorphic) if and only if they have the same canonical double cover; (ii) the canonical double cover of a graph $G$ is disconnected if and only if $G$ is bipartite; and (iii) when $G$ is bipartite, the canonical double cover of $G$ is simply two disjoint copies of $G$ .", "Therefore, for $H$ to have the same canonical double cover as $G$ , it must consist of two components isomorphic to $K$ , where $G$ is the canonical double cover of $K$ .", "This gives Theorem REF .", "And moreover, from these remarks we also see that the only bipartite graphs for which there are non-isomorphic graphs with the same neighbourhood hypergraph are those which are canonical double covers.", "The Realisability Problem restricted to bipartite graphs therefore becomes: given a bipartite graph $G$ , is there a graph $K$ such that $G$ is the canonical double cover of $K$ ?", "A result in this direction was proved in [14], where graphs whose canonical double covers are Cayley graphs are characterised So it seems that the TF-isomorphism point of view can give a new handle on some of these problems.", "We intend to pursue this line of research in a forthcoming work." ], [ "Acknowledgement", "We are grateful to M. Muzychuk for first pointing out to us the usefulness of considering TF-isomorphisms as isomorphisms between incidence structures." ], [ "Alternating double covers and an equivalence relation on arcs", "Let $G$ be any mixed graph.", "Consider the relation $R$ on the set $A(G)$ defined by: $xRy$ if and only if $x$ and $y$ are the first and last arcs of an A-trail of $G$ .", "Clearly $xRx$ since any given arc is the first and also the last arc of an A-trail containing only one arc.", "If $xRy$ then $yRx$ since if $x$ is the first arc of an A-trail, then $y$ is the last arc and vice-versa.", "Now suppose that $xRy$ and $yRz$ .", "If $x$ is the first arc of an A-trail $P$ , then $y$ is the last arc of the $P$ .", "Then if $y$ is the first arc of an A-trail $Q$ and $z$ is the last arc of $Q$ , then the set-theoretical union of $P$ and $Q$ is an A-trail which has $x$ as first arc and $z$ as last arc.", "Lemma 5.1 Let $G$ be a connected graph.", "Then every two edges of G are joined by trails of both odd and even length if and only if $G$ is not bipartite.", "Let $G$ be non-bipartite.", "Take any two edges $e_1$ , $ e_2$ and fix an odd cycle $C$ of $G$ and two vertices $v_1$ , $v_2$ of $C$ .", "Choose two trails $P_1$ , $P_2$ joining $e_1$ , $e_2$ with $v_1$ , $v_2$ respectively.", "Let $P^{\\prime }$ and $P^{\\prime \\prime }$ the two trails that join $v_1$ and $v_2$ using the edges of $C$ .", "Then there are two trails joining $e_1$ , $e_2$ , namely $P_1,\\ P^{\\prime },\\ P_2$ and $P_1,\\ P^{\\prime \\prime },\\ P_2$ .", "One of them has odd length and the other even because if $P^{\\prime \\prime }$ is odd, $P^{\\prime }$ is even and vice versa and therefore, the inclusion of one instead of the other switches parity.", "Conversely, suppose that there are trails of odd and even length between two fixed edges.", "In the subgraph induced by these paths, the vertices cannot be partitioned in two distinct colour classes and hence this subgraph must be non-bipartite and hence must contain an odd circuit.", "Hence $G$ contains an odd circuit and is also non-bipartite.", "$\\Box $ Corollary 5.2 Let $G$ be a connected graph.", "Then $R$ has one equivalence class if $G$ is not bipartite and two if it is bipartite.", "Suppose that $G$ is non-bipartite.", "Consider two arcs $x_1$ and $x_2$ and take the corresponding edges as $e_1$ and $e_2$ .", "Start from $x_1$ .", "On each edge of trails joining $e_1$ and $e_2$ , choose the arc to obtain an A-trail.", "Note that before this process may continue for all edges except at most $e_2$ .", "When $e_2$ is reached, the arc corresponding to edge incident with $e_2$ may form an A-trail of order 2 or a directed path.", "But this depends on whether the concerned trail has odd or even length, so one of the two will give a whole A-trail containing both $x$ and $y$ .", "On the other hand, if $G$ is bipartite, given that $x$ and $y$ form a directed path, which always happens, then every trail joining $x$ and $y$ will be of even length; but an A-trail of even length is open and can't allow directed paths.", "$\\Box $ Each equivalence class of $R$ is a set, to which one can naturally associate an A-connected sub-digraph, whose arcs are the elements of the class and whose vertices are those incident to at least one of such arcs.", "In general, the relation $R$ may yield any number of classes, not just one or two as in the case of graphs, as we shall see in Theorem REF below.", "If $v$ is any vertex, two different arcs that have $v$ as a first vertex form an A-trail; the same can be said for two different arcs having $v$ as a head.", "Therefore, the arcs incident with $v$ belong to only one class or two.", "In the latter case, we say that $v$ is a frontier vertex.", "Let $F(G)$ be the set of all frontier vertices of $G$ .", "In view of Corollary REF , if $G$ is a graph then $F(G)$ is either empty (if $G$ is not bipartite) or $F(G)$ $=$ $V$$(G)$ (if $G$ is bipartite).", "The proof of the next result is straightforward.", "Proposition 5.3 Let $G$ be a connected mixed graph.", "The following are equivalent: (i) All classes of $R$ are singletons.", "(ii) All A-trails of $G$ are singletons.", "(iii) Each vertex of $G$ has both in-degree and out-degree less than or equal to 1.", "(iv) $G$ is a directed path or a directed cycle.", "$\\Box $ Proposition 5.4 Let $G$ be a connected mixed graph.", "Then $R$ has only one class if and only if the set $F(G)$ is empty.", "The condition is clearly necessary, for if $v$ were an element of $F(G)$ then by definition we would have at least two different classes.", "On the other hand, if there is more than one class, let $x$ and $y$ be arcs that belong to different classes.", "Since $G$ is connected, there is a trail $P$ that joins $x$ and $y$ .", "Somewhere in $P$ there must be $x^{\\prime }$ and $y^{\\prime }$ that belong to different classes and are incident with a vertex $v$ .", "Thus $v \\in F(G)$ .", "$\\Box $ Proposition 5.5 Let $G$ be a connected mixed graph.", "Then $F(G)$ is empty or $F(G)$ $=$ $V$$(G)$ or $F(G)$ is a disconnected set of the underlying graph.", "We can assume that $F(G)$ is a proper subset of $V$$(G)$ .", "By Proposition REF , there are at least two classes for $R$ .", "Letting $x$ , $y$ be elements of different classes, by the same argument as in Proposition REF we infer that each trial joining $x$ and $y$ must pass through a vertex $v\\in F(G)$ .", "Therefore, removing $F(G)$ the arcs $x$ and $y$ end up in different connected components.", "$\\Box $ Theorem 5.6 For every pair $(m,k)$ of positive integers, there exists a mixed graph on which the equivalence relation $R$ induces $m$ classes and having $k$ frontier vertices if and only if $m-1\\le k$ .", "Let us first construct a mixed graph with $m$ classes and $k$ frontier vertices whenever $m-1 \\le k$ .", "Note that if $m-1=k$ a directed path satisfies the statement (each class consists of a single arc).", "The same holds for $m-2=k$ and a directed cycle.", "Assume then that $m-3\\le k$ .", "Consider the 4-set $\\lbrace a,b,c,d\\rbrace $ and consider $m-2$ mixed graphs $H_i$ for $i=1,\\ \\dots ,m-2$ where $V$$(H_i)$ $=$ $\\lbrace (a,i),(b,i),(c,i),(d,i),(a,i+1)\\rbrace $ and $A(H_i)$ contains all the arcs of the triangle $(b,i)$ , $(c,i)$ , $(d,i)$ , plus the additional arcs $((a,i),(b,i))$ and $((d,i),(a,i+1))$ .", "Take any connected bipartite graph $K$ with $k-m+2$ vertices and fix a vertex $u$ of $K$ .", "Let $L$ be the digraph consisting of the single arc $(u,(a,1))$ .", "Let $G$ be the (standard graph-theoretical) union of $K$ , $L$ , $H_1,\\ \\dots , H_{m-2}$ .", "Then $G$ is a connected mixed graph.", "The classes for $R$ in $G$ are: (i) the class of $K$ containing the arcs incident to $u$ ; (ii) the class of $K$ containing the arcs incident from $u$ , together with the extra arc $(u(a,1))$ ; (iii) each of the $H_i$ 's for $i=1,\\dots ,\\ m-2$ .", "Hence, their number is $m$ .", "Moreover, $F(G)=V(K) \\cup \\lbrace (a,1),(a,2),...,(a,m-2)\\rbrace $ , then $\|F(G)\|=(k-m+2)+(m-2)=k$ .", "Therefore, for all cases where the stated inequality holds, there is a mixed graph $G$ as claimed.", "Conversely, consider now any mixed graph $G$ and define a graph $X$ such that $V$$(X)$ $=$ $V$$(G)/R$ (that is, the set of classes of $R$ in $G$ ) and two vertices are adjacent when the associated mixed graphs share a frontier vertex.", "Then $m=\|X\|$ , while the number $k^{\\prime }$ of edges of $X$ is less or equal to $k=\|F(G)\|$ (because two classes might share more than a frontier vertex).", "The known inequality $m-1\\le k^{\\prime }$ implies $m-1 \\le k$ as claimed.", "$\\Box $ As remarked earlier, a strongly bipartite digraph can be associated with each equivalence class of $R$ .", "Now let these strongly bipartite digraphs $D_{1}, D_{2}, \\dots , D_{k}$ corresponding to the different classes of $R$ obtained from the mixed graph $G$ .", "Let any vertex $u$ of $V(G)$ which appears as a source in $D_{i}$ be labelled $u_{0}$ and let any vertex $v$ of $V(G)$ which appears as a sink in $D_{i}$ be labelled $v_{1}$ .", "Therefore, an arc $(u,v)$ in $D_{i}$ now becomes $(u_{0},v_{1})$ .", "It turns out that the strongly bipartite digraph consisting of the components $D_{i}$ labelled this way is ADC$(G)$ which we have already defined earlier.", "Figure REF shows an example which may be used to illustrate the following remarks which highlight certain properties of $\\mbox{\\textbf {ADC}}(G)$ in relation to the mixed graph $G$ : 1.", "We know that, for any vertex $u$ of $G$ , all incoming arcs $(x,u)$ of $G$ are in the same component of ADC$(G)$ and similarly all outgoing arcs $(u,x)$ of $G$ are in the same component of ADC$(G)$ .", "Therefore $u_0$ if present in ADC$(G)$ , cannot appear in two different components.", "Similarly for $u_1$ .", "However, as we see in examples below, $u_0,$ and $u_1$ can, in some cases, appear in the same component and they can, in other cases, appear in different components.", "In particular, if $G$ is a bipartite graph they appear in different components as shown in Figure REF $(ii)$ and if $G$ is a non-bipartite graph, they are in the same component as shown in Figure REF $(ii)$ .", "2.", "ADC$(G)$ is a strongly bipartite digraph.", "3.", "By definition, there is no $u_0$ in ADC($G$ ) if $u$ is a sink in $G$ , and there is no $u_1$ if $u$ is a source.", "Figure: ADC(G)\\mbox{\\textbf {ADC}}(G) obtained from a digraph GG." ], [ "TF-isomorphisms and mixed graph covers", "The following result can be seen as a corollary to Theorem REF and the proof is easy since the IDC of a mixed graph $G$ can be obtained from $\\mbox{\\textbf {ADC}}(G)$ simply by removing the directions of the arcs and isolated vertices are irrelevant.", "Here we give an direct proof because it will help us in later constructions.", "Theorem 6.1 Let $G$ , $H$ be mixed graphs.", "The $G$ and $H$ are TF-isomorphic if and only if $\\mbox{\\textbf {\\emph {ADC}}}(G)$ and $\\mbox{\\textbf {\\emph {ADC}}}(H)$ are isomorphic.", "Let $(\\alpha ,\\beta )$ be a TF-isomorphism from $G$ to $H$ .", "Let $(u,v)$ be an arc of $G$ .", "First note that if $(\\alpha (u),\\beta (v))$ is an arc of $H$ , then $(u_{0},v_{1})$ is an arc of $\\mbox{\\textbf {ADC}}(G)$ and $(\\alpha (u)_{0},\\beta (u)_{1})$ is an arc of $\\mbox{\\textbf {ADC}}(H)$ .", "Let $f$ be a map from $V$$(\\mbox{\\textbf {ADC}}(G))$ to $V$$(\\mbox{\\textbf {ADC}}(H))$ such that $f: u_{0} \\mapsto x_{0}$ if $x = \\alpha (u)$ and $f: v_{1} \\mapsto y_{1}$ if $y =\\beta (v)$ .", "Consider any arc $(u,v)$ of $G$ and consider the corresponding arc $(u_{0},v_{1})$ in $A(\\mbox{\\textbf {ADC}}(G))$ .", "Let $(\\alpha ,\\beta )(u,v) = (x,y)$ .", "Then by definition $f$ takes $(u_{0},v_{1})$ to $(x_{0},y_{1})$ in $A(\\mbox{\\textbf {ADC}}(H))$ .", "The function $f$ maps arcs of $\\mbox{\\textbf {ADC}}(G)$ to arcs of $\\mbox{\\textbf {ADC}}(H)$ and it is clearly bijective.", "Hence, $f$ is an isomorphism from $\\mbox{\\textbf {ADC}}(G)$ to $\\mbox{\\textbf {ADC}}(H)$ .", "Now suppose that $\\mbox{\\textbf {ADC}}(G)$ and $\\mbox{\\textbf {ADC}}(H)$ are isomorphic.", "This implies that there exists a map $f$ such that $f(u_{0},v_{1})$ $=$ $(x_{0},y_{1})$ .", "Note that the arcs must always start from a vertex whose label has 0 as subscript and incident to a vertex whose label has 1 as subscript, by virtue of the construction presented above.", "Define $\\alpha $ , $\\beta $ from $V$$(G)$ to $V$$(H)$ as follows.", "Let $\\alpha (u) = x$ if $f(u_{0}) = x_{0}$ where $u \\in $ $V$$(G)$ and $x \\in $ $V$$(H)$ and let $\\beta (v) = y$ if $f(v_{1}) = y_{1}$ where $v \\in $ $V$$(G)$ and $y \\in $ $V$$(H)$ .", "Then $(\\alpha ,\\beta )$ takes any arc $(u,v) \\in $ $A(G)$ to some $(x,y)$ in $A(H)$ .", "This two-fold mapping is bijective and hence $(\\alpha ,\\beta )$ is a TF-isomorphism from $G$ to $H$ .", "$\\Box $ Corollary 6.2 Let $(\\alpha ,\\beta )$ be a TF-isomorphism from a mixed graph $G$ to a mixed graph $H$ .", "Then there exists an isomorphism $f_{\\alpha ,\\beta }$ from $\\mbox{\\textbf {\\emph {ADC}}}(G)$ to $\\mbox{\\textbf {\\emph {ADC}}}(H)$ such that $f_{\\alpha ,\\beta }(u_{0},v_{1}) = (x_{0},y_{1})$ if and only if $x = \\alpha (u)$ and $y =\\beta (v)$ for some TF-isomorphism $(\\alpha ,\\beta )$ from $G$ to $H$ .", "The result follows from the proof of Theorem REF .", "$\\Box $ Refer to Figure REF .", "An isomorphism $f$ from $\\mbox{\\textbf {ADC}}(G)$ to $\\mbox{\\textbf {ADC}}(H)$ and the corresponding maps $\\alpha $ and $\\beta $ from V$(G)$ onto V$(H)$ , which are derived from $f$ as described in the proof of Theorem REF , are given below.", "$f: 1_{0} \\mapsto 1_{0}^{\\prime } & \\ & f: 1_{1} \\mapsto 1_{1}^{\\prime }\\\\f: 2_{1} \\mapsto 2_{0}^{\\prime } & \\ & f: 2_{1} \\mapsto 3_{1}^{\\prime }\\\\f: 3_{0} \\mapsto 3_{0}^{\\prime } & \\ & f: 3_{1} \\mapsto 2_{1}^{\\prime }\\\\f: 4_{0} \\mapsto 6_{0}^{\\prime } &\\ & f: 4_{1} \\mapsto 5_{1}^{\\prime }\\\\f: 5_{0} \\mapsto 7_{0}^{\\prime } &\\ & f: 5_{1} \\mapsto 4_{1}^{\\prime }\\\\f: 6_{0} \\mapsto 5_{0}^{\\prime } & \\ & f: 6_{1} \\mapsto 6_{1}^{\\prime }\\\\f: 7_{0} \\mapsto 4_{0}^{\\prime } & \\ & f: 7_{1} \\mapsto 7_{1}^{\\prime }$ $\\alpha : 1 \\mapsto 1^{\\prime } & \\ & \\beta : 1 \\mapsto 1^{\\prime }\\\\\\alpha : 2 \\mapsto 2^{\\prime } & \\ & \\beta : 2 \\mapsto 3^{\\prime }\\\\\\alpha : 3 \\mapsto 3^{\\prime } & \\ & \\beta : 3 \\mapsto 2^{\\prime }\\\\\\alpha : 4 \\mapsto 6^{\\prime } &\\ & \\beta : 4 \\mapsto 5^{\\prime }\\\\\\alpha : 5 \\mapsto 7^{\\prime } &\\ & \\beta : 5 \\mapsto 4^{\\prime }\\\\\\alpha : 6 \\mapsto 5^{\\prime } & \\ & \\beta : 6 \\mapsto 6^{\\prime }\\\\\\alpha : 7 \\mapsto 4^{\\prime } & \\ & \\beta : 7 \\mapsto 7^{\\prime }$ Figure REF shows a digraph $G$ and its alternating double cover $\\mbox{\\textbf {ADC}}(G)$ which in this case has three components, namely $D_{1}$ , $D_{2}$ and $D_{3}$ .", "Figure REF also shows how the components of $\\mbox{\\textbf {ADC}}(G)$ can be combined by associating vertices of the form $u_{0}$ with vertices of the form $v_{1}$ , irrespective of whether $u=v$ or $u \\ne v$ , to form $G$ or other digraphs such as $G_{1}$ , $G_{2}$ and $G_{3}$ having the same number of vertices as $G$ .", "It is easy to check that $G$ , $G_{1}$ , $G_{2}$ and $G_{3}$ are pairwise two-fold isomorphic as expected from the result of Theorem REF since each of these digraphs have the same number of vertices and have isomorphic ADCs.", "Proposition 6.3 (i) A digraph $H$ is isomorphic to $\\mbox{\\emph {\\textbf {ADC}}}(G)$ for some $G$ if and only if $H$ is strongly bipartite.", "(ii) For every digraph $G$ , ${\\mbox{\\emph {\\textbf {ADC}}}}(\\mbox{\\emph {\\textbf {ADC}}}(G))$ is isomorphic to $\\mbox{\\emph {\\textbf {ADC}}}(G)$ .", "We already know that the condition stated in $(i)$ is necessary in order to have $H$ isomorphic to some $\\mbox{\\textbf {ADC}}(G)$ .", "Conversely, if $H$ has this property, define map $f:$ $V(H)$ $\\rightarrow V (\\mbox{\\textbf {ADC}}(H))$ as follows: $f(u) = u_{0}$ if $u$ is a source, $f(u) = u_{1}$ if $u$ is a sink.", "Clearly $f$ is a bijection If $(u,v)$ is an arc of $H$ then by our assumption $u$ is a source and $v$ is a sink of $H$ .", "Then $(u_{0},v_{1})$ $=$ $(f(u),f(v))$ is an arc of $\\mbox{\\textbf {ADC}}(H)$ .", "Likewise, each arc of $\\mbox{\\textbf {ADC}}(H)$ takes the form $(u_{0},v_{1})$ , with $u$ source and $v$ sink of $H$ and hence $(u_{0},v_{1})$ is the image of $(u,v)$ under $f$ .", "This proves that $f$ is an isomorphism from $H$ to $\\mbox{\\textbf {ADC}}(H)$ , so $(i)$ is satisfied with $G=H$ .", "Now $(ii)$ is a straightforward consequence of $(i)$ , taking $H = \\mbox{\\textbf {ADC}}(G)$ .", "$\\Box $ Figure: GG and HH are TF-isomorphic graphs and have isomorphic ADCs." ], [ "Two-fold orbitals", "Let $\\mathbf {\\Gamma } \\le \\mathcal {S} = S_{\\mbox{\\scriptsize {$\|V\|$}}} \\times S_{\\mbox{\\scriptsize {$\|V\|$}}}$ .", "For a fixed element $(u,v)$ of $V\\times V$ let $ \\mathbf {\\Gamma }(u,v) = \\lbrace (\\alpha (u),\\beta (v)\\ \|\\ (\\alpha ,\\beta ) \\in \\mathbf {\\Gamma }\\rbrace .", "$ The set $\\mathbf {\\Gamma }(u,v)$ is called a two-fold orbital or TF-orbital.", "A two-fold orbital is the set of arcs of a digraph $G$ having vertex set $V$ which we call two-fold orbital digraph or TF-orbital digraph.", "If for every arc $(x,y)$ in $\\mathbf {\\Gamma }(u,v)$ , the oppositely directed arc $(y,x)$ is also contained in $\\mathbf {\\Gamma }(u,v)$ , then $G$ is a two-fold orbital graph or TF-orbital graph.", "This generalisation of the well-known concept of orbital (di)graph has been discussed in [9].", "Proposition 7.1 Let $G$ be a strongly bipartite digraph.", "Then (i) There is a homomorphism $\\psi $ of Aut$^{\\mbox{\\tiny \\textbf {{TF}}}}(G)$ onto Aut$(G)$ .", "(ii) If $G$ is a TF-orbital digraph, then it is also an orbital digraph.", "If $(\\alpha ,\\beta )$ is a TF-automorphism of $G$ , define $\\psi (\\alpha ,\\beta )$ $=$ $f:$ $V(G) \\rightarrow $ $V(G)$ as follows: $f(u) = \\alpha (u)$ if $u$ is a source and $f(u)=\\beta (u)$ if $u$ is a sink.", "Since $\\alpha $ preserves sources then $f$ takes sources to sources.", "Similarly, since $\\beta $ preserves sinks, then $f$ takes sinks to sinks.", "Since both $\\alpha $ and $\\beta $ are permutations, the restrictions of $f$ to the set of sources and to the set of sinks are also permutations.", "Hence $f$ is a permutation of V$(G)$ .", "Given any arc $(u,v)$ of $G$ , note that $(\\alpha ,\\beta )$ takes $(u,v)$ to $(\\alpha (u),\\beta (v))$ , which is equal to $(f(u),f(v))$ because $u$ is a source and $v$ is a sink.", "Hence $f$ is an automorphism of $G$ .", "so $\\psi $ maps $\\mbox{Aut}^{\\mbox{{\\tiny {\\textbf {TF}}}}}(G)$$\\ $ to Aut$(G)$ .", "A direct computation proves that $\\psi $ is a group homomorphism, hence $(i)$ holds.", "Assume now that $G = \\mathbf {\\Gamma }(u,v)$ for some $\\mathbf {\\Gamma }$ .", "Then $\\mathbf {\\Gamma }$ is a subgroup of $\\mbox{Aut}^{\\mbox{{\\tiny {\\textbf {TF}}}}}(G)$$\\ $ and $\\psi (\\mathbf {\\Gamma })$ is a subgroup of Aut $G$ .", "Each arc of $G$ takes the form $(\\alpha (u),\\beta (v))$ , where $(\\alpha ,\\beta )$ $\\in $ $\\mathbf {\\Gamma }$ and $u$ , $v$ are a source and a sink respectively.", "Letting $f = \\psi (\\alpha ,\\beta )$ this arc is $(f(u),f(v))$ , so it belongs to the orbital digraph $\\psi (\\mathbf {\\Gamma })(u,v)$ .", "This proves that $G$ is contained in this orbital digraph.", "The opposite inclusion can be shown the same way, so that $G = \\psi (\\Gamma )(u,v)$ and $(ii)$ follows.", "$\\Box $ Corollary 7.2 Let $G$ be a strongly bipartite digraph.", "Then $G$ is a two-fold orbital digraph if and only if ${\\emph {\\mbox{\\textbf {CDC}}(G)}}$ is an orbital digraph.", "By Proposition REF , $G$ and $\\mbox{\\textbf {ADC}}(G)$ are isomorphic.", "If either of them is a TF-orbital, then of course the same holds for the other one, but by Proposition REF in this case these TF-orbitals are both orbitals.", "$\\Box $" ], [ "Conclusion", "We believe that TF-isomorphisms is a relatively new concept.", "The only other author who considered them was Zelinka in a short paper motivated by the concept of isotopy in semigroups [22], [23].", "Our papers ([9] and [10]) are the first attempts at a systematic study of TF-isomorphisms.", "In this paper we have shown close links between TF-isomorphisms and double covers, and how the decomposition of a particular double cover can be used to obtain TF-isomorphic graphs.", "We have also seen that TF-isomorphisms give a new angle for looking at some older problems in graph theory.", "But does the notion of TF-isomorphism add anything new to these older questions?", "We believe that it does.", "For example, in [11] we prove this result which explains instability of graphs in terms of TF-automorphisms.", "Theorem 8.1 Let Aut$^{\\mbox{\\tiny \\textbf {{TF}}}}(G)$ be the group of TF-automorphisms of a mixed graph $G$ .", "Then Aut$({\\emph {\\mbox{\\textbf {CDC}}(G)}})$ is isomorphic to the semi-direct product Aut$^{\\mbox{\\tiny \\textbf {{TF}}}}(G)\\rtimes \\mathbb {Z}_2$ .", "Therefore $G$ is unstable if and only if it has a non-trivial TF-automorphism.", "$\\Box $ Also, it is not very likely that looking at these questions without the notion of TF-isomorphisms would lead one to the notion of A-trails, a technique which we feel is very useful, or the construction of asymmetric graphs with a non-trivial TF-isomorphism, an interesting notion which would be not so natural to formulate using only matrix methods, say.", "Some results and proofs are clearer in the TF-isomorphism setting.", "For example, in some of the papers cited we find this result about graph reconstruction from neighbourhoods.", "Theorem 8.2 ([1]) If $G$ is a connected bipartite graph, then any nonisomorphic graph $H$ with the same neighbourhood family as $G$ must be a disconnected graph with two components which themselves have identical neighbourhood hypergraphs.", "$\\Box $ From the TF-isomorphism point of view, this result follows from three very basic facts: (i) two graphs have the same neighbourhood family (equivalent to being TF-isomorphic) if and only if they have the same canonical double cover; (ii) the canonical double cover of a graph $G$ is disconnected if and only if $G$ is bipartite; and (iii) when $G$ is bipartite, the canonical double cover of $G$ is simply two disjoint copies of $G$ .", "Therefore, for $H$ to have the same canonical double cover as $G$ , it must consist of two components isomorphic to $K$ , where $G$ is the canonical double cover of $K$ .", "This gives Theorem REF .", "And moreover, from these remarks we also see that the only bipartite graphs for which there are non-isomorphic graphs with the same neighbourhood hypergraph are those which are canonical double covers.", "The Realisability Problem restricted to bipartite graphs therefore becomes: given a bipartite graph $G$ , is there a graph $K$ such that $G$ is the canonical double cover of $K$ ?", "A result in this direction was proved in [14], where graphs whose canonical double covers are Cayley graphs are characterised So it seems that the TF-isomorphism point of view can give a new handle on some of these problems.", "We intend to pursue this line of research in a forthcoming work." ], [ "Acknowledgement", "We are grateful to M. Muzychuk for first pointing out to us the usefulness of considering TF-isomorphisms as isomorphisms between incidence structures." ], [ "TF-isomorphisms and mixed graph covers", "The following result can be seen as a corollary to Theorem REF and the proof is easy since the IDC of a mixed graph $G$ can be obtained from $\\mbox{\\textbf {ADC}}(G)$ simply by removing the directions of the arcs and isolated vertices are irrelevant.", "Here we give an direct proof because it will help us in later constructions.", "Theorem 6.1 Let $G$ , $H$ be mixed graphs.", "The $G$ and $H$ are TF-isomorphic if and only if $\\mbox{\\textbf {\\emph {ADC}}}(G)$ and $\\mbox{\\textbf {\\emph {ADC}}}(H)$ are isomorphic.", "Let $(\\alpha ,\\beta )$ be a TF-isomorphism from $G$ to $H$ .", "Let $(u,v)$ be an arc of $G$ .", "First note that if $(\\alpha (u),\\beta (v))$ is an arc of $H$ , then $(u_{0},v_{1})$ is an arc of $\\mbox{\\textbf {ADC}}(G)$ and $(\\alpha (u)_{0},\\beta (u)_{1})$ is an arc of $\\mbox{\\textbf {ADC}}(H)$ .", "Let $f$ be a map from $V$$(\\mbox{\\textbf {ADC}}(G))$ to $V$$(\\mbox{\\textbf {ADC}}(H))$ such that $f: u_{0} \\mapsto x_{0}$ if $x = \\alpha (u)$ and $f: v_{1} \\mapsto y_{1}$ if $y =\\beta (v)$ .", "Consider any arc $(u,v)$ of $G$ and consider the corresponding arc $(u_{0},v_{1})$ in $A(\\mbox{\\textbf {ADC}}(G))$ .", "Let $(\\alpha ,\\beta )(u,v) = (x,y)$ .", "Then by definition $f$ takes $(u_{0},v_{1})$ to $(x_{0},y_{1})$ in $A(\\mbox{\\textbf {ADC}}(H))$ .", "The function $f$ maps arcs of $\\mbox{\\textbf {ADC}}(G)$ to arcs of $\\mbox{\\textbf {ADC}}(H)$ and it is clearly bijective.", "Hence, $f$ is an isomorphism from $\\mbox{\\textbf {ADC}}(G)$ to $\\mbox{\\textbf {ADC}}(H)$ .", "Now suppose that $\\mbox{\\textbf {ADC}}(G)$ and $\\mbox{\\textbf {ADC}}(H)$ are isomorphic.", "This implies that there exists a map $f$ such that $f(u_{0},v_{1})$ $=$ $(x_{0},y_{1})$ .", "Note that the arcs must always start from a vertex whose label has 0 as subscript and incident to a vertex whose label has 1 as subscript, by virtue of the construction presented above.", "Define $\\alpha $ , $\\beta $ from $V$$(G)$ to $V$$(H)$ as follows.", "Let $\\alpha (u) = x$ if $f(u_{0}) = x_{0}$ where $u \\in $ $V$$(G)$ and $x \\in $ $V$$(H)$ and let $\\beta (v) = y$ if $f(v_{1}) = y_{1}$ where $v \\in $ $V$$(G)$ and $y \\in $ $V$$(H)$ .", "Then $(\\alpha ,\\beta )$ takes any arc $(u,v) \\in $ $A(G)$ to some $(x,y)$ in $A(H)$ .", "This two-fold mapping is bijective and hence $(\\alpha ,\\beta )$ is a TF-isomorphism from $G$ to $H$ .", "$\\Box $ Corollary 6.2 Let $(\\alpha ,\\beta )$ be a TF-isomorphism from a mixed graph $G$ to a mixed graph $H$ .", "Then there exists an isomorphism $f_{\\alpha ,\\beta }$ from $\\mbox{\\textbf {\\emph {ADC}}}(G)$ to $\\mbox{\\textbf {\\emph {ADC}}}(H)$ such that $f_{\\alpha ,\\beta }(u_{0},v_{1}) = (x_{0},y_{1})$ if and only if $x = \\alpha (u)$ and $y =\\beta (v)$ for some TF-isomorphism $(\\alpha ,\\beta )$ from $G$ to $H$ .", "The result follows from the proof of Theorem REF .", "$\\Box $ Refer to Figure REF .", "An isomorphism $f$ from $\\mbox{\\textbf {ADC}}(G)$ to $\\mbox{\\textbf {ADC}}(H)$ and the corresponding maps $\\alpha $ and $\\beta $ from V$(G)$ onto V$(H)$ , which are derived from $f$ as described in the proof of Theorem REF , are given below.", "$f: 1_{0} \\mapsto 1_{0}^{\\prime } & \\ & f: 1_{1} \\mapsto 1_{1}^{\\prime }\\\\f: 2_{1} \\mapsto 2_{0}^{\\prime } & \\ & f: 2_{1} \\mapsto 3_{1}^{\\prime }\\\\f: 3_{0} \\mapsto 3_{0}^{\\prime } & \\ & f: 3_{1} \\mapsto 2_{1}^{\\prime }\\\\f: 4_{0} \\mapsto 6_{0}^{\\prime } &\\ & f: 4_{1} \\mapsto 5_{1}^{\\prime }\\\\f: 5_{0} \\mapsto 7_{0}^{\\prime } &\\ & f: 5_{1} \\mapsto 4_{1}^{\\prime }\\\\f: 6_{0} \\mapsto 5_{0}^{\\prime } & \\ & f: 6_{1} \\mapsto 6_{1}^{\\prime }\\\\f: 7_{0} \\mapsto 4_{0}^{\\prime } & \\ & f: 7_{1} \\mapsto 7_{1}^{\\prime }$ $\\alpha : 1 \\mapsto 1^{\\prime } & \\ & \\beta : 1 \\mapsto 1^{\\prime }\\\\\\alpha : 2 \\mapsto 2^{\\prime } & \\ & \\beta : 2 \\mapsto 3^{\\prime }\\\\\\alpha : 3 \\mapsto 3^{\\prime } & \\ & \\beta : 3 \\mapsto 2^{\\prime }\\\\\\alpha : 4 \\mapsto 6^{\\prime } &\\ & \\beta : 4 \\mapsto 5^{\\prime }\\\\\\alpha : 5 \\mapsto 7^{\\prime } &\\ & \\beta : 5 \\mapsto 4^{\\prime }\\\\\\alpha : 6 \\mapsto 5^{\\prime } & \\ & \\beta : 6 \\mapsto 6^{\\prime }\\\\\\alpha : 7 \\mapsto 4^{\\prime } & \\ & \\beta : 7 \\mapsto 7^{\\prime }$ Figure REF shows a digraph $G$ and its alternating double cover $\\mbox{\\textbf {ADC}}(G)$ which in this case has three components, namely $D_{1}$ , $D_{2}$ and $D_{3}$ .", "Figure REF also shows how the components of $\\mbox{\\textbf {ADC}}(G)$ can be combined by associating vertices of the form $u_{0}$ with vertices of the form $v_{1}$ , irrespective of whether $u=v$ or $u \\ne v$ , to form $G$ or other digraphs such as $G_{1}$ , $G_{2}$ and $G_{3}$ having the same number of vertices as $G$ .", "It is easy to check that $G$ , $G_{1}$ , $G_{2}$ and $G_{3}$ are pairwise two-fold isomorphic as expected from the result of Theorem REF since each of these digraphs have the same number of vertices and have isomorphic ADCs.", "Proposition 6.3 (i) A digraph $H$ is isomorphic to $\\mbox{\\emph {\\textbf {ADC}}}(G)$ for some $G$ if and only if $H$ is strongly bipartite.", "(ii) For every digraph $G$ , ${\\mbox{\\emph {\\textbf {ADC}}}}(\\mbox{\\emph {\\textbf {ADC}}}(G))$ is isomorphic to $\\mbox{\\emph {\\textbf {ADC}}}(G)$ .", "We already know that the condition stated in $(i)$ is necessary in order to have $H$ isomorphic to some $\\mbox{\\textbf {ADC}}(G)$ .", "Conversely, if $H$ has this property, define map $f:$ $V(H)$ $\\rightarrow V (\\mbox{\\textbf {ADC}}(H))$ as follows: $f(u) = u_{0}$ if $u$ is a source, $f(u) = u_{1}$ if $u$ is a sink.", "Clearly $f$ is a bijection If $(u,v)$ is an arc of $H$ then by our assumption $u$ is a source and $v$ is a sink of $H$ .", "Then $(u_{0},v_{1})$ $=$ $(f(u),f(v))$ is an arc of $\\mbox{\\textbf {ADC}}(H)$ .", "Likewise, each arc of $\\mbox{\\textbf {ADC}}(H)$ takes the form $(u_{0},v_{1})$ , with $u$ source and $v$ sink of $H$ and hence $(u_{0},v_{1})$ is the image of $(u,v)$ under $f$ .", "This proves that $f$ is an isomorphism from $H$ to $\\mbox{\\textbf {ADC}}(H)$ , so $(i)$ is satisfied with $G=H$ .", "Now $(ii)$ is a straightforward consequence of $(i)$ , taking $H = \\mbox{\\textbf {ADC}}(G)$ .", "$\\Box $ Figure: GG and HH are TF-isomorphic graphs and have isomorphic ADCs." ], [ "Two-fold orbitals", "Let $\\mathbf {\\Gamma } \\le \\mathcal {S} = S_{\\mbox{\\scriptsize {$\|V\|$}}} \\times S_{\\mbox{\\scriptsize {$\|V\|$}}}$ .", "For a fixed element $(u,v)$ of $V\\times V$ let $ \\mathbf {\\Gamma }(u,v) = \\lbrace (\\alpha (u),\\beta (v)\\ \|\\ (\\alpha ,\\beta ) \\in \\mathbf {\\Gamma }\\rbrace .", "$ The set $\\mathbf {\\Gamma }(u,v)$ is called a two-fold orbital or TF-orbital.", "A two-fold orbital is the set of arcs of a digraph $G$ having vertex set $V$ which we call two-fold orbital digraph or TF-orbital digraph.", "If for every arc $(x,y)$ in $\\mathbf {\\Gamma }(u,v)$ , the oppositely directed arc $(y,x)$ is also contained in $\\mathbf {\\Gamma }(u,v)$ , then $G$ is a two-fold orbital graph or TF-orbital graph.", "This generalisation of the well-known concept of orbital (di)graph has been discussed in [9].", "Proposition 7.1 Let $G$ be a strongly bipartite digraph.", "Then (i) There is a homomorphism $\\psi $ of Aut$^{\\mbox{\\tiny \\textbf {{TF}}}}(G)$ onto Aut$(G)$ .", "(ii) If $G$ is a TF-orbital digraph, then it is also an orbital digraph.", "If $(\\alpha ,\\beta )$ is a TF-automorphism of $G$ , define $\\psi (\\alpha ,\\beta )$ $=$ $f:$ $V(G) \\rightarrow $ $V(G)$ as follows: $f(u) = \\alpha (u)$ if $u$ is a source and $f(u)=\\beta (u)$ if $u$ is a sink.", "Since $\\alpha $ preserves sources then $f$ takes sources to sources.", "Similarly, since $\\beta $ preserves sinks, then $f$ takes sinks to sinks.", "Since both $\\alpha $ and $\\beta $ are permutations, the restrictions of $f$ to the set of sources and to the set of sinks are also permutations.", "Hence $f$ is a permutation of V$(G)$ .", "Given any arc $(u,v)$ of $G$ , note that $(\\alpha ,\\beta )$ takes $(u,v)$ to $(\\alpha (u),\\beta (v))$ , which is equal to $(f(u),f(v))$ because $u$ is a source and $v$ is a sink.", "Hence $f$ is an automorphism of $G$ .", "so $\\psi $ maps $\\mbox{Aut}^{\\mbox{{\\tiny {\\textbf {TF}}}}}(G)$$\\ $ to Aut$(G)$ .", "A direct computation proves that $\\psi $ is a group homomorphism, hence $(i)$ holds.", "Assume now that $G = \\mathbf {\\Gamma }(u,v)$ for some $\\mathbf {\\Gamma }$ .", "Then $\\mathbf {\\Gamma }$ is a subgroup of $\\mbox{Aut}^{\\mbox{{\\tiny {\\textbf {TF}}}}}(G)$$\\ $ and $\\psi (\\mathbf {\\Gamma })$ is a subgroup of Aut $G$ .", "Each arc of $G$ takes the form $(\\alpha (u),\\beta (v))$ , where $(\\alpha ,\\beta )$ $\\in $ $\\mathbf {\\Gamma }$ and $u$ , $v$ are a source and a sink respectively.", "Letting $f = \\psi (\\alpha ,\\beta )$ this arc is $(f(u),f(v))$ , so it belongs to the orbital digraph $\\psi (\\mathbf {\\Gamma })(u,v)$ .", "This proves that $G$ is contained in this orbital digraph.", "The opposite inclusion can be shown the same way, so that $G = \\psi (\\Gamma )(u,v)$ and $(ii)$ follows.", "$\\Box $ Corollary 7.2 Let $G$ be a strongly bipartite digraph.", "Then $G$ is a two-fold orbital digraph if and only if ${\\emph {\\mbox{\\textbf {CDC}}(G)}}$ is an orbital digraph.", "By Proposition REF , $G$ and $\\mbox{\\textbf {ADC}}(G)$ are isomorphic.", "If either of them is a TF-orbital, then of course the same holds for the other one, but by Proposition REF in this case these TF-orbitals are both orbitals.", "$\\Box $" ], [ "Conclusion", "We believe that TF-isomorphisms is a relatively new concept.", "The only other author who considered them was Zelinka in a short paper motivated by the concept of isotopy in semigroups [22], [23].", "Our papers ([9] and [10]) are the first attempts at a systematic study of TF-isomorphisms.", "In this paper we have shown close links between TF-isomorphisms and double covers, and how the decomposition of a particular double cover can be used to obtain TF-isomorphic graphs.", "We have also seen that TF-isomorphisms give a new angle for looking at some older problems in graph theory.", "But does the notion of TF-isomorphism add anything new to these older questions?", "We believe that it does.", "For example, in [11] we prove this result which explains instability of graphs in terms of TF-automorphisms.", "Theorem 8.1 Let Aut$^{\\mbox{\\tiny \\textbf {{TF}}}}(G)$ be the group of TF-automorphisms of a mixed graph $G$ .", "Then Aut$({\\emph {\\mbox{\\textbf {CDC}}(G)}})$ is isomorphic to the semi-direct product Aut$^{\\mbox{\\tiny \\textbf {{TF}}}}(G)\\rtimes \\mathbb {Z}_2$ .", "Therefore $G$ is unstable if and only if it has a non-trivial TF-automorphism.", "$\\Box $ Also, it is not very likely that looking at these questions without the notion of TF-isomorphisms would lead one to the notion of A-trails, a technique which we feel is very useful, or the construction of asymmetric graphs with a non-trivial TF-isomorphism, an interesting notion which would be not so natural to formulate using only matrix methods, say.", "Some results and proofs are clearer in the TF-isomorphism setting.", "For example, in some of the papers cited we find this result about graph reconstruction from neighbourhoods.", "Theorem 8.2 ([1]) If $G$ is a connected bipartite graph, then any nonisomorphic graph $H$ with the same neighbourhood family as $G$ must be a disconnected graph with two components which themselves have identical neighbourhood hypergraphs.", "$\\Box $ From the TF-isomorphism point of view, this result follows from three very basic facts: (i) two graphs have the same neighbourhood family (equivalent to being TF-isomorphic) if and only if they have the same canonical double cover; (ii) the canonical double cover of a graph $G$ is disconnected if and only if $G$ is bipartite; and (iii) when $G$ is bipartite, the canonical double cover of $G$ is simply two disjoint copies of $G$ .", "Therefore, for $H$ to have the same canonical double cover as $G$ , it must consist of two components isomorphic to $K$ , where $G$ is the canonical double cover of $K$ .", "This gives Theorem REF .", "And moreover, from these remarks we also see that the only bipartite graphs for which there are non-isomorphic graphs with the same neighbourhood hypergraph are those which are canonical double covers.", "The Realisability Problem restricted to bipartite graphs therefore becomes: given a bipartite graph $G$ , is there a graph $K$ such that $G$ is the canonical double cover of $K$ ?", "A result in this direction was proved in [14], where graphs whose canonical double covers are Cayley graphs are characterised So it seems that the TF-isomorphism point of view can give a new handle on some of these problems.", "We intend to pursue this line of research in a forthcoming work." ], [ "Acknowledgement", "We are grateful to M. Muzychuk for first pointing out to us the usefulness of considering TF-isomorphisms as isomorphisms between incidence structures." ], [ "Two-fold orbitals", "Let $\\mathbf {\\Gamma } \\le \\mathcal {S} = S_{\\mbox{\\scriptsize {$\|V\|$}}} \\times S_{\\mbox{\\scriptsize {$\|V\|$}}}$ .", "For a fixed element $(u,v)$ of $V\\times V$ let $ \\mathbf {\\Gamma }(u,v) = \\lbrace (\\alpha (u),\\beta (v)\\ \|\\ (\\alpha ,\\beta ) \\in \\mathbf {\\Gamma }\\rbrace .", "$ The set $\\mathbf {\\Gamma }(u,v)$ is called a two-fold orbital or TF-orbital.", "A two-fold orbital is the set of arcs of a digraph $G$ having vertex set $V$ which we call two-fold orbital digraph or TF-orbital digraph.", "If for every arc $(x,y)$ in $\\mathbf {\\Gamma }(u,v)$ , the oppositely directed arc $(y,x)$ is also contained in $\\mathbf {\\Gamma }(u,v)$ , then $G$ is a two-fold orbital graph or TF-orbital graph.", "This generalisation of the well-known concept of orbital (di)graph has been discussed in [9].", "Proposition 7.1 Let $G$ be a strongly bipartite digraph.", "Then (i) There is a homomorphism $\\psi $ of Aut$^{\\mbox{\\tiny \\textbf {{TF}}}}(G)$ onto Aut$(G)$ .", "(ii) If $G$ is a TF-orbital digraph, then it is also an orbital digraph.", "If $(\\alpha ,\\beta )$ is a TF-automorphism of $G$ , define $\\psi (\\alpha ,\\beta )$ $=$ $f:$ $V(G) \\rightarrow $ $V(G)$ as follows: $f(u) = \\alpha (u)$ if $u$ is a source and $f(u)=\\beta (u)$ if $u$ is a sink.", "Since $\\alpha $ preserves sources then $f$ takes sources to sources.", "Similarly, since $\\beta $ preserves sinks, then $f$ takes sinks to sinks.", "Since both $\\alpha $ and $\\beta $ are permutations, the restrictions of $f$ to the set of sources and to the set of sinks are also permutations.", "Hence $f$ is a permutation of V$(G)$ .", "Given any arc $(u,v)$ of $G$ , note that $(\\alpha ,\\beta )$ takes $(u,v)$ to $(\\alpha (u),\\beta (v))$ , which is equal to $(f(u),f(v))$ because $u$ is a source and $v$ is a sink.", "Hence $f$ is an automorphism of $G$ .", "so $\\psi $ maps $\\mbox{Aut}^{\\mbox{{\\tiny {\\textbf {TF}}}}}(G)$$\\ $ to Aut$(G)$ .", "A direct computation proves that $\\psi $ is a group homomorphism, hence $(i)$ holds.", "Assume now that $G = \\mathbf {\\Gamma }(u,v)$ for some $\\mathbf {\\Gamma }$ .", "Then $\\mathbf {\\Gamma }$ is a subgroup of $\\mbox{Aut}^{\\mbox{{\\tiny {\\textbf {TF}}}}}(G)$$\\ $ and $\\psi (\\mathbf {\\Gamma })$ is a subgroup of Aut $G$ .", "Each arc of $G$ takes the form $(\\alpha (u),\\beta (v))$ , where $(\\alpha ,\\beta )$ $\\in $ $\\mathbf {\\Gamma }$ and $u$ , $v$ are a source and a sink respectively.", "Letting $f = \\psi (\\alpha ,\\beta )$ this arc is $(f(u),f(v))$ , so it belongs to the orbital digraph $\\psi (\\mathbf {\\Gamma })(u,v)$ .", "This proves that $G$ is contained in this orbital digraph.", "The opposite inclusion can be shown the same way, so that $G = \\psi (\\Gamma )(u,v)$ and $(ii)$ follows.", "$\\Box $ Corollary 7.2 Let $G$ be a strongly bipartite digraph.", "Then $G$ is a two-fold orbital digraph if and only if ${\\emph {\\mbox{\\textbf {CDC}}(G)}}$ is an orbital digraph.", "By Proposition REF , $G$ and $\\mbox{\\textbf {ADC}}(G)$ are isomorphic.", "If either of them is a TF-orbital, then of course the same holds for the other one, but by Proposition REF in this case these TF-orbitals are both orbitals.", "$\\Box $" ], [ "Conclusion", "We believe that TF-isomorphisms is a relatively new concept.", "The only other author who considered them was Zelinka in a short paper motivated by the concept of isotopy in semigroups [22], [23].", "Our papers ([9] and [10]) are the first attempts at a systematic study of TF-isomorphisms.", "In this paper we have shown close links between TF-isomorphisms and double covers, and how the decomposition of a particular double cover can be used to obtain TF-isomorphic graphs.", "We have also seen that TF-isomorphisms give a new angle for looking at some older problems in graph theory.", "But does the notion of TF-isomorphism add anything new to these older questions?", "We believe that it does.", "For example, in [11] we prove this result which explains instability of graphs in terms of TF-automorphisms.", "Theorem 8.1 Let Aut$^{\\mbox{\\tiny \\textbf {{TF}}}}(G)$ be the group of TF-automorphisms of a mixed graph $G$ .", "Then Aut$({\\emph {\\mbox{\\textbf {CDC}}(G)}})$ is isomorphic to the semi-direct product Aut$^{\\mbox{\\tiny \\textbf {{TF}}}}(G)\\rtimes \\mathbb {Z}_2$ .", "Therefore $G$ is unstable if and only if it has a non-trivial TF-automorphism.", "$\\Box $ Also, it is not very likely that looking at these questions without the notion of TF-isomorphisms would lead one to the notion of A-trails, a technique which we feel is very useful, or the construction of asymmetric graphs with a non-trivial TF-isomorphism, an interesting notion which would be not so natural to formulate using only matrix methods, say.", "Some results and proofs are clearer in the TF-isomorphism setting.", "For example, in some of the papers cited we find this result about graph reconstruction from neighbourhoods.", "Theorem 8.2 ([1]) If $G$ is a connected bipartite graph, then any nonisomorphic graph $H$ with the same neighbourhood family as $G$ must be a disconnected graph with two components which themselves have identical neighbourhood hypergraphs.", "$\\Box $ From the TF-isomorphism point of view, this result follows from three very basic facts: (i) two graphs have the same neighbourhood family (equivalent to being TF-isomorphic) if and only if they have the same canonical double cover; (ii) the canonical double cover of a graph $G$ is disconnected if and only if $G$ is bipartite; and (iii) when $G$ is bipartite, the canonical double cover of $G$ is simply two disjoint copies of $G$ .", "Therefore, for $H$ to have the same canonical double cover as $G$ , it must consist of two components isomorphic to $K$ , where $G$ is the canonical double cover of $K$ .", "This gives Theorem REF .", "And moreover, from these remarks we also see that the only bipartite graphs for which there are non-isomorphic graphs with the same neighbourhood hypergraph are those which are canonical double covers.", "The Realisability Problem restricted to bipartite graphs therefore becomes: given a bipartite graph $G$ , is there a graph $K$ such that $G$ is the canonical double cover of $K$ ?", "A result in this direction was proved in [14], where graphs whose canonical double covers are Cayley graphs are characterised So it seems that the TF-isomorphism point of view can give a new handle on some of these problems.", "We intend to pursue this line of research in a forthcoming work." ], [ "Acknowledgement", "We are grateful to M. Muzychuk for first pointing out to us the usefulness of considering TF-isomorphisms as isomorphisms between incidence structures." ], [ "Conclusion", "We believe that TF-isomorphisms is a relatively new concept.", "The only other author who considered them was Zelinka in a short paper motivated by the concept of isotopy in semigroups [22], [23].", "Our papers ([9] and [10]) are the first attempts at a systematic study of TF-isomorphisms.", "In this paper we have shown close links between TF-isomorphisms and double covers, and how the decomposition of a particular double cover can be used to obtain TF-isomorphic graphs.", "We have also seen that TF-isomorphisms give a new angle for looking at some older problems in graph theory.", "But does the notion of TF-isomorphism add anything new to these older questions?", "We believe that it does.", "For example, in [11] we prove this result which explains instability of graphs in terms of TF-automorphisms.", "Theorem 8.1 Let Aut$^{\\mbox{\\tiny \\textbf {{TF}}}}(G)$ be the group of TF-automorphisms of a mixed graph $G$ .", "Then Aut$({\\emph {\\mbox{\\textbf {CDC}}(G)}})$ is isomorphic to the semi-direct product Aut$^{\\mbox{\\tiny \\textbf {{TF}}}}(G)\\rtimes \\mathbb {Z}_2$ .", "Therefore $G$ is unstable if and only if it has a non-trivial TF-automorphism.", "$\\Box $ Also, it is not very likely that looking at these questions without the notion of TF-isomorphisms would lead one to the notion of A-trails, a technique which we feel is very useful, or the construction of asymmetric graphs with a non-trivial TF-isomorphism, an interesting notion which would be not so natural to formulate using only matrix methods, say.", "Some results and proofs are clearer in the TF-isomorphism setting.", "For example, in some of the papers cited we find this result about graph reconstruction from neighbourhoods.", "Theorem 8.2 ([1]) If $G$ is a connected bipartite graph, then any nonisomorphic graph $H$ with the same neighbourhood family as $G$ must be a disconnected graph with two components which themselves have identical neighbourhood hypergraphs.", "$\\Box $ From the TF-isomorphism point of view, this result follows from three very basic facts: (i) two graphs have the same neighbourhood family (equivalent to being TF-isomorphic) if and only if they have the same canonical double cover; (ii) the canonical double cover of a graph $G$ is disconnected if and only if $G$ is bipartite; and (iii) when $G$ is bipartite, the canonical double cover of $G$ is simply two disjoint copies of $G$ .", "Therefore, for $H$ to have the same canonical double cover as $G$ , it must consist of two components isomorphic to $K$ , where $G$ is the canonical double cover of $K$ .", "This gives Theorem REF .", "And moreover, from these remarks we also see that the only bipartite graphs for which there are non-isomorphic graphs with the same neighbourhood hypergraph are those which are canonical double covers.", "The Realisability Problem restricted to bipartite graphs therefore becomes: given a bipartite graph $G$ , is there a graph $K$ such that $G$ is the canonical double cover of $K$ ?", "A result in this direction was proved in [14], where graphs whose canonical double covers are Cayley graphs are characterised So it seems that the TF-isomorphism point of view can give a new handle on some of these problems.", "We intend to pursue this line of research in a forthcoming work." ], [ "Acknowledgement", "We are grateful to M. Muzychuk for first pointing out to us the usefulness of considering TF-isomorphisms as isomorphisms between incidence structures.", "We are grateful to M. Muzychuk for first pointing out to us the usefulness of considering TF-isomorphisms as isomorphisms between incidence structures." ] ]	1403.0342
[ [ "AC-KBO Revisited" ], [ "Abstract Equational theories that contain axioms expressing associativity and commutativity (AC) of certain operators are ubiquitous.", "Theorem proving methods in such theories rely on well-founded orders that are compatible with the AC axioms.", "In this paper we consider various definitions of AC-compatible Knuth-Bendix orders.", "The orders of Steinbach and of Korovin and Voronkov are revisited.", "The former is enhanced to a more powerful version, and we modify the latter to amend its lack of monotonicity on non-ground terms.", "We further present new complexity results.", "An extension reflecting the recent proposal of subterm coefficients in standard Knuth-Bendix orders is also given.", "The various orders are compared on problems in termination and completion." ], [ "Introduction", "Associative and commutative (AC) operators appear in many applications, e.g.", "in automated reasoning with respect to algebraic structures such as commutative groups or rings.", "We are interested in proving termination of term rewrite systems with AC symbols.", "AC termination is important when deciding validity in equational theories with AC operators by means of completion.", "Several termination methods for plain rewriting have been extended to deal with AC symbols.", "BL87 presented a characterization of polynomial interpretations that ensures compatibility with the AC axioms.", "There have been numerous papers on extending the recursive path order (RPO) of D82 to deal with AC symbols, starting with the associative path order of BP85 and culminating in the fully syntactic AC-RPO of R02.", "Several authors [13], [17], [7], [1] adapted the influential dependency pair method of AG00 to AC rewriting.", "We are aware of only two papers on AC extensions of the order (KBO) of KB70.", "In this paper we revisit these orders and present yet another AC-compatible KBO.", "S90 presented a first version, which comes with the restriction that AC symbols are minimal in the precedence.", "By incorporating ideas of [19], KV03b presented a version without this restriction.", "Actually, they present two versions.", "One is defined on ground terms and another one on arbitrary terms.", "For (automatically) proving AC termination of rewrite systems, an AC-compatible order on arbitrary terms is required.Any AC-compatible reduction order $\\mathrel {\\succ }_\\mathrm {g}$ on ground terms can trivially be extended to arbitrary terms by defining $s \\mathrel {\\succ }t$ if and only if $s\\sigma \\mathrel {\\succ }_\\mathrm {g} t\\sigma $ for all grounding substitutions $\\sigma $ .", "This is, however, only of (mild) theoretical interest.", "We show that the second order of KV03b lacks the monotonicity property which is required by the definition of simplification orders.", "Nevertheless we prove that the order is sound for proving termination by extending it to an AC-compatible simplification order.", "We furthermore present a simpler variant of this latter order which properly extends the order of S90.", "In particular, Steinbach's order is a correct AC-compatible simplification order, contrary to what is claimed in [9].", "We also present new complexity results which confirm that AC rewriting is much more involved than plain rewriting.", "Apart from these theoretical contributions, we implemented the various AC-compatible KBOs to compare them also experimentally.", "The remainder of this paper is organized as follows.", "After recalling basic concepts of rewriting modulo AC and orders, we revisit Steinbach's order in Section .", "Section is devoted to the two orders of Korovin and Voronkov.", "We present a first version of our AC-compatible KBO in Section , also giving the non-trivial proof that it has the required properties.", "(The proofs in [9] are limited to the order on ground terms.)", "In Section we consider the complexity of the membership and orientation decision problems for the various orders.", "In Section we compare AC-KBO with AC-RPO.", "In Section our order is strengthened with subterm coefficients.", "In order to show effectiveness of these orders experimental data is provided in Section .", "The paper is concluded in Section .", "This article is an updated and extended version of [25].", "Our earlier results on complexity are extended by showing that the orientability problems for different versions of AC-KBO are in NP.", "Moreover, we include a comparison with AC-RPO, which we present in a slightly simplified manner compared to [19].", "Due to space limitations, some proofs can be found in the online appendix." ], [ "Preliminaries", "We assume familiarity with rewriting and termination.", "Throughout this paper we deal with rewrite systems over a set $\\mathcal {V}$ of variables and a finite signature $\\mathcal {F}$ together with a designated subset $\\mathcal {F}_\\mathrm {\\mathsf {AC}}$ of binary AC symbols.", "The congruence relation induced by the equations $f(x,y) \\approx f(y,x)$ and $f(f(x,y),z) \\approx f(x,f(y,z))$ for all $f \\in \\mathcal {F}_\\mathrm {\\mathsf {AC}}$ is denoted by $=_\\mathrm {\\mathsf {AC}}$ .", "A term rewrite system (TRS for short) $\\mathcal {R}$ is AC terminating if the relation ${=_\\mathrm {\\mathsf {AC}}} \\cdot {\\rightarrow _\\mathcal {R}} \\cdot {=_\\mathrm {\\mathsf {AC}}}$ is well-founded.", "In this paper AC termination is established by AC-compatible simplification orders $\\mathrel {\\succ }$ , which are strict orders (i.e., irreflexive and transitive relations) closed under contexts and substitutions that have the subterm property $f({t_1},\\dots ,{t_{n}}) \\mathrel {\\succ }t_i$ for all $1 \\leqslant i \\leqslant n$ and satisfy ${=_\\mathrm {\\mathsf {AC}}} \\cdot {\\mathrel {\\succ }} \\cdot {=_\\mathrm {\\mathsf {AC}}} \\subseteq {\\mathrel {\\succ }}$ .", "A strict order $\\mathrel {\\succ }$ is AC-total if $s \\mathrel {\\succ }t$ , $t \\mathrel {\\succ }s$ or $s =_\\mathrm {\\mathsf {AC}}t$ , for all ground terms $s$ and $t$ .", "A pair $({\\mathrel {\\succsim }},{\\mathrel {\\succ }})$ consisting of a preorder $\\mathrel {\\succsim }$ and a strict order $\\mathrel {\\succ }$ is said to be an order pair if the compatibility condition ${\\mathrel {\\succsim }\\cdot \\mathrel {\\succ }\\cdot \\mathrel {\\succsim }} \\subseteq {\\mathrel {\\succ }}$ holds.", "Definition 2.1 Let $\\mathrel {\\succ }$ be a strict order and $\\mathrel {\\succsim }$ be a preorder on a set $A$ .", "The lexicographic extensions $\\mathrel {\\succ }^\\mathsf {lex}$ and $\\mathrel {\\succsim }^\\mathsf {lex}$ are defined as follows: $\\vec{x} \\mathrel {\\succsim }^\\mathsf {lex}\\vec{y}$ if $\\vec{x} \\sqsupset _k^\\mathsf {lex}\\vec{y}$ for some $1 \\leqslant k \\leqslant n$ , $\\vec{x} \\mathrel {\\succ }^\\mathsf {lex}\\vec{y}$ if $\\vec{x} \\sqsupset _k^\\mathsf {lex}\\vec{y}$ for some $1 \\leqslant k < n$ .", "Here $\\vec{x} = ({x_1},\\dots ,{x_{n}})$ , $\\vec{y} = ({y_1},\\dots ,{y_{n}})$ , and $\\vec{x} \\sqsupset _k^\\mathsf {lex}\\vec{y}$ denotes the following condition: $x_i \\mathrel {\\succsim }y_i$ for all $i \\leqslant k$ and either $k < n$ and $x_{k+1} \\mathrel {\\succ }y_{k+1}$ or $k = n$ .", "The multiset extensions $\\mathrel {\\succ }^\\mathsf {mul}$ and $\\mathrel {\\succsim }^\\mathsf {mul}$ are defined as follows: $M \\mathrel {\\succsim }^\\mathsf {mul}N$ if $M \\sqsupset _k^\\mathsf {mul}N$ for some $0 \\leqslant k \\leqslant \\min (m,n)$ , $M \\mathrel {\\succ }^\\mathsf {mul}N$ if $M \\sqsupset _k^\\mathsf {mul}N$ for some $0 \\leqslant k \\leqslant \\min (m-1,n)$ .", "Here $M \\sqsupset _k^\\mathsf {mul}N$ if $M$ and $N$ consist of ${x_1},\\dots ,{x_{m}}$ and ${y_1},\\dots ,{y_{n}}$ respectively such that $x_j \\mathrel {\\succsim }y_j$ for all $j \\leqslant k$ , and for every $k < j \\leqslant n$ there is some $k < i \\leqslant m$ with $x_i \\mathrel {\\succ }y_j$ .", "Note that these extended relations depend on both $\\mathrel {\\succsim }$ and $\\mathrel {\\succ }$ .", "The following result is folklore; a recent formalization of multiset extensions in Isabelle/HOL is presented in [22].", "Theorem 2.2 If $(\\mathrel {\\succsim },\\mathrel {\\succ })$ is an order pair then $(\\mathrel {\\succsim }^\\mathsf {lex},\\mathrel {\\succ }^\\mathsf {lex})$ and $(\\mathrel {\\succsim }^\\mathsf {mul},\\mathrel {\\succ }^\\mathsf {mul})$ are order pairs.", "$\\Box $" ], [ "Steinbach's Order", "In this section we recall the AC-compatible KBO $>_\\mathrm {\\mathsf {S}}$ of S90, which reduces to the standard KBO if AC symbols are absent.The version in [21] is slightly more general, since non-AC function symbols can have arbitrary status.", "To simplify the discussion, we do not consider status in this paper.", "The order $>_\\mathrm {\\mathsf {S}}$ depends on a precedence and an admissible weight function.", "A precedence $>$ is a strict order on $\\mathcal {F}$ .", "A weight function $(w,w_0)$ for a signature $\\mathcal {F}$ consists of a mapping $w\\colon \\mathcal {F}\\rightarrow \\mathbb {N}$ and a constant $w_0 > 0$ such that $w(c) \\geqslant w_0$ for every constant $c \\in \\mathcal {F}$ .", "The weight of a term $t$ is recursively computed as follows: $w(t) ={\\left\\lbrace \\begin{array}{ll}w_0 & \\text{if $t \\in \\mathcal {V}$} \\\\\\displaystyle w(f) + \\smash[b]{\\sum _{1 \\leqslant i \\leqslant n}} w(t_i) &\\text{if $t = f({t_1},\\dots ,{t_{n}})$}\\end{array}\\right.", "}$ A weight function $(w,w_0)$ is admissible for $>$ if every unary $f$ with $w(f) = 0$ satisfies $f > g$ for all function symbols $g$ different from $f$ .", "Throughout this paper we assume admissibility.", "The topflattening [19] of a term $t$ with respect to an AC symbol $f$ is the multiset ${\\triangledown _{\\!f}}(t)$ defined inductively as follows: ${\\triangledown _{\\!f}}(t) ={\\left\\lbrace \\begin{array}{ll}\\lbrace t \\rbrace & \\text{if $\\mathsf {root}(t) \\ne f$} \\\\{\\triangledown _{\\!f}}(t_1) \\uplus {\\triangledown _{\\!f}}(t_2) & \\text{if $t = f(t_1,t_2)$}\\end{array}\\right.", "}$ Definition 3.1 Let $>$ be a precedence and $(w,w_0)$ a weight function.", "The order $>_\\mathrm {\\mathsf {S}}$ is inductively defined as follows: $s >_\\mathrm {\\mathsf {S}}t$ if $\|s\|_x \\geqslant \|t\|_x$ for all $x \\in \\mathcal {V}$ and either $w(s) > w(t)$ , or $w(s) = w(t)$ and one of the following alternatives holds: $s = f^k(t)$ and $t \\in \\mathcal {V}$ for some $k > 0$ , $s = f({s_1},\\dots ,{s_{n}})$ , $t = g({t_1},\\dots ,{t_{m}})$ , and $f > g$ , $s = f({s_1},\\dots ,{s_{n}})$ , $t = f({t_1},\\dots ,{t_{n}})$ , $f \\notin \\mathcal {F}_\\mathrm {\\mathsf {AC}}$ , $({s_1},\\dots ,{s_{n}}) >_\\mathrm {\\mathsf {S}}^\\mathsf {lex}({t_1},\\dots ,{t_{n}})$ , $s = f(s_1,s_2)$ , $t = f(t_1,t_2)$ , $f \\in \\mathcal {F}_\\mathrm {\\mathsf {AC}}$ , and ${\\triangledown _{\\!f}}(s) >_\\mathrm {\\mathsf {S}}^\\mathsf {mul}{\\triangledown _{\\!f}}(t)$ .", "The relation $=_\\mathrm {\\mathsf {AC}}$ is used as preorder in $>_\\mathrm {\\mathsf {S}}^\\mathsf {lex}$ and $>_\\mathrm {\\mathsf {S}}^\\mathsf {mul}$ .", "Cases 0–2 are the same as in the standard Knuth-Bendix order.", "In case 3 terms rooted by the same AC symbol $f$ are treated by comparing their topflattenings in the multiset extension of $>_\\mathrm {\\mathsf {S}}$ .", "Example 3.2 Consider the signature $\\mathcal {F}= \\lbrace \\mathsf {a}, \\mathsf {f}, + \\rbrace $ with ${+} \\in \\mathcal {F}_\\mathrm {\\mathsf {AC}}$ , precedence $\\mathsf {f} > \\mathsf {a} > +$ and admissible weight function $(w,w_0)$ with $w(\\mathsf {f}) = w(+) = 0$ and $w_0 = w(\\mathsf {a}) = 1$ .", "Let $\\mathcal {R}_1$ be the following ground TRS: $\\mathsf {f}(\\mathsf {a} + \\mathsf {a}) &\\rightarrow \\mathsf {f}(\\mathsf {a}) + \\mathsf {f}(\\mathsf {a})$ $\\mathsf {a} + \\mathsf {f}(\\mathsf {f}(\\mathsf {a})) &\\rightarrow \\mathsf {f}(\\mathsf {a}) + \\mathsf {f}(\\mathsf {a})$ For $1 \\leqslant i \\leqslant 2$ , let $\\ell _i$ and $r_i$ be the left- and right-hand side of rule $(i)$ , $S_i = {\\triangledown _{\\!+}}(\\ell _i)$ and $T_i = {\\triangledown _{\\!+}}(r_i)$ .", "Both rules vacuously satisfy the variable condition.", "We have $w(\\ell _1) = 2 = w(r_1)$ and $\\mathsf {f} > +$ , so $\\ell _1 >_\\mathrm {\\mathsf {S}}r_1$ holds by case 1.", "We have $w(\\ell _2) = 2 = w(r_2)$ , $S_2 = \\lbrace \\mathsf {a}, \\mathsf {f}(\\mathsf {f}(\\mathsf {a})) \\rbrace $ , and $T_2 = \\lbrace \\mathsf {f}(\\mathsf {a}), \\mathsf {f}(\\mathsf {a}) \\rbrace $ .", "Since $\\mathsf {f}(\\mathsf {a}) >_\\mathrm {\\mathsf {S}}\\mathsf {a}$ holds by case 1, $\\mathsf {f}(\\mathsf {f}(\\mathsf {a})) >_\\mathrm {\\mathsf {S}}\\mathsf {f}(\\mathsf {a})$ holds by case 2, and therefore $\\ell _2 >_\\mathrm {\\mathsf {S}}r_2$ by case 3.", "Theorem 3.3 (S90) If every symbol in $\\mathcal {F}_\\mathrm {\\mathsf {AC}}$ is minimal with respect to $>$ then $>_\\mathrm {\\mathsf {S}}$ is an AC-compatible simplification order.In [21] AC symbols are further required to have weight 0 because terms are flattened.", "Our version of $>_\\mathrm {\\mathsf {S}}$ does not impose this restriction due to the use of topflattening.", "In Section we reproveThe counterexample in [9] against the monotonicity of $>_\\mathrm {\\mathsf {S}}$ is invalid as the condition that AC symbols are minimal in the precedence is not satisfied.", "Theorem REF by showing that $>_\\mathrm {\\mathsf {S}}$ is a special case of our new AC-compatible Knuth-Bendix order." ], [ "Korovin and Voronkov's Orders", "In this section we recall the orders of KV03b.", "The first one is defined on ground terms.", "The difference with $>_\\mathrm {\\mathsf {S}}$ is that in case 3 of the definition a further case analysis is performed based on terms in $S$ and $T$ whose root symbols are not smaller than $f$ in the precedence.", "Rather than recursively comparing these terms with the order being defined, a lighter non-recursive version is used in which the weights and root symbols are considered.", "This is formally defined below.", "Given a multiset $T$ of terms, a function symbol $f$ , and a binary relation $R$ on function symbols, we define the following submultisets of $T$ : $T{\\upharpoonright }_\\mathcal {V}&= \\lbrace x \\in T \\mid x \\in \\mathcal {V}\\rbrace &{T}{\\upharpoonright }^{R}_{f} &= \\lbrace t \\in T \\setminus \\mathcal {V}\\mid \\mathsf {root}(t) \\mathrel {R} f \\rbrace $ Definition 4.1 Let $>$ be a precedence and $(w,w_0)$ a weight function.Here we do not impose totality on precedences, cf.", "[9].", "See also Example REF .", "First we define the auxiliary relations $=_\\mathsf {kv}$ and $>_\\mathsf {kv}$ on ground terms as follows: $s =_\\mathsf {kv}t$ if $w(s) = w(t)$ and $\\mathsf {root}(s) = \\mathsf {root}(t)$ , $s >_\\mathsf {kv}t$ if either $w(s) > w(t)$ or both $w(s) = w(t)$ and $\\mathsf {root}(s) > \\mathsf {root}(t)$ .", "The order $>_\\mathrm {\\mathsf {KV}}$ is inductively defined on ground terms as follows: $s >_\\mathrm {\\mathsf {KV}}t$ if either $w(s) > w(t)$ , or $w(s) = w(t)$ and one of the following alternatives holds: $s = f({s_1},\\dots ,{s_{n}})$ , $t = g({t_1},\\dots ,{t_{m}})$ , and $f > g$ , $s = f({s_1},\\dots ,{s_{n}})$ , $t = f({t_1},\\dots ,{t_{n}})$ , $f \\notin \\mathcal {F}_\\mathrm {\\mathsf {AC}}$ , $({s_1},\\dots ,{s_{n}}) >_\\mathrm {\\mathsf {KV}}^\\mathsf {lex}({t_1},\\dots ,{t_{n}})$ , $s = f(s_1,s_2)$ , $t = f(t_1,t_2)$ , $f \\in \\mathcal {F}_\\mathrm {\\mathsf {AC}}$ , and for $S = {\\triangledown _{\\!f}}(s)$ and $T = {\\triangledown _{\\!f}}(t)$ (a) ${S}{\\upharpoonright }^{\\smash{\\nless }}_{f} >_\\mathsf {kv}^\\mathsf {mul}{T}{\\upharpoonright }^{\\smash{\\nless }}_{f}$ , or (b) ${S}{\\upharpoonright }^{\\smash{\\nless }}_{f} =_\\mathsf {kv}^\\mathsf {mul}{T}{\\upharpoonright }^{\\smash{\\nless }}_{f}$ and $\|S\| > \|T\|$ , or (c) ${S}{\\upharpoonright }^{\\smash{\\nless }}_{f} =_\\mathsf {kv}^\\mathsf {mul}{T}{\\upharpoonright }^{\\smash{\\nless }}_{f}$ , $\|S\| = \|T\|$ , and $S >_\\mathrm {\\mathsf {KV}}^\\mathsf {mul}T$ .", "Here $=_\\mathrm {\\mathsf {AC}}$ is used as preorder in $>_\\mathrm {\\mathsf {KV}}^\\mathsf {lex}$ and $>_\\mathrm {\\mathsf {KV}}^\\mathsf {mul}$ whereas $=_\\mathsf {kv}$ is used in $>_\\mathsf {kv}^\\mathsf {mul}$ .", "Only in cases 2 and 3(c) the order $>_\\mathrm {\\mathsf {KV}}$ is used recursively.", "In case 3 terms rooted by the same AC symbol $f$ are compared by extracting from the topflattenings $S$ and $T$ the multisets ${S}{\\upharpoonright }^{\\smash{\\nless }}_{f}$ and ${T}{\\upharpoonright }^{\\smash{\\nless }}_{f}$ consisting of all terms rooted by a function symbol not smaller than $f$ in the precedence.", "If ${S}{\\upharpoonright }^{\\smash{\\nless }}_{f}$ is larger than ${T}{\\upharpoonright }^{\\smash{\\nless }}_{f}$ in the multiset extension of $>_\\mathsf {kv}$ , we conclude in case 3(a).", "Otherwise the multisets must be equal (with respect to $=_\\mathsf {kv}^\\mathsf {mul}$ ).", "If $S$ has more terms than $T$ , we conclude in case 3(b).", "In the final case 3(c) $S$ and $T$ have the same number of terms and we compare $S$ and $T$ in the multiset extension of $>_\\mathrm {\\mathsf {KV}}$ .", "Theorem 4.2 (KV03b) The order $>_\\mathrm {\\mathsf {KV}}$ is an AC-compatible simplification order on ground terms.", "If $>$ is total then $>_\\mathrm {\\mathsf {KV}}$ is AC-total on ground terms.", "The two orders $>_\\mathrm {\\mathsf {KV}}$ and $>_\\mathrm {\\mathsf {S}}$ are incomparable on ground TRSs.", "Example 4.3 Consider again the ground TRS $\\mathcal {R}_1$ of Example REF .", "To orient rule (1) with $>_\\mathrm {\\mathsf {KV}}$ , the weight of the unary function symbol $\\mathsf {f}$ must be 0 and admissibility demands $\\mathsf {f} > \\mathsf {a}$ and $\\mathsf {f} > +$ .", "Hence rule (1) is handled by case 1 of the definition.", "For rule (2), the multisets $S = \\lbrace \\mathsf {a}, \\mathsf {f}(\\mathsf {f}(\\mathsf {a})) \\rbrace $ and $T = \\lbrace \\mathsf {f}(\\mathsf {a}), \\mathsf {f}(\\mathsf {a}) \\rbrace $ are compared in case 3.", "We have ${S}{\\upharpoonright }^{\\smash{\\nless }}_{+} = \\lbrace \\mathsf {f}(\\mathsf {f}(\\mathsf {a})) \\rbrace $ if $+ > \\mathsf {a}$ and ${S}{\\upharpoonright }^{\\smash{\\nless }}_{+} = S$ otherwise.", "In both cases we have ${T}{\\upharpoonright }^{\\smash{\\nless }}_{+} = T$ .", "Note that neither $\\mathsf {a} >_\\mathsf {kv}\\mathsf {f}(\\mathsf {a})$ nor $\\mathsf {f}(\\mathsf {f}(\\mathsf {a})) >_\\mathsf {kv}\\mathsf {f}(\\mathsf {a})$ holds.", "Hence case 3(a) does not apply.", "But also cases 3(b) and 3(c) are not applicable as $\\mathsf {f}(\\mathsf {f}(\\mathsf {a})) =_\\mathsf {kv}\\mathsf {f}(\\mathsf {a})$ and $\\mathsf {a} \\ne _\\mathsf {kv}\\mathsf {f}(\\mathsf {a})$ .", "Hence, independent of the choice of $>$ , $\\mathcal {R}_1$ cannot be proved terminating by $>_\\mathrm {\\mathsf {KV}}$ .", "Conversely, the TRS $\\mathcal {R}_2$ resulting from reversing rule (2) in $\\mathcal {R}_1$ can be proved terminating by $>_\\mathrm {\\mathsf {KV}}$ but not by $>_\\mathrm {\\mathsf {S}}$ .", "Next we present the second order of KV03b, the extension of $>_\\mathrm {\\mathsf {KV}}$ to non-ground terms.", "Since it coincides with $>_\\mathrm {\\mathsf {KV}}$ on ground terms, we use the same notation for the order.", "In case 3 of the following definition, also variables appearing in the topflattenings $S$ and $T$ are taken into account in the first multiset comparison.", "Given a relation $\\mathrel {R}$ on terms, we write $S \\mathrel {R}^f T$ for ${S}{\\upharpoonright }^{\\smash{\\nless }}_{f}\\,\\mathrel {R}^\\mathsf {mul}\\,{T}{\\upharpoonright }^{\\smash{\\nless }}_{f}\\uplus T{\\upharpoonright }_\\mathcal {V}- S{\\upharpoonright }_\\mathcal {V}$ Note that $\\mathrel {R}^f$ depends on a precedence $>$ .", "Whenever we use $\\mathrel {R}^f$ , $>$ is defined.", "Definition 4.4 Let $>$ be a precedence and $(w,w_0)$ a weight function.", "The orders $=_\\mathsf {kv}$ and $>_\\mathsf {kv}$ are extended to non-ground terms as follows: $s =_\\mathsf {kv}t$ if $\|s\|_x = \|t\|_x$ for all $x \\in \\mathcal {V}$ , $w(s) = w(t)$ and $\\mathsf {root}(s) = \\mathsf {root}(t)$ , $s >_\\mathsf {kv}t$ if $\|s\|_x \\geqslant \|t\|_x$ for all $x \\in \\mathcal {V}$ and either $w(s) > w(t)$ or both $w(s) = w(t)$ and $\\mathsf {root}(s) > \\mathsf {root}(t)$ .", "Some tricky features of the relations $=_\\mathsf {kv}$ and $>_\\mathsf {kv}$ are illustrated below.", "Example 4.5 Let $\\mathsf {c}$ be a constant and $\\mathsf {f}$ a unary symbol.", "We have $\\mathsf {f}(\\mathsf {c}) >_\\mathsf {kv}\\mathsf {c}$ whenever admissibility is assumed: If $w(\\mathsf {f}) > 0$ then $w(\\mathsf {f}(\\mathsf {c})) > w(\\mathsf {c})$ , and if $w(\\mathsf {f}) = 0$ then admissibility imposes $\\mathsf {f} > \\mathsf {c}$ .", "On the other hand, $\\mathsf {f}(x) >_\\mathsf {kv}x$ holds only if $w(\\mathsf {f}) > 0$ , since $\\mathsf {f} \\ngtr x$ .", "Furthermore, $\\mathsf {f}(x) =_\\mathsf {kv}x$ does not hold as $\\mathsf {f} \\ne x$ .", "Example 4.6 Let $\\mathsf {c}$ be a constant with $w(\\mathsf {c}) = w_0$ , $\\mathsf {f}$ a unary symbol, and $\\mathsf {g}$ a non-AC binary symbol.", "We do not have $\\ell = \\mathsf {g}(\\mathsf {f}(\\mathsf {c}),x) >_\\mathsf {kv}\\mathsf {g}(\\mathsf {c},\\mathsf {f}(\\mathsf {c})) = r$ since $w(\\ell ) = w(r)$ and $\\mathsf {root}(\\ell ) = \\mathsf {root}(r) = \\mathsf {g}$ .", "On the other hand, $\\ell =_\\mathsf {kv}r$ also does not hold since the condition “$\|s\|_x = \|t\|_x$ for all $x \\in \\mathcal {V}$ ” is not satisfied.", "Now the non-ground version of $>_\\mathrm {\\mathsf {KV}}$ is defined as follows.", "Definition 4.7 Let $>$ be a precedence and $(w,w_0)$ a weight function.", "The order $>_\\mathrm {\\mathsf {KV}}$ is inductively defined as follows: $s >_\\mathrm {\\mathsf {KV}}t$ if $\|s\|_x \\geqslant \|t\|_x$ for all $x \\in \\mathcal {V}$ and either $w(s) > w(t)$ , or $w(s) = w(t)$ and one of the following alternatives holds: $s = f^k(t)$ and $t \\in \\mathcal {V}$ for some $k > 0$ , $s = f({s_1},\\dots ,{s_{n}})$ , $t = g({t_1},\\dots ,{t_{m}})$ , and $f > g$ , $s = f({s_1},\\dots ,{s_{n}})$ , $t = f({t_1},\\dots ,{t_{n}})$ , $f \\notin \\mathcal {F}_\\mathrm {\\mathsf {AC}}$ , $({s_1},\\dots ,{s_{n}}) >_\\mathrm {\\mathsf {KV}}^\\mathsf {lex}({t_1},\\dots ,{t_{n}})$ , $s = f(s_1,s_2)$ , $t = f(t_1,t_2)$ , $f \\in \\mathcal {F}_\\mathrm {\\mathsf {AC}}$ , and for $S = {\\triangledown _{\\!f}}(s)$ and $T = {\\triangledown _{\\!f}}(t)$ (a) $S >_\\mathsf {kv}^f T$ , or (b) $S =_\\mathsf {kv}^f T$ and $\|S\| > \|T\|$ , or (c) $S =_\\mathsf {kv}^f T$ , $\|S\| = \|T\|$ , and $S >_\\mathrm {\\mathsf {KV}}^\\mathsf {mul}T$ .", "Here $=_\\mathrm {\\mathsf {AC}}$ is used as preorder in $>_\\mathrm {\\mathsf {KV}}^\\mathsf {lex}$ and $>_\\mathrm {\\mathsf {KV}}^\\mathsf {mul}$ whereas $=_\\mathsf {kv}$ is used in $>_\\mathsf {kv}^\\mathsf {mul}$ .", "Contrary to what is claimed in [9], the order $>_\\mathrm {\\mathsf {KV}}$ of Definition REF is not a simplification order because it lacks the monotonicity property (i.e., $>_\\mathrm {\\mathsf {KV}}$ is not closed under contexts), as shown in the following examples.", "Example 4.8 We continue Example REF by adding an AC symbol $+$ .", "We obviously have $\\mathsf {f}(x) >_\\mathrm {\\mathsf {KV}}x$ .", "However, $\\mathsf {f}(x) + y >_\\mathrm {\\mathsf {KV}}x + y$ does not hold if $w(\\mathsf {f}) = 0$ .", "Let $S &= {\\triangledown _{\\!+}}(s) = \\lbrace \\mathsf {f}(x), y \\rbrace &T &= {\\triangledown _{\\!+}}(t) = \\lbrace x, y \\rbrace $ We have ${S}{\\upharpoonright }^{\\smash{\\nless }}_{+} = \\lbrace \\mathsf {f}(x) \\rbrace $ , and ${T}{\\upharpoonright }^{\\smash{\\nless }}_{+} \\cup T{\\upharpoonright }_\\mathcal {V}- S{\\upharpoonright }_\\mathcal {V}=\\lbrace x \\rbrace $ .", "As shown in Example REF , neither $\\mathsf {f}(x) >_\\mathsf {kv}x$ nor $\\mathsf {f}(x) =_\\mathsf {kv}x$ holds.", "Hence none of the cases 3(a,b,c) of Definition REF can be applied.", "Note that the use of a unary function of weight 0 is not crucial.", "The following example illustrates that the non-ground version of $>_\\mathrm {\\mathsf {KV}}$ need not be closed under contexts, even if there is no unary symbol of weight zero.", "Example 4.9 We continue Example REF by adding an AC symbol $+$ with $\\mathsf {g} > + > \\mathsf {c}$ .", "We have $\\ell = \\mathsf {g}(\\mathsf {f}(\\mathsf {c}),x) >_\\mathrm {\\mathsf {KV}}\\mathsf {g}(\\mathsf {c},\\mathsf {f}(\\mathsf {c})) = r$ by case 2.", "However, $s = \\ell + \\mathsf {c} >_\\mathrm {\\mathsf {KV}}r + \\mathsf {c} = t$ does not hold.", "Let $S &= {\\triangledown _{\\!+}}(s) = \\lbrace \\ell , \\mathsf {c} \\rbrace &T &= {\\triangledown _{\\!+}}(t) = \\lbrace r, \\mathsf {c} \\rbrace $ We have ${S}{\\upharpoonright }^{\\smash{\\nless }}_{+} = \\lbrace \\ell \\rbrace $ , ${T}{\\upharpoonright }^{\\smash{\\nless }}_{+} = \\lbrace r \\rbrace $ , and $S{\\upharpoonright }_\\mathcal {V}= T{\\upharpoonright }_\\mathcal {V}= \\varnothing $ .", "As shown in Example REF , $\\ell >_\\mathsf {kv}r$ does not hold.", "Hence case 3(a) in Definition REF does not apply.", "But also $\\ell =_\\mathsf {kv}r$ does not hold, excluding 3(b) and 3(c).", "These examples do not refute the soundness of $>_\\mathrm {\\mathsf {KV}}$ for proving AC termination; note that e.g.", "in Example REF also $x + y >_\\mathrm {\\mathsf {KV}}\\mathsf {f}(x) + y$ does not hold.", "We prove soundness by extending $>_\\mathrm {\\mathsf {KV}}$ to $>_\\mathrm {\\mathsf {KV^{\\prime }}}$ which has all desired properties.", "Definition 4.10 The order $>_\\mathrm {\\mathsf {KV^{\\prime }}}$ is obtained as in Definition REF after replacing $=_\\mathsf {kv}^f$ by $\\geqslant _{\\mathsf {kv^{\\prime }}}^f$ in cases 3(b) and 3(c), and using $\\geqslant _{\\mathsf {kv^{\\prime }}}$ as preorder in $>_\\mathsf {kv}^\\mathsf {mul}$ in case 3(a).", "Here the relation $\\geqslant _{\\mathsf {kv^{\\prime }}}$ is defined as follows: $s \\geqslant _{\\mathsf {kv^{\\prime }}}t$ if $\|s\|_x \\geqslant \|t\|_x$ for all $x \\in \\mathcal {V}$ and either $w(s) > w(t)$ , or $w(s) = w(t)$ and either $\\mathsf {root}(s) \\geqslant \\mathsf {root}(t)$ or $t \\in \\mathcal {V}$ .", "Note that $\\geqslant _{\\mathsf {kv^{\\prime }}}$ is a preorder that contains $=_\\mathrm {\\mathsf {AC}}$ .", "Example 4.11 Consider again Example REF .", "We have $\\mathsf {f}(x) \\geqslant _{\\mathsf {kv^{\\prime }}}x$ due to the new possibility “$t \\in \\mathcal {V}$ ”.", "We have $\\mathsf {f}(x) + y >_\\mathrm {\\mathsf {KV^{\\prime }}}x + y$ because now case 3(c) applies: ${S}{\\upharpoonright }^{\\smash{\\nless }}_{+} = \\lbrace \\mathsf {f}(x) \\rbrace \\geqslant _{\\mathsf {kv^{\\prime }}}^\\mathsf {mul}\\lbrace x \\rbrace = {T}{\\upharpoonright }^{\\smash{\\nless }}_{+} \\uplus T{\\upharpoonright }_\\mathcal {V}-S{\\upharpoonright }_\\mathcal {V}$ , $\|S\| = 2 = \|T\|$ , and $S = \\lbrace \\mathsf {f}(x), y \\rbrace >_\\mathrm {\\mathsf {KV^{\\prime }}}^\\mathsf {mul}\\lbrace x, y \\rbrace = T$ because $\\mathsf {f}(x) >_\\mathrm {\\mathsf {KV^{\\prime }}}x$ .", "Analogously, we have $\\ell + \\mathsf {c} >_\\mathrm {\\mathsf {KV^{\\prime }}}r + \\mathsf {c}$ for Example REF .", "The proof of the following result can be found in the online appendix.", "Theorem 4.12 The order $>_\\mathrm {\\mathsf {KV^{\\prime }}}$ is an AC-compatible simplification order.", "Since the inclusion ${>_\\mathrm {\\mathsf {KV}}} \\subseteq {>_\\mathrm {\\mathsf {KV^{\\prime }}}}$ obviously holds, it follows that $>_\\mathrm {\\mathsf {KV}}$ is a sound method for establishing AC termination, despite the lack of monotonicity." ], [ "AC-KBO", "In this section we present another AC-compatible simplification order.", "In contrast to $>_\\mathrm {\\mathsf {KV^{\\prime }}}$ , our new order $>_\\mathrm {\\mathsf {ACKBO}}$ contains $>_\\mathrm {\\mathsf {S}}$ .", "Moreover, its definition is simpler than $>_\\mathrm {\\mathsf {KV^{\\prime }}}$ since we avoid the use of an auxiliary order in case 3.", "In the next section we show that $>_\\mathrm {\\mathsf {ACKBO}}$ is decidable in polynomial-time, whereas the membership decision problem for $>_\\mathrm {\\mathsf {KV^{\\prime }}}$ is NP-complete.", "Hence it will be used as the basis for the extension discussed in Section .", "Definition 5.1 Let $>$ be a precedence and $(w,w_0)$ a weight function.", "We define $>_\\mathrm {\\mathsf {ACKBO}}$ inductively as follows: $s >_\\mathrm {\\mathsf {ACKBO}}t$ if $\|s\|_x \\geqslant \|t\|_x$ for all $x \\in \\mathcal {V}$ and either $w(s) > w(t)$ , or $w(s) = w(t)$ and one of the following alternatives holds: $s = f^k(t)$ and $t \\in \\mathcal {V}$ for some $k > 0$ , $s = f({s_1},\\dots ,{s_{n}})$ , $t = g({t_1},\\dots ,{t_{m}})$ , and $f > g$ , $s = f({s_1},\\dots ,{s_{n}})$ , $t = f({t_1},\\dots ,{t_{n}})$ , $f \\notin \\mathcal {F}_\\mathrm {\\mathsf {AC}}$ , $({s_1},\\dots ,{s_{n}}) >_\\mathrm {\\mathsf {ACKBO}}^\\mathsf {lex}({t_1},\\dots ,{t_{n}})$ , $s = f(s_1,s_2)$ , $t = f(t_1,t_2)$ , $f \\in \\mathcal {F}_\\mathrm {\\mathsf {AC}}$ , and for $S = {\\triangledown _{\\!f}}(s)$ and $T = {\\triangledown _{\\!f}}(t)$ (a) $S >_\\mathrm {\\mathsf {ACKBO}}^f T$ , or (b) $S =_\\mathrm {\\mathsf {AC}}^f T$ , and $\|S\| > \|T\|$ , or (c) $S =_\\mathrm {\\mathsf {AC}}^f T$ , $\|S\| = \|T\|$ , and ${S}{\\upharpoonright }^{\\smash{<}}_{f} >_\\mathrm {\\mathsf {ACKBO}}^\\mathsf {mul}{T}{\\upharpoonright }^{\\smash{<}}_{f}$ .", "The relation $=_\\mathrm {\\mathsf {AC}}$ is used as preorder in $>_\\mathrm {\\mathsf {ACKBO}}^\\mathsf {lex}$ and $>_\\mathrm {\\mathsf {ACKBO}}^\\mathsf {mul}$ .", "Note that, in contrast to $>_\\mathrm {\\mathsf {KV}}$ , in case 3(c) we compare the multisets ${S}{\\upharpoonright }^{\\smash{<}}_{f}$ and ${T}{\\upharpoonright }^{\\smash{<}}_{f}$ rather than $S$ and $T$ in the multiset extension of $>_\\mathrm {\\mathsf {ACKBO}}$ .", "Steinbach's order is a special case of the order defined above.", "Theorem 5.2 If every AC symbol has minimal precedence then ${>_\\mathrm {\\mathsf {S}}} = {>_\\mathrm {\\mathsf {ACKBO}}}$ .", "Suppose that every function symbol in $\\mathcal {F}_\\mathrm {\\mathsf {AC}}$ is minimal with respect to $>$ .", "We show that $s >_\\mathrm {\\mathsf {S}}t$ if and only if $s >_\\mathrm {\\mathsf {ACKBO}}t$ by induction on $s$ .", "It is clearly sufficient to consider case 3 in Definition REF and cases 3(a,b,c) in Definition REF .", "So let $s = f(s_1,s_2)$ and $t = f(t_1,t_2)$ such that $w(s) = w(t)$ and $f \\in \\mathcal {F}_\\mathrm {\\mathsf {AC}}$ .", "Let $S = {\\triangledown _{\\!f}}(s)$ and $T = {\\triangledown _{\\!f}}(t)$ .", "Let $s >_\\mathrm {\\mathsf {S}}t$ by case 3.", "We have $S >_\\mathrm {\\mathsf {S}}^\\mathsf {mul}T$ .", "Since $S >_\\mathrm {\\mathsf {S}}^\\mathsf {mul}T$ involves only comparisons $s^{\\prime } >_\\mathrm {\\mathsf {S}}t^{\\prime }$ for subterms $s^{\\prime }$ of $s$ , the induction hypothesis yields $S >_\\mathrm {\\mathsf {ACKBO}}^\\mathsf {mul}T$ .", "Because $f$ is minimal in $>$ , $S = {S}{\\upharpoonright }^{\\smash{\\nless }}_{f} \\uplus S{\\upharpoonright _\\mathcal {V}}$ and $T = {T}{\\upharpoonright }^{\\smash{\\nless }}_{f} \\uplus T{\\upharpoonright _\\mathcal {V}}$ .", "For no elements $u \\in S{\\upharpoonright _\\mathcal {V}}$ and $v \\in {T}{\\upharpoonright }^{\\smash{\\nless }}_{f}$ , $u >_\\mathrm {\\mathsf {ACKBO}}v$ or $u =_\\mathrm {\\mathsf {AC}}v$ holds.", "Hence $S >_\\mathrm {\\mathsf {ACKBO}}^\\mathsf {mul}T$ implies $S >_\\mathrm {\\mathsf {ACKBO}}^f T$ or both $S =_\\mathrm {\\mathsf {AC}}^f T$ and $S{\\upharpoonright _\\mathcal {V}} \\supsetneq T{\\upharpoonright _\\mathcal {V}}$ .", "In the former case $s >_\\mathrm {\\mathsf {ACKBO}}t$ is due to case 3(a) in Definition REF .", "In the latter case we have $\|S\| > \|T\|$ and $s >_\\mathrm {\\mathsf {ACKBO}}t$ follows by case 3(b).", "Let $s >_\\mathrm {\\mathsf {ACKBO}}t$ by applying one of the cases 3(a,b,c) in Definition REF .", "Suppose 3(a) applies.", "Then we have $S >_\\mathrm {\\mathsf {ACKBO}}^f T$ .", "Since $f$ is minimal in $>$ , ${S}{\\upharpoonright }^{\\smash{\\nless }}_{f} = S - S{\\upharpoonright }_\\mathcal {V}$ and ${T}{\\upharpoonright }^{\\smash{\\nless }}_{f} \\uplus T{\\upharpoonright }_\\mathcal {V}= T$ .", "Hence $S >_\\mathrm {\\mathsf {ACKBO}}^\\mathsf {mul}(T - S{\\upharpoonright }_\\mathcal {V}) \\uplus S{\\upharpoonright }_\\mathcal {V}\\supseteq T$ .", "We obtain $S >_\\mathrm {\\mathsf {S}}^\\mathsf {mul}T$ from the induction hypothesis and thus case 3 in Definition REF applies.", "Suppose 3(b) applies.", "Analogous to the previous case, the inclusion $S =_\\mathrm {\\mathsf {AC}}^\\mathsf {mul}(T - S{\\upharpoonright }_\\mathcal {V}) \\uplus S{\\upharpoonright }_\\mathcal {V}\\supseteq T$ holds.", "Since $\|S\| > \|T\|$ , $S =_\\mathrm {\\mathsf {AC}}^\\mathsf {mul}T$ is not possible.", "Thus $(T - S{\\upharpoonright }_\\mathcal {V}) \\uplus S{\\upharpoonright }_\\mathcal {V}\\supsetneq T$ and hence $S >_\\mathrm {\\mathsf {S}}^\\mathsf {mul}T$ .", "If case 3(c) applies then ${S}{\\upharpoonright }^{\\smash{<}}_{f} >_\\mathrm {\\mathsf {ACKBO}}^\\mathsf {mul}{T}{\\upharpoonright }^{\\smash{<}}_{f}$ .", "This is impossible since both sides are empty as $f$ is minimal in $>$ .", "$\\Box $ The following example shows that $>_\\mathrm {\\mathsf {ACKBO}}$ is a proper extension of $>_\\mathrm {\\mathsf {S}}$ and incomparable with $>_\\mathrm {\\mathsf {KV^{\\prime }}}$ .", "Example 5.3 Consider the TRS $\\mathcal {R}_3$ consisting of the rules $\\mathsf {f}(x+y) &\\rightarrow \\mathsf {f}(x)+y&\\mathsf {h}(\\mathsf {a},\\mathsf {b}) &\\rightarrow \\mathsf {h}(\\mathsf {b},\\mathsf {a})&\\mathsf {h}(\\mathsf {g}(\\mathsf {a}),\\mathsf {a}) &\\rightarrow \\mathsf {h}(\\mathsf {a},\\mathsf {g}(\\mathsf {b}))\\\\\\mathsf {g}(x)+y &\\rightarrow \\mathsf {g}(x+y)&\\mathsf {h}(\\mathsf {a},\\mathsf {g}(\\mathsf {g}(\\mathsf {a}))) &\\rightarrow \\mathsf {h}(\\mathsf {g}(\\mathsf {a}),\\mathsf {f}(\\mathsf {a}))&\\mathsf {h}(\\mathsf {g}(\\mathsf {a}),\\mathsf {b}) &\\rightarrow \\mathsf {h}(\\mathsf {a},\\mathsf {g}(\\mathsf {a}))\\\\\\mathsf {f}(\\mathsf {a})+\\mathsf {g}(\\mathsf {b}) &\\rightarrow \\mathsf {f}(\\mathsf {b})+\\mathsf {g}(\\mathsf {a})$ over the signature $\\lbrace {+}, \\mathsf {f}, \\mathsf {g}, \\mathsf {h}, \\mathsf {a}, \\mathsf {b} \\rbrace $ with ${+} \\in \\mathcal {F}_\\mathrm {\\mathsf {AC}}$ .", "Consider the precedence $\\mathsf {f} > {+} > \\mathsf {g} > \\mathsf {a} > \\mathsf {b} > \\mathsf {h}$ together with the admissible weight function $(w,w_0)$ with $w({+}) &= w(\\mathsf {h}) = 0&w(\\mathsf {f}) &= w(\\mathsf {a}) = w(\\mathsf {b}) = w_0 = 1&w(\\mathsf {g}) &= 2$ The interesting rule is $\\mathsf {f}(\\mathsf {a})+\\mathsf {g}(\\mathsf {b}) \\rightarrow \\mathsf {f}(\\mathsf {b})+\\mathsf {g}(\\mathsf {a})$ .", "For $S = {\\triangledown _{\\!\\,+}}(\\mathsf {f}(\\mathsf {a})+\\mathsf {g}(\\mathsf {b}))$ and $T = {\\triangledown _{\\!\\,+}}(\\mathsf {f}(\\mathsf {b})+\\mathsf {g}(\\mathsf {a}))$ the multisets $S^{\\prime } = {S}{\\upharpoonright }^{\\smash{\\nless }}_{+} = \\lbrace \\mathsf {f}(\\mathsf {a}) \\rbrace $ and $T^{\\prime } = {T}{\\upharpoonright }^{\\smash{\\nless }}_{+} \\uplus T{\\upharpoonright }_\\mathcal {V}-S{\\upharpoonright }_\\mathcal {V}= \\lbrace \\mathsf {f}(\\mathsf {b}) \\rbrace $ satisfy $S^{\\prime } >_\\mathrm {\\mathsf {ACKBO}}^\\mathsf {mul}T^{\\prime }$ as $\\mathsf {f}(\\mathsf {a}) >_\\mathrm {\\mathsf {ACKBO}}\\mathsf {f}(\\mathsf {b})$ , so that case 3(a) of Definition REF applies.", "All other rules are oriented from left to right by both $>_\\mathrm {\\mathsf {KV^{\\prime }}}$ and $>_\\mathrm {\\mathsf {ACKBO}}$ , and they enforce a precedence and weight function which are identical (or very similar) to the one given above.", "Since $>_\\mathrm {\\mathsf {KV^{\\prime }}}$ orients the rule $\\mathsf {f}(\\mathsf {a})+\\mathsf {g}(\\mathsf {b}) \\rightarrow \\mathsf {f}(\\mathsf {b})+\\mathsf {g}(\\mathsf {a})$ from right to left, $\\mathcal {R}_3$ cannot be compatible with $>_\\mathrm {\\mathsf {KV^{\\prime }}}$ .", "It is easy to see that the rule $\\mathsf {g}(x)+y \\rightarrow \\mathsf {g}(x+y)$ requires $+ >\\mathsf {g}$ , and hence $>_\\mathrm {\\mathsf {S}}$ cannot be applied.", "Figure: Comparison.Fig.", "REF summarizes the relationships between the orders introduced so far.", "In the following, we show that $>_\\mathrm {\\mathsf {ACKBO}}$ is an AC-compatible simplification order.", "As a consequence, correctness of $>_\\mathrm {\\mathsf {S}}$ (i.e., Theorem REF ) is concluded by Theorem REF .", "In the online appendix we prove the following property.", "Lemma 5.4 The pair $({=_\\mathrm {\\mathsf {AC}}},{>_\\mathrm {\\mathsf {ACKBO}}})$ is an order pair.", "The subterm property is an easy consequence of transitivity and admissibility.", "Lemma 5.5 The order $>_\\mathrm {\\mathsf {ACKBO}}$ has the subterm property.", "$\\Box $ Next we prove that $>_\\mathrm {\\mathsf {ACKBO}}$ is closed under contexts.", "The following lemma is an auxiliary result needed for its proof.", "In order to reuse this lemma for the correctness proof of $>_\\mathrm {\\mathsf {KV^{\\prime }}}$ in the online appendix, we prove it in an abstract setting.", "Lemma 5.6 Let $({\\mathrel {\\succsim }},{\\mathrel {\\succ }})$ be an order pair and $f \\in \\mathcal {F}_\\mathrm {\\mathsf {AC}}$ with $f(u,v) \\mathrel {\\succ }u, v$ for all terms $u$ and $v$ .", "If $s \\mathrel {\\succsim }t$ then $\\lbrace s \\rbrace \\mathrel {\\succsim }^\\mathsf {mul}{\\triangledown _{\\!f}}(t)$ or $\\lbrace s \\rbrace \\mathrel {\\succ }^\\mathsf {mul}{\\triangledown _{\\!f}}(t)$ .", "If $s \\mathrel {\\succ }t$ then $\\lbrace s \\rbrace \\mathrel {\\succ }^\\mathsf {mul}{\\triangledown _{\\!f}}(t)$ .", "Let ${\\triangledown _{\\!f}}(t) = \\lbrace {t_1},\\dots ,{t_{m}} \\rbrace $ .", "If $m = 1$ then ${\\triangledown _{\\!f}}(t) = \\lbrace t \\rbrace $ and the lemma holds trivially.", "Otherwise we get $t \\mathrel {\\succ }t_j$ for all $1 \\leqslant j \\leqslant m$ by recursively applying the assumption.", "Hence $s \\mathrel {\\succ }t_j$ by the transitivity of $\\mathrel {\\succ }$ or the compatibility of $\\mathrel {\\succ }$ and $\\mathrel {\\succsim }$ .", "We conclude that $\\lbrace s \\rbrace \\mathrel {\\succ }^\\mathsf {mul}{\\triangledown _{\\!f}}(t)$ .", "In the following proof of closure under contexts, admissibility is essential.", "This is in contrast to the corresponding result for standard KBO.", "Lemma 5.7 If $(w,w_0)$ is admissible for $>$ then $>_\\mathrm {\\mathsf {ACKBO}}$ is closed under contexts.", "Suppose $s >_\\mathrm {\\mathsf {ACKBO}}t$ .", "We consider the context $h(\\Box ,u)$ with $h \\in \\mathcal {F}_\\mathrm {\\mathsf {AC}}$ and $u$ an arbitrary term, and prove that $s^{\\prime } = h(s,u) >_\\mathrm {\\mathsf {ACKBO}}h(t,u) = t^{\\prime }$ .", "Closure under contexts of $>_\\mathrm {\\mathsf {ACKBO}}$ follows then by induction; contexts rooted by a non-AC symbol are handled as in the proof for standard KBO.", "If $w(s) > w(t)$ then obviously $w(s^{\\prime }) > w(t^{\\prime })$ .", "So we assume $w(s) =w(t)$ .", "Let $S = {\\triangledown _{\\!h}}(s)$ , $T = {\\triangledown _{\\!h}}(t)$ , and $U = {\\triangledown _{\\!h}}(u)$ .", "Note that ${\\triangledown _{\\!h}}(s^{\\prime }) = S \\uplus U$ and ${\\triangledown _{\\!h}}(t^{\\prime }) = T \\uplus U$ .", "Because $>_\\mathrm {\\mathsf {ACKBO}}^\\mathsf {mul}$ is closed under multiset sum, it suffices to show that one of the cases 3(a,b,c) of Definition REF holds for $S$ and $T$ .", "Let $f = \\mathsf {root}(s)$ and $g = \\mathsf {root}(t)$ .", "We distinguish the following cases.", "Suppose $f \\nleq h$ .", "We have $S = {S}{\\upharpoonright }^{\\smash{\\nless }}_{h} = \\lbrace s \\rbrace $ , and from Lemmata REF and REF we obtain $S >_\\mathrm {\\mathsf {ACKBO}}^\\mathsf {mul}T$ .", "Since $T$ is a superset of ${T}{\\upharpoonright }^{\\smash{\\nless }}_{h} \\uplus T{\\upharpoonright }_\\mathcal {V}- S{\\upharpoonright }_\\mathcal {V}$ , 3(a) applies.", "Suppose $f = h > g$ .", "We have ${T}{\\upharpoonright }^{\\smash{\\nless }}_{h} \\uplus T{\\upharpoonright }_\\mathcal {V}= \\varnothing $ .", "If ${S}{\\upharpoonright }^{\\smash{\\nless }}_{h} \\ne \\varnothing $ , then 3(a) applies.", "Otherwise, since AC symbols are binary and $T = \\lbrace t \\rbrace $ , $\|S\| \\geqslant 2 > 1 = \|T\|$ .", "Hence 3(b) applies.", "If $f = g = h$ then $s >_\\mathrm {\\mathsf {ACKBO}}t$ must be derived by one of the cases 3(a,b,c) for $S$ and $T$ .", "Suppose $f, g < h$ .", "We have ${S}{\\upharpoonright }^{\\smash{\\nless }}_{h} = {T}{\\upharpoonright }^{\\smash{\\nless }}_{h} \\uplus T{\\upharpoonright }_\\mathcal {V}=\\varnothing $ , $\|S\| = \|T\| = 1$ , and ${S}{\\upharpoonright }^{\\smash{<}}_{h} = \\lbrace s \\rbrace >_\\mathrm {\\mathsf {ACKBO}}^\\mathsf {mul}\\lbrace t \\rbrace = {T}{\\upharpoonright }^{\\smash{<}}_{h}$ .", "Hence 3(c) holds.", "Note that $f \\geqslant g$ since $w(s) = w(t)$ and $s >_\\mathrm {\\mathsf {ACKBO}}t$ .", "Moreover, if $t \\in \\mathcal {V}$ then $s = f^k(t)$ for some $k > 0$ with $w(f) = 0$ , which entails $f > h$ due to the admissibility assumption.", "Closure under substitutions is the trickiest part since by substituting AC-rooted terms for variables that appear in the topflattening of a term, the structure of the term changes.", "In the proof, the multisets $\\lbrace t \\in T \\mid t \\notin \\mathcal {V}\\rbrace $ , $\\lbrace t\\sigma \\mid t \\in T \\rbrace $ , and $\\lbrace {\\triangledown _{\\!f}}(t) \\mid t \\in T \\rbrace $ are denoted by $T{\\upharpoonright }_\\mathcal {F}$ , $T\\sigma $ , and ${\\triangledown _{\\!f}}(T)$ , respectively.", "Lemma 5.8 Let $>$ be a precedence, $f \\in \\mathcal {F}_\\mathrm {\\mathsf {AC}}$ , and $({\\mathrel {\\succsim }},{\\mathrel {\\succ }})$ an order pair on terms such that $\\mathrel {\\succsim }$ and $\\mathrel {\\succ }$ are closed under substitutions and $f(x,y) \\mathrel {\\succ }x, y$ .", "Consider terms $s$ and $t$ such that $S = {\\triangledown _{\\!f}}(s)$ , $T = {\\triangledown _{\\!f}}(t)$ , $S^{\\prime } = {\\triangledown _{\\!f}}(s\\sigma )$ , and $T^{\\prime } = {\\triangledown _{\\!f}}(t\\sigma )$ .", "If $S \\mathrel {\\succ }^f T$ then $S^{\\prime } \\mathrel {\\succ }^f T^{\\prime }$ .", "If $S \\mathrel {\\succsim }^f T$ then $S^{\\prime } \\mathrel {\\succ }^f T^{\\prime }$ or $S^{\\prime } \\mathrel {\\succsim }^f T^{\\prime }$ .", "In the latter case $\|S\| - \|T\| \\leqslant \|S^{\\prime }\| - \|T^{\\prime }\|$ and ${S^{\\prime }}{\\upharpoonright }^{\\smash{<}}_{f} \\mathrel {\\succ }^\\mathsf {mul}{T^{\\prime }}{\\upharpoonright }^{\\smash{<}}_{f}$ whenever ${S}{\\upharpoonright }^{\\smash{<}}_{f} \\mathrel {\\succ }^\\mathsf {mul}{T}{\\upharpoonright }^{\\smash{<}}_{f}$ .", "Let $v$ be an arbitrary term.", "By the assumption on $\\mathrel {\\succ }$ we have either $\\lbrace v \\rbrace = {\\triangledown _{\\!f}}(v)$ or both $\\lbrace v \\rbrace \\mathrel {\\succ }^\\mathsf {mul}{\\triangledown _{\\!f}}(v)$ and $1 < \|{\\triangledown _{\\!f}}(v)\|$ .", "Hence, for any set $V$ of terms, either $V = {\\triangledown _{\\!f}}(V)$ or both $V \\mathrel {\\succ }^\\mathsf {mul}{\\triangledown _{\\!f}}(V)$ and $\|V\| < \|{\\triangledown _{\\!f}}(V)\|$ .", "Moreover, for $V = {\\triangledown _{\\!f}}(v)$ , the following equalities hold: ${{\\triangledown _{\\!f}}(v\\sigma )}{\\upharpoonright }^{\\smash{\\nless }}_{f} & ={V}{\\upharpoonright }^{\\smash{\\nless }}_{f}\\sigma \\uplus {{\\triangledown _{\\!f}}(V{\\upharpoonright }_\\mathcal {V}\\sigma )}{\\upharpoonright }^{\\smash{\\nless }}_{f}&{\\triangledown _{\\!f}}(v\\sigma ){\\upharpoonright }_\\mathcal {V}& ={\\triangledown _{\\!f}}(V{\\upharpoonright }_\\mathcal {V}\\sigma ){\\upharpoonright }_\\mathcal {V}$ To prove the lemma, assume $S \\mathrel {R}^f T$ for ${\\mathrel {R}} \\in \\lbrace {\\mathrel {\\succsim }}, {\\mathrel {\\succ }} \\rbrace $ .", "We have ${S}{\\upharpoonright }^{\\smash{\\nless }}_{f} \\mathrel {R}^\\mathsf {mul}{T}{\\upharpoonright }^{\\smash{\\nless }}_{f} \\uplus U$ where $U = (T - S){\\upharpoonright }_\\mathcal {V}$ .", "Since multiset extensions preserve closure under substitutions, ${S}{\\upharpoonright }^{\\smash{\\nless }}_{f}\\sigma \\, \\mathrel {R}^\\mathsf {mul}\\,{T}{\\upharpoonright }^{\\smash{\\nless }}_{f}\\sigma \\uplus U\\sigma $ follows.", "Using the above (in)equalities, we obtain ${S^{\\prime }}{\\upharpoonright }^{\\smash{\\nless }}_{f}&=^{\\phantom{\\mathsf {mul}}}{S}{\\upharpoonright }^{\\smash{\\nless }}_{f}\\sigma \\uplus {{\\triangledown _{\\!f}}(S{\\upharpoonright }_\\mathcal {V}\\sigma )}{\\upharpoonright }^{\\smash{\\nless }}_{f}\\\\&\\mathrel {R}^\\mathsf {mul}{T}{\\upharpoonright }^{\\smash{\\nless }}_{f}\\sigma \\uplus {{\\triangledown _{\\!f}}(S{\\upharpoonright }_\\mathcal {V}\\sigma )}{\\upharpoonright }^{\\smash{\\nless }}_{f}\\uplus U\\sigma \\\\&\\mathrel {O}^{\\phantom{\\mathsf {mul}}}{T}{\\upharpoonright }^{\\smash{\\nless }}_{f}\\sigma \\uplus {{\\triangledown _{\\!f}}(S{\\upharpoonright }_\\mathcal {V}\\sigma )}{\\upharpoonright }^{\\smash{\\nless }}_{f}\\uplus {\\triangledown _{\\!f}}(U\\sigma )\\\\&=^{\\phantom{\\mathsf {mul}}}{T}{\\upharpoonright }^{\\smash{\\nless }}_{f}\\sigma \\uplus {{\\triangledown _{\\!f}}(S{\\upharpoonright }_\\mathcal {V}\\sigma )}{\\upharpoonright }^{\\smash{\\nless }}_{f}\\uplus {\\triangledown _{\\!f}}(U\\sigma ){\\upharpoonright }_\\mathcal {V}\\uplus {{\\triangledown _{\\!f}}(U\\sigma )}{\\upharpoonright }^{\\smash{\\nless }}_{f}\\uplus {{\\triangledown _{\\!f}}(U\\sigma )}{\\upharpoonright }^{\\smash{<}}_{f}\\\\&\\mathrel {P}^{\\phantom{\\mathsf {mul}}}{T}{\\upharpoonright }^{\\smash{\\nless }}_{f}\\sigma \\uplus {{\\triangledown _{\\!f}}(T{\\upharpoonright }_\\mathcal {V}\\sigma )}{\\upharpoonright }^{\\smash{\\nless }}_{f}\\uplus {\\triangledown _{\\!f}}(U\\sigma ){\\upharpoonright }_\\mathcal {V}\\\\&=^{\\phantom{\\mathsf {mul}}}{T}{\\upharpoonright }^{\\smash{\\nless }}_{f}\\sigma \\uplus {{\\triangledown _{\\!f}}(T{\\upharpoonright }_\\mathcal {V}\\sigma )}{\\upharpoonright }^{\\smash{\\nless }}_{f}\\uplus {\\triangledown _{\\!f}}(T{\\upharpoonright }_\\mathcal {V}\\sigma ){\\upharpoonright }_\\mathcal {V}- {\\triangledown _{\\!f}}(S{\\upharpoonright }_\\mathcal {V}\\sigma ){\\upharpoonright }_\\mathcal {V}\\\\&=^{\\phantom{\\mathsf {mul}}}{T^{\\prime }}{\\upharpoonright }^{\\smash{\\nless }}_{f} \\uplus T^{\\prime }{\\upharpoonright }_\\mathcal {V}- S^{\\prime }{\\upharpoonright }_\\mathcal {V}$ Here $O$ denotes $=$ if $U\\sigma = {\\triangledown _{\\!f}}(U\\sigma )$ and $\\mathrel {\\succ }^\\mathsf {mul}$ if $\|U\\sigma \| < \|{\\triangledown _{\\!f}}(U\\sigma )\|$ , while $P$ denotes $=$ if ${U\\sigma }{\\upharpoonright }^{\\smash{<}}_{f} = \\varnothing $ and $\\supsetneq $ otherwise.", "Since $({\\mathrel {\\succsim }^\\mathsf {mul}},{\\mathrel {\\succ }^\\mathsf {mul}})$ is an order pair with ${\\supseteq } \\subseteq {\\mathrel {\\succsim }^\\mathsf {mul}}$ and ${\\supsetneq } \\subseteq {\\mathrel {\\succ }^\\mathsf {mul}}$ , we obtain $S^{\\prime } \\mathrel {R}^f T^{\\prime }$ .", "It remains to show 2.", "If $S^{\\prime } \\mathrel {\\nsucc }^f T^{\\prime }$ then $O$ and $P$ are both $=$ and thus $U\\sigma = {\\triangledown _{\\!f}}(U\\sigma )$ and ${U\\sigma }{\\upharpoonright }^{\\smash{<}}_{f} = \\varnothing $ .", "Let $X = S{\\upharpoonright }_\\mathcal {V}\\cap T{\\upharpoonright }_\\mathcal {V}$ .", "We have $U = T{\\upharpoonright }_\\mathcal {V}- X$ .", "Since $\|W{\\upharpoonright }_\\mathcal {F}\\sigma \| = \|W{\\upharpoonright }_\\mathcal {F}\|$ and $\|W\| \\leqslant \|{\\triangledown _{\\!f}}(W)\|$ for an arbitrary set $W$ of terms, we have $\|S^{\\prime }\| \\geqslant \|S\| - \|X\| + \|{\\triangledown _{\\!f}}(X \\sigma )\|$ .", "From $\|U\\sigma \| = \|U\| = \|T{\\upharpoonright }_\\mathcal {V}\| - \|X\|$ we obtain $\|T^{\\prime }\| = \|T{\\upharpoonright }_\\mathcal {F}\\sigma \| + \|{\\triangledown _{\\!f}}(U\\sigma )\| +\|{\\triangledown _{\\!f}}(X \\sigma )\| = \|T\| - \|X\| + \|{\\triangledown _{\\!f}}(X \\sigma )\|$ Hence $\|S\| - \|T\| \\leqslant \|S^{\\prime }\| - \|T^{\\prime }\|$ as desired.", "Suppose ${S}{\\upharpoonright }^{\\smash{<}}_{f} \\mathrel {\\succ }^\\mathsf {mul}{T}{\\upharpoonright }^{\\smash{<}}_{f}$ .", "From ${U\\sigma }{\\upharpoonright }^{\\smash{<}}_{f} = \\varnothing $ we infer ${T{\\upharpoonright }_\\mathcal {V}\\sigma }{\\upharpoonright }^{\\smash{<}}_{f} \\subseteq {S{\\upharpoonright }_\\mathcal {V}\\sigma }{\\upharpoonright }^{\\smash{<}}_{f}$ .", "Because ${S^{\\prime }}{\\upharpoonright }^{\\smash{<}}_{f} = {S}{\\upharpoonright }^{\\smash{<}}_{f}\\sigma \\uplus {S{\\upharpoonright }_\\mathcal {V}\\sigma }{\\upharpoonright }^{\\smash{<}}_{f}$ and ${T^{\\prime }}{\\upharpoonright }^{\\smash{<}}_{f} = {T}{\\upharpoonright }^{\\smash{<}}_{f}\\sigma \\uplus {T{\\upharpoonright }_\\mathcal {V}\\sigma }{\\upharpoonright }^{\\smash{<}}_{f}$ , closure under substitutions of $\\mathrel {\\succ }^\\mathsf {mul}$ (which it inherits from $\\mathrel {\\succ }$ and $\\mathrel {\\succsim }$ ) yields the desired ${S^{\\prime }}{\\upharpoonright }^{\\smash{<}}_{f} \\mathrel {\\succ }^\\mathsf {mul}{T^{\\prime }}{\\upharpoonright }^{\\smash{<}}_{f}$ .", "$\\Box $ Lemma 5.9 $>_\\mathrm {\\mathsf {ACKBO}}$ is closed under substitutions.", "If $s >_\\mathrm {\\mathsf {ACKBO}}t$ is obtained by cases 0 or 1 in Definition REF , the proof for standard KBO goes through.", "If 3(a) or 3(b) is used to obtain $s >_\\mathrm {\\mathsf {ACKBO}}t$ , according to Lemma REF one of these cases also applies to $s\\sigma >_\\mathrm {\\mathsf {ACKBO}}t\\sigma $ .", "The final case is 3(c).", "So ${\\triangledown _{\\!f}}(s){\\upharpoonright }_f^<>_\\mathrm {\\mathsf {ACKBO}}^\\mathsf {mul}{\\triangledown _{\\!f}}(t){\\upharpoonright }_f^<$ .", "Suppose $s\\sigma >_\\mathrm {\\mathsf {ACKBO}}t\\sigma $ cannot be obtained by 3(a) or 3(b).", "Lemma REF (2) yields $\|{{\\triangledown _{\\!f}}(s\\sigma )}\| = \|{{\\triangledown _{\\!f}}(t\\sigma )}\|$ and ${\\triangledown _{\\!f}}(s\\sigma ){\\upharpoonright }_f^< >_\\mathrm {\\mathsf {ACKBO}}^\\mathsf {mul}{\\triangledown _{\\!f}}(t\\sigma ){\\upharpoonright }_f^<$ .", "Hence case 3(c) is applicable to obtain $s\\sigma >_\\mathrm {\\mathsf {ACKBO}}t\\sigma $ .", "We arrive at the main theorem of this section.", "Theorem 5.10 The order $>_\\mathrm {\\mathsf {ACKBO}}$ is an AC-compatible simplification order.", "$\\Box $ Since we deal with finite non-variadic signatures, simplification orders are well-founded.", "The following example shows that AC-KBO is not incremental, i.e., orientability is not necessarily preserved when the precedence is extended.", "This is in contrast to the AC-RPO of R02.", "However, this is not necessarily a disadvantage; actually, the example shows that by allowing partial precedences more TRSs can be proved to be AC terminating using AC-KBO.", "Example 5.11 Consider the TRS $\\mathcal {R}$ consisting of the rules $\\mathsf {a} \\circ (\\mathsf {b} \\bullet \\mathsf {c})&\\rightarrow \\mathsf {b} \\circ \\mathsf {f}(\\mathsf {a} \\bullet \\mathsf {c}) &\\mathsf {a} \\bullet (\\mathsf {b} \\circ \\mathsf {c})&\\rightarrow \\mathsf {b} \\bullet \\mathsf {f}(\\mathsf {a} \\circ \\mathsf {c})$ over the signature $\\mathcal {F}= \\lbrace \\mathsf {a}, \\mathsf {b}, \\mathsf {c}, \\mathsf {f}, {\\circ }, {\\bullet } \\rbrace $ with ${\\circ }, {\\bullet } \\in \\mathcal {F}_\\mathrm {\\mathsf {AC}}$ .", "By taking the precedence $\\mathsf {f} > \\mathsf {a}, \\mathsf {b}, \\mathsf {c}, {\\circ }, {\\bullet }$ and admissible weight function $(w,w_0)$ with $w(\\mathsf {f}) &= w({\\circ }) = w({\\bullet }) = 0 &w_0 &= w(\\mathsf {a}) = w(\\mathsf {c}) = 1 &w(\\mathsf {b}) &= 2$ the resulting $>_\\mathrm {\\mathsf {ACKBO}}$ orients both rules from left to right.", "It is essential that $\\circ $ and $\\bullet $ are incomparable in the precedence: We must have $w(\\mathsf {f}) = 0$ , so $\\mathsf {f} > \\mathsf {a}, \\mathsf {b}, \\mathsf {c}, {\\circ }, {\\bullet }$ is enforced by admissibility.", "If ${\\circ } > {\\bullet }$ then the first rule can only be oriented from left to right if $\\mathsf {a} >_\\mathrm {\\mathsf {ACKBO}}\\mathsf {f}(\\mathsf {a} \\bullet \\mathsf {c})$ holds, which contradicts the subterm property.", "If ${\\bullet } > {\\circ }$ then we use the second rule to obtain the impossible $\\mathsf {a} >_\\mathrm {\\mathsf {ACKBO}}\\mathsf {f}(\\mathsf {a} \\circ \\mathsf {c})$ .", "Similarly, $\\mathcal {R}$ is also orientable by $>_\\mathrm {\\mathsf {KV^{\\prime }}}$ but we must adopt a non-total precedence.", "The easy proof of the final theorem in this section can be found in the online appendix.", "Theorem 5.12 If $>$ is total then $>_\\mathrm {\\mathsf {ACKBO}}$ is AC-total on ground terms." ], [ "Complexity", "In this section we discuss complexity issues for the orders defined in the preceding sections.", "We start with the membership problem: Given two terms $s$ and $t$ , a weight function, and a precedence, does $s > t$ hold?", "For plain KBO this problem is known to be decidable in linear time [15].", "For $>_{\\mathrm {\\mathsf {S}}}$ , $>_\\mathrm {\\mathsf {KV}}$ , and $>_\\mathrm {\\mathsf {ACKBO}}$ we show the problem to be decidable in polynomial time, but we start with the unexpected result that $>_\\mathrm {\\mathsf {KV^{\\prime }}}$ membership is NP-complete.", "For NP-hardness we use the reduction technique of [Theorem 4.2]TAN12.", "Theorem 6.1 The decision problem for $>_\\mathrm {\\mathsf {KV^{\\prime }}}$ is NP-complete.", "We start with NP-hardness.", "It is sufficient to show NP-hardness of deciding $S >_{\\mathsf {kv^{\\prime }}}^{\\mathsf {mul}} T$ since we can easily construct terms $s$ and $t$ such that $S >_{\\mathsf {kv^{\\prime }}}^{\\mathsf {mul}} T$ if and only if $s >_\\mathrm {\\mathsf {KV^{\\prime }}}t$ .", "To wit, for $S = \\lbrace {s_1},\\dots ,{s_{n}} \\rbrace $ and $T = \\lbrace {t_1},\\dots ,{t_{m}} \\rbrace $ we introduce an AC symbol $\\circ $ and constants $\\mathsf {c}$ and $\\mathsf {d}$ such that $\\circ > \\mathsf {c}, \\mathsf {d}$ and define $s &= s_1 \\circ \\dots \\circ s_n \\circ \\mathsf {c} &t &= t_1 \\circ \\dots \\circ t_m \\circ \\mathsf {d} \\circ \\mathsf {d}$ The weights of $\\mathsf {c}$ and $\\mathsf {d}$ should be chosen so that $w(s) = w(t)$ .", "If $S >_{\\mathsf {kv^{\\prime }}}^\\mathsf {mul}T$ then case 3(a) applies for $s >_\\mathrm {\\mathsf {KV^{\\prime }}}t$ .", "Otherwise, $S \\geqslant _{\\mathsf {kv^{\\prime }}}^\\mathsf {mul}T$ implies $n = m$ and thus $\|{\\triangledown _{\\!\\circ }}(s)\| < \|{\\triangledown _{\\!\\circ }}(t)\|$ .", "Hence neither case 3(b) nor 3(c) applies.", "We reduce a non-empty CNF SAT problem $\\phi = \\lbrace C_1, \\dots , C_m \\rbrace $ over propositional variables ${x_1},\\dots ,{x_{n}}$ to the decision problem $S_\\phi >_{\\mathsf {kv^{\\prime }}}^\\mathsf {mul}T_\\phi $ .", "The multisets $S_\\phi $ and $T_\\phi $ will consist of terms in $\\mathcal {T}(\\lbrace \\mathsf {a}, \\mathsf {f} \\rbrace ,\\lbrace {x_1},\\dots ,{x_{n}},{y_1},\\dots ,{y_{m}} \\rbrace )$ , where $\\mathsf {a}$ is a constant with $w(\\mathsf {a}) = w_0$ and $\\mathsf {f}$ has arity $m+1$ .", "For each $1 \\leqslant j \\leqslant m$ and literal $l$ , we define $s_j(l) = {\\left\\lbrace \\begin{array}{ll}y_j & \\text{if $l \\in C_j$} \\\\\\mathsf {a} & \\text{otherwise}\\end{array}\\right.", "}$ Moreover, for each $1 \\leqslant i \\leqslant n$ we define $t_i^+ &= \\mathsf {f}(x_i, s_1(x_i), \\dots , s_m(x_i)) &t_i^- &= \\mathsf {f}(x_i, s_1(\\lnot x_i), \\dots , s_m(\\lnot x_i))$ and $t_i = \\mathsf {f}(x_i, \\mathsf {a}, \\dots , \\mathsf {a})$ .", "Note that $w(t_i^+) = w(t_i^-) = w(t_i) > w(y_j)$ for all $1 \\leqslant i \\leqslant n$ and $1 \\leqslant j \\leqslant m$ .", "Finally, we define $S_\\phi &= \\lbrace t_1^+, t_1^-, \\dots , t_n^+, t_n^- \\rbrace &T_\\phi &= \\lbrace {t_1},\\dots ,{t_{n}}, {y_1},\\dots ,{y_{m}} \\rbrace $ Note that for every $1 \\leqslant i \\leqslant n$ there is no $s \\in S_\\phi $ such that $s >_\\mathsf {kv}t_i$ .", "Hence $S_\\phi >_{\\mathsf {kv^{\\prime }}}^\\mathsf {mul}T_\\phi $ if and only if $S_\\phi $ can be written as $\\lbrace {s_1},\\dots ,{s_{n}}, {s^{\\prime }_1},\\dots ,{s^{\\prime }_{n}} \\rbrace $ such that $s_i \\geqslant _{\\mathsf {kv^{\\prime }}}t_i$ for all $1 \\leqslant i \\leqslant n$ , and for all $1 \\leqslant j \\leqslant m$ there exists an $1 \\leqslant i \\leqslant n$ such that $s^{\\prime }_i >_\\mathsf {kv}y_j$ .", "It is easy to see that the only candidates for $s_i$ are $t_i^+$ and $t_i^-$ .", "Now suppose $S_\\phi >_{\\mathsf {kv^{\\prime }}}^\\mathsf {mul}T_\\phi $ with $S_\\phi $ written as above.", "Consider the assignment $\\alpha $ defined as follows: $\\alpha (x_i)$ is true if and only if $s_i = t_i^-$ .", "We claim that $\\alpha $ satisfies every $C_j \\in \\phi $ .", "We know that there exists $1 \\leqslant i \\leqslant n$ such that $s^{\\prime }_i >_\\mathsf {kv}y_j$ and thus also $y_j \\in \\mathcal {V}\\mathrm {ar}(s^{\\prime }_i)$ .", "This is only possible if $x_i \\in C_j$ (when $s^{\\prime }_i = t_i^+$ ) or $\\lnot x_i \\in C_j$ (when $s^{\\prime }_i = t_i^-$ ).", "Hence, by construction of $\\alpha $ , $\\alpha $ satisfies $C_j$ .", "Conversely, suppose $\\alpha $ satisfies $\\phi $ .", "Let $s^{\\prime }_i = t_i^+$ and $s_i = t_i^-$ if $\\alpha (x_i)$ is true and $s^{\\prime }_i = t_i^-$ and $s_i = t_i^+$ if $\\alpha (x_i)$ is false.", "We trivially have $s_i \\geqslant _{\\mathsf {kv^{\\prime }}}t_i$ for all $1 \\leqslant i \\leqslant n$ .", "Moreover, for each $1 \\leqslant j \\leqslant m$ , $C_j$ contains a literal $l = (\\lnot ) x_i$ such that $\\alpha (l)$ is true.", "By construction, $y_j \\in \\mathcal {V}\\mathrm {ar}(s^{\\prime }_i)$ and thus $s^{\\prime }_i >_\\mathsf {kv}y_j$ .", "Since $\\phi $ is non-empty, $m > 0$ and hence $S_\\phi >_{\\mathsf {kv^{\\prime }}}^\\mathsf {mul}T_\\phi $ as desired.", "To obtain NP-completeness we need to show membership in NP, which is easy; one just guesses how the terms in the various multisets relate to each other in order to satisfy the multiset comparisons in the definition of $>_\\mathrm {\\mathsf {KV^{\\prime }}}$ .", "Next we show that the complexity of deciding $>_\\mathrm {\\mathsf {KV}}$ and $>_\\mathrm {\\mathsf {ACKBO}}$ for given weights and precedence is decidable in polynomial time.", "Given a sequence $S = {s_1},\\dots ,{s_{n}}$ and an index $1 \\leqslant i \\leqslant n$ , we denote by $S[t]_i$ the sequence obtained by replacing $s_i$ with $t$ in $S$ , and by $S[\\,]_i$ the sequence obtained by removing $s_i$ from $S$ .", "Moreover, we write $\\lbrace S \\rbrace $ as a shorthand for the multiset $\\lbrace {s_1},\\dots ,{s_{n}} \\rbrace $ .", "Lemma 6.2 Let $(\\mathrel {\\succsim },\\mathrel {\\succ })$ be an order pair such that ${\\sim } := {\\mathrel {\\succsim }} \\setminus {\\mathrel {\\succ }}$ is symmetric.", "If $s \\sim t$ then $M \\mathrel {\\uplus } \\lbrace s \\rbrace \\mathrel {\\succ }^\\mathsf {mul}N \\mathrel {\\uplus } \\lbrace t \\rbrace $ and $M \\mathrel {\\succ }^\\mathsf {mul}N$ are equivalent.", "We only show that $M \\mathrel {\\uplus } \\lbrace s \\rbrace \\mathrel {\\succ }^\\mathsf {mul}N \\mathrel {\\uplus } \\lbrace t \\rbrace $ implies $M \\mathrel {\\succ }^\\mathsf {mul}N$ , since the other direction is trivial.", "So suppose $M \\uplus \\lbrace s \\rbrace \\sqsupset _k^\\mathsf {mul}N \\uplus \\lbrace t \\rbrace $ , where sequences $S = {s_1},\\dots ,{s_{m}}$ and $T = {t_1},\\dots ,{t_{n}}$ satisfy the conditions for $\\sqsupset _k^\\mathsf {mul}$ in Definition REF .", "Because we have $\\lbrace S\\rbrace = M \\uplus \\lbrace s\\rbrace $ and $\\lbrace T\\rbrace = N \\uplus \\lbrace t\\rbrace $ , there are indices $i$ and $j$ such that $s = s_i$ and $t = t_j$ .", "In order to establish $M \\mathrel {\\succ }^\\mathsf {mul}N$ we distinguish four cases.", "If $i, j \\leqslant k$ then $s_j \\mathrel {\\succsim }t_j = t \\sim s = s_i \\mathrel {\\succsim }t_i$ and thus $ \\lbrace S[s_j]_i[\\,]_j \\rbrace \\sqsupset _{k-1}^\\mathsf {mul}\\lbrace T[\\,]_j \\rbrace $ .", "If $i \\leqslant k < j$ then there exists some $l > k$ such that $s_l \\mathrel {\\succ }t_j = t \\sim s = s_i \\mathrel {\\succsim }t_i$ .", "Therefore, $\\lbrace S[\\,]_i \\rbrace \\sqsupset _{k-1}^\\mathsf {mul}\\lbrace T[t_i]_j[\\,]_i \\rbrace $ .", "If $j \\leqslant k < i$ then $s_j \\mathrel {\\succsim }t_j = t \\sim s = s_i$ and thus $s_j \\mathrel {\\succ }t_l$ for every $l > k$ such that $s_i \\mathrel {\\succ }t_l$ .", "Hence $\\lbrace S[s_j]_i[\\,]_j \\rbrace \\sqsupset _{k-1}^\\mathsf {mul}\\lbrace T[\\,]_j \\rbrace $ .", "The remaining case $k < i, j$ is analogous to the previous case, and we obtain $\\lbrace S[\\,]_i \\rbrace \\sqsupset _k^\\mathsf {mul}\\lbrace T[\\,]_j \\rbrace $ .", "Because $\\lbrace S[s_j]_i[\\,]_j \\rbrace = \\lbrace S[\\,]_i \\rbrace = M$ and $\\lbrace T[t_i]_j[\\,]_i \\rbrace = \\lbrace T[\\,]_j \\rbrace = N$ hold, in all cases $M \\mathrel {\\succ }^\\mathsf {mul}N$ is concluded.", "$\\Box $ Lemma 6.3 Let $(\\mathrel {\\succsim },\\mathrel {\\succ })$ be an order pair such that ${\\sim } := {\\mathrel {\\succsim }} \\setminus {\\mathrel {\\succ }}$ is symmetric and the decision problems for $\\mathrel {\\succsim }$ and $\\mathrel {\\succ }$ are in P. Then the decision problem for $\\mathrel {\\succ }^\\mathsf {mul}$ is in P. Suppose we want to decide whether two multisets $S$ and $T$ satisfy $S \\mathrel {\\succ }^\\mathsf {mul}T$ .", "We first check if there exists a pair $(s,t) \\in S \\times T$ such that $s \\sim t$ , which can be done by testing $s \\mathrel {\\succsim }t$ and $s \\mathrel {\\nsucc }t$ at most $\|S\| \\times \|T\|$ times.", "If such a pair is found then according to Lemma REF , the problem is reduced to $S - \\lbrace s \\rbrace \\mathrel {\\succ }^\\mathsf {mul}T - \\lbrace t \\rbrace $ .", "Otherwise, we check for each $t \\in T$ whether there exists $s \\in S$ such that $s \\mathrel {\\succ }t$ , which can be done by testing $s \\mathrel {\\succ }t$ at most $\|S\| \\times \|T\|$ times.", "Using the above lemma, we obtain the following result by a straightforward induction argument.", "Corollary 6.4 The decision problems for $>_\\mathrm {\\mathsf {ACKBO}}$ , $>_\\mathrm {\\mathsf {KV}}$ , and $>_\\mathrm {\\mathsf {S}}$ belong to P. $\\Box $ Next we address the complexity of the important orientability problem: Given a TRS $\\mathcal {R}$ , do there exist a weight function and a precedence such that the rules of $\\mathcal {R}$ are oriented from left to right with respect to the order under consideration?", "It is well-known [10] that KBO orientability is decidable in polynomial time.", "We show that $>_\\mathrm {\\mathsf {KV}}$ and $>_\\mathrm {\\mathsf {ACKBO}}$ orientability are NP-complete even for ground TRSs.", "First we show NP-hardness of $>_\\mathrm {\\mathsf {KV}}$ orientability by a reduction from SAT.", "Let $\\phi = \\lbrace {C_1},\\dots ,{C_{n}} \\rbrace $ be a CNF SAT problem over propositional variables ${p_1},\\dots ,{p_{m}}$ .", "We consider the signature $\\mathcal {F}_\\phi $ consisting of an AC symbol $+$ , constants $\\mathsf {c}$ and ${\\mathsf {d}_1},\\dots ,{\\mathsf {d}_{n}}$ , and unary function symbols ${p_1},\\dots ,{p_{m}}$ , $\\mathsf {a}$ , $\\mathsf {b}$ , and $\\mathsf {e}_i^j$ for all $i \\in \\lbrace 1, \\dots , n \\rbrace $ and $j \\in \\lbrace 0, \\dots , m \\rbrace $ .", "We define a ground TRS $\\mathcal {R}_\\phi $ on $\\mathcal {T}(\\mathcal {F}_\\phi )$ such that $>_\\mathrm {\\mathsf {KV}}$ orients $\\mathcal {R}_\\phi $ if and only if $\\phi $ is satisfiable.", "The TRS $\\mathcal {R}_\\phi $ will contain the following base system $\\mathcal {R}_0$ that enforces certain constraints on the precedence and the weight function: $\\mathsf {a}(\\mathsf {c}+ \\mathsf {c}) \\rightarrow \\mathsf {a}(\\mathsf {c}) + \\mathsf {c}\\qquad \\mathsf {b}(\\mathsf {c}) + \\mathsf {c}\\rightarrow \\mathsf {b}(\\mathsf {c}+ \\mathsf {c}) \\qquad \\mathsf {a}(\\mathsf {b}(\\mathsf {b}(\\mathsf {c}))) \\rightarrow \\mathsf {b}(\\mathsf {a}(\\mathsf {a}(\\mathsf {c}))) \\\\\\mathsf {a}(p_1(\\mathsf {c})) \\rightarrow \\mathsf {b}(p_2(\\mathsf {c})) \\qquad \\cdots \\qquad \\mathsf {a}(p_m(\\mathsf {c})) \\rightarrow \\mathsf {b}(\\mathsf {a}(\\mathsf {c})) \\qquad \\mathsf {a}(\\mathsf {a}(\\mathsf {c})) \\rightarrow \\mathsf {b}(p_1(\\mathsf {c}))$ Lemma 6.5 The order $>_\\mathrm {\\mathsf {KV}}$ is compatible with $\\mathcal {R}_0$ if and only if $\\mathsf {a}> + > \\mathsf {b}$ and $w(\\mathsf {a}) = w(\\mathsf {b}) = w(p_j)$ for all $1 \\leqslant j \\leqslant m$ .", "$\\Box $ Consider the clause $C_i$ of the form $\\lbrace {p^{\\prime }_1},\\dots ,{p^{\\prime }_{k}}, {\\lnot p^{\\prime \\prime }_1},\\dots ,{\\lnot p^{\\prime \\prime }_{l}} \\rbrace $ .", "Let $U$ , $U^{\\prime }$ , $V$ , and $W$ denote the following multisets: $U &= \\lbrace p_1^{\\prime }(\\mathsf {b}(\\mathsf {d}_i)), \\dots , p_k^{\\prime }(\\mathsf {b}(\\mathsf {d}_i)) \\rbrace &V &= \\lbrace p_0^{\\prime \\prime }(\\mathsf {e}_i^{0,1}), \\dots , p_{l-1}^{\\prime \\prime }(\\mathsf {e}_i^{l-1,l}),p_l^{\\prime \\prime }(\\mathsf {e}_i^{l,0}) \\rbrace \\\\U^{\\prime } &= \\lbrace \\mathsf {b}(p_1^{\\prime }(\\mathsf {d}_i)), \\dots , \\mathsf {b}(p_k^{\\prime }(\\mathsf {d}_i)) \\rbrace &W &= \\lbrace p_0^{\\prime \\prime }(\\mathsf {e}_i^{0,0}), \\dots , p_l^{\\prime \\prime }(\\mathsf {e}_i^{l,l}) \\rbrace $ where we write $p^{\\prime \\prime }_0$ for $\\mathsf {a}$ and $\\mathsf {e}_i^{j,k}$ for $\\mathsf {e}_i^j(\\mathsf {e}_i^k(\\mathsf {c}))$ .", "The TRS $\\mathcal {R}_\\phi $ is defined as the union of $\\mathcal {R}_0$ and $\\lbrace \\ell _i \\rightarrow r_i \\mid 1 \\leqslant i \\leqslant n \\rbrace $ with $\\ell _i & = \\mathsf {b}(\\mathsf {b}(\\mathsf {c}+ \\mathsf {c})) +\\textstyle \\sum U + \\sum V&r_i & = \\mathsf {b}(\\mathsf {c}) + \\mathsf {b}(\\mathsf {c}) +\\textstyle \\sum U^{\\prime } + \\sum W$ Note that the symbols $\\mathsf {d}_i$ and $\\mathsf {e}_i^0, \\dots , \\mathsf {e}_i^l$ are specific to the rule $\\ell _i \\rightarrow r_i$ .", "Example 6.6 Consider a clause $C_1 = \\lbrace x, \\lnot y, \\lnot z \\rbrace $ .", "We have $\\ell _1 &=\\mathsf {b}(\\mathsf {b}(\\mathsf {c}+ \\mathsf {c})) +x(\\mathsf {b}(\\mathsf {d}_i)) +\\mathsf {a}(\\mathsf {e}_1^0(\\mathsf {e}_1^1(\\mathsf {c}))) +y(\\mathsf {e}_1^1(\\mathsf {e}_1^2(\\mathsf {c}))) +z(\\mathsf {e}_1^2(\\mathsf {e}_1^0(\\mathsf {c}))) \\\\r_1 &=\\mathsf {b}(\\mathsf {c}) + \\mathsf {b}(\\mathsf {c}) +\\mathsf {b}(x(\\mathsf {d}_i)) +\\mathsf {a}(\\mathsf {e}_1^0(\\mathsf {e}_1^0(\\mathsf {c}))) +y(\\mathsf {e}_1^1(\\mathsf {e}_1^1(\\mathsf {c}))) +z(\\mathsf {e}_1^2(\\mathsf {e}_1^2(\\mathsf {c})))$ Note that $x$ , $y$ , and $z$ are unary function symbols.", "We have $w(\\ell _1) = w(r_1)$ for any weight function $w$ .", "Suppose $\\mathsf {a}> + > \\mathsf {b}$ and $w(\\mathsf {a}) = w(\\mathsf {b}) = w(x) = w(y) = w(z)$ .", "We consider a number of cases, depending on the order of $x$ , $y$ , $z$ , and $+$ in the precedence.", "If $x, y, z > +$ (i.e., $x$ , $y$ , and $z$ are assigned true) then $\\ell _1 >_\\mathrm {\\mathsf {KV}}r_1$ can be satisfied by choosing $w(\\mathsf {d}_1)$ large enough such that $w(x(\\mathsf {b}(\\mathsf {d}_1))) > w(t)$ for all $t \\in {{\\triangledown _{\\!+}}(r_1)}{\\upharpoonright }^{\\smash{>}}_{+}$ , where ${{\\triangledown _{\\!+}}(\\ell _1)}{\\upharpoonright }^{\\smash{>}}_{+} &~=~ \\lbrace x(\\mathsf {b}(\\mathsf {d}_1)),\\mathsf {a}(\\mathsf {e}_1^0(\\mathsf {e}_1^1(\\mathsf {c}))),y(\\mathsf {e}_1^1(\\mathsf {e}_1^2(\\mathsf {c}))),z(\\mathsf {e}_1^2(\\mathsf {e}_1^0(\\mathsf {c}))) \\rbrace \\\\{{\\triangledown _{\\!+}}(r_1)}{\\upharpoonright }^{\\smash{>}}_{+} &~=~ \\lbrace \\phantom{x(\\mathsf {b}(\\mathsf {d}_1)),{}}\\mathsf {a}(\\mathsf {e}_1^0(\\mathsf {e}_1^0(\\mathsf {c}))),y(\\mathsf {e}_1^1(\\mathsf {e}_1^1(\\mathsf {c}))),z(\\mathsf {e}_1^2(\\mathsf {e}_1^2(\\mathsf {c}))) \\rbrace \\multicolumn{2}{l}{\\text{On the other hand, if $y, z > + > x$ (i.e., $x$ is falsified)then $\\ell _1 >_\\mathrm {\\mathsf {KV}}r_1$ is not satisfiable;no matter how we assign weights to $\\mathsf {e}_1^0$, $\\mathsf {e}_1^1$, and $\\mathsf {e}_1^2$,a term in ${\\triangledown _{\\!+}}(r_1)$ has the maximum weight, where}}\\\\{{\\triangledown _{\\!+}}(\\ell _1)}{\\upharpoonright }^{\\smash{>}}_{+} &~=~ \\lbrace \\mathsf {a}(\\mathsf {e}_1^0(\\mathsf {e}_1^1(\\mathsf {c}))),y(\\mathsf {e}_1^1(\\mathsf {e}_1^2(\\mathsf {c}))),z(\\mathsf {e}_1^2(\\mathsf {e}_1^0(\\mathsf {c}))) \\rbrace \\\\{{\\triangledown _{\\!+}}(r_1)}{\\upharpoonright }^{\\smash{>}}_{+} &~=~ \\lbrace \\mathsf {a}(\\mathsf {e}_1^0(\\mathsf {e}_1^0(\\mathsf {c}))),y(\\mathsf {e}_1^1(\\mathsf {e}_1^1(\\mathsf {c}))),z(\\mathsf {e}_1^2(\\mathsf {e}_1^2(\\mathsf {c}))) \\rbrace \\multicolumn{2}{l}{\\text{However, if $y > + > x, z$ (i.e.\\ $z$ is falsified) then$\\ell _1 >_\\mathrm {\\mathsf {KV}}r_1$ can be satisfied by choosing $w(\\mathsf {e}_1^2)$ largeenough, where}}\\\\{{\\triangledown _{\\!+}}(\\ell _1)}{\\upharpoonright }^{\\smash{>}}_{+} &~=~ \\lbrace \\mathsf {a}(\\mathsf {e}_1^0(\\mathsf {e}_1^1(\\mathsf {c}))),y(\\mathsf {e}_1^1(\\mathsf {e}_1^2(\\mathsf {c}))) \\rbrace \\\\{{\\triangledown _{\\!+}}(r_1)}{\\upharpoonright }^{\\smash{>}}_{+} &~=~ \\lbrace \\mathsf {a}(\\mathsf {e}_1^0(\\mathsf {e}_1^0(\\mathsf {c}))),y(\\mathsf {e}_1^1(\\mathsf {e}_1^1(\\mathsf {c}))) \\rbrace \\multicolumn{2}{l}{\\text{Similarly, if $+ > x, y, z$ then $\\ell _1 >_\\mathrm {\\mathsf {KV}}r_1$ can be satisfied bychoosing $w(\\mathsf {e}_1^1)$ large enough, where}}\\\\{{\\triangledown _{\\!+}}(\\ell _1)}{\\upharpoonright }^{\\smash{>}}_{+} &~=~ \\lbrace \\mathsf {a}(\\mathsf {e}_1^0(\\mathsf {e}_1^1(\\mathsf {c}))) \\rbrace \\\\{{\\triangledown _{\\!+}}(r_1)}{\\upharpoonright }^{\\smash{>}}_{+} &~=~ \\lbrace \\mathsf {a}(\\mathsf {e}_1^0(\\mathsf {e}_1^0(\\mathsf {c}))) \\rbrace $ Lemma 6.7 Let $\\mathsf {a}> + > \\mathsf {b}$ .", "Then, $\\mathcal {R}_\\phi \\subseteq {>_\\mathrm {\\mathsf {KV}}}$ for some $(w, w_0)$ if and only if for every $i$ there is some $p$ such that $p \\in C_i$ with $p \\nless +$ or $\\lnot p \\in C_i$ with $+ > p$ .", "For the “if” direction we reason as follows.", "Consider a (partial) weight function $w$ such that $w(\\mathsf {a}) = w(\\mathsf {b}) = w(p_j)$ for all $1 \\leqslant j \\leqslant m$ .", "We obtain $\\mathcal {R}_0 \\subseteq {>_\\mathrm {\\mathsf {KV}}}$ from Lemma REF .", "Furthermore, consider $C_i = \\lbrace p^{\\prime }_1,\\dots ,p^{\\prime }_k, \\lnot p^{\\prime \\prime }_1,\\dots , \\lnot p^{\\prime \\prime }_l \\rbrace $ and $\\ell _i$ , $r_i$ , $U$ , $V$ and $W$ defined above.", "Let $L = {\\triangledown _{\\!+}}(\\ell _i)$ and $R = {\\triangledown _{\\!+}}(r_i)$ .", "We clearly have ${L}{\\upharpoonright }^{\\smash{\\nless }}_{+} = {U}{\\upharpoonright }^{\\smash{\\nless }}_{+} \\cup {V}{\\upharpoonright }^{\\smash{\\nless }}_{+}$ and ${R}{\\upharpoonright }^{\\smash{\\nless }}_{+} = {W}{\\upharpoonright }^{\\smash{\\nless }}_{+}$ .", "It is easy to show that $w(\\ell _i) = w(r_i)$ .", "We show $\\ell _i >_\\mathrm {\\mathsf {KV}}r_i$ by distinguishing two cases.", "First suppose that $p^{\\prime }_j \\nless +$ for some $1 \\leqslant j \\leqslant k$ .", "We have $p^{\\prime }_j(\\mathsf {b}(\\mathsf {d}_i)) \\in {U}{\\upharpoonright }^{\\smash{\\nless }}_{+}$ .", "Extend the weight function $w$ such that $w(\\mathsf {d}_i) = 1 + 2 \\cdot \\max \\,\\lbrace w(\\mathsf {e}_i^0), \\dots , w(\\mathsf {e}_i^l) \\rbrace $ Then $p^{\\prime }_j(\\mathsf {b}(\\mathsf {d}_i)) >_\\mathsf {kv}t$ for all terms $t \\in W$ and hence ${L}{\\upharpoonright }^{\\smash{\\nless }}_{+} >_\\mathsf {kv}^\\mathsf {mul}{R}{\\upharpoonright }^{\\smash{\\nless }}_{+}$ .", "Therefore $\\ell _i >_\\mathrm {\\mathsf {KV}}r_i$ by case 3(a).", "Otherwise, ${U}{\\upharpoonright }^{\\smash{\\nless }}_{+} = \\varnothing $ holds.", "By assumption $+ > p^{\\prime \\prime }_j$ for some $1 \\leqslant j \\leqslant l$ .", "Consider the smallest $m$ such that $+ > p^{\\prime \\prime }_m$ .", "Extend the weight function $w$ such that $w(\\mathsf {e}_i^m) = 1 + 2 \\cdot \\max \\,\\lbrace w(\\mathsf {e}_i^j) \\mid j \\ne m \\rbrace $ Then $w(p^{\\prime \\prime }_{m-1}(\\mathsf {e}_i^{m-1,m})) > w(p^{\\prime \\prime }_j(\\mathsf {e}_i^{j,j}))$ for all $j \\ne m$ .", "From $p^{\\prime \\prime }_{m-1} > +$ we infer $p^{\\prime \\prime }_{m-1}(\\mathsf {e}_i^{m-1,m}) \\in {V}{\\upharpoonright }^{\\smash{\\nless }}_{+}$ .", "(Note that $p_{m-1}^{\\prime \\prime } = \\mathsf {a}> +$ if $m = 1$ .)", "By definition of $m$ , $p^{\\prime \\prime }_m(\\mathsf {e}_i^{m,m}) \\notin {W}{\\upharpoonright }^{\\smash{\\nless }}_{+}$ .", "It follows that ${L}{\\upharpoonright }^{\\smash{\\nless }}_{+} >_\\mathsf {kv}^\\mathsf {mul}{R}{\\upharpoonright }^{\\smash{\\nless }}_{+}$ and thus $\\ell _i >_\\mathrm {\\mathsf {KV}}r_i$ by case 3(a).", "Next we prove the “only if” direction.", "So suppose there exists a weight function $w$ such that $\\mathcal {R}_\\phi \\subseteq {>_\\mathrm {\\mathsf {KV}}}$ .", "We obtain $w(\\mathsf {a}) = w(\\mathsf {b}) = w(p_j)$ for all $1 \\leqslant j \\leqslant m$ from Lemma REF .", "It follows that $w(\\ell _i) = w(r_i)$ for every $C_i \\in \\phi $ .", "Suppose for a proof by contradiction that there exists $C_i \\in \\phi $ such that $+ > p$ for all $p \\in C_i$ and $p \\nless +$ whenever $\\lnot p \\in C_i$ .", "So ${L}{\\upharpoonright }^{\\smash{\\nless }}_{+} = V$ and ${R}{\\upharpoonright }^{\\smash{\\nless }}_{+} = W$ .", "Since $\|R\| = \|L\| + 1$ , we must have $\\ell _i >_\\mathrm {\\mathsf {KV}}r_i$ by case 3(a) and thus $V >_\\mathsf {kv}W$ .", "Let $s$ be a term in $V$ of maximal weight.", "We must have $w(s) \\geqslant w(t)$ for all terms $t \\in W$ .", "By construction of the terms in $V$ and $W$ , this is only possible if all symbols $\\mathsf {e}_i^j$ have the same weight.", "It follows that all terms in $V$ and $W$ have the same weight.", "Since $\|V\| = \|W\|$ and for every term $s^{\\prime } \\in V$ there exists a unique term $t^{\\prime } \\in W$ with $\\mathsf {root}(s^{\\prime }) = \\mathsf {root}(t^{\\prime })$ , we conclude $V =_\\mathsf {kv}W$ , which provides the desired contradiction.", "After these preliminaries we are ready to prove NP-hardness.", "Theorem 6.8 The (ground) orientability problem for $>_\\mathrm {\\mathsf {KV}}$ is NP-hard.", "It is sufficient to prove that a CNF formula $\\phi = \\lbrace {C_1},\\dots ,{C_{n}} \\rbrace $ is satisfiable if and only if the corresponding $\\mathcal {R}_\\phi $ is orientable by $>_\\mathrm {\\mathsf {KV}}$ .", "Note that the size of $\\mathcal {R}_\\phi $ is linear in the size of $\\phi $ .", "First suppose that $\\phi $ is satisfiable.", "Let $\\alpha $ be a satisfying assignment for the atoms ${p_1},\\dots ,{p_{m}}$ .", "Define the precedence $>$ as follows: $\\mathsf {a}> + > \\mathsf {b}$ and $p_j > +$ if $\\alpha (p_j)$ is true and $+ > p_j$ if $\\alpha (p_j)$ is false.", "Then $\\mathcal {R}_\\phi \\subseteq {>_\\mathrm {\\mathsf {KV}}}$ follows from Lemma REF .", "Conversely, if $\\mathcal {R}_\\phi $ is compatible with $>_\\mathrm {\\mathsf {KV}}$ then we define an assignment $\\alpha $ for the atoms in $\\phi $ as follows: $\\alpha (p)$ is true if $p \\nless +$ and $\\alpha (p)$ is false if $+ > p$ .", "We claim that $\\alpha $ satisfies $\\phi $ .", "Let $C_i$ be a clause in $\\phi $ .", "According to Lemma REF , $p \\nless +$ for one of the atoms $p$ in $C_i$ or $+ > p$ for one of the negative literals $\\lnot p$ in $C_i$ .", "Hence $\\alpha $ satisfies $C_i$ by definition.", "We can show NP-hardness of $>_\\mathrm {\\mathsf {ACKBO}}$ by adapting the above construction accordingly, as shown in Appendix REF .", "Theorem 6.9 The (ground) orientability problem for $>_\\mathrm {\\mathsf {ACKBO}}$ is NP-hard.", "$\\Box $ The NP-hardness results of Theorems REF and REF can be strengthened to NP-completeness.", "This is not entirely trivial because there are infinitely many different weight functions to consider.", "Lemma 6.10 The orientability problems for $>_\\mathrm {\\mathsf {ACKBO}}$ and $>_\\mathrm {\\mathsf {KV}}$ belong to NP.", "[Proof (sketch)] We sketch the proof for $>_\\mathrm {\\mathsf {ACKBO}}$ .", "With minor modifications the result for $>_\\mathrm {\\mathsf {KV}}$ is obtained.", "For each rule $\\ell \\rightarrow r$ of a given TRS $\\mathcal {R}$ we guess which choices are made in the definition of $>_\\mathrm {\\mathsf {ACKBO}}$ when evaluating $\\ell >_\\mathrm {\\mathsf {ACKBO}}r$ .", "In particular, we do not guess the weight function, but rather the comparison ($=$ or $>$ ) of the weights of certain subterms of $\\ell $ and $r$ .", "These comparisons are transformed into constraints on the weight function by symbolically evaluating the weight expressions.", "We add the constraints stemming from the definition of the weight function.", "The resulting problem is a conjunction of linear constraints over unknowns (the weights of the function symbols and $w_0$ ) over the integers.", "It is well-known [20] that solving such a linear program over the rationals can be done in polynomial time.", "If there is a solution we check the admissibility condition and well-foundedness of the precedence.", "(If an integer valued weight function is desired, one can simply multiply the weights by the least common multiple of their denominators.", "This induces the same weight order on terms and does not affect the admissibility condition.)", "Since there are polynomially (in the size of the compared terms) many choices in the definition of $>_\\mathrm {\\mathsf {ACKBO}}$ and each choice can be checked for correctness in polynomial time, membership in NP follows.", "Corollary 6.11 The orientability problems for $>_\\mathrm {\\mathsf {ACKBO}}$ and $>_\\mathrm {\\mathsf {KV}}$ are NP-complete.", "$\\Box $ The NP-hardness proofs of $>_\\mathrm {\\mathsf {KV}}$ and $>_\\mathrm {\\mathsf {ACKBO}}$ orientability given earlier do not extend to $>_\\mathrm {\\mathsf {S}}$ since the latter requires that AC symbols are minimal in the precedence.", "We conjecture that the orientability problem for $>_\\mathrm {\\mathsf {S}}$ belongs to P." ], [ "AC-RPO", "In this section we compare AC-KBO with AC-RPO [19].", "Since the latter is incremental [19], we restrict the discussion to total precedences.", "Definition 7.1 Let $>$ be a precedence and $t = f(u,v)$ such that $f \\in \\mathcal {F}_\\mathrm {\\mathsf {AC}}$ and ${\\triangledown _{\\!f}}(t) = \\lbrace {t_1},\\dots ,{t_{n}} \\rbrace $ .", "We write $t \\vartriangleright ^{\\!f}_{\\!\\mathsf {emb}} u$ for all terms $u$ such that ${\\triangledown _{\\!f}}(u) = \\lbrace t_1,\\ldots ,t_{i-1},s_j,t_{i+1},\\ldots ,t_n \\rbrace $ for some $t_i = g({s_1},\\dots ,{s_{m}})$ with $f > g$ and $1 \\leqslant j \\leqslant m$ .", "Using previously introduced notations, AC-RPO can be defined as follows.", "Definition 7.2 Let $>$ be a precedence and let $\\mathcal {F}\\setminus \\mathcal {F}_\\mathrm {\\mathsf {AC}}= \\mathcal {F}_{\\mathsf {mul}} \\uplus \\mathcal {F}_{\\mathsf {lex}}$ .", "We define $>_\\mathrm {\\mathsf {ACRPO}}$ inductively as follows: $s >_\\mathrm {\\mathsf {ACRPO}}t$ if one of the following conditions holds: $s = f({s_1},\\dots ,{s_{n}})$ and $s_i \\geqslant _\\mathrm {\\mathsf {ACRPO}}t$ for some $1 \\leqslant i \\leqslant n$ , $s = f({s_1},\\dots ,{s_{n}})$ , $t = g({t_1},\\dots ,{t_{m}})$ , $f > g$ , and $s >_\\mathrm {\\mathsf {ACRPO}}t_j$ for all $1 \\leqslant j \\leqslant m$ , $s = f({s_1},\\dots ,{s_{n}})$ , $t = f({t_1},\\dots ,{t_{n}})$ , $f \\notin \\mathcal {F}_\\mathrm {\\mathsf {AC}}$ , $s >_\\mathrm {\\mathsf {ACRPO}}t_j$ for all $1 \\leqslant j \\leqslant n$ , and either (a) $f \\in \\mathcal {F}_{\\textsf {lex}}$ and $({s_1},\\dots ,{s_{n}}) >_\\mathrm {\\mathsf {ACRPO}}^\\mathsf {lex}({t_1},\\dots ,{t_{n}})$ , or (b) $f \\in \\mathcal {F}_{\\textsf {mul}}$ and $\\lbrace {s_1},\\dots ,{s_{n}} \\rbrace >_\\mathrm {\\mathsf {ACRPO}}^\\mathsf {mul}\\lbrace {t_1},\\dots ,{t_{n}} \\rbrace $ , $s = f(s_1,s_2)$ , $t = f(t_1,t_2)$ , $f \\in \\mathcal {F}_\\mathrm {\\mathsf {AC}}$ , and $s^{\\prime } \\geqslant _\\mathrm {\\mathsf {ACRPO}}t$ for some $s^{\\prime }$ such that $s \\vartriangleright ^{\\!f}_{\\!\\mathsf {emb}} s^{\\prime }$ , $s = f(s_1,s_2)$ , $t = f(t_1,t_2)$ , $f \\in \\mathcal {F}_\\mathrm {\\mathsf {AC}}$ , $s >_\\mathrm {\\mathsf {ACRPO}}t^{\\prime }$ for all $t^{\\prime }$ such that $t \\vartriangleright ^{\\!f}_{\\!\\mathsf {emb}} t^{\\prime }$ , and for $S = {\\triangledown _{\\!f}}(s)$ and $T = {\\triangledown _{\\!f}}(t)$ (a) $S >_\\mathrm {\\mathsf {ACRPO}}^f T$ , (b) $S =_\\mathrm {\\mathsf {AC}}^f T$ and $\|S\| > \|T\|$ , or (c) $S =_\\mathrm {\\mathsf {AC}}^f T$ , $\|S\| = \|T\|$ , and ${S}{\\upharpoonright }^{\\smash{<}}_{f} >_\\mathrm {\\mathsf {ACRPO}}^{\\mathsf {mul}} {T}{\\upharpoonright }^{\\smash{<}}_{f}$ .", "The relation $=_\\mathrm {\\mathsf {AC}}$ is used as preorder in $>_\\mathrm {\\mathsf {ACRPO}}^\\mathsf {lex}$ and $>_\\mathrm {\\mathsf {ACRPO}}^\\mathsf {mul}$ , and as equivalence relation in $\\geqslant _\\mathrm {\\mathsf {ACRPO}}$ .", "Example 7.3 Consider the TRS $\\mathcal {R}$ consisting of the rules $\\mathsf {f}(x) + \\mathsf {g}(x) &\\rightarrow \\mathsf {g}(x) + (\\mathsf {g}(x) + \\mathsf {g}(x)) &\\mathsf {f}(x) &\\rightarrow \\mathsf {g}(x) + \\mathsf {a}$ over the signature $\\mathcal {F}= \\lbrace \\mathsf {f}, \\mathsf {g}, +, \\mathsf {a} \\rbrace $ with $+ \\in \\mathcal {F}_\\mathrm {\\mathsf {AC}}$ .", "Let $\\mathcal {R}^{\\prime }$ be the TRS obtained from $\\mathcal {R}$ by reverting the first rule.", "When using AC-RPO with precedence $\\mathsf {f} > + > \\mathsf {g} > \\mathsf {a}$ , both rules in $\\mathcal {R}$ can be oriented from left to right.", "Since the second rule requires $\\mathsf {f} > +$ and $\\mathsf {f} > \\mathsf {g}$ , termination of $\\mathcal {R}^{\\prime }$ cannot be shown with AC-RPO.", "In contrast, AC-KBO cannot orient $\\mathcal {R}$ due to the variable condition.", "But the precedence $\\mathsf {g} > + > \\mathsf {f} > \\mathsf {a}$ and admissible weight function $(w,w_0)$ with $w(+) = 0$ , $w_0 = w(\\mathsf {g}) = w(\\mathsf {a}) = 1$ and $w(\\mathsf {f}) = 3$ allows the resulting $>_\\mathrm {\\mathsf {ACKBO}}$ to orient both rules of $\\mathcal {R}^{\\prime }$ .", "Case 4 in Definition REF differs from the original version in [19] in that we used notions introduced for AC-KBO.", "We now recall the original definition and prove the two versions equivalent in Lemma REF .", "Definition 7.4 For $S = \\lbrace {s_1},\\dots ,{s_{n}} \\rbrace $ let $\\#(S) = \\#(s_1) + \\cdots + \\#(s_n)$ where $\\#(s_i) = s_i$ for $s_i \\in \\mathcal {V}$ and $\\#(s_i) = 1$ otherwise.", "Then $\\#(S) > \\#(T)$ ($\\#(S) \\geqslant \\#(T)$ ) is defined via comparison of linear polynomials over the positive integers.", "Let $>$ be a total precedence.", "The order $>_\\mathrm {\\mathsf {ACRPO^{\\prime }}}$ is inductively defined as in Definition REF , but with case 4 as follows: 4$^{\\prime }$ .", "$s = f(s_1,s_2)$ , $t = f(t_1,t_2)$ , $f \\in \\mathcal {F}_\\mathrm {\\mathsf {AC}}$ , $s >_{\\mathrm {\\mathsf {ACRPO^{\\prime }}}} t^{\\prime }$ for all $t^{\\prime }$ such that $t \\vartriangleright ^{\\!f}_{\\!\\mathsf {emb}} t^{\\prime }$ , ${S}{\\upharpoonright }^{\\smash{>}}_{f} \\uplus S{\\upharpoonright }_\\mathcal {V}\\geqslant _{\\mathrm {\\mathsf {ACRPO^{\\prime }}}}^\\mathsf {mul}{T}{\\upharpoonright }^{\\smash{>}}_{f} \\uplus T{\\upharpoonright }_\\mathcal {V}$ for $S = {\\triangledown _{\\!f}}(s)$ and $T = {\\triangledown _{\\!f}}(t)$ , and (a) ${S}{\\upharpoonright }^{\\smash{>}}_{f} >_{\\mathrm {\\mathsf {ACRPO^{\\prime }}}}^\\mathsf {mul}{T}{\\upharpoonright }^{\\smash{>}}_{f}$ , or (b) $\\#(S) > \\#(T)$ , or (c) $\\#(S) \\geqslant \\#(T)$ , and $S >_{\\mathrm {\\mathsf {ACRPO^{\\prime }}}}^\\mathsf {mul}T$ .", "The proof of the following correspondence can be found in the online appendix.", "Lemma 7.5 Let $>$ be a total precedence.", "We have $s >_\\mathrm {\\mathsf {ACRPO}}t$ if and only if $s >_\\mathrm {\\mathsf {ACRPO^{\\prime }}}t$ .", "It is known that both orientability and membership are NP-hard for the multiset path order [11].", "It is not hard to adapt these proofs to LPO, and NP-hardness for the case of RPO is an easy consequence.", "In contrast to AC-KBO, a straightforward application of the definition of AC-RPO (in particular case 4 of Definition REF ) may generate an exponential number of subproblems, as illustrated by the following example.", "Example 7.6 Consider the signature $\\mathcal {F}= \\lbrace \\mathsf {f}, \\mathsf {g}, \\mathsf {h}, \\circ \\rbrace $ with $\\circ \\in \\mathcal {F}_\\mathrm {\\mathsf {AC}}$ and precedence $\\mathsf {f} > {\\circ } > \\mathsf {g} > \\mathsf {h}$ .", "Let $t = x \\circ y$ and $t_n = t\\sigma ^n$ for the substitution $\\sigma = \\lbrace x \\mapsto \\mathsf {g}(x) \\circ \\mathsf {h}(y), y \\mapsto \\mathsf {h}(y) \\rbrace $ .", "The size of $t_n$ is quadratic in $n$ but the number of terms $u$ that satisfy $t_n \\mathrel {(\\vartriangleright ^{\\!\\circ }_{\\!\\mathsf {emb}})^+} u$ is exponential in $n$ .", "Now suppose one wants to decide whether $\\mathsf {f}(x) \\circ \\mathsf {f}(y) >_\\mathrm {\\mathsf {ACRPO}}t_n$ holds.", "Only case 4(a) is applicable but in order to conclude orientability, case 4(a) needs to be applied recursively in order to verify $\\mathsf {f}(x) \\circ \\mathsf {f}(x) >_\\mathrm {\\mathsf {ACRPO}}u$ for the exponentially many terms $u$ such that $t_n \\mathrel {(\\vartriangleright ^{\\!\\circ }_{\\!\\mathsf {emb}})^+} u$ ." ], [ "Subterm Coefficients", "Subterm coefficients were introduced in [16] in order to cope with rewrite rules like $\\mathsf {f}(x) \\rightarrow \\mathsf {g}(x,x)$ which violate the variable condition.", "A subterm coefficient function is a partial mapping $\\mathit {sc}\\colon \\mathcal {F}\\times \\mathbb {N}\\rightarrow \\mathbb {N}$ such that for a function symbol $f$ of arity $n$ we have $\\mathit {sc}(f,i) > 0$ for all $1 \\leqslant i \\leqslant n$ .", "Given a weight function $(w,w_0)$ and a subterm coefficient function $\\mathit {sc}$ , the weight of a term is inductively defined as follows: $w(t) ={\\left\\lbrace \\begin{array}{ll}w_0 & \\text{if $t \\in \\mathcal {V}$} \\\\\\displaystyle w(f) + \\smash[b]{\\sum _{1 \\leqslant i \\leqslant n}}\\mathit {sc}(f,i) \\cdot w(t_i) & \\text{if $t = f({t_1},\\dots ,{t_{n}})$}\\end{array}\\right.", "}$ The variable coefficient $\\mathsf {vc}(x,t)$ of a variable $x$ in a term $t$ is inductively defined as follows: $\\mathsf {vc}(x,t) = {\\left\\lbrace \\begin{array}{ll}1 & \\text{if $t = x$} \\\\0 & \\text{if $t \\in \\mathcal {V}\\setminus \\lbrace x \\rbrace $} \\\\\\displaystyle \\smash[b]{\\sum _{1 \\leqslant i \\leqslant n}}\\mathit {sc}(f,i) \\cdot \\mathsf {vc}(x,t_i) & \\text{if $t = f({t_1},\\dots ,{t_{n}})$}\\end{array}\\right.", "}$ Definition 8.1 The order $>_\\mathrm {\\mathsf {ACKBO}}^\\mathit {sc}$ is obtained from Definition REF by replacing the condition “ $\|s\|_x \\geqslant \|t\|_x$ for all $x \\in \\mathcal {V}$ ” with “ $\\mathsf {vc}(x,s) \\geqslant \\mathsf {vc}(x,t)$ for all $x \\in \\mathcal {V}$ ” and using the modified weight function introduced above.", "In order to guarantee AC compatibility of $>_\\mathrm {\\mathsf {ACKBO}}^\\mathit {sc}$ , the subterm coefficient function $\\mathit {sc}$ has to assign the value 1 to arguments of AC symbols.", "This follows by considering the terms $t \\circ (u \\circ v)$ and $(t \\circ u) \\circ v$ for an AC symbol $\\circ $ with $\\mathit {sc}({\\circ },1) = m$ and $\\mathit {sc}({\\circ },2) = n$ .", "We have $w(t \\circ (u \\circ v))&= 2 \\cdot w(\\circ ) + m \\cdot w(t) + mn \\cdot w(u) + n^2 \\cdot w(v)\\\\w((t \\circ u) \\circ v)&= 2 \\cdot w(\\circ ) + m^2 \\cdot w(t) + mn \\cdot w(u) + n \\cdot w(v)$ Since $w(t \\circ (u \\circ v)) = w((t \\circ u) \\circ v)$ must hold for all possible terms $t$ , $u$ , and $v$ , it follows that $m = m^2$ and $n^2 = n$ , implying $m = n = 1$ .This condition is also obtained by restricting [4] to linear polynomials.", "The proof of the following theorem is very similar to the one of Theorem REF and hence omitted.", "Theorem 8.2 If $\\mathit {sc}(f,1) = \\mathit {sc}(f,2) = 1$ for every function symbol $f \\in \\mathcal {F}_\\mathrm {\\mathsf {AC}}$ then $>_\\mathrm {\\mathsf {ACKBO}}^\\mathit {sc}$ is an AC-compatible simplification order.", "$\\Box $ Subterm coefficients can be viewed as linear interpretations.", "L79 suggested to use polynomial interpretations for the weight function of KBO.", "A general framework for the use of arbitrary well-founded algebras in connection with KBO is described in [18].", "These developments can be lifted to the AC setting with little effort.", "Example 8.3 Consider the following TRS $\\mathcal {R}$ with $\\circ \\in \\mathcal {F}_\\mathrm {\\mathsf {AC}}$ : $\\mathsf {f}(\\mathsf {0}, x \\circ x) &\\rightarrow x \\\\\\mathsf {f}(x,\\mathsf {s}(y)) &\\rightarrow \\mathsf {f}(x \\circ y, \\mathsf {0})$ $\\mathsf {f}(\\mathsf {s}(x),y) &\\rightarrow \\mathsf {f}(x \\circ y, \\mathsf {0}) \\\\\\mathsf {f}(x \\circ y, \\mathsf {0}) &\\rightarrow \\mathsf {f}(x,\\mathsf {0}) \\circ \\mathsf {f}(y,\\mathsf {0})$ Termination of $\\mathcal {R}$ was shown using AC dependency pairs in [12].", "Consider a precedence $\\mathsf {f} > \\circ > \\mathsf {s} > \\mathsf {0}$ , and weights and subterm coefficients given by $w_0 = 1$ and the following interpretation $\\mathcal {A}$ , mapping function symbols in $\\mathcal {F}$ to linear polynomials over $\\mathbb {N}$ : $\\mathsf {s}_\\mathcal {A}(x) &= x + 6 &\\mathsf {f}_\\mathcal {A}(x,y) &= 4x + 4y + 5 &x \\circ _{\\!\\mathcal {A}} y &= x + y + 3 &\\mathsf {0}_\\mathcal {A}&= 1$ It is easy to check that the first three rules result in a weight decrease.", "The left- and right-hand side of rule $(4)$ are both interpreted as $4x+4y+21$ , so both terms have weight 29, but since $\\mathsf {f} > \\circ $ we conclude termination of $\\mathcal {R}$ from case 1 in Definition REF (REF ).", "Note that termination of $\\mathcal {R}$ cannot be shown by AC-RPO or any of the previously considered versions of AC-KBO." ], [ "Experiments", "We ran experiments on a server equipped with eight dual-core AMD Opteron$^{\\mbox{\\begin{scriptsize}®\\end{scriptsize}}}$ processors 885 running at a clock rate of 2.6GHz with 64GB of main memory.", "The different versions of AC-KBO considered in this paper as well as AC-RPO [19] were implemented on top of TTT2 using encodings in SAT/SMT.", "These encodings resemble those for standard KBO [26] and transfinite KBO [24].", "The encoding of multiset extensions of order pairs are based on [5], but careful modifications were required to deal with submultisets induced by the precedence.", "Table: Experiments on 145 termination and 67 completion problems.For termination experiments, our test set comprises all AC problems in the Termination Problem Data Base 9.0,http://termination-portal.org/wiki/TPDB all examples in this paper, some further problems harvested from the literature, and constraint systems produced by the completion tool mkbtt [23] (145 TRSs in total).", "The timeout was set to 60 seconds.", "The results are summarized in Table REF , where we list for each order the number of successful termination proofs, the total time, and the number of timeouts (column $\\infty $ ).", "The `orientability' column directly applies the order to orient all the rules.", "Although AC-RPO succeeds on more input problems, termination of 9 TRSs could only be established by (variants of) AC-KBO.", "We found that our definition of AC-KBO is about equally powerful as Korovin and Voronkov's order, but both are considerably more useful than Steinbach's version.", "When it comes to proving termination, we did not observe a difference between Definitions REF and REF .", "Subterm coefficients clearly increase the success rate, although efficiency is affected.", "In all settings partial precedences were allowed.", "The `AC-DP' column applies the order in the AC-dependency pair framework of [1], in combination with argument filterings and usable rules.", "Here AC symbols in dependency pairs are unmarked, as proposed in [17].", "In this setting the variants of AC-KBO become considerably more powerful and competitive to AC-RPO, since argument filterings relax the variable condition, as pointed out in [26].", "For completion experiments, we ran the normalized completion tool mkbtt with AC-RPO and the variants of AC-KBO for termination checks on 67 equational systems collected from the literature.", "The overall timeout was set to 60 seconds, the timeout for each termination check to 1.5 seconds.", "The `completion' column in Table REF summarizes our results, listing for each order the number of successful completions, the total time, and the number of timeouts.", "It should be noted that the results do not change if the overall timeout is increased to 600 seconds.", "For several of these input problems it is actually unknown whether an AC-convergent system exists.", "All experimental details, source code, and TTT2 binaries are available online.http://cl-informatik.uibk.ac.at/software/ackbo The following example can be completed using AC-KBO, whereas AC-RPO does not succeed.", "Example 9.1 Consider the following TRS $\\mathcal {R}$ [17] for addition of binary numbers: $\\# + \\mathsf {0} &\\rightarrow \\# &x\\mathsf {0} + y\\mathsf {0} &\\rightarrow (x + y)\\mathsf {0} &x\\mathsf {1} + y\\mathsf {1} &\\rightarrow (x + y + \\#\\mathsf {1})\\mathsf {0} \\\\x + \\# &\\rightarrow x &x\\mathsf {0} + y\\mathsf {1} &\\rightarrow (x + y)\\mathsf {1}$ Here ${+} \\in \\mathcal {F}_\\mathrm {\\mathsf {AC}}$ , $\\mathsf {0}$ and $\\mathsf {1}$ are unary operators in postfix notation, and $\\#$ denotes the empty bit sequence.", "For example, $\\#\\mathsf {100}$ represents the number 4.", "This TRS is not compatible with AC-RPO but AC termination can easily be shown by AC-KBO, for instance with the weight function $(w,w_0)$ with $w(\\mathsf {+}) = 0$ , $w_0 = w(\\mathsf {0}) = w(\\#) = 1$ , and $w(\\mathsf {1}) = 3$ .", "It can be completed into an AC-convergent TRS using AC-KBO." ], [ "Conclusion", "We revisited the two variants of AC-compatible extensions of KBO.", "We extended the first version $>_\\mathrm {\\mathsf {S}}$ introduced by Steinbach [21] to a new version $>_\\mathrm {\\mathsf {ACKBO}}$ , and presented a rigorous correctness proof.", "By this we conclude correctness of $>_\\mathrm {\\mathsf {S}}$ , which had been put in doubt in [9].", "We also modified the order $>_\\mathrm {\\mathsf {KV}}$ by KV03b to a new version $>_\\mathrm {\\mathsf {KV^{\\prime }}}$ which is monotone on non-ground terms, in contrast to $>_\\mathrm {\\mathsf {KV}}$ .", "We further presented several complexity results regarding these variants (see Table REF ).", "While a polynomial time algorithm is known for the orientability problem of standard KBO [10], the problem becomes NP-complete even for the ground version of $>_\\mathrm {\\mathsf {KV}}$ , as well as for our $>_\\mathrm {\\mathsf {ACKBO}}$ .", "Somewhat unexpectedly, even deciding $>_\\mathrm {\\mathsf {KV^{\\prime }}}$ is NP-complete while deciding standard KBO is linear [15].", "In contrast, the membership problem is polynomial-time decidable for our $>_\\mathrm {\\mathsf {ACKBO}}$ .", "Finally, we implemented these variants of AC-compatible KBO as well as the AC-dependency pair framework of ALM10.", "We presented full experimental results both for termination proving and normalized completion.", "Table: Complexity results (KV is the ground version of > 𝖪𝖵 >_\\mathrm {\\mathsf {KV}})." ], [ "Acknowledgments.", "We are grateful to Konstantin Korovin for discussions and the reviewers of the conference version [25] for their detailed comments which helped to improve the presentation.", "René Thiemann suggested the proof of Lemma REF .", "First we show that $({=_\\mathrm {\\mathsf {AC}}},{>_\\mathrm {\\mathsf {ACKBO}}})$ is an order pair.", "To facilitate the proof, we decompose $>_\\mathrm {\\mathsf {ACKBO}}$ into several orders.", "We write $s >_{01} t$ if $\|s\|_x \\geqslant \|t\|_x$ for all $x \\in \\mathcal {V}$ and either $w(s) > w(t)$ or $w(s) = w(t)$ and case 0 or case 1 of Definition REF applies, $s >_{23,k} t$ if $\|s\|, \|t\| \\leqslant k$ , $\|s\|_x \\geqslant \|t\|_x$ for all $x \\in \\mathcal {V}$ , $w(s) = w(t)$ , and case 2 or case 3 applies.", "The union of $>_{01}$ and $>_{23,k}$ is denoted by $>_k$ .", "The next lemma states straightforward properties.", "Lemma A.1 The following statements hold: ${>_\\mathrm {\\mathsf {ACKBO}}} \\:=\\: \\bigcup \\,\\lbrace {>_k} \\mid k \\in \\mathbb {N}\\rbrace $ , $({=_\\mathrm {\\mathsf {AC}}}, {>_{01}})$ is an order pair, and $({>_{01} \\cdot >_k}) \\cup ({>_k \\cdot >_{01}})\\:\\subseteq \\: {>_{01}}$ .", "The inclusion from right to left is obvious from the definition.", "For the inclusion from left to right, suppose $s >_\\mathrm {\\mathsf {ACKBO}}t$ .", "If either $w(s) > w(t)$ , or $w(s) = w(t)$ and case 0 or case 1 of Definition REF applies, then trivially $s >_{01} t$ .", "If case 2 or case 3 applies, then $s >_{23,k} t$ for any $k$ with $k \\geqslant \\max (\|s\|,\|t\|)$ .", "First we show that $>_{01}$ is transitive.", "Suppose $s >_{01} t >_{01} u$ .", "If $w(s) > w(t)$ or $w(t) > w(u)$ , then $w(s) > w(u)$ and $s >_{01} u$ .", "Hence suppose $w(s) = w(t) = w(u)$ .", "Since $s, t \\notin \\mathcal {V}$ , we may write $s = f({s_1},\\dots ,{s_{n}})$ and $t = g({t_1},\\dots ,{t_{m}})$ with $f > g$ .", "Because of admissibility, $g$ is not a unary symbol with $w(g) = 0$ .", "Thus $u \\notin \\mathcal {V}$ , and we may write $u = h({u_1},\\dots ,{u_{l}})$ with $g > h$ .", "By the transitivity of $>$ we obtain $s >_{01} u$ .", "The irreflexivity of $>_{01}$ is obvious from the definition.", "It remains to show the compatibility condition ${=_\\mathrm {\\mathsf {AC}}} \\cdot {>_{01}} \\cdot {=_\\mathrm {\\mathsf {AC}}} \\subseteq {>_{01}}$ .", "This easily follows from the fact that $w(s) = w(t)$ and $\\mathsf {root}(s) = \\mathsf {root}(t)$ whenever $s =_\\mathrm {\\mathsf {AC}}t$ .", "Suppose $s = f({s_1},\\dots ,{s_{n}}) >_{01} t = g({t_1},\\dots ,{t_{m}}) >_k u$ .", "If $t >_{01} u$ then $s >_{01} u$ follows from the transitivity of $>_{01}$ .", "Suppose $t >_{23,k} u$ .", "So $w(t) = w(u)$ .", "Thus $w(s) > w(u)$ if $w(s) > w(t)$ , and case 1 applies if $w(s) = w(t)$ .", "The inclusion ${>_k} \\cdot {>_{01}} \\subseteq {>_k}$ is proved in exactly the same way.", "$\\Box $ Lemma A.2 Let $>$ be a precedence, $f \\in \\mathcal {F}$ , and $({\\mathrel {\\succsim }},{\\mathrel {\\succ }})$ an order pair on terms.", "Then $({\\mathrel {\\succsim }^f},{\\mathrel {\\succ }^f})$ is an order pair.", "We first prove compatibility.", "Suppose $S \\mathrel {\\succsim }^f T \\mathrel {\\succ }^f U$ .", "From $T \\mathrel {\\succ }^f U$ we infer that ${T}{\\upharpoonright }^{\\smash{\\nless }}_{f} \\uplus T{\\upharpoonright }_\\mathcal {V}\\mathrel {\\succ }^\\mathsf {mul}{U}{\\upharpoonright }^{\\smash{\\nless }}_{f} \\uplus U{\\upharpoonright }_\\mathcal {V}$ .", "Hence ${S}{\\upharpoonright }^{\\smash{\\nless }}_{f} \\mathrel {\\succ }^\\mathsf {mul}{U}{\\upharpoonright }^{\\smash{\\nless }}_{f} \\uplus U{\\upharpoonright }_\\mathcal {V}-S{\\upharpoonright }_\\mathcal {V}$ follows from $S \\mathrel {\\succsim }^f T$ .", "Hence also $S \\mathrel {({\\mathrel {\\succsim }} \\cdot {\\mathrel {\\succ }})}^f U$ .", "We obtain the desired $S \\mathrel {\\succ }^f U$ from the compatibility of $\\mathrel {\\succsim }$ and $\\mathrel {\\succ }$ .", "Transitivity of $\\mathrel {\\succsim }^f$ and $\\mathrel {\\succ }^f$ is obtained in a very similar way.", "Reflexivity of $\\mathrel {\\succsim }^f$ and irreflexivity of $\\mathrel {\\succ }^f$ are obvious.", "We employ the following simple criterion to construct order pairs, which enables us to prove correctness in a modular way.", "Lemma A.3 Let $({\\mathrel {\\succsim }},{\\mathrel {\\succ }_k})$ be order pairs for $k \\in \\mathbb {N}$ with ${\\mathrel {\\succ }_k} \\subseteq {\\mathrel {\\succ }_{k+1}}$ .", "If $\\mathrel {\\succ }$ is the union of all $\\mathrel {\\succ }_k$ then $({\\mathrel {\\succsim }},{\\mathrel {\\succ }})$ is an order pair.", "The relation $\\mathrel {\\succsim }$ is a preorder by assumption.", "Suppose $s \\mathrel {\\succ }t \\mathrel {\\succ }u$ .", "By assumption there exist $k$ and $l$ such that $s \\mathrel {\\succ }_k t \\mathrel {\\succ }_l u$ .", "Let $m = \\max (k,l)$ .", "We obtain $s \\mathrel {\\succ }_m t \\mathrel {\\succ }_m u$ from the assumptions of the lemma and hence $s \\mathrel {\\succ }_m u$ follows from the fact that $({\\mathrel {\\succsim }},{\\mathrel {\\succ }_m})$ is an order pair.", "Compatibility is an immediate consequence of the assumptions and the irreflexivity of $\\mathrel {\\succ }$ is obtained by an easy induction proof.", "[Proof of Lemma REF ] According to Lemmata REF and REF (1), it is sufficient to prove that $({=_\\mathrm {\\mathsf {AC}}},{>_k})$ is an order pair for all $k \\in \\mathbb {N}$ .", "Due to Lemma REF (2,3) it suffices to prove that $({=_\\mathrm {\\mathsf {AC}}},{>_{23,k}})$ is an order pair, which follows by using induction on $k$ in combination with Lemma REF and Theorem REF .", "[Proof of Theorem REF ] Let $\\mathcal {T}_k$ denote the set of ground terms of size at most $k$ .", "We use induction on $k \\geqslant 1$ to show that $>_\\mathrm {\\mathsf {ACKBO}}$ is AC-total on $\\mathcal {T}_k$ .", "Let $s, t \\in \\mathcal {T}_k$ .", "We consider the case where $w(s) = w(t)$ and $\\mathsf {root}(s) = \\mathsf {root}(t) = f \\in \\mathcal {F}_\\mathrm {\\mathsf {AC}}$ .", "The other cases follow as for standard KBO.", "Let $S = {\\triangledown _{\\!f}}(s)$ and $T = {\\triangledown _{\\!f}}(t)$ .", "Clearly $S$ and $T$ are multisets over $\\mathcal {T}_{k-1}$ .", "According to the induction hypothesis, $>_\\mathrm {\\mathsf {ACKBO}}$ is AC-total on $\\mathcal {T}_{k-1}$ and since multiset extension preserves AC totality, $>_\\mathrm {\\mathsf {ACKBO}}^\\mathsf {mul}$ is AC-total on multisets over $\\mathcal {T}_{k-1}$ .", "Hence for any pair of multisets $U$ and $V$ over $\\mathcal {T}_{k-1}$ , either $U >_\\mathrm {\\mathsf {ACKBO}}^\\mathsf {mul}V\\quad \\text{or}\\quad V >_\\mathrm {\\mathsf {ACKBO}}^\\mathsf {mul}U\\quad \\text{or}\\quad U =_\\mathrm {\\mathsf {AC}}^\\mathsf {mul}V$ Because the precedence $>$ is total and $S$ and $T$ contain neither variables nor terms with $f$ as their root symbol, we have $S &= {S}{\\upharpoonright }^{\\smash{\\nless }}_{f} \\cup {S}{\\upharpoonright }^{\\smash{<}}_{f} = {S}{\\upharpoonright }^{\\smash{>}}_{f} \\cup {S}{\\upharpoonright }^{\\smash{<}}_{f} &T &= {T}{\\upharpoonright }^{\\smash{\\nless }}_{f} \\cup {T}{\\upharpoonright }^{\\smash{<}}_{f} = {T}{\\upharpoonright }^{\\smash{>}}_{f} \\cup {T}{\\upharpoonright }^{\\smash{<}}_{f}$ If ${S}{\\upharpoonright }^{\\smash{>}}_{f} >_\\mathrm {\\mathsf {ACKBO}}^\\mathsf {mul}{T}{\\upharpoonright }^{\\smash{>}}_{f}$ or ${T}{\\upharpoonright }^{\\smash{>}}_{f} >_\\mathrm {\\mathsf {ACKBO}}^\\mathsf {mul}{S}{\\upharpoonright }^{\\smash{>}}_{f}$ then case 3(a) of Definition REF is applicable to derive either $s >_\\mathrm {\\mathsf {ACKBO}}t$ or $t >_\\mathrm {\\mathsf {ACKBO}}s$ .", "Otherwise we must have ${S}{\\upharpoonright }^{\\smash{>}}_{f} =_\\mathrm {\\mathsf {AC}}^\\mathsf {mul}{T}{\\upharpoonright }^{\\smash{>}}_{f}$ by AC-totality.", "If $\|S\| > \|T\|$ then we obtain $s >_\\mathrm {\\mathsf {ACKBO}}t$ by case 3(b).", "Similarly, $\|S\| < \|T\|$ gives rise to $t >_\\mathrm {\\mathsf {ACKBO}}s$ .", "In the remaining case we have both ${S}{\\upharpoonright }^{\\smash{>}}_{f} =_\\mathrm {\\mathsf {AC}}^\\mathsf {mul}{T}{\\upharpoonright }^{\\smash{>}}_{f}$ and $\|S\| = \|T\|$ .", "Using case 3(c) of Definition REF we obtain $s >_\\mathrm {\\mathsf {ACKBO}}t$ when ${S}{\\upharpoonright }^{\\smash{<}}_{f} >_\\mathrm {\\mathsf {ACKBO}}^\\mathsf {mul}{T}{\\upharpoonright }^{\\smash{<}}_{f}$ and $t >_\\mathrm {\\mathsf {ACKBO}}s$ when ${T}{\\upharpoonright }^{\\smash{<}}_{f} >_\\mathrm {\\mathsf {ACKBO}}^\\mathsf {mul}{S}{\\upharpoonright }^{\\smash{<}}_{f}$ .", "By AC totality there is one case remaining: ${S}{\\upharpoonright }^{\\smash{<}}_{f} =_\\mathrm {\\mathsf {AC}}^\\mathsf {mul}{T}{\\upharpoonright }^{\\smash{<}}_{f}$ .", "Combined with ${S}{\\upharpoonright }^{\\smash{>}}_{f} =_\\mathrm {\\mathsf {AC}}^\\mathsf {mul}{T}{\\upharpoonright }^{\\smash{>}}_{f}$ we obtain $S =_\\mathrm {\\mathsf {AC}}^\\mathsf {mul}T$ .", "We may write $S = \\lbrace {s_1},\\dots ,{s_{n}} \\rbrace $ and $T = \\lbrace {t_1},\\dots ,{t_{n}} \\rbrace $ such that $s_i =_\\mathrm {\\mathsf {AC}}t_i$ for all $1 \\leqslant i \\leqslant n$ .", "Since $f$ is an AC symbol, $s =_\\mathrm {\\mathsf {AC}}f(s_1,f(\\dots , s_n)\\dots )$ and $t =_\\mathrm {\\mathsf {AC}}f(t_1,f(\\dots , t_n)\\dots )$ , from which we conclude $s =_\\mathrm {\\mathsf {AC}}t$ ." ], [ "Correctness of $>_\\mathrm {\\mathsf {KV^{\\prime }}}$", "We prove that $>_\\mathrm {\\mathsf {KV^{\\prime }}}$ is an AC-compatible simplification order.", "The proof mimics the one given in Sections and REF for $>_\\mathrm {\\mathsf {ACKBO}}$ , but there are some subtle differences.", "The easy proof of the following lemma is omitted.", "Lemma A.4 The pairs $({=_\\mathrm {\\mathsf {AC}}},{>_\\mathsf {kv}})$ and $({\\geqslant _{\\mathsf {kv^{\\prime }}}},{>_\\mathsf {kv}})$ are order pairs.", "$\\Box $ Lemma A.5 The pair $({=_\\mathrm {\\mathsf {AC}}},{>_\\mathrm {\\mathsf {KV^{\\prime }}}})$ is an order pair.", "Similar to the proof of Lemma REF , except for case 3 of Definition REF , where we need Lemma REF and Theorem REF .", "The subterm property follows exactly as in the proof of Lemma REF ; note that the relation $>_{01}$ has the subterm property, and we obviously have ${>_{01}} \\subseteq {>_\\mathrm {\\mathsf {KV^{\\prime }}}}$ .", "Lemma A.6 The order $>_\\mathrm {\\mathsf {KV^{\\prime }}}$ has the subterm property.", "$\\Box $ Lemma A.7 The order $>_\\mathrm {\\mathsf {KV^{\\prime }}}$ is closed under contexts.", "Suppose $s >_\\mathrm {\\mathsf {KV^{\\prime }}}t$ .", "We follow the proof for $>_\\mathrm {\\mathsf {ACKBO}}$ in Lemma REF and consider here the case that $w(s) = w(t)$ .", "We will show that one of the cases 3(a,b,c) in Definition REF (REF ) is applicable to $S = {\\triangledown _{\\!h}}(s)$ and $T = {\\triangledown _{\\!h}}(t)$ .", "Let $f = \\mathsf {root}(s)$ and $g = \\mathsf {root}(t)$ .", "The proof proceeds by case splitting according to the derivation of $s >_\\mathrm {\\mathsf {KV^{\\prime }}}t$ .", "Suppose $s = f^k(t)$ with $k > 0$ and $t \\in \\mathcal {V}$ .", "Admissibility enforces $f > h$ and thus ${S}{\\upharpoonright }^{\\smash{\\nless }}_{h} = \\lbrace s \\rbrace \\geqslant _{\\mathsf {kv^{\\prime }}}^\\mathsf {mul}\\lbrace t \\rbrace $ .", "We have $\|S\| = \|T\| = 1$ and $S >_\\mathrm {\\mathsf {KV^{\\prime }}}^\\mathsf {mul}T$ .", "Hence 3(c) applies.", "(This case breaks down for $>_\\mathrm {\\mathsf {KV}}$ .)", "Suppose $f = g \\notin \\mathcal {F}_\\mathrm {\\mathsf {AC}}$ .", "We have $S \\geqslant _{\\mathsf {kv^{\\prime }}}^\\mathsf {mul}T$ , $\|S\| = \|T\| = 1$ , and $S = \\lbrace s \\rbrace >_\\mathrm {\\mathsf {KV^{\\prime }}}^\\mathsf {mul}\\lbrace t \\rbrace = T$ .", "Hence 3(c) applies.", "The remaining cases are similar to the proof of Lemma REF , except that we use Lemma REF with $({\\geqslant _{\\mathsf {kv^{\\prime }}}},{>_\\mathsf {kv}})$ .", "$\\Box $ For closure under substitutions we need to extend Lemma REF with the following case: If $S \\mathrel {\\succsim }^f T$ and $S^{\\prime } \\mathrel {\\nsucc }^f T^{\\prime }$ then $S^{\\prime } - T^{\\prime } \\supseteq S\\sigma - T\\sigma $ and $T\\sigma - S\\sigma \\supseteq T^{\\prime } - S^{\\prime }$ .", "We continue the proof of Lemma REF .", "From ${\\triangledown _{\\!f}}(U\\sigma ) = U\\sigma $ we infer that $T^{\\prime } = T{\\upharpoonright }_\\mathcal {F}\\sigma \\uplus U\\sigma \\uplus {\\triangledown _{\\!f}}(X\\sigma )$ .", "On the other hand, $S^{\\prime } = S{\\upharpoonright }_\\mathcal {F}\\sigma \\uplus {\\triangledown _{\\!f}}(Y\\sigma ) \\uplus {\\triangledown _{\\!f}}(X\\sigma )$ with $Y = S{\\upharpoonright }_\\mathcal {V}- X$ .", "Hence $T^{\\prime } - S^{\\prime }~ &\\subseteq ~T{\\upharpoonright }_\\mathcal {F}\\sigma \\uplus U\\sigma - S{\\upharpoonright }_\\mathcal {F}\\sigma \\\\&=~T{\\upharpoonright }_\\mathcal {F}\\sigma \\uplus U\\sigma \\uplus X\\sigma -(S{\\upharpoonright }_\\mathcal {F}\\uplus X\\sigma ) \\\\&\\subseteq ~T\\sigma - S\\sigma $ and $S^{\\prime } - T^{\\prime }~ &\\supseteq ~S{\\upharpoonright }_\\mathcal {F}\\sigma - T{\\upharpoonright }_\\mathcal {F}\\sigma - U\\sigma \\\\&=~S{\\upharpoonright }_\\mathcal {F}\\sigma \\uplus X\\sigma -(T{\\upharpoonright }_\\mathcal {F}\\uplus U\\sigma \\uplus X\\sigma ) \\\\&\\supseteq ~S\\sigma - T\\sigma $ establishing the desired inclusions.", "$\\Box $ Lemma A.8 The order $>_\\mathrm {\\mathsf {KV^{\\prime }}}$ is closed under substitutions.", "By induction on $\|s\|$ we verify that $s >_\\mathrm {\\mathsf {KV^{\\prime }}}t$ implies $s\\sigma >_\\mathrm {\\mathsf {KV^{\\prime }}}t\\sigma $ .", "If $s >_\\mathrm {\\mathsf {KV^{\\prime }}}t$ is derived by one of the cases 0, 1, 2, 3(a) or 3(b) in Definition REF (REF ), the proof of Lemma REF goes through.", "So suppose that $s >_\\mathrm {\\mathsf {KV^{\\prime }}}t$ is derived by case 3(c) and further suppose that $s\\sigma >_\\mathrm {\\mathsf {KV^{\\prime }}}t\\sigma $ can be derived neither by case 3(a) nor 3(b).", "By definition we have ${\\triangledown _{\\!f}}(s) >_\\mathrm {\\mathsf {KV^{\\prime }}}^\\mathsf {mul}{\\triangledown _{\\!f}}(t)$ .", "This is equivalentThis property is well-known for standard multiset extensions (involving a single strict order).", "It is also not difficult to prove for the multiset extension defined in Definition REF .", "to ${\\triangledown _{\\!f}}(s) - {\\triangledown _{\\!f}}(t) >_\\mathrm {\\mathsf {KV^{\\prime }}}^\\mathsf {mul}{\\triangledown _{\\!f}}(t) - {\\triangledown _{\\!f}}(s)$ We obtain ${\\triangledown _{\\!f}}(s)\\sigma - {\\triangledown _{\\!f}}(t)\\sigma >_\\mathrm {\\mathsf {KV^{\\prime }}}^\\mathsf {mul}{\\triangledown _{\\!f}}(t)\\sigma - {\\triangledown _{\\!f}}(s)\\sigma $ from the induction hypothesis and thus ${\\triangledown _{\\!f}}(s\\sigma ) - {\\triangledown _{\\!f}}(t\\sigma ) >_\\mathrm {\\mathsf {KV^{\\prime }}}^\\mathsf {mul}{\\triangledown _{\\!f}}(t\\sigma ) - {\\triangledown _{\\!f}}(s\\sigma )$ by Lemma REF (1).", "Using the earlier equivalence, we infer ${\\triangledown _{\\!f}}(s\\sigma ) >_\\mathrm {\\mathsf {KV^{\\prime }}}^\\mathsf {mul}{\\triangledown _{\\!f}}(t\\sigma )$ and hence case 3(c) applies to obtain the desired $s\\sigma >_\\mathrm {\\mathsf {KV^{\\prime }}}t\\sigma $ .", "The combination of the above results proves Theorem REF ." ], [ "NP-Hardness of AC-KBO", "Next we show NP-hardness of the orientability problem for $>_\\mathrm {\\mathsf {ACKBO}}$ .", "To this end we introduce the TRS $\\mathcal {R}_0^{\\prime }$ consisting of the rules $\\mathsf {a}(p_1(\\mathsf {c})) \\rightarrow p_1(\\mathsf {a}(\\mathsf {c}))\\qquad \\cdots \\qquad \\mathsf {a}(p_m(\\mathsf {c})) \\rightarrow p_m(\\mathsf {a}(\\mathsf {c}))$ together with a rule $\\mathsf {e}_i^0(\\mathsf {e}_i^1(\\mathsf {c})) \\rightarrow \\mathsf {e}_i^1(\\mathsf {e}_i^0(\\mathsf {c}))$ for each clause $C_i$ that contains a negative literal.", "The next property is immediate.", "Lemma A.9 If $\\mathcal {R}_0^{\\prime } \\subseteq {>_\\mathrm {\\mathsf {ACKBO}}}$ then $\\mathsf {e}_i^0 > \\mathsf {e}_i^1$ for all $1 \\leqslant i \\leqslant n$ and $\\mathsf {a}> p_j$ for all $1 \\leqslant j \\leqslant m$ .", "$\\Box $ The TRS $\\mathcal {R}_0 \\cup \\mathcal {R}_0^{\\prime } \\cup \\lbrace \\ell _i \\rightarrow r_i \\mid 1 \\leqslant i \\leqslant n \\rbrace $ is denoted by $\\mathcal {R}_\\phi ^{\\prime }$ .", "Lemma A.10 Suppose $\\mathsf {a}> + > \\mathsf {b}$ and the consequence of Lemma REF holds.", "Then $\\mathcal {R}_\\phi ^{\\prime } \\subseteq {>_\\mathrm {\\mathsf {ACKBO}}}$ for some $(w, w_0)$ if and only if for every $i$ there is some $p$ such that $p \\in C_i$ with $p \\nless +$ or $\\lnot p \\in C_i$ with $+ > p$ .", "The “if” direction is analogous to Lemma REF .", "Let us prove the “only if” direction by contradiction.", "Suppose $+ > p^{\\prime }_j$ for all $1 \\leqslant j \\leqslant k$ , $p^{\\prime \\prime }_j \\nless +$ for all $1 \\leqslant j \\leqslant l$ , and $\\mathcal {R}_\\phi ^{\\prime } \\subseteq {>_\\mathrm {\\mathsf {ACKBO}}}$ .", "As discussed in the proof of Lemma REF , for the multisets $V$ and $W$ on page we obtain $V >_\\mathrm {\\mathsf {ACKBO}}^\\mathsf {mul}W$ and all terms in $V$ and $W$ have the same weight.", "With the help of Lemma REF we infer that $\\mathsf {a}(\\mathsf {e}_i^0(\\mathsf {e}_i^0(\\mathsf {c}))) \\in W$ is greater than every other term in $V$ and $W$ .", "This contradicts $V >_\\mathrm {\\mathsf {ACKBO}}^\\mathsf {mul}W$ .", "Using Lemmata REF and REF , Theorem REF can now be proved in the same way as Theorem REF ." ], [ "AC-RPO", "[Proof of Lemma REF ] Because of totality of the precedence, ${S}{\\upharpoonright }^{\\smash{\\lnot <}}_{f}$ is identified with ${S}{\\upharpoonright }^{\\smash{>}}_{f}$ in the sequel.", "First suppose $s >_\\mathrm {\\mathsf {ACRPO}}t$ holds by case 4.", "We may assume that $>_\\mathrm {\\mathsf {ACRPO}}$ and $>_\\mathrm {\\mathsf {ACRPO^{\\prime }}}$ coincide on smaller terms.", "The conditions on $\\vartriangleright ^{\\!f}_{\\!\\mathsf {emb}}$ are obviously the same.", "We distinguish which case applies.", "We have ${S}{\\upharpoonright }^{\\smash{>}}_{f} >_\\mathrm {\\mathsf {ACRPO}}^\\mathsf {mul}{T}{\\upharpoonright }^{\\smash{>}}_{f} \\uplus T{\\upharpoonright }_\\mathcal {V}-S{\\upharpoonright }_\\mathcal {V}$ and thus both ${S}{\\upharpoonright }^{\\smash{>}}_{f} \\uplus S{\\upharpoonright }_\\mathcal {V}\\geqslant _\\mathrm {\\mathsf {ACRPO}}^\\mathsf {mul}{T}{\\upharpoonright }^{\\smash{>}}_{f} \\uplus T{\\upharpoonright }_\\mathcal {V}$ and ${S}{\\upharpoonright }^{\\smash{>}}_{f} >_\\mathrm {\\mathsf {ACRPO}}^\\mathsf {mul}{T}{\\upharpoonright }^{\\smash{>}}_{f}$ .", "So case 4$^{\\prime }$ (a) is applicable.", "We have $\|S\| > \|T\|$ and $S =_\\mathrm {\\mathsf {AC}}^f T$ , i.e., ${S}{\\upharpoonright }^{\\smash{>}}_{f} =_\\mathrm {\\mathsf {AC}}^\\mathsf {mul}{T}{\\upharpoonright }^{\\smash{>}}_{f} \\uplus T{\\upharpoonright }_\\mathcal {V}-S{\\upharpoonright }_\\mathcal {V}$ , and in particular $T{\\upharpoonright }_\\mathcal {V}\\subseteq S{\\upharpoonright }_\\mathcal {V}$ .", "Thus ${S}{\\upharpoonright }^{\\smash{>}}_{f} \\uplus S{\\upharpoonright }_\\mathcal {V}\\geqslant _\\mathrm {\\mathsf {ACRPO}}^\\mathsf {mul}{T}{\\upharpoonright }^{\\smash{>}}_{f} \\uplus T{\\upharpoonright }_\\mathcal {V}$ holds.", "Since $T{\\upharpoonright }_\\mathcal {V}\\subseteq S{\\upharpoonright }_\\mathcal {V}$ and $\|S\| > \|T\|$ imply $\\#(S) > \\#(T)$ , case 4$^{\\prime }$ (b) applies.", "We obtain ${S}{\\upharpoonright }^{\\smash{>}}_{f} \\uplus S{\\upharpoonright }_\\mathcal {V}\\geqslant _\\mathrm {\\mathsf {ACRPO}}^\\mathsf {mul}{T}{\\upharpoonright }^{\\smash{>}}_{f} \\uplus T{\\upharpoonright }_\\mathcal {V}$ as in case 4(b).", "Together with $\|S\| = \|T\|$ this implies $\\#(S) \\geqslant \\#(T)$ .", "As $S = {S}{\\upharpoonright }^{\\smash{>}}_{f} \\uplus S{\\upharpoonright }_\\mathcal {V}\\uplus {S}{\\upharpoonright }^{\\smash{<}}_{f}$ and similar for $T$ , we obtain $S >_\\mathrm {\\mathsf {ACRPO}}^\\mathsf {mul}T$ from the assumption ${S}{\\upharpoonright }^{\\smash{<}}_{f} >_\\mathrm {\\mathsf {ACRPO}}^\\mathsf {mul}{T}{\\upharpoonright }^{\\smash{<}}_{f}$ .", "Hence case 4$^{\\prime }$ (c) is applicable.", "Now let $s >_\\mathrm {\\mathsf {ACRPO^{\\prime }}}t$ by case 4$^{\\prime }$ .", "Again we assume that $>_\\mathrm {\\mathsf {ACRPO}}$ and $>_\\mathrm {\\mathsf {ACRPO^{\\prime }}}$ coincide on smaller terms.", "We have ${S}{\\upharpoonright }^{\\smash{>}}_{f} \\uplus S{\\upharpoonright }_\\mathcal {V}\\geqslant _\\mathrm {\\mathsf {ACRPO}}^\\mathsf {mul}{T}{\\upharpoonright }^{\\smash{>}}_{f} \\uplus T{\\upharpoonright }_\\mathcal {V}$ ($\\ast $ ).", "We have ${S}{\\upharpoonright }^{\\smash{>}}_{f} >_\\mathrm {\\mathsf {ACRPO}}^\\mathsf {mul}{T}{\\upharpoonright }^{\\smash{>}}_{f}$ .", "Suppose $S \\lnot >_\\mathrm {\\mathsf {ACRPO}}^f T$ , i.e., ${S}{\\upharpoonright }^{\\smash{>}}_{f} >_\\mathrm {\\mathsf {ACRPO}}^\\mathsf {mul}{T}{\\upharpoonright }^{\\smash{>}}_{f} \\uplus T{\\upharpoonright }_\\mathcal {V}-S{\\upharpoonright }_\\mathcal {V}$ does not hold.", "This is only possible if there is some variable $x \\in T{\\upharpoonright }_\\mathcal {V}- S{\\upharpoonright }_\\mathcal {V}$ for which there is no term $s^{\\prime } \\in {S}{\\upharpoonright }^{\\smash{>}}_{f}$ with $s^{\\prime } >_\\mathrm {\\mathsf {ACRPO}}x$ .", "This however contradicts ($\\ast $ ), so $S >_\\mathrm {\\mathsf {ACRPO}}^f T$ holds and case 4(a) applies.", "If ${S}{\\upharpoonright }^{\\smash{>}}_{f} >_\\mathrm {\\mathsf {ACRPO}}^\\mathsf {mul}{T}{\\upharpoonright }^{\\smash{>}}_{f}$ holds then case 4(a) applies by the reasoning in case 4$^{\\prime }$ (a).", "Otherwise, due to ($\\ast $ ) we must have $S =_\\mathrm {\\mathsf {AC}}^f T$ .", "Since $\\#(S) > \\#(T)$ implies $\|S\| > \|T\|$ , case 4(b) applies.", "If $\\#(S) > \\#(T)$ is satisfied we argue as in the preceding case.", "Otherwise $\\#(S) \\geqslant \\#(T)$ and $\\#(S) \\ngtr \\#(T)$ .", "This implies both $\|S\| = \|T\|$ and $S{\\upharpoonright }_\\mathcal {V}\\supseteq T{\\upharpoonright }_\\mathcal {V}$ .", "We obtain $S =_\\mathrm {\\mathsf {AC}}^f T$ as in case 4$^{\\prime }$ (b).", "From the assumption $S >_\\mathrm {\\mathsf {ACRPO}}^\\mathsf {mul}T$ we infer ${S}{\\upharpoonright }^{\\smash{<}}_{f} >_\\mathrm {\\mathsf {ACRPO}}^\\mathsf {mul}{T}{\\upharpoonright }^{\\smash{<}}_{f}$ and thus case 4(c) applies.", "$\\Box $" ] ]	1403.0406
[ [ "The automorphism group of a shift of subquadratic growth" ], [ "Abstract For a subshift over a finite alphabet, a measure of the complexity of the system is obtained by counting the number of nonempty cylinder sets of length $n$.", "When this complexity grows exponentially, the automorphism group has been shown to be large for various classes of subshifts.", "In contrast, we show that subquadratic growth of the complexity implies that for a topologically transitive shift $X$, the automorphism group $\\Aut(X)$ is small: if $H$ is the subgroup of $\\Aut(X)$ generated by the shift, then $\\Aut(X)/H$ is periodic." ], [ "Introduction", "In this note, we study the group of automorphisms of a subshift.", "More precisely, if $\\mathcal {A}$ is a finite alphabet with the discrete topology and we endow $\\mathcal {A}^{\\mathbb {Z}}$ with the product topology, a closed set $X\\subseteq \\mathcal {A}^{\\mathbb {Z}}$ is called a subshift it is invariant under the left shift map $\\sigma \\colon \\mathcal {A}^{\\mathbb {Z}}\\rightarrow \\mathcal {A}^{\\mathbb {Z}}$ that acts on $x\\in \\mathcal {A}^{\\mathbb {Z}}$ by $(\\sigma x)(i+1):=x(i)$ for all $i\\in \\mathbb {Z}$ .", "An automorphism of $(X,\\sigma )$ is a homeomorphism $\\varphi \\colon X\\rightarrow X$ that commutes with $\\sigma $ .", "We denote the group of automorphisms of $(X, \\sigma )$ by ${\\rm Aut}(X)$ .", "There are numerous theorems showing that the automorphism group can be extremely large for different classes of subshifts.", "The first such result was proven by Curtis, Hedlund and Lyndon (see Hedlund [7]), who showed that ${\\rm Aut}(\\mathcal {A}^{\\mathbb {Z}})$ contains isomorphic copies of any finite group and also contains two involutions whose product has infinite order.", "For mixing one dimensional subshifts of finite type (of which $\\mathcal {A}^{\\mathbb {Z}}$ is an example), Boyle, Lind and Rudolph [2] showed that the automorphism group contains the free group on two generators, the direct sum of countably many copies of $\\mathbb {Z}$ , and the direct sum of every countable collection of finite groups.", "These theorems show that ${\\rm Aut}(X)$ is large, and are proven by constructing automorphisms that generate subgroups with prescribed properties.", "In contrast, we are interested in placing restrictions on ${\\rm Aut}(X)$ , showing that for certain classes of subshifts, the automorphism group can not contain certain structures, and so we need a different approach.", "There are cases in which the automorphism group can be characterized.", "For example, Host and Parreau [5] gave a complete description for primitive substitutions of constant length and this was generalized in Salo and Törmä [14].", "Olli [11] described the automorphism group of Sturmian shifts, and generalizations are given in [4].", "For each result showing that the automorphism group is large, there is a notion of complexity associated with the system and this complexity is large.", "More precisely, for a shift system, let $P_X(n)$ denote the number of nonempty cylinder sets of length $n$ .", "For the full shift and for mixing subshifts of finite type, this complexity grows exponentially and this growth is an important ingredient in the constructions.", "We study the opposite situation, where the complexity has slow growth and we show that this places strong restrictions on ${\\rm Aut}(X)$ .", "In particular, we are interested in shifts $(X,\\sigma )$ for which $P_X(n)$ grows subquadratically (see Section for the precise definition of the growth).", "To state our main theorem, recall that a (possibly infinite) group is periodic if every element has finite order.", "We show: Theorem 1.1 Suppose $(X,\\sigma )$ is a topologically transitive shift of subquadratic growth and let $H$ be the subgroup of ${\\rm Aut}(X)$ generated by $\\sigma $ .", "Then ${\\rm Aut}(X)/H$ is a periodic group.", "The collection of shifts of subquadratic growth includes many examples that arise naturally in symbolic dynamics and in the combinatorics of words.", "The theorem applies to Sturmian sfhits, and more generally to Arnoux-Rauzy shifts [1] (which have linear complexity) and linearly recurrent systems [3].", "A theorem of Pansiot [12] shows that for a purely morphic shift $X$ , $P_X$ is one of $\\Theta (1)$ , $\\Theta (n)$ , $\\Theta (n\\log \\log n)$ , $\\Theta (n\\log n)$ , or $\\Theta (n^2)$ , where $\\Theta $ is the asymptotic growth rate, and all but the last class have subquadratic growth.", "For more extensive literature on shift systems, see for example [6].", "We conclude with a brief comment on the ideas in the proof of Theorem REF .", "Instead of working in the one dimensional setting, we use the one dimensional automorphisms to produce colorings of $\\mathbb {Z}^2$ .", "By the complexity assumption on ${\\rm Aut}(X)$ , we can apply a theorem of Quas and Zamboni showing that these colorings are simple.", "We then use this information on the two dimensional colorings to deduce the one dimensional result." ], [ "One dimensional shifts", "Throughout we assume that $\\mathcal {A}$ is a finite set.", "For $x\\in \\mathcal {A}^{\\mathbb {Z}}$ , we write $x = (x(i)\\colon i\\in \\mathbb {Z})$ and let $x(i)$ denote the element of $\\mathcal {A}$ that $x$ assigns to $i\\in \\mathbb {Z}$ .", "The shift map $\\sigma \\colon \\mathcal {A}^{\\mathbb {Z}}\\rightarrow \\mathcal {A}^{\\mathbb {Z}}$ is defined by $(\\sigma x)(i):=x(i+1)$ .", "With respect to the metric $d(x,y):=2^{-\\min \\lbrace \|i\|\\colon x(i)\\ne y(i)\\rbrace },$ $\\mathcal {A}^{\\mathbb {Z}}$ is compact and $\\sigma $ is a homeomorphism.", "If $F\\subset \\mathbb {Z}$ is a finite set and $\\beta \\in \\mathcal {A}^F$ , then the cylinder set $[F;\\beta ]$ is defined as $[F;\\beta ] := \\lbrace x\\in \\mathcal {A}^\\mathbb {Z}\\colon x(i) = \\beta (i) \\text{ for all } i\\in F\\rbrace .$ The collection of all cylinder sets is a basis for the topology of $\\mathcal {A}^{\\mathbb {Z}}$ .", "A closed, $\\sigma $ -invariant set $X\\subseteq \\mathcal {A}^{\\mathbb {Z}}$ is called a subshift.", "The group of all homeomorphisms from $X$ to itself that commute with $\\sigma $ is called the automorphism group of $X$ and is denoted ${\\rm Aut}(X)$ .", "A classical result of Hedlund [7] says that if $\\varphi \\in {\\rm Aut}(X)$ , then $\\varphi $ is a sliding block code, meaning that there exists $N_{\\varphi }\\in \\mathbb {N}$ such that for all $x\\in X$ and all $i\\in \\mathbb {Z}$ , $(\\varphi x)(i)$ is determined entirely by $(x_{i-N_{\\varphi }}, x_{i-N_{\\varphi }+1},\\dots ,x_{i+N_{\\varphi }-1},x_{i+N_{\\varphi }})\\in \\mathcal {A}^{2N_{\\varphi }+1}$ .", "An automorphism $\\varphi $ has range $N$ if $N_{\\varphi }$ can be chosen to be $N$ .", "As a measure of the complexity of a given subshift $X$ , the block complexity function $P_X\\colon \\mathbb {N}\\rightarrow \\mathbb {N}$ is defined by $P_X(n)=\\bigl \\vert \\lbrace \\beta \\in \\mathcal {A}^{B_n}\\colon [\\beta , B_n] \\cap X\\ne \\emptyset \\rbrace \\bigr \\vert ,$ where $B_n :=\\lbrace x\\in \\mathbb {Z}\\colon 0\\le x< n\\rbrace $ .", "Defining the complexity $P_x(n)$ to be the number of configurations in a window of size $n$ in some fixed $x\\in X$ , we have that $P_X(n) \\ge \\sup _{x\\in X} P_x(n),$ and equality holds when the subshift $X$ is transitive.", "It is well-known that $P_X(n)$ is sub-multiplicative and so the topological entropy $h_{top}(X)$ of $X$ defined by $h_{top}(X):=\\lim _{n\\rightarrow \\infty }\\frac{\\log (P_X(n))}{n},$ is well-defined (see, for example [8]).", "For subshifts whose topological entropy is zero, one can study the upper polynomial growth rate of $X$ defined by $\\overline{P}(X):=\\limsup _{n\\rightarrow \\infty }\\frac{\\log (P_X(n))}{\\log (n)}\\in [0,\\infty ]$ and the lower polynomial growth rate of $X$ given by $\\underline{P}(X):=\\liminf _{n\\rightarrow \\infty }\\frac{\\log (P_X(n))}{\\log (n)}\\in [0,\\infty ].$ The classical Morse-Hedlund Theorem [9] states that $x\\in X$ is periodic if and only there exists some $n\\in \\mathbb {N}$ such that $P_x(n)\\le n$ .", "It follows immediately that if $X$ contains at least one aperiodic element (that is, at least one $x\\in X$ for which $\\sigma ^ix\\ne \\sigma ^jx$ for any $i\\ne j$ ), then $\\underline{P}(X)\\ge 1$ ." ], [ "Two dimensional shifts", "With minor modifications, these notions extend to higher dimensions.", "We only need the results in two dimensions and so only state the generalizations in this setting.", "If $\\eta \\in \\mathcal {A}^{\\mathbb {Z}^2}$ , let $\\eta (i,j)$ denote the entry $\\eta $ assigns to $(i,j)\\in \\mathbb {Z}^2$ .", "With respect to the metric $d(\\eta _1,\\eta _2)=2^{-\\min \\lbrace \\Vert \\vec{v}\\Vert \\colon \\eta _1(\\vec{v})\\ne \\eta _2(\\vec{v})\\rbrace },$ the space $\\mathcal {A}^{\\mathbb {Z}^2}$ is compact.", "If $F\\subset \\mathbb {Z}^2$ is finite and $\\beta \\colon F\\rightarrow \\mathcal {A}$ , then the cylinder set $[F;\\beta ]$ is defined as $[F;\\beta ]:=\\lbrace \\eta \\in \\mathcal {A}^{\\mathbb {Z}^2}\\colon \\eta (i,j)=\\beta (i,j)\\text{ for all }(i,j)\\in F\\rbrace .$ As for $\\mathcal {A}^{\\mathbb {Z}}$ , the cylinder sets form a basis for the topology of $\\mathcal {A}^{\\mathbb {Z}^2}$ .", "We define the left-shift $S\\colon \\mathcal {A}^{\\mathbb {Z}^2}\\rightarrow \\mathcal {A}^{\\mathbb {Z}^2}$ by $(S\\eta )(i,j):=x(i+1,j)$ and the down-shift $T\\colon \\mathcal {A}^{\\mathbb {Z}^2}\\rightarrow \\mathcal {A}^{\\mathbb {Z}^2}$ by $(T\\eta )(i,j):=x(i,j+1).$ These maps commute and both are homeomorphisms of $\\mathcal {A}^{\\mathbb {Z}^2}$ .", "For $\\eta \\in \\mathcal {A}^{\\mathbb {Z}^2}$ , we denote the $\\mathbb {Z}^2$ -orbit of $\\eta $ by $\\mathcal {O}(\\eta ):=\\lbrace S^aT^b\\eta \\colon (a,b)\\in \\mathbb {Z}^2\\rbrace $ and let $\\overline{\\mathcal {O}}(\\eta )$ denote the closure of $\\mathcal {O}(x)$ in $\\mathcal {A}^{\\mathbb {Z}^2}$ (note that the $\\mathbb {Z}^2$ action by the shifts $S$ and $T$ is implicit in this notation).", "A closed subset $Y\\subseteq \\mathcal {A}^{\\mathbb {Z}^2}$ is a subshift of $\\mathcal {A}^{\\mathbb {Z}^2}$ if it is both $S$ -invariant and $T$ -invariant.", "In particular, for any fixed $\\eta \\in \\mathcal {A}^{\\mathbb {Z}^2}$ the set $\\overline{\\mathcal {O}}(\\eta )$ is a subshift." ], [ "Automorphisms and $\\mathbb {Z}^2$ configurations", "Suppose $\\varphi \\in {\\rm Aut}(X)$ and $x\\in X$ .", "We define an element of $\\mathcal {A}^{\\mathbb {Z}^2}$ by: $\\eta _{\\varphi ,x}(i,j):=(\\varphi ^jx)_i.$ For fixed $\\varphi \\in {\\rm Aut}(X)$ , the inclusion map $\\imath _{\\varphi }\\colon X\\rightarrow \\mathcal {A}^{\\mathbb {Z}^2}$ given by $\\imath _{\\varphi }(x):=\\eta _{\\varphi ,x}$ is a homeomorphism from $X$ to $\\imath _{\\varphi }(X)$ and satisfies $\\imath _{\\varphi }\\circ \\sigma =S\\circ \\imath _{\\varphi }$ and $\\imath _{\\varphi }\\circ \\varphi =T\\circ \\imath _{\\varphi }$ .", "That is, $\\imath _{\\varphi }$ is a topological conjugacy between the $\\mathbb {Z}^2$ -dynamical system $(X,\\sigma ,\\varphi )$ and the $\\mathbb {Z}^2$ -dynamical system $(\\imath _{\\varphi }(X),S,T)$ .", "(Note that the joint action of $\\sigma $ and $\\varphi $ on $X$ is a $\\mathbb {Z}^2$ -dynamical system, as $\\sigma $ commutes with $\\varphi $ .)", "For a subshift $Y\\subseteq \\mathcal {A}^{\\mathbb {Z}^2}$ , the rectangular complexity function $P_Y\\colon \\mathbb {N}\\times \\mathbb {N}\\rightarrow \\mathbb {N}$ is defined by $P_Y(n,k):=\\left\|\\lbrace \\beta \\in \\mathcal {A}^{R_{n,k}}\\colon [\\beta ; R_{n,k}]\\cap Y\\ne \\emptyset \\rbrace \\right\|$ where $R_{n,k}:=\\lbrace (x,y)\\in \\mathbb {Z}^2\\colon 0\\le x<n\\text{ and }0\\le y<k\\rbrace $ .", "(Again, we could have defined this for elements $y\\in Y$ and taken a supremum and equality of these complexities holds for transitive systems.)", "For $x\\in X$ , the crucial relationship between the (one-dimensional) block complexity $P_X$ and the (two-dimensional) rectangular complexity $P_{\\overline{\\mathcal {O}}(\\imath _{\\varphi }(x))}$ is the following: Lemma 2.1 Suppose $X$ is a shift and $\\varphi ,\\varphi ^{-1}\\in {\\rm Aut}(X)$ are block codes of range $N$ .", "Then for any $x\\in X$ , we have $P_{\\overline{\\mathcal {O}}(\\eta _{\\varphi ,x})}(n,k)\\le P_X(2Nk-2N+n).$ Suppose $[\\beta ;R_{n,k}]\\cap \\imath _{\\varphi }(X)\\ne \\emptyset $ .", "Since $\\imath _{\\varphi }$ is a topological conjugacy between $(X,\\sigma ,\\varphi )$ and $(\\imath _{\\varphi }(X),S,T)$ , there exists $x\\in X$ such that the restriction of $\\eta _{\\varphi ,x}$ to $R_{n,k}$ is $\\beta $ .", "Since $\\varphi $ is a block code of range $N$ , the word $(x_{-N-i_1},x_{-N-i_1+1},\\dots ,x_{N+i_2-1},x_{N+i_2})$ determines $(\\varphi x)_j$ for all $-i_1\\le j\\le i_2$ .", "It follows inductively that the word $(x_{-tN-i_1},x_{-tN-i_1+1},\\dots ,x_{tN+i_2-1},x_{tN+i_2})$ determines $(\\varphi ^rx)_j$ for $1\\le r\\le t$ and $(t-r)N-i_1\\le j\\le (t-r)N+i_2$ .", "In particular, the word $(x_{-(k-1)N},x_{-(k-1)N+1},\\dots ,x_{(k-1)N+n-1},x_{(k-1)N+n-1})$ determines $(\\varphi ^rx)_j$ for all $0\\le r<k$ and all $0\\le j<n$ .", "This means that the restriction of $\\eta _{\\varphi ,x}$ to the set $R_{n,k}$ is determined by the restriction of $\\eta _{\\varphi ,x}$ to the set $\\lbrace (x,0)\\in \\mathbb {Z}^2\\colon -(k-1)N\\le x\\le (k-1)N+n-1\\rbrace $ .", "By definition of $\\eta _{\\varphi ,x}$ , this is determined by the word $(x_{-(k-1)N},x_{-(k-1)N+1},\\dots ,x_{(k-1)N+n-1},x_{(k-1)N+n-1}).$ The number of distinct colorings of this form is $P_X(2(k-1)N+n)$ and so the number of distinct $\\beta \\colon R_{n,k}\\rightarrow \\mathcal {A}$ for which $[\\beta ;R_{n,k}]\\ne \\emptyset $ is at most $P_X(2Nk-2N+n)$ .", "It follows from Lemma REF that if $\\liminf _{n\\rightarrow \\infty }\\frac{P_X(n)}{n^2}=0,$ then $\\liminf _{n\\rightarrow \\infty }\\frac{P_{\\overline{\\mathcal {O}}(\\imath _{\\varphi }(x))}(n,n)}{n^2}=0.$ Definition 2.2 We say that a shift $X\\subseteq \\mathcal {A}^{\\mathbb {Z}}$ satisfying (REF ) is a shift of subquadratic growth.", "Remark 2.3 We remark that the condition that the shift X has subquadratic growth is related to a statement about the lower polynomial growth rate of $X$ .", "If $\\underline{P}(X)<2$ , then $X$ has subquadratic growth.", "On the other hand, if $X$ has subquadratic growth, then $\\underline{P}(X)\\le 2$ .", "If $\\underline{P}(X)=2$ , then $X$ may or may not have subquadratic growth." ], [ "Shifts of subquadratic growth", "If $\\varphi \\in {\\rm Aut}(X)$ , then as $x\\in X$ varies, our main tool to study the $\\mathbb {Z}^2$ -configurations that arise as $\\eta _{\\varphi ,x}$ (as defined in (REF )) is the following theorem of Quas and Zamboni: Theorem 3.1 (Quas and Zamboni [13]) Suppose $\\eta \\in \\mathcal {A}^{\\mathbb {Z}^2}$ and there exists $n,k\\in \\mathbb {N}$ such that $P_{\\eta }(n,k)\\le nk/16$ .", "Then there is a finite set $F\\subset \\mathbb {Z}^2\\setminus \\lbrace (0,0)\\rbrace $ (that depends only on $n$ and $k$ ) and a vector $\\vec{v}\\in F$ such that $\\eta (\\vec{x}+\\vec{v})=\\eta (\\vec{x})$ for all $\\vec{x}\\in \\mathbb {Z}^2$ .", "Although this is not the way their theorem is stated, by checking through the cases in the proof, this is exactly what they show in Theorem 4 in [13].", "We use this to show: Lemma 3.2 Suppose $X$ is a shift of subquadratic growth and $\\varphi \\in {\\rm Aut}(X)$ .", "Then there exists a finite set $F\\subset \\mathbb {Z}^2\\setminus \\lbrace (0,0)\\rbrace $ (which depends only on $\\varphi $ and $X$ ) such that for all $x\\in X$ , there exists $(a_{(\\varphi ,x)}, b_{(\\varphi ,x)})\\in F$ such that $S^{a_{(\\varphi ,x)}}T^{b_{(\\varphi ,x)}}\\imath _{\\varphi }(x)=\\imath _{\\varphi }(x)$ .", "Suppose $\\liminf _{n\\rightarrow \\infty }\\frac{P_X(n)}{n^2}=0.$ Let $N$ be the range of the block code $\\varphi $ .", "Find the smallest $n_1\\in \\mathbb {N}$ for which $P_X(2N(n_1-1)+n_1)\\le n_1^2/16.$ By Theorem REF , there exists a finite set $F\\subset \\mathbb {Z}^2\\setminus \\lbrace (0,0)\\rbrace $ (which depends only on $n_1$ and hence only on the subshift $X$ ) such that if $\\eta \\in \\mathcal {A}^{\\mathbb {Z}^2}$ satisfies $P_{\\eta }(n_1,n_1)\\le n_1^2/16$ , then there exists $\\vec{v}\\in F$ for which $\\eta (\\vec{x}+\\vec{v})=\\eta (\\vec{x})$ for all $\\vec{x}\\in \\mathbb {Z}^2$ .", "Let $x\\in X$ be fixed.", "By (REF ), $P_{\\overline{\\mathcal {O}}(\\imath _{\\varphi }(x))}(n_1,n_1)\\le n_1^2/16.$ Thus for some $(a_{\\varphi ,x},b_{\\varphi ,x})\\in F$ , we have that $S^{a_{\\varphi ,x}}T^{b_{\\varphi ,x}}\\imath _{\\varphi }(x)=\\imath _{\\varphi }(x)$ .", "Lemma 3.3 Suppose $X$ is a topologically transitive shift of subquadratic growth and let $\\varphi \\in {\\rm Aut}(X)$ .", "Then there exists a vector $(a_{\\varphi },b_{\\varphi })\\in \\mathbb {Z}^2\\setminus \\lbrace (0,0)\\rbrace $ such that for all $x\\in X$ , $S^{a_{\\varphi }}T^{b_{\\varphi }}\\imath _{\\varphi }(x)=\\imath _{\\varphi }(x)$ .", "By Lemma REF there exists a finite set $F\\subseteq \\mathbb {Z}^2\\setminus \\lbrace (0,0)\\rbrace $ such that for all $x\\in X$ there exists $\\vec{v}\\in F$ such that $\\imath _{\\varphi }(x)$ is periodic with period vector $\\vec{v}$ .", "For each $x\\in X$ let $V_x:=\\lbrace \\vec{v}\\in F\\colon \\vec{v}\\text{ is a period vector of }\\imath _{\\varphi }(x)\\rbrace .$ Since $V_x\\subseteq F$ and $F$ is finite, there exists $M\\in \\mathbb {N}$ such that whenever $V_x$ contains two linearly independent vectors (so that $\\imath _{\\varphi }(x)$ is doubly periodic) the vertical period of $\\imath _{\\varphi }(x)$ is at most $M$ .", "Therefore, if $V_x$ contains two linearly independent vectors for all $x\\in X$ , then $\\imath _{\\varphi }(x)$ is vertically periodic with period vector $(0,M!", ")$ for all $x\\in X$ .", "In this case, the vector $(a_{\\varphi },b_{\\varphi })=(0,M!", ")$ satisfies the conclusion of the lemma.", "We are left with showing that if there exists $x\\in X$ such that all of the vectors in $V_x$ are collinear, then there exists $(a_{\\varphi },b_{\\varphi })\\in F$ such that $\\imath _{\\varphi }(x)$ is periodic with period $(a_{\\varphi },b_{\\varphi })$ for all $x\\in X$ .", "Let $B:=\\lbrace x\\in X\\colon \\dim ({\\rm Span}(V_x))=1\\rbrace $ be the set of “bad points” in $X$ .", "For each $x\\in B$ let $v(x)$ be a shortest nonzero integer vector that spans ${\\rm Span}(V_x)$ (there are two possible choices).", "Fix some $x_0\\in B$ and let $\\vec{v}=v(x_0)$ .", "There are two cases to consider: Case 1: Suppose that $v(x)$ is collinear with $v(y)$ for any $x,y\\in B$ .", "Fix $x_0\\in B$ and let $\\vec{v}\\in \\mathbb {Z}^2\\setminus \\lbrace (0,0)\\rbrace $ be a shortest integer vector parallel to $v(x_0)$ (there are two possible choices).", "Then for all $x\\in B$ , there exists $n_x\\in \\mathbb {Z}$ such that $v(x)=n_x\\cdot v_{x_0}$ .", "Since $v(x)\\in F$ and $F$ is finite, $\\lbrace n_x\\colon x\\in B\\rbrace $ is bounded.", "For all $y\\in X\\setminus B$ the coloring $\\imath _{\\varphi }(y)$ is doubly periodic.", "For each such $y$ , choose $n_y\\in \\mathbb {Z}$ such that $n_y\\cdot \\vec{v}$ is a shortest nonzero period vector for $\\imath _{\\varphi }(y)$ parallel to $\\vec{v}$ .", "Since $\\imath _{\\varphi }(y)$ has two linearly independent period vectors in $F$ , the set $\\lbrace n_y\\colon y\\in X\\setminus B\\rbrace $ is bounded.", "Therefore if $N$ is the least common multiple of $\\lbrace \|n_z\|\\colon z\\in X\\rbrace $ , then $N\\cdot \\vec{v}$ is a period vector for $\\imath _{\\varphi }(z)$ for all $z\\in X$ .", "In this case, set $(a_{\\varphi },b_{\\varphi }):=N\\cdot \\vec{v}$ .", "Case 2: Suppose there exist $x_1, x_2\\in B$ such that $v(x_1)$ is not collinear with $v(x_2)$ .", "We obtain a contradiction in this case, thereby completing the proof of the lemma.", "Since $\\dim ({\\rm Span}(V_{x_1}))=1$ , for any $\\vec{w}\\in F\\setminus V_{x_1}$ there exists $\\vec{y}_{\\vec{w}}\\in \\mathbb {Z}^2$ such that $\\eta _{\\varphi ,x_1}(\\vec{y}_{\\vec{w}})\\ne \\eta _{\\varphi ,x_1}(\\vec{y}_{\\vec{w}}+\\vec{w}).$ Choose $N_1\\in \\mathbb {N}$ such that the restriction of $\\eta _{\\varphi ,x_1}$ to the set $\\lbrace (x,0)\\colon -N_1\\le x\\le N_1\\rbrace $ determines $\\eta _{\\varphi ,x_1}(\\vec{y}_{\\vec{w}})$ and $\\eta _{\\varphi ,x_1}(\\vec{y}_{\\vec{w}}+\\vec{w})$ for all $\\vec{w}\\in F\\setminus V_{x_1}$ .", "Similarly, for $\\vec{w}\\in F\\setminus V_{x_2}$ , there exists $\\vec{z}_{\\vec{w}}\\in \\mathbb {Z}^2$ such that $\\eta _{\\varphi ,x_2}(\\vec{z}_{\\vec{w}})\\ne \\eta _{\\varphi ,x_2}(\\vec{z}_{\\vec{w}}+\\vec{w}).$ Choose $N_2\\in \\mathbb {N}$ such that the restriction of $\\eta _{\\varphi ,x_2}$ to the set $\\lbrace (x,0)\\colon -N_2\\le x\\le N_2\\rbrace $ determines $\\eta _{\\varphi ,x_2}(\\vec{z}_{\\vec{w}})$ and $\\eta _{\\varphi ,x_2}(\\vec{z}_{\\vec{w}}+\\vec{w})$ for all $\\vec{w}\\in F\\setminus V_{x_2}$ .", "By topological transitivity of $(X,\\sigma )$ , there exists $\\xi \\in X$ and $a,b\\in \\mathbb {Z}$ such that $\\xi (i-a)&=& x_1(i)\\text{ for all }-N_1\\le i\\le N_1; \\\\\\xi (i-b)&=& x_2(i)\\text{ for all }-N_2\\le i\\le N_2.$ Therefore for any $\\vec{w}\\in F\\setminus V_{x_1}$ we have $\\eta _{\\varphi ,\\xi }(\\vec{y}_{\\vec{w}}-(a,0))&=&\\eta _{\\varphi ,x_1}(\\vec{y}_{\\vec{w}}); \\\\\\eta _{\\varphi ,\\xi }(\\vec{y}_{\\vec{w}}+\\vec{w}-(a,0))&=&\\eta _{\\varphi ,x_1}(\\vec{y}_{\\vec{w}}+\\vec{w}),$ and for any $\\vec{w}\\in F\\setminus V_{x_2}$ we have $\\eta _{\\varphi ,\\xi }(\\vec{y}_{\\vec{w}}-(b,0))&=&\\eta _{\\varphi ,x_1}(\\vec{y}_{\\vec{w}}); \\\\\\eta _{\\varphi ,\\xi }(\\vec{y}_{\\vec{w}}+\\vec{w}-(b,0))&=&\\eta _{\\varphi ,x_1}(\\vec{y}_{\\vec{w}}+\\vec{w}).$ By Lemma REF , $\\eta _{\\varphi ,\\xi }$ is periodic and its period vector lies in $F$ .", "Combining equations (REF ), (REF ), and () we see this vector is not an element of the set $F\\setminus V_{x_1}$ .", "Similarly, by combining equations (REF ), (REF ), and (), we see this vector is not in the set $F\\setminus V_{x_2}$ .", "Since $V_{x_1}\\cap V_{x_2}=\\emptyset $ , we obtain the desired contradiction.", "We use this lemma to complete the proof of Theorem REF : Suppose $X$ is a shift of subquadratic growth and let $\\varphi \\in {\\rm Aut}(X)$ .", "By Lemma REF , there exists $(a_{\\varphi },b_{\\varphi })\\in \\mathbb {Z}^2\\setminus \\lbrace (0,0)\\rbrace $ such that $S^{a_{\\varphi }}T^{b_{\\varphi }}\\imath _{\\varphi }(x)=\\imath _{\\varphi }(x)$ for all $x\\in X$ .", "Since $\\imath _{\\varphi }$ is a topological conjugacy between $(X,\\sigma ,\\varphi )$ and $(\\imath _{\\varphi }(X),S,T)$ , we have that $\\sigma ^{a_{\\varphi }}\\varphi ^{b_{\\varphi }}x=x$ for all $x\\in X$ and so $\\varphi ^{b_{\\varphi }}=\\sigma ^{-a_{\\varphi }}$ .", "Thus if $H$ is the subgroup of ${\\rm Aut}(X)$ generated by the powers of $\\sigma $ , the projection of $\\varphi ^{b_{\\varphi }}$ to ${\\rm Aut}(X)/H$ is the identity.", "Since this argument can be applied to any $\\varphi \\in {\\rm Aut}(X)$ (where the parameters $a_{\\varphi }$ and $b_{\\varphi }$ depend on $\\varphi $ ), it follows that ${\\rm Aut}(X)/H$ is a periodic group." ] ]	1403.0238
[ [ "Decline of the current quadrupole moment during the merger phase of\n binary black hole coalescence" ], [ "Abstract Utilizing the tools of tendex and vortex, we study the highly dynamic plunge and merger phases of several $\\pi$-symmetric binary black hole coalescences.", "In particular, we observe a decline of the strength of the current quadrupole moment as compared to that of the mass quadrupole moment during the merger phase, contrary to a naive estimate according to the dependence of these moments on the separation between the black holes.", "We further show that this decline of the current quadrupole moment is achieved through the remnants of the two individual spins becoming nearly aligned or anti-aligned with the total angular momentum.", "We also speculate on the implication of our observations for achieving a consistency between the electric and magnetic parity quasinormal modes." ], [ "Introduction ", "Binary black hole (BBH) coalescences constitute one of the most promising types of gravitational wave sources for the network of detectors, such as the Advanced LIGO [1], Virgo [2], [3], GEO [4], and KAGRA [5].", "Beyond an initial detection, gravitational wave astronomy is also on the horizon (see e.g.", "[6]), and so it is important to know what dynamics of the astronomical system underlies the inspiral, merger and ringdown stages of a BBH waveform, and can therefore be studied using that waveform.", "For example, the merger phase (defined here as between the formation of the common apparent horizon, i.e.", "merger, and the beginning of the quasinormal mode ringdown) dynamics are interesting because they reflect strong gravity behaviors and correspond to a large gravitational wave amplitude.", "Table: Initial parameters for the BBH simulations.", "In all the simulations, the black holes are initially on the xx axis, and the orbital plane is spanned by the xx and yy axes, while the total angular momentum is in the 𝐳 ^\\mathbf {\\hat{z}} direction (the hat signifies unit magnitude).", "We also include the mass and spin of the final remnant black holes in the bottom two rows.The particular aspect of the merger phase dynamics we examine is the decline (not necessarily disappearance) of the current quadrupole moment relative to the mass quadrupole moment in the near zone.", "For our study, we will rely on $\\pi $ -symmetric simulations such as the superkick (equal-mass BBH with anti-aligned spins in the orbital plane) BBH coalescence previously examined in Ref.", "[7] for demonstration, which we will refer to as the SK simulation.", "By $\\pi $ -symmetry, we mean that the system is invariant under a $\\pi $ -rotation around an axis orthogonal to the initial orbital plane [8].", "This symmetry brings about significant simplifications that are useful for us.", "We will also utilize other $\\pi $ -symmetric simulations (not necessarily superkick configurations) whose initial parameters are similar to those of SK, aside from the initial spin orientations.", "The details of the initial parameters for these simulations are given in Table REF .", "In this paper, weak field and/or perturbative expressions are utilized to help build intuition and aid in the formulation of qualitative arguments.", "However, we will use the tools of tendex and vortex, which are non-perturbative and valid in strong fields, to examine the numerical simulations.", "We begin by examining the analytical predictions for the tendex and vortex fields generated by the mass and current quadrupoles in Sec. .", "We then use the knowledge gained to study these quantities in the SK simulation, and show that the current quadrupole declines in relative importance against the mass quadrupole during the merger phase.", "In Sec.", ", we propose the mechanism through which the current quadrupole makes its exit, namely that the remnants of the individual spins become (nearly) aligned or anti-aligned with the total angular momentum.", "In Sec.", ", we directly visualize the movements of these remnant spins using the horizon vorticity, which appear to be in agreement with our proposal.", "Finally in Sec.", ", we speculate on the implication of our observations in terms of helping the electric and magnetic parity quasinormal modes (QNM) achieve equality in their frequencies.", "Note that the spin-total angular momentum alignment or anti-alignment considered here is not the same as the spin-flip examined in, for example, Refs.", "[10], [9], which considers the difference between the spin of the remnant black hole and the pre-merger individual spins (i.e.", "a comparison between different entities), and is simply a result of the former acquiring much of the pre-merger orbital angular momentum [9].", "Our discussion is, however, a comparison between the individual (remnant) spins with their earlier selves.", "In the formulas below, the spacetime indices are written in the front part of the Latin alphabet, while the spatial indices use the middle part of that alphabet.", "We will use bold-face font for vectors and tensors, and adopt geometrized units with $c=1=G$ .", "All the simulations and visualizations are performed with the Spectral Einstein Code (SpEC) [11] infrastructure." ], [ "Vorticity from the mass and current quadrupoles ", "Given a timelike vector field $\\mathbf {u}$ normal to a foliation of the spacetime by spatial hypersurfaces, the tendex $\\mathbf {\\mathcal {E}}$ and vortex $\\mathbf {\\mathcal {B}}$ fields are spatial tensors defined by $\\mathcal {E}_{ij} + i \\mathcal {B}_{ij} = h_i{}^e h_j{}^f \\left(C_{ecfd} - i{}^*C_{ecfd} \\right) u^c u^d,$ where $C_{abcd}$ is the Weyl curvature tensor, $h_{ab}=g_{ab}+ u_a u_b$ is the projection operator into the spatial hypersurfaces with $g_{ab}$ being the spacetime metric, and the Hodge dual operates on the first two indices.", "Because the Weyl curvature tensor can be decomposed into and be reconstructed from the tendex and vortex fields, we can see these fields as representations of the spacetime geometry.", "The eigenvalues of $\\mathbf {\\mathcal {E}}$ and $\\mathbf {\\mathcal {B}}$ are called tendicities and vorticities.", "Because the tendex and vortex tensors are $3\\times 3$ matrices at each field location, there are three branches of tendicities and vorticities.", "From the discussions in Sec.", "VI of Ref.", "[12], we know that in the wave zone, one of the branches is associated with the Coulomb background piece of the Weyl curvature tensor, in the sense that the tendicity and vorticity are the real and imaginary parts of the Newman-Penrose (NP) scalar $\\Psi _2$ (see Ref.", "[13] for more details on interpreting $\\Psi _2$ as the Coulomb background, and $\\Psi _4$ as the outgoing transverse radiation).", "We will refer to this branch as the Coulomb branch, even in the near zone.", "The other two branches weave into the gravitational wavefront in the sense of Figs.", "7 and 8 of Ref.", "[12], as well as Ref. [14].", "Another definition we will need is the horizon vorticity.", "Given the spatial normal $\\mathbf {N}$ to an apparent or event horizon, the horizon vorticity $\\mathcal {B}_{NN}$ is defined by $\\mathcal {B}_{NN} \\equiv \\mathcal {B}_{ij}N^iN^j$ .", "For the rest of the section, we will mostly specialize to the Coulomb branch vorticity field generated by the mass and current quadrupoles, although as the discussions are centered on symmetries, they would work with the other two branches as well.", "The mass quadrupole contains only orbital motion contribution; and is given by $\\mathcal {I}_{jk} = \\left(\\sum _{A} m_A x_{Aj}x_{Ak} \\right)^{\\text{STF}},$ where $A \\in \\lbrace 1,2\\rbrace $ labels the black holes and $\\text{STF}$ stands for taking the symmetric, trace-free part.", "The current quadrupole moment is given by $\\mathcal {S}_{jk} = \\left( \\sum _{A} x_{Aj}J^{\\rm tot}_{Ak} \\right)^{\\text{STF}},$ with $\\mathbf {J}^{\\rm tot}$ being the total angular momentum that has two components: the orbital angular momentum and the spins $\\mathbf {J}_A^{\\rm tot} = \\mathbf {J}_A^{\\rm orb} + \\mathbf {J}_A^{\\rm spin} = \\mathbf {x}_A \\times \\mathbf {p}_A + \\mathbf {S}_A.$ For the $\\pi $ -symmetric simulations we consider, the $\\mathbf {J}^{\\rm orb}_A$ are the same for the two black holes, but the $\\mathbf {x}_A$ are opposite, so the orbital contribution from the two black holes cancel out in Eq.", "(REF ).", "The spin contribution, on the other hand, have opposite $\\mathbb {P}\\mathbf {S}_A$ for the two black holes ($\\mathbb {P}$ is the projection operator into the orbital plane, which for the post-merger context will refer to the equatorial plane of the remnant black hole), and therefore does not need to vanish.", "Furthermore, the total orbital angular momentum $\\sum _A \\mathbf {J}_A^{\\rm orb}$ and the total spin $\\sum _A \\mathbf {S}_A$ are both collinear with $\\mathbf {J}_A^{\\rm tot}$ .", "Note that Eqs.", "(REF ) and (REF ) are in the STF notation of Ref.", "[15], [16], [17], which has been summed over $m$ .", "For our simulations, $\\pi $ -symmetry suppresses the $m=\\pm 1$ modes, and even though there can be some small $m=0$ mode contribution in the waveforms, we are most interested in the $m=\\pm 2$ modes.", "To approximate the $\\mathbf {\\mathcal {I}}$ and $\\mathbf {\\mathcal {S}}$ generating such modes in our simulations, we can use the quasi-Newtonian formula $\\mathbf {\\mathcal {I}} = \\frac{MR^2}{8}\\begin{pmatrix}\\cos (2\\Omega \\tilde{t}) + \\frac{1}{3} & \\sin (2\\Omega \\tilde{t}) & 0 \\\\\\sin (2\\Omega \\tilde{t}) & - \\cos (2\\Omega \\tilde{t}) + \\frac{1}{3} & 0 \\\\0 & 0 & -\\frac{2}{3} \\end{pmatrix}$ on a Cartesian coordinate system $(x,y,z)$ with $(x,y)$ spanning the orbital plane.", "The quantity $M$ is the total mass, $R$ is the separation between the black holes, and $\\Omega =\\sqrt{M/R^3}$ is the Newtonian orbital angular frequency.", "Note that we have replaced the time $t$ in a purely Newtonian expression by the retarded time $\\tilde{t}$ .", "For the current quadrupole, there are a few interesting configurations.", "First of all, when the spins are constant, anti-parallel to each other and in the orbital plane, we have $\\mathbf {\\mathcal {S}} = \\frac{R S}{2} \\begin{pmatrix}\\frac{4}{3}\\cos (\\Omega \\tilde{t}) & \\sin (\\Omega \\tilde{t}) & 0 \\\\\\sin (\\Omega \\tilde{t}) & -\\frac{2}{3}\\cos (\\Omega \\tilde{t}) & 0 \\\\0& 0 & -\\frac{2}{3}\\cos (\\Omega \\tilde{t})\\end{pmatrix},$ where $S$ is the shared magnitude of the individual spins.", "Note that as the spins don't precess, there is only one $\\Omega $ factor in Eq.", "(REF ) coming from the $\\mathbf {x_A}$ term in Eq.", "(REF ), so the current quadrupole will evolve at the orbital frequency, instead of twice the orbital frequency like the mass quadrupole.", "If however, the spins precess also at frequency $\\Omega $ , we would then have the current quadrupole evolving at a frequency of $2\\Omega $ .", "For example, in the simple case where the spins are anti-parallel, locked to orthogonal directions to the line linking the black holes, and confined to the orbital plane, we have $\\mathbf {\\mathcal {S}} = \\frac{R S}{2} \\begin{pmatrix}-\\sin (2\\Omega \\tilde{t}) & \\cos (2\\Omega \\tilde{t}) & 0 \\\\\\cos (2\\Omega \\tilde{t}) & \\sin (2\\Omega \\tilde{t}) & 0 \\\\0& 0 & 0\\end{pmatrix}.$ Another useful result is for the case when spin $\\mathbf {S}_A$ is locked to the $-\\mathbf {x}_A$ direction, where $\\mathbf {\\mathcal {S}} = -\\frac{R S}{2} \\begin{pmatrix}\\cos (2\\Omega \\tilde{t}) + \\frac{1}{3} & \\sin (2\\Omega \\tilde{t}) & 0 \\\\\\sin (2\\Omega \\tilde{t}) &-\\cos (2\\Omega \\tilde{t})+ \\frac{1}{3} & 0 \\\\0& 0 & -\\frac{2}{3}\\end{pmatrix}.$ Finally, we note that if the two spins are aligned with each other, such as in the AA simulation (spins anti-aligned with the orbital angular momentum), then we suffer from the same effect that diminishes orbital contribution to $\\mathbf {\\mathcal {S}}$ : the $\\mathbf {S}_A$ are the same for the two black holes, while their $\\mathbf {x}_A$ are opposite, so the overall current quadrupole vanishes.", "For the SK, SK- and SK$\\bot $ simulations (spins initially in the orbital plane, see Table REF ), the current quadrupole moment is non-vanishing during inspiral, and can be approximated by Eq.", "(REF ) during the early part of inspiral.", "Towards later stages of inspiral, the spin precession frequency increases and $\\mathbf {\\mathcal {S}}$ is somewhere between Eq.", "(REF ), (REF ) and (REF ).", "We now develop some tools for tracking how $\\mathbf {\\mathcal {S}}$ evolves in, e.g.", "the SK simulation, during the merger phase.", "The tendex and vortex fields corresponding to the current quadrupole $\\mathbf {\\mathcal {S}}$ , in weak gravity, with a source region smaller than the gravitational wavelength, are given in Ref.", "[12] as $\\mathcal {B}_{ij} &=& \\frac{2}{3}\\left[-\\left(\\frac{\\mathcal {S}_{pq}}{r}\\right)_{,pqij} + \\epsilon _{ipq} \\left(\\frac{\\ddot{\\mathcal {S}}_{pl}}{r}\\right)_{,qk}\\epsilon _{jlk} \\right.", "\\\\&&\\left.", "+ 2\\left(\\frac{\\ddot{\\mathcal {S}}_{p(i}}{r} \\right)_{,j)p}- \\left(\\frac{\\ddddot{\\mathcal {S}}_{ij}}{r} \\right) \\right],$ $\\mathcal {E}_{ij} = \\frac{4}{3} \\epsilon _{pq(i}\\left[-\\left(\\frac{\\dot{\\mathcal {S}}_{pk}}{r}\\right)_{,j)kq}+\\left(\\frac{\\dddot{\\mathcal {S}}_{j)p}}{r}\\right)_{,q}\\right],$ where repeated indices are summed over, and the overdot denotes time derivatives.", "Roughly, each time derivative introduces an $\\Omega $ factor, while each spatial derivative can introduce either an $1/r$ factor when operating on explicit $r$ 's in Eq.", "(REF ) and (REF ), or an $\\Omega $ factor through the retarded time.", "In the near zone, where $r < $ ($$ is the reduced wavelength), the spatial derivatives that generate $1/r$ factors are favorable, so terms with more spatial derivatives are more dominant, and the strength of the tendex and vortex fields are determined by the first terms in Eqs.", "(REF ) and (REF ).", "The ratio of the strength between them is proportional to $/r$ .", "When $r > $ (in the wave zone), it is preferable for spatial derivatives to introduce an $\\Omega $ factor instead, and all the terms in the sums contribute equally.", "The result in the case of Eq.", "(REF ) is essentially the same as a transverse-traceless projection operator acting on the four time derivative term multiplied by $-2$ [18].", "In this case, the $\\mathbf {\\mathcal {B}}$ and $\\mathbf {\\mathcal {E}}$ fields are of the same strength, as one would expect from them being sustained by mutual induction in a gravitational wave [12].", "The $\\mathbf {\\mathcal {E}}$ and $\\mathbf {\\mathcal {B}}$ fields generated by the mass quadrupole $\\mathbf {\\mathcal {I}}$ are the mirror image, and are given by $\\mathcal {B}_{ij} = \\epsilon _{pq(i}\\left[\\left(\\frac{\\dot{\\mathcal {I}}_{pk}}{r}\\right)_{,j)kq}-\\left(\\frac{\\dddot{\\mathcal {I}}_{j)p}}{r}\\right)_{,q}\\right],$ $\\mathcal {E}_{ij} &=& \\frac{1}{2}\\left[-\\left(\\frac{\\mathcal {I}_{pq}}{r}\\right)_{,pqij} + \\epsilon _{ipq} \\left(\\frac{\\ddot{\\mathcal {I}}_{pl}}{r}\\right)_{,qk}\\epsilon _{jlk} \\right.", "\\\\&&\\left.", "+ 2\\left(\\frac{\\ddot{\\mathcal {I}}_{p(i}}{r} \\right)_{,j)p}- \\left(\\frac{ \\ddddot{\\mathcal {I}}_{ij}}{r} \\right) \\right].$ Figure: (a): Two Coulomb branch vorticity contours from the mass quadrupole according to Eq. ().", "The red and blue contours correspond to a pair of opposite vorticity values, with the red being ++ve.", "(b): Coulomb branch vorticity contours from the current quadrupole given by Eq.", "().The symmetry between Eqs.", "(REF ), (REF ) and Eqs.", "(REF ), (REF ) allows for the definition of a complex quadrupole moment $\\mathcal {M}_{ij}= \\frac{4}{3} \\frac{\\mathcal {S}_{ij}}{r}-i \\frac{\\mathcal {I}_{ij}}{r},$ and then the tendex and vortex fields are given by the unified expression $\\mathcal {E}_{ij} + i \\mathcal {B}_{ij} &=& \\epsilon _{pq(i}\\left[ -\\dot{\\mathcal {M}}_{pk,j)kq} +\\dddot{\\mathcal {M}}_{j)p,q}\\right] \\\\&& + \\frac{i}{2}\\left[-\\mathcal {M}_{pq,pqij}+\\epsilon _{ipq}\\ddot{\\mathcal {M}}_{pm,qn} \\epsilon _{jmn} \\right.", "\\\\&& \\left.", "+ 2\\ddot{\\mathcal {M}}_{p(i,j)p}+\\ddddot{\\mathcal {M}}_{ij}\\right].$ We now turn to examine the symmetry properties of the $\\mathbf {\\mathcal {B}}$ field generated by Eqs.", "(REF ) and (REF ).", "Following Ref.", "[19], we define a positive/negative (abbreviated to $+$ ve/$-$ ve below) parity tensor field to be one that does not/does change sign under a reflection against the origin.", "Note that the parity operation we consider applies to the field location coordinates (e.g.", "$r$ in Eq.", "REF ), and not the source (black hole) locations or motions (e.g.", "$\\mathbf {x}_A$ or $\\mathbf {S}_A$ in Eq.", "REF ), which can be seen formally as existing in a separate internal vector space.", "This is akin to applying the parity transformation to only the $x$ coordinate of a Green function $G(x,x^{\\prime })$ while leaving $x^{\\prime }$ unaffected.", "Therefore, even though quantities like $\\mathbf {x}_A$ are polar-vectors in that internal space, they, together with axial-vectors in the internal space, behave as axial-vectors under our parity transformation.", "Subsequently both $\\mathbf {\\mathcal {S}}$ and $\\mathbf {\\mathcal {I}}$ have $+$ ve parity, as do $\\mathbf {\\mathcal {B}}$ in Eq.", "(REF ) and $\\mathbf {\\mathcal {E}}$ in Eq.", "(REF ) that have even numbers of derivatives, while $\\mathbf {\\mathcal {E}}$ in Eq.", "(REF ) and $\\mathbf {\\mathcal {B}}$ in Eq.", "(REF ) take on $-$ ve parity.", "We show in Fig.", "REF , the Coulomb branch vorticity contours for $\\mathbf {\\mathcal {B}}$ as given by Eqs.", "(REF ) and (REF ).", "In addition to parity, our $\\mathbf {\\mathcal {B}}$ fields are $\\pi $ -symmetric by construction.", "So by combining a $\\pi $ -rotation with a parity transformation, we arrive at reflection anti-symmetry/symmetry against the orbital plane, for the mass/current quadrupole generated vorticity.", "Furthermore, for the $m=\\pm 2$ modes we are considering, there is a $\\pi /2$ -rotation antisymmetry for both $\\mathbf {\\mathcal {I}}$ and $\\mathbf {\\mathcal {S}}$ generated vorticity, as evident from Fig.", "REF .", "We will call the combination of a parity transformation with a $\\pi /2$ -rotation the “skew-reflection”, and then the mass/current quadrupole generated vorticity is skew-reflection symmetric/anti-symmetric.", "Now consider an axisymmetric dipolar vorticity which also has a $-$ ve parity, such as that generated by the orbital angular momentum or the spin of the post-merger final remnant black hole.", "It would be reflection anti-symmetric, as well as skew-reflection anti-symmetric.", "Therefore, a combination of the mass quadrupolar and the dipolar vorticities would have a definitive reflection anti-symmetry, but has no definitive skew-reflection symmetry property.", "Subsequently, the contours of opposite vorticities will be aligned with each other (in terms of rotation against the $\\mathbf {J}^{\\rm tot}$ axis) across the orbital plane.", "On the other hand, when we combine the current quadrupolar vorticity with the dipolar vorticity, we will have definite skew-reflection anti-symmetry, but no definitive reflection symmetry property.", "Therefore, the contours of opposite vorticities will be misaligned by $\\pi /2$ instead.", "This conclusion is demonstrated graphically in the top two rows of Fig.", "REF , where contours of opposite vorticity are represented in red and blue.", "When constructing this figure, the dipole contribution is approximated as the vorticity field of a Kerr black hole in the Kerr-Schild coordinates.", "When we combine the contributions from both quadrupoles as well as the dipole, the red and blue spiraling arms subtend a misalignment angle between 0 and $\\pi /2$ (possessing neither definitive reflection symmetry nor definitive skew-reflection symmetry), as shown in the third row of Fig.", "REF .", "Note that although we have been utilizing weak gravity expressions in this section to construct examples, the qualitative symmetry considerations should remain valid in strong gravity, where this misalignment angle can serve as a convenient measure of the relative strength between the two types of quadrupoles.", "Aside from a non-vanishing misalignment angle, another indicator for the existence of a current quadrupole contribution is that the contours can slice through the orbital plane (see Fig.", "REF (b) and Fig.", "REF (e)), which is allowed by the skew-reflection anti-symmetry.", "On the other hand, reflection antisymmetry prevents the contours with non-vanishing vorticity from intersecting the orbital plane (see Fig.", "REF (a) and Fig.", "REF (a)).", "Finally, note that when both types of quadrupoles are present, the red and blue contours do not need to share the same size and/or shape (see Fig.", "REF (e) and (f)), with the difference between them dependent on the relative strength of these quadrupoles, as well as their relative phase.", "Figure: The analytically constructed contours of opposite vorticity.", "(a)-(b): Two contours of opposite vorticity from the mass quadrupole (Eq. )", "plus a current dipole.The contours, in particular the spiraling arms, are aligned with each other across the orbital plane.", "(c)-(d): Similar contours from the current quadrupole (Eq. )", "plus the dipole.The contours are misaligned by π/2\\pi /2, and obey skew-reflection antisymmetry.", "(e)-(f): Both current and mass quadrupoles are included in addition to the dipole.The contours are misaligned by an angle between 0 and π/2\\pi /2, breaking both reflection and skew-reflection antisymmetry.Left column: side views of the contours (not at the same angle).Right column: top views (looking down 𝐉 tot \\mathbf {J}^{\\rm tot}) of the contours.Figure: The numerically observed contours of opposite vorticity in the simulations.", "(a)-(d): The time evolution of the Coulomb branch vorticity during the merger phase for the SK simulation.", "The blue and red surfaces are contours of the same absolute vorticity value but of the opposite signs.", "The rows correspond to different times.", "Over time, the red and blue contours become more aligned with each other in their orientation.", "(e)-(f): The Coulomb branch vorticity contours for the AA simulation, at the same time delay from merger as (a) and (b).", "The lack of current quadrupole moment ensures the alignment of the red and blue contours.Left column: side views of the contours (not at the same angle).Right column: top views (looking down 𝐉 tot \\mathbf {J}^{\\rm tot}) of the contours.We now plot the time evolution (during the merger phase) of the opposite vorticity contours for the SK simulation in the top two rows of Fig.", "REF , and observe an increase of alignment over time, suggesting a decreasing current quadrupole moment.", "We also note that the blue spiraling arms in Fig.", "REF (a) slice through the orbital plane, while their counterparts in Fig.", "REF (c) do not behave in the same way.", "Furthermore, the blue arms are larger in spatial extend initially, but reduce to be of similar sizes as the red arms later.", "These observations are also consistent with a declining current quadrupole contribution.", "For comparison, we also plot in the bottom row of Fig.", "REF , the contours for the AA simulation at the same time delay from merger as in Fig.", "REF (a) and (b), where as the current quadrupole moment vanishes according to Eq.", "(REF ), the contours are already exactly aligned." ], [ "Avenue for the exit of the current quadrupole ", "If we hold spin magnitude $S$ constant, and keep spin directions tangential to the orbital plane in a $\\pi $ -symmetry configuration, then a comparison between Eq.", "(REF ) and Eqs.", "(REF )-(REF ) suggests that as $R$ decreases, the relative strength of $\\mathbf {\\mathcal {S}}$ as compared to $\\mathbf {\\mathcal {I}}$ should increase.", "Using the values for $M$ and $S$ in Table REF , we find that $\\mathbf {\\mathcal {S}}$ and $\\mathbf {\\mathcal {I}}$ should become similar in magnitude when $R \\approx 2$ , or around merger time.", "Furthermore, if $r < $ for the $r$ range we are interested in (e.g.", "the range plotted in Fig.", "REF ), then the first terms dominate in Eqs.", "(REF ) and (REF ), and the ratio of $\\mathbf {\\mathcal {B}}$ field strength as generated by $\\mathbf {\\mathcal {S}}$ to that generated by $\\mathbf {\\mathcal {I}}$ is multiplied by a factor of $/r >1$ on top of the strength ratio between $\\mathbf {\\mathcal {S}}$ and $\\mathbf {\\mathcal {I}}$ .", "On the other hand, when $r > $ , all the terms in Eqs.", "(REF ) and (REF ) contribute, so the additional factor is around 1 because all the terms introduce an $\\Omega ^4$ factor.", "In other words, the current quadrupole is either more effective or equally effective at generating the vortex field as compared to the mass quadrupole.", "This conclusion should remain true in a strong gravitational field, as $\\mathbf {\\mathcal {B}}$ can be seen as the primary field generated by $\\mathbf {\\mathcal {S}}$ , while only a secondary field for $\\mathbf {\\mathcal {I}}$ that is induced by the time variation of $\\mathbf {\\mathcal {E}}$ (see Sec.", "VI C and Sec.", "VI D 1 of Ref.", "[12] for further discussions and examples).", "Consequently, as the absolute magnitude of $\\mathbf {\\mathcal {S}}$ catches up to or even overtakes that of $\\mathbf {\\mathcal {I}}$ during the merger phase, we should not see its influence in vorticity decline in the fashion of Fig.", "REF .", "Therefore, the current quadrupole must somehow diminish during the merger phase.", "It would vanish if the equal and opposite spins of the two individual black holes simply annihilate each other, but this does not explain why spins annihilate faster than the two masses merge (i.e.", "why the current quadrupole declines faster than the mass quadrupole).", "In other words, we need the current quadrupole to reduce faster than the signature of the individual black holes disappears.", "Such a scenario is possible if the individual spins experience a re-orientation into configurations that produce near-vanishing $\\mathbf {\\mathcal {S}}$ , even as the spins themselves are still non-vanishing.", "This can be achieved if the spins move to become nearly aligned with each other, which according to Eq.", "(REF ) would result in a nearly vanishing $\\mathbf {\\mathcal {S}}$ .", "Above, we have used the term “individual spins” in a generalized sense.", "Even before merger, the spins of the individual black holes are really reflections of the spacetime dynamics outside of these holes, as the characteristic modes for the Einstein equation are strictly outgoing at the apparent horizons (which are inside the event horizons).", "The merger would not instantaneously remove the near-zone dynamics that were underlying the individual spins, so one may regard the continuation of such dynamics as a kind of remnant spin (the word “remnant” will be omitted frequently for brevity).", "In the tendex and vortex language, one may say that these remnant spins are the vorticity and tendicity of the spacetime that were associated with the spins before merger, but would not instantly dissipate upon the formation of the common apparent horizon.", "Similarly, the tendicity and the vorticity of the spacetime that were associated with the orbital motion of the individual black holes before merger continue to evolve post-merger, and provide a remnant orbital motion.", "We note that the remnant spins considered here belong to the defunct individual black holes, and are not the spin of the remnant black hole.", "Due to the lack of analytical descriptions for highly dynamic regimes, we will rely on the available perturbative expressions to aid our qualitative arguments in the rest of this section.", "Although we are likely pushing these expressions beyond their reasonable range of validity, we will only be interested in the qualitative features of the spacetime they expose, and not their quantitative accuracy.", "In order to achieve alignment, it is required that the spins be lifted out of the orbital plane, either through spin-orbit coupling or spin-spin coupling, because the $\\pi $ -symmetry forbids the spins from being aligned when they are confined to the orbital plane.", "The spin-orbit coupling is given by the leading order PN expression [20], [21], [22], [23] $\\mathbf {\\dot{S}}_1 = \\frac{1}{R^3}\\left(2+\\frac{3}{2}\\frac{m_2}{m_1}\\right)\\left(\\mathbf {L}_N \\times \\mathbf {S_1} \\right),$ in the center-of-mass frame, where $\\mathbf {L}_N$ is the orbital angular momentum at the Newtonian order $\\mathbf {L}_N = \\mu \\mathbf {R} \\times \\dot{\\mathbf {R}}$ with $\\mu $ being the reduced mass and $\\mathbf {R} =\\mathbf {x}_1 -\\mathbf {x}_2$ , and also $R$ being the magnitude of $\\mathbf {R}$ .", "For the post-merger context, we will instead take $\\mathbf {L}_N$ to be $\\mathbf {J}^{\\rm tot}$ minus the remnant individual spins.", "We note that the $\\mathbf {\\dot{S}}_1$ in Eq.", "(REF ) cannot point out of the orbital plane and will only generate spin precession within it.", "The spin-spin coupling, on the other hand, can create a torque pointing out of the orbital plane.", "The leading order expression for spin-spin coupling is given by [20], [21], [22], [23] $\\dot{\\mathbf {S}}_1 = -\\frac{1}{R^3}\\left[\\mathbf {S}_2 - 3(\\mathbf {n}\\cdot \\mathbf {S}_2)\\mathbf {n} \\right] \\times \\mathbf {S}_1,$ in the center-of-mass frame, and $\\mathbf {n}$ is defined as $\\mathbf {R}/R$ .", "For the SK configuration, $\\mathbf {S}_2 \\times \\mathbf {S}_1 =0$ when the spins are in the orbital plane, as they must be anti-parallel by $\\pi $ -symmetry, but $\\mathbf {S}_2$ is not required to be transverse (especially when the spins are not locked to the orbital motion), so $\\mathbf {n}\\cdot \\mathbf {S}_2 \\ne 0$ and there is a $\\mathbf {\\dot{S}}$ in the direction orthogonal to the orbital plane.", "Normally, this direction is not constant over an orbital cycle for a pair of spins not locked to the orbital motion, so the spin-spin coupling effect does not accumulate significantly during early inspiral, but as the merger phase will not take up a whole cycle, we need not worry about cancellations.", "On the other hand, the directions of $\\mathbf {\\dot{S}}_1$ and $\\mathbf {\\dot{S}}_2$ are the same, so as desired, the spins for our SK configuration either both move upwards (more aligned with $\\mathbf {J}^{\\rm tot}$ ) or both move downwards (more anti-aligned with $\\mathbf {J}^{\\rm tot}$ ), retaining the $\\pi $ -symmetry.", "According to Eq.", "(REF ), the spins do not need to move towards spin-spin alignment.", "In fact, the $\\pi $ -symmetry enforces $\\mathbb {P}\\mathbf {S}_1 = -\\mathbb {P}\\mathbf {S}_2$ , so unless $\\mathbb {P}\\mathbf {S}_1 = \\mathbb {P}\\mathbf {S}_2 = 0$ , the spins would never be aligned.", "The situation would change however if we include the radiation reaction.", "Because gravitational waves drain dynamical energy, radiation reaction should push the spin orientations towards an energetically favorable equilibrium configuration, where the spin precession due to both spin-spin and spin-orbit coupling ceases.", "From Eq.", "REF , it is clear that the spin-orbit induced precession would shut off only when $\\mathbf {S}_1$ and $\\mathbf {S}_2$ are orthogonal to the orbital plane, or in other words are collinear with $\\mathbf {J}^{\\rm tot}$ .", "Furthermore, Eq.", "(REF ) shows that the spin-spin coupling generated spin precession also stops for such configurations.", "In addition, when the spins are either both aligned or both anti-aligned with $\\mathbf {J}^{\\rm tot}$ (henceforth referred to as spin-$\\mathbf {J}^{\\rm tot}$ alignment or anti-alignment), the current quadrupole will vanish.", "For this postulate to work, a necessary condition is that the spin-$\\mathbf {J}^{\\rm tot}$ alignment or anti-alignment configuration should correspond to a local minimum in energy.", "To this end, we note that the potential energy associated with the spin-spin interaction at leading order is given by Refs.", "[20], [21], [22], [23] as $U_{\\text{SS}}=-\\frac{1}{R^3}\\left[\\mathbf {S}_1 \\cdot \\mathbf {S}_2 -3\\left(\\mathbf {S}_1 \\cdot \\mathbf {n} \\right) \\left(\\mathbf {S}_2 \\cdot \\mathbf {n} \\right) \\right],$ while the potential for the spin-orbit coupling is [20], [21], [22], [23] $U_{\\text{SO}} = \\frac{1}{R^3}\\mathbf {L}_N \\cdot \\left(\\left(2+\\frac{3m_1}{2m_2}\\right)\\mathbf {S}_2 + \\left(2+\\frac{3m_2}{2m_1}\\right)\\mathbf {S}_1 \\right).$ The absolute minimum of $U_{\\text{SS}}$ is achieved when $\\mathbf {S}_1$ and $\\mathbf {S}_2$ are anti-parallel and collinear with $\\mathbf {n}$ , as the second term in the square bracket of Eq.", "(REF ) favors anti-parallel orientations and dominates over the parallel-orientation favoring first term, due to its extra factor of 3.", "This is also an equilibrium configuration for Eq.", "(REF ), but does not lead to a small current quadrupole moment, as shown by Eq.", "(REF ).", "For the spin-$\\mathbf {J}^{\\rm tot}$ alignment or anti-alignment equilibrium configurations that we are interested in, we have $\\phi _1 = \\phi _2 = 0$ or $\\pi $ , and $\\theta _1 = \\theta _2 = \\pi /2$ , with $\\theta _1$ being the angle $\\mathbf {S}_1$ spans with $\\mathbf {n}$ and $\\phi _1$ the angle between $\\mathbf {J}^{\\rm tot}$ and the projection of $\\mathbf {S}_1$ into the plane orthogonal to $\\mathbf {n}$ .", "The angles $\\theta _2$ and $\\phi _2$ are defined similarly for $\\mathbf {S}_2$ .", "It is easy to verify that all first derivatives of $U_{\\text{SS}}$ against the angles vanish for these configurations, so they are indeed critical/equilibrium points.", "However, the eigenvalues of the Hessian are $\\lbrace 0,2,3,-1\\rbrace \|\\mathbf {S}_1\|\|\\mathbf {S}_2\|/R^3$ and not all positive, so they are not (local) minima of the potential energy.", "When we add in the potential $U_{\\text{SO}}$ , which achieves its absolute minimum at the spin-$\\mathbf {J}^{\\rm tot}$ anti-alignment configuration, the eigenvalues of the Hessian (of $U_{\\text{SO}} + U_{\\text{SS}}$ ) for this configuration become $\\frac{1}{4R^3} \\left\\lbrace 7{L}_N, 7{L}_N-1, 7{L}_N + 2, 7{L}_N + 3 \\right\\rbrace ,$ where we have taken $\|\\mathbf {S}_1\|=\|\\mathbf {S}_2\|=1/2$ and $\\mathbf {L}_N = L_N \\mathbf {\\hat{z}}$ to simplify expressions.", "For our $M \\approx 1$ simulations, and using the Newtonian expression for $\\mathbf {L}_N$ , spin-$\\mathbf {J}^{\\rm tot}$ anti-alignment configuration is a local minimum as long as $R >1/49$ M. The spin-$\\mathbf {J}^{\\rm tot}$ alignment configuration, on the other hand, has eigenvalues $\\frac{1}{4R^3} \\left\\lbrace -7{L}_N, -7{L}_N-1, -7{L}_N + 2, -7{L}_N + 3 \\right\\rbrace ,$ and is therefore not a local minimum unless ${L}_N$ is sufficiently negative.", "When we add in the next-to-leading-order PN expressions for $U_{\\text{SO}}$ [25], [24], [26], [27], [28], we acquire extra multiplicative factors onto $\\mathbf {L}_N$ that can reverse the sign of the effective ${L}_N$ at small $r$ , and make spin-$\\mathbf {J}^{\\rm tot}$ alignment configuration a local minimum (effective $\\mathbf {L}_N$ in $U_{\\text{SO}}$ is reversed, but $\\mathbf {J}^{\\rm tot}$ is not, so anti-alignment with the effective $\\mathbf {L}_N$ now results in an alignment with $\\mathbf {J}^{\\rm tot}$ ).", "The $U_{\\text{SO}}$ that includes both leading and next-to-leading order PN contributions can be deduced from the spin precession equation (Eqs.", "61-64 in Ref.", "[26]) $\\dot{\\mathbf {\\bar{S}}}_1 = \\mathbf {H}_1 \\times \\mathbf {\\bar{S}}_1$ in a general frame, where we can regard $\\mathbf {H}_1 R^3 / (2+3m_2/2m_1)$ as an effective $\\mathbf {L}_N$ for $\\mathbf {\\bar{S}}_1$ , and $\\mathbf {S}_1$ 's contribution to $U_{\\text{SO}}$ is $\\mathbf {H}_1 \\cdot \\mathbf {\\bar{S}}_1$ .", "The quantities appearing in Eq.", "(REF ) are $\\mathbf {\\bar{S}}_1 &=& \\left( 1- \\frac{\\mathbf {\\bar{v}}^2_1}{2} - \\frac{\\mathbf {\\bar{v}}^4_1}{8} \\right) \\mathbf {S}_1 \\\\&&+ \\frac{1}{2}\\mathbf {\\bar{v}}_1(\\mathbf {\\bar{v}}_1\\cdot \\mathbf {S}_1)\\left( 1+ \\frac{1}{4}\\mathbf {\\bar{v}}^2_1\\right) \\\\\\mathbf {\\bar{v}}_1 &=& \\frac{\\left(1+m_2/R\\right) \\mathbf {v}_1 - 2(m_2/R) \\mathbf {v}_2}{1-m_2/R}$ and more importantly $\\mathbf {H}_1 &=& \\frac{m_2}{R^3}\\left[\\left( \\frac{3}{2} + \\frac{1}{8} \\mathbf {v}^2_1 + \\mathbf {v}^2_2 - \\mathbf {v}_1 \\cdot \\mathbf {v}_2 - \\frac{9}{4}(\\mathbf {v}_2\\cdot \\mathbf {n})^2 + \\frac{1}{2R}(m_1 -m_2) \\right) \\mathbf {L}_1 \\right.", "\\\\&& \\left.", "+ \\left( -2 - 2 \\mathbf {v}^2_2 + 2\\mathbf {v}_1 \\cdot \\mathbf {v}_2 + 3(\\mathbf {v}_2\\cdot \\mathbf {n})^2 + \\frac{1}{2R}(2m_1 +3 m_2) \\right) \\mathbf {L}_2- \\frac{1}{2} (\\mathbf {v}_2 \\cdot \\mathbf {R}) \\mathbf {v}_1 \\times \\mathbf {v}_2 \\right] $ with $\\mathbf {L}_1 = \\mathbf {R} \\times \\mathbf {v}_1$ and $\\mathbf {L}_2 = \\mathbf {R} \\times \\mathbf {v}_2$ .", "Note $\\mathbf {L}_2$ is in the opposite direction to particle 2's orbital angular momentum, so the $(1/2R)(2m_1 +3 m_2)$ term in Eq.", "(REF ) could reverse the direction of the effective $\\mathbf {L}_N$ when $R$ is small.", "An interesting complication is the fact that the equilibrium direction as determined by $\\mathbf {H}_1$ is for $\\mathbf {\\bar{S}}_1$ that has a directional difference with $\\mathbf {S}_1$ from the term proportional to $\\mathbf {\\bar{v}}_1$ in Eq.", "(REF ), and that we can also have directional adjustments in $\\mathbf {H}_1$ to make it deviate from the $\\pm \\mathbf {J}^{\\rm tot}$ directions.", "Such frame-dependent adjustments may prevent a perfect alignment or anti-alignment of the spins with $\\mathbf {J}^{\\rm tot}$ , and so the current quadrupole will not vanish completely.", "We will speculate on the significance of this complication in Sec. .", "Throwing away the crutch of perturbative expressions, we really only need the qualitative statements, that the spin-$\\mathbf {J}^{\\rm tot}$ near-alignment or near-antialignment configurations are energetically favorable for the spin-orbit coupling, and that mechanisms like the spin-spin coupling exist that can lift the spins out of the orbital plane, to remain true in the strong field regime.", "The perturbative expressions have hinted that it may be possible to meet these requirements, but strong field expressions are needed for quantitative assessments.", "Figure: A contour of ℜ(Ψ 4 )\\Re (\\Psi _4) for the SK simulation immediately after the merger, with Ψ 4 \\Psi _4 extracted on the quasi-Kinnersley tetrad.", "Also shown is the common apparent horizon colored by the horizon vorticity ℬ NN \\mathcal {B}_{NN}.", "The ℜ(Ψ 4 )\\Re (\\Psi _4) contour connects to certain blue horizon vorticity patches.These are the negative ℬ NN \\mathcal {B}_{NN} counterparts to the C ' C^{\\prime } patch of Fig.", "(d) below, which we will show in Sec.", "to be a direct manifestation of the remnant spins.", "Note that there is some visualization complication near the polar directions, where we don't have any collocation points, so the contour shown there is an ill-constructed interpolation.Another condition for our postulate to work is that the remnant spin dynamics should be efficient radiators during the merger phase, such that the spins experience a significant reaction.", "In contrast, when modeling early inspiral, the radiation reaction felt by the spins is usually neglected [23], [29].", "If we plot the distribution of $\\Psi _4$ near the common apparent horizon, we should see a close association between the high intensity regions of $\\Psi _4$ and entities that can be interpreted as representing the remnant spins.", "To verify this, we adopt the quasi-Kinnersley tetrad described in Refs.", "[31], [32], [33], [34], [35], [30].", "Our particular version of the tetrad follows Ref.", "[30] and suffers from some numerical noise because of the third derivative of the metric required for its construction.", "However, because we are now examining the region very close to the remnant black hole, the mixing of $\\Psi _2$ into $\\Psi _4$ under a simple simulation-coordinate based tetrad will overwhelm the interesting features.", "The quasi-Kinnersley tetrad avoids this problem by correctly identifying the gravitational wave propagation direction [30].", "Specifically, the tetrad bases correspond directly to the super-Poynting vector, so that the $\\Psi _4$ extracted under this tetrad retains a simple relationship with the energy flux even in the near zone.", "In Fig.", "REF , we plot a large absolute value, and thus close to the radiating source, contour of $\\Re (\\Psi _4)$ .", "It is clear that this contour attaches to certain blue horizon vorticity patches (contours of even higher $\|\\Re (\\Psi _4)\|$ are observed to be confined to regions closer to these patches), which we will now show (in Sec. )", "to be direct manifestations of the remnant spins." ], [ "Observing the remnant spins ", "One possible way to visualize the (remnant) spins directly is through the horizon vorticity $\\mathcal {B}_{NN}$ .", "This quantity is closely related to the spin measurement expressed as an integral over the horizon [36], [37], [38], namely [39] $S = \\frac{1}{8\\pi } \\oint \\mathcal {X} \\zeta dA,$ where $dA$ is the area element on the horizon, and for the event horizons, $\\mathcal {X}$ is $-2\\mathcal {B}_{NN}$ plus some spin coefficient corrections that vanish in a stationary limit (see Refs.", "[7], [39]).", "We will use the apparent horizons in this paper, which coincide with the event horizons in the stationary limit [40], but are not teleological and therefore more widely utilized in numerical simulations.", "The quantity $\\zeta $ in Eq.", "(REF ) is determined by an eigenvalue problem on the horizon, and is essentially an $l=1$ spherical harmonic [41].", "In other words, the spin is given by the dipole part of the horizon vorticity $\\mathcal {B}_{NN}$ .", "For example, during early inspiral, the $\\mathcal {B}_{NN}$ pattern on each individual horizon is dominated by the spin of that black hole, forming a dipolar shape like those seen for the Kerr black holes in Ref. [39].", "This is shown in Fig.", "REF (a), and can be used to identify the spin directions.", "Close to and past merger, aside from the overall $l=1$ harmonic-weighted integral in Eq.", "(REF ), there are many interesting finer details that can be seen from the $\\mathcal {B}_{NN}$ plots for our various simulations shown in Fig.", "REF .", "The panels (c)-(d), (e)-(f) and (g)-(h) depict the horizon vorticity patterns for the SK, SK- and SK$\\bot $ simulations, respectively, while panel (b) shows the pattern for the AA simulation.", "The panels (b), (d) and (f) show the common apparent horizon, while the rest of the panels are for the two individual horizons.", "At the end of the inspiral stage, the horizon vorticity picks up a visible contribution from the mass quadrupole-induced $\\mathbf {\\mathcal {B}}$ field (e.g.", "patches $A$ and $B$ in Fig.", "REF (c) for the SK simulation), while the spin contributions are also present (e.g.", "patch $C$ and a blue patch at the back that is blocked from view in Fig.", "REF (c)).", "The spin patches like $C$ are smooth continuations of the dipolar patches we see during early inspiral in Fig.", "REF (a), while mass quadrupole patches $A$ and $B$ only appear shortly before merger, but grow quickly to have larger $\|\\mathcal {B}_{NN}\|$ than the spin patches.", "Post-merger, the horizon vorticity for the SK simulation is shown in Fig.", "REF (d), where in addition to a dipolar contribution from the spin of the remnant black hole (similar to the $F$ patch in Fig.", "REF (b) for the AA simulation), there are also visible patches that can be interpreted as the continuation of the pre-merger spin patches.", "For example, the region $C^{\\prime }$ in Fig.", "REF (d) corresponds to patch $C$ in Fig.", "REF (c), while the patches $D$ and $E$ in Fig.", "REF (b) correspond to the continuation of the pre-merger individual spins in the AA simulation that were anti-aligned with $\\mathbf {J}^{\\rm tot}$ .", "Such finer details in $\\mathcal {B}_{NN}$ on the common apparent horizon thus provide us with a more concrete manifestation of the abstract remnant spins discussed earlier.", "We can now use the spin patches to track the pre- and post-merger (remnant) spin dynamics.", "First of all, we note that the SK- simulation is the same as the SK simulation aside from a reversal of the individual spins.", "Since Eq.", "REF is invariant under $\\mathbf {S_1} \\rightarrow -\\mathbf {S}_1$ and $\\mathbf {S}_2 \\rightarrow -\\mathbf {S}_2$ , we should observe the spins lifting up into the same side of the orbital plane for these two simulations Provided that the black holes merge at similar orbital configurations for the two simulations, which appears to be the case.", "For example, the merger time is the same as shown in the time labels of Fig.", "REF (c)/(d) and (e)/(f).. A comparison between Fig.", "REF (c)/(d) and (e)/(f) confirms this expectation, and provides us with some confidence that the spin-spin coupling is indeed responsible for generating the $S^z_1$ and $S^z_2$ components at merger.", "Using the numerical values for the spins (measured essentially with Eq.", "REF on each individual apparent horizon) and the numerical trajectories of the black holes in the simulation coordinates, we can further make a prediction for the $z$ components of the spins by integrating Eqs.", "(REF ) and (REF ).", "Namely, we calculate the $\\dot{S}^z_1$ values at a dense collection of times for one of the spins according to Eqs.", "(REF ) and (REF ), using the numerically measured $\\mathbf {S}_1$ , $\\mathbf {S}_2$ and $\\mathbf {n}$ , before adding these $\\dot{S}^z_1$ increments up into a predicted history for $S^z_1$ .", "The results are shown in Fig.", "REF .", "We see a steep rise of $S_1^z$ for the SK, SK- and SK$\\bot $ simulations towards merger, matching our $\\mathcal {B}_{NN}$ spin patch observations.", "In particular, the prediction is for a dimensionless spin with $S^z_1 \\approx 0.2$ just before the merger for all three simulations, which translates into an angle of $\\sin ^{-1}(0.2/0.5)\\approx 0.4$ that $\\mathbf {S}_1$ spans with the orbital plane.", "Taking the SK$\\bot $ simulation for example, the line connecting the centers of the $+$ ve $\\mathcal {B}_{NN}$ and $-$ ve $\\mathcal {B}_{NN}$ spin patches for this simulation (see Fig.", "REF (g) and (h)) spans an angle of $0.368$ with the orbital plane, which is fairly close to the prediction.", "The closeness between these two numbers is somewhat surprising, in that the simulation gauge is not the same as the harmonic gauge used for the PN calculations, and that we are operating in a regime close to merger.", "To further test the quality of the PN prediction, we produce the predicted histories for $S^x_1$ and $S^y_1$ using the same prescription, and compare them to their numerically computed counterparts in Fig.", "REF , which also show general agreement.", "The situation is different when we compare the predicted and the numerically computed $S^z_1$ , as the latter lacks the steep rise just before merger that is also seen in spin patches.", "To understand this, recall that the numerical spin values are calculated as integrals that are essentially Eq.", "(REF ).", "Such an integration over the entire horizon does not distinguish the spin patches like $C$ and $G$ in Fig.", "REF (c) and (e) from the mass quadrupole induced patches $A$ and $B$ as in Fig.", "REF (c).", "As the mass quadrupole patches on each apparent horizon resembles a spin pointing in the $-\\mathbf {\\hat{z}}$ direction at merger time (see Fig.", "REF (c), (e), (g) and (h)), the spin measurement $S^z_1$ according to Eq.", "(REF ) could be negative even when the actual $S^z_1$ is positive.", "On the other hand, the measurements on $S^x_1$ and $S^y_1$ are less affected by this contamination.", "Furthermore, it is required that the first derivative of the scalar $\\zeta $ in Eq.", "(REF ) should be a rotation generating approximate Killing vector [42], [43], [44], [45], which may not be applicable when we approach the highly non-stationary merger phase.", "One interesting feature we have observed with our simulations is that at the merger, the spins seem to have been preferentially lifted out of the orbital plane towards the spin-$\\mathbf {J}^{\\rm tot}$ alignment side (see Fig.", "REF (d)).", "This trend also continues post-merger, as shown in Fig.", "REF for the SK simulation (similar behavior is observed for SK- and SK$\\bot $ ), until a spin-$\\mathbf {J}^{\\rm tot}$ near-alignment is achieved, in agreement with our proposal in Sec. .", "The SK$\\bot $ simulation is particularly interesting in that its initial spin orientations are chosen such that if the black holes merge at exactly the same orbital configuration as the SK and SK- cases, we would have $\\dot{S}^z_1 < 0$ at merger according to Eq.", "(REF ).", "Instead, the black holes merge with $\\dot{S}^z_1 > 0$ (see Fig.", "REF (g), (h) and Fig.", "REF (c), (d)) at an earlier time.", "Therefore, based on our small sample of simulations, there appears to be a preference towards spin-$\\mathbf {J}^{\\rm tot}$ alignment at merger, which is not reversed post-merger.", "The exceptional case is the AA simulation, where the spins remain anti-aligned with $\\mathbf {J}^{\\rm tot}$ throughout the inspiral and merger phases, perhaps because the spins are stuck in a (possibly unstable) equilibrium configuration.", "Figure: The post-merger (during the merger phase) time evolution of the horizon vorticity ℬ NN \\mathcal {B}_{NN} for the SK simulation.", "The white curves are the contours of ℬ NN \\mathcal {B}_{NN}.", "We have also drawn black lines connecting the centers of the spin patches for both remnant spins.", "Over time, the spin patches become more aligned with the 𝐉 tot \\mathbf {J}^{\\rm tot} direction.One possible explanation for the preference at merger time is provided by the spins' influence on the orbital motion, through the extra acceleration [20], [21], [22], [23], [46], [47], [48] $\\mathbf {a} &=& -\\frac{3}{\\mu R^4}\\left[ \\mathbf {n} (\\mathbf {S}_1\\cdot \\mathbf {S}_2 ) + \\mathbf {S}_1 (\\mathbf {n}\\cdot \\mathbf {S}_2) + \\mathbf {S}_2 (\\mathbf {n}\\cdot \\mathbf {S}_1) \\right.", "\\\\&& \\left.", "- 5 \\mathbf {n}(\\mathbf {n}\\cdot \\mathbf {S}_1)(\\mathbf {n}\\cdot \\mathbf {S}_2) \\right]$ they impose on the relative one body equivalence to the two body motion.", "In Eq.", "(REF ), we have only shown the spin-spin contribution, as we are interested in the beginning of the final ascend of $S_1^z$ and $S_2^z$ , and the spin-orbit contribution is small for our $\\pi $ -symmetric simulations when $S^z_1$ and $S^z_2$ are still small (i.e.", "$\\mathbf {S}_1+\\mathbf {S}_2 \\approx 0$ ).", "It is plausible that the BBH coalescence progresses quickly towards merger when ${a}_n \\equiv \\mathbf {a} \\cdot \\mathbf {n} < 0$ (radially pulling the black holes closer) and ${a}_t \\equiv \\mathbf {a} \\cdot (\\mathbf {\\hat{z}} \\times \\mathbf {n}) < 0$ (slowing down the transverse orbital motion of the black holes), such that the instantaneous impact parameter is altered in the direction conducive to merger.", "We note that only the second and third terms in the square bracket of Eq.", "(REF ) contribute to $a_t$ , and these two terms have the same value in our $\\mathbf {S}_1 \\approx -\\mathbf {S}_2$ context.", "Therefore we have $a_t &=& -\\frac{6}{\\mu R^4} \\left[(\\mathbf {\\hat{z}} \\times \\mathbf {n}) \\cdot \\mathbf {S}_1\\right] (\\mathbf {n}\\cdot \\mathbf {S}_2) \\\\&=& -\\frac{6}{\\mu R^4}\\mathbf {\\hat{z}} \\cdot \\left[(\\mathbf {n}\\cdot \\mathbf {S}_2) \\mathbf {n}\\times \\mathbf {S}_1\\right].$ Comparing Eq.", "(REF ) with Eq.", "(REF ), we arrive at $a_t = -\\frac{2}{\\mu R} \\dot{S}^z_1,$ so that regions of $a_t < 0$ always correspond to $\\dot{S}^z_1 > 0$ , and there would subsequently be a statistical preference for spin-$\\mathbf {J}^{\\rm tot}$ alignment at merger." ], [ "Discussion ", "We note that an observation in this paper may help achieve a consistency between the QNM frequencies.", "First recall that an electric/magnetic parity quasinormal mode is defined to be one whose corresponding metric perturbation has the parity matching the sign of $(-1)^l$ /$(-1)^{l+1}$ .", "Because the $\\mathbf {\\mathcal {B}}$ /$\\mathbf {\\mathcal {E}}$ field has the opposite/same parity to the underlying metric perturbations [19], the mass/current quadrupole would then excite the electric/magnetic parity $l=2$ QNMs (see also Ref.", "[49], as well as Ref.", "[19] for discussions on the similarities between the mass/current quadrupole generated $\\mathbf {\\mathcal {E}}$ and $\\mathbf {\\mathcal {B}}$ fields and those associated with the electric/magnetic parity QNMs).", "The parity properties of the various quantities are summarized in Table REF .", "The electric and magnetic parity QNMs are degenerate in that they share the same complex frequency (see e.g.", "Sec.", "IC3 of Ref.", "[19] and Sec.", "VB1 of Ref. [49]).", "This has an interesting consequence, as was noted by Ref.", "[49] when motivating spin-locking in the superkick configurations.", "Namely, when QNMs of both parities are present, the current quadrupole should evolve at the same frequency as the mass quadrupole at the end of merger (just before the onset of the QNM ringdown phase).", "For the superkick configurations, because the mass quadrupole evolves at twice the orbital frequency, while the current quadrupole's frequency is essentially a sum of the orbital and the spin precession frequencies, this further implies that the spin precession frequency must lock onto the orbital frequency [49].", "Table: The parity of various quantities generated by the mass and current quadrupole moments.A robust mechanism must be present for this locking to occur.", "A calculation using the leading order PN spin-orbit coupling expression (REF ) for a $\\pi -$ symmetric superkick configuration yields [49], [8] $\\dot{\\beta }(t) = \\frac{7M}{8R(t)}\\Omega (t),$ where $\\dot{\\beta }$ is the spin precession frequency.", "Therefore, as the black holes move closer to each other, $\\dot{\\beta }$ can approach $\\Omega $ , and with a mixed use of gauge [49], equalize with it [8].", "However, this equality is broken again when $R(t)$ reduces further.", "So instead of locking, we have only a momentary coincidence.", "Another mechanism for locking is proposed by Ref.", "[49], which considers geodetic precession in black hole perturbation theory.", "This alternative provides a stronger precession $\\dot{\\beta }(t) = \\frac{3M}{R(t)}\\Omega (t),$ but is otherwise similar to the leading order PN spin-orbit coupling result.", "Without invoking further dynamics, one would then be forced to make the inference that the BBH QNM ringdown begins, and that the spacetime dynamics that can be construed as two individual black holes approaching each other, ceases, precisely at the $R(t)$ that gives $\\dot{\\beta }=\\Omega $ .", "Note that Eqs.", "(REF ) and (REF ) do not depend on the magnitude of the spins, so infinitesimal spins would appear to still play a vital role in the transition into the QNM ringdown phase.", "So additional dynamics are likely involved.", "For example, if the magnetic parity QNMs are completely absent, so that their frequencies become irrelevant, then the consistency would be achieved by default.", "Perhaps more interestingly, as discussed in relation to the next-to-leading-order PN corrections to $U_{\\text{SO}}$ in Sec.", ", the energetically favorable orientation of the spins have a frame-dependent offset from the directions of $\\pm \\mathbf {J}^{\\rm tot}$ .", "As this offset can evolve at the frequency of $\\Omega $ when the precession of $\\mathbf {S}_1$ is locked onto the orbital motion, we could in principle have a self-consistent sustained locking (as opposed to a momentary coincidence) scenario if the spins are kept in these orientations by, for example, the same dynamics that drove them there in the first place.", "In other words, the mechanism responsible for the decline of the current quadrupole moment may also be responsible for locking its frequency to the desired value.", "In this case, the magnetic parity QNMs will not need to vanish in the ringdown waveform.", "However, we are operating in a regime where the PN expressions are not expected to remain fully valid.", "To see whether similar effects are actually present in a strong field and fast motion setting, we plan to carry out further studies at a later date.", "The author is grateful to Robert Owen for carrying out a previous incarnation of the SK simulation.", "We thank David Nichols, Sean McWilliams, Aaron Zimmerman and Robert Owen for carefully reading a draft of the paper, and providing valuable suggestions.", "We would also like to thank Yanbei Chen, Eliu Huerta, Lawrence Kidder, Geoffrey Lovelace, Harald Pfeiffer, Mark Scheel and Kip Thorne for useful discussions and help on references.", "The simulations and visualizations in this work are performed on the WVU computer clusters Spruce Knob and Mountaineer, and the Caltech cluster Zwicky funded by the Sherman Fairchild Foundation and the NSF MRI-R2 grant PHY-0960291." ] ]	1403.0512
[ [ "First measurement of $\\sigma_8$ using supernova magnitudes only" ], [ "Abstract A method was recently proposed which allows the conversion of the weak-lensing effects in the supernova Hubble diagram from noise into signal.", "Such signal is sensitive to the growth of structure in the universe, and in particular can be used as a measurement of $\\sigma_8$ which is independent from more traditional methods such as those based on the CMB, cosmic shear or cluster abundance.", "We extend here that analysis to allow for intrinsic non-Gaussianities in the supernova PDF, and discuss how this can be best modelled using the Bayes Factor.", "Although it was shown that a precise measurement of $\\sigma_8$ requires ~$10^5$ supernovae, current data already allows an important proof of principle.", "In particular we make use of the 732 supernovae with z < 1 of the recent JLA catalog and show that a simple treatment of intrinsic non-Gaussianities with a couple of nuisance parameters is enough for our method to yield the values $\\sigma_8 = 0.84^{+0.28}_{-0.65}$ or $\\sigma_8 < 1.45$ at a $2\\sigma$ confidence level.", "This result is consistent with mock simulations and it is also in agreement with independent measurements and presents the first ever measurement of $\\sigma_8$ using supernova magnitudes alone." ], [ "Introduction", "Type Ia supernovae (SNeIa) are arguably the most important and reliable estimators of extragalactic distances.", "As it is well know, they provided the first solid evidence of the present cosmological acceleration [30], [28].", "Since this discovery a large effort has been devoted to testing and improving the calibration of the SNeIa and to correcting their light curves in order to understand and control systematics [25], [11], [7], [31].", "As their light comes from high redshifts (up to $z\\simeq 2$ ) gravitational lensing from intervening matter is expected to play an important role.", "The correction induced by lensing will in fact become a major source of uncertainty when richer and deeper SNeIa catalogs are compiled in the next years.", "The Large Synoptic Survey Telescope (LSST) project plans for instance to collect over a million SNeIa in ten years [1], roughly a thousand-fold increase from number of SNeIa observed so far.", "A great effort is therefore being put forward to better understand this and avoid biases; see e.g.", "[19], [2], [34], [10], [9], [4], [39].", "Gravitational lensing changes the intrinsic distribution function of the SNeIa magnitudes, increasing the scatter and introducing non-Gaussianity.", "In [27], we have obtained the lensing variance, skewness and kurtosis of the SNeIa distribution via sGL, a fast simulation method developed in [21], [22], [23].", "When confronted to $N$ -body simulations sGL was shown to be very accurate up to $z \\simeq 1.5$ , with the advantage of results being given as function of the relevant cosmological parameters.", "They also were in very good agreement with observational data [18], [26], [20] and with other recent independent theoretical estimations [4].", "These fits can be employed to take into account the lensing extra scatter for any value of the cosmological parameters and also to model the lensing non-Gaussianity.", "This fact was explored in [29] where we proposed to use these accurate determinations of the lensing moments to measure cosmological parameters, following the ideas first discussed in [6], [16], [37] and later further developed in [12].", "We showed that by using not just the variance of the lensing signal but the third and fourth order moments as well, a more precise and robust measurement was possible.", "In a $\\Lambda $ CDM scenario it was verified that the most sensitive cosmological parameters to supernova lensing were $\\Omega _{m0}$ and $\\sigma _8$ .", "Now since the former is already tightly constrained by the measurement of the supernova magnitudes themselves (i.e., by the first moment of the distribution), the most important new information gained was that pertaining to $\\sigma _8$ .", "In particular it was shown that $\\sigma _8$ could be measured by the LSST survey to within 3–7%, a value that is competitive with usual methods based on cosmic shear, cosmic microwave background (CMB) or cluster abundance, and completely independent of these.", "In particular, it does not rely on measuring galaxy shapes and is thus immune to the systematics associated to the cross-correlation of intrinsic galaxy ellipticities.", "Also, it does not require to extrapolate the amplitude $\\sigma _8$ from recombination epoch to today, as with the CMB technique, nor to make assumptions on the threshold of formation of structures that is needed when employing galaxy clusters.", "It also complements the method proposed in [15], to wit correlating nearby supernova magnitudes with their positions to obtain their peculiar velocity correlations, which is also sensitive to $\\sigma _8$ .", "Here we extend on previous works on two fronts.", "First, we generalize the method to include intrinsic non-Gaussianities in the SNeIa distributions (that is, excluding all lensing effects).", "We do so by employing one nuisance parameter for each central moment of the distribution.", "We then argue that this is the most straightforward extension of the standard supernova analysis and that a more complicated parametrization should only be used if data itself demands it; the Bayes Factor is a nice and simple way to decide which parametrization to use.", "Second, we apply the above generalized procedure to two real supernova catalogs: the recently published combined SDSS-II and SNLS 3-year results [8], dubbed the Joint Lightcurve Analysis (JLA) catalog and the older standard SNLS 3-year catalog (SNLS3) [11].", "We find that the method works as is, even though data is usually not treated for systematics that affect the higher moments.", "We thus obtain the first measurement of $\\sigma _8$ from supernova magnitudes alone.", "This letter is organized as follows.", "In Section we summarize our methodology.", "In Section we show how the Bayed Factor can be used to best model the SNeIa probability distribution function (PDF), and in Section we apply our method to real data.", "Finally, we conclude in Section ." ], [ "The Method of the Moments", "Here we summarize the main point of the method-of-the-moments (MeMo), originally discussed in [29].", "In a nutshell, the idea is to use the scatter in the Hubble diagram to measure $\\lbrace \\Omega _{m0}, \\sigma _8\\rbrace $ by measuring the mean $\\mu _{1}^{\\prime }$ and the first three central moments (which we will collectively refer to simply as $\\mu _{1-4}$ ).", "The moments of the lensing PDF $\\mu _{1-4,\\rm lens}$ were originally obtained from turboGL and accurate fitting functions were made available in [27].", "They are related to the full (observed) central moments $\\mu _{1-4}$ by $\\mu _{2} & \\;\\equiv \\;\\sigma _{{\\rm tot}}^{2}\\;=\\;\\sigma _{{\\rm lens}}^{2}+\\sigma _{\\rm int}^{2}\\,,\\\\\\mu _{3} & \\;=\\;\\mu _{3,{\\rm lens}} + \\mu _{3,\\rm int}\\,,\\\\\\mu _{4} & \\;=\\;\\mu _{4,{\\rm lens}}+6\\,\\sigma _{{\\rm lens}}^{2}\\,\\sigma _{\\rm int}^{2} + 3 \\, \\sigma _{\\rm int}^{4} + \\mu _{4,\\rm int}\\,,$ where $\\lbrace \\sigma _{\\rm int},\\mu _{3,\\rm int},\\mu _{4,\\rm int}\\rbrace $ are the “intrinsic” SNeIa dispersions, which we define including any experimental contributions.", "The number of moments to be used in this analysis is in principle arbitrary as each new moment adds information.", "However, it was shown in [29] that for supernovae almost all of the information is already included using $\\mu _{1-4}$ (and a very good fraction of it already in $\\mu _{1-3}$ ).", "The MeMo likelihood at each redshift bin is obtained directly from the first four moments $\\mu _{1-4}$ : $&L_{\\rm MeMo}(\\Omega _{m0}, \\sigma _8, \\lbrace \\sigma _{{\\rm int},j}\\rbrace ) = \\exp \\bigg ( - \\frac{1}{2} \\sum _{j}^{{\\rm bins}} \\chi _{j}^2 \\bigg ) \\,, \\\\&\\chi ^2_j = \\big (\\mu -\\mu _{\\rm data}\\big )^t \\;\\Sigma _j^{-1}\\; \\big (\\mu -\\mu _{\\rm data}\\big ) \\,, \\\\&\\mu = \\lbrace \\mu _1^{\\prime },\\,\\mu _2,\\,\\mu _3,\\,\\mu _4 \\rbrace \\,,$ where the vector $\\mu (z_{j},\\sigma _{8},\\Omega _{m0}, \\sigma _{\\rm int})$ is the theoretical prediction for the moments, and its second-to-fourth components are defined in (REF )–().", "The mean $\\mu _{1}^{\\prime }$ is the theoretical distance modulus.", "The quantity $\\mu _{\\rm data}(z_{j})$ is the vector of fiducial or measured (sample) moments.", "In forecasts it is $\\mu (z_{j},\\sigma _{8},\\Omega _{m0}, \\sigma _{\\rm int})$ evaluated at the fiducial model, while for real data it is best to use unbiased estimators of the central moments (sometimes called $h$ -statistics, see [13]).", "For instance for the third moment $\\mu _{3,{\\rm data}}(z_{j}) & = \\sum _{k} N_j \\frac{ \\big [m_{k,j}-\\mu _{1,{\\rm data}}^{\\prime }(z_{j}) \\big ]^3}{(N_j-1)(N_j-2)} \\,, $ where $m_{k,j}$ are the SNeIa distance moduli observed in the redshift bin centered at $z_{j}$ .", "The covariance matrix $\\Sigma $ is built using the fiducial (or observed) moments and therefore does not depend explicitly on cosmology (but it does on $z$ ).", "The full covariance matrix for $\\mu _{1-4}$ , which appears in (), can be found in [29].", "Note that the estimators found in [29] are in fact biased estimators, which only converge to the unbiased ones in the limit of large number of data points in each bin.", "For forecasts, such as the ones carried out in [29] this is irrelevant, but for real data here employed we find small but non-negligible corrections due to the fact that most bins have less than 50 SNeIa.", "Note that for such a small number of data points there are also small corrections to the full covariance matrix, the computation of which is straightforward using computer algebra software (we employed the Mathematica package MathStatica) but the result is too large to present here explicitly.", "Figure: All 10 independent terms in the MeMo likelihood.", "The diagonal plots depict the measured central momenta together with the weak lensing prediction for 2 values of σ 8 \\sigma _8: the CMB fiducial (σ 8 =0.8\\sigma _8=0.8, dashed orange) and σ 8 =1.6\\sigma _8=1.6 (dot-dashed blue), which we exclude at 2.3σ2.3\\sigma .", "In the off-diagonal cases data and model intermix, so instead we plot Σ XY \\Sigma _{XY}:points above (below) zero increase (decrease) the χ 2 \\chi ^2.Although in some cases the modelling of the intrinsic non-Gaussianities as extra moments constant in zz (using 2 nuisance parameters) looks simplistic, the get χ 2 \\,\\chi ^2/d.o.f.", "= 1.31.3.", "However, the last bin is an outlier, so we remove it and get a very good χ 2 \\,\\chi ^2/d.o.f.", "= 1.061.06." ], [ "Dealing with the Intrinsic Supernova PDF with the Bayes Factor", "When the MeMo was originally proposed in [29] the assumptions made about the intrinsic supernovae dispersion was at the same time both conservative and aggressive.", "The SNeIa were allowed to have a dispersion which in one hand had a different $\\sigma _{\\rm int}(z)$ in each redshift bin, but in the other was assumed to be Gaussian in each bin.", "Real SNeIa data may nevertheless contain non-Gaussianities which are not due to lensing, either intrinsic or due to systematics and/or to the lightcurve fitting procedures.", "Here we generalize the method to include non-Gaussianities in the form of intrinsic third and fourth central moments.", "However, if we allowed all three parameters to be free in every bin we would have no less than 30 nuisance parameters to marginalize over!", "Clearly this is too conservative, and instead we can do much better by following the same prescription used for the standard supernova analysis, which uses only $\\mu _1^{\\prime }$ .", "In that case, the supernova give the distance modulus up to a single nuisance parameter $M$ , which describes the intrinsic magnitude of the supernovae, and which is assumed to be constant in $z$ .", "In fact, a fine tuned $M(z)$ is able to fit all supernova data without any need for a cosmological constant or accelerated expansion.", "Clearly this is a contrived scenario, and cosmologist find it best to keep $M$ as a constant parameter and interpret supernovae data as an indication of cosmic acceleration.", "The same approach is probably best also for lensing, and we should only go beyond constant $\\sigma _{\\rm int}$ , $\\mu _{3,{\\rm int}}$ and $\\mu _{4,{\\rm int}}$ if data demands it.", "In fact, for both catalogs here employed $\\mu _{4,{\\rm int}} = 0$ was either the preferred value or very close to it, so for simplicity henceforth we assume, unless otherwise stated, that $\\mu _{4,{\\rm int}} = 0$ .", "This has only a small effect on the end results.", "The best way to decide whether additional nuisance parameters are necessary is through the Bayes Factor ($B_{12}$ ) [35], [36], [38], which is just a ratio of the so-called “evidences” of two models.", "The evidence is just the integral of the posterior over all data, and is usually neglected in parameter estimations.", "It is nevertheless very useful to compare models because it not only prefers models that fit best the data but has also a built-in “Occam's Razor” property.", "It is usually employed in conjunction with the Jeffrey's scale to decide which model is best.", "Here we went further and converted probabilities, given by $1 /(1+\\exp \|B_{12}\|)$ , into $\\sigma -$ levels assuming Gaussian errors (i.e., $\\,0.32 \\rightarrow 1\\sigma $ , $\\,0.05 \\rightarrow 2\\sigma $ , $\\,0.003 \\rightarrow 3\\sigma $ and so forth).", "We believe this makes it simpler to interpret the results.", "We thus computed $B_{12}$ for real data in order to decide which is the best way to parametrize the intrinsic dispersion of the SNeIa.", "We conclude that a constant $\\sigma _{\\rm int}$ and $\\mu _{3,\\rm int}$ is favored over $\\sigma _{\\rm int}(z)$ and $\\mu _{3,\\rm int}(z)$ .", "The results are in table REF .", "For future data from the Dark Energy Survey (DES), we did a similar test this time assuming a constant $\\sigma _{\\rm int}$ and $\\mu _{3,\\rm int}$ as fiducial.", "The results clearly show that if that is the case, data will strongly favor the simpler model.", "It is possible that more complex modelling of intrinsic non-Gaussianity will be needed in the future for very large catalogs such as the one from LSST [1], but this can be tested as above.", "We also tested the MeMo for the Union 2.1 catalog [33].", "However, we found that for the complete catalog we could not get a good fit (too high $\\chi ^2$ /d.o.f.).", "This may be due to the fact that it is a compilation of SNeIa from many different surveys.", "Although care was taken to homogenize the catalog (and that a recent blind search for systematics in [3], [17] found no evidence of any), the focus has always been on $\\mu _1^{\\prime }$ , whereas here the lensing signal comes from higher moments.", "Figure: Left: Posteriors on σ 8 \\sigma _8 from JLA and SNLS3 data after marginalizing over all other parameters.", "We assume that both σ int \\sigma _{\\rm int} and μ 3, int \\mu _{3,\\rm int} are constant in redshift and that μ 4, int =0\\mu _{4,\\rm int}= 0.", "The solid dark red curve is the posterior using the real JLA data; the green long-dashed curve is the same for SNLS 3-year data; the orange dotted curves are 5 different forecasts using mock catalogs with the same number and redshift distribution of SNeIa as the JLA catalog.", "Right: Same for {σ 8 ,Ω m0 }\\lbrace \\sigma _8,\\Omega _{m0}\\rbrace for JLA." ], [ "Measuring $\\sigma _8$ with JLA and SNLS3 supernova catalogs", "In this Section we apply the method for data with $z \\le 1$ in two supernova catalogs: SNLS3 (460 supernovae) and JLA (732 supernovae).", "The reason for the cutoff at $z=1$ is that both catalogs have too few supernovae beyond that, making it pointless and error prone any attempt to compute the central moments in that range.", "We employ a simple binning of the data in 10 redshift bins of 0.1 width.", "Since the distance modulus change inside each bin is significant, care must be taken when computing the central moments.", "One cannot use $m_{k,j}$ in (REF ) directly as the measured distance moduli of each supernova.", "Instead, $m_{k,j}$ should be evaluated as the distance modulus at $z_j$ at the bin center plus the deviation $\\Delta m_{k,j}$ with relation to the best fit curve $m_{\\rm best}(z)$ .", "In other words: $m_{k,j} \\;\\equiv \\; m_{k,z_k}^{\\rm catalog} - m_{\\rm best}(z_k) + m_{\\rm best}(z_j) \\,.$ Moreover, since current data does not put tight constraints in $\\sigma _8$ , we extended the numerical simulations in [27] for a broader range of values, namely $0<\\sigma _8<2$ .", "Figure REF depicts all 10 central moment terms in the likelihood, together with the expectation due to lensing assuming two different values of $\\sigma _8$ .", "Figure REF [left] depicts the marginalized posterior of $\\sigma _8$ for the JLA and SNLS3 data, together with 5 mock catalogs with the same number and redshift distribution of SNeIa as the JLA catalog.", "Figure REF [right] shows the marginalized posterior of $\\lbrace \\sigma _8, \\Omega _{m0}\\rbrace $ for the JLA catalog.", "For JLA the last $z$ bin (with only 26 SNeIa) is an outlier, so we removed it.", "We then get $\\,\\sigma _8 = 0.84^{+0.28}_{-0.65} \\,$ or that $\\,\\sigma _8 < 1.45\\,$ at a $2\\sigma $ confidence level.", "The overall $\\,\\chi ^2$ /d.o.f.", "is a very good $1.06$ (if we kept the last bin, $\\,\\chi ^2$ /d.o.f.", "= $1.3$ ).", "For the mock catalogs we use as fiducial values for the moments of the intrinsic SNeIa PDF the values obtained in the best-fit of the JLA catalog.", "It is interesting to note that even for the older SNLS3 catalog one can gets $\\,\\sigma _8 = 0.93^{+0.24}_{-0.72} \\,$ or that $\\,\\sigma _8 < 1.49$ .", "This is the first time information on cosmological perturbations is obtained from SNeIa data alone.", "Figure REF shows the marginalized likelihoods for the intrinsic moments (our nuisance parameters).", "In both catalogs $\\mu _{4,\\rm int}= 0$ is well inside $1\\sigma $ .", "For $\\mu _3$ , for JLA one has $\\mu _{3,{\\rm int}} = (0.8\\pm 2.7) \\times 10^{-4}$ , while for SNLS3 we find $\\mu _{3,{\\rm int}} = (6.1\\pm 1.9) \\times 10^{-4}$ .", "Figure: Intrinsic moments, in magnitudes, for the JLA (solid curves) and SNLS3 (dashed) catalogs.", "For μ 3 \\mu _3, JLA accepts μ 3, int =0\\mu _{3,{\\rm int}} = 0, while for SNLS μ 3, int =(6.1±1.9)×10 -4 \\,\\mu _{3,{\\rm int}} = (6.1\\pm 1.9) \\times 10^{-4}.", "Note that in both catalogs μ 4, int =0\\mu _{4,{\\rm int}} = 0 is well inside 1σ1\\sigma .In table REF we compute the evidence for lensing in JLA, SNLS3 and future surveys in detecting lensing signal.", "We find that JLA can only give a very faint hint at the existence of lensing ($0.9\\sigma $ ), and even that only when using all 4 moments.", "In fact, using only the variance as usually done in the literature, this faint hint disappears completely, which is consistent with the results in [24].", "This is better understood in our forecasts for future DES and LSST data (using $10^5$ SNeIa) where one can clearly see that adding the third and fourth moments increases the evidence for lensing.", "For these forecasts we assume intrinsic Gaussianity with $\\sigma _{\\rm int}=0.12$ mag as our fiducial model.", "Table: Model comparison between supernovae with lensing (model 1) and without lensing (model 2)" ], [ "Discussion", "In this letter we obtained the first constraints for $\\sigma _8$ from SNeIa data alone.", "In other words, without need to cross-correlate SNeIa with matter distribution data, as done for instance in [32].", "In order to obtain such bounds we used two nuisance parameters to cope with intrinsic scatter and skewness in the data.", "In principle one can use also a third nuisance parameter for the kurtosis, but data showed no need of it.", "In fact, for the JLA catalog even $\\mu _{3,{\\rm int}}$ could be set to zero, but we chose to leave it and marginalize over to get more conservative results.", "Nevertheless, although the obtained bounds for $\\sigma _8$ are very broad and systematics may be present, the consistency of the data with our mocks serves as an important validation of the method and opens up a new avenue in cosmology.", "In the future in order to best use this lensing information it is important to study whether experimental details or data reduction methods introduce systematics in the form of non-Gaussianities.", "Moreover, here we made use of the inferred SNeIa distances directly from JLA and SNLS3 catalogs.", "It would be interesting to check in detail whether including the $\\sigma _8$ dependence due to lensing in the lightcurve fitter itself (i.e., simultaneously with the stretch and color corrections) significantly affects any of the results.", "It is clear that other similar tests can be employed with our methods.", "For instance, one can fix completely the cosmology at, say, the CMB values and just do a hypothesis test on the data as a consistency check with lensing predictions.", "Other interesting possibilities would be instead to use SNeIa lensing to test either the power spectrum directly [5] or the halo models [14], but both require re-deriving our estimates for the central moments." ], [ "Acknowledgment", "It is a pleasure to thank Luca Amendola, Marcos Lima, Martin Makler, Valerio Marra, Ben Metcalf, Alessio Notari and Ribamar Reis for fruitful discussions.", "MQ is grateful to Brazilian research agencies CNPq and FAPERJ for support." ] ]	1403.0293
[ [ "Plasmon mediated non-photochemical nucleation of nanoparticles by\n circularly polarized light" ], [ "Abstract We predict nucleation of pancake shaped metallic nanoparticles having plasmonic frequencies in resonance with a non-absorbed circularly polarized electromagnetic field.", "We show that the same field can induce nucleation of randomly oriented needle shaped particles.", "The probabilities of these shapes are estimated vs. field frequency and strength, material parameters, and temperature.", "This constitutes a quantitative model of non-photochemical laser induced nucleation (NPLIN) consistent with the observed particle geometry.", "Our results open a venue to nucleation of nanoparticles of desirable shapes controlled by the field frequency and polarization." ], [ "Introduction", "Accelerated nucleation in response to laser or dc electric fields has been observed in a number of systems.", "[1], [2], [3], [4], [5], [6], [7], [8], [9] It is typically attributed to the field-induced polarization of the new phase particles that suppresses the nucleation barrier.", "The corresponding theoretical work described mostly static field effects on nucleation.", "[4], [12], [10], [11] It was realized recently [13], [14] that non-absorbed ac electric fields can affect nucleation differently due to plasmonic excitations in metallic nuclei.", "Their resonance interaction with ac fields make nucleation barriers frequency dependent.", "For linearly polarized ac fields, that interaction leads to nucleation of strongly asymmetric needle shaped metal particles (prolate spheroids) with frequency dependent dimensions aligned to polarization.", "[13] The corresponding energy gain can be significant enough to change the phase equilibrium thus allowing particles that would not form in zero fields.", "We emphasize that the above work implies nucleation (induced by dc or low frequency ac fields) when no significant light absorption is possible.", "Such nucleation is therefore due to the electric field effects not related to any photochemical transformations.", "Following seminal work [1] by Garetz et.", "al., these phenomena are often referred to as non-photochemical laser induced nucleation (NPLIN).", "It was found experimentally, but not understood theoretically, that shapes of the NPLIN nucleated particles depend on the light polarization: the polarization aligned needle shaped nuclei for the case of linear polarization, vs. pancake or randomly oriented needle shaped nuclei under a circularly polarized light.", "[15], [16] Here we consider nucleation of metallic nanoparticles in the field of a circularly polarized light.", "The electric vector executing a circle perpendicular to the path of propagation will induce circular electron polarization; hence, nuclei of circular cross-section become energetically favorable.", "Here, we show that such nuclei of pancake shape, illustrated in Fig.", "REF , can have plasmonic resonance amplifying their polarization and exponentially increasing nucleation rates.", "On the other hand, as a superposition of two linearly polarized components, any circularly polarized light can trigger nucleation of the needle shaped particles characteristic of linearly polarized fields.", "Unlike that previously known case, such particles will be created with random orientations perpendicular to the path of propagation of a circularly polarized light.", "Therefore, nucleation of both the pancake and needle shaped nuclei must be considered for the case of circularly polarized light.", "Figure: Polarization plane (x,y)(x,y) cross-sections of nucleus shapes maximizing field induced electric dipoles.", "Bold arrows represent the electric field vectors.", "Left: needle shaped prolate spheroid for the case of linear polarization.", "Two arrows correspond to electric field vectors separated by half a period of the ac field oscillations.", "Right: pancake shaped oblate spheroid for the case of circular polarization.", "Dashed circle with arrow shows the electric field rotation.This paper is organized as follows.", "In Sec.", "we describe the general treatment of electric field effects on nucleation.", "Sec.", "presents qualitative arguments explaining the results of this paper.", "The corresponding quantitative analysis is given in Sec.", ".", "Sec.", "briefly describes a particularly interesting case of nucleation in the proximity of phase transitions.", "The numerical estimates are given in Sec.", ".", "Finally, Sec.", "contains general discussion and conclusions." ], [ "Electric field effect in classical nucleation theory", "Following the classical nucleation theory, [17], [18], [12] the system free energy is the sum of the bulk and surface contributions of the new phase particle, and an electric field dependent term, $F_E$ , $F=F_E+\\mu V+\\sigma A.$ Here, $\\mu $ is the difference in chemical potential (per volume) due to nucleation, and $\\sigma $ is the surface tension, $V$ and $A$ are the particle volume and area.", "The case of $\\mu <0$ corresponds to a metastable system in which nucleation is expected without external field; $\\mu >0$ describes the case where metal particles are energetically unfavorable in zero field, yet, as shown below, they can appear in a sufficient electromagnetic field.", "$F_E$ represents the polarization energy gain due to an induced dipole, [4], [12], [10], [11]${\\bf p}=\\alpha {\\bf E}$ in field ${\\bf E}$ .", "For a particle of polarizability $\\alpha $ in a dielectric host with permittivity $\\epsilon $ , one can write, [12] $F_E=-\\epsilon \\alpha E^2.$ The factor $\\epsilon $ makes Eq.", "(REF ) different from the energy of a dipole in an external field.", "It reflects the contributions from all charges in the system, including those responsible for the field.", "[12], [10], [11] We consider ac fields of frequency $\\omega \\gg \\omega _{at}$ , where $\\omega _{at}\\sim 10^{13}$ s$^{-1}$ is the characteristic frequency of atomic vibrations; hence, polarization of predominantly electronic origin.", "The corresponding energy, proportional to $-{\\bf p\\cdot E}$ , will oscillate in time with frequency $\\omega $ .", "According to the standard procedure of adiabatic (Born-Oppenheimer) approximation, its time average can be treated as a contribution to the potential energy of the atomic subsystem.", "Using the known recipe for time averages, [19] the latter becomes, $F_E=-\\epsilon \\frac{E^2}{2}{\\rm Re}(\\alpha ),$ where $E$ is now understood as the field amplitude and ${\\rm Re} (\\alpha )$ represents the real part of the polarizability.", "Here we limit ourselves to the case of normal light incidence.", "The field vector of circularly polarized light, is ${\\bf E}=({\\bf i}+i{\\bf j})E_0\\exp [i(kz-\\omega t)]$ where ${\\bf i}$ and ${\\bf j}$ are unit vectors along $x$ and $y$ axes respectively, and $z$ axis is along the path of light propagation; $k$ and $\\omega $ are the wave number and frequency, and $i=\\sqrt{-1}$ .", "Eq.", "(REF ) then yields, $F_E=-\\epsilon \\frac{E^2}{2}[{\\rm Re}(\\alpha _x)+{\\rm Re}(\\alpha _y)],$ where $\\alpha _x$ and $\\alpha _y=\\alpha _x\\equiv \\alpha $ are polarizabilities along $x$ and $y$ axes.", "To introduce useful notations, we recall the case of a spherical metallic nucleus of radius $R$ in a static field where $\\alpha = R^3$ , $V=4 \\pi R^3/3$ , $A=4\\pi R^2$ .", "Assuming $\\mu <0$ , the nucleation barrier $W$ and radius $R$ are determined by the maximum of $F$ in Eq.", "(REF ), $W=W_0(1+\\xi /2)^{-2}\\quad {\\rm when}\\quad R=R_0(1+\\xi /2)^{-1}.$ The corresponding zero-field quantities and the dimensionless field strength parameter $\\xi $ are, $W_0=\\frac{16\\pi }{3}\\frac{\\sigma ^3}{\\mu ^2},\\quad R_0=\\frac{2\\sigma }{\|\\mu \|},\\quad \\xi =\\frac{\\epsilon E^2R_0^3}{W_0}.$ Their ballpark values are $W_0\\sim 1$ eV, $R_0\\sim 1$ nm, and $\\xi \\ll 1$ for a moderate field of $E=30$ kV/cm and $\\epsilon \\sim 3-10$ ." ], [ "Qualitative analysis", "All results of this work can be obtained qualitatively.", "We start with noting the high static polarizability, $\\alpha _{\\rm stat}\\sim (R_{\\perp }/R_{\\parallel })V\\sim R_{\\perp }^3\\gg V,$ of pancake shaped particles, e.g., an oblate spheroid or a cylinder of height $2R_{\\parallel }$ and radius $R_{\\perp }$ aligned to the field (Fig.", "REF ).", "Indeed, the field-induced charges, $\\pm q$ , induced at the opposite poles are estimated from the balance of forces, $q^2/R_{\\perp }^2=qE$ , which gives the dipole moment $p\\sim R_{\\perp }q\\sim ER_{\\perp }^3\\sim V(R_{\\perp }/R_{\\parallel })E\\equiv \\alpha _{\\rm stat} E$ .", "Another amplification factor contributing to the high dynamic polarizability of pancake shaped metal particles is due to its plasmonic excitations.", "The plasmonic resonance in such a particle can be qualitatively explained by considering dipole oscillations of electrons in its volume.", "Shifting the negative electron charges over small distance, $x\\ll R_{\\perp }$ along $R_{\\perp }$ , deposits charges $\\pm q$ on the two halves of the spheroid where $q\\sim R_{\\parallel }R_{\\perp }xNe$ , and $N$ is the electron concentration.", "These charges exert forces $\\sim qe/R_{\\perp }^2$ on individual electrons on the opposite side.", "Interpreting the latter as the restoring forces $m\\omega ^2x$ yields the resonant frequency $\\omega \\sim \\omega _{\\rm pl}\\sqrt{R_{\\parallel }/R_{\\perp }}$ , where $m$ is the electron mass and we have used the standard definition of the electron plasma frequency, $\\omega _{\\rm pl}=\\sqrt{4\\pi Ne^2/m}$ .", "The above resonance plasmonic frequency $\\omega $ has been experimentally observed in light scattering.", "[20] The maximum polarizability corresponds to the frequency of plasmonic resonance.", "Since the resonance amplitude is by the quality factor $Q\\gg 1$ greater than its static value, $\\alpha _{\\rm stat}$ from Eq.", "(REF ) must be multiplied by $Q$ in order to obtain the maximum (resonance) polarizability.", "Since $Q=\\omega \\tau $ where $\\tau $ is the electron relaxation time, one gets the maximum dynamic polarizability $\\alpha \\sim V(\\omega _{\\rm pl}/\\omega )^2\\omega \\tau \\sim R_{\\perp }^3(\\omega \\tau )\\gg V.$ This coincides, to a numerical coefficient with the result of rigorous treatment in Eq.", "(REF ) below.", "The latter gigantic increase in polarizability takes us to the major prediction of this work: an ac field of frequency $\\omega $ can drive the nucleation of pancake-shaped particles with resonant aspect ratio $R_{\\perp }/R_{\\parallel } \\sim (\\omega _p/\\omega )^2\\gg 1$ .", "Taking into account the amplification ratio in Eq.", "(REF ), an approximate result for the ac resonant nucleation barrier can be guessed from the known static result, [4] or even from Eq.", "(REF ), with $\\xi \\rightarrow (\\omega \\tau )\\xi \\gg 1$ , which yields, $W\\sim (1 /\\omega \\tau )^2(W_0/E^2R_0^3)^2W_0.$ To within the accuracy of a numerical multiplier, this coincides with the exact result in Eq.", "(REF ) below." ], [ "Formal consideration", "Our rigorous analysis begins with the polarizability of a spheroid along its major axes,[21] $\\alpha _{j}= \\frac{V}{4\\pi }\\frac{\\epsilon _p-\\epsilon }{\\epsilon +n_j(\\epsilon _p- \\epsilon )}.$ Here, $\\epsilon _p$ is the dielectric permittivity of a metal particle, $\\epsilon $ , assumed to be a real number, is that of the medium, and $n_j$ is the depolarizing factor of the spheroid in $j$ -th direction (see Fig.", "REF ).", "The frequency dependent dielectric permittivity of a metal is represented as, $\\epsilon _p=1-\\frac{\\omega _{\\rm pl}^2}{\\omega ^2}+i\\frac{\\omega _{\\rm pl}^2}{\\omega ^3\\tau }$ where the relaxation time, $\\tau \\gg \\omega ^{-1}\\gg \\omega _{\\rm pl}^{-1}$ .", "For a strongly asymmetric oblate spheroid with semi-axes $R_{\\perp }$ and $R_{\\parallel }\\ll R_{\\perp }$ respectively perpendicular and parallel to the propagation path, and eccentricity, $\\eta =R_{\\perp }/R_{\\parallel }\\gg 1$ , the depolarizing factors are given by $n_x=n_y\\equiv n _{\\perp }=\\frac{\\pi R_{\\parallel }}{4R_{\\perp }}\\ll 1.$ The volume and area of that spheroid are given by, $V=4\\pi R_{\\perp }^3/3\\eta \\approx 16R_{\\perp }^3n_{\\perp }/3,\\quad \\textrm {and}\\quad A\\approx 2\\pi R_{\\perp }^2.$ Figure: Oblate spheroid with semi-axes R ∥ R_{\\parallel } and R ⊥ R_{\\perp }.Using Eqs.", "(REF - REF ) with Eq.", "(REF ) yields, ${\\rm Re}(\\alpha )=\\frac{V}{4\\pi }\\frac{(n _{\\perp }-n_{\\omega })+bn_{\\perp }}{(n_{\\perp }-n_{\\omega })^2+bn_{\\perp }^2},$ where $b=1/(\\omega \\tau )^2\\ll 1$ and, $n_{\\omega }=\\frac{\\epsilon \\omega ^2}{\\omega _{\\rm pl}^2+(\\epsilon -1)\\omega ^2}\\approx \\frac{\\epsilon \\omega ^2}{\\omega _{\\rm pl}^2}\\ll 1.$ The polarizability, ${\\rm Re}(\\alpha )$ , has a sharp maximum when, $n_{\\perp } =\\frac{1+\\sqrt{b}}{1+b}n_{\\omega }\\approx n_{\\omega }(1+\\sqrt{b}),$ which reflects the presence of the plasmonic resonance.", "Given the sharpness of the resonance, all other $n$ -dependent quantities, in particular the spheroid volume and area, can be evaluated at $n_{\\perp }=n_{\\omega }$ .", "That yields the maximum (resonance) polarizability, $\\left[{\\rm Re}(\\alpha )\\right]_{max}\\approx \\frac{V}{8\\pi }\\frac{\\omega _{\\rm pl}^2\\tau }{\\omega \\epsilon },$ consistent with our earlier estimate in Eq.", "(REF ).", "Including the above electrostatic contribution given by Eqs.", "(REF ), (REF ) and normalizing the free energy in Eq.", "(REF ) with respect to the classical barrier $W_0$ , yields $\\frac{F}{W_0}=-\\frac{8\\epsilon }{\\pi }\\left(\\frac{R_{\\perp }}{R_0}\\right)^3\\left(\\frac{\\omega }{\\omega _{\\rm pl}}\\right)^2\\left[\\left(\\frac{E}{E_{\\omega }}\\right)^2 \\pm 1\\right]+\\frac{3}{2}\\left(\\frac{R_{\\perp }}{R_0}\\right)^2.$ Here $\\pm $ corresponds to the cases when the original phase is metastable ($+$ ) and stable ($-$ ), and $E_{\\omega }\\equiv E_c\\sqrt{\\frac{\\omega }{\\omega _{\\rm pl}^2\\tau }},\\quad E_c\\equiv \\sqrt{\\frac{12W_0}{R_0^3}}.$ Note that in the case of stable systems [$+$ sign in Eq.", "(REF )], nucleation is impossible in zero field, however it becomes possible in a moderately high applied fields $E>E_{\\omega }$ ; see estimates in Sec.", ".", "Optimizing free energy in Eq.", "(REF ) with respect to $R_{\\perp }/R_0$ yields the nucleation radius and barrier, $\\frac{R_{\\perp }}{R_0}&=&\\frac{\\pi }{8\\epsilon }\\left(\\frac{\\omega _{\\rm pl}}{\\omega }\\right)^2\\left[\\left(\\frac{E}{E_{\\omega }}\\right)^2 \\pm 1\\right]^{-1},\\nonumber \\\\\\frac{W}{W_0}&=&\\frac{\\pi ^2}{192\\epsilon ^2}\\left(\\frac{\\omega _{\\rm pl}}{\\omega }\\right)^4\\left[\\left(\\frac{E}{E_{\\omega }}\\right)^2 \\pm 1\\right]^{-2}.$ These equations are valid as long as the expression in square brackets remains positive, i. e. for metastable systems in arbitrary fields $E$ or for stable systems in strong enough fields $E>E_{\\omega }$ .", "For the case of field dominated nucleation, $E\\gg E_{\\omega }$ these expressions reduce to the following, $\\frac{R_{\\perp }}{R_0}&=&\\frac{3\\pi }{2\\epsilon }\\frac{1}{\\omega \\tau }\\left(\\frac{E_c}{E}\\right)^2,\\nonumber \\\\\\frac{W}{W_0}&=&\\frac{\\pi ^2}{192\\epsilon ^2}\\left(\\frac{1}{\\omega \\tau }\\right)^2\\left(\\frac{E_c}{E}\\right)^4.$ The latter nucleation barrier should be compared to that for needle shaped particles, [13] which, in the current notations, is given by $\\frac{W}{W_0}=\\frac{\\pi ^3}{16\\epsilon ^2}\\sqrt{\\frac{\\Lambda }{\\epsilon }}\\frac{\\omega }{\\omega _{\\rm pl}^3\\tau ^2}\\left(\\frac{E_c}{E}\\right)^4$ where $\\Lambda \\approx \\ln \\left(2\\omega _{\\rm pl}/\\sqrt{\\epsilon }\\omega \\right)-1\\gg 1.$ [We have taken into account that $E_c$ defined in Ref.", "karpov2012a is by the factor $\\sqrt{2/\\epsilon }$ different from that in Eq.", "(REF ).]", "We observe that the nucleation barrier of pancake shaped particles is lower than that of the needle shaped ones when $\\frac{\\omega }{\\omega _{\\rm pl}}>\\left(\\frac{1}{12\\pi }\\sqrt{\\frac{\\epsilon }{\\Lambda }}\\right)^{1/3}.$" ], [ "Plasmonic mediated nucleation in the proximity of a phase transition", "Similar to the findings in Refs.", "karpov2012a,karpov2012b this theory applies most easily to a system close to a bulk phase transition.", "At temperature $T$ close to $T_c$ , the chemical potential $\\mu =\\mu _0(1-T/T_c)$ ; hence classical nucleation radius and barrier become, $R_0=R_{00}(1-T/T_c)^{-1},\\quad W_0=W_{00}(1-T/T_c)^{-2},$ and $E_c=E_{c0}(1-T/T_c)^{1/2}$ , where $R_{00}$ , $W_{00}$ , and $E_{c0}$ are obtained from their definitions in Eqs.", "(REF ) and (REF ) with $\\mu =\\mu _0$ .", "This allows macroscopically large $R_0$ , consistent with CNT; also, it corresponds to lower $E_c$ , making the plasmonic nucleation easier to observe.", "Using the above scaling, the results for particle nucleation length and barrier turn out to be temperature independent: they retain their form of Eq.", "(REF ) with the trivial substitutions, $R_0\\rightarrow R_{00},\\quad W_0\\rightarrow W_{00}, \\quad E_c\\rightarrow E_{c0}$ .", "This is in striking difference with the zero field classical nucleation theory, which predicts a diverging nucleation barrier $W_0\\propto (1-T/T_c)^{-2}$ [cf.", "Eq.", "(REF )].", "The conclusion of temperature independent nucleation barrier remains valid for the case of static field induced nucleation.", "[22] The distinctive feature of temperature independent barrier can serve as an evidence of field induced nucleation that can take place even at relatively weak field $E\\gtrsim E_{c0}(1-T/T_c)^{1/2}$ when $T\\rightarrow T_c$ ." ], [ "Numerical estimates", "To a large part, suitable numerical estimates here are the same as that of Ref.", "karpov2012a dealing with needle shape particle nucleation.", "We recall that the $Q$ -factor in Eq.", "(REF ) can be represented[23] as $\\omega _p\\tau = 160/(\\sqrt{Na_B^3}\\rho )\\sim 10^3$ , where $a_B$ is the Bohr radius, $1/\\sqrt{Na_B^3}$ is in the range of 5-10, and the resistivity, $\\rho $ , (in units of $\\mu \\Omega $ cm) is smaller than unity.", "In nanoparticles, surface scattering can decrease that product, [24], [25], [26] which is reflected in the available data [20] showing that plasmonic line widths can be comparable to their resonance frequencies.", "In fact, the available data presented in Ref.", "grigorchuk2011 show that in a broad range of nanoparticle sizes, $\\omega \\tau \\sim 3$ where omega stands for the resonance (plasmonic) frequency.", "To estimate $E_c$ in Eq.", "(REF ), consider $R_0\\sim 3$ nm, $W_0\\sim 2$ eV, and $\\epsilon =16$ typical of e. g. conductive (crystalline) nuclei in the prototype phase change material Ge$_2$ Sb$_2$ Te$_5$ (see Refs.", "nardone2012,agarwal2011 and references therein) and other metal nuclei, [29], [12] which yields $E_c\\sim 10^6$ V/cm.", "The other multipliers in Eq.", "(REF ) make $E_{\\omega }$ much lower, say $E_{\\omega }\\sim 30$ kV/cm, corresponding to laser power density $P\\sim 10 $ mW/$\\mu $ m$^2$ , an order of magnitude below that used with DVD burners.", "Assuming lower $\\epsilon \\sim 3-5$ for nucleation in a liquid or a glass will increase $P$ by a factor of $16/\\epsilon $ keeping it rather low.", "Finally, assuming $\\omega \\tau \\sim 3$ , Eq.", "(REF ) predicts the field dominated pancake nucleation barrier $W$ lower than that of classical spherical nuclei $W_0$ when $E>0.1E_c\\sim 10^5$ V/cm, achievable with moderate power lasers.", "All the above estimates become more favorable to plasmonic mediated nucleation in a proximity of phase transition.", "The case of VaO$_2$ can provide a relevant example [22] with its low $T_c$ and $1-T/T_c\\approx 0.2$ at room temperature.", "Corresponding to the above $E_c$ , $\\omega \\tau \\sim 1$ , and reasonable $E\\lesssim 0.1E_c$ yields the long semi-axis estimate $R_{\\perp }\\sim 10-100$ nm, well above the quantum range of very small particle sizes ($\\lesssim 1$ nm).", "We conclude that there exists a range of laser frequencies an powers, in which pancake shaped particles nucleate easier than both the needle-shaped and spherical particles.", "The characteristic pancake radii $R_{\\perp }$ are expected to be greater than $\\sim 10$ nm.", "The most restrictive condition of their dominance is that the frequency of a circularly polarized light is by a numerical factor lower than the plasma frequency; the region of yet lower frequencies will favor nucleation of randomly oriented needle shaped particles perpendicular to the light propagation path.", "We would like to mention here the earlier work [14] predicting pancake shapes for nucleation of voids in metal skin layers.", "While the approach and final results for nucleation rates in Ref.", "karpov2012b remain valid, the accompanying figure (Fig.", "2 in Ref.", "karpov2012b) does not match them.", "The discrepancy is that that figure presents the case of linear polarization parallel to the spheroid long axis, contrary to the accompanying calculations assuming the short axis polarization.", "Hence, the calculated rate [14] describes the nucleation of oblate spheroidal voids perpendicular to the metal surface under the light of normal incidence.", "Alternatively, it can describe the nucleation of oblate spheroidal voids parallel to the metal surface due to a linearly TM-polarized light of graze incidence provided that no surface plasmon polariton modes are excited." ], [ "Discussion and conclusions", "Nucleation of needle shaped nanoparticles with aspect ratio frequency governed by linearly polarized light was observed in Ref.", "ouacha2005 The case of circular polarization was studied in Refs.", "garetz2002,sun2008 where disk (or pancake) shaped particles were observed in NPLIN experiments.", "However, the final products of the latter observations were dielectric (rather than here considered metallic) nanoparticles; hence, no direct comparison with the present theory is possible.", "It was argued [5] that the observed dielectric particles in NPLIN experiments result from subsequent structural transformations of the originally metal nuclei.", "Assuming the latter secondary process, our theory provides an explanation of why circularly polarized light creates pancake shaped particles that are not observed with the linear polarization.", "In support, we would like to mention that the mechanism of precursor metal particle nucleation remains so far the only one that explains the observed extremely high NPLIN rates; [5] the underlying physics is that metal particle polarization is by several orders of magnitude higher than that of dielectric ones.", "It follows then that direct verification of here presented theory would be possible if both the linear and circular laser beam polarizations were used in the experiments [6], [30] that earlier discovered the frequency dependent effect of linearly polarized laser beams on metal particle nucleation.", "In conclusion, we predicted a phenomenon of plasmonic mediated nucleation of pancake shaped metallic nanoparticles under non-absorbed circularly polarized light.", "They can dominate nucleation at frequencies not too low compared to the plasmon frequency.", "At low enough frequencies, the circularly polarized light is predicted to induce nucleation of randomly oriented (in the polarization plane) needle shaped metal nanoparticles.", "These predictions open a venue to nucleation of nanoparticles of desirable shapes controlled by the frequency and polarization of a non-absorbed light.", "Useful discussions with A. V. Subashiev, M. Nardone, and D. Shvydka are greatly appreciated." ] ]	1403.0038
[ [ "Polar catastrophe in ultra-thin limit: A case of rare-earth perovskite\n LaNiO3" ], [ "Abstract We address the fundamental issue of growth of perovskite ultra-thin films under the condition of a strong polar mismatch at the heterointerface exemplified by the growth of a correlated metal LaNiO$_3$ on the band insulator SrTiO$_3$ along the pseudo cubic [111] direction.", "While in general the metallic LaNiO$_3$ film can effectively screen this polarity mismatch, we establish that in the ultra-thin limit, films are insulating in nature and require additional chemical and structural reconstruction to compensate for such mismatch.", "A combination of in-situ reflection high-energy electron diffraction recorded during the growth, X-ray diffraction, and synchrotron based resonant X-ray spectroscopy reveal the formation of a chemical phase La$_2$Ni$_2$O$_5$ (Ni$^{2+}$) for a few unit-cell thick films.", "First-principles layer-resolved calculations of the potential energy across the nominal LaNiO$_3$/SrTiO$_3$ interface confirm that the oxygen vacancies can efficiently reduce the electric field at the interface." ], [ "Methods", "Experimental Techniques: LaNiO$_3$ thin films with different thicknesses were grown on high-quality STO (111) and LAO (111) substrates (Crystec, Germany) by pulsed laser interval deposition [27], [38], [40] (laser frequency: 18 Hz).", "In order to avoid additional surface defects on substrate, formed by the chemical treatment for achieving single termination [48], as received substrates were used.", "50 mTorr partial pressure of oxygen was maintained during the growth and all of the grown samples were subsequently post annealed in-situ for 30 min in 1 atm of ultra pure oxygen at growth temperature (670$^\\circ $ C), which was found to be essential to maintain correct oxygen stoichiometry for the (001) oriented LNO/LAO heterostructures[27].", "The films were characterized ex-situ by laboratory-based XRD (Panalytical Xpert Pro MRD [Panalytical, Almelo]).", "Ni $L_{3,2}$ edge and O $K$ edge XAS spectra were taken at room temperature at the 4-ID-C beam line of the Advanced Photon Source at Argonne National Laboratory.", "Electrical d.c. transport characterization was performed on a commercial physical properties measurement system (PPMS) with van der Paw geometry.", "Theoretical Methods: LDA calculations on 15uc STO/$n-$ uc LNO multilayer slabs oriented along the (111) direction were carried out with the Vienna ab initio simulation package (VASP) [49] using projector-augmented waves [50], [51].", "Total energies have been calculated for the stoichiometric structure with a 6x6x1 k-point sampling, with full structural relaxation and dipole corrections.", "The slabs have a 20 Å - 25 Å vacuum in both sides to minimize artificial interactions among the periodic images.", "In computing oxygen vacancies, total electronic energies were calculated by removing one oxygen (for each calculation) from different layers with full structural atomic relaxation and fixed lattice vectors.", "J. C. was supported by DOD-ARO under Grant No.", "0402-17291 and DOE under Grant no.", "0402 81814-21-0000.", "Work at the Advanced Photon Source, Argonne was supported by the U.S. Department of Energy, Office of Science under Grant No.", "DEAC02-06CH11357.", "S. M. thanks M. Hawkridge for the help in XRD measurements and S. B.-L. thanks L. Bellaiche and H. Fufor discussions and funding from Arkansas Biosciences Institute.", "Calculations were carried out at TACC (Stampede, Grant XSEDE TG-PHY090002) and Razor (Arkansas)." ] ]	1403.0149
[ [ "Sources of n-type conductivity in GaInO3" ], [ "Abstract Using hybrid density functional theory, we investigated formation energies and transition energies of possible donor-like defects in GaInO3, with the aim of exploring the sources of the experimentally observed n-type conductivity in this material.", "We predicted that O vacancies are deep donors; interstitial Ga and In are shallow donors but with rather high formation energies (>2.5 eV).", "Thus these intrinsic defects cannot cause high levels of n-type conductivity.", "However, ubiquitous H impurities existing in samples can act as shallow donors.", "As for extrinsic dopants, substitutional Sn and Ge are shown to act as effective donor dopants and can give rise to highly n-type conductive GaInO3; while substitutional N behaviors as a compensating center.", "Our results provide a consistent explanation of experimental observations." ], [ "Introduction", "Transparent conducting oxides (TCOs) are unique materials which combine concomitant electrical conductivity and optical transparency in a single material.", "Thus they currently play an important role in a wide range of optoelectronic devices, such as solar cells, flat panel displays and light emitting diodes.", "[1], [2], [3], [4], [5], [6], [7], [8] The ideal TCOs should have a carrier concentration on the order of 10$^{20}$ cm$^{-3}$ and a band-gap energy above 3.1 eV to ensure transparency to visible light.", "Recently, multicomponent oxide semiconductors have been attracting much attention as new TCOs.", "[9], [10], [11], [8] Monoclinic GaInO$_3$ is a promising TCO due to its excellent optical transmission characteristics.", "[12], [13], [14], [15] It shows a very low optical absorption coefficient on the order of a few hundreds cm$^{-1}$ which is significantly lower those of ITO, ZnO:Al and SnO$_2$ :F in the visible region.", "It has a refractive index of around 1.65 which matches well with that of glass ($\\sim $ 1.5).", "Its experimental band-gap is about 3.4 eV.", "Additionally, it can be well coated on transparent substrate such as glass, fused silica, plastic, and semiconductors.", "Even in polycrystalline sample, the resistivity is comparable to conventional wide-band-gap transparent conductors such as indium tin oxide, while exhibiting superior light transmission.", "Particularly in the blue wave length region of the visible spectrum, it exhibits superior light transmission.", "The high quality n-type GaInO$_3$ samples with conductivities of over 300 ($\\Omega \\cdot $ cm)$^{-1}$ through doping Ge and/or Sn have been experimentally synthesized by Phillips and Minami et al.", "respectively.", "[13], [14] They also observed that the carrier concentrations vary strongly with oxygen partial pressure p(O$_2$ ) and concluded that oxygen vacancy (V$_\\text{O}$ ) might play a key role as a native donor-like defect present in n-type GaInO$_3$ .", "However, it is generally accepted that V$_\\text{O}$ cannot produce free electrons due to their deep donor levels even in high concentrations of V$_\\text{O}$ in many TCOs, such as in ZnO,[16], [17], [18] SnO$_2$[19], In$_2$ O$_3$[20].", "In contrast, ubiquitous H impurities might be responsible for the unintentional n-type conductivity in these materials.", "[22], [23], [24], [25], [26] The structural, bonding, electronic and optical properties of GaInO$_3$ have been investigated in our previous ab-initio studies.", "[21] Despite extremely high n-type conductivity in GaInO$_3$ , to date, the origin mechanism of electron carriers is still unclear.", "Hence, an atomistic detailed understanding on the donor-like intrinsic and extrinsic defects possibly forming in GaInO$_3$ is necessary.", "In the present work, we investigated formation energies and transition levels of intrinsic and extrinsic defects which might be responsible for the n-type conductivity based on the hybrid density functional theory.", "[27], [28], [29] The recent development of hybrid density functional theory can yield the experimental band gap values,[30], [31], [32] and thus provides more reliable description on formation energies and transition levels of defects in semiconductors.", "[17], [18], [19], [20] We demonstrated that (i) O vacancy and interstitial In as well as interstitial Ga are not responsible for the experimentally observed n-type conductivity of GaInO$_3$ ; (ii) incorporation of H, Sn and Ge impurities act as shallow donors, which can provide a consistent explanation of experimental observations.", "(iii) substitutional N on O site acts as a compensating center in n-type GaInO$_3$ .", "The remainder of this paper is organized as follows.", "In Sec.", "II, the details of methodology and computational details are described.", "Sec.", "III presents our calculated formation energies and transition energies of various donor-like defects in GaInO$_3$ .", "Finally, a short summary is given in Sec.", "IV." ], [ "Methodology", "Our total energy and electronic structure calculations were carried out within a revised Heyd-Scuseria-Ernzerhof (HSE06) range-separated hybrid functional[29], [33] as implemented in VASP code.", "[34], [35] In the HSE06 approach, the screening parameter $\\mu $ =0.2 Å$^{-1}$ and the Hartree-Fock (HF) mixing parameter $\\alpha $ =28% which means 28% HF exchange with 72% GGA of Perdew, Burke and Ernzerhof (PBE) [36] exchange were chosen to well reproduce the experimental band gap ($\\sim $ 3.4 eV) of GaInO$_3$ .", "The core-valence interaction was described by the frozen-core projector augmented wave (PAW) method.", "[37], [38] The electronic wave functions were expanded in a plane wave basis with a cutoff of 400 eV.", "The semicore d electrons of both Ga and In atoms were treated as core electrons.", "Test calculations show that the calculated formation energies of defects differ by less 0.1 eV/atom than those of the corresponding configurations in which the d electrons were included as valence electrons.", "As seen in Fig.", "REF (a), the monoclinic GaInO$_3$ which has a c2/m space group is characterized by four lattice parameters: three vectors (a, b and c) and the angle $\\beta $ between a and c lattices.", "[39], [12], [13] Our previous studies predicted that a, b, c and $\\beta $ are 12.96 Å, 3.20 Å, 6.01 Å and 77.89 $^\\circ $ respectively, with a calculated formation energy of -8.43 eV per formula unit.", "[21] The local structure of GaInO$_3$ is shown in Fig.", "REF (b), one can find that all Ga (In) atoms site tetrahedrally (octahedrally) coordinated.", "There are three nonequivalent O atoms and we denote them as O(i), O(ii) and O(iii) respectively.", "The O(i) is threefold coordinated surrounded by two In and one Ga atoms; the O(ii) is also threefold coordinated surrounded by one In and two Ga atoms; while the O(iii) is fourfold coordinated surrounded by three In and one Ga atoms.", "A more detailed discussions regarding the structural and electronic properties of GaInO$_3$ were given in our previous work.", "[21] Figure: (Color online) (a) Schematic polyhedral representation of GaInO 3 _3 conventional cell; (b) local structure of three nonequivalent O atoms.", "Green, blue and red balls represent Ga, In and O atoms respectively.The defective systems were modeled by adding (removing) an atom to (from) a 1$\\times $ 4$\\times $ 2 supercell consisting of 160 atoms.", "A 2$\\times $ 2$\\times $ 2 k-point mesh within Monkhorst-Pack scheme [40] was applied to the Brillouin-zone integrations in total-energy calculations.", "The internal coordinates in the defective supercells were relaxed to reduce the residual force on each atom to less than 0.02 eV$\\cdot $ Å$^{\\text{-1}}$ .", "All defect calculations were spin-polarized.", "To investigate the source of n-conductivity in GaInO$_3$ , the intrinsic donor-like defects, including oxygen vacancy (V$_\\text{O}$ ), interstitial Ga (Ga$_i$ ) and In (In$_i$ ) were considered in the present work.", "As for extrinsic impurities, previous experimental findings have shown that substitutional Sn on In sites (Sn$_\\text{In}$ ) and Ge on Ga sites (Ge$_\\text{Ga}$ ) are effective n-type dopants.", "[13] Additionally, the incorporation of H and N impurities were also explored since they might act as ubiquitous or purposeful impurities during the growth of GaInO$_3$ .", "There are several possible interstitial sites due to the low symmetry of monoclinic structure.", "Here we adopted the most favorable interstitial configuration in the $\\beta $ -Ga$_2$ O$_3$ as discussed in the previous works.", "[41], [42] In the charged-defect calculations, a uniform background charge was added to keep the global charge neutrality of supercell.", "The formation energy of a charged defect is defined as: [43] $\\begin{split}\\Delta E^f_D(\\alpha ,q)=E_{tot}(\\alpha ,q)-E_{tot}(host,0)-\\sum n_{\\alpha }(\\mu _{\\alpha }^{0}+\\mu _{\\alpha }) \\\\+q(\\mu _{e}+\\epsilon _{v})+E_{corr}[q],\\end{split}$ where $E_{tot}(\\alpha ,q)$ and $E_{tot}(host,0)$ are the total energies of the supercells with and without defect.", "n$_\\alpha $ is the number of atoms of species $\\alpha $ added to (n$_\\alpha $ >0) or removed from (n$_\\alpha $ <0) the perfect supercell to create defect.", "$\\mu _{\\alpha }^{0}$ is the atomic chemical potential of species $\\alpha $ which is equal to the total energy of per atom in its most stable elemental phase, namely, $\\alpha $ -Ga, tetragonal-In, $\\alpha $ -Sn, Ge, O$_2$ , H$_2$ and N$_2$ .", "$\\mu _{\\alpha }$ is relative chemical potential referenced to the corresponding $\\mu _{\\alpha }^{0}$ .", "q is the charge state of defect and $\\mu _{e}$ is electron chemical potential in reference to the host valence band maximum (VBM).", "Therefore, $\\mu _{e}$ can vary between zero and the host band-gap E$_g$ .", "The final term accounts for both the alignment of the electrostatic potential between the bulk and defective charged supercells, as well as the finite-size effects resulting from the long-range Coulomb interaction of charged defects in a homogeneous neutralizing background, as outlined by Freysoldt et al.", "[44], using a calculated dielectric constant of 10.2.", "[21] The chemical potential $\\mu _{\\alpha }$ can vary from O-rich to O-poor limits depending on the growth conditions.", "The chemical potentials of Ga, In and O atoms are subject to their lower bounds satisfied by the constraint $\\mu _{\\text{Ga}}$ +$\\mu _{\\text{In}}$ +3$\\mu _{\\text{O}}$ =$\\Delta $H$_f$ (GaInO$_3$ ), where $\\Delta $H$_f$ (GaInO$_3$ ) is the formation energy of GaInO$_3$ .", "They are subject to the upper bounds $\\mu _{\\text{O}}$$\\le $ 0 (O-rich limit), $\\mu _{\\text{Ga}}$ $\\le $ 0 as well as $\\mu _{\\text{In}}$ $\\le $ 0 (O-poor limit).", "In addition, we examined In$_{2}$ O$_3$ and Ga$_{2}$ O$_3$ as limiting phases and found that they do not affect our conclusions.", "For an extrinsic impurity A (A=Ga, Sn, N and N), $\\mu _{\\text{A}}$ is limited by the formation of its corresponding solid (gaseous) elemental phase.", "Additionally, $\\mu _{\\text{A}}$ and $\\mu _{\\text{O}}$ are further limited by the formation of secondary phases A$_m$ O$_n$ , namely, $\\mu _{A}\\le 0, \\\\ \\mu _\\text{In}+\\mu _\\text{Ga}+3\\mu _\\text{O}=\\Delta H_{f}(\\text{GaInO}_{3}), \\\\ m\\mu _\\text{A}+n\\mu _\\text{O}\\le \\Delta H_{f}(\\text{A}_{m}\\text{O}_{n}).$ We take substitutional Sn on In site as an example.", "To avoid the formation of secondary phase SnO$_2$ , $\\mu _{\\text{Sn}}$ +2$\\mu _{\\text{O}}$ $\\le $ $\\Delta $H$_f$ (SnO$_2$ ).", "The O-poor limit (supposing that both In and Sn are rich) is characterized by $\\mu _{\\text{In}}$ =0, $\\mu _{\\text{O}}$ =$\\frac{1}{3}$$\\Delta $H$_f$ (GaInO$_3$ ), and $\\mu _{\\text{Sn}}$ <$\\Delta $H$_f$ (SnO$_2$ )-$\\frac{2}{3}$$\\Delta $H$_f$ (GaInO$_3$ ) as well as $\\mu _{\\text{Sn}}$ <0 (to avoid the segregation of $\\alpha $ -Sn); while the O-rich limit is characterized by $\\mu _{\\text{O}}$ =0, $\\mu _{\\text{In}}$ =$\\Delta $H$_f$ (GaInO$_3$ ), and $\\mu _{\\text{Sn}}$ <$\\Delta $H$_f$ (SnO$_2$ ) as well as $\\mu _{\\text{Sn}}$ <0.", "It is worth mentioning that the HSE06 calculated formation energies of these complexes depend on the HF mixing parameter $\\alpha $ .", "Our calculations show that HSE06 ($\\alpha $ =28%) gives a value of -7.30 eV for $\\Delta $H$_f$ (In$_2$ O$_3$ ); while HSE06 ($\\alpha $ =32%) predicts a value of -9.53 eV,[45] which is quite consistent with the experimental data of -9.60 eV.", "[46] Thus, the available experimental formation energies of A$_m$ O$_n$ , together with HSE06 ($\\alpha $ =28%) calculated $\\mu _{\\alpha }^{0}$ were adopted to determine the stability of various defects.", "In other words, the absolute value of the chemical potential $\\mu _{\\alpha }^{abs}$ is equal to the HSE06 calculated $\\mu _{\\alpha }$ plus the $\\mu _{\\alpha }^{0}$ determined from the experimental formation energies of A$_m$ O$_n$ .", "The defect transition (ionization) energy level $\\epsilon _{\\alpha }$ (q/$\\emph {q}^{\\prime }$ ) is defined as the Fermi-level (E$_\\text{F}$ ) position for which the formation energies of these charge states are equal for the same defect, $\\epsilon _{\\alpha }(q/q^{\\prime })=[\\Delta E^f_D(\\alpha ,q)-\\Delta E^f_D(\\alpha ,q^{\\prime })]/(q^{\\prime }-q).$ Specifically, the defect is stable in the charge state q when the E$_\\text{F}$ is below $\\epsilon _{\\alpha }(q/q^{\\prime })$ , while the defect is stable in the charge state q$^{\\prime }$ for the E$_\\text{F}$ positions above $\\epsilon _{\\alpha }(q/q^{\\prime })$ ." ], [ "Results and discussion", "In semiconductors and insulators, the defect-levels induced by impurities or defects are either located in the band gaps, or resonant inside the continuous host bands.", "Similar to what was done in our previous studies,[47], [48] a semiquantitative model which describes the single particle defect levels for all the considered neutral defects is proposed and displayed in Fig.", "REF , with the aim of determining the possible charge states and sketchily catching the conductive characteristic of various defects in GaInO$_3$ .", "One can find that V$_\\text{O}$ introduces one doubly-occupied level locating around the host middle gap.", "Thus its possible charge states could vary from 0 to 2+, implying that V$_\\text{O}$ is a donor-like defect.", "It is worth mentioning that the positions of defect levels would be changed over the charge state for a given defect.", "The local magnetic moments of V$_\\text{O}^{0}$ , V$_\\text{O}^{1+}$ and V$_\\text{O}^{2+}$ are predicted to be 0 $\\mu _B$ , 1 $\\mu _B$ and 0 $\\mu _B$ respectively, based on the filling of electrons on this defect level.", "This is in good agreement with the calculated findings.", "Considering that the resulting defect level of V$_\\text{O}$ lies deep inside the band gap, V$_\\text{O}$ is expected to be a deep defect and the wave functions of defect states are predicted to be localized around V$_\\text{O}$ and/or its neighbors, showing an atomic-like characteristic.", "These speculations will be confirmed later by investigating the charge-density distribution together with transition energy levels of V$_\\text{O}$ .", "The neutral Ga$_i$ creates two singly-occupied levels in the spin-up component, one singly-occupied and one singly-unoccupied levels in the spin-down component.", "Thus, its possible charge states could range from 1- to 3+.", "However, it is expected that the formation of Ga$_i^{1-}$ (acting as an acceptor) is energetically unfavored as the electron affinity of Ga ion is relatively low.", "A similar behavior is found for In$_i$ due to the same valence electron configuration with Ga$_i$ .", "From Fig.", "REF we see that the neutral Ge$_\\text{Ga}$ , Sn$_\\text{In}$ , H$_\\text{O}$ and H$_{i}$ introduce one singly-occupied defect level above the host conduction band minimum (CBM) independently.", "Since the host CBM is lower in energy than these defect levels, the electrons introduced by these impurities will drop to the CBM and occupy the perturbed conduction states.", "In this case, a delocalized state showing a host-band-like character is created.", "The system consisting of one of the above-mentioned defects has an odd number of total electrons and carries a total magnetic moments of 1 $\\mu _B$ .", "An occupied level resonant inside the bottom of the host conduction band is the signature of a shallow donor that exhibits hydrogenic effective-mass like characteristics.", "[49] This delocalized electron at the CBM is loosely bound to the donor whose core is now in the charge state of 1+.", "We expect that Ge$_\\text{Ga}^{1+}$ , Sn$_\\text{In}^{1+}$ , H$_\\text{O}^{1+}$ and H$_{i}^{1+}$ are energetically favorable when the electron chemical potential $\\mu _{e}$ is below the CBM.", "In other words, these defect will act as donors and be stable in the 1+ charge state for all positions of the Fermi energy E$_\\text{F}$ in the band gap.", "The neutral N$_\\text{O}$ is observed to create three singly-occupied levels above the VBM and one singly-unoccupied level just below the CBM.", "The neutral N$_i$ induces two singly-occupied levels in the spin-up channel above the VBM.", "Interestingly, both N$_\\text{O}$ and N$_i$ introduce several localized states in the host forbidden bands far below the VBM (not shown in Fig.", "REF ).", "These defects states will not be further discussed as they do not contribute to the conductivity of GaInO$_3$ .", "Figure: Semiquantitative single particle defect levels for the neutral intrinsic and extrinsic defects in GaInO 3 _3.", "The filled dots (•\\bullet ) and open dots (∘\\circ ) indicate electrons and holes.", "The ↑\\uparrow and ↓\\downarrow represents spin-up and spin-down components respectively.The calculated formation energies of V$_\\text{O}$ , Ga$_i$ and In$_i$ as a function of electron chemical potential $\\mu _{e}$ are displayed in Fig.", "REF .", "For a given value of $\\mu _{e}$ , only the energetically stable charge state (with the lowest formation energy) of a specified defect is presented.", "The Fermi energies at which the slopes change correspond to the positions of thermodynamic transition levels.", "One can note that the calculated transition levels $\\epsilon $ (2+/0) of V$_\\text{O}$ are located between 1.2 and 1.6 eV below CBM depending on the site of V$_\\text{O}$ .", "This implies that V$_\\text{O}$ acts as a deep donor.", "The V$_\\text{O}^{1+}$ defects are observed to be not stable for any E$_\\text{F}$ position.", "The reason is attributed to the negative-U behavior which lies in the large difference in lattice relaxations between different charge states of V$_\\text{O}$ .", "Taken as a whole, one can find that the formation energies and transition energies of oxygen vacancies on three nonequivalent O sites are lightly different due to their distinct local surroundings.", "The oxygen vacancy on the O(iii) site, henceforth labelled as V$_{\\text{O(iii)}}$ , is the most favorable configuration with a little deeper level of 1.6 eV below CBM.", "The behaviors of the remaining V$_{\\text{O(i)}}$ and V$_{\\text{O(ii)}}$ are almost indistinguishable.", "Our results on oxygen vacancies are similar to those obtained for $\\beta $ -Ga$_2$ O$_3$ .", "[42] In contrast, both Ga$_i$ and In$_i$ act as shallow donors with $\\epsilon $ (1+/0) ionization energies of around 0.1 eV and 0.2 eV above CBM respectively.", "However, their formation energies are more than 2.5 eV even under n-type conditions, in the most favorable O-poor limit.", "This suggests that the concentration of Ga$_i$ and In$_i$ should be negligible under equilibrium growth conditions.", "Based on these calculated results, we conclude that the native donor-like defects could not explain the origin of n-type conductivity in GaInO$_3$ .", "On the other hand, note that oxygen vacancies and cation interstitial defects are energetically stable in the 2+ and 3+ charge states respectively, with the calculated formation energies as low as -3.0 eV when the E$_\\text{F}$ is close to the VBM under O-poor growth conditions.", "This means that the concentrations of these native donors are high enough to certainly compensate the p-type conductivity of GaInO$_3$ that one wants to create.", "The formation of these hole-compensating defects can be suppressed by growing in the O-rich limit.", "More advanced experimental methods, such as nonequilibrium growth techniques may further minimize self-compensation effects in p-type doping GaInO$_3$ .", "Figure: (Color online) Formation energies of Ga i _\\text{i}, In i _\\text{i} and three nonequivalent O vacancies (V O _\\text{O}) in GaInO 3 _3 as a function of Fermi level under (a) extreme oxygen-rich and (b) extreme oxygen-poor conditions.", "For the V O _\\text{O}, three nonequivalent O vacancies are labeled as V O(i) _\\text{O(i)}, V O(ii) _\\text{O(ii)} and V O(iii) _\\text{O(iii)}.", "The VBM is set to zero.As for extrinsic impurities, previous experimental findings have shown that Sn and Ge prefer occupying the In and Ga sites respectively.", "[13] This is attributed to the close ionic radii of Sn (Ge) with In (Ga).", "The calculated shallowest transition levels are $\\epsilon $ (1+/0)=3.4 eV for Ge$_\\text{Ga}$ and $\\epsilon $ (1+/0)=3.5 eV for Sn$_\\text{In}$ , lying just above the CBM.", "This means that both substitutional Sn on In sites and substitutional Ge on Ga sites are highly effective donor dopants, especially the former with a calculated formation energy of around -1.5 eV under n-type and O-rich conditions.", "Our findings are in good agreement with experiments which indicate that both In and Ge can significantly enhance the n-conductivity in GaInO$_3$ samples.", "[13] The transition energies $\\epsilon $ (0/1-) of substitutional N defects on three nonequivalent O sites are higher than 1.5 eV above the VBM.", "Clearly, N$_\\text{O}$ is a deep acceptor and will not enable p-type conductivity in GaInO$_3$ .", "A very similar behavior has been reported for the N dopant in ZnO.", "[50] However, we note that N$_\\text{O(i)}$ is stable in the 1- charge state with a calculated formation energy of around 0.8 eV for the E$_\\text{F}$ near the host CBM.", "It is also found that N$_\\text{O}$ defects have formation energies comparable to, or even lower than those of Ge$_\\text{Ga}$ and Sn$_\\text{In}$ under O-rich conditions but becomes energetically less favorable under O-rich conditions.", "This suggests that N$_\\text{O}$ can act as an electron killer and compensate the n-type conductivity of GaInO$_3$ .", "This explains the experimentally observed the decrease trend on the electronic conductivity when the samples were annealed in nitrogen partial pressure.", "By comparison, N$_i$ is always energetically stable in the neutral state with a rather high formation energy of 6.5 eV, regardless of the position of the E$_\\text{F}$ .", "This indicates that N$_i$ is electrically inactive and its concentration should be negligible under equilibrium conditions.", "As shown in Fig.", "REF , all H$_\\text{O}$ defects in the different configurations act as shallow donors with the (1+/0) thermodynamic transition levels very close to the CBM.", "Their formation energies are lower under O-poor conditions than under O-rich ones, explaining that the concentration of H$_\\text{O}$ in the samples increases with the decrease of oxygen partial pressure p(O$_2$ ) during growth.", "[13], [14] It is also expected that H impurities will fall into the oxygen vacancy sites, and thus H$_\\text{O}$ can enhance the electrical conductivity of GaInO$_3$ under oxygen reducing (O-poor) conditions.", "Besides, H$_i$ is observed to be energetically more stable than H$_\\text{O}$ , yielding a transition level (+1/0)=3.5 eV, just above the CBM.", "Hence H$_i$ also behaves exclusively as a shallow donor for any E$_\\text{F}$ value ranging from the VBM to the CBM.", "In consideration of the fact that both H$_i$ and H$_\\text{O}$ serve as shallow donors, the post growth annealing in hydrogen partial pressure could help to reduce the resistivity of n-type GaInO$_3$ samples, as was observed in experiments.", "Nevertheless, one can find that the positively charged H impurities have formation energies of less than 0.8 eV for the E$_\\text{F}$ close to VBM, implying that H impurities might act as hole compensating centers in acceptor-doped GaInO$_3$ .", "Figure: (Color online) Formation energies of Ge, Sn, H and N impurities in GaInO 3 _3 as a function of Fermi level under (a) extreme oxygen-rich and (b) extreme oxygen-poor conditions.", "The VBM is set to zero.To gain a deeper understanding of the defect states from the real space point of view, we take H$_\\text{O}$ and V$_\\text{O}$ as examples and plot the wavefunction squared of defect levels induced by them.", "As we have discussed above, H$_\\text{O}$ is a typical shallow donor.", "From the results depicted in Fig.", "REF we see that the wave functions of H$_\\text{O}$ defect states distribute over all O atoms, showing O-2s like and rather delocalized characteristics.", "The CBM of GaInO$_3$ was observed to be mainly derived from O-2s states.", "[21] This confirms that the defect level of H$_\\text{O}$ is resonant inside the bottom of the conduction band as schematized in Fig.", "REF .", "As expected, the wave functions of V$_\\text{O}$ mainly localize at the oxygen vacancy and its seven next nearest-neighbor oxygen atoms, showing a highly-localized characteristic as V$_\\text{O}$ is a deep donor.", "A similar behavior has been reported for V$_\\text{O}$ in ZnO.", "[18] Figure: (Color online) Wavefunction squared for H O _\\text{O} acting as a typical shallow donor in the charge states of (a) 0, (b) 1+.", "Wavefunction squared for V O _\\text{O}, an example of the typical atomic-like deep donor in the charge states of (c) 0 and (d) 2+.", "The charge density isosurfaces are shown at 20% of their maximum value.", "Green, blue, red and black balls represent Ga, In, O and H atoms respectively.Considering that the formation energies of charged defects depend on the position of the E$_F$ in the host band gap which is sensitive to the choice of HF mixing parameter $\\alpha $ , we take V$_\\text{O}$ and N$_\\text{O}$ as examples to investigate the roles of $\\alpha $ in their stability and conductivity.", "From the results reported in Fig.", "REF (a), we find that the calculated formation energies of V$_\\text{O}$ differ less than 0.4 eV for $\\alpha $ =15% and 28%.", "In contrast, the magnitude of band gap significantly decreases from 3.4 eV for $\\alpha $ =28% to 2.5 eV for $\\alpha $ =15%.", "The underestimation of band gap using HSE06 ($\\alpha $ =15%) method leads to shallower transition levels of $\\epsilon $ (0/1-)=0.99 eV for N$_\\text{O}$ and $\\epsilon $ (2+/0)=1.49 eV for V$_\\text{O}$ when referred to the corresponding calculated VBM.", "To have a better understanding of the origins of these observed trends in the transition levels, we plot the transition levels on an absolute energy scale, e.g., referenced to the vacuum level, in Fig.", "REF (b).", "we note that $\\alpha $ has almost negligible effects on the transition levels of V$_\\text{O}$ and N$_\\text{O}$ defects, with respect to the vacuum level.", "Nevertheless, the host VBM moves upward, while the CBM moves downward.", "Consequently, the host band gap decreases along with $\\alpha $ .", "In addition, the magnitude of band offset on valence band is found to be more significantly than that on conduction band.", "This implies that the transition levels of acceptors should be more sensitive to the choice of $\\alpha $ than those of donors in GaInO$_3$ .", "In a word, the transition levels of acceptors and donors become shallow when reducing the value of $\\alpha $ from 28% to 15%.", "The rigid shifts of the host VBM and CBM are primarily responsible for the shallower transition levels which are calculated by using HSE06 ($\\alpha $ =15%) approach.", "Figure: (Color online) (a) Formation energies of substitutional N and O vacancy as a function of Fermi level under extreme oxygen-rich condition, and (b) transition energy levels referenced to the vacuum level using HSE06 (α\\alpha =28%) and HSE06 (α\\alpha =15%) methods respectively.", "The blue region represents the HSE06 (α\\alpha =15%) calculated band gap." ], [ "Summary", "In summary, we performed first-principles calculations based on hybrid density functional theory to systematically explore the behaviors of possible donor-like defects which are ubiquitous or deliberately incorporated into GaInO$_3$ during the synthesis processes.", "We found that the native defects including O vacancies and interstitial Ga as well as interstitial In cannot contribute to the n-type conductivity of GaInO$_3$ as they are either deep donors or have negligible concentrations.", "In contrast, our results suggest that Ge, Sn and H impurities act as shallow donors with low formation energies and they are most likely the sources of the n-type conductivity observed in the experiments; while substitutional N acts as a compensating center in n-type GaInO$_3$ .", "We thank Profs.", "Wen-Tong Geng and Yu-Jun Zhao for providing valuable suggestions.", "Dr. Wang acknowledges the support of the Natural Science Foundation of Shaanxi Province, China (Grant No.", "2013JQ1021).", "Prof. Kawazoe is thankful to the Russian Megagrant Project No.14.B25.31.0030 “New energy technologies and energy carriers” for supporting the present research.", "The calculations were performed on the HITACHI SR16000 supercomputer at the Institute for Materials Research of Tohoku University, Japan." ] ]	1403.0218
[ [ "Universal wave functions structure in mixed systems" ], [ "Abstract When a regular classical system is perturbed, non-linear resonances appear as prescribed by the KAM and Poincar\\`{e}-Birkhoff theorems.", "Manifestations of this classical phenomena to the morphologies of quantum wave functions are studied in this letter.", "We reveal a systematic formation of an universal structure of localized wave functions in systems with mixed classical dynamics.", "Unperturbed states that live around invariant tori are mixed when they collide in an avoided crossing if their quantum numbers differ in a multiple to the order of the classical resonance.", "At the avoided crossing eigenstates are localized in the island chain or in the vicinity of the unstable periodic orbit corresponding to the resonance.", "The difference of the quantum numbers determines the excitation of the localized states which is reveled using the zeros of the Husimi distribution." ], [ "Universal wave functions structure in mixed systems Diego A. Wisniacki Departamento de Física and IFIBA, FCEyN, UBA Ciudad Universitaria, Pabellón 1, Ciudad Universitaria, 1428 Buenos Aires, Argentina.", "When a regular classical system is perturbed, non-linear resonances appear as prescribed by the KAM and Poincarè-Birkhoff theorems.", "Manifestations of this classical phenomena to the morphologies of quantum wave functions are studied in this letter.", "We reveal a systematic formation of an universal structure of localized wave functions in systems with mixed classical dynamics.", "Unperturbed states that live around invariant tori are mixed when they collide in an avoided crossing if their quantum numbers differ in a multiple to the order of the classical resonance.", "At the avoided crossing eigenstates are localized in the island chain or in the vicinity of the unstable periodic orbit corresponding to the resonance.", "The difference of the quantum numbers determines the excitation of the localized states which is reveled using the zeros of the Husimi distribution.", "05.45.Mt, 03.65.Sq Introduction: Hamiltonian classical systems have a variety of dynamical behaviors [1].", "On one side are integrable systems with conserved quantities as degrees of freedom resulting in constrained dynamics around invariant tori.", "On the other extreme, chaotic systems are characterized by properties of mixing and ergodicity.", "The phase space is dynamically filled and only constrained by the conservation of the energy.", "The quantum mechanics of these dynamical systems have been intensively studied in the last forty years and the correspondence between classical and quantum mechanics has been established with solid grounds [2], [3].", "It is unusual that a generic system belongs to those extreme cases as it would display mixed dynamics where chaos coexist with regions of regular motion.", "The dynamics of mixed systems are more subtle mainly because regular and chaotic regions are connected by fractal boundaries.", "A standard way to understand this complex dynamics is perturbing an integrable system.", "Its response to weak perturbations has been completely understood in terms of the celebrated Kolmogorov- Arnold-Moser (KAM) and the Poincarè-Birkhoff (PB) theorems [4], [5].", "The KAM theorem states that depending on the rationality of the frequencies of the motion, some of the invariant tori are deformed and survive while others are destroyed.", "The consequence of the fate of rational tori is the survival of an equal even number of stable (elliptic) and unstable (hyperbolic) periodic orbits (PB theorem).", "In their vicinity of an stable orbit, a chain of islands of regularity surrounded by a chaotic sea are developed [1].", "This classical structures, usually called nonlinear resonance, have important influences in various phenomena from chemical systems, solid state physics to nano optics [6], [7], [8].", "Almost ubiquitous, the quantum mechanics of nearly regular and mixed systems is much less known than the integrable or chaotic cases.", "The relation between nonlinear resonances and avoided crossings (ACs) observed in the spectra of quantum systems has been a subject of several studies in the past [9].", "Using a semiclassical approach, nonlinear resonance were shown to be responsible for energy levels approach each other closely, exhibiting avoided crossings, instead of crossings as happen in integrable systems [10], [11].", "More recently, Brodier, Schlagheck, and Ullmo [12], have developed a semiclassical theory of resonance assisted tunneling showing the coupling interaction term between states localized in invariant tori.", "Interesting enough, a selection rule emerges because the interaction only occurs between states with quantum number that differ in a multiple of the order of the resonance.", "So, it is expected that the appearance of a nonlinear resonance is revealed in the quantum spectra with series of ACs of states localized in tori with quantum number that fulfil the selection rule.", "In fact, this has been shown in Ref.", "[13] when the multiple is equal to one.", "In this letter we go one step forward and show such a systematic for ACs between states with quantum number that differ in a multiple greater that one.", "We compute the series of ACs corresponding to a resonance of a paradigmatic system of quantum chaos studies and show that the states in the center of AC have a surpassing structure: one is an excited state localized the island chain and the other in the associated periodic orbit.", "This shows that a classical nonlinear resonance imprints clear signatures in the wave function morphologies.", "The eigenstates at the AC are carefully analyzed using the Husimi distribution in phase space and their zeros [14].", "The excitations of the localized structure are related with the number of zeros in each island or in the vicinity of the unstable periodic orbit (PO) associate to the resonance.", "The systematic of ACs generated by a nonlinear resonance is a fertile field to test the building block of the semiclassical theory of resonance assisted tunneling, that is, the interaction coupling term between states localized in invariant tori [12].", "For these reason, we have studied the behavior of the series of ACs varying the value of the Planck constant $\\hbar $ .", "We clearly show an unexpected result that the semiclassical expression for the interaction coupling term works better in the deep quantum regime seeing signs of an improvement of this theory seems necessary.", "The model: We study the Harper map in the unit square as a model system, $p_{n+1} & = & p_n + k \\sin (2\\pi q_n) \\qquad ({\\rm mod}\\; 1), \\nonumber \\\\q_{n+1} & = & q_n - k \\sin (2\\pi p_{n+1}) \\qquad ({\\rm mod}\\; 1),$ where $k$ is a parameter that measures the strength of the perturbation.", "This map can be understood as the stroboscopic version of the flow corresponding to the (kicked) Hamiltonian $H(p,q,t) = - \\frac{k}{2\\pi } \\cos (2\\pi p) - \\frac{k}{2\\pi } \\cos (2\\pi q)\\sum _n \\delta (t-n).$ The Harper map comprises all the essential ingredients of mixed dynamics and is extremely simple from a numerical point of view.", "For very small $k $ , the dynamics described by the map is essentially regular, that is, the phase space is covered by invariant tori.", "As $k$ gets bigger, non linear resonances (islands) start to appear following the KAM and PB theorems.", "The system presents a mixed dynamics with regions of regularity around the origin and the corners coexisting with chaos as shown in the bottom panels of Fig.", "REF and REF .", "For $k> 0.63$ there are no remaining visible regular islands due to chaotic dynamics cover all the phase space.", "The quantum mechanics of the Harper map is described by the unitary time-evolution operator [16], [17] $\\hat{U_k} = \\exp [{\\rm i}N k \\cos (2\\pi \\hat{q})] \\; \\exp [{\\rm i}N k \\cos (2\\pi \\hat{p})],$ with $N=(2\\pi \\hbar )^{-1}$ , that is, a Hilbert space of $N$ dimensions for a fixed value of $\\hbar $ .", "This is due to the quantization on the torus which implies that the wave function should be periodic in both position and momentum representations.", "The semiclassical limit is reached as $N$ takes increasing values.", "For a fixed value of $N$ , the spectrum of eigenphases $\\phi _i(k)$ and eigenfunctions $\|\\psi _i(k)\\rangle $ of the evolution operator of the quantum map are obtaining by diagonalization of Eq.", "REF .", "The characteristics of $\|\\psi _i(k)\\rangle $ are analyzed using the Husimi distribution [18].", "The Husimi representation of an eigenstate of a quantum map is a quasiprobablity distribution in phase space that has exactly $N$ zeros in the unite square [14].", "Eigenphases $\\phi _i(k)$ change linearly for very small strength of the perturbation $k$ and the Husimi distribution of the eigenstates $\|\\psi _i(k)\\rangle $ are localized in the vicinity of invariant tori [16], [14].", "Eigenstates with negative slope are centered in $(q,p)=(1/2,1/2)$ and in $(q,p)=(0,0)$ for positive slope.", "A bigger absolute value of the slope implies that the state is nearer to the periodic point $(q,p)=(1/2,1/2)$ or $(q,p)=(0,0)$ .", "The states with maximun absolute value of the slope resemble a gaussian distribution centered in the mentioned periodic points and corresponds to label 1 (negative slope) and $N$ (positive slope).", "Exited states has $n$ zeros inside the region of maximum probability, being $n+1$ its label for negative slope and $ (N-(n+1)$ for positive slope.", "Method: The influence of a non-linear resonance r:s to quantum maps is uncovered using the following numerical procedure.", "We consider a serie of eigenstates $\|\\phi _i(k)\\rangle $ with $i=1,...i_{max}$ ($i_{max}< N/2$ ) for very small perturbation $k=k_0$ .", "and we associate for each state $i$ a perturbed one with $k=k_0+\\delta k$ , if the overlap $\\langle \\phi _i(k_0) \|\\phi _j(k_0+\\delta k)\\rangle $ is the maximum of all $j$ .", "Then, this procedure is repeated for perturbations $k=k_0+n \\delta k$ with $n=2,...n_{max}$ an integer.", "Thus, we have associated a serie of perturbed states and eigenphases with the unperturbed one.", "From now on, we refer quasistate $i$ to the serie of perturbed states associated to $\|\\phi _i(k_0)\\rangle $ and quantum number to the label $i$ of the quasistate.", "If we join lines through the eigenphases of each quasistate, we can establish where two of the quasistates have a crossing.", "In the vicinity of that intersection we find an AC of eigenstates that has the localized properties of the states at $k=k_0$ .", "If the dimensions $N$ of the Hilbert space is small enough, the previous procedure can be done by visual inspection of the spectra as a function of the perturbation $k$ .", "The value of $\\delta k$ is crucial for the success of the procedure: if it is small and for an $n$ , $k=k=k_0+n \\delta k$ falls close to an AC, the quasistate loses the localization properties related with the unperturbed state and the method fails.", "On the contrary, if $\\delta k$ is very large the method also fails because the phase of the quasistate has an erratic development and hence it is not possible to find its crossing with other quasistates.", "Once we have computed the quasistates for a serie of unperturbed eigenstates $\|\\phi _i(k_0)\\rangle $ , we obtain the position of the corresponding ACs looking at the intersection of the eigenphases $i$ with $j$ .", "If we are considering a non-linear resonance r:s with $r$ the number of islands, we compute the series of ACs for quasistates with quantum numbers $i$ and $j$ that $\\Delta n=\|i-j\|=lr$ with $l$ an integer.", "In Ref.", "[13] the series of AC for two different nonlinear resonances of the Harper map were found for $l=1$ using visual inspection of the spectra.", "This was feasible due to the small value of the dimension of the Hilbert space $N$ .", "In the following, we show that using the described method it is possible to find the series of ACs for greater $l$ and $N$ .", "Figure: (Color online) (a) and (c) Husimi distribution of the states at the center of an AC obtained from the intersection ofquasistates 9 and 15 for a Hilbert space with N=160N=160.", "The zeros of the distribution are plotted with ×\\times and ◯\\bigcirc .", "In the inset (middle) of (a) and (c) the region of the spectra where the AC take place is plotted.", "The states before and after the AC are labeled A, B, C, D and are also plotted in the insets of (a) and (c).", "In (e) the zeros of the Husimi distributions (a) and (c) are plotted with the classical phase space at k=0.1822k=0.1822.", "(b) and (d) Husimi distribution of the states at the center of the AC obtained from the intersection of quasistates 19 and 25 for N=300N=300.", "In (f) the zeros of the Husimi distributions (d) and (d) are plotted with a classical phase space at k=0.1833k=0.1833 In the inset of (d)-((f) a blow up of a part of panel (f) is plotted to see the location of the zeros of the Husimi distribution in more detail.", "Results: The Harper map is a mixed system that has an usual regular to chaotic transition.", "As the perturbation strength $k$ grows, non-linear resonances get bigger as the surrounding chaotic layers and eventually disappear covered by the chaotic sea.", "As can be seen in the classical phase space showed in the bottom panels of Fig.", "REF and REF , the resonaces 6:1 reach the largest size [see also Ref.", "[13], [12]].", "Other resonances as 8:1, 10:1 and 14:1 take up an appreciable regions of phase space.", "Our main goal is to disentangle the influence of non-linear resonance to the egenfunctions of a mixed system.", "We focus on the resonance 6:1 of the Harper map.", "Using the method described below, we have found the ACs associated to the intersections of the eigenphases of quasistates with quantum number that differs in a multiple of 6, the order of the resonance 6:1, that is, for $\\Delta n=6l$ , with $l=1,2$ and 3.", "The calculations are done for three sizes of the Hilbert space $N=80$ , 160 and 300.", "As an examples, in Fig.", "REF and REF we show the morphologies of the eigenstates in these ACs.", "In Fig.", "REF the behavior of the wave functions in vicinity of an AC obtained from quasistates with quantum number that differ in 6 is exhibited.", "Left panels corresponds to a Hilbert space with $N=160$ and right panels to $N=300$ .", "In the insets of the left panels of Fig.", "REF we can see the Husimi distributions of eigenfunctions before and after the AC that was obtained from the intersection of quasistates 9 and 15.", "In the central inset of Fig.", "REF (a) and (c) we show the region of the spectra where the AC take place.", "The eigenphases of the states that have the AC are plotted in red lines.", "As it is usual, the states exchange their distributions upon AC [$(A) \\leftrightarrow (D)$ and $(B) \\leftrightarrow (C)$ ].", "But surprising enough, the distributions at the center of the AC are highly localized.", "One state has the maximum of the probability of the Husimi distribution in the vicinity of the island chain and have 6 zeros near the corresponding unstable PO [see Fig.", "REF (a)].", "The other state [Fig.", "REF (c)], is localized in the vicinity of the unstable PO and one zero in the center of each island is observed.", "This is better displayed in Fig.", "REF (e) where a part of the classical phase space and the zeros of the Husimi distribution of states of Fig.", "REF (a) and (c) are shown.", "The zeros of the spates (a) and (b) are plotted with $\\times $ and with $\\bigcirc $ for (c) and (d).", "In the right panels of Fig.", "REF , we show a similar example for $N=300$ and quasistates 19 and 25.", "A blow up of the vicinity of an islands of the resonance $6:1$ is shown in the inset in the middle of panels (d) and (f) of Fig.", "REF .", "When the AC is between states with quantum number that differ in a multiple (greater that one) of the number of islands in the chain, the morphologies of the eigenstates are more impressive.", "As an example of this phenomenon, in Fig.", "REF we show the Husimi distributions at the center of AC between states with quantum numbers that differ in 12 [Fig.", "REF (a),(c) and (e)] and 18 [Fig.", "REF (b), (d) and (f)].", "As can be seen in the cases displayed in Fig.", "REF , one of the states is localized in the island chain, but in Fig.", "REF (a) there is one zero inside each island and two zeros for Fig.", "REF (b).", "This fact point out that these states are excited states of the island chain.", "The other states, Fig.", "REF (c) and (d) are localized in the corresponding unstable PO and the zeros are accumulated inside the islands.", "This fact resemble the behavior of scar functions builded for chaotic system [20], [19].", "To see the location of the zeros in more detail, in Fig.", "REF (e) and (f) we plot a part of the classical phase space and the zeros of the Husimi distributions of panels (a)-(c) and (b)-(d).", "We have seen that when $\\Delta n=24$ the structure of the wave functions at the AC have the same systematic with one more zero in each island of the chain.", "In summary, in the center of an AC with $\\Delta n=6 l$ , one of the states is localized in the vicinity of the island chain and has $l-1$ zeros in each island, whereas the other state has the maximum probability around the unstable PO of the resonance and has $l$ zeros in each island.", "It is important to note that in Figs.", "REF and REF only the zeros of the Husimi distribution that are inside or over the regions of maximum probability are plotted.", "Other zeros that lie outside these regions and have exponential small probability are not displayed.", "Figure: (Color online)Husimi distribution of the eigenstates at the center of an AC obtained with the intersection of quasistates 15 and 27 [(a) and (c)] and 26 and 44 [(b) and (d)] .", "The zeros of the distribution are also plotted with ×\\times and ◯\\bigcirc .", "In (e) and (f) the zeros of the Husimi distributions [(a), (c) and (b), (d)] are plotted with the corresponding part of the classical phase space at k=0.1809k=0.1809.", "and k=0.1978k=0.1978.The semiclassical theory of resonance assisted tunneling predicts a coupling strength between quasi modes located on opposite sides of a nonlinear resonance and therefore an eigenphases difference $\\Delta \\phi $ for the ACs considered before.", "This theory was recently developed and applied in several situations [12], [21], [22] .", "The starting point is the classical secular perturbation theory which allows to construct an effective time independent Hamiltonian that describes the local dynamics near a r:s resonance of the map, $H_{r:s} \\simeq H_0(I_{r:s})+\\sum _{l=1}^{\\infty } V_{r,l}(I_{r:s})\\cos (l r \\theta +\\phi _l)$ with $H_0(I_{r:s})$ an integrable approximation of the Hamiltonian of the map [12].", "This effective Hamiltonian entails a selection rule that an eigenstate of the unperturbed Hamiltonian of a quantum number $n$ can be coupled to another state of a quantum number $n + lr$ (l an integer) with a strength proportional to $V_{r,l}$ .", "$V_{r,l} e^{i \\varphi _l} = \\frac{1}{i\\pi rs \\tau }\\int _0^{2\\pi } \\exp (-i r l \\theta ) \\; \\delta I_{r:s}(\\theta ) d\\theta $ $\\delta I_{r:s}(\\theta )$ being given by $\\delta I_{r:s}(\\theta ) = I^{(-1)}(I_{r:s},\\theta )-I_{r:s},$ where $I^{(-1)}(I_{r:s},\\theta )$ is the action variable obtained by applying the inverse Poincaré map to $(I,\\theta )$ .", "The interaction coupling strength [Eq.", "REF ] is numerically computed following the next setps.", "First, the resonant periodic torus is found, and its action $I_{r:s}$ calculated.", "Then, it is computed a large number of points $(q_i,p_i)$ and its corresponding angle variable $\\theta _i$ belonging to the resonant tours .", "We applied a back-propagation for these points with the exact inverse map.", "The associated perturbed action of each point $I^{(-1)}(I_{r:s},\\theta _i)$ is computed by numerical propagation in a complete cycle.", "Using these quantities the coupling interaction $V_{r,l}$ is calculated with Eq.", "REF .", "Finally, a semiclassical approximation of the eigenphases differences $\\Delta \\phi $ produced by a resonace $r:s$ for AC that comes from the intersection of quasistates eigenphases with $\\Delta n=l r$ results $\\Delta \\phi \\approx \|V_{r,l}\|/2 \\hbar .$ Figure: (Color online:) Eigenphase difference Δφ\\Delta \\phi (scaled with ℏ\\hbar ) as a function of the perturbation strength kk for ACs associated withthe nonlinear resonance 6:1 of the Harper map.", "Symbols corresponds to ACs obtained from the quantum spectra.For Δn=6\\Delta n=6 it is used ×\\times for N=80N=80,□\\Box for N=160N=160 and ▪\\blacksquare for N=300N=300.", "Δn=12\\Delta n=12 is plotted with⋄\\Diamond for N=80N=80, △\\bigtriangleup for N=160N=160 and ▴\\blacktriangle for N=300N=300.", "In the inset Δφ\\Delta \\phi corresponds to ACs with Δn=18\\Delta n=18 with△\\bigtriangleup for N=80N=80, ◯\\bigcirc for N=160N=160 and □\\square for N=300N=300.The semiclassical prediction of the eigenphase difference Eq.", "is plotted with lines:m=1m=1 solid, m=2m=2 dashed and m=3m=3 dotted (in the inset).We have computed the semiclassical approximation of the eigenphase difference for the resonance 6:1 with $l=1,2$ and 3 as a function of the strength $k$ .", "The integral was done using the 7-point Newton-Cotes formula and with an integrable approximation of the Harper Hamiltonian [Eq.", "(REF )] up to 5th order that was obtained using the Baker-Campbell-Hausdorff formula in Eq.", "REF and the semiclassical relation between quantum commutator and the Poisson brackets [23].", "In Fig.", "REF , the semiclassical approximation of eigenphase difference $\\Delta \\phi $ is plotted with lines ($l=1$ solid, $l=2$ dashed and $l=3$ dotted).", "The eigenphases differences $\\Delta \\phi $ for the ACs between states with $\\Delta n=6l$ , with $l=1,2$ and 3 are plotted with symbols.", "These eigenphases differences were obtained from the quantum spectra using the method presented before for three values of the number of states of the Hilbert space $N=80$ , 160 and 300.", "In Fig.", "REF we can see that, contrary to the expectations, the semiclassical approximation works very nice for all $l$ only in the case of $N=80$ , the minimum value of the considered number of states of the Hilbert space.", "This fact indicates that semiclassical expression of $\\Delta \\phi $ [Eq.", "REF ] works in the deep quantum regime and separates from quantum results as $N$ increase.", "The discrepancies for large $N$ is becoming an strong evidence that the semiclassical theory of the interaction coupling needs an improvement [12].", "We note that similar behavior was observed in the tunneling-induced level splittings of a very simple one dimensional model computed with the semiclassical theory of resonance assisted tunneling[22].", "In Fig.", "REF we show an unexpected scaling of the $\\Delta \\phi $ for $l=2$ and $l=3$ .", "Eigenphase difference scales with $\\hbar ^3$ for $l=2$ and with $\\hbar ^5$ for $l=3$ .", "This could indicate the existence of an effective $\\hbar $ that depends on $l$ and could be a clue in the improvement of the semiclassical theory of interaction coupling strength [Eq.", "REF ] .", "Figure: (Color online:) Scaled eigenphase difference Δφ\\Delta \\phi as a function of the perturbation strength kk for ACswith Δn=12\\Delta n=12 and Δn=18\\Delta n=18.", "The scaling factor is ℏ 3 \\hbar ^3 forΔn=12\\Delta n=12 and ℏ 5 \\hbar ^5 for Δn=18\\Delta n=18.", "Symbols are the same as Fig.", "Final Remarks: In this letter we have shown that a classical non linear resonance imprints a systematic influence in the quantum eigenvalues and eigenfunctions of a mixed system.", "We have found an universal structure embedded in the spectra: states localized in tori interact in AC if the quantum numbers differ in a multiple of the order of the resonance.", "These series of AC are observed when a parameter of the system is varied producing the development of the resonance characterized by a chain of islands.", "Surprisingly, eigenstates in the middle of the AC has a particular morphology.", "One state is localized in the vicinity of the unstable PO associated to the resonance.", "The other state is localized on the island chain.", "The difference of the quantum numbers of the unperturbed states that are localized in tori and interact in the AC determines the distribution of the zeros of the Husimi function of the states.", "These findings could be of importance in the design of optical micro cavities[15], [24], [25].", "In those devices it is desirable to obtain a specific directional light emission that could be accomplished tuning a system parameter to reach to an AC where the states have a desirable localization.", "We have compared the eigenphases gaps of the AC with a semiclassical prediction based on the theory of resonance assisted tunneling [12].", "We have shown that the semiclassical prediction deviate from the quantum results as we reach the semiclassical limit.", "This unexpected result indicate that an improvement of the this theory is needed.", "We thank Marcos Saraceno for stimulating discussions.", "Financial support from from ANCyPT (PICT 2010-1556), UBACyT, and CONICET is acknowledged." ] ]	1403.0275
[ [ "The Role of Feedback in Shaping the Structure of the Interstellar Medium" ], [ "Abstract We present an analysis of the role of feedback in shaping the neutral hydrogen (HI) content of simulated disc galaxies.", "For our analysis, we have used two realisations of two separate Milky Way-like (~L*) discs - one employing a conservative feedback scheme (MUGS), the other significantly more energetic (MaGICC).", "To quantify the impact of these schemes, we generate zeroth moment (surface density) maps of the inferred HI distribution; construct power spectra associated with the underlying structure of the simulated cold ISM, in addition to their radial surface density and velocity dispersion profiles.", "Our results are compared with a parallel, self-consistent, analysis of empirical data from THINGS (The HI Nearby Galaxy Survey).", "Single power-law fits (P~k^gamma) to the power spectra of the stronger-feedback (MaGICC) runs (over spatial scales corresponding to 0.5 kpc to 20 kpc) result in slopes consistent with those seen in the THINGS sample (gamma = -2.5).", "The weaker-feedback (MUGS) runs exhibit shallower power law slopes (gamma = -1.2).", "The power spectra of the MaGICC simulations are more consistent though with a two-component fit, with a flatter distribution of power on larger scales (i.e., gamma = -1.4 for scales in excess of 2 kpc) and a steeper slope on scales below 1 kpc (gamma = -5), qualitatively consistent with empirical claims, as well as our earlier work on dwarf discs.", "The radial HI surface density profiles of the MaGICC discs show a clear exponential behaviour, while those of the MUGS suite are essentially flat; both behaviours are encountered in nature, although the THINGS sample is more consistent with our stronger (MaGICC) feedback runs." ], [ "Introduction", "The feedback of energy into the interstellar medium (ISM) is a fundamental factor in shaping the morphology, kinematics, and chemistry of galaxies, both in nature and in their simulated analogues [29], [12], [23], [13], [4], [22], [8], [14].", "Perhaps the single-most frustrating impediment to realising accurate realisations of simulated galaxies is the spatial `mismatch' between the sub-pc scale on which star formation and feedback operates, and the 10s to 100s of pc scale accessible to modellers within a cosmological framework.", "Attempts to better constrain `sub-grid' physics, on a macroscopic scale, have driven the field for more than a decade, and will likely continue to do so into the foreseeable future.", "The efficiency and mechanism by which energy from massive stars (both explosive energy deposition from supernovae and pre-explosion radiation energy), cosmic rays, and magnetic fields couple to the ISM can be constrained indirectly via an array of empirical probes, including (but not limited to) stellar halo [3] and disc [20] metallicity distribution functions, statistical measures of galaxy light compactness, asymmetry, and clumpiness [13], stellar disc age-velocity dispersion relations [15], rotation curves and density profiles of dwarf galaxies [17], low- and high-redshift `global' scaling relations [4], background QSO probes of the ionised circum-galactic medium [28], and the spatial distribution of metals (e.g., abundance gradients and age-metallicity relations) throughout the stellar disc [19], [11].", "In [21], we explored an alternate means by which to assess the efficacy of energy feedback schemes within a cosmological context: specifically, the predicted distribution of structural `power' encoded within the underlying cold gas of late-type dwarf galaxies.", "Empirically, star forming dwarfs present steep spatial power-law spectra ($P$$\\propto $$k$$^\\gamma $ ) for their cold gas, with $\\gamma $$<$$-3$ on spatial scales $\\mathrel {\\hbox{t}o 0pt{}\\hss }$$$$1~kpc \\cite {Stan99,Combes2012}, consistent with theslope expected when HI density fluctuations dominate the ISM structure,rather than turbulent velocity fluctuations (which dominate when isolating`thin^{\\prime } velocity slices).", "Our simulated (dwarf) disc galaxies showedsimilarly steep ISM power-law spectra, albeit deviating somewhat from thesimple, single, power-law seen by Stanimirovic et~al.$ In comparison, the cold gas of late-type giant galaxies appears to possess a more complex distribution of structural power.", "[9] demonstrate that while such massive discs also present comparably steep (if not steeper) power spectra on smaller scales ($\\gamma $$\\sim $$-$ 3, for $\\mathrel {\\hbox{t}o 0pt{}\\hss }$$$$1kpc), there is a strong tendency for the power to `flatten^{\\prime } tosignificantly shallower slopes on larger scales ($ $\\sim $ -$1.5, for$$$ $$$2~kpc).", "Dutta et~al.", "propose a scenario in which the steeper powerlaw component is driven by three-dimensional turbulence in the ISM on scalessmaller than a given galaxy^{\\prime }s scaleheight, while the flatter component isdriven by two-dimensional turbulence in the plane of the galaxy^{\\prime }s disc.$ In what follows, we build upon our earlier work on dwarf galaxies [21], utilising the Fourier domain approach outlined by [25], but now applied to a set of four simulated massive ($\\sim $ L$\\star $ ) disc systems.", "The simulations have each been realised with both conventional (i.e., moderate) and enhanced (i.e., strong/efficient) energy feedback.", "The impact of the feedback prescriptions upon the distribution of power in the ISM of their respective neutral hydrogen (HI) discs will be used, in an attempt to constrain the uncertain implementation of sub-grid physics.", "HI moment maps will be generated for each simulation and (for consistency) massive disc from The HI Nearby Galaxy Survey (THINGS: [31]), to make a fairer comparison with the observational data.", "In §, the basic properties of the simulations are reviewed, including the means by which the HI moment maps, and associated Fourier domain power spectra, were analysed.", "The resulting radial surface density profiles, velocity dispersion profiles, and distributions of power in the corresponding cold interstellar media are described in §.", "Our conclusions are presented in §.", "Two $\\sim $ L$\\star $ disc galaxies (g1536; g15784), drawn from the McMaster Unbiased Galaxy Survey (MUGS: [26]) and realised with the Smoothed Particle Hydrodynamics (SPH) code Gasoline [30], form the primary inputs to our analysis.The role of feedback in shaping the abundance gradients, metallicity distribution functions, and age-metallicity relations of these same four realisations has been presented recently by [11].", "Two variants for each disc were generated, one employing `conventional' feedback (MUGS) and one using our `enhanced' feedback scheme (MaGICC: Making Galaxies In a Cosmological Context - [4]; [28]).To link the simulation nomenclature with their earlier appearances in the literature, the MUGS variants of g1536 and g15784 are as first presented by [26], and analysed subsequently by [19] and [5], while the MaGICC variant of g1536 corresponds to the `Fiducial' run in [27] (itself, essentially the same as SG5LR, as first described by [4]).", "These four massive disc simulations are referred to henceforth as g1536-MUGS, g1536-MaGICC, g15784-MUGS, and g15784-MaGICC, and form the primary suite to which the subsequent analysis has been employed.", "To provide a bridge to our earlier study of the ISM power spectra of dwarf galaxies [21], we have analysed an ancillary set of three simulated low-mass discs (§3.4).", "An in-depth discussion of the MUGS and MaGICC star formation and feedback prescriptions are provided in the aforementioned works, although a brief summary of the key characteristics follows now.", "The MUGS runs assume a thermal feedback scheme in which 4$\\times $ 10$^{50}$ erg per supernova (SN) is made available to heat the surrounding ISM (`conventional'), while the MaGICC runs use 10$^{51}$ erg/SN (`enhanced').", "The MUGS simulations employ a [16] initial mass function (IMF), while MaGICC use the more `top-heavy' [6] form.The MUGS runs assumed that the global metallicity Z$\\equiv $ O+Fe, while those of MaGICC assume Z$\\equiv $ O+Fe+C+N+Ne+Mg+Si.", "Radiation energy feedback from massive stars during their pre-SN phase (lasting $\\sim $ 4 Myr) is included in the MaGICC runs, although it should be emphasised that the effective coupling efficiency is $<$ 1% [4], [27].", "For both MUGS and MaGICC, cooling is disabled for gas particles situated within a blast region of size $\\sim $ 100 pc, for a time period of $\\sim $ 10 Myr.", "Star formation is restricted to regions which are both sufficiently cool and dense (MUGS: $>$ 1 cm$^{-3}$ ; MaGICC: $>$ 9 cm$^{−3}$ ).", "Metal diffusion [24] is included in all runs.", "Supplementing the above four massive disc simulations, we have included three lower mass dwarf discs: (a) SG2 and SG3 [4] were realised with the same star formation and feedback schemes as the MaGICC versions of g1536 and g15784, respectively; the only difference lies in their initial conditions, where the former have been `scaled-down' by an order of magnitude in mass; (b) DG1 [12], the low mass dwarf that formed the basis of our earlier work [21]." ], [ "Analysis", "The analysis which follows is based upon a comparison of the HI gas properies of the MUGS+MaGICC simulations with their empirical `analogues', drawn from The HI Nearby Galaxy Survey (THINGS: [31]).", "We `view' the simulations face-on and restrict the comparison to massive discs from THINGS which are also close to face-on.", "In practice, this has meant limiting the analysis to the same sub-sample as that used by [9].", "In contrast, our earlier work [21] focussed on low-mass dwarf galaxies, rather than massive discs; in that study, we found that the index of the simulated ISM power spectrum ($\\gamma $ , where $P$$\\propto $$k^\\gamma $ ) was consistent, to first order, with that observed in dwarfs (on spatial scales $\\mathrel {\\hbox{t}o 0pt{}\\hss }$$$$1~kpc) such as the Small MagellanicCloud (i.e., $ $\\sim $ -$3.2).", "Besides determining the slope of theISM power spectra for our new suite of massive disc galaxy simulations, wewill present the radial HI surface density and velocity dispersion profiles,and contrast them with empirical data from the literature, in a furtherattempt to shed light on the role of feedback in shaping theircharacteristics.$ In what follows, we make use of zeroth- (surface density) and second- (velocity dispersion) moment maps of each simulation's HI distribution (viewed, face-on), realised with the image processing package Tipsy.www-hpcc.astro.washington.edu/tools/tipsy/tipsy.html The redshift $z$ =0 snapshots for each galaxy are first centred and aligned such that the angular momentum vector of the disc is aligned with the $z$ -axis, and the neutral hydrogen fraction of each SPH particle inferred under the assumption of combined photo- and collisional-ionisation equilibrium.", "From the zeroth- (second-) moment maps, radial HI surface density (velocity dispersion) profiles were generated for each simulation and (near) face-on, late-type, disc from THINGS.", "Individual results for each will be presented in §.", "It is worth noting that out of the THINGS galaxies presented in Figure REF , NGCs 3031, 5236, 5457 and 6946 are more extended than the VLA primary beam, resulting potentially in missing larger-scale information [31].", "After [25] and [21], the Fourier Transform of each of the aforementioned zeroth-moment HI maps (both simulations and empirical THINGS data) was taken, with circular annuli in Fourier space then employed to derive the average power in the structure of the ISM on different spatial scales." ], [ "Moment Maps", "The zeroth-moment HI maps for our four simulated $\\sim $ L$\\star $ late-type discs are shown in Fig REF , with the two MaGICC (MUGS) variants shown in the upper (lower) panels.", "Each panel spans 100$\\times $ 100 kpc.", "The `dynamic range' in HI column density in each panel is $\\sim $ 10$^{19}$ cm$^{-2}$ to $\\sim $ 10$^{21}$ cm$^{-2}$ - i.e., (roughly) the current observational lower and upper limits for HI (21cm) detection [2].", "Figure: Zeroth-moment HI maps for our four simulated ∼\\sim L☆\\star late-type discs: g1536-MaGICC (upper-left); g15784-MaGICC (upper-right);g1536-MUGS (bottom-left); g15784-MUGS (bottom-right).", "Each panel spans100×\\times 100 kpc, with a column density range of 10 19 ^{19} cm -2 ^{-2}to 10 21 ^{21} cm -2 ^{-2} (comparable to the limits imposed by 21cm surveyssuch as THINGS).Even a cursory inspection of Fig REF suggests that the enhanced feedback employed within MaGICC results in significantly more extended HI discs, relative to the conventional feedback treatment within MUGS.", "Similary, at these column densities, the eye is drawn to the enhanced structure on larger scales seen in the MaGICC runs (relative to the more locally `confined' structure seen in MUGS).", "Both points will be returned to below in a more quantitative sense." ], [ "Radial Surface Density Profiles", "From the face-on moment zero maps of Fig REF , radial HI surface density profiles were generated.", "These are reflected in Fig REF with the MUGS and MaGICC variants for g1536 (g15784) shown in the left (right) panel.", "As for Fig REF , the dynamic range has been limited to $\\mathrel {\\hbox{t}o 0pt{}\\hss }$$$ 1019$~cm$ -2$ ($$$ $$ 105$~M$$/kpc$ 2$), to reflectthe (typical) limiting 21cm detection limit in surveys such as THINGS;conversely, the horizontal line in each panel corresponds to the empiricalHI upper limit (also from THINGS) of $ 106.9$~M$$~kpc$ -2$.$ Figure: Radial HI surface density profiles for the g1536 (left panel)and g15784 (right panel) simulations.", "The plus symbols represent the MaGICCruns and the solid lines correspond to the MUGS runs.", "The solidhorizontal line in each panel correspond to the empirical HI upper limitfrom .The MaGICC discs (plus symbols in both panels) possess exponential surface density profiles (in HI) with $\\sim $ 6$-$ 8 kpc scalelengths.", "Conversely, the MUGS realisations are clearly more `compact', with essentially `flat' radial HI surface density profiles (each with $\\sim $ 10$^{20}$ cm$^{-2}$ , independent of galactocentric radius), with an extremely `sharp' HI edge at $\\sim $ 12$-$ 15 kpc.", "At a limiting HI (21cm) column density of $\\sim $ 10$^{19}$ cm$^{-2}$ , the MaGICC discs are $\\sim $ 2$-$ 3$\\times $ more extended than their MUGS analogues.", "At first glance, in terms of both radial dependence and amplitude, the HI surface density profiles of the MaGICC discs resemble very closely those shown in Fig 23 of O'Brien et al (2010).", "It is important to bear in mind though that the O'Brien et al.", "profiles were inferred (necessarily) from observations of edge-on discs.", "Our analysis of the simulations is restricted to face-on orientations, and so a fairer comparison would be to the sample of [1], who derived both HI and H$_2$ surface density profiles for a sample of face-on galaxies observed by THINGS.", "[1] show that the HI in such disc galaxies is distributed more uniformly, in terms of surface density, out to $\\sim $ 10 kpc, with (roughly) only a factor of $\\sim $ 3 decline in going to a galactocentric radius of $\\sim $ 20 kpc.", "This is consistent with the flatter gradient seen for the MUGS simulations, albeit the issue of their aforementioned overly truncated `edges' remains.", "Because we cannot resolve the transition from HI to H$_2$ in our simulations, some fraction of what is labelled as `HI' in Fig 2 (at least within the inner 5$-$ 10 kpc, for the MaGICC simulations, where the surface density is close to, or exceeds, the empirical upper limit for HI in nature) could certainly be misidentified H$_2$ , and so our inner gradients would be somewhat flatter than presented and therefore more consistent with the profiles of Bigiel & Blitz for radii $\\mathrel {\\hbox{t}o 0pt{}\\hss }$$$$10~kpc.", "Our predicted HI surface densitygradients in the $ 10-$20~kpc range are (on average) somewhat steeperthan the typical galaxy from Bigiel \\& Blitz (in the sameradial range - see their Fig~1a), but certainly lie within $ 1$of the distribution.", "In that sense, the extended nature and (outer disc)exponential profiles of the MaGICC simulations are more consistent withthose encountered in nature.", "\\rm $" ], [ "Radial Velocity Dispersion Profiles", "The radial HI velocity dispersion profiles derived from the second-moment maps (Fig REF ) present fairly 'flat' trends with increasing galactocentric distance, save for perhaps g15784, with $\\sigma $ decreasing typically by $\\sim $ 50% in going from the inner disc to a galactocentric radius of $\\sim $ 10 kpc; the profiles for the dwarfs (SG2 and SG3) are flat over this radial range, consistent with the dwarfs shown in Fig 3 of Pilkington et al (2011).", "Here, as the second-moment maps are for face-on viewing angles, the velocity dispersions quoted in Fig REF are equivalent to $\\sigma _{\\rm W}$ .", "We are only showing the velocity dispersion profiles within the star-forming parts of the disks (i.e., radii $\\mathrel {\\hbox{t}o 0pt{}\\hss }$$$ $r_{25}$ for the massive MaGICC and MUGS discs, and $\\mathrel {\\hbox{t}o 0pt{}\\hss }$$$ 2r25$ for the lower mass dwarfs SG2 and SG3, where$ r25$ is the isophotal radius corresponding to 25~mag~arcsec$ -2$).As such, the dispersions being $ 20-$100\\% higher than the`characteristic^{\\prime } value outside the star-forming disc ($$10~km/s:Tamburro et~al 2009) is not entirely unexpected.$ The thee main conclusions to take from this part of the analysis are that: (i) the profiles and amplitudes for the velocity dispersions of the cold gas within the star-forming region of the four massive MUGS and MaGICC discs overlap with those encountered in nature (Fig 1 of Tamburro et al.", "); (ii) the flat profiles of the two dwarfs (SG2 and SG3) are more problematic, consistent with the conclusions of Pilkington et al.", "(2011), and reflecting a limitation of our inability to resolve molecular hydrogen processes on these scales; (iii) the amplitudes of the MaGICC variants, relative to their MUGS counterparts, are $\\sim $ 50% higher (although both are within the range encountered in nature); such a result is not entirely unexpected, given the significantly enhanced feedback associated with the MaGICC runs." ], [ "Power Spectra", "As noted in §REF , power spectra were derived from each of the simulated and empirical (THINGS) HI moment-zero maps, by averaging in circular annuli in frequency space after Fourier transforming the images.", "The technique is identical to that employed by [25] and [21].", "While alternate approaches certainly exist (cf.", "[9]), we are more concerned here with adopting a homogeneous approach for both the simulations and the data, rather than necessarily inter-comparing the various techniques available.", "Figure: Power spectra for the four ∼\\sim L☆\\star MaGICC and MUGSsimulations (upper four spectra), two `dwarf' variants of g1536 andg15784 (SG2 and SG3, respectively), and the low-mass dwarf DG1, from.", "The inset within the panel links the symbol withthe relevant simulation.", "The ordinate represents arbitrary units ofspatial power, as the relative distribution (rather than absolute) isthe focus of this work; each spectrum has been offset with respect tothe next, for ease of viewing.Fig REF shows the power spectra for both the MaGICC and MUGS variants of g1536 and g15784 simulations, as well as their respective dwarf galaxy analogs, SG2 and SG3.", "For each of the four massive discs' spectra, single power-law fits are shown (solid curves) for the spatial scales over which the fits were derived ($\\sim $ 0.6$-$ 2 kpc).", "It should be emphasised that the lower limit on the spatial scale over which these fits were made corresponds to twice the softening length employed in the simulations; while an argument could be made to extending to somewhat smaller scales, we felt it prudent to be conservative in our analyses.", "What can hopefully be appreciated from a cursory analysis of Fig REF is the relatively enhanced power on sub-kpc scales seen in MUGS (conventional feedback) realisations, compared with the their MaGICC (enhanced feedback) analogues.", "This is reflected in the single power-law slopes itemised in the inset to the panel (which are weighted heavily by the more `numerous' higher frequency `bins' on sub-kpc scales), which are meant to be illustrative here, rather than represent the formal `best fit' to the data.", "Broadly speaking, the power spectra are roughly an order of magnitude steeper when using the MaGICC feedback scheme, as opposed to that of MUGS - i.e., it appears that the stronger feedback shifts the ISM power from predominantly `small' ($\\mathrel {\\hbox{t}o 0pt{}\\hss }$$$$1~kpc) to large ($$$ $$$2~kpc) spatial scales.", "\\rm $ We next extended our analysis to lower mass, late-type, systems, including the two dwarf variants to g1536-MaGICC and g15784-MaGICC (referred to as SG2 and SG3, as per [4]).", "We also performed an independent re-analysis of the dwarf (DG1) that formed the basis of our earlier work [21].", "The inclusion of these three `dwarfs' allows us to push the analysis to somewhat smaller spatial scales, while still working within a framework of `enhanced' feedback.", "The power spectra for all seven systems are shown in Fig REF .", "An important conclusion to be drawn from this figure (and associated quoted single power-law fits within the inset to the panel) is that on $\\sim $ sub-kpc scales, the power spectra slopes of the three dwarfs (SG2, SG3, DG1) are steeper ($-$ 3.5$\\mathrel {\\hbox{t}o 0pt{}\\hss }$$$ $\\gamma $ $$ $$ $-$ ) then their more massive analogues.", "Figure: Power spectra for the two MaGICC simulations, and two selectedfrom the empirical THINGS dataset (NGC 628 and 3184).", "All other detailsare as per the caption to Fig .We then compared the predicted power spectra from the two $\\sim $ L$\\star $ discs realised with the enhanced MaGICC feedback scheme, with those derived from galaxies from the THINGS database; the full database is shown in Fig REF , but for succinctness, we only show the power spectra for NGC 628 and 3184 (which were chosen, in part, because they were the closest to face-on, matching, by construct, the MaGICC simulations), alongside the MaGICC discs, in Fig REF .", "In terms of formal single power-law fits to these spectra, the MaGICC and (selected) THINGS galaxies are very similar (as shown by the quoted slopes within the inset to the panel).", "Having said that, as already alluded to in relation to Fig REF , the MaGICC spectra do not appear entirely consistent with a single power-law, instead presenting evidence for something of a `break' in the structural power, on the scales of $\\sim $ 1$-$ 2 kpc (being flatter on larger scales, and steeper on smaller scales, a point to which we return below).", "Figure: Power spectra of g1536-MaGICC, g15784-MaGICC, and NGC 2403,respectively (from left to right).", "Each spectrum appears (in aqualitative sense) inconsistent with a single power-law fit;two-component fits, with a shallower (steeper) slope on larger (smaller)scales, are suggested, although the `knee' in the spectra occurs ondifferent scales for the simulations (∼\\sim 2 kpc), as opposed to thatof NGC 2403 (∼\\sim 0.5 kpc).Inspection of Figs REF and REF suggests that single power law fits are not necessarily the best option.", "In Fig REF , we show the result of performing two-component fits to both the MaGICC data and a selected galaxy from THINGS (NGC 2403, chosen as it is the THINGS galaxy whose power spectrum looks like it would suit a 2-component fit best).", "In a qualitative sense, the behaviour is not dissimilar - i.e., both the MaGICC simulations and NGC 2403 show flatter power spectra on larger scales, compared with smaller scales, although the transition from `flat' to `steep' occurs at $\\sim $ 2 kpc in the simulations, as opposed to $\\sim $ 0.5 kpc in NGC 2403.", "This seems to be consistent with the idea posed by [9] that there is a steep power-law component on smaller scales driven by 3-dimensional turbulent motions, which flattens at larger spatial scales.", "At these larger scales, 2-dimensional turbulent motions begin to dominate within the plane of the galactic disc.", "The steepening of the power spectra on small spatial scales observed in the power spectra of the MaGICC large discs is also seen in work undertaken by [10] in their work on the LMC.", "Power spectra have been generated for the 17 THINGS galaxies employed in the analysis of [9]; these are provided in the accompanying Appendix as Fig 7.", "The majority have slopes on the order of $\\gamma $$\\sim $$-$ 2.3 to $-$ 2.8, with two exceptions: NGC 3031 ($\\gamma $$\\sim $$-$ 0.9) and NGC 3521 ($\\gamma $$\\sim $$-$ 3.3).", "Much as for the simulations, the point associated with the largest spatial scales in each panel should be viewed with some skepticism, as edge effects do come into play (i.e., the `edge' of the HI disc is `seen' as a high power `scale' against an almost noise-free background).", "We have presented an analysis of the cold gas and HI content of simulated discs with both 'standard' (MUGS) and `enhanced' (MaGICC) energy feedback schemes, as well as re-scaled dwarf variants of the massive (MaGICC) simulations.", "Radial density profiles were generated for the MUGS and MaGICC $\\sim $ L$\\star $ variants of g1536 and g15784 (Fig 2).", "These were generated using their respective zeroth HI moment maps; the weaker feedback associated with MUGS resulted in very flat radial HI distributions, with sharp cut-offs at galactocentric radii of $\\sim $ 12$-$ 15 kpc, while the stronger feedback associated with MaGICC resulted in HI discs with exponential surface density profiles (with scalelengths of $\\sim $ 6$-$ 8 kpc) which were $\\sim $ 2$-$ 3$\\times $ more extended (at an HI column density limit of $\\sim $ 10$^{19}$ cm$^{-2}$ ).", "The exponential profiles exhibited by the enhanced feedback runs are consistent with the typical profile observed in nature [2], [18].", "The majority of the THINGS radial density profiles show evidence of exponential components, indicating that the MaGICC simulations distribute the column density in a way that better matches observational evidence.", "The power spectra generated for the massive ($\\sim $ L$\\star $ ) discs with enhanced (MaGICC) feedback are steeper than their weaker (MUGS) feedback counterparts.", "In other words, the stronger feedback shifts the power in ISM from smaller scales to larger scales.", "Forcing a single component power-law to the MaGICC spectra yields slopes consistent with similarly forced single component fits to the empirical THINGS spectra.", "also well-described by a single component power law; having said that, the MaGICC spectra are more consistent with a two-component structure, with a steeper slope on sub-kpc spatial scales, flattening to shallower slopes on larger scales.", "The massive discs realised with the MUGS feedback scheme are both shallower than MaGICC, but also well-fit with a single power-law across all spatial scales.", "The dwarf galaxies realised in our work with enhanced feedback possess steeper slopes than their more massive counterparts, with values that are in agreement with [25] and [21].", "It is arguable that several of the THINGS power spectra warrant multiple-component fits (namely NGC 2403, 3031, 3184, 3198 and 7793) and the multi-component fits performed on NGC 2403 and the two large disc MaGICC galaxy power spectra indicate that the large-scale slopes agree well, whereas the small scale slopes differ largely.", "This indicates that the MaGICC feedback scheme distributes HI structures on a scale that is comparable to those of observational results, but there is a lack of small-scale structure.", "It is apparent that there is no 1:1 match to the THINGS data from either the MUGS or MaGICC feedback schemes, but MaGICC appears to fare better than the MUGS feedback scheme from a single-component fit in an average sense.", "The lack of a 1:1 relation may be largely due to the challenges in converting from 'cold gas' to 'HI' as well as a lack of exactly face-on systems observed in nature and in the THINGS survey." ], [ "Acknowledgments", "BKG acknowledges the support of the UK’s Science & Technology Facilities Council (ST/J001341/1).", "KP acknowledges the support of STFC through its PhD Studentship programme (ST/F007701/1).", "The generous allocation of resources from STFC’s DiRAC Facility (COSMOS: Galactic Archaeology), the DEISA consortium, co-funded through EU FP6 project RI-031513 and the FP7 project RI-222919 (through the DEISA Extreme Computing Initiative), the PRACE-2IP Project (FP7 RI-283493), and the University of Central Lancashire’s High Performance Computing Facility.", "We present here the ISM power spectra for the 17 THINGS galaxies used in this work.", "The inset to each panel includes the galaxy name, the weighting scheme employed (RO=robust), and the best-fit (single component) power-law slope.", "Figure: Power spectra for all the THINGS galaxies analysed in this work; names of the galaxies are listed on their corresponding plots along with the power law slope value.", "The power law slope is plotted over the points as a solid line." ] ]	1403.0488
[ [ "Parameter Estimation of Gravitational Waves from Precessing BH-NS\n Inspirals with higher harmonics" ], [ "Abstract Precessing black hole-neutron star (BH-NS) binaries produce a rich gravitational wave signal, encoding the binary's nature and inspiral kinematics.", "Using the lalinference\\_mcmc Markov-chain Monte Carlo parameter estimation code, we use two fiducial examples to illustrate how the geometry and kinematics are encoded into the modulated gravitational wave signal, using coordinates well-adapted to precession.", "Even for precessing binaries, we show the performance of detailed parameter estimation can be estimated by \"effective\" estimates: comparisons of a prototype signal with its nearest neighbors, adopting a fixed sky location and idealized two-detector network.", "We use detailed and effective approaches to show higher harmonics provide nonzero but small local improvement when estimating the parameters of precessing BH-NS binaries.", "That said, we show higher harmonics can improve parameter estimation accuracy for precessing binaries ruling out approximately-degenerate source orientations.", "Our work illustrates quantities gravitational wave measurements can provide, such as reliable component masses and the precise orientation of a precessing short gamma ray burst progenitor relative to the line of sight.", "\"Effective\" estimates may provide a simple way to estimate trends in the performance of parameter estimation for generic precessing BH-NS binaries in next-generation detectors.", "For example, our results suggest that the orbital chirp rate, precession rate, and precession geometry are roughly-independent observables, defining natural variables to organize correlations in the high-dimensional BH-NS binary parameter space." ], [ "Introduction", "Ground based gravitational wave detector networks (notably LIGO [1] and Virgo [2]) are sensitive to the relatively well understood signal from the lowest-mass compact binaries $M=m_1+m_2\\le 16 M_\\odot $ [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14].", "Strong signals permit high-precision constraints on binary parameters, particularly when the binary precesses.", "Precession arises only from spin-orbit misalignment; occurs on a distinctive timescale between the inspiral and orbit; and produces distinctive polarization and phase modulations [17], [15], [16].", "As a result, the complicated gravitational wave signal from precessing binaries is unusually rich, allowing high-precision constraints on multiple parameters, notably the (misaligned) spin [19], [18].", "Measurements of the spin orientations alone could provide insight into processes that misalign spins and orbits, such as supernova kicks [20], [21], or realign them, such as tides and post-Newtonian resonances [22].", "More broadly, gravitational waves constrain the pre-merger orbital plane and total angular momentum direction, both of which may correlate with the presence, beaming, and light curve [23], [24], [25] of any post-merger ultrarelativistic blastwave (e.g, short GRB) [26].", "Moreover, spin-orbit coupling strongly influences orbital decay and hence the overall gravitational wave phase: the accuracy with which most other parameters can be determined is limited by knowledge of BH spins [28], [29], [18], [27].", "Precession is known to break this degeneracy [30], [31], [32], [18], [33], [19].", "In sum, the rich gravitational waves emitted from a precessing binary allow higher-precision measurements of individual neutron star masses, black hole masses, and and black hole spins, enabling constraints on their distribution across multiple events.", "In conjunction with electromagnetic measurements, the complexity of a fully precessing gravitational wave signal may enable correlated electromagnetic and gravitational wave measurements to much more tightly constrain the central engine of short gamma ray bursts.", "Interpreting gravitational wave data requires systematically comparing all possible candidate signals to the data, constructing a Bayesian posterior probability distribution for candidate binary parameters [19], [37], [38], [40], [41], [39], [35], [34], [36].", "Owing to the complexity and multimodality of these posteriors, successful strategies adopt two elements: a well-tested generic algorithm for parameter estimation, such as variants of Markov Chain Monte Carlo or nested sampling; and deep insight into the structure of possible gravitational wave signals, to ensure efficient and complete coverage of all possible options [42], [43].", "Owing both to the relatively large number of parameters needed to specify a precessing binary's orbit and to the seemingly-complicated evolution, Bayesian parameter estimation methods have only recently able to efficiently draw inferences about gravitational waves from precessing sources [43].", "These improvements mirror and draw upon a greater theoretical appreciation of the surprisingly simple dynamics and gravitational waves from precessing binaries, both in the post-Newtonian limit [17], [45], [44], [16], [46] and strong field [47], [48], [50], [51], [52], [49].", "For our purposes, these insights have suggested particularly well-adapted coordinates with which to express the dynamics and gravitational waves from precessing BH-NS binaries, enabling more efficient and easily understood calculations.", "In particular, these coordinates have been previously applied to estimate how well BH-NS parameters can be measured by ground-based detectors [18].", "In this work, we will present the first detailed parameter estimation calculations which fully benefit from these insights into precessing dynamics.", "In short, we will review the natural parameters to describe the gravitational wave signal; demonstrate how well they can be measured, for a handful of selected examples; and interpret our posteriors using simple, easily-generalized analytic and geometric arguments.", "As a concrete objective, following prior work [18], [27] we will explore whether higher harmonics break degeneracies and provide additional information about black hole-neutron star binaries.", "In the absence of precession, higher harmonics are known to break degeneracies and improve sky localization, particularly for LISA [30], [31], [32].", "That said, these and other studies also suggest that higher harmonics provide relatively little additional information about generic precessing binaries, over and above the leading-order quadrupole radiation [18], [32].", "For example, for two fiducial nonprecessing and two fiducial precessing signals, [18], henceforth denoted COOKL, provide concrete predictions for how well detailed parameter estimation strategies should perform, for a specific waveform model.", "A previous work [27], henceforth denoted OFOCKL, demonstrated these simple predictions accurately reproduced the results of detailed parameter estimation strategies.", "In this work, we report on detailed parameter estimation for the two fiducial precessing signals described in COOKL.", "As with nonprecessing binaries, we find higher harmonics seem to provide significant insight into geometric parameters, in this case the projection of the orbital angular momentum direction on the plane of the sky.", "As this orientation could conceivably correlate with properties of associated electromagnetic counterparts, higher harmonics may have a nontrivial role in the interpretation of coordinated electromagnetic and gravitational wave observations.", "This paper is organized as follows.", "In Section we describe the gravitational wave signal from precessing BH-NS binaries, emphasizing suitable coordinates for the spins (i.e., defined at $100\\, {\\rm Hz}$ , relative to the total angular momentum direction) and the waveform (i.e., exploiting the corotating frame to decompose the signal into three timescales: orbit, precession, and inspiral).", "Our description of gravitational waves from precessing BH-NS binaries follows [15], henceforth denoted BLO, and [16], henceforth denoted LO.", "Next, in Section we describe how we created synthetic data consistent with the two fiducial precessing signals described in COOKL in gaussian noise; reconstructed a best estimate (“posterior distribution”) for the possible precessing source parameters consistent with that signal; and compared those predictions with semianalytic estimates.", "These semianalytic estimates generalize work by COOKL, approximating the full response of a multidetector network with a simpler but more easily understood expression.", "Using simple analytic arguments, we describe how to reproduce our full numerical and semianalytic results using a simple separation of scales and physics: orbital cycles, precession cycles, and geometry.", "The success of these arguments can be extrapolated to regimes well outside its limited scope, allowing simple predictions for the performance of precessing parameter estimation.", "We conclude in Section .", "For the benefit of experts, in Appendix we discuss the numerical stability and separability of our effective Fisher matrix.", "The kinematics of precessing binaries are well described in [17], BLO, and LO; see, e.g., Eq.", "(10) in BLO.", "In brief, the orbit contracts in the instantaneous orbital plane on a long timescale $1/\\Omega _{rad}$ over many orbital periods $1/\\Omega _\\phi $ .", "On an intermediate timescale $1/\\Omega _{prec}$ , due to spin-orbit coupling the angular momenta precess around the total angular momentum direction, which remains nearly constant.", "On timescales $1/\\Omega _{\\rm prec}$ between $1/\\Omega _{\\phi }$ and $1/\\Omega _{rad}$ , the orbital angular momentum traces out a “precession cone”.", "For this reason, we adopt coordinates at $100 \\, {\\rm Hz}$ which describe the orientation of all angular momenta relative to $\\vec{J}$ ; our coordinates are identical to those used in BLO, LO, and COOKL.", "Relative to a frame with $\\hat{z}$ oriented along the line of sight, the total, orbital, and spin angular momenta are described by the vectors:Strictly speaking, the total angular momentum $\\vec{J}$ precesses [17].", "For the results described in this work, we always adopt the total angular momentum direction evaluated at $f=100 \\, {\\rm Hz}$ .", "$\\hat{J} &= \\sin \\theta _{JN} \\cos \\psi _J \\hat{x} + \\sin \\theta _{JN} \\sin \\psi _J \\hat{y} + \\cos \\theta _{JN} \\hat{z} \\\\\\hat{L} & = \\sin \\iota \\cos \\psi _L \\hat{x} + \\sin \\iota \\sin \\psi _L \\hat{y} + \\cos \\iota \\hat{z} \\\\\\hat{S}_1 & = \\sin \\theta _1 \\cos (\\psi _L+\\phi _1) \\hat{x} + \\sin \\theta _1 \\sin ( \\psi _L+\\phi _1) \\hat{y} + \\cos \\theta _1 \\hat{z}$ where in this and subsequent expressions we restrict to a binary with a single spin (i.e., $\\vec{S}_2=0$ ).", "Because the orbital angular momentum evolves along a cone, precessing around $\\hat{J}$ , we prefer to describe the orbital and spin angular momenta in frame aligned with the total angular momentum $\\hat{z}^{\\prime }=\\hat{J}$ : $\\hat{L} = \\sin \\beta _{JL} \\cos \\alpha _{JL} \\hat{x}^{\\prime } + \\sin \\beta _{JL} \\sin \\alpha _{JL} \\hat{y}^{\\prime } + \\cos \\beta _{JL} \\hat{J}$ where the frame is defined so $\\hat{y}^{\\prime }$ is perpendicular to $\\hat{N}$ as in Figure REF [3]: $\\hat{y}^{\\prime } &= - \\frac{\\hat{N}\\times \\hat{J}}{\|\\hat{N}\\times \\hat{J}\|}\\;, \\quad \\;\\hat{x}^{\\prime } = \\hat{y}^{\\prime }\\times \\hat{J} = \\frac{\\hat{N} - \\hat{J}(\\hat{J}\\cdot \\hat{N})}{\|\\hat{N}\\times \\hat{J}\|}$ In this phase convention for $\\alpha _{JL}$ , the zero of $\\alpha _{JL}$ is one of the two points when $\\hat{L},\\hat{J},\\hat{N}$ are all in a common plane, sharing a common direction in the plane of the sky.", "Transforming between these two representations for $\\hat{L}$ is straightforward.", "For example, given $\\hat{N},\\hat{L}$ and $\\hat{J}$ , we identify $\\alpha $ and $\\beta _{JL}$ via $\\beta _{JL} &= \\cos ^{-1} \\hat{J} \\cdot \\hat{L} \\\\\\alpha _{JL}& =\\text{arg} \\hat{J}\\cdot [ \\frac{\\hat{L} \\times (\\hat{x}^{\\prime }+ i \\hat{y}^{\\prime })}{i\\sin \\beta _{JL}}] $ The spin angular momentum direction is determined from the direction of $\\hat{L}$ , the direction of $\\hat{J}$ , and the angle $\\theta _{LS}$ between $\\hat{S}_1$ and $\\hat{L}$ : $\\hat{S}_1 &= \\sin (\\beta _{JL}-\\theta _{LS}) \\cos \\alpha _{JL} \\hat{x}^{\\prime } + \\sin (\\beta _{JL}-\\theta _{LS}) \\sin \\alpha _{JL} \\hat{y}^{\\prime }\\nonumber \\\\& + \\cos (\\beta _{JL}-\\theta _{LS}) \\hat{J}$ Finally, the opening angle $\\beta _{JL}$ and the angle $\\theta _{LS}$ are related.", "Using the ratio of $\\vec{S}_1$ to the Newtonian angular momentum $\\vec{L}=\\mu \\vec{r}\\times \\vec{v}$ as a parameter: $\\gamma (t) &\\equiv & \|\\vec{S}_1\|/\|\\vec{L}(t)\| = \\frac{\\chi _1 m_1^2}{\\eta M\\sqrt{M r(t)} } = \\frac{m_1 \\chi _1}{m_2} v$ Using this parameter, the opening angle $\\beta _{JL}$ of the precession cone (denoted $\\lambda _L$ in ACST) can be expressed trigonometrically as $\\beta _{JL}(t)& \\equiv & \\arccos \\hat{J}\\cdot \\hat{L}= \\arccos \\frac{1+\\kappa \\gamma }{\\sqrt{1+2\\kappa \\gamma +\\gamma ^2}}$ where $\\kappa = \\cos \\theta _{LS} = \\hat{L}\\cdot \\hat{S}_1$ .", "Most BH-NS binaries' angular momenta evolve via simple precession: $\\alpha $ increases nearly uniformly on the precession timescale, producing several precession cycles in band [Eq.", "(9) in BLO], while $\\beta _{JL}$ increases slowly on the inspiral timescale, changing opening angle only slightly [Fig.", "1 in BLO]; see Figure REF .", "As described in OFOCKL, we evolve the angular momenta evolve according to expressions derived from general relativity in the post-Newtonian, adiabatic, orbit-averaged limit, an approximation presented in [17] and described [53].", "Though some literature adopts a purely Hamiltonian approach to characterize spin precession [54], [55], [56], [57], [58], [59], this orbit-averaged approach is usually adopted when simulating gravitational waves from precessing binaries [6], [13], [22].", "In this work we adopt two fiducial precessing BH-NS binaries, with intrinsic and extrinsic parameters specified in Tables REF and REF .", "Figure REF shows how each binary precesses around the total angular momentum direction $\\hat{J}$ and in the plane of the sky.", "For this mass ratio, the opening angle $\\beta _{JL}$ adopted in consistent with randomly-oriented BH spin; see, e.g., Eq.", "(REF ) and Fig.", "4 of LO.", "For this sky location, our simplified three-detector gravitational wave network has comparable sensitivity to both linear (or both circular) polarizations." ], [ "Gravitational waves from precessing binaries", "Precession introduces modulations onto the “carrier signal” produced by the secular decay of the orbit over time.", "BLO and LO provide a compact summary of the associated signal, in the time and frequency domain.", "In a frame aligned with the total angular momentum, several harmonics $h_{lm}$ are significant: $h_+ - i h_\\times = \\sum _{lm} h_{lm} Y^{(-2)}_{lm}$ where the harmonics $h_{lm}$ are provided and described in the literature [13].", "By “significant”, we mean that harmonics have nontrivial power $\\rho _{lm}$ : $\\rho _{lm}^2 = 2\\int _{-\\infty }^{\\infty } \\frac{\|\\tilde{h}_{lm}\|^2}{S_h(f)}$ where $S_h$ is the fiducial initial LIGO design noise power spectrum.", "These precession-induced modulations are most easily understood in a corotating frame, as in LO[47], [60], [44], [50], [51], [52], [49]: $h_{lm} = \\sum _{m^{\\prime }}D^l_{mm^{\\prime }}(\\alpha _{JL},\\beta _{JL},\\gamma )h_{lm^{\\prime }}^{\\rm ROT}$ where $\\gamma = -\\int d\\alpha \\cos \\beta _{JL}$ and where $D^l_{mm^{\\prime }}$ is a Wigner D-matrix.", "In this expression, $h_{lm}^{\\rm ROT}$ is the gravitational wave signal emitted by a binary with instantaneous angular momentum along the $\\hat{L}$ axis.", "In the low-velocity limit, $h_{lm}^{\\rm ROT}$ is dominated by leading-order radiation and hence by equal-magnitude $(l,m)=(2,\\pm 2)$ modes.", "Due to spin-orbit precession with $\\beta _{JL} \\ne 0$ , however, these harmonics are mixed.", "When $\\beta _{JL}$ is greater than tens of degrees, then in the simulation frame all $h_{lm}$ are generally present and significant.", "To illustrate that gravitational wave emission from a precessing binary requires several harmonics $h_{lm}$ to describe it when $\\beta _{JL} >0$ , we evaluate $\\rho _{2m}$ , conservatively assuming only the $(2,\\pm 2)$ corotating-frame modes are nonzero: $\\rho _{2m}^2 &\\simeq [\\rho _{2,2}^{\\rm ROT}]^2 \|d^2_{2,m}(\\beta _{JL})\|^2 + [\\rho _{2,-2}^{\\rm ROT}]^2 \|d^2_{-2,m}(\\beta _{JL})\|^2\\nonumber \\\\& =\\rho _{2,2}^{\\rm ROT} [\|d^2_{2,m}(\\beta _{JL})\|^2 + \|d^2_{-2,m}(\\beta _{JL})\|^2]$ where we use orthogonality of the corotating-frame $(2,\\pm 2)$ modes.", "Figure REF shows that except for a small region $\\beta _{JL} \\simeq 0$ , several harmonics contribute significantly to the amplitude along generic lines of sight, with $\\rho _{2m}/\\rho _{22}\\gtrsim 0.1$ .", "At this level, these harmonics change the signal significantly, both in overall amplitude ($\\rho ^2/2 * 0.1^2 \\simeq \\rho ^2 0.05$ ) and in fit to candidate data.", "Gravitational waves from precessing BH-NS binaries are modulated in amplitude, phase, and polarization.", "A generic precessing source oscillates between emitting preferentially right-handed and preferentially left-handed radiation along any line of sight; see [47].", "For the scenario adopted here, however, the orbital angular momentum almost always preferentially points towards the observer ($\\hat{L}\\cdot \\hat{N}\\gtrsim 0$ ), so gravitational waves emitted along the line of sight are principally right handed for almost all time.", "Figure: Harmonic amplitude versus opening angle: A plot ofρ 2m 2 /(ρ 22 ROT ) 2 \\rho _{2m}^2/(\\rho _{22}^{\\rm ROT})^2 predicted by Eq.", "() for m=2m=2 (blue), 1 (red), and0 (yellow)." ], [ "Symmetry and degeneracy", "Gravitational waves are spin 2: the spin-weight $-2$ expression $h=h_+-i h_\\times $ transforms as $h\\rightarrow h \\exp (-2i\\psi )$ under a rotation by $\\psi $ around the propagation axis.", "Any gravitational wave signal is unchanged by rotating the binary by $\\pi $ around the propagation direction.", "This exact discrete symmetry insures two physically distinct binaries can produce the same gravitational wave signal and can never be distinguished.", "For this reason, our MCMCs evaluate the polarization angle $\\psi _L$ only over a half-domain $[0,\\pi ]$ : the remaining half-angle space follows from symmetry.", "Physically, however, at least two physically distinct spin, $\\hat{L}$ , and $\\hat{J}$ configurations produce the same best fitting gravitational wave signal.", "These two spin configurations can be part of the same probability contour or two discrete islands.", "For circularly polarized nonprecessing sources, these two spin configurations blur together: the gravitational wave signal is independent of $\\psi _L-\\phi _{orb}$ (for example), preventing independent measurement of $\\psi _L$ ; see, e.g., the top left panel in Figure 4 of OFOCKL.", "For generic nonprecessing sources, however, higher harmonics generally isolate the best-fitting gravitational wave signal and thus polarization angles $\\psi _L$ , producing two distinct and exactly degenerate islands of probability over $\\psi _L\\in [0,2\\pi ]$ ; see OFOCKL.", "For generic precessing sources, the same degeneracy applied to the total angular momentum produces two distinct, exactly degenerate choices for the direction $\\psi _J$ of $\\hat{J}$ in the plane of the sky.", "This degeneracy cannot be broken.", "However, at leading order, precessing binaries can still be degenerate in $\\psi _L\\pm \\phi _{\\rm orb}$ , in the absence of higher harmonics.", "As we will see below, this degeneracy can be broken, ruling out discrete choices for $\\psi _L$ .", "To construct synthetic data containing a signal, to interpret that signal, and to compare interpretations from different simulations to each other and to theory, we adopt the same methods as used in OFOCKL.", "Specifically, to determine the shape of each posterior, we employ the lalsimulation and lalinference [19], [34] code libraries developed by the LIGO Scientific Collaboration and Virgo collaboration.", "As in OFOCKL, we adopt a fiducial 3-detector network: initial LIGO and Virgo, with analytic gaussian noise power spectrum provided by their Eqs.", "(1-2).", "In contrast to the simplified, purely single-spin discussion adopted in Section to describe the kinematics of the physical signal in the data, the model used to interpret the data allows for nonzero, generic spin on both compact objects.", "That said, because compact object spin scales as the mass squared times the dimensionless spin parameter ($S=m^2 \\chi $ ), in our high-mass-ratio systems the small neutron star's spin has minimal dynamical impact.", "Our simulations show gravitational waves provide almost no information about the neutron star's spin magnitude or direction.", "For the purposes of simplicity, we will omit further mention of the smaller spin henceforth.", "Table: One-dimensional parameter errors: Measurement accuracy σ x \\sigma _x forxx one of several intrinsic (ℳ c ,η,χ 1 {{\\cal M}_c},\\eta , \\chi _1), extrinsic (ψ ± ,t,RA,DEC\\psi _{\\pm },t,RA,DEC), and precession-geometry (α JL ,β JL ,θ JN \\alpha _{JL},\\beta _{JL},\\theta _{JN}) parameters.The extrinsic parameters are the event timett; the sky position measured in RA and DEC; and the sky area AA, estimated using the 2×22\\times 2 covariance matrixΣ ab \\Sigma _{ab} on the sky via π\|Σ\|\\pi \|\\Sigma \|.", "The precession cone parameters are as described in Figure : the precession phase α JL \\alpha _{JL} at the reference frequency; the precession coneopening angle β JL \\beta _{JL}; and the viewing angle θ JN \\theta _{JN}." ], [ "Intrinsic parameters", "As shown in Figure REF , the intrinsic parameters of our relatively loud ($\\rho \\simeq 20$ ) fiducial binaries are extremely well-constrained.", "For example, the neutron star's mass, black hole's mass, and black hole spin are all relatively well-measured, compared to the accuracy of existing measurements and hypothesized distributions of these parameters [61], [62], [63].", "Higher harmonics provide relatively little additional information about these parameters.", "Applied to an even simpler idealized problem – a similar source known to be directly overhead two orthogonal detectors – the effective fisher matrix procedure of COOKL produces qualitatively similar results, notably reproducing relatively minimal impact from higher harmonics.", "Given the simplifications adopted, the effective fisher matrix predictions inevitably disagree quantitatively with our detailed Monte Carlo calculations, particularly regarding multidimensional correlations.", "We nonetheless expect the effective fisher matrix to correctly identify scales and trends in parameter estimation; moreover, being amenable to analysis, this simple construct allows us to develop and validate simple interpretations for why some parameters can be measured as well as they are.", "As a concrete example, we can explain Figure REF .", "The time-dependent orbital phase depends on the black hole spin, principally through the “aligned component” $\\hat{L}\\cdot \\vec{S}_1$ [28].", "As discussed in COOKL and OFOCKL, the “aligned component” cannot be easily distinguished from the mass ratio in the absence of precession.", "Spin-orbit precession breaks this degeneracy, allowing significantly tighter constraints on the mass ratio of our precessing binary.", "In our particular example, comparing Fig 3 in OFOCKL to our Figure REF , we can measure $\\eta $ and hence the smaller mass roughly three times more accurately at the same signal amplitude.", "As seen in the bottom panel of Figure REF , both the spin-orbit misalignment $\\hat{L}\\cdot \\hat{S}_1$ and spin magnitude $\\chi _1$ remain individually poorly-constrained.", "As a concrete example, our ability to measure $\\chi _1$ for this precessing binary is comparable to the accuracy possible for a similar nonprecessing binary [OFOCKL].", "One correlated combination of $\\hat{L}\\cdot \\hat{S}_1$ and $\\chi _1$ is well-constrained: the combination that enters into the precession rate.", "In Figure REF we show contours of constant precession cone opening angle ($\\beta _{JL}$ ) and constant precession rate [LO Eq.", "(7-8)] $\\Omega _p = \\frac{\|J\|}{2r^3} =\\eta \\left(2+\\frac{3 m_2}{2m_1} \\right)v^5\\sqrt{1+2\\kappa \\gamma +\\gamma ^2}$ When evaluating these expressions, we estimate $\\gamma \\simeq 1.85 \\chi _1$ [Eq.", "(REF )], so the contours shown correspond to $\\cos \\beta _{JL} =0.65,0.7,0.75$ and $\\sqrt{1+2\\kappa \\gamma +\\gamma ^2} = 2.2, 2.4,2.6$ .", "As expected, the presence of several precession cycles allows us to relatively tightly constrain the precession rate.", "Future gravitational wave detectors, being sensitive to longer signals and hence more precession cycles, can be expected to even more tightly constrain this combination.", "By contrast, as described below, the precession geometry $\\beta _{JL}$ is relatively poorly constrained, with error independent of the number of orbital or precession cycles.With relatively few precession cycles in our study, the discrepancy between these two measurement accuracies is fairly small.", "However, when advanced instruments with longer waveforms can probe more precession cycles, we expect this simple argument will explain dominant correlations.", "Figure: Estimating astrophysical parameters (C): For ourfiducial binary C, the solid and dotted lines show an estimated 90% confidence interval with and without higher harmonics, respectively; colors indicate different noiserealizations; and the (nearly indistinguishable) thick solid and dashed lines shows an approximate effective Fishermatrix result, with and without higher harmonics, not accounting for the constraint imposed by χ 1 <1\\chi _1<1.", "Results for case A are qualitatively and quantitatively similar.The different panels show different two-dimensional projections of the astrophysically relevant parameters of a merging BH-NSbinary: the binary mass ratio, black hole spin, and degree of spin-orbit misalignment κ≡L ^·S ^ 1 \\kappa \\equiv \\hat{L}\\cdot \\hat{S}_1.Top,center panels: The masses and spin magnitude of the binary can be measured very reliably,consistent with a single gaussian distribution in four dimensions.", "The analytic predictions produced by an effectiveFisher matrix agree qualitatively but not quantitatively with our simulations.Bottom panel: To guide the eye, the posterior versus χ 1 \\chi _1 and L ^·S ^ 1 \\hat{L}\\cdot \\hat{S}_1 is compared withcontours of constant β JL =cos -1 0.65,0.7,0.75\\beta _{JL}=\\cos ^{-1}0.65, 0.7, 0.75 (precession cone opening angle; dotted black) and Ω p \\Omega _p [Eq.", "()](precession rate; solid black).", "The precession rate is relatively wellconstrained by the presence of several (≃7\\simeq 7) precession cycles available in data, while the geometry is relatively poorly constrained,relative to the whole χ 1 \\chi _1 vs L ^·S ^ 1 \\hat{L}\\cdot \\hat{S}_1 plane." ], [ "Geometry", "As expected analytically and demonstrated by Figure REF , precession-induced modulations encode the orientation of the various angular momenta relative to the line of sight.", "For our loud fiducial signal, the individual spin components can be well-constrained.", "Equivalently, because our fiducial source performs many precession cycles about a wide precession cone and because that source is viewed along a generic line of sight, we can tightly constrain the precession cone's geometry: its opening angle; its orientation relative to the line of sight; and even the precise precession phase, measured either by $\\cos \\iota $ or $\\alpha _{JL}$ .", "The effective Fisher matrix provides a reliable estimate of how well these parameters can be measured; see Table REF and Figure REF .", "Figure: Source geometry: Angular momenta (C,A):For case C (top panels) and case A (bottom panels), the posterior for the “precession cone” (path of the angular momentum direction), expressed using theprecession cone representation.", "This figure demonstrates that both the path (θ JN ,β JL \\theta _{JN},\\beta _{JL}) and instantaneousorientation (α JL ,ι\\alpha _{JL},\\iota ) of the orbital angular momentum can be well-determinedAs in Figure , colors indicate different noise realizations; solid and dotted lines indicatethe neglect or use of higher harmonics; the green point shows the actual value; and the solid gray path shows thetrajectory of LL over one precession cycle.Left panels: The precession angle α JL \\alpha _{JL} of LL around JJ.", "For comparison, the greenpoints show the simulated values; when present, the solid blue path shows variables covered in one precession cycle.Roughly speaking, the precession phase can be measured with relative accuracy tens ofpercent at this signal amplitude ρ\\rho .Right panels: Illustration that both the opening angle β JL \\beta _{JL} of the precession cone and the angleθ JN \\theta _{JN} between the line of sight and J ^\\hat{J} can be measured accurately." ], [ "Comparison to and interpretation of analytic predictions", "COOKL presented an effective Fisher matrix for two fiducial precessing binaries, adopting a specific post-Newtonian model to evolve the orbit.", "Following OFOCKL, we adopt a refined post-Newtonian model, including higher-order spin terms.", "In the Supplementary Material, available online, we provide a revised effective Fisher matrix, including the contribution from these terms.", "Table REF summarizes key features of this seven-dimensional effective Fisher matrix for case A.", "As noted above, the two-dimensional marginalized predictions are in good qualitative agreement.", "The one-dimensional marginalized predictions agree surprisingly well with our simulations [Table REF ].", "Since the ingredients of the effective Fisher matrix are fully under our analytic control, we can directly assess what factors drive measurement accuracy in each parameter.", "Table: Properties of precessing effective Fisher matrix: Quantities derived from the normalized effectiveFisher matrix Γ ^\\hat{\\Gamma }, as provided in the supplementary information: the eigenvalues λ k \\lambda _k andone-dimensional parameter measurement accuracies Γ ^ -1 /ρ 2 \\sqrt{\\hat{\\Gamma }^{-1}/\\rho ^2} evaluated for ρ=20\\rho =20.", "(As we only compute Fisher matrices after marginalizing over ψ\\psi or φ ref \\phi _{\\rm ref}, we provide only 7 eigenvaluesand independent parameter measurement errors at a time.", ")First and foremost, as in COOKL, this effective Fisher matrix has a hierarchy of scales and eigenvalues, with decreasing measurement error: $ {{\\cal M}_c},\\eta ,\\chi , \\ldots $ .", "Unlike nonprecessing binaries, this hierarchy does not clearly split between well-constrained intrinsic parameters ($ {{\\cal M}_c},\\eta ,\\chi _1$ ) and poorly-constrained geometric parameters (everything else); for example, as seen in Table REF , the eigenvalues of the Fisher matrix span a continuous range of scales.", "The scales in the Fisher matrix are intimately tied to timescales and angular scales in the outgoing signal.", "The largest eigenvalues of the Fisher matrix are set by the shortest timescales: the orbital timescale, and changes to the orbital phase versus time.", "These scales control measurement of $ {{\\cal M}_c},\\eta , L\\cdot a$ and set the reference event time and phase.", "Qualitatively speaking, we measure these parameters well because good matches require the orbital phase to be aligned over a wide range in time.", "We measure the reference waveform phase reliably because each waveform must be properly aligned.", "For this reason, parameters related to orbital phase (i.e., $ {{\\cal M}_c}$ ) can be measured to order $1/\\sqrt{N_{cycles}}$ , times suitable powers of $v$ to account for the post-Newtonian order at which those terms influence the orbital phase.", "The next-shortest scales are precession scales: changes to the zero of precession phase, and how precession phase accumulates with time.", "Qualitatively speaking, we can measure the reference precession phase reliably because each precession cycle needs to be in phase.", "Due to spin-orbit precession, our fiducial BH-NS binaries will undergo $N_{\\rm cycles}\\simeq 10 $ (30) amplitude and phase modulations in band, as seen by an initial (advanced) detector [BLO Eq.", "(9), for an angular-momentum-dominated binary, with $\|L\|>\|S\|$ ] $N_P &\\simeq & \\int _{\\pi f_{min}}^{\\pi f_{max}} d f_{orb} \\frac{dt}{d f_{orb}} \\Omega _p \\nonumber \\\\&=& \\frac{5}{96}(2+1.5 \\frac{m_2}{m_1}) [(M\\pi f_{min})^{-1} - (M\\pi f_{max})^{-1}] \\nonumber \\\\&\\approx & \\frac{10 (1+0.75 m_2/m_1) }{M/10 M_\\odot } (f_{\\rm min}/50\\, {\\rm Hz})^{-1}$ This estimate agrees favorably with the roughly 7 precession cycles performed by our spin-dominated ($\|S\|>\|L\|$ ) binary between 30 and 500 Hz [Figure REF ].", "For this reason, parameters tied to spin-orbit precession rate $\\Omega _p$ (i.e., $\\eta $ ) will be measured to a relative accuracy $1/\\sqrt{N_p}$ .", "Applied to the mass ratio, this estimate leads to the surprisingly successful estimate $\\sigma _{\\eta } \\simeq O(1)\\times \\frac{\\eta }{\\sqrt{N_p}\\rho } \\simeq O(1)\\times 1.6\\times 10^{-3}$ i.e., roughly $1/\\sqrt{N_P}$ times smaller than the measurement accuracy possible without breaking the spin-mass ratio degeneracy.", "While some parameters change the rate at which orbital and precession phase accumulate, other reference phases simply fix the geometry.", "For example, a shift in the precession phase at some reference frequency (i.e., $\\alpha (f=100\\, {\\rm Hz})$ ) leads to a correlated shift in the precession and hence gravitational wave phase in each precession cycle.", "In other words, like our ability to measure the orbital phase at some time, our ability to measure the reference precession phase is essentially independent of the number of orbital or precession cycles, solely reflecting geometric factors.", "We expect the accuracy with which these purely geometric parameters $x$ can be determined can be estimated from first principles.", "To order of magnitude, we expect Fisher matrix components $\\Gamma _{xx}$ comparable to $\\Delta x^2$ , where $\\Delta x$ is the parameter's range.", "For example, the angular parameters ($\\beta _{JL},\\theta _{JN},\\alpha ,\\phi $ ) should be measured to within $\\sigma _{\\rm angle} \\simeq \\frac{(2\\pi )}{\\sqrt{12} \\rho } \\simeq 0.09 \\, {\\rm rad}$ where the factor $\\sqrt{12}$ is the standard deviation of a uniform distribution over $[0,1]$ .", "This simple order-of-magnitude estimate compares favorably to the Fisher matrix results shown in Table REF and to our full numerical simulations [Figure REF and Table REF ].", "This naive estimate ignores all dependence on precession geometry; in general, all geometric factors are tied directly to the magnitude of precession-induced modulations, which grow increasingly significant for larger misalignment, roughly in proportion to $\\cos \\beta _{JL}$ .", "This estimate for how well geometric angles can be measured should break down for nearly end-over-end precession ($\\beta _{JL} \\rightarrow \\pi /2$ ).", "Nearly end-over-end precession requires extreme fine tuning; is associated with transitional precession; and is correlated with rapid change in $\\beta _{JL}$ [BLO].", "We anticipate a different set of approximations will be required to address this limit." ], [ "Relative role of higher harmonics ", "To this point, both our analytic and numerical calculations suggest higher harmonics provide relatively little additional information about intrinsic and extrinsic parameters.", "That said, as illustrated by Figure REF , higher harmonics do break a discrete degeneracy, determining the orientation of $\\hat{L}$ on the plane of the sky at $f=100\\, {\\rm Hz}$ up to a rotation by $\\pi $ .", "OFOCKL used the evidence to demonstrate conclusively that higher harmonics had no additional impact, beyond improving knowledge of one parameter.", "Given expected systematic uncertainties in the evidence, at the present time we do not feel we can make as robust and global a statement.", "That said, all of our one- and two-dimensional marginalized posteriors support the same conclusion: higher harmonics provide little new information, aside from breaking one global degeneracy.", "Figure: Angular momentum direction on the sky (C) : Projection of the orbital angular momentum direction(L ^\\hat{L}) on the plane of the sky at f=100 Hz f=100\\, {\\rm Hz}; compare to Figure .This figure demonstrates that the individual angular momenta to bewell-constrained to two discrete regions and that higher harmonics allow us to distinguish between the twoalternatives; and (3) that the precession cone is well-determined, at the accuracy level expected from the number ofprecession cycles.As in Figure , colors indicate different noise realizations; solid and dotted lines indicatethe neglect or use of higher harmonics; and the green point shows the expected solution." ], [ "Timing, sky location, and distance", "As seen in Figure REF , precessing binaries do not have the strong source orientation versus distance degeneracy that plagues nonprecessing binaries: because they emit distinctively different multi-harmonic signals in each direction, both the distance and emission direction can be tightly constrained.", "Conversely, the sky location of precessing binaries can be determined to little better than the sky location of a nonprecessing binary with comparable signal amplitude; compare, for example, Table REF and Figures REF against the corresponding figures in OFOCKL.", "Figure: Distance and inclination degeneracy broken (C): Posterior probability contours in distance andinclination.Finally, the event time can be marginally better determined for a precessing than for a nonprecessing binary.", "This accuracy may be of interest for multimessenger observations of gamma ray bursts." ], [ "Advanced versus initial instruments", "All the discussion above assumed first-generation instrumental sensitivity.", "For comparison and to further validate our estimates, we have also done one calculation using the expected sensitivity of second-generation instruments [64], [65].", "In this calculation, the source (event C) has been placed at a larger distance ($d=298.7 \\, {\\rm Mpc}$ ) to produce the same network SNR.", "Also unlike the analysis above, we have for simplicity assumed the smaller compact object has no spin.", "Table REF shows the resulting one-dimensional measurement accuracies, compared against a concrete simulation.", "All results agree with the expected scalings, as described previously.", "First and foremost, all geometric quantities ($RA, DEC,\\alpha _{JL},\\beta _{JL},\\theta _{JN}$ ) and time can be measured to the same accuracy as in initial instruments, at fixed SNR.", "Second, quanitites that influence the orbital decay – chirp mass, mass ratio, and spin – are all measured more precisely, because more gravitational wave cycles contribute to detection with advanced instruments.", "Finally, as illustrated by Figure REF , quantities that reflect precession-induced modulation - the precession rate $\\Omega _p$ and precession cone angle $\\beta _{JL}$ misalignment – are at best measured marginally more accurately, reflecting the relatively small increase in number of observationally-accessible precession cycles for advanced detectors [Eqs.", "(REF and REF ].", "As shown by the bottom panel of Figure REF , this small increase in sensitivity is comparable to the typical effect of different noise realizations.", "Figure: Estimating astrophysical parameters with advanced detectors:Like the bottom panel of Figure , but using advanced instruments; see Table .Table: Parameter estimation with initial and advanced instruments:Like Table , measurement accuracy σ x \\sigma _x for several intrinsic and extrinsicparameters.", "The first row provides results for initial-scale instruments, duplicating an entry in Table.", "The second row provides results for advanced detectors, operating at design sensitivity.At fixed signal amplitude, most geometric quantities can be measured to fixed accuracy, independent of detectorsensitivity.", "Quantities impacting the orbital phase versus time (mass, mass ratio, and spin) are more accuratelymeasured with advanced instruments, with their access to lower frequencies and hence more cycles." ], [ "Conclusions", "In this work we performed detailed parameter estimation for two selected BH-NS binaries, explained several features in terms of the binary's kinematics and geometry, and compared our results against analytic predictions using the methods of [18], [27].", "First, despite adopting a relatively low-sensitivity initial-detector network for consistency with prior work, we find by example that parameter estimation of precessing binaries can draw astrophysically interesting conclusions.", "Since our study adopted relatively band-limited initial detector noise spectra, we expect advanced interferometers [66], [67] will perform at least as well (if not better) at fixed SNR.", "For our fiducial binaries, the mass parameters are constrained well enough to definitively say if it is a BH-NS binary (as opposed to BH-BH); the mass parameters are constrained better than similar non-precessing binaries; and several parameters related to the spin and orientation of the binary can be measured with reasonable accuracy.", "Second and more importantly, we were able to explain our results qualitatively and often quantitatively using far simpler, often analytic calculations.", "Building on prior work by BLO, LO, and others [47], we argued precession introduced distinctive amplitude, phase, and polarization modulations on a precession timescale, effectively providing another information channel independent from the usual inspiral-scale channel found in non-precessing binaries.", "Though our study targeted only two specific configurations, we anticipate many of our arguments explaining the measurement accuracy of various parameters can be extrapolated to other binary configurations and advanced detectors.", "The effective Fisher matrix approach of COOKL and OFOCKL provides a computationally-efficient means to undertake such extrapolations.", "Third and finally, we demonstrated that for this mass range and orientation, higher harmonics have minimal local but significant global impact.", "For our systems, we found higher harmonics broke a degeneracy in the orientation of $\\hat{L}$ at our reference frequency (100 Hz), but otherwise had negligible impact on the estimation of any other parameters.", "Due to the relatively limited calculations of spin effects in post-Newtonian theory, all inferences regarding black hole spin necessarily come with significant systematic limitations.", "For example, [68] imply that poorly-constrained spin-dependent contributions to the orbital phase versus time could significantly impact parameter estimation of nonprecessing black hole-neutron star binaries.", "Fortunately, the leading-order precession equations and physics are relatively well-determined.", "For example, the amplitude of precession-induced modulations is set by the relative magnitude and misalignment of $\\vec{L}$ and $\\vec{S}_1$ .", "In our opinion, the leading-order symmetry-breaking effects of precession are less likely to be susceptible to systematic error than high-order corrections to the orbital phase.", "Significantly more study would be needed to validate this hypothesis.", "Robust though these correlations may be, the quantities gravitational wave measurements naturally provide (chirp mass; precession rate; geometry) rarely correspond to astrophysical questions.", "We have demonstrated by example that measurements of relatively strong gravitational wave signals can distinguish individual component masses and spins to astrophysical interesting accuracy [Fig.", "REF ].", "Given the accuracy and number of measurements gravitational waves will provide, compared to existing astrophysical experience [69], [70], [71], [72], [73], these measurements should transform our understanding of the lives and deaths of massive stars.", "Ignoring correlations, gravitational wave measurements seem to only relatively weakly constrain spin-orbit misalignent [Fig.", "REF ], a proxy for several processes including supernova kicks and stellar dynamics.", "That said, gravitational wave measurements should strongly constrain the precession rate, a known expression of spins, masses, and spin-orbit misalignment.", "Formation models which make nontrivial predictions about both spin magnitude and misalignment might therefore be put to a strong test with gravitational wave measurements." ], [ "Acknowledgements", "This material is based upon work supported by the National Science Foundation under Grant No.", "PHY-0970074, PHY-0923409, PHY-1126812, and PHY-1307429.", "ROS acknowledges support from the UWM Research Growth Initiative.", "BF is supported by an NSF fellowship DGE-0824162.", "VR was supported by a Richard Chase Tolman fellowship at the California Institute of Technology.", "HSC, CK and CHL are supported in part by the National Research Foundation Grant funded by the Korean Government (No.", "NRF-2011-220-C00029) and the Global Science Experimental Data Hub Center (GSDC) at KISTI.", "HSC and CHL are supported in part by the BAERI Nuclear R & D program (No.", "M20808740002).", "This work uses computing resources both at KISTI and CIERA, the latter funded by NSF PHY-1126812." ], [ "Separation of scales and mutual information", "On physical grounds, we expect the timescales and modulations produced by precession to separate, allowing roughly independent measurements of orbital- and precession-rate-related parameters (i.e., $ {{\\cal M}_c},\\eta ,a, \\beta _{JL}$ ) and purely geometric parameters ($\\alpha ,\\phi ,\\theta $ ).", "To assess this hypothesis quantitatively, we evaluate the mutual information between the two subspaces.", "For a gaussian distribution described by a covariance matrix $\\Gamma $ , the mutual information between two subspaces $A,B$ is [OFOCKL Eq.", "(31)]: $I(A,B) = -\\frac{1}{2}\\ln \\frac{\|\\Gamma \|}{\|\\Gamma _A\|\|\\Gamma _B\|}$ Table REF shows that after marginalizing out orbital phase ($\\phi _{\\rm ref}$ ), the mutual information between orbital-phase-related parameters ($ {{\\cal M}_c},\\eta ,a,\\beta _{JL}$ ) and geometric parameters is small but nonzero (0.31): the two subspaces are weakly correlated.", "By comparison, the mutual information $I(a,c\|B)$ between two intrinsic parameters $a,c$ in $A =\\lbrace {{\\cal M}_c},\\eta ,\\chi _1\\rbrace $ is large.", "Finally, after marginalizing out all other parameters, the mutual information between $\\alpha _{JL}$ and $(\\theta _{JN})$ is small, as expected given the different forms in which these quantities enter into the outgoing gravitational wave signal." ], [ "Regularizing calculations with a prior", "Due to the wide range of eigenvalues and poor condition number, all Fisher matrices are prone to numerical instability in high dimension.", "Additionally, due to physical near-degeneracies, the error ellipsoid derived from the Fisher matrix alone may extend significantly outside the prior range; see, e.g., examples in [74].", "Following convention, to insure our results are stable to physical limitations, we derive parameter measurements accuracies $\\Sigma =\\Gamma ^{-1}/\\rho ^2$ by combining the signal amplitude $\\rho $ , the normalized effective Fisher matrix $\\hat{\\Gamma }^{\\rm eff}$ provided above, and a prior $\\Gamma ^{\\rm prior}$ : $\\Gamma ^{\\rm eff}_{\\lambda } &\\equiv \\rho ^2 \\hat{\\Gamma }^{\\rm eff} + \\lambda \\Gamma ^{\\rm prior} \\\\\\Gamma ^{\\rm prior} & = e_{\\eta }\\otimes e_{\\eta } + \\frac{e_a\\otimes e_a}{20} +\\frac{1}{(2\\pi )^2}[e_\\beta \\otimes e_\\beta +e_\\theta \\otimes e_\\theta +e_\\alpha \\otimes e_\\alpha +e_\\phi \\otimes e_\\phi ]$ As expected given the eigenvalues and signal amplitude, this prior has no significant impact on our calculations.", "In particular, the eigenvalues and parameter measurement accuracies reported in the text are unchanged if this weak prior is included." ] ]	1403.0544
[ [ "Construction of Lagrangian and Hamiltonian Structures starting from one\n Constant of Motion" ], [ "Abstract The problem of the construction of Lagrangian and Hamiltonian structures starting from two first order equations of motion is presented.", "This new approach requires the knowledge of one (time independent) constant of motion for the dynamical system only.", "The Hamiltonian and Lagrangian structures are constructed, the Hamilton Jacobi equation is then written and solved and the second (time dependent) constant of the motion for the problem is explicitly exhibited." ], [ "Introduction", "Consider a system ${S^1}_{2n}$ of $2n$ first order differential equations for $2n$ variables $x^a$ , $\\dot{x}^a = f^a (x^b) ,\\ \\ \\ \\ a, b = 1, 2, 3,..., 2n $ A Hamiltonian structure for system ${S^1}_{2n}$ consists of a Hamiltonian $H$ and a Poisson Bracket relation defined in terms of an antisymmetric matrix $J^{ab}$ which satisfy the antisymmetry condition (see, for instance [1], [2], [3]) $J^{ab} = - J^{ba} $ the Jacobi identity $J^{ab}_{,d} J^{dc} + J^{bc}_{,d} J^{da} + J^{ca}_{,d} J^{db} \\equiv 0 $ and Hamilton equations $f^a(x^c)=J^{ab}\\frac{\\partial H}{\\partial x^b}\\equiv [x^a,H]$ where the Poisson Bracket $[A,B]$ for any pair of dynamical variables $A(x^a)$ and $B(x^b)$ is defined by $[A, B]\\equiv \\frac{\\partial A}{\\partial x^a}J^{ab}\\frac{\\partial B}{\\partial x^b}=-[B, A]$ A Hamiltonian system ${S^1}_{2n}$ of $2n$ first order differential equations is said to be (Liouville) integrable if $n$ constants of motion $C_i,\\ (i=1, 2, 3,..., n)$ are known and they are in involution [4], [5], $[C_i, C_j]=0,\\ \\forall \\ \\lbrace i,j\\rbrace $ i.e., the Poisson Brackets of all possible pairs of constants vanish.", "In order to test Liouville integrability then, the system needs to be cast in Hamiltonian form with well defined Poisson Bracket relations.", "In the usual case, if a Lagrangian for a regular system is known, then the Hamiltonian structure (canocical momenta, a Hamiltonian and Poisson Bracket relations) may be constructed using standard textbook procedures (see, [6], for instance).", "Nevertheless, if a Lagrangian is not known (or if it fails to exist) the usual textbook procedure is no longer applicable.", "Techniques to construct a Hamiltonian structure have been devised and they require the knowledge of at least one constant of the motion and symmetry vectors of the differential system [2], [3].", "In this article we present new results regarding the case $n=1$ where one constant of motion allows for the construction of the Hamiltonian and Lagrangian structures as well as the complete integration of the problem.", "Solutions obtained previously require the knowledge of two constants of motion [7], [8], [9], [10] or a constant of motion and a symmetry vector [2], [3].", "The new procedure is applied to motion of projectiles subject to air drag as well as to other dynamical equations." ], [ "Construction of a Hamiltonian Structure", "Consider a system ${S^1}_{2}$ of two first order differential equations for two variables $x^a$ [7], [8], [9], [10], $\\dot{x}^a = f^a (x^b).\\ \\ \\ \\ a, b = 1, 2 $ Assume one time independent constant of motion $C_1=C_1 (x^b)$ is known.", "The (time independent) constant of motion $C_1(x^b)$ satisfies $\\frac{\\partial C_1}{\\partial x^a} f^a (x^b) \\equiv 0.\\ \\ \\ \\ a, b =1, 2 $ The construction of a Hamiltonian structure for system (REF ) requires the knowledge of a Hamiltonian $H$ and a Poisson Bracket antisymmetric matrix $J^{ab}$ .", "Nevertheless, in dimension 2 there is essentially one antisymmetric matrix, therefore, $J^{ab}= \\left( \\begin{array}{ccc}0 & \\mu (x^b) \\\\-\\mu (x^b) & 0 \\\\\\end{array} \\right),$ where the function $\\mu (x^b)$ is determined by the Hamilton equations condition $f^a(x^c)=J^{ab}\\frac{\\partial C_1}{\\partial x^b},$ where the choice $H=C_1$ has been made.", "Note that due to (REF ) the gradient of $C_1$ is orthogonal to $f$ so that $J^{ab}\\frac{\\partial C_1}{\\partial x^b}$ is parallel to $f^a$ in a two dimensional space, thus condition (REF ) determines the function $\\mu (x^b)$ uniquely.", "It is worth mentioning that in a two dimensional space the Jacobi identity (REF ) is always satisfied by any antisymmetric matrix.", "Therefore, if a time independent constant of motion $C_1(x^b)$ for system (REF ) is known, the Hamiltonian structure is defined by choosing the Hamiltonian $H=C_1$ and the Jacobi matrix $J^{ab}$ is completely determined by the antisymmetry requirement (REF ) and the Hamilton equations condition (REF ).", "Furthermore, Liouville integrability criterion is always met with one constant of motion in a two dimensional phase space." ], [ "Construction of a Lagrangian Structure", "Hamilton equations (REF ) may be rewritten as $\\dot{x}^a = J^{ab}\\frac{\\partial C_1}{\\partial x^b}$ Introduce the antisymmetric Lagrange Brackets matrix ${\\sigma }_{ab}= -{\\sigma }_{ba}$ $J^{ab}{\\sigma }_{bc}=-{{\\delta }^a}_c $ which is (up to a sign) the matrix inverse of the Poisson Brackets matrix $J^{ab}$ .", "Left multiply (REF ) by ${\\sigma }_{ca}$ , to get the Lagrangian form of Hamilton equations, ${\\sigma }_{ca} \\dot{x}^a +\\frac{\\partial C_1}{\\partial x^c}=0.", "$ In matrix form ${\\sigma }_{ab}$ is ${\\sigma }_{ab}= \\left( \\begin{array}{ccc}0 & \\frac{1}{\\mu (x^b)} \\\\-\\frac{1}{\\mu (x^b)} & 0 \\\\\\end{array} \\right).$ Consider the Lagrangian $L = L({x}^a, \\dot{x}^b)$ $L = L({x}^a, \\dot{x}^b)= l_1 ({x}^a) \\dot{x}^1 - C_1 ({x}^a),$ where $l_1 ({x}^a)$ is defined by $\\frac{\\partial l_1}{\\partial {x}^2} = \\frac{1}{\\mu }.", "$ The Euler Lagrange equations for (REF ) are $\\frac{\\partial l_1}{\\partial {x}^2}\\dot{x}^2 + \\frac{\\partial C_1}{\\partial {x}^1}=0 $ and $-\\frac{\\partial l_1}{\\partial {x}^2}\\dot{x}^1 + \\frac{\\partial C_1}{\\partial {x}^2}=0 $ which are identical to (REF ) and equivalent to (REF ) once (REF ) and (REF ) are taken into account.", "Note that $l_1 ({x}^a)$ is determined only up to the addition of an arbitrary function $f_1 ({x}^1)$ .", "This addition modifies the Lagrangian (REF ) by a total time derivative, which means that equations (REF ) and (REF ) remain invariant under the change.", "Now define phase space variables $q$ and $p$ by $q\\equiv x^1, $ and $p\\equiv l_1 ({x}^a).", "$ It is a straightforward matter to compute the Poisson Bracket $[q, p] = 1, $ using (REF ), (REF ) and (REF ), to realize that $q$ and $p$ are canonically conjugated variables.", "Note that the non uniqueness in the definition of $l_1 ({x}^a)$ generates canonical transformations in $p$ leaving $q$ invariant." ], [ "Construction of the Hamilton–Jacobi equation and the general solution of the system", "Consider now the inverse transformation of (REF ), (REF ) given by $x^1 = q, $ and $x^2 = g(q,p), $ where the last equation is obtained by solving (REF ) for $x^2$ .", "We have used the fact that $\\frac{\\partial l_1}{\\partial x^2} \\ne 0, $ so that one can always solve (REF ) for $x^2$ .", "Define now the Hamiltonian ${H(q,p)}$ by $H(q,p) = C_1(q,g(q,p)), $ Therefore, the Hamilton–Jacobi equation is $H(q,\\frac{\\partial S}{\\partial q})+ \\frac{\\partial S}{\\partial t}= 0, $ or $H(q,\\frac{\\partial W}{\\partial q})= C_1, $ where $S (q, C_1, t) = W (q, C_1) - C_1 t $ The second (explicitly time dependent) constant of motion $C_2$ for the system is obtained by solving (REF ) for $W (q, C_1)$ , (i.e., solving for $\\frac{\\partial W}{\\partial q}$ and integrating) and defining $C_2= \\frac{\\partial S}{\\partial C_1}= \\frac{\\partial W}{\\partial C_1} - t, $ thus one gets the general solution to the problem (up to computing an integral)." ], [ "One second order equation", "Consider the dynamics of a system defined by $\\ddot{q}= F(q)G(\\dot{q}) $ One second order equation may, of course, be written as a two dimensional first order system.", "Define $x^1 \\equiv q, $ and $x^2 \\equiv \\dot{q}.", "$ The equations of motion ${\\dot{x}}^1= x^2 $ and ${\\dot{x}}^2= F(x^1)G(x^2) $ are equivalent to (REF ) and definitions (REF ) and (REF ).", "A time independent constant of motion $C_1(q, \\dot{q})$ for (REF ) is given by $C_1(q, \\dot{q}) = - \\int {F(q) dq} +\\int {\\frac{\\dot{q}}{G(\\dot{q})}d\\dot{q}}.", "$ Therefore, the Hamiltonian $H$ is $H(x^1, x^2) = - \\int {F(x^1) dx^1} +\\int {\\frac{x^2}{G(x^2)} d x^2},$ and the Poisson Bracket matrix can be written as $J^{ab}= \\left( \\begin{array}{ccc}0 & G(x^2) \\\\- G(x^2) & 0 \\\\\\end{array} \\right)$ to reproduce (REF ) and (REF ).", "The momentum $p$ is given by $p=\\int \\frac{dx^2}{G(x^2)} $ One can now proceed following the steps described in Section IV.", "To be more concrete, consider a few explicit functional forms for $G(\\dot{q})$ In this case (as a matter of fact, for any constant $G(\\dot{q})$ ), one gets that $C_1$ is the energy, $p=x^2$ and the Hamilton–Jacobi equation is $\\frac{1}{2}\\left({\\frac{\\partial S}{\\partial q}}\\right)^2-\\int F(q)dq + \\frac{\\partial S}{\\partial t}= 0, $ where $S(q, C_1, t) = W(q, C_1)- C_1 t, $ and $W(q, C_1)$ satisfies $\\frac{1}{2}\\left({\\frac{\\partial W}{\\partial q}}\\right)^2-\\int F(q)dq = C_1.", "$ The second (time dependent) constant of motion $C_2$ is given by $C_2= \\frac{\\partial S(q, C_1, t)}{\\partial C_1} = -t + \\int \\frac{dq}{{\\sqrt{2(C_1+\\int F(q^{\\prime }) dq^{\\prime })}}}, $ which completes the solution." ], [ "$G(\\dot{q})=\\dot{q}$", "Now $C_1= - \\int {F(q) dq} +\\dot{q} ,$ and $p= \\int \\frac{d\\dot{q} }{\\dot{q} } = \\ln \\dot{q} .$ The Hamitonian is $H(q, p)= - \\int {F(q) dq} +e^p ,$ and the Hamilton–Jacobi equation is $e^\\frac{\\partial S}{\\partial q}- \\int {F(q) dq} + \\frac{\\partial S}{\\partial t}= 0.", "$ The solution to the Hamilton–Jacobi equation is $S(q, C_1, t) = \\int dq\\ {\\ln \\left(C_1+ \\int {F(q^{\\prime }) dq^{\\prime }}\\right)}- C_1 t, $ The second constant $C_2$ is $C_2= \\frac{\\partial S(q, C_1, t)}{\\partial C_1} = -t + \\int \\frac{dq}{{(C_1+\\int F(q^{\\prime }) dq^{\\prime })}} $ which completes the solution.", "The trajectories of projectiles moving on a medium in which drag forces are not negligible are of potential practical importance to ballistics and sports, for instance, in addition to their interest associated to theoretical problems related to solving ordinary differential equations and constructing Lagrangian and Hamiltonian structures as well as the associated Hamilton–Jacobi equation.", "Parabolic motion is the well known solution for the trajectory of a projectile moving on the surface of the Earth neglecting air resistance.", "The actual motion in the presence of a viscous medium is, of course, different when drag forces are important.", "Parker dealt with such a problem in an article published in 1977 [11] where the general solution as well as applications to different regimes of projectile motion were presented.", "Recently, a different approach to solve the same problem was presented by Shouryya Ray, an Indian born german highschool student but there does not seem to be any published record of his findings.", "Nevertheless, see http://bit.ly/KLgGYd Some time ago, it was commonplace to state that dissipative problems were out of the realm of Lagrangian and/or Hamiltonian descriptions.", "Nevertheless, in the last decades, different procedures to relate symmetries and conserved quantities to the construction of Lagrangian and Hamiltonian structures have been devised [2], [3], [12], [13].", "Consider a projectile moving near the surface of the Earth subject to gravity and air drag force proportional to the square of its speed.", "Let $u$ and $v$ be the horizontal and vertical components of its velocity, respectively.", "The equations of motion are (following http://bit.ly/KLgGYd) $\\dot{u}= - a u\\sqrt{u^2+v^2},\\ and$ $\\dot{v}= - a v\\sqrt{u^2+v^2}-g,\\ $ where $a$ and $g$ are constants.", "Multiply (REF ) times $v$ and (REF ) times $u$ and subtract to get $v \\dot{u}- u \\dot{v} = g u$ or $\\frac{d}{dt}({\\frac{v}{u}}) = - \\frac{g}{u}$ Now multiply (REF ) times $\\dot{v}$ and (REF ) times $\\dot{u}$ and subtract to get $-a( u \\dot{v}- v \\dot{u}) \\sqrt{u^2+v^2}+ g \\dot{u}=0$ or $a \\ \\sqrt{1+({\\frac{v}{u}})^2}\\ \\frac{d}{dt}({\\frac{v}{u}})- g\\ \\frac{\\dot{u}}{u^3}=0$ Define $q_1\\equiv {\\frac{v}{u}} $ and $q_2\\equiv - {\\frac{1}{u}} $ to get $\\dot{q_1}= g\\ q_2$ and $a \\ \\sqrt{1+{q_1}^2}\\ \\dot{q_1}+\\ \\frac{1}{2}\\ g\\ \\frac{d({q_2}^2)}{dt} =\\ 0 $ Equation (REF ) may be readily integrated to yield a constant of motion $C_1$ $C_1 =\\frac{1}{2} \\left[ a \\left(q_1 \\sqrt{1+{q_1}^2}+ {sinh}^{-1}(q_1)\\right)+\\ g\\ {q_2}^2 \\right].$ As a matter of fact, a Hamiltonian structure defined by canonical variables $q\\equiv q_1$ , $p\\equiv q_2$ and Hamiltonian $C_1(q,p)$ is such that Hamilton equations for this system are equivalent to the equations of motion (REF ) and (REF ) as it can be straightforwardly realized.", "In fact, one Lagrangian $L$ for equations (REF ) and (REF ) may be written as $L= p\\ \\dot{q} - H (q, p) $ The Hamilton–Jacobi equation for such a system is $\\frac{1}{2} \\left[ \\ g\\ \\left({\\frac{\\partial S}{\\partial q}}\\right)^2+ a \\left(q \\sqrt{1+{q}^2}+ {sinh}^{-1} (q)\\right)\\right]+ \\frac{\\partial S}{\\partial t} =0.$ Therefore, as usual, $S(q,E,t)= W(q,E)-Et $ where $\\frac{1}{2} \\left[ \\ g\\ \\left({\\frac{\\partial W}{\\partial q}}\\right)^2+ a \\left(q \\sqrt{1+{q}^2}+ {sinh}^{-1} (q)\\right)\\right]=E.$ Solve for $W$ to get $W = \\int dq \\sqrt{\\frac{\\left(2E-a \\left(q \\sqrt{1+{q}^2}+{sinh}^{-1}(q)\\right)\\right)}{g}}$ and the time dependent constant of motion $t_0$ is $t_0= -\\frac{\\partial S}{\\partial E}$ $t_0= t-\\int \\frac{dq}{\\sqrt{\\left(2gE-ag \\left(q \\sqrt{1+{q}^2}+{sinh}^{-1}(q)\\right)\\right)}},$ which completes the solution of the problem." ], [ "Conclusions", "The usual approach to Lagrangian and Hamiltonian dynamics assumes the knowledge of a Lagrangian from which the (Euler–Lagrange) equations of motion, the canonical momenta, the Poisson Brackets relations, the Hamiltonian, Hamilton's and Hamilton–Jacobi equations are derived (and sometimes solved).", "One tool to test integrability of the dynamical differential equations is Liouville's theorem based on the structures described above.", "The quantization of the classical system may also be achieved using these exact same structures.", "In other words, the Lagrangian is an extremely powerful tool which allows us to construct all the dynamical entities which are needed to study a classical dynamical system and to quantize it.", "In this article a different approach is presented.", "The building blocks are the equations of motion (no prior knowledge of a Lagrangian or a Hamiltonian structure is assumed) and one time independent constant of motion for a two dimensional first order system.", "These ingredients are enough to construct a Hamiltonian, Poisson Brackets relations, a Lagrangian, a canonical momentum, Hamilton's and Hamilton–Jacobi equations as well as the second constant of the motion of the problem, therefore solving the Inverse Problem of the Calculus of Variations and getting the general solution to the differential system.", "Previous approaches to the solution of these problems require prior knowledge of two constants of motion [7], [10] or of a constant of motion and a symmetry vector [2], which amounts to knowing the general solution of the problem.", "Therefore, the approach presented here represents a real progress compared to previous methods in the sense that less stringent requirements are needed and the solution to the problem is explicitly provided (instead of required)." ] ]	1403.0557
[ [ "Chemistry and Radiative Transfer of Water in Cold, Dense Clouds" ], [ "Abstract The Herschel Space Observatory's recent detections of water vapor in the cold, dense cloud L1544 allow a direct comparison between observations and chemical models for oxygen species in conditions just before star formation.", "We explain a chemical model for gas phase water, simplified for the limited number of reactions or processes that are active in extreme cold ($<$ 15 K).", "In this model, water is removed from the gas phase by freezing onto grains and by photodissociation.", "Water is formed as ice on the surface of dust grains from O and OH and released into the gas phase by photodesorption.", "The reactions are fast enough with respect to the slow dynamical evolution of L1544 that the gas phase water is in equilibrium for the local conditions thoughout the cloud.", "We explain the paradoxical radiative transfer of the H$_2$O ($1_{10}-1_{01}$) line.", "Despite discouragingly high optical depth caused by the large Einstein A coefficient, the subcritical excitation in the cold, rarefied H$_2$ causes the line brightness to scale linearly with column density.", "Thus the water line can provide information on the chemical and dynamical processes in the darkest region in the center of a cold, dense cloud.", "The inverse P-Cygni profile of the observed water line generally indicates a contracting cloud.", "This profile is reproduced with a dynamical model of slow contraction from unstable quasi-static hydrodynamic equilibrium (an unstable Bonnor-Ebert sphere)." ], [ "Introduction", "Observations of water vapor in the interstellar medium (ISM) by the Infrared Space Observatory [40] and the Submillimeter Wave Astronomy Satellite (SWAS) [2] show general agreement with chemical models for warm ($> 300$ K) conditions in the ISM [30], [33].", "However, in cold conditions, most of the water is frozen onto dust grains [42], [41], and the production of water occurs mainly on the grain surfaces.", "In order to test chemical models that include grain-surface chemistry we used the Heterodyne Instrument for the Far-Infrared (HIFI) [10] on the Herschel Space Observatory to observe the H$_2$ O ($1_{10}-1_{01}$ ) line in the cold, dense cloud L1544 [7], [6].", "The first of these two Herschel observations was made with the wide-band spectrometer (WBS) and detected water vapor in absorption against the weak continuum radiation of dust in the cloud.", "Follow-up observations with higher spectral resolution and sensitivity, made with the high resolution spectrometer (HRS), confirmed the absorption and detected a blue-shifted emission line that was predicted by theoretical modeling [7], but too narrow to be seen by the WBS in the first observation.", "With the better constraints provided by the second observation, we improved the chemical and radiative transfer modeling in our previous papers.", "We modified the radiative transfer code MOLLIE to calculate the line emission in the approximation that the molecule is sub-critically excited.", "This assumes that the collision rate is so slow that every excitation leads immediately to a radiative de-excitation and the production of one photon which escapes the cloud, possibly after many absorptions and re-emissions, before another excitation.", "The emission behaves as if the line were optically thin with the line brightness proportional to the column density.", "This approximation can be correct even at very high optical depth as long as the excitation rate is slow enough, C $<$ A/$\\tau $ , where C is the collision rate, A is the spontaneous emission rate and $\\tau $ the optical depth [28].", "[6] presented the observations and the results of this modeling.", "In this paper, we discuss in detail the theory behind the modeling.", "A comparison of the spectral line observation with theory requires three models.", "First, we require a hydrodynamical model to describe the density, velocity, and temperature across the cloud.", "We use a model of slow contraction in quasi-static unstable equilibrium that we developed in our previous research [23], [22].", "Second, we require a chemical model to predict the molecular abundance across the varying conditions in the cloud.", "Following the philosophy for simplified chemical networks in [21] or [4], we extract from a general chemical model for photo-dissociation regions [16] a subset of reactions expected to be active in cold conditions, principally grain-surface reactions as well as freeze-out and photodissociation.", "Third, we require a radiative transfer model to generate a simulated molecular line.", "We modify our non-LTE radiative transfer code MOLLIE to use the escape probability approximation.", "This allows better control of the solution in extreme optical depth.", "The three models are described in more detail in three sections below.", "The relevant equations are included in the appendices.", "Given their importance as the nurseries of star formation, the small ($< 0.5$ pc), cold ($<15$ K), dense ($n>10^3$ cm$^{-3}$ ) clouds in low-mass star ($< 2$ M$_\\odot $ ) forming regions such as Taurus are widely studied [3], [11].", "Observations show a unique simplicity.", "They contain no internal sources of heat, stars or protostars.", "Their internal turbulence is subsonic, barely broadening their molecular line widths above thermal [32].", "With most of their internal energy in simple thermal energy, and the weak turbulence just a perturbation [20], [5], the observed density structure approximates the solution of the Lane-Emden equation for hydrostatic equilibrium [27], [19].", "Correspondingly, most are nearly spherical with an average aspect ratio of about 1.5 [17].", "They are heated from the outside both by cosmic rays and by the UV background of starlight and are cooled from the inside by long wavelength molecular line and dust continuum radiation [13].", "Because of their simplicity, we understand the structure and dynamics of these small, cold, dense clouds better than any other molecular clouds in the interstellar medium.", "They are therefore uniquely useful as a laboratory for testing hypotheses of more complex phenomena such as the chemistry of molecular gas." ], [ "Structure and dynamics", "Our physical model for cold, dense clouds is computed with a spherical Lagrangian hydrodynamic code with the gas temperature set by radiative equilibrium between heating by external starlight and cosmic rays and cooling by molecular line and dust radiation.", "The theory is discussed in [23] and [21].", "In our previous research [22], we generated a dynamical model for the particular case of L1544 by comparing observations and snapshots in time out of a theoretical model for the contraction toward star formation.", "We began the hydrodynamic evolution with a 10 M$_\\odot $ Bonnor-Ebert (BE) sphere with a central density of $10^4$ cm$^{-3}$ in unstable dynamical equilibrium and in radiative equilibrium with an external UV field of one Habing flux.", "In the early stages of contraction, the cloud evolves most rapidly in the center.", "As long as the velocities remain subsonic, the evolving density profile closely follows a sequence of spherical equilibria or BE spheres with increasing central densities.", "We compared modeled CO and N$_2$ H$^+$ spectra during the contraction against those observed in L1544 and determined that the stage of contraction that best matches the data has a central density of $1\\times 10^7$ cm$^{-3}$ and a maximum inward velocity just about the sound speed [22].", "Figure REF shows the density and velocity at this time along with the H$_2$ O abundance and temperature.", "In the present investigation we modify our numerical hydrodynamic code to include cooling by atomic oxygen.", "This improves the accuracy of the calculated gas temperature in the photodissociation region outside the molecular cloud.", "The equations governing the cooling by the fine structure lines of atomic oxygen are presented in the appendix." ], [ "Chemistry of H$_2$ O in cold conditions", "The cold conditions in L1544 allow us to simplify the chemical model for gas phase water.", "We include the four oxygen-bearing species most abundant in cold, dark clouds, O, OH, H$_2$ O gas, and H$_2$ O ice.", "Even though all three gas phase molecules may freeze onto the grains, we consider only one species of ice because the formation of water from OH and the formation of OH from O are rapid enough on the grain surface that most of the ice is in the form of H$_2$ O.", "To provide a back reaction for the freeze-out of atomic oxygen and preserve detailed balance, we arbitrarily assign a desorption rate for atomic O equal to that of H$_2$ O even though the production of atomic O from H$_2$ O ice is not indicated.", "Our simplified model is shown in figure REF .", "The resulting abundances, calculated as equilibria between creation and destruction, are shown in figures REF and REF .", "Figure REF shows the abundances near the photodissociation region (PDR) boundary as a function of the visual extinction, $A_V$ .", "Figure REF shows the abundances against the log of the radius to emphasize the center.", "Gas phase water is created by UV photodesorption of water ice which also creates gas phase OH in a ratio H$_2$ O/OH = 2 [16].", "In the outer part of the cloud, the UV radiation derives from the background field of external starlight.", "The inward attenuation of the UV flux is modeled from the visual extinction as $\\exp {(-1.8 {\\rm A_V})}$ .", "In the interior where all the external UV radiation has been attenuated, the only UV radiation is generated by cosmic ray strikes on H$_2$ .", "In our previous paper [6], we set this secondary UV radiation to $1\\times 10^{-3}$ times the Habing flux (G$_0$ =1) [16].", "In our current model, we use a lower level, $1\\times 10^{-4}$ , that is more consistent with estimated rates [35].", "The difference in abundance for the two rates is shown in figure REF .", "H$_2$ O and OH are removed from the gas phase by UV photodissociation and by freezing onto dust grains.", "To preserve detailed balance with the photodissociation we include the back reactions, the gas phase production of H$_2$ O, O + H$_2$ $\\rightarrow $ OH and OH + H$_2$ $\\rightarrow $ H$_2$ O, even though these are not expected to be important in cold gas.", "Removal of gas phase water by freeze-out is important in the interior where the higher gas density increases the dust-gas collision rate, and hence the freeze-out rate.", "We assume that the gas-phase ion-neutral reactions that lead to the production of water are less important at cold temperatures ($< 15$ K) than the reactions that produce water on the surfaces of ice-coated dust grains.", "Thus, we do not include gas-phase ion-neutral reactions in the model.", "This is valid if the oxygen is quickly removed from the gas-phase by freeze-out and efficiently converted into water ice on the grain surface.", "By leaving out CO, we avoid coupling in the carbon chemistry.", "Although we already have a simple model for the carbon chemistry [21], [22], we prefer to keep our oxygen model as simple as possible.", "This could create an error of a factor of a few in the abundance of the oxygen species.", "Carbon is one-third as abundant as oxygen, and in certain conditions CO is the dominant carbon molecule.", "Therefore as much as one-third of the oxygen could potentially be bound in CO.", "Ignoring O$_2$ is less of a problem.", "Created primarily by the reaction of OH with atomic oxygen, O$_2$ tends to closely follow the abundance of OH.", "Since the amount of oxygen in OH should be 1% or less (figure REF ), the abundance of O$_2$ does not affect the abundances of the other oxygen species, O, OH, and H$_2$ O.", "Figures REF and REF compare the abundances from our simplified network with those from the more complex network of [16] (courtesy of E. Bergin) that includes gas-phase neutral-neutral and ion-neutral reactions.", "In this calculation, we hold the cloud at the same time in its dynamical evolution and allow the chemistry to evolve for 10 Myr from the assumed starting conditions in which all species are atomic and neutral.", "Both models generally agree.", "The gas-phase water peaks in a region near the boundary.", "Here there is enough external UV to rapidly desorb the water from the ice, but not so much as to dissociate all the molecules.", "Further inward, the abundance of water falls as the gas density and the dust-gas collision rate (freeze-out rate) both increase while the photodesorption rate decreases with the attenuation of the UV radiation.", "At high A$\\rm _v$ , the water is desorbed only by cosmic rays and the UV radiation they produce in collisions with H$_2$ .", "The general agreement between the two models suggests that the simple model includes the processes that are significant in the cold environment.", "The rate equations for the processes selected for the simplified model are listed in the appendix (§).", "Our simple model calculates equilibrium abundances.", "We can estimate the equilibrium time scale from the combined rates for creation and destruction [8], $t = \\frac{t_{creation}t_{destruction}}{t_{creation} + t_{destruction}}$ where the time scales are the inverses of the rates.", "Figure REF shows the equilibrium time scales for each species as a function of radius.", "These may be compared with the time for the hydrodynamic evolution.", "A cloud with a mass of 10 M$_\\odot $ and a central density of $2\\times 10^6$ cm$^{-3}$ has a free-fall time, $t_{ff}=0.03$ Myr using the central density in the standard equation whereas the sound crossing time is about 2 Myr [22].", "Because the chemical time scales are all shorter than the dynamical time scales the chemistry reaches equilibrium before the conditions, density, temperature, and UV flux change.", "In this estimate of the time scale for chemical evolution, we are asking whether the oxygen chemistry in the contracting molecular cloud can maintain equilibrium as the cloud evolves dynamically.", "This is different from the question of how long it would take for the chemistry to equilibrate if the gas were held at molecular conditions but evolving from an atomic state.", "Figure: Model of a slowly contracting cloud in quasi-staticunstable equilibrium.", "The log of the density profile incm -3 ^{-3} is shown in blue (dotted line), the fractionalabundance of H 2 _2O with respectto H 2 _2 is shown in green (dashed line),the velocity as the black (solid) line, andthe gas temperature as the red (dot-dashed) line.The model spectrum is shown in figure .Figure: Simplified model of the oxygen chemistry in a cold cloud.The model includes 3 gas-phase species and H 2 _2O ice.The significant reactions at cold temperatures (T <300<300 K) are thefreeze-out of molecules colliding with dust grains, cosmic ray andphotodesorption of the ice, and photodissociation of the gas phase molecules.Figure: Abundances of oxygen species as a function of A V A_Vfor the model of L1544 based on aslowly contracting Bonnor-Ebert sphere.", "The figure emphasizes the variationof abundances in the PDR at the edge.The figure comparesthe abundances for the physical conditions in (figure ) fromtwo models: (dashed lines) (courtesy E. Bergin);and our simplified model (figure ).Figure shows abundances from the samemodels but plotted against log radius to emphasize the variationsin the center.Figure: Abundances of oxygen species, same as figure ,except plotted against the log of the radius rather than visual extinction.This figure emphasizes the variations in abundance in the center.The figure shows the H 2 _2O abundance calculated with our simplified modelusing two values for the cosmic ray-induced UV photodesorption(equation ).", "The solid green line shows the abundance calculatedwith factor α=10 -4 \\alpha = 10^{-4}.", "The dotted line shows the abundance calculatedwith factor α=10 -3 \\alpha = 10^{-3}.", "The abundance calculated with the model assumes α=10 -3 \\alpha = 10^{-3} (dashed line).Figure: Time scales for chemical equilibrium.", "From top to bottom, the three linesshow the equilibration time scales for H 2 _2O, OH, and O calculated fromequation and the reaction rates in the appendix." ], [ "Radiative Transfer", "We use our radiative transfer code MOLLIE [25], [24] to compute model H$_2$ O spectra to compare with the Herschel observation.", "Here we encounter an interesting question.", "The large Einstein A coefficient of the H$_2$ O ($1_{10}-1_{01}$ ) line results in optical depths across the cloud of several hundred to a thousand depending on excitation.", "High optical depths generally result in radiative trapping and enhanced excitation of the line.", "In this case, the line brightness could have a non-linear relationship to the column density.", "For example, the line could be saturated.", "On the other hand, the large Einstein A means that the critical density for collisional de-excitation is quite high ($1\\times 10^8$ cm$^{-3}$ ) at the temperatures $< 15$ K, higher than the maximum density ($1\\times 10^7$ cm$^{-3}$ ) in our dynamical model of L1544.", "This suggests that the line emission should be proportional to the column density.", "This question was addressed by [28] who proposed a solution using the escape probability approximation [18].", "They assumed a two level molecule, equal statistical weights in both levels, and the mean radiation field, $\\bar{J}$ , set by the escape probability, $\\beta $ , $ \\bar{J} = J_0\\beta + (1-\\beta )S$ where $J_0$ is the continuum from dust and the cosmic microwave background, $S$ is the line source function, and $ \\beta = (1-\\exp {(-\\tau )}) / \\tau .$ After a satisfying bout with three pages of elementary algebra and some further minor approximations, they show that as long as $C< A/\\tau $ , the line brightness is linearly dependent on the column density, no matter whether the optical depth is low or high, provided that the line is not too bright.", "To determine whether the water emission line brightness in L1544 has a non-linear or linear dependence, we numerically solve the equations for the two-level molecule with no approximations other than the escape probability and plot the result.", "Figures REF and REF show the dependence of the antenna temperature on the density for low and high densities respectively.", "Since the column density, the optical depth, and the ratio C/A are all linearly dependent on the density, any of these may be used on the abcissa.", "The latter two are shown just above the axis.", "Figure REF shows that the antenna temperature of the water line emission is linearly dependent on the column density even at high density or high optical depth.", "Figure REF shows that the linear relation breaks down when C/A is no longer small.", "The densities in both figures show that the water line emission in L1544 is in the linear regime.", "Figure: The dependence of the observed antennatemperature of the H 2 _2O (1 10 -1 01 1_{10} - 1_{01}) lineon the H 2 _2 number density (cm -3 ^{-3}).Because the optical depth and theratio of the collision rate to spontaneous emission rate (C/A) areboth linearly dependent on the density, the abscissa can be labeled inthese units as well.", "Both are shown above the axis.", "The antennatemperature is linearly dependent on the density or column density evenat very high optical depth as long as the ratio C/A is small.Figure: The dependence of the observed antenna temperature of the H 2 _2O line(1 10 -1 01 1_{10} - 1_{01}) on the number density.Same as figure but at higher densities where theratio C/A is no longer small and thedependence of the antenna temperature on the density is no longer linear.For an intuitive explanation, suppose that a photon is absorbed on average once per optical depth of one.", "A photon may be absorbed and another re-emitted many times in escaping a cloud of high optical depth.", "The time scale for each de-excitation is $A^{-1}$ .", "Therefore, the time that it takes a photon to escape the cloud is $\\tau /A$ .", "As long as this time is shorter than the collisional excitation time ($1/C$ ), then on average, an emitted photon will escape the cloud before another photon is created by the next collisional excitation event and radiative de-excitation.", "In this case, the line remains subcritically excited.", "The molecules are in the lower state almost all the time.", "This is the same condition that would prevail if the cloud were optically thin ($\\bar{J}=0$ or $\\beta =1$ ).", "On this basis, in our earlier paper we determined the emissivity and opacity of the H$_2$ O line in L1544 by setting $\\bar{J}=0$ [6].", "This approximation was earlier adopted in analyzing water emission observed by the SWAS satellite [37] where it is referred to as \"effectively optically thin\".", "In this current paper, we seek an improved estimate of $\\bar{J} > 0$ and $\\beta < 1$ by using the escape probability formalism as suggested by [28].", "We determine $\\beta $ using the local velocity gradient as given by our hydrodynamical model along with the local opacity using the Sobolev or large velocity gradient (LVG) approximation [18].", "We use the 6-ray approximation for the angle averaging.", "We allow for one free scaling parameter on $\\beta $ to match the modeled emission line brightness to the observation.", "We scale the LVG opacity by 1/2.", "Because the opacity, column density, and line brightness, are all linearly related, the scaling could be considered to derive from any or any combination of these parameters.", "Given all the uncertain parameters, for example the mean grain cross-section which also affects the line brightness (appendix ), this factor of 2 is not significant.", "An alternative method to calculate the excitation is the accelerated $\\Lambda $ -iteration algorithm (ALI).", "We do not know if this method is reliable with the extremely high optical depth, several hundred to a thousand.", "$\\Lambda $ -iteration generally converges, but whether it converges to the correct solution cannot be determined from the algorithm itself [31].", "The excitation may be uncertain, but analysis with the escape probability method allows us to understand the effect of the uncertainty.", "For example, because we know that the dependence of the line brightness on the opacity or optical depth is linear, we can say that any uncertainty in excitation results in the same percentage uncertainty in the abundance of the chemical model, or the pathlength of the structural model.", "Once $\\bar{J}$ is determined everywhere in the cloud, the equations of statistical equilibrium are solved to determine the emissivity and opacity.", "These are then used in the radiative transfer equation to produce the simulated spectral line emission and absorption.", "This calculation is done in MOLLIE in the same way as if $\\bar{J}$ were determined by any other means, for example, by $\\Lambda $ -iteration.", "Both the emissivity and opacity depend on frequency through the Doppler shifted line profile function [18] that varies as a function of position in the cloud.", "We use a line profile function that is the thermal width plus a microturbulent Gaussian broadening of 0.08 km s$^{-1}$ derived from our CO modeling [22].", "By the approximation of complete frequency redistribution [31], both have the same frequency dependence.", "This also implies that each photon emitted after an absorption event has no memory of the frequency of the absorbed photon.", "It is emitted with the frequency probability distribution described by the line profile function Doppler shifted by the local velocity along the direction of emission.", "We also assume complete redistribution in angle.", "Figure REF shows the modeled line profile against the observed profile.", "The V$_{LSR}$ is assumed to be 7.16 kms$^{-1}$ , slightly different than 7.2 kms$^{-1}$ used in [6].", "The lower value is chosen here as the best fit to the H$_2$ O observation.", "The combination of blue-shifted emission and red-shifted absorption is the inverse P-Cygni profile characteristic of contraction, with the emission and absorption split by the inward gas motion in the front and rear of the cloud.", "The absorption against the dust continuum is unambiguously from the front side indicating contraction rather than expansion.", "This profile has also been seen in other molecules in other low-mass cold, dense clouds, with the absorption against the dust continuum [12].", "In L1544, because the inward velocities are below the sound speed, and the H$_2$ O line width is just larger than thermal, the emission is shifted with respect to the absorption by less than a line width.", "In the observations, what appears to be a blue-shifted emission line is just the blue shoulder and wing of the complete emission, most of which is brighter, redder and wider than the observed emission.", "Our model also shows weaker emission to the red of the absorption line.", "This emission is from inward moving gas in the front side of the contracting zone.", "Again most of the emission is absorbed by the envelope and only the blue shoulder of the line is seen.", "The asymmetry between the red and blue emission comes about because the absorbing envelope, which is on the front side of the cloud, is closer in velocity to inward flowing gas (red) on the front side of the contraction.", "This is the same effect that produces the blue asymmetric or double-peaked line profiles seen in contraction in molecular lines without such significant envelope absorption [1].", "The model shows more red emission than is seen in the observations.", "This red emission may be absorbed by foreground gas that is not in the model.", "Figure 1 of [6] shows additional red shifted absorption in H$_2$ O and red shifted emission in CO, both centered around 9 kms$^{-1}$ .", "The blue wing of this red shifted water line may be absorbing the red wing of the emission from the dense cloud.", "If L1544 were static, no inward contraction, the emission from the center would be at the same frequency as the envelope.", "Because of the extremely high optical depth, the absorption line is saturated and would absorb all the emission.", "We would see only the absorption line.", "The depth of the absorption line is set by brightness of the dust continuum which is weak (0.011 K) and not by the optical depth of the line which is high (few hundred to a thousand).", "In the current radiative transfer calculation, we also use a slightly different collisional excitation rate than before.", "The collisional rates for ortho-H$_2$ O are different with ortho and para-H$_2$ .", "In our previous paper [6] we modeled the H$_2$ ortho-to-para ratio as a lower limit 1:1 or higher.", "Here we assume that almost all the hydrogen, 99.9%, is in the para state.", "This is suggested by recent chemical models that require a low ortho-to-para ratio to produce the high deuterium fraction observed in cold, dense clouds.", "[26], [36]." ], [ "Interpretation", "The shape of the line profile (figure REF ) is unaffected by any uncertainty in the excitation which scales the emission across the spectrum.", "The absorption is saturated and does not scale with the excitation.", "Because of the very high critical density for collisional de-excitation, we know that the line emission is generated only in the densest gas ($>10^6$ cm$^{-3}$ ) within a few thousand AU of the center.", "Thus the observation of the inverse P-Cygni profile seen in H$_2$ O confirms the model for quasi-hydrostatic contraction with the highest velocities near the center (figure REF ).", "The chemical model requires external UV to create the gas phase water by photodesorption.", "This confirms the physical model of L1544 as a molecular cloud bounded by a photodissociation region.", "The UV flux necessarily creates a higher temperature, up to about 100 K at the boundary by photoelectric heating.", "This helps maintain the pressure balance at the boundary consistent with the model of a BE sphere." ], [ "Uncertainties", "The comparison of the simulated and observed spectral line involves three models each with multiple parameters.", "Unavoidably the choice of parameters in any one of the three models affects not only the choice of other parameters in the other two models but also the interpretation.", "It would be a mistake to focus on the uncertainties in any one of the models to the exclusion of the others.", "For example, because of the linear relationship between the line brightness, the optical depth, and the opacity, uncertainties in the excitation, pathlength, and abundance, have equal effect on the spectrum.", "A factor of two uncertainty in the excitation can be compensated by a factor of two in the pathlength or a factor of two in the abundance of H$_2$ O.", "The pathlength is unknown.", "On the plane of the sky, L1544 has an axial ratio of 2:1, but we are using a spherical model for the cloud.", "Our rates in the chemical model involve estimation of the surface density of sites for desorption and the covering fraction of water ice on the grains.", "The latter is assumed to be one even though we know that CO and methane ice, not included in the simple model, make up a significant fraction of the ice mantle.", "The radiative excitation, parameterized as $\\beta $ in the escape probability is also uncertain because of the competing effects of high optical depth and subcritical excitation.", "On a linear plot, a factor of two difference in the brightness of the simulated and observed spectral line looks to be a damning discrepancy.", "However, there is at least this much uncertainty in each of the three models and this does not significantly affect the conclusions of the study, namely that the cloud can be modeled as a slowly contracting BE sphere bounded by a photodissociation region with the gas phase water abundance set by grain surface reactions.", "In this paper, we concentrate on the observation of H$_2$ O, but there are also other constraints that define the model.", "These are both observational and theoretical.", "In an earlier paper, we showed how observations of CO and N$_2$ H$^+$ define the physical model with the two spectral lines giving us information on the outer and inner regions of the cloud respectively.", "In this regard, the water emission gives us information in the central few thousand AU of the cloud where the density approaches or exceeds the critical density for de-excitation.", "This small volume of rapid inflow and high density does not much affect the N$_2$ H$^+$ spectrum which is generated in a much larger volume, and has no affect at all on the CO spectrum.", "A successful model for L1544 has to satisfy the constraints of all the data.", "On the theoretical side, there is an infinite space of combinations of abundance, density, velocity, and temperature that would form models that match the data.", "Only models that are physically motivated are of interest.", "It may be tempting to change the abundances, velocities, or densities arbitrarily, but this is unlikely to be a useful exercise giving the infinite possibilities.", "A successful model for L1544 has to be relevant to plausible theory.", "There is a natural prejudice for more complex models that in principle contain more details.", "The goal of our simplified models is to enhance our understanding of the most significant phenomena.", "In our research on cold, dense clouds, spanning a number of papers, we have developed simplified models for the density and temperature structure, for the dynamics including oscillations, for the CO chemistry, and in this paper simplified models for H$_2$ O chemistry and radiative transfer.", "Each of these models isolates one or a few key physical processes and shows how they generate the observables and operate to control the evolution toward star formation.", "Figure: Observed spectrumof H 2 _2O (1 10 -1 01 _{10} - 1_{01}) (black lines with crosses)compared with modeled spectrum (simple red line)for slow contraction at the time thatthe central density reaches 1×10 7 1 \\times 10^7 cm -3 ^{-3}.The model structure is shown in figure ." ], [ "Conclusions", "A simplified chemical model for cold oxygen chemistry primarily by grain surface reactions is verified by comparing the simulated spectrum of the H$_2$ O ($1_{10}-1_{01}$ ) line against an observation of water vapor in L1544 made with HIFI spectrometer on the Herschel Space Observatory.", "This model reproduces the observed spectrum of H$_2$ O, and also approximates the abundances calculated by a more complete model that includes gas-phase neutral-neutral and ion-neutral reactions.", "The gas phase water is released from ice grains by ultraviolet (UV) photodesorption.", "The UV radiation derives from two sources: external starlight and collisions of cosmic rays with molecular hydrogen.", "The latter may be important deep inside the cloud where the visual extinction is high enough ($>50$ mag) to block out the external UV radiation.", "Water is removed from the gas phase by photodissociation and freeze-out onto grains.", "The former is important at the boundary where the UV from external starlight is intense enough to create a photodissociation region.", "Here, atomic oxygen replaces water as the most abundant oxygen species.", "In the center where the external UV radiation is completely attenuated, freeze-out is the significant loss mechanism.", "Time dependent chemistry is not required to match the observations because the time scale for the chemical processes is short compared to the dynamical time scale.", "The molecular cloud L1544 is bounded by a photodissociation region.", "The water emission derives only from the central few thousand AU where the gas density approaches the critical density for collisional de-excitation of the water line.", "In the model of hydrostatic equilibrium, the gas density in the center is rising with decreasing radius more steeply than the abundance of water is decreasing by freeze-out.", "Thus the water spectrum provides unique information on the dynamics in the very center.", "The large Einstein A coefficient ($3\\times 10^{-3}$ s$^{-1}$ ) of the 557 GHz H$_2$ O ($1_{10}-1_{01}$ ) line results in extremely high optical depth, several hundred to a thousand.", "However, the density ($< 10^7$ cm$^{-3}$ ) and temperature ($<15$ K) are low enough that the line is subcritically excited.", "The result is that the line brightness under these conditions is directly proportional to the column density." ], [ "Acknowledgements", "The authors acknowledge Simon Bruderer, Fabien Daniel, Michiel Hogerheijde, Joe Mottram, Floris van der Tak for interesting discussions on the radiative transfer of water.", "PC acknowledges the financial support of the European Research Council (ERC; project PALs 320620), of successive rolling grants awarded by the UK Science and Technology Funding Council.", "JR acknowledges the financial support of the Submillimeter Array Telescope." ], [ "Cooling by atomic oxygen fine structure lines", "The fine structures lines of C$^+$ and atomic O are the major coolants in the diffuse ($n < 1000$ cm$^{-3}$ ), photodissociated gas around the molecular clouds.", "The more important coolant at temperatures less than 100 K is C$^+$ .", "At higher temperatures, oxygen becomes increasingly important in the energy balance.", "The reason is that the 63.2 and 145.6 $\\mu $ m fine structure lines of atomic oxygen have upper states $^3$ P$_1$ and $^3$ P$_0$ that are at 228 K and 326 K above ground, considerably higher than the 92 K of the upper state, $^2$ P$_{3/2}$ of the 157.6 $\\mu $ m fine structure line of C$^+$ .", "The cooling by atomic oxygen is simple to model because atomic oxygen is a product of photodissociation and is therefore abundant only in gas with low A$_{\\rm v}$ implying gas densities below the critical densities for collisional de-excitation, 6400 and 3400 cm$^{-3}$ for the 63.2 and 145.6 $\\mu $ m lines respectively [38].", "At this density, we assume that the optically thin approximation applies.", "In this case, every collisional excitation to an upper state of the fine structure lines results in spontaneous emission that escapes the cloud and cools the gas, $\\Lambda _{\\rm O} = n({\\rm O})n({\\rm H_2}) (E_{21}C_{21} + E_{20}C_{20})\\ \\ {\\rm ergs}\\ {\\rm cm}^{-3}\\ {\\rm s}^{-1}$ where the upward collision rates are, $C_{21} = 1.4\\times 10^{-8}\\frac{g_1}{g_2} C_{12} \\exp {(-E_{21} / kT)} \\sqrt{T}\\ \\ {\\rm cm}^3 {\\rm s}^{-1}$ $C_{20} = 1.4\\times 10^{-8}\\frac{g_0}{g_2} C_{02} \\exp {(-E_{20} / kT)} \\sqrt{T}\\ \\ {\\rm cm}^3 {\\rm s}^{-1}.$ and the statistical weights are $g_2 = 5$ , $g_1 = 3$ , and $g_0 = 1$ and the transition energies are $E_{12}/k = 228$ K and $E_{02}/k = 326$ K." ], [ "Freeze-out", "Molecules freeze onto dust grains, sticking when they collide.", "This process is easily modeled.", "We follow [21] to calculate the collision timescale.", "The time scale for depletion onto dust may be estimated as [34], $\\tau _{on} = (S_0 R_{dg} n({\\rm H_2}) \\sigma V_T)^{-1} \\ {\\rm s}$ Here $S_0$ is the sticking coefficient, with $S_0=1$ meaning that the colliding molecule always sticks to the dust in each collision; $R_{dg}$ is the ratio of the number density of dust grains relative to molecular hydrogen; $\\sigma $ is the mean cross-section of the dust grains; and $V_T$ is the relative velocity between the grains and the gas.", "If the grains have a power law distribution of sizes with the number of grains of each size scaling as the -3.5 power of their radii [29], then we can estimate their mean cross-section as, $\\langle \\sigma \\rangle = \\bigg (\\int ^{a_2}_{a_1} n(a)da\\bigg )^{-1}\\int ^{a_2}_{a_1} n(a) \\sigma (a) da,$ where $a_1$ and $a_2$ are the minimum and maximum grain sizes.", "If $a_1 = 0.005$ $\\mu $ m and $a_2 = 0.3$ $\\mu $ m, then $\\langle \\sigma \\rangle = 3.4\\times 10^{-4}$ $\\mu {\\rm m}^2$ .", "Similarly, the ratio of the number densities of dust and gas may be estimated by computing the mean mass of a dust grain and assuming the standard gas-to-dust mass ratio of 100.", "If the density of the dust is 2 grams cm$^{-3}$ , then the ratio of number densities is $R_{dg} = 4\\times 10^{-10}$ .", "Consistent with [21], our model has a slightly lower value for the grain cross-section, $1.4 \\times 10^{-21}$ cm$^2$ , than [16], $\\sigma _h = 2 \\times 10^{-21}$ cm$^2$ .", "Both values are per hydrogen nucleus (2H$_2$ + H).", "Because the ice forms and desorbs off the grain surfaces, larger values of the average cross-section result in fewer molecules in the gas phase.", "The actual properties of grains in cold clouds are somewhat uncertain.", "The relative velocity due to thermal motion is, $V_T = \\bigg ( {{8kT}\\over {\\pi \\mu }} \\bigg )^{1/2},$ where $T$ is the temperature and $\\mu $ the molecular weight.", "The freeze-out rate for species $i$ is, $f_i = \\tau _{on}^{-1} n(H_2) \\ \\ {\\rm cm}^{-3}\\ {\\rm s}^{-1}$" ], [ "Gas-phase reactions", "The neutral-neutral molecular and photodissociation reactions are from [39].", "The reaction rate $k_1$ for ${\\rm O + H_2} \\rightarrow {\\rm OH + H}$ is, $k_1 = 3.1 \\times 10^{-13} \\ (T/300)^{2.7} \\exp {(-3150/T)} \\ \\ {\\rm cm}^{3}\\ \\ {\\rm s}^{-1}$ The reaction $k_2$ for ${\\rm OH + H_2} \\rightarrow {\\rm H_2O + H}$ is, $k_2 = 2.0 \\times 10^{-12} (T/300)^{1.57}\\exp {(-1736/T)} \\ \\ {\\rm cm}^{3}\\ \\ {\\rm s}^{-1}$ OH + H has a rate, $5.3 \\times 10^{-18} (T/300)^{-5.22} \\exp { (-90/T) }.$ All three of these reactions have an activation barrier and are irrelevant at temperatures below 300 K. The photodissociation rate for the destruction of OH and the formation of O is, $P_1 = 3.5 \\times 10^{-10} G_0 \\exp {(-1.7 A_V)} \\ \\ {\\rm s}^{-1}$ and the rate for the destruction of H$_2$ O and formation of OH is, $P_2 = 5.9 \\times 10^{-10} G_0 \\exp {(-1.7 A_V)} \\\\ \\ {\\rm s}^{-1}.$ The unitless parameter $G_0 = 1$ corresponds to the average local interstellar radiation field in the FUV band [15].", "$A_V$ is the visual extinction." ], [ "Desorption", "The desorption rates are from [16].", "The total desorption rate includes thermal desorption, photodesorption, and desorption by cosmic rays.", "We use equation 2 from [16] for the rate for thermal desorption, $D_{Th} = 1.6 \\times 10^{11} \\bigg ( \\frac{E_i}{k} \\bigg )^{1/2}\\bigg (\\frac{m_H }{ m_i}\\bigg )^{1/2}\\exp {\\bigg (\\frac{-E_i}{kT_{gr}}\\bigg )} \\ \\ {\\rm s}^{-1} \\ \\ {\\rm molecule}^{-1}$ where $E_i/k$ , the adsorption energy is 800, 1300, and 5770 K for O, OH, and H$_2$ O respectively, and m$_i$ /m$_{\\rm H}$ is the weight of the species with respect to H. The thermal desorption rate for water is negligible at the temperatures ($<15$ K) of cold, dense clouds.", "For the cosmic-ray desorption rate, we use equation 8 from [16].", "We include only the cosmic-ray desorption rate for H$_2$ O, $D_{CR} = 4.4\\times 10^{-17} {\\rm molecule}^{-1} {\\rm s}^{-1}.$ Both the thermal desorption rate and the cosmic ray desorption rate in units of molecule$^{-1}$ s$^{-1}$ are multiplied by the number of molecules on the surface of grains per molecule of H$_2$ which is $ N_s f_s A_{gr} R_{dg} $ where $N_s = 10^{15} $ cm$^{-2}$ is the number of desorption sites per cm$^2$ on the grain surface [16], $f_s = 1$ is the fraction of the grain surface covered by ice, the average surface area of a grain is 4 times the grain cross-section, $A_{gr} = 4\\sigma = 4 \\times 3.4 \\times 10^{-4}$ $\\mu $ m $^2$ [21], and the dust-to-gas ratio $R_{dg} = 4\\times 10^{-10}$ [21].", "The photodesorption rates are from equations 6 and 7 [16], $D_{UV} = G_0 F_0 Y_i f_i \\ \\exp {(-1.8A_V)}\\ {\\rm s}^{-1}$ where $F_0 = 10^8$ is the number of UV photons per Habing flux, and $Y_i = 10^{-3}$ and $2\\times 10^{-3}$ are the photodesorption yields per UV photon per second for the production of OH and H$_2$ O respectively from table 1 of [16].", "We assume that all the ice is H$_2$ O and follow [16] in assuming that the photodesorption of this water ice results in twice as much OH as H$_2$ O in the gas phase.", "The desorption of water ice does not result in the production of gas phase oxygen, and we have no oxygen ice in our model.", "To provide a back reaction to the freeze-out of atomic oxygen, we arbitrarily assign a desorption rate equal to that of water.", "In regions of high extinction ($A_V > 4$ ) this results in a gas phase abundance of atomic oxygen that is approximately the same as predicted by [16].", "This is $< 0.001$ of the total oxygen and has no effect on the other abundances.", "In the outer part of the cloud where the UV flux is higher ($A_V < 4$ ) most of the atomic oxygen derives from photodissociation.", "Here the UV desorption off grains is insignificant.", "Additional desorption is caused by the UV photons emitted by hydrogen excitation by energetic electrons released in the ionization of hydrogen by cosmic rays.", "We follow [35] and scale this process as $10^{-4}$ of one Habing flux, $G_0 = 1$ , so that, $D_{CR\\ UV} = \\alpha G_0 F_0 Y_i f_i \\ {s^{-1}}$ with $\\alpha = 10^{-4}$ ." ], [ "Equilibrium", "In equilibrium, the rate equations in matrix form are, $\\begin{array}{lll}\\begin{pmatrix}-(f_{O} + k_1) &P_{1} &0 &0 \\\\k_1 &-(f_{OH} + P_1) &P_{2} &0 \\\\0 &k_2 &-(f_{H_2O} + P_2) &D_{H_2O} \\\\f_{O} &f_{OH} & f_{H_2O} &-(D_{OH} + D_{H_2O}) \\\\1 &1 &1 &1 \\\\\\end{pmatrix}\\begin{pmatrix}O \\\\OH \\\\H_2O \\\\ICE \\\\\\end{pmatrix} =\\begin{pmatrix}0 \\\\0 \\\\0 \\\\0 \\\\1 \\\\\\end{pmatrix}\\end{array}$ where the last row is the conservation equation for oxygen among all the species.", "As written, this system is overdetermined, but can be solved by dropping any one of the first 4 rows." ], [ "H$_2$ O ortho-para ratio", "Since the ortho state of H$_2$ O is 24K above the para state, the O/P ratio in thermal equilibrium is very small at lower temperatures [16].", "However, when the water molecule is formed, created from OH on the grain surface for example, it is formed in the ratio of the available quantum states, ortho:para 3:1.", "The ortho and para states of H$_2$ O equilibrate by collisions with H or H$_2$ .", "If the chemical equilibrium time scale is much shorter than the thermal equilibrium time scale, the O/P ratio will not deviate much from 3:1.", "Observations generally show ratios close to 3:1 [41].", "We have not found previous research on the equilibration of H$_2$ O, but an appreciation of the time scale can be estimated from previous research on the equilibration of the ortho and para states of molecular hydrogen.", "The dissociation energies of H-H and OH-H are not too different nor the collisional cross-sections of the molecules.", "[9] and [14] suggest three processes for the equilibration of the ortho and para states of H$_2$ are: (1) gas phase H exchange, (2) gas phase paramagnetic conversion with H$_2$ , and (3) H exchange on a surface.", "We assume that these same processes are applicable to the water.", "The rates for these processes scale with the gas density through the collision rate and scale as the inverse exponential of the temperature.", "Scaling the rates for H$_2$ from the conditions in the atmosphere of Jupiter to rarefied, cold gas of the interstellar medium (10 K and $10^6$ cm$^{-3}$ ) the time scales for these processes are all $> 1$ Gyr.", "In contrast, the chemical time scale is very much shorter (figure REF ) throughout the cloud.", "In this model, water is dissociated in the gas phase by photodissociation and also coming off the grain surfaces by photodesorption in which gas phase OH is produced twice as often as gas phase H$_2$ O.", "The equilibrium comparison between ortho-para equilibration and chemistry may not be needed because the equilibration time scale exceeds the expected life times of the cold, dense, clouds." ] ]	1403.0155
[ [ "Stochastic differential equation with jumps for multi-type continuous\n state and continuous time branching processes with immigration" ], [ "Abstract A multi-type continuous state and continuous time branching process with immigration satisfying some moment conditions is identified as a pathwise unique strong solution of certain stochastic differential equation with jumps." ], [ "Introduction", "Continuous state and continuous time branching processes with immigration (CBI processes) arise as high density limits of Galton–Watson branching processes with immigration, see, e.g., Li [13] without immigration and Li [12] with immigration.", "A single-type continuous state and continuous time branching process (CB process) is a non-negative Markov process with a branching property.", "This class of processes has been first introduced by Jiřina [9] both in discrete and continuous times.", "As a generalization of CB processes, Kawazu and Watanabe [11] introduced the more general class of CBI processes, where immigrants may come from outer sources.", "They defined a single-type CBI process as an $[0,\\infty ]$ -valued Markov process with $\\infty $ as a trap in terms of Laplace transforms, see [11].", "An analytic characterization of CBI processes was also presented by giving the explicit form of the corresponding non-negative strongly continuous contraction semigroup, see [11].", "Further, limit theorems for Galton-Watson branching processes with immigration towards CBI processes were also investigated, see [11].", "Dawson and Li [2] proved that a general single-type CBI process is the pathwise unique strong solution of a stochastic differential equation (SDE) with jumps driven by Wiener processes and Poisson random measures.", "Watanabe [16] introduced two-type CB processes as $[0,\\infty )^2$ -valued Markov processes satisfying a branching property.", "He characterized them in an analytic way by giving the explicit form of the infinitesimal generator of the corresponding non-negative strongly continuous contraction semigroup, see Watanabe [16].", "Fittipaldi and Fontbona [5] represented a (sub)-critical continuous time and continuous state branching process conditioned to never be extinct as a pathwise unique strong solution of an appropriate SDE with jumps.", "It was also shown that a two-type diffusion CB process can be obtained as a pathwise unique strong solution of an SDE (without jumps), see Watanabe [16].", "Recently, for a special two-type (not necessarily diffusion) CBI process (with a special immigration mechanism), an SDE with jumps (a special case of the SDE (REF ) given later on) has already been presented by Ma [14] together with the existence of a pathwise unique $[0,\\infty )^2$ -valued strong solution of this SDE.", "For a comparison of our results with those of Ma [14], see Section .", "The aim of the present paper is to derive and study an SDE with jumps for a general multi-type CBI process.", "Next, we give an overview of the structure of the paper by explaining some of its technical merits and including some sort of preview of the types of results which are proved.", "In Section we recall some facts about CBI processes (e.g., set of admissible parameters, infinitesimal generator) with special emphasis on their identification (under some moment conditions) as special immigration superprocesses.", "This identification turns out to be very important since it is the starting point for deriving a formula for the expectation and an SDE with jumps for a general multi-type CBI process (see the proofs of Lemma REF and Theorem REF ).", "In Section we formulate an SDE with jumps and, under the same moment conditions, we prove that this SDE admits an $[0,\\infty )^d$ -valued weak solution which is unique in the sense of probability law among $[0,\\infty )^d$ -valued weak solutions.", "The idea behind of deriving such an SDE goes back to a result of Li [13] that an immigration superprocess can be represented as a sum of a continuous local martingale, a purely discontinuous local martingale and a drift term.", "In our special case, this purely discontinuous local martingale takes the form $\\int _0^t \\int _{[0,\\infty )^d\\setminus \\lbrace {0}\\rbrace } {z}\\widetilde{N}_0(\\mathrm {d}s,\\mathrm {d}{z})$ , $t\\geqslant 0$ , with some (not necessarily Poisson) random measure $N_0(\\mathrm {d}s,\\mathrm {d}{z})$ on $(0,\\infty )\\times ([0,\\infty )^d\\setminus \\lbrace {0}\\rbrace )$ , where $\\widetilde{N}_0(\\mathrm {d}s,\\mathrm {d}{z})$ denotes the compensation of $N_0(\\mathrm {d}s,\\mathrm {d}{z})$ .", "The next key step is that the integral $\\int _0^t \\int _{[0,\\infty )^d\\setminus \\lbrace {0}\\rbrace } {z}\\widetilde{N}_0(\\mathrm {d}s,\\mathrm {d}{z})$ can be rewritten as an appropriate sum of integrals with respect to a Possion and compensated Poisson random measures, and some additional drift term, due to a representation theorem of right continuous martingales, see, e.g., Ikeda and Watanabe [7].", "We also prove that any $[0,\\infty )^d$ -valued weak solution of this SDE is a CBI process, see Theorem REF .", "For the proof of Theorem REF , we need a formula for the first moment of a CBI process, see Lemma REF .", "The proof of Lemma REF is based on a formula for expectation of immigration superprocesses, see Li [13].", "In Section we prove that, under the same moment conditions, there is a pathwise unique $[0,\\infty )^d$ -valued strong solution to the SDE (REF ) and the solution is a CBI process, see Theorem REF .", "For the proof, we need a comparison theorem for the SDE (REF ) (see, Lemma REF ), which, in particular, yields that pathwise uniqueness holds for the SDE (REF ) among $[0,\\infty )^d$ -valued weak solutions.", "The ideas of the proof of Lemma REF follow those of Theorem 3.1 of Ma [14], which are adaptations of those of Theorem 5.5 of Fu and Li [6].", "More precisely, we derive an upper bound for an appropriate deterministic function of the difference of two $[0,\\infty )^d$ -valued weak solutions of the SDE (REF ) and then apply Gronwall's inequality.", "In Section we specialize our SDE (REF ) to dimension 1 and 2, respectively, which enables us to compare our results with those of Dawson and Li [2] (single-type) and Ma [14] (two-type), respectively.", "Moreover, we discuss a special case of the SDE (REF ) with $\\nu = 0$ , $\\mu _i = 0$ , $i \\in \\lbrace 1, \\ldots , d\\rbrace $ , i.e., without integrals with respect to (compensated) Poisson random measures (corresponding to the so-called multi-factor Cox-Ingersoll-Ross process if ${B}$ is diagonal, see, e.g., Jagannathan et al.", "[8]), and another special case with ${c}= {0}$ , i.e., without integral with respect to a Wiener process.", "In Appendix we present some facts about extensions of probability spaces.", "Finally, we mention that our work goes beyond that of Ma [14] in the sense that we consider general multi-type CBI processes with arbitrary branching and immigration mechanisms instead of two-type CBI processes with a special immigration mechanism, and we carefully present some missing details in the proofs of Ma [14] for the general multi-type case such as the application of Theorem 9.18 in Li [13] and of Theorem 7.4 in Chapter II in Ikeda and Watanabe [7].", "Further, in a companion paper we established Yamada-Watanabe type results for SDEs with jumps that are needed in the proof of Theorem REF (existence of pathwise unique strong solution of the SDE (REF )).", "We point out that Ma [14] implicitly used these results without proving or referring to them." ], [ "Multi-type CBI processes", "Let $\\mathbb {Z}_+$ , $\\mathbb {N}$ , $\\mathbb {R}$ , $\\mathbb {R}_+$ and $\\mathbb {R}_{++}$ denote the set of non-negative integers, positive integers, real numbers, non-negative real numbers and positive real numbers, respectively.", "For $x , y \\in \\mathbb {R}$ , we will use the notations $x \\wedge y := \\min \\lbrace x, y\\rbrace $ and $x^+:= \\max \\lbrace 0, x\\rbrace $ .", "By $\\Vert {x}\\Vert $ and $\\Vert {A}\\Vert $ , we denote the Euclidean norm of a vector ${x}\\in \\mathbb {R}^d$ and the induced matrix norm of a matrix ${A}\\in \\mathbb {R}^{d\\times d}$ , respectively.", "The natural basis in $\\mathbb {R}^d$ and the Borel $\\sigma $ -algebras on $\\mathbb {R}^d$ and on $\\mathbb {R}_+^d$ will be denoted by ${e}_1$ , ..., ${e}_d$ , and by ${\\mathcal {B}}(\\mathbb {R}^d)$ and ${\\mathcal {B}}(\\mathbb {R}_+^d)$ , respectively.", "The $d$ -dimensional unit matrix is denoted by ${I}_d$ .", "For ${x}= (x_i)_{i\\in \\lbrace 1,\\ldots ,d\\rbrace } \\in \\mathbb {R}^d$ and ${y}= (y_i)_{i\\in \\lbrace 1,\\ldots ,d\\rbrace } \\in \\mathbb {R}^d$ , we will use the notation ${x}\\leqslant {y}$ indicating that $x_i \\leqslant y_i$ for all $i \\in \\lbrace 1, \\ldots , d\\rbrace $ .", "By $C^2_\\mathrm {c}(\\mathbb {R}_+^d,\\mathbb {R})$ we denote the set of twice continuously differentiable real-valued functions on $\\mathbb {R}_+^d$ with compact support.", "Throughout this paper, we make the conventions $\\int _a^b := \\int _{(a,b]}$ and $\\int _a^\\infty := \\int _{(a,\\infty )}$ for any $a, b \\in \\mathbb {R}$ with $a < b$ .", "Definition.", "2.1 A matrix ${A}= (a_{i,j})_{i,j\\in \\lbrace 1,\\ldots ,d\\rbrace } \\in \\mathbb {R}^{d\\times d}$ is called essentially non-negative if $a_{i,j} \\in \\mathbb {R}_+$ whenever $i, j \\in \\lbrace 1,\\ldots ,d\\rbrace $ with $i \\ne j$ , i.e., if ${A}$ has non-negative off-diagonal entries.", "The set of essentially non-negative $d \\times d$ matrices will be denoted by $\\mathbb {R}^{d\\times d}_{(+)}$ .", "Definition.", "2.2 A tuple $(d, {c}, {\\beta }, {B}, \\nu , {\\mu })$ is called a set of admissible parameters if $d \\in \\mathbb {N}$ , ${c}= (c_i)_{i\\in \\lbrace 1,\\ldots ,d\\rbrace } \\in \\mathbb {R}_+^d$ , ${\\beta }= (\\beta _i)_{i\\in \\lbrace 1,\\ldots ,d\\rbrace } \\in \\mathbb {R}_+^d$ , ${B}= (b_{i,j})_{i,j\\in \\lbrace 1,\\ldots ,d\\rbrace } \\in \\mathbb {R}^{d \\times d}_{(+)}$ , $\\nu $ is a Borel measure on $U_d := \\mathbb {R}_+^d \\setminus \\lbrace {0}\\rbrace $ satisfying $\\int _{U_d} (1\\wedge \\Vert {z}\\Vert ) \\, \\nu (\\mathrm {d}{z}) < \\infty $ , ${\\mu }= (\\mu _1, \\ldots , \\mu _d)$ , where, for each $i \\in \\lbrace 1, \\ldots , d\\rbrace $ , $\\mu _i$ is a Borel measure on $U_d$ satisfying $\\int _{U_d}\\left[ \\Vert {z}\\Vert \\wedge \\Vert {z}\\Vert ^2+ \\sum _{j \\in \\lbrace 1, \\ldots , d\\rbrace \\setminus \\lbrace i\\rbrace } z_j\\;\\right] \\mu _i(\\mathrm {d}{z})<\\infty .$ Remark.", "2.3 Our Definition REF of the set of admissible parameters is a special case of Definition 2.6 in Duffie et al.", "[4], which is suitable for all affine processes.", "Namely, one should take $m = d$ , $n = 0$ and zero killing rate in Definition 2.6 in Duffie et al.", "[4] noting also that part (v) of our Definition REF is equivalent to the corresponding one $\\int _{U_d} \\sum _{i=1}^d (1 \\wedge z_i) \\, \\nu (\\mathrm {d}{z}) < \\infty $ in Definition 2.6 in Duffie et al. [4].", "Indeed, $1 \\wedge \\Vert {z}\\Vert \\leqslant 1 \\wedge \\left( \\sum _{i=1}^d z_i \\right)\\leqslant \\sum _{i=1}^d (1 \\wedge z_i)\\leqslant d (1 \\wedge \\Vert {z}\\Vert )$ for all ${z}= (z_1, \\ldots , z_d) \\in \\mathbb {R}_+^d$ .", "Further, for all $i\\in \\lbrace 1,\\ldots ,d\\rbrace $ , the condition (REF ) is equivalent to $\\int _{U_d}\\left[ (1 \\wedge z_i)^2+ \\sum _{j \\in \\lbrace 1, \\ldots , d\\rbrace \\setminus \\lbrace i\\rbrace } (1 \\wedge z_j) \\right]\\mu _i(\\mathrm {d}{z})< \\infty \\quad \\text{and} \\quad \\int _{U_d} \\Vert {z}\\Vert \\mathbb {1}_{\\lbrace \\Vert {z}\\Vert \\geqslant 1\\rbrace }\\,\\mu _i(\\mathrm {d}{z})<\\infty .$ Indeed, if (REF ) holds, then $\\int _{U_d} \\Vert {z}\\Vert \\mathbb {1}_{\\lbrace \\Vert {z}\\Vert \\geqslant 1\\rbrace }\\,\\mu _i(\\mathrm {d}{z})= \\int _{U_d} (\\Vert {z}\\Vert \\wedge \\Vert {z}\\Vert ^2)\\mathbb {1}_{\\lbrace \\Vert {z}\\Vert \\geqslant 1\\rbrace }\\,\\mu _i(\\mathrm {d}{z})<\\infty $ , and using that $z_i\\leqslant \\Vert {z}\\Vert $ and $(1\\wedge z_i)^2 = (1\\wedge z_i)^2\\mathbb {1}_{\\lbrace \\Vert {z}\\Vert \\leqslant 1\\rbrace } + (1\\wedge z_i)^2\\mathbb {1}_{\\lbrace \\Vert {z}\\Vert > 1\\rbrace }\\leqslant \\Vert {z}\\Vert ^2 \\mathbb {1}_{\\lbrace \\Vert {z}\\Vert \\leqslant 1\\rbrace } + \\Vert {z}\\Vert \\mathbb {1}_{\\lbrace \\Vert {z}\\Vert > 1\\rbrace }= \\Vert {z}\\Vert \\wedge \\Vert {z}\\Vert ^2$ , $i\\in \\lbrace 1,\\ldots ,d\\rbrace $ , we have (REF ).", "If (REF ) holds, then, using again $z_j\\leqslant \\Vert {z}\\Vert $ , $j\\in \\lbrace 1,\\ldots ,d\\rbrace $ , we have $&\\int _{U_d}\\left[ \\Vert {z}\\Vert \\wedge \\Vert {z}\\Vert ^2+ \\sum _{j \\in \\lbrace 1, \\ldots , d\\rbrace \\setminus \\lbrace i\\rbrace } z_j\\;\\right] \\mu _i(\\mathrm {d}{z}) \\\\& = \\int _{U_d}\\left[ \\Vert {z}\\Vert ^2 + \\sum _{j \\in \\lbrace 1, \\ldots , d\\rbrace \\setminus \\lbrace i\\rbrace } z_j\\;\\right] \\mathbb {1}_{\\lbrace \\Vert {z}\\Vert < 1 \\rbrace } \\mu _i(\\mathrm {d}{z})+ \\int _{U_d}\\left[ \\Vert {z}\\Vert + \\sum _{j \\in \\lbrace 1, \\ldots , d\\rbrace \\setminus \\lbrace i\\rbrace } z_j\\;\\right] \\mathbb {1}_{\\lbrace \\Vert {z}\\Vert \\geqslant 1 \\rbrace } \\mu _i(\\mathrm {d}{z}) \\\\&\\leqslant \\int _{U_d}\\left[ z_i^2 + 2\\sum _{j \\in \\lbrace 1, \\ldots , d\\rbrace \\setminus \\lbrace i\\rbrace } z_j\\;\\right] \\mathbb {1}_{\\lbrace \\Vert {z}\\Vert < 1 \\rbrace } \\mu _i(\\mathrm {d}{z})+ \\int _{U_d} \\Vert {z}\\Vert \\mathbb {1}_{\\lbrace \\Vert {z}\\Vert \\geqslant 1\\rbrace }\\,\\mu _i(\\mathrm {d}{z})\\\\&\\quad + \\sum _{j \\in \\lbrace 1, \\ldots , d\\rbrace \\setminus \\lbrace i\\rbrace } \\int _{U_d} \\Vert {z}\\Vert \\mathbb {1}_{\\lbrace \\Vert {z}\\Vert \\geqslant 1\\rbrace }\\mu _i(\\mathrm {d}{z}) < \\infty , \\qquad i\\in \\lbrace 1,\\ldots ,d\\rbrace ,$ yielding (REF ).", "Note that, here the finiteness of the first integral in (REF ) is nothing else but condition (2.11) in Definition 2.6 in Duffie et al.", "[4], and the finiteness of the second integral in (REF ) is an additional condition that we assume compared to Duffie et al.", "[4], its role is explained in Remark REF .", "$\\Box $ Theorem.", "2.4 Let $(d, {c}, {\\beta }, {B}, \\nu , {\\mu })$ be a set of admissible parameters in the sense of Definition REF .", "Then there exists a unique conservative transition semigroup $(P_t)_{t\\in \\mathbb {R}_+}$ acting on the Banach space (endowed with the supremum norm) of real-valued bounded Borel-measurable functions on the state space $\\mathbb {R}_+^d$ such that its infinitesimal generator is $\\begin{split}({\\mathcal {A}}f)({x})&= \\sum _{i=1}^d c_i x_i f_{i,i}^{\\prime \\prime }({x})+ \\langle {\\beta }+ {B}{x}, {f}^{\\prime }({x}) \\rangle + \\int _{U_d} \\bigl ( f({x}+ {z}) - f({x}) \\bigr ) \\, \\nu (\\mathrm {d}{z}) \\\\&\\phantom{\\quad }+ \\sum _{i=1}^dx_i\\int _{U_d}\\bigl ( f({x}+ {z}) - f({x}) - f^{\\prime }_i({x}) (1 \\wedge z_i) \\bigr )\\, \\mu _i(\\mathrm {d}{z})\\end{split}$ for $f \\in C^2_\\mathrm {c}(\\mathbb {R}_+^d,\\mathbb {R})$ and ${x}\\in \\mathbb {R}_+^d$ , where $f_i^{\\prime }$ and $f_{i,i}^{\\prime \\prime }$ , $i \\in \\lbrace 1, \\ldots , d\\rbrace $ , denote the first and second order partial derivatives of $f$ with respect to its $i$ -th variable, respectively, and ${f}^{\\prime }({x}) := (f_1^{\\prime }({x}), \\ldots , f_d^{\\prime }({x}))^\\top $ .", "Moreover, the Laplace transform of the transition semigroup $(P_t)_{t\\in \\mathbb {R}_+}$ has a representation $\\int _{\\mathbb {R}_+^d} \\mathrm {e}^{- \\langle {\\lambda }, {y}\\rangle } P_t({x}, \\mathrm {d}{y})= \\mathrm {e}^{- \\langle {x}, {v}(t, {\\lambda }) \\rangle - \\int _0^t \\psi ({v}(s, {\\lambda })) \\, \\mathrm {d}s} ,\\qquad {x}\\in \\mathbb {R}_+^d, \\quad {\\lambda }\\in \\mathbb {R}_+^d , \\quad t \\in \\mathbb {R}_+ ,$ where, for any ${\\lambda }\\in \\mathbb {R}_+^d$ , the continuously differentiable function $\\mathbb {R}_+ \\ni t \\mapsto {v}(t, {\\lambda })= (v_1(t, {\\lambda }), \\ldots , v_d(t, {\\lambda }))^\\top \\in \\mathbb {R}_+^d$ is the unique locally bounded solution to the system of differential equations $\\partial _t v_i(t, {\\lambda }) = - \\varphi _i({v}(t, {\\lambda })) , \\qquad v_i(0, {\\lambda }) = \\lambda _i , \\qquad i \\in \\lbrace 1, \\ldots , d\\rbrace ,$ with $\\varphi _i({\\lambda }):= c_i \\lambda _i^2 - \\langle {B}{e}_i, {\\lambda }\\rangle + \\int _{U_d}\\bigl ( \\mathrm {e}^{- \\langle {\\lambda }, {z}\\rangle } - 1+ \\lambda _i (1 \\wedge z_i) \\bigr )\\, \\mu _i(\\mathrm {d}{z})$ for ${\\lambda }\\in \\mathbb {R}_+^d$ and $i \\in \\lbrace 1, \\ldots , d\\rbrace $ , and $\\psi ({\\lambda }):= \\langle {\\beta }, {\\lambda }\\rangle - \\int _{U_d}\\bigl ( \\mathrm {e}^{- \\langle {\\lambda }, {z}\\rangle } - 1 \\bigr )\\, \\nu (\\mathrm {d}{z}) , \\qquad {\\lambda }\\in \\mathbb {R}_+^d .$ Further, the function $\\mathbb {R}_+\\times \\mathbb {R}_+^d\\ni (t,{\\lambda }) \\mapsto {v}(t, {\\lambda })$ is continuous.", "Remark.", "2.5 This theorem is a special case of Theorem 2.7 of Duffie et al.", "[4] with $m = d$ , $n = 0$ and zero killing rate.", "The unique existence of a locally bounded solution to the system of differential equations (REF ) is proved by Li [13].", "Here, we point out that the moment condition given in part (vi) in Definition REF (which is stronger than the one (2.11) in Definition 2.6 in Duffie et al.", "[4]) ensures that the semigroup $(P_t)_{t\\in \\mathbb {R}_+}$ is conservative (we do not need the one-point compactification of $\\mathbb {R}_+^d$ ), see Duffie et al.", "[4] and Li [13].", "For the continuity of the function $\\mathbb {R}_+\\times \\mathbb {R}_+^d\\ni (t,{\\lambda }) \\mapsto {v}(t, {\\lambda })$ , see Duffie et al. [4].", "Finally, we note that the infinitesimal generator (REF ) can be rewritten in another equivalent form, see formula (REF ) in Lemma REF .", "$\\Box $ Definition.", "2.6 A conservative Markov process with state space $\\mathbb {R}_+^d$ and with transition semigroup $(P_t)_{t\\in \\mathbb {R}_+}$ given in Theorem REF is called a multi-type CBI process with parameters $(d, {c}, {\\beta }, {B}, \\nu , {\\mu })$ .", "In what follows, we will identify a multi-type CBI process $({X}_t)_{t\\in \\mathbb {R}_+}$ with parameters $(d, {c}, {\\beta }, {B}, \\nu , {\\mu })$ under a moment condition as a special immigration superprocess.", "First we parametrize the family of immigration superprocesses for which Theorem 9.18 in Li [13] is valid.", "We will use some notations of the book of Li [13].", "For a locally compact separable metric space $E$ , let us introduce the following function spaces: $B(E)$ is the space of bounded real-valued Borel functions on $E$ , $B(E)^+$ is the space of bounded non-negative real-valued Borel functions on $E$ , $C(E)$ is the space of bounded continuous real-valued functions on $E$ , $C(E)^+$ is the space of bounded continuous non-negative real-valued functions on $E$ , $C_0(E)$ is the space of continuous real-valued functions on $E$ vanishing at infinity.", "Let $M(E)$ denote the space of finite Borel measures on $E$ .", "We write $\\mu (f) := \\int _E f(x) \\, \\mu (\\mathrm {d}x)$ for the integral of a function $f : E \\rightarrow \\mathbb {R}$ with respect to a measure $\\mu \\in M(E)$ if the integral exists.", "Definition.", "2.7 A tuple $\\bigl ( E, (R_t)_{t\\in \\mathbb {R}_+}, c, \\beta , b, B, H_1, H_2 \\bigr )$ is called a set of admissible parameters if $E$ is a locally compact separable metric space, $(R_t)_{t\\in \\mathbb {R}_+}$ is the transition semigroup of a Hunt process $\\xi = \\bigl ( \\Omega , {\\mathcal {G}}, ({\\mathcal {G}}_t)_{t\\in \\mathbb {R}_+}, (\\xi _t)_{t\\in \\mathbb {R}_+},(\\theta _t)_{t\\in \\mathbb {R}_+}, (\\operatorname{\\mathbb {P}}_x)_{x\\in E} \\bigr )$ with values in $E$ (see, e.g., Li [13]) such that $(R_t)_{t\\in \\mathbb {R}_+}$ preserves $C_0(E)$ , and $\\mathbb {R}_+ \\ni t \\mapsto R_tf \\in C_0(E)$ is continuous in the supremum norm for every $f \\in C_0(E)$ , $c \\in C(E)^+$ , $\\beta \\in M(E)$ , $b \\in C(E)$ , $H_1$ is a finite measure on $M(E)^\\circ := M(E) \\setminus \\lbrace 0\\rbrace $ (where 0 denotes the null measure) satisfying $\\int _{M(E)^\\circ } \\kappa (1) \\, H_1(\\mathrm {d}\\kappa ) < \\infty $ , $B(x, \\mathrm {d}y)$ is a bounded kernel on $E$ (i.e., from $E$ to $E$ ) and $H_2(x, \\mathrm {d}\\kappa )$ is a $\\sigma $ -finite kernel from $E$ to $M(E)^\\circ $ such that $E \\ni x \\mapsto \\big ( \\kappa (1) \\wedge \\kappa (1)^2 \\big ) H_2(x, \\mathrm {d}\\kappa )$ is continuous with respect to the topology of weak convergence in $M(E)^\\circ $ , and the operators $f \\mapsto \\int _{M(E)^\\circ }\\big ( \\kappa (f) \\wedge \\kappa (f)^2 \\big ) H_2(\\cdot , \\mathrm {d}\\kappa )\\qquad \\text{and} \\qquad f \\mapsto \\gamma (\\cdot , f)$ preserve $C_0(E)^+$ , where the kernel $\\gamma (x, \\mathrm {d}y)$ on $E$ is defined by $\\gamma (x, \\mathrm {d}y):= B(x, \\mathrm {d}y) + \\int _{M(E)^\\circ } \\kappa _x(\\mathrm {d}y) \\, H_2(x, \\mathrm {d}\\kappa ) ,$ where $\\kappa _x(\\mathrm {d}y)$ denotes the restriction of $\\kappa (\\mathrm {d}y)$ to $E \\setminus \\lbrace x\\rbrace $ , and by $\\gamma (\\cdot , f)$ we mean the function $E \\ni x \\mapsto \\gamma (x, f) := \\int _E f(y) \\, \\gamma (x, \\mathrm {d}y)$ .", "Remark.", "2.8 Note that Condition (2.25) in Li [13] readily follows from (vii) of Definition REF , since a function in $C_0(E)$ is bounded, hence $\\sup _{x\\in E}\\int _{M(E)^\\circ }\\big [ \\kappa (1) \\wedge \\kappa (1)^2 \\big ] H_2(x, \\mathrm {d}\\kappa )< \\infty , \\qquad \\sup _{x\\in E} \\int _{M(E)^\\circ } \\kappa _x(1) \\, H_2(x, \\mathrm {d}\\kappa )\\leqslant \\sup _{x\\in E} \\gamma (x, 1) < \\infty ,$ where we used that $B(x, 1) \\in \\mathbb {R}_+$ for all $x \\in E$ .", "$\\Box $ Theorem.", "2.9 Let $\\bigl ( E, (R_t)_{t\\in \\mathbb {R}_+}, c, \\beta , b, B, H_1, H_2 \\bigr )$ be a set of admissible parameters in the sense of Definition REF .", "Then there exists a unique transition semigroup $(Q_t)_{t\\in \\mathbb {R}_+}$ acting on the Banach space (endowed with the supremum norm) of real-valued bounded Borel-measurable functions on the state space $M(E)$ such that its infinitesimal generator is $\\begin{split}({\\mathcal {A}}F)(\\mu )&= \\int _E c(x) F^{\\prime \\prime }(\\mu ; x) \\, \\mu (\\mathrm {d}x)+ \\int _E\\bigl ( AF^{\\prime }(\\mu ; x) + \\gamma (x, F^{\\prime }(\\mu ; \\cdot ))- b(x) F^{\\prime }(\\mu ; x) \\bigr )\\mu (\\mathrm {d}x) \\\\&\\quad + \\int _E F^{\\prime }(\\mu ; x) \\, \\beta (\\mathrm {d}x)+ \\int _{M(E)^\\circ }\\bigl ( F(\\mu + \\kappa ) - F(\\mu ) \\bigr ) H_1(\\mathrm {d}\\kappa ) \\\\&\\quad + \\int _E\\left( \\int _{M(E)^\\circ }\\bigl ( F(\\mu + \\kappa ) - F(\\mu )- \\kappa (F^{\\prime }(\\mu ; \\cdot ))\\bigr )\\, H_2(x, \\mathrm {d}\\kappa ) \\right)\\mu (\\mathrm {d}x) ,\\end{split}$ for $\\mu \\in M(E)$ and functions $F : M(E) \\rightarrow \\mathbb {R}$ of the form $F(\\mu ) = G(\\mu (f_1), \\ldots , \\mu (f_n))$ , where $n \\in \\mathbb {N}$ , $G \\in C^2(\\mathbb {R}^n, \\mathbb {R})$ , and $f_1, \\ldots , f_n \\in D_0(A)$ , where $A$ denotes the strong generator of $(R_t)_{t\\in \\mathbb {R}_+}$ defined by $Af(x) := \\lim _{t\\downarrow 0} \\frac{R_tf(x) - f(x)}{t} , \\qquad x \\in E ,$ where the limit is taken in the supremum norm, and the domain $D_0(A)$ of $A$ is the totality of functions $f \\in C_0(E)$ for which the above limit exists, $F^{\\prime }(\\mu ; x):= \\lim _{\\varepsilon \\downarrow 0}\\frac{F(\\mu + \\varepsilon \\delta _x) - F(\\mu )}{\\varepsilon } , \\qquad \\mu \\in M(E) , \\quad x \\in E ,$ and $F^{\\prime \\prime }(\\mu ; x)$ is defined by the limit with $F(\\cdot )$ replaced by $F^{\\prime }(\\cdot ; x)$ .", "Moreover, the Laplace transform of the transition semigroup $(Q_t)_{t\\in \\mathbb {R}_+}$ has a representation $\\int _{M(E)} \\mathrm {e}^{- \\kappa (f)} Q_t(\\mu , \\mathrm {d}\\kappa )= \\mathrm {e}^{- \\mu (V_tf) - \\int _0^t I(V_sf) \\, \\mathrm {d}s} ,\\qquad \\mu \\in M(E), \\quad f \\in B(E)^+ , \\quad t \\in \\mathbb {R}_+ ,$ where, for any $x \\in E$ and $f \\in B(E)^+$ , the continuously differentiable function $\\mathbb {R}_+ \\ni t \\mapsto V_tf(x) \\in \\mathbb {R}_+$ is the unique locally bounded solution to the integral evolution equation $V_tf(x) = R_tf(x)- \\int _0^t\\left( \\int _E \\phi (y, V_sf) \\, R_{t-s}(x, \\mathrm {d}y) \\right) \\mathrm {d}s ,\\qquad t \\in \\mathbb {R}_+ ,$ with $\\phi (x, f):= c(x) f(x)^2 + b(x) f(x) - \\int _E f(y) \\, B(x, \\mathrm {d}y)+ \\int _{M(E)^\\circ }\\bigl ( \\mathrm {e}^{- \\kappa (f)} - 1 + \\kappa (\\lbrace x\\rbrace ) f(x) \\bigr )\\, H_2(x, \\mathrm {d}\\kappa )$ for $x \\in E$ and $f \\in B(E)^+$ , and $I(f):= \\beta (f)+ \\int _{M(E)^\\circ }\\bigl ( 1 - \\mathrm {e}^{- \\kappa (f)} \\bigr ) \\, H_1(\\mathrm {d}\\kappa ) , \\qquad f \\in B(E)^+ .$ Proof.", "Formula (REF ), which is, in fact, formula (9.18) in Li [13], defines a transition semigroup of an immigration superprocess corresponding to the skew convolution semigroup given by (9.7) in Li [13].", "Theorem 9.18 in Li [13] yields that the infinitesimal generator of the immigration superprocess in question has the form given in (REF ), and the unicity of the transition semigroup.", "$\\Box $ Definition.", "2.10 A Markov process with state space $M(E)$ and with transition semigroup $(Q_t)_{t\\in \\mathbb {R}_+}$ given in Theorem REF is called an immigration superprocess with state space $M(E)$ with parameters $\\bigl ( E, (R_t)_{t\\in \\mathbb {R}_+}, c, \\beta , b, B, H_1, H_2 \\bigr )$ .", "In what follows, we identify a multi-type CBI process $({X}_t)_{t\\in \\mathbb {R}_+}$ with parameters $(d, {c}, {\\beta }, {B}, \\nu , {\\mu })$ under the moment condition $\\int _{U_d} \\Vert {z}\\Vert \\mathbb {1}_{\\lbrace \\Vert {z}\\Vert \\geqslant 1\\rbrace } \\, \\nu (\\mathrm {d}{z}) < \\infty ,$ as a special immigration superprocess.", "First we introduce the modified parameters $\\widetilde{{\\beta }}:= (\\widetilde{\\beta }_i)_{i\\in \\lbrace 1,\\ldots ,d\\rbrace }$ , $\\widetilde{{B}}:= (\\widetilde{b}_{i,j})_{i,j\\in \\lbrace 1,\\ldots ,d\\rbrace }$ and ${D}:= (d_{i,j})_{i,j\\in \\lbrace 1,\\ldots ,d\\rbrace }$ given by $\\widetilde{{\\beta }}:= {\\beta }+ \\int _{U_d} {z}\\, \\nu (\\mathrm {d}{z}) , \\qquad \\widetilde{b}_{i,j} := b_{i,j} + \\int _{U_d} (z_i - \\delta _{i,j})^+ \\, \\mu _j(\\mathrm {d}{z}), \\\\d_{i,j} := \\widetilde{b}_{i,j} - \\int _{U_d} z_i \\mathbb {1}_{\\lbrace \\Vert {z}\\Vert \\geqslant 1\\rbrace } \\, \\mu _j(\\mathrm {d}{z}) ,$ with $\\delta _{i,j} := 1$ if $i = j$ , and $\\delta _{i,j} := 0$ if $i \\ne j$ .", "The moment condition (REF ) together with the fact that $\\nu $ and ${\\mu }$ satisfy Definition REF imply $\\widetilde{{\\beta }}\\in \\mathbb {R}_+^d$ , $\\widetilde{{B}}\\in \\mathbb {R}^{d \\times d}_{(+)}$ and ${D}\\in \\mathbb {R}^{d \\times d}_{(+)}$ .", "Indeed, $\\int _{U_d} \\Vert {z}\\Vert \\, \\nu (\\mathrm {d}{z})= \\int _{U_d} (1 \\wedge \\Vert {z}\\Vert ) \\mathbb {1}_{\\lbrace \\Vert {z}\\Vert <1\\rbrace } \\, \\nu (\\mathrm {d}{z})+ \\int _{U_d} \\Vert {z}\\Vert \\mathbb {1}_{\\lbrace \\Vert {z}\\Vert \\geqslant 1\\rbrace } \\, \\nu (\\mathrm {d}{z})< \\infty $ by part (v) of Definition REF and (REF ).", "Moreover, for all $i \\in \\lbrace 1, \\ldots , d\\rbrace $ , $\\int _{U_d} (z_i - 1)^+ \\, \\mu _i(\\mathrm {d}{z})\\leqslant \\int _{U_d} z_i \\mathbb {1}_{\\lbrace z_i\\geqslant 1\\rbrace } \\, \\mu _i(\\mathrm {d}{z})\\leqslant \\int _{U_d} \\Vert {z}\\Vert \\mathbb {1}_{\\lbrace \\Vert {z}\\Vert \\geqslant 1\\rbrace } \\, \\mu _i(\\mathrm {d}{z})< \\infty $ by $z_i \\leqslant \\Vert {z}\\Vert $ , ${z}\\in \\mathbb {R}_+^d$ , and (REF ).", "Further, for all $i, j \\in \\lbrace 1, \\ldots , d\\rbrace $ , $i \\ne j$ , $\\begin{split}\\int _{U_d} z_i \\, \\mu _j(\\mathrm {d}{z})&= \\int _{U_d} z_i \\mathbb {1}_{\\lbrace z_i<1\\rbrace } \\, \\mu _j(\\mathrm {d}{z})+ \\int _{U_d} z_i \\mathbb {1}_{\\lbrace z_i\\geqslant 1\\rbrace } \\, \\mu _j(\\mathrm {d}{z}) \\\\&\\leqslant \\int _{U_d} (1 \\wedge z_i) \\, \\mu _j(\\mathrm {d}{z})+ \\int _{U_d} \\Vert {z}\\Vert \\mathbb {1}_{\\lbrace \\Vert {z}\\Vert \\geqslant 1\\rbrace } \\, \\mu _j(\\mathrm {d}{z})< \\infty \\end{split}$ by $z_i \\leqslant \\Vert {z}\\Vert $ , ${z}\\in \\mathbb {R}_+^d$ , part (vi) of Definition REF and (REF ).", "Finally, $d_{i,j}$ is well-defined for all $i, j \\in \\lbrace 1, \\ldots , d\\rbrace $ because of (REF ), and, for all $i, j \\in \\lbrace 1, \\ldots , d\\rbrace $ , $i \\ne j$ , $d_{i,j} = b_{i,j} + \\int _{U_d} z_i \\, \\mu _j(\\mathrm {d}z)- \\int _{U_d} z_i \\mathbb {1}_{\\lbrace \\Vert {z}\\Vert \\geqslant 1\\rbrace } \\, \\mu _j(\\mathrm {d}{z})= b_{i,j} + \\int _{U_d} z_i \\mathbb {1}_{\\lbrace \\Vert {z}\\Vert <1\\rbrace } \\, \\mu _j(\\mathrm {d}{z})\\in \\mathbb {R}_+ .$ Note also that for all $j \\in \\lbrace 1, \\ldots , d\\rbrace $ , $\\int _{U_d} \\Vert {z}\\Vert ^2 \\mathbb {1}_{\\lbrace \\Vert {z}\\Vert <1\\rbrace } \\, \\mu _j(\\mathrm {d}{z})\\leqslant \\int _{U_d}\\biggl ( z_j^2 + \\sum _{k\\in \\lbrace 1,\\ldots ,d\\rbrace \\setminus \\lbrace j\\rbrace } z_k \\biggr )\\mathbb {1}_{\\lbrace \\Vert {z}\\Vert <1\\rbrace } \\, \\mu _j(\\mathrm {d}{z})< \\infty $ by $z_i \\leqslant \\Vert {z}\\Vert $ , ${z}\\in \\mathbb {R}_+^d$ , part (vi) of Definition REF and (REF ).", "For the discrete metric space $E := \\lbrace 1, \\ldots , d\\rbrace $ , we have the following identifications: $B(E)$ , $C(E)$ and $C_0(E)$ can be identified with $\\mathbb {R}^d$ , since a function $f : E \\rightarrow \\mathbb {R}$ can be identified with the vector $(f(1), \\ldots , f(d))^\\top \\in \\mathbb {R}^d$ , $B(E)^+$ and $C(E)^+$ can be identified with $\\mathbb {R}_+^d$ , $M(E)$ can be identified with $\\mathbb {R}_+^d$ , since a finite Borel measure $\\mu $ on $E$ can be identified with the vector $(\\mu (\\lbrace 1\\rbrace ), \\ldots , \\mu (\\lbrace d\\rbrace ))^\\top \\in \\mathbb {R}_+^d$ , for $\\mu \\in M(E)$ and $f \\in B(E)$ , the integral $\\mu (f) = \\int _E f(x) \\, \\mu (\\mathrm {d}x) = \\sum _{i=1}^d f(i) \\mu (\\lbrace i\\rbrace )$ can be identified with the usual Euclidean inner product $\\langle \\mu , f \\rangle $ in $\\mathbb {R}^d$ , $M(E)^\\circ $ can be identified with $U_d$ .", "If $(\\Omega , {\\mathcal {F}}, \\operatorname{\\mathbb {P}})$ is a probability space, then, by $\\operatorname{\\mathbb {P}}$ -null sets from a sub $\\sigma $ -algebra ${\\mathcal {H}}\\subset {\\mathcal {F}}$ , we mean the elements of the set $\\lbrace A \\subset \\Omega \\,: \\, \\exists B \\in {\\mathcal {H}}\\;\\text{\\ such that \\ $A \\subset B$ \\ and \\ $\\operatorname{\\mathbb {P}}(B) = 0$} \\rbrace .$ A filtered probability space $(\\Omega , {\\mathcal {F}}, ({\\mathcal {F}}_t)_{t\\in \\mathbb {R}_+}, \\operatorname{\\mathbb {P}})$ is said to satisfy the usual hypotheses if $({\\mathcal {F}}_t)_{t\\in \\mathbb {R}_+}$ is right continuous and ${\\mathcal {F}}_0$ contains all the $\\operatorname{\\mathbb {P}}$ -null sets in ${\\mathcal {F}}$ .", "Lemma.", "2.11 Let $(d, {c}, {\\beta }, {B}, \\nu , {\\mu })$ be a set of admissible parameters in the sense of Definition REF satisfying the moment condition (REF ).", "Then $\\bigl ( E, (R_t)_{t\\in \\mathbb {R}_+}, c, \\beta , b, B, H_1, H_2 \\bigr )$ is a set of admissible parameters in the sense of Definition REF , where $E := \\lbrace 1, \\ldots , d\\rbrace $ with the discrete metric, $(R_t)_{t\\in \\mathbb {R}_+}$ is the transition semigroup given by $R_tf := f$ , $f \\in B(E)$ , $t \\in \\mathbb {R}_+$ , $c \\in B(E)^+$ is given by $c(i) := c_i$ , $i \\in E$ , $\\beta \\in M(E)$ is given by $\\beta (\\lbrace i\\rbrace ) := \\beta _i$ , $i \\in E$ , $b \\in B(E)$ , is given by $b(i) := - \\widetilde{b}_{i,i}$ , $i \\in E$ , $B(x, \\mathrm {d}y)$ is the kernel on $E$ given by $B(i, \\lbrace i\\rbrace ) := 0$ for $i \\in \\lbrace 1, \\ldots , d\\rbrace $ and $B(i, \\lbrace j\\rbrace ) := b_{j,i}$ for $i, j \\in \\lbrace 1, \\ldots , d\\rbrace $ with $i \\ne j$ , $H_1$ is the measure on $M(E)^\\circ $ identified with the measure $\\nu $ on $U_d$ , $H_2(x, \\mathrm {d}\\kappa )$ is the kernel from $E$ to $M(E)^\\circ $ such that the measure $H_2(i, \\cdot )$ on $M(E)^\\circ $ is identified with the measure $\\mu _i$ on $U_d$ for each $i \\in \\lbrace 1, \\ldots , d\\rbrace $ .", "If $(\\Omega , {\\mathcal {F}}, ({\\mathcal {F}}_t)_{t\\in \\mathbb {R}_+}, \\operatorname{\\mathbb {P}})$ is a filtered probability space satisfying the usual hypotheses and $(Y_t)_{t\\in \\mathbb {R}_+}$ is a càdlàg immigration superprocess with parameters $\\bigl ( E, (R_t)_{t\\in \\mathbb {R}_+}, c, \\beta , b, B, H_1, H_2 \\bigr )$ satisfying $\\operatorname{\\mathbb {E}}(Y_0(1)) < \\infty $ and adapted to $({\\mathcal {F}}_t)_{t\\in \\mathbb {R}_+}$ , then ${X}_t := (Y_t(\\lbrace 1\\rbrace ), \\ldots , Y_t(\\lbrace d\\rbrace ))^\\top $ , $t \\in \\mathbb {R}_+$ , is a multi-type CBI process with parameters $(d, {c}, {\\beta }, {B}, \\nu , {\\mu })$ satisfying $\\operatorname{\\mathbb {E}}(\\Vert {X}_0\\Vert ) < \\infty $ .", "The infinitesimal generator (REF ) of $({X}_t)_{t\\in \\mathbb {R}_+}$ can also be written in the form $\\begin{split}({\\mathcal {A}}_{{X}} f)({x})&= \\sum _{i=1}^d c_i x_i f_{i,i}^{\\prime \\prime }(x)+ \\sum _{i=1}^dx_i\\int _{U_d}\\bigl (f({x}+ {z}) - f({x}) - \\langle {z}, {f}^{\\prime }({x}) \\rangle \\bigr )\\, \\mu _i(\\mathrm {d}{z}) \\\\&\\quad + \\langle {\\beta }+ \\widetilde{{B}}{x}, {f}^{\\prime }({x}) \\rangle + \\int _{U_d} \\bigl (f({x}+ {z}) - f({x})\\bigr ) \\, \\nu (\\mathrm {d}{z})\\end{split}$ for $f \\in C^2_\\mathrm {c}(\\mathbb {R}_+^d,\\mathbb {R})$ and ${x}\\in \\mathbb {R}_+^d$ .", "Proof.", "The discrete metric space $\\lbrace 1, \\ldots , d\\rbrace $ is trivially a locally compact separable metric space.", "Clearly, $R_tf := f$ , $f \\in B(E)$ , $t \\in \\mathbb {R}_+$ , is the transition semigroup of the Hunt process $\\xi = \\bigl ( \\Omega , {\\mathcal {G}}, ({\\mathcal {G}}_t)_{t\\in \\mathbb {R}_+}, (\\xi _t)_{t\\in \\mathbb {R}_+},(\\theta _t)_{t\\in \\mathbb {R}_+}, (\\operatorname{\\mathbb {P}}_x)_{x\\in E} \\bigr )$ with $\\Omega = \\lbrace 1, \\ldots , d\\rbrace $ , ${\\mathcal {G}}= {\\mathcal {G}}_t = 2^\\Omega $ , $t \\in \\mathbb {R}_+$ , $\\xi _t(\\omega ) = \\theta _t(\\omega ) = \\omega $ , $\\omega \\in \\Omega $ , $t \\in \\mathbb {R}_+$ , $\\operatorname{\\mathbb {P}}_x = \\delta _x$ , $x \\in \\lbrace 1, \\ldots , d\\rbrace $ .", "Moreover, $(R_t)_{t\\in \\mathbb {R}_+}$ trivially satisfies (ii) of Definition REF , and (iii), (iv) and (v) of Definition REF trivially hold.", "Further (vi) of Definition REF also holds, since $\\int _{U_d} \\left(\\sum _{i=1}^d z_i\\right) \\nu (\\mathrm {d}{z}) < \\infty $ follows from (REF ) by $z_i \\leqslant \\Vert {z}\\Vert $ , ${z}\\in \\mathbb {R}_+^d$ , $i \\in \\lbrace 1, \\ldots , d\\rbrace $ .", "The kernel $B(x, \\mathrm {d}y)$ on $E$ is bounded, since $\\sup _{x\\in E} B(x, E)= \\max _{i\\in \\lbrace 1,\\ldots ,d\\rbrace } \\sum _{j\\in \\lbrace 1,\\ldots ,d\\rbrace \\setminus \\lbrace i\\rbrace } b_{j,i}< \\infty $ .", "On the dicrete metric space $\\lbrace 1, \\ldots , d\\rbrace $ every function is continuous, hence $E \\ni x \\mapsto \\big ( \\kappa (1) \\wedge \\kappa (1)^2 \\big ) H_2(x, \\mathrm {d}\\kappa )$ is continuous with respect to the topology of weak convergence in $M(E)^\\circ $ .", "In order to show that the operator $f \\mapsto \\int _{M(E)^\\circ }\\big ( \\kappa (f) \\wedge \\kappa (f)^2 \\big ) H_2(\\cdot , \\mathrm {d}\\kappa )$ preserve $C_0(E)^+$ , it suffices to observe that for each ${\\lambda }\\in \\mathbb {R}_+^d$ and $i \\in \\lbrace 1, \\ldots , d\\rbrace $ , we have $\\int _{U_d}\\left( \\langle {\\lambda }, {z}\\rangle \\wedge \\langle {\\lambda }, {z}\\rangle ^2 \\right)\\mu _i(\\mathrm {d}{z}) \\in \\mathbb {R}_+ ,$ which follows from the estimate $\\int _{U_d}&\\left( \\langle {\\lambda }, {z}\\rangle \\wedge \\langle {\\lambda }, {z}\\rangle ^2 \\right)\\mu _i(\\mathrm {d}{z})\\leqslant \\int _{U_d}\\bigl [ (\\Vert {\\lambda }\\Vert \\Vert {z}\\Vert ) \\wedge (\\Vert {\\lambda }\\Vert \\Vert {z}\\Vert )^2 \\bigr ]\\, \\mu _i(\\mathrm {d}{z})\\leqslant c_{\\lambda }\\int _{U_d} (\\Vert {z}\\Vert \\wedge \\Vert {z}\\Vert ^2) \\, \\mu _i(\\mathrm {d}{z}) \\\\&= c_{\\lambda }\\int _{U_d}(\\Vert {z}\\Vert \\wedge \\Vert {z}\\Vert ^2) \\mathbb {1}_{\\lbrace \\Vert {z}\\Vert \\leqslant 1\\rbrace }\\, \\mu _i(\\mathrm {d}{z})+ c_{\\lambda }\\int _{U_d}(\\Vert {z}\\Vert \\wedge \\Vert {z}\\Vert ^2) \\mathbb {1}_{\\lbrace \\Vert {z}\\Vert >1\\rbrace }\\, \\mu _i(\\mathrm {d}{z}) \\\\&= c_{\\lambda }\\int _{U_d} \\Vert {z}\\Vert ^2 \\mathbb {1}_{\\lbrace \\Vert {z}\\Vert \\leqslant 1\\rbrace } \\, \\mu _i(\\mathrm {d}{z})+ c_{\\lambda }\\int _{U_d} \\Vert {z}\\Vert \\mathbb {1}_{\\lbrace \\Vert {z}\\Vert >1\\rbrace } \\, \\mu _i(\\mathrm {d}{z})< \\infty $ with $c_{\\lambda }:= \\max \\lbrace \\Vert {\\lambda }\\Vert , \\Vert {\\lambda }\\Vert ^2\\rbrace $ by (REF ) and (REF ).", "In order to show that the operator $f \\mapsto \\gamma (\\cdot , f)$ preserves $C_0(E)^+$ , it suffices to observe that for each ${\\lambda }= (\\lambda _1, \\ldots , \\lambda _d)^\\top \\in \\mathbb {R}_+^d$ and $i \\in \\lbrace 1, \\ldots , d\\rbrace $ , we have $\\sum _{j=1}^d \\lambda _j B(i, \\lbrace j\\rbrace )+ \\sum _{j\\in \\lbrace 1,\\ldots ,d\\rbrace \\setminus \\lbrace i\\rbrace } \\lambda _j \\int _{U_d} z_j \\, \\mu _j(\\mathrm {d}{z})\\in \\mathbb {R}_+ ,$ which follows from (REF ).", "Consequently, $\\bigl ( E, (R_t)_{t\\in \\mathbb {R}_+}, c, \\beta , b, B, H_1, H_2 \\bigr )$ is a set of admissible parameters in the sense of Definition REF .", "By Theorem REF , we have $\\operatorname{\\mathbb {E}}(\\mathrm {e}^{-Y_t(f)} \\,\|\\,Y_0 = \\mu )= \\int _{M(E)^\\circ } \\mathrm {e}^{-\\kappa (f)} \\, Q_t(\\mu , \\mathrm {d}\\kappa )= \\mathrm {e}^{- \\mu (V_tf) - \\int _0^t I(V_sf) \\, \\mathrm {d}s}$ for $\\mu \\in M(E)$ , $f \\in B(E)^+$ and $t \\in \\mathbb {R}_+$ , hence we obtain $\\operatorname{\\mathbb {E}}(\\mathrm {e}^{-\\langle {\\lambda },{X}_t\\rangle } \\,\|\\,{X}_0 = {x})= \\mathrm {e}^{-\\langle {x},{v}(t,{\\lambda })\\rangle -\\int _0^t\\psi ({v}(s,{\\lambda }))\\,\\mathrm {d}s} ,\\qquad {x}, {\\lambda }\\in \\mathbb {R}_+^d, \\quad t \\in \\mathbb {R}_+ ,$ where, for any $i \\in \\lbrace 1, \\ldots , d\\rbrace $ and ${\\lambda }\\in \\mathbb {R}_+^d$ , the function $\\mathbb {R}_+ \\ni t \\mapsto {v}(t, {\\lambda }) = (v_1(t, {\\lambda }), \\ldots , v_d(t, {\\lambda }))$ is the unique locally bounded solution to the integral evolution equation $v_i(t, {\\lambda })= \\lambda _i - \\int _0^t \\varphi _i({v}(s, {\\lambda })) \\, \\mathrm {d}s , \\qquad t \\in \\mathbb {R}_+ , \\quad i \\in \\lbrace 1, \\ldots , d\\rbrace ,$ with $\\varphi _i({\\lambda }):= c_i \\lambda _i^2 - \\widetilde{b}_{i,i} \\lambda _i- \\sum _{j\\in \\lbrace 1,\\ldots ,d\\rbrace \\setminus \\lbrace i\\rbrace } \\lambda _j b_{j,i}+ \\int _{U_d}(\\mathrm {e}^{-\\langle {\\lambda },{z}\\rangle } - 1 + \\lambda _i z_i) \\, \\mu _i(\\mathrm {d}{z})$ for ${\\lambda }\\in \\mathbb {R}_+^d$ and $i \\in \\lbrace 1, \\ldots , d\\rbrace $ , and $\\psi ({\\lambda }):= \\langle {\\beta }, {\\lambda }\\rangle + \\int _{U_d} (1 - \\mathrm {e}^{-\\langle {\\lambda },{z}\\rangle }) \\, \\nu (\\mathrm {d}{z}) , \\qquad {\\lambda }\\in \\mathbb {R}_+^d .$ We have $\\varphi _i({\\lambda })= c_i \\lambda _i^2 - \\langle \\widetilde{{B}}{e}_i, {\\lambda }\\rangle + \\int _{U_d}(\\mathrm {e}^{-\\langle {\\lambda },{z}\\rangle } - 1 + \\langle {\\lambda }, {z}\\rangle )\\, \\mu _i(\\mathrm {d}{z}) ,$ since, by (REF ), $&\\varphi _i({\\lambda }) - c_i \\lambda _i^2 + \\langle \\widetilde{{B}}{e}_i, {\\lambda }\\rangle - \\int _{U_d}(\\mathrm {e}^{-\\langle {\\lambda },{z}\\rangle } - 1 + \\langle {\\lambda }, {z}\\rangle )\\, \\mu _i(\\mathrm {d}{z}) \\\\&= - \\widetilde{b}_{i,i} \\lambda _i - \\sum _{j\\in \\lbrace 1,\\ldots ,d\\rbrace \\setminus \\lbrace i\\rbrace } \\lambda _j b_{j,i}+ \\sum _{j=1}^d \\lambda _j \\widetilde{b}_{j,i}- \\sum _{j\\in \\lbrace 1,\\ldots ,d\\rbrace \\setminus \\lbrace i\\rbrace } \\lambda _j \\int _{U_d} z_j \\, \\mu _i(\\mathrm {d}{z})= 0 .$ Moreover, we can write the functions $\\varphi _i$ , $i \\in \\lbrace 1, \\ldots , d\\rbrace $ , in the form $\\varphi _i({\\lambda })= c_i \\lambda _i^2 - \\langle {B}{e}_i, {\\lambda }\\rangle + \\int _{U_d}\\bigl ( \\mathrm {e}^{- \\langle {\\lambda }, {z}\\rangle } - 1+ \\lambda _i (1 \\wedge z_i) \\bigr )\\, \\mu _i(\\mathrm {d}{z})$ for ${\\lambda }= (\\lambda _1, \\ldots , \\lambda _d)^\\top \\in \\mathbb {R}_+^d$ and $i \\in \\lbrace 1, \\ldots , d\\rbrace $ .", "Indeed, by (REF ) and (REF ), $\\varphi _i({\\lambda }) - c_i \\lambda _i^2 &+ \\langle {B}{e}_i, {\\lambda }\\rangle - \\int _{U_d}\\bigl ( \\mathrm {e}^{- \\langle {\\lambda }, {z}\\rangle } - 1+ \\lambda _i (1 \\wedge z_i) \\bigr )\\, \\mu _i(\\mathrm {d}{z}) \\\\&= \\langle ({B}- \\widetilde{{B}}) {e}_i, {\\lambda }\\rangle - \\int _{U_d}\\bigl ( \\lambda _i(1 \\wedge z_i) - \\langle {\\lambda }, {z}\\rangle \\bigr )\\, \\mu _i(\\mathrm {d}{z}) \\\\&= - \\lambda _i \\int _{U_d} (z_i-1)^+ \\, \\mu _i(\\mathrm {d}{z})- \\int _{U_d}\\bigl ( \\lambda _i(1 \\wedge z_i) - \\lambda _i z_i \\bigr )\\, \\mu _i(\\mathrm {d}{z})= 0 .$ By Theorem REF , $({X}_t)_{t\\in \\mathbb {R}_+}$ is a multi-type CBI with parameters $(d, {c}, {\\beta }, {B}, \\nu , {\\mu })$ satisfying $\\operatorname{\\mathbb {E}}(\\Vert {X}_0\\Vert ) < \\infty $ .", "Finally, (REF ) follows from $&({\\mathcal {A}}_{{X}} f)({x}) - \\sum _{i=1}^d c_i x_i f_{i,i}^{\\prime \\prime }({x})- \\sum _{i=1}^dx_i\\int _{U_d}\\bigl (f({x}+ {z}) - f({x}) - \\langle {z}, {f}^{\\prime }({x}) \\rangle \\bigr )\\, \\mu _i(\\mathrm {d}{z}) \\\\&- \\langle {\\beta }+ \\widetilde{{B}}{x}, {f}^{\\prime }({x}) \\rangle - \\int _{U_d}\\bigl (f({x}+ {z}) - f({x})\\bigr ) \\, \\nu (\\mathrm {d}{z}) \\\\&= \\sum _{i=1}^dx_i\\int _{U_d}\\bigl ( \\langle {z}, {f}^{\\prime }({x}) \\rangle - f_i^{\\prime }({x}) (1 \\wedge z_i) \\bigr )\\, \\mu _i(\\mathrm {d}{z})- \\langle (\\widetilde{{B}}- {B}) {x}, {f}^{\\prime }({x}) \\rangle \\\\&= \\sum _{i=1}^dx_i \\int _{U_d}\\biggl ( f_i^{\\prime }({x}) ( z_i - (1 \\wedge z_i) )+ \\sum _{j\\in \\lbrace 1,\\ldots ,d\\rbrace \\setminus \\lbrace i\\rbrace } z_j f_j^{\\prime }({x}) \\biggr )\\mu _i(\\mathrm {d}{z}) \\\\&\\phantom{=\\;}- \\sum _{i=1}^d\\sum _{j=1}^dx_j f_i^{\\prime }({x}) \\int _{U_d} (z_i-\\delta _{i,j})^+ \\, \\mu _j(\\mathrm {d}{z})= 0 .$ using (REF ), (REF ) and (REF ).", "$\\Box $" ], [ "Multi-type CBI process as a weak solution of an SDE", "Let ${\\mathcal {R}}:= \\bigcup _{j=0}^d {\\mathcal {R}}_j$ , where ${\\mathcal {R}}_j$ , $j \\in \\lbrace 0, 1, \\ldots , d\\rbrace $ , are disjoint sets given by ${\\mathcal {R}}_0 := U_d \\times \\lbrace ({0}, 0) \\rbrace ^d\\subset \\mathbb {R}_+^d \\times (\\mathbb {R}_+^d \\times \\mathbb {R}_+)^d ,$ and ${\\mathcal {R}}_j := \\lbrace {0}\\rbrace \\times H_{j,1} \\times \\cdots \\times H_{j,d}\\subset \\mathbb {R}_+^d \\times (\\mathbb {R}_+^d \\times \\mathbb {R}_+)^d , \\qquad j \\in \\lbrace 1, \\ldots , d\\rbrace ,$ where $H_{j,i} := {\\left\\lbrace \\begin{array}{ll}U_d \\times U_1 & \\text{if \\ $i = j$,} \\\\\\lbrace ({0}, 0) \\rbrace & \\text{if \\ $i \\ne j$.}\\end{array}\\right.", "}$ (Recall that $U_1 = \\mathbb {R}_{++}$ .)", "Let $m$ be the uniquely defined measure on $V := \\mathbb {R}_+^d \\times (\\mathbb {R}_+^d \\times \\mathbb {R}_+)^d$ such that $m(V \\setminus {\\mathcal {R}}) = 0$ and its restrictions on ${\\mathcal {R}}_j$ , $j \\in \\lbrace 0, 1, \\ldots , d\\rbrace $ , are $m\|_{{\\mathcal {R}}_0}(\\mathrm {d}{r}) = \\nu (\\mathrm {d}{r}) , \\qquad m\|_{{\\mathcal {R}}_j}(\\mathrm {d}{z}, \\mathrm {d}u) = \\mu _j(\\mathrm {d}{z}) \\, \\mathrm {d}u ,\\quad j \\in \\lbrace 1, \\ldots , d\\rbrace ,$ where we identify ${\\mathcal {R}}_0$ with $U_d$ and ${\\mathcal {R}}_1$ , ..., ${\\mathcal {R}}_d$ with $U_d \\times U_1$ in a natural way.", "Using again this identification, let $h : \\mathbb {R}^d \\times V \\rightarrow \\mathbb {R}_+^d$ be defined by $h({x}, {r}):= {\\left\\lbrace \\begin{array}{ll}{r}, & \\text{if \\ ${x}\\in \\mathbb {R}_+^d$, \\ ${r}\\in {\\mathcal {R}}_0$,} \\\\{z}\\mathbb {1}_{\\lbrace u \\leqslant x_j\\rbrace } ,& \\text{if \\ ${x}= (x_1, \\ldots , x_d)^\\top \\in \\mathbb {R}_+^d$,\\ ${r}= ({z}, u) \\in {\\mathcal {R}}_j$, \\ $j \\in \\lbrace 1, \\ldots , d\\rbrace $,} \\\\{0}, & \\text{otherwise.}\\end{array}\\right.", "}$ Consider the decomposition ${\\mathcal {R}}= V_0 \\cup V_1$ , where $V_0 := \\bigcup _{j=1}^d {\\mathcal {R}}_{j,0}$ and $V_1 := {\\mathcal {R}}_0 \\cup \\bigl ( \\bigcup _{j=1}^d {\\mathcal {R}}_{j,1} \\bigr )$ with ${\\mathcal {R}}_{j,k} := \\lbrace {0}\\rbrace \\times H_{j,1,k} \\times \\cdots \\times H_{j,d,k}$ , $j \\in \\lbrace 1, \\ldots , d\\rbrace $ , $k \\in \\lbrace 0, 1\\rbrace $ , and $H_{j,i,k} := {\\left\\lbrace \\begin{array}{ll}U_{d,k} \\times U_1 & \\text{if \\ $i = j$,} \\\\\\lbrace ({0}, 0) \\rbrace & \\text{if \\ $i \\ne j$,}\\end{array}\\right.}", "\\qquad U_{d,k} := {\\left\\lbrace \\begin{array}{ll}\\lbrace {z}\\in U_d : \\Vert {z}\\Vert < 1 \\rbrace & \\text{if \\ $k = 0$,} \\\\\\lbrace {z}\\in U_d : \\Vert {z}\\Vert \\geqslant 1 \\rbrace & \\text{if \\ $k = 1$.}\\end{array}\\right.", "}$ Then the sets $V_0$ and $V_1$ are disjoint, and the function $h$ can be decomposed in the form $h = f + g$ with $f({x}, {r}) := h({x}, {r}) \\mathbb {1}_{V_0}({r}) , \\qquad g({x}, {r}) := h({x}, {r}) \\mathbb {1}_{V_1}({r}) , \\qquad ({x}, {r}) \\in \\mathbb {R}^d \\times V .$ Let $(d, {c}, {\\beta }, {B}, \\nu , {\\mu })$ be a set of admissible parameters in the sense of Definition REF such that the moment condition (REF ) holds.", "Let us consider the $d$ -dimensional SDE $\\begin{aligned}{X}_t&= {X}_0 + \\int _0^t b({X}_s) \\, \\mathrm {d}s+ \\int _0^t \\sigma ({X}_s) \\, \\mathrm {d}{W}_s \\\\&\\quad + \\int _0^t \\int _{V_0} f({X}_{s-}, {r}) \\, \\widetilde{N}(\\mathrm {d}s, \\mathrm {d}{r})+ \\int _0^t \\int _{V_1} g({X}_{s-}, {r}) \\, N(\\mathrm {d}s, \\mathrm {d}{r}) , \\qquad t \\in \\mathbb {R}_+ ,\\end{aligned}$ where the functions $b : \\mathbb {R}^d \\rightarrow \\mathbb {R}^d$ and $\\sigma : \\mathbb {R}^d \\rightarrow \\mathbb {R}^{d\\times d}$ are defined by $b({x}) := {\\beta }+ {D}{x}, \\qquad \\sigma ({x}) := \\sum _{i=1}^d \\sqrt{2 c_i x_i^+} \\, {e}_i {e}_i^\\top , \\qquad {x}\\in \\mathbb {R}^d ,$ ${D}$ is defined in (), $({W}_t)_{t\\in \\mathbb {R}_+}$ is a $d$ -dimensional standard Brownian motion, $N(\\mathrm {d}s, \\mathrm {d}{r})$ is a Poisson random measure on $\\mathbb {R}_{++} \\times V$ with intensity measure $\\mathrm {d}s \\, m(\\mathrm {d}{r})$ , and $\\widetilde{N}(\\mathrm {d}s, \\mathrm {d}{r}) := N(\\mathrm {d}s, \\mathrm {d}{r}) - \\mathrm {d}s \\, m(\\mathrm {d}{r})$ .", "For a short review on point measures and point processes needed for this paper, see, e.g., Barczy et al.", "[1].", "Definition.", "3.1 Let $n$ be a probability measure on $(\\mathbb {R}_+^d, {\\mathcal {B}}(\\mathbb {R}_+^d))$ .", "An $\\mathbb {R}_+^d$ -valued weak solution of the SDE (REF ) with initial distribution $n$ is a tuple $\\bigl ( \\Omega , {\\mathcal {F}}, ({\\mathcal {F}}_t)_{t\\in \\mathbb {R}_+}, \\operatorname{\\mathbb {P}}, {W}, p, {X}\\bigr )$ , where $(\\Omega , {\\mathcal {F}}, ({\\mathcal {F}}_t)_{t\\in \\mathbb {R}_+}, \\operatorname{\\mathbb {P}})$ is a filtered probability space satisfying the usual hypotheses; $({W}_t)_{t\\in \\mathbb {R}_+}$ is a $d$ -dimensional standard $({\\mathcal {F}}_t)_{t\\in \\mathbb {R}_+}$ -Brownian motion; $p$ is a stationary $({\\mathcal {F}}_t)_{t\\in \\mathbb {R}_+}$ -Poisson point process on $V$ with characteristic measure $m$ given in (REF ); $({X}_t)_{t\\in \\mathbb {R}_+}$ is an $\\mathbb {R}_+^d$ -valued $({\\mathcal {F}}_t)_{t\\in \\mathbb {R}_+}$ -adapted càdlàg process such that (a) the distribution of ${X}_0$ is $n$ , (b) $\\operatorname{\\mathbb {P}}\\bigl (\\int _0^t\\big (\\Vert b({X}_s)\\Vert + \\Vert \\sigma ({X}_s)\\Vert ^2 \\big )\\, \\mathrm {d}s< \\infty \\bigr ) = 1$ for all $t \\in \\mathbb {R}_+$ , (c) $\\operatorname{\\mathbb {P}}\\bigl ( \\int _0^t\\int _{V_0} \\Vert f({X}_s, {r})\\Vert ^2 \\, \\mathrm {d}s \\, m(\\mathrm {d}{r})< \\infty \\bigr )= 1$ for all $t \\in \\mathbb {R}_+$ , (d) $\\operatorname{\\mathbb {P}}\\bigl ( \\int _0^t \\int _{V_1} \\Vert g({X}_{s-}, {r})\\Vert \\, N(\\mathrm {d}s, \\mathrm {d}{r})< \\infty \\bigr )= 1$ for all $t \\in \\mathbb {R}_+$ , where $N(\\mathrm {d}s, \\mathrm {d}{r})$ is the counting measure of $p$ on $\\mathbb {R}_{++} \\times V$ , (e) equation (REF ) holds $\\operatorname{\\mathbb {P}}$ -a.s., where $\\widetilde{N}(\\mathrm {d}s, \\mathrm {d}{r}) := N(\\mathrm {d}s, \\mathrm {d}{r}) - \\mathrm {d}s \\, m(\\mathrm {d}{r})$ .", "For the definitions of an $({\\mathcal {F}}_t)_{t\\in \\mathbb {R}_+}$ -Brownian motion and an $({\\mathcal {F}}_t)_{t\\in \\mathbb {R}_+}$ -Poisson point process, see, e.g., Ikeda and Watanabe [7].", "Remark.", "3.2 If conditions (D1)–(D3) and (D4)(b)–(d) are satisfied, then the mappings $\\mathbb {R}_+ \\times V_0 \\times \\Omega \\ni (s, {r}, \\omega )\\mapsto f({X}_{s-}(\\omega ), {r}) \\in \\mathbb {R}^d$ and $\\mathbb {R}_+ \\times V_1 \\times \\Omega \\ni (s, {r}, \\omega )\\mapsto g({X}_{s-}(\\omega ), {r}) \\in \\mathbb {R}^d$ are in the (multidimensional versions of the) classes ${F}_p^{2,loc}$ and ${F}_p$ , respectively, defined in Ikeda and Watanabe [7], the integrals in (REF ) are well-defined and have càdlàg modifications as functions of $t$ , see, e.g., Barczy et al.", "[1].", "Moreover, if $\\operatorname{\\mathbb {E}}\\bigl (\\int _0^t \\Vert {X}_s\\Vert \\, \\mathrm {d}s\\bigr ) < \\infty $ for all $t \\in \\mathbb {R}_+$ , and the moment condition (REF ) holds, then conditions (D4)(b)–(d) are satisfied, and the mappings $\\mathbb {R}_+ \\times V_0 \\times \\Omega \\ni (s, {r}, \\omega )\\mapsto f({X}_{s-}(\\omega ), {r}) \\in \\mathbb {R}^d$ and $\\mathbb {R}_+ \\times V_1 \\times \\Omega \\ni (s, {r}, \\omega )\\mapsto g({X}_{s-}(\\omega ), {r}) \\in \\mathbb {R}^d$ are in the (multidimensional versions of the) smaller classes ${F}_p^2$ and ${F}_p^1$ , respectively, defined in Ikeda and Watanabe [7].", "Indeed, with the notation ${X}_s = (X_{s,1}, \\ldots , X_{s,d})^\\top $ , $s \\in \\mathbb {R}_+$ , $\\operatorname{\\mathbb {E}}\\left( \\int _0^t \\int _{V_0}\\Vert f({X}_s, {r})\\Vert ^2 \\, \\mathrm {d}s \\, m(\\mathrm {d}{r}) \\right)&= \\sum _{j=1}^d\\operatorname{\\mathbb {E}}\\left( \\int _0^t \\int _{U_d} \\int _{U_1}\\Vert {z}\\Vert ^2 \\mathbb {1}_{\\lbrace \\Vert {z}\\Vert <1\\rbrace } \\mathbb {1}_{\\lbrace u\\leqslant X_{s,j} \\rbrace }\\, \\mathrm {d}s \\, \\mu _j(\\mathrm {d}{z}) \\, \\mathrm {d}u \\right) \\\\&= \\sum _{j=1}^d\\operatorname{\\mathbb {E}}\\left( \\int _0^t X_{s,j} \\, \\mathrm {d}s \\right)\\int _{U_d} \\Vert {z}\\Vert ^2 \\mathbb {1}_{\\lbrace \\Vert {z}\\Vert <1\\rbrace } \\, \\mu _j(\\mathrm {d}{z})< \\infty $ by (REF ), and $&\\operatorname{\\mathbb {E}}\\left( \\int _0^t \\int _{V_1}\\Vert g({X}_s, {r})\\Vert \\, \\mathrm {d}s \\, m(\\mathrm {d}{r}) \\right) \\\\&= \\int _0^t \\int _{U_d} \\Vert {r}\\Vert \\, \\mathrm {d}s \\, \\nu (\\mathrm {d}{r})+ \\sum _{j=1}^d\\operatorname{\\mathbb {E}}\\left( \\int _0^t \\int _{U_d} \\int _{U_1}\\Vert {z}\\Vert \\mathbb {1}_{\\lbrace \\Vert {z}\\Vert \\geqslant 1\\rbrace }\\mathbb {1}_{\\lbrace u\\leqslant X_{s,j} \\rbrace }\\, \\mathrm {d}s \\, \\mu _j(\\mathrm {d}{z}) \\, \\mathrm {d}u \\right)$ $= t \\int _{U_d} \\Vert {r}\\Vert \\, \\nu (\\mathrm {d}{r})+ \\sum _{j=1}^d \\operatorname{\\mathbb {E}}\\left( \\int _0^t X_{s,j} \\, \\mathrm {d}s \\right)\\int _{U_d} \\Vert {z}\\Vert \\mathbb {1}_{\\lbrace \\Vert {z}\\Vert \\geqslant 1\\rbrace } \\, \\mu _j(\\mathrm {d}{z})< \\infty $ by (REF ) and (REF ).", "Note that if $({X}_t)_{t\\in \\mathbb {R}_+}$ is a CBI process with $\\operatorname{\\mathbb {E}}(\\Vert {X}_0\\Vert ) < \\infty $ satisfying the moment condition (REF ), then $\\operatorname{\\mathbb {E}}\\bigl (\\int _0^t \\Vert {X}_s\\Vert \\, \\mathrm {d}s\\bigr ) < \\infty $ for all $t \\in \\mathbb {R}_+$ , see Lemma REF .", "$\\Box $ Remark.", "3.3 Note that if conditions (D1)–(D3) are satisfied, then ${W}$ and $p$ are automatically independent according to Theorem 6.3 in Chapter II of Ikeda and Watanabe [7], since the intensity measure $\\mathrm {d}s \\, m(\\mathrm {d}{r})$ of $p$ is deterministic.", "Moreover, if $\\bigl ( \\Omega , {\\mathcal {F}}, ({\\mathcal {F}}_t)_{t\\in \\mathbb {R}_+}, \\operatorname{\\mathbb {P}}, {W}, p, {X}\\bigr )$ is an $\\mathbb {R}_+^d$ -valued weak solution of the SDE (REF ), then ${\\mathcal {F}}_0$ , ${W}$ and $p$ are mutually independent, and hence ${X}_0$ , ${W}$ and $p$ are mutually independent as well, see, e.g., Barczy et al.", "[1].", "$\\Box $ Lemma.", "3.4 Let $({X}_t)_{t\\in \\mathbb {R}_+}$ be a CBI process with parameters $(d, {c}, {\\beta }, {B}, \\nu , {\\mu })$ and with initial distribution $n$ satisfying $\\int _{\\mathbb {R}_+^d} \\Vert {z}\\Vert \\, n(\\mathrm {d}{z}) < \\infty $ .", "Suppose that the moment condition (REF ) holds.", "Then $\\operatorname{\\mathbb {E}}({X}_t)= \\mathrm {e}^{t\\widetilde{{B}}} \\operatorname{\\mathbb {E}}({X}_0)+ \\left( \\int _0^t \\mathrm {e}^{u\\widetilde{{B}}} \\, \\mathrm {d}u \\right) \\widetilde{{\\beta }}, \\qquad t \\in \\mathbb {R}_+ ,$ where $\\widetilde{{B}}\\in \\mathbb {R}^{d \\times d}_{(+)}$ and $\\widetilde{{\\beta }}\\in \\mathbb {R}_+^d$ are defined in (REF ).", "In particular, $\\int _0^t \\operatorname{\\mathbb {E}}(\\Vert {X}_s\\Vert ) \\, \\mathrm {d}s < \\infty $ for all $t \\in \\mathbb {R}_+$ .", "Proof.", "By the tower rule for conditional expectations, it suffices to show $\\operatorname{\\mathbb {E}}({X}_t \\,\|\\,{X}_0)= \\mathrm {e}^{t\\widetilde{{B}}} {X}_0 + \\left( \\int _0^t \\mathrm {e}^{u\\widetilde{{B}}} \\, \\mathrm {d}u \\right) \\widetilde{{\\beta }},\\qquad t \\in \\mathbb {R}_+ ,$ where the conditional expectation $\\operatorname{\\mathbb {E}}({X}_t \\,\|\\,{X}_0) \\in [0, \\infty ]^d$ is meant in the generalized sense, see, e.g., Stroock [15].", "In order to show (REF ), it is enough to check that for a CBI process $({X}_t)_{t\\in \\mathbb {R}_+}$ with initial value ${X}_0 = {x}\\in \\mathbb {R}_+^d$ , we have $\\operatorname{\\mathbb {E}}({X}_t)= \\mathrm {e}^{t\\widetilde{{B}}} {x}+ \\left( \\int _0^t \\mathrm {e}^{u\\widetilde{{B}}} \\, \\mathrm {d}u \\right) \\widetilde{{\\beta }},\\qquad t \\in \\mathbb {R}_+ , \\quad {x}\\in \\mathbb {R}_+^d .$ Indeed, let $\\phi _n : \\mathbb {R}_+^d \\rightarrow \\mathbb {R}_+^d$ , $n \\in \\mathbb {N}$ , be simple functions such that $\\phi _n({y}) \\uparrow {y}$ as $n \\rightarrow \\infty $ for all ${y}\\in \\mathbb {R}_+^d$ .", "Then, by the (multidimensional version of the) monotone convergence theorem for (generalized) conditional expectations, see, e.g., Stroock [15], we obtain $\\operatorname{\\mathbb {E}}(\\phi _n({X}_t) \\,\|\\,{X}_0) \\uparrow \\operatorname{\\mathbb {E}}({X}_t \\,\|\\,{X}_0)$ as $n \\rightarrow \\infty $ $\\operatorname{\\mathbb {P}}$ -almost surely.", "For each $B \\in {\\mathcal {B}}(\\mathbb {R}^d)$ , we have $\\operatorname{\\mathbb {E}}(\\mathbb {1}_B({X}_t) \\,\|\\,{X}_0)= \\operatorname{\\mathbb {P}}({X}_t \\in B \\,\|\\,{X}_0)= \\int _{\\mathbb {R}_+^d} \\mathbb {1}_B({y}) \\, P_t({X}_0, \\mathrm {d}{y}) ,$ hence $\\operatorname{\\mathbb {E}}(\\phi _n({X}_t) \\,\|\\,{X}_0)= \\int _{\\mathbb {R}_+^d} \\phi _n({y}) \\, P_t({X}_0, \\mathrm {d}{y})$ .", "By the (multidimensional version of the) monotone convergence theorem, $\\int _{\\mathbb {R}_+^d} \\phi _n({y}) \\, P_t({X}_0, \\mathrm {d}{y})\\uparrow \\int _{\\mathbb {R}_+^d} {y}\\, P_t({X}_0, \\mathrm {d}{y})$ as $n \\rightarrow \\infty $ .", "By (REF ), we get $\\operatorname{\\mathbb {E}}({X}_t \\,\|\\,{X}_0)= \\int _{\\mathbb {R}_+^d} {y}\\, P_t({X}_0, \\mathrm {d}{y})= \\mathrm {e}^{t\\widetilde{{B}}} {X}_0 + \\left( \\int _0^t \\mathrm {e}^{u\\widetilde{{B}}} \\, \\mathrm {d}u \\right) \\widetilde{{\\beta }},$ hence we conclude (REF ).", "In order to show (REF ), we are going to apply Proposition 9.11 of Li [13] for the immigration superprocess given in Lemma REF .", "For each $f \\in B(E)$ and $i \\in E$ , the function $\\mathbb {R}_+ \\ni t \\mapsto \\pi _tf(i)$ is the unique locally bounded solution to the linear evolution equation (2.35) in Li [13] taking the form $&\\pi _tf(i)= f(i) + \\int _0^t \\gamma (i, \\pi _sf) \\, \\mathrm {d}s- \\int _0^t b(i) \\pi _sf(i) \\, \\mathrm {d}s \\\\&= f(i)+ \\int _0^t \\left( \\sum _{j=1}^d \\pi _sf(j) \\gamma (i, \\lbrace j\\rbrace ) \\right) \\mathrm {d}s- \\int _0^t b(i) \\pi _sf(i) \\, \\mathrm {d}s= f(i)+ \\int _0^t \\left( \\sum _{j=1}^d \\pi _sf(j) \\widetilde{b}_{j,i} \\right) \\mathrm {d}s ,$ where we used $R_tf = f$ for $f \\in B(E)$ and $t \\in \\mathbb {R}_+$ , $b(i) = - \\widetilde{b}_{i,i}$ and $\\gamma (i, \\lbrace i\\rbrace ) = B(i, \\lbrace i\\rbrace ) = 0$ for $i \\in \\lbrace 1, \\ldots , d\\rbrace $ , and $\\gamma (i, \\lbrace j\\rbrace )= B(i, \\lbrace j\\rbrace ) + \\int _{U_d} z_j \\, \\mu _i(\\mathrm {d}{z})= b_{j,i} + \\int _{U_d} z_j \\, \\mu _i(\\mathrm {d}{z})= \\widetilde{b}_{j,i}$ for $i, j \\in \\lbrace 1, \\ldots , d\\rbrace $ with $i \\ne j$ .", "The functions $\\mathbb {R}_+ \\ni t \\mapsto \\pi _tf(i)$ , $f \\in B(E)$ , $i \\in \\lbrace 1, \\ldots , d\\rbrace $ , can be identified with the functions $\\mathbb {R}_+ \\ni t \\mapsto \\pi _i(t, {\\lambda })$ , ${\\lambda }\\in \\mathbb {R}^d$ , $i \\in \\lbrace 1, \\ldots , d\\rbrace $ , which are the unique locally bounded solution to the linear evolution equations $\\pi _i(t,{\\lambda })= \\lambda _i+ \\int _0^t \\langle \\widetilde{{B}}{e}_i, \\pi (s, {\\lambda }) \\rangle \\, \\mathrm {d}s ,\\qquad t \\in \\mathbb {R}_+ , \\quad i \\in \\lbrace 1, \\ldots , d\\rbrace , \\quad {\\lambda }\\in \\mathbb {R}^d .$ Consequently, the functions $\\mathbb {R}_+ \\ni t \\mapsto \\pi (t, {\\lambda }) := (\\pi _1(t, {\\lambda }), \\ldots , \\pi _d(t, {\\lambda }))$ , ${\\lambda }\\in \\mathbb {R}^d$ , satisfies ${\\pi }(t, {\\lambda })= {\\lambda }+ \\int _0^t \\widetilde{{B}}^\\top \\!\\!", "{\\pi }(s, {\\lambda }) \\, \\mathrm {d}s ,\\qquad t \\in \\mathbb {R}_+ , \\quad {\\lambda }\\in \\mathbb {R}^d ,$ and hence ${\\pi }(t, {\\lambda }) = \\mathrm {e}^{t\\widetilde{{B}}^\\top } \\!", "{\\lambda },\\qquad t \\in \\mathbb {R}_+ , \\quad {\\lambda }\\in \\mathbb {R}^d .$ The functional $B(E) \\ni f \\mapsto \\Gamma (f)= \\eta (f) + \\int _{M(E)^\\circ } \\kappa (f) \\, H_1(\\mathrm {d}\\kappa )$ of [13] can be identified with the functional $\\mathbb {R}^d \\ni {x}\\mapsto {x}^\\top {\\beta }+ \\int _{U_d} {x}^\\top {z}\\, \\nu (\\mathrm {d}{z})= {x}^\\top \\widetilde{{\\beta }}$ .", "Hence Proposition 9.11 of Li [13] implies $\\langle {\\lambda }, \\operatorname{\\mathbb {E}}({X}_t)\\rangle = \\langle \\mathrm {e}^{t\\widetilde{{B}}^\\top } \\!", "{\\lambda }, {x}\\rangle + \\left(\\int _0^t (\\mathrm {e}^{s\\widetilde{{B}}^\\top }{\\lambda })^\\top \\,\\mathrm {d}s \\right) \\widetilde{{\\beta }}= \\left\\langle {\\lambda }, \\mathrm {e}^{t\\widetilde{{B}}} {x}+ \\left(\\int _0^t \\mathrm {e}^{s\\widetilde{{B}}}\\,\\mathrm {d}s \\right) \\widetilde{{\\beta }}\\right\\rangle $ for $t \\in \\mathbb {R}_+$ and ${\\lambda }\\in \\mathbb {R}^d$ , which yields (REF ).", "$\\Box $ Remark.", "3.5 We call the attention that in the proof of the forthcoming Theorem REF , which states existence of an $\\mathbb {R}_+^d$ -valued weak solution of the SDE (REF ), we will extensively use that for a CBI process $({X}_t)_{t\\in \\mathbb {R}_+}$ with parameters $(d, {c}, {\\beta }, {B}, \\nu , {\\mu })$ satisfying $\\operatorname{\\mathbb {E}}(\\Vert {X}_0\\Vert ) < \\infty $ and the moment condition (REF ), we have $\\int _0^t \\operatorname{\\mathbb {E}}(\\Vert {X}_s\\Vert ) \\, \\mathrm {d}s < \\infty $ , $t \\in \\mathbb {R}_+$ , proved in Lemma REF .", "We point out that in the proof of Lemma REF we can not use the SDE (REF ), since at that point it has not yet been proved that a CBI process is a solution of this SDE.", "This drives us back to Definition REF of CBI processes in the proof of Lemma REF .", "Having proved that a CBI process is a solution of the SDE (REF ), one could give another proof of Lemma REF (roughly speaking by taking expectations via localization argument).", "$\\Box $ Definition.", "3.6 We say that uniqueness in the sense of probability law holds for the SDE (REF ) among $\\mathbb {R}_+^d$ -valued weak solutions if whenever $\\bigl ( \\Omega , {\\mathcal {F}}, ({\\mathcal {F}}_t)_{t\\in \\mathbb {R}_+}, \\operatorname{\\mathbb {P}}, {W}, p, {X}\\bigr )$ and $\\bigl ( \\widetilde{\\Omega }, \\widetilde{{\\mathcal {F}}}, (\\widetilde{{\\mathcal {F}}}_t)_{t\\in \\mathbb {R}_+}, \\widetilde{\\operatorname{\\mathbb {P}}}, \\widetilde{{W}}, \\widetilde{p}, \\widetilde{{X}}\\bigr )$ are $\\mathbb {R}_+^d$ -valued weak solutions of the SDE (REF ) such that $\\operatorname{\\mathbb {P}}({X}_0 \\in B) = \\widetilde{\\operatorname{\\mathbb {P}}}(\\widetilde{{X}}_0 \\in B)$ for all $B \\in {\\mathcal {B}}(\\mathbb {R}^d)$ , then $\\operatorname{\\mathbb {P}}({X}\\in C) = \\widetilde{\\operatorname{\\mathbb {P}}}(\\widetilde{{X}}\\in C)$ for all $C \\in {\\mathcal {D}}(\\mathbb {R}_+, \\mathbb {R}^d)$ .", "Theorem.", "3.7 Let $(d, {c}, {\\beta }, {B}, \\nu , {\\mu })$ be a set of admissible parameters in the sense of Definition REF such that the moment condition (REF ) holds.", "Then for any probability measure $n$ on $(\\mathbb {R}_+^d, {\\mathcal {B}}(\\mathbb {R}_+^d))$ with $\\int _{\\mathbb {R}^d_+} \\Vert {z}\\Vert \\, n(\\mathrm {d}{z}) < \\infty $ , the SDE (REF ) admits an $\\mathbb {R}_+^d$ -valued weak solution with initial distribution $n$ which is unique in the sense of probability law among $\\mathbb {R}_+^d$ -valued weak solutions.", "Moreover, any $\\mathbb {R}_+^d$ -valued weak solution is a CBI process with parameters $(d, {c}, {\\beta }, {B}, \\nu , {\\mu })$ .", "Proof.", "Suppose that $({X}_t)_{t\\in \\mathbb {R}_+}$ is a càdlàg realization of a CBI process with parameters $(d, {c}, {\\beta }, {B}, \\nu , {\\mu })$ on a probability space $(\\Omega , {\\mathcal {F}}, \\operatorname{\\mathbb {P}})$ having initial distribution $n$ , i.e., $({X}_t)_{t\\in \\mathbb {R}_+}$ is a time homogeneous Markov process having càdlàg trajectories and the same finite dimensional distributions as a CBI process with parameters $(d, {c}, {\\beta }, {B}, \\nu , {\\mu })$ having initial distribution $n$ (such a realization exists due to Theorem 9.15 in Li [13]).", "Let ${\\mathcal {F}}_t := \\bigcap _{\\varepsilon >0} \\sigma \\left({\\mathcal {F}}_{t+\\varepsilon }^{X}\\cup {\\mathcal {N}}\\right) ,\\qquad t \\in \\mathbb {R}_+ ,$ where ${\\mathcal {N}}$ denotes the collection of null sets under the probability measure $\\operatorname{\\mathbb {P}}$ , and $({\\mathcal {F}}_t^{X})_{t\\in \\mathbb {R}_+}$ stands for the natural filtration generated by the process $({X}_t)_{t\\in \\mathbb {R}_+}$ , hence the filtered probability space $(\\Omega , {\\mathcal {F}}, ({\\mathcal {F}}_t)_{t\\in \\mathbb {R}_+}, \\operatorname{\\mathbb {P}})$ satisfies the usual hypotheses.", "By the equivalence of parts (3) and (4) of Theorem 9.18 of Li [13] applied to the immigration superprocess given in Lemma REF , we conclude that the process $({X}_t)_{t\\in \\mathbb {R}_+}$ has no negative jumps, the (not necessarily Poisson) random measure $N_0(\\mathrm {d}s, \\mathrm {d}{z}):= \\sum _{u\\in \\mathbb {R}_{++}}\\mathbb {1}_{\\lbrace {X}_u\\ne {X}_{u-}\\rbrace } \\delta _{(u,{X}_u-{X}_{u-})} (\\mathrm {d}s, \\mathrm {d}{z})$ on $\\mathbb {R}_{++} \\times U_d$ has predictable compensator $\\widehat{N}_0(\\mathrm {d}s, \\mathrm {d}{z}):= \\sum _{j=1}^d X_{s-,j} \\, \\mathrm {d}s \\, \\mu _j(\\mathrm {d}{z}) + \\mathrm {d}s \\, \\nu (\\mathrm {d}{z}) ,$ and ${X}_t - {X}_0- \\int _0^t \\bigl ( \\widetilde{{\\beta }}+ \\widetilde{{B}}{X}_s \\bigr ) \\, \\mathrm {d}s- \\int _0^t \\int _{U_d} {z}\\, \\widetilde{N}_0(\\mathrm {d}s, \\mathrm {d}{z}) , \\qquad t \\in \\mathbb {R}_+ ,$ is a continuous locally square integrable martingale starting from ${0}\\in \\mathbb {R}^d$ with quadratic variation process $\\left( 2 \\delta _{i,j} c_i\\int _0^t X_{s,i} \\, \\mathrm {d}s \\right)_{i,j\\in \\lbrace 1,\\ldots ,d\\rbrace } , \\qquad t \\in \\mathbb {R}_+ ,$ where $\\widetilde{N}_0(\\mathrm {d}s, \\mathrm {d}{z}) := N_0(\\mathrm {d}s, \\mathrm {d}{z}) - \\widehat{N}_0(\\mathrm {d}s, \\mathrm {d}{z})$ .", "Indeed, first, note that $R_tf=f$ , $t\\in \\mathbb {R}_+$ , $f\\in B(E)$ , yields that the strong generator of $(R_t)_{t\\in \\mathbb {R}_+}$ is identically 0, i.e., $A=0$ , see Li [13].", "Using $b(i) = - \\widetilde{b}_{i,i}$ and $\\gamma (i, \\lbrace i\\rbrace ) = B(i, \\lbrace i\\rbrace ) = 0$ for $i \\in \\lbrace 1, \\ldots , d\\rbrace $ and $\\gamma (i, \\lbrace j\\rbrace ) = \\widetilde{b}_{j,i}$ for $i, j \\in \\lbrace 1, \\ldots , d\\rbrace $ with $i \\ne j$ (see, (REF )), the function $B(E)\\ni f\\mapsto Af+\\gamma f-bf$ of Li [13] can be identified with the function $E\\ni i \\mapsto \\sum _{j=1}^d f(j)\\gamma (i,\\lbrace j\\rbrace ) - b(i)f(i)= \\sum _{j=1}^d \\widetilde{b}_{j,i}f(j).$ Recalling that the functional $B(E) \\ni f \\mapsto \\Gamma (f)= \\eta (f) + \\int _{M(E)^\\circ } \\kappa (f) \\, H(\\mathrm {d}\\kappa )$ is identified with the functional $\\mathbb {R}^d \\ni {x}\\mapsto {x}^\\top \\widetilde{{\\beta }}$ (see, the end of the proof of Lemma REF ), Theorem 9.18 of Li [13] yields that for each ${w}= (w_1, \\ldots , w_d)^\\top \\in \\mathbb {R}^d$ , the process $({w}^\\top {X}_t)_{t\\in \\mathbb {R}_+}$ has no negative jumps, and ${w}^\\top {X}_t - {w}^\\top {X}_0- \\int _0^t\\bigl ( {w}^\\top \\widetilde{{\\beta }}+ {w}^\\top \\widetilde{{B}}{X}_s \\bigr ) \\, \\mathrm {d}s- \\int _0^t \\int _{U_d} {w}^\\top {z}\\, \\widetilde{N}_0(\\mathrm {d}s, \\mathrm {d}{z}) ,\\qquad t \\in \\mathbb {R}_+ ,$ is a continuous locally square integrable martingale strating from $0 \\in \\mathbb {R}$ with quadratic variation process $\\langle {w}^\\top {X}\\rangle _t= 2 \\sum _{i=1}^d c_i w_i^2 \\int _0^t X_{s,i} \\, \\mathrm {d}s , \\qquad t \\in \\mathbb {R}_+ .$ Further, by polarization identity, for all ${w}, \\widetilde{{w}}\\in \\mathbb {R}^d$ , the cross quadratic variation process of $({w}^\\top {X}_t)_{t\\in \\mathbb {R}_+}$ and $(\\widetilde{{w}}^\\top {X}_t)_{t\\in \\mathbb {R}_+}$ takes the form $\\langle {w}^\\top {X}, \\widetilde{{w}}^\\top {X}\\rangle _t&= \\frac{1}{4}\\left( \\langle ({w}+ \\widetilde{{w}})^\\top {X}\\rangle _t- \\langle ({w}- \\widetilde{{w}})^\\top {X}\\rangle _t \\right) \\\\&= \\frac{1}{4}\\biggl ( 2 \\sum _{i=1}^d c_i (w_i + \\widetilde{w}_i)^2 \\int _0^t X_{s,i} \\, \\mathrm {d}s- 2 \\sum _{i=1}^d c_i (w_i - \\widetilde{w}_i)^2 \\int _0^t X_{s,i} \\, \\mathrm {d}s\\biggr )\\\\&= 2 \\sum _{i=1}^d c_i w_i \\widetilde{w}_i \\int _0^t X_{s,i} \\, \\mathrm {d}s, \\qquad t \\in \\mathbb {R}_+ .$ We note that the integral $\\int _0^t \\int _{U_d} {z}\\, \\widetilde{N}_0(\\mathrm {d}s, \\mathrm {d}{z})$ is well-defined, since ${z}= {z}\\mathbb {1}_{\\lbrace \\Vert {z}\\Vert <1\\rbrace } + {z}\\mathbb {1}_{\\lbrace \\Vert {z}\\Vert \\geqslant 1\\rbrace }$ , ${z}\\in U_d$ , and the functions $\\mathbb {R}_+ \\times U_d \\times \\Omega \\ni (s, {z}, \\omega )\\mapsto {z}\\mathbb {1}_{\\lbrace \\Vert {z}\\Vert <1\\rbrace }$ and $\\mathbb {R}_+ \\times U_d \\times \\Omega \\ni (s,{z},\\omega )\\mapsto {z}\\mathbb {1}_{\\lbrace \\Vert {z}\\Vert \\geqslant 1\\rbrace }$ belong to the classes ${F}_{p_0}^2$ and ${F}_{p_0}^1$ , respectively, where $p_0$ denotes the point process on $U_d$ with counting measure $N_0(\\mathrm {d}s, \\mathrm {d}{z})$ , i.e., $p_0(u) := {X}_u - {X}_{u-}$ for $u \\in D(p_0)$ with $D(p_0) := \\lbrace u \\in \\mathbb {R}_{++} : {X}_u \\ne {X}_{u-} \\rbrace $ .", "Indeed, $&\\operatorname{\\mathbb {E}}\\left(\\int _0^t \\int _{U_d}\\Vert {z}\\Vert ^2 \\mathbb {1}_{\\lbrace \\Vert {z}\\Vert <1\\rbrace } \\, \\widehat{N}_0(\\mathrm {d}s, \\mathrm {d}{z})\\right) \\\\&= \\int _0^t \\int _{U_d}\\Vert {z}\\Vert ^2 \\mathbb {1}_{\\lbrace \\Vert {z}\\Vert <1\\rbrace } \\, \\mathrm {d}s \\, \\nu (\\mathrm {d}{z})+ \\sum _{j=1}^d\\int _0^t \\int _{U_d}\\Vert {z}\\Vert ^2 \\mathbb {1}_{\\lbrace \\Vert {z}\\Vert <1\\rbrace } \\operatorname{\\mathbb {E}}(X_{s,j}) \\, \\mathrm {d}s \\, \\mu _j(\\mathrm {d}{z}) \\\\&\\leqslant t \\int _{U_d} \\Vert {z}\\Vert \\, \\nu (\\mathrm {d}{z})+ \\sum _{j=1}^d\\int _0^t \\operatorname{\\mathbb {E}}(X_{s,j}) \\, \\mathrm {d}s\\int _{U_d} \\Vert {z}\\Vert ^2 \\mathbb {1}_{\\lbrace \\Vert {z}\\Vert <1\\rbrace } \\, \\mu _j(\\mathrm {d}{z})< \\infty $ by Lemma REF and the inequalities (REF ) and (REF ), and $&\\operatorname{\\mathbb {E}}\\left(\\int _0^t \\int _{U_d}\\Vert {z}\\Vert \\mathbb {1}_{\\lbrace \\Vert {z}\\Vert \\geqslant 1\\rbrace } \\, \\widehat{N}_0(\\mathrm {d}s, \\mathrm {d}{z})\\right) \\\\&= \\int _0^t \\int _{U_d}\\Vert {z}\\Vert \\mathbb {1}_{\\lbrace \\Vert {z}\\Vert \\geqslant 1\\rbrace } \\, \\mathrm {d}s \\, \\nu (\\mathrm {d}{z})+ \\sum _{j=1}^d\\int _0^t \\int _{U_d}\\Vert {z}\\Vert \\mathbb {1}_{\\lbrace \\Vert {z}\\Vert \\geqslant 1\\rbrace } \\operatorname{\\mathbb {E}}(X_{s,j}) \\, \\mathrm {d}s \\, \\mu _j(\\mathrm {d}{z})$ $&\\leqslant t \\int _{U_d} \\Vert {z}\\Vert \\, \\nu (\\mathrm {d}{z})+ \\sum _{j=1}^d\\int _0^t \\operatorname{\\mathbb {E}}(X_{s,j}) \\, \\mathrm {d}s\\int _{U_d} \\Vert {z}\\Vert \\mathbb {1}_{\\lbrace \\Vert {z}\\Vert \\geqslant 1\\rbrace } \\, \\mu _j(\\mathrm {d}{z})< \\infty $ by Lemma REF and the inequalities (REF ) and (REF ).", "Using that $\\operatorname{\\mathbb {P}}\\big (\\int _0^t X_{s,i} \\, \\mathrm {d}s < \\infty \\big ) = 1$ , $i \\in \\lbrace 1, \\ldots , d\\rbrace $ (since ${X}$ has càdlàg trajectories almost surely), by choosing ${w}= {e}_j$ , $j \\in \\lbrace 1, \\ldots , d\\rbrace $ , a representation theorem for continuous locally square integrable martingales (see, e.g., Ikeda and Watanabe [7]) yields ${X}_t= {X}_0 + \\int _0^t \\bigl ( \\widetilde{{\\beta }}+ \\widetilde{{B}}{X}_s \\bigr ) \\, \\mathrm {d}s+ \\sum _{i=1}^d {e}_i \\int _0^t \\sqrt{2c_i X_{s,i}} \\, \\mathrm {d}W_{s,i}+ \\int _0^t \\int _{U_d} {z}\\, \\widetilde{N}_0(\\mathrm {d}s, \\mathrm {d}{z})$ for all $t \\in \\mathbb {R}_+$ , $\\widetilde{\\operatorname{\\mathbb {P}}}$ -almost surely on an extension $\\bigl (\\widetilde{\\Omega }, \\widetilde{{\\mathcal {F}}}, (\\widetilde{{\\mathcal {F}}}_t)_{t\\in \\mathbb {R}_+}, \\widetilde{\\operatorname{\\mathbb {P}}}\\bigr )$ of the filtered probability space $(\\Omega , {\\mathcal {F}}, ({\\mathcal {F}}_t)_{t\\in \\mathbb {R}_+}, \\operatorname{\\mathbb {P}})$ (see Definition REF ), and $(W_{t,1}, \\ldots , W_{t,d})_{t\\in \\mathbb {R}_+}$ is a $d$ -dimensional $(\\widetilde{{\\mathcal {F}}}_t)_{t\\in \\mathbb {R}_+}$ -Brownian motion.", "We note that, with a little abuse of notation, the extended random variables on the extension $\\bigl (\\widetilde{\\Omega }, \\widetilde{{\\mathcal {F}}}, (\\widetilde{{\\mathcal {F}}}_t)_{t\\in \\mathbb {R}_+}, \\widetilde{\\operatorname{\\mathbb {P}}}\\bigr )$ are denoted in the same way as the original ones.", "Let $\\widetilde{{\\mathcal {G}}}_t := \\bigcap _{\\varepsilon >0} \\sigma \\left(\\widetilde{{\\mathcal {F}}}_{t+\\varepsilon } \\cup \\widetilde{{\\mathcal {N}}}\\right) ,\\qquad t \\in \\mathbb {R}_+ ,$ where $\\widetilde{{\\mathcal {N}}}$ denotes the collection of null sets under the probability measure $\\widetilde{\\operatorname{\\mathbb {P}}}$ .", "Then the filtered probability space $(\\widetilde{\\Omega }, \\widetilde{{\\mathcal {F}}}, (\\widetilde{{\\mathcal {G}}}_t)_{t\\in \\mathbb {R}_+}, \\widetilde{\\operatorname{\\mathbb {P}}})$ satisfies the usual hypotheses, and by Lemma REF , $(W_{t,1}, \\ldots , W_{t,d})_{t\\in \\mathbb {R}_+}$ is a $d$ -dimensional $(\\widetilde{{\\mathcal {G}}}_t)_{t\\in \\mathbb {R}_+}$ -Brownian motion.", "The aim of the following discussion is to show, by the representation theorem of Ikeda and Watanabe [7], that the SDE (REF ) holds on an extension of the original probability space.", "The predictable compensator of the random measure $N_0(\\mathrm {d}s, \\mathrm {d}{z})$ can be written in the form $\\widehat{N}_0(\\mathrm {d}s, \\mathrm {d}{z}) = \\mathrm {d}s \\, q(s, \\mathrm {d}{z})$ , where $q(s, \\mathrm {d}{z}) := \\sum _{j=1}^d X_{s-,j} \\, \\mu _j(\\mathrm {d}{z}) + \\nu (\\mathrm {d}{z}) .$ Let $\\Theta : \\mathbb {R}_+ \\times V \\times \\widetilde{\\Omega }\\rightarrow U_d \\cup \\lbrace {0}\\rbrace = \\mathbb {R}_+^d$ be defined by $\\Theta (s, {r}, \\widetilde{\\omega }) := h({X}_{s-}(\\widetilde{\\omega }), {r}) , \\qquad (s, {r}, \\widetilde{\\omega }) \\in \\mathbb {R}_+ \\times V \\times \\widetilde{\\Omega }.$ (Note, that $\\Delta = {0}$ in the notation of Ikeda and Watanabe [7].)", "Then condition (7.26) on page 93 in Ikeda and Watanabe [7] holds, since for all $s\\in \\mathbb {R}_+$ , $\\widetilde{\\omega }\\in \\widetilde{\\Omega }$ , and $B \\in {\\mathcal {B}}(U_d)$ , we have $&m(\\lbrace {r}\\in V : \\Theta (s, {r}, \\widetilde{\\omega }) \\in B \\rbrace )= \\sum _{i=0}^d m(\\lbrace {r}\\in {\\mathcal {R}}_i : \\Theta (s, {r}, \\widetilde{\\omega }) \\in B \\rbrace ) \\\\&= \\sum _{i=1}^d(\\mu _i \\times \\ell )\\big ( \\lbrace ({z}, u) \\in {\\mathcal {R}}_i: {z}\\mathbb {1}_{\\lbrace u\\leqslant X_{s-,i}(\\widetilde{\\omega })\\rbrace } \\in B \\rbrace \\big )+ \\nu \\big ( \\lbrace {r}\\in {\\mathcal {R}}_0 : {r}\\in B \\rbrace \\big )$ $&= \\sum _{i=1}^d X_{s-,i}(\\widetilde{\\omega }) \\, \\mu _i(B) + \\nu (B)= q(s, B)(\\widetilde{\\omega }) ,$ where $\\ell $ denotes the Lebesgue measure on $\\mathbb {R}_{++}$ , and we used that ${0}\\notin B$ .", "By Theorem II.7.4 in Ikeda and Watanabe [7], on an extension $\\bigl (\\widetilde{\\widetilde{\\Omega }}, \\widetilde{\\widetilde{{\\mathcal {F}}}}, (\\widetilde{\\widetilde{{\\mathcal {F}}}}_t)_{t\\in \\mathbb {R}_+}, \\widetilde{\\widetilde{\\operatorname{\\mathbb {P}}}}\\bigr )$ of $(\\widetilde{\\Omega }, \\widetilde{{\\mathcal {F}}}, (\\widetilde{{\\mathcal {G}}}_t)_{t\\in \\mathbb {R}_+}, \\widetilde{\\operatorname{\\mathbb {P}}})$ , there is a stationary $(\\widetilde{\\widetilde{{\\mathcal {F}}}}_t)_{t\\in \\mathbb {R}_+}$ -Poisson point process $p$ on $V$ with characteristic measure $m$ such that $N_0\\bigl ( (0, t] \\times B \\bigr )&= \\int _0^t \\int _V \\mathbb {1}_B(\\Theta (s, {r})) \\, N(\\mathrm {d}s, \\mathrm {d}{r}) \\\\&= \\# \\lbrace s \\in D(p) : s \\in (0, t] , \\, \\Theta (s, p(s)) \\in B \\rbrace \\qquad \\text{$\\widetilde{\\widetilde{\\operatorname{\\mathbb {P}}}}$-a.s.}$ for all $B \\in {\\mathcal {B}}(U_d)$ , where $N(\\mathrm {d}s, \\mathrm {d}{r})$ denotes the counting measure of $p$ , and $D(p)$ is the domain of $p$ being a countable subset of $\\mathbb {R}_{++}$ such that $\\lbrace s \\in D(p) : s \\in (0, t] , \\, p(s) \\in B\\rbrace $ is finite for all $t \\in \\mathbb {R}_+$ and compact subsets $B \\in {\\mathcal {B}}(U_d)$ .", "Then, by Lemma REF , $(W_{t,1}, \\ldots , W_{t,d})_{t\\in \\mathbb {R}_+}$ is a $d$ -dimensional $(\\widetilde{\\widetilde{{\\mathcal {F}}}}_t)_{t\\in \\mathbb {R}_+}$ -Brownian motion.", "Let $\\widetilde{\\widetilde{{\\mathcal {G}}}}_t := \\bigcap _{\\varepsilon >0} \\sigma \\left(\\widetilde{\\widetilde{{\\mathcal {F}}}}_{t+\\varepsilon } \\cup \\widetilde{\\widetilde{{\\mathcal {N}}}}\\right) ,\\qquad t \\in \\mathbb {R}_+ ,$ where $\\widetilde{\\widetilde{{\\mathcal {N}}}}$ denotes the collection of null sets under the probability measure $\\widetilde{\\widetilde{\\operatorname{\\mathbb {P}}}}$ .", "Then the filtered probability space $(\\widetilde{\\widetilde{\\Omega }}, \\widetilde{\\widetilde{{\\mathcal {F}}}}, (\\widetilde{\\widetilde{{\\mathcal {G}}}}_t)_{t\\in \\mathbb {R}_+}, \\widetilde{\\widetilde{\\operatorname{\\mathbb {P}}}})$ satisfies the usual hypotheses.", "By Lemma REF , $(W_{t,1}, \\ldots , W_{t,d})_{t\\in \\mathbb {R}_+}$ is a $d$ -dimensional $(\\widetilde{\\widetilde{{\\mathcal {G}}}}_t)_{t\\in \\mathbb {R}_+}$ -Brownian motion, and $p$ is a stationary $(\\widetilde{\\widetilde{{\\mathcal {G}}}}_t)_{t\\in \\mathbb {R}_+}$ -Poisson point process on $V$ with characteristic measure $m$ .", "Consequently, $\\# \\lbrace s \\in D(p_0) : s \\in (0, t] , \\, p_0(s) \\in B \\rbrace = \\# \\lbrace s \\in D(p) : s \\in (0, t] , \\, h({X}_{s-}, p(s)) \\in B \\rbrace $ for all $B \\in {\\mathcal {B}}(U_d)$ .", "Using this representation, we will calculate $\\int _0^t \\int _{U_d} {z}\\, \\widetilde{N}_0(\\mathrm {d}s, \\mathrm {d}{z})$ , $t \\in \\mathbb {R}_+$ .", "First observe that $\\int _0^t \\int _{U_d} {z}\\, \\widetilde{N}_0(\\mathrm {d}s, \\mathrm {d}{z})= \\int _0^t \\int _{U_d} {z}\\mathbb {1}_{\\lbrace \\Vert {z}\\Vert \\geqslant 1\\rbrace } \\, \\widetilde{N}_0(\\mathrm {d}s, \\mathrm {d}{z})+ \\int _0^t \\int _{U_d} {z}\\mathbb {1}_{\\lbrace \\Vert {z}\\Vert <1\\rbrace } \\, \\widetilde{N}_0(\\mathrm {d}s, \\mathrm {d}{z}) .$ Since the function $\\mathbb {R}_+ \\times U_d \\times \\Omega \\ni (s, {z}, \\omega )\\mapsto {z}\\mathbb {1}_{\\lbrace \\Vert {z}\\Vert \\geqslant 1\\rbrace }$ belongs to the class ${F}_{p_0}^1$ , by Ikeda and Watanabe [7], we obtain $\\int _0^t \\int _{U_d} {z}\\mathbb {1}_{\\lbrace \\Vert {z}\\Vert \\geqslant 1\\rbrace } \\, \\widetilde{N}_0(\\mathrm {d}s, \\mathrm {d}{z})= \\int _0^t \\int _{U_d} {z}\\mathbb {1}_{\\lbrace \\Vert {z}\\Vert \\geqslant 1\\rbrace } \\, N_0(\\mathrm {d}s, \\mathrm {d}{z})- \\int _0^t \\int _{U_d} {z}\\mathbb {1}_{\\lbrace \\Vert {z}\\Vert \\geqslant 1\\rbrace } \\, \\widehat{N}_0(\\mathrm {d}s, \\mathrm {d}{z}) .$ Applying (REF ), we obtain $&\\int _0^t \\int _{U_d} {z}\\mathbb {1}_{\\lbrace \\Vert {z}\\Vert \\geqslant 1\\rbrace } \\, N_0(\\mathrm {d}s, \\mathrm {d}{z})= \\sum _{s\\in D(p_0)\\cap (0,t]} p_0(s) \\mathbb {1}_{\\lbrace \\Vert p_0(s)\\Vert \\geqslant 1\\rbrace } \\\\&= \\sum _{s\\in D(p)\\cap (0,t]} h({X}_{s-}, p(s)) \\mathbb {1}_{\\lbrace \\Vert h({X}_{s-}, p(s))\\Vert \\geqslant 1\\rbrace }= \\int _0^t \\int _Vh({X}_{s-}, {r}) \\mathbb {1}_{\\lbrace \\Vert h({X}_{s-}, {r})\\Vert \\geqslant 1\\rbrace }\\, N(\\mathrm {d}s, \\mathrm {d}{r}) \\\\&= \\int _0^t \\int _{{\\mathcal {R}}_0} {r}\\mathbb {1}_{\\lbrace \\Vert {r}\\Vert \\geqslant 1\\rbrace } \\, N(\\mathrm {d}s, \\mathrm {d}{r})+ \\sum _{j=1}^d\\int _0^t \\int _{{\\mathcal {R}}_j}{z}\\mathbb {1}_{\\lbrace \\Vert {z}\\Vert \\geqslant 1\\rbrace } \\mathbb {1}_{\\lbrace u\\leqslant X_{s-,j}\\rbrace }\\, N(\\mathrm {d}s, \\mathrm {d}{r}) \\\\&= \\int _0^t \\int _{V_1} g({X}_{s-}, {r}) \\, N(\\mathrm {d}s, \\mathrm {d}{r})- \\int _0^t \\int _{{\\mathcal {R}}_0} {r}\\mathbb {1}_{\\lbrace \\Vert {r}\\Vert <1\\rbrace } \\, N(\\mathrm {d}s, \\mathrm {d}{r}) .$ Here we used that the function $\\mathbb {R}_+ \\times U_d \\times \\widetilde{\\widetilde{\\Omega }}\\ni (s, {z}, \\widetilde{\\widetilde{\\omega }})\\mapsto {z}\\mathbb {1}_{\\lbrace \\Vert {z}\\Vert \\geqslant 1\\rbrace }$ belongs to the class ${F}_{p_0}^1$ , hence the function $\\mathbb {R}_+ \\times V \\times \\widetilde{\\widetilde{\\Omega }}\\ni (s, {r}, \\widetilde{\\widetilde{\\omega }})\\mapsto h({X}_{s-}(\\widetilde{\\widetilde{\\omega }}), {r}) \\mathbb {1}_{\\lbrace \\Vert h({X}_{s-}(\\widetilde{\\widetilde{\\omega }}), {r})\\Vert \\geqslant 1\\rbrace }$ belongs to the class ${F}_p^1$ , and function $\\mathbb {R}_+ \\times V \\times \\widetilde{\\widetilde{\\Omega }}\\ni (s, {r}, \\widetilde{\\widetilde{\\omega }})\\mapsto {r}\\mathbb {1}_{\\lbrace \\Vert {r}\\Vert <1\\rbrace } \\mathbb {1}_{{\\mathcal {R}}_0}({r})$ also belongs to the class ${F}_p^1$ (due to (REF )), thus the function $\\mathbb {R}_+ \\times V \\times \\widetilde{\\widetilde{\\Omega }}\\ni (s, {r}, \\widetilde{\\widetilde{\\omega }})\\mapsto g({X}_{s-}(\\widetilde{\\widetilde{\\omega }}), {r})$ belongs to the class ${F}_p^1$ as well.", "Moreover, $&\\int _0^t \\int _{U_d} {z}\\mathbb {1}_{\\lbrace \\Vert {z}\\Vert \\geqslant 1\\rbrace } \\, \\widehat{N}_0(\\mathrm {d}s, \\mathrm {d}{z})= \\int _0^t \\int _{U_d} {z}\\mathbb {1}_{\\lbrace \\Vert {z}\\Vert \\geqslant 1\\rbrace } \\, \\mathrm {d}s \\, \\nu (\\mathrm {d}{z})+ \\sum _{j=1}^d\\int _0^t \\int _{U_d}{z}\\mathbb {1}_{\\lbrace \\Vert {z}\\Vert \\geqslant 1\\rbrace } X_{s,j} \\, \\mathrm {d}s \\, \\mu _j(\\mathrm {d}{z}) \\\\&= \\int _0^t \\int _{U_d} {r}\\mathbb {1}_{\\lbrace \\Vert {r}\\Vert \\geqslant 1\\rbrace } \\, \\mathrm {d}s \\, \\nu (\\mathrm {d}{r})+ \\sum _{j=1}^d\\int _0^t X_{s,j}\\,\\mathrm {d}s \\int _{U_d}{z}\\mathbb {1}_{\\lbrace \\Vert {z}\\Vert \\geqslant 1\\rbrace } \\, \\mu _j(\\mathrm {d}{z}) .$ Let ${\\mathcal {M}}_2$ denote the complete metric space of square integrable right continuous $d$ -dimensional martingales on $(\\widetilde{\\widetilde{\\Omega }}, \\widetilde{\\widetilde{{\\mathcal {F}}}}, \\widetilde{\\widetilde{\\operatorname{\\mathbb {P}}}})$ with respect to $(\\widetilde{\\widetilde{{\\mathcal {F}}}}_t)_{t\\in \\mathbb {R}_+}$ starting from 0, see, e.g., Ikeda and Watanabe [7].", "The function $\\mathbb {R}_+ \\times U_d \\times \\widetilde{\\widetilde{\\Omega }}\\ni (s, {z}, \\widetilde{\\widetilde{\\omega }})\\mapsto {z}\\mathbb {1}_{\\lbrace \\Vert {z}\\Vert <1\\rbrace }$ belongs to the class ${F}_{p_0}^2$ , hence, by Ikeda and Watanabe [7], the process $\\bigl (\\int _0^t \\int _{U_d}{z}\\mathbb {1}_{\\lbrace \\Vert {z}\\Vert <1\\rbrace } \\, \\widetilde{N}_0(\\mathrm {d}s, \\mathrm {d}{z})\\bigr )_{t\\in \\mathbb {R}_+}$ belongs to the space ${\\mathcal {M}}_2$ .", "Moreover, by Ikeda and Watanabe [7], $\\int _0^t \\int _{U_d} {z}\\mathbb {1}_{\\lbrace \\Vert {z}\\Vert <1\\rbrace } \\, \\widetilde{N}_0(\\mathrm {d}s, \\mathrm {d}{z})$ is the limit of the sequence $\\int _0^t \\int _{U_d}{z}\\mathbb {1}_{\\lbrace \\frac{1}{n}\\leqslant \\Vert {z}\\Vert <1\\rbrace } \\, \\widetilde{N}_0(\\mathrm {d}s, \\mathrm {d}{z})$ , $n \\in \\mathbb {N}$ , in ${\\mathcal {M}}_2$ as $n \\rightarrow \\infty $ .", "For all $n \\in \\mathbb {N}$ , the mapping $\\mathbb {R}_+ \\times U_d \\times \\widetilde{\\widetilde{\\Omega }}\\ni (s, {z}, \\widetilde{\\widetilde{\\omega }})\\mapsto {z}\\mathbb {1}_{\\lbrace \\frac{1}{n}\\leqslant \\Vert {z}\\Vert <1\\rbrace }$ belongs to the class ${F}_{p_0}^1 \\cap {F}_{p_0}^2$ , hence we obtain $\\int _0^t \\int _{U_d}{z}\\mathbb {1}_{\\lbrace \\frac{1}{n}\\leqslant \\Vert {z}\\Vert <1\\rbrace } \\, \\widetilde{N}_0(\\mathrm {d}s, \\mathrm {d}{z})= \\int _0^t \\int _{U_d} {z}\\mathbb {1}_{\\lbrace \\frac{1}{n}\\leqslant \\Vert {z}\\Vert <1\\rbrace } \\, N_0(\\mathrm {d}s, \\mathrm {d}{z})- \\int _0^t \\int _{U_d}{z}\\mathbb {1}_{\\lbrace \\frac{1}{n}\\leqslant \\Vert {z}\\Vert <1\\rbrace } \\, \\widehat{N}_0(\\mathrm {d}s, \\mathrm {d}{z}) .$ Similarly as above, $&\\int _0^t \\int _{U_d}{z}\\mathbb {1}_{\\lbrace \\frac{1}{n}\\leqslant \\Vert {z}\\Vert <1\\rbrace } \\, N_0(\\mathrm {d}s, \\mathrm {d}{z}) \\\\&= \\int _0^t \\int _{{\\mathcal {R}}_0}{r}\\mathbb {1}_{\\lbrace \\frac{1}{n}\\leqslant \\Vert {r}\\Vert <1\\rbrace } \\, N(\\mathrm {d}s, \\mathrm {d}{r})+ \\sum _{j=1}^d\\int _0^t \\int _{{\\mathcal {R}}_j}{z}\\mathbb {1}_{\\lbrace \\frac{1}{n}\\leqslant \\Vert {z}\\Vert <1\\rbrace } \\mathbb {1}_{\\lbrace u\\leqslant X_{s-,j}\\rbrace }\\, N(\\mathrm {d}s, \\mathrm {d}{r})$ and $&\\int _0^t \\int _{U_d}{z}\\mathbb {1}_{\\lbrace \\frac{1}{n}\\leqslant \\Vert {z}\\Vert <1\\rbrace } \\, \\widehat{N}_0(\\mathrm {d}s, \\mathrm {d}{z}) \\\\&= \\int _0^t \\int _{U_d}{r}\\mathbb {1}_{\\lbrace \\frac{1}{n}\\leqslant \\Vert {r}\\Vert <1\\rbrace } \\, \\mathrm {d}s \\, \\nu (\\mathrm {d}{r})+ \\sum _{j=1}^d\\int _0^t \\int _{U_d} \\int _{U_1}{z}\\mathbb {1}_{\\lbrace \\frac{1}{n}\\leqslant \\Vert {z}\\Vert <1\\rbrace } \\mathbb {1}_{\\lbrace u\\leqslant X_{s,j}\\rbrace }\\, \\mathrm {d}s \\, \\mu _j(\\mathrm {d}{z}) \\, \\mathrm {d}u .$ Consequently, $&\\int _0^t \\int _{U_d}{z}\\mathbb {1}_{\\lbrace \\frac{1}{n}\\leqslant \\Vert {z}\\Vert <1\\rbrace } \\, \\widetilde{N}_0(\\mathrm {d}s, \\mathrm {d}{z}) \\\\&= \\int _0^t \\int _{{\\mathcal {R}}_0}{r}\\mathbb {1}_{\\lbrace \\frac{1}{n}\\leqslant \\Vert {r}\\Vert <1\\rbrace } \\, \\widetilde{N}(\\mathrm {d}s, \\mathrm {d}{r})+ \\sum _{j=1}^d\\int _0^t \\int _{{\\mathcal {R}}_j}{z}\\mathbb {1}_{\\lbrace \\frac{1}{n}\\leqslant \\Vert {z}\\Vert <1\\rbrace } \\mathbb {1}_{\\lbrace u\\leqslant X_{s-,j}\\rbrace }\\, \\widetilde{N}(\\mathrm {d}s, \\mathrm {d}{r}) .$ Taking the limit in ${\\mathcal {M}}_2$ as $n \\rightarrow \\infty $ , we conclude $&\\int _0^t \\int _{U_d} {z}\\mathbb {1}_{\\lbrace \\Vert {z}\\Vert <1\\rbrace } \\, \\widetilde{N}_0(\\mathrm {d}s, \\mathrm {d}{z}) \\\\&= \\int _0^t \\int _{{\\mathcal {R}}_0}{r}\\mathbb {1}_{\\lbrace \\Vert {r}\\Vert <1\\rbrace } \\, \\widetilde{N}(\\mathrm {d}s, \\mathrm {d}{r})+ \\sum _{j=1}^d\\int _0^t \\int _{{\\mathcal {R}}_j}{z}\\mathbb {1}_{\\lbrace \\Vert {z}\\Vert <1\\rbrace } \\mathbb {1}_{\\lbrace u\\leqslant X_{s-,j}\\rbrace }\\, \\widetilde{N}(\\mathrm {d}s, \\mathrm {d}{r}) \\\\&= \\int _0^t \\int _{{\\mathcal {R}}_0}{r}\\mathbb {1}_{\\lbrace \\Vert {r}\\Vert <1\\rbrace } \\, \\widetilde{N}(\\mathrm {d}s, \\mathrm {d}{r})+ \\int _0^t \\int _{V_0} f({X}_{s-}, {r}) \\, \\widetilde{N}(\\mathrm {d}s, \\mathrm {d}{r}) .$ Summarizing, we conclude $\\int _0^t \\int _{U_d} {z}\\, \\widetilde{N}_0(\\mathrm {d}s, \\mathrm {d}{z})&= \\int _0^t \\int _{V_1} g({X}_{s-}, {r}) \\, N(\\mathrm {d}s, \\mathrm {d}{r})- \\int _0^t \\int _{{\\mathcal {R}}_0} {r}\\mathbb {1}_{\\lbrace \\Vert {r}\\Vert <1\\rbrace } \\, N(\\mathrm {d}s, \\mathrm {d}{r}) \\\\&\\quad - \\int _0^t \\int _{U_d} {r}\\mathbb {1}_{\\lbrace \\Vert {r}\\Vert \\geqslant 1\\rbrace } \\, \\mathrm {d}s \\, \\nu (\\mathrm {d}{r})- \\sum _{j=1}^d\\int _0^t X_{s,j} \\, \\mathrm {d}s\\int _{U_d} {z}\\mathbb {1}_{\\lbrace \\Vert {z}\\Vert \\geqslant 1\\rbrace } \\, \\mu _j(\\mathrm {d}{z}) \\\\&\\quad + \\int _0^t \\int _{{\\mathcal {R}}_0}{r}\\mathbb {1}_{\\lbrace \\Vert {r}\\Vert <1\\rbrace } \\, \\widetilde{N}(\\mathrm {d}s, \\mathrm {d}{r})+ \\int _0^t \\int _{V_0} f({X}_{s-}, {r}) \\, \\widetilde{N}(\\mathrm {d}s, \\mathrm {d}{r}) \\\\&= \\int _0^t \\int _{V_0} f({X}_{s-}, {r}) \\, \\widetilde{N}(\\mathrm {d}s, \\mathrm {d}{r})+ \\int _0^t \\int _{V_1} g({X}_{s-}, {r}) \\, N(\\mathrm {d}s, \\mathrm {d}{r}) \\\\&\\quad - \\int _0^t \\int _{U_d} {r}\\, \\mathrm {d}s \\, \\nu (\\mathrm {d}{r})- \\sum _{j=1}^d\\int _0^t X_{s,j} \\, \\mathrm {d}s\\int _{U_d} {z}\\mathbb {1}_{\\lbrace \\Vert {z}\\Vert \\geqslant 1\\rbrace } \\, \\mu _j(\\mathrm {d}{z}) .$ This proves that the SDE (REF ) holds $\\widetilde{\\widetilde{\\operatorname{\\mathbb {P}}}}$ -almost surely, since $\\begin{aligned}&\\int _0^t \\bigl ( \\widetilde{{\\beta }}+ \\widetilde{{B}}{X}_s \\bigr ) \\mathrm {d}s- \\int _0^t \\int _{U_d} {r}\\, \\mathrm {d}s \\, \\nu (\\mathrm {d}{r})- \\sum _{j=1}^d\\int _0^t X_{s,j} \\, \\mathrm {d}s\\int _{U_d} {z}\\mathbb {1}_{\\lbrace \\Vert {z}\\Vert \\geqslant 1\\rbrace } \\, \\mu _j(\\mathrm {d}{z}) \\\\&= \\widetilde{{\\beta }}t + \\widetilde{{B}}\\int _0^t {X}_s \\, \\mathrm {d}s- t \\int _{U_d} {r}\\, \\nu (\\mathrm {d}{r})- \\sum _{j=1}^d\\int _0^t X_{s,j} \\, \\mathrm {d}s\\int _{U_d} {z}\\mathbb {1}_{\\lbrace \\Vert {z}\\Vert \\geqslant 1\\rbrace } \\, \\mu _j(\\mathrm {d}{z}) \\\\&= \\left( {\\beta }+ \\int _{U_d} {r}\\, \\nu (\\mathrm {d}{r}) \\right) t+ {D}\\int _0^t {X}_s \\, \\mathrm {d}s+ \\sum _{j=1}^d\\int _{U_d} {z}\\mathbb {1}_{\\lbrace \\Vert {z}\\Vert \\geqslant 1\\rbrace } \\, \\mu _j(\\mathrm {d}{z})\\int _0^t X_{s,j} \\, \\mathrm {d}s \\\\&\\quad - t \\int _{U_d} {r}\\, \\nu (\\mathrm {d}{r})- \\sum _{j=1}^d\\int _0^t X_{s,j} \\, \\mathrm {d}s\\int _{U_d} {z}\\mathbb {1}_{\\lbrace \\Vert {z}\\Vert \\geqslant 1\\rbrace } \\, \\mu _j(\\mathrm {d}{z})= \\int _0^t \\bigl ( {\\beta }+ {D}{X}_s \\bigr ) \\mathrm {d}s .\\end{aligned}$ The aim of the following discussion is to show that $\\bigl ( \\widetilde{\\widetilde{\\Omega }}, \\widetilde{\\widetilde{{\\mathcal {F}}}}, (\\widetilde{\\widetilde{{\\mathcal {G}}}}_t)_{t\\in \\mathbb {R}_+}, \\widetilde{\\widetilde{\\operatorname{\\mathbb {P}}}}, {W}, p, {X}\\bigr )$ is an $\\mathbb {R}_+^d$ -valued weak solution to the SDE (REF ).", "Recall that the filtered probability space $(\\widetilde{\\widetilde{\\Omega }}, \\widetilde{\\widetilde{{\\mathcal {F}}}}, (\\widetilde{\\widetilde{{\\mathcal {G}}}}_t)_{t\\in \\mathbb {R}_+}, \\widetilde{\\widetilde{\\operatorname{\\mathbb {P}}}})$ satisfies the usual hypotheses, and by Lemma REF , $(W_{t,1}, \\ldots , W_{t,d})_{t\\in \\mathbb {R}_+}$ is a $d$ -dimensional $(\\widetilde{\\widetilde{{\\mathcal {G}}}}_t)_{t\\in \\mathbb {R}_+}$ -Brownian motion, and $p$ is a stationary $(\\widetilde{\\widetilde{{\\mathcal {G}}}}_t)_{t\\in \\mathbb {R}_+}$ -Poisson point process on $V$ with characteristic measure $m$ .", "Since $({X}_t)_{t\\in \\mathbb {R}_+}$ is $\\mathbb {R}_+^d$ -valued and has càdlàg trajectories on the original probability space $(\\Omega , {\\mathcal {F}}, \\operatorname{\\mathbb {P}})$ , by the definition of an extension of a probability space (see Definition REF ), the extended process (which is denoted by ${X}$ as well) on the extended probability space is $\\mathbb {R}_+^d$ -valued and admits càdlàg trajectories as well.", "By Remark REF , the process $({X}_t)_{t\\in \\mathbb {R}_+}$ is $(\\widetilde{\\widetilde{{\\mathcal {G}}}}_t)_{t\\in \\mathbb {R}_+}$ -adapted, and clearly, the distribution of ${X}_0$ is $n$ .", "Since $({X}_t)_{t\\in \\mathbb {R}_+}$ has càdlàg trajectories, (D4)(b) holds.", "Since the process $\\bigl (\\int _0^t \\int _{V_0}f({X}_{s-}, {r}) \\, \\widetilde{N}(\\mathrm {d}s, \\mathrm {d}{r})\\bigr )_{t\\in \\mathbb {R}_+}$ belongs to the space ${\\mathcal {M}}_2$ , we have $&\\operatorname{\\mathbb {\\widetilde{\\widetilde{E}}}}\\left( \\left\\Vert \\int _0^t \\int _{V_0}f({X}_{s-}, {r}) \\, \\widetilde{N}(\\mathrm {d}s, \\mathrm {d}{r})\\right\\Vert ^2 \\right) \\\\&= \\sum _{j=1}^d\\operatorname{\\mathbb {\\widetilde{\\widetilde{E}}}}\\left( \\int _0^t \\int _{U_d} \\int _{U_1}\\Vert {z}\\Vert ^2 \\mathbb {1}_{\\lbrace \\Vert {z}\\Vert <1\\rbrace } \\mathbb {1}_{\\lbrace u\\leqslant X_{s,j}\\rbrace }\\, \\mathrm {d}s \\, \\mu _j(\\mathrm {d}{z}) \\, \\mathrm {d}u \\right)< \\infty \\vspace{-8.53581pt}$ by Ikeda and Watanabe [7], which yields (D4)(c).", "We have already checked that (D4)(d) and (D4)(e) are satisfied.", "Now we turn to prove the uniqueness in the sense of probability law for the SDE (REF ) among $\\mathbb {R}_+^d$ -valued weak solutions.", "If $\\bigl ( \\Omega , {\\mathcal {F}}, ({\\mathcal {F}}_t)_{t\\in \\mathbb {R}_+}, \\operatorname{\\mathbb {P}}, {W}, p, {X}\\bigr )$ is an $\\mathbb {R}_+^d$ -valued weak solution to the SDE (REF ), then for each $G \\in C_\\mathrm {c}^2(\\mathbb {R}, \\mathbb {R})$ and ${w}= (w_1, \\ldots , w_d)^\\top \\in \\mathbb {R}^d$ , by Itô's formula for $F({x}) := G({w}^\\top {x})$ , ${x}= (x_1, \\ldots , x_d)^\\top \\in \\mathbb {R}^d$ , with $\\partial _{x_k} F({x}) = G^{\\prime }({w}^\\top {x}) w_k$ , $\\partial _{x_k} \\partial _{x_\\ell } F({x}) = G^{\\prime \\prime }({w}^\\top {x}) w_k w_\\ell $ , $k, \\ell \\in \\lbrace 1, \\ldots , d\\rbrace $ , we have $G({w}^\\top {X}_t) = G({w}^\\top {X}_0) + \\sum _{\\ell =1}^6 I_\\ell (t) , \\qquad t \\in \\mathbb {R}_+ ,\\vspace{-2.84526pt}$ where $I_1(t)&:= \\int _0^tG^{\\prime }({w}^\\top {X}_s) {w}^\\top ({\\beta }+ {D}{X}_s) \\, \\mathrm {d}s , \\\\I_2(t)&:= \\sum _{j=1}^d\\int _0^t w_j G^{\\prime }({w}^\\top {X}_s) \\sqrt{2 c_j X_{s,j}} \\, \\mathrm {d}W_{s,j} , \\\\I_3(t)&:= \\sum _{j=1}^d \\int _0^t w_j^2 G^{\\prime \\prime }({w}^\\top {X}_s) c_j X_{s,j} \\, \\mathrm {d}s , \\\\I_4(t)&:= \\int _0^t \\int _{V_0}\\bigl [ G({w}^\\top {X}_{s-} + {w}^\\top f({X}_{s-}, {r}))- G({w}^\\top {X}_{s-}) \\bigr ]\\, \\widetilde{N}(\\mathrm {d}s, \\mathrm {d}{r}) , \\\\I_5(t)&:= \\int _0^t \\int _{V_0}\\bigl [ G({w}^\\top {X}_s + {w}^\\top f({X}_{s-}, {r}))- G({w}^\\top {X}_s) \\\\&\\phantom{:= \\int _0^t \\int _{{\\mathcal {R}}_j} \\bigl (}- G^{\\prime }({w}^\\top {X}_s) {w}^\\top f({X}_{s-}, {r}) \\bigr ]\\, \\mathrm {d}s \\, m(\\mathrm {d}{r}) ,$ $I_6(t)&:= \\int _0^t \\int _{V_1}\\bigl [ G({w}^\\top {X}_{s-} + {w}^\\top g({X}_{s-}, {r}))- G({w}^\\top {X}_{s-}) \\bigr ]\\, N(\\mathrm {d}s, \\mathrm {d}{r}) .$ The last integral can be written as $I_6(t) = I_{6,1}(t) + I_{6,2}(t)$ , where $I_{6,1}(t)&:= \\int _0^t \\int _{V_1}\\bigl [ G({w}^\\top {X}_{s-} + {w}^\\top g({X}_{s-}, {r}))- G({w}^\\top {X}_{s-}) \\bigr ]\\, \\widetilde{N}(\\mathrm {d}s, \\mathrm {d}{r}) , \\\\I_{6,2}(t)&:= \\int _0^t \\int _{V_1}\\bigl [ G({w}^\\top {X}_s + {w}^\\top g({X}_s, {r}))- G({w}^\\top {X}_s) \\bigr ]\\,\\mathrm {d}s \\, m(\\mathrm {d}{r}) ,$ since $\\begin{split}&\\operatorname{\\mathbb {E}}\\left( \\int _0^t \\int _{V_1}\\bigl \| G({w}^\\top {X}_{s-} + {w}^\\top g({X}_{s-}, {r}))- G({w}^\\top {X}_{s-}) \\bigr \|\\, \\mathrm {d}s \\, m(\\mathrm {d}{r}) \\right) \\\\&= \\operatorname{\\mathbb {E}}\\left( \\int _0^t \\int _{U_d}\\bigl \| G({w}^\\top {X}_s + {w}^\\top {r})- G({w}^\\top {X}_s) \\bigr \|\\, \\mathrm {d}s \\, \\nu (\\mathrm {d}{r}) \\right) \\\\&\\quad + \\sum _{j=1}^d\\operatorname{\\mathbb {E}}\\left( \\int _0^t \\int _{U_d} \\int _{U_1}\\bigl \| G({w}^\\top {X}_s + {w}^\\top {z}\\mathbb {1}_{\\lbrace u\\leqslant X_{s,j}\\rbrace })- G({w}^\\top {X}_s) \\bigr \| \\mathbb {1}_{\\lbrace \\Vert {z}\\Vert \\geqslant 1\\rbrace }\\, \\mathrm {d}s \\, \\mu _j(\\mathrm {d}{z}) \\, \\mathrm {d}u \\right)\\\\&< \\infty ,\\end{split}$ i.e., for all ${w}\\in \\mathbb {R}^d$ , the function $\\mathbb {R}_+\\times V_1\\times \\Omega \\ni (s,{r},\\omega )\\mapsto G({w}^\\top {X}_{s-}(\\omega ) + {w}^\\top g({X}_{s-}(\\omega ), {r})) - G({w}^\\top {X}_{s-}(\\omega ))$ belongs to the class ${F}_p^1$ .", "Indeed, by mean value theorem and (REF ), there exists some $\\theta _0 = \\theta _0({w}, {X}_s, {r}) \\in [0, 1]$ such that $&\\operatorname{\\mathbb {E}}\\left( \\int _0^t \\int _{U_d}\\bigl \| G({w}^\\top {X}_s + {w}^\\top {r})- G({w}^\\top {X}_s) \\bigr \|\\, \\mathrm {d}s \\, \\nu (\\mathrm {d}{r}) \\right) \\\\& = \\operatorname{\\mathbb {E}}\\left( \\int _0^t \\int _{U_d}\\bigl \| G^{\\prime }({w}^\\top {X}_s + \\theta _0 {w}^\\top {r}) \\bigr \|\|{w}^\\top {r}\|\\, \\mathrm {d}s \\, \\nu (\\mathrm {d}{r}) \\right)\\leqslant \\Vert {w}\\Vert \\sup _{x\\in \\mathbb {R}} \|G^{\\prime }(x)\| \\int _{U_d} \\Vert {r}\\Vert \\, \\nu (\\mathrm {d}{r})< \\infty $ due to that $G^{\\prime }$ is bounded.", "In a similar way, there exists some $\\theta = \\theta ({w}, {X}_s, {z}) \\in [0, 1]$ such that for each $j \\in \\lbrace 1, \\ldots , d\\rbrace $ , $&\\operatorname{\\mathbb {E}}\\left( \\int _0^t \\int _{U_d} \\int _{U_1}\\bigl \| G({w}^\\top {X}_s + {w}^\\top {z}\\mathbb {1}_{\\lbrace u\\leqslant X_{s,j}\\rbrace })- G({w}^\\top {X}_s) \\bigr \|\\mathbb {1}_{\\lbrace \\Vert {z}\\Vert \\geqslant 1\\rbrace }\\, \\mathrm {d}s \\, \\mu _j(\\mathrm {d}{z}) \\, \\mathrm {d}u \\right) \\\\&\\qquad =\\operatorname{\\mathbb {E}}\\left( \\int _0^t \\int _{U_d} \\int _{U_1}\\bigl \| G({w}^\\top {X}_s + {w}^\\top {z})- G({w}^\\top {X}_s) \\bigr \|\\mathbb {1}_{\\lbrace u\\leqslant X_{s,j}\\rbrace } \\mathbb {1}_{\\lbrace \\Vert {z}\\Vert \\geqslant 1\\rbrace }\\, \\mathrm {d}s \\, \\mu _j(\\mathrm {d}{z}) \\, \\mathrm {d}u \\right) \\\\&\\qquad = \\operatorname{\\mathbb {E}}\\left( \\int _0^t \\int _{U_d} \\int _{U_1}\\bigl \| G^{\\prime }({w}^\\top {X}_s+ \\theta {w}^\\top {z}) \\bigr \|\|{w}^\\top {z}\| \\mathbb {1}_{\\lbrace u\\leqslant X_{s,j}\\rbrace } \\mathbb {1}_{\\lbrace \\Vert {z}\\Vert \\geqslant 1\\rbrace }\\, \\mathrm {d}s \\, \\mu _j(\\mathrm {d}{r}) \\, \\mathrm {d}u \\right) \\\\&\\qquad \\leqslant \\Vert {w}\\Vert \\sup _{x\\in \\mathbb {R}} \|G^{\\prime }(x)\| \\int _0^t \\operatorname{\\mathbb {E}}(X_{s,j}) \\, \\mathrm {d}s\\int _{U_d} \\Vert {z}\\Vert \\mathbb {1}_{\\lbrace \\Vert {z}\\Vert \\geqslant 1\\rbrace } \\, \\mu _j(\\mathrm {d}{z})< \\infty $ due to that $G^{\\prime }$ is bounded, Lemma REF (which can be applied since $\\int _{\\mathbb {R}_+^d} \\Vert {z}\\Vert \\, n(\\mathrm {d}{z}) < \\infty $ ) and the moment condition (REF ).", "In what follows, we identify some of these integrals with some terms in part (5) of Theorem 9.18 of Li [13].", "We have $I_1(t) &= \\int _0^t G^{\\prime }({w}^\\top {X}_s) {w}^\\top {\\beta }\\, \\mathrm {d}s+ \\int _0^t G^{\\prime }({w}^\\top {X}_s) {w}^\\top \\widetilde{{B}}{X}_s \\, \\mathrm {d}s \\\\&\\quad - \\sum _{i=1}^d \\sum _{j=1}^d\\int _0^t G^{\\prime }({w}^\\top {X}_s) w_i X_{s,j} \\, \\mathrm {d}s\\int _{U_d} z_i \\mathbb {1}_{\\lbrace \\Vert {z}\\Vert \\geqslant 1 \\rbrace } \\, \\mu _j(\\mathrm {d}{z}) ,$ where the first two terms on the right hand side can be identified with $\\int _0^t G^{\\prime }(Y_s(f)) \\eta (f) \\, \\mathrm {d}s$ and $\\int _0^t G^{\\prime }(Y_s(f)) Y_s(Af+\\gamma f - bf) \\, \\mathrm {d}s$ (see, (REF )).", "The sum of the third term on the right hand side and $I_{6,2}(t) + I_5(t)$ can be written in the form $&- \\sum _{j=1}^d\\int _0^t \\int _{U_d} \\int _{U_1}G^{\\prime }({w}^\\top {X}_s) {w}^\\top {z}\\mathbb {1}_{\\lbrace \\Vert {z}\\Vert \\geqslant 1\\rbrace }\\mathbb {1}_{\\lbrace u\\leqslant X_{s,j}\\rbrace }\\, \\mathrm {d}s \\, \\mu _j(\\mathrm {d}{z}) \\, \\mathrm {d}u \\\\&+ \\int _0^t \\int _{U_d}\\bigl [ G({w}^\\top {X}_s + {w}^\\top {r}) - G({w}^\\top {X}_s) \\bigr ]\\, \\mathrm {d}s \\, \\nu (\\mathrm {d}{r}) \\\\&+ \\sum _{j=1}^d\\int _0^t \\int _{U_d} \\int _{U_1}\\bigl [ G({w}^\\top {X}_s + {w}^\\top {z}\\mathbb {1}_{\\lbrace u\\leqslant X_{s,j}\\rbrace })- G({w}^\\top {X}_s) \\bigr ]\\mathbb {1}_{\\lbrace \\Vert {z}\\Vert \\geqslant 1\\rbrace }\\, \\mathrm {d}s \\, \\mu _j(\\mathrm {d}{z}) \\, \\mathrm {d}u \\\\&+ \\sum _{j=1}^d\\int _0^t \\int _{U_d} \\int _{U_1}\\bigl [ G({w}^\\top {X}_s + {w}^\\top {z}\\mathbb {1}_{\\lbrace u\\leqslant X_{s,j}\\rbrace })- G({w}^\\top {X}_s) \\\\&\\phantom{+ \\sum _{j=1}^d \\int _0^t \\int _{U_d} \\int _{U_1} \\bigl [}- G^{\\prime }({w}^\\top {X}_s) {w}^\\top {z}\\mathbb {1}_{\\lbrace u\\leqslant X_{s,j}\\rbrace } \\bigr ]\\mathbb {1}_{\\lbrace \\Vert {z}\\Vert <1\\rbrace }\\, \\mathrm {d}s \\, \\mu _j(\\mathrm {d}{z}) \\, \\mathrm {d}u \\\\&= \\int _0^t \\int _{U_d}\\bigl [ G({w}^\\top {X}_s + {w}^\\top {r}) - G({w}^\\top {X}_s) \\bigr ]\\, \\mathrm {d}s \\, \\nu (\\mathrm {d}{r}) \\\\&\\quad + \\sum _{j=1}^d\\int _0^t \\int _{U_d} \\int _{U_1}\\bigl [ G({w}^\\top {X}_s + {w}^\\top {z}\\mathbb {1}_{\\lbrace u\\leqslant X_{s,j}\\rbrace })- G({w}^\\top {X}_s) \\\\&\\phantom{\\quad + \\sum _{j=1}^d \\int _0^t \\int _{U_d} \\int _{U_1} \\bigl [}- G^{\\prime }({w}^\\top {X}_s) {w}^\\top {z}\\mathbb {1}_{\\lbrace u\\leqslant X_{s,j}\\rbrace } \\bigr ]\\, \\mathrm {d}s \\, \\mu _j(\\mathrm {d}{z}) \\, \\mathrm {d}u ,$ which can be identified with $&\\int _0^t \\int _{M(E)^\\circ }[G(Y_s(f) + \\kappa (f)) - G(Y_s(f))] \\, H(\\mathrm {d}\\kappa ) \\, \\mathrm {d}s \\\\&+ \\int _0^t \\int _E Y_s(\\mathrm {d}x) \\int _{M(E)^\\circ }[G(Y_s(f) + \\kappa (f)) - G(Y_s(f)) - \\kappa (f) G^{\\prime }(Y_s(f))]\\, H(x, \\mathrm {d}\\kappa ) \\, \\mathrm {d}s .$ The integral $I_3(t)$ can be identified with $\\int _0^t G^{\\prime \\prime }(Y_s(f)) Y_s(cf^2) \\, \\mathrm {d}s$ .", "Next we show that the process $(I_2(t) + I_4(t) + I_{6,1}(t))_{t\\in \\mathbb {R}_+}$ is a continuous local martingale.", "Since $G^{\\prime }$ is bounded and ${X}$ has càdlàg trajectories, we have $\\operatorname{\\mathbb {P}}(\\int _0^t w_j^2 G^{\\prime }({w}^\\top {X}_s)^2 \\, 2 c_j X_{s,j} \\, \\mathrm {d}s < \\infty )= 1$ for all $t \\in \\mathbb {R}_+$ and $j \\in \\lbrace 1, \\ldots , d\\rbrace $ , hence $(I_2(t))_{t\\in \\mathbb {R}_+}$ is a continuous local martingale (see, e.g., Karatzas and Shreve [10]).", "In order to prove that $(I_4(t))_{t\\in \\mathbb {R}_+}$ is a martingale, by page 62 in Ikeda and Watanabe [7], it is enough to check that $\\operatorname{\\mathbb {E}}\\left( \\int _0^t \\int _{V_0}\\bigl \| G({w}^\\top {X}_s + {w}^\\top f({X}_s, {r}))- G({w}^\\top {X}_s) \\bigr \|^2\\,\\mathrm {d}s \\, m(\\mathrm {d}{r}) \\right)< \\infty .$ By mean value theorem, there exists some $\\vartheta _0 = \\vartheta _0({w}, {X}_s, {z}) \\in [0, 1]$ such that for each $j \\in \\lbrace 1, \\ldots , d\\rbrace $ , $&\\operatorname{\\mathbb {E}}\\left( \\int _0^t \\int _{U_d} \\int _{U_1}\\bigl \| G({w}^\\top {X}_s + {w}^\\top {z}\\mathbb {1}_{\\lbrace u\\leqslant X_{s,j}\\rbrace })- G({w}^\\top {X}_s) \\bigr \|^2\\mathbb {1}_{\\lbrace \\Vert {z}\\Vert <1\\rbrace }\\, \\mathrm {d}s \\, \\mu _j(\\mathrm {d}{z}) \\, \\mathrm {d}u \\right) \\\\&\\qquad = \\operatorname{\\mathbb {E}}\\left( \\int _0^t \\int _{U_d} \\int _{U_1}\\bigl \| G^{\\prime }({w}^\\top {X}_s+ \\vartheta _0 {w}^\\top {z}) \\bigr \|^2({w}^\\top {z})^2 \\mathbb {1}_{\\lbrace u\\leqslant X_{s,j}\\rbrace }\\mathbb {1}_{\\lbrace \\Vert {z}\\Vert <1\\rbrace }\\, \\mathrm {d}s \\, \\mu _j(\\mathrm {d}{z}) \\, \\mathrm {d}u \\right) \\\\&\\qquad \\leqslant \\Vert {w}\\Vert ^2 \\sup _{x\\in \\mathbb {R}} \|G^{\\prime }(x)\|^2 \\int _0^t \\operatorname{\\mathbb {E}}(X_{s,j}) \\, \\mathrm {d}s\\int _{U_d} \\Vert {z}\\Vert ^2 \\mathbb {1}_{\\lbrace \\Vert {z}\\Vert <1\\rbrace } \\, \\mu _j(\\mathrm {d}{z})< \\infty $ due to that $G^{\\prime }$ is bounded, Lemma REF and (REF ).", "Hence $(I_4(t))_{t\\in \\mathbb {R}_+}$ is a martingale.", "Further, by (REF ) and page 62 in Ikeda and Watanabe [7], we get $(I_{6,1}(t))_{t\\in \\mathbb {R}_+}$ is a martingale.", "Consequently, by Theorem 9.18 of Li [13], $({X}_t)_{t\\in \\mathbb {R}_+}$ is a CBI process with parameters $(d, {c}, {\\beta }, {B}, \\nu , {\\mu })$ .", "This yields the uniqueness in the sense of probability law for the SDE (REF ) among $\\mathbb {R}_+^d$ -valued weak solutions, and that any $\\mathbb {R}_+^d$ -valued weak solution is a CBI process with parameters $(d, {c}, {\\beta }, {B}, \\nu , {\\mu })$ as well.", "$\\Box $" ], [ "Multi-type CBI process as a strong solution of an SDE", "Definition.", "4.1 We say that pathwise uniqueness holds for the SDE (REF ) among $\\mathbb {R}_+^d$ -valued weak solutions if whenever $\\bigl ( \\Omega , {\\mathcal {F}}, ({\\mathcal {F}}_t)_{t\\in \\mathbb {R}_+}, \\operatorname{\\mathbb {P}}, {W}, p, {X}\\bigr )$ and $\\bigl ( \\Omega , {\\mathcal {F}}, ({\\mathcal {F}}_t)_{t\\in \\mathbb {R}_+}, \\operatorname{\\mathbb {P}}, {W}, p, \\widetilde{{X}}\\bigr )$ are $\\mathbb {R}_+^d$ -valued weak solutions of the SDE (REF ) such that $\\operatorname{\\mathbb {P}}({X}_0 = \\widetilde{{X}}_0) = 1$ , then $\\operatorname{\\mathbb {P}}(\\text{${X}_t = \\widetilde{{X}}_t$ \\ for all \\ $t \\in \\mathbb {R}_+$}) = 1$ .", "Next we prove a comparison theorem for the SDE (REF ) in ${\\beta }$ .", "Lemma.", "4.2 Let $(d, {c}, {\\beta }, {B}, \\nu , {\\mu })$ be a set of admissible parameters in the sense of Definition REF such that the moment condition (REF ) holds.", "Suppose that ${\\beta }^{\\prime } \\in \\mathbb {R}_+^d$ with ${\\beta }\\leqslant {\\beta }^{\\prime }$ .", "Let $\\bigl ( \\Omega , {\\mathcal {F}}, ({\\mathcal {F}}_t)_{t\\in \\mathbb {R}_+}, \\operatorname{\\mathbb {P}}, {W}, p, {X}\\bigr )$ and $\\bigl ( \\Omega , {\\mathcal {F}}, ({\\mathcal {F}}_t)_{t\\in \\mathbb {R}_+}, \\operatorname{\\mathbb {P}}, {W}, p, {X}^{\\prime } \\bigr )$ be $\\mathbb {R}_+^d$ -valued weak solutions of the SDE (REF ) with ${\\beta }$ and ${\\beta }^{\\prime }$ , respectively.", "Then $\\operatorname{\\mathbb {P}}({X}_0 \\leqslant {X}_0^{\\prime }) = 1$ implies $\\operatorname{\\mathbb {P}}(\\text{${X}_t \\leqslant {X}_t^{\\prime }$ \\ for all \\ $t \\in \\mathbb {R}_+$}) = 1$ .", "Particularly, pathwise uniqueness holds for the SDE (REF ) among $\\mathbb {R}_+^d$ -valued weak solutions.", "Proof.", "We follow the ideas of the proof of Theorem 3.1 of Ma [14], which is an adaptation of that of Theorem 5.5 of Fu and Li [6].", "There is a sequence $\\phi _k : \\mathbb {R}\\rightarrow \\mathbb {R}_+$ , $k \\in \\mathbb {N}$ , of twice continuously differentiable functions such that $\\phi _k(z) \\uparrow z^+$ as $k \\rightarrow \\infty $ ; $\\phi _k^{\\prime }(z) \\in [0, 1]$ for all $z \\in \\mathbb {R}_+$ and $k \\in \\mathbb {N}$ ; $\\phi _k^{\\prime }(z) = \\phi _k(z) = 0$ whenever $-z \\in \\mathbb {R}_+$ and $k \\in \\mathbb {N}$ ; $\\phi _k^{\\prime \\prime }(x - y) (\\sqrt{x} - \\sqrt{y})^2 \\leqslant 2/k$ for all $x, y \\in \\mathbb {R}_+$ and $k \\in \\mathbb {N}$ .", "For a construction of such functions, see, e.g., the proof of Theorem 3.1 of Ma [14].", "Let ${Y}_t := {X}_t - {X}_t^{\\prime }$ for all $t \\in \\mathbb {R}_+$ .", "By (REF ), and using that $\\int _0^t\\int _{{\\mathcal {R}}_0} g({X}_{s-},{r})\\, N(\\mathrm {d}s,\\mathrm {d}{r})= \\int _0^t\\int _{{\\mathcal {R}}_0} {r}\\, N(\\mathrm {d}s,\\mathrm {d}{r})= \\int _0^t\\int _{{\\mathcal {R}}_0} g({X}_{s-}^{\\prime },{r})\\, N(\\mathrm {d}s,\\mathrm {d}{r}),$ we have $Y_{t,i}&= Y_{0,i}+ \\int _0^t\\Bigl ( \\beta _i - \\beta _i^{\\prime } + {e}_i^\\top {D}{Y}_s \\Bigr )\\, \\mathrm {d}s+ \\int _0^t\\sqrt{2c_i} \\Bigl (\\sqrt{X_{s,i}} - \\sqrt{X_{s,i}^{\\prime }}\\Bigr ) \\, \\mathrm {d}W_{s,i} \\\\&\\quad + \\sum _{j=1}^d\\int _0^t \\int _{{\\mathcal {R}}_{j,0}}\\bigl (\\mathbb {1}_{\\lbrace u\\leqslant X_{s-,j}\\rbrace } - \\mathbb {1}_{\\lbrace u\\leqslant X_{s-,j}^{\\prime }\\rbrace }\\bigr )z_i \\mathbb {1}_{\\lbrace \\Vert {z}\\Vert <1\\rbrace }\\, \\widetilde{N}(\\mathrm {d}s, \\mathrm {d}{r}) \\\\&\\quad + \\sum _{j=1}^d\\int _0^t \\int _{{\\mathcal {R}}_{j,1}}\\bigl (\\mathbb {1}_{\\lbrace u\\leqslant X_{s-,j}\\rbrace } - \\mathbb {1}_{\\lbrace u\\leqslant X_{s-,j}^{\\prime }\\rbrace }\\bigr )z_i \\mathbb {1}_{\\lbrace \\Vert {z}\\Vert \\geqslant 1\\rbrace }\\, N(\\mathrm {d}s, \\mathrm {d}{r})$ for all $t \\in \\mathbb {R}_+$ and $i \\in \\lbrace 1, \\ldots , d\\rbrace $ .", "For each $m \\in \\mathbb {N}$ , put $\\tau _m := \\inf \\Bigl \\lbrace t \\in \\mathbb {R}_+: \\max _{i \\in \\lbrace 1, \\ldots , d\\rbrace } \\max \\lbrace X_{t,i}, X_{t,i}^{\\prime }\\rbrace \\geqslant m \\Bigr \\rbrace .$ By Itô's formula (which can be used since ${X}$ and ${X}^{\\prime }$ are adapted to the same filtration $({\\mathcal {F}}_t)_{t\\in \\mathbb {R}_+}$ ), we obtain $\\phi _k(Y_{t\\wedge \\tau _m,i}) = \\phi _k(Y_{0,i}) + \\sum _{\\ell =1}^6 I_{i,m,k,\\ell }(t)$ for all $t \\in \\mathbb {R}_+$ , $i \\in \\lbrace 1, \\ldots , d\\rbrace $ and $k, m \\in \\mathbb {N}$ , where $&I_{i,m,k,1}(t):= \\int _0^{t\\wedge \\tau _m}\\phi _k^{\\prime }(Y_{s,i})\\Bigl ( \\beta _i - \\beta _i^{\\prime } + {e}_i^\\top {D}{Y}_s \\Bigr )\\, \\mathrm {d}s , \\\\&I_{i,m,k,2}(t):= \\int _0^{t\\wedge \\tau _m}\\phi _k^{\\prime }(Y_{s,i}) \\sqrt{2c_i} \\Bigl (\\sqrt{X_{s,i}} - \\sqrt{X_{s,i}^{\\prime }}\\Bigr )\\, \\mathrm {d}W_{s,i} , \\\\&I_{i,m,k,3}(t):= \\frac{1}{2}\\int _0^{t\\wedge \\tau _m}\\phi _k^{\\prime \\prime }(Y_{s,i}) 2 c_i \\Bigl (\\sqrt{X_{s,i}} - \\sqrt{X_{s,i}^{\\prime }}\\Bigr )^2\\, \\mathrm {d}s ,\\\\&I_{i,m,k,4}(t):= \\sum _{j=1}^d\\int _0^{t\\wedge \\tau _m} \\int _{{\\mathcal {R}}_{j,0}}\\Bigl [ \\phi _k\\bigl ( Y_{s-,i}+ (\\mathbb {1}_{\\lbrace u\\leqslant X_{s-,j}\\rbrace }- \\mathbb {1}_{\\lbrace u\\leqslant X_{s-,j}^{\\prime }\\rbrace }) z_i \\bigr )- \\phi _k(Y_{s-,i}) \\Bigr ]\\mathbb {1}_{\\lbrace \\Vert {z}\\Vert <1\\rbrace }\\widetilde{N}(\\mathrm {d}s, \\mathrm {d}{r}) \\\\&= \\sum _{j=1}^d\\int _0^{t\\wedge \\tau _m} \\int _{{\\mathcal {R}}_{j,0}}\\Bigl [ \\phi _k(Y_{s-,i} + z_i) - \\phi _k(Y_{s-,i}) \\Bigr ]\\mathbb {1}_{\\lbrace \\Vert {z}\\Vert <1\\rbrace }\\mathbb {1}_{\\lbrace X_{s-,j}^{\\prime }<u\\leqslant X_{s-,j}\\rbrace } \\mathbb {1}_{\\lbrace Y_{s-,j}>0\\rbrace }\\, \\widetilde{N}(\\mathrm {d}s, \\mathrm {d}{r}) \\\\&\\quad + \\sum _{j=1}^d\\int _0^{t\\wedge \\tau _m} \\int _{{\\mathcal {R}}_{j,0}}\\Bigl [ \\phi _k(Y_{s-,i} - z_i) - \\phi _k(Y_{s-,i}) \\Bigr ]\\mathbb {1}_{\\lbrace \\Vert {z}\\Vert <1\\rbrace }\\mathbb {1}_{\\lbrace X_{s-,j}<u\\leqslant X_{s-,j}^{\\prime }\\rbrace } \\mathbb {1}_{\\lbrace Y_{s-,j}<0\\rbrace }\\, \\widetilde{N}(\\mathrm {d}s, \\mathrm {d}{r}) ,$ $&I_{i,m,k,5}(t):= \\sum _{j=1}^d\\int _0^{t\\wedge \\tau _m} \\int _{U_d} \\int _{U_1}\\Bigl [ \\phi _k\\bigl (Y_{s-,i}+ (\\mathbb {1}_{\\lbrace u\\leqslant X_{s-,j}\\rbrace }- \\mathbb {1}_{\\lbrace u\\leqslant X_{s-,j}^{\\prime }\\rbrace }) z_i \\bigr )- \\phi _k(Y_{s-,i}) \\\\&\\phantom{I_{i,m,k,5}(t):= \\sum _{j=1}^d \\int _0^{t\\wedge \\tau _m} \\int _{U_d} \\int _{U_1} \\Bigl [}- \\phi _k^{\\prime }(Y_{s-,i})(\\mathbb {1}_{\\lbrace u\\leqslant X_{s-,j}\\rbrace } - \\mathbb {1}_{\\lbrace u\\leqslant X_{s-,j}^{\\prime }\\rbrace }) z_i \\Bigr ]\\mathbb {1}_{\\lbrace \\Vert {z}\\Vert <1\\rbrace }\\, \\mathrm {d}s \\, \\mu _j(\\mathrm {d}{z}) \\, \\mathrm {d}u \\\\&= \\sum _{j=1}^d\\int _0^{t\\wedge \\tau _m} \\int _{U_d} \\int _{U_1}\\Bigl [ \\phi _k(Y_{s-,i} + z_i) - \\phi _k(Y_{s-,i})- \\phi _k^{\\prime }(Y_{s-,i}) z_i \\Bigr ]\\\\&\\phantom{= \\sum _{j=1}^d \\int _0^{t\\wedge \\tau _m} \\int _{U_d} \\int _{U_1}\\; }\\;\\times \\mathbb {1}_{\\lbrace \\Vert {z}\\Vert <1\\rbrace }\\mathbb {1}_{\\lbrace X_{s-,j}^{\\prime }<u\\leqslant X_{s-,j}\\rbrace } \\mathbb {1}_{\\lbrace Y_{s-,j}>0\\rbrace }\\, \\mathrm {d}s \\, \\mu _j(\\mathrm {d}{z}) \\, \\mathrm {d}u \\\\&\\quad + \\sum _{j=1}^d\\int _0^{t\\wedge \\tau _m} \\int _{U_d} \\int _{U_1}\\Bigl [ \\phi _k(Y_{s-,i} - z_i) - \\phi _k(Y_{s-,i})+ \\phi _k^{\\prime }(Y_{s-,i}) z_i \\Bigr ]\\\\&\\phantom{= +\\sum _{j=1}^d \\int _0^{t\\wedge \\tau _m} \\int _{U_d} \\int _{U_1}\\; }\\;\\times \\mathbb {1}_{\\lbrace \\Vert {z}\\Vert <1\\rbrace }\\mathbb {1}_{\\lbrace X_{s-,j}<u\\leqslant X_{s-,j}^{\\prime }\\rbrace } \\mathbb {1}_{\\lbrace Y_{s-,j}<0\\rbrace }\\, \\mathrm {d}s \\, \\mu _j(\\mathrm {d}{z}) \\, \\mathrm {d}u , \\\\&I_{i,m,k,6}(t):= \\sum _{j=1}^d\\int _0^{t\\wedge \\tau _m} \\int _{{\\mathcal {R}}_{j,1}}\\Bigl [ \\phi _k\\bigl ( Y_{s-,i}+ (\\mathbb {1}_{\\lbrace u\\leqslant X_{s-,j}\\rbrace }- \\mathbb {1}_{\\lbrace u\\leqslant X_{s-,j}^{\\prime }\\rbrace }) z_i \\bigr )- \\phi _k(Y_{s-,i}) \\Bigr ]\\mathbb {1}_{\\lbrace \\Vert {z}\\Vert \\geqslant 1\\rbrace }N(\\mathrm {d}s, \\mathrm {d}{r}) ,$ where we used that $\\mathbb {1}_{\\lbrace u\\leqslant X_{s-,j}\\rbrace } - \\mathbb {1}_{\\lbrace u\\leqslant X_{s-,j}^{\\prime }\\rbrace }= {\\left\\lbrace \\begin{array}{ll}1 & \\text{if \\ $Y_{s-,j} > 0$ \\ and \\ $X_{s-,j}^{\\prime } < u \\leqslant X_{s-,j}$,} \\\\-1 & \\text{if \\ $Y_{s-,j} < 0$ \\ and \\ $X_{s-,j} < u \\leqslant X_{s-,j}^{\\prime }$,} \\\\0 & \\text{otherwise.}\\end{array}\\right.", "}$ Using formula (3.8) in Chapter II in Ikeda and Watanabe [7], the last integral can be written as $I_{i,m,k,6}(t) = I_{i,m,k,6,1}(t) + I_{i,m,k,6,2}(t)$ , where $I_{i,m,k,6,1}(t):= \\sum _{j=1}^d\\int _0^{t\\wedge \\tau _m} \\int _{{\\mathcal {R}}_{j,1}}\\Bigl [ \\phi _k\\bigl ( Y_{s-,i}+ (\\mathbb {1}_{\\lbrace u\\leqslant X_{s-,j}\\rbrace }- \\mathbb {1}_{\\lbrace u\\leqslant X_{s-,j}^{\\prime }\\rbrace }) z_i \\bigr )- \\phi _k(Y_{s-,i}) \\Bigr ]\\mathbb {1}_{\\lbrace \\Vert {z}\\Vert \\geqslant 1\\rbrace }\\widetilde{N}(\\mathrm {d}s, \\mathrm {d}{r})$ $I_{i,m,k,6,2}(t):= \\sum _{j=1}^d\\int _0^{t\\wedge \\tau _m} \\int _{U_d} \\int _{U_1}\\Bigl [ \\phi _k\\bigl ( Y_{s-,i}+ (\\mathbb {1}_{\\lbrace u\\leqslant X_{s-,j}\\rbrace }- \\mathbb {1}_{\\lbrace u\\leqslant X_{s-,j}^{\\prime }\\rbrace }) z_i \\bigr )- \\phi _k(Y_{s-,i}) \\Bigr ]\\mathbb {1}_{\\lbrace \\Vert {z}\\Vert \\geqslant 1\\rbrace }\\, \\mathrm {d}s \\, \\mu _j(\\mathrm {d}{z}) \\, \\mathrm {d}u ,$ since, for each $j \\in \\lbrace 1, \\ldots , d\\rbrace $ , $&\\operatorname{\\mathbb {E}}\\left( \\int _0^{t\\wedge \\tau _m} \\int _{U_d} \\int _{U_1}\\Bigl \| \\phi _k\\bigl ( Y_{s-,i}+ (\\mathbb {1}_{\\lbrace u\\leqslant X_{s-,j}\\rbrace }- \\mathbb {1}_{\\lbrace u\\leqslant X_{s-,j}^{\\prime }\\rbrace }) z_i \\bigr )- \\phi _k(Y_{s-,i}) \\Bigr \|\\mathbb {1}_{\\lbrace \\Vert {z}\\Vert \\geqslant 1\\rbrace }\\, \\mathrm {d}s \\, \\mu _j(\\mathrm {d}{z}) \\, \\mathrm {d}u \\right) \\\\&=\\operatorname{\\mathbb {E}}\\left( \\int _0^{t\\wedge \\tau _m} \\int _{U_d} \\int _{U_1}\\Bigl \| \\phi _k(Y_{s-,i} + z_i) - \\phi _k(Y_{s-,i}) \\Bigr \|\\mathbb {1}_{\\lbrace \\Vert {z}\\Vert \\geqslant 1\\rbrace } \\mathbb {1}_{\\lbrace X_{s-,j}^{\\prime }<u\\leqslant X_{s-,j}\\rbrace }\\mathbb {1}_{\\lbrace Y_{s-,j}>0\\rbrace }\\, \\mathrm {d}s \\, \\mu _j(\\mathrm {d}{z}) \\, \\mathrm {d}u \\right) \\\\&\\quad + \\operatorname{\\mathbb {E}}\\left( \\int _0^{t\\wedge \\tau _m} \\int _{U_d} \\int _{U_1}\\Bigl \| \\phi _k(Y_{s-,i} - z_i) - \\phi _k(Y_{s-,i}) \\Bigr \|\\mathbb {1}_{\\lbrace \\Vert {z}\\Vert \\geqslant 1\\rbrace } \\mathbb {1}_{\\lbrace X_{s-,j}<u\\leqslant X_{s-,j}^{\\prime }\\rbrace }\\mathbb {1}_{\\lbrace Y_{s-,j}<0\\rbrace }\\, \\mathrm {d}s \\, \\mu _j(\\mathrm {d}{z}) \\, \\mathrm {d}u \\right)$ $&\\leqslant \\operatorname{\\mathbb {E}}\\left( \\int _0^{t\\wedge \\tau _m} \\int _{U_d}z_i \\mathbb {1}_{\\lbrace \\Vert {z}\\Vert \\geqslant 1\\rbrace } \|Y_{s-,j}\|\\, \\mathrm {d}s \\, \\mu _j(\\mathrm {d}{z}) \\right)\\leqslant 2 m t \\int _{U_d} z_i \\mathbb {1}_{\\lbrace \\Vert {z}\\Vert \\geqslant 1\\rbrace } \\, \\mu _j(\\mathrm {d}{z})< \\infty ,$ where we used that, by properties (ii) and (iii) of the function $\\phi _k$ , we have $\\phi _k^{\\prime }(u) \\in [0, 1]$ for all $u \\in \\mathbb {R}$ , and hence, by mean value theorem, $- z \\leqslant \\phi _k(y - z) - \\phi _k(y) \\leqslant 0 \\leqslant \\phi _k(y + z) - \\phi _k(y)\\leqslant z ,\\qquad y \\in \\mathbb {R}, \\quad z \\in \\mathbb {R}_+ , \\quad k \\in \\mathbb {N}.$ One can check that the process $\\left(I_{i,m,k,2}(t) + I_{i,m,k,4}(t)+ I_{i,m,k,6,1}(t)\\right)_{t\\in \\mathbb {R}_+}$ is a martingale.", "Indeed, by properties (ii) and (iii) of the function $\\phi _k$ and the definition of $\\tau _m$ , $\\operatorname{\\mathbb {E}}\\left( \\int _0^{t\\wedge \\tau _m}\\left(\\phi _k^{\\prime }(Y_{s,i}) \\sqrt{2c_i}\\Bigl (\\sqrt{X_{s,i}} - \\sqrt{X_{s,i}^{\\prime }}\\Bigr )\\right)^2\\, \\mathrm {d}s \\right)&\\leqslant 2c_i \\operatorname{\\mathbb {E}}\\left( \\int _0^{t\\wedge \\tau _m}(X_{s,i} + X_{s,i}^{\\prime }) \\, \\mathrm {d}s \\right) \\\\&\\leqslant 4 c_i m t < \\infty ,$ hence, by Ikeda and Watanabe [7], $\\left(I_{i,m,k,2}(t)\\right)_{t\\in \\mathbb {R}_+}$ is a martingale.", "Next we show $\\operatorname{\\mathbb {E}}\\left( \\int _0^{t\\wedge \\tau _m} \\int _{U_d} \\int _{U_1}\\left\| \\phi _k(Y_{s-,i} + z_i) - \\phi _k(Y_{s-,i}) \\right\|^2\\mathbb {1}_{\\lbrace \\Vert {z}\\Vert <1\\rbrace }\\mathbb {1}_{\\lbrace X_{s-,j}^{\\prime }<u\\leqslant X_{s-,j}\\rbrace } \\mathbb {1}_{\\lbrace Y_{s-,j}>0\\rbrace }\\, \\mathrm {d}s \\, \\mu _j(\\mathrm {d}{z}) \\, \\mathrm {d}u \\right)< \\infty ,$ and $\\operatorname{\\mathbb {E}}\\left( \\int _0^{t\\wedge \\tau _m} \\int _{U_d} \\int _{U_1}\\left\| \\phi _k(Y_{s-,i} - z_i) - \\phi _k(Y_{s-,i}) \\right\|^2\\mathbb {1}_{\\lbrace \\Vert {z}\\Vert <1\\rbrace }\\mathbb {1}_{\\lbrace X_{s-,j}<u\\leqslant X_{s-,j}^{\\prime }\\rbrace } \\mathbb {1}_{\\lbrace Y_{s-,j}<0\\rbrace }\\, \\mathrm {d}s \\, \\mu _j(\\mathrm {d}{z}) \\, \\mathrm {d}u \\right)< \\infty $ for all $j \\in \\lbrace 1, \\ldots , d\\rbrace $ , which yield that the functions $\\mathbb {R}_+\\times U_d\\times U_1\\times \\Omega \\ni (s,{z},u,\\omega )\\mapsto & (\\phi _k(Y_{s-,i}(\\omega ) + z_i) - \\phi _k(Y_{s-,i}(\\omega )))\\mathbb {1}_{\\lbrace \\Vert {z}\\Vert <1\\rbrace } \\\\& \\times \\mathbb {1}_{\\lbrace X_{s-,j}^{\\prime }(\\omega )<u\\leqslant X_{s-,j}(\\omega )\\rbrace } \\mathbb {1}_{\\lbrace Y_{s-,j}(\\omega )>0\\rbrace }\\mathbb {1}_{\\lbrace s\\leqslant \\tau _m(\\omega )\\rbrace }$ and $\\mathbb {R}_+\\times U_d\\times U_1\\times \\Omega \\ni (s,{z},u,\\omega )\\mapsto & (\\phi _k(Y_{s-,i}(\\omega ) - z_i) - \\phi _k(Y_{s-,i}(\\omega )))\\mathbb {1}_{\\lbrace \\Vert {z}\\Vert <1\\rbrace } \\\\& \\times \\mathbb {1}_{\\lbrace X_{s-,j}(\\omega )<u\\leqslant X_{s-,j}^{\\prime }(\\omega )\\rbrace } \\mathbb {1}_{\\lbrace Y_{s-,j}(\\omega )<0\\rbrace }\\mathbb {1}_{\\lbrace s\\leqslant \\tau _m(\\omega )\\rbrace }$ belong to the class ${F}_p^2$ , and then $\\left(I_{i,m,k,4}(t)\\right)_{t\\in \\mathbb {R}_+}$ is a martingale, again by page 62 in Ikeda and Watanabe [7].", "By (REF ) and (REF ), $\\operatorname{\\mathbb {E}}\\left( \\int _0^{t\\wedge \\tau _m} \\int _{U_d} \\int _{U_1}\\left\| \\phi _k(Y_{s-,i} + z_i) - \\phi _k(Y_{s-,i}) \\right\|^2\\mathbb {1}_{\\lbrace \\Vert {z}\\Vert <1\\rbrace }\\mathbb {1}_{\\lbrace X_{s-,j}^{\\prime }<u\\leqslant X_{s-,j}\\rbrace } \\mathbb {1}_{\\lbrace Y_{s-,j}>0\\rbrace }\\, \\mathrm {d}s \\, \\mu _j(\\mathrm {d}{z}) \\, \\mathrm {d}u \\right) \\\\\\begin{aligned}&\\leqslant \\operatorname{\\mathbb {E}}\\left( \\int _0^{t\\wedge \\tau _m} \\int _{U_d} \\int _{U_1}z_i^2 \\mathbb {1}_{\\lbrace \\Vert {z}\\Vert <1\\rbrace } \\mathbb {1}_{\\lbrace X_{s-,j}^{\\prime }<u\\leqslant X_{s-,j}\\rbrace }\\mathbb {1}_{\\lbrace Y_{s-,j}>0\\rbrace }\\, \\mathrm {d}s \\, \\mu _j(\\mathrm {d}{z}) \\, \\mathrm {d}u \\right) \\\\&= \\operatorname{\\mathbb {E}}\\left( \\int _0^{t\\wedge \\tau _m} \\int _{U_d}z_i^2 \\mathbb {1}_{\\lbrace \\Vert {z}\\Vert <1\\rbrace } Y_{s-,j} \\mathbb {1}_{\\lbrace Y_{s-,j}>0\\rbrace }\\, \\mathrm {d}s \\, \\mu _j(\\mathrm {d}{z}) \\right)\\leqslant 2mt\\int _{U_d} z_i^2 \\mathbb {1}_{\\lbrace \\Vert {z}\\Vert <1\\rbrace } \\, \\mu _j(\\mathrm {d}{z})< \\infty .\\end{aligned}$ In the same way one can get the finiteness of the other expectation.", "Finally, we show $\\operatorname{\\mathbb {E}}\\left( \\int _0^{t\\wedge \\tau _m} \\int _{U_d} \\int _{U_1}\\left\| \\phi _k(Y_{s-,i} + z_i) - \\phi _k(Y_{s-,i}) \\right\|\\mathbb {1}_{\\lbrace \\Vert {z}\\Vert \\geqslant 1\\rbrace }\\mathbb {1}_{\\lbrace X_{s-,j}^{\\prime }<u\\leqslant X_{s-,j}\\rbrace } \\mathbb {1}_{\\lbrace Y_{s-,j}>0\\rbrace }\\, \\mathrm {d}s \\, \\mu _j(\\mathrm {d}{z}) \\, \\mathrm {d}u \\right)< \\infty ,$ and $\\operatorname{\\mathbb {E}}\\left( \\int _0^{t\\wedge \\tau _m} \\int _{U_d} \\int _{U_1}\\left\| \\phi _k(Y_{s-,i} - z_i) - \\phi _k(Y_{s-,i}) \\right\|\\mathbb {1}_{\\lbrace \\Vert {z}\\Vert \\geqslant 1\\rbrace }\\mathbb {1}_{\\lbrace X_{s-,j}<u\\leqslant X_{s-,j}^{\\prime }\\rbrace } \\mathbb {1}_{\\lbrace Y_{s-,j}<0\\rbrace }\\, \\mathrm {d}s \\, \\mu _j(\\mathrm {d}{z}) \\, \\mathrm {d}u \\right)< \\infty $ for all $j \\in \\lbrace 1, \\ldots , d\\rbrace $ , which yield that the functions $\\mathbb {R}_+\\times U_d\\times U_1\\times \\Omega \\ni (s,{z},u,\\omega )\\mapsto & ( \\phi _k(Y_{s-,i}(\\omega ) + z_i) - \\phi _k(Y_{s-,i}(\\omega )) )\\mathbb {1}_{\\lbrace \\Vert {z}\\Vert \\geqslant 1\\rbrace }\\\\&\\times \\mathbb {1}_{\\lbrace X_{s-,j}^{\\prime }(\\omega )<u\\leqslant X_{s-,j}(\\omega )\\rbrace } \\mathbb {1}_{\\lbrace Y_{s-,j}(\\omega )>0\\rbrace }\\mathbb {1}_{\\lbrace s\\leqslant \\tau _m(\\omega )\\rbrace }$ and $\\mathbb {R}_+\\times U_d\\times U_1\\times \\Omega \\ni (s,{z},u,\\omega )\\mapsto & \\left( \\phi _k(Y_{s-,i}(\\omega ) - z_i) - \\phi _k(Y_{s-,i}(\\omega )) \\right)\\mathbb {1}_{\\lbrace \\Vert {z}\\Vert \\geqslant 1\\rbrace }\\\\&\\times \\mathbb {1}_{\\lbrace X_{s-,j}(\\omega )<u\\leqslant X_{s-,j}^{\\prime }(\\omega )\\rbrace } \\mathbb {1}_{\\lbrace Y_{s-,j}(\\omega )<0\\rbrace }\\mathbb {1}_{\\lbrace s\\leqslant \\tau _m(\\omega )\\rbrace }$ belong to the class ${F}_p^1$ , and then $\\left(I_{i,m,k,6,1}(t)\\right)_{t\\in \\mathbb {R}_+}$ is a martingale, again by Ikeda and Watanabe [7].", "By (REF ) and (REF ), $\\operatorname{\\mathbb {E}}\\left( \\int _0^{t\\wedge \\tau _m} \\int _{U_d} \\int _{U_1}\\left\| \\phi _k(Y_{s-,i} + z_i) - \\phi _k(Y_{s-,i}) \\right\|\\mathbb {1}_{\\lbrace \\Vert {z}\\Vert \\geqslant 1\\rbrace }\\mathbb {1}_{\\lbrace X_{s-,j}^{\\prime }<u\\leqslant X_{s-,j}\\rbrace } \\mathbb {1}_{\\lbrace Y_{s-,j}>0\\rbrace }\\, \\mathrm {d}s \\, \\mu _j(\\mathrm {d}{z}) \\, \\mathrm {d}u \\right) \\\\\\begin{aligned}&\\leqslant \\operatorname{\\mathbb {E}}\\left( \\int _0^{t\\wedge \\tau _m} \\int _{U_d} \\int _{U_1}z_i \\mathbb {1}_{\\lbrace \\Vert {z}\\Vert \\geqslant 1\\rbrace } \\mathbb {1}_{\\lbrace X_{s-,j}^{\\prime }<u\\leqslant X_{s-,j}\\rbrace }\\mathbb {1}_{\\lbrace Y_{s-,j}>0\\rbrace }\\, \\mathrm {d}s \\, \\mu _j(\\mathrm {d}{z}) \\, \\mathrm {d}u \\right) \\\\&= \\operatorname{\\mathbb {E}}\\left( \\int _0^{t\\wedge \\tau _m} \\int _{U_d}z_i \\mathbb {1}_{\\lbrace \\Vert {z}\\Vert \\geqslant 1\\rbrace } Y_{s-,j} \\mathbb {1}_{\\lbrace Y_{s-,j}>0\\rbrace }\\, \\mathrm {d}s \\, \\mu _j(\\mathrm {d}{z}) \\right)\\leqslant 2mt\\int _{U_d} z_i \\mathbb {1}_{\\lbrace \\Vert {z}\\Vert \\geqslant 1\\rbrace } \\, \\mu _j(\\mathrm {d}{z})< \\infty ,\\end{aligned}$ and the finiteness of the other expectation can be shown in the same way.", "Using the assumption ${\\beta }\\leqslant {\\beta }^{\\prime }$ , the property that the matrix ${D}$ has non-negative off-diagonal entries and the properties (ii) and (iii), we obtain $I_{i,m,k,1}(t)&= \\int _0^{t\\wedge \\tau _m}\\phi _k^{\\prime }(Y_{s,i})\\biggl ( \\beta _i - \\beta _i^{\\prime } + \\sum _{j=1}^d d_{i,j} Y_{s,j} \\biggr )\\, \\mathrm {d}s \\\\&\\leqslant \\int _0^{t\\wedge \\tau _m}\\phi _k^{\\prime }(Y_{s,i})\\biggl ( d_{i,i} Y_{s,i}+ \\sum _{j\\in \\lbrace 1,\\ldots ,d\\rbrace \\setminus \\lbrace i\\rbrace } d_{i,j} Y_{s,j}^+ \\biggr )\\mathbb {1}_{\\mathbb {R}_+}(Y_{s,i}) \\, \\mathrm {d}s \\\\&\\leqslant \\int _0^{t\\wedge \\tau _m}\\biggl ( \|d_{i,i}\| Y_{s,i}^++ \\sum _{j\\in \\lbrace 1,\\ldots ,d\\rbrace \\setminus \\lbrace i\\rbrace } d_{i,j} Y_{s,j}^+ \\biggr )\\, \\mathrm {d}s= \\sum _{j=1}^d \|d_{i,j}\| \\int _0^{t\\wedge \\tau _m} Y_{s,j}^+ \\, \\mathrm {d}s .$ By (iv), $I_{i,m,k,3}(t) \\leqslant (t\\wedge \\tau _m) c_i \\frac{2}{k} \\leqslant \\frac{2 c_i t}{k} .$ Now we estimate $&I_{i,m,k,5}(t)= \\sum _{j=1}^d\\int _0^{t\\wedge \\tau _m} \\int _{U_d}\\Bigl [ \\phi _k(Y_{s-,i} + z_i) - \\phi _k(Y_{s-,i})- \\phi _k^{\\prime }(Y_{s-,i}) z_i \\Bigr ]\\mathbb {1}_{\\lbrace \\Vert {z}\\Vert <1\\rbrace } Y_{s-,j} \\mathbb {1}_{\\lbrace Y_{s-,j}>0\\rbrace }\\, \\mathrm {d}s \\, \\mu _j(\\mathrm {d}{z}) \\\\&\\qquad + \\sum _{j=1}^d\\int _0^{t\\wedge \\tau _m} \\int _{U_d}\\Bigl [ \\phi _k(Y_{s-,i} - z_i) - \\phi _k(Y_{s-,i})+ \\phi _k^{\\prime }(Y_{s-,i}) z_i \\Bigr ]\\mathbb {1}_{\\lbrace \\Vert {z}\\Vert <1\\rbrace } (-Y_{s-,j}) \\mathbb {1}_{\\lbrace Y_{s-,j}<0\\rbrace }\\, \\mathrm {d}s \\, \\mu _j(\\mathrm {d}{z}) .$ By (REF ) and (iii), we obtain $\\int _0^{t\\wedge \\tau _m} \\int _{U_d}\\Bigl [ \\phi _k(Y_{s-,i} - z_i) - \\phi _k(Y_{s-,i}) + \\phi _k^{\\prime }(Y_{s-,i}) z_i \\Bigr ]\\mathbb {1}_{\\lbrace \\Vert {z}\\Vert <1\\rbrace } (-Y_{s-,j}) \\mathbb {1}_{\\lbrace Y_{s-,j}<0\\rbrace }\\, \\mathrm {d}s \\, \\mu _j(\\mathrm {d}{z})\\leqslant 0$ for all $i, j \\in \\lbrace 1, \\ldots , d\\rbrace $ .", "By (REF ), $\\int _{U_d} z_i \\mathbb {1}_{\\lbrace \\Vert {z}\\Vert <1\\rbrace } \\, \\mu _j(\\mathrm {d}z) < \\infty $ for all $i, j \\in \\lbrace 1, \\ldots , d\\rbrace $ with $i \\ne j$ , hence using (iii), we obtain $I_{i,m,k,5}(t)&\\leqslant \\int _0^{t\\wedge \\tau _m} \\int _{U_d}\\Bigl [ \\phi _k(Y_{s-,i} + z_i) - \\phi _k(Y_{s-,i})- \\phi _k^{\\prime }(Y_{s-,i}) z_i \\Bigr ]\\mathbb {1}_{\\lbrace \\Vert {z}\\Vert <1\\rbrace } Y_{s-,i}^+ \\, \\mathrm {d}s \\, \\mu _i(\\mathrm {d}{z}) \\\\&\\quad + \\sum _{j\\in \\lbrace 1,\\ldots ,d\\rbrace \\setminus \\lbrace i\\rbrace }\\int _0^{t\\wedge \\tau _m} \\int _{U_d}\\Bigl [ \\phi _k(Y_{s-,i} + z_i) - \\phi _k(Y_{s-,i}) \\Bigr ]\\mathbb {1}_{\\lbrace \\Vert {z}\\Vert <1\\rbrace } Y_{s-,j}^+ \\, \\mathrm {d}s \\, \\mu _j(\\mathrm {d}{z}) .$ By (REF ), for $i \\ne j$ , $\\int _0^{t\\wedge \\tau _m} \\int _{U_d}\\Bigl [ \\phi _k(Y_{s-,i} + z_i) - \\phi _k(Y_{s-,i}) \\Bigr ]\\mathbb {1}_{\\lbrace \\Vert {z}\\Vert <1\\rbrace } Y_{s-,j}^+ \\, \\mathrm {d}s \\, \\mu _j(\\mathrm {d}{z}) \\\\\\leqslant \\int _0^{t\\wedge \\tau _m} Y_{s,j}^+ \\, \\mathrm {d}s\\int _{U_d} z_i \\mathbb {1}_{\\lbrace \\Vert {z}\\Vert <1\\rbrace } \\, \\mu _j(\\mathrm {d}{z}) .$ Applying (iv) with $y = 0$ , we have $x \\phi _k^{\\prime \\prime }(x) \\leqslant 2/k$ for all $x \\in \\mathbb {R}_+$ and $k \\in \\mathbb {N}$ .", "By Taylor's theorem, for all $y \\in \\mathbb {R}_{++}$ , $z \\in \\mathbb {R}_+$ and $k \\in \\mathbb {N}$ , there exists some $\\vartheta = \\vartheta (y, z) \\in [0,1]$ such that $\\phi _k(y + z) - \\phi _k(y) - \\phi _k^{\\prime }(y) z= \\phi _k^{\\prime \\prime }(y + \\vartheta z) \\frac{z^2}{2}\\leqslant \\frac{2 z^2}{2k(y + \\vartheta z)}\\leqslant \\frac{z^2}{k y} .$ Hence, using (REF ), we obtain $&\\int _0^{t\\wedge \\tau _m} \\int _{U_d}\\Bigl [ \\phi _k(Y_{s-,i} + z_i) - \\phi _k(Y_{s-,i}) - \\phi _k^{\\prime }(Y_{s-,i}) z_i \\Bigr ]\\mathbb {1}_{\\lbrace \\Vert {z}\\Vert <1\\rbrace } Y_{s-,i}^+ \\, \\mathrm {d}s \\, \\mu _i(\\mathrm {d}{z}) \\\\&\\qquad \\qquad \\leqslant \\int _0^{t\\wedge \\tau _m} \\int _{U_d}\\frac{z_i^2}{k Y_{s-,i}} \\mathbb {1}_{\\lbrace Y_{s-,i}>0\\rbrace }\\mathbb {1}_{\\lbrace \\Vert {z}\\Vert <1\\rbrace } Y_{s-,i}^+ \\, \\mathrm {d}s \\, \\mu _i(\\mathrm {d}{z})\\leqslant \\frac{t}{k} \\int _{U_d} z_i^2 \\mathbb {1}_{\\lbrace \\Vert {z}\\Vert <1\\rbrace } \\, \\mu _i(\\mathrm {d}{z}) .$ Using (REF ), one can easily check that $I_{i,m,k,6,2}(t)&= \\sum _{j=1}^d\\int _0^{t\\wedge \\tau _m} \\int _{U_d}\\Bigl [ \\phi _k(Y_{s-,i} + z_i) - \\phi _k(Y_{s-,i}) \\Bigr ]\\mathbb {1}_{\\lbrace \\Vert {z}\\Vert \\geqslant 1\\rbrace } Y_{s-,j} \\mathbb {1}_{\\lbrace Y_{s-,j}>0\\rbrace }\\, \\mathrm {d}s \\, \\mu _j(\\mathrm {d}{z}) \\\\&\\quad + \\sum _{j=1}^d\\int _0^{t\\wedge \\tau _m} \\int _{U_d}\\Bigl [ \\phi _k(Y_{s-,i} - z_i) - \\phi _k(Y_{s-,i}) \\Bigr ]\\mathbb {1}_{\\lbrace \\Vert {z}\\Vert \\geqslant 1\\rbrace } (-Y_{s-,j}) \\mathbb {1}_{\\lbrace Y_{s-,j}<0\\rbrace }\\, \\mathrm {d}s \\, \\mu _j(\\mathrm {d}{z}) .$ By (REF ), we obtain $\\int _0^{t\\wedge \\tau _m} \\int _{U_d}\\Bigl [ \\phi _k(Y_{s-,i} - z_i) - \\phi _k(Y_{s-,i}) \\Bigr ]\\mathbb {1}_{\\lbrace \\Vert {z}\\Vert \\geqslant 1\\rbrace } (-Y_{s-,j}) \\mathbb {1}_{\\lbrace Y_{s-,j}<0\\rbrace }\\, \\mathrm {d}s \\, \\mu _j(\\mathrm {d}{z})\\leqslant 0$ for all $i, j \\in \\lbrace 1, \\ldots , d\\rbrace $ .", "By (REF ), $\\int _{U_d} z_i \\mathbb {1}_{\\lbrace \\Vert {z}\\Vert \\geqslant 1\\rbrace } \\, \\mu _j(\\mathrm {d}z) < \\infty $ for all $i, j \\in \\lbrace 1, \\ldots , d\\rbrace $ , thus applying (REF ), we obtain $I_{i,m,k,6,2}(t)&\\leqslant \\sum _{j=1}^d\\int _0^{t\\wedge \\tau _m} \\int _{U_d}\\Bigl [ \\phi _k(Y_{s-,i} + z_i) - \\phi _k(Y_{s-,i}) \\Bigr ]\\mathbb {1}_{\\lbrace \\Vert {z}\\Vert \\geqslant 1\\rbrace } Y_{s-,j}^+ \\, \\mathrm {d}s \\, \\mu _j(\\mathrm {d}{z}) \\\\&\\leqslant \\sum _{j=1}^d\\int _0^{t\\wedge \\tau _m} Y_{s,j}^+ \\, \\mathrm {d}s\\int _{U_d} z_i \\mathbb {1}_{\\lbrace \\Vert {z}\\Vert \\geqslant 1\\rbrace } \\, \\mu _j(\\mathrm {d}{z}) .$ Summarizing, we have $\\begin{split}\\phi _k(Y_{t\\wedge \\tau _m,i})&\\leqslant \\phi _k(Y_{0,i})+ C_i \\sum _{j=1}^d \\int _0^{t\\wedge \\tau _m} Y_{s,j}^+ \\, \\mathrm {d}s+ \\frac{2 c_i t}{k}+ \\frac{t}{k}\\int _{U_d} z_i^2 \\mathbb {1}_{\\lbrace \\Vert {z}\\Vert <1\\rbrace } \\, \\mu _i(\\mathrm {d}{z}) \\\\&\\quad + I_{i,m,k,2}(t) + I_{i,m,k,4}(t) + I_{i,m,k,6,1}(t) , \\qquad t \\in \\mathbb {R}_+ ,\\end{split}$ where $C_i := \\max _{j\\in \\lbrace 1,\\ldots ,d\\rbrace } \|d_{i,j}\|+ \\max _{j\\in \\lbrace 1,\\ldots ,d\\rbrace \\setminus \\lbrace i\\rbrace } \\int _{U_d} z_i \\, \\mu _j(\\mathrm {d}{z})+ \\int _{U_d} z_i \\mathbb {1}_{\\lbrace \\Vert {z}\\Vert \\geqslant 1\\rbrace } \\, \\mu _i(\\mathrm {d}{z}) .$ By (iii), we obtain $\\operatorname{\\mathbb {P}}(\\phi _k(Y_{0,i}) \\leqslant 0) = 1$ , $i \\in \\lbrace 1, \\ldots , d\\rbrace $ .", "By (i), the non-negativeness of $\\phi _k$ and monotone convergence theorem yield $\\operatorname{\\mathbb {E}}(\\phi _k(Y_{t\\wedge \\tau _m,i})) \\rightarrow \\operatorname{\\mathbb {E}}(Y_{t\\wedge \\tau _m,i}^+)$ as $k \\rightarrow \\infty $ for all $t \\in \\mathbb {R}_+$ , $m\\in \\mathbb {N}$ , and $i \\in \\lbrace 1, \\ldots , d\\rbrace $ .", "We have $\\int _0^{t\\wedge \\tau _m} Y_{s,j}^+ \\, \\mathrm {d}s\\leqslant \\int _0^t Y_{s\\wedge \\tau _m,j}^+ \\, \\mathrm {d}s$ , hence taking the expectation of (REF ) and letting $k \\rightarrow \\infty $ , we obtain $\\operatorname{\\mathbb {E}}\\biggl (\\sum _{i=1}^d Y_{t\\wedge \\tau _m,i}^+\\biggr )\\leqslant C \\int _0^t \\operatorname{\\mathbb {E}}\\biggl (\\sum _{i=1}^d Y_{s\\wedge \\tau _m,i}^+\\biggr ) \\mathrm {d}s ,$ with $C := \\sum _{i=1}^d C_i$ .", "By Gronwall's inequality, we conclude $\\operatorname{\\mathbb {E}}\\biggl (\\sum _{i=1}^d Y_{t\\wedge \\tau _m,i}^+\\biggr ) = 0$ for all $t \\in \\mathbb {R}_+$ and $m \\in \\mathbb {N}$ .", "Hence $\\operatorname{\\mathbb {P}}({X}_{t\\wedge \\tau _m,i} \\leqslant {X}^{\\prime }_{t\\wedge \\tau _m,i} )=1$ for all $t \\in \\mathbb {R}_+$ , $m \\in \\mathbb {N}$ and $i \\in \\lbrace 1, \\ldots , d\\rbrace $ , and then $\\operatorname{\\mathbb {P}}(\\text{${X}_{t\\wedge \\tau _m,i} \\leqslant {X}^{\\prime }_{t\\wedge \\tau _m,i}$ for all$m \\in \\mathbb {N}$})= 1$ for all $t \\in \\mathbb {R}_+$ and $i \\in \\lbrace 1, \\ldots , d\\rbrace $ .", "Since ${X}$ and ${X}^{\\prime }$ have càdlàg trajectories, these trajectories are bounded almost surely on $[0, T]$ for all $T \\in \\mathbb {R}_+$ , hence $\\tau _m \\stackrel{{\\mathrm {a.s.}}}{\\longrightarrow }\\infty $ as $m \\rightarrow \\infty $ .", "This yields $\\operatorname{\\mathbb {P}}({X}_t \\leqslant {X}^{\\prime }_t )=1$ for all $t \\in \\mathbb {R}_+$ .", "Since the set of non-negative rational numbers $\\mathbb {Q}_+$ is countable, we obtain $\\operatorname{\\mathbb {P}}(\\text{${X}_t \\leqslant {X}^{\\prime }_t$ for all $t \\in \\mathbb {Q}_+$}) = 1$ .", "Using again that ${X}$ and ${X}^{\\prime }$ have càdlàg trajectories almost surely, we get $\\operatorname{\\mathbb {P}}(\\text{${X}_t \\leqslant {X}_t^{\\prime }$ for all $t \\in \\mathbb {R}_+$}) = 1$ .", "$\\Box $ Remark.", "4.3 We note that Dawson and Li [3] provided a comparison theorem for SDEs with jumps in a much more general setting, but only for 1-dimensional processes.", "$\\Box $ Consider the following objects: a probability space $(\\Omega , {\\mathcal {F}}, \\operatorname{\\mathbb {P}})$ ; a $d$ -dimensional standard Brownian motion $({W}_t)_{t\\in \\mathbb {R}_+}$ ; a stationary Poisson point process $p$ on $V$ with characteristic measure $m$ given in (REF ); a random vector ${\\xi }$ with values in $\\mathbb {R}_+^d$ , independent of ${W}$ and $p$ .", "Remark.", "4.4 Note that if conditions (E1)–(E4) are satisfied, then ${\\xi }$ , ${W}$ and $p$ are automatically mutually independent according to Remark REF .", "$\\Box $ Provided that the objects (E1)–(E4) are given, let $({\\mathcal {F}}^{{\\xi },{W}\\!,\\,p}_t)_{t\\in \\mathbb {R}_+}$ be the augmented filtration generated by ${\\xi }$ , ${W}$ and $p$ , i.e., for each $t \\in \\mathbb {R}_+$ , ${\\mathcal {F}}^{{\\xi },{W}\\!,\\,p}_t$ is the $\\sigma $ -field generated by $\\sigma ({\\xi }; \\, {W}_s, s\\in [0,t]; \\, p(s), s\\in (0,t]\\cap D(p))$ and by the $\\operatorname{\\mathbb {P}}$ -null sets from $\\sigma ({\\xi }; \\, {W}_s, s\\in \\mathbb {R}_+; \\, p(s), s\\in \\mathbb {R}_{++}\\cap D(p))$ (which is similar to the definition in Karatzas and Shreve [10]).", "One can check that $({\\mathcal {F}}^{{\\xi },{W}\\!,\\,p}_t)_{t\\in \\mathbb {R}_+}$ satisfies the usual hypotheses, $({W}_t)_{t\\in \\mathbb {R}_+}$ is a standard $({\\mathcal {F}}^{{\\xi },{W}\\!,\\,p}_t)_{t\\in \\mathbb {R}_+}$ -Brownian motion, and $p$ is a stationary $({\\mathcal {F}}^{{\\xi },{W}\\!,\\,p}_t)_{t\\in \\mathbb {R}_+}$ -Poisson point process on $V$ with characteristic measure $m$ , see, e.g., Barczy et al.", "[1].", "Definition.", "4.5 Suppose that the objects (E1)–(E4) are given.", "An $\\mathbb {R}_+^d$ -valued strong solution of the SDE (REF ) on $(\\Omega , {\\mathcal {F}}, \\operatorname{\\mathbb {P}})$ and with respect to the standard Brownian motion ${W}$ , the stationary Poisson point process $p$ and initial value ${\\xi }$ , is an $\\mathbb {R}_+^d$ -valued $({\\mathcal {F}}^{{\\xi },{W}\\!,\\,p}_t)_{t\\in \\mathbb {R}_+}$ -adapted càdlàg process $({X}_t)_{t \\in \\mathbb {R}_+}$ with $\\operatorname{\\mathbb {P}}({X}_0 = {\\xi }) = 1$ satisfying (D4)(b)–(e).", "Clearly, if $({X}_t)_{t \\in \\mathbb {R}_+}$ is an $\\mathbb {R}_+^d$ -valued strong solution, then $\\bigl ( \\Omega , {\\mathcal {F}}, ({\\mathcal {F}}^{{\\xi },{W}\\!,\\,p}_t)_{t\\in \\mathbb {R}_+}, \\operatorname{\\mathbb {P}}, {W}, p,{X}\\bigr )$ is an $\\mathbb {R}_+^d$ -valued weak solution.", "Theorem.", "4.6 Let $(d, {c}, {\\beta }, {B}, \\nu , {\\mu })$ be a set of admissible parameters in the sense of Definition REF such that the moment condition (REF ) holds.", "Suppose that objects (E1)–(E4) are given.", "If $\\operatorname{\\mathbb {E}}(\\Vert {\\xi }\\Vert ) < \\infty $ , then there is a pathwise unique $\\mathbb {R}_+^d$ -valued strong solution to the SDE (REF ) with initial value ${\\xi }$ , and the solution is a CBI process with parameters $(d, {c}, {\\beta }, {B}, \\nu , {\\mu })$ .", "Proof.", "The pathwise uniqueness among $\\mathbb {R}_+^d$ -valued weak solutions follows from Lemma REF .", "Then, by Theorem 5.5 in Barczy et al.", "[1] (Yamada-Watanabe type result for SDEs with jumps) and Theorem REF , we conclude that the SDE (REF ) has a pathwise unique $\\mathbb {R}_+^d$ -valued strong solution.", "$\\Box $" ], [ "Special cases", "In this section we specialize our results to dimension 1 and 2.", "Moreover, we consider a special case of the SDE (REF ) with $\\nu = 0$ , $\\mu _i = 0$ , $i \\in \\lbrace 1, \\ldots , d\\rbrace $ , i.e., without integrals with respect to (compensated) Poisson random measures, and another special case with ${c}= {0}$ , i.e., without integral with respect to a Wiener process.", "First we rewrite the SDE (REF ) in a form which is more comparable with the results of Li [13] (one-dimensional case) and Ma [14] (two-dimensional case).", "For each $j \\in \\lbrace 0, 1, \\ldots , d\\rbrace $ , the thinning $p_j$ of $p$ onto ${\\mathcal {R}}_j$ is again a stationary $({\\mathcal {F}}_t)_{t\\in \\mathbb {R}_+}$ -Poisson point process on ${\\mathcal {R}}_j$ , and its characteristic measure is the restriction $m\|_{{\\mathcal {R}}_j}$ of $m$ onto ${\\mathcal {R}}_j$ (this can be checked calculating its conditional Laplace transform, see Ikeda and Watanabe [7]).", "Using these Poisson point processes, we obtain the useful decomposition $\\begin{aligned}&\\int _0^t \\int _{V_0} f({X}_{s-}, {r}) \\, \\widetilde{N}(\\mathrm {d}s, \\mathrm {d}{r})+ \\int _0^t \\int _{V_1} g({X}_{s-}, {r}) \\, N(\\mathrm {d}s, \\mathrm {d}{r}) \\\\&= \\sum _{j=1}^d\\int _0^t \\int _{{\\mathcal {R}}_{j,0}}{z}\\mathbb {1}_{\\lbrace u\\leqslant X_{s-,j}\\rbrace }\\, \\widetilde{N}_j(\\mathrm {d}s, \\mathrm {d}{r}) \\\\&\\quad + \\sum _{j=1}^d\\int _0^t \\int _{{\\mathcal {R}}_{j,1}}{z}\\mathbb {1}_{\\lbrace u\\leqslant X_{s-,j}\\rbrace }\\, N_j(\\mathrm {d}s, \\mathrm {d}{r})+ \\int _0^t \\int _{{\\mathcal {R}}_0}{r}\\, M(\\mathrm {d}s, \\mathrm {d}{r}) ,\\end{aligned}$ where, for each $j \\in \\lbrace 1, \\ldots , d\\rbrace $ , $N_j(\\mathrm {d}s, \\mathrm {d}{r})$ is the counting measure of $p_j$ on $\\mathbb {R}_{++} \\times {\\mathcal {R}}_j$ , $\\widetilde{N}_j(\\mathrm {d}s, \\mathrm {d}{r}):= N_j(\\mathrm {d}s, \\mathrm {d}{r}) - \\mathrm {d}s \\, (\\mu _j(\\mathrm {d}{z}) \\, \\mathrm {d}u)$ , and $M(\\mathrm {d}s, \\mathrm {d}{r})$ is the counting measure of $p_0$ on $\\mathbb {R}_{++} \\times {\\mathcal {R}}_0$ .", "Indeed, $\\begin{aligned}\\int _0^t \\int _{{\\mathcal {R}}^{\\prime }} F(s, {r}) \\, \\widetilde{N}(\\mathrm {d}s, \\mathrm {d}{r})&= \\int _0^t \\int _{{\\mathcal {R}}^{\\prime }} F(s, {r}) \\, \\widetilde{N}^{\\prime }(\\mathrm {d}s, \\mathrm {d}{r}) ,\\qquad F \\in {F}_p^{2,loc} , \\\\\\int _0^t \\int _{{\\mathcal {R}}^{\\prime }} G(s, {r}) \\, N(\\mathrm {d}s, \\mathrm {d}{r})&= \\int _0^t \\int _{{\\mathcal {R}}^{\\prime }} G(s, {r}) \\, N^{\\prime }(\\mathrm {d}s, \\mathrm {d}{r}) , \\qquad G \\in {F}_p ,\\end{aligned}$ are valid for the thinning $p^{\\prime }$ of $p$ onto any measurable subset ${\\mathcal {R}}^{\\prime } \\subset {\\mathcal {R}}$ , where $N^{\\prime }(\\mathrm {d}s, \\mathrm {d}{r})$ denotes the counting measure of the stationary $({\\mathcal {F}}_t)_{t\\in \\mathbb {R}_+}$ -Poisson point process $p^{\\prime }$ , and $\\widetilde{N}^{\\prime }(\\mathrm {d}s, \\mathrm {d}{r}):= N^{\\prime }(\\mathrm {d}s, \\mathrm {d}{r}) - \\mathbb {1}_{\\lbrace {r}\\in {\\mathcal {R}}^{\\prime }\\rbrace } \\mathrm {d}s \\, m(\\mathrm {d}{r})$ .", "Remark that for any $\\mathbb {R}_+^d$ -valued weak solution of the SDE (REF ), the Brownian motion ${W}$ and the stationary Poisson point processes $p_j$ , $j \\in \\lbrace 0, 1, \\ldots , d\\rbrace $ are mutually independent according again to Theorem 6.3 in Chapter II of Ikeda and Watanabe [7].", "Indeed, the intensity measures of $p_j$ , $j \\in \\lbrace 0, 1, \\ldots , d\\rbrace $ , are deterministic, and condition (6.11) of this theorem is satisfied, because $p_j$ , $j \\in \\lbrace 0, 1, \\ldots , d\\rbrace $ , live on disjoint subsets of ${\\mathcal {R}}$ .", "For $d = 1$ , applying (REF ), the SDE (REF ) takes the form $X_t&= X_0 + \\int _0^t (\\beta + d X_s) \\, \\mathrm {d}s+ \\int _0^t \\sqrt{2 c X_s^+} \\, \\mathrm {d}W_s \\\\&\\quad + \\int _0^t \\int _{{\\mathcal {R}}_{1,0}} z \\mathbb {1}_{\\lbrace u\\leqslant X_{s-}\\rbrace } \\, \\widetilde{N}_1(\\mathrm {d}s, \\mathrm {d}r)+ \\int _0^t \\int _{{\\mathcal {R}}_{1,1}} z \\mathbb {1}_{\\lbrace u\\leqslant X_{s-}\\rbrace } \\, N_1(\\mathrm {d}s, \\mathrm {d}r)+ \\int _0^t \\int _{{\\mathcal {R}}_0} r \\, M(\\mathrm {d}s, \\mathrm {d}r)$ for $t\\in \\mathbb {R}_+$ , where $\\beta \\in \\mathbb {R}_+$ , $d = \\widetilde{b}- \\int _0^\\infty z \\mathbb {1}_{\\lbrace z\\geqslant 1\\rbrace } \\, \\mu _1(\\mathrm {d}z)$ , $\\widetilde{b}= b + \\int _0^\\infty (z - 1)^+ \\, \\mu _1(\\mathrm {d}z)$ , $b \\in \\mathbb {R}$ , $c \\in \\mathbb {R}_+$ , ${\\mathcal {R}}_{1,0} = \\lbrace 0\\rbrace \\times \\lbrace z \\in \\mathbb {R}_{++} : z < 1\\rbrace \\times \\mathbb {R}_{++}$ , ${\\mathcal {R}}_{1,1} = \\lbrace 0\\rbrace \\times \\lbrace z \\in \\mathbb {R}_{++} : z \\geqslant 1\\rbrace \\times \\mathbb {R}_{++}$ , ${\\mathcal {R}}_0 = \\mathbb {R}_{++} \\times \\lbrace (0, 0)\\rbrace $ .", "We have $I_0&:=\\int _0^t \\int _{{\\mathcal {R}}_{1,0}} z \\mathbb {1}_{\\lbrace u\\leqslant X_{s-}\\rbrace } \\, \\widetilde{N}_1(\\mathrm {d}s, \\mathrm {d}r)= \\int _0^t \\int _0^\\infty \\int _0^\\infty z \\mathbb {1}_{\\lbrace z<1\\rbrace } \\mathbb {1}_{\\lbrace u\\leqslant X_{s-}\\rbrace }\\, \\widetilde{\\overline{N}}_1(\\mathrm {d}s, \\mathrm {d}z, \\mathrm {d}u) , \\\\I_1&:= \\int _0^t \\int _{{\\mathcal {R}}_{1,1}} z \\mathbb {1}_{\\lbrace u\\leqslant X_{s-}\\rbrace } \\, N_1(\\mathrm {d}s, \\mathrm {d}r)= \\int _0^t \\int _0^\\infty \\int _0^\\infty z \\mathbb {1}_{\\lbrace z\\geqslant 1\\rbrace } \\mathbb {1}_{\\lbrace u\\leqslant X_{s-}\\rbrace }\\, \\overline{N}_1(\\mathrm {d}s, \\mathrm {d}z, \\mathrm {d}u) , \\\\I_2&:= \\int _0^t \\int _{{\\mathcal {R}}_0} r \\, M(\\mathrm {d}s, \\mathrm {d}r)= \\int _0^t \\int _0^\\infty z \\, \\overline{M}(\\mathrm {d}s, \\mathrm {d}z) ,$ where $\\overline{N}_1$ and $\\overline{M}$ are Poisson random measures on $\\mathbb {R}_{++} \\times \\mathbb {R}_{++}^2$ and on $\\mathbb {R}_{++} \\times \\mathbb {R}_{++}$ with intensity measures $\\mathrm {d}s \\, \\mu _1(\\mathrm {d}z) \\, \\mathrm {d}u$ and $\\mathrm {d}s \\, \\nu (\\mathrm {d}z)$ , respectively, and $\\widetilde{\\overline{N}}_1(\\mathrm {d}s, \\mathrm {d}z, \\mathrm {d}u):= \\overline{N}_1(\\mathrm {d}s, \\mathrm {d}z, \\mathrm {d}u) - \\mathrm {d}s \\, \\mu _1(\\mathrm {d}z) \\, \\mathrm {d}u$ .", "Under the moment conditions (REF ), $I_0 + I_1= \\int _0^t \\int _0^\\infty \\int _0^\\infty z \\mathbb {1}_{\\lbrace u\\leqslant X_{s-}\\rbrace } \\, \\widetilde{\\overline{N}}_1(\\mathrm {d}s, \\mathrm {d}z, \\mathrm {d}u)+ \\int _0^t X_s \\, \\mathrm {d}s\\int _0^\\infty z \\mathbb {1}_{\\lbrace z\\geqslant 1\\rbrace } \\, \\mu _1(\\mathrm {d}z) .$ Consequently, the SDE (REF ) can be rewritten in the form $X_t&= X_0 + \\int _0^t (\\beta + \\widetilde{b}X_s) \\, \\mathrm {d}s+ \\int _0^t \\sqrt{2 c X_s^+} \\, \\mathrm {d}W_s \\\\&\\quad + \\int _0^t \\int _0^\\infty \\int _0^\\infty z \\mathbb {1}_{\\lbrace u\\leqslant X_{s-}\\rbrace } \\, \\widetilde{\\overline{N}}_1(\\mathrm {d}s, \\mathrm {d}z, \\mathrm {d}u)+ \\int _0^t \\int _0^\\infty z \\, \\overline{M}(\\mathrm {d}s, \\mathrm {d}z) ,\\qquad t\\in \\mathbb {R}_+,$ hence, taking into account the form (REF ) of the infinitesimal generator of the process $(X_t)_{t\\in \\mathbb {R}_+}$ , we obtain equation (9.46) of Li [13].", "In a similar way, for $d = 2$ , applying (REF ), the SDE (REF ) takes the form ${X}_t&= {X}_0 + \\int _0^t ({\\beta }+ {D}{X}_s) \\, \\mathrm {d}s+ \\sum _{i=1}^2\\int _0^t \\sqrt{2 c_i X_{s,i}^+} {e}_i {e}_i^\\top \\, \\mathrm {d}{W}_s+ \\int _0^t \\int _{{\\mathcal {R}}_0} {r}\\, M(\\mathrm {d}s, \\mathrm {d}{r}) \\\\&\\quad + \\sum _{j=1}^2\\int _0^t \\int _{{\\mathcal {R}}_{j,0}}{z}\\mathbb {1}_{\\lbrace u\\leqslant X_{s-,j}\\rbrace } \\, \\widetilde{N}_j(\\mathrm {d}s, \\mathrm {d}{r})+ \\sum _{j=1}^2\\int _0^t \\int _{{\\mathcal {R}}_{j,1}}{z}\\mathbb {1}_{\\lbrace u\\leqslant X_{s-,j}\\rbrace } \\, N_j(\\mathrm {d}s, \\mathrm {d}{r})$ for $t\\in \\mathbb {R}_+$ , where ${\\beta }\\in \\mathbb {R}_+^2$ , ${D}$ is given in (REF ), $(c_1, c_2)^\\top \\in \\mathbb {R}_+^2$ , ${\\mathcal {R}}_0 &= U_2 \\times \\lbrace (0, 0, 0)\\rbrace \\times \\lbrace (0, 0, 0)\\rbrace , \\\\{\\mathcal {R}}_{1,0} &= \\lbrace (0, 0)\\rbrace \\times \\lbrace {z}\\in U_2 : \\Vert {z}\\Vert < 1 \\rbrace \\times \\mathbb {R}_{++} \\times \\lbrace (0, 0, 0)\\rbrace , \\\\{\\mathcal {R}}_{2,0} &= \\lbrace (0, 0)\\rbrace \\times \\lbrace (0, 0, 0)\\rbrace \\times \\lbrace {z}\\in U_2 : \\Vert {z}\\Vert < 1 \\rbrace \\times \\mathbb {R}_{++} , \\\\{\\mathcal {R}}_{1,1} &= \\lbrace (0, 0)\\rbrace \\times \\lbrace {z}\\in U_2 : \\Vert {z}\\Vert \\geqslant 1 \\rbrace \\times \\mathbb {R}_{++} \\times \\lbrace (0, 0, 0)\\rbrace , \\\\{\\mathcal {R}}_{2,1} &= \\lbrace (0, 0)\\rbrace \\times \\lbrace (0, 0, 0)\\rbrace \\times \\lbrace {z}\\in U_2 : \\Vert {z}\\Vert \\geqslant 1 \\rbrace \\times \\mathbb {R}_{++} .$ For each $j \\in \\lbrace 1, 2\\rbrace $ , we have $I_{j,0}&:=\\int _0^t \\int _{{\\mathcal {R}}_{j,0}}{z}\\mathbb {1}_{\\lbrace u\\leqslant X_{s-,j}\\rbrace } \\, \\widetilde{N}_j(\\mathrm {d}s, \\mathrm {d}{r})= \\int _0^t \\int _{U_2} \\int _0^\\infty {z}\\mathbb {1}_{\\lbrace \\Vert {z}\\Vert <1\\rbrace } \\mathbb {1}_{\\lbrace u\\leqslant X_{s-,j}\\rbrace }\\, \\widetilde{\\overline{N}}_j(\\mathrm {d}s, \\mathrm {d}{z}, \\mathrm {d}u) , \\\\I_{j,1}&:= \\int _0^t \\int _{{\\mathcal {R}}_{j,1}}{z}\\mathbb {1}_{\\lbrace u\\leqslant X_{s-,j}\\rbrace } \\, N_j(\\mathrm {d}s, \\mathrm {d}{r})= \\int _0^t \\int _{U_2} \\int _0^\\infty {z}\\mathbb {1}_{\\lbrace \\Vert {z}\\Vert \\geqslant 1\\rbrace } \\mathbb {1}_{\\lbrace u\\leqslant X_{s-,j}\\rbrace }\\, \\overline{N}_j(\\mathrm {d}s, \\mathrm {d}{z}, \\mathrm {d}u) , \\\\I_2&:= \\int _0^t \\int _{{\\mathcal {R}}_0} {r}\\, M(\\mathrm {d}s, \\mathrm {d}{r})= \\int _0^t \\int _{U_2} {z}\\, \\overline{M}(\\mathrm {d}s, \\mathrm {d}{z}) ,$ where $\\overline{N}_j$ and $\\overline{M}$ are Poisson random measures on $\\mathbb {R}_{++} \\times U_2 \\times \\mathbb {R}_{++}$ and on $\\mathbb {R}_{++} \\times U_2$ with intensity measures $\\mathrm {d}s \\, \\mu _j(\\mathrm {d}{z}) \\, \\mathrm {d}u$ and $\\mathrm {d}s \\, \\nu (\\mathrm {d}{z})$ , respectively, and $\\widetilde{\\overline{N}}_j(\\mathrm {d}s, \\mathrm {d}{z}, \\mathrm {d}u):= \\overline{N}_j(\\mathrm {d}s, \\mathrm {d}{z}, \\mathrm {d}u) - \\mathrm {d}s \\, \\mu _j(\\mathrm {d}{z}) \\, \\mathrm {d}u$ .", "Under the moment conditions (REF ), $I_{j,0} + I_{j,1}= \\int _0^t \\int _{U_2} \\int _0^\\infty {z}\\mathbb {1}_{\\lbrace u\\leqslant X_{s-,j}\\rbrace } \\, \\widetilde{\\overline{N}}_j(\\mathrm {d}s, \\mathrm {d}{z}, \\mathrm {d}u)+ \\int _0^t X_{s,j} \\, \\mathrm {d}s\\int _{U_2} {z}\\mathbb {1}_{\\lbrace \\Vert {z}\\Vert \\geqslant 1\\rbrace } \\, \\mu _j(\\mathrm {d}{z}) .$ Consequently, the SDE (REF ) can be rewritten in the form ${X}_t&= {X}_0 + \\int _0^t ({\\beta }+ \\widetilde{{B}}{X}_s) \\, \\mathrm {d}s+ \\sum _{i=1}^2\\int _0^t \\sqrt{2 c_i X_{s,i}^+} \\, \\mathrm {d}W_{s,i} \\, {e}_i \\\\&\\quad + \\sum _{j=1}^2\\int _0^t \\int _{U_2} \\int _0^\\infty {z}\\mathbb {1}_{\\lbrace u\\leqslant X_{s-,j}\\rbrace } \\, \\widetilde{\\overline{N}}_j(\\mathrm {d}s, \\mathrm {d}{z}, \\mathrm {d}u)+ \\int _0^t \\int _{U_2} {z}\\, \\overline{M}(\\mathrm {d}s, \\mathrm {d}{z}) , \\qquad t\\in \\mathbb {R}_+.$ Due to (REF ), we have $X_{t,1} & = X_{0,1}+ \\int _0^t \\left(\\beta _1 + \\widetilde{b}_{1,1} X_{s,1}+ \\left(\\widetilde{b}_{1,2} - \\int _{U_2} z_1\\, \\mu _2(\\mathrm {d}{z}) \\right) X_{s,2}\\right)\\,\\mathrm {d}s+ \\int _0^t\\sqrt{2c_1 X_{s,1}^+}\\,\\mathrm {d}W_{s,1}\\\\&\\quad + \\int _0^t \\int _{U_2} \\int _0^\\infty z_1 \\mathbb {1}_{\\lbrace u\\leqslant X_{s-,1}\\rbrace } \\, \\widetilde{\\overline{N}}_1(\\mathrm {d}s, \\mathrm {d}{z}, \\mathrm {d}u)+ \\int _0^t \\int _{U_2} \\int _0^\\infty z_1 \\mathbb {1}_{\\lbrace u\\leqslant X_{s-,2}\\rbrace } \\, \\overline{N}_2(\\mathrm {d}s, \\mathrm {d}{z}, \\mathrm {d}u) \\\\&\\quad + \\int _0^t \\int _{U_2} z_1 \\, \\overline{M}(\\mathrm {d}s, \\mathrm {d}{z}) , \\qquad t\\in \\mathbb {R}_+,$ and $X_{t,2} &= X_{0,2}+ \\int _0^t \\left(\\beta _2 + \\left(\\widetilde{b}_{2,1} - \\int _{U_2} z_2 \\,\\mu _1(\\mathrm {d}{z}) \\right) X_{s,1}+ \\widetilde{b}_{2,2} X_{s,2}\\right)\\,\\mathrm {d}s+ \\int _0^t\\sqrt{2c_2 X_{s,2}^+}\\,\\mathrm {d}W_{s,2}\\\\&\\quad + \\int _0^t \\int _{U_2} \\int _0^\\infty z_2 \\mathbb {1}_{\\lbrace u\\leqslant X_{s-,2}\\rbrace } \\, \\widetilde{\\overline{N}}_2(\\mathrm {d}s, \\mathrm {d}{z}, \\mathrm {d}u)+ \\int _0^t \\int _{U_2} \\int _0^\\infty z_2 \\mathbb {1}_{\\lbrace u\\leqslant X_{s-,1}\\rbrace } \\, \\overline{N}_1(\\mathrm {d}s, \\mathrm {d}{z}, \\mathrm {d}u)\\\\&\\quad + \\int _0^t \\int _{U_2} z_2 \\, \\overline{M}(\\mathrm {d}s, \\mathrm {d}{z}) , \\qquad t\\in \\mathbb {R}_+.$ In the special case $\\nu = 0$ , we obtain equations (2.1) and (2.2) of Ma [14].", "Indeed, due to (REF ), one can rewrite the infinitesimal generator (REF ) of the process $({X}_t)_{t\\in \\mathbb {R}_+}$ in the following form $({\\mathcal {A}}_{{X}} f)({x})&= \\sum _{i=1}^2 c_i x_i f_{i,i}^{\\prime \\prime }(x)+ \\sum _{i=1}^2x_i\\int _{U_2}\\bigl (f({x}+ {z}) - f({x}) - z_if_i^{\\prime }(x)\\bigr )\\, \\mu _i(\\mathrm {d}{z}) \\\\&\\quad + \\langle {\\beta }+ \\widetilde{{B}}{x}, {f}^{\\prime }({x}) \\rangle + \\int _{U_2} \\bigl (f({x}+ {z}) - f({x})\\bigr ) \\, \\nu (\\mathrm {d}{z})\\\\&\\quad - x_1f_2^{\\prime }(x)\\int _{U_2}z_2\\,\\mu _1(\\mathrm {d}z)- x_2f_1^{\\prime }(x)\\int _{U_2}z_1\\,\\mu _2(\\mathrm {d}z) \\\\&= \\sum _{i=1}^2 c_i x_i f_{i,i}^{\\prime \\prime }(x)+ \\sum _{i=1}^2x_i\\int _{U_2}\\bigl (f({x}+ {z}) - f({x}) - z_if_i^{\\prime }(x)\\bigr )\\, \\mu _i(\\mathrm {d}{z}) \\\\&\\quad + \\langle {\\beta }+ \\widetilde{\\widetilde{{B}}}{x}, {f}^{\\prime }({x}) \\rangle + \\int _{U_2} \\bigl (f({x}+ {z}) - f({x})\\bigr ) \\, \\nu (\\mathrm {d}{z})$ for $f \\in C^2_\\mathrm {c}(\\mathbb {R}_+^d,\\mathbb {R})$ and ${x}\\in \\mathbb {R}_+^d$ , where $\\widetilde{\\widetilde{{B}}}:= \\begin{bmatrix}\\widetilde{b}_{1,1} & \\widetilde{b}_{1,2} - \\int _{U_2}z_1\\,\\mu _2(\\mathrm {d}{z}) \\\\[1mm]\\widetilde{b}_{2,1} - \\int _{U_2}z_2\\,\\mu _1(\\mathrm {d}{z}) & \\widetilde{b}_{2,2}\\end{bmatrix}.$ This form of the infinitesimal generator ${\\mathcal {A}}_{{X}}$ is readily comparable with the corresponding one in Ma [14].", "In what follows, we consider a special form of the SDE (REF ) without integrals with respect to (compensated) Poisson random measures.", "Namely, if $\\nu = 0$ , $\\mu _i = 0$ , $i \\in \\lbrace 1, \\ldots , d\\rbrace $ , then the SDE (REF ) takes the form ${X}_t&= {X}_0 + \\int _0^t b({X}_s) \\, \\mathrm {d}s+ \\int _0^t \\sigma ({X}_s) \\, \\mathrm {d}{W}_s \\\\& = {X}_0 + \\int _0^t ({\\beta }+ {B}{X}_s)\\,\\mathrm {d}s+ \\sum _{i=1}^d \\int _0^t \\sqrt{2 c_i X_{s,i}} {e}_i {e}_i^\\top \\,\\mathrm {d}{W}_s, \\qquad t \\in \\mathbb {R}_+ ,$ and consequently, $X_{t,i} = \\int _0^t \\Bigg (\\beta _i + \\sum _{j=1}^d b_{i,j} X_{s,j}\\Bigg )\\mathrm {d}t+ \\int _0^t \\sqrt{2c_i X_{s,i}}\\,\\mathrm {d}W_{s,i},\\qquad t\\in \\mathbb {R}_+, \\quad i\\in \\lbrace 1,\\ldots ,d\\rbrace .$ If ${B}$ is diagonal, then the process $({X}_t)_{t\\in \\mathbb {R}_+}$ is known to be a multi-factor Cox-Ingersoll-Ross process, see, e.g., Jagannathan et al.", "[8].", "Finally, Theorem REF is valid also if the SDE (REF ) does not contain integral with respect to a Wiener process, i.e., if ${c}= {0}$ .", "We note that in the proof of Theorem REF we applied Theorem 7.1' in Chapter II of Ikeda and Watanabe [7], which is valid in case ${c}= {0}$ as well.", "Appendix" ], [ "Extension of a probability space", "We recall the definition of extensions of probability spaces, see, e.g., Ikeda and Watanabe [7].", "Definition.", "A.1 We say that a filtered probability space $(\\widetilde{\\Omega }, \\widetilde{{\\mathcal {F}}}, (\\widetilde{{\\mathcal {F}}}_t)_{t\\in \\mathbb {R}_+}, \\widetilde{\\operatorname{\\mathbb {P}}})$ is an extension of a filtered probability space $(\\Omega , {\\mathcal {F}}, ({\\mathcal {F}}_t)_{t\\in \\mathbb {R}_+}, \\operatorname{\\mathbb {P}})$ , if there exists an $\\widetilde{{\\mathcal {F}}}/{\\mathcal {F}}$ -measurable mapping $\\pi :\\widetilde{\\Omega }\\rightarrow \\Omega $ such that $\\pi ^{-1}({\\mathcal {F}}_t) \\subset \\widetilde{{\\mathcal {F}}}_t$ for all $t \\in \\mathbb {R}_+$ , $\\operatorname{\\mathbb {P}}(A) = \\widetilde{\\operatorname{\\mathbb {P}}}(\\pi ^{-1}(A))$ for all $A \\in {\\mathcal {F}}$ , and $\\operatorname{\\mathbb {\\widetilde{E}}}(\\widetilde{X}\\,\|\\,\\widetilde{{\\mathcal {F}}}_t)(\\widetilde{\\omega }) = \\operatorname{\\mathbb {E}}(X \\,\|\\,{\\mathcal {F}}_t)(\\pi (\\widetilde{\\omega }))$ $\\widetilde{\\operatorname{\\mathbb {P}}}$ -almost surely for each essentially bounded (${\\mathcal {F}}/{\\mathcal {B}}(\\mathbb {R}^d)$ -measurable) random variable $X : \\Omega \\rightarrow \\mathbb {R}^d$ , where we set $\\widetilde{X}(\\widetilde{\\omega }) := X(\\pi (\\widetilde{\\omega }))$ , $\\widetilde{\\omega }\\in \\widetilde{\\Omega }$ .", "Remark.", "A.2 With the notations of Definition REF , if $({X}_t)_{t\\in \\mathbb {R}_+}$ is an $\\mathbb {R}^d$ -valued $({\\mathcal {F}}_t)_{t\\in \\mathbb {R}_+}$ -adapted stochastic process, then $(\\widetilde{{X}}_t)_{t\\in \\mathbb {R}_+}$ is $(\\widetilde{{\\mathcal {F}}}_t)_{t\\in \\mathbb {R}_+}$ -adapted.", "Indeed, for each $t \\in \\mathbb {R}_+$ and $B \\in {\\mathcal {B}}(\\mathbb {R}^d)$ , we have $\\widetilde{{X}}_t^{-1}(B)= \\lbrace \\widetilde{\\omega }\\in \\widetilde{\\Omega }: \\widetilde{{X}}_t(\\widetilde{\\omega }) \\in B\\rbrace = \\lbrace \\widetilde{\\omega }\\in \\widetilde{\\Omega }: {X}_t(\\pi (\\widetilde{\\omega })) \\in B\\rbrace = \\pi ^{-1}({X}_t^{-1}(B))\\in \\widetilde{{\\mathcal {F}}}_t ,$ since ${X}_t^{-1}(B) \\in {\\mathcal {F}}_t$ .", "$\\Box $ Lemma.", "A.3 Let $(\\Omega , {\\mathcal {F}}, ({\\mathcal {F}}_t)_{t\\in \\mathbb {R}_+}, \\operatorname{\\mathbb {P}})$ be a filtered probability space, and let $({W}_t)_{t\\in \\mathbb {R}_+}$ be a $d$ -dimensional $({\\mathcal {F}}_t)_{t\\in \\mathbb {R}_+}$ -Brownian motion.", "Let $(\\widetilde{\\Omega }, \\widetilde{{\\mathcal {F}}}, (\\widetilde{{\\mathcal {F}}}_t)_{t\\in \\mathbb {R}_+}, \\widetilde{\\operatorname{\\mathbb {P}}})$ be an extension of $(\\Omega , {\\mathcal {F}}, ({\\mathcal {F}}_t)_{t\\in \\mathbb {R}_+}, \\operatorname{\\mathbb {P}})$ with the mapping $\\pi :\\widetilde{\\Omega }\\rightarrow \\Omega $ .", "Let $\\widetilde{{W}}_t(\\widetilde{\\omega }) := {W}_t(\\pi (\\widetilde{\\omega }))$ for all $\\widetilde{\\omega }\\in \\widetilde{\\Omega }$ and $t \\in \\mathbb {R}_+$ .", "Then $(\\widetilde{{W}}_t)_{t\\in \\mathbb {R}_+}$ is a $d$ -dimensional $(\\widetilde{{\\mathcal {F}}}_t)_{t\\in \\mathbb {R}_+}$ -Brownian motion.", "Proof.", "According to Ikeda and Watanabe [7], we have to check that the process $(\\widetilde{{W}}_t)_{t\\in \\mathbb {R}_+}$ has continuous trajectories, it is $(\\widetilde{{\\mathcal {F}}}_t)_{t\\in \\mathbb {R}_+}$ -adapted, and satisfies $\\operatorname{\\mathbb {\\widetilde{E}}}(\\exp \\lbrace \\mathrm {i}\\langle {u}, \\widetilde{{W}}_t - \\widetilde{{W}}_s\\rangle \\rbrace \\,\|\\,\\widetilde{{\\mathcal {F}}}_s)= \\mathrm {e}^{-(t-s)\\Vert {u}\\Vert ^2/2} \\qquad \\text{$\\widetilde{\\operatorname{\\mathbb {P}}}$-almost surely}$ for every ${u}\\in \\mathbb {R}^d$ and $s, t \\in \\mathbb {R}_+$ with $s < t$ .", "Clearly, $\\mathbb {R}_+ \\ni t \\mapsto \\widetilde{{W}}_t(\\widetilde{\\omega }) = {W}_t(\\pi (\\widetilde{\\omega }))$ is continuous for all $\\widetilde{\\omega }\\in \\widetilde{\\Omega }$ .", "By Remark REF , $(\\widetilde{{W}}_t)_{t\\in \\mathbb {R}_+}$ is $(\\widetilde{{\\mathcal {F}}}_t)_{t\\in \\mathbb {R}_+}$ -adapted.", "Finally, for every ${u}\\in \\mathbb {R}^d$ and $s, t \\in \\mathbb {R}_+$ with $s < t$ , $\\operatorname{\\mathbb {\\widetilde{E}}}(\\exp \\lbrace \\mathrm {i}\\langle {u}, \\widetilde{{W}}_t - \\widetilde{{W}}_s\\rangle \\rbrace \\,\|\\,\\widetilde{{\\mathcal {F}}}_s)(\\widetilde{\\omega })= \\operatorname{\\mathbb {E}}(\\exp \\lbrace \\mathrm {i}\\langle {u}, {W}_t - {W}_s\\rangle \\rbrace \\,\|\\,{\\mathcal {F}}_s)(\\pi (\\widetilde{\\omega }))= \\mathrm {e}^{-(t-s)\\Vert {u}\\Vert ^2/2}$ $\\widetilde{\\operatorname{\\mathbb {P}}}$ -almost surely, since we have $\\xi (\\omega ) = c$ $\\operatorname{\\mathbb {P}}$ -almost surely with $\\xi := \\operatorname{\\mathbb {E}}(\\exp \\lbrace \\mathrm {i}\\langle {u}, {W}_t - {W}_s\\rangle \\rbrace \\,\|\\,{\\mathcal {F}}_s)$ and $c := \\mathrm {e}^{-(t-s)\\Vert {u}\\Vert ^2/2}$ , which implies $\\xi (\\pi (\\widetilde{\\omega })) = c$ $\\widetilde{\\operatorname{\\mathbb {P}}}$ -almost surely, because $\\widetilde{\\operatorname{\\mathbb {P}}}(\\lbrace \\widetilde{\\omega }\\in \\widetilde{\\Omega }: \\xi (\\pi (\\widetilde{\\omega })) = c \\rbrace )= \\widetilde{\\operatorname{\\mathbb {P}}}(\\pi ^{-1}(\\xi ^{-1}(\\lbrace c\\rbrace ))) = \\operatorname{\\mathbb {P}}(\\xi ^{-1}(\\lbrace c\\rbrace )) = 1$ .", "$\\Box $ Lemma.", "A.4 Let $(\\Omega , {\\mathcal {F}}, ({\\mathcal {F}}_t)_{t\\in \\mathbb {R}_+}, \\operatorname{\\mathbb {P}})$ be a filtered probability space, let $({W}_t)_{t\\in \\mathbb {R}_+}$ be a $d$ -dimensional $({\\mathcal {F}}_t)_{t\\in \\mathbb {R}_+}$ -Brownian motion, and let $p$ be a stationary $({\\mathcal {F}}_t)_{t\\in \\mathbb {R}_+}$ -Poisson point process on $V = \\mathbb {R}_+^d \\times (\\mathbb {R}_+^d \\times \\mathbb {R}_+)^d$ with characteristic measure $m$ , where $m$ is given in (REF ).", "Let ${\\mathcal {G}}_t := \\bigcap _{\\varepsilon >0} \\sigma \\left({\\mathcal {F}}_{t+\\varepsilon } \\cup {\\mathcal {N}}\\right) ,\\qquad t \\in \\mathbb {R}_+ ,$ where ${\\mathcal {N}}$ denotes the collection of null sets under the probability measure $\\operatorname{\\mathbb {P}}$ .", "Then $({W}_t)_{t\\in \\mathbb {R}_+}$ is a $d$ -dimensional $({\\mathcal {G}}_t)_{t\\in \\mathbb {R}_+}$ -Brownian motion, and $p$ is a stationary $({\\mathcal {G}}_t)_{t\\in \\mathbb {R}_+}$ -Poisson point process on $V$ with characteristic measure $m$ .", "Proof.", "The proof is essentially the same as the proof of Lemma A.5 in Barczy et al. [1].", "$\\Box $" ], [ "Acknowledgements", "We would like to thank the referee for his/her comments that helped us to improve the presentation of the Introduction." ] ]	1403.0245
[ [ "Lighting up topological insulators: large surface photocurrents from\n magnetic superlattices" ], [ "Abstract The gapless surface states of topological insulators (TI) can potentially be used to detect and harvest low-frequency infrared light.", "Nonetheless, it was shown that significant surface photocurrents due to light with frequency below the bulk gap are rather hard to produce.", "Here we demonstrate that a periodic magnetic pattern added to the surface dramatically enhances surface photocurrents in TI's.", "Moreover, the sensitivity of this set-up to the wavelength of the incident light can be optimized by tuning the geometry of the magnetic pattern.", "The ability to produce substantial photocurrents on TI surfaces from mid-range and far-infrared light could be used in photovoltaic applications, as well as for detection of micrometer wavelength radiation.", "For light of wavelength greater than 15$\\mu$m we estimate that at room temperature, a detector based on the effect we describe can have a specific detectivity as high as 10$^7$ cm$\\sqrt{\\text{Hz}}$/W (i.e.", "10$^9$ Jones).", "The device can therefore operate at much larger wavelengths than existing infrared detectors, while maintaining a comparable figure of merit.er wavelength radiation." ], [ "Introduction", "Light-matter interactions are central to modern science and technology.", "It is the principle at the heart of many solid-state material probes, and at the same time, it is an important ingredient in our energy economy, particularly through photovoltaic harvesting of solar energy.", "A challenging problem of solar energy is how to harness the infrared (IR) part of the spectrum.", "This could apply to the solar radiation, as well as to Earth's radiation, which is almost exclusively in the infrared, and continuously has the same energy flux as the sun [1].", "Similarly, electric mid and far infrared detection is essentially limited to a single type of material: HgCdTe alloys.", "Additional platforms are likely to be competitive in certain temperature and frequency regimes.", "Efforts to extend the spectrum accessible in photovoltaics concentrated on new low band-gap materials; organics [2], [3], [4], [5], as well as carbon nanotubes [6], [7], [8] were shown capable of IR harvesting, albeit with a small efficiency.", "Another approach utilizes plasmonics as an intermediate step between IR and currents in a semiconductor [9], [10], [11], [12].", "When mentioning new materials for IR harversting, topological insulators [13], [14], [15], [16], [17] immediately come to mind.", "On the one hand, they have a unique response to electromagnetic fields [18], [19], [20].", "More importantly, their mid-gap surface states exhibiting spin-momentum locking raised hopes that surface photocurrents could easily be produce by irradiation with circularly-polarized light.", "These hopes have gone unfulfilled.", "Even when including a series of modifications to the band structure likely to appear in real materials, such as warping, band curvature, or a uniform magnetic field, the photocurrents produced in response to sub-bandgap light were shown to be remarkably minute, even when a high-intensity laser is considered [21], [22].", "The only scheme for producing a photo-voltage so far relied on the unique thermoelectric effects associated with a Dirac cone dispersion [23].", "In this manuscript we describe how to turn a topological insulator surface with a simple Dirac dispersion into a photocurrent rectifier.", "We show that by adding a magnetic coating with a spatially periodic magnetic texture, the TI produces a significant surface photocurrent in response to circularly polarized light in the IR regime.", "This effect should, in principle, allow making diode-free IR sensitive photocells from topological insulator films.", "We discuss application of the effect to room temperature infra-red detection, and show that it can lead to a detector operating at much larger wavelengths then those available with existing technologies.", "Beyond such applications, the effect can be used to investigate the unique properties of TI surfaces using non-ionizing light (as in [24]).", "The paper is organized as follows.", "In Sec.", ", we give a description of the device and summarize our main results.", "In Sec.", "we present the model describing the magnetically patterned TI surface.", "The symmetries of the model are discussed in Sec. .", "In Sec.", "we derive the equations describing the photocurrent response of the device.", "We consider the implications of the symmetries on the photocurrent response (Sec.", "REF ), and find the conditions under which a large photocurrent response is obtained.", "A perturbative calculation of the photocurrent response is given in Sec.", "REF .", "Our main results for the frequency dependent photocurrent response of the device are given in Sec. .", "Several applications of the device, and in particular, room temperature infra-red detection are discussed in Sec. .", "We close with concluding remarks in Sec.", "." ], [ "The proposed device and summary of the main results", "The device we propose and analyze in this paper consists of a bulk three dimensional topological insulator, whose surface is coated with stripes of magnetic material, see Fig.", "REF .", "We consider magnetic stripes which are evenly spaced.", "The stripes' spacing defines a wave vector ${\\bf q}$ in the plane of the surface and normal to the orientation of the stripes.", "Via their magnetic coupling to the electrons in the surface state of the TI, the magnetic stripes break symmetries which suppress the photocurrents in their absence.", "Thereby, the magnetic stripes dramatically enhance the photocurrent response of the TI's surface.", "The magnetic stripes are taken to be magnetically ordered in the same direction.", "As we explain in Sec.", ", the direction of the stripes' magnetization needs to have non-zero components both normal to the surface as well as along the vector ${\\bf q}$ .", "The photocurrents flow parallel to the direction of the stripes (perpendicular to ${\\bf q}$ ), as shown in Sec. .", "The photocurrent response of the device can be described by a dimensionless, frequency dependent response function $\\eta (\\omega )$ .", "In Sec.", "we demonstrate a key feature of $\\eta (\\omega )$ : it exhibits a strong maximum at frequency $\\omega \\approx 1.7v_F \|{\\bf q}\|$ , where $v_F$ is the velocity associated the Dirac cone.", "This result has significant implications in future applications of the proposed device: the frequency corresponding to the peak sensitivity of the device can be tuned by appropriately choosing the spacing of the magnetic stripes.", "In Sec.", "REF , we analyze the performance of this set-up at finite temperature and with the chemical potential tuned away from the Dirac point.", "This analysis gives an “operational” region for the device: we show that the performance of the device is not significantly reduced for temperatures up to $\\hbar v_F \|{\\bf q}\|$ , which could translate to $300K$ in practical realizations.", "Similarity, we show that deviations of the chemical potential from the Dirac point do not significantly hinder the the performance, as long as they remain below $\\hbar v_F \|{\\bf q}\|$ .", "Quantitative estimates for the photocurrent response in several applications are given in Sec. .", "We estimate that the two dimensional photocurrent density resulting from illumination with sunlight could reach $10^{-8}\\frac{A}{m}$ .", "Illumination with a conventional laser beam can yield currents of the order of $10^{-4}\\frac{A}{m}$ .", "A particularly appealing application of the device is room temperature detection of infra-red radiation.", "We explore the potential of this system to detect black-body radiation emitted at a variety of different source temperatures.", "We conclude that the device may be able to detect black-body radiation of objects at room temperature while itself being at a comparable temperature.", "Finally, we explore several theoretical figures of merit for the device as a room temperature IR detector.", "In particular we calculate the device's external quantum efficiency and its specific detectivity, which gives its normalized signal to noise ratio [25] .", "Near room temperature and with peak sensitivity tuned to wavelengths near $15\\mu $ m we estimate a quantum efficiency of 0.01$\\%$ and a specific detectivity $\\sim 10^{7}$ cm$\\sqrt{\\text{Hz}}$ /W, before any device optimization takes place.", "Such a detectivity compares well with the detectivity of current room temperature photo-detectors [26], which can usually only detect up to $10\\mu \\mathrm {m}$ [27], [28], [29], [26], [30], [31].", "Importantly, the proposed device has the potential to be functional for wavelengths greater than $15\\mu \\mathrm {m}$ .", "Our findings therefore support the idea that this set-up may be promising for room temperature detection of long wavelength infrared radiation.", "Figure: Proposed scheme for achieving a photovoltaic effect on a topological-insulator surface, coated by a magnetic grating.", "When the magnetization (depicted as yellow arrows)breaks both rotation and reflection symmetries, circularly polarized light induces a photocurrent (green) in the direction parallel to the stripes." ], [ "Minimal model for surface photocurrent rectification", "Our photocurrent rectification scheme emerges from the minimal model of a surface of a 3D topological insulator (TI).", "With the surface lying in the $xy$ plane, the Hamiltonian describing the surface electrons is $H_{0} & = & v_{F}\\left(p_x\\sigma ^y-p_y\\sigma ^x\\right),$ where $\\sigma ^{x},\\sigma ^{y}$ are Pauli matrices, and $\\mathbf {p}=(p_x,p_y)=\\left(\\frac{\\hbar }{i}\\frac{\\partial }{\\partial x},\\,\\frac{\\hbar }{i}\\frac{\\partial }{\\partial y}\\right)$ .", "This model is clearly time-reversal and rotationally invariant, ${\\cal T}H_{0}{\\cal T}^{-1}=U_{\\phi }H_{0}U_{\\phi }^{\\dagger }=H_{0}$ , with the symmetry operators ${\\cal T}=i\\sigma _{y}K,\\hspace{28.45274pt} U_{\\phi }=e^{i\\phi \\left(\\sigma ^z/2+L_z/\\hbar \\right)}.", "$ Here, $K$ denotes complex conjugation, $L_z=x p_y-y p_x$ is the orbital angular momentum normal to the surface, and $\\phi $ the angle of rotation.", "These two symmetries immediately imply no current response to incident light at normal incidence.", "Time-reversal invariance requires that the incident beam is circularly polarized to see any response.", "Since circularly polarized light, however, has no preferred direction on the surface, the rotational symmetry rules out any net photocurrent from forming.", "In materials such as $\\mathrm {Bi_{3}Se}_{2}$ , the lattice structure reduces the full $SO(2)$ rotational symmetry to a $C_{3}$ symmetry, with $\\phi =2\\pi /3$ in Eq.", "(REF ).", "This allows $H_0$ to have a trigonal warping term [32].", "However even with the reduced symmetry, no photocurrents are possible [21], [22].", "Next, we consider a magnetic grating structure deposited on the surface, see Fig.", "REF .", "Consider strips of a ferromagnetic material set parallel to the $y$ axis, and placed periodically with a wave number $\\mathbf {q}=(q,0)$ .", "We model the proximity-induced ferromagnetic interaction on the surface electrons by $V= \\mathbf {u}\\cdot {\\mbox{$\\sigma $}}\\cos (q x).", "$ Once the magnetic structure is introduced, it is convenient to enumerate the eigenstates of the full Hamiltonian, $H=H_0+V$ within the reduced Brillouin zone (BZ) in terms of the quasi-momentum $k_{x}\\in [-\\frac{q}{2},\\frac{q}{2}]$ in x-direction, using Greek indices to denote the bands.", "Thus denote the eigenstates as $\\left\| k_x,k_y;\\alpha \\right>$ .", "We use the convention that conduction bands are enumerated by $\\alpha >0$ and valence bands by $\\alpha <0$ , as illustrated in Fig.", "REF" ], [ "Symmetry considerations of the modified surface", "The addition of the magnetic strips on the surface alters its symmetries.", "Time-reversal symmetry remains, as long as we consider a modified operator which concatenates time reversal with a spatial translation: $\\tilde{\\cal T}={\\cal T}M$ with $MxM^{\\dagger }=x+\\pi /q$ .", "The eigenstates and energies transform as $\\left\| -k_x,-k_y;\\alpha \\right>=\\tilde{\\cal T}\\left\| k_x,k_y;\\alpha \\right>,E_{-k_x,-k_y;\\alpha }=E_{k_x,k_y;\\alpha }$ Particle-hole symmetry is also present.", "First define $\\Pi _a$ as the spatial reflection operator about the $a=x,y$ directions, e.g., $\\Pi _x x \\Pi _x=-x$ .", "Now, ${\\cal C}=\\Pi _x\\Pi _y {\\cal T}$ implements: $\\left\| k_x,k_y;-\\alpha \\right>={\\cal C}\\left\| k_x,k_y;\\alpha \\right>,E_{k_x,k_y;\\alpha }=-E_{k_x,k_y;-\\alpha }.$ Additional symmetries appear restricted due to the arbitrary form of $V$ .", "Nonetheless, a gauge transformation allows us to cancel an arbitrary $u_y$ component of $V$ , and allows additional mirror symmetries.", "Define the gauge transformation $G=\\exp {\\left(i\\frac{u_y}{ \\hbar v_F q}\\sin qx\\right)} .$ It is easy to verify that $\\tilde{H}=G H G^{-1}&=&H- u_y\\sigma ^y\\cos qx\\nonumber \\\\&=&H_0+(u_x\\sigma ^x+u_z\\sigma ^z)\\cos qx.$ With $u_y$ eliminated, we can construct the mirror transformation ${\\cal P}_x=\\Pi _y K.$ The only term that can possibly be affected by this compounded transformation is actually invariant, $\\begin{array}{c}\\Pi _y K ł[\\frac{\\hbar v_F}{i}ł(-\\sigma ^x\\partial _y)]K^{-1}\\Pi _y\\\\=\\Pi _y ł[\\frac{\\hbar v_F}{i}ł(\\sigma ^x\\partial _y)]\\Pi _y=\\frac{\\hbar v_F}{i}ł(-\\sigma ^x\\partial _y),\\end{array}$ so that ${\\cal P}_x\\tilde{H}{\\cal P}_x^{-1}=\\tilde{H}$ .", "Since complex conjugation imposes $(k_x,k_y)\\rightarrow (-k_x,-k_y)$ , and $\\Pi _y$ reverses $-k_y$ back to $k_y$ , we have: $\\left\| -k_x,k_y;\\alpha \\right>={\\cal P}_x\\left\| k_x,k_y;\\alpha \\right>, E_{-k_x,k_y;\\alpha }=E_{k_x,k_y;\\alpha }.$ By compounding ${\\cal P}_x$ with time reversal, $\\tilde{\\cal T}$ , we also obtain a reflection about the x-axis: $\\tilde{\\cal T}{\\cal P}_x:(k_x,k_y)\\rightarrow (k_x,-k_y)$ .", "Below we will first discuss the equations describing the photocurrent response of the device, and then consider the consequences of the symmetries on the resulting photocurrent." ], [ "Calculation of the photocurrent response", "Within Fermi's golden rule, we expect that the photocurrent response to a particular frequency will be quadratic in the photon field.", "We restrict ourselves to normally incident photons, at frequencies which allow us to approximate the vector potential as spatially uniform, $\\mathbf {A}(\\mathbf {x},t)={\\rm Re}\\mathbf {A}_{0}(\\omega )e^{i\\omega t}$ .", "Then, the $k,m,n=x,y$ component of the photocurrent is quite generally given by: $j_{k}(\\omega )=\\frac{e\\tau }{4\\hbar \\omega ^{2}} E_{m}(\\omega ){\\cal Q}_{kmn}(\\mathbf {\\omega })E_{n}^{}(\\omega ).", "$ Here, repeated indices are summed, ${\\bf E}(\\omega )=-i\\omega {\\bf A_0}$ .", "Also, in Eq.", "(REF ) we assume that the current decays on a time scale $\\tau $ .", "Quite remarkably, in the presence of a periodic structure of magnetic strips lying along the y-axis, we will find that there is only one independent element of ${\\cal Q}_{kmn}(\\mathbf {\\omega })$ which is nonzero: ${\\cal Q}(\\omega )_{y,x,y}={\\cal Q}(\\omega )_{y,y,x}^{}=-{\\cal Q}(\\omega )_{y,y,x}.$ To calculate $ {\\cal Q}_{kmn}(\\mathbf {\\omega })$ , we first write the surface photon-electron interaction, which we assume follows the minimal coupling prescription: $\\hat{H}_{int}=e\\frac{\\partial H_{0}}{\\partial \\mathbf {p}}\\cdot \\mathbf {A}(\\mathrm {\\mathbf {x}},t).$ The presence of the photon field can either excite electrons to a higher sub band or allow them to relax to a lower sub band through emission.", "Taking these possibilities into account we have the following result for ${\\cal Q}_{kmn}(\\omega )$ ${\\cal Q}_{kmn}(\\omega )=\\int \\frac{d^2k}{(2\\pi )^2}\\sum _{\\alpha ,\\beta }{\\cal Q}_{kmn}^{\\alpha \\beta }(\\mathbf {k},\\omega , T, \\mu ),$ where ${\\cal Q}_{kmn}^{\\alpha \\beta }(\\mathbf {k},\\omega , T,\\mu )$ describes the excitation/relaxation of electrons at momentum $\\mathbf {k}$ and temperature $T$ .", "An application of Fermi's golden rule yields: $\\begin{array}{c}{\\cal Q}_{kmn}^{\\alpha \\beta }(\\mathbf {k,\\omega }, T,\\mu ) = {\\hat{\\mathbf {}{x}_{k}}\\cdot \\left(\\mathbf {v}_{\\mathbf {k}}^{(\\alpha )}-\\mathbf {v_{k}^{(\\beta )}}\\right)M_{mn}^{\\alpha \\beta }(\\mathbf {k})\\\\\\times 2\\pi \\mathrm {\\delta (E_{\\mathbf {k}}^{(\\alpha )}-E_{\\mathbf {k}}^{(\\beta )}-\\omega )}(n^0_{\\mathbf {k},\\beta }-n^0_{\\mathbf {k},\\alpha }),}\\end{array}where n^0_{\\mathbf {k},\\beta } is a Fermi function at a temperature T and chemical potential \\mu and the velocities in the band \\alpha are given by \\mathbf {v}_{\\mathbf {k}}^{(\\alpha )}=\\left< \\mathbf {k};\\alpha \\right\|\\frac{\\partial H_{0}}{\\partial \\mathbf {p}}\\left\| \\mathbf {k};\\alpha \\right>, and the matrix elements are given by\\begin{equation}M_{mn}^{\\alpha \\beta }(\\mathbf {k})=\\left< \\mathbf {k},\\alpha \\right\|\\Gamma _{m}\\left\| \\mathbf {k},\\beta \\right>\\left< \\mathbf {k},\\beta \\right\|\\Gamma _{n}^{\\dagger }\\left\| \\mathbf {k},\\alpha \\right>,\\end{equation}with \\Gamma _{m}=e\\frac{\\partial H_{0}}{\\partial \\mathbf {p}}\\cdot \\hat{\\mathbf {x}}_{m}.From this definition it is clear that M_{mn}^{\\alpha \\beta }(\\mathbf {k})=\\left(M_{nm}^{\\alpha \\beta }(\\mathbf {k})\\right)^, i.e., it is hermitian.", "At zero temperature and with the chemical potential tuned to the Dirac point the valance sub bands are entirely full and the conduction sub bands completely empty.", "In this case only electron excitation is possible and we may excite electrons from any sub band of the valence band to any sub band of the conduction band, and at any momentum.", "Therefore:\\begin{equation}{\\cal Q}_{kmn}(\\omega )=\\int \\frac{d^2k}{(2\\pi )^2}\\sum _{\\alpha >0,\\beta <0}{\\cal Q}_{kmn}^{\\alpha \\beta }(\\mathbf {k},\\omega ),\\end{equation}where {\\cal Q}_{kmn}^{\\alpha \\beta }(\\mathbf {k},\\omega )={\\cal Q}_{kmn}^{\\alpha \\beta }(\\mathbf {k,\\omega }, T\\rightarrow 0, \\mu =0) accounts for excitations from the valence band \\beta <0 to the conduction band \\alpha >0 at momentum \\mathbf {k}.", "It is this limiting case that we will develop first, and then move on to discuss how temperature and chemical potential effect these results.", "Setting T\\rightarrow 0 and \\mu =0 in Eq.\\ (\\ref {qdef2}) now gives\\begin{equation}\\begin{array}{c}{\\cal Q}_{kmn}^{\\alpha \\beta }(\\mathbf {k,\\omega }) = {\\hat{\\mathbf {}{x}_{k}}\\cdot \\left(\\mathbf {v}_{\\mathbf {k}}^{(\\alpha )}-\\mathbf {v_{k}^{(\\beta )}}\\right)M_{mn}^{\\alpha \\beta }(\\mathbf {k})\\\\\\times 2\\pi \\mathrm {\\delta (E_{\\mathbf {k}}^{(\\alpha )}-E_{\\mathbf {k}}^{(\\beta )}-\\omega )},}\\end{array}\\end{equation}$" ], [ "Implications of the symmetries", "The calculation of the photocurrent response in the presence of the magnetic texture can now follow.", "Significant simplifications can be made by taking into account the symmetries discussed in Sec. .", "We first define ${\\cal \\widetilde{Q}}_{kmn}^{\\alpha \\beta }(\\mathbf {k})=\\sum _{\\sigma ,\\sigma ^{\\prime }=\\pm 1}{\\cal Q}_{kmn}^{\\alpha \\beta }(\\sigma k_{x},\\sigma ^{\\prime }k_{y}),$ which sums the contributions of the four mirror-related momenta, $\\left(\\pm k_{x,}\\pm k_{y}\\right)$ , and is defined for $k_{x},k_{y}>0$ .", "This definition takes into account all symmetry-related current cancellations.", "Assuming that $u_y$ has been gauged away, we use Eqs.", "(REF ) and (REF ) to connect the contributions arising from the four momenta $(\\pm k_x,\\,\\pm k_y)$ .", "Due to these symmetries, along with the particle-hole transformation, Eq.", "(REF ), we have: $\\begin{array}{c}v_{x}^{\\alpha }(\\sigma k_{x},\\sigma ^{\\prime }k_{y})=\\sigma v_{x}^{\\alpha }(k_{x},k_{y}),\\\\v_{y}^{\\alpha }(\\sigma k_{x},\\sigma ^{\\prime } k_{y})=\\sigma ^{\\prime } v_{y}^{\\alpha }(k_{x},k_{y}),\\\\\\mathbf {v}^{\\alpha }(\\mathbf {k})=-\\mathbf {v}^{-\\alpha }(\\mathbf {k})\\end{array}$ for $\\sigma ,\\sigma ^{\\prime }=\\pm 1$ .", "Figure: Effective bandstructure of the proposed heterostructure, cut along the line k y =0k_{y}=0 (units such that \|𝐪\|=1\|\\mathbf {q}\|=1).", "Band indices are shown on the left.", "Transitions yielding a negative (positive) contribution to 𝒬 yxy (ω){\\cal Q}_{yxy}(\\omega ) are shown in red, blue (purple, green).", "The corresponding momentum dependent 𝒬 yxy α,β (ω){\\cal Q}^{\\alpha ,\\beta }_{yxy}(\\omega ) are given in Fig.", "of the appendix.The same symmetries, applied to the matrix elements yield the relations $\\tilde{\\cal T}:\\,\\,\\,M_{mn}^{\\alpha \\beta }(\\mathbf {-k}) & = & M_{nm}^{\\alpha \\beta }(\\mathbf {k})\\nonumber \\\\{\\cal P}_x:\\,\\,\\,M_{xy}^{\\alpha \\beta }(-k_{x},k_{y}) & = & -M_{yx}^{\\alpha \\beta }(\\mathbf {k})\\nonumber \\\\{\\cal P}_x:\\,\\,\\,M_{nn}^{\\alpha \\beta }(-k_{x},k_{y}) & = & M_{nn}^{\\alpha \\beta }(\\mathbf {k}).$ The diagonal elements, $M_{nn}^{\\alpha \\beta }(\\mathbf {k})$ , are the same at all four points $(\\pm k_x,\\,\\pm k_y)$ .", "This makes the contribution of these points to a current in any direction cancel identically, since the velocities obey the mirror symmetries in Eq.", "(REF ).", "From Eqs.", "(REF ) and (REF ), we find that the only nonzero elements of the tensor ${\\cal \\widetilde{Q}}_{kmn}^{\\alpha \\beta }(\\mathbf {k})$ are ${\\cal \\widetilde{Q}}_{yxy}^{\\alpha \\beta }(\\mathbf {k})&=&8\\pi i \\left[(v_y^{\|\\alpha \|}(\\mathbf {k})+ v_y^{\|\\beta \|}(\\mathbf {k})\\right] \\mathrm {Im}\\left[M_{xy}^{\\alpha \\beta }(\\mathbf {k})\\right]\\\\ \\nonumber &\\times &\\mathrm {\\delta (E_{\\mathbf {k}}^{(\\alpha )}-E_{\\mathbf {k}}^{(\\beta )}-\\omega )}$ and ${\\cal \\widetilde{Q}}_{yyx}^{\\alpha \\beta }(\\mathbf {k})=-{\\cal \\widetilde{Q}}_{yxy}^{\\alpha \\beta }(\\mathbf {k})$ .", "These conclusions confirm our claim regarding the photo-response tensor, ${\\cal Q}_{kmn}(\\omega )$ defined in Eq.", "(): It has only one independent nonzero contribution, ${\\cal Q}_{yxy}(\\omega )=-{\\cal Q}_{yyx}(\\omega )$ , which is imaginary.", "This implies that the current in the $x$ direction vanishes, i.e., the photocurrent is always parallel to the magnetic pattern.", "Furthermore, this current is only induced by the circular component of the incident light.", "This result also lets us determine what magnetic patterning vector $\\mathbf {u}$ is necessary for a finite response.", "As it turns out, having either $u_x=0$ or $u_z=0$ leads to $\\mathrm {Im}M_{xy}^{\\alpha \\beta }=0$ , and to a vanishing response.", "To see this, consider the composite transformation $U=\\Pi _x\\Pi _y\\sigma ^z \\tilde{\\cal T}$ .", "The first part of the transformation, $\\Pi _x\\Pi _y\\sigma ^z$ , implements a $\\pi $ rotation on the bare model, $H_0$ , and leaves it invariant.", "If $u_x=u_y=0$ , then $H^{(z)}=H_0+u_z\\sigma ^z\\cos (qx)$ is also invariant this transformation.", "$\\tilde{\\cal T}$ then leaves $H^{(z)}$ invariant, and reverses momentum directions.", "Together, they make an anti-unitary transformation which leaves the momentum $\\mathbf {p}$ invariant.", "Its effect on the transition matrix is $M^{\\alpha \\beta }_{mn}(\\mathbf {k})=M^{\\alpha \\beta }_{nm}(\\mathbf {k})=M^{\\alpha \\beta }_{mn}(\\mathbf {k})^$ .", "The same relation is obtained also for the case $u_z=0$ with a finite $u_x$ , with $UM$ used instead of $U$ (with M the half-period translation operator).", "Thus both $u_z$ and $u_x$ must be finite for a finite photo-response." ], [ "Perturbative photocurrent calculation", "The summed momentum-specific photocurrent contributions, ${\\cal \\widetilde{Q}}_{kmn}^{\\alpha \\beta }(\\mathbf {k})$ , can be found analytically to lowest order in the strength of the magnetic texture.", "To do so, we expand the eigenstates of the Hamiltonian that appear in the definition of $M_{mn}^{\\alpha \\beta }(\\mathbf {k})$ in Eq.", "(), and also separate the current inducing processes ${\\cal \\widetilde{Q}}_{kmn}^{\\alpha \\beta }(\\mathbf {k})$ according to channels of interband scattering.", "In terms of momenta in the extended BZ, the possible scattering processes to order $V^2$ are ${\\bf k}\\rightarrow {\\bf k}+q$ , ${\\bf k}\\rightarrow {\\bf k}-q$ , and ${\\bf k}\\rightarrow {\\bf k}$ .", "The resulting photocurrent can be written as ${\\cal \\widetilde{Q}}^{\\rm ext}_{yxy}(\\mathbf {k}) = 2\\pi u_x u_z(e v_F)^2 \\sum _{\\lambda =0,+,-} F_\\lambda (\\mathbf {k})\\delta _\\lambda (\\omega ,\\mathbf {k}) ,$ where the functions $F_\\lambda $ , $\\lambda =+,-,0$ account for the above scattering processes and are given by $F_0(\\mathbf {k}) & = & v_{F}\\frac{-512ik_{y}^{2}k_{x}^{2}}{\|\\mathbf {k}\|^{2}(-4k_{x}^{2}+\\mathbf {q}^{2})^{2}}\\nonumber \\\\F_{\\pm }(\\mathbf {k}) & = & v_{F}\\frac{8ik_{y}^{2}q^{2}}{\|\\mathbf {k}\|^2\\,\|\\mathbf {k\\pm q}\|^2(\|\\mathbf {k \\pm q}\|-\|\\mathbf {k}\|){}^{2}}.$ The delta functions in Eq.", "(REF ) were abbreviated to $\\delta _\\lambda (\\omega ,\\mathbf {k})=\\delta (E_{\\mathbf {k}+\\lambda {\\bf q}}+E_{\\mathbf {k}}-\\omega )$ .", "The momentum integrated response tensor becomes ${\\cal Q}_{kmn}(\\omega )=\\int \\frac{d^2k}{2\\pi } {\\cal \\widetilde{Q}}^{\\rm ext}_{kmn}(\\mathbf {k})$ , where the integral is taken over the $k_x,k_y>0$ quadrant of the extended BZ The mapping between the band index $\\alpha $ and momentum ${\\bf k}$ in the reduced BZ, and the extended zone momenta is given by ${\\bf k};\\alpha \\rightarrow {\\bf k}-(-1)^{\\alpha }\\lfloor \\frac{\\alpha }{2}\\rfloor {\\bf q}$ .", "Note that the divergences cancel between $F_0({\\bf k})$ and $F_\\pm ({\\bf k})$ (see the appendix for more information)." ], [ "Results: Photocurrent response of the proposed device", "Our results are best expressed in terms of the intensity, $I$ , of the light field.", "For a coherent monochromatic circularly polarized wave with electric-field amplitude $E_0$ , we have $I=\\epsilon _0 c E_0^2$ .", "This yields the current response: $j_y(\\omega )=\\frac{e^3 v_F^2 q\\tau }{\\epsilon _0 c \\hbar ^2}\\frac{I}{\\omega ^2}\\eta (\\omega )$ in terms of the dimensionless frequency-dependent response, $\\eta (\\omega )$ , defined by ${\\cal Q}_{yxy}(\\omega )=2\\frac{e^2 v_F^2 q}{\\hbar }\\eta (\\omega )$ .", "For a continuous spectrum with intensity per unit angular frequency, containing both circular polarizations, we write $I(\\omega )d\\omega = 2\\epsilon _0 c \|{\\bf E}(\\omega )\|^2$ .", "The total current response is then: $j_y=\\frac{e^3 v_F^2 q\\tau }{2\\epsilon _0 c \\hbar ^2}\\int \\limits _0^{\\Omega }\\frac{I(\\omega )}{\\omega ^2}\\eta (\\omega )d\\omega ,$ where $\\Omega $ is the high-frequency cutoff.", "Figure: The dimensionless response function η(ω)\\eta (\\omega ), for u x /ℏv F q=u z /ℏv F q=0.1,0.2,0.3u_{x}/\\hbar v_{F}q=u_{z}/\\hbar v_{F}q=0.1,0.2,0.3 (purple, blue and red, respectively).", "The horizontal axis gives the frequency ω\\omega in units of v F qv_{F}q.", "The inset shows the saturation value η\\eta of the response function at high frequencies ω≫v F q\\omega \\gg v_F q, as a function of u/ℏv F qu/\\hbar v_F q, with u=u x =u z u=u_x=u_z.Figure: Device performance for various temperatures and chemical potential strengths.", "Left: η\\eta plotted over a space of TT and ω\\omega values with μ=0\\mu =0, middle: η\\eta plotted for a small but finite temperature of T=10 -3 ℏv F q/k B T=10^{-3}\\hbar v_F q/k_B over a space of ω\\omega and μ\\mu values, and right: the peak value of η\\eta for various different chemical potential strengths and at different temperatures.", "In all plots we have fixed the strength of the magnetic coupling such that u x =u z =0.3ℏv F qu_x=u_z=0.3\\hbar v_Fq.", "Notice that η\\eta remains large up until k B T∼ℏv F qk_BT\\sim \\hbar v_F q indicating that the proposed device may still be functional at large temperatures.", "Similarly, η\\eta is nonzero for a wide range of chemical potentials showcasing the freedom this device has in where its Fermi level is set.Fig.", "REF displays our numerical results for the dimensionless response $\\eta (\\omega )$ for three magnetic patterning strengths.", "We make three observations: (1) Most of the contribution to the current density arises from frequencies $\\omega >v_{F}q$ .", "(2) For $\\omega \\gg v_F q$ , the dimensionless response $\\eta (\\omega )$ approaches a constant.", "(3) $\\eta (\\omega )$ changes sign at $\\Omega ^{}\\sim v_{F}q$ .", "The latter observation can also be deduced from Eq.", "(REF ), as $F_0(\\mathbf {k})$ differs in sign from the two other contributions, and dominates below $\\Omega ^{}$ .", "Further intuition for the origin of the photocurrent distributions can be gained by studying the momentum-specific response $\\tilde{{\\cal Q}}_{yxy}^{\\alpha \\beta }(\\mathbf {k},\\omega )$ .", "These are plotted in Fig.", "REF of the appendix, which demonstrates that the momenta contributing to the photocurrent are uniformly distributed in the reduced BZ.", "Furthermore, in agreement with the perturbative results, the sign change of $\\eta (\\omega )$ is shown to arise due to processes involving scattering by momentum $\\pm {\\bf q}$ (indicated in red and blue in Fig.", "REF ) which dominate for $\\omega >\\Omega ^$ ; they contribute with opposite sign to momentum conserving processes (green and purple in Fig.", "REF ), which dominate at low frequencies." ], [ " Temperature and chemical potential dependence of the photocurrent response", "Next let us illuminate the potential operating regime of such a device.", "More specifically, let us address the question of functionality of the above device over a range of temperatures and chemical potential.", "Towards this end, we have evaluated the dimensionless response, $\\eta (\\omega )$ , at several operating temperatures and with the chemical potential tuned away from the Dirac point.", "Our results of this calculation are summarized in Fig.", "REF .", "By studying Fig.", "REF we learn several important factors for the operation of the device.", "First, by examining Fig.", "REF a we see that for temperatures $k_BT\\le \\hbar v_F q$ the features of $\\eta (\\omega )$ are not significantly changed; rather, increasing the temperature in this range seems only to moderately suppress $\\eta (\\omega )$ .", "Consequently, the peak of $\\eta (\\omega )$ is suppressed by about $60\\%$ at $k_BT=\\hbar v_Fq$ .", "For $k_BT>\\hbar v_F q$ the peak becomes very flattened out and is lost.", "Second, in Fig.", "REF we study the effect of the chemical potential.", "We see that for $\\mu <\\hbar v_Fq$ ($\\mu =0$ signifies the Dirac point), tuning the chemical potential away from the Dirac point leads to an overall moderate suppression of $\\eta (\\omega )$ but has little effect on its functional form.", "Furthermore, we see that the value of $\\eta (\\omega )$ becomes almost entirely “turned off\" after a critical value of $\\mu \\simeq \\pm \\omega /2$ .", "A heuristic understanding for this behaviour is as follows.", "At zero temperature all states with energy above $\\mu $ are empty, and those with energy below $\\mu $ are occupied.", "Moreover, as discussed previously, the system is particle-hole symmetric.", "A major contribution to $\\eta $ comes from electrons at energy $-\\omega /2$ being excited into states at energy $\\omega /2$ .", "As $\\mu $ is tuned away from zero this is changed very little until it reaches $\\omega /2$ (or $-\\omega /2$ ).", "At this point the transition from $-\\omega /2$ to $\\omega /2$ is no longer possible because the state at energy $\\omega /2$ (-$\\omega /2$ ) is full (empty).", "Thus the value of $\\eta $ is largely suppressed after this point.", "Finally, Fig.", "REF c shows the behaviour of the peak $\\eta $ value as a function of $\\mu $ and $T$ .", "This plot summarizes our main conclusion of this part of the paper: the zero temperature, zero chemical potential results are not significantly changed provided $k_BT$ and $\\mu $ are within $\\hbar v_Fq$ of zero.", "For $T$ and $\\mu $ outside of this region the current is highly suppressed.", "This gives the appropriate operational region for such a device.", "In a typical TI one can expect $v_F\\sim 10^5$ m/s.", "With this and a reasonable grating pitch of $q\\sim 10^8$ m$^{-1}$ the operational temperature scale is set at approximately $380K$ .", "In the above discussion we treated the effects of finite temperature by considering its effect on the electronic occupation of the surface states.", "Another effect will come from phonon scattering at finite temperature.", "Scattering from phonons will lift the momentum conservation conditions assumed above.", "The strength of electron-phonon interactions on the surface of a TI is presently an active area of research [33], [34], [35], [36], [37], [38], [39], [40].", "The role that phonons will play in this device is also an open issue.", "Intuitively one may expect that the phonons will scatter photoexcited electrons thereby reducing the photocurrent.", "It may, however, be possible to see an analogue of the phonon assisted transitions recently predicted in graphene [41].", "We leave a rigorous consideration of these two situations to future work.", "In our present treatment all scattering effects are incorporated into the relaxation time $\\tau $ .", "At low temperatures phonon modes are frozen out and scattering should be dominated by disorder Due to the Dirac dispersion of TI surface electrons, Coulomb interactions are unable to relax a current when the chemical potential is near the neutrality point, see ref.", "[50].", "As working estimates we take $\\tau \\sim 1$ ps at low temperatures and $\\tau \\sim 0.1ps$ near room temperature [33], [34]." ], [ " Applications", "Here we outline several appealing practical aspects of this device.", "We begin by calculating some representative photocurrents for the illumination of the device by particular radiation sources.", "We move on to discuss the “tunability\" of the device with $q$ and consider its application to room temperature black body detection.", "We close with a theoretical treatment of the figures of merit for the device as a room temperature IR detector.", "We show that at room temperature, the normalized signal to noise ratio (specific detectivity) is comparable with present technologies.", "Importantly, the device can achieve this signal to noise ratio for wavelengths which go beyond those accessible for current technologies." ], [ " Solar energy", "An appealing application of the magnetically patterned surface is solar energy harvesting, particularly in the IR range.", "The intensity spectrum of the sun, for low frequencies, is approximately given by the Rayleigh-Jeans law, $I=\\frac{k_BT_{sun}}{4\\pi ^2c^2}\\omega ^2$ .", "At the Earth's distance from the sun, at normal incidence we expect this to be suppressed by $\\left(R_{sun}/R_{Sun-Earth}\\right)^2\\approx 2\\cdot 10^{-5}$ .", "Combined, this yields the 2d closed-circuit current expected for normally incident sunlight: $j_{y}^{(solar)}\\approx \\frac{e^3 v_F^2 q\\tau }{2\\epsilon _0 c \\hbar ^3} \\frac{k_BT_{sun}}{4\\pi ^2c^2}\\left(\\frac{R_{sun}}{R_{Sun-Earth}}\\right)^2 E_{gap}\\eta _\\infty ,$ where $E_{gap}$ is the bandgap of the topological insulator hosting the Dirac cone, and $\\eta _\\infty $ is the constant characterizing $\\eta (\\omega )$ at frequencies $\\omega \\gg v_F q$ .", "We use a scattering timescale of $\\tau =1ps$ , a typical bandgap of $E_{gap}\\approx 0.3 eV$ , and a wavenumber for the magnetic structure $q=10^{8}m^{-1}$ .", "For low magnetic coupling, we obtain $\\eta \\approx 0.0345 (\\frac{u}{\\hbar v_{F}q})^{2}$ , see Fig.", "REF .", "Taking a typical Fermi velocity of $v_{F}=5\\cdot 10^{5}\\frac{m}{sec}$ , we use $\\eta _\\infty =0.01$ which corresponds to a magnetic coupling of about 17meV.", "The above parameters yield $j_y^{(solar)}\\approx 4\\eta \\times 10^{-7} A/m$" ], [ "Laser induced photocurrents", "The effect can also easily be explored using monochromatic laser light.", "Using the same parameters as above, Eq.", "(REF ) yields: $j_y\\approx 2\\cdot 10^{21}\\frac{I}{\\omega ^2}\\eta (\\omega )\\;\\;\\frac{A m}{J sec}$ For laser light of intensity $I=10^5W/m^2$ [24] at angular frequency $\\omega =3\\cdot 10^{14}s^{-1}$ , with $\\eta (\\omega )\\sim .1$ this yields $j_y\\sim 2\\cdot 10^{-4}A/m$ ." ], [ " Room temperature detection of infrared radiation", "A particularly appealing application of the device is detection of infrared radiation.", "We now look at the question of optimal detection of thermal radiation for different emitter and device temperature.", "Our results show that the device can serve as an efficient room temperature detector of IR radiation.", "For the purposes of this discussion we will assume the radiation comes from a black body in equilibrium with its environment at a temperature $T_{BB}$ .", "Such an object radiates at intensity $I(\\omega , T_{BB})= \\frac{1}{4\\pi ^2} \\frac{\\hbar \\omega ^3}{c^2} \\frac{1}{e^{\\hbar \\omega /k_BT_{BB}}-1}$ , which has a maximum at frequency $\\omega _{peak} = b k_B T_{BB}/\\hbar $ and $b=2.8$ .", "Keeping this fact in mind, we now point out the following desirable quality of the our proposed set-up: the frequency that the device is most sensitive to can be tuned by changing the grating pitch $q$ , since the peak in $\\eta (\\omega )$ occurs at $\\omega \\simeq 1.7v_F q$ , see Fig.", "REF .", "Note that this observation is very insensitive to temperature.", "Given this we now imagine fabricating our device such that the peak in $\\eta (\\omega )$ and the peak in the black-body spectrum coincide, this requires that we set $q=\\omega _{peak}/(1.7 \\hbar v_F)$ .", "Figure: η peak \\eta _{peak} as a function of the ratio of device temperature to black body temperature (T device /T BB T_{device}/T_{BB}).", "Here we have also fixed u x =u z =0.3(ℏv F q)u_x=u_z=0.3(\\hbar v_F q).", "The figure shows that up to 55% of η peak \\eta _{peak} remains intact when T device =T BB T_{device}=T_{BB}.", "This implies that the proposed device may be able to detect radiation from a black body at temperature T BB T_{BB} while itself being at this temperature.We now gauge the performance of the system for differences in temperature between the device and the radiation source.", "We define $\\eta _{peak}(T_{device}/T_{BB})$ as $\\eta (\\omega _{peak})$ when the device is set with $q=\\omega _{peak}/(1.7v_F)$ and is operated at a temperature $T_{device}$ and chemical potential $\\mu =0$ .", "We plot $\\eta _{peak}(T_{device}/T_{BB})$ in Fig.", "REF .", "As expected, the function decreases with $T_{device}/T_{BB}$ .", "Interestingly, we see that for $T_{device}/T_{BB}\\simeq 1$ , i.e., a device and black body at similar temperatures, nearly $55\\%$ of the peak value of $\\eta (\\omega _{peak})$ remains.", "This is of particular interest to room temperature detection of IR radiation, where both the device and the black body are near the same temperature and where the black-body radiation lies within the gap of the TI.", "We now move on to discus the figures of merit [42], [43], [25] for the detector we have described above.", "The first is the external quantum efficiency.", "This figure of merit quantifies the optical absorption of the device and is defined as $E_{Q} = \\frac{\\hbar \\omega }{e} R_{I}$ where the responsivity, $R_I$ , is given by $R_I=i_{photo}/(A I_{incident})$ , where $I_{incident}$ is the intensity of the incident radiation and $A=L_xL_y$ is the area of the device absorbing this radiation.", "In our device $L_y$ is the length parallel to the stripes and $L_x$ the length perpendicular to them.", "By defining the dimensionless frequency $\\omega = v_F q \\bar{\\omega }$ we can write ${E_{Q} = \\frac{e^2}{2\\hbar c \\epsilon _0} \\frac{ v_F \\tau }{L_y} \\frac{\\eta (\\bar{\\omega })}{\\bar{\\omega }} }$ There are several interesting pieces of information in this expression.", "First, we see that reducting $L_y$ leads to a higher quantum efficiency.", "Second, similar to the discussion above, the frequency at which the device has the highest quantum efficiency is completely tunable with the grating wave vector $q$ .", "In other words, this frequency scale is not set by a band gap as it is in traditional semiconductor based detectors.", "Third, the above is a result for a single device.", "We could in principle layer thin films of this device in order to multiply the efficiency; the incident light not absorbed by one layer has the potential to be absorbed by other layers.", "Finally, at room temperate an optimized value of $E_Q\\sim 0.01\\%$ is obtained using $\\tau =0.1$ ps and $L_y=100nm$ .", "This value is independent of the wavelength $\\lambda $ of the incident radiation, if the device's grating pitch $q$ is set to optimize $E_Q$ .", "As a comparison, the room temperature detector proposed in Ref.", "[26], functional near $\\lambda \\simeq 10.6\\mu $ m, has a quantum efficiency $\\sim 0.01\\%$ as well.", "Table: Specific detectivities of the device proposed in this paper compared to other devices.", "For our calculations we have used the estimates L x =1mmL_x=1mm, L y =100nmL_y=100nm, I incident ∼10 4 I_{incident}\\sim 10^{4}W/m 2 ^2 and R D ∼10 5 ΩR_D\\sim 10^5\\Omega The second figure of merit we wish to consider is the specific detectivity of the device.", "One issue with a photodetector is differentiating a photo-induced current from other “dark\" currents, i.e.", "those created by noise.", "Here we will call these noise currents $i_{noise}$ .", "In principle one would like the signal-to-noise ratio $i_{photo}/i_{noise}$ to be large.", "In practice, it is useful to define something called the specific detectivity, $D^=\\sqrt{A\\Delta f} {R_I}/{i_{noise}}$ An alternative, but equivalent, definition of the specific detectivity is the reciprocal of a measure called the noise-equivalent power normalized by the square root of the sensor's area and frequency bandwidth, in symbols $D^=\\sqrt{A\\Delta f}/\\text{NEP}$ with $\\text{NEP}$ the noise-equivalent power.", "The noise-equivalent power is the incident power required to have a signal-to-noise ratio of 1, where $A$ and $R_I$ are the area and responsivity that we defined previously and $\\Delta f$ is the range of operational frequencies of the device used to measure current (used here for illustration only, as it will ultimately cancel out).", "We will assume our system is prone to shot and thermal noise which gives rise to[43] $i_{noise}=\\sqrt{\\left(2ei_{induced}+\\frac{4k_BT}{R_D}\\right)\\Delta f}$ .", "Here, $R_D$ is the resistance of the device, and $i_{induced}$ is the current from sources other than noise (e.g.", "photocurrent and/or the current from a bias etc.).", "With this model, and assuming the only induced current in the device is the photocurrent, we find the specific detectivity of our proposed device is $D^ &=& \\frac{e^2}{2\\hbar c \\epsilon _0} \\frac{\\tau }{\\hbar q} \\sqrt{\\frac{L_x}{L_y}} \\frac{\\eta (\\bar{\\omega })}{\\bar{\\omega }^2} \\\\ \\nonumber &\\times & \\left[\\frac{4k_BT}{e^2 R_D} + \\frac{e^2}{\\hbar c \\epsilon _0} \\frac{\\tau }{\\hbar q } \\frac{\\eta (\\bar{\\omega })}{\\bar{\\omega }^2} L_xI_{incident}\\right]^{-1/2}$ where $I_{incident}$ is the incident intensity of radiation, and we have assumed a monochromatic source of light.", "Similar to the external quantum efficiency, the detectivity of this device only depends on the frequency of incident radiation through $\\eta (\\bar{\\omega })$ and as such can optimized by choosing $q$ .", "Second, near room temperature the first term on the second line of Eq.", "(REF ) dominates and we see $D^$ scales with $\\sqrt{L_x/L_y}$ and so having a “rectangular\" device which is large in the $x$ direction compared to the $y$ direction is most beneficial.", "Third, we again note that the above expression is for a single device.", "One could imagine engineering a layered geometry of many of these devices.", "The signal current would increase with the number of layers whereas the noise would scale as the square root of these layers.", "Thus overall $D^$ should scale like the square root of the number of layers.", "The utility of $D^$ is that it enables a comparison of performance across different detector technologies.", "We present such a comparison in Table REF , where we give results for the detectivity of our proposed device (for several different values of $q$ ) alongside $D^$ for several other high temperature IR detectors.", "The table demonstrates that the specific detectivity of the proposed device compares well with recent measurements in other technologies capable of detecting IR radiation at room temperature[27], [28], [29], [26], [30], [31], [44].", "Importantly, the proposed device achieves these values at large wavelengths, which are beyond reach for these technologies.", "In fact, note that for the proposed device, $D^$ grows with increasing $\\lambda $ .", "This is a very desirable property for building a room temperature mid and far-IR detector[42].", "Finally, we remind the reader that the above numerical estimates do not include any of the possible device optimization routes outlined above." ], [ "Conclusions", "The unique properties of the surfaces of topological insulators beg to be translated into practical applications.", "The lack of a generic photocurrent response on such surfaces so far has stifled the possibility of applications in light detection and photovoltaics.", "In this manuscript we demonstrated how surface magnetic patterning employs the spin-orbit locking, and allows for a substantial photocurrent response even to low-intensity sources such as the low-energy solar spectrum.", "The surface is naturally sensitive to photon energies below the bandgap of $0.3eV$ , as opposed to semiconductor based photovoltaics, which require energies that exceed the material's bandgap.", "As such, this effect can be used for detection of micrometer wavelength radiation - a range with limited electric detection schemes.", "Our estimates give a specific detectivity of $\\sim 10^{7}$ cm$\\sqrt{\\text{Hz}}$ /W at $15\\mu $ m and room temperature, with the ability to go to higher wavelengths by adjusting the separation between magnetic strips.", "This value of the specific detectivity has room for further optimization by, e.g.", "creating a layered device.", "Present technologies with comparable room temperature detectivities are confined to wavelengths $\\le 10\\mu $ m and therefore the proposed device represents a significant potential advancement in mid-IR and far-IR detection.", "We note that magnetic coating of topological insulators has been experimentally demonstrated in refs.", "[46], [47], [48], as well as studied numerically using first principle calculations [49].", "The use of magnetic insulators, such as the ones used in [48], will be advantageous in order to minimize effects such as absorption by the magnetic coating and electron doping of the TI surface.", "Many aspects remain unexplored.", "To understand how the TI surface could be harnessed for solar energy harvesting, we need to understand what the natural open-circuit voltage is.", "In addition, we have only provided a crude account of disorder and phonon scattering effects on the surface, and completely ignored the possibility of bulk contributions at high frequencies.", "Lastly, we are confident that the magnitude of the effect could be improved by optimizing our device by using other magnetic patterns, or different materials.", "For instance, we expect that a similar affect will exist in arrays of 2d topological insulator strips, e.g., HgTe/CdTe heterostructures, put in an in-plane spatially varying field.", "We intend to explore at least some of these issues in future work." ], [ "Acknowledgments", "We acknowledge financial support from NSF through DMR-1410435, the Packard Foundation, the IQIM - an NSF center funded in part by the Gordon and Betty Moore Foundation - and especially DARPA through FENA (Caltech), through SPP 1666 of the Deutsche Forschungsgemeinschaft and the Helmholtz Virtual Institute “New States of Matter and Their Excitations” (Berlin).", "NL acknowledges support from the CIG Marie Curie grant, the Bi-National Science Foundation and I-Core: the Israeli Excellence Center \"Circle of Light\" (Technion).", "AF acknowledges support from the Natural Sciences and Engineering Research Council of Canada through the Vanier Canada Graduate Scholarships program and McGill HPC supercomputing resources.", "Finally, GR and FvO acknowledge the hospitality of the Aspen Center for Physics where a portion of this work was completed." ], [ "PHOTOCURRENT DUE TO THE SOLAR SPECTRUM", "We begin with the photocurrent induced by the incident radiation $j=e\\int \\frac{d^{2}k}{(2\\pi )^{2}}\\sum _{\\alpha }\\left[\\mathbf {v}_{\\alpha ,\\mathbf {k}}(n_{\\mathbf {k},\\alpha }-n_{\\mathbf {k},\\alpha }^{0})\\right]$ where $n_{\\mathbf {k},\\alpha }^{0}$ is the equilibrium distribution function, and $n_{\\mathbf {k},\\alpha }$ is the distribution function induced by the incident light.", "Here we derive the result for the chemical potential is at the Dirac point and zero temperature.", "The generalization to finite chemical potential and temperatures will be discussed later.", "We model the relaxation of the system within the relaxation time approximation.", "Working at $\\mu =k_BT=0$ all negative energy states are occupied and all positive energy states are vacant in the absence of the light.", "Owing to this the light must excite a negative energy electron into a positive energy state.", "This ultimately leads to the results $(n_{\\mathbf {k},\\alpha }-n_{\\mathbf {k},\\alpha }^{0})=\\tau \\sum _{\\beta <0}\\Gamma (\\mathbf {k},\\beta \\rightarrow \\mathbf {k},\\alpha )(n_{\\mathbf {k},\\beta }^{0}-n_{\\mathbf {k},\\alpha }^{0}),\\qquad \\alpha >0$ and $(n_{\\mathbf {k},-\\beta }-n_{\\mathbf {k},-\\beta }^{0})=-(n_{\\mathbf {k},\\beta }-n_{\\mathbf {k},\\beta }^{0}),\\qquad \\beta >0.$ where $\\Gamma (\\mathbf {k},\\beta \\rightarrow \\mathbf {k},\\alpha )$ is the transition rate for an electron to move from state $(\\mathbf {k},\\beta )$ to state $(\\mathbf {k},\\alpha )$ .", "The above considerations give us $j=e\\tau \\int \\frac{d^{2}k}{(2\\pi )^{2}}\\sum _{\\alpha >0,\\beta <0}\\left[\\mathbf {v}_{\\alpha ,\\mathbf {k}}\\Gamma (\\mathbf {k},\\beta \\rightarrow \\mathbf {k},\\alpha )-\\mathbf {v_{\\beta ,\\mathbf {k}}}\\Gamma (\\mathbf {k},\\beta \\rightarrow \\mathbf {k},\\alpha )\\right].$ As an approximation of the transition rates we use Fermi's golden rule which gives $\\Gamma (\\mathbf {k},\\beta \\rightarrow \\mathbf {k},\\alpha )=\\frac{\|\\langle \\mathbf {k},\\alpha \|H_{int}(\\omega )\|\\mathbf {k},\\beta \\rangle \|^{2}}{\\hbar }2\\pi \\delta (E_{\\mathbf {k},\\alpha }-E_{\\mathbf {k},\\beta }-\\omega )$ for a time dependent Hamiltonian containing a single frequency.", "The interaction hamiltonian is written as $ \\hat{H}_{int}=e\\frac{\\partial H_{0}}{\\partial \\mathbf {p}}\\cdot \\mathbf {A}(\\mathrm {\\mathbf {x}},t) $ We assume a circularly polarized light: $\\mathbf {E}(t)=E_c(\\hat{x} \\cos \\omega t+ \\hat{y}\\sin \\omega t)$ This corresponds to a vector potential $\\begin{array}{c}\\mathbf {A}(\\omega )=\\frac{E_c}{\\omega }(\\hat{x}\\sin \\omega t-\\hat{y}\\cos \\omega t)\\\\=\\frac{1}{2i}\\frac{E_c}{\\omega }((\\hat{x}-i\\hat{y})e^{i\\omega t}-(\\hat{x}+i\\hat{y})e^{-i\\omega t}),\\end{array}$ The current response to this field is: $j_{k}=\\frac{e\\tau }{\\hbar \\omega ^2}E_{m}(\\omega ){\\cal Q}_{kmn}(\\mathbf {\\omega })E_{n}^{}(\\omega ).$ where $E_x(\\pm \\omega )=E_c/2,\\,E_y(\\omega )=\\pm \\frac{E_c}{2i}$ .", "The ${\\cal Q}_{kmn}(\\omega )$ tensor, is given by integrating over the momentum resolved ${\\cal Q}^{\\alpha \\beta }_{kmn}(\\mathbf {k},\\omega )$ as ${\\cal Q}_{kmn}(\\omega )=\\int \\frac{dk_xdk_y}{(2\\pi )^2} \\sum _{\\alpha >0,\\beta <0}{\\cal Q}_{kmn}^{\\alpha \\beta }(\\mathbf {k},\\omega ).", "$ The momentum resolved ${\\cal Q}^{\\alpha \\beta }_{kmn}(\\mathbf {k},\\omega )$ are in turn given by ${\\cal Q}_{kmn}^{\\alpha \\beta }(\\mathbf {k,\\omega }) & = & \\hat{\\mathbf {x}_{k}}\\cdot \\left(\\mathbf {v}_{\\mathbf {k}}^{(\\alpha )}-\\mathbf {v_{k}}^{(\\beta )}\\right)M_{mn}^{\\alpha \\beta }(\\mathbf {k})\\nonumber \\\\ & & \\times 2\\pi \\mathrm {\\delta (E_{\\mathbf {k}}^{(\\alpha )}-E_{\\mathbf {k}}^{(\\beta )}-\\hbar \\omega )},$ The matrix elements are given by $M_{mn}^{\\alpha \\beta }(\\mathbf {k})=\\left(e v_F\\right)^2\\langle \\mathbf {k},\\alpha \|\\sigma _{m}\|\\mathbf {k},\\beta \\rangle \\langle \\mathbf {k},\\beta \|\\sigma _{n}^{\\dagger }\|\\mathbf {k},\\alpha \\rangle .$ In the following, we carry out the calculation for a non-monochromatic source of light, which has an intensity ditribution as a function of angular frequency, $I(\\omega )$ .", "The monochromatic limit is easy to extract, by setting $I(\\omega )$ to be proportional to a delta-function.", "The intensity of light at a given frequency with amplitude $E_x$ and $E_y$ is: $I(\\omega )d\\omega =\\frac{1}{2}\\epsilon _0 c (E_x^2+E_y^2)$ The $1/2$ comes from averaging the $\\cos ^2(\\omega t),\\,\\sin ^2(\\omega t)$ over time.", "For the two circular polarizations of light this gives: $I(\\omega )d\\omega =\\frac{1}{2}\\epsilon _0 c (2E_{c+}^2+2E_{c-}^2)=\\epsilon _0 c (E_{c+}^2+E_{c-}^2)=2\\epsilon _0 c E_{c+}^2$ where we assumed that the two circular polarizations have the same amplitude.", "So our circular polarization in terms of the solar intensity is: $E_c^2=\\frac{1}{2\\epsilon _0 c}I(\\omega )d\\omega .$ Collecting all the coefficients, and using the property of the tensor ${\\cal Q}_{kmn}$ , we get $j_{y}=\\frac{e\\tau }{{2}\\hbar c \\epsilon _0}\\int d\\omega {2}{\\cal Q}_{yxy}(\\mathbf {\\omega })\\frac{1}{4}\\frac{I(\\omega )}{\\omega ^2}.$ We now define $\\eta (\\omega )$ as a dimensionless quantity that encodes the photocurrent respnse as a function of frequency, which also contains all the intrinsic numerical factors: $\\eta (\\omega )=\\frac{1}{2}\\frac{1}{\\left(e v_F\\right)^2}\\int \\limits _{-q/2}^{q/2}\\frac{dk_{x}}{2\\pi q}\\int \\limits _{-\\infty }^{\\infty }\\frac{d\\tilde{k}_{y}}{2\\pi q}\\hbar v_F q\\frac{{\\cal Q}_{yxy}(\\mathbf {k,\\,\\omega })}{v_F},$ Using this quantity in Eq.", "(REF ) , we get: $j_{y}=\\frac{e^{3}\\tau v_{F}^2q}{2c\\epsilon _{0}\\hbar ^2}\\int d\\omega \\eta (\\omega )\\frac{I(\\omega )}{\\omega ^{2}}$ Now let us substitute $I(\\omega )$ for the sun.", "For a black body at Temperature $T$ , the black-body luminosity per $\\omega $ is: $I(\\omega ,T)=\\frac{1}{4\\pi ^2}\\frac{\\hbar \\omega ^3}{c^{2}}\\frac{1}{\\exp (\\hbar \\omega /k_{B}T)-1}$ where $I(\\nu ,T)$ is the energy per unit time (or the power) radiated per unit area of emitting surface in the normal direction per unit solid angle per unit frequency by a black body at temperature T. The power per unit area arriving at the earth, and assuming normal incidence is: $I_{SE}(\\omega ,T_{sun})=\\frac{R_{sun}^{2}}{R_{earth}^{2}}I(\\omega ,T_{sun})$ For low frequencies, we can approximate the black-body spectrum as $I(\\omega ,T)=\\frac{k_{B}T\\omega ^{2}}{4 \\pi ^2 c^{2}}$ which is the Rayleigh-Jeans law.", "Inserting this into Eq.", "(REF ), and taking $\\eta (\\omega )=\\eta $ (appropriate for large frequencies), we get $j_{y}=\\frac{e^{3}\\tau v_{F}^{3}q^{2}}{2c\\epsilon _{0}\\hbar ^2}\\eta {\\cal I}_{0}\\frac{\\omega _{max}}{v_F q}$ where the constant ${\\cal I}_{0}=\\frac{k_{B}T_{sun}}{c^{2}(2\\pi )^2}\\left(\\frac{R_{sun}^{2}}{R_{earth}^{2}}\\right)$ In order to extend the above analysis to finite temperature and chemical potential we must make an observation which ultimately lead to a simple modification of the formula above.", "For a system at finite temperature and with the chemical potential at an arbitrary point the incident light can excite or relax (through absorption or emission) electrons from any initial state to any final state.", "This must be accounted for in our model for the steady state $n_{\\mathbf {k},\\alpha }$ .", "This physical considerations lead to a description identical to the one above, provided we use the following modified form for $\\mathcal {Q}^{\\alpha \\beta }_{kmn}(\\mathbf {k}, \\omega )$ $\\mathcal {Q}^{\\alpha \\beta }_{kmn}(\\mathbf {k}, \\omega , T)=\\mathcal {Q}^{\\alpha \\beta }_{kmn}(\\mathbf {k}, \\omega )(n_{\\mathbf {k},\\beta }^{0}-n_{\\mathbf {k},\\alpha }^{0})$ Figure: Top left: Effective bandstructure of the proposed heterostructure, cut along the line k y =0k_{y}=0 (units such that \|𝐪\|=1\|\\mathbf {q}\|=1).", "Band indices are shown on the left.", "Transitions contributing to the summed momentum specific tensor 𝒬 kmn αβ (𝐤){\\cal Q}_{kmn}^{\\alpha \\beta }(\\mathbf {k}) are depicted by arrows.", "(a)-(d) Numerical results for ∫dω𝒬 ˜ yxy αβ (𝐤)\\int d\\omega \\tilde{{\\cal Q}}_{yxy}^{\\alpha \\beta }(\\mathbf {k}) , in units of e 2 v F 3 c 2 \\frac{e^{2}v_{F}^{3}}{c^{2}}.", "The colors indicate the photon frequency of the transition as determined by the δ\\delta functions in Eq.", "().Panels (a) and (b) contain the tensors for the transitions (α,β)=(1,-1)(\\alpha ,\\beta )=(1,-1) and (2,-2)(2,-2),respectively, in which the excitation leaves the electron momentum unchanged.", "These transitions indicated by green and purplearrows, respectively, in the top left panel renormalize the conduction and valence bands.", "(c) Same for (α,β)=(2,-1)(\\alpha ,\\beta )=(2,-1) and (1,-2)(1,-2) (blue arrows in the top left panel).", "(d) Same for (α,β)=(3,-1)(\\alpha ,\\beta )=(3,-1) and (1,-3)(1,-3) (red arrows)." ], [ "MOMENTUM SPECIFIC RESPONSE", "In order to get some intuition for the origin of the photocurrent contributions, we study the momentum-specific response $\\tilde{{\\cal Q}}_{yxy}^{\\alpha \\beta }(\\mathbf {k},\\omega )$ .", "This quantity allows us to understand which parts of the BZ contribute most to the effect.", "This function is plotted in Fig.", "REF (a-d), where in addition to the response as a function of momentum, the photon energy responsible for the transition at each momentum is encoded in the color.", "We see that the effect is not exclusively due to the edges of the BZ.", "Rather, the contribution is uniformly distributed in momentum space, validating a perturbative perspective on the effects of the magnetic surface texture.", "Fig.", "REF also demonstrates that the sign change of $\\eta (\\omega )$ arises due to a sign difference between: (i) contributions of processes involving scattering by momentum $\\pm {\\bf q}$ (indicated in red and blue in Fig.", "REF ) which dominate for $\\omega > v_F q$ and (ii) contributions of momentum conserving processes (green and purple in Fig.", "REF ) which dominate for $\\omega < v_F q$" ], [ "PERTURBATIVE ANALYSIS OF THE PHOTOCURRENT RESPONSE", "In the following, we shall calculate the response tensor, accounting for the surface magnetic pattern within second order perturbation theory.", "This can be most conveniently expressed using momenta in the extended Brillouin zone.", "Denoting by $\|\\psi ^{(0)}({\\bf k},\\alpha )\\rangle $ the eigenstates of $H_0$ (without the magnetic structure), we expand the eigenstates in second order perturbation theory in $V=V^+ + V^-=(\\textbf {u}\\cdot \\vec{\\sigma })e^{i{\\bf q}\\cdot r}+h.c.$ , as $\|\\psi ({\\bf k},\\alpha )\\rangle = \|\\psi ^{(0)}({\\bf k},\\alpha )\\rangle + \|\\psi ^{(1)}({\\bf k},\\alpha )\\rangle +\|\\psi ^{(2)}({\\bf k},\\alpha )\\rangle $ with $\\alpha =c,v$ for conduction and valence bands.", "The first and second order corrections are given by $\|\\psi ^{(1)}({\\bf k},\\alpha )\\rangle =\\sum _{\\sigma =\\pm }\\frac{1}{E^\\alpha _{\\bf k}-H({\\bf k}+\\sigma {\\bf q})}V^\\sigma \|\\psi ^{(0)}({\\bf k},\\alpha )\\rangle $ and $\|\\psi ^{(2)}({\\bf k},v)\\rangle &=&P^c_{\\bf k}\\sum _{\\sigma =\\pm }\\frac{1}{E^v_{\\bf k}-E^c_{\\bf k}}V^{\\sigma \\dagger }\\frac{1}{E^v_{\\bf k}-H({\\bf k}+\\sigma {\\bf q})}V^\\sigma \|\\psi ^{(0)}({\\bf k},v)\\rangle .\\nonumber \\\\\|\\psi ^{(2)}({\\bf k},c)\\rangle &=&P^v_{\\bf k}\\sum _{\\sigma =\\pm }\\frac{1}{E^c_{\\bf k}-E^v_{\\bf k}}V^{\\sigma \\dagger }\\frac{1}{E^c_{\\bf k}-H({\\bf k}+\\sigma {\\bf q})}V^\\sigma \|\\psi ^{(0)}({\\bf k},c)\\rangle .", "\\nonumber \\\\$ where $P_{\\bf k}^c$ is a projector on the conduction band state with momentum ${\\bf k}$ .", "In second order perturbation theory, the total photocurrent response can be written as ${\\cal Q}^{\\rm ext}_{yxy}(\\mathbf {k}) = 2\\pi (e v_F)^2 \\sum _{\\lambda =0,+,-}\\hat{\\textbf {y}}\\cdot \\left(v^c_{{\\bf k}+\\lambda {\\bf q}}-v^v_{{\\bf k}}\\right) M_{xy}^{\\lambda }(\\mathbf {k}) \\delta _\\lambda (\\omega ,\\mathbf {k})$ where $v^v_{\\bf k}$ and $v^c_{\\bf k}$ denote the velocities in the conduction and valence bands, and the delta functions in Eq.", "(REF ) were abbreviated to $\\delta _\\lambda (\\omega ,\\mathbf {k})=\\delta (E^c_{\\mathbf {k}+\\lambda {\\bf q}}+E^v_{\\mathbf {k}}-\\omega )$ .", "The index $\\lambda =0,+1,-1$ denotes process which correspond to $ {\\bf k},v \\rightarrow {\\bf k},c$ , $ {\\bf k},v \\rightarrow {\\bf k}+{\\bf q},c$ and $ {\\bf k},v \\rightarrow {\\bf k}-{\\bf q},c$ , respectively.", "Our goal is to calculate the matrix elements: $M_{mn}^{\\lambda }(\\mathbf {k})=\\left< \\psi (\\mathbf {k}+\\lambda {\\bf q},c) \\right\|\\sigma _m\|\\psi ({\\bf k},v)\\rangle \\left< \\psi (\\mathbf {k},v) \\right\|\\sigma _n\|\\psi ({\\bf k}+\\lambda {\\bf q},c)\\rangle ,$ to second order in $V$ .", "First, we describe transitions from $ {\\bf k},v \\rightarrow {\\bf k}+{\\bf q},c$ in the extended BZ.", "Two substitutions of $\|\\psi ^{(1)}({\\bf k},v)\\rangle $ from Eq.", "(REF ) into Eq.", "(REF ) yield $M^{+}_{mn}({\\bf k})=\\sum _{r,s}u_r u^_s\\left[F^{v,v}_{mrsn}({\\bf k})+F^{v,c}_{mrns}({\\bf k})+F^{c,v}_{rmsn}({\\bf k})+F^{c,c}_{rmns}({\\bf k})\\right]$ where $F^{\\alpha \\beta }_{mnrs}({\\bf k})={\\rm Tr}\\left[P_{{\\bf k}+{\\bf q}}^c\\sigma _{m}R_+^\\alpha ({\\bf k})\\sigma _{n}P_{\\bf k}^v\\sigma _{r}R_+^\\beta ({\\bf k})\\sigma _{s}\\right]$ with $\\alpha ,\\beta =v,c$ and $R_\\pm ^v({\\bf k})&=&\\frac{1}{E^v_{\\bf k}-H({\\bf k}\\pm {\\bf q})}\\nonumber \\\\R_\\pm ^c({\\bf k})&=&\\frac{1}{E^c_{{\\bf k}\\pm {\\bf q}}-H({\\bf k})}$ Note the permutation of the indices in Eq.", "(REF ).", "Next we compute the matrix elements for transitions which in the extended BZ, correspond to transitions ${\\bf k}\\rightarrow {\\bf k}-{\\bf q}$ .", "By taking ${\\bf q}\\rightarrow -{\\bf q}$ in Eq.", "(REF , $M_{mn}^{-}(\\mathbf {k})=\\sum _{r,s}u^_r u_s \\left[B^{v,v}_{mrsn}({\\bf k})+B^{v,c}_{mrns}({\\bf k})+B^{c,v}_{rmsn}({\\bf k})+B^{c,c}_{rmns}({\\bf k})\\right]$ with $B^{\\alpha \\beta }_{mnrs}({\\bf k})={\\rm Tr}\\left[P_{{\\bf k}-{\\bf q}}^c\\sigma _{m}R_-^\\alpha ({\\bf k})\\sigma _{n}P_{\\bf k}^v\\sigma _{r}R_-^\\beta ({\\bf k})\\sigma _{s}\\right].$ Next, we calculate the elements $M^{0}_{mn}$ which correspond to transitions ${\\bf k},v \\rightarrow {\\bf k},c$ .", "These can give a non zero contribution to the current in second order perturbation theory due to the renormalization of the bands, c.f.", "Eq.", "(REF ).", "This yields $M^0_{mn}({\\bf k})&=&\\sum _{r,s}u_r^* u_s\\Big \\lbrace \\frac{1}{E_{\\bf k}^v-E_{\\bf k}^c}\\Big (W^{+}_{nmrs}+W^{-}_{nmsr}+(W^{+}_{mnsr})^\\dagger +(W^{-}_{mnrs})^\\dagger \\Big )\\nonumber \\\\&\\phantom{=}&\\phantom{blahblahbl}-\\frac{1}{E_{\\bf k}^v-E_{\\bf k}^c}\\Big (\\tilde{W}^{+}_{mnrs}+\\tilde{W}^{-}_{mnsr}+(\\tilde{W}^{+}_{nmsr})^\\dagger +(\\tilde{W}^{-}_{nmrs})^\\dagger \\Big )\\nonumber \\\\&\\phantom{=}&\\phantom{blahblahbl}+Z^{+}_{mnrs}+Z^{-}_{mnsr}+(Z^{+}_{nmsr})^\\dagger +(Z^{-}_{nmrs})^\\dagger \\Big \\rbrace ,$ where $W^{\\rho }_{mnrs}&=&{\\rm Tr}\\left[P_{\\bf k}^c\\sigma _m P_{\\bf k}^c \\sigma _n P_{\\bf k}^v \\sigma _r R^v_\\rho ({\\bf k})\\sigma _s\\right],\\nonumber \\\\\\tilde{W}^{\\rho }_{mnrs}&=&{\\rm Tr}\\left[P_{\\bf k}^v\\sigma _m P_{\\bf k}^v \\sigma _n P_{\\bf k}^c \\sigma _r \\tilde{R}^c_{\\rho }({\\bf k})\\sigma _s\\right],\\nonumber \\\\Z^{\\rho }_{mnrs}&=&{\\rm Tr}\\left[P_{\\bf k}^c\\sigma _m P_{\\bf k}^v \\sigma _r R^v_\\rho ({\\bf k}) \\sigma _n \\tilde{R}^c_{\\rho }({\\bf k})\\sigma _s\\right],\\nonumber \\\\$ and where we have introduced the notation $\\tilde{R}^c_\\rho ({\\bf k})=\\frac{1}{E^c_{{\\bf k}}-H({\\bf k}\\pm {\\bf q})}.$ In Eq.", "(REF ), the first (second) term arises due to the second order corrections to the valence (conduction) states at momentum ${\\bf k}$ , c.f.", "first (second) line in Eq.", "(REF ).", "The third term in Eq.", "(REF ) arises due to first order corrections (as in Eq.", "(REF )) to both the valence and conduction bands.", "To make a connection with the results presented in the main text, we would like to sum over momenta in the four quadrants of the BZ, and obtain the the momentum summed response tensor, ${\\cal \\widetilde{Q}}^{\\rm ext}_{yxy}(\\mathbf {k}) = \\sum _{\\sigma ,\\sigma ^{\\prime }=\\pm }{\\cal Q}^{\\rm ext}_{yxy}(\\sigma k_x,\\sigma ^{\\prime } k_y)$ Note that the energy differences obey the symmetries $E^c_{(k_x,k_y)+\\lambda {\\bf q}}-E^v_{(k_x,k_y)}=E^c_{(-k_x,k_y)-\\lambda {\\bf q}}-E^v_{(-k_x,k_y)},$ and the velocities obey the symmetries appearing in Eq.", "(17) of the main text.", "Using these symmetries, it is natural to define the functions $F_\\lambda ({\\bf k})$ which were used in Eq.", "(20) the main text, $F_\\lambda ({\\bf k})=\\sum _{\\sigma ,\\sigma ^{\\prime }=\\pm }M^{\\lambda \\cdot \\sigma }_{xy}(\\sigma k_x,\\sigma ^{\\prime } k_y)\\left(v_y^c({\\bf k}+\\lambda {\\bf q})-v_y^v({\\bf k})\\right)\\sigma ^{\\prime }$ The functions $F_\\lambda ({\\bf k})$ sum the matrix elements for the four transitions $(k_x,\\pm k_y) \\rightarrow (k_x,\\pm k_y) +\\lambda {\\bf q}$ , and $(-k_x,\\pm k_y) \\rightarrow (-k_x,\\pm k_y) -\\lambda {\\bf q}$ .", "These transitions occur at the same photon frequency, by Eq.", "(REF ).", "Therefore, using the functions $F_\\lambda ({\\bf k})$ , Eq.", "(REF ) can be written as ${\\cal \\widetilde{Q}}^{\\rm ext}_{yxy}(\\mathbf {k}) = 2\\pi {\\textrm {R}e}\\left\\lbrace {u_x u^_z }\\right\\rbrace (e v_F)^2 \\sum _{\\lambda =0,+,-} F_\\lambda (\\mathbf {k})\\delta _\\lambda (\\omega ,\\mathbf {k})$" ], [ "Second order perturbation theory in the reduced Brillouin zone scheme", "In this section, we will make the connetion between the response tensor ${\\cal \\widetilde{Q}}^{\\rm ext}_{kmn}(\\mathbf {k})$ obtained in second order perturbation theory, and the response tensor ${\\cal \\widetilde{Q}}^{\\rm \\alpha \\beta }_{kmn}(\\mathbf {k})$ for the reduced Brillouin zone.", "First, we note the relation between the unperturbed eigenstates in the reduced BZ, which we denote by $\|{\\bf k},\\alpha \\rangle $ , with $\\alpha $ a positive (negative) integer for bands with $E>0$ ($E<0$ ), to those in the extended BZ, which we denote by $\|\\psi ^{(0)}({\\bf k},a\\rangle $ , with $a=v,c$ .", "We will be interested only in the quadrant with $k_x,k_y>0$ due to the symmetries discussed in the main text.", "$\\left\|{\\bf k},\\alpha \\right\\rangle \\rightarrow \\left\|\\psi ^{(0)}\\left({\\bf k}-(-1)^{\\alpha }\\lfloor \\frac{\\alpha }{2}\\rfloor {\\bf q},a\\right)\\right\\rangle ,\\qquad $ where in the above equation, set $a=c$ for $\\alpha >0$ and $a=v$ for $\\alpha <0$ .", "Table: Mapping between the response tensors in the reduced Brillouin zone 𝒬 ˜ yxy αβ (𝐤){\\cal \\widetilde{Q}}^{\\rm \\alpha \\beta }_{yxy}(\\mathbf {k}) and the results obtained in second order perturbation theory.", "Only the values for the pairs (α,β)(\\alpha ,\\beta ) that have non zero rate in second order perturbation theory are shown.", "Note that the functions 𝒬 ˜ yxy αβ (𝐤){\\cal \\widetilde{Q}}^{\\rm \\alpha \\beta }_{yxy}(\\mathbf {k}) are defined for momenta 𝐤{\\bf k} in the k x >0,k y >0k_x>0,k_y>0 of the reduced Brillouin zone.", "The value of these functions, in second order perturbation theory, corresponds to 𝒬 ˜ λ (𝐤 E ){\\cal \\widetilde{Q}}^\\lambda ({\\bf k}_E), where λ\\lambda and k E k_E take the values shown in the table.", "In the left two columns, the nn is an integer such that n≥0n\\ge 0.For the response second order perturbation theory, it is convenient to define each of the terms appearing in Eq.", "(REF ) as ${\\cal \\widetilde{Q}}^{\\lambda }(\\mathbf {k}) = 2\\pi {\\textrm {R}e}\\left\\lbrace {u_x u^_z }\\right\\rbrace (e v_F)^2 F_\\lambda (\\mathbf {k})\\delta _\\lambda (\\omega ,\\mathbf {k})$ From Eq.", "(REF ), we get a map between the response tensors ${\\cal \\widetilde{Q}}^{\\rm \\alpha \\beta }_{kmn}(\\mathbf {k})$ defined in the $k_x>0,k_y>0$ quadrant of the reduced Brillouin zone, to the processes corresponding to ${\\cal \\widetilde{Q}}^\\lambda ({\\bf k}_E)$ in Eq.", "(REF ), where ${\\bf k}_E$ takes value in the $k_x>0,k_y>0$ quadrant of the extended Brillouin zone.", "This map is constructed such that both ${\\bf k}_E$ and $\\lambda $ are functions of ${\\bf k}$ , $\\alpha $ and $\\beta $ .", "This map is given explicitly in Table REF ." ] ]	1403.0010

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